[{"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Finite.lean", "full_name": "Set.Finite.biUnion", "start": [798, 1], "end": [803, 19], "traced_tactics": [{"tactic": "classical\n  cases hs\n  haveI := fintypeBiUnion s t fun i hi => (ht i hi).fintype\n  apply toFinite", "annotated_tactic": ["classical\n    cases hs\n    haveI := <a>fintypeBiUnion</a> s t fun i hi => (ht i hi).<a>fintype</a>\n    apply <a>toFinite</a>", [{"full_name": "Set.fintypeBiUnion", "def_path": "Mathlib/Data/Set/Finite.lean", "def_pos": [392, 5], "def_end_pos": [392, 19]}, {"full_name": "Set.Finite.fintype", "def_path": "Mathlib/Data/Set/Finite.lean", "def_pos": [103, 29], "def_end_pos": [103, 43]}, {"full_name": "Set.toFinite", "def_path": "Mathlib/Data/Set/Finite.lean", "def_pos": [82, 9], "def_end_pos": [82, 17]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b9\u271d : Sort w\n\u03b3 : Type x\n\u03b9 : Type u_1\ns : Set \u03b9\nhs : Set.Finite s\nt : \u03b9 \u2192 Set \u03b1\nht : \u2200 (i : \u03b9), i \u2208 s \u2192 Set.Finite (t i)\n\u22a2 Set.Finite (\u22c3 i \u2208 s, t i)", "state_after": "no goals"}, {"tactic": "cases hs", "annotated_tactic": ["cases hs", []], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b9\u271d : Sort w\n\u03b3 : Type x\n\u03b9 : Type u_1\ns : Set \u03b9\nhs : Set.Finite s\nt : \u03b9 \u2192 Set \u03b1\nht : \u2200 (i : \u03b9), i \u2208 s \u2192 Set.Finite (t i)\n\u22a2 Set.Finite (\u22c3 i \u2208 s, t i)", "state_after": "case intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9\u271d : Sort w\n\u03b3 : Type x\n\u03b9 : Type u_1\ns : Set \u03b9\nt : \u03b9 \u2192 Set \u03b1\nht : \u2200 (i : \u03b9), i \u2208 s \u2192 Set.Finite (t i)\na\u271d : Fintype \u2191s\n\u22a2 Set.Finite (\u22c3 i \u2208 s, t i)"}, {"tactic": "haveI := fintypeBiUnion s t fun i hi => (ht i hi).fintype", "annotated_tactic": ["haveI := <a>fintypeBiUnion</a> s t fun i hi => (ht i hi).<a>fintype</a>", [{"full_name": "Set.fintypeBiUnion", "def_path": "Mathlib/Data/Set/Finite.lean", "def_pos": [392, 5], "def_end_pos": [392, 19]}, {"full_name": "Set.Finite.fintype", "def_path": "Mathlib/Data/Set/Finite.lean", "def_pos": [103, 29], "def_end_pos": [103, 43]}]], "state_before": "case intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9\u271d : Sort w\n\u03b3 : Type x\n\u03b9 : Type u_1\ns : Set \u03b9\nt : \u03b9 \u2192 Set \u03b1\nht : \u2200 (i : \u03b9), i \u2208 s \u2192 Set.Finite (t i)\na\u271d : Fintype \u2191s\n\u22a2 Set.Finite (\u22c3 i \u2208 s, t i)", "state_after": "case intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9\u271d : Sort w\n\u03b3 : Type x\n\u03b9 : Type u_1\ns : Set \u03b9\nt : \u03b9 \u2192 Set \u03b1\nht : \u2200 (i : \u03b9), i \u2208 s \u2192 Set.Finite (t i)\na\u271d : Fintype \u2191s\nthis : Fintype \u2191(\u22c3 x \u2208 s, t x)\n\u22a2 Set.Finite (\u22c3 i \u2208 s, t i)"}, {"tactic": "apply toFinite", "annotated_tactic": ["apply <a>toFinite</a>", [{"full_name": "Set.toFinite", "def_path": "Mathlib/Data/Set/Finite.lean", "def_pos": [82, 9], "def_end_pos": [82, 17]}]], "state_before": "case intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9\u271d : Sort w\n\u03b3 : Type x\n\u03b9 : Type u_1\ns : Set \u03b9\nt : \u03b9 \u2192 Set \u03b1\nht : \u2200 (i : \u03b9), i \u2208 s \u2192 Set.Finite (t i)\na\u271d : Fintype \u2191s\nthis : Fintype \u2191(\u22c3 x \u2208 s, t x)\n\u22a2 Set.Finite (\u22c3 i \u2208 s, t i)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Martingale/Upcrossing.lean", "full_name": "MeasureTheory.mul_upcrossingsBefore_le", "start": [599, 1], "end": [652, 67], "traced_tactics": [{"tactic": "by_cases hN : N = 0", "annotated_tactic": ["by_cases hN : N = 0", []], "state_before": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\nhf : a \u2264 f N \u03c9\nhab : a < b\n\u22a2 (b - a) * \u2191(upcrossingsBefore a b f N \u03c9) \u2264 \u2211 k in Finset.range N, upcrossingStrat a b f N k \u03c9 * (f (k + 1) - f k) \u03c9", "state_after": "case pos\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\nhf : a \u2264 f N \u03c9\nhab : a < b\nhN : N = 0\n\u22a2 (b - a) * \u2191(upcrossingsBefore a b f N \u03c9) \u2264 \u2211 k in Finset.range N, upcrossingStrat a b f N k \u03c9 * (f (k + 1) - f k) \u03c9\n\ncase neg\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\nhf : a \u2264 f N \u03c9\nhab : a < b\nhN : \u00acN = 0\n\u22a2 (b - a) * \u2191(upcrossingsBefore a b f N \u03c9) \u2264 \u2211 k in Finset.range N, upcrossingStrat a b f N k \u03c9 * (f (k + 1) - f k) \u03c9"}, {"tactic": "simp_rw [upcrossingStrat, Finset.sum_mul, \u2190\n  Set.indicator_mul_left _ _ (fun x \u21a6 (f (x + 1) - f x) \u03c9), Pi.one_apply, Pi.sub_apply, one_mul]", "annotated_tactic": ["simp_rw [<a>upcrossingStrat</a>, <a>Finset.sum_mul</a>, \u2190\n    <a>Set.indicator_mul_left</a> _ _ (fun x \u21a6 (f (x + 1) - f x) \u03c9), <a>Pi.one_apply</a>, <a>Pi.sub_apply</a>, <a>one_mul</a>]", [{"full_name": "MeasureTheory.upcrossingStrat", "def_path": "Mathlib/Probability/Martingale/Upcrossing.lean", "def_pos": [368, 19], "def_end_pos": [368, 34]}, {"full_name": "Finset.sum_mul", "def_path": "Mathlib/Algebra/BigOperators/Ring.lean", "def_pos": [51, 9], "def_end_pos": [51, 16]}, {"full_name": "Set.indicator_mul_left", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [714, 9], "def_end_pos": [714, 27]}, {"full_name": "Pi.one_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [47, 9], "def_end_pos": [47, 18]}, {"full_name": "Pi.sub_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [200, 3], "def_end_pos": [200, 14]}, {"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [464, 9], "def_end_pos": [464, 16]}]], "state_before": "case neg\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\nhf : a \u2264 f N \u03c9\nhab : a < b\nhN : \u00acN = 0\n\u22a2 (b - a) * \u2191(upcrossingsBefore a b f N \u03c9) \u2264 \u2211 k in Finset.range N, upcrossingStrat a b f N k \u03c9 * (f (k + 1) - f k) \u03c9", "state_after": "case neg\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\nhf : a \u2264 f N \u03c9\nhab : a < b\nhN : \u00acN = 0\n\u22a2 (b - a) * \u2191(upcrossingsBefore a b f N \u03c9) \u2264\n    \u2211 x in Finset.range N,\n      \u2211 x_1 in Finset.range N,\n        Set.indicator (Set.Ico (lowerCrossingTime a b f N x_1 \u03c9) (upperCrossingTime a b f N (x_1 + 1) \u03c9))\n          (fun a => f (a + 1) \u03c9 - f a \u03c9) x"}, {"tactic": "rw [Finset.sum_comm]", "annotated_tactic": ["rw [<a>Finset.sum_comm</a>]", [{"full_name": "Finset.sum_comm", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [712, 3], "def_end_pos": [712, 14]}]], "state_before": "case neg\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\nhf : a \u2264 f N \u03c9\nhab : a < b\nhN : \u00acN = 0\n\u22a2 (b - a) * \u2191(upcrossingsBefore a b f N \u03c9) \u2264\n    \u2211 x in Finset.range N,\n      \u2211 x_1 in Finset.range N,\n        Set.indicator (Set.Ico (lowerCrossingTime a b f N x_1 \u03c9) (upperCrossingTime a b f N (x_1 + 1) \u03c9))\n          (fun a => f (a + 1) \u03c9 - f a \u03c9) x", "state_after": "case neg\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\nhf : a \u2264 f N \u03c9\nhab : a < b\nhN : \u00acN = 0\n\u22a2 (b - a) * \u2191(upcrossingsBefore a b f N \u03c9) \u2264\n    \u2211 y in Finset.range N,\n      \u2211 x in Finset.range N,\n        Set.indicator (Set.Ico (lowerCrossingTime a b f N y \u03c9) (upperCrossingTime a b f N (y + 1) \u03c9))\n          (fun a => f (a + 1) \u03c9 - f a \u03c9) x"}, {"tactic": "simp_rw [h\u2081]", "annotated_tactic": ["simp_rw [h\u2081]", []], "state_before": "case neg\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\nhf : a \u2264 f N \u03c9\nhab : a < b\nhN : \u00acN = 0\nh\u2081 :\n  \u2200 (k : \u2115),\n    \u2211 n in Finset.range N,\n        Set.indicator (Set.Ico (lowerCrossingTime a b f N k \u03c9) (upperCrossingTime a b f N (k + 1) \u03c9))\n          (fun m => f (m + 1) \u03c9 - f m \u03c9) n =\n      stoppedValue f (upperCrossingTime a b f N (k + 1)) \u03c9 - stoppedValue f (lowerCrossingTime a b f N k) \u03c9\n\u22a2 (b - a) * \u2191(upcrossingsBefore a b f N \u03c9) \u2264\n    \u2211 y in Finset.range N,\n      \u2211 x in Finset.range N,\n        Set.indicator (Set.Ico (lowerCrossingTime a b f N y \u03c9) (upperCrossingTime a b f N (y + 1) \u03c9))\n          (fun a => f (a + 1) \u03c9 - f a \u03c9) x", "state_after": "case neg\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\nhf : a \u2264 f N \u03c9\nhab : a < b\nhN : \u00acN = 0\nh\u2081 :\n  \u2200 (k : \u2115),\n    \u2211 n in Finset.range N,\n        Set.indicator (Set.Ico (lowerCrossingTime a b f N k \u03c9) (upperCrossingTime a b f N (k + 1) \u03c9))\n          (fun m => f (m + 1) \u03c9 - f m \u03c9) n =\n      stoppedValue f (upperCrossingTime a b f N (k + 1)) \u03c9 - stoppedValue f (lowerCrossingTime a b f N k) \u03c9\n\u22a2 (b - a) * \u2191(upcrossingsBefore a b f N \u03c9) \u2264\n    \u2211 x in Finset.range N,\n      (stoppedValue f (upperCrossingTime a b f N (x + 1)) \u03c9 - stoppedValue f (lowerCrossingTime a b f N x) \u03c9)"}, {"tactic": "refine' le_trans _ h\u2082", "annotated_tactic": ["refine' <a>le_trans</a> _ h\u2082", [{"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}]], "state_before": "case neg\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\nhf : a \u2264 f N \u03c9\nhab : a < b\nhN : \u00acN = 0\nh\u2081 :\n  \u2200 (k : \u2115),\n    \u2211 n in Finset.range N,\n        Set.indicator (Set.Ico (lowerCrossingTime a b f N k \u03c9) (upperCrossingTime a b f N (k + 1) \u03c9))\n          (fun m => f (m + 1) \u03c9 - f m \u03c9) n =\n      stoppedValue f (upperCrossingTime a b f N (k + 1)) \u03c9 - stoppedValue f (lowerCrossingTime a b f N k) \u03c9\nh\u2082 :\n  \u2211 _k in Finset.range (upcrossingsBefore a b f N \u03c9), (b - a) \u2264\n    \u2211 k in Finset.range N,\n      (stoppedValue f (upperCrossingTime a b f N (k + 1)) \u03c9 - stoppedValue f (lowerCrossingTime a b f N k) \u03c9)\n\u22a2 (b - a) * \u2191(upcrossingsBefore a b f N \u03c9) \u2264\n    \u2211 x in Finset.range N,\n      (stoppedValue f (upperCrossingTime a b f N (x + 1)) \u03c9 - stoppedValue f (lowerCrossingTime a b f N x) \u03c9)", "state_after": "case neg\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\nhf : a \u2264 f N \u03c9\nhab : a < b\nhN : \u00acN = 0\nh\u2081 :\n  \u2200 (k : \u2115),\n    \u2211 n in Finset.range N,\n        Set.indicator (Set.Ico (lowerCrossingTime a b f N k \u03c9) (upperCrossingTime a b f N (k + 1) \u03c9))\n          (fun m => f (m + 1) \u03c9 - f m \u03c9) n =\n      stoppedValue f (upperCrossingTime a b f N (k + 1)) \u03c9 - stoppedValue f (lowerCrossingTime a b f N k) \u03c9\nh\u2082 :\n  \u2211 _k in Finset.range (upcrossingsBefore a b f N \u03c9), (b - a) \u2264\n    \u2211 k in Finset.range N,\n      (stoppedValue f (upperCrossingTime a b f N (k + 1)) \u03c9 - stoppedValue f (lowerCrossingTime a b f N k) \u03c9)\n\u22a2 (b - a) * \u2191(upcrossingsBefore a b f N \u03c9) \u2264 \u2211 _k in Finset.range (upcrossingsBefore a b f N \u03c9), (b - a)"}, {"tactic": "rw [Finset.sum_const, Finset.card_range, nsmul_eq_mul, mul_comm]", "annotated_tactic": ["rw [<a>Finset.sum_const</a>, <a>Finset.card_range</a>, <a>nsmul_eq_mul</a>, <a>mul_comm</a>]", [{"full_name": "Finset.sum_const", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [1440, 3], "def_end_pos": [1440, 14]}, {"full_name": "Finset.card_range", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [177, 9], "def_end_pos": [177, 19]}, {"full_name": "nsmul_eq_mul", "def_path": "Mathlib/Algebra/GroupPower/Lemmas.lean", "def_pos": [509, 9], "def_end_pos": [509, 21]}, {"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [302, 9], "def_end_pos": [302, 17]}]], "state_before": "case neg\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\nhf : a \u2264 f N \u03c9\nhab : a < b\nhN : \u00acN = 0\nh\u2081 :\n  \u2200 (k : \u2115),\n    \u2211 n in Finset.range N,\n        Set.indicator (Set.Ico (lowerCrossingTime a b f N k \u03c9) (upperCrossingTime a b f N (k + 1) \u03c9))\n          (fun m => f (m + 1) \u03c9 - f m \u03c9) n =\n      stoppedValue f (upperCrossingTime a b f N (k + 1)) \u03c9 - stoppedValue f (lowerCrossingTime a b f N k) \u03c9\nh\u2082 :\n  \u2211 _k in Finset.range (upcrossingsBefore a b f N \u03c9), (b - a) \u2264\n    \u2211 k in Finset.range N,\n      (stoppedValue f (upperCrossingTime a b f N (k + 1)) \u03c9 - stoppedValue f (lowerCrossingTime a b f N k) \u03c9)\n\u22a2 (b - a) * \u2191(upcrossingsBefore a b f N \u03c9) \u2264 \u2211 _k in Finset.range (upcrossingsBefore a b f N \u03c9), (b - a)", "state_after": "no goals"}, {"tactic": "simp [hN]", "annotated_tactic": ["simp [hN]", []], "state_before": "case pos\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\nhf : a \u2264 f N \u03c9\nhab : a < b\nhN : N = 0\n\u22a2 (b - a) * \u2191(upcrossingsBefore a b f N \u03c9) \u2264 \u2211 k in Finset.range N, upcrossingStrat a b f N k \u03c9 * (f (k + 1) - f k) \u03c9", "state_after": "no goals"}, {"tactic": "intro k", "annotated_tactic": ["intro k", []], "state_before": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\nhf : a \u2264 f N \u03c9\nhab : a < b\nhN : \u00acN = 0\n\u22a2 \u2200 (k : \u2115),\n    \u2211 n in Finset.range N,\n        Set.indicator (Set.Ico (lowerCrossingTime a b f N k \u03c9) (upperCrossingTime a b f N (k + 1) \u03c9))\n          (fun m => f (m + 1) \u03c9 - f m \u03c9) n =\n      stoppedValue f (upperCrossingTime a b f N (k + 1)) \u03c9 - stoppedValue f (lowerCrossingTime a b f N k) \u03c9", "state_after": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\nhf : a \u2264 f N \u03c9\nhab : a < b\nhN : \u00acN = 0\nk : \u2115\n\u22a2 \u2211 n in Finset.range N,\n      Set.indicator (Set.Ico (lowerCrossingTime a b f N k \u03c9) (upperCrossingTime a b f N (k + 1) \u03c9))\n        (fun m => f (m + 1) \u03c9 - f m \u03c9) n =\n    stoppedValue f (upperCrossingTime a b f N (k + 1)) \u03c9 - stoppedValue f (lowerCrossingTime a b f N k) \u03c9"}, {"tactic": "rw [Finset.sum_indicator_eq_sum_filter, (_ : Finset.filter (fun i => i \u2208 Set.Ico\n  (lowerCrossingTime a b f N k \u03c9) (upperCrossingTime a b f N (k + 1) \u03c9)) (Finset.range N) =\n  Finset.Ico (lowerCrossingTime a b f N k \u03c9) (upperCrossingTime a b f N (k + 1) \u03c9)),\n  Finset.sum_Ico_eq_add_neg _ lowerCrossingTime_le_upperCrossingTime_succ,\n  Finset.sum_range_sub fun n => f n \u03c9, Finset.sum_range_sub fun n => f n \u03c9, neg_sub,\n  sub_add_sub_cancel]", "annotated_tactic": ["rw [<a>Finset.sum_indicator_eq_sum_filter</a>, (_ : <a>Finset.filter</a> (fun i => i \u2208 <a>Set.Ico</a>\n      (<a>lowerCrossingTime</a> a b f N k \u03c9) (<a>upperCrossingTime</a> a b f N (k + 1) \u03c9)) (<a>Finset.range</a> N) =\n      <a>Finset.Ico</a> (<a>lowerCrossingTime</a> a b f N k \u03c9) (<a>upperCrossingTime</a> a b f N (k + 1) \u03c9)),\n      <a>Finset.sum_Ico_eq_add_neg</a> _ <a>lowerCrossingTime_le_upperCrossingTime_succ</a>,\n      <a>Finset.sum_range_sub</a> fun n => f n \u03c9, <a>Finset.sum_range_sub</a> fun n => f n \u03c9, <a>neg_sub</a>,\n      <a>sub_add_sub_cancel</a>]", [{"full_name": "Finset.sum_indicator_eq_sum_filter", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [648, 3], "def_end_pos": [648, 14]}, {"full_name": "Finset.filter", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2691, 5], "def_end_pos": [2691, 11]}, {"full_name": "Set.Ico", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [49, 5], "def_end_pos": [49, 8]}, {"full_name": "MeasureTheory.lowerCrossingTime", "def_path": "Mathlib/Probability/Martingale/Upcrossing.lean", "def_pos": [148, 19], "def_end_pos": [148, 36]}, {"full_name": "MeasureTheory.upperCrossingTime", "def_path": "Mathlib/Probability/Martingale/Upcrossing.lean", "def_pos": [139, 19], "def_end_pos": [139, 36]}, {"full_name": "Finset.range", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3027, 5], "def_end_pos": [3027, 10]}, {"full_name": "Finset.Ico", "def_path": "Mathlib/Order/LocallyFinite.lean", "def_pos": [309, 5], "def_end_pos": [309, 8]}, {"full_name": "MeasureTheory.lowerCrossingTime", "def_path": "Mathlib/Probability/Martingale/Upcrossing.lean", "def_pos": [148, 19], "def_end_pos": [148, 36]}, {"full_name": "MeasureTheory.upperCrossingTime", "def_path": "Mathlib/Probability/Martingale/Upcrossing.lean", "def_pos": [139, 19], "def_end_pos": [139, 36]}, {"full_name": "Finset.sum_Ico_eq_add_neg", "def_path": "Mathlib/Algebra/BigOperators/Intervals.lean", "def_pos": [95, 3], "def_end_pos": [95, 14]}, {"full_name": "MeasureTheory.lowerCrossingTime_le_upperCrossingTime_succ", "def_path": "Mathlib/Probability/Martingale/Upcrossing.lean", "def_pos": [207, 9], "def_end_pos": [207, 52]}, {"full_name": "Finset.sum_range_sub", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [1397, 3], "def_end_pos": [1397, 14]}, {"full_name": "Finset.sum_range_sub", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [1397, 3], "def_end_pos": [1397, 14]}, {"full_name": "neg_sub", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [400, 3], "def_end_pos": [400, 14]}, {"full_name": "sub_add_sub_cancel", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [789, 30], "def_end_pos": [789, 48]}]], "state_before": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\nhf : a \u2264 f N \u03c9\nhab : a < b\nhN : \u00acN = 0\nk : \u2115\n\u22a2 \u2211 n in Finset.range N,\n      Set.indicator (Set.Ico (lowerCrossingTime a b f N k \u03c9) (upperCrossingTime a b f N (k + 1) \u03c9))\n        (fun m => f (m + 1) \u03c9 - f m \u03c9) n =\n    stoppedValue f (upperCrossingTime a b f N (k + 1)) \u03c9 - stoppedValue f (lowerCrossingTime a b f N k) \u03c9", "state_after": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\nhf : a \u2264 f N \u03c9\nhab : a < b\nhN : \u00acN = 0\nk : \u2115\n\u22a2 f (upperCrossingTime a b f N (k + 1) \u03c9) \u03c9 - f (lowerCrossingTime a b f N k \u03c9) \u03c9 =\n    stoppedValue f (upperCrossingTime a b f N (k + 1)) \u03c9 - stoppedValue f (lowerCrossingTime a b f N k) \u03c9\n\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\nhf : a \u2264 f N \u03c9\nhab : a < b\nhN : \u00acN = 0\nk : \u2115\n\u22a2 Finset.filter (fun i => i \u2208 Set.Ico (lowerCrossingTime a b f N k \u03c9) (upperCrossingTime a b f N (k + 1) \u03c9))\n      (Finset.range N) =\n    Finset.Ico (lowerCrossingTime a b f N k \u03c9) (upperCrossingTime a b f N (k + 1) \u03c9)"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\nhf : a \u2264 f N \u03c9\nhab : a < b\nhN : \u00acN = 0\nk : \u2115\n\u22a2 f (upperCrossingTime a b f N (k + 1) \u03c9) \u03c9 - f (lowerCrossingTime a b f N k \u03c9) \u03c9 =\n    stoppedValue f (upperCrossingTime a b f N (k + 1)) \u03c9 - stoppedValue f (lowerCrossingTime a b f N k) \u03c9", "state_after": "no goals"}, {"tactic": "ext i", "annotated_tactic": ["ext i", []], "state_before": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\nhf : a \u2264 f N \u03c9\nhab : a < b\nhN : \u00acN = 0\nk : \u2115\n\u22a2 Finset.filter (fun i => i \u2208 Set.Ico (lowerCrossingTime a b f N k \u03c9) (upperCrossingTime a b f N (k + 1) \u03c9))\n      (Finset.range N) =\n    Finset.Ico (lowerCrossingTime a b f N k \u03c9) (upperCrossingTime a b f N (k + 1) \u03c9)", "state_after": "case a\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\nhf : a \u2264 f N \u03c9\nhab : a < b\nhN : \u00acN = 0\nk i : \u2115\n\u22a2 i \u2208\n      Finset.filter (fun i => i \u2208 Set.Ico (lowerCrossingTime a b f N k \u03c9) (upperCrossingTime a b f N (k + 1) \u03c9))\n        (Finset.range N) \u2194\n    i \u2208 Finset.Ico (lowerCrossingTime a b f N k \u03c9) (upperCrossingTime a b f N (k + 1) \u03c9)"}, {"tactic": "simp only [Set.mem_Ico, Finset.mem_filter, Finset.mem_range, Finset.mem_Ico,\n  and_iff_right_iff_imp, and_imp]", "annotated_tactic": ["simp only [<a>Set.mem_Ico</a>, <a>Finset.mem_filter</a>, <a>Finset.mem_range</a>, <a>Finset.mem_Ico</a>,\n        <a>and_iff_right_iff_imp</a>, <a>and_imp</a>]", [{"full_name": "Set.mem_Ico", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [121, 9], "def_end_pos": [121, 16]}, {"full_name": "Finset.mem_filter", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2708, 9], "def_end_pos": [2708, 19]}, {"full_name": "Finset.mem_range", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3037, 9], "def_end_pos": [3037, 18]}, {"full_name": "Finset.mem_Ico", "def_path": "Mathlib/Order/LocallyFinite.lean", "def_pos": [331, 9], "def_end_pos": [331, 16]}, {"full_name": "and_iff_right_iff_imp", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [211, 17], "def_end_pos": [211, 38]}, {"full_name": "and_imp", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [313, 17], "def_end_pos": [313, 24]}]], "state_before": "case a\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\nhf : a \u2264 f N \u03c9\nhab : a < b\nhN : \u00acN = 0\nk i : \u2115\n\u22a2 i \u2208\n      Finset.filter (fun i => i \u2208 Set.Ico (lowerCrossingTime a b f N k \u03c9) (upperCrossingTime a b f N (k + 1) \u03c9))\n        (Finset.range N) \u2194\n    i \u2208 Finset.Ico (lowerCrossingTime a b f N k \u03c9) (upperCrossingTime a b f N (k + 1) \u03c9)", "state_after": "case a\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\nhf : a \u2264 f N \u03c9\nhab : a < b\nhN : \u00acN = 0\nk i : \u2115\n\u22a2 lowerCrossingTime a b f N k \u03c9 \u2264 i \u2192 i < upperCrossingTime a b f N (k + 1) \u03c9 \u2192 i < N"}, {"tactic": "exact fun _ h => lt_of_lt_of_le h upperCrossingTime_le", "annotated_tactic": ["exact fun _ h => <a>lt_of_lt_of_le</a> h <a>upperCrossingTime_le</a>", [{"full_name": "lt_of_lt_of_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [115, 9], "def_end_pos": [115, 23]}, {"full_name": "MeasureTheory.upperCrossingTime_le", "def_path": "Mathlib/Probability/Martingale/Upcrossing.lean", "def_pos": [187, 9], "def_end_pos": [187, 29]}]], "state_before": "case a\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\nhf : a \u2264 f N \u03c9\nhab : a < b\nhN : \u00acN = 0\nk i : \u2115\n\u22a2 lowerCrossingTime a b f N k \u03c9 \u2264 i \u2192 i < upperCrossingTime a b f N (k + 1) \u03c9 \u2192 i < N", "state_after": "no goals"}, {"tactic": "refine' Finset.sum_le_sum fun i hi =>\n  le_sub_of_le_upcrossingsBefore (zero_lt_iff.2 hN) hab _", "annotated_tactic": ["refine' <a>Finset.sum_le_sum</a> fun i hi =>\n          <a>le_sub_of_le_upcrossingsBefore</a> (<a>zero_lt_iff</a>.2 hN) hab _", [{"full_name": "Finset.sum_le_sum", "def_path": "Mathlib/Algebra/BigOperators/Order.lean", "def_pos": [111, 15], "def_end_pos": [111, 25]}, {"full_name": "MeasureTheory.le_sub_of_le_upcrossingsBefore", "def_path": "Mathlib/Probability/Martingale/Upcrossing.lean", "def_pos": [579, 9], "def_end_pos": [579, 39]}, {"full_name": "zero_lt_iff", "def_path": "Mathlib/Algebra/Order/WithZero.lean", "def_pos": [106, 9], "def_end_pos": [106, 20]}]], "state_before": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\nhf : a \u2264 f N \u03c9\nhab : a < b\nhN : \u00acN = 0\nh\u2081 :\n  \u2200 (k : \u2115),\n    \u2211 n in Finset.range N,\n        Set.indicator (Set.Ico (lowerCrossingTime a b f N k \u03c9) (upperCrossingTime a b f N (k + 1) \u03c9))\n          (fun m => f (m + 1) \u03c9 - f m \u03c9) n =\n      stoppedValue f (upperCrossingTime a b f N (k + 1)) \u03c9 - stoppedValue f (lowerCrossingTime a b f N k) \u03c9\n\u22a2 \u2211 _k in Finset.range (upcrossingsBefore a b f N \u03c9), (b - a) \u2264\n    \u2211 k in Finset.range (upcrossingsBefore a b f N \u03c9),\n      (stoppedValue f (upperCrossingTime a b f N (k + 1)) \u03c9 - stoppedValue f (lowerCrossingTime a b f N k) \u03c9)", "state_after": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\nhf : a \u2264 f N \u03c9\nhab : a < b\nhN : \u00acN = 0\nh\u2081 :\n  \u2200 (k : \u2115),\n    \u2211 n in Finset.range N,\n        Set.indicator (Set.Ico (lowerCrossingTime a b f N k \u03c9) (upperCrossingTime a b f N (k + 1) \u03c9))\n          (fun m => f (m + 1) \u03c9 - f m \u03c9) n =\n      stoppedValue f (upperCrossingTime a b f N (k + 1)) \u03c9 - stoppedValue f (lowerCrossingTime a b f N k) \u03c9\ni : \u2115\nhi : i \u2208 Finset.range (upcrossingsBefore a b f N \u03c9)\n\u22a2 i < upcrossingsBefore a b f N \u03c9"}, {"tactic": "rwa [Finset.mem_range] at hi", "annotated_tactic": ["rwa [<a>Finset.mem_range</a>] at hi", [{"full_name": "Finset.mem_range", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3037, 9], "def_end_pos": [3037, 18]}]], "state_before": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\nhf : a \u2264 f N \u03c9\nhab : a < b\nhN : \u00acN = 0\nh\u2081 :\n  \u2200 (k : \u2115),\n    \u2211 n in Finset.range N,\n        Set.indicator (Set.Ico (lowerCrossingTime a b f N k \u03c9) (upperCrossingTime a b f N (k + 1) \u03c9))\n          (fun m => f (m + 1) \u03c9 - f m \u03c9) n =\n      stoppedValue f (upperCrossingTime a b f N (k + 1)) \u03c9 - stoppedValue f (lowerCrossingTime a b f N k) \u03c9\ni : \u2115\nhi : i \u2208 Finset.range (upcrossingsBefore a b f N \u03c9)\n\u22a2 i < upcrossingsBefore a b f N \u03c9", "state_after": "no goals"}, {"tactic": "refine' Finset.sum_le_sum_of_subset_of_nonneg\n  (Finset.range_subset.2 (upcrossingsBefore_le f \u03c9 hab)) fun i _ hi => _", "annotated_tactic": ["refine' <a>Finset.sum_le_sum_of_subset_of_nonneg</a>\n          (<a>Finset.range_subset</a>.2 (<a>upcrossingsBefore_le</a> f \u03c9 hab)) fun i _ hi => _", [{"full_name": "Finset.sum_le_sum_of_subset_of_nonneg", "def_path": "Mathlib/Algebra/BigOperators/Order.lean", "def_pos": [156, 15], "def_end_pos": [156, 45]}, {"full_name": "Finset.range_subset", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3075, 9], "def_end_pos": [3075, 21]}, {"full_name": "MeasureTheory.upcrossingsBefore_le", "def_path": "Mathlib/Probability/Martingale/Upcrossing.lean", "def_pos": [482, 9], "def_end_pos": [482, 29]}]], "state_before": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\nhf : a \u2264 f N \u03c9\nhab : a < b\nhN : \u00acN = 0\nh\u2081 :\n  \u2200 (k : \u2115),\n    \u2211 n in Finset.range N,\n        Set.indicator (Set.Ico (lowerCrossingTime a b f N k \u03c9) (upperCrossingTime a b f N (k + 1) \u03c9))\n          (fun m => f (m + 1) \u03c9 - f m \u03c9) n =\n      stoppedValue f (upperCrossingTime a b f N (k + 1)) \u03c9 - stoppedValue f (lowerCrossingTime a b f N k) \u03c9\n\u22a2 \u2211 k in Finset.range (upcrossingsBefore a b f N \u03c9),\n      (stoppedValue f (upperCrossingTime a b f N (k + 1)) \u03c9 - stoppedValue f (lowerCrossingTime a b f N k) \u03c9) \u2264\n    \u2211 k in Finset.range N,\n      (stoppedValue f (upperCrossingTime a b f N (k + 1)) \u03c9 - stoppedValue f (lowerCrossingTime a b f N k) \u03c9)", "state_after": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\nhf : a \u2264 f N \u03c9\nhab : a < b\nhN : \u00acN = 0\nh\u2081 :\n  \u2200 (k : \u2115),\n    \u2211 n in Finset.range N,\n        Set.indicator (Set.Ico (lowerCrossingTime a b f N k \u03c9) (upperCrossingTime a b f N (k + 1) \u03c9))\n          (fun m => f (m + 1) \u03c9 - f m \u03c9) n =\n      stoppedValue f (upperCrossingTime a b f N (k + 1)) \u03c9 - stoppedValue f (lowerCrossingTime a b f N k) \u03c9\ni : \u2115\nx\u271d : i \u2208 Finset.range N\nhi : \u00aci \u2208 Finset.range (upcrossingsBefore a b f N \u03c9)\n\u22a2 0 \u2264 stoppedValue f (upperCrossingTime a b f N (i + 1)) \u03c9 - stoppedValue f (lowerCrossingTime a b f N i) \u03c9"}, {"tactic": "by_cases hi' : i = upcrossingsBefore a b f N \u03c9", "annotated_tactic": ["by_cases hi' : i = <a>upcrossingsBefore</a> a b f N \u03c9", [{"full_name": "MeasureTheory.upcrossingsBefore", "def_path": "Mathlib/Probability/Martingale/Upcrossing.lean", "def_pos": [450, 19], "def_end_pos": [450, 36]}]], "state_before": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\nhf : a \u2264 f N \u03c9\nhab : a < b\nhN : \u00acN = 0\nh\u2081 :\n  \u2200 (k : \u2115),\n    \u2211 n in Finset.range N,\n        Set.indicator (Set.Ico (lowerCrossingTime a b f N k \u03c9) (upperCrossingTime a b f N (k + 1) \u03c9))\n          (fun m => f (m + 1) \u03c9 - f m \u03c9) n =\n      stoppedValue f (upperCrossingTime a b f N (k + 1)) \u03c9 - stoppedValue f (lowerCrossingTime a b f N k) \u03c9\ni : \u2115\nx\u271d : i \u2208 Finset.range N\nhi : \u00aci \u2208 Finset.range (upcrossingsBefore a b f N \u03c9)\n\u22a2 0 \u2264 stoppedValue f (upperCrossingTime a b f N (i + 1)) \u03c9 - stoppedValue f (lowerCrossingTime a b f N i) \u03c9", "state_after": "case pos\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\nhf : a \u2264 f N \u03c9\nhab : a < b\nhN : \u00acN = 0\nh\u2081 :\n  \u2200 (k : \u2115),\n    \u2211 n in Finset.range N,\n        Set.indicator (Set.Ico (lowerCrossingTime a b f N k \u03c9) (upperCrossingTime a b f N (k + 1) \u03c9))\n          (fun m => f (m + 1) \u03c9 - f m \u03c9) n =\n      stoppedValue f (upperCrossingTime a b f N (k + 1)) \u03c9 - stoppedValue f (lowerCrossingTime a b f N k) \u03c9\ni : \u2115\nx\u271d : i \u2208 Finset.range N\nhi : \u00aci \u2208 Finset.range (upcrossingsBefore a b f N \u03c9)\nhi' : i = upcrossingsBefore a b f N \u03c9\n\u22a2 0 \u2264 stoppedValue f (upperCrossingTime a b f N (i + 1)) \u03c9 - stoppedValue f (lowerCrossingTime a b f N i) \u03c9\n\ncase neg\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\nhf : a \u2264 f N \u03c9\nhab : a < b\nhN : \u00acN = 0\nh\u2081 :\n  \u2200 (k : \u2115),\n    \u2211 n in Finset.range N,\n        Set.indicator (Set.Ico (lowerCrossingTime a b f N k \u03c9) (upperCrossingTime a b f N (k + 1) \u03c9))\n          (fun m => f (m + 1) \u03c9 - f m \u03c9) n =\n      stoppedValue f (upperCrossingTime a b f N (k + 1)) \u03c9 - stoppedValue f (lowerCrossingTime a b f N k) \u03c9\ni : \u2115\nx\u271d : i \u2208 Finset.range N\nhi : \u00aci \u2208 Finset.range (upcrossingsBefore a b f N \u03c9)\nhi' : \u00aci = upcrossingsBefore a b f N \u03c9\n\u22a2 0 \u2264 stoppedValue f (upperCrossingTime a b f N (i + 1)) \u03c9 - stoppedValue f (lowerCrossingTime a b f N i) \u03c9"}, {"tactic": "subst hi'", "annotated_tactic": ["subst hi'", []], "state_before": "case pos\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\nhf : a \u2264 f N \u03c9\nhab : a < b\nhN : \u00acN = 0\nh\u2081 :\n  \u2200 (k : \u2115),\n    \u2211 n in Finset.range N,\n        Set.indicator (Set.Ico (lowerCrossingTime a b f N k \u03c9) (upperCrossingTime a b f N (k + 1) \u03c9))\n          (fun m => f (m + 1) \u03c9 - f m \u03c9) n =\n      stoppedValue f (upperCrossingTime a b f N (k + 1)) \u03c9 - stoppedValue f (lowerCrossingTime a b f N k) \u03c9\ni : \u2115\nx\u271d : i \u2208 Finset.range N\nhi : \u00aci \u2208 Finset.range (upcrossingsBefore a b f N \u03c9)\nhi' : i = upcrossingsBefore a b f N \u03c9\n\u22a2 0 \u2264 stoppedValue f (upperCrossingTime a b f N (i + 1)) \u03c9 - stoppedValue f (lowerCrossingTime a b f N i) \u03c9", "state_after": "case pos\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\nhf : a \u2264 f N \u03c9\nhab : a < b\nhN : \u00acN = 0\nh\u2081 :\n  \u2200 (k : \u2115),\n    \u2211 n in Finset.range N,\n        Set.indicator (Set.Ico (lowerCrossingTime a b f N k \u03c9) (upperCrossingTime a b f N (k + 1) \u03c9))\n          (fun m => f (m + 1) \u03c9 - f m \u03c9) n =\n      stoppedValue f (upperCrossingTime a b f N (k + 1)) \u03c9 - stoppedValue f (lowerCrossingTime a b f N k) \u03c9\nx\u271d : upcrossingsBefore a b f N \u03c9 \u2208 Finset.range N\nhi : \u00acupcrossingsBefore a b f N \u03c9 \u2208 Finset.range (upcrossingsBefore a b f N \u03c9)\n\u22a2 0 \u2264\n    stoppedValue f (upperCrossingTime a b f N (upcrossingsBefore a b f N \u03c9 + 1)) \u03c9 -\n      stoppedValue f (lowerCrossingTime a b f N (upcrossingsBefore a b f N \u03c9)) \u03c9"}, {"tactic": "simp only [stoppedValue]", "annotated_tactic": ["simp only [<a>stoppedValue</a>]", [{"full_name": "MeasureTheory.stoppedValue", "def_path": "Mathlib/Probability/Process/Stopping.lean", "def_pos": [768, 5], "def_end_pos": [768, 17]}]], "state_before": "case pos\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\nhf : a \u2264 f N \u03c9\nhab : a < b\nhN : \u00acN = 0\nh\u2081 :\n  \u2200 (k : \u2115),\n    \u2211 n in Finset.range N,\n        Set.indicator (Set.Ico (lowerCrossingTime a b f N k \u03c9) (upperCrossingTime a b f N (k + 1) \u03c9))\n          (fun m => f (m + 1) \u03c9 - f m \u03c9) n =\n      stoppedValue f (upperCrossingTime a b f N (k + 1)) \u03c9 - stoppedValue f (lowerCrossingTime a b f N k) \u03c9\nx\u271d : upcrossingsBefore a b f N \u03c9 \u2208 Finset.range N\nhi : \u00acupcrossingsBefore a b f N \u03c9 \u2208 Finset.range (upcrossingsBefore a b f N \u03c9)\n\u22a2 0 \u2264\n    stoppedValue f (upperCrossingTime a b f N (upcrossingsBefore a b f N \u03c9 + 1)) \u03c9 -\n      stoppedValue f (lowerCrossingTime a b f N (upcrossingsBefore a b f N \u03c9)) \u03c9", "state_after": "case pos\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\nhf : a \u2264 f N \u03c9\nhab : a < b\nhN : \u00acN = 0\nh\u2081 :\n  \u2200 (k : \u2115),\n    \u2211 n in Finset.range N,\n        Set.indicator (Set.Ico (lowerCrossingTime a b f N k \u03c9) (upperCrossingTime a b f N (k + 1) \u03c9))\n          (fun m => f (m + 1) \u03c9 - f m \u03c9) n =\n      stoppedValue f (upperCrossingTime a b f N (k + 1)) \u03c9 - stoppedValue f (lowerCrossingTime a b f N k) \u03c9\nx\u271d : upcrossingsBefore a b f N \u03c9 \u2208 Finset.range N\nhi : \u00acupcrossingsBefore a b f N \u03c9 \u2208 Finset.range (upcrossingsBefore a b f N \u03c9)\n\u22a2 0 \u2264\n    f (upperCrossingTime a b f N (upcrossingsBefore a b f N \u03c9 + 1) \u03c9) \u03c9 -\n      f (lowerCrossingTime a b f N (upcrossingsBefore a b f N \u03c9) \u03c9) \u03c9"}, {"tactic": "rw [upperCrossingTime_eq_of_upcrossingsBefore_lt hab (Nat.lt_succ_self _)]", "annotated_tactic": ["rw [<a>upperCrossingTime_eq_of_upcrossingsBefore_lt</a> hab (<a>Nat.lt_succ_self</a> _)]", [{"full_name": "MeasureTheory.upperCrossingTime_eq_of_upcrossingsBefore_lt", "def_path": "Mathlib/Probability/Martingale/Upcrossing.lean", "def_pos": [476, 9], "def_end_pos": [476, 53]}, {"full_name": "Nat.lt_succ_self", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [294, 9], "def_end_pos": [294, 21]}]], "state_before": "case pos\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\nhf : a \u2264 f N \u03c9\nhab : a < b\nhN : \u00acN = 0\nh\u2081 :\n  \u2200 (k : \u2115),\n    \u2211 n in Finset.range N,\n        Set.indicator (Set.Ico (lowerCrossingTime a b f N k \u03c9) (upperCrossingTime a b f N (k + 1) \u03c9))\n          (fun m => f (m + 1) \u03c9 - f m \u03c9) n =\n      stoppedValue f (upperCrossingTime a b f N (k + 1)) \u03c9 - stoppedValue f (lowerCrossingTime a b f N k) \u03c9\nx\u271d : upcrossingsBefore a b f N \u03c9 \u2208 Finset.range N\nhi : \u00acupcrossingsBefore a b f N \u03c9 \u2208 Finset.range (upcrossingsBefore a b f N \u03c9)\n\u22a2 0 \u2264\n    f (upperCrossingTime a b f N (upcrossingsBefore a b f N \u03c9 + 1) \u03c9) \u03c9 -\n      f (lowerCrossingTime a b f N (upcrossingsBefore a b f N \u03c9) \u03c9) \u03c9", "state_after": "case pos\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\nhf : a \u2264 f N \u03c9\nhab : a < b\nhN : \u00acN = 0\nh\u2081 :\n  \u2200 (k : \u2115),\n    \u2211 n in Finset.range N,\n        Set.indicator (Set.Ico (lowerCrossingTime a b f N k \u03c9) (upperCrossingTime a b f N (k + 1) \u03c9))\n          (fun m => f (m + 1) \u03c9 - f m \u03c9) n =\n      stoppedValue f (upperCrossingTime a b f N (k + 1)) \u03c9 - stoppedValue f (lowerCrossingTime a b f N k) \u03c9\nx\u271d : upcrossingsBefore a b f N \u03c9 \u2208 Finset.range N\nhi : \u00acupcrossingsBefore a b f N \u03c9 \u2208 Finset.range (upcrossingsBefore a b f N \u03c9)\n\u22a2 0 \u2264 f N \u03c9 - f (lowerCrossingTime a b f N (upcrossingsBefore a b f N \u03c9) \u03c9) \u03c9"}, {"tactic": "by_cases heq : lowerCrossingTime a b f N (upcrossingsBefore a b f N \u03c9) \u03c9 = N", "annotated_tactic": ["by_cases heq : <a>lowerCrossingTime</a> a b f N (<a>upcrossingsBefore</a> a b f N \u03c9) \u03c9 = N", [{"full_name": "MeasureTheory.lowerCrossingTime", "def_path": "Mathlib/Probability/Martingale/Upcrossing.lean", "def_pos": [148, 19], "def_end_pos": [148, 36]}, {"full_name": "MeasureTheory.upcrossingsBefore", "def_path": "Mathlib/Probability/Martingale/Upcrossing.lean", "def_pos": [450, 19], "def_end_pos": [450, 36]}]], "state_before": "case pos\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\nhf : a \u2264 f N \u03c9\nhab : a < b\nhN : \u00acN = 0\nh\u2081 :\n  \u2200 (k : \u2115),\n    \u2211 n in Finset.range N,\n        Set.indicator (Set.Ico (lowerCrossingTime a b f N k \u03c9) (upperCrossingTime a b f N (k + 1) \u03c9))\n          (fun m => f (m + 1) \u03c9 - f m \u03c9) n =\n      stoppedValue f (upperCrossingTime a b f N (k + 1)) \u03c9 - stoppedValue f (lowerCrossingTime a b f N k) \u03c9\nx\u271d : upcrossingsBefore a b f N \u03c9 \u2208 Finset.range N\nhi : \u00acupcrossingsBefore a b f N \u03c9 \u2208 Finset.range (upcrossingsBefore a b f N \u03c9)\n\u22a2 0 \u2264 f N \u03c9 - f (lowerCrossingTime a b f N (upcrossingsBefore a b f N \u03c9) \u03c9) \u03c9", "state_after": "case pos\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\nhf : a \u2264 f N \u03c9\nhab : a < b\nhN : \u00acN = 0\nh\u2081 :\n  \u2200 (k : \u2115),\n    \u2211 n in Finset.range N,\n        Set.indicator (Set.Ico (lowerCrossingTime a b f N k \u03c9) (upperCrossingTime a b f N (k + 1) \u03c9))\n          (fun m => f (m + 1) \u03c9 - f m \u03c9) n =\n      stoppedValue f (upperCrossingTime a b f N (k + 1)) \u03c9 - stoppedValue f (lowerCrossingTime a b f N k) \u03c9\nx\u271d : upcrossingsBefore a b f N \u03c9 \u2208 Finset.range N\nhi : \u00acupcrossingsBefore a b f N \u03c9 \u2208 Finset.range (upcrossingsBefore a b f N \u03c9)\nheq : lowerCrossingTime a b f N (upcrossingsBefore a b f N \u03c9) \u03c9 = N\n\u22a2 0 \u2264 f N \u03c9 - f (lowerCrossingTime a b f N (upcrossingsBefore a b f N \u03c9) \u03c9) \u03c9\n\ncase neg\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\nhf : a \u2264 f N \u03c9\nhab : a < b\nhN : \u00acN = 0\nh\u2081 :\n  \u2200 (k : \u2115),\n    \u2211 n in Finset.range N,\n        Set.indicator (Set.Ico (lowerCrossingTime a b f N k \u03c9) (upperCrossingTime a b f N (k + 1) \u03c9))\n          (fun m => f (m + 1) \u03c9 - f m \u03c9) n =\n      stoppedValue f (upperCrossingTime a b f N (k + 1)) \u03c9 - stoppedValue f (lowerCrossingTime a b f N k) \u03c9\nx\u271d : upcrossingsBefore a b f N \u03c9 \u2208 Finset.range N\nhi : \u00acupcrossingsBefore a b f N \u03c9 \u2208 Finset.range (upcrossingsBefore a b f N \u03c9)\nheq : \u00aclowerCrossingTime a b f N (upcrossingsBefore a b f N \u03c9) \u03c9 = N\n\u22a2 0 \u2264 f N \u03c9 - f (lowerCrossingTime a b f N (upcrossingsBefore a b f N \u03c9) \u03c9) \u03c9"}, {"tactic": "rw [heq, sub_self]", "annotated_tactic": ["rw [heq, <a>sub_self</a>]", [{"full_name": "sub_self", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [734, 30], "def_end_pos": [734, 38]}]], "state_before": "case pos\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\nhf : a \u2264 f N \u03c9\nhab : a < b\nhN : \u00acN = 0\nh\u2081 :\n  \u2200 (k : \u2115),\n    \u2211 n in Finset.range N,\n        Set.indicator (Set.Ico (lowerCrossingTime a b f N k \u03c9) (upperCrossingTime a b f N (k + 1) \u03c9))\n          (fun m => f (m + 1) \u03c9 - f m \u03c9) n =\n      stoppedValue f (upperCrossingTime a b f N (k + 1)) \u03c9 - stoppedValue f (lowerCrossingTime a b f N k) \u03c9\nx\u271d : upcrossingsBefore a b f N \u03c9 \u2208 Finset.range N\nhi : \u00acupcrossingsBefore a b f N \u03c9 \u2208 Finset.range (upcrossingsBefore a b f N \u03c9)\nheq : lowerCrossingTime a b f N (upcrossingsBefore a b f N \u03c9) \u03c9 = N\n\u22a2 0 \u2264 f N \u03c9 - f (lowerCrossingTime a b f N (upcrossingsBefore a b f N \u03c9) \u03c9) \u03c9", "state_after": "no goals"}, {"tactic": "rw [sub_nonneg]", "annotated_tactic": ["rw [<a>sub_nonneg</a>]", [{"full_name": "sub_nonneg", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [720, 30], "def_end_pos": [720, 40]}]], "state_before": "case neg\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\nhf : a \u2264 f N \u03c9\nhab : a < b\nhN : \u00acN = 0\nh\u2081 :\n  \u2200 (k : \u2115),\n    \u2211 n in Finset.range N,\n        Set.indicator (Set.Ico (lowerCrossingTime a b f N k \u03c9) (upperCrossingTime a b f N (k + 1) \u03c9))\n          (fun m => f (m + 1) \u03c9 - f m \u03c9) n =\n      stoppedValue f (upperCrossingTime a b f N (k + 1)) \u03c9 - stoppedValue f (lowerCrossingTime a b f N k) \u03c9\nx\u271d : upcrossingsBefore a b f N \u03c9 \u2208 Finset.range N\nhi : \u00acupcrossingsBefore a b f N \u03c9 \u2208 Finset.range (upcrossingsBefore a b f N \u03c9)\nheq : \u00aclowerCrossingTime a b f N (upcrossingsBefore a b f N \u03c9) \u03c9 = N\n\u22a2 0 \u2264 f N \u03c9 - f (lowerCrossingTime a b f N (upcrossingsBefore a b f N \u03c9) \u03c9) \u03c9", "state_after": "case neg\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\nhf : a \u2264 f N \u03c9\nhab : a < b\nhN : \u00acN = 0\nh\u2081 :\n  \u2200 (k : \u2115),\n    \u2211 n in Finset.range N,\n        Set.indicator (Set.Ico (lowerCrossingTime a b f N k \u03c9) (upperCrossingTime a b f N (k + 1) \u03c9))\n          (fun m => f (m + 1) \u03c9 - f m \u03c9) n =\n      stoppedValue f (upperCrossingTime a b f N (k + 1)) \u03c9 - stoppedValue f (lowerCrossingTime a b f N k) \u03c9\nx\u271d : upcrossingsBefore a b f N \u03c9 \u2208 Finset.range N\nhi : \u00acupcrossingsBefore a b f N \u03c9 \u2208 Finset.range (upcrossingsBefore a b f N \u03c9)\nheq : \u00aclowerCrossingTime a b f N (upcrossingsBefore a b f N \u03c9) \u03c9 = N\n\u22a2 f (lowerCrossingTime a b f N (upcrossingsBefore a b f N \u03c9) \u03c9) \u03c9 \u2264 f N \u03c9"}, {"tactic": "exact le_trans (stoppedValue_lowerCrossingTime heq) hf", "annotated_tactic": ["exact <a>le_trans</a> (<a>stoppedValue_lowerCrossingTime</a> heq) hf", [{"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}, {"full_name": "MeasureTheory.stoppedValue_lowerCrossingTime", "def_path": "Mathlib/Probability/Martingale/Upcrossing.lean", "def_pos": [231, 9], "def_end_pos": [231, 39]}]], "state_before": "case neg\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\nhf : a \u2264 f N \u03c9\nhab : a < b\nhN : \u00acN = 0\nh\u2081 :\n  \u2200 (k : \u2115),\n    \u2211 n in Finset.range N,\n        Set.indicator (Set.Ico (lowerCrossingTime a b f N k \u03c9) (upperCrossingTime a b f N (k + 1) \u03c9))\n          (fun m => f (m + 1) \u03c9 - f m \u03c9) n =\n      stoppedValue f (upperCrossingTime a b f N (k + 1)) \u03c9 - stoppedValue f (lowerCrossingTime a b f N k) \u03c9\nx\u271d : upcrossingsBefore a b f N \u03c9 \u2208 Finset.range N\nhi : \u00acupcrossingsBefore a b f N \u03c9 \u2208 Finset.range (upcrossingsBefore a b f N \u03c9)\nheq : \u00aclowerCrossingTime a b f N (upcrossingsBefore a b f N \u03c9) \u03c9 = N\n\u22a2 f (lowerCrossingTime a b f N (upcrossingsBefore a b f N \u03c9) \u03c9) \u03c9 \u2264 f N \u03c9", "state_after": "no goals"}, {"tactic": "rw [sub_eq_zero_of_upcrossingsBefore_lt hab]", "annotated_tactic": ["rw [<a>sub_eq_zero_of_upcrossingsBefore_lt</a> hab]", [{"full_name": "MeasureTheory.sub_eq_zero_of_upcrossingsBefore_lt", "def_path": "Mathlib/Probability/Martingale/Upcrossing.lean", "def_pos": [588, 9], "def_end_pos": [588, 44]}]], "state_before": "case neg\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\nhf : a \u2264 f N \u03c9\nhab : a < b\nhN : \u00acN = 0\nh\u2081 :\n  \u2200 (k : \u2115),\n    \u2211 n in Finset.range N,\n        Set.indicator (Set.Ico (lowerCrossingTime a b f N k \u03c9) (upperCrossingTime a b f N (k + 1) \u03c9))\n          (fun m => f (m + 1) \u03c9 - f m \u03c9) n =\n      stoppedValue f (upperCrossingTime a b f N (k + 1)) \u03c9 - stoppedValue f (lowerCrossingTime a b f N k) \u03c9\ni : \u2115\nx\u271d : i \u2208 Finset.range N\nhi : \u00aci \u2208 Finset.range (upcrossingsBefore a b f N \u03c9)\nhi' : \u00aci = upcrossingsBefore a b f N \u03c9\n\u22a2 0 \u2264 stoppedValue f (upperCrossingTime a b f N (i + 1)) \u03c9 - stoppedValue f (lowerCrossingTime a b f N i) \u03c9", "state_after": "case neg\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\nhf : a \u2264 f N \u03c9\nhab : a < b\nhN : \u00acN = 0\nh\u2081 :\n  \u2200 (k : \u2115),\n    \u2211 n in Finset.range N,\n        Set.indicator (Set.Ico (lowerCrossingTime a b f N k \u03c9) (upperCrossingTime a b f N (k + 1) \u03c9))\n          (fun m => f (m + 1) \u03c9 - f m \u03c9) n =\n      stoppedValue f (upperCrossingTime a b f N (k + 1)) \u03c9 - stoppedValue f (lowerCrossingTime a b f N k) \u03c9\ni : \u2115\nx\u271d : i \u2208 Finset.range N\nhi : \u00aci \u2208 Finset.range (upcrossingsBefore a b f N \u03c9)\nhi' : \u00aci = upcrossingsBefore a b f N \u03c9\n\u22a2 upcrossingsBefore a b f N \u03c9 < i"}, {"tactic": "rw [Finset.mem_range, not_lt] at hi", "annotated_tactic": ["rw [<a>Finset.mem_range</a>, <a>not_lt</a>] at hi", [{"full_name": "Finset.mem_range", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3037, 9], "def_end_pos": [3037, 18]}, {"full_name": "not_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [368, 9], "def_end_pos": [368, 15]}]], "state_before": "case neg\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\nhf : a \u2264 f N \u03c9\nhab : a < b\nhN : \u00acN = 0\nh\u2081 :\n  \u2200 (k : \u2115),\n    \u2211 n in Finset.range N,\n        Set.indicator (Set.Ico (lowerCrossingTime a b f N k \u03c9) (upperCrossingTime a b f N (k + 1) \u03c9))\n          (fun m => f (m + 1) \u03c9 - f m \u03c9) n =\n      stoppedValue f (upperCrossingTime a b f N (k + 1)) \u03c9 - stoppedValue f (lowerCrossingTime a b f N k) \u03c9\ni : \u2115\nx\u271d : i \u2208 Finset.range N\nhi : \u00aci \u2208 Finset.range (upcrossingsBefore a b f N \u03c9)\nhi' : \u00aci = upcrossingsBefore a b f N \u03c9\n\u22a2 upcrossingsBefore a b f N \u03c9 < i", "state_after": "case neg\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\nhf : a \u2264 f N \u03c9\nhab : a < b\nhN : \u00acN = 0\nh\u2081 :\n  \u2200 (k : \u2115),\n    \u2211 n in Finset.range N,\n        Set.indicator (Set.Ico (lowerCrossingTime a b f N k \u03c9) (upperCrossingTime a b f N (k + 1) \u03c9))\n          (fun m => f (m + 1) \u03c9 - f m \u03c9) n =\n      stoppedValue f (upperCrossingTime a b f N (k + 1)) \u03c9 - stoppedValue f (lowerCrossingTime a b f N k) \u03c9\ni : \u2115\nx\u271d : i \u2208 Finset.range N\nhi : upcrossingsBefore a b f N \u03c9 \u2264 i\nhi' : \u00aci = upcrossingsBefore a b f N \u03c9\n\u22a2 upcrossingsBefore a b f N \u03c9 < i"}, {"tactic": "exact lt_of_le_of_ne hi (Ne.symm hi')", "annotated_tactic": ["exact <a>lt_of_le_of_ne</a> hi (<a>Ne.symm</a> hi')", [{"full_name": "lt_of_le_of_ne", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [196, 9], "def_end_pos": [196, 23]}, {"full_name": "Ne.symm", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [575, 9], "def_end_pos": [575, 16]}]], "state_before": "case neg\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\nhf : a \u2264 f N \u03c9\nhab : a < b\nhN : \u00acN = 0\nh\u2081 :\n  \u2200 (k : \u2115),\n    \u2211 n in Finset.range N,\n        Set.indicator (Set.Ico (lowerCrossingTime a b f N k \u03c9) (upperCrossingTime a b f N (k + 1) \u03c9))\n          (fun m => f (m + 1) \u03c9 - f m \u03c9) n =\n      stoppedValue f (upperCrossingTime a b f N (k + 1)) \u03c9 - stoppedValue f (lowerCrossingTime a b f N k) \u03c9\ni : \u2115\nx\u271d : i \u2208 Finset.range N\nhi : upcrossingsBefore a b f N \u03c9 \u2264 i\nhi' : \u00aci = upcrossingsBefore a b f N \u03c9\n\u22a2 upcrossingsBefore a b f N \u03c9 < i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Hausdorff.lean", "full_name": "MeasureTheory.OuterMeasure.mkMetric'.eq_iSup_nat", "start": [302, 1], "end": [306, 71], "traced_tactics": [{"tactic": "ext1 s", "annotated_tactic": ["ext1 s", []], "state_before": "\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nm\u271d : Set X \u2192 \u211d\u22650\u221e\nr : \u211d\u22650\u221e\n\u03bc : OuterMeasure X\ns : Set X\nm : Set X \u2192 \u211d\u22650\u221e\n\u22a2 mkMetric' m = \u2a06 n, pre m (\u2191n)\u207b\u00b9", "state_after": "case h\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nm\u271d : Set X \u2192 \u211d\u22650\u221e\nr : \u211d\u22650\u221e\n\u03bc : OuterMeasure X\ns\u271d : Set X\nm : Set X \u2192 \u211d\u22650\u221e\ns : Set X\n\u22a2 \u2191(mkMetric' m) s = \u2191(\u2a06 n, pre m (\u2191n)\u207b\u00b9) s"}, {"tactic": "rw [iSup_apply]", "annotated_tactic": ["rw [<a>iSup_apply</a>]", [{"full_name": "MeasureTheory.OuterMeasure.iSup_apply", "def_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "def_pos": [412, 9], "def_end_pos": [412, 19]}]], "state_before": "case h\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nm\u271d : Set X \u2192 \u211d\u22650\u221e\nr : \u211d\u22650\u221e\n\u03bc : OuterMeasure X\ns\u271d : Set X\nm : Set X \u2192 \u211d\u22650\u221e\ns : Set X\n\u22a2 \u2191(mkMetric' m) s = \u2191(\u2a06 n, pre m (\u2191n)\u207b\u00b9) s", "state_after": "case h\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nm\u271d : Set X \u2192 \u211d\u22650\u221e\nr : \u211d\u22650\u221e\n\u03bc : OuterMeasure X\ns\u271d : Set X\nm : Set X \u2192 \u211d\u22650\u221e\ns : Set X\n\u22a2 \u2191(mkMetric' m) s = \u2a06 i, \u2191(pre m (\u2191i)\u207b\u00b9) s"}, {"tactic": "refine' tendsto_nhds_unique (mkMetric'.tendsto_pre_nat m s)\n  (tendsto_atTop_iSup fun k l hkl => mkMetric'.mono_pre_nat m hkl s)", "annotated_tactic": ["refine' <a>tendsto_nhds_unique</a> (<a>mkMetric'.tendsto_pre_nat</a> m s)\n    (<a>tendsto_atTop_iSup</a> fun k l hkl => <a>mkMetric'.mono_pre_nat</a> m hkl s)", [{"full_name": "tendsto_nhds_unique", "def_path": "Mathlib/Topology/Separation.lean", "def_pos": [994, 9], "def_end_pos": [994, 28]}, {"full_name": "MeasureTheory.OuterMeasure.mkMetric'.tendsto_pre_nat", "def_path": "Mathlib/MeasureTheory/Measure/Hausdorff.lean", "def_pos": [295, 9], "def_end_pos": [295, 24]}, {"full_name": "tendsto_atTop_iSup", "def_path": "Mathlib/Topology/Algebra/Order/MonotoneConvergence.lean", "def_pos": [156, 9], "def_end_pos": [156, 27]}, {"full_name": "MeasureTheory.OuterMeasure.mkMetric'.mono_pre_nat", "def_path": "Mathlib/MeasureTheory/Measure/Hausdorff.lean", "def_pos": [284, 9], "def_end_pos": [284, 21]}]], "state_before": "case h\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nm\u271d : Set X \u2192 \u211d\u22650\u221e\nr : \u211d\u22650\u221e\n\u03bc : OuterMeasure X\ns\u271d : Set X\nm : Set X \u2192 \u211d\u22650\u221e\ns : Set X\n\u22a2 \u2191(mkMetric' m) s = \u2a06 i, \u2191(pre m (\u2191i)\u207b\u00b9) s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "full_name": "MeasureTheory.SimpleFunc.sum_measure_preimage_singleton", "start": [211, 11], "end": [213, 70], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/PDeriv.lean", "full_name": "MvPolynomial.pderiv_one", "start": [87, 1], "end": [87, 77], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Process/Stopping.lean", "full_name": "MeasureTheory.stoppedProcess_eq'", "start": [1099, 1], "end": [1109, 72], "traced_tactics": [{"tactic": "rw [stoppedProcess_eq, this, Finset.sum_range_succ_comm, \u2190 add_assoc]", "annotated_tactic": ["rw [<a>stoppedProcess_eq</a>, this, <a>Finset.sum_range_succ_comm</a>, \u2190 <a>add_assoc</a>]", [{"full_name": "MeasureTheory.stoppedProcess_eq", "def_path": "Mathlib/Probability/Process/Stopping.lean", "def_pos": [1092, 9], "def_end_pos": [1092, 26]}, {"full_name": "Finset.sum_range_succ_comm", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [1212, 3], "def_end_pos": [1212, 14]}, {"full_name": "add_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [263, 3], "def_end_pos": [263, 14]}]], "state_before": "\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\nf : Filtration \u2115 m\nu : \u2115 \u2192 \u03a9 \u2192 \u03b2\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : AddCommMonoid \u03b2\nn : \u2115\nthis : Set.indicator {a | n \u2264 \u03c4 a} (u n) = Set.indicator {a | n + 1 \u2264 \u03c4 a} (u n) + Set.indicator {a | \u03c4 a = n} (u n)\n\u22a2 stoppedProcess u \u03c4 n =\n    Set.indicator {a | n + 1 \u2264 \u03c4 a} (u n) + \u2211 i in Finset.range (n + 1), Set.indicator {a | \u03c4 a = i} (u i)", "state_after": "no goals"}, {"tactic": "ext x", "annotated_tactic": ["ext x", []], "state_before": "\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\nf : Filtration \u2115 m\nu : \u2115 \u2192 \u03a9 \u2192 \u03b2\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : AddCommMonoid \u03b2\nn : \u2115\n\u22a2 Set.indicator {a | n \u2264 \u03c4 a} (u n) = Set.indicator {a | n + 1 \u2264 \u03c4 a} (u n) + Set.indicator {a | \u03c4 a = n} (u n)", "state_after": "case h\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\nf : Filtration \u2115 m\nu : \u2115 \u2192 \u03a9 \u2192 \u03b2\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : AddCommMonoid \u03b2\nn : \u2115\nx : \u03a9\n\u22a2 Set.indicator {a | n \u2264 \u03c4 a} (u n) x = (Set.indicator {a | n + 1 \u2264 \u03c4 a} (u n) + Set.indicator {a | \u03c4 a = n} (u n)) x"}, {"tactic": "rw [add_comm, Pi.add_apply, \u2190 Set.indicator_union_of_not_mem_inter]", "annotated_tactic": ["rw [<a>add_comm</a>, <a>Pi.add_apply</a>, \u2190 <a>Set.indicator_union_of_not_mem_inter</a>]", [{"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [301, 3], "def_end_pos": [301, 14]}, {"full_name": "Pi.add_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [82, 3], "def_end_pos": [82, 14]}, {"full_name": "Set.indicator_union_of_not_mem_inter", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [370, 3], "def_end_pos": [370, 14]}]], "state_before": "case h\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\nf : Filtration \u2115 m\nu : \u2115 \u2192 \u03a9 \u2192 \u03b2\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : AddCommMonoid \u03b2\nn : \u2115\nx : \u03a9\n\u22a2 Set.indicator {a | n \u2264 \u03c4 a} (u n) x = (Set.indicator {a | n + 1 \u2264 \u03c4 a} (u n) + Set.indicator {a | \u03c4 a = n} (u n)) x", "state_after": "case h\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\nf : Filtration \u2115 m\nu : \u2115 \u2192 \u03a9 \u2192 \u03b2\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : AddCommMonoid \u03b2\nn : \u2115\nx : \u03a9\n\u22a2 Set.indicator {a | n \u2264 \u03c4 a} (u n) x = Set.indicator ({a | \u03c4 a = n} \u222a {a | n + 1 \u2264 \u03c4 a}) (u n) x\n\ncase h.h\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\nf : Filtration \u2115 m\nu : \u2115 \u2192 \u03a9 \u2192 \u03b2\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : AddCommMonoid \u03b2\nn : \u2115\nx : \u03a9\n\u22a2 \u00acx \u2208 {a | \u03c4 a = n} \u2229 {a | n + 1 \u2264 \u03c4 a}"}, {"tactic": "simp_rw [@eq_comm _ _ n, @le_iff_eq_or_lt _ _ n, Nat.succ_le_iff]", "annotated_tactic": ["simp_rw [@<a>eq_comm</a> _ _ n, @<a>le_iff_eq_or_lt</a> _ _ n, <a>Nat.succ_le_iff</a>]", [{"full_name": "eq_comm", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [104, 9], "def_end_pos": [104, 16]}, {"full_name": "le_iff_eq_or_lt", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [390, 9], "def_end_pos": [390, 24]}, {"full_name": "Nat.succ_le_iff", "def_path": "Mathlib/Data/Nat/Basic.lean", "def_pos": [211, 9], "def_end_pos": [211, 20]}]], "state_before": "case h\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\nf : Filtration \u2115 m\nu : \u2115 \u2192 \u03a9 \u2192 \u03b2\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : AddCommMonoid \u03b2\nn : \u2115\nx : \u03a9\n\u22a2 Set.indicator {a | n \u2264 \u03c4 a} (u n) x = Set.indicator ({a | \u03c4 a = n} \u222a {a | n + 1 \u2264 \u03c4 a}) (u n) x", "state_after": "case h\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\nf : Filtration \u2115 m\nu : \u2115 \u2192 \u03a9 \u2192 \u03b2\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : AddCommMonoid \u03b2\nn : \u2115\nx : \u03a9\n\u22a2 Set.indicator {a | n = \u03c4 a \u2228 n < \u03c4 a} (u n) x = Set.indicator ({a | n = \u03c4 a} \u222a {a | n < \u03c4 a}) (u n) x"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case h\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\nf : Filtration \u2115 m\nu : \u2115 \u2192 \u03a9 \u2192 \u03b2\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : AddCommMonoid \u03b2\nn : \u2115\nx : \u03a9\n\u22a2 Set.indicator {a | n = \u03c4 a \u2228 n < \u03c4 a} (u n) x = Set.indicator ({a | n = \u03c4 a} \u222a {a | n < \u03c4 a}) (u n) x", "state_after": "no goals"}, {"tactic": "rintro \u27e8h\u2081, h\u2082\u27e9", "annotated_tactic": ["rintro \u27e8h\u2081, h\u2082\u27e9", []], "state_before": "case h.h\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\nf : Filtration \u2115 m\nu : \u2115 \u2192 \u03a9 \u2192 \u03b2\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : AddCommMonoid \u03b2\nn : \u2115\nx : \u03a9\n\u22a2 \u00acx \u2208 {a | \u03c4 a = n} \u2229 {a | n + 1 \u2264 \u03c4 a}", "state_after": "case h.h.intro\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\nf : Filtration \u2115 m\nu : \u2115 \u2192 \u03a9 \u2192 \u03b2\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : AddCommMonoid \u03b2\nn : \u2115\nx : \u03a9\nh\u2081 : x \u2208 {a | \u03c4 a = n}\nh\u2082 : x \u2208 {a | n + 1 \u2264 \u03c4 a}\n\u22a2 False"}, {"tactic": "exact (Nat.succ_le_iff.1 h\u2082).ne h\u2081.symm", "annotated_tactic": ["exact (<a>Nat.succ_le_iff</a>.1 h\u2082).<a>ne</a> h\u2081.symm", [{"full_name": "Nat.succ_le_iff", "def_path": "Mathlib/Data/Nat/Basic.lean", "def_pos": [211, 9], "def_end_pos": [211, 20]}, {"full_name": "LT.lt.ne", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [152, 7], "def_end_pos": [152, 15]}]], "state_before": "case h.h.intro\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\nf : Filtration \u2115 m\nu : \u2115 \u2192 \u03a9 \u2192 \u03b2\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : AddCommMonoid \u03b2\nn : \u2115\nx : \u03a9\nh\u2081 : x \u2208 {a | \u03c4 a = n}\nh\u2082 : x \u2208 {a | n + 1 \u2264 \u03c4 a}\n\u22a2 False", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Constructions/Prod/Basic.lean", "full_name": "MeasureTheory.Measure.ae_ae_of_ae_prod", "start": [456, 1], "end": [458, 33], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Primrec.lean", "full_name": "Primrec.option_casesOn", "start": [610, 1], "end": [619, 55], "traced_tactics": [{"tactic": "cases' o a with b <;> simp [encodek]", "annotated_tactic": ["cases' o a with b <;> simp [<a>encodek</a>]", [{"full_name": "Encodable.encodek", "def_path": "Mathlib/Logic/Encodable/Basic.lean", "def_pos": [53, 3], "def_end_pos": [53, 10]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03c3 : Type u_5\ninst\u271d\u2074 : Primcodable \u03b1\ninst\u271d\u00b3 : Primcodable \u03b2\ninst\u271d\u00b2 : Primcodable \u03b3\ninst\u271d\u00b9 : Primcodable \u03b4\ninst\u271d : Primcodable \u03c3\no : \u03b1 \u2192 Option \u03b2\nf : \u03b1 \u2192 \u03c3\ng : \u03b1 \u2192 \u03b2 \u2192 \u03c3\nho : Primrec o\nhf : Primrec f\nhg : Primrec\u2082 g\na : \u03b1\n\u22a2 (Nat.casesOn (encode (o a)) (encode (f a)) fun b =>\n      Nat.pred (encode (Option.bind (decode (encode (a, b).1)) fun a => Option.map (g a) (decode b)))) =\n    encode (Option.casesOn (o a) (f a) (g a))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/ZMod/Quotient.lean", "full_name": "AddAction.zmultiplesQuotientStabilizerEquiv_symm_apply", "start": [111, 1], "end": [114, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/Funext.lean", "full_name": "MvPolynomial.funext_fin", "start": [30, 9], "end": [42, 22], "traced_tactics": [{"tactic": "induction' n with n ih", "annotated_tactic": ["induction' n with n ih", []], "state_before": "R : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : Infinite R\nn : \u2115\np : MvPolynomial (Fin n) R\nh : \u2200 (x : Fin n \u2192 R), \u2191(eval x) p = 0\n\u22a2 p = 0", "state_after": "case zero\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : Infinite R\nn : \u2115\np\u271d : MvPolynomial (Fin n) R\nh\u271d : \u2200 (x : Fin n \u2192 R), \u2191(eval x) p\u271d = 0\np : MvPolynomial (Fin Nat.zero) R\nh : \u2200 (x : Fin Nat.zero \u2192 R), \u2191(eval x) p = 0\n\u22a2 p = 0\n\ncase succ\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : Infinite R\nn\u271d : \u2115\np\u271d : MvPolynomial (Fin n\u271d) R\nh\u271d : \u2200 (x : Fin n\u271d \u2192 R), \u2191(eval x) p\u271d = 0\nn : \u2115\nih : \u2200 {p : MvPolynomial (Fin n) R}, (\u2200 (x : Fin n \u2192 R), \u2191(eval x) p = 0) \u2192 p = 0\np : MvPolynomial (Fin (Nat.succ n)) R\nh : \u2200 (x : Fin (Nat.succ n) \u2192 R), \u2191(eval x) p = 0\n\u22a2 p = 0"}, {"tactic": "apply (MvPolynomial.isEmptyRingEquiv R (Fin 0)).injective", "annotated_tactic": ["apply (<a>MvPolynomial.isEmptyRingEquiv</a> R (<a>Fin</a> 0)).<a>injective</a>", [{"full_name": "MvPolynomial.isEmptyRingEquiv", "def_path": "Mathlib/Data/MvPolynomial/Equiv.lean", "def_pos": [228, 5], "def_end_pos": [228, 21]}, {"full_name": "Fin", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1745, 11], "def_end_pos": [1745, 14]}, {"full_name": "RingEquiv.injective", "def_path": "Mathlib/Algebra/Ring/Equiv.lean", "def_pos": [346, 19], "def_end_pos": [346, 28]}]], "state_before": "case zero\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : Infinite R\nn : \u2115\np\u271d : MvPolynomial (Fin n) R\nh\u271d : \u2200 (x : Fin n \u2192 R), \u2191(eval x) p\u271d = 0\np : MvPolynomial (Fin Nat.zero) R\nh : \u2200 (x : Fin Nat.zero \u2192 R), \u2191(eval x) p = 0\n\u22a2 p = 0", "state_after": "case zero.a\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : Infinite R\nn : \u2115\np\u271d : MvPolynomial (Fin n) R\nh\u271d : \u2200 (x : Fin n \u2192 R), \u2191(eval x) p\u271d = 0\np : MvPolynomial (Fin Nat.zero) R\nh : \u2200 (x : Fin Nat.zero \u2192 R), \u2191(eval x) p = 0\n\u22a2 \u2191(isEmptyRingEquiv R (Fin 0)) p = \u2191(isEmptyRingEquiv R (Fin 0)) 0"}, {"tactic": "rw [RingEquiv.map_zero]", "annotated_tactic": ["rw [<a>RingEquiv.map_zero</a>]", [{"full_name": "RingEquiv.map_zero", "def_path": "Mathlib/Algebra/Ring/Equiv.lean", "def_pos": [444, 19], "def_end_pos": [444, 27]}]], "state_before": "case zero.a\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : Infinite R\nn : \u2115\np\u271d : MvPolynomial (Fin n) R\nh\u271d : \u2200 (x : Fin n \u2192 R), \u2191(eval x) p\u271d = 0\np : MvPolynomial (Fin Nat.zero) R\nh : \u2200 (x : Fin Nat.zero \u2192 R), \u2191(eval x) p = 0\n\u22a2 \u2191(isEmptyRingEquiv R (Fin 0)) p = \u2191(isEmptyRingEquiv R (Fin 0)) 0", "state_after": "case zero.a\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : Infinite R\nn : \u2115\np\u271d : MvPolynomial (Fin n) R\nh\u271d : \u2200 (x : Fin n \u2192 R), \u2191(eval x) p\u271d = 0\np : MvPolynomial (Fin Nat.zero) R\nh : \u2200 (x : Fin Nat.zero \u2192 R), \u2191(eval x) p = 0\n\u22a2 \u2191(isEmptyRingEquiv R (Fin 0)) p = 0"}, {"tactic": "convert h finZeroElim", "annotated_tactic": ["convert h <a>finZeroElim</a>", [{"full_name": "finZeroElim", "def_path": "Mathlib/Data/Fin/Basic.lean", "def_pos": [84, 5], "def_end_pos": [84, 16]}]], "state_before": "case zero.a\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : Infinite R\nn : \u2115\np\u271d : MvPolynomial (Fin n) R\nh\u271d : \u2200 (x : Fin n \u2192 R), \u2191(eval x) p\u271d = 0\np : MvPolynomial (Fin Nat.zero) R\nh : \u2200 (x : Fin Nat.zero \u2192 R), \u2191(eval x) p = 0\n\u22a2 \u2191(isEmptyRingEquiv R (Fin 0)) p = 0", "state_after": "no goals"}, {"tactic": "apply (finSuccEquiv R n).injective", "annotated_tactic": ["apply (<a>finSuccEquiv</a> R n).<a>injective</a>", [{"full_name": "MvPolynomial.finSuccEquiv", "def_path": "Mathlib/Data/MvPolynomial/Equiv.lean", "def_pos": [316, 5], "def_end_pos": [316, 17]}, {"full_name": "AlgEquiv.injective", "def_path": "Mathlib/Algebra/Algebra/Equiv.lean", "def_pos": [284, 19], "def_end_pos": [284, 28]}]], "state_before": "case succ\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : Infinite R\nn\u271d : \u2115\np\u271d : MvPolynomial (Fin n\u271d) R\nh\u271d : \u2200 (x : Fin n\u271d \u2192 R), \u2191(eval x) p\u271d = 0\nn : \u2115\nih : \u2200 {p : MvPolynomial (Fin n) R}, (\u2200 (x : Fin n \u2192 R), \u2191(eval x) p = 0) \u2192 p = 0\np : MvPolynomial (Fin (Nat.succ n)) R\nh : \u2200 (x : Fin (Nat.succ n) \u2192 R), \u2191(eval x) p = 0\n\u22a2 p = 0", "state_after": "case succ.a\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : Infinite R\nn\u271d : \u2115\np\u271d : MvPolynomial (Fin n\u271d) R\nh\u271d : \u2200 (x : Fin n\u271d \u2192 R), \u2191(eval x) p\u271d = 0\nn : \u2115\nih : \u2200 {p : MvPolynomial (Fin n) R}, (\u2200 (x : Fin n \u2192 R), \u2191(eval x) p = 0) \u2192 p = 0\np : MvPolynomial (Fin (Nat.succ n)) R\nh : \u2200 (x : Fin (Nat.succ n) \u2192 R), \u2191(eval x) p = 0\n\u22a2 \u2191(finSuccEquiv R n) p = \u2191(finSuccEquiv R n) 0"}, {"tactic": "simp only [AlgEquiv.map_zero]", "annotated_tactic": ["simp only [<a>AlgEquiv.map_zero</a>]", [{"full_name": "AlgEquiv.map_zero", "def_path": "Mathlib/Algebra/Algebra/Equiv.lean", "def_pos": [218, 19], "def_end_pos": [218, 27]}]], "state_before": "case succ.a\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : Infinite R\nn\u271d : \u2115\np\u271d : MvPolynomial (Fin n\u271d) R\nh\u271d : \u2200 (x : Fin n\u271d \u2192 R), \u2191(eval x) p\u271d = 0\nn : \u2115\nih : \u2200 {p : MvPolynomial (Fin n) R}, (\u2200 (x : Fin n \u2192 R), \u2191(eval x) p = 0) \u2192 p = 0\np : MvPolynomial (Fin (Nat.succ n)) R\nh : \u2200 (x : Fin (Nat.succ n) \u2192 R), \u2191(eval x) p = 0\n\u22a2 \u2191(finSuccEquiv R n) p = \u2191(finSuccEquiv R n) 0", "state_after": "case succ.a\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : Infinite R\nn\u271d : \u2115\np\u271d : MvPolynomial (Fin n\u271d) R\nh\u271d : \u2200 (x : Fin n\u271d \u2192 R), \u2191(eval x) p\u271d = 0\nn : \u2115\nih : \u2200 {p : MvPolynomial (Fin n) R}, (\u2200 (x : Fin n \u2192 R), \u2191(eval x) p = 0) \u2192 p = 0\np : MvPolynomial (Fin (Nat.succ n)) R\nh : \u2200 (x : Fin (Nat.succ n) \u2192 R), \u2191(eval x) p = 0\n\u22a2 \u2191(finSuccEquiv R n) p = 0"}, {"tactic": "refine Polynomial.funext fun q => ?_", "annotated_tactic": ["refine <a>Polynomial.funext</a> fun q => ?_", [{"full_name": "Polynomial.funext", "def_path": "Mathlib/Data/Polynomial/RingDivision.lean", "def_pos": [906, 9], "def_end_pos": [906, 15]}]], "state_before": "case succ.a\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : Infinite R\nn\u271d : \u2115\np\u271d : MvPolynomial (Fin n\u271d) R\nh\u271d : \u2200 (x : Fin n\u271d \u2192 R), \u2191(eval x) p\u271d = 0\nn : \u2115\nih : \u2200 {p : MvPolynomial (Fin n) R}, (\u2200 (x : Fin n \u2192 R), \u2191(eval x) p = 0) \u2192 p = 0\np : MvPolynomial (Fin (Nat.succ n)) R\nh : \u2200 (x : Fin (Nat.succ n) \u2192 R), \u2191(eval x) p = 0\n\u22a2 \u2191(finSuccEquiv R n) p = 0", "state_after": "case succ.a\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : Infinite R\nn\u271d : \u2115\np\u271d : MvPolynomial (Fin n\u271d) R\nh\u271d : \u2200 (x : Fin n\u271d \u2192 R), \u2191(eval x) p\u271d = 0\nn : \u2115\nih : \u2200 {p : MvPolynomial (Fin n) R}, (\u2200 (x : Fin n \u2192 R), \u2191(eval x) p = 0) \u2192 p = 0\np : MvPolynomial (Fin (Nat.succ n)) R\nh : \u2200 (x : Fin (Nat.succ n) \u2192 R), \u2191(eval x) p = 0\nq : MvPolynomial (Fin n) R\n\u22a2 Polynomial.eval q (\u2191(finSuccEquiv R n) p) = Polynomial.eval q 0"}, {"tactic": "rw [Polynomial.eval_zero]", "annotated_tactic": ["rw [<a>Polynomial.eval_zero</a>]", [{"full_name": "Polynomial.eval_zero", "def_path": "Mathlib/Data/Polynomial/Eval.lean", "def_pos": [383, 9], "def_end_pos": [383, 18]}]], "state_before": "case succ.a\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : Infinite R\nn\u271d : \u2115\np\u271d : MvPolynomial (Fin n\u271d) R\nh\u271d : \u2200 (x : Fin n\u271d \u2192 R), \u2191(eval x) p\u271d = 0\nn : \u2115\nih : \u2200 {p : MvPolynomial (Fin n) R}, (\u2200 (x : Fin n \u2192 R), \u2191(eval x) p = 0) \u2192 p = 0\np : MvPolynomial (Fin (Nat.succ n)) R\nh : \u2200 (x : Fin (Nat.succ n) \u2192 R), \u2191(eval x) p = 0\nq : MvPolynomial (Fin n) R\n\u22a2 Polynomial.eval q (\u2191(finSuccEquiv R n) p) = Polynomial.eval q 0", "state_after": "case succ.a\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : Infinite R\nn\u271d : \u2115\np\u271d : MvPolynomial (Fin n\u271d) R\nh\u271d : \u2200 (x : Fin n\u271d \u2192 R), \u2191(eval x) p\u271d = 0\nn : \u2115\nih : \u2200 {p : MvPolynomial (Fin n) R}, (\u2200 (x : Fin n \u2192 R), \u2191(eval x) p = 0) \u2192 p = 0\np : MvPolynomial (Fin (Nat.succ n)) R\nh : \u2200 (x : Fin (Nat.succ n) \u2192 R), \u2191(eval x) p = 0\nq : MvPolynomial (Fin n) R\n\u22a2 Polynomial.eval q (\u2191(finSuccEquiv R n) p) = 0"}, {"tactic": "apply ih fun x => ?_", "annotated_tactic": ["apply ih fun x => ?_", []], "state_before": "case succ.a\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : Infinite R\nn\u271d : \u2115\np\u271d : MvPolynomial (Fin n\u271d) R\nh\u271d : \u2200 (x : Fin n\u271d \u2192 R), \u2191(eval x) p\u271d = 0\nn : \u2115\nih : \u2200 {p : MvPolynomial (Fin n) R}, (\u2200 (x : Fin n \u2192 R), \u2191(eval x) p = 0) \u2192 p = 0\np : MvPolynomial (Fin (Nat.succ n)) R\nh : \u2200 (x : Fin (Nat.succ n) \u2192 R), \u2191(eval x) p = 0\nq : MvPolynomial (Fin n) R\n\u22a2 Polynomial.eval q (\u2191(finSuccEquiv R n) p) = 0", "state_after": "R : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : Infinite R\nn\u271d : \u2115\np\u271d : MvPolynomial (Fin n\u271d) R\nh\u271d : \u2200 (x : Fin n\u271d \u2192 R), \u2191(eval x) p\u271d = 0\nn : \u2115\nih : \u2200 {p : MvPolynomial (Fin n) R}, (\u2200 (x : Fin n \u2192 R), \u2191(eval x) p = 0) \u2192 p = 0\np : MvPolynomial (Fin (Nat.succ n)) R\nh : \u2200 (x : Fin (Nat.succ n) \u2192 R), \u2191(eval x) p = 0\nq : MvPolynomial (Fin n) R\nx : Fin n \u2192 R\n\u22a2 \u2191(eval x) (Polynomial.eval q (\u2191(finSuccEquiv R n) p)) = 0"}, {"tactic": "calc _ = _ := eval_polynomial_eval_finSuccEquiv p _\n     _ = 0 := h _", "annotated_tactic": ["calc _ = _ := <a>eval_polynomial_eval_finSuccEquiv</a> p _\n         _ = 0 := h _", [{"full_name": "MvPolynomial.eval_polynomial_eval_finSuccEquiv", "def_path": "Mathlib/Data/MvPolynomial/Polynomial.lean", "def_pos": [30, 9], "def_end_pos": [30, 42]}]], "state_before": "R : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : Infinite R\nn\u271d : \u2115\np\u271d : MvPolynomial (Fin n\u271d) R\nh\u271d : \u2200 (x : Fin n\u271d \u2192 R), \u2191(eval x) p\u271d = 0\nn : \u2115\nih : \u2200 {p : MvPolynomial (Fin n) R}, (\u2200 (x : Fin n \u2192 R), \u2191(eval x) p = 0) \u2192 p = 0\np : MvPolynomial (Fin (Nat.succ n)) R\nh : \u2200 (x : Fin (Nat.succ n) \u2192 R), \u2191(eval x) p = 0\nq : MvPolynomial (Fin n) R\nx : Fin n \u2192 R\n\u22a2 \u2191(eval x) (Polynomial.eval q (\u2191(finSuccEquiv R n) p)) = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "full_name": "measure_le_lintegral_thickenedIndicatorAux", "start": [1351, 1], "end": [1357, 45], "traced_tactics": [{"tactic": "convert_to lintegral \u03bc (E.indicator fun _ => (1 : \u211d\u22650\u221e)) \u2264 lintegral \u03bc (thickenedIndicatorAux \u03b4 E)", "annotated_tactic": ["convert_to <a>lintegral</a> \u03bc (E.indicator fun _ => (1 : \u211d\u22650\u221e)) \u2264 <a>lintegral</a> \u03bc (<a>thickenedIndicatorAux</a> \u03b4 E)", [{"full_name": "MeasureTheory.lintegral", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [60, 17], "def_end_pos": [60, 26]}, {"full_name": "MeasureTheory.lintegral", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [60, 17], "def_end_pos": [60, 26]}, {"full_name": "thickenedIndicatorAux", "def_path": "Mathlib/Topology/MetricSpace/ThickenedIndicator.lean", "def_pos": [54, 5], "def_end_pos": [54, 26]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nE\u271d : Type u_3\nF : Type u_4\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup E\u271d\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03bc : Measure \u03b1\nE : Set \u03b1\nE_mble : MeasurableSet E\n\u03b4 : \u211d\n\u22a2 \u2191\u2191\u03bc E \u2264 \u222b\u207b (a : \u03b1), thickenedIndicatorAux \u03b4 E a \u2202\u03bc", "state_after": "case h.e'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE\u271d : Type u_3\nF : Type u_4\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup E\u271d\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03bc : Measure \u03b1\nE : Set \u03b1\nE_mble : MeasurableSet E\n\u03b4 : \u211d\n\u22a2 \u2191\u2191\u03bc E = lintegral \u03bc (indicator E fun x => 1)\n\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE\u271d : Type u_3\nF : Type u_4\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup E\u271d\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03bc : Measure \u03b1\nE : Set \u03b1\nE_mble : MeasurableSet E\n\u03b4 : \u211d\n\u22a2 lintegral \u03bc (indicator E fun x => 1) \u2264 lintegral \u03bc (thickenedIndicatorAux \u03b4 E)"}, {"tactic": "rw [lintegral_indicator _ E_mble]", "annotated_tactic": ["rw [<a>lintegral_indicator</a> _ E_mble]", [{"full_name": "MeasureTheory.lintegral_indicator", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [762, 9], "def_end_pos": [762, 28]}]], "state_before": "case h.e'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE\u271d : Type u_3\nF : Type u_4\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup E\u271d\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03bc : Measure \u03b1\nE : Set \u03b1\nE_mble : MeasurableSet E\n\u03b4 : \u211d\n\u22a2 \u2191\u2191\u03bc E = lintegral \u03bc (indicator E fun x => 1)", "state_after": "case h.e'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE\u271d : Type u_3\nF : Type u_4\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup E\u271d\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03bc : Measure \u03b1\nE : Set \u03b1\nE_mble : MeasurableSet E\n\u03b4 : \u211d\n\u22a2 \u2191\u2191\u03bc E = \u222b\u207b (a : \u03b1) in E, 1 \u2202\u03bc"}, {"tactic": "simp only [lintegral_one, Measure.restrict_apply, MeasurableSet.univ, univ_inter]", "annotated_tactic": ["simp only [<a>lintegral_one</a>, <a>Measure.restrict_apply</a>, <a>MeasurableSet.univ</a>, <a>univ_inter</a>]", [{"full_name": "MeasureTheory.lintegral_one", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [149, 9], "def_end_pos": [149, 22]}, {"full_name": "MeasureTheory.Measure.restrict_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1533, 9], "def_end_pos": [1533, 23]}, {"full_name": "MeasurableSet.univ", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [101, 19], "def_end_pos": [101, 37]}, {"full_name": "Set.univ_inter", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1017, 9], "def_end_pos": [1017, 19]}]], "state_before": "case h.e'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE\u271d : Type u_3\nF : Type u_4\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup E\u271d\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03bc : Measure \u03b1\nE : Set \u03b1\nE_mble : MeasurableSet E\n\u03b4 : \u211d\n\u22a2 \u2191\u2191\u03bc E = \u222b\u207b (a : \u03b1) in E, 1 \u2202\u03bc", "state_after": "no goals"}, {"tactic": "apply lintegral_mono", "annotated_tactic": ["apply <a>lintegral_mono</a>", [{"full_name": "MeasureTheory.lintegral_mono", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [99, 9], "def_end_pos": [99, 23]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nE\u271d : Type u_3\nF : Type u_4\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup E\u271d\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03bc : Measure \u03b1\nE : Set \u03b1\nE_mble : MeasurableSet E\n\u03b4 : \u211d\n\u22a2 lintegral \u03bc (indicator E fun x => 1) \u2264 lintegral \u03bc (thickenedIndicatorAux \u03b4 E)", "state_after": "case hfg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE\u271d : Type u_3\nF : Type u_4\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup E\u271d\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03bc : Measure \u03b1\nE : Set \u03b1\nE_mble : MeasurableSet E\n\u03b4 : \u211d\n\u22a2 (fun a => indicator E (fun x => 1) a) \u2264 fun a => thickenedIndicatorAux \u03b4 E a"}, {"tactic": "apply indicator_le_thickenedIndicatorAux", "annotated_tactic": ["apply <a>indicator_le_thickenedIndicatorAux</a>", [{"full_name": "indicator_le_thickenedIndicatorAux", "def_path": "Mathlib/Topology/MetricSpace/ThickenedIndicator.lean", "def_pos": [110, 9], "def_end_pos": [110, 43]}]], "state_before": "case hfg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE\u271d : Type u_3\nF : Type u_4\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup E\u271d\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03bc : Measure \u03b1\nE : Set \u03b1\nE_mble : MeasurableSet E\n\u03b4 : \u211d\n\u22a2 (fun a => indicator E (fun x => 1) a) \u2264 fun a => thickenedIndicatorAux \u03b4 E a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "full_name": "Substring.ValidFor.extract", "start": [923, 1], "end": [934, 25], "traced_tactics": [{"tactic": "simp [Substring.extract]", "annotated_tactic": ["simp [<a>Substring.extract</a>]", [{"full_name": "Substring.extract", "def_path": "lake-packages/lean4/src/lean/Init/Data/String/Basic.lean", "def_pos": [590, 15], "def_end_pos": [590, 22]}]], "state_before": "l m r ml mm mr : List Char\nb e : Pos\n\u22a2 \u2203 l' r',\n    ValidFor l' mm r'\n      (Substring.extract\n        { str := { data := l ++ (ml ++ mm ++ mr) ++ r }, startPos := { byteIdx := utf8Len l },\n          stopPos := { byteIdx := utf8Len l + utf8Len (ml ++ mm ++ mr) } }\n        { byteIdx := utf8Len ml } { byteIdx := utf8Len ml + utf8Len mm })", "state_after": "l m r ml mm mr : List Char\nb e : Pos\n\u22a2 \u2203 l' r',\n    ValidFor l' mm r'\n      (if utf8Len ml + utf8Len mm \u2264 utf8Len ml then { str := \"\", startPos := 0, stopPos := 0 }\n      else\n        { str := { data := l ++ (ml ++ (mm ++ (mr ++ r))) },\n          startPos :=\n            Pos.min { byteIdx := utf8Len l + (utf8Len ml + (utf8Len mm + utf8Len mr)) }\n              ({ byteIdx := utf8Len l } + { byteIdx := utf8Len ml }),\n          stopPos :=\n            Pos.min { byteIdx := utf8Len l + (utf8Len ml + (utf8Len mm + utf8Len mr)) }\n              ({ byteIdx := utf8Len l } + { byteIdx := utf8Len ml + utf8Len mm }) })"}, {"tactic": "split", "annotated_tactic": ["split", []], "state_before": "l m r ml mm mr : List Char\nb e : Pos\n\u22a2 \u2203 l' r',\n    ValidFor l' mm r'\n      (if utf8Len ml + utf8Len mm \u2264 utf8Len ml then { str := \"\", startPos := 0, stopPos := 0 }\n      else\n        { str := { data := l ++ (ml ++ (mm ++ (mr ++ r))) },\n          startPos :=\n            Pos.min { byteIdx := utf8Len l + (utf8Len ml + (utf8Len mm + utf8Len mr)) }\n              ({ byteIdx := utf8Len l } + { byteIdx := utf8Len ml }),\n          stopPos :=\n            Pos.min { byteIdx := utf8Len l + (utf8Len ml + (utf8Len mm + utf8Len mr)) }\n              ({ byteIdx := utf8Len l } + { byteIdx := utf8Len ml + utf8Len mm }) })", "state_after": "case inl\nl m r ml mm mr : List Char\nb e : Pos\nh\u271d : utf8Len ml + utf8Len mm \u2264 utf8Len ml\n\u22a2 \u2203 l' r', ValidFor l' mm r' { str := \"\", startPos := 0, stopPos := 0 }\n\ncase inr\nl m r ml mm mr : List Char\nb e : Pos\nh\u271d : \u00acutf8Len ml + utf8Len mm \u2264 utf8Len ml\n\u22a2 \u2203 l' r',\n    ValidFor l' mm r'\n      { str := { data := l ++ (ml ++ (mm ++ (mr ++ r))) },\n        startPos :=\n          Pos.min { byteIdx := utf8Len l + (utf8Len ml + (utf8Len mm + utf8Len mr)) }\n            ({ byteIdx := utf8Len l } + { byteIdx := utf8Len ml }),\n        stopPos :=\n          Pos.min { byteIdx := utf8Len l + (utf8Len ml + (utf8Len mm + utf8Len mr)) }\n            ({ byteIdx := utf8Len l } + { byteIdx := utf8Len ml + utf8Len mm }) }"}, {"tactic": "next h =>\nrw [utf8Len_eq_zero.1 <| Nat.le_zero.1 <| (Nat.add_le_add_iff_left _ _ 0).1 h]\nexact \u27e8[], [], \u27e8\u27e9\u27e9", "annotated_tactic": ["next h =>\n      rw [<a>utf8Len_eq_zero</a>.1 <| <a>Nat.le_zero</a>.1 <| (<a>Nat.add_le_add_iff_left</a> _ _ 0).1 h]\n      exact \u27e8[], [], \u27e8\u27e9\u27e9", [{"full_name": "String.utf8Len_eq_zero", "def_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "def_pos": [73, 17], "def_end_pos": [73, 32]}, {"full_name": "Nat.le_zero", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [206, 9], "def_end_pos": [206, 16]}, {"full_name": "Nat.add_le_add_iff_left", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [296, 19], "def_end_pos": [296, 38]}]], "state_before": "case inl\nl m r ml mm mr : List Char\nb e : Pos\nh\u271d : utf8Len ml + utf8Len mm \u2264 utf8Len ml\n\u22a2 \u2203 l' r', ValidFor l' mm r' { str := \"\", startPos := 0, stopPos := 0 }", "state_after": "no goals"}, {"tactic": "rw [utf8Len_eq_zero.1 <| Nat.le_zero.1 <| (Nat.add_le_add_iff_left _ _ 0).1 h]", "annotated_tactic": ["rw [<a>utf8Len_eq_zero</a>.1 <| <a>Nat.le_zero</a>.1 <| (<a>Nat.add_le_add_iff_left</a> _ _ 0).1 h]", [{"full_name": "String.utf8Len_eq_zero", "def_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "def_pos": [73, 17], "def_end_pos": [73, 32]}, {"full_name": "Nat.le_zero", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [206, 9], "def_end_pos": [206, 16]}, {"full_name": "Nat.add_le_add_iff_left", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [296, 19], "def_end_pos": [296, 38]}]], "state_before": "l m r ml mm mr : List Char\nb e : Pos\nh : utf8Len ml + utf8Len mm \u2264 utf8Len ml\n\u22a2 \u2203 l' r', ValidFor l' mm r' { str := \"\", startPos := 0, stopPos := 0 }", "state_after": "l m r ml mm mr : List Char\nb e : Pos\nh : utf8Len ml + utf8Len mm \u2264 utf8Len ml\n\u22a2 \u2203 l' r', ValidFor l' [] r' { str := \"\", startPos := 0, stopPos := 0 }"}, {"tactic": "exact \u27e8[], [], \u27e8\u27e9\u27e9", "annotated_tactic": ["exact \u27e8[], [], \u27e8\u27e9\u27e9", []], "state_before": "l m r ml mm mr : List Char\nb e : Pos\nh : utf8Len ml + utf8Len mm \u2264 utf8Len ml\n\u22a2 \u2203 l' r', ValidFor l' [] r' { str := \"\", startPos := 0, stopPos := 0 }", "state_after": "no goals"}, {"tactic": "next h =>\nrefine \u27e8l ++ ml, mr ++ r, .of_eq _ (by simp) ?_ ?_\u27e9 <;>\n  simp [Nat.min_eq_min] <;> rw [Nat.min_eq_right] <;>\n  try simp [Nat.add_le_add_iff_left, Nat.le_add_right]\nrw [Nat.add_assoc]", "annotated_tactic": ["next h =>\n      refine \u27e8l ++ ml, mr ++ r, .of_eq _ (by simp) ?_ ?_\u27e9 <;>\n        simp [<a>Nat.min_eq_min</a>] <;> rw [<a>Nat.min_eq_right</a>] <;>\n        try simp [<a>Nat.add_le_add_iff_left</a>, <a>Nat.le_add_right</a>]\n      rw [<a>Nat.add_assoc</a>]", [{"full_name": "Nat.min_eq_min", "def_path": "lake-packages/std/Std/Data/Nat/Init/Lemmas.lean", "def_pos": [27, 19], "def_end_pos": [27, 29]}, {"full_name": "Nat.min_eq_right", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [464, 19], "def_end_pos": [464, 31]}, {"full_name": "Nat.add_le_add_iff_left", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [296, 19], "def_end_pos": [296, 38]}, {"full_name": "Nat.le_add_right", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [340, 9], "def_end_pos": [340, 21]}, {"full_name": "Nat.add_assoc", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [138, 19], "def_end_pos": [138, 28]}]], "state_before": "case inr\nl m r ml mm mr : List Char\nb e : Pos\nh\u271d : \u00acutf8Len ml + utf8Len mm \u2264 utf8Len ml\n\u22a2 \u2203 l' r',\n    ValidFor l' mm r'\n      { str := { data := l ++ (ml ++ (mm ++ (mr ++ r))) },\n        startPos :=\n          Pos.min { byteIdx := utf8Len l + (utf8Len ml + (utf8Len mm + utf8Len mr)) }\n            ({ byteIdx := utf8Len l } + { byteIdx := utf8Len ml }),\n        stopPos :=\n          Pos.min { byteIdx := utf8Len l + (utf8Len ml + (utf8Len mm + utf8Len mr)) }\n            ({ byteIdx := utf8Len l } + { byteIdx := utf8Len ml + utf8Len mm }) }", "state_after": "no goals"}, {"tactic": "refine \u27e8l ++ ml, mr ++ r, .of_eq _ (by simp) ?_ ?_\u27e9 <;>\n  simp [Nat.min_eq_min] <;> rw [Nat.min_eq_right] <;>\n  try simp [Nat.add_le_add_iff_left, Nat.le_add_right]", "annotated_tactic": ["refine \u27e8l ++ ml, mr ++ r, .of_eq _ (by simp) ?_ ?_\u27e9 <;>\n        simp [<a>Nat.min_eq_min</a>] <;> rw [<a>Nat.min_eq_right</a>] <;>\n        try simp [<a>Nat.add_le_add_iff_left</a>, <a>Nat.le_add_right</a>]", [{"full_name": "Nat.min_eq_min", "def_path": "lake-packages/std/Std/Data/Nat/Init/Lemmas.lean", "def_pos": [27, 19], "def_end_pos": [27, 29]}, {"full_name": "Nat.min_eq_right", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [464, 19], "def_end_pos": [464, 31]}, {"full_name": "Nat.add_le_add_iff_left", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [296, 19], "def_end_pos": [296, 38]}, {"full_name": "Nat.le_add_right", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [340, 9], "def_end_pos": [340, 21]}]], "state_before": "l m r ml mm mr : List Char\nb e : Pos\nh : \u00acutf8Len ml + utf8Len mm \u2264 utf8Len ml\n\u22a2 \u2203 l' r',\n    ValidFor l' mm r'\n      { str := { data := l ++ (ml ++ (mm ++ (mr ++ r))) },\n        startPos :=\n          Pos.min { byteIdx := utf8Len l + (utf8Len ml + (utf8Len mm + utf8Len mr)) }\n            ({ byteIdx := utf8Len l } + { byteIdx := utf8Len ml }),\n        stopPos :=\n          Pos.min { byteIdx := utf8Len l + (utf8Len ml + (utf8Len mm + utf8Len mr)) }\n            ({ byteIdx := utf8Len l } + { byteIdx := utf8Len ml + utf8Len mm }) }", "state_after": "case refine_2\nl m r ml mm mr : List Char\nb e : Pos\nh : \u00acutf8Len ml + utf8Len mm \u2264 utf8Len ml\n\u22a2 utf8Len l + (utf8Len ml + utf8Len mm) = utf8Len l + utf8Len ml + utf8Len mm"}, {"tactic": "rw [Nat.add_assoc]", "annotated_tactic": ["rw [<a>Nat.add_assoc</a>]", [{"full_name": "Nat.add_assoc", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [138, 19], "def_end_pos": [138, 28]}]], "state_before": "case refine_2\nl m r ml mm mr : List Char\nb e : Pos\nh : \u00acutf8Len ml + utf8Len mm \u2264 utf8Len ml\n\u22a2 utf8Len l + (utf8Len ml + utf8Len mm) = utf8Len l + utf8Len ml + utf8Len mm", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "l m r ml mm mr : List Char\nb e : Pos\nh : \u00acutf8Len ml + utf8Len mm \u2264 utf8Len ml\n\u22a2 { str := { data := l ++ (ml ++ (mm ++ (mr ++ r))) },\n          startPos :=\n            Pos.min { byteIdx := utf8Len l + (utf8Len ml + (utf8Len mm + utf8Len mr)) }\n              ({ byteIdx := utf8Len l } + { byteIdx := utf8Len ml }),\n          stopPos :=\n            Pos.min { byteIdx := utf8Len l + (utf8Len ml + (utf8Len mm + utf8Len mr)) }\n              ({ byteIdx := utf8Len l } + { byteIdx := utf8Len ml + utf8Len mm }) }.str.data =\n    l ++ ml ++ mm ++ (mr ++ r)", "state_after": "no goals"}, {"tactic": "simp [Nat.add_le_add_iff_left, Nat.le_add_right]", "annotated_tactic": ["simp [<a>Nat.add_le_add_iff_left</a>, <a>Nat.le_add_right</a>]", [{"full_name": "Nat.add_le_add_iff_left", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [296, 19], "def_end_pos": [296, 38]}, {"full_name": "Nat.le_add_right", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [340, 9], "def_end_pos": [340, 21]}]], "state_before": "case refine_2\nl m r ml mm mr : List Char\nb e : Pos\nh : \u00acutf8Len ml + utf8Len mm \u2264 utf8Len ml\n\u22a2 utf8Len l + (utf8Len ml + utf8Len mm) \u2264 utf8Len l + (utf8Len ml + (utf8Len mm + utf8Len mr))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Martingale/Upcrossing.lean", "full_name": "MeasureTheory.upperCrossingTime_lt_of_le_upcrossingsBefore", "start": [468, 1], "end": [473, 50], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "full_name": "measurable_iUnionLift", "start": [824, 1], "end": [829, 100], "traced_tactics": [{"tactic": "rw [preimage_iUnionLift]", "annotated_tactic": ["rw [<a>preimage_iUnionLift</a>]", [{"full_name": "Set.preimage_iUnionLift", "def_path": "Mathlib/Data/Set/UnionLift.lean", "def_pos": [79, 9], "def_end_pos": [79, 28]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u\u03b9\ns\u271d t\u271d u : Set \u03b1\nm : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d : Countable \u03b9\nt : \u03b9 \u2192 Set \u03b1\nf : (i : \u03b9) \u2192 \u2191(t i) \u2192 \u03b2\nhtf :\n  \u2200 (i j : \u03b9) (x : \u03b1) (hxi : x \u2208 t i) (hxj : x \u2208 t j),\n    f i { val := x, property := hxi } = f j { val := x, property := hxj }\nT : Set \u03b1\nhT : T \u2286 \u22c3 i, t i\nhtm : \u2200 (i : \u03b9), MeasurableSet (t i)\nhfm : \u2200 (i : \u03b9), Measurable (f i)\ns : Set \u03b2\nhs : MeasurableSet s\n\u22a2 MeasurableSet (iUnionLift t f htf T hT \u207b\u00b9' s)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u\u03b9\ns\u271d t\u271d u : Set \u03b1\nm : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d : Countable \u03b9\nt : \u03b9 \u2192 Set \u03b1\nf : (i : \u03b9) \u2192 \u2191(t i) \u2192 \u03b2\nhtf :\n  \u2200 (i j : \u03b9) (x : \u03b1) (hxi : x \u2208 t i) (hxj : x \u2208 t j),\n    f i { val := x, property := hxi } = f j { val := x, property := hxj }\nT : Set \u03b1\nhT : T \u2286 \u22c3 i, t i\nhtm : \u2200 (i : \u03b9), MeasurableSet (t i)\nhfm : \u2200 (i : \u03b9), Measurable (f i)\ns : Set \u03b2\nhs : MeasurableSet s\n\u22a2 MeasurableSet (inclusion hT \u207b\u00b9' \u22c3 i, inclusion (_ : t i \u2286 \u22c3 i, t i) '' (f i \u207b\u00b9' s))"}, {"tactic": "exact .preimage (.iUnion fun i => .image_inclusion _ (htm _) (hfm i hs)) (measurable_inclusion _)", "annotated_tactic": ["exact .preimage (.iUnion fun i => .image_inclusion _ (htm _) (hfm i hs)) (<a>measurable_inclusion</a> _)", [{"full_name": "measurable_inclusion", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [626, 9], "def_end_pos": [626, 29]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u\u03b9\ns\u271d t\u271d u : Set \u03b1\nm : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d : Countable \u03b9\nt : \u03b9 \u2192 Set \u03b1\nf : (i : \u03b9) \u2192 \u2191(t i) \u2192 \u03b2\nhtf :\n  \u2200 (i j : \u03b9) (x : \u03b1) (hxi : x \u2208 t i) (hxj : x \u2208 t j),\n    f i { val := x, property := hxi } = f j { val := x, property := hxj }\nT : Set \u03b1\nhT : T \u2286 \u22c3 i, t i\nhtm : \u2200 (i : \u03b9), MeasurableSet (t i)\nhfm : \u2200 (i : \u03b9), Measurable (f i)\ns : Set \u03b2\nhs : MeasurableSet s\n\u22a2 MeasurableSet (inclusion hT \u207b\u00b9' \u22c3 i, inclusion (_ : t i \u2286 \u22c3 i, t i) '' (f i \u207b\u00b9' s))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/List/Basic.lean", "full_name": "List.join_eq_joinTR", "start": [91, 10], "end": [92, 60], "traced_tactics": [{"tactic": "funext \u03b1 l", "annotated_tactic": ["funext \u03b1 l", []], "state_before": "\u22a2 @join = @joinTR", "state_after": "case h.h\n\u03b1 : Type u_1\nl : List (List \u03b1)\n\u22a2 join l = joinTR l"}, {"tactic": "rw [\u2190 List.bind_id, List.bind_eq_bindTR]", "annotated_tactic": ["rw [\u2190 <a>List.bind_id</a>, <a>List.bind_eq_bindTR</a>]", [{"full_name": "List.bind_id", "def_path": "lake-packages/std/Std/Data/List/Init/Lemmas.lean", "def_pos": [110, 17], "def_end_pos": [110, 24]}, {"full_name": "List.bind_eq_bindTR", "def_path": "lake-packages/std/Std/Data/List/Basic.lean", "def_pos": [81, 18], "def_end_pos": [81, 32]}]], "state_before": "case h.h\n\u03b1 : Type u_1\nl : List (List \u03b1)\n\u22a2 join l = joinTR l", "state_after": "case h.h\n\u03b1 : Type u_1\nl : List (List \u03b1)\n\u22a2 bindTR l id = joinTR l"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case h.h\n\u03b1 : Type u_1\nl : List (List \u03b1)\n\u22a2 bindTR l id = joinTR l", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "full_name": "MeasurableSpace.comap_not", "start": [1840, 9], "end": [1842, 62], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/Equiv.lean", "full_name": "MvPolynomial.finSuccEquiv_support", "start": [447, 1], "end": [458, 76], "traced_tactics": [{"tactic": "ext i", "annotated_tactic": ["ext i", []], "state_before": "R : Type u\nS\u2081 : Type v\nS\u2082 : Type w\nS\u2083 : Type x\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\nn : \u2115\nf : MvPolynomial (Fin (n + 1)) R\n\u22a2 Polynomial.support (\u2191(finSuccEquiv R n) f) = Finset.image (fun m => \u2191m 0) (support f)", "state_after": "case a\nR : Type u\nS\u2081 : Type v\nS\u2082 : Type w\nS\u2083 : Type x\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\nn : \u2115\nf : MvPolynomial (Fin (n + 1)) R\ni : \u2115\n\u22a2 i \u2208 Polynomial.support (\u2191(finSuccEquiv R n) f) \u2194 i \u2208 Finset.image (fun m => \u2191m 0) (support f)"}, {"tactic": "rw [Polynomial.mem_support_iff, Finset.mem_image, nonzero_iff_exists]", "annotated_tactic": ["rw [<a>Polynomial.mem_support_iff</a>, <a>Finset.mem_image</a>, <a>nonzero_iff_exists</a>]", [{"full_name": "Polynomial.mem_support_iff", "def_path": "Mathlib/Data/Polynomial/Basic.lean", "def_pos": [732, 9], "def_end_pos": [732, 24]}, {"full_name": "Finset.mem_image", "def_path": "Mathlib/Data/Finset/Image.lean", "def_pos": [330, 9], "def_end_pos": [330, 18]}, {"full_name": "Finsupp.nonzero_iff_exists", "def_path": "Mathlib/Data/Finsupp/Defs.lean", "def_pos": [222, 9], "def_end_pos": [222, 27]}]], "state_before": "case a\nR : Type u\nS\u2081 : Type v\nS\u2082 : Type w\nS\u2083 : Type x\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\nn : \u2115\nf : MvPolynomial (Fin (n + 1)) R\ni : \u2115\n\u22a2 i \u2208 Polynomial.support (\u2191(finSuccEquiv R n) f) \u2194 i \u2208 Finset.image (fun m => \u2191m 0) (support f)", "state_after": "case a\nR : Type u\nS\u2081 : Type v\nS\u2082 : Type w\nS\u2083 : Type x\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\nn : \u2115\nf : MvPolynomial (Fin (n + 1)) R\ni : \u2115\n\u22a2 (\u2203 a, \u2191(Polynomial.coeff (\u2191(finSuccEquiv R n) f) i) a \u2260 0) \u2194 \u2203 a, a \u2208 support f \u2227 \u2191a 0 = i"}, {"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "case a\nR : Type u\nS\u2081 : Type v\nS\u2082 : Type w\nS\u2083 : Type x\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\nn : \u2115\nf : MvPolynomial (Fin (n + 1)) R\ni : \u2115\n\u22a2 (\u2203 a, \u2191(Polynomial.coeff (\u2191(finSuccEquiv R n) f) i) a \u2260 0) \u2194 \u2203 a, a \u2208 support f \u2227 \u2191a 0 = i", "state_after": "case a.mp\nR : Type u\nS\u2081 : Type v\nS\u2082 : Type w\nS\u2083 : Type x\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\nn : \u2115\nf : MvPolynomial (Fin (n + 1)) R\ni : \u2115\n\u22a2 (\u2203 a, \u2191(Polynomial.coeff (\u2191(finSuccEquiv R n) f) i) a \u2260 0) \u2192 \u2203 a, a \u2208 support f \u2227 \u2191a 0 = i\n\ncase a.mpr\nR : Type u\nS\u2081 : Type v\nS\u2082 : Type w\nS\u2083 : Type x\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\nn : \u2115\nf : MvPolynomial (Fin (n + 1)) R\ni : \u2115\n\u22a2 (\u2203 a, a \u2208 support f \u2227 \u2191a 0 = i) \u2192 \u2203 a, \u2191(Polynomial.coeff (\u2191(finSuccEquiv R n) f) i) a \u2260 0"}, {"tactic": "rintro \u27e8m, hm\u27e9", "annotated_tactic": ["rintro \u27e8m, hm\u27e9", []], "state_before": "case a.mp\nR : Type u\nS\u2081 : Type v\nS\u2082 : Type w\nS\u2083 : Type x\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\nn : \u2115\nf : MvPolynomial (Fin (n + 1)) R\ni : \u2115\n\u22a2 (\u2203 a, \u2191(Polynomial.coeff (\u2191(finSuccEquiv R n) f) i) a \u2260 0) \u2192 \u2203 a, a \u2208 support f \u2227 \u2191a 0 = i", "state_after": "case a.mp.intro\nR : Type u\nS\u2081 : Type v\nS\u2082 : Type w\nS\u2083 : Type x\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\nn : \u2115\nf : MvPolynomial (Fin (n + 1)) R\ni : \u2115\nm : Fin n \u2192\u2080 \u2115\nhm : \u2191(Polynomial.coeff (\u2191(finSuccEquiv R n) f) i) m \u2260 0\n\u22a2 \u2203 a, a \u2208 support f \u2227 \u2191a 0 = i"}, {"tactic": "refine' \u27e8cons i m, _, cons_zero _ _\u27e9", "annotated_tactic": ["refine' \u27e8<a>cons</a> i m, _, <a>cons_zero</a> _ _\u27e9", [{"full_name": "Finsupp.cons", "def_path": "Mathlib/Data/Finsupp/Fin.lean", "def_pos": [35, 5], "def_end_pos": [35, 9]}, {"full_name": "Finsupp.cons_zero", "def_path": "Mathlib/Data/Finsupp/Fin.lean", "def_pos": [44, 9], "def_end_pos": [44, 18]}]], "state_before": "case a.mp.intro\nR : Type u\nS\u2081 : Type v\nS\u2082 : Type w\nS\u2083 : Type x\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\nn : \u2115\nf : MvPolynomial (Fin (n + 1)) R\ni : \u2115\nm : Fin n \u2192\u2080 \u2115\nhm : \u2191(Polynomial.coeff (\u2191(finSuccEquiv R n) f) i) m \u2260 0\n\u22a2 \u2203 a, a \u2208 support f \u2227 \u2191a 0 = i", "state_after": "case a.mp.intro\nR : Type u\nS\u2081 : Type v\nS\u2082 : Type w\nS\u2083 : Type x\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\nn : \u2115\nf : MvPolynomial (Fin (n + 1)) R\ni : \u2115\nm : Fin n \u2192\u2080 \u2115\nhm : \u2191(Polynomial.coeff (\u2191(finSuccEquiv R n) f) i) m \u2260 0\n\u22a2 cons i m \u2208 support f"}, {"tactic": "rw [\u2190 support_coeff_finSuccEquiv]", "annotated_tactic": ["rw [\u2190 <a>support_coeff_finSuccEquiv</a>]", [{"full_name": "MvPolynomial.support_coeff_finSuccEquiv", "def_path": "Mathlib/Data/MvPolynomial/Equiv.lean", "def_pos": [418, 9], "def_end_pos": [418, 35]}]], "state_before": "case a.mp.intro\nR : Type u\nS\u2081 : Type v\nS\u2082 : Type w\nS\u2083 : Type x\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\nn : \u2115\nf : MvPolynomial (Fin (n + 1)) R\ni : \u2115\nm : Fin n \u2192\u2080 \u2115\nhm : \u2191(Polynomial.coeff (\u2191(finSuccEquiv R n) f) i) m \u2260 0\n\u22a2 cons i m \u2208 support f", "state_after": "case a.mp.intro\nR : Type u\nS\u2081 : Type v\nS\u2082 : Type w\nS\u2083 : Type x\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\nn : \u2115\nf : MvPolynomial (Fin (n + 1)) R\ni : \u2115\nm : Fin n \u2192\u2080 \u2115\nhm : \u2191(Polynomial.coeff (\u2191(finSuccEquiv R n) f) i) m \u2260 0\n\u22a2 m \u2208 support (Polynomial.coeff (\u2191(finSuccEquiv R n) f) i)"}, {"tactic": "simpa using hm", "annotated_tactic": ["simpa using hm", []], "state_before": "case a.mp.intro\nR : Type u\nS\u2081 : Type v\nS\u2082 : Type w\nS\u2083 : Type x\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\nn : \u2115\nf : MvPolynomial (Fin (n + 1)) R\ni : \u2115\nm : Fin n \u2192\u2080 \u2115\nhm : \u2191(Polynomial.coeff (\u2191(finSuccEquiv R n) f) i) m \u2260 0\n\u22a2 m \u2208 support (Polynomial.coeff (\u2191(finSuccEquiv R n) f) i)", "state_after": "no goals"}, {"tactic": "rintro \u27e8m, h, rfl\u27e9", "annotated_tactic": ["rintro \u27e8m, h, rfl\u27e9", []], "state_before": "case a.mpr\nR : Type u\nS\u2081 : Type v\nS\u2082 : Type w\nS\u2083 : Type x\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\nn : \u2115\nf : MvPolynomial (Fin (n + 1)) R\ni : \u2115\n\u22a2 (\u2203 a, a \u2208 support f \u2227 \u2191a 0 = i) \u2192 \u2203 a, \u2191(Polynomial.coeff (\u2191(finSuccEquiv R n) f) i) a \u2260 0", "state_after": "case a.mpr.intro.intro\nR : Type u\nS\u2081 : Type v\nS\u2082 : Type w\nS\u2083 : Type x\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\nn : \u2115\nf : MvPolynomial (Fin (n + 1)) R\nm : Fin (n + 1) \u2192\u2080 \u2115\nh : m \u2208 support f\n\u22a2 \u2203 a, \u2191(Polynomial.coeff (\u2191(finSuccEquiv R n) f) (\u2191m 0)) a \u2260 0"}, {"tactic": "refine' \u27e8tail m, _\u27e9", "annotated_tactic": ["refine' \u27e8<a>tail</a> m, _\u27e9", [{"full_name": "Finsupp.tail", "def_path": "Mathlib/Data/Finsupp/Fin.lean", "def_pos": [30, 5], "def_end_pos": [30, 9]}]], "state_before": "case a.mpr.intro.intro\nR : Type u\nS\u2081 : Type v\nS\u2082 : Type w\nS\u2083 : Type x\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\nn : \u2115\nf : MvPolynomial (Fin (n + 1)) R\nm : Fin (n + 1) \u2192\u2080 \u2115\nh : m \u2208 support f\n\u22a2 \u2203 a, \u2191(Polynomial.coeff (\u2191(finSuccEquiv R n) f) (\u2191m 0)) a \u2260 0", "state_after": "case a.mpr.intro.intro\nR : Type u\nS\u2081 : Type v\nS\u2082 : Type w\nS\u2083 : Type x\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\nn : \u2115\nf : MvPolynomial (Fin (n + 1)) R\nm : Fin (n + 1) \u2192\u2080 \u2115\nh : m \u2208 support f\n\u22a2 \u2191(Polynomial.coeff (\u2191(finSuccEquiv R n) f) (\u2191m 0)) (tail m) \u2260 0"}, {"tactic": "rwa [\u2190 coeff, \u2190 mem_support_iff, support_coeff_finSuccEquiv, cons_tail]", "annotated_tactic": ["rwa [\u2190 <a>coeff</a>, \u2190 <a>mem_support_iff</a>, <a>support_coeff_finSuccEquiv</a>, <a>cons_tail</a>]", [{"full_name": "MvPolynomial.coeff", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [580, 5], "def_end_pos": [580, 10]}, {"full_name": "MvPolynomial.mem_support_iff", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [587, 9], "def_end_pos": [587, 24]}, {"full_name": "MvPolynomial.support_coeff_finSuccEquiv", "def_path": "Mathlib/Data/MvPolynomial/Equiv.lean", "def_pos": [418, 9], "def_end_pos": [418, 35]}, {"full_name": "Finsupp.cons_tail", "def_path": "Mathlib/Data/Finsupp/Fin.lean", "def_pos": [60, 9], "def_end_pos": [60, 18]}]], "state_before": "case a.mpr.intro.intro\nR : Type u\nS\u2081 : Type v\nS\u2082 : Type w\nS\u2083 : Type x\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\nn : \u2115\nf : MvPolynomial (Fin (n + 1)) R\nm : Fin (n + 1) \u2192\u2080 \u2115\nh : m \u2208 support f\n\u22a2 \u2191(Polynomial.coeff (\u2191(finSuccEquiv R n) f) (\u2191m 0)) (tail m) \u2260 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Martingale/Basic.lean", "full_name": "MeasureTheory.Submartingale.pos", "start": [280, 11], "end": [281, 47], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Lattice.lean", "full_name": "Finset.max_erase_ne_self", "start": [1642, 1], "end": [1647, 29], "traced_tactics": [{"tactic": "by_cases s0 : (s.erase x).Nonempty", "annotated_tactic": ["by_cases s0 : (s.erase x).<a>Nonempty</a>", [{"full_name": "Finset.Nonempty", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [474, 15], "def_end_pos": [474, 23]}]], "state_before": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Type u_5\n\u03ba : Type u_6\ninst\u271d : LinearOrder \u03b1\ns\u271d : Finset \u03b1\nH : Finset.Nonempty s\u271d\nx : \u03b1\ns : Finset \u03b1\n\u22a2 Finset.max (erase s x) \u2260 \u2191x", "state_after": "case pos\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Type u_5\n\u03ba : Type u_6\ninst\u271d : LinearOrder \u03b1\ns\u271d : Finset \u03b1\nH : Finset.Nonempty s\u271d\nx : \u03b1\ns : Finset \u03b1\ns0 : Finset.Nonempty (erase s x)\n\u22a2 Finset.max (erase s x) \u2260 \u2191x\n\ncase neg\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Type u_5\n\u03ba : Type u_6\ninst\u271d : LinearOrder \u03b1\ns\u271d : Finset \u03b1\nH : Finset.Nonempty s\u271d\nx : \u03b1\ns : Finset \u03b1\ns0 : \u00acFinset.Nonempty (erase s x)\n\u22a2 Finset.max (erase s x) \u2260 \u2191x"}, {"tactic": "refine' ne_of_eq_of_ne (coe_max' s0).symm _", "annotated_tactic": ["refine' <a>ne_of_eq_of_ne</a> (<a>coe_max'</a> s0).<a>symm</a> _", [{"full_name": "ne_of_eq_of_ne", "def_path": "Mathlib/Init/CCLemmas.lean", "def_pos": [118, 9], "def_end_pos": [118, 23]}, {"full_name": "Finset.coe_max'", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [1606, 9], "def_end_pos": [1606, 17]}, {"full_name": "Eq.symm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [310, 9], "def_end_pos": [310, 16]}]], "state_before": "case pos\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Type u_5\n\u03ba : Type u_6\ninst\u271d : LinearOrder \u03b1\ns\u271d : Finset \u03b1\nH : Finset.Nonempty s\u271d\nx : \u03b1\ns : Finset \u03b1\ns0 : Finset.Nonempty (erase s x)\n\u22a2 Finset.max (erase s x) \u2260 \u2191x", "state_after": "case pos\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Type u_5\n\u03ba : Type u_6\ninst\u271d : LinearOrder \u03b1\ns\u271d : Finset \u03b1\nH : Finset.Nonempty s\u271d\nx : \u03b1\ns : Finset \u03b1\ns0 : Finset.Nonempty (erase s x)\n\u22a2 \u2191(max' (erase s x) s0) \u2260 \u2191x"}, {"tactic": "exact WithBot.coe_eq_coe.not.mpr (max'_erase_ne_self _)", "annotated_tactic": ["exact WithBot.coe_eq_coe.not.mpr (<a>max'_erase_ne_self</a> _)", [{"full_name": "Finset.max'_erase_ne_self", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [1634, 9], "def_end_pos": [1634, 27]}]], "state_before": "case pos\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Type u_5\n\u03ba : Type u_6\ninst\u271d : LinearOrder \u03b1\ns\u271d : Finset \u03b1\nH : Finset.Nonempty s\u271d\nx : \u03b1\ns : Finset \u03b1\ns0 : Finset.Nonempty (erase s x)\n\u22a2 \u2191(max' (erase s x) s0) \u2260 \u2191x", "state_after": "no goals"}, {"tactic": "rw [not_nonempty_iff_eq_empty.mp s0, max_empty]", "annotated_tactic": ["rw [not_nonempty_iff_eq_empty.mp s0, <a>max_empty</a>]", [{"full_name": "Finset.max_empty", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [1256, 9], "def_end_pos": [1256, 18]}]], "state_before": "case neg\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Type u_5\n\u03ba : Type u_6\ninst\u271d : LinearOrder \u03b1\ns\u271d : Finset \u03b1\nH : Finset.Nonempty s\u271d\nx : \u03b1\ns : Finset \u03b1\ns0 : \u00acFinset.Nonempty (erase s x)\n\u22a2 Finset.max (erase s x) \u2260 \u2191x", "state_after": "case neg\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Type u_5\n\u03ba : Type u_6\ninst\u271d : LinearOrder \u03b1\ns\u271d : Finset \u03b1\nH : Finset.Nonempty s\u271d\nx : \u03b1\ns : Finset \u03b1\ns0 : \u00acFinset.Nonempty (erase s x)\n\u22a2 \u22a5 \u2260 \u2191x"}, {"tactic": "exact WithBot.bot_ne_coe", "annotated_tactic": ["exact <a>WithBot.bot_ne_coe</a>", [{"full_name": "WithBot.bot_ne_coe", "def_path": "Mathlib/Order/WithBot.lean", "def_pos": [86, 9], "def_end_pos": [86, 19]}]], "state_before": "case neg\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Type u_5\n\u03ba : Type u_6\ninst\u271d : LinearOrder \u03b1\ns\u271d : Finset \u03b1\nH : Finset.Nonempty s\u271d\nx : \u03b1\ns : Finset \u03b1\ns0 : \u00acFinset.Nonempty (erase s x)\n\u22a2 \u22a5 \u2260 \u2191x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Independence/Basic.lean", "full_name": "ProbabilityTheory.IndepSets.bInter", "start": [311, 1], "end": [314, 28], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/LocallyFinite.lean", "full_name": "Set.infinite_iff_exists_lt", "start": [894, 1], "end": [895, 61], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Decomposition/RadonNikodym.lean", "full_name": "MeasureTheory.SignedMeasure.withDensity\u1d65_rnDeriv_eq", "start": [87, 1], "end": [105, 35], "traced_tactics": [{"tactic": "rw [absolutelyContinuous_ennreal_iff, (_ : \u03bc.toENNRealVectorMeasure.ennrealToMeasure = \u03bc),\n  totalVariation_absolutelyContinuous_iff] at h", "annotated_tactic": ["rw [<a>absolutelyContinuous_ennreal_iff</a>, (_ : \u03bc.toENNRealVectorMeasure.ennrealToMeasure = \u03bc),\n    <a>totalVariation_absolutelyContinuous_iff</a>] at h", [{"full_name": "MeasureTheory.SignedMeasure.absolutelyContinuous_ennreal_iff", "def_path": "Mathlib/MeasureTheory/Decomposition/Jordan.lean", "def_pos": [516, 9], "def_end_pos": [516, 41]}, {"full_name": "MeasureTheory.SignedMeasure.totalVariation_absolutelyContinuous_iff", "def_path": "Mathlib/MeasureTheory/Decomposition/Jordan.lean", "def_pos": [532, 9], "def_end_pos": [532, 48]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nh : s \u226a\u1d65 toENNRealVectorMeasure \u03bc\n\u22a2 withDensity\u1d65 \u03bc (rnDeriv s \u03bc) = s", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nh : (toJordanDecomposition s).posPart \u226a \u03bc \u2227 (toJordanDecomposition s).negPart \u226a \u03bc\n\u22a2 withDensity\u1d65 \u03bc (rnDeriv s \u03bc) = s\n\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nh : totalVariation s \u226a ennrealToMeasure (toENNRealVectorMeasure \u03bc)\n\u22a2 ennrealToMeasure (toENNRealVectorMeasure \u03bc) = \u03bc"}, {"tactic": "ext1 i hi", "annotated_tactic": ["ext1 i hi", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nh : (toJordanDecomposition s).posPart \u226a \u03bc \u2227 (toJordanDecomposition s).negPart \u226a \u03bc\n\u22a2 withDensity\u1d65 \u03bc (rnDeriv s \u03bc) = s", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nh : (toJordanDecomposition s).posPart \u226a \u03bc \u2227 (toJordanDecomposition s).negPart \u226a \u03bc\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 \u2191(withDensity\u1d65 \u03bc (rnDeriv s \u03bc)) i = \u2191s i"}, {"tactic": "rw [withDensity\u1d65_apply (integrable_rnDeriv _ _) hi, rnDeriv, integral_sub,\n  withDensity_rnDeriv_toReal_eq h.1 hi, withDensity_rnDeriv_toReal_eq h.2 hi]", "annotated_tactic": ["rw [<a>withDensity\u1d65_apply</a> (<a>integrable_rnDeriv</a> _ _) hi, <a>rnDeriv</a>, <a>integral_sub</a>,\n      <a>withDensity_rnDeriv_toReal_eq</a> h.1 hi, <a>withDensity_rnDeriv_toReal_eq</a> h.2 hi]", [{"full_name": "MeasureTheory.withDensity\u1d65_apply", "def_path": "Mathlib/MeasureTheory/Measure/WithDensityVectorMeasure.lean", "def_pos": [60, 9], "def_end_pos": [60, 27]}, {"full_name": "MeasureTheory.SignedMeasure.integrable_rnDeriv", "def_path": "Mathlib/MeasureTheory/Decomposition/Lebesgue.lean", "def_pos": [916, 9], "def_end_pos": [916, 27]}, {"full_name": "MeasureTheory.SignedMeasure.rnDeriv", "def_path": "Mathlib/MeasureTheory/Decomposition/Lebesgue.lean", "def_pos": [894, 5], "def_end_pos": [894, 12]}, {"full_name": "MeasureTheory.integral_sub", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [901, 9], "def_end_pos": [901, 21]}, {"full_name": "MeasureTheory.Measure.withDensity_rnDeriv_toReal_eq", "def_path": "Mathlib/MeasureTheory/Decomposition/RadonNikodym.lean", "def_pos": [70, 9], "def_end_pos": [70, 38]}, {"full_name": "MeasureTheory.Measure.withDensity_rnDeriv_toReal_eq", "def_path": "Mathlib/MeasureTheory/Decomposition/RadonNikodym.lean", "def_pos": [70, 9], "def_end_pos": [70, 38]}]], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nh : (toJordanDecomposition s).posPart \u226a \u03bc \u2227 (toJordanDecomposition s).negPart \u226a \u03bc\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 \u2191(withDensity\u1d65 \u03bc (rnDeriv s \u03bc)) i = \u2191s i", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nh : (toJordanDecomposition s).posPart \u226a \u03bc \u2227 (toJordanDecomposition s).negPart \u226a \u03bc\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 ENNReal.toReal (\u2191\u2191(toJordanDecomposition s).posPart i) - ENNReal.toReal (\u2191\u2191(toJordanDecomposition s).negPart i) = \u2191s i\n\ncase h.hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nh : (toJordanDecomposition s).posPart \u226a \u03bc \u2227 (toJordanDecomposition s).negPart \u226a \u03bc\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 Integrable fun x => ENNReal.toReal (Measure.rnDeriv (toJordanDecomposition s).posPart \u03bc x)\n\ncase h.hg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nh : (toJordanDecomposition s).posPart \u226a \u03bc \u2227 (toJordanDecomposition s).negPart \u226a \u03bc\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 Integrable fun x => ENNReal.toReal (Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x)"}, {"tactic": "conv_rhs => rw [\u2190 s.toSignedMeasure_toJordanDecomposition]", "annotated_tactic": ["conv_rhs => rw [\u2190 s.toSignedMeasure_toJordanDecomposition]", []], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nh : (toJordanDecomposition s).posPart \u226a \u03bc \u2227 (toJordanDecomposition s).negPart \u226a \u03bc\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 ENNReal.toReal (\u2191\u2191(toJordanDecomposition s).posPart i) - ENNReal.toReal (\u2191\u2191(toJordanDecomposition s).negPart i) = \u2191s i", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nh : (toJordanDecomposition s).posPart \u226a \u03bc \u2227 (toJordanDecomposition s).negPart \u226a \u03bc\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 ENNReal.toReal (\u2191\u2191(toJordanDecomposition s).posPart i) - ENNReal.toReal (\u2191\u2191(toJordanDecomposition s).negPart i) =\n    \u2191(JordanDecomposition.toSignedMeasure (toJordanDecomposition s)) i"}, {"tactic": "erw [VectorMeasure.sub_apply]", "annotated_tactic": ["erw [<a>VectorMeasure.sub_apply</a>]", [{"full_name": "MeasureTheory.VectorMeasure.sub_apply", "def_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "def_pos": [370, 9], "def_end_pos": [370, 18]}]], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nh : (toJordanDecomposition s).posPart \u226a \u03bc \u2227 (toJordanDecomposition s).negPart \u226a \u03bc\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 ENNReal.toReal (\u2191\u2191(toJordanDecomposition s).posPart i) - ENNReal.toReal (\u2191\u2191(toJordanDecomposition s).negPart i) =\n    \u2191(JordanDecomposition.toSignedMeasure (toJordanDecomposition s)) i", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nh : (toJordanDecomposition s).posPart \u226a \u03bc \u2227 (toJordanDecomposition s).negPart \u226a \u03bc\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 ENNReal.toReal (\u2191\u2191(toJordanDecomposition s).posPart i) - ENNReal.toReal (\u2191\u2191(toJordanDecomposition s).negPart i) =\n    \u2191(toSignedMeasure (toJordanDecomposition s).posPart) i - \u2191(toSignedMeasure (toJordanDecomposition s).negPart) i"}, {"tactic": "rw [toSignedMeasure_apply_measurable hi, toSignedMeasure_apply_measurable hi]", "annotated_tactic": ["rw [<a>toSignedMeasure_apply_measurable</a> hi, <a>toSignedMeasure_apply_measurable</a> hi]", [{"full_name": "MeasureTheory.Measure.toSignedMeasure_apply_measurable", "def_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "def_pos": [435, 9], "def_end_pos": [435, 41]}, {"full_name": "MeasureTheory.Measure.toSignedMeasure_apply_measurable", "def_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "def_pos": [435, 9], "def_end_pos": [435, 41]}]], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nh : (toJordanDecomposition s).posPart \u226a \u03bc \u2227 (toJordanDecomposition s).negPart \u226a \u03bc\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 ENNReal.toReal (\u2191\u2191(toJordanDecomposition s).posPart i) - ENNReal.toReal (\u2191\u2191(toJordanDecomposition s).negPart i) =\n    \u2191(toSignedMeasure (toJordanDecomposition s).posPart) i - \u2191(toSignedMeasure (toJordanDecomposition s).negPart) i", "state_after": "no goals"}, {"tactic": "rw [\u2190 integrableOn_univ]", "annotated_tactic": ["rw [\u2190 <a>integrableOn_univ</a>]", [{"full_name": "MeasureTheory.integrableOn_univ", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [110, 9], "def_end_pos": [110, 26]}]], "state_before": "case h.hg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nh : (toJordanDecomposition s).posPart \u226a \u03bc \u2227 (toJordanDecomposition s).negPart \u226a \u03bc\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 Integrable fun x => ENNReal.toReal (Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x)", "state_after": "case h.hg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nh : (toJordanDecomposition s).posPart \u226a \u03bc \u2227 (toJordanDecomposition s).negPart \u226a \u03bc\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 IntegrableOn (fun x => ENNReal.toReal (Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x)) Set.univ"}, {"tactic": "refine' \u27e8_, hasFiniteIntegral_toReal_of_lintegral_ne_top _\u27e9", "annotated_tactic": ["refine' \u27e8_, <a>hasFiniteIntegral_toReal_of_lintegral_ne_top</a> _\u27e9", [{"full_name": "MeasureTheory.hasFiniteIntegral_toReal_of_lintegral_ne_top", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [266, 9], "def_end_pos": [266, 53]}]], "state_before": "case h.hg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nh : (toJordanDecomposition s).posPart \u226a \u03bc \u2227 (toJordanDecomposition s).negPart \u226a \u03bc\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 IntegrableOn (fun x => ENNReal.toReal (Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x)) Set.univ", "state_after": "case h.hg.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nh : (toJordanDecomposition s).posPart \u226a \u03bc \u2227 (toJordanDecomposition s).negPart \u226a \u03bc\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 AEStronglyMeasurable (fun x => ENNReal.toReal (Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x))\n    (Measure.restrict \u03bc Set.univ)\n\ncase h.hg.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nh : (toJordanDecomposition s).posPart \u226a \u03bc \u2227 (toJordanDecomposition s).negPart \u226a \u03bc\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 \u222b\u207b (x : \u03b1) in Set.univ, Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x \u2202\u03bc \u2260 \u22a4"}, {"tactic": "apply Measurable.aestronglyMeasurable", "annotated_tactic": ["apply <a>Measurable.aestronglyMeasurable</a>", [{"full_name": "Measurable.aestronglyMeasurable", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1285, 9], "def_end_pos": [1285, 47]}]], "state_before": "case h.hg.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nh : (toJordanDecomposition s).posPart \u226a \u03bc \u2227 (toJordanDecomposition s).negPart \u226a \u03bc\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 AEStronglyMeasurable (fun x => ENNReal.toReal (Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x))\n    (Measure.restrict \u03bc Set.univ)", "state_after": "case h.hg.refine'_1.hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nh : (toJordanDecomposition s).posPart \u226a \u03bc \u2227 (toJordanDecomposition s).negPart \u226a \u03bc\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 Measurable fun x => ENNReal.toReal (Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x)"}, {"tactic": "measurability", "annotated_tactic": ["measurability", []], "state_before": "case h.hg.refine'_1.hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nh : (toJordanDecomposition s).posPart \u226a \u03bc \u2227 (toJordanDecomposition s).negPart \u226a \u03bc\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 Measurable fun x => ENNReal.toReal (Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x)", "state_after": "no goals"}, {"tactic": "rw [set_lintegral_univ]", "annotated_tactic": ["rw [<a>set_lintegral_univ</a>]", [{"full_name": "MeasureTheory.set_lintegral_univ", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [645, 9], "def_end_pos": [645, 27]}]], "state_before": "case h.hg.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nh : (toJordanDecomposition s).posPart \u226a \u03bc \u2227 (toJordanDecomposition s).negPart \u226a \u03bc\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 \u222b\u207b (x : \u03b1) in Set.univ, Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x \u2202\u03bc \u2260 \u22a4", "state_after": "case h.hg.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nh : (toJordanDecomposition s).posPart \u226a \u03bc \u2227 (toJordanDecomposition s).negPart \u226a \u03bc\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x \u2202\u03bc \u2260 \u22a4"}, {"tactic": "exact (lintegral_rnDeriv_lt_top _ _).ne", "annotated_tactic": ["exact (<a>lintegral_rnDeriv_lt_top</a> _ _).<a>ne</a>", [{"full_name": "MeasureTheory.Measure.lintegral_rnDeriv_lt_top", "def_path": "Mathlib/MeasureTheory/Decomposition/Lebesgue.lean", "def_pos": [204, 9], "def_end_pos": [204, 33]}, {"full_name": "LT.lt.ne", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [152, 7], "def_end_pos": [152, 15]}]], "state_before": "case h.hg.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nh : (toJordanDecomposition s).posPart \u226a \u03bc \u2227 (toJordanDecomposition s).negPart \u226a \u03bc\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x \u2202\u03bc \u2260 \u22a4", "state_after": "no goals"}, {"tactic": "exact equivMeasure.right_inv \u03bc", "annotated_tactic": ["exact equivMeasure.right_inv \u03bc", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nh : totalVariation s \u226a ennrealToMeasure (toENNRealVectorMeasure \u03bc)\n\u22a2 ennrealToMeasure (toENNRealVectorMeasure \u03bc) = \u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "full_name": "MeasureTheory.set_lintegral_max", "start": [1274, 1], "end": [1278, 58], "traced_tactics": [{"tactic": "rw [lintegral_max hf hg, restrict_restrict, restrict_restrict, inter_comm s, inter_comm s]", "annotated_tactic": ["rw [<a>lintegral_max</a> hf hg, <a>restrict_restrict</a>, <a>restrict_restrict</a>, <a>inter_comm</a> s, <a>inter_comm</a> s]", [{"full_name": "MeasureTheory.lintegral_max", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [1263, 9], "def_end_pos": [1263, 22]}, {"full_name": "MeasureTheory.Measure.restrict_restrict", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1645, 9], "def_end_pos": [1645, 26]}, {"full_name": "MeasureTheory.Measure.restrict_restrict", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1645, 9], "def_end_pos": [1645, 26]}, {"full_name": "Set.inter_comm", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [940, 9], "def_end_pos": [940, 19]}, {"full_name": "Set.inter_comm", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [940, 9], "def_end_pos": [940, 19]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\ns : Set \u03b1\n\u22a2 \u222b\u207b (x : \u03b1) in s, max (f x) (g x) \u2202\u03bc =\n    \u222b\u207b (x : \u03b1) in s \u2229 {x | f x \u2264 g x}, g x \u2202\u03bc + \u222b\u207b (x : \u03b1) in s \u2229 {x | g x < f x}, f x \u2202\u03bc", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\ns : Set \u03b1\n\u22a2 MeasurableSet {x | g x < f x}\n\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\ns : Set \u03b1\n\u22a2 MeasurableSet {x | f x \u2264 g x}"}, {"tactic": "exacts [measurableSet_lt hg hf, measurableSet_le hf hg]", "annotated_tactic": ["exacts [<a>measurableSet_lt</a> hg hf, <a>measurableSet_le</a> hf hg]", [{"full_name": "measurableSet_lt", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [616, 9], "def_end_pos": [616, 25]}, {"full_name": "measurableSet_le", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [559, 9], "def_end_pos": [559, 25]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\ns : Set \u03b1\n\u22a2 MeasurableSet {x | g x < f x}\n\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\ns : Set \u03b1\n\u22a2 MeasurableSet {x | f x \u2264 g x}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/Average.lean", "full_name": "MeasureTheory.measure_le_setAverage_pos", "start": [469, 1], "end": [484, 84], "traced_tactics": [{"tactic": "refine' pos_iff_ne_zero.2 fun H => _", "annotated_tactic": ["refine' <a>pos_iff_ne_zero</a>.2 fun H => _", [{"full_name": "pos_iff_ne_zero", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [243, 3], "def_end_pos": [243, 14]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t N : Set \u03b1\nf : \u03b1 \u2192 \u211d\nh\u03bc : \u2191\u2191\u03bc s \u2260 0\nh\u03bc\u2081 : \u2191\u2191\u03bc s \u2260 \u22a4\nhf : IntegrableOn f s\n\u22a2 0 < \u2191\u2191\u03bc {x | x \u2208 s \u2227 f x \u2264 \u2a0d (a : \u03b1) in s, f a \u2202\u03bc}", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t N : Set \u03b1\nf : \u03b1 \u2192 \u211d\nh\u03bc : \u2191\u2191\u03bc s \u2260 0\nh\u03bc\u2081 : \u2191\u2191\u03bc s \u2260 \u22a4\nhf : IntegrableOn f s\nH : \u2191\u2191\u03bc {x | x \u2208 s \u2227 f x \u2264 \u2a0d (a : \u03b1) in s, f a \u2202\u03bc} = 0\n\u22a2 False"}, {"tactic": "replace H : (\u03bc.restrict s) {x | f x \u2264 \u2a0d a in s, f a \u2202\u03bc} = 0", "annotated_tactic": ["replace H : (\u03bc.restrict s) {x | f x \u2264 \u2a0d a in s, f a \u2202\u03bc} = 0", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t N : Set \u03b1\nf : \u03b1 \u2192 \u211d\nh\u03bc : \u2191\u2191\u03bc s \u2260 0\nh\u03bc\u2081 : \u2191\u2191\u03bc s \u2260 \u22a4\nhf : IntegrableOn f s\nH : \u2191\u2191\u03bc {x | x \u2208 s \u2227 f x \u2264 \u2a0d (a : \u03b1) in s, f a \u2202\u03bc} = 0\n\u22a2 False", "state_after": "case H\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t N : Set \u03b1\nf : \u03b1 \u2192 \u211d\nh\u03bc : \u2191\u2191\u03bc s \u2260 0\nh\u03bc\u2081 : \u2191\u2191\u03bc s \u2260 \u22a4\nhf : IntegrableOn f s\nH : \u2191\u2191\u03bc {x | x \u2208 s \u2227 f x \u2264 \u2a0d (a : \u03b1) in s, f a \u2202\u03bc} = 0\n\u22a2 \u2191\u2191(Measure.restrict \u03bc s) {x | f x \u2264 \u2a0d (a : \u03b1) in s, f a \u2202\u03bc} = 0\n\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t N : Set \u03b1\nf : \u03b1 \u2192 \u211d\nh\u03bc : \u2191\u2191\u03bc s \u2260 0\nh\u03bc\u2081 : \u2191\u2191\u03bc s \u2260 \u22a4\nhf : IntegrableOn f s\nH : \u2191\u2191(Measure.restrict \u03bc s) {x | f x \u2264 \u2a0d (a : \u03b1) in s, f a \u2202\u03bc} = 0\n\u22a2 False"}, {"tactic": "haveI := Fact.mk h\u03bc\u2081.lt_top", "annotated_tactic": ["haveI := <a>Fact.mk</a> h\u03bc\u2081.lt_top", [{"full_name": "Fact.mk", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [115, 12], "def_end_pos": [115, 22]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t N : Set \u03b1\nf : \u03b1 \u2192 \u211d\nh\u03bc : \u2191\u2191\u03bc s \u2260 0\nh\u03bc\u2081 : \u2191\u2191\u03bc s \u2260 \u22a4\nhf : IntegrableOn f s\nH : \u2191\u2191(Measure.restrict \u03bc s) {x | f x \u2264 \u2a0d (a : \u03b1) in s, f a \u2202\u03bc} = 0\n\u22a2 False", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t N : Set \u03b1\nf : \u03b1 \u2192 \u211d\nh\u03bc : \u2191\u2191\u03bc s \u2260 0\nh\u03bc\u2081 : \u2191\u2191\u03bc s \u2260 \u22a4\nhf : IntegrableOn f s\nH : \u2191\u2191(Measure.restrict \u03bc s) {x | f x \u2264 \u2a0d (a : \u03b1) in s, f a \u2202\u03bc} = 0\nthis : Fact (\u2191\u2191\u03bc s < \u22a4)\n\u22a2 False"}, {"tactic": "refine' (integral_sub_average (\u03bc.restrict s) f).not_gt _", "annotated_tactic": ["refine' (<a>integral_sub_average</a> (\u03bc.restrict s) f).<a>not_gt</a> _", [{"full_name": "MeasureTheory.integral_sub_average", "def_path": "Mathlib/MeasureTheory/Integral/Average.lean", "def_pos": [408, 9], "def_end_pos": [408, 29]}, {"full_name": "Eq.not_gt", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [246, 9], "def_end_pos": [246, 15]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t N : Set \u03b1\nf : \u03b1 \u2192 \u211d\nh\u03bc : \u2191\u2191\u03bc s \u2260 0\nh\u03bc\u2081 : \u2191\u2191\u03bc s \u2260 \u22a4\nhf : IntegrableOn f s\nH : \u2191\u2191(Measure.restrict \u03bc s) {x | f x \u2264 \u2a0d (a : \u03b1) in s, f a \u2202\u03bc} = 0\nthis : Fact (\u2191\u2191\u03bc s < \u22a4)\n\u22a2 False", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t N : Set \u03b1\nf : \u03b1 \u2192 \u211d\nh\u03bc : \u2191\u2191\u03bc s \u2260 0\nh\u03bc\u2081 : \u2191\u2191\u03bc s \u2260 \u22a4\nhf : IntegrableOn f s\nH : \u2191\u2191(Measure.restrict \u03bc s) {x | f x \u2264 \u2a0d (a : \u03b1) in s, f a \u2202\u03bc} = 0\nthis : Fact (\u2191\u2191\u03bc s < \u22a4)\n\u22a2 0 < \u222b (x : \u03b1) in s, f x - \u2a0d (a : \u03b1) in s, f a \u2202\u03bc \u2202\u03bc"}, {"tactic": "refine' (set_integral_pos_iff_support_of_nonneg_ae _ _).2 _", "annotated_tactic": ["refine' (<a>set_integral_pos_iff_support_of_nonneg_ae</a> _ _).2 _", [{"full_name": "MeasureTheory.set_integral_pos_iff_support_of_nonneg_ae", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [590, 9], "def_end_pos": [590, 50]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t N : Set \u03b1\nf : \u03b1 \u2192 \u211d\nh\u03bc : \u2191\u2191\u03bc s \u2260 0\nh\u03bc\u2081 : \u2191\u2191\u03bc s \u2260 \u22a4\nhf : IntegrableOn f s\nH : \u2191\u2191(Measure.restrict \u03bc s) {x | f x \u2264 \u2a0d (a : \u03b1) in s, f a \u2202\u03bc} = 0\nthis : Fact (\u2191\u2191\u03bc s < \u22a4)\n\u22a2 0 < \u222b (x : \u03b1) in s, f x - \u2a0d (a : \u03b1) in s, f a \u2202\u03bc \u2202\u03bc", "state_after": "case refine'_1\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t N : Set \u03b1\nf : \u03b1 \u2192 \u211d\nh\u03bc : \u2191\u2191\u03bc s \u2260 0\nh\u03bc\u2081 : \u2191\u2191\u03bc s \u2260 \u22a4\nhf : IntegrableOn f s\nH : \u2191\u2191(Measure.restrict \u03bc s) {x | f x \u2264 \u2a0d (a : \u03b1) in s, f a \u2202\u03bc} = 0\nthis : Fact (\u2191\u2191\u03bc s < \u22a4)\n\u22a2 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => f x - \u2a0d (a : \u03b1) in s, f a \u2202\u03bc\n\ncase refine'_2\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t N : Set \u03b1\nf : \u03b1 \u2192 \u211d\nh\u03bc : \u2191\u2191\u03bc s \u2260 0\nh\u03bc\u2081 : \u2191\u2191\u03bc s \u2260 \u22a4\nhf : IntegrableOn f s\nH : \u2191\u2191(Measure.restrict \u03bc s) {x | f x \u2264 \u2a0d (a : \u03b1) in s, f a \u2202\u03bc} = 0\nthis : Fact (\u2191\u2191\u03bc s < \u22a4)\n\u22a2 IntegrableOn (fun x => f x - \u2a0d (a : \u03b1) in s, f a \u2202\u03bc) s\n\ncase refine'_3\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t N : Set \u03b1\nf : \u03b1 \u2192 \u211d\nh\u03bc : \u2191\u2191\u03bc s \u2260 0\nh\u03bc\u2081 : \u2191\u2191\u03bc s \u2260 \u22a4\nhf : IntegrableOn f s\nH : \u2191\u2191(Measure.restrict \u03bc s) {x | f x \u2264 \u2a0d (a : \u03b1) in s, f a \u2202\u03bc} = 0\nthis : Fact (\u2191\u2191\u03bc s < \u22a4)\n\u22a2 0 < \u2191\u2191\u03bc ((support fun x => f x - \u2a0d (a : \u03b1) in s, f a \u2202\u03bc) \u2229 s)"}, {"tactic": "rwa [restrict_apply\u2080, inter_comm]", "annotated_tactic": ["rwa [<a>restrict_apply\u2080</a>, <a>inter_comm</a>]", [{"full_name": "MeasureTheory.Measure.restrict_apply\u2080", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1521, 9], "def_end_pos": [1521, 24]}, {"full_name": "Set.inter_comm", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [940, 9], "def_end_pos": [940, 19]}]], "state_before": "case H\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t N : Set \u03b1\nf : \u03b1 \u2192 \u211d\nh\u03bc : \u2191\u2191\u03bc s \u2260 0\nh\u03bc\u2081 : \u2191\u2191\u03bc s \u2260 \u22a4\nhf : IntegrableOn f s\nH : \u2191\u2191\u03bc {x | x \u2208 s \u2227 f x \u2264 \u2a0d (a : \u03b1) in s, f a \u2202\u03bc} = 0\n\u22a2 \u2191\u2191(Measure.restrict \u03bc s) {x | f x \u2264 \u2a0d (a : \u03b1) in s, f a \u2202\u03bc} = 0", "state_after": "case H\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t N : Set \u03b1\nf : \u03b1 \u2192 \u211d\nh\u03bc : \u2191\u2191\u03bc s \u2260 0\nh\u03bc\u2081 : \u2191\u2191\u03bc s \u2260 \u22a4\nhf : IntegrableOn f s\nH : \u2191\u2191\u03bc {x | x \u2208 s \u2227 f x \u2264 \u2a0d (a : \u03b1) in s, f a \u2202\u03bc} = 0\n\u22a2 NullMeasurableSet {x | f x \u2264 \u2a0d (a : \u03b1) in s, f a \u2202\u03bc}"}, {"tactic": "exact AEStronglyMeasurable.nullMeasurableSet_le hf.1 aestronglyMeasurable_const", "annotated_tactic": ["exact <a>AEStronglyMeasurable.nullMeasurableSet_le</a> hf.1 <a>aestronglyMeasurable_const</a>", [{"full_name": "MeasureTheory.AEStronglyMeasurable.nullMeasurableSet_le", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1550, 9], "def_end_pos": [1550, 29]}, {"full_name": "MeasureTheory.aestronglyMeasurable_const", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1155, 9], "def_end_pos": [1155, 35]}]], "state_before": "case H\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t N : Set \u03b1\nf : \u03b1 \u2192 \u211d\nh\u03bc : \u2191\u2191\u03bc s \u2260 0\nh\u03bc\u2081 : \u2191\u2191\u03bc s \u2260 \u22a4\nhf : IntegrableOn f s\nH : \u2191\u2191\u03bc {x | x \u2208 s \u2227 f x \u2264 \u2a0d (a : \u03b1) in s, f a \u2202\u03bc} = 0\n\u22a2 NullMeasurableSet {x | f x \u2264 \u2a0d (a : \u03b1) in s, f a \u2202\u03bc}", "state_after": "no goals"}, {"tactic": "refine' eq_bot_mono (measure_mono fun x hx => _) H", "annotated_tactic": ["refine' <a>eq_bot_mono</a> (<a>measure_mono</a> fun x hx => _) H", [{"full_name": "eq_bot_mono", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [367, 9], "def_end_pos": [367, 20]}, {"full_name": "MeasureTheory.measure_mono", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [193, 9], "def_end_pos": [193, 21]}]], "state_before": "case refine'_1\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t N : Set \u03b1\nf : \u03b1 \u2192 \u211d\nh\u03bc : \u2191\u2191\u03bc s \u2260 0\nh\u03bc\u2081 : \u2191\u2191\u03bc s \u2260 \u22a4\nhf : IntegrableOn f s\nH : \u2191\u2191(Measure.restrict \u03bc s) {x | f x \u2264 \u2a0d (a : \u03b1) in s, f a \u2202\u03bc} = 0\nthis : Fact (\u2191\u2191\u03bc s < \u22a4)\n\u22a2 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => f x - \u2a0d (a : \u03b1) in s, f a \u2202\u03bc", "state_after": "case refine'_1\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t N : Set \u03b1\nf : \u03b1 \u2192 \u211d\nh\u03bc : \u2191\u2191\u03bc s \u2260 0\nh\u03bc\u2081 : \u2191\u2191\u03bc s \u2260 \u22a4\nhf : IntegrableOn f s\nH : \u2191\u2191(Measure.restrict \u03bc s) {x | f x \u2264 \u2a0d (a : \u03b1) in s, f a \u2202\u03bc} = 0\nthis : Fact (\u2191\u2191\u03bc s < \u22a4)\nx : \u03b1\nhx : x \u2208 {x | (fun x => OfNat.ofNat 0 x \u2264 (fun x => f x - \u2a0d (a : \u03b1) in s, f a \u2202\u03bc) x) x}\u1d9c\n\u22a2 x \u2208 {x | f x \u2264 \u2a0d (a : \u03b1) in s, f a \u2202\u03bc}"}, {"tactic": "simp only [Pi.zero_apply, sub_nonneg, mem_compl_iff, mem_setOf_eq, not_le] at hx", "annotated_tactic": ["simp only [<a>Pi.zero_apply</a>, <a>sub_nonneg</a>, <a>mem_compl_iff</a>, <a>mem_setOf_eq</a>, <a>not_le</a>] at hx", [{"full_name": "Pi.zero_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [46, 3], "def_end_pos": [46, 14]}, {"full_name": "sub_nonneg", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [720, 30], "def_end_pos": [720, 40]}, {"full_name": "Set.mem_compl_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1658, 9], "def_end_pos": [1658, 22]}, {"full_name": "Set.mem_setOf_eq", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [256, 29], "def_end_pos": [256, 41]}, {"full_name": "not_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [373, 9], "def_end_pos": [373, 15]}]], "state_before": "case refine'_1\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t N : Set \u03b1\nf : \u03b1 \u2192 \u211d\nh\u03bc : \u2191\u2191\u03bc s \u2260 0\nh\u03bc\u2081 : \u2191\u2191\u03bc s \u2260 \u22a4\nhf : IntegrableOn f s\nH : \u2191\u2191(Measure.restrict \u03bc s) {x | f x \u2264 \u2a0d (a : \u03b1) in s, f a \u2202\u03bc} = 0\nthis : Fact (\u2191\u2191\u03bc s < \u22a4)\nx : \u03b1\nhx : x \u2208 {x | (fun x => OfNat.ofNat 0 x \u2264 (fun x => f x - \u2a0d (a : \u03b1) in s, f a \u2202\u03bc) x) x}\u1d9c\n\u22a2 x \u2208 {x | f x \u2264 \u2a0d (a : \u03b1) in s, f a \u2202\u03bc}", "state_after": "case refine'_1\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t N : Set \u03b1\nf : \u03b1 \u2192 \u211d\nh\u03bc : \u2191\u2191\u03bc s \u2260 0\nh\u03bc\u2081 : \u2191\u2191\u03bc s \u2260 \u22a4\nhf : IntegrableOn f s\nH : \u2191\u2191(Measure.restrict \u03bc s) {x | f x \u2264 \u2a0d (a : \u03b1) in s, f a \u2202\u03bc} = 0\nthis : Fact (\u2191\u2191\u03bc s < \u22a4)\nx : \u03b1\nhx : f x < \u2a0d (a : \u03b1) in s, f a \u2202\u03bc\n\u22a2 x \u2208 {x | f x \u2264 \u2a0d (a : \u03b1) in s, f a \u2202\u03bc}"}, {"tactic": "exact hx.le", "annotated_tactic": ["exact hx.le", []], "state_before": "case refine'_1\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t N : Set \u03b1\nf : \u03b1 \u2192 \u211d\nh\u03bc : \u2191\u2191\u03bc s \u2260 0\nh\u03bc\u2081 : \u2191\u2191\u03bc s \u2260 \u22a4\nhf : IntegrableOn f s\nH : \u2191\u2191(Measure.restrict \u03bc s) {x | f x \u2264 \u2a0d (a : \u03b1) in s, f a \u2202\u03bc} = 0\nthis : Fact (\u2191\u2191\u03bc s < \u22a4)\nx : \u03b1\nhx : f x < \u2a0d (a : \u03b1) in s, f a \u2202\u03bc\n\u22a2 x \u2208 {x | f x \u2264 \u2a0d (a : \u03b1) in s, f a \u2202\u03bc}", "state_after": "no goals"}, {"tactic": "exact hf.sub (integrableOn_const.2 <| Or.inr <| lt_top_iff_ne_top.2 h\u03bc\u2081)", "annotated_tactic": ["exact hf.sub (<a>integrableOn_const</a>.2 <| <a>Or.inr</a> <| <a>lt_top_iff_ne_top</a>.2 h\u03bc\u2081)", [{"full_name": "MeasureTheory.integrableOn_const", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [119, 9], "def_end_pos": [119, 27]}, {"full_name": "Or.inr", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [519, 5], "def_end_pos": [519, 8]}, {"full_name": "lt_top_iff_ne_top", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [173, 9], "def_end_pos": [173, 26]}]], "state_before": "case refine'_2\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t N : Set \u03b1\nf : \u03b1 \u2192 \u211d\nh\u03bc : \u2191\u2191\u03bc s \u2260 0\nh\u03bc\u2081 : \u2191\u2191\u03bc s \u2260 \u22a4\nhf : IntegrableOn f s\nH : \u2191\u2191(Measure.restrict \u03bc s) {x | f x \u2264 \u2a0d (a : \u03b1) in s, f a \u2202\u03bc} = 0\nthis : Fact (\u2191\u2191\u03bc s < \u22a4)\n\u22a2 IntegrableOn (fun x => f x - \u2a0d (a : \u03b1) in s, f a \u2202\u03bc) s", "state_after": "no goals"}, {"tactic": "rwa [pos_iff_ne_zero, inter_comm, \u2190 diff_compl, \u2190 diff_inter_self_eq_diff, measure_diff_null]", "annotated_tactic": ["rwa [<a>pos_iff_ne_zero</a>, <a>inter_comm</a>, \u2190 <a>diff_compl</a>, \u2190 <a>diff_inter_self_eq_diff</a>, <a>measure_diff_null</a>]", [{"full_name": "pos_iff_ne_zero", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [243, 3], "def_end_pos": [243, 14]}, {"full_name": "Set.inter_comm", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [940, 9], "def_end_pos": [940, 19]}, {"full_name": "Set.diff_compl", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1986, 9], "def_end_pos": [1986, 19]}, {"full_name": "Set.diff_inter_self_eq_diff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [2058, 9], "def_end_pos": [2058, 32]}, {"full_name": "MeasureTheory.measure_diff_null", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [239, 9], "def_end_pos": [239, 26]}]], "state_before": "case refine'_3\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t N : Set \u03b1\nf : \u03b1 \u2192 \u211d\nh\u03bc : \u2191\u2191\u03bc s \u2260 0\nh\u03bc\u2081 : \u2191\u2191\u03bc s \u2260 \u22a4\nhf : IntegrableOn f s\nH : \u2191\u2191(Measure.restrict \u03bc s) {x | f x \u2264 \u2a0d (a : \u03b1) in s, f a \u2202\u03bc} = 0\nthis : Fact (\u2191\u2191\u03bc s < \u22a4)\n\u22a2 0 < \u2191\u2191\u03bc ((support fun x => f x - \u2a0d (a : \u03b1) in s, f a \u2202\u03bc) \u2229 s)", "state_after": "case refine'_3\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t N : Set \u03b1\nf : \u03b1 \u2192 \u211d\nh\u03bc : \u2191\u2191\u03bc s \u2260 0\nh\u03bc\u2081 : \u2191\u2191\u03bc s \u2260 \u22a4\nhf : IntegrableOn f s\nH : \u2191\u2191(Measure.restrict \u03bc s) {x | f x \u2264 \u2a0d (a : \u03b1) in s, f a \u2202\u03bc} = 0\nthis : Fact (\u2191\u2191\u03bc s < \u22a4)\n\u22a2 \u2191\u2191\u03bc ((support fun x => f x - \u2a0d (a : \u03b1) in s, f a \u2202\u03bc)\u1d9c \u2229 s) = 0"}, {"tactic": "refine' eq_bot_mono (measure_mono _) (measure_inter_eq_zero_of_restrict H)", "annotated_tactic": ["refine' <a>eq_bot_mono</a> (<a>measure_mono</a> _) (<a>measure_inter_eq_zero_of_restrict</a> H)", [{"full_name": "eq_bot_mono", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [367, 9], "def_end_pos": [367, 20]}, {"full_name": "MeasureTheory.measure_mono", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [193, 9], "def_end_pos": [193, 21]}, {"full_name": "MeasureTheory.Measure.measure_inter_eq_zero_of_restrict", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1674, 9], "def_end_pos": [1674, 42]}]], "state_before": "case refine'_3\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t N : Set \u03b1\nf : \u03b1 \u2192 \u211d\nh\u03bc : \u2191\u2191\u03bc s \u2260 0\nh\u03bc\u2081 : \u2191\u2191\u03bc s \u2260 \u22a4\nhf : IntegrableOn f s\nH : \u2191\u2191(Measure.restrict \u03bc s) {x | f x \u2264 \u2a0d (a : \u03b1) in s, f a \u2202\u03bc} = 0\nthis : Fact (\u2191\u2191\u03bc s < \u22a4)\n\u22a2 \u2191\u2191\u03bc ((support fun x => f x - \u2a0d (a : \u03b1) in s, f a \u2202\u03bc)\u1d9c \u2229 s) = 0", "state_after": "case refine'_3\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t N : Set \u03b1\nf : \u03b1 \u2192 \u211d\nh\u03bc : \u2191\u2191\u03bc s \u2260 0\nh\u03bc\u2081 : \u2191\u2191\u03bc s \u2260 \u22a4\nhf : IntegrableOn f s\nH : \u2191\u2191(Measure.restrict \u03bc s) {x | f x \u2264 \u2a0d (a : \u03b1) in s, f a \u2202\u03bc} = 0\nthis : Fact (\u2191\u2191\u03bc s < \u22a4)\n\u22a2 (support fun x => f x - \u2a0d (a : \u03b1) in s, f a \u2202\u03bc)\u1d9c \u2229 s \u2286 {x | f x \u2264 \u2a0d (a : \u03b1) in s, f a \u2202\u03bc} \u2229 s"}, {"tactic": "exact inter_subset_inter_left _ fun a ha => (sub_eq_zero.1 <| of_not_not ha).le", "annotated_tactic": ["exact <a>inter_subset_inter_left</a> _ fun a ha => (<a>sub_eq_zero</a>.1 <| <a>of_not_not</a> ha).<a>le</a>", [{"full_name": "Set.inter_subset_inter_left", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1027, 9], "def_end_pos": [1027, 32]}, {"full_name": "sub_eq_zero", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [801, 3], "def_end_pos": [801, 14]}, {"full_name": "of_not_not", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [250, 9], "def_end_pos": [250, 19]}, {"full_name": "Eq.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [159, 7], "def_end_pos": [159, 12]}]], "state_before": "case refine'_3\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t N : Set \u03b1\nf : \u03b1 \u2192 \u211d\nh\u03bc : \u2191\u2191\u03bc s \u2260 0\nh\u03bc\u2081 : \u2191\u2191\u03bc s \u2260 \u22a4\nhf : IntegrableOn f s\nH : \u2191\u2191(Measure.restrict \u03bc s) {x | f x \u2264 \u2a0d (a : \u03b1) in s, f a \u2202\u03bc} = 0\nthis : Fact (\u2191\u2191\u03bc s < \u22a4)\n\u22a2 (support fun x => f x - \u2a0d (a : \u03b1) in s, f a \u2202\u03bc)\u1d9c \u2229 s \u2286 {x | f x \u2264 \u2a0d (a : \u03b1) in s, f a \u2202\u03bc} \u2229 s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "full_name": "MeasureTheory.Measure.FiniteAtFilter.exists_mem_basis", "start": [3257, 1], "end": [3259, 76], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Array/Lemmas.lean", "full_name": "Array.set_set", "start": [110, 1], "end": [111, 86], "traced_tactics": [{"tactic": "simp [i.2]", "annotated_tactic": ["simp [i.2]", []], "state_before": "\u03b1 : Type ?u.17292\na : Array \u03b1\ni : Fin (size a)\nv v' : \u03b1\n\u22a2 i.val < size (set a i v)", "state_after": "no goals"}, {"tactic": "simp [set, List.set_set]", "annotated_tactic": ["simp [<a>set</a>, <a>List.set_set</a>]", [{"full_name": "Array.set", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2587, 5], "def_end_pos": [2587, 14]}, {"full_name": "List.set_set", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [947, 9], "def_end_pos": [947, 16]}]], "state_before": "\u03b1 : Type u_1\na : Array \u03b1\ni : Fin (size a)\nv v' : \u03b1\n\u22a2 set (set a i v) { val := i.val, isLt := (_ : i.val < size (set a i v)) } v' = set a i v'", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "full_name": "Substring.ValidFor.take", "start": [908, 1], "end": [916, 49], "traced_tactics": [{"tactic": "have : Substring.nextn {..} .. = _ := h.nextn (m\u2081 := []) n", "annotated_tactic": ["have : <a>Substring.nextn</a> {..} .. = _ := h.nextn (m\u2081 := []) n", [{"full_name": "Substring.nextn", "def_path": "lake-packages/lean4/src/lean/Init/Data/String/Basic.lean", "def_pos": [558, 5], "def_end_pos": [558, 10]}]], "state_before": "l m r : List Char\ns : Substring\nh : ValidFor l m r s\nn : Nat\n\u22a2 ValidFor l (List.take n m) (List.drop n m ++ r) (Substring.take s n)", "state_after": "l m r : List Char\ns : Substring\nh : ValidFor l m r s\nn : Nat\nthis :\n  Substring.nextn { str := s.str, startPos := s.startPos, stopPos := s.stopPos } n { byteIdx := utf8Len [] } =\n    { byteIdx := utf8Len [] + utf8Len (List.take n m) }\n\u22a2 ValidFor l (List.take n m) (List.drop n m ++ r) (Substring.take s n)"}, {"tactic": "simp at this", "annotated_tactic": ["simp at this", []], "state_before": "l m r : List Char\ns : Substring\nh : ValidFor l m r s\nn : Nat\nthis :\n  Substring.nextn { str := s.str, startPos := s.startPos, stopPos := s.stopPos } n { byteIdx := utf8Len [] } =\n    { byteIdx := utf8Len [] + utf8Len (List.take n m) }\n\u22a2 ValidFor l (List.take n m) (List.drop n m ++ r) (Substring.take s n)", "state_after": "l m r : List Char\ns : Substring\nh : ValidFor l m r s\nn : Nat\nthis :\n  Substring.nextn { str := s.str, startPos := s.startPos, stopPos := s.stopPos } n 0 =\n    { byteIdx := utf8Len (List.take n m) }\n\u22a2 ValidFor l (List.take n m) (List.drop n m ++ r) (Substring.take s n)"}, {"tactic": "simp [Substring.take, this]", "annotated_tactic": ["simp [<a>Substring.take</a>, this]", [{"full_name": "Substring.take", "def_path": "lake-packages/lean4/src/lean/Init/Data/String/Basic.lean", "def_pos": [581, 15], "def_end_pos": [581, 19]}]], "state_before": "l m r : List Char\ns : Substring\nh : ValidFor l m r s\nn : Nat\nthis :\n  Substring.nextn { str := s.str, startPos := s.startPos, stopPos := s.stopPos } n 0 =\n    { byteIdx := utf8Len (List.take n m) }\n\u22a2 ValidFor l (List.take n m) (List.drop n m ++ r) (Substring.take s n)", "state_after": "l m r : List Char\ns : Substring\nh : ValidFor l m r s\nn : Nat\nthis :\n  Substring.nextn { str := s.str, startPos := s.startPos, stopPos := s.stopPos } n 0 =\n    { byteIdx := utf8Len (List.take n m) }\n\u22a2 ValidFor l (List.take n m) (List.drop n m ++ r)\n    { str := s.str, startPos := s.startPos, stopPos := s.startPos + { byteIdx := utf8Len (List.take n m) } }"}, {"tactic": "simp [h.str, h.startPos, h.stopPos]", "annotated_tactic": ["simp [h.str, h.startPos, h.stopPos]", []], "state_before": "l m r : List Char\ns : Substring\nh : ValidFor l m r s\nn : Nat\nthis :\n  Substring.nextn { str := s.str, startPos := s.startPos, stopPos := s.stopPos } n 0 =\n    { byteIdx := utf8Len (List.take n m) }\n\u22a2 ValidFor l (List.take n m) (List.drop n m ++ r)\n    { str := s.str, startPos := s.startPos, stopPos := s.startPos + { byteIdx := utf8Len (List.take n m) } }", "state_after": "l m r : List Char\ns : Substring\nh : ValidFor l m r s\nn : Nat\nthis :\n  Substring.nextn { str := s.str, startPos := s.startPos, stopPos := s.stopPos } n 0 =\n    { byteIdx := utf8Len (List.take n m) }\n\u22a2 ValidFor l (List.take n m) (List.drop n m ++ r)\n    { str := { data := l ++ (m ++ r) }, startPos := { byteIdx := utf8Len l },\n      stopPos := { byteIdx := utf8Len l } + { byteIdx := utf8Len (List.take n m) } }"}, {"tactic": "rw [\u2190 List.take_append_drop n m] at h", "annotated_tactic": ["rw [\u2190 <a>List.take_append_drop</a> n m] at h", [{"full_name": "List.take_append_drop", "def_path": "lake-packages/std/Std/Data/List/Init/Lemmas.lean", "def_pos": [138, 17], "def_end_pos": [138, 33]}]], "state_before": "l m r : List Char\ns : Substring\nh : ValidFor l m r s\nn : Nat\nthis :\n  Substring.nextn { str := s.str, startPos := s.startPos, stopPos := s.stopPos } n 0 =\n    { byteIdx := utf8Len (List.take n m) }\n\u22a2 ValidFor l (List.take n m) (List.drop n m ++ r)\n    { str := { data := l ++ (m ++ r) }, startPos := { byteIdx := utf8Len l },\n      stopPos := { byteIdx := utf8Len l } + { byteIdx := utf8Len (List.take n m) } }", "state_after": "l m r : List Char\ns : Substring\nn : Nat\nh : ValidFor l (List.take n m ++ List.drop n m) r s\nthis :\n  Substring.nextn { str := s.str, startPos := s.startPos, stopPos := s.stopPos } n 0 =\n    { byteIdx := utf8Len (List.take n m) }\n\u22a2 ValidFor l (List.take n m) (List.drop n m ++ r)\n    { str := { data := l ++ (m ++ r) }, startPos := { byteIdx := utf8Len l },\n      stopPos := { byteIdx := utf8Len l } + { byteIdx := utf8Len (List.take n m) } }"}, {"tactic": "refine .of_eq _ ?_ (by simp) (by simp)", "annotated_tactic": ["refine .of_eq _ ?_ (by simp) (by simp)", []], "state_before": "l m r : List Char\ns : Substring\nn : Nat\nh : ValidFor l (List.take n m ++ List.drop n m) r s\nthis :\n  Substring.nextn { str := s.str, startPos := s.startPos, stopPos := s.stopPos } n 0 =\n    { byteIdx := utf8Len (List.take n m) }\n\u22a2 ValidFor l (List.take n m) (List.drop n m ++ r)\n    { str := { data := l ++ (m ++ r) }, startPos := { byteIdx := utf8Len l },\n      stopPos := { byteIdx := utf8Len l } + { byteIdx := utf8Len (List.take n m) } }", "state_after": "l m r : List Char\ns : Substring\nn : Nat\nh : ValidFor l (List.take n m ++ List.drop n m) r s\nthis :\n  Substring.nextn { str := s.str, startPos := s.startPos, stopPos := s.stopPos } n 0 =\n    { byteIdx := utf8Len (List.take n m) }\n\u22a2 { str := { data := l ++ (m ++ r) }, startPos := { byteIdx := utf8Len l },\n          stopPos := { byteIdx := utf8Len l } + { byteIdx := utf8Len (List.take n m) } }.str.data =\n    l ++ List.take n m ++ (List.drop n m ++ r)"}, {"tactic": "conv => lhs; rw [\u2190 List.take_append_drop n m]", "annotated_tactic": ["conv => lhs; rw [\u2190 <a>List.take_append_drop</a> n m]", [{"full_name": "List.take_append_drop", "def_path": "lake-packages/std/Std/Data/List/Init/Lemmas.lean", "def_pos": [138, 17], "def_end_pos": [138, 33]}]], "state_before": "l m r : List Char\ns : Substring\nn : Nat\nh : ValidFor l (List.take n m ++ List.drop n m) r s\nthis :\n  Substring.nextn { str := s.str, startPos := s.startPos, stopPos := s.stopPos } n 0 =\n    { byteIdx := utf8Len (List.take n m) }\n\u22a2 { str := { data := l ++ (m ++ r) }, startPos := { byteIdx := utf8Len l },\n          stopPos := { byteIdx := utf8Len l } + { byteIdx := utf8Len (List.take n m) } }.str.data =\n    l ++ List.take n m ++ (List.drop n m ++ r)", "state_after": "l m r : List Char\ns : Substring\nn : Nat\nh : ValidFor l (List.take n m ++ List.drop n m) r s\nthis :\n  Substring.nextn { str := s.str, startPos := s.startPos, stopPos := s.stopPos } n 0 =\n    { byteIdx := utf8Len (List.take n m) }\n\u22a2 { str := { data := l ++ (List.take n m ++ List.drop n m ++ r) }, startPos := { byteIdx := utf8Len l },\n          stopPos :=\n            { byteIdx := utf8Len l } +\n              { byteIdx := utf8Len (List.take n (List.take n m ++ List.drop n m)) } }.str.data =\n    l ++ List.take n m ++ (List.drop n m ++ r)"}, {"tactic": "simp [-List.take_append_drop, Nat.add_assoc]", "annotated_tactic": ["simp [-<a>List.take_append_drop</a>, <a>Nat.add_assoc</a>]", [{"full_name": "List.take_append_drop", "def_path": "lake-packages/std/Std/Data/List/Init/Lemmas.lean", "def_pos": [138, 17], "def_end_pos": [138, 33]}, {"full_name": "Nat.add_assoc", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [138, 19], "def_end_pos": [138, 28]}]], "state_before": "l m r : List Char\ns : Substring\nn : Nat\nh : ValidFor l (List.take n m ++ List.drop n m) r s\nthis :\n  Substring.nextn { str := s.str, startPos := s.startPos, stopPos := s.stopPos } n 0 =\n    { byteIdx := utf8Len (List.take n m) }\n\u22a2 { str := { data := l ++ (List.take n m ++ List.drop n m ++ r) }, startPos := { byteIdx := utf8Len l },\n          stopPos :=\n            { byteIdx := utf8Len l } +\n              { byteIdx := utf8Len (List.take n (List.take n m ++ List.drop n m)) } }.str.data =\n    l ++ List.take n m ++ (List.drop n m ++ r)", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "l m r : List Char\ns : Substring\nn : Nat\nh : ValidFor l (List.take n m ++ List.drop n m) r s\nthis :\n  Substring.nextn { str := s.str, startPos := s.startPos, stopPos := s.stopPos } n 0 =\n    { byteIdx := utf8Len (List.take n m) }\n\u22a2 { str := { data := l ++ (m ++ r) }, startPos := { byteIdx := utf8Len l },\n          stopPos := { byteIdx := utf8Len l } + { byteIdx := utf8Len (List.take n m) } }.startPos.byteIdx =\n    utf8Len l", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "l m r : List Char\ns : Substring\nn : Nat\nh : ValidFor l (List.take n m ++ List.drop n m) r s\nthis :\n  Substring.nextn { str := s.str, startPos := s.startPos, stopPos := s.stopPos } n 0 =\n    { byteIdx := utf8Len (List.take n m) }\n\u22a2 { str := { data := l ++ (m ++ r) }, startPos := { byteIdx := utf8Len l },\n          stopPos := { byteIdx := utf8Len l } + { byteIdx := utf8Len (List.take n m) } }.stopPos.byteIdx =\n    utf8Len l + utf8Len (List.take n m)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean", "full_name": "MeasureTheory.AECover.lintegral_tendsto_of_countably_generated", "start": [376, 1], "end": [379, 88], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Lattice.lean", "full_name": "Finset.set_biInter_finset_image", "start": [2151, 1], "end": [2153, 20], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Independence/Basic.lean", "full_name": "ProbabilityTheory.indep_iSup_of_disjoint", "start": [413, 1], "end": [416, 49], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/RBMap/Alter.lean", "full_name": "Std.RBNode.Path.zoom_fill'", "start": [48, 1], "end": [52, 79], "traced_tactics": [{"tactic": "induction t generalizing path with\n| nil => rfl\n| node _ _ _ _ iha ihb => unfold zoom; split <;> [apply iha; apply ihb; rfl]", "annotated_tactic": ["induction t generalizing path with\n  | <a>nil</a> => rfl\n  | <a>node</a> _ _ _ _ iha ihb => unfold <a>zoom</a>; split <;> [apply iha; apply ihb; rfl]", [{"full_name": "Std.RBNode.nil", "def_path": "lake-packages/std/Std/Data/RBMap/Basic.lean", "def_pos": [46, 5], "def_end_pos": [46, 8]}, {"full_name": "Std.RBNode.node", "def_path": "lake-packages/std/Std/Data/RBMap/Basic.lean", "def_pos": [50, 5], "def_end_pos": [50, 9]}, {"full_name": "Std.RBNode.zoom", "def_path": "lake-packages/std/Std/Data/RBMap/Basic.lean", "def_pos": [451, 19], "def_end_pos": [451, 23]}]], "state_before": "\u03b1 : Type u_1\ncut : \u03b1 \u2192 Ordering\nt : RBNode \u03b1\npath : Path \u03b1\n\u22a2 fill' (zoom cut t path) = fill path t", "state_after": "no goals"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case nil\n\u03b1 : Type u_1\ncut : \u03b1 \u2192 Ordering\npath : Path \u03b1\n\u22a2 fill' (zoom cut nil path) = fill path nil", "state_after": "no goals"}, {"tactic": "unfold zoom", "annotated_tactic": ["unfold <a>zoom</a>", [{"full_name": "Std.RBNode.zoom", "def_path": "lake-packages/std/Std/Data/RBMap/Basic.lean", "def_pos": [451, 19], "def_end_pos": [451, 23]}]], "state_before": "case node\n\u03b1 : Type u_1\ncut : \u03b1 \u2192 Ordering\nc\u271d : RBColor\nl\u271d : RBNode \u03b1\nv\u271d : \u03b1\nr\u271d : RBNode \u03b1\niha : \u2200 (path : Path \u03b1), fill' (zoom cut l\u271d path) = fill path l\u271d\nihb : \u2200 (path : Path \u03b1), fill' (zoom cut r\u271d path) = fill path r\u271d\npath : Path \u03b1\n\u22a2 fill' (zoom cut (node c\u271d l\u271d v\u271d r\u271d) path) = fill path (node c\u271d l\u271d v\u271d r\u271d)", "state_after": "case node\n\u03b1 : Type u_1\ncut : \u03b1 \u2192 Ordering\nc\u271d : RBColor\nl\u271d : RBNode \u03b1\nv\u271d : \u03b1\nr\u271d : RBNode \u03b1\niha : \u2200 (path : Path \u03b1), fill' (zoom cut l\u271d path) = fill path l\u271d\nihb : \u2200 (path : Path \u03b1), fill' (zoom cut r\u271d path) = fill path r\u271d\npath : Path \u03b1\n\u22a2 fill'\n      (match cut v\u271d with\n      | Ordering.lt => zoom cut l\u271d (left c\u271d path v\u271d r\u271d)\n      | Ordering.gt => zoom cut r\u271d (right c\u271d l\u271d v\u271d path)\n      | Ordering.eq => (node c\u271d l\u271d v\u271d r\u271d, path)) =\n    fill path (node c\u271d l\u271d v\u271d r\u271d)"}, {"tactic": "split <;> [apply iha; apply ihb; rfl]", "annotated_tactic": ["split <;> [apply iha; apply ihb; rfl]", []], "state_before": "case node\n\u03b1 : Type u_1\ncut : \u03b1 \u2192 Ordering\nc\u271d : RBColor\nl\u271d : RBNode \u03b1\nv\u271d : \u03b1\nr\u271d : RBNode \u03b1\niha : \u2200 (path : Path \u03b1), fill' (zoom cut l\u271d path) = fill path l\u271d\nihb : \u2200 (path : Path \u03b1), fill' (zoom cut r\u271d path) = fill path r\u271d\npath : Path \u03b1\n\u22a2 fill'\n      (match cut v\u271d with\n      | Ordering.lt => zoom cut l\u271d (left c\u271d path v\u271d r\u271d)\n      | Ordering.gt => zoom cut r\u271d (right c\u271d l\u271d v\u271d path)\n      | Ordering.eq => (node c\u271d l\u271d v\u271d r\u271d, path)) =\n    fill path (node c\u271d l\u271d v\u271d r\u271d)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "full_name": "indicator_ae_eq_of_ae_eq_set", "start": [4512, 1], "end": [4513, 35], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "full_name": "intervalIntegrable_const", "start": [125, 1], "end": [127, 62], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/FinEnum.lean", "full_name": "FinEnum.mem_pi", "start": [230, 1], "end": [245, 12], "traced_tactics": [{"tactic": "induction' xs with xs_hd xs_tl xs_ih <;> simp [pi, -List.map_eq_map, monad_norm, functor_norm]", "annotated_tactic": ["induction' xs with xs_hd xs_tl xs_ih <;> simp [<a>pi</a>, -<a>List.map_eq_map</a>, monad_norm, functor_norm]", [{"full_name": "FinEnum.pi", "def_path": "Mathlib/Data/FinEnum.lean", "def_pos": [224, 5], "def_end_pos": [224, 7]}, {"full_name": "List.map_eq_map", "def_path": "Mathlib/Data/List/Basic.lean", "def_pos": [1765, 9], "def_end_pos": [1765, 19]}]], "state_before": "\u03b1 : Type u\n\u03b2\u271d : \u03b1 \u2192 Type v\n\u03b2 : \u03b1 \u2192 Type (max u u_1)\ninst\u271d\u00b9 : FinEnum \u03b1\ninst\u271d : (a : \u03b1) \u2192 FinEnum (\u03b2 a)\nxs : List \u03b1\nf : (a : \u03b1) \u2192 a \u2208 xs \u2192 \u03b2 a\n\u22a2 f \u2208 pi xs fun x => toList (\u03b2 x)", "state_after": "case nil\n\u03b1 : Type u\n\u03b2\u271d : \u03b1 \u2192 Type v\n\u03b2 : \u03b1 \u2192 Type (max u u_1)\ninst\u271d\u00b9 : FinEnum \u03b1\ninst\u271d : (a : \u03b1) \u2192 FinEnum (\u03b2 a)\nxs : List \u03b1\nf\u271d : (a : \u03b1) \u2192 a \u2208 xs \u2192 \u03b2 a\nf : (a : \u03b1) \u2192 a \u2208 [] \u2192 \u03b2 a\n\u22a2 f = fun x h => (_ : False).elim\n\ncase cons\n\u03b1 : Type u\n\u03b2\u271d : \u03b1 \u2192 Type v\n\u03b2 : \u03b1 \u2192 Type (max u u_1)\ninst\u271d\u00b9 : FinEnum \u03b1\ninst\u271d : (a : \u03b1) \u2192 FinEnum (\u03b2 a)\nxs : List \u03b1\nf\u271d : (a : \u03b1) \u2192 a \u2208 xs \u2192 \u03b2 a\nxs_hd : \u03b1\nxs_tl : List \u03b1\nxs_ih : \u2200 (f : (a : \u03b1) \u2192 a \u2208 xs_tl \u2192 \u03b2 a), f \u2208 pi xs_tl fun x => toList (\u03b2 x)\nf : (a : \u03b1) \u2192 a \u2208 xs_hd :: xs_tl \u2192 \u03b2 a\n\u22a2 \u2203 a, (\u2203 a_1, a = Pi.cons xs_hd xs_tl a_1) \u2227 \u2203 a_1, (a_1 \u2208 pi xs_tl fun x => toList (\u03b2 x)) \u2227 f = a a_1"}, {"tactic": "ext a \u27e8\u27e9", "annotated_tactic": ["ext a \u27e8\u27e9", []], "state_before": "case nil\n\u03b1 : Type u\n\u03b2\u271d : \u03b1 \u2192 Type v\n\u03b2 : \u03b1 \u2192 Type (max u u_1)\ninst\u271d\u00b9 : FinEnum \u03b1\ninst\u271d : (a : \u03b1) \u2192 FinEnum (\u03b2 a)\nxs : List \u03b1\nf\u271d : (a : \u03b1) \u2192 a \u2208 xs \u2192 \u03b2 a\nf : (a : \u03b1) \u2192 a \u2208 [] \u2192 \u03b2 a\n\u22a2 f = fun x h => (_ : False).elim", "state_after": "no goals"}, {"tactic": "exists Pi.cons xs_hd xs_tl (f _ (List.mem_cons_self _ _))", "annotated_tactic": ["exists <a>Pi.cons</a> xs_hd xs_tl (f _ (<a>List.mem_cons_self</a> _ _))", [{"full_name": "FinEnum.Pi.cons", "def_path": "Mathlib/Data/FinEnum.lean", "def_pos": [212, 5], "def_end_pos": [212, 12]}, {"full_name": "List.mem_cons_self", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [66, 9], "def_end_pos": [66, 22]}]], "state_before": "case cons\n\u03b1 : Type u\n\u03b2\u271d : \u03b1 \u2192 Type v\n\u03b2 : \u03b1 \u2192 Type (max u u_1)\ninst\u271d\u00b9 : FinEnum \u03b1\ninst\u271d : (a : \u03b1) \u2192 FinEnum (\u03b2 a)\nxs : List \u03b1\nf\u271d : (a : \u03b1) \u2192 a \u2208 xs \u2192 \u03b2 a\nxs_hd : \u03b1\nxs_tl : List \u03b1\nxs_ih : \u2200 (f : (a : \u03b1) \u2192 a \u2208 xs_tl \u2192 \u03b2 a), f \u2208 pi xs_tl fun x => toList (\u03b2 x)\nf : (a : \u03b1) \u2192 a \u2208 xs_hd :: xs_tl \u2192 \u03b2 a\n\u22a2 \u2203 a, (\u2203 a_1, a = Pi.cons xs_hd xs_tl a_1) \u2227 \u2203 a_1, (a_1 \u2208 pi xs_tl fun x => toList (\u03b2 x)) \u2227 f = a a_1", "state_after": "case cons\n\u03b1 : Type u\n\u03b2\u271d : \u03b1 \u2192 Type v\n\u03b2 : \u03b1 \u2192 Type (max u u_1)\ninst\u271d\u00b9 : FinEnum \u03b1\ninst\u271d : (a : \u03b1) \u2192 FinEnum (\u03b2 a)\nxs : List \u03b1\nf\u271d : (a : \u03b1) \u2192 a \u2208 xs \u2192 \u03b2 a\nxs_hd : \u03b1\nxs_tl : List \u03b1\nxs_ih : \u2200 (f : (a : \u03b1) \u2192 a \u2208 xs_tl \u2192 \u03b2 a), f \u2208 pi xs_tl fun x => toList (\u03b2 x)\nf : (a : \u03b1) \u2192 a \u2208 xs_hd :: xs_tl \u2192 \u03b2 a\n\u22a2 (\u2203 a, Pi.cons xs_hd xs_tl (f xs_hd (_ : xs_hd \u2208 xs_hd :: xs_tl)) = Pi.cons xs_hd xs_tl a) \u2227\n    \u2203 a, (a \u2208 pi xs_tl fun x => toList (\u03b2 x)) \u2227 f = Pi.cons xs_hd xs_tl (f xs_hd (_ : xs_hd \u2208 xs_hd :: xs_tl)) a"}, {"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "case cons\n\u03b1 : Type u\n\u03b2\u271d : \u03b1 \u2192 Type v\n\u03b2 : \u03b1 \u2192 Type (max u u_1)\ninst\u271d\u00b9 : FinEnum \u03b1\ninst\u271d : (a : \u03b1) \u2192 FinEnum (\u03b2 a)\nxs : List \u03b1\nf\u271d : (a : \u03b1) \u2192 a \u2208 xs \u2192 \u03b2 a\nxs_hd : \u03b1\nxs_tl : List \u03b1\nxs_ih : \u2200 (f : (a : \u03b1) \u2192 a \u2208 xs_tl \u2192 \u03b2 a), f \u2208 pi xs_tl fun x => toList (\u03b2 x)\nf : (a : \u03b1) \u2192 a \u2208 xs_hd :: xs_tl \u2192 \u03b2 a\n\u22a2 (\u2203 a, Pi.cons xs_hd xs_tl (f xs_hd (_ : xs_hd \u2208 xs_hd :: xs_tl)) = Pi.cons xs_hd xs_tl a) \u2227\n    \u2203 a, (a \u2208 pi xs_tl fun x => toList (\u03b2 x)) \u2227 f = Pi.cons xs_hd xs_tl (f xs_hd (_ : xs_hd \u2208 xs_hd :: xs_tl)) a", "state_after": "case cons.left\n\u03b1 : Type u\n\u03b2\u271d : \u03b1 \u2192 Type v\n\u03b2 : \u03b1 \u2192 Type (max u u_1)\ninst\u271d\u00b9 : FinEnum \u03b1\ninst\u271d : (a : \u03b1) \u2192 FinEnum (\u03b2 a)\nxs : List \u03b1\nf\u271d : (a : \u03b1) \u2192 a \u2208 xs \u2192 \u03b2 a\nxs_hd : \u03b1\nxs_tl : List \u03b1\nxs_ih : \u2200 (f : (a : \u03b1) \u2192 a \u2208 xs_tl \u2192 \u03b2 a), f \u2208 pi xs_tl fun x => toList (\u03b2 x)\nf : (a : \u03b1) \u2192 a \u2208 xs_hd :: xs_tl \u2192 \u03b2 a\n\u22a2 \u2203 a, Pi.cons xs_hd xs_tl (f xs_hd (_ : xs_hd \u2208 xs_hd :: xs_tl)) = Pi.cons xs_hd xs_tl a\n\ncase cons.right\n\u03b1 : Type u\n\u03b2\u271d : \u03b1 \u2192 Type v\n\u03b2 : \u03b1 \u2192 Type (max u u_1)\ninst\u271d\u00b9 : FinEnum \u03b1\ninst\u271d : (a : \u03b1) \u2192 FinEnum (\u03b2 a)\nxs : List \u03b1\nf\u271d : (a : \u03b1) \u2192 a \u2208 xs \u2192 \u03b2 a\nxs_hd : \u03b1\nxs_tl : List \u03b1\nxs_ih : \u2200 (f : (a : \u03b1) \u2192 a \u2208 xs_tl \u2192 \u03b2 a), f \u2208 pi xs_tl fun x => toList (\u03b2 x)\nf : (a : \u03b1) \u2192 a \u2208 xs_hd :: xs_tl \u2192 \u03b2 a\n\u22a2 \u2203 a, (a \u2208 pi xs_tl fun x => toList (\u03b2 x)) \u2227 f = Pi.cons xs_hd xs_tl (f xs_hd (_ : xs_hd \u2208 xs_hd :: xs_tl)) a"}, {"tactic": "exact \u27e8_, rfl\u27e9", "annotated_tactic": ["exact \u27e8_, <a>rfl</a>\u27e9", [{"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "case cons.left\n\u03b1 : Type u\n\u03b2\u271d : \u03b1 \u2192 Type v\n\u03b2 : \u03b1 \u2192 Type (max u u_1)\ninst\u271d\u00b9 : FinEnum \u03b1\ninst\u271d : (a : \u03b1) \u2192 FinEnum (\u03b2 a)\nxs : List \u03b1\nf\u271d : (a : \u03b1) \u2192 a \u2208 xs \u2192 \u03b2 a\nxs_hd : \u03b1\nxs_tl : List \u03b1\nxs_ih : \u2200 (f : (a : \u03b1) \u2192 a \u2208 xs_tl \u2192 \u03b2 a), f \u2208 pi xs_tl fun x => toList (\u03b2 x)\nf : (a : \u03b1) \u2192 a \u2208 xs_hd :: xs_tl \u2192 \u03b2 a\n\u22a2 \u2203 a, Pi.cons xs_hd xs_tl (f xs_hd (_ : xs_hd \u2208 xs_hd :: xs_tl)) = Pi.cons xs_hd xs_tl a\n\ncase cons.right\n\u03b1 : Type u\n\u03b2\u271d : \u03b1 \u2192 Type v\n\u03b2 : \u03b1 \u2192 Type (max u u_1)\ninst\u271d\u00b9 : FinEnum \u03b1\ninst\u271d : (a : \u03b1) \u2192 FinEnum (\u03b2 a)\nxs : List \u03b1\nf\u271d : (a : \u03b1) \u2192 a \u2208 xs \u2192 \u03b2 a\nxs_hd : \u03b1\nxs_tl : List \u03b1\nxs_ih : \u2200 (f : (a : \u03b1) \u2192 a \u2208 xs_tl \u2192 \u03b2 a), f \u2208 pi xs_tl fun x => toList (\u03b2 x)\nf : (a : \u03b1) \u2192 a \u2208 xs_hd :: xs_tl \u2192 \u03b2 a\n\u22a2 \u2203 a, (a \u2208 pi xs_tl fun x => toList (\u03b2 x)) \u2227 f = Pi.cons xs_hd xs_tl (f xs_hd (_ : xs_hd \u2208 xs_hd :: xs_tl)) a", "state_after": "case cons.right\n\u03b1 : Type u\n\u03b2\u271d : \u03b1 \u2192 Type v\n\u03b2 : \u03b1 \u2192 Type (max u u_1)\ninst\u271d\u00b9 : FinEnum \u03b1\ninst\u271d : (a : \u03b1) \u2192 FinEnum (\u03b2 a)\nxs : List \u03b1\nf\u271d : (a : \u03b1) \u2192 a \u2208 xs \u2192 \u03b2 a\nxs_hd : \u03b1\nxs_tl : List \u03b1\nxs_ih : \u2200 (f : (a : \u03b1) \u2192 a \u2208 xs_tl \u2192 \u03b2 a), f \u2208 pi xs_tl fun x => toList (\u03b2 x)\nf : (a : \u03b1) \u2192 a \u2208 xs_hd :: xs_tl \u2192 \u03b2 a\n\u22a2 \u2203 a, (a \u2208 pi xs_tl fun x => toList (\u03b2 x)) \u2227 f = Pi.cons xs_hd xs_tl (f xs_hd (_ : xs_hd \u2208 xs_hd :: xs_tl)) a"}, {"tactic": "exists Pi.tail f", "annotated_tactic": ["exists <a>Pi.tail</a> f", [{"full_name": "FinEnum.Pi.tail", "def_path": "Mathlib/Data/FinEnum.lean", "def_pos": [219, 5], "def_end_pos": [219, 12]}]], "state_before": "case cons.right\n\u03b1 : Type u\n\u03b2\u271d : \u03b1 \u2192 Type v\n\u03b2 : \u03b1 \u2192 Type (max u u_1)\ninst\u271d\u00b9 : FinEnum \u03b1\ninst\u271d : (a : \u03b1) \u2192 FinEnum (\u03b2 a)\nxs : List \u03b1\nf\u271d : (a : \u03b1) \u2192 a \u2208 xs \u2192 \u03b2 a\nxs_hd : \u03b1\nxs_tl : List \u03b1\nxs_ih : \u2200 (f : (a : \u03b1) \u2192 a \u2208 xs_tl \u2192 \u03b2 a), f \u2208 pi xs_tl fun x => toList (\u03b2 x)\nf : (a : \u03b1) \u2192 a \u2208 xs_hd :: xs_tl \u2192 \u03b2 a\n\u22a2 \u2203 a, (a \u2208 pi xs_tl fun x => toList (\u03b2 x)) \u2227 f = Pi.cons xs_hd xs_tl (f xs_hd (_ : xs_hd \u2208 xs_hd :: xs_tl)) a", "state_after": "case cons.right\n\u03b1 : Type u\n\u03b2\u271d : \u03b1 \u2192 Type v\n\u03b2 : \u03b1 \u2192 Type (max u u_1)\ninst\u271d\u00b9 : FinEnum \u03b1\ninst\u271d : (a : \u03b1) \u2192 FinEnum (\u03b2 a)\nxs : List \u03b1\nf\u271d : (a : \u03b1) \u2192 a \u2208 xs \u2192 \u03b2 a\nxs_hd : \u03b1\nxs_tl : List \u03b1\nxs_ih : \u2200 (f : (a : \u03b1) \u2192 a \u2208 xs_tl \u2192 \u03b2 a), f \u2208 pi xs_tl fun x => toList (\u03b2 x)\nf : (a : \u03b1) \u2192 a \u2208 xs_hd :: xs_tl \u2192 \u03b2 a\n\u22a2 (Pi.tail f \u2208 pi xs_tl fun x => toList (\u03b2 x)) \u2227\n    f = Pi.cons xs_hd xs_tl (f xs_hd (_ : xs_hd \u2208 xs_hd :: xs_tl)) (Pi.tail f)"}, {"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "case cons.right\n\u03b1 : Type u\n\u03b2\u271d : \u03b1 \u2192 Type v\n\u03b2 : \u03b1 \u2192 Type (max u u_1)\ninst\u271d\u00b9 : FinEnum \u03b1\ninst\u271d : (a : \u03b1) \u2192 FinEnum (\u03b2 a)\nxs : List \u03b1\nf\u271d : (a : \u03b1) \u2192 a \u2208 xs \u2192 \u03b2 a\nxs_hd : \u03b1\nxs_tl : List \u03b1\nxs_ih : \u2200 (f : (a : \u03b1) \u2192 a \u2208 xs_tl \u2192 \u03b2 a), f \u2208 pi xs_tl fun x => toList (\u03b2 x)\nf : (a : \u03b1) \u2192 a \u2208 xs_hd :: xs_tl \u2192 \u03b2 a\n\u22a2 (Pi.tail f \u2208 pi xs_tl fun x => toList (\u03b2 x)) \u2227\n    f = Pi.cons xs_hd xs_tl (f xs_hd (_ : xs_hd \u2208 xs_hd :: xs_tl)) (Pi.tail f)", "state_after": "case cons.right.left\n\u03b1 : Type u\n\u03b2\u271d : \u03b1 \u2192 Type v\n\u03b2 : \u03b1 \u2192 Type (max u u_1)\ninst\u271d\u00b9 : FinEnum \u03b1\ninst\u271d : (a : \u03b1) \u2192 FinEnum (\u03b2 a)\nxs : List \u03b1\nf\u271d : (a : \u03b1) \u2192 a \u2208 xs \u2192 \u03b2 a\nxs_hd : \u03b1\nxs_tl : List \u03b1\nxs_ih : \u2200 (f : (a : \u03b1) \u2192 a \u2208 xs_tl \u2192 \u03b2 a), f \u2208 pi xs_tl fun x => toList (\u03b2 x)\nf : (a : \u03b1) \u2192 a \u2208 xs_hd :: xs_tl \u2192 \u03b2 a\n\u22a2 Pi.tail f \u2208 pi xs_tl fun x => toList (\u03b2 x)\n\ncase cons.right.right\n\u03b1 : Type u\n\u03b2\u271d : \u03b1 \u2192 Type v\n\u03b2 : \u03b1 \u2192 Type (max u u_1)\ninst\u271d\u00b9 : FinEnum \u03b1\ninst\u271d : (a : \u03b1) \u2192 FinEnum (\u03b2 a)\nxs : List \u03b1\nf\u271d : (a : \u03b1) \u2192 a \u2208 xs \u2192 \u03b2 a\nxs_hd : \u03b1\nxs_tl : List \u03b1\nxs_ih : \u2200 (f : (a : \u03b1) \u2192 a \u2208 xs_tl \u2192 \u03b2 a), f \u2208 pi xs_tl fun x => toList (\u03b2 x)\nf : (a : \u03b1) \u2192 a \u2208 xs_hd :: xs_tl \u2192 \u03b2 a\n\u22a2 f = Pi.cons xs_hd xs_tl (f xs_hd (_ : xs_hd \u2208 xs_hd :: xs_tl)) (Pi.tail f)"}, {"tactic": "apply xs_ih", "annotated_tactic": ["apply xs_ih", []], "state_before": "case cons.right.left\n\u03b1 : Type u\n\u03b2\u271d : \u03b1 \u2192 Type v\n\u03b2 : \u03b1 \u2192 Type (max u u_1)\ninst\u271d\u00b9 : FinEnum \u03b1\ninst\u271d : (a : \u03b1) \u2192 FinEnum (\u03b2 a)\nxs : List \u03b1\nf\u271d : (a : \u03b1) \u2192 a \u2208 xs \u2192 \u03b2 a\nxs_hd : \u03b1\nxs_tl : List \u03b1\nxs_ih : \u2200 (f : (a : \u03b1) \u2192 a \u2208 xs_tl \u2192 \u03b2 a), f \u2208 pi xs_tl fun x => toList (\u03b2 x)\nf : (a : \u03b1) \u2192 a \u2208 xs_hd :: xs_tl \u2192 \u03b2 a\n\u22a2 Pi.tail f \u2208 pi xs_tl fun x => toList (\u03b2 x)", "state_after": "no goals"}, {"tactic": "ext x h", "annotated_tactic": ["ext x h", []], "state_before": "case cons.right.right\n\u03b1 : Type u\n\u03b2\u271d : \u03b1 \u2192 Type v\n\u03b2 : \u03b1 \u2192 Type (max u u_1)\ninst\u271d\u00b9 : FinEnum \u03b1\ninst\u271d : (a : \u03b1) \u2192 FinEnum (\u03b2 a)\nxs : List \u03b1\nf\u271d : (a : \u03b1) \u2192 a \u2208 xs \u2192 \u03b2 a\nxs_hd : \u03b1\nxs_tl : List \u03b1\nxs_ih : \u2200 (f : (a : \u03b1) \u2192 a \u2208 xs_tl \u2192 \u03b2 a), f \u2208 pi xs_tl fun x => toList (\u03b2 x)\nf : (a : \u03b1) \u2192 a \u2208 xs_hd :: xs_tl \u2192 \u03b2 a\n\u22a2 f = Pi.cons xs_hd xs_tl (f xs_hd (_ : xs_hd \u2208 xs_hd :: xs_tl)) (Pi.tail f)", "state_after": "case cons.right.right.h.h\n\u03b1 : Type u\n\u03b2\u271d : \u03b1 \u2192 Type v\n\u03b2 : \u03b1 \u2192 Type (max u u_1)\ninst\u271d\u00b9 : FinEnum \u03b1\ninst\u271d : (a : \u03b1) \u2192 FinEnum (\u03b2 a)\nxs : List \u03b1\nf\u271d : (a : \u03b1) \u2192 a \u2208 xs \u2192 \u03b2 a\nxs_hd : \u03b1\nxs_tl : List \u03b1\nxs_ih : \u2200 (f : (a : \u03b1) \u2192 a \u2208 xs_tl \u2192 \u03b2 a), f \u2208 pi xs_tl fun x => toList (\u03b2 x)\nf : (a : \u03b1) \u2192 a \u2208 xs_hd :: xs_tl \u2192 \u03b2 a\nx : \u03b1\nh : x \u2208 xs_hd :: xs_tl\n\u22a2 f x h = Pi.cons xs_hd xs_tl (f xs_hd (_ : xs_hd \u2208 xs_hd :: xs_tl)) (Pi.tail f) x h"}, {"tactic": "simp only [Pi.cons]", "annotated_tactic": ["simp only [<a>Pi.cons</a>]", [{"full_name": "FinEnum.Pi.cons", "def_path": "Mathlib/Data/FinEnum.lean", "def_pos": [212, 5], "def_end_pos": [212, 12]}]], "state_before": "case cons.right.right.h.h\n\u03b1 : Type u\n\u03b2\u271d : \u03b1 \u2192 Type v\n\u03b2 : \u03b1 \u2192 Type (max u u_1)\ninst\u271d\u00b9 : FinEnum \u03b1\ninst\u271d : (a : \u03b1) \u2192 FinEnum (\u03b2 a)\nxs : List \u03b1\nf\u271d : (a : \u03b1) \u2192 a \u2208 xs \u2192 \u03b2 a\nxs_hd : \u03b1\nxs_tl : List \u03b1\nxs_ih : \u2200 (f : (a : \u03b1) \u2192 a \u2208 xs_tl \u2192 \u03b2 a), f \u2208 pi xs_tl fun x => toList (\u03b2 x)\nf : (a : \u03b1) \u2192 a \u2208 xs_hd :: xs_tl \u2192 \u03b2 a\nx : \u03b1\nh : x \u2208 xs_hd :: xs_tl\n\u22a2 f x h = Pi.cons xs_hd xs_tl (f xs_hd (_ : xs_hd \u2208 xs_hd :: xs_tl)) (Pi.tail f) x h", "state_after": "case cons.right.right.h.h\n\u03b1 : Type u\n\u03b2\u271d : \u03b1 \u2192 Type v\n\u03b2 : \u03b1 \u2192 Type (max u u_1)\ninst\u271d\u00b9 : FinEnum \u03b1\ninst\u271d : (a : \u03b1) \u2192 FinEnum (\u03b2 a)\nxs : List \u03b1\nf\u271d : (a : \u03b1) \u2192 a \u2208 xs \u2192 \u03b2 a\nxs_hd : \u03b1\nxs_tl : List \u03b1\nxs_ih : \u2200 (f : (a : \u03b1) \u2192 a \u2208 xs_tl \u2192 \u03b2 a), f \u2208 pi xs_tl fun x => toList (\u03b2 x)\nf : (a : \u03b1) \u2192 a \u2208 xs_hd :: xs_tl \u2192 \u03b2 a\nx : \u03b1\nh : x \u2208 xs_hd :: xs_tl\n\u22a2 f x h =\n    if h' : x = xs_hd then cast (_ : \u03b2 xs_hd = \u03b2 x) (f xs_hd (_ : xs_hd \u2208 xs_hd :: xs_tl))\n    else Pi.tail f x (_ : x \u2208 xs_tl)"}, {"tactic": "split_ifs", "annotated_tactic": ["split_ifs", []], "state_before": "case cons.right.right.h.h\n\u03b1 : Type u\n\u03b2\u271d : \u03b1 \u2192 Type v\n\u03b2 : \u03b1 \u2192 Type (max u u_1)\ninst\u271d\u00b9 : FinEnum \u03b1\ninst\u271d : (a : \u03b1) \u2192 FinEnum (\u03b2 a)\nxs : List \u03b1\nf\u271d : (a : \u03b1) \u2192 a \u2208 xs \u2192 \u03b2 a\nxs_hd : \u03b1\nxs_tl : List \u03b1\nxs_ih : \u2200 (f : (a : \u03b1) \u2192 a \u2208 xs_tl \u2192 \u03b2 a), f \u2208 pi xs_tl fun x => toList (\u03b2 x)\nf : (a : \u03b1) \u2192 a \u2208 xs_hd :: xs_tl \u2192 \u03b2 a\nx : \u03b1\nh : x \u2208 xs_hd :: xs_tl\n\u22a2 f x h =\n    if h' : x = xs_hd then cast (_ : \u03b2 xs_hd = \u03b2 x) (f xs_hd (_ : xs_hd \u2208 xs_hd :: xs_tl))\n    else Pi.tail f x (_ : x \u2208 xs_tl)", "state_after": "case pos\n\u03b1 : Type u\n\u03b2\u271d : \u03b1 \u2192 Type v\n\u03b2 : \u03b1 \u2192 Type (max u u_1)\ninst\u271d\u00b9 : FinEnum \u03b1\ninst\u271d : (a : \u03b1) \u2192 FinEnum (\u03b2 a)\nxs : List \u03b1\nf\u271d : (a : \u03b1) \u2192 a \u2208 xs \u2192 \u03b2 a\nxs_hd : \u03b1\nxs_tl : List \u03b1\nxs_ih : \u2200 (f : (a : \u03b1) \u2192 a \u2208 xs_tl \u2192 \u03b2 a), f \u2208 pi xs_tl fun x => toList (\u03b2 x)\nf : (a : \u03b1) \u2192 a \u2208 xs_hd :: xs_tl \u2192 \u03b2 a\nx : \u03b1\nh : x \u2208 xs_hd :: xs_tl\nh\u271d : x = xs_hd\n\u22a2 f x h = cast (_ : \u03b2 xs_hd = \u03b2 x) (f xs_hd (_ : xs_hd \u2208 xs_hd :: xs_tl))\n\ncase neg\n\u03b1 : Type u\n\u03b2\u271d : \u03b1 \u2192 Type v\n\u03b2 : \u03b1 \u2192 Type (max u u_1)\ninst\u271d\u00b9 : FinEnum \u03b1\ninst\u271d : (a : \u03b1) \u2192 FinEnum (\u03b2 a)\nxs : List \u03b1\nf\u271d : (a : \u03b1) \u2192 a \u2208 xs \u2192 \u03b2 a\nxs_hd : \u03b1\nxs_tl : List \u03b1\nxs_ih : \u2200 (f : (a : \u03b1) \u2192 a \u2208 xs_tl \u2192 \u03b2 a), f \u2208 pi xs_tl fun x => toList (\u03b2 x)\nf : (a : \u03b1) \u2192 a \u2208 xs_hd :: xs_tl \u2192 \u03b2 a\nx : \u03b1\nh : x \u2208 xs_hd :: xs_tl\nh\u271d : \u00acx = xs_hd\n\u22a2 f x h = Pi.tail f x (_ : x \u2208 xs_tl)"}, {"tactic": "subst x", "annotated_tactic": ["subst x", []], "state_before": "case pos\n\u03b1 : Type u\n\u03b2\u271d : \u03b1 \u2192 Type v\n\u03b2 : \u03b1 \u2192 Type (max u u_1)\ninst\u271d\u00b9 : FinEnum \u03b1\ninst\u271d : (a : \u03b1) \u2192 FinEnum (\u03b2 a)\nxs : List \u03b1\nf\u271d : (a : \u03b1) \u2192 a \u2208 xs \u2192 \u03b2 a\nxs_hd : \u03b1\nxs_tl : List \u03b1\nxs_ih : \u2200 (f : (a : \u03b1) \u2192 a \u2208 xs_tl \u2192 \u03b2 a), f \u2208 pi xs_tl fun x => toList (\u03b2 x)\nf : (a : \u03b1) \u2192 a \u2208 xs_hd :: xs_tl \u2192 \u03b2 a\nx : \u03b1\nh : x \u2208 xs_hd :: xs_tl\nh\u271d : x = xs_hd\n\u22a2 f x h = cast (_ : \u03b2 xs_hd = \u03b2 x) (f xs_hd (_ : xs_hd \u2208 xs_hd :: xs_tl))", "state_after": "case pos\n\u03b1 : Type u\n\u03b2\u271d : \u03b1 \u2192 Type v\n\u03b2 : \u03b1 \u2192 Type (max u u_1)\ninst\u271d\u00b9 : FinEnum \u03b1\ninst\u271d : (a : \u03b1) \u2192 FinEnum (\u03b2 a)\nxs : List \u03b1\nf\u271d : (a : \u03b1) \u2192 a \u2208 xs \u2192 \u03b2 a\nxs_hd : \u03b1\nxs_tl : List \u03b1\nxs_ih : \u2200 (f : (a : \u03b1) \u2192 a \u2208 xs_tl \u2192 \u03b2 a), f \u2208 pi xs_tl fun x => toList (\u03b2 x)\nf : (a : \u03b1) \u2192 a \u2208 xs_hd :: xs_tl \u2192 \u03b2 a\nh : xs_hd \u2208 xs_hd :: xs_tl\n\u22a2 f xs_hd h = cast (_ : \u03b2 xs_hd = \u03b2 xs_hd) (f xs_hd (_ : xs_hd \u2208 xs_hd :: xs_tl))"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case pos\n\u03b1 : Type u\n\u03b2\u271d : \u03b1 \u2192 Type v\n\u03b2 : \u03b1 \u2192 Type (max u u_1)\ninst\u271d\u00b9 : FinEnum \u03b1\ninst\u271d : (a : \u03b1) \u2192 FinEnum (\u03b2 a)\nxs : List \u03b1\nf\u271d : (a : \u03b1) \u2192 a \u2208 xs \u2192 \u03b2 a\nxs_hd : \u03b1\nxs_tl : List \u03b1\nxs_ih : \u2200 (f : (a : \u03b1) \u2192 a \u2208 xs_tl \u2192 \u03b2 a), f \u2208 pi xs_tl fun x => toList (\u03b2 x)\nf : (a : \u03b1) \u2192 a \u2208 xs_hd :: xs_tl \u2192 \u03b2 a\nh : xs_hd \u2208 xs_hd :: xs_tl\n\u22a2 f xs_hd h = cast (_ : \u03b2 xs_hd = \u03b2 xs_hd) (f xs_hd (_ : xs_hd \u2208 xs_hd :: xs_tl))", "state_after": "no goals"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case neg\n\u03b1 : Type u\n\u03b2\u271d : \u03b1 \u2192 Type v\n\u03b2 : \u03b1 \u2192 Type (max u u_1)\ninst\u271d\u00b9 : FinEnum \u03b1\ninst\u271d : (a : \u03b1) \u2192 FinEnum (\u03b2 a)\nxs : List \u03b1\nf\u271d : (a : \u03b1) \u2192 a \u2208 xs \u2192 \u03b2 a\nxs_hd : \u03b1\nxs_tl : List \u03b1\nxs_ih : \u2200 (f : (a : \u03b1) \u2192 a \u2208 xs_tl \u2192 \u03b2 a), f \u2208 pi xs_tl fun x => toList (\u03b2 x)\nf : (a : \u03b1) \u2192 a \u2208 xs_hd :: xs_tl \u2192 \u03b2 a\nx : \u03b1\nh : x \u2208 xs_hd :: xs_tl\nh\u271d : \u00acx = xs_hd\n\u22a2 f x h = Pi.tail f x (_ : x \u2208 xs_tl)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Constructions/Pi.lean", "full_name": "MeasureTheory.volume_measurePreserving_piCongrLeft", "start": [762, 1], "end": [765, 51], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Image.lean", "full_name": "Set.image_swap_eq_preimage_swap", "start": [1040, 1], "end": [1041, 76], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "full_name": "exists_stronglyMeasurable_limit_of_tendsto_ae", "start": [1685, 1], "end": [1697, 18], "traced_tactics": [{"tactic": "borelize \u03b2", "annotated_tactic": ["borelize \u03b2", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b3 : Countable \u03b9\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\nf\u271d g : \u03b1 \u2192 \u03b2\ninst\u271d : PseudoMetrizableSpace \u03b2\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nh_ae_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) atTop (\ud835\udcdd l)\n\u22a2 \u2203 f_lim hf_lim_meas, \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (f_lim x))", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b3 : Countable \u03b9\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\nf\u271d g : \u03b1 \u2192 \u03b2\ninst\u271d : PseudoMetrizableSpace \u03b2\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nh_ae_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) atTop (\ud835\udcdd l)\nthis\u271d\u00b9 : MeasurableSpace \u03b2 := borel \u03b2\nthis\u271d : BorelSpace \u03b2\n\u22a2 \u2203 f_lim hf_lim_meas, \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (f_lim x))"}, {"tactic": "obtain \u27e8g, _, hg\u27e9 :\n  \u2203 (g : \u03b1 \u2192 \u03b2) (_ : Measurable g), \u2200\u1d50 x \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x)) :=\n  measurable_limit_of_tendsto_metrizable_ae (fun n => (hf n).aemeasurable) h_ae_tendsto", "annotated_tactic": ["obtain \u27e8g, _, hg\u27e9 :\n    \u2203 (g : \u03b1 \u2192 \u03b2) (_ : <a>Measurable</a> g), \u2200\u1d50 x \u2202\u03bc, <a>Tendsto</a> (fun n => f n x) <a>atTop</a> (\ud835\udcdd (g x)) :=\n    <a>measurable_limit_of_tendsto_metrizable_ae</a> (fun n => (hf n).<a>aemeasurable</a>) h_ae_tendsto", [{"full_name": "Measurable", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [535, 5], "def_end_pos": [535, 15]}, {"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2939, 5], "def_end_pos": [2939, 12]}, {"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}, {"full_name": "measurable_limit_of_tendsto_metrizable_ae", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Metrizable.lean", "def_pos": [146, 9], "def_end_pos": [146, 50]}, {"full_name": "MeasureTheory.AEStronglyMeasurable.aemeasurable", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1220, 19], "def_end_pos": [1220, 31]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b3 : Countable \u03b9\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\nf\u271d g : \u03b1 \u2192 \u03b2\ninst\u271d : PseudoMetrizableSpace \u03b2\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nh_ae_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) atTop (\ud835\udcdd l)\nthis\u271d\u00b9 : MeasurableSpace \u03b2 := borel \u03b2\nthis\u271d : BorelSpace \u03b2\n\u22a2 \u2203 f_lim hf_lim_meas, \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (f_lim x))", "state_after": "case intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b3 : Countable \u03b9\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\ninst\u271d : PseudoMetrizableSpace \u03b2\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nh_ae_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) atTop (\ud835\udcdd l)\nthis\u271d\u00b9 : MeasurableSpace \u03b2 := borel \u03b2\nthis\u271d : BorelSpace \u03b2\ng : \u03b1 \u2192 \u03b2\nw\u271d : Measurable g\nhg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u22a2 \u2203 f_lim hf_lim_meas, \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (f_lim x))"}, {"tactic": "have Hg : AEStronglyMeasurable g \u03bc := aestronglyMeasurable_of_tendsto_ae _ hf hg", "annotated_tactic": ["have Hg : <a>AEStronglyMeasurable</a> g \u03bc := <a>aestronglyMeasurable_of_tendsto_ae</a> _ hf hg", [{"full_name": "MeasureTheory.AEStronglyMeasurable", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [93, 5], "def_end_pos": [93, 25]}, {"full_name": "aestronglyMeasurable_of_tendsto_ae", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1664, 9], "def_end_pos": [1664, 50]}]], "state_before": "case intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b3 : Countable \u03b9\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\ninst\u271d : PseudoMetrizableSpace \u03b2\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nh_ae_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) atTop (\ud835\udcdd l)\nthis\u271d\u00b9 : MeasurableSpace \u03b2 := borel \u03b2\nthis\u271d : BorelSpace \u03b2\ng : \u03b1 \u2192 \u03b2\nw\u271d : Measurable g\nhg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u22a2 \u2203 f_lim hf_lim_meas, \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (f_lim x))", "state_after": "case intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b3 : Countable \u03b9\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\ninst\u271d : PseudoMetrizableSpace \u03b2\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nh_ae_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) atTop (\ud835\udcdd l)\nthis\u271d\u00b9 : MeasurableSpace \u03b2 := borel \u03b2\nthis\u271d : BorelSpace \u03b2\ng : \u03b1 \u2192 \u03b2\nw\u271d : Measurable g\nhg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\nHg : AEStronglyMeasurable g \u03bc\n\u22a2 \u2203 f_lim hf_lim_meas, \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (f_lim x))"}, {"tactic": "refine' \u27e8Hg.mk g, Hg.stronglyMeasurable_mk, _\u27e9", "annotated_tactic": ["refine' \u27e8Hg.mk g, Hg.stronglyMeasurable_mk, _\u27e9", []], "state_before": "case intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b3 : Countable \u03b9\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\ninst\u271d : PseudoMetrizableSpace \u03b2\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nh_ae_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) atTop (\ud835\udcdd l)\nthis\u271d\u00b9 : MeasurableSpace \u03b2 := borel \u03b2\nthis\u271d : BorelSpace \u03b2\ng : \u03b1 \u2192 \u03b2\nw\u271d : Measurable g\nhg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\nHg : AEStronglyMeasurable g \u03bc\n\u22a2 \u2203 f_lim hf_lim_meas, \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (f_lim x))", "state_after": "case intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b3 : Countable \u03b9\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\ninst\u271d : PseudoMetrizableSpace \u03b2\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nh_ae_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) atTop (\ud835\udcdd l)\nthis\u271d\u00b9 : MeasurableSpace \u03b2 := borel \u03b2\nthis\u271d : BorelSpace \u03b2\ng : \u03b1 \u2192 \u03b2\nw\u271d : Measurable g\nhg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\nHg : AEStronglyMeasurable g \u03bc\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (AEStronglyMeasurable.mk g Hg x))"}, {"tactic": "filter_upwards [hg, Hg.ae_eq_mk] with x hx h'x", "annotated_tactic": ["filter_upwards [hg, Hg.ae_eq_mk] with x hx h'x", []], "state_before": "case intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b3 : Countable \u03b9\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\ninst\u271d : PseudoMetrizableSpace \u03b2\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nh_ae_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) atTop (\ud835\udcdd l)\nthis\u271d\u00b9 : MeasurableSpace \u03b2 := borel \u03b2\nthis\u271d : BorelSpace \u03b2\ng : \u03b1 \u2192 \u03b2\nw\u271d : Measurable g\nhg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\nHg : AEStronglyMeasurable g \u03bc\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (AEStronglyMeasurable.mk g Hg x))", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b3 : Countable \u03b9\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\ninst\u271d : PseudoMetrizableSpace \u03b2\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nh_ae_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) atTop (\ud835\udcdd l)\nthis\u271d\u00b9 : MeasurableSpace \u03b2 := borel \u03b2\nthis\u271d : BorelSpace \u03b2\ng : \u03b1 \u2192 \u03b2\nw\u271d : Measurable g\nhg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\nHg : AEStronglyMeasurable g \u03bc\nx : \u03b1\nhx : Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\nh'x : g x = AEStronglyMeasurable.mk g Hg x\n\u22a2 Tendsto (fun n => f n x) atTop (\ud835\udcdd (AEStronglyMeasurable.mk g Hg x))"}, {"tactic": "rwa [h'x] at hx", "annotated_tactic": ["rwa [h'x] at hx", []], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b3 : Countable \u03b9\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\ninst\u271d : PseudoMetrizableSpace \u03b2\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nh_ae_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) atTop (\ud835\udcdd l)\nthis\u271d\u00b9 : MeasurableSpace \u03b2 := borel \u03b2\nthis\u271d : BorelSpace \u03b2\ng : \u03b1 \u2192 \u03b2\nw\u271d : Measurable g\nhg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\nHg : AEStronglyMeasurable g \u03bc\nx : \u03b1\nhx : Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\nh'x : g x = AEStronglyMeasurable.mk g Hg x\n\u22a2 Tendsto (fun n => f n x) atTop (\ud835\udcdd (AEStronglyMeasurable.mk g Hg x))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Hausdorff.lean", "full_name": "MeasureTheory.OuterMeasure.mkMetric'.tendsto_pre", "start": [288, 1], "end": [292, 58], "traced_tactics": [{"tactic": "rw [\u2190 map_coe_Ioi_atBot, tendsto_map'_iff]", "annotated_tactic": ["rw [\u2190 <a>map_coe_Ioi_atBot</a>, <a>tendsto_map'_iff</a>]", [{"full_name": "map_coe_Ioi_atBot", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [2599, 9], "def_end_pos": [2599, 26]}, {"full_name": "Filter.tendsto_map'_iff", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [3056, 9], "def_end_pos": [3056, 25]}]], "state_before": "\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nm\u271d : Set X \u2192 \u211d\u22650\u221e\nr : \u211d\u22650\u221e\n\u03bc : OuterMeasure X\ns\u271d : Set X\nm : Set X \u2192 \u211d\u22650\u221e\ns : Set X\n\u22a2 Tendsto (fun r => \u2191(pre m r) s) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191(mkMetric' m) s))", "state_after": "\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nm\u271d : Set X \u2192 \u211d\u22650\u221e\nr : \u211d\u22650\u221e\n\u03bc : OuterMeasure X\ns\u271d : Set X\nm : Set X \u2192 \u211d\u22650\u221e\ns : Set X\n\u22a2 Tendsto ((fun r => \u2191(pre m r) s) \u2218 Subtype.val) atBot (\ud835\udcdd (\u2191(mkMetric' m) s))"}, {"tactic": "simp only [mkMetric', OuterMeasure.iSup_apply, iSup_subtype']", "annotated_tactic": ["simp only [<a>mkMetric'</a>, <a>OuterMeasure.iSup_apply</a>, <a>iSup_subtype'</a>]", [{"full_name": "MeasureTheory.OuterMeasure.mkMetric'", "def_path": "Mathlib/MeasureTheory/Measure/Hausdorff.lean", "def_pos": [258, 5], "def_end_pos": [258, 14]}, {"full_name": "MeasureTheory.OuterMeasure.iSup_apply", "def_path": 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"Mathlib/MeasureTheory/Function/AEEqFun.lean", "full_name": "MeasureTheory.AEEqFun.coeFn_abs", "start": [940, 1], "end": [944, 36], "traced_tactics": [{"tactic": "simp_rw [abs_eq_sup_neg]", "annotated_tactic": ["simp_rw [<a>abs_eq_sup_neg</a>]", [{"full_name": "abs_eq_sup_neg", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [33, 3], "def_end_pos": [33, 14]}]], "state_before": "\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2078 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2077 : TopologicalSpace \u03b2\u271d\ninst\u271d\u2076 : TopologicalSpace \u03b3\ninst\u271d\u2075 : TopologicalSpace \u03b4\n\u03b2 : Type u_5\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : Lattice \u03b2\ninst\u271d\u00b2 : TopologicalLattice \u03b2\ninst\u271d\u00b9 : AddGroup \u03b2\ninst\u271d : TopologicalAddGroup \u03b2\nf : \u03b1 \u2192\u2098[\u03bc] \u03b2\n\u22a2 \u2191|f| =\u1d50[\u03bc] fun x => 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"def_pos": [530, 9], "def_end_pos": [530, 18]}, {"full_name": "MeasureTheory.AEEqFun.coeFn_neg", "def_path": "Mathlib/MeasureTheory/Function/AEEqFun.lean", "def_pos": [795, 3], "def_end_pos": [795, 14]}]], "state_before": "\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2078 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2077 : TopologicalSpace \u03b2\u271d\ninst\u271d\u2076 : TopologicalSpace \u03b3\ninst\u271d\u2075 : TopologicalSpace \u03b4\n\u03b2 : Type u_5\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : Lattice \u03b2\ninst\u271d\u00b2 : TopologicalLattice \u03b2\ninst\u271d\u00b9 : AddGroup \u03b2\ninst\u271d : TopologicalAddGroup \u03b2\nf : \u03b1 \u2192\u2098[\u03bc] \u03b2\n\u22a2 \u2191(f \u2294 -f) =\u1d50[\u03bc] fun x => \u2191f x \u2294 -\u2191f x", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2078 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2077 : TopologicalSpace \u03b2\u271d\ninst\u271d\u2076 : TopologicalSpace \u03b3\ninst\u271d\u2075 : TopologicalSpace \u03b4\n\u03b2 : Type u_5\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : Lattice \u03b2\ninst\u271d\u00b2 : TopologicalLattice \u03b2\ninst\u271d\u00b9 : AddGroup \u03b2\ninst\u271d : TopologicalAddGroup \u03b2\nf : \u03b1 \u2192\u2098[\u03bc] \u03b2\nx : \u03b1\nhx_sup : \u2191(f \u2294 -f) x = \u2191f x \u2294 \u2191(-f) x\nhx_neg : \u2191(-f) x = (-\u2191f) x\n\u22a2 \u2191(f \u2294 -f) x = \u2191f x \u2294 -\u2191f x"}, {"tactic": "rw [hx_sup, hx_neg, Pi.neg_apply]", "annotated_tactic": ["rw [hx_sup, hx_neg, <a>Pi.neg_apply</a>]", [{"full_name": "Pi.neg_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [170, 3], "def_end_pos": [170, 14]}]], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2078 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2077 : TopologicalSpace \u03b2\u271d\ninst\u271d\u2076 : TopologicalSpace \u03b3\ninst\u271d\u2075 : TopologicalSpace \u03b4\n\u03b2 : Type u_5\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : Lattice \u03b2\ninst\u271d\u00b2 : TopologicalLattice \u03b2\ninst\u271d\u00b9 : AddGroup \u03b2\ninst\u271d : TopologicalAddGroup \u03b2\nf : \u03b1 \u2192\u2098[\u03bc] \u03b2\nx : \u03b1\nhx_sup : \u2191(f \u2294 -f) x = \u2191f x \u2294 \u2191(-f) x\nhx_neg : \u2191(-f) x = (-\u2191f) x\n\u22a2 \u2191(f \u2294 -f) x = \u2191f x \u2294 -\u2191f x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Constructions/Polish.lean", "full_name": "MeasureTheory.analyticSet_range_of_polishSpace", "start": [173, 1], "end": [182, 23], "traced_tactics": [{"tactic": "cases isEmpty_or_nonempty \u03b2", "annotated_tactic": ["cases <a>isEmpty_or_nonempty</a> \u03b2", [{"full_name": "isEmpty_or_nonempty", "def_path": "Mathlib/Logic/IsEmpty.lean", "def_pos": [207, 9], "def_end_pos": [207, 28]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03b2 : Type u_3\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : PolishSpace \u03b2\nf : \u03b2 \u2192 \u03b1\nf_cont : Continuous f\n\u22a2 AnalyticSet (range f)", "state_after": "case inl\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03b2 : Type u_3\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : PolishSpace \u03b2\nf : \u03b2 \u2192 \u03b1\nf_cont : Continuous f\nh\u271d : IsEmpty \u03b2\n\u22a2 AnalyticSet (range f)\n\ncase inr\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03b2 : Type u_3\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : PolishSpace \u03b2\nf : \u03b2 \u2192 \u03b1\nf_cont : Continuous f\nh\u271d : Nonempty \u03b2\n\u22a2 AnalyticSet (range f)"}, {"tactic": "rw [range_eq_empty]", "annotated_tactic": ["rw [<a>range_eq_empty</a>]", [{"full_name": "Set.range_eq_empty", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [780, 9], "def_end_pos": [780, 23]}]], "state_before": "case inl\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03b2 : Type u_3\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : PolishSpace \u03b2\nf : \u03b2 \u2192 \u03b1\nf_cont : Continuous f\nh\u271d : IsEmpty \u03b2\n\u22a2 AnalyticSet (range f)", "state_after": "case inl\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03b2 : Type u_3\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : PolishSpace \u03b2\nf : \u03b2 \u2192 \u03b1\nf_cont : Continuous f\nh\u271d : IsEmpty \u03b2\n\u22a2 AnalyticSet \u2205"}, {"tactic": "exact analyticSet_empty", "annotated_tactic": ["exact <a>analyticSet_empty</a>", [{"full_name": "MeasureTheory.analyticSet_empty", "def_path": "Mathlib/MeasureTheory/Constructions/Polish.lean", "def_pos": [168, 9], "def_end_pos": [168, 26]}]], "state_before": "case inl\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03b2 : Type u_3\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : PolishSpace \u03b2\nf : \u03b2 \u2192 \u03b1\nf_cont : Continuous f\nh\u271d : IsEmpty \u03b2\n\u22a2 AnalyticSet \u2205", "state_after": "no goals"}, {"tactic": "rw [AnalyticSet]", "annotated_tactic": ["rw [<a>AnalyticSet</a>]", [{"full_name": "MeasureTheory.AnalyticSet", "def_path": "Mathlib/MeasureTheory/Constructions/Polish.lean", "def_pos": [164, 17], "def_end_pos": [164, 28]}]], "state_before": "case inr\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03b2 : Type u_3\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : PolishSpace \u03b2\nf : \u03b2 \u2192 \u03b1\nf_cont : Continuous f\nh\u271d : Nonempty \u03b2\n\u22a2 AnalyticSet (range f)", "state_after": "case inr\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03b2 : Type u_3\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : PolishSpace \u03b2\nf : \u03b2 \u2192 \u03b1\nf_cont : Continuous f\nh\u271d : Nonempty \u03b2\n\u22a2 range f = \u2205 \u2228 \u2203 f_1, Continuous f_1 \u2227 range f_1 = range f"}, {"tactic": "obtain \u27e8g, g_cont, hg\u27e9 : \u2203 g : (\u2115 \u2192 \u2115) \u2192 \u03b2, Continuous g \u2227 Surjective g :=\n  exists_nat_nat_continuous_surjective \u03b2", "annotated_tactic": ["obtain \u27e8g, g_cont, hg\u27e9 : \u2203 g : (\u2115 \u2192 \u2115) \u2192 \u03b2, <a>Continuous</a> g \u2227 <a>Surjective</a> g :=\n      <a>exists_nat_nat_continuous_surjective</a> \u03b2", [{"full_name": "Continuous", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [1591, 11], "def_end_pos": [1591, 21]}, {"full_name": "Function.Surjective", "def_path": "Mathlib/Init/Function.lean", "def_pos": [119, 5], "def_end_pos": [119, 15]}, {"full_name": "PolishSpace.exists_nat_nat_continuous_surjective", "def_path": "Mathlib/Topology/MetricSpace/Polish.lean", "def_pos": [145, 9], "def_end_pos": [145, 45]}]], "state_before": "case inr\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03b2 : Type u_3\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : PolishSpace \u03b2\nf : \u03b2 \u2192 \u03b1\nf_cont : Continuous f\nh\u271d : Nonempty \u03b2\n\u22a2 range f = \u2205 \u2228 \u2203 f_1, Continuous f_1 \u2227 range f_1 = range f", "state_after": "case inr.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03b2 : Type u_3\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : PolishSpace \u03b2\nf : \u03b2 \u2192 \u03b1\nf_cont : Continuous f\nh\u271d : Nonempty \u03b2\ng : (\u2115 \u2192 \u2115) \u2192 \u03b2\ng_cont : Continuous g\nhg : Surjective g\n\u22a2 range f = \u2205 \u2228 \u2203 f_1, Continuous f_1 \u2227 range f_1 = range f"}, {"tactic": "refine' Or.inr \u27e8f \u2218 g, f_cont.comp g_cont, _\u27e9", "annotated_tactic": ["refine' <a>Or.inr</a> \u27e8f \u2218 g, f_cont.comp g_cont, _\u27e9", [{"full_name": "Or.inr", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [519, 5], "def_end_pos": [519, 8]}]], "state_before": "case inr.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03b2 : Type u_3\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : PolishSpace \u03b2\nf : \u03b2 \u2192 \u03b1\nf_cont : Continuous f\nh\u271d : Nonempty \u03b2\ng : (\u2115 \u2192 \u2115) \u2192 \u03b2\ng_cont : Continuous g\nhg : Surjective g\n\u22a2 range f = \u2205 \u2228 \u2203 f_1, Continuous f_1 \u2227 range f_1 = range f", "state_after": "case inr.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03b2 : Type u_3\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : PolishSpace \u03b2\nf : \u03b2 \u2192 \u03b1\nf_cont : Continuous f\nh\u271d : Nonempty \u03b2\ng : (\u2115 \u2192 \u2115) \u2192 \u03b2\ng_cont : Continuous g\nhg : Surjective g\n\u22a2 range (f \u2218 g) = range f"}, {"tactic": "rw [hg.range_comp]", "annotated_tactic": ["rw [hg.range_comp]", []], "state_before": "case inr.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03b2 : Type u_3\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : PolishSpace \u03b2\nf : \u03b2 \u2192 \u03b1\nf_cont : Continuous f\nh\u271d : Nonempty \u03b2\ng : (\u2115 \u2192 \u2115) \u2192 \u03b2\ng_cont : Continuous g\nhg : Surjective g\n\u22a2 range (f \u2218 g) = range f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "full_name": "String.all_iff", "start": [742, 1], "end": [742, 94], "traced_tactics": [{"tactic": "simp [all_eq]", "annotated_tactic": ["simp [<a>all_eq</a>]", [{"full_name": "String.all_eq", "def_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "def_pos": [739, 9], "def_end_pos": [739, 15]}]], "state_before": "s : String\np : Char \u2192 Bool\n\u22a2 all s p = true \u2194 \u2200 (c : Char), c \u2208 s.data \u2192 p c = true", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "full_name": "MeasureTheory.snorm'_add_le", "start": [772, 1], "end": [780, 93], "traced_tactics": [{"tactic": "refine' ENNReal.rpow_le_rpow _ (by simp [le_trans zero_le_one hq1] : 0 \u2264 1 / q)", "annotated_tactic": ["refine' <a>ENNReal.rpow_le_rpow</a> _ (by simp [<a>le_trans</a> <a>zero_le_one</a> hq1] : 0 \u2264 1 / q)", [{"full_name": "ENNReal.rpow_le_rpow", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [642, 9], "def_end_pos": [642, 21]}, {"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}, {"full_name": "zero_le_one", "def_path": "Mathlib/Algebra/Order/ZeroLEOne.lean", "def_pos": [26, 15], "def_end_pos": [26, 26]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf g : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhg : AEStronglyMeasurable g \u03bc\nhq1 : 1 \u2264 q\n\u22a2 (\u222b\u207b (a : \u03b1), \u2191\u2016(f + g) a\u2016\u208a ^ q \u2202\u03bc) ^ (1 / q) \u2264\n    (\u222b\u207b (a : \u03b1), ((fun a => \u2191\u2016f a\u2016\u208a) + fun a => \u2191\u2016g a\u2016\u208a) a ^ q \u2202\u03bc) ^ (1 / q)", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf g : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhg : AEStronglyMeasurable g \u03bc\nhq1 : 1 \u2264 q\n\u22a2 \u222b\u207b (a : \u03b1), \u2191\u2016(f + g) a\u2016\u208a ^ q \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), ((fun a => \u2191\u2016f a\u2016\u208a) + fun a => \u2191\u2016g a\u2016\u208a) a ^ q \u2202\u03bc"}, {"tactic": "refine' lintegral_mono fun a => ENNReal.rpow_le_rpow _ (le_trans zero_le_one hq1)", "annotated_tactic": ["refine' <a>lintegral_mono</a> fun a => <a>ENNReal.rpow_le_rpow</a> _ (<a>le_trans</a> <a>zero_le_one</a> hq1)", [{"full_name": "MeasureTheory.lintegral_mono", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [99, 9], "def_end_pos": [99, 23]}, {"full_name": "ENNReal.rpow_le_rpow", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [642, 9], "def_end_pos": [642, 21]}, {"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}, {"full_name": "zero_le_one", "def_path": "Mathlib/Algebra/Order/ZeroLEOne.lean", "def_pos": [26, 15], "def_end_pos": [26, 26]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf g : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhg : AEStronglyMeasurable g \u03bc\nhq1 : 1 \u2264 q\n\u22a2 \u222b\u207b (a : \u03b1), \u2191\u2016(f + g) a\u2016\u208a ^ q \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), ((fun a => \u2191\u2016f a\u2016\u208a) + fun a => \u2191\u2016g a\u2016\u208a) a ^ q \u2202\u03bc", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf g : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhg : AEStronglyMeasurable g \u03bc\nhq1 : 1 \u2264 q\na : \u03b1\n\u22a2 \u2191\u2016(f + g) a\u2016\u208a \u2264 ((fun a => \u2191\u2016f a\u2016\u208a) + fun a => \u2191\u2016g a\u2016\u208a) a"}, {"tactic": "simp only [Pi.add_apply, \u2190 ENNReal.coe_add, ENNReal.coe_le_coe, nnnorm_add_le]", "annotated_tactic": ["simp only [<a>Pi.add_apply</a>, \u2190 <a>ENNReal.coe_add</a>, <a>ENNReal.coe_le_coe</a>, <a>nnnorm_add_le</a>]", [{"full_name": "Pi.add_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [82, 3], "def_end_pos": [82, 14]}, {"full_name": "ENNReal.coe_add", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [386, 28], "def_end_pos": [386, 35]}, {"full_name": "ENNReal.coe_le_coe", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [349, 28], "def_end_pos": [349, 38]}, {"full_name": "nnnorm_add_le", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [946, 15], "def_end_pos": [946, 28]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf g : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhg : AEStronglyMeasurable g \u03bc\nhq1 : 1 \u2264 q\na : \u03b1\n\u22a2 \u2191\u2016(f + g) a\u2016\u208a \u2264 ((fun a => \u2191\u2016f a\u2016\u208a) + fun a => \u2191\u2016g a\u2016\u208a) a", "state_after": "no goals"}, {"tactic": "simp [le_trans zero_le_one hq1]", "annotated_tactic": ["simp [<a>le_trans</a> <a>zero_le_one</a> hq1]", [{"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}, {"full_name": "zero_le_one", "def_path": "Mathlib/Algebra/Order/ZeroLEOne.lean", "def_pos": [26, 15], "def_end_pos": [26, 26]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf g : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhg : AEStronglyMeasurable g \u03bc\nhq1 : 1 \u2264 q\n\u22a2 0 \u2264 1 / q", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Haar/Basic.lean", "full_name": "MeasureTheory.Measure.haar.le_index_mul", "start": [185, 1], "end": [196, 53], "traced_tactics": [{"tactic": "obtain \u27e8s, h1s, h2s\u27e9 := index_elim K.isCompact K\u2080.interior_nonempty", "annotated_tactic": ["obtain \u27e8s, h1s, h2s\u27e9 := <a>index_elim</a> K.isCompact K\u2080.interior_nonempty", [{"full_name": "MeasureTheory.Measure.haar.index_elim", "def_path": "Mathlib/MeasureTheory/Measure/Haar/Basic.lean", "def_pos": [178, 9], "def_end_pos": [178, 19]}]], "state_before": "G : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\n\u22a2 index (\u2191K) V \u2264 index \u2191K \u2191K\u2080 * index (\u2191K\u2080) V", "state_after": "case intro.intro\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\ns : Finset G\nh1s : \u2191K \u2286 \u22c3 g \u2208 s, (fun h => g * h) \u207b\u00b9' \u2191K\u2080\nh2s : Finset.card s = index \u2191K \u2191K\u2080\n\u22a2 index (\u2191K) V \u2264 index \u2191K \u2191K\u2080 * index (\u2191K\u2080) V"}, {"tactic": "obtain \u27e8t, h1t, h2t\u27e9 := index_elim K\u2080.isCompact hV", "annotated_tactic": ["obtain \u27e8t, h1t, h2t\u27e9 := <a>index_elim</a> K\u2080.isCompact hV", [{"full_name": "MeasureTheory.Measure.haar.index_elim", "def_path": "Mathlib/MeasureTheory/Measure/Haar/Basic.lean", "def_pos": [178, 9], "def_end_pos": [178, 19]}]], "state_before": "case intro.intro\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\ns : Finset G\nh1s : \u2191K \u2286 \u22c3 g \u2208 s, (fun h => g * h) \u207b\u00b9' \u2191K\u2080\nh2s : Finset.card s = index \u2191K \u2191K\u2080\n\u22a2 index (\u2191K) V \u2264 index \u2191K \u2191K\u2080 * index (\u2191K\u2080) V", "state_after": "case intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\ns : Finset G\nh1s : \u2191K \u2286 \u22c3 g \u2208 s, (fun h => g * h) \u207b\u00b9' \u2191K\u2080\nh2s : Finset.card s = index \u2191K \u2191K\u2080\nt : Finset G\nh1t : \u2191K\u2080 \u2286 \u22c3 g \u2208 t, (fun h => g * h) \u207b\u00b9' V\nh2t : Finset.card t = index (\u2191K\u2080) V\n\u22a2 index (\u2191K) V \u2264 index \u2191K \u2191K\u2080 * index (\u2191K\u2080) V"}, {"tactic": "rw [\u2190 h2s, \u2190 h2t, mul_comm]", "annotated_tactic": ["rw [\u2190 h2s, \u2190 h2t, <a>mul_comm</a>]", [{"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [302, 9], "def_end_pos": [302, 17]}]], "state_before": "case intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\ns : Finset G\nh1s : \u2191K \u2286 \u22c3 g \u2208 s, (fun h => g * h) \u207b\u00b9' \u2191K\u2080\nh2s : Finset.card s = index \u2191K \u2191K\u2080\nt : Finset G\nh1t : \u2191K\u2080 \u2286 \u22c3 g \u2208 t, (fun h => g * h) \u207b\u00b9' V\nh2t : Finset.card t = index (\u2191K\u2080) V\n\u22a2 index (\u2191K) V \u2264 index \u2191K \u2191K\u2080 * index (\u2191K\u2080) V", "state_after": "case intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\ns : Finset G\nh1s : \u2191K \u2286 \u22c3 g \u2208 s, (fun h => g * h) \u207b\u00b9' \u2191K\u2080\nh2s : Finset.card s = index \u2191K \u2191K\u2080\nt : Finset G\nh1t : \u2191K\u2080 \u2286 \u22c3 g \u2208 t, (fun h => g * h) \u207b\u00b9' V\nh2t : Finset.card t = index (\u2191K\u2080) V\n\u22a2 index (\u2191K) V \u2264 Finset.card t * Finset.card s"}, {"tactic": "refine' le_trans _ Finset.card_mul_le", "annotated_tactic": ["refine' <a>le_trans</a> _ <a>Finset.card_mul_le</a>", [{"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}, {"full_name": "Finset.card_mul_le", "def_path": "Mathlib/Data/Finset/Pointwise.lean", "def_pos": [339, 9], "def_end_pos": [339, 20]}]], "state_before": "case intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\ns : Finset G\nh1s : \u2191K \u2286 \u22c3 g \u2208 s, (fun h => g * h) \u207b\u00b9' \u2191K\u2080\nh2s : Finset.card s = index \u2191K \u2191K\u2080\nt : Finset G\nh1t : \u2191K\u2080 \u2286 \u22c3 g \u2208 t, (fun h => g * h) \u207b\u00b9' V\nh2t : Finset.card t = index (\u2191K\u2080) V\n\u22a2 index (\u2191K) V \u2264 Finset.card t * Finset.card s", "state_after": "case intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\ns : Finset G\nh1s : \u2191K \u2286 \u22c3 g \u2208 s, (fun h => g * h) \u207b\u00b9' \u2191K\u2080\nh2s : Finset.card s = index \u2191K \u2191K\u2080\nt : Finset G\nh1t : \u2191K\u2080 \u2286 \u22c3 g \u2208 t, (fun h => g * h) \u207b\u00b9' V\nh2t : Finset.card t = index (\u2191K\u2080) V\n\u22a2 index (\u2191K) V \u2264 Finset.card (t * s)"}, {"tactic": "apply Nat.sInf_le", "annotated_tactic": ["apply <a>Nat.sInf_le</a>", [{"full_name": "Nat.sInf_le", "def_path": "Mathlib/Data/Nat/Lattice.lean", "def_pos": [80, 19], "def_end_pos": [80, 26]}]], "state_before": "case intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\ns : Finset G\nh1s : \u2191K \u2286 \u22c3 g \u2208 s, (fun h => g * h) \u207b\u00b9' \u2191K\u2080\nh2s : Finset.card s = index \u2191K \u2191K\u2080\nt : Finset G\nh1t : \u2191K\u2080 \u2286 \u22c3 g \u2208 t, (fun h => g * h) \u207b\u00b9' V\nh2t : Finset.card t = index (\u2191K\u2080) V\n\u22a2 index (\u2191K) V \u2264 Finset.card (t * s)", "state_after": "case intro.intro.intro.intro.hm\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\ns : Finset G\nh1s : \u2191K \u2286 \u22c3 g \u2208 s, (fun h => g * h) \u207b\u00b9' \u2191K\u2080\nh2s : Finset.card s = index \u2191K \u2191K\u2080\nt : Finset G\nh1t : \u2191K\u2080 \u2286 \u22c3 g \u2208 t, (fun h => g * h) \u207b\u00b9' V\nh2t : Finset.card t = index (\u2191K\u2080) V\n\u22a2 Finset.card (t * s) \u2208 Finset.card '' {t | \u2191K \u2286 \u22c3 g \u2208 t, (fun h => g * h) \u207b\u00b9' V}"}, {"tactic": "refine' \u27e8_, _, rfl\u27e9", "annotated_tactic": ["refine' \u27e8_, _, <a>rfl</a>\u27e9", [{"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "case intro.intro.intro.intro.hm\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\ns : Finset G\nh1s : \u2191K \u2286 \u22c3 g \u2208 s, (fun h => g * h) \u207b\u00b9' \u2191K\u2080\nh2s : Finset.card s = index \u2191K \u2191K\u2080\nt : Finset G\nh1t : \u2191K\u2080 \u2286 \u22c3 g \u2208 t, (fun h => g * h) \u207b\u00b9' V\nh2t : Finset.card t = index (\u2191K\u2080) V\n\u22a2 Finset.card (t * s) \u2208 Finset.card '' {t | \u2191K \u2286 \u22c3 g \u2208 t, (fun h => g * h) \u207b\u00b9' V}", "state_after": "case intro.intro.intro.intro.hm\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\ns : Finset G\nh1s : \u2191K \u2286 \u22c3 g \u2208 s, (fun h => g * h) \u207b\u00b9' \u2191K\u2080\nh2s : Finset.card s = index \u2191K \u2191K\u2080\nt : Finset G\nh1t : \u2191K\u2080 \u2286 \u22c3 g \u2208 t, (fun h => g * h) \u207b\u00b9' V\nh2t : Finset.card t = index (\u2191K\u2080) V\n\u22a2 t * s \u2208 {t | \u2191K \u2286 \u22c3 g \u2208 t, (fun h => g * h) \u207b\u00b9' V}"}, {"tactic": "rw [mem_setOf_eq]", "annotated_tactic": ["rw [<a>mem_setOf_eq</a>]", [{"full_name": "Set.mem_setOf_eq", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [256, 29], "def_end_pos": [256, 41]}]], "state_before": "case intro.intro.intro.intro.hm\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\ns : Finset G\nh1s : \u2191K \u2286 \u22c3 g \u2208 s, (fun h => g * h) \u207b\u00b9' \u2191K\u2080\nh2s : Finset.card s = index \u2191K \u2191K\u2080\nt : Finset G\nh1t : \u2191K\u2080 \u2286 \u22c3 g \u2208 t, (fun h => g * h) \u207b\u00b9' V\nh2t : Finset.card t = index (\u2191K\u2080) V\n\u22a2 t * s \u2208 {t | \u2191K \u2286 \u22c3 g \u2208 t, (fun h => g * h) \u207b\u00b9' V}", "state_after": "case intro.intro.intro.intro.hm\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\ns : Finset G\nh1s : \u2191K \u2286 \u22c3 g \u2208 s, (fun h => g * h) \u207b\u00b9' \u2191K\u2080\nh2s : Finset.card s = index \u2191K \u2191K\u2080\nt : Finset G\nh1t : \u2191K\u2080 \u2286 \u22c3 g \u2208 t, (fun h => g * h) \u207b\u00b9' V\nh2t : Finset.card t = index (\u2191K\u2080) V\n\u22a2 \u2191K \u2286 \u22c3 g \u2208 t * s, (fun h => g * h) \u207b\u00b9' V"}, {"tactic": "refine' Subset.trans h1s _", "annotated_tactic": ["refine' <a>Subset.trans</a> h1s _", [{"full_name": "Set.Subset.trans", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [362, 9], "def_end_pos": [362, 21]}]], "state_before": "case intro.intro.intro.intro.hm\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\ns : Finset G\nh1s : \u2191K \u2286 \u22c3 g \u2208 s, (fun h => g * h) \u207b\u00b9' \u2191K\u2080\nh2s : Finset.card s = index \u2191K \u2191K\u2080\nt : Finset G\nh1t : \u2191K\u2080 \u2286 \u22c3 g \u2208 t, (fun h => g * h) \u207b\u00b9' V\nh2t : Finset.card t = index (\u2191K\u2080) V\n\u22a2 \u2191K \u2286 \u22c3 g \u2208 t * s, (fun h => g * h) \u207b\u00b9' V", "state_after": "case intro.intro.intro.intro.hm\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\ns : Finset G\nh1s : \u2191K \u2286 \u22c3 g \u2208 s, (fun h => g * h) \u207b\u00b9' \u2191K\u2080\nh2s : Finset.card s = index \u2191K \u2191K\u2080\nt : Finset G\nh1t : \u2191K\u2080 \u2286 \u22c3 g \u2208 t, (fun h => g * h) \u207b\u00b9' V\nh2t : Finset.card t = index (\u2191K\u2080) V\n\u22a2 \u22c3 g \u2208 s, (fun h => g * h) \u207b\u00b9' \u2191K\u2080 \u2286 \u22c3 g \u2208 t * s, (fun h => g * h) \u207b\u00b9' V"}, {"tactic": "apply iUnion\u2082_subset", "annotated_tactic": ["apply <a>iUnion\u2082_subset</a>", [{"full_name": "Set.iUnion\u2082_subset", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [395, 9], "def_end_pos": [395, 23]}]], "state_before": "case intro.intro.intro.intro.hm\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\ns : Finset G\nh1s : \u2191K \u2286 \u22c3 g \u2208 s, (fun h => g * h) \u207b\u00b9' \u2191K\u2080\nh2s : Finset.card s = index \u2191K \u2191K\u2080\nt : Finset G\nh1t : \u2191K\u2080 \u2286 \u22c3 g \u2208 t, (fun h => g * h) \u207b\u00b9' V\nh2t : Finset.card t = index (\u2191K\u2080) V\n\u22a2 \u22c3 g \u2208 s, (fun h => g * h) \u207b\u00b9' \u2191K\u2080 \u2286 \u22c3 g \u2208 t * s, (fun h => g * h) \u207b\u00b9' V", "state_after": "case intro.intro.intro.intro.hm.h\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\ns : Finset G\nh1s : \u2191K \u2286 \u22c3 g \u2208 s, (fun h => g * h) \u207b\u00b9' \u2191K\u2080\nh2s : Finset.card s = index \u2191K \u2191K\u2080\nt : Finset G\nh1t : \u2191K\u2080 \u2286 \u22c3 g \u2208 t, (fun h => g * h) \u207b\u00b9' V\nh2t : Finset.card t = index (\u2191K\u2080) V\n\u22a2 \u2200 (i : G), i \u2208 s \u2192 (fun h => i * h) \u207b\u00b9' \u2191K\u2080 \u2286 \u22c3 g \u2208 t * s, (fun h => g * h) \u207b\u00b9' V"}, {"tactic": "intro g\u2081 hg\u2081", "annotated_tactic": ["intro g\u2081 hg\u2081", []], "state_before": "case intro.intro.intro.intro.hm.h\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\ns : Finset G\nh1s : \u2191K \u2286 \u22c3 g \u2208 s, (fun h => g * h) \u207b\u00b9' \u2191K\u2080\nh2s : Finset.card s = index \u2191K \u2191K\u2080\nt : Finset G\nh1t : \u2191K\u2080 \u2286 \u22c3 g \u2208 t, (fun h => g * h) \u207b\u00b9' V\nh2t : Finset.card t = index (\u2191K\u2080) V\n\u22a2 \u2200 (i : G), i \u2208 s \u2192 (fun h => i * h) \u207b\u00b9' \u2191K\u2080 \u2286 \u22c3 g \u2208 t * s, (fun h => g * h) \u207b\u00b9' V", "state_after": "case intro.intro.intro.intro.hm.h\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\ns : Finset G\nh1s : \u2191K \u2286 \u22c3 g \u2208 s, (fun h => g * h) \u207b\u00b9' \u2191K\u2080\nh2s : Finset.card s = index \u2191K \u2191K\u2080\nt : Finset G\nh1t : \u2191K\u2080 \u2286 \u22c3 g \u2208 t, (fun h => g * h) \u207b\u00b9' V\nh2t : Finset.card t = index (\u2191K\u2080) V\ng\u2081 : G\nhg\u2081 : g\u2081 \u2208 s\n\u22a2 (fun h => g\u2081 * h) \u207b\u00b9' \u2191K\u2080 \u2286 \u22c3 g \u2208 t * s, (fun h => g * h) \u207b\u00b9' V"}, {"tactic": "rw [preimage_subset_iff]", "annotated_tactic": ["rw [<a>preimage_subset_iff</a>]", [{"full_name": "Set.preimage_subset_iff", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [573, 9], "def_end_pos": [573, 28]}]], "state_before": "case intro.intro.intro.intro.hm.h\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\ns : Finset G\nh1s : \u2191K \u2286 \u22c3 g \u2208 s, (fun h => g * h) \u207b\u00b9' \u2191K\u2080\nh2s : Finset.card s = index \u2191K \u2191K\u2080\nt : Finset G\nh1t : \u2191K\u2080 \u2286 \u22c3 g \u2208 t, (fun h => g * h) \u207b\u00b9' V\nh2t : Finset.card t = index (\u2191K\u2080) V\ng\u2081 : G\nhg\u2081 : g\u2081 \u2208 s\n\u22a2 (fun h => g\u2081 * h) \u207b\u00b9' \u2191K\u2080 \u2286 \u22c3 g \u2208 t * s, (fun h => g * h) \u207b\u00b9' V", "state_after": "case intro.intro.intro.intro.hm.h\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\ns : Finset G\nh1s : \u2191K \u2286 \u22c3 g \u2208 s, (fun h => g * h) \u207b\u00b9' \u2191K\u2080\nh2s : Finset.card s = index \u2191K \u2191K\u2080\nt : Finset G\nh1t : \u2191K\u2080 \u2286 \u22c3 g \u2208 t, (fun h => g * h) \u207b\u00b9' V\nh2t : Finset.card t = index (\u2191K\u2080) V\ng\u2081 : G\nhg\u2081 : g\u2081 \u2208 s\n\u22a2 \u2200 (a : G), g\u2081 * a \u2208 \u2191K\u2080 \u2192 a \u2208 \u22c3 g \u2208 t * s, (fun h => g * h) \u207b\u00b9' V"}, {"tactic": "intro g\u2082 hg\u2082", "annotated_tactic": ["intro g\u2082 hg\u2082", []], "state_before": "case intro.intro.intro.intro.hm.h\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\ns : Finset G\nh1s : \u2191K \u2286 \u22c3 g \u2208 s, (fun h => g * h) \u207b\u00b9' \u2191K\u2080\nh2s : Finset.card s = index \u2191K \u2191K\u2080\nt : Finset G\nh1t : \u2191K\u2080 \u2286 \u22c3 g \u2208 t, (fun h => g * h) \u207b\u00b9' V\nh2t : Finset.card t = index (\u2191K\u2080) V\ng\u2081 : G\nhg\u2081 : g\u2081 \u2208 s\n\u22a2 \u2200 (a : G), g\u2081 * a \u2208 \u2191K\u2080 \u2192 a \u2208 \u22c3 g \u2208 t * s, (fun h => g * h) \u207b\u00b9' V", "state_after": "case intro.intro.intro.intro.hm.h\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\ns : Finset G\nh1s : \u2191K \u2286 \u22c3 g \u2208 s, (fun h => g * h) \u207b\u00b9' \u2191K\u2080\nh2s : Finset.card s = index \u2191K \u2191K\u2080\nt : Finset G\nh1t : \u2191K\u2080 \u2286 \u22c3 g \u2208 t, (fun h => g * h) \u207b\u00b9' V\nh2t : Finset.card t = index (\u2191K\u2080) V\ng\u2081 : G\nhg\u2081 : g\u2081 \u2208 s\ng\u2082 : G\nhg\u2082 : g\u2081 * g\u2082 \u2208 \u2191K\u2080\n\u22a2 g\u2082 \u2208 \u22c3 g \u2208 t * s, (fun h => g * h) \u207b\u00b9' V"}, {"tactic": "have := h1t hg\u2082", "annotated_tactic": ["have := h1t hg\u2082", []], "state_before": "case intro.intro.intro.intro.hm.h\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\ns : Finset G\nh1s : \u2191K \u2286 \u22c3 g \u2208 s, (fun h => g * h) \u207b\u00b9' \u2191K\u2080\nh2s : Finset.card s = index \u2191K \u2191K\u2080\nt : Finset G\nh1t : \u2191K\u2080 \u2286 \u22c3 g \u2208 t, (fun h => g * h) \u207b\u00b9' V\nh2t : Finset.card t = index (\u2191K\u2080) V\ng\u2081 : G\nhg\u2081 : g\u2081 \u2208 s\ng\u2082 : G\nhg\u2082 : g\u2081 * g\u2082 \u2208 \u2191K\u2080\n\u22a2 g\u2082 \u2208 \u22c3 g \u2208 t * s, (fun h => g * h) \u207b\u00b9' V", "state_after": "case intro.intro.intro.intro.hm.h\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\ns : Finset G\nh1s : \u2191K \u2286 \u22c3 g \u2208 s, (fun h => g * h) \u207b\u00b9' \u2191K\u2080\nh2s : Finset.card s = index \u2191K \u2191K\u2080\nt : Finset G\nh1t : \u2191K\u2080 \u2286 \u22c3 g \u2208 t, (fun h => g * h) \u207b\u00b9' V\nh2t : Finset.card t = index (\u2191K\u2080) V\ng\u2081 : G\nhg\u2081 : g\u2081 \u2208 s\ng\u2082 : G\nhg\u2082 : g\u2081 * g\u2082 \u2208 \u2191K\u2080\nthis : g\u2081 * g\u2082 \u2208 \u22c3 g \u2208 t, (fun h => g * h) \u207b\u00b9' V\n\u22a2 g\u2082 \u2208 \u22c3 g \u2208 t * s, (fun h => g * h) \u207b\u00b9' V"}, {"tactic": "rcases this with \u27e8_, \u27e8g\u2083, rfl\u27e9, A, \u27e8hg\u2083, rfl\u27e9, h2V\u27e9", "annotated_tactic": ["rcases this with \u27e8_, \u27e8g\u2083, rfl\u27e9, A, \u27e8hg\u2083, rfl\u27e9, h2V\u27e9", []], "state_before": "case intro.intro.intro.intro.hm.h\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\ns : Finset G\nh1s : \u2191K \u2286 \u22c3 g \u2208 s, (fun h => g * h) \u207b\u00b9' \u2191K\u2080\nh2s : Finset.card s = index \u2191K \u2191K\u2080\nt : Finset G\nh1t : \u2191K\u2080 \u2286 \u22c3 g \u2208 t, (fun h => g * h) \u207b\u00b9' V\nh2t : Finset.card t = index (\u2191K\u2080) V\ng\u2081 : G\nhg\u2081 : g\u2081 \u2208 s\ng\u2082 : G\nhg\u2082 : g\u2081 * g\u2082 \u2208 \u2191K\u2080\nthis : g\u2081 * g\u2082 \u2208 \u22c3 g \u2208 t, (fun h => g * h) \u207b\u00b9' V\n\u22a2 g\u2082 \u2208 \u22c3 g \u2208 t * s, (fun h => g * h) \u207b\u00b9' V", "state_after": "case intro.intro.intro.intro.hm.h.intro.intro.intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\ns : Finset G\nh1s : \u2191K \u2286 \u22c3 g \u2208 s, (fun h => g * h) \u207b\u00b9' \u2191K\u2080\nh2s : Finset.card s = index \u2191K \u2191K\u2080\nt : Finset G\nh1t : \u2191K\u2080 \u2286 \u22c3 g \u2208 t, (fun h => g * h) \u207b\u00b9' V\nh2t : Finset.card t = index (\u2191K\u2080) V\ng\u2081 : G\nhg\u2081 : g\u2081 \u2208 s\ng\u2082 : G\nhg\u2082 : g\u2081 * g\u2082 \u2208 \u2191K\u2080\ng\u2083 : G\nhg\u2083 : g\u2083 \u2208 t\nh2V : g\u2081 * g\u2082 \u2208 (fun h => (fun h => g\u2083 * h) \u207b\u00b9' V) hg\u2083\n\u22a2 g\u2082 \u2208 \u22c3 g \u2208 t * s, (fun h => g * h) \u207b\u00b9' V"}, {"tactic": "rw [mem_preimage, \u2190 mul_assoc] at h2V", "annotated_tactic": ["rw [<a>mem_preimage</a>, \u2190 <a>mul_assoc</a>] at h2V", [{"full_name": "Set.mem_preimage", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [64, 9], "def_end_pos": [64, 21]}, {"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [264, 9], "def_end_pos": [264, 18]}]], "state_before": "case intro.intro.intro.intro.hm.h.intro.intro.intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\ns : Finset G\nh1s : \u2191K \u2286 \u22c3 g \u2208 s, (fun h => g * h) \u207b\u00b9' \u2191K\u2080\nh2s : Finset.card s = index \u2191K \u2191K\u2080\nt : Finset G\nh1t : \u2191K\u2080 \u2286 \u22c3 g \u2208 t, (fun h => g * h) \u207b\u00b9' V\nh2t : Finset.card t = index (\u2191K\u2080) V\ng\u2081 : G\nhg\u2081 : g\u2081 \u2208 s\ng\u2082 : G\nhg\u2082 : g\u2081 * g\u2082 \u2208 \u2191K\u2080\ng\u2083 : G\nhg\u2083 : g\u2083 \u2208 t\nh2V : g\u2081 * g\u2082 \u2208 (fun h => (fun h => g\u2083 * h) \u207b\u00b9' V) hg\u2083\n\u22a2 g\u2082 \u2208 \u22c3 g \u2208 t * s, (fun h => g * h) \u207b\u00b9' V", "state_after": "case intro.intro.intro.intro.hm.h.intro.intro.intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\ns : Finset G\nh1s : \u2191K \u2286 \u22c3 g \u2208 s, (fun h => g * h) \u207b\u00b9' \u2191K\u2080\nh2s : Finset.card s = index \u2191K \u2191K\u2080\nt : Finset G\nh1t : \u2191K\u2080 \u2286 \u22c3 g \u2208 t, (fun h => g * h) \u207b\u00b9' V\nh2t : Finset.card t = index (\u2191K\u2080) V\ng\u2081 : G\nhg\u2081 : g\u2081 \u2208 s\ng\u2082 : G\nhg\u2082 : g\u2081 * g\u2082 \u2208 \u2191K\u2080\ng\u2083 : G\nhg\u2083 : g\u2083 \u2208 t\nh2V : g\u2083 * g\u2081 * g\u2082 \u2208 V\n\u22a2 g\u2082 \u2208 \u22c3 g \u2208 t * s, (fun h => g * h) \u207b\u00b9' V"}, {"tactic": "exact mem_biUnion (Finset.mul_mem_mul hg\u2083 hg\u2081) h2V", "annotated_tactic": ["exact <a>mem_biUnion</a> (<a>Finset.mul_mem_mul</a> hg\u2083 hg\u2081) h2V", [{"full_name": "Set.mem_biUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [966, 9], "def_end_pos": [966, 20]}, {"full_name": "Finset.mul_mem_mul", "def_path": "Mathlib/Data/Finset/Pointwise.lean", "def_pos": [333, 9], "def_end_pos": [333, 20]}]], "state_before": "case intro.intro.intro.intro.hm.h.intro.intro.intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\ns : Finset G\nh1s : \u2191K \u2286 \u22c3 g \u2208 s, (fun h => g * h) \u207b\u00b9' \u2191K\u2080\nh2s : Finset.card s = index \u2191K \u2191K\u2080\nt : Finset G\nh1t : \u2191K\u2080 \u2286 \u22c3 g \u2208 t, (fun h => g * h) \u207b\u00b9' V\nh2t : Finset.card t = index (\u2191K\u2080) V\ng\u2081 : G\nhg\u2081 : g\u2081 \u2208 s\ng\u2082 : G\nhg\u2082 : g\u2081 * g\u2082 \u2208 \u2191K\u2080\ng\u2083 : G\nhg\u2083 : g\u2083 \u2208 t\nh2V : g\u2083 * g\u2081 * g\u2082 \u2208 V\n\u22a2 g\u2082 \u2208 \u22c3 g \u2208 t * s, (fun h => g * h) \u207b\u00b9' V", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Density.lean", "full_name": "MeasureTheory.pdf.IsUniform.mul_pdf_integrable", "start": [365, 1], "end": [386, 60], "traced_tactics": [{"tactic": "by_cases hsupp : volume s = \u221e", "annotated_tactic": ["by_cases hsupp : <a>volume</a> s = \u221e", [{"full_name": "MeasureTheory.MeasureSpace.volume", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [663, 3], "def_end_pos": [663, 9]}]], "state_before": "\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b9 : MeasurableSpace E\nm : MeasurableSpace \u03a9\n\u2119 : Measure \u03a9\n\u03bc : Measure E\nX : \u03a9 \u2192 \u211d\ns : Set \u211d\nhms : MeasurableSet s\nhns : \u2191\u2191volume s \u2260 0\ninst\u271d : IsFiniteMeasure \u2119\nhcs : IsCompact s\nhuX : IsUniform X s \u2119\n\u22a2 Integrable fun x => x * ENNReal.toReal (pdf X \u2119 x)", "state_after": "case pos\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b9 : MeasurableSpace E\nm : MeasurableSpace \u03a9\n\u2119 : Measure \u03a9\n\u03bc : Measure E\nX : \u03a9 \u2192 \u211d\ns : Set \u211d\nhms : MeasurableSet s\nhns : \u2191\u2191volume s \u2260 0\ninst\u271d : IsFiniteMeasure \u2119\nhcs : IsCompact s\nhuX : IsUniform X s \u2119\nhsupp : \u2191\u2191volume s = \u22a4\n\u22a2 Integrable fun x => x * ENNReal.toReal (pdf X \u2119 x)\n\ncase neg\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b9 : MeasurableSpace E\nm : MeasurableSpace \u03a9\n\u2119 : Measure \u03a9\n\u03bc : Measure E\nX : \u03a9 \u2192 \u211d\ns : Set \u211d\nhms : MeasurableSet s\nhns : \u2191\u2191volume s \u2260 0\ninst\u271d : IsFiniteMeasure \u2119\nhcs : IsCompact s\nhuX : IsUniform X s \u2119\nhsupp : \u00ac\u2191\u2191volume s = \u22a4\n\u22a2 Integrable fun x => x * ENNReal.toReal (pdf X \u2119 x)"}, {"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "case neg\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b9 : MeasurableSpace E\nm : MeasurableSpace \u03a9\n\u2119 : Measure \u03a9\n\u03bc : Measure E\nX : \u03a9 \u2192 \u211d\ns : Set \u211d\nhms : MeasurableSet s\nhns : \u2191\u2191volume s \u2260 0\ninst\u271d : IsFiniteMeasure \u2119\nhcs : IsCompact s\nhuX : IsUniform X s \u2119\nhsupp : \u00ac\u2191\u2191volume s = \u22a4\n\u22a2 Integrable fun x => x * ENNReal.toReal (pdf X \u2119 x)", "state_after": "case neg.left\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b9 : MeasurableSpace E\nm : MeasurableSpace \u03a9\n\u2119 : Measure \u03a9\n\u03bc : Measure E\nX : \u03a9 \u2192 \u211d\ns : Set \u211d\nhms : MeasurableSet s\nhns : \u2191\u2191volume s \u2260 0\ninst\u271d : IsFiniteMeasure \u2119\nhcs : IsCompact s\nhuX : IsUniform X s \u2119\nhsupp : \u00ac\u2191\u2191volume s = \u22a4\n\u22a2 AEStronglyMeasurable (fun x => x * ENNReal.toReal (pdf X \u2119 x)) volume\n\ncase neg.right\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b9 : MeasurableSpace E\nm : MeasurableSpace \u03a9\n\u2119 : Measure \u03a9\n\u03bc : Measure E\nX : \u03a9 \u2192 \u211d\ns : Set \u211d\nhms : MeasurableSet s\nhns : \u2191\u2191volume s \u2260 0\ninst\u271d : IsFiniteMeasure \u2119\nhcs : IsCompact s\nhuX : IsUniform X s \u2119\nhsupp : \u00ac\u2191\u2191volume s = \u22a4\n\u22a2 HasFiniteIntegral fun x => x * ENNReal.toReal (pdf X \u2119 x)"}, {"tactic": "refine' hasFiniteIntegral_mul huX _", "annotated_tactic": ["refine' <a>hasFiniteIntegral_mul</a> huX _", [{"full_name": "MeasureTheory.pdf.hasFiniteIntegral_mul", "def_path": "Mathlib/Probability/Density.lean", "def_pos": [292, 9], "def_end_pos": [292, 30]}]], "state_before": "case neg.right\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b9 : MeasurableSpace E\nm : MeasurableSpace \u03a9\n\u2119 : Measure \u03a9\n\u03bc : Measure E\nX : \u03a9 \u2192 \u211d\ns : Set \u211d\nhms : MeasurableSet s\nhns : \u2191\u2191volume s \u2260 0\ninst\u271d : IsFiniteMeasure \u2119\nhcs : IsCompact s\nhuX : IsUniform X s \u2119\nhsupp : \u00ac\u2191\u2191volume s = \u22a4\n\u22a2 HasFiniteIntegral fun x => x * ENNReal.toReal (pdf X \u2119 x)", "state_after": "case neg.right\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b9 : MeasurableSpace E\nm : MeasurableSpace \u03a9\n\u2119 : Measure \u03a9\n\u03bc : Measure E\nX : \u03a9 \u2192 \u211d\ns : Set \u211d\nhms : MeasurableSet s\nhns : \u2191\u2191volume s \u2260 0\ninst\u271d : IsFiniteMeasure \u2119\nhcs : IsCompact s\nhuX : IsUniform X s \u2119\nhsupp : \u00ac\u2191\u2191volume s = \u22a4\n\u22a2 \u222b\u207b (x : \u211d), \u2191\u2016x\u2016\u208a * Set.indicator s ((\u2191\u2191volume s)\u207b\u00b9 \u2022 1) x \u2260 \u22a4"}, {"tactic": "set ind := (volume s)\u207b\u00b9 \u2022 (1 : \u211d \u2192 \u211d\u22650\u221e)", "annotated_tactic": ["set ind := (<a>volume</a> s)\u207b\u00b9 \u2022 (1 : \u211d \u2192 \u211d\u22650\u221e)", [{"full_name": "MeasureTheory.MeasureSpace.volume", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [663, 3], "def_end_pos": [663, 9]}]], "state_before": "case neg.right\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b9 : MeasurableSpace E\nm : MeasurableSpace \u03a9\n\u2119 : Measure \u03a9\n\u03bc : Measure E\nX : \u03a9 \u2192 \u211d\ns : Set \u211d\nhms : MeasurableSet s\nhns : \u2191\u2191volume s \u2260 0\ninst\u271d : IsFiniteMeasure \u2119\nhcs : IsCompact s\nhuX : IsUniform X s \u2119\nhsupp : \u00ac\u2191\u2191volume s = \u22a4\n\u22a2 \u222b\u207b (x : \u211d), \u2191\u2016x\u2016\u208a * Set.indicator s ((\u2191\u2191volume s)\u207b\u00b9 \u2022 1) x \u2260 \u22a4", "state_after": "case neg.right\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b9 : MeasurableSpace E\nm : MeasurableSpace \u03a9\n\u2119 : Measure \u03a9\n\u03bc : Measure E\nX : \u03a9 \u2192 \u211d\ns : Set \u211d\nhms : MeasurableSet s\nhns : \u2191\u2191volume s \u2260 0\ninst\u271d : IsFiniteMeasure \u2119\nhcs : IsCompact s\nhuX : IsUniform X s \u2119\nhsupp : \u00ac\u2191\u2191volume s = \u22a4\nind : \u211d \u2192 \u211d\u22650\u221e := (\u2191\u2191volume s)\u207b\u00b9 \u2022 1\n\u22a2 \u222b\u207b (x : \u211d), \u2191\u2016x\u2016\u208a * Set.indicator s ind x \u2260 \u22a4"}, {"tactic": "have : \u2200 x, \u2191\u2016x\u2016\u208a * s.indicator ind x = s.indicator (fun x => \u2016x\u2016\u208a * ind x) x := fun x =>\n  (s.indicator_mul_right (fun x => \u2191\u2016x\u2016\u208a) ind).symm", "annotated_tactic": ["have : \u2200 x, \u2191\u2016x\u2016\u208a * s.indicator ind x = s.indicator (fun x => \u2016x\u2016\u208a * ind x) x := fun x =>\n    (s.indicator_mul_right (fun x => \u2191\u2016x\u2016\u208a) ind).<a>symm</a>", [{"full_name": "Eq.symm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [310, 9], "def_end_pos": [310, 16]}]], "state_before": "case neg.right\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b9 : MeasurableSpace E\nm : MeasurableSpace \u03a9\n\u2119 : Measure \u03a9\n\u03bc : Measure E\nX : \u03a9 \u2192 \u211d\ns : Set \u211d\nhms : MeasurableSet s\nhns : \u2191\u2191volume s \u2260 0\ninst\u271d : IsFiniteMeasure \u2119\nhcs : IsCompact s\nhuX : IsUniform X s \u2119\nhsupp : \u00ac\u2191\u2191volume s = \u22a4\nind : \u211d \u2192 \u211d\u22650\u221e := (\u2191\u2191volume s)\u207b\u00b9 \u2022 1\n\u22a2 \u222b\u207b (x : \u211d), \u2191\u2016x\u2016\u208a * Set.indicator s ind x \u2260 \u22a4", "state_after": "case neg.right\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b9 : MeasurableSpace E\nm : MeasurableSpace \u03a9\n\u2119 : Measure \u03a9\n\u03bc : Measure E\nX : \u03a9 \u2192 \u211d\ns : Set \u211d\nhms : MeasurableSet s\nhns : \u2191\u2191volume s \u2260 0\ninst\u271d : IsFiniteMeasure \u2119\nhcs : IsCompact s\nhuX : IsUniform X s \u2119\nhsupp : \u00ac\u2191\u2191volume s = \u22a4\nind : \u211d \u2192 \u211d\u22650\u221e := (\u2191\u2191volume s)\u207b\u00b9 \u2022 1\nthis : \u2200 (x : \u211d), \u2191\u2016x\u2016\u208a * Set.indicator s ind x = Set.indicator s (fun x => \u2191\u2016x\u2016\u208a * ind x) x\n\u22a2 \u222b\u207b (x : \u211d), \u2191\u2016x\u2016\u208a * Set.indicator s ind x \u2260 \u22a4"}, {"tactic": "simp only [this, lintegral_indicator _ hms, mul_one, Algebra.id.smul_eq_mul, Pi.one_apply,\n  Pi.smul_apply]", "annotated_tactic": ["simp only [this, <a>lintegral_indicator</a> _ hms, <a>mul_one</a>, <a>Algebra.id.smul_eq_mul</a>, <a>Pi.one_apply</a>,\n    <a>Pi.smul_apply</a>]", [{"full_name": "MeasureTheory.lintegral_indicator", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [762, 9], "def_end_pos": [762, 28]}, {"full_name": "mul_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [470, 9], "def_end_pos": [470, 16]}, {"full_name": "Algebra.id.smul_eq_mul", "def_path": "Mathlib/Algebra/Algebra/Basic.lean", "def_pos": [453, 9], "def_end_pos": [453, 20]}, {"full_name": "Pi.one_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [47, 9], "def_end_pos": [47, 18]}, {"full_name": "Pi.smul_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [116, 60], "def_end_pos": [116, 70]}]], "state_before": "case neg.right\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b9 : MeasurableSpace E\nm : MeasurableSpace \u03a9\n\u2119 : Measure \u03a9\n\u03bc : Measure E\nX : \u03a9 \u2192 \u211d\ns : Set \u211d\nhms : MeasurableSet s\nhns : \u2191\u2191volume s \u2260 0\ninst\u271d : IsFiniteMeasure \u2119\nhcs : IsCompact s\nhuX : IsUniform X s \u2119\nhsupp : \u00ac\u2191\u2191volume s = \u22a4\nind : \u211d \u2192 \u211d\u22650\u221e := (\u2191\u2191volume s)\u207b\u00b9 \u2022 1\nthis : \u2200 (x : \u211d), \u2191\u2016x\u2016\u208a * Set.indicator s ind x = Set.indicator s (fun x => \u2191\u2016x\u2016\u208a * ind x) x\n\u22a2 \u222b\u207b (x : \u211d), \u2191\u2016x\u2016\u208a * Set.indicator s ind x \u2260 \u22a4", "state_after": "case neg.right\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b9 : MeasurableSpace E\nm : MeasurableSpace \u03a9\n\u2119 : Measure \u03a9\n\u03bc : Measure E\nX : \u03a9 \u2192 \u211d\ns : Set \u211d\nhms : MeasurableSet s\nhns : \u2191\u2191volume s \u2260 0\ninst\u271d : IsFiniteMeasure \u2119\nhcs : IsCompact s\nhuX : IsUniform X s \u2119\nhsupp : \u00ac\u2191\u2191volume s = \u22a4\nind : \u211d \u2192 \u211d\u22650\u221e := (\u2191\u2191volume s)\u207b\u00b9 \u2022 1\nthis : \u2200 (x : \u211d), \u2191\u2016x\u2016\u208a * Set.indicator s ind x = Set.indicator s (fun x => \u2191\u2016x\u2016\u208a * ind x) x\n\u22a2 \u222b\u207b (x : \u211d) in s, \u2191\u2016x\u2016\u208a * (\u2191\u2191volume s)\u207b\u00b9 \u2260 \u22a4"}, {"tactic": "rw [lintegral_mul_const _ measurable_nnnorm.coe_nnreal_ennreal]", "annotated_tactic": ["rw [<a>lintegral_mul_const</a> _ measurable_nnnorm.coe_nnreal_ennreal]", [{"full_name": "MeasureTheory.lintegral_mul_const", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [725, 9], "def_end_pos": [725, 28]}]], "state_before": "case neg.right\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b9 : MeasurableSpace E\nm : MeasurableSpace \u03a9\n\u2119 : Measure \u03a9\n\u03bc : Measure E\nX : \u03a9 \u2192 \u211d\ns : Set \u211d\nhms : MeasurableSet s\nhns : \u2191\u2191volume s \u2260 0\ninst\u271d : IsFiniteMeasure \u2119\nhcs : IsCompact s\nhuX : IsUniform X s \u2119\nhsupp : \u00ac\u2191\u2191volume s = \u22a4\nind : \u211d \u2192 \u211d\u22650\u221e := (\u2191\u2191volume s)\u207b\u00b9 \u2022 1\nthis : \u2200 (x : \u211d), \u2191\u2016x\u2016\u208a * Set.indicator s ind x = Set.indicator s (fun x => \u2191\u2016x\u2016\u208a * ind x) x\n\u22a2 \u222b\u207b (x : \u211d) in s, \u2191\u2016x\u2016\u208a * (\u2191\u2191volume s)\u207b\u00b9 \u2260 \u22a4", "state_after": "case neg.right\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b9 : MeasurableSpace E\nm : MeasurableSpace \u03a9\n\u2119 : Measure \u03a9\n\u03bc : Measure E\nX : \u03a9 \u2192 \u211d\ns : Set \u211d\nhms : MeasurableSet s\nhns : \u2191\u2191volume s \u2260 0\ninst\u271d : IsFiniteMeasure \u2119\nhcs : IsCompact s\nhuX : IsUniform X s \u2119\nhsupp : \u00ac\u2191\u2191volume s = \u22a4\nind : \u211d \u2192 \u211d\u22650\u221e := (\u2191\u2191volume s)\u207b\u00b9 \u2022 1\nthis : \u2200 (x : \u211d), \u2191\u2016x\u2016\u208a * Set.indicator s ind x = Set.indicator s (fun x => \u2191\u2016x\u2016\u208a * ind x) x\n\u22a2 (\u222b\u207b (a : \u211d) in s, \u2191\u2016a\u2016\u208a) * (\u2191\u2191volume s)\u207b\u00b9 \u2260 \u22a4"}, {"tactic": "refine' (ENNReal.mul_lt_top (set_lintegral_lt_top_of_isCompact hsupp hcs continuous_nnnorm).ne\n  (ENNReal.inv_lt_top.2 (pos_iff_ne_zero.mpr hns)).ne).ne", "annotated_tactic": ["refine' (<a>ENNReal.mul_lt_top</a> (<a>set_lintegral_lt_top_of_isCompact</a> hsupp hcs <a>continuous_nnnorm</a>).<a>ne</a>\n    (<a>ENNReal.inv_lt_top</a>.2 (pos_iff_ne_zero.mpr hns)).<a>ne</a>).<a>ne</a>", [{"full_name": "ENNReal.mul_lt_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [612, 9], "def_end_pos": [612, 19]}, {"full_name": "MeasureTheory.set_lintegral_lt_top_of_isCompact", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [1553, 9], "def_end_pos": [1553, 42]}, {"full_name": "continuous_nnnorm", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [1151, 36], "def_end_pos": [1151, 53]}, {"full_name": "LT.lt.ne", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [152, 7], "def_end_pos": [152, 15]}, {"full_name": "ENNReal.inv_lt_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1456, 9], "def_end_pos": [1456, 19]}, {"full_name": "LT.lt.ne", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [152, 7], "def_end_pos": [152, 15]}, {"full_name": "LT.lt.ne", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [152, 7], "def_end_pos": [152, 15]}]], "state_before": "case neg.right\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b9 : MeasurableSpace E\nm : MeasurableSpace \u03a9\n\u2119 : Measure \u03a9\n\u03bc : Measure E\nX : \u03a9 \u2192 \u211d\ns : Set \u211d\nhms : MeasurableSet s\nhns : \u2191\u2191volume s \u2260 0\ninst\u271d : IsFiniteMeasure \u2119\nhcs : IsCompact s\nhuX : IsUniform X s \u2119\nhsupp : \u00ac\u2191\u2191volume s = \u22a4\nind : \u211d \u2192 \u211d\u22650\u221e := (\u2191\u2191volume s)\u207b\u00b9 \u2022 1\nthis : \u2200 (x : \u211d), \u2191\u2016x\u2016\u208a * Set.indicator s ind x = Set.indicator s (fun x => \u2191\u2016x\u2016\u208a * ind x) x\n\u22a2 (\u222b\u207b (a : \u211d) in s, \u2191\u2016a\u2016\u208a) * (\u2191\u2191volume s)\u207b\u00b9 \u2260 \u22a4", "state_after": "no goals"}, {"tactic": "have : pdf X \u2119 =\u1d50[volume] 0 := by\n  refine' ae_eq_trans huX _\n  simp [hsupp, ae_eq_refl]", "annotated_tactic": ["have : <a>pdf</a> X \u2119 =\u1d50[<a>volume</a>] 0 := by\n      refine' <a>ae_eq_trans</a> huX _\n      simp [hsupp, <a>ae_eq_refl</a>]", [{"full_name": "MeasureTheory.pdf", "def_path": "Mathlib/Probability/Density.lean", "def_pos": [82, 5], "def_end_pos": [82, 8]}, {"full_name": "MeasureTheory.MeasureSpace.volume", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [663, 3], "def_end_pos": [663, 9]}, {"full_name": "MeasureTheory.ae_eq_trans", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [444, 9], "def_end_pos": [444, 20]}, {"full_name": "MeasureTheory.ae_eq_refl", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [436, 9], "def_end_pos": [436, 19]}]], "state_before": "case pos\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b9 : MeasurableSpace E\nm : MeasurableSpace \u03a9\n\u2119 : Measure \u03a9\n\u03bc : Measure E\nX : \u03a9 \u2192 \u211d\ns : Set \u211d\nhms : MeasurableSet s\nhns : \u2191\u2191volume s \u2260 0\ninst\u271d : IsFiniteMeasure \u2119\nhcs : IsCompact s\nhuX : IsUniform X s \u2119\nhsupp : \u2191\u2191volume s = \u22a4\n\u22a2 Integrable fun x => x * ENNReal.toReal (pdf X \u2119 x)", "state_after": "case pos\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b9 : MeasurableSpace E\nm : MeasurableSpace \u03a9\n\u2119 : Measure \u03a9\n\u03bc : Measure E\nX : \u03a9 \u2192 \u211d\ns : Set \u211d\nhms : MeasurableSet s\nhns : \u2191\u2191volume s \u2260 0\ninst\u271d : IsFiniteMeasure \u2119\nhcs : IsCompact s\nhuX : IsUniform X s \u2119\nhsupp : \u2191\u2191volume s = \u22a4\nthis : pdf X \u2119 =\u1da0[ae volume] 0\n\u22a2 Integrable fun x => x * ENNReal.toReal (pdf X \u2119 x)"}, {"tactic": "refine' Integrable.congr (integrable_zero _ _ _) _", "annotated_tactic": ["refine' <a>Integrable.congr</a> (<a>integrable_zero</a> _ _ _) _", [{"full_name": "MeasureTheory.Integrable.congr", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [492, 9], "def_end_pos": [492, 25]}, {"full_name": "MeasureTheory.integrable_zero", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [662, 9], "def_end_pos": [662, 24]}]], "state_before": "case pos\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b9 : MeasurableSpace E\nm : MeasurableSpace \u03a9\n\u2119 : Measure \u03a9\n\u03bc : Measure E\nX : \u03a9 \u2192 \u211d\ns : Set \u211d\nhms : MeasurableSet s\nhns : \u2191\u2191volume s \u2260 0\ninst\u271d : IsFiniteMeasure \u2119\nhcs : IsCompact s\nhuX : IsUniform X s \u2119\nhsupp : \u2191\u2191volume s = \u22a4\nthis : pdf X \u2119 =\u1da0[ae volume] 0\n\u22a2 Integrable fun x => x * ENNReal.toReal (pdf X \u2119 x)", "state_after": "case pos\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b9 : MeasurableSpace E\nm : MeasurableSpace \u03a9\n\u2119 : Measure \u03a9\n\u03bc : Measure E\nX : \u03a9 \u2192 \u211d\ns : Set \u211d\nhms : MeasurableSet s\nhns : \u2191\u2191volume s \u2260 0\ninst\u271d : IsFiniteMeasure \u2119\nhcs : IsCompact s\nhuX : IsUniform X s \u2119\nhsupp : \u2191\u2191volume s = \u22a4\nthis : pdf X \u2119 =\u1da0[ae volume] 0\n\u22a2 (fun x => 0) =\u1da0[ae volume] fun x => x * ENNReal.toReal (pdf X \u2119 x)"}, {"tactic": "rw [(by simp : (fun x => 0 : \u211d \u2192 \u211d) = fun x => x * (0 : \u211d\u22650\u221e).toReal)]", "annotated_tactic": ["rw [(by simp : (fun x => 0 : \u211d \u2192 \u211d) = fun x => x * (0 : \u211d\u22650\u221e).toReal)]", []], "state_before": "case pos\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b9 : MeasurableSpace E\nm : MeasurableSpace \u03a9\n\u2119 : Measure \u03a9\n\u03bc : Measure E\nX : \u03a9 \u2192 \u211d\ns : Set \u211d\nhms : MeasurableSet s\nhns : \u2191\u2191volume s \u2260 0\ninst\u271d : IsFiniteMeasure \u2119\nhcs : IsCompact s\nhuX : IsUniform X s \u2119\nhsupp : \u2191\u2191volume s = \u22a4\nthis : pdf X \u2119 =\u1da0[ae volume] 0\n\u22a2 (fun x => 0) =\u1da0[ae volume] fun x => x * ENNReal.toReal (pdf X \u2119 x)", "state_after": "case pos\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b9 : MeasurableSpace E\nm : MeasurableSpace \u03a9\n\u2119 : Measure \u03a9\n\u03bc : Measure E\nX : \u03a9 \u2192 \u211d\ns : Set \u211d\nhms : MeasurableSet s\nhns : \u2191\u2191volume s \u2260 0\ninst\u271d : IsFiniteMeasure \u2119\nhcs : IsCompact s\nhuX : IsUniform X s \u2119\nhsupp : \u2191\u2191volume s = \u22a4\nthis : pdf X \u2119 =\u1da0[ae volume] 0\n\u22a2 (fun x => x * ENNReal.toReal 0) =\u1da0[ae volume] fun x => x * ENNReal.toReal (pdf X \u2119 x)"}, {"tactic": "refine'\n  Filter.EventuallyEq.mul (ae_eq_refl _) (Filter.EventuallyEq.fun_comp this.symm ENNReal.toReal)", "annotated_tactic": ["refine'\n      <a>Filter.EventuallyEq.mul</a> (<a>ae_eq_refl</a> _) (<a>Filter.EventuallyEq.fun_comp</a> this.symm <a>ENNReal.toReal</a>)", [{"full_name": "Filter.EventuallyEq.mul", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1531, 9], "def_end_pos": [1531, 25]}, {"full_name": "MeasureTheory.ae_eq_refl", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [436, 9], "def_end_pos": [436, 19]}, {"full_name": "Filter.EventuallyEq.fun_comp", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1520, 9], "def_end_pos": [1520, 30]}, {"full_name": "ENNReal.toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [168, 15], "def_end_pos": [168, 21]}]], "state_before": "case pos\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b9 : MeasurableSpace E\nm : MeasurableSpace \u03a9\n\u2119 : Measure \u03a9\n\u03bc : Measure E\nX : \u03a9 \u2192 \u211d\ns : Set \u211d\nhms : MeasurableSet s\nhns : \u2191\u2191volume s \u2260 0\ninst\u271d : IsFiniteMeasure \u2119\nhcs : IsCompact s\nhuX : IsUniform X s \u2119\nhsupp : \u2191\u2191volume s = \u22a4\nthis : pdf X \u2119 =\u1da0[ae volume] 0\n\u22a2 (fun x => x * ENNReal.toReal 0) =\u1da0[ae volume] fun x => x * ENNReal.toReal (pdf X \u2119 x)", "state_after": "no goals"}, {"tactic": "refine' ae_eq_trans huX _", "annotated_tactic": ["refine' <a>ae_eq_trans</a> huX _", [{"full_name": "MeasureTheory.ae_eq_trans", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [444, 9], "def_end_pos": [444, 20]}]], "state_before": "\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b9 : MeasurableSpace E\nm : MeasurableSpace \u03a9\n\u2119 : Measure \u03a9\n\u03bc : Measure E\nX : \u03a9 \u2192 \u211d\ns : Set \u211d\nhms : MeasurableSet s\nhns : \u2191\u2191volume s \u2260 0\ninst\u271d : IsFiniteMeasure \u2119\nhcs : IsCompact s\nhuX : IsUniform X s \u2119\nhsupp : \u2191\u2191volume s = \u22a4\n\u22a2 pdf X \u2119 =\u1da0[ae volume] 0", "state_after": "\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b9 : MeasurableSpace E\nm : MeasurableSpace \u03a9\n\u2119 : Measure \u03a9\n\u03bc : Measure E\nX : \u03a9 \u2192 \u211d\ns : Set \u211d\nhms : MeasurableSet s\nhns : \u2191\u2191volume s \u2260 0\ninst\u271d : IsFiniteMeasure \u2119\nhcs : IsCompact s\nhuX : IsUniform X s \u2119\nhsupp : \u2191\u2191volume s = \u22a4\n\u22a2 Set.indicator s ((\u2191\u2191volume s)\u207b\u00b9 \u2022 1) =\u1da0[ae volume] 0"}, {"tactic": "simp [hsupp, ae_eq_refl]", "annotated_tactic": ["simp [hsupp, <a>ae_eq_refl</a>]", [{"full_name": "MeasureTheory.ae_eq_refl", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [436, 9], "def_end_pos": [436, 19]}]], "state_before": "\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b9 : MeasurableSpace E\nm : MeasurableSpace \u03a9\n\u2119 : Measure \u03a9\n\u03bc : Measure E\nX : \u03a9 \u2192 \u211d\ns : Set \u211d\nhms : MeasurableSet s\nhns : \u2191\u2191volume s \u2260 0\ninst\u271d : IsFiniteMeasure \u2119\nhcs : IsCompact s\nhuX : IsUniform X s \u2119\nhsupp : \u2191\u2191volume s = \u22a4\n\u22a2 Set.indicator s ((\u2191\u2191volume s)\u207b\u00b9 \u2022 1) =\u1da0[ae volume] 0", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b9 : MeasurableSpace E\nm : MeasurableSpace \u03a9\n\u2119 : Measure \u03a9\n\u03bc : Measure E\nX : \u03a9 \u2192 \u211d\ns : Set \u211d\nhms : MeasurableSet s\nhns : \u2191\u2191volume s \u2260 0\ninst\u271d : IsFiniteMeasure \u2119\nhcs : IsCompact s\nhuX : IsUniform X s \u2119\nhsupp : \u2191\u2191volume s = \u22a4\nthis : pdf X \u2119 =\u1da0[ae volume] 0\n\u22a2 (fun x => 0) = fun x => x * ENNReal.toReal 0", "state_after": "no goals"}, {"tactic": "exact aestronglyMeasurable_id.mul\n  (measurable_pdf X \u2119).aemeasurable.ennreal_toReal.aestronglyMeasurable", "annotated_tactic": ["exact aestronglyMeasurable_id.mul\n      (<a>measurable_pdf</a> X \u2119).aemeasurable.ennreal_toReal.aestronglyMeasurable", [{"full_name": "MeasureTheory.measurable_pdf", "def_path": "Mathlib/Probability/Density.lean", "def_pos": [109, 9], "def_end_pos": [109, 23]}]], "state_before": "case neg.left\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b9 : MeasurableSpace E\nm : MeasurableSpace \u03a9\n\u2119 : Measure \u03a9\n\u03bc : Measure E\nX : \u03a9 \u2192 \u211d\ns : Set \u211d\nhms : MeasurableSet s\nhns : \u2191\u2191volume s \u2260 0\ninst\u271d : IsFiniteMeasure \u2119\nhcs : IsCompact s\nhuX : IsUniform X s \u2119\nhsupp : \u00ac\u2191\u2191volume s = \u22a4\n\u22a2 AEStronglyMeasurable (fun x => x * ENNReal.toReal (pdf X \u2119 x)) volume", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "full_name": "MeasurableEmbedding.comap_preimage", "start": [4144, 11], "end": [4147, 69], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "full_name": "List.findIdx_get?_eq_get_of_exists", "start": [1440, 1], "end": [1442, 46], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "full_name": "List.drop_subset", "start": [1904, 1], "end": [1905, 28], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "full_name": "Filter.Tendsto.eventually_intervalIntegrable", "start": [428, 1], "end": [432, 75], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Sum.lean", "full_name": "Finset.disjSum_mono", "start": [82, 1], "end": [83, 76], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Group/Arithmetic.lean", "full_name": "measurable_const_smul_iff\u2080", "start": [756, 1], "end": [758, 46], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "full_name": "MeasureTheory.OuterMeasure.iUnion_nat_of_monotone_of_tsum_ne_top", "start": [193, 1], "end": [211, 54], "traced_tactics": [{"tactic": "refine' m.iUnion_of_tendsto_zero atTop _", "annotated_tactic": ["refine' m.iUnion_of_tendsto_zero <a>atTop</a> _", [{"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm\u271d m : OuterMeasure \u03b1\ns : \u2115 \u2192 Set \u03b1\nh_mono : \u2200 (n : \u2115), s n \u2286 s (n + 1)\nh0 : \u2211' (k : \u2115), \u2191m (s (k + 1) \\ s k) \u2260 \u22a4\ninst\u271d : (i : \u2115) \u2192 DecidablePred fun x => x \u2208 s i\n\u22a2 \u2191m (\u22c3 n, s n) = \u2a06 n, \u2191m (s n)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm\u271d m : OuterMeasure \u03b1\ns : \u2115 \u2192 Set \u03b1\nh_mono : \u2200 (n : \u2115), s n \u2286 s (n + 1)\nh0 : \u2211' (k : \u2115), \u2191m (s (k + 1) \\ s k) \u2260 \u22a4\ninst\u271d : (i : \u2115) \u2192 DecidablePred fun x => x \u2208 s i\n\u22a2 Tendsto (fun k => \u2191m ((\u22c3 n, s n) \\ s k)) atTop (\ud835\udcdd 0)"}, {"tactic": "refine' tendsto_nhds_bot_mono' (ENNReal.tendsto_sum_nat_add _ h0) fun n => _", "annotated_tactic": ["refine' <a>tendsto_nhds_bot_mono'</a> (<a>ENNReal.tendsto_sum_nat_add</a> _ h0) fun n => _", [{"full_name": "tendsto_nhds_bot_mono'", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [1164, 9], "def_end_pos": [1164, 31]}, {"full_name": "ENNReal.tendsto_sum_nat_add", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [1255, 9], "def_end_pos": [1255, 28]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm\u271d m : OuterMeasure \u03b1\ns : \u2115 \u2192 Set \u03b1\nh_mono : \u2200 (n : \u2115), s n \u2286 s (n + 1)\nh0 : \u2211' (k : \u2115), \u2191m (s (k + 1) \\ s k) \u2260 \u22a4\ninst\u271d : (i : \u2115) \u2192 DecidablePred fun x => x \u2208 s i\n\u22a2 Tendsto (fun k => \u2191m ((\u22c3 n, s n) \\ s k)) atTop (\ud835\udcdd 0)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm\u271d m : OuterMeasure \u03b1\ns : \u2115 \u2192 Set \u03b1\nh_mono : \u2200 (n : \u2115), s n \u2286 s (n + 1)\nh0 : \u2211' (k : \u2115), \u2191m (s (k + 1) \\ s k) \u2260 \u22a4\ninst\u271d : (i : \u2115) \u2192 DecidablePred fun x => x \u2208 s i\nn : \u2115\n\u22a2 \u2191m ((\u22c3 n, s n) \\ s n) \u2264 \u2211' (k : \u2115), \u2191m (s (k + n + 1) \\ s (k + n))"}, {"tactic": "refine' (m.mono _).trans (m.iUnion _)", "annotated_tactic": ["refine' (m.mono _).<a>trans</a> (m.iUnion _)", [{"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [120, 7], "def_end_pos": [120, 18]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm\u271d m : OuterMeasure \u03b1\ns : \u2115 \u2192 Set \u03b1\nh_mono : \u2200 (n : \u2115), s n \u2286 s (n + 1)\nh0 : \u2211' (k : \u2115), \u2191m (s (k + 1) \\ s k) \u2260 \u22a4\ninst\u271d : (i : \u2115) \u2192 DecidablePred fun x => x \u2208 s i\nn : \u2115\n\u22a2 \u2191m ((\u22c3 n, s n) \\ s n) \u2264 \u2211' (k : \u2115), \u2191m (s (k + n + 1) \\ s (k + n))", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm\u271d m : OuterMeasure \u03b1\ns : \u2115 \u2192 Set \u03b1\nh_mono : \u2200 (n : \u2115), s n \u2286 s (n + 1)\nh0 : \u2211' (k : \u2115), \u2191m (s (k + 1) \\ s k) \u2260 \u22a4\ninst\u271d : (i : \u2115) \u2192 DecidablePred fun x => x \u2208 s i\nn : \u2115\n\u22a2 (\u22c3 n, s n) \\ s n \u2286 \u22c3 i, s (i + n + 1) \\ s (i + n)"}, {"tactic": "have h' : Monotone s := @monotone_nat_of_le_succ (Set \u03b1) _ _ h_mono", "annotated_tactic": ["have h' : <a>Monotone</a> s := @<a>monotone_nat_of_le_succ</a> (<a>Set</a> \u03b1) _ _ h_mono", [{"full_name": "Monotone", "def_path": "Mathlib/Order/Monotone/Basic.lean", "def_pos": [77, 5], "def_end_pos": [77, 13]}, {"full_name": "monotone_nat_of_le_succ", "def_path": "Mathlib/Order/Monotone/Basic.lean", "def_pos": [1025, 9], "def_end_pos": [1025, 32]}, {"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm\u271d m : OuterMeasure \u03b1\ns : \u2115 \u2192 Set \u03b1\nh_mono : \u2200 (n : \u2115), s n \u2286 s (n + 1)\nh0 : \u2211' (k : \u2115), \u2191m (s (k + 1) \\ s k) \u2260 \u22a4\ninst\u271d : (i : \u2115) \u2192 DecidablePred fun x => x \u2208 s i\nn : \u2115\n\u22a2 (\u22c3 n, s n) \\ s n \u2286 \u22c3 i, s (i + n + 1) \\ s (i + n)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm\u271d m : OuterMeasure \u03b1\ns : \u2115 \u2192 Set \u03b1\nh_mono : \u2200 (n : \u2115), s n \u2286 s (n + 1)\nh0 : \u2211' (k : \u2115), \u2191m (s (k + 1) \\ s k) \u2260 \u22a4\ninst\u271d : (i : \u2115) \u2192 DecidablePred fun x => x \u2208 s i\nn : \u2115\nh' : Monotone s\n\u22a2 (\u22c3 n, s n) \\ s n \u2286 \u22c3 i, s (i + n + 1) \\ s (i + n)"}, {"tactic": "simp only [diff_subset_iff, iUnion_subset_iff]", "annotated_tactic": ["simp only [<a>diff_subset_iff</a>, <a>iUnion_subset_iff</a>]", [{"full_name": "Set.diff_subset_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1948, 9], "def_end_pos": [1948, 24]}, {"full_name": "Set.iUnion_subset_iff", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [411, 9], "def_end_pos": [411, 26]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm\u271d m : OuterMeasure \u03b1\ns : \u2115 \u2192 Set \u03b1\nh_mono : \u2200 (n : \u2115), s n \u2286 s (n + 1)\nh0 : \u2211' (k : \u2115), \u2191m (s (k + 1) \\ s k) \u2260 \u22a4\ninst\u271d : (i : \u2115) \u2192 DecidablePred fun x => x \u2208 s i\nn : \u2115\nh' : Monotone s\n\u22a2 (\u22c3 n, s n) \\ s n \u2286 \u22c3 i, s (i + n + 1) \\ s (i + n)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm\u271d m : OuterMeasure \u03b1\ns : \u2115 \u2192 Set \u03b1\nh_mono : \u2200 (n : \u2115), s n \u2286 s (n + 1)\nh0 : \u2211' (k : \u2115), \u2191m (s (k + 1) \\ s k) \u2260 \u22a4\ninst\u271d : (i : \u2115) \u2192 DecidablePred fun x => x \u2208 s i\nn : \u2115\nh' : Monotone s\n\u22a2 \u2200 (i : \u2115), s i \u2286 s n \u222a \u22c3 i, s (i + n + 1) \\ s (i + n)"}, {"tactic": "intro i x hx", "annotated_tactic": ["intro i x hx", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm\u271d m : OuterMeasure \u03b1\ns : \u2115 \u2192 Set \u03b1\nh_mono : \u2200 (n : \u2115), s n \u2286 s (n + 1)\nh0 : \u2211' (k : \u2115), \u2191m (s (k + 1) \\ s k) \u2260 \u22a4\ninst\u271d : (i : \u2115) \u2192 DecidablePred fun x => x \u2208 s i\nn : \u2115\nh' : Monotone s\n\u22a2 \u2200 (i : \u2115), s i \u2286 s n \u222a \u22c3 i, s (i + n + 1) \\ s (i + n)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm\u271d m : OuterMeasure \u03b1\ns : \u2115 \u2192 Set \u03b1\nh_mono : \u2200 (n : \u2115), s n \u2286 s (n + 1)\nh0 : \u2211' (k : \u2115), \u2191m (s (k + 1) \\ s k) \u2260 \u22a4\ninst\u271d : (i : \u2115) \u2192 DecidablePred fun x => x \u2208 s i\nn : \u2115\nh' : Monotone s\ni : \u2115\nx : \u03b1\nhx : x \u2208 s i\n\u22a2 x \u2208 s n \u222a \u22c3 i, s (i + n + 1) \\ s (i + n)"}, {"tactic": "have : \u2203i, x \u2208 s i := by exists i", "annotated_tactic": ["have : \u2203i, x \u2208 s i := by exists i", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm\u271d m : OuterMeasure \u03b1\ns : \u2115 \u2192 Set \u03b1\nh_mono : \u2200 (n : \u2115), s n \u2286 s (n + 1)\nh0 : \u2211' (k : \u2115), \u2191m (s (k + 1) \\ s k) \u2260 \u22a4\ninst\u271d : (i : \u2115) \u2192 DecidablePred fun x => x \u2208 s i\nn : \u2115\nh' : Monotone s\ni : \u2115\nx : \u03b1\nhx : x \u2208 s i\n\u22a2 x \u2208 s n \u222a \u22c3 i, s (i + n + 1) \\ s (i + n)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm\u271d m : OuterMeasure \u03b1\ns : \u2115 \u2192 Set \u03b1\nh_mono : \u2200 (n : \u2115), s n \u2286 s (n + 1)\nh0 : \u2211' (k : \u2115), \u2191m (s (k + 1) \\ s k) \u2260 \u22a4\ninst\u271d : (i : \u2115) \u2192 DecidablePred fun x => x \u2208 s i\nn : \u2115\nh' : Monotone s\ni : \u2115\nx : \u03b1\nhx : x \u2208 s i\nthis : \u2203 i, x \u2208 s i\n\u22a2 x \u2208 s n \u222a \u22c3 i, s (i + n + 1) \\ s (i + n)"}, {"tactic": "rcases Nat.findX this with \u27e8j, hj, hlt\u27e9", "annotated_tactic": ["rcases <a>Nat.findX</a> this with \u27e8j, hj, hlt\u27e9", [{"full_name": "Nat.findX", "def_path": "Mathlib/Init/Data/Nat/Lemmas.lean", "def_pos": [691, 15], "def_end_pos": [691, 20]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm\u271d m : OuterMeasure \u03b1\ns : \u2115 \u2192 Set \u03b1\nh_mono : \u2200 (n : \u2115), s n \u2286 s (n + 1)\nh0 : \u2211' (k : \u2115), \u2191m (s (k + 1) \\ s k) \u2260 \u22a4\ninst\u271d : (i : \u2115) \u2192 DecidablePred fun x => x \u2208 s i\nn : \u2115\nh' : Monotone s\ni : \u2115\nx : \u03b1\nhx : x \u2208 s i\nthis : \u2203 i, x \u2208 s i\n\u22a2 x \u2208 s n \u222a \u22c3 i, s (i + n + 1) \\ s (i + n)", "state_after": "case mk.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm\u271d m : OuterMeasure \u03b1\ns : \u2115 \u2192 Set \u03b1\nh_mono : \u2200 (n : \u2115), s n \u2286 s (n + 1)\nh0 : \u2211' (k : \u2115), \u2191m (s (k + 1) \\ s k) \u2260 \u22a4\ninst\u271d : (i : \u2115) \u2192 DecidablePred fun x => x \u2208 s i\nn : \u2115\nh' : Monotone s\ni : \u2115\nx : \u03b1\nhx : x \u2208 s i\nthis : \u2203 i, x \u2208 s i\nj : \u2115\nhj : x \u2208 s j\nhlt : \u2200 (m : \u2115), m < j \u2192 \u00acx \u2208 s m\n\u22a2 x \u2208 s n \u222a \u22c3 i, s (i + n + 1) \\ s (i + n)"}, {"tactic": "clear hx i", "annotated_tactic": ["clear hx i", []], "state_before": "case mk.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm\u271d m : OuterMeasure \u03b1\ns : \u2115 \u2192 Set \u03b1\nh_mono : \u2200 (n : \u2115), s n \u2286 s (n + 1)\nh0 : \u2211' (k : \u2115), \u2191m (s (k + 1) \\ s k) \u2260 \u22a4\ninst\u271d : (i : \u2115) \u2192 DecidablePred fun x => x \u2208 s i\nn : \u2115\nh' : Monotone s\ni : \u2115\nx : \u03b1\nhx : x \u2208 s i\nthis : \u2203 i, x \u2208 s i\nj : \u2115\nhj : x \u2208 s j\nhlt : \u2200 (m : \u2115), m < j \u2192 \u00acx \u2208 s m\n\u22a2 x \u2208 s n \u222a \u22c3 i, s (i + n + 1) \\ s (i + n)", "state_after": "case mk.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm\u271d m : OuterMeasure \u03b1\ns : \u2115 \u2192 Set \u03b1\nh_mono : \u2200 (n : \u2115), s n \u2286 s (n + 1)\nh0 : \u2211' (k : \u2115), \u2191m (s (k + 1) \\ s k) \u2260 \u22a4\ninst\u271d : (i : \u2115) \u2192 DecidablePred fun x => x \u2208 s i\nn : \u2115\nh' : Monotone s\nx : \u03b1\nthis : \u2203 i, x \u2208 s i\nj : \u2115\nhj : x \u2208 s j\nhlt : \u2200 (m : \u2115), m < j \u2192 \u00acx \u2208 s m\n\u22a2 x \u2208 s n \u222a \u22c3 i, s (i + n + 1) \\ s (i + n)"}, {"tactic": "cases' le_or_lt j n with hjn hnj", "annotated_tactic": ["cases' <a>le_or_lt</a> j n with hjn hnj", [{"full_name": "le_or_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [340, 9], "def_end_pos": [340, 17]}]], "state_before": "case mk.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm\u271d m : OuterMeasure \u03b1\ns : \u2115 \u2192 Set \u03b1\nh_mono : \u2200 (n : \u2115), s n \u2286 s (n + 1)\nh0 : \u2211' (k : \u2115), \u2191m (s (k + 1) \\ s k) \u2260 \u22a4\ninst\u271d : (i : \u2115) \u2192 DecidablePred fun x => x \u2208 s i\nn : \u2115\nh' : Monotone s\nx : \u03b1\nthis : \u2203 i, x \u2208 s i\nj : \u2115\nhj : x \u2208 s j\nhlt : \u2200 (m : \u2115), m < j \u2192 \u00acx \u2208 s m\n\u22a2 x \u2208 s n \u222a \u22c3 i, s (i + n + 1) \\ s (i + n)", "state_after": "case mk.intro.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm\u271d m : OuterMeasure \u03b1\ns : \u2115 \u2192 Set \u03b1\nh_mono : \u2200 (n : \u2115), s n \u2286 s (n + 1)\nh0 : \u2211' (k : \u2115), \u2191m (s (k + 1) \\ s k) \u2260 \u22a4\ninst\u271d : (i : \u2115) \u2192 DecidablePred fun x => x \u2208 s i\nn : \u2115\nh' : Monotone s\nx : \u03b1\nthis : \u2203 i, x \u2208 s i\nj : \u2115\nhj : x \u2208 s j\nhlt : \u2200 (m : \u2115), m < j \u2192 \u00acx \u2208 s m\nhjn : j \u2264 n\n\u22a2 x \u2208 s n \u222a \u22c3 i, s (i + n + 1) \\ s (i + n)\n\ncase mk.intro.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm\u271d m : OuterMeasure \u03b1\ns : \u2115 \u2192 Set \u03b1\nh_mono : \u2200 (n : \u2115), s n \u2286 s (n + 1)\nh0 : \u2211' (k : \u2115), \u2191m (s (k + 1) \\ s k) \u2260 \u22a4\ninst\u271d : (i : \u2115) \u2192 DecidablePred fun x => x \u2208 s i\nn : \u2115\nh' : Monotone s\nx : \u03b1\nthis : \u2203 i, x \u2208 s i\nj : \u2115\nhj : x \u2208 s j\nhlt : \u2200 (m : \u2115), m < j \u2192 \u00acx \u2208 s m\nhnj : n < j\n\u22a2 x \u2208 s n \u222a \u22c3 i, s (i + n + 1) \\ s (i + n)"}, {"tactic": "have : j - (n + 1) + n + 1 = j := by rw [add_assoc, tsub_add_cancel_of_le hnj.nat_succ_le]", "annotated_tactic": ["have : j - (n + 1) + n + 1 = j := by rw [<a>add_assoc</a>, <a>tsub_add_cancel_of_le</a> hnj.nat_succ_le]", [{"full_name": "add_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [263, 3], "def_end_pos": [263, 14]}, {"full_name": "tsub_add_cancel_of_le", "def_path": "Mathlib/Algebra/Order/Sub/Canonical.lean", "def_pos": [30, 9], "def_end_pos": [30, 30]}]], "state_before": "case mk.intro.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm\u271d m : OuterMeasure \u03b1\ns : \u2115 \u2192 Set \u03b1\nh_mono : \u2200 (n : \u2115), s n \u2286 s (n + 1)\nh0 : \u2211' (k : \u2115), \u2191m (s (k + 1) \\ s k) \u2260 \u22a4\ninst\u271d : (i : \u2115) \u2192 DecidablePred fun x => x \u2208 s i\nn : \u2115\nh' : Monotone s\nx : \u03b1\nthis : \u2203 i, x \u2208 s i\nj : \u2115\nhj : x \u2208 s j\nhlt : \u2200 (m : \u2115), m < j \u2192 \u00acx \u2208 s m\nhnj : n < j\n\u22a2 x \u2208 s n \u222a \u22c3 i, s (i + n + 1) \\ s (i + n)", "state_after": "case mk.intro.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm\u271d m : OuterMeasure \u03b1\ns : \u2115 \u2192 Set \u03b1\nh_mono : \u2200 (n : \u2115), s n \u2286 s (n + 1)\nh0 : \u2211' (k : \u2115), \u2191m (s (k + 1) \\ s k) \u2260 \u22a4\ninst\u271d : (i : \u2115) \u2192 DecidablePred fun x => x \u2208 s i\nn : \u2115\nh' : Monotone s\nx : \u03b1\nthis\u271d : \u2203 i, x \u2208 s i\nj : \u2115\nhj : x \u2208 s j\nhlt : \u2200 (m : \u2115), m < j \u2192 \u00acx \u2208 s m\nhnj : n < j\nthis : j - (n + 1) + n + 1 = j\n\u22a2 x \u2208 s n \u222a \u22c3 i, s (i + n + 1) \\ s (i + n)"}, {"tactic": "refine' Or.inr (mem_iUnion.2 \u27e8j - (n + 1), _, hlt _ _\u27e9)", "annotated_tactic": ["refine' <a>Or.inr</a> (<a>mem_iUnion</a>.2 \u27e8j - (n + 1), _, hlt _ _\u27e9)", [{"full_name": "Or.inr", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [519, 5], "def_end_pos": [519, 8]}, {"full_name": "Set.mem_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [201, 9], "def_end_pos": [201, 19]}]], "state_before": "case mk.intro.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm\u271d m : OuterMeasure \u03b1\ns : \u2115 \u2192 Set \u03b1\nh_mono : \u2200 (n : \u2115), s n \u2286 s (n + 1)\nh0 : \u2211' (k : \u2115), \u2191m (s (k + 1) \\ s k) \u2260 \u22a4\ninst\u271d : (i : \u2115) \u2192 DecidablePred fun x => x \u2208 s i\nn : \u2115\nh' : Monotone s\nx : \u03b1\nthis\u271d : \u2203 i, x \u2208 s i\nj : \u2115\nhj : x \u2208 s j\nhlt : \u2200 (m : \u2115), m < j \u2192 \u00acx \u2208 s m\nhnj : n < j\nthis : j - (n + 1) + n + 1 = j\n\u22a2 x \u2208 s n \u222a \u22c3 i, s (i + n + 1) \\ s (i + n)", "state_after": "case mk.intro.inr.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm\u271d m : OuterMeasure \u03b1\ns : \u2115 \u2192 Set \u03b1\nh_mono : \u2200 (n : \u2115), s n \u2286 s (n + 1)\nh0 : \u2211' (k : \u2115), \u2191m (s (k + 1) \\ s k) \u2260 \u22a4\ninst\u271d : (i : \u2115) \u2192 DecidablePred fun x => x \u2208 s i\nn : \u2115\nh' : Monotone s\nx : \u03b1\nthis\u271d : \u2203 i, x \u2208 s i\nj : \u2115\nhj : x \u2208 s j\nhlt : \u2200 (m : \u2115), m < j \u2192 \u00acx \u2208 s m\nhnj : n < j\nthis : j - (n + 1) + n + 1 = j\n\u22a2 x \u2208 s (j - (n + 1) + n + 1)\n\ncase mk.intro.inr.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm\u271d m : OuterMeasure \u03b1\ns : \u2115 \u2192 Set \u03b1\nh_mono : \u2200 (n : \u2115), s n \u2286 s (n + 1)\nh0 : \u2211' (k : \u2115), \u2191m (s (k + 1) \\ s k) \u2260 \u22a4\ninst\u271d : (i : \u2115) \u2192 DecidablePred fun x => x \u2208 s i\nn : \u2115\nh' : Monotone s\nx : \u03b1\nthis\u271d : \u2203 i, x \u2208 s i\nj : \u2115\nhj : x \u2208 s j\nhlt : \u2200 (m : \u2115), m < j \u2192 \u00acx \u2208 s m\nhnj : n < j\nthis : j - (n + 1) + n + 1 = j\n\u22a2 j - (n + 1) + n < j"}, {"tactic": "exists i", "annotated_tactic": ["exists i", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm\u271d m : OuterMeasure \u03b1\ns : \u2115 \u2192 Set \u03b1\nh_mono : \u2200 (n : \u2115), s n \u2286 s (n + 1)\nh0 : \u2211' (k : \u2115), \u2191m (s (k + 1) \\ s k) \u2260 \u22a4\ninst\u271d : (i : \u2115) \u2192 DecidablePred fun x => x \u2208 s i\nn : \u2115\nh' : Monotone s\ni : \u2115\nx : \u03b1\nhx : x \u2208 s i\n\u22a2 \u2203 i, x \u2208 s i", "state_after": "no goals"}, {"tactic": "exact Or.inl (h' hjn hj)", "annotated_tactic": ["exact <a>Or.inl</a> (h' hjn hj)", [{"full_name": "Or.inl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [517, 5], "def_end_pos": [517, 8]}]], "state_before": "case mk.intro.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm\u271d m : OuterMeasure \u03b1\ns : \u2115 \u2192 Set \u03b1\nh_mono : \u2200 (n : \u2115), s n \u2286 s (n + 1)\nh0 : \u2211' (k : \u2115), \u2191m (s (k + 1) \\ s k) \u2260 \u22a4\ninst\u271d : (i : \u2115) \u2192 DecidablePred fun x => x \u2208 s i\nn : \u2115\nh' : Monotone s\nx : \u03b1\nthis : \u2203 i, x \u2208 s i\nj : \u2115\nhj : x \u2208 s j\nhlt : \u2200 (m : \u2115), m < j \u2192 \u00acx \u2208 s m\nhjn : j \u2264 n\n\u22a2 x \u2208 s n \u222a \u22c3 i, s (i + n + 1) \\ s (i + n)", "state_after": "no goals"}, {"tactic": "rw [add_assoc, tsub_add_cancel_of_le hnj.nat_succ_le]", "annotated_tactic": ["rw [<a>add_assoc</a>, <a>tsub_add_cancel_of_le</a> hnj.nat_succ_le]", [{"full_name": "add_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [263, 3], "def_end_pos": [263, 14]}, {"full_name": "tsub_add_cancel_of_le", "def_path": "Mathlib/Algebra/Order/Sub/Canonical.lean", "def_pos": [30, 9], "def_end_pos": [30, 30]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm\u271d m : OuterMeasure \u03b1\ns : \u2115 \u2192 Set \u03b1\nh_mono : \u2200 (n : \u2115), s n \u2286 s (n + 1)\nh0 : \u2211' (k : \u2115), \u2191m (s (k + 1) \\ s k) \u2260 \u22a4\ninst\u271d : (i : \u2115) \u2192 DecidablePred fun x => x \u2208 s i\nn : \u2115\nh' : Monotone s\nx : \u03b1\nthis : \u2203 i, x \u2208 s i\nj : \u2115\nhj : x \u2208 s j\nhlt : \u2200 (m : \u2115), m < j \u2192 \u00acx \u2208 s m\nhnj : n < j\n\u22a2 j - (n + 1) + n + 1 = j", "state_after": "no goals"}, {"tactic": "rwa [this]", "annotated_tactic": ["rwa [this]", []], "state_before": "case mk.intro.inr.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm\u271d m : OuterMeasure \u03b1\ns : \u2115 \u2192 Set \u03b1\nh_mono : \u2200 (n : \u2115), s n \u2286 s (n + 1)\nh0 : \u2211' (k : \u2115), \u2191m (s (k + 1) \\ s k) \u2260 \u22a4\ninst\u271d : (i : \u2115) \u2192 DecidablePred fun x => x \u2208 s i\nn : \u2115\nh' : Monotone s\nx : \u03b1\nthis\u271d : \u2203 i, x \u2208 s i\nj : \u2115\nhj : x \u2208 s j\nhlt : \u2200 (m : \u2115), m < j \u2192 \u00acx \u2208 s m\nhnj : n < j\nthis : j - (n + 1) + n + 1 = j\n\u22a2 x \u2208 s (j - (n + 1) + n + 1)", "state_after": "no goals"}, {"tactic": "rw [\u2190 Nat.succ_le_iff, Nat.succ_eq_add_one, this]", "annotated_tactic": ["rw [\u2190 <a>Nat.succ_le_iff</a>, <a>Nat.succ_eq_add_one</a>, this]", [{"full_name": "Nat.succ_le_iff", "def_path": "Mathlib/Data/Nat/Basic.lean", "def_pos": [211, 9], "def_end_pos": [211, 20]}, {"full_name": "Nat.succ_eq_add_one", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [128, 9], "def_end_pos": [128, 24]}]], "state_before": "case mk.intro.inr.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm\u271d m : OuterMeasure \u03b1\ns : \u2115 \u2192 Set \u03b1\nh_mono : \u2200 (n : \u2115), s n \u2286 s (n + 1)\nh0 : \u2211' (k : \u2115), \u2191m (s (k + 1) \\ s k) \u2260 \u22a4\ninst\u271d : (i : \u2115) \u2192 DecidablePred fun x => x \u2208 s i\nn : \u2115\nh' : Monotone s\nx : \u03b1\nthis\u271d : \u2203 i, x \u2208 s i\nj : \u2115\nhj : x \u2208 s j\nhlt : \u2200 (m : \u2115), m < j \u2192 \u00acx \u2208 s m\nhnj : n < j\nthis : j - (n + 1) + n + 1 = j\n\u22a2 j - (n + 1) + n < j", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Martingale/Upcrossing.lean", "full_name": "MeasureTheory.upcrossingsBefore_le", "start": [482, 1], "end": [489, 73], "traced_tactics": [{"tactic": "by_cases hN : N = 0", "annotated_tactic": ["by_cases hN : N = 0", []], "state_before": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf\u271d : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9\u271d : \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nhab : a < b\n\u22a2 upcrossingsBefore a b f N \u03c9 \u2264 N", "state_after": "case pos\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf\u271d : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9\u271d : \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nhab : a < b\nhN : N = 0\n\u22a2 upcrossingsBefore a b f N \u03c9 \u2264 N\n\ncase neg\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf\u271d : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9\u271d : \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nhab : a < b\nhN : \u00acN = 0\n\u22a2 upcrossingsBefore a b f N \u03c9 \u2264 N"}, {"tactic": "subst hN", "annotated_tactic": ["subst hN", []], "state_before": "case pos\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf\u271d : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9\u271d : \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nhab : a < b\nhN : N = 0\n\u22a2 upcrossingsBefore a b f N \u03c9 \u2264 N", "state_after": "case pos\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf\u271d : \u2115 \u2192 \u03a9 \u2192 \u211d\nn m : \u2115\n\u03c9\u271d : \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nhab : a < b\n\u22a2 upcrossingsBefore a b f 0 \u03c9 \u2264 0"}, {"tactic": "rw [upcrossingsBefore_zero]", "annotated_tactic": ["rw [<a>upcrossingsBefore_zero</a>]", [{"full_name": "MeasureTheory.upcrossingsBefore_zero", "def_path": "Mathlib/Probability/Martingale/Upcrossing.lean", "def_pos": [460, 9], "def_end_pos": [460, 31]}]], "state_before": "case pos\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf\u271d : \u2115 \u2192 \u03a9 \u2192 \u211d\nn m : \u2115\n\u03c9\u271d : \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nhab : a < b\n\u22a2 upcrossingsBefore a b f 0 \u03c9 \u2264 0", "state_after": "no goals"}, {"tactic": "refine' csSup_le \u27e80, zero_lt_iff.2 hN\u27e9 fun n (hn : _ < N) => _", "annotated_tactic": ["refine' <a>csSup_le</a> \u27e80, <a>zero_lt_iff</a>.2 hN\u27e9 fun n (hn : _ < N) => _", [{"full_name": "csSup_le", "def_path": "Mathlib/Order/ConditionallyCompleteLattice/Basic.lean", "def_pos": [461, 9], "def_end_pos": [461, 17]}, {"full_name": "zero_lt_iff", "def_path": "Mathlib/Algebra/Order/WithZero.lean", "def_pos": [106, 9], "def_end_pos": [106, 20]}]], "state_before": "case neg\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf\u271d : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9\u271d : \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nhab : a < b\nhN : \u00acN = 0\n\u22a2 upcrossingsBefore a b f N \u03c9 \u2264 N", "state_after": "case neg\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf\u271d : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n\u271d m : \u2115\n\u03c9\u271d : \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nhab : a < b\nhN : \u00acN = 0\nn : \u2115\nhn : upperCrossingTime a b f N n \u03c9 < N\n\u22a2 n \u2264 N"}, {"tactic": "by_contra hnN", "annotated_tactic": ["by_contra hnN", []], "state_before": "case neg\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf\u271d : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n\u271d m : \u2115\n\u03c9\u271d : \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nhab : a < b\nhN : \u00acN = 0\nn : \u2115\nhn : upperCrossingTime a b f N n \u03c9 < N\n\u22a2 n \u2264 N", "state_after": "case neg\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf\u271d : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n\u271d m : \u2115\n\u03c9\u271d : \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nhab : a < b\nhN : \u00acN = 0\nn : \u2115\nhn : upperCrossingTime a b f N n \u03c9 < N\nhnN : \u00acn \u2264 N\n\u22a2 False"}, {"tactic": "exact hn.ne (upperCrossingTime_eq_of_bound_le hab (not_le.1 hnN).le)", "annotated_tactic": ["exact hn.ne (<a>upperCrossingTime_eq_of_bound_le</a> hab (<a>not_le</a>.1 hnN).<a>le</a>)", [{"full_name": "MeasureTheory.upperCrossingTime_eq_of_bound_le", "def_path": "Mathlib/Probability/Martingale/Upcrossing.lean", "def_pos": [329, 9], "def_end_pos": [329, 41]}, {"full_name": "not_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [373, 9], "def_end_pos": [373, 15]}, {"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [142, 7], "def_end_pos": [142, 15]}]], "state_before": "case neg\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf\u271d : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n\u271d m : \u2115\n\u03c9\u271d : \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nhab : a < b\nhN : \u00acN = 0\nn : \u2115\nhn : upperCrossingTime a b f N n \u03c9 < N\nhnN : \u00acn \u2264 N\n\u22a2 False", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Kernel/Disintegration.lean", "full_name": "ProbabilityTheory.eq_condKernel_of_measure_eq_compProd", "start": [530, 1], "end": [592, 50], "traced_tactics": [{"tactic": "obtain \u27e8f, hf\u27e9 := exists_measurableEmbedding_real \u03a9", "annotated_tactic": ["obtain \u27e8f, hf\u27e9 := <a>exists_measurableEmbedding_real</a> \u03a9", [{"full_name": "MeasureTheory.exists_measurableEmbedding_real", "def_path": "Mathlib/MeasureTheory/Constructions/Polish.lean", "def_pos": [1072, 9], "def_end_pos": [1072, 40]}]], "state_before": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1, \u2191\u03ba x = \u2191(Measure.condKernel \u03c1) x", "state_after": "case intro\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1, \u2191\u03ba x = \u2191(Measure.condKernel \u03c1) x"}, {"tactic": "set \u03c1' : Measure (\u03b1 \u00d7 \u211d) := \u03c1.map (Prod.map id f) with h\u03c1'def", "annotated_tactic": ["set \u03c1' : <a>Measure</a> (\u03b1 \u00d7 \u211d) := \u03c1.map (<a>Prod.map</a> <a>id</a> f) with h\u03c1'def", [{"full_name": "MeasureTheory.Measure", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [74, 11], "def_end_pos": [74, 18]}, {"full_name": "Prod.map", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [1043, 5], "def_end_pos": [1043, 13]}, {"full_name": "id", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [33, 15], "def_end_pos": [33, 17]}]], "state_before": "case intro\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1, \u2191\u03ba x = \u2191(Measure.condKernel \u03c1) x", "state_after": "case intro\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1, \u2191\u03ba x = \u2191(Measure.condKernel \u03c1) x"}, {"tactic": "have h\u03c1' : \u03c1'.fst = \u03c1.fst", "annotated_tactic": ["have h\u03c1' : \u03c1'.fst = \u03c1.fst", []], "state_before": "case intro\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1, \u2191\u03ba x = \u2191(Measure.condKernel \u03c1) x", "state_after": "case h\u03c1'\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\n\u22a2 Measure.fst \u03c1' = Measure.fst \u03c1\n\ncase intro\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1, \u2191\u03ba x = \u2191(Measure.condKernel \u03c1) x"}, {"tactic": "have h\u03c1'' : \u2200\u1d50 x \u2202\u03c1'.fst, kernel.map \u03ba f hf.measurable x = \u03c1'.condKernel x", "annotated_tactic": ["have h\u03c1'' : \u2200\u1d50 x \u2202\u03c1'.fst, <a>kernel.map</a> \u03ba f hf.measurable x = \u03c1'.condKernel x", [{"full_name": "ProbabilityTheory.kernel.map", "def_path": "Mathlib/Probability/Kernel/Composition.lean", "def_pos": [573, 19], "def_end_pos": [573, 22]}]], "state_before": "case intro\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1, \u2191\u03ba x = \u2191(Measure.condKernel \u03c1) x", "state_after": "case h\u03c1''\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1', \u2191(kernel.map \u03ba f (_ : Measurable f)) x = \u2191(Measure.condKernel \u03c1') x\n\ncase intro\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1\nh\u03c1'' : \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1', \u2191(kernel.map \u03ba f (_ : Measurable f)) x = \u2191(Measure.condKernel \u03c1') x\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1, \u2191\u03ba x = \u2191(Measure.condKernel \u03c1) x"}, {"tactic": "rw [h\u03c1'] at h\u03c1''", "annotated_tactic": ["rw [h\u03c1'] at h\u03c1''", []], "state_before": "case intro\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1\nh\u03c1'' : \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1', \u2191(kernel.map \u03ba f (_ : Measurable f)) x = \u2191(Measure.condKernel \u03c1') x\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1, \u2191\u03ba x = \u2191(Measure.condKernel \u03c1) x", "state_after": "case intro\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1\nh\u03c1'' : \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1, \u2191(kernel.map \u03ba f (_ : Measurable f)) x = \u2191(Measure.condKernel \u03c1') x\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1, \u2191\u03ba x = \u2191(Measure.condKernel \u03c1) x"}, {"tactic": "suffices : \u2200\u1d50 x \u2202\u03c1.fst, \u2200 s, MeasurableSet s \u2192\n  ((\u03c1.map (Prod.map id f)).condKernel x) s = (\u03c1.condKernel x) (f \u207b\u00b9' s)", "annotated_tactic": ["suffices : \u2200\u1d50 x \u2202\u03c1.fst, \u2200 s, <a>MeasurableSet</a> s \u2192\n    ((\u03c1.map (<a>Prod.map</a> <a>id</a> f)).<a>condKernel</a> x) s = (\u03c1.condKernel x) (f \u207b\u00b9' s)", [{"full_name": "MeasurableSet", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [64, 5], "def_end_pos": [64, 18]}, {"full_name": "Prod.map", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [1043, 5], "def_end_pos": [1043, 13]}, {"full_name": "id", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [33, 15], "def_end_pos": [33, 17]}, {"full_name": "MeasureTheory.Measure.condKernel", "def_path": "Mathlib/Probability/Kernel/Disintegration.lean", "def_pos": [349, 31], "def_end_pos": [349, 70]}]], "state_before": "case intro\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1\nh\u03c1'' : \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1, \u2191(kernel.map \u03ba f (_ : Measurable f)) x = \u2191(Measure.condKernel \u03c1') x\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1, \u2191\u03ba x = \u2191(Measure.condKernel \u03c1) x", "state_after": "case intro\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1\nh\u03c1'' : \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1, \u2191(kernel.map \u03ba f (_ : Measurable f)) x = \u2191(Measure.condKernel \u03c1') x\nthis :\n  \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1,\n    \u2200 (s : Set \u211d),\n      MeasurableSet s \u2192\n        \u2191\u2191(\u2191(Measure.condKernel (Measure.map (Prod.map id f) \u03c1)) x) s = \u2191\u2191(\u2191(Measure.condKernel \u03c1) x) (f \u207b\u00b9' s)\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1, \u2191\u03ba x = \u2191(Measure.condKernel \u03c1) x\n\ncase this\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1\nh\u03c1'' : \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1, \u2191(kernel.map \u03ba f (_ : Measurable f)) x = \u2191(Measure.condKernel \u03c1') x\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1,\n    \u2200 (s : Set \u211d),\n      MeasurableSet s \u2192\n        \u2191\u2191(\u2191(Measure.condKernel (Measure.map (Prod.map id f) \u03c1)) x) s = \u2191\u2191(\u2191(Measure.condKernel \u03c1) x) (f \u207b\u00b9' s)"}, {"tactic": "have hprod : (\u03c1.map (Prod.map id f)).fst = \u03c1.fst", "annotated_tactic": ["have hprod : (\u03c1.map (<a>Prod.map</a> <a>id</a> f)).<a>fst</a> = \u03c1.fst", [{"full_name": "Prod.map", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [1043, 5], "def_end_pos": [1043, 13]}, {"full_name": "id", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [33, 15], "def_end_pos": [33, 17]}, {"full_name": "MeasureTheory.Measure.fst", "def_path": "Mathlib/MeasureTheory/Constructions/Prod/Basic.lean", "def_pos": [910, 19], "def_end_pos": [910, 22]}]], "state_before": "case this\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1\nh\u03c1'' : \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1, \u2191(kernel.map \u03ba f (_ : Measurable f)) x = \u2191(Measure.condKernel \u03c1') x\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1,\n    \u2200 (s : Set \u211d),\n      MeasurableSet s \u2192\n        \u2191\u2191(\u2191(Measure.condKernel (Measure.map (Prod.map id f) \u03c1)) x) s = \u2191\u2191(\u2191(Measure.condKernel \u03c1) x) (f \u207b\u00b9' s)", "state_after": "case hprod\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1\nh\u03c1'' : \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1, \u2191(kernel.map \u03ba f (_ : Measurable f)) x = \u2191(Measure.condKernel \u03c1') x\n\u22a2 Measure.fst (Measure.map (Prod.map id f) \u03c1) = Measure.fst \u03c1\n\ncase this\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1\nh\u03c1'' : \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1, \u2191(kernel.map \u03ba f (_ : Measurable f)) x = \u2191(Measure.condKernel \u03c1') x\nhprod : Measure.fst (Measure.map (Prod.map id f) \u03c1) = Measure.fst \u03c1\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1,\n    \u2200 (s : Set \u211d),\n      MeasurableSet s \u2192\n        \u2191\u2191(\u2191(Measure.condKernel (Measure.map (Prod.map id f) \u03c1)) x) s = \u2191\u2191(\u2191(Measure.condKernel \u03c1) x) (f \u207b\u00b9' s)"}, {"tactic": "suffices : \u03c1.map (Prod.map id f) =\n  (kernel.const Unit (\u03c1.map (Prod.map id f)).fst \u2297\u2096\n    kernel.prodMkLeft Unit (kernel.map (Measure.condKernel \u03c1) f hf.measurable)) ()", "annotated_tactic": ["suffices : \u03c1.map (<a>Prod.map</a> <a>id</a> f) =\n    (<a>kernel.const</a> <a>Unit</a> (\u03c1.map (<a>Prod.map</a> <a>id</a> f)).<a>fst</a> \u2297\u2096\n      <a>kernel.prodMkLeft</a> <a>Unit</a> (<a>kernel.map</a> (<a>Measure.condKernel</a> \u03c1) f hf.measurable)) ()", [{"full_name": "Prod.map", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [1043, 5], "def_end_pos": [1043, 13]}, {"full_name": "id", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [33, 15], "def_end_pos": [33, 17]}, {"full_name": "ProbabilityTheory.kernel.const", "def_path": "Mathlib/Probability/Kernel/Basic.lean", "def_pos": [439, 5], "def_end_pos": [439, 10]}, {"full_name": "Unit", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [129, 8], "def_end_pos": [129, 12]}, {"full_name": "Prod.map", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [1043, 5], "def_end_pos": [1043, 13]}, {"full_name": "id", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [33, 15], "def_end_pos": [33, 17]}, {"full_name": "MeasureTheory.Measure.fst", "def_path": "Mathlib/MeasureTheory/Constructions/Prod/Basic.lean", "def_pos": [910, 19], "def_end_pos": [910, 22]}, {"full_name": "ProbabilityTheory.kernel.prodMkLeft", "def_path": "Mathlib/Probability/Kernel/Composition.lean", "def_pos": [679, 5], "def_end_pos": [679, 15]}, {"full_name": "Unit", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [129, 8], "def_end_pos": [129, 12]}, {"full_name": "ProbabilityTheory.kernel.map", "def_path": "Mathlib/Probability/Kernel/Composition.lean", "def_pos": [573, 19], "def_end_pos": [573, 22]}, {"full_name": "MeasureTheory.Measure.condKernel", "def_path": "Mathlib/Probability/Kernel/Disintegration.lean", "def_pos": [349, 31], "def_end_pos": [349, 70]}]], "state_before": "case this\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1\nh\u03c1'' : \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1, \u2191(kernel.map \u03ba f (_ : Measurable f)) x = \u2191(Measure.condKernel \u03c1') x\nhprod : Measure.fst (Measure.map (Prod.map id f) \u03c1) = Measure.fst \u03c1\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1,\n    \u2200 (s : Set \u211d),\n      MeasurableSet s \u2192\n        \u2191\u2191(\u2191(Measure.condKernel (Measure.map (Prod.map id f) \u03c1)) x) s = \u2191\u2191(\u2191(Measure.condKernel \u03c1) x) (f \u207b\u00b9' s)", "state_after": "case this\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1\nh\u03c1'' : \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1, \u2191(kernel.map \u03ba f (_ : Measurable f)) x = \u2191(Measure.condKernel \u03c1') x\nhprod : Measure.fst (Measure.map (Prod.map id f) \u03c1) = Measure.fst \u03c1\nthis :\n  Measure.map (Prod.map id f) \u03c1 =\n    \u2191(kernel.const Unit (Measure.fst (Measure.map (Prod.map id f) \u03c1)) \u2297\u2096\n          kernel.prodMkLeft Unit (kernel.map (Measure.condKernel \u03c1) f (_ : Measurable f)))\n      ()\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1,\n    \u2200 (s : Set \u211d),\n      MeasurableSet s \u2192\n        \u2191\u2191(\u2191(Measure.condKernel (Measure.map (Prod.map id f) \u03c1)) x) s = \u2191\u2191(\u2191(Measure.condKernel \u03c1) x) (f \u207b\u00b9' s)\n\ncase this\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1\nh\u03c1'' : \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1, \u2191(kernel.map \u03ba f (_ : Measurable f)) x = \u2191(Measure.condKernel \u03c1') x\nhprod : Measure.fst (Measure.map (Prod.map id f) \u03c1) = Measure.fst \u03c1\n\u22a2 Measure.map (Prod.map id f) \u03c1 =\n    \u2191(kernel.const Unit (Measure.fst (Measure.map (Prod.map id f) \u03c1)) \u2297\u2096\n          kernel.prodMkLeft Unit (kernel.map (Measure.condKernel \u03c1) f (_ : Measurable f)))\n      ()"}, {"tactic": "ext s hs", "annotated_tactic": ["ext s hs", []], "state_before": "case this\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1\nh\u03c1'' : \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1, \u2191(kernel.map \u03ba f (_ : Measurable f)) x = \u2191(Measure.condKernel \u03c1') x\nhprod : Measure.fst (Measure.map (Prod.map id f) \u03c1) = Measure.fst \u03c1\n\u22a2 Measure.map (Prod.map id f) \u03c1 =\n    \u2191(kernel.const Unit (Measure.fst (Measure.map (Prod.map id f) \u03c1)) \u2297\u2096\n          kernel.prodMkLeft Unit (kernel.map (Measure.condKernel \u03c1) f (_ : Measurable f)))\n      ()", "state_after": "case this.h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1\nh\u03c1'' : \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1, \u2191(kernel.map \u03ba f (_ : Measurable f)) x = \u2191(Measure.condKernel \u03c1') x\nhprod : Measure.fst (Measure.map (Prod.map id f) \u03c1) = Measure.fst \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(Measure.map (Prod.map id f) \u03c1) s =\n    \u2191\u2191(\u2191(kernel.const Unit (Measure.fst (Measure.map (Prod.map id f) \u03c1)) \u2297\u2096\n                kernel.prodMkLeft Unit (kernel.map (Measure.condKernel \u03c1) f (_ : Measurable f)))\n            ())\n      s"}, {"tactic": "have hinteq : \u2200 x, (\u03c1.condKernel x).map f {c | (x, c) \u2208 s} =\n    \u03c1.condKernel x {c | (x, c) \u2208 Prod.map id f \u207b\u00b9' s}", "annotated_tactic": ["have hinteq : \u2200 x, (\u03c1.condKernel x).<a>map</a> f {c | (x, c) \u2208 s} =\n      \u03c1.condKernel x {c | (x, c) \u2208 <a>Prod.map</a> <a>id</a> f \u207b\u00b9' s}", [{"full_name": "MeasureTheory.Measure.map", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1163, 17], "def_end_pos": [1163, 20]}, {"full_name": "Prod.map", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [1043, 5], "def_end_pos": [1043, 13]}, {"full_name": "id", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [33, 15], "def_end_pos": [33, 17]}]], "state_before": "case this.h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1\nh\u03c1'' : \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1, \u2191(kernel.map \u03ba f (_ : Measurable f)) x = \u2191(Measure.condKernel \u03c1') x\nhprod : Measure.fst (Measure.map (Prod.map id f) \u03c1) = Measure.fst \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(Measure.map (Prod.map id f) \u03c1) s =\n    \u2191\u2191(\u2191(kernel.const Unit (Measure.fst (Measure.map (Prod.map id f) \u03c1)) \u2297\u2096\n                kernel.prodMkLeft Unit (kernel.map (Measure.condKernel \u03c1) f (_ : Measurable f)))\n            ())\n      s", "state_after": "case hinteq\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1\nh\u03c1'' : \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1, \u2191(kernel.map \u03ba f (_ : Measurable f)) x = \u2191(Measure.condKernel \u03c1') x\nhprod : Measure.fst (Measure.map (Prod.map id f) \u03c1) = Measure.fst \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\n\u22a2 \u2200 (x : \u03b1),\n    \u2191\u2191(Measure.map f (\u2191(Measure.condKernel \u03c1) x)) {c | (x, c) \u2208 s} =\n      \u2191\u2191(\u2191(Measure.condKernel \u03c1) x) {c | (x, c) \u2208 Prod.map id f \u207b\u00b9' s}\n\ncase this.h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1\nh\u03c1'' : \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1, \u2191(kernel.map \u03ba f (_ : Measurable f)) x = \u2191(Measure.condKernel \u03c1') x\nhprod : Measure.fst (Measure.map (Prod.map id f) \u03c1) = Measure.fst \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nhinteq :\n  \u2200 (x : \u03b1),\n    \u2191\u2191(Measure.map f (\u2191(Measure.condKernel \u03c1) x)) {c | (x, c) \u2208 s} =\n      \u2191\u2191(\u2191(Measure.condKernel \u03c1) x) {c | (x, c) \u2208 Prod.map id f \u207b\u00b9' s}\n\u22a2 \u2191\u2191(Measure.map (Prod.map id f) \u03c1) s =\n    \u2191\u2191(\u2191(kernel.const Unit (Measure.fst (Measure.map (Prod.map id f) \u03c1)) \u2297\u2096\n                kernel.prodMkLeft Unit (kernel.map (Measure.condKernel \u03c1) f (_ : Measurable f)))\n            ())\n      s"}, {"tactic": "simp only [hprod, kernel.compProd_apply _ _ _ hs, kernel.prodMkLeft_apply,\n  kernel.map_apply _ hf.measurable, hinteq, Set.mem_preimage, Prod_map, id_eq,\n  kernel.lintegral_const]", "annotated_tactic": ["simp only [hprod, <a>kernel.compProd_apply</a> _ _ _ hs, <a>kernel.prodMkLeft_apply</a>,\n    <a>kernel.map_apply</a> _ hf.measurable, hinteq, <a>Set.mem_preimage</a>, <a>Prod_map</a>, <a>id_eq</a>,\n    <a>kernel.lintegral_const</a>]", [{"full_name": "ProbabilityTheory.kernel.compProd_apply", "def_path": "Mathlib/Probability/Kernel/Composition.lean", "def_pos": [242, 9], "def_end_pos": [242, 23]}, {"full_name": "ProbabilityTheory.kernel.prodMkLeft_apply", "def_path": "Mathlib/Probability/Kernel/Composition.lean", "def_pos": [685, 9], "def_end_pos": [685, 25]}, {"full_name": "ProbabilityTheory.kernel.map_apply", "def_path": "Mathlib/Probability/Kernel/Composition.lean", "def_pos": [578, 9], "def_end_pos": [578, 18]}, {"full_name": "Set.mem_preimage", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [64, 9], "def_end_pos": [64, 21]}, {"full_name": "Prod_map", "def_path": "Mathlib/Data/Prod/Basic.lean", "def_pos": [25, 9], "def_end_pos": [25, 17]}, {"full_name": "id_eq", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [284, 17], "def_end_pos": [284, 22]}, {"full_name": "ProbabilityTheory.kernel.lintegral_const", "def_path": "Mathlib/Probability/Kernel/Basic.lean", "def_pos": [460, 9], "def_end_pos": [460, 24]}]], "state_before": "case this.h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1\nh\u03c1'' : \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1, \u2191(kernel.map \u03ba f (_ : Measurable f)) x = \u2191(Measure.condKernel \u03c1') x\nhprod : Measure.fst (Measure.map (Prod.map id f) \u03c1) = Measure.fst \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nhinteq :\n  \u2200 (x : \u03b1),\n    \u2191\u2191(Measure.map f (\u2191(Measure.condKernel \u03c1) x)) {c | (x, c) \u2208 s} =\n      \u2191\u2191(\u2191(Measure.condKernel \u03c1) x) {c | (x, c) \u2208 Prod.map id f \u207b\u00b9' s}\n\u22a2 \u2191\u2191(Measure.map (Prod.map id f) \u03c1) s =\n    \u2191\u2191(\u2191(kernel.const Unit (Measure.fst (Measure.map (Prod.map id f) \u03c1)) \u2297\u2096\n                kernel.prodMkLeft Unit (kernel.map (Measure.condKernel \u03c1) f (_ : Measurable f)))\n            ())\n      s", "state_after": "case this.h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1\nh\u03c1'' : \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1, \u2191(kernel.map \u03ba f (_ : Measurable f)) x = \u2191(Measure.condKernel \u03c1') x\nhprod : Measure.fst (Measure.map (Prod.map id f) \u03c1) = Measure.fst \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nhinteq :\n  \u2200 (x : \u03b1),\n    \u2191\u2191(Measure.map f (\u2191(Measure.condKernel \u03c1) x)) {c | (x, c) \u2208 s} =\n      \u2191\u2191(\u2191(Measure.condKernel \u03c1) x) {c | (x, c) \u2208 Prod.map id f \u207b\u00b9' s}\n\u22a2 \u2191\u2191(Measure.map (Prod.map id f) \u03c1) s = \u222b\u207b (x : \u03b1), \u2191\u2191(\u2191(Measure.condKernel \u03c1) x) {c | (x, f c) \u2208 s} \u2202Measure.fst \u03c1"}, {"tactic": "rw [Measure.map_apply (measurable_id.prod_map hf.measurable) hs, \u2190 lintegral_condKernel_mem]", "annotated_tactic": ["rw [<a>Measure.map_apply</a> (measurable_id.prod_map hf.measurable) hs, \u2190 <a>lintegral_condKernel_mem</a>]", [{"full_name": "MeasureTheory.Measure.map_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1236, 9], "def_end_pos": [1236, 18]}, {"full_name": "ProbabilityTheory.lintegral_condKernel_mem", "def_path": "Mathlib/Probability/Kernel/Disintegration.lean", "def_pos": [386, 9], "def_end_pos": [386, 33]}]], "state_before": "case this.h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1\nh\u03c1'' : \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1, \u2191(kernel.map \u03ba f (_ : Measurable f)) x = \u2191(Measure.condKernel \u03c1') x\nhprod : Measure.fst (Measure.map (Prod.map id f) \u03c1) = Measure.fst \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nhinteq :\n  \u2200 (x : \u03b1),\n    \u2191\u2191(Measure.map f (\u2191(Measure.condKernel \u03c1) x)) {c | (x, c) \u2208 s} =\n      \u2191\u2191(\u2191(Measure.condKernel \u03c1) x) {c | (x, c) \u2208 Prod.map id f \u207b\u00b9' s}\n\u22a2 \u2191\u2191(Measure.map (Prod.map id f) \u03c1) s = \u222b\u207b (x : \u03b1), \u2191\u2191(\u2191(Measure.condKernel \u03c1) x) {c | (x, f c) \u2208 s} \u2202Measure.fst \u03c1", "state_after": "case this.h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1\nh\u03c1'' : \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1, \u2191(kernel.map \u03ba f (_ : Measurable f)) x = \u2191(Measure.condKernel \u03c1') x\nhprod : Measure.fst (Measure.map (Prod.map id f) \u03c1) = Measure.fst \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nhinteq :\n  \u2200 (x : \u03b1),\n    \u2191\u2191(Measure.map f (\u2191(Measure.condKernel \u03c1) x)) {c | (x, c) \u2208 s} =\n      \u2191\u2191(\u2191(Measure.condKernel \u03c1) x) {c | (x, c) \u2208 Prod.map id f \u207b\u00b9' s}\n\u22a2 \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(Measure.condKernel \u03c1) a) {x | (a, x) \u2208 Prod.map id f \u207b\u00b9' s} \u2202Measure.fst \u03c1 =\n    \u222b\u207b (x : \u03b1), \u2191\u2191(\u2191(Measure.condKernel \u03c1) x) {c | (x, f c) \u2208 s} \u2202Measure.fst \u03c1\n\ncase this.h.hs\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1\nh\u03c1'' : \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1, \u2191(kernel.map \u03ba f (_ : Measurable f)) x = \u2191(Measure.condKernel \u03c1') x\nhprod : Measure.fst (Measure.map (Prod.map id f) \u03c1) = Measure.fst \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nhinteq :\n  \u2200 (x : \u03b1),\n    \u2191\u2191(Measure.map f (\u2191(Measure.condKernel \u03c1) x)) {c | (x, c) \u2208 s} =\n      \u2191\u2191(\u2191(Measure.condKernel \u03c1) x) {c | (x, c) \u2208 Prod.map id f \u207b\u00b9' s}\n\u22a2 MeasurableSet (Prod.map id f \u207b\u00b9' s)"}, {"tactic": "ext s hs", "annotated_tactic": ["ext s hs", []], "state_before": "case h\u03c1'\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\n\u22a2 Measure.fst \u03c1' = Measure.fst \u03c1", "state_after": "case h\u03c1'.h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(Measure.fst \u03c1') s = \u2191\u2191(Measure.fst \u03c1) s"}, {"tactic": "rw [h\u03c1'def, Measure.fst_apply, Measure.fst_apply, Measure.map_apply]", "annotated_tactic": ["rw [h\u03c1'def, <a>Measure.fst_apply</a>, <a>Measure.fst_apply</a>, <a>Measure.map_apply</a>]", [{"full_name": "MeasureTheory.Measure.fst_apply", "def_path": "Mathlib/MeasureTheory/Constructions/Prod/Basic.lean", "def_pos": [914, 9], "def_end_pos": [914, 18]}, {"full_name": "MeasureTheory.Measure.fst_apply", "def_path": "Mathlib/MeasureTheory/Constructions/Prod/Basic.lean", "def_pos": [914, 9], "def_end_pos": [914, 18]}, {"full_name": "MeasureTheory.Measure.map_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1236, 9], "def_end_pos": [1236, 18]}]], "state_before": "case h\u03c1'.h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(Measure.fst \u03c1') s = \u2191\u2191(Measure.fst \u03c1) s", "state_after": "case h\u03c1'.h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191\u2191\u03c1 (Prod.map id f \u207b\u00b9' (Prod.fst \u207b\u00b9' s)) = \u2191\u2191\u03c1 (Prod.fst \u207b\u00b9' s)\n\ncase h\u03c1'.h.hf\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 Measurable (Prod.map id f)\n\ncase h\u03c1'.h.hs\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 MeasurableSet (Prod.fst \u207b\u00b9' s)\n\ncase h\u03c1'.h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 MeasurableSet s\n\ncase h\u03c1'.h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 MeasurableSet s"}, {"tactic": "exacts [rfl, Measurable.prod measurable_fst <| hf.measurable.comp measurable_snd,\n  measurable_fst hs, hs, hs]", "annotated_tactic": ["exacts [<a>rfl</a>, <a>Measurable.prod</a> <a>measurable_fst</a> <| hf.measurable.comp <a>measurable_snd</a>,\n      <a>measurable_fst</a> hs, hs, hs]", [{"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}, {"full_name": "Measurable.prod", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [714, 9], "def_end_pos": [714, 24]}, {"full_name": "measurable_fst", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [692, 9], "def_end_pos": [692, 23]}, {"full_name": "measurable_snd", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [698, 9], "def_end_pos": [698, 23]}, {"full_name": "measurable_fst", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [692, 9], "def_end_pos": [692, 23]}]], "state_before": "case h\u03c1'.h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191\u2191\u03c1 (Prod.map id f \u207b\u00b9' (Prod.fst \u207b\u00b9' s)) = \u2191\u2191\u03c1 (Prod.fst \u207b\u00b9' s)\n\ncase h\u03c1'.h.hf\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 Measurable (Prod.map id f)\n\ncase h\u03c1'.h.hs\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 MeasurableSet (Prod.fst \u207b\u00b9' s)\n\ncase h\u03c1'.h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 MeasurableSet s\n\ncase h\u03c1'.h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 MeasurableSet s", "state_after": "no goals"}, {"tactic": "refine' eq_condKernel_of_measure_eq_compProd_real \u03c1' (kernel.map \u03ba f hf.measurable) _", "annotated_tactic": ["refine' <a>eq_condKernel_of_measure_eq_compProd_real</a> \u03c1' (<a>kernel.map</a> \u03ba f hf.measurable) _", [{"full_name": "ProbabilityTheory.eq_condKernel_of_measure_eq_compProd_real", "def_path": "Mathlib/Probability/Kernel/Disintegration.lean", "def_pos": [508, 7], "def_end_pos": [508, 48]}, {"full_name": "ProbabilityTheory.kernel.map", "def_path": "Mathlib/Probability/Kernel/Composition.lean", "def_pos": [573, 19], "def_end_pos": [573, 22]}]], "state_before": "case h\u03c1''\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1', \u2191(kernel.map \u03ba f (_ : Measurable f)) x = \u2191(Measure.condKernel \u03c1') x", "state_after": "case h\u03c1''\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1\n\u22a2 \u03c1' = \u2191(kernel.const Unit (Measure.fst \u03c1') \u2297\u2096 kernel.prodMkLeft Unit (kernel.map \u03ba f (_ : Measurable f))) ()"}, {"tactic": "ext s hs", "annotated_tactic": ["ext s hs", []], "state_before": "case h\u03c1''\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1\n\u22a2 \u03c1' = \u2191(kernel.const Unit (Measure.fst \u03c1') \u2297\u2096 kernel.prodMkLeft Unit (kernel.map \u03ba f (_ : Measurable f))) ()", "state_after": "case h\u03c1''.h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\n\u22a2 \u2191\u2191\u03c1' s = \u2191\u2191(\u2191(kernel.const Unit (Measure.fst \u03c1') \u2297\u2096 kernel.prodMkLeft Unit (kernel.map \u03ba f (_ : Measurable f))) ()) s"}, {"tactic": "simp only [Measure.map_apply (measurable_id.prod_map hf.measurable) hs]", "annotated_tactic": ["simp only [<a>Measure.map_apply</a> (measurable_id.prod_map hf.measurable) hs]", [{"full_name": "MeasureTheory.Measure.map_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1236, 9], "def_end_pos": [1236, 18]}]], "state_before": "case h\u03c1''.h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\n\u22a2 \u2191\u2191\u03c1' s = \u2191\u2191(\u2191(kernel.const Unit (Measure.fst \u03c1') \u2297\u2096 kernel.prodMkLeft Unit (kernel.map \u03ba f (_ : Measurable f))) ()) s", "state_after": "case h\u03c1''.h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\n\u22a2 \u2191\u2191\u03c1 (Prod.map id f \u207b\u00b9' s) =\n    \u2191\u2191(\u2191(kernel.const Unit (Measure.fst (Measure.map (Prod.map id f) \u03c1)) \u2297\u2096\n                kernel.prodMkLeft Unit (kernel.map \u03ba f (_ : Measurable f)))\n            ())\n      s"}, {"tactic": "conv_lhs => congr; rw [h\u03ba]", "annotated_tactic": ["conv_lhs => congr; rw [h\u03ba]", []], "state_before": "case h\u03c1''.h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\n\u22a2 \u2191\u2191\u03c1 (Prod.map id f \u207b\u00b9' s) =\n    \u2191\u2191(\u2191(kernel.const Unit (Measure.fst (Measure.map (Prod.map id f) \u03c1)) \u2297\u2096\n                kernel.prodMkLeft Unit (kernel.map \u03ba f (_ : Measurable f)))\n            ())\n      s", "state_after": "case h\u03c1''.h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(\u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()) (Prod.map id f \u207b\u00b9' s) =\n    \u2191\u2191(\u2191(kernel.const Unit (Measure.fst (Measure.map (Prod.map id f) \u03c1)) \u2297\u2096\n                kernel.prodMkLeft Unit (kernel.map \u03ba f (_ : Measurable f)))\n            ())\n      s"}, {"tactic": "rw [kernel.compProd_apply _ _ _ hs, kernel.compProd_apply _ _ _\n  (measurable_id.prod_map hf.measurable hs), (_ : (\u03c1.map (Prod.map id f)).fst = \u03c1.fst)]", "annotated_tactic": ["rw [<a>kernel.compProd_apply</a> _ _ _ hs, <a>kernel.compProd_apply</a> _ _ _\n      (measurable_id.prod_map hf.measurable hs), (_ : (\u03c1.map (<a>Prod.map</a> <a>id</a> f)).<a>fst</a> = \u03c1.fst)]", [{"full_name": "ProbabilityTheory.kernel.compProd_apply", "def_path": "Mathlib/Probability/Kernel/Composition.lean", "def_pos": [242, 9], "def_end_pos": [242, 23]}, {"full_name": "ProbabilityTheory.kernel.compProd_apply", "def_path": "Mathlib/Probability/Kernel/Composition.lean", "def_pos": [242, 9], "def_end_pos": [242, 23]}, {"full_name": "Prod.map", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [1043, 5], "def_end_pos": [1043, 13]}, {"full_name": "id", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [33, 15], "def_end_pos": [33, 17]}, {"full_name": "MeasureTheory.Measure.fst", "def_path": "Mathlib/MeasureTheory/Constructions/Prod/Basic.lean", "def_pos": [910, 19], "def_end_pos": [910, 22]}]], "state_before": "case h\u03c1''.h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(\u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()) (Prod.map id f \u207b\u00b9' s) =\n    \u2191\u2191(\u2191(kernel.const Unit (Measure.fst (Measure.map (Prod.map id f) \u03c1)) \u2297\u2096\n                kernel.prodMkLeft Unit (kernel.map \u03ba f (_ : Measurable f)))\n            ())\n      s", "state_after": "case h\u03c1''.h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\n\u22a2 \u222b\u207b (b : \u03b1),\n      \u2191\u2191(\u2191(kernel.prodMkLeft Unit \u03ba) ((), b))\n        {c | (b, c) \u2208 Prod.map id f \u207b\u00b9' s} \u2202\u2191(kernel.const Unit (Measure.fst \u03c1)) () =\n    \u222b\u207b (b : \u03b1),\n      \u2191\u2191(\u2191(kernel.prodMkLeft Unit (kernel.map \u03ba f (_ : Measurable f))) ((), b))\n        {c | (b, c) \u2208 s} \u2202\u2191(kernel.const Unit (Measure.fst \u03c1)) ()\n\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\n\u22a2 Measure.fst (Measure.map (Prod.map id f) \u03c1) = Measure.fst \u03c1"}, {"tactic": "congr", "annotated_tactic": ["congr", []], "state_before": "case h\u03c1''.h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\n\u22a2 \u222b\u207b (b : \u03b1),\n      \u2191\u2191(\u2191(kernel.prodMkLeft Unit \u03ba) ((), b))\n        {c | (b, c) \u2208 Prod.map id f \u207b\u00b9' s} \u2202\u2191(kernel.const Unit (Measure.fst \u03c1)) () =\n    \u222b\u207b (b : \u03b1),\n      \u2191\u2191(\u2191(kernel.prodMkLeft Unit (kernel.map \u03ba f (_ : Measurable f))) ((), b))\n        {c | (b, c) \u2208 s} \u2202\u2191(kernel.const Unit (Measure.fst \u03c1)) ()", "state_after": "case h\u03c1''.h.e_f\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\n\u22a2 (fun b => \u2191\u2191(\u2191(kernel.prodMkLeft Unit \u03ba) ((), b)) {c | (b, c) \u2208 Prod.map id f \u207b\u00b9' s}) = fun b =>\n    \u2191\u2191(\u2191(kernel.prodMkLeft Unit (kernel.map \u03ba f (_ : Measurable f))) ((), b)) {c | (b, c) \u2208 s}"}, {"tactic": "ext x", "annotated_tactic": ["ext x", []], "state_before": "case h\u03c1''.h.e_f\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\n\u22a2 (fun b => \u2191\u2191(\u2191(kernel.prodMkLeft Unit \u03ba) ((), b)) {c | (b, c) \u2208 Prod.map id f \u207b\u00b9' s}) = fun b =>\n    \u2191\u2191(\u2191(kernel.prodMkLeft Unit (kernel.map \u03ba f (_ : Measurable f))) ((), b)) {c | (b, c) \u2208 s}", "state_after": "case h\u03c1''.h.e_f.h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nx : \u03b1\n\u22a2 \u2191\u2191(\u2191(kernel.prodMkLeft Unit \u03ba) ((), x)) {c | (x, c) \u2208 Prod.map id f \u207b\u00b9' s} =\n    \u2191\u2191(\u2191(kernel.prodMkLeft Unit (kernel.map \u03ba f (_ : Measurable f))) ((), x)) {c | (x, c) \u2208 s}"}, {"tactic": "simp only [Set.mem_preimage, Prod_map, id_eq, kernel.prodMkLeft_apply, kernel.map_apply]", "annotated_tactic": ["simp only [<a>Set.mem_preimage</a>, <a>Prod_map</a>, <a>id_eq</a>, <a>kernel.prodMkLeft_apply</a>, <a>kernel.map_apply</a>]", [{"full_name": "Set.mem_preimage", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [64, 9], "def_end_pos": [64, 21]}, {"full_name": "Prod_map", "def_path": "Mathlib/Data/Prod/Basic.lean", "def_pos": [25, 9], "def_end_pos": [25, 17]}, {"full_name": "id_eq", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [284, 17], "def_end_pos": [284, 22]}, {"full_name": "ProbabilityTheory.kernel.prodMkLeft_apply", "def_path": "Mathlib/Probability/Kernel/Composition.lean", "def_pos": [685, 9], "def_end_pos": [685, 25]}, {"full_name": "ProbabilityTheory.kernel.map_apply", "def_path": "Mathlib/Probability/Kernel/Composition.lean", "def_pos": [578, 9], "def_end_pos": [578, 18]}]], "state_before": "case h\u03c1''.h.e_f.h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nx : \u03b1\n\u22a2 \u2191\u2191(\u2191(kernel.prodMkLeft Unit \u03ba) ((), x)) {c | (x, c) \u2208 Prod.map id f \u207b\u00b9' s} =\n    \u2191\u2191(\u2191(kernel.prodMkLeft Unit (kernel.map \u03ba f (_ : Measurable f))) ((), x)) {c | (x, c) \u2208 s}", "state_after": "case h\u03c1''.h.e_f.h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nx : \u03b1\n\u22a2 \u2191\u2191(\u2191\u03ba x) {c | (x, f c) \u2208 s} = \u2191\u2191(Measure.map f (\u2191\u03ba x)) {c | (x, c) \u2208 s}"}, {"tactic": "rw [Measure.map_apply hf.measurable]", "annotated_tactic": ["rw [<a>Measure.map_apply</a> hf.measurable]", [{"full_name": "MeasureTheory.Measure.map_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1236, 9], "def_end_pos": [1236, 18]}]], "state_before": "case h\u03c1''.h.e_f.h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nx : \u03b1\n\u22a2 \u2191\u2191(\u2191\u03ba x) {c | (x, f c) \u2208 s} = \u2191\u2191(Measure.map f (\u2191\u03ba x)) {c | (x, c) \u2208 s}", "state_after": "case h\u03c1''.h.e_f.h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nx : \u03b1\n\u22a2 \u2191\u2191(\u2191\u03ba x) {c | (x, f c) \u2208 s} = \u2191\u2191(\u2191\u03ba x) (f \u207b\u00b9' {c | (x, c) \u2208 s})\n\ncase h\u03c1''.h.e_f.h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nx : \u03b1\n\u22a2 MeasurableSet {c | (x, c) \u2208 s}"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case h\u03c1''.h.e_f.h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nx : \u03b1\n\u22a2 \u2191\u2191(\u2191\u03ba x) {c | (x, f c) \u2208 s} = \u2191\u2191(\u2191\u03ba x) (f \u207b\u00b9' {c | (x, c) \u2208 s})", "state_after": "no goals"}, {"tactic": "exact measurable_prod_mk_left hs", "annotated_tactic": ["exact <a>measurable_prod_mk_left</a> hs", [{"full_name": "measurable_prod_mk_left", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [736, 9], "def_end_pos": [736, 32]}]], "state_before": "case h\u03c1''.h.e_f.h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nx : \u03b1\n\u22a2 MeasurableSet {c | (x, c) \u2208 s}", "state_after": "no goals"}, {"tactic": "rw [Measure.fst_map_prod_mk]", "annotated_tactic": ["rw [<a>Measure.fst_map_prod_mk</a>]", [{"full_name": "MeasureTheory.Measure.fst_map_prod_mk", "def_path": "Mathlib/MeasureTheory/Constructions/Prod/Basic.lean", "def_pos": [951, 9], "def_end_pos": [951, 24]}]], "state_before": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\n\u22a2 Measure.fst (Measure.map (Prod.map id f) \u03c1) = Measure.fst \u03c1", "state_after": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\n\u22a2 Measure.map (fun a => (Prod.map id f a).1) \u03c1 = Measure.fst \u03c1\n\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\n\u22a2 Measurable fun a => (Prod.map id f a).2"}, {"tactic": "simp only [Prod_map, id_eq]", "annotated_tactic": ["simp only [<a>Prod_map</a>, <a>id_eq</a>]", [{"full_name": "Prod_map", "def_path": "Mathlib/Data/Prod/Basic.lean", "def_pos": [25, 9], "def_end_pos": [25, 17]}, {"full_name": "id_eq", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [284, 17], "def_end_pos": [284, 22]}]], "state_before": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\n\u22a2 Measure.map (fun a => (Prod.map id f a).1) \u03c1 = Measure.fst \u03c1", "state_after": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\n\u22a2 Measure.map (fun a => a.1) \u03c1 = Measure.fst \u03c1"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\n\u22a2 Measure.map (fun a => a.1) \u03c1 = Measure.fst \u03c1", "state_after": "no goals"}, {"tactic": "exact (hf.measurable.comp measurable_snd)", "annotated_tactic": ["exact (hf.measurable.comp <a>measurable_snd</a>)", [{"full_name": "measurable_snd", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [698, 9], "def_end_pos": [698, 23]}]], "state_before": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\n\u22a2 Measurable fun a => (Prod.map id f a).2", "state_after": "no goals"}, {"tactic": "filter_upwards [h\u03c1'', this] with x hx h", "annotated_tactic": ["filter_upwards [h\u03c1'', this] with x hx h", []], "state_before": "case intro\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1\nh\u03c1'' : \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1, \u2191(kernel.map \u03ba f (_ : Measurable f)) x = \u2191(Measure.condKernel \u03c1') x\nthis :\n  \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1,\n    \u2200 (s : Set \u211d),\n      MeasurableSet s \u2192\n        \u2191\u2191(\u2191(Measure.condKernel (Measure.map (Prod.map id f) \u03c1)) x) s = \u2191\u2191(\u2191(Measure.condKernel \u03c1) x) (f \u207b\u00b9' s)\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1, \u2191\u03ba x = \u2191(Measure.condKernel \u03c1) x", "state_after": "case h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1\nh\u03c1'' : \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1, \u2191(kernel.map \u03ba f (_ : Measurable f)) x = \u2191(Measure.condKernel \u03c1') x\nthis :\n  \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1,\n    \u2200 (s : Set \u211d),\n      MeasurableSet s \u2192\n        \u2191\u2191(\u2191(Measure.condKernel (Measure.map (Prod.map id f) \u03c1)) x) s = \u2191\u2191(\u2191(Measure.condKernel \u03c1) x) (f \u207b\u00b9' s)\nx : \u03b1\nhx : \u2191(kernel.map \u03ba f (_ : Measurable f)) x = \u2191(Measure.condKernel \u03c1') x\nh :\n  \u2200 (s : Set \u211d),\n    MeasurableSet s \u2192\n      \u2191\u2191(\u2191(Measure.condKernel (Measure.map (Prod.map id f) \u03c1)) x) s = \u2191\u2191(\u2191(Measure.condKernel \u03c1) x) (f \u207b\u00b9' s)\n\u22a2 \u2191\u03ba x = \u2191(Measure.condKernel \u03c1) x"}, {"tactic": "rw [kernel.map_apply] at hx", "annotated_tactic": ["rw [<a>kernel.map_apply</a>] at hx", [{"full_name": "ProbabilityTheory.kernel.map_apply", "def_path": "Mathlib/Probability/Kernel/Composition.lean", "def_pos": [578, 9], "def_end_pos": [578, 18]}]], "state_before": "case h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1\nh\u03c1'' : \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1, \u2191(kernel.map \u03ba f (_ : Measurable f)) x = \u2191(Measure.condKernel \u03c1') x\nthis :\n  \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1,\n    \u2200 (s : Set \u211d),\n      MeasurableSet s \u2192\n        \u2191\u2191(\u2191(Measure.condKernel (Measure.map (Prod.map id f) \u03c1)) x) s = \u2191\u2191(\u2191(Measure.condKernel \u03c1) x) (f \u207b\u00b9' s)\nx : \u03b1\nhx : \u2191(kernel.map \u03ba f (_ : Measurable f)) x = \u2191(Measure.condKernel \u03c1') x\nh :\n  \u2200 (s : Set \u211d),\n    MeasurableSet s \u2192\n      \u2191\u2191(\u2191(Measure.condKernel (Measure.map (Prod.map id f) \u03c1)) x) s = \u2191\u2191(\u2191(Measure.condKernel \u03c1) x) (f \u207b\u00b9' s)\n\u22a2 \u2191\u03ba x = \u2191(Measure.condKernel \u03c1) x", "state_after": "case h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1\nh\u03c1'' : \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1, \u2191(kernel.map \u03ba f (_ : Measurable f)) x = \u2191(Measure.condKernel \u03c1') x\nthis :\n  \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1,\n    \u2200 (s : Set \u211d),\n      MeasurableSet s \u2192\n        \u2191\u2191(\u2191(Measure.condKernel (Measure.map (Prod.map id f) \u03c1)) x) s = \u2191\u2191(\u2191(Measure.condKernel \u03c1) x) (f \u207b\u00b9' s)\nx : \u03b1\nhx : Measure.map f (\u2191\u03ba x) = \u2191(Measure.condKernel \u03c1') x\nh :\n  \u2200 (s : Set \u211d),\n    MeasurableSet s \u2192\n      \u2191\u2191(\u2191(Measure.condKernel (Measure.map (Prod.map id f) \u03c1)) x) s = \u2191\u2191(\u2191(Measure.condKernel \u03c1) x) (f \u207b\u00b9' s)\n\u22a2 \u2191\u03ba x = \u2191(Measure.condKernel \u03c1) x"}, {"tactic": "ext s hs", "annotated_tactic": ["ext s hs", []], "state_before": "case h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1\nh\u03c1'' : \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1, \u2191(kernel.map \u03ba f (_ : Measurable f)) x = \u2191(Measure.condKernel \u03c1') x\nthis :\n  \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1,\n    \u2200 (s : Set \u211d),\n      MeasurableSet s \u2192\n        \u2191\u2191(\u2191(Measure.condKernel (Measure.map (Prod.map id f) \u03c1)) x) s = \u2191\u2191(\u2191(Measure.condKernel \u03c1) x) (f \u207b\u00b9' s)\nx : \u03b1\nhx : Measure.map f (\u2191\u03ba x) = \u2191(Measure.condKernel \u03c1') x\nh :\n  \u2200 (s : Set \u211d),\n    MeasurableSet s \u2192\n      \u2191\u2191(\u2191(Measure.condKernel (Measure.map (Prod.map id f) \u03c1)) x) s = \u2191\u2191(\u2191(Measure.condKernel \u03c1) x) (f \u207b\u00b9' s)\n\u22a2 \u2191\u03ba x = \u2191(Measure.condKernel \u03c1) x", "state_after": "case h.h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1\nh\u03c1'' : \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1, \u2191(kernel.map \u03ba f (_ : Measurable f)) x = \u2191(Measure.condKernel \u03c1') x\nthis :\n  \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1,\n    \u2200 (s : Set \u211d),\n      MeasurableSet s \u2192\n        \u2191\u2191(\u2191(Measure.condKernel (Measure.map (Prod.map id f) \u03c1)) x) s = \u2191\u2191(\u2191(Measure.condKernel \u03c1) x) (f \u207b\u00b9' s)\nx : \u03b1\nhx : Measure.map f (\u2191\u03ba x) = \u2191(Measure.condKernel \u03c1') x\nh :\n  \u2200 (s : Set \u211d),\n    MeasurableSet s \u2192\n      \u2191\u2191(\u2191(Measure.condKernel (Measure.map (Prod.map id f) \u03c1)) x) s = \u2191\u2191(\u2191(Measure.condKernel \u03c1) x) (f \u207b\u00b9' s)\ns : Set \u03a9\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(\u2191\u03ba x) s = \u2191\u2191(\u2191(Measure.condKernel \u03c1) x) s"}, {"tactic": "rw [\u2190 Set.preimage_image_eq s hf.injective,\n  \u2190 Measure.map_apply hf.measurable <| hf.measurableSet_image.2 hs, hx,\n  h _ <| hf.measurableSet_image.2 hs]", "annotated_tactic": ["rw [\u2190 <a>Set.preimage_image_eq</a> s hf.injective,\n      \u2190 <a>Measure.map_apply</a> hf.measurable <| hf.measurableSet_image.2 hs, hx,\n      h _ <| hf.measurableSet_image.2 hs]", [{"full_name": "Set.preimage_image_eq", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [510, 9], "def_end_pos": [510, 26]}, {"full_name": "MeasureTheory.Measure.map_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1236, 9], "def_end_pos": [1236, 18]}]], "state_before": "case h.h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1\nh\u03c1'' : \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1, \u2191(kernel.map \u03ba f (_ : Measurable f)) x = \u2191(Measure.condKernel \u03c1') x\nthis :\n  \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1,\n    \u2200 (s : Set \u211d),\n      MeasurableSet s \u2192\n        \u2191\u2191(\u2191(Measure.condKernel (Measure.map (Prod.map id f) \u03c1)) x) s = \u2191\u2191(\u2191(Measure.condKernel \u03c1) x) (f \u207b\u00b9' s)\nx : \u03b1\nhx : Measure.map f (\u2191\u03ba x) = \u2191(Measure.condKernel \u03c1') x\nh :\n  \u2200 (s : Set \u211d),\n    MeasurableSet s \u2192\n      \u2191\u2191(\u2191(Measure.condKernel (Measure.map (Prod.map id f) \u03c1)) x) s = \u2191\u2191(\u2191(Measure.condKernel \u03c1) x) (f \u207b\u00b9' s)\ns : Set \u03a9\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(\u2191\u03ba x) s = \u2191\u2191(\u2191(Measure.condKernel \u03c1) x) s", "state_after": "no goals"}, {"tactic": "ext s hs", "annotated_tactic": ["ext s hs", []], "state_before": "case hprod\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1\nh\u03c1'' : \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1, \u2191(kernel.map \u03ba f (_ : Measurable f)) x = \u2191(Measure.condKernel \u03c1') x\n\u22a2 Measure.fst (Measure.map (Prod.map id f) \u03c1) = Measure.fst \u03c1", "state_after": "case hprod.h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1\nh\u03c1'' : \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1, \u2191(kernel.map \u03ba f (_ : Measurable f)) x = \u2191(Measure.condKernel \u03c1') x\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(Measure.fst (Measure.map (Prod.map id f) \u03c1)) s = \u2191\u2191(Measure.fst \u03c1) s"}, {"tactic": "rw [Measure.fst_apply hs,\n  Measure.map_apply (measurable_id.prod_map hf.measurable) (measurable_fst hs),\n  \u2190 Set.preimage_comp, Measure.fst_apply hs]", "annotated_tactic": ["rw [<a>Measure.fst_apply</a> hs,\n      <a>Measure.map_apply</a> (measurable_id.prod_map hf.measurable) (<a>measurable_fst</a> hs),\n      \u2190 <a>Set.preimage_comp</a>, <a>Measure.fst_apply</a> hs]", [{"full_name": "MeasureTheory.Measure.fst_apply", "def_path": "Mathlib/MeasureTheory/Constructions/Prod/Basic.lean", "def_pos": [914, 9], "def_end_pos": [914, 18]}, {"full_name": "MeasureTheory.Measure.map_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1236, 9], "def_end_pos": [1236, 18]}, {"full_name": "measurable_fst", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [692, 9], "def_end_pos": [692, 23]}, {"full_name": "Set.preimage_comp", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [163, 9], "def_end_pos": [163, 22]}, {"full_name": "MeasureTheory.Measure.fst_apply", "def_path": "Mathlib/MeasureTheory/Constructions/Prod/Basic.lean", "def_pos": [914, 9], "def_end_pos": [914, 18]}]], "state_before": "case hprod.h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1\nh\u03c1'' : \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1, \u2191(kernel.map \u03ba f (_ : Measurable f)) x = \u2191(Measure.condKernel \u03c1') x\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(Measure.fst (Measure.map (Prod.map id f) \u03c1)) s = \u2191\u2191(Measure.fst \u03c1) s", "state_after": "case hprod.h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1\nh\u03c1'' : \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1, \u2191(kernel.map \u03ba f (_ : Measurable f)) x = \u2191(Measure.condKernel \u03c1') x\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191\u2191\u03c1 (Prod.fst \u2218 Prod.map id f \u207b\u00b9' s) = \u2191\u2191\u03c1 (Prod.fst \u207b\u00b9' s)"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case hprod.h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1\nh\u03c1'' : \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1, \u2191(kernel.map \u03ba f (_ : Measurable f)) x = \u2191(Measure.condKernel \u03c1') x\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191\u2191\u03c1 (Prod.fst \u2218 Prod.map id f \u207b\u00b9' s) = \u2191\u2191\u03c1 (Prod.fst \u207b\u00b9' s)", "state_after": "no goals"}, {"tactic": "have heq := eq_condKernel_of_measure_eq_compProd_real _ _ this", "annotated_tactic": ["have heq := <a>eq_condKernel_of_measure_eq_compProd_real</a> _ _ this", [{"full_name": "ProbabilityTheory.eq_condKernel_of_measure_eq_compProd_real", "def_path": "Mathlib/Probability/Kernel/Disintegration.lean", "def_pos": [508, 7], "def_end_pos": [508, 48]}]], "state_before": "case this\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1\nh\u03c1'' : \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1, \u2191(kernel.map \u03ba f (_ : Measurable f)) x = \u2191(Measure.condKernel \u03c1') x\nhprod : Measure.fst (Measure.map (Prod.map id f) \u03c1) = Measure.fst \u03c1\nthis :\n  Measure.map (Prod.map id f) \u03c1 =\n    \u2191(kernel.const Unit (Measure.fst (Measure.map (Prod.map id f) \u03c1)) \u2297\u2096\n          kernel.prodMkLeft Unit (kernel.map (Measure.condKernel \u03c1) f (_ : Measurable f)))\n      ()\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1,\n    \u2200 (s : Set \u211d),\n      MeasurableSet s \u2192\n        \u2191\u2191(\u2191(Measure.condKernel (Measure.map (Prod.map id f) \u03c1)) x) s = \u2191\u2191(\u2191(Measure.condKernel \u03c1) x) (f \u207b\u00b9' s)", "state_after": "case this\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1\nh\u03c1'' : \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1, \u2191(kernel.map \u03ba f (_ : Measurable f)) x = \u2191(Measure.condKernel \u03c1') x\nhprod : Measure.fst (Measure.map (Prod.map id f) \u03c1) = Measure.fst \u03c1\nthis :\n  Measure.map (Prod.map id f) \u03c1 =\n    \u2191(kernel.const Unit (Measure.fst (Measure.map (Prod.map id f) \u03c1)) \u2297\u2096\n          kernel.prodMkLeft Unit (kernel.map (Measure.condKernel \u03c1) f (_ : Measurable f)))\n      ()\nheq :\n  \u2200\u1d50 (x : \u03b1) \u2202Measure.fst (Measure.map (Prod.map id f) \u03c1),\n    \u2191(kernel.map (Measure.condKernel \u03c1) f (_ : Measurable f)) x =\n      \u2191(Measure.condKernel (Measure.map (Prod.map id f) \u03c1)) x\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1,\n    \u2200 (s : Set \u211d),\n      MeasurableSet s \u2192\n        \u2191\u2191(\u2191(Measure.condKernel (Measure.map (Prod.map id f) \u03c1)) x) s = \u2191\u2191(\u2191(Measure.condKernel \u03c1) x) (f \u207b\u00b9' s)"}, {"tactic": "rw [hprod] at heq", "annotated_tactic": ["rw [hprod] at heq", []], "state_before": "case this\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1\nh\u03c1'' : \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1, \u2191(kernel.map \u03ba f (_ : Measurable f)) x = \u2191(Measure.condKernel \u03c1') x\nhprod : Measure.fst (Measure.map (Prod.map id f) \u03c1) = Measure.fst \u03c1\nthis :\n  Measure.map (Prod.map id f) \u03c1 =\n    \u2191(kernel.const Unit (Measure.fst (Measure.map (Prod.map id f) \u03c1)) \u2297\u2096\n          kernel.prodMkLeft Unit (kernel.map (Measure.condKernel \u03c1) f (_ : Measurable f)))\n      ()\nheq :\n  \u2200\u1d50 (x : \u03b1) \u2202Measure.fst (Measure.map (Prod.map id f) \u03c1),\n    \u2191(kernel.map (Measure.condKernel \u03c1) f (_ : Measurable f)) x =\n      \u2191(Measure.condKernel (Measure.map (Prod.map id f) \u03c1)) x\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1,\n    \u2200 (s : Set \u211d),\n      MeasurableSet s \u2192\n        \u2191\u2191(\u2191(Measure.condKernel (Measure.map (Prod.map id f) \u03c1)) x) s = \u2191\u2191(\u2191(Measure.condKernel \u03c1) x) (f \u207b\u00b9' s)", "state_after": "case this\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1\nh\u03c1'' : \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1, \u2191(kernel.map \u03ba f (_ : Measurable f)) x = \u2191(Measure.condKernel \u03c1') x\nhprod : Measure.fst (Measure.map (Prod.map id f) \u03c1) = Measure.fst \u03c1\nthis :\n  Measure.map (Prod.map id f) \u03c1 =\n    \u2191(kernel.const Unit (Measure.fst (Measure.map (Prod.map id f) \u03c1)) \u2297\u2096\n          kernel.prodMkLeft Unit (kernel.map (Measure.condKernel \u03c1) f (_ : Measurable f)))\n      ()\nheq :\n  \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1,\n    \u2191(kernel.map (Measure.condKernel \u03c1) f (_ : Measurable f)) x =\n      \u2191(Measure.condKernel (Measure.map (Prod.map id f) \u03c1)) x\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1,\n    \u2200 (s : Set \u211d),\n      MeasurableSet s \u2192\n        \u2191\u2191(\u2191(Measure.condKernel (Measure.map (Prod.map id f) \u03c1)) x) s = \u2191\u2191(\u2191(Measure.condKernel \u03c1) x) (f \u207b\u00b9' s)"}, {"tactic": "filter_upwards [heq] with x hx s hs", "annotated_tactic": ["filter_upwards [heq] with x hx s hs", []], "state_before": "case this\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1\nh\u03c1'' : \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1, \u2191(kernel.map \u03ba f (_ : Measurable f)) x = \u2191(Measure.condKernel \u03c1') x\nhprod : Measure.fst (Measure.map (Prod.map id f) \u03c1) = Measure.fst \u03c1\nthis :\n  Measure.map (Prod.map id f) \u03c1 =\n    \u2191(kernel.const Unit (Measure.fst (Measure.map (Prod.map id f) \u03c1)) \u2297\u2096\n          kernel.prodMkLeft Unit (kernel.map (Measure.condKernel \u03c1) f (_ : Measurable f)))\n      ()\nheq :\n  \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1,\n    \u2191(kernel.map (Measure.condKernel \u03c1) f (_ : Measurable f)) x =\n      \u2191(Measure.condKernel (Measure.map (Prod.map id f) \u03c1)) x\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1,\n    \u2200 (s : Set \u211d),\n      MeasurableSet s \u2192\n        \u2191\u2191(\u2191(Measure.condKernel (Measure.map (Prod.map id f) \u03c1)) x) s = \u2191\u2191(\u2191(Measure.condKernel \u03c1) x) (f \u207b\u00b9' s)", "state_after": "case h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1\nh\u03c1'' : \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1, \u2191(kernel.map \u03ba f (_ : Measurable f)) x = \u2191(Measure.condKernel \u03c1') x\nhprod : Measure.fst (Measure.map (Prod.map id f) \u03c1) = Measure.fst \u03c1\nthis :\n  Measure.map (Prod.map id f) \u03c1 =\n    \u2191(kernel.const Unit (Measure.fst (Measure.map (Prod.map id f) \u03c1)) \u2297\u2096\n          kernel.prodMkLeft Unit (kernel.map (Measure.condKernel \u03c1) f (_ : Measurable f)))\n      ()\nheq :\n  \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1,\n    \u2191(kernel.map (Measure.condKernel \u03c1) f (_ : Measurable f)) x =\n      \u2191(Measure.condKernel (Measure.map (Prod.map id f) \u03c1)) x\nx : \u03b1\nhx :\n  \u2191(kernel.map (Measure.condKernel \u03c1) f (_ : Measurable f)) x = \u2191(Measure.condKernel (Measure.map (Prod.map id f) \u03c1)) x\ns : Set \u211d\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(\u2191(Measure.condKernel (Measure.map (Prod.map id f) \u03c1)) x) s = \u2191\u2191(\u2191(Measure.condKernel \u03c1) x) (f \u207b\u00b9' s)"}, {"tactic": "rw [\u2190 hx, kernel.map_apply, Measure.map_apply hf.measurable hs]", "annotated_tactic": ["rw [\u2190 hx, <a>kernel.map_apply</a>, <a>Measure.map_apply</a> hf.measurable hs]", [{"full_name": "ProbabilityTheory.kernel.map_apply", "def_path": "Mathlib/Probability/Kernel/Composition.lean", "def_pos": [578, 9], "def_end_pos": [578, 18]}, {"full_name": "MeasureTheory.Measure.map_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1236, 9], "def_end_pos": [1236, 18]}]], "state_before": "case h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1\nh\u03c1'' : \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1, \u2191(kernel.map \u03ba f (_ : Measurable f)) x = \u2191(Measure.condKernel \u03c1') x\nhprod : Measure.fst (Measure.map (Prod.map id f) \u03c1) = Measure.fst \u03c1\nthis :\n  Measure.map (Prod.map id f) \u03c1 =\n    \u2191(kernel.const Unit (Measure.fst (Measure.map (Prod.map id f) \u03c1)) \u2297\u2096\n          kernel.prodMkLeft Unit (kernel.map (Measure.condKernel \u03c1) f (_ : Measurable f)))\n      ()\nheq :\n  \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1,\n    \u2191(kernel.map (Measure.condKernel \u03c1) f (_ : Measurable f)) x =\n      \u2191(Measure.condKernel (Measure.map (Prod.map id f) \u03c1)) x\nx : \u03b1\nhx :\n  \u2191(kernel.map (Measure.condKernel \u03c1) f (_ : Measurable f)) x = \u2191(Measure.condKernel (Measure.map (Prod.map id f) \u03c1)) x\ns : Set \u211d\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(\u2191(Measure.condKernel (Measure.map (Prod.map id f) \u03c1)) x) s = \u2191\u2191(\u2191(Measure.condKernel \u03c1) x) (f \u207b\u00b9' s)", "state_after": "no goals"}, {"tactic": "intro x", "annotated_tactic": ["intro x", []], "state_before": "case hinteq\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1\nh\u03c1'' : \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1, \u2191(kernel.map \u03ba f (_ : Measurable f)) x = \u2191(Measure.condKernel \u03c1') x\nhprod : Measure.fst (Measure.map (Prod.map id f) \u03c1) = Measure.fst \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\n\u22a2 \u2200 (x : \u03b1),\n    \u2191\u2191(Measure.map f (\u2191(Measure.condKernel \u03c1) x)) {c | (x, c) \u2208 s} =\n      \u2191\u2191(\u2191(Measure.condKernel \u03c1) x) {c | (x, c) \u2208 Prod.map id f \u207b\u00b9' s}", "state_after": "case hinteq\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1\nh\u03c1'' : \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1, \u2191(kernel.map \u03ba f (_ : Measurable f)) x = \u2191(Measure.condKernel \u03c1') x\nhprod : Measure.fst (Measure.map (Prod.map id f) \u03c1) = Measure.fst \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nx : \u03b1\n\u22a2 \u2191\u2191(Measure.map f (\u2191(Measure.condKernel \u03c1) x)) {c | (x, c) \u2208 s} =\n    \u2191\u2191(\u2191(Measure.condKernel \u03c1) x) {c | (x, c) \u2208 Prod.map id f \u207b\u00b9' s}"}, {"tactic": "rw [Measure.map_apply hf.measurable]", "annotated_tactic": ["rw [<a>Measure.map_apply</a> hf.measurable]", [{"full_name": "MeasureTheory.Measure.map_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1236, 9], "def_end_pos": [1236, 18]}]], "state_before": "case hinteq\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1\nh\u03c1'' : \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1, \u2191(kernel.map \u03ba f (_ : Measurable f)) x = \u2191(Measure.condKernel \u03c1') x\nhprod : Measure.fst (Measure.map (Prod.map id f) \u03c1) = Measure.fst \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nx : \u03b1\n\u22a2 \u2191\u2191(Measure.map f (\u2191(Measure.condKernel \u03c1) x)) {c | (x, c) \u2208 s} =\n    \u2191\u2191(\u2191(Measure.condKernel \u03c1) x) {c | (x, c) \u2208 Prod.map id f \u207b\u00b9' s}", "state_after": "case hinteq\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1\nh\u03c1'' : \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1, \u2191(kernel.map \u03ba f (_ : Measurable f)) x = \u2191(Measure.condKernel \u03c1') x\nhprod : Measure.fst (Measure.map (Prod.map id f) \u03c1) = Measure.fst \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nx : \u03b1\n\u22a2 \u2191\u2191(\u2191(Measure.condKernel \u03c1) x) (f \u207b\u00b9' {c | (x, c) \u2208 s}) =\n    \u2191\u2191(\u2191(Measure.condKernel \u03c1) x) {c | (x, c) \u2208 Prod.map id f \u207b\u00b9' s}\n\ncase hinteq\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1\nh\u03c1'' : \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1, \u2191(kernel.map \u03ba f (_ : Measurable f)) x = \u2191(Measure.condKernel \u03c1') x\nhprod : Measure.fst (Measure.map (Prod.map id f) \u03c1) = Measure.fst \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nx : \u03b1\n\u22a2 MeasurableSet {c | (x, c) \u2208 s}"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case hinteq\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1\nh\u03c1'' : \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1, \u2191(kernel.map \u03ba f (_ : Measurable f)) x = \u2191(Measure.condKernel \u03c1') x\nhprod : Measure.fst (Measure.map (Prod.map id f) \u03c1) = Measure.fst \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nx : \u03b1\n\u22a2 \u2191\u2191(\u2191(Measure.condKernel \u03c1) x) (f \u207b\u00b9' {c | (x, c) \u2208 s}) =\n    \u2191\u2191(\u2191(Measure.condKernel \u03c1) x) {c | (x, c) \u2208 Prod.map id f \u207b\u00b9' s}", "state_after": "no goals"}, {"tactic": "exact measurable_prod_mk_left hs", "annotated_tactic": ["exact <a>measurable_prod_mk_left</a> hs", [{"full_name": "measurable_prod_mk_left", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [736, 9], "def_end_pos": [736, 32]}]], "state_before": "case hinteq\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1\nh\u03c1'' : \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1, \u2191(kernel.map \u03ba f (_ : Measurable f)) x = \u2191(Measure.condKernel \u03c1') x\nhprod : Measure.fst (Measure.map (Prod.map id f) \u03c1) = Measure.fst \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nx : \u03b1\n\u22a2 MeasurableSet {c | (x, c) \u2208 s}", "state_after": "no goals"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case this.h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1\nh\u03c1'' : \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1, \u2191(kernel.map \u03ba f (_ : Measurable f)) x = \u2191(Measure.condKernel \u03c1') x\nhprod : Measure.fst (Measure.map (Prod.map id f) \u03c1) = Measure.fst \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nhinteq :\n  \u2200 (x : \u03b1),\n    \u2191\u2191(Measure.map f (\u2191(Measure.condKernel \u03c1) x)) {c | (x, c) \u2208 s} =\n      \u2191\u2191(\u2191(Measure.condKernel \u03c1) x) {c | (x, c) \u2208 Prod.map id f \u207b\u00b9' s}\n\u22a2 \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(Measure.condKernel \u03c1) a) {x | (a, x) \u2208 Prod.map id f \u207b\u00b9' s} \u2202Measure.fst \u03c1 =\n    \u222b\u207b (x : \u03b1), \u2191\u2191(\u2191(Measure.condKernel \u03c1) x) {c | (x, f c) \u2208 s} \u2202Measure.fst \u03c1", "state_after": "no goals"}, {"tactic": "exact measurable_id.prod_map hf.measurable hs", "annotated_tactic": ["exact measurable_id.prod_map hf.measurable hs", []], "state_before": "case this.h.hs\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1\nh\u03c1'' : \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1, \u2191(kernel.map \u03ba f (_ : Measurable f)) x = \u2191(Measure.condKernel \u03c1') x\nhprod : Measure.fst (Measure.map (Prod.map id f) \u03c1) = Measure.fst \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nhinteq :\n  \u2200 (x : \u03b1),\n    \u2191\u2191(Measure.map f (\u2191(Measure.condKernel \u03c1) x)) {c | (x, c) \u2208 s} =\n      \u2191\u2191(\u2191(Measure.condKernel \u03c1) x) {c | (x, c) \u2208 Prod.map id f \u207b\u00b9' s}\n\u22a2 MeasurableSet (Prod.map id f \u207b\u00b9' s)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Primrec.lean", "full_name": "Primrec\u2082.curry", "start": [462, 1], "end": [463, 41], "traced_tactics": [{"tactic": "rw [\u2190 uncurry, Function.uncurry_curry]", "annotated_tactic": ["rw [\u2190 <a>uncurry</a>, <a>Function.uncurry_curry</a>]", [{"full_name": "Primrec\u2082.uncurry", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [458, 9], "def_end_pos": [458, 16]}, {"full_name": "Function.uncurry_curry", "def_path": "Mathlib/Init/Function.lean", "def_pos": [226, 9], "def_end_pos": [226, 22]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03c3 : Type u_3\ninst\u271d\u00b2 : Primcodable \u03b1\ninst\u271d\u00b9 : Primcodable \u03b2\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u00d7 \u03b2 \u2192 \u03c3\n\u22a2 Primrec\u2082 (Function.curry f) \u2194 Primrec f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Ackermann.lean", "full_name": "ack_strict_mono_left'", "start": [202, 9], "end": [216, 53], "traced_tactics": [{"tactic": "simpa using one_lt_ack_succ_right m 0", "annotated_tactic": ["simpa using <a>one_lt_ack_succ_right</a> m 0", [{"full_name": "one_lt_ack_succ_right", "def_path": "Mathlib/Computability/Ackermann.lean", "def_pos": [127, 9], "def_end_pos": [127, 30]}]], "state_before": "m : \u2115\n_h : 0 < m + 1\n\u22a2 ack 0 0 < ack (m + 1) 0", "state_after": "no goals"}, {"tactic": "rw [ack_zero, ack_succ_succ]", "annotated_tactic": ["rw [<a>ack_zero</a>, <a>ack_succ_succ</a>]", [{"full_name": "ack_zero", "def_path": "Mathlib/Computability/Ackermann.lean", "def_pos": [70, 9], "def_end_pos": [70, 17]}, {"full_name": "ack_succ_succ", "def_path": "Mathlib/Computability/Ackermann.lean", "def_pos": [78, 9], "def_end_pos": [78, 22]}]], "state_before": "m n : \u2115\nh : 0 < m + 1\n\u22a2 ack 0 (n + 1) < ack (m + 1) (n + 1)", "state_after": "m n : \u2115\nh : 0 < m + 1\n\u22a2 n + 1 + 1 < ack m (ack (m + 1) n)"}, {"tactic": "apply lt_of_le_of_lt (le_trans _ <| add_le_add_left (add_add_one_le_ack _ _) m) (add_lt_ack _ _)", "annotated_tactic": ["apply <a>lt_of_le_of_lt</a> (<a>le_trans</a> _ <| <a>add_le_add_left</a> (<a>add_add_one_le_ack</a> _ _) m) (<a>add_lt_ack</a> _ _)", [{"full_name": "lt_of_le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [122, 9], "def_end_pos": [122, 23]}, {"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}, {"full_name": "add_le_add_left", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [49, 15], "def_end_pos": [49, 30]}, {"full_name": "add_add_one_le_ack", "def_path": "Mathlib/Computability/Ackermann.lean", "def_pos": [189, 9], "def_end_pos": [189, 27]}, {"full_name": "add_lt_ack", "def_path": "Mathlib/Computability/Ackermann.lean", "def_pos": [175, 9], "def_end_pos": [175, 19]}]], "state_before": "m n : \u2115\nh : 0 < m + 1\n\u22a2 n + 1 + 1 < ack m (ack (m + 1) n)", "state_after": "m n : \u2115\nh : 0 < m + 1\n\u22a2 n + 1 + 1 \u2264 m + (m + 1 + n + 1)"}, {"tactic": "linarith", "annotated_tactic": ["linarith", []], "state_before": "m n : \u2115\nh : 0 < m + 1\n\u22a2 n + 1 + 1 \u2264 m + (m + 1 + n + 1)", "state_after": "no goals"}, {"tactic": "simpa using ack_strict_mono_left' 1 ((add_lt_add_iff_right 1).1 h)", "annotated_tactic": ["simpa using ack_strict_mono_left' 1 ((<a>add_lt_add_iff_right</a> 1).1 h)", [{"full_name": "add_lt_add_iff_right", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [112, 3], "def_end_pos": [112, 14]}]], "state_before": "m\u2081 m\u2082 : \u2115\nh : m\u2081 + 1 < m\u2082 + 1\n\u22a2 ack (m\u2081 + 1) 0 < ack (m\u2082 + 1) 0", "state_after": "no goals"}, {"tactic": "rw [ack_succ_succ, ack_succ_succ]", "annotated_tactic": ["rw [<a>ack_succ_succ</a>, <a>ack_succ_succ</a>]", [{"full_name": "ack_succ_succ", "def_path": "Mathlib/Computability/Ackermann.lean", "def_pos": [78, 9], "def_end_pos": [78, 22]}, {"full_name": "ack_succ_succ", "def_path": "Mathlib/Computability/Ackermann.lean", "def_pos": [78, 9], "def_end_pos": [78, 22]}]], "state_before": "m\u2081 m\u2082 n : \u2115\nh : m\u2081 + 1 < m\u2082 + 1\n\u22a2 ack (m\u2081 + 1) (n + 1) < ack (m\u2082 + 1) (n + 1)", "state_after": "m\u2081 m\u2082 n : \u2115\nh : m\u2081 + 1 < m\u2082 + 1\n\u22a2 ack m\u2081 (ack (m\u2081 + 1) n) < ack m\u2082 (ack (m\u2082 + 1) n)"}, {"tactic": "exact\n  (ack_strict_mono_left' _ <| (add_lt_add_iff_right 1).1 h).trans\n    (ack_strictMono_right _ <| ack_strict_mono_left' n h)", "annotated_tactic": ["exact\n      (ack_strict_mono_left' _ <| (<a>add_lt_add_iff_right</a> 1).1 h).<a>trans</a>\n        (<a>ack_strictMono_right</a> _ <| ack_strict_mono_left' n h)", [{"full_name": "add_lt_add_iff_right", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [112, 3], "def_end_pos": [112, 14]}, {"full_name": "LT.lt.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [144, 7], "def_end_pos": [144, 18]}, {"full_name": "ack_strictMono_right", "def_path": "Mathlib/Computability/Ackermann.lean", "def_pos": [136, 9], "def_end_pos": [136, 29]}]], "state_before": "m\u2081 m\u2082 n : \u2115\nh : m\u2081 + 1 < m\u2082 + 1\n\u22a2 ack m\u2081 (ack (m\u2081 + 1) n) < ack m\u2082 (ack (m\u2082 + 1) n)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Decomposition/Jordan.lean", "full_name": "MeasureTheory.SignedMeasure.toJordanDecomposition_eq", "start": [488, 1], "end": [490, 48], "traced_tactics": [{"tactic": "rw [h, toJordanDecomposition_toSignedMeasure]", "annotated_tactic": ["rw [h, <a>toJordanDecomposition_toSignedMeasure</a>]", [{"full_name": "MeasureTheory.JordanDecomposition.toJordanDecomposition_toSignedMeasure", "def_path": "Mathlib/MeasureTheory/Decomposition/Jordan.lean", "def_pos": [426, 9], "def_end_pos": [426, 46]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\nj : JordanDecomposition \u03b1\nh : s = toSignedMeasure j\n\u22a2 toJordanDecomposition s = j", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean", "full_name": "MeasureTheory.condexpIndL1_smul", "start": [212, 1], "end": [219, 42], "traced_tactics": [{"tactic": "by_cases hs : MeasurableSet s", "annotated_tactic": ["by_cases hs : <a>MeasurableSet</a> s", [{"full_name": "MeasurableSet", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [64, 5], "def_end_pos": [64, 18]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : CompleteSpace F'\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \u211d G'\ninst\u271d\u00b2 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nc : \u211d\nx : G\n\u22a2 condexpIndL1 hm \u03bc s (c \u2022 x) = c \u2022 condexpIndL1 hm \u03bc s x", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : CompleteSpace F'\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \u211d G'\ninst\u271d\u00b2 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nc : \u211d\nx : G\nhs : MeasurableSet s\n\u22a2 condexpIndL1 hm \u03bc s (c \u2022 x) = c \u2022 condexpIndL1 hm \u03bc s x\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : CompleteSpace F'\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \u211d G'\ninst\u271d\u00b2 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nc : \u211d\nx : G\nhs : \u00acMeasurableSet s\n\u22a2 condexpIndL1 hm \u03bc s (c \u2022 x) = c \u2022 condexpIndL1 hm \u03bc s x"}, {"tactic": "swap", "annotated_tactic": ["swap", []], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : CompleteSpace F'\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \u211d G'\ninst\u271d\u00b2 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nc : \u211d\nx : G\nhs : MeasurableSet s\n\u22a2 condexpIndL1 hm \u03bc s (c \u2022 x) = c \u2022 condexpIndL1 hm \u03bc s x\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : CompleteSpace F'\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \u211d G'\ninst\u271d\u00b2 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nc : \u211d\nx : G\nhs : \u00acMeasurableSet s\n\u22a2 condexpIndL1 hm \u03bc s (c \u2022 x) = c \u2022 condexpIndL1 hm \u03bc s x", "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : CompleteSpace F'\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \u211d G'\ninst\u271d\u00b2 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nc : \u211d\nx : G\nhs : \u00acMeasurableSet s\n\u22a2 condexpIndL1 hm \u03bc s (c \u2022 x) = c \u2022 condexpIndL1 hm \u03bc s x\n\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : CompleteSpace F'\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \u211d G'\ninst\u271d\u00b2 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nc : \u211d\nx : G\nhs : MeasurableSet s\n\u22a2 condexpIndL1 hm \u03bc s (c \u2022 x) = c \u2022 condexpIndL1 hm \u03bc s x"}, {"tactic": "by_cases h\u03bcs : \u03bc s = \u221e", "annotated_tactic": ["by_cases h\u03bcs : \u03bc s = \u221e", []], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : CompleteSpace F'\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \u211d G'\ninst\u271d\u00b2 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nc : \u211d\nx : G\nhs : MeasurableSet s\n\u22a2 condexpIndL1 hm \u03bc s (c \u2022 x) = c \u2022 condexpIndL1 hm \u03bc s x", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : CompleteSpace F'\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \u211d G'\ninst\u271d\u00b2 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nc : \u211d\nx : G\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s = \u22a4\n\u22a2 condexpIndL1 hm \u03bc s (c \u2022 x) = c \u2022 condexpIndL1 hm \u03bc s x\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : CompleteSpace F'\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \u211d G'\ninst\u271d\u00b2 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nc : \u211d\nx : G\nhs : MeasurableSet s\nh\u03bcs : \u00ac\u2191\u2191\u03bc s = \u22a4\n\u22a2 condexpIndL1 hm \u03bc s (c \u2022 x) = c \u2022 condexpIndL1 hm \u03bc s x"}, {"tactic": "simp_rw [condexpIndL1_of_not_measurableSet hs]", "annotated_tactic": ["simp_rw [<a>condexpIndL1_of_not_measurableSet</a> hs]", [{"full_name": "MeasureTheory.condexpIndL1_of_not_measurableSet", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean", "def_pos": [197, 9], "def_end_pos": [197, 42]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : CompleteSpace F'\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \u211d G'\ninst\u271d\u00b2 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nc : \u211d\nx : G\nhs : \u00acMeasurableSet s\n\u22a2 condexpIndL1 hm \u03bc s (c \u2022 x) = c \u2022 condexpIndL1 hm \u03bc s x", "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : CompleteSpace F'\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \u211d G'\ninst\u271d\u00b2 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nc : \u211d\nx : G\nhs : \u00acMeasurableSet s\n\u22a2 0 = c \u2022 0"}, {"tactic": "rw [smul_zero]", "annotated_tactic": ["rw [<a>smul_zero</a>]", [{"full_name": "smul_zero", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [732, 9], "def_end_pos": [732, 18]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : CompleteSpace F'\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \u211d G'\ninst\u271d\u00b2 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nc : \u211d\nx : G\nhs : \u00acMeasurableSet s\n\u22a2 0 = c \u2022 0", "state_after": "no goals"}, {"tactic": "simp_rw [condexpIndL1_of_measure_eq_top h\u03bcs]", "annotated_tactic": ["simp_rw [<a>condexpIndL1_of_measure_eq_top</a> h\u03bcs]", [{"full_name": "MeasureTheory.condexpIndL1_of_measure_eq_top", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean", "def_pos": [192, 9], "def_end_pos": [192, 39]}]], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : CompleteSpace F'\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \u211d G'\ninst\u271d\u00b2 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nc : \u211d\nx : G\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s = \u22a4\n\u22a2 condexpIndL1 hm \u03bc s (c \u2022 x) = c \u2022 condexpIndL1 hm \u03bc s x", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : CompleteSpace F'\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \u211d G'\ninst\u271d\u00b2 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nc : \u211d\nx : G\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s = \u22a4\n\u22a2 0 = c \u2022 0"}, {"tactic": "rw [smul_zero]", "annotated_tactic": ["rw [<a>smul_zero</a>]", [{"full_name": "smul_zero", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [732, 9], "def_end_pos": [732, 18]}]], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : CompleteSpace F'\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \u211d G'\ninst\u271d\u00b2 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nc : \u211d\nx : G\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s = \u22a4\n\u22a2 0 = c \u2022 0", "state_after": "no goals"}, {"tactic": "simp_rw [condexpIndL1_of_measurableSet_of_measure_ne_top hs h\u03bcs]", "annotated_tactic": ["simp_rw [<a>condexpIndL1_of_measurableSet_of_measure_ne_top</a> hs h\u03bcs]", [{"full_name": "MeasureTheory.condexpIndL1_of_measurableSet_of_measure_ne_top", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean", "def_pos": [187, 9], "def_end_pos": [187, 56]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : CompleteSpace F'\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \u211d G'\ninst\u271d\u00b2 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nc : \u211d\nx : G\nhs : MeasurableSet s\nh\u03bcs : \u00ac\u2191\u2191\u03bc s = \u22a4\n\u22a2 condexpIndL1 hm \u03bc s (c \u2022 x) = c \u2022 condexpIndL1 hm \u03bc s x", "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : CompleteSpace F'\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \u211d G'\ninst\u271d\u00b2 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nc : \u211d\nx : G\nhs : MeasurableSet s\nh\u03bcs : \u00ac\u2191\u2191\u03bc s = \u22a4\n\u22a2 condexpIndL1Fin hm hs h\u03bcs (c \u2022 x) = c \u2022 condexpIndL1Fin hm hs h\u03bcs x"}, {"tactic": "exact condexpIndL1Fin_smul hs h\u03bcs c x", "annotated_tactic": ["exact <a>condexpIndL1Fin_smul</a> hs h\u03bcs c x", [{"full_name": "MeasureTheory.condexpIndL1Fin_smul", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean", "def_pos": [105, 9], "def_end_pos": [105, 29]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : CompleteSpace F'\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \u211d G'\ninst\u271d\u00b2 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nc : \u211d\nx : G\nhs : MeasurableSet s\nh\u03bcs : \u00ac\u2191\u2191\u03bc s = \u22a4\n\u22a2 condexpIndL1Fin hm hs h\u03bcs (c \u2022 x) = c \u2022 condexpIndL1Fin hm hs h\u03bcs x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/OpenPos.lean", "full_name": "MeasureTheory.Measure.eqOn_Icc_of_ae_eq", "start": [203, 1], "end": [206, 65], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "full_name": "MeasureTheory.setToFun_smul_left'", "start": [1336, 1], "end": [1342, 45], "traced_tactics": [{"tactic": "by_cases hf : Integrable f \u03bc", "annotated_tactic": ["by_cases hf : <a>Integrable</a> f \u03bc", [{"full_name": "MeasureTheory.Integrable", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [442, 5], "def_end_pos": [442, 15]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc T' C'\nc : \u211d\nh_smul : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T' s = c \u2022 T s\nf : \u03b1 \u2192 E\n\u22a2 setToFun \u03bc T' hT' f = c \u2022 setToFun \u03bc T hT f", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc T' C'\nc : \u211d\nh_smul : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T' s = c \u2022 T s\nf : \u03b1 \u2192 E\nhf : Integrable f\n\u22a2 setToFun \u03bc T' hT' f = c \u2022 setToFun \u03bc T hT f\n\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc T' C'\nc : \u211d\nh_smul : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T' s = c \u2022 T s\nf : \u03b1 \u2192 E\nhf : \u00acIntegrable f\n\u22a2 setToFun \u03bc T' hT' f = c \u2022 setToFun \u03bc T hT f"}, {"tactic": "simp_rw [setToFun_eq _ hf, L1.setToL1_smul_left' hT hT' c h_smul]", "annotated_tactic": ["simp_rw [<a>setToFun_eq</a> _ hf, <a>L1.setToL1_smul_left'</a> hT hT' c h_smul]", [{"full_name": "MeasureTheory.setToFun_eq", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [1276, 9], "def_end_pos": [1276, 20]}, {"full_name": "MeasureTheory.L1.setToL1_smul_left'", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [1116, 9], "def_end_pos": [1116, 27]}]], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc T' C'\nc : \u211d\nh_smul : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T' s = c \u2022 T s\nf : \u03b1 \u2192 E\nhf : Integrable f\n\u22a2 setToFun \u03bc T' hT' f = c \u2022 setToFun \u03bc T hT f", "state_after": "no goals"}, {"tactic": "simp_rw [setToFun_undef _ hf, smul_zero]", "annotated_tactic": ["simp_rw [<a>setToFun_undef</a> _ hf, <a>smul_zero</a>]", [{"full_name": "MeasureTheory.setToFun_undef", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [1286, 9], "def_end_pos": [1286, 23]}, {"full_name": "smul_zero", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [732, 9], "def_end_pos": [732, 18]}]], "state_before": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc T' C'\nc : \u211d\nh_smul : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T' s = c \u2022 T s\nf : \u03b1 \u2192 E\nhf : \u00acIntegrable f\n\u22a2 setToFun \u03bc T' hT' f = c \u2022 setToFun \u03bc T hT f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Primrec.lean", "full_name": "Primrec.list_headI", "start": [1044, 1], "end": [1045, 69], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Haar/OfBasis.lean", "full_name": "parallelepiped_single", "start": [139, 1], "end": [167, 34], "traced_tactics": [{"tactic": "ext x", "annotated_tactic": ["ext x", []], "state_before": "\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2076 : Fintype \u03b9\ninst\u271d\u2075 : Fintype \u03b9'\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \u211d E\ninst\u271d\u00b2 : AddCommGroup F\ninst\u271d\u00b9 : Module \u211d F\ninst\u271d : DecidableEq \u03b9\na : \u03b9 \u2192 \u211d\n\u22a2 (parallelepiped fun i => Pi.single i (a i)) = uIcc 0 a", "state_after": "case h\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2076 : Fintype \u03b9\ninst\u271d\u2075 : Fintype \u03b9'\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \u211d E\ninst\u271d\u00b2 : AddCommGroup F\ninst\u271d\u00b9 : Module \u211d F\ninst\u271d : DecidableEq \u03b9\na x : \u03b9 \u2192 \u211d\n\u22a2 (x \u2208 parallelepiped fun i => Pi.single i (a i)) \u2194 x \u2208 uIcc 0 a"}, {"tactic": "simp_rw [Set.uIcc, mem_parallelepiped_iff, Set.mem_Icc, Pi.le_def, \u2190 forall_and, Pi.inf_apply,\n  Pi.sup_apply, \u2190 Pi.single_smul', Pi.one_apply, Pi.zero_apply, \u2190 Pi.smul_apply',\n  Finset.univ_sum_single (_ : \u03b9 \u2192 \u211d)]", "annotated_tactic": ["simp_rw [<a>Set.uIcc</a>, <a>mem_parallelepiped_iff</a>, <a>Set.mem_Icc</a>, <a>Pi.le_def</a>, \u2190 <a>forall_and</a>, <a>Pi.inf_apply</a>,\n    <a>Pi.sup_apply</a>, \u2190 <a>Pi.single_smul'</a>, <a>Pi.one_apply</a>, <a>Pi.zero_apply</a>, \u2190 <a>Pi.smul_apply'</a>,\n    <a>Finset.univ_sum_single</a> (_ : \u03b9 \u2192 \u211d)]", [{"full_name": "Set.uIcc", "def_path": "Mathlib/Data/Set/Intervals/UnorderedInterval.lean", "def_pos": [54, 5], "def_end_pos": [54, 9]}, {"full_name": "mem_parallelepiped_iff", "def_path": "Mathlib/MeasureTheory/Measure/Haar/OfBasis.lean", "def_pos": [48, 9], "def_end_pos": [48, 31]}, {"full_name": "Set.mem_Icc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [131, 9], "def_end_pos": [131, 16]}, {"full_name": "Pi.le_def", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [814, 9], "def_end_pos": [814, 18]}, {"full_name": "forall_and", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [426, 9], "def_end_pos": [426, 19]}, {"full_name": "Pi.inf_apply", "def_path": "Mathlib/Order/Lattice.lean", "def_pos": [1025, 9], "def_end_pos": [1025, 18]}, {"full_name": "Pi.sup_apply", "def_path": "Mathlib/Order/Lattice.lean", "def_pos": [1013, 9], "def_end_pos": [1013, 18]}, {"full_name": "Pi.single_smul'", "def_path": "Mathlib/GroupTheory/GroupAction/Pi.lean", "def_pos": [188, 9], "def_end_pos": [188, 21]}, {"full_name": "Pi.one_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [47, 9], "def_end_pos": [47, 18]}, {"full_name": "Pi.zero_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [46, 3], "def_end_pos": [46, 14]}, {"full_name": "Pi.smul_apply'", "def_path": "Mathlib/GroupTheory/GroupAction/Pi.lean", "def_pos": [44, 9], "def_end_pos": [44, 20]}, {"full_name": "Finset.univ_sum_single", "def_path": "Mathlib/Algebra/BigOperators/Pi.lean", "def_pos": [77, 3], "def_end_pos": [77, 14]}]], "state_before": "case h\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2076 : Fintype \u03b9\ninst\u271d\u2075 : Fintype \u03b9'\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \u211d E\ninst\u271d\u00b2 : AddCommGroup F\ninst\u271d\u00b9 : Module \u211d F\ninst\u271d : DecidableEq \u03b9\na x : \u03b9 \u2192 \u211d\n\u22a2 (x \u2208 parallelepiped fun i => Pi.single i (a i)) \u2194 x \u2208 uIcc 0 a", "state_after": "case h\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2076 : Fintype \u03b9\ninst\u271d\u2075 : Fintype \u03b9'\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \u211d E\ninst\u271d\u00b2 : AddCommGroup F\ninst\u271d\u00b9 : Module \u211d F\ninst\u271d : DecidableEq \u03b9\na x : \u03b9 \u2192 \u211d\n\u22a2 (\u2203 t h, x = fun i => (t \u2022 a) i) \u2194 \u2200 (x_1 : \u03b9), 0 \u2293 a x_1 \u2264 x x_1 \u2227 x x_1 \u2264 0 \u2294 a x_1"}, {"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "case h\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2076 : Fintype \u03b9\ninst\u271d\u2075 : Fintype \u03b9'\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \u211d E\ninst\u271d\u00b2 : AddCommGroup F\ninst\u271d\u00b9 : Module \u211d F\ninst\u271d : DecidableEq \u03b9\na x : \u03b9 \u2192 \u211d\n\u22a2 (\u2203 t h, x = fun i => (t \u2022 a) i) \u2194 \u2200 (x_1 : \u03b9), 0 \u2293 a x_1 \u2264 x x_1 \u2227 x x_1 \u2264 0 \u2294 a x_1", "state_after": "case h.mp\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2076 : Fintype \u03b9\ninst\u271d\u2075 : Fintype \u03b9'\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \u211d E\ninst\u271d\u00b2 : AddCommGroup F\ninst\u271d\u00b9 : Module \u211d F\ninst\u271d : DecidableEq \u03b9\na x : \u03b9 \u2192 \u211d\n\u22a2 (\u2203 t h, x = fun i => (t \u2022 a) i) \u2192 \u2200 (x_1 : \u03b9), 0 \u2293 a x_1 \u2264 x x_1 \u2227 x x_1 \u2264 0 \u2294 a x_1\n\ncase h.mpr\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2076 : Fintype \u03b9\ninst\u271d\u2075 : Fintype \u03b9'\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \u211d E\ninst\u271d\u00b2 : AddCommGroup F\ninst\u271d\u00b9 : Module \u211d F\ninst\u271d : DecidableEq \u03b9\na x : \u03b9 \u2192 \u211d\n\u22a2 (\u2200 (x_1 : \u03b9), 0 \u2293 a x_1 \u2264 x x_1 \u2227 x x_1 \u2264 0 \u2294 a x_1) \u2192 \u2203 t h, x = fun i => (t \u2022 a) i"}, {"tactic": "rintro \u27e8t, ht, rfl\u27e9 i", "annotated_tactic": ["rintro \u27e8t, ht, rfl\u27e9 i", []], "state_before": "case h.mp\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2076 : Fintype \u03b9\ninst\u271d\u2075 : Fintype \u03b9'\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \u211d E\ninst\u271d\u00b2 : AddCommGroup F\ninst\u271d\u00b9 : Module \u211d F\ninst\u271d : DecidableEq \u03b9\na x : \u03b9 \u2192 \u211d\n\u22a2 (\u2203 t h, x = fun i => (t \u2022 a) i) \u2192 \u2200 (x_1 : \u03b9), 0 \u2293 a x_1 \u2264 x x_1 \u2227 x x_1 \u2264 0 \u2294 a x_1", "state_after": "case h.mp.intro.intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2076 : Fintype \u03b9\ninst\u271d\u2075 : Fintype \u03b9'\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \u211d E\ninst\u271d\u00b2 : AddCommGroup F\ninst\u271d\u00b9 : Module \u211d F\ninst\u271d : DecidableEq \u03b9\na t : \u03b9 \u2192 \u211d\nht : \u2200 (x : \u03b9), 0 \u2264 t x \u2227 t x \u2264 1\ni : \u03b9\n\u22a2 0 \u2293 a i \u2264 (fun i => (t \u2022 a) i) i \u2227 (fun i => (t \u2022 a) i) i \u2264 0 \u2294 a i"}, {"tactic": "specialize ht i", "annotated_tactic": ["specialize ht i", []], "state_before": "case h.mp.intro.intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2076 : Fintype \u03b9\ninst\u271d\u2075 : Fintype \u03b9'\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \u211d E\ninst\u271d\u00b2 : AddCommGroup F\ninst\u271d\u00b9 : Module \u211d F\ninst\u271d : DecidableEq \u03b9\na t : \u03b9 \u2192 \u211d\nht : \u2200 (x : \u03b9), 0 \u2264 t x \u2227 t x \u2264 1\ni : \u03b9\n\u22a2 0 \u2293 a i \u2264 (fun i => (t \u2022 a) i) i \u2227 (fun i => (t \u2022 a) i) i \u2264 0 \u2294 a i", "state_after": "case h.mp.intro.intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2076 : Fintype \u03b9\ninst\u271d\u2075 : Fintype \u03b9'\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \u211d E\ninst\u271d\u00b2 : AddCommGroup F\ninst\u271d\u00b9 : Module \u211d F\ninst\u271d : DecidableEq \u03b9\na t : \u03b9 \u2192 \u211d\ni : \u03b9\nht : 0 \u2264 t i \u2227 t i \u2264 1\n\u22a2 0 \u2293 a i \u2264 (fun i => (t \u2022 a) i) i \u2227 (fun i => (t \u2022 a) i) i \u2264 0 \u2294 a i"}, {"tactic": "simp_rw [smul_eq_mul, Pi.mul_apply]", "annotated_tactic": ["simp_rw [<a>smul_eq_mul</a>, <a>Pi.mul_apply</a>]", [{"full_name": "smul_eq_mul", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [93, 9], "def_end_pos": [93, 20]}, {"full_name": "Pi.mul_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [83, 9], "def_end_pos": [83, 18]}]], "state_before": "case h.mp.intro.intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2076 : Fintype \u03b9\ninst\u271d\u2075 : Fintype \u03b9'\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \u211d E\ninst\u271d\u00b2 : AddCommGroup F\ninst\u271d\u00b9 : Module \u211d F\ninst\u271d : DecidableEq \u03b9\na t : \u03b9 \u2192 \u211d\ni : \u03b9\nht : 0 \u2264 t i \u2227 t i \u2264 1\n\u22a2 0 \u2293 a i \u2264 (fun i => (t \u2022 a) i) i \u2227 (fun i => (t \u2022 a) i) i \u2264 0 \u2294 a i", "state_after": "case h.mp.intro.intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2076 : Fintype \u03b9\ninst\u271d\u2075 : Fintype \u03b9'\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \u211d E\ninst\u271d\u00b2 : AddCommGroup F\ninst\u271d\u00b9 : Module \u211d F\ninst\u271d : DecidableEq \u03b9\na t : \u03b9 \u2192 \u211d\ni : \u03b9\nht : 0 \u2264 t i \u2227 t i \u2264 1\n\u22a2 0 \u2293 a i \u2264 t i * a i \u2227 t i * a i \u2264 0 \u2294 a i"}, {"tactic": "cases' le_total (a i) 0 with hai hai", "annotated_tactic": ["cases' <a>le_total</a> (a i) 0 with hai hai", [{"full_name": "le_total", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [295, 9], "def_end_pos": [295, 17]}]], "state_before": "case h.mp.intro.intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2076 : Fintype \u03b9\ninst\u271d\u2075 : Fintype \u03b9'\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \u211d E\ninst\u271d\u00b2 : AddCommGroup F\ninst\u271d\u00b9 : Module \u211d F\ninst\u271d : DecidableEq \u03b9\na t : \u03b9 \u2192 \u211d\ni : \u03b9\nht : 0 \u2264 t i \u2227 t i \u2264 1\n\u22a2 0 \u2293 a i \u2264 t i * a i \u2227 t i * a i \u2264 0 \u2294 a i", "state_after": "case h.mp.intro.intro.inl\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2076 : Fintype \u03b9\ninst\u271d\u2075 : Fintype \u03b9'\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \u211d E\ninst\u271d\u00b2 : AddCommGroup F\ninst\u271d\u00b9 : Module \u211d F\ninst\u271d : DecidableEq \u03b9\na t : \u03b9 \u2192 \u211d\ni : \u03b9\nht : 0 \u2264 t i \u2227 t i \u2264 1\nhai : a i \u2264 0\n\u22a2 0 \u2293 a i \u2264 t i * a i \u2227 t i * a i \u2264 0 \u2294 a i\n\ncase h.mp.intro.intro.inr\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2076 : Fintype \u03b9\ninst\u271d\u2075 : Fintype \u03b9'\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \u211d E\ninst\u271d\u00b2 : AddCommGroup F\ninst\u271d\u00b9 : Module \u211d F\ninst\u271d : DecidableEq \u03b9\na t : \u03b9 \u2192 \u211d\ni : \u03b9\nht : 0 \u2264 t i \u2227 t i \u2264 1\nhai : 0 \u2264 a i\n\u22a2 0 \u2293 a i \u2264 t i * a i \u2227 t i * a i \u2264 0 \u2294 a i"}, {"tactic": "rw [sup_eq_left.mpr hai, inf_eq_right.mpr hai]", "annotated_tactic": ["rw [sup_eq_left.mpr hai, inf_eq_right.mpr hai]", []], "state_before": "case h.mp.intro.intro.inl\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2076 : Fintype \u03b9\ninst\u271d\u2075 : Fintype \u03b9'\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \u211d E\ninst\u271d\u00b2 : AddCommGroup F\ninst\u271d\u00b9 : Module \u211d F\ninst\u271d : DecidableEq \u03b9\na t : \u03b9 \u2192 \u211d\ni : \u03b9\nht : 0 \u2264 t i \u2227 t i \u2264 1\nhai : a i \u2264 0\n\u22a2 0 \u2293 a i \u2264 t i * a i \u2227 t i * a i \u2264 0 \u2294 a i", "state_after": "case h.mp.intro.intro.inl\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2076 : Fintype \u03b9\ninst\u271d\u2075 : Fintype \u03b9'\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \u211d E\ninst\u271d\u00b2 : AddCommGroup F\ninst\u271d\u00b9 : Module \u211d F\ninst\u271d : DecidableEq \u03b9\na t : \u03b9 \u2192 \u211d\ni : \u03b9\nht : 0 \u2264 t i \u2227 t i \u2264 1\nhai : a i \u2264 0\n\u22a2 a i \u2264 t i * a i \u2227 t i * a i \u2264 0"}, {"tactic": "exact \u27e8le_mul_of_le_one_left hai ht.2, mul_nonpos_of_nonneg_of_nonpos ht.1 hai\u27e9", "annotated_tactic": ["exact \u27e8<a>le_mul_of_le_one_left</a> hai ht.2, <a>mul_nonpos_of_nonneg_of_nonpos</a> ht.1 hai\u27e9", [{"full_name": "le_mul_of_le_one_left", "def_path": "Mathlib/Algebra/Order/Ring/Defs.lean", "def_pos": [394, 9], "def_end_pos": [394, 30]}, {"full_name": "mul_nonpos_of_nonneg_of_nonpos", "def_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "def_pos": [383, 9], "def_end_pos": [383, 39]}]], "state_before": "case h.mp.intro.intro.inl\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2076 : Fintype \u03b9\ninst\u271d\u2075 : Fintype \u03b9'\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \u211d E\ninst\u271d\u00b2 : AddCommGroup F\ninst\u271d\u00b9 : Module \u211d F\ninst\u271d : DecidableEq \u03b9\na t : \u03b9 \u2192 \u211d\ni : \u03b9\nht : 0 \u2264 t i \u2227 t i \u2264 1\nhai : a i \u2264 0\n\u22a2 a i \u2264 t i * a i \u2227 t i * a i \u2264 0", "state_after": "no goals"}, {"tactic": "rw [sup_eq_right.mpr hai, inf_eq_left.mpr hai]", "annotated_tactic": ["rw [sup_eq_right.mpr hai, inf_eq_left.mpr hai]", []], "state_before": "case h.mp.intro.intro.inr\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2076 : Fintype \u03b9\ninst\u271d\u2075 : Fintype \u03b9'\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \u211d E\ninst\u271d\u00b2 : AddCommGroup F\ninst\u271d\u00b9 : Module \u211d F\ninst\u271d : DecidableEq \u03b9\na t : \u03b9 \u2192 \u211d\ni : \u03b9\nht : 0 \u2264 t i \u2227 t i \u2264 1\nhai : 0 \u2264 a i\n\u22a2 0 \u2293 a i \u2264 t i * a i \u2227 t i * a i \u2264 0 \u2294 a i", "state_after": "case h.mp.intro.intro.inr\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2076 : Fintype \u03b9\ninst\u271d\u2075 : Fintype \u03b9'\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \u211d E\ninst\u271d\u00b2 : AddCommGroup F\ninst\u271d\u00b9 : Module \u211d F\ninst\u271d : DecidableEq \u03b9\na t : \u03b9 \u2192 \u211d\ni : \u03b9\nht : 0 \u2264 t i \u2227 t i \u2264 1\nhai : 0 \u2264 a i\n\u22a2 0 \u2264 t i * a i \u2227 t i * a i \u2264 a i"}, {"tactic": "exact \u27e8mul_nonneg ht.1 hai, mul_le_of_le_one_left hai ht.2\u27e9", "annotated_tactic": ["exact \u27e8<a>mul_nonneg</a> ht.1 hai, <a>mul_le_of_le_one_left</a> hai ht.2\u27e9", [{"full_name": "mul_nonneg", "def_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "def_pos": [380, 7], "def_end_pos": [380, 17]}, {"full_name": "mul_le_of_le_one_left", "def_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "def_pos": [666, 9], "def_end_pos": [666, 30]}]], "state_before": "case h.mp.intro.intro.inr\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2076 : Fintype \u03b9\ninst\u271d\u2075 : Fintype \u03b9'\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \u211d E\ninst\u271d\u00b2 : AddCommGroup F\ninst\u271d\u00b9 : Module \u211d F\ninst\u271d : DecidableEq \u03b9\na t : \u03b9 \u2192 \u211d\ni : \u03b9\nht : 0 \u2264 t i \u2227 t i \u2264 1\nhai : 0 \u2264 a i\n\u22a2 0 \u2264 t i * a i \u2227 t i * a i \u2264 a i", "state_after": "no goals"}, {"tactic": "intro h", "annotated_tactic": ["intro h", []], "state_before": "case h.mpr\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2076 : Fintype \u03b9\ninst\u271d\u2075 : Fintype \u03b9'\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \u211d E\ninst\u271d\u00b2 : AddCommGroup F\ninst\u271d\u00b9 : Module \u211d F\ninst\u271d : DecidableEq \u03b9\na x : \u03b9 \u2192 \u211d\n\u22a2 (\u2200 (x_1 : \u03b9), 0 \u2293 a x_1 \u2264 x x_1 \u2227 x x_1 \u2264 0 \u2294 a x_1) \u2192 \u2203 t h, x = fun i => (t \u2022 a) i", "state_after": "case h.mpr\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2076 : Fintype \u03b9\ninst\u271d\u2075 : Fintype \u03b9'\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \u211d E\ninst\u271d\u00b2 : AddCommGroup F\ninst\u271d\u00b9 : Module \u211d F\ninst\u271d : DecidableEq \u03b9\na x : \u03b9 \u2192 \u211d\nh : \u2200 (x_1 : \u03b9), 0 \u2293 a x_1 \u2264 x x_1 \u2227 x x_1 \u2264 0 \u2294 a x_1\n\u22a2 \u2203 t h, x = fun i => (t \u2022 a) i"}, {"tactic": "refine' \u27e8fun i => x i / a i, fun i => _, funext fun i => _\u27e9", "annotated_tactic": ["refine' \u27e8fun i => x i / a i, fun i => _, <a>funext</a> fun i => _\u27e9", [{"full_name": "funext", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [1555, 9], "def_end_pos": [1555, 15]}]], "state_before": "case h.mpr\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2076 : Fintype \u03b9\ninst\u271d\u2075 : Fintype \u03b9'\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \u211d E\ninst\u271d\u00b2 : AddCommGroup F\ninst\u271d\u00b9 : Module \u211d F\ninst\u271d : DecidableEq \u03b9\na x : \u03b9 \u2192 \u211d\nh : \u2200 (x_1 : \u03b9), 0 \u2293 a x_1 \u2264 x x_1 \u2227 x x_1 \u2264 0 \u2294 a x_1\n\u22a2 \u2203 t h, x = fun i => (t \u2022 a) i", "state_after": "case h.mpr.refine'_1\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2076 : Fintype \u03b9\ninst\u271d\u2075 : Fintype \u03b9'\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \u211d E\ninst\u271d\u00b2 : AddCommGroup F\ninst\u271d\u00b9 : Module \u211d F\ninst\u271d : DecidableEq \u03b9\na x : \u03b9 \u2192 \u211d\nh : \u2200 (x_1 : \u03b9), 0 \u2293 a x_1 \u2264 x x_1 \u2227 x x_1 \u2264 0 \u2294 a x_1\ni : \u03b9\n\u22a2 0 \u2264 (fun i => x i / a i) i \u2227 (fun i => x i / a i) i \u2264 1\n\ncase h.mpr.refine'_2\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2076 : Fintype \u03b9\ninst\u271d\u2075 : Fintype \u03b9'\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \u211d E\ninst\u271d\u00b2 : AddCommGroup F\ninst\u271d\u00b9 : Module \u211d F\ninst\u271d : DecidableEq \u03b9\na x : \u03b9 \u2192 \u211d\nh : \u2200 (x_1 : \u03b9), 0 \u2293 a x_1 \u2264 x x_1 \u2227 x x_1 \u2264 0 \u2294 a x_1\ni : \u03b9\n\u22a2 x i = ((fun i => x i / a i) \u2022 a) i"}, {"tactic": "specialize h i", "annotated_tactic": ["specialize h i", []], "state_before": "case h.mpr.refine'_1\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2076 : Fintype \u03b9\ninst\u271d\u2075 : Fintype \u03b9'\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \u211d E\ninst\u271d\u00b2 : AddCommGroup F\ninst\u271d\u00b9 : Module \u211d F\ninst\u271d : DecidableEq \u03b9\na x : \u03b9 \u2192 \u211d\nh : \u2200 (x_1 : \u03b9), 0 \u2293 a x_1 \u2264 x x_1 \u2227 x x_1 \u2264 0 \u2294 a x_1\ni : \u03b9\n\u22a2 0 \u2264 (fun i => x i / a i) i \u2227 (fun i => x i / a i) i \u2264 1", "state_after": "case h.mpr.refine'_1\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2076 : Fintype \u03b9\ninst\u271d\u2075 : Fintype \u03b9'\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \u211d E\ninst\u271d\u00b2 : AddCommGroup F\ninst\u271d\u00b9 : Module \u211d F\ninst\u271d : DecidableEq \u03b9\na x : \u03b9 \u2192 \u211d\ni : \u03b9\nh : 0 \u2293 a i \u2264 x i \u2227 x i \u2264 0 \u2294 a i\n\u22a2 0 \u2264 (fun i => x i / a i) i \u2227 (fun i => x i / a i) i \u2264 1"}, {"tactic": "cases' le_total (a i) 0 with hai hai", "annotated_tactic": ["cases' <a>le_total</a> (a i) 0 with hai hai", [{"full_name": "le_total", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [295, 9], "def_end_pos": [295, 17]}]], "state_before": "case h.mpr.refine'_1\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2076 : Fintype \u03b9\ninst\u271d\u2075 : Fintype \u03b9'\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \u211d E\ninst\u271d\u00b2 : AddCommGroup F\ninst\u271d\u00b9 : Module \u211d F\ninst\u271d : DecidableEq \u03b9\na x : \u03b9 \u2192 \u211d\ni : \u03b9\nh : 0 \u2293 a i \u2264 x i \u2227 x i \u2264 0 \u2294 a i\n\u22a2 0 \u2264 (fun i => x i / a i) i \u2227 (fun i => x i / a i) i \u2264 1", "state_after": "case h.mpr.refine'_1.inl\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2076 : Fintype \u03b9\ninst\u271d\u2075 : Fintype \u03b9'\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \u211d E\ninst\u271d\u00b2 : AddCommGroup F\ninst\u271d\u00b9 : Module \u211d F\ninst\u271d : DecidableEq \u03b9\na x : \u03b9 \u2192 \u211d\ni : \u03b9\nh : 0 \u2293 a i \u2264 x i \u2227 x i \u2264 0 \u2294 a i\nhai : a i \u2264 0\n\u22a2 0 \u2264 (fun i => x i / a i) i \u2227 (fun i => x i / a i) i \u2264 1\n\ncase h.mpr.refine'_1.inr\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2076 : Fintype \u03b9\ninst\u271d\u2075 : Fintype \u03b9'\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \u211d E\ninst\u271d\u00b2 : AddCommGroup F\ninst\u271d\u00b9 : Module \u211d F\ninst\u271d : DecidableEq \u03b9\na x : \u03b9 \u2192 \u211d\ni : \u03b9\nh : 0 \u2293 a i \u2264 x i \u2227 x i \u2264 0 \u2294 a i\nhai : 0 \u2264 a i\n\u22a2 0 \u2264 (fun i => x i / a i) i \u2227 (fun i => x i / a i) i \u2264 1"}, {"tactic": "rw [sup_eq_left.mpr hai, inf_eq_right.mpr hai] at h", "annotated_tactic": ["rw [sup_eq_left.mpr hai, inf_eq_right.mpr hai] at h", []], "state_before": "case h.mpr.refine'_1.inl\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2076 : Fintype \u03b9\ninst\u271d\u2075 : Fintype \u03b9'\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \u211d E\ninst\u271d\u00b2 : AddCommGroup F\ninst\u271d\u00b9 : Module \u211d F\ninst\u271d : DecidableEq \u03b9\na x : \u03b9 \u2192 \u211d\ni : \u03b9\nh : 0 \u2293 a i \u2264 x i \u2227 x i \u2264 0 \u2294 a i\nhai : a i \u2264 0\n\u22a2 0 \u2264 (fun i => x i / a i) i \u2227 (fun i => x i / a i) i \u2264 1", "state_after": "case h.mpr.refine'_1.inl\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2076 : Fintype \u03b9\ninst\u271d\u2075 : Fintype \u03b9'\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \u211d E\ninst\u271d\u00b2 : AddCommGroup F\ninst\u271d\u00b9 : Module \u211d F\ninst\u271d : DecidableEq \u03b9\na x : \u03b9 \u2192 \u211d\ni : \u03b9\nh : a i \u2264 x i \u2227 x i \u2264 0\nhai : a i \u2264 0\n\u22a2 0 \u2264 (fun i => x i / a i) i \u2227 (fun i => x i / a i) i \u2264 1"}, {"tactic": "exact \u27e8div_nonneg_of_nonpos h.2 hai, div_le_one_of_ge h.1 hai\u27e9", "annotated_tactic": ["exact \u27e8<a>div_nonneg_of_nonpos</a> h.2 hai, <a>div_le_one_of_ge</a> h.1 hai\u27e9", [{"full_name": "div_nonneg_of_nonpos", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [694, 9], "def_end_pos": [694, 29]}, {"full_name": "div_le_one_of_ge", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [750, 9], "def_end_pos": [750, 25]}]], "state_before": "case h.mpr.refine'_1.inl\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2076 : Fintype \u03b9\ninst\u271d\u2075 : Fintype \u03b9'\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \u211d E\ninst\u271d\u00b2 : AddCommGroup F\ninst\u271d\u00b9 : Module \u211d F\ninst\u271d : DecidableEq \u03b9\na x : \u03b9 \u2192 \u211d\ni : \u03b9\nh : a i \u2264 x i \u2227 x i \u2264 0\nhai : a i \u2264 0\n\u22a2 0 \u2264 (fun i => x i / a i) i \u2227 (fun i => x i / a i) i \u2264 1", "state_after": "no goals"}, {"tactic": "rw [sup_eq_right.mpr hai, inf_eq_left.mpr hai] at h", "annotated_tactic": ["rw [sup_eq_right.mpr hai, inf_eq_left.mpr hai] at h", []], "state_before": "case h.mpr.refine'_1.inr\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2076 : Fintype \u03b9\ninst\u271d\u2075 : Fintype \u03b9'\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \u211d E\ninst\u271d\u00b2 : AddCommGroup F\ninst\u271d\u00b9 : Module \u211d F\ninst\u271d : DecidableEq \u03b9\na x : \u03b9 \u2192 \u211d\ni : \u03b9\nh : 0 \u2293 a i \u2264 x i \u2227 x i \u2264 0 \u2294 a i\nhai : 0 \u2264 a i\n\u22a2 0 \u2264 (fun i => x i / a i) i \u2227 (fun i => x i / a i) i \u2264 1", "state_after": "case h.mpr.refine'_1.inr\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2076 : Fintype \u03b9\ninst\u271d\u2075 : Fintype \u03b9'\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \u211d E\ninst\u271d\u00b2 : AddCommGroup F\ninst\u271d\u00b9 : Module \u211d F\ninst\u271d : DecidableEq \u03b9\na x : \u03b9 \u2192 \u211d\ni : \u03b9\nh : 0 \u2264 x i \u2227 x i \u2264 a i\nhai : 0 \u2264 a i\n\u22a2 0 \u2264 (fun i => x i / a i) i \u2227 (fun i => x i / a i) i \u2264 1"}, {"tactic": "exact \u27e8div_nonneg h.1 hai, div_le_one_of_le h.2 hai\u27e9", "annotated_tactic": ["exact \u27e8<a>div_nonneg</a> h.1 hai, <a>div_le_one_of_le</a> h.2 hai\u27e9", [{"full_name": "div_nonneg", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [94, 9], "def_end_pos": [94, 19]}, {"full_name": "div_le_one_of_le", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [242, 9], "def_end_pos": [242, 25]}]], "state_before": "case h.mpr.refine'_1.inr\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2076 : Fintype \u03b9\ninst\u271d\u2075 : Fintype \u03b9'\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \u211d E\ninst\u271d\u00b2 : AddCommGroup F\ninst\u271d\u00b9 : Module \u211d F\ninst\u271d : DecidableEq \u03b9\na x : \u03b9 \u2192 \u211d\ni : \u03b9\nh : 0 \u2264 x i \u2227 x i \u2264 a i\nhai : 0 \u2264 a i\n\u22a2 0 \u2264 (fun i => x i / a i) i \u2227 (fun i => x i / a i) i \u2264 1", "state_after": "no goals"}, {"tactic": "specialize h i", "annotated_tactic": ["specialize h i", []], "state_before": "case h.mpr.refine'_2\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2076 : Fintype \u03b9\ninst\u271d\u2075 : Fintype \u03b9'\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \u211d E\ninst\u271d\u00b2 : AddCommGroup F\ninst\u271d\u00b9 : Module \u211d F\ninst\u271d : DecidableEq \u03b9\na x : \u03b9 \u2192 \u211d\nh : \u2200 (x_1 : \u03b9), 0 \u2293 a x_1 \u2264 x x_1 \u2227 x x_1 \u2264 0 \u2294 a x_1\ni : \u03b9\n\u22a2 x i = ((fun i => x i / a i) \u2022 a) i", "state_after": "case h.mpr.refine'_2\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2076 : Fintype \u03b9\ninst\u271d\u2075 : Fintype \u03b9'\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \u211d E\ninst\u271d\u00b2 : AddCommGroup F\ninst\u271d\u00b9 : Module \u211d F\ninst\u271d : DecidableEq \u03b9\na x : \u03b9 \u2192 \u211d\ni : \u03b9\nh : 0 \u2293 a i \u2264 x i \u2227 x i \u2264 0 \u2294 a i\n\u22a2 x i = ((fun i => x i / a i) \u2022 a) i"}, {"tactic": "simp only [smul_eq_mul, Pi.mul_apply]", "annotated_tactic": ["simp only [<a>smul_eq_mul</a>, <a>Pi.mul_apply</a>]", [{"full_name": "smul_eq_mul", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [93, 9], "def_end_pos": [93, 20]}, {"full_name": "Pi.mul_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [83, 9], "def_end_pos": [83, 18]}]], "state_before": "case h.mpr.refine'_2\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2076 : Fintype \u03b9\ninst\u271d\u2075 : Fintype \u03b9'\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \u211d E\ninst\u271d\u00b2 : AddCommGroup F\ninst\u271d\u00b9 : Module \u211d F\ninst\u271d : DecidableEq \u03b9\na x : \u03b9 \u2192 \u211d\ni : \u03b9\nh : 0 \u2293 a i \u2264 x i \u2227 x i \u2264 0 \u2294 a i\n\u22a2 x i = ((fun i => x i / a i) \u2022 a) i", "state_after": "case h.mpr.refine'_2\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2076 : Fintype \u03b9\ninst\u271d\u2075 : Fintype \u03b9'\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \u211d E\ninst\u271d\u00b2 : AddCommGroup F\ninst\u271d\u00b9 : Module \u211d F\ninst\u271d : DecidableEq \u03b9\na x : \u03b9 \u2192 \u211d\ni : \u03b9\nh : 0 \u2293 a i \u2264 x i \u2227 x i \u2264 0 \u2294 a i\n\u22a2 x i = x i / a i * a i"}, {"tactic": "cases' eq_or_ne (a i) 0 with hai hai", "annotated_tactic": ["cases' <a>eq_or_ne</a> (a i) 0 with hai hai", [{"full_name": "eq_or_ne", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [209, 9], "def_end_pos": [209, 17]}]], "state_before": "case h.mpr.refine'_2\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2076 : Fintype \u03b9\ninst\u271d\u2075 : Fintype \u03b9'\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \u211d E\ninst\u271d\u00b2 : AddCommGroup F\ninst\u271d\u00b9 : Module \u211d F\ninst\u271d : DecidableEq \u03b9\na x : \u03b9 \u2192 \u211d\ni : \u03b9\nh : 0 \u2293 a i \u2264 x i \u2227 x i \u2264 0 \u2294 a i\n\u22a2 x i = x i / a i * a i", "state_after": "case h.mpr.refine'_2.inl\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2076 : Fintype \u03b9\ninst\u271d\u2075 : Fintype \u03b9'\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \u211d E\ninst\u271d\u00b2 : AddCommGroup F\ninst\u271d\u00b9 : Module \u211d F\ninst\u271d : DecidableEq \u03b9\na x : \u03b9 \u2192 \u211d\ni : \u03b9\nh : 0 \u2293 a i \u2264 x i \u2227 x i \u2264 0 \u2294 a i\nhai : a i = 0\n\u22a2 x i = x i / a i * a i\n\ncase h.mpr.refine'_2.inr\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2076 : Fintype \u03b9\ninst\u271d\u2075 : Fintype \u03b9'\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \u211d E\ninst\u271d\u00b2 : AddCommGroup F\ninst\u271d\u00b9 : Module \u211d F\ninst\u271d : DecidableEq \u03b9\na x : \u03b9 \u2192 \u211d\ni : \u03b9\nh : 0 \u2293 a i \u2264 x i \u2227 x i \u2264 0 \u2294 a i\nhai : a i \u2260 0\n\u22a2 x i = x i / a i * a i"}, {"tactic": "rw [hai, inf_idem, sup_idem, \u2190 le_antisymm_iff] at h", "annotated_tactic": ["rw [hai, <a>inf_idem</a>, <a>sup_idem</a>, \u2190 <a>le_antisymm_iff</a>] at h", [{"full_name": "inf_idem", "def_path": "Mathlib/Order/Lattice.lean", "def_pos": [493, 9], "def_end_pos": [493, 17]}, {"full_name": "sup_idem", "def_path": "Mathlib/Order/Lattice.lean", "def_pos": [244, 9], "def_end_pos": [244, 17]}, {"full_name": "le_antisymm_iff", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [192, 9], "def_end_pos": [192, 24]}]], "state_before": "case h.mpr.refine'_2.inl\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2076 : Fintype \u03b9\ninst\u271d\u2075 : Fintype \u03b9'\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \u211d E\ninst\u271d\u00b2 : AddCommGroup F\ninst\u271d\u00b9 : Module \u211d F\ninst\u271d : DecidableEq \u03b9\na x : \u03b9 \u2192 \u211d\ni : \u03b9\nh : 0 \u2293 a i \u2264 x i \u2227 x i \u2264 0 \u2294 a i\nhai : a i = 0\n\u22a2 x i = x i / a i * a i", "state_after": "case h.mpr.refine'_2.inl\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2076 : Fintype \u03b9\ninst\u271d\u2075 : Fintype \u03b9'\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \u211d E\ninst\u271d\u00b2 : AddCommGroup F\ninst\u271d\u00b9 : Module \u211d F\ninst\u271d : DecidableEq \u03b9\na x : \u03b9 \u2192 \u211d\ni : \u03b9\nh : 0 = x i\nhai : a i = 0\n\u22a2 x i = x i / a i * a i"}, {"tactic": "rw [hai, \u2190 h, zero_div, zero_mul]", "annotated_tactic": ["rw [hai, \u2190 h, <a>zero_div</a>, <a>zero_mul</a>]", [{"full_name": "zero_div", "def_path": "Mathlib/Algebra/GroupWithZero/Basic.lean", "def_pos": [291, 9], "def_end_pos": [291, 17]}, {"full_name": "MulZeroClass.zero_mul", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [36, 3], "def_end_pos": [36, 11]}]], "state_before": "case h.mpr.refine'_2.inl\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2076 : Fintype \u03b9\ninst\u271d\u2075 : Fintype \u03b9'\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \u211d E\ninst\u271d\u00b2 : AddCommGroup F\ninst\u271d\u00b9 : Module \u211d F\ninst\u271d : DecidableEq \u03b9\na x : \u03b9 \u2192 \u211d\ni : \u03b9\nh : 0 = x i\nhai : a i = 0\n\u22a2 x i = x i / a i * a i", "state_after": "no goals"}, {"tactic": "rw [div_mul_cancel _ hai]", "annotated_tactic": ["rw [<a>div_mul_cancel</a> _ hai]", [{"full_name": "div_mul_cancel", "def_path": "Mathlib/Algebra/GroupWithZero/Units/Lemmas.lean", "def_pos": [66, 9], "def_end_pos": [66, 23]}]], "state_before": "case h.mpr.refine'_2.inr\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2076 : Fintype \u03b9\ninst\u271d\u2075 : Fintype \u03b9'\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \u211d E\ninst\u271d\u00b2 : AddCommGroup F\ninst\u271d\u00b9 : Module \u211d F\ninst\u271d : DecidableEq \u03b9\na x : \u03b9 \u2192 \u211d\ni : \u03b9\nh : 0 \u2293 a i \u2264 x i \u2227 x i \u2264 0 \u2294 a i\nhai : a i \u2260 0\n\u22a2 x i = x i / a i * a i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Sign.lean", "full_name": "exists_signed_sum'", "start": [517, 1], "end": [529, 8], "traced_tactics": [{"tactic": "obtain \u27e8\u03b2, _, sgn, g, hg, h\u03b2, hf\u27e9 := exists_signed_sum s f", "annotated_tactic": ["obtain \u27e8\u03b2, _, sgn, g, hg, h\u03b2, hf\u27e9 := <a>exists_signed_sum</a> s f", [{"full_name": "exists_signed_sum", "def_path": "Mathlib/Data/Sign.lean", "def_pos": [506, 9], "def_end_pos": [506, 26]}]], "state_before": "\u03b1\u271d \u03b1 : Type u_1\ninst\u271d\u00b9 : Nonempty \u03b1\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\nf : \u03b1 \u2192 \u2124\nn : \u2115\nh : \u2211 i in s, Int.natAbs (f i) \u2264 n\n\u22a2 \u2203 \u03b2 x sgn g,\n    (\u2200 (b : \u03b2), \u00acg b \u2208 s \u2192 sgn b = 0) \u2227\n      Fintype.card \u03b2 = n \u2227 \u2200 (a : \u03b1), a \u2208 s \u2192 (\u2211 i : \u03b2, if g i = a then \u2191(sgn i) else 0) = f a", "state_after": "case intro.intro.intro.intro.intro.intro\n\u03b1\u271d \u03b1 : Type u_1\ninst\u271d\u00b9 : Nonempty \u03b1\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\nf : \u03b1 \u2192 \u2124\nn : \u2115\nh : \u2211 i in s, Int.natAbs (f i) \u2264 n\n\u03b2 : Type u_1\nw\u271d : Fintype \u03b2\nsgn : \u03b2 \u2192 SignType\ng : \u03b2 \u2192 \u03b1\nhg : \u2200 (b : \u03b2), g b \u2208 s\nh\u03b2 : Fintype.card \u03b2 = \u2211 a in s, Int.natAbs (f a)\nhf : \u2200 (a : \u03b1), a \u2208 s \u2192 (\u2211 b : \u03b2, if g b = a then \u2191(sgn b) else 0) = f a\n\u22a2 \u2203 \u03b2 x sgn g,\n    (\u2200 (b : \u03b2), \u00acg b \u2208 s \u2192 sgn b = 0) \u2227\n      Fintype.card \u03b2 = n \u2227 \u2200 (a : \u03b1), a \u2208 s \u2192 (\u2211 i : \u03b2, if g i = a then \u2191(sgn i) else 0) = f a"}, {"tactic": "refine'\n  \u27e8Sum \u03b2 (Fin (n - \u2211 i in s, (f i).natAbs)), inferInstance, Sum.elim sgn 0,\n    Sum.elim g (Classical.arbitrary (Fin (n - Finset.sum s fun i => Int.natAbs (f i)) \u2192 \u03b1)),\n      _, by simp [h\u03b2, h], fun a ha => by simp [hf _ ha]\u27e9", "annotated_tactic": ["refine'\n    \u27e8<a>Sum</a> \u03b2 (<a>Fin</a> (n - \u2211 i in s, (f i).<a>natAbs</a>)), <a>inferInstance</a>, <a>Sum.elim</a> sgn 0,\n      <a>Sum.elim</a> g (<a>Classical.arbitrary</a> (<a>Fin</a> (n - <a>Finset.sum</a> s fun i => <a>Int.natAbs</a> (f i)) \u2192 \u03b1)),\n        _, by simp [h\u03b2, h], fun a ha => by simp [hf _ ha]\u27e9", [{"full_name": "Sum", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [102, 11], "def_end_pos": [102, 14]}, {"full_name": "Fin", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1745, 11], "def_end_pos": [1745, 14]}, {"full_name": "Int.natAbs", "def_path": "lake-packages/lean4/src/lean/Init/Data/Int/Basic.lean", "def_pos": [242, 5], "def_end_pos": [242, 11]}, {"full_name": "inferInstance", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [86, 8], "def_end_pos": [86, 21]}, {"full_name": "Sum.elim", "def_path": "lake-packages/std/Std/Data/Sum/Basic.lean", "def_pos": [94, 15], "def_end_pos": [94, 19]}, {"full_name": "Sum.elim", "def_path": "lake-packages/std/Std/Data/Sum/Basic.lean", "def_pos": [94, 15], "def_end_pos": [94, 19]}, {"full_name": "Classical.arbitrary", "def_path": "Mathlib/Logic/Nonempty.lean", "def_pos": [134, 29], "def_end_pos": [134, 48]}, {"full_name": "Fin", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1745, 11], "def_end_pos": [1745, 14]}, {"full_name": "Finset.sum", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [59, 3], "def_end_pos": [59, 14]}, {"full_name": "Int.natAbs", "def_path": "lake-packages/lean4/src/lean/Init/Data/Int/Basic.lean", "def_pos": [242, 5], "def_end_pos": [242, 11]}]], "state_before": "case intro.intro.intro.intro.intro.intro\n\u03b1\u271d \u03b1 : Type u_1\ninst\u271d\u00b9 : Nonempty \u03b1\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\nf : \u03b1 \u2192 \u2124\nn : \u2115\nh : \u2211 i in s, Int.natAbs (f i) \u2264 n\n\u03b2 : Type u_1\nw\u271d : Fintype \u03b2\nsgn : \u03b2 \u2192 SignType\ng : \u03b2 \u2192 \u03b1\nhg : \u2200 (b : \u03b2), g b \u2208 s\nh\u03b2 : Fintype.card \u03b2 = \u2211 a in s, Int.natAbs (f a)\nhf : \u2200 (a : \u03b1), a \u2208 s \u2192 (\u2211 b : \u03b2, if g b = a then \u2191(sgn b) else 0) = f a\n\u22a2 \u2203 \u03b2 x sgn g,\n    (\u2200 (b : \u03b2), \u00acg b \u2208 s \u2192 sgn b = 0) \u2227\n      Fintype.card \u03b2 = n \u2227 \u2200 (a : \u03b1), a \u2208 s \u2192 (\u2211 i : \u03b2, if g i = a then \u2191(sgn i) else 0) = f a", "state_after": "case intro.intro.intro.intro.intro.intro\n\u03b1\u271d \u03b1 : Type u_1\ninst\u271d\u00b9 : Nonempty \u03b1\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\nf : \u03b1 \u2192 \u2124\nn : \u2115\nh : \u2211 i in s, Int.natAbs (f i) \u2264 n\n\u03b2 : Type u_1\nw\u271d : Fintype \u03b2\nsgn : \u03b2 \u2192 SignType\ng : \u03b2 \u2192 \u03b1\nhg : \u2200 (b : \u03b2), g b \u2208 s\nh\u03b2 : Fintype.card \u03b2 = \u2211 a in s, Int.natAbs (f a)\nhf : \u2200 (a : \u03b1), a \u2208 s \u2192 (\u2211 b : \u03b2, if g b = a then \u2191(sgn b) else 0) = f a\n\u22a2 \u2200 (b : \u03b2 \u2295 Fin (n - \u2211 i in s, Int.natAbs (f i))),\n    \u00acSum.elim g (Classical.arbitrary (Fin (n - \u2211 i in s, Int.natAbs (f i)) \u2192 \u03b1)) b \u2208 s \u2192 Sum.elim sgn 0 b = 0"}, {"tactic": "rintro (b | b) hb", "annotated_tactic": ["rintro (b | b) hb", []], "state_before": "case intro.intro.intro.intro.intro.intro\n\u03b1\u271d \u03b1 : Type u_1\ninst\u271d\u00b9 : Nonempty \u03b1\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\nf : \u03b1 \u2192 \u2124\nn : \u2115\nh : \u2211 i in s, Int.natAbs (f i) \u2264 n\n\u03b2 : Type u_1\nw\u271d : Fintype \u03b2\nsgn : \u03b2 \u2192 SignType\ng : \u03b2 \u2192 \u03b1\nhg : \u2200 (b : \u03b2), g b \u2208 s\nh\u03b2 : Fintype.card \u03b2 = \u2211 a in s, Int.natAbs (f a)\nhf : \u2200 (a : \u03b1), a \u2208 s \u2192 (\u2211 b : \u03b2, if g b = a then \u2191(sgn b) else 0) = f a\n\u22a2 \u2200 (b : \u03b2 \u2295 Fin (n - \u2211 i in s, Int.natAbs (f i))),\n    \u00acSum.elim g (Classical.arbitrary (Fin (n - \u2211 i in s, Int.natAbs (f i)) \u2192 \u03b1)) b \u2208 s \u2192 Sum.elim sgn 0 b = 0", "state_after": "case intro.intro.intro.intro.intro.intro.inl\n\u03b1\u271d \u03b1 : Type u_1\ninst\u271d\u00b9 : Nonempty \u03b1\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\nf : \u03b1 \u2192 \u2124\nn : \u2115\nh : \u2211 i in s, Int.natAbs (f i) \u2264 n\n\u03b2 : Type u_1\nw\u271d : Fintype \u03b2\nsgn : \u03b2 \u2192 SignType\ng : \u03b2 \u2192 \u03b1\nhg : \u2200 (b : \u03b2), g b \u2208 s\nh\u03b2 : Fintype.card \u03b2 = \u2211 a in s, Int.natAbs (f a)\nhf : \u2200 (a : \u03b1), a \u2208 s \u2192 (\u2211 b : \u03b2, if g b = a then \u2191(sgn b) else 0) = f a\nb : \u03b2\nhb : \u00acSum.elim g (Classical.arbitrary (Fin (n - \u2211 i in s, Int.natAbs (f i)) \u2192 \u03b1)) (Sum.inl b) \u2208 s\n\u22a2 Sum.elim sgn 0 (Sum.inl b) = 0\n\ncase intro.intro.intro.intro.intro.intro.inr\n\u03b1\u271d \u03b1 : Type u_1\ninst\u271d\u00b9 : Nonempty \u03b1\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\nf : \u03b1 \u2192 \u2124\nn : \u2115\nh : \u2211 i in s, Int.natAbs (f i) \u2264 n\n\u03b2 : Type u_1\nw\u271d : Fintype \u03b2\nsgn : \u03b2 \u2192 SignType\ng : \u03b2 \u2192 \u03b1\nhg : \u2200 (b : \u03b2), g b \u2208 s\nh\u03b2 : Fintype.card \u03b2 = \u2211 a in s, Int.natAbs (f a)\nhf : \u2200 (a : \u03b1), a \u2208 s \u2192 (\u2211 b : \u03b2, if g b = a then \u2191(sgn b) else 0) = f a\nb : Fin (n - \u2211 i in s, Int.natAbs (f i))\nhb : \u00acSum.elim g (Classical.arbitrary (Fin (n - \u2211 i in s, Int.natAbs (f i)) \u2192 \u03b1)) (Sum.inr b) \u2208 s\n\u22a2 Sum.elim sgn 0 (Sum.inr b) = 0"}, {"tactic": "simp [h\u03b2, h]", "annotated_tactic": ["simp [h\u03b2, h]", []], "state_before": "\u03b1\u271d \u03b1 : Type u_1\ninst\u271d\u00b9 : Nonempty \u03b1\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\nf : \u03b1 \u2192 \u2124\nn : \u2115\nh : \u2211 i in s, Int.natAbs (f i) \u2264 n\n\u03b2 : Type u_1\nw\u271d : Fintype \u03b2\nsgn : \u03b2 \u2192 SignType\ng : \u03b2 \u2192 \u03b1\nhg : \u2200 (b : \u03b2), g b \u2208 s\nh\u03b2 : Fintype.card \u03b2 = \u2211 a in s, Int.natAbs (f a)\nhf : \u2200 (a : \u03b1), a \u2208 s \u2192 (\u2211 b : \u03b2, if g b = a then \u2191(sgn b) else 0) = f a\n\u22a2 Fintype.card (\u03b2 \u2295 Fin (n - \u2211 i in s, Int.natAbs (f i))) = n", "state_after": "no goals"}, {"tactic": "simp [hf _ ha]", "annotated_tactic": ["simp [hf _ ha]", []], "state_before": "\u03b1\u271d \u03b1 : Type u_1\ninst\u271d\u00b9 : Nonempty \u03b1\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\nf : \u03b1 \u2192 \u2124\nn : \u2115\nh : \u2211 i in s, Int.natAbs (f i) \u2264 n\n\u03b2 : Type u_1\nw\u271d : Fintype \u03b2\nsgn : \u03b2 \u2192 SignType\ng : \u03b2 \u2192 \u03b1\nhg : \u2200 (b : \u03b2), g b \u2208 s\nh\u03b2 : Fintype.card \u03b2 = \u2211 a in s, Int.natAbs (f a)\nhf : \u2200 (a : \u03b1), a \u2208 s \u2192 (\u2211 b : \u03b2, if g b = a then \u2191(sgn b) else 0) = f a\na : \u03b1\nha : a \u2208 s\n\u22a2 (\u2211 i : \u03b2 \u2295 Fin (n - \u2211 i in s, Int.natAbs (f i)),\n      if Sum.elim g (Classical.arbitrary (Fin (n - \u2211 i in s, Int.natAbs (f i)) \u2192 \u03b1)) i = a then \u2191(Sum.elim sgn 0 i)\n      else 0) =\n    f a", "state_after": "no goals"}, {"tactic": "cases hb (hg _)", "annotated_tactic": ["cases hb (hg _)", []], "state_before": "case intro.intro.intro.intro.intro.intro.inl\n\u03b1\u271d \u03b1 : Type u_1\ninst\u271d\u00b9 : Nonempty \u03b1\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\nf : \u03b1 \u2192 \u2124\nn : \u2115\nh : \u2211 i in s, Int.natAbs (f i) \u2264 n\n\u03b2 : Type u_1\nw\u271d : Fintype \u03b2\nsgn : \u03b2 \u2192 SignType\ng : \u03b2 \u2192 \u03b1\nhg : \u2200 (b : \u03b2), g b \u2208 s\nh\u03b2 : Fintype.card \u03b2 = \u2211 a in s, Int.natAbs (f a)\nhf : \u2200 (a : \u03b1), a \u2208 s \u2192 (\u2211 b : \u03b2, if g b = a then \u2191(sgn b) else 0) = f a\nb : \u03b2\nhb : \u00acSum.elim g (Classical.arbitrary (Fin (n - \u2211 i in s, Int.natAbs (f i)) \u2192 \u03b1)) (Sum.inl b) \u2208 s\n\u22a2 Sum.elim sgn 0 (Sum.inl b) = 0", "state_after": "no goals"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case intro.intro.intro.intro.intro.intro.inr\n\u03b1\u271d \u03b1 : Type u_1\ninst\u271d\u00b9 : Nonempty \u03b1\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\nf : \u03b1 \u2192 \u2124\nn : \u2115\nh : \u2211 i in s, Int.natAbs (f i) \u2264 n\n\u03b2 : Type u_1\nw\u271d : Fintype \u03b2\nsgn : \u03b2 \u2192 SignType\ng : \u03b2 \u2192 \u03b1\nhg : \u2200 (b : \u03b2), g b \u2208 s\nh\u03b2 : Fintype.card \u03b2 = \u2211 a in s, Int.natAbs (f a)\nhf : \u2200 (a : \u03b1), a \u2208 s \u2192 (\u2211 b : \u03b2, if g b = a then \u2191(sgn b) else 0) = f a\nb : Fin (n - \u2211 i in s, Int.natAbs (f i))\nhb : \u00acSum.elim g (Classical.arbitrary (Fin (n - \u2211 i in s, Int.natAbs (f i)) \u2192 \u03b1)) (Sum.inr b) \u2208 s\n\u22a2 Sum.elim sgn 0 (Sum.inr b) = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "full_name": "MeasureTheory.Lp.ext_iff", "start": [167, 1], "end": [168, 39], "traced_tactics": [{"tactic": "rw [h]", "annotated_tactic": ["rw [h]", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf g : { x // x \u2208 Lp E p }\nh : f = g\n\u22a2 \u2191\u2191f =\u1d50[\u03bc] \u2191\u2191g", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/AEEqFun.lean", "full_name": "MeasureTheory.AEEqFun.coeFn_comp\u2082Measurable", "start": [428, 1], "end": [431, 17], "traced_tactics": [{"tactic": "rw [comp\u2082Measurable_eq_mk]", "annotated_tactic": ["rw [<a>comp\u2082Measurable_eq_mk</a>]", [{"full_name": "MeasureTheory.AEEqFun.comp\u2082Measurable_eq_mk", "def_path": "Mathlib/MeasureTheory/Function/AEEqFun.lean", "def_pos": [420, 9], "def_end_pos": [420, 30]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9\u2074 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u00b2 : TopologicalSpace \u03b3\ninst\u271d\u00b9\u00b9 : TopologicalSpace \u03b4\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b2\ninst\u271d\u2079 : PseudoMetrizableSpace \u03b2\ninst\u271d\u2078 : BorelSpace \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b3\ninst\u271d\u2076 : PseudoMetrizableSpace \u03b3\ninst\u271d\u2075 : BorelSpace \u03b3\ninst\u271d\u2074 : SecondCountableTopologyEither \u03b2 \u03b3\ninst\u271d\u00b3 : MeasurableSpace \u03b4\ninst\u271d\u00b2 : PseudoMetrizableSpace \u03b4\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b4\ninst\u271d : SecondCountableTopology \u03b4\ng : \u03b2 \u2192 \u03b3 \u2192 \u03b4\nhg : Measurable (uncurry g)\nf\u2081 : \u03b1 \u2192\u2098[\u03bc] \u03b2\nf\u2082 : \u03b1 \u2192\u2098[\u03bc] \u03b3\n\u22a2 \u2191(comp\u2082Measurable g hg f\u2081 f\u2082) =\u1d50[\u03bc] fun a => g (\u2191f\u2081 a) (\u2191f\u2082 a)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9\u2074 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u00b2 : TopologicalSpace \u03b3\ninst\u271d\u00b9\u00b9 : TopologicalSpace \u03b4\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b2\ninst\u271d\u2079 : PseudoMetrizableSpace \u03b2\ninst\u271d\u2078 : BorelSpace \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b3\ninst\u271d\u2076 : PseudoMetrizableSpace \u03b3\ninst\u271d\u2075 : BorelSpace \u03b3\ninst\u271d\u2074 : SecondCountableTopologyEither \u03b2 \u03b3\ninst\u271d\u00b3 : MeasurableSpace \u03b4\ninst\u271d\u00b2 : PseudoMetrizableSpace \u03b4\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b4\ninst\u271d : SecondCountableTopology \u03b4\ng : \u03b2 \u2192 \u03b3 \u2192 \u03b4\nhg : Measurable (uncurry g)\nf\u2081 : \u03b1 \u2192\u2098[\u03bc] \u03b2\nf\u2082 : \u03b1 \u2192\u2098[\u03bc] \u03b3\n\u22a2 \u2191(mk (fun a => g (\u2191f\u2081 a) (\u2191f\u2082 a)) (_ : AEStronglyMeasurable (uncurry g \u2218 fun x => (\u2191f\u2081 x, \u2191f\u2082 x)) \u03bc)) =\u1d50[\u03bc] fun a =>\n    g (\u2191f\u2081 a) (\u2191f\u2082 a)"}, {"tactic": "apply coeFn_mk", "annotated_tactic": ["apply <a>coeFn_mk</a>", [{"full_name": "MeasureTheory.AEEqFun.coeFn_mk", "def_path": "Mathlib/MeasureTheory/Function/AEEqFun.lean", "def_pos": [182, 9], "def_end_pos": [182, 17]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9\u2074 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u00b2 : TopologicalSpace \u03b3\ninst\u271d\u00b9\u00b9 : TopologicalSpace \u03b4\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b2\ninst\u271d\u2079 : PseudoMetrizableSpace \u03b2\ninst\u271d\u2078 : BorelSpace \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b3\ninst\u271d\u2076 : PseudoMetrizableSpace \u03b3\ninst\u271d\u2075 : BorelSpace \u03b3\ninst\u271d\u2074 : SecondCountableTopologyEither \u03b2 \u03b3\ninst\u271d\u00b3 : MeasurableSpace \u03b4\ninst\u271d\u00b2 : PseudoMetrizableSpace \u03b4\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b4\ninst\u271d : SecondCountableTopology \u03b4\ng : \u03b2 \u2192 \u03b3 \u2192 \u03b4\nhg : Measurable (uncurry g)\nf\u2081 : \u03b1 \u2192\u2098[\u03bc] \u03b2\nf\u2082 : \u03b1 \u2192\u2098[\u03bc] \u03b3\n\u22a2 \u2191(mk (fun a => g (\u2191f\u2081 a) (\u2191f\u2082 a)) (_ : AEStronglyMeasurable (uncurry g \u2218 fun x => (\u2191f\u2081 x, \u2191f\u2082 x)) \u03bc)) =\u1d50[\u03bc] fun a =>\n    g (\u2191f\u2081 a) (\u2191f\u2082 a)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "full_name": "MeasureTheory.lintegral_liminf_le'", "start": [1020, 1], "end": [1029, 87], "traced_tactics": [{"tactic": "simp only [liminf_eq_iSup_iInf_of_nat]", "annotated_tactic": ["simp only [<a>liminf_eq_iSup_iInf_of_nat</a>]", [{"full_name": "Filter.liminf_eq_iSup_iInf_of_nat", "def_path": "Mathlib/Order/LiminfLimsup.lean", "def_pos": [855, 9], "def_end_pos": [855, 35]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nh_meas : \u2200 (n : \u2115), AEMeasurable (f n)\n\u22a2 \u222b\u207b (a : \u03b1), liminf (fun n => f n a) atTop \u2202\u03bc = \u222b\u207b (a : \u03b1), \u2a06 n, \u2a05 i, \u2a05 (_ : i \u2265 n), f i a \u2202\u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL2.lean", "full_name": "MeasureTheory.norm_condexpL2_le_one", "start": [95, 1], "end": [97, 33], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/RegularExpressions.lean", "full_name": "RegularExpression.rmatch_iff_matches'", "start": [350, 1], "end": [395, 12], "traced_tactics": [{"tactic": "intro x", "annotated_tactic": ["intro x", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\nP : RegularExpression \u03b1\n\u22a2 \u2200 (x : List \u03b1), rmatch P x = true \u2194 x \u2208 matches' P", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\nP : RegularExpression \u03b1\nx : List \u03b1\n\u22a2 rmatch P x = true \u2194 x \u2208 matches' P"}, {"tactic": "induction P generalizing x", "annotated_tactic": ["induction P generalizing x", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\nP : RegularExpression \u03b1\nx : List \u03b1\n\u22a2 rmatch P x = true \u2194 x \u2208 matches' P", "state_after": "case zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\nx : List \u03b1\n\u22a2 rmatch zero x = true \u2194 x \u2208 matches' zero\n\ncase epsilon\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\nx : List \u03b1\n\u22a2 rmatch epsilon x = true \u2194 x \u2208 matches' epsilon\n\ncase char\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b a\u271d : \u03b1\nx : List \u03b1\n\u22a2 rmatch (char a\u271d) x = true \u2194 x \u2208 matches' (char a\u271d)\n\ncase plus\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\na\u271d\u00b9 a\u271d : RegularExpression \u03b1\na_ih\u271d\u00b9 : \u2200 (x : List \u03b1), rmatch a\u271d\u00b9 x = true \u2194 x \u2208 matches' a\u271d\u00b9\na_ih\u271d : \u2200 (x : List \u03b1), rmatch a\u271d x = true \u2194 x \u2208 matches' a\u271d\nx : List \u03b1\n\u22a2 rmatch (plus a\u271d\u00b9 a\u271d) x = true \u2194 x \u2208 matches' (plus a\u271d\u00b9 a\u271d)\n\ncase comp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\na\u271d\u00b9 a\u271d : RegularExpression \u03b1\na_ih\u271d\u00b9 : \u2200 (x : List \u03b1), rmatch a\u271d\u00b9 x = true \u2194 x \u2208 matches' a\u271d\u00b9\na_ih\u271d : \u2200 (x : List \u03b1), rmatch a\u271d x = true \u2194 x \u2208 matches' a\u271d\nx : List \u03b1\n\u22a2 rmatch (comp a\u271d\u00b9 a\u271d) x = true \u2194 x \u2208 matches' (comp a\u271d\u00b9 a\u271d)\n\ncase star\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\na\u271d : RegularExpression \u03b1\na_ih\u271d : \u2200 (x : List \u03b1), rmatch a\u271d x = true \u2194 x \u2208 matches' a\u271d\nx : List \u03b1\n\u22a2 rmatch (star a\u271d) x = true \u2194 x \u2208 matches' (star a\u271d)"}, {"tactic": "all_goals\n  try rw [zero_def]\n  try rw [one_def]\n  try rw [plus_def]\n  try rw [comp_def]", "annotated_tactic": ["all_goals\n    try rw [<a>zero_def</a>]\n    try rw [<a>one_def</a>]\n    try rw [<a>plus_def</a>]\n    try rw [<a>comp_def</a>]", [{"full_name": "RegularExpression.zero_def", "def_path": "Mathlib/Computability/RegularExpressions.lean", "def_pos": [87, 9], "def_end_pos": [87, 17]}, {"full_name": "RegularExpression.one_def", "def_path": "Mathlib/Computability/RegularExpressions.lean", "def_pos": [92, 9], "def_end_pos": [92, 16]}, {"full_name": "RegularExpression.plus_def", "def_path": "Mathlib/Computability/RegularExpressions.lean", "def_pos": [97, 9], "def_end_pos": [97, 17]}, {"full_name": "RegularExpression.comp_def", "def_path": "Mathlib/Computability/RegularExpressions.lean", "def_pos": [102, 9], "def_end_pos": [102, 17]}]], "state_before": "case zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\nx : List \u03b1\n\u22a2 rmatch zero x = true \u2194 x \u2208 matches' zero\n\ncase epsilon\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\nx : List \u03b1\n\u22a2 rmatch epsilon x = true \u2194 x \u2208 matches' epsilon\n\ncase char\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b a\u271d : \u03b1\nx : List \u03b1\n\u22a2 rmatch (char a\u271d) x = true \u2194 x \u2208 matches' (char a\u271d)\n\ncase plus\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\na\u271d\u00b9 a\u271d : RegularExpression \u03b1\na_ih\u271d\u00b9 : \u2200 (x : List \u03b1), rmatch a\u271d\u00b9 x = true \u2194 x \u2208 matches' a\u271d\u00b9\na_ih\u271d : \u2200 (x : List \u03b1), rmatch a\u271d x = true \u2194 x \u2208 matches' a\u271d\nx : List \u03b1\n\u22a2 rmatch (plus a\u271d\u00b9 a\u271d) x = true \u2194 x \u2208 matches' (plus a\u271d\u00b9 a\u271d)\n\ncase comp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\na\u271d\u00b9 a\u271d : RegularExpression \u03b1\na_ih\u271d\u00b9 : \u2200 (x : List \u03b1), rmatch a\u271d\u00b9 x = true \u2194 x \u2208 matches' a\u271d\u00b9\na_ih\u271d : \u2200 (x : List \u03b1), rmatch a\u271d x = true \u2194 x \u2208 matches' a\u271d\nx : List \u03b1\n\u22a2 rmatch (comp a\u271d\u00b9 a\u271d) x = true \u2194 x \u2208 matches' (comp a\u271d\u00b9 a\u271d)\n\ncase star\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\na\u271d : RegularExpression \u03b1\na_ih\u271d : \u2200 (x : List \u03b1), rmatch a\u271d x = true \u2194 x \u2208 matches' a\u271d\nx : List \u03b1\n\u22a2 rmatch (star a\u271d) x = true \u2194 x \u2208 matches' (star a\u271d)", "state_after": "case zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\nx : List \u03b1\n\u22a2 rmatch 0 x = true \u2194 x \u2208 matches' 0\n\ncase epsilon\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\nx : List \u03b1\n\u22a2 rmatch 1 x = true \u2194 x \u2208 matches' 1\n\ncase char\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b a\u271d : \u03b1\nx : List \u03b1\n\u22a2 rmatch (char a\u271d) x = true \u2194 x \u2208 matches' (char a\u271d)\n\ncase plus\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\na\u271d\u00b9 a\u271d : RegularExpression \u03b1\na_ih\u271d\u00b9 : \u2200 (x : List \u03b1), rmatch a\u271d\u00b9 x = true \u2194 x \u2208 matches' a\u271d\u00b9\na_ih\u271d : \u2200 (x : List \u03b1), rmatch a\u271d x = true \u2194 x \u2208 matches' a\u271d\nx : List \u03b1\n\u22a2 rmatch (a\u271d\u00b9 + a\u271d) x = true \u2194 x \u2208 matches' (a\u271d\u00b9 + a\u271d)\n\ncase comp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\na\u271d\u00b9 a\u271d : RegularExpression \u03b1\na_ih\u271d\u00b9 : \u2200 (x : List \u03b1), rmatch a\u271d\u00b9 x = true \u2194 x \u2208 matches' a\u271d\u00b9\na_ih\u271d : \u2200 (x : List \u03b1), rmatch a\u271d x = true \u2194 x \u2208 matches' a\u271d\nx : List \u03b1\n\u22a2 rmatch (a\u271d\u00b9 * a\u271d) x = true \u2194 x \u2208 matches' (a\u271d\u00b9 * a\u271d)\n\ncase star\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\na\u271d : RegularExpression \u03b1\na_ih\u271d : \u2200 (x : List \u03b1), rmatch a\u271d x = true \u2194 x \u2208 matches' a\u271d\nx : List \u03b1\n\u22a2 rmatch (star a\u271d) x = true \u2194 x \u2208 matches' (star a\u271d)"}, {"tactic": "case zero =>\n  rw [zero_rmatch]\n  tauto", "annotated_tactic": ["case zero =>\n    rw [<a>zero_rmatch</a>]\n    tauto", [{"full_name": "RegularExpression.zero_rmatch", "def_path": "Mathlib/Computability/RegularExpressions.lean", "def_pos": [216, 9], "def_end_pos": [216, 20]}]], "state_before": "case zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\nx : List \u03b1\n\u22a2 rmatch 0 x = true \u2194 x \u2208 matches' 0\n\ncase epsilon\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\nx : List \u03b1\n\u22a2 rmatch 1 x = true \u2194 x \u2208 matches' 1\n\ncase char\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b a\u271d : \u03b1\nx : List \u03b1\n\u22a2 rmatch (char a\u271d) x = true \u2194 x \u2208 matches' (char a\u271d)\n\ncase plus\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\na\u271d\u00b9 a\u271d : RegularExpression \u03b1\na_ih\u271d\u00b9 : \u2200 (x : List \u03b1), rmatch a\u271d\u00b9 x = true \u2194 x \u2208 matches' a\u271d\u00b9\na_ih\u271d : \u2200 (x : List \u03b1), rmatch a\u271d x = true \u2194 x \u2208 matches' a\u271d\nx : List \u03b1\n\u22a2 rmatch (a\u271d\u00b9 + a\u271d) x = true \u2194 x \u2208 matches' (a\u271d\u00b9 + a\u271d)\n\ncase comp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\na\u271d\u00b9 a\u271d : RegularExpression \u03b1\na_ih\u271d\u00b9 : \u2200 (x : List \u03b1), rmatch a\u271d\u00b9 x = true \u2194 x \u2208 matches' a\u271d\u00b9\na_ih\u271d : \u2200 (x : List \u03b1), rmatch a\u271d x = true \u2194 x \u2208 matches' a\u271d\nx : List \u03b1\n\u22a2 rmatch (a\u271d\u00b9 * a\u271d) x = true \u2194 x \u2208 matches' (a\u271d\u00b9 * a\u271d)\n\ncase star\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\na\u271d : RegularExpression \u03b1\na_ih\u271d : \u2200 (x : List \u03b1), rmatch a\u271d x = true \u2194 x \u2208 matches' a\u271d\nx : List \u03b1\n\u22a2 rmatch (star a\u271d) x = true \u2194 x \u2208 matches' (star a\u271d)", "state_after": "case epsilon\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\nx : List \u03b1\n\u22a2 rmatch 1 x = true \u2194 x \u2208 matches' 1\n\ncase char\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b a\u271d : \u03b1\nx : List \u03b1\n\u22a2 rmatch (char a\u271d) x = true \u2194 x \u2208 matches' (char a\u271d)\n\ncase plus\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\na\u271d\u00b9 a\u271d : RegularExpression \u03b1\na_ih\u271d\u00b9 : \u2200 (x : List \u03b1), rmatch a\u271d\u00b9 x = true \u2194 x \u2208 matches' a\u271d\u00b9\na_ih\u271d : \u2200 (x : List \u03b1), rmatch a\u271d x = true \u2194 x \u2208 matches' a\u271d\nx : List \u03b1\n\u22a2 rmatch (a\u271d\u00b9 + a\u271d) x = true \u2194 x \u2208 matches' (a\u271d\u00b9 + a\u271d)\n\ncase comp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\na\u271d\u00b9 a\u271d : RegularExpression \u03b1\na_ih\u271d\u00b9 : \u2200 (x : List \u03b1), rmatch a\u271d\u00b9 x = true \u2194 x \u2208 matches' a\u271d\u00b9\na_ih\u271d : \u2200 (x : List \u03b1), rmatch a\u271d x = true \u2194 x \u2208 matches' a\u271d\nx : List \u03b1\n\u22a2 rmatch (a\u271d\u00b9 * a\u271d) x = true \u2194 x \u2208 matches' (a\u271d\u00b9 * a\u271d)\n\ncase star\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\na\u271d : RegularExpression \u03b1\na_ih\u271d : \u2200 (x : List \u03b1), rmatch a\u271d x = true \u2194 x \u2208 matches' a\u271d\nx : List \u03b1\n\u22a2 rmatch (star a\u271d) x = true \u2194 x \u2208 matches' (star a\u271d)"}, {"tactic": "case epsilon =>\n  rw [one_rmatch_iff]\n  rfl", "annotated_tactic": ["case epsilon =>\n    rw [<a>one_rmatch_iff</a>]\n    rfl", [{"full_name": "RegularExpression.one_rmatch_iff", "def_path": "Mathlib/Computability/RegularExpressions.lean", "def_pos": [220, 9], "def_end_pos": [220, 23]}]], "state_before": "case epsilon\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\nx : List \u03b1\n\u22a2 rmatch 1 x = true \u2194 x \u2208 matches' 1\n\ncase char\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b a\u271d : \u03b1\nx : List \u03b1\n\u22a2 rmatch (char a\u271d) x = true \u2194 x \u2208 matches' (char a\u271d)\n\ncase plus\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\na\u271d\u00b9 a\u271d : RegularExpression \u03b1\na_ih\u271d\u00b9 : \u2200 (x : List \u03b1), rmatch a\u271d\u00b9 x = true \u2194 x \u2208 matches' a\u271d\u00b9\na_ih\u271d : \u2200 (x : List \u03b1), rmatch a\u271d x = true \u2194 x \u2208 matches' a\u271d\nx : List \u03b1\n\u22a2 rmatch (a\u271d\u00b9 + a\u271d) x = true \u2194 x \u2208 matches' (a\u271d\u00b9 + a\u271d)\n\ncase comp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\na\u271d\u00b9 a\u271d : RegularExpression \u03b1\na_ih\u271d\u00b9 : \u2200 (x : List \u03b1), rmatch a\u271d\u00b9 x = true \u2194 x \u2208 matches' a\u271d\u00b9\na_ih\u271d : \u2200 (x : List \u03b1), rmatch a\u271d x = true \u2194 x \u2208 matches' a\u271d\nx : List \u03b1\n\u22a2 rmatch (a\u271d\u00b9 * a\u271d) x = true \u2194 x \u2208 matches' (a\u271d\u00b9 * a\u271d)\n\ncase star\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\na\u271d : RegularExpression \u03b1\na_ih\u271d : \u2200 (x : List \u03b1), rmatch a\u271d x = true \u2194 x \u2208 matches' a\u271d\nx : List \u03b1\n\u22a2 rmatch (star a\u271d) x = true \u2194 x \u2208 matches' (star a\u271d)", "state_after": "case char\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b a\u271d : \u03b1\nx : List \u03b1\n\u22a2 rmatch (char a\u271d) x = true \u2194 x \u2208 matches' (char a\u271d)\n\ncase plus\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\na\u271d\u00b9 a\u271d : RegularExpression \u03b1\na_ih\u271d\u00b9 : \u2200 (x : List \u03b1), rmatch a\u271d\u00b9 x = true \u2194 x \u2208 matches' a\u271d\u00b9\na_ih\u271d : \u2200 (x : List \u03b1), rmatch a\u271d x = true \u2194 x \u2208 matches' a\u271d\nx : List \u03b1\n\u22a2 rmatch (a\u271d\u00b9 + a\u271d) x = true \u2194 x \u2208 matches' (a\u271d\u00b9 + a\u271d)\n\ncase comp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\na\u271d\u00b9 a\u271d : RegularExpression \u03b1\na_ih\u271d\u00b9 : \u2200 (x : List \u03b1), rmatch a\u271d\u00b9 x = true \u2194 x \u2208 matches' a\u271d\u00b9\na_ih\u271d : \u2200 (x : List \u03b1), rmatch a\u271d x = true \u2194 x \u2208 matches' a\u271d\nx : List \u03b1\n\u22a2 rmatch (a\u271d\u00b9 * a\u271d) x = true \u2194 x \u2208 matches' (a\u271d\u00b9 * a\u271d)\n\ncase star\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\na\u271d : RegularExpression \u03b1\na_ih\u271d : \u2200 (x : List \u03b1), rmatch a\u271d x = true \u2194 x \u2208 matches' a\u271d\nx : List \u03b1\n\u22a2 rmatch (star a\u271d) x = true \u2194 x \u2208 matches' (star a\u271d)"}, {"tactic": "case char =>\n  rw [char_rmatch_iff]\n  rfl", "annotated_tactic": ["case char =>\n    rw [<a>char_rmatch_iff</a>]\n    rfl", [{"full_name": "RegularExpression.char_rmatch_iff", "def_path": "Mathlib/Computability/RegularExpressions.lean", "def_pos": [224, 9], "def_end_pos": [224, 24]}]], "state_before": "case char\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b a\u271d : \u03b1\nx : List \u03b1\n\u22a2 rmatch (char a\u271d) x = true \u2194 x \u2208 matches' (char a\u271d)\n\ncase plus\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\na\u271d\u00b9 a\u271d : RegularExpression \u03b1\na_ih\u271d\u00b9 : \u2200 (x : List \u03b1), rmatch a\u271d\u00b9 x = true \u2194 x \u2208 matches' a\u271d\u00b9\na_ih\u271d : \u2200 (x : List \u03b1), rmatch a\u271d x = true \u2194 x \u2208 matches' a\u271d\nx : List \u03b1\n\u22a2 rmatch (a\u271d\u00b9 + a\u271d) x = true \u2194 x \u2208 matches' (a\u271d\u00b9 + a\u271d)\n\ncase comp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\na\u271d\u00b9 a\u271d : RegularExpression \u03b1\na_ih\u271d\u00b9 : \u2200 (x : List \u03b1), rmatch a\u271d\u00b9 x = true \u2194 x \u2208 matches' a\u271d\u00b9\na_ih\u271d : \u2200 (x : List \u03b1), rmatch a\u271d x = true \u2194 x \u2208 matches' a\u271d\nx : List \u03b1\n\u22a2 rmatch (a\u271d\u00b9 * a\u271d) x = true \u2194 x \u2208 matches' (a\u271d\u00b9 * a\u271d)\n\ncase star\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\na\u271d : RegularExpression \u03b1\na_ih\u271d : \u2200 (x : List \u03b1), rmatch a\u271d x = true \u2194 x \u2208 matches' a\u271d\nx : List \u03b1\n\u22a2 rmatch (star a\u271d) x = true \u2194 x \u2208 matches' (star a\u271d)", "state_after": "case plus\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\na\u271d\u00b9 a\u271d : RegularExpression \u03b1\na_ih\u271d\u00b9 : \u2200 (x : List \u03b1), rmatch a\u271d\u00b9 x = true \u2194 x \u2208 matches' a\u271d\u00b9\na_ih\u271d : \u2200 (x : List \u03b1), rmatch a\u271d x = true \u2194 x \u2208 matches' a\u271d\nx : List \u03b1\n\u22a2 rmatch (a\u271d\u00b9 + a\u271d) x = true \u2194 x \u2208 matches' (a\u271d\u00b9 + a\u271d)\n\ncase comp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\na\u271d\u00b9 a\u271d : RegularExpression \u03b1\na_ih\u271d\u00b9 : \u2200 (x : List \u03b1), rmatch a\u271d\u00b9 x = true \u2194 x \u2208 matches' a\u271d\u00b9\na_ih\u271d : \u2200 (x : List \u03b1), rmatch a\u271d x = true \u2194 x \u2208 matches' a\u271d\nx : List \u03b1\n\u22a2 rmatch (a\u271d\u00b9 * a\u271d) x = true \u2194 x \u2208 matches' (a\u271d\u00b9 * a\u271d)\n\ncase star\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\na\u271d : RegularExpression \u03b1\na_ih\u271d : \u2200 (x : List \u03b1), rmatch a\u271d x = true \u2194 x \u2208 matches' a\u271d\nx : List \u03b1\n\u22a2 rmatch (star a\u271d) x = true \u2194 x \u2208 matches' (star a\u271d)"}, {"tactic": "case plus _ _ ih\u2081 ih\u2082 =>\n  rw [add_rmatch_iff, ih\u2081, ih\u2082]\n  rfl", "annotated_tactic": ["case plus _ _ ih\u2081 ih\u2082 =>\n    rw [<a>add_rmatch_iff</a>, ih\u2081, ih\u2082]\n    rfl", [{"full_name": "RegularExpression.add_rmatch_iff", "def_path": "Mathlib/Computability/RegularExpressions.lean", "def_pos": [238, 9], "def_end_pos": [238, 23]}]], "state_before": "case plus\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\na\u271d\u00b9 a\u271d : RegularExpression \u03b1\na_ih\u271d\u00b9 : \u2200 (x : List \u03b1), rmatch a\u271d\u00b9 x = true \u2194 x \u2208 matches' a\u271d\u00b9\na_ih\u271d : \u2200 (x : List \u03b1), rmatch a\u271d x = true \u2194 x \u2208 matches' a\u271d\nx : List \u03b1\n\u22a2 rmatch (a\u271d\u00b9 + a\u271d) x = true \u2194 x \u2208 matches' (a\u271d\u00b9 + a\u271d)\n\ncase comp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\na\u271d\u00b9 a\u271d : RegularExpression \u03b1\na_ih\u271d\u00b9 : \u2200 (x : List \u03b1), rmatch a\u271d\u00b9 x = true \u2194 x \u2208 matches' a\u271d\u00b9\na_ih\u271d : \u2200 (x : List \u03b1), rmatch a\u271d x = true \u2194 x \u2208 matches' a\u271d\nx : List \u03b1\n\u22a2 rmatch (a\u271d\u00b9 * a\u271d) x = true \u2194 x \u2208 matches' (a\u271d\u00b9 * a\u271d)\n\ncase star\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\na\u271d : RegularExpression \u03b1\na_ih\u271d : \u2200 (x : List \u03b1), rmatch a\u271d x = true \u2194 x \u2208 matches' a\u271d\nx : List \u03b1\n\u22a2 rmatch (star a\u271d) x = true \u2194 x \u2208 matches' (star a\u271d)", "state_after": "case comp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\na\u271d\u00b9 a\u271d : RegularExpression \u03b1\na_ih\u271d\u00b9 : \u2200 (x : List \u03b1), rmatch a\u271d\u00b9 x = true \u2194 x \u2208 matches' a\u271d\u00b9\na_ih\u271d : \u2200 (x : List \u03b1), rmatch a\u271d x = true \u2194 x \u2208 matches' a\u271d\nx : List \u03b1\n\u22a2 rmatch (a\u271d\u00b9 * a\u271d) x = true \u2194 x \u2208 matches' (a\u271d\u00b9 * a\u271d)\n\ncase star\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\na\u271d : RegularExpression \u03b1\na_ih\u271d : \u2200 (x : List \u03b1), rmatch a\u271d x = true \u2194 x \u2208 matches' a\u271d\nx : List \u03b1\n\u22a2 rmatch (star a\u271d) x = true \u2194 x \u2208 matches' (star a\u271d)"}, {"tactic": "try rw [zero_def]", "annotated_tactic": ["try rw [<a>zero_def</a>]", [{"full_name": "RegularExpression.zero_def", "def_path": "Mathlib/Computability/RegularExpressions.lean", "def_pos": [87, 9], "def_end_pos": [87, 17]}]], "state_before": "case zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\nx : List \u03b1\n\u22a2 rmatch zero x = true \u2194 x \u2208 matches' zero", "state_after": "case zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\nx : List \u03b1\n\u22a2 rmatch 0 x = true \u2194 x \u2208 matches' 0"}, {"tactic": "try rw [one_def]", "annotated_tactic": ["try rw [<a>one_def</a>]", [{"full_name": "RegularExpression.one_def", "def_path": "Mathlib/Computability/RegularExpressions.lean", "def_pos": [92, 9], "def_end_pos": [92, 16]}]], "state_before": "case epsilon\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\nx : List \u03b1\n\u22a2 rmatch epsilon x = true \u2194 x \u2208 matches' epsilon", "state_after": "case epsilon\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\nx : List \u03b1\n\u22a2 rmatch 1 x = true \u2194 x \u2208 matches' 1"}, {"tactic": "try rw [plus_def]", "annotated_tactic": ["try rw [<a>plus_def</a>]", [{"full_name": "RegularExpression.plus_def", "def_path": "Mathlib/Computability/RegularExpressions.lean", "def_pos": [97, 9], "def_end_pos": [97, 17]}]], "state_before": "case plus\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\na\u271d\u00b9 a\u271d : RegularExpression \u03b1\na_ih\u271d\u00b9 : \u2200 (x : List \u03b1), rmatch a\u271d\u00b9 x = true \u2194 x \u2208 matches' a\u271d\u00b9\na_ih\u271d : \u2200 (x : List \u03b1), rmatch a\u271d x = true \u2194 x \u2208 matches' a\u271d\nx : List \u03b1\n\u22a2 rmatch (plus a\u271d\u00b9 a\u271d) x = true \u2194 x \u2208 matches' (plus a\u271d\u00b9 a\u271d)", "state_after": "case plus\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\na\u271d\u00b9 a\u271d : RegularExpression \u03b1\na_ih\u271d\u00b9 : \u2200 (x : List \u03b1), rmatch a\u271d\u00b9 x = true \u2194 x \u2208 matches' a\u271d\u00b9\na_ih\u271d : \u2200 (x : List \u03b1), rmatch a\u271d x = true \u2194 x \u2208 matches' a\u271d\nx : List \u03b1\n\u22a2 rmatch (a\u271d\u00b9 + a\u271d) x = true \u2194 x \u2208 matches' (a\u271d\u00b9 + a\u271d)"}, {"tactic": "try rw [comp_def]", "annotated_tactic": ["try rw [<a>comp_def</a>]", [{"full_name": "RegularExpression.comp_def", "def_path": "Mathlib/Computability/RegularExpressions.lean", "def_pos": [102, 9], "def_end_pos": [102, 17]}]], "state_before": "case comp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\na\u271d\u00b9 a\u271d : RegularExpression \u03b1\na_ih\u271d\u00b9 : \u2200 (x : List \u03b1), rmatch a\u271d\u00b9 x = true \u2194 x \u2208 matches' a\u271d\u00b9\na_ih\u271d : \u2200 (x : List \u03b1), rmatch a\u271d x = true \u2194 x \u2208 matches' a\u271d\nx : List \u03b1\n\u22a2 rmatch (comp a\u271d\u00b9 a\u271d) x = true \u2194 x \u2208 matches' (comp a\u271d\u00b9 a\u271d)", "state_after": "case comp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\na\u271d\u00b9 a\u271d : RegularExpression \u03b1\na_ih\u271d\u00b9 : \u2200 (x : List \u03b1), rmatch a\u271d\u00b9 x = true \u2194 x \u2208 matches' a\u271d\u00b9\na_ih\u271d : \u2200 (x : List \u03b1), rmatch a\u271d x = true \u2194 x \u2208 matches' a\u271d\nx : List \u03b1\n\u22a2 rmatch (a\u271d\u00b9 * a\u271d) x = true \u2194 x \u2208 matches' (a\u271d\u00b9 * a\u271d)"}, {"tactic": "rw [zero_def]", "annotated_tactic": ["rw [<a>zero_def</a>]", [{"full_name": "RegularExpression.zero_def", "def_path": "Mathlib/Computability/RegularExpressions.lean", "def_pos": [87, 9], "def_end_pos": [87, 17]}]], "state_before": "case zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\nx : List \u03b1\n\u22a2 rmatch zero x = true \u2194 x \u2208 matches' zero", "state_after": "case zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\nx : List \u03b1\n\u22a2 rmatch 0 x = true \u2194 x \u2208 matches' 0"}, {"tactic": "rw [one_def]", "annotated_tactic": ["rw [<a>one_def</a>]", [{"full_name": "RegularExpression.one_def", "def_path": "Mathlib/Computability/RegularExpressions.lean", "def_pos": [92, 9], "def_end_pos": [92, 16]}]], "state_before": "case epsilon\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\nx : List \u03b1\n\u22a2 rmatch epsilon x = true \u2194 x \u2208 matches' epsilon", "state_after": "case epsilon\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\nx : List \u03b1\n\u22a2 rmatch 1 x = true \u2194 x \u2208 matches' 1"}, {"tactic": "rw [plus_def]", "annotated_tactic": ["rw [<a>plus_def</a>]", [{"full_name": "RegularExpression.plus_def", "def_path": "Mathlib/Computability/RegularExpressions.lean", "def_pos": [97, 9], "def_end_pos": [97, 17]}]], "state_before": "case plus\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\na\u271d\u00b9 a\u271d : RegularExpression \u03b1\na_ih\u271d\u00b9 : \u2200 (x : List \u03b1), rmatch a\u271d\u00b9 x = true \u2194 x \u2208 matches' a\u271d\u00b9\na_ih\u271d : \u2200 (x : List \u03b1), rmatch a\u271d x = true \u2194 x \u2208 matches' a\u271d\nx : List \u03b1\n\u22a2 rmatch (plus a\u271d\u00b9 a\u271d) x = true \u2194 x \u2208 matches' (plus a\u271d\u00b9 a\u271d)", "state_after": "case plus\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\na\u271d\u00b9 a\u271d : RegularExpression \u03b1\na_ih\u271d\u00b9 : \u2200 (x : List \u03b1), rmatch a\u271d\u00b9 x = true \u2194 x \u2208 matches' a\u271d\u00b9\na_ih\u271d : \u2200 (x : List \u03b1), rmatch a\u271d x = true \u2194 x \u2208 matches' a\u271d\nx : List \u03b1\n\u22a2 rmatch (a\u271d\u00b9 + a\u271d) x = true \u2194 x \u2208 matches' (a\u271d\u00b9 + a\u271d)"}, {"tactic": "rw [comp_def]", "annotated_tactic": ["rw [<a>comp_def</a>]", [{"full_name": "RegularExpression.comp_def", "def_path": "Mathlib/Computability/RegularExpressions.lean", "def_pos": [102, 9], "def_end_pos": [102, 17]}]], "state_before": "case comp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\na\u271d\u00b9 a\u271d : RegularExpression \u03b1\na_ih\u271d\u00b9 : \u2200 (x : List \u03b1), rmatch a\u271d\u00b9 x = true \u2194 x \u2208 matches' a\u271d\u00b9\na_ih\u271d : \u2200 (x : List \u03b1), rmatch a\u271d x = true \u2194 x \u2208 matches' a\u271d\nx : List \u03b1\n\u22a2 rmatch (comp a\u271d\u00b9 a\u271d) x = true \u2194 x \u2208 matches' (comp a\u271d\u00b9 a\u271d)", "state_after": "case comp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\na\u271d\u00b9 a\u271d : RegularExpression \u03b1\na_ih\u271d\u00b9 : \u2200 (x : List \u03b1), rmatch a\u271d\u00b9 x = true \u2194 x \u2208 matches' a\u271d\u00b9\na_ih\u271d : \u2200 (x : List \u03b1), rmatch a\u271d x = true \u2194 x \u2208 matches' a\u271d\nx : List \u03b1\n\u22a2 rmatch (a\u271d\u00b9 * a\u271d) x = true \u2194 x \u2208 matches' (a\u271d\u00b9 * a\u271d)"}, {"tactic": "rw [zero_rmatch]", "annotated_tactic": ["rw [<a>zero_rmatch</a>]", [{"full_name": "RegularExpression.zero_rmatch", "def_path": "Mathlib/Computability/RegularExpressions.lean", "def_pos": [216, 9], "def_end_pos": [216, 20]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\nx : List \u03b1\n\u22a2 rmatch 0 x = true \u2194 x \u2208 matches' 0", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\nx : List \u03b1\n\u22a2 false = true \u2194 x \u2208 matches' 0"}, {"tactic": "tauto", "annotated_tactic": ["tauto", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\nx : List \u03b1\n\u22a2 false = true \u2194 x \u2208 matches' 0", "state_after": "no goals"}, {"tactic": "rw [one_rmatch_iff]", "annotated_tactic": ["rw [<a>one_rmatch_iff</a>]", [{"full_name": "RegularExpression.one_rmatch_iff", "def_path": "Mathlib/Computability/RegularExpressions.lean", "def_pos": [220, 9], "def_end_pos": [220, 23]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\nx : List \u03b1\n\u22a2 rmatch 1 x = true \u2194 x \u2208 matches' 1", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\nx : List \u03b1\n\u22a2 x = [] \u2194 x \u2208 matches' 1"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\nx : List \u03b1\n\u22a2 x = [] \u2194 x \u2208 matches' 1", "state_after": "no goals"}, {"tactic": "rw [char_rmatch_iff]", "annotated_tactic": ["rw [<a>char_rmatch_iff</a>]", [{"full_name": "RegularExpression.char_rmatch_iff", "def_path": "Mathlib/Computability/RegularExpressions.lean", "def_pos": [224, 9], "def_end_pos": [224, 24]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b a\u271d : \u03b1\nx : List \u03b1\n\u22a2 rmatch (char a\u271d) x = true \u2194 x \u2208 matches' (char a\u271d)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b a\u271d : \u03b1\nx : List \u03b1\n\u22a2 x = [a\u271d] \u2194 x \u2208 matches' (char a\u271d)"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b a\u271d : \u03b1\nx : List \u03b1\n\u22a2 x = [a\u271d] \u2194 x \u2208 matches' (char a\u271d)", "state_after": "no goals"}, {"tactic": "rw [add_rmatch_iff, ih\u2081, ih\u2082]", "annotated_tactic": ["rw [<a>add_rmatch_iff</a>, ih\u2081, ih\u2082]", [{"full_name": "RegularExpression.add_rmatch_iff", "def_path": "Mathlib/Computability/RegularExpressions.lean", "def_pos": [238, 9], "def_end_pos": [238, 23]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\na\u271d\u00b9 a\u271d : RegularExpression \u03b1\nih\u2081 : \u2200 (x : List \u03b1), rmatch a\u271d\u00b9 x = true \u2194 x \u2208 matches' a\u271d\u00b9\nih\u2082 : \u2200 (x : List \u03b1), rmatch a\u271d x = true \u2194 x \u2208 matches' a\u271d\nx : List \u03b1\n\u22a2 rmatch (a\u271d\u00b9 + a\u271d) x = true \u2194 x \u2208 matches' (a\u271d\u00b9 + a\u271d)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\na\u271d\u00b9 a\u271d : RegularExpression \u03b1\nih\u2081 : \u2200 (x : List \u03b1), rmatch a\u271d\u00b9 x = true \u2194 x \u2208 matches' a\u271d\u00b9\nih\u2082 : \u2200 (x : List \u03b1), rmatch a\u271d x = true \u2194 x \u2208 matches' a\u271d\nx : List \u03b1\n\u22a2 x \u2208 matches' a\u271d\u00b9 \u2228 x \u2208 matches' a\u271d \u2194 x \u2208 matches' (a\u271d\u00b9 + a\u271d)"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\na\u271d\u00b9 a\u271d : RegularExpression \u03b1\nih\u2081 : \u2200 (x : List \u03b1), rmatch a\u271d\u00b9 x = true \u2194 x \u2208 matches' a\u271d\u00b9\nih\u2082 : \u2200 (x : List \u03b1), rmatch a\u271d x = true \u2194 x \u2208 matches' a\u271d\nx : List \u03b1\n\u22a2 x \u2208 matches' a\u271d\u00b9 \u2228 x \u2208 matches' a\u271d \u2194 x \u2208 matches' (a\u271d\u00b9 + a\u271d)", "state_after": "no goals"}, {"tactic": "simp only [mul_rmatch_iff, comp_def, Language.mul_def, exists_and_left, Set.mem_image2,\n  Set.image_prod]", "annotated_tactic": ["simp only [<a>mul_rmatch_iff</a>, <a>comp_def</a>, <a>Language.mul_def</a>, <a>exists_and_left</a>, <a>Set.mem_image2</a>,\n      <a>Set.image_prod</a>]", [{"full_name": "RegularExpression.mul_rmatch_iff", "def_path": "Mathlib/Computability/RegularExpressions.lean", "def_pos": [247, 9], "def_end_pos": [247, 23]}, {"full_name": "RegularExpression.comp_def", "def_path": "Mathlib/Computability/RegularExpressions.lean", "def_pos": [102, 9], "def_end_pos": [102, 17]}, {"full_name": "Language.mul_def", "def_path": "Mathlib/Computability/Language.lean", "def_pos": [78, 9], "def_end_pos": [78, 16]}, {"full_name": "exists_and_left", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [465, 17], "def_end_pos": [465, 32]}, {"full_name": "Set.mem_image2", "def_path": "Mathlib/Data/Set/NAry.lean", "def_pos": [40, 9], "def_end_pos": [40, 19]}, {"full_name": "Set.image_prod", "def_path": "Mathlib/Data/Set/NAry.lean", "def_pos": [98, 7], "def_end_pos": [98, 17]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\nP Q : RegularExpression \u03b1\nih\u2081 : \u2200 (x : List \u03b1), rmatch P x = true \u2194 x \u2208 matches' P\nih\u2082 : \u2200 (x : List \u03b1), rmatch Q x = true \u2194 x \u2208 matches' Q\nx : List \u03b1\n\u22a2 rmatch (P * Q) x = true \u2194 x \u2208 matches' (P * Q)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\nP Q : RegularExpression \u03b1\nih\u2081 : \u2200 (x : List \u03b1), rmatch P x = true \u2194 x \u2208 matches' P\nih\u2082 : \u2200 (x : List \u03b1), rmatch Q x = true \u2194 x \u2208 matches' Q\nx : List \u03b1\n\u22a2 (\u2203 t u, x = t ++ u \u2227 rmatch P t = true \u2227 rmatch Q u = true) \u2194 x \u2208 matches' (P * Q)"}, {"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\nP Q : RegularExpression \u03b1\nih\u2081 : \u2200 (x : List \u03b1), rmatch P x = true \u2194 x \u2208 matches' P\nih\u2082 : \u2200 (x : List \u03b1), rmatch Q x = true \u2194 x \u2208 matches' Q\nx : List \u03b1\n\u22a2 (\u2203 t u, x = t ++ u \u2227 rmatch P t = true \u2227 rmatch Q u = true) \u2194 x \u2208 matches' (P * Q)", "state_after": "case mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\nP Q : RegularExpression \u03b1\nih\u2081 : \u2200 (x : List \u03b1), rmatch P x = true \u2194 x \u2208 matches' P\nih\u2082 : \u2200 (x : List \u03b1), rmatch Q x = true \u2194 x \u2208 matches' Q\nx : List \u03b1\n\u22a2 (\u2203 t u, x = t ++ u \u2227 rmatch P t = true \u2227 rmatch Q u = true) \u2192 x \u2208 matches' (P * Q)\n\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\nP Q : RegularExpression \u03b1\nih\u2081 : \u2200 (x : List \u03b1), rmatch P x = true \u2194 x \u2208 matches' P\nih\u2082 : \u2200 (x : List \u03b1), rmatch Q x = true \u2194 x \u2208 matches' Q\nx : List \u03b1\n\u22a2 x \u2208 matches' (P * Q) \u2192 \u2203 t u, x = t ++ u \u2227 rmatch P t = true \u2227 rmatch Q u = true"}, {"tactic": "rintro \u27e8x, y, hsum, hmatch\u2081, hmatch\u2082\u27e9", "annotated_tactic": ["rintro \u27e8x, y, hsum, hmatch\u2081, hmatch\u2082\u27e9", []], "state_before": "case mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\nP Q : RegularExpression \u03b1\nih\u2081 : \u2200 (x : List \u03b1), rmatch P x = true \u2194 x \u2208 matches' P\nih\u2082 : \u2200 (x : List \u03b1), rmatch Q x = true \u2194 x \u2208 matches' Q\nx : List \u03b1\n\u22a2 (\u2203 t u, x = t ++ u \u2227 rmatch P t = true \u2227 rmatch Q u = true) \u2192 x \u2208 matches' (P * Q)", "state_after": "case mp.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\nP Q : RegularExpression \u03b1\nih\u2081 : \u2200 (x : List \u03b1), rmatch P x = true \u2194 x \u2208 matches' P\nih\u2082 : \u2200 (x : List \u03b1), rmatch Q x = true \u2194 x \u2208 matches' Q\nx\u271d x y : List \u03b1\nhsum : x\u271d = x ++ y\nhmatch\u2081 : rmatch P x = true\nhmatch\u2082 : rmatch Q y = true\n\u22a2 x\u271d \u2208 matches' (P * Q)"}, {"tactic": "rw [ih\u2081] at hmatch\u2081", "annotated_tactic": ["rw [ih\u2081] at hmatch\u2081", []], "state_before": "case mp.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\nP Q : RegularExpression \u03b1\nih\u2081 : \u2200 (x : List \u03b1), rmatch P x = true \u2194 x \u2208 matches' P\nih\u2082 : \u2200 (x : List \u03b1), rmatch Q x = true \u2194 x \u2208 matches' Q\nx\u271d x y : List \u03b1\nhsum : x\u271d = x ++ y\nhmatch\u2081 : rmatch P x = true\nhmatch\u2082 : rmatch Q y = true\n\u22a2 x\u271d \u2208 matches' (P * Q)", "state_after": "case mp.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\nP Q : RegularExpression \u03b1\nih\u2081 : \u2200 (x : List \u03b1), rmatch P x = true \u2194 x \u2208 matches' P\nih\u2082 : \u2200 (x : List \u03b1), rmatch Q x = true \u2194 x \u2208 matches' Q\nx\u271d x y : List \u03b1\nhsum : x\u271d = x ++ y\nhmatch\u2081 : x \u2208 matches' P\nhmatch\u2082 : rmatch Q y = true\n\u22a2 x\u271d \u2208 matches' (P * Q)"}, {"tactic": "rw [ih\u2082] at hmatch\u2082", "annotated_tactic": ["rw [ih\u2082] at hmatch\u2082", []], "state_before": "case mp.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\nP Q : RegularExpression \u03b1\nih\u2081 : \u2200 (x : List \u03b1), rmatch P x = true \u2194 x \u2208 matches' P\nih\u2082 : \u2200 (x : List \u03b1), rmatch Q x = true \u2194 x \u2208 matches' Q\nx\u271d x y : List \u03b1\nhsum : x\u271d = x ++ y\nhmatch\u2081 : x \u2208 matches' P\nhmatch\u2082 : rmatch Q y = true\n\u22a2 x\u271d \u2208 matches' (P * Q)", "state_after": "case mp.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\nP Q : RegularExpression \u03b1\nih\u2081 : \u2200 (x : List \u03b1), rmatch P x = true \u2194 x \u2208 matches' P\nih\u2082 : \u2200 (x : List \u03b1), rmatch Q x = true \u2194 x \u2208 matches' Q\nx\u271d x y : List \u03b1\nhsum : x\u271d = x ++ y\nhmatch\u2081 : x \u2208 matches' P\nhmatch\u2082 : y \u2208 matches' Q\n\u22a2 x\u271d \u2208 matches' (P * Q)"}, {"tactic": "exact \u27e8x, y, hmatch\u2081, hmatch\u2082, hsum.symm\u27e9", "annotated_tactic": ["exact \u27e8x, y, hmatch\u2081, hmatch\u2082, hsum.symm\u27e9", []], "state_before": "case mp.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\nP Q : RegularExpression \u03b1\nih\u2081 : \u2200 (x : List \u03b1), rmatch P x = true \u2194 x \u2208 matches' P\nih\u2082 : \u2200 (x : List \u03b1), rmatch Q x = true \u2194 x \u2208 matches' Q\nx\u271d x y : List \u03b1\nhsum : x\u271d = x ++ y\nhmatch\u2081 : x \u2208 matches' P\nhmatch\u2082 : y \u2208 matches' Q\n\u22a2 x\u271d \u2208 matches' (P * Q)", "state_after": "no goals"}, {"tactic": "rintro \u27e8x, y, hmatch\u2081, hmatch\u2082, hsum\u27e9", "annotated_tactic": ["rintro \u27e8x, y, hmatch\u2081, hmatch\u2082, hsum\u27e9", []], "state_before": "case mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\nP Q : RegularExpression \u03b1\nih\u2081 : \u2200 (x : List \u03b1), rmatch P x = true \u2194 x \u2208 matches' P\nih\u2082 : \u2200 (x : List \u03b1), rmatch Q x = true \u2194 x \u2208 matches' Q\nx : List \u03b1\n\u22a2 x \u2208 matches' (P * Q) \u2192 \u2203 t u, x = t ++ u \u2227 rmatch P t = true \u2227 rmatch Q u = true", "state_after": "case mpr.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\nP Q : RegularExpression \u03b1\nih\u2081 : \u2200 (x : List \u03b1), rmatch P x = true \u2194 x \u2208 matches' P\nih\u2082 : \u2200 (x : List \u03b1), rmatch Q x = true \u2194 x \u2208 matches' Q\nx\u271d x y : List \u03b1\nhmatch\u2081 : x \u2208 matches' P\nhmatch\u2082 : y \u2208 matches' Q\nhsum : (fun x x_1 => x ++ x_1) x y = x\u271d\n\u22a2 \u2203 t u, x\u271d = t ++ u \u2227 rmatch P t = true \u2227 rmatch Q u = true"}, {"tactic": "rw [\u2190 ih\u2081] at hmatch\u2081", "annotated_tactic": ["rw [\u2190 ih\u2081] at hmatch\u2081", []], "state_before": "case mpr.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\nP Q : RegularExpression \u03b1\nih\u2081 : \u2200 (x : List \u03b1), rmatch P x = true \u2194 x \u2208 matches' P\nih\u2082 : \u2200 (x : List \u03b1), rmatch Q x = true \u2194 x \u2208 matches' Q\nx\u271d x y : List \u03b1\nhmatch\u2081 : x \u2208 matches' P\nhmatch\u2082 : y \u2208 matches' Q\nhsum : (fun x x_1 => x ++ x_1) x y = x\u271d\n\u22a2 \u2203 t u, x\u271d = t ++ u \u2227 rmatch P t = true \u2227 rmatch Q u = true", "state_after": "case mpr.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\nP Q : RegularExpression \u03b1\nih\u2081 : \u2200 (x : List \u03b1), rmatch P x = true \u2194 x \u2208 matches' P\nih\u2082 : \u2200 (x : List \u03b1), rmatch Q x = true \u2194 x \u2208 matches' Q\nx\u271d x y : List \u03b1\nhmatch\u2081 : rmatch P x = true\nhmatch\u2082 : y \u2208 matches' Q\nhsum : (fun x x_1 => x ++ x_1) x y = x\u271d\n\u22a2 \u2203 t u, x\u271d = t ++ u \u2227 rmatch P t = true \u2227 rmatch Q u = true"}, {"tactic": "rw [\u2190 ih\u2082] at hmatch\u2082", "annotated_tactic": ["rw [\u2190 ih\u2082] at hmatch\u2082", []], "state_before": "case mpr.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\nP Q : RegularExpression \u03b1\nih\u2081 : \u2200 (x : List \u03b1), rmatch P x = true \u2194 x \u2208 matches' P\nih\u2082 : \u2200 (x : List \u03b1), rmatch Q x = true \u2194 x \u2208 matches' Q\nx\u271d x y : List \u03b1\nhmatch\u2081 : rmatch P x = true\nhmatch\u2082 : y \u2208 matches' Q\nhsum : (fun x x_1 => x ++ x_1) x y = x\u271d\n\u22a2 \u2203 t u, x\u271d = t ++ u \u2227 rmatch P t = true \u2227 rmatch Q u = true", "state_after": "case mpr.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\nP Q : RegularExpression \u03b1\nih\u2081 : \u2200 (x : List \u03b1), rmatch P x = true \u2194 x \u2208 matches' P\nih\u2082 : \u2200 (x : List \u03b1), rmatch Q x = true \u2194 x \u2208 matches' Q\nx\u271d x y : List \u03b1\nhmatch\u2081 : rmatch P x = true\nhmatch\u2082 : rmatch Q y = true\nhsum : (fun x x_1 => x ++ x_1) x y = x\u271d\n\u22a2 \u2203 t u, x\u271d = t ++ u \u2227 rmatch P t = true \u2227 rmatch Q u = true"}, {"tactic": "exact \u27e8x, y, hsum.symm, hmatch\u2081, hmatch\u2082\u27e9", "annotated_tactic": ["exact \u27e8x, y, hsum.symm, hmatch\u2081, hmatch\u2082\u27e9", []], "state_before": "case mpr.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\nP Q : RegularExpression \u03b1\nih\u2081 : \u2200 (x : List \u03b1), rmatch P x = true \u2194 x \u2208 matches' P\nih\u2082 : \u2200 (x : List \u03b1), rmatch Q x = true \u2194 x \u2208 matches' Q\nx\u271d x y : List \u03b1\nhmatch\u2081 : rmatch P x = true\nhmatch\u2082 : rmatch Q y = true\nhsum : (fun x x_1 => x ++ x_1) x y = x\u271d\n\u22a2 \u2203 t u, x\u271d = t ++ u \u2227 rmatch P t = true \u2227 rmatch Q u = true", "state_after": "no goals"}, {"tactic": "rw [star_rmatch_iff]", "annotated_tactic": ["rw [<a>star_rmatch_iff</a>]", [{"full_name": "RegularExpression.star_rmatch_iff", "def_path": "Mathlib/Computability/RegularExpressions.lean", "def_pos": [293, 9], "def_end_pos": [293, 24]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\na\u271d : RegularExpression \u03b1\nih : \u2200 (x : List \u03b1), rmatch a\u271d x = true \u2194 x \u2208 matches' a\u271d\nx : List \u03b1\n\u22a2 rmatch (star a\u271d) x = true \u2194 x \u2208 matches' (star a\u271d)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\na\u271d : RegularExpression \u03b1\nih : \u2200 (x : List \u03b1), rmatch a\u271d x = true \u2194 x \u2208 matches' a\u271d\nx : List \u03b1\n\u22a2 (\u2203 S, x = join S \u2227 \u2200 (t : List \u03b1), t \u2208 S \u2192 t \u2260 [] \u2227 rmatch a\u271d t = true) \u2194 x \u2208 matches' (star a\u271d)"}, {"tactic": "simp only [ne_eq, matches', Language.kstar_def_nonempty, mem_setOf_eq]", "annotated_tactic": ["simp only [<a>ne_eq</a>, <a>matches'</a>, <a>Language.kstar_def_nonempty</a>, <a>mem_setOf_eq</a>]", [{"full_name": "ne_eq", "def_path": "lake-packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [76, 17], "def_end_pos": [76, 22]}, {"full_name": "RegularExpression.matches'", "def_path": "Mathlib/Computability/RegularExpressions.lean", "def_pos": [110, 5], "def_end_pos": [110, 13]}, {"full_name": "Language.kstar_def_nonempty", "def_path": "Mathlib/Computability/Language.lean", "def_pos": [176, 9], "def_end_pos": [176, 27]}, {"full_name": "Set.mem_setOf_eq", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [256, 29], "def_end_pos": [256, 41]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\na\u271d : RegularExpression \u03b1\nih : \u2200 (x : List \u03b1), rmatch a\u271d x = true \u2194 x \u2208 matches' a\u271d\nx : List \u03b1\n\u22a2 (\u2203 S, x = join S \u2227 \u2200 (t : List \u03b1), t \u2208 S \u2192 t \u2260 [] \u2227 rmatch a\u271d t = true) \u2194 x \u2208 matches' (star a\u271d)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\na\u271d : RegularExpression \u03b1\nih : \u2200 (x : List \u03b1), rmatch a\u271d x = true \u2194 x \u2208 matches' a\u271d\nx : List \u03b1\n\u22a2 (\u2203 S, x = join S \u2227 \u2200 (t : List \u03b1), t \u2208 S \u2192 \u00act = [] \u2227 rmatch a\u271d t = true) \u2194\n    x \u2208 {x | \u2203 S, x = join S \u2227 \u2200 (y : List \u03b1), y \u2208 S \u2192 y \u2208 matches' a\u271d \u2227 \u00acy = []}"}, {"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\na\u271d : RegularExpression \u03b1\nih : \u2200 (x : List \u03b1), rmatch a\u271d x = true \u2194 x \u2208 matches' a\u271d\nx : List \u03b1\n\u22a2 (\u2203 S, x = join S \u2227 \u2200 (t : List \u03b1), t \u2208 S \u2192 \u00act = [] \u2227 rmatch a\u271d t = true) \u2194\n    x \u2208 {x | \u2203 S, x = join S \u2227 \u2200 (y : List \u03b1), y \u2208 S \u2192 y \u2208 matches' a\u271d \u2227 \u00acy = []}", "state_after": "case mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\na\u271d : RegularExpression \u03b1\nih : \u2200 (x : List \u03b1), rmatch a\u271d x = true \u2194 x \u2208 matches' a\u271d\nx : List \u03b1\n\u22a2 (\u2203 S, x = join S \u2227 \u2200 (t : List \u03b1), t \u2208 S \u2192 \u00act = [] \u2227 rmatch a\u271d t = true) \u2192\n    x \u2208 {x | \u2203 S, x = join S \u2227 \u2200 (y : List \u03b1), y \u2208 S \u2192 y \u2208 matches' a\u271d \u2227 \u00acy = []}\n\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\na\u271d : RegularExpression \u03b1\nih : \u2200 (x : List \u03b1), rmatch a\u271d x = true \u2194 x \u2208 matches' a\u271d\nx : List \u03b1\n\u22a2 x \u2208 {x | \u2203 S, x = join S \u2227 \u2200 (y : List \u03b1), y \u2208 S \u2192 y \u2208 matches' a\u271d \u2227 \u00acy = []} \u2192\n    \u2203 S, x = join S \u2227 \u2200 (t : List \u03b1), t \u2208 S \u2192 \u00act = [] \u2227 rmatch a\u271d t = true"}, {"tactic": "all_goals\n  rintro \u27e8S, hx, hS\u27e9\n  refine' \u27e8S, hx, _\u27e9\n  intro y\n  specialize hS y", "annotated_tactic": ["all_goals\n      rintro \u27e8S, hx, hS\u27e9\n      refine' \u27e8S, hx, _\u27e9\n      intro y\n      specialize hS y", []], "state_before": "case mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\na\u271d : RegularExpression \u03b1\nih : \u2200 (x : List \u03b1), rmatch a\u271d x = true \u2194 x \u2208 matches' a\u271d\nx : List \u03b1\n\u22a2 (\u2203 S, x = join S \u2227 \u2200 (t : List \u03b1), t \u2208 S \u2192 \u00act = [] \u2227 rmatch a\u271d t = true) \u2192\n    x \u2208 {x | \u2203 S, x = join S \u2227 \u2200 (y : List \u03b1), y \u2208 S \u2192 y \u2208 matches' a\u271d \u2227 \u00acy = []}\n\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\na\u271d : RegularExpression \u03b1\nih : \u2200 (x : List \u03b1), rmatch a\u271d x = true \u2194 x \u2208 matches' a\u271d\nx : List \u03b1\n\u22a2 x \u2208 {x | \u2203 S, x = join S \u2227 \u2200 (y : List \u03b1), y \u2208 S \u2192 y \u2208 matches' a\u271d \u2227 \u00acy = []} \u2192\n    \u2203 S, x = join S \u2227 \u2200 (t : List \u03b1), t \u2208 S \u2192 \u00act = [] \u2227 rmatch a\u271d t = true", "state_after": "case mp.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\na\u271d : RegularExpression \u03b1\nih : \u2200 (x : List \u03b1), rmatch a\u271d x = true \u2194 x \u2208 matches' a\u271d\nx : List \u03b1\nS : List (List \u03b1)\nhx : x = join S\ny : List \u03b1\nhS : y \u2208 S \u2192 \u00acy = [] \u2227 rmatch a\u271d y = true\n\u22a2 y \u2208 S \u2192 y \u2208 matches' a\u271d \u2227 \u00acy = []\n\ncase mpr.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\na\u271d : RegularExpression \u03b1\nih : \u2200 (x : List \u03b1), rmatch a\u271d x = true \u2194 x \u2208 matches' a\u271d\nx : List \u03b1\nS : List (List \u03b1)\nhx : x = join S\ny : List \u03b1\nhS : y \u2208 S \u2192 y \u2208 matches' a\u271d \u2227 \u00acy = []\n\u22a2 y \u2208 S \u2192 \u00acy = [] \u2227 rmatch a\u271d y = true"}, {"tactic": "rintro \u27e8S, hx, hS\u27e9", "annotated_tactic": ["rintro \u27e8S, hx, hS\u27e9", []], "state_before": "case mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\na\u271d : RegularExpression \u03b1\nih : \u2200 (x : List \u03b1), rmatch a\u271d x = true \u2194 x \u2208 matches' a\u271d\nx : List \u03b1\n\u22a2 x \u2208 {x | \u2203 S, x = join S \u2227 \u2200 (y : List \u03b1), y \u2208 S \u2192 y \u2208 matches' a\u271d \u2227 \u00acy = []} \u2192\n    \u2203 S, x = join S \u2227 \u2200 (t : List \u03b1), t \u2208 S \u2192 \u00act = [] \u2227 rmatch a\u271d t = true", "state_after": "case mpr.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\na\u271d : RegularExpression \u03b1\nih : \u2200 (x : List \u03b1), rmatch a\u271d x = true \u2194 x \u2208 matches' a\u271d\nx : List \u03b1\nS : List (List \u03b1)\nhx : x = join S\nhS : \u2200 (y : List \u03b1), y \u2208 S \u2192 y \u2208 matches' a\u271d \u2227 \u00acy = []\n\u22a2 \u2203 S, x = join S \u2227 \u2200 (t : List \u03b1), t \u2208 S \u2192 \u00act = [] \u2227 rmatch a\u271d t = true"}, {"tactic": "refine' \u27e8S, hx, _\u27e9", "annotated_tactic": ["refine' \u27e8S, hx, _\u27e9", []], "state_before": "case mpr.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\na\u271d : RegularExpression \u03b1\nih : \u2200 (x : List \u03b1), rmatch a\u271d x = true \u2194 x \u2208 matches' a\u271d\nx : List \u03b1\nS : List (List \u03b1)\nhx : x = join S\nhS : \u2200 (y : List \u03b1), y \u2208 S \u2192 y \u2208 matches' a\u271d \u2227 \u00acy = []\n\u22a2 \u2203 S, x = join S \u2227 \u2200 (t : List \u03b1), t \u2208 S \u2192 \u00act = [] \u2227 rmatch a\u271d t = true", "state_after": "case mpr.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\na\u271d : RegularExpression \u03b1\nih : \u2200 (x : List \u03b1), rmatch a\u271d x = true \u2194 x \u2208 matches' a\u271d\nx : List \u03b1\nS : List (List \u03b1)\nhx : x = join S\nhS : \u2200 (y : List \u03b1), y \u2208 S \u2192 y \u2208 matches' a\u271d \u2227 \u00acy = []\n\u22a2 \u2200 (t : List \u03b1), t \u2208 S \u2192 \u00act = [] \u2227 rmatch a\u271d t = true"}, {"tactic": "intro y", "annotated_tactic": ["intro y", []], "state_before": "case mpr.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\na\u271d : RegularExpression \u03b1\nih : \u2200 (x : List \u03b1), rmatch a\u271d x = true \u2194 x \u2208 matches' a\u271d\nx : List \u03b1\nS : List (List \u03b1)\nhx : x = join S\nhS : \u2200 (y : List \u03b1), y \u2208 S \u2192 y \u2208 matches' a\u271d \u2227 \u00acy = []\n\u22a2 \u2200 (t : List \u03b1), t \u2208 S \u2192 \u00act = [] \u2227 rmatch a\u271d t = true", "state_after": "case mpr.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\na\u271d : RegularExpression \u03b1\nih : \u2200 (x : List \u03b1), rmatch a\u271d x = true \u2194 x \u2208 matches' a\u271d\nx : List \u03b1\nS : List (List \u03b1)\nhx : x = join S\nhS : \u2200 (y : List \u03b1), y \u2208 S \u2192 y \u2208 matches' a\u271d \u2227 \u00acy = []\ny : List \u03b1\n\u22a2 y \u2208 S \u2192 \u00acy = [] \u2227 rmatch a\u271d y = true"}, {"tactic": "specialize hS y", "annotated_tactic": ["specialize hS y", []], "state_before": "case mpr.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\na\u271d : RegularExpression \u03b1\nih : \u2200 (x : List \u03b1), rmatch a\u271d x = true \u2194 x \u2208 matches' a\u271d\nx : List \u03b1\nS : List (List \u03b1)\nhx : x = join S\nhS : \u2200 (y : List \u03b1), y \u2208 S \u2192 y \u2208 matches' a\u271d \u2227 \u00acy = []\ny : List \u03b1\n\u22a2 y \u2208 S \u2192 \u00acy = [] \u2227 rmatch a\u271d y = true", "state_after": "case mpr.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\na\u271d : RegularExpression \u03b1\nih : \u2200 (x : List \u03b1), rmatch a\u271d x = true \u2194 x \u2208 matches' a\u271d\nx : List \u03b1\nS : List (List \u03b1)\nhx : x = join S\ny : List \u03b1\nhS : y \u2208 S \u2192 y \u2208 matches' a\u271d \u2227 \u00acy = []\n\u22a2 y \u2208 S \u2192 \u00acy = [] \u2227 rmatch a\u271d y = true"}, {"tactic": "rw [\u2190 ih y]", "annotated_tactic": ["rw [\u2190 ih y]", []], "state_before": "case mp.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\na\u271d : RegularExpression \u03b1\nih : \u2200 (x : List \u03b1), rmatch a\u271d x = true \u2194 x \u2208 matches' a\u271d\nx : List \u03b1\nS : List (List \u03b1)\nhx : x = join S\ny : List \u03b1\nhS : y \u2208 S \u2192 \u00acy = [] \u2227 rmatch a\u271d y = true\n\u22a2 y \u2208 S \u2192 y \u2208 matches' a\u271d \u2227 \u00acy = []", "state_after": "case mp.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\na\u271d : RegularExpression \u03b1\nih : \u2200 (x : List \u03b1), rmatch a\u271d x = true \u2194 x \u2208 matches' a\u271d\nx : List \u03b1\nS : List (List \u03b1)\nhx : x = join S\ny : List \u03b1\nhS : y \u2208 S \u2192 \u00acy = [] \u2227 rmatch a\u271d y = true\n\u22a2 y \u2208 S \u2192 rmatch a\u271d y = true \u2227 \u00acy = []"}, {"tactic": "tauto", "annotated_tactic": ["tauto", []], "state_before": "case mp.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\na\u271d : RegularExpression \u03b1\nih : \u2200 (x : List \u03b1), rmatch a\u271d x = true \u2194 x \u2208 matches' a\u271d\nx : List \u03b1\nS : List (List \u03b1)\nhx : x = join S\ny : List \u03b1\nhS : y \u2208 S \u2192 \u00acy = [] \u2227 rmatch a\u271d y = true\n\u22a2 y \u2208 S \u2192 rmatch a\u271d y = true \u2227 \u00acy = []", "state_after": "no goals"}, {"tactic": "rw [ih y]", "annotated_tactic": ["rw [ih y]", []], "state_before": "case mpr.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\na\u271d : RegularExpression \u03b1\nih : \u2200 (x : List \u03b1), rmatch a\u271d x = true \u2194 x \u2208 matches' a\u271d\nx : List \u03b1\nS : List (List \u03b1)\nhx : x = join S\ny : List \u03b1\nhS : y \u2208 S \u2192 y \u2208 matches' a\u271d \u2227 \u00acy = []\n\u22a2 y \u2208 S \u2192 \u00acy = [] \u2227 rmatch a\u271d y = true", "state_after": "case mpr.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\na\u271d : RegularExpression \u03b1\nih : \u2200 (x : List \u03b1), rmatch a\u271d x = true \u2194 x \u2208 matches' a\u271d\nx : List \u03b1\nS : List (List \u03b1)\nhx : x = join S\ny : List \u03b1\nhS : y \u2208 S \u2192 y \u2208 matches' a\u271d \u2227 \u00acy = []\n\u22a2 y \u2208 S \u2192 \u00acy = [] \u2227 y \u2208 matches' a\u271d"}, {"tactic": "tauto", "annotated_tactic": ["tauto", []], "state_before": "case mpr.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\na\u271d : RegularExpression \u03b1\nih : \u2200 (x : List \u03b1), rmatch a\u271d x = true \u2194 x \u2208 matches' a\u271d\nx : List \u03b1\nS : List (List \u03b1)\nhx : x = join S\ny : List \u03b1\nhS : y \u2208 S \u2192 y \u2208 matches' a\u271d \u2227 \u00acy = []\n\u22a2 y \u2208 S \u2192 \u00acy = [] \u2227 y \u2208 matches' a\u271d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/TMToPartrec.lean", "full_name": "Turing.PartrecToTM2.tr_supports", "start": [2085, 1], "end": [2086, 93], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/LocallyFinite.lean", "full_name": "Finset.image_add_left_Ico", "start": [1142, 1], "end": [1143, 78], "traced_tactics": [{"tactic": "rw [\u2190 map_add_left_Ico, map_eq_image, addLeftEmbedding, Embedding.coeFn_mk]", "annotated_tactic": ["rw [\u2190 <a>map_add_left_Ico</a>, <a>map_eq_image</a>, <a>addLeftEmbedding</a>, <a>Embedding.coeFn_mk</a>]", [{"full_name": "Finset.map_add_left_Ico", "def_path": "Mathlib/Data/Finset/LocallyFinite.lean", "def_pos": [1093, 9], "def_end_pos": [1093, 25]}, {"full_name": "Finset.map_eq_image", "def_path": "Mathlib/Data/Finset/Image.lean", "def_pos": [342, 9], "def_end_pos": [342, 21]}, {"full_name": "addLeftEmbedding", "def_path": "Mathlib/Algebra/Hom/Embedding.lean", "def_pos": [23, 3], "def_end_pos": [23, 14]}, {"full_name": "Function.Embedding.coeFn_mk", "def_path": "Mathlib/Logic/Embedding/Basic.lean", "def_pos": [115, 9], "def_end_pos": [115, 17]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : OrderedCancelAddCommMonoid \u03b1\ninst\u271d\u00b2 : ExistsAddOfLE \u03b1\ninst\u271d\u00b9 : LocallyFiniteOrder \u03b1\ninst\u271d : DecidableEq \u03b1\na b c : \u03b1\n\u22a2 image ((fun x x_1 => x + x_1) c) (Ico a b) = Ico (c + a) (c + b)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Martingale/Upcrossing.lean", "full_name": "MeasureTheory.Submartingale.mul_integral_upcrossingsBefore_le_integral_pos_part", "start": [738, 1], "end": [745, 66], "traced_tactics": [{"tactic": "by_cases hab : a < b", "annotated_tactic": ["by_cases hab : a < b", []], "state_before": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na\u271d b\u271d : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN\u271d n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\ninst\u271d : IsFiniteMeasure \u03bc\na b : \u211d\nhf : Submartingale f \u2131 \u03bc\nN : \u2115\n\u22a2 (b - a) * \u222b (x : \u03a9), \u2191(upcrossingsBefore a b f N x) \u2202\u03bc \u2264 \u222b (x : \u03a9), (fun \u03c9 => (f N \u03c9 - a)\u207a) x \u2202\u03bc", "state_after": "case pos\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na\u271d b\u271d : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN\u271d n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\ninst\u271d : IsFiniteMeasure \u03bc\na b : \u211d\nhf : Submartingale f \u2131 \u03bc\nN : \u2115\nhab : a < b\n\u22a2 (b - a) * \u222b (x : \u03a9), \u2191(upcrossingsBefore a b f N x) \u2202\u03bc \u2264 \u222b (x : \u03a9), (fun \u03c9 => (f N \u03c9 - a)\u207a) x \u2202\u03bc\n\ncase neg\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na\u271d b\u271d : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN\u271d n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\ninst\u271d : IsFiniteMeasure \u03bc\na b : \u211d\nhf : Submartingale f \u2131 \u03bc\nN : \u2115\nhab : \u00aca < b\n\u22a2 (b - a) * \u222b (x : \u03a9), \u2191(upcrossingsBefore a b f N x) \u2202\u03bc \u2264 \u222b (x : \u03a9), (fun \u03c9 => (f N \u03c9 - a)\u207a) x \u2202\u03bc"}, {"tactic": "exact mul_integral_upcrossingsBefore_le_integral_pos_part_aux hf hab", "annotated_tactic": ["exact <a>mul_integral_upcrossingsBefore_le_integral_pos_part_aux</a> hf hab", [{"full_name": "MeasureTheory.mul_integral_upcrossingsBefore_le_integral_pos_part_aux", "def_path": "Mathlib/Probability/Martingale/Upcrossing.lean", "def_pos": [723, 9], "def_end_pos": [723, 64]}]], "state_before": "case pos\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na\u271d b\u271d : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN\u271d n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\ninst\u271d : IsFiniteMeasure \u03bc\na b : \u211d\nhf : Submartingale f \u2131 \u03bc\nN : \u2115\nhab : a < b\n\u22a2 (b - a) * \u222b (x : \u03a9), \u2191(upcrossingsBefore a b f N x) \u2202\u03bc \u2264 \u222b (x : \u03a9), (fun \u03c9 => (f N \u03c9 - a)\u207a) x \u2202\u03bc", "state_after": "no goals"}, {"tactic": "rw [not_lt, \u2190 sub_nonpos] at hab", "annotated_tactic": ["rw [<a>not_lt</a>, \u2190 <a>sub_nonpos</a>] at hab", [{"full_name": "not_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [368, 9], "def_end_pos": [368, 15]}, {"full_name": "sub_nonpos", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [730, 30], "def_end_pos": [730, 40]}]], "state_before": "case neg\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na\u271d b\u271d : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN\u271d n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\ninst\u271d : IsFiniteMeasure \u03bc\na b : \u211d\nhf : Submartingale f \u2131 \u03bc\nN : \u2115\nhab : \u00aca < b\n\u22a2 (b - a) * \u222b (x : \u03a9), \u2191(upcrossingsBefore a b f N x) \u2202\u03bc \u2264 \u222b (x : \u03a9), (fun \u03c9 => (f N \u03c9 - a)\u207a) x \u2202\u03bc", "state_after": "case neg\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na\u271d b\u271d : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN\u271d n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\ninst\u271d : IsFiniteMeasure \u03bc\na b : \u211d\nhf : Submartingale f \u2131 \u03bc\nN : \u2115\nhab\u271d : b \u2264 a\nhab : b - a \u2264 0\n\u22a2 (b - a) * \u222b (x : \u03a9), \u2191(upcrossingsBefore a b f N x) \u2202\u03bc \u2264 \u222b (x : \u03a9), (fun \u03c9 => (f N \u03c9 - a)\u207a) x \u2202\u03bc"}, {"tactic": "exact le_trans (mul_nonpos_of_nonpos_of_nonneg hab (integral_nonneg fun \u03c9 => Nat.cast_nonneg _))\n  (integral_nonneg fun \u03c9 => LatticeOrderedGroup.pos_nonneg _)", "annotated_tactic": ["exact <a>le_trans</a> (<a>mul_nonpos_of_nonpos_of_nonneg</a> hab (<a>integral_nonneg</a> fun \u03c9 => <a>Nat.cast_nonneg</a> _))\n      (<a>integral_nonneg</a> fun \u03c9 => <a>LatticeOrderedGroup.pos_nonneg</a> _)", [{"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}, {"full_name": "mul_nonpos_of_nonpos_of_nonneg", "def_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "def_pos": [392, 9], "def_end_pos": [392, 39]}, {"full_name": "MeasureTheory.integral_nonneg", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1251, 9], "def_end_pos": [1251, 24]}, {"full_name": "Nat.cast_nonneg", "def_path": "Mathlib/Data/Nat/Cast/Order.lean", "def_pos": [44, 9], "def_end_pos": [44, 20]}, {"full_name": "MeasureTheory.integral_nonneg", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1251, 9], "def_end_pos": [1251, 24]}, {"full_name": "LatticeOrderedGroup.pos_nonneg", "def_path": "Mathlib/Algebra/Order/LatticeGroup.lean", "def_pos": [200, 15], "def_end_pos": [200, 25]}]], "state_before": "case neg\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na\u271d b\u271d : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN\u271d n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\ninst\u271d : IsFiniteMeasure \u03bc\na b : \u211d\nhf : Submartingale f \u2131 \u03bc\nN : \u2115\nhab\u271d : b \u2264 a\nhab : b - a \u2264 0\n\u22a2 (b - a) * \u222b (x : \u03a9), \u2191(upcrossingsBefore a b f N x) \u2202\u03bc \u2264 \u222b (x : \u03a9), (fun \u03c9 => (f N \u03c9 - a)\u207a) x \u2202\u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "full_name": "MeasureTheory.L1.SimpleFunc.norm_setToL1SCLM_le'", "start": [949, 1], "end": [951, 38], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Pointwise/Interval.lean", "full_name": "Set.preimage_add_const_uIcc", "start": [431, 1], "end": [432, 60], "traced_tactics": [{"tactic": "simpa only [add_comm] using preimage_const_add_uIcc a b c", "annotated_tactic": ["simpa only [<a>add_comm</a>] using <a>preimage_const_add_uIcc</a> a b c", [{"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [301, 3], "def_end_pos": [301, 14]}, {"full_name": "Set.preimage_const_add_uIcc", "def_path": "Mathlib/Data/Set/Pointwise/Interval.lean", "def_pos": [426, 9], "def_end_pos": [426, 32]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : LinearOrderedAddCommGroup \u03b1\na b c d : \u03b1\n\u22a2 (fun x => x + a) \u207b\u00b9' [[b, c]] = [[b - a, c - a]]", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "full_name": "String.toString_toSubstring", "start": [483, 9], "end": [484, 24], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "full_name": "MeasureTheory.snorm_le_snorm_top_mul_snorm", "start": [1357, 1], "end": [1398, 30], "traced_tactics": [{"tactic": "by_cases hp_top : p = \u221e", "annotated_tactic": ["by_cases hp_top : p = \u221e", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\n\u22a2 snorm (fun x => b (f x) (g x)) p \u03bc \u2264 snorm f \u22a4 \u03bc * snorm g p \u03bc", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhp_top : p = \u22a4\n\u22a2 snorm (fun x => b (f x) (g x)) p \u03bc \u2264 snorm f \u22a4 \u03bc * snorm g p \u03bc\n\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhp_top : \u00acp = \u22a4\n\u22a2 snorm (fun x => b (f x) (g x)) p \u03bc \u2264 snorm f \u22a4 \u03bc * snorm g p \u03bc"}, {"tactic": "by_cases hp_zero : p = 0", "annotated_tactic": ["by_cases hp_zero : p = 0", []], "state_before": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhp_top : \u00acp = \u22a4\n\u22a2 snorm (fun x => b (f x) (g x)) p \u03bc \u2264 snorm f \u22a4 \u03bc * snorm g p \u03bc", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhp_top : \u00acp = \u22a4\nhp_zero : p = 0\n\u22a2 snorm (fun x => b (f x) (g x)) p \u03bc \u2264 snorm f \u22a4 \u03bc * snorm g p \u03bc\n\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhp_top : \u00acp = \u22a4\nhp_zero : \u00acp = 0\n\u22a2 snorm (fun x => b (f x) (g x)) p \u03bc \u2264 snorm f \u22a4 \u03bc * snorm g p \u03bc"}, {"tactic": "simp_rw [snorm_eq_lintegral_rpow_nnnorm hp_zero hp_top, snorm_exponent_top, snormEssSup]", "annotated_tactic": ["simp_rw [<a>snorm_eq_lintegral_rpow_nnnorm</a> hp_zero hp_top, <a>snorm_exponent_top</a>, <a>snormEssSup</a>]", [{"full_name": "MeasureTheory.snorm_eq_lintegral_rpow_nnnorm", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [92, 9], "def_end_pos": [92, 39]}, {"full_name": "MeasureTheory.snorm_exponent_top", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [103, 9], "def_end_pos": [103, 27]}, {"full_name": "MeasureTheory.snormEssSup", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [78, 5], "def_end_pos": [78, 16]}]], "state_before": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhp_top : \u00acp = \u22a4\nhp_zero : \u00acp = 0\n\u22a2 snorm (fun x => b (f x) (g x)) p \u03bc \u2264 snorm f \u22a4 \u03bc * snorm g p \u03bc", "state_after": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhp_top : \u00acp = \u22a4\nhp_zero : \u00acp = 0\n\u22a2 (\u222b\u207b (x : \u03b1), \u2191\u2016b (f x) (g x)\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) ^ (1 / ENNReal.toReal p) \u2264\n    essSup (fun x => \u2191\u2016f x\u2016\u208a) \u03bc * (\u222b\u207b (x : \u03b1), \u2191\u2016g x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) ^ (1 / ENNReal.toReal p)"}, {"tactic": "simp_rw [hp_top, snorm_exponent_top]", "annotated_tactic": ["simp_rw [hp_top, <a>snorm_exponent_top</a>]", [{"full_name": "MeasureTheory.snorm_exponent_top", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [103, 9], "def_end_pos": [103, 27]}]], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhp_top : p = \u22a4\n\u22a2 snorm (fun x => b (f x) (g x)) p \u03bc \u2264 snorm f \u22a4 \u03bc * snorm g p \u03bc", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhp_top : p = \u22a4\n\u22a2 snormEssSup (fun x => b (f x) (g x)) \u03bc \u2264 snormEssSup f \u03bc * snormEssSup g \u03bc"}, {"tactic": "refine' le_trans (essSup_mono_ae <| h.mono fun a ha => _) (ENNReal.essSup_mul_le _ _)", "annotated_tactic": ["refine' <a>le_trans</a> (<a>essSup_mono_ae</a> <| h.mono fun a ha => _) (<a>ENNReal.essSup_mul_le</a> _ _)", [{"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}, {"full_name": "essSup_mono_ae", "def_path": "Mathlib/MeasureTheory/Function/EssSup.lean", "def_pos": [154, 9], "def_end_pos": [154, 23]}, {"full_name": "ENNReal.essSup_mul_le", "def_path": "Mathlib/MeasureTheory/Function/EssSup.lean", "def_pos": [317, 9], "def_end_pos": [317, 22]}]], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhp_top : p = \u22a4\n\u22a2 snormEssSup (fun x => b (f x) (g x)) \u03bc \u2264 snormEssSup f \u03bc * snormEssSup g \u03bc", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhp_top : p = \u22a4\na : \u03b1\nha : \u2016b (f a) (g a)\u2016\u208a \u2264 \u2016f a\u2016\u208a * \u2016g a\u2016\u208a\n\u22a2 (fun x => \u2191\u2016(fun x => b (f x) (g x)) x\u2016\u208a) a \u2264 ((fun x => \u2191\u2016f x\u2016\u208a) * fun x => \u2191\u2016g x\u2016\u208a) a"}, {"tactic": "simp_rw [Pi.mul_apply, \u2190 ENNReal.coe_mul, ENNReal.coe_le_coe]", "annotated_tactic": ["simp_rw [<a>Pi.mul_apply</a>, \u2190 <a>ENNReal.coe_mul</a>, <a>ENNReal.coe_le_coe</a>]", [{"full_name": "Pi.mul_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [83, 9], "def_end_pos": [83, 18]}, {"full_name": "ENNReal.coe_mul", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [390, 9], "def_end_pos": [390, 16]}, {"full_name": "ENNReal.coe_le_coe", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [349, 28], "def_end_pos": [349, 38]}]], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhp_top : p = \u22a4\na : \u03b1\nha : \u2016b (f a) (g a)\u2016\u208a \u2264 \u2016f a\u2016\u208a * \u2016g a\u2016\u208a\n\u22a2 (fun x => \u2191\u2016(fun x => b (f x) (g x)) x\u2016\u208a) a \u2264 ((fun x => \u2191\u2016f x\u2016\u208a) * fun x => \u2191\u2016g x\u2016\u208a) a", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhp_top : p = \u22a4\na : \u03b1\nha : \u2016b (f a) (g a)\u2016\u208a \u2264 \u2016f a\u2016\u208a * \u2016g a\u2016\u208a\n\u22a2 \u2016b (f a) (g a)\u2016\u208a \u2264 \u2016f a\u2016\u208a * \u2016g a\u2016\u208a"}, {"tactic": "exact ha", "annotated_tactic": ["exact ha", []], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhp_top : p = \u22a4\na : \u03b1\nha : \u2016b (f a) (g a)\u2016\u208a \u2264 \u2016f a\u2016\u208a * \u2016g a\u2016\u208a\n\u22a2 \u2016b (f a) (g a)\u2016\u208a \u2264 \u2016f a\u2016\u208a * \u2016g a\u2016\u208a", "state_after": "no goals"}, {"tactic": "simp only [hp_zero, snorm_exponent_zero, mul_zero, le_zero_iff]", "annotated_tactic": ["simp only [hp_zero, <a>snorm_exponent_zero</a>, <a>mul_zero</a>, <a>le_zero_iff</a>]", [{"full_name": "MeasureTheory.snorm_exponent_zero", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [176, 9], "def_end_pos": [176, 28]}, {"full_name": "MulZeroClass.mul_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [38, 3], "def_end_pos": [38, 11]}, {"full_name": "le_zero_iff", "def_path": "Mathlib/Algebra/Order/WithZero.lean", "def_pos": [102, 9], "def_end_pos": [102, 20]}]], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhp_top : \u00acp = \u22a4\nhp_zero : p = 0\n\u22a2 snorm (fun x => b (f x) (g x)) p \u03bc \u2264 snorm f \u22a4 \u03bc * snorm g p \u03bc", "state_after": "no goals"}, {"tactic": "refine' ENNReal.rpow_le_rpow _ (one_div_nonneg.mpr ENNReal.toReal_nonneg)", "annotated_tactic": ["refine' <a>ENNReal.rpow_le_rpow</a> _ (one_div_nonneg.mpr <a>ENNReal.toReal_nonneg</a>)", [{"full_name": "ENNReal.rpow_le_rpow", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [642, 9], "def_end_pos": [642, 21]}, {"full_name": "ENNReal.toReal_nonneg", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [221, 17], "def_end_pos": [221, 30]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhp_top : \u00acp = \u22a4\nhp_zero : \u00acp = 0\n\u22a2 (\u222b\u207b (x : \u03b1), \u2191\u2016b (f x) (g x)\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) ^ (1 / ENNReal.toReal p) \u2264\n    (\u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a ^ ENNReal.toReal p * \u2191\u2016g x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) ^ (1 / ENNReal.toReal p)", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhp_top : \u00acp = \u22a4\nhp_zero : \u00acp = 0\n\u22a2 \u222b\u207b (x : \u03b1), \u2191\u2016b (f x) (g x)\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc \u2264\n    \u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a ^ ENNReal.toReal p * \u2191\u2016g x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc"}, {"tactic": "refine' lintegral_mono_ae (h.mono fun a ha => _)", "annotated_tactic": ["refine' <a>lintegral_mono_ae</a> (h.mono fun a ha => _)", [{"full_name": "MeasureTheory.lintegral_mono_ae", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [265, 9], "def_end_pos": [265, 26]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhp_top : \u00acp = \u22a4\nhp_zero : \u00acp = 0\n\u22a2 \u222b\u207b (x : \u03b1), \u2191\u2016b (f x) (g x)\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc \u2264\n    \u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a ^ ENNReal.toReal p * \u2191\u2016g x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhp_top : \u00acp = \u22a4\nhp_zero : \u00acp = 0\na : \u03b1\nha : \u2016b (f a) (g a)\u2016\u208a \u2264 \u2016f a\u2016\u208a * \u2016g a\u2016\u208a\n\u22a2 \u2191\u2016b (f a) (g a)\u2016\u208a ^ ENNReal.toReal p \u2264 \u2191\u2016f a\u2016\u208a ^ ENNReal.toReal p * \u2191\u2016g a\u2016\u208a ^ ENNReal.toReal p"}, {"tactic": "rw [\u2190 ENNReal.mul_rpow_of_nonneg _ _ ENNReal.toReal_nonneg]", "annotated_tactic": ["rw [\u2190 <a>ENNReal.mul_rpow_of_nonneg</a> _ _ <a>ENNReal.toReal_nonneg</a>]", [{"full_name": "ENNReal.mul_rpow_of_nonneg", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [594, 9], "def_end_pos": [594, 27]}, {"full_name": "ENNReal.toReal_nonneg", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [221, 17], "def_end_pos": [221, 30]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhp_top : \u00acp = \u22a4\nhp_zero : \u00acp = 0\na : \u03b1\nha : \u2016b (f a) (g a)\u2016\u208a \u2264 \u2016f a\u2016\u208a * \u2016g a\u2016\u208a\n\u22a2 \u2191\u2016b (f a) (g a)\u2016\u208a ^ ENNReal.toReal p \u2264 \u2191\u2016f a\u2016\u208a ^ ENNReal.toReal p * \u2191\u2016g a\u2016\u208a ^ ENNReal.toReal p", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhp_top : \u00acp = \u22a4\nhp_zero : \u00acp = 0\na : \u03b1\nha : \u2016b (f a) (g a)\u2016\u208a \u2264 \u2016f a\u2016\u208a * \u2016g a\u2016\u208a\n\u22a2 \u2191\u2016b (f a) (g a)\u2016\u208a ^ ENNReal.toReal p \u2264 (\u2191\u2016f a\u2016\u208a * \u2191\u2016g a\u2016\u208a) ^ ENNReal.toReal p"}, {"tactic": "refine' ENNReal.rpow_le_rpow _ ENNReal.toReal_nonneg", "annotated_tactic": ["refine' <a>ENNReal.rpow_le_rpow</a> _ <a>ENNReal.toReal_nonneg</a>", [{"full_name": "ENNReal.rpow_le_rpow", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [642, 9], "def_end_pos": [642, 21]}, {"full_name": "ENNReal.toReal_nonneg", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [221, 17], "def_end_pos": [221, 30]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhp_top : \u00acp = \u22a4\nhp_zero : \u00acp = 0\na : \u03b1\nha : \u2016b (f a) (g a)\u2016\u208a \u2264 \u2016f a\u2016\u208a * \u2016g a\u2016\u208a\n\u22a2 \u2191\u2016b (f a) (g a)\u2016\u208a ^ ENNReal.toReal p \u2264 (\u2191\u2016f a\u2016\u208a * \u2191\u2016g a\u2016\u208a) ^ ENNReal.toReal p", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhp_top : \u00acp = \u22a4\nhp_zero : \u00acp = 0\na : \u03b1\nha : \u2016b (f a) (g a)\u2016\u208a \u2264 \u2016f a\u2016\u208a * \u2016g a\u2016\u208a\n\u22a2 \u2191\u2016b (f a) (g a)\u2016\u208a \u2264 \u2191\u2016f a\u2016\u208a * \u2191\u2016g a\u2016\u208a"}, {"tactic": "rw [\u2190 ENNReal.coe_mul, ENNReal.coe_le_coe]", "annotated_tactic": ["rw [\u2190 <a>ENNReal.coe_mul</a>, <a>ENNReal.coe_le_coe</a>]", [{"full_name": "ENNReal.coe_mul", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [390, 9], "def_end_pos": [390, 16]}, {"full_name": "ENNReal.coe_le_coe", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [349, 28], "def_end_pos": [349, 38]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhp_top : \u00acp = \u22a4\nhp_zero : \u00acp = 0\na : \u03b1\nha : \u2016b (f a) (g a)\u2016\u208a \u2264 \u2016f a\u2016\u208a * \u2016g a\u2016\u208a\n\u22a2 \u2191\u2016b (f a) (g a)\u2016\u208a \u2264 \u2191\u2016f a\u2016\u208a * \u2191\u2016g a\u2016\u208a", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhp_top : \u00acp = \u22a4\nhp_zero : \u00acp = 0\na : \u03b1\nha : \u2016b (f a) (g a)\u2016\u208a \u2264 \u2016f a\u2016\u208a * \u2016g a\u2016\u208a\n\u22a2 \u2016b (f a) (g a)\u2016\u208a \u2264 \u2016f a\u2016\u208a * \u2016g a\u2016\u208a"}, {"tactic": "exact ha", "annotated_tactic": ["exact ha", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhp_top : \u00acp = \u22a4\nhp_zero : \u00acp = 0\na : \u03b1\nha : \u2016b (f a) (g a)\u2016\u208a \u2264 \u2016f a\u2016\u208a * \u2016g a\u2016\u208a\n\u22a2 \u2016b (f a) (g a)\u2016\u208a \u2264 \u2016f a\u2016\u208a * \u2016g a\u2016\u208a", "state_after": "no goals"}, {"tactic": "refine' ENNReal.rpow_le_rpow _ _", "annotated_tactic": ["refine' <a>ENNReal.rpow_le_rpow</a> _ _", [{"full_name": "ENNReal.rpow_le_rpow", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [642, 9], "def_end_pos": [642, 21]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhp_top : \u00acp = \u22a4\nhp_zero : \u00acp = 0\n\u22a2 (\u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a ^ ENNReal.toReal p * \u2191\u2016g x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) ^ (1 / ENNReal.toReal p) \u2264\n    (\u222b\u207b (x : \u03b1), essSup (fun x => \u2191\u2016f x\u2016\u208a) \u03bc ^ ENNReal.toReal p * \u2191\u2016g x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) ^\n      (1 / ENNReal.toReal p)", "state_after": "case refine'_1\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhp_top : \u00acp = \u22a4\nhp_zero : \u00acp = 0\n\u22a2 \u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a ^ ENNReal.toReal p * \u2191\u2016g x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc \u2264\n    \u222b\u207b (x : \u03b1), essSup (fun x => \u2191\u2016f x\u2016\u208a) \u03bc ^ ENNReal.toReal p * \u2191\u2016g x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc\n\ncase refine'_2\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhp_top : \u00acp = \u22a4\nhp_zero : \u00acp = 0\n\u22a2 0 \u2264 1 / ENNReal.toReal p"}, {"tactic": "swap", "annotated_tactic": ["swap", []], "state_before": "case refine'_1\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhp_top : \u00acp = \u22a4\nhp_zero : \u00acp = 0\n\u22a2 \u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a ^ ENNReal.toReal p * \u2191\u2016g x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc \u2264\n    \u222b\u207b (x : \u03b1), essSup (fun x => \u2191\u2016f x\u2016\u208a) \u03bc ^ ENNReal.toReal p * \u2191\u2016g x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc\n\ncase refine'_2\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhp_top : \u00acp = \u22a4\nhp_zero : \u00acp = 0\n\u22a2 0 \u2264 1 / ENNReal.toReal p", "state_after": "case refine'_2\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhp_top : \u00acp = \u22a4\nhp_zero : \u00acp = 0\n\u22a2 0 \u2264 1 / ENNReal.toReal p\n\ncase refine'_1\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhp_top : \u00acp = \u22a4\nhp_zero : \u00acp = 0\n\u22a2 \u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a ^ ENNReal.toReal p * \u2191\u2016g x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc \u2264\n    \u222b\u207b (x : \u03b1), essSup (fun x => \u2191\u2016f x\u2016\u208a) \u03bc ^ ENNReal.toReal p * \u2191\u2016g x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc"}, {"tactic": "refine' lintegral_mono_ae _", "annotated_tactic": ["refine' <a>lintegral_mono_ae</a> _", [{"full_name": "MeasureTheory.lintegral_mono_ae", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [265, 9], "def_end_pos": [265, 26]}]], "state_before": "case refine'_1\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhp_top : \u00acp = \u22a4\nhp_zero : \u00acp = 0\n\u22a2 \u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a ^ ENNReal.toReal p * \u2191\u2016g x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc \u2264\n    \u222b\u207b (x : \u03b1), essSup (fun x => \u2191\u2016f x\u2016\u208a) \u03bc ^ ENNReal.toReal p * \u2191\u2016g x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc", "state_after": "case refine'_1\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhp_top : \u00acp = \u22a4\nhp_zero : \u00acp = 0\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc,\n    \u2191\u2016f a\u2016\u208a ^ ENNReal.toReal p * \u2191\u2016g a\u2016\u208a ^ ENNReal.toReal p \u2264\n      essSup (fun x => \u2191\u2016f x\u2016\u208a) \u03bc ^ ENNReal.toReal p * \u2191\u2016g a\u2016\u208a ^ ENNReal.toReal p"}, {"tactic": "filter_upwards [@ENNReal.ae_le_essSup _ _ \u03bc fun x => (\u2016f x\u2016\u208a : \u211d\u22650\u221e)] with x hx", "annotated_tactic": ["filter_upwards [@<a>ENNReal.ae_le_essSup</a> _ _ \u03bc fun x => (\u2016f x\u2016\u208a : \u211d\u22650\u221e)] with x hx", [{"full_name": "ENNReal.ae_le_essSup", "def_path": "Mathlib/MeasureTheory/Function/EssSup.lean", "def_pos": [304, 9], "def_end_pos": [304, 21]}]], "state_before": "case refine'_1\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhp_top : \u00acp = \u22a4\nhp_zero : \u00acp = 0\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc,\n    \u2191\u2016f a\u2016\u208a ^ ENNReal.toReal p * \u2191\u2016g a\u2016\u208a ^ ENNReal.toReal p \u2264\n      essSup (fun x => \u2191\u2016f x\u2016\u208a) \u03bc ^ ENNReal.toReal p * \u2191\u2016g a\u2016\u208a ^ ENNReal.toReal p", "state_after": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhp_top : \u00acp = \u22a4\nhp_zero : \u00acp = 0\nx : \u03b1\nhx : \u2191\u2016f x\u2016\u208a \u2264 essSup (fun x => \u2191\u2016f x\u2016\u208a) \u03bc\n\u22a2 \u2191\u2016f x\u2016\u208a ^ ENNReal.toReal p * \u2191\u2016g x\u2016\u208a ^ ENNReal.toReal p \u2264\n    essSup (fun x => \u2191\u2016f x\u2016\u208a) \u03bc ^ ENNReal.toReal p * \u2191\u2016g x\u2016\u208a ^ ENNReal.toReal p"}, {"tactic": "exact mul_le_mul_right' (ENNReal.rpow_le_rpow hx ENNReal.toReal_nonneg) _", "annotated_tactic": ["exact <a>mul_le_mul_right'</a> (<a>ENNReal.rpow_le_rpow</a> hx <a>ENNReal.toReal_nonneg</a>) _", [{"full_name": "mul_le_mul_right'", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [67, 9], "def_end_pos": [67, 26]}, {"full_name": "ENNReal.rpow_le_rpow", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [642, 9], "def_end_pos": [642, 21]}, {"full_name": "ENNReal.toReal_nonneg", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [221, 17], "def_end_pos": [221, 30]}]], "state_before": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhp_top : \u00acp = \u22a4\nhp_zero : \u00acp = 0\nx : \u03b1\nhx : \u2191\u2016f x\u2016\u208a \u2264 essSup (fun x => \u2191\u2016f x\u2016\u208a) \u03bc\n\u22a2 \u2191\u2016f x\u2016\u208a ^ ENNReal.toReal p * \u2191\u2016g x\u2016\u208a ^ ENNReal.toReal p \u2264\n    essSup (fun x => \u2191\u2016f x\u2016\u208a) \u03bc ^ ENNReal.toReal p * \u2191\u2016g x\u2016\u208a ^ ENNReal.toReal p", "state_after": "no goals"}, {"tactic": "rw [one_div_nonneg]", "annotated_tactic": ["rw [<a>one_div_nonneg</a>]", [{"full_name": "one_div_nonneg", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [81, 9], "def_end_pos": [81, 23]}]], "state_before": "case refine'_2\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhp_top : \u00acp = \u22a4\nhp_zero : \u00acp = 0\n\u22a2 0 \u2264 1 / ENNReal.toReal p", "state_after": "case refine'_2\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhp_top : \u00acp = \u22a4\nhp_zero : \u00acp = 0\n\u22a2 0 \u2264 ENNReal.toReal p"}, {"tactic": "exact ENNReal.toReal_nonneg", "annotated_tactic": ["exact <a>ENNReal.toReal_nonneg</a>", [{"full_name": "ENNReal.toReal_nonneg", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [221, 17], "def_end_pos": [221, 30]}]], "state_before": "case refine'_2\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhp_top : \u00acp = \u22a4\nhp_zero : \u00acp = 0\n\u22a2 0 \u2264 ENNReal.toReal p", "state_after": "no goals"}, {"tactic": "rw [lintegral_const_mul'']", "annotated_tactic": ["rw [<a>lintegral_const_mul''</a>]", [{"full_name": "MeasureTheory.lintegral_const_mul''", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [691, 9], "def_end_pos": [691, 30]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhp_top : \u00acp = \u22a4\nhp_zero : \u00acp = 0\n\u22a2 (\u222b\u207b (x : \u03b1), essSup (fun x => \u2191\u2016f x\u2016\u208a) \u03bc ^ ENNReal.toReal p * \u2191\u2016g x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) ^\n      (1 / ENNReal.toReal p) =\n    essSup (fun x => \u2191\u2016f x\u2016\u208a) \u03bc * (\u222b\u207b (x : \u03b1), \u2191\u2016g x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) ^ (1 / ENNReal.toReal p)", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhp_top : \u00acp = \u22a4\nhp_zero : \u00acp = 0\n\u22a2 (essSup (fun x => \u2191\u2016f x\u2016\u208a) \u03bc ^ ENNReal.toReal p * \u222b\u207b (a : \u03b1), \u2191\u2016g a\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) ^\n      (1 / ENNReal.toReal p) =\n    essSup (fun x => \u2191\u2016f x\u2016\u208a) \u03bc * (\u222b\u207b (x : \u03b1), \u2191\u2016g x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) ^ (1 / ENNReal.toReal p)\n\ncase hf\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhp_top : \u00acp = \u22a4\nhp_zero : \u00acp = 0\n\u22a2 AEMeasurable fun x => \u2191\u2016g x\u2016\u208a ^ ENNReal.toReal p"}, {"tactic": "swap", "annotated_tactic": ["swap", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhp_top : \u00acp = \u22a4\nhp_zero : \u00acp = 0\n\u22a2 (essSup (fun x => \u2191\u2016f x\u2016\u208a) \u03bc ^ ENNReal.toReal p * \u222b\u207b (a : \u03b1), \u2191\u2016g a\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) ^\n      (1 / ENNReal.toReal p) =\n    essSup (fun x => \u2191\u2016f x\u2016\u208a) \u03bc * (\u222b\u207b (x : \u03b1), \u2191\u2016g x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) ^ (1 / ENNReal.toReal p)\n\ncase hf\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhp_top : \u00acp = \u22a4\nhp_zero : \u00acp = 0\n\u22a2 AEMeasurable fun x => \u2191\u2016g x\u2016\u208a ^ ENNReal.toReal p", "state_after": "case hf\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhp_top : \u00acp = \u22a4\nhp_zero : \u00acp = 0\n\u22a2 AEMeasurable fun x => \u2191\u2016g x\u2016\u208a ^ ENNReal.toReal p\n\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhp_top : \u00acp = \u22a4\nhp_zero : \u00acp = 0\n\u22a2 (essSup (fun x => \u2191\u2016f x\u2016\u208a) \u03bc ^ ENNReal.toReal p * \u222b\u207b (a : \u03b1), \u2191\u2016g a\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) ^\n      (1 / ENNReal.toReal p) =\n    essSup (fun x => \u2191\u2016f x\u2016\u208a) \u03bc * (\u222b\u207b (x : \u03b1), \u2191\u2016g x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) ^ (1 / ENNReal.toReal p)"}, {"tactic": "rw [ENNReal.mul_rpow_of_nonneg]", "annotated_tactic": ["rw [<a>ENNReal.mul_rpow_of_nonneg</a>]", [{"full_name": "ENNReal.mul_rpow_of_nonneg", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [594, 9], "def_end_pos": [594, 27]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhp_top : \u00acp = \u22a4\nhp_zero : \u00acp = 0\n\u22a2 (essSup (fun x => \u2191\u2016f x\u2016\u208a) \u03bc ^ ENNReal.toReal p * \u222b\u207b (a : \u03b1), \u2191\u2016g a\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) ^\n      (1 / ENNReal.toReal p) =\n    essSup (fun x => \u2191\u2016f x\u2016\u208a) \u03bc * (\u222b\u207b (x : \u03b1), \u2191\u2016g x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) ^ (1 / ENNReal.toReal p)", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhp_top : \u00acp = \u22a4\nhp_zero : \u00acp = 0\n\u22a2 (essSup (fun x => \u2191\u2016f x\u2016\u208a) \u03bc ^ ENNReal.toReal p) ^ (1 / ENNReal.toReal p) *\n      (\u222b\u207b (a : \u03b1), \u2191\u2016g a\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) ^ (1 / ENNReal.toReal p) =\n    essSup (fun x => \u2191\u2016f x\u2016\u208a) \u03bc * (\u222b\u207b (x : \u03b1), \u2191\u2016g x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) ^ (1 / ENNReal.toReal p)\n\ncase hz\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhp_top : \u00acp = \u22a4\nhp_zero : \u00acp = 0\n\u22a2 0 \u2264 1 / ENNReal.toReal p"}, {"tactic": "swap", "annotated_tactic": ["swap", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhp_top : \u00acp = \u22a4\nhp_zero : \u00acp = 0\n\u22a2 (essSup (fun x => \u2191\u2016f x\u2016\u208a) \u03bc ^ ENNReal.toReal p) ^ (1 / ENNReal.toReal p) *\n      (\u222b\u207b (a : \u03b1), \u2191\u2016g a\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) ^ (1 / ENNReal.toReal p) =\n    essSup (fun x => \u2191\u2016f x\u2016\u208a) \u03bc * (\u222b\u207b (x : \u03b1), \u2191\u2016g x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) ^ (1 / ENNReal.toReal p)\n\ncase hz\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhp_top : \u00acp = \u22a4\nhp_zero : \u00acp = 0\n\u22a2 0 \u2264 1 / ENNReal.toReal p", "state_after": "case hz\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhp_top : \u00acp = \u22a4\nhp_zero : \u00acp = 0\n\u22a2 0 \u2264 1 / ENNReal.toReal p\n\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhp_top : \u00acp = \u22a4\nhp_zero : \u00acp = 0\n\u22a2 (essSup (fun x => \u2191\u2016f x\u2016\u208a) \u03bc ^ ENNReal.toReal p) ^ (1 / ENNReal.toReal p) *\n      (\u222b\u207b (a : \u03b1), \u2191\u2016g a\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) ^ (1 / ENNReal.toReal p) =\n    essSup (fun x => \u2191\u2016f x\u2016\u208a) \u03bc * (\u222b\u207b (x : \u03b1), \u2191\u2016g x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) ^ (1 / ENNReal.toReal p)"}, {"tactic": "rw [\u2190 ENNReal.rpow_mul, one_div, mul_inv_cancel, ENNReal.rpow_one]", "annotated_tactic": ["rw [\u2190 <a>ENNReal.rpow_mul</a>, <a>one_div</a>, <a>mul_inv_cancel</a>, <a>ENNReal.rpow_one</a>]", [{"full_name": "ENNReal.rpow_mul", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [532, 9], "def_end_pos": [532, 17]}, {"full_name": "one_div", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [318, 9], "def_end_pos": [318, 16]}, {"full_name": "mul_inv_cancel", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [194, 15], "def_end_pos": [194, 29]}, {"full_name": "ENNReal.rpow_one", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [450, 9], "def_end_pos": [450, 17]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhp_top : \u00acp = \u22a4\nhp_zero : \u00acp = 0\n\u22a2 (essSup (fun x => \u2191\u2016f x\u2016\u208a) \u03bc ^ ENNReal.toReal p) ^ (1 / ENNReal.toReal p) *\n      (\u222b\u207b (a : \u03b1), \u2191\u2016g a\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) ^ (1 / ENNReal.toReal p) =\n    essSup (fun x => \u2191\u2016f x\u2016\u208a) \u03bc * (\u222b\u207b (x : \u03b1), \u2191\u2016g x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) ^ (1 / ENNReal.toReal p)", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhp_top : \u00acp = \u22a4\nhp_zero : \u00acp = 0\n\u22a2 ENNReal.toReal p \u2260 0"}, {"tactic": "rw [Ne.def, ENNReal.toReal_eq_zero_iff, not_or]", "annotated_tactic": ["rw [<a>Ne.def</a>, <a>ENNReal.toReal_eq_zero_iff</a>, <a>not_or</a>]", [{"full_name": "Ne.def", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [59, 9], "def_end_pos": [59, 15]}, {"full_name": "ENNReal.toReal_eq_zero_iff", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [274, 9], "def_end_pos": [274, 27]}, {"full_name": "not_or", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [340, 9], "def_end_pos": [340, 15]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhp_top : \u00acp = \u22a4\nhp_zero : \u00acp = 0\n\u22a2 ENNReal.toReal p \u2260 0", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhp_top : \u00acp = \u22a4\nhp_zero : \u00acp = 0\n\u22a2 \u00acp = 0 \u2227 \u00acp = \u22a4"}, {"tactic": "exact \u27e8hp_zero, hp_top\u27e9", "annotated_tactic": ["exact \u27e8hp_zero, hp_top\u27e9", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhp_top : \u00acp = \u22a4\nhp_zero : \u00acp = 0\n\u22a2 \u00acp = 0 \u2227 \u00acp = \u22a4", "state_after": "no goals"}, {"tactic": "exact hg.nnnorm.aemeasurable.coe_nnreal_ennreal.pow aemeasurable_const", "annotated_tactic": ["exact hg.nnnorm.aemeasurable.coe_nnreal_ennreal.pow <a>aemeasurable_const</a>", [{"full_name": "aemeasurable_const", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [746, 9], "def_end_pos": [746, 27]}]], "state_before": "case hf\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhp_top : \u00acp = \u22a4\nhp_zero : \u00acp = 0\n\u22a2 AEMeasurable fun x => \u2191\u2016g x\u2016\u208a ^ ENNReal.toReal p", "state_after": "no goals"}, {"tactic": "rw [one_div_nonneg]", "annotated_tactic": ["rw [<a>one_div_nonneg</a>]", [{"full_name": "one_div_nonneg", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [81, 9], "def_end_pos": [81, 23]}]], "state_before": "case hz\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhp_top : \u00acp = \u22a4\nhp_zero : \u00acp = 0\n\u22a2 0 \u2264 1 / ENNReal.toReal p", "state_after": "case hz\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhp_top : \u00acp = \u22a4\nhp_zero : \u00acp = 0\n\u22a2 0 \u2264 ENNReal.toReal p"}, {"tactic": "exact ENNReal.toReal_nonneg", "annotated_tactic": ["exact <a>ENNReal.toReal_nonneg</a>", [{"full_name": "ENNReal.toReal_nonneg", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [221, 17], "def_end_pos": [221, 30]}]], "state_before": "case hz\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhp_top : \u00acp = \u22a4\nhp_zero : \u00acp = 0\n\u22a2 0 \u2264 ENNReal.toReal p", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/TuringMachine.lean", "full_name": "Turing.TM0.Machine.map_step", "start": [1155, 1], "end": [1166, 10], "traced_tactics": [{"tactic": "unfold step Machine.map Cfg.map", "annotated_tactic": ["unfold <a>step</a> <a>Machine.map</a> <a>Cfg.map</a>", [{"full_name": "Turing.TM0.step", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1076, 5], "def_end_pos": [1076, 9]}, {"full_name": "Turing.TM0.Machine.map", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1151, 5], "def_end_pos": [1151, 16]}, {"full_name": "Turing.TM0.Cfg.map", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1142, 5], "def_end_pos": [1142, 12]}]], "state_before": "\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u0393' : Type u_2\ninst\u271d\u00b2 : Inhabited \u0393'\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u039b' : Type u_4\ninst\u271d : Inhabited \u039b'\nM : Machine \u0393 \u039b\nf\u2081 : PointedMap \u0393 \u0393'\nf\u2082 : PointedMap \u0393' \u0393\ng\u2081 : \u039b \u2192 \u039b'\ng\u2082 : \u039b' \u2192 \u039b\nS : Set \u039b\nf\u2082\u2081 : Function.RightInverse f\u2081.f f\u2082.f\ng\u2082\u2081 : \u2200 (q : \u039b), q \u2208 S \u2192 g\u2082 (g\u2081 q) = q\nq : \u039b\nT : Tape \u0393\nh : { q := q, Tape := T }.q \u2208 S\n\u22a2 Option.map (Cfg.map f\u2081 g\u2081) (step M { q := q, Tape := T }) =\n    step (map M f\u2081 f\u2082 g\u2081 g\u2082) (Cfg.map f\u2081 g\u2081 { q := q, Tape := T })", "state_after": "\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u0393' : Type u_2\ninst\u271d\u00b2 : Inhabited \u0393'\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u039b' : Type u_4\ninst\u271d : Inhabited \u039b'\nM : Machine \u0393 \u039b\nf\u2081 : PointedMap \u0393 \u0393'\nf\u2082 : PointedMap \u0393' \u0393\ng\u2081 : \u039b \u2192 \u039b'\ng\u2082 : \u039b' \u2192 \u039b\nS : Set \u039b\nf\u2082\u2081 : Function.RightInverse f\u2081.f f\u2082.f\ng\u2082\u2081 : \u2200 (q : \u039b), q \u2208 S \u2192 g\u2082 (g\u2081 q) = q\nq : \u039b\nT : Tape \u0393\nh : { q := q, Tape := T }.q \u2208 S\n\u22a2 Option.map\n      (fun x =>\n        match x with\n        | { q := q, Tape := T } => { q := g\u2081 q, Tape := Tape.map f\u2081 T })\n      (match { q := q, Tape := T } with\n      | { q := q, Tape := T } =>\n        Option.map\n          (fun x =>\n            match x with\n            | (q', a) =>\n              { q := q',\n                Tape :=\n                  match a with\n                  | Stmt.move d => Tape.move d T\n                  | Stmt.write a => Tape.write a T })\n          (M q T.head)) =\n    match\n      match { q := q, Tape := T } with\n      | { q := q, Tape := T } => { q := g\u2081 q, Tape := Tape.map f\u2081 T } with\n    | { q := q, Tape := T } =>\n      Option.map\n        (fun x =>\n          match x with\n          | (q', a) =>\n            { q := q',\n              Tape :=\n                match a with\n                | Stmt.move d => Tape.move d T\n                | Stmt.write a => Tape.write a T })\n        (match q, T.head with\n        | q, l => Option.map (Prod.map g\u2081 (Stmt.map f\u2081)) (M (g\u2082 q) (PointedMap.f f\u2082 l)))"}, {"tactic": "simp only [Turing.Tape.map_fst, g\u2082\u2081 q h, f\u2082\u2081 _]", "annotated_tactic": ["simp only [<a>Turing.Tape.map_fst</a>, g\u2082\u2081 q h, f\u2082\u2081 _]", [{"full_name": "Turing.Tape.map_fst", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [687, 9], "def_end_pos": [687, 21]}]], "state_before": "\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u0393' : Type u_2\ninst\u271d\u00b2 : Inhabited \u0393'\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u039b' : Type u_4\ninst\u271d : Inhabited \u039b'\nM : Machine \u0393 \u039b\nf\u2081 : PointedMap \u0393 \u0393'\nf\u2082 : PointedMap \u0393' \u0393\ng\u2081 : \u039b \u2192 \u039b'\ng\u2082 : \u039b' \u2192 \u039b\nS : Set \u039b\nf\u2082\u2081 : Function.RightInverse f\u2081.f f\u2082.f\ng\u2082\u2081 : \u2200 (q : \u039b), q \u2208 S \u2192 g\u2082 (g\u2081 q) = q\nq : \u039b\nT : Tape \u0393\nh : { q := q, Tape := T }.q \u2208 S\n\u22a2 Option.map\n      (fun x =>\n        match x with\n        | { q := q, Tape := T } => { q := g\u2081 q, Tape := Tape.map f\u2081 T })\n      (match { q := q, Tape := T } with\n      | { q := q, Tape := T } =>\n        Option.map\n          (fun x =>\n            match x with\n            | (q', a) =>\n              { q := q',\n                Tape :=\n                  match a with\n                  | Stmt.move d => Tape.move d T\n                  | Stmt.write a => Tape.write a T })\n          (M q T.head)) =\n    match\n      match { q := q, Tape := T } with\n      | { q := q, Tape := T } => { q := g\u2081 q, Tape := Tape.map f\u2081 T } with\n    | { q := q, Tape := T } =>\n      Option.map\n        (fun x =>\n          match x with\n          | (q', a) =>\n            { q := q',\n              Tape :=\n                match a with\n                | Stmt.move d => Tape.move d T\n                | Stmt.write a => Tape.write a T })\n        (match q, T.head with\n        | q, l => Option.map (Prod.map g\u2081 (Stmt.map f\u2081)) (M (g\u2082 q) (PointedMap.f f\u2082 l)))", "state_after": "\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u0393' : Type u_2\ninst\u271d\u00b2 : Inhabited \u0393'\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u039b' : Type u_4\ninst\u271d : Inhabited \u039b'\nM : Machine \u0393 \u039b\nf\u2081 : PointedMap \u0393 \u0393'\nf\u2082 : PointedMap \u0393' \u0393\ng\u2081 : \u039b \u2192 \u039b'\ng\u2082 : \u039b' \u2192 \u039b\nS : Set \u039b\nf\u2082\u2081 : Function.RightInverse f\u2081.f f\u2082.f\ng\u2082\u2081 : \u2200 (q : \u039b), q \u2208 S \u2192 g\u2082 (g\u2081 q) = q\nq : \u039b\nT : Tape \u0393\nh : { q := q, Tape := T }.q \u2208 S\n\u22a2 Option.map (fun x => { q := g\u2081 x.q, Tape := Tape.map f\u2081 x.Tape })\n      (Option.map\n        (fun x =>\n          { q := x.1,\n            Tape :=\n              match x.2 with\n              | Stmt.move d => Tape.move d T\n              | Stmt.write a => Tape.write a T })\n        (M q T.head)) =\n    Option.map\n      (fun x =>\n        { q := x.1,\n          Tape :=\n            match x.2 with\n            | Stmt.move d => Tape.move d (Tape.map f\u2081 T)\n            | Stmt.write a => Tape.write a (Tape.map f\u2081 T) })\n      (Option.map (Prod.map g\u2081 (Stmt.map f\u2081)) (M q T.head))"}, {"tactic": "rcases M q T.1 with (_ | \u27e8q', d | a\u27e9)", "annotated_tactic": ["rcases M q T.1 with (_ | \u27e8q', d | a\u27e9)", []], "state_before": "\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u0393' : Type u_2\ninst\u271d\u00b2 : Inhabited \u0393'\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u039b' : Type u_4\ninst\u271d : Inhabited \u039b'\nM : Machine \u0393 \u039b\nf\u2081 : PointedMap \u0393 \u0393'\nf\u2082 : PointedMap \u0393' \u0393\ng\u2081 : \u039b \u2192 \u039b'\ng\u2082 : \u039b' \u2192 \u039b\nS : Set \u039b\nf\u2082\u2081 : Function.RightInverse f\u2081.f f\u2082.f\ng\u2082\u2081 : \u2200 (q : \u039b), q \u2208 S \u2192 g\u2082 (g\u2081 q) = q\nq : \u039b\nT : Tape \u0393\nh : { q := q, Tape := T }.q \u2208 S\n\u22a2 Option.map (fun x => { q := g\u2081 x.q, Tape := Tape.map f\u2081 x.Tape })\n      (Option.map\n        (fun x =>\n          { q := x.1,\n            Tape :=\n              match x.2 with\n              | Stmt.move d => Tape.move d T\n              | Stmt.write a => Tape.write a T })\n        (M q T.head)) =\n    Option.map\n      (fun x =>\n        { q := x.1,\n          Tape :=\n            match x.2 with\n            | Stmt.move d => Tape.move d (Tape.map f\u2081 T)\n            | Stmt.write a => Tape.write a (Tape.map f\u2081 T) })\n      (Option.map (Prod.map g\u2081 (Stmt.map f\u2081)) (M q T.head))", "state_after": "case none\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u0393' : Type u_2\ninst\u271d\u00b2 : Inhabited \u0393'\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u039b' : Type u_4\ninst\u271d : Inhabited \u039b'\nM : Machine \u0393 \u039b\nf\u2081 : PointedMap \u0393 \u0393'\nf\u2082 : PointedMap \u0393' \u0393\ng\u2081 : \u039b \u2192 \u039b'\ng\u2082 : \u039b' \u2192 \u039b\nS : Set \u039b\nf\u2082\u2081 : Function.RightInverse f\u2081.f f\u2082.f\ng\u2082\u2081 : \u2200 (q : \u039b), q \u2208 S \u2192 g\u2082 (g\u2081 q) = q\nq : \u039b\nT : Tape \u0393\nh : { q := q, Tape := T }.q \u2208 S\n\u22a2 Option.map (fun x => { q := g\u2081 x.q, Tape := Tape.map f\u2081 x.Tape })\n      (Option.map\n        (fun x =>\n          { q := x.1,\n            Tape :=\n              match x.2 with\n              | Stmt.move d => Tape.move d T\n              | Stmt.write a => Tape.write a T })\n        none) =\n    Option.map\n      (fun x =>\n        { q := x.1,\n          Tape :=\n            match x.2 with\n            | Stmt.move d => Tape.move d (Tape.map f\u2081 T)\n            | Stmt.write a => Tape.write a (Tape.map f\u2081 T) })\n      (Option.map (Prod.map g\u2081 (Stmt.map f\u2081)) none)\n\ncase some.mk.move\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u0393' : Type u_2\ninst\u271d\u00b2 : Inhabited \u0393'\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u039b' : Type u_4\ninst\u271d : Inhabited \u039b'\nM : Machine \u0393 \u039b\nf\u2081 : PointedMap \u0393 \u0393'\nf\u2082 : PointedMap \u0393' \u0393\ng\u2081 : \u039b \u2192 \u039b'\ng\u2082 : \u039b' \u2192 \u039b\nS : Set \u039b\nf\u2082\u2081 : Function.RightInverse f\u2081.f f\u2082.f\ng\u2082\u2081 : \u2200 (q : \u039b), q \u2208 S \u2192 g\u2082 (g\u2081 q) = q\nq : \u039b\nT : Tape \u0393\nh : { q := q, Tape := T }.q \u2208 S\nq' : \u039b\nd : Dir\n\u22a2 Option.map (fun x => { q := g\u2081 x.q, Tape := Tape.map f\u2081 x.Tape })\n      (Option.map\n        (fun x =>\n          { q := x.1,\n            Tape :=\n              match x.2 with\n              | Stmt.move d => Tape.move d T\n              | Stmt.write a => Tape.write a T })\n        (some (q', Stmt.move d))) =\n    Option.map\n      (fun x =>\n        { q := x.1,\n          Tape :=\n            match x.2 with\n            | Stmt.move d => Tape.move d (Tape.map f\u2081 T)\n            | Stmt.write a => Tape.write a (Tape.map f\u2081 T) })\n      (Option.map (Prod.map g\u2081 (Stmt.map f\u2081)) (some (q', Stmt.move d)))\n\ncase some.mk.write\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u0393' : Type u_2\ninst\u271d\u00b2 : Inhabited \u0393'\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u039b' : Type u_4\ninst\u271d : Inhabited \u039b'\nM : Machine \u0393 \u039b\nf\u2081 : PointedMap \u0393 \u0393'\nf\u2082 : PointedMap \u0393' \u0393\ng\u2081 : \u039b \u2192 \u039b'\ng\u2082 : \u039b' \u2192 \u039b\nS : Set \u039b\nf\u2082\u2081 : Function.RightInverse f\u2081.f f\u2082.f\ng\u2082\u2081 : \u2200 (q : \u039b), q \u2208 S \u2192 g\u2082 (g\u2081 q) = q\nq : \u039b\nT : Tape \u0393\nh : { q := q, Tape := T }.q \u2208 S\nq' : \u039b\na : \u0393\n\u22a2 Option.map (fun x => { q := g\u2081 x.q, Tape := Tape.map f\u2081 x.Tape })\n      (Option.map\n        (fun x =>\n          { q := x.1,\n            Tape :=\n              match x.2 with\n              | Stmt.move d => Tape.move d T\n              | Stmt.write a => Tape.write a T })\n        (some (q', Stmt.write a))) =\n    Option.map\n      (fun x =>\n        { q := x.1,\n          Tape :=\n            match x.2 with\n            | Stmt.move d => Tape.move d (Tape.map f\u2081 T)\n            | Stmt.write a => Tape.write a (Tape.map f\u2081 T) })\n      (Option.map (Prod.map g\u2081 (Stmt.map f\u2081)) (some (q', Stmt.write a)))"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case none\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u0393' : Type u_2\ninst\u271d\u00b2 : Inhabited \u0393'\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u039b' : Type u_4\ninst\u271d : Inhabited \u039b'\nM : Machine \u0393 \u039b\nf\u2081 : PointedMap \u0393 \u0393'\nf\u2082 : PointedMap \u0393' \u0393\ng\u2081 : \u039b \u2192 \u039b'\ng\u2082 : \u039b' \u2192 \u039b\nS : Set \u039b\nf\u2082\u2081 : Function.RightInverse f\u2081.f f\u2082.f\ng\u2082\u2081 : \u2200 (q : \u039b), q \u2208 S \u2192 g\u2082 (g\u2081 q) = q\nq : \u039b\nT : Tape \u0393\nh : { q := q, Tape := T }.q \u2208 S\n\u22a2 Option.map (fun x => { q := g\u2081 x.q, Tape := Tape.map f\u2081 x.Tape })\n      (Option.map\n        (fun x =>\n          { q := x.1,\n            Tape :=\n              match x.2 with\n              | Stmt.move d => Tape.move d T\n              | Stmt.write a => Tape.write a T })\n        none) =\n    Option.map\n      (fun x =>\n        { q := x.1,\n          Tape :=\n            match x.2 with\n            | Stmt.move d => Tape.move d (Tape.map f\u2081 T)\n            | Stmt.write a => Tape.write a (Tape.map f\u2081 T) })\n      (Option.map (Prod.map g\u2081 (Stmt.map f\u2081)) none)", "state_after": "no goals"}, {"tactic": "simp only [step, Cfg.map, Option.map_some', Tape.map_move f\u2081]", "annotated_tactic": ["simp only [<a>step</a>, <a>Cfg.map</a>, <a>Option.map_some'</a>, <a>Tape.map_move</a> f\u2081]", [{"full_name": "Turing.TM0.step", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1076, 5], "def_end_pos": [1076, 9]}, {"full_name": "Turing.TM0.Cfg.map", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1142, 5], "def_end_pos": [1142, 12]}, {"full_name": "Option.map_some'", "def_path": "lake-packages/std/Std/Data/Option/Init/Lemmas.lean", "def_pos": [20, 17], "def_end_pos": [20, 26]}, {"full_name": "Turing.Tape.map_move", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [711, 9], "def_end_pos": [711, 22]}]], "state_before": "case some.mk.move\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u0393' : Type u_2\ninst\u271d\u00b2 : Inhabited \u0393'\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u039b' : Type u_4\ninst\u271d : Inhabited \u039b'\nM : Machine \u0393 \u039b\nf\u2081 : PointedMap \u0393 \u0393'\nf\u2082 : PointedMap \u0393' \u0393\ng\u2081 : \u039b \u2192 \u039b'\ng\u2082 : \u039b' \u2192 \u039b\nS : Set \u039b\nf\u2082\u2081 : Function.RightInverse f\u2081.f f\u2082.f\ng\u2082\u2081 : \u2200 (q : \u039b), q \u2208 S \u2192 g\u2082 (g\u2081 q) = q\nq : \u039b\nT : Tape \u0393\nh : { q := q, Tape := T }.q \u2208 S\nq' : \u039b\nd : Dir\n\u22a2 Option.map (fun x => { q := g\u2081 x.q, Tape := Tape.map f\u2081 x.Tape })\n      (Option.map\n        (fun x =>\n          { q := x.1,\n            Tape :=\n              match x.2 with\n              | Stmt.move d => Tape.move d T\n              | Stmt.write a => Tape.write a T })\n        (some (q', Stmt.move d))) =\n    Option.map\n      (fun x =>\n        { q := x.1,\n          Tape :=\n            match x.2 with\n            | Stmt.move d => Tape.move d (Tape.map f\u2081 T)\n            | Stmt.write a => Tape.write a (Tape.map f\u2081 T) })\n      (Option.map (Prod.map g\u2081 (Stmt.map f\u2081)) (some (q', Stmt.move d)))", "state_after": "case some.mk.move\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u0393' : Type u_2\ninst\u271d\u00b2 : Inhabited \u0393'\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u039b' : Type u_4\ninst\u271d : Inhabited \u039b'\nM : Machine \u0393 \u039b\nf\u2081 : PointedMap \u0393 \u0393'\nf\u2082 : PointedMap \u0393' \u0393\ng\u2081 : \u039b \u2192 \u039b'\ng\u2082 : \u039b' \u2192 \u039b\nS : Set \u039b\nf\u2082\u2081 : Function.RightInverse f\u2081.f f\u2082.f\ng\u2082\u2081 : \u2200 (q : \u039b), q \u2208 S \u2192 g\u2082 (g\u2081 q) = q\nq : \u039b\nT : Tape \u0393\nh : { q := q, Tape := T }.q \u2208 S\nq' : \u039b\nd : Dir\n\u22a2 some { q := g\u2081 q', Tape := Tape.move d (Tape.map f\u2081 T) } =\n    some\n      { q := (Prod.map g\u2081 (Stmt.map f\u2081) (q', Stmt.move d)).1,\n        Tape :=\n          match (Prod.map g\u2081 (Stmt.map f\u2081) (q', Stmt.move d)).2 with\n          | Stmt.move d => Tape.move d (Tape.map f\u2081 T)\n          | Stmt.write a => Tape.write a (Tape.map f\u2081 T) }"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case some.mk.move\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u0393' : Type u_2\ninst\u271d\u00b2 : Inhabited \u0393'\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u039b' : Type u_4\ninst\u271d : Inhabited \u039b'\nM : Machine \u0393 \u039b\nf\u2081 : PointedMap \u0393 \u0393'\nf\u2082 : PointedMap \u0393' \u0393\ng\u2081 : \u039b \u2192 \u039b'\ng\u2082 : \u039b' \u2192 \u039b\nS : Set \u039b\nf\u2082\u2081 : Function.RightInverse f\u2081.f f\u2082.f\ng\u2082\u2081 : \u2200 (q : \u039b), q \u2208 S \u2192 g\u2082 (g\u2081 q) = q\nq : \u039b\nT : Tape \u0393\nh : { q := q, Tape := T }.q \u2208 S\nq' : \u039b\nd : Dir\n\u22a2 some { q := g\u2081 q', Tape := Tape.move d (Tape.map f\u2081 T) } =\n    some\n      { q := (Prod.map g\u2081 (Stmt.map f\u2081) (q', Stmt.move d)).1,\n        Tape :=\n          match (Prod.map g\u2081 (Stmt.map f\u2081) (q', Stmt.move d)).2 with\n          | Stmt.move d => Tape.move d (Tape.map f\u2081 T)\n          | Stmt.write a => Tape.write a (Tape.map f\u2081 T) }", "state_after": "no goals"}, {"tactic": "simp only [step, Cfg.map, Option.map_some', Tape.map_write]", "annotated_tactic": ["simp only [<a>step</a>, <a>Cfg.map</a>, <a>Option.map_some'</a>, <a>Tape.map_write</a>]", [{"full_name": "Turing.TM0.step", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1076, 5], "def_end_pos": [1076, 9]}, {"full_name": "Turing.TM0.Cfg.map", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1142, 5], "def_end_pos": [1142, 12]}, {"full_name": "Option.map_some'", "def_path": "lake-packages/std/Std/Data/Option/Init/Lemmas.lean", "def_pos": [20, 17], "def_end_pos": [20, 26]}, {"full_name": "Turing.Tape.map_write", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [693, 9], "def_end_pos": [693, 23]}]], "state_before": "case some.mk.write\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u0393' : Type u_2\ninst\u271d\u00b2 : Inhabited \u0393'\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u039b' : Type u_4\ninst\u271d : Inhabited \u039b'\nM : Machine \u0393 \u039b\nf\u2081 : PointedMap \u0393 \u0393'\nf\u2082 : PointedMap \u0393' \u0393\ng\u2081 : \u039b \u2192 \u039b'\ng\u2082 : \u039b' \u2192 \u039b\nS : Set \u039b\nf\u2082\u2081 : Function.RightInverse f\u2081.f f\u2082.f\ng\u2082\u2081 : \u2200 (q : \u039b), q \u2208 S \u2192 g\u2082 (g\u2081 q) = q\nq : \u039b\nT : Tape \u0393\nh : { q := q, Tape := T }.q \u2208 S\nq' : \u039b\na : \u0393\n\u22a2 Option.map (fun x => { q := g\u2081 x.q, Tape := Tape.map f\u2081 x.Tape })\n      (Option.map\n        (fun x =>\n          { q := x.1,\n            Tape :=\n              match x.2 with\n              | Stmt.move d => Tape.move d T\n              | Stmt.write a => Tape.write a T })\n        (some (q', Stmt.write a))) =\n    Option.map\n      (fun x =>\n        { q := x.1,\n          Tape :=\n            match x.2 with\n            | Stmt.move d => Tape.move d (Tape.map f\u2081 T)\n            | Stmt.write a => Tape.write a (Tape.map f\u2081 T) })\n      (Option.map (Prod.map g\u2081 (Stmt.map f\u2081)) (some (q', Stmt.write a)))", "state_after": "case some.mk.write\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u0393' : Type u_2\ninst\u271d\u00b2 : Inhabited \u0393'\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u039b' : Type u_4\ninst\u271d : Inhabited \u039b'\nM : Machine \u0393 \u039b\nf\u2081 : PointedMap \u0393 \u0393'\nf\u2082 : PointedMap \u0393' \u0393\ng\u2081 : \u039b \u2192 \u039b'\ng\u2082 : \u039b' \u2192 \u039b\nS : Set \u039b\nf\u2082\u2081 : Function.RightInverse f\u2081.f f\u2082.f\ng\u2082\u2081 : \u2200 (q : \u039b), q \u2208 S \u2192 g\u2082 (g\u2081 q) = q\nq : \u039b\nT : Tape \u0393\nh : { q := q, Tape := T }.q \u2208 S\nq' : \u039b\na : \u0393\n\u22a2 some { q := g\u2081 q', Tape := Tape.write (PointedMap.f f\u2081 a) (Tape.map f\u2081 T) } =\n    some\n      { q := (Prod.map g\u2081 (Stmt.map f\u2081) (q', Stmt.write a)).1,\n        Tape :=\n          match (Prod.map g\u2081 (Stmt.map f\u2081) (q', Stmt.write a)).2 with\n          | Stmt.move d => Tape.move d (Tape.map f\u2081 T)\n          | Stmt.write a => Tape.write a (Tape.map f\u2081 T) }"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case some.mk.write\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u0393' : Type u_2\ninst\u271d\u00b2 : Inhabited \u0393'\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u039b' : Type u_4\ninst\u271d : Inhabited \u039b'\nM : Machine \u0393 \u039b\nf\u2081 : PointedMap \u0393 \u0393'\nf\u2082 : PointedMap \u0393' \u0393\ng\u2081 : \u039b \u2192 \u039b'\ng\u2082 : \u039b' \u2192 \u039b\nS : Set \u039b\nf\u2082\u2081 : Function.RightInverse f\u2081.f f\u2082.f\ng\u2082\u2081 : \u2200 (q : \u039b), q \u2208 S \u2192 g\u2082 (g\u2081 q) = q\nq : \u039b\nT : Tape \u0393\nh : { q := q, Tape := T }.q \u2208 S\nq' : \u039b\na : \u0393\n\u22a2 some { q := g\u2081 q', Tape := Tape.write (PointedMap.f f\u2081 a) (Tape.map f\u2081 T) } =\n    some\n      { q := (Prod.map g\u2081 (Stmt.map f\u2081) (q', Stmt.write a)).1,\n        Tape :=\n          match (Prod.map g\u2081 (Stmt.map f\u2081) (q', Stmt.write a)).2 with\n          | Stmt.move d => Tape.move d (Tape.map f\u2081 T)\n          | Stmt.write a => Tape.write a (Tape.map f\u2081 T) }", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean", "full_name": "Real.volume_pi_closedBall", "start": [264, 8], "end": [267, 64], "traced_tactics": [{"tactic": "simp only [MeasureTheory.volume_pi_closedBall a hr, volume_closedBall, Finset.prod_const]", "annotated_tactic": ["simp only [<a>MeasureTheory.volume_pi_closedBall</a> a hr, <a>volume_closedBall</a>, <a>Finset.prod_const</a>]", [{"full_name": "MeasureTheory.volume_pi_closedBall", "def_path": "Mathlib/MeasureTheory/Constructions/Pi.lean", "def_pos": [687, 9], "def_end_pos": [687, 29]}, {"full_name": "Real.volume_closedBall", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean", "def_pos": [113, 9], "def_end_pos": [113, 26]}, {"full_name": "Finset.prod_const", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [1441, 9], "def_end_pos": [1441, 19]}]], "state_before": "\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\na : \u03b9 \u2192 \u211d\nr : \u211d\nhr : 0 \u2264 r\n\u22a2 \u2191\u2191volume (Metric.closedBall a r) = ofReal ((2 * r) ^ Fintype.card \u03b9)", "state_after": "\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\na : \u03b9 \u2192 \u211d\nr : \u211d\nhr : 0 \u2264 r\n\u22a2 ofReal (2 * r) ^ Finset.card Finset.univ = ofReal ((2 * r) ^ Fintype.card \u03b9)"}, {"tactic": "exact (ENNReal.ofReal_pow (mul_nonneg zero_le_two hr) _).symm", "annotated_tactic": ["exact (<a>ENNReal.ofReal_pow</a> (<a>mul_nonneg</a> <a>zero_le_two</a> hr) _).<a>symm</a>", [{"full_name": "ENNReal.ofReal_pow", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2235, 9], "def_end_pos": [2235, 19]}, {"full_name": "mul_nonneg", "def_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "def_pos": [380, 7], "def_end_pos": [380, 17]}, {"full_name": "zero_le_two", "def_path": "Mathlib/Algebra/Order/Monoid/NatCast.lean", "def_pos": [32, 7], "def_end_pos": [32, 18]}, {"full_name": "Eq.symm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [310, 9], "def_end_pos": [310, 16]}]], "state_before": "\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\na : \u03b9 \u2192 \u211d\nr : \u211d\nhr : 0 \u2264 r\n\u22a2 ofReal (2 * r) ^ Finset.card Finset.univ = ofReal ((2 * r) ^ Fintype.card \u03b9)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/L2Space.lean", "full_name": "MeasureTheory.mem\u2112p_two_iff_integrable_sq", "start": [56, 1], "end": [59, 7], "traced_tactics": [{"tactic": "convert mem\u2112p_two_iff_integrable_sq_norm hf using 3", "annotated_tactic": ["convert <a>mem\u2112p_two_iff_integrable_sq_norm</a> hf using 3", [{"full_name": "MeasureTheory.mem\u2112p_two_iff_integrable_sq_norm", "def_path": "Mathlib/MeasureTheory/Function/L2Space.lean", "def_pos": [48, 9], "def_end_pos": [48, 41]}]], "state_before": "\u03b1 : Type u_1\nF : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup F\nf : \u03b1 \u2192 \u211d\nhf : AEStronglyMeasurable f \u03bc\n\u22a2 Mem\u2112p f 2 \u2194 Integrable fun x => f x ^ 2", "state_after": "case h.e'_2.h.e'_5.h\n\u03b1 : Type u_1\nF : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup F\nf : \u03b1 \u2192 \u211d\nhf : AEStronglyMeasurable f \u03bc\nx\u271d : \u03b1\n\u22a2 f x\u271d ^ 2 = \u2016f x\u271d\u2016 ^ 2"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case h.e'_2.h.e'_5.h\n\u03b1 : Type u_1\nF : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup F\nf : \u03b1 \u2192 \u211d\nhf : AEStronglyMeasurable f \u03bc\nx\u271d : \u03b1\n\u22a2 f x\u271d ^ 2 = \u2016f x\u271d\u2016 ^ 2", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/PFun.lean", "full_name": "PFun.fix_fwd", "start": [322, 1], "end": [323, 46], "traced_tactics": [{"tactic": "rwa [\u2190 fix_fwd_eq ha']", "annotated_tactic": ["rwa [\u2190 <a>fix_fwd_eq</a> ha']", [{"full_name": "PFun.fix_fwd_eq", "def_path": "Mathlib/Data/PFun.lean", "def_pos": [312, 9], "def_end_pos": [312, 19]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\nb : \u03b2\na a' : \u03b1\nhb : b \u2208 fix f a\nha' : Sum.inr a' \u2208 f a\n\u22a2 b \u2208 fix f a'", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Intervals/Pi.lean", "full_name": "Set.piecewise_mem_Icc'", "start": [54, 1], "end": [56, 76], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Group/Measure.lean", "full_name": "MeasureTheory.map_div_right_ae", "start": [354, 1], "end": [356, 93], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/ZMod/Basic.lean", "full_name": "ZMod.natAbs_valMinAbs_neg", "start": [1082, 1], "end": [1085, 54], "traced_tactics": [{"tactic": "by_cases h2a : 2 * a.val = n", "annotated_tactic": ["by_cases h2a : 2 * a.val = n", []], "state_before": "n : \u2115\na : ZMod n\n\u22a2 Int.natAbs (valMinAbs (-a)) = Int.natAbs (valMinAbs a)", "state_after": "case pos\nn : \u2115\na : ZMod n\nh2a : 2 * val a = n\n\u22a2 Int.natAbs (valMinAbs (-a)) = Int.natAbs (valMinAbs a)\n\ncase neg\nn : \u2115\na : ZMod n\nh2a : \u00ac2 * val a = n\n\u22a2 Int.natAbs (valMinAbs (-a)) = Int.natAbs (valMinAbs a)"}, {"tactic": "rw [a.neg_eq_self_iff.2 (Or.inr h2a)]", "annotated_tactic": ["rw [a.neg_eq_self_iff.2 (<a>Or.inr</a> h2a)]", [{"full_name": "Or.inr", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [519, 5], "def_end_pos": [519, 8]}]], "state_before": "case pos\nn : \u2115\na : ZMod n\nh2a : 2 * val a = n\n\u22a2 Int.natAbs (valMinAbs (-a)) = Int.natAbs (valMinAbs a)", "state_after": "no goals"}, {"tactic": "rw [valMinAbs_neg_of_ne_half h2a, Int.natAbs_neg]", "annotated_tactic": ["rw [<a>valMinAbs_neg_of_ne_half</a> h2a, <a>Int.natAbs_neg</a>]", [{"full_name": "ZMod.valMinAbs_neg_of_ne_half", "def_path": "Mathlib/Data/ZMod/Basic.lean", "def_pos": [1069, 9], "def_end_pos": [1069, 33]}, {"full_name": "Int.natAbs_neg", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [157, 17], "def_end_pos": [157, 27]}]], "state_before": "case neg\nn : \u2115\na : ZMod n\nh2a : \u00ac2 * val a = n\n\u22a2 Int.natAbs (valMinAbs (-a)) = Int.natAbs (valMinAbs a)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Pairwise.lean", "full_name": "List.pairwise_disjoint_of_coe_toFinset_pairwiseDisjoint", "start": [92, 1], "end": [95, 42], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Sub.lean", "full_name": "MeasureTheory.Measure.sub_apply", "start": [71, 1], "end": [96, 40], "traced_tactics": [{"tactic": "let measure_sub : Measure \u03b1 := MeasureTheory.Measure.ofMeasurable\n  (fun (t : Set \u03b1) (_ : MeasurableSet t) => \u03bc t - \u03bd t) (by simp)\n  (by\n    intro g h_meas h_disj; simp only; rw [ENNReal.tsum_sub]\n    repeat' rw [\u2190 MeasureTheory.measure_iUnion h_disj h_meas]\n    exacts [MeasureTheory.measure_ne_top _ _, fun i => h\u2082 _ (h_meas _)])", "annotated_tactic": ["let measure_sub : <a>Measure</a> \u03b1 := <a>MeasureTheory.Measure.ofMeasurable</a>\n    (fun (t : <a>Set</a> \u03b1) (_ : <a>MeasurableSet</a> t) => \u03bc t - \u03bd t) (by simp)\n    (by\n      intro g h_meas h_disj; simp only; rw [<a>ENNReal.tsum_sub</a>]\n      repeat' rw [\u2190 <a>MeasureTheory.measure_iUnion</a> h_disj h_meas]\n      exacts [<a>MeasureTheory.measure_ne_top</a> _ _, fun i => h\u2082 _ (h_meas _)])", [{"full_name": "MeasureTheory.Measure", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [74, 11], "def_end_pos": [74, 18]}, {"full_name": "MeasureTheory.Measure.ofMeasurable", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [103, 5], "def_end_pos": [103, 17]}, {"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}, {"full_name": "MeasurableSet", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [64, 5], "def_end_pos": [64, 18]}, {"full_name": "ENNReal.tsum_sub", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [975, 9], "def_end_pos": [975, 17]}, {"full_name": "MeasureTheory.measure_iUnion", "def_path": "Mathlib/MeasureTheory/Measure/NullMeasurable.lean", "def_pos": [272, 9], "def_end_pos": [272, 23]}, {"full_name": "MeasureTheory.measure_ne_top", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2875, 9], "def_end_pos": [2875, 23]}]], "state_before": "\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\ninst\u271d : IsFiniteMeasure \u03bd\nh\u2081 : MeasurableSet s\nh\u2082 : \u03bd \u2264 \u03bc\n\u22a2 \u2191\u2191(\u03bc - \u03bd) s = \u2191\u2191\u03bc s - \u2191\u2191\u03bd s", "state_after": "\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\ninst\u271d : IsFiniteMeasure \u03bd\nh\u2081 : MeasurableSet s\nh\u2082 : \u03bd \u2264 \u03bc\nmeasure_sub : Measure \u03b1 :=\n  ofMeasurable (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (_ : \u2191\u2191\u03bc \u2205 - \u2191\u2191\u03bd \u2205 = 0)\n    (_ :\n      \u2200 \u2983g : \u2115 \u2192 Set \u03b1\u2984 (h_meas : \u2200 (i : \u2115), MeasurableSet (g i)),\n        Pairwise (Disjoint on g) \u2192\n          (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (\u22c3 i, g i) (_ : MeasurableSet (\u22c3 b, g b)) =\n            \u2211' (i : \u2115), (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (g i) (_ : MeasurableSet (g i)))\n\u22a2 \u2191\u2191(\u03bc - \u03bd) s = \u2191\u2191\u03bc s - \u2191\u2191\u03bd s"}, {"tactic": "have h_measure_sub_add : \u03bd + measure_sub = \u03bc := by\n  ext1 t h_t_measurable_set\n  simp only [Pi.add_apply, coe_add]\n  rw [MeasureTheory.Measure.ofMeasurable_apply _ h_t_measurable_set, add_comm,\n    tsub_add_cancel_of_le (h\u2082 t h_t_measurable_set)]", "annotated_tactic": ["have h_measure_sub_add : \u03bd + measure_sub = \u03bc := by\n    ext1 t h_t_measurable_set\n    simp only [<a>Pi.add_apply</a>, <a>coe_add</a>]\n    rw [<a>MeasureTheory.Measure.ofMeasurable_apply</a> _ h_t_measurable_set, <a>add_comm</a>,\n      <a>tsub_add_cancel_of_le</a> (h\u2082 t h_t_measurable_set)]", [{"full_name": "Pi.add_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [82, 3], "def_end_pos": [82, 14]}, {"full_name": "MeasureTheory.Measure.coe_add", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [794, 9], "def_end_pos": [794, 16]}, {"full_name": "MeasureTheory.Measure.ofMeasurable_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [120, 9], "def_end_pos": [120, 27]}, {"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [301, 3], "def_end_pos": [301, 14]}, {"full_name": "tsub_add_cancel_of_le", "def_path": "Mathlib/Algebra/Order/Sub/Canonical.lean", "def_pos": [30, 9], "def_end_pos": [30, 30]}]], "state_before": "\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\ninst\u271d : IsFiniteMeasure \u03bd\nh\u2081 : MeasurableSet s\nh\u2082 : \u03bd \u2264 \u03bc\nmeasure_sub : Measure \u03b1 :=\n  ofMeasurable (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (_ : \u2191\u2191\u03bc \u2205 - \u2191\u2191\u03bd \u2205 = 0)\n    (_ :\n      \u2200 \u2983g : \u2115 \u2192 Set \u03b1\u2984 (h_meas : \u2200 (i : \u2115), MeasurableSet (g i)),\n        Pairwise (Disjoint on g) \u2192\n          (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (\u22c3 i, g i) (_ : MeasurableSet (\u22c3 b, g b)) =\n            \u2211' (i : \u2115), (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (g i) (_ : MeasurableSet (g i)))\n\u22a2 \u2191\u2191(\u03bc - \u03bd) s = \u2191\u2191\u03bc s - \u2191\u2191\u03bd s", "state_after": "\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\ninst\u271d : IsFiniteMeasure \u03bd\nh\u2081 : MeasurableSet s\nh\u2082 : \u03bd \u2264 \u03bc\nmeasure_sub : Measure \u03b1 :=\n  ofMeasurable (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (_ : \u2191\u2191\u03bc \u2205 - \u2191\u2191\u03bd \u2205 = 0)\n    (_ :\n      \u2200 \u2983g : \u2115 \u2192 Set \u03b1\u2984 (h_meas : \u2200 (i : \u2115), MeasurableSet (g i)),\n        Pairwise (Disjoint on g) \u2192\n          (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (\u22c3 i, g i) (_ : MeasurableSet (\u22c3 b, g b)) =\n            \u2211' (i : \u2115), (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (g i) (_ : MeasurableSet (g i)))\nh_measure_sub_add : \u03bd + measure_sub = \u03bc\n\u22a2 \u2191\u2191(\u03bc - \u03bd) s = \u2191\u2191\u03bc s - \u2191\u2191\u03bd s"}, {"tactic": "rw [h_measure_sub_eq]", "annotated_tactic": ["rw [h_measure_sub_eq]", []], "state_before": "\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\ninst\u271d : IsFiniteMeasure \u03bd\nh\u2081 : MeasurableSet s\nh\u2082 : \u03bd \u2264 \u03bc\nmeasure_sub : Measure \u03b1 :=\n  ofMeasurable (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (_ : \u2191\u2191\u03bc \u2205 - \u2191\u2191\u03bd \u2205 = 0)\n    (_ :\n      \u2200 \u2983g : \u2115 \u2192 Set \u03b1\u2984 (h_meas : \u2200 (i : \u2115), MeasurableSet (g i)),\n        Pairwise (Disjoint on g) \u2192\n          (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (\u22c3 i, g i) (_ : MeasurableSet (\u22c3 b, g b)) =\n            \u2211' (i : \u2115), (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (g i) (_ : MeasurableSet (g i)))\nh_measure_sub_add : \u03bd + measure_sub = \u03bc\nh_measure_sub_eq : \u03bc - \u03bd = measure_sub\n\u22a2 \u2191\u2191(\u03bc - \u03bd) s = \u2191\u2191\u03bc s - \u2191\u2191\u03bd s", "state_after": "\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\ninst\u271d : IsFiniteMeasure \u03bd\nh\u2081 : MeasurableSet s\nh\u2082 : \u03bd \u2264 \u03bc\nmeasure_sub : Measure \u03b1 :=\n  ofMeasurable (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (_ : \u2191\u2191\u03bc \u2205 - \u2191\u2191\u03bd \u2205 = 0)\n    (_ :\n      \u2200 \u2983g : \u2115 \u2192 Set \u03b1\u2984 (h_meas : \u2200 (i : \u2115), MeasurableSet (g i)),\n        Pairwise (Disjoint on g) \u2192\n          (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (\u22c3 i, g i) (_ : MeasurableSet (\u22c3 b, g b)) =\n            \u2211' (i : \u2115), (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (g i) (_ : MeasurableSet (g i)))\nh_measure_sub_add : \u03bd + measure_sub = \u03bc\nh_measure_sub_eq : \u03bc - \u03bd = measure_sub\n\u22a2 \u2191\u2191measure_sub s = \u2191\u2191\u03bc s - \u2191\u2191\u03bd s"}, {"tactic": "apply Measure.ofMeasurable_apply _ h\u2081", "annotated_tactic": ["apply <a>Measure.ofMeasurable_apply</a> _ h\u2081", [{"full_name": "MeasureTheory.Measure.ofMeasurable_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [120, 9], "def_end_pos": [120, 27]}]], "state_before": "\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\ninst\u271d : IsFiniteMeasure \u03bd\nh\u2081 : MeasurableSet s\nh\u2082 : \u03bd \u2264 \u03bc\nmeasure_sub : Measure \u03b1 :=\n  ofMeasurable (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (_ : \u2191\u2191\u03bc \u2205 - \u2191\u2191\u03bd \u2205 = 0)\n    (_ :\n      \u2200 \u2983g : \u2115 \u2192 Set \u03b1\u2984 (h_meas : \u2200 (i : \u2115), MeasurableSet (g i)),\n        Pairwise (Disjoint on g) \u2192\n          (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (\u22c3 i, g i) (_ : MeasurableSet (\u22c3 b, g b)) =\n            \u2211' (i : \u2115), (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (g i) (_ : MeasurableSet (g i)))\nh_measure_sub_add : \u03bd + measure_sub = \u03bc\nh_measure_sub_eq : \u03bc - \u03bd = measure_sub\n\u22a2 \u2191\u2191measure_sub s = \u2191\u2191\u03bc s - \u2191\u2191\u03bd s", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\ninst\u271d : IsFiniteMeasure \u03bd\nh\u2081 : MeasurableSet s\nh\u2082 : \u03bd \u2264 \u03bc\n\u22a2 (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) \u2205 (_ : MeasurableSet \u2205) = 0", "state_after": "no goals"}, {"tactic": "intro g h_meas h_disj", "annotated_tactic": ["intro g h_meas h_disj", []], "state_before": "\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\ninst\u271d : IsFiniteMeasure \u03bd\nh\u2081 : MeasurableSet s\nh\u2082 : \u03bd \u2264 \u03bc\n\u22a2 \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (h : \u2200 (i : \u2115), MeasurableSet (f i)),\n    Pairwise (Disjoint on f) \u2192\n      (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (\u22c3 i, f i) (_ : MeasurableSet (\u22c3 b, f b)) =\n        \u2211' (i : \u2115), (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (f i) (_ : MeasurableSet (f i))", "state_after": "\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\ninst\u271d : IsFiniteMeasure \u03bd\nh\u2081 : MeasurableSet s\nh\u2082 : \u03bd \u2264 \u03bc\ng : \u2115 \u2192 Set \u03b1\nh_meas : \u2200 (i : \u2115), MeasurableSet (g i)\nh_disj : Pairwise (Disjoint on g)\n\u22a2 (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (\u22c3 i, g i) (_ : MeasurableSet (\u22c3 b, g b)) =\n    \u2211' (i : \u2115), (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (g i) (_ : MeasurableSet (g i))"}, {"tactic": "simp only", "annotated_tactic": ["simp only", []], "state_before": "\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\ninst\u271d : IsFiniteMeasure \u03bd\nh\u2081 : MeasurableSet s\nh\u2082 : \u03bd \u2264 \u03bc\ng : \u2115 \u2192 Set \u03b1\nh_meas : \u2200 (i : \u2115), MeasurableSet (g i)\nh_disj : Pairwise (Disjoint on g)\n\u22a2 (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (\u22c3 i, g i) (_ : MeasurableSet (\u22c3 b, g b)) =\n    \u2211' (i : \u2115), (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (g i) (_ : MeasurableSet (g i))", "state_after": "\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\ninst\u271d : IsFiniteMeasure \u03bd\nh\u2081 : MeasurableSet s\nh\u2082 : \u03bd \u2264 \u03bc\ng : \u2115 \u2192 Set \u03b1\nh_meas : \u2200 (i : \u2115), MeasurableSet (g i)\nh_disj : Pairwise (Disjoint on g)\n\u22a2 \u2191\u2191\u03bc (\u22c3 i, g i) - \u2191\u2191\u03bd (\u22c3 i, g i) = \u2211' (i : \u2115), (\u2191\u2191\u03bc (g i) - \u2191\u2191\u03bd (g i))"}, {"tactic": "rw [ENNReal.tsum_sub]", "annotated_tactic": ["rw [<a>ENNReal.tsum_sub</a>]", [{"full_name": "ENNReal.tsum_sub", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [975, 9], "def_end_pos": [975, 17]}]], "state_before": "\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\ninst\u271d : IsFiniteMeasure \u03bd\nh\u2081 : MeasurableSet s\nh\u2082 : \u03bd \u2264 \u03bc\ng : \u2115 \u2192 Set \u03b1\nh_meas : \u2200 (i : \u2115), MeasurableSet (g i)\nh_disj : Pairwise (Disjoint on g)\n\u22a2 \u2191\u2191\u03bc (\u22c3 i, g i) - \u2191\u2191\u03bd (\u22c3 i, g i) = \u2211' (i : \u2115), (\u2191\u2191\u03bc (g i) - \u2191\u2191\u03bd (g i))", "state_after": "\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\ninst\u271d : IsFiniteMeasure \u03bd\nh\u2081 : MeasurableSet s\nh\u2082 : \u03bd \u2264 \u03bc\ng : \u2115 \u2192 Set \u03b1\nh_meas : \u2200 (i : \u2115), MeasurableSet (g i)\nh_disj : Pairwise (Disjoint on g)\n\u22a2 \u2191\u2191\u03bc (\u22c3 i, g i) - \u2191\u2191\u03bd (\u22c3 i, g i) = \u2211' (i : \u2115), \u2191\u2191\u03bc (g i) - \u2211' (i : \u2115), \u2191\u2191\u03bd (g i)\n\ncase h\u2081\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\ninst\u271d : IsFiniteMeasure \u03bd\nh\u2081 : MeasurableSet s\nh\u2082 : \u03bd \u2264 \u03bc\ng : \u2115 \u2192 Set \u03b1\nh_meas : \u2200 (i : \u2115), MeasurableSet (g i)\nh_disj : Pairwise (Disjoint on g)\n\u22a2 \u2211' (i : \u2115), \u2191\u2191\u03bd (g i) \u2260 \u22a4\n\ncase h\u2082\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\ninst\u271d : IsFiniteMeasure \u03bd\nh\u2081 : MeasurableSet s\nh\u2082 : \u03bd \u2264 \u03bc\ng : \u2115 \u2192 Set \u03b1\nh_meas : \u2200 (i : \u2115), MeasurableSet (g i)\nh_disj : Pairwise (Disjoint on g)\n\u22a2 (fun i => \u2191\u2191\u03bd (g i)) \u2264 fun i => \u2191\u2191\u03bc (g i)"}, {"tactic": "repeat' rw [\u2190 MeasureTheory.measure_iUnion h_disj h_meas]", "annotated_tactic": ["repeat' rw [\u2190 <a>MeasureTheory.measure_iUnion</a> h_disj h_meas]", [{"full_name": "MeasureTheory.measure_iUnion", "def_path": "Mathlib/MeasureTheory/Measure/NullMeasurable.lean", "def_pos": [272, 9], "def_end_pos": [272, 23]}]], "state_before": "\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\ninst\u271d : IsFiniteMeasure \u03bd\nh\u2081 : MeasurableSet s\nh\u2082 : \u03bd \u2264 \u03bc\ng : \u2115 \u2192 Set \u03b1\nh_meas : \u2200 (i : \u2115), MeasurableSet (g i)\nh_disj : Pairwise (Disjoint on g)\n\u22a2 \u2191\u2191\u03bc (\u22c3 i, g i) - \u2191\u2191\u03bd (\u22c3 i, g i) = \u2211' (i : \u2115), \u2191\u2191\u03bc (g i) - \u2211' (i : \u2115), \u2191\u2191\u03bd (g i)\n\ncase h\u2081\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\ninst\u271d : IsFiniteMeasure \u03bd\nh\u2081 : MeasurableSet s\nh\u2082 : \u03bd \u2264 \u03bc\ng : \u2115 \u2192 Set \u03b1\nh_meas : \u2200 (i : \u2115), MeasurableSet (g i)\nh_disj : Pairwise (Disjoint on g)\n\u22a2 \u2211' (i : \u2115), \u2191\u2191\u03bd (g i) \u2260 \u22a4\n\ncase h\u2082\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\ninst\u271d : IsFiniteMeasure \u03bd\nh\u2081 : MeasurableSet s\nh\u2082 : \u03bd \u2264 \u03bc\ng : \u2115 \u2192 Set \u03b1\nh_meas : \u2200 (i : \u2115), MeasurableSet (g i)\nh_disj : Pairwise (Disjoint on g)\n\u22a2 (fun i => \u2191\u2191\u03bd (g i)) \u2264 fun i => \u2191\u2191\u03bc (g i)", "state_after": "case h\u2081\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\ninst\u271d : IsFiniteMeasure \u03bd\nh\u2081 : MeasurableSet s\nh\u2082 : \u03bd \u2264 \u03bc\ng : \u2115 \u2192 Set \u03b1\nh_meas : \u2200 (i : \u2115), MeasurableSet (g i)\nh_disj : Pairwise (Disjoint on g)\n\u22a2 \u2191\u2191\u03bd (\u22c3 i, g i) \u2260 \u22a4\n\ncase h\u2082\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\ninst\u271d : IsFiniteMeasure \u03bd\nh\u2081 : MeasurableSet s\nh\u2082 : \u03bd \u2264 \u03bc\ng : \u2115 \u2192 Set \u03b1\nh_meas : \u2200 (i : \u2115), MeasurableSet (g i)\nh_disj : Pairwise (Disjoint on g)\n\u22a2 (fun i => \u2191\u2191\u03bd (g i)) \u2264 fun i => \u2191\u2191\u03bc (g i)"}, {"tactic": "exacts [MeasureTheory.measure_ne_top _ _, fun i => h\u2082 _ (h_meas _)]", "annotated_tactic": ["exacts [<a>MeasureTheory.measure_ne_top</a> _ _, fun i => h\u2082 _ (h_meas _)]", [{"full_name": "MeasureTheory.measure_ne_top", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2875, 9], "def_end_pos": [2875, 23]}]], "state_before": "case h\u2081\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\ninst\u271d : IsFiniteMeasure \u03bd\nh\u2081 : MeasurableSet s\nh\u2082 : \u03bd \u2264 \u03bc\ng : \u2115 \u2192 Set \u03b1\nh_meas : \u2200 (i : \u2115), MeasurableSet (g i)\nh_disj : Pairwise (Disjoint on g)\n\u22a2 \u2191\u2191\u03bd (\u22c3 i, g i) \u2260 \u22a4\n\ncase h\u2082\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\ninst\u271d : IsFiniteMeasure \u03bd\nh\u2081 : MeasurableSet s\nh\u2082 : \u03bd \u2264 \u03bc\ng : \u2115 \u2192 Set \u03b1\nh_meas : \u2200 (i : \u2115), MeasurableSet (g i)\nh_disj : Pairwise (Disjoint on g)\n\u22a2 (fun i => \u2191\u2191\u03bd (g i)) \u2264 fun i => \u2191\u2191\u03bc (g i)", "state_after": "no goals"}, {"tactic": "rw [\u2190 MeasureTheory.measure_iUnion h_disj h_meas]", "annotated_tactic": ["rw [\u2190 <a>MeasureTheory.measure_iUnion</a> h_disj h_meas]", [{"full_name": "MeasureTheory.measure_iUnion", "def_path": "Mathlib/MeasureTheory/Measure/NullMeasurable.lean", "def_pos": [272, 9], "def_end_pos": [272, 23]}]], "state_before": "case h\u2081\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\ninst\u271d : IsFiniteMeasure \u03bd\nh\u2081 : MeasurableSet s\nh\u2082 : \u03bd \u2264 \u03bc\ng : \u2115 \u2192 Set \u03b1\nh_meas : \u2200 (i : \u2115), MeasurableSet (g i)\nh_disj : Pairwise (Disjoint on g)\n\u22a2 \u2211' (i : \u2115), \u2191\u2191\u03bd (g i) \u2260 \u22a4", "state_after": "case h\u2081\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\ninst\u271d : IsFiniteMeasure \u03bd\nh\u2081 : MeasurableSet s\nh\u2082 : \u03bd \u2264 \u03bc\ng : \u2115 \u2192 Set \u03b1\nh_meas : \u2200 (i : \u2115), MeasurableSet (g i)\nh_disj : Pairwise (Disjoint on g)\n\u22a2 \u2191\u2191\u03bd (\u22c3 i, g i) \u2260 \u22a4"}, {"tactic": "ext1 t h_t_measurable_set", "annotated_tactic": ["ext1 t h_t_measurable_set", []], "state_before": "\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\ninst\u271d : IsFiniteMeasure \u03bd\nh\u2081 : MeasurableSet s\nh\u2082 : \u03bd \u2264 \u03bc\nmeasure_sub : Measure \u03b1 :=\n  ofMeasurable (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (_ : \u2191\u2191\u03bc \u2205 - \u2191\u2191\u03bd \u2205 = 0)\n    (_ :\n      \u2200 \u2983g : \u2115 \u2192 Set \u03b1\u2984 (h_meas : \u2200 (i : \u2115), MeasurableSet (g i)),\n        Pairwise (Disjoint on g) \u2192\n          (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (\u22c3 i, g i) (_ : MeasurableSet (\u22c3 b, g b)) =\n            \u2211' (i : \u2115), (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (g i) (_ : MeasurableSet (g i)))\n\u22a2 \u03bd + measure_sub = \u03bc", "state_after": "case h\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\ninst\u271d : IsFiniteMeasure \u03bd\nh\u2081 : MeasurableSet s\nh\u2082 : \u03bd \u2264 \u03bc\nmeasure_sub : Measure \u03b1 :=\n  ofMeasurable (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (_ : \u2191\u2191\u03bc \u2205 - \u2191\u2191\u03bd \u2205 = 0)\n    (_ :\n      \u2200 \u2983g : \u2115 \u2192 Set \u03b1\u2984 (h_meas : \u2200 (i : \u2115), MeasurableSet (g i)),\n        Pairwise (Disjoint on g) \u2192\n          (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (\u22c3 i, g i) (_ : MeasurableSet (\u22c3 b, g b)) =\n            \u2211' (i : \u2115), (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (g i) (_ : MeasurableSet (g i)))\nt : Set \u03b1\nh_t_measurable_set : MeasurableSet t\n\u22a2 \u2191\u2191(\u03bd + measure_sub) t = \u2191\u2191\u03bc t"}, {"tactic": "simp only [Pi.add_apply, coe_add]", "annotated_tactic": ["simp only [<a>Pi.add_apply</a>, <a>coe_add</a>]", [{"full_name": "Pi.add_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [82, 3], "def_end_pos": [82, 14]}, {"full_name": "MeasureTheory.Measure.coe_add", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [794, 9], "def_end_pos": [794, 16]}]], "state_before": "case h\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\ninst\u271d : IsFiniteMeasure \u03bd\nh\u2081 : MeasurableSet s\nh\u2082 : \u03bd \u2264 \u03bc\nmeasure_sub : Measure \u03b1 :=\n  ofMeasurable (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (_ : \u2191\u2191\u03bc \u2205 - \u2191\u2191\u03bd \u2205 = 0)\n    (_ :\n      \u2200 \u2983g : \u2115 \u2192 Set \u03b1\u2984 (h_meas : \u2200 (i : \u2115), MeasurableSet (g i)),\n        Pairwise (Disjoint on g) \u2192\n          (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (\u22c3 i, g i) (_ : MeasurableSet (\u22c3 b, g b)) =\n            \u2211' (i : \u2115), (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (g i) (_ : MeasurableSet (g i)))\nt : Set \u03b1\nh_t_measurable_set : MeasurableSet t\n\u22a2 \u2191\u2191(\u03bd + measure_sub) t = \u2191\u2191\u03bc t", "state_after": "case h\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\ninst\u271d : IsFiniteMeasure \u03bd\nh\u2081 : MeasurableSet s\nh\u2082 : \u03bd \u2264 \u03bc\nmeasure_sub : Measure \u03b1 :=\n  ofMeasurable (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (_ : \u2191\u2191\u03bc \u2205 - \u2191\u2191\u03bd \u2205 = 0)\n    (_ :\n      \u2200 \u2983g : \u2115 \u2192 Set \u03b1\u2984 (h_meas : \u2200 (i : \u2115), MeasurableSet (g i)),\n        Pairwise (Disjoint on g) \u2192\n          (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (\u22c3 i, g i) (_ : MeasurableSet (\u22c3 b, g b)) =\n            \u2211' (i : \u2115), (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (g i) (_ : MeasurableSet (g i)))\nt : Set \u03b1\nh_t_measurable_set : MeasurableSet t\n\u22a2 \u2191\u2191\u03bd t +\n      \u2191\u2191(ofMeasurable (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (_ : \u2191\u2191\u03bc \u2205 - \u2191\u2191\u03bd \u2205 = 0)\n              (_ :\n                \u2200 \u2983g : \u2115 \u2192 Set \u03b1\u2984 (h_meas : \u2200 (i : \u2115), MeasurableSet (g i)),\n                  Pairwise (Disjoint on g) \u2192\n                    (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (\u22c3 i, g i) (_ : MeasurableSet (\u22c3 b, g b)) =\n                      \u2211' (i : \u2115), (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (g i) (_ : MeasurableSet (g i))))\n        t =\n    \u2191\u2191\u03bc t"}, {"tactic": "rw [MeasureTheory.Measure.ofMeasurable_apply _ h_t_measurable_set, add_comm,\n  tsub_add_cancel_of_le (h\u2082 t h_t_measurable_set)]", "annotated_tactic": ["rw [<a>MeasureTheory.Measure.ofMeasurable_apply</a> _ h_t_measurable_set, <a>add_comm</a>,\n      <a>tsub_add_cancel_of_le</a> (h\u2082 t h_t_measurable_set)]", [{"full_name": "MeasureTheory.Measure.ofMeasurable_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [120, 9], "def_end_pos": [120, 27]}, {"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [301, 3], "def_end_pos": [301, 14]}, {"full_name": "tsub_add_cancel_of_le", "def_path": "Mathlib/Algebra/Order/Sub/Canonical.lean", "def_pos": [30, 9], "def_end_pos": [30, 30]}]], "state_before": "case h\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\ninst\u271d : IsFiniteMeasure \u03bd\nh\u2081 : MeasurableSet s\nh\u2082 : \u03bd \u2264 \u03bc\nmeasure_sub : Measure \u03b1 :=\n  ofMeasurable (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (_ : \u2191\u2191\u03bc \u2205 - \u2191\u2191\u03bd \u2205 = 0)\n    (_ :\n      \u2200 \u2983g : \u2115 \u2192 Set \u03b1\u2984 (h_meas : \u2200 (i : \u2115), MeasurableSet (g i)),\n        Pairwise (Disjoint on g) \u2192\n          (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (\u22c3 i, g i) (_ : MeasurableSet (\u22c3 b, g b)) =\n            \u2211' (i : \u2115), (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (g i) (_ : MeasurableSet (g i)))\nt : Set \u03b1\nh_t_measurable_set : MeasurableSet t\n\u22a2 \u2191\u2191\u03bd t +\n      \u2191\u2191(ofMeasurable (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (_ : \u2191\u2191\u03bc \u2205 - \u2191\u2191\u03bd \u2205 = 0)\n              (_ :\n                \u2200 \u2983g : \u2115 \u2192 Set \u03b1\u2984 (h_meas : \u2200 (i : \u2115), MeasurableSet (g i)),\n                  Pairwise (Disjoint on g) \u2192\n                    (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (\u22c3 i, g i) (_ : MeasurableSet (\u22c3 b, g b)) =\n                      \u2211' (i : \u2115), (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (g i) (_ : MeasurableSet (g i))))\n        t =\n    \u2191\u2191\u03bc t", "state_after": "no goals"}, {"tactic": "rw [MeasureTheory.Measure.sub_def]", "annotated_tactic": ["rw [<a>MeasureTheory.Measure.sub_def</a>]", [{"full_name": "MeasureTheory.Measure.sub_def", "def_path": "Mathlib/MeasureTheory/Measure/Sub.lean", "def_pos": [39, 9], "def_end_pos": [39, 16]}]], "state_before": "\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\ninst\u271d : IsFiniteMeasure \u03bd\nh\u2081 : MeasurableSet s\nh\u2082 : \u03bd \u2264 \u03bc\nmeasure_sub : Measure \u03b1 :=\n  ofMeasurable (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (_ : \u2191\u2191\u03bc \u2205 - \u2191\u2191\u03bd \u2205 = 0)\n    (_ :\n      \u2200 \u2983g : \u2115 \u2192 Set \u03b1\u2984 (h_meas : \u2200 (i : \u2115), MeasurableSet (g i)),\n        Pairwise (Disjoint on g) \u2192\n          (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (\u22c3 i, g i) (_ : MeasurableSet (\u22c3 b, g b)) =\n            \u2211' (i : \u2115), (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (g i) (_ : MeasurableSet (g i)))\nh_measure_sub_add : \u03bd + measure_sub = \u03bc\n\u22a2 \u03bc - \u03bd = measure_sub", "state_after": "\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\ninst\u271d : IsFiniteMeasure \u03bd\nh\u2081 : MeasurableSet s\nh\u2082 : \u03bd \u2264 \u03bc\nmeasure_sub : Measure \u03b1 :=\n  ofMeasurable (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (_ : \u2191\u2191\u03bc \u2205 - \u2191\u2191\u03bd \u2205 = 0)\n    (_ :\n      \u2200 \u2983g : \u2115 \u2192 Set \u03b1\u2984 (h_meas : \u2200 (i : \u2115), MeasurableSet (g i)),\n        Pairwise (Disjoint on g) \u2192\n          (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (\u22c3 i, g i) (_ : MeasurableSet (\u22c3 b, g b)) =\n            \u2211' (i : \u2115), (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (g i) (_ : MeasurableSet (g i)))\nh_measure_sub_add : \u03bd + measure_sub = \u03bc\n\u22a2 sInf {d | \u03bc \u2264 d + \u03bd} = measure_sub"}, {"tactic": "apply le_antisymm", "annotated_tactic": ["apply <a>le_antisymm</a>", [{"full_name": "le_antisymm", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [188, 9], "def_end_pos": [188, 20]}]], "state_before": "\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\ninst\u271d : IsFiniteMeasure \u03bd\nh\u2081 : MeasurableSet s\nh\u2082 : \u03bd \u2264 \u03bc\nmeasure_sub : Measure \u03b1 :=\n  ofMeasurable (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (_ : \u2191\u2191\u03bc \u2205 - \u2191\u2191\u03bd \u2205 = 0)\n    (_ :\n      \u2200 \u2983g : \u2115 \u2192 Set \u03b1\u2984 (h_meas : \u2200 (i : \u2115), MeasurableSet (g i)),\n        Pairwise (Disjoint on g) \u2192\n          (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (\u22c3 i, g i) (_ : MeasurableSet (\u22c3 b, g b)) =\n            \u2211' (i : \u2115), (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (g i) (_ : MeasurableSet (g i)))\nh_measure_sub_add : \u03bd + measure_sub = \u03bc\n\u22a2 sInf {d | \u03bc \u2264 d + \u03bd} = measure_sub", "state_after": "case a\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\ninst\u271d : IsFiniteMeasure \u03bd\nh\u2081 : MeasurableSet s\nh\u2082 : \u03bd \u2264 \u03bc\nmeasure_sub : Measure \u03b1 :=\n  ofMeasurable (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (_ : \u2191\u2191\u03bc \u2205 - \u2191\u2191\u03bd \u2205 = 0)\n    (_ :\n      \u2200 \u2983g : \u2115 \u2192 Set \u03b1\u2984 (h_meas : \u2200 (i : \u2115), MeasurableSet (g i)),\n        Pairwise (Disjoint on g) \u2192\n          (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (\u22c3 i, g i) (_ : MeasurableSet (\u22c3 b, g b)) =\n            \u2211' (i : \u2115), (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (g i) (_ : MeasurableSet (g i)))\nh_measure_sub_add : \u03bd + measure_sub = \u03bc\n\u22a2 sInf {d | \u03bc \u2264 d + \u03bd} \u2264 measure_sub\n\ncase a\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\ninst\u271d : IsFiniteMeasure \u03bd\nh\u2081 : MeasurableSet s\nh\u2082 : \u03bd \u2264 \u03bc\nmeasure_sub : Measure \u03b1 :=\n  ofMeasurable (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (_ : \u2191\u2191\u03bc \u2205 - \u2191\u2191\u03bd \u2205 = 0)\n    (_ :\n      \u2200 \u2983g : \u2115 \u2192 Set \u03b1\u2984 (h_meas : \u2200 (i : \u2115), MeasurableSet (g i)),\n        Pairwise (Disjoint on g) \u2192\n          (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (\u22c3 i, g i) (_ : MeasurableSet (\u22c3 b, g b)) =\n            \u2211' (i : \u2115), (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (g i) (_ : MeasurableSet (g i)))\nh_measure_sub_add : \u03bd + measure_sub = \u03bc\n\u22a2 measure_sub \u2264 sInf {d | \u03bc \u2264 d + \u03bd}"}, {"tactic": "apply le_sInf", "annotated_tactic": ["apply <a>le_sInf</a>", [{"full_name": "le_sInf", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [269, 9], "def_end_pos": [269, 16]}]], "state_before": "case a\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\ninst\u271d : IsFiniteMeasure \u03bd\nh\u2081 : MeasurableSet s\nh\u2082 : \u03bd \u2264 \u03bc\nmeasure_sub : Measure \u03b1 :=\n  ofMeasurable (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (_ : \u2191\u2191\u03bc \u2205 - \u2191\u2191\u03bd \u2205 = 0)\n    (_ :\n      \u2200 \u2983g : \u2115 \u2192 Set \u03b1\u2984 (h_meas : \u2200 (i : \u2115), MeasurableSet (g i)),\n        Pairwise (Disjoint on g) \u2192\n          (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (\u22c3 i, g i) (_ : MeasurableSet (\u22c3 b, g b)) =\n            \u2211' (i : \u2115), (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (g i) (_ : MeasurableSet (g i)))\nh_measure_sub_add : \u03bd + measure_sub = \u03bc\n\u22a2 measure_sub \u2264 sInf {d | \u03bc \u2264 d + \u03bd}", "state_after": "case a.a\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\ninst\u271d : IsFiniteMeasure \u03bd\nh\u2081 : MeasurableSet s\nh\u2082 : \u03bd \u2264 \u03bc\nmeasure_sub : Measure \u03b1 :=\n  ofMeasurable (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (_ : \u2191\u2191\u03bc \u2205 - \u2191\u2191\u03bd \u2205 = 0)\n    (_ :\n      \u2200 \u2983g : \u2115 \u2192 Set \u03b1\u2984 (h_meas : \u2200 (i : \u2115), MeasurableSet (g i)),\n        Pairwise (Disjoint on g) \u2192\n          (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (\u22c3 i, g i) (_ : MeasurableSet (\u22c3 b, g b)) =\n            \u2211' (i : \u2115), (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (g i) (_ : MeasurableSet (g i)))\nh_measure_sub_add : \u03bd + measure_sub = \u03bc\n\u22a2 \u2200 (b : Measure \u03b1), b \u2208 {d | \u03bc \u2264 d + \u03bd} \u2192 measure_sub \u2264 b"}, {"tactic": "intro d h_d", "annotated_tactic": ["intro d h_d", []], "state_before": "case a.a\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\ninst\u271d : IsFiniteMeasure \u03bd\nh\u2081 : MeasurableSet s\nh\u2082 : \u03bd \u2264 \u03bc\nmeasure_sub : Measure \u03b1 :=\n  ofMeasurable (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (_ : \u2191\u2191\u03bc \u2205 - \u2191\u2191\u03bd \u2205 = 0)\n    (_ :\n      \u2200 \u2983g : \u2115 \u2192 Set \u03b1\u2984 (h_meas : \u2200 (i : \u2115), MeasurableSet (g i)),\n        Pairwise (Disjoint on g) \u2192\n          (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (\u22c3 i, g i) (_ : MeasurableSet (\u22c3 b, g b)) =\n            \u2211' (i : \u2115), (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (g i) (_ : MeasurableSet (g i)))\nh_measure_sub_add : \u03bd + measure_sub = \u03bc\n\u22a2 \u2200 (b : Measure \u03b1), b \u2208 {d | \u03bc \u2264 d + \u03bd} \u2192 measure_sub \u2264 b", "state_after": "case a.a\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\ninst\u271d : IsFiniteMeasure \u03bd\nh\u2081 : MeasurableSet s\nh\u2082 : \u03bd \u2264 \u03bc\nmeasure_sub : Measure \u03b1 :=\n  ofMeasurable (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (_ : \u2191\u2191\u03bc \u2205 - \u2191\u2191\u03bd \u2205 = 0)\n    (_ :\n      \u2200 \u2983g : \u2115 \u2192 Set \u03b1\u2984 (h_meas : \u2200 (i : \u2115), MeasurableSet (g i)),\n        Pairwise (Disjoint on g) \u2192\n          (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (\u22c3 i, g i) (_ : MeasurableSet (\u22c3 b, g b)) =\n            \u2211' (i : \u2115), (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (g i) (_ : MeasurableSet (g i)))\nh_measure_sub_add : \u03bd + measure_sub = \u03bc\nd : Measure \u03b1\nh_d : d \u2208 {d | \u03bc \u2264 d + \u03bd}\n\u22a2 measure_sub \u2264 d"}, {"tactic": "rw [\u2190 h_measure_sub_add, mem_setOf_eq, add_comm d] at h_d", "annotated_tactic": ["rw [\u2190 h_measure_sub_add, <a>mem_setOf_eq</a>, <a>add_comm</a> d] at h_d", [{"full_name": "Set.mem_setOf_eq", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [256, 29], "def_end_pos": [256, 41]}, {"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [301, 3], "def_end_pos": [301, 14]}]], "state_before": "case a.a\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\ninst\u271d : IsFiniteMeasure \u03bd\nh\u2081 : MeasurableSet s\nh\u2082 : \u03bd \u2264 \u03bc\nmeasure_sub : Measure \u03b1 :=\n  ofMeasurable (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (_ : \u2191\u2191\u03bc \u2205 - \u2191\u2191\u03bd \u2205 = 0)\n    (_ :\n      \u2200 \u2983g : \u2115 \u2192 Set \u03b1\u2984 (h_meas : \u2200 (i : \u2115), MeasurableSet (g i)),\n        Pairwise (Disjoint on g) \u2192\n          (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (\u22c3 i, g i) (_ : MeasurableSet (\u22c3 b, g b)) =\n            \u2211' (i : \u2115), (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (g i) (_ : MeasurableSet (g i)))\nh_measure_sub_add : \u03bd + measure_sub = \u03bc\nd : Measure \u03b1\nh_d : d \u2208 {d | \u03bc \u2264 d + \u03bd}\n\u22a2 measure_sub \u2264 d", "state_after": "case a.a\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\ninst\u271d : IsFiniteMeasure \u03bd\nh\u2081 : MeasurableSet s\nh\u2082 : \u03bd \u2264 \u03bc\nmeasure_sub : Measure \u03b1 :=\n  ofMeasurable (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (_ : \u2191\u2191\u03bc \u2205 - \u2191\u2191\u03bd \u2205 = 0)\n    (_ :\n      \u2200 \u2983g : \u2115 \u2192 Set \u03b1\u2984 (h_meas : \u2200 (i : \u2115), MeasurableSet (g i)),\n        Pairwise (Disjoint on g) \u2192\n          (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (\u22c3 i, g i) (_ : MeasurableSet (\u22c3 b, g b)) =\n            \u2211' (i : \u2115), (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (g i) (_ : MeasurableSet (g i)))\nh_measure_sub_add : \u03bd + measure_sub = \u03bc\nd : Measure \u03b1\nh_d : \u03bd + measure_sub \u2264 \u03bd + d\n\u22a2 measure_sub \u2264 d"}, {"tactic": "apply Measure.le_of_add_le_add_left h_d", "annotated_tactic": ["apply <a>Measure.le_of_add_le_add_left</a> h_d", [{"full_name": "MeasureTheory.Measure.le_of_add_le_add_left", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2970, 9], "def_end_pos": [2970, 38]}]], "state_before": "case a.a\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\ninst\u271d : IsFiniteMeasure \u03bd\nh\u2081 : MeasurableSet s\nh\u2082 : \u03bd \u2264 \u03bc\nmeasure_sub : Measure \u03b1 :=\n  ofMeasurable (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (_ : \u2191\u2191\u03bc \u2205 - \u2191\u2191\u03bd \u2205 = 0)\n    (_ :\n      \u2200 \u2983g : \u2115 \u2192 Set \u03b1\u2984 (h_meas : \u2200 (i : \u2115), MeasurableSet (g i)),\n        Pairwise (Disjoint on g) \u2192\n          (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (\u22c3 i, g i) (_ : MeasurableSet (\u22c3 b, g b)) =\n            \u2211' (i : \u2115), (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (g i) (_ : MeasurableSet (g i)))\nh_measure_sub_add : \u03bd + measure_sub = \u03bc\nd : Measure \u03b1\nh_d : \u03bd + measure_sub \u2264 \u03bd + d\n\u22a2 measure_sub \u2264 d", "state_after": "no goals"}, {"tactic": "apply sInf_le", "annotated_tactic": ["apply <a>sInf_le</a>", [{"full_name": "sInf_le", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [265, 9], "def_end_pos": [265, 16]}]], "state_before": "case a\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\ninst\u271d : IsFiniteMeasure \u03bd\nh\u2081 : MeasurableSet s\nh\u2082 : \u03bd \u2264 \u03bc\nmeasure_sub : Measure \u03b1 :=\n  ofMeasurable (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (_ : \u2191\u2191\u03bc \u2205 - \u2191\u2191\u03bd \u2205 = 0)\n    (_ :\n      \u2200 \u2983g : \u2115 \u2192 Set \u03b1\u2984 (h_meas : \u2200 (i : \u2115), MeasurableSet (g i)),\n        Pairwise (Disjoint on g) \u2192\n          (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (\u22c3 i, g i) (_ : MeasurableSet (\u22c3 b, g b)) =\n            \u2211' (i : \u2115), (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (g i) (_ : MeasurableSet (g i)))\nh_measure_sub_add : \u03bd + measure_sub = \u03bc\n\u22a2 sInf {d | \u03bc \u2264 d + \u03bd} \u2264 measure_sub", "state_after": "case a.a\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\ninst\u271d : IsFiniteMeasure \u03bd\nh\u2081 : MeasurableSet s\nh\u2082 : \u03bd \u2264 \u03bc\nmeasure_sub : Measure \u03b1 :=\n  ofMeasurable (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (_ : \u2191\u2191\u03bc \u2205 - \u2191\u2191\u03bd \u2205 = 0)\n    (_ :\n      \u2200 \u2983g : \u2115 \u2192 Set \u03b1\u2984 (h_meas : \u2200 (i : \u2115), MeasurableSet (g i)),\n        Pairwise (Disjoint on g) \u2192\n          (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (\u22c3 i, g i) (_ : MeasurableSet (\u22c3 b, g b)) =\n            \u2211' (i : \u2115), (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (g i) (_ : MeasurableSet (g i)))\nh_measure_sub_add : \u03bd + measure_sub = \u03bc\n\u22a2 measure_sub \u2208 {d | \u03bc \u2264 d + \u03bd}"}, {"tactic": "simp [le_refl, add_comm, h_measure_sub_add]", "annotated_tactic": ["simp [<a>le_refl</a>, <a>add_comm</a>, h_measure_sub_add]", [{"full_name": "le_refl", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [50, 9], "def_end_pos": [50, 16]}, {"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [301, 3], "def_end_pos": [301, 14]}]], "state_before": "case a.a\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\ninst\u271d : IsFiniteMeasure \u03bd\nh\u2081 : MeasurableSet s\nh\u2082 : \u03bd \u2264 \u03bc\nmeasure_sub : Measure \u03b1 :=\n  ofMeasurable (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (_ : \u2191\u2191\u03bc \u2205 - \u2191\u2191\u03bd \u2205 = 0)\n    (_ :\n      \u2200 \u2983g : \u2115 \u2192 Set \u03b1\u2984 (h_meas : \u2200 (i : \u2115), MeasurableSet (g i)),\n        Pairwise (Disjoint on g) \u2192\n          (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (\u22c3 i, g i) (_ : MeasurableSet (\u22c3 b, g b)) =\n            \u2211' (i : \u2115), (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (g i) (_ : MeasurableSet (g i)))\nh_measure_sub_add : \u03bd + measure_sub = \u03bc\n\u22a2 measure_sub \u2208 {d | \u03bc \u2264 d + \u03bd}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean", "full_name": "MeasureTheory.AECover.inter_restrict", "start": [288, 1], "end": [292, 34], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/AEDisjoint.lean", "full_name": "MeasureTheory.AEDisjoint.of_null_left", "start": [151, 1], "end": [152, 36], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "full_name": "Nat.pow_right_comm", "start": [832, 11], "end": [833, 34], "traced_tactics": [{"tactic": "rw [\u2190Nat.pow_mul, Nat.pow_mul']", "annotated_tactic": ["rw [\u2190<a>Nat.pow_mul</a>, <a>Nat.pow_mul'</a>]", [{"full_name": "Nat.pow_mul", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [824, 19], "def_end_pos": [824, 26]}, {"full_name": "Nat.pow_mul'", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [829, 19], "def_end_pos": [829, 27]}]], "state_before": "a m n : Nat\n\u22a2 (a ^ m) ^ n = (a ^ n) ^ m", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Decomposition/Jordan.lean", "full_name": "MeasureTheory.SignedMeasure.absolutelyContinuous_ennreal_iff", "start": [516, 1], "end": [529, 48], "traced_tactics": [{"tactic": "constructor <;> intro h", "annotated_tactic": ["constructor <;> intro h", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc : VectorMeasure \u03b1 \u211d\u22650\u221e\n\u22a2 s \u226a\u1d65 \u03bc \u2194 totalVariation s \u226a VectorMeasure.ennrealToMeasure \u03bc", "state_after": "case mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc : VectorMeasure \u03b1 \u211d\u22650\u221e\nh : s \u226a\u1d65 \u03bc\n\u22a2 totalVariation s \u226a VectorMeasure.ennrealToMeasure \u03bc\n\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc : VectorMeasure \u03b1 \u211d\u22650\u221e\nh : totalVariation s \u226a VectorMeasure.ennrealToMeasure \u03bc\n\u22a2 s \u226a\u1d65 \u03bc"}, {"tactic": "refine' Measure.AbsolutelyContinuous.mk fun S hS\u2081 hS\u2082 => _", "annotated_tactic": ["refine' <a>Measure.AbsolutelyContinuous.mk</a> fun S hS\u2081 hS\u2082 => _", [{"full_name": "MeasureTheory.Measure.AbsolutelyContinuous.mk", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2135, 9], "def_end_pos": [2135, 11]}]], "state_before": "case mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc : VectorMeasure \u03b1 \u211d\u22650\u221e\nh : s \u226a\u1d65 \u03bc\n\u22a2 totalVariation s \u226a VectorMeasure.ennrealToMeasure \u03bc", "state_after": "case mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc : VectorMeasure \u03b1 \u211d\u22650\u221e\nh : s \u226a\u1d65 \u03bc\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : \u2191\u2191(VectorMeasure.ennrealToMeasure \u03bc) S = 0\n\u22a2 \u2191\u2191(totalVariation s) S = 0"}, {"tactic": "obtain \u27e8i, hi\u2081, hi\u2082, hi\u2083, hpos, hneg\u27e9 := s.toJordanDecomposition_spec", "annotated_tactic": ["obtain \u27e8i, hi\u2081, hi\u2082, hi\u2083, hpos, hneg\u27e9 := s.toJordanDecomposition_spec", []], "state_before": "case mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc : VectorMeasure \u03b1 \u211d\u22650\u221e\nh : s \u226a\u1d65 \u03bc\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : \u2191\u2191(VectorMeasure.ennrealToMeasure \u03bc) S = 0\n\u22a2 \u2191\u2191(totalVariation s) S = 0", "state_after": "case mp.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc : VectorMeasure \u03b1 \u211d\u22650\u221e\nh : s \u226a\u1d65 \u03bc\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : \u2191\u2191(VectorMeasure.ennrealToMeasure \u03bc) S = 0\ni : Set \u03b1\nhi\u2081 : MeasurableSet i\nhi\u2082 : VectorMeasure.restrict 0 i \u2264 VectorMeasure.restrict s i\nhi\u2083 : VectorMeasure.restrict s i\u1d9c \u2264 VectorMeasure.restrict 0 i\u1d9c\nhpos : (toJordanDecomposition s).posPart = toMeasureOfZeroLE s i hi\u2081 hi\u2082\nhneg : (toJordanDecomposition s).negPart = toMeasureOfLEZero s i\u1d9c (_ : MeasurableSet i\u1d9c) hi\u2083\n\u22a2 \u2191\u2191(totalVariation s) S = 0"}, {"tactic": "rw [totalVariation, Measure.add_apply, hpos, hneg, toMeasureOfZeroLE_apply _ _ _ hS\u2081,\n  toMeasureOfLEZero_apply _ _ _ hS\u2081]", "annotated_tactic": ["rw [<a>totalVariation</a>, <a>Measure.add_apply</a>, hpos, hneg, <a>toMeasureOfZeroLE_apply</a> _ _ _ hS\u2081,\n      <a>toMeasureOfLEZero_apply</a> _ _ _ hS\u2081]", [{"full_name": "MeasureTheory.SignedMeasure.totalVariation", "def_path": "Mathlib/MeasureTheory/Decomposition/Jordan.lean", "def_pos": [494, 5], "def_end_pos": [494, 19]}, {"full_name": "MeasureTheory.Measure.add_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [798, 9], "def_end_pos": [798, 18]}, {"full_name": "MeasureTheory.SignedMeasure.toMeasureOfZeroLE_apply", "def_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "def_pos": [1349, 9], "def_end_pos": [1349, 32]}, {"full_name": "MeasureTheory.SignedMeasure.toMeasureOfLEZero_apply", "def_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "def_pos": [1363, 9], "def_end_pos": [1363, 32]}]], "state_before": "case mp.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc : VectorMeasure \u03b1 \u211d\u22650\u221e\nh : s \u226a\u1d65 \u03bc\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : \u2191\u2191(VectorMeasure.ennrealToMeasure \u03bc) S = 0\ni : Set \u03b1\nhi\u2081 : MeasurableSet i\nhi\u2082 : VectorMeasure.restrict 0 i \u2264 VectorMeasure.restrict s i\nhi\u2083 : VectorMeasure.restrict s i\u1d9c \u2264 VectorMeasure.restrict 0 i\u1d9c\nhpos : (toJordanDecomposition s).posPart = toMeasureOfZeroLE s i hi\u2081 hi\u2082\nhneg : (toJordanDecomposition s).negPart = toMeasureOfLEZero s i\u1d9c (_ : MeasurableSet i\u1d9c) hi\u2083\n\u22a2 \u2191\u2191(totalVariation s) S = 0", "state_after": "case mp.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc : VectorMeasure \u03b1 \u211d\u22650\u221e\nh : s \u226a\u1d65 \u03bc\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : \u2191\u2191(VectorMeasure.ennrealToMeasure \u03bc) S = 0\ni : Set \u03b1\nhi\u2081 : MeasurableSet i\nhi\u2082 : VectorMeasure.restrict 0 i \u2264 VectorMeasure.restrict s i\nhi\u2083 : VectorMeasure.restrict s i\u1d9c \u2264 VectorMeasure.restrict 0 i\u1d9c\nhpos : (toJordanDecomposition s).posPart = toMeasureOfZeroLE s i hi\u2081 hi\u2082\nhneg : (toJordanDecomposition s).negPart = toMeasureOfLEZero s i\u1d9c (_ : MeasurableSet i\u1d9c) hi\u2083\n\u22a2 \u2191{ val := \u2191s (i \u2229 S), property := (_ : 0 \u2264 \u2191s (i \u2229 S)) } +\n      \u2191{ val := -\u2191s (i\u1d9c \u2229 S), property := (_ : 0 \u2264 -\u2191s (i\u1d9c \u2229 S)) } =\n    0"}, {"tactic": "rw [\u2190 VectorMeasure.AbsolutelyContinuous.ennrealToMeasure] at h", "annotated_tactic": ["rw [\u2190 <a>VectorMeasure.AbsolutelyContinuous.ennrealToMeasure</a>] at h", [{"full_name": "MeasureTheory.VectorMeasure.AbsolutelyContinuous.ennrealToMeasure", "def_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "def_pos": [1144, 9], "def_end_pos": [1144, 25]}]], "state_before": "case mp.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc : VectorMeasure \u03b1 \u211d\u22650\u221e\nh : s \u226a\u1d65 \u03bc\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : \u2191\u2191(VectorMeasure.ennrealToMeasure \u03bc) S = 0\ni : Set \u03b1\nhi\u2081 : MeasurableSet i\nhi\u2082 : VectorMeasure.restrict 0 i \u2264 VectorMeasure.restrict s i\nhi\u2083 : VectorMeasure.restrict s i\u1d9c \u2264 VectorMeasure.restrict 0 i\u1d9c\nhpos : (toJordanDecomposition s).posPart = toMeasureOfZeroLE s i hi\u2081 hi\u2082\nhneg : (toJordanDecomposition s).negPart = toMeasureOfLEZero s i\u1d9c (_ : MeasurableSet i\u1d9c) hi\u2083\n\u22a2 \u2191{ val := \u2191s (i \u2229 S), property := (_ : 0 \u2264 \u2191s (i \u2229 S)) } +\n      \u2191{ val := -\u2191s (i\u1d9c \u2229 S), property := (_ : 0 \u2264 -\u2191s (i\u1d9c \u2229 S)) } =\n    0", "state_after": "case mp.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc : VectorMeasure \u03b1 \u211d\u22650\u221e\nh : \u2200 \u2983s_1 : Set \u03b1\u2984, \u2191\u2191(VectorMeasure.ennrealToMeasure \u03bc) s_1 = 0 \u2192 \u2191s s_1 = 0\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : \u2191\u2191(VectorMeasure.ennrealToMeasure \u03bc) S = 0\ni : Set \u03b1\nhi\u2081 : MeasurableSet i\nhi\u2082 : VectorMeasure.restrict 0 i \u2264 VectorMeasure.restrict s i\nhi\u2083 : VectorMeasure.restrict s i\u1d9c \u2264 VectorMeasure.restrict 0 i\u1d9c\nhpos : (toJordanDecomposition s).posPart = toMeasureOfZeroLE s i hi\u2081 hi\u2082\nhneg : (toJordanDecomposition s).negPart = toMeasureOfLEZero s i\u1d9c (_ : MeasurableSet i\u1d9c) hi\u2083\n\u22a2 \u2191{ val := \u2191s (i \u2229 S), property := (_ : 0 \u2264 \u2191s (i \u2229 S)) } +\n      \u2191{ val := -\u2191s (i\u1d9c \u2229 S), property := (_ : 0 \u2264 -\u2191s (i\u1d9c \u2229 S)) } =\n    0"}, {"tactic": "simp [h (measure_mono_null (i.inter_subset_right S) hS\u2082),\n  h (measure_mono_null (i\u1d9c.inter_subset_right S) hS\u2082), \u2190 NNReal.eq_iff]", "annotated_tactic": ["simp [h (<a>measure_mono_null</a> (i.inter_subset_right S) hS\u2082),\n      h (<a>measure_mono_null</a> (i\u1d9c.<a>inter_subset_right</a> S) hS\u2082), \u2190 <a>NNReal.eq_iff</a>]", [{"full_name": "MeasureTheory.measure_mono_null", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [197, 9], "def_end_pos": [197, 26]}, {"full_name": "MeasureTheory.measure_mono_null", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [197, 9], "def_end_pos": [197, 26]}, {"full_name": "Set.inter_subset_right", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [969, 9], "def_end_pos": [969, 27]}, {"full_name": "NNReal.eq_iff", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [99, 19], "def_end_pos": [99, 25]}]], "state_before": "case mp.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc : VectorMeasure \u03b1 \u211d\u22650\u221e\nh : \u2200 \u2983s_1 : Set \u03b1\u2984, \u2191\u2191(VectorMeasure.ennrealToMeasure \u03bc) s_1 = 0 \u2192 \u2191s s_1 = 0\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : \u2191\u2191(VectorMeasure.ennrealToMeasure \u03bc) S = 0\ni : Set \u03b1\nhi\u2081 : MeasurableSet i\nhi\u2082 : VectorMeasure.restrict 0 i \u2264 VectorMeasure.restrict s i\nhi\u2083 : VectorMeasure.restrict s i\u1d9c \u2264 VectorMeasure.restrict 0 i\u1d9c\nhpos : (toJordanDecomposition s).posPart = toMeasureOfZeroLE s i hi\u2081 hi\u2082\nhneg : (toJordanDecomposition s).negPart = toMeasureOfLEZero s i\u1d9c (_ : MeasurableSet i\u1d9c) hi\u2083\n\u22a2 \u2191{ val := \u2191s (i \u2229 S), property := (_ : 0 \u2264 \u2191s (i \u2229 S)) } +\n      \u2191{ val := -\u2191s (i\u1d9c \u2229 S), property := (_ : 0 \u2264 -\u2191s (i\u1d9c \u2229 S)) } =\n    0", "state_after": "no goals"}, {"tactic": "refine' VectorMeasure.AbsolutelyContinuous.mk fun S hS\u2081 hS\u2082 => _", "annotated_tactic": ["refine' <a>VectorMeasure.AbsolutelyContinuous.mk</a> fun S hS\u2081 hS\u2082 => _", [{"full_name": "MeasureTheory.VectorMeasure.AbsolutelyContinuous.mk", "def_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "def_pos": [1077, 9], "def_end_pos": [1077, 11]}]], "state_before": "case mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc : VectorMeasure \u03b1 \u211d\u22650\u221e\nh : totalVariation s \u226a VectorMeasure.ennrealToMeasure \u03bc\n\u22a2 s \u226a\u1d65 \u03bc", "state_after": "case mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc : VectorMeasure \u03b1 \u211d\u22650\u221e\nh : totalVariation s \u226a VectorMeasure.ennrealToMeasure \u03bc\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : \u2191\u03bc S = 0\n\u22a2 \u2191s S = 0"}, {"tactic": "rw [\u2190 VectorMeasure.ennrealToMeasure_apply hS\u2081] at hS\u2082", "annotated_tactic": ["rw [\u2190 <a>VectorMeasure.ennrealToMeasure_apply</a> hS\u2081] at hS\u2082", [{"full_name": "MeasureTheory.VectorMeasure.ennrealToMeasure_apply", "def_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "def_pos": [534, 9], "def_end_pos": [534, 31]}]], "state_before": "case mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc : VectorMeasure \u03b1 \u211d\u22650\u221e\nh : totalVariation s \u226a VectorMeasure.ennrealToMeasure \u03bc\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : \u2191\u03bc S = 0\n\u22a2 \u2191s S = 0", "state_after": "case mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc : VectorMeasure \u03b1 \u211d\u22650\u221e\nh : totalVariation s \u226a VectorMeasure.ennrealToMeasure \u03bc\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : \u2191\u2191(VectorMeasure.ennrealToMeasure \u03bc) S = 0\n\u22a2 \u2191s S = 0"}, {"tactic": "exact null_of_totalVariation_zero s (h hS\u2082)", "annotated_tactic": ["exact <a>null_of_totalVariation_zero</a> s (h hS\u2082)", [{"full_name": "MeasureTheory.SignedMeasure.null_of_totalVariation_zero", "def_path": "Mathlib/MeasureTheory/Decomposition/Jordan.lean", "def_pos": [506, 9], "def_end_pos": [506, 36]}]], "state_before": "case mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc : VectorMeasure \u03b1 \u211d\u22650\u221e\nh : totalVariation s \u226a VectorMeasure.ennrealToMeasure \u03bc\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : \u2191\u2191(VectorMeasure.ennrealToMeasure \u03bc) S = 0\n\u22a2 \u2191s S = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Integration.lean", "full_name": "ProbabilityTheory.lintegral_mul_eq_lintegral_mul_lintegral_of_indepFun'", "start": [120, 1], "end": [128, 62], "traced_tactics": [{"tactic": "have fg_ae : f * g =\u1d50[\u03bc] h_meas_f.mk _ * h_meas_g.mk _ := h_meas_f.ae_eq_mk.mul h_meas_g.ae_eq_mk", "annotated_tactic": ["have fg_ae : f * g =\u1d50[\u03bc] h_meas_f.mk _ * h_meas_g.mk _ := h_meas_f.ae_eq_mk.mul h_meas_g.ae_eq_mk", []], "state_before": "\u03a9 : Type u_1\nm\u03a9 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nf g : \u03a9 \u2192 \u211d\u22650\u221e\nX Y : \u03a9 \u2192 \u211d\nh_meas_f : AEMeasurable f\nh_meas_g : AEMeasurable g\nh_indep_fun : IndepFun f g\n\u22a2 \u222b\u207b (\u03c9 : \u03a9), (f * g) \u03c9 \u2202\u03bc = (\u222b\u207b (\u03c9 : \u03a9), f \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), g \u03c9 \u2202\u03bc", "state_after": "\u03a9 : Type u_1\nm\u03a9 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nf g : \u03a9 \u2192 \u211d\u22650\u221e\nX Y : \u03a9 \u2192 \u211d\nh_meas_f : AEMeasurable f\nh_meas_g : AEMeasurable g\nh_indep_fun : IndepFun f g\nfg_ae : f * g =\u1d50[\u03bc] AEMeasurable.mk f h_meas_f * AEMeasurable.mk g h_meas_g\n\u22a2 \u222b\u207b (\u03c9 : \u03a9), (f * g) \u03c9 \u2202\u03bc = (\u222b\u207b (\u03c9 : \u03a9), f \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), g \u03c9 \u2202\u03bc"}, {"tactic": "rw [lintegral_congr_ae h_meas_f.ae_eq_mk, lintegral_congr_ae h_meas_g.ae_eq_mk,\n  lintegral_congr_ae fg_ae]", "annotated_tactic": ["rw [<a>lintegral_congr_ae</a> h_meas_f.ae_eq_mk, <a>lintegral_congr_ae</a> h_meas_g.ae_eq_mk,\n    <a>lintegral_congr_ae</a> fg_ae]", [{"full_name": "MeasureTheory.lintegral_congr_ae", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [304, 9], "def_end_pos": [304, 27]}, {"full_name": "MeasureTheory.lintegral_congr_ae", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [304, 9], "def_end_pos": [304, 27]}, {"full_name": "MeasureTheory.lintegral_congr_ae", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [304, 9], "def_end_pos": [304, 27]}]], "state_before": "\u03a9 : Type u_1\nm\u03a9 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nf g : \u03a9 \u2192 \u211d\u22650\u221e\nX Y : \u03a9 \u2192 \u211d\nh_meas_f : AEMeasurable f\nh_meas_g : AEMeasurable g\nh_indep_fun : IndepFun f g\nfg_ae : f * g =\u1d50[\u03bc] AEMeasurable.mk f h_meas_f * AEMeasurable.mk g h_meas_g\n\u22a2 \u222b\u207b (\u03c9 : \u03a9), (f * g) \u03c9 \u2202\u03bc = (\u222b\u207b (\u03c9 : \u03a9), f \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), g \u03c9 \u2202\u03bc", "state_after": "\u03a9 : Type u_1\nm\u03a9 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nf g : \u03a9 \u2192 \u211d\u22650\u221e\nX Y : \u03a9 \u2192 \u211d\nh_meas_f : AEMeasurable f\nh_meas_g : AEMeasurable g\nh_indep_fun : IndepFun f g\nfg_ae : f * g =\u1d50[\u03bc] AEMeasurable.mk f h_meas_f * AEMeasurable.mk g h_meas_g\n\u22a2 \u222b\u207b (a : \u03a9), (AEMeasurable.mk f h_meas_f * AEMeasurable.mk g h_meas_g) a \u2202\u03bc =\n    (\u222b\u207b (a : \u03a9), AEMeasurable.mk f h_meas_f a \u2202\u03bc) * \u222b\u207b (a : \u03a9), AEMeasurable.mk g h_meas_g a \u2202\u03bc"}, {"tactic": "apply lintegral_mul_eq_lintegral_mul_lintegral_of_indepFun h_meas_f.measurable_mk\n    h_meas_g.measurable_mk", "annotated_tactic": ["apply <a>lintegral_mul_eq_lintegral_mul_lintegral_of_indepFun</a> h_meas_f.measurable_mk\n      h_meas_g.measurable_mk", [{"full_name": "ProbabilityTheory.lintegral_mul_eq_lintegral_mul_lintegral_of_indepFun", "def_path": "Mathlib/Probability/Integration.lean", "def_pos": [109, 9], "def_end_pos": [109, 61]}]], "state_before": "\u03a9 : Type u_1\nm\u03a9 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nf g : \u03a9 \u2192 \u211d\u22650\u221e\nX Y : \u03a9 \u2192 \u211d\nh_meas_f : AEMeasurable f\nh_meas_g : AEMeasurable g\nh_indep_fun : IndepFun f g\nfg_ae : f * g =\u1d50[\u03bc] AEMeasurable.mk f h_meas_f * AEMeasurable.mk g h_meas_g\n\u22a2 \u222b\u207b (a : \u03a9), (AEMeasurable.mk f h_meas_f * AEMeasurable.mk g h_meas_g) a \u2202\u03bc =\n    (\u222b\u207b (a : \u03a9), AEMeasurable.mk f h_meas_f a \u2202\u03bc) * \u222b\u207b (a : \u03a9), AEMeasurable.mk g h_meas_g a \u2202\u03bc", "state_after": "\u03a9 : Type u_1\nm\u03a9 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nf g : \u03a9 \u2192 \u211d\u22650\u221e\nX Y : \u03a9 \u2192 \u211d\nh_meas_f : AEMeasurable f\nh_meas_g : AEMeasurable g\nh_indep_fun : IndepFun f g\nfg_ae : f * g =\u1d50[\u03bc] AEMeasurable.mk f h_meas_f * AEMeasurable.mk g h_meas_g\n\u22a2 IndepFun (AEMeasurable.mk f h_meas_f) (AEMeasurable.mk g h_meas_g)"}, {"tactic": "exact h_indep_fun.ae_eq h_meas_f.ae_eq_mk h_meas_g.ae_eq_mk", "annotated_tactic": ["exact h_indep_fun.ae_eq h_meas_f.ae_eq_mk h_meas_g.ae_eq_mk", []], "state_before": "\u03a9 : Type u_1\nm\u03a9 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nf g : \u03a9 \u2192 \u211d\u22650\u221e\nX Y : \u03a9 \u2192 \u211d\nh_meas_f : AEMeasurable f\nh_meas_g : AEMeasurable g\nh_indep_fun : IndepFun f g\nfg_ae : f * g =\u1d50[\u03bc] AEMeasurable.mk f h_meas_f * AEMeasurable.mk g h_meas_g\n\u22a2 IndepFun (AEMeasurable.mk f h_meas_f) (AEMeasurable.mk g h_meas_g)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Pointwise.lean", "full_name": "Finset.one_mem_div_iff", "start": [1112, 1], "end": [1113, 63], "traced_tactics": [{"tactic": "rw [\u2190 mem_coe, \u2190 disjoint_coe, coe_div, Set.one_mem_div_iff]", "annotated_tactic": ["rw [\u2190 <a>mem_coe</a>, \u2190 <a>disjoint_coe</a>, <a>coe_div</a>, <a>Set.one_mem_div_iff</a>]", [{"full_name": "Finset.mem_coe", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [208, 9], "def_end_pos": [208, 16]}, {"full_name": "Finset.disjoint_coe", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1004, 9], "def_end_pos": [1004, 21]}, {"full_name": "Finset.coe_div", "def_path": "Mathlib/Data/Finset/Pointwise.lean", "def_pos": [555, 9], "def_end_pos": [555, 16]}, {"full_name": "Set.one_mem_div_iff", "def_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "def_pos": [1162, 9], "def_end_pos": [1162, 24]}]], "state_before": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u2074 : DecidableEq \u03b1\ninst\u271d\u00b3 : DecidableEq \u03b2\ninst\u271d\u00b2 : Group \u03b1\ninst\u271d\u00b9 : DivisionMonoid \u03b2\ninst\u271d : MonoidHomClass F \u03b1 \u03b2\nf : F\ns t : Finset \u03b1\na b : \u03b1\n\u22a2 1 \u2208 s / t \u2194 \u00acDisjoint s t", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Variance.lean", "full_name": "ProbabilityTheory.evariance_def'", "start": [245, 1], "end": [262, 43], "traced_tactics": [{"tactic": "by_cases h\u2112 : Mem\u2112p X 2", "annotated_tactic": ["by_cases h\u2112 : <a>Mem\u2112p</a> X 2", [{"full_name": "MeasureTheory.Mem\u2112p", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [108, 5], "def_end_pos": [108, 10]}]], "state_before": "\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhX : AEStronglyMeasurable X \u2119\n\u22a2 evariance X \u2119 = (\u222b\u207b (\u03c9 : \u03a9), \u2191(\u2016X \u03c9\u2016\u208a ^ 2)) - ENNReal.ofReal ((\u222b (a : \u03a9), X a) ^ 2)", "state_after": "case pos\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhX : AEStronglyMeasurable X \u2119\nh\u2112 : Mem\u2112p X 2\n\u22a2 evariance X \u2119 = (\u222b\u207b (\u03c9 : \u03a9), \u2191(\u2016X \u03c9\u2016\u208a ^ 2)) - ENNReal.ofReal ((\u222b (a : \u03a9), X a) ^ 2)\n\ncase neg\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhX : AEStronglyMeasurable X \u2119\nh\u2112 : \u00acMem\u2112p X 2\n\u22a2 evariance X \u2119 = (\u222b\u207b (\u03c9 : \u03a9), \u2191(\u2016X \u03c9\u2016\u208a ^ 2)) - ENNReal.ofReal ((\u222b (a : \u03a9), X a) ^ 2)"}, {"tactic": "rw [\u2190 h\u2112.ofReal_variance_eq, variance_def' h\u2112, ENNReal.ofReal_sub _ (sq_nonneg _)]", "annotated_tactic": ["rw [\u2190 h\u2112.ofReal_variance_eq, <a>variance_def'</a> h\u2112, <a>ENNReal.ofReal_sub</a> _ (<a>sq_nonneg</a> _)]", [{"full_name": "ProbabilityTheory.variance_def'", "def_path": "Mathlib/Probability/Variance.lean", "def_pos": [210, 9], "def_end_pos": [210, 22]}, {"full_name": "ENNReal.ofReal_sub", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2181, 9], "def_end_pos": [2181, 19]}, {"full_name": "sq_nonneg", "def_path": "Mathlib/Algebra/GroupPower/Order.lean", "def_pos": [645, 9], "def_end_pos": [645, 18]}]], "state_before": "case pos\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhX : AEStronglyMeasurable X \u2119\nh\u2112 : Mem\u2112p X 2\n\u22a2 evariance X \u2119 = (\u222b\u207b (\u03c9 : \u03a9), \u2191(\u2016X \u03c9\u2016\u208a ^ 2)) - ENNReal.ofReal ((\u222b (a : \u03a9), X a) ^ 2)", "state_after": "case pos\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhX : AEStronglyMeasurable X \u2119\nh\u2112 : Mem\u2112p X 2\n\u22a2 ENNReal.ofReal (\u222b (a : \u03a9), (X ^ 2) a) - ENNReal.ofReal ((\u222b (a : \u03a9), X a) ^ 2) =\n    (\u222b\u207b (\u03c9 : \u03a9), \u2191(\u2016X \u03c9\u2016\u208a ^ 2)) - ENNReal.ofReal ((\u222b (a : \u03a9), X a) ^ 2)"}, {"tactic": "congr", "annotated_tactic": ["congr", []], "state_before": "case pos\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhX : AEStronglyMeasurable X \u2119\nh\u2112 : Mem\u2112p X 2\n\u22a2 ENNReal.ofReal (\u222b (a : \u03a9), (X ^ 2) a) - ENNReal.ofReal ((\u222b (a : \u03a9), X a) ^ 2) =\n    (\u222b\u207b (\u03c9 : \u03a9), \u2191(\u2016X \u03c9\u2016\u208a ^ 2)) - ENNReal.ofReal ((\u222b (a : \u03a9), X a) ^ 2)", "state_after": "case pos.e_a\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhX : AEStronglyMeasurable X \u2119\nh\u2112 : Mem\u2112p X 2\n\u22a2 ENNReal.ofReal (\u222b (a : \u03a9), (X ^ 2) a) = \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2016X \u03c9\u2016\u208a ^ 2)"}, {"tactic": "rw [lintegral_coe_eq_integral]", "annotated_tactic": ["rw [<a>lintegral_coe_eq_integral</a>]", [{"full_name": "MeasureTheory.lintegral_coe_eq_integral", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1208, 9], "def_end_pos": [1208, 34]}]], "state_before": "case pos.e_a\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhX : AEStronglyMeasurable X \u2119\nh\u2112 : Mem\u2112p X 2\n\u22a2 ENNReal.ofReal (\u222b (a : \u03a9), (X ^ 2) a) = \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2016X \u03c9\u2016\u208a ^ 2)", "state_after": "case pos.e_a\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhX : AEStronglyMeasurable X \u2119\nh\u2112 : Mem\u2112p X 2\n\u22a2 ENNReal.ofReal (\u222b (a : \u03a9), (X ^ 2) a) = ENNReal.ofReal (\u222b (a : \u03a9), \u2191(\u2016X a\u2016\u208a ^ 2))\n\ncase pos.e_a.hfi\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhX : AEStronglyMeasurable X \u2119\nh\u2112 : Mem\u2112p X 2\n\u22a2 Integrable fun x => \u2191(\u2016X x\u2016\u208a ^ 2)"}, {"tactic": "congr 2 with \u03c9", "annotated_tactic": ["congr 2 with \u03c9", []], "state_before": "case pos.e_a\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhX : AEStronglyMeasurable X \u2119\nh\u2112 : Mem\u2112p X 2\n\u22a2 ENNReal.ofReal (\u222b (a : \u03a9), (X ^ 2) a) = ENNReal.ofReal (\u222b (a : \u03a9), \u2191(\u2016X a\u2016\u208a ^ 2))", "state_after": "case pos.e_a.e_r.e_f.h\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhX : AEStronglyMeasurable X \u2119\nh\u2112 : Mem\u2112p X 2\n\u03c9 : \u03a9\n\u22a2 (X ^ 2) \u03c9 = \u2191(\u2016X \u03c9\u2016\u208a ^ 2)"}, {"tactic": "simp only [Pi.pow_apply, NNReal.coe_pow, coe_nnnorm, Real.norm_eq_abs, Even.pow_abs even_two]", "annotated_tactic": ["simp only [<a>Pi.pow_apply</a>, <a>NNReal.coe_pow</a>, <a>coe_nnnorm</a>, <a>Real.norm_eq_abs</a>, <a>Even.pow_abs</a> <a>even_two</a>]", [{"full_name": "Pi.pow_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [117, 9], "def_end_pos": [117, 18]}, {"full_name": "NNReal.coe_pow", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [298, 9], "def_end_pos": [298, 16]}, {"full_name": "coe_nnnorm", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [905, 41], "def_end_pos": [905, 51]}, {"full_name": "Real.norm_eq_abs", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [1761, 9], "def_end_pos": [1761, 20]}, {"full_name": "Even.pow_abs", "def_path": "Mathlib/Algebra/Parity.lean", "def_pos": [521, 9], "def_end_pos": [521, 21]}, {"full_name": "even_two", "def_path": "Mathlib/Algebra/Parity.lean", "def_pos": [274, 9], "def_end_pos": [274, 17]}]], "state_before": "case pos.e_a.e_r.e_f.h\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhX : AEStronglyMeasurable X \u2119\nh\u2112 : Mem\u2112p X 2\n\u03c9 : \u03a9\n\u22a2 (X ^ 2) \u03c9 = \u2191(\u2016X \u03c9\u2016\u208a ^ 2)", "state_after": "no goals"}, {"tactic": "exact h\u2112.abs.integrable_sq", "annotated_tactic": ["exact h\u2112.abs.integrable_sq", []], "state_before": "case pos.e_a.hfi\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhX : AEStronglyMeasurable X \u2119\nh\u2112 : Mem\u2112p X 2\n\u22a2 Integrable fun x => \u2191(\u2016X x\u2016\u208a ^ 2)", "state_after": "no goals"}, {"tactic": "symm", "annotated_tactic": ["symm", []], "state_before": "case neg\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhX : AEStronglyMeasurable X \u2119\nh\u2112 : \u00acMem\u2112p X 2\n\u22a2 evariance X \u2119 = (\u222b\u207b (\u03c9 : \u03a9), \u2191(\u2016X \u03c9\u2016\u208a ^ 2)) - ENNReal.ofReal ((\u222b (a : \u03a9), X a) ^ 2)", "state_after": "case neg\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhX : AEStronglyMeasurable X \u2119\nh\u2112 : \u00acMem\u2112p X 2\n\u22a2 (\u222b\u207b (\u03c9 : \u03a9), \u2191(\u2016X \u03c9\u2016\u208a ^ 2)) - ENNReal.ofReal ((\u222b (a : \u03a9), X a) ^ 2) = evariance X \u2119"}, {"tactic": "rw [evariance_eq_top hX h\u2112, ENNReal.sub_eq_top_iff]", "annotated_tactic": ["rw [<a>evariance_eq_top</a> hX h\u2112, <a>ENNReal.sub_eq_top_iff</a>]", [{"full_name": "ProbabilityTheory.evariance_eq_top", "def_path": "Mathlib/Probability/Variance.lean", "def_pos": [77, 9], "def_end_pos": [77, 25]}, {"full_name": "ENNReal.sub_eq_top_iff", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1149, 17], "def_end_pos": [1149, 31]}]], "state_before": "case neg\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhX : AEStronglyMeasurable X \u2119\nh\u2112 : \u00acMem\u2112p X 2\n\u22a2 (\u222b\u207b (\u03c9 : \u03a9), \u2191(\u2016X \u03c9\u2016\u208a ^ 2)) - ENNReal.ofReal ((\u222b (a : \u03a9), X a) ^ 2) = evariance X \u2119", "state_after": "case neg\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhX : AEStronglyMeasurable X \u2119\nh\u2112 : \u00acMem\u2112p X 2\n\u22a2 \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2016X \u03c9\u2016\u208a ^ 2) = \u22a4 \u2227 ENNReal.ofReal ((\u222b (a : \u03a9), X a) ^ 2) \u2260 \u22a4"}, {"tactic": "refine' \u27e8_, ENNReal.ofReal_ne_top\u27e9", "annotated_tactic": ["refine' \u27e8_, <a>ENNReal.ofReal_ne_top</a>\u27e9", [{"full_name": "ENNReal.ofReal_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [311, 17], "def_end_pos": [311, 30]}]], "state_before": "case neg\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhX : AEStronglyMeasurable X \u2119\nh\u2112 : \u00acMem\u2112p X 2\n\u22a2 \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2016X \u03c9\u2016\u208a ^ 2) = \u22a4 \u2227 ENNReal.ofReal ((\u222b (a : \u03a9), X a) ^ 2) \u2260 \u22a4", "state_after": "case neg\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhX : AEStronglyMeasurable X \u2119\nh\u2112 : \u00acMem\u2112p X 2\n\u22a2 \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2016X \u03c9\u2016\u208a ^ 2) = \u22a4"}, {"tactic": "rw [Mem\u2112p, not_and] at h\u2112", "annotated_tactic": ["rw [<a>Mem\u2112p</a>, <a>not_and</a>] at h\u2112", [{"full_name": "MeasureTheory.Mem\u2112p", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [108, 5], "def_end_pos": [108, 10]}, {"full_name": "not_and", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [316, 17], "def_end_pos": [316, 24]}]], "state_before": "case neg\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhX : AEStronglyMeasurable X \u2119\nh\u2112 : \u00acMem\u2112p X 2\n\u22a2 \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2016X \u03c9\u2016\u208a ^ 2) = \u22a4", "state_after": "case neg\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhX : AEStronglyMeasurable X \u2119\nh\u2112 : AEStronglyMeasurable X \u2119 \u2192 \u00acsnorm X 2 \u2119 < \u22a4\n\u22a2 \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2016X \u03c9\u2016\u208a ^ 2) = \u22a4"}, {"tactic": "specialize h\u2112 hX", "annotated_tactic": ["specialize h\u2112 hX", []], "state_before": "case neg\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhX : AEStronglyMeasurable X \u2119\nh\u2112 : AEStronglyMeasurable X \u2119 \u2192 \u00acsnorm X 2 \u2119 < \u22a4\n\u22a2 \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2016X \u03c9\u2016\u208a ^ 2) = \u22a4", "state_after": "case neg\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhX : AEStronglyMeasurable X \u2119\nh\u2112 : \u00acsnorm X 2 \u2119 < \u22a4\n\u22a2 \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2016X \u03c9\u2016\u208a ^ 2) = \u22a4"}, {"tactic": "simp only [snorm_eq_lintegral_rpow_nnnorm two_ne_zero ENNReal.two_ne_top, not_lt, top_le_iff,\n  coe_two, one_div, ENNReal.rpow_eq_top_iff, inv_lt_zero, inv_pos, and_true_iff,\n  or_iff_not_imp_left, not_and_or, zero_lt_two] at h\u2112", "annotated_tactic": ["simp only [<a>snorm_eq_lintegral_rpow_nnnorm</a> <a>two_ne_zero</a> <a>ENNReal.two_ne_top</a>, <a>not_lt</a>, <a>top_le_iff</a>,\n      <a>coe_two</a>, <a>one_div</a>, <a>ENNReal.rpow_eq_top_iff</a>, <a>inv_lt_zero</a>, <a>inv_pos</a>, <a>and_true_iff</a>,\n      <a>or_iff_not_imp_left</a>, <a>not_and_or</a>, <a>zero_lt_two</a>] at h\u2112", [{"full_name": "MeasureTheory.snorm_eq_lintegral_rpow_nnnorm", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [92, 9], "def_end_pos": [92, 39]}, {"full_name": "two_ne_zero", "def_path": "Mathlib/Algebra/NeZero.lean", "def_pos": [62, 7], "def_end_pos": [62, 18]}, {"full_name": "ENNReal.two_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [431, 9], "def_end_pos": [431, 19]}, {"full_name": "not_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [368, 9], "def_end_pos": [368, 15]}, {"full_name": "top_le_iff", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [157, 9], "def_end_pos": [157, 19]}, {"full_name": "_private.Mathlib.Probability.Variance.0.ProbabilityTheory.coe_two", "def_path": "Mathlib/Probability/Variance.lean", "def_pos": [49, 15], "def_end_pos": [49, 22]}, {"full_name": "one_div", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [318, 9], "def_end_pos": [318, 16]}, {"full_name": "ENNReal.rpow_eq_top_iff", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [476, 9], "def_end_pos": [476, 24]}, {"full_name": "inv_lt_zero", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [66, 9], "def_end_pos": [66, 20]}, {"full_name": "inv_pos", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [49, 9], "def_end_pos": [49, 16]}, {"full_name": "and_true_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [145, 9], "def_end_pos": [145, 21]}, {"full_name": "or_iff_not_imp_left", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [360, 9], "def_end_pos": [360, 28]}, {"full_name": "not_and_or", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [473, 9], "def_end_pos": [473, 19]}, {"full_name": "zero_lt_two", "def_path": "Mathlib/Algebra/Order/Monoid/NatCast.lean", "def_pos": [71, 15], "def_end_pos": [71, 26]}]], "state_before": "case neg\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhX : AEStronglyMeasurable X \u2119\nh\u2112 : \u00acsnorm X 2 \u2119 < \u22a4\n\u22a2 \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2016X \u03c9\u2016\u208a ^ 2) = \u22a4", "state_after": "case neg\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhX : AEStronglyMeasurable X \u2119\nh\u2112 : (\u00ac\u00ac\u222b\u207b (x : \u03a9), \u2191\u2016X x\u2016\u208a ^ 2 = 0 \u2192 0 \u2264 2) \u2192 \u222b\u207b (x : \u03a9), \u2191\u2016X x\u2016\u208a ^ 2 = \u22a4\n\u22a2 \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2016X \u03c9\u2016\u208a ^ 2) = \u22a4"}, {"tactic": "exact_mod_cast h\u2112 fun _ => zero_le_two", "annotated_tactic": ["exact_mod_cast h\u2112 fun _ => <a>zero_le_two</a>", [{"full_name": "zero_le_two", "def_path": "Mathlib/Algebra/Order/Monoid/NatCast.lean", "def_pos": [32, 7], "def_end_pos": [32, 18]}]], "state_before": "case neg\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhX : AEStronglyMeasurable X \u2119\nh\u2112 : (\u00ac\u00ac\u222b\u207b (x : \u03a9), \u2191\u2016X x\u2016\u208a ^ 2 = 0 \u2192 0 \u2264 2) \u2192 \u222b\u207b (x : \u03a9), \u2191\u2016X x\u2016\u208a ^ 2 = \u22a4\n\u22a2 \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2016X \u03c9\u2016\u208a ^ 2) = \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Group/FundamentalDomain.lean", "full_name": "MeasureTheory.IsFundamentalDomain.measure_zero_of_invariant", "start": [307, 1], "end": [309, 41], "traced_tactics": [{"tactic": "rw [measure_eq_tsum h]", "annotated_tactic": ["rw [<a>measure_eq_tsum</a> h]", [{"full_name": "MeasureTheory.IsFundamentalDomain.measure_eq_tsum", "def_path": "Mathlib/MeasureTheory/Group/FundamentalDomain.lean", "def_pos": [300, 9], "def_end_pos": [300, 24]}]], "state_before": "G : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u2070 : Group G\ninst\u271d\u2079 : Group H\ninst\u271d\u2078 : MulAction G \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MulAction H \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : NormedAddCommGroup E\ns t\u271d : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : MeasurableSpace G\ninst\u271d\u00b2 : MeasurableSMul G \u03b1\ninst\u271d\u00b9 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d : Countable G\n\u03bd : Measure \u03b1\nh : IsFundamentalDomain G s\nt : Set \u03b1\nht : \u2200 (g : G), g \u2022 t = t\nhts : \u2191\u2191\u03bc (t \u2229 s) = 0\n\u22a2 \u2191\u2191\u03bc t = 0", "state_after": "G : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u2070 : Group G\ninst\u271d\u2079 : Group H\ninst\u271d\u2078 : MulAction G \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MulAction H \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : NormedAddCommGroup E\ns t\u271d : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : MeasurableSpace G\ninst\u271d\u00b2 : MeasurableSMul G \u03b1\ninst\u271d\u00b9 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d : Countable G\n\u03bd : Measure \u03b1\nh : IsFundamentalDomain G s\nt : Set \u03b1\nht : \u2200 (g : G), g \u2022 t = t\nhts : \u2191\u2191\u03bc (t \u2229 s) = 0\n\u22a2 \u2211' (g : G), \u2191\u2191\u03bc (g \u2022 t \u2229 s) = 0"}, {"tactic": "simp [ht, hts]", "annotated_tactic": ["simp [ht, hts]", []], "state_before": "G : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u2070 : Group G\ninst\u271d\u2079 : Group H\ninst\u271d\u2078 : MulAction G \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MulAction H \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : NormedAddCommGroup E\ns t\u271d : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : MeasurableSpace G\ninst\u271d\u00b2 : MeasurableSMul G \u03b1\ninst\u271d\u00b9 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d : Countable G\n\u03bd : Measure \u03b1\nh : IsFundamentalDomain G s\nt : Set \u03b1\nht : \u2200 (g : G), g \u2022 t = t\nhts : \u2191\u2191\u03bc (t \u2229 s) = 0\n\u22a2 \u2211' (g : G), \u2191\u2191\u03bc (g \u2022 t \u2229 s) = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Int/GCD.lean", "full_name": "Int.lcm_self", "start": [479, 1], "end": [481, 21], "traced_tactics": [{"tactic": "rw [Int.lcm]", "annotated_tactic": ["rw [<a>Int.lcm</a>]", [{"full_name": "Int.lcm", "def_path": "Mathlib/Data/Int/GCD.lean", "def_pos": [235, 5], "def_end_pos": [235, 8]}]], "state_before": "i : \u2124\n\u22a2 lcm i i = natAbs i", "state_after": "i : \u2124\n\u22a2 Nat.lcm (natAbs i) (natAbs i) = natAbs i"}, {"tactic": "apply Nat.lcm_self", "annotated_tactic": ["apply <a>Nat.lcm_self</a>", [{"full_name": "Nat.lcm_self", "def_path": "lake-packages/std/Std/Data/Nat/Gcd.lean", "def_pos": [207, 17], "def_end_pos": [207, 25]}]], "state_before": "i : \u2124\n\u22a2 Nat.lcm (natAbs i) (natAbs i) = natAbs i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/Variables.lean", "full_name": "MvPolynomial.vars_map", "start": [454, 1], "end": [454, 88], "traced_tactics": [{"tactic": "classical simp [vars_def, degrees_map]", "annotated_tactic": ["classical simp [<a>vars_def</a>, <a>degrees_map</a>]", [{"full_name": "MvPolynomial.vars_def", "def_path": "Mathlib/Data/MvPolynomial/Variables.lean", "def_pos": [279, 9], "def_end_pos": [279, 17]}, {"full_name": "MvPolynomial.degrees_map", "def_path": "Mathlib/Data/MvPolynomial/Variables.lean", "def_pos": [226, 9], "def_end_pos": [226, 20]}]], "state_before": "R : Type u\nS : Type v\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nr : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u00b9 : CommSemiring R\np q : MvPolynomial \u03c3 R\ninst\u271d : CommSemiring S\nf : R \u2192+* S\n\u22a2 vars (\u2191(map f) p) \u2286 vars p", "state_after": "no goals"}, {"tactic": "simp [vars_def, degrees_map]", "annotated_tactic": ["simp [<a>vars_def</a>, <a>degrees_map</a>]", [{"full_name": "MvPolynomial.vars_def", "def_path": "Mathlib/Data/MvPolynomial/Variables.lean", "def_pos": [279, 9], "def_end_pos": [279, 17]}, {"full_name": "MvPolynomial.degrees_map", "def_path": "Mathlib/Data/MvPolynomial/Variables.lean", "def_pos": [226, 9], "def_end_pos": [226, 20]}]], "state_before": "R : Type u\nS : Type v\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nr : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u00b9 : CommSemiring R\np q : MvPolynomial \u03c3 R\ninst\u271d : CommSemiring S\nf : R \u2192+* S\n\u22a2 vars (\u2191(map f) p) \u2286 vars p", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "full_name": "MeasureTheory.AEStronglyMeasurable.aestronglyMeasurable_uIoc_iff", "start": [1759, 1], "end": [1764, 56], "traced_tactics": [{"tactic": "rw [uIoc_eq_union, aestronglyMeasurable_union_iff]", "annotated_tactic": ["rw [<a>uIoc_eq_union</a>, <a>aestronglyMeasurable_union_iff</a>]", [{"full_name": "Set.uIoc_eq_union", "def_path": "Mathlib/Data/Set/Intervals/UnorderedInterval.lean", "def_pos": [293, 7], "def_end_pos": [293, 20]}, {"full_name": "aestronglyMeasurable_union_iff", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1753, 9], "def_end_pos": [1753, 46]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u2074 : Countable \u03b9\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\nf\u271d g : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : PseudoMetrizableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\n\u22a2 AEStronglyMeasurable f (Measure.restrict \u03bc (\u0399 a b)) \u2194\n    AEStronglyMeasurable f (Measure.restrict \u03bc (Ioc a b)) \u2227 AEStronglyMeasurable f (Measure.restrict \u03bc (Ioc b a))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Basic.lean", "full_name": "Set.empty_ssubset_singleton", "start": [1325, 1], "end": [1326, 39], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/List/Count.lean", "full_name": "List.count_eq_zero", "start": [176, 1], "end": [177, 55], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Kernel/MeasurableIntegral.lean", "full_name": "MeasureTheory.StronglyMeasurable.integral_kernel_prod_right", "start": [257, 1], "end": [302, 49], "traced_tactics": [{"tactic": "borelize E", "annotated_tactic": ["borelize E", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na : \u03b1\nE : Type u_4\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\n\u22a2 StronglyMeasurable fun x => \u222b (y : \u03b2), f x y \u2202\u2191\u03ba x", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na : \u03b1\nE : Type u_4\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\n\u22a2 StronglyMeasurable fun x => \u222b (y : \u03b2), f x y \u2202\u2191\u03ba x"}, {"tactic": "haveI : TopologicalSpace.SeparableSpace (range (uncurry f) \u222a {0} : Set E) :=\n  hf.separableSpace_range_union_singleton", "annotated_tactic": ["haveI : <a>TopologicalSpace.SeparableSpace</a> (<a>range</a> (<a>uncurry</a> f) \u222a {0} : <a>Set</a> E) :=\n    hf.separableSpace_range_union_singleton", [{"full_name": "TopologicalSpace.SeparableSpace", "def_path": "Mathlib/Topology/Bases.lean", "def_pos": [313, 17], "def_end_pos": [313, 31]}, {"full_name": "Set.range", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [668, 5], "def_end_pos": [668, 10]}, {"full_name": "Function.uncurry", "def_path": "Mathlib/Init/Function.lean", "def_pos": [217, 5], "def_end_pos": [217, 12]}, {"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na : \u03b1\nE : Type u_4\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\n\u22a2 StronglyMeasurable fun x => \u222b (y : \u03b2), f x y \u2202\u2191\u03ba x", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na : \u03b1\nE : Type u_4\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : TopologicalSpace.SeparableSpace \u2191(range (uncurry f) \u222a {0})\n\u22a2 StronglyMeasurable fun x => \u222b (y : \u03b2), f x y \u2202\u2191\u03ba x"}, {"tactic": "let s : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n  SimpleFunc.approxOn _ hf.measurable (range (uncurry f) \u222a {0}) 0 (by simp)", "annotated_tactic": ["let s : \u2115 \u2192 <a>SimpleFunc</a> (\u03b1 \u00d7 \u03b2) E :=\n    <a>SimpleFunc.approxOn</a> _ hf.measurable (<a>range</a> (<a>uncurry</a> f) \u222a {0}) 0 (by simp)", [{"full_name": "MeasureTheory.SimpleFunc", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [44, 11], "def_end_pos": [44, 21]}, {"full_name": "MeasureTheory.SimpleFunc.approxOn", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDense.lean", "def_pos": [127, 19], "def_end_pos": [127, 27]}, {"full_name": "Set.range", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [668, 5], "def_end_pos": [668, 10]}, {"full_name": "Function.uncurry", "def_path": "Mathlib/Init/Function.lean", "def_pos": [217, 5], "def_end_pos": [217, 12]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na : \u03b1\nE : Type u_4\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : TopologicalSpace.SeparableSpace \u2191(range (uncurry f) \u222a {0})\n\u22a2 StronglyMeasurable fun x => \u222b (y : \u03b2), f x y \u2202\u2191\u03ba x", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na : \u03b1\nE : Type u_4\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : TopologicalSpace.SeparableSpace \u2191(range (uncurry f) \u222a {0})\ns : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n  SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n    (_ : 0 \u2208 range (uncurry f) \u222a {0})\n\u22a2 StronglyMeasurable fun x => \u222b (y : \u03b2), f x y \u2202\u2191\u03ba x"}, {"tactic": "let s' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => (s n).comp (Prod.mk x) measurable_prod_mk_left", "annotated_tactic": ["let s' : \u2115 \u2192 \u03b1 \u2192 <a>SimpleFunc</a> \u03b2 E := fun n x => (s n).<a>comp</a> (<a>Prod.mk</a> x) <a>measurable_prod_mk_left</a>", [{"full_name": "MeasureTheory.SimpleFunc", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [44, 11], "def_end_pos": [44, 21]}, {"full_name": "MeasureTheory.SimpleFunc.comp", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [330, 5], "def_end_pos": [330, 9]}, {"full_name": "Prod.mk", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [466, 16], "def_end_pos": [466, 41]}, {"full_name": "measurable_prod_mk_left", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [736, 9], "def_end_pos": [736, 32]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na : \u03b1\nE : Type u_4\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : TopologicalSpace.SeparableSpace \u2191(range (uncurry f) \u222a {0})\ns : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n  SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n    (_ : 0 \u2208 range (uncurry f) \u222a {0})\n\u22a2 StronglyMeasurable fun x => \u222b (y : \u03b2), f x y \u2202\u2191\u03ba x", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na : \u03b1\nE : Type u_4\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : TopologicalSpace.SeparableSpace \u2191(range (uncurry f) \u222a {0})\ns : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n  SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n    (_ : 0 \u2208 range (uncurry f) \u222a {0})\ns' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x))\n\u22a2 StronglyMeasurable fun x => \u222b (y : \u03b2), f x y \u2202\u2191\u03ba x"}, {"tactic": "let f' : \u2115 \u2192 \u03b1 \u2192 E := fun n =>\n  {x | Integrable (f x) (\u03ba x)}.indicator fun x => (s' n x).integral (\u03ba x)", "annotated_tactic": ["let f' : \u2115 \u2192 \u03b1 \u2192 E := fun n =>\n    {x | <a>Integrable</a> (f x) (\u03ba x)}.<a>indicator</a> fun x => (s' n x).<a>integral</a> (\u03ba x)", [{"full_name": "MeasureTheory.Integrable", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [442, 5], "def_end_pos": [442, 15]}, {"full_name": "Set.indicator", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [46, 3], "def_end_pos": [46, 14]}, {"full_name": "MeasureTheory.SimpleFunc.integral", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [309, 5], "def_end_pos": [309, 13]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na : \u03b1\nE : Type u_4\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : TopologicalSpace.SeparableSpace \u2191(range (uncurry f) \u222a {0})\ns : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n  SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n    (_ : 0 \u2208 range (uncurry f) \u222a {0})\ns' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x))\n\u22a2 StronglyMeasurable fun x => \u222b (y : \u03b2), f x y \u2202\u2191\u03ba x", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na : \u03b1\nE : Type u_4\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : TopologicalSpace.SeparableSpace \u2191(range (uncurry f) \u222a {0})\ns : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n  SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n    (_ : 0 \u2208 range (uncurry f) \u222a {0})\ns' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x))\nf' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral (\u2191\u03ba x) (s' n x)\n\u22a2 StronglyMeasurable fun x => \u222b (y : \u03b2), f x y \u2202\u2191\u03ba x"}, {"tactic": "have hf' : \u2200 n, StronglyMeasurable (f' n) := by\n  intro n; refine' StronglyMeasurable.indicator _ (measurableSet_kernel_integrable hf)\n  have : \u2200 x, ((s' n x).range.filter fun x => x \u2260 0) \u2286 (s n).range := by\n    intro x; refine' Finset.Subset.trans (Finset.filter_subset _ _) _; intro y\n    simp_rw [SimpleFunc.mem_range]; rintro \u27e8z, rfl\u27e9; exact \u27e8(x, z), rfl\u27e9\n  simp only [SimpleFunc.integral_eq_sum_of_subset (this _)]\n  refine' Finset.stronglyMeasurable_sum _ fun x _ => _\n  refine' (Measurable.ennreal_toReal _).stronglyMeasurable.smul_const _\n  simp (config := { singlePass := true }) only [SimpleFunc.coe_comp, preimage_comp]\n  apply kernel.measurable_kernel_prod_mk_left\n  exact (s n).measurableSet_fiber x", "annotated_tactic": ["have hf' : \u2200 n, <a>StronglyMeasurable</a> (f' n) := by\n    intro n; refine' <a>StronglyMeasurable.indicator</a> _ (<a>measurableSet_kernel_integrable</a> hf)\n    have : \u2200 x, ((s' n x).range.filter fun x => x \u2260 0) \u2286 (s n).<a>range</a> := by\n      intro x; refine' <a>Finset.Subset.trans</a> (<a>Finset.filter_subset</a> _ _) _; intro y\n      simp_rw [<a>SimpleFunc.mem_range</a>]; rintro \u27e8z, rfl\u27e9; exact \u27e8(x, z), <a>rfl</a>\u27e9\n    simp only [<a>SimpleFunc.integral_eq_sum_of_subset</a> (this _)]\n    refine' <a>Finset.stronglyMeasurable_sum</a> _ fun x _ => _\n    refine' (<a>Measurable.ennreal_toReal</a> _).stronglyMeasurable.smul_const _\n    simp (config := { singlePass := <a>true</a> }) only [<a>SimpleFunc.coe_comp</a>, <a>preimage_comp</a>]\n    apply <a>kernel.measurable_kernel_prod_mk_left</a>\n    exact (s n).<a>measurableSet_fiber</a> x", [{"full_name": "MeasureTheory.StronglyMeasurable", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [78, 5], "def_end_pos": [78, 23]}, {"full_name": "MeasureTheory.StronglyMeasurable.indicator", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [825, 19], "def_end_pos": [825, 28]}, {"full_name": "ProbabilityTheory.measurableSet_kernel_integrable", "def_path": "Mathlib/Probability/Kernel/MeasurableIntegral.lean", "def_pos": [242, 9], "def_end_pos": [242, 40]}, {"full_name": "MeasureTheory.SimpleFunc.range", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [103, 15], "def_end_pos": [103, 20]}, {"full_name": "Finset.Subset.trans", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [351, 9], "def_end_pos": [351, 21]}, {"full_name": "Finset.filter_subset", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2701, 9], "def_end_pos": [2701, 22]}, {"full_name": "MeasureTheory.SimpleFunc.mem_range", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [108, 9], "def_end_pos": [108, 18]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}, {"full_name": "MeasureTheory.SimpleFunc.integral_eq_sum_of_subset", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [329, 9], "def_end_pos": [329, 34]}, {"full_name": "Finset.stronglyMeasurable_sum", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [618, 3], "def_end_pos": [618, 14]}, {"full_name": "Measurable.ennreal_toReal", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [2123, 9], "def_end_pos": [2123, 34]}, {"full_name": "Bool.true", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [549, 5], "def_end_pos": [549, 9]}, {"full_name": "MeasureTheory.SimpleFunc.coe_comp", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [337, 9], "def_end_pos": [337, 17]}, {"full_name": "Set.preimage_comp", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [163, 9], "def_end_pos": [163, 22]}, {"full_name": "ProbabilityTheory.kernel.measurable_kernel_prod_mk_left", "def_path": "Mathlib/Probability/Kernel/MeasurableIntegral.lean", "def_pos": [100, 9], "def_end_pos": [100, 39]}, {"full_name": "MeasureTheory.SimpleFunc.measurableSet_fiber", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [80, 9], "def_end_pos": [80, 28]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na : \u03b1\nE : Type u_4\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : TopologicalSpace.SeparableSpace \u2191(range (uncurry f) \u222a {0})\ns : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n  SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n    (_ : 0 \u2208 range (uncurry f) \u222a {0})\ns' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x))\nf' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral (\u2191\u03ba x) (s' n x)\n\u22a2 StronglyMeasurable fun x => \u222b (y : \u03b2), f x y \u2202\u2191\u03ba x", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na : \u03b1\nE : Type u_4\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : TopologicalSpace.SeparableSpace \u2191(range (uncurry f) \u222a {0})\ns : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n  SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n    (_ : 0 \u2208 range (uncurry f) \u222a {0})\ns' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x))\nf' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral (\u2191\u03ba x) (s' n x)\nhf' : \u2200 (n : \u2115), StronglyMeasurable (f' n)\n\u22a2 StronglyMeasurable fun x => \u222b (y : \u03b2), f x y \u2202\u2191\u03ba x"}, {"tactic": "exact stronglyMeasurable_of_tendsto _ hf' h2f'", "annotated_tactic": ["exact <a>stronglyMeasurable_of_tendsto</a> _ hf' h2f'", [{"full_name": "stronglyMeasurable_of_tendsto", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [733, 9], "def_end_pos": [733, 45]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na : \u03b1\nE : Type u_4\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : TopologicalSpace.SeparableSpace \u2191(range (uncurry f) \u222a {0})\ns : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n  SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n    (_ : 0 \u2208 range (uncurry f) \u222a {0})\ns' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x))\nf' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral (\u2191\u03ba x) (s' n x)\nhf' : \u2200 (n : \u2115), StronglyMeasurable (f' n)\nh2f' : Tendsto f' atTop (\ud835\udcdd fun x => \u222b (y : \u03b2), f x y \u2202\u2191\u03ba x)\n\u22a2 StronglyMeasurable fun x => \u222b (y : \u03b2), f x y \u2202\u2191\u03ba x", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na : \u03b1\nE : Type u_4\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : TopologicalSpace.SeparableSpace \u2191(range (uncurry f) \u222a {0})\n\u22a2 0 \u2208 range (uncurry f) \u222a {0}", "state_after": "no goals"}, {"tactic": "intro n", "annotated_tactic": ["intro n", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na : \u03b1\nE : Type u_4\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : TopologicalSpace.SeparableSpace \u2191(range (uncurry f) \u222a {0})\ns : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n  SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n    (_ : 0 \u2208 range (uncurry f) \u222a {0})\ns' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x))\nf' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral (\u2191\u03ba x) (s' n x)\n\u22a2 \u2200 (n : \u2115), StronglyMeasurable (f' n)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na : \u03b1\nE : Type u_4\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : TopologicalSpace.SeparableSpace \u2191(range (uncurry f) \u222a {0})\ns : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n  SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n    (_ : 0 \u2208 range (uncurry f) \u222a {0})\ns' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x))\nf' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral (\u2191\u03ba x) (s' n x)\nn : \u2115\n\u22a2 StronglyMeasurable (f' n)"}, {"tactic": "refine' StronglyMeasurable.indicator _ (measurableSet_kernel_integrable hf)", "annotated_tactic": ["refine' <a>StronglyMeasurable.indicator</a> _ (<a>measurableSet_kernel_integrable</a> hf)", [{"full_name": "MeasureTheory.StronglyMeasurable.indicator", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [825, 19], "def_end_pos": [825, 28]}, {"full_name": "ProbabilityTheory.measurableSet_kernel_integrable", "def_path": "Mathlib/Probability/Kernel/MeasurableIntegral.lean", "def_pos": [242, 9], "def_end_pos": [242, 40]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na : \u03b1\nE : Type u_4\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : TopologicalSpace.SeparableSpace \u2191(range (uncurry f) \u222a {0})\ns : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n  SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n    (_ : 0 \u2208 range (uncurry f) \u222a {0})\ns' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x))\nf' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral (\u2191\u03ba x) (s' n x)\nn : \u2115\n\u22a2 StronglyMeasurable (f' n)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na : \u03b1\nE : Type u_4\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : TopologicalSpace.SeparableSpace \u2191(range (uncurry f) \u222a {0})\ns : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n  SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n    (_ : 0 \u2208 range (uncurry f) \u222a {0})\ns' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x))\nf' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral (\u2191\u03ba x) (s' n x)\nn : \u2115\n\u22a2 StronglyMeasurable fun x => SimpleFunc.integral (\u2191\u03ba x) (s' n x)"}, {"tactic": "have : \u2200 x, ((s' n x).range.filter fun x => x \u2260 0) \u2286 (s n).range := by\n  intro x; refine' Finset.Subset.trans (Finset.filter_subset _ _) _; intro y\n  simp_rw [SimpleFunc.mem_range]; rintro \u27e8z, rfl\u27e9; exact \u27e8(x, z), rfl\u27e9", "annotated_tactic": ["have : \u2200 x, ((s' n x).range.filter fun x => x \u2260 0) \u2286 (s n).<a>range</a> := by\n      intro x; refine' <a>Finset.Subset.trans</a> (<a>Finset.filter_subset</a> _ _) _; intro y\n      simp_rw [<a>SimpleFunc.mem_range</a>]; rintro \u27e8z, rfl\u27e9; exact \u27e8(x, z), <a>rfl</a>\u27e9", [{"full_name": "MeasureTheory.SimpleFunc.range", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [103, 15], "def_end_pos": [103, 20]}, {"full_name": "Finset.Subset.trans", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [351, 9], "def_end_pos": [351, 21]}, {"full_name": "Finset.filter_subset", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2701, 9], "def_end_pos": [2701, 22]}, {"full_name": "MeasureTheory.SimpleFunc.mem_range", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [108, 9], "def_end_pos": [108, 18]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na : \u03b1\nE : Type u_4\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : TopologicalSpace.SeparableSpace \u2191(range (uncurry f) \u222a {0})\ns : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n  SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n    (_ : 0 \u2208 range (uncurry f) \u222a {0})\ns' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x))\nf' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral (\u2191\u03ba x) (s' n x)\nn : \u2115\n\u22a2 StronglyMeasurable fun x => SimpleFunc.integral (\u2191\u03ba x) (s' n x)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na : \u03b1\nE : Type u_4\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b2 : MeasurableSpace E := borel E\nthis\u271d\u00b9 : BorelSpace E\nthis\u271d : TopologicalSpace.SeparableSpace \u2191(range (uncurry f) \u222a {0})\ns : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n  SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n    (_ : 0 \u2208 range (uncurry f) \u222a {0})\ns' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x))\nf' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral (\u2191\u03ba x) (s' n x)\nn : \u2115\nthis : \u2200 (x : \u03b1), Finset.filter (fun x => x \u2260 0) (SimpleFunc.range (s' n x)) \u2286 SimpleFunc.range (s n)\n\u22a2 StronglyMeasurable fun x => SimpleFunc.integral (\u2191\u03ba x) (s' n x)"}, {"tactic": "simp only [SimpleFunc.integral_eq_sum_of_subset (this _)]", "annotated_tactic": ["simp only [<a>SimpleFunc.integral_eq_sum_of_subset</a> (this _)]", [{"full_name": "MeasureTheory.SimpleFunc.integral_eq_sum_of_subset", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [329, 9], "def_end_pos": [329, 34]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na : \u03b1\nE : Type u_4\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b2 : MeasurableSpace E := borel E\nthis\u271d\u00b9 : BorelSpace E\nthis\u271d : TopologicalSpace.SeparableSpace \u2191(range (uncurry f) \u222a {0})\ns : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n  SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n    (_ : 0 \u2208 range (uncurry f) \u222a {0})\ns' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x))\nf' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral (\u2191\u03ba x) (s' n x)\nn : \u2115\nthis : \u2200 (x : \u03b1), Finset.filter (fun x => x \u2260 0) (SimpleFunc.range (s' n x)) \u2286 SimpleFunc.range (s n)\n\u22a2 StronglyMeasurable fun x => SimpleFunc.integral (\u2191\u03ba x) (s' n x)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na : \u03b1\nE : Type u_4\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b2 : MeasurableSpace E := borel E\nthis\u271d\u00b9 : BorelSpace E\nthis\u271d : TopologicalSpace.SeparableSpace \u2191(range (uncurry f) \u222a {0})\ns : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n  SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n    (_ : 0 \u2208 range (uncurry f) \u222a {0})\ns' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x))\nf' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral (\u2191\u03ba x) (s' n x)\nn : \u2115\nthis : \u2200 (x : \u03b1), Finset.filter (fun x => x \u2260 0) (SimpleFunc.range (s' n x)) \u2286 SimpleFunc.range (s n)\n\u22a2 StronglyMeasurable fun x =>\n    Finset.sum\n      (SimpleFunc.range\n        (SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n          (_ : 0 \u2208 range (uncurry f) \u222a {0}) n))\n      fun x_1 =>\n      ENNReal.toReal\n          (\u2191\u2191(\u2191\u03ba x)\n            (\u2191(SimpleFunc.comp\n                  (SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n                    (_ : 0 \u2208 range (uncurry f) \u222a {0}) n)\n                  (Prod.mk x) (_ : Measurable (Prod.mk x))) \u207b\u00b9'\n              {x_1})) \u2022\n        x_1"}, {"tactic": "refine' Finset.stronglyMeasurable_sum _ fun x _ => _", "annotated_tactic": ["refine' <a>Finset.stronglyMeasurable_sum</a> _ fun x _ => _", [{"full_name": "Finset.stronglyMeasurable_sum", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [618, 3], "def_end_pos": [618, 14]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na : \u03b1\nE : Type u_4\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b2 : MeasurableSpace E := borel E\nthis\u271d\u00b9 : BorelSpace E\nthis\u271d : TopologicalSpace.SeparableSpace \u2191(range (uncurry f) \u222a {0})\ns : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n  SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n    (_ : 0 \u2208 range (uncurry f) \u222a {0})\ns' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x))\nf' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral (\u2191\u03ba x) (s' n x)\nn : \u2115\nthis : \u2200 (x : \u03b1), Finset.filter (fun x => x \u2260 0) (SimpleFunc.range (s' n x)) \u2286 SimpleFunc.range (s n)\n\u22a2 StronglyMeasurable fun x =>\n    Finset.sum\n      (SimpleFunc.range\n        (SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n          (_ : 0 \u2208 range (uncurry f) \u222a {0}) n))\n      fun x_1 =>\n      ENNReal.toReal\n          (\u2191\u2191(\u2191\u03ba x)\n            (\u2191(SimpleFunc.comp\n                  (SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n                    (_ : 0 \u2208 range (uncurry f) \u222a {0}) n)\n                  (Prod.mk x) (_ : Measurable (Prod.mk x))) \u207b\u00b9'\n              {x_1})) \u2022\n        x_1", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na : \u03b1\nE : Type u_4\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b2 : MeasurableSpace E := borel E\nthis\u271d\u00b9 : BorelSpace E\nthis\u271d : TopologicalSpace.SeparableSpace \u2191(range (uncurry f) \u222a {0})\ns : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n  SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n    (_ : 0 \u2208 range (uncurry f) \u222a {0})\ns' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x))\nf' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral (\u2191\u03ba x) (s' n x)\nn : \u2115\nthis : \u2200 (x : \u03b1), Finset.filter (fun x => x \u2260 0) (SimpleFunc.range (s' n x)) \u2286 SimpleFunc.range (s n)\nx : E\nx\u271d :\n  x \u2208\n    SimpleFunc.range\n      (SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n        (_ : 0 \u2208 range (uncurry f) \u222a {0}) n)\n\u22a2 StronglyMeasurable fun x_1 =>\n    ENNReal.toReal\n        (\u2191\u2191(\u2191\u03ba x_1)\n          (\u2191(SimpleFunc.comp\n                (SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n                  (_ : 0 \u2208 range (uncurry f) \u222a {0}) n)\n                (Prod.mk x_1) (_ : Measurable (Prod.mk x_1))) \u207b\u00b9'\n            {x})) \u2022\n      x"}, {"tactic": "refine' (Measurable.ennreal_toReal _).stronglyMeasurable.smul_const _", "annotated_tactic": ["refine' (<a>Measurable.ennreal_toReal</a> _).stronglyMeasurable.smul_const _", [{"full_name": "Measurable.ennreal_toReal", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [2123, 9], "def_end_pos": [2123, 34]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na : \u03b1\nE : Type u_4\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b2 : MeasurableSpace E := borel E\nthis\u271d\u00b9 : BorelSpace E\nthis\u271d : TopologicalSpace.SeparableSpace \u2191(range (uncurry f) \u222a {0})\ns : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n  SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n    (_ : 0 \u2208 range (uncurry f) \u222a {0})\ns' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x))\nf' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral (\u2191\u03ba x) (s' n x)\nn : \u2115\nthis : \u2200 (x : \u03b1), Finset.filter (fun x => x \u2260 0) (SimpleFunc.range (s' n x)) \u2286 SimpleFunc.range (s n)\nx : E\nx\u271d :\n  x \u2208\n    SimpleFunc.range\n      (SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n        (_ : 0 \u2208 range (uncurry f) \u222a {0}) n)\n\u22a2 StronglyMeasurable fun x_1 =>\n    ENNReal.toReal\n        (\u2191\u2191(\u2191\u03ba x_1)\n          (\u2191(SimpleFunc.comp\n                (SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n                  (_ : 0 \u2208 range (uncurry f) \u222a {0}) n)\n                (Prod.mk x_1) (_ : Measurable (Prod.mk x_1))) \u207b\u00b9'\n            {x})) \u2022\n      x", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na : \u03b1\nE : Type u_4\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b2 : MeasurableSpace E := borel E\nthis\u271d\u00b9 : BorelSpace E\nthis\u271d : TopologicalSpace.SeparableSpace \u2191(range (uncurry f) \u222a {0})\ns : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n  SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n    (_ : 0 \u2208 range (uncurry f) \u222a {0})\ns' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x))\nf' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral (\u2191\u03ba x) (s' n x)\nn : \u2115\nthis : \u2200 (x : \u03b1), Finset.filter (fun x => x \u2260 0) (SimpleFunc.range (s' n x)) \u2286 SimpleFunc.range (s n)\nx : E\nx\u271d :\n  x \u2208\n    SimpleFunc.range\n      (SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n        (_ : 0 \u2208 range (uncurry f) \u222a {0}) n)\n\u22a2 Measurable fun x_1 =>\n    \u2191\u2191(\u2191\u03ba x_1)\n      (\u2191(SimpleFunc.comp\n            (SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n              (_ : 0 \u2208 range (uncurry f) \u222a {0}) n)\n            (Prod.mk x_1) (_ : Measurable (Prod.mk x_1))) \u207b\u00b9'\n        {x})"}, {"tactic": "simp (config := { singlePass := true }) only [SimpleFunc.coe_comp, preimage_comp]", "annotated_tactic": ["simp (config := { singlePass := <a>true</a> }) only [<a>SimpleFunc.coe_comp</a>, <a>preimage_comp</a>]", [{"full_name": "Bool.true", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [549, 5], "def_end_pos": [549, 9]}, {"full_name": "MeasureTheory.SimpleFunc.coe_comp", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [337, 9], "def_end_pos": [337, 17]}, {"full_name": "Set.preimage_comp", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [163, 9], "def_end_pos": [163, 22]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na : \u03b1\nE : Type u_4\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b2 : MeasurableSpace E := borel E\nthis\u271d\u00b9 : BorelSpace E\nthis\u271d : TopologicalSpace.SeparableSpace \u2191(range (uncurry f) \u222a {0})\ns : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n  SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n    (_ : 0 \u2208 range (uncurry f) \u222a {0})\ns' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x))\nf' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral (\u2191\u03ba x) (s' n x)\nn : \u2115\nthis : \u2200 (x : \u03b1), Finset.filter (fun x => x \u2260 0) (SimpleFunc.range (s' n x)) \u2286 SimpleFunc.range (s n)\nx : E\nx\u271d :\n  x \u2208\n    SimpleFunc.range\n      (SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n        (_ : 0 \u2208 range (uncurry f) \u222a {0}) n)\n\u22a2 Measurable fun x_1 =>\n    \u2191\u2191(\u2191\u03ba x_1)\n      (\u2191(SimpleFunc.comp\n            (SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n              (_ : 0 \u2208 range (uncurry f) \u222a {0}) n)\n            (Prod.mk x_1) (_ : Measurable (Prod.mk x_1))) \u207b\u00b9'\n        {x})", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na : \u03b1\nE : Type u_4\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b2 : MeasurableSpace E := borel E\nthis\u271d\u00b9 : BorelSpace E\nthis\u271d : TopologicalSpace.SeparableSpace \u2191(range (uncurry f) \u222a {0})\ns : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n  SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n    (_ : 0 \u2208 range (uncurry f) \u222a {0})\ns' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x))\nf' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral (\u2191\u03ba x) (s' n x)\nn : \u2115\nthis : \u2200 (x : \u03b1), Finset.filter (fun x => x \u2260 0) (SimpleFunc.range (s' n x)) \u2286 SimpleFunc.range (s n)\nx : E\nx\u271d :\n  x \u2208\n    SimpleFunc.range\n      (SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n        (_ : 0 \u2208 range (uncurry f) \u222a {0}) n)\n\u22a2 Measurable fun x_1 =>\n    \u2191\u2191(\u2191\u03ba x_1)\n      (Prod.mk x_1 \u207b\u00b9'\n        (\u2191(SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n              (_ : 0 \u2208 range (uncurry f) \u222a {0}) n) \u207b\u00b9'\n          {x}))"}, {"tactic": "apply kernel.measurable_kernel_prod_mk_left", "annotated_tactic": ["apply <a>kernel.measurable_kernel_prod_mk_left</a>", [{"full_name": "ProbabilityTheory.kernel.measurable_kernel_prod_mk_left", "def_path": "Mathlib/Probability/Kernel/MeasurableIntegral.lean", "def_pos": [100, 9], "def_end_pos": [100, 39]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na : \u03b1\nE : Type u_4\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b2 : MeasurableSpace E := borel E\nthis\u271d\u00b9 : BorelSpace E\nthis\u271d : TopologicalSpace.SeparableSpace \u2191(range (uncurry f) \u222a {0})\ns : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n  SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n    (_ : 0 \u2208 range (uncurry f) \u222a {0})\ns' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x))\nf' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral (\u2191\u03ba x) (s' n x)\nn : \u2115\nthis : \u2200 (x : \u03b1), Finset.filter (fun x => x \u2260 0) (SimpleFunc.range (s' n x)) \u2286 SimpleFunc.range (s n)\nx : E\nx\u271d :\n  x \u2208\n    SimpleFunc.range\n      (SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n        (_ : 0 \u2208 range (uncurry f) \u222a {0}) n)\n\u22a2 Measurable fun x_1 =>\n    \u2191\u2191(\u2191\u03ba x_1)\n      (Prod.mk x_1 \u207b\u00b9'\n        (\u2191(SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n              (_ : 0 \u2208 range (uncurry f) \u222a {0}) n) \u207b\u00b9'\n          {x}))", "state_after": "case ht\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na : \u03b1\nE : Type u_4\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b2 : MeasurableSpace E := borel E\nthis\u271d\u00b9 : BorelSpace E\nthis\u271d : TopologicalSpace.SeparableSpace \u2191(range (uncurry f) \u222a {0})\ns : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n  SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n    (_ : 0 \u2208 range (uncurry f) \u222a {0})\ns' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x))\nf' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral (\u2191\u03ba x) (s' n x)\nn : \u2115\nthis : \u2200 (x : \u03b1), Finset.filter (fun x => x \u2260 0) (SimpleFunc.range (s' n x)) \u2286 SimpleFunc.range (s n)\nx : E\nx\u271d :\n  x \u2208\n    SimpleFunc.range\n      (SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n        (_ : 0 \u2208 range (uncurry f) \u222a {0}) n)\n\u22a2 MeasurableSet\n    (\u2191(SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n          (_ : 0 \u2208 range (uncurry f) \u222a {0}) n) \u207b\u00b9'\n      {x})"}, {"tactic": "exact (s n).measurableSet_fiber x", "annotated_tactic": ["exact (s n).<a>measurableSet_fiber</a> x", [{"full_name": "MeasureTheory.SimpleFunc.measurableSet_fiber", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [80, 9], "def_end_pos": [80, 28]}]], "state_before": "case ht\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na : \u03b1\nE : Type u_4\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b2 : MeasurableSpace E := borel E\nthis\u271d\u00b9 : BorelSpace E\nthis\u271d : TopologicalSpace.SeparableSpace \u2191(range (uncurry f) \u222a {0})\ns : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n  SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n    (_ : 0 \u2208 range (uncurry f) \u222a {0})\ns' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x))\nf' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral (\u2191\u03ba x) (s' n x)\nn : \u2115\nthis : \u2200 (x : \u03b1), Finset.filter (fun x => x \u2260 0) (SimpleFunc.range (s' n x)) \u2286 SimpleFunc.range (s n)\nx : E\nx\u271d :\n  x \u2208\n    SimpleFunc.range\n      (SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n        (_ : 0 \u2208 range (uncurry f) \u222a {0}) n)\n\u22a2 MeasurableSet\n    (\u2191(SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n          (_ : 0 \u2208 range (uncurry f) \u222a {0}) n) \u207b\u00b9'\n      {x})", "state_after": "no goals"}, {"tactic": "intro x", "annotated_tactic": ["intro x", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na : \u03b1\nE : Type u_4\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : TopologicalSpace.SeparableSpace \u2191(range (uncurry f) \u222a {0})\ns : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n  SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n    (_ : 0 \u2208 range (uncurry f) \u222a {0})\ns' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x))\nf' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral (\u2191\u03ba x) (s' n x)\nn : \u2115\n\u22a2 \u2200 (x : \u03b1), Finset.filter (fun x => x \u2260 0) (SimpleFunc.range (s' n x)) \u2286 SimpleFunc.range (s n)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na : \u03b1\nE : Type u_4\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : TopologicalSpace.SeparableSpace \u2191(range (uncurry f) \u222a {0})\ns : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n  SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n    (_ : 0 \u2208 range (uncurry f) \u222a {0})\ns' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x))\nf' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral (\u2191\u03ba x) (s' n x)\nn : \u2115\nx : \u03b1\n\u22a2 Finset.filter (fun x => x \u2260 0) (SimpleFunc.range (s' n x)) \u2286 SimpleFunc.range (s n)"}, {"tactic": "refine' Finset.Subset.trans (Finset.filter_subset _ _) _", "annotated_tactic": ["refine' <a>Finset.Subset.trans</a> (<a>Finset.filter_subset</a> _ _) _", [{"full_name": "Finset.Subset.trans", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [351, 9], "def_end_pos": [351, 21]}, {"full_name": "Finset.filter_subset", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2701, 9], "def_end_pos": [2701, 22]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na : \u03b1\nE : Type u_4\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : TopologicalSpace.SeparableSpace \u2191(range (uncurry f) \u222a {0})\ns : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n  SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n    (_ : 0 \u2208 range (uncurry f) \u222a {0})\ns' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x))\nf' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral (\u2191\u03ba x) (s' n x)\nn : \u2115\nx : \u03b1\n\u22a2 Finset.filter (fun x => x \u2260 0) (SimpleFunc.range (s' n x)) \u2286 SimpleFunc.range (s n)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na : \u03b1\nE : Type u_4\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : TopologicalSpace.SeparableSpace \u2191(range (uncurry f) \u222a {0})\ns : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n  SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n    (_ : 0 \u2208 range (uncurry f) \u222a {0})\ns' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x))\nf' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral (\u2191\u03ba x) (s' n x)\nn : \u2115\nx : \u03b1\n\u22a2 SimpleFunc.range (s' n x) \u2286 SimpleFunc.range (s n)"}, {"tactic": "intro y", "annotated_tactic": ["intro y", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na : \u03b1\nE : Type u_4\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : TopologicalSpace.SeparableSpace \u2191(range (uncurry f) \u222a {0})\ns : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n  SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n    (_ : 0 \u2208 range (uncurry f) \u222a {0})\ns' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x))\nf' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral (\u2191\u03ba x) (s' n x)\nn : \u2115\nx : \u03b1\n\u22a2 SimpleFunc.range (s' n x) \u2286 SimpleFunc.range (s n)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na : \u03b1\nE : Type u_4\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : TopologicalSpace.SeparableSpace \u2191(range (uncurry f) \u222a {0})\ns : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n  SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n    (_ : 0 \u2208 range (uncurry f) \u222a {0})\ns' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x))\nf' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral (\u2191\u03ba x) (s' n x)\nn : \u2115\nx : \u03b1\ny : E\n\u22a2 y \u2208 SimpleFunc.range (s' n x) \u2192 y \u2208 SimpleFunc.range (s n)"}, {"tactic": "simp_rw [SimpleFunc.mem_range]", "annotated_tactic": ["simp_rw [<a>SimpleFunc.mem_range</a>]", [{"full_name": "MeasureTheory.SimpleFunc.mem_range", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [108, 9], "def_end_pos": [108, 18]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na : \u03b1\nE : Type u_4\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : TopologicalSpace.SeparableSpace \u2191(range (uncurry f) \u222a {0})\ns : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n  SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n    (_ : 0 \u2208 range (uncurry f) \u222a {0})\ns' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x))\nf' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral (\u2191\u03ba x) (s' n x)\nn : \u2115\nx : \u03b1\ny : E\n\u22a2 y \u2208 SimpleFunc.range (s' n x) \u2192 y \u2208 SimpleFunc.range (s n)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na : \u03b1\nE : Type u_4\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : TopologicalSpace.SeparableSpace \u2191(range (uncurry f) \u222a {0})\ns : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n  SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n    (_ : 0 \u2208 range (uncurry f) \u222a {0})\ns' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x))\nf' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral (\u2191\u03ba x) (s' n x)\nn : \u2115\nx : \u03b1\ny : E\n\u22a2 y \u2208\n      range\n        \u2191(SimpleFunc.comp\n            (SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n              (_ : 0 \u2208 range (uncurry f) \u222a {0}) n)\n            (Prod.mk x) (_ : Measurable (Prod.mk x))) \u2192\n    y \u2208\n      range\n        \u2191(SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n            (_ : 0 \u2208 range (uncurry f) \u222a {0}) n)"}, {"tactic": "rintro \u27e8z, rfl\u27e9", "annotated_tactic": ["rintro \u27e8z, rfl\u27e9", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na : \u03b1\nE : Type u_4\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : TopologicalSpace.SeparableSpace \u2191(range (uncurry f) \u222a {0})\ns : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n  SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n    (_ : 0 \u2208 range (uncurry f) \u222a {0})\ns' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x))\nf' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral (\u2191\u03ba x) (s' n x)\nn : \u2115\nx : \u03b1\ny : E\n\u22a2 y \u2208\n      range\n        \u2191(SimpleFunc.comp\n            (SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n              (_ : 0 \u2208 range (uncurry f) \u222a {0}) n)\n            (Prod.mk x) (_ : Measurable (Prod.mk x))) \u2192\n    y \u2208\n      range\n        \u2191(SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n            (_ : 0 \u2208 range (uncurry f) \u222a {0}) n)", "state_after": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na : \u03b1\nE : Type u_4\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : TopologicalSpace.SeparableSpace \u2191(range (uncurry f) \u222a {0})\ns : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n  SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n    (_ : 0 \u2208 range (uncurry f) \u222a {0})\ns' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x))\nf' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral (\u2191\u03ba x) (s' n x)\nn : \u2115\nx : \u03b1\nz : \u03b2\n\u22a2 \u2191(SimpleFunc.comp\n          (SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n            (_ : 0 \u2208 range (uncurry f) \u222a {0}) n)\n          (Prod.mk x) (_ : Measurable (Prod.mk x)))\n      z \u2208\n    range\n      \u2191(SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n          (_ : 0 \u2208 range (uncurry f) \u222a {0}) n)"}, {"tactic": "exact \u27e8(x, z), rfl\u27e9", "annotated_tactic": ["exact \u27e8(x, z), <a>rfl</a>\u27e9", [{"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na : \u03b1\nE : Type u_4\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : TopologicalSpace.SeparableSpace \u2191(range (uncurry f) \u222a {0})\ns : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n  SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n    (_ : 0 \u2208 range (uncurry f) \u222a {0})\ns' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x))\nf' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral (\u2191\u03ba x) (s' n x)\nn : \u2115\nx : \u03b1\nz : \u03b2\n\u22a2 \u2191(SimpleFunc.comp\n          (SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n            (_ : 0 \u2208 range (uncurry f) \u222a {0}) n)\n          (Prod.mk x) (_ : Measurable (Prod.mk x)))\n      z \u2208\n    range\n      \u2191(SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n          (_ : 0 \u2208 range (uncurry f) \u222a {0}) n)", "state_after": "no goals"}, {"tactic": "rw [tendsto_pi_nhds]", "annotated_tactic": ["rw [<a>tendsto_pi_nhds</a>]", [{"full_name": "tendsto_pi_nhds", "def_path": "Mathlib/Topology/Constructions.lean", "def_pos": [1231, 9], "def_end_pos": [1231, 24]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na : \u03b1\nE : Type u_4\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : TopologicalSpace.SeparableSpace \u2191(range (uncurry f) \u222a {0})\ns : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n  SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n    (_ : 0 \u2208 range (uncurry f) \u222a {0})\ns' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x))\nf' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral (\u2191\u03ba x) (s' n x)\nhf' : \u2200 (n : \u2115), StronglyMeasurable (f' n)\n\u22a2 Tendsto f' atTop (\ud835\udcdd fun x => \u222b (y : \u03b2), f x y \u2202\u2191\u03ba x)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na : \u03b1\nE : Type u_4\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : TopologicalSpace.SeparableSpace \u2191(range (uncurry f) \u222a {0})\ns : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n  SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n    (_ : 0 \u2208 range (uncurry f) \u222a {0})\ns' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x))\nf' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral (\u2191\u03ba x) (s' n x)\nhf' : \u2200 (n : \u2115), StronglyMeasurable (f' n)\n\u22a2 \u2200 (x : \u03b1), Tendsto (fun i => f' i x) atTop (\ud835\udcdd (\u222b (y : \u03b2), f x y \u2202\u2191\u03ba x))"}, {"tactic": "intro x", "annotated_tactic": ["intro x", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na : \u03b1\nE : Type u_4\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : TopologicalSpace.SeparableSpace \u2191(range (uncurry f) \u222a {0})\ns : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n  SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n    (_ : 0 \u2208 range (uncurry f) \u222a {0})\ns' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x))\nf' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral (\u2191\u03ba x) (s' n x)\nhf' : \u2200 (n : \u2115), StronglyMeasurable (f' n)\n\u22a2 \u2200 (x : \u03b1), Tendsto (fun i => f' i x) atTop (\ud835\udcdd (\u222b (y : \u03b2), f x y \u2202\u2191\u03ba x))", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na : \u03b1\nE : Type u_4\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : TopologicalSpace.SeparableSpace \u2191(range (uncurry f) \u222a {0})\ns : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n  SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n    (_ : 0 \u2208 range (uncurry f) \u222a {0})\ns' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x))\nf' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral (\u2191\u03ba x) (s' n x)\nhf' : \u2200 (n : \u2115), StronglyMeasurable (f' n)\nx : \u03b1\n\u22a2 Tendsto (fun i => f' i x) atTop (\ud835\udcdd (\u222b (y : \u03b2), f x y \u2202\u2191\u03ba x))"}, {"tactic": "by_cases hfx : Integrable (f x) (\u03ba x)", "annotated_tactic": ["by_cases hfx : <a>Integrable</a> (f x) (\u03ba x)", [{"full_name": "MeasureTheory.Integrable", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [442, 5], "def_end_pos": [442, 15]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na : \u03b1\nE : Type u_4\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : TopologicalSpace.SeparableSpace \u2191(range (uncurry f) \u222a {0})\ns : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n  SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n    (_ : 0 \u2208 range (uncurry f) \u222a {0})\ns' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x))\nf' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral (\u2191\u03ba x) (s' n x)\nhf' : \u2200 (n : \u2115), StronglyMeasurable (f' n)\nx : \u03b1\n\u22a2 Tendsto (fun i => f' i x) atTop (\ud835\udcdd (\u222b (y : \u03b2), f x y \u2202\u2191\u03ba x))", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na : \u03b1\nE : Type u_4\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : TopologicalSpace.SeparableSpace \u2191(range (uncurry f) \u222a {0})\ns : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n  SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n    (_ : 0 \u2208 range (uncurry f) \u222a {0})\ns' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x))\nf' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral (\u2191\u03ba x) (s' n x)\nhf' : \u2200 (n : \u2115), StronglyMeasurable (f' n)\nx : \u03b1\nhfx : Integrable (f x)\n\u22a2 Tendsto (fun i => f' i x) atTop (\ud835\udcdd (\u222b (y : \u03b2), f x y \u2202\u2191\u03ba x))\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na : \u03b1\nE : Type u_4\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : TopologicalSpace.SeparableSpace \u2191(range (uncurry f) \u222a {0})\ns : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n  SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n    (_ : 0 \u2208 range (uncurry f) \u222a {0})\ns' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x))\nf' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral (\u2191\u03ba x) (s' n x)\nhf' : \u2200 (n : \u2115), StronglyMeasurable (f' n)\nx : \u03b1\nhfx : \u00acIntegrable (f x)\n\u22a2 Tendsto (fun i => f' i x) atTop (\ud835\udcdd (\u222b (y : \u03b2), f x y \u2202\u2191\u03ba x))"}, {"tactic": "have : \u2200 n, Integrable (s' n x) (\u03ba x) := by\n  intro n; apply (hfx.norm.add hfx.norm).mono' (s' n x).aestronglyMeasurable\n  apply eventually_of_forall; intro y\n  simp_rw [SimpleFunc.coe_comp]; exact SimpleFunc.norm_approxOn_zero_le _ _ (x, y) n", "annotated_tactic": ["have : \u2200 n, <a>Integrable</a> (s' n x) (\u03ba x) := by\n        intro n; apply (hfx.norm.add hfx.norm).<a>mono'</a> (s' n x).<a>aestronglyMeasurable</a>\n        apply <a>eventually_of_forall</a>; intro y\n        simp_rw [<a>SimpleFunc.coe_comp</a>]; exact <a>SimpleFunc.norm_approxOn_zero_le</a> _ _ (x, y) n", [{"full_name": "MeasureTheory.Integrable", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [442, 5], "def_end_pos": [442, 15]}, {"full_name": "MeasureTheory.Integrable.mono'", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [476, 9], "def_end_pos": [476, 25]}, {"full_name": "MeasureTheory.SimpleFunc.aestronglyMeasurable", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1188, 9], "def_end_pos": [1188, 40]}, {"full_name": "Filter.eventually_of_forall", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1111, 9], "def_end_pos": [1111, 29]}, {"full_name": "MeasureTheory.SimpleFunc.coe_comp", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [337, 9], "def_end_pos": [337, 17]}, {"full_name": "MeasureTheory.SimpleFunc.norm_approxOn_zero_le", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "def_pos": [86, 9], "def_end_pos": [86, 30]}]], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na : \u03b1\nE : Type u_4\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : TopologicalSpace.SeparableSpace \u2191(range (uncurry f) \u222a {0})\ns : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n  SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n    (_ : 0 \u2208 range (uncurry f) \u222a {0})\ns' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x))\nf' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral (\u2191\u03ba x) (s' n x)\nhf' : \u2200 (n : \u2115), StronglyMeasurable (f' n)\nx : \u03b1\nhfx : Integrable (f x)\n\u22a2 Tendsto (fun i => f' i x) atTop (\ud835\udcdd (\u222b (y : \u03b2), f x y \u2202\u2191\u03ba x))", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na : \u03b1\nE : Type u_4\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b2 : MeasurableSpace E := borel E\nthis\u271d\u00b9 : BorelSpace E\nthis\u271d : TopologicalSpace.SeparableSpace \u2191(range (uncurry f) \u222a {0})\ns : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n  SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n    (_ : 0 \u2208 range (uncurry f) \u222a {0})\ns' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x))\nf' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral (\u2191\u03ba x) (s' n x)\nhf' : \u2200 (n : \u2115), StronglyMeasurable (f' n)\nx : \u03b1\nhfx : Integrable (f x)\nthis : \u2200 (n : \u2115), Integrable \u2191(s' n x)\n\u22a2 Tendsto (fun i => f' i x) atTop (\ud835\udcdd (\u222b (y : \u03b2), f x y \u2202\u2191\u03ba x))"}, {"tactic": "simp only [ hfx, SimpleFunc.integral_eq_integral _ (this _), indicator_of_mem,\n  mem_setOf_eq]", "annotated_tactic": ["simp only [ hfx, <a>SimpleFunc.integral_eq_integral</a> _ (this _), <a>indicator_of_mem</a>,\n        <a>mem_setOf_eq</a>]", [{"full_name": "MeasureTheory.SimpleFunc.integral_eq_integral", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1396, 9], "def_end_pos": [1396, 40]}, {"full_name": "Set.indicator_of_mem", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [67, 3], "def_end_pos": [67, 14]}, {"full_name": "Set.mem_setOf_eq", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [256, 29], "def_end_pos": [256, 41]}]], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na : \u03b1\nE : Type u_4\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b2 : MeasurableSpace E := borel E\nthis\u271d\u00b9 : BorelSpace E\nthis\u271d : TopologicalSpace.SeparableSpace \u2191(range (uncurry f) \u222a {0})\ns : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n  SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n    (_ : 0 \u2208 range (uncurry f) \u222a {0})\ns' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x))\nf' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral (\u2191\u03ba x) (s' n x)\nhf' : \u2200 (n : \u2115), StronglyMeasurable (f' n)\nx : \u03b1\nhfx : Integrable (f x)\nthis : \u2200 (n : \u2115), Integrable \u2191(s' n x)\n\u22a2 Tendsto (fun i => f' i x) atTop (\ud835\udcdd (\u222b (y : \u03b2), f x y \u2202\u2191\u03ba x))", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na : \u03b1\nE : Type u_4\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b2 : MeasurableSpace E := borel E\nthis\u271d\u00b9 : BorelSpace E\nthis\u271d : TopologicalSpace.SeparableSpace \u2191(range (uncurry f) \u222a {0})\ns : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n  SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n    (_ : 0 \u2208 range (uncurry f) \u222a {0})\ns' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x))\nf' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral (\u2191\u03ba x) (s' n x)\nhf' : \u2200 (n : \u2115), StronglyMeasurable (f' n)\nx : \u03b1\nhfx : Integrable (f x)\nthis : \u2200 (n : \u2115), Integrable \u2191(s' n x)\n\u22a2 Tendsto\n    (fun i =>\n      \u222b (x_1 : \u03b2),\n        \u2191(SimpleFunc.comp\n              (SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n                (_ : 0 \u2208 range (uncurry f) \u222a {0}) i)\n              (Prod.mk x) (_ : Measurable (Prod.mk x)))\n          x_1 \u2202\u2191\u03ba x)\n    atTop (\ud835\udcdd (\u222b (y : \u03b2), f x y \u2202\u2191\u03ba x))"}, {"tactic": "refine'\n  tendsto_integral_of_dominated_convergence (fun y => \u2016f x y\u2016 + \u2016f x y\u2016)\n    (fun n => (s' n x).aestronglyMeasurable) (hfx.norm.add hfx.norm) _ _", "annotated_tactic": ["refine'\n        <a>tendsto_integral_of_dominated_convergence</a> (fun y => \u2016f x y\u2016 + \u2016f x y\u2016)\n          (fun n => (s' n x).<a>aestronglyMeasurable</a>) (hfx.norm.add hfx.norm) _ _", [{"full_name": "MeasureTheory.tendsto_integral_of_dominated_convergence", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1023, 9], "def_end_pos": [1023, 50]}, {"full_name": "MeasureTheory.SimpleFunc.aestronglyMeasurable", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1188, 9], "def_end_pos": [1188, 40]}]], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na : \u03b1\nE : Type u_4\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b2 : MeasurableSpace E := borel E\nthis\u271d\u00b9 : BorelSpace E\nthis\u271d : TopologicalSpace.SeparableSpace \u2191(range (uncurry f) \u222a {0})\ns : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n  SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n    (_ : 0 \u2208 range (uncurry f) \u222a {0})\ns' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x))\nf' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral (\u2191\u03ba x) (s' n x)\nhf' : \u2200 (n : \u2115), StronglyMeasurable (f' n)\nx : \u03b1\nhfx : Integrable (f x)\nthis : \u2200 (n : \u2115), Integrable \u2191(s' n x)\n\u22a2 Tendsto\n    (fun i =>\n      \u222b (x_1 : \u03b2),\n        \u2191(SimpleFunc.comp\n              (SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n                (_ : 0 \u2208 range (uncurry f) \u222a {0}) i)\n              (Prod.mk x) (_ : Measurable (Prod.mk x)))\n          x_1 \u2202\u2191\u03ba x)\n    atTop (\ud835\udcdd (\u222b (y : \u03b2), f x y \u2202\u2191\u03ba x))", "state_after": "case pos.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na : \u03b1\nE : Type u_4\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b2 : MeasurableSpace E := borel E\nthis\u271d\u00b9 : BorelSpace E\nthis\u271d : TopologicalSpace.SeparableSpace \u2191(range (uncurry f) \u222a {0})\ns : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n  SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n    (_ : 0 \u2208 range (uncurry f) \u222a {0})\ns' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x))\nf' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral (\u2191\u03ba x) (s' n x)\nhf' : \u2200 (n : \u2115), StronglyMeasurable (f' n)\nx : \u03b1\nhfx : Integrable (f x)\nthis : \u2200 (n : \u2115), Integrable \u2191(s' n x)\n\u22a2 \u2200 (n : \u2115),\n    \u2200\u1d50 (a : \u03b2) \u2202\u2191\u03ba x,\n      \u2016\u2191(SimpleFunc.comp\n                (SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n                  (_ : 0 \u2208 range (uncurry f) \u222a {0}) n)\n                (Prod.mk x) (_ : Measurable (Prod.mk x)))\n            a\u2016 \u2264\n        (fun y => \u2016f x y\u2016 + \u2016f x y\u2016) a\n\ncase pos.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na : \u03b1\nE : Type u_4\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b2 : MeasurableSpace E := borel E\nthis\u271d\u00b9 : BorelSpace E\nthis\u271d : TopologicalSpace.SeparableSpace \u2191(range (uncurry f) \u222a {0})\ns : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n  SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n    (_ : 0 \u2208 range (uncurry f) \u222a {0})\ns' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x))\nf' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral (\u2191\u03ba x) (s' n x)\nhf' : \u2200 (n : \u2115), StronglyMeasurable (f' n)\nx : \u03b1\nhfx : Integrable (f x)\nthis : \u2200 (n : \u2115), Integrable \u2191(s' n x)\n\u22a2 \u2200\u1d50 (a : \u03b2) \u2202\u2191\u03ba x,\n    Tendsto\n      (fun n =>\n        \u2191(SimpleFunc.comp\n              (SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n                (_ : 0 \u2208 range (uncurry f) \u222a {0}) n)\n              (Prod.mk x) (_ : Measurable (Prod.mk x)))\n          a)\n      atTop (\ud835\udcdd (f x a))"}, {"tactic": "intro n", "annotated_tactic": ["intro n", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na : \u03b1\nE : Type u_4\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : TopologicalSpace.SeparableSpace \u2191(range (uncurry f) \u222a {0})\ns : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n  SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n    (_ : 0 \u2208 range (uncurry f) \u222a {0})\ns' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x))\nf' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral (\u2191\u03ba x) (s' n x)\nhf' : \u2200 (n : \u2115), StronglyMeasurable (f' n)\nx : \u03b1\nhfx : Integrable (f x)\n\u22a2 \u2200 (n : \u2115), Integrable \u2191(s' n x)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na : \u03b1\nE : Type u_4\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : TopologicalSpace.SeparableSpace \u2191(range (uncurry f) \u222a {0})\ns : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n  SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n    (_ : 0 \u2208 range (uncurry f) \u222a {0})\ns' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x))\nf' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral (\u2191\u03ba x) (s' n x)\nhf' : \u2200 (n : \u2115), StronglyMeasurable (f' n)\nx : \u03b1\nhfx : Integrable (f x)\nn : \u2115\n\u22a2 Integrable \u2191(s' n x)"}, {"tactic": "apply (hfx.norm.add hfx.norm).mono' (s' n x).aestronglyMeasurable", "annotated_tactic": ["apply (hfx.norm.add hfx.norm).<a>mono'</a> (s' n x).<a>aestronglyMeasurable</a>", [{"full_name": "MeasureTheory.Integrable.mono'", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [476, 9], "def_end_pos": [476, 25]}, {"full_name": "MeasureTheory.SimpleFunc.aestronglyMeasurable", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1188, 9], "def_end_pos": [1188, 40]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na : \u03b1\nE : Type u_4\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : TopologicalSpace.SeparableSpace \u2191(range (uncurry f) \u222a {0})\ns : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n  SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n    (_ : 0 \u2208 range (uncurry f) \u222a {0})\ns' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x))\nf' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral (\u2191\u03ba x) (s' n x)\nhf' : \u2200 (n : \u2115), StronglyMeasurable (f' n)\nx : \u03b1\nhfx : Integrable (f x)\nn : \u2115\n\u22a2 Integrable \u2191(s' n x)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na : \u03b1\nE : Type u_4\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : TopologicalSpace.SeparableSpace \u2191(range (uncurry f) \u222a {0})\ns : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n  SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n    (_ : 0 \u2208 range (uncurry f) \u222a {0})\ns' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x))\nf' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral (\u2191\u03ba x) (s' n x)\nhf' : \u2200 (n : \u2115), StronglyMeasurable (f' n)\nx : \u03b1\nhfx : Integrable (f x)\nn : \u2115\n\u22a2 \u2200\u1d50 (a : \u03b2) \u2202\u2191\u03ba x, \u2016\u2191(s' n x) a\u2016 \u2264 ((fun a => \u2016f x a\u2016) + fun a => \u2016f x a\u2016) a"}, {"tactic": "apply eventually_of_forall", "annotated_tactic": ["apply <a>eventually_of_forall</a>", [{"full_name": "Filter.eventually_of_forall", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1111, 9], "def_end_pos": [1111, 29]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na : \u03b1\nE : Type u_4\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : TopologicalSpace.SeparableSpace \u2191(range (uncurry f) \u222a {0})\ns : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n  SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n    (_ : 0 \u2208 range (uncurry f) \u222a {0})\ns' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x))\nf' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral (\u2191\u03ba x) (s' n x)\nhf' : \u2200 (n : \u2115), StronglyMeasurable (f' n)\nx : \u03b1\nhfx : Integrable (f x)\nn : \u2115\n\u22a2 \u2200\u1d50 (a : \u03b2) \u2202\u2191\u03ba x, \u2016\u2191(s' n x) a\u2016 \u2264 ((fun a => \u2016f x a\u2016) + fun a => \u2016f x a\u2016) a", "state_after": "case hp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na : \u03b1\nE : Type u_4\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : TopologicalSpace.SeparableSpace \u2191(range (uncurry f) \u222a {0})\ns : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n  SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n    (_ : 0 \u2208 range (uncurry f) \u222a {0})\ns' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x))\nf' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral (\u2191\u03ba x) (s' n x)\nhf' : \u2200 (n : \u2115), StronglyMeasurable (f' n)\nx : \u03b1\nhfx : Integrable (f x)\nn : \u2115\n\u22a2 \u2200 (x_1 : \u03b2), \u2016\u2191(s' n x) x_1\u2016 \u2264 ((fun a => \u2016f x a\u2016) + fun a => \u2016f x a\u2016) x_1"}, {"tactic": "intro y", "annotated_tactic": ["intro y", []], "state_before": "case hp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na : \u03b1\nE : Type u_4\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : TopologicalSpace.SeparableSpace \u2191(range (uncurry f) \u222a {0})\ns : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n  SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n    (_ : 0 \u2208 range (uncurry f) \u222a {0})\ns' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x))\nf' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral (\u2191\u03ba x) (s' n x)\nhf' : \u2200 (n : \u2115), StronglyMeasurable (f' n)\nx : \u03b1\nhfx : Integrable (f x)\nn : \u2115\n\u22a2 \u2200 (x_1 : \u03b2), \u2016\u2191(s' n x) x_1\u2016 \u2264 ((fun a => \u2016f x a\u2016) + fun a => \u2016f x a\u2016) x_1", "state_after": "case hp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na : \u03b1\nE : Type u_4\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : TopologicalSpace.SeparableSpace \u2191(range (uncurry f) \u222a {0})\ns : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n  SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n    (_ : 0 \u2208 range (uncurry f) \u222a {0})\ns' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x))\nf' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral (\u2191\u03ba x) (s' n x)\nhf' : \u2200 (n : \u2115), StronglyMeasurable (f' n)\nx : \u03b1\nhfx : Integrable (f x)\nn : \u2115\ny : \u03b2\n\u22a2 \u2016\u2191(s' n x) y\u2016 \u2264 ((fun a => \u2016f x a\u2016) + fun a => \u2016f x a\u2016) y"}, {"tactic": "simp_rw [SimpleFunc.coe_comp]", "annotated_tactic": ["simp_rw [<a>SimpleFunc.coe_comp</a>]", [{"full_name": "MeasureTheory.SimpleFunc.coe_comp", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [337, 9], "def_end_pos": [337, 17]}]], "state_before": "case hp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na : \u03b1\nE : Type u_4\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : TopologicalSpace.SeparableSpace \u2191(range (uncurry f) \u222a {0})\ns : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n  SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n    (_ : 0 \u2208 range (uncurry f) \u222a {0})\ns' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x))\nf' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral (\u2191\u03ba x) (s' n x)\nhf' : \u2200 (n : \u2115), StronglyMeasurable (f' n)\nx : \u03b1\nhfx : Integrable (f x)\nn : \u2115\ny : \u03b2\n\u22a2 \u2016\u2191(s' n x) y\u2016 \u2264 ((fun a => \u2016f x a\u2016) + fun a => \u2016f x a\u2016) y", "state_after": "case hp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na : \u03b1\nE : Type u_4\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : TopologicalSpace.SeparableSpace \u2191(range (uncurry f) \u222a {0})\ns : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n  SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n    (_ : 0 \u2208 range (uncurry f) \u222a {0})\ns' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x))\nf' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral (\u2191\u03ba x) (s' n x)\nhf' : \u2200 (n : \u2115), StronglyMeasurable (f' n)\nx : \u03b1\nhfx : Integrable (f x)\nn : \u2115\ny : \u03b2\n\u22a2 \u2016(\u2191(SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n              (_ : 0 \u2208 range (uncurry f) \u222a {0}) n) \u2218\n          Prod.mk x)\n        y\u2016 \u2264\n    ((fun a => \u2016f x a\u2016) + fun a => \u2016f x a\u2016) y"}, {"tactic": "exact SimpleFunc.norm_approxOn_zero_le _ _ (x, y) n", "annotated_tactic": ["exact <a>SimpleFunc.norm_approxOn_zero_le</a> _ _ (x, y) n", [{"full_name": "MeasureTheory.SimpleFunc.norm_approxOn_zero_le", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "def_pos": [86, 9], "def_end_pos": [86, 30]}]], "state_before": "case hp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na : \u03b1\nE : Type u_4\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : TopologicalSpace.SeparableSpace \u2191(range (uncurry f) \u222a {0})\ns : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n  SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n    (_ : 0 \u2208 range (uncurry f) \u222a {0})\ns' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x))\nf' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral (\u2191\u03ba x) (s' n x)\nhf' : \u2200 (n : \u2115), StronglyMeasurable (f' n)\nx : \u03b1\nhfx : Integrable (f x)\nn : \u2115\ny : \u03b2\n\u22a2 \u2016(\u2191(SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n              (_ : 0 \u2208 range (uncurry f) \u222a {0}) n) \u2218\n          Prod.mk x)\n        y\u2016 \u2264\n    ((fun a => \u2016f x a\u2016) + fun a => \u2016f x a\u2016) y", "state_after": "no goals"}, {"tactic": "exact fun n => eventually_of_forall fun y =>\n  SimpleFunc.norm_approxOn_zero_le hf.measurable (by simp) (x, y) n", "annotated_tactic": ["exact fun n => <a>eventually_of_forall</a> fun y =>\n          <a>SimpleFunc.norm_approxOn_zero_le</a> hf.measurable (by simp) (x, y) n", [{"full_name": "Filter.eventually_of_forall", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1111, 9], "def_end_pos": [1111, 29]}, {"full_name": "MeasureTheory.SimpleFunc.norm_approxOn_zero_le", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "def_pos": [86, 9], "def_end_pos": [86, 30]}]], "state_before": "case pos.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na : \u03b1\nE : Type u_4\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b2 : MeasurableSpace E := borel E\nthis\u271d\u00b9 : BorelSpace E\nthis\u271d : TopologicalSpace.SeparableSpace \u2191(range (uncurry f) \u222a {0})\ns : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n  SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n    (_ : 0 \u2208 range (uncurry f) \u222a {0})\ns' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x))\nf' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral (\u2191\u03ba x) (s' n x)\nhf' : \u2200 (n : \u2115), StronglyMeasurable (f' n)\nx : \u03b1\nhfx : Integrable (f x)\nthis : \u2200 (n : \u2115), Integrable \u2191(s' n x)\n\u22a2 \u2200 (n : \u2115),\n    \u2200\u1d50 (a : \u03b2) \u2202\u2191\u03ba x,\n      \u2016\u2191(SimpleFunc.comp\n                (SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n                  (_ : 0 \u2208 range (uncurry f) \u222a {0}) n)\n                (Prod.mk x) (_ : Measurable (Prod.mk x)))\n            a\u2016 \u2264\n        (fun y => \u2016f x y\u2016 + \u2016f x y\u2016) a", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na : \u03b1\nE : Type u_4\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b2 : MeasurableSpace E := borel E\nthis\u271d\u00b9 : BorelSpace E\nthis\u271d : TopologicalSpace.SeparableSpace \u2191(range (uncurry f) \u222a {0})\ns : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n  SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n    (_ : 0 \u2208 range (uncurry f) \u222a {0})\ns' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x))\nf' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral (\u2191\u03ba x) (s' n x)\nhf' : \u2200 (n : \u2115), StronglyMeasurable (f' n)\nx : \u03b1\nhfx : Integrable (f x)\nthis : \u2200 (n : \u2115), Integrable \u2191(s' n x)\nn : \u2115\ny : \u03b2\n\u22a2 0 \u2208 range (uncurry f) \u222a {0}", "state_after": "no goals"}, {"tactic": "refine' eventually_of_forall fun y => SimpleFunc.tendsto_approxOn hf.measurable (by simp) _", "annotated_tactic": ["refine' <a>eventually_of_forall</a> fun y => <a>SimpleFunc.tendsto_approxOn</a> hf.measurable (by simp) _", [{"full_name": "Filter.eventually_of_forall", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1111, 9], "def_end_pos": [1111, 29]}, {"full_name": "MeasureTheory.SimpleFunc.tendsto_approxOn", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDense.lean", "def_pos": [154, 9], "def_end_pos": [154, 25]}]], "state_before": "case pos.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na : \u03b1\nE : Type u_4\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b2 : MeasurableSpace E := borel E\nthis\u271d\u00b9 : BorelSpace E\nthis\u271d : TopologicalSpace.SeparableSpace \u2191(range (uncurry f) \u222a {0})\ns : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n  SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n    (_ : 0 \u2208 range (uncurry f) \u222a {0})\ns' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x))\nf' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral (\u2191\u03ba x) (s' n x)\nhf' : \u2200 (n : \u2115), StronglyMeasurable (f' n)\nx : \u03b1\nhfx : Integrable (f x)\nthis : \u2200 (n : \u2115), Integrable \u2191(s' n x)\n\u22a2 \u2200\u1d50 (a : \u03b2) \u2202\u2191\u03ba x,\n    Tendsto\n      (fun n =>\n        \u2191(SimpleFunc.comp\n              (SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n                (_ : 0 \u2208 range (uncurry f) \u222a {0}) n)\n              (Prod.mk x) (_ : Measurable (Prod.mk x)))\n          a)\n      atTop (\ud835\udcdd (f x a))", "state_after": "case pos.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na : \u03b1\nE : Type u_4\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b2 : MeasurableSpace E := borel E\nthis\u271d\u00b9 : BorelSpace E\nthis\u271d : TopologicalSpace.SeparableSpace \u2191(range (uncurry f) \u222a {0})\ns : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n  SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n    (_ : 0 \u2208 range (uncurry f) \u222a {0})\ns' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x))\nf' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral (\u2191\u03ba x) (s' n x)\nhf' : \u2200 (n : \u2115), StronglyMeasurable (f' n)\nx : \u03b1\nhfx : Integrable (f x)\nthis : \u2200 (n : \u2115), Integrable \u2191(s' n x)\ny : \u03b2\n\u22a2 uncurry f (x, y) \u2208 closure (range (uncurry f) \u222a {0})"}, {"tactic": "apply subset_closure", "annotated_tactic": ["apply <a>subset_closure</a>", [{"full_name": "subset_closure", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [435, 9], "def_end_pos": [435, 23]}]], "state_before": "case pos.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na : \u03b1\nE : Type u_4\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b2 : MeasurableSpace E := borel E\nthis\u271d\u00b9 : BorelSpace E\nthis\u271d : TopologicalSpace.SeparableSpace \u2191(range (uncurry f) \u222a {0})\ns : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n  SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n    (_ : 0 \u2208 range (uncurry f) \u222a {0})\ns' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x))\nf' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral (\u2191\u03ba x) (s' n x)\nhf' : \u2200 (n : \u2115), StronglyMeasurable (f' n)\nx : \u03b1\nhfx : Integrable (f x)\nthis : \u2200 (n : \u2115), Integrable \u2191(s' n x)\ny : \u03b2\n\u22a2 uncurry f (x, y) \u2208 closure (range (uncurry f) \u222a {0})", "state_after": "case pos.refine'_2.a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na : \u03b1\nE : Type u_4\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b2 : MeasurableSpace E := borel E\nthis\u271d\u00b9 : BorelSpace E\nthis\u271d : TopologicalSpace.SeparableSpace \u2191(range (uncurry f) \u222a {0})\ns : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n  SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n    (_ : 0 \u2208 range (uncurry f) \u222a {0})\ns' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x))\nf' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral (\u2191\u03ba x) (s' n x)\nhf' : \u2200 (n : \u2115), StronglyMeasurable (f' n)\nx : \u03b1\nhfx : Integrable (f x)\nthis : \u2200 (n : \u2115), Integrable \u2191(s' n x)\ny : \u03b2\n\u22a2 uncurry f (x, y) \u2208 range (uncurry f) \u222a {0}"}, {"tactic": "simp [-uncurry_apply_pair]", "annotated_tactic": ["simp [-<a>uncurry_apply_pair</a>]", [{"full_name": "Function.uncurry_apply_pair", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [817, 9], "def_end_pos": [817, 27]}]], "state_before": "case pos.refine'_2.a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na : \u03b1\nE : Type u_4\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b2 : MeasurableSpace E := borel E\nthis\u271d\u00b9 : BorelSpace E\nthis\u271d : TopologicalSpace.SeparableSpace \u2191(range (uncurry f) \u222a {0})\ns : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n  SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n    (_ : 0 \u2208 range (uncurry f) \u222a {0})\ns' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x))\nf' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral (\u2191\u03ba x) (s' n x)\nhf' : \u2200 (n : \u2115), StronglyMeasurable (f' n)\nx : \u03b1\nhfx : Integrable (f x)\nthis : \u2200 (n : \u2115), Integrable \u2191(s' n x)\ny : \u03b2\n\u22a2 uncurry f (x, y) \u2208 range (uncurry f) \u222a {0}", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na : \u03b1\nE : Type u_4\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b2 : MeasurableSpace E := borel E\nthis\u271d\u00b9 : BorelSpace E\nthis\u271d : TopologicalSpace.SeparableSpace \u2191(range (uncurry f) \u222a {0})\ns : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n  SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n    (_ : 0 \u2208 range (uncurry f) \u222a {0})\ns' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x))\nf' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral (\u2191\u03ba x) (s' n x)\nhf' : \u2200 (n : \u2115), StronglyMeasurable (f' n)\nx : \u03b1\nhfx : Integrable (f x)\nthis : \u2200 (n : \u2115), Integrable \u2191(s' n x)\ny : \u03b2\n\u22a2 0 \u2208 range (uncurry f) \u222a {0}", "state_after": "no goals"}, {"tactic": "simp [hfx, integral_undef]", "annotated_tactic": ["simp [hfx, <a>integral_undef</a>]", [{"full_name": "MeasureTheory.integral_undef", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [836, 9], "def_end_pos": [836, 23]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na : \u03b1\nE : Type u_4\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : TopologicalSpace.SeparableSpace \u2191(range (uncurry f) \u222a {0})\ns : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n  SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n    (_ : 0 \u2208 range (uncurry f) \u222a {0})\ns' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x))\nf' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral (\u2191\u03ba x) (s' n x)\nhf' : \u2200 (n : \u2115), StronglyMeasurable (f' n)\nx : \u03b1\nhfx : \u00acIntegrable (f x)\n\u22a2 Tendsto (fun i => f' i x) atTop (\ud835\udcdd (\u222b (y : \u03b2), f x y \u2202\u2191\u03ba x))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "full_name": "MeasurableSpace.measurableSet_iInf", "start": [495, 1], "end": [497, 50], "traced_tactics": [{"tactic": "rw [iInf, measurableSet_sInf, forall_range_iff]", "annotated_tactic": ["rw [<a>iInf</a>, <a>measurableSet_sInf</a>, <a>forall_range_iff</a>]", [{"full_name": "iInf", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [83, 5], "def_end_pos": [83, 9]}, {"full_name": "MeasurableSpace.measurableSet_sInf", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [490, 9], "def_end_pos": [490, 27]}, {"full_name": "Set.forall_range_iff", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [684, 9], "def_end_pos": [684, 25]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9\u271d : Sort u_6\ns\u271d t u : Set \u03b1\n\u03b9 : Sort u_7\nm : \u03b9 \u2192 MeasurableSpace \u03b1\ns : Set \u03b1\n\u22a2 MeasurableSet s \u2194 \u2200 (i : \u03b9), MeasurableSet s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Card.lean", "full_name": "Set.Finite.exists_injOn_of_encard_le", "start": [417, 1], "end": [440, 29], "traced_tactics": [{"tactic": "obtain (rfl | h | \u27e8a, has, -\u27e9) := s.eq_empty_or_encard_eq_top_or_encard_diff_singleton_lt", "annotated_tactic": ["obtain (rfl | h | \u27e8a, has, -\u27e9) := s.eq_empty_or_encard_eq_top_or_encard_diff_singleton_lt", []], "state_before": "\u03b1 : Type u_2\n\u03b2 : Type u_1\ns\u271d\u00b9 t\u271d\u00b9 s\u271d : Set \u03b1\nt\u271d : Set \u03b2\nf : \u03b1 \u2192 \u03b2\ninst\u271d : Nonempty \u03b2\ns : Set \u03b1\nt : Set \u03b2\nhs : Set.Finite s\nhle : encard s \u2264 encard t\n\u22a2 \u2203 f, s \u2286 f \u207b\u00b9' t \u2227 InjOn f s", "state_after": "case inl\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ns\u271d t\u271d\u00b9 s : Set \u03b1\nt\u271d : Set \u03b2\nf : \u03b1 \u2192 \u03b2\ninst\u271d : Nonempty \u03b2\nt : Set \u03b2\nhs : Set.Finite \u2205\nhle : encard \u2205 \u2264 encard t\n\u22a2 \u2203 f, \u2205 \u2286 f \u207b\u00b9' t \u2227 InjOn f \u2205\n\ncase inr.inl\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ns\u271d\u00b9 t\u271d\u00b9 s\u271d : Set \u03b1\nt\u271d : Set \u03b2\nf : \u03b1 \u2192 \u03b2\ninst\u271d : Nonempty \u03b2\ns : Set \u03b1\nt : Set \u03b2\nhs : Set.Finite s\nhle : encard s \u2264 encard t\nh : encard s = \u22a4\n\u22a2 \u2203 f, s \u2286 f \u207b\u00b9' t \u2227 InjOn f s\n\ncase inr.inr.intro.intro\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ns\u271d\u00b9 t\u271d\u00b9 s\u271d : Set \u03b1\nt\u271d : Set \u03b2\nf : \u03b1 \u2192 \u03b2\ninst\u271d : Nonempty \u03b2\ns : Set \u03b1\nt : Set \u03b2\nhs : Set.Finite s\nhle : encard s \u2264 encard t\na : \u03b1\nhas : a \u2208 s\n\u22a2 \u2203 f, s \u2286 f \u207b\u00b9' t \u2227 InjOn f s"}, {"tactic": "obtain \u27e8b, hbt\u27e9 := encard_pos.1 ((encard_pos.2 \u27e8_, has\u27e9).trans_le hle)", "annotated_tactic": ["obtain \u27e8b, hbt\u27e9 := <a>encard_pos</a>.1 ((<a>encard_pos</a>.2 \u27e8_, has\u27e9).<a>trans_le</a> hle)", [{"full_name": "Set.encard_pos", "def_path": "Mathlib/Data/Set/Card.lean", "def_pos": [106, 17], "def_end_pos": [106, 27]}, {"full_name": "Set.encard_pos", "def_path": "Mathlib/Data/Set/Card.lean", "def_pos": [106, 17], "def_end_pos": [106, 27]}, {"full_name": "LT.lt.trans_le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [148, 7], "def_end_pos": [148, 21]}]], "state_before": "case inr.inr.intro.intro\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ns\u271d\u00b9 t\u271d\u00b9 s\u271d : Set \u03b1\nt\u271d : Set \u03b2\nf : \u03b1 \u2192 \u03b2\ninst\u271d : Nonempty \u03b2\ns : Set \u03b1\nt : Set \u03b2\nhs : Set.Finite s\nhle : encard s \u2264 encard t\na : \u03b1\nhas : a \u2208 s\n\u22a2 \u2203 f, s \u2286 f \u207b\u00b9' t \u2227 InjOn f s", "state_after": "case inr.inr.intro.intro.intro\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ns\u271d\u00b9 t\u271d\u00b9 s\u271d : Set \u03b1\nt\u271d : Set \u03b2\nf : \u03b1 \u2192 \u03b2\ninst\u271d : Nonempty \u03b2\ns : Set \u03b1\nt : Set \u03b2\nhs : Set.Finite s\nhle : encard s \u2264 encard t\na : \u03b1\nhas : a \u2208 s\nb : \u03b2\nhbt : b \u2208 t\n\u22a2 \u2203 f, s \u2286 f \u207b\u00b9' t \u2227 InjOn f s"}, {"tactic": "have hle' : (s \\ {a}).encard \u2264 (t \\ {b}).encard", "annotated_tactic": ["have hle' : (s \\ {a}).<a>encard</a> \u2264 (t \\ {b}).<a>encard</a>", [{"full_name": "Set.encard", "def_path": "Mathlib/Data/Set/Card.lean", "def_pos": [66, 19], "def_end_pos": [66, 25]}, {"full_name": "Set.encard", "def_path": "Mathlib/Data/Set/Card.lean", "def_pos": [66, 19], "def_end_pos": [66, 25]}]], "state_before": "case inr.inr.intro.intro.intro\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ns\u271d\u00b9 t\u271d\u00b9 s\u271d : Set \u03b1\nt\u271d : Set \u03b2\nf : \u03b1 \u2192 \u03b2\ninst\u271d : Nonempty \u03b2\ns : Set \u03b1\nt : Set \u03b2\nhs : Set.Finite s\nhle : encard s \u2264 encard t\na : \u03b1\nhas : a \u2208 s\nb : \u03b2\nhbt : b \u2208 t\n\u22a2 \u2203 f, s \u2286 f \u207b\u00b9' t \u2227 InjOn f s", "state_after": "case hle'\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ns\u271d\u00b9 t\u271d\u00b9 s\u271d : Set \u03b1\nt\u271d : Set \u03b2\nf : \u03b1 \u2192 \u03b2\ninst\u271d : Nonempty \u03b2\ns : Set \u03b1\nt : Set \u03b2\nhs : Set.Finite s\nhle : encard s \u2264 encard t\na : \u03b1\nhas : a \u2208 s\nb : \u03b2\nhbt : b \u2208 t\n\u22a2 encard (s \\ {a}) \u2264 encard (t \\ {b})\n\ncase inr.inr.intro.intro.intro\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ns\u271d\u00b9 t\u271d\u00b9 s\u271d : Set \u03b1\nt\u271d : Set \u03b2\nf : \u03b1 \u2192 \u03b2\ninst\u271d : Nonempty \u03b2\ns : Set \u03b1\nt : Set \u03b2\nhs : Set.Finite s\nhle : encard s \u2264 encard t\na : \u03b1\nhas : a \u2208 s\nb : \u03b2\nhbt : b \u2208 t\nhle' : encard (s \\ {a}) \u2264 encard (t \\ {b})\n\u22a2 \u2203 f, s \u2286 f \u207b\u00b9' t \u2227 InjOn f s"}, {"tactic": "obtain \u27e8f\u2080, hf\u2080s, hinj\u27e9 := exists_injOn_of_encard_le (hs.diff {a}) hle'", "annotated_tactic": ["obtain \u27e8f\u2080, hf\u2080s, hinj\u27e9 := exists_injOn_of_encard_le (hs.diff {a}) hle'", []], "state_before": "case inr.inr.intro.intro.intro\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ns\u271d\u00b9 t\u271d\u00b9 s\u271d : Set \u03b1\nt\u271d : Set \u03b2\nf : \u03b1 \u2192 \u03b2\ninst\u271d : Nonempty \u03b2\ns : Set \u03b1\nt : Set \u03b2\nhs : Set.Finite s\nhle : encard s \u2264 encard t\na : \u03b1\nhas : a \u2208 s\nb : \u03b2\nhbt : b \u2208 t\nhle' : encard (s \\ {a}) \u2264 encard (t \\ {b})\n\u22a2 \u2203 f, s \u2286 f \u207b\u00b9' t \u2227 InjOn f s", "state_after": "case inr.inr.intro.intro.intro.intro.intro\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ns\u271d\u00b9 t\u271d\u00b9 s\u271d : Set \u03b1\nt\u271d : Set \u03b2\nf : \u03b1 \u2192 \u03b2\ninst\u271d : Nonempty \u03b2\ns : Set \u03b1\nt : Set \u03b2\nhs : Set.Finite s\nhle : encard s \u2264 encard t\na : \u03b1\nhas : a \u2208 s\nb : \u03b2\nhbt : b \u2208 t\nhle' : encard (s \\ {a}) \u2264 encard (t \\ {b})\nf\u2080 : \u03b1 \u2192 \u03b2\nhf\u2080s : s \\ {a} \u2286 f\u2080 \u207b\u00b9' (t \\ {b})\nhinj : InjOn f\u2080 (s \\ {a})\n\u22a2 \u2203 f, s \u2286 f \u207b\u00b9' t \u2227 InjOn f s"}, {"tactic": "simp only [preimage_diff, subset_def, mem_diff, mem_singleton_iff, mem_preimage, and_imp] at hf\u2080s", "annotated_tactic": ["simp only [<a>preimage_diff</a>, <a>subset_def</a>, <a>mem_diff</a>, <a>mem_singleton_iff</a>, <a>mem_preimage</a>, <a>and_imp</a>] at hf\u2080s", [{"full_name": "Set.preimage_diff", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [102, 9], "def_end_pos": [102, 22]}, {"full_name": "Set.subset_def", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [345, 9], "def_end_pos": [345, 19]}, {"full_name": "Set.mem_diff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1819, 9], "def_end_pos": [1819, 17]}, {"full_name": "Set.mem_singleton_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1273, 9], "def_end_pos": [1273, 26]}, {"full_name": "Set.mem_preimage", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [64, 9], "def_end_pos": [64, 21]}, {"full_name": "and_imp", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [313, 17], "def_end_pos": [313, 24]}]], "state_before": "case inr.inr.intro.intro.intro.intro.intro\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ns\u271d\u00b9 t\u271d\u00b9 s\u271d : Set \u03b1\nt\u271d : Set \u03b2\nf : \u03b1 \u2192 \u03b2\ninst\u271d : Nonempty \u03b2\ns : Set \u03b1\nt : Set \u03b2\nhs : Set.Finite s\nhle : encard s \u2264 encard t\na : \u03b1\nhas : a \u2208 s\nb : \u03b2\nhbt : b \u2208 t\nhle' : encard (s \\ {a}) \u2264 encard (t \\ {b})\nf\u2080 : \u03b1 \u2192 \u03b2\nhf\u2080s : s \\ {a} \u2286 f\u2080 \u207b\u00b9' (t \\ {b})\nhinj : InjOn f\u2080 (s \\ {a})\n\u22a2 \u2203 f, s \u2286 f \u207b\u00b9' t \u2227 InjOn f s", "state_after": "case inr.inr.intro.intro.intro.intro.intro\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ns\u271d\u00b9 t\u271d\u00b9 s\u271d : Set \u03b1\nt\u271d : Set \u03b2\nf : \u03b1 \u2192 \u03b2\ninst\u271d : Nonempty \u03b2\ns : Set \u03b1\nt : Set \u03b2\nhs : Set.Finite s\nhle : encard s \u2264 encard t\na : \u03b1\nhas : a \u2208 s\nb : \u03b2\nhbt : b \u2208 t\nhle' : encard (s \\ {a}) \u2264 encard (t \\ {b})\nf\u2080 : \u03b1 \u2192 \u03b2\nhinj : InjOn f\u2080 (s \\ {a})\nhf\u2080s : \u2200 (x : \u03b1), x \u2208 s \u2192 \u00acx = a \u2192 f\u2080 x \u2208 t \u2227 \u00acf\u2080 x = b\n\u22a2 \u2203 f, s \u2286 f \u207b\u00b9' t \u2227 InjOn f s"}, {"tactic": "use Function.update f\u2080 a b", "annotated_tactic": ["use <a>Function.update</a> f\u2080 a b", [{"full_name": "Function.update", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [550, 5], "def_end_pos": [550, 11]}]], "state_before": "case inr.inr.intro.intro.intro.intro.intro\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ns\u271d\u00b9 t\u271d\u00b9 s\u271d : Set \u03b1\nt\u271d : Set \u03b2\nf : \u03b1 \u2192 \u03b2\ninst\u271d : Nonempty \u03b2\ns : Set \u03b1\nt : Set \u03b2\nhs : Set.Finite s\nhle : encard s \u2264 encard t\na : \u03b1\nhas : a \u2208 s\nb : \u03b2\nhbt : b \u2208 t\nhle' : encard (s \\ {a}) \u2264 encard (t \\ {b})\nf\u2080 : \u03b1 \u2192 \u03b2\nhinj : InjOn f\u2080 (s \\ {a})\nhf\u2080s : \u2200 (x : \u03b1), x \u2208 s \u2192 \u00acx = a \u2192 f\u2080 x \u2208 t \u2227 \u00acf\u2080 x = b\n\u22a2 \u2203 f, s \u2286 f \u207b\u00b9' t \u2227 InjOn f s", "state_after": "case h\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ns\u271d\u00b9 t\u271d\u00b9 s\u271d : Set \u03b1\nt\u271d : Set \u03b2\nf : \u03b1 \u2192 \u03b2\ninst\u271d : Nonempty \u03b2\ns : Set \u03b1\nt : Set \u03b2\nhs : Set.Finite s\nhle : encard s \u2264 encard t\na : \u03b1\nhas : a \u2208 s\nb : \u03b2\nhbt : b \u2208 t\nhle' : encard (s \\ {a}) \u2264 encard (t \\ {b})\nf\u2080 : \u03b1 \u2192 \u03b2\nhinj : InjOn f\u2080 (s \\ {a})\nhf\u2080s : \u2200 (x : \u03b1), x \u2208 s \u2192 \u00acx = a \u2192 f\u2080 x \u2208 t \u2227 \u00acf\u2080 x = b\n\u22a2 s \u2286 Function.update f\u2080 a b \u207b\u00b9' t \u2227 InjOn (Function.update f\u2080 a b) s"}, {"tactic": "rw [\u2190insert_eq_of_mem has, \u2190insert_diff_singleton, injOn_insert (fun h \u21a6 h.2 rfl)]", "annotated_tactic": ["rw [\u2190<a>insert_eq_of_mem</a> has, \u2190<a>insert_diff_singleton</a>, <a>injOn_insert</a> (fun h \u21a6 h.2 <a>rfl</a>)]", [{"full_name": "Set.insert_eq_of_mem", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1152, 9], "def_end_pos": [1152, 25]}, {"full_name": "Set.insert_diff_singleton", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [2078, 9], "def_end_pos": [2078, 30]}, {"full_name": "Set.injOn_insert", "def_path": "Mathlib/Data/Set/Function.lean", "def_pos": [654, 9], "def_end_pos": [654, 21]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "case h\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ns\u271d\u00b9 t\u271d\u00b9 s\u271d : Set \u03b1\nt\u271d : Set \u03b2\nf : \u03b1 \u2192 \u03b2\ninst\u271d : Nonempty \u03b2\ns : Set \u03b1\nt : Set \u03b2\nhs : Set.Finite s\nhle : encard s \u2264 encard t\na : \u03b1\nhas : a \u2208 s\nb : \u03b2\nhbt : b \u2208 t\nhle' : encard (s \\ {a}) \u2264 encard (t \\ {b})\nf\u2080 : \u03b1 \u2192 \u03b2\nhinj : InjOn f\u2080 (s \\ {a})\nhf\u2080s : \u2200 (x : \u03b1), x \u2208 s \u2192 \u00acx = a \u2192 f\u2080 x \u2208 t \u2227 \u00acf\u2080 x = b\n\u22a2 s \u2286 Function.update f\u2080 a b \u207b\u00b9' t \u2227 InjOn (Function.update f\u2080 a b) s", "state_after": "case h\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ns\u271d\u00b9 t\u271d\u00b9 s\u271d : Set \u03b1\nt\u271d : Set \u03b2\nf : \u03b1 \u2192 \u03b2\ninst\u271d : Nonempty \u03b2\ns : Set \u03b1\nt : Set \u03b2\nhs : Set.Finite s\nhle : encard s \u2264 encard t\na : \u03b1\nhas : a \u2208 s\nb : \u03b2\nhbt : b \u2208 t\nhle' : encard (s \\ {a}) \u2264 encard (t \\ {b})\nf\u2080 : \u03b1 \u2192 \u03b2\nhinj : InjOn f\u2080 (s \\ {a})\nhf\u2080s : \u2200 (x : \u03b1), x \u2208 s \u2192 \u00acx = a \u2192 f\u2080 x \u2208 t \u2227 \u00acf\u2080 x = b\n\u22a2 insert a (s \\ {a}) \u2286 Function.update f\u2080 a b \u207b\u00b9' t \u2227\n    InjOn (Function.update f\u2080 a b) (s \\ {a}) \u2227 \u00acFunction.update f\u2080 a b a \u2208 Function.update f\u2080 a b '' (s \\ {a})"}, {"tactic": "simp only [mem_diff, mem_singleton_iff, not_true, and_false, insert_diff_singleton, subset_def,\n  mem_insert_iff, mem_preimage, ne_eq, Function.update_apply, forall_eq_or_imp, ite_true, and_imp,\n  mem_image, ite_eq_left_iff, not_exists, not_and, not_forall, exists_prop, and_iff_right hbt]", "annotated_tactic": ["simp only [<a>mem_diff</a>, <a>mem_singleton_iff</a>, <a>not_true</a>, <a>and_false</a>, <a>insert_diff_singleton</a>, <a>subset_def</a>,\n    <a>mem_insert_iff</a>, <a>mem_preimage</a>, <a>ne_eq</a>, <a>Function.update_apply</a>, <a>forall_eq_or_imp</a>, <a>ite_true</a>, <a>and_imp</a>,\n    <a>mem_image</a>, <a>ite_eq_left_iff</a>, <a>not_exists</a>, <a>not_and</a>, <a>not_forall</a>, <a>exists_prop</a>, <a>and_iff_right</a> hbt]", [{"full_name": "Set.mem_diff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1819, 9], "def_end_pos": [1819, 17]}, {"full_name": "Set.mem_singleton_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1273, 9], "def_end_pos": [1273, 26]}, {"full_name": "not_true", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [80, 17], "def_end_pos": [80, 25]}, {"full_name": "and_false", "def_path": "lake-packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [85, 17], "def_end_pos": [85, 26]}, {"full_name": "Set.insert_diff_singleton", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [2078, 9], "def_end_pos": [2078, 30]}, {"full_name": "Set.subset_def", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [345, 9], "def_end_pos": [345, 19]}, {"full_name": "Set.mem_insert_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1147, 9], "def_end_pos": [1147, 23]}, {"full_name": "Set.mem_preimage", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [64, 9], "def_end_pos": [64, 21]}, {"full_name": "ne_eq", "def_path": "lake-packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [76, 17], "def_end_pos": [76, 22]}, {"full_name": "Function.update_apply", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [565, 9], "def_end_pos": [565, 21]}, {"full_name": "forall_eq_or_imp", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [473, 17], "def_end_pos": [473, 33]}, {"full_name": "ite_true", "def_path": "lake-packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [77, 17], "def_end_pos": [77, 25]}, {"full_name": "and_imp", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [313, 17], "def_end_pos": [313, 24]}, {"full_name": "Set.mem_image", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [231, 9], "def_end_pos": [231, 18]}, {"full_name": "ite_eq_left_iff", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [1159, 17], "def_end_pos": [1159, 32]}, {"full_name": "not_exists", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [422, 17], "def_end_pos": [422, 27]}, {"full_name": "not_and", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [316, 17], "def_end_pos": [316, 24]}, {"full_name": "Classical.not_forall", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [686, 9], "def_end_pos": [686, 19]}, {"full_name": "exists_prop", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [485, 17], "def_end_pos": [485, 28]}, {"full_name": "and_iff_right", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [206, 9], "def_end_pos": [206, 22]}]], "state_before": "case h\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ns\u271d\u00b9 t\u271d\u00b9 s\u271d : Set \u03b1\nt\u271d : Set \u03b2\nf : \u03b1 \u2192 \u03b2\ninst\u271d : Nonempty \u03b2\ns : Set \u03b1\nt : Set \u03b2\nhs : Set.Finite s\nhle : encard s \u2264 encard t\na : \u03b1\nhas : a \u2208 s\nb : \u03b2\nhbt : b \u2208 t\nhle' : encard (s \\ {a}) \u2264 encard (t \\ {b})\nf\u2080 : \u03b1 \u2192 \u03b2\nhinj : InjOn f\u2080 (s \\ {a})\nhf\u2080s : \u2200 (x : \u03b1), x \u2208 s \u2192 \u00acx = a \u2192 f\u2080 x \u2208 t \u2227 \u00acf\u2080 x = b\n\u22a2 insert a (s \\ {a}) \u2286 Function.update f\u2080 a b \u207b\u00b9' t \u2227\n    InjOn (Function.update f\u2080 a b) (s \\ {a}) \u2227 \u00acFunction.update f\u2080 a b a \u2208 Function.update f\u2080 a b '' (s \\ {a})", "state_after": "case h\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ns\u271d\u00b9 t\u271d\u00b9 s\u271d : Set \u03b1\nt\u271d : Set \u03b2\nf : \u03b1 \u2192 \u03b2\ninst\u271d : Nonempty \u03b2\ns : Set \u03b1\nt : Set \u03b2\nhs : Set.Finite s\nhle : encard s \u2264 encard t\na : \u03b1\nhas : a \u2208 s\nb : \u03b2\nhbt : b \u2208 t\nhle' : encard (s \\ {a}) \u2264 encard (t \\ {b})\nf\u2080 : \u03b1 \u2192 \u03b2\nhinj : InjOn f\u2080 (s \\ {a})\nhf\u2080s : \u2200 (x : \u03b1), x \u2208 s \u2192 \u00acx = a \u2192 f\u2080 x \u2208 t \u2227 \u00acf\u2080 x = b\n\u22a2 (\u2200 (a_1 : \u03b1), a_1 \u2208 s \u2192 (if a_1 = a then b else f\u2080 a_1) \u2208 t) \u2227\n    InjOn (Function.update f\u2080 a b) (s \\ {a}) \u2227 \u2200 (x : \u03b1), x \u2208 s \u2192 \u00acx = a \u2192 \u00acx = a \u2227 \u00acf\u2080 x = b"}, {"tactic": "refine \u27e8?_, ?_, fun x hxs hxa \u21a6 \u27e8hxa, (hf\u2080s x hxs hxa).2\u27e9\u27e9", "annotated_tactic": ["refine \u27e8?_, ?_, fun x hxs hxa \u21a6 \u27e8hxa, (hf\u2080s x hxs hxa).2\u27e9\u27e9", []], "state_before": "case h\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ns\u271d\u00b9 t\u271d\u00b9 s\u271d : Set \u03b1\nt\u271d : Set \u03b2\nf : \u03b1 \u2192 \u03b2\ninst\u271d : Nonempty \u03b2\ns : Set \u03b1\nt : Set \u03b2\nhs : Set.Finite s\nhle : encard s \u2264 encard t\na : \u03b1\nhas : a \u2208 s\nb : \u03b2\nhbt : b \u2208 t\nhle' : encard (s \\ {a}) \u2264 encard (t \\ {b})\nf\u2080 : \u03b1 \u2192 \u03b2\nhinj : InjOn f\u2080 (s \\ {a})\nhf\u2080s : \u2200 (x : \u03b1), x \u2208 s \u2192 \u00acx = a \u2192 f\u2080 x \u2208 t \u2227 \u00acf\u2080 x = b\n\u22a2 (\u2200 (a_1 : \u03b1), a_1 \u2208 s \u2192 (if a_1 = a then b else f\u2080 a_1) \u2208 t) \u2227\n    InjOn (Function.update f\u2080 a b) (s \\ {a}) \u2227 \u2200 (x : \u03b1), x \u2208 s \u2192 \u00acx = a \u2192 \u00acx = a \u2227 \u00acf\u2080 x = b", "state_after": "case h.refine_1\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ns\u271d\u00b9 t\u271d\u00b9 s\u271d : Set \u03b1\nt\u271d : Set \u03b2\nf : \u03b1 \u2192 \u03b2\ninst\u271d : Nonempty \u03b2\ns : Set \u03b1\nt : Set \u03b2\nhs : Set.Finite s\nhle : encard s \u2264 encard t\na : \u03b1\nhas : a \u2208 s\nb : \u03b2\nhbt : b \u2208 t\nhle' : encard (s \\ {a}) \u2264 encard (t \\ {b})\nf\u2080 : \u03b1 \u2192 \u03b2\nhinj : InjOn f\u2080 (s \\ {a})\nhf\u2080s : \u2200 (x : \u03b1), x \u2208 s \u2192 \u00acx = a \u2192 f\u2080 x \u2208 t \u2227 \u00acf\u2080 x = b\n\u22a2 \u2200 (a_1 : \u03b1), a_1 \u2208 s \u2192 (if a_1 = a then b else f\u2080 a_1) \u2208 t\n\ncase h.refine_2\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ns\u271d\u00b9 t\u271d\u00b9 s\u271d : Set \u03b1\nt\u271d : Set \u03b2\nf : \u03b1 \u2192 \u03b2\ninst\u271d : Nonempty \u03b2\ns : Set \u03b1\nt : Set \u03b2\nhs : Set.Finite s\nhle : encard s \u2264 encard t\na : \u03b1\nhas : a \u2208 s\nb : \u03b2\nhbt : b \u2208 t\nhle' : encard (s \\ {a}) \u2264 encard (t \\ {b})\nf\u2080 : \u03b1 \u2192 \u03b2\nhinj : InjOn f\u2080 (s \\ {a})\nhf\u2080s : \u2200 (x : \u03b1), x \u2208 s \u2192 \u00acx = a \u2192 f\u2080 x \u2208 t \u2227 \u00acf\u2080 x = b\n\u22a2 InjOn (Function.update f\u2080 a b) (s \\ {a})"}, {"tactic": "exact InjOn.congr hinj (fun x \u27e8_, hxa\u27e9 \u21a6 by rwa [Function.update_noteq])", "annotated_tactic": ["exact <a>InjOn.congr</a> hinj (fun x \u27e8_, hxa\u27e9 \u21a6 by rwa [<a>Function.update_noteq</a>])", [{"full_name": "Set.InjOn.congr", "def_path": "Mathlib/Data/Set/Function.lean", "def_pos": [631, 9], "def_end_pos": [631, 20]}, {"full_name": "Function.update_noteq", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [560, 9], "def_end_pos": [560, 21]}]], "state_before": "case h.refine_2\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ns\u271d\u00b9 t\u271d\u00b9 s\u271d : Set \u03b1\nt\u271d : Set \u03b2\nf : \u03b1 \u2192 \u03b2\ninst\u271d : Nonempty \u03b2\ns : Set \u03b1\nt : Set \u03b2\nhs : Set.Finite s\nhle : encard s \u2264 encard t\na : \u03b1\nhas : a \u2208 s\nb : \u03b2\nhbt : b \u2208 t\nhle' : encard (s \\ {a}) \u2264 encard (t \\ {b})\nf\u2080 : \u03b1 \u2192 \u03b2\nhinj : InjOn f\u2080 (s \\ {a})\nhf\u2080s : \u2200 (x : \u03b1), x \u2208 s \u2192 \u00acx = a \u2192 f\u2080 x \u2208 t \u2227 \u00acf\u2080 x = b\n\u22a2 InjOn (Function.update f\u2080 a b) (s \\ {a})", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case inl\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ns\u271d t\u271d\u00b9 s : Set \u03b1\nt\u271d : Set \u03b2\nf : \u03b1 \u2192 \u03b2\ninst\u271d : Nonempty \u03b2\nt : Set \u03b2\nhs : Set.Finite \u2205\nhle : encard \u2205 \u2264 encard t\n\u22a2 \u2203 f, \u2205 \u2286 f \u207b\u00b9' t \u2227 InjOn f \u2205", "state_after": "no goals"}, {"tactic": "exact (encard_ne_top_iff.mpr hs h).elim", "annotated_tactic": ["exact (encard_ne_top_iff.mpr hs h).<a>elim</a>", [{"full_name": "False.elim", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [223, 21], "def_end_pos": [223, 31]}]], "state_before": "case inr.inl\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ns\u271d\u00b9 t\u271d\u00b9 s\u271d : Set \u03b1\nt\u271d : Set \u03b2\nf : \u03b1 \u2192 \u03b2\ninst\u271d : Nonempty \u03b2\ns : Set \u03b1\nt : Set \u03b2\nhs : Set.Finite s\nhle : encard s \u2264 encard t\nh : encard s = \u22a4\n\u22a2 \u2203 f, s \u2286 f \u207b\u00b9' t \u2227 InjOn f s", "state_after": "no goals"}, {"tactic": "rwa [\u2190WithTop.add_le_add_iff_right WithTop.one_ne_top,\nencard_diff_singleton_add_one has, encard_diff_singleton_add_one hbt]", "annotated_tactic": ["rwa [\u2190<a>WithTop.add_le_add_iff_right</a> <a>WithTop.one_ne_top</a>,\n    <a>encard_diff_singleton_add_one</a> has, <a>encard_diff_singleton_add_one</a> hbt]", [{"full_name": "WithTop.add_le_add_iff_right", "def_path": "Mathlib/Algebra/Order/Monoid/WithTop.lean", "def_pos": [284, 19], "def_end_pos": [284, 39]}, {"full_name": "WithTop.one_ne_top", "def_path": "Mathlib/Algebra/Order/Monoid/WithTop.lean", "def_pos": [103, 9], "def_end_pos": [103, 19]}, {"full_name": "Set.encard_diff_singleton_add_one", "def_path": "Mathlib/Data/Set/Card.lean", "def_pos": [243, 9], "def_end_pos": [243, 38]}, {"full_name": "Set.encard_diff_singleton_add_one", "def_path": "Mathlib/Data/Set/Card.lean", "def_pos": [243, 9], "def_end_pos": [243, 38]}]], "state_before": "case hle'\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ns\u271d\u00b9 t\u271d\u00b9 s\u271d : Set \u03b1\nt\u271d : Set \u03b2\nf : \u03b1 \u2192 \u03b2\ninst\u271d : Nonempty \u03b2\ns : Set \u03b1\nt : Set \u03b2\nhs : Set.Finite s\nhle : encard s \u2264 encard t\na : \u03b1\nhas : a \u2208 s\nb : \u03b2\nhbt : b \u2208 t\n\u22a2 encard (s \\ {a}) \u2264 encard (t \\ {b})", "state_after": "no goals"}, {"tactic": "rintro x hx", "annotated_tactic": ["rintro x hx", []], "state_before": "case h.refine_1\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ns\u271d\u00b9 t\u271d\u00b9 s\u271d : Set \u03b1\nt\u271d : Set \u03b2\nf : \u03b1 \u2192 \u03b2\ninst\u271d : Nonempty \u03b2\ns : Set \u03b1\nt : Set \u03b2\nhs : Set.Finite s\nhle : encard s \u2264 encard t\na : \u03b1\nhas : a \u2208 s\nb : \u03b2\nhbt : b \u2208 t\nhle' : encard (s \\ {a}) \u2264 encard (t \\ {b})\nf\u2080 : \u03b1 \u2192 \u03b2\nhinj : InjOn f\u2080 (s \\ {a})\nhf\u2080s : \u2200 (x : \u03b1), x \u2208 s \u2192 \u00acx = a \u2192 f\u2080 x \u2208 t \u2227 \u00acf\u2080 x = b\n\u22a2 \u2200 (a_1 : \u03b1), a_1 \u2208 s \u2192 (if a_1 = a then b else f\u2080 a_1) \u2208 t", "state_after": "case h.refine_1\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ns\u271d\u00b9 t\u271d\u00b9 s\u271d : Set \u03b1\nt\u271d : Set \u03b2\nf : \u03b1 \u2192 \u03b2\ninst\u271d : Nonempty \u03b2\ns : Set \u03b1\nt : Set \u03b2\nhs : Set.Finite s\nhle : encard s \u2264 encard t\na : \u03b1\nhas : a \u2208 s\nb : \u03b2\nhbt : b \u2208 t\nhle' : encard (s \\ {a}) \u2264 encard (t \\ {b})\nf\u2080 : \u03b1 \u2192 \u03b2\nhinj : InjOn f\u2080 (s \\ {a})\nhf\u2080s : \u2200 (x : \u03b1), x \u2208 s \u2192 \u00acx = a \u2192 f\u2080 x \u2208 t \u2227 \u00acf\u2080 x = b\nx : \u03b1\nhx : x \u2208 s\n\u22a2 (if x = a then b else f\u2080 x) \u2208 t"}, {"tactic": "split_ifs with h", "annotated_tactic": ["split_ifs with h", []], "state_before": "case h.refine_1\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ns\u271d\u00b9 t\u271d\u00b9 s\u271d : Set \u03b1\nt\u271d : Set \u03b2\nf : \u03b1 \u2192 \u03b2\ninst\u271d : Nonempty \u03b2\ns : Set \u03b1\nt : Set \u03b2\nhs : Set.Finite s\nhle : encard s \u2264 encard t\na : \u03b1\nhas : a \u2208 s\nb : \u03b2\nhbt : b \u2208 t\nhle' : encard (s \\ {a}) \u2264 encard (t \\ {b})\nf\u2080 : \u03b1 \u2192 \u03b2\nhinj : InjOn f\u2080 (s \\ {a})\nhf\u2080s : \u2200 (x : \u03b1), x \u2208 s \u2192 \u00acx = a \u2192 f\u2080 x \u2208 t \u2227 \u00acf\u2080 x = b\nx : \u03b1\nhx : x \u2208 s\n\u22a2 (if x = a then b else f\u2080 x) \u2208 t", "state_after": "case pos\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ns\u271d\u00b9 t\u271d\u00b9 s\u271d : Set \u03b1\nt\u271d : Set \u03b2\nf : \u03b1 \u2192 \u03b2\ninst\u271d : Nonempty \u03b2\ns : Set \u03b1\nt : Set \u03b2\nhs : Set.Finite s\nhle : encard s \u2264 encard t\na : \u03b1\nhas : a \u2208 s\nb : \u03b2\nhbt : b \u2208 t\nhle' : encard (s \\ {a}) \u2264 encard (t \\ {b})\nf\u2080 : \u03b1 \u2192 \u03b2\nhinj : InjOn f\u2080 (s \\ {a})\nhf\u2080s : \u2200 (x : \u03b1), x \u2208 s \u2192 \u00acx = a \u2192 f\u2080 x \u2208 t \u2227 \u00acf\u2080 x = b\nx : \u03b1\nhx : x \u2208 s\nh : x = a\n\u22a2 b \u2208 t\n\ncase neg\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ns\u271d\u00b9 t\u271d\u00b9 s\u271d : Set \u03b1\nt\u271d : Set \u03b2\nf : \u03b1 \u2192 \u03b2\ninst\u271d : Nonempty \u03b2\ns : Set \u03b1\nt : Set \u03b2\nhs : Set.Finite s\nhle : encard s \u2264 encard t\na : \u03b1\nhas : a \u2208 s\nb : \u03b2\nhbt : b \u2208 t\nhle' : encard (s \\ {a}) \u2264 encard (t \\ {b})\nf\u2080 : \u03b1 \u2192 \u03b2\nhinj : InjOn f\u2080 (s \\ {a})\nhf\u2080s : \u2200 (x : \u03b1), x \u2208 s \u2192 \u00acx = a \u2192 f\u2080 x \u2208 t \u2227 \u00acf\u2080 x = b\nx : \u03b1\nhx : x \u2208 s\nh : \u00acx = a\n\u22a2 f\u2080 x \u2208 t"}, {"tactic": "assumption", "annotated_tactic": ["assumption", []], "state_before": "case pos\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ns\u271d\u00b9 t\u271d\u00b9 s\u271d : Set \u03b1\nt\u271d : Set \u03b2\nf : \u03b1 \u2192 \u03b2\ninst\u271d : Nonempty \u03b2\ns : Set \u03b1\nt : Set \u03b2\nhs : Set.Finite s\nhle : encard s \u2264 encard t\na : \u03b1\nhas : a \u2208 s\nb : \u03b2\nhbt : b \u2208 t\nhle' : encard (s \\ {a}) \u2264 encard (t \\ {b})\nf\u2080 : \u03b1 \u2192 \u03b2\nhinj : InjOn f\u2080 (s \\ {a})\nhf\u2080s : \u2200 (x : \u03b1), x \u2208 s \u2192 \u00acx = a \u2192 f\u2080 x \u2208 t \u2227 \u00acf\u2080 x = b\nx : \u03b1\nhx : x \u2208 s\nh : x = a\n\u22a2 b \u2208 t\n\ncase neg\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ns\u271d\u00b9 t\u271d\u00b9 s\u271d : Set \u03b1\nt\u271d : Set \u03b2\nf : \u03b1 \u2192 \u03b2\ninst\u271d : Nonempty \u03b2\ns : Set \u03b1\nt : Set \u03b2\nhs : Set.Finite s\nhle : encard s \u2264 encard t\na : \u03b1\nhas : a \u2208 s\nb : \u03b2\nhbt : b \u2208 t\nhle' : encard (s \\ {a}) \u2264 encard (t \\ {b})\nf\u2080 : \u03b1 \u2192 \u03b2\nhinj : InjOn f\u2080 (s \\ {a})\nhf\u2080s : \u2200 (x : \u03b1), x \u2208 s \u2192 \u00acx = a \u2192 f\u2080 x \u2208 t \u2227 \u00acf\u2080 x = b\nx : \u03b1\nhx : x \u2208 s\nh : \u00acx = a\n\u22a2 f\u2080 x \u2208 t", "state_after": "case neg\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ns\u271d\u00b9 t\u271d\u00b9 s\u271d : Set \u03b1\nt\u271d : Set \u03b2\nf : \u03b1 \u2192 \u03b2\ninst\u271d : Nonempty \u03b2\ns : Set \u03b1\nt : Set \u03b2\nhs : Set.Finite s\nhle : encard s \u2264 encard t\na : \u03b1\nhas : a \u2208 s\nb : \u03b2\nhbt : b \u2208 t\nhle' : encard (s \\ {a}) \u2264 encard (t \\ {b})\nf\u2080 : \u03b1 \u2192 \u03b2\nhinj : InjOn f\u2080 (s \\ {a})\nhf\u2080s : \u2200 (x : \u03b1), x \u2208 s \u2192 \u00acx = a \u2192 f\u2080 x \u2208 t \u2227 \u00acf\u2080 x = b\nx : \u03b1\nhx : x \u2208 s\nh : \u00acx = a\n\u22a2 f\u2080 x \u2208 t"}, {"tactic": "exact (hf\u2080s x hx h).1", "annotated_tactic": ["exact (hf\u2080s x hx h).1", []], "state_before": "case neg\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ns\u271d\u00b9 t\u271d\u00b9 s\u271d : Set \u03b1\nt\u271d : Set \u03b2\nf : \u03b1 \u2192 \u03b2\ninst\u271d : Nonempty \u03b2\ns : Set \u03b1\nt : Set \u03b2\nhs : Set.Finite s\nhle : encard s \u2264 encard t\na : \u03b1\nhas : a \u2208 s\nb : \u03b2\nhbt : b \u2208 t\nhle' : encard (s \\ {a}) \u2264 encard (t \\ {b})\nf\u2080 : \u03b1 \u2192 \u03b2\nhinj : InjOn f\u2080 (s \\ {a})\nhf\u2080s : \u2200 (x : \u03b1), x \u2208 s \u2192 \u00acx = a \u2192 f\u2080 x \u2208 t \u2227 \u00acf\u2080 x = b\nx : \u03b1\nhx : x \u2208 s\nh : \u00acx = a\n\u22a2 f\u2080 x \u2208 t", "state_after": "no goals"}, {"tactic": "rwa [Function.update_noteq]", "annotated_tactic": ["rwa [<a>Function.update_noteq</a>]", [{"full_name": "Function.update_noteq", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [560, 9], "def_end_pos": [560, 21]}]], "state_before": "\u03b1 : Type u_2\n\u03b2 : Type u_1\ns\u271d\u00b9 t\u271d\u00b9 s\u271d : Set \u03b1\nt\u271d : Set \u03b2\nf : \u03b1 \u2192 \u03b2\ninst\u271d : Nonempty \u03b2\ns : Set \u03b1\nt : Set \u03b2\nhs : Set.Finite s\nhle : encard s \u2264 encard t\na : \u03b1\nhas : a \u2208 s\nb : \u03b2\nhbt : b \u2208 t\nhle' : encard (s \\ {a}) \u2264 encard (t \\ {b})\nf\u2080 : \u03b1 \u2192 \u03b2\nhinj : InjOn f\u2080 (s \\ {a})\nhf\u2080s : \u2200 (x : \u03b1), x \u2208 s \u2192 \u00acx = a \u2192 f\u2080 x \u2208 t \u2227 \u00acf\u2080 x = b\nx : \u03b1\nx\u271d : x \u2208 s \\ {a}\nleft\u271d : x \u2208 s\nhxa : \u00acx \u2208 {a}\n\u22a2 f\u2080 x = Function.update f\u2080 a b x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "full_name": "MeasureTheory.snorm_le_snorm_of_exponent_le", "start": [1118, 1], "end": [1120, 90], "traced_tactics": [{"tactic": "simp [measure_univ]", "annotated_tactic": ["simp [<a>measure_univ</a>]", [{"full_name": "MeasureTheory.IsProbabilityMeasure.measure_univ", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3027, 3], "def_end_pos": [3027, 15]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\np q : \u211d\u22650\u221e\nhpq : p \u2264 q\ninst\u271d : IsProbabilityMeasure \u03bc\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\n\u22a2 snorm f q \u03bc * \u2191\u2191\u03bc Set.univ ^ (1 / ENNReal.toReal p - 1 / ENNReal.toReal q) = snorm f q \u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Kernel/Basic.lean", "full_name": "ProbabilityTheory.kernel.sum_comm", "start": [255, 1], "end": [257, 54], "traced_tactics": [{"tactic": "ext a s", "annotated_tactic": ["ext a s", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : Countable \u03b9\n\u03ba : \u03b9 \u2192 \u03b9 \u2192 { x // x \u2208 kernel \u03b1 \u03b2 }\n\u22a2 (kernel.sum fun n => kernel.sum (\u03ba n)) = kernel.sum fun m => kernel.sum fun n => \u03ba n m", "state_after": "case h.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : Countable \u03b9\n\u03ba : \u03b9 \u2192 \u03b9 \u2192 { x // x \u2208 kernel \u03b1 \u03b2 }\na : \u03b1\ns : Set \u03b2\na\u271d : MeasurableSet s\n\u22a2 \u2191\u2191(\u2191(kernel.sum fun n => kernel.sum (\u03ba n)) a) s = \u2191\u2191(\u2191(kernel.sum fun m => kernel.sum fun n => \u03ba n m) a) s"}, {"tactic": "simp_rw [sum_apply]", "annotated_tactic": ["simp_rw [<a>sum_apply</a>]", [{"full_name": "ProbabilityTheory.kernel.sum_apply", "def_path": "Mathlib/Probability/Kernel/Basic.lean", "def_pos": [239, 9], "def_end_pos": [239, 18]}]], "state_before": "case h.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : Countable \u03b9\n\u03ba : \u03b9 \u2192 \u03b9 \u2192 { x // x \u2208 kernel \u03b1 \u03b2 }\na : \u03b1\ns : Set \u03b2\na\u271d : MeasurableSet s\n\u22a2 \u2191\u2191(\u2191(kernel.sum fun n => kernel.sum (\u03ba n)) a) s = \u2191\u2191(\u2191(kernel.sum fun m => kernel.sum fun n => \u03ba n m) a) s", "state_after": "case h.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : Countable \u03b9\n\u03ba : \u03b9 \u2192 \u03b9 \u2192 { x // x \u2208 kernel \u03b1 \u03b2 }\na : \u03b1\ns : Set \u03b2\na\u271d : MeasurableSet s\n\u22a2 \u2191\u2191(Measure.sum fun n => Measure.sum fun n_1 => \u2191(\u03ba n n_1) a) s =\n    \u2191\u2191(Measure.sum fun n => Measure.sum fun n_1 => \u2191(\u03ba n_1 n) a) s"}, {"tactic": "rw [Measure.sum_comm]", "annotated_tactic": ["rw [<a>Measure.sum_comm</a>]", [{"full_name": "MeasureTheory.Measure.sum_comm", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2014, 9], "def_end_pos": [2014, 17]}]], "state_before": "case h.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : Countable \u03b9\n\u03ba : \u03b9 \u2192 \u03b9 \u2192 { x // x \u2208 kernel \u03b1 \u03b2 }\na : \u03b1\ns : Set \u03b2\na\u271d : MeasurableSet s\n\u22a2 \u2191\u2191(Measure.sum fun n => Measure.sum fun n_1 => \u2191(\u03ba n n_1) a) s =\n    \u2191\u2191(Measure.sum fun n => Measure.sum fun n_1 => \u2191(\u03ba n_1 n) a) s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean", "full_name": "MeasureTheory.norm_condexpIndL1Fin_le", "start": [128, 1], "end": [144, 55], "traced_tactics": [{"tactic": "have : 0 \u2264 \u222b a : \u03b1, \u2016condexpIndL1Fin hm hs h\u03bcs x a\u2016 \u2202\u03bc :=\n  integral_nonneg fun a => norm_nonneg _", "annotated_tactic": ["have : 0 \u2264 \u222b a : \u03b1, \u2016<a>condexpIndL1Fin</a> hm hs h\u03bcs x a\u2016 \u2202\u03bc :=\n    <a>integral_nonneg</a> fun a => <a>norm_nonneg</a> _", [{"full_name": "MeasureTheory.condexpIndL1Fin", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean", "def_pos": [73, 5], "def_end_pos": [73, 20]}, {"full_name": "MeasureTheory.integral_nonneg", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1251, 9], "def_end_pos": [1251, 24]}, {"full_name": "norm_nonneg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [500, 30], "def_end_pos": [500, 41]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : CompleteSpace F'\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \u211d G'\ninst\u271d\u00b2 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx : G\n\u22a2 \u2016condexpIndL1Fin hm hs h\u03bcs x\u2016 \u2264 ENNReal.toReal (\u2191\u2191\u03bc s) * \u2016x\u2016", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : CompleteSpace F'\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \u211d G'\ninst\u271d\u00b2 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx : G\nthis : 0 \u2264 \u222b (a : \u03b1), \u2016\u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) a\u2016 \u2202\u03bc\n\u22a2 \u2016condexpIndL1Fin hm hs h\u03bcs x\u2016 \u2264 ENNReal.toReal (\u2191\u2191\u03bc s) * \u2016x\u2016"}, {"tactic": "rw [L1.norm_eq_integral_norm, \u2190 ENNReal.toReal_ofReal (norm_nonneg x), \u2190 ENNReal.toReal_mul, \u2190\n  ENNReal.toReal_ofReal this,\n  ENNReal.toReal_le_toReal ENNReal.ofReal_ne_top (ENNReal.mul_ne_top h\u03bcs ENNReal.ofReal_ne_top),\n  ofReal_integral_norm_eq_lintegral_nnnorm]", "annotated_tactic": ["rw [<a>L1.norm_eq_integral_norm</a>, \u2190 <a>ENNReal.toReal_ofReal</a> (<a>norm_nonneg</a> x), \u2190 <a>ENNReal.toReal_mul</a>, \u2190\n    <a>ENNReal.toReal_ofReal</a> this,\n    <a>ENNReal.toReal_le_toReal</a> <a>ENNReal.ofReal_ne_top</a> (<a>ENNReal.mul_ne_top</a> h\u03bcs <a>ENNReal.ofReal_ne_top</a>),\n    <a>ofReal_integral_norm_eq_lintegral_nnnorm</a>]", [{"full_name": "MeasureTheory.L1.norm_eq_integral_norm", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1298, 9], "def_end_pos": [1298, 33]}, {"full_name": "ENNReal.toReal_ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [191, 9], "def_end_pos": [191, 22]}, {"full_name": "norm_nonneg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [500, 30], "def_end_pos": [500, 41]}, {"full_name": "ENNReal.toReal_mul", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2296, 9], "def_end_pos": [2296, 19]}, {"full_name": "ENNReal.toReal_ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [191, 9], "def_end_pos": [191, 22]}, {"full_name": "ENNReal.toReal_le_toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2036, 9], "def_end_pos": [2036, 25]}, {"full_name": "ENNReal.ofReal_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [311, 17], "def_end_pos": [311, 30]}, {"full_name": "ENNReal.mul_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [615, 9], "def_end_pos": [615, 19]}, {"full_name": "ENNReal.ofReal_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [311, 17], "def_end_pos": [311, 30]}, {"full_name": "MeasureTheory.ofReal_integral_norm_eq_lintegral_nnnorm", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1188, 9], "def_end_pos": [1188, 49]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : CompleteSpace F'\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \u211d G'\ninst\u271d\u00b2 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx : G\nthis : 0 \u2264 \u222b (a : \u03b1), \u2016\u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) a\u2016 \u2202\u03bc\n\u22a2 \u2016condexpIndL1Fin hm hs h\u03bcs x\u2016 \u2264 ENNReal.toReal (\u2191\u2191\u03bc s) * \u2016x\u2016", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : CompleteSpace F'\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \u211d G'\ninst\u271d\u00b2 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx : G\nthis : 0 \u2264 \u222b (a : \u03b1), \u2016\u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) a\u2016 \u2202\u03bc\n\u22a2 \u222b\u207b (x_1 : \u03b1), \u2191\u2016\u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) x_1\u2016\u208a \u2202\u03bc \u2264 \u2191\u2191\u03bc s * ENNReal.ofReal \u2016x\u2016\n\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : CompleteSpace F'\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \u211d G'\ninst\u271d\u00b2 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx : G\nthis : 0 \u2264 \u222b (a : \u03b1), \u2016\u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) a\u2016 \u2202\u03bc\n\u22a2 Integrable fun a => \u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) a"}, {"tactic": "swap", "annotated_tactic": ["swap", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : CompleteSpace F'\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \u211d G'\ninst\u271d\u00b2 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx : G\nthis : 0 \u2264 \u222b (a : \u03b1), \u2016\u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) a\u2016 \u2202\u03bc\n\u22a2 \u222b\u207b (x_1 : \u03b1), \u2191\u2016\u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) x_1\u2016\u208a \u2202\u03bc \u2264 \u2191\u2191\u03bc s * ENNReal.ofReal \u2016x\u2016\n\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : CompleteSpace F'\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \u211d G'\ninst\u271d\u00b2 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx : G\nthis : 0 \u2264 \u222b (a : \u03b1), \u2016\u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) a\u2016 \u2202\u03bc\n\u22a2 Integrable fun a => \u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) a", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : CompleteSpace F'\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \u211d G'\ninst\u271d\u00b2 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx : G\nthis : 0 \u2264 \u222b (a : \u03b1), \u2016\u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) a\u2016 \u2202\u03bc\n\u22a2 Integrable fun a => \u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) a\n\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : CompleteSpace F'\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \u211d G'\ninst\u271d\u00b2 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx : G\nthis : 0 \u2264 \u222b (a : \u03b1), \u2016\u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) a\u2016 \u2202\u03bc\n\u22a2 \u222b\u207b (x_1 : \u03b1), \u2191\u2016\u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) x_1\u2016\u208a \u2202\u03bc \u2264 \u2191\u2191\u03bc s * ENNReal.ofReal \u2016x\u2016"}, {"tactic": "have h_eq :\n  \u222b\u207b a, \u2016condexpIndL1Fin hm hs h\u03bcs x a\u2016\u208a \u2202\u03bc = \u222b\u207b a, \u2016condexpIndSMul hm hs h\u03bcs x a\u2016\u208a \u2202\u03bc := by\n  refine' lintegral_congr_ae _\n  refine' (condexpIndL1Fin_ae_eq_condexpIndSMul hm hs h\u03bcs x).mono fun z hz => _\n  dsimp only\n  rw [hz]", "annotated_tactic": ["have h_eq :\n    \u222b\u207b a, \u2016<a>condexpIndL1Fin</a> hm hs h\u03bcs x a\u2016\u208a \u2202\u03bc = \u222b\u207b a, \u2016<a>condexpIndSMul</a> hm hs h\u03bcs x a\u2016\u208a \u2202\u03bc := by\n    refine' <a>lintegral_congr_ae</a> _\n    refine' (<a>condexpIndL1Fin_ae_eq_condexpIndSMul</a> hm hs h\u03bcs x).<a>mono</a> fun z hz => _\n    dsimp only\n    rw [hz]", [{"full_name": "MeasureTheory.condexpIndL1Fin", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean", "def_pos": [73, 5], "def_end_pos": [73, 20]}, {"full_name": "MeasureTheory.condexpIndSMul", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL2.lean", "def_pos": [384, 19], "def_end_pos": [384, 33]}, {"full_name": "MeasureTheory.lintegral_congr_ae", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [304, 9], "def_end_pos": [304, 27]}, {"full_name": "MeasureTheory.condexpIndL1Fin_ae_eq_condexpIndSMul", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean", "def_pos": [78, 9], "def_end_pos": [78, 45]}, {"full_name": "Filter.Eventually.mono", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1140, 9], "def_end_pos": [1140, 24]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : CompleteSpace F'\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \u211d G'\ninst\u271d\u00b2 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx : G\nthis : 0 \u2264 \u222b (a : \u03b1), \u2016\u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) a\u2016 \u2202\u03bc\n\u22a2 \u222b\u207b (x_1 : \u03b1), \u2191\u2016\u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) x_1\u2016\u208a \u2202\u03bc \u2264 \u2191\u2191\u03bc s * ENNReal.ofReal \u2016x\u2016", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : CompleteSpace F'\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \u211d G'\ninst\u271d\u00b2 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx : G\nthis : 0 \u2264 \u222b (a : \u03b1), \u2016\u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) a\u2016 \u2202\u03bc\nh_eq : \u222b\u207b (a : \u03b1), \u2191\u2016\u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) a\u2016\u208a \u2202\u03bc = \u222b\u207b (a : \u03b1), \u2191\u2016\u2191\u2191(condexpIndSMul hm hs h\u03bcs x) a\u2016\u208a \u2202\u03bc\n\u22a2 \u222b\u207b (x_1 : \u03b1), \u2191\u2016\u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) x_1\u2016\u208a \u2202\u03bc \u2264 \u2191\u2191\u03bc s * ENNReal.ofReal \u2016x\u2016"}, {"tactic": "rw [h_eq, ofReal_norm_eq_coe_nnnorm]", "annotated_tactic": ["rw [h_eq, <a>ofReal_norm_eq_coe_nnnorm</a>]", [{"full_name": "ofReal_norm_eq_coe_nnnorm", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [999, 15], "def_end_pos": [999, 40]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : CompleteSpace F'\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \u211d G'\ninst\u271d\u00b2 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx : G\nthis : 0 \u2264 \u222b (a : \u03b1), \u2016\u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) a\u2016 \u2202\u03bc\nh_eq : \u222b\u207b (a : \u03b1), \u2191\u2016\u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) a\u2016\u208a \u2202\u03bc = \u222b\u207b (a : \u03b1), \u2191\u2016\u2191\u2191(condexpIndSMul hm hs h\u03bcs x) a\u2016\u208a \u2202\u03bc\n\u22a2 \u222b\u207b (x_1 : \u03b1), \u2191\u2016\u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) x_1\u2016\u208a \u2202\u03bc \u2264 \u2191\u2191\u03bc s * ENNReal.ofReal \u2016x\u2016", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : CompleteSpace F'\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \u211d G'\ninst\u271d\u00b2 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx : G\nthis : 0 \u2264 \u222b (a : \u03b1), \u2016\u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) a\u2016 \u2202\u03bc\nh_eq : \u222b\u207b (a : \u03b1), \u2191\u2016\u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) a\u2016\u208a \u2202\u03bc = \u222b\u207b (a : \u03b1), \u2191\u2016\u2191\u2191(condexpIndSMul hm hs h\u03bcs x) a\u2016\u208a \u2202\u03bc\n\u22a2 \u222b\u207b (a : \u03b1), \u2191\u2016\u2191\u2191(condexpIndSMul hm hs h\u03bcs x) a\u2016\u208a \u2202\u03bc \u2264 \u2191\u2191\u03bc s * \u2191\u2016x\u2016\u208a"}, {"tactic": "exact lintegral_nnnorm_condexpIndSMul_le hm hs h\u03bcs x", "annotated_tactic": ["exact <a>lintegral_nnnorm_condexpIndSMul_le</a> hm hs h\u03bcs x", [{"full_name": "MeasureTheory.lintegral_nnnorm_condexpIndSMul_le", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL2.lean", "def_pos": [441, 9], "def_end_pos": [441, 43]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : CompleteSpace F'\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \u211d G'\ninst\u271d\u00b2 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx : G\nthis : 0 \u2264 \u222b (a : \u03b1), \u2016\u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) a\u2016 \u2202\u03bc\nh_eq : \u222b\u207b (a : \u03b1), \u2191\u2016\u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) a\u2016\u208a \u2202\u03bc = \u222b\u207b (a : \u03b1), \u2191\u2016\u2191\u2191(condexpIndSMul hm hs h\u03bcs x) a\u2016\u208a \u2202\u03bc\n\u22a2 \u222b\u207b (a : \u03b1), \u2191\u2016\u2191\u2191(condexpIndSMul hm hs h\u03bcs x) a\u2016\u208a \u2202\u03bc \u2264 \u2191\u2191\u03bc s * \u2191\u2016x\u2016\u208a", "state_after": "no goals"}, {"tactic": "rw [\u2190 mem\u2112p_one_iff_integrable]", "annotated_tactic": ["rw [\u2190 <a>mem\u2112p_one_iff_integrable</a>]", [{"full_name": "MeasureTheory.mem\u2112p_one_iff_integrable", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [453, 9], "def_end_pos": [453, 33]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : CompleteSpace F'\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \u211d G'\ninst\u271d\u00b2 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx : G\nthis : 0 \u2264 \u222b (a : \u03b1), \u2016\u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) a\u2016 \u2202\u03bc\n\u22a2 Integrable fun a => \u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) a", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : CompleteSpace F'\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \u211d G'\ninst\u271d\u00b2 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx : G\nthis : 0 \u2264 \u222b (a : \u03b1), \u2016\u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) a\u2016 \u2202\u03bc\n\u22a2 Mem\u2112p (fun a => \u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) a) 1"}, {"tactic": "exact Lp.mem\u2112p _", "annotated_tactic": ["exact <a>Lp.mem\u2112p</a> _", [{"full_name": "MeasureTheory.Lp.mem\u2112p", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [216, 19], "def_end_pos": [216, 24]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : CompleteSpace F'\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \u211d G'\ninst\u271d\u00b2 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx : G\nthis : 0 \u2264 \u222b (a : \u03b1), \u2016\u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) a\u2016 \u2202\u03bc\n\u22a2 Mem\u2112p (fun a => \u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) a) 1", "state_after": "no goals"}, {"tactic": "refine' lintegral_congr_ae _", "annotated_tactic": ["refine' <a>lintegral_congr_ae</a> _", [{"full_name": "MeasureTheory.lintegral_congr_ae", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [304, 9], "def_end_pos": [304, 27]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : CompleteSpace F'\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \u211d G'\ninst\u271d\u00b2 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx : G\nthis : 0 \u2264 \u222b (a : \u03b1), \u2016\u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) a\u2016 \u2202\u03bc\n\u22a2 \u222b\u207b (a : \u03b1), \u2191\u2016\u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) a\u2016\u208a \u2202\u03bc = \u222b\u207b (a : \u03b1), \u2191\u2016\u2191\u2191(condexpIndSMul hm hs h\u03bcs x) a\u2016\u208a \u2202\u03bc", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : CompleteSpace F'\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \u211d G'\ninst\u271d\u00b2 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx : G\nthis : 0 \u2264 \u222b (a : \u03b1), \u2016\u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) a\u2016 \u2202\u03bc\n\u22a2 (fun a => \u2191\u2016\u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) a\u2016\u208a) =\u1d50[\u03bc] fun a => \u2191\u2016\u2191\u2191(condexpIndSMul hm hs h\u03bcs x) a\u2016\u208a"}, {"tactic": "refine' (condexpIndL1Fin_ae_eq_condexpIndSMul hm hs h\u03bcs x).mono fun z hz => _", "annotated_tactic": ["refine' (<a>condexpIndL1Fin_ae_eq_condexpIndSMul</a> hm hs h\u03bcs x).<a>mono</a> fun z hz => _", [{"full_name": "MeasureTheory.condexpIndL1Fin_ae_eq_condexpIndSMul", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean", "def_pos": [78, 9], "def_end_pos": [78, 45]}, {"full_name": "Filter.Eventually.mono", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1140, 9], "def_end_pos": [1140, 24]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : CompleteSpace F'\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \u211d G'\ninst\u271d\u00b2 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx : G\nthis : 0 \u2264 \u222b (a : \u03b1), \u2016\u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) a\u2016 \u2202\u03bc\n\u22a2 (fun a => \u2191\u2016\u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) a\u2016\u208a) =\u1d50[\u03bc] fun a => \u2191\u2016\u2191\u2191(condexpIndSMul hm hs h\u03bcs x) a\u2016\u208a", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : CompleteSpace F'\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \u211d G'\ninst\u271d\u00b2 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx : G\nthis : 0 \u2264 \u222b (a : \u03b1), \u2016\u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) a\u2016 \u2202\u03bc\nz : \u03b1\nhz : \u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) z = \u2191\u2191(condexpIndSMul hm hs h\u03bcs x) z\n\u22a2 (fun a => \u2191\u2016\u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) a\u2016\u208a) z = (fun a => \u2191\u2016\u2191\u2191(condexpIndSMul hm hs h\u03bcs x) a\u2016\u208a) z"}, {"tactic": "dsimp only", "annotated_tactic": ["dsimp only", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : CompleteSpace F'\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \u211d G'\ninst\u271d\u00b2 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx : G\nthis : 0 \u2264 \u222b (a : \u03b1), \u2016\u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) a\u2016 \u2202\u03bc\nz : \u03b1\nhz : \u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) z = \u2191\u2191(condexpIndSMul hm hs h\u03bcs x) z\n\u22a2 (fun a => \u2191\u2016\u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) a\u2016\u208a) z = (fun a => \u2191\u2016\u2191\u2191(condexpIndSMul hm hs h\u03bcs x) a\u2016\u208a) z", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : CompleteSpace F'\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \u211d G'\ninst\u271d\u00b2 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx : G\nthis : 0 \u2264 \u222b (a : \u03b1), \u2016\u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) a\u2016 \u2202\u03bc\nz : \u03b1\nhz : \u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) z = \u2191\u2191(condexpIndSMul hm hs h\u03bcs x) z\n\u22a2 \u2191\u2016\u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) z\u2016\u208a = \u2191\u2016\u2191\u2191(condexpIndSMul hm hs h\u03bcs x) z\u2016\u208a"}, {"tactic": "rw [hz]", "annotated_tactic": ["rw [hz]", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : CompleteSpace F'\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \u211d G'\ninst\u271d\u00b2 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx : G\nthis : 0 \u2264 \u222b (a : \u03b1), \u2016\u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) a\u2016 \u2202\u03bc\nz : \u03b1\nhz : \u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) z = \u2191\u2191(condexpIndSMul hm hs h\u03bcs x) z\n\u22a2 \u2191\u2016\u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) z\u2016\u208a = \u2191\u2016\u2191\u2191(condexpIndSMul hm hs h\u03bcs x) z\u2016\u208a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Martingale/BorelCantelli.lean", "full_name": "MeasureTheory.BorelCantelli.predictablePart_process_ae_eq", "start": [297, 1], "end": [302, 33], "traced_tactics": [{"tactic": "have := martingalePart_process_ae_eq \u2131 \u03bc s n", "annotated_tactic": ["have := <a>martingalePart_process_ae_eq</a> \u2131 \u03bc s n", [{"full_name": "MeasureTheory.BorelCantelli.martingalePart_process_ae_eq", "def_path": "Mathlib/Probability/Martingale/BorelCantelli.lean", "def_pos": [289, 9], "def_end_pos": [289, 37]}]], "state_before": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc\u271d : Measure \u03a9\n\u2131\u271d : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nr : \u211d\nR : \u211d\u22650\ns\u271d : \u2115 \u2192 Set \u03a9\n\u2131 : Filtration \u2115 m0\n\u03bc : Measure \u03a9\ns : \u2115 \u2192 Set \u03a9\nn : \u2115\n\u22a2 predictablePart (process s) \u2131 \u03bc n = \u2211 k in Finset.range n, \u03bc[Set.indicator (s (k + 1)) 1|\u2191\u2131 k]", "state_after": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc\u271d : Measure \u03a9\n\u2131\u271d : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nr : \u211d\nR : \u211d\u22650\ns\u271d : \u2115 \u2192 Set \u03a9\n\u2131 : Filtration \u2115 m0\n\u03bc : Measure \u03a9\ns : \u2115 \u2192 Set \u03a9\nn : \u2115\nthis :\n  martingalePart (process s) \u2131 \u03bc n =\n    \u2211 k in Finset.range n, (Set.indicator (s (k + 1)) 1 - \u03bc[Set.indicator (s (k + 1)) 1|\u2191\u2131 k])\n\u22a2 predictablePart (process s) \u2131 \u03bc n = \u2211 k in Finset.range n, \u03bc[Set.indicator (s (k + 1)) 1|\u2191\u2131 k]"}, {"tactic": "simp_rw [martingalePart, process, Finset.sum_sub_distrib] at this", "annotated_tactic": ["simp_rw [<a>martingalePart</a>, <a>process</a>, <a>Finset.sum_sub_distrib</a>] at this", [{"full_name": "MeasureTheory.martingalePart", "def_path": "Mathlib/Probability/Martingale/Centering.lean", "def_pos": [66, 19], "def_end_pos": [66, 33]}, {"full_name": "MeasureTheory.BorelCantelli.process", "def_path": "Mathlib/Probability/Martingale/BorelCantelli.lean", "def_pos": [275, 19], "def_end_pos": [275, 26]}, {"full_name": "Finset.sum_sub_distrib", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [1820, 3], "def_end_pos": [1820, 14]}]], "state_before": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc\u271d : Measure \u03a9\n\u2131\u271d : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nr : \u211d\nR : \u211d\u22650\ns\u271d : \u2115 \u2192 Set \u03a9\n\u2131 : Filtration \u2115 m0\n\u03bc : Measure \u03a9\ns : \u2115 \u2192 Set \u03a9\nn : \u2115\nthis :\n  martingalePart (process s) \u2131 \u03bc n =\n    \u2211 k in Finset.range n, (Set.indicator (s (k + 1)) 1 - \u03bc[Set.indicator (s (k + 1)) 1|\u2191\u2131 k])\n\u22a2 predictablePart (process s) \u2131 \u03bc n = \u2211 k in Finset.range n, \u03bc[Set.indicator (s (k + 1)) 1|\u2191\u2131 k]", "state_after": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc\u271d : Measure \u03a9\n\u2131\u271d : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nr : \u211d\nR : \u211d\u22650\ns\u271d : \u2115 \u2192 Set \u03a9\n\u2131 : Filtration \u2115 m0\n\u03bc : Measure \u03a9\ns : \u2115 \u2192 Set \u03a9\nn : \u2115\nthis :\n  \u2211 k in Finset.range n, Set.indicator (s (k + 1)) 1 - predictablePart (process s) \u2131 \u03bc n =\n    \u2211 k in Finset.range n, Set.indicator (s (k + 1)) 1 - \u2211 x in Finset.range n, \u03bc[Set.indicator (s (x + 1)) 1|\u2191\u2131 x]\n\u22a2 predictablePart (process s) \u2131 \u03bc n = \u2211 k in Finset.range n, \u03bc[Set.indicator (s (k + 1)) 1|\u2191\u2131 k]"}, {"tactic": "exact sub_right_injective this", "annotated_tactic": ["exact <a>sub_right_injective</a> this", [{"full_name": "sub_right_injective", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [720, 3], "def_end_pos": [720, 14]}]], "state_before": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc\u271d : Measure \u03a9\n\u2131\u271d : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nr : \u211d\nR : \u211d\u22650\ns\u271d : \u2115 \u2192 Set \u03a9\n\u2131 : Filtration \u2115 m0\n\u03bc : Measure \u03a9\ns : \u2115 \u2192 Set \u03a9\nn : \u2115\nthis :\n  \u2211 k in Finset.range n, Set.indicator (s (k + 1)) 1 - predictablePart (process s) \u2131 \u03bc n =\n    \u2211 k in Finset.range n, Set.indicator (s (k + 1)) 1 - \u2211 x in Finset.range n, \u03bc[Set.indicator (s (x + 1)) 1|\u2191\u2131 x]\n\u22a2 predictablePart (process s) \u2131 \u03bc n = \u2211 k in Finset.range n, \u03bc[Set.indicator (s (k + 1)) 1|\u2191\u2131 k]", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/PFun.lean", "full_name": "PFun.core_inter", "start": [480, 1], "end": [481, 23], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "full_name": "intervalIntegral.integral_ofReal", "start": [655, 8], "end": [657, 34], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "full_name": "MeasureTheory.inducedOuterMeasure_eq_extend'", "start": [1459, 1], "end": [1461, 89], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Halting.lean", "full_name": "Nat.Partrec'.comp\u2081", "start": [362, 1], "end": [364, 55], "traced_tactics": [{"tactic": "simpa using hf.comp' (Partrec'.cons hg Partrec'.nil)", "annotated_tactic": ["simpa using hf.comp' (<a>Partrec'.cons</a> hg <a>Partrec'.nil</a>)", [{"full_name": "Nat.Partrec'.cons", "def_path": "Mathlib/Computability/Halting.lean", "def_pos": [349, 19], "def_end_pos": [349, 23]}, {"full_name": "Nat.Partrec'.nil", "def_path": "Mathlib/Computability/Halting.lean", "def_pos": [346, 19], "def_end_pos": [346, 22]}]], "state_before": "n : \u2115\nf : \u2115 \u2192. \u2115\ng : Vector \u2115 n \u2192 \u2115\nhf : Partrec' fun v => f (Vector.head v)\nhg : Partrec' \u2191g\n\u22a2 Partrec' fun v => f (g v)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/TorusIntegral.lean", "full_name": "torusIntegral_dim0", "start": [211, 1], "end": [215, 88], "traced_tactics": [{"tactic": "simp only [torusIntegral, Fin.prod_univ_zero, one_smul,\n  Subsingleton.elim (fun _ : Fin 0 => 2 * \u03c0) 0, Icc_self, Measure.restrict_singleton, volume_pi,\n  integral_smul_measure, integral_dirac, Measure.pi_of_empty (fun _ : Fin 0 \u21a6 volume) 0,\n  Measure.dirac_apply_of_mem (mem_singleton _), Subsingleton.elim (torusMap c R 0) c]", "annotated_tactic": ["simp only [<a>torusIntegral</a>, <a>Fin.prod_univ_zero</a>, <a>one_smul</a>,\n    <a>Subsingleton.elim</a> (fun _ : <a>Fin</a> 0 => 2 * \u03c0) 0, 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: (Fin n \u2192 \u2102) \u2192 E\nc\u271d : Fin n \u2192 \u2102\nR\u271d : Fin n \u2192 \u211d\nf : (Fin 0 \u2192 \u2102) \u2192 E\nc : Fin 0 \u2192 \u2102\nR : Fin 0 \u2192 \u211d\n\u22a2 (\u222f (x : Fin 0 \u2192 \u2102) in T(c, R), f x) = f c", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Fold.lean", "full_name": "Finset.fold_sup_bot_singleton", "start": [206, 1], "end": [208, 31], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/List/Basic.lean", "full_name": "List.replicateTR_loop_eq", "start": [210, 1], "end": [213, 87], "traced_tactics": [{"tactic": "rw [\u2190 replicateTR_loop_replicate_eq _ 1 n, replicate, replicate,\nreplicateTR.loop, replicateTR_loop_eq n, replicateTR_loop_eq n, append_assoc]", "annotated_tactic": ["rw [\u2190 <a>replicateTR_loop_replicate_eq</a> _ 1 n, <a>replicate</a>, <a>replicate</a>,\n    <a>replicateTR.loop</a>, replicateTR_loop_eq n, replicateTR_loop_eq n, <a>append_assoc</a>]", [{"full_name": "List.replicateTR_loop_replicate_eq", "def_path": "lake-packages/lean4/src/lean/Init/Data/List/Basic.lean", "def_pos": [747, 9], "def_end_pos": [747, 38]}, {"full_name": "List.replicate", "def_path": "lake-packages/lean4/src/lean/Init/Data/List/Basic.lean", "def_pos": [736, 13], "def_end_pos": [736, 22]}, {"full_name": "List.replicate", "def_path": "lake-packages/lean4/src/lean/Init/Data/List/Basic.lean", "def_pos": [736, 13], "def_end_pos": [736, 22]}, {"full_name": "List.replicateTR.loop", "def_path": "lake-packages/lean4/src/lean/Init/Data/List/Basic.lean", "def_pos": [742, 11], "def_end_pos": [742, 15]}, {"full_name": "List.append_assoc", "def_path": "lake-packages/lean4/src/lean/Init/Data/List/Basic.lean", "def_pos": [103, 9], "def_end_pos": [103, 21]}]], "state_before": "\u03b1\u271d : 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9], "end": [1209, 73], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Kernel/CondCdf.lean", "full_name": "ProbabilityTheory.condCdf'_eq_condCdfRat", "start": [715, 1], "end": [725, 48], "traced_tactics": [{"tactic": "rw [\u2190 inf_gt_condCdfRat \u03c1 a r, condCdf']", "annotated_tactic": ["rw [\u2190 <a>inf_gt_condCdfRat</a> \u03c1 a r, <a>condCdf'</a>]", [{"full_name": "ProbabilityTheory.inf_gt_condCdfRat", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [668, 9], "def_end_pos": [668, 26]}, {"full_name": "ProbabilityTheory.condCdf'", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [707, 31], "def_end_pos": [707, 39]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nr : \u211a\n\u22a2 condCdf' \u03c1 a \u2191r = condCdfRat \u03c1 a r", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nr : \u211a\n\u22a2 \u2a05 r_1, condCdfRat \u03c1 a \u2191r_1 = \u2a05 r_1, condCdfRat \u03c1 a \u2191r_1"}, {"tactic": "refine' Equiv.iInf_congr _ _", "annotated_tactic": ["refine' <a>Equiv.iInf_congr</a> _ _", [{"full_name": "Equiv.iInf_congr", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [775, 19], "def_end_pos": [775, 35]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nr : \u211a\n\u22a2 \u2a05 r_1, condCdfRat \u03c1 a \u2191r_1 = \u2a05 r_1, condCdfRat \u03c1 a \u2191r_1", "state_after": "case refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nr : \u211a\n\u22a2 { r' // \u2191r < \u2191r' } \u2243 \u2191(Ioi r)\n\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nr : \u211a\n\u22a2 \u2200 (x : { r' // \u2191r < \u2191r' }), condCdfRat \u03c1 a \u2191(\u2191?refine'_1 x) = condCdfRat \u03c1 a \u2191x"}, {"tactic": "exact\n  { toFun := fun t => \u27e8t.1, by exact_mod_cast t.2\u27e9\n    invFun := fun t => \u27e8t.1, by exact_mod_cast t.2\u27e9\n    left_inv := fun t => by simp only [Subtype.coe_eta]\n    right_inv := fun t => by simp only [Subtype.coe_eta] }", "annotated_tactic": ["exact\n      { toFun := fun t => \u27e8t.1, by exact_mod_cast t.2\u27e9\n        invFun := fun t => \u27e8t.1, by exact_mod_cast t.2\u27e9\n        left_inv := fun t => by simp only [<a>Subtype.coe_eta</a>]\n        right_inv := fun t => by simp only [<a>Subtype.coe_eta</a>] }", [{"full_name": "Subtype.coe_eta", "def_path": "Mathlib/Data/Subtype.lean", "def_pos": [95, 9], "def_end_pos": [95, 16]}, {"full_name": "Subtype.coe_eta", "def_path": "Mathlib/Data/Subtype.lean", "def_pos": [95, 9], "def_end_pos": [95, 16]}]], "state_before": "case refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nr : \u211a\n\u22a2 { r' // \u2191r < \u2191r' } \u2243 \u2191(Ioi r)", "state_after": "no goals"}, {"tactic": "exact_mod_cast t.2", "annotated_tactic": ["exact_mod_cast t.2", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nr : \u211a\nt : { r' // \u2191r < \u2191r' }\n\u22a2 \u2191t \u2208 Ioi r", "state_after": "no goals"}, {"tactic": "exact_mod_cast t.2", "annotated_tactic": ["exact_mod_cast t.2", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nr : \u211a\nt : \u2191(Ioi r)\n\u22a2 \u2191r < \u2191\u2191t", "state_after": "no goals"}, {"tactic": "simp only [Subtype.coe_eta]", "annotated_tactic": ["simp only [<a>Subtype.coe_eta</a>]", [{"full_name": "Subtype.coe_eta", "def_path": "Mathlib/Data/Subtype.lean", "def_pos": [95, 9], "def_end_pos": [95, 16]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nr : \u211a\nt : { r' // \u2191r < \u2191r' }\n\u22a2 (fun t => { val := \u2191t, property := (_ : \u2191r < \u2191\u2191t) }) ((fun t => { val := \u2191t, property := (_ : r < \u2191t) }) t) = t", "state_after": "no goals"}, {"tactic": "simp only [Subtype.coe_eta]", "annotated_tactic": ["simp only [<a>Subtype.coe_eta</a>]", [{"full_name": "Subtype.coe_eta", "def_path": "Mathlib/Data/Subtype.lean", "def_pos": [95, 9], "def_end_pos": [95, 16]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nr : \u211a\nt : \u2191(Ioi r)\n\u22a2 (fun t => { val := \u2191t, property := (_ : r < \u2191t) }) ((fun t => { val := \u2191t, property := (_ : \u2191r < \u2191\u2191t) }) t) = t", "state_after": "no goals"}, {"tactic": "intro t", "annotated_tactic": ["intro t", []], "state_before": "case refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nr : \u211a\n\u22a2 \u2200 (x : { r' // \u2191r < \u2191r' }),\n    condCdfRat \u03c1 a\n        \u2191(\u2191{ toFun := fun t => { val := \u2191t, property := (_ : r < \u2191t) },\n                invFun := fun t => { val := \u2191t, property := (_ : \u2191r < \u2191\u2191t) },\n                left_inv :=\n                  (_ :\n                    \u2200 (t : { r' // \u2191r < \u2191r' }),\n                      { val := \u2191t, property := (_ : \u2191r < \u2191\u2191((fun t => { val := \u2191t, property := (_ : r < \u2191t) }) t)) } =\n                        t),\n                right_inv :=\n                  (_ :\n                    \u2200 (t : \u2191(Ioi r)),\n                      { val := \u2191t, property := (_ : r < \u2191((fun t => { val := \u2191t, property := (_ : \u2191r < \u2191\u2191t) }) t)) } =\n                        t) }\n            x) =\n      condCdfRat \u03c1 a \u2191x", "state_after": "case refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nr : \u211a\nt : { r' // \u2191r < \u2191r' }\n\u22a2 condCdfRat \u03c1 a\n      \u2191(\u2191{ toFun := fun t => { val := \u2191t, property := (_ : r < \u2191t) },\n              invFun := fun t => { val := \u2191t, property := (_ : \u2191r < \u2191\u2191t) },\n              left_inv :=\n                (_ :\n                  \u2200 (t : { r' // \u2191r < \u2191r' }),\n                    { val := \u2191t, property := (_ : \u2191r < \u2191\u2191((fun t => { val := \u2191t, property := (_ : r < \u2191t) }) t)) } = t),\n              right_inv :=\n                (_ :\n                  \u2200 (t : \u2191(Ioi r)),\n                    { val := \u2191t, property := (_ : r < \u2191((fun t => { val := \u2191t, property := (_ : \u2191r < \u2191\u2191t) }) t)) } =\n                      t) }\n          t) =\n    condCdfRat \u03c1 a \u2191t"}, {"tactic": "simp only [Equiv.coe_fn_mk, Subtype.coe_mk]", "annotated_tactic": ["simp only [<a>Equiv.coe_fn_mk</a>, <a>Subtype.coe_mk</a>]", [{"full_name": "Equiv.coe_fn_mk", "def_path": "Mathlib/Logic/Equiv/Defs.lean", "def_pos": [108, 17], "def_end_pos": [108, 26]}, {"full_name": "Subtype.coe_mk", "def_path": "Mathlib/Data/Subtype.lean", "def_pos": [99, 9], "def_end_pos": [99, 15]}]], "state_before": "case refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nr : \u211a\nt : { r' // \u2191r < \u2191r' }\n\u22a2 condCdfRat \u03c1 a\n      \u2191(\u2191{ toFun := fun t => { val := \u2191t, property := (_ : r < \u2191t) },\n              invFun := fun t => { val := \u2191t, property := (_ : \u2191r < \u2191\u2191t) },\n              left_inv :=\n                (_ :\n                  \u2200 (t : { r' // \u2191r < \u2191r' }),\n                    { val := \u2191t, property := (_ : \u2191r < \u2191\u2191((fun t => { val := \u2191t, property := (_ : r < \u2191t) }) t)) } = t),\n              right_inv :=\n                (_ :\n                  \u2200 (t : \u2191(Ioi r)),\n                    { val := \u2191t, property := (_ : r < \u2191((fun t => { val := \u2191t, property := (_ : \u2191r < \u2191\u2191t) }) t)) } =\n                      t) }\n          t) =\n    condCdfRat \u03c1 a \u2191t", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "full_name": "intervalIntegral.inv_smul_integral_comp_add_div", "start": [793, 1], "end": [795, 59], "traced_tactics": [{"tactic": "by_cases hc : c = 0 <;> simp [hc, integral_comp_add_div]", "annotated_tactic": ["by_cases hc : c = 0 <;> simp [hc, <a>integral_comp_add_div</a>]", [{"full_name": "intervalIntegral.integral_comp_add_div", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [787, 9], "def_end_pos": [787, 30]}]], "state_before": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\na b c\u271d d\u271d : \u211d\nf : \u211d \u2192 E\nc d : \u211d\n\u22a2 c\u207b\u00b9 \u2022 \u222b (x : \u211d) in a..b, f (d + x / c) = \u222b (x : \u211d) in d + a / c..d + b / c, f x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Classes/Order.lean", "full_name": "Std.TransCmp.lt_trans", "start": [70, 1], "end": [71, 46], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "Finset.filter_union_filter_neg_eq", "start": [3012, 1], "end": [3014, 74], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Int/Lemmas.lean", "full_name": "Int.natAbs_inj_of_nonneg_of_nonneg", "start": [61, 1], "end": [62, 83], "traced_tactics": [{"tactic": "rw [\u2190 sq_eq_sq ha hb, \u2190 natAbs_eq_iff_sq_eq]", "annotated_tactic": ["rw [\u2190 <a>sq_eq_sq</a> ha hb, \u2190 <a>natAbs_eq_iff_sq_eq</a>]", [{"full_name": "sq_eq_sq", "def_path": "Mathlib/Algebra/GroupPower/Order.lean", "def_pos": [610, 9], "def_end_pos": [610, 17]}, {"full_name": "Int.natAbs_eq_iff_sq_eq", "def_path": "Mathlib/Data/Int/Lemmas.lean", "def_pos": [46, 9], "def_end_pos": [46, 28]}]], "state_before": "a\u271d b\u271d : \u2124\nn : \u2115\na b : \u2124\nha : 0 \u2264 a\nhb : 0 \u2264 b\n\u22a2 natAbs a = natAbs b \u2194 a = b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Basic.lean", "full_name": "Set.inter_symmDiff_distrib_right", "start": [2142, 1], "end": [2143, 35], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "full_name": "MeasureTheory.union_ae_eq_right_iff_ae_subset", "start": [319, 1], "end": [320, 50], "traced_tactics": [{"tactic": "rw [union_comm, union_ae_eq_left_iff_ae_subset]", "annotated_tactic": ["rw [<a>union_comm</a>, <a>union_ae_eq_left_iff_ae_subset</a>]", [{"full_name": "Set.union_comm", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [786, 9], "def_end_pos": [786, 19]}, {"full_name": "MeasureTheory.union_ae_eq_left_iff_ae_subset", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [309, 9], "def_end_pos": [309, 39]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns s\u2081 s\u2082 t : Set \u03b1\n\u22a2 s \u222a t =\u1d50[\u03bc] t \u2194 s \u2264\u1d50[\u03bc] t", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean", "full_name": "Real.volume_pi_ball", "start": [257, 8], "end": [260, 67], "traced_tactics": [{"tactic": "simp only [MeasureTheory.volume_pi_ball a hr, volume_ball, Finset.prod_const]", "annotated_tactic": ["simp only [<a>MeasureTheory.volume_pi_ball</a> a hr, <a>volume_ball</a>, <a>Finset.prod_const</a>]", [{"full_name": "MeasureTheory.volume_pi_ball", "def_path": "Mathlib/MeasureTheory/Constructions/Pi.lean", "def_pos": [681, 9], "def_end_pos": [681, 23]}, {"full_name": "Real.volume_ball", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean", "def_pos": [108, 9], "def_end_pos": [108, 20]}, {"full_name": "Finset.prod_const", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [1441, 9], "def_end_pos": [1441, 19]}]], "state_before": "\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\na : \u03b9 \u2192 \u211d\nr : \u211d\nhr : 0 < r\n\u22a2 \u2191\u2191volume (Metric.ball a r) = ofReal ((2 * r) ^ Fintype.card \u03b9)", "state_after": "\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\na : \u03b9 \u2192 \u211d\nr : \u211d\nhr : 0 < r\n\u22a2 ofReal (2 * r) ^ Finset.card Finset.univ = ofReal ((2 * r) ^ Fintype.card \u03b9)"}, {"tactic": "exact (ENNReal.ofReal_pow (mul_nonneg zero_le_two hr.le) _).symm", "annotated_tactic": ["exact (<a>ENNReal.ofReal_pow</a> (<a>mul_nonneg</a> <a>zero_le_two</a> hr.le) _).<a>symm</a>", [{"full_name": "ENNReal.ofReal_pow", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2235, 9], "def_end_pos": [2235, 19]}, {"full_name": "mul_nonneg", "def_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "def_pos": [380, 7], "def_end_pos": [380, 17]}, {"full_name": "zero_le_two", "def_path": "Mathlib/Algebra/Order/Monoid/NatCast.lean", "def_pos": [32, 7], "def_end_pos": [32, 18]}, {"full_name": "Eq.symm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [310, 9], "def_end_pos": [310, 16]}]], "state_before": "\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\na : \u03b9 \u2192 \u211d\nr : \u211d\nhr : 0 < r\n\u22a2 ofReal (2 * r) ^ Finset.card Finset.univ = ofReal ((2 * r) ^ Fintype.card \u03b9)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Nat/Gcd.lean", "full_name": "Nat.Coprime.gcd_mul_left_cancel", "start": [270, 1], "end": [275, 33], "traced_tactics": [{"tactic": "rw [Coprime, Nat.gcd_assoc, H.symm.gcd_eq_one, gcd_one_right]", "annotated_tactic": ["rw [<a>Coprime</a>, <a>Nat.gcd_assoc</a>, H.symm.gcd_eq_one, <a>gcd_one_right</a>]", [{"full_name": "Nat.Coprime", "def_path": "lake-packages/std/Std/Data/Nat/Gcd.lean", "def_pos": [33, 18], "def_end_pos": [33, 25]}, {"full_name": "Nat.gcd_assoc", "def_path": "lake-packages/std/Std/Data/Nat/Gcd.lean", "def_pos": [71, 9], "def_end_pos": [71, 18]}, {"full_name": "Nat.gcd_one_right", "def_path": "lake-packages/std/Std/Data/Nat/Gcd.lean", "def_pos": [82, 17], "def_end_pos": [82, 30]}]], "state_before": "k n m : Nat\nH : Coprime k n\n\u22a2 Coprime (gcd (k * m) n) k", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Covering/LiminfLimsup.lean", "full_name": "blimsup_cthickening_ae_eq_blimsup_thickening", "start": [233, 1], "end": [244, 68], "traced_tactics": [{"tactic": "refine' eventuallyLE_antisymm_iff.mpr \u27e8_, HasSubset.Subset.eventuallyLE (_ : _ \u2264 _)\u27e9", "annotated_tactic": ["refine' eventuallyLE_antisymm_iff.mpr \u27e8_, <a>HasSubset.Subset.eventuallyLE</a> (_ : _ \u2264 _)\u27e9", [{"full_name": "HasSubset.Subset.eventuallyLE", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [3239, 9], "def_end_pos": [3239, 38]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nr : \u2115 \u2192 \u211d\nhr : Tendsto r atTop (\ud835\udcdd 0)\nhr' : \u2200\u1da0 (i : \u2115) in atTop, p i \u2192 0 < r i\n\u22a2 blimsup (fun i => cthickening (r i) (s i)) atTop p =\u1d50[\u03bc] blimsup (fun i => thickening (r i) (s i)) atTop p", "state_after": "case refine'_1\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nr : \u2115 \u2192 \u211d\nhr : Tendsto r atTop (\ud835\udcdd 0)\nhr' : \u2200\u1da0 (i : \u2115) in atTop, p i \u2192 0 < r i\n\u22a2 blimsup (fun i => cthickening (r i) (s i)) atTop p \u2264\u1d50[\u03bc] blimsup (fun i => thickening (r i) (s i)) atTop p\n\ncase refine'_2\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nr : \u2115 \u2192 \u211d\nhr : Tendsto r atTop (\ud835\udcdd 0)\nhr' : \u2200\u1da0 (i : \u2115) in atTop, p i \u2192 0 < r i\n\u22a2 blimsup (fun i => thickening (r i) (s i)) atTop p \u2264 blimsup (fun i => cthickening (r i) (s i)) atTop p"}, {"tactic": "rw [eventuallyLE_congr (blimsup_cthickening_mul_ae_eq \u03bc p s (@one_half_pos \u211d _) r hr).symm\n  EventuallyEq.rfl]", "annotated_tactic": ["rw [<a>eventuallyLE_congr</a> (<a>blimsup_cthickening_mul_ae_eq</a> \u03bc p s (@<a>one_half_pos</a> \u211d _) r hr).<a>symm</a>\n      <a>EventuallyEq.rfl</a>]", [{"full_name": "Filter.eventuallyLE_congr", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1650, 9], "def_end_pos": [1650, 27]}, {"full_name": "blimsup_cthickening_mul_ae_eq", "def_path": "Mathlib/MeasureTheory/Covering/LiminfLimsup.lean", "def_pos": [194, 9], "def_end_pos": [194, 38]}, {"full_name": "one_half_pos", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [508, 9], "def_end_pos": [508, 21]}, {"full_name": "Filter.EventuallyEq.symm", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1498, 9], "def_end_pos": [1498, 26]}, {"full_name": "Filter.EventuallyEq.rfl", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1493, 9], "def_end_pos": [1493, 25]}]], "state_before": "case refine'_1\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nr : \u2115 \u2192 \u211d\nhr : Tendsto r atTop (\ud835\udcdd 0)\nhr' : \u2200\u1da0 (i : \u2115) in atTop, p i \u2192 0 < r i\n\u22a2 blimsup (fun i => cthickening (r i) (s i)) atTop p \u2264\u1d50[\u03bc] blimsup (fun i => thickening (r i) (s i)) atTop p", "state_after": "case refine'_1\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nr : \u2115 \u2192 \u211d\nhr : Tendsto r atTop (\ud835\udcdd 0)\nhr' : \u2200\u1da0 (i : \u2115) in atTop, p i \u2192 0 < r i\n\u22a2 blimsup (fun i => cthickening (1 / 2 * r i) (s i)) atTop p \u2264\u1d50[\u03bc] blimsup (fun i => thickening (r i) (s i)) atTop p"}, {"tactic": "apply HasSubset.Subset.eventuallyLE", "annotated_tactic": ["apply <a>HasSubset.Subset.eventuallyLE</a>", [{"full_name": "HasSubset.Subset.eventuallyLE", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [3239, 9], "def_end_pos": [3239, 38]}]], "state_before": "case refine'_1\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nr : \u2115 \u2192 \u211d\nhr : Tendsto r atTop (\ud835\udcdd 0)\nhr' : \u2200\u1da0 (i : \u2115) in atTop, p i \u2192 0 < r i\n\u22a2 blimsup (fun i => cthickening (1 / 2 * r i) (s i)) atTop p \u2264\u1d50[\u03bc] blimsup (fun i => thickening (r i) (s i)) atTop p", "state_after": "case refine'_1.h\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nr : \u2115 \u2192 \u211d\nhr : Tendsto r atTop (\ud835\udcdd 0)\nhr' : \u2200\u1da0 (i : \u2115) in atTop, p i \u2192 0 < r i\n\u22a2 blimsup (fun i => cthickening (1 / 2 * r i) (s i)) atTop p \u2286 blimsup (fun i => thickening (r i) (s i)) atTop p"}, {"tactic": "change _ \u2264 _", "annotated_tactic": ["change _ \u2264 _", []], "state_before": "case refine'_1.h\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nr : \u2115 \u2192 \u211d\nhr : Tendsto r atTop (\ud835\udcdd 0)\nhr' : \u2200\u1da0 (i : \u2115) in atTop, p i \u2192 0 < r i\n\u22a2 blimsup (fun i => cthickening (1 / 2 * r i) (s i)) atTop p \u2286 blimsup (fun i => thickening (r i) (s i)) atTop p", "state_after": "case refine'_1.h\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nr : \u2115 \u2192 \u211d\nhr : Tendsto r atTop (\ud835\udcdd 0)\nhr' : \u2200\u1da0 (i : \u2115) in atTop, p i \u2192 0 < r i\n\u22a2 blimsup (fun i => cthickening (1 / 2 * r i) (s i)) atTop p \u2264 blimsup (fun i => thickening (r i) (s i)) atTop p"}, {"tactic": "refine' mono_blimsup' (hr'.mono fun i hi pi => cthickening_subset_thickening' (hi pi) _ (s i))", "annotated_tactic": ["refine' <a>mono_blimsup'</a> (hr'.mono fun i hi pi => <a>cthickening_subset_thickening'</a> (hi pi) _ (s i))", [{"full_name": "Filter.mono_blimsup'", "def_path": "Mathlib/Order/LiminfLimsup.lean", "def_pos": [962, 9], "def_end_pos": [962, 22]}, {"full_name": "Metric.cthickening_subset_thickening'", "def_path": "Mathlib/Topology/MetricSpace/HausdorffDistance.lean", "def_pos": [1122, 9], "def_end_pos": [1122, 39]}]], "state_before": "case refine'_1.h\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nr : \u2115 \u2192 \u211d\nhr : Tendsto r atTop (\ud835\udcdd 0)\nhr' : \u2200\u1da0 (i : \u2115) in atTop, p i \u2192 0 < r i\n\u22a2 blimsup (fun i => cthickening (1 / 2 * r i) (s i)) atTop p \u2264 blimsup (fun i => thickening (r i) (s i)) atTop p", "state_after": "case refine'_1.h\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nr : \u2115 \u2192 \u211d\nhr : Tendsto r atTop (\ud835\udcdd 0)\nhr' : \u2200\u1da0 (i : \u2115) in atTop, p i \u2192 0 < r i\ni : \u2115\nhi : p i \u2192 0 < r i\npi : p i\n\u22a2 1 / 2 * r i < r i"}, {"tactic": "nlinarith [hi pi]", "annotated_tactic": ["nlinarith [hi pi]", []], "state_before": "case refine'_1.h\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nr : \u2115 \u2192 \u211d\nhr : Tendsto r atTop (\ud835\udcdd 0)\nhr' : \u2200\u1da0 (i : \u2115) in atTop, p i \u2192 0 < r i\ni : \u2115\nhi : p i \u2192 0 < r i\npi : p i\n\u22a2 1 / 2 * r i < r i", "state_after": "no goals"}, {"tactic": "exact mono_blimsup fun i _ => thickening_subset_cthickening _ _", "annotated_tactic": ["exact <a>mono_blimsup</a> fun i _ => <a>thickening_subset_cthickening</a> _ _", [{"full_name": "Filter.mono_blimsup", "def_path": "Mathlib/Order/LiminfLimsup.lean", "def_pos": [966, 9], "def_end_pos": [966, 21]}, {"full_name": "Metric.thickening_subset_cthickening", "def_path": "Mathlib/Topology/MetricSpace/HausdorffDistance.lean", "def_pos": [1129, 9], "def_end_pos": [1129, 38]}]], "state_before": "case refine'_2\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nr : \u2115 \u2192 \u211d\nhr : Tendsto r atTop (\ud835\udcdd 0)\nhr' : \u2200\u1da0 (i : \u2115) in atTop, p i \u2192 0 < r i\n\u22a2 blimsup (fun i => thickening (r i) (s i)) atTop p \u2264 blimsup (fun i => cthickening (r i) (s i)) atTop p", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean", "full_name": "Basis.parallelepiped_eq_map", "start": [83, 1], "end": [91, 25], "traced_tactics": [{"tactic": "classical\nrw [\u2190 Basis.parallelepiped_basisFun, \u2190 Basis.parallelepiped_map]\ncongr\next; simp only [map_apply, Pi.basisFun_apply, equivFun_symm_apply, LinearMap.stdBasis_apply',\n  Finset.sum_univ_ite]", "annotated_tactic": ["classical\n  rw [\u2190 <a>Basis.parallelepiped_basisFun</a>, \u2190 <a>Basis.parallelepiped_map</a>]\n  congr\n  ext; simp only [<a>map_apply</a>, <a>Pi.basisFun_apply</a>, <a>equivFun_symm_apply</a>, <a>LinearMap.stdBasis_apply'</a>,\n    <a>Finset.sum_univ_ite</a>]", [{"full_name": "Basis.parallelepiped_basisFun", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean", "def_pos": [74, 9], "def_end_pos": [74, 38]}, {"full_name": "Basis.parallelepiped_map", "def_path": "Mathlib/MeasureTheory/Measure/Haar/OfBasis.lean", "def_pos": [206, 9], "def_end_pos": [206, 33]}, {"full_name": "Basis.map_apply", "def_path": "Mathlib/LinearAlgebra/Basis.lean", "def_pos": [354, 9], "def_end_pos": [354, 18]}, {"full_name": "Pi.basisFun_apply", "def_path": "Mathlib/LinearAlgebra/StdBasis.lean", "def_pos": [280, 9], "def_end_pos": [280, 23]}, {"full_name": "Basis.equivFun_symm_apply", "def_path": "Mathlib/LinearAlgebra/Basis.lean", "def_pos": [915, 9], "def_end_pos": [915, 34]}, {"full_name": "LinearMap.stdBasis_apply'", "def_path": "Mathlib/LinearAlgebra/StdBasis.lean", "def_pos": [57, 9], "def_end_pos": [57, 24]}, {"full_name": "Finset.sum_univ_ite", "def_path": "Mathlib/LinearAlgebra/FinsuppVectorSpace.lean", "def_pos": [185, 9], "def_end_pos": [185, 35]}]], "state_before": "\u03b9 : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : Fintype \u03b9\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nb : Basis \u03b9 \u211d E\n\u22a2 parallelepiped b =\n    PositiveCompacts.map \u2191(LinearEquiv.symm (equivFun b)) (_ : Continuous \u2191(ContinuousLinearEquiv.symm (equivFunL b)))\n      (_ : IsOpenMap \u2191(ContinuousLinearEquiv.symm (equivFunL b))) (PositiveCompacts.piIcc01 \u03b9)", "state_after": "no goals"}, {"tactic": "rw [\u2190 Basis.parallelepiped_basisFun, \u2190 Basis.parallelepiped_map]", "annotated_tactic": ["rw [\u2190 <a>Basis.parallelepiped_basisFun</a>, \u2190 <a>Basis.parallelepiped_map</a>]", [{"full_name": "Basis.parallelepiped_basisFun", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean", "def_pos": [74, 9], "def_end_pos": [74, 38]}, {"full_name": "Basis.parallelepiped_map", "def_path": "Mathlib/MeasureTheory/Measure/Haar/OfBasis.lean", "def_pos": [206, 9], "def_end_pos": [206, 33]}]], "state_before": "\u03b9 : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : Fintype \u03b9\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nb : Basis \u03b9 \u211d E\n\u22a2 parallelepiped b =\n    PositiveCompacts.map \u2191(LinearEquiv.symm (equivFun b)) (_ : Continuous \u2191(ContinuousLinearEquiv.symm (equivFunL b)))\n      (_ : IsOpenMap \u2191(ContinuousLinearEquiv.symm (equivFunL b))) (PositiveCompacts.piIcc01 \u03b9)", "state_after": "\u03b9 : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : Fintype \u03b9\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nb : Basis \u03b9 \u211d E\n\u22a2 parallelepiped b = parallelepiped (Basis.map (Pi.basisFun \u211d \u03b9) (LinearEquiv.symm (equivFun b)))"}, {"tactic": "congr", "annotated_tactic": ["congr", []], "state_before": "\u03b9 : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : Fintype \u03b9\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nb : Basis \u03b9 \u211d E\n\u22a2 parallelepiped b = parallelepiped (Basis.map (Pi.basisFun \u211d \u03b9) (LinearEquiv.symm (equivFun b)))", "state_after": "case e_b\n\u03b9 : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : Fintype \u03b9\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nb : Basis \u03b9 \u211d E\n\u22a2 b = Basis.map (Pi.basisFun \u211d \u03b9) (LinearEquiv.symm (equivFun b))"}, {"tactic": "ext", "annotated_tactic": ["ext", []], "state_before": "case e_b\n\u03b9 : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : Fintype \u03b9\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nb : Basis \u03b9 \u211d E\n\u22a2 b = Basis.map (Pi.basisFun \u211d \u03b9) (LinearEquiv.symm (equivFun b))", "state_after": "case e_b.a\n\u03b9 : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : Fintype \u03b9\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nb : Basis \u03b9 \u211d E\ni\u271d : \u03b9\n\u22a2 \u2191b i\u271d = \u2191(Basis.map (Pi.basisFun \u211d \u03b9) (LinearEquiv.symm (equivFun b))) i\u271d"}, {"tactic": "simp only [map_apply, Pi.basisFun_apply, equivFun_symm_apply, LinearMap.stdBasis_apply',\nFinset.sum_univ_ite]", "annotated_tactic": ["simp only [<a>map_apply</a>, <a>Pi.basisFun_apply</a>, <a>equivFun_symm_apply</a>, <a>LinearMap.stdBasis_apply'</a>,\n    <a>Finset.sum_univ_ite</a>]", [{"full_name": "Basis.map_apply", "def_path": "Mathlib/LinearAlgebra/Basis.lean", "def_pos": [354, 9], "def_end_pos": [354, 18]}, {"full_name": "Pi.basisFun_apply", "def_path": "Mathlib/LinearAlgebra/StdBasis.lean", "def_pos": [280, 9], "def_end_pos": [280, 23]}, {"full_name": "Basis.equivFun_symm_apply", "def_path": "Mathlib/LinearAlgebra/Basis.lean", "def_pos": [915, 9], "def_end_pos": [915, 34]}, {"full_name": "LinearMap.stdBasis_apply'", "def_path": "Mathlib/LinearAlgebra/StdBasis.lean", "def_pos": [57, 9], "def_end_pos": [57, 24]}, {"full_name": "Finset.sum_univ_ite", "def_path": "Mathlib/LinearAlgebra/FinsuppVectorSpace.lean", "def_pos": [185, 9], "def_end_pos": [185, 35]}]], "state_before": "case e_b.a\n\u03b9 : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : Fintype \u03b9\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nb : Basis \u03b9 \u211d E\ni\u271d : \u03b9\n\u22a2 \u2191b i\u271d = \u2191(Basis.map (Pi.basisFun \u211d \u03b9) (LinearEquiv.symm (equivFun b))) i\u271d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/FundThmCalculus.lean", "full_name": "intervalIntegral.sub_le_integral_of_hasDeriv_right_of_le_Ico", "start": [1020, 1], "end": [1110, 75], "traced_tactics": [{"tactic": "refine' le_of_forall_pos_le_add fun \u03b5 \u03b5pos => _", "annotated_tactic": ["refine' <a>le_of_forall_pos_le_add</a> fun \u03b5 \u03b5pos => _", [{"full_name": "le_of_forall_pos_le_add", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [72, 3], "def_end_pos": [72, 14]}]], "state_before": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\na b : \u211d\nhab : a \u2264 b\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x\n\u22a2 g b - g a \u2264 \u222b (y : \u211d) in a..b, \u03c6 y", "state_after": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\na b : \u211d\nhab : a \u2264 b\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u22a2 g b - g a \u2264 (\u222b (y : \u211d) in a..b, \u03c6 y) + \u03b5"}, {"tactic": "rcases exists_lt_lowerSemicontinuous_integral_lt \u03c6 \u03c6int \u03b5pos with\n  \u27e8G', f_lt_G', G'cont, G'int, G'lt_top, hG'\u27e9", "annotated_tactic": ["rcases <a>exists_lt_lowerSemicontinuous_integral_lt</a> \u03c6 \u03c6int \u03b5pos with\n    \u27e8G', f_lt_G', G'cont, G'int, G'lt_top, hG'\u27e9", [{"full_name": "MeasureTheory.exists_lt_lowerSemicontinuous_integral_lt", "def_path": "Mathlib/MeasureTheory/Integral/VitaliCaratheodory.lean", "def_pos": [457, 9], "def_end_pos": [457, 50]}]], "state_before": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\na b : \u211d\nhab : a \u2264 b\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u22a2 g b - g a \u2264 (\u222b (y : \u211d) in a..b, \u03c6 y) + \u03b5", "state_after": "case intro.intro.intro.intro.intro\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\na b : \u211d\nhab : a \u2264 b\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nG' : \u211d \u2192 EReal\nf_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x\nG'cont : LowerSemicontinuous G'\nG'int : Integrable fun x => EReal.toReal (G' x)\nG'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4\nhG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5\n\u22a2 g b - g a \u2264 (\u222b (y : \u211d) in a..b, \u03c6 y) + \u03b5"}, {"tactic": "set s := {t | g t - g a \u2264 \u222b u in a..t, (G' u).toReal} \u2229 Icc a b", "annotated_tactic": ["set s := {t | g t - g a \u2264 \u222b u in a..t, (G' u).<a>toReal</a>} \u2229 <a>Icc</a> a b", [{"full_name": "EReal.toReal", "def_path": "Mathlib/Data/Real/EReal.lean", "def_pos": [254, 5], "def_end_pos": [254, 11]}, {"full_name": "Set.Icc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [59, 5], "def_end_pos": [59, 8]}]], "state_before": "case intro.intro.intro.intro.intro\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\na b : \u211d\nhab : a \u2264 b\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nG' : \u211d \u2192 EReal\nf_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x\nG'cont : LowerSemicontinuous G'\nG'int : Integrable fun x => EReal.toReal (G' x)\nG'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4\nhG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5\n\u22a2 g b - g a \u2264 (\u222b (y : \u211d) in a..b, \u03c6 y) + \u03b5", "state_after": "case intro.intro.intro.intro.intro\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\na b : \u211d\nhab : a \u2264 b\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nG' : \u211d \u2192 EReal\nf_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x\nG'cont : LowerSemicontinuous G'\nG'int : Integrable fun x => EReal.toReal (G' x)\nG'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4\nhG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5\ns : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b\n\u22a2 g b - g a \u2264 (\u222b (y : \u211d) in a..b, \u03c6 y) + \u03b5"}, {"tactic": "have s_closed : IsClosed s := by\n  have : ContinuousOn (fun t => (g t - g a, \u222b u in a..t, (G' u).toReal)) (Icc a b) := by\n    rw [\u2190 uIcc_of_le hab] at G'int hcont \u22a2\n    exact (hcont.sub continuousOn_const).prod (continuousOn_primitive_interval G'int)\n  simp only [inter_comm]\n  exact this.preimage_closed_of_closed isClosed_Icc OrderClosedTopology.isClosed_le'", "annotated_tactic": ["have s_closed : <a>IsClosed</a> s := by\n    have : <a>ContinuousOn</a> (fun t => (g t - g a, \u222b u in a..t, (G' u).<a>toReal</a>)) (<a>Icc</a> a b) := by\n      rw [\u2190 <a>uIcc_of_le</a> hab] at G'int hcont \u22a2\n      exact (hcont.sub <a>continuousOn_const</a>).<a>prod</a> (<a>continuousOn_primitive_interval</a> G'int)\n    simp only [<a>inter_comm</a>]\n    exact this.preimage_closed_of_closed <a>isClosed_Icc</a> <a>OrderClosedTopology.isClosed_le'</a>", [{"full_name": "IsClosed", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [196, 7], "def_end_pos": [196, 15]}, {"full_name": "ContinuousOn", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [532, 5], "def_end_pos": [532, 17]}, {"full_name": "EReal.toReal", "def_path": "Mathlib/Data/Real/EReal.lean", "def_pos": [254, 5], "def_end_pos": [254, 11]}, {"full_name": "Set.Icc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [59, 5], "def_end_pos": [59, 8]}, {"full_name": "Set.uIcc_of_le", "def_path": "Mathlib/Data/Set/Intervals/UnorderedInterval.lean", "def_pos": [69, 7], "def_end_pos": [69, 17]}, {"full_name": "continuousOn_const", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [1025, 9], "def_end_pos": [1025, 27]}, {"full_name": "ContinuousOn.prod", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [1113, 9], "def_end_pos": [1113, 26]}, {"full_name": "intervalIntegral.continuousOn_primitive_interval", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [1225, 9], "def_end_pos": [1225, 40]}, {"full_name": "Set.inter_comm", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [940, 9], "def_end_pos": [940, 19]}, {"full_name": "isClosed_Icc", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [214, 9], "def_end_pos": [214, 21]}, {"full_name": "OrderClosedTopology.isClosed_le'", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [113, 3], "def_end_pos": [113, 15]}]], "state_before": "case intro.intro.intro.intro.intro\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\na b : \u211d\nhab : a \u2264 b\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nG' : \u211d \u2192 EReal\nf_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x\nG'cont : LowerSemicontinuous G'\nG'int : Integrable fun x => EReal.toReal (G' x)\nG'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4\nhG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5\ns : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b\n\u22a2 g b - g a \u2264 (\u222b (y : \u211d) in a..b, \u03c6 y) + \u03b5", "state_after": "case intro.intro.intro.intro.intro\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\na b : \u211d\nhab : a \u2264 b\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nG' : \u211d \u2192 EReal\nf_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x\nG'cont : LowerSemicontinuous G'\nG'int : Integrable fun x => EReal.toReal (G' x)\nG'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4\nhG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5\ns : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b\ns_closed : IsClosed s\n\u22a2 g b - g a \u2264 (\u222b (y : \u211d) in a..b, \u03c6 y) + \u03b5"}, {"tactic": "have : ContinuousOn (fun t => (g t - g a, \u222b u in a..t, (G' u).toReal)) (Icc a b) := by\n  rw [\u2190 uIcc_of_le hab] at G'int hcont \u22a2\n  exact (hcont.sub continuousOn_const).prod (continuousOn_primitive_interval G'int)", "annotated_tactic": ["have : <a>ContinuousOn</a> (fun t => (g t - g a, \u222b u in a..t, (G' u).<a>toReal</a>)) (<a>Icc</a> a b) := by\n      rw [\u2190 <a>uIcc_of_le</a> hab] at G'int hcont \u22a2\n      exact (hcont.sub <a>continuousOn_const</a>).<a>prod</a> (<a>continuousOn_primitive_interval</a> G'int)", [{"full_name": "ContinuousOn", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [532, 5], "def_end_pos": [532, 17]}, {"full_name": "EReal.toReal", "def_path": "Mathlib/Data/Real/EReal.lean", "def_pos": [254, 5], "def_end_pos": [254, 11]}, {"full_name": "Set.Icc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [59, 5], "def_end_pos": [59, 8]}, {"full_name": "Set.uIcc_of_le", "def_path": "Mathlib/Data/Set/Intervals/UnorderedInterval.lean", "def_pos": [69, 7], "def_end_pos": [69, 17]}, {"full_name": "continuousOn_const", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [1025, 9], "def_end_pos": [1025, 27]}, {"full_name": "ContinuousOn.prod", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [1113, 9], "def_end_pos": [1113, 26]}, {"full_name": "intervalIntegral.continuousOn_primitive_interval", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [1225, 9], "def_end_pos": [1225, 40]}]], "state_before": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\na b : \u211d\nhab : a \u2264 b\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nG' : \u211d \u2192 EReal\nf_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x\nG'cont : LowerSemicontinuous G'\nG'int : Integrable fun x => EReal.toReal (G' x)\nG'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4\nhG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5\ns : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b\n\u22a2 IsClosed s", "state_after": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\na b : \u211d\nhab : a \u2264 b\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nG' : \u211d \u2192 EReal\nf_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x\nG'cont : LowerSemicontinuous G'\nG'int : Integrable fun x => EReal.toReal (G' x)\nG'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4\nhG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5\ns : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b\nthis : ContinuousOn (fun t => (g t - g a, \u222b (u : \u211d) in a..t, EReal.toReal (G' u))) (Icc a b)\n\u22a2 IsClosed s"}, {"tactic": "simp only [inter_comm]", "annotated_tactic": ["simp only [<a>inter_comm</a>]", [{"full_name": "Set.inter_comm", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [940, 9], "def_end_pos": [940, 19]}]], "state_before": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\na b : \u211d\nhab : a \u2264 b\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nG' : \u211d \u2192 EReal\nf_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x\nG'cont : LowerSemicontinuous G'\nG'int : Integrable fun x => EReal.toReal (G' x)\nG'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4\nhG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5\ns : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b\nthis : ContinuousOn (fun t => (g t - g a, \u222b (u : \u211d) in a..t, EReal.toReal (G' u))) (Icc a b)\n\u22a2 IsClosed s", "state_after": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\na b : \u211d\nhab : a \u2264 b\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nG' : \u211d \u2192 EReal\nf_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x\nG'cont : LowerSemicontinuous G'\nG'int : Integrable fun x => EReal.toReal (G' x)\nG'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4\nhG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5\ns : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b\nthis : ContinuousOn (fun t => (g t - g a, \u222b (u : \u211d) in a..t, EReal.toReal (G' u))) (Icc a b)\n\u22a2 IsClosed (Icc a b \u2229 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)})"}, {"tactic": "exact this.preimage_closed_of_closed isClosed_Icc OrderClosedTopology.isClosed_le'", "annotated_tactic": ["exact this.preimage_closed_of_closed <a>isClosed_Icc</a> <a>OrderClosedTopology.isClosed_le'</a>", [{"full_name": "isClosed_Icc", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [214, 9], "def_end_pos": [214, 21]}, {"full_name": "OrderClosedTopology.isClosed_le'", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [113, 3], "def_end_pos": [113, 15]}]], "state_before": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\na b : \u211d\nhab : a \u2264 b\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nG' : \u211d \u2192 EReal\nf_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x\nG'cont : LowerSemicontinuous G'\nG'int : Integrable fun x => EReal.toReal (G' x)\nG'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4\nhG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5\ns : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b\nthis : ContinuousOn (fun t => (g t - g a, \u222b (u : \u211d) in a..t, EReal.toReal (G' u))) (Icc a b)\n\u22a2 IsClosed (Icc a b \u2229 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)})", "state_after": "no goals"}, {"tactic": "rw [\u2190 uIcc_of_le hab] at G'int hcont \u22a2", "annotated_tactic": ["rw [\u2190 <a>uIcc_of_le</a> hab] at G'int hcont \u22a2", [{"full_name": "Set.uIcc_of_le", "def_path": "Mathlib/Data/Set/Intervals/UnorderedInterval.lean", "def_pos": [69, 7], "def_end_pos": [69, 17]}]], "state_before": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\na b : \u211d\nhab : a \u2264 b\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nG' : \u211d \u2192 EReal\nf_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x\nG'cont : LowerSemicontinuous G'\nG'int : Integrable fun x => EReal.toReal (G' x)\nG'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4\nhG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5\ns : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b\n\u22a2 ContinuousOn (fun t => (g t - g a, \u222b (u : \u211d) in a..t, EReal.toReal (G' u))) (Icc a b)", "state_after": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\na b : \u211d\nhab : a \u2264 b\nhcont : ContinuousOn g [[a, b]]\nhderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nG' : \u211d \u2192 EReal\nf_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x\nG'cont : LowerSemicontinuous G'\nG'int : Integrable fun x => EReal.toReal (G' x)\nG'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4\nhG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5\ns : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b\n\u22a2 ContinuousOn (fun t => (g t - g a, \u222b (u : \u211d) in a..t, EReal.toReal (G' u))) [[a, b]]"}, {"tactic": "exact (hcont.sub continuousOn_const).prod (continuousOn_primitive_interval G'int)", "annotated_tactic": ["exact (hcont.sub <a>continuousOn_const</a>).<a>prod</a> (<a>continuousOn_primitive_interval</a> G'int)", [{"full_name": "continuousOn_const", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [1025, 9], "def_end_pos": [1025, 27]}, {"full_name": "ContinuousOn.prod", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [1113, 9], "def_end_pos": [1113, 26]}, {"full_name": "intervalIntegral.continuousOn_primitive_interval", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [1225, 9], "def_end_pos": [1225, 40]}]], "state_before": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\na b : \u211d\nhab : a \u2264 b\nhcont : ContinuousOn g [[a, b]]\nhderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nG' : \u211d \u2192 EReal\nf_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x\nG'cont : LowerSemicontinuous G'\nG'int : Integrable fun x => EReal.toReal (G' x)\nG'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4\nhG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5\ns : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b\n\u22a2 ContinuousOn (fun t => (g t - g a, \u222b (u : \u211d) in a..t, EReal.toReal (G' u))) [[a, b]]", "state_after": "no goals"}, {"tactic": "refine s_closed.Icc_subset_of_forall_exists_gt\n  (by simp only [integral_same, mem_setOf_eq, sub_self, le_rfl]) fun t ht v t_lt_v => ?_", "annotated_tactic": ["refine s_closed.Icc_subset_of_forall_exists_gt\n      (by simp only [<a>integral_same</a>, <a>mem_setOf_eq</a>, <a>sub_self</a>, <a>le_rfl</a>]) fun t ht v t_lt_v => ?_", [{"full_name": "intervalIntegral.integral_same", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [470, 9], "def_end_pos": [470, 22]}, {"full_name": "Set.mem_setOf_eq", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [256, 29], "def_end_pos": [256, 41]}, {"full_name": "sub_self", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [734, 30], "def_end_pos": [734, 38]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}]], "state_before": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\na b : \u211d\nhab : a \u2264 b\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nG' : \u211d \u2192 EReal\nf_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x\nG'cont : LowerSemicontinuous G'\nG'int : Integrable fun x => EReal.toReal (G' x)\nG'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4\nhG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5\ns : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b\ns_closed : IsClosed s\n\u22a2 Icc a b \u2286 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)}", "state_after": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\na b : \u211d\nhab : a \u2264 b\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nG' : \u211d \u2192 EReal\nf_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x\nG'cont : LowerSemicontinuous G'\nG'int : Integrable fun x => EReal.toReal (G' x)\nG'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4\nhG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5\ns : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b\ns_closed : IsClosed s\nt : \u211d\nht : t \u2208 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ico a b\nv : \u211d\nt_lt_v : v \u2208 Ioi t\n\u22a2 Set.Nonempty ({t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ioc t v)"}, {"tactic": "obtain \u27e8y, g'_lt_y', y_lt_G'\u27e9 : \u2203 y : \u211d, (g' t : EReal) < y \u2227 (y : EReal) < G' t :=\n  EReal.lt_iff_exists_real_btwn.1 ((EReal.coe_le_coe_iff.2 (h\u03c6g t ht.2)).trans_lt (f_lt_G' t))", "annotated_tactic": ["obtain \u27e8y, g'_lt_y', y_lt_G'\u27e9 : \u2203 y : \u211d, (g' t : <a>EReal</a>) < y \u2227 (y : <a>EReal</a>) < G' t :=\n      <a>EReal.lt_iff_exists_real_btwn</a>.1 ((<a>EReal.coe_le_coe_iff</a>.2 (h\u03c6g t ht.2)).<a>trans_lt</a> (f_lt_G' t))", [{"full_name": "EReal", "def_path": "Mathlib/Data/Real/EReal.lean", "def_pos": [57, 5], "def_end_pos": [57, 10]}, {"full_name": "EReal", "def_path": "Mathlib/Data/Real/EReal.lean", "def_pos": [57, 5], "def_end_pos": [57, 10]}, {"full_name": "EReal.lt_iff_exists_real_btwn", "def_path": "Mathlib/Data/Real/EReal.lean", "def_pos": [617, 9], "def_end_pos": [617, 32]}, {"full_name": "EReal.coe_le_coe_iff", "def_path": "Mathlib/Data/Real/EReal.lean", "def_pos": [102, 19], "def_end_pos": [102, 33]}, {"full_name": "LE.le.trans_lt", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [124, 7], "def_end_pos": [124, 21]}]], "state_before": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\na b : \u211d\nhab : a \u2264 b\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nG' : \u211d \u2192 EReal\nf_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x\nG'cont : LowerSemicontinuous G'\nG'int : Integrable fun x => EReal.toReal (G' x)\nG'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4\nhG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5\ns : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b\ns_closed : IsClosed s\nt : \u211d\nht : t \u2208 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ico a b\nv : \u211d\nt_lt_v : v \u2208 Ioi t\n\u22a2 Set.Nonempty ({t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ioc t v)", "state_after": "case intro.intro\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\na b : \u211d\nhab : a \u2264 b\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nG' : \u211d \u2192 EReal\nf_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x\nG'cont : LowerSemicontinuous G'\nG'int : Integrable fun x => EReal.toReal (G' x)\nG'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4\nhG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5\ns : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b\ns_closed : IsClosed s\nt : \u211d\nht : t \u2208 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ico a b\nv : \u211d\nt_lt_v : v \u2208 Ioi t\ny : \u211d\ng'_lt_y' : \u2191(g' t) < \u2191y\ny_lt_G' : \u2191y < G' t\n\u22a2 Set.Nonempty ({t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ioc t v)"}, {"tactic": "have I2 : \u2200\u1da0 u in \ud835\udcdd[>] t, g u - g t \u2264 (u - t) * y := by\n  have g'_lt_y : g' t < y := EReal.coe_lt_coe_iff.1 g'_lt_y'\n  filter_upwards [(hderiv t \u27e8ht.2.1, ht.2.2\u27e9).limsup_slope_le' (not_mem_Ioi.2 le_rfl) g'_lt_y,\n    self_mem_nhdsWithin] with u hu t_lt_u\n  have := mul_le_mul_of_nonneg_left hu.le (sub_pos.2 t_lt_u.out).le\n  rwa [\u2190 smul_eq_mul, sub_smul_slope] at this", "annotated_tactic": ["have I2 : \u2200\u1da0 u in \ud835\udcdd[>] t, g u - g t \u2264 (u - t) * y := by\n      have g'_lt_y : g' t < y := <a>EReal.coe_lt_coe_iff</a>.1 g'_lt_y'\n      filter_upwards [(hderiv t \u27e8ht.2.1, ht.2.2\u27e9).<a>limsup_slope_le'</a> (<a>not_mem_Ioi</a>.2 <a>le_rfl</a>) g'_lt_y,\n        <a>self_mem_nhdsWithin</a>] with u hu t_lt_u\n      have := <a>mul_le_mul_of_nonneg_left</a> hu.le (<a>sub_pos</a>.2 t_lt_u.out).<a>le</a>\n      rwa [\u2190 <a>smul_eq_mul</a>, <a>sub_smul_slope</a>] at this", [{"full_name": "EReal.coe_lt_coe_iff", "def_path": "Mathlib/Data/Real/EReal.lean", "def_pos": [107, 19], "def_end_pos": [107, 33]}, {"full_name": "HasDerivWithinAt.limsup_slope_le'", "def_path": "Mathlib/Analysis/Calculus/Deriv/Slope.lean", "def_pos": [178, 9], "def_end_pos": [178, 42]}, {"full_name": "Set.not_mem_Ioi", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [1065, 9], "def_end_pos": [1065, 20]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}, {"full_name": "self_mem_nhdsWithin", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [151, 9], "def_end_pos": [151, 28]}, {"full_name": "mul_le_mul_of_nonneg_left", "def_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "def_pos": [152, 9], "def_end_pos": [152, 34]}, {"full_name": "sub_pos", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [883, 30], "def_end_pos": [883, 37]}, {"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [142, 7], "def_end_pos": [142, 15]}, {"full_name": "smul_eq_mul", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [93, 9], "def_end_pos": [93, 20]}, {"full_name": "sub_smul_slope", "def_path": "Mathlib/LinearAlgebra/AffineSpace/Slope.lean", "def_pos": [56, 9], "def_end_pos": [56, 23]}]], "state_before": "case intro.intro\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\na b : \u211d\nhab : a \u2264 b\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nG' : \u211d \u2192 EReal\nf_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x\nG'cont : LowerSemicontinuous G'\nG'int : Integrable fun x => EReal.toReal (G' x)\nG'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4\nhG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5\ns : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b\ns_closed : IsClosed s\nt : \u211d\nht : t \u2208 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ico a b\nv : \u211d\nt_lt_v : v \u2208 Ioi t\ny : \u211d\ng'_lt_y' : \u2191(g' t) < \u2191y\ny_lt_G' : \u2191y < G' t\nI1 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, (u - t) * y \u2264 \u222b (w : \u211d) in t..u, EReal.toReal (G' w)\n\u22a2 Set.Nonempty ({t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ioc t v)", "state_after": "case intro.intro\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\na b : \u211d\nhab : a \u2264 b\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nG' : \u211d \u2192 EReal\nf_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x\nG'cont : LowerSemicontinuous G'\nG'int : Integrable fun x => EReal.toReal (G' x)\nG'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4\nhG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5\ns : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b\ns_closed : IsClosed s\nt : \u211d\nht : t \u2208 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ico a b\nv : \u211d\nt_lt_v : v \u2208 Ioi t\ny : \u211d\ng'_lt_y' : \u2191(g' t) < \u2191y\ny_lt_G' : \u2191y < G' t\nI1 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, (u - t) * y \u2264 \u222b (w : \u211d) in t..u, EReal.toReal (G' w)\nI2 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, g u - g t \u2264 (u - t) * y\n\u22a2 Set.Nonempty ({t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ioc t v)"}, {"tactic": "have I3 : \u2200\u1da0 u in \ud835\udcdd[>] t, g u - g t \u2264 \u222b w in t..u, (G' w).toReal := by\n  filter_upwards [I1, I2] with u hu1 hu2 using hu2.trans hu1", "annotated_tactic": ["have I3 : \u2200\u1da0 u in \ud835\udcdd[>] t, g u - g t \u2264 \u222b w in t..u, (G' w).<a>toReal</a> := by\n      filter_upwards [I1, I2] with u hu1 hu2 using hu2.trans hu1", [{"full_name": "EReal.toReal", "def_path": "Mathlib/Data/Real/EReal.lean", "def_pos": [254, 5], "def_end_pos": [254, 11]}]], "state_before": "case intro.intro\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\na b : \u211d\nhab : a \u2264 b\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nG' : \u211d \u2192 EReal\nf_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x\nG'cont : LowerSemicontinuous G'\nG'int : Integrable fun x => EReal.toReal (G' x)\nG'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4\nhG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5\ns : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b\ns_closed : IsClosed s\nt : \u211d\nht : t \u2208 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ico a b\nv : \u211d\nt_lt_v : v \u2208 Ioi t\ny : \u211d\ng'_lt_y' : \u2191(g' t) < \u2191y\ny_lt_G' : \u2191y < G' t\nI1 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, (u - t) * y \u2264 \u222b (w : \u211d) in t..u, EReal.toReal (G' w)\nI2 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, g u - g t \u2264 (u - t) * y\n\u22a2 Set.Nonempty ({t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ioc t v)", "state_after": "case intro.intro\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\na b : \u211d\nhab : a \u2264 b\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nG' : \u211d \u2192 EReal\nf_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x\nG'cont : LowerSemicontinuous G'\nG'int : Integrable fun x => EReal.toReal (G' x)\nG'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4\nhG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5\ns : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b\ns_closed : IsClosed s\nt : \u211d\nht : t \u2208 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ico a b\nv : \u211d\nt_lt_v : v \u2208 Ioi t\ny : \u211d\ng'_lt_y' : \u2191(g' t) < \u2191y\ny_lt_G' : \u2191y < G' t\nI1 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, (u - t) * y \u2264 \u222b (w : \u211d) in t..u, EReal.toReal (G' w)\nI2 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, g u - g t \u2264 (u - t) * y\nI3 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, g u - g t \u2264 \u222b (w : \u211d) in t..u, EReal.toReal (G' w)\n\u22a2 Set.Nonempty ({t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ioc t v)"}, {"tactic": "have I4 : \u2200\u1da0 u in \ud835\udcdd[>] t, u \u2208 Ioc t (min v b) := by\n  refine' mem_nhdsWithin_Ioi_iff_exists_Ioc_subset.2 \u27e8min v b, _, Subset.rfl\u27e9\n  simp only [lt_min_iff, mem_Ioi]\n  exact \u27e8t_lt_v, ht.2.2\u27e9", "annotated_tactic": ["have I4 : \u2200\u1da0 u in \ud835\udcdd[>] t, u \u2208 <a>Ioc</a> t (<a>min</a> v b) := by\n      refine' <a>mem_nhdsWithin_Ioi_iff_exists_Ioc_subset</a>.2 \u27e8<a>min</a> v b, _, <a>Subset.rfl</a>\u27e9\n      simp only [<a>lt_min_iff</a>, <a>mem_Ioi</a>]\n      exact \u27e8t_lt_v, ht.2.2\u27e9", [{"full_name": "Set.Ioc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [69, 5], "def_end_pos": [69, 8]}, {"full_name": "Min.min", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1103, 3], "def_end_pos": [1103, 6]}, {"full_name": "mem_nhdsWithin_Ioi_iff_exists_Ioc_subset", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [1733, 9], "def_end_pos": [1733, 49]}, {"full_name": "Min.min", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1103, 3], "def_end_pos": [1103, 6]}, {"full_name": "Set.Subset.rfl", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [357, 9], "def_end_pos": [357, 19]}, {"full_name": "lt_min_iff", "def_path": "Mathlib/Order/MinMax.lean", "def_pos": [53, 9], "def_end_pos": [53, 19]}, {"full_name": "Set.mem_Ioi", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [151, 9], "def_end_pos": [151, 16]}]], "state_before": "case intro.intro\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\na b : \u211d\nhab : a \u2264 b\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nG' : \u211d \u2192 EReal\nf_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x\nG'cont : LowerSemicontinuous G'\nG'int : Integrable fun x => EReal.toReal (G' x)\nG'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4\nhG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5\ns : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b\ns_closed : IsClosed s\nt : \u211d\nht : t \u2208 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ico a b\nv : \u211d\nt_lt_v : v \u2208 Ioi t\ny : \u211d\ng'_lt_y' : \u2191(g' t) < \u2191y\ny_lt_G' : \u2191y < G' t\nI1 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, (u - t) * y \u2264 \u222b (w : \u211d) in t..u, EReal.toReal (G' w)\nI2 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, g u - g t \u2264 (u - t) * y\nI3 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, g u - g t \u2264 \u222b (w : \u211d) in t..u, EReal.toReal (G' w)\n\u22a2 Set.Nonempty ({t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ioc t v)", "state_after": "case intro.intro\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\na b : \u211d\nhab : a \u2264 b\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nG' : \u211d \u2192 EReal\nf_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x\nG'cont : LowerSemicontinuous G'\nG'int : Integrable fun x => EReal.toReal (G' x)\nG'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4\nhG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5\ns : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b\ns_closed : IsClosed s\nt : \u211d\nht : t \u2208 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ico a b\nv : \u211d\nt_lt_v : v \u2208 Ioi t\ny : \u211d\ng'_lt_y' : \u2191(g' t) < \u2191y\ny_lt_G' : \u2191y < G' t\nI1 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, (u - t) * y \u2264 \u222b (w : \u211d) in t..u, EReal.toReal (G' w)\nI2 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, g u - g t \u2264 (u - t) * y\nI3 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, g u - g t \u2264 \u222b (w : \u211d) in t..u, EReal.toReal (G' w)\nI4 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, u \u2208 Ioc t (min v b)\n\u22a2 Set.Nonempty ({t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ioc t v)"}, {"tactic": "rcases (I3.and I4).exists with \u27e8x, hx, h'x\u27e9", "annotated_tactic": ["rcases (I3.and I4).<a>exists</a> with \u27e8x, hx, h'x\u27e9", [{"full_name": "Filter.Eventually.exists", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1308, 9], "def_end_pos": [1308, 26]}]], "state_before": "case intro.intro\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\na b : \u211d\nhab : a \u2264 b\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nG' : \u211d \u2192 EReal\nf_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x\nG'cont : LowerSemicontinuous G'\nG'int : Integrable fun x => EReal.toReal (G' x)\nG'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4\nhG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5\ns : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b\ns_closed : IsClosed s\nt : \u211d\nht : t \u2208 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ico a b\nv : \u211d\nt_lt_v : v \u2208 Ioi t\ny : \u211d\ng'_lt_y' : \u2191(g' t) < \u2191y\ny_lt_G' : \u2191y < G' t\nI1 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, (u - t) * y \u2264 \u222b (w : \u211d) in t..u, EReal.toReal (G' w)\nI2 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, g u - g t \u2264 (u - t) * y\nI3 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, g u - g t \u2264 \u222b (w : \u211d) in t..u, EReal.toReal (G' w)\nI4 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, u \u2208 Ioc t (min v b)\n\u22a2 Set.Nonempty ({t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ioc t v)", "state_after": "case intro.intro.intro.intro\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\na b : \u211d\nhab : a \u2264 b\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nG' : \u211d \u2192 EReal\nf_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x\nG'cont : LowerSemicontinuous G'\nG'int : Integrable fun x => EReal.toReal (G' x)\nG'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4\nhG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5\ns : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b\ns_closed : IsClosed s\nt : \u211d\nht : t \u2208 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ico a b\nv : \u211d\nt_lt_v : v \u2208 Ioi t\ny : \u211d\ng'_lt_y' : \u2191(g' t) < \u2191y\ny_lt_G' : \u2191y < G' t\nI1 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, (u - t) * y \u2264 \u222b (w : \u211d) in t..u, EReal.toReal (G' w)\nI2 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, g u - g t \u2264 (u - t) * y\nI3 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, g u - g t \u2264 \u222b (w : \u211d) in t..u, EReal.toReal (G' w)\nI4 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, u \u2208 Ioc t (min v b)\nx : \u211d\nhx : g x - g t \u2264 \u222b (w : \u211d) in t..x, EReal.toReal (G' w)\nh'x : x \u2208 Ioc t (min v b)\n\u22a2 Set.Nonempty ({t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ioc t v)"}, {"tactic": "refine' \u27e8x, _, Ioc_subset_Ioc le_rfl (min_le_left _ _) h'x\u27e9", "annotated_tactic": ["refine' \u27e8x, _, <a>Ioc_subset_Ioc</a> <a>le_rfl</a> (<a>min_le_left</a> _ _) h'x\u27e9", [{"full_name": "Set.Ioc_subset_Ioc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [480, 9], "def_end_pos": [480, 23]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}, {"full_name": "min_le_left", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [33, 9], "def_end_pos": [33, 20]}]], "state_before": "case intro.intro.intro.intro\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\na b : \u211d\nhab : a \u2264 b\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nG' : \u211d \u2192 EReal\nf_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x\nG'cont : LowerSemicontinuous G'\nG'int : Integrable fun x => EReal.toReal (G' x)\nG'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4\nhG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5\ns : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b\ns_closed : IsClosed s\nt : \u211d\nht : t \u2208 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ico a b\nv : \u211d\nt_lt_v : v \u2208 Ioi t\ny : \u211d\ng'_lt_y' : \u2191(g' t) < \u2191y\ny_lt_G' : \u2191y < G' t\nI1 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, (u - t) * y \u2264 \u222b (w : \u211d) in t..u, EReal.toReal (G' w)\nI2 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, g u - g t \u2264 (u - t) * y\nI3 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, g u - g t \u2264 \u222b (w : \u211d) in t..u, EReal.toReal (G' w)\nI4 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, u \u2208 Ioc t (min v b)\nx : \u211d\nhx : g x - g t \u2264 \u222b (w : \u211d) in t..x, EReal.toReal (G' w)\nh'x : x \u2208 Ioc t (min v b)\n\u22a2 Set.Nonempty ({t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ioc t v)", "state_after": "case intro.intro.intro.intro\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\na b : \u211d\nhab : a \u2264 b\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nG' : \u211d \u2192 EReal\nf_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x\nG'cont : LowerSemicontinuous G'\nG'int : Integrable fun x => EReal.toReal (G' x)\nG'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4\nhG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5\ns : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b\ns_closed : IsClosed s\nt : \u211d\nht : t \u2208 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ico a b\nv : \u211d\nt_lt_v : v \u2208 Ioi t\ny : \u211d\ng'_lt_y' : \u2191(g' t) < \u2191y\ny_lt_G' : \u2191y < G' t\nI1 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, (u - t) * y \u2264 \u222b (w : \u211d) in t..u, EReal.toReal (G' w)\nI2 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, g u - g t \u2264 (u - t) * y\nI3 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, g u - g t \u2264 \u222b (w : \u211d) in t..u, EReal.toReal (G' w)\nI4 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, u \u2208 Ioc t (min v b)\nx : \u211d\nhx : g x - g t \u2264 \u222b (w : \u211d) in t..x, EReal.toReal (G' w)\nh'x : x \u2208 Ioc t (min v b)\n\u22a2 x \u2208 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)}"}, {"tactic": "simp only [integral_same, mem_setOf_eq, sub_self, le_rfl]", "annotated_tactic": ["simp only [<a>integral_same</a>, <a>mem_setOf_eq</a>, <a>sub_self</a>, <a>le_rfl</a>]", [{"full_name": "intervalIntegral.integral_same", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [470, 9], "def_end_pos": [470, 22]}, {"full_name": "Set.mem_setOf_eq", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [256, 29], "def_end_pos": [256, 41]}, {"full_name": "sub_self", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [734, 30], "def_end_pos": [734, 38]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}]], "state_before": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\na b : \u211d\nhab : a \u2264 b\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nG' : \u211d \u2192 EReal\nf_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x\nG'cont : LowerSemicontinuous G'\nG'int : Integrable fun x => EReal.toReal (G' x)\nG'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4\nhG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5\ns : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b\ns_closed : IsClosed s\n\u22a2 a \u2208 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)}", "state_after": "no goals"}, {"tactic": "have B : \u2200\u1da0 u in \ud835\udcdd t, (y : EReal) < G' u := G'cont.lowerSemicontinuousAt _ _ y_lt_G'", "annotated_tactic": ["have B : \u2200\u1da0 u in \ud835\udcdd t, (y : <a>EReal</a>) < G' u := G'cont.lowerSemicontinuousAt _ _ y_lt_G'", [{"full_name": "EReal", "def_path": "Mathlib/Data/Real/EReal.lean", "def_pos": [57, 5], "def_end_pos": [57, 10]}]], "state_before": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\na b : \u211d\nhab : a \u2264 b\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nG' : \u211d \u2192 EReal\nf_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x\nG'cont : LowerSemicontinuous G'\nG'int : Integrable fun x => EReal.toReal (G' x)\nG'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4\nhG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5\ns : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b\ns_closed : IsClosed s\nt : \u211d\nht : t \u2208 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ico a b\nv : \u211d\nt_lt_v : v \u2208 Ioi t\ny : \u211d\ng'_lt_y' : \u2191(g' t) < \u2191y\ny_lt_G' : \u2191y < G' t\n\u22a2 \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, (u - t) * y \u2264 \u222b (w : \u211d) in t..u, EReal.toReal (G' w)", "state_after": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\na b : \u211d\nhab : a \u2264 b\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nG' : \u211d \u2192 EReal\nf_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x\nG'cont : LowerSemicontinuous G'\nG'int : Integrable fun x => EReal.toReal (G' x)\nG'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4\nhG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5\ns : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b\ns_closed : IsClosed s\nt : \u211d\nht : t \u2208 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ico a b\nv : \u211d\nt_lt_v : v \u2208 Ioi t\ny : \u211d\ng'_lt_y' : \u2191(g' t) < \u2191y\ny_lt_G' : \u2191y < G' t\nB : \u2200\u1da0 (u : \u211d) in \ud835\udcdd t, \u2191y < G' u\n\u22a2 \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, (u - t) * y \u2264 \u222b (w : \u211d) in t..u, EReal.toReal (G' w)"}, {"tactic": "rcases mem_nhds_iff_exists_Ioo_subset.1 B with \u27e8m, M, \u27e8hm, hM\u27e9, H\u27e9", "annotated_tactic": ["rcases <a>mem_nhds_iff_exists_Ioo_subset</a>.1 B with \u27e8m, M, \u27e8hm, hM\u27e9, H\u27e9", [{"full_name": "mem_nhds_iff_exists_Ioo_subset", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [1309, 9], "def_end_pos": [1309, 39]}]], "state_before": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\na b : \u211d\nhab : a \u2264 b\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nG' : \u211d \u2192 EReal\nf_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x\nG'cont : LowerSemicontinuous G'\nG'int : Integrable fun x => EReal.toReal (G' x)\nG'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4\nhG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5\ns : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b\ns_closed : IsClosed s\nt : \u211d\nht : t \u2208 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ico a b\nv : \u211d\nt_lt_v : v \u2208 Ioi t\ny : \u211d\ng'_lt_y' : \u2191(g' t) < \u2191y\ny_lt_G' : \u2191y < G' t\nB : \u2200\u1da0 (u : \u211d) in \ud835\udcdd t, \u2191y < G' u\n\u22a2 \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, (u - t) * y \u2264 \u222b (w : \u211d) in t..u, EReal.toReal (G' w)", "state_after": "case intro.intro.intro.intro\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\na b : \u211d\nhab : a \u2264 b\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nG' : \u211d \u2192 EReal\nf_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x\nG'cont : LowerSemicontinuous G'\nG'int : Integrable fun x => EReal.toReal (G' x)\nG'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4\nhG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5\ns : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b\ns_closed : IsClosed s\nt : \u211d\nht : t \u2208 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ico a b\nv : \u211d\nt_lt_v : v \u2208 Ioi t\ny : \u211d\ng'_lt_y' : \u2191(g' t) < \u2191y\ny_lt_G' : \u2191y < G' t\nB : \u2200\u1da0 (u : \u211d) in \ud835\udcdd t, \u2191y < G' u\nm M : \u211d\nH : Ioo m M \u2286 {x | (fun u => \u2191y < G' u) x}\nhm : m < t\nhM : t < M\n\u22a2 \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, (u - t) * y \u2264 \u222b (w : \u211d) in t..u, EReal.toReal (G' w)"}, {"tactic": "have : Ioo t (min M b) \u2208 \ud835\udcdd[>] t := Ioo_mem_nhdsWithin_Ioi' (lt_min hM ht.right.right)", "annotated_tactic": ["have : <a>Ioo</a> t (<a>min</a> M b) \u2208 \ud835\udcdd[>] t := <a>Ioo_mem_nhdsWithin_Ioi'</a> (<a>lt_min</a> hM ht.right.right)", [{"full_name": "Set.Ioo", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [44, 5], "def_end_pos": [44, 8]}, {"full_name": "Min.min", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1103, 3], "def_end_pos": [1103, 6]}, {"full_name": "Ioo_mem_nhdsWithin_Ioi'", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [433, 9], "def_end_pos": [433, 32]}, {"full_name": "lt_min", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [159, 9], "def_end_pos": [159, 15]}]], "state_before": "case intro.intro.intro.intro\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\na b : \u211d\nhab : a \u2264 b\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nG' : \u211d \u2192 EReal\nf_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x\nG'cont : LowerSemicontinuous G'\nG'int : Integrable fun x => EReal.toReal (G' x)\nG'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4\nhG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5\ns : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b\ns_closed : IsClosed s\nt : \u211d\nht : t \u2208 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ico a b\nv : \u211d\nt_lt_v : v \u2208 Ioi t\ny : \u211d\ng'_lt_y' : \u2191(g' t) < \u2191y\ny_lt_G' : \u2191y < G' t\nB : \u2200\u1da0 (u : \u211d) in \ud835\udcdd t, \u2191y < G' u\nm M : \u211d\nH : Ioo m M \u2286 {x | (fun u => \u2191y < G' u) x}\nhm : m < t\nhM : t < M\n\u22a2 \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, (u - t) * y \u2264 \u222b (w : \u211d) in t..u, EReal.toReal (G' w)", "state_after": "case intro.intro.intro.intro\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\na b : \u211d\nhab : a \u2264 b\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nG' : \u211d \u2192 EReal\nf_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x\nG'cont : LowerSemicontinuous G'\nG'int : Integrable fun x => EReal.toReal (G' x)\nG'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4\nhG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5\ns : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b\ns_closed : IsClosed s\nt : \u211d\nht : t \u2208 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ico a b\nv : \u211d\nt_lt_v : v \u2208 Ioi t\ny : \u211d\ng'_lt_y' : \u2191(g' t) < \u2191y\ny_lt_G' : \u2191y < G' t\nB : \u2200\u1da0 (u : \u211d) in \ud835\udcdd t, \u2191y < G' u\nm M : \u211d\nH : Ioo m M \u2286 {x | (fun u => \u2191y < G' u) x}\nhm : m < t\nhM : t < M\nthis : Ioo t (min M b) \u2208 \ud835\udcdd[Ioi t] t\n\u22a2 \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, (u - t) * y \u2264 \u222b (w : \u211d) in t..u, EReal.toReal (G' w)"}, {"tactic": "filter_upwards [this] with u hu", "annotated_tactic": ["filter_upwards [this] with u hu", []], "state_before": "case intro.intro.intro.intro\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\na b : \u211d\nhab : a \u2264 b\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nG' : \u211d \u2192 EReal\nf_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x\nG'cont : LowerSemicontinuous G'\nG'int : Integrable fun x => EReal.toReal (G' x)\nG'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4\nhG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5\ns : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b\ns_closed : IsClosed s\nt : \u211d\nht : t \u2208 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ico a b\nv : \u211d\nt_lt_v : v \u2208 Ioi t\ny : \u211d\ng'_lt_y' : \u2191(g' t) < \u2191y\ny_lt_G' : \u2191y < G' t\nB : \u2200\u1da0 (u : \u211d) in \ud835\udcdd t, \u2191y < G' u\nm M : \u211d\nH : Ioo m M \u2286 {x | (fun u => \u2191y < G' u) x}\nhm : m < t\nhM : t < M\nthis : Ioo t (min M b) \u2208 \ud835\udcdd[Ioi t] t\n\u22a2 \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, (u - t) * y \u2264 \u222b (w : \u211d) in t..u, EReal.toReal (G' w)", "state_after": "case h\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\na b : \u211d\nhab : a \u2264 b\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nG' : \u211d \u2192 EReal\nf_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x\nG'cont : LowerSemicontinuous G'\nG'int : Integrable fun x => EReal.toReal (G' x)\nG'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4\nhG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5\ns : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b\ns_closed : IsClosed s\nt : \u211d\nht : t \u2208 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ico a b\nv : \u211d\nt_lt_v : v \u2208 Ioi t\ny : \u211d\ng'_lt_y' : \u2191(g' t) < \u2191y\ny_lt_G' : \u2191y < G' t\nB : \u2200\u1da0 (u : \u211d) in \ud835\udcdd t, \u2191y < G' u\nm M : \u211d\nH : Ioo m M \u2286 {x | (fun u => \u2191y < G' u) x}\nhm : m < t\nhM : t < M\nthis : Ioo t (min M b) \u2208 \ud835\udcdd[Ioi t] t\nu : \u211d\nhu : u \u2208 Ioo t (min M b)\n\u22a2 (u - t) * y \u2264 \u222b (w : \u211d) in t..u, EReal.toReal (G' w)"}, {"tactic": "have I : Icc t u \u2286 Icc a b := Icc_subset_Icc ht.2.1 (hu.2.le.trans (min_le_right _ _))", "annotated_tactic": ["have I : <a>Icc</a> t u \u2286 <a>Icc</a> a b := <a>Icc_subset_Icc</a> ht.2.1 (hu.2.le.trans (<a>min_le_right</a> _ _))", [{"full_name": "Set.Icc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [59, 5], "def_end_pos": [59, 8]}, {"full_name": "Set.Icc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [59, 5], "def_end_pos": [59, 8]}, {"full_name": "Set.Icc_subset_Icc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [455, 9], "def_end_pos": [455, 23]}, {"full_name": "min_le_right", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [40, 9], "def_end_pos": [40, 21]}]], "state_before": "case h\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\na b : \u211d\nhab : a \u2264 b\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nG' : \u211d \u2192 EReal\nf_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x\nG'cont : LowerSemicontinuous G'\nG'int : Integrable fun x => EReal.toReal (G' x)\nG'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4\nhG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5\ns : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b\ns_closed : IsClosed s\nt : \u211d\nht : t \u2208 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ico a b\nv : \u211d\nt_lt_v : v \u2208 Ioi t\ny : \u211d\ng'_lt_y' : \u2191(g' t) < \u2191y\ny_lt_G' : \u2191y < G' t\nB : \u2200\u1da0 (u : \u211d) in \ud835\udcdd t, \u2191y < G' u\nm M : \u211d\nH : Ioo m M \u2286 {x | (fun u => \u2191y < G' u) x}\nhm : m < t\nhM : t < M\nthis : Ioo t (min M b) \u2208 \ud835\udcdd[Ioi t] t\nu : \u211d\nhu : u \u2208 Ioo t (min M b)\n\u22a2 (u - t) * y \u2264 \u222b (w : \u211d) in t..u, EReal.toReal (G' w)", "state_after": "case h\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\na b : \u211d\nhab : a \u2264 b\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nG' : \u211d \u2192 EReal\nf_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x\nG'cont : LowerSemicontinuous G'\nG'int : Integrable fun x => EReal.toReal (G' x)\nG'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4\nhG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5\ns : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b\ns_closed : IsClosed s\nt : \u211d\nht : t \u2208 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ico a b\nv : \u211d\nt_lt_v : v \u2208 Ioi t\ny : \u211d\ng'_lt_y' : \u2191(g' t) < \u2191y\ny_lt_G' : \u2191y < G' t\nB : \u2200\u1da0 (u : \u211d) in \ud835\udcdd t, \u2191y < G' u\nm M : \u211d\nH : Ioo m M \u2286 {x | (fun u => \u2191y < G' u) x}\nhm : m < t\nhM : t < M\nthis : Ioo t (min M b) \u2208 \ud835\udcdd[Ioi t] t\nu : \u211d\nhu : u \u2208 Ioo t (min M b)\nI : Icc t u \u2286 Icc a b\n\u22a2 (u - t) * y \u2264 \u222b (w : \u211d) in t..u, EReal.toReal (G' w)"}, {"tactic": "simp only [hu.left.le, MeasureTheory.integral_const, Algebra.id.smul_eq_mul, sub_nonneg,\n  MeasurableSet.univ, Real.volume_Icc, Measure.restrict_apply, univ_inter,\n  ENNReal.toReal_ofReal]", "annotated_tactic": ["simp only [hu.left.le, <a>MeasureTheory.integral_const</a>, <a>Algebra.id.smul_eq_mul</a>, <a>sub_nonneg</a>,\n            <a>MeasurableSet.univ</a>, <a>Real.volume_Icc</a>, <a>Measure.restrict_apply</a>, <a>univ_inter</a>,\n            <a>ENNReal.toReal_ofReal</a>]", [{"full_name": "MeasureTheory.integral_const", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1409, 9], "def_end_pos": [1409, 23]}, {"full_name": "Algebra.id.smul_eq_mul", "def_path": "Mathlib/Algebra/Algebra/Basic.lean", "def_pos": [453, 9], "def_end_pos": [453, 20]}, {"full_name": "sub_nonneg", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [720, 30], "def_end_pos": [720, 40]}, {"full_name": "MeasurableSet.univ", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [101, 19], "def_end_pos": [101, 37]}, {"full_name": "Real.volume_Icc", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean", "def_pos": [84, 9], "def_end_pos": [84, 19]}, {"full_name": "MeasureTheory.Measure.restrict_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1533, 9], "def_end_pos": [1533, 23]}, {"full_name": "Set.univ_inter", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1017, 9], "def_end_pos": [1017, 19]}, {"full_name": "ENNReal.toReal_ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [191, 9], "def_end_pos": [191, 22]}]], "state_before": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\na b : \u211d\nhab : a \u2264 b\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nG' : \u211d \u2192 EReal\nf_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x\nG'cont : LowerSemicontinuous G'\nG'int : Integrable fun x => EReal.toReal (G' x)\nG'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4\nhG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5\ns : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b\ns_closed : IsClosed s\nt : \u211d\nht : t \u2208 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ico a b\nv : \u211d\nt_lt_v : v \u2208 Ioi t\ny : \u211d\ng'_lt_y' : \u2191(g' t) < \u2191y\ny_lt_G' : \u2191y < G' t\nB : \u2200\u1da0 (u : \u211d) in \ud835\udcdd t, \u2191y < G' u\nm M : \u211d\nH : Ioo m M \u2286 {x | (fun u => \u2191y < G' u) x}\nhm : m < t\nhM : t < M\nthis : Ioo t (min M b) \u2208 \ud835\udcdd[Ioi t] t\nu : \u211d\nhu : u \u2208 Ioo t (min M b)\nI : Icc t u \u2286 Icc a b\n\u22a2 (u - t) * y = \u222b (x : \u211d) in Icc t u, y", "state_after": "no goals"}, {"tactic": "rw [intervalIntegral.integral_of_le hu.1.le, \u2190 integral_Icc_eq_integral_Ioc]", "annotated_tactic": ["rw [<a>intervalIntegral.integral_of_le</a> hu.1.<a>le</a>, \u2190 <a>integral_Icc_eq_integral_Ioc</a>]", [{"full_name": "intervalIntegral.integral_of_le", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [465, 9], "def_end_pos": [465, 23]}, {"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [142, 7], "def_end_pos": [142, 15]}, {"full_name": "MeasureTheory.integral_Icc_eq_integral_Ioc", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [675, 9], "def_end_pos": [675, 37]}]], "state_before": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\na b : \u211d\nhab : a \u2264 b\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nG' : \u211d \u2192 EReal\nf_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x\nG'cont : LowerSemicontinuous G'\nG'int : Integrable fun x => EReal.toReal (G' x)\nG'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4\nhG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5\ns : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b\ns_closed : IsClosed s\nt : \u211d\nht : t \u2208 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ico a b\nv : \u211d\nt_lt_v : v \u2208 Ioi t\ny : \u211d\ng'_lt_y' : \u2191(g' t) < \u2191y\ny_lt_G' : \u2191y < G' t\nB : \u2200\u1da0 (u : \u211d) in \ud835\udcdd t, \u2191y < G' u\nm M : \u211d\nH : Ioo m M \u2286 {x | (fun u => \u2191y < G' u) x}\nhm : m < t\nhM : t < M\nthis : Ioo t (min M b) \u2208 \ud835\udcdd[Ioi t] t\nu : \u211d\nhu : u \u2208 Ioo t (min M b)\nI : Icc t u \u2286 Icc a b\n\u22a2 \u222b (x : \u211d) in Icc t u, y \u2264 \u222b (w : \u211d) in t..u, EReal.toReal (G' w)", "state_after": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\na b : \u211d\nhab : a \u2264 b\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nG' : \u211d \u2192 EReal\nf_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x\nG'cont : LowerSemicontinuous G'\nG'int : Integrable fun x => EReal.toReal (G' x)\nG'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4\nhG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5\ns : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b\ns_closed : IsClosed s\nt : \u211d\nht : t \u2208 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ico a b\nv : \u211d\nt_lt_v : v \u2208 Ioi t\ny : \u211d\ng'_lt_y' : \u2191(g' t) < \u2191y\ny_lt_G' : \u2191y < G' t\nB : \u2200\u1da0 (u : \u211d) in \ud835\udcdd t, \u2191y < G' u\nm M : \u211d\nH : Ioo m M \u2286 {x | (fun u => \u2191y < G' u) x}\nhm : m < t\nhM : t < M\nthis : Ioo t (min M b) \u2208 \ud835\udcdd[Ioi t] t\nu : \u211d\nhu : u \u2208 Ioo t (min M b)\nI : Icc t u \u2286 Icc a b\n\u22a2 \u222b (x : \u211d) in Icc t u, y \u2264 \u222b (t : \u211d) in Icc t u, EReal.toReal (G' t)"}, {"tactic": "apply set_integral_mono_ae_restrict", "annotated_tactic": ["apply <a>set_integral_mono_ae_restrict</a>", [{"full_name": "MeasureTheory.set_integral_mono_ae_restrict", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [712, 9], "def_end_pos": [712, 38]}]], "state_before": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\na b : \u211d\nhab : a \u2264 b\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nG' : \u211d \u2192 EReal\nf_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x\nG'cont : LowerSemicontinuous G'\nG'int : Integrable fun x => EReal.toReal (G' x)\nG'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4\nhG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5\ns : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b\ns_closed : IsClosed s\nt : \u211d\nht : t \u2208 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ico a b\nv : \u211d\nt_lt_v : v \u2208 Ioi t\ny : \u211d\ng'_lt_y' : \u2191(g' t) < \u2191y\ny_lt_G' : \u2191y < G' t\nB : \u2200\u1da0 (u : \u211d) in \ud835\udcdd t, \u2191y < G' u\nm M : \u211d\nH : Ioo m M \u2286 {x | (fun u => \u2191y < G' u) x}\nhm : m < t\nhM : t < M\nthis : Ioo t (min M b) \u2208 \ud835\udcdd[Ioi t] t\nu : \u211d\nhu : u \u2208 Ioo t (min M b)\nI : Icc t u \u2286 Icc a b\n\u22a2 \u222b (x : \u211d) in Icc t u, y \u2264 \u222b (t : \u211d) in Icc t u, EReal.toReal (G' t)", "state_after": "case hf\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\na b : \u211d\nhab : a \u2264 b\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nG' : \u211d \u2192 EReal\nf_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x\nG'cont : LowerSemicontinuous G'\nG'int : Integrable fun x => EReal.toReal (G' x)\nG'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4\nhG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5\ns : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b\ns_closed : IsClosed s\nt : \u211d\nht : t \u2208 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ico a b\nv : \u211d\nt_lt_v : v \u2208 Ioi t\ny : \u211d\ng'_lt_y' : \u2191(g' t) < \u2191y\ny_lt_G' : \u2191y < G' t\nB : \u2200\u1da0 (u : \u211d) in \ud835\udcdd t, \u2191y < G' u\nm M : \u211d\nH : Ioo m M \u2286 {x | (fun u => \u2191y < G' u) x}\nhm : m < t\nhM : t < M\nthis : Ioo t (min M b) \u2208 \ud835\udcdd[Ioi t] t\nu : \u211d\nhu : u \u2208 Ioo t (min M b)\nI : Icc t u \u2286 Icc a b\n\u22a2 IntegrableOn (fun a => y) (Icc t u)\n\ncase hg\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\na b : \u211d\nhab : a \u2264 b\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nG' : \u211d \u2192 EReal\nf_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x\nG'cont : LowerSemicontinuous G'\nG'int : Integrable fun x => EReal.toReal (G' x)\nG'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4\nhG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5\ns : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b\ns_closed : IsClosed s\nt : \u211d\nht : t \u2208 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ico a b\nv : \u211d\nt_lt_v : v \u2208 Ioi t\ny : \u211d\ng'_lt_y' : \u2191(g' t) < \u2191y\ny_lt_G' : \u2191y < G' t\nB : \u2200\u1da0 (u : \u211d) in \ud835\udcdd t, \u2191y < G' u\nm M : \u211d\nH : Ioo m M \u2286 {x | (fun u => \u2191y < G' u) x}\nhm : m < t\nhM : t < M\nthis : Ioo t (min M b) \u2208 \ud835\udcdd[Ioi t] t\nu : \u211d\nhu : u \u2208 Ioo t (min M b)\nI : Icc t u \u2286 Icc a b\n\u22a2 IntegrableOn (fun a => EReal.toReal (G' a)) (Icc t u)\n\ncase h\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\na b : \u211d\nhab : a \u2264 b\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nG' : \u211d \u2192 EReal\nf_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x\nG'cont : LowerSemicontinuous G'\nG'int : Integrable fun x => EReal.toReal (G' x)\nG'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4\nhG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5\ns : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b\ns_closed : IsClosed s\nt : \u211d\nht : t \u2208 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ico a b\nv : \u211d\nt_lt_v : v \u2208 Ioi t\ny : \u211d\ng'_lt_y' : \u2191(g' t) < \u2191y\ny_lt_G' : \u2191y < G' t\nB : \u2200\u1da0 (u : \u211d) in \ud835\udcdd t, \u2191y < G' u\nm M : \u211d\nH : Ioo m M \u2286 {x | (fun u => \u2191y < G' u) x}\nhm : m < t\nhM : t < M\nthis : Ioo t (min M b) \u2208 \ud835\udcdd[Ioi t] t\nu : \u211d\nhu : u \u2208 Ioo t (min M b)\nI : Icc t u \u2286 Icc a b\n\u22a2 (fun a => y) \u2264\u1d50[Measure.restrict volume (Icc t u)] fun a => EReal.toReal (G' a)"}, {"tactic": "simp only [integrableOn_const, Real.volume_Icc, ENNReal.ofReal_lt_top, or_true_iff]", "annotated_tactic": ["simp only [<a>integrableOn_const</a>, <a>Real.volume_Icc</a>, <a>ENNReal.ofReal_lt_top</a>, <a>or_true_iff</a>]", [{"full_name": "MeasureTheory.integrableOn_const", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [119, 9], "def_end_pos": [119, 27]}, {"full_name": "Real.volume_Icc", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean", "def_pos": [84, 9], "def_end_pos": [84, 19]}, {"full_name": "ENNReal.ofReal_lt_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [314, 17], "def_end_pos": [314, 30]}, {"full_name": "or_true_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [184, 9], "def_end_pos": [184, 20]}]], "state_before": "case hf\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\na b : \u211d\nhab : a \u2264 b\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nG' : \u211d \u2192 EReal\nf_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x\nG'cont : LowerSemicontinuous G'\nG'int : Integrable fun x => EReal.toReal (G' x)\nG'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4\nhG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5\ns : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b\ns_closed : IsClosed s\nt : \u211d\nht : t \u2208 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ico a b\nv : \u211d\nt_lt_v : v \u2208 Ioi t\ny : \u211d\ng'_lt_y' : \u2191(g' t) < \u2191y\ny_lt_G' : \u2191y < G' t\nB : \u2200\u1da0 (u : \u211d) in \ud835\udcdd t, \u2191y < G' u\nm M : \u211d\nH : Ioo m M \u2286 {x | (fun u => \u2191y < G' u) x}\nhm : m < t\nhM : t < M\nthis : Ioo t (min M b) \u2208 \ud835\udcdd[Ioi t] t\nu : \u211d\nhu : u \u2208 Ioo t (min M b)\nI : Icc t u \u2286 Icc a b\n\u22a2 IntegrableOn (fun a => y) (Icc t u)", "state_after": "no goals"}, {"tactic": "exact IntegrableOn.mono_set G'int I", "annotated_tactic": ["exact <a>IntegrableOn.mono_set</a> G'int I", [{"full_name": "MeasureTheory.IntegrableOn.mono_set", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [127, 9], "def_end_pos": [127, 30]}]], "state_before": "case hg\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\na b : \u211d\nhab : a \u2264 b\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nG' : \u211d \u2192 EReal\nf_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x\nG'cont : LowerSemicontinuous G'\nG'int : Integrable fun x => EReal.toReal (G' x)\nG'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4\nhG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5\ns : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b\ns_closed : IsClosed s\nt : \u211d\nht : t \u2208 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ico a b\nv : \u211d\nt_lt_v : v \u2208 Ioi t\ny : \u211d\ng'_lt_y' : \u2191(g' t) < \u2191y\ny_lt_G' : \u2191y < G' t\nB : \u2200\u1da0 (u : \u211d) in \ud835\udcdd t, \u2191y < G' u\nm M : \u211d\nH : Ioo m M \u2286 {x | (fun u => \u2191y < G' u) x}\nhm : m < t\nhM : t < M\nthis : Ioo t (min M b) \u2208 \ud835\udcdd[Ioi t] t\nu : \u211d\nhu : u \u2208 Ioo t (min M b)\nI : Icc t u \u2286 Icc a b\n\u22a2 IntegrableOn (fun a => EReal.toReal (G' a)) (Icc t u)", "state_after": "no goals"}, {"tactic": "have C1 : \u2200\u1d50 x : \u211d \u2202volume.restrict (Icc t u), G' x < \u221e :=\n  ae_mono (Measure.restrict_mono I le_rfl) G'lt_top", "annotated_tactic": ["have C1 : \u2200\u1d50 x : \u211d \u2202volume.restrict (<a>Icc</a> t u), G' x < \u221e :=\n              <a>ae_mono</a> (<a>Measure.restrict_mono</a> I <a>le_rfl</a>) G'lt_top", [{"full_name": "Set.Icc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [59, 5], "def_end_pos": [59, 8]}, {"full_name": "MeasureTheory.ae_mono", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2456, 9], "def_end_pos": [2456, 16]}, {"full_name": "MeasureTheory.Measure.restrict_mono", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1550, 9], "def_end_pos": [1550, 22]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}]], "state_before": "case h\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\na b : \u211d\nhab : a \u2264 b\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nG' : \u211d \u2192 EReal\nf_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x\nG'cont : LowerSemicontinuous G'\nG'int : Integrable fun x => EReal.toReal (G' x)\nG'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4\nhG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5\ns : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b\ns_closed : IsClosed s\nt : \u211d\nht : t \u2208 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ico a b\nv : \u211d\nt_lt_v : v \u2208 Ioi t\ny : \u211d\ng'_lt_y' : \u2191(g' t) < \u2191y\ny_lt_G' : \u2191y < G' t\nB : \u2200\u1da0 (u : \u211d) in \ud835\udcdd t, \u2191y < G' u\nm M : \u211d\nH : Ioo m M \u2286 {x | (fun u => \u2191y < G' u) x}\nhm : m < t\nhM : t < M\nthis : Ioo t (min M b) \u2208 \ud835\udcdd[Ioi t] t\nu : \u211d\nhu : u \u2208 Ioo t (min M b)\nI : Icc t u \u2286 Icc a b\n\u22a2 (fun a => y) \u2264\u1d50[Measure.restrict volume (Icc t u)] fun a => EReal.toReal (G' a)", "state_after": "case h\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\na b : \u211d\nhab : a \u2264 b\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nG' : \u211d \u2192 EReal\nf_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x\nG'cont : LowerSemicontinuous G'\nG'int : Integrable fun x => EReal.toReal (G' x)\nG'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4\nhG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5\ns : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b\ns_closed : IsClosed s\nt : \u211d\nht : t \u2208 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ico a b\nv : \u211d\nt_lt_v : v \u2208 Ioi t\ny : \u211d\ng'_lt_y' : \u2191(g' t) < \u2191y\ny_lt_G' : \u2191y < G' t\nB : \u2200\u1da0 (u : \u211d) in \ud835\udcdd t, \u2191y < G' u\nm M : \u211d\nH : Ioo m M \u2286 {x | (fun u => \u2191y < G' u) x}\nhm : m < t\nhM : t < M\nthis : Ioo t (min M b) \u2208 \ud835\udcdd[Ioi t] t\nu : \u211d\nhu : u \u2208 Ioo t (min M b)\nI : Icc t u \u2286 Icc a b\nC1 : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc t u), G' x < \u2191\u22a4\n\u22a2 (fun a => y) \u2264\u1d50[Measure.restrict volume (Icc t u)] fun a => EReal.toReal (G' a)"}, {"tactic": "have C2 : \u2200\u1d50 x : \u211d \u2202volume.restrict (Icc t u), x \u2208 Icc t u :=\n  ae_restrict_mem measurableSet_Icc", "annotated_tactic": ["have C2 : \u2200\u1d50 x : \u211d \u2202volume.restrict (<a>Icc</a> t u), x \u2208 <a>Icc</a> t u :=\n              <a>ae_restrict_mem</a> <a>measurableSet_Icc</a>", [{"full_name": "Set.Icc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [59, 5], "def_end_pos": [59, 8]}, {"full_name": "Set.Icc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [59, 5], "def_end_pos": [59, 8]}, {"full_name": "MeasureTheory.ae_restrict_mem", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2586, 9], "def_end_pos": [2586, 24]}, {"full_name": "measurableSet_Icc", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [520, 9], "def_end_pos": [520, 26]}]], "state_before": "case h\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\na b : \u211d\nhab : a \u2264 b\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nG' : \u211d \u2192 EReal\nf_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x\nG'cont : LowerSemicontinuous G'\nG'int : Integrable fun x => EReal.toReal (G' x)\nG'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4\nhG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5\ns : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b\ns_closed : IsClosed s\nt : \u211d\nht : t \u2208 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ico a b\nv : \u211d\nt_lt_v : v \u2208 Ioi t\ny : \u211d\ng'_lt_y' : \u2191(g' t) < \u2191y\ny_lt_G' : \u2191y < G' t\nB : \u2200\u1da0 (u : \u211d) in \ud835\udcdd t, \u2191y < G' u\nm M : \u211d\nH : Ioo m M \u2286 {x | (fun u => \u2191y < G' u) x}\nhm : m < t\nhM : t < M\nthis : Ioo t (min M b) \u2208 \ud835\udcdd[Ioi t] t\nu : \u211d\nhu : u \u2208 Ioo t (min M b)\nI : Icc t u \u2286 Icc a b\nC1 : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc t u), G' x < \u2191\u22a4\n\u22a2 (fun a => y) \u2264\u1d50[Measure.restrict volume (Icc t u)] fun a => EReal.toReal (G' a)", "state_after": "case h\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\na b : \u211d\nhab : a \u2264 b\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nG' : \u211d \u2192 EReal\nf_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x\nG'cont : LowerSemicontinuous G'\nG'int : Integrable fun x => EReal.toReal (G' x)\nG'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4\nhG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5\ns : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b\ns_closed : IsClosed s\nt : \u211d\nht : t \u2208 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ico a b\nv : \u211d\nt_lt_v : v \u2208 Ioi t\ny : \u211d\ng'_lt_y' : \u2191(g' t) < \u2191y\ny_lt_G' : \u2191y < G' t\nB : \u2200\u1da0 (u : \u211d) in \ud835\udcdd t, \u2191y < G' u\nm M : \u211d\nH : Ioo m M \u2286 {x | (fun u => \u2191y < G' u) x}\nhm : m < t\nhM : t < M\nthis : Ioo t (min M b) \u2208 \ud835\udcdd[Ioi t] t\nu : \u211d\nhu : u \u2208 Ioo t (min M b)\nI : Icc t u \u2286 Icc a b\nC1 : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc t u), G' x < \u2191\u22a4\nC2 : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc t u), x \u2208 Icc t u\n\u22a2 (fun a => y) \u2264\u1d50[Measure.restrict volume (Icc t u)] fun a => EReal.toReal (G' a)"}, {"tactic": "filter_upwards [C1, C2] with x G'x hx", "annotated_tactic": ["filter_upwards [C1, C2] with x G'x hx", []], "state_before": "case h\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\na b : \u211d\nhab : a \u2264 b\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nG' : \u211d \u2192 EReal\nf_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x\nG'cont : LowerSemicontinuous G'\nG'int : Integrable fun x => EReal.toReal (G' x)\nG'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4\nhG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5\ns : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b\ns_closed : IsClosed s\nt : \u211d\nht : t \u2208 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ico a b\nv : \u211d\nt_lt_v : v \u2208 Ioi t\ny : \u211d\ng'_lt_y' : \u2191(g' t) < \u2191y\ny_lt_G' : \u2191y < G' t\nB : \u2200\u1da0 (u : \u211d) in \ud835\udcdd t, \u2191y < G' u\nm M : \u211d\nH : Ioo m M \u2286 {x | (fun u => \u2191y < G' u) x}\nhm : m < t\nhM : t < M\nthis : Ioo t (min M b) \u2208 \ud835\udcdd[Ioi t] t\nu : \u211d\nhu : u \u2208 Ioo t (min M b)\nI : Icc t u \u2286 Icc a b\nC1 : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc t u), G' x < \u2191\u22a4\nC2 : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc t u), x \u2208 Icc t u\n\u22a2 (fun a => y) \u2264\u1d50[Measure.restrict volume (Icc t u)] fun a => EReal.toReal (G' a)", "state_after": "case h\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\na b : \u211d\nhab : a \u2264 b\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nG' : \u211d \u2192 EReal\nf_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x\nG'cont : LowerSemicontinuous G'\nG'int : Integrable fun x => EReal.toReal (G' x)\nG'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4\nhG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5\ns : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b\ns_closed : IsClosed s\nt : \u211d\nht : t \u2208 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ico a b\nv : \u211d\nt_lt_v : v \u2208 Ioi t\ny : \u211d\ng'_lt_y' : \u2191(g' t) < \u2191y\ny_lt_G' : \u2191y < G' t\nB : \u2200\u1da0 (u : \u211d) in \ud835\udcdd t, \u2191y < G' u\nm M : \u211d\nH : Ioo m M \u2286 {x | (fun u => \u2191y < G' u) x}\nhm : m < t\nhM : t < M\nthis : Ioo t (min M b) \u2208 \ud835\udcdd[Ioi t] t\nu : \u211d\nhu : u \u2208 Ioo t (min M b)\nI : Icc t u \u2286 Icc a b\nC1 : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc t u), G' x < \u2191\u22a4\nC2 : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc t u), x \u2208 Icc t u\nx : \u211d\nG'x : G' x < \u2191\u22a4\nhx : x \u2208 Icc t u\n\u22a2 y \u2264 EReal.toReal (G' x)"}, {"tactic": "apply EReal.coe_le_coe_iff.1", "annotated_tactic": ["apply <a>EReal.coe_le_coe_iff</a>.1", [{"full_name": "EReal.coe_le_coe_iff", "def_path": "Mathlib/Data/Real/EReal.lean", "def_pos": [102, 19], "def_end_pos": [102, 33]}]], "state_before": "case h\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\na b : \u211d\nhab : a \u2264 b\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nG' : \u211d \u2192 EReal\nf_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x\nG'cont : LowerSemicontinuous G'\nG'int : Integrable fun x => EReal.toReal (G' x)\nG'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4\nhG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5\ns : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b\ns_closed : IsClosed s\nt : \u211d\nht : t \u2208 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ico a b\nv : \u211d\nt_lt_v : v \u2208 Ioi t\ny : \u211d\ng'_lt_y' : \u2191(g' t) < \u2191y\ny_lt_G' : \u2191y < G' t\nB : \u2200\u1da0 (u : \u211d) in \ud835\udcdd t, \u2191y < G' u\nm M : \u211d\nH : Ioo m M \u2286 {x | (fun u => \u2191y < G' u) x}\nhm : m < t\nhM : t < M\nthis : Ioo t (min M b) \u2208 \ud835\udcdd[Ioi t] t\nu : \u211d\nhu : u \u2208 Ioo t (min M b)\nI : Icc t u \u2286 Icc a b\nC1 : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc t u), G' x < \u2191\u22a4\nC2 : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc t u), x \u2208 Icc t u\nx : \u211d\nG'x : G' x < \u2191\u22a4\nhx : x \u2208 Icc t u\n\u22a2 y \u2264 EReal.toReal (G' x)", "state_after": "case h\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\na b : \u211d\nhab : a \u2264 b\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nG' : \u211d \u2192 EReal\nf_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x\nG'cont : LowerSemicontinuous G'\nG'int : Integrable fun x => EReal.toReal (G' x)\nG'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4\nhG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5\ns : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b\ns_closed : IsClosed s\nt : \u211d\nht : t \u2208 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ico a b\nv : \u211d\nt_lt_v : v \u2208 Ioi t\ny : \u211d\ng'_lt_y' : \u2191(g' t) < \u2191y\ny_lt_G' : \u2191y < G' t\nB : \u2200\u1da0 (u : \u211d) in \ud835\udcdd t, \u2191y < G' u\nm M : \u211d\nH : Ioo m M \u2286 {x | (fun u => \u2191y < G' u) x}\nhm : m < t\nhM : t < M\nthis : Ioo t (min M b) \u2208 \ud835\udcdd[Ioi t] t\nu : \u211d\nhu : u \u2208 Ioo t (min M b)\nI : Icc t u \u2286 Icc a b\nC1 : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc t u), G' x < \u2191\u22a4\nC2 : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc t u), x \u2208 Icc t u\nx : \u211d\nG'x : G' x < \u2191\u22a4\nhx : x \u2208 Icc t u\n\u22a2 \u2191y \u2264 \u2191(EReal.toReal (G' x))"}, {"tactic": "have : x \u2208 Ioo m M := by\n  simp only [hm.trans_le hx.left,\n    (hx.right.trans_lt hu.right).trans_le (min_le_left M b), mem_Ioo, and_self_iff]", "annotated_tactic": ["have : x \u2208 <a>Ioo</a> m M := by\n              simp only [hm.trans_le hx.left,\n                (hx.right.trans_lt hu.right).<a>trans_le</a> (<a>min_le_left</a> M b), <a>mem_Ioo</a>, <a>and_self_iff</a>]", [{"full_name": "Set.Ioo", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [44, 5], "def_end_pos": [44, 8]}, {"full_name": "LT.lt.trans_le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [148, 7], "def_end_pos": [148, 21]}, {"full_name": "min_le_left", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [33, 9], "def_end_pos": [33, 20]}, {"full_name": "Set.mem_Ioo", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [116, 9], "def_end_pos": [116, 16]}, {"full_name": "and_self_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [155, 9], "def_end_pos": [155, 21]}]], "state_before": "case h\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\na b : \u211d\nhab : a \u2264 b\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nG' : \u211d \u2192 EReal\nf_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x\nG'cont : LowerSemicontinuous G'\nG'int : Integrable fun x => EReal.toReal (G' x)\nG'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4\nhG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5\ns : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b\ns_closed : IsClosed s\nt : \u211d\nht : t \u2208 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ico a b\nv : \u211d\nt_lt_v : v \u2208 Ioi t\ny : \u211d\ng'_lt_y' : \u2191(g' t) < \u2191y\ny_lt_G' : \u2191y < G' t\nB : \u2200\u1da0 (u : \u211d) in \ud835\udcdd t, \u2191y < G' u\nm M : \u211d\nH : Ioo m M \u2286 {x | (fun u => \u2191y < G' u) x}\nhm : m < t\nhM : t < M\nthis : Ioo t (min M b) \u2208 \ud835\udcdd[Ioi t] t\nu : \u211d\nhu : u \u2208 Ioo t (min M b)\nI : Icc t u \u2286 Icc a b\nC1 : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc t u), G' x < \u2191\u22a4\nC2 : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc t u), x \u2208 Icc t u\nx : \u211d\nG'x : G' x < \u2191\u22a4\nhx : x \u2208 Icc t u\n\u22a2 \u2191y \u2264 \u2191(EReal.toReal (G' x))", "state_after": "case h\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\na b : \u211d\nhab : a \u2264 b\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nG' : \u211d \u2192 EReal\nf_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x\nG'cont : LowerSemicontinuous G'\nG'int : Integrable fun x => EReal.toReal (G' x)\nG'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4\nhG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5\ns : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b\ns_closed : IsClosed s\nt : \u211d\nht : t \u2208 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ico a b\nv : \u211d\nt_lt_v : v \u2208 Ioi t\ny : \u211d\ng'_lt_y' : \u2191(g' t) < \u2191y\ny_lt_G' : \u2191y < G' t\nB : \u2200\u1da0 (u : \u211d) in \ud835\udcdd t, \u2191y < G' u\nm M : \u211d\nH : Ioo m M \u2286 {x | (fun u => \u2191y < G' u) x}\nhm : m < t\nhM : t < M\nthis\u271d : Ioo t (min M b) \u2208 \ud835\udcdd[Ioi t] t\nu : \u211d\nhu : u \u2208 Ioo t (min M b)\nI : Icc t u \u2286 Icc a b\nC1 : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc t u), G' x < \u2191\u22a4\nC2 : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc t u), x \u2208 Icc t u\nx : \u211d\nG'x : G' x < \u2191\u22a4\nhx : x \u2208 Icc t u\nthis : x \u2208 Ioo m M\n\u22a2 \u2191y \u2264 \u2191(EReal.toReal (G' x))"}, {"tactic": "refine (H this).out.le.trans_eq ?_", "annotated_tactic": ["refine (H this).out.le.trans_eq ?_", []], "state_before": "case h\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\na b : \u211d\nhab : a \u2264 b\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nG' : \u211d \u2192 EReal\nf_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x\nG'cont : LowerSemicontinuous G'\nG'int : Integrable fun x => EReal.toReal (G' x)\nG'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4\nhG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5\ns : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b\ns_closed : IsClosed s\nt : \u211d\nht : t \u2208 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ico a b\nv : \u211d\nt_lt_v : v \u2208 Ioi t\ny : \u211d\ng'_lt_y' : \u2191(g' t) < \u2191y\ny_lt_G' : \u2191y < G' t\nB : \u2200\u1da0 (u : \u211d) in \ud835\udcdd t, \u2191y < G' u\nm M : \u211d\nH : Ioo m M \u2286 {x | (fun u => \u2191y < G' u) x}\nhm : m < t\nhM : t < M\nthis\u271d : Ioo t (min M b) \u2208 \ud835\udcdd[Ioi t] t\nu : \u211d\nhu : u \u2208 Ioo t (min M b)\nI : Icc t u \u2286 Icc a b\nC1 : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc t u), G' x < \u2191\u22a4\nC2 : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc t u), x \u2208 Icc t u\nx : \u211d\nG'x : G' x < \u2191\u22a4\nhx : x \u2208 Icc t u\nthis : x \u2208 Ioo m M\n\u22a2 \u2191y \u2264 \u2191(EReal.toReal (G' x))", "state_after": "case h\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\na b : \u211d\nhab : a \u2264 b\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nG' : \u211d \u2192 EReal\nf_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x\nG'cont : LowerSemicontinuous G'\nG'int : Integrable fun x => EReal.toReal (G' x)\nG'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4\nhG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5\ns : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b\ns_closed : IsClosed s\nt : \u211d\nht : t \u2208 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ico a b\nv : \u211d\nt_lt_v : v \u2208 Ioi t\ny : \u211d\ng'_lt_y' : \u2191(g' t) < \u2191y\ny_lt_G' : \u2191y < G' t\nB : \u2200\u1da0 (u : \u211d) in \ud835\udcdd t, \u2191y < G' u\nm M : \u211d\nH : Ioo m M \u2286 {x | (fun u => \u2191y < G' u) x}\nhm : m < t\nhM : t < M\nthis\u271d : Ioo t (min M b) \u2208 \ud835\udcdd[Ioi t] t\nu : \u211d\nhu : u \u2208 Ioo t (min M b)\nI : Icc t u \u2286 Icc a b\nC1 : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc t u), G' x < \u2191\u22a4\nC2 : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc t u), x \u2208 Icc t u\nx : \u211d\nG'x : G' x < \u2191\u22a4\nhx : x \u2208 Icc t u\nthis : x \u2208 Ioo m M\n\u22a2 G' x = \u2191(EReal.toReal (G' x))"}, {"tactic": "exact (EReal.coe_toReal G'x.ne (f_lt_G' x).ne_bot).symm", "annotated_tactic": ["exact (<a>EReal.coe_toReal</a> G'x.ne (f_lt_G' x).<a>ne_bot</a>).<a>symm</a>", [{"full_name": "EReal.coe_toReal", "def_path": "Mathlib/Data/Real/EReal.lean", "def_pos": [413, 9], "def_end_pos": [413, 19]}, {"full_name": "LT.lt.ne_bot", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [324, 7], "def_end_pos": [324, 19]}, {"full_name": "Eq.symm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [310, 9], "def_end_pos": [310, 16]}]], "state_before": "case h\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\na b : \u211d\nhab : a \u2264 b\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nG' : \u211d \u2192 EReal\nf_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x\nG'cont : LowerSemicontinuous G'\nG'int : Integrable fun x => EReal.toReal (G' x)\nG'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4\nhG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5\ns : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b\ns_closed : IsClosed s\nt : \u211d\nht : t \u2208 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ico a b\nv : \u211d\nt_lt_v : v \u2208 Ioi t\ny : \u211d\ng'_lt_y' : \u2191(g' t) < \u2191y\ny_lt_G' : \u2191y < G' t\nB : \u2200\u1da0 (u : \u211d) in \ud835\udcdd t, \u2191y < G' u\nm M : \u211d\nH : Ioo m M \u2286 {x | (fun u => \u2191y < G' u) x}\nhm : m < t\nhM : t < M\nthis\u271d : Ioo t (min M b) \u2208 \ud835\udcdd[Ioi t] t\nu : \u211d\nhu : u \u2208 Ioo t (min M b)\nI : Icc t u \u2286 Icc a b\nC1 : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc t u), G' x < \u2191\u22a4\nC2 : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc t u), x \u2208 Icc t u\nx : \u211d\nG'x : G' x < \u2191\u22a4\nhx : x \u2208 Icc t u\nthis : x \u2208 Ioo m M\n\u22a2 G' x = \u2191(EReal.toReal (G' x))", "state_after": "no goals"}, {"tactic": "simp only [hm.trans_le hx.left,\n  (hx.right.trans_lt hu.right).trans_le (min_le_left M b), mem_Ioo, and_self_iff]", "annotated_tactic": ["simp only [hm.trans_le hx.left,\n                (hx.right.trans_lt hu.right).<a>trans_le</a> (<a>min_le_left</a> M b), <a>mem_Ioo</a>, <a>and_self_iff</a>]", [{"full_name": "LT.lt.trans_le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [148, 7], "def_end_pos": [148, 21]}, {"full_name": "min_le_left", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [33, 9], "def_end_pos": [33, 20]}, {"full_name": "Set.mem_Ioo", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [116, 9], "def_end_pos": [116, 16]}, {"full_name": "and_self_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [155, 9], "def_end_pos": [155, 21]}]], "state_before": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\na b : \u211d\nhab : a \u2264 b\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nG' : \u211d \u2192 EReal\nf_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x\nG'cont : LowerSemicontinuous G'\nG'int : Integrable fun x => EReal.toReal (G' x)\nG'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4\nhG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5\ns : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b\ns_closed : IsClosed s\nt : \u211d\nht : t \u2208 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ico a b\nv : \u211d\nt_lt_v : v \u2208 Ioi t\ny : \u211d\ng'_lt_y' : \u2191(g' t) < \u2191y\ny_lt_G' : \u2191y < G' t\nB : \u2200\u1da0 (u : \u211d) in \ud835\udcdd t, \u2191y < G' u\nm M : \u211d\nH : Ioo m M \u2286 {x | (fun u => \u2191y < G' u) x}\nhm : m < t\nhM : t < M\nthis : Ioo t (min M b) \u2208 \ud835\udcdd[Ioi t] t\nu : \u211d\nhu : u \u2208 Ioo t (min M b)\nI : Icc t u \u2286 Icc a b\nC1 : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc t u), G' x < \u2191\u22a4\nC2 : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc t u), x \u2208 Icc t u\nx : \u211d\nG'x : G' x < \u2191\u22a4\nhx : x \u2208 Icc t u\n\u22a2 x \u2208 Ioo m M", "state_after": "no goals"}, {"tactic": "have g'_lt_y : g' t < y := EReal.coe_lt_coe_iff.1 g'_lt_y'", "annotated_tactic": ["have g'_lt_y : g' t < y := <a>EReal.coe_lt_coe_iff</a>.1 g'_lt_y'", [{"full_name": "EReal.coe_lt_coe_iff", "def_path": "Mathlib/Data/Real/EReal.lean", "def_pos": [107, 19], "def_end_pos": [107, 33]}]], "state_before": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\na b : \u211d\nhab : a \u2264 b\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nG' : \u211d \u2192 EReal\nf_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x\nG'cont : LowerSemicontinuous G'\nG'int : Integrable fun x => EReal.toReal (G' x)\nG'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4\nhG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5\ns : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b\ns_closed : IsClosed s\nt : \u211d\nht : t \u2208 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ico a b\nv : \u211d\nt_lt_v : v \u2208 Ioi t\ny : \u211d\ng'_lt_y' : \u2191(g' t) < \u2191y\ny_lt_G' : \u2191y < G' t\nI1 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, (u - t) * y \u2264 \u222b (w : \u211d) in t..u, EReal.toReal (G' w)\n\u22a2 \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, g u - g t \u2264 (u - t) * y", "state_after": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\na b : \u211d\nhab : a \u2264 b\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nG' : \u211d \u2192 EReal\nf_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x\nG'cont : LowerSemicontinuous G'\nG'int : Integrable fun x => EReal.toReal (G' x)\nG'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4\nhG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5\ns : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b\ns_closed : IsClosed s\nt : \u211d\nht : t \u2208 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ico a b\nv : \u211d\nt_lt_v : v \u2208 Ioi t\ny : \u211d\ng'_lt_y' : \u2191(g' t) < \u2191y\ny_lt_G' : \u2191y < G' t\nI1 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, (u - t) * y \u2264 \u222b (w : \u211d) in t..u, EReal.toReal (G' w)\ng'_lt_y : g' t < y\n\u22a2 \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, g u - g t \u2264 (u - t) * y"}, {"tactic": "filter_upwards [(hderiv t \u27e8ht.2.1, ht.2.2\u27e9).limsup_slope_le' (not_mem_Ioi.2 le_rfl) g'_lt_y,\n  self_mem_nhdsWithin] with u hu t_lt_u", "annotated_tactic": ["filter_upwards [(hderiv t \u27e8ht.2.1, ht.2.2\u27e9).<a>limsup_slope_le'</a> (<a>not_mem_Ioi</a>.2 <a>le_rfl</a>) g'_lt_y,\n        <a>self_mem_nhdsWithin</a>] with u hu t_lt_u", [{"full_name": "HasDerivWithinAt.limsup_slope_le'", "def_path": "Mathlib/Analysis/Calculus/Deriv/Slope.lean", "def_pos": [178, 9], "def_end_pos": [178, 42]}, {"full_name": "Set.not_mem_Ioi", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [1065, 9], "def_end_pos": [1065, 20]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}, {"full_name": "self_mem_nhdsWithin", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [151, 9], "def_end_pos": [151, 28]}]], "state_before": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\na b : \u211d\nhab : a \u2264 b\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nG' : \u211d \u2192 EReal\nf_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x\nG'cont : LowerSemicontinuous G'\nG'int : Integrable fun x => EReal.toReal (G' x)\nG'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4\nhG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5\ns : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b\ns_closed : IsClosed s\nt : \u211d\nht : t \u2208 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ico a b\nv : \u211d\nt_lt_v : v \u2208 Ioi t\ny : \u211d\ng'_lt_y' : \u2191(g' t) < \u2191y\ny_lt_G' : \u2191y < G' t\nI1 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, (u - t) * y \u2264 \u222b (w : \u211d) in t..u, EReal.toReal (G' w)\ng'_lt_y : g' t < y\n\u22a2 \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, g u - g t \u2264 (u - t) * y", "state_after": "case h\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\na b : \u211d\nhab : a \u2264 b\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nG' : \u211d \u2192 EReal\nf_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x\nG'cont : LowerSemicontinuous G'\nG'int : Integrable fun x => EReal.toReal (G' x)\nG'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4\nhG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5\ns : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b\ns_closed : IsClosed s\nt : \u211d\nht : t \u2208 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ico a b\nv : \u211d\nt_lt_v : v \u2208 Ioi t\ny : \u211d\ng'_lt_y' : \u2191(g' t) < \u2191y\ny_lt_G' : \u2191y < G' t\nI1 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, (u - t) * y \u2264 \u222b (w : \u211d) in t..u, EReal.toReal (G' w)\ng'_lt_y : g' t < y\nu : \u211d\nhu : slope g t u < y\nt_lt_u : u \u2208 Ioi t\n\u22a2 g u - g t \u2264 (u - t) * y"}, {"tactic": "have := mul_le_mul_of_nonneg_left hu.le (sub_pos.2 t_lt_u.out).le", "annotated_tactic": ["have := <a>mul_le_mul_of_nonneg_left</a> hu.le (<a>sub_pos</a>.2 t_lt_u.out).<a>le</a>", [{"full_name": "mul_le_mul_of_nonneg_left", "def_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "def_pos": [152, 9], "def_end_pos": [152, 34]}, {"full_name": "sub_pos", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [883, 30], "def_end_pos": [883, 37]}, {"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [142, 7], "def_end_pos": [142, 15]}]], "state_before": "case h\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\na b : \u211d\nhab : a \u2264 b\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nG' : \u211d \u2192 EReal\nf_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x\nG'cont : LowerSemicontinuous G'\nG'int : Integrable fun x => EReal.toReal (G' x)\nG'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4\nhG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5\ns : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b\ns_closed : IsClosed s\nt : \u211d\nht : t \u2208 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ico a b\nv : \u211d\nt_lt_v : v \u2208 Ioi t\ny : \u211d\ng'_lt_y' : \u2191(g' t) < \u2191y\ny_lt_G' : \u2191y < G' t\nI1 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, (u - t) * y \u2264 \u222b (w : \u211d) in t..u, EReal.toReal (G' w)\ng'_lt_y : g' t < y\nu : \u211d\nhu : slope g t u < y\nt_lt_u : u \u2208 Ioi t\n\u22a2 g u - g t \u2264 (u - t) * y", "state_after": "case h\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\na b : \u211d\nhab : a \u2264 b\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nG' : \u211d \u2192 EReal\nf_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x\nG'cont : LowerSemicontinuous G'\nG'int : Integrable fun x => EReal.toReal (G' x)\nG'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4\nhG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5\ns : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b\ns_closed : IsClosed s\nt : \u211d\nht : t \u2208 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ico a b\nv : \u211d\nt_lt_v : v \u2208 Ioi t\ny : \u211d\ng'_lt_y' : \u2191(g' t) < \u2191y\ny_lt_G' : \u2191y < G' t\nI1 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, (u - t) * y \u2264 \u222b (w : \u211d) in t..u, EReal.toReal (G' w)\ng'_lt_y : g' t < y\nu : \u211d\nhu : slope g t u < y\nt_lt_u : u \u2208 Ioi t\nthis : (u - t) * slope g t u \u2264 (u - t) * y\n\u22a2 g u - g t \u2264 (u - t) * y"}, {"tactic": "rwa [\u2190 smul_eq_mul, sub_smul_slope] at this", "annotated_tactic": ["rwa [\u2190 <a>smul_eq_mul</a>, <a>sub_smul_slope</a>] at this", [{"full_name": "smul_eq_mul", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [93, 9], "def_end_pos": [93, 20]}, {"full_name": "sub_smul_slope", "def_path": "Mathlib/LinearAlgebra/AffineSpace/Slope.lean", "def_pos": [56, 9], "def_end_pos": [56, 23]}]], "state_before": "case h\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\na b : \u211d\nhab : a \u2264 b\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nG' : \u211d \u2192 EReal\nf_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x\nG'cont : LowerSemicontinuous G'\nG'int : Integrable fun x => EReal.toReal (G' x)\nG'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4\nhG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5\ns : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b\ns_closed : IsClosed s\nt : \u211d\nht : t \u2208 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ico a b\nv : \u211d\nt_lt_v : v \u2208 Ioi t\ny : \u211d\ng'_lt_y' : \u2191(g' t) < \u2191y\ny_lt_G' : \u2191y < G' t\nI1 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, (u - t) * y \u2264 \u222b (w : \u211d) in t..u, EReal.toReal (G' w)\ng'_lt_y : g' t < y\nu : \u211d\nhu : slope g t u < y\nt_lt_u : u \u2208 Ioi t\nthis : (u - t) * slope g t u \u2264 (u - t) * y\n\u22a2 g u - g t \u2264 (u - t) * y", "state_after": "no goals"}, {"tactic": "filter_upwards [I1, I2] with u hu1 hu2 using hu2.trans hu1", "annotated_tactic": ["filter_upwards [I1, I2] with u hu1 hu2 using hu2.trans hu1", []], "state_before": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\na b : \u211d\nhab : a \u2264 b\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nG' : \u211d \u2192 EReal\nf_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x\nG'cont : LowerSemicontinuous G'\nG'int : Integrable fun x => EReal.toReal (G' x)\nG'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4\nhG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5\ns : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b\ns_closed : IsClosed s\nt : \u211d\nht : t \u2208 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ico a b\nv : \u211d\nt_lt_v : v \u2208 Ioi t\ny : \u211d\ng'_lt_y' : \u2191(g' t) < \u2191y\ny_lt_G' : \u2191y < G' t\nI1 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, (u - t) * y \u2264 \u222b (w : \u211d) in t..u, EReal.toReal (G' w)\nI2 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, g u - g t \u2264 (u - t) * y\n\u22a2 \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, g u - g t \u2264 \u222b (w : \u211d) in t..u, EReal.toReal (G' w)", "state_after": "no goals"}, {"tactic": "refine' mem_nhdsWithin_Ioi_iff_exists_Ioc_subset.2 \u27e8min v b, _, Subset.rfl\u27e9", "annotated_tactic": ["refine' <a>mem_nhdsWithin_Ioi_iff_exists_Ioc_subset</a>.2 \u27e8<a>min</a> v b, _, <a>Subset.rfl</a>\u27e9", [{"full_name": "mem_nhdsWithin_Ioi_iff_exists_Ioc_subset", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [1733, 9], "def_end_pos": [1733, 49]}, {"full_name": "Min.min", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1103, 3], "def_end_pos": [1103, 6]}, {"full_name": "Set.Subset.rfl", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [357, 9], "def_end_pos": [357, 19]}]], "state_before": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\na b : \u211d\nhab : a \u2264 b\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nG' : \u211d \u2192 EReal\nf_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x\nG'cont : LowerSemicontinuous G'\nG'int : Integrable fun x => EReal.toReal (G' x)\nG'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4\nhG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5\ns : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b\ns_closed : IsClosed s\nt : \u211d\nht : t \u2208 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ico a b\nv : \u211d\nt_lt_v : v \u2208 Ioi t\ny : \u211d\ng'_lt_y' : \u2191(g' t) < \u2191y\ny_lt_G' : \u2191y < G' t\nI1 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, (u - t) * y \u2264 \u222b (w : \u211d) in t..u, EReal.toReal (G' w)\nI2 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, g u - g t \u2264 (u - t) * y\nI3 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, g u - g t \u2264 \u222b (w : \u211d) in t..u, EReal.toReal (G' w)\n\u22a2 \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, u \u2208 Ioc t (min v b)", "state_after": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\na b : \u211d\nhab : a \u2264 b\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nG' : \u211d \u2192 EReal\nf_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x\nG'cont : LowerSemicontinuous G'\nG'int : Integrable fun x => EReal.toReal (G' x)\nG'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4\nhG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5\ns : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b\ns_closed : IsClosed s\nt : \u211d\nht : t \u2208 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ico a b\nv : \u211d\nt_lt_v : v \u2208 Ioi t\ny : \u211d\ng'_lt_y' : \u2191(g' t) < \u2191y\ny_lt_G' : \u2191y < G' t\nI1 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, (u - t) * y \u2264 \u222b (w : \u211d) in t..u, EReal.toReal (G' w)\nI2 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, g u - g t \u2264 (u - t) * y\nI3 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, g u - g t \u2264 \u222b (w : \u211d) in t..u, EReal.toReal (G' w)\n\u22a2 min v b \u2208 Ioi t"}, {"tactic": "simp only [lt_min_iff, mem_Ioi]", "annotated_tactic": ["simp only [<a>lt_min_iff</a>, <a>mem_Ioi</a>]", [{"full_name": "lt_min_iff", "def_path": "Mathlib/Order/MinMax.lean", "def_pos": [53, 9], "def_end_pos": [53, 19]}, {"full_name": "Set.mem_Ioi", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [151, 9], "def_end_pos": [151, 16]}]], "state_before": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\na b : \u211d\nhab : a \u2264 b\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nG' : \u211d \u2192 EReal\nf_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x\nG'cont : LowerSemicontinuous G'\nG'int : Integrable fun x => EReal.toReal (G' x)\nG'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4\nhG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5\ns : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b\ns_closed : IsClosed s\nt : \u211d\nht : t \u2208 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ico a b\nv : \u211d\nt_lt_v : v \u2208 Ioi t\ny : \u211d\ng'_lt_y' : \u2191(g' t) < \u2191y\ny_lt_G' : \u2191y < G' t\nI1 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, (u - t) * y \u2264 \u222b (w : \u211d) in t..u, EReal.toReal (G' w)\nI2 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, g u - g t \u2264 (u - t) * y\nI3 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, g u - g t \u2264 \u222b (w : \u211d) in t..u, EReal.toReal (G' w)\n\u22a2 min v b \u2208 Ioi t", "state_after": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\na b : \u211d\nhab : a \u2264 b\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nG' : \u211d \u2192 EReal\nf_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x\nG'cont : LowerSemicontinuous G'\nG'int : Integrable fun x => EReal.toReal (G' x)\nG'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4\nhG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5\ns : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b\ns_closed : IsClosed s\nt : \u211d\nht : t \u2208 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ico a b\nv : \u211d\nt_lt_v : v \u2208 Ioi t\ny : \u211d\ng'_lt_y' : \u2191(g' t) < \u2191y\ny_lt_G' : \u2191y < G' t\nI1 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, (u - t) * y \u2264 \u222b (w : \u211d) in t..u, EReal.toReal (G' w)\nI2 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, g u - g t \u2264 (u - t) * y\nI3 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, g u - g t \u2264 \u222b (w : \u211d) in t..u, EReal.toReal (G' w)\n\u22a2 t < v \u2227 t < b"}, {"tactic": "exact \u27e8t_lt_v, ht.2.2\u27e9", "annotated_tactic": ["exact \u27e8t_lt_v, ht.2.2\u27e9", []], "state_before": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\na b : \u211d\nhab : a \u2264 b\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nG' : \u211d \u2192 EReal\nf_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x\nG'cont : LowerSemicontinuous G'\nG'int : Integrable fun x => EReal.toReal (G' x)\nG'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4\nhG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5\ns : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b\ns_closed : IsClosed s\nt : \u211d\nht : t \u2208 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ico a b\nv : \u211d\nt_lt_v : v \u2208 Ioi t\ny : \u211d\ng'_lt_y' : \u2191(g' t) < \u2191y\ny_lt_G' : \u2191y < G' t\nI1 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, (u - t) * y \u2264 \u222b (w : \u211d) in t..u, EReal.toReal (G' w)\nI2 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, g u - g t \u2264 (u - t) * y\nI3 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, g u - g t \u2264 \u222b (w : \u211d) in t..u, EReal.toReal (G' w)\n\u22a2 t < v \u2227 t < b", "state_after": "no goals"}, {"tactic": "abel", "annotated_tactic": ["abel", []], "state_before": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\na b : \u211d\nhab : a \u2264 b\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nG' : \u211d \u2192 EReal\nf_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x\nG'cont : LowerSemicontinuous G'\nG'int : Integrable fun x => EReal.toReal (G' x)\nG'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4\nhG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5\ns : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b\ns_closed : IsClosed s\nt : \u211d\nht : t \u2208 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ico a b\nv : \u211d\nt_lt_v : v \u2208 Ioi t\ny : \u211d\ng'_lt_y' : \u2191(g' t) < \u2191y\ny_lt_G' : \u2191y < G' t\nI1 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, (u - t) * y \u2264 \u222b (w : \u211d) in t..u, EReal.toReal (G' w)\nI2 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, g u - g t \u2264 (u - t) * y\nI3 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, g u - g t \u2264 \u222b (w : \u211d) in t..u, EReal.toReal (G' w)\nI4 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, u \u2208 Ioc t (min v b)\nx : \u211d\nhx : g x - g t \u2264 \u222b (w : \u211d) in t..x, EReal.toReal (G' w)\nh'x : x \u2208 Ioc t (min v b)\n\u22a2 g x - g a = g t - g a + (g x - g t)", "state_after": "no goals"}, {"tactic": "apply integral_add_adjacent_intervals", "annotated_tactic": ["apply <a>integral_add_adjacent_intervals</a>", [{"full_name": "intervalIntegral.integral_add_adjacent_intervals", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [900, 9], "def_end_pos": [900, 40]}]], "state_before": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\na b : \u211d\nhab : a \u2264 b\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nG' : \u211d \u2192 EReal\nf_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x\nG'cont : LowerSemicontinuous G'\nG'int : Integrable fun x => EReal.toReal (G' x)\nG'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4\nhG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5\ns : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b\ns_closed : IsClosed s\nt : \u211d\nht : t \u2208 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ico a b\nv : \u211d\nt_lt_v : v \u2208 Ioi t\ny : \u211d\ng'_lt_y' : \u2191(g' t) < \u2191y\ny_lt_G' : \u2191y < G' t\nI1 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, (u - t) * y \u2264 \u222b (w : \u211d) in t..u, EReal.toReal (G' w)\nI2 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, g u - g t \u2264 (u - t) * y\nI3 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, g u - g t \u2264 \u222b (w : \u211d) in t..u, EReal.toReal (G' w)\nI4 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, u \u2208 Ioc t (min v b)\nx : \u211d\nhx : g x - g t \u2264 \u222b (w : \u211d) in t..x, EReal.toReal (G' w)\nh'x : x \u2208 Ioc t (min v b)\n\u22a2 (\u222b (w : \u211d) in a..t, EReal.toReal (G' w)) + \u222b (w : \u211d) in t..x, EReal.toReal (G' w) =\n    \u222b (w : \u211d) in a..x, EReal.toReal (G' w)", "state_after": "case hab\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\na b : \u211d\nhab : a \u2264 b\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nG' : \u211d \u2192 EReal\nf_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x\nG'cont : LowerSemicontinuous G'\nG'int : Integrable fun x => EReal.toReal (G' x)\nG'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4\nhG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5\ns : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b\ns_closed : IsClosed s\nt : \u211d\nht : t \u2208 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ico a b\nv : \u211d\nt_lt_v : v \u2208 Ioi t\ny : \u211d\ng'_lt_y' : \u2191(g' t) < \u2191y\ny_lt_G' : \u2191y < G' t\nI1 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, (u - t) * y \u2264 \u222b (w : \u211d) in t..u, EReal.toReal (G' w)\nI2 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, g u - g t \u2264 (u - t) * y\nI3 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, g u - g t \u2264 \u222b (w : \u211d) in t..u, EReal.toReal (G' w)\nI4 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, u \u2208 Ioc t (min v b)\nx : \u211d\nhx : g x - g t \u2264 \u222b (w : \u211d) in t..x, EReal.toReal (G' w)\nh'x : x \u2208 Ioc t (min v b)\n\u22a2 IntervalIntegrable (fun x => EReal.toReal (G' x)) volume a t\n\ncase hbc\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\na b : \u211d\nhab : a \u2264 b\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nG' : \u211d \u2192 EReal\nf_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x\nG'cont : LowerSemicontinuous G'\nG'int : Integrable fun x => EReal.toReal (G' x)\nG'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4\nhG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5\ns : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b\ns_closed : IsClosed s\nt : \u211d\nht : t \u2208 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ico a b\nv : \u211d\nt_lt_v : v \u2208 Ioi t\ny : \u211d\ng'_lt_y' : \u2191(g' t) < \u2191y\ny_lt_G' : \u2191y < G' t\nI1 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, (u - t) * y \u2264 \u222b (w : \u211d) in t..u, EReal.toReal (G' w)\nI2 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, g u - g t \u2264 (u - t) * y\nI3 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, g u - g t \u2264 \u222b (w : \u211d) in t..u, EReal.toReal (G' w)\nI4 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, u \u2208 Ioc t (min v b)\nx : \u211d\nhx : g x - g t \u2264 \u222b (w : \u211d) in t..x, EReal.toReal (G' w)\nh'x : x \u2208 Ioc t (min v b)\n\u22a2 IntervalIntegrable (fun x => EReal.toReal (G' x)) volume t x"}, {"tactic": "rw [intervalIntegrable_iff_integrable_Ioc_of_le ht.2.1]", "annotated_tactic": ["rw [<a>intervalIntegrable_iff_integrable_Ioc_of_le</a> ht.2.1]", [{"full_name": "intervalIntegrable_iff_integrable_Ioc_of_le", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [91, 9], "def_end_pos": [91, 52]}]], "state_before": "case hab\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\na b : \u211d\nhab : a \u2264 b\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nG' : \u211d \u2192 EReal\nf_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x\nG'cont : LowerSemicontinuous G'\nG'int : Integrable fun x => EReal.toReal (G' x)\nG'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4\nhG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5\ns : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b\ns_closed : IsClosed s\nt : \u211d\nht : t \u2208 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ico a b\nv : \u211d\nt_lt_v : v \u2208 Ioi t\ny : \u211d\ng'_lt_y' : \u2191(g' t) < \u2191y\ny_lt_G' : \u2191y < G' t\nI1 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, (u - t) * y \u2264 \u222b (w : \u211d) in t..u, EReal.toReal (G' w)\nI2 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, g u - g t \u2264 (u - t) * y\nI3 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, g u - g t \u2264 \u222b (w : \u211d) in t..u, EReal.toReal (G' w)\nI4 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, u \u2208 Ioc t (min v b)\nx : \u211d\nhx : g x - g t \u2264 \u222b (w : \u211d) in t..x, EReal.toReal (G' w)\nh'x : x \u2208 Ioc t (min v b)\n\u22a2 IntervalIntegrable (fun x => EReal.toReal (G' x)) volume a t", "state_after": "case hab\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\na b : \u211d\nhab : a \u2264 b\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nG' : \u211d \u2192 EReal\nf_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x\nG'cont : LowerSemicontinuous G'\nG'int : Integrable fun x => EReal.toReal (G' x)\nG'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4\nhG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5\ns : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b\ns_closed : IsClosed s\nt : \u211d\nht : t \u2208 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ico a b\nv : \u211d\nt_lt_v : v \u2208 Ioi t\ny : \u211d\ng'_lt_y' : \u2191(g' t) < \u2191y\ny_lt_G' : \u2191y < G' t\nI1 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, (u - t) * y \u2264 \u222b (w : \u211d) in t..u, EReal.toReal (G' w)\nI2 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, g u - g t \u2264 (u - t) * y\nI3 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, g u - g t \u2264 \u222b (w : \u211d) in t..u, EReal.toReal (G' w)\nI4 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, u \u2208 Ioc t (min v b)\nx : \u211d\nhx : g x - g t \u2264 \u222b (w : \u211d) in t..x, EReal.toReal (G' w)\nh'x : x \u2208 Ioc t (min v b)\n\u22a2 IntegrableOn (fun x => EReal.toReal (G' x)) (Ioc a t)"}, {"tactic": "exact IntegrableOn.mono_set G'int\n  (Ioc_subset_Icc_self.trans (Icc_subset_Icc le_rfl ht.2.2.le))", "annotated_tactic": ["exact <a>IntegrableOn.mono_set</a> G'int\n            (Ioc_subset_Icc_self.trans (<a>Icc_subset_Icc</a> <a>le_rfl</a> ht.2.2.<a>le</a>))", [{"full_name": "MeasureTheory.IntegrableOn.mono_set", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [127, 9], "def_end_pos": [127, 30]}, {"full_name": "Set.Icc_subset_Icc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [455, 9], "def_end_pos": [455, 23]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}, {"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [142, 7], "def_end_pos": [142, 15]}]], "state_before": "case hab\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\na b : \u211d\nhab : a \u2264 b\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nG' : \u211d \u2192 EReal\nf_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x\nG'cont : LowerSemicontinuous G'\nG'int : Integrable fun x => EReal.toReal (G' x)\nG'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4\nhG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5\ns : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b\ns_closed : IsClosed s\nt : \u211d\nht : t \u2208 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ico a b\nv : \u211d\nt_lt_v : v \u2208 Ioi t\ny : \u211d\ng'_lt_y' : \u2191(g' t) < \u2191y\ny_lt_G' : \u2191y < G' t\nI1 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, (u - t) * y \u2264 \u222b (w : \u211d) in t..u, EReal.toReal (G' w)\nI2 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, g u - g t \u2264 (u - t) * y\nI3 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, g u - g t \u2264 \u222b (w : \u211d) in t..u, EReal.toReal (G' w)\nI4 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, u \u2208 Ioc t (min v b)\nx : \u211d\nhx : g x - g t \u2264 \u222b (w : \u211d) in t..x, EReal.toReal (G' w)\nh'x : x \u2208 Ioc t (min v b)\n\u22a2 IntegrableOn (fun x => EReal.toReal (G' x)) (Ioc a t)", "state_after": "no goals"}, {"tactic": "rw [intervalIntegrable_iff_integrable_Ioc_of_le h'x.1.le]", "annotated_tactic": ["rw [<a>intervalIntegrable_iff_integrable_Ioc_of_le</a> h'x.1.<a>le</a>]", [{"full_name": "intervalIntegrable_iff_integrable_Ioc_of_le", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [91, 9], "def_end_pos": [91, 52]}, {"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [142, 7], "def_end_pos": [142, 15]}]], "state_before": "case hbc\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\na b : \u211d\nhab : a \u2264 b\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nG' : \u211d \u2192 EReal\nf_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x\nG'cont : LowerSemicontinuous G'\nG'int : Integrable fun x => EReal.toReal (G' x)\nG'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4\nhG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5\ns : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b\ns_closed : IsClosed s\nt : \u211d\nht : t \u2208 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ico a b\nv : \u211d\nt_lt_v : v \u2208 Ioi t\ny : \u211d\ng'_lt_y' : \u2191(g' t) < \u2191y\ny_lt_G' : \u2191y < G' t\nI1 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, (u - t) * y \u2264 \u222b (w : \u211d) in t..u, EReal.toReal (G' w)\nI2 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, g u - g t \u2264 (u - t) * y\nI3 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, g u - g t \u2264 \u222b (w : \u211d) in t..u, EReal.toReal (G' w)\nI4 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, u \u2208 Ioc t (min v b)\nx : \u211d\nhx : g x - g t \u2264 \u222b (w : \u211d) in t..x, EReal.toReal (G' w)\nh'x : x \u2208 Ioc t (min v b)\n\u22a2 IntervalIntegrable (fun x => EReal.toReal (G' x)) volume t x", "state_after": "case hbc\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\na b : \u211d\nhab : a \u2264 b\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nG' : \u211d \u2192 EReal\nf_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x\nG'cont : LowerSemicontinuous G'\nG'int : Integrable fun x => EReal.toReal (G' x)\nG'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4\nhG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5\ns : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b\ns_closed : IsClosed s\nt : \u211d\nht : t \u2208 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ico a b\nv : \u211d\nt_lt_v : v \u2208 Ioi t\ny : \u211d\ng'_lt_y' : \u2191(g' t) < \u2191y\ny_lt_G' : \u2191y < G' t\nI1 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, (u - t) * y \u2264 \u222b (w : \u211d) in t..u, EReal.toReal (G' w)\nI2 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, g u - g t \u2264 (u - t) * y\nI3 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, g u - g t \u2264 \u222b (w : \u211d) in t..u, EReal.toReal (G' w)\nI4 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, u \u2208 Ioc t (min v b)\nx : \u211d\nhx : g x - g t \u2264 \u222b (w : \u211d) in t..x, EReal.toReal (G' w)\nh'x : x \u2208 Ioc t (min v b)\n\u22a2 IntegrableOn (fun x => EReal.toReal (G' x)) (Ioc t x)"}, {"tactic": "apply IntegrableOn.mono_set G'int", "annotated_tactic": ["apply <a>IntegrableOn.mono_set</a> G'int", [{"full_name": "MeasureTheory.IntegrableOn.mono_set", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [127, 9], "def_end_pos": [127, 30]}]], "state_before": "case hbc\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\na b : \u211d\nhab : a \u2264 b\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nG' : \u211d \u2192 EReal\nf_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x\nG'cont : LowerSemicontinuous G'\nG'int : Integrable fun x => EReal.toReal (G' x)\nG'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4\nhG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5\ns : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b\ns_closed : IsClosed s\nt : \u211d\nht : t \u2208 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ico a b\nv : \u211d\nt_lt_v : v \u2208 Ioi t\ny : \u211d\ng'_lt_y' : \u2191(g' t) < \u2191y\ny_lt_G' : \u2191y < G' t\nI1 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, (u - t) * y \u2264 \u222b (w : \u211d) in t..u, EReal.toReal (G' w)\nI2 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, g u - g t \u2264 (u - t) * y\nI3 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, g u - g t \u2264 \u222b (w : \u211d) in t..u, EReal.toReal (G' w)\nI4 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, u \u2208 Ioc t (min v b)\nx : \u211d\nhx : g x - g t \u2264 \u222b (w : \u211d) in t..x, EReal.toReal (G' w)\nh'x : x \u2208 Ioc t (min v b)\n\u22a2 IntegrableOn (fun x => EReal.toReal (G' x)) (Ioc t x)", "state_after": "case hbc\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\na b : \u211d\nhab : a \u2264 b\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nG' : \u211d \u2192 EReal\nf_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x\nG'cont : LowerSemicontinuous G'\nG'int : Integrable fun x => EReal.toReal (G' x)\nG'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4\nhG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5\ns : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b\ns_closed : IsClosed s\nt : \u211d\nht : t \u2208 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ico a b\nv : \u211d\nt_lt_v : v \u2208 Ioi t\ny : \u211d\ng'_lt_y' : \u2191(g' t) < \u2191y\ny_lt_G' : \u2191y < G' t\nI1 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, (u - t) * y \u2264 \u222b (w : \u211d) in t..u, EReal.toReal (G' w)\nI2 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, g u - g t \u2264 (u - t) * y\nI3 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, g u - g t \u2264 \u222b (w : \u211d) in t..u, EReal.toReal (G' w)\nI4 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, u \u2208 Ioc t (min v b)\nx : \u211d\nhx : g x - g t \u2264 \u222b (w : \u211d) in t..x, EReal.toReal (G' w)\nh'x : x \u2208 Ioc t (min v b)\n\u22a2 Ioc t x \u2286 Icc a b"}, {"tactic": "exact Ioc_subset_Icc_self.trans (Icc_subset_Icc ht.2.1 (h'x.2.trans (min_le_right _ _)))", "annotated_tactic": ["exact Ioc_subset_Icc_self.trans (<a>Icc_subset_Icc</a> ht.2.1 (h'x.2.<a>trans</a> (<a>min_le_right</a> _ _)))", [{"full_name": "Set.Icc_subset_Icc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [455, 9], "def_end_pos": [455, 23]}, {"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [120, 7], "def_end_pos": [120, 18]}, {"full_name": "min_le_right", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [40, 9], "def_end_pos": [40, 21]}]], "state_before": "case hbc\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\na b : \u211d\nhab : a \u2264 b\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nG' : \u211d \u2192 EReal\nf_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x\nG'cont : LowerSemicontinuous G'\nG'int : Integrable fun x => EReal.toReal (G' x)\nG'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4\nhG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5\ns : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b\ns_closed : IsClosed s\nt : \u211d\nht : t \u2208 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ico a b\nv : \u211d\nt_lt_v : v \u2208 Ioi t\ny : \u211d\ng'_lt_y' : \u2191(g' t) < \u2191y\ny_lt_G' : \u2191y < G' t\nI1 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, (u - t) * y \u2264 \u222b (w : \u211d) in t..u, EReal.toReal (G' w)\nI2 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, g u - g t \u2264 (u - t) * y\nI3 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, g u - g t \u2264 \u222b (w : \u211d) in t..u, EReal.toReal (G' w)\nI4 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, u \u2208 Ioc t (min v b)\nx : \u211d\nhx : g x - g t \u2264 \u222b (w : \u211d) in t..x, EReal.toReal (G' w)\nh'x : x \u2208 Ioc t (min v b)\n\u22a2 Ioc t x \u2286 Icc a b", "state_after": "no goals"}, {"tactic": "rw [intervalIntegral.integral_of_le hab]", "annotated_tactic": ["rw [<a>intervalIntegral.integral_of_le</a> hab]", [{"full_name": "intervalIntegral.integral_of_le", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [465, 9], "def_end_pos": [465, 23]}]], "state_before": "case h.e'_4.h.e'_5\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\na b : \u211d\nhab : a \u2264 b\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nG' : \u211d \u2192 EReal\nf_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x\nG'cont : LowerSemicontinuous G'\nG'int : Integrable fun x => EReal.toReal (G' x)\nG'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4\nhG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5\ns : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b\ns_closed : IsClosed s\nmain : Icc a b \u2286 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)}\n\u22a2 \u222b (y : \u211d) in a..b, \u03c6 y = \u222b (x : \u211d) in Icc a b, \u03c6 x", "state_after": "case h.e'_4.h.e'_5\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\na b : \u211d\nhab : a \u2264 b\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nG' : \u211d \u2192 EReal\nf_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x\nG'cont : LowerSemicontinuous G'\nG'int : Integrable fun x => EReal.toReal (G' x)\nG'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4\nhG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5\ns : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b\ns_closed : IsClosed s\nmain : Icc a b \u2286 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)}\n\u22a2 \u222b (x : \u211d) in Ioc a b, \u03c6 x = \u222b (x : \u211d) in Icc a b, \u03c6 x"}, {"tactic": "simp only [integral_Icc_eq_integral_Ioc', Real.volume_singleton]", "annotated_tactic": ["simp only [<a>integral_Icc_eq_integral_Ioc'</a>, <a>Real.volume_singleton</a>]", [{"full_name": "MeasureTheory.integral_Icc_eq_integral_Ioc'", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [638, 9], "def_end_pos": [638, 38]}, {"full_name": "Real.volume_singleton", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean", "def_pos": [96, 9], "def_end_pos": [96, 25]}]], "state_before": "case h.e'_4.h.e'_5\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\na b : \u211d\nhab : a \u2264 b\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nG' : \u211d \u2192 EReal\nf_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x\nG'cont : LowerSemicontinuous G'\nG'int : Integrable fun x => EReal.toReal (G' x)\nG'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4\nhG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5\ns : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b\ns_closed : IsClosed s\nmain : Icc a b \u2286 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)}\n\u22a2 \u222b (x : \u211d) in Ioc a b, \u03c6 x = \u222b (x : \u211d) in Icc a b, \u03c6 x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "Multiset.toFinset_bind_dedup", "start": [3270, 1], "end": [3271, 95], "traced_tactics": [{"tactic": "simp_rw [toFinset, dedup_bind_dedup]", "annotated_tactic": ["simp_rw [<a>toFinset</a>, <a>dedup_bind_dedup</a>]", [{"full_name": "Multiset.toFinset", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3181, 5], "def_end_pos": [3181, 13]}, {"full_name": "Multiset.dedup_bind_dedup", "def_path": "Mathlib/Data/Multiset/Dedup.lean", "def_pos": [99, 9], "def_end_pos": [99, 25]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : DecidableEq \u03b1\ns t : Multiset \u03b1\ninst\u271d : DecidableEq \u03b2\nm : Multiset \u03b1\nf : \u03b1 \u2192 Multiset \u03b2\n\u22a2 toFinset (bind (dedup m) f) = toFinset (bind m f)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "full_name": "MeasureTheory.integral_tsum", "start": [1537, 1], "end": [1567, 48], "traced_tactics": [{"tactic": "by_cases hG : CompleteSpace G", "annotated_tactic": ["by_cases hG : <a>CompleteSpace</a> G", [{"full_name": "CompleteSpace", "def_path": "Mathlib/Topology/UniformSpace/Cauchy.lean", "def_pos": [397, 7], "def_end_pos": [397, 20]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\nG : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b9 : Type u_7\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 G\nhf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\nhf' : \u2211' (i : \u03b9), \u222b\u207b (a : \u03b1), \u2191\u2016f i a\u2016\u208a \u2202\u03bc \u2260 \u22a4\n\u22a2 \u222b (a : \u03b1), \u2211' (i : \u03b9), f i a \u2202\u03bc = \u2211' (i : \u03b9), \u222b (a : \u03b1), f i a \u2202\u03bc", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\nG : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b9 : Type u_7\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 G\nhf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\nhf' : \u2211' (i : \u03b9), \u222b\u207b (a : \u03b1), \u2191\u2016f i a\u2016\u208a \u2202\u03bc \u2260 \u22a4\nhG : CompleteSpace G\n\u22a2 \u222b (a : \u03b1), \u2211' (i : \u03b9), f i a \u2202\u03bc = \u2211' (i : \u03b9), \u222b (a : \u03b1), f i a \u2202\u03bc\n\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\nG : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b9 : Type u_7\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 G\nhf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\nhf' : \u2211' (i : \u03b9), \u222b\u207b (a : \u03b1), \u2191\u2016f i a\u2016\u208a \u2202\u03bc \u2260 \u22a4\nhG : \u00acCompleteSpace G\n\u22a2 \u222b (a : \u03b1), \u2211' (i : \u03b9), f i a \u2202\u03bc = \u2211' (i : \u03b9), \u222b (a : \u03b1), f i a \u2202\u03bc"}, {"tactic": "swap", "annotated_tactic": ["swap", []], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\nG : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b9 : Type u_7\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 G\nhf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\nhf' : \u2211' (i : \u03b9), \u222b\u207b (a : \u03b1), \u2191\u2016f i a\u2016\u208a \u2202\u03bc \u2260 \u22a4\nhG : CompleteSpace G\n\u22a2 \u222b (a : \u03b1), \u2211' (i : \u03b9), f i a \u2202\u03bc = \u2211' (i : \u03b9), \u222b (a : \u03b1), f i a \u2202\u03bc\n\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\nG : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b9 : Type u_7\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 G\nhf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\nhf' : \u2211' (i : \u03b9), \u222b\u207b (a : \u03b1), \u2191\u2016f i a\u2016\u208a \u2202\u03bc \u2260 \u22a4\nhG : \u00acCompleteSpace G\n\u22a2 \u222b (a : \u03b1), \u2211' (i : \u03b9), f i a \u2202\u03bc = \u2211' (i : \u03b9), \u222b (a : \u03b1), f i a \u2202\u03bc", "state_after": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\nG : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b9 : Type u_7\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 G\nhf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\nhf' : \u2211' (i : \u03b9), \u222b\u207b (a : \u03b1), \u2191\u2016f i a\u2016\u208a \u2202\u03bc \u2260 \u22a4\nhG : \u00acCompleteSpace G\n\u22a2 \u222b (a : \u03b1), \u2211' (i : \u03b9), f i a \u2202\u03bc = \u2211' (i : \u03b9), \u222b (a : \u03b1), f i a \u2202\u03bc\n\ncase pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\nG : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b9 : Type u_7\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 G\nhf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\nhf' : \u2211' (i : \u03b9), \u222b\u207b (a : \u03b1), \u2191\u2016f i a\u2016\u208a \u2202\u03bc \u2260 \u22a4\nhG : CompleteSpace G\n\u22a2 \u222b (a : \u03b1), \u2211' (i : \u03b9), f i a \u2202\u03bc = \u2211' (i : \u03b9), \u222b (a : \u03b1), f i a \u2202\u03bc"}, {"tactic": "have hf'' : \u2200 i, AEMeasurable (fun x => (\u2016f i x\u2016\u208a : \u211d\u22650\u221e)) \u03bc := fun i => (hf i).ennnorm", "annotated_tactic": ["have hf'' : \u2200 i, <a>AEMeasurable</a> (fun x => (\u2016f i x\u2016\u208a : \u211d\u22650\u221e)) \u03bc := fun i => (hf i).<a>ennnorm</a>", [{"full_name": "AEMeasurable", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [708, 5], "def_end_pos": [708, 17]}, {"full_name": "MeasureTheory.AEStronglyMeasurable.ennnorm", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1491, 19], "def_end_pos": [1491, 26]}]], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\nG : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b9 : Type u_7\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 G\nhf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\nhf' : \u2211' (i : \u03b9), \u222b\u207b (a : \u03b1), \u2191\u2016f i a\u2016\u208a \u2202\u03bc \u2260 \u22a4\nhG : CompleteSpace G\n\u22a2 \u222b (a : \u03b1), \u2211' (i : \u03b9), f i a \u2202\u03bc = \u2211' (i : \u03b9), \u222b (a : \u03b1), f i a \u2202\u03bc", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\nG : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b9 : Type u_7\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 G\nhf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\nhf' : \u2211' (i : \u03b9), \u222b\u207b (a : \u03b1), \u2191\u2016f i a\u2016\u208a \u2202\u03bc \u2260 \u22a4\nhG : CompleteSpace G\nhf'' : \u2200 (i : \u03b9), AEMeasurable fun x => \u2191\u2016f i x\u2016\u208a\n\u22a2 \u222b (a : \u03b1), \u2211' (i : \u03b9), f i a \u2202\u03bc = \u2211' (i : \u03b9), \u222b (a : \u03b1), f i a \u2202\u03bc"}, {"tactic": "have hhh : \u2200\u1d50 a : \u03b1 \u2202\u03bc, Summable fun n => (\u2016f n a\u2016\u208a : \u211d) := by\n  rw [\u2190 lintegral_tsum hf''] at hf'\n  refine' (ae_lt_top' (AEMeasurable.ennreal_tsum hf'') hf').mono _\n  intro x hx\n  rw [\u2190 ENNReal.tsum_coe_ne_top_iff_summable_coe]\n  exact hx.ne", "annotated_tactic": ["have hhh : \u2200\u1d50 a : \u03b1 \u2202\u03bc, <a>Summable</a> fun n => (\u2016f n a\u2016\u208a : \u211d) := by\n    rw [\u2190 <a>lintegral_tsum</a> hf''] at hf'\n    refine' (<a>ae_lt_top'</a> (<a>AEMeasurable.ennreal_tsum</a> hf'') hf').<a>mono</a> _\n    intro x hx\n    rw [\u2190 <a>ENNReal.tsum_coe_ne_top_iff_summable_coe</a>]\n    exact hx.ne", [{"full_name": "Summable", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/Basic.lean", "def_pos": [62, 5], "def_end_pos": [62, 13]}, {"full_name": "MeasureTheory.lintegral_tsum", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [1184, 9], "def_end_pos": [1184, 23]}, {"full_name": "MeasureTheory.ae_lt_top'", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [1535, 9], "def_end_pos": [1535, 19]}, {"full_name": "AEMeasurable.ennreal_tsum", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [2158, 9], "def_end_pos": [2158, 34]}, {"full_name": "Filter.Eventually.mono", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1140, 9], "def_end_pos": [1140, 24]}, {"full_name": "ENNReal.tsum_coe_ne_top_iff_summable_coe", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [1069, 9], "def_end_pos": [1069, 41]}]], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\nG : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b9 : Type u_7\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 G\nhf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\nhf' : \u2211' (i : \u03b9), \u222b\u207b (a : \u03b1), \u2191\u2016f i a\u2016\u208a \u2202\u03bc \u2260 \u22a4\nhG : CompleteSpace G\nhf'' : \u2200 (i : \u03b9), AEMeasurable fun x => \u2191\u2016f i x\u2016\u208a\n\u22a2 \u222b (a : \u03b1), \u2211' (i : \u03b9), f i a \u2202\u03bc = \u2211' (i : \u03b9), \u222b (a : \u03b1), f i a \u2202\u03bc", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\nG : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b9 : Type u_7\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 G\nhf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\nhf' : \u2211' (i : \u03b9), \u222b\u207b (a : \u03b1), \u2191\u2016f i a\u2016\u208a \u2202\u03bc \u2260 \u22a4\nhG : CompleteSpace G\nhf'' : \u2200 (i : \u03b9), AEMeasurable fun x => \u2191\u2016f i x\u2016\u208a\nhhh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Summable fun n => \u2191\u2016f n a\u2016\u208a\n\u22a2 \u222b (a : \u03b1), \u2211' (i : \u03b9), f i a \u2202\u03bc = \u2211' (i : \u03b9), \u222b (a : \u03b1), f i a \u2202\u03bc"}, {"tactic": "convert (MeasureTheory.hasSum_integral_of_dominated_convergence (fun i a => \u2016f i a\u2016\u208a) hf _ hhh\n        \u27e8_, _\u27e9 _).tsum_eq.symm", "annotated_tactic": ["convert (<a>MeasureTheory.hasSum_integral_of_dominated_convergence</a> (fun i a => \u2016f i a\u2016\u208a) hf _ hhh\n          \u27e8_, _\u27e9 _).tsum_eq.symm", [{"full_name": "MeasureTheory.hasSum_integral_of_dominated_convergence", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1049, 9], "def_end_pos": [1049, 49]}]], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\nG : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b9 : Type u_7\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 G\nhf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\nhf' : \u2211' (i : \u03b9), \u222b\u207b (a : \u03b1), \u2191\u2016f i a\u2016\u208a \u2202\u03bc \u2260 \u22a4\nhG : CompleteSpace G\nhf'' : \u2200 (i : \u03b9), AEMeasurable fun x => \u2191\u2016f i x\u2016\u208a\nhhh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Summable fun n => \u2191\u2016f n a\u2016\u208a\n\u22a2 \u222b (a : \u03b1), \u2211' (i : \u03b9), f i a \u2202\u03bc = \u2211' (i : \u03b9), \u222b (a : \u03b1), f i a \u2202\u03bc", "state_after": "case pos.convert_2\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\nG : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b9 : Type u_7\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 G\nhf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\nhf' : \u2211' (i : \u03b9), \u222b\u207b (a : \u03b1), \u2191\u2016f i a\u2016\u208a \u2202\u03bc \u2260 \u22a4\nhG : CompleteSpace G\nhf'' : \u2200 (i : \u03b9), AEMeasurable fun x => \u2191\u2016f i x\u2016\u208a\nhhh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Summable fun n => \u2191\u2016f n a\u2016\u208a\n\u22a2 \u2200 (n : \u03b9), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016f n a\u2016 \u2264 (fun i a => \u2191\u2016f i a\u2016\u208a) n a\n\ncase pos.convert_3\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\nG : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b9 : Type u_7\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 G\nhf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\nhf' : \u2211' (i : \u03b9), \u222b\u207b (a : \u03b1), \u2191\u2016f i a\u2016\u208a \u2202\u03bc \u2260 \u22a4\nhG : CompleteSpace G\nhf'' : \u2200 (i : \u03b9), AEMeasurable fun x => \u2191\u2016f i x\u2016\u208a\nhhh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Summable fun n => \u2191\u2016f n a\u2016\u208a\n\u22a2 AEStronglyMeasurable (fun a => \u2211' (n : \u03b9), (fun i a => \u2191\u2016f i a\u2016\u208a) n a) \u03bc\n\ncase pos.convert_4\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\nG : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b9 : Type u_7\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 G\nhf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\nhf' : \u2211' (i : \u03b9), \u222b\u207b (a : \u03b1), \u2191\u2016f i a\u2016\u208a \u2202\u03bc \u2260 \u22a4\nhG : CompleteSpace G\nhf'' : \u2200 (i : \u03b9), AEMeasurable fun x => \u2191\u2016f i x\u2016\u208a\nhhh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Summable fun n => \u2191\u2016f n a\u2016\u208a\n\u22a2 HasFiniteIntegral fun a => \u2211' (n : \u03b9), (fun i a => \u2191\u2016f i a\u2016\u208a) n a\n\ncase pos.convert_5\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\nG : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b9 : Type u_7\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 G\nhf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\nhf' : \u2211' (i : \u03b9), \u222b\u207b (a : \u03b1), \u2191\u2016f i a\u2016\u208a \u2202\u03bc \u2260 \u22a4\nhG : CompleteSpace G\nhf'' : \u2200 (i : \u03b9), AEMeasurable fun x => \u2191\u2016f i x\u2016\u208a\nhhh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Summable fun n => \u2191\u2016f n a\u2016\u208a\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasSum (fun n => f n a) (\u2211' (i : \u03b9), f i a)"}, {"tactic": "simp [integral, hG]", "annotated_tactic": ["simp [<a>integral</a>, hG]", [{"full_name": "MeasureTheory.integral", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [791, 17], "def_end_pos": [791, 25]}]], "state_before": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\nG : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b9 : Type u_7\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 G\nhf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\nhf' : \u2211' (i : \u03b9), \u222b\u207b (a : \u03b1), \u2191\u2016f i a\u2016\u208a \u2202\u03bc \u2260 \u22a4\nhG : \u00acCompleteSpace G\n\u22a2 \u222b (a : \u03b1), \u2211' (i : \u03b9), f i a \u2202\u03bc = \u2211' (i : \u03b9), \u222b (a : \u03b1), f i a \u2202\u03bc", "state_after": "no goals"}, {"tactic": "rw [\u2190 lintegral_tsum hf''] at hf'", "annotated_tactic": ["rw [\u2190 <a>lintegral_tsum</a> hf''] at hf'", [{"full_name": "MeasureTheory.lintegral_tsum", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [1184, 9], "def_end_pos": [1184, 23]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\nG : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b9 : Type u_7\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 G\nhf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\nhf' : \u2211' (i : \u03b9), \u222b\u207b (a : \u03b1), \u2191\u2016f i a\u2016\u208a \u2202\u03bc \u2260 \u22a4\nhG : CompleteSpace G\nhf'' : \u2200 (i : \u03b9), AEMeasurable fun x => \u2191\u2016f i x\u2016\u208a\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Summable fun n => \u2191\u2016f n a\u2016\u208a", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\nG : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b9 : Type u_7\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 G\nhf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\nhf' : \u222b\u207b (a : \u03b1), \u2211' (i : \u03b9), \u2191\u2016f i a\u2016\u208a \u2202\u03bc \u2260 \u22a4\nhG : CompleteSpace G\nhf'' : \u2200 (i : \u03b9), AEMeasurable fun x => \u2191\u2016f i x\u2016\u208a\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Summable fun n => \u2191\u2016f n a\u2016\u208a"}, {"tactic": "refine' (ae_lt_top' (AEMeasurable.ennreal_tsum hf'') hf').mono _", "annotated_tactic": ["refine' (<a>ae_lt_top'</a> (<a>AEMeasurable.ennreal_tsum</a> hf'') hf').<a>mono</a> _", [{"full_name": "MeasureTheory.ae_lt_top'", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [1535, 9], "def_end_pos": [1535, 19]}, {"full_name": "AEMeasurable.ennreal_tsum", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [2158, 9], "def_end_pos": [2158, 34]}, {"full_name": "Filter.Eventually.mono", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1140, 9], "def_end_pos": [1140, 24]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\nG : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b9 : Type u_7\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 G\nhf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\nhf' : \u222b\u207b (a : \u03b1), \u2211' (i : \u03b9), \u2191\u2016f i a\u2016\u208a \u2202\u03bc \u2260 \u22a4\nhG : CompleteSpace G\nhf'' : \u2200 (i : \u03b9), AEMeasurable fun x => \u2191\u2016f i x\u2016\u208a\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Summable fun n => \u2191\u2016f n a\u2016\u208a", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\nG : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b9 : Type u_7\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 G\nhf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\nhf' : \u222b\u207b (a : \u03b1), \u2211' (i : \u03b9), \u2191\u2016f i a\u2016\u208a \u2202\u03bc \u2260 \u22a4\nhG : CompleteSpace G\nhf'' : \u2200 (i : \u03b9), AEMeasurable fun x => \u2191\u2016f i x\u2016\u208a\n\u22a2 \u2200 (x : \u03b1), \u2211' (i : \u03b9), \u2191\u2016f i x\u2016\u208a < \u22a4 \u2192 Summable fun n => \u2191\u2016f n x\u2016\u208a"}, {"tactic": "intro x hx", "annotated_tactic": ["intro x hx", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\nG : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b9 : Type u_7\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 G\nhf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\nhf' : \u222b\u207b (a : \u03b1), \u2211' (i : \u03b9), \u2191\u2016f i a\u2016\u208a \u2202\u03bc \u2260 \u22a4\nhG : CompleteSpace G\nhf'' : \u2200 (i : \u03b9), AEMeasurable fun x => \u2191\u2016f i x\u2016\u208a\n\u22a2 \u2200 (x : \u03b1), \u2211' (i : \u03b9), \u2191\u2016f i x\u2016\u208a < \u22a4 \u2192 Summable fun n => \u2191\u2016f n x\u2016\u208a", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\nG : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b9 : Type u_7\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 G\nhf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\nhf' : \u222b\u207b (a : \u03b1), \u2211' (i : \u03b9), \u2191\u2016f i a\u2016\u208a \u2202\u03bc \u2260 \u22a4\nhG : CompleteSpace G\nhf'' : \u2200 (i : \u03b9), AEMeasurable fun x => \u2191\u2016f i x\u2016\u208a\nx : \u03b1\nhx : \u2211' (i : \u03b9), \u2191\u2016f i x\u2016\u208a < \u22a4\n\u22a2 Summable fun n => \u2191\u2016f n x\u2016\u208a"}, {"tactic": "rw [\u2190 ENNReal.tsum_coe_ne_top_iff_summable_coe]", "annotated_tactic": ["rw [\u2190 <a>ENNReal.tsum_coe_ne_top_iff_summable_coe</a>]", [{"full_name": "ENNReal.tsum_coe_ne_top_iff_summable_coe", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [1069, 9], "def_end_pos": [1069, 41]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\nG : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b9 : Type u_7\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 G\nhf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\nhf' : \u222b\u207b (a : \u03b1), \u2211' (i : \u03b9), \u2191\u2016f i a\u2016\u208a \u2202\u03bc \u2260 \u22a4\nhG : CompleteSpace G\nhf'' : \u2200 (i : \u03b9), AEMeasurable fun x => \u2191\u2016f i x\u2016\u208a\nx : \u03b1\nhx : \u2211' (i : \u03b9), \u2191\u2016f i x\u2016\u208a < \u22a4\n\u22a2 Summable fun n => \u2191\u2016f n x\u2016\u208a", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\nG : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b9 : Type u_7\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 G\nhf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\nhf' : \u222b\u207b (a : \u03b1), \u2211' (i : \u03b9), \u2191\u2016f i a\u2016\u208a \u2202\u03bc \u2260 \u22a4\nhG : CompleteSpace G\nhf'' : \u2200 (i : \u03b9), AEMeasurable fun x => \u2191\u2016f i x\u2016\u208a\nx : \u03b1\nhx : \u2211' (i : \u03b9), \u2191\u2016f i x\u2016\u208a < \u22a4\n\u22a2 \u2211' (a : \u03b9), \u2191\u2016f a x\u2016\u208a \u2260 \u22a4"}, {"tactic": "exact hx.ne", "annotated_tactic": ["exact hx.ne", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\nG : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b9 : Type u_7\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 G\nhf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\nhf' : \u222b\u207b (a : \u03b1), \u2211' (i : \u03b9), \u2191\u2016f i a\u2016\u208a \u2202\u03bc \u2260 \u22a4\nhG : CompleteSpace G\nhf'' : \u2200 (i : \u03b9), AEMeasurable fun x => \u2191\u2016f i x\u2016\u208a\nx : \u03b1\nhx : \u2211' (i : \u03b9), \u2191\u2016f i x\u2016\u208a < \u22a4\n\u22a2 \u2211' (a : \u03b9), \u2191\u2016f a x\u2016\u208a \u2260 \u22a4", "state_after": "no goals"}, {"tactic": "intro n", "annotated_tactic": ["intro n", []], "state_before": "case pos.convert_2\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\nG : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b9 : Type u_7\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 G\nhf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\nhf' : \u2211' (i : \u03b9), \u222b\u207b (a : \u03b1), \u2191\u2016f i a\u2016\u208a \u2202\u03bc \u2260 \u22a4\nhG : CompleteSpace G\nhf'' : \u2200 (i : \u03b9), AEMeasurable fun x => \u2191\u2016f i x\u2016\u208a\nhhh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Summable fun n => \u2191\u2016f n a\u2016\u208a\n\u22a2 \u2200 (n : \u03b9), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016f n a\u2016 \u2264 (fun i a => \u2191\u2016f i a\u2016\u208a) n a", "state_after": "case pos.convert_2\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\nG : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b9 : Type u_7\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 G\nhf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\nhf' : \u2211' (i : \u03b9), \u222b\u207b (a : \u03b1), \u2191\u2016f i a\u2016\u208a \u2202\u03bc \u2260 \u22a4\nhG : CompleteSpace G\nhf'' : \u2200 (i : \u03b9), AEMeasurable fun x => \u2191\u2016f i x\u2016\u208a\nhhh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Summable fun n => \u2191\u2016f n a\u2016\u208a\nn : \u03b9\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016f n a\u2016 \u2264 (fun i a => \u2191\u2016f i a\u2016\u208a) n a"}, {"tactic": "filter_upwards with x", "annotated_tactic": ["filter_upwards with x", []], "state_before": "case pos.convert_2\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\nG : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b9 : Type u_7\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 G\nhf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\nhf' : \u2211' (i : \u03b9), \u222b\u207b (a : \u03b1), \u2191\u2016f i a\u2016\u208a \u2202\u03bc \u2260 \u22a4\nhG : CompleteSpace G\nhf'' : \u2200 (i : \u03b9), AEMeasurable fun x => \u2191\u2016f i x\u2016\u208a\nhhh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Summable fun n => \u2191\u2016f n a\u2016\u208a\nn : \u03b9\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016f n a\u2016 \u2264 (fun i a => \u2191\u2016f i a\u2016\u208a) n a", "state_after": "case pos.convert_2.h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\nG : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b9 : Type u_7\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 G\nhf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\nhf' : \u2211' (i : \u03b9), \u222b\u207b (a : \u03b1), \u2191\u2016f i a\u2016\u208a \u2202\u03bc \u2260 \u22a4\nhG : CompleteSpace G\nhf'' : \u2200 (i : \u03b9), AEMeasurable fun x => \u2191\u2016f i x\u2016\u208a\nhhh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Summable fun n => \u2191\u2016f n a\u2016\u208a\nn : \u03b9\nx : \u03b1\n\u22a2 \u2016f n x\u2016 \u2264 \u2191\u2016f n x\u2016\u208a"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case pos.convert_2.h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\nG : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b9 : Type u_7\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 G\nhf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\nhf' : \u2211' (i : \u03b9), \u222b\u207b (a : \u03b1), \u2191\u2016f i a\u2016\u208a \u2202\u03bc \u2260 \u22a4\nhG : CompleteSpace G\nhf'' : \u2200 (i : \u03b9), AEMeasurable fun x => \u2191\u2016f i x\u2016\u208a\nhhh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Summable fun n => \u2191\u2016f n a\u2016\u208a\nn : \u03b9\nx : \u03b1\n\u22a2 \u2016f n x\u2016 \u2264 \u2191\u2016f n x\u2016\u208a", "state_after": "no goals"}, {"tactic": "simp_rw [\u2190 NNReal.coe_tsum]", "annotated_tactic": ["simp_rw [\u2190 <a>NNReal.coe_tsum</a>]", [{"full_name": "NNReal.coe_tsum", "def_path": "Mathlib/Topology/Instances/NNReal.lean", "def_pos": [184, 9], "def_end_pos": [184, 17]}]], "state_before": "case pos.convert_3\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\nG : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b9 : Type u_7\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 G\nhf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\nhf' : \u2211' (i : \u03b9), \u222b\u207b (a : \u03b1), \u2191\u2016f i a\u2016\u208a \u2202\u03bc \u2260 \u22a4\nhG : CompleteSpace G\nhf'' : \u2200 (i : \u03b9), AEMeasurable fun x => \u2191\u2016f i x\u2016\u208a\nhhh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Summable fun n => \u2191\u2016f n a\u2016\u208a\n\u22a2 AEStronglyMeasurable (fun a => \u2211' (n : \u03b9), (fun i a => \u2191\u2016f i a\u2016\u208a) n a) \u03bc", "state_after": "case pos.convert_3\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\nG : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b9 : Type u_7\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 G\nhf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\nhf' : \u2211' (i : \u03b9), \u222b\u207b (a : \u03b1), \u2191\u2016f i a\u2016\u208a \u2202\u03bc \u2260 \u22a4\nhG : CompleteSpace G\nhf'' : \u2200 (i : \u03b9), AEMeasurable fun x => \u2191\u2016f i x\u2016\u208a\nhhh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Summable fun n => \u2191\u2016f n a\u2016\u208a\n\u22a2 AEStronglyMeasurable (fun a => \u2191(\u2211' (a_1 : \u03b9), \u2016f a_1 a\u2016\u208a)) \u03bc"}, {"tactic": "rw [aestronglyMeasurable_iff_aemeasurable]", "annotated_tactic": ["rw [<a>aestronglyMeasurable_iff_aemeasurable</a>]", [{"full_name": "aestronglyMeasurable_iff_aemeasurable", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1464, 9], "def_end_pos": [1464, 53]}]], "state_before": "case pos.convert_3\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\nG : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b9 : Type u_7\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 G\nhf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\nhf' : \u2211' (i : \u03b9), \u222b\u207b (a : \u03b1), \u2191\u2016f i a\u2016\u208a \u2202\u03bc \u2260 \u22a4\nhG : CompleteSpace G\nhf'' : \u2200 (i : \u03b9), AEMeasurable fun x => \u2191\u2016f i x\u2016\u208a\nhhh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Summable fun n => \u2191\u2016f n a\u2016\u208a\n\u22a2 AEStronglyMeasurable (fun a => \u2191(\u2211' (a_1 : \u03b9), \u2016f a_1 a\u2016\u208a)) \u03bc", "state_after": "case pos.convert_3\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\nG : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b9 : Type u_7\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 G\nhf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\nhf' : \u2211' (i : \u03b9), \u222b\u207b (a : \u03b1), \u2191\u2016f i a\u2016\u208a \u2202\u03bc \u2260 \u22a4\nhG : CompleteSpace G\nhf'' : \u2200 (i : \u03b9), AEMeasurable fun x => \u2191\u2016f i x\u2016\u208a\nhhh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Summable fun n => \u2191\u2016f n a\u2016\u208a\n\u22a2 AEMeasurable fun a => \u2191(\u2211' (a_1 : \u03b9), \u2016f a_1 a\u2016\u208a)"}, {"tactic": "apply AEMeasurable.coe_nnreal_real", "annotated_tactic": ["apply <a>AEMeasurable.coe_nnreal_real</a>", [{"full_name": "AEMeasurable.coe_nnreal_real", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [1982, 9], "def_end_pos": [1982, 37]}]], "state_before": "case pos.convert_3\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\nG : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b9 : Type u_7\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 G\nhf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\nhf' : \u2211' (i : \u03b9), \u222b\u207b (a : \u03b1), \u2191\u2016f i a\u2016\u208a \u2202\u03bc \u2260 \u22a4\nhG : CompleteSpace G\nhf'' : \u2200 (i : \u03b9), AEMeasurable fun x => \u2191\u2016f i x\u2016\u208a\nhhh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Summable fun n => \u2191\u2016f n a\u2016\u208a\n\u22a2 AEMeasurable fun a => \u2191(\u2211' (a_1 : \u03b9), \u2016f a_1 a\u2016\u208a)", "state_after": "case pos.convert_3.hf\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\nG : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b9 : Type u_7\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 G\nhf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\nhf' : \u2211' (i : \u03b9), \u222b\u207b (a : \u03b1), \u2191\u2016f i a\u2016\u208a \u2202\u03bc \u2260 \u22a4\nhG : CompleteSpace G\nhf'' : \u2200 (i : \u03b9), AEMeasurable fun x => \u2191\u2016f i x\u2016\u208a\nhhh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Summable fun n => \u2191\u2016f n a\u2016\u208a\n\u22a2 AEMeasurable fun x => \u2211' (a : \u03b9), \u2016f a x\u2016\u208a"}, {"tactic": "apply AEMeasurable.nnreal_tsum", "annotated_tactic": ["apply <a>AEMeasurable.nnreal_tsum</a>", [{"full_name": "AEMeasurable.nnreal_tsum", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [2166, 9], "def_end_pos": [2166, 33]}]], "state_before": "case pos.convert_3.hf\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\nG : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b9 : Type u_7\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 G\nhf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\nhf' : \u2211' (i : \u03b9), \u222b\u207b (a : \u03b1), \u2191\u2016f i a\u2016\u208a \u2202\u03bc \u2260 \u22a4\nhG : CompleteSpace G\nhf'' : \u2200 (i : \u03b9), AEMeasurable fun x => \u2191\u2016f i x\u2016\u208a\nhhh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Summable fun n => \u2191\u2016f n a\u2016\u208a\n\u22a2 AEMeasurable fun x => \u2211' (a : \u03b9), \u2016f a x\u2016\u208a", "state_after": "case pos.convert_3.hf.h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\nG : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b9 : Type u_7\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 G\nhf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\nhf' : \u2211' (i : \u03b9), \u222b\u207b (a : \u03b1), \u2191\u2016f i a\u2016\u208a \u2202\u03bc \u2260 \u22a4\nhG : CompleteSpace G\nhf'' : \u2200 (i : \u03b9), AEMeasurable fun x => \u2191\u2016f i x\u2016\u208a\nhhh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Summable fun n => \u2191\u2016f n a\u2016\u208a\n\u22a2 \u2200 (i : \u03b9), AEMeasurable fun x => \u2016f i x\u2016\u208a"}, {"tactic": "exact fun i => (hf i).nnnorm.aemeasurable", "annotated_tactic": ["exact fun i => (hf i).nnnorm.aemeasurable", []], "state_before": "case pos.convert_3.hf.h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\nG : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b9 : Type u_7\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 G\nhf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\nhf' : \u2211' (i : \u03b9), \u222b\u207b (a : \u03b1), \u2191\u2016f i a\u2016\u208a \u2202\u03bc \u2260 \u22a4\nhG : CompleteSpace G\nhf'' : \u2200 (i : \u03b9), AEMeasurable fun x => \u2191\u2016f i x\u2016\u208a\nhhh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Summable fun n => \u2191\u2016f n a\u2016\u208a\n\u22a2 \u2200 (i : \u03b9), AEMeasurable fun x => \u2016f i x\u2016\u208a", "state_after": "no goals"}, {"tactic": "dsimp [HasFiniteIntegral]", "annotated_tactic": ["dsimp [<a>HasFiniteIntegral</a>]", [{"full_name": "MeasureTheory.HasFiniteIntegral", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [106, 5], "def_end_pos": [106, 22]}]], "state_before": "case pos.convert_4\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\nG : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b9 : Type u_7\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 G\nhf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\nhf' : \u2211' (i : \u03b9), \u222b\u207b (a : \u03b1), \u2191\u2016f i a\u2016\u208a \u2202\u03bc \u2260 \u22a4\nhG : CompleteSpace G\nhf'' : \u2200 (i : \u03b9), AEMeasurable fun x => \u2191\u2016f i x\u2016\u208a\nhhh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Summable fun n => \u2191\u2016f n a\u2016\u208a\n\u22a2 HasFiniteIntegral fun a => \u2211' (n : \u03b9), (fun i a => \u2191\u2016f i a\u2016\u208a) n a", "state_after": "case pos.convert_4\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\nG : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b9 : Type u_7\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 G\nhf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\nhf' : \u2211' (i : \u03b9), \u222b\u207b (a : \u03b1), \u2191\u2016f i a\u2016\u208a \u2202\u03bc \u2260 \u22a4\nhG : CompleteSpace G\nhf'' : \u2200 (i : \u03b9), AEMeasurable fun x => \u2191\u2016f i x\u2016\u208a\nhhh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Summable fun n => \u2191\u2016f n a\u2016\u208a\n\u22a2 \u222b\u207b (a : \u03b1), \u2191\u2016\u2211' (n : \u03b9), \u2016f n a\u2016\u2016\u208a \u2202\u03bc < \u22a4"}, {"tactic": "have : \u222b\u207b a, \u2211' n, \u2016f n a\u2016\u208a \u2202\u03bc < \u22a4 := by rwa [lintegral_tsum hf'', lt_top_iff_ne_top]", "annotated_tactic": ["have : \u222b\u207b a, \u2211' n, \u2016f n a\u2016\u208a \u2202\u03bc < \u22a4 := by rwa [<a>lintegral_tsum</a> hf'', <a>lt_top_iff_ne_top</a>]", [{"full_name": "MeasureTheory.lintegral_tsum", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [1184, 9], "def_end_pos": [1184, 23]}, {"full_name": "lt_top_iff_ne_top", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [173, 9], "def_end_pos": [173, 26]}]], "state_before": "case pos.convert_4\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\nG : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b9 : Type u_7\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 G\nhf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\nhf' : \u2211' (i : \u03b9), \u222b\u207b (a : \u03b1), \u2191\u2016f i a\u2016\u208a \u2202\u03bc \u2260 \u22a4\nhG : CompleteSpace G\nhf'' : \u2200 (i : \u03b9), AEMeasurable fun x => \u2191\u2016f i x\u2016\u208a\nhhh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Summable fun n => \u2191\u2016f n a\u2016\u208a\n\u22a2 \u222b\u207b (a : \u03b1), \u2191\u2016\u2211' (n : \u03b9), \u2016f n a\u2016\u2016\u208a \u2202\u03bc < \u22a4", "state_after": "case pos.convert_4\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\nG : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b9 : Type u_7\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 G\nhf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\nhf' : \u2211' (i : \u03b9), \u222b\u207b (a : \u03b1), \u2191\u2016f i a\u2016\u208a \u2202\u03bc \u2260 \u22a4\nhG : CompleteSpace G\nhf'' : \u2200 (i : \u03b9), AEMeasurable fun x => \u2191\u2016f i x\u2016\u208a\nhhh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Summable fun n => \u2191\u2016f n a\u2016\u208a\nthis : \u222b\u207b (a : \u03b1), \u2211' (n : \u03b9), \u2191\u2016f n a\u2016\u208a \u2202\u03bc < \u22a4\n\u22a2 \u222b\u207b (a : \u03b1), \u2191\u2016\u2211' (n : \u03b9), \u2016f n a\u2016\u2016\u208a \u2202\u03bc < \u22a4"}, {"tactic": "convert this using 1", "annotated_tactic": ["convert this using 1", []], "state_before": "case pos.convert_4\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\nG : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b9 : Type u_7\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 G\nhf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\nhf' : \u2211' (i : \u03b9), \u222b\u207b (a : \u03b1), \u2191\u2016f i a\u2016\u208a \u2202\u03bc \u2260 \u22a4\nhG : CompleteSpace G\nhf'' : \u2200 (i : \u03b9), AEMeasurable fun x => \u2191\u2016f i x\u2016\u208a\nhhh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Summable fun n => \u2191\u2016f n a\u2016\u208a\nthis : \u222b\u207b (a : \u03b1), \u2211' (n : \u03b9), \u2191\u2016f n a\u2016\u208a \u2202\u03bc < \u22a4\n\u22a2 \u222b\u207b (a : \u03b1), \u2191\u2016\u2211' (n : \u03b9), \u2016f n a\u2016\u2016\u208a \u2202\u03bc < \u22a4", "state_after": "case h.e'_3\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\nG : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b9 : Type u_7\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 G\nhf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\nhf' : \u2211' (i : \u03b9), \u222b\u207b (a : \u03b1), \u2191\u2016f i a\u2016\u208a \u2202\u03bc \u2260 \u22a4\nhG : CompleteSpace G\nhf'' : \u2200 (i : \u03b9), AEMeasurable fun x => \u2191\u2016f i x\u2016\u208a\nhhh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Summable fun n => \u2191\u2016f n a\u2016\u208a\nthis : \u222b\u207b (a : \u03b1), \u2211' (n : \u03b9), \u2191\u2016f n a\u2016\u208a \u2202\u03bc < \u22a4\n\u22a2 \u222b\u207b (a : \u03b1), \u2191\u2016\u2211' (n : \u03b9), \u2016f n a\u2016\u2016\u208a \u2202\u03bc = \u222b\u207b (a : \u03b1), \u2211' (n : \u03b9), \u2191\u2016f n a\u2016\u208a \u2202\u03bc"}, {"tactic": "apply lintegral_congr_ae", "annotated_tactic": ["apply <a>lintegral_congr_ae</a>", [{"full_name": "MeasureTheory.lintegral_congr_ae", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [304, 9], "def_end_pos": [304, 27]}]], "state_before": "case h.e'_3\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\nG : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b9 : Type u_7\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 G\nhf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\nhf' : \u2211' (i : \u03b9), \u222b\u207b (a : \u03b1), \u2191\u2016f i a\u2016\u208a \u2202\u03bc \u2260 \u22a4\nhG : CompleteSpace G\nhf'' : \u2200 (i : \u03b9), AEMeasurable fun x => \u2191\u2016f i x\u2016\u208a\nhhh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Summable fun n => \u2191\u2016f n a\u2016\u208a\nthis : \u222b\u207b (a : \u03b1), \u2211' (n : \u03b9), \u2191\u2016f n a\u2016\u208a \u2202\u03bc < \u22a4\n\u22a2 \u222b\u207b (a : \u03b1), \u2191\u2016\u2211' (n : \u03b9), \u2016f n a\u2016\u2016\u208a \u2202\u03bc = \u222b\u207b (a : \u03b1), \u2211' (n : \u03b9), \u2191\u2016f n a\u2016\u208a \u2202\u03bc", "state_after": "case h.e'_3.h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\nG : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b9 : Type u_7\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 G\nhf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\nhf' : \u2211' (i : \u03b9), \u222b\u207b (a : \u03b1), \u2191\u2016f i a\u2016\u208a \u2202\u03bc \u2260 \u22a4\nhG : CompleteSpace G\nhf'' : \u2200 (i : \u03b9), AEMeasurable fun x => \u2191\u2016f i x\u2016\u208a\nhhh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Summable fun n => \u2191\u2016f n a\u2016\u208a\nthis : \u222b\u207b (a : \u03b1), \u2211' (n : \u03b9), \u2191\u2016f n a\u2016\u208a \u2202\u03bc < \u22a4\n\u22a2 (fun a => \u2191\u2016\u2211' (n : \u03b9), \u2016f n a\u2016\u2016\u208a) =\u1d50[\u03bc] fun a => \u2211' (n : \u03b9), \u2191\u2016f n a\u2016\u208a"}, {"tactic": "simp_rw [\u2190 coe_nnnorm, \u2190 NNReal.coe_tsum, NNReal.nnnorm_eq]", "annotated_tactic": ["simp_rw [\u2190 <a>coe_nnnorm</a>, \u2190 <a>NNReal.coe_tsum</a>, <a>NNReal.nnnorm_eq</a>]", [{"full_name": "coe_nnnorm", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [905, 41], "def_end_pos": [905, 51]}, {"full_name": "NNReal.coe_tsum", "def_path": "Mathlib/Topology/Instances/NNReal.lean", "def_pos": [184, 9], "def_end_pos": [184, 17]}, {"full_name": "NNReal.nnnorm_eq", "def_path": "Mathlib/Analysis/Normed/Field/Basic.lean", "def_pos": [830, 9], "def_end_pos": [830, 18]}]], "state_before": "case h.e'_3.h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\nG : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b9 : Type u_7\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 G\nhf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\nhf' : \u2211' (i : \u03b9), \u222b\u207b (a : \u03b1), \u2191\u2016f i a\u2016\u208a \u2202\u03bc \u2260 \u22a4\nhG : CompleteSpace G\nhf'' : \u2200 (i : \u03b9), AEMeasurable fun x => \u2191\u2016f i x\u2016\u208a\nhhh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Summable fun n => \u2191\u2016f n a\u2016\u208a\nthis : \u222b\u207b (a : \u03b1), \u2211' (n : \u03b9), \u2191\u2016f n a\u2016\u208a \u2202\u03bc < \u22a4\n\u22a2 (fun a => \u2191\u2016\u2211' (n : \u03b9), \u2016f n a\u2016\u2016\u208a) =\u1d50[\u03bc] fun a => \u2211' (n : \u03b9), \u2191\u2016f n a\u2016\u208a", "state_after": "case h.e'_3.h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\nG : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b9 : Type u_7\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 G\nhf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\nhf' : \u2211' (i : \u03b9), \u222b\u207b (a : \u03b1), \u2191\u2016f i a\u2016\u208a \u2202\u03bc \u2260 \u22a4\nhG : CompleteSpace G\nhf'' : \u2200 (i : \u03b9), AEMeasurable fun x => \u2191\u2016f i x\u2016\u208a\nhhh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Summable fun n => \u2191\u2016f n a\u2016\u208a\nthis : \u222b\u207b (a : \u03b1), \u2211' (n : \u03b9), \u2191\u2016f n a\u2016\u208a \u2202\u03bc < \u22a4\n\u22a2 (fun a => \u2191(\u2211' (a_1 : \u03b9), \u2016f a_1 a\u2016\u208a)) =\u1d50[\u03bc] fun a => \u2211' (n : \u03b9), \u2191\u2016f n a\u2016\u208a"}, {"tactic": "filter_upwards [hhh] with a ha", "annotated_tactic": ["filter_upwards [hhh] with a ha", []], "state_before": "case h.e'_3.h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\nG : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b9 : Type u_7\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 G\nhf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\nhf' : \u2211' (i : \u03b9), \u222b\u207b (a : \u03b1), \u2191\u2016f i a\u2016\u208a \u2202\u03bc \u2260 \u22a4\nhG : CompleteSpace G\nhf'' : \u2200 (i : \u03b9), AEMeasurable fun x => \u2191\u2016f i x\u2016\u208a\nhhh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Summable fun n => \u2191\u2016f n a\u2016\u208a\nthis : \u222b\u207b (a : \u03b1), \u2211' (n : \u03b9), \u2191\u2016f n a\u2016\u208a \u2202\u03bc < \u22a4\n\u22a2 (fun a => \u2191(\u2211' (a_1 : \u03b9), \u2016f a_1 a\u2016\u208a)) =\u1d50[\u03bc] fun a => \u2211' (n : \u03b9), \u2191\u2016f n a\u2016\u208a", "state_after": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\nG : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b9 : Type u_7\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 G\nhf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\nhf' : \u2211' (i : \u03b9), \u222b\u207b (a : \u03b1), \u2191\u2016f i a\u2016\u208a \u2202\u03bc \u2260 \u22a4\nhG : CompleteSpace G\nhf'' : \u2200 (i : \u03b9), AEMeasurable fun x => \u2191\u2016f i x\u2016\u208a\nhhh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Summable fun n => \u2191\u2016f n a\u2016\u208a\nthis : \u222b\u207b (a : \u03b1), \u2211' (n : \u03b9), \u2191\u2016f n a\u2016\u208a \u2202\u03bc < \u22a4\na : \u03b1\nha : Summable fun n => \u2191\u2016f n a\u2016\u208a\n\u22a2 \u2191(\u2211' (a_1 : \u03b9), \u2016f a_1 a\u2016\u208a) = \u2211' (n : \u03b9), \u2191\u2016f n a\u2016\u208a"}, {"tactic": "exact ENNReal.coe_tsum (NNReal.summable_coe.mp ha)", "annotated_tactic": ["exact <a>ENNReal.coe_tsum</a> (NNReal.summable_coe.mp ha)", [{"full_name": "ENNReal.coe_tsum", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [768, 19], "def_end_pos": [768, 27]}]], "state_before": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\nG : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b9 : Type u_7\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 G\nhf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\nhf' : \u2211' (i : \u03b9), \u222b\u207b (a : \u03b1), \u2191\u2016f i a\u2016\u208a \u2202\u03bc \u2260 \u22a4\nhG : CompleteSpace G\nhf'' : \u2200 (i : \u03b9), AEMeasurable fun x => \u2191\u2016f i x\u2016\u208a\nhhh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Summable fun n => \u2191\u2016f n a\u2016\u208a\nthis : \u222b\u207b (a : \u03b1), \u2211' (n : \u03b9), \u2191\u2016f n a\u2016\u208a \u2202\u03bc < \u22a4\na : \u03b1\nha : Summable fun n => \u2191\u2016f n a\u2016\u208a\n\u22a2 \u2191(\u2211' (a_1 : \u03b9), \u2016f a_1 a\u2016\u208a) = \u2211' (n : \u03b9), \u2191\u2016f n a\u2016\u208a", "state_after": "no goals"}, {"tactic": "rwa [lintegral_tsum hf'', lt_top_iff_ne_top]", "annotated_tactic": ["rwa [<a>lintegral_tsum</a> hf'', <a>lt_top_iff_ne_top</a>]", [{"full_name": "MeasureTheory.lintegral_tsum", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [1184, 9], "def_end_pos": [1184, 23]}, {"full_name": "lt_top_iff_ne_top", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [173, 9], "def_end_pos": [173, 26]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\nG : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b9 : Type u_7\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 G\nhf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\nhf' : \u2211' (i : \u03b9), \u222b\u207b (a : \u03b1), \u2191\u2016f i a\u2016\u208a \u2202\u03bc \u2260 \u22a4\nhG : CompleteSpace G\nhf'' : \u2200 (i : \u03b9), AEMeasurable fun x => \u2191\u2016f i x\u2016\u208a\nhhh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Summable fun n => \u2191\u2016f n a\u2016\u208a\n\u22a2 \u222b\u207b (a : \u03b1), \u2211' (n : \u03b9), \u2191\u2016f n a\u2016\u208a \u2202\u03bc < \u22a4", "state_after": "no goals"}, {"tactic": "filter_upwards [hhh] with x hx", "annotated_tactic": ["filter_upwards [hhh] with x hx", []], "state_before": "case pos.convert_5\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\nG : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b9 : Type u_7\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 G\nhf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\nhf' : \u2211' (i : \u03b9), \u222b\u207b (a : \u03b1), \u2191\u2016f i a\u2016\u208a \u2202\u03bc \u2260 \u22a4\nhG : CompleteSpace G\nhf'' : \u2200 (i : \u03b9), AEMeasurable fun x => \u2191\u2016f i x\u2016\u208a\nhhh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Summable fun n => \u2191\u2016f n a\u2016\u208a\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasSum (fun n => f n a) (\u2211' (i : \u03b9), f i a)", "state_after": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\nG : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b9 : Type u_7\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 G\nhf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\nhf' : \u2211' (i : \u03b9), \u222b\u207b (a : \u03b1), \u2191\u2016f i a\u2016\u208a \u2202\u03bc \u2260 \u22a4\nhG : CompleteSpace G\nhf'' : \u2200 (i : \u03b9), AEMeasurable fun x => \u2191\u2016f i x\u2016\u208a\nhhh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Summable fun n => \u2191\u2016f n a\u2016\u208a\nx : \u03b1\nhx : Summable fun n => \u2191\u2016f n x\u2016\u208a\n\u22a2 HasSum (fun n => f n x) (\u2211' (i : \u03b9), f i x)"}, {"tactic": "exact (summable_of_summable_norm hx).hasSum", "annotated_tactic": ["exact (<a>summable_of_summable_norm</a> hx).<a>hasSum</a>", [{"full_name": "summable_of_summable_norm", "def_path": "Mathlib/Analysis/Normed/Group/InfiniteSum.lean", "def_pos": [173, 9], "def_end_pos": [173, 34]}, {"full_name": "Summable.hasSum", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/Basic.lean", "def_pos": [206, 9], "def_end_pos": [206, 24]}]], "state_before": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\nG : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b9 : Type u_7\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 G\nhf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\nhf' : \u2211' (i : \u03b9), \u222b\u207b (a : \u03b1), \u2191\u2016f i a\u2016\u208a \u2202\u03bc \u2260 \u22a4\nhG : CompleteSpace G\nhf'' : \u2200 (i : \u03b9), AEMeasurable fun x => \u2191\u2016f i x\u2016\u208a\nhhh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Summable fun n => \u2191\u2016f n a\u2016\u208a\nx : \u03b1\nhx : Summable fun n => \u2191\u2016f n x\u2016\u208a\n\u22a2 HasSum (fun n => f n x) (\u2211' (i : \u03b9), f i x)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Part.lean", "full_name": "Part.induction_on", "start": [366, 11], "end": [369, 36], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "Finset.mem_inter", "start": [1601, 1], "end": [1602, 14], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/Basic.lean", "full_name": "MvPolynomial.aevalTower_comp_C", "start": [1636, 1], "end": [1637, 34], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Sigma.lean", "full_name": "Set.image_sigmaMk_preimage_sigmaMap", "start": [42, 1], "end": [49, 22], "traced_tactics": [{"tactic": "refine' (image_sigmaMk_preimage_sigmaMap_subset f g i s).antisymm _", "annotated_tactic": ["refine' (<a>image_sigmaMk_preimage_sigmaMap_subset</a> f g i s).<a>antisymm</a> _", [{"full_name": "Set.image_sigmaMk_preimage_sigmaMap_subset", "def_path": "Mathlib/Data/Set/Sigma.lean", "def_pos": [36, 9], "def_end_pos": [36, 47]}, {"full_name": "HasSubset.Subset.antisymm", "def_path": "Mathlib/Order/RelClasses.lean", "def_pos": [667, 7], "def_end_pos": [667, 32]}]], "state_before": "\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : \u03b9 \u2192 Type u_3\n\u03b2\u271d : \u03b9 \u2192 Type u_4\ns\u271d s\u2081 s\u2082 : Set \u03b9\nt t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\nu : Set ((i : \u03b9) \u00d7 \u03b1 i)\nx : (i : \u03b9) \u00d7 \u03b1 i\ni\u271d j : \u03b9\na : \u03b1 i\u271d\n\u03b2 : \u03b9' \u2192 Type u_5\nf : \u03b9 \u2192 \u03b9'\nhf : Function.Injective f\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b2 (f i)\ni : \u03b9\ns : Set (\u03b2 (f i))\n\u22a2 Sigma.mk i '' (g i \u207b\u00b9' s) = Sigma.map f g \u207b\u00b9' (Sigma.mk (f i) '' s)", "state_after": "\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : \u03b9 \u2192 Type u_3\n\u03b2\u271d : \u03b9 \u2192 Type u_4\ns\u271d s\u2081 s\u2082 : Set \u03b9\nt t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\nu : Set ((i : \u03b9) \u00d7 \u03b1 i)\nx : (i : \u03b9) \u00d7 \u03b1 i\ni\u271d j : \u03b9\na : \u03b1 i\u271d\n\u03b2 : \u03b9' \u2192 Type u_5\nf : \u03b9 \u2192 \u03b9'\nhf : Function.Injective f\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b2 (f i)\ni : \u03b9\ns : Set (\u03b2 (f i))\n\u22a2 Sigma.map f g \u207b\u00b9' (Sigma.mk (f i) '' s) \u2286 Sigma.mk i '' (g i \u207b\u00b9' s)"}, {"tactic": "rintro \u27e8j, x\u27e9 \u27e8y, hys, hxy\u27e9", "annotated_tactic": ["rintro \u27e8j, x\u27e9 \u27e8y, hys, hxy\u27e9", []], "state_before": "\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : \u03b9 \u2192 Type u_3\n\u03b2\u271d : \u03b9 \u2192 Type u_4\ns\u271d s\u2081 s\u2082 : Set \u03b9\nt t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\nu : Set ((i : \u03b9) \u00d7 \u03b1 i)\nx : (i : \u03b9) \u00d7 \u03b1 i\ni\u271d j : \u03b9\na : \u03b1 i\u271d\n\u03b2 : \u03b9' \u2192 Type u_5\nf : \u03b9 \u2192 \u03b9'\nhf : Function.Injective f\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b2 (f i)\ni : \u03b9\ns : Set (\u03b2 (f i))\n\u22a2 Sigma.map f g \u207b\u00b9' (Sigma.mk (f i) '' s) \u2286 Sigma.mk i '' (g i \u207b\u00b9' s)", "state_after": "case mk.intro.intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : \u03b9 \u2192 Type u_3\n\u03b2\u271d : \u03b9 \u2192 Type u_4\ns\u271d s\u2081 s\u2082 : Set \u03b9\nt t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\nu : Set ((i : \u03b9) \u00d7 \u03b1 i)\nx\u271d : (i : \u03b9) \u00d7 \u03b1 i\ni\u271d j\u271d : \u03b9\na : \u03b1 i\u271d\n\u03b2 : \u03b9' \u2192 Type u_5\nf : \u03b9 \u2192 \u03b9'\nhf : Function.Injective f\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b2 (f i)\ni : \u03b9\ns : Set (\u03b2 (f i))\nj : \u03b9\nx : \u03b1 j\ny : \u03b2 (f i)\nhys : y \u2208 s\nhxy : { fst := f i, snd := y } = Sigma.map f g { fst := j, snd := x }\n\u22a2 { fst := j, snd := x } \u2208 Sigma.mk i '' (g i \u207b\u00b9' s)"}, {"tactic": "simp only [hf.eq_iff, Sigma.map, Sigma.ext_iff] at hxy", "annotated_tactic": ["simp only [hf.eq_iff, <a>Sigma.map</a>, <a>Sigma.ext_iff</a>] at hxy", [{"full_name": "Sigma.map", "def_path": "Mathlib/Data/Sigma/Basic.lean", "def_pos": [113, 5], "def_end_pos": [113, 8]}, {"full_name": "Sigma.ext_iff", "def_path": "Mathlib/Data/Sigma/Basic.lean", "def_pos": [72, 9], "def_end_pos": [72, 16]}]], "state_before": "case mk.intro.intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : \u03b9 \u2192 Type u_3\n\u03b2\u271d : \u03b9 \u2192 Type u_4\ns\u271d s\u2081 s\u2082 : Set \u03b9\nt t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\nu : Set ((i : \u03b9) \u00d7 \u03b1 i)\nx\u271d : (i : \u03b9) \u00d7 \u03b1 i\ni\u271d j\u271d : \u03b9\na : \u03b1 i\u271d\n\u03b2 : \u03b9' \u2192 Type u_5\nf : \u03b9 \u2192 \u03b9'\nhf : Function.Injective f\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b2 (f i)\ni : \u03b9\ns : Set (\u03b2 (f i))\nj : \u03b9\nx : \u03b1 j\ny : \u03b2 (f i)\nhys : y \u2208 s\nhxy : { fst := f i, snd := y } = Sigma.map f g { fst := j, snd := x }\n\u22a2 { fst := j, snd := x } \u2208 Sigma.mk i '' (g i \u207b\u00b9' s)", "state_after": "case mk.intro.intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : \u03b9 \u2192 Type u_3\n\u03b2\u271d : \u03b9 \u2192 Type u_4\ns\u271d s\u2081 s\u2082 : Set \u03b9\nt t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\nu : Set ((i : \u03b9) \u00d7 \u03b1 i)\nx\u271d : (i : \u03b9) \u00d7 \u03b1 i\ni\u271d j\u271d : \u03b9\na : \u03b1 i\u271d\n\u03b2 : \u03b9' \u2192 Type u_5\nf : \u03b9 \u2192 \u03b9'\nhf : Function.Injective f\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b2 (f i)\ni : \u03b9\ns : Set (\u03b2 (f i))\nj : \u03b9\nx : \u03b1 j\ny : \u03b2 (f i)\nhys : y \u2208 s\nhxy : i = j \u2227 HEq y (g j x)\n\u22a2 { fst := j, snd := x } \u2208 Sigma.mk i '' (g i \u207b\u00b9' s)"}, {"tactic": "rcases hxy with \u27e8rfl, hxy\u27e9", "annotated_tactic": ["rcases hxy with \u27e8rfl, hxy\u27e9", []], "state_before": "case mk.intro.intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : \u03b9 \u2192 Type u_3\n\u03b2\u271d : \u03b9 \u2192 Type u_4\ns\u271d s\u2081 s\u2082 : Set \u03b9\nt t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\nu : Set ((i : \u03b9) \u00d7 \u03b1 i)\nx\u271d : (i : \u03b9) \u00d7 \u03b1 i\ni\u271d j\u271d : \u03b9\na : \u03b1 i\u271d\n\u03b2 : \u03b9' \u2192 Type u_5\nf : \u03b9 \u2192 \u03b9'\nhf : Function.Injective f\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b2 (f i)\ni : \u03b9\ns : Set (\u03b2 (f i))\nj : \u03b9\nx : \u03b1 j\ny : \u03b2 (f i)\nhys : y \u2208 s\nhxy : i = j \u2227 HEq y (g j x)\n\u22a2 { fst := j, snd := x } \u2208 Sigma.mk i '' (g i \u207b\u00b9' s)", "state_after": "case mk.intro.intro.intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : \u03b9 \u2192 Type u_3\n\u03b2\u271d : \u03b9 \u2192 Type u_4\ns\u271d s\u2081 s\u2082 : Set \u03b9\nt t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\nu : Set ((i : \u03b9) \u00d7 \u03b1 i)\nx\u271d : (i : \u03b9) \u00d7 \u03b1 i\ni\u271d j : \u03b9\na : \u03b1 i\u271d\n\u03b2 : \u03b9' \u2192 Type u_5\nf : \u03b9 \u2192 \u03b9'\nhf : Function.Injective f\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b2 (f i)\ni : \u03b9\ns : Set (\u03b2 (f i))\ny : \u03b2 (f i)\nhys : y \u2208 s\nx : \u03b1 i\nhxy : HEq y (g i x)\n\u22a2 { fst := i, snd := x } \u2208 Sigma.mk i '' (g i \u207b\u00b9' s)"}, {"tactic": "rw [heq_iff_eq] at hxy", "annotated_tactic": ["rw [<a>heq_iff_eq</a>] at hxy", [{"full_name": "heq_iff_eq", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [703, 9], "def_end_pos": [703, 19]}]], "state_before": "case mk.intro.intro.intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : \u03b9 \u2192 Type u_3\n\u03b2\u271d : \u03b9 \u2192 Type u_4\ns\u271d s\u2081 s\u2082 : Set \u03b9\nt t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\nu : Set ((i : \u03b9) \u00d7 \u03b1 i)\nx\u271d : (i : \u03b9) \u00d7 \u03b1 i\ni\u271d j : \u03b9\na : \u03b1 i\u271d\n\u03b2 : \u03b9' \u2192 Type u_5\nf : \u03b9 \u2192 \u03b9'\nhf : Function.Injective f\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b2 (f i)\ni : \u03b9\ns : Set (\u03b2 (f i))\ny : \u03b2 (f i)\nhys : y \u2208 s\nx : \u03b1 i\nhxy : HEq y (g i x)\n\u22a2 { fst := i, snd := x } \u2208 Sigma.mk i '' (g i \u207b\u00b9' s)", "state_after": "case mk.intro.intro.intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : \u03b9 \u2192 Type u_3\n\u03b2\u271d : \u03b9 \u2192 Type u_4\ns\u271d s\u2081 s\u2082 : Set \u03b9\nt t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\nu : Set ((i : \u03b9) \u00d7 \u03b1 i)\nx\u271d : (i : \u03b9) \u00d7 \u03b1 i\ni\u271d j : \u03b9\na : \u03b1 i\u271d\n\u03b2 : \u03b9' \u2192 Type u_5\nf : \u03b9 \u2192 \u03b9'\nhf : Function.Injective f\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b2 (f i)\ni : \u03b9\ns : Set (\u03b2 (f i))\ny : \u03b2 (f i)\nhys : y \u2208 s\nx : \u03b1 i\nhxy : y = g i x\n\u22a2 { fst := i, snd := x } \u2208 Sigma.mk i '' (g i \u207b\u00b9' s)"}, {"tactic": "subst y", "annotated_tactic": ["subst y", []], "state_before": "case mk.intro.intro.intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : \u03b9 \u2192 Type u_3\n\u03b2\u271d : \u03b9 \u2192 Type u_4\ns\u271d s\u2081 s\u2082 : Set \u03b9\nt t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\nu : Set ((i : \u03b9) \u00d7 \u03b1 i)\nx\u271d : (i : \u03b9) \u00d7 \u03b1 i\ni\u271d j : \u03b9\na : \u03b1 i\u271d\n\u03b2 : \u03b9' \u2192 Type u_5\nf : \u03b9 \u2192 \u03b9'\nhf : Function.Injective f\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b2 (f i)\ni : \u03b9\ns : Set (\u03b2 (f i))\ny : \u03b2 (f i)\nhys : y \u2208 s\nx : \u03b1 i\nhxy : y = g i x\n\u22a2 { fst := i, snd := x } \u2208 Sigma.mk i '' (g i \u207b\u00b9' s)", "state_after": "case mk.intro.intro.intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : \u03b9 \u2192 Type u_3\n\u03b2\u271d : \u03b9 \u2192 Type u_4\ns\u271d s\u2081 s\u2082 : Set \u03b9\nt t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\nu : Set ((i : \u03b9) \u00d7 \u03b1 i)\nx\u271d : (i : \u03b9) \u00d7 \u03b1 i\ni\u271d j : \u03b9\na : \u03b1 i\u271d\n\u03b2 : \u03b9' \u2192 Type u_5\nf : \u03b9 \u2192 \u03b9'\nhf : Function.Injective f\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b2 (f i)\ni : \u03b9\ns : Set (\u03b2 (f i))\nx : \u03b1 i\nhys : g i x \u2208 s\n\u22a2 { fst := i, snd := x } \u2208 Sigma.mk i '' (g i \u207b\u00b9' s)"}, {"tactic": "exact \u27e8x, hys, rfl\u27e9", "annotated_tactic": ["exact \u27e8x, hys, <a>rfl</a>\u27e9", [{"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "case mk.intro.intro.intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : \u03b9 \u2192 Type u_3\n\u03b2\u271d : \u03b9 \u2192 Type u_4\ns\u271d s\u2081 s\u2082 : Set \u03b9\nt t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\nu : Set ((i : \u03b9) \u00d7 \u03b1 i)\nx\u271d : (i : \u03b9) \u00d7 \u03b1 i\ni\u271d j : \u03b9\na : \u03b1 i\u271d\n\u03b2 : \u03b9' \u2192 Type u_5\nf : \u03b9 \u2192 \u03b9'\nhf : Function.Injective f\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b2 (f i)\ni : \u03b9\ns : Set (\u03b2 (f i))\nx : \u03b1 i\nhys : g i x \u2208 s\n\u22a2 { fst := i, snd := x } \u2208 Sigma.mk i '' (g i \u207b\u00b9' s)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "full_name": "MeasureTheory.L1.setToL1_eq_setToL1'", "start": [1030, 1], "end": [1033, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Int/Lemmas.lean", "full_name": "Int.natAbs_le_iff_sq_le", "start": [56, 1], "end": [58, 34], "traced_tactics": [{"tactic": "rw [sq, sq]", "annotated_tactic": ["rw [<a>sq</a>, <a>sq</a>]", [{"full_name": "sq", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [106, 7], "def_end_pos": [106, 9]}, {"full_name": "sq", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [106, 7], "def_end_pos": [106, 9]}]], "state_before": "a\u271d b\u271d : \u2124\nn : \u2115\na b : \u2124\n\u22a2 natAbs a \u2264 natAbs b \u2194 a ^ 2 \u2264 b ^ 2", "state_after": "a\u271d b\u271d : \u2124\nn : \u2115\na b : \u2124\n\u22a2 natAbs a \u2264 natAbs b \u2194 a * a \u2264 b * b"}, {"tactic": "exact natAbs_le_iff_mul_self_le", "annotated_tactic": ["exact <a>natAbs_le_iff_mul_self_le</a>", [{"full_name": "Int.natAbs_le_iff_mul_self_le", "def_path": "Mathlib/Data/Int/Order/Lemmas.lean", "def_pos": [41, 9], "def_end_pos": [41, 34]}]], "state_before": "a\u271d b\u271d : \u2124\nn : \u2115\na b : \u2124\n\u22a2 natAbs a \u2264 natAbs b \u2194 a * a \u2264 b * b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean", "full_name": "MeasureTheory.integrable_condexp", "start": [210, 1], "end": [216, 75], "traced_tactics": [{"tactic": "by_cases hm : m \u2264 m0", "annotated_tactic": ["by_cases hm : m \u2264 m0", []], "state_before": "\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\n\u22a2 Integrable (\u03bc[f|m])", "state_after": "case pos\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nhm : m \u2264 m0\n\u22a2 Integrable (\u03bc[f|m])\n\ncase neg\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nhm : \u00acm \u2264 m0\n\u22a2 Integrable (\u03bc[f|m])"}, {"tactic": "swap", "annotated_tactic": ["swap", []], "state_before": "case pos\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nhm : m \u2264 m0\n\u22a2 Integrable (\u03bc[f|m])\n\ncase neg\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nhm : \u00acm \u2264 m0\n\u22a2 Integrable (\u03bc[f|m])", "state_after": "case neg\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nhm : \u00acm \u2264 m0\n\u22a2 Integrable (\u03bc[f|m])\n\ncase pos\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nhm : m \u2264 m0\n\u22a2 Integrable (\u03bc[f|m])"}, {"tactic": "by_cases h\u03bcm : SigmaFinite (\u03bc.trim hm)", "annotated_tactic": ["by_cases h\u03bcm : <a>SigmaFinite</a> (\u03bc.trim hm)", [{"full_name": "MeasureTheory.SigmaFinite", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3289, 7], "def_end_pos": [3289, 18]}]], "state_before": "case pos\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nhm : m \u2264 m0\n\u22a2 Integrable (\u03bc[f|m])", "state_after": "case pos\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nhm : m \u2264 m0\nh\u03bcm : SigmaFinite (Measure.trim \u03bc hm)\n\u22a2 Integrable (\u03bc[f|m])\n\ncase neg\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nhm : m \u2264 m0\nh\u03bcm : \u00acSigmaFinite (Measure.trim \u03bc hm)\n\u22a2 Integrable (\u03bc[f|m])"}, {"tactic": "swap", "annotated_tactic": ["swap", []], "state_before": "case pos\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nhm : m \u2264 m0\nh\u03bcm : SigmaFinite (Measure.trim \u03bc hm)\n\u22a2 Integrable (\u03bc[f|m])\n\ncase neg\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nhm : m \u2264 m0\nh\u03bcm : \u00acSigmaFinite (Measure.trim \u03bc hm)\n\u22a2 Integrable (\u03bc[f|m])", "state_after": "case neg\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nhm : m \u2264 m0\nh\u03bcm : \u00acSigmaFinite (Measure.trim \u03bc hm)\n\u22a2 Integrable (\u03bc[f|m])\n\ncase pos\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nhm : m \u2264 m0\nh\u03bcm : SigmaFinite (Measure.trim \u03bc hm)\n\u22a2 Integrable (\u03bc[f|m])"}, {"tactic": "haveI : SigmaFinite (\u03bc.trim hm) := h\u03bcm", "annotated_tactic": ["haveI : <a>SigmaFinite</a> (\u03bc.trim hm) := h\u03bcm", [{"full_name": "MeasureTheory.SigmaFinite", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3289, 7], "def_end_pos": [3289, 18]}]], "state_before": "case pos\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nhm : m \u2264 m0\nh\u03bcm : SigmaFinite (Measure.trim \u03bc hm)\n\u22a2 Integrable (\u03bc[f|m])", "state_after": "case pos\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nhm : m \u2264 m0\nh\u03bcm this : SigmaFinite (Measure.trim \u03bc hm)\n\u22a2 Integrable (\u03bc[f|m])"}, {"tactic": "exact (integrable_condexpL1 f).congr (condexp_ae_eq_condexpL1 hm f).symm", "annotated_tactic": ["exact (<a>integrable_condexpL1</a> f).<a>congr</a> (<a>condexp_ae_eq_condexpL1</a> hm f).<a>symm</a>", [{"full_name": "MeasureTheory.integrable_condexpL1", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean", "def_pos": [551, 9], "def_end_pos": [551, 29]}, {"full_name": "MeasureTheory.Integrable.congr", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [492, 9], "def_end_pos": [492, 25]}, {"full_name": "MeasureTheory.condexp_ae_eq_condexpL1", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean", "def_pos": [136, 9], "def_end_pos": [136, 32]}, {"full_name": "Filter.EventuallyEq.symm", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1498, 9], "def_end_pos": [1498, 26]}]], "state_before": "case pos\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nhm : m \u2264 m0\nh\u03bcm this : SigmaFinite (Measure.trim \u03bc hm)\n\u22a2 Integrable (\u03bc[f|m])", "state_after": "no goals"}, {"tactic": "rw [condexp_of_not_le hm]", "annotated_tactic": ["rw [<a>condexp_of_not_le</a> hm]", [{"full_name": "MeasureTheory.condexp_of_not_le", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean", "def_pos": [106, 9], "def_end_pos": [106, 26]}]], "state_before": "case neg\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nhm : \u00acm \u2264 m0\n\u22a2 Integrable (\u03bc[f|m])", "state_after": "case neg\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nhm : \u00acm \u2264 m0\n\u22a2 Integrable 0"}, {"tactic": "exact integrable_zero _ _ _", "annotated_tactic": ["exact <a>integrable_zero</a> _ _ _", [{"full_name": "MeasureTheory.integrable_zero", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [662, 9], "def_end_pos": [662, 24]}]], "state_before": "case neg\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nhm : \u00acm \u2264 m0\n\u22a2 Integrable 0", "state_after": "no goals"}, {"tactic": "rw [condexp_of_not_sigmaFinite hm h\u03bcm]", "annotated_tactic": ["rw [<a>condexp_of_not_sigmaFinite</a> hm h\u03bcm]", [{"full_name": "MeasureTheory.condexp_of_not_sigmaFinite", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean", "def_pos": [109, 9], "def_end_pos": [109, 35]}]], "state_before": "case neg\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nhm : m \u2264 m0\nh\u03bcm : \u00acSigmaFinite (Measure.trim \u03bc hm)\n\u22a2 Integrable (\u03bc[f|m])", "state_after": "case neg\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nhm : m \u2264 m0\nh\u03bcm : \u00acSigmaFinite (Measure.trim \u03bc hm)\n\u22a2 Integrable 0"}, {"tactic": "exact integrable_zero _ _ _", "annotated_tactic": ["exact <a>integrable_zero</a> _ _ _", [{"full_name": "MeasureTheory.integrable_zero", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [662, 9], "def_end_pos": [662, 24]}]], "state_before": "case neg\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nhm : m \u2264 m0\nh\u03bcm : \u00acSigmaFinite (Measure.trim \u03bc hm)\n\u22a2 Integrable 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Pointwise.lean", "full_name": "Finset.subset_smul_finset_iff\u2080", "start": [2084, 1], "end": [2085, 62], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Martingale/OptionalSampling.lean", "full_name": "MeasureTheory.Martingale.condexp_stoppedValue_stopping_time_ae_eq_restrict_le", "start": [163, 1], "end": [189, 76], "traced_tactics": [{"tactic": "rw [ae_eq_restrict_iff_indicator_ae_eq\n  (h\u03c4.measurableSpace_le _ (h\u03c4.measurableSet_le_stopping_time h\u03c3))]", "annotated_tactic": ["rw [<a>ae_eq_restrict_iff_indicator_ae_eq</a>\n    (h\u03c4.measurableSpace_le _ (h\u03c4.measurableSet_le_stopping_time h\u03c3))]", [{"full_name": "ae_eq_restrict_iff_indicator_ae_eq", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [4520, 9], "def_end_pos": [4520, 43]}]], "state_before": "\u03a9 : Type u_1\nE : Type u_2\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b9 : CompleteSpace E\n\u03b9 : Type u_3\ninst\u271d\u00b9\u2070 : LinearOrder \u03b9\ninst\u271d\u2079 : LocallyFiniteOrder \u03b9\ninst\u271d\u2078 : OrderBot \u03b9\ninst\u271d\u2077 : TopologicalSpace \u03b9\ninst\u271d\u2076 : DiscreteTopology \u03b9\ninst\u271d\u2075 : MeasurableSpace \u03b9\ninst\u271d\u2074 : BorelSpace \u03b9\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : SecondCountableTopology E\n\u2131 : Filtration \u03b9 m\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nf : \u03b9 \u2192 \u03a9 \u2192 E\ni n : \u03b9\nh : Martingale f \u2131 \u03bc\nh\u03c4 : IsStoppingTime \u2131 \u03c4\nh\u03c3 : IsStoppingTime \u2131 \u03c3\ninst\u271d : SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace h\u03c3 \u2264 m))\nh\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n\n\u22a2 \u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace h\u03c3] =\u1d50[Measure.restrict \u03bc {x | \u03c4 x \u2264 \u03c3 x}] stoppedValue f \u03c4", "state_after": "\u03a9 : Type u_1\nE : Type u_2\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b9 : CompleteSpace E\n\u03b9 : Type u_3\ninst\u271d\u00b9\u2070 : LinearOrder \u03b9\ninst\u271d\u2079 : LocallyFiniteOrder \u03b9\ninst\u271d\u2078 : OrderBot \u03b9\ninst\u271d\u2077 : TopologicalSpace \u03b9\ninst\u271d\u2076 : DiscreteTopology \u03b9\ninst\u271d\u2075 : MeasurableSpace \u03b9\ninst\u271d\u2074 : BorelSpace \u03b9\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : SecondCountableTopology E\n\u2131 : Filtration \u03b9 m\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nf : \u03b9 \u2192 \u03a9 \u2192 E\ni n : \u03b9\nh : Martingale f \u2131 \u03bc\nh\u03c4 : IsStoppingTime \u2131 \u03c4\nh\u03c3 : IsStoppingTime \u2131 \u03c3\ninst\u271d : SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace h\u03c3 \u2264 m))\nh\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n\n\u22a2 Set.indicator {\u03c9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9} (\u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace h\u03c3]) =\u1d50[\u03bc]\n    Set.indicator {\u03c9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9} (stoppedValue f \u03c4)"}, {"tactic": "refine' (condexp_indicator (integrable_stoppedValue \u03b9 h\u03c4 h.integrable h\u03c4_le)\n  (h\u03c4.measurableSet_stopping_time_le h\u03c3)).symm.trans _", "annotated_tactic": ["refine' (<a>condexp_indicator</a> (<a>integrable_stoppedValue</a> \u03b9 h\u03c4 h.integrable h\u03c4_le)\n    (h\u03c4.measurableSet_stopping_time_le h\u03c3)).symm.trans _", [{"full_name": "MeasureTheory.condexp_indicator", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Indicator.lean", "def_pos": [75, 9], "def_end_pos": [75, 26]}, {"full_name": "MeasureTheory.integrable_stoppedValue", "def_path": "Mathlib/Probability/Process/Stopping.lean", "def_pos": [975, 9], "def_end_pos": [975, 32]}]], "state_before": "\u03a9 : Type u_1\nE : Type u_2\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b9 : CompleteSpace E\n\u03b9 : Type u_3\ninst\u271d\u00b9\u2070 : LinearOrder \u03b9\ninst\u271d\u2079 : LocallyFiniteOrder \u03b9\ninst\u271d\u2078 : OrderBot \u03b9\ninst\u271d\u2077 : TopologicalSpace \u03b9\ninst\u271d\u2076 : DiscreteTopology \u03b9\ninst\u271d\u2075 : MeasurableSpace \u03b9\ninst\u271d\u2074 : BorelSpace \u03b9\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : SecondCountableTopology E\n\u2131 : Filtration \u03b9 m\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nf : \u03b9 \u2192 \u03a9 \u2192 E\ni n : \u03b9\nh : Martingale f \u2131 \u03bc\nh\u03c4 : IsStoppingTime \u2131 \u03c4\nh\u03c3 : IsStoppingTime \u2131 \u03c3\ninst\u271d : SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace h\u03c3 \u2264 m))\nh\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n\n\u22a2 Set.indicator {\u03c9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9} (\u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace h\u03c3]) =\u1d50[\u03bc]\n    Set.indicator {\u03c9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9} (stoppedValue f \u03c4)", "state_after": "\u03a9 : Type u_1\nE : Type u_2\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b9 : CompleteSpace E\n\u03b9 : Type u_3\ninst\u271d\u00b9\u2070 : LinearOrder \u03b9\ninst\u271d\u2079 : LocallyFiniteOrder \u03b9\ninst\u271d\u2078 : OrderBot \u03b9\ninst\u271d\u2077 : TopologicalSpace \u03b9\ninst\u271d\u2076 : DiscreteTopology \u03b9\ninst\u271d\u2075 : MeasurableSpace \u03b9\ninst\u271d\u2074 : BorelSpace \u03b9\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : SecondCountableTopology E\n\u2131 : Filtration \u03b9 m\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nf : \u03b9 \u2192 \u03a9 \u2192 E\ni n : \u03b9\nh : Martingale f \u2131 \u03bc\nh\u03c4 : IsStoppingTime \u2131 \u03c4\nh\u03c3 : IsStoppingTime \u2131 \u03c3\ninst\u271d : SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace h\u03c3 \u2264 m))\nh\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n\n\u22a2 \u03bc[Set.indicator {\u03c9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9} (stoppedValue (fun n => f n) \u03c4)|IsStoppingTime.measurableSpace h\u03c3] =\u1d50[\u03bc]\n    Set.indicator {\u03c9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9} (stoppedValue f \u03c4)"}, {"tactic": "have h_int :\n    Integrable ({\u03c9 : \u03a9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9}.indicator (stoppedValue (fun n : \u03b9 => f n) \u03c4)) \u03bc := by\n  refine' (integrable_stoppedValue \u03b9 h\u03c4 h.integrable h\u03c4_le).indicator _\n  exact h\u03c4.measurableSpace_le _ (h\u03c4.measurableSet_le_stopping_time h\u03c3)", "annotated_tactic": ["have h_int :\n      <a>Integrable</a> ({\u03c9 : \u03a9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9}.<a>indicator</a> (<a>stoppedValue</a> (fun n : \u03b9 => f n) \u03c4)) \u03bc := by\n    refine' (<a>integrable_stoppedValue</a> \u03b9 h\u03c4 h.integrable h\u03c4_le).<a>indicator</a> _\n    exact h\u03c4.measurableSpace_le _ (h\u03c4.measurableSet_le_stopping_time h\u03c3)", [{"full_name": "MeasureTheory.Integrable", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [442, 5], "def_end_pos": [442, 15]}, {"full_name": "Set.indicator", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [46, 3], "def_end_pos": [46, 14]}, {"full_name": "MeasureTheory.stoppedValue", "def_path": "Mathlib/Probability/Process/Stopping.lean", "def_pos": [768, 5], "def_end_pos": [768, 17]}, {"full_name": "MeasureTheory.integrable_stoppedValue", "def_path": "Mathlib/Probability/Process/Stopping.lean", "def_pos": [975, 9], "def_end_pos": [975, 32]}, {"full_name": "MeasureTheory.Integrable.indicator", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [268, 9], "def_end_pos": [268, 29]}]], "state_before": "\u03a9 : Type u_1\nE : Type u_2\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b9 : CompleteSpace E\n\u03b9 : Type u_3\ninst\u271d\u00b9\u2070 : LinearOrder \u03b9\ninst\u271d\u2079 : LocallyFiniteOrder \u03b9\ninst\u271d\u2078 : OrderBot \u03b9\ninst\u271d\u2077 : TopologicalSpace \u03b9\ninst\u271d\u2076 : DiscreteTopology \u03b9\ninst\u271d\u2075 : MeasurableSpace \u03b9\ninst\u271d\u2074 : BorelSpace \u03b9\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : SecondCountableTopology E\n\u2131 : Filtration \u03b9 m\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nf : \u03b9 \u2192 \u03a9 \u2192 E\ni n : \u03b9\nh : Martingale f \u2131 \u03bc\nh\u03c4 : IsStoppingTime \u2131 \u03c4\nh\u03c3 : IsStoppingTime \u2131 \u03c3\ninst\u271d : SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace h\u03c3 \u2264 m))\nh\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n\n\u22a2 \u03bc[Set.indicator {\u03c9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9} (stoppedValue (fun n => f n) \u03c4)|IsStoppingTime.measurableSpace h\u03c3] =\u1d50[\u03bc]\n    Set.indicator {\u03c9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9} (stoppedValue f \u03c4)", "state_after": "\u03a9 : Type u_1\nE : Type u_2\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b9 : CompleteSpace E\n\u03b9 : Type u_3\ninst\u271d\u00b9\u2070 : LinearOrder \u03b9\ninst\u271d\u2079 : LocallyFiniteOrder \u03b9\ninst\u271d\u2078 : OrderBot \u03b9\ninst\u271d\u2077 : TopologicalSpace \u03b9\ninst\u271d\u2076 : DiscreteTopology \u03b9\ninst\u271d\u2075 : MeasurableSpace \u03b9\ninst\u271d\u2074 : BorelSpace \u03b9\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : SecondCountableTopology E\n\u2131 : Filtration \u03b9 m\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nf : \u03b9 \u2192 \u03a9 \u2192 E\ni n : \u03b9\nh : Martingale f \u2131 \u03bc\nh\u03c4 : IsStoppingTime \u2131 \u03c4\nh\u03c3 : IsStoppingTime \u2131 \u03c3\ninst\u271d : SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace h\u03c3 \u2264 m))\nh\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n\nh_int : Integrable (Set.indicator {\u03c9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9} (stoppedValue (fun n => f n) \u03c4))\n\u22a2 \u03bc[Set.indicator {\u03c9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9} (stoppedValue (fun n => f n) \u03c4)|IsStoppingTime.measurableSpace h\u03c3] =\u1d50[\u03bc]\n    Set.indicator {\u03c9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9} (stoppedValue f \u03c4)"}, {"tactic": "exact condexp_of_aestronglyMeasurable' h\u03c3.measurableSpace_le h_meas h_int", "annotated_tactic": ["exact <a>condexp_of_aestronglyMeasurable'</a> h\u03c3.measurableSpace_le h_meas h_int", [{"full_name": "MeasureTheory.condexp_of_aestronglyMeasurable'", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean", "def_pos": [203, 9], "def_end_pos": [203, 41]}]], "state_before": "\u03a9 : Type u_1\nE : Type u_2\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b9 : CompleteSpace E\n\u03b9 : Type u_3\ninst\u271d\u00b9\u2070 : LinearOrder \u03b9\ninst\u271d\u2079 : LocallyFiniteOrder \u03b9\ninst\u271d\u2078 : OrderBot \u03b9\ninst\u271d\u2077 : TopologicalSpace \u03b9\ninst\u271d\u2076 : DiscreteTopology \u03b9\ninst\u271d\u2075 : MeasurableSpace \u03b9\ninst\u271d\u2074 : BorelSpace \u03b9\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : SecondCountableTopology E\n\u2131 : Filtration \u03b9 m\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nf : \u03b9 \u2192 \u03a9 \u2192 E\ni n : \u03b9\nh : Martingale f \u2131 \u03bc\nh\u03c4 : IsStoppingTime \u2131 \u03c4\nh\u03c3 : IsStoppingTime \u2131 \u03c3\ninst\u271d : SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace h\u03c3 \u2264 m))\nh\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n\nh_int : Integrable (Set.indicator {\u03c9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9} (stoppedValue (fun n => f n) \u03c4))\nh_meas :\n  AEStronglyMeasurable' (IsStoppingTime.measurableSpace h\u03c3)\n    (Set.indicator {\u03c9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9} (stoppedValue (fun n => f n) \u03c4)) \u03bc\n\u22a2 \u03bc[Set.indicator {\u03c9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9} (stoppedValue (fun n => f n) \u03c4)|IsStoppingTime.measurableSpace h\u03c3] =\u1d50[\u03bc]\n    Set.indicator {\u03c9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9} (stoppedValue f \u03c4)", "state_after": "no goals"}, {"tactic": "refine' (integrable_stoppedValue \u03b9 h\u03c4 h.integrable h\u03c4_le).indicator _", "annotated_tactic": ["refine' (<a>integrable_stoppedValue</a> \u03b9 h\u03c4 h.integrable h\u03c4_le).<a>indicator</a> _", [{"full_name": "MeasureTheory.integrable_stoppedValue", "def_path": "Mathlib/Probability/Process/Stopping.lean", "def_pos": [975, 9], "def_end_pos": [975, 32]}, {"full_name": "MeasureTheory.Integrable.indicator", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [268, 9], "def_end_pos": [268, 29]}]], "state_before": "\u03a9 : Type u_1\nE : Type u_2\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b9 : CompleteSpace E\n\u03b9 : Type u_3\ninst\u271d\u00b9\u2070 : LinearOrder \u03b9\ninst\u271d\u2079 : LocallyFiniteOrder \u03b9\ninst\u271d\u2078 : OrderBot \u03b9\ninst\u271d\u2077 : TopologicalSpace \u03b9\ninst\u271d\u2076 : DiscreteTopology \u03b9\ninst\u271d\u2075 : MeasurableSpace \u03b9\ninst\u271d\u2074 : BorelSpace \u03b9\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : SecondCountableTopology E\n\u2131 : Filtration \u03b9 m\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nf : \u03b9 \u2192 \u03a9 \u2192 E\ni n : \u03b9\nh : Martingale f \u2131 \u03bc\nh\u03c4 : IsStoppingTime \u2131 \u03c4\nh\u03c3 : IsStoppingTime \u2131 \u03c3\ninst\u271d : SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace h\u03c3 \u2264 m))\nh\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n\n\u22a2 Integrable (Set.indicator {\u03c9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9} (stoppedValue (fun n => f n) \u03c4))", "state_after": "\u03a9 : Type u_1\nE : Type u_2\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b9 : CompleteSpace E\n\u03b9 : Type u_3\ninst\u271d\u00b9\u2070 : LinearOrder \u03b9\ninst\u271d\u2079 : LocallyFiniteOrder \u03b9\ninst\u271d\u2078 : OrderBot \u03b9\ninst\u271d\u2077 : TopologicalSpace \u03b9\ninst\u271d\u2076 : DiscreteTopology \u03b9\ninst\u271d\u2075 : MeasurableSpace \u03b9\ninst\u271d\u2074 : BorelSpace \u03b9\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : SecondCountableTopology E\n\u2131 : Filtration \u03b9 m\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nf : \u03b9 \u2192 \u03a9 \u2192 E\ni n : \u03b9\nh : Martingale f \u2131 \u03bc\nh\u03c4 : IsStoppingTime \u2131 \u03c4\nh\u03c3 : IsStoppingTime \u2131 \u03c3\ninst\u271d : SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace h\u03c3 \u2264 m))\nh\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n\n\u22a2 MeasurableSet {\u03c9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9}"}, {"tactic": "exact h\u03c4.measurableSpace_le _ (h\u03c4.measurableSet_le_stopping_time h\u03c3)", "annotated_tactic": ["exact h\u03c4.measurableSpace_le _ (h\u03c4.measurableSet_le_stopping_time h\u03c3)", []], "state_before": "\u03a9 : Type u_1\nE : Type u_2\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b9 : CompleteSpace E\n\u03b9 : Type u_3\ninst\u271d\u00b9\u2070 : LinearOrder \u03b9\ninst\u271d\u2079 : LocallyFiniteOrder \u03b9\ninst\u271d\u2078 : OrderBot \u03b9\ninst\u271d\u2077 : TopologicalSpace \u03b9\ninst\u271d\u2076 : DiscreteTopology \u03b9\ninst\u271d\u2075 : MeasurableSpace \u03b9\ninst\u271d\u2074 : BorelSpace \u03b9\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : SecondCountableTopology E\n\u2131 : Filtration \u03b9 m\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nf : \u03b9 \u2192 \u03a9 \u2192 E\ni n : \u03b9\nh : Martingale f \u2131 \u03bc\nh\u03c4 : IsStoppingTime \u2131 \u03c4\nh\u03c3 : IsStoppingTime \u2131 \u03c3\ninst\u271d : SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace h\u03c3 \u2264 m))\nh\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n\n\u22a2 MeasurableSet {\u03c9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9}", "state_after": "no goals"}, {"tactic": "refine' StronglyMeasurable.aeStronglyMeasurable' _", "annotated_tactic": ["refine' <a>StronglyMeasurable.aeStronglyMeasurable'</a> _", [{"full_name": "MeasureTheory.StronglyMeasurable.aeStronglyMeasurable'", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean", "def_pos": [138, 9], "def_end_pos": [138, 49]}]], "state_before": "\u03a9 : Type u_1\nE : Type u_2\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b9 : CompleteSpace E\n\u03b9 : Type u_3\ninst\u271d\u00b9\u2070 : LinearOrder \u03b9\ninst\u271d\u2079 : LocallyFiniteOrder \u03b9\ninst\u271d\u2078 : OrderBot \u03b9\ninst\u271d\u2077 : TopologicalSpace \u03b9\ninst\u271d\u2076 : DiscreteTopology \u03b9\ninst\u271d\u2075 : MeasurableSpace \u03b9\ninst\u271d\u2074 : BorelSpace \u03b9\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : SecondCountableTopology E\n\u2131 : Filtration \u03b9 m\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nf : \u03b9 \u2192 \u03a9 \u2192 E\ni n : \u03b9\nh : Martingale f \u2131 \u03bc\nh\u03c4 : IsStoppingTime \u2131 \u03c4\nh\u03c3 : IsStoppingTime \u2131 \u03c3\ninst\u271d : SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace h\u03c3 \u2264 m))\nh\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n\nh_int : Integrable (Set.indicator {\u03c9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9} (stoppedValue (fun n => f n) \u03c4))\n\u22a2 AEStronglyMeasurable' (IsStoppingTime.measurableSpace h\u03c3)\n    (Set.indicator {\u03c9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9} (stoppedValue (fun n => f n) \u03c4)) \u03bc", "state_after": "\u03a9 : Type u_1\nE : Type u_2\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b9 : CompleteSpace E\n\u03b9 : Type u_3\ninst\u271d\u00b9\u2070 : LinearOrder \u03b9\ninst\u271d\u2079 : LocallyFiniteOrder \u03b9\ninst\u271d\u2078 : OrderBot \u03b9\ninst\u271d\u2077 : TopologicalSpace \u03b9\ninst\u271d\u2076 : DiscreteTopology \u03b9\ninst\u271d\u2075 : MeasurableSpace \u03b9\ninst\u271d\u2074 : BorelSpace \u03b9\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : SecondCountableTopology E\n\u2131 : Filtration \u03b9 m\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nf : \u03b9 \u2192 \u03a9 \u2192 E\ni n : \u03b9\nh : Martingale f \u2131 \u03bc\nh\u03c4 : IsStoppingTime \u2131 \u03c4\nh\u03c3 : IsStoppingTime \u2131 \u03c3\ninst\u271d : SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace h\u03c3 \u2264 m))\nh\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n\nh_int : Integrable (Set.indicator {\u03c9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9} (stoppedValue (fun n => f n) \u03c4))\n\u22a2 StronglyMeasurable (Set.indicator {\u03c9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9} (stoppedValue (fun n => f n) \u03c4))"}, {"tactic": "refine' StronglyMeasurable.stronglyMeasurable_of_measurableSpace_le_on\n  (h\u03c4.measurableSet_le_stopping_time h\u03c3) _ _ _", "annotated_tactic": ["refine' <a>StronglyMeasurable.stronglyMeasurable_of_measurableSpace_le_on</a>\n      (h\u03c4.measurableSet_le_stopping_time h\u03c3) _ _ _", [{"full_name": "MeasureTheory.StronglyMeasurable.stronglyMeasurable_of_measurableSpace_le_on", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [906, 9], "def_end_pos": [906, 52]}]], "state_before": "\u03a9 : Type u_1\nE : Type u_2\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b9 : CompleteSpace E\n\u03b9 : Type u_3\ninst\u271d\u00b9\u2070 : LinearOrder \u03b9\ninst\u271d\u2079 : LocallyFiniteOrder \u03b9\ninst\u271d\u2078 : OrderBot \u03b9\ninst\u271d\u2077 : TopologicalSpace \u03b9\ninst\u271d\u2076 : DiscreteTopology \u03b9\ninst\u271d\u2075 : MeasurableSpace \u03b9\ninst\u271d\u2074 : BorelSpace \u03b9\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : SecondCountableTopology E\n\u2131 : Filtration \u03b9 m\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nf : \u03b9 \u2192 \u03a9 \u2192 E\ni n : \u03b9\nh : Martingale f \u2131 \u03bc\nh\u03c4 : IsStoppingTime \u2131 \u03c4\nh\u03c3 : IsStoppingTime \u2131 \u03c3\ninst\u271d : SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace h\u03c3 \u2264 m))\nh\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n\nh_int : Integrable (Set.indicator {\u03c9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9} (stoppedValue (fun n => f n) \u03c4))\n\u22a2 StronglyMeasurable (Set.indicator {\u03c9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9} (stoppedValue (fun n => f n) \u03c4))", "state_after": "case refine'_1\n\u03a9 : Type u_1\nE : Type u_2\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b9 : CompleteSpace E\n\u03b9 : Type u_3\ninst\u271d\u00b9\u2070 : LinearOrder \u03b9\ninst\u271d\u2079 : LocallyFiniteOrder \u03b9\ninst\u271d\u2078 : OrderBot \u03b9\ninst\u271d\u2077 : TopologicalSpace \u03b9\ninst\u271d\u2076 : DiscreteTopology \u03b9\ninst\u271d\u2075 : MeasurableSpace \u03b9\ninst\u271d\u2074 : BorelSpace \u03b9\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : SecondCountableTopology E\n\u2131 : Filtration \u03b9 m\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nf : \u03b9 \u2192 \u03a9 \u2192 E\ni n : \u03b9\nh : Martingale f \u2131 \u03bc\nh\u03c4 : IsStoppingTime \u2131 \u03c4\nh\u03c3 : IsStoppingTime \u2131 \u03c3\ninst\u271d : SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace h\u03c3 \u2264 m))\nh\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n\nh_int : Integrable (Set.indicator {\u03c9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9} (stoppedValue (fun n => f n) \u03c4))\n\u22a2 \u2200 (t : Set \u03a9), MeasurableSet ({\u03c9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9} \u2229 t) \u2192 MeasurableSet ({\u03c9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9} \u2229 t)\n\ncase refine'_2\n\u03a9 : Type u_1\nE : Type u_2\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b9 : CompleteSpace E\n\u03b9 : Type u_3\ninst\u271d\u00b9\u2070 : LinearOrder \u03b9\ninst\u271d\u2079 : LocallyFiniteOrder \u03b9\ninst\u271d\u2078 : OrderBot \u03b9\ninst\u271d\u2077 : TopologicalSpace \u03b9\ninst\u271d\u2076 : DiscreteTopology \u03b9\ninst\u271d\u2075 : MeasurableSpace \u03b9\ninst\u271d\u2074 : BorelSpace \u03b9\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : SecondCountableTopology E\n\u2131 : Filtration \u03b9 m\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nf : \u03b9 \u2192 \u03a9 \u2192 E\ni n : \u03b9\nh : Martingale f \u2131 \u03bc\nh\u03c4 : IsStoppingTime \u2131 \u03c4\nh\u03c3 : IsStoppingTime \u2131 \u03c3\ninst\u271d : SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace h\u03c3 \u2264 m))\nh\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n\nh_int : Integrable (Set.indicator {\u03c9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9} (stoppedValue (fun n => f n) \u03c4))\n\u22a2 StronglyMeasurable (Set.indicator {\u03c9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9} (stoppedValue (fun n => f n) \u03c4))\n\ncase refine'_3\n\u03a9 : Type u_1\nE : Type u_2\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b9 : CompleteSpace E\n\u03b9 : Type u_3\ninst\u271d\u00b9\u2070 : LinearOrder \u03b9\ninst\u271d\u2079 : LocallyFiniteOrder \u03b9\ninst\u271d\u2078 : OrderBot \u03b9\ninst\u271d\u2077 : TopologicalSpace \u03b9\ninst\u271d\u2076 : DiscreteTopology \u03b9\ninst\u271d\u2075 : MeasurableSpace \u03b9\ninst\u271d\u2074 : BorelSpace \u03b9\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : SecondCountableTopology E\n\u2131 : Filtration \u03b9 m\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nf : \u03b9 \u2192 \u03a9 \u2192 E\ni n : \u03b9\nh : Martingale f \u2131 \u03bc\nh\u03c4 : IsStoppingTime \u2131 \u03c4\nh\u03c3 : IsStoppingTime \u2131 \u03c3\ninst\u271d : SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace h\u03c3 \u2264 m))\nh\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n\nh_int : Integrable (Set.indicator {\u03c9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9} (stoppedValue (fun n => f n) \u03c4))\n\u22a2 \u2200 (x : \u03a9), \u00acx \u2208 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9} \u2192 Set.indicator {\u03c9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9} (stoppedValue (fun n => f n) \u03c4) x = 0"}, {"tactic": "intro t ht", "annotated_tactic": ["intro t ht", []], "state_before": "case refine'_1\n\u03a9 : Type u_1\nE : Type u_2\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b9 : CompleteSpace E\n\u03b9 : Type u_3\ninst\u271d\u00b9\u2070 : LinearOrder \u03b9\ninst\u271d\u2079 : LocallyFiniteOrder \u03b9\ninst\u271d\u2078 : OrderBot \u03b9\ninst\u271d\u2077 : TopologicalSpace \u03b9\ninst\u271d\u2076 : DiscreteTopology \u03b9\ninst\u271d\u2075 : MeasurableSpace \u03b9\ninst\u271d\u2074 : BorelSpace \u03b9\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : SecondCountableTopology E\n\u2131 : Filtration \u03b9 m\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nf : \u03b9 \u2192 \u03a9 \u2192 E\ni n : \u03b9\nh : Martingale f \u2131 \u03bc\nh\u03c4 : IsStoppingTime \u2131 \u03c4\nh\u03c3 : IsStoppingTime \u2131 \u03c3\ninst\u271d : SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace h\u03c3 \u2264 m))\nh\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n\nh_int : Integrable (Set.indicator {\u03c9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9} (stoppedValue (fun n => f n) \u03c4))\n\u22a2 \u2200 (t : Set \u03a9), MeasurableSet ({\u03c9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9} \u2229 t) \u2192 MeasurableSet ({\u03c9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9} \u2229 t)", "state_after": "case refine'_1\n\u03a9 : Type u_1\nE : Type u_2\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b9 : CompleteSpace E\n\u03b9 : Type u_3\ninst\u271d\u00b9\u2070 : LinearOrder \u03b9\ninst\u271d\u2079 : LocallyFiniteOrder \u03b9\ninst\u271d\u2078 : OrderBot \u03b9\ninst\u271d\u2077 : TopologicalSpace \u03b9\ninst\u271d\u2076 : DiscreteTopology \u03b9\ninst\u271d\u2075 : MeasurableSpace \u03b9\ninst\u271d\u2074 : BorelSpace \u03b9\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : SecondCountableTopology E\n\u2131 : Filtration \u03b9 m\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nf : \u03b9 \u2192 \u03a9 \u2192 E\ni n : \u03b9\nh : Martingale f \u2131 \u03bc\nh\u03c4 : IsStoppingTime \u2131 \u03c4\nh\u03c3 : IsStoppingTime \u2131 \u03c3\ninst\u271d : SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace h\u03c3 \u2264 m))\nh\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n\nh_int : Integrable (Set.indicator {\u03c9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9} (stoppedValue (fun n => f n) \u03c4))\nt : Set \u03a9\nht : MeasurableSet ({\u03c9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9} \u2229 t)\n\u22a2 MeasurableSet ({\u03c9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9} \u2229 t)"}, {"tactic": "rw [Set.inter_comm _ t] at ht \u22a2", "annotated_tactic": ["rw [<a>Set.inter_comm</a> _ t] at ht \u22a2", [{"full_name": "Set.inter_comm", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [940, 9], "def_end_pos": [940, 19]}]], "state_before": "case refine'_1\n\u03a9 : Type u_1\nE : Type u_2\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b9 : CompleteSpace E\n\u03b9 : Type u_3\ninst\u271d\u00b9\u2070 : LinearOrder \u03b9\ninst\u271d\u2079 : LocallyFiniteOrder \u03b9\ninst\u271d\u2078 : OrderBot \u03b9\ninst\u271d\u2077 : TopologicalSpace \u03b9\ninst\u271d\u2076 : DiscreteTopology \u03b9\ninst\u271d\u2075 : MeasurableSpace \u03b9\ninst\u271d\u2074 : BorelSpace \u03b9\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : SecondCountableTopology E\n\u2131 : Filtration \u03b9 m\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nf : \u03b9 \u2192 \u03a9 \u2192 E\ni n : \u03b9\nh : Martingale f \u2131 \u03bc\nh\u03c4 : IsStoppingTime \u2131 \u03c4\nh\u03c3 : IsStoppingTime \u2131 \u03c3\ninst\u271d : SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace h\u03c3 \u2264 m))\nh\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n\nh_int : Integrable (Set.indicator {\u03c9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9} (stoppedValue (fun n => f n) \u03c4))\nt : Set \u03a9\nht : MeasurableSet ({\u03c9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9} \u2229 t)\n\u22a2 MeasurableSet ({\u03c9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9} \u2229 t)", "state_after": "case refine'_1\n\u03a9 : Type u_1\nE : Type u_2\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b9 : CompleteSpace E\n\u03b9 : Type u_3\ninst\u271d\u00b9\u2070 : LinearOrder \u03b9\ninst\u271d\u2079 : LocallyFiniteOrder \u03b9\ninst\u271d\u2078 : OrderBot \u03b9\ninst\u271d\u2077 : TopologicalSpace \u03b9\ninst\u271d\u2076 : DiscreteTopology \u03b9\ninst\u271d\u2075 : MeasurableSpace \u03b9\ninst\u271d\u2074 : BorelSpace \u03b9\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : SecondCountableTopology E\n\u2131 : Filtration \u03b9 m\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nf : \u03b9 \u2192 \u03a9 \u2192 E\ni n : \u03b9\nh : Martingale f \u2131 \u03bc\nh\u03c4 : IsStoppingTime \u2131 \u03c4\nh\u03c3 : IsStoppingTime \u2131 \u03c3\ninst\u271d : SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace h\u03c3 \u2264 m))\nh\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n\nh_int : Integrable (Set.indicator {\u03c9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9} (stoppedValue (fun n => f n) \u03c4))\nt : Set \u03a9\nht : MeasurableSet (t \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9})\n\u22a2 MeasurableSet (t \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9})"}, {"tactic": "rw [h\u03c4.measurableSet_inter_le_iff h\u03c3, IsStoppingTime.measurableSet_min_iff h\u03c4 h\u03c3] at ht", "annotated_tactic": ["rw [h\u03c4.measurableSet_inter_le_iff h\u03c3, <a>IsStoppingTime.measurableSet_min_iff</a> h\u03c4 h\u03c3] at ht", [{"full_name": "MeasureTheory.IsStoppingTime.measurableSet_min_iff", "def_path": "Mathlib/Probability/Process/Stopping.lean", "def_pos": [606, 9], "def_end_pos": [606, 30]}]], "state_before": "case refine'_1\n\u03a9 : Type u_1\nE : Type u_2\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b9 : CompleteSpace E\n\u03b9 : Type u_3\ninst\u271d\u00b9\u2070 : LinearOrder \u03b9\ninst\u271d\u2079 : LocallyFiniteOrder \u03b9\ninst\u271d\u2078 : OrderBot \u03b9\ninst\u271d\u2077 : TopologicalSpace \u03b9\ninst\u271d\u2076 : DiscreteTopology \u03b9\ninst\u271d\u2075 : MeasurableSpace \u03b9\ninst\u271d\u2074 : BorelSpace \u03b9\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : SecondCountableTopology E\n\u2131 : Filtration \u03b9 m\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nf : \u03b9 \u2192 \u03a9 \u2192 E\ni n : \u03b9\nh : Martingale f \u2131 \u03bc\nh\u03c4 : IsStoppingTime \u2131 \u03c4\nh\u03c3 : IsStoppingTime \u2131 \u03c3\ninst\u271d : SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace h\u03c3 \u2264 m))\nh\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n\nh_int : Integrable (Set.indicator {\u03c9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9} (stoppedValue (fun n => f n) \u03c4))\nt : Set \u03a9\nht : MeasurableSet (t \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9})\n\u22a2 MeasurableSet (t \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9})", "state_after": "case refine'_1\n\u03a9 : Type u_1\nE : Type u_2\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b9 : CompleteSpace E\n\u03b9 : Type u_3\ninst\u271d\u00b9\u2070 : LinearOrder \u03b9\ninst\u271d\u2079 : LocallyFiniteOrder \u03b9\ninst\u271d\u2078 : OrderBot \u03b9\ninst\u271d\u2077 : TopologicalSpace \u03b9\ninst\u271d\u2076 : DiscreteTopology \u03b9\ninst\u271d\u2075 : MeasurableSpace \u03b9\ninst\u271d\u2074 : BorelSpace \u03b9\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : SecondCountableTopology E\n\u2131 : Filtration \u03b9 m\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nf : \u03b9 \u2192 \u03a9 \u2192 E\ni n : \u03b9\nh : Martingale f \u2131 \u03bc\nh\u03c4 : IsStoppingTime \u2131 \u03c4\nh\u03c3 : IsStoppingTime \u2131 \u03c3\ninst\u271d : SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace h\u03c3 \u2264 m))\nh\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n\nh_int : Integrable (Set.indicator {\u03c9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9} (stoppedValue (fun n => f n) \u03c4))\nt : Set \u03a9\nht : MeasurableSet (t \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9}) \u2227 MeasurableSet (t \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9})\n\u22a2 MeasurableSet (t \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9})"}, {"tactic": "exact ht.2", "annotated_tactic": ["exact ht.2", []], "state_before": "case refine'_1\n\u03a9 : Type u_1\nE : Type u_2\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b9 : CompleteSpace E\n\u03b9 : Type u_3\ninst\u271d\u00b9\u2070 : LinearOrder \u03b9\ninst\u271d\u2079 : LocallyFiniteOrder \u03b9\ninst\u271d\u2078 : OrderBot \u03b9\ninst\u271d\u2077 : TopologicalSpace \u03b9\ninst\u271d\u2076 : DiscreteTopology \u03b9\ninst\u271d\u2075 : MeasurableSpace \u03b9\ninst\u271d\u2074 : BorelSpace \u03b9\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : SecondCountableTopology E\n\u2131 : Filtration \u03b9 m\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nf : \u03b9 \u2192 \u03a9 \u2192 E\ni n : \u03b9\nh : Martingale f \u2131 \u03bc\nh\u03c4 : IsStoppingTime \u2131 \u03c4\nh\u03c3 : IsStoppingTime \u2131 \u03c3\ninst\u271d : SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace h\u03c3 \u2264 m))\nh\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n\nh_int : Integrable (Set.indicator {\u03c9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9} (stoppedValue (fun n => f n) \u03c4))\nt : Set \u03a9\nht : MeasurableSet (t \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9}) \u2227 MeasurableSet (t \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9})\n\u22a2 MeasurableSet (t \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9})", "state_after": "no goals"}, {"tactic": "refine' StronglyMeasurable.indicator _ (h\u03c4.measurableSet_le_stopping_time h\u03c3)", "annotated_tactic": ["refine' <a>StronglyMeasurable.indicator</a> _ (h\u03c4.measurableSet_le_stopping_time h\u03c3)", [{"full_name": "MeasureTheory.StronglyMeasurable.indicator", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [825, 19], "def_end_pos": [825, 28]}]], "state_before": "case refine'_2\n\u03a9 : Type u_1\nE : Type u_2\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b9 : CompleteSpace E\n\u03b9 : Type u_3\ninst\u271d\u00b9\u2070 : LinearOrder \u03b9\ninst\u271d\u2079 : LocallyFiniteOrder \u03b9\ninst\u271d\u2078 : OrderBot \u03b9\ninst\u271d\u2077 : TopologicalSpace \u03b9\ninst\u271d\u2076 : DiscreteTopology \u03b9\ninst\u271d\u2075 : MeasurableSpace \u03b9\ninst\u271d\u2074 : BorelSpace \u03b9\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : SecondCountableTopology E\n\u2131 : Filtration \u03b9 m\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nf : \u03b9 \u2192 \u03a9 \u2192 E\ni n : \u03b9\nh : Martingale f \u2131 \u03bc\nh\u03c4 : IsStoppingTime \u2131 \u03c4\nh\u03c3 : IsStoppingTime \u2131 \u03c3\ninst\u271d : SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace h\u03c3 \u2264 m))\nh\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n\nh_int : Integrable (Set.indicator {\u03c9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9} (stoppedValue (fun n => f n) \u03c4))\n\u22a2 StronglyMeasurable (Set.indicator {\u03c9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9} (stoppedValue (fun n => f n) \u03c4))", "state_after": "case refine'_2\n\u03a9 : Type u_1\nE : Type u_2\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b9 : CompleteSpace E\n\u03b9 : Type u_3\ninst\u271d\u00b9\u2070 : LinearOrder \u03b9\ninst\u271d\u2079 : LocallyFiniteOrder \u03b9\ninst\u271d\u2078 : OrderBot \u03b9\ninst\u271d\u2077 : TopologicalSpace \u03b9\ninst\u271d\u2076 : DiscreteTopology \u03b9\ninst\u271d\u2075 : MeasurableSpace \u03b9\ninst\u271d\u2074 : BorelSpace \u03b9\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : SecondCountableTopology E\n\u2131 : Filtration \u03b9 m\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nf : \u03b9 \u2192 \u03a9 \u2192 E\ni n : \u03b9\nh : Martingale f \u2131 \u03bc\nh\u03c4 : IsStoppingTime \u2131 \u03c4\nh\u03c3 : IsStoppingTime \u2131 \u03c3\ninst\u271d : SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace h\u03c3 \u2264 m))\nh\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n\nh_int : Integrable (Set.indicator {\u03c9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9} (stoppedValue (fun n => f n) \u03c4))\n\u22a2 StronglyMeasurable (stoppedValue (fun n => f n) \u03c4)"}, {"tactic": "refine' Measurable.stronglyMeasurable _", "annotated_tactic": ["refine' <a>Measurable.stronglyMeasurable</a> _", [{"full_name": "Measurable.stronglyMeasurable", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [653, 9], "def_end_pos": [653, 45]}]], "state_before": "case refine'_2\n\u03a9 : Type u_1\nE : Type u_2\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b9 : CompleteSpace E\n\u03b9 : Type u_3\ninst\u271d\u00b9\u2070 : LinearOrder \u03b9\ninst\u271d\u2079 : LocallyFiniteOrder \u03b9\ninst\u271d\u2078 : OrderBot \u03b9\ninst\u271d\u2077 : TopologicalSpace \u03b9\ninst\u271d\u2076 : DiscreteTopology \u03b9\ninst\u271d\u2075 : MeasurableSpace \u03b9\ninst\u271d\u2074 : BorelSpace \u03b9\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : SecondCountableTopology E\n\u2131 : Filtration \u03b9 m\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nf : \u03b9 \u2192 \u03a9 \u2192 E\ni n : \u03b9\nh : Martingale f \u2131 \u03bc\nh\u03c4 : IsStoppingTime \u2131 \u03c4\nh\u03c3 : IsStoppingTime \u2131 \u03c3\ninst\u271d : SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace h\u03c3 \u2264 m))\nh\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n\nh_int : Integrable (Set.indicator {\u03c9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9} (stoppedValue (fun n => f n) \u03c4))\n\u22a2 StronglyMeasurable (stoppedValue (fun n => f n) \u03c4)", "state_after": "case refine'_2\n\u03a9 : Type u_1\nE : Type u_2\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b9 : CompleteSpace E\n\u03b9 : Type u_3\ninst\u271d\u00b9\u2070 : LinearOrder \u03b9\ninst\u271d\u2079 : LocallyFiniteOrder \u03b9\ninst\u271d\u2078 : OrderBot \u03b9\ninst\u271d\u2077 : TopologicalSpace \u03b9\ninst\u271d\u2076 : DiscreteTopology \u03b9\ninst\u271d\u2075 : MeasurableSpace \u03b9\ninst\u271d\u2074 : BorelSpace \u03b9\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : SecondCountableTopology E\n\u2131 : Filtration \u03b9 m\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nf : \u03b9 \u2192 \u03a9 \u2192 E\ni n : \u03b9\nh : Martingale f \u2131 \u03bc\nh\u03c4 : IsStoppingTime \u2131 \u03c4\nh\u03c3 : IsStoppingTime \u2131 \u03c3\ninst\u271d : SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace h\u03c3 \u2264 m))\nh\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n\nh_int : Integrable (Set.indicator {\u03c9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9} (stoppedValue (fun n => f n) \u03c4))\n\u22a2 Measurable (stoppedValue (fun n => f n) \u03c4)"}, {"tactic": "exact measurable_stoppedValue h.adapted.progMeasurable_of_discrete h\u03c4", "annotated_tactic": ["exact <a>measurable_stoppedValue</a> h.adapted.progMeasurable_of_discrete h\u03c4", [{"full_name": "MeasureTheory.measurable_stoppedValue", "def_path": "Mathlib/Probability/Process/Stopping.lean", "def_pos": [871, 9], "def_end_pos": [871, 32]}]], "state_before": "case refine'_2\n\u03a9 : Type u_1\nE : Type u_2\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b9 : CompleteSpace E\n\u03b9 : Type u_3\ninst\u271d\u00b9\u2070 : LinearOrder \u03b9\ninst\u271d\u2079 : LocallyFiniteOrder \u03b9\ninst\u271d\u2078 : OrderBot \u03b9\ninst\u271d\u2077 : TopologicalSpace \u03b9\ninst\u271d\u2076 : DiscreteTopology \u03b9\ninst\u271d\u2075 : MeasurableSpace \u03b9\ninst\u271d\u2074 : BorelSpace \u03b9\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : SecondCountableTopology E\n\u2131 : Filtration \u03b9 m\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nf : \u03b9 \u2192 \u03a9 \u2192 E\ni n : \u03b9\nh : Martingale f \u2131 \u03bc\nh\u03c4 : IsStoppingTime \u2131 \u03c4\nh\u03c3 : IsStoppingTime \u2131 \u03c3\ninst\u271d : SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace h\u03c3 \u2264 m))\nh\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n\nh_int : Integrable (Set.indicator {\u03c9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9} (stoppedValue (fun n => f n) \u03c4))\n\u22a2 Measurable (stoppedValue (fun n => f n) \u03c4)", "state_after": "no goals"}, {"tactic": "intro x hx", "annotated_tactic": ["intro x hx", []], "state_before": "case refine'_3\n\u03a9 : Type u_1\nE : Type u_2\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b9 : CompleteSpace E\n\u03b9 : Type u_3\ninst\u271d\u00b9\u2070 : LinearOrder \u03b9\ninst\u271d\u2079 : LocallyFiniteOrder \u03b9\ninst\u271d\u2078 : OrderBot \u03b9\ninst\u271d\u2077 : TopologicalSpace \u03b9\ninst\u271d\u2076 : DiscreteTopology \u03b9\ninst\u271d\u2075 : MeasurableSpace \u03b9\ninst\u271d\u2074 : BorelSpace \u03b9\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : SecondCountableTopology E\n\u2131 : Filtration \u03b9 m\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nf : \u03b9 \u2192 \u03a9 \u2192 E\ni n : \u03b9\nh : Martingale f \u2131 \u03bc\nh\u03c4 : IsStoppingTime \u2131 \u03c4\nh\u03c3 : IsStoppingTime \u2131 \u03c3\ninst\u271d : SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace h\u03c3 \u2264 m))\nh\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n\nh_int : Integrable (Set.indicator {\u03c9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9} (stoppedValue (fun n => f n) \u03c4))\n\u22a2 \u2200 (x : \u03a9), \u00acx \u2208 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9} \u2192 Set.indicator {\u03c9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9} (stoppedValue (fun n => f n) \u03c4) x = 0", "state_after": "case refine'_3\n\u03a9 : Type u_1\nE : Type u_2\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b9 : CompleteSpace E\n\u03b9 : Type u_3\ninst\u271d\u00b9\u2070 : LinearOrder \u03b9\ninst\u271d\u2079 : LocallyFiniteOrder \u03b9\ninst\u271d\u2078 : OrderBot \u03b9\ninst\u271d\u2077 : TopologicalSpace \u03b9\ninst\u271d\u2076 : DiscreteTopology \u03b9\ninst\u271d\u2075 : MeasurableSpace \u03b9\ninst\u271d\u2074 : BorelSpace \u03b9\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : SecondCountableTopology E\n\u2131 : Filtration \u03b9 m\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nf : \u03b9 \u2192 \u03a9 \u2192 E\ni n : \u03b9\nh : Martingale f \u2131 \u03bc\nh\u03c4 : IsStoppingTime \u2131 \u03c4\nh\u03c3 : IsStoppingTime \u2131 \u03c3\ninst\u271d : SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace h\u03c3 \u2264 m))\nh\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n\nh_int : Integrable (Set.indicator {\u03c9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9} (stoppedValue (fun n => f n) \u03c4))\nx : \u03a9\nhx : \u00acx \u2208 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9}\n\u22a2 Set.indicator {\u03c9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9} (stoppedValue (fun n => f n) \u03c4) x = 0"}, {"tactic": "simp only [hx, Set.indicator_of_not_mem, not_false_iff]", "annotated_tactic": ["simp only [hx, <a>Set.indicator_of_not_mem</a>, <a>not_false_iff</a>]", [{"full_name": "Set.indicator_of_not_mem", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [73, 3], "def_end_pos": [73, 14]}, {"full_name": "not_false_iff", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [82, 9], "def_end_pos": [82, 22]}]], "state_before": "case refine'_3\n\u03a9 : Type u_1\nE : Type u_2\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b9 : CompleteSpace E\n\u03b9 : Type u_3\ninst\u271d\u00b9\u2070 : LinearOrder \u03b9\ninst\u271d\u2079 : LocallyFiniteOrder \u03b9\ninst\u271d\u2078 : OrderBot \u03b9\ninst\u271d\u2077 : TopologicalSpace \u03b9\ninst\u271d\u2076 : DiscreteTopology \u03b9\ninst\u271d\u2075 : MeasurableSpace \u03b9\ninst\u271d\u2074 : BorelSpace \u03b9\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : SecondCountableTopology E\n\u2131 : Filtration \u03b9 m\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nf : \u03b9 \u2192 \u03a9 \u2192 E\ni n : \u03b9\nh : Martingale f \u2131 \u03bc\nh\u03c4 : IsStoppingTime \u2131 \u03c4\nh\u03c3 : IsStoppingTime \u2131 \u03c3\ninst\u271d : SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace h\u03c3 \u2264 m))\nh\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n\nh_int : Integrable (Set.indicator {\u03c9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9} (stoppedValue (fun n => f n) \u03c4))\nx : \u03a9\nhx : \u00acx \u2208 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9}\n\u22a2 Set.indicator {\u03c9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9} (stoppedValue (fun n => f n) \u03c4) x = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/PDeriv.lean", "full_name": "MvPolynomial.pderiv_eq_zero_of_not_mem_vars", "start": [108, 1], "end": [110, 96], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "full_name": "MeasurableEquiv.map_apply_eq_iff_map_symm_apply_eq", "start": [4246, 1], "end": [4247, 73], "traced_tactics": [{"tactic": "rw [\u2190 (map_measurableEquiv_injective e).eq_iff, map_map_symm, eq_comm]", "annotated_tactic": ["rw [\u2190 (<a>map_measurableEquiv_injective</a> e).<a>eq_iff</a>, <a>map_map_symm</a>, <a>eq_comm</a>]", [{"full_name": "MeasurableEquiv.map_measurableEquiv_injective", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [4240, 9], "def_end_pos": [4240, 38]}, {"full_name": "Function.Injective.eq_iff", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [95, 9], "def_end_pos": [95, 25]}, {"full_name": "MeasurableEquiv.map_map_symm", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [4236, 9], "def_end_pos": [4236, 21]}, {"full_name": "eq_comm", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [104, 9], "def_end_pos": [104, 16]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\n\u03bc : Measure \u03b1\n\u03bd : Measure \u03b2\ne : \u03b1 \u2243\u1d50 \u03b2\n\u22a2 Measure.map (\u2191e) \u03bc = \u03bd \u2194 Measure.map (\u2191(symm e)) \u03bd = \u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Constructions/Pi.lean", "full_name": "MeasureTheory.Measure.pi_pi", "start": [394, 1], "end": [396, 27], "traced_tactics": [{"tactic": "haveI : Encodable \u03b9 := Fintype.toEncodable \u03b9", "annotated_tactic": ["haveI : <a>Encodable</a> \u03b9 := <a>Fintype.toEncodable</a> \u03b9", [{"full_name": "Encodable", "def_path": "Mathlib/Logic/Encodable/Basic.lean", "def_pos": [45, 7], "def_end_pos": [45, 16]}, {"full_name": "Fintype.toEncodable", "def_path": "Mathlib/Logic/Equiv/List.lean", "def_pos": [143, 19], "def_end_pos": [143, 45]}]], "state_before": "\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : \u03b9 \u2192 Type u_3\ninst\u271d\u00b2 : Fintype \u03b9\nm : (i : \u03b9) \u2192 OuterMeasure (\u03b1 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 MeasurableSpace (\u03b1 i)\n\u03bc : (i : \u03b9) \u2192 Measure (\u03b1 i)\ninst\u271d : \u2200 (i : \u03b9), SigmaFinite (\u03bc i)\ns : (i : \u03b9) \u2192 Set (\u03b1 i)\n\u22a2 \u2191\u2191(Measure.pi \u03bc) (Set.pi univ s) = \u220f i : \u03b9, \u2191\u2191(\u03bc i) (s i)", "state_after": "\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : \u03b9 \u2192 Type u_3\ninst\u271d\u00b2 : Fintype \u03b9\nm : (i : \u03b9) \u2192 OuterMeasure (\u03b1 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 MeasurableSpace (\u03b1 i)\n\u03bc : (i : \u03b9) \u2192 Measure (\u03b1 i)\ninst\u271d : \u2200 (i : \u03b9), SigmaFinite (\u03bc i)\ns : (i : \u03b9) \u2192 Set (\u03b1 i)\nthis : Encodable \u03b9\n\u22a2 \u2191\u2191(Measure.pi \u03bc) (Set.pi univ s) = \u220f i : \u03b9, \u2191\u2191(\u03bc i) (s i)"}, {"tactic": "rw [\u2190 pi'_eq_pi, pi'_pi]", "annotated_tactic": ["rw [\u2190 <a>pi'_eq_pi</a>, <a>pi'_pi</a>]", [{"full_name": "MeasureTheory.Measure.pi'_eq_pi", "def_path": "Mathlib/MeasureTheory/Constructions/Pi.lean", "def_pos": [389, 9], "def_end_pos": [389, 18]}, {"full_name": "MeasureTheory.Measure.pi'_pi", "def_path": "Mathlib/MeasureTheory/Constructions/Pi.lean", "def_pos": [274, 9], "def_end_pos": [274, 15]}]], "state_before": "\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : \u03b9 \u2192 Type u_3\ninst\u271d\u00b2 : Fintype \u03b9\nm : (i : \u03b9) \u2192 OuterMeasure (\u03b1 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 MeasurableSpace (\u03b1 i)\n\u03bc : (i : \u03b9) \u2192 Measure (\u03b1 i)\ninst\u271d : \u2200 (i : \u03b9), SigmaFinite (\u03bc i)\ns : (i : \u03b9) \u2192 Set (\u03b1 i)\nthis : Encodable \u03b9\n\u22a2 \u2191\u2191(Measure.pi \u03bc) (Set.pi univ s) = \u220f i : \u03b9, \u2191\u2191(\u03bc i) (s i)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Part.lean", "full_name": "Part.mem_coe", "start": [351, 1], "end": [352, 15], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "full_name": "ContinuousLinearMap.add_compLpL", "start": [1187, 1], "end": [1188, 96], "traced_tactics": [{"tactic": "ext1 f", "annotated_tactic": ["ext1 f", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedAddCommGroup G\ng : E \u2192 F\nc : \u211d\u22650\n\ud835\udd5c : Type u_5\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : Fact (1 \u2264 p)\nL L' : E \u2192L[\ud835\udd5c] F\n\u22a2 compLpL p \u03bc (L + L') = compLpL p \u03bc L + compLpL p \u03bc L'", "state_after": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedAddCommGroup G\ng : E \u2192 F\nc : \u211d\u22650\n\ud835\udd5c : Type u_5\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : Fact (1 \u2264 p)\nL L' : E \u2192L[\ud835\udd5c] F\nf : { x // x \u2208 Lp E p }\n\u22a2 \u2191(compLpL p \u03bc (L + L')) f = \u2191(compLpL p \u03bc L + compLpL p \u03bc L') f"}, {"tactic": "exact add_compLp L L' f", "annotated_tactic": ["exact <a>add_compLp</a> L L' f", [{"full_name": "ContinuousLinearMap.add_compLp", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [1126, 9], "def_end_pos": [1126, 19]}]], "state_before": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedAddCommGroup G\ng : E \u2192 F\nc : \u211d\u22650\n\ud835\udd5c : Type u_5\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : Fact (1 \u2264 p)\nL L' : E \u2192L[\ud835\udd5c] F\nf : { x // x \u2208 Lp E p }\n\u22a2 \u2191(compLpL p \u03bc (L + L')) f = \u2191(compLpL p \u03bc L + compLpL p \u03bc L') f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Kernel/CondDistrib.lean", "full_name": "ProbabilityTheory.condexp_prod_ae_eq_integral_condDistrib'", "start": [227, 1], "end": [247, 81], "traced_tactics": [{"tactic": "have hf_int' : Integrable (fun a => f (X a, Y a)) \u03bc :=\n  (integrable_map_measure hf_int.1 (hX.aemeasurable.prod_mk hY)).mp hf_int", "annotated_tactic": ["have hf_int' : <a>Integrable</a> (fun a => f (X a, Y a)) \u03bc :=\n    (<a>integrable_map_measure</a> hf_int.1 (hX.aemeasurable.prod_mk hY)).<a>mp</a> hf_int", [{"full_name": "MeasureTheory.Integrable", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [442, 5], "def_end_pos": [442, 15]}, {"full_name": "MeasureTheory.integrable_map_measure", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [610, 9], "def_end_pos": [610, 31]}, {"full_name": "Iff.mp", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [90, 3], "def_end_pos": [90, 5]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nF : Type u_4\ninst\u271d\u2078 : TopologicalSpace \u03a9\ninst\u271d\u2077 : MeasurableSpace \u03a9\ninst\u271d\u2076 : PolishSpace \u03a9\ninst\u271d\u2075 : BorelSpace \u03a9\ninst\u271d\u2074 : Nonempty \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup F\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : IsFiniteMeasure \u03bc\nX : \u03b1 \u2192 \u03b2\nY : \u03b1 \u2192 \u03a9\nm\u03b2 : MeasurableSpace \u03b2\ns : Set \u03a9\nt : Set \u03b2\nf : \u03b2 \u00d7 \u03a9 \u2192 F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nhX : Measurable X\nhY : AEMeasurable Y\nhf_int : Integrable f\n\u22a2 \u03bc[fun a => f (X a, Y a)|MeasurableSpace.comap X m\u03b2] =\u1d50[\u03bc] fun a => \u222b (y : \u03a9), f (X a, y) \u2202\u2191(condDistrib Y X \u03bc) (X a)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nF : Type u_4\ninst\u271d\u2078 : TopologicalSpace \u03a9\ninst\u271d\u2077 : MeasurableSpace \u03a9\ninst\u271d\u2076 : PolishSpace \u03a9\ninst\u271d\u2075 : BorelSpace \u03a9\ninst\u271d\u2074 : Nonempty \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup F\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : IsFiniteMeasure \u03bc\nX : \u03b1 \u2192 \u03b2\nY : \u03b1 \u2192 \u03a9\nm\u03b2 : MeasurableSpace \u03b2\ns : Set \u03a9\nt : Set \u03b2\nf : \u03b2 \u00d7 \u03a9 \u2192 F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nhX : Measurable X\nhY : AEMeasurable Y\nhf_int : Integrable f\nhf_int' : Integrable fun a => f (X a, Y a)\n\u22a2 \u03bc[fun a => f (X a, Y a)|MeasurableSpace.comap X m\u03b2] =\u1d50[\u03bc] fun a => \u222b (y : \u03a9), f (X a, y) \u2202\u2191(condDistrib Y X \u03bc) (X a)"}, {"tactic": "refine' (ae_eq_condexp_of_forall_set_integral_eq hX.comap_le hf_int' (fun s _ _ => _) _ _).symm", "annotated_tactic": ["refine' (<a>ae_eq_condexp_of_forall_set_integral_eq</a> hX.comap_le hf_int' (fun s _ _ => _) _ _).<a>symm</a>", [{"full_name": "MeasureTheory.ae_eq_condexp_of_forall_set_integral_eq", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean", "def_pos": [237, 9], "def_end_pos": [237, 48]}, {"full_name": "Filter.EventuallyEq.symm", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1498, 9], "def_end_pos": [1498, 26]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nF : Type u_4\ninst\u271d\u2078 : TopologicalSpace \u03a9\ninst\u271d\u2077 : MeasurableSpace \u03a9\ninst\u271d\u2076 : PolishSpace \u03a9\ninst\u271d\u2075 : BorelSpace \u03a9\ninst\u271d\u2074 : Nonempty \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup F\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : IsFiniteMeasure \u03bc\nX : \u03b1 \u2192 \u03b2\nY : \u03b1 \u2192 \u03a9\nm\u03b2 : MeasurableSpace \u03b2\ns : Set \u03a9\nt : Set \u03b2\nf : \u03b2 \u00d7 \u03a9 \u2192 F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nhX : Measurable X\nhY : AEMeasurable Y\nhf_int : Integrable f\nhf_int' : Integrable fun a => f (X a, Y a)\n\u22a2 \u03bc[fun a => f (X a, Y a)|MeasurableSpace.comap X m\u03b2] =\u1d50[\u03bc] fun a => \u222b (y : \u03a9), f (X a, y) \u2202\u2191(condDistrib Y X \u03bc) (X a)", "state_after": "case refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nF : Type u_4\ninst\u271d\u2078 : TopologicalSpace \u03a9\ninst\u271d\u2077 : MeasurableSpace \u03a9\ninst\u271d\u2076 : PolishSpace \u03a9\ninst\u271d\u2075 : BorelSpace \u03a9\ninst\u271d\u2074 : Nonempty \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup F\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : IsFiniteMeasure \u03bc\nX : \u03b1 \u2192 \u03b2\nY : \u03b1 \u2192 \u03a9\nm\u03b2 : MeasurableSpace \u03b2\ns\u271d : Set \u03a9\nt : Set \u03b2\nf : \u03b2 \u00d7 \u03a9 \u2192 F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nhX : Measurable X\nhY : AEMeasurable Y\nhf_int : Integrable f\nhf_int' : Integrable fun a => f (X a, Y a)\ns : Set \u03b1\nx\u271d\u00b9 : MeasurableSet s\nx\u271d : \u2191\u2191\u03bc s < \u22a4\n\u22a2 IntegrableOn (fun a => \u222b (y : \u03a9), f (X a, y) \u2202\u2191(condDistrib Y X \u03bc) (X a)) s\n\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nF : Type u_4\ninst\u271d\u2078 : TopologicalSpace \u03a9\ninst\u271d\u2077 : MeasurableSpace \u03a9\ninst\u271d\u2076 : PolishSpace \u03a9\ninst\u271d\u2075 : BorelSpace \u03a9\ninst\u271d\u2074 : Nonempty \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup F\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : IsFiniteMeasure \u03bc\nX : \u03b1 \u2192 \u03b2\nY : \u03b1 \u2192 \u03a9\nm\u03b2 : MeasurableSpace \u03b2\ns : Set \u03a9\nt : Set \u03b2\nf : \u03b2 \u00d7 \u03a9 \u2192 F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nhX : Measurable X\nhY : AEMeasurable Y\nhf_int : Integrable f\nhf_int' : Integrable fun a => f (X a, Y a)\n\u22a2 \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u222b (y : \u03a9), f (X x, y) \u2202\u2191(condDistrib Y X \u03bc) (X x) \u2202\u03bc = \u222b (x : \u03b1) in s, f (X x, Y x) \u2202\u03bc\n\ncase refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nF : Type u_4\ninst\u271d\u2078 : TopologicalSpace \u03a9\ninst\u271d\u2077 : MeasurableSpace \u03a9\ninst\u271d\u2076 : PolishSpace \u03a9\ninst\u271d\u2075 : BorelSpace \u03a9\ninst\u271d\u2074 : Nonempty \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup F\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : IsFiniteMeasure \u03bc\nX : \u03b1 \u2192 \u03b2\nY : \u03b1 \u2192 \u03a9\nm\u03b2 : MeasurableSpace \u03b2\ns : Set \u03a9\nt : Set \u03b2\nf : \u03b2 \u00d7 \u03a9 \u2192 F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nhX : Measurable X\nhY : AEMeasurable Y\nhf_int : Integrable f\nhf_int' : Integrable fun a => f (X a, Y a)\n\u22a2 AEStronglyMeasurable' (MeasurableSpace.comap X m\u03b2) (fun a => \u222b (y : \u03a9), f (X a, y) \u2202\u2191(condDistrib Y X \u03bc) (X a)) \u03bc"}, {"tactic": "exact (hf_int.integral_condDistrib hX.aemeasurable hY).integrableOn", "annotated_tactic": ["exact (hf_int.integral_condDistrib hX.aemeasurable hY).<a>integrableOn</a>", [{"full_name": "MeasureTheory.Integrable.integrableOn", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [163, 9], "def_end_pos": [163, 32]}]], "state_before": "case refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nF : Type u_4\ninst\u271d\u2078 : TopologicalSpace \u03a9\ninst\u271d\u2077 : MeasurableSpace \u03a9\ninst\u271d\u2076 : PolishSpace \u03a9\ninst\u271d\u2075 : BorelSpace \u03a9\ninst\u271d\u2074 : Nonempty \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup F\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : IsFiniteMeasure \u03bc\nX : \u03b1 \u2192 \u03b2\nY : \u03b1 \u2192 \u03a9\nm\u03b2 : MeasurableSpace \u03b2\ns\u271d : Set \u03a9\nt : Set \u03b2\nf : \u03b2 \u00d7 \u03a9 \u2192 F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nhX : Measurable X\nhY : AEMeasurable Y\nhf_int : Integrable f\nhf_int' : Integrable fun a => f (X a, Y a)\ns : Set \u03b1\nx\u271d\u00b9 : MeasurableSet s\nx\u271d : \u2191\u2191\u03bc s < \u22a4\n\u22a2 IntegrableOn (fun a => \u222b (y : \u03a9), f (X a, y) \u2202\u2191(condDistrib Y X \u03bc) (X a)) s", "state_after": "no goals"}, {"tactic": "rintro s \u27e8t, ht, rfl\u27e9 _", "annotated_tactic": ["rintro s \u27e8t, ht, rfl\u27e9 _", []], "state_before": "case refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nF : Type u_4\ninst\u271d\u2078 : TopologicalSpace \u03a9\ninst\u271d\u2077 : MeasurableSpace \u03a9\ninst\u271d\u2076 : PolishSpace \u03a9\ninst\u271d\u2075 : BorelSpace \u03a9\ninst\u271d\u2074 : Nonempty \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup F\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : IsFiniteMeasure \u03bc\nX : \u03b1 \u2192 \u03b2\nY : \u03b1 \u2192 \u03a9\nm\u03b2 : MeasurableSpace \u03b2\ns : Set \u03a9\nt : Set \u03b2\nf : \u03b2 \u00d7 \u03a9 \u2192 F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nhX : Measurable X\nhY : AEMeasurable Y\nhf_int : Integrable f\nhf_int' : Integrable fun a => f (X a, Y a)\n\u22a2 \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u222b (y : \u03a9), f (X x, y) \u2202\u2191(condDistrib Y X \u03bc) (X x) \u2202\u03bc = \u222b (x : \u03b1) in s, f (X x, Y x) \u2202\u03bc", "state_after": "case refine'_2.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nF : Type u_4\ninst\u271d\u2078 : TopologicalSpace \u03a9\ninst\u271d\u2077 : MeasurableSpace \u03a9\ninst\u271d\u2076 : PolishSpace \u03a9\ninst\u271d\u2075 : BorelSpace \u03a9\ninst\u271d\u2074 : Nonempty \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup F\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : IsFiniteMeasure \u03bc\nX : \u03b1 \u2192 \u03b2\nY : \u03b1 \u2192 \u03a9\nm\u03b2 : MeasurableSpace \u03b2\ns : Set \u03a9\nt\u271d : Set \u03b2\nf : \u03b2 \u00d7 \u03a9 \u2192 F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nhX : Measurable X\nhY : AEMeasurable Y\nhf_int : Integrable f\nhf_int' : Integrable fun a => f (X a, Y a)\nt : Set \u03b2\nht : MeasurableSet t\na\u271d : \u2191\u2191\u03bc (X \u207b\u00b9' t) < \u22a4\n\u22a2 \u222b (x : \u03b1) in X \u207b\u00b9' t, \u222b (y : \u03a9), f (X x, y) \u2202\u2191(condDistrib Y X \u03bc) (X x) \u2202\u03bc = \u222b (x : \u03b1) in X \u207b\u00b9' t, f (X x, Y x) \u2202\u03bc"}, {"tactic": "change \u222b a in X \u207b\u00b9' t, ((fun x' => \u222b y, f (x', y) \u2202(condDistrib Y X \u03bc) x') \u2218 X) a \u2202\u03bc =\n  \u222b a in X \u207b\u00b9' t, f (X a, Y a) \u2202\u03bc", "annotated_tactic": ["change \u222b a in X \u207b\u00b9' t, ((fun x' => \u222b y, f (x', y) \u2202(<a>condDistrib</a> Y X \u03bc) x') \u2218 X) a \u2202\u03bc =\n      \u222b a in X \u207b\u00b9' t, f (X a, Y a) \u2202\u03bc", [{"full_name": "ProbabilityTheory.condDistrib", "def_path": "Mathlib/Probability/Kernel/CondDistrib.lean", "def_pos": [62, 31], "def_end_pos": [62, 42]}]], "state_before": "case refine'_2.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nF : Type u_4\ninst\u271d\u2078 : TopologicalSpace \u03a9\ninst\u271d\u2077 : MeasurableSpace \u03a9\ninst\u271d\u2076 : PolishSpace \u03a9\ninst\u271d\u2075 : BorelSpace \u03a9\ninst\u271d\u2074 : Nonempty \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup F\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : IsFiniteMeasure \u03bc\nX : \u03b1 \u2192 \u03b2\nY : \u03b1 \u2192 \u03a9\nm\u03b2 : MeasurableSpace \u03b2\ns : Set \u03a9\nt\u271d : Set \u03b2\nf : \u03b2 \u00d7 \u03a9 \u2192 F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nhX : Measurable X\nhY : AEMeasurable Y\nhf_int : Integrable f\nhf_int' : Integrable fun a => f (X a, Y a)\nt : Set \u03b2\nht : MeasurableSet t\na\u271d : \u2191\u2191\u03bc (X \u207b\u00b9' t) < \u22a4\n\u22a2 \u222b (x : \u03b1) in X \u207b\u00b9' t, \u222b (y : \u03a9), f (X x, y) \u2202\u2191(condDistrib Y X \u03bc) (X x) \u2202\u03bc = \u222b (x : \u03b1) in X \u207b\u00b9' t, f (X x, Y x) \u2202\u03bc", "state_after": "case refine'_2.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nF : Type u_4\ninst\u271d\u2078 : TopologicalSpace \u03a9\ninst\u271d\u2077 : MeasurableSpace \u03a9\ninst\u271d\u2076 : PolishSpace \u03a9\ninst\u271d\u2075 : BorelSpace \u03a9\ninst\u271d\u2074 : Nonempty \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup F\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : IsFiniteMeasure \u03bc\nX : \u03b1 \u2192 \u03b2\nY : \u03b1 \u2192 \u03a9\nm\u03b2 : MeasurableSpace \u03b2\ns : Set \u03a9\nt\u271d : Set \u03b2\nf : \u03b2 \u00d7 \u03a9 \u2192 F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nhX : Measurable X\nhY : AEMeasurable Y\nhf_int : Integrable f\nhf_int' : Integrable fun a => f (X a, Y a)\nt : Set \u03b2\nht : MeasurableSet t\na\u271d : \u2191\u2191\u03bc (X \u207b\u00b9' t) < \u22a4\n\u22a2 \u222b (a : \u03b1) in X \u207b\u00b9' t, ((fun x' => \u222b (y : \u03a9), f (x', y) \u2202\u2191(condDistrib Y X \u03bc) x') \u2218 X) a \u2202\u03bc =\n    \u222b (a : \u03b1) in X \u207b\u00b9' t, f (X a, Y a) \u2202\u03bc"}, {"tactic": "simp only [Function.comp_apply]", "annotated_tactic": ["simp only [<a>Function.comp_apply</a>]", [{"full_name": "Function.comp_apply", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [33, 17], "def_end_pos": [33, 36]}]], "state_before": "case refine'_2.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nF : Type u_4\ninst\u271d\u2078 : TopologicalSpace \u03a9\ninst\u271d\u2077 : MeasurableSpace \u03a9\ninst\u271d\u2076 : PolishSpace \u03a9\ninst\u271d\u2075 : BorelSpace \u03a9\ninst\u271d\u2074 : Nonempty \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup F\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : IsFiniteMeasure \u03bc\nX : \u03b1 \u2192 \u03b2\nY : \u03b1 \u2192 \u03a9\nm\u03b2 : MeasurableSpace \u03b2\ns : Set \u03a9\nt\u271d : Set \u03b2\nf : \u03b2 \u00d7 \u03a9 \u2192 F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nhX : Measurable X\nhY : AEMeasurable Y\nhf_int : Integrable f\nhf_int' : Integrable fun a => f (X a, Y a)\nt : Set \u03b2\nht : MeasurableSet t\na\u271d : \u2191\u2191\u03bc (X \u207b\u00b9' t) < \u22a4\n\u22a2 \u222b (a : \u03b1) in X \u207b\u00b9' t, ((fun x' => \u222b (y : \u03a9), f (x', y) \u2202\u2191(condDistrib Y X \u03bc) x') \u2218 X) a \u2202\u03bc =\n    \u222b (a : \u03b1) in X \u207b\u00b9' t, f (X a, Y a) \u2202\u03bc", "state_after": "case refine'_2.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nF : Type u_4\ninst\u271d\u2078 : TopologicalSpace \u03a9\ninst\u271d\u2077 : MeasurableSpace \u03a9\ninst\u271d\u2076 : PolishSpace \u03a9\ninst\u271d\u2075 : BorelSpace \u03a9\ninst\u271d\u2074 : Nonempty \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup F\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : IsFiniteMeasure \u03bc\nX : \u03b1 \u2192 \u03b2\nY : \u03b1 \u2192 \u03a9\nm\u03b2 : MeasurableSpace \u03b2\ns : Set \u03a9\nt\u271d : Set \u03b2\nf : \u03b2 \u00d7 \u03a9 \u2192 F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nhX : Measurable X\nhY : AEMeasurable Y\nhf_int : Integrable f\nhf_int' : Integrable fun a => f (X a, Y a)\nt : Set \u03b2\nht : MeasurableSet t\na\u271d : \u2191\u2191\u03bc (X \u207b\u00b9' t) < \u22a4\n\u22a2 \u222b (a : \u03b1) in X \u207b\u00b9' t, \u222b (y : \u03a9), f (X a, y) \u2202\u2191(condDistrib Y X \u03bc) (X a) \u2202\u03bc = \u222b (a : \u03b1) in X \u207b\u00b9' t, f (X a, Y a) \u2202\u03bc"}, {"tactic": "rw [\u2190 integral_map hX.aemeasurable (f := fun x' => \u222b y, f (x', y) \u2202(condDistrib Y X \u03bc) x')]", "annotated_tactic": ["rw [\u2190 <a>integral_map</a> hX.aemeasurable (f := fun x' => \u222b y, f (x', y) \u2202(<a>condDistrib</a> Y X \u03bc) x')]", [{"full_name": "MeasureTheory.integral_map", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1610, 9], "def_end_pos": [1610, 21]}, {"full_name": "ProbabilityTheory.condDistrib", "def_path": "Mathlib/Probability/Kernel/CondDistrib.lean", "def_pos": [62, 31], "def_end_pos": [62, 42]}]], "state_before": "case refine'_2.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nF : Type u_4\ninst\u271d\u2078 : TopologicalSpace \u03a9\ninst\u271d\u2077 : MeasurableSpace \u03a9\ninst\u271d\u2076 : PolishSpace \u03a9\ninst\u271d\u2075 : BorelSpace \u03a9\ninst\u271d\u2074 : Nonempty \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup F\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : IsFiniteMeasure \u03bc\nX : \u03b1 \u2192 \u03b2\nY : \u03b1 \u2192 \u03a9\nm\u03b2 : MeasurableSpace \u03b2\ns : Set \u03a9\nt\u271d : Set \u03b2\nf : \u03b2 \u00d7 \u03a9 \u2192 F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nhX : Measurable X\nhY : AEMeasurable Y\nhf_int : Integrable f\nhf_int' : Integrable fun a => f (X a, Y a)\nt : Set \u03b2\nht : MeasurableSet t\na\u271d : \u2191\u2191\u03bc (X \u207b\u00b9' t) < \u22a4\n\u22a2 \u222b (a : \u03b1) in X \u207b\u00b9' t, \u222b (y : \u03a9), f (X a, y) \u2202\u2191(condDistrib Y X \u03bc) (X a) \u2202\u03bc = \u222b (a : \u03b1) in X \u207b\u00b9' t, f (X a, Y a) \u2202\u03bc", "state_after": "case refine'_2.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nF : Type u_4\ninst\u271d\u2078 : TopologicalSpace \u03a9\ninst\u271d\u2077 : MeasurableSpace \u03a9\ninst\u271d\u2076 : PolishSpace \u03a9\ninst\u271d\u2075 : BorelSpace \u03a9\ninst\u271d\u2074 : Nonempty \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup F\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : IsFiniteMeasure \u03bc\nX : \u03b1 \u2192 \u03b2\nY : \u03b1 \u2192 \u03a9\nm\u03b2 : MeasurableSpace \u03b2\ns : Set \u03a9\nt\u271d : Set \u03b2\nf : \u03b2 \u00d7 \u03a9 \u2192 F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nhX : Measurable X\nhY : AEMeasurable Y\nhf_int : Integrable f\nhf_int' : Integrable fun a => f (X a, Y a)\nt : Set \u03b2\nht : MeasurableSet t\na\u271d : \u2191\u2191\u03bc (X \u207b\u00b9' t) < \u22a4\n\u22a2 \u222b (y : \u03b2), \u222b (y_1 : \u03a9), f (y, y_1) \u2202\u2191(condDistrib Y X \u03bc) y \u2202Measure.map X (Measure.restrict \u03bc (X \u207b\u00b9' t)) =\n    \u222b (a : \u03b1) in X \u207b\u00b9' t, f (X a, Y a) \u2202\u03bc\n\ncase refine'_2.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nF : Type u_4\ninst\u271d\u2078 : TopologicalSpace \u03a9\ninst\u271d\u2077 : MeasurableSpace \u03a9\ninst\u271d\u2076 : PolishSpace \u03a9\ninst\u271d\u2075 : BorelSpace \u03a9\ninst\u271d\u2074 : Nonempty \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup F\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : IsFiniteMeasure \u03bc\nX : \u03b1 \u2192 \u03b2\nY : \u03b1 \u2192 \u03a9\nm\u03b2 : MeasurableSpace \u03b2\ns : Set \u03a9\nt\u271d : Set \u03b2\nf : \u03b2 \u00d7 \u03a9 \u2192 F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nhX : Measurable X\nhY : AEMeasurable Y\nhf_int : Integrable f\nhf_int' : Integrable fun a => f (X a, Y a)\nt : Set \u03b2\nht : MeasurableSet t\na\u271d : \u2191\u2191\u03bc (X \u207b\u00b9' t) < \u22a4\n\u22a2 AEStronglyMeasurable (fun x' => \u222b (y : \u03a9), f (x', y) \u2202\u2191(condDistrib Y X \u03bc) x')\n    (Measure.map X (Measure.restrict \u03bc (X \u207b\u00b9' t)))"}, {"tactic": "swap", "annotated_tactic": ["swap", []], "state_before": "case refine'_2.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nF : Type u_4\ninst\u271d\u2078 : TopologicalSpace \u03a9\ninst\u271d\u2077 : MeasurableSpace \u03a9\ninst\u271d\u2076 : PolishSpace \u03a9\ninst\u271d\u2075 : BorelSpace \u03a9\ninst\u271d\u2074 : Nonempty \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup F\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : IsFiniteMeasure \u03bc\nX : \u03b1 \u2192 \u03b2\nY : \u03b1 \u2192 \u03a9\nm\u03b2 : MeasurableSpace \u03b2\ns : Set \u03a9\nt\u271d : Set \u03b2\nf : \u03b2 \u00d7 \u03a9 \u2192 F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nhX : Measurable X\nhY : AEMeasurable Y\nhf_int : Integrable f\nhf_int' : Integrable fun a => f (X a, Y a)\nt : Set \u03b2\nht : MeasurableSet t\na\u271d : \u2191\u2191\u03bc (X \u207b\u00b9' t) < \u22a4\n\u22a2 \u222b (y : \u03b2), \u222b (y_1 : \u03a9), f (y, y_1) \u2202\u2191(condDistrib Y X \u03bc) y \u2202Measure.map X (Measure.restrict \u03bc (X \u207b\u00b9' t)) =\n    \u222b (a : \u03b1) in X \u207b\u00b9' t, f (X a, Y a) \u2202\u03bc\n\ncase refine'_2.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nF : Type u_4\ninst\u271d\u2078 : TopologicalSpace \u03a9\ninst\u271d\u2077 : MeasurableSpace \u03a9\ninst\u271d\u2076 : PolishSpace \u03a9\ninst\u271d\u2075 : BorelSpace \u03a9\ninst\u271d\u2074 : Nonempty \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup F\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : IsFiniteMeasure \u03bc\nX : \u03b1 \u2192 \u03b2\nY : \u03b1 \u2192 \u03a9\nm\u03b2 : MeasurableSpace \u03b2\ns : Set \u03a9\nt\u271d : Set \u03b2\nf : \u03b2 \u00d7 \u03a9 \u2192 F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nhX : Measurable X\nhY : AEMeasurable Y\nhf_int : Integrable f\nhf_int' : Integrable fun a => f (X a, Y a)\nt : Set \u03b2\nht : MeasurableSet t\na\u271d : \u2191\u2191\u03bc (X \u207b\u00b9' t) < \u22a4\n\u22a2 AEStronglyMeasurable (fun x' => \u222b (y : \u03a9), f (x', y) \u2202\u2191(condDistrib Y X \u03bc) x')\n    (Measure.map X (Measure.restrict \u03bc (X \u207b\u00b9' t)))", "state_after": "case refine'_2.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nF : Type u_4\ninst\u271d\u2078 : TopologicalSpace \u03a9\ninst\u271d\u2077 : MeasurableSpace \u03a9\ninst\u271d\u2076 : PolishSpace \u03a9\ninst\u271d\u2075 : BorelSpace \u03a9\ninst\u271d\u2074 : Nonempty \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup F\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : IsFiniteMeasure \u03bc\nX : \u03b1 \u2192 \u03b2\nY : \u03b1 \u2192 \u03a9\nm\u03b2 : MeasurableSpace \u03b2\ns : Set \u03a9\nt\u271d : Set \u03b2\nf : \u03b2 \u00d7 \u03a9 \u2192 F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nhX : Measurable X\nhY : AEMeasurable Y\nhf_int : Integrable f\nhf_int' : Integrable fun a => f (X a, Y a)\nt : Set \u03b2\nht : MeasurableSet t\na\u271d : \u2191\u2191\u03bc (X \u207b\u00b9' t) < \u22a4\n\u22a2 AEStronglyMeasurable (fun x' => \u222b (y : \u03a9), f (x', y) \u2202\u2191(condDistrib Y X \u03bc) x')\n    (Measure.map X (Measure.restrict \u03bc (X \u207b\u00b9' t)))\n\ncase refine'_2.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nF : Type u_4\ninst\u271d\u2078 : TopologicalSpace \u03a9\ninst\u271d\u2077 : MeasurableSpace \u03a9\ninst\u271d\u2076 : PolishSpace \u03a9\ninst\u271d\u2075 : BorelSpace \u03a9\ninst\u271d\u2074 : Nonempty \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup F\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : IsFiniteMeasure \u03bc\nX : \u03b1 \u2192 \u03b2\nY : \u03b1 \u2192 \u03a9\nm\u03b2 : MeasurableSpace \u03b2\ns : Set \u03a9\nt\u271d : Set \u03b2\nf : \u03b2 \u00d7 \u03a9 \u2192 F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nhX : Measurable X\nhY : AEMeasurable Y\nhf_int : Integrable f\nhf_int' : Integrable fun a => f (X a, Y a)\nt : Set \u03b2\nht : MeasurableSet t\na\u271d : \u2191\u2191\u03bc (X \u207b\u00b9' t) < \u22a4\n\u22a2 \u222b (y : \u03b2), \u222b (y_1 : \u03a9), f (y, y_1) \u2202\u2191(condDistrib Y X \u03bc) y \u2202Measure.map X (Measure.restrict \u03bc (X \u207b\u00b9' t)) =\n    \u222b (a : \u03b1) in X \u207b\u00b9' t, f (X a, Y a) \u2202\u03bc"}, {"tactic": "rw [\u2190 Measure.restrict_map hX ht, \u2190 Measure.fst_map_prod_mk\u2080 hY, condDistrib,\n  set_integral_condKernel_univ_right ht hf_int.integrableOn,\n  set_integral_map (ht.prod MeasurableSet.univ) hf_int.1 (hX.aemeasurable.prod_mk hY),\n  mk_preimage_prod, preimage_univ, inter_univ]", "annotated_tactic": ["rw [\u2190 <a>Measure.restrict_map</a> hX ht, \u2190 <a>Measure.fst_map_prod_mk\u2080</a> hY, <a>condDistrib</a>,\n      <a>set_integral_condKernel_univ_right</a> ht hf_int.integrableOn,\n      <a>set_integral_map</a> (ht.prod <a>MeasurableSet.univ</a>) hf_int.1 (hX.aemeasurable.prod_mk hY),\n      <a>mk_preimage_prod</a>, <a>preimage_univ</a>, <a>inter_univ</a>]", [{"full_name": "MeasureTheory.Measure.restrict_map", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1790, 9], "def_end_pos": [1790, 21]}, {"full_name": "MeasureTheory.Measure.fst_map_prod_mk\u2080", "def_path": "Mathlib/MeasureTheory/Constructions/Prod/Basic.lean", "def_pos": [939, 9], "def_end_pos": [939, 25]}, {"full_name": "ProbabilityTheory.condDistrib", "def_path": "Mathlib/Probability/Kernel/CondDistrib.lean", "def_pos": [62, 31], "def_end_pos": [62, 42]}, {"full_name": "ProbabilityTheory.set_integral_condKernel_univ_right", "def_path": "Mathlib/Probability/Kernel/Disintegration.lean", "def_pos": [467, 9], "def_end_pos": [467, 43]}, {"full_name": "MeasureTheory.set_integral_map", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [511, 9], "def_end_pos": [511, 25]}, {"full_name": "MeasurableSet.univ", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [101, 19], "def_end_pos": [101, 37]}, {"full_name": "Set.mk_preimage_prod", "def_path": "Mathlib/Data/Set/Prod.lean", "def_pos": [244, 9], "def_end_pos": [244, 25]}, {"full_name": "Set.preimage_univ", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [78, 9], "def_end_pos": [78, 22]}, {"full_name": "Set.inter_univ", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1012, 9], "def_end_pos": [1012, 19]}]], "state_before": "case refine'_2.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nF : Type u_4\ninst\u271d\u2078 : TopologicalSpace \u03a9\ninst\u271d\u2077 : MeasurableSpace \u03a9\ninst\u271d\u2076 : PolishSpace \u03a9\ninst\u271d\u2075 : BorelSpace \u03a9\ninst\u271d\u2074 : Nonempty \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup F\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : IsFiniteMeasure \u03bc\nX : \u03b1 \u2192 \u03b2\nY : \u03b1 \u2192 \u03a9\nm\u03b2 : MeasurableSpace \u03b2\ns : Set \u03a9\nt\u271d : Set \u03b2\nf : \u03b2 \u00d7 \u03a9 \u2192 F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nhX : Measurable X\nhY : AEMeasurable Y\nhf_int : Integrable f\nhf_int' : Integrable fun a => f (X a, Y a)\nt : Set \u03b2\nht : MeasurableSet t\na\u271d : \u2191\u2191\u03bc (X \u207b\u00b9' t) < \u22a4\n\u22a2 \u222b (y : \u03b2), \u222b (y_1 : \u03a9), f (y, y_1) \u2202\u2191(condDistrib Y X \u03bc) y \u2202Measure.map X (Measure.restrict \u03bc (X \u207b\u00b9' t)) =\n    \u222b (a : \u03b1) in X \u207b\u00b9' t, f (X a, Y a) \u2202\u03bc", "state_after": "no goals"}, {"tactic": "rw [\u2190 Measure.restrict_map hX ht]", "annotated_tactic": ["rw [\u2190 <a>Measure.restrict_map</a> hX ht]", [{"full_name": "MeasureTheory.Measure.restrict_map", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1790, 9], "def_end_pos": [1790, 21]}]], "state_before": "case refine'_2.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nF : Type u_4\ninst\u271d\u2078 : TopologicalSpace \u03a9\ninst\u271d\u2077 : MeasurableSpace \u03a9\ninst\u271d\u2076 : PolishSpace \u03a9\ninst\u271d\u2075 : BorelSpace \u03a9\ninst\u271d\u2074 : Nonempty \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup F\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : IsFiniteMeasure \u03bc\nX : \u03b1 \u2192 \u03b2\nY : \u03b1 \u2192 \u03a9\nm\u03b2 : MeasurableSpace \u03b2\ns : Set \u03a9\nt\u271d : Set \u03b2\nf : \u03b2 \u00d7 \u03a9 \u2192 F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nhX : Measurable X\nhY : AEMeasurable Y\nhf_int : Integrable f\nhf_int' : Integrable fun a => f (X a, Y a)\nt : Set \u03b2\nht : MeasurableSet t\na\u271d : \u2191\u2191\u03bc (X \u207b\u00b9' t) < \u22a4\n\u22a2 AEStronglyMeasurable (fun x' => \u222b (y : \u03a9), f (x', y) \u2202\u2191(condDistrib Y X \u03bc) x')\n    (Measure.map X (Measure.restrict \u03bc (X \u207b\u00b9' t)))", "state_after": "case refine'_2.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nF : Type u_4\ninst\u271d\u2078 : TopologicalSpace \u03a9\ninst\u271d\u2077 : MeasurableSpace \u03a9\ninst\u271d\u2076 : PolishSpace \u03a9\ninst\u271d\u2075 : BorelSpace \u03a9\ninst\u271d\u2074 : Nonempty \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup F\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : IsFiniteMeasure \u03bc\nX : \u03b1 \u2192 \u03b2\nY : \u03b1 \u2192 \u03a9\nm\u03b2 : MeasurableSpace \u03b2\ns : Set \u03a9\nt\u271d : Set \u03b2\nf : \u03b2 \u00d7 \u03a9 \u2192 F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nhX : Measurable X\nhY : AEMeasurable Y\nhf_int : Integrable f\nhf_int' : Integrable fun a => f (X a, Y a)\nt : Set \u03b2\nht : MeasurableSet t\na\u271d : \u2191\u2191\u03bc (X \u207b\u00b9' t) < \u22a4\n\u22a2 AEStronglyMeasurable (fun x' => \u222b (y : \u03a9), f (x', y) \u2202\u2191(condDistrib Y X \u03bc) x') (Measure.restrict (Measure.map X \u03bc) t)"}, {"tactic": "exact (hf_int.1.integral_condDistrib_map hY).restrict", "annotated_tactic": ["exact (hf_int.1.<a>integral_condDistrib_map</a> hY).<a>restrict</a>", [{"full_name": "MeasureTheory.AEStronglyMeasurable.integral_condDistrib_map", "def_path": "Mathlib/Probability/Kernel/CondDistrib.lean", "def_pos": [89, 9], "def_end_pos": [89, 75]}, {"full_name": "MeasureTheory.AEStronglyMeasurable.restrict", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1252, 19], "def_end_pos": [1252, 27]}]], "state_before": "case refine'_2.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nF : Type u_4\ninst\u271d\u2078 : TopologicalSpace \u03a9\ninst\u271d\u2077 : MeasurableSpace \u03a9\ninst\u271d\u2076 : PolishSpace \u03a9\ninst\u271d\u2075 : BorelSpace \u03a9\ninst\u271d\u2074 : Nonempty \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup F\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : IsFiniteMeasure \u03bc\nX : \u03b1 \u2192 \u03b2\nY : \u03b1 \u2192 \u03a9\nm\u03b2 : MeasurableSpace \u03b2\ns : Set \u03a9\nt\u271d : Set \u03b2\nf : \u03b2 \u00d7 \u03a9 \u2192 F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nhX : Measurable X\nhY : AEMeasurable Y\nhf_int : Integrable f\nhf_int' : Integrable fun a => f (X a, Y a)\nt : Set \u03b2\nht : MeasurableSet t\na\u271d : \u2191\u2191\u03bc (X \u207b\u00b9' t) < \u22a4\n\u22a2 AEStronglyMeasurable (fun x' => \u222b (y : \u03a9), f (x', y) \u2202\u2191(condDistrib Y X \u03bc) x') (Measure.restrict (Measure.map X \u03bc) t)", "state_after": "no goals"}, {"tactic": "exact aestronglyMeasurable'_integral_condDistrib hX.aemeasurable hY hf_int.1", "annotated_tactic": ["exact <a>aestronglyMeasurable'_integral_condDistrib</a> hX.aemeasurable hY hf_int.1", [{"full_name": "ProbabilityTheory.aestronglyMeasurable'_integral_condDistrib", "def_path": "Mathlib/Probability/Kernel/CondDistrib.lean", "def_pos": [101, 9], "def_end_pos": [101, 51]}]], "state_before": "case refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nF : Type u_4\ninst\u271d\u2078 : TopologicalSpace \u03a9\ninst\u271d\u2077 : MeasurableSpace \u03a9\ninst\u271d\u2076 : PolishSpace \u03a9\ninst\u271d\u2075 : BorelSpace \u03a9\ninst\u271d\u2074 : Nonempty \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup F\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : IsFiniteMeasure \u03bc\nX : \u03b1 \u2192 \u03b2\nY : \u03b1 \u2192 \u03a9\nm\u03b2 : MeasurableSpace \u03b2\ns : Set \u03a9\nt : Set \u03b2\nf : \u03b2 \u00d7 \u03a9 \u2192 F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nhX : Measurable X\nhY : AEMeasurable Y\nhf_int : Integrable f\nhf_int' : Integrable fun a => f (X a, Y a)\n\u22a2 AEStronglyMeasurable' (MeasurableSpace.comap X m\u03b2) (fun a => \u222b (y : \u03a9), f (X a, y) \u2202\u2191(condDistrib Y X \u03bc) (X a)) \u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Logic.lean", "full_name": "Classical.not_forall", "start": [686, 1], "end": [687, 23], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Sym.lean", "full_name": "Finset.sym_singleton", "start": [177, 1], "end": [180, 87], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/PartrecCode.lean", "full_name": "Nat.Partrec.Code.eval_eq_rfindOpt", "start": [1154, 1], "end": [1157, 40], "traced_tactics": [{"tactic": "refine' evaln_complete.trans (Nat.rfindOpt_mono _).symm", "annotated_tactic": ["refine' evaln_complete.trans (<a>Nat.rfindOpt_mono</a> _).<a>symm</a>", [{"full_name": "Nat.rfindOpt_mono", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [146, 9], "def_end_pos": [146, 22]}, {"full_name": "Iff.symm", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [671, 9], "def_end_pos": [671, 17]}]], "state_before": "c : Code\nn x : \u2115\n\u22a2 x \u2208 eval c n \u2194 x \u2208 rfindOpt fun k => evaln k c n", "state_after": "c : Code\nn x : \u2115\n\u22a2 \u2200 {a m n_1 : \u2115}, m \u2264 n_1 \u2192 a \u2208 evaln m c n \u2192 a \u2208 evaln n_1 c n"}, {"tactic": "intro a m n hl", "annotated_tactic": ["intro a m n hl", []], "state_before": "c : Code\nn x : \u2115\n\u22a2 \u2200 {a m n_1 : \u2115}, m \u2264 n_1 \u2192 a \u2208 evaln m c n \u2192 a \u2208 evaln n_1 c n", "state_after": "c : Code\nn\u271d x a m n : \u2115\nhl : m \u2264 n\n\u22a2 a \u2208 evaln m c n\u271d \u2192 a \u2208 evaln n c n\u271d"}, {"tactic": "apply evaln_mono hl", "annotated_tactic": ["apply <a>evaln_mono</a> hl", [{"full_name": "Nat.Partrec.Code.evaln_mono", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [783, 9], "def_end_pos": [783, 19]}]], "state_before": "c : Code\nn\u271d x a m n : \u2115\nhl : m \u2264 n\n\u22a2 a \u2208 evaln m c n\u271d \u2192 a \u2208 evaln n c n\u271d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/List/Pairwise.lean", "full_name": "List.pairwise_middle", "start": [114, 1], "end": [118, 34], "traced_tactics": [{"tactic": "show Pairwise R (l\u2081 ++ ([a] ++ l\u2082)) \u2194 Pairwise R ([a] ++ l\u2081 ++ l\u2082)", "annotated_tactic": ["show <a>Pairwise</a> R (l\u2081 ++ ([a] ++ l\u2082)) \u2194 <a>Pairwise</a> R ([a] ++ l\u2081 ++ l\u2082)", [{"full_name": "List.Pairwise", "def_path": "lake-packages/std/Std/Data/List/Basic.lean", "def_pos": [1118, 11], "def_end_pos": [1118, 19]}, {"full_name": "List.Pairwise", "def_path": "lake-packages/std/Std/Data/List/Basic.lean", "def_pos": [1118, 11], "def_end_pos": [1118, 19]}]], "state_before": "\u03b1 : Type u_1\nR : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u2200 {x y : \u03b1}, R x y \u2192 R y x\na : \u03b1\nl\u2081 l\u2082 : List \u03b1\n\u22a2 Pairwise R (l\u2081 ++ a :: l\u2082) \u2194 Pairwise R (a :: (l\u2081 ++ l\u2082))", "state_after": "\u03b1 : Type u_1\nR : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u2200 {x y : \u03b1}, R x y \u2192 R y x\na : \u03b1\nl\u2081 l\u2082 : List \u03b1\n\u22a2 Pairwise R (l\u2081 ++ ([a] ++ l\u2082)) \u2194 Pairwise R ([a] ++ l\u2081 ++ l\u2082)"}, {"tactic": "rw [\u2190 append_assoc, pairwise_append, @pairwise_append _ _ ([a] ++ l\u2081), pairwise_append_comm s]", "annotated_tactic": ["rw [\u2190 <a>append_assoc</a>, <a>pairwise_append</a>, @<a>pairwise_append</a> _ _ ([a] ++ l\u2081), <a>pairwise_append_comm</a> s]", [{"full_name": "List.append_assoc", "def_path": "lake-packages/lean4/src/lean/Init/Data/List/Basic.lean", "def_pos": [103, 9], "def_end_pos": [103, 21]}, {"full_name": "List.pairwise_append", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [1457, 9], "def_end_pos": [1457, 24]}, {"full_name": "List.pairwise_append", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [1457, 9], "def_end_pos": [1457, 24]}, {"full_name": "List.pairwise_append_comm", "def_path": "lake-packages/std/Std/Data/List/Pairwise.lean", "def_pos": [108, 9], "def_end_pos": [108, 29]}]], "state_before": "\u03b1 : Type u_1\nR : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u2200 {x y : \u03b1}, R x y \u2192 R y x\na : \u03b1\nl\u2081 l\u2082 : List \u03b1\n\u22a2 Pairwise R (l\u2081 ++ ([a] ++ l\u2082)) \u2194 Pairwise R ([a] ++ l\u2081 ++ l\u2082)", "state_after": "\u03b1 : Type u_1\nR : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u2200 {x y : \u03b1}, R x y \u2192 R y x\na : \u03b1\nl\u2081 l\u2082 : List \u03b1\n\u22a2 (Pairwise (fun {x y} => R x y) ([a] ++ l\u2081) \u2227\n      Pairwise R l\u2082 \u2227 \u2200 (a_1 : \u03b1), a_1 \u2208 l\u2081 ++ [a] \u2192 \u2200 (b : \u03b1), b \u2208 l\u2082 \u2192 R a_1 b) \u2194\n    Pairwise R ([a] ++ l\u2081) \u2227 Pairwise R l\u2082 \u2227 \u2200 (a_1 : \u03b1), a_1 \u2208 [a] ++ l\u2081 \u2192 \u2200 (b : \u03b1), b \u2208 l\u2082 \u2192 R a_1 b"}, {"tactic": "simp only [mem_append, or_comm]", "annotated_tactic": ["simp only [<a>mem_append</a>, <a>or_comm</a>]", [{"full_name": "List.mem_append", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [137, 17], "def_end_pos": [137, 27]}, {"full_name": "or_comm", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [263, 9], "def_end_pos": [263, 16]}]], "state_before": "\u03b1 : Type u_1\nR : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u2200 {x y : \u03b1}, R x y \u2192 R y x\na : \u03b1\nl\u2081 l\u2082 : List \u03b1\n\u22a2 (Pairwise (fun {x y} => R x y) ([a] ++ l\u2081) \u2227\n      Pairwise R l\u2082 \u2227 \u2200 (a_1 : \u03b1), a_1 \u2208 l\u2081 ++ [a] \u2192 \u2200 (b : \u03b1), b \u2208 l\u2082 \u2192 R a_1 b) \u2194\n    Pairwise R ([a] ++ l\u2081) \u2227 Pairwise R l\u2082 \u2227 \u2200 (a_1 : \u03b1), a_1 \u2208 [a] ++ l\u2081 \u2192 \u2200 (b : \u03b1), b \u2208 l\u2082 \u2192 R a_1 b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Holor.lean", "full_name": "Holor.slice_eq", "start": [250, 1], "end": [259, 72], "traced_tactics": [{"tactic": "rw [\u2190 hiis]", "annotated_tactic": ["rw [\u2190 hiis]", []], "state_before": "\u03b1 : Type\nd : \u2115\nds ds\u2081 ds\u2082 ds\u2083 : List \u2115\nx y : Holor \u03b1 (d :: ds)\nh : slice x = slice y\nt : HolorIndex (d :: ds)\ni : \u2115\nis : List \u2115\nhiis : \u2191t = i :: is\n\u22a2 Forall\u2082 (fun x x_1 => x < x_1) (i :: is) (d :: ds)", "state_after": "\u03b1 : Type\nd : \u2115\nds ds\u2081 ds\u2082 ds\u2083 : List \u2115\nx y : Holor \u03b1 (d :: ds)\nh : slice x = slice y\nt : HolorIndex (d :: ds)\ni : \u2115\nis : List \u2115\nhiis : \u2191t = i :: is\n\u22a2 Forall\u2082 (fun x x_1 => x < x_1) (\u2191t) (d :: ds)"}, {"tactic": "exact t.2", "annotated_tactic": ["exact t.2", []], "state_before": "\u03b1 : Type\nd : \u2115\nds ds\u2081 ds\u2082 ds\u2083 : List \u2115\nx y : Holor \u03b1 (d :: ds)\nh : slice x = slice y\nt : HolorIndex (d :: ds)\ni : \u2115\nis : List \u2115\nhiis : \u2191t = i :: is\n\u22a2 Forall\u2082 (fun x x_1 => x < x_1) (\u2191t) (d :: ds)", "state_after": "no goals"}, {"tactic": "rw [h]", "annotated_tactic": ["rw [h]", []], "state_before": "\u03b1 : Type\nd : \u2115\nds ds\u2081 ds\u2082 ds\u2083 : List \u2115\nx y : Holor \u03b1 (d :: ds)\nh : slice x = slice y\nt : HolorIndex (d :: ds)\ni : \u2115\nis : List \u2115\nhiis : \u2191t = i :: is\nhiisdds : Forall\u2082 (fun x x_1 => x < x_1) (i :: is) (d :: ds)\nhid : i < d\nhisds : Forall\u2082 (fun x x_1 => x < x_1) is ds\n\u22a2 slice x i hid { val := is, property := hisds } = slice y i hid { val := is, property := hisds }", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "full_name": "MeasureTheory.setToFun_congr_left'", "start": [1304, 1], "end": [1309, 34], "traced_tactics": [{"tactic": "by_cases hf : Integrable f \u03bc", "annotated_tactic": ["by_cases hf : <a>Integrable</a> f \u03bc", [{"full_name": "MeasureTheory.Integrable", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [442, 5], "def_end_pos": [442, 15]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc T' C'\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T s = T' s\nf : \u03b1 \u2192 E\n\u22a2 setToFun \u03bc T hT f = setToFun \u03bc T' hT' f", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc T' C'\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T s = T' s\nf : \u03b1 \u2192 E\nhf : Integrable f\n\u22a2 setToFun \u03bc T hT f = setToFun \u03bc T' hT' f\n\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc T' C'\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T s = T' s\nf : \u03b1 \u2192 E\nhf : \u00acIntegrable f\n\u22a2 setToFun \u03bc T hT f = setToFun \u03bc T' hT' f"}, {"tactic": "simp_rw [setToFun_eq _ hf, L1.setToL1_congr_left' T T' hT hT' h]", "annotated_tactic": ["simp_rw [<a>setToFun_eq</a> _ hf, <a>L1.setToL1_congr_left'</a> T T' hT hT' h]", [{"full_name": "MeasureTheory.setToFun_eq", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [1276, 9], "def_end_pos": [1276, 20]}, {"full_name": "MeasureTheory.L1.setToL1_congr_left'", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [1065, 9], "def_end_pos": [1065, 28]}]], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc T' C'\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T s = T' s\nf : \u03b1 \u2192 E\nhf : Integrable f\n\u22a2 setToFun \u03bc T hT f = setToFun \u03bc T' hT' f", "state_after": "no goals"}, {"tactic": "simp_rw [setToFun_undef _ hf]", "annotated_tactic": ["simp_rw [<a>setToFun_undef</a> _ hf]", [{"full_name": "MeasureTheory.setToFun_undef", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [1286, 9], "def_end_pos": [1286, 23]}]], "state_before": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc T' C'\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T s = T' s\nf : \u03b1 \u2192 E\nhf : \u00acIntegrable f\n\u22a2 setToFun \u03bc T hT f = setToFun \u03bc T' hT' f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Int/GCD.lean", "full_name": "Int.lcm_zero_right", "start": [461, 1], "end": [463, 27], "traced_tactics": [{"tactic": "rw [Int.lcm]", "annotated_tactic": ["rw [<a>Int.lcm</a>]", [{"full_name": "Int.lcm", "def_path": "Mathlib/Data/Int/GCD.lean", "def_pos": [235, 5], "def_end_pos": [235, 8]}]], "state_before": "i : \u2124\n\u22a2 lcm i 0 = 0", "state_after": "i : \u2124\n\u22a2 Nat.lcm (natAbs i) (natAbs 0) = 0"}, {"tactic": "apply Nat.lcm_zero_right", "annotated_tactic": ["apply <a>Nat.lcm_zero_right</a>", [{"full_name": "Nat.lcm_zero_right", "def_path": "lake-packages/std/Std/Data/Nat/Gcd.lean", "def_pos": [201, 17], "def_end_pos": [201, 31]}]], "state_before": "i : \u2124\n\u22a2 Nat.lcm (natAbs i) (natAbs 0) = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/FundThmCalculus.lean", "full_name": "intervalIntegral.integrableOn_deriv_of_nonneg", "start": [1290, 1], "end": [1293, 94], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "full_name": "String.revFind_of_valid", "start": [364, 1], "end": [366, 51], "traced_tactics": [{"tactic": "simpa using revFindAux_of_valid p s.1.reverse []", "annotated_tactic": ["simpa using <a>revFindAux_of_valid</a> p s.1.<a>reverse</a> []", [{"full_name": "String.revFindAux_of_valid", "def_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "def_pos": [353, 9], "def_end_pos": [353, 28]}, {"full_name": "List.reverse", "def_path": "lake-packages/lean4/src/lean/Init/Data/List/Basic.lean", "def_pos": [54, 5], "def_end_pos": [54, 12]}]], "state_before": "p : Char \u2192 Bool\ns : String\n\u22a2 revFind s p =\n    Option.map (fun x => { byteIdx := utf8Len x }) (List.tail? (List.dropWhile (fun x => !p x) (List.reverse s.data)))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/List/Basic.lean", "full_name": "List.zipWith_eq_zipWithTR", "start": [180, 10], "end": [185, 28], "traced_tactics": [{"tactic": "funext \u03b1 \u03b2 \u03b3 f as bs", "annotated_tactic": ["funext \u03b1 \u03b2 \u03b3 f as bs", []], "state_before": "\u22a2 @zipWith = @zipWithTR", "state_after": "case h.h.h.h.h.h\n\u03b1 : Type u_3\n\u03b2 : Type u_2\n\u03b3 : Type u_1\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nas : List \u03b1\nbs : List \u03b2\n\u22a2 zipWith f as bs = zipWithTR f as bs"}, {"tactic": "exact (go as bs #[]).symm", "annotated_tactic": ["exact (go as bs #[]).<a>symm</a>", [{"full_name": "Eq.symm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [310, 9], "def_end_pos": [310, 16]}]], "state_before": "case h.h.h.h.h.h\n\u03b1 : Type u_3\n\u03b2 : Type u_2\n\u03b3 : Type u_1\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nas : List \u03b1\nbs : List \u03b2\n\u22a2 zipWith f as bs = zipWithTR f as bs", "state_after": "no goals"}, {"tactic": "simp [zipWithTR.go, zipWith]", "annotated_tactic": ["simp [<a>zipWithTR.go</a>, <a>zipWith</a>]", [{"full_name": "List.zipWithTR.go", "def_path": "lake-packages/std/Std/Data/List/Basic.lean", "def_pos": [176, 3], "def_end_pos": [176, 5]}, {"full_name": "List.zipWith", "def_path": "lake-packages/lean4/src/lean/Init/Data/List/Basic.lean", "def_pos": [548, 19], "def_end_pos": [548, 26]}]], "state_before": "\u03b1 : Type u_3\n\u03b2 : Type u_2\n\u03b3 : Type u_1\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nas : List \u03b1\nbs : List \u03b2\nhead\u271d : \u03b1\ntail\u271d : List \u03b1\nacc : Array \u03b3\n\u22a2 zipWithTR.go f (head\u271d :: tail\u271d) [] acc = acc.data ++ zipWith f (head\u271d :: tail\u271d) []", "state_after": "no goals"}, {"tactic": "simp [zipWithTR.go, zipWith, go as bs]", "annotated_tactic": ["simp [<a>zipWithTR.go</a>, <a>zipWith</a>, go as bs]", [{"full_name": "List.zipWithTR.go", "def_path": "lake-packages/std/Std/Data/List/Basic.lean", "def_pos": [176, 3], "def_end_pos": [176, 5]}, {"full_name": "List.zipWith", "def_path": "lake-packages/lean4/src/lean/Init/Data/List/Basic.lean", "def_pos": [548, 19], "def_end_pos": [548, 26]}]], "state_before": "\u03b1 : Type u_3\n\u03b2 : Type u_2\n\u03b3 : Type u_1\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nas\u271d : List \u03b1\nbs\u271d : List \u03b2\na : \u03b1\nas : List \u03b1\nb : \u03b2\nbs : List \u03b2\nacc : Array \u03b3\n\u22a2 zipWithTR.go f (a :: as) (b :: bs) acc = acc.data ++ zipWith f (a :: as) (b :: bs)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finmap.lean", "full_name": "Finmap.lookup_erase_ne", "start": [451, 1], "end": [452, 50], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Vector/MapLemmas.lean", "full_name": "Vector.mapAccumr\u2082_eq_map\u2082", "start": [252, 1], "end": [259, 64], "traced_tactics": [{"tactic": "rw[Vector.map\u2082_eq_mapAccumr\u2082]", "annotated_tactic": ["rw[<a>Vector.map\u2082_eq_mapAccumr\u2082</a>]", [{"full_name": "Vector.map\u2082_eq_mapAccumr\u2082", "def_path": "Mathlib/Data/Vector/MapLemmas.lean", "def_pos": [243, 19], "def_end_pos": [243, 37]}]], "state_before": "\u03b1 : Type\nn : \u2115\n\u03b2 : Type\nxs : Vector \u03b1 n\nys : Vector \u03b2 n\n\u03c3 \u03b3 : Type\nf : \u03b1 \u2192 \u03b2 \u2192 \u03c3 \u2192 \u03c3 \u00d7 \u03b3\ns\u2080 : \u03c3\nS : Set \u03c3\nh\u2080 : s\u2080 \u2208 S\nclosure : \u2200 (a : \u03b1) (b : \u03b2) (s : \u03c3), s \u2208 S \u2192 (f a b s).1 \u2208 S\nout : \u2200 (a : \u03b1) (b : \u03b2) (s s' : \u03c3), s \u2208 S \u2192 s' \u2208 S \u2192 (f a b s).2 = (f a b s').2\n\u22a2 (mapAccumr\u2082 f xs ys s\u2080).2 = map\u2082 (fun x x_1 => (f x x_1 s\u2080).2) xs ys", "state_after": "\u03b1 : Type\nn : \u2115\n\u03b2 : Type\nxs : Vector \u03b1 n\nys : Vector \u03b2 n\n\u03c3 \u03b3 : Type\nf : \u03b1 \u2192 \u03b2 \u2192 \u03c3 \u2192 \u03c3 \u00d7 \u03b3\ns\u2080 : \u03c3\nS : Set \u03c3\nh\u2080 : s\u2080 \u2208 S\nclosure : \u2200 (a : \u03b1) (b : \u03b2) (s : \u03c3), s \u2208 S \u2192 (f a b s).1 \u2208 S\nout : \u2200 (a : \u03b1) (b : \u03b2) (s s' : \u03c3), s \u2208 S \u2192 s' \u2208 S \u2192 (f a b s).2 = (f a b s').2\n\u22a2 (mapAccumr\u2082 f xs ys s\u2080).2 = (mapAccumr\u2082 (fun x y x_1 => ((), (f x y s\u2080).2)) xs ys ()).2"}, {"tactic": "apply mapAccumr\u2082_bisim_tail", "annotated_tactic": ["apply <a>mapAccumr\u2082_bisim_tail</a>", [{"full_name": "Vector.mapAccumr\u2082_bisim_tail", "def_path": "Mathlib/Data/Vector/MapLemmas.lean", "def_pos": [205, 9], "def_end_pos": [205, 30]}]], "state_before": "\u03b1 : Type\nn : \u2115\n\u03b2 : Type\nxs : Vector \u03b1 n\nys : Vector \u03b2 n\n\u03c3 \u03b3 : Type\nf : \u03b1 \u2192 \u03b2 \u2192 \u03c3 \u2192 \u03c3 \u00d7 \u03b3\ns\u2080 : \u03c3\nS : Set \u03c3\nh\u2080 : s\u2080 \u2208 S\nclosure : \u2200 (a : \u03b1) (b : \u03b2) (s : \u03c3), s \u2208 S \u2192 (f a b s).1 \u2208 S\nout : \u2200 (a : \u03b1) (b : \u03b2) (s s' : \u03c3), s \u2208 S \u2192 s' \u2208 S \u2192 (f a b s).2 = (f a b s').2\n\u22a2 (mapAccumr\u2082 f xs ys s\u2080).2 = (mapAccumr\u2082 (fun x y x_1 => ((), (f x y s\u2080).2)) xs ys ()).2", "state_after": "case h\n\u03b1 : Type\nn : \u2115\n\u03b2 : Type\nxs : Vector \u03b1 n\nys : Vector \u03b2 n\n\u03c3 \u03b3 : Type\nf : \u03b1 \u2192 \u03b2 \u2192 \u03c3 \u2192 \u03c3 \u00d7 \u03b3\ns\u2080 : \u03c3\nS : Set \u03c3\nh\u2080 : s\u2080 \u2208 S\nclosure : \u2200 (a : \u03b1) (b : \u03b2) (s : \u03c3), s \u2208 S \u2192 (f a b s).1 \u2208 S\nout : \u2200 (a : \u03b1) (b : \u03b2) (s s' : \u03c3), s \u2208 S \u2192 s' \u2208 S \u2192 (f a b s).2 = (f a b s').2\n\u22a2 \u2203 R,\n    R s\u2080 () \u2227\n      \u2200 {s : \u03c3} {q : Unit} (a : \u03b1) (b : \u03b2),\n        R s q \u2192 R (f a b s).1 ((), (f a b s\u2080).2).1 \u2227 (f a b s).2 = ((), (f a b s\u2080).2).2"}, {"tactic": "use fun s _ => s \u2208 S, h\u2080", "annotated_tactic": ["use fun s _ => s \u2208 S, h\u2080", []], "state_before": "case h\n\u03b1 : Type\nn : \u2115\n\u03b2 : Type\nxs : Vector \u03b1 n\nys : Vector \u03b2 n\n\u03c3 \u03b3 : Type\nf : \u03b1 \u2192 \u03b2 \u2192 \u03c3 \u2192 \u03c3 \u00d7 \u03b3\ns\u2080 : \u03c3\nS : Set \u03c3\nh\u2080 : s\u2080 \u2208 S\nclosure : \u2200 (a : \u03b1) (b : \u03b2) (s : \u03c3), s \u2208 S \u2192 (f a b s).1 \u2208 S\nout : \u2200 (a : \u03b1) (b : \u03b2) (s s' : \u03c3), s \u2208 S \u2192 s' \u2208 S \u2192 (f a b s).2 = (f a b s').2\n\u22a2 \u2203 R,\n    R s\u2080 () \u2227\n      \u2200 {s : \u03c3} {q : Unit} (a : \u03b1) (b : \u03b2),\n        R s q \u2192 R (f a b s).1 ((), (f a b s\u2080).2).1 \u2227 (f a b s).2 = ((), (f a b s\u2080).2).2", "state_after": "case right\n\u03b1 : Type\nn : \u2115\n\u03b2 : Type\nxs : Vector \u03b1 n\nys : Vector \u03b2 n\n\u03c3 \u03b3 : Type\nf : \u03b1 \u2192 \u03b2 \u2192 \u03c3 \u2192 \u03c3 \u00d7 \u03b3\ns\u2080 : \u03c3\nS : Set \u03c3\nh\u2080 : s\u2080 \u2208 S\nclosure : \u2200 (a : \u03b1) (b : \u03b2) (s : \u03c3), s \u2208 S \u2192 (f a b s).1 \u2208 S\nout : \u2200 (a : \u03b1) (b : \u03b2) (s s' : \u03c3), s \u2208 S \u2192 s' \u2208 S \u2192 (f a b s).2 = (f a b s').2\n\u22a2 \u2200 {s : \u03c3} {q : Unit} (a : \u03b1) (b : \u03b2), s \u2208 S \u2192 (f a b s).1 \u2208 S \u2227 (f a b s).2 = ((), (f a b s\u2080).2).2"}, {"tactic": "exact @fun s _q a b h => \u27e8closure a b s h, out a b s s\u2080 h h\u2080\u27e9", "annotated_tactic": ["exact @fun s _q a b h => \u27e8closure a b s h, out a b s s\u2080 h h\u2080\u27e9", []], "state_before": "case right\n\u03b1 : Type\nn : \u2115\n\u03b2 : Type\nxs : Vector \u03b1 n\nys : Vector \u03b2 n\n\u03c3 \u03b3 : Type\nf : \u03b1 \u2192 \u03b2 \u2192 \u03c3 \u2192 \u03c3 \u00d7 \u03b3\ns\u2080 : \u03c3\nS : Set \u03c3\nh\u2080 : s\u2080 \u2208 S\nclosure : \u2200 (a : \u03b1) (b : \u03b2) (s : \u03c3), s \u2208 S \u2192 (f a b s).1 \u2208 S\nout : \u2200 (a : \u03b1) (b : \u03b2) (s s' : \u03c3), s \u2208 S \u2192 s' \u2208 S \u2192 (f a b s).2 = (f a b s').2\n\u22a2 \u2200 {s : \u03c3} {q : Unit} (a : \u03b1) (b : \u03b2), s \u2208 S \u2192 (f a b s).1 \u2208 S \u2227 (f a b s).2 = ((), (f a b s\u2080).2).2", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Pointwise/Interval.lean", "full_name": "Set.Iio_add_bij", "start": [415, 1], "end": [416, 73], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "full_name": "AEMeasurable.isGLB", "start": [1236, 1], "end": [1239, 38], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Haar/OfBasis.lean", "full_name": "parallelepiped_comp_equiv", "start": [62, 1], "end": [80, 84], "traced_tactics": [{"tactic": "simp only [parallelepiped]", "annotated_tactic": ["simp only [<a>parallelepiped</a>]", [{"full_name": "parallelepiped", "def_path": "Mathlib/MeasureTheory/Measure/Haar/OfBasis.lean", "def_pos": [44, 5], "def_end_pos": [44, 19]}]], "state_before": "\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2075 : Fintype \u03b9\ninst\u271d\u2074 : Fintype \u03b9'\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \u211d E\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \u211d F\nv : \u03b9 \u2192 E\ne : \u03b9' \u2243 \u03b9\n\u22a2 parallelepiped (v \u2218 \u2191e) = parallelepiped v", "state_after": "\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2075 : Fintype \u03b9\ninst\u271d\u2074 : Fintype \u03b9'\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \u211d E\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \u211d F\nv : \u03b9 \u2192 E\ne : \u03b9' \u2243 \u03b9\n\u22a2 (fun a => \u2211 x : \u03b9', a x \u2022 (v \u2218 \u2191e) x) '' Icc 0 1 = (fun t => \u2211 i : \u03b9, t i \u2022 v i) '' Icc 0 1"}, {"tactic": "let K : (\u03b9' \u2192 \u211d) \u2243 (\u03b9 \u2192 \u211d) := Equiv.piCongrLeft' (fun _a : \u03b9' => \u211d) e", "annotated_tactic": ["let K : (\u03b9' \u2192 \u211d) \u2243 (\u03b9 \u2192 \u211d) := <a>Equiv.piCongrLeft'</a> (fun _a : \u03b9' => \u211d) e", [{"full_name": "Equiv.piCongrLeft'", "def_path": "Mathlib/Logic/Equiv/Basic.lean", "def_pos": [1823, 5], "def_end_pos": [1823, 17]}]], "state_before": "\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2075 : Fintype \u03b9\ninst\u271d\u2074 : Fintype \u03b9'\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \u211d E\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \u211d F\nv : \u03b9 \u2192 E\ne : \u03b9' \u2243 \u03b9\n\u22a2 (fun a => \u2211 x : \u03b9', a x \u2022 (v \u2218 \u2191e) x) '' Icc 0 1 = (fun t => \u2211 i : \u03b9, t i \u2022 v i) '' Icc 0 1", "state_after": "\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2075 : Fintype \u03b9\ninst\u271d\u2074 : Fintype \u03b9'\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \u211d E\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \u211d F\nv : \u03b9 \u2192 E\ne : \u03b9' \u2243 \u03b9\nK : (\u03b9' \u2192 \u211d) \u2243 (\u03b9 \u2192 \u211d) := Equiv.piCongrLeft' (fun _a => \u211d) e\n\u22a2 (fun a => \u2211 x : \u03b9', a x \u2022 (v \u2218 \u2191e) x) '' Icc 0 1 = (fun t => \u2211 i : \u03b9, t i \u2022 v i) '' Icc 0 1"}, {"tactic": "rw [this, \u2190 image_comp]", "annotated_tactic": ["rw [this, \u2190 <a>image_comp</a>]", [{"full_name": "Set.image_comp", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [293, 9], "def_end_pos": [293, 19]}]], "state_before": "\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2075 : Fintype \u03b9\ninst\u271d\u2074 : Fintype \u03b9'\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \u211d E\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \u211d F\nv : \u03b9 \u2192 E\ne : \u03b9' \u2243 \u03b9\nK : (\u03b9' \u2192 \u211d) \u2243 (\u03b9 \u2192 \u211d) := Equiv.piCongrLeft' (fun _a => \u211d) e\nthis : Icc 0 1 = \u2191K '' Icc 0 1\n\u22a2 (fun a => \u2211 x : \u03b9', a x \u2022 (v \u2218 \u2191e) x) '' Icc 0 1 = (fun t => \u2211 i : \u03b9, t i \u2022 v i) '' Icc 0 1", "state_after": "\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2075 : Fintype \u03b9\ninst\u271d\u2074 : Fintype \u03b9'\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \u211d E\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \u211d F\nv : \u03b9 \u2192 E\ne : \u03b9' \u2243 \u03b9\nK : (\u03b9' \u2192 \u211d) \u2243 (\u03b9 \u2192 \u211d) := Equiv.piCongrLeft' (fun _a => \u211d) e\nthis : Icc 0 1 = \u2191K '' Icc 0 1\n\u22a2 (fun a => \u2211 x : \u03b9', a x \u2022 (v \u2218 \u2191e) x) '' Icc 0 1 = (fun t => \u2211 i : \u03b9, t i \u2022 v i) \u2218 \u2191K '' Icc 0 1"}, {"tactic": "congr 1 with x", "annotated_tactic": ["congr 1 with x", []], "state_before": "\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2075 : Fintype \u03b9\ninst\u271d\u2074 : Fintype \u03b9'\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \u211d E\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \u211d F\nv : \u03b9 \u2192 E\ne : \u03b9' \u2243 \u03b9\nK : (\u03b9' \u2192 \u211d) \u2243 (\u03b9 \u2192 \u211d) := Equiv.piCongrLeft' (fun _a => \u211d) e\nthis : Icc 0 1 = \u2191K '' Icc 0 1\n\u22a2 (fun a => \u2211 x : \u03b9', a x \u2022 (v \u2218 \u2191e) x) '' Icc 0 1 = (fun t => \u2211 i : \u03b9, t i \u2022 v i) \u2218 \u2191K '' Icc 0 1", "state_after": "case h\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2075 : Fintype \u03b9\ninst\u271d\u2074 : Fintype \u03b9'\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \u211d E\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \u211d F\nv : \u03b9 \u2192 E\ne : \u03b9' \u2243 \u03b9\nK : (\u03b9' \u2192 \u211d) \u2243 (\u03b9 \u2192 \u211d) := Equiv.piCongrLeft' (fun _a => \u211d) e\nthis : Icc 0 1 = \u2191K '' Icc 0 1\nx : E\n\u22a2 x \u2208 (fun a => \u2211 x : \u03b9', a x \u2022 (v \u2218 \u2191e) x) '' Icc 0 1 \u2194 x \u2208 (fun t => \u2211 i : \u03b9, t i \u2022 v i) \u2218 \u2191K '' Icc 0 1"}, {"tactic": "have := fun z : \u03b9' \u2192 \u211d => e.symm.sum_comp fun i => z i \u2022 v (e i)", "annotated_tactic": ["have := fun z : \u03b9' \u2192 \u211d => e.symm.sum_comp fun i => z i \u2022 v (e i)", []], "state_before": "case h\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2075 : Fintype \u03b9\ninst\u271d\u2074 : Fintype \u03b9'\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \u211d E\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \u211d F\nv : \u03b9 \u2192 E\ne : \u03b9' \u2243 \u03b9\nK : (\u03b9' \u2192 \u211d) \u2243 (\u03b9 \u2192 \u211d) := Equiv.piCongrLeft' (fun _a => \u211d) e\nthis : Icc 0 1 = \u2191K '' Icc 0 1\nx : E\n\u22a2 x \u2208 (fun a => \u2211 x : \u03b9', a x \u2022 (v \u2218 \u2191e) x) '' Icc 0 1 \u2194 x \u2208 (fun t => \u2211 i : \u03b9, t i \u2022 v i) \u2218 \u2191K '' Icc 0 1", "state_after": "case h\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2075 : Fintype \u03b9\ninst\u271d\u2074 : Fintype \u03b9'\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \u211d E\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \u211d F\nv : \u03b9 \u2192 E\ne : \u03b9' \u2243 \u03b9\nK : (\u03b9' \u2192 \u211d) \u2243 (\u03b9 \u2192 \u211d) := Equiv.piCongrLeft' (fun _a => \u211d) e\nthis\u271d : Icc 0 1 = \u2191K '' Icc 0 1\nx : E\nthis : \u2200 (z : \u03b9' \u2192 \u211d), \u2211 i : \u03b9, z (\u2191e.symm i) \u2022 v (\u2191e (\u2191e.symm i)) = \u2211 i : \u03b9', z i \u2022 v (\u2191e i)\n\u22a2 x \u2208 (fun a => \u2211 x : \u03b9', a x \u2022 (v \u2218 \u2191e) x) '' Icc 0 1 \u2194 x \u2208 (fun t => \u2211 i : \u03b9, t i \u2022 v i) \u2218 \u2191K '' Icc 0 1"}, {"tactic": "simp_rw [Equiv.apply_symm_apply] at this", "annotated_tactic": ["simp_rw [<a>Equiv.apply_symm_apply</a>] at this", [{"full_name": "Equiv.apply_symm_apply", "def_path": "Mathlib/Logic/Equiv/Defs.lean", "def_pos": [280, 17], "def_end_pos": [280, 33]}]], "state_before": "case h\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2075 : Fintype \u03b9\ninst\u271d\u2074 : Fintype \u03b9'\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \u211d E\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \u211d F\nv : \u03b9 \u2192 E\ne : \u03b9' \u2243 \u03b9\nK : (\u03b9' \u2192 \u211d) \u2243 (\u03b9 \u2192 \u211d) := Equiv.piCongrLeft' (fun _a => \u211d) e\nthis\u271d : Icc 0 1 = \u2191K '' Icc 0 1\nx : E\nthis : \u2200 (z : \u03b9' \u2192 \u211d), \u2211 i : \u03b9, z (\u2191e.symm i) \u2022 v (\u2191e (\u2191e.symm i)) = \u2211 i : \u03b9', z i \u2022 v (\u2191e i)\n\u22a2 x \u2208 (fun a => \u2211 x : \u03b9', a x \u2022 (v \u2218 \u2191e) x) '' Icc 0 1 \u2194 x \u2208 (fun t => \u2211 i : \u03b9, t i \u2022 v i) \u2218 \u2191K '' Icc 0 1", "state_after": "case h\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2075 : Fintype \u03b9\ninst\u271d\u2074 : Fintype \u03b9'\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \u211d E\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \u211d F\nv : \u03b9 \u2192 E\ne : \u03b9' \u2243 \u03b9\nK : (\u03b9' \u2192 \u211d) \u2243 (\u03b9 \u2192 \u211d) := Equiv.piCongrLeft' (fun _a => \u211d) e\nthis\u271d : Icc 0 1 = \u2191K '' Icc 0 1\nx : E\nthis : \u2200 (z : \u03b9' \u2192 \u211d), \u2211 x : \u03b9, z (\u2191e.symm x) \u2022 v x = \u2211 x : \u03b9', z x \u2022 v (\u2191e x)\n\u22a2 x \u2208 (fun a => \u2211 x : \u03b9', a x \u2022 (v \u2218 \u2191e) x) '' Icc 0 1 \u2194 x \u2208 (fun t => \u2211 i : \u03b9, t i \u2022 v i) \u2218 \u2191K '' Icc 0 1"}, {"tactic": "simp_rw [Function.comp_apply, mem_image, mem_Icc, Equiv.piCongrLeft'_apply, this]", "annotated_tactic": ["simp_rw [<a>Function.comp_apply</a>, <a>mem_image</a>, <a>mem_Icc</a>, <a>Equiv.piCongrLeft'_apply</a>, this]", [{"full_name": "Function.comp_apply", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [33, 17], "def_end_pos": [33, 36]}, {"full_name": "Set.mem_image", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [231, 9], "def_end_pos": [231, 18]}, {"full_name": "Set.mem_Icc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [131, 9], "def_end_pos": [131, 16]}, {"full_name": "Equiv.piCongrLeft'_apply", "def_path": "Mathlib/Logic/Equiv/Basic.lean", "def_pos": [1822, 3], "def_end_pos": [1822, 8]}]], "state_before": "case h\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2075 : Fintype \u03b9\ninst\u271d\u2074 : Fintype \u03b9'\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \u211d E\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \u211d F\nv : \u03b9 \u2192 E\ne : \u03b9' \u2243 \u03b9\nK : (\u03b9' \u2192 \u211d) \u2243 (\u03b9 \u2192 \u211d) := Equiv.piCongrLeft' (fun _a => \u211d) e\nthis\u271d : Icc 0 1 = \u2191K '' Icc 0 1\nx : E\nthis : \u2200 (z : \u03b9' \u2192 \u211d), \u2211 x : \u03b9, z (\u2191e.symm x) \u2022 v x = \u2211 x : \u03b9', z x \u2022 v (\u2191e x)\n\u22a2 x \u2208 (fun a => \u2211 x : \u03b9', a x \u2022 (v \u2218 \u2191e) x) '' Icc 0 1 \u2194 x \u2208 (fun t => \u2211 i : \u03b9, t i \u2022 v i) \u2218 \u2191K '' Icc 0 1", "state_after": "no goals"}, {"tactic": "rw [\u2190 Equiv.preimage_eq_iff_eq_image]", "annotated_tactic": ["rw [\u2190 <a>Equiv.preimage_eq_iff_eq_image</a>]", [{"full_name": "Equiv.preimage_eq_iff_eq_image", "def_path": "Mathlib/Logic/Equiv/Set.lean", "def_pos": [131, 9], "def_end_pos": [131, 33]}]], "state_before": "\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2075 : Fintype \u03b9\ninst\u271d\u2074 : Fintype \u03b9'\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \u211d E\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \u211d F\nv : \u03b9 \u2192 E\ne : \u03b9' \u2243 \u03b9\nK : (\u03b9' \u2192 \u211d) \u2243 (\u03b9 \u2192 \u211d) := Equiv.piCongrLeft' (fun _a => \u211d) e\n\u22a2 Icc 0 1 = \u2191K '' Icc 0 1", "state_after": "\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2075 : Fintype \u03b9\ninst\u271d\u2074 : Fintype \u03b9'\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \u211d E\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \u211d F\nv : \u03b9 \u2192 E\ne : \u03b9' \u2243 \u03b9\nK : (\u03b9' \u2192 \u211d) \u2243 (\u03b9 \u2192 \u211d) := Equiv.piCongrLeft' (fun _a => \u211d) e\n\u22a2 \u2191K \u207b\u00b9' Icc 0 1 = Icc 0 1"}, {"tactic": "ext x", "annotated_tactic": ["ext x", []], "state_before": "\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2075 : Fintype \u03b9\ninst\u271d\u2074 : Fintype \u03b9'\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \u211d E\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \u211d F\nv : \u03b9 \u2192 E\ne : \u03b9' \u2243 \u03b9\nK : (\u03b9' \u2192 \u211d) \u2243 (\u03b9 \u2192 \u211d) := Equiv.piCongrLeft' (fun _a => \u211d) e\n\u22a2 \u2191K \u207b\u00b9' Icc 0 1 = Icc 0 1", "state_after": "case h\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2075 : Fintype \u03b9\ninst\u271d\u2074 : Fintype \u03b9'\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \u211d E\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \u211d F\nv : \u03b9 \u2192 E\ne : \u03b9' \u2243 \u03b9\nK : (\u03b9' \u2192 \u211d) \u2243 (\u03b9 \u2192 \u211d) := Equiv.piCongrLeft' (fun _a => \u211d) e\nx : \u03b9' \u2192 \u211d\n\u22a2 x \u2208 \u2191K \u207b\u00b9' Icc 0 1 \u2194 x \u2208 Icc 0 1"}, {"tactic": "simp only [mem_preimage, mem_Icc, Pi.le_def, Pi.zero_apply, Equiv.piCongrLeft'_apply,\n  Pi.one_apply]", "annotated_tactic": ["simp only [<a>mem_preimage</a>, <a>mem_Icc</a>, <a>Pi.le_def</a>, <a>Pi.zero_apply</a>, <a>Equiv.piCongrLeft'_apply</a>,\n      <a>Pi.one_apply</a>]", [{"full_name": "Set.mem_preimage", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [64, 9], "def_end_pos": [64, 21]}, {"full_name": "Set.mem_Icc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [131, 9], "def_end_pos": [131, 16]}, {"full_name": "Pi.le_def", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [814, 9], "def_end_pos": [814, 18]}, {"full_name": "Pi.zero_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [46, 3], "def_end_pos": [46, 14]}, {"full_name": "Equiv.piCongrLeft'_apply", "def_path": "Mathlib/Logic/Equiv/Basic.lean", "def_pos": [1822, 3], "def_end_pos": [1822, 8]}, {"full_name": "Pi.one_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [47, 9], "def_end_pos": [47, 18]}]], "state_before": "case h\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2075 : Fintype \u03b9\ninst\u271d\u2074 : Fintype \u03b9'\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \u211d E\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \u211d F\nv : \u03b9 \u2192 E\ne : \u03b9' \u2243 \u03b9\nK : (\u03b9' \u2192 \u211d) \u2243 (\u03b9 \u2192 \u211d) := Equiv.piCongrLeft' (fun _a => \u211d) e\nx : \u03b9' \u2192 \u211d\n\u22a2 x \u2208 \u2191K \u207b\u00b9' Icc 0 1 \u2194 x \u2208 Icc 0 1", "state_after": "case h\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2075 : Fintype \u03b9\ninst\u271d\u2074 : Fintype \u03b9'\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \u211d E\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \u211d F\nv : \u03b9 \u2192 E\ne : \u03b9' \u2243 \u03b9\nK : (\u03b9' \u2192 \u211d) \u2243 (\u03b9 \u2192 \u211d) := Equiv.piCongrLeft' (fun _a => \u211d) e\nx : \u03b9' \u2192 \u211d\n\u22a2 ((\u2200 (i : \u03b9), 0 \u2264 x (\u2191e.symm i)) \u2227 \u2200 (i : \u03b9), x (\u2191e.symm i) \u2264 1) \u2194 (\u2200 (i : \u03b9'), 0 \u2264 x i) \u2227 \u2200 (i : \u03b9'), x i \u2264 1"}, {"tactic": "refine'\n  \u27e8fun h => \u27e8fun i => _, fun i => _\u27e9, fun h =>\n    \u27e8fun i => h.1 (e.symm i), fun i => h.2 (e.symm i)\u27e9\u27e9", "annotated_tactic": ["refine'\n      \u27e8fun h => \u27e8fun i => _, fun i => _\u27e9, fun h =>\n        \u27e8fun i => h.1 (e.symm i), fun i => h.2 (e.symm i)\u27e9\u27e9", []], "state_before": "case h\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2075 : Fintype \u03b9\ninst\u271d\u2074 : Fintype \u03b9'\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \u211d E\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \u211d F\nv : \u03b9 \u2192 E\ne : \u03b9' \u2243 \u03b9\nK : (\u03b9' \u2192 \u211d) \u2243 (\u03b9 \u2192 \u211d) := Equiv.piCongrLeft' (fun _a => \u211d) e\nx : \u03b9' \u2192 \u211d\n\u22a2 ((\u2200 (i : \u03b9), 0 \u2264 x (\u2191e.symm i)) \u2227 \u2200 (i : \u03b9), x (\u2191e.symm i) \u2264 1) \u2194 (\u2200 (i : \u03b9'), 0 \u2264 x i) \u2227 \u2200 (i : \u03b9'), x i \u2264 1", "state_after": "case h.refine'_1\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2075 : Fintype \u03b9\ninst\u271d\u2074 : Fintype \u03b9'\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \u211d E\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \u211d F\nv : \u03b9 \u2192 E\ne : \u03b9' \u2243 \u03b9\nK : (\u03b9' \u2192 \u211d) \u2243 (\u03b9 \u2192 \u211d) := Equiv.piCongrLeft' (fun _a => \u211d) e\nx : \u03b9' \u2192 \u211d\nh : (\u2200 (i : \u03b9), 0 \u2264 x (\u2191e.symm i)) \u2227 \u2200 (i : \u03b9), x (\u2191e.symm i) \u2264 1\ni : \u03b9'\n\u22a2 0 \u2264 x i\n\ncase h.refine'_2\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2075 : Fintype \u03b9\ninst\u271d\u2074 : Fintype \u03b9'\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \u211d E\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \u211d F\nv : \u03b9 \u2192 E\ne : \u03b9' \u2243 \u03b9\nK : (\u03b9' \u2192 \u211d) \u2243 (\u03b9 \u2192 \u211d) := Equiv.piCongrLeft' (fun _a => \u211d) e\nx : \u03b9' \u2192 \u211d\nh : (\u2200 (i : \u03b9), 0 \u2264 x (\u2191e.symm i)) \u2227 \u2200 (i : \u03b9), x (\u2191e.symm i) \u2264 1\ni : \u03b9'\n\u22a2 x i \u2264 1"}, {"tactic": "simpa only [Equiv.symm_apply_apply] using h.1 (e i)", "annotated_tactic": ["simpa only [<a>Equiv.symm_apply_apply</a>] using h.1 (e i)", [{"full_name": "Equiv.symm_apply_apply", "def_path": "Mathlib/Logic/Equiv/Defs.lean", "def_pos": [283, 17], "def_end_pos": [283, 33]}]], "state_before": "case h.refine'_1\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2075 : Fintype \u03b9\ninst\u271d\u2074 : Fintype \u03b9'\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \u211d E\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \u211d F\nv : \u03b9 \u2192 E\ne : \u03b9' \u2243 \u03b9\nK : (\u03b9' \u2192 \u211d) \u2243 (\u03b9 \u2192 \u211d) := Equiv.piCongrLeft' (fun _a => \u211d) e\nx : \u03b9' \u2192 \u211d\nh : (\u2200 (i : \u03b9), 0 \u2264 x (\u2191e.symm i)) \u2227 \u2200 (i : \u03b9), x (\u2191e.symm i) \u2264 1\ni : \u03b9'\n\u22a2 0 \u2264 x i", "state_after": "no goals"}, {"tactic": "simpa only [Equiv.symm_apply_apply] using h.2 (e i)", "annotated_tactic": ["simpa only [<a>Equiv.symm_apply_apply</a>] using h.2 (e i)", [{"full_name": "Equiv.symm_apply_apply", "def_path": "Mathlib/Logic/Equiv/Defs.lean", "def_pos": [283, 17], "def_end_pos": [283, 33]}]], "state_before": "case h.refine'_2\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2075 : Fintype \u03b9\ninst\u271d\u2074 : Fintype \u03b9'\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \u211d E\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \u211d F\nv : \u03b9 \u2192 E\ne : \u03b9' \u2243 \u03b9\nK : (\u03b9' \u2192 \u211d) \u2243 (\u03b9 \u2192 \u211d) := Equiv.piCongrLeft' (fun _a => \u211d) e\nx : \u03b9' \u2192 \u211d\nh : (\u2200 (i : \u03b9), 0 \u2264 x (\u2191e.symm i)) \u2227 \u2200 (i : \u03b9), x (\u2191e.symm i) \u2264 1\ni : \u03b9'\n\u22a2 x i \u2264 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Constructions/Polish.lean", "full_name": "MeasurableSet.image_of_measurable_injOn", "start": [838, 1], "end": [856, 62], "traced_tactics": [{"tactic": "letI := upgradeStandardBorel \u03b3", "annotated_tactic": ["letI := <a>upgradeStandardBorel</a> \u03b3", [{"full_name": "upgradeStandardBorel", "def_path": "Mathlib/MeasureTheory/Constructions/Polish.lean", "def_pos": [93, 5], "def_end_pos": [93, 25]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\nt\u03b2 : TopologicalSpace \u03b2\ninst\u271d\u2075 : T2Space \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2\ns : Set \u03b3\nf : \u03b3 \u2192 \u03b2\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b3\ninst\u271d\u00b9 : StandardBorelSpace \u03b3\ninst\u271d : SecondCountableTopology \u03b2\nhs : MeasurableSet s\nf_meas : Measurable f\nf_inj : InjOn f s\n\u22a2 MeasurableSet (f '' s)", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\nt\u03b2 : TopologicalSpace \u03b2\ninst\u271d\u2075 : T2Space \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2\ns : Set \u03b3\nf : \u03b3 \u2192 \u03b2\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b3\ninst\u271d\u00b9 : StandardBorelSpace \u03b3\ninst\u271d : SecondCountableTopology \u03b2\nhs : MeasurableSet s\nf_meas : Measurable f\nf_inj : InjOn f s\nthis : UpgradedStandardBorel \u03b3 := upgradeStandardBorel \u03b3\n\u22a2 MeasurableSet (f '' s)"}, {"tactic": "let t\u03b3 : TopologicalSpace \u03b3 := inferInstance", "annotated_tactic": ["let t\u03b3 : <a>TopologicalSpace</a> \u03b3 := <a>inferInstance</a>", [{"full_name": "TopologicalSpace", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [70, 7], "def_end_pos": [70, 23]}, {"full_name": "inferInstance", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [86, 8], "def_end_pos": [86, 21]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\nt\u03b2 : TopologicalSpace \u03b2\ninst\u271d\u2075 : T2Space \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2\ns : Set \u03b3\nf : \u03b3 \u2192 \u03b2\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b3\ninst\u271d\u00b9 : StandardBorelSpace \u03b3\ninst\u271d : SecondCountableTopology \u03b2\nhs : MeasurableSet s\nf_meas : Measurable f\nf_inj : InjOn f s\nthis : UpgradedStandardBorel \u03b3 := upgradeStandardBorel \u03b3\n\u22a2 MeasurableSet (f '' s)", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\nt\u03b2 : TopologicalSpace \u03b2\ninst\u271d\u2075 : T2Space \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2\ns : Set \u03b3\nf : \u03b3 \u2192 \u03b2\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b3\ninst\u271d\u00b9 : StandardBorelSpace \u03b3\ninst\u271d : SecondCountableTopology \u03b2\nhs : MeasurableSet s\nf_meas : Measurable f\nf_inj : InjOn f s\nthis : UpgradedStandardBorel \u03b3 := upgradeStandardBorel \u03b3\nt\u03b3 : TopologicalSpace \u03b3 := inferInstance\n\u22a2 MeasurableSet (f '' s)"}, {"tactic": "obtain \u27e8t', t't, f_cont, t'_polish\u27e9 :\n  \u2203 t' : TopologicalSpace \u03b3, t' \u2264 t\u03b3 \u2227 @Continuous \u03b3 \u03b2 t' t\u03b2 f \u2227 @PolishSpace \u03b3 t' :=\n  f_meas.exists_continuous", "annotated_tactic": ["obtain \u27e8t', t't, f_cont, t'_polish\u27e9 :\n    \u2203 t' : <a>TopologicalSpace</a> \u03b3, t' \u2264 t\u03b3 \u2227 @<a>Continuous</a> \u03b3 \u03b2 t' t\u03b2 f \u2227 @<a>PolishSpace</a> \u03b3 t' :=\n    f_meas.exists_continuous", [{"full_name": "TopologicalSpace", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [70, 7], "def_end_pos": [70, 23]}, {"full_name": "Continuous", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [1591, 11], "def_end_pos": [1591, 21]}, {"full_name": "PolishSpace", "def_path": "Mathlib/Topology/MetricSpace/Polish.lean", "def_pos": [65, 7], "def_end_pos": [65, 18]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\nt\u03b2 : TopologicalSpace \u03b2\ninst\u271d\u2075 : T2Space \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2\ns : Set \u03b3\nf : \u03b3 \u2192 \u03b2\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b3\ninst\u271d\u00b9 : StandardBorelSpace \u03b3\ninst\u271d : SecondCountableTopology \u03b2\nhs : MeasurableSet s\nf_meas : Measurable f\nf_inj : InjOn f s\nthis : UpgradedStandardBorel \u03b3 := upgradeStandardBorel \u03b3\nt\u03b3 : TopologicalSpace \u03b3 := inferInstance\n\u22a2 MeasurableSet (f '' s)", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\nt\u03b2 : TopologicalSpace \u03b2\ninst\u271d\u2075 : T2Space \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2\ns : Set \u03b3\nf : \u03b3 \u2192 \u03b2\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b3\ninst\u271d\u00b9 : StandardBorelSpace \u03b3\ninst\u271d : SecondCountableTopology \u03b2\nhs : MeasurableSet s\nf_meas : Measurable f\nf_inj : InjOn f s\nthis : UpgradedStandardBorel \u03b3 := upgradeStandardBorel \u03b3\nt\u03b3 : TopologicalSpace \u03b3 := inferInstance\nt' : TopologicalSpace \u03b3\nt't : t' \u2264 t\u03b3\nf_cont : Continuous f\nt'_polish : PolishSpace \u03b3\n\u22a2 MeasurableSet (f '' s)"}, {"tactic": "have M : MeasurableSet[@borel \u03b3 t'] s :=\n  @Continuous.measurable \u03b3 \u03b3 t' (@borel \u03b3 t')\n    (@BorelSpace.opensMeasurable \u03b3 t' (@borel \u03b3 t') (@BorelSpace.mk _ _ (borel \u03b3) rfl))\n    t\u03b3 _ _ _ (continuous_id_of_le t't) s hs", "annotated_tactic": ["have M : MeasurableSet[@<a>borel</a> \u03b3 t'] s :=\n    @<a>Continuous.measurable</a> \u03b3 \u03b3 t' (@<a>borel</a> \u03b3 t')\n      (@<a>BorelSpace.opensMeasurable</a> \u03b3 t' (@<a>borel</a> \u03b3 t') (@<a>BorelSpace.mk</a> _ _ (<a>borel</a> \u03b3) <a>rfl</a>))\n      t\u03b3 _ _ _ (<a>continuous_id_of_le</a> t't) s hs", [{"full_name": "borel", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [59, 5], "def_end_pos": [59, 10]}, {"full_name": "Continuous.measurable", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [891, 9], "def_end_pos": [891, 30]}, {"full_name": "borel", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [59, 5], "def_end_pos": [59, 10]}, {"full_name": "BorelSpace.opensMeasurable", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [268, 28], "def_end_pos": [268, 54]}, {"full_name": "borel", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [59, 5], "def_end_pos": [59, 10]}, {"full_name": "BorelSpace.mk", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [202, 18], "def_end_pos": [202, 70]}, {"full_name": "borel", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [59, 5], "def_end_pos": [59, 10]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}, {"full_name": "continuous_id_of_le", "def_path": "Mathlib/Topology/Order.lean", "def_pos": [847, 9], "def_end_pos": [847, 28]}]], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\nt\u03b2 : TopologicalSpace \u03b2\ninst\u271d\u2075 : T2Space \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2\ns : Set \u03b3\nf : \u03b3 \u2192 \u03b2\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b3\ninst\u271d\u00b9 : StandardBorelSpace \u03b3\ninst\u271d : SecondCountableTopology \u03b2\nhs : MeasurableSet s\nf_meas : Measurable f\nf_inj : InjOn f s\nthis : UpgradedStandardBorel \u03b3 := upgradeStandardBorel \u03b3\nt\u03b3 : TopologicalSpace \u03b3 := inferInstance\nt' : TopologicalSpace \u03b3\nt't : t' \u2264 t\u03b3\nf_cont : Continuous f\nt'_polish : PolishSpace \u03b3\n\u22a2 MeasurableSet (f '' s)", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\nt\u03b2 : TopologicalSpace \u03b2\ninst\u271d\u2075 : T2Space \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2\ns : Set \u03b3\nf : \u03b3 \u2192 \u03b2\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b3\ninst\u271d\u00b9 : StandardBorelSpace \u03b3\ninst\u271d : SecondCountableTopology \u03b2\nhs : MeasurableSet s\nf_meas : Measurable f\nf_inj : InjOn f s\nthis : UpgradedStandardBorel \u03b3 := upgradeStandardBorel \u03b3\nt\u03b3 : TopologicalSpace \u03b3 := inferInstance\nt' : TopologicalSpace \u03b3\nt't : t' \u2264 t\u03b3\nf_cont : Continuous f\nt'_polish : PolishSpace \u03b3\nM : MeasurableSet s\n\u22a2 MeasurableSet (f '' s)"}, {"tactic": "exact\n  @MeasurableSet.image_of_continuousOn_injOn \u03b3\n    \u03b2 _ _ _  s f _ t' t'_polish (@borel \u03b3 t') (@BorelSpace.mk _ _ (borel \u03b3) rfl)\n    M (@Continuous.continuousOn \u03b3 \u03b2 t' t\u03b2 f s f_cont) f_inj", "annotated_tactic": ["exact\n    @<a>MeasurableSet.image_of_continuousOn_injOn</a> \u03b3\n      \u03b2 _ _ _  s f _ t' t'_polish (@<a>borel</a> \u03b3 t') (@<a>BorelSpace.mk</a> _ _ (<a>borel</a> \u03b3) <a>rfl</a>)\n      M (@<a>Continuous.continuousOn</a> \u03b3 \u03b2 t' t\u03b2 f s f_cont) f_inj", [{"full_name": "MeasurableSet.image_of_continuousOn_injOn", "def_path": "Mathlib/MeasureTheory/Constructions/Polish.lean", "def_pos": [823, 9], "def_end_pos": [823, 57]}, {"full_name": "borel", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [59, 5], "def_end_pos": [59, 10]}, {"full_name": "BorelSpace.mk", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [202, 18], "def_end_pos": [202, 70]}, {"full_name": "borel", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [59, 5], "def_end_pos": [59, 10]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}, {"full_name": "Continuous.continuousOn", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [947, 9], "def_end_pos": [947, 32]}]], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\nt\u03b2 : TopologicalSpace \u03b2\ninst\u271d\u2075 : T2Space \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2\ns : Set \u03b3\nf : \u03b3 \u2192 \u03b2\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b3\ninst\u271d\u00b9 : StandardBorelSpace \u03b3\ninst\u271d : SecondCountableTopology \u03b2\nhs : MeasurableSet s\nf_meas : Measurable f\nf_inj : InjOn f s\nthis : UpgradedStandardBorel \u03b3 := upgradeStandardBorel \u03b3\nt\u03b3 : TopologicalSpace \u03b3 := inferInstance\nt' : TopologicalSpace \u03b3\nt't : t' \u2264 t\u03b3\nf_cont : Continuous f\nt'_polish : PolishSpace \u03b3\nM : MeasurableSet s\n\u22a2 MeasurableSet (f '' s)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Primrec.lean", "full_name": "Primrec.nat_rec", "start": [559, 1], "end": [576, 24], "traced_tactics": [{"tactic": "simp only [Nat.unpaired, id_eq, Nat.unpair_pair, decode_prod_val, decode_nat,\n  Option.some_bind, Option.map_map, Option.map_some']", "annotated_tactic": ["simp only [<a>Nat.unpaired</a>, <a>id_eq</a>, <a>Nat.unpair_pair</a>, <a>decode_prod_val</a>, <a>decode_nat</a>,\n        <a>Option.some_bind</a>, <a>Option.map_map</a>, <a>Option.map_some'</a>]", [{"full_name": "Nat.unpaired", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [70, 5], "def_end_pos": [70, 13]}, {"full_name": "id_eq", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [284, 17], "def_end_pos": [284, 22]}, {"full_name": "Nat.unpair_pair", "def_path": "Mathlib/Data/Nat/Pairing.lean", "def_pos": [65, 9], "def_end_pos": [65, 20]}, {"full_name": "Encodable.decode_prod_val", "def_path": "Mathlib/Logic/Encodable/Basic.lean", "def_pos": [385, 9], "def_end_pos": [385, 24]}, {"full_name": "Encodable.decode_nat", "def_path": "Mathlib/Logic/Encodable/Basic.lean", "def_pos": [130, 9], "def_end_pos": [130, 19]}, {"full_name": "Option.some_bind", "def_path": "lake-packages/std/Std/Data/Option/Init/Lemmas.lean", "def_pos": [23, 17], "def_end_pos": [23, 26]}, {"full_name": "Option.map_map", "def_path": "lake-packages/std/Std/Data/Option/Lemmas.lean", "def_pos": [145, 17], "def_end_pos": [145, 24]}, {"full_name": "Option.map_some'", "def_path": "lake-packages/std/Std/Data/Option/Init/Lemmas.lean", "def_pos": [20, 17], "def_end_pos": [20, 26]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03c3 : Type u_5\ninst\u271d\u2074 : Primcodable \u03b1\ninst\u271d\u00b3 : Primcodable \u03b2\ninst\u271d\u00b2 : Primcodable \u03b3\ninst\u271d\u00b9 : Primcodable \u03b4\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u2115 \u00d7 \u03b2 \u2192 \u03b2\nhf : Primrec f\nhg : Primrec\u2082 g\nn : \u2115\n\u22a2 Nat.unpaired\n      (fun z n =>\n        Nat.casesOn n 0 fun y =>\n          Nat.unpaired\n            (fun z n =>\n              Nat.rec (encode (Option.map f (decode z)))\n                (fun y IH =>\n                  encode\n                    (Option.map (fun p => g p.1 p.2)\n                      (decode\n                        (Nat.pair (Nat.unpair (Nat.pair z (Nat.pair y IH))).1\n                          (Nat.pair (Nat.unpair (Nat.unpair (Nat.pair z (Nat.pair y IH))).2).1\n                            (Nat.pred (Nat.unpair (Nat.unpair (Nat.pair z (Nat.pair y IH))).2).2))))))\n                n)\n            (Nat.pair (Nat.unpair (Nat.pair z y)).2 (Nat.unpair (Nat.unpair (Nat.pair z y)).1).2))\n      (Nat.pair (id n) (encode (decode (Nat.unpair n).1))) =\n    Nat.unpaired\n      (fun m n =>\n        encode\n          (Option.bind (decode m) fun a => Option.map (fun n => Nat.rec (f a) (fun n IH => g a (n, IH)) n) (decode n)))\n      n", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03c3 : Type u_5\ninst\u271d\u2074 : Primcodable \u03b1\ninst\u271d\u00b3 : Primcodable \u03b2\ninst\u271d\u00b2 : Primcodable \u03b3\ninst\u271d\u00b9 : Primcodable \u03b4\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u2115 \u00d7 \u03b2 \u2192 \u03b2\nhf : Primrec f\nhg : Primrec\u2082 g\nn : \u2115\n\u22a2 Nat.rec 0\n      (fun n_1 n_ih =>\n        Nat.rec (encode (Option.map f (decode n_1)))\n          (fun y IH =>\n            encode\n              (Option.map (fun p => g p.1 p.2)\n                (Option.bind (decode n_1) fun a => Option.map (Prod.mk a \u2218 Prod.mk y) (decode (Nat.pred IH)))))\n          (Nat.unpair n).2)\n      (encode (decode (Nat.unpair n).1)) =\n    encode\n      (Option.bind (decode (Nat.unpair n).1) fun a => some (Nat.rec (f a) (fun n IH => g a (n, IH)) (Nat.unpair n).2))"}, {"tactic": "cases' @decode \u03b1 _ n.unpair.1 with a", "annotated_tactic": ["cases' @<a>decode</a> \u03b1 _ n.unpair.1 with a", [{"full_name": "Encodable.decode", "def_path": "Mathlib/Logic/Encodable/Basic.lean", "def_pos": [51, 3], "def_end_pos": [51, 9]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03c3 : Type u_5\ninst\u271d\u2074 : Primcodable \u03b1\ninst\u271d\u00b3 : Primcodable \u03b2\ninst\u271d\u00b2 : Primcodable \u03b3\ninst\u271d\u00b9 : Primcodable \u03b4\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u2115 \u00d7 \u03b2 \u2192 \u03b2\nhf : Primrec f\nhg : Primrec\u2082 g\nn : \u2115\n\u22a2 Nat.rec 0\n      (fun n_1 n_ih =>\n        Nat.rec (encode (Option.map f (decode n_1)))\n          (fun y IH =>\n            encode\n              (Option.map (fun p => g p.1 p.2)\n                (Option.bind (decode n_1) fun a => Option.map (Prod.mk a \u2218 Prod.mk y) (decode (Nat.pred IH)))))\n          (Nat.unpair n).2)\n      (encode (decode (Nat.unpair n).1)) =\n    encode\n      (Option.bind (decode (Nat.unpair n).1) fun a => some (Nat.rec (f a) (fun n IH => g a (n, IH)) (Nat.unpair n).2))", "state_after": "case none\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03c3 : Type u_5\ninst\u271d\u2074 : Primcodable \u03b1\ninst\u271d\u00b3 : Primcodable \u03b2\ninst\u271d\u00b2 : Primcodable \u03b3\ninst\u271d\u00b9 : Primcodable \u03b4\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u2115 \u00d7 \u03b2 \u2192 \u03b2\nhf : Primrec f\nhg : Primrec\u2082 g\nn : \u2115\n\u22a2 Nat.rec 0\n      (fun n_1 n_ih =>\n        Nat.rec (encode (Option.map f (decode n_1)))\n          (fun y IH =>\n            encode\n              (Option.map (fun p => g p.1 p.2)\n                (Option.bind (decode n_1) fun a => Option.map (Prod.mk a \u2218 Prod.mk y) (decode (Nat.pred IH)))))\n          (Nat.unpair n).2)\n      (encode none) =\n    encode (Option.bind none fun a => some (Nat.rec (f a) (fun n IH => g a (n, IH)) (Nat.unpair n).2))\n\ncase some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03c3 : Type u_5\ninst\u271d\u2074 : Primcodable \u03b1\ninst\u271d\u00b3 : Primcodable \u03b2\ninst\u271d\u00b2 : Primcodable \u03b3\ninst\u271d\u00b9 : Primcodable \u03b4\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u2115 \u00d7 \u03b2 \u2192 \u03b2\nhf : Primrec f\nhg : Primrec\u2082 g\nn : \u2115\na : \u03b1\n\u22a2 Nat.rec 0\n      (fun n_1 n_ih =>\n        Nat.rec (encode (Option.map f (decode n_1)))\n          (fun y IH =>\n            encode\n              (Option.map (fun p => g p.1 p.2)\n                (Option.bind (decode n_1) fun a => Option.map (Prod.mk a \u2218 Prod.mk y) (decode (Nat.pred IH)))))\n          (Nat.unpair n).2)\n      (encode (some a)) =\n    encode (Option.bind (some a) fun a => some (Nat.rec (f a) (fun n IH => g a (n, IH)) (Nat.unpair n).2))"}, {"tactic": "simp only [encode_some, encodek, Option.map_some', Option.some_bind, Option.map_map]", "annotated_tactic": ["simp only [<a>encode_some</a>, <a>encodek</a>, <a>Option.map_some'</a>, <a>Option.some_bind</a>, <a>Option.map_map</a>]", [{"full_name": "Encodable.encode_some", "def_path": "Mathlib/Logic/Encodable/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 20]}, {"full_name": "Encodable.encodek", "def_path": "Mathlib/Logic/Encodable/Basic.lean", "def_pos": [53, 3], "def_end_pos": [53, 10]}, {"full_name": "Option.map_some'", "def_path": "lake-packages/std/Std/Data/Option/Init/Lemmas.lean", "def_pos": [20, 17], "def_end_pos": [20, 26]}, {"full_name": "Option.some_bind", "def_path": "lake-packages/std/Std/Data/Option/Init/Lemmas.lean", "def_pos": [23, 17], "def_end_pos": [23, 26]}, {"full_name": "Option.map_map", "def_path": "lake-packages/std/Std/Data/Option/Lemmas.lean", "def_pos": [145, 17], "def_end_pos": [145, 24]}]], "state_before": "case some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03c3 : Type u_5\ninst\u271d\u2074 : Primcodable \u03b1\ninst\u271d\u00b3 : Primcodable \u03b2\ninst\u271d\u00b2 : Primcodable \u03b3\ninst\u271d\u00b9 : Primcodable \u03b4\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u2115 \u00d7 \u03b2 \u2192 \u03b2\nhf : Primrec f\nhg : Primrec\u2082 g\nn : \u2115\na : \u03b1\n\u22a2 Nat.rec 0\n      (fun n_1 n_ih =>\n        Nat.rec (encode (Option.map f (decode n_1)))\n          (fun y IH =>\n            encode\n              (Option.map (fun p => g p.1 p.2)\n                (Option.bind (decode n_1) fun a => Option.map (Prod.mk a \u2218 Prod.mk y) (decode (Nat.pred IH)))))\n          (Nat.unpair n).2)\n      (encode (some a)) =\n    encode (Option.bind (some a) fun a => some (Nat.rec (f a) (fun n IH => g a (n, IH)) (Nat.unpair n).2))", "state_after": "case some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03c3 : Type u_5\ninst\u271d\u2074 : Primcodable \u03b1\ninst\u271d\u00b3 : Primcodable \u03b2\ninst\u271d\u00b2 : Primcodable \u03b3\ninst\u271d\u00b9 : Primcodable \u03b4\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u2115 \u00d7 \u03b2 \u2192 \u03b2\nhf : Primrec f\nhg : Primrec\u2082 g\nn : \u2115\na : \u03b1\n\u22a2 Nat.rec (Nat.succ (encode (f a)))\n      (fun y IH => encode (Option.map ((fun p => g p.1 p.2) \u2218 Prod.mk a \u2218 Prod.mk y) (decode (Nat.pred IH))))\n      (Nat.unpair n).2 =\n    Nat.succ (encode (Nat.rec (f a) (fun n IH => g a (n, IH)) (Nat.unpair n).2))"}, {"tactic": "induction' n.unpair.2 with m <;> simp [encodek]", "annotated_tactic": ["induction' n.unpair.2 with m <;> simp [<a>encodek</a>]", [{"full_name": "Encodable.encodek", "def_path": "Mathlib/Logic/Encodable/Basic.lean", "def_pos": [53, 3], "def_end_pos": [53, 10]}]], "state_before": "case some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03c3 : Type u_5\ninst\u271d\u2074 : Primcodable \u03b1\ninst\u271d\u00b3 : Primcodable \u03b2\ninst\u271d\u00b2 : Primcodable \u03b3\ninst\u271d\u00b9 : Primcodable \u03b4\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u2115 \u00d7 \u03b2 \u2192 \u03b2\nhf : Primrec f\nhg : Primrec\u2082 g\nn : \u2115\na : \u03b1\n\u22a2 Nat.rec (Nat.succ (encode (f a)))\n      (fun y IH => encode (Option.map ((fun p => g p.1 p.2) \u2218 Prod.mk a \u2218 Prod.mk y) (decode (Nat.pred IH))))\n      (Nat.unpair n).2 =\n    Nat.succ (encode (Nat.rec (f a) (fun n IH => g a (n, IH)) (Nat.unpair n).2))", "state_after": "case some.succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03c3 : Type u_5\ninst\u271d\u2074 : Primcodable \u03b1\ninst\u271d\u00b3 : Primcodable \u03b2\ninst\u271d\u00b2 : Primcodable \u03b3\ninst\u271d\u00b9 : Primcodable \u03b4\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u2115 \u00d7 \u03b2 \u2192 \u03b2\nhf : Primrec f\nhg : Primrec\u2082 g\nn : \u2115\na : \u03b1\nm : \u2115\nn_ih\u271d :\n  Nat.rec (Nat.succ (encode (f a)))\n      (fun y IH => encode (Option.map ((fun p => g p.1 p.2) \u2218 Prod.mk a \u2218 Prod.mk y) (decode (Nat.pred IH)))) m =\n    Nat.succ (encode (Nat.rec (f a) (fun n IH => g a (n, IH)) m))\n\u22a2 encode\n      (Option.map ((fun p => g p.1 p.2) \u2218 Prod.mk a \u2218 Prod.mk m)\n        (decode\n          (Nat.pred\n            (Nat.rec (Nat.succ (encode (f a)))\n              (fun y IH => encode (Option.map ((fun p => g p.1 p.2) \u2218 Prod.mk a \u2218 Prod.mk y) (decode (Nat.pred IH))))\n              m)))) =\n    Nat.succ (encode (g a (m, Nat.rec (f a) (fun n IH => g a (n, IH)) m)))"}, {"tactic": "simp [*, encodek]", "annotated_tactic": ["simp [*, <a>encodek</a>]", [{"full_name": "Encodable.encodek", "def_path": "Mathlib/Logic/Encodable/Basic.lean", "def_pos": [53, 3], "def_end_pos": [53, 10]}]], "state_before": "case some.succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03c3 : Type u_5\ninst\u271d\u2074 : Primcodable \u03b1\ninst\u271d\u00b3 : Primcodable \u03b2\ninst\u271d\u00b2 : Primcodable \u03b3\ninst\u271d\u00b9 : Primcodable \u03b4\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u2115 \u00d7 \u03b2 \u2192 \u03b2\nhf : Primrec f\nhg : Primrec\u2082 g\nn : \u2115\na : \u03b1\nm : \u2115\nn_ih\u271d :\n  Nat.rec (Nat.succ (encode (f a)))\n      (fun y IH => encode (Option.map ((fun p => g p.1 p.2) \u2218 Prod.mk a \u2218 Prod.mk y) (decode (Nat.pred IH)))) m =\n    Nat.succ (encode (Nat.rec (f a) (fun n IH => g a (n, IH)) m))\n\u22a2 encode\n      (Option.map ((fun p => g p.1 p.2) \u2218 Prod.mk a \u2218 Prod.mk m)\n        (decode\n          (Nat.pred\n            (Nat.rec (Nat.succ (encode (f a)))\n              (fun y IH => encode (Option.map ((fun p => g p.1 p.2) \u2218 Prod.mk a \u2218 Prod.mk y) (decode (Nat.pred IH))))\n              m)))) =\n    Nat.succ (encode (g a (m, Nat.rec (f a) (fun n IH => g a (n, IH)) m)))", "state_after": "no goals"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case none\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03c3 : Type u_5\ninst\u271d\u2074 : Primcodable \u03b1\ninst\u271d\u00b3 : Primcodable \u03b2\ninst\u271d\u00b2 : Primcodable \u03b3\ninst\u271d\u00b9 : Primcodable \u03b4\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u2115 \u00d7 \u03b2 \u2192 \u03b2\nhf : Primrec f\nhg : Primrec\u2082 g\nn : \u2115\n\u22a2 Nat.rec 0\n      (fun n_1 n_ih =>\n        Nat.rec (encode (Option.map f (decode n_1)))\n          (fun y IH =>\n            encode\n              (Option.map (fun p => g p.1 p.2)\n                (Option.bind (decode n_1) fun a => Option.map (Prod.mk a \u2218 Prod.mk y) (decode (Nat.pred IH)))))\n          (Nat.unpair n).2)\n      (encode none) =\n    encode (Option.bind none fun a => some (Nat.rec (f a) (fun n IH => g a (n, IH)) (Nat.unpair n).2))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean", "full_name": "MeasureTheory.Measure.addHaar_submodule", "start": [179, 1], "end": [203, 16], "traced_tactics": [{"tactic": "obtain \u27e8x, hx\u27e9 : \u2203 x, x \u2209 s := by\n  simpa only [Submodule.eq_top_iff', not_exists, Ne.def, not_forall] using hs", "annotated_tactic": ["obtain \u27e8x, hx\u27e9 : \u2203 x, x \u2209 s := by\n    simpa only [<a>Submodule.eq_top_iff'</a>, <a>not_exists</a>, <a>Ne.def</a>, <a>not_forall</a>] using hs", [{"full_name": "Submodule.eq_top_iff'", "def_path": "Mathlib/Algebra/Module/Submodule/Lattice.lean", "def_pos": [184, 9], "def_end_pos": [184, 20]}, {"full_name": "not_exists", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [422, 17], "def_end_pos": [422, 27]}, {"full_name": "Ne.def", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [59, 9], "def_end_pos": [59, 15]}, {"full_name": "Classical.not_forall", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [686, 9], "def_end_pos": [686, 19]}]], "state_before": "E : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\ns : Submodule \u211d E\nhs : s \u2260 \u22a4\n\u22a2 \u2191\u2191\u03bc \u2191s = 0", "state_after": "case intro\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\ns : Submodule \u211d E\nhs : s \u2260 \u22a4\nx : E\nhx : \u00acx \u2208 s\n\u22a2 \u2191\u2191\u03bc \u2191s = 0"}, {"tactic": "obtain \u27e8c, cpos, cone\u27e9 : \u2203 c : \u211d, 0 < c \u2227 c < 1 := \u27e81 / 2, by norm_num, by norm_num\u27e9", "annotated_tactic": ["obtain \u27e8c, cpos, cone\u27e9 : \u2203 c : \u211d, 0 < c \u2227 c < 1 := \u27e81 / 2, by norm_num, by norm_num\u27e9", []], "state_before": "case intro\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\ns : Submodule \u211d E\nhs : s \u2260 \u22a4\nx : E\nhx : \u00acx \u2208 s\n\u22a2 \u2191\u2191\u03bc \u2191s = 0", "state_after": "case intro.intro.intro\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\ns : Submodule \u211d E\nhs : s \u2260 \u22a4\nx : E\nhx : \u00acx \u2208 s\nc : \u211d\ncpos : 0 < c\ncone : c < 1\n\u22a2 \u2191\u2191\u03bc \u2191s = 0"}, {"tactic": "have A : IsBounded (range fun n : \u2115 => c ^ n \u2022 x) :=\n  have : Tendsto (fun n : \u2115 => c ^ n \u2022 x) atTop (\ud835\udcdd ((0 : \u211d) \u2022 x)) :=\n    (tendsto_pow_atTop_nhds_0_of_lt_1 cpos.le cone).smul_const x\n  isBounded_range_of_tendsto _ this", "annotated_tactic": ["have A : <a>IsBounded</a> (<a>range</a> fun n : \u2115 => c ^ n \u2022 x) :=\n    have : <a>Tendsto</a> (fun n : \u2115 => c ^ n \u2022 x) <a>atTop</a> (\ud835\udcdd ((0 : \u211d) \u2022 x)) :=\n      (<a>tendsto_pow_atTop_nhds_0_of_lt_1</a> cpos.le cone).<a>smul_const</a> x\n    <a>isBounded_range_of_tendsto</a> _ this", [{"full_name": "Bornology.IsBounded", "def_path": "Mathlib/Topology/Bornology/Basic.lean", "def_pos": [140, 5], "def_end_pos": [140, 14]}, {"full_name": "Set.range", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [668, 5], "def_end_pos": [668, 10]}, {"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2939, 5], "def_end_pos": [2939, 12]}, {"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}, {"full_name": "tendsto_pow_atTop_nhds_0_of_lt_1", "def_path": "Mathlib/Analysis/SpecificLimits/Basic.lean", "def_pos": [110, 9], "def_end_pos": [110, 41]}, {"full_name": "Filter.Tendsto.smul_const", "def_path": "Mathlib/Topology/Algebra/MulAction.lean", "def_pos": [91, 9], "def_end_pos": [91, 34]}, {"full_name": "Metric.isBounded_range_of_tendsto", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [2523, 9], "def_end_pos": [2523, 35]}]], "state_before": "case intro.intro.intro\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\ns : Submodule \u211d E\nhs : s \u2260 \u22a4\nx : E\nhx : \u00acx \u2208 s\nc : \u211d\ncpos : 0 < c\ncone : c < 1\n\u22a2 \u2191\u2191\u03bc \u2191s = 0", "state_after": "case intro.intro.intro\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\ns : Submodule \u211d E\nhs : s \u2260 \u22a4\nx : E\nhx : \u00acx \u2208 s\nc : \u211d\ncpos : 0 < c\ncone : c < 1\nA : Bornology.IsBounded (range fun n => c ^ n \u2022 x)\n\u22a2 \u2191\u2191\u03bc \u2191s = 0"}, {"tactic": "apply addHaar_eq_zero_of_disjoint_translates \u03bc _ A _\n  (Submodule.closed_of_finiteDimensional s).measurableSet", "annotated_tactic": ["apply <a>addHaar_eq_zero_of_disjoint_translates</a> \u03bc _ A _\n    (<a>Submodule.closed_of_finiteDimensional</a> s).<a>measurableSet</a>", [{"full_name": "MeasureTheory.Measure.addHaar_eq_zero_of_disjoint_translates", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean", "def_pos": [160, 9], "def_end_pos": [160, 47]}, {"full_name": "Submodule.closed_of_finiteDimensional", "def_path": "Mathlib/Topology/Algebra/Module/FiniteDimension.lean", "def_pos": [540, 9], "def_end_pos": [540, 46]}, {"full_name": "IsClosed.measurableSet", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [340, 9], "def_end_pos": [340, 31]}]], "state_before": "case intro.intro.intro\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\ns : Submodule \u211d E\nhs : s \u2260 \u22a4\nx : E\nhx : \u00acx \u2208 s\nc : \u211d\ncpos : 0 < c\ncone : c < 1\nA : Bornology.IsBounded (range fun n => c ^ n \u2022 x)\n\u22a2 \u2191\u2191\u03bc \u2191s = 0", "state_after": "E : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\ns : Submodule \u211d E\nhs : s \u2260 \u22a4\nx : E\nhx : \u00acx \u2208 s\nc : \u211d\ncpos : 0 < c\ncone : c < 1\nA : Bornology.IsBounded (range fun n => c ^ n \u2022 x)\n\u22a2 Pairwise (Disjoint on fun n => {c ^ n \u2022 x} + \u2191s)"}, {"tactic": "intro m n hmn", "annotated_tactic": ["intro m n hmn", []], "state_before": "E : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\ns : Submodule \u211d E\nhs : s \u2260 \u22a4\nx : E\nhx : \u00acx \u2208 s\nc : \u211d\ncpos : 0 < c\ncone : c < 1\nA : Bornology.IsBounded (range fun n => c ^ n \u2022 x)\n\u22a2 Pairwise (Disjoint on fun n => {c ^ n \u2022 x} + \u2191s)", "state_after": "E : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\ns : Submodule \u211d E\nhs : s \u2260 \u22a4\nx : E\nhx : \u00acx \u2208 s\nc : \u211d\ncpos : 0 < c\ncone : c < 1\nA : Bornology.IsBounded (range fun n => c ^ n \u2022 x)\nm n : \u2115\nhmn : m \u2260 n\n\u22a2 (Disjoint on fun n => {c ^ n \u2022 x} + \u2191s) m n"}, {"tactic": "simp only [Function.onFun, image_add_left, singleton_add, disjoint_left, mem_preimage,\n  SetLike.mem_coe]", "annotated_tactic": ["simp only [<a>Function.onFun</a>, <a>image_add_left</a>, <a>singleton_add</a>, <a>disjoint_left</a>, <a>mem_preimage</a>,\n    <a>SetLike.mem_coe</a>]", [{"full_name": "Function.onFun", "def_path": "Mathlib/Init/Function.lean", "def_pos": [49, 5], "def_end_pos": [49, 10]}, {"full_name": "Set.image_add_left", "def_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "def_pos": [1198, 3], "def_end_pos": [1198, 14]}, {"full_name": "Set.singleton_add", "def_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "def_pos": [402, 3], "def_end_pos": [402, 14]}, {"full_name": "Set.disjoint_left", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1546, 9], "def_end_pos": [1546, 22]}, {"full_name": "Set.mem_preimage", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [64, 9], "def_end_pos": [64, 21]}, {"full_name": "SetLike.mem_coe", "def_path": "Mathlib/Data/SetLike/Basic.lean", "def_pos": [163, 9], "def_end_pos": [163, 16]}]], "state_before": "E : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\ns : Submodule \u211d E\nhs : s \u2260 \u22a4\nx : E\nhx : \u00acx \u2208 s\nc : \u211d\ncpos : 0 < c\ncone : c < 1\nA : Bornology.IsBounded (range fun n => c ^ n \u2022 x)\nm n : \u2115\nhmn : m \u2260 n\n\u22a2 (Disjoint on fun n => {c ^ n \u2022 x} + \u2191s) m n", "state_after": "E : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\ns : Submodule \u211d E\nhs : s \u2260 \u22a4\nx : E\nhx : \u00acx \u2208 s\nc : \u211d\ncpos : 0 < c\ncone : c < 1\nA : Bornology.IsBounded (range fun n => c ^ n \u2022 x)\nm n : \u2115\nhmn : m \u2260 n\n\u22a2 \u2200 \u2983a : E\u2984, -(c ^ m \u2022 x) + a \u2208 s \u2192 \u00ac-(c ^ n \u2022 x) + a \u2208 s"}, {"tactic": "intro y hym hyn", "annotated_tactic": ["intro y hym hyn", []], "state_before": "E : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\ns : Submodule \u211d E\nhs : s \u2260 \u22a4\nx : E\nhx : \u00acx \u2208 s\nc : \u211d\ncpos : 0 < c\ncone : c < 1\nA : Bornology.IsBounded (range fun n => c ^ n \u2022 x)\nm n : \u2115\nhmn : m \u2260 n\n\u22a2 \u2200 \u2983a : E\u2984, -(c ^ m \u2022 x) + a \u2208 s \u2192 \u00ac-(c ^ n \u2022 x) + a \u2208 s", "state_after": "E : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\ns : Submodule \u211d E\nhs : s \u2260 \u22a4\nx : E\nhx : \u00acx \u2208 s\nc : \u211d\ncpos : 0 < c\ncone : c < 1\nA : Bornology.IsBounded (range fun n => c ^ n \u2022 x)\nm n : \u2115\nhmn : m \u2260 n\ny : E\nhym : -(c ^ m \u2022 x) + y \u2208 s\nhyn : -(c ^ n \u2022 x) + y \u2208 s\n\u22a2 False"}, {"tactic": "have A : (c ^ n - c ^ m) \u2022 x \u2208 s := by\n  convert s.sub_mem hym hyn using 1\n  simp only [sub_smul, neg_sub_neg, add_sub_add_right_eq_sub]", "annotated_tactic": ["have A : (c ^ n - c ^ m) \u2022 x \u2208 s := by\n    convert s.sub_mem hym hyn using 1\n    simp only [<a>sub_smul</a>, <a>neg_sub_neg</a>, <a>add_sub_add_right_eq_sub</a>]", [{"full_name": "sub_smul", "def_path": "Mathlib/Algebra/Module/Basic.lean", "def_pos": [299, 9], "def_end_pos": [299, 17]}, {"full_name": "neg_sub_neg", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [506, 3], "def_end_pos": [506, 14]}, {"full_name": "add_sub_add_right_eq_sub", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [750, 3], "def_end_pos": [750, 14]}]], "state_before": "E : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\ns : Submodule \u211d E\nhs : s \u2260 \u22a4\nx : E\nhx : \u00acx \u2208 s\nc : \u211d\ncpos : 0 < c\ncone : c < 1\nA : Bornology.IsBounded (range fun n => c ^ n \u2022 x)\nm n : \u2115\nhmn : m \u2260 n\ny : E\nhym : -(c ^ m \u2022 x) + y \u2208 s\nhyn : -(c ^ n \u2022 x) + y \u2208 s\n\u22a2 False", "state_after": "E : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\ns : Submodule \u211d E\nhs : s \u2260 \u22a4\nx : E\nhx : \u00acx \u2208 s\nc : \u211d\ncpos : 0 < c\ncone : c < 1\nA\u271d : Bornology.IsBounded (range fun n => c ^ n \u2022 x)\nm n : \u2115\nhmn : m \u2260 n\ny : E\nhym : -(c ^ m \u2022 x) + y \u2208 s\nhyn : -(c ^ n \u2022 x) + y \u2208 s\nA : (c ^ n - c ^ m) \u2022 x \u2208 s\n\u22a2 False"}, {"tactic": "have H : c ^ n - c ^ m \u2260 0 := by\n  simpa only [sub_eq_zero, Ne.def] using (strictAnti_pow cpos cone).injective.ne hmn.symm", "annotated_tactic": ["have H : c ^ n - c ^ m \u2260 0 := by\n    simpa only [<a>sub_eq_zero</a>, <a>Ne.def</a>] using (<a>strictAnti_pow</a> cpos cone).injective.ne hmn.symm", [{"full_name": "sub_eq_zero", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [801, 3], "def_end_pos": [801, 14]}, {"full_name": "Ne.def", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [59, 9], "def_end_pos": [59, 15]}, {"full_name": "strictAnti_pow", "def_path": "Mathlib/Algebra/GroupPower/Order.lean", "def_pos": [508, 9], "def_end_pos": [508, 23]}]], "state_before": "E : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\ns : Submodule \u211d E\nhs : s \u2260 \u22a4\nx : E\nhx : \u00acx \u2208 s\nc : \u211d\ncpos : 0 < c\ncone : c < 1\nA\u271d : Bornology.IsBounded (range fun n => c ^ n \u2022 x)\nm n : \u2115\nhmn : m \u2260 n\ny : E\nhym : -(c ^ m \u2022 x) + y \u2208 s\nhyn : -(c ^ n \u2022 x) + y \u2208 s\nA : (c ^ n - c ^ m) \u2022 x \u2208 s\n\u22a2 False", "state_after": "E : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\ns : Submodule \u211d E\nhs : s \u2260 \u22a4\nx : E\nhx : \u00acx \u2208 s\nc : \u211d\ncpos : 0 < c\ncone : c < 1\nA\u271d : Bornology.IsBounded (range fun n => c ^ n \u2022 x)\nm n : \u2115\nhmn : m \u2260 n\ny : E\nhym : -(c ^ m \u2022 x) + y \u2208 s\nhyn : -(c ^ n \u2022 x) + y \u2208 s\nA : (c ^ n - c ^ m) \u2022 x \u2208 s\nH : c ^ n - c ^ m \u2260 0\n\u22a2 False"}, {"tactic": "have : x \u2208 s := by\n  convert s.smul_mem (c ^ n - c ^ m)\u207b\u00b9 A\n  rw [smul_smul, inv_mul_cancel H, one_smul]", "annotated_tactic": ["have : x \u2208 s := by\n    convert s.smul_mem (c ^ n - c ^ m)\u207b\u00b9 A\n    rw [<a>smul_smul</a>, <a>inv_mul_cancel</a> H, <a>one_smul</a>]", [{"full_name": "smul_smul", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [484, 9], "def_end_pos": [484, 18]}, {"full_name": "inv_mul_cancel", "def_path": "Mathlib/Algebra/GroupWithZero/NeZero.lean", "def_pos": [55, 9], "def_end_pos": [55, 23]}, {"full_name": "one_smul", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [492, 9], "def_end_pos": [492, 17]}]], "state_before": "E : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\ns : Submodule \u211d E\nhs : s \u2260 \u22a4\nx : E\nhx : \u00acx \u2208 s\nc : \u211d\ncpos : 0 < c\ncone : c < 1\nA\u271d : Bornology.IsBounded (range fun n => c ^ n \u2022 x)\nm n : \u2115\nhmn : m \u2260 n\ny : E\nhym : -(c ^ m \u2022 x) + y \u2208 s\nhyn : -(c ^ n \u2022 x) + y \u2208 s\nA : (c ^ n - c ^ m) \u2022 x \u2208 s\nH : c ^ n - c ^ m \u2260 0\n\u22a2 False", "state_after": "E : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\ns : Submodule \u211d E\nhs : s \u2260 \u22a4\nx : E\nhx : \u00acx \u2208 s\nc : \u211d\ncpos : 0 < c\ncone : c < 1\nA\u271d : Bornology.IsBounded (range fun n => c ^ n \u2022 x)\nm n : \u2115\nhmn : m \u2260 n\ny : E\nhym : -(c ^ m \u2022 x) + y \u2208 s\nhyn : -(c ^ n \u2022 x) + y \u2208 s\nA : (c ^ n - c ^ m) \u2022 x \u2208 s\nH : c ^ n - c ^ m \u2260 0\nthis : x \u2208 s\n\u22a2 False"}, {"tactic": "exact hx this", "annotated_tactic": ["exact hx this", []], "state_before": "E : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\ns : Submodule \u211d E\nhs : s \u2260 \u22a4\nx : E\nhx : \u00acx \u2208 s\nc : \u211d\ncpos : 0 < c\ncone : c < 1\nA\u271d : Bornology.IsBounded (range fun n => c ^ n \u2022 x)\nm n : \u2115\nhmn : m \u2260 n\ny : E\nhym : -(c ^ m \u2022 x) + y \u2208 s\nhyn : -(c ^ n \u2022 x) + y \u2208 s\nA : (c ^ n - c ^ m) \u2022 x \u2208 s\nH : c ^ n - c ^ m \u2260 0\nthis : x \u2208 s\n\u22a2 False", "state_after": "no goals"}, {"tactic": "simpa only [Submodule.eq_top_iff', not_exists, Ne.def, not_forall] using hs", "annotated_tactic": ["simpa only [<a>Submodule.eq_top_iff'</a>, <a>not_exists</a>, <a>Ne.def</a>, <a>not_forall</a>] using hs", [{"full_name": "Submodule.eq_top_iff'", "def_path": "Mathlib/Algebra/Module/Submodule/Lattice.lean", "def_pos": [184, 9], "def_end_pos": [184, 20]}, {"full_name": "not_exists", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [422, 17], "def_end_pos": [422, 27]}, {"full_name": "Ne.def", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [59, 9], "def_end_pos": [59, 15]}, {"full_name": "Classical.not_forall", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [686, 9], "def_end_pos": [686, 19]}]], "state_before": "E : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\ns : Submodule \u211d E\nhs : s \u2260 \u22a4\n\u22a2 \u2203 x, \u00acx \u2208 s", "state_after": "no goals"}, {"tactic": "norm_num", "annotated_tactic": ["norm_num", []], "state_before": "E : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\ns : Submodule \u211d E\nhs : s \u2260 \u22a4\nx : E\nhx : \u00acx \u2208 s\n\u22a2 0 < 1 / 2", "state_after": "no goals"}, {"tactic": "norm_num", "annotated_tactic": ["norm_num", []], "state_before": "E : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\ns : Submodule \u211d E\nhs : s \u2260 \u22a4\nx : E\nhx : \u00acx \u2208 s\n\u22a2 1 / 2 < 1", "state_after": "no goals"}, {"tactic": "convert s.sub_mem hym hyn using 1", "annotated_tactic": ["convert s.sub_mem hym hyn using 1", []], "state_before": "E : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\ns : Submodule \u211d E\nhs : s \u2260 \u22a4\nx : E\nhx : \u00acx \u2208 s\nc : \u211d\ncpos : 0 < c\ncone : c < 1\nA : Bornology.IsBounded (range fun n => c ^ n \u2022 x)\nm n : \u2115\nhmn : m \u2260 n\ny : E\nhym : -(c ^ m \u2022 x) + y \u2208 s\nhyn : -(c ^ n \u2022 x) + y \u2208 s\n\u22a2 (c ^ n - c ^ m) \u2022 x \u2208 s", "state_after": "case h.e'_4\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\ns : Submodule \u211d E\nhs : s \u2260 \u22a4\nx : E\nhx : \u00acx \u2208 s\nc : \u211d\ncpos : 0 < c\ncone : c < 1\nA : Bornology.IsBounded (range fun n => c ^ n \u2022 x)\nm n : \u2115\nhmn : m \u2260 n\ny : E\nhym : -(c ^ m \u2022 x) + y \u2208 s\nhyn : -(c ^ n \u2022 x) + y \u2208 s\n\u22a2 (c ^ n - c ^ m) \u2022 x = -(c ^ m \u2022 x) + y - (-(c ^ n \u2022 x) + y)"}, {"tactic": "simp only [sub_smul, neg_sub_neg, add_sub_add_right_eq_sub]", "annotated_tactic": ["simp only [<a>sub_smul</a>, <a>neg_sub_neg</a>, <a>add_sub_add_right_eq_sub</a>]", [{"full_name": "sub_smul", "def_path": "Mathlib/Algebra/Module/Basic.lean", "def_pos": [299, 9], "def_end_pos": [299, 17]}, {"full_name": "neg_sub_neg", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [506, 3], "def_end_pos": [506, 14]}, {"full_name": "add_sub_add_right_eq_sub", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [750, 3], "def_end_pos": [750, 14]}]], "state_before": "case h.e'_4\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\ns : Submodule \u211d E\nhs : s \u2260 \u22a4\nx : E\nhx : \u00acx \u2208 s\nc : \u211d\ncpos : 0 < c\ncone : c < 1\nA : Bornology.IsBounded (range fun n => c ^ n \u2022 x)\nm n : \u2115\nhmn : m \u2260 n\ny : E\nhym : -(c ^ m \u2022 x) + y \u2208 s\nhyn : -(c ^ n \u2022 x) + y \u2208 s\n\u22a2 (c ^ n - c ^ m) \u2022 x = -(c ^ m \u2022 x) + y - (-(c ^ n \u2022 x) + y)", "state_after": "no goals"}, {"tactic": "simpa only [sub_eq_zero, Ne.def] using (strictAnti_pow cpos cone).injective.ne hmn.symm", "annotated_tactic": ["simpa only [<a>sub_eq_zero</a>, <a>Ne.def</a>] using (<a>strictAnti_pow</a> cpos cone).injective.ne hmn.symm", [{"full_name": "sub_eq_zero", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [801, 3], "def_end_pos": [801, 14]}, {"full_name": "Ne.def", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [59, 9], "def_end_pos": [59, 15]}, {"full_name": "strictAnti_pow", "def_path": "Mathlib/Algebra/GroupPower/Order.lean", "def_pos": [508, 9], "def_end_pos": [508, 23]}]], "state_before": "E : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\ns : Submodule \u211d E\nhs : s \u2260 \u22a4\nx : E\nhx : \u00acx \u2208 s\nc : \u211d\ncpos : 0 < c\ncone : c < 1\nA\u271d : Bornology.IsBounded (range fun n => c ^ n \u2022 x)\nm n : \u2115\nhmn : m \u2260 n\ny : E\nhym : -(c ^ m \u2022 x) + y \u2208 s\nhyn : -(c ^ n \u2022 x) + y \u2208 s\nA : (c ^ n - c ^ m) \u2022 x \u2208 s\n\u22a2 c ^ n - c ^ m \u2260 0", "state_after": "no goals"}, {"tactic": "convert s.smul_mem (c ^ n - c ^ m)\u207b\u00b9 A", "annotated_tactic": ["convert s.smul_mem (c ^ n - c ^ m)\u207b\u00b9 A", []], "state_before": "E : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\ns : Submodule \u211d E\nhs : s \u2260 \u22a4\nx : E\nhx : \u00acx \u2208 s\nc : \u211d\ncpos : 0 < c\ncone : c < 1\nA\u271d : Bornology.IsBounded (range fun n => c ^ n \u2022 x)\nm n : \u2115\nhmn : m \u2260 n\ny : E\nhym : -(c ^ m \u2022 x) + y \u2208 s\nhyn : -(c ^ n \u2022 x) + y \u2208 s\nA : (c ^ n - c ^ m) \u2022 x \u2208 s\nH : c ^ n - c ^ m \u2260 0\n\u22a2 x \u2208 s", "state_after": "case h.e'_4\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\ns : Submodule \u211d E\nhs : s \u2260 \u22a4\nx : E\nhx : \u00acx \u2208 s\nc : \u211d\ncpos : 0 < c\ncone : c < 1\nA\u271d : Bornology.IsBounded (range fun n => c ^ n \u2022 x)\nm n : \u2115\nhmn : m \u2260 n\ny : E\nhym : -(c ^ m \u2022 x) + y \u2208 s\nhyn : -(c ^ n \u2022 x) + y \u2208 s\nA : (c ^ n - c ^ m) \u2022 x \u2208 s\nH : c ^ n - c ^ m \u2260 0\n\u22a2 x = (c ^ n - c ^ m)\u207b\u00b9 \u2022 (c ^ n - c ^ m) \u2022 x"}, {"tactic": "rw [smul_smul, inv_mul_cancel H, one_smul]", "annotated_tactic": ["rw [<a>smul_smul</a>, <a>inv_mul_cancel</a> H, <a>one_smul</a>]", [{"full_name": "smul_smul", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [484, 9], "def_end_pos": [484, 18]}, {"full_name": "inv_mul_cancel", "def_path": "Mathlib/Algebra/GroupWithZero/NeZero.lean", "def_pos": [55, 9], "def_end_pos": [55, 23]}, {"full_name": "one_smul", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [492, 9], "def_end_pos": [492, 17]}]], "state_before": "case h.e'_4\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\ns : Submodule \u211d E\nhs : s \u2260 \u22a4\nx : E\nhx : \u00acx \u2208 s\nc : \u211d\ncpos : 0 < c\ncone : c < 1\nA\u271d : Bornology.IsBounded (range fun n => c ^ n \u2022 x)\nm n : \u2115\nhmn : m \u2260 n\ny : E\nhym : -(c ^ m \u2022 x) + y \u2208 s\nhyn : -(c ^ n \u2022 x) + y \u2208 s\nA : (c ^ n - c ^ m) \u2022 x \u2208 s\nH : c ^ n - c ^ m \u2260 0\n\u22a2 x = (c ^ n - c ^ m)\u207b\u00b9 \u2022 (c ^ n - c ^ m) \u2022 x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Pairwise/Lattice.lean", "full_name": "Set.pairwiseDisjoint_prod_left", "start": [110, 1], "end": [119, 84], "traced_tactics": [{"tactic": "refine'\n      \u27e8fun h => \u27e8fun i hi j hj hij => _, fun i hi j hj hij => _\u27e9, fun h => h.1.prod_left h.2\u27e9 <;>\n    simp_rw [Function.onFun, iSup_disjoint_iff, disjoint_iSup_iff] <;>\n  intro i' hi' j' hj'", "annotated_tactic": ["refine'\n        \u27e8fun h => \u27e8fun i hi j hj hij => _, fun i hi j hj hij => _\u27e9, fun h => h.1.<a>prod_left</a> h.2\u27e9 <;>\n      simp_rw [<a>Function.onFun</a>, <a>iSup_disjoint_iff</a>, <a>disjoint_iSup_iff</a>] <;>\n    intro i' hi' j' hj'", [{"full_name": "Set.PairwiseDisjoint.prod_left", "def_path": "Mathlib/Data/Set/Pairwise/Lattice.lean", "def_pos": [89, 9], "def_end_pos": [89, 35]}, {"full_name": "Function.onFun", "def_path": "Mathlib/Init/Function.lean", "def_pos": [49, 5], "def_end_pos": [49, 10]}, {"full_name": "iSup_disjoint_iff", "def_path": "Mathlib/Order/CompleteBooleanAlgebra.lean", "def_pos": [239, 9], "def_end_pos": [239, 26]}, {"full_name": "disjoint_iSup_iff", "def_path": "Mathlib/Order/CompleteBooleanAlgebra.lean", "def_pos": [243, 9], "def_end_pos": [243, 26]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03ba : Sort u_6\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : Frame \u03b1\ns : Set \u03b9\nt : Set \u03b9'\nf : \u03b9 \u00d7 \u03b9' \u2192 \u03b1\n\u22a2 PairwiseDisjoint (s \u00d7\u02e2 t) f \u2194\n    (PairwiseDisjoint s fun i => \u2a06 i' \u2208 t, f (i, i')) \u2227 PairwiseDisjoint t fun i' => \u2a06 i \u2208 s, f (i, i')", "state_after": "case refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03ba : Sort u_6\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : Frame \u03b1\ns : Set \u03b9\nt : Set \u03b9'\nf : \u03b9 \u00d7 \u03b9' \u2192 \u03b1\nh : PairwiseDisjoint (s \u00d7\u02e2 t) f\ni : \u03b9\nhi : i \u2208 s\nj : \u03b9\nhj : j \u2208 s\nhij : i \u2260 j\ni' : \u03b9'\nhi' : i' \u2208 t\nj' : \u03b9'\nhj' : j' \u2208 t\n\u22a2 Disjoint (f (i, i')) (f (j, j'))\n\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03ba : Sort u_6\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : Frame \u03b1\ns : Set \u03b9\nt : Set \u03b9'\nf : \u03b9 \u00d7 \u03b9' \u2192 \u03b1\nh : PairwiseDisjoint (s \u00d7\u02e2 t) f\ni : \u03b9'\nhi : i \u2208 t\nj : \u03b9'\nhj : j \u2208 t\nhij : i \u2260 j\ni' : \u03b9\nhi' : i' \u2208 s\nj' : \u03b9\nhj' : j' \u2208 s\n\u22a2 Disjoint (f (i', i)) (f (j', j))"}, {"tactic": "exact h (mk_mem_prod hi hi') (mk_mem_prod hj hj') (ne_of_apply_ne Prod.fst hij)", "annotated_tactic": ["exact h (<a>mk_mem_prod</a> hi hi') (<a>mk_mem_prod</a> hj hj') (<a>ne_of_apply_ne</a> <a>Prod.fst</a> hij)", [{"full_name": "Set.mk_mem_prod", "def_path": "Mathlib/Data/Set/Prod.lean", "def_pos": [66, 9], "def_end_pos": [66, 20]}, {"full_name": "Set.mk_mem_prod", "def_path": "Mathlib/Data/Set/Prod.lean", "def_pos": [66, 9], "def_end_pos": [66, 20]}, {"full_name": "ne_of_apply_ne", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [520, 9], "def_end_pos": [520, 23]}, {"full_name": "Prod.fst", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [468, 3], "def_end_pos": [468, 6]}]], "state_before": "case refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03ba : Sort u_6\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : Frame \u03b1\ns : Set \u03b9\nt : Set \u03b9'\nf : \u03b9 \u00d7 \u03b9' \u2192 \u03b1\nh : PairwiseDisjoint (s \u00d7\u02e2 t) f\ni : \u03b9\nhi : i \u2208 s\nj : \u03b9\nhj : j \u2208 s\nhij : i \u2260 j\ni' : \u03b9'\nhi' : i' \u2208 t\nj' : \u03b9'\nhj' : j' \u2208 t\n\u22a2 Disjoint (f (i, i')) (f (j, j'))", "state_after": "no goals"}, {"tactic": "exact h (mk_mem_prod hi' hi) (mk_mem_prod hj' hj) (ne_of_apply_ne Prod.snd hij)", "annotated_tactic": ["exact h (<a>mk_mem_prod</a> hi' hi) (<a>mk_mem_prod</a> hj' hj) (<a>ne_of_apply_ne</a> <a>Prod.snd</a> hij)", [{"full_name": "Set.mk_mem_prod", "def_path": "Mathlib/Data/Set/Prod.lean", "def_pos": [66, 9], "def_end_pos": [66, 20]}, {"full_name": "Set.mk_mem_prod", "def_path": "Mathlib/Data/Set/Prod.lean", "def_pos": [66, 9], "def_end_pos": [66, 20]}, {"full_name": "ne_of_apply_ne", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [520, 9], "def_end_pos": [520, 23]}, {"full_name": "Prod.snd", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [470, 3], "def_end_pos": [470, 6]}]], "state_before": "case refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03ba : Sort u_6\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : Frame \u03b1\ns : Set \u03b9\nt : Set \u03b9'\nf : \u03b9 \u00d7 \u03b9' \u2192 \u03b1\nh : PairwiseDisjoint (s \u00d7\u02e2 t) f\ni : \u03b9'\nhi : i \u2208 t\nj : \u03b9'\nhj : j \u2208 t\nhij : i \u2260 j\ni' : \u03b9\nhi' : i' \u2208 s\nj' : \u03b9\nhj' : j' \u2208 s\n\u22a2 Disjoint (f (i', i)) (f (j', j))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/NAry.lean", "full_name": "Finset.image\u2082_congr'", "start": [230, 1], "end": [231, 36], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Kernel/CondCdf.lean", "full_name": "ProbabilityTheory.condCdf_le_one", "start": [794, 1], "end": [799, 17], "traced_tactics": [{"tactic": "obtain \u27e8r, hrx\u27e9 := exists_rat_gt x", "annotated_tactic": ["obtain \u27e8r, hrx\u27e9 := <a>exists_rat_gt</a> x", [{"full_name": "exists_rat_gt", "def_path": "Mathlib/Algebra/Order/Archimedean.lean", "def_pos": [253, 9], "def_end_pos": [253, 22]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nx : \u211d\n\u22a2 \u2191(condCdf \u03c1 a) x \u2264 1", "state_after": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nx : \u211d\nr : \u211a\nhrx : x < \u2191r\n\u22a2 \u2191(condCdf \u03c1 a) x \u2264 1"}, {"tactic": "rw [\u2190 StieltjesFunction.iInf_rat_gt_eq]", "annotated_tactic": ["rw [\u2190 <a>StieltjesFunction.iInf_rat_gt_eq</a>]", [{"full_name": "StieltjesFunction.iInf_rat_gt_eq", "def_path": "Mathlib/MeasureTheory/Measure/Stieltjes.lean", "def_pos": [82, 9], "def_end_pos": [82, 23]}]], "state_before": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nx : \u211d\nr : \u211a\nhrx : x < \u2191r\n\u22a2 \u2191(condCdf \u03c1 a) x \u2264 1", "state_after": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nx : \u211d\nr : \u211a\nhrx : x < \u2191r\n\u22a2 \u2a05 r, \u2191(condCdf \u03c1 a) \u2191\u2191r \u2264 1"}, {"tactic": "simp_rw [condCdf_eq_condCdfRat]", "annotated_tactic": ["simp_rw [<a>condCdf_eq_condCdfRat</a>]", [{"full_name": "ProbabilityTheory.condCdf_eq_condCdfRat", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [783, 9], "def_end_pos": [783, 30]}]], "state_before": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nx : \u211d\nr : \u211a\nhrx : x < \u2191r\n\u22a2 \u2a05 r, \u2191(condCdf \u03c1 a) \u2191\u2191r \u2264 1", "state_after": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nx : \u211d\nr : \u211a\nhrx : x < \u2191r\n\u22a2 \u2a05 r, condCdfRat \u03c1 a \u2191r \u2264 1"}, {"tactic": "refine' ciInf_le_of_le (bddBelow_range_condCdfRat_gt \u03c1 a x) _ (condCdfRat_le_one _ _ _)", "annotated_tactic": ["refine' <a>ciInf_le_of_le</a> (<a>bddBelow_range_condCdfRat_gt</a> \u03c1 a x) _ (<a>condCdfRat_le_one</a> _ _ _)", [{"full_name": "ciInf_le_of_le", "def_path": "Mathlib/Order/ConditionallyCompleteLattice/Basic.lean", "def_pos": [836, 9], "def_end_pos": [836, 23]}, {"full_name": "ProbabilityTheory.bddBelow_range_condCdfRat_gt", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [736, 9], "def_end_pos": [736, 37]}, {"full_name": "ProbabilityTheory.condCdfRat_le_one", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [616, 9], "def_end_pos": [616, 26]}]], "state_before": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nx : \u211d\nr : \u211a\nhrx : x < \u2191r\n\u22a2 \u2a05 r, condCdfRat \u03c1 a \u2191r \u2264 1", "state_after": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nx : \u211d\nr : \u211a\nhrx : x < \u2191r\n\u22a2 { r' // x < \u2191r' }"}, {"tactic": "exact \u27e8r, hrx\u27e9", "annotated_tactic": ["exact \u27e8r, hrx\u27e9", []], "state_before": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nx : \u211d\nr : \u211a\nhrx : x < \u2191r\n\u22a2 { r' // x < \u2191r' }", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Finite.lean", "full_name": "Set.Finite.disjoint_toFinset", "start": [247, 8], "end": [249, 49], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Image.lean", "full_name": "Set.preimage_inl_range_inr", "start": [943, 1], "end": [944, 44], "traced_tactics": [{"tactic": "rw [\u2190 image_univ, preimage_inl_image_inr]", "annotated_tactic": ["rw [\u2190 <a>image_univ</a>, <a>preimage_inl_image_inr</a>]", [{"full_name": "Set.image_univ", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [718, 9], "def_end_pos": [718, 19]}, {"full_name": "Set.preimage_inl_image_inr", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [931, 9], "def_end_pos": [931, 31]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\nf : \u03b9 \u2192 \u03b1\ns t : Set \u03b1\n\u22a2 Sum.inl \u207b\u00b9' range Sum.inr = \u2205", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Lattice.lean", "full_name": "Finset.inf'_congr", "start": [1024, 1], "end": [1026, 32], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Finite.lean", "full_name": "Set.iUnion_pi_of_monotone", "start": [1554, 1], "end": [1560, 36], "traced_tactics": [{"tactic": "simp only [pi_def, biInter_eq_iInter, preimage_iUnion]", "annotated_tactic": ["simp only [<a>pi_def</a>, <a>biInter_eq_iInter</a>, <a>preimage_iUnion</a>]", [{"full_name": "Set.pi_def", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [2163, 9], "def_end_pos": [2163, 15]}, {"full_name": "Set.biInter_eq_iInter", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [1015, 9], "def_end_pos": [1015, 26]}, {"full_name": "Set.preimage_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [1854, 9], "def_end_pos": [1854, 24]}]], "state_before": "\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b9\u271d : Sort w\n\u03b3 : Type x\n\u03b9 : Type u_1\n\u03b9' : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b9'\ninst\u271d : Nonempty \u03b9'\n\u03b1 : \u03b9 \u2192 Type u_3\nI : Set \u03b9\ns : (i : \u03b9) \u2192 \u03b9' \u2192 Set (\u03b1 i)\nhI : Set.Finite I\nhs : \u2200 (i : \u03b9), i \u2208 I \u2192 Monotone (s i)\n\u22a2 (\u22c3 j, pi I fun i => s i j) = pi I fun i => \u22c3 j, s i j", "state_after": "\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b9\u271d : Sort w\n\u03b3 : Type x\n\u03b9 : Type u_1\n\u03b9' : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b9'\ninst\u271d : Nonempty \u03b9'\n\u03b1 : \u03b9 \u2192 Type u_3\nI : Set \u03b9\ns : (i : \u03b9) \u2192 \u03b9' \u2192 Set (\u03b1 i)\nhI : Set.Finite I\nhs : \u2200 (i : \u03b9), i \u2208 I \u2192 Monotone (s i)\n\u22a2 \u22c3 j, \u22c2 x, eval \u2191x \u207b\u00b9' s (\u2191x) j = \u22c2 x, \u22c3 i, eval \u2191x \u207b\u00b9' s (\u2191x) i"}, {"tactic": "haveI := hI.fintype.finite", "annotated_tactic": ["haveI := hI.fintype.finite", []], "state_before": "\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b9\u271d : Sort w\n\u03b3 : Type x\n\u03b9 : Type u_1\n\u03b9' : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b9'\ninst\u271d : Nonempty \u03b9'\n\u03b1 : \u03b9 \u2192 Type u_3\nI : Set \u03b9\ns : (i : \u03b9) \u2192 \u03b9' \u2192 Set (\u03b1 i)\nhI : Set.Finite I\nhs : \u2200 (i : \u03b9), i \u2208 I \u2192 Monotone (s i)\n\u22a2 \u22c3 j, \u22c2 x, eval \u2191x \u207b\u00b9' s (\u2191x) j = \u22c2 x, \u22c3 i, eval \u2191x \u207b\u00b9' s (\u2191x) i", "state_after": "\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b9\u271d : Sort w\n\u03b3 : Type x\n\u03b9 : Type u_1\n\u03b9' : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b9'\ninst\u271d : Nonempty \u03b9'\n\u03b1 : \u03b9 \u2192 Type u_3\nI : Set \u03b9\ns : (i : \u03b9) \u2192 \u03b9' \u2192 Set (\u03b1 i)\nhI : Set.Finite I\nhs : \u2200 (i : \u03b9), i \u2208 I \u2192 Monotone (s i)\nthis : Finite \u2191I\n\u22a2 \u22c3 j, \u22c2 x, eval \u2191x \u207b\u00b9' s (\u2191x) j = \u22c2 x, \u22c3 i, eval \u2191x \u207b\u00b9' s (\u2191x) i"}, {"tactic": "refine' iUnion_iInter_of_monotone (\u03b9' := \u03b9') (fun (i : I) j\u2081 j\u2082 h => _)", "annotated_tactic": ["refine' <a>iUnion_iInter_of_monotone</a> (\u03b9' := \u03b9') (fun (i : I) j\u2081 j\u2082 h => _)", [{"full_name": "Set.iUnion_iInter_of_monotone", "def_path": "Mathlib/Data/Set/Finite.lean", "def_pos": [1527, 9], "def_end_pos": [1527, 34]}]], "state_before": "\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b9\u271d : Sort w\n\u03b3 : Type x\n\u03b9 : Type u_1\n\u03b9' : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b9'\ninst\u271d : Nonempty \u03b9'\n\u03b1 : \u03b9 \u2192 Type u_3\nI : Set \u03b9\ns : (i : \u03b9) \u2192 \u03b9' \u2192 Set (\u03b1 i)\nhI : Set.Finite I\nhs : \u2200 (i : \u03b9), i \u2208 I \u2192 Monotone (s i)\nthis : Finite \u2191I\n\u22a2 \u22c3 j, \u22c2 x, eval \u2191x \u207b\u00b9' s (\u2191x) j = \u22c2 x, \u22c3 i, eval \u2191x \u207b\u00b9' s (\u2191x) i", "state_after": "\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b9\u271d : Sort w\n\u03b3 : Type x\n\u03b9 : Type u_1\n\u03b9' : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b9'\ninst\u271d : Nonempty \u03b9'\n\u03b1 : \u03b9 \u2192 Type u_3\nI : Set \u03b9\ns : (i : \u03b9) \u2192 \u03b9' \u2192 Set (\u03b1 i)\nhI : Set.Finite I\nhs : \u2200 (i : \u03b9), i \u2208 I \u2192 Monotone (s i)\nthis : Finite \u2191I\ni : \u2191I\nj\u2081 j\u2082 : \u03b9'\nh : j\u2081 \u2264 j\u2082\n\u22a2 eval \u2191i \u207b\u00b9' s (\u2191i) j\u2081 \u2264 eval \u2191i \u207b\u00b9' s (\u2191i) j\u2082"}, {"tactic": "exact preimage_mono <| hs i i.2 h", "annotated_tactic": ["exact <a>preimage_mono</a> <| hs i i.2 h", [{"full_name": "Set.preimage_mono", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [74, 9], "def_end_pos": [74, 22]}]], "state_before": "\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b9\u271d : Sort w\n\u03b3 : Type x\n\u03b9 : Type u_1\n\u03b9' : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b9'\ninst\u271d : Nonempty \u03b9'\n\u03b1 : \u03b9 \u2192 Type u_3\nI : Set \u03b9\ns : (i : \u03b9) \u2192 \u03b9' \u2192 Set (\u03b1 i)\nhI : Set.Finite I\nhs : \u2200 (i : \u03b9), i \u2208 I \u2192 Monotone (s i)\nthis : Finite \u2191I\ni : \u2191I\nj\u2081 j\u2082 : \u03b9'\nh : j\u2081 \u2264 j\u2082\n\u22a2 eval \u2191i \u207b\u00b9' s (\u2191i) j\u2081 \u2264 eval \u2191i \u207b\u00b9' s (\u2191i) j\u2082", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/TuringMachine.lean", "full_name": "Turing.PointedMap.mk_val", "start": [362, 1], "end": [364, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Lattice.lean", "full_name": "Finset.min'_insert", "start": [1571, 1], "end": [1575, 38], "traced_tactics": [{"tactic": "rw [coe_insert, min_comm]", "annotated_tactic": ["rw [<a>coe_insert</a>, <a>min_comm</a>]", [{"full_name": "Finset.coe_insert", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1113, 9], "def_end_pos": [1113, 19]}, {"full_name": "min_comm", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [80, 9], "def_end_pos": [80, 17]}]], "state_before": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Type u_5\n\u03ba : Type u_6\ninst\u271d : LinearOrder \u03b1\ns\u271d : Finset \u03b1\nH\u271d : Finset.Nonempty s\u271d\nx a : \u03b1\ns : Finset \u03b1\nH : Finset.Nonempty s\n\u22a2 IsLeast (\u2191(insert a s)) (min (min' s H) a)", "state_after": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Type u_5\n\u03ba : Type u_6\ninst\u271d : LinearOrder \u03b1\ns\u271d : Finset \u03b1\nH\u271d : Finset.Nonempty s\u271d\nx a : \u03b1\ns : Finset \u03b1\nH : Finset.Nonempty s\n\u22a2 IsLeast (insert a \u2191s) (min a (min' s H))"}, {"tactic": "exact (isLeast_min' _ _).insert _", "annotated_tactic": ["exact (<a>isLeast_min'</a> _ _).<a>insert</a> _", [{"full_name": "Finset.isLeast_min'", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [1427, 9], "def_end_pos": [1427, 21]}, {"full_name": "IsLeast.insert", "def_path": "Mathlib/Order/Bounds/Basic.lean", "def_pos": [964, 19], "def_end_pos": [964, 33]}]], "state_before": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Type u_5\n\u03ba : Type u_6\ninst\u271d : LinearOrder \u03b1\ns\u271d : Finset \u03b1\nH\u271d : Finset.Nonempty s\u271d\nx a : \u03b1\ns : Finset \u03b1\nH : Finset.Nonempty s\n\u22a2 IsLeast (insert a \u2191s) (min a (min' s H))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "full_name": "MeasureTheory.snorm_norm_rpow", "start": [518, 1], "end": [546, 45], "traced_tactics": [{"tactic": "by_cases h0 : p = 0", "annotated_tactic": ["by_cases h0 : p = 0", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nhq_pos : 0 < q\n\u22a2 snorm (fun x => \u2016f x\u2016 ^ q) p \u03bc = snorm f (p * ENNReal.ofReal q) \u03bc ^ q", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nhq_pos : 0 < q\nh0 : p = 0\n\u22a2 snorm (fun x => \u2016f x\u2016 ^ q) p \u03bc = snorm f (p * ENNReal.ofReal q) \u03bc ^ q\n\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nhq_pos : 0 < q\nh0 : \u00acp = 0\n\u22a2 snorm (fun x => \u2016f x\u2016 ^ q) p \u03bc = snorm f (p * ENNReal.ofReal q) \u03bc ^ q"}, {"tactic": "by_cases hp_top : p = \u221e", "annotated_tactic": ["by_cases hp_top : p = \u221e", []], "state_before": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nhq_pos : 0 < q\nh0 : \u00acp = 0\n\u22a2 snorm (fun x => \u2016f x\u2016 ^ q) p \u03bc = snorm f (p * ENNReal.ofReal q) \u03bc ^ q", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nhq_pos : 0 < q\nh0 : \u00acp = 0\nhp_top : p = \u22a4\n\u22a2 snorm (fun x => \u2016f x\u2016 ^ q) p \u03bc = snorm f (p * ENNReal.ofReal q) \u03bc ^ q\n\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nhq_pos : 0 < q\nh0 : \u00acp = 0\nhp_top : \u00acp = \u22a4\n\u22a2 snorm (fun x => \u2016f x\u2016 ^ q) p \u03bc = snorm f (p * ENNReal.ofReal q) \u03bc ^ q"}, {"tactic": "rw [snorm_eq_snorm' h0 hp_top, snorm_eq_snorm' _ _]", "annotated_tactic": ["rw [<a>snorm_eq_snorm'</a> h0 hp_top, <a>snorm_eq_snorm'</a> _ _]", [{"full_name": "MeasureTheory.snorm_eq_snorm'", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [88, 9], "def_end_pos": [88, 24]}, {"full_name": "MeasureTheory.snorm_eq_snorm'", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [88, 9], "def_end_pos": [88, 24]}]], "state_before": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nhq_pos : 0 < q\nh0 : \u00acp = 0\nhp_top : \u00acp = \u22a4\n\u22a2 snorm (fun x => \u2016f x\u2016 ^ q) p \u03bc = snorm f (p * ENNReal.ofReal q) \u03bc ^ q", "state_after": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nhq_pos : 0 < q\nh0 : \u00acp = 0\nhp_top : \u00acp = \u22a4\n\u22a2 snorm' (fun x => \u2016f x\u2016 ^ q) (ENNReal.toReal p) \u03bc = snorm' f (ENNReal.toReal (p * ENNReal.ofReal q)) \u03bc ^ q\n\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nhq_pos : 0 < q\nh0 : \u00acp = 0\nhp_top : \u00acp = \u22a4\n\u22a2 p * ENNReal.ofReal q \u2260 0\n\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nhq_pos : 0 < q\nh0 : \u00acp = 0\nhp_top : \u00acp = \u22a4\n\u22a2 p * ENNReal.ofReal q \u2260 \u22a4"}, {"tactic": "swap", "annotated_tactic": ["swap", []], "state_before": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nhq_pos : 0 < q\nh0 : \u00acp = 0\nhp_top : \u00acp = \u22a4\n\u22a2 snorm' (fun x => \u2016f x\u2016 ^ q) (ENNReal.toReal p) \u03bc = snorm' f (ENNReal.toReal (p * ENNReal.ofReal q)) \u03bc ^ q\n\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nhq_pos : 0 < q\nh0 : \u00acp = 0\nhp_top : \u00acp = \u22a4\n\u22a2 p * ENNReal.ofReal q \u2260 0\n\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nhq_pos : 0 < q\nh0 : \u00acp = 0\nhp_top : \u00acp = \u22a4\n\u22a2 p * ENNReal.ofReal q \u2260 \u22a4", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nhq_pos : 0 < q\nh0 : \u00acp = 0\nhp_top : \u00acp = \u22a4\n\u22a2 p * ENNReal.ofReal q \u2260 0\n\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nhq_pos : 0 < q\nh0 : \u00acp = 0\nhp_top : \u00acp = \u22a4\n\u22a2 snorm' (fun x => \u2016f x\u2016 ^ q) (ENNReal.toReal p) \u03bc = snorm' f (ENNReal.toReal (p * ENNReal.ofReal q)) \u03bc ^ q\n\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nhq_pos : 0 < q\nh0 : \u00acp = 0\nhp_top : \u00acp = \u22a4\n\u22a2 p * ENNReal.ofReal q \u2260 \u22a4"}, {"tactic": "swap", "annotated_tactic": ["swap", []], "state_before": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nhq_pos : 0 < q\nh0 : \u00acp = 0\nhp_top : \u00acp = \u22a4\n\u22a2 snorm' (fun x => \u2016f x\u2016 ^ q) (ENNReal.toReal p) \u03bc = snorm' f (ENNReal.toReal (p * ENNReal.ofReal q)) \u03bc ^ q\n\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nhq_pos : 0 < q\nh0 : \u00acp = 0\nhp_top : \u00acp = \u22a4\n\u22a2 p * ENNReal.ofReal q \u2260 \u22a4", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nhq_pos : 0 < q\nh0 : \u00acp = 0\nhp_top : \u00acp = \u22a4\n\u22a2 p * ENNReal.ofReal q \u2260 \u22a4\n\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nhq_pos : 0 < q\nh0 : \u00acp = 0\nhp_top : \u00acp = \u22a4\n\u22a2 snorm' (fun x => \u2016f x\u2016 ^ q) (ENNReal.toReal p) \u03bc = snorm' f (ENNReal.toReal (p * ENNReal.ofReal q)) \u03bc ^ q"}, {"tactic": "rw [ENNReal.toReal_mul, ENNReal.toReal_ofReal hq_pos.le]", "annotated_tactic": ["rw [<a>ENNReal.toReal_mul</a>, <a>ENNReal.toReal_ofReal</a> hq_pos.le]", [{"full_name": "ENNReal.toReal_mul", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2296, 9], "def_end_pos": [2296, 19]}, {"full_name": "ENNReal.toReal_ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [191, 9], "def_end_pos": [191, 22]}]], "state_before": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nhq_pos : 0 < q\nh0 : \u00acp = 0\nhp_top : \u00acp = \u22a4\n\u22a2 snorm' (fun x => \u2016f x\u2016 ^ q) (ENNReal.toReal p) \u03bc = snorm' f (ENNReal.toReal (p * ENNReal.ofReal q)) \u03bc ^ q", "state_after": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nhq_pos : 0 < q\nh0 : \u00acp = 0\nhp_top : \u00acp = \u22a4\n\u22a2 snorm' (fun x => \u2016f x\u2016 ^ q) (ENNReal.toReal p) \u03bc = snorm' f (ENNReal.toReal p * q) \u03bc ^ q"}, {"tactic": "exact snorm'_norm_rpow f p.toReal q hq_pos", "annotated_tactic": ["exact <a>snorm'_norm_rpow</a> f p.toReal q hq_pos", [{"full_name": "MeasureTheory.snorm'_norm_rpow", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [505, 9], "def_end_pos": [505, 25]}]], "state_before": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nhq_pos : 0 < q\nh0 : \u00acp = 0\nhp_top : \u00acp = \u22a4\n\u22a2 snorm' (fun x => \u2016f x\u2016 ^ q) (ENNReal.toReal p) \u03bc = snorm' f (ENNReal.toReal p * q) \u03bc ^ q", "state_after": "no goals"}, {"tactic": "simp [h0, ENNReal.zero_rpow_of_pos hq_pos]", "annotated_tactic": ["simp [h0, <a>ENNReal.zero_rpow_of_pos</a> hq_pos]", [{"full_name": "ENNReal.zero_rpow_of_pos", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [401, 9], "def_end_pos": [401, 25]}]], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nhq_pos : 0 < q\nh0 : p = 0\n\u22a2 snorm (fun x => \u2016f x\u2016 ^ q) p \u03bc = snorm f (p * ENNReal.ofReal q) \u03bc ^ q", "state_after": "no goals"}, {"tactic": "simp only [hp_top, snorm_exponent_top, ENNReal.top_mul', hq_pos.not_le, ENNReal.ofReal_eq_zero,\n  if_false, snorm_exponent_top, snormEssSup]", "annotated_tactic": ["simp only [hp_top, <a>snorm_exponent_top</a>, <a>ENNReal.top_mul'</a>, hq_pos.not_le, <a>ENNReal.ofReal_eq_zero</a>,\n      <a>if_false</a>, <a>snorm_exponent_top</a>, <a>snormEssSup</a>]", [{"full_name": "MeasureTheory.snorm_exponent_top", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [103, 9], "def_end_pos": [103, 27]}, {"full_name": "ENNReal.top_mul'", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [583, 9], "def_end_pos": [583, 17]}, {"full_name": "ENNReal.ofReal_eq_zero", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2170, 9], "def_end_pos": [2170, 23]}, {"full_name": "if_false", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [729, 17], "def_end_pos": [729, 25]}, {"full_name": "MeasureTheory.snorm_exponent_top", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [103, 9], "def_end_pos": [103, 27]}, {"full_name": "MeasureTheory.snormEssSup", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [78, 5], "def_end_pos": [78, 16]}]], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nhq_pos : 0 < q\nh0 : \u00acp = 0\nhp_top : p = \u22a4\n\u22a2 snorm (fun x => \u2016f x\u2016 ^ q) p \u03bc = snorm f (p * ENNReal.ofReal q) \u03bc ^ q", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nhq_pos : 0 < q\nh0 : \u00acp = 0\nhp_top : p = \u22a4\n\u22a2 essSup (fun x => \u2191\u2016\u2016f x\u2016 ^ q\u2016\u208a) \u03bc = essSup (fun x => \u2191\u2016f x\u2016\u208a) \u03bc ^ q"}, {"tactic": "have h_rpow :\n  essSup (fun x : \u03b1 => (\u2016\u2016f x\u2016 ^ q\u2016\u208a : \u211d\u22650\u221e)) \u03bc =\n    essSup (fun x : \u03b1 => (\u2016f x\u2016\u208a : \u211d\u22650\u221e) ^ q) \u03bc := by\n  congr\n  ext1 x\n  conv_rhs => rw [\u2190 nnnorm_norm]\n  rw [ENNReal.coe_rpow_of_nonneg _ hq_pos.le, ENNReal.coe_eq_coe]\n  ext\n  push_cast\n  rw [Real.norm_rpow_of_nonneg (norm_nonneg _)]", "annotated_tactic": ["have h_rpow :\n      <a>essSup</a> (fun x : \u03b1 => (\u2016\u2016f x\u2016 ^ q\u2016\u208a : \u211d\u22650\u221e)) \u03bc =\n        <a>essSup</a> (fun x : \u03b1 => (\u2016f x\u2016\u208a : \u211d\u22650\u221e) ^ q) \u03bc := by\n      congr\n      ext1 x\n      conv_rhs => rw [\u2190 <a>nnnorm_norm</a>]\n      rw [<a>ENNReal.coe_rpow_of_nonneg</a> _ hq_pos.le, <a>ENNReal.coe_eq_coe</a>]\n      ext\n      push_cast\n      rw [<a>Real.norm_rpow_of_nonneg</a> (<a>norm_nonneg</a> _)]", [{"full_name": "essSup", "def_path": "Mathlib/MeasureTheory/Function/EssSup.lean", "def_pos": [44, 5], "def_end_pos": [44, 11]}, {"full_name": "essSup", "def_path": "Mathlib/MeasureTheory/Function/EssSup.lean", "def_pos": [44, 5], "def_end_pos": [44, 11]}, {"full_name": "nnnorm_norm", "def_path": "Mathlib/Analysis/Normed/Field/Basic.lean", "def_pos": [842, 9], "def_end_pos": [842, 20]}, {"full_name": "ENNReal.coe_rpow_of_nonneg", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [436, 9], "def_end_pos": [436, 27]}, {"full_name": "ENNReal.coe_eq_coe", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [346, 28], "def_end_pos": [346, 38]}, {"full_name": "Real.norm_rpow_of_nonneg", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [166, 9], "def_end_pos": [166, 28]}, {"full_name": "norm_nonneg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [500, 30], "def_end_pos": [500, 41]}]], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nhq_pos : 0 < q\nh0 : \u00acp = 0\nhp_top : p = \u22a4\n\u22a2 essSup (fun x => \u2191\u2016\u2016f x\u2016 ^ q\u2016\u208a) \u03bc = essSup (fun x => \u2191\u2016f x\u2016\u208a) \u03bc ^ q", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nhq_pos : 0 < q\nh0 : \u00acp = 0\nhp_top : p = \u22a4\nh_rpow : essSup (fun x => \u2191\u2016\u2016f x\u2016 ^ q\u2016\u208a) \u03bc = essSup (fun x => \u2191\u2016f x\u2016\u208a ^ q) \u03bc\n\u22a2 essSup (fun x => \u2191\u2016\u2016f x\u2016 ^ q\u2016\u208a) \u03bc = essSup (fun x => \u2191\u2016f x\u2016\u208a) \u03bc ^ q"}, {"tactic": "rw [h_rpow]", "annotated_tactic": ["rw [h_rpow]", []], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nhq_pos : 0 < q\nh0 : \u00acp = 0\nhp_top : p = \u22a4\nh_rpow : essSup (fun x => \u2191\u2016\u2016f x\u2016 ^ q\u2016\u208a) \u03bc = essSup (fun x => \u2191\u2016f x\u2016\u208a ^ q) \u03bc\n\u22a2 essSup (fun x => \u2191\u2016\u2016f x\u2016 ^ q\u2016\u208a) \u03bc = essSup (fun x => \u2191\u2016f x\u2016\u208a) \u03bc ^ q", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nhq_pos : 0 < q\nh0 : \u00acp = 0\nhp_top : p = \u22a4\nh_rpow : essSup (fun x => \u2191\u2016\u2016f x\u2016 ^ q\u2016\u208a) \u03bc = essSup (fun x => \u2191\u2016f x\u2016\u208a ^ q) \u03bc\n\u22a2 essSup (fun x => \u2191\u2016f x\u2016\u208a ^ q) \u03bc = essSup (fun x => \u2191\u2016f x\u2016\u208a) \u03bc ^ q"}, {"tactic": "have h_rpow_mono := ENNReal.strictMono_rpow_of_pos hq_pos", "annotated_tactic": ["have h_rpow_mono := <a>ENNReal.strictMono_rpow_of_pos</a> hq_pos", [{"full_name": "ENNReal.strictMono_rpow_of_pos", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [612, 9], "def_end_pos": [612, 31]}]], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nhq_pos : 0 < q\nh0 : \u00acp = 0\nhp_top : p = \u22a4\nh_rpow : essSup (fun x => \u2191\u2016\u2016f x\u2016 ^ q\u2016\u208a) \u03bc = essSup (fun x => \u2191\u2016f x\u2016\u208a ^ q) \u03bc\n\u22a2 essSup (fun x => \u2191\u2016f x\u2016\u208a ^ q) \u03bc = essSup (fun x => \u2191\u2016f x\u2016\u208a) \u03bc ^ q", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nhq_pos : 0 < q\nh0 : \u00acp = 0\nhp_top : p = \u22a4\nh_rpow : essSup (fun x => \u2191\u2016\u2016f x\u2016 ^ q\u2016\u208a) \u03bc = essSup (fun x => \u2191\u2016f x\u2016\u208a ^ q) \u03bc\nh_rpow_mono : StrictMono fun x => x ^ q\n\u22a2 essSup (fun x => \u2191\u2016f x\u2016\u208a ^ q) \u03bc = essSup (fun x => \u2191\u2016f x\u2016\u208a) \u03bc ^ q"}, {"tactic": "have h_rpow_surj := (ENNReal.rpow_left_bijective hq_pos.ne.symm).2", "annotated_tactic": ["have h_rpow_surj := (<a>ENNReal.rpow_left_bijective</a> hq_pos.ne.symm).2", [{"full_name": "ENNReal.rpow_left_bijective", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [845, 9], "def_end_pos": [845, 28]}]], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nhq_pos : 0 < q\nh0 : \u00acp = 0\nhp_top : p = \u22a4\nh_rpow : essSup (fun x => \u2191\u2016\u2016f x\u2016 ^ q\u2016\u208a) \u03bc = essSup (fun x => \u2191\u2016f x\u2016\u208a ^ q) \u03bc\nh_rpow_mono : StrictMono fun x => x ^ q\n\u22a2 essSup (fun x => \u2191\u2016f x\u2016\u208a ^ q) \u03bc = essSup (fun x => \u2191\u2016f x\u2016\u208a) \u03bc ^ q", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nhq_pos : 0 < q\nh0 : \u00acp = 0\nhp_top : p = \u22a4\nh_rpow : essSup (fun x => \u2191\u2016\u2016f x\u2016 ^ q\u2016\u208a) \u03bc = essSup (fun x => \u2191\u2016f x\u2016\u208a ^ q) \u03bc\nh_rpow_mono : StrictMono fun x => x ^ q\nh_rpow_surj : Function.Surjective fun y => y ^ q\n\u22a2 essSup (fun x => \u2191\u2016f x\u2016\u208a ^ q) \u03bc = essSup (fun x => \u2191\u2016f x\u2016\u208a) \u03bc ^ q"}, {"tactic": "let iso := h_rpow_mono.orderIsoOfSurjective _ h_rpow_surj", "annotated_tactic": ["let iso := h_rpow_mono.orderIsoOfSurjective _ h_rpow_surj", []], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nhq_pos : 0 < q\nh0 : \u00acp = 0\nhp_top : p = \u22a4\nh_rpow : essSup (fun x => \u2191\u2016\u2016f x\u2016 ^ q\u2016\u208a) \u03bc = essSup (fun x => \u2191\u2016f x\u2016\u208a ^ q) \u03bc\nh_rpow_mono : StrictMono fun x => x ^ q\nh_rpow_surj : Function.Surjective fun y => y ^ q\n\u22a2 essSup (fun x => \u2191\u2016f x\u2016\u208a ^ q) \u03bc = essSup (fun x => \u2191\u2016f x\u2016\u208a) \u03bc ^ q", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nhq_pos : 0 < q\nh0 : \u00acp = 0\nhp_top : p = \u22a4\nh_rpow : essSup (fun x => \u2191\u2016\u2016f x\u2016 ^ q\u2016\u208a) \u03bc = essSup (fun x => \u2191\u2016f x\u2016\u208a ^ q) \u03bc\nh_rpow_mono : StrictMono fun x => x ^ q\nh_rpow_surj : Function.Surjective fun y => y ^ q\niso : \u211d\u22650\u221e \u2243o \u211d\u22650\u221e := StrictMono.orderIsoOfSurjective (fun x => x ^ q) h_rpow_mono h_rpow_surj\n\u22a2 essSup (fun x => \u2191\u2016f x\u2016\u208a ^ q) \u03bc = essSup (fun x => \u2191\u2016f x\u2016\u208a) \u03bc ^ q"}, {"tactic": "exact (iso.essSup_apply (fun x => (\u2016f x\u2016\u208a : \u211d\u22650\u221e)) \u03bc).symm", "annotated_tactic": ["exact (iso.essSup_apply (fun x => (\u2016f x\u2016\u208a : \u211d\u22650\u221e)) \u03bc).<a>symm</a>", [{"full_name": "Eq.symm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [310, 9], "def_end_pos": [310, 16]}]], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nhq_pos : 0 < q\nh0 : \u00acp = 0\nhp_top : p = \u22a4\nh_rpow : essSup (fun x => \u2191\u2016\u2016f x\u2016 ^ q\u2016\u208a) \u03bc = essSup (fun x => \u2191\u2016f x\u2016\u208a ^ q) \u03bc\nh_rpow_mono : StrictMono fun x => x ^ q\nh_rpow_surj : Function.Surjective fun y => y ^ q\niso : \u211d\u22650\u221e \u2243o \u211d\u22650\u221e := StrictMono.orderIsoOfSurjective (fun x => x ^ q) h_rpow_mono h_rpow_surj\n\u22a2 essSup (fun x => \u2191\u2016f x\u2016\u208a ^ q) \u03bc = essSup (fun x => \u2191\u2016f x\u2016\u208a) \u03bc ^ q", "state_after": "no goals"}, {"tactic": "congr", "annotated_tactic": ["congr", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nhq_pos : 0 < q\nh0 : \u00acp = 0\nhp_top : p = \u22a4\n\u22a2 essSup (fun x => \u2191\u2016\u2016f x\u2016 ^ q\u2016\u208a) \u03bc = essSup (fun x => \u2191\u2016f x\u2016\u208a ^ q) \u03bc", "state_after": "case e_f\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nhq_pos : 0 < q\nh0 : \u00acp = 0\nhp_top : p = \u22a4\n\u22a2 (fun x => \u2191\u2016\u2016f x\u2016 ^ q\u2016\u208a) = fun x => \u2191\u2016f x\u2016\u208a ^ q"}, {"tactic": "ext1 x", "annotated_tactic": ["ext1 x", []], "state_before": "case e_f\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nhq_pos : 0 < q\nh0 : \u00acp = 0\nhp_top : p = \u22a4\n\u22a2 (fun x => \u2191\u2016\u2016f x\u2016 ^ q\u2016\u208a) = fun x => \u2191\u2016f x\u2016\u208a ^ q", "state_after": "case e_f.h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nhq_pos : 0 < q\nh0 : \u00acp = 0\nhp_top : p = \u22a4\nx : \u03b1\n\u22a2 \u2191\u2016\u2016f x\u2016 ^ q\u2016\u208a = \u2191\u2016f x\u2016\u208a ^ q"}, {"tactic": "conv_rhs => rw [\u2190 nnnorm_norm]", "annotated_tactic": ["conv_rhs => rw [\u2190 <a>nnnorm_norm</a>]", [{"full_name": "nnnorm_norm", "def_path": "Mathlib/Analysis/Normed/Field/Basic.lean", "def_pos": [842, 9], "def_end_pos": [842, 20]}]], "state_before": "case e_f.h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nhq_pos : 0 < q\nh0 : \u00acp = 0\nhp_top : p = \u22a4\nx : \u03b1\n\u22a2 \u2191\u2016\u2016f x\u2016 ^ q\u2016\u208a = \u2191\u2016f x\u2016\u208a ^ q", "state_after": "case e_f.h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nhq_pos : 0 < q\nh0 : \u00acp = 0\nhp_top : p = \u22a4\nx : \u03b1\n\u22a2 \u2191\u2016\u2016f x\u2016 ^ q\u2016\u208a = \u2191\u2016\u2016f x\u2016\u2016\u208a ^ q"}, {"tactic": "rw [ENNReal.coe_rpow_of_nonneg _ hq_pos.le, ENNReal.coe_eq_coe]", "annotated_tactic": ["rw [<a>ENNReal.coe_rpow_of_nonneg</a> _ hq_pos.le, <a>ENNReal.coe_eq_coe</a>]", [{"full_name": "ENNReal.coe_rpow_of_nonneg", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [436, 9], "def_end_pos": [436, 27]}, {"full_name": "ENNReal.coe_eq_coe", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [346, 28], "def_end_pos": [346, 38]}]], "state_before": "case e_f.h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nhq_pos : 0 < q\nh0 : \u00acp = 0\nhp_top : p = \u22a4\nx : \u03b1\n\u22a2 \u2191\u2016\u2016f x\u2016 ^ q\u2016\u208a = \u2191\u2016\u2016f x\u2016\u2016\u208a ^ q", "state_after": "case e_f.h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nhq_pos : 0 < q\nh0 : \u00acp = 0\nhp_top : p = \u22a4\nx : \u03b1\n\u22a2 \u2016\u2016f x\u2016 ^ q\u2016\u208a = \u2016\u2016f x\u2016\u2016\u208a ^ q"}, {"tactic": "ext", "annotated_tactic": ["ext", []], "state_before": "case e_f.h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nhq_pos : 0 < q\nh0 : \u00acp = 0\nhp_top : p = \u22a4\nx : \u03b1\n\u22a2 \u2016\u2016f x\u2016 ^ q\u2016\u208a = \u2016\u2016f x\u2016\u2016\u208a ^ q", "state_after": "case e_f.h.a\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nhq_pos : 0 < q\nh0 : \u00acp = 0\nhp_top : p = \u22a4\nx : \u03b1\n\u22a2 \u2191\u2016\u2016f x\u2016 ^ q\u2016\u208a = \u2191(\u2016\u2016f x\u2016\u2016\u208a ^ q)"}, {"tactic": "push_cast", "annotated_tactic": ["push_cast", []], "state_before": "case e_f.h.a\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nhq_pos : 0 < q\nh0 : \u00acp = 0\nhp_top : p = \u22a4\nx : \u03b1\n\u22a2 \u2191\u2016\u2016f x\u2016 ^ q\u2016\u208a = \u2191(\u2016\u2016f x\u2016\u2016\u208a ^ q)", "state_after": "case e_f.h.a\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nhq_pos : 0 < q\nh0 : \u00acp = 0\nhp_top : p = \u22a4\nx : \u03b1\n\u22a2 \u2016\u2016f x\u2016 ^ q\u2016 = \u2016\u2016f x\u2016\u2016 ^ q"}, {"tactic": "rw [Real.norm_rpow_of_nonneg (norm_nonneg _)]", "annotated_tactic": ["rw [<a>Real.norm_rpow_of_nonneg</a> (<a>norm_nonneg</a> _)]", [{"full_name": "Real.norm_rpow_of_nonneg", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [166, 9], "def_end_pos": [166, 28]}, {"full_name": "norm_nonneg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [500, 30], "def_end_pos": [500, 41]}]], "state_before": "case e_f.h.a\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nhq_pos : 0 < q\nh0 : \u00acp = 0\nhp_top : p = \u22a4\nx : \u03b1\n\u22a2 \u2016\u2016f x\u2016 ^ q\u2016 = \u2016\u2016f x\u2016\u2016 ^ q", "state_after": "no goals"}, {"tactic": "refine' mul_ne_zero h0 _", "annotated_tactic": ["refine' <a>mul_ne_zero</a> h0 _", [{"full_name": "mul_ne_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Basic.lean", "def_pos": [88, 9], "def_end_pos": [88, 20]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nhq_pos : 0 < q\nh0 : \u00acp = 0\nhp_top : \u00acp = \u22a4\n\u22a2 p * ENNReal.ofReal q \u2260 0", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nhq_pos : 0 < q\nh0 : \u00acp = 0\nhp_top : \u00acp = \u22a4\n\u22a2 ENNReal.ofReal q \u2260 0"}, {"tactic": "rwa [Ne.def, ENNReal.ofReal_eq_zero, not_le]", "annotated_tactic": ["rwa [<a>Ne.def</a>, <a>ENNReal.ofReal_eq_zero</a>, <a>not_le</a>]", [{"full_name": "Ne.def", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [59, 9], "def_end_pos": [59, 15]}, {"full_name": "ENNReal.ofReal_eq_zero", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2170, 9], "def_end_pos": [2170, 23]}, {"full_name": "not_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [373, 9], "def_end_pos": [373, 15]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nhq_pos : 0 < q\nh0 : \u00acp = 0\nhp_top : \u00acp = \u22a4\n\u22a2 ENNReal.ofReal q \u2260 0", "state_after": "no goals"}, {"tactic": "exact ENNReal.mul_ne_top hp_top ENNReal.ofReal_ne_top", "annotated_tactic": ["exact <a>ENNReal.mul_ne_top</a> hp_top <a>ENNReal.ofReal_ne_top</a>", [{"full_name": "ENNReal.mul_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [615, 9], "def_end_pos": [615, 19]}, {"full_name": "ENNReal.ofReal_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [311, 17], "def_end_pos": [311, 30]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nhq_pos : 0 < q\nh0 : \u00acp = 0\nhp_top : \u00acp = \u22a4\n\u22a2 p * ENNReal.ofReal q \u2260 \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Setoid/Partition.lean", "full_name": "Setoid.eqv_class_mem'", "start": [157, 1], "end": [160, 20], "traced_tactics": [{"tactic": "convert @Setoid.eqv_class_mem _ _ H x using 3", "annotated_tactic": ["convert @<a>Setoid.eqv_class_mem</a> _ _ H x using 3", [{"full_name": "Setoid.eqv_class_mem", "def_path": "Mathlib/Data/Setoid/Partition.lean", "def_pos": [150, 9], "def_end_pos": [150, 22]}]], "state_before": "\u03b1 : Type u_1\nc : Set (Set \u03b1)\nH : \u2200 (a : \u03b1), \u2203! b x, a \u2208 b\nx : \u03b1\n\u22a2 {y | Rel (mkClasses c H) x y} \u2208 c", "state_after": "case h.e'_4.h.e'_2.h.a\n\u03b1 : Type u_1\nc : Set (Set \u03b1)\nH : \u2200 (a : \u03b1), \u2203! b x, a \u2208 b\nx x\u271d : \u03b1\n\u22a2 Rel (mkClasses c H) x x\u271d \u2194 Rel (mkClasses c H) x\u271d x"}, {"tactic": "rw [Setoid.comm']", "annotated_tactic": ["rw [<a>Setoid.comm'</a>]", [{"full_name": "Setoid.comm'", "def_path": "Mathlib/Data/Setoid/Basic.lean", "def_pos": [90, 9], "def_end_pos": [90, 14]}]], "state_before": "case h.e'_4.h.e'_2.h.a\n\u03b1 : Type u_1\nc : Set (Set \u03b1)\nH : \u2200 (a : \u03b1), \u2203! b x, a \u2208 b\nx x\u271d : \u03b1\n\u22a2 Rel (mkClasses c H) x x\u271d \u2194 Rel (mkClasses c H) x\u271d x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Function.lean", "full_name": "Set.EqOn.piecewise_ite'", "start": [1494, 1], "end": [1496, 27], "traced_tactics": [{"tactic": "simp [eqOn_piecewise, *]", "annotated_tactic": ["simp [<a>eqOn_piecewise</a>, *]", [{"full_name": "Set.eqOn_piecewise", "def_path": "Mathlib/Data/Set/Function.lean", "def_pos": [1488, 9], "def_end_pos": [1488, 23]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03c0 : \u03b1 \u2192 Type u_5\n\u03b4 : \u03b1 \u2192 Sort u_6\ns : Set \u03b1\nf\u271d g\u271d : (i : \u03b1) \u2192 \u03b4 i\ninst\u271d : (j : \u03b1) \u2192 Decidable (j \u2208 s)\nf f' g : \u03b1 \u2192 \u03b2\nt t' : Set \u03b1\nh : EqOn f g (t \u2229 s)\nh' : EqOn f' g (t' \u2229 s\u1d9c)\n\u22a2 EqOn (piecewise s f f') g (Set.ite s t t')", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Dirac.lean", "full_name": "MeasureTheory.Measure.map_eq_sum", "start": [77, 1], "end": [82, 91], "traced_tactics": [{"tactic": "ext1 s hs", "annotated_tactic": ["ext1 s hs", []], "state_before": "\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ns : Set \u03b1\ninst\u271d\u00b9 : Countable \u03b2\ninst\u271d : MeasurableSingletonClass \u03b2\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : Measurable f\n\u22a2 map f \u03bc = sum fun b => \u2191\u2191\u03bc (f \u207b\u00b9' {b}) \u2022 dirac b", "state_after": "case h\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ns\u271d : Set \u03b1\ninst\u271d\u00b9 : Countable \u03b2\ninst\u271d : MeasurableSingletonClass \u03b2\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : Measurable f\ns : Set \u03b2\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(map f \u03bc) s = \u2191\u2191(sum fun b => \u2191\u2191\u03bc (f \u207b\u00b9' {b}) \u2022 dirac b) s"}, {"tactic": "have : \u2200 y \u2208 s, MeasurableSet (f \u207b\u00b9' {y}) := fun y _ => hf (measurableSet_singleton _)", "annotated_tactic": ["have : \u2200 y \u2208 s, <a>MeasurableSet</a> (f \u207b\u00b9' {y}) := fun y _ => hf (<a>measurableSet_singleton</a> _)", [{"full_name": "MeasurableSet", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [64, 5], "def_end_pos": [64, 18]}, {"full_name": "MeasurableSingletonClass.measurableSet_singleton", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [269, 3], "def_end_pos": [269, 26]}]], "state_before": "case h\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ns\u271d : Set \u03b1\ninst\u271d\u00b9 : Countable \u03b2\ninst\u271d : MeasurableSingletonClass \u03b2\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : Measurable f\ns : Set \u03b2\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(map f \u03bc) s = \u2191\u2191(sum fun b => \u2191\u2191\u03bc (f \u207b\u00b9' {b}) \u2022 dirac b) s", "state_after": "case h\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ns\u271d : Set \u03b1\ninst\u271d\u00b9 : Countable \u03b2\ninst\u271d : MeasurableSingletonClass \u03b2\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : Measurable f\ns : Set \u03b2\nhs : MeasurableSet s\nthis : \u2200 (y : \u03b2), y \u2208 s \u2192 MeasurableSet (f \u207b\u00b9' {y})\n\u22a2 \u2191\u2191(map f \u03bc) s = \u2191\u2191(sum fun b => \u2191\u2191\u03bc (f \u207b\u00b9' {b}) \u2022 dirac b) s"}, {"tactic": "simp [\u2190 tsum_measure_preimage_singleton (to_countable s) this, *,\n  tsum_subtype s fun b => \u03bc (f \u207b\u00b9' {b}), \u2190 indicator_mul_right s fun b => \u03bc (f \u207b\u00b9' {b})]", "annotated_tactic": ["simp [\u2190 <a>tsum_measure_preimage_singleton</a> (<a>to_countable</a> s) this, *,\n    <a>tsum_subtype</a> s fun b => \u03bc (f \u207b\u00b9' {b}), \u2190 <a>indicator_mul_right</a> s fun b => \u03bc (f \u207b\u00b9' {b})]", [{"full_name": "MeasureTheory.tsum_measure_preimage_singleton", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [222, 9], "def_end_pos": [222, 40]}, {"full_name": "Set.to_countable", "def_path": "Mathlib/Data/Set/Countable.lean", "def_pos": [41, 9], "def_end_pos": [41, 21]}, {"full_name": "tsum_subtype", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/Basic.lean", "def_pos": [615, 9], "def_end_pos": [615, 21]}, {"full_name": "Set.indicator_mul_right", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [722, 9], "def_end_pos": [722, 28]}]], "state_before": "case h\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ns\u271d : Set \u03b1\ninst\u271d\u00b9 : Countable \u03b2\ninst\u271d : MeasurableSingletonClass \u03b2\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : Measurable f\ns : Set \u03b2\nhs : MeasurableSet s\nthis : \u2200 (y : \u03b2), y \u2208 s \u2192 MeasurableSet (f \u207b\u00b9' {y})\n\u22a2 \u2191\u2191(map f \u03bc) s = \u2191\u2191(sum fun b => \u2191\u2191\u03bc (f \u207b\u00b9' {b}) \u2022 dirac b) s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "full_name": "Int.mul_nonneg_of_nonpos_of_nonpos", "start": [1213, 11], "end": [1216, 29], "traced_tactics": [{"tactic": "have : 0 * b \u2264 a * b := Int.mul_le_mul_of_nonpos_right ha hb", "annotated_tactic": ["have : 0 * b \u2264 a * b := <a>Int.mul_le_mul_of_nonpos_right</a> ha hb", [{"full_name": "Int.mul_le_mul_of_nonpos_right", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [1207, 19], "def_end_pos": [1207, 45]}]], "state_before": "a b : Int\nha : a \u2264 0\nhb : b \u2264 0\n\u22a2 0 \u2264 a * b", "state_after": "a b : Int\nha : a \u2264 0\nhb : b \u2264 0\nthis : 0 * b \u2264 a * b\n\u22a2 0 \u2264 a * b"}, {"tactic": "rwa [Int.zero_mul] at this", "annotated_tactic": ["rwa [<a>Int.zero_mul</a>] at this", [{"full_name": "Int.zero_mul", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [409, 27], "def_end_pos": [409, 35]}]], "state_before": "a b : Int\nha : a \u2264 0\nhb : b \u2264 0\nthis : 0 * b \u2264 a * b\n\u22a2 0 \u2264 a * b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/Supported.lean", "full_name": "MvPolynomial.supported_empty", "start": [107, 1], "end": [107, 89], "traced_tactics": [{"tactic": "simp [supported_eq_adjoin_X]", "annotated_tactic": ["simp [<a>supported_eq_adjoin_X</a>]", [{"full_name": "MvPolynomial.supported_eq_adjoin_X", "def_path": "Mathlib/Data/MvPolynomial/Supported.lean", "def_pos": [97, 9], "def_end_pos": [97, 30]}]], "state_before": "\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u\nS : Type v\nr : R\ne : \u2115\nn m : \u03c3\ninst\u271d : CommSemiring R\np q : MvPolynomial \u03c3 R\ns t : Set \u03c3\n\u22a2 supported R \u2205 = \u22a5", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Kernel/WithDensity.lean", "full_name": "ProbabilityTheory.kernel.withDensity_add_left", "start": [92, 1], "end": [99, 18], "traced_tactics": [{"tactic": "by_cases hf : Measurable (Function.uncurry f)", "annotated_tactic": ["by_cases hf : <a>Measurable</a> (<a>Function.uncurry</a> f)", [{"full_name": "Measurable", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [535, 5], "def_end_pos": [535, 15]}, {"full_name": "Function.uncurry", "def_path": "Mathlib/Init/Function.lean", "def_pos": [217, 5], "def_end_pos": [217, 12]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\n\u03ba \u03b7 : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\n\u22a2 withDensity (\u03ba + \u03b7) f = withDensity \u03ba f + withDensity \u03b7 f", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\n\u03ba \u03b7 : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\nhf : Measurable (Function.uncurry f)\n\u22a2 withDensity (\u03ba + \u03b7) f = withDensity \u03ba f + withDensity \u03b7 f\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\n\u03ba \u03b7 : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\nhf : \u00acMeasurable (Function.uncurry f)\n\u22a2 withDensity (\u03ba + \u03b7) f = withDensity \u03ba f + withDensity \u03b7 f"}, {"tactic": "ext a s", "annotated_tactic": ["ext a s", []], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\n\u03ba \u03b7 : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\nhf : Measurable (Function.uncurry f)\n\u22a2 withDensity (\u03ba + \u03b7) f = withDensity \u03ba f + withDensity \u03b7 f", "state_after": "case pos.h.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\n\u03ba \u03b7 : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\nhf : Measurable (Function.uncurry f)\na : \u03b1\ns : Set \u03b2\na\u271d : MeasurableSet s\n\u22a2 \u2191\u2191(\u2191(withDensity (\u03ba + \u03b7) f) a) s = \u2191\u2191(\u2191(withDensity \u03ba f + withDensity \u03b7 f) a) s"}, {"tactic": "simp only [kernel.withDensity_apply _ hf, coeFn_add, Pi.add_apply, withDensity_add_measure,\n  Measure.add_apply]", "annotated_tactic": ["simp only [<a>kernel.withDensity_apply</a> _ hf, <a>coeFn_add</a>, <a>Pi.add_apply</a>, <a>withDensity_add_measure</a>,\n      <a>Measure.add_apply</a>]", [{"full_name": "ProbabilityTheory.kernel.withDensity_apply", "def_path": "Mathlib/Probability/Kernel/WithDensity.lean", "def_pos": [61, 19], "def_end_pos": [61, 36]}, {"full_name": "ProbabilityTheory.kernel.coeFn_add", "def_path": "Mathlib/Probability/Kernel/Basic.lean", "def_pos": [83, 9], "def_end_pos": [83, 18]}, {"full_name": "Pi.add_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [82, 3], "def_end_pos": [82, 14]}, {"full_name": "MeasureTheory.withDensity_add_measure", "def_path": "Mathlib/MeasureTheory/Measure/WithDensity.lean", "def_pos": [77, 9], "def_end_pos": [77, 32]}, {"full_name": "MeasureTheory.Measure.add_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [798, 9], "def_end_pos": [798, 18]}]], "state_before": "case pos.h.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\n\u03ba \u03b7 : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\nhf : Measurable (Function.uncurry f)\na : \u03b1\ns : Set \u03b2\na\u271d : MeasurableSet s\n\u22a2 \u2191\u2191(\u2191(withDensity (\u03ba + \u03b7) f) a) s = \u2191\u2191(\u2191(withDensity \u03ba f + withDensity \u03b7 f) a) s", "state_after": "no goals"}, {"tactic": "simp_rw [withDensity_of_not_measurable _ hf]", "annotated_tactic": ["simp_rw [<a>withDensity_of_not_measurable</a> _ hf]", [{"full_name": "ProbabilityTheory.kernel.withDensity_of_not_measurable", "def_path": "Mathlib/Probability/Kernel/WithDensity.lean", "def_pos": [57, 9], "def_end_pos": [57, 38]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\n\u03ba \u03b7 : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\nhf : \u00acMeasurable (Function.uncurry f)\n\u22a2 withDensity (\u03ba + \u03b7) f = withDensity \u03ba f + withDensity \u03b7 f", "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\n\u03ba \u03b7 : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\nhf : \u00acMeasurable (Function.uncurry f)\n\u22a2 0 = 0 + 0"}, {"tactic": "rw [zero_add]", "annotated_tactic": ["rw [<a>zero_add</a>]", [{"full_name": "zero_add", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [463, 3], "def_end_pos": [463, 14]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\n\u03ba \u03b7 : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\nhf : \u00acMeasurable (Function.uncurry f)\n\u22a2 0 = 0 + 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Basic.lean", "full_name": "Set.mem_ite_univ_right", "start": [2220, 1], "end": [2222, 39], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Num/Lemmas.lean", "full_name": "PosNum.of_to_nat", "start": [524, 1], "end": [525, 13], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/WithDensity.lean", "full_name": "MeasureTheory.withDensity_apply_eq_zero", "start": [189, 1], "end": [220, 56], "traced_tactics": [{"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhf : Measurable f\n\u22a2 \u2191\u2191(withDensity \u03bc f) s = 0 \u2194 \u2191\u2191\u03bc ({x | f x \u2260 0} \u2229 s) = 0", "state_after": "case mp\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhf : Measurable f\n\u22a2 \u2191\u2191(withDensity \u03bc f) s = 0 \u2192 \u2191\u2191\u03bc ({x | f x \u2260 0} \u2229 s) = 0\n\ncase mpr\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhf : Measurable f\n\u22a2 \u2191\u2191\u03bc ({x | f x \u2260 0} \u2229 s) = 0 \u2192 \u2191\u2191(withDensity \u03bc f) s = 0"}, {"tactic": "intro hs", "annotated_tactic": ["intro hs", []], "state_before": "case mp\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhf : Measurable f\n\u22a2 \u2191\u2191(withDensity \u03bc f) s = 0 \u2192 \u2191\u2191\u03bc ({x | f x \u2260 0} \u2229 s) = 0", "state_after": "case mp\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhf : Measurable f\nhs : \u2191\u2191(withDensity \u03bc f) s = 0\n\u22a2 \u2191\u2191\u03bc ({x | f x \u2260 0} \u2229 s) = 0"}, {"tactic": "let t := toMeasurable (\u03bc.withDensity f) s", "annotated_tactic": ["let t := <a>toMeasurable</a> (\u03bc.withDensity f) s", [{"full_name": "MeasureTheory.toMeasurable", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [626, 17], "def_end_pos": [626, 29]}]], "state_before": "case mp\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhf : Measurable f\nhs : \u2191\u2191(withDensity \u03bc f) s = 0\n\u22a2 \u2191\u2191\u03bc ({x | f x \u2260 0} \u2229 s) = 0", "state_after": "case mp\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhf : Measurable f\nhs : \u2191\u2191(withDensity \u03bc f) s = 0\nt : Set \u03b1 := toMeasurable (withDensity \u03bc f) s\n\u22a2 \u2191\u2191\u03bc ({x | f x \u2260 0} \u2229 s) = 0"}, {"tactic": "apply measure_mono_null (inter_subset_inter_right _ (subset_toMeasurable (\u03bc.withDensity f) s))", "annotated_tactic": ["apply <a>measure_mono_null</a> (<a>inter_subset_inter_right</a> _ (<a>subset_toMeasurable</a> (\u03bc.withDensity f) s))", [{"full_name": "MeasureTheory.measure_mono_null", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [197, 9], "def_end_pos": [197, 26]}, {"full_name": "Set.inter_subset_inter_right", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1032, 9], "def_end_pos": [1032, 33]}, {"full_name": "MeasureTheory.subset_toMeasurable", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [633, 9], "def_end_pos": [633, 28]}]], "state_before": "case mp\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhf : Measurable f\nhs : \u2191\u2191(withDensity \u03bc f) s = 0\nt : Set \u03b1 := toMeasurable (withDensity \u03bc f) s\n\u22a2 \u2191\u2191\u03bc ({x | f x \u2260 0} \u2229 s) = 0", "state_after": "case mp\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhf : Measurable f\nhs : \u2191\u2191(withDensity \u03bc f) s = 0\nt : Set \u03b1 := toMeasurable (withDensity \u03bc f) s\n\u22a2 \u2191\u2191\u03bc ({x | f x \u2260 0} \u2229 toMeasurable (withDensity \u03bc f) s) = 0"}, {"tactic": "have A : \u03bc.withDensity f t = 0 := by rw [measure_toMeasurable, hs]", "annotated_tactic": ["have A : \u03bc.withDensity f t = 0 := by rw [<a>measure_toMeasurable</a>, hs]", [{"full_name": "MeasureTheory.measure_toMeasurable", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [653, 9], "def_end_pos": [653, 29]}]], "state_before": "case mp\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhf : Measurable f\nhs : \u2191\u2191(withDensity \u03bc f) s = 0\nt : Set \u03b1 := toMeasurable (withDensity \u03bc f) s\n\u22a2 \u2191\u2191\u03bc ({x | f x \u2260 0} \u2229 toMeasurable (withDensity \u03bc f) s) = 0", "state_after": "case mp\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhf : Measurable f\nhs : \u2191\u2191(withDensity \u03bc f) s = 0\nt : Set \u03b1 := toMeasurable (withDensity \u03bc f) s\nA : \u2191\u2191(withDensity \u03bc f) t = 0\n\u22a2 \u2191\u2191\u03bc ({x | f x \u2260 0} \u2229 toMeasurable (withDensity \u03bc f) s) = 0"}, {"tactic": "rw [withDensity_apply f (measurableSet_toMeasurable _ s), lintegral_eq_zero_iff hf,\n  EventuallyEq, ae_restrict_iff, ae_iff] at A", "annotated_tactic": ["rw [<a>withDensity_apply</a> f (<a>measurableSet_toMeasurable</a> _ s), <a>lintegral_eq_zero_iff</a> hf,\n      <a>EventuallyEq</a>, <a>ae_restrict_iff</a>, <a>ae_iff</a>] at A", [{"full_name": "MeasureTheory.withDensity_apply", "def_path": "Mathlib/MeasureTheory/Measure/WithDensity.lean", "def_pos": [39, 9], "def_end_pos": [39, 26]}, {"full_name": "MeasureTheory.measurableSet_toMeasurable", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [645, 9], "def_end_pos": [645, 35]}, {"full_name": "MeasureTheory.lintegral_eq_zero_iff", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [898, 9], "def_end_pos": [898, 30]}, {"full_name": "Filter.EventuallyEq", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1438, 5], "def_end_pos": [1438, 17]}, {"full_name": "MeasureTheory.ae_restrict_iff", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2560, 9], "def_end_pos": [2560, 24]}, {"full_name": "MeasureTheory.ae_iff", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [388, 9], "def_end_pos": [388, 15]}]], "state_before": "case mp\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhf : Measurable f\nhs : \u2191\u2191(withDensity \u03bc f) s = 0\nt : Set \u03b1 := toMeasurable (withDensity \u03bc f) s\nA : \u2191\u2191(withDensity \u03bc f) t = 0\n\u22a2 \u2191\u2191\u03bc ({x | f x \u2260 0} \u2229 toMeasurable (withDensity \u03bc f) s) = 0", "state_after": "case mp\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhf : Measurable f\nhs : \u2191\u2191(withDensity \u03bc f) s = 0\nt : Set \u03b1 := toMeasurable (withDensity \u03bc f) s\nA : \u2191\u2191\u03bc {a | \u00ac(a \u2208 toMeasurable (withDensity \u03bc f) s \u2192 f a = OfNat.ofNat 0 a)} = 0\n\u22a2 \u2191\u2191\u03bc ({x | f x \u2260 0} \u2229 toMeasurable (withDensity \u03bc f) s) = 0\n\ncase mp\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhf : Measurable f\nhs : \u2191\u2191(withDensity \u03bc f) s = 0\nt : Set \u03b1 := toMeasurable (withDensity \u03bc f) s\nA : \u2200\u1d50 (x : \u03b1) \u2202restrict \u03bc (toMeasurable (withDensity \u03bc f) s), f x = OfNat.ofNat 0 x\n\u22a2 MeasurableSet {x | f x = OfNat.ofNat 0 x}"}, {"tactic": "swap", "annotated_tactic": ["swap", []], "state_before": "case mp\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhf : Measurable f\nhs : \u2191\u2191(withDensity \u03bc f) s = 0\nt : Set \u03b1 := toMeasurable (withDensity \u03bc f) s\nA : \u2191\u2191\u03bc {a | \u00ac(a \u2208 toMeasurable (withDensity \u03bc f) s \u2192 f a = OfNat.ofNat 0 a)} = 0\n\u22a2 \u2191\u2191\u03bc ({x | f x \u2260 0} \u2229 toMeasurable (withDensity \u03bc f) s) = 0\n\ncase mp\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhf : Measurable f\nhs : \u2191\u2191(withDensity \u03bc f) s = 0\nt : Set \u03b1 := toMeasurable (withDensity \u03bc f) s\nA : \u2200\u1d50 (x : \u03b1) \u2202restrict \u03bc (toMeasurable (withDensity \u03bc f) s), f x = OfNat.ofNat 0 x\n\u22a2 MeasurableSet {x | f x = OfNat.ofNat 0 x}", "state_after": "case mp\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhf : Measurable f\nhs : \u2191\u2191(withDensity \u03bc f) s = 0\nt : Set \u03b1 := toMeasurable (withDensity \u03bc f) s\nA : \u2200\u1d50 (x : \u03b1) \u2202restrict \u03bc (toMeasurable (withDensity \u03bc f) s), f x = OfNat.ofNat 0 x\n\u22a2 MeasurableSet {x | f x = OfNat.ofNat 0 x}\n\ncase mp\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhf : Measurable f\nhs : \u2191\u2191(withDensity \u03bc f) s = 0\nt : Set \u03b1 := toMeasurable (withDensity \u03bc f) s\nA : \u2191\u2191\u03bc {a | \u00ac(a \u2208 toMeasurable (withDensity \u03bc f) s \u2192 f a = OfNat.ofNat 0 a)} = 0\n\u22a2 \u2191\u2191\u03bc ({x | f x \u2260 0} \u2229 toMeasurable (withDensity \u03bc f) s) = 0"}, {"tactic": "simp only [Pi.zero_apply, mem_setOf_eq, Filter.mem_mk] at A", "annotated_tactic": ["simp only [<a>Pi.zero_apply</a>, <a>mem_setOf_eq</a>, <a>Filter.mem_mk</a>] at A", [{"full_name": "Pi.zero_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [46, 3], "def_end_pos": [46, 14]}, {"full_name": "Set.mem_setOf_eq", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [256, 29], "def_end_pos": [256, 41]}, {"full_name": "Filter.mem_mk", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [111, 19], "def_end_pos": [111, 25]}]], "state_before": "case mp\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhf : Measurable f\nhs : \u2191\u2191(withDensity \u03bc f) s = 0\nt : Set \u03b1 := toMeasurable (withDensity \u03bc f) s\nA : \u2191\u2191\u03bc {a | \u00ac(a \u2208 toMeasurable (withDensity \u03bc f) s \u2192 f a = OfNat.ofNat 0 a)} = 0\n\u22a2 \u2191\u2191\u03bc ({x | f x \u2260 0} \u2229 toMeasurable (withDensity \u03bc f) s) = 0", "state_after": "case mp\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhf : Measurable f\nhs : \u2191\u2191(withDensity \u03bc f) s = 0\nt : Set \u03b1 := toMeasurable (withDensity \u03bc f) s\nA : \u2191\u2191\u03bc {a | \u00ac(a \u2208 toMeasurable (withDensity \u03bc f) s \u2192 f a = 0)} = 0\n\u22a2 \u2191\u2191\u03bc ({x | f x \u2260 0} \u2229 toMeasurable (withDensity \u03bc f) s) = 0"}, {"tactic": "convert A using 2", "annotated_tactic": ["convert A using 2", []], "state_before": "case mp\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhf : Measurable f\nhs : \u2191\u2191(withDensity \u03bc f) s = 0\nt : Set \u03b1 := toMeasurable (withDensity \u03bc f) s\nA : \u2191\u2191\u03bc {a | \u00ac(a \u2208 toMeasurable (withDensity \u03bc f) s \u2192 f a = 0)} = 0\n\u22a2 \u2191\u2191\u03bc ({x | f x \u2260 0} \u2229 toMeasurable (withDensity \u03bc f) s) = 0", "state_after": "case h.e'_2.h.e'_3\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhf : Measurable f\nhs : \u2191\u2191(withDensity \u03bc f) s = 0\nt : Set \u03b1 := toMeasurable (withDensity \u03bc f) s\nA : \u2191\u2191\u03bc {a | \u00ac(a \u2208 toMeasurable (withDensity \u03bc f) s \u2192 f a = 0)} = 0\n\u22a2 {x | f x \u2260 0} \u2229 toMeasurable (withDensity \u03bc f) s = {a | \u00ac(a \u2208 toMeasurable (withDensity \u03bc f) s \u2192 f a = 0)}"}, {"tactic": "ext x", "annotated_tactic": ["ext x", []], "state_before": "case h.e'_2.h.e'_3\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhf : Measurable f\nhs : \u2191\u2191(withDensity \u03bc f) s = 0\nt : Set \u03b1 := toMeasurable (withDensity \u03bc f) s\nA : \u2191\u2191\u03bc {a | \u00ac(a \u2208 toMeasurable (withDensity \u03bc f) s \u2192 f a = 0)} = 0\n\u22a2 {x | f x \u2260 0} \u2229 toMeasurable (withDensity \u03bc f) s = {a | \u00ac(a \u2208 toMeasurable (withDensity \u03bc f) s \u2192 f a = 0)}", "state_after": "case h.e'_2.h.e'_3.h\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhf : Measurable f\nhs : \u2191\u2191(withDensity \u03bc f) s = 0\nt : Set \u03b1 := toMeasurable (withDensity \u03bc f) s\nA : \u2191\u2191\u03bc {a | \u00ac(a \u2208 toMeasurable (withDensity \u03bc f) s \u2192 f a = 0)} = 0\nx : \u03b1\n\u22a2 x \u2208 {x | f x \u2260 0} \u2229 toMeasurable (withDensity \u03bc f) s \u2194 x \u2208 {a | \u00ac(a \u2208 toMeasurable (withDensity \u03bc f) s \u2192 f a = 0)}"}, {"tactic": "simp only [and_comm, exists_prop, mem_inter_iff, iff_self_iff, mem_setOf_eq, mem_compl_iff,\n  not_forall]", "annotated_tactic": ["simp only [<a>and_comm</a>, <a>exists_prop</a>, <a>mem_inter_iff</a>, <a>iff_self_iff</a>, <a>mem_setOf_eq</a>, <a>mem_compl_iff</a>,\n      <a>not_forall</a>]", [{"full_name": "and_comm", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [159, 9], "def_end_pos": [159, 17]}, {"full_name": "exists_prop", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [485, 17], "def_end_pos": [485, 28]}, {"full_name": "Set.mem_inter_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [909, 9], "def_end_pos": [909, 22]}, {"full_name": "iff_self_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [207, 9], "def_end_pos": [207, 21]}, {"full_name": "Set.mem_setOf_eq", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [256, 29], "def_end_pos": [256, 41]}, {"full_name": "Set.mem_compl_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1658, 9], "def_end_pos": [1658, 22]}, {"full_name": "Classical.not_forall", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [686, 9], "def_end_pos": [686, 19]}]], "state_before": "case h.e'_2.h.e'_3.h\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhf : Measurable f\nhs : \u2191\u2191(withDensity \u03bc f) s = 0\nt : Set \u03b1 := toMeasurable (withDensity \u03bc f) s\nA : \u2191\u2191\u03bc {a | \u00ac(a \u2208 toMeasurable (withDensity \u03bc f) s \u2192 f a = 0)} = 0\nx : \u03b1\n\u22a2 x \u2208 {x | f x \u2260 0} \u2229 toMeasurable (withDensity \u03bc f) s \u2194 x \u2208 {a | \u00ac(a \u2208 toMeasurable (withDensity \u03bc f) s \u2192 f a = 0)}", "state_after": "no goals"}, {"tactic": "rw [measure_toMeasurable, hs]", "annotated_tactic": ["rw [<a>measure_toMeasurable</a>, hs]", [{"full_name": "MeasureTheory.measure_toMeasurable", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [653, 9], "def_end_pos": [653, 29]}]], "state_before": "\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhf : Measurable f\nhs : \u2191\u2191(withDensity \u03bc f) s = 0\nt : Set \u03b1 := toMeasurable (withDensity \u03bc f) s\n\u22a2 \u2191\u2191(withDensity \u03bc f) t = 0", "state_after": "no goals"}, {"tactic": "exact hf (measurableSet_singleton 0)", "annotated_tactic": ["exact hf (<a>measurableSet_singleton</a> 0)", [{"full_name": "MeasurableSingletonClass.measurableSet_singleton", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [269, 3], "def_end_pos": [269, 26]}]], "state_before": "case mp\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhf : Measurable f\nhs : \u2191\u2191(withDensity \u03bc f) s = 0\nt : Set \u03b1 := toMeasurable (withDensity \u03bc f) s\nA : \u2200\u1d50 (x : \u03b1) \u2202restrict \u03bc (toMeasurable (withDensity \u03bc f) s), f x = OfNat.ofNat 0 x\n\u22a2 MeasurableSet {x | f x = OfNat.ofNat 0 x}", "state_after": "no goals"}, {"tactic": "intro hs", "annotated_tactic": ["intro hs", []], "state_before": "case mpr\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhf : Measurable f\n\u22a2 \u2191\u2191\u03bc ({x | f x \u2260 0} \u2229 s) = 0 \u2192 \u2191\u2191(withDensity \u03bc f) s = 0", "state_after": "case mpr\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhf : Measurable f\nhs : \u2191\u2191\u03bc ({x | f x \u2260 0} \u2229 s) = 0\n\u22a2 \u2191\u2191(withDensity \u03bc f) s = 0"}, {"tactic": "let t := toMeasurable \u03bc ({ x | f x \u2260 0 } \u2229 s)", "annotated_tactic": ["let t := <a>toMeasurable</a> \u03bc ({ x | f x \u2260 0 } \u2229 s)", [{"full_name": "MeasureTheory.toMeasurable", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [626, 17], "def_end_pos": [626, 29]}]], "state_before": "case mpr\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhf : Measurable f\nhs : \u2191\u2191\u03bc ({x | f x \u2260 0} \u2229 s) = 0\n\u22a2 \u2191\u2191(withDensity \u03bc f) s = 0", "state_after": "case mpr\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhf : Measurable f\nhs : \u2191\u2191\u03bc ({x | f x \u2260 0} \u2229 s) = 0\nt : Set \u03b1 := toMeasurable \u03bc ({x | f x \u2260 0} \u2229 s)\n\u22a2 \u2191\u2191(withDensity \u03bc f) s = 0"}, {"tactic": "apply measure_mono_null A (measure_union_null _ _)", "annotated_tactic": ["apply <a>measure_mono_null</a> A (<a>measure_union_null</a> _ _)", [{"full_name": "MeasureTheory.measure_mono_null", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [197, 9], "def_end_pos": [197, 26]}, {"full_name": "MeasureTheory.measure_union_null", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [302, 9], "def_end_pos": [302, 27]}]], "state_before": "case mpr\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhf : Measurable f\nhs : \u2191\u2191\u03bc ({x | f x \u2260 0} \u2229 s) = 0\nt : Set \u03b1 := toMeasurable \u03bc ({x | f x \u2260 0} \u2229 s)\nA : s \u2286 t \u222a {x | f x = 0}\n\u22a2 \u2191\u2191(withDensity \u03bc f) s = 0", "state_after": "\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhf : Measurable f\nhs : \u2191\u2191\u03bc ({x | f x \u2260 0} \u2229 s) = 0\nt : Set \u03b1 := toMeasurable \u03bc ({x | f x \u2260 0} \u2229 s)\nA : s \u2286 t \u222a {x | f x = 0}\n\u22a2 \u2191\u2191(withDensity \u03bc f) t = 0\n\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhf : Measurable f\nhs : \u2191\u2191\u03bc ({x | f x \u2260 0} \u2229 s) = 0\nt : Set \u03b1 := toMeasurable \u03bc ({x | f x \u2260 0} \u2229 s)\nA : s \u2286 t \u222a {x | f x = 0}\n\u22a2 \u2191\u2191(withDensity \u03bc f) {x | f x = 0} = 0"}, {"tactic": "intro x hx", "annotated_tactic": ["intro x hx", []], "state_before": "\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhf : Measurable f\nhs : \u2191\u2191\u03bc ({x | f x \u2260 0} \u2229 s) = 0\nt : Set \u03b1 := toMeasurable \u03bc ({x | f x \u2260 0} \u2229 s)\n\u22a2 s \u2286 t \u222a {x | f x = 0}", "state_after": "\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhf : Measurable f\nhs : \u2191\u2191\u03bc ({x | f x \u2260 0} \u2229 s) = 0\nt : Set \u03b1 := toMeasurable \u03bc ({x | f x \u2260 0} \u2229 s)\nx : \u03b1\nhx : x \u2208 s\n\u22a2 x \u2208 t \u222a {x | f x = 0}"}, {"tactic": "rcases eq_or_ne (f x) 0 with (fx | fx)", "annotated_tactic": ["rcases <a>eq_or_ne</a> (f x) 0 with (fx | fx)", [{"full_name": "eq_or_ne", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [209, 9], "def_end_pos": [209, 17]}]], "state_before": "\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhf : Measurable f\nhs : \u2191\u2191\u03bc ({x | f x \u2260 0} \u2229 s) = 0\nt : Set \u03b1 := toMeasurable \u03bc ({x | f x \u2260 0} \u2229 s)\nx : \u03b1\nhx : x \u2208 s\n\u22a2 x \u2208 t \u222a {x | f x = 0}", "state_after": "case inl\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhf : Measurable f\nhs : \u2191\u2191\u03bc ({x | f x \u2260 0} \u2229 s) = 0\nt : Set \u03b1 := toMeasurable \u03bc ({x | f x \u2260 0} \u2229 s)\nx : \u03b1\nhx : x \u2208 s\nfx : f x = 0\n\u22a2 x \u2208 t \u222a {x | f x = 0}\n\ncase inr\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhf : Measurable f\nhs : \u2191\u2191\u03bc ({x | f x \u2260 0} \u2229 s) = 0\nt : Set \u03b1 := toMeasurable \u03bc ({x | f x \u2260 0} \u2229 s)\nx : \u03b1\nhx : x \u2208 s\nfx : f x \u2260 0\n\u22a2 x \u2208 t \u222a {x | f x = 0}"}, {"tactic": "simp only [fx, mem_union, mem_setOf_eq, eq_self_iff_true, or_true_iff]", "annotated_tactic": ["simp only [fx, <a>mem_union</a>, <a>mem_setOf_eq</a>, <a>eq_self_iff_true</a>, <a>or_true_iff</a>]", [{"full_name": "Set.mem_union", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [767, 9], "def_end_pos": [767, 18]}, {"full_name": "Set.mem_setOf_eq", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [256, 29], "def_end_pos": [256, 41]}, {"full_name": "eq_self_iff_true", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [86, 9], "def_end_pos": [86, 25]}, {"full_name": "or_true_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [184, 9], "def_end_pos": [184, 20]}]], "state_before": "case inl\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhf : Measurable f\nhs : \u2191\u2191\u03bc ({x | f x \u2260 0} \u2229 s) = 0\nt : Set \u03b1 := toMeasurable \u03bc ({x | f x \u2260 0} \u2229 s)\nx : \u03b1\nhx : x \u2208 s\nfx : f x = 0\n\u22a2 x \u2208 t \u222a {x | f x = 0}", "state_after": "no goals"}, {"tactic": "left", "annotated_tactic": ["left", []], "state_before": "case inr\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhf : Measurable f\nhs : \u2191\u2191\u03bc ({x | f x \u2260 0} \u2229 s) = 0\nt : Set \u03b1 := toMeasurable \u03bc ({x | f x \u2260 0} \u2229 s)\nx : \u03b1\nhx : x \u2208 s\nfx : f x \u2260 0\n\u22a2 x \u2208 t \u222a {x | f x = 0}", "state_after": "case inr.h\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhf : Measurable f\nhs : \u2191\u2191\u03bc ({x | f x \u2260 0} \u2229 s) = 0\nt : Set \u03b1 := toMeasurable \u03bc ({x | f x \u2260 0} \u2229 s)\nx : \u03b1\nhx : x \u2208 s\nfx : f x \u2260 0\n\u22a2 x \u2208 t"}, {"tactic": "apply subset_toMeasurable _ _", "annotated_tactic": ["apply <a>subset_toMeasurable</a> _ _", [{"full_name": "MeasureTheory.subset_toMeasurable", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [633, 9], "def_end_pos": [633, 28]}]], "state_before": "case inr.h\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhf : Measurable f\nhs : \u2191\u2191\u03bc ({x | f x \u2260 0} \u2229 s) = 0\nt : Set \u03b1 := toMeasurable \u03bc ({x | f x \u2260 0} \u2229 s)\nx : \u03b1\nhx : x \u2208 s\nfx : f x \u2260 0\n\u22a2 x \u2208 t", "state_after": "case inr.h.a\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhf : Measurable f\nhs : \u2191\u2191\u03bc ({x | f x \u2260 0} \u2229 s) = 0\nt : Set \u03b1 := toMeasurable \u03bc ({x | f x \u2260 0} \u2229 s)\nx : \u03b1\nhx : x \u2208 s\nfx : f x \u2260 0\n\u22a2 x \u2208 {x | f x \u2260 0} \u2229 s"}, {"tactic": "exact \u27e8fx, hx\u27e9", "annotated_tactic": ["exact \u27e8fx, hx\u27e9", []], "state_before": "case inr.h.a\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhf : Measurable f\nhs : \u2191\u2191\u03bc ({x | f x \u2260 0} \u2229 s) = 0\nt : Set \u03b1 := toMeasurable \u03bc ({x | f x \u2260 0} \u2229 s)\nx : \u03b1\nhx : x \u2208 s\nfx : f x \u2260 0\n\u22a2 x \u2208 {x | f x \u2260 0} \u2229 s", "state_after": "no goals"}, {"tactic": "apply withDensity_absolutelyContinuous", "annotated_tactic": ["apply <a>withDensity_absolutelyContinuous</a>", [{"full_name": "MeasureTheory.withDensity_absolutelyContinuous", "def_path": "Mathlib/MeasureTheory/Measure/WithDensity.lean", "def_pos": [117, 9], "def_end_pos": [117, 41]}]], "state_before": "\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhf : Measurable f\nhs : \u2191\u2191\u03bc ({x | f x \u2260 0} \u2229 s) = 0\nt : Set \u03b1 := toMeasurable \u03bc ({x | f x \u2260 0} \u2229 s)\nA : s \u2286 t \u222a {x | f x = 0}\n\u22a2 \u2191\u2191(withDensity \u03bc f) t = 0", "state_after": "case a\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhf : Measurable f\nhs : \u2191\u2191\u03bc ({x | f x \u2260 0} \u2229 s) = 0\nt : Set \u03b1 := toMeasurable \u03bc ({x | f x \u2260 0} \u2229 s)\nA : s \u2286 t \u222a {x | f x = 0}\n\u22a2 \u2191\u2191\u03bc t = 0"}, {"tactic": "rwa [measure_toMeasurable]", "annotated_tactic": ["rwa [<a>measure_toMeasurable</a>]", [{"full_name": "MeasureTheory.measure_toMeasurable", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [653, 9], "def_end_pos": [653, 29]}]], "state_before": "case a\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhf : Measurable f\nhs : \u2191\u2191\u03bc ({x | f x \u2260 0} \u2229 s) = 0\nt : Set \u03b1 := toMeasurable \u03bc ({x | f x \u2260 0} \u2229 s)\nA : s \u2286 t \u222a {x | f x = 0}\n\u22a2 \u2191\u2191\u03bc t = 0", "state_after": "no goals"}, {"tactic": "have M : MeasurableSet { x : \u03b1 | f x = 0 } := hf (measurableSet_singleton _)", "annotated_tactic": ["have M : <a>MeasurableSet</a> { x : \u03b1 | f x = 0 } := hf (<a>measurableSet_singleton</a> _)", [{"full_name": "MeasurableSet", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [64, 5], "def_end_pos": [64, 18]}, {"full_name": "MeasurableSingletonClass.measurableSet_singleton", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [269, 3], "def_end_pos": [269, 26]}]], "state_before": "\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhf : Measurable f\nhs : \u2191\u2191\u03bc ({x | f x \u2260 0} \u2229 s) = 0\nt : Set \u03b1 := toMeasurable \u03bc ({x | f x \u2260 0} \u2229 s)\nA : s \u2286 t \u222a {x | f x = 0}\n\u22a2 \u2191\u2191(withDensity \u03bc f) {x | f x = 0} = 0", "state_after": "\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhf : Measurable f\nhs : \u2191\u2191\u03bc ({x | f x \u2260 0} \u2229 s) = 0\nt : Set \u03b1 := toMeasurable \u03bc ({x | f x \u2260 0} \u2229 s)\nA : s \u2286 t \u222a {x | f x = 0}\nM : MeasurableSet {x | f x = 0}\n\u22a2 \u2191\u2191(withDensity \u03bc f) {x | f x = 0} = 0"}, {"tactic": "rw [withDensity_apply _ M, lintegral_eq_zero_iff hf]", "annotated_tactic": ["rw [<a>withDensity_apply</a> _ M, <a>lintegral_eq_zero_iff</a> hf]", [{"full_name": "MeasureTheory.withDensity_apply", "def_path": "Mathlib/MeasureTheory/Measure/WithDensity.lean", "def_pos": [39, 9], "def_end_pos": [39, 26]}, {"full_name": "MeasureTheory.lintegral_eq_zero_iff", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [898, 9], "def_end_pos": [898, 30]}]], "state_before": "\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhf : Measurable f\nhs : \u2191\u2191\u03bc ({x | f x \u2260 0} \u2229 s) = 0\nt : Set \u03b1 := toMeasurable \u03bc ({x | f x \u2260 0} \u2229 s)\nA : s \u2286 t \u222a {x | f x = 0}\nM : MeasurableSet {x | f x = 0}\n\u22a2 \u2191\u2191(withDensity \u03bc f) {x | f x = 0} = 0", "state_after": "\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhf : Measurable f\nhs : \u2191\u2191\u03bc ({x | f x \u2260 0} \u2229 s) = 0\nt : Set \u03b1 := toMeasurable \u03bc ({x | f x \u2260 0} \u2229 s)\nA : s \u2286 t \u222a {x | f x = 0}\nM : MeasurableSet {x | f x = 0}\n\u22a2 f =\u1da0[ae (restrict \u03bc {x | f x = 0})] 0"}, {"tactic": "filter_upwards [ae_restrict_mem M]", "annotated_tactic": ["filter_upwards [<a>ae_restrict_mem</a> M]", [{"full_name": "MeasureTheory.ae_restrict_mem", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2586, 9], "def_end_pos": [2586, 24]}]], "state_before": "\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhf : Measurable f\nhs : \u2191\u2191\u03bc ({x | f x \u2260 0} \u2229 s) = 0\nt : Set \u03b1 := toMeasurable \u03bc ({x | f x \u2260 0} \u2229 s)\nA : s \u2286 t \u222a {x | f x = 0}\nM : MeasurableSet {x | f x = 0}\n\u22a2 f =\u1da0[ae (restrict \u03bc {x | f x = 0})] 0", "state_after": "case h\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhf : Measurable f\nhs : \u2191\u2191\u03bc ({x | f x \u2260 0} \u2229 s) = 0\nt : Set \u03b1 := toMeasurable \u03bc ({x | f x \u2260 0} \u2229 s)\nA : s \u2286 t \u222a {x | f x = 0}\nM : MeasurableSet {x | f x = 0}\n\u22a2 \u2200 (a : \u03b1), f a = 0 \u2192 f a = OfNat.ofNat 0 a"}, {"tactic": "simp only [imp_self, Pi.zero_apply, imp_true_iff]", "annotated_tactic": ["simp only [<a>imp_self</a>, <a>Pi.zero_apply</a>, <a>imp_true_iff</a>]", [{"full_name": "imp_self", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [122, 17], "def_end_pos": [122, 25]}, {"full_name": "Pi.zero_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [46, 3], "def_end_pos": [46, 14]}, {"full_name": "imp_true_iff", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [116, 9], "def_end_pos": [116, 21]}]], "state_before": "case h\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhf : Measurable f\nhs : \u2191\u2191\u03bc ({x | f x \u2260 0} \u2229 s) = 0\nt : Set \u03b1 := toMeasurable \u03bc ({x | f x \u2260 0} \u2229 s)\nA : s \u2286 t \u222a {x | f x = 0}\nM : MeasurableSet {x | f x = 0}\n\u22a2 \u2200 (a : \u03b1), f a = 0 \u2192 f a = OfNat.ofNat 0 a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/NoncommProd.lean", "full_name": "Multiset.noncommFold_eq_fold", "start": [103, 1], "end": [106, 7], "traced_tactics": [{"tactic": "induction s using Quotient.inductionOn", "annotated_tactic": ["induction s using <a>Quotient.inductionOn</a>", [{"full_name": "Quotient.inductionOn", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [1367, 19], "def_end_pos": [1367, 30]}]], "state_before": "F : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\nassoc : IsAssociative \u03b1 op\ns : Multiset \u03b1\ninst\u271d : IsCommutative \u03b1 op\na : \u03b1\n\u22a2 noncommFold op s (_ : \u2200 (x : \u03b1), x \u2208 {x | x \u2208 s} \u2192 \u2200 (y : \u03b1), y \u2208 {x | x \u2208 s} \u2192 x \u2260 y \u2192 op x y = op y x) a =\n    fold op a s", "state_after": "case h\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\nassoc : IsAssociative \u03b1 op\ninst\u271d : IsCommutative \u03b1 op\na : \u03b1\na\u271d : List \u03b1\n\u22a2 noncommFold op (Quotient.mk (List.isSetoid \u03b1) a\u271d)\n      (_ :\n        \u2200 (x : \u03b1),\n          x \u2208 {x | x \u2208 Quotient.mk (List.isSetoid \u03b1) a\u271d} \u2192\n            \u2200 (y : \u03b1), y \u2208 {x | x \u2208 Quotient.mk (List.isSetoid \u03b1) a\u271d} \u2192 x \u2260 y \u2192 op x y = op y x)\n      a =\n    fold op a (Quotient.mk (List.isSetoid \u03b1) a\u271d)"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case h\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\nassoc : IsAssociative \u03b1 op\ninst\u271d : IsCommutative \u03b1 op\na : \u03b1\na\u271d : List \u03b1\n\u22a2 noncommFold op (Quotient.mk (List.isSetoid \u03b1) a\u271d)\n      (_ :\n        \u2200 (x : \u03b1),\n          x \u2208 {x | x \u2208 Quotient.mk (List.isSetoid \u03b1) a\u271d} \u2192\n            \u2200 (y : \u03b1), y \u2208 {x | x \u2208 Quotient.mk (List.isSetoid \u03b1) a\u271d} \u2192 x \u2260 y \u2192 op x y = op y x)\n      a =\n    fold op a (Quotient.mk (List.isSetoid \u03b1) a\u271d)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/LocallyFinite.lean", "full_name": "Finset.Ioi_disjUnion_Iio", "start": [902, 1], "end": [905, 17], "traced_tactics": [{"tactic": "ext", "annotated_tactic": ["ext", []], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : LinearOrder \u03b1\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : LocallyFiniteOrderTop \u03b1\ninst\u271d : LocallyFiniteOrderBot \u03b1\na : \u03b1\n\u22a2 disjUnion (Ioi a) (Iio a) (_ : Disjoint (Ioi a) (Iio a)) = {a}\u1d9c", "state_after": "case a\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : LinearOrder \u03b1\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : LocallyFiniteOrderTop \u03b1\ninst\u271d : LocallyFiniteOrderBot \u03b1\na a\u271d : \u03b1\n\u22a2 a\u271d \u2208 disjUnion (Ioi a) (Iio a) (_ : Disjoint (Ioi a) (Iio a)) \u2194 a\u271d \u2208 {a}\u1d9c"}, {"tactic": "simp [eq_comm]", "annotated_tactic": ["simp [<a>eq_comm</a>]", [{"full_name": "eq_comm", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [104, 9], "def_end_pos": [104, 16]}]], "state_before": "case a\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : LinearOrder \u03b1\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : LocallyFiniteOrderTop \u03b1\ninst\u271d : LocallyFiniteOrderBot \u03b1\na a\u271d : \u03b1\n\u22a2 a\u271d \u2208 disjUnion (Ioi a) (Iio a) (_ : Disjoint (Ioi a) (Iio a)) \u2194 a\u271d \u2208 {a}\u1d9c", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "full_name": "MeasureTheory.setToFun_neg", "start": [1392, 1], "end": [1398, 37], "traced_tactics": [{"tactic": "by_cases hf : Integrable f \u03bc", "annotated_tactic": ["by_cases hf : <a>Integrable</a> f \u03bc", [{"full_name": "MeasureTheory.Integrable", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [442, 5], "def_end_pos": [442, 15]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\nf : \u03b1 \u2192 E\n\u22a2 setToFun \u03bc T hT (-f) = -setToFun \u03bc T hT f", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\nf : \u03b1 \u2192 E\nhf : Integrable f\n\u22a2 setToFun \u03bc T hT (-f) = -setToFun \u03bc T hT f\n\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : 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NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\nf : \u03b1 \u2192 E\nhf : Integrable f\n\u22a2 setToFun \u03bc T hT (-f) = -setToFun \u03bc T hT f", "state_after": "no goals"}, {"tactic": "rw [setToFun_undef hT hf, setToFun_undef hT, neg_zero]", "annotated_tactic": ["rw [<a>setToFun_undef</a> hT hf, <a>setToFun_undef</a> hT, <a>neg_zero</a>]", [{"full_name": "MeasureTheory.setToFun_undef", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [1286, 9], "def_end_pos": [1286, 23]}, {"full_name": "MeasureTheory.setToFun_undef", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [1286, 9], "def_end_pos": [1286, 23]}, {"full_name": "neg_zero", "def_path": 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[595, 1], "end": [604, 39], "traced_tactics": [{"tactic": "refine' Subtype.eq _", "annotated_tactic": ["refine' <a>Subtype.eq</a> _", [{"full_name": "Subtype.eq", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [960, 19], "def_end_pos": [960, 21]}]], "state_before": "n : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na\u271d a b : \u03b1\ni j : Fin (n + 1)\nh : i \u2264 j\nl : List \u03b1\nhl : List.length l = n\n\u22a2 insertNth b (Fin.succ j) (insertNth a i { val := l, property := hl }) =\n    insertNth a (Fin.castSucc i) (insertNth b j { val := l, property := hl })", "state_after": "n : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na\u271d a b : \u03b1\ni j : Fin (n + 1)\nh : i \u2264 j\nl : List \u03b1\nhl : List.length l = n\n\u22a2 \u2191(insertNth b (Fin.succ j) (insertNth a i { val := l, property := hl })) =\n    \u2191(insertNth a (Fin.castSucc i) (insertNth b j { val := l, property := hl }))"}, {"tactic": "simp only [insertNth_val, Fin.val_succ, Fin.castSucc, Fin.coe_castAdd]", "annotated_tactic": ["simp only [<a>insertNth_val</a>, <a>Fin.val_succ</a>, <a>Fin.castSucc</a>, <a>Fin.coe_castAdd</a>]", [{"full_name": "Vector.insertNth_val", "def_path": "Mathlib/Data/Vector/Basic.lean", "def_pos": [559, 9], "def_end_pos": [559, 22]}, {"full_name": "Fin.val_succ", "def_path": "lake-packages/std/Std/Data/Fin/Lemmas.lean", "def_pos": [211, 17], "def_end_pos": [211, 25]}, {"full_name": "Fin.castSucc", "def_path": "lake-packages/std/Std/Data/Fin/Basic.lean", "def_pos": [30, 15], "def_end_pos": [30, 23]}, {"full_name": "Fin.coe_castAdd", "def_path": "lake-packages/std/Std/Data/Fin/Lemmas.lean", "def_pos": [308, 17], "def_end_pos": [308, 28]}]], "state_before": "n : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na\u271d a b : \u03b1\ni j : Fin (n + 1)\nh : i \u2264 j\nl : List \u03b1\nhl : List.length l = n\n\u22a2 \u2191(insertNth b (Fin.succ j) (insertNth a i { val := l, property := hl })) =\n    \u2191(insertNth a (Fin.castSucc i) (insertNth b j { val := l, property := hl }))", "state_after": "n : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na\u271d a b : \u03b1\ni j : Fin (n + 1)\nh : i \u2264 j\nl : List \u03b1\nhl : List.length l = n\n\u22a2 List.insertNth (\u2191j + 1) b (List.insertNth (\u2191i) a l) = List.insertNth (\u2191i) a (List.insertNth (\u2191j) b l)"}, {"tactic": "apply List.insertNth_comm", "annotated_tactic": ["apply <a>List.insertNth_comm</a>", [{"full_name": "List.insertNth_comm", "def_path": "Mathlib/Data/List/Basic.lean", "def_pos": [1583, 9], "def_end_pos": [1583, 23]}]], "state_before": "n : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na\u271d a b : \u03b1\ni j : Fin (n + 1)\nh : i \u2264 j\nl : List \u03b1\nhl : List.length l = n\n\u22a2 List.insertNth (\u2191j + 1) b (List.insertNth (\u2191i) a l) = List.insertNth (\u2191i) a (List.insertNth (\u2191j) b l)", "state_after": "case x\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na\u271d a b : \u03b1\ni j : Fin (n + 1)\nh : i \u2264 j\nl : List \u03b1\nhl : List.length l = n\n\u22a2 \u2191i \u2264 \u2191j\n\ncase x\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na\u271d a b : \u03b1\ni j : Fin (n + 1)\nh : i \u2264 j\nl : List \u03b1\nhl : List.length l = n\n\u22a2 \u2191j \u2264 List.length l"}, {"tactic": "assumption", "annotated_tactic": ["assumption", []], "state_before": "case x\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na\u271d a b : \u03b1\ni j : Fin (n + 1)\nh : i \u2264 j\nl : List \u03b1\nhl : List.length l = n\n\u22a2 \u2191i \u2264 \u2191j", "state_after": "no goals"}, {"tactic": "rw [hl]", "annotated_tactic": ["rw [hl]", []], "state_before": "case x\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na\u271d a b : \u03b1\ni j : Fin (n + 1)\nh : i \u2264 j\nl : List \u03b1\nhl : List.length l = n\n\u22a2 \u2191j \u2264 List.length l", "state_after": "case x\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na\u271d a b : \u03b1\ni j : Fin (n + 1)\nh : i \u2264 j\nl : List \u03b1\nhl : List.length l = n\n\u22a2 \u2191j \u2264 n"}, {"tactic": "exact Nat.le_of_succ_le_succ j.2", "annotated_tactic": ["exact <a>Nat.le_of_succ_le_succ</a> j.2", [{"full_name": "Nat.le_of_succ_le_succ", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1628, 9], "def_end_pos": [1628, 31]}]], "state_before": "case x\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na\u271d a b : \u03b1\ni j : Fin (n + 1)\nh : i \u2264 j\nl : List \u03b1\nhl : List.length l = n\n\u22a2 \u2191j \u2264 n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "full_name": "String.Pos.zero_addString_eq", "start": [149, 1], "end": [150, 32], "traced_tactics": [{"tactic": "rw [\u2190 zero_addString_byteIdx]", "annotated_tactic": ["rw [\u2190 <a>zero_addString_byteIdx</a>]", [{"full_name": "String.Pos.zero_addString_byteIdx", "def_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "def_pos": [143, 9], "def_end_pos": [143, 31]}]], "state_before": "s : String\n\u22a2 0 + s = { byteIdx := utf8ByteSize s }", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Card.lean", "full_name": "Set.ncard_eq_of_bijective", "start": [736, 1], "end": [743, 18], "traced_tactics": [{"tactic": "rw [ncard_eq_toFinset_card _ hs]", "annotated_tactic": ["rw [<a>ncard_eq_toFinset_card</a> _ hs]", [{"full_name": "Set.ncard_eq_toFinset_card", "def_path": "Mathlib/Data/Set/Card.lean", "def_pos": [489, 9], "def_end_pos": [489, 31]}]], "state_before": "\u03b1 : Type u_1\ns t : Set \u03b1\nn : \u2115\nf : (i : \u2115) \u2192 i < n \u2192 \u03b1\nhf : \u2200 (a : \u03b1), a \u2208 s \u2192 \u2203 i h, f i h = a\nhf' : \u2200 (i : \u2115) (h : i < n), f i h \u2208 s\nf_inj : \u2200 (i j : \u2115) (hi : i < n) (hj : j < n), f i hi = f j hj \u2192 i = j\nhs : autoParam (Set.Finite s) _auto\u271d\n\u22a2 ncard s = n", "state_after": "\u03b1 : Type u_1\ns t : Set \u03b1\nn : \u2115\nf : (i : \u2115) \u2192 i < n \u2192 \u03b1\nhf : \u2200 (a : \u03b1), a \u2208 s \u2192 \u2203 i h, f i h = a\nhf' : \u2200 (i : \u2115) (h : i < n), f i h \u2208 s\nf_inj : \u2200 (i j : \u2115) (hi : i < n) (hj : j < n), f i hi = f j hj \u2192 i = j\nhs : autoParam (Set.Finite s) _auto\u271d\n\u22a2 Finset.card (Finite.toFinset hs) = n"}, {"tactic": "apply Finset.card_eq_of_bijective", "annotated_tactic": ["apply <a>Finset.card_eq_of_bijective</a>", [{"full_name": "Finset.card_eq_of_bijective", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [301, 9], "def_end_pos": [301, 29]}]], "state_before": "\u03b1 : Type u_1\ns t : Set \u03b1\nn : \u2115\nf : (i : \u2115) \u2192 i < n \u2192 \u03b1\nhf : \u2200 (a : \u03b1), a \u2208 s \u2192 \u2203 i h, f i h = a\nhf' : \u2200 (i : \u2115) (h : i < n), f i h \u2208 s\nf_inj : \u2200 (i j : \u2115) (hi : i < n) (hj : j < n), f i hi = f j hj \u2192 i = j\nhs : autoParam (Set.Finite s) _auto\u271d\n\u22a2 Finset.card (Finite.toFinset hs) = n", "state_after": "case hf\n\u03b1 : Type u_1\ns t : Set \u03b1\nn : \u2115\nf : (i : \u2115) \u2192 i < n \u2192 \u03b1\nhf : \u2200 (a : \u03b1), a \u2208 s \u2192 \u2203 i h, f i h = a\nhf' : \u2200 (i : \u2115) (h : i < n), f i h \u2208 s\nf_inj : \u2200 (i j : \u2115) (hi : i < n) (hj : j < n), f i hi = f j hj \u2192 i = j\nhs : autoParam (Set.Finite s) _auto\u271d\n\u22a2 \u2200 (a : \u03b1), a \u2208 Finite.toFinset hs \u2192 \u2203 i h, ?f i h = a\n\ncase hf'\n\u03b1 : Type u_1\ns t : Set \u03b1\nn : \u2115\nf : (i : \u2115) \u2192 i < n \u2192 \u03b1\nhf : \u2200 (a : \u03b1), a \u2208 s \u2192 \u2203 i h, f i h = a\nhf' : \u2200 (i : \u2115) (h : i < n), f i h \u2208 s\nf_inj : \u2200 (i j : \u2115) (hi : i < n) (hj : j < n), f i hi = f j hj \u2192 i = j\nhs : autoParam (Set.Finite s) _auto\u271d\n\u22a2 \u2200 (i : \u2115) (h : i < n), ?f i h \u2208 Finite.toFinset hs\n\ncase f_inj\n\u03b1 : Type u_1\ns t : Set \u03b1\nn : \u2115\nf : (i : \u2115) \u2192 i < n \u2192 \u03b1\nhf : \u2200 (a : \u03b1), a \u2208 s \u2192 \u2203 i h, f i h = a\nhf' : \u2200 (i : \u2115) (h : i < n), f i h \u2208 s\nf_inj : \u2200 (i j : \u2115) (hi : i < n) (hj : j < n), f i hi = f j hj \u2192 i = j\nhs : autoParam (Set.Finite s) _auto\u271d\n\u22a2 \u2200 (i j : \u2115) (hi : i < n) (hj : j < n), ?f i hi = ?f j hj \u2192 i = j\n\ncase f\n\u03b1 : Type u_1\ns t : Set \u03b1\nn : \u2115\nf : (i : \u2115) \u2192 i < n \u2192 \u03b1\nhf : \u2200 (a : \u03b1), a \u2208 s \u2192 \u2203 i h, f i h = a\nhf' : \u2200 (i : \u2115) (h : i < n), f i h \u2208 s\nf_inj : \u2200 (i j : \u2115) (hi : i < n) (hj : j < n), f i hi = f j hj \u2192 i = j\nhs : autoParam (Set.Finite s) _auto\u271d\n\u22a2 (i : \u2115) \u2192 i < n \u2192 \u03b1"}, {"tactic": "all_goals simpa", "annotated_tactic": ["all_goals simpa", []], "state_before": "case hf\n\u03b1 : Type u_1\ns t : Set \u03b1\nn : \u2115\nf : (i : \u2115) \u2192 i < n \u2192 \u03b1\nhf : \u2200 (a : \u03b1), a \u2208 s \u2192 \u2203 i h, f i h = a\nhf' : \u2200 (i : \u2115) (h : i < n), f i h \u2208 s\nf_inj : \u2200 (i j : \u2115) (hi : i < n) (hj : j < n), f i hi = f j hj \u2192 i = j\nhs : autoParam (Set.Finite s) _auto\u271d\n\u22a2 \u2200 (a : \u03b1), a \u2208 Finite.toFinset hs \u2192 \u2203 i h, ?f i h = a\n\ncase hf'\n\u03b1 : Type u_1\ns t : Set \u03b1\nn : \u2115\nf : (i : \u2115) \u2192 i < n \u2192 \u03b1\nhf : \u2200 (a : \u03b1), a \u2208 s \u2192 \u2203 i h, f i h = a\nhf' : \u2200 (i : \u2115) (h : i < n), f i h \u2208 s\nf_inj : \u2200 (i j : \u2115) (hi : i < n) (hj : j < n), f i hi = f j hj \u2192 i = j\nhs : autoParam (Set.Finite s) _auto\u271d\n\u22a2 \u2200 (i : \u2115) (h : i < n), ?f i h \u2208 Finite.toFinset hs\n\ncase f_inj\n\u03b1 : Type u_1\ns t : Set \u03b1\nn : \u2115\nf : (i : \u2115) \u2192 i < n \u2192 \u03b1\nhf : \u2200 (a : \u03b1), a \u2208 s \u2192 \u2203 i h, f i h = a\nhf' : \u2200 (i : \u2115) (h : i < n), f i h \u2208 s\nf_inj : \u2200 (i j : \u2115) (hi : i < n) (hj : j < n), f i hi = f j hj \u2192 i = j\nhs : autoParam (Set.Finite s) _auto\u271d\n\u22a2 \u2200 (i j : \u2115) (hi : i < n) (hj : j < n), ?f i hi = ?f j hj \u2192 i = j\n\ncase f\n\u03b1 : Type u_1\ns t : Set \u03b1\nn : \u2115\nf : (i : \u2115) \u2192 i < n \u2192 \u03b1\nhf : \u2200 (a : \u03b1), a \u2208 s \u2192 \u2203 i h, f i h = a\nhf' : \u2200 (i : \u2115) (h : i < n), f i h \u2208 s\nf_inj : \u2200 (i j : \u2115) (hi : i < n) (hj : j < n), f i hi = f j hj \u2192 i = j\nhs : autoParam (Set.Finite s) _auto\u271d\n\u22a2 (i : \u2115) \u2192 i < n \u2192 \u03b1", "state_after": "no goals"}, {"tactic": "simpa", "annotated_tactic": ["simpa", []], "state_before": "case f_inj\n\u03b1 : Type u_1\ns t : Set \u03b1\nn : \u2115\nf : (i : \u2115) \u2192 i < n \u2192 \u03b1\nhf : \u2200 (a : \u03b1), a \u2208 s \u2192 \u2203 i h, f i h = a\nhf' : \u2200 (i : \u2115) (h : i < n), f i h \u2208 s\nf_inj : \u2200 (i j : \u2115) (hi : i < n) (hj : j < n), f i hi = f j hj \u2192 i = j\nhs : autoParam (Set.Finite s) _auto\u271d\n\u22a2 \u2200 (i j : \u2115) (hi : i < n) (hj : j < n), f i hi = f j hj \u2192 i = j", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/Equiv.lean", "full_name": "MvPolynomial.mapEquiv_trans", "start": [116, 1], "end": [120, 15], "traced_tactics": [{"tactic": "simp only [RingEquiv.coe_trans, comp_apply, mapEquiv_apply, RingEquiv.coe_ringHom_trans,\n  map_map]", "annotated_tactic": ["simp only [<a>RingEquiv.coe_trans</a>, <a>comp_apply</a>, <a>mapEquiv_apply</a>, <a>RingEquiv.coe_ringHom_trans</a>,\n      <a>map_map</a>]", [{"full_name": "RingEquiv.coe_trans", "def_path": "Mathlib/Algebra/Ring/Equiv.lean", "def_pos": [328, 9], "def_end_pos": [328, 18]}, {"full_name": "Function.comp_apply", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [33, 17], "def_end_pos": [33, 36]}, {"full_name": "MvPolynomial.mapEquiv_apply", "def_path": "Mathlib/Data/MvPolynomial/Equiv.lean", "def_pos": [94, 9], "def_end_pos": [94, 14]}, {"full_name": "RingEquiv.coe_ringHom_trans", "def_path": "Mathlib/Algebra/Ring/Equiv.lean", "def_pos": [569, 9], "def_end_pos": [569, 26]}, {"full_name": "MvPolynomial.map_map", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [1251, 9], "def_end_pos": [1251, 16]}]], "state_before": "R : Type u\nS\u2081 : Type v\nS\u2082 : Type w\nS\u2083 : Type x\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne\u271d : \u2115\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommSemiring S\u2081\ninst\u271d\u00b9 : CommSemiring S\u2082\ninst\u271d : CommSemiring S\u2083\ne : S\u2081 \u2243+* S\u2082\nf : S\u2082 \u2243+* S\u2083\np : MvPolynomial \u03c3 S\u2081\n\u22a2 \u2191(RingEquiv.trans (mapEquiv \u03c3 e) (mapEquiv \u03c3 f)) p = \u2191(mapEquiv \u03c3 (RingEquiv.trans e f)) p", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Primrec.lean", "full_name": "Primrec.nat_add", "start": [666, 1], "end": [667, 39], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/PImage.lean", "full_name": "Finset.pimage_union", "start": [100, 1], "end": [102, 56], "traced_tactics": [{"tactic": "simp only [coe_pimage, coe_union, \u2190 PFun.image_union]", "annotated_tactic": ["simp only [<a>coe_pimage</a>, <a>coe_union</a>, \u2190 <a>PFun.image_union</a>]", [{"full_name": "Finset.coe_pimage", "def_path": "Mathlib/Data/Finset/PImage.lean", "def_pos": [71, 9], "def_end_pos": [71, 19]}, {"full_name": "Finset.coe_union", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1399, 9], "def_end_pos": [1399, 18]}, {"full_name": "PFun.image_union", "def_path": "Mathlib/Data/PFun.lean", "def_pos": [417, 9], "def_end_pos": [417, 20]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : DecidableEq \u03b2\nf g : \u03b1 \u2192. \u03b2\ninst\u271d\u00b2 : (x : \u03b1) \u2192 Decidable (f x).Dom\ninst\u271d\u00b9 : (x : \u03b1) \u2192 Decidable (g x).Dom\ns t : Finset \u03b1\nb : \u03b2\ninst\u271d : DecidableEq \u03b1\n\u22a2 \u2191(pimage f (s \u222a t)) = \u2191(pimage f s \u222a pimage f t)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Part.lean", "full_name": "Part.mod_get_eq", "start": [797, 1], "end": [799, 27], "traced_tactics": [{"tactic": "simp [mod_def]", "annotated_tactic": ["simp [<a>mod_def</a>]", [{"full_name": "Part.mod_def", "def_path": "Mathlib/Data/Part.lean", "def_pos": [705, 9], "def_end_pos": [705, 16]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : Mod \u03b1\na b : Part \u03b1\nhab : (a % b).Dom\n\u22a2 get (a % b) hab = get a (_ : a.Dom) % get b (_ : b.Dom)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : Mod \u03b1\na b : Part \u03b1\nhab : (a % b).Dom\n\u22a2 get (Part.bind a fun y => map (fun x => y % x) b) (_ : (Part.bind a fun y => map (fun x => y % x) b).Dom) =\n    get a (_ : a.Dom) % get b (_ : b.Dom)"}, {"tactic": "aesop", "annotated_tactic": ["aesop", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : Mod \u03b1\na b : Part \u03b1\nhab : (a % b).Dom\n\u22a2 get (Part.bind a fun y => map (fun x => y % x) b) (_ : (Part.bind a fun y => map (fun x => y % x) b).Dom) =\n    get a (_ : a.Dom) % get b (_ : b.Dom)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean", "full_name": "MeasureTheory.Measure.addHaar_eq_zero_of_disjoint_translates_aux", "start": [142, 1], "end": [155, 40], "traced_tactics": [{"tactic": "by_contra h", "annotated_tactic": ["by_contra h", []], "state_before": "E : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\ns : Set E\nu : \u2115 \u2192 E\nsb : Bornology.IsBounded s\nhu : Bornology.IsBounded (range u)\nhs : Pairwise (Disjoint on fun n => {u n} + s)\nh's : MeasurableSet s\n\u22a2 \u2191\u2191\u03bc s = 0", "state_after": "E : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\ns : Set E\nu : \u2115 \u2192 E\nsb : Bornology.IsBounded s\nhu : Bornology.IsBounded (range u)\nhs : Pairwise (Disjoint on fun n => {u n} + s)\nh's : MeasurableSet s\nh : \u00ac\u2191\u2191\u03bc s = 0\n\u22a2 False"}, {"tactic": "apply lt_irrefl \u221e", "annotated_tactic": ["apply <a>lt_irrefl</a> \u221e", [{"full_name": "lt_irrefl", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [79, 9], "def_end_pos": [79, 18]}]], "state_before": "E : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\ns : Set E\nu : \u2115 \u2192 E\nsb : Bornology.IsBounded s\nhu : Bornology.IsBounded (range u)\nhs : Pairwise (Disjoint on fun n => {u n} + s)\nh's : MeasurableSet s\nh : \u00ac\u2191\u2191\u03bc s = 0\n\u22a2 False", "state_after": "E : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\ns : Set E\nu : \u2115 \u2192 E\nsb : Bornology.IsBounded s\nhu : Bornology.IsBounded (range u)\nhs : Pairwise (Disjoint on fun n => {u n} + s)\nh's : MeasurableSet s\nh : \u00ac\u2191\u2191\u03bc s = 0\n\u22a2 \u22a4 < \u22a4"}, {"tactic": "calc\n  \u221e = \u2211' _ : \u2115, \u03bc s := (ENNReal.tsum_const_eq_top_of_ne_zero h).symm\n  _ = \u2211' n : \u2115, \u03bc ({u n} + s) := by\n    congr 1; ext1 n; simp only [image_add_left, measure_preimage_add, singleton_add]\n  _ = \u03bc (\u22c3 n, {u n} + s) := Eq.symm <| measure_iUnion hs fun n => by\n    simpa only [image_add_left, singleton_add] using measurable_id.const_add _ h's\n  _ = \u03bc (range u + s) := by rw [\u2190 iUnion_add, iUnion_singleton_eq_range]\n  _ < \u221e := (hu.add sb).measure_lt_top", "annotated_tactic": ["calc\n    \u221e = \u2211' _ : \u2115, \u03bc s := (<a>ENNReal.tsum_const_eq_top_of_ne_zero</a> h).<a>symm</a>\n    _ = \u2211' n : \u2115, \u03bc ({u n} + s) := by\n      congr 1; ext1 n; simp only [<a>image_add_left</a>, <a>measure_preimage_add</a>, <a>singleton_add</a>]\n    _ = \u03bc (\u22c3 n, {u n} + s) := <a>Eq.symm</a> <| <a>measure_iUnion</a> hs fun n => by\n      simpa only [<a>image_add_left</a>, <a>singleton_add</a>] using measurable_id.const_add _ h's\n    _ = \u03bc (<a>range</a> u + s) := by rw [\u2190 <a>iUnion_add</a>, <a>iUnion_singleton_eq_range</a>]\n    _ < \u221e := (hu.add sb).<a>measure_lt_top</a>", [{"full_name": "ENNReal.tsum_const_eq_top_of_ne_zero", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [882, 9], "def_end_pos": [882, 37]}, {"full_name": "Eq.symm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [310, 9], "def_end_pos": [310, 16]}, {"full_name": "Set.image_add_left", "def_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "def_pos": [1198, 3], "def_end_pos": [1198, 14]}, {"full_name": "MeasureTheory.measure_preimage_add", "def_path": "Mathlib/MeasureTheory/Group/Measure.lean", "def_pos": [317, 3], "def_end_pos": [317, 14]}, {"full_name": "Set.singleton_add", "def_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "def_pos": [402, 3], "def_end_pos": [402, 14]}, {"full_name": "Eq.symm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [310, 9], "def_end_pos": [310, 16]}, {"full_name": "MeasureTheory.measure_iUnion", "def_path": "Mathlib/MeasureTheory/Measure/NullMeasurable.lean", "def_pos": [272, 9], "def_end_pos": [272, 23]}, {"full_name": "Set.image_add_left", "def_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "def_pos": [1198, 3], "def_end_pos": [1198, 14]}, {"full_name": "Set.singleton_add", "def_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "def_pos": [402, 3], "def_end_pos": [402, 14]}, {"full_name": "Set.range", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [668, 5], "def_end_pos": [668, 10]}, {"full_name": "Set.iUnion_add", "def_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "def_pos": [487, 3], "def_end_pos": [487, 14]}, {"full_name": "Set.iUnion_singleton_eq_range", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [1441, 9], "def_end_pos": [1441, 34]}, {"full_name": "Bornology.IsBounded.measure_lt_top", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3865, 9], "def_end_pos": [3865, 50]}]], "state_before": "E : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\ns : Set E\nu : \u2115 \u2192 E\nsb : Bornology.IsBounded s\nhu : Bornology.IsBounded (range u)\nhs : Pairwise (Disjoint on fun n => {u n} + s)\nh's : MeasurableSet s\nh : \u00ac\u2191\u2191\u03bc s = 0\n\u22a2 \u22a4 < \u22a4", "state_after": "no goals"}, {"tactic": "congr 1", "annotated_tactic": ["congr 1", []], "state_before": "E : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\ns : Set E\nu : \u2115 \u2192 E\nsb : Bornology.IsBounded s\nhu : Bornology.IsBounded (range u)\nhs : Pairwise (Disjoint on fun n => {u n} + s)\nh's : MeasurableSet s\nh : \u00ac\u2191\u2191\u03bc s = 0\n\u22a2 \u2211' (x : \u2115), \u2191\u2191\u03bc s = \u2211' (n : \u2115), \u2191\u2191\u03bc ({u n} + s)", "state_after": "case e_f\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\ns : Set E\nu : \u2115 \u2192 E\nsb : Bornology.IsBounded s\nhu : Bornology.IsBounded (range u)\nhs : Pairwise (Disjoint on fun n => {u n} + s)\nh's : MeasurableSet s\nh : \u00ac\u2191\u2191\u03bc s = 0\n\u22a2 (fun x => \u2191\u2191\u03bc s) = fun n => \u2191\u2191\u03bc ({u n} + s)"}, {"tactic": "ext1 n", "annotated_tactic": ["ext1 n", []], "state_before": "case e_f\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\ns : Set E\nu : \u2115 \u2192 E\nsb : Bornology.IsBounded s\nhu : Bornology.IsBounded (range u)\nhs : Pairwise (Disjoint on fun n => {u n} + s)\nh's : MeasurableSet s\nh : \u00ac\u2191\u2191\u03bc s = 0\n\u22a2 (fun x => \u2191\u2191\u03bc s) = fun n => \u2191\u2191\u03bc ({u n} + s)", "state_after": "case e_f.h\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\ns : Set E\nu : \u2115 \u2192 E\nsb : Bornology.IsBounded s\nhu : Bornology.IsBounded (range u)\nhs : Pairwise (Disjoint on fun n => {u n} + s)\nh's : MeasurableSet s\nh : \u00ac\u2191\u2191\u03bc s = 0\nn : \u2115\n\u22a2 \u2191\u2191\u03bc s = \u2191\u2191\u03bc ({u n} + s)"}, {"tactic": "simp only [image_add_left, measure_preimage_add, singleton_add]", "annotated_tactic": ["simp only [<a>image_add_left</a>, <a>measure_preimage_add</a>, <a>singleton_add</a>]", [{"full_name": "Set.image_add_left", "def_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "def_pos": [1198, 3], "def_end_pos": [1198, 14]}, {"full_name": "MeasureTheory.measure_preimage_add", "def_path": "Mathlib/MeasureTheory/Group/Measure.lean", "def_pos": [317, 3], "def_end_pos": [317, 14]}, {"full_name": "Set.singleton_add", "def_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "def_pos": [402, 3], "def_end_pos": [402, 14]}]], "state_before": "case e_f.h\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\ns : Set E\nu : \u2115 \u2192 E\nsb : Bornology.IsBounded s\nhu : Bornology.IsBounded (range u)\nhs : Pairwise (Disjoint on fun n => {u n} + s)\nh's : MeasurableSet s\nh : \u00ac\u2191\u2191\u03bc s = 0\nn : \u2115\n\u22a2 \u2191\u2191\u03bc s = \u2191\u2191\u03bc ({u n} + s)", "state_after": "no goals"}, {"tactic": "simpa only [image_add_left, singleton_add] using measurable_id.const_add _ h's", "annotated_tactic": ["simpa only [<a>image_add_left</a>, <a>singleton_add</a>] using measurable_id.const_add _ h's", [{"full_name": "Set.image_add_left", "def_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "def_pos": [1198, 3], "def_end_pos": [1198, 14]}, {"full_name": "Set.singleton_add", "def_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "def_pos": [402, 3], "def_end_pos": [402, 14]}]], "state_before": "E : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\ns : Set E\nu : \u2115 \u2192 E\nsb : Bornology.IsBounded s\nhu : Bornology.IsBounded (range u)\nhs : Pairwise (Disjoint on fun n => {u n} + s)\nh's : MeasurableSet s\nh : \u00ac\u2191\u2191\u03bc s = 0\nn : \u2115\n\u22a2 MeasurableSet ({u n} + s)", "state_after": "no goals"}, {"tactic": "rw [\u2190 iUnion_add, iUnion_singleton_eq_range]", "annotated_tactic": ["rw [\u2190 <a>iUnion_add</a>, <a>iUnion_singleton_eq_range</a>]", [{"full_name": "Set.iUnion_add", "def_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "def_pos": [487, 3], "def_end_pos": [487, 14]}, {"full_name": "Set.iUnion_singleton_eq_range", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [1441, 9], "def_end_pos": [1441, 34]}]], "state_before": "E : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\ns : Set E\nu : \u2115 \u2192 E\nsb : Bornology.IsBounded s\nhu : Bornology.IsBounded (range u)\nhs : Pairwise (Disjoint on fun n => {u n} + s)\nh's : MeasurableSet s\nh : \u00ac\u2191\u2191\u03bc s = 0\n\u22a2 \u2191\u2191\u03bc (\u22c3 n, {u n} + s) = \u2191\u2191\u03bc (range u + s)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Vector/Basic.lean", "full_name": "Vector.get_map\u2082", "start": [799, 1], "end": [809, 36], "traced_tactics": [{"tactic": "clear * - v\u2081 v\u2082", "annotated_tactic": ["clear * - v\u2081 v\u2082", []], "state_before": "\u03b1 : Type u_2\n\u03b2 : Type u_3\nn : \u2115\nxs : Vector \u03b1 n\nys : Vector \u03b2 n\n\u03b3 : Type u_1\nv\u2081 : Vector \u03b1 n\nv\u2082 : Vector \u03b2 n\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ni : Fin n\n\u22a2 get (map\u2082 f v\u2081 v\u2082) i = f (get v\u2081 i) (get v\u2082 i)", "state_after": "\u03b1 : Type u_2\n\u03b2 : Type u_3\nn : \u2115\n\u03b3 : Type u_1\nv\u2081 : Vector \u03b1 n\nv\u2082 : Vector \u03b2 n\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ni : Fin n\n\u22a2 get (map\u2082 f v\u2081 v\u2082) i = f (get v\u2081 i) (get v\u2082 i)"}, {"tactic": "induction v\u2081, v\u2082 using inductionOn\u2082", "annotated_tactic": ["induction v\u2081, v\u2082 using <a>inductionOn\u2082</a>", [{"full_name": "Vector.inductionOn\u2082", "def_path": "Mathlib/Data/Vector/Basic.lean", "def_pos": [475, 5], "def_end_pos": [475, 17]}]], "state_before": "\u03b1 : Type u_2\n\u03b2 : Type u_3\nn : \u2115\n\u03b3 : Type u_1\nv\u2081 : Vector \u03b1 n\nv\u2082 : Vector \u03b2 n\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ni : Fin n\n\u22a2 get (map\u2082 f v\u2081 v\u2082) i = f (get v\u2081 i) (get v\u2082 i)", "state_after": "case nil\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nn : \u2115\n\u03b3 : Type u_1\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ni : Fin 0\n\u22a2 get (map\u2082 f nil nil) i = f (get nil i) (get nil i)\n\ncase cons\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nn : \u2115\n\u03b3 : Type u_1\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nn\u271d : \u2115\na\u271d\u00b9 : \u03b1\nb\u271d : \u03b2\nx\u271d : Vector \u03b1 n\u271d\ny\u271d : Vector \u03b2 n\u271d\na\u271d : \u2200 (i : Fin n\u271d), get (map\u2082 f x\u271d y\u271d) i = f (get x\u271d i) (get y\u271d i)\ni : Fin (Nat.succ n\u271d)\n\u22a2 get (map\u2082 f (a\u271d\u00b9 ::\u1d65 x\u271d) (b\u271d ::\u1d65 y\u271d)) i = f (get (a\u271d\u00b9 ::\u1d65 x\u271d) i) (get (b\u271d ::\u1d65 y\u271d) i)"}, {"tactic": "case nil =>\n  exact Fin.elim0 i", "annotated_tactic": ["case nil =>\n    exact <a>Fin.elim0</a> i", [{"full_name": "Fin.elim0", "def_path": "lake-packages/lean4/src/lean/Init/Data/Fin/Basic.lean", "def_pos": [18, 5], "def_end_pos": [18, 10]}]], "state_before": "case nil\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nn : \u2115\n\u03b3 : Type u_1\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ni : Fin 0\n\u22a2 get (map\u2082 f nil nil) i = f (get nil i) (get nil i)\n\ncase cons\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nn : \u2115\n\u03b3 : Type u_1\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nn\u271d : \u2115\na\u271d\u00b9 : \u03b1\nb\u271d : \u03b2\nx\u271d : Vector \u03b1 n\u271d\ny\u271d : Vector \u03b2 n\u271d\na\u271d : \u2200 (i : Fin n\u271d), get (map\u2082 f x\u271d y\u271d) i = f (get x\u271d i) (get y\u271d i)\ni : Fin (Nat.succ n\u271d)\n\u22a2 get (map\u2082 f (a\u271d\u00b9 ::\u1d65 x\u271d) (b\u271d ::\u1d65 y\u271d)) i = f (get (a\u271d\u00b9 ::\u1d65 x\u271d) i) (get (b\u271d ::\u1d65 y\u271d) i)", "state_after": "case cons\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nn : \u2115\n\u03b3 : Type u_1\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nn\u271d : \u2115\na\u271d\u00b9 : \u03b1\nb\u271d : \u03b2\nx\u271d : Vector \u03b1 n\u271d\ny\u271d : Vector \u03b2 n\u271d\na\u271d : \u2200 (i : Fin n\u271d), get (map\u2082 f x\u271d y\u271d) i = f (get x\u271d i) (get y\u271d i)\ni : Fin (Nat.succ n\u271d)\n\u22a2 get (map\u2082 f (a\u271d\u00b9 ::\u1d65 x\u271d) (b\u271d ::\u1d65 y\u271d)) i = f (get (a\u271d\u00b9 ::\u1d65 x\u271d) i) (get (b\u271d ::\u1d65 y\u271d) i)"}, {"tactic": "exact Fin.elim0 i", "annotated_tactic": ["exact <a>Fin.elim0</a> i", [{"full_name": "Fin.elim0", "def_path": "lake-packages/lean4/src/lean/Init/Data/Fin/Basic.lean", "def_pos": [18, 5], "def_end_pos": [18, 10]}]], "state_before": "\u03b1 : Type u_2\n\u03b2 : Type u_3\nn : \u2115\n\u03b3 : Type u_1\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ni : Fin 0\n\u22a2 get (map\u2082 f nil nil) i = f (get nil i) (get nil i)", "state_after": "no goals"}, {"tactic": "rw [map\u2082_cons]", "annotated_tactic": ["rw [<a>map\u2082_cons</a>]", [{"full_name": "Vector.map\u2082_cons", "def_path": "Mathlib/Data/Vector/Basic.lean", "def_pos": [143, 9], "def_end_pos": [143, 18]}]], "state_before": "\u03b1 : Type u_2\n\u03b2 : Type u_3\nn : \u2115\n\u03b3 : Type u_1\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nn\u271d : \u2115\nx : \u03b1\nxs : \u03b2\ny : Vector \u03b1 n\u271d\nys : Vector \u03b2 n\u271d\nih : \u2200 (i : Fin n\u271d), get (map\u2082 f y ys) i = f (get y i) (get ys i)\ni : Fin (Nat.succ n\u271d)\n\u22a2 get (map\u2082 f (x ::\u1d65 y) (xs ::\u1d65 ys)) i = f (get (x ::\u1d65 y) i) (get (xs ::\u1d65 ys) i)", "state_after": "\u03b1 : Type u_2\n\u03b2 : Type u_3\nn : \u2115\n\u03b3 : Type u_1\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nn\u271d : \u2115\nx : \u03b1\nxs : \u03b2\ny : Vector \u03b1 n\u271d\nys : Vector \u03b2 n\u271d\nih : \u2200 (i : Fin n\u271d), get (map\u2082 f y ys) i = f (get y i) (get ys i)\ni : Fin (Nat.succ n\u271d)\n\u22a2 get (f x xs ::\u1d65 map\u2082 f y ys) i = f (get (x ::\u1d65 y) i) (get (xs ::\u1d65 ys) i)"}, {"tactic": "cases i using Fin.cases", "annotated_tactic": ["cases i using <a>Fin.cases</a>", [{"full_name": "Fin.cases", "def_path": "lake-packages/std/Std/Data/Fin/Lemmas.lean", "def_pos": [613, 21], "def_end_pos": [613, 26]}]], "state_before": "\u03b1 : Type u_2\n\u03b2 : Type u_3\nn : \u2115\n\u03b3 : Type u_1\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nn\u271d : \u2115\nx : \u03b1\nxs : \u03b2\ny : Vector \u03b1 n\u271d\nys : Vector \u03b2 n\u271d\nih : \u2200 (i : Fin n\u271d), get (map\u2082 f y ys) i = f (get y i) (get ys i)\ni : Fin (Nat.succ n\u271d)\n\u22a2 get (f x xs ::\u1d65 map\u2082 f y ys) i = f (get (x ::\u1d65 y) i) (get (xs ::\u1d65 ys) i)", "state_after": "case zero\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nn : \u2115\n\u03b3 : Type u_1\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nn\u271d : \u2115\nx : \u03b1\nxs : \u03b2\ny : Vector \u03b1 n\u271d\nys : Vector \u03b2 n\u271d\nih : \u2200 (i : Fin n\u271d), get (map\u2082 f y ys) i = f (get y i) (get ys i)\n\u22a2 get (f x xs ::\u1d65 map\u2082 f y ys) 0 = f (get (x ::\u1d65 y) 0) (get (xs ::\u1d65 ys) 0)\n\ncase succ\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nn : \u2115\n\u03b3 : Type u_1\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nn\u271d : \u2115\nx : \u03b1\nxs : \u03b2\ny : Vector \u03b1 n\u271d\nys : Vector \u03b2 n\u271d\nih : \u2200 (i : Fin n\u271d), get (map\u2082 f y ys) i = f (get y i) (get ys i)\ni\u271d : Fin n\u271d\n\u22a2 get (f x xs ::\u1d65 map\u2082 f y ys) (Fin.succ i\u271d) = f (get (x ::\u1d65 y) (Fin.succ i\u271d)) (get (xs ::\u1d65 ys) (Fin.succ i\u271d))"}, {"tactic": "simp only [get_zero, head_cons]", "annotated_tactic": ["simp only [<a>get_zero</a>, <a>head_cons</a>]", [{"full_name": "Vector.get_zero", "def_path": "Mathlib/Data/Vector/Basic.lean", "def_pos": [264, 9], "def_end_pos": [264, 17]}, {"full_name": "Vector.head_cons", "def_path": "Mathlib/Data/Vector.lean", "def_pos": [64, 9], "def_end_pos": [64, 18]}]], "state_before": "case zero\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nn : \u2115\n\u03b3 : Type u_1\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nn\u271d : \u2115\nx : \u03b1\nxs : \u03b2\ny : Vector \u03b1 n\u271d\nys : Vector \u03b2 n\u271d\nih : \u2200 (i : Fin n\u271d), get (map\u2082 f y ys) i = f (get y i) (get ys i)\n\u22a2 get (f x xs ::\u1d65 map\u2082 f y ys) 0 = f (get (x ::\u1d65 y) 0) (get (xs ::\u1d65 ys) 0)", "state_after": "no goals"}, {"tactic": "simp only [get_cons_succ, ih]", "annotated_tactic": ["simp only [<a>get_cons_succ</a>, ih]", [{"full_name": "Vector.get_cons_succ", "def_path": "Mathlib/Data/Vector/Basic.lean", "def_pos": [285, 9], "def_end_pos": [285, 22]}]], "state_before": "case succ\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nn : \u2115\n\u03b3 : Type u_1\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nn\u271d : \u2115\nx : \u03b1\nxs : \u03b2\ny : Vector \u03b1 n\u271d\nys : Vector \u03b2 n\u271d\nih : \u2200 (i : Fin n\u271d), get (map\u2082 f y ys) i = f (get y i) (get ys i)\ni\u271d : Fin n\u271d\n\u22a2 get (f x xs ::\u1d65 map\u2082 f y ys) (Fin.succ i\u271d) = f (get (x ::\u1d65 y) (Fin.succ i\u271d)) (get (xs ::\u1d65 ys) (Fin.succ i\u271d))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/TMToPartrec.lean", "full_name": "Turing.ToPartrec.Code.exists_code", "start": [285, 1], "end": [389, 57], "traced_tactics": [{"tactic": "induction' hf with n f hf", "annotated_tactic": ["induction' hf with n f hf", []], "state_before": "n : \u2115\nf : Vector \u2115 n \u2192. \u2115\nhf : Nat.Partrec' f\n\u22a2 \u2203 c, \u2200 (v : Vector \u2115 n), eval c \u2191v = pure <$> f v", "state_after": "case prim\nn\u271d : \u2115\nf\u271d : Vector \u2115 n\u271d \u2192. \u2115\nn : \u2115\nf : Vector \u2115 n \u2192 \u2115\nhf : Nat.Primrec' f\n\u22a2 \u2203 c, \u2200 (v : Vector \u2115 n), eval c \u2191v = pure <$> \u2191f v\n\ncase comp\nn : \u2115\nf : Vector \u2115 n \u2192. \u2115\nm\u271d n\u271d : \u2115\nf\u271d : Vector \u2115 n\u271d \u2192. \u2115\ng\u271d : Fin n\u271d \u2192 Vector \u2115 m\u271d \u2192. \u2115\na\u271d\u00b9 : Nat.Partrec' f\u271d\na\u271d : \u2200 (i : Fin n\u271d), Nat.Partrec' (g\u271d i)\na_ih\u271d\u00b9 : \u2203 c, \u2200 (v : Vector \u2115 n\u271d), eval c \u2191v = pure <$> f\u271d v\na_ih\u271d : \u2200 (i : Fin n\u271d), \u2203 c, \u2200 (v : Vector \u2115 m\u271d), eval c \u2191v = pure <$> g\u271d i v\n\u22a2 \u2203 c, \u2200 (v : Vector \u2115 m\u271d), eval c \u2191v = pure <$> (fun v => (Vector.mOfFn fun i => g\u271d i v) >>= f\u271d) v\n\ncase rfind\nn : \u2115\nf : Vector \u2115 n \u2192. \u2115\nn\u271d : \u2115\nf\u271d : Vector \u2115 (n\u271d + 1) \u2192 \u2115\na\u271d : Nat.Partrec' \u2191f\u271d\na_ih\u271d : \u2203 c, \u2200 (v : Vector \u2115 (n\u271d + 1)), eval c \u2191v = pure <$> \u2191f\u271d v\n\u22a2 \u2203 c, \u2200 (v : Vector \u2115 n\u271d), eval c \u2191v = pure <$> (fun v => Nat.rfind fun n => Part.some (decide (f\u271d (n ::\u1d65 v) = 0))) v"}, {"tactic": "induction hf", "annotated_tactic": ["induction hf", []], "state_before": "case prim\nn\u271d : \u2115\nf\u271d : Vector \u2115 n\u271d \u2192. \u2115\nn : \u2115\nf : Vector \u2115 n \u2192 \u2115\nhf : Nat.Primrec' f\n\u22a2 \u2203 c, \u2200 (v : Vector \u2115 n), eval c \u2191v = pure <$> \u2191f v\n\ncase comp\nn : \u2115\nf : Vector \u2115 n \u2192. \u2115\nm\u271d n\u271d : \u2115\nf\u271d : Vector \u2115 n\u271d \u2192. \u2115\ng\u271d : Fin n\u271d \u2192 Vector \u2115 m\u271d \u2192. \u2115\na\u271d\u00b9 : Nat.Partrec' f\u271d\na\u271d : \u2200 (i : Fin n\u271d), Nat.Partrec' (g\u271d i)\na_ih\u271d\u00b9 : \u2203 c, \u2200 (v : Vector \u2115 n\u271d), eval c \u2191v = pure <$> f\u271d v\na_ih\u271d : \u2200 (i : Fin n\u271d), \u2203 c, \u2200 (v : Vector \u2115 m\u271d), eval c \u2191v = pure <$> g\u271d i v\n\u22a2 \u2203 c, \u2200 (v : Vector \u2115 m\u271d), eval c \u2191v = pure <$> (fun v => (Vector.mOfFn fun i => g\u271d i v) >>= f\u271d) v\n\ncase rfind\nn : \u2115\nf : Vector \u2115 n \u2192. \u2115\nn\u271d : \u2115\nf\u271d : Vector \u2115 (n\u271d + 1) \u2192 \u2115\na\u271d : Nat.Partrec' \u2191f\u271d\na_ih\u271d : \u2203 c, \u2200 (v : Vector \u2115 (n\u271d + 1)), eval c \u2191v = pure <$> \u2191f\u271d v\n\u22a2 \u2203 c, \u2200 (v : Vector \u2115 n\u271d), eval c \u2191v = pure <$> (fun v => Nat.rfind fun n => Part.some (decide (f\u271d (n ::\u1d65 v) = 0))) v", "state_after": "case prim.zero\nn\u271d : \u2115\nf\u271d : Vector \u2115 n\u271d \u2192. \u2115\nn : \u2115\nf : Vector \u2115 n \u2192 \u2115\n\u22a2 \u2203 c, \u2200 (v : Vector \u2115 0), eval c \u2191v = pure <$> (\u2191fun x => 0) v\n\ncase prim.succ\nn\u271d : \u2115\nf\u271d : Vector \u2115 n\u271d \u2192. \u2115\nn : \u2115\nf : Vector \u2115 n \u2192 \u2115\n\u22a2 \u2203 c, \u2200 (v : Vector \u2115 1), eval c \u2191v = pure <$> (\u2191fun v => Nat.succ (Vector.head v)) v\n\ncase prim.get\nn\u271d\u00b9 : \u2115\nf\u271d : Vector \u2115 n\u271d\u00b9 \u2192. \u2115\nn : \u2115\nf : Vector \u2115 n \u2192 \u2115\nn\u271d : \u2115\ni\u271d : Fin n\u271d\n\u22a2 \u2203 c, \u2200 (v : Vector \u2115 n\u271d), eval c \u2191v = pure <$> (\u2191fun v => Vector.get v i\u271d) v\n\ncase prim.comp\nn\u271d\u00b9 : \u2115\nf\u271d\u00b9 : Vector \u2115 n\u271d\u00b9 \u2192. \u2115\nn : \u2115\nf : Vector \u2115 n \u2192 \u2115\nm\u271d n\u271d : \u2115\nf\u271d : Vector \u2115 n\u271d \u2192 \u2115\ng\u271d : Fin n\u271d \u2192 Vector \u2115 m\u271d \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec' f\u271d\na\u271d : \u2200 (i : Fin n\u271d), Nat.Primrec' (g\u271d i)\na_ih\u271d\u00b9 : \u2203 c, \u2200 (v : Vector \u2115 n\u271d), eval c \u2191v = pure <$> \u2191f\u271d v\na_ih\u271d : \u2200 (i : Fin n\u271d), \u2203 c, \u2200 (v : Vector \u2115 m\u271d), eval c \u2191v = pure <$> \u2191(g\u271d i) v\n\u22a2 \u2203 c, \u2200 (v : Vector \u2115 m\u271d), eval c \u2191v = pure <$> (\u2191fun a => f\u271d (Vector.ofFn fun i => g\u271d i a)) v\n\ncase prim.prec\nn\u271d\u00b9 : \u2115\nf\u271d\u00b9 : Vector \u2115 n\u271d\u00b9 \u2192. \u2115\nn : \u2115\nf : Vector \u2115 n \u2192 \u2115\nn\u271d : \u2115\nf\u271d : Vector \u2115 n\u271d \u2192 \u2115\ng\u271d : Vector \u2115 (n\u271d + 2) \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec' f\u271d\na\u271d : Nat.Primrec' g\u271d\na_ih\u271d\u00b9 : \u2203 c, \u2200 (v : Vector \u2115 n\u271d), eval c \u2191v = pure <$> \u2191f\u271d v\na_ih\u271d : \u2203 c, \u2200 (v : Vector \u2115 (n\u271d + 2)), eval c \u2191v = pure <$> \u2191g\u271d v\n\u22a2 \u2203 c,\n    \u2200 (v : Vector \u2115 (n\u271d + 1)),\n      eval c \u2191v =\n        pure <$>\n          (\u2191fun v => Nat.rec (f\u271d (Vector.tail v)) (fun y IH => g\u271d (y ::\u1d65 IH ::\u1d65 Vector.tail v)) (Vector.head v)) v\n\ncase comp\nn : \u2115\nf : Vector \u2115 n \u2192. \u2115\nm\u271d n\u271d : \u2115\nf\u271d : Vector \u2115 n\u271d \u2192. \u2115\ng\u271d : Fin n\u271d \u2192 Vector \u2115 m\u271d \u2192. \u2115\na\u271d\u00b9 : Nat.Partrec' f\u271d\na\u271d : \u2200 (i : Fin n\u271d), Nat.Partrec' (g\u271d i)\na_ih\u271d\u00b9 : \u2203 c, \u2200 (v : Vector \u2115 n\u271d), eval c \u2191v = pure <$> f\u271d v\na_ih\u271d : \u2200 (i : Fin n\u271d), \u2203 c, \u2200 (v : Vector \u2115 m\u271d), eval c \u2191v = pure <$> g\u271d i v\n\u22a2 \u2203 c, \u2200 (v : Vector \u2115 m\u271d), eval c \u2191v = pure <$> (fun v => (Vector.mOfFn fun i => g\u271d i v) >>= f\u271d) v\n\ncase rfind\nn : \u2115\nf : Vector \u2115 n \u2192. \u2115\nn\u271d : \u2115\nf\u271d : Vector \u2115 (n\u271d + 1) \u2192 \u2115\na\u271d : Nat.Partrec' \u2191f\u271d\na_ih\u271d : \u2203 c, \u2200 (v : Vector \u2115 (n\u271d + 1)), eval c \u2191v = pure <$> \u2191f\u271d v\n\u22a2 \u2203 c, \u2200 (v : Vector \u2115 n\u271d), eval c \u2191v = pure <$> (fun v => Nat.rfind fun n => Part.some (decide (f\u271d (n ::\u1d65 v) = 0))) v"}, {"tactic": "case prim.zero => exact \u27e8zero', fun \u27e8[], _\u27e9 => rfl\u27e9", "annotated_tactic": ["case prim.zero => exact \u27e8<a>zero'</a>, fun \u27e8[], _\u27e9 => <a>rfl</a>\u27e9", [{"full_name": "Turing.ToPartrec.Code.zero'", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [76, 5], "def_end_pos": [76, 10]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "case prim.zero\nn\u271d : \u2115\nf\u271d : Vector \u2115 n\u271d \u2192. \u2115\nn : \u2115\nf : Vector \u2115 n \u2192 \u2115\n\u22a2 \u2203 c, \u2200 (v : Vector \u2115 0), eval c \u2191v = pure <$> (\u2191fun x => 0) v\n\ncase prim.succ\nn\u271d : \u2115\nf\u271d : Vector \u2115 n\u271d \u2192. \u2115\nn : \u2115\nf : Vector \u2115 n \u2192 \u2115\n\u22a2 \u2203 c, \u2200 (v : Vector \u2115 1), eval c \u2191v = pure <$> (\u2191fun v => Nat.succ (Vector.head v)) v\n\ncase prim.get\nn\u271d\u00b9 : \u2115\nf\u271d : Vector \u2115 n\u271d\u00b9 \u2192. \u2115\nn : \u2115\nf : Vector \u2115 n \u2192 \u2115\nn\u271d : \u2115\ni\u271d : Fin n\u271d\n\u22a2 \u2203 c, \u2200 (v : Vector \u2115 n\u271d), eval c \u2191v = pure <$> (\u2191fun v => Vector.get v i\u271d) v\n\ncase prim.comp\nn\u271d\u00b9 : \u2115\nf\u271d\u00b9 : Vector \u2115 n\u271d\u00b9 \u2192. \u2115\nn : \u2115\nf : Vector \u2115 n \u2192 \u2115\nm\u271d n\u271d : \u2115\nf\u271d : Vector \u2115 n\u271d \u2192 \u2115\ng\u271d : Fin n\u271d \u2192 Vector \u2115 m\u271d \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec' f\u271d\na\u271d : \u2200 (i : Fin n\u271d), Nat.Primrec' (g\u271d i)\na_ih\u271d\u00b9 : \u2203 c, \u2200 (v : Vector \u2115 n\u271d), eval c \u2191v = pure <$> \u2191f\u271d v\na_ih\u271d : \u2200 (i : Fin n\u271d), \u2203 c, \u2200 (v : Vector \u2115 m\u271d), eval c \u2191v = pure <$> \u2191(g\u271d i) v\n\u22a2 \u2203 c, \u2200 (v : Vector \u2115 m\u271d), eval c \u2191v = pure <$> (\u2191fun a => f\u271d (Vector.ofFn fun i => g\u271d i a)) v\n\ncase prim.prec\nn\u271d\u00b9 : \u2115\nf\u271d\u00b9 : Vector \u2115 n\u271d\u00b9 \u2192. \u2115\nn : \u2115\nf : Vector \u2115 n \u2192 \u2115\nn\u271d : \u2115\nf\u271d : Vector \u2115 n\u271d \u2192 \u2115\ng\u271d : Vector \u2115 (n\u271d + 2) \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec' f\u271d\na\u271d : Nat.Primrec' g\u271d\na_ih\u271d\u00b9 : \u2203 c, \u2200 (v : Vector \u2115 n\u271d), eval c \u2191v = pure <$> \u2191f\u271d v\na_ih\u271d : \u2203 c, \u2200 (v : Vector \u2115 (n\u271d + 2)), eval c \u2191v = pure <$> \u2191g\u271d v\n\u22a2 \u2203 c,\n    \u2200 (v : Vector \u2115 (n\u271d + 1)),\n      eval c \u2191v =\n        pure <$>\n          (\u2191fun v => Nat.rec (f\u271d (Vector.tail v)) (fun y IH => g\u271d (y ::\u1d65 IH ::\u1d65 Vector.tail v)) (Vector.head v)) v\n\ncase comp\nn : \u2115\nf : Vector \u2115 n \u2192. \u2115\nm\u271d n\u271d : \u2115\nf\u271d : Vector \u2115 n\u271d \u2192. \u2115\ng\u271d : Fin n\u271d \u2192 Vector \u2115 m\u271d \u2192. \u2115\na\u271d\u00b9 : Nat.Partrec' f\u271d\na\u271d : \u2200 (i : Fin n\u271d), Nat.Partrec' (g\u271d i)\na_ih\u271d\u00b9 : \u2203 c, \u2200 (v : Vector \u2115 n\u271d), eval c \u2191v = pure <$> f\u271d v\na_ih\u271d : \u2200 (i : Fin n\u271d), \u2203 c, \u2200 (v : Vector \u2115 m\u271d), eval c \u2191v = pure <$> g\u271d i v\n\u22a2 \u2203 c, \u2200 (v : Vector \u2115 m\u271d), eval c \u2191v = pure <$> (fun v => (Vector.mOfFn fun i => g\u271d i v) >>= f\u271d) v\n\ncase rfind\nn : \u2115\nf : Vector \u2115 n \u2192. \u2115\nn\u271d : \u2115\nf\u271d : Vector \u2115 (n\u271d + 1) \u2192 \u2115\na\u271d : Nat.Partrec' \u2191f\u271d\na_ih\u271d : \u2203 c, \u2200 (v : Vector \u2115 (n\u271d + 1)), eval c \u2191v = pure <$> \u2191f\u271d v\n\u22a2 \u2203 c, \u2200 (v : Vector \u2115 n\u271d), eval c \u2191v = pure <$> (fun v => Nat.rfind fun n => Part.some (decide (f\u271d (n ::\u1d65 v) = 0))) v", "state_after": "case prim.succ\nn\u271d : \u2115\nf\u271d : Vector \u2115 n\u271d \u2192. \u2115\nn : \u2115\nf : Vector \u2115 n \u2192 \u2115\n\u22a2 \u2203 c, \u2200 (v : Vector \u2115 1), eval c \u2191v = pure <$> (\u2191fun v => Nat.succ (Vector.head v)) v\n\ncase prim.get\nn\u271d\u00b9 : \u2115\nf\u271d : Vector \u2115 n\u271d\u00b9 \u2192. \u2115\nn : \u2115\nf : Vector \u2115 n \u2192 \u2115\nn\u271d : \u2115\ni\u271d : Fin n\u271d\n\u22a2 \u2203 c, \u2200 (v : Vector \u2115 n\u271d), eval c \u2191v = pure <$> (\u2191fun v => Vector.get v i\u271d) v\n\ncase prim.comp\nn\u271d\u00b9 : \u2115\nf\u271d\u00b9 : Vector \u2115 n\u271d\u00b9 \u2192. \u2115\nn : \u2115\nf : Vector \u2115 n \u2192 \u2115\nm\u271d n\u271d : \u2115\nf\u271d : Vector \u2115 n\u271d \u2192 \u2115\ng\u271d : Fin n\u271d \u2192 Vector \u2115 m\u271d \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec' f\u271d\na\u271d : \u2200 (i : Fin n\u271d), Nat.Primrec' (g\u271d i)\na_ih\u271d\u00b9 : \u2203 c, \u2200 (v : Vector \u2115 n\u271d), eval c \u2191v = pure <$> \u2191f\u271d v\na_ih\u271d : \u2200 (i : Fin n\u271d), \u2203 c, \u2200 (v : Vector \u2115 m\u271d), eval c \u2191v = pure <$> \u2191(g\u271d i) v\n\u22a2 \u2203 c, \u2200 (v : Vector \u2115 m\u271d), eval c \u2191v = pure <$> (\u2191fun a => f\u271d (Vector.ofFn fun i => g\u271d i a)) v\n\ncase prim.prec\nn\u271d\u00b9 : \u2115\nf\u271d\u00b9 : Vector \u2115 n\u271d\u00b9 \u2192. \u2115\nn : \u2115\nf : Vector \u2115 n \u2192 \u2115\nn\u271d : \u2115\nf\u271d : Vector \u2115 n\u271d \u2192 \u2115\ng\u271d : Vector \u2115 (n\u271d + 2) \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec' f\u271d\na\u271d : Nat.Primrec' g\u271d\na_ih\u271d\u00b9 : \u2203 c, \u2200 (v : Vector \u2115 n\u271d), eval c \u2191v = pure <$> \u2191f\u271d v\na_ih\u271d : \u2203 c, \u2200 (v : Vector \u2115 (n\u271d + 2)), eval c \u2191v = pure <$> \u2191g\u271d v\n\u22a2 \u2203 c,\n    \u2200 (v : Vector \u2115 (n\u271d + 1)),\n      eval c \u2191v =\n        pure <$>\n          (\u2191fun v => Nat.rec (f\u271d (Vector.tail v)) (fun y IH => g\u271d (y ::\u1d65 IH ::\u1d65 Vector.tail v)) (Vector.head v)) v\n\ncase comp\nn : \u2115\nf : Vector \u2115 n \u2192. \u2115\nm\u271d n\u271d : \u2115\nf\u271d : Vector \u2115 n\u271d \u2192. \u2115\ng\u271d : Fin n\u271d \u2192 Vector \u2115 m\u271d \u2192. \u2115\na\u271d\u00b9 : Nat.Partrec' f\u271d\na\u271d : \u2200 (i : Fin n\u271d), Nat.Partrec' (g\u271d i)\na_ih\u271d\u00b9 : \u2203 c, \u2200 (v : Vector \u2115 n\u271d), eval c \u2191v = pure <$> f\u271d v\na_ih\u271d : \u2200 (i : Fin n\u271d), \u2203 c, \u2200 (v : Vector \u2115 m\u271d), eval c \u2191v = pure <$> g\u271d i v\n\u22a2 \u2203 c, \u2200 (v : Vector \u2115 m\u271d), eval c \u2191v = pure <$> (fun v => (Vector.mOfFn fun i => g\u271d i v) >>= f\u271d) v\n\ncase rfind\nn : \u2115\nf : Vector \u2115 n \u2192. \u2115\nn\u271d : \u2115\nf\u271d : Vector \u2115 (n\u271d + 1) \u2192 \u2115\na\u271d : Nat.Partrec' \u2191f\u271d\na_ih\u271d : \u2203 c, \u2200 (v : Vector \u2115 (n\u271d + 1)), eval c \u2191v = pure <$> \u2191f\u271d v\n\u22a2 \u2203 c, \u2200 (v : Vector \u2115 n\u271d), eval c \u2191v = pure <$> (fun v => Nat.rfind fun n => Part.some (decide (f\u271d (n ::\u1d65 v) = 0))) v"}, {"tactic": "case prim.succ => exact \u27e8succ, fun \u27e8[v], _\u27e9 => rfl\u27e9", "annotated_tactic": ["case prim.succ => exact \u27e8<a>succ</a>, fun \u27e8[v], _\u27e9 => <a>rfl</a>\u27e9", [{"full_name": "Turing.ToPartrec.Code.succ", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [77, 5], "def_end_pos": [77, 9]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "case prim.succ\nn\u271d : \u2115\nf\u271d : Vector \u2115 n\u271d \u2192. \u2115\nn : \u2115\nf : Vector \u2115 n \u2192 \u2115\n\u22a2 \u2203 c, \u2200 (v : Vector \u2115 1), eval c \u2191v = pure <$> (\u2191fun v => Nat.succ (Vector.head v)) v\n\ncase prim.get\nn\u271d\u00b9 : \u2115\nf\u271d : Vector \u2115 n\u271d\u00b9 \u2192. \u2115\nn : \u2115\nf : Vector \u2115 n \u2192 \u2115\nn\u271d : \u2115\ni\u271d : Fin n\u271d\n\u22a2 \u2203 c, \u2200 (v : Vector \u2115 n\u271d), eval c \u2191v = pure <$> (\u2191fun v => Vector.get v i\u271d) v\n\ncase prim.comp\nn\u271d\u00b9 : \u2115\nf\u271d\u00b9 : Vector \u2115 n\u271d\u00b9 \u2192. \u2115\nn : \u2115\nf : Vector \u2115 n \u2192 \u2115\nm\u271d n\u271d : \u2115\nf\u271d : Vector \u2115 n\u271d \u2192 \u2115\ng\u271d : Fin n\u271d \u2192 Vector \u2115 m\u271d \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec' f\u271d\na\u271d : \u2200 (i : Fin n\u271d), Nat.Primrec' (g\u271d i)\na_ih\u271d\u00b9 : \u2203 c, \u2200 (v : Vector \u2115 n\u271d), eval c \u2191v = pure <$> \u2191f\u271d v\na_ih\u271d : \u2200 (i : Fin n\u271d), \u2203 c, \u2200 (v : Vector \u2115 m\u271d), eval c \u2191v = pure <$> \u2191(g\u271d i) v\n\u22a2 \u2203 c, \u2200 (v : Vector \u2115 m\u271d), eval c \u2191v = pure <$> (\u2191fun a => f\u271d (Vector.ofFn fun i => g\u271d i a)) v\n\ncase prim.prec\nn\u271d\u00b9 : \u2115\nf\u271d\u00b9 : Vector \u2115 n\u271d\u00b9 \u2192. \u2115\nn : \u2115\nf : Vector \u2115 n \u2192 \u2115\nn\u271d : \u2115\nf\u271d : Vector \u2115 n\u271d \u2192 \u2115\ng\u271d : Vector \u2115 (n\u271d + 2) \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec' f\u271d\na\u271d : Nat.Primrec' g\u271d\na_ih\u271d\u00b9 : \u2203 c, \u2200 (v : Vector \u2115 n\u271d), eval c \u2191v = pure <$> \u2191f\u271d v\na_ih\u271d : \u2203 c, \u2200 (v : Vector \u2115 (n\u271d + 2)), eval c \u2191v = pure <$> \u2191g\u271d v\n\u22a2 \u2203 c,\n    \u2200 (v : Vector \u2115 (n\u271d + 1)),\n      eval c \u2191v =\n        pure <$>\n          (\u2191fun v => Nat.rec (f\u271d (Vector.tail v)) (fun y IH => g\u271d (y ::\u1d65 IH ::\u1d65 Vector.tail v)) (Vector.head v)) v\n\ncase comp\nn : \u2115\nf : Vector \u2115 n \u2192. \u2115\nm\u271d n\u271d : \u2115\nf\u271d : Vector \u2115 n\u271d \u2192. \u2115\ng\u271d : Fin n\u271d \u2192 Vector \u2115 m\u271d \u2192. \u2115\na\u271d\u00b9 : Nat.Partrec' f\u271d\na\u271d : \u2200 (i : Fin n\u271d), Nat.Partrec' (g\u271d i)\na_ih\u271d\u00b9 : \u2203 c, \u2200 (v : Vector \u2115 n\u271d), eval c \u2191v = pure <$> f\u271d v\na_ih\u271d : \u2200 (i : Fin n\u271d), \u2203 c, \u2200 (v : Vector \u2115 m\u271d), eval c \u2191v = pure <$> g\u271d i v\n\u22a2 \u2203 c, \u2200 (v : Vector \u2115 m\u271d), eval c \u2191v = pure <$> (fun v => (Vector.mOfFn fun i => g\u271d i v) >>= f\u271d) v\n\ncase rfind\nn : \u2115\nf : Vector \u2115 n \u2192. \u2115\nn\u271d : \u2115\nf\u271d : Vector \u2115 (n\u271d + 1) \u2192 \u2115\na\u271d : Nat.Partrec' \u2191f\u271d\na_ih\u271d : \u2203 c, \u2200 (v : Vector \u2115 (n\u271d + 1)), eval c \u2191v = pure <$> \u2191f\u271d v\n\u22a2 \u2203 c, \u2200 (v : Vector \u2115 n\u271d), eval c \u2191v = pure <$> (fun v => Nat.rfind fun n => Part.some (decide (f\u271d (n ::\u1d65 v) = 0))) v", "state_after": "case prim.get\nn\u271d\u00b9 : \u2115\nf\u271d : Vector \u2115 n\u271d\u00b9 \u2192. \u2115\nn : \u2115\nf : Vector \u2115 n \u2192 \u2115\nn\u271d : \u2115\ni\u271d : Fin n\u271d\n\u22a2 \u2203 c, \u2200 (v : Vector \u2115 n\u271d), eval c \u2191v = pure <$> (\u2191fun v => Vector.get v i\u271d) v\n\ncase prim.comp\nn\u271d\u00b9 : \u2115\nf\u271d\u00b9 : Vector \u2115 n\u271d\u00b9 \u2192. \u2115\nn : \u2115\nf : Vector \u2115 n \u2192 \u2115\nm\u271d n\u271d : \u2115\nf\u271d : Vector \u2115 n\u271d \u2192 \u2115\ng\u271d : Fin n\u271d \u2192 Vector \u2115 m\u271d \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec' f\u271d\na\u271d : \u2200 (i : Fin n\u271d), Nat.Primrec' (g\u271d i)\na_ih\u271d\u00b9 : \u2203 c, \u2200 (v : Vector \u2115 n\u271d), eval c \u2191v = pure <$> \u2191f\u271d v\na_ih\u271d : \u2200 (i : Fin n\u271d), \u2203 c, \u2200 (v : Vector \u2115 m\u271d), eval c \u2191v = pure <$> \u2191(g\u271d i) v\n\u22a2 \u2203 c, \u2200 (v : Vector \u2115 m\u271d), eval c \u2191v = pure <$> (\u2191fun a => f\u271d (Vector.ofFn fun i => g\u271d i a)) v\n\ncase prim.prec\nn\u271d\u00b9 : \u2115\nf\u271d\u00b9 : Vector \u2115 n\u271d\u00b9 \u2192. \u2115\nn : \u2115\nf : Vector \u2115 n \u2192 \u2115\nn\u271d : \u2115\nf\u271d : Vector \u2115 n\u271d \u2192 \u2115\ng\u271d : Vector \u2115 (n\u271d + 2) \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec' f\u271d\na\u271d : Nat.Primrec' g\u271d\na_ih\u271d\u00b9 : \u2203 c, \u2200 (v : Vector \u2115 n\u271d), eval c \u2191v = pure <$> \u2191f\u271d v\na_ih\u271d : \u2203 c, \u2200 (v : Vector \u2115 (n\u271d + 2)), eval c \u2191v = pure <$> \u2191g\u271d v\n\u22a2 \u2203 c,\n    \u2200 (v : Vector \u2115 (n\u271d + 1)),\n      eval c \u2191v =\n        pure <$>\n          (\u2191fun v => Nat.rec (f\u271d (Vector.tail v)) (fun y IH => g\u271d (y ::\u1d65 IH ::\u1d65 Vector.tail v)) (Vector.head v)) v\n\ncase comp\nn : \u2115\nf : Vector \u2115 n \u2192. \u2115\nm\u271d n\u271d : \u2115\nf\u271d : Vector \u2115 n\u271d \u2192. \u2115\ng\u271d : Fin n\u271d \u2192 Vector \u2115 m\u271d \u2192. \u2115\na\u271d\u00b9 : Nat.Partrec' f\u271d\na\u271d : \u2200 (i : Fin n\u271d), Nat.Partrec' (g\u271d i)\na_ih\u271d\u00b9 : \u2203 c, \u2200 (v : Vector \u2115 n\u271d), eval c \u2191v = pure <$> f\u271d v\na_ih\u271d : \u2200 (i : Fin n\u271d), \u2203 c, \u2200 (v : Vector \u2115 m\u271d), eval c \u2191v = pure <$> g\u271d i v\n\u22a2 \u2203 c, \u2200 (v : Vector \u2115 m\u271d), eval c \u2191v = pure <$> (fun v => (Vector.mOfFn fun i => g\u271d i v) >>= f\u271d) v\n\ncase rfind\nn : \u2115\nf : Vector \u2115 n \u2192. \u2115\nn\u271d : \u2115\nf\u271d : Vector \u2115 (n\u271d + 1) \u2192 \u2115\na\u271d : Nat.Partrec' \u2191f\u271d\na_ih\u271d : \u2203 c, \u2200 (v : Vector \u2115 (n\u271d + 1)), eval c \u2191v = pure <$> \u2191f\u271d v\n\u22a2 \u2203 c, \u2200 (v : Vector \u2115 n\u271d), eval c \u2191v = pure <$> (fun v => Nat.rfind fun n => Part.some (decide (f\u271d (n ::\u1d65 v) = 0))) v"}, {"tactic": "case prim.comp m n f g hf hg IHf IHg =>\n  simpa [Part.bind_eq_bind] using exists_code.comp IHf IHg", "annotated_tactic": ["case prim.comp m n f g hf hg IHf IHg =>\n    simpa [<a>Part.bind_eq_bind</a>] using <a>exists_code.comp</a> IHf IHg", [{"full_name": "Part.bind_eq_bind", "def_path": "Mathlib/Data/Part.lean", "def_pos": [614, 9], "def_end_pos": [614, 21]}, {"full_name": "Turing.ToPartrec.Code.exists_code.comp", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [264, 9], "def_end_pos": [264, 25]}]], "state_before": "case prim.comp\nn\u271d\u00b9 : \u2115\nf\u271d\u00b9 : Vector \u2115 n\u271d\u00b9 \u2192. \u2115\nn : \u2115\nf : Vector \u2115 n \u2192 \u2115\nm\u271d n\u271d : \u2115\nf\u271d : Vector \u2115 n\u271d \u2192 \u2115\ng\u271d : Fin n\u271d \u2192 Vector \u2115 m\u271d \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec' f\u271d\na\u271d : \u2200 (i : Fin n\u271d), Nat.Primrec' (g\u271d i)\na_ih\u271d\u00b9 : \u2203 c, \u2200 (v : Vector \u2115 n\u271d), eval c \u2191v = pure <$> \u2191f\u271d v\na_ih\u271d : \u2200 (i : Fin n\u271d), \u2203 c, \u2200 (v : Vector \u2115 m\u271d), eval c \u2191v = pure <$> \u2191(g\u271d i) v\n\u22a2 \u2203 c, \u2200 (v : Vector \u2115 m\u271d), eval c \u2191v = pure <$> (\u2191fun a => f\u271d (Vector.ofFn fun i => g\u271d i a)) v\n\ncase prim.prec\nn\u271d\u00b9 : \u2115\nf\u271d\u00b9 : Vector \u2115 n\u271d\u00b9 \u2192. \u2115\nn : \u2115\nf : Vector \u2115 n \u2192 \u2115\nn\u271d : \u2115\nf\u271d : Vector \u2115 n\u271d \u2192 \u2115\ng\u271d : Vector \u2115 (n\u271d + 2) \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec' f\u271d\na\u271d : Nat.Primrec' g\u271d\na_ih\u271d\u00b9 : \u2203 c, \u2200 (v : Vector \u2115 n\u271d), eval c \u2191v = pure <$> \u2191f\u271d v\na_ih\u271d : \u2203 c, \u2200 (v : Vector \u2115 (n\u271d + 2)), eval c \u2191v = pure <$> \u2191g\u271d v\n\u22a2 \u2203 c,\n    \u2200 (v : Vector \u2115 (n\u271d + 1)),\n      eval c \u2191v =\n        pure <$>\n          (\u2191fun v => Nat.rec (f\u271d (Vector.tail v)) (fun y IH => g\u271d (y ::\u1d65 IH ::\u1d65 Vector.tail v)) (Vector.head v)) v\n\ncase comp\nn : \u2115\nf : Vector \u2115 n \u2192. \u2115\nm\u271d n\u271d : \u2115\nf\u271d : Vector \u2115 n\u271d \u2192. \u2115\ng\u271d : Fin n\u271d \u2192 Vector \u2115 m\u271d \u2192. \u2115\na\u271d\u00b9 : Nat.Partrec' f\u271d\na\u271d : \u2200 (i : Fin n\u271d), Nat.Partrec' (g\u271d i)\na_ih\u271d\u00b9 : \u2203 c, \u2200 (v : Vector \u2115 n\u271d), eval c \u2191v = pure <$> f\u271d v\na_ih\u271d : \u2200 (i : Fin n\u271d), \u2203 c, \u2200 (v : Vector \u2115 m\u271d), eval c \u2191v = pure <$> g\u271d i v\n\u22a2 \u2203 c, \u2200 (v : Vector \u2115 m\u271d), eval c \u2191v = pure <$> (fun v => (Vector.mOfFn fun i => g\u271d i v) >>= f\u271d) v\n\ncase rfind\nn : \u2115\nf : Vector \u2115 n \u2192. \u2115\nn\u271d : \u2115\nf\u271d : Vector \u2115 (n\u271d + 1) \u2192 \u2115\na\u271d : Nat.Partrec' \u2191f\u271d\na_ih\u271d : \u2203 c, \u2200 (v : Vector \u2115 (n\u271d + 1)), eval c \u2191v = pure <$> \u2191f\u271d v\n\u22a2 \u2203 c, \u2200 (v : Vector \u2115 n\u271d), eval c \u2191v = pure <$> (fun v => Nat.rfind fun n => Part.some (decide (f\u271d (n ::\u1d65 v) = 0))) v", "state_after": "case prim.prec\nn\u271d\u00b9 : \u2115\nf\u271d\u00b9 : Vector \u2115 n\u271d\u00b9 \u2192. \u2115\nn : \u2115\nf : Vector \u2115 n \u2192 \u2115\nn\u271d : \u2115\nf\u271d : Vector \u2115 n\u271d \u2192 \u2115\ng\u271d : Vector \u2115 (n\u271d + 2) \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec' f\u271d\na\u271d : Nat.Primrec' g\u271d\na_ih\u271d\u00b9 : \u2203 c, \u2200 (v : Vector \u2115 n\u271d), eval c \u2191v = pure <$> \u2191f\u271d v\na_ih\u271d : \u2203 c, \u2200 (v : Vector \u2115 (n\u271d + 2)), eval c \u2191v = pure <$> \u2191g\u271d v\n\u22a2 \u2203 c,\n    \u2200 (v : Vector \u2115 (n\u271d + 1)),\n      eval c \u2191v =\n        pure <$>\n          (\u2191fun v => Nat.rec (f\u271d (Vector.tail v)) (fun y IH => g\u271d (y ::\u1d65 IH ::\u1d65 Vector.tail v)) (Vector.head v)) v\n\ncase comp\nn : \u2115\nf : Vector \u2115 n \u2192. \u2115\nm\u271d n\u271d : \u2115\nf\u271d : Vector \u2115 n\u271d \u2192. \u2115\ng\u271d : Fin n\u271d \u2192 Vector \u2115 m\u271d \u2192. \u2115\na\u271d\u00b9 : Nat.Partrec' f\u271d\na\u271d : \u2200 (i : Fin n\u271d), Nat.Partrec' (g\u271d i)\na_ih\u271d\u00b9 : \u2203 c, \u2200 (v : Vector \u2115 n\u271d), eval c \u2191v = pure <$> f\u271d v\na_ih\u271d : \u2200 (i : Fin n\u271d), \u2203 c, \u2200 (v : Vector \u2115 m\u271d), eval c \u2191v = pure <$> g\u271d i v\n\u22a2 \u2203 c, \u2200 (v : Vector \u2115 m\u271d), eval c \u2191v = pure <$> (fun v => (Vector.mOfFn fun i => g\u271d i v) >>= f\u271d) v\n\ncase rfind\nn : \u2115\nf : Vector \u2115 n \u2192. \u2115\nn\u271d : \u2115\nf\u271d : Vector \u2115 (n\u271d + 1) \u2192 \u2115\na\u271d : Nat.Partrec' \u2191f\u271d\na_ih\u271d : \u2203 c, \u2200 (v : Vector \u2115 (n\u271d + 1)), eval c \u2191v = pure <$> \u2191f\u271d v\n\u22a2 \u2203 c, \u2200 (v : Vector \u2115 n\u271d), eval c \u2191v = pure <$> (fun v => Nat.rfind fun n => Part.some (decide (f\u271d (n ::\u1d65 v) = 0))) v"}, {"tactic": "case comp m n f g _ _ IHf IHg => exact exists_code.comp IHf IHg", "annotated_tactic": ["case comp m n f g _ _ IHf IHg => exact <a>exists_code.comp</a> IHf IHg", [{"full_name": "Turing.ToPartrec.Code.exists_code.comp", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [264, 9], "def_end_pos": [264, 25]}]], "state_before": "case comp\nn : \u2115\nf : Vector \u2115 n \u2192. \u2115\nm\u271d n\u271d : \u2115\nf\u271d : Vector \u2115 n\u271d \u2192. \u2115\ng\u271d : Fin n\u271d \u2192 Vector \u2115 m\u271d \u2192. \u2115\na\u271d\u00b9 : Nat.Partrec' f\u271d\na\u271d : \u2200 (i : Fin n\u271d), Nat.Partrec' (g\u271d i)\na_ih\u271d\u00b9 : \u2203 c, \u2200 (v : Vector \u2115 n\u271d), eval c \u2191v = pure <$> f\u271d v\na_ih\u271d : \u2200 (i : Fin n\u271d), \u2203 c, \u2200 (v : Vector \u2115 m\u271d), eval c \u2191v = pure <$> g\u271d i v\n\u22a2 \u2203 c, \u2200 (v : Vector \u2115 m\u271d), eval c \u2191v = pure <$> (fun v => (Vector.mOfFn fun i => g\u271d i v) >>= f\u271d) v\n\ncase rfind\nn : \u2115\nf : Vector \u2115 n \u2192. \u2115\nn\u271d : \u2115\nf\u271d : Vector \u2115 (n\u271d + 1) \u2192 \u2115\na\u271d : Nat.Partrec' \u2191f\u271d\na_ih\u271d : \u2203 c, \u2200 (v : Vector \u2115 (n\u271d + 1)), eval c \u2191v = pure <$> \u2191f\u271d v\n\u22a2 \u2203 c, \u2200 (v : Vector \u2115 n\u271d), eval c \u2191v = pure <$> (fun v => Nat.rfind fun n => Part.some (decide (f\u271d (n ::\u1d65 v) = 0))) v", "state_after": "case rfind\nn : \u2115\nf : Vector \u2115 n \u2192. \u2115\nn\u271d : \u2115\nf\u271d : Vector \u2115 (n\u271d + 1) \u2192 \u2115\na\u271d : Nat.Partrec' \u2191f\u271d\na_ih\u271d : \u2203 c, \u2200 (v : Vector \u2115 (n\u271d + 1)), eval c \u2191v = pure <$> \u2191f\u271d v\n\u22a2 \u2203 c, \u2200 (v : Vector \u2115 n\u271d), eval c \u2191v = pure <$> (fun v => Nat.rfind fun n => Part.some (decide (f\u271d (n ::\u1d65 v) = 0))) v"}, {"tactic": "exact \u27e8zero', fun \u27e8[], _\u27e9 => rfl\u27e9", "annotated_tactic": ["exact \u27e8<a>zero'</a>, fun \u27e8[], _\u27e9 => <a>rfl</a>\u27e9", [{"full_name": "Turing.ToPartrec.Code.zero'", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [76, 5], "def_end_pos": [76, 10]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "n\u271d : \u2115\nf\u271d : Vector \u2115 n\u271d \u2192. \u2115\nn : \u2115\nf : Vector \u2115 n \u2192 \u2115\n\u22a2 \u2203 c, \u2200 (v : Vector \u2115 0), eval c \u2191v = pure <$> (\u2191fun x => 0) v", "state_after": "no goals"}, {"tactic": "exact \u27e8succ, fun \u27e8[v], _\u27e9 => rfl\u27e9", "annotated_tactic": ["exact \u27e8<a>succ</a>, fun \u27e8[v], _\u27e9 => <a>rfl</a>\u27e9", [{"full_name": "Turing.ToPartrec.Code.succ", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [77, 5], "def_end_pos": [77, 9]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "n\u271d : \u2115\nf\u271d : Vector \u2115 n\u271d \u2192. \u2115\nn : \u2115\nf : Vector \u2115 n \u2192 \u2115\n\u22a2 \u2203 c, \u2200 (v : Vector \u2115 1), eval c \u2191v = pure <$> (\u2191fun v => Nat.succ (Vector.head v)) v", "state_after": "no goals"}, {"tactic": "refine' Fin.succRec (fun n => _) (fun n i IH => _) i", "annotated_tactic": ["refine' <a>Fin.succRec</a> (fun n => _) (fun n i IH => _) i", [{"full_name": "Fin.succRec", "def_path": "lake-packages/std/Std/Data/Fin/Lemmas.lean", "def_pos": [555, 21], "def_end_pos": [555, 28]}]], "state_before": "n\u271d\u00b9 : \u2115\nf\u271d : Vector \u2115 n\u271d\u00b9 \u2192. \u2115\nn\u271d : \u2115\nf : Vector \u2115 n\u271d \u2192 \u2115\nn : \u2115\ni : Fin n\n\u22a2 \u2203 c, \u2200 (v : Vector \u2115 n), eval c \u2191v = pure <$> (\u2191fun v => Vector.get v i) v", "state_after": "case refine'_1\nn\u271d\u00b2 : \u2115\nf\u271d : Vector \u2115 n\u271d\u00b2 \u2192. \u2115\nn\u271d\u00b9 : \u2115\nf : Vector \u2115 n\u271d\u00b9 \u2192 \u2115\nn\u271d : \u2115\ni : Fin n\u271d\nn : \u2115\n\u22a2 \u2203 c, \u2200 (v : Vector \u2115 (Nat.succ n)), eval c \u2191v = pure <$> (\u2191fun v => Vector.get v 0) v\n\ncase refine'_2\nn\u271d\u00b2 : \u2115\nf\u271d : Vector \u2115 n\u271d\u00b2 \u2192. \u2115\nn\u271d\u00b9 : \u2115\nf : Vector \u2115 n\u271d\u00b9 \u2192 \u2115\nn\u271d : \u2115\ni\u271d : Fin n\u271d\nn : \u2115\ni : Fin n\nIH : \u2203 c, \u2200 (v : Vector \u2115 n), eval c \u2191v = pure <$> (\u2191fun v => Vector.get v i) v\n\u22a2 \u2203 c, \u2200 (v : Vector \u2115 (Nat.succ n)), eval c \u2191v = pure <$> (\u2191fun v => Vector.get v (Fin.succ i)) v"}, {"tactic": "exact \u27e8head, fun \u27e8List.cons a as, _\u27e9 => by simp [Bind.bind]; rfl\u27e9", "annotated_tactic": ["exact \u27e8<a>head</a>, fun \u27e8<a>List.cons</a> a as, _\u27e9 => by simp [<a>Bind.bind</a>]; rfl\u27e9", [{"full_name": "Turing.ToPartrec.Code.head", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [187, 5], "def_end_pos": [187, 9]}, {"full_name": "List.cons", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2187, 5], "def_end_pos": [2187, 9]}, {"full_name": "Bind.bind", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2662, 3], "def_end_pos": [2662, 7]}]], "state_before": "case refine'_1\nn\u271d\u00b2 : \u2115\nf\u271d : Vector \u2115 n\u271d\u00b2 \u2192. \u2115\nn\u271d\u00b9 : \u2115\nf : Vector \u2115 n\u271d\u00b9 \u2192 \u2115\nn\u271d : \u2115\ni : Fin n\u271d\nn : \u2115\n\u22a2 \u2203 c, \u2200 (v : Vector \u2115 (Nat.succ n)), eval c \u2191v = pure <$> (\u2191fun v => Vector.get v 0) v", "state_after": "no goals"}, {"tactic": "simp [Bind.bind]", "annotated_tactic": ["simp [<a>Bind.bind</a>]", [{"full_name": "Bind.bind", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2662, 3], "def_end_pos": [2662, 7]}]], "state_before": "n\u271d\u00b2 : \u2115\nf\u271d : Vector \u2115 n\u271d\u00b2 \u2192. \u2115\nn\u271d\u00b9 : \u2115\nf : Vector \u2115 n\u271d\u00b9 \u2192 \u2115\nn\u271d : \u2115\ni : Fin n\u271d\nn : \u2115\nx\u271d : Vector \u2115 (Nat.succ n)\na : \u2115\nas : List \u2115\nproperty\u271d : List.length (a :: as) = Nat.succ n\n\u22a2 eval head \u2191{ val := a :: as, property := property\u271d } =\n    pure <$> (\u2191fun v => Vector.get v 0) { val := a :: as, property := property\u271d }", "state_after": "n\u271d\u00b2 : \u2115\nf\u271d : Vector \u2115 n\u271d\u00b2 \u2192. \u2115\nn\u271d\u00b9 : \u2115\nf : Vector \u2115 n\u271d\u00b9 \u2192 \u2115\nn\u271d : \u2115\ni : Fin n\u271d\nn : \u2115\nx\u271d : Vector \u2115 (Nat.succ n)\na : \u2115\nas : List \u2115\nproperty\u271d : List.length (a :: as) = Nat.succ n\n\u22a2 Part.some [a] = pure <$> Part.some (Vector.head { val := a :: as, property := property\u271d })"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "n\u271d\u00b2 : \u2115\nf\u271d : Vector \u2115 n\u271d\u00b2 \u2192. \u2115\nn\u271d\u00b9 : \u2115\nf : Vector \u2115 n\u271d\u00b9 \u2192 \u2115\nn\u271d : \u2115\ni : Fin n\u271d\nn : \u2115\nx\u271d : Vector \u2115 (Nat.succ n)\na : \u2115\nas : List \u2115\nproperty\u271d : List.length (a :: as) = Nat.succ n\n\u22a2 Part.some [a] = pure <$> Part.some (Vector.head { val := a :: as, property := property\u271d })", "state_after": "no goals"}, {"tactic": "obtain \u27e8c, h\u27e9 := IH", "annotated_tactic": ["obtain \u27e8c, h\u27e9 := IH", []], "state_before": "case refine'_2\nn\u271d\u00b2 : \u2115\nf\u271d : Vector \u2115 n\u271d\u00b2 \u2192. \u2115\nn\u271d\u00b9 : \u2115\nf : Vector \u2115 n\u271d\u00b9 \u2192 \u2115\nn\u271d : \u2115\ni\u271d : Fin n\u271d\nn : \u2115\ni : Fin n\nIH : \u2203 c, \u2200 (v : Vector \u2115 n), eval c \u2191v = pure <$> (\u2191fun v => Vector.get v i) v\n\u22a2 \u2203 c, \u2200 (v : Vector \u2115 (Nat.succ n)), eval c \u2191v = pure <$> (\u2191fun v => Vector.get v (Fin.succ i)) v", "state_after": "case refine'_2.intro\nn\u271d\u00b2 : \u2115\nf\u271d : Vector \u2115 n\u271d\u00b2 \u2192. \u2115\nn\u271d\u00b9 : \u2115\nf : Vector \u2115 n\u271d\u00b9 \u2192 \u2115\nn\u271d : \u2115\ni\u271d : Fin n\u271d\nn : \u2115\ni : Fin n\nc : Code\nh : \u2200 (v : Vector \u2115 n), eval c \u2191v = pure <$> (\u2191fun v => Vector.get v i) v\n\u22a2 \u2203 c, \u2200 (v : Vector \u2115 (Nat.succ n)), eval c \u2191v = pure <$> (\u2191fun v => Vector.get v (Fin.succ i)) v"}, {"tactic": "exact \u27e8c.comp tail, fun v => by simpa [\u2190 Vector.get_tail, Bind.bind] using h v.tail\u27e9", "annotated_tactic": ["exact \u27e8c.comp <a>tail</a>, fun v => by simpa [\u2190 <a>Vector.get_tail</a>, <a>Bind.bind</a>] using h v.tail\u27e9", [{"full_name": "Turing.ToPartrec.Code.tail", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [78, 5], "def_end_pos": [78, 9]}, {"full_name": "Vector.get_tail", "def_path": "Mathlib/Data/Vector/Basic.lean", "def_pos": [167, 9], "def_end_pos": [167, 17]}, {"full_name": "Bind.bind", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2662, 3], "def_end_pos": [2662, 7]}]], "state_before": "case refine'_2.intro\nn\u271d\u00b2 : \u2115\nf\u271d : Vector \u2115 n\u271d\u00b2 \u2192. \u2115\nn\u271d\u00b9 : \u2115\nf : Vector \u2115 n\u271d\u00b9 \u2192 \u2115\nn\u271d : \u2115\ni\u271d : Fin n\u271d\nn : \u2115\ni : Fin n\nc : Code\nh : \u2200 (v : Vector \u2115 n), eval c \u2191v = pure <$> (\u2191fun v => Vector.get v i) v\n\u22a2 \u2203 c, \u2200 (v : Vector \u2115 (Nat.succ n)), eval c \u2191v = pure <$> (\u2191fun v => Vector.get v (Fin.succ i)) v", "state_after": "no goals"}, {"tactic": "simpa [\u2190 Vector.get_tail, Bind.bind] using h v.tail", "annotated_tactic": ["simpa [\u2190 <a>Vector.get_tail</a>, <a>Bind.bind</a>] using h v.tail", [{"full_name": "Vector.get_tail", "def_path": "Mathlib/Data/Vector/Basic.lean", "def_pos": [167, 9], "def_end_pos": [167, 17]}, {"full_name": "Bind.bind", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2662, 3], "def_end_pos": [2662, 7]}]], "state_before": "n\u271d\u00b2 : \u2115\nf\u271d : Vector \u2115 n\u271d\u00b2 \u2192. \u2115\nn\u271d\u00b9 : \u2115\nf : Vector \u2115 n\u271d\u00b9 \u2192 \u2115\nn\u271d : \u2115\ni\u271d : Fin n\u271d\nn : \u2115\ni : Fin n\nc : Code\nh : \u2200 (v : Vector \u2115 n), eval c \u2191v = pure <$> (\u2191fun v => Vector.get v i) v\nv : Vector \u2115 (Nat.succ n)\n\u22a2 eval (comp c tail) \u2191v = pure <$> (\u2191fun v => Vector.get v (Fin.succ i)) v", "state_after": "no goals"}, {"tactic": "simpa [Part.bind_eq_bind] using exists_code.comp IHf IHg", "annotated_tactic": ["simpa [<a>Part.bind_eq_bind</a>] using <a>exists_code.comp</a> IHf IHg", [{"full_name": "Part.bind_eq_bind", "def_path": "Mathlib/Data/Part.lean", "def_pos": [614, 9], "def_end_pos": [614, 21]}, {"full_name": "Turing.ToPartrec.Code.exists_code.comp", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [264, 9], "def_end_pos": [264, 25]}]], "state_before": "n\u271d\u00b9 : \u2115\nf\u271d\u00b9 : Vector \u2115 n\u271d\u00b9 \u2192. \u2115\nn\u271d : \u2115\nf\u271d : Vector \u2115 n\u271d \u2192 \u2115\nm n : \u2115\nf : Vector \u2115 n \u2192 \u2115\ng : Fin n \u2192 Vector \u2115 m \u2192 \u2115\nhf : Nat.Primrec' f\nhg : \u2200 (i : Fin n), Nat.Primrec' (g i)\nIHf : \u2203 c, \u2200 (v : Vector \u2115 n), eval c \u2191v = pure <$> \u2191f v\nIHg : \u2200 (i : Fin n), \u2203 c, \u2200 (v : Vector \u2115 m), eval c \u2191v = pure <$> \u2191(g i) v\n\u22a2 \u2203 c, \u2200 (v : Vector \u2115 m), eval c \u2191v = pure <$> (\u2191fun a => f (Vector.ofFn fun i => g i a)) v", "state_after": "no goals"}, {"tactic": "obtain \u27e8cf, hf\u27e9 := IHf", "annotated_tactic": ["obtain \u27e8cf, hf\u27e9 := IHf", []], "state_before": "n\u271d\u00b9 : \u2115\nf\u271d\u00b9 : Vector \u2115 n\u271d\u00b9 \u2192. \u2115\nn\u271d : \u2115\nf\u271d : Vector \u2115 n\u271d \u2192 \u2115\nn : \u2115\nf : Vector \u2115 n \u2192 \u2115\ng : Vector \u2115 (n + 2) \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec' f\na\u271d : Nat.Primrec' g\nIHf : \u2203 c, \u2200 (v : Vector \u2115 n), eval c \u2191v = pure <$> \u2191f v\nIHg : \u2203 c, \u2200 (v : Vector \u2115 (n + 2)), eval c \u2191v = pure <$> \u2191g v\n\u22a2 \u2203 c,\n    \u2200 (v : Vector \u2115 (n + 1)),\n      eval c \u2191v =\n        pure <$> (\u2191fun v => Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) (Vector.head v)) v", "state_after": "case intro\nn\u271d\u00b9 : \u2115\nf\u271d\u00b9 : Vector \u2115 n\u271d\u00b9 \u2192. \u2115\nn\u271d : \u2115\nf\u271d : Vector \u2115 n\u271d \u2192 \u2115\nn : \u2115\nf : Vector \u2115 n \u2192 \u2115\ng : Vector \u2115 (n + 2) \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec' f\na\u271d : Nat.Primrec' g\nIHg : \u2203 c, \u2200 (v : Vector \u2115 (n + 2)), eval c \u2191v = pure <$> \u2191g v\ncf : Code\nhf : \u2200 (v : Vector \u2115 n), eval cf \u2191v = pure <$> \u2191f v\n\u22a2 \u2203 c,\n    \u2200 (v : Vector \u2115 (n + 1)),\n      eval c \u2191v =\n        pure <$> (\u2191fun v => Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) (Vector.head v)) v"}, {"tactic": "obtain \u27e8cg, hg\u27e9 := IHg", "annotated_tactic": ["obtain \u27e8cg, hg\u27e9 := IHg", []], "state_before": "case intro\nn\u271d\u00b9 : \u2115\nf\u271d\u00b9 : Vector \u2115 n\u271d\u00b9 \u2192. \u2115\nn\u271d : \u2115\nf\u271d : Vector \u2115 n\u271d \u2192 \u2115\nn : \u2115\nf : Vector \u2115 n \u2192 \u2115\ng : Vector \u2115 (n + 2) \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec' f\na\u271d : Nat.Primrec' g\nIHg : \u2203 c, \u2200 (v : Vector \u2115 (n + 2)), eval c \u2191v = pure <$> \u2191g v\ncf : Code\nhf : \u2200 (v : Vector \u2115 n), eval cf \u2191v = pure <$> \u2191f v\n\u22a2 \u2203 c,\n    \u2200 (v : Vector \u2115 (n + 1)),\n      eval c \u2191v =\n        pure <$> (\u2191fun v => Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) (Vector.head v)) v", "state_after": "case intro.intro\nn\u271d\u00b9 : \u2115\nf\u271d\u00b9 : Vector \u2115 n\u271d\u00b9 \u2192. \u2115\nn\u271d : \u2115\nf\u271d : Vector \u2115 n\u271d \u2192 \u2115\nn : \u2115\nf : Vector \u2115 n \u2192 \u2115\ng : Vector \u2115 (n + 2) \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec' f\na\u271d : Nat.Primrec' g\ncf : Code\nhf : \u2200 (v : Vector \u2115 n), eval cf \u2191v = pure <$> \u2191f v\ncg : Code\nhg : \u2200 (v : Vector \u2115 (n + 2)), eval cg \u2191v = pure <$> \u2191g v\n\u22a2 \u2203 c,\n    \u2200 (v : Vector \u2115 (n + 1)),\n      eval c \u2191v =\n        pure <$> (\u2191fun v => Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) (Vector.head v)) v"}, {"tactic": "simp only [Part.map_eq_map, Part.map_some, PFun.coe_val] at hf hg", "annotated_tactic": ["simp only [<a>Part.map_eq_map</a>, <a>Part.map_some</a>, <a>PFun.coe_val</a>] at hf hg", [{"full_name": "Part.map_eq_map", "def_path": "Mathlib/Data/Part.lean", "def_pos": [609, 9], "def_end_pos": [609, 19]}, {"full_name": "Part.map_some", "def_path": "Mathlib/Data/Part.lean", "def_pos": [457, 9], "def_end_pos": [457, 17]}, {"full_name": "PFun.coe_val", "def_path": "Mathlib/Data/PFun.lean", "def_pos": [144, 9], "def_end_pos": [144, 16]}]], "state_before": "case intro.intro\nn\u271d\u00b9 : \u2115\nf\u271d\u00b9 : Vector \u2115 n\u271d\u00b9 \u2192. \u2115\nn\u271d : \u2115\nf\u271d : Vector \u2115 n\u271d \u2192 \u2115\nn : \u2115\nf : Vector \u2115 n \u2192 \u2115\ng : Vector \u2115 (n + 2) \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec' f\na\u271d : Nat.Primrec' g\ncf : Code\nhf : \u2200 (v : Vector \u2115 n), eval cf \u2191v = pure <$> \u2191f v\ncg : Code\nhg : \u2200 (v : Vector \u2115 (n + 2)), eval cg \u2191v = pure <$> \u2191g v\n\u22a2 \u2203 c,\n    \u2200 (v : Vector \u2115 (n + 1)),\n      eval c \u2191v =\n        pure <$> (\u2191fun v => Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) (Vector.head v)) v", "state_after": "case intro.intro\nn\u271d\u00b9 : \u2115\nf\u271d\u00b9 : Vector \u2115 n\u271d\u00b9 \u2192. \u2115\nn\u271d : \u2115\nf\u271d : Vector \u2115 n\u271d \u2192 \u2115\nn : \u2115\nf : Vector \u2115 n \u2192 \u2115\ng : Vector \u2115 (n + 2) \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec' f\na\u271d : Nat.Primrec' g\ncf cg : Code\nhf : \u2200 (v : Vector \u2115 n), eval cf \u2191v = Part.some (pure (f v))\nhg : \u2200 (v : Vector \u2115 (n + 2)), eval cg \u2191v = Part.some (pure (g v))\n\u22a2 \u2203 c,\n    \u2200 (v : Vector \u2115 (n + 1)),\n      eval c \u2191v =\n        pure <$> (\u2191fun v => Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) (Vector.head v)) v"}, {"tactic": "refine' \u27e8prec cf cg, fun v => _\u27e9", "annotated_tactic": ["refine' \u27e8<a>prec</a> cf cg, fun v => _\u27e9", [{"full_name": "Turing.ToPartrec.Code.prec", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [252, 5], "def_end_pos": [252, 9]}]], "state_before": "case intro.intro\nn\u271d\u00b9 : \u2115\nf\u271d\u00b9 : Vector \u2115 n\u271d\u00b9 \u2192. \u2115\nn\u271d : \u2115\nf\u271d : Vector \u2115 n\u271d \u2192 \u2115\nn : \u2115\nf : Vector \u2115 n \u2192 \u2115\ng : Vector \u2115 (n + 2) \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec' f\na\u271d : Nat.Primrec' g\ncf cg : Code\nhf : \u2200 (v : Vector \u2115 n), eval cf \u2191v = Part.some (pure (f v))\nhg : \u2200 (v : Vector \u2115 (n + 2)), eval cg \u2191v = Part.some (pure (g v))\n\u22a2 \u2203 c,\n    \u2200 (v : Vector \u2115 (n + 1)),\n      eval c \u2191v =\n        pure <$> (\u2191fun v => Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) (Vector.head v)) v", "state_after": "case intro.intro\nn\u271d\u00b9 : \u2115\nf\u271d\u00b9 : Vector \u2115 n\u271d\u00b9 \u2192. \u2115\nn\u271d : \u2115\nf\u271d : Vector \u2115 n\u271d \u2192 \u2115\nn : \u2115\nf : Vector \u2115 n \u2192 \u2115\ng : Vector \u2115 (n + 2) \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec' f\na\u271d : Nat.Primrec' g\ncf cg : Code\nhf : \u2200 (v : Vector \u2115 n), eval cf \u2191v = Part.some (pure (f v))\nhg : \u2200 (v : Vector \u2115 (n + 2)), eval cg \u2191v = Part.some (pure (g v))\nv : Vector \u2115 (n + 1)\n\u22a2 eval (prec cf cg) \u2191v =\n    pure <$> (\u2191fun v => Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) (Vector.head v)) v"}, {"tactic": "rw [\u2190 v.cons_head_tail]", "annotated_tactic": ["rw [\u2190 v.cons_head_tail]", []], "state_before": "case intro.intro\nn\u271d\u00b9 : \u2115\nf\u271d\u00b9 : Vector \u2115 n\u271d\u00b9 \u2192. \u2115\nn\u271d : \u2115\nf\u271d : Vector \u2115 n\u271d \u2192 \u2115\nn : \u2115\nf : Vector \u2115 n \u2192 \u2115\ng : Vector \u2115 (n + 2) \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec' f\na\u271d : Nat.Primrec' g\ncf cg : Code\nhf : \u2200 (v : Vector \u2115 n), eval cf \u2191v = Part.some (pure (f v))\nhg : \u2200 (v : Vector \u2115 (n + 2)), eval cg \u2191v = Part.some (pure (g v))\nv : Vector \u2115 (n + 1)\n\u22a2 eval (prec cf cg) \u2191v =\n    pure <$> (\u2191fun v => Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) (Vector.head v)) v", "state_after": "case intro.intro\nn\u271d\u00b9 : \u2115\nf\u271d\u00b9 : Vector \u2115 n\u271d\u00b9 \u2192. \u2115\nn\u271d : \u2115\nf\u271d : Vector \u2115 n\u271d \u2192 \u2115\nn : \u2115\nf : Vector \u2115 n \u2192 \u2115\ng : Vector \u2115 (n + 2) \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec' f\na\u271d : Nat.Primrec' g\ncf cg : Code\nhf : \u2200 (v : Vector \u2115 n), eval cf \u2191v = Part.some (pure (f v))\nhg : \u2200 (v : Vector \u2115 (n + 2)), eval cg \u2191v = Part.some (pure (g v))\nv : Vector \u2115 (n + 1)\n\u22a2 eval (prec cf cg) \u2191(Vector.head v ::\u1d65 Vector.tail v) =\n    pure <$>\n      (\u2191fun v => Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) (Vector.head v))\n        (Vector.head v ::\u1d65 Vector.tail v)"}, {"tactic": "specialize hf v.tail", "annotated_tactic": ["specialize hf v.tail", []], "state_before": "case intro.intro\nn\u271d\u00b9 : \u2115\nf\u271d\u00b9 : Vector \u2115 n\u271d\u00b9 \u2192. \u2115\nn\u271d : \u2115\nf\u271d : Vector \u2115 n\u271d \u2192 \u2115\nn : \u2115\nf : Vector \u2115 n \u2192 \u2115\ng : Vector \u2115 (n + 2) \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec' f\na\u271d : Nat.Primrec' g\ncf cg : Code\nhf : \u2200 (v : Vector \u2115 n), eval cf \u2191v = Part.some (pure (f v))\nhg : \u2200 (v : Vector \u2115 (n + 2)), eval cg \u2191v = Part.some (pure (g v))\nv : Vector \u2115 (n + 1)\n\u22a2 eval (prec cf cg) \u2191(Vector.head v ::\u1d65 Vector.tail v) =\n    pure <$>\n      (\u2191fun v => Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) (Vector.head v))\n        (Vector.head v ::\u1d65 Vector.tail v)", "state_after": "case intro.intro\nn\u271d\u00b9 : \u2115\nf\u271d\u00b9 : Vector \u2115 n\u271d\u00b9 \u2192. \u2115\nn\u271d : \u2115\nf\u271d : Vector \u2115 n\u271d \u2192 \u2115\nn : \u2115\nf : Vector \u2115 n \u2192 \u2115\ng : Vector \u2115 (n + 2) \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec' f\na\u271d : Nat.Primrec' g\ncf cg : Code\nhg : \u2200 (v : Vector \u2115 (n + 2)), eval cg \u2191v = Part.some (pure (g v))\nv : Vector \u2115 (n + 1)\nhf : eval cf \u2191(Vector.tail v) = Part.some (pure (f (Vector.tail v)))\n\u22a2 eval (prec cf cg) \u2191(Vector.head v ::\u1d65 Vector.tail v) =\n    pure <$>\n      (\u2191fun v => Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) (Vector.head v))\n        (Vector.head v ::\u1d65 Vector.tail v)"}, {"tactic": "replace hg := fun a b => hg (a ::\u1d65 b ::\u1d65 v.tail)", "annotated_tactic": ["replace hg := fun a b => hg (a ::\u1d65 b ::\u1d65 v.tail)", []], "state_before": "case intro.intro\nn\u271d\u00b9 : \u2115\nf\u271d\u00b9 : Vector \u2115 n\u271d\u00b9 \u2192. \u2115\nn\u271d : \u2115\nf\u271d : Vector \u2115 n\u271d \u2192 \u2115\nn : \u2115\nf : Vector \u2115 n \u2192 \u2115\ng : Vector \u2115 (n + 2) \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec' f\na\u271d : Nat.Primrec' g\ncf cg : Code\nhg : \u2200 (v : Vector \u2115 (n + 2)), eval cg \u2191v = Part.some (pure (g v))\nv : Vector \u2115 (n + 1)\nhf : eval cf \u2191(Vector.tail v) = Part.some (pure (f (Vector.tail v)))\n\u22a2 eval (prec cf cg) \u2191(Vector.head v ::\u1d65 Vector.tail v) =\n    pure <$>\n      (\u2191fun v => Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) (Vector.head v))\n        (Vector.head v ::\u1d65 Vector.tail v)", "state_after": "case intro.intro\nn\u271d\u00b9 : \u2115\nf\u271d\u00b9 : Vector \u2115 n\u271d\u00b9 \u2192. \u2115\nn\u271d : \u2115\nf\u271d : Vector \u2115 n\u271d \u2192 \u2115\nn : \u2115\nf : Vector \u2115 n \u2192 \u2115\ng : Vector \u2115 (n + 2) \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec' f\na\u271d : Nat.Primrec' g\ncf cg : Code\nv : Vector \u2115 (n + 1)\nhf : eval cf \u2191(Vector.tail v) = Part.some (pure (f (Vector.tail v)))\nhg : \u2200 (a b : \u2115), eval cg \u2191(a ::\u1d65 b ::\u1d65 Vector.tail v) = Part.some (pure (g (a ::\u1d65 b ::\u1d65 Vector.tail v)))\n\u22a2 eval (prec cf cg) \u2191(Vector.head v ::\u1d65 Vector.tail v) =\n    pure <$>\n      (\u2191fun v => Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) (Vector.head v))\n        (Vector.head v ::\u1d65 Vector.tail v)"}, {"tactic": "simp only [Vector.cons_val, Vector.tail_val] at hf hg", "annotated_tactic": ["simp only [<a>Vector.cons_val</a>, <a>Vector.tail_val</a>] at hf hg", [{"full_name": "Vector.cons_val", "def_path": "Mathlib/Data/Vector/Basic.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}, {"full_name": "Vector.tail_val", "def_path": "Mathlib/Data/Vector/Basic.lean", "def_pos": [182, 9], "def_end_pos": [182, 17]}]], "state_before": "case intro.intro\nn\u271d\u00b9 : \u2115\nf\u271d\u00b9 : Vector \u2115 n\u271d\u00b9 \u2192. \u2115\nn\u271d : \u2115\nf\u271d : Vector \u2115 n\u271d \u2192 \u2115\nn : \u2115\nf : Vector \u2115 n \u2192 \u2115\ng : Vector \u2115 (n + 2) \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec' f\na\u271d : Nat.Primrec' g\ncf cg : Code\nv : Vector \u2115 (n + 1)\nhf : eval cf \u2191(Vector.tail v) = Part.some (pure (f (Vector.tail v)))\nhg : \u2200 (a b : \u2115), eval cg \u2191(a ::\u1d65 b ::\u1d65 Vector.tail v) = Part.some (pure (g (a ::\u1d65 b ::\u1d65 Vector.tail v)))\n\u22a2 eval (prec cf cg) \u2191(Vector.head v ::\u1d65 Vector.tail v) =\n    pure <$>\n      (\u2191fun v => Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) (Vector.head v))\n        (Vector.head v ::\u1d65 Vector.tail v)", "state_after": "case intro.intro\nn\u271d\u00b9 : \u2115\nf\u271d\u00b9 : Vector \u2115 n\u271d\u00b9 \u2192. \u2115\nn\u271d : \u2115\nf\u271d : Vector \u2115 n\u271d \u2192 \u2115\nn : \u2115\nf : Vector \u2115 n \u2192 \u2115\ng : Vector \u2115 (n + 2) \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec' f\na\u271d : Nat.Primrec' g\ncf cg : Code\nv : Vector \u2115 (n + 1)\nhf : eval cf (List.tail \u2191v) = Part.some (pure (f (Vector.tail v)))\nhg : \u2200 (a b : \u2115), eval cg (a :: b :: List.tail \u2191v) = Part.some (pure (g (a ::\u1d65 b ::\u1d65 Vector.tail v)))\n\u22a2 eval (prec cf cg) \u2191(Vector.head v ::\u1d65 Vector.tail v) =\n    pure <$>\n      (\u2191fun v => Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) (Vector.head v))\n        (Vector.head v ::\u1d65 Vector.tail v)"}, {"tactic": "simp only [Part.map_eq_map, Part.map_some, Vector.cons_val, Vector.tail_cons, Vector.head_cons,\n  PFun.coe_val, Vector.tail_val]", "annotated_tactic": ["simp only [<a>Part.map_eq_map</a>, <a>Part.map_some</a>, <a>Vector.cons_val</a>, <a>Vector.tail_cons</a>, <a>Vector.head_cons</a>,\n      <a>PFun.coe_val</a>, <a>Vector.tail_val</a>]", [{"full_name": "Part.map_eq_map", "def_path": "Mathlib/Data/Part.lean", "def_pos": [609, 9], "def_end_pos": [609, 19]}, {"full_name": "Part.map_some", "def_path": "Mathlib/Data/Part.lean", "def_pos": [457, 9], "def_end_pos": [457, 17]}, {"full_name": "Vector.cons_val", "def_path": "Mathlib/Data/Vector/Basic.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}, {"full_name": "Vector.tail_cons", "def_path": "Mathlib/Data/Vector.lean", "def_pos": [75, 9], "def_end_pos": [75, 18]}, {"full_name": "Vector.head_cons", "def_path": "Mathlib/Data/Vector.lean", "def_pos": [64, 9], "def_end_pos": [64, 18]}, {"full_name": "PFun.coe_val", "def_path": "Mathlib/Data/PFun.lean", "def_pos": [144, 9], "def_end_pos": [144, 16]}, {"full_name": "Vector.tail_val", "def_path": "Mathlib/Data/Vector/Basic.lean", "def_pos": [182, 9], "def_end_pos": [182, 17]}]], "state_before": "case intro.intro\nn\u271d\u00b9 : \u2115\nf\u271d\u00b9 : Vector \u2115 n\u271d\u00b9 \u2192. \u2115\nn\u271d : \u2115\nf\u271d : Vector \u2115 n\u271d \u2192 \u2115\nn : \u2115\nf : Vector \u2115 n \u2192 \u2115\ng : Vector \u2115 (n + 2) \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec' f\na\u271d : Nat.Primrec' g\ncf cg : Code\nv : Vector \u2115 (n + 1)\nhf : eval cf (List.tail \u2191v) = Part.some (pure (f (Vector.tail v)))\nhg : \u2200 (a b : \u2115), eval cg (a :: b :: List.tail \u2191v) = Part.some (pure (g (a ::\u1d65 b ::\u1d65 Vector.tail v)))\n\u22a2 eval (prec cf cg) \u2191(Vector.head v ::\u1d65 Vector.tail v) =\n    pure <$>\n      (\u2191fun v => Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) (Vector.head v))\n        (Vector.head v ::\u1d65 Vector.tail v)", "state_after": "case intro.intro\nn\u271d\u00b9 : \u2115\nf\u271d\u00b9 : Vector \u2115 n\u271d\u00b9 \u2192. \u2115\nn\u271d : \u2115\nf\u271d : Vector \u2115 n\u271d \u2192 \u2115\nn : \u2115\nf : Vector \u2115 n \u2192 \u2115\ng : Vector \u2115 (n + 2) \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec' f\na\u271d : Nat.Primrec' g\ncf cg : Code\nv : Vector \u2115 (n + 1)\nhf : eval cf (List.tail \u2191v) = Part.some (pure (f (Vector.tail v)))\nhg : \u2200 (a b : \u2115), eval cg (a :: b :: List.tail \u2191v) = Part.some (pure (g (a ::\u1d65 b ::\u1d65 Vector.tail v)))\n\u22a2 eval (prec cf cg) (Vector.head v :: List.tail \u2191v) =\n    Part.some (pure (Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) (Vector.head v)))"}, {"tactic": "simp only [\u2190 Part.pure_eq_some] at hf hg \u22a2", "annotated_tactic": ["simp only [\u2190 <a>Part.pure_eq_some</a>] at hf hg \u22a2", [{"full_name": "Part.pure_eq_some", "def_path": "Mathlib/Data/Part.lean", "def_pos": [599, 9], "def_end_pos": [599, 21]}]], "state_before": "case intro.intro\nn\u271d\u00b9 : \u2115\nf\u271d\u00b9 : Vector \u2115 n\u271d\u00b9 \u2192. \u2115\nn\u271d : \u2115\nf\u271d : Vector \u2115 n\u271d \u2192 \u2115\nn : \u2115\nf : Vector \u2115 n \u2192 \u2115\ng : Vector \u2115 (n + 2) \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec' f\na\u271d : Nat.Primrec' g\ncf cg : Code\nv : Vector \u2115 (n + 1)\nhf : eval cf (List.tail \u2191v) = Part.some (pure (f (Vector.tail v)))\nhg : \u2200 (a b : \u2115), eval cg (a :: b :: List.tail \u2191v) = Part.some (pure (g (a ::\u1d65 b ::\u1d65 Vector.tail v)))\n\u22a2 eval (prec cf cg) (Vector.head v :: List.tail \u2191v) =\n    Part.some (pure (Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) (Vector.head v)))", "state_after": "case intro.intro\nn\u271d\u00b9 : \u2115\nf\u271d\u00b9 : Vector \u2115 n\u271d\u00b9 \u2192. \u2115\nn\u271d : \u2115\nf\u271d : Vector \u2115 n\u271d \u2192 \u2115\nn : \u2115\nf : Vector \u2115 n \u2192 \u2115\ng : Vector \u2115 (n + 2) \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec' f\na\u271d : Nat.Primrec' g\ncf cg : Code\nv : Vector \u2115 (n + 1)\nhf : eval cf (List.tail \u2191v) = pure (pure (f (Vector.tail v)))\nhg : \u2200 (a b : \u2115), eval cg (a :: b :: List.tail \u2191v) = pure (pure (g (a ::\u1d65 b ::\u1d65 Vector.tail v)))\n\u22a2 eval (prec cf cg) (Vector.head v :: List.tail \u2191v) =\n    pure (pure (Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) (Vector.head v)))"}, {"tactic": "induction' v.head with n _ <;>\n  simp [prec, hf, Part.bind_assoc, \u2190 Part.bind_some_eq_map, Part.bind_some,\n    show \u2200 x, pure x = [x] from fun _ => rfl, Bind.bind, Functor.map]", "annotated_tactic": ["induction' v.head with n _ <;>\n      simp [<a>prec</a>, hf, <a>Part.bind_assoc</a>, \u2190 <a>Part.bind_some_eq_map</a>, <a>Part.bind_some</a>,\n        show \u2200 x, <a>pure</a> x = [x] from fun _ => <a>rfl</a>, <a>Bind.bind</a>, <a>Functor.map</a>]", [{"full_name": "Turing.ToPartrec.Code.prec", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [252, 5], "def_end_pos": [252, 9]}, {"full_name": "Part.bind_assoc", "def_path": "Mathlib/Data/Part.lean", "def_pos": [540, 9], "def_end_pos": [540, 19]}, {"full_name": "Part.bind_some_eq_map", "def_path": "Mathlib/Data/Part.lean", "def_pos": [526, 9], "def_end_pos": [526, 25]}, {"full_name": "Part.bind_some", "def_path": "Mathlib/Data/Part.lean", "def_pos": [518, 9], "def_end_pos": [518, 18]}, {"full_name": "Pure.pure", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2670, 3], "def_end_pos": [2670, 7]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}, {"full_name": "Bind.bind", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2662, 3], "def_end_pos": [2662, 7]}, {"full_name": "Functor.map", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2685, 3], "def_end_pos": [2685, 6]}]], "state_before": "case intro.intro\nn\u271d\u00b9 : \u2115\nf\u271d\u00b9 : Vector \u2115 n\u271d\u00b9 \u2192. \u2115\nn\u271d : \u2115\nf\u271d : Vector \u2115 n\u271d \u2192 \u2115\nn : \u2115\nf : Vector \u2115 n \u2192 \u2115\ng : Vector \u2115 (n + 2) \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec' f\na\u271d : Nat.Primrec' g\ncf cg : Code\nv : Vector \u2115 (n + 1)\nhf : eval cf (List.tail \u2191v) = pure (pure (f (Vector.tail v)))\nhg : \u2200 (a b : \u2115), eval cg (a :: b :: List.tail \u2191v) = pure (pure (g (a ::\u1d65 b ::\u1d65 Vector.tail v)))\n\u22a2 eval (prec cf cg) (Vector.head v :: List.tail \u2191v) =\n    pure (pure (Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) (Vector.head v)))", "state_after": "case intro.intro.succ\nn\u271d\u00b2 : \u2115\nf\u271d\u00b9 : Vector \u2115 n\u271d\u00b2 \u2192. \u2115\nn\u271d\u00b9 : \u2115\nf\u271d : Vector \u2115 n\u271d\u00b9 \u2192 \u2115\nn\u271d : \u2115\nf : Vector \u2115 n\u271d \u2192 \u2115\ng : Vector \u2115 (n\u271d + 2) \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec' f\na\u271d : Nat.Primrec' g\ncf cg : Code\nv : Vector \u2115 (n\u271d + 1)\nhf : eval cf (List.tail \u2191v) = pure (pure (f (Vector.tail v)))\nhg : \u2200 (a b : \u2115), eval cg (a :: b :: List.tail \u2191v) = pure (pure (g (a ::\u1d65 b ::\u1d65 Vector.tail v)))\nn : \u2115\nn_ih\u271d :\n  eval (prec cf cg) (n :: List.tail \u2191v) =\n    pure (pure (Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) n))\n\u22a2 (Part.bind\n      (PFun.fix\n        (fun v =>\n          Part.bind (eval cg (List.headI v :: List.tail (List.tail v))) fun x =>\n            Part.some\n              (if List.headI (List.tail v) = 0 then\n                Sum.inl\n                  (Nat.succ (List.headI v) ::\n                    Nat.pred (List.headI (List.tail v)) :: List.headI x :: List.tail (List.tail (List.tail v)))\n              else\n                Sum.inr\n                  (Nat.succ (List.headI v) ::\n                    Nat.pred (List.headI (List.tail v)) :: List.headI x :: List.tail (List.tail (List.tail v)))))\n        (0 :: n :: f (Vector.tail v) :: List.tail \u2191v))\n      fun x => Part.some [List.headI (List.tail (List.tail x))]) =\n    Part.some [g (n ::\u1d65 Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) n ::\u1d65 Vector.tail v)]"}, {"tactic": "suffices \u2200 a b, a + b = n \u2192\n  (n.succ :: 0 ::\n    g (n ::\u1d65 Nat.rec (f v.tail) (fun y IH => g (y ::\u1d65 IH ::\u1d65 v.tail)) n ::\u1d65 v.tail) ::\n        v.val.tail : List \u2115) \u2208\n    PFun.fix\n      (fun v : List \u2115 => Part.bind (cg.eval (v.headI :: v.tail.tail))\n        (fun x => Part.some (if v.tail.headI = 0\n          then Sum.inl\n            (v.headI.succ :: v.tail.headI.pred :: x.headI :: v.tail.tail.tail : List \u2115)\n          else Sum.inr\n            (v.headI.succ :: v.tail.headI.pred :: x.headI :: v.tail.tail.tail))))\n      (a :: b :: Nat.rec (f v.tail) (fun y IH => g (y ::\u1d65 IH ::\u1d65 v.tail)) a :: v.val.tail) by\n  erw [Part.eq_some_iff.2 (this 0 n (zero_add n))]\n  simp only [List.headI, Part.bind_some, List.tail_cons]", "annotated_tactic": ["suffices \u2200 a b, a + b = n \u2192\n      (n.succ :: 0 ::\n        g (n ::\u1d65 <a>Nat.rec</a> (f v.tail) (fun y IH => g (y ::\u1d65 IH ::\u1d65 v.tail)) n ::\u1d65 v.tail) ::\n            v.val.tail : <a>List</a> \u2115) \u2208\n        <a>PFun.fix</a>\n          (fun v : <a>List</a> \u2115 => <a>Part.bind</a> (cg.eval (v.headI :: v.tail.tail))\n            (fun x => <a>Part.some</a> (if v.tail.headI = 0\n              then <a>Sum.inl</a>\n                (v.headI.succ :: v.tail.headI.pred :: x.headI :: v.tail.tail.tail : <a>List</a> \u2115)\n              else <a>Sum.inr</a>\n                (v.headI.succ :: v.tail.headI.pred :: x.headI :: v.tail.tail.tail))))\n          (a :: b :: <a>Nat.rec</a> (f v.tail) (fun y IH => g (y ::\u1d65 IH ::\u1d65 v.tail)) a :: v.val.tail) by\n      erw [<a>Part.eq_some_iff</a>.2 (this 0 n (<a>zero_add</a> n))]\n      simp only [<a>List.headI</a>, <a>Part.bind_some</a>, <a>List.tail_cons</a>]", [{"full_name": "Nat.rec", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1038, 11], "def_end_pos": [1038, 14]}, {"full_name": "List", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2182, 11], "def_end_pos": [2182, 15]}, {"full_name": "PFun.fix", "def_path": "Mathlib/Data/PFun.lean", "def_pos": [250, 5], "def_end_pos": [250, 8]}, {"full_name": "List", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2182, 11], "def_end_pos": [2182, 15]}, {"full_name": "Part.bind", "def_path": "Mathlib/Data/Part.lean", "def_pos": [428, 15], "def_end_pos": [428, 19]}, {"full_name": "Part.some", "def_path": "Mathlib/Data/Part.lean", "def_pos": [135, 5], "def_end_pos": [135, 9]}, {"full_name": "Sum.inl", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [104, 5], "def_end_pos": [104, 8]}, {"full_name": "List", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2182, 11], "def_end_pos": [2182, 15]}, {"full_name": "Sum.inr", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [106, 5], "def_end_pos": [106, 8]}, {"full_name": "Nat.rec", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1038, 11], "def_end_pos": [1038, 14]}, {"full_name": "Part.eq_some_iff", "def_path": "Mathlib/Data/Part.lean", "def_pos": [174, 9], "def_end_pos": [174, 20]}, {"full_name": "zero_add", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [463, 3], "def_end_pos": [463, 14]}, {"full_name": "List.headI", "def_path": "Mathlib/Init/Data/List/Basic.lean", "def_pos": [39, 5], "def_end_pos": [39, 10]}, {"full_name": "Part.bind_some", "def_path": "Mathlib/Data/Part.lean", "def_pos": [518, 9], "def_end_pos": [518, 18]}, {"full_name": "List.tail_cons", "def_path": "lake-packages/std/Std/Data/List/Basic.lean", "def_pos": [316, 17], "def_end_pos": [316, 26]}]], "state_before": "case intro.intro.succ\nn\u271d\u00b2 : \u2115\nf\u271d\u00b9 : Vector \u2115 n\u271d\u00b2 \u2192. \u2115\nn\u271d\u00b9 : \u2115\nf\u271d : Vector \u2115 n\u271d\u00b9 \u2192 \u2115\nn\u271d : \u2115\nf : Vector \u2115 n\u271d \u2192 \u2115\ng : Vector \u2115 (n\u271d + 2) \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec' f\na\u271d : Nat.Primrec' g\ncf cg : Code\nv : Vector \u2115 (n\u271d + 1)\nhf : eval cf (List.tail \u2191v) = pure (pure (f (Vector.tail v)))\nhg : \u2200 (a b : \u2115), eval cg (a :: b :: List.tail \u2191v) = pure (pure (g (a ::\u1d65 b ::\u1d65 Vector.tail v)))\nn : \u2115\nn_ih\u271d :\n  eval (prec cf cg) (n :: List.tail \u2191v) =\n    pure (pure (Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) n))\n\u22a2 (Part.bind\n      (PFun.fix\n        (fun v =>\n          Part.bind (eval cg (List.headI v :: List.tail (List.tail v))) fun x =>\n            Part.some\n              (if List.headI (List.tail v) = 0 then\n                Sum.inl\n                  (Nat.succ (List.headI v) ::\n                    Nat.pred (List.headI (List.tail v)) :: List.headI x :: List.tail (List.tail (List.tail v)))\n              else\n                Sum.inr\n                  (Nat.succ (List.headI v) ::\n                    Nat.pred (List.headI (List.tail v)) :: List.headI x :: List.tail (List.tail (List.tail v)))))\n        (0 :: n :: f (Vector.tail v) :: List.tail \u2191v))\n      fun x => Part.some [List.headI (List.tail (List.tail x))]) =\n    Part.some [g (n ::\u1d65 Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) n ::\u1d65 Vector.tail v)]", "state_after": "case intro.intro.succ\nn\u271d\u00b2 : \u2115\nf\u271d\u00b9 : Vector \u2115 n\u271d\u00b2 \u2192. \u2115\nn\u271d\u00b9 : \u2115\nf\u271d : Vector \u2115 n\u271d\u00b9 \u2192 \u2115\nn\u271d : \u2115\nf : Vector \u2115 n\u271d \u2192 \u2115\ng : Vector \u2115 (n\u271d + 2) \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec' f\na\u271d : Nat.Primrec' g\ncf cg : Code\nv : Vector \u2115 (n\u271d + 1)\nhf : eval cf (List.tail \u2191v) = pure (pure (f (Vector.tail v)))\nhg : \u2200 (a b : \u2115), eval cg (a :: b :: List.tail \u2191v) = pure (pure (g (a ::\u1d65 b ::\u1d65 Vector.tail v)))\nn : \u2115\nn_ih\u271d :\n  eval (prec cf cg) (n :: List.tail \u2191v) =\n    pure (pure (Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) n))\n\u22a2 \u2200 (a b : \u2115),\n    a + b = n \u2192\n      Nat.succ n ::\n          0 ::\n            g (n ::\u1d65 Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) n ::\u1d65 Vector.tail v) ::\n              List.tail \u2191v \u2208\n        PFun.fix\n          (fun v =>\n            Part.bind (eval cg (List.headI v :: List.tail (List.tail v))) fun x =>\n              Part.some\n                (if List.headI (List.tail v) = 0 then\n                  Sum.inl\n                    (Nat.succ (List.headI v) ::\n                      Nat.pred (List.headI (List.tail v)) :: List.headI x :: List.tail (List.tail (List.tail v)))\n                else\n                  Sum.inr\n                    (Nat.succ (List.headI v) ::\n                      Nat.pred (List.headI (List.tail v)) :: List.headI x :: List.tail (List.tail (List.tail v)))))\n          (a :: b :: Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a :: List.tail \u2191v)"}, {"tactic": "intro a b e", "annotated_tactic": ["intro a b e", []], "state_before": "case intro.intro.succ\nn\u271d\u00b2 : \u2115\nf\u271d\u00b9 : Vector \u2115 n\u271d\u00b2 \u2192. \u2115\nn\u271d\u00b9 : \u2115\nf\u271d : Vector \u2115 n\u271d\u00b9 \u2192 \u2115\nn\u271d : \u2115\nf : Vector \u2115 n\u271d \u2192 \u2115\ng : Vector \u2115 (n\u271d + 2) \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec' f\na\u271d : Nat.Primrec' g\ncf cg : Code\nv : Vector \u2115 (n\u271d + 1)\nhf : eval cf (List.tail \u2191v) = pure (pure (f (Vector.tail v)))\nhg : \u2200 (a b : \u2115), eval cg (a :: b :: List.tail \u2191v) = pure (pure (g (a ::\u1d65 b ::\u1d65 Vector.tail v)))\nn : \u2115\nn_ih\u271d :\n  eval (prec cf cg) (n :: List.tail \u2191v) =\n    pure (pure (Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) n))\n\u22a2 \u2200 (a b : \u2115),\n    a + b = n \u2192\n      Nat.succ n ::\n          0 ::\n            g (n ::\u1d65 Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) n ::\u1d65 Vector.tail v) ::\n              List.tail \u2191v \u2208\n        PFun.fix\n          (fun v =>\n            Part.bind (eval cg (List.headI v :: List.tail (List.tail v))) fun x =>\n              Part.some\n                (if List.headI (List.tail v) = 0 then\n                  Sum.inl\n                    (Nat.succ (List.headI v) ::\n                      Nat.pred (List.headI (List.tail v)) :: List.headI x :: List.tail (List.tail (List.tail v)))\n                else\n                  Sum.inr\n                    (Nat.succ (List.headI v) ::\n                      Nat.pred (List.headI (List.tail v)) :: List.headI x :: List.tail (List.tail (List.tail v)))))\n          (a :: b :: Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a :: List.tail \u2191v)", "state_after": "case intro.intro.succ\nn\u271d\u00b2 : \u2115\nf\u271d\u00b9 : Vector \u2115 n\u271d\u00b2 \u2192. \u2115\nn\u271d\u00b9 : \u2115\nf\u271d : Vector \u2115 n\u271d\u00b9 \u2192 \u2115\nn\u271d : \u2115\nf : Vector \u2115 n\u271d \u2192 \u2115\ng : Vector \u2115 (n\u271d + 2) \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec' f\na\u271d : Nat.Primrec' g\ncf cg : Code\nv : Vector \u2115 (n\u271d + 1)\nhf : eval cf (List.tail \u2191v) = pure (pure (f (Vector.tail v)))\nhg : \u2200 (a b : \u2115), eval cg (a :: b :: List.tail \u2191v) = pure (pure (g (a ::\u1d65 b ::\u1d65 Vector.tail v)))\nn : \u2115\nn_ih\u271d :\n  eval (prec cf cg) (n :: List.tail \u2191v) =\n    pure (pure (Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) n))\na b : \u2115\ne : a + b = n\n\u22a2 Nat.succ n ::\n      0 ::\n        g (n ::\u1d65 Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) n ::\u1d65 Vector.tail v) ::\n          List.tail \u2191v \u2208\n    PFun.fix\n      (fun v =>\n        Part.bind (eval cg (List.headI v :: List.tail (List.tail v))) fun x =>\n          Part.some\n            (if List.headI (List.tail v) = 0 then\n              Sum.inl\n                (Nat.succ (List.headI v) ::\n                  Nat.pred (List.headI (List.tail v)) :: List.headI x :: List.tail (List.tail (List.tail v)))\n            else\n              Sum.inr\n                (Nat.succ (List.headI v) ::\n                  Nat.pred (List.headI (List.tail v)) :: List.headI x :: List.tail (List.tail (List.tail v)))))\n      (a :: b :: Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a :: List.tail \u2191v)"}, {"tactic": "induction' b with b IH generalizing a", "annotated_tactic": ["induction' b with b IH generalizing a", []], "state_before": "case intro.intro.succ\nn\u271d\u00b2 : \u2115\nf\u271d\u00b9 : Vector \u2115 n\u271d\u00b2 \u2192. \u2115\nn\u271d\u00b9 : \u2115\nf\u271d : Vector \u2115 n\u271d\u00b9 \u2192 \u2115\nn\u271d : \u2115\nf : Vector \u2115 n\u271d \u2192 \u2115\ng : Vector \u2115 (n\u271d + 2) \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec' f\na\u271d : Nat.Primrec' g\ncf cg : Code\nv : Vector \u2115 (n\u271d + 1)\nhf : eval cf (List.tail \u2191v) = pure (pure (f (Vector.tail v)))\nhg : \u2200 (a b : \u2115), eval cg (a :: b :: List.tail \u2191v) = pure (pure (g (a ::\u1d65 b ::\u1d65 Vector.tail v)))\nn : \u2115\nn_ih\u271d :\n  eval (prec cf cg) (n :: List.tail \u2191v) =\n    pure (pure (Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) n))\na b : \u2115\ne : a + b = n\n\u22a2 Nat.succ n ::\n      0 ::\n        g (n ::\u1d65 Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) n ::\u1d65 Vector.tail v) ::\n          List.tail \u2191v \u2208\n    PFun.fix\n      (fun v =>\n        Part.bind (eval cg (List.headI v :: List.tail (List.tail v))) fun x =>\n          Part.some\n            (if List.headI (List.tail v) = 0 then\n              Sum.inl\n                (Nat.succ (List.headI v) ::\n                  Nat.pred (List.headI (List.tail v)) :: List.headI x :: List.tail (List.tail (List.tail v)))\n            else\n              Sum.inr\n                (Nat.succ (List.headI v) ::\n                  Nat.pred (List.headI (List.tail v)) :: List.headI x :: List.tail (List.tail (List.tail v)))))\n      (a :: b :: Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a :: List.tail \u2191v)", "state_after": "case intro.intro.succ.zero\nn\u271d\u00b2 : \u2115\nf\u271d\u00b9 : Vector \u2115 n\u271d\u00b2 \u2192. \u2115\nn\u271d\u00b9 : \u2115\nf\u271d : Vector \u2115 n\u271d\u00b9 \u2192 \u2115\nn\u271d : \u2115\nf : Vector \u2115 n\u271d \u2192 \u2115\ng : Vector \u2115 (n\u271d + 2) \u2192 \u2115\na\u271d\u00b2 : Nat.Primrec' f\na\u271d\u00b9 : Nat.Primrec' g\ncf cg : Code\nv : Vector \u2115 (n\u271d + 1)\nhf : eval cf (List.tail \u2191v) = pure (pure (f (Vector.tail v)))\nhg : \u2200 (a b : \u2115), eval cg (a :: b :: List.tail \u2191v) = pure (pure (g (a ::\u1d65 b ::\u1d65 Vector.tail v)))\nn : \u2115\nn_ih\u271d :\n  eval (prec cf cg) (n :: List.tail \u2191v) =\n    pure (pure (Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) n))\na\u271d b : \u2115\ne\u271d : a\u271d + b = n\na : \u2115\ne : a + Nat.zero = n\n\u22a2 Nat.succ n ::\n      0 ::\n        g (n ::\u1d65 Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) n ::\u1d65 Vector.tail v) ::\n          List.tail \u2191v \u2208\n    PFun.fix\n      (fun v =>\n        Part.bind (eval cg (List.headI v :: List.tail (List.tail v))) fun x =>\n          Part.some\n            (if List.headI (List.tail v) = 0 then\n              Sum.inl\n                (Nat.succ (List.headI v) ::\n                  Nat.pred (List.headI (List.tail v)) :: List.headI x :: List.tail (List.tail (List.tail v)))\n            else\n              Sum.inr\n                (Nat.succ (List.headI v) ::\n                  Nat.pred (List.headI (List.tail v)) :: List.headI x :: List.tail (List.tail (List.tail v)))))\n      (a :: Nat.zero :: Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a :: List.tail \u2191v)\n\ncase intro.intro.succ.succ\nn\u271d\u00b2 : \u2115\nf\u271d\u00b9 : Vector \u2115 n\u271d\u00b2 \u2192. \u2115\nn\u271d\u00b9 : \u2115\nf\u271d : Vector \u2115 n\u271d\u00b9 \u2192 \u2115\nn\u271d : \u2115\nf : Vector \u2115 n\u271d \u2192 \u2115\ng : Vector \u2115 (n\u271d + 2) \u2192 \u2115\na\u271d\u00b2 : Nat.Primrec' f\na\u271d\u00b9 : Nat.Primrec' g\ncf cg : Code\nv : Vector \u2115 (n\u271d + 1)\nhf : eval cf (List.tail \u2191v) = pure (pure (f (Vector.tail v)))\nhg : \u2200 (a b : \u2115), eval cg (a :: b :: List.tail \u2191v) = pure (pure (g (a ::\u1d65 b ::\u1d65 Vector.tail v)))\nn : \u2115\nn_ih\u271d :\n  eval (prec cf cg) (n :: List.tail \u2191v) =\n    pure (pure (Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) n))\na\u271d b\u271d : \u2115\ne\u271d : a\u271d + b\u271d = n\nb : \u2115\nIH :\n  \u2200 (a : \u2115),\n    a + b = n \u2192\n      Nat.succ n ::\n          0 ::\n            g (n ::\u1d65 Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) n ::\u1d65 Vector.tail v) ::\n              List.tail \u2191v \u2208\n        PFun.fix\n          (fun v =>\n            Part.bind (eval cg (List.headI v :: List.tail (List.tail v))) fun x =>\n              Part.some\n                (if List.headI (List.tail v) = 0 then\n                  Sum.inl\n                    (Nat.succ (List.headI v) ::\n                      Nat.pred (List.headI (List.tail v)) :: List.headI x :: List.tail (List.tail (List.tail v)))\n                else\n                  Sum.inr\n                    (Nat.succ (List.headI v) ::\n                      Nat.pred (List.headI (List.tail v)) :: List.headI x :: List.tail (List.tail (List.tail v)))))\n          (a :: b :: Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a :: List.tail \u2191v)\na : \u2115\ne : a + Nat.succ b = n\n\u22a2 Nat.succ n ::\n      0 ::\n        g (n ::\u1d65 Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) n ::\u1d65 Vector.tail v) ::\n          List.tail \u2191v \u2208\n    PFun.fix\n      (fun v =>\n        Part.bind (eval cg (List.headI v :: List.tail (List.tail v))) fun x =>\n          Part.some\n            (if List.headI (List.tail v) = 0 then\n              Sum.inl\n                (Nat.succ (List.headI v) ::\n                  Nat.pred (List.headI (List.tail v)) :: List.headI x :: List.tail (List.tail (List.tail v)))\n            else\n              Sum.inr\n                (Nat.succ (List.headI v) ::\n                  Nat.pred (List.headI (List.tail v)) :: List.headI x :: List.tail (List.tail (List.tail v)))))\n      (a :: Nat.succ b :: Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a :: List.tail \u2191v)"}, {"tactic": "erw [Part.eq_some_iff.2 (this 0 n (zero_add n))]", "annotated_tactic": ["erw [<a>Part.eq_some_iff</a>.2 (this 0 n (<a>zero_add</a> n))]", [{"full_name": "Part.eq_some_iff", "def_path": "Mathlib/Data/Part.lean", "def_pos": [174, 9], "def_end_pos": [174, 20]}, {"full_name": "zero_add", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [463, 3], "def_end_pos": [463, 14]}]], "state_before": "n\u271d\u00b2 : \u2115\nf\u271d\u00b9 : Vector \u2115 n\u271d\u00b2 \u2192. \u2115\nn\u271d\u00b9 : \u2115\nf\u271d : Vector \u2115 n\u271d\u00b9 \u2192 \u2115\nn\u271d : \u2115\nf : Vector \u2115 n\u271d \u2192 \u2115\ng : Vector \u2115 (n\u271d + 2) \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec' f\na\u271d : Nat.Primrec' g\ncf cg : Code\nv : Vector \u2115 (n\u271d + 1)\nhf : eval cf (List.tail \u2191v) = pure (pure (f (Vector.tail v)))\nhg : \u2200 (a b : \u2115), eval cg (a :: b :: List.tail \u2191v) = pure (pure (g (a ::\u1d65 b ::\u1d65 Vector.tail v)))\nn : \u2115\nn_ih\u271d :\n  eval (prec cf cg) (n :: List.tail \u2191v) =\n    pure (pure (Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) n))\nthis :\n  \u2200 (a b : \u2115),\n    a + b = n \u2192\n      Nat.succ n ::\n          0 ::\n            g (n ::\u1d65 Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) n ::\u1d65 Vector.tail v) ::\n              List.tail \u2191v \u2208\n        PFun.fix\n          (fun v =>\n            Part.bind (eval cg (List.headI v :: List.tail (List.tail v))) fun x =>\n              Part.some\n                (if List.headI (List.tail v) = 0 then\n                  Sum.inl\n                    (Nat.succ (List.headI v) ::\n                      Nat.pred (List.headI (List.tail v)) :: List.headI x :: List.tail (List.tail (List.tail v)))\n                else\n                  Sum.inr\n                    (Nat.succ (List.headI v) ::\n                      Nat.pred (List.headI (List.tail v)) :: List.headI x :: List.tail (List.tail (List.tail v)))))\n          (a :: b :: Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a :: List.tail \u2191v)\n\u22a2 (Part.bind\n      (PFun.fix\n        (fun v =>\n          Part.bind (eval cg (List.headI v :: List.tail (List.tail v))) fun x =>\n            Part.some\n              (if List.headI (List.tail v) = 0 then\n                Sum.inl\n                  (Nat.succ (List.headI v) ::\n                    Nat.pred (List.headI (List.tail v)) :: List.headI x :: List.tail (List.tail (List.tail v)))\n              else\n                Sum.inr\n                  (Nat.succ (List.headI v) ::\n                    Nat.pred (List.headI (List.tail v)) :: List.headI x :: List.tail (List.tail (List.tail v)))))\n        (0 :: n :: f (Vector.tail v) :: List.tail \u2191v))\n      fun x => Part.some [List.headI (List.tail (List.tail x))]) =\n    Part.some [g (n ::\u1d65 Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) n ::\u1d65 Vector.tail v)]", "state_after": "n\u271d\u00b2 : \u2115\nf\u271d\u00b9 : Vector \u2115 n\u271d\u00b2 \u2192. \u2115\nn\u271d\u00b9 : \u2115\nf\u271d : Vector \u2115 n\u271d\u00b9 \u2192 \u2115\nn\u271d : \u2115\nf : Vector \u2115 n\u271d \u2192 \u2115\ng : Vector \u2115 (n\u271d + 2) \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec' f\na\u271d : Nat.Primrec' g\ncf cg : Code\nv : Vector \u2115 (n\u271d + 1)\nhf : eval cf (List.tail \u2191v) = pure (pure (f (Vector.tail v)))\nhg : \u2200 (a b : \u2115), eval cg (a :: b :: List.tail \u2191v) = pure (pure (g (a ::\u1d65 b ::\u1d65 Vector.tail v)))\nn : \u2115\nn_ih\u271d :\n  eval (prec cf cg) (n :: List.tail \u2191v) =\n    pure (pure (Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) n))\nthis :\n  \u2200 (a b : \u2115),\n    a + b = n \u2192\n      Nat.succ n ::\n          0 ::\n            g (n ::\u1d65 Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) n ::\u1d65 Vector.tail v) ::\n              List.tail \u2191v \u2208\n        PFun.fix\n          (fun v =>\n            Part.bind (eval cg (List.headI v :: List.tail (List.tail v))) fun x =>\n              Part.some\n                (if List.headI (List.tail v) = 0 then\n                  Sum.inl\n                    (Nat.succ (List.headI v) ::\n                      Nat.pred (List.headI (List.tail v)) :: List.headI x :: List.tail (List.tail (List.tail v)))\n                else\n                  Sum.inr\n                    (Nat.succ (List.headI v) ::\n                      Nat.pred (List.headI (List.tail v)) :: List.headI x :: List.tail (List.tail (List.tail v)))))\n          (a :: b :: Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a :: List.tail \u2191v)\n\u22a2 (Part.bind\n      (Part.some\n        (Nat.succ n ::\n          0 ::\n            g (n ::\u1d65 Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) n ::\u1d65 Vector.tail v) ::\n              List.tail \u2191v))\n      fun x => Part.some [List.headI (List.tail (List.tail x))]) =\n    Part.some [g (n ::\u1d65 Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) n ::\u1d65 Vector.tail v)]"}, {"tactic": "simp only [List.headI, Part.bind_some, List.tail_cons]", "annotated_tactic": ["simp only [<a>List.headI</a>, <a>Part.bind_some</a>, <a>List.tail_cons</a>]", [{"full_name": "List.headI", "def_path": "Mathlib/Init/Data/List/Basic.lean", "def_pos": [39, 5], "def_end_pos": [39, 10]}, {"full_name": "Part.bind_some", "def_path": "Mathlib/Data/Part.lean", "def_pos": [518, 9], "def_end_pos": [518, 18]}, {"full_name": "List.tail_cons", "def_path": "lake-packages/std/Std/Data/List/Basic.lean", "def_pos": [316, 17], "def_end_pos": [316, 26]}]], "state_before": "n\u271d\u00b2 : \u2115\nf\u271d\u00b9 : Vector \u2115 n\u271d\u00b2 \u2192. \u2115\nn\u271d\u00b9 : \u2115\nf\u271d : Vector \u2115 n\u271d\u00b9 \u2192 \u2115\nn\u271d : \u2115\nf : Vector \u2115 n\u271d \u2192 \u2115\ng : Vector \u2115 (n\u271d + 2) \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec' f\na\u271d : Nat.Primrec' g\ncf cg : Code\nv : Vector \u2115 (n\u271d + 1)\nhf : eval cf (List.tail \u2191v) = pure (pure (f (Vector.tail v)))\nhg : \u2200 (a b : \u2115), eval cg (a :: b :: List.tail \u2191v) = pure (pure (g (a ::\u1d65 b ::\u1d65 Vector.tail v)))\nn : \u2115\nn_ih\u271d :\n  eval (prec cf cg) (n :: List.tail \u2191v) =\n    pure (pure (Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) n))\nthis :\n  \u2200 (a b : \u2115),\n    a + b = n \u2192\n      Nat.succ n ::\n          0 ::\n            g (n ::\u1d65 Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) n ::\u1d65 Vector.tail v) ::\n              List.tail \u2191v \u2208\n        PFun.fix\n          (fun v =>\n            Part.bind (eval cg (List.headI v :: List.tail (List.tail v))) fun x =>\n              Part.some\n                (if List.headI (List.tail v) = 0 then\n                  Sum.inl\n                    (Nat.succ (List.headI v) ::\n                      Nat.pred (List.headI (List.tail v)) :: List.headI x :: List.tail (List.tail (List.tail v)))\n                else\n                  Sum.inr\n                    (Nat.succ (List.headI v) ::\n                      Nat.pred (List.headI (List.tail v)) :: List.headI x :: List.tail (List.tail (List.tail v)))))\n          (a :: b :: Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a :: List.tail \u2191v)\n\u22a2 (Part.bind\n      (Part.some\n        (Nat.succ n ::\n          0 ::\n            g (n ::\u1d65 Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) n ::\u1d65 Vector.tail v) ::\n              List.tail \u2191v))\n      fun x => Part.some [List.headI (List.tail (List.tail x))]) =\n    Part.some [g (n ::\u1d65 Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) n ::\u1d65 Vector.tail v)]", "state_after": "no goals"}, {"tactic": "refine' PFun.mem_fix_iff.2 (Or.inl <| Part.eq_some_iff.1 _)", "annotated_tactic": ["refine' <a>PFun.mem_fix_iff</a>.2 (<a>Or.inl</a> <| <a>Part.eq_some_iff</a>.1 _)", [{"full_name": "PFun.mem_fix_iff", "def_path": "Mathlib/Data/PFun.lean", "def_pos": [266, 9], "def_end_pos": [266, 20]}, {"full_name": "Or.inl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [517, 5], "def_end_pos": [517, 8]}, {"full_name": "Part.eq_some_iff", "def_path": "Mathlib/Data/Part.lean", "def_pos": [174, 9], "def_end_pos": [174, 20]}]], "state_before": "case intro.intro.succ.zero\nn\u271d\u00b2 : \u2115\nf\u271d\u00b9 : Vector \u2115 n\u271d\u00b2 \u2192. \u2115\nn\u271d\u00b9 : \u2115\nf\u271d : Vector \u2115 n\u271d\u00b9 \u2192 \u2115\nn\u271d : \u2115\nf : Vector \u2115 n\u271d \u2192 \u2115\ng : Vector \u2115 (n\u271d + 2) \u2192 \u2115\na\u271d\u00b2 : Nat.Primrec' f\na\u271d\u00b9 : Nat.Primrec' g\ncf cg : Code\nv : Vector \u2115 (n\u271d + 1)\nhf : eval cf (List.tail \u2191v) = pure (pure (f (Vector.tail v)))\nhg : \u2200 (a b : \u2115), eval cg (a :: b :: List.tail \u2191v) = pure (pure (g (a ::\u1d65 b ::\u1d65 Vector.tail v)))\nn : \u2115\nn_ih\u271d :\n  eval (prec cf cg) (n :: List.tail \u2191v) =\n    pure (pure (Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) n))\na\u271d b : \u2115\ne\u271d : a\u271d + b = n\na : \u2115\ne : a + Nat.zero = n\n\u22a2 Nat.succ n ::\n      0 ::\n        g (n ::\u1d65 Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) n ::\u1d65 Vector.tail v) ::\n          List.tail \u2191v \u2208\n    PFun.fix\n      (fun v =>\n        Part.bind (eval cg (List.headI v :: List.tail (List.tail v))) fun x =>\n          Part.some\n            (if List.headI (List.tail v) = 0 then\n              Sum.inl\n                (Nat.succ (List.headI v) ::\n                  Nat.pred (List.headI (List.tail v)) :: List.headI x :: List.tail (List.tail (List.tail v)))\n            else\n              Sum.inr\n                (Nat.succ (List.headI v) ::\n                  Nat.pred (List.headI (List.tail v)) :: List.headI x :: List.tail (List.tail (List.tail v)))))\n      (a :: Nat.zero :: Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a :: List.tail \u2191v)", "state_after": "case intro.intro.succ.zero\nn\u271d\u00b2 : \u2115\nf\u271d\u00b9 : Vector \u2115 n\u271d\u00b2 \u2192. \u2115\nn\u271d\u00b9 : \u2115\nf\u271d : Vector \u2115 n\u271d\u00b9 \u2192 \u2115\nn\u271d : \u2115\nf : Vector \u2115 n\u271d \u2192 \u2115\ng : Vector \u2115 (n\u271d + 2) \u2192 \u2115\na\u271d\u00b2 : Nat.Primrec' f\na\u271d\u00b9 : Nat.Primrec' g\ncf cg : Code\nv : Vector \u2115 (n\u271d + 1)\nhf : eval cf (List.tail \u2191v) = pure (pure (f (Vector.tail v)))\nhg : \u2200 (a b : \u2115), eval cg (a :: b :: List.tail \u2191v) = pure (pure (g (a ::\u1d65 b ::\u1d65 Vector.tail v)))\nn : \u2115\nn_ih\u271d :\n  eval (prec cf cg) (n :: List.tail \u2191v) =\n    pure (pure (Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) n))\na\u271d b : \u2115\ne\u271d : a\u271d + b = n\na : \u2115\ne : a + Nat.zero = n\n\u22a2 (Part.bind\n      (eval cg\n        (List.headI\n            (a ::\n              Nat.zero :: Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a :: List.tail \u2191v) ::\n          List.tail\n            (List.tail\n              (a ::\n                Nat.zero ::\n                  Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a :: List.tail \u2191v))))\n      fun x =>\n      Part.some\n        (if\n            List.headI\n                (List.tail\n                  (a ::\n                    Nat.zero ::\n                      Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a :: List.tail \u2191v)) =\n              0 then\n          Sum.inl\n            (Nat.succ\n                (List.headI\n                  (a ::\n                    Nat.zero ::\n                      Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a :: List.tail \u2191v)) ::\n              Nat.pred\n                  (List.headI\n                    (List.tail\n                      (a ::\n                        Nat.zero ::\n                          Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a ::\n                            List.tail \u2191v))) ::\n                List.headI x ::\n                  List.tail\n                    (List.tail\n                      (List.tail\n                        (a ::\n                          Nat.zero ::\n                            Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a ::\n                              List.tail \u2191v))))\n        else\n          Sum.inr\n            (Nat.succ\n                (List.headI\n                  (a ::\n                    Nat.zero ::\n                      Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a :: List.tail \u2191v)) ::\n              Nat.pred\n                  (List.headI\n                    (List.tail\n                      (a ::\n                        Nat.zero ::\n                          Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a ::\n                            List.tail \u2191v))) ::\n                List.headI x ::\n                  List.tail\n                    (List.tail\n                      (List.tail\n                        (a ::\n                          Nat.zero ::\n                            Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a ::\n                              List.tail \u2191v)))))) =\n    Part.some\n      (Sum.inl\n        (Nat.succ n ::\n          0 ::\n            g (n ::\u1d65 Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) n ::\u1d65 Vector.tail v) ::\n              List.tail \u2191v))"}, {"tactic": "simp only [hg, \u2190 e, Part.bind_some, List.tail_cons, pure]", "annotated_tactic": ["simp only [hg, \u2190 e, <a>Part.bind_some</a>, <a>List.tail_cons</a>, <a>pure</a>]", [{"full_name": "Part.bind_some", "def_path": "Mathlib/Data/Part.lean", "def_pos": [518, 9], "def_end_pos": [518, 18]}, {"full_name": "List.tail_cons", "def_path": "lake-packages/std/Std/Data/List/Basic.lean", "def_pos": [316, 17], "def_end_pos": [316, 26]}, {"full_name": "Pure.pure", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2670, 3], "def_end_pos": [2670, 7]}]], "state_before": "case intro.intro.succ.zero\nn\u271d\u00b2 : \u2115\nf\u271d\u00b9 : Vector \u2115 n\u271d\u00b2 \u2192. \u2115\nn\u271d\u00b9 : \u2115\nf\u271d : Vector \u2115 n\u271d\u00b9 \u2192 \u2115\nn\u271d : \u2115\nf : Vector \u2115 n\u271d \u2192 \u2115\ng : Vector \u2115 (n\u271d + 2) \u2192 \u2115\na\u271d\u00b2 : Nat.Primrec' f\na\u271d\u00b9 : Nat.Primrec' g\ncf cg : Code\nv : Vector \u2115 (n\u271d + 1)\nhf : eval cf (List.tail \u2191v) = pure (pure (f (Vector.tail v)))\nhg : \u2200 (a b : \u2115), eval cg (a :: b :: List.tail \u2191v) = pure (pure (g (a ::\u1d65 b ::\u1d65 Vector.tail v)))\nn : \u2115\nn_ih\u271d :\n  eval (prec cf cg) (n :: List.tail \u2191v) =\n    pure (pure (Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) n))\na\u271d b : \u2115\ne\u271d : a\u271d + b = n\na : \u2115\ne : a + Nat.zero = n\n\u22a2 (Part.bind\n      (eval cg\n        (List.headI\n            (a ::\n              Nat.zero :: Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a :: List.tail \u2191v) ::\n          List.tail\n            (List.tail\n              (a ::\n                Nat.zero ::\n                  Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a :: List.tail \u2191v))))\n      fun x =>\n      Part.some\n        (if\n            List.headI\n                (List.tail\n                  (a ::\n                    Nat.zero ::\n                      Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a :: List.tail \u2191v)) =\n              0 then\n          Sum.inl\n            (Nat.succ\n                (List.headI\n                  (a ::\n                    Nat.zero ::\n                      Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a :: List.tail \u2191v)) ::\n              Nat.pred\n                  (List.headI\n                    (List.tail\n                      (a ::\n                        Nat.zero ::\n                          Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a ::\n                            List.tail \u2191v))) ::\n                List.headI x ::\n                  List.tail\n                    (List.tail\n                      (List.tail\n                        (a ::\n                          Nat.zero ::\n                            Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a ::\n                              List.tail \u2191v))))\n        else\n          Sum.inr\n            (Nat.succ\n                (List.headI\n                  (a ::\n                    Nat.zero ::\n                      Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a :: List.tail \u2191v)) ::\n              Nat.pred\n                  (List.headI\n                    (List.tail\n                      (a ::\n                        Nat.zero ::\n                          Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a ::\n                            List.tail \u2191v))) ::\n                List.headI x ::\n                  List.tail\n                    (List.tail\n                      (List.tail\n                        (a ::\n                          Nat.zero ::\n                            Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a ::\n                              List.tail \u2191v)))))) =\n    Part.some\n      (Sum.inl\n        (Nat.succ n ::\n          0 ::\n            g (n ::\u1d65 Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) n ::\u1d65 Vector.tail v) ::\n              List.tail \u2191v))", "state_after": "case intro.intro.succ.zero\nn\u271d\u00b2 : \u2115\nf\u271d\u00b9 : Vector \u2115 n\u271d\u00b2 \u2192. \u2115\nn\u271d\u00b9 : \u2115\nf\u271d : Vector \u2115 n\u271d\u00b9 \u2192 \u2115\nn\u271d : \u2115\nf : Vector \u2115 n\u271d \u2192 \u2115\ng : Vector \u2115 (n\u271d + 2) \u2192 \u2115\na\u271d\u00b2 : Nat.Primrec' f\na\u271d\u00b9 : Nat.Primrec' g\ncf cg : Code\nv : Vector \u2115 (n\u271d + 1)\nhf : eval cf (List.tail \u2191v) = pure (pure (f (Vector.tail v)))\nhg : \u2200 (a b : \u2115), eval cg (a :: b :: List.tail \u2191v) = pure (pure (g (a ::\u1d65 b ::\u1d65 Vector.tail v)))\nn : \u2115\nn_ih\u271d :\n  eval (prec cf cg) (n :: List.tail \u2191v) =\n    pure (pure (Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) n))\na\u271d b : \u2115\ne\u271d : a\u271d + b = n\na : \u2115\ne : a + Nat.zero = n\n\u22a2 Part.some\n      (if\n          List.headI\n              (Nat.zero :: Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a :: List.tail \u2191v) =\n            0 then\n        Sum.inl\n          (Nat.succ\n              (List.headI\n                (a ::\n                  Nat.zero ::\n                    Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a :: List.tail \u2191v)) ::\n            Nat.pred\n                (List.headI\n                  (Nat.zero ::\n                    Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a :: List.tail \u2191v)) ::\n              List.headI\n                  (List.ret\n                    (g\n                      (List.headI\n                          (a ::\n                            Nat.zero ::\n                              Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a ::\n                                List.tail \u2191v) ::\u1d65\n                        Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a ::\u1d65\n                          Vector.tail v))) ::\n                List.tail \u2191v)\n      else\n        Sum.inr\n          (Nat.succ\n              (List.headI\n                (a ::\n                  Nat.zero ::\n                    Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a :: List.tail \u2191v)) ::\n            Nat.pred\n                (List.headI\n                  (Nat.zero ::\n                    Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a :: List.tail \u2191v)) ::\n              List.headI\n                  (List.ret\n                    (g\n                      (List.headI\n                          (a ::\n                            Nat.zero ::\n                              Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a ::\n                                List.tail \u2191v) ::\u1d65\n                        Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a ::\u1d65\n                          Vector.tail v))) ::\n                List.tail \u2191v)) =\n    Part.some\n      (Sum.inl\n        (Nat.succ (a + Nat.zero) ::\n          0 ::\n            g\n                ((a + Nat.zero) ::\u1d65\n                  Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) (a + Nat.zero) ::\u1d65\n                    Vector.tail v) ::\n              List.tail \u2191v))"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case intro.intro.succ.zero\nn\u271d\u00b2 : \u2115\nf\u271d\u00b9 : Vector \u2115 n\u271d\u00b2 \u2192. \u2115\nn\u271d\u00b9 : \u2115\nf\u271d : Vector \u2115 n\u271d\u00b9 \u2192 \u2115\nn\u271d : \u2115\nf : Vector \u2115 n\u271d \u2192 \u2115\ng : Vector \u2115 (n\u271d + 2) \u2192 \u2115\na\u271d\u00b2 : Nat.Primrec' f\na\u271d\u00b9 : Nat.Primrec' g\ncf cg : Code\nv : Vector \u2115 (n\u271d + 1)\nhf : eval cf (List.tail \u2191v) = pure (pure (f (Vector.tail v)))\nhg : \u2200 (a b : \u2115), eval cg (a :: b :: List.tail \u2191v) = pure (pure (g (a ::\u1d65 b ::\u1d65 Vector.tail v)))\nn : \u2115\nn_ih\u271d :\n  eval (prec cf cg) (n :: List.tail \u2191v) =\n    pure (pure (Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) n))\na\u271d b : \u2115\ne\u271d : a\u271d + b = n\na : \u2115\ne : a + Nat.zero = n\n\u22a2 Part.some\n      (if\n          List.headI\n              (Nat.zero :: Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a :: List.tail \u2191v) =\n            0 then\n        Sum.inl\n          (Nat.succ\n              (List.headI\n                (a ::\n                  Nat.zero ::\n                    Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a :: List.tail \u2191v)) ::\n            Nat.pred\n                (List.headI\n                  (Nat.zero ::\n                    Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a :: List.tail \u2191v)) ::\n              List.headI\n                  (List.ret\n                    (g\n                      (List.headI\n                          (a ::\n                            Nat.zero ::\n                              Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a ::\n                                List.tail \u2191v) ::\u1d65\n                        Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a ::\u1d65\n                          Vector.tail v))) ::\n                List.tail \u2191v)\n      else\n        Sum.inr\n          (Nat.succ\n              (List.headI\n                (a ::\n                  Nat.zero ::\n                    Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a :: List.tail \u2191v)) ::\n            Nat.pred\n                (List.headI\n                  (Nat.zero ::\n                    Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a :: List.tail \u2191v)) ::\n              List.headI\n                  (List.ret\n                    (g\n                      (List.headI\n                          (a ::\n                            Nat.zero ::\n                              Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a ::\n                                List.tail \u2191v) ::\u1d65\n                        Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a ::\u1d65\n                          Vector.tail v))) ::\n                List.tail \u2191v)) =\n    Part.some\n      (Sum.inl\n        (Nat.succ (a + Nat.zero) ::\n          0 ::\n            g\n                ((a + Nat.zero) ::\u1d65\n                  Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) (a + Nat.zero) ::\u1d65\n                    Vector.tail v) ::\n              List.tail \u2191v))", "state_after": "no goals"}, {"tactic": "refine' PFun.mem_fix_iff.2 (Or.inr \u27e8_, _, IH (a + 1) (by rwa [add_right_comm])\u27e9)", "annotated_tactic": ["refine' <a>PFun.mem_fix_iff</a>.2 (<a>Or.inr</a> \u27e8_, _, IH (a + 1) (by rwa [<a>add_right_comm</a>])\u27e9)", [{"full_name": "PFun.mem_fix_iff", "def_path": "Mathlib/Data/PFun.lean", "def_pos": [266, 9], "def_end_pos": [266, 20]}, {"full_name": "Or.inr", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [519, 5], "def_end_pos": [519, 8]}, {"full_name": "add_right_comm", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [101, 3], "def_end_pos": [101, 14]}]], "state_before": "case intro.intro.succ.succ\nn\u271d\u00b2 : \u2115\nf\u271d\u00b9 : Vector \u2115 n\u271d\u00b2 \u2192. \u2115\nn\u271d\u00b9 : \u2115\nf\u271d : Vector \u2115 n\u271d\u00b9 \u2192 \u2115\nn\u271d : \u2115\nf : Vector \u2115 n\u271d \u2192 \u2115\ng : Vector \u2115 (n\u271d + 2) \u2192 \u2115\na\u271d\u00b2 : Nat.Primrec' f\na\u271d\u00b9 : Nat.Primrec' g\ncf cg : Code\nv : Vector \u2115 (n\u271d + 1)\nhf : eval cf (List.tail \u2191v) = pure (pure (f (Vector.tail v)))\nhg : \u2200 (a b : \u2115), eval cg (a :: b :: List.tail \u2191v) = pure (pure (g (a ::\u1d65 b ::\u1d65 Vector.tail v)))\nn : \u2115\nn_ih\u271d :\n  eval (prec cf cg) (n :: List.tail \u2191v) =\n    pure (pure (Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) n))\na\u271d b\u271d : \u2115\ne\u271d : a\u271d + b\u271d = n\nb : \u2115\nIH :\n  \u2200 (a : \u2115),\n    a + b = n \u2192\n      Nat.succ n ::\n          0 ::\n            g (n ::\u1d65 Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) n ::\u1d65 Vector.tail v) ::\n              List.tail \u2191v \u2208\n        PFun.fix\n          (fun v =>\n            Part.bind (eval cg (List.headI v :: List.tail (List.tail v))) fun x =>\n              Part.some\n                (if List.headI (List.tail v) = 0 then\n                  Sum.inl\n                    (Nat.succ (List.headI v) ::\n                      Nat.pred (List.headI (List.tail v)) :: List.headI x :: List.tail (List.tail (List.tail v)))\n                else\n                  Sum.inr\n                    (Nat.succ (List.headI v) ::\n                      Nat.pred (List.headI (List.tail v)) :: List.headI x :: List.tail (List.tail (List.tail v)))))\n          (a :: b :: Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a :: List.tail \u2191v)\na : \u2115\ne : a + Nat.succ b = n\n\u22a2 Nat.succ n ::\n      0 ::\n        g (n ::\u1d65 Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) n ::\u1d65 Vector.tail v) ::\n          List.tail \u2191v \u2208\n    PFun.fix\n      (fun v =>\n        Part.bind (eval cg (List.headI v :: List.tail (List.tail v))) fun x =>\n          Part.some\n            (if List.headI (List.tail v) = 0 then\n              Sum.inl\n                (Nat.succ (List.headI v) ::\n                  Nat.pred (List.headI (List.tail v)) :: List.headI x :: List.tail (List.tail (List.tail v)))\n            else\n              Sum.inr\n                (Nat.succ (List.headI v) ::\n                  Nat.pred (List.headI (List.tail v)) :: List.headI x :: List.tail (List.tail (List.tail v)))))\n      (a :: Nat.succ b :: Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a :: List.tail \u2191v)", "state_after": "case intro.intro.succ.succ\nn\u271d\u00b2 : \u2115\nf\u271d\u00b9 : Vector \u2115 n\u271d\u00b2 \u2192. \u2115\nn\u271d\u00b9 : \u2115\nf\u271d : Vector \u2115 n\u271d\u00b9 \u2192 \u2115\nn\u271d : \u2115\nf : Vector \u2115 n\u271d \u2192 \u2115\ng : Vector \u2115 (n\u271d + 2) \u2192 \u2115\na\u271d\u00b2 : Nat.Primrec' f\na\u271d\u00b9 : Nat.Primrec' g\ncf cg : Code\nv : Vector \u2115 (n\u271d + 1)\nhf : eval cf (List.tail \u2191v) = pure (pure (f (Vector.tail v)))\nhg : \u2200 (a b : \u2115), eval cg (a :: b :: List.tail \u2191v) = pure (pure (g (a ::\u1d65 b ::\u1d65 Vector.tail v)))\nn : \u2115\nn_ih\u271d :\n  eval (prec cf cg) (n :: List.tail \u2191v) =\n    pure (pure (Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) n))\na\u271d b\u271d : \u2115\ne\u271d : a\u271d + b\u271d = n\nb : \u2115\nIH :\n  \u2200 (a : \u2115),\n    a + b = n \u2192\n      Nat.succ n ::\n          0 ::\n            g (n ::\u1d65 Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) n ::\u1d65 Vector.tail v) ::\n              List.tail \u2191v \u2208\n        PFun.fix\n          (fun v =>\n            Part.bind (eval cg (List.headI v :: List.tail (List.tail v))) fun x =>\n              Part.some\n                (if List.headI (List.tail v) = 0 then\n                  Sum.inl\n                    (Nat.succ (List.headI v) ::\n                      Nat.pred (List.headI (List.tail v)) :: List.headI x :: List.tail (List.tail (List.tail v)))\n                else\n                  Sum.inr\n                    (Nat.succ (List.headI v) ::\n                      Nat.pred (List.headI (List.tail v)) :: List.headI x :: List.tail (List.tail (List.tail v)))))\n          (a :: b :: Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a :: List.tail \u2191v)\na : \u2115\ne : a + Nat.succ b = n\n\u22a2 Sum.inr\n      ((a + 1) ::\n        b :: Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) (a + 1) :: List.tail \u2191v) \u2208\n    Part.bind\n      (eval cg\n        (List.headI\n            (a ::\n              Nat.succ b ::\n                Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a :: List.tail \u2191v) ::\n          List.tail\n            (List.tail\n              (a ::\n                Nat.succ b ::\n                  Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a :: List.tail \u2191v))))\n      fun x =>\n      Part.some\n        (if\n            List.headI\n                (List.tail\n                  (a ::\n                    Nat.succ b ::\n                      Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a :: List.tail \u2191v)) =\n              0 then\n          Sum.inl\n            (Nat.succ\n                (List.headI\n                  (a ::\n                    Nat.succ b ::\n                      Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a :: List.tail \u2191v)) ::\n              Nat.pred\n                  (List.headI\n                    (List.tail\n                      (a ::\n                        Nat.succ b ::\n                          Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a ::\n                            List.tail \u2191v))) ::\n                List.headI x ::\n                  List.tail\n                    (List.tail\n                      (List.tail\n                        (a ::\n                          Nat.succ b ::\n                            Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a ::\n                              List.tail \u2191v))))\n        else\n          Sum.inr\n            (Nat.succ\n                (List.headI\n                  (a ::\n                    Nat.succ b ::\n                      Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a :: List.tail \u2191v)) ::\n              Nat.pred\n                  (List.headI\n                    (List.tail\n                      (a ::\n                        Nat.succ b ::\n                          Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a ::\n                            List.tail \u2191v))) ::\n                List.headI x ::\n                  List.tail\n                    (List.tail\n                      (List.tail\n                        (a ::\n                          Nat.succ b ::\n                            Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a ::\n                              List.tail \u2191v)))))"}, {"tactic": "simp only [hg, eval, Part.bind_some, Nat.rec_add_one, List.tail_nil, List.tail_cons, pure]", "annotated_tactic": ["simp only [hg, <a>eval</a>, <a>Part.bind_some</a>, <a>Nat.rec_add_one</a>, <a>List.tail_nil</a>, <a>List.tail_cons</a>, <a>pure</a>]", [{"full_name": "Turing.ToPartrec.Code.eval", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [119, 5], "def_end_pos": [119, 14]}, {"full_name": "Part.bind_some", "def_path": "Mathlib/Data/Part.lean", "def_pos": [518, 9], "def_end_pos": [518, 18]}, {"full_name": "Nat.rec_add_one", "def_path": "Mathlib/Data/Nat/Basic.lean", "def_pos": [410, 9], "def_end_pos": [410, 20]}, {"full_name": "List.tail_nil", "def_path": "lake-packages/std/Std/Data/List/Basic.lean", "def_pos": [315, 17], "def_end_pos": [315, 25]}, {"full_name": "List.tail_cons", "def_path": "lake-packages/std/Std/Data/List/Basic.lean", "def_pos": [316, 17], "def_end_pos": [316, 26]}, {"full_name": "Pure.pure", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2670, 3], "def_end_pos": [2670, 7]}]], "state_before": "case intro.intro.succ.succ\nn\u271d\u00b2 : \u2115\nf\u271d\u00b9 : Vector \u2115 n\u271d\u00b2 \u2192. \u2115\nn\u271d\u00b9 : \u2115\nf\u271d : Vector \u2115 n\u271d\u00b9 \u2192 \u2115\nn\u271d : \u2115\nf : Vector \u2115 n\u271d \u2192 \u2115\ng : Vector \u2115 (n\u271d + 2) \u2192 \u2115\na\u271d\u00b2 : Nat.Primrec' f\na\u271d\u00b9 : Nat.Primrec' g\ncf cg : Code\nv : Vector \u2115 (n\u271d + 1)\nhf : eval cf (List.tail \u2191v) = pure (pure (f (Vector.tail v)))\nhg : \u2200 (a b : \u2115), eval cg (a :: b :: List.tail \u2191v) = pure (pure (g (a ::\u1d65 b ::\u1d65 Vector.tail v)))\nn : \u2115\nn_ih\u271d :\n  eval (prec cf cg) (n :: List.tail \u2191v) =\n    pure (pure (Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) n))\na\u271d b\u271d : \u2115\ne\u271d : a\u271d + b\u271d = n\nb : \u2115\nIH :\n  \u2200 (a : \u2115),\n    a + b = n \u2192\n      Nat.succ n ::\n          0 ::\n            g (n ::\u1d65 Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) n ::\u1d65 Vector.tail v) ::\n              List.tail \u2191v \u2208\n        PFun.fix\n          (fun v =>\n            Part.bind (eval cg (List.headI v :: List.tail (List.tail v))) fun x =>\n              Part.some\n                (if List.headI (List.tail v) = 0 then\n                  Sum.inl\n                    (Nat.succ (List.headI v) ::\n                      Nat.pred (List.headI (List.tail v)) :: List.headI x :: List.tail (List.tail (List.tail v)))\n                else\n                  Sum.inr\n                    (Nat.succ (List.headI v) ::\n                      Nat.pred (List.headI (List.tail v)) :: List.headI x :: List.tail (List.tail (List.tail v)))))\n          (a :: b :: Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a :: List.tail \u2191v)\na : \u2115\ne : a + Nat.succ b = n\n\u22a2 Sum.inr\n      ((a + 1) ::\n        b :: Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) (a + 1) :: List.tail \u2191v) \u2208\n    Part.bind\n      (eval cg\n        (List.headI\n            (a ::\n              Nat.succ b ::\n                Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a :: List.tail \u2191v) ::\n          List.tail\n            (List.tail\n              (a ::\n                Nat.succ b ::\n                  Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a :: List.tail \u2191v))))\n      fun x =>\n      Part.some\n        (if\n            List.headI\n                (List.tail\n                  (a ::\n                    Nat.succ b ::\n                      Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a :: List.tail \u2191v)) =\n              0 then\n          Sum.inl\n            (Nat.succ\n                (List.headI\n                  (a ::\n                    Nat.succ b ::\n                      Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a :: List.tail \u2191v)) ::\n              Nat.pred\n                  (List.headI\n                    (List.tail\n                      (a ::\n                        Nat.succ b ::\n                          Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a ::\n                            List.tail \u2191v))) ::\n                List.headI x ::\n                  List.tail\n                    (List.tail\n                      (List.tail\n                        (a ::\n                          Nat.succ b ::\n                            Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a ::\n                              List.tail \u2191v))))\n        else\n          Sum.inr\n            (Nat.succ\n                (List.headI\n                  (a ::\n                    Nat.succ b ::\n                      Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a :: List.tail \u2191v)) ::\n              Nat.pred\n                  (List.headI\n                    (List.tail\n                      (a ::\n                        Nat.succ b ::\n                          Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a ::\n                            List.tail \u2191v))) ::\n                List.headI x ::\n                  List.tail\n                    (List.tail\n                      (List.tail\n                        (a ::\n                          Nat.succ b ::\n                            Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a ::\n                              List.tail \u2191v)))))", "state_after": "case intro.intro.succ.succ\nn\u271d\u00b2 : \u2115\nf\u271d\u00b9 : Vector \u2115 n\u271d\u00b2 \u2192. \u2115\nn\u271d\u00b9 : \u2115\nf\u271d : Vector \u2115 n\u271d\u00b9 \u2192 \u2115\nn\u271d : \u2115\nf : Vector \u2115 n\u271d \u2192 \u2115\ng : Vector \u2115 (n\u271d + 2) \u2192 \u2115\na\u271d\u00b2 : Nat.Primrec' f\na\u271d\u00b9 : Nat.Primrec' g\ncf cg : Code\nv : Vector \u2115 (n\u271d + 1)\nhf : eval cf (List.tail \u2191v) = pure (pure (f (Vector.tail v)))\nhg : \u2200 (a b : \u2115), eval cg (a :: b :: List.tail \u2191v) = pure (pure (g (a ::\u1d65 b ::\u1d65 Vector.tail v)))\nn : \u2115\nn_ih\u271d :\n  eval (prec cf cg) (n :: List.tail \u2191v) =\n    pure (pure (Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) n))\na\u271d b\u271d : \u2115\ne\u271d : a\u271d + b\u271d = n\nb : \u2115\nIH :\n  \u2200 (a : \u2115),\n    a + b = n \u2192\n      Nat.succ n ::\n          0 ::\n            g (n ::\u1d65 Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) n ::\u1d65 Vector.tail v) ::\n              List.tail \u2191v \u2208\n        PFun.fix\n          (fun v =>\n            Part.bind (eval cg (List.headI v :: List.tail (List.tail v))) fun x =>\n              Part.some\n                (if List.headI (List.tail v) = 0 then\n                  Sum.inl\n                    (Nat.succ (List.headI v) ::\n                      Nat.pred (List.headI (List.tail v)) :: List.headI x :: List.tail (List.tail (List.tail v)))\n                else\n                  Sum.inr\n                    (Nat.succ (List.headI v) ::\n                      Nat.pred (List.headI (List.tail v)) :: List.headI x :: List.tail (List.tail (List.tail v)))))\n          (a :: b :: Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a :: List.tail \u2191v)\na : \u2115\ne : a + Nat.succ b = n\n\u22a2 Sum.inr\n      ((a + 1) ::\n        b ::\n          g (a ::\u1d65 Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a ::\u1d65 Vector.tail v) ::\n            List.tail \u2191v) \u2208\n    Part.some\n      (if\n          List.headI\n              (Nat.succ b ::\n                Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a :: List.tail \u2191v) =\n            0 then\n        Sum.inl\n          (Nat.succ\n              (List.headI\n                (a ::\n                  Nat.succ b ::\n                    Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a :: List.tail \u2191v)) ::\n            Nat.pred\n                (List.headI\n                  (Nat.succ b ::\n                    Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a :: List.tail \u2191v)) ::\n              List.headI\n                  (List.ret\n                    (g\n                      (List.headI\n                          (a ::\n                            Nat.succ b ::\n                              Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a ::\n                                List.tail \u2191v) ::\u1d65\n                        Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a ::\u1d65\n                          Vector.tail v))) ::\n                List.tail \u2191v)\n      else\n        Sum.inr\n          (Nat.succ\n              (List.headI\n                (a ::\n                  Nat.succ b ::\n                    Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a :: List.tail \u2191v)) ::\n            Nat.pred\n                (List.headI\n                  (Nat.succ b ::\n                    Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a :: List.tail \u2191v)) ::\n              List.headI\n                  (List.ret\n                    (g\n                      (List.headI\n                          (a ::\n                            Nat.succ b ::\n                              Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a ::\n                                List.tail \u2191v) ::\u1d65\n                        Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a ::\u1d65\n                          Vector.tail v))) ::\n                List.tail \u2191v))"}, {"tactic": "exact Part.mem_some_iff.2 rfl", "annotated_tactic": ["exact <a>Part.mem_some_iff</a>.2 <a>rfl</a>", [{"full_name": "Part.mem_some_iff", "def_path": "Mathlib/Data/Part.lean", "def_pos": [170, 9], "def_end_pos": [170, 21]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "case intro.intro.succ.succ\nn\u271d\u00b2 : \u2115\nf\u271d\u00b9 : Vector \u2115 n\u271d\u00b2 \u2192. \u2115\nn\u271d\u00b9 : \u2115\nf\u271d : Vector \u2115 n\u271d\u00b9 \u2192 \u2115\nn\u271d : \u2115\nf : Vector \u2115 n\u271d \u2192 \u2115\ng : Vector \u2115 (n\u271d + 2) \u2192 \u2115\na\u271d\u00b2 : Nat.Primrec' f\na\u271d\u00b9 : Nat.Primrec' g\ncf cg : Code\nv : Vector \u2115 (n\u271d + 1)\nhf : eval cf (List.tail \u2191v) = pure (pure (f (Vector.tail v)))\nhg : \u2200 (a b : \u2115), eval cg (a :: b :: List.tail \u2191v) = pure (pure (g (a ::\u1d65 b ::\u1d65 Vector.tail v)))\nn : \u2115\nn_ih\u271d :\n  eval (prec cf cg) (n :: List.tail \u2191v) =\n    pure (pure (Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) n))\na\u271d b\u271d : \u2115\ne\u271d : a\u271d + b\u271d = n\nb : \u2115\nIH :\n  \u2200 (a : \u2115),\n    a + b = n \u2192\n      Nat.succ n ::\n          0 ::\n            g (n ::\u1d65 Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) n ::\u1d65 Vector.tail v) ::\n              List.tail \u2191v \u2208\n        PFun.fix\n          (fun v =>\n            Part.bind (eval cg (List.headI v :: List.tail (List.tail v))) fun x =>\n              Part.some\n                (if List.headI (List.tail v) = 0 then\n                  Sum.inl\n                    (Nat.succ (List.headI v) ::\n                      Nat.pred (List.headI (List.tail v)) :: List.headI x :: List.tail (List.tail (List.tail v)))\n                else\n                  Sum.inr\n                    (Nat.succ (List.headI v) ::\n                      Nat.pred (List.headI (List.tail v)) :: List.headI x :: List.tail (List.tail (List.tail v)))))\n          (a :: b :: Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a :: List.tail \u2191v)\na : \u2115\ne : a + Nat.succ b = n\n\u22a2 Sum.inr\n      ((a + 1) ::\n        b ::\n          g (a ::\u1d65 Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a ::\u1d65 Vector.tail v) ::\n            List.tail \u2191v) \u2208\n    Part.some\n      (if\n          List.headI\n              (Nat.succ b ::\n                Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a :: List.tail \u2191v) =\n            0 then\n        Sum.inl\n          (Nat.succ\n              (List.headI\n                (a ::\n                  Nat.succ b ::\n                    Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a :: List.tail \u2191v)) ::\n            Nat.pred\n                (List.headI\n                  (Nat.succ b ::\n                    Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a :: List.tail \u2191v)) ::\n              List.headI\n                  (List.ret\n                    (g\n                      (List.headI\n                          (a ::\n                            Nat.succ b ::\n                              Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a ::\n                                List.tail \u2191v) ::\u1d65\n                        Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a ::\u1d65\n                          Vector.tail v))) ::\n                List.tail \u2191v)\n      else\n        Sum.inr\n          (Nat.succ\n              (List.headI\n                (a ::\n                  Nat.succ b ::\n                    Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a :: List.tail \u2191v)) ::\n            Nat.pred\n                (List.headI\n                  (Nat.succ b ::\n                    Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a :: List.tail \u2191v)) ::\n              List.headI\n                  (List.ret\n                    (g\n                      (List.headI\n                          (a ::\n                            Nat.succ b ::\n                              Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a ::\n                                List.tail \u2191v) ::\u1d65\n                        Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a ::\u1d65\n                          Vector.tail v))) ::\n                List.tail \u2191v))", "state_after": "no goals"}, {"tactic": "rwa [add_right_comm]", "annotated_tactic": ["rwa [<a>add_right_comm</a>]", [{"full_name": "add_right_comm", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [101, 3], "def_end_pos": [101, 14]}]], "state_before": "n\u271d\u00b2 : \u2115\nf\u271d\u00b9 : Vector \u2115 n\u271d\u00b2 \u2192. \u2115\nn\u271d\u00b9 : \u2115\nf\u271d : Vector \u2115 n\u271d\u00b9 \u2192 \u2115\nn\u271d : \u2115\nf : Vector \u2115 n\u271d \u2192 \u2115\ng : Vector \u2115 (n\u271d + 2) \u2192 \u2115\na\u271d\u00b2 : Nat.Primrec' f\na\u271d\u00b9 : Nat.Primrec' g\ncf cg : Code\nv : Vector \u2115 (n\u271d + 1)\nhf : eval cf (List.tail \u2191v) = pure (pure (f (Vector.tail v)))\nhg : \u2200 (a b : \u2115), eval cg (a :: b :: List.tail \u2191v) = pure (pure (g (a ::\u1d65 b ::\u1d65 Vector.tail v)))\nn : \u2115\nn_ih\u271d :\n  eval (prec cf cg) (n :: List.tail \u2191v) =\n    pure (pure (Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) n))\na\u271d b\u271d : \u2115\ne\u271d : a\u271d + b\u271d = n\nb : \u2115\nIH :\n  \u2200 (a : \u2115),\n    a + b = n \u2192\n      Nat.succ n ::\n          0 ::\n            g (n ::\u1d65 Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) n ::\u1d65 Vector.tail v) ::\n              List.tail \u2191v \u2208\n        PFun.fix\n          (fun v =>\n            Part.bind (eval cg (List.headI v :: List.tail (List.tail v))) fun x =>\n              Part.some\n                (if List.headI (List.tail v) = 0 then\n                  Sum.inl\n                    (Nat.succ (List.headI v) ::\n                      Nat.pred (List.headI (List.tail v)) :: List.headI x :: List.tail (List.tail (List.tail v)))\n                else\n                  Sum.inr\n                    (Nat.succ (List.headI v) ::\n                      Nat.pred (List.headI (List.tail v)) :: List.headI x :: List.tail (List.tail (List.tail v)))))\n          (a :: b :: Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a :: List.tail \u2191v)\na : \u2115\ne : a + Nat.succ b = n\n\u22a2 a + 1 + b = n", "state_after": "no goals"}, {"tactic": "exact exists_code.comp IHf IHg", "annotated_tactic": ["exact <a>exists_code.comp</a> IHf IHg", [{"full_name": "Turing.ToPartrec.Code.exists_code.comp", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [264, 9], "def_end_pos": [264, 25]}]], "state_before": "n\u271d : \u2115\nf\u271d : Vector \u2115 n\u271d \u2192. \u2115\nm n : \u2115\nf : Vector \u2115 n \u2192. \u2115\ng : Fin n \u2192 Vector \u2115 m \u2192. \u2115\na\u271d\u00b9 : Nat.Partrec' f\na\u271d : \u2200 (i : Fin n), Nat.Partrec' (g i)\nIHf : \u2203 c, \u2200 (v : Vector \u2115 n), eval c \u2191v = pure <$> f v\nIHg : \u2200 (i : Fin n), \u2203 c, \u2200 (v : Vector \u2115 m), eval c \u2191v = pure <$> g i v\n\u22a2 \u2203 c, \u2200 (v : Vector \u2115 m), eval c \u2191v = pure <$> (fun v => (Vector.mOfFn fun i => g i v) >>= f) v", "state_after": "no goals"}, {"tactic": "obtain \u27e8cf, hf\u27e9 := IHf", "annotated_tactic": ["obtain \u27e8cf, hf\u27e9 := IHf", []], "state_before": "n\u271d : \u2115\nf\u271d : Vector \u2115 n\u271d \u2192. \u2115\nn : \u2115\nf : Vector \u2115 (n + 1) \u2192 \u2115\na\u271d : Nat.Partrec' \u2191f\nIHf : \u2203 c, \u2200 (v : Vector \u2115 (n + 1)), eval c \u2191v = pure <$> \u2191f v\n\u22a2 \u2203 c, \u2200 (v : Vector \u2115 n), eval c \u2191v = pure <$> (fun v => Nat.rfind fun n_1 => Part.some (decide (f (n_1 ::\u1d65 v) = 0))) v", "state_after": "case intro\nn\u271d : \u2115\nf\u271d : Vector \u2115 n\u271d \u2192. \u2115\nn : \u2115\nf : Vector \u2115 (n + 1) \u2192 \u2115\na\u271d : Nat.Partrec' \u2191f\ncf : Code\nhf : \u2200 (v : Vector \u2115 (n + 1)), eval cf \u2191v = pure <$> \u2191f v\n\u22a2 \u2203 c, \u2200 (v : Vector \u2115 n), eval c \u2191v = pure <$> (fun v => Nat.rfind fun n_1 => Part.some (decide (f (n_1 ::\u1d65 v) = 0))) v"}, {"tactic": "refine' \u27e8rfind cf, fun v => _\u27e9", "annotated_tactic": ["refine' \u27e8<a>rfind</a> cf, fun v => _\u27e9", [{"full_name": "Turing.ToPartrec.Code.rfind", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [226, 5], "def_end_pos": [226, 10]}]], "state_before": "case intro\nn\u271d : \u2115\nf\u271d : Vector \u2115 n\u271d \u2192. \u2115\nn : \u2115\nf : Vector \u2115 (n + 1) \u2192 \u2115\na\u271d : Nat.Partrec' \u2191f\ncf : Code\nhf : \u2200 (v : Vector \u2115 (n + 1)), eval cf \u2191v = pure <$> \u2191f v\n\u22a2 \u2203 c, \u2200 (v : Vector \u2115 n), eval c \u2191v = pure <$> (fun v => Nat.rfind fun n_1 => Part.some (decide (f (n_1 ::\u1d65 v) = 0))) v", "state_after": "case intro\nn\u271d : \u2115\nf\u271d : Vector \u2115 n\u271d \u2192. \u2115\nn : \u2115\nf : Vector \u2115 (n + 1) \u2192 \u2115\na\u271d : Nat.Partrec' \u2191f\ncf : Code\nhf : \u2200 (v : Vector \u2115 (n + 1)), eval cf \u2191v = pure <$> \u2191f v\nv : Vector \u2115 n\n\u22a2 eval (rfind cf) \u2191v = pure <$> (fun v => Nat.rfind fun n_1 => Part.some (decide (f (n_1 ::\u1d65 v) = 0))) v"}, {"tactic": "replace hf := fun a => hf (a ::\u1d65 v)", "annotated_tactic": ["replace hf := fun a => hf (a ::\u1d65 v)", []], "state_before": "case intro\nn\u271d : \u2115\nf\u271d : Vector \u2115 n\u271d \u2192. \u2115\nn : \u2115\nf : Vector \u2115 (n + 1) \u2192 \u2115\na\u271d : Nat.Partrec' \u2191f\ncf : Code\nhf : \u2200 (v : Vector \u2115 (n + 1)), eval cf \u2191v = pure <$> \u2191f v\nv : Vector \u2115 n\n\u22a2 eval (rfind cf) \u2191v = pure <$> (fun v => Nat.rfind fun n_1 => Part.some (decide (f (n_1 ::\u1d65 v) = 0))) v", "state_after": "case intro\nn\u271d : \u2115\nf\u271d : Vector \u2115 n\u271d \u2192. \u2115\nn : \u2115\nf : Vector \u2115 (n + 1) \u2192 \u2115\na\u271d : Nat.Partrec' \u2191f\ncf : Code\nv : Vector \u2115 n\nhf : \u2200 (a : \u2115), eval cf \u2191(a ::\u1d65 v) = pure <$> \u2191f (a ::\u1d65 v)\n\u22a2 eval (rfind cf) \u2191v = pure <$> (fun v => Nat.rfind fun n_1 => Part.some (decide (f (n_1 ::\u1d65 v) = 0))) v"}, {"tactic": "simp only [Part.map_eq_map, Part.map_some, Vector.cons_val, PFun.coe_val,\n  show \u2200 x, pure x = [x] from fun _ => rfl] at hf \u22a2", "annotated_tactic": ["simp only [<a>Part.map_eq_map</a>, <a>Part.map_some</a>, <a>Vector.cons_val</a>, <a>PFun.coe_val</a>,\n      show \u2200 x, <a>pure</a> x = [x] from fun _ => <a>rfl</a>] at hf \u22a2", [{"full_name": "Part.map_eq_map", "def_path": "Mathlib/Data/Part.lean", "def_pos": [609, 9], "def_end_pos": [609, 19]}, {"full_name": "Part.map_some", "def_path": "Mathlib/Data/Part.lean", "def_pos": [457, 9], "def_end_pos": [457, 17]}, {"full_name": "Vector.cons_val", "def_path": "Mathlib/Data/Vector/Basic.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}, {"full_name": "PFun.coe_val", "def_path": "Mathlib/Data/PFun.lean", "def_pos": [144, 9], "def_end_pos": [144, 16]}, {"full_name": "Pure.pure", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2670, 3], "def_end_pos": [2670, 7]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "case intro\nn\u271d : \u2115\nf\u271d : Vector \u2115 n\u271d \u2192. \u2115\nn : \u2115\nf : Vector \u2115 (n + 1) \u2192 \u2115\na\u271d : Nat.Partrec' \u2191f\ncf : Code\nv : Vector \u2115 n\nhf : \u2200 (a : \u2115), eval cf \u2191(a ::\u1d65 v) = pure <$> \u2191f (a ::\u1d65 v)\n\u22a2 eval (rfind cf) \u2191v = pure <$> (fun v => Nat.rfind fun n_1 => Part.some (decide (f (n_1 ::\u1d65 v) = 0))) v", "state_after": "case intro\nn\u271d : \u2115\nf\u271d : Vector \u2115 n\u271d \u2192. \u2115\nn : \u2115\nf : Vector \u2115 (n + 1) \u2192 \u2115\na\u271d : Nat.Partrec' \u2191f\ncf : Code\nv : Vector \u2115 n\nhf : \u2200 (a : \u2115), eval cf (a :: \u2191v) = Part.some [f (a ::\u1d65 v)]\n\u22a2 eval (rfind cf) \u2191v = Part.map pure (Nat.rfind fun n_1 => Part.some (decide (f (n_1 ::\u1d65 v) = 0)))"}, {"tactic": "refine' Part.ext fun x => _", "annotated_tactic": ["refine' <a>Part.ext</a> fun x => _", [{"full_name": "Part.ext", "def_path": "Mathlib/Data/Part.lean", "def_pos": [116, 9], "def_end_pos": [116, 12]}]], "state_before": "case intro\nn\u271d : \u2115\nf\u271d : Vector \u2115 n\u271d \u2192. \u2115\nn : \u2115\nf : Vector \u2115 (n + 1) \u2192 \u2115\na\u271d : Nat.Partrec' \u2191f\ncf : Code\nv : Vector \u2115 n\nhf : \u2200 (a : \u2115), eval cf (a :: \u2191v) = Part.some [f (a ::\u1d65 v)]\n\u22a2 eval (rfind cf) \u2191v = Part.map pure (Nat.rfind fun n_1 => Part.some (decide (f (n_1 ::\u1d65 v) = 0)))", "state_after": "case intro\nn\u271d : \u2115\nf\u271d : Vector \u2115 n\u271d \u2192. \u2115\nn : \u2115\nf : Vector \u2115 (n + 1) \u2192 \u2115\na\u271d : Nat.Partrec' \u2191f\ncf : Code\nv : Vector \u2115 n\nhf : \u2200 (a : \u2115), eval cf (a :: \u2191v) = Part.some [f (a ::\u1d65 v)]\nx : List \u2115\n\u22a2 x \u2208 eval (rfind cf) \u2191v \u2194 x \u2208 Part.map pure (Nat.rfind fun n_1 => Part.some (decide (f (n_1 ::\u1d65 v) = 0)))"}, {"tactic": "simp only [rfind, Part.bind_eq_bind, Part.pure_eq_some, Part.map_eq_map, Part.bind_some,\n  exists_prop, cons_eval, comp_eval, fix_eval, tail_eval, succ_eval, zero'_eval,\n  List.headI_nil, List.headI_cons, pred_eval, Part.map_some, Bool.false_eq_decide_iff,\n  Part.mem_bind_iff, List.length, Part.mem_map_iff, Nat.mem_rfind, List.tail_nil,\n  List.tail_cons, Bool.true_eq_decide_iff, Part.mem_some_iff, Part.map_bind]", "annotated_tactic": ["simp only [<a>rfind</a>, <a>Part.bind_eq_bind</a>, <a>Part.pure_eq_some</a>, <a>Part.map_eq_map</a>, <a>Part.bind_some</a>,\n      <a>exists_prop</a>, <a>cons_eval</a>, <a>comp_eval</a>, <a>fix_eval</a>, <a>tail_eval</a>, <a>succ_eval</a>, <a>zero'_eval</a>,\n      <a>List.headI_nil</a>, <a>List.headI_cons</a>, <a>pred_eval</a>, <a>Part.map_some</a>, <a>Bool.false_eq_decide_iff</a>,\n      <a>Part.mem_bind_iff</a>, <a>List.length</a>, <a>Part.mem_map_iff</a>, <a>Nat.mem_rfind</a>, <a>List.tail_nil</a>,\n      <a>List.tail_cons</a>, <a>Bool.true_eq_decide_iff</a>, <a>Part.mem_some_iff</a>, <a>Part.map_bind</a>]", [{"full_name": "Turing.ToPartrec.Code.rfind", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [226, 5], "def_end_pos": [226, 10]}, {"full_name": "Part.bind_eq_bind", "def_path": "Mathlib/Data/Part.lean", "def_pos": [614, 9], "def_end_pos": [614, 21]}, {"full_name": "Part.pure_eq_some", "def_path": "Mathlib/Data/Part.lean", "def_pos": [599, 9], "def_end_pos": [599, 21]}, {"full_name": "Part.map_eq_map", "def_path": "Mathlib/Data/Part.lean", "def_pos": [609, 9], "def_end_pos": [609, 19]}, {"full_name": "Part.bind_some", "def_path": "Mathlib/Data/Part.lean", "def_pos": [518, 9], "def_end_pos": [518, 18]}, {"full_name": "exists_prop", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [485, 17], "def_end_pos": [485, 28]}, {"full_name": "Turing.ToPartrec.Code.cons_eval", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [149, 9], "def_end_pos": [149, 18]}, {"full_name": "Turing.ToPartrec.Code.comp_eval", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [155, 9], "def_end_pos": [155, 18]}, {"full_name": "Turing.ToPartrec.Code.fix_eval", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [163, 9], "def_end_pos": [163, 17]}, {"full_name": "Turing.ToPartrec.Code.tail_eval", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [146, 9], "def_end_pos": [146, 18]}, {"full_name": "Turing.ToPartrec.Code.succ_eval", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [143, 9], "def_end_pos": [143, 18]}, {"full_name": "Turing.ToPartrec.Code.zero'_eval", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [140, 9], "def_end_pos": [140, 19]}, {"full_name": "List.headI_nil", "def_path": "Mathlib/Init/Data/List/Basic.lean", "def_pos": [44, 17], "def_end_pos": [44, 26]}, {"full_name": "List.headI_cons", "def_path": "Mathlib/Init/Data/List/Basic.lean", "def_pos": [45, 17], "def_end_pos": [45, 27]}, {"full_name": "Turing.ToPartrec.Code.pred_eval", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [211, 9], "def_end_pos": [211, 18]}, {"full_name": "Part.map_some", "def_path": "Mathlib/Data/Part.lean", "def_pos": [457, 9], "def_end_pos": [457, 17]}, {"full_name": "Bool.false_eq_decide_iff", "def_path": "Mathlib/Data/Bool/Basic.lean", "def_pos": [58, 9], "def_end_pos": [58, 28]}, {"full_name": "Part.mem_bind_iff", "def_path": "Mathlib/Data/Part.lean", "def_pos": [494, 9], "def_end_pos": [494, 21]}, {"full_name": "List.length", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2232, 5], "def_end_pos": [2232, 16]}, {"full_name": "Part.mem_map_iff", "def_path": "Mathlib/Data/Part.lean", "def_pos": [445, 9], "def_end_pos": [445, 20]}, {"full_name": "Nat.mem_rfind", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [102, 9], "def_end_pos": [102, 18]}, {"full_name": "List.tail_nil", "def_path": "lake-packages/std/Std/Data/List/Basic.lean", "def_pos": [315, 17], "def_end_pos": [315, 25]}, {"full_name": "List.tail_cons", "def_path": "lake-packages/std/Std/Data/List/Basic.lean", "def_pos": [316, 17], "def_end_pos": [316, 26]}, {"full_name": "Bool.true_eq_decide_iff", "def_path": "Mathlib/Data/Bool/Basic.lean", "def_pos": [53, 9], "def_end_pos": [53, 27]}, {"full_name": "Part.mem_some_iff", "def_path": "Mathlib/Data/Part.lean", "def_pos": [170, 9], "def_end_pos": [170, 21]}, {"full_name": "Part.map_bind", "def_path": "Mathlib/Data/Part.lean", "def_pos": [554, 9], "def_end_pos": [554, 17]}]], "state_before": "case intro\nn\u271d : \u2115\nf\u271d : Vector \u2115 n\u271d \u2192. \u2115\nn : \u2115\nf : Vector \u2115 (n + 1) \u2192 \u2115\na\u271d : Nat.Partrec' \u2191f\ncf : Code\nv : Vector \u2115 n\nhf : \u2200 (a : \u2115), eval cf (a :: \u2191v) = Part.some [f (a ::\u1d65 v)]\nx : List \u2115\n\u22a2 x \u2208 eval (rfind cf) \u2191v \u2194 x \u2208 Part.map pure (Nat.rfind fun n_1 => Part.some (decide (f (n_1 ::\u1d65 v) = 0)))", "state_after": "case intro\nn\u271d : \u2115\nf\u271d : Vector \u2115 n\u271d \u2192. \u2115\nn : \u2115\nf : Vector \u2115 (n + 1) \u2192 \u2115\na\u271d : Nat.Partrec' \u2191f\ncf : Code\nv : Vector \u2115 n\nhf : \u2200 (a : \u2115), eval cf (a :: \u2191v) = Part.some [f (a ::\u1d65 v)]\nx : List \u2115\n\u22a2 (\u2203 a,\n      a \u2208\n          PFun.fix\n            (fun v =>\n              Part.bind (eval cf v) fun y =>\n                Part.some\n                  (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)\n                  else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))\n            (0 :: \u2191v) \u2227\n        x = [Nat.pred (List.headI a)]) \u2194\n    \u2203 a, (f (a ::\u1d65 v) = 0 \u2227 \u2200 {m : \u2115}, m < a \u2192 \u00acf (m ::\u1d65 v) = 0) \u2227 pure a = x"}, {"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "case intro\nn\u271d : \u2115\nf\u271d : Vector \u2115 n\u271d \u2192. \u2115\nn : \u2115\nf : Vector \u2115 (n + 1) \u2192 \u2115\na\u271d : Nat.Partrec' \u2191f\ncf : Code\nv : Vector \u2115 n\nhf : \u2200 (a : \u2115), eval cf (a :: \u2191v) = Part.some [f (a ::\u1d65 v)]\nx : List \u2115\n\u22a2 (\u2203 a,\n      a \u2208\n          PFun.fix\n            (fun v =>\n              Part.bind (eval cf v) fun y =>\n                Part.some\n                  (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)\n                  else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))\n            (0 :: \u2191v) \u2227\n        x = [Nat.pred (List.headI a)]) \u2194\n    \u2203 a, (f (a ::\u1d65 v) = 0 \u2227 \u2200 {m : \u2115}, m < a \u2192 \u00acf (m ::\u1d65 v) = 0) \u2227 pure a = x", "state_after": "case intro.mp\nn\u271d : \u2115\nf\u271d : Vector \u2115 n\u271d \u2192. \u2115\nn : \u2115\nf : Vector \u2115 (n + 1) \u2192 \u2115\na\u271d : Nat.Partrec' \u2191f\ncf : Code\nv : Vector \u2115 n\nhf : \u2200 (a : \u2115), eval cf (a :: \u2191v) = Part.some [f (a ::\u1d65 v)]\nx : List \u2115\n\u22a2 (\u2203 a,\n      a \u2208\n          PFun.fix\n            (fun v =>\n              Part.bind (eval cf v) fun y =>\n                Part.some\n                  (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)\n                  else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))\n            (0 :: \u2191v) \u2227\n        x = [Nat.pred (List.headI a)]) \u2192\n    \u2203 a, (f (a ::\u1d65 v) = 0 \u2227 \u2200 {m : \u2115}, m < a \u2192 \u00acf (m ::\u1d65 v) = 0) \u2227 pure a = x\n\ncase intro.mpr\nn\u271d : \u2115\nf\u271d : Vector \u2115 n\u271d \u2192. \u2115\nn : \u2115\nf : Vector \u2115 (n + 1) \u2192 \u2115\na\u271d : Nat.Partrec' \u2191f\ncf : Code\nv : Vector \u2115 n\nhf : \u2200 (a : \u2115), eval cf (a :: \u2191v) = Part.some [f (a ::\u1d65 v)]\nx : List \u2115\n\u22a2 (\u2203 a, (f (a ::\u1d65 v) = 0 \u2227 \u2200 {m : \u2115}, m < a \u2192 \u00acf (m ::\u1d65 v) = 0) \u2227 pure a = x) \u2192\n    \u2203 a,\n      a \u2208\n          PFun.fix\n            (fun v =>\n              Part.bind (eval cf v) fun y =>\n                Part.some\n                  (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)\n                  else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))\n            (0 :: \u2191v) \u2227\n        x = [Nat.pred (List.headI a)]"}, {"tactic": "rintro \u27e8v', h1, rfl\u27e9", "annotated_tactic": ["rintro \u27e8v', h1, rfl\u27e9", []], "state_before": "case intro.mp\nn\u271d : \u2115\nf\u271d : Vector \u2115 n\u271d \u2192. \u2115\nn : \u2115\nf : Vector \u2115 (n + 1) \u2192 \u2115\na\u271d : Nat.Partrec' \u2191f\ncf : Code\nv : Vector \u2115 n\nhf : \u2200 (a : \u2115), eval cf (a :: \u2191v) = Part.some [f (a ::\u1d65 v)]\nx : List \u2115\n\u22a2 (\u2203 a,\n      a \u2208\n          PFun.fix\n            (fun v =>\n              Part.bind (eval cf v) fun y =>\n                Part.some\n                  (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)\n                  else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))\n            (0 :: \u2191v) \u2227\n        x = [Nat.pred (List.headI a)]) \u2192\n    \u2203 a, (f (a ::\u1d65 v) = 0 \u2227 \u2200 {m : \u2115}, m < a \u2192 \u00acf (m ::\u1d65 v) = 0) \u2227 pure a = x", "state_after": "case intro.mp.intro.intro\nn\u271d : \u2115\nf\u271d : Vector \u2115 n\u271d \u2192. \u2115\nn : \u2115\nf : Vector \u2115 (n + 1) \u2192 \u2115\na\u271d : Nat.Partrec' \u2191f\ncf : Code\nv : Vector \u2115 n\nhf : \u2200 (a : \u2115), eval cf (a :: \u2191v) = Part.some [f (a ::\u1d65 v)]\nv' : List \u2115\nh1 :\n  v' \u2208\n    PFun.fix\n      (fun v =>\n        Part.bind (eval cf v) fun y =>\n          Part.some\n            (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)\n            else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))\n      (0 :: \u2191v)\n\u22a2 \u2203 a, (f (a ::\u1d65 v) = 0 \u2227 \u2200 {m : \u2115}, m < a \u2192 \u00acf (m ::\u1d65 v) = 0) \u2227 pure a = [Nat.pred (List.headI v')]"}, {"tactic": "suffices \u2200 v\u2081 : List \u2115, v' \u2208 PFun.fix\n  (fun v => (cf.eval v).bind fun y => Part.some <|\n    if y.headI = 0 then Sum.inl (v.headI.succ :: v.tail)\n      else Sum.inr (v.headI.succ :: v.tail)) v\u2081 \u2192\n  \u2200 n, (v\u2081 = n :: v.val) \u2192 (\u2200 m < n, \u00acf (m ::\u1d65 v) = 0) \u2192\n    \u2203 a : \u2115,\n      (f (a ::\u1d65 v) = 0 \u2227 \u2200 {m : \u2115}, m < a \u2192 \u00acf (m ::\u1d65 v) = 0) \u2227 [a] = [v'.headI.pred]\n  by exact this _ h1 0 rfl (by rintro _ \u27e8\u27e9)", "annotated_tactic": ["suffices \u2200 v\u2081 : <a>List</a> \u2115, v' \u2208 <a>PFun.fix</a>\n        (fun v => (cf.eval v).<a>bind</a> fun y => <a>Part.some</a> <|\n          if y.headI = 0 then <a>Sum.inl</a> (v.headI.succ :: v.tail)\n            else <a>Sum.inr</a> (v.headI.succ :: v.tail)) v\u2081 \u2192\n        \u2200 n, (v\u2081 = n :: v.val) \u2192 (\u2200 m < n, \u00acf (m ::\u1d65 v) = 0) \u2192\n          \u2203 a : \u2115,\n            (f (a ::\u1d65 v) = 0 \u2227 \u2200 {m : \u2115}, m < a \u2192 \u00acf (m ::\u1d65 v) = 0) \u2227 [a] = [v'.headI.pred]\n        by exact this _ h1 0 <a>rfl</a> (by rintro _ \u27e8\u27e9)", [{"full_name": "List", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2182, 11], "def_end_pos": [2182, 15]}, {"full_name": "PFun.fix", "def_path": "Mathlib/Data/PFun.lean", "def_pos": [250, 5], "def_end_pos": [250, 8]}, {"full_name": "Part.bind", "def_path": "Mathlib/Data/Part.lean", "def_pos": [428, 15], "def_end_pos": [428, 19]}, {"full_name": "Part.some", "def_path": "Mathlib/Data/Part.lean", "def_pos": [135, 5], "def_end_pos": [135, 9]}, {"full_name": "Sum.inl", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [104, 5], "def_end_pos": [104, 8]}, {"full_name": "Sum.inr", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [106, 5], "def_end_pos": [106, 8]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "case intro.mp.intro.intro\nn\u271d : \u2115\nf\u271d : Vector \u2115 n\u271d \u2192. \u2115\nn : \u2115\nf : Vector \u2115 (n + 1) \u2192 \u2115\na\u271d : Nat.Partrec' \u2191f\ncf : Code\nv : Vector \u2115 n\nhf : \u2200 (a : \u2115), eval cf (a :: \u2191v) = Part.some [f (a ::\u1d65 v)]\nv' : List \u2115\nh1 :\n  v' \u2208\n    PFun.fix\n      (fun v =>\n        Part.bind (eval cf v) fun y =>\n          Part.some\n            (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)\n            else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))\n      (0 :: \u2191v)\n\u22a2 \u2203 a, (f (a ::\u1d65 v) = 0 \u2227 \u2200 {m : \u2115}, m < a \u2192 \u00acf (m ::\u1d65 v) = 0) \u2227 pure a = [Nat.pred (List.headI v')]", "state_after": "case intro.mp.intro.intro\nn\u271d : \u2115\nf\u271d : Vector \u2115 n\u271d \u2192. \u2115\nn : \u2115\nf : Vector \u2115 (n + 1) \u2192 \u2115\na\u271d : Nat.Partrec' \u2191f\ncf : Code\nv : Vector \u2115 n\nhf : \u2200 (a : \u2115), eval cf (a :: \u2191v) = Part.some [f (a ::\u1d65 v)]\nv' : List \u2115\nh1 :\n  v' \u2208\n    PFun.fix\n      (fun v =>\n        Part.bind (eval cf v) fun y =>\n          Part.some\n            (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)\n            else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))\n      (0 :: \u2191v)\n\u22a2 \u2200 (v\u2081 : List \u2115),\n    v' \u2208\n        PFun.fix\n          (fun v =>\n            Part.bind (eval cf v) fun y =>\n              Part.some\n                (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)\n                else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))\n          v\u2081 \u2192\n      \u2200 (n_1 : \u2115),\n        v\u2081 = n_1 :: \u2191v \u2192\n          (\u2200 (m : \u2115), m < n_1 \u2192 \u00acf (m ::\u1d65 v) = 0) \u2192\n            \u2203 a, (f (a ::\u1d65 v) = 0 \u2227 \u2200 {m : \u2115}, m < a \u2192 \u00acf (m ::\u1d65 v) = 0) \u2227 [a] = [Nat.pred (List.headI v')]"}, {"tactic": "clear h1", "annotated_tactic": ["clear h1", []], "state_before": "case intro.mp.intro.intro\nn\u271d : \u2115\nf\u271d : Vector \u2115 n\u271d \u2192. \u2115\nn : \u2115\nf : Vector \u2115 (n + 1) \u2192 \u2115\na\u271d : Nat.Partrec' \u2191f\ncf : Code\nv : Vector \u2115 n\nhf : \u2200 (a : \u2115), eval cf (a :: \u2191v) = Part.some [f (a ::\u1d65 v)]\nv' : List \u2115\nh1 :\n  v' \u2208\n    PFun.fix\n      (fun v =>\n        Part.bind (eval cf v) fun y =>\n          Part.some\n            (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)\n            else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))\n      (0 :: \u2191v)\n\u22a2 \u2200 (v\u2081 : List \u2115),\n    v' \u2208\n        PFun.fix\n          (fun v =>\n            Part.bind (eval cf v) fun y =>\n              Part.some\n                (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)\n                else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))\n          v\u2081 \u2192\n      \u2200 (n_1 : \u2115),\n        v\u2081 = n_1 :: \u2191v \u2192\n          (\u2200 (m : \u2115), m < n_1 \u2192 \u00acf (m ::\u1d65 v) = 0) \u2192\n            \u2203 a, (f (a ::\u1d65 v) = 0 \u2227 \u2200 {m : \u2115}, m < a \u2192 \u00acf (m ::\u1d65 v) = 0) \u2227 [a] = [Nat.pred (List.headI v')]", "state_after": "case intro.mp.intro.intro\nn\u271d : \u2115\nf\u271d : Vector \u2115 n\u271d \u2192. \u2115\nn : \u2115\nf : Vector \u2115 (n + 1) \u2192 \u2115\na\u271d : Nat.Partrec' \u2191f\ncf : Code\nv : Vector \u2115 n\nhf : \u2200 (a : \u2115), eval cf (a :: \u2191v) = Part.some [f (a ::\u1d65 v)]\nv' : List \u2115\n\u22a2 \u2200 (v\u2081 : List \u2115),\n    v' \u2208\n        PFun.fix\n          (fun v =>\n            Part.bind (eval cf v) fun y =>\n              Part.some\n                (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)\n                else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))\n          v\u2081 \u2192\n      \u2200 (n_1 : \u2115),\n        v\u2081 = n_1 :: \u2191v \u2192\n          (\u2200 (m : \u2115), m < n_1 \u2192 \u00acf (m ::\u1d65 v) = 0) \u2192\n            \u2203 a, (f (a ::\u1d65 v) = 0 \u2227 \u2200 {m : \u2115}, m < a \u2192 \u00acf (m ::\u1d65 v) = 0) \u2227 [a] = [Nat.pred (List.headI v')]"}, {"tactic": "intro v\u2080 h1", "annotated_tactic": ["intro v\u2080 h1", []], "state_before": "case intro.mp.intro.intro\nn\u271d : \u2115\nf\u271d : Vector \u2115 n\u271d \u2192. \u2115\nn : \u2115\nf : Vector \u2115 (n + 1) \u2192 \u2115\na\u271d : Nat.Partrec' \u2191f\ncf : Code\nv : Vector \u2115 n\nhf : \u2200 (a : \u2115), eval cf (a :: \u2191v) = Part.some [f (a ::\u1d65 v)]\nv' : List \u2115\n\u22a2 \u2200 (v\u2081 : List \u2115),\n    v' \u2208\n        PFun.fix\n          (fun v =>\n            Part.bind (eval cf v) fun y =>\n              Part.some\n                (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)\n                else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))\n          v\u2081 \u2192\n      \u2200 (n_1 : \u2115),\n        v\u2081 = n_1 :: \u2191v \u2192\n          (\u2200 (m : \u2115), m < n_1 \u2192 \u00acf (m ::\u1d65 v) = 0) \u2192\n            \u2203 a, (f (a ::\u1d65 v) = 0 \u2227 \u2200 {m : \u2115}, m < a \u2192 \u00acf (m ::\u1d65 v) = 0) \u2227 [a] = [Nat.pred (List.headI v')]", "state_after": "case intro.mp.intro.intro\nn\u271d : \u2115\nf\u271d : Vector \u2115 n\u271d \u2192. \u2115\nn : \u2115\nf : Vector \u2115 (n + 1) \u2192 \u2115\na\u271d : Nat.Partrec' \u2191f\ncf : Code\nv : Vector \u2115 n\nhf : \u2200 (a : \u2115), eval cf (a :: \u2191v) = Part.some [f (a ::\u1d65 v)]\nv' v\u2080 : List \u2115\nh1 :\n  v' \u2208\n    PFun.fix\n      (fun v =>\n        Part.bind (eval cf v) fun y =>\n          Part.some\n            (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)\n            else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))\n      v\u2080\n\u22a2 \u2200 (n_1 : \u2115),\n    v\u2080 = n_1 :: \u2191v \u2192\n      (\u2200 (m : \u2115), m < n_1 \u2192 \u00acf (m ::\u1d65 v) = 0) \u2192\n        \u2203 a, (f (a ::\u1d65 v) = 0 \u2227 \u2200 {m : \u2115}, m < a \u2192 \u00acf (m ::\u1d65 v) = 0) \u2227 [a] = [Nat.pred (List.headI v')]"}, {"tactic": "refine' PFun.fixInduction h1 fun v\u2081 h2 IH => _", "annotated_tactic": ["refine' <a>PFun.fixInduction</a> h1 fun v\u2081 h2 IH => _", [{"full_name": "PFun.fixInduction", "def_path": "Mathlib/Data/PFun.lean", "def_pos": [328, 5], "def_end_pos": [328, 17]}]], "state_before": "case intro.mp.intro.intro\nn\u271d : \u2115\nf\u271d : Vector \u2115 n\u271d \u2192. \u2115\nn : \u2115\nf : Vector \u2115 (n + 1) \u2192 \u2115\na\u271d : Nat.Partrec' \u2191f\ncf : Code\nv : Vector \u2115 n\nhf : \u2200 (a : \u2115), eval cf (a :: \u2191v) = Part.some [f (a ::\u1d65 v)]\nv' v\u2080 : List \u2115\nh1 :\n  v' \u2208\n    PFun.fix\n      (fun v =>\n        Part.bind (eval cf v) fun y =>\n          Part.some\n            (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)\n            else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))\n      v\u2080\n\u22a2 \u2200 (n_1 : \u2115),\n    v\u2080 = n_1 :: \u2191v \u2192\n      (\u2200 (m : \u2115), m < n_1 \u2192 \u00acf (m ::\u1d65 v) = 0) \u2192\n        \u2203 a, (f (a ::\u1d65 v) = 0 \u2227 \u2200 {m : \u2115}, m < a \u2192 \u00acf (m ::\u1d65 v) = 0) \u2227 [a] = [Nat.pred (List.headI v')]", "state_after": "case intro.mp.intro.intro\nn\u271d : \u2115\nf\u271d : Vector \u2115 n\u271d \u2192. \u2115\nn : \u2115\nf : Vector \u2115 (n + 1) \u2192 \u2115\na\u271d : Nat.Partrec' \u2191f\ncf : Code\nv : Vector \u2115 n\nhf : \u2200 (a : \u2115), eval cf (a :: \u2191v) = Part.some [f (a ::\u1d65 v)]\nv' v\u2080 : List \u2115\nh1 :\n  v' \u2208\n    PFun.fix\n      (fun v =>\n        Part.bind (eval cf v) fun y =>\n          Part.some\n            (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)\n            else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))\n      v\u2080\nv\u2081 : List \u2115\nh2 :\n  v' \u2208\n    PFun.fix\n      (fun v =>\n        Part.bind (eval cf v) fun y =>\n          Part.some\n            (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)\n            else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))\n      v\u2081\nIH :\n  \u2200 (a'' : List \u2115),\n    (Sum.inr a'' \u2208\n        Part.bind (eval cf v\u2081) fun y =>\n          Part.some\n            (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v\u2081) :: List.tail v\u2081)\n            else Sum.inr (Nat.succ (List.headI v\u2081) :: List.tail v\u2081))) \u2192\n      \u2200 (n_1 : \u2115),\n        a'' = n_1 :: \u2191v \u2192\n          (\u2200 (m : \u2115), m < n_1 \u2192 \u00acf (m ::\u1d65 v) = 0) \u2192\n            \u2203 a, (f (a ::\u1d65 v) = 0 \u2227 \u2200 {m : \u2115}, m < a \u2192 \u00acf (m ::\u1d65 v) = 0) \u2227 [a] = [Nat.pred (List.headI v')]\n\u22a2 \u2200 (n_1 : \u2115),\n    v\u2081 = n_1 :: \u2191v \u2192\n      (\u2200 (m : \u2115), m < n_1 \u2192 \u00acf (m ::\u1d65 v) = 0) \u2192\n        \u2203 a, (f (a ::\u1d65 v) = 0 \u2227 \u2200 {m : \u2115}, m < a \u2192 \u00acf (m ::\u1d65 v) = 0) \u2227 [a] = [Nat.pred (List.headI v')]"}, {"tactic": "clear h1", "annotated_tactic": ["clear h1", []], "state_before": "case intro.mp.intro.intro\nn\u271d : \u2115\nf\u271d : Vector \u2115 n\u271d \u2192. \u2115\nn : \u2115\nf : Vector \u2115 (n + 1) \u2192 \u2115\na\u271d : Nat.Partrec' \u2191f\ncf : Code\nv : Vector \u2115 n\nhf : \u2200 (a : \u2115), eval cf (a :: \u2191v) = Part.some [f (a ::\u1d65 v)]\nv' v\u2080 : List \u2115\nh1 :\n  v' \u2208\n    PFun.fix\n      (fun v =>\n        Part.bind (eval cf v) fun y =>\n          Part.some\n            (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)\n            else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))\n      v\u2080\nv\u2081 : List \u2115\nh2 :\n  v' \u2208\n    PFun.fix\n      (fun v =>\n        Part.bind (eval cf v) fun y =>\n          Part.some\n            (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)\n            else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))\n      v\u2081\nIH :\n  \u2200 (a'' : List \u2115),\n    (Sum.inr a'' \u2208\n        Part.bind (eval cf v\u2081) fun y =>\n          Part.some\n            (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v\u2081) :: List.tail v\u2081)\n            else Sum.inr (Nat.succ (List.headI v\u2081) :: List.tail v\u2081))) \u2192\n      \u2200 (n_1 : \u2115),\n        a'' = n_1 :: \u2191v \u2192\n          (\u2200 (m : \u2115), m < n_1 \u2192 \u00acf (m ::\u1d65 v) = 0) \u2192\n            \u2203 a, (f (a ::\u1d65 v) = 0 \u2227 \u2200 {m : \u2115}, m < a \u2192 \u00acf (m ::\u1d65 v) = 0) \u2227 [a] = [Nat.pred (List.headI v')]\n\u22a2 \u2200 (n_1 : \u2115),\n    v\u2081 = n_1 :: \u2191v \u2192\n      (\u2200 (m : \u2115), m < n_1 \u2192 \u00acf (m ::\u1d65 v) = 0) \u2192\n        \u2203 a, (f (a ::\u1d65 v) = 0 \u2227 \u2200 {m : \u2115}, m < a \u2192 \u00acf (m ::\u1d65 v) = 0) \u2227 [a] = [Nat.pred (List.headI v')]", "state_after": "case intro.mp.intro.intro\nn\u271d : \u2115\nf\u271d : Vector \u2115 n\u271d \u2192. \u2115\nn : \u2115\nf : Vector \u2115 (n + 1) \u2192 \u2115\na\u271d : Nat.Partrec' \u2191f\ncf : Code\nv : Vector \u2115 n\nhf : \u2200 (a : \u2115), eval cf (a :: \u2191v) = Part.some [f (a ::\u1d65 v)]\nv' v\u2080 v\u2081 : List \u2115\nh2 :\n  v' \u2208\n    PFun.fix\n      (fun v =>\n        Part.bind (eval cf v) fun y =>\n          Part.some\n            (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)\n            else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))\n      v\u2081\nIH :\n  \u2200 (a'' : List \u2115),\n    (Sum.inr a'' \u2208\n        Part.bind (eval cf v\u2081) fun y =>\n          Part.some\n            (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v\u2081) :: List.tail v\u2081)\n            else Sum.inr (Nat.succ (List.headI v\u2081) :: List.tail v\u2081))) \u2192\n      \u2200 (n_1 : \u2115),\n        a'' = n_1 :: \u2191v \u2192\n          (\u2200 (m : \u2115), m < n_1 \u2192 \u00acf (m ::\u1d65 v) = 0) \u2192\n            \u2203 a, (f (a ::\u1d65 v) = 0 \u2227 \u2200 {m : \u2115}, m < a \u2192 \u00acf (m ::\u1d65 v) = 0) \u2227 [a] = [Nat.pred (List.headI v')]\n\u22a2 \u2200 (n_1 : \u2115),\n    v\u2081 = n_1 :: \u2191v \u2192\n      (\u2200 (m : \u2115), m < n_1 \u2192 \u00acf (m ::\u1d65 v) = 0) \u2192\n        \u2203 a, (f (a ::\u1d65 v) = 0 \u2227 \u2200 {m : \u2115}, m < a \u2192 \u00acf (m ::\u1d65 v) = 0) \u2227 [a] = [Nat.pred (List.headI v')]"}, {"tactic": "rintro n rfl hm", "annotated_tactic": ["rintro n rfl hm", []], "state_before": "case intro.mp.intro.intro\nn\u271d : \u2115\nf\u271d : Vector \u2115 n\u271d \u2192. \u2115\nn : \u2115\nf : Vector \u2115 (n + 1) \u2192 \u2115\na\u271d : Nat.Partrec' \u2191f\ncf : Code\nv : Vector \u2115 n\nhf : \u2200 (a : \u2115), eval cf (a :: \u2191v) = Part.some [f (a ::\u1d65 v)]\nv' v\u2080 v\u2081 : List \u2115\nh2 :\n  v' \u2208\n    PFun.fix\n      (fun v =>\n        Part.bind (eval cf v) fun y =>\n          Part.some\n            (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)\n            else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))\n      v\u2081\nIH :\n  \u2200 (a'' : List \u2115),\n    (Sum.inr a'' \u2208\n        Part.bind (eval cf v\u2081) fun y =>\n          Part.some\n            (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v\u2081) :: List.tail v\u2081)\n            else Sum.inr (Nat.succ (List.headI v\u2081) :: List.tail v\u2081))) \u2192\n      \u2200 (n_1 : \u2115),\n        a'' = n_1 :: \u2191v \u2192\n          (\u2200 (m : \u2115), m < n_1 \u2192 \u00acf (m ::\u1d65 v) = 0) \u2192\n            \u2203 a, (f (a ::\u1d65 v) = 0 \u2227 \u2200 {m : \u2115}, m < a \u2192 \u00acf (m ::\u1d65 v) = 0) \u2227 [a] = [Nat.pred (List.headI v')]\n\u22a2 \u2200 (n_1 : \u2115),\n    v\u2081 = n_1 :: \u2191v \u2192\n      (\u2200 (m : \u2115), m < n_1 \u2192 \u00acf (m ::\u1d65 v) = 0) \u2192\n        \u2203 a, (f (a ::\u1d65 v) = 0 \u2227 \u2200 {m : \u2115}, m < a \u2192 \u00acf (m ::\u1d65 v) = 0) \u2227 [a] = [Nat.pred (List.headI v')]", "state_after": "case intro.mp.intro.intro\nn\u271d\u00b9 : \u2115\nf\u271d : Vector \u2115 n\u271d\u00b9 \u2192. \u2115\nn\u271d : \u2115\nf : Vector \u2115 (n\u271d + 1) \u2192 \u2115\na\u271d : Nat.Partrec' \u2191f\ncf : Code\nv : Vector \u2115 n\u271d\nhf : \u2200 (a : \u2115), eval cf (a :: \u2191v) = Part.some [f (a ::\u1d65 v)]\nv' v\u2080 : List \u2115\nn : \u2115\nh2 :\n  v' \u2208\n    PFun.fix\n      (fun v =>\n        Part.bind (eval cf v) fun y =>\n          Part.some\n            (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)\n            else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))\n      (n :: \u2191v)\nIH :\n  \u2200 (a'' : List \u2115),\n    (Sum.inr a'' \u2208\n        Part.bind (eval cf (n :: \u2191v)) fun y =>\n          Part.some\n            (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v))\n            else Sum.inr (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v)))) \u2192\n      \u2200 (n : \u2115),\n        a'' = n :: \u2191v \u2192\n          (\u2200 (m : \u2115), m < n \u2192 \u00acf (m ::\u1d65 v) = 0) \u2192\n            \u2203 a, (f (a ::\u1d65 v) = 0 \u2227 \u2200 {m : \u2115}, m < a \u2192 \u00acf (m ::\u1d65 v) = 0) \u2227 [a] = [Nat.pred (List.headI v')]\nhm : \u2200 (m : \u2115), m < n \u2192 \u00acf (m ::\u1d65 v) = 0\n\u22a2 \u2203 a, (f (a ::\u1d65 v) = 0 \u2227 \u2200 {m : \u2115}, m < a \u2192 \u00acf (m ::\u1d65 v) = 0) \u2227 [a] = [Nat.pred (List.headI v')]"}, {"tactic": "have := PFun.mem_fix_iff.1 h2", "annotated_tactic": ["have := <a>PFun.mem_fix_iff</a>.1 h2", [{"full_name": "PFun.mem_fix_iff", "def_path": "Mathlib/Data/PFun.lean", "def_pos": [266, 9], "def_end_pos": [266, 20]}]], "state_before": "case intro.mp.intro.intro\nn\u271d\u00b9 : \u2115\nf\u271d : Vector \u2115 n\u271d\u00b9 \u2192. \u2115\nn\u271d : \u2115\nf : Vector \u2115 (n\u271d + 1) \u2192 \u2115\na\u271d : Nat.Partrec' \u2191f\ncf : Code\nv : Vector \u2115 n\u271d\nhf : \u2200 (a : \u2115), eval cf (a :: \u2191v) = Part.some [f (a ::\u1d65 v)]\nv' v\u2080 : List \u2115\nn : \u2115\nh2 :\n  v' \u2208\n    PFun.fix\n      (fun v =>\n        Part.bind (eval cf v) fun y =>\n          Part.some\n            (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)\n            else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))\n      (n :: \u2191v)\nIH :\n  \u2200 (a'' : List \u2115),\n    (Sum.inr a'' \u2208\n        Part.bind (eval cf (n :: \u2191v)) fun y =>\n          Part.some\n            (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v))\n            else Sum.inr (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v)))) \u2192\n      \u2200 (n : \u2115),\n        a'' = n :: \u2191v \u2192\n          (\u2200 (m : \u2115), m < n \u2192 \u00acf (m ::\u1d65 v) = 0) \u2192\n            \u2203 a, (f (a ::\u1d65 v) = 0 \u2227 \u2200 {m : \u2115}, m < a \u2192 \u00acf (m ::\u1d65 v) = 0) \u2227 [a] = [Nat.pred (List.headI v')]\nhm : \u2200 (m : \u2115), m < n \u2192 \u00acf (m ::\u1d65 v) = 0\n\u22a2 \u2203 a, (f (a ::\u1d65 v) = 0 \u2227 \u2200 {m : \u2115}, m < a \u2192 \u00acf (m ::\u1d65 v) = 0) \u2227 [a] = [Nat.pred (List.headI v')]", "state_after": "case intro.mp.intro.intro\nn\u271d\u00b9 : \u2115\nf\u271d : Vector \u2115 n\u271d\u00b9 \u2192. \u2115\nn\u271d : \u2115\nf : Vector \u2115 (n\u271d + 1) \u2192 \u2115\na\u271d : Nat.Partrec' \u2191f\ncf : Code\nv : Vector \u2115 n\u271d\nhf : \u2200 (a : \u2115), eval cf (a :: \u2191v) = Part.some [f (a ::\u1d65 v)]\nv' v\u2080 : List \u2115\nn : \u2115\nh2 :\n  v' \u2208\n    PFun.fix\n      (fun v =>\n        Part.bind (eval cf v) fun y =>\n          Part.some\n            (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)\n            else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))\n      (n :: \u2191v)\nIH :\n  \u2200 (a'' : List \u2115),\n    (Sum.inr a'' \u2208\n        Part.bind (eval cf (n :: \u2191v)) fun y =>\n          Part.some\n            (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v))\n            else Sum.inr (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v)))) \u2192\n      \u2200 (n : \u2115),\n        a'' = n :: \u2191v \u2192\n          (\u2200 (m : \u2115), m < n \u2192 \u00acf (m ::\u1d65 v) = 0) \u2192\n            \u2203 a, (f (a ::\u1d65 v) = 0 \u2227 \u2200 {m : \u2115}, m < a \u2192 \u00acf (m ::\u1d65 v) = 0) \u2227 [a] = [Nat.pred (List.headI v')]\nhm : \u2200 (m : \u2115), m < n \u2192 \u00acf (m ::\u1d65 v) = 0\nthis :\n  (Sum.inl v' \u2208\n      Part.bind (eval cf (n :: \u2191v)) fun y =>\n        Part.some\n          (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v))\n          else Sum.inr (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v)))) \u2228\n    \u2203 a',\n      (Sum.inr a' \u2208\n          Part.bind (eval cf (n :: \u2191v)) fun y =>\n            Part.some\n              (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v))\n              else Sum.inr (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v)))) \u2227\n        v' \u2208\n          PFun.fix\n            (fun v =>\n              Part.bind (eval cf v) fun y =>\n                Part.some\n                  (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)\n                  else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))\n            a'\n\u22a2 \u2203 a, (f (a ::\u1d65 v) = 0 \u2227 \u2200 {m : \u2115}, m < a \u2192 \u00acf (m ::\u1d65 v) = 0) \u2227 [a] = [Nat.pred (List.headI v')]"}, {"tactic": "simp only [hf, Part.bind_some] at this", "annotated_tactic": ["simp only [hf, <a>Part.bind_some</a>] at this", [{"full_name": "Part.bind_some", "def_path": "Mathlib/Data/Part.lean", "def_pos": [518, 9], "def_end_pos": [518, 18]}]], "state_before": "case intro.mp.intro.intro\nn\u271d\u00b9 : \u2115\nf\u271d : Vector \u2115 n\u271d\u00b9 \u2192. \u2115\nn\u271d : \u2115\nf : Vector \u2115 (n\u271d + 1) \u2192 \u2115\na\u271d : Nat.Partrec' \u2191f\ncf : Code\nv : Vector \u2115 n\u271d\nhf : \u2200 (a : \u2115), eval cf (a :: \u2191v) = Part.some [f (a ::\u1d65 v)]\nv' v\u2080 : List \u2115\nn : \u2115\nh2 :\n  v' \u2208\n    PFun.fix\n      (fun v =>\n        Part.bind (eval cf v) fun y =>\n          Part.some\n            (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)\n            else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))\n      (n :: \u2191v)\nIH :\n  \u2200 (a'' : List \u2115),\n    (Sum.inr a'' \u2208\n        Part.bind (eval cf (n :: \u2191v)) fun y =>\n          Part.some\n            (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v))\n            else Sum.inr (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v)))) \u2192\n      \u2200 (n : \u2115),\n        a'' = n :: \u2191v \u2192\n          (\u2200 (m : \u2115), m < n \u2192 \u00acf (m ::\u1d65 v) = 0) \u2192\n            \u2203 a, (f (a ::\u1d65 v) = 0 \u2227 \u2200 {m : \u2115}, m < a \u2192 \u00acf (m ::\u1d65 v) = 0) \u2227 [a] = [Nat.pred (List.headI v')]\nhm : \u2200 (m : \u2115), m < n \u2192 \u00acf (m ::\u1d65 v) = 0\nthis :\n  (Sum.inl v' \u2208\n      Part.bind (eval cf (n :: \u2191v)) fun y =>\n        Part.some\n          (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v))\n          else Sum.inr (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v)))) \u2228\n    \u2203 a',\n      (Sum.inr a' \u2208\n          Part.bind (eval cf (n :: \u2191v)) fun y =>\n            Part.some\n              (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v))\n              else Sum.inr (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v)))) \u2227\n        v' \u2208\n          PFun.fix\n            (fun v =>\n              Part.bind (eval cf v) fun y =>\n                Part.some\n                  (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)\n                  else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))\n            a'\n\u22a2 \u2203 a, (f (a ::\u1d65 v) = 0 \u2227 \u2200 {m : \u2115}, m < a \u2192 \u00acf (m ::\u1d65 v) = 0) \u2227 [a] = [Nat.pred (List.headI v')]", "state_after": "case intro.mp.intro.intro\nn\u271d\u00b9 : \u2115\nf\u271d : Vector \u2115 n\u271d\u00b9 \u2192. \u2115\nn\u271d : \u2115\nf : Vector \u2115 (n\u271d + 1) \u2192 \u2115\na\u271d : Nat.Partrec' \u2191f\ncf : Code\nv : Vector \u2115 n\u271d\nhf : \u2200 (a : \u2115), eval cf (a :: \u2191v) = Part.some [f (a ::\u1d65 v)]\nv' v\u2080 : List \u2115\nn : \u2115\nh2 :\n  v' \u2208\n    PFun.fix\n      (fun v =>\n        Part.bind (eval cf v) fun y =>\n          Part.some\n            (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)\n            else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))\n      (n :: \u2191v)\nIH :\n  \u2200 (a'' : List \u2115),\n    (Sum.inr a'' \u2208\n        Part.bind (eval cf (n :: \u2191v)) fun y =>\n          Part.some\n            (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v))\n            else Sum.inr (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v)))) \u2192\n      \u2200 (n : \u2115),\n        a'' = n :: \u2191v \u2192\n          (\u2200 (m : \u2115), m < n \u2192 \u00acf (m ::\u1d65 v) = 0) \u2192\n            \u2203 a, (f (a ::\u1d65 v) = 0 \u2227 \u2200 {m : \u2115}, m < a \u2192 \u00acf (m ::\u1d65 v) = 0) \u2227 [a] = [Nat.pred (List.headI v')]\nhm : \u2200 (m : \u2115), m < n \u2192 \u00acf (m ::\u1d65 v) = 0\nthis :\n  Sum.inl v' \u2208\n      Part.some\n        (if List.headI [f (n ::\u1d65 v)] = 0 then Sum.inl (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v))\n        else Sum.inr (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v))) \u2228\n    \u2203 a',\n      Sum.inr a' \u2208\n          Part.some\n            (if List.headI [f (n ::\u1d65 v)] = 0 then Sum.inl (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v))\n            else Sum.inr (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v))) \u2227\n        v' \u2208\n          PFun.fix\n            (fun v =>\n              Part.bind (eval cf v) fun y =>\n                Part.some\n                  (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)\n                  else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))\n            a'\n\u22a2 \u2203 a, (f (a ::\u1d65 v) = 0 \u2227 \u2200 {m : \u2115}, m < a \u2192 \u00acf (m ::\u1d65 v) = 0) \u2227 [a] = [Nat.pred (List.headI v')]"}, {"tactic": "split_ifs at this with h", "annotated_tactic": ["split_ifs at this with h", []], "state_before": "case intro.mp.intro.intro\nn\u271d\u00b9 : \u2115\nf\u271d : Vector \u2115 n\u271d\u00b9 \u2192. \u2115\nn\u271d : \u2115\nf : Vector \u2115 (n\u271d + 1) \u2192 \u2115\na\u271d : Nat.Partrec' \u2191f\ncf : Code\nv : Vector \u2115 n\u271d\nhf : \u2200 (a : \u2115), eval cf (a :: \u2191v) = Part.some [f (a ::\u1d65 v)]\nv' v\u2080 : List \u2115\nn : \u2115\nh2 :\n  v' \u2208\n    PFun.fix\n      (fun v =>\n        Part.bind (eval cf v) fun y =>\n          Part.some\n            (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)\n            else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))\n      (n :: \u2191v)\nIH :\n  \u2200 (a'' : List \u2115),\n    (Sum.inr a'' \u2208\n        Part.bind (eval cf (n :: \u2191v)) fun y =>\n          Part.some\n            (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v))\n            else Sum.inr (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v)))) \u2192\n      \u2200 (n : \u2115),\n        a'' = n :: \u2191v \u2192\n          (\u2200 (m : \u2115), m < n \u2192 \u00acf (m ::\u1d65 v) = 0) \u2192\n            \u2203 a, (f (a ::\u1d65 v) = 0 \u2227 \u2200 {m : \u2115}, m < a \u2192 \u00acf (m ::\u1d65 v) = 0) \u2227 [a] = [Nat.pred (List.headI v')]\nhm : \u2200 (m : \u2115), m < n \u2192 \u00acf (m ::\u1d65 v) = 0\nthis :\n  Sum.inl v' \u2208\n      Part.some\n        (if List.headI [f (n ::\u1d65 v)] = 0 then Sum.inl (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v))\n        else Sum.inr (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v))) \u2228\n    \u2203 a',\n      Sum.inr a' \u2208\n          Part.some\n            (if List.headI [f (n ::\u1d65 v)] = 0 then Sum.inl (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v))\n            else Sum.inr (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v))) \u2227\n        v' \u2208\n          PFun.fix\n            (fun v =>\n              Part.bind (eval cf v) fun y =>\n                Part.some\n                  (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)\n                  else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))\n            a'\n\u22a2 \u2203 a, (f (a ::\u1d65 v) = 0 \u2227 \u2200 {m : \u2115}, m < a \u2192 \u00acf (m ::\u1d65 v) = 0) \u2227 [a] = [Nat.pred (List.headI v')]", "state_after": "case pos\nn\u271d\u00b9 : \u2115\nf\u271d : Vector \u2115 n\u271d\u00b9 \u2192. \u2115\nn\u271d : \u2115\nf : Vector \u2115 (n\u271d + 1) \u2192 \u2115\na\u271d : Nat.Partrec' \u2191f\ncf : Code\nv : Vector \u2115 n\u271d\nhf : \u2200 (a : \u2115), eval cf (a :: \u2191v) = Part.some [f (a ::\u1d65 v)]\nv' v\u2080 : List \u2115\nn : \u2115\nh2 :\n  v' \u2208\n    PFun.fix\n      (fun v =>\n        Part.bind (eval cf v) fun y =>\n          Part.some\n            (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)\n            else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))\n      (n :: \u2191v)\nIH :\n  \u2200 (a'' : List \u2115),\n    (Sum.inr a'' \u2208\n        Part.bind (eval cf (n :: \u2191v)) fun y =>\n          Part.some\n            (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v))\n            else Sum.inr (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v)))) \u2192\n      \u2200 (n : \u2115),\n        a'' = n :: \u2191v \u2192\n          (\u2200 (m : \u2115), m < n \u2192 \u00acf (m ::\u1d65 v) = 0) \u2192\n            \u2203 a, (f (a ::\u1d65 v) = 0 \u2227 \u2200 {m : \u2115}, m < a \u2192 \u00acf (m ::\u1d65 v) = 0) \u2227 [a] = [Nat.pred (List.headI v')]\nhm : \u2200 (m : \u2115), m < n \u2192 \u00acf (m ::\u1d65 v) = 0\nh : List.headI [f (n ::\u1d65 v)] = 0\nthis :\n  Sum.inl v' \u2208 Part.some (Sum.inl (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v))) \u2228\n    \u2203 a',\n      Sum.inr a' \u2208 Part.some (Sum.inl (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v))) \u2227\n        v' \u2208\n          PFun.fix\n            (fun v =>\n              Part.bind (eval cf v) fun y =>\n                Part.some\n                  (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)\n                  else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))\n            a'\n\u22a2 \u2203 a, (f (a ::\u1d65 v) = 0 \u2227 \u2200 {m : \u2115}, m < a \u2192 \u00acf (m ::\u1d65 v) = 0) \u2227 [a] = [Nat.pred (List.headI v')]\n\ncase neg\nn\u271d\u00b9 : \u2115\nf\u271d : Vector \u2115 n\u271d\u00b9 \u2192. \u2115\nn\u271d : \u2115\nf : Vector \u2115 (n\u271d + 1) \u2192 \u2115\na\u271d : Nat.Partrec' \u2191f\ncf : Code\nv : Vector \u2115 n\u271d\nhf : \u2200 (a : \u2115), eval cf (a :: \u2191v) = Part.some [f (a ::\u1d65 v)]\nv' v\u2080 : List \u2115\nn : \u2115\nh2 :\n  v' \u2208\n    PFun.fix\n      (fun v =>\n        Part.bind (eval cf v) fun y =>\n          Part.some\n            (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)\n            else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))\n      (n :: \u2191v)\nIH :\n  \u2200 (a'' : List \u2115),\n    (Sum.inr a'' \u2208\n        Part.bind (eval cf (n :: \u2191v)) fun y =>\n          Part.some\n            (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v))\n            else Sum.inr (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v)))) \u2192\n      \u2200 (n : \u2115),\n        a'' = n :: \u2191v \u2192\n          (\u2200 (m : \u2115), m < n \u2192 \u00acf (m ::\u1d65 v) = 0) \u2192\n            \u2203 a, (f (a ::\u1d65 v) = 0 \u2227 \u2200 {m : \u2115}, m < a \u2192 \u00acf (m ::\u1d65 v) = 0) \u2227 [a] = [Nat.pred (List.headI v')]\nhm : \u2200 (m : \u2115), m < n \u2192 \u00acf (m ::\u1d65 v) = 0\nh : \u00acList.headI [f (n ::\u1d65 v)] = 0\nthis :\n  Sum.inl v' \u2208 Part.some (Sum.inr (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v))) \u2228\n    \u2203 a',\n      Sum.inr a' \u2208 Part.some (Sum.inr (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v))) \u2227\n        v' \u2208\n          PFun.fix\n            (fun v =>\n              Part.bind (eval cf v) fun y =>\n                Part.some\n                  (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)\n                  else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))\n            a'\n\u22a2 \u2203 a, (f (a ::\u1d65 v) = 0 \u2227 \u2200 {m : \u2115}, m < a \u2192 \u00acf (m ::\u1d65 v) = 0) \u2227 [a] = [Nat.pred (List.headI v')]"}, {"tactic": "exact this _ h1 0 rfl (by rintro _ \u27e8\u27e9)", "annotated_tactic": ["exact this _ h1 0 <a>rfl</a> (by rintro _ \u27e8\u27e9)", [{"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "n\u271d : \u2115\nf\u271d : Vector \u2115 n\u271d \u2192. \u2115\nn : \u2115\nf : Vector \u2115 (n + 1) \u2192 \u2115\na\u271d : Nat.Partrec' \u2191f\ncf : Code\nv : Vector \u2115 n\nhf : \u2200 (a : \u2115), eval cf (a :: \u2191v) = Part.some [f (a ::\u1d65 v)]\nv' : List \u2115\nh1 :\n  v' \u2208\n    PFun.fix\n      (fun v =>\n        Part.bind (eval cf v) fun y =>\n          Part.some\n            (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)\n            else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))\n      (0 :: \u2191v)\nthis :\n  \u2200 (v\u2081 : List \u2115),\n    v' \u2208\n        PFun.fix\n          (fun v =>\n            Part.bind (eval cf v) fun y =>\n              Part.some\n                (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)\n                else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))\n          v\u2081 \u2192\n      \u2200 (n_1 : \u2115),\n        v\u2081 = n_1 :: \u2191v \u2192\n          (\u2200 (m : \u2115), m < n_1 \u2192 \u00acf (m ::\u1d65 v) = 0) \u2192\n            \u2203 a, (f (a ::\u1d65 v) = 0 \u2227 \u2200 {m : \u2115}, m < a \u2192 \u00acf (m ::\u1d65 v) = 0) \u2227 [a] = [Nat.pred (List.headI v')]\n\u22a2 \u2203 a, (f (a ::\u1d65 v) = 0 \u2227 \u2200 {m : \u2115}, m < a \u2192 \u00acf (m ::\u1d65 v) = 0) \u2227 pure a = [Nat.pred (List.headI v')]", "state_after": "no goals"}, {"tactic": "rintro _ \u27e8\u27e9", "annotated_tactic": ["rintro _ \u27e8\u27e9", []], "state_before": "n\u271d : \u2115\nf\u271d : Vector \u2115 n\u271d \u2192. \u2115\nn : \u2115\nf : Vector \u2115 (n + 1) \u2192 \u2115\na\u271d : Nat.Partrec' \u2191f\ncf : Code\nv : Vector \u2115 n\nhf : \u2200 (a : \u2115), eval cf (a :: \u2191v) = Part.some [f (a ::\u1d65 v)]\nv' : List \u2115\nh1 :\n  v' \u2208\n    PFun.fix\n      (fun v =>\n        Part.bind (eval cf v) fun y =>\n          Part.some\n            (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)\n            else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))\n      (0 :: \u2191v)\nthis :\n  \u2200 (v\u2081 : List \u2115),\n    v' \u2208\n        PFun.fix\n          (fun v =>\n            Part.bind (eval cf v) fun y =>\n              Part.some\n                (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)\n                else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))\n          v\u2081 \u2192\n      \u2200 (n_1 : \u2115),\n        v\u2081 = n_1 :: \u2191v \u2192\n          (\u2200 (m : \u2115), m < n_1 \u2192 \u00acf (m ::\u1d65 v) = 0) \u2192\n            \u2203 a, (f (a ::\u1d65 v) = 0 \u2227 \u2200 {m : \u2115}, m < a \u2192 \u00acf (m ::\u1d65 v) = 0) \u2227 [a] = [Nat.pred (List.headI v')]\n\u22a2 \u2200 (m : \u2115), m < 0 \u2192 \u00acf (m ::\u1d65 v) = 0", "state_after": "no goals"}, {"tactic": "simp only [List.headI_nil, List.headI_cons, exists_false, or_false_iff, Part.mem_some_iff,\n  List.tail_cons, false_and_iff, Sum.inl.injEq] at this", "annotated_tactic": ["simp only [<a>List.headI_nil</a>, <a>List.headI_cons</a>, <a>exists_false</a>, <a>or_false_iff</a>, <a>Part.mem_some_iff</a>,\n          <a>List.tail_cons</a>, <a>false_and_iff</a>, Sum.inl.injEq] at this", [{"full_name": "List.headI_nil", "def_path": "Mathlib/Init/Data/List/Basic.lean", "def_pos": [44, 17], "def_end_pos": [44, 26]}, {"full_name": "List.headI_cons", "def_path": "Mathlib/Init/Data/List/Basic.lean", "def_pos": [45, 17], "def_end_pos": [45, 27]}, {"full_name": "exists_false", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [433, 17], "def_end_pos": [433, 29]}, {"full_name": "or_false_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [188, 9], "def_end_pos": [188, 21]}, {"full_name": "Part.mem_some_iff", "def_path": "Mathlib/Data/Part.lean", "def_pos": [170, 9], "def_end_pos": [170, 21]}, {"full_name": "List.tail_cons", "def_path": "lake-packages/std/Std/Data/List/Basic.lean", "def_pos": [316, 17], "def_end_pos": [316, 26]}, {"full_name": "false_and_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [151, 9], "def_end_pos": [151, 22]}]], "state_before": "case pos\nn\u271d\u00b9 : \u2115\nf\u271d : Vector \u2115 n\u271d\u00b9 \u2192. \u2115\nn\u271d : \u2115\nf : Vector \u2115 (n\u271d + 1) \u2192 \u2115\na\u271d : Nat.Partrec' \u2191f\ncf : Code\nv : Vector \u2115 n\u271d\nhf : \u2200 (a : \u2115), eval cf (a :: \u2191v) = Part.some [f (a ::\u1d65 v)]\nv' v\u2080 : List \u2115\nn : \u2115\nh2 :\n  v' \u2208\n    PFun.fix\n      (fun v =>\n        Part.bind (eval cf v) fun y =>\n          Part.some\n            (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)\n            else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))\n      (n :: \u2191v)\nIH :\n  \u2200 (a'' : List \u2115),\n    (Sum.inr a'' \u2208\n        Part.bind (eval cf (n :: \u2191v)) fun y =>\n          Part.some\n            (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v))\n            else Sum.inr (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v)))) \u2192\n      \u2200 (n : \u2115),\n        a'' = n :: \u2191v \u2192\n          (\u2200 (m : \u2115), m < n \u2192 \u00acf (m ::\u1d65 v) = 0) \u2192\n            \u2203 a, (f (a ::\u1d65 v) = 0 \u2227 \u2200 {m : \u2115}, m < a \u2192 \u00acf (m ::\u1d65 v) = 0) \u2227 [a] = [Nat.pred (List.headI v')]\nhm : \u2200 (m : \u2115), m < n \u2192 \u00acf (m ::\u1d65 v) = 0\nh : List.headI [f (n ::\u1d65 v)] = 0\nthis :\n  Sum.inl v' \u2208 Part.some (Sum.inl (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v))) \u2228\n    \u2203 a',\n      Sum.inr a' \u2208 Part.some (Sum.inl (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v))) \u2227\n        v' \u2208\n          PFun.fix\n            (fun v =>\n              Part.bind (eval cf v) fun y =>\n                Part.some\n                  (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)\n                  else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))\n            a'\n\u22a2 \u2203 a, (f (a ::\u1d65 v) = 0 \u2227 \u2200 {m : \u2115}, m < a \u2192 \u00acf (m ::\u1d65 v) = 0) \u2227 [a] = [Nat.pred (List.headI v')]", "state_after": "case pos\nn\u271d\u00b9 : \u2115\nf\u271d : Vector \u2115 n\u271d\u00b9 \u2192. \u2115\nn\u271d : \u2115\nf : Vector \u2115 (n\u271d + 1) \u2192 \u2115\na\u271d : Nat.Partrec' \u2191f\ncf : Code\nv : Vector \u2115 n\u271d\nhf : \u2200 (a : \u2115), eval cf (a :: \u2191v) = Part.some [f (a ::\u1d65 v)]\nv' v\u2080 : List \u2115\nn : \u2115\nh2 :\n  v' \u2208\n    PFun.fix\n      (fun v =>\n        Part.bind (eval cf v) fun y =>\n          Part.some\n            (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)\n            else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))\n      (n :: \u2191v)\nIH :\n  \u2200 (a'' : List \u2115),\n    (Sum.inr a'' \u2208\n        Part.bind (eval cf (n :: \u2191v)) fun y =>\n          Part.some\n            (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v))\n            else Sum.inr (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v)))) \u2192\n      \u2200 (n : \u2115),\n        a'' = n :: \u2191v \u2192\n          (\u2200 (m : \u2115), m < n \u2192 \u00acf (m ::\u1d65 v) = 0) \u2192\n            \u2203 a, (f (a ::\u1d65 v) = 0 \u2227 \u2200 {m : \u2115}, m < a \u2192 \u00acf (m ::\u1d65 v) = 0) \u2227 [a] = [Nat.pred (List.headI v')]\nhm : \u2200 (m : \u2115), m < n \u2192 \u00acf (m ::\u1d65 v) = 0\nh : List.headI [f (n ::\u1d65 v)] = 0\nthis : v' = Nat.succ n :: \u2191v\n\u22a2 \u2203 a, (f (a ::\u1d65 v) = 0 \u2227 \u2200 {m : \u2115}, m < a \u2192 \u00acf (m ::\u1d65 v) = 0) \u2227 [a] = [Nat.pred (List.headI v')]"}, {"tactic": "subst this", "annotated_tactic": ["subst this", []], "state_before": "case pos\nn\u271d\u00b9 : \u2115\nf\u271d : Vector \u2115 n\u271d\u00b9 \u2192. \u2115\nn\u271d : \u2115\nf : Vector \u2115 (n\u271d + 1) \u2192 \u2115\na\u271d : Nat.Partrec' \u2191f\ncf : Code\nv : Vector \u2115 n\u271d\nhf : \u2200 (a : \u2115), eval cf (a :: \u2191v) = Part.some [f (a ::\u1d65 v)]\nv' v\u2080 : List \u2115\nn : \u2115\nh2 :\n  v' \u2208\n    PFun.fix\n      (fun v =>\n        Part.bind (eval cf v) fun y =>\n          Part.some\n            (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)\n            else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))\n      (n :: \u2191v)\nIH :\n  \u2200 (a'' : List \u2115),\n    (Sum.inr a'' \u2208\n        Part.bind (eval cf (n :: \u2191v)) fun y =>\n          Part.some\n            (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v))\n            else Sum.inr (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v)))) \u2192\n      \u2200 (n : \u2115),\n        a'' = n :: \u2191v \u2192\n          (\u2200 (m : \u2115), m < n \u2192 \u00acf (m ::\u1d65 v) = 0) \u2192\n            \u2203 a, (f (a ::\u1d65 v) = 0 \u2227 \u2200 {m : \u2115}, m < a \u2192 \u00acf (m ::\u1d65 v) = 0) \u2227 [a] = [Nat.pred (List.headI v')]\nhm : \u2200 (m : \u2115), m < n \u2192 \u00acf (m ::\u1d65 v) = 0\nh : List.headI [f (n ::\u1d65 v)] = 0\nthis : v' = Nat.succ n :: \u2191v\n\u22a2 \u2203 a, (f (a ::\u1d65 v) = 0 \u2227 \u2200 {m : \u2115}, m < a \u2192 \u00acf (m ::\u1d65 v) = 0) \u2227 [a] = [Nat.pred (List.headI v')]", "state_after": "case pos\nn\u271d\u00b9 : \u2115\nf\u271d : Vector \u2115 n\u271d\u00b9 \u2192. \u2115\nn\u271d : \u2115\nf : Vector \u2115 (n\u271d + 1) \u2192 \u2115\na\u271d : Nat.Partrec' \u2191f\ncf : Code\nv : Vector \u2115 n\u271d\nhf : \u2200 (a : \u2115), eval cf (a :: \u2191v) = Part.some [f (a ::\u1d65 v)]\nv\u2080 : List \u2115\nn : \u2115\nhm : \u2200 (m : \u2115), m < n \u2192 \u00acf (m ::\u1d65 v) = 0\nh : List.headI [f (n ::\u1d65 v)] = 0\nh2 :\n  Nat.succ n :: \u2191v \u2208\n    PFun.fix\n      (fun v =>\n        Part.bind (eval cf v) fun y =>\n          Part.some\n            (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)\n            else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))\n      (n :: \u2191v)\nIH :\n  \u2200 (a'' : List \u2115),\n    (Sum.inr a'' \u2208\n        Part.bind (eval cf (n :: \u2191v)) fun y =>\n          Part.some\n            (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v))\n            else Sum.inr (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v)))) \u2192\n      \u2200 (n_1 : \u2115),\n        a'' = n_1 :: \u2191v \u2192\n          (\u2200 (m : \u2115), m < n_1 \u2192 \u00acf (m ::\u1d65 v) = 0) \u2192\n            \u2203 a,\n              (f (a ::\u1d65 v) = 0 \u2227 \u2200 {m : \u2115}, m < a \u2192 \u00acf (m ::\u1d65 v) = 0) \u2227 [a] = [Nat.pred (List.headI (Nat.succ n :: \u2191v))]\n\u22a2 \u2203 a, (f (a ::\u1d65 v) = 0 \u2227 \u2200 {m : \u2115}, m < a \u2192 \u00acf (m ::\u1d65 v) = 0) \u2227 [a] = [Nat.pred (List.headI (Nat.succ n :: \u2191v))]"}, {"tactic": "exact \u27e8_, \u27e8h, @(hm)\u27e9, rfl\u27e9", "annotated_tactic": ["exact \u27e8_, \u27e8h, @(hm)\u27e9, <a>rfl</a>\u27e9", [{"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "case pos\nn\u271d\u00b9 : \u2115\nf\u271d : Vector \u2115 n\u271d\u00b9 \u2192. \u2115\nn\u271d : \u2115\nf : Vector \u2115 (n\u271d + 1) \u2192 \u2115\na\u271d : Nat.Partrec' \u2191f\ncf : Code\nv : Vector \u2115 n\u271d\nhf : \u2200 (a : \u2115), eval cf (a :: \u2191v) = Part.some [f (a ::\u1d65 v)]\nv\u2080 : List \u2115\nn : \u2115\nhm : \u2200 (m : \u2115), m < n \u2192 \u00acf (m ::\u1d65 v) = 0\nh : List.headI [f (n ::\u1d65 v)] = 0\nh2 :\n  Nat.succ n :: \u2191v \u2208\n    PFun.fix\n      (fun v =>\n        Part.bind (eval cf v) fun y =>\n          Part.some\n            (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)\n            else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))\n      (n :: \u2191v)\nIH :\n  \u2200 (a'' : List \u2115),\n    (Sum.inr a'' \u2208\n        Part.bind (eval cf (n :: \u2191v)) fun y =>\n          Part.some\n            (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v))\n            else Sum.inr (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v)))) \u2192\n      \u2200 (n_1 : \u2115),\n        a'' = n_1 :: \u2191v \u2192\n          (\u2200 (m : \u2115), m < n_1 \u2192 \u00acf (m ::\u1d65 v) = 0) \u2192\n            \u2203 a,\n              (f (a ::\u1d65 v) = 0 \u2227 \u2200 {m : \u2115}, m < a \u2192 \u00acf (m ::\u1d65 v) = 0) \u2227 [a] = [Nat.pred (List.headI (Nat.succ n :: \u2191v))]\n\u22a2 \u2203 a, (f (a ::\u1d65 v) = 0 \u2227 \u2200 {m : \u2115}, m < a \u2192 \u00acf (m ::\u1d65 v) = 0) \u2227 [a] = [Nat.pred (List.headI (Nat.succ n :: \u2191v))]", "state_after": "no goals"}, {"tactic": "refine' IH (n.succ::v.val) (by simp_all) _ rfl fun m h' => _", "annotated_tactic": ["refine' IH (n.succ::v.val) (by simp_all) _ <a>rfl</a> fun m h' => _", [{"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "case neg\nn\u271d\u00b9 : \u2115\nf\u271d : Vector \u2115 n\u271d\u00b9 \u2192. \u2115\nn\u271d : \u2115\nf : Vector \u2115 (n\u271d + 1) \u2192 \u2115\na\u271d : Nat.Partrec' \u2191f\ncf : Code\nv : Vector \u2115 n\u271d\nhf : \u2200 (a : \u2115), eval cf (a :: \u2191v) = Part.some [f (a ::\u1d65 v)]\nv' v\u2080 : List \u2115\nn : \u2115\nh2 :\n  v' \u2208\n    PFun.fix\n      (fun v =>\n        Part.bind (eval cf v) fun y =>\n          Part.some\n            (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)\n            else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))\n      (n :: \u2191v)\nIH :\n  \u2200 (a'' : List \u2115),\n    (Sum.inr a'' \u2208\n        Part.bind (eval cf (n :: \u2191v)) fun y =>\n          Part.some\n            (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v))\n            else Sum.inr (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v)))) \u2192\n      \u2200 (n : \u2115),\n        a'' = n :: \u2191v \u2192\n          (\u2200 (m : \u2115), m < n \u2192 \u00acf (m ::\u1d65 v) = 0) \u2192\n            \u2203 a, (f (a ::\u1d65 v) = 0 \u2227 \u2200 {m : \u2115}, m < a \u2192 \u00acf (m ::\u1d65 v) = 0) \u2227 [a] = [Nat.pred (List.headI v')]\nhm : \u2200 (m : \u2115), m < n \u2192 \u00acf (m ::\u1d65 v) = 0\nh : \u00acList.headI [f (n ::\u1d65 v)] = 0\nthis :\n  Sum.inl v' \u2208 Part.some (Sum.inr (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v))) \u2228\n    \u2203 a',\n      Sum.inr a' \u2208 Part.some (Sum.inr (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v))) \u2227\n        v' \u2208\n          PFun.fix\n            (fun v =>\n              Part.bind (eval cf v) fun y =>\n                Part.some\n                  (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)\n                  else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))\n            a'\n\u22a2 \u2203 a, (f (a ::\u1d65 v) = 0 \u2227 \u2200 {m : \u2115}, m < a \u2192 \u00acf (m ::\u1d65 v) = 0) \u2227 [a] = [Nat.pred (List.headI v')]", "state_after": "case neg\nn\u271d\u00b9 : \u2115\nf\u271d : Vector \u2115 n\u271d\u00b9 \u2192. \u2115\nn\u271d : \u2115\nf : Vector \u2115 (n\u271d + 1) \u2192 \u2115\na\u271d : Nat.Partrec' \u2191f\ncf : Code\nv : Vector \u2115 n\u271d\nhf : \u2200 (a : \u2115), eval cf (a :: \u2191v) = Part.some [f (a ::\u1d65 v)]\nv' v\u2080 : List \u2115\nn : \u2115\nh2 :\n  v' \u2208\n    PFun.fix\n      (fun v =>\n        Part.bind (eval cf v) fun y =>\n          Part.some\n            (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)\n            else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))\n      (n :: \u2191v)\nIH :\n  \u2200 (a'' : List \u2115),\n    (Sum.inr a'' \u2208\n        Part.bind (eval cf (n :: \u2191v)) fun y =>\n          Part.some\n            (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v))\n            else Sum.inr (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v)))) \u2192\n      \u2200 (n : \u2115),\n        a'' = n :: \u2191v \u2192\n          (\u2200 (m : \u2115), m < n \u2192 \u00acf (m ::\u1d65 v) = 0) \u2192\n            \u2203 a, (f (a ::\u1d65 v) = 0 \u2227 \u2200 {m : \u2115}, m < a \u2192 \u00acf (m ::\u1d65 v) = 0) \u2227 [a] = [Nat.pred (List.headI v')]\nhm : \u2200 (m : \u2115), m < n \u2192 \u00acf (m ::\u1d65 v) = 0\nh : \u00acList.headI [f (n ::\u1d65 v)] = 0\nthis :\n  Sum.inl v' \u2208 Part.some (Sum.inr (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v))) \u2228\n    \u2203 a',\n      Sum.inr a' \u2208 Part.some (Sum.inr (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v))) \u2227\n        v' \u2208\n          PFun.fix\n            (fun v =>\n              Part.bind (eval cf v) fun y =>\n                Part.some\n                  (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)\n                  else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))\n            a'\nm : \u2115\nh' : m < Nat.succ n\n\u22a2 \u00acf (m ::\u1d65 v) = 0"}, {"tactic": "obtain h | rfl := Nat.lt_succ_iff_lt_or_eq.1 h'", "annotated_tactic": ["obtain h | rfl := <a>Nat.lt_succ_iff_lt_or_eq</a>.1 h'", [{"full_name": "Nat.lt_succ_iff_lt_or_eq", "def_path": "Mathlib/Data/Nat/Basic.lean", "def_pos": [375, 9], "def_end_pos": [375, 29]}]], "state_before": "case neg\nn\u271d\u00b9 : \u2115\nf\u271d : Vector \u2115 n\u271d\u00b9 \u2192. \u2115\nn\u271d : \u2115\nf : Vector \u2115 (n\u271d + 1) \u2192 \u2115\na\u271d : Nat.Partrec' \u2191f\ncf : Code\nv : Vector \u2115 n\u271d\nhf : \u2200 (a : \u2115), eval cf (a :: \u2191v) = Part.some [f (a ::\u1d65 v)]\nv' v\u2080 : List \u2115\nn : \u2115\nh2 :\n  v' \u2208\n    PFun.fix\n      (fun v =>\n        Part.bind (eval cf v) fun y =>\n          Part.some\n            (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)\n            else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))\n      (n :: \u2191v)\nIH :\n  \u2200 (a'' : List \u2115),\n    (Sum.inr a'' \u2208\n        Part.bind (eval cf (n :: \u2191v)) fun y =>\n          Part.some\n            (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v))\n            else Sum.inr (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v)))) \u2192\n      \u2200 (n : \u2115),\n        a'' = n :: \u2191v \u2192\n          (\u2200 (m : \u2115), m < n \u2192 \u00acf (m ::\u1d65 v) = 0) \u2192\n            \u2203 a, (f (a ::\u1d65 v) = 0 \u2227 \u2200 {m : \u2115}, m < a \u2192 \u00acf (m ::\u1d65 v) = 0) \u2227 [a] = [Nat.pred (List.headI v')]\nhm : \u2200 (m : \u2115), m < n \u2192 \u00acf (m ::\u1d65 v) = 0\nh : \u00acList.headI [f (n ::\u1d65 v)] = 0\nthis :\n  Sum.inl v' \u2208 Part.some (Sum.inr (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v))) \u2228\n    \u2203 a',\n      Sum.inr a' \u2208 Part.some (Sum.inr (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v))) \u2227\n        v' \u2208\n          PFun.fix\n            (fun v =>\n              Part.bind (eval cf v) fun y =>\n                Part.some\n                  (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)\n                  else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))\n            a'\nm : \u2115\nh' : m < Nat.succ n\n\u22a2 \u00acf (m ::\u1d65 v) = 0", "state_after": "case neg.inl\nn\u271d\u00b9 : \u2115\nf\u271d : Vector \u2115 n\u271d\u00b9 \u2192. \u2115\nn\u271d : \u2115\nf : Vector \u2115 (n\u271d + 1) \u2192 \u2115\na\u271d : Nat.Partrec' \u2191f\ncf : Code\nv : Vector \u2115 n\u271d\nhf : \u2200 (a : \u2115), eval cf (a :: \u2191v) = Part.some [f (a ::\u1d65 v)]\nv' v\u2080 : List \u2115\nn : \u2115\nh2 :\n  v' \u2208\n    PFun.fix\n      (fun v =>\n        Part.bind (eval cf v) fun y =>\n          Part.some\n            (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)\n            else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))\n      (n :: \u2191v)\nIH :\n  \u2200 (a'' : List \u2115),\n    (Sum.inr a'' \u2208\n        Part.bind (eval cf (n :: \u2191v)) fun y =>\n          Part.some\n            (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v))\n            else Sum.inr (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v)))) \u2192\n      \u2200 (n : \u2115),\n        a'' = n :: \u2191v \u2192\n          (\u2200 (m : \u2115), m < n \u2192 \u00acf (m ::\u1d65 v) = 0) \u2192\n            \u2203 a, (f (a ::\u1d65 v) = 0 \u2227 \u2200 {m : \u2115}, m < a \u2192 \u00acf (m ::\u1d65 v) = 0) \u2227 [a] = [Nat.pred (List.headI v')]\nhm : \u2200 (m : \u2115), m < n \u2192 \u00acf (m ::\u1d65 v) = 0\nh\u271d : \u00acList.headI [f (n ::\u1d65 v)] = 0\nthis :\n  Sum.inl v' \u2208 Part.some (Sum.inr (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v))) \u2228\n    \u2203 a',\n      Sum.inr a' \u2208 Part.some (Sum.inr (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v))) \u2227\n        v' \u2208\n          PFun.fix\n            (fun v =>\n              Part.bind (eval cf v) fun y =>\n                Part.some\n                  (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)\n                  else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))\n            a'\nm : \u2115\nh' : m < Nat.succ n\nh : m < n\n\u22a2 \u00acf (m ::\u1d65 v) = 0\n\ncase neg.inr\nn\u271d : \u2115\nf\u271d : Vector \u2115 n\u271d \u2192. \u2115\nn : \u2115\nf : Vector \u2115 (n + 1) \u2192 \u2115\na\u271d : Nat.Partrec' \u2191f\ncf : Code\nv : Vector \u2115 n\nhf : \u2200 (a : \u2115), eval cf (a :: \u2191v) = Part.some [f (a ::\u1d65 v)]\nv' v\u2080 : List \u2115\nm : \u2115\nh2 :\n  v' \u2208\n    PFun.fix\n      (fun v =>\n        Part.bind (eval cf v) fun y =>\n          Part.some\n            (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)\n            else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))\n      (m :: \u2191v)\nIH :\n  \u2200 (a'' : List \u2115),\n    (Sum.inr a'' \u2208\n        Part.bind (eval cf (m :: \u2191v)) fun y =>\n          Part.some\n            (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI (m :: \u2191v)) :: List.tail (m :: \u2191v))\n            else Sum.inr (Nat.succ (List.headI (m :: \u2191v)) :: List.tail (m :: \u2191v)))) \u2192\n      \u2200 (n_1 : \u2115),\n        a'' = n_1 :: \u2191v \u2192\n          (\u2200 (m : \u2115), m < n_1 \u2192 \u00acf (m ::\u1d65 v) = 0) \u2192\n            \u2203 a, (f (a ::\u1d65 v) = 0 \u2227 \u2200 {m : \u2115}, m < a \u2192 \u00acf (m ::\u1d65 v) = 0) \u2227 [a] = [Nat.pred (List.headI v')]\nhm : \u2200 (m_1 : \u2115), m_1 < m \u2192 \u00acf (m_1 ::\u1d65 v) = 0\nh : \u00acList.headI [f (m ::\u1d65 v)] = 0\nthis :\n  Sum.inl v' \u2208 Part.some (Sum.inr (Nat.succ (List.headI (m :: \u2191v)) :: List.tail (m :: \u2191v))) \u2228\n    \u2203 a',\n      Sum.inr a' \u2208 Part.some (Sum.inr (Nat.succ (List.headI (m :: \u2191v)) :: List.tail (m :: \u2191v))) \u2227\n        v' \u2208\n          PFun.fix\n            (fun v =>\n              Part.bind (eval cf v) fun y =>\n                Part.some\n                  (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)\n                  else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))\n            a'\nh' : m < Nat.succ m\n\u22a2 \u00acf (m ::\u1d65 v) = 0"}, {"tactic": "exacts [hm _ h, h]", "annotated_tactic": ["exacts [hm _ h, h]", []], "state_before": "case neg.inl\nn\u271d\u00b9 : \u2115\nf\u271d : Vector \u2115 n\u271d\u00b9 \u2192. \u2115\nn\u271d : \u2115\nf : Vector \u2115 (n\u271d + 1) \u2192 \u2115\na\u271d : Nat.Partrec' \u2191f\ncf : Code\nv : Vector \u2115 n\u271d\nhf : \u2200 (a : \u2115), eval cf (a :: \u2191v) = Part.some [f (a ::\u1d65 v)]\nv' v\u2080 : List \u2115\nn : \u2115\nh2 :\n  v' \u2208\n    PFun.fix\n      (fun v =>\n        Part.bind (eval cf v) fun y =>\n          Part.some\n            (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)\n            else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))\n      (n :: \u2191v)\nIH :\n  \u2200 (a'' : List \u2115),\n    (Sum.inr a'' \u2208\n        Part.bind (eval cf (n :: \u2191v)) fun y =>\n          Part.some\n            (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v))\n            else Sum.inr (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v)))) \u2192\n      \u2200 (n : \u2115),\n        a'' = n :: \u2191v \u2192\n          (\u2200 (m : \u2115), m < n \u2192 \u00acf (m ::\u1d65 v) = 0) \u2192\n            \u2203 a, (f (a ::\u1d65 v) = 0 \u2227 \u2200 {m : \u2115}, m < a \u2192 \u00acf (m ::\u1d65 v) = 0) \u2227 [a] = [Nat.pred (List.headI v')]\nhm : \u2200 (m : \u2115), m < n \u2192 \u00acf (m ::\u1d65 v) = 0\nh\u271d : \u00acList.headI [f (n ::\u1d65 v)] = 0\nthis :\n  Sum.inl v' \u2208 Part.some (Sum.inr (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v))) \u2228\n    \u2203 a',\n      Sum.inr a' \u2208 Part.some (Sum.inr (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v))) \u2227\n        v' \u2208\n          PFun.fix\n            (fun v =>\n              Part.bind (eval cf v) fun y =>\n                Part.some\n                  (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)\n                  else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))\n            a'\nm : \u2115\nh' : m < Nat.succ n\nh : m < n\n\u22a2 \u00acf (m ::\u1d65 v) = 0\n\ncase neg.inr\nn\u271d : \u2115\nf\u271d : Vector \u2115 n\u271d \u2192. \u2115\nn : \u2115\nf : Vector \u2115 (n + 1) \u2192 \u2115\na\u271d : Nat.Partrec' \u2191f\ncf : Code\nv : Vector \u2115 n\nhf : \u2200 (a : \u2115), eval cf (a :: \u2191v) = Part.some [f (a ::\u1d65 v)]\nv' v\u2080 : List \u2115\nm : \u2115\nh2 :\n  v' \u2208\n    PFun.fix\n      (fun v =>\n        Part.bind (eval cf v) fun y =>\n          Part.some\n            (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)\n            else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))\n      (m :: \u2191v)\nIH :\n  \u2200 (a'' : List \u2115),\n    (Sum.inr a'' \u2208\n        Part.bind (eval cf (m :: \u2191v)) fun y =>\n          Part.some\n            (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI (m :: \u2191v)) :: List.tail (m :: \u2191v))\n            else Sum.inr (Nat.succ (List.headI (m :: \u2191v)) :: List.tail (m :: \u2191v)))) \u2192\n      \u2200 (n_1 : \u2115),\n        a'' = n_1 :: \u2191v \u2192\n          (\u2200 (m : \u2115), m < n_1 \u2192 \u00acf (m ::\u1d65 v) = 0) \u2192\n            \u2203 a, (f (a ::\u1d65 v) = 0 \u2227 \u2200 {m : \u2115}, m < a \u2192 \u00acf (m ::\u1d65 v) = 0) \u2227 [a] = [Nat.pred (List.headI v')]\nhm : \u2200 (m_1 : \u2115), m_1 < m \u2192 \u00acf (m_1 ::\u1d65 v) = 0\nh : \u00acList.headI [f (m ::\u1d65 v)] = 0\nthis :\n  Sum.inl v' \u2208 Part.some (Sum.inr (Nat.succ (List.headI (m :: \u2191v)) :: List.tail (m :: \u2191v))) \u2228\n    \u2203 a',\n      Sum.inr a' \u2208 Part.some (Sum.inr (Nat.succ (List.headI (m :: \u2191v)) :: List.tail (m :: \u2191v))) \u2227\n        v' \u2208\n          PFun.fix\n            (fun v =>\n              Part.bind (eval cf v) fun y =>\n                Part.some\n                  (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)\n                  else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))\n            a'\nh' : m < Nat.succ m\n\u22a2 \u00acf (m ::\u1d65 v) = 0", "state_after": "no goals"}, {"tactic": "simp_all", "annotated_tactic": ["simp_all", []], "state_before": "n\u271d\u00b9 : \u2115\nf\u271d : Vector \u2115 n\u271d\u00b9 \u2192. \u2115\nn\u271d : \u2115\nf : Vector \u2115 (n\u271d + 1) \u2192 \u2115\na\u271d : Nat.Partrec' \u2191f\ncf : Code\nv : Vector \u2115 n\u271d\nhf : \u2200 (a : \u2115), eval cf (a :: \u2191v) = Part.some [f (a ::\u1d65 v)]\nv' v\u2080 : List \u2115\nn : \u2115\nh2 :\n  v' \u2208\n    PFun.fix\n      (fun v =>\n        Part.bind (eval cf v) fun y =>\n          Part.some\n            (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)\n            else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))\n      (n :: \u2191v)\nIH :\n  \u2200 (a'' : List \u2115),\n    (Sum.inr a'' \u2208\n        Part.bind (eval cf (n :: \u2191v)) fun y =>\n          Part.some\n            (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v))\n            else Sum.inr (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v)))) \u2192\n      \u2200 (n : \u2115),\n        a'' = n :: \u2191v \u2192\n          (\u2200 (m : \u2115), m < n \u2192 \u00acf (m ::\u1d65 v) = 0) \u2192\n            \u2203 a, (f (a ::\u1d65 v) = 0 \u2227 \u2200 {m : \u2115}, m < a \u2192 \u00acf (m ::\u1d65 v) = 0) \u2227 [a] = [Nat.pred (List.headI v')]\nhm : \u2200 (m : \u2115), m < n \u2192 \u00acf (m ::\u1d65 v) = 0\nh : \u00acList.headI [f (n ::\u1d65 v)] = 0\nthis :\n  Sum.inl v' \u2208 Part.some (Sum.inr (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v))) \u2228\n    \u2203 a',\n      Sum.inr a' \u2208 Part.some (Sum.inr (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v))) \u2227\n        v' \u2208\n          PFun.fix\n            (fun v =>\n              Part.bind (eval cf v) fun y =>\n                Part.some\n                  (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)\n                  else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))\n            a'\n\u22a2 Sum.inr (Nat.succ n :: \u2191v) \u2208\n    Part.bind (eval cf (n :: \u2191v)) fun y =>\n      Part.some\n        (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v))\n        else Sum.inr (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v)))", "state_after": "no goals"}, {"tactic": "rintro \u27e8n, \u27e8hn, hm\u27e9, rfl\u27e9", "annotated_tactic": ["rintro \u27e8n, \u27e8hn, hm\u27e9, rfl\u27e9", []], "state_before": "case intro.mpr\nn\u271d : \u2115\nf\u271d : Vector \u2115 n\u271d \u2192. \u2115\nn : \u2115\nf : Vector \u2115 (n + 1) \u2192 \u2115\na\u271d : Nat.Partrec' \u2191f\ncf : Code\nv : Vector \u2115 n\nhf : \u2200 (a : \u2115), eval cf (a :: \u2191v) = Part.some [f (a ::\u1d65 v)]\nx : List \u2115\n\u22a2 (\u2203 a, (f (a ::\u1d65 v) = 0 \u2227 \u2200 {m : \u2115}, m < a \u2192 \u00acf (m ::\u1d65 v) = 0) \u2227 pure a = x) \u2192\n    \u2203 a,\n      a \u2208\n          PFun.fix\n            (fun v =>\n              Part.bind (eval cf v) fun y =>\n                Part.some\n                  (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)\n                  else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))\n            (0 :: \u2191v) \u2227\n        x = [Nat.pred (List.headI a)]", "state_after": "case intro.mpr.intro.intro.intro\nn\u271d\u00b9 : \u2115\nf\u271d : Vector \u2115 n\u271d\u00b9 \u2192. \u2115\nn\u271d : \u2115\nf : Vector \u2115 (n\u271d + 1) \u2192 \u2115\na\u271d : Nat.Partrec' \u2191f\ncf : Code\nv : Vector \u2115 n\u271d\nhf : \u2200 (a : \u2115), eval cf (a :: \u2191v) = Part.some [f (a ::\u1d65 v)]\nn : \u2115\nhn : f (n ::\u1d65 v) = 0\nhm : \u2200 {m : \u2115}, m < n \u2192 \u00acf (m ::\u1d65 v) = 0\n\u22a2 \u2203 a,\n    a \u2208\n        PFun.fix\n          (fun v =>\n            Part.bind (eval cf v) fun y =>\n              Part.some\n                (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)\n                else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))\n          (0 :: \u2191v) \u2227\n      pure n = [Nat.pred (List.headI a)]"}, {"tactic": "refine' \u27e8n.succ::v.1, _, rfl\u27e9", "annotated_tactic": ["refine' \u27e8n.succ::v.1, _, <a>rfl</a>\u27e9", [{"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "case intro.mpr.intro.intro.intro\nn\u271d\u00b9 : \u2115\nf\u271d : Vector \u2115 n\u271d\u00b9 \u2192. \u2115\nn\u271d : \u2115\nf : Vector \u2115 (n\u271d + 1) \u2192 \u2115\na\u271d : Nat.Partrec' \u2191f\ncf : Code\nv : Vector \u2115 n\u271d\nhf : \u2200 (a : \u2115), eval cf (a :: \u2191v) = Part.some [f (a ::\u1d65 v)]\nn : \u2115\nhn : f (n ::\u1d65 v) = 0\nhm : \u2200 {m : \u2115}, m < n \u2192 \u00acf (m ::\u1d65 v) = 0\n\u22a2 \u2203 a,\n    a \u2208\n        PFun.fix\n          (fun v =>\n            Part.bind (eval cf v) fun y =>\n              Part.some\n                (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)\n                else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))\n          (0 :: \u2191v) \u2227\n      pure n = [Nat.pred (List.headI a)]", "state_after": "case intro.mpr.intro.intro.intro\nn\u271d\u00b9 : \u2115\nf\u271d : Vector \u2115 n\u271d\u00b9 \u2192. \u2115\nn\u271d : \u2115\nf : Vector \u2115 (n\u271d + 1) \u2192 \u2115\na\u271d : Nat.Partrec' \u2191f\ncf : Code\nv : Vector \u2115 n\u271d\nhf : \u2200 (a : \u2115), eval cf (a :: \u2191v) = Part.some [f (a ::\u1d65 v)]\nn : \u2115\nhn : f (n ::\u1d65 v) = 0\nhm : \u2200 {m : \u2115}, m < n \u2192 \u00acf (m ::\u1d65 v) = 0\n\u22a2 Nat.succ n :: \u2191v \u2208\n    PFun.fix\n      (fun v =>\n        Part.bind (eval cf v) fun y =>\n          Part.some\n            (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)\n            else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))\n      (0 :: \u2191v)"}, {"tactic": "have : (n.succ::v.1 : List \u2115) \u2208\n  PFun.fix (fun v =>\n    (cf.eval v).bind fun y =>\n      Part.some <|\n        if y.headI = 0 then Sum.inl (v.headI.succ :: v.tail)\n          else Sum.inr (v.headI.succ :: v.tail))\n      (n::v.val) :=\n  PFun.mem_fix_iff.2 (Or.inl (by simp [hf, hn]))", "annotated_tactic": ["have : (n.succ::v.1 : <a>List</a> \u2115) \u2208\n        <a>PFun.fix</a> (fun v =>\n          (cf.eval v).<a>bind</a> fun y =>\n            <a>Part.some</a> <|\n              if y.headI = 0 then <a>Sum.inl</a> (v.headI.succ :: v.tail)\n                else <a>Sum.inr</a> (v.headI.succ :: v.tail))\n            (n::v.val) :=\n        <a>PFun.mem_fix_iff</a>.2 (<a>Or.inl</a> (by simp [hf, hn]))", [{"full_name": "List", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2182, 11], "def_end_pos": [2182, 15]}, {"full_name": "PFun.fix", "def_path": "Mathlib/Data/PFun.lean", "def_pos": [250, 5], "def_end_pos": [250, 8]}, {"full_name": "Part.bind", "def_path": "Mathlib/Data/Part.lean", "def_pos": [428, 15], "def_end_pos": [428, 19]}, {"full_name": "Part.some", "def_path": "Mathlib/Data/Part.lean", "def_pos": [135, 5], "def_end_pos": [135, 9]}, {"full_name": "Sum.inl", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [104, 5], "def_end_pos": [104, 8]}, {"full_name": "Sum.inr", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [106, 5], "def_end_pos": [106, 8]}, {"full_name": "PFun.mem_fix_iff", "def_path": "Mathlib/Data/PFun.lean", "def_pos": [266, 9], "def_end_pos": [266, 20]}, {"full_name": "Or.inl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [517, 5], "def_end_pos": [517, 8]}]], "state_before": "case intro.mpr.intro.intro.intro\nn\u271d\u00b9 : \u2115\nf\u271d : Vector \u2115 n\u271d\u00b9 \u2192. \u2115\nn\u271d : \u2115\nf : Vector \u2115 (n\u271d + 1) \u2192 \u2115\na\u271d : Nat.Partrec' \u2191f\ncf : Code\nv : Vector \u2115 n\u271d\nhf : \u2200 (a : \u2115), eval cf (a :: \u2191v) = Part.some [f (a ::\u1d65 v)]\nn : \u2115\nhn : f (n ::\u1d65 v) = 0\nhm : \u2200 {m : \u2115}, m < n \u2192 \u00acf (m ::\u1d65 v) = 0\n\u22a2 Nat.succ n :: \u2191v \u2208\n    PFun.fix\n      (fun v =>\n        Part.bind (eval cf v) fun y =>\n          Part.some\n            (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)\n            else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))\n      (0 :: \u2191v)", "state_after": "case intro.mpr.intro.intro.intro\nn\u271d\u00b9 : \u2115\nf\u271d : Vector \u2115 n\u271d\u00b9 \u2192. \u2115\nn\u271d : \u2115\nf : Vector \u2115 (n\u271d + 1) \u2192 \u2115\na\u271d : Nat.Partrec' \u2191f\ncf : Code\nv : Vector \u2115 n\u271d\nhf : \u2200 (a : \u2115), eval cf (a :: \u2191v) = Part.some [f (a ::\u1d65 v)]\nn : \u2115\nhn : f (n ::\u1d65 v) = 0\nhm : \u2200 {m : \u2115}, m < n \u2192 \u00acf (m ::\u1d65 v) = 0\nthis :\n  Nat.succ n :: \u2191v \u2208\n    PFun.fix\n      (fun v =>\n        Part.bind (eval cf v) fun y =>\n          Part.some\n            (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)\n            else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))\n      (n :: \u2191v)\n\u22a2 Nat.succ n :: \u2191v \u2208\n    PFun.fix\n      (fun v =>\n        Part.bind (eval cf v) fun y =>\n          Part.some\n            (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)\n            else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))\n      (0 :: \u2191v)"}, {"tactic": "generalize (n.succ :: v.1 : List \u2115) = w at this \u22a2", "annotated_tactic": ["generalize (n.succ :: v.1 : <a>List</a> \u2115) = w at this \u22a2", [{"full_name": "List", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2182, 11], "def_end_pos": [2182, 15]}]], "state_before": "case intro.mpr.intro.intro.intro\nn\u271d\u00b9 : \u2115\nf\u271d : Vector \u2115 n\u271d\u00b9 \u2192. \u2115\nn\u271d : \u2115\nf : Vector \u2115 (n\u271d + 1) \u2192 \u2115\na\u271d : Nat.Partrec' \u2191f\ncf : Code\nv : Vector \u2115 n\u271d\nhf : \u2200 (a : \u2115), eval cf (a :: \u2191v) = Part.some [f (a ::\u1d65 v)]\nn : \u2115\nhn : f (n ::\u1d65 v) = 0\nhm : \u2200 {m : \u2115}, m < n \u2192 \u00acf (m ::\u1d65 v) = 0\nthis :\n  Nat.succ n :: \u2191v \u2208\n    PFun.fix\n      (fun v =>\n        Part.bind (eval cf v) fun y =>\n          Part.some\n            (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)\n            else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))\n      (n :: \u2191v)\n\u22a2 Nat.succ n :: \u2191v \u2208\n    PFun.fix\n      (fun v =>\n        Part.bind (eval cf v) fun y =>\n          Part.some\n            (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)\n            else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))\n      (0 :: \u2191v)", "state_after": "case intro.mpr.intro.intro.intro\nn\u271d\u00b9 : \u2115\nf\u271d : Vector \u2115 n\u271d\u00b9 \u2192. \u2115\nn\u271d : \u2115\nf : Vector \u2115 (n\u271d + 1) \u2192 \u2115\na\u271d : Nat.Partrec' \u2191f\ncf : Code\nv : Vector \u2115 n\u271d\nhf : \u2200 (a : \u2115), eval cf (a :: \u2191v) = Part.some [f (a ::\u1d65 v)]\nn : \u2115\nhn : f (n ::\u1d65 v) = 0\nhm : \u2200 {m : \u2115}, m < n \u2192 \u00acf (m ::\u1d65 v) = 0\nw : List \u2115\nthis :\n  w \u2208\n    PFun.fix\n      (fun v =>\n        Part.bind (eval cf v) fun y =>\n          Part.some\n            (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)\n            else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))\n      (n :: \u2191v)\n\u22a2 w \u2208\n    PFun.fix\n      (fun v =>\n        Part.bind (eval cf v) fun y =>\n          Part.some\n            (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)\n            else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))\n      (0 :: \u2191v)"}, {"tactic": "clear hn", "annotated_tactic": ["clear hn", []], "state_before": "case intro.mpr.intro.intro.intro\nn\u271d\u00b9 : \u2115\nf\u271d : Vector \u2115 n\u271d\u00b9 \u2192. \u2115\nn\u271d : \u2115\nf : Vector \u2115 (n\u271d + 1) \u2192 \u2115\na\u271d : Nat.Partrec' \u2191f\ncf : Code\nv : Vector \u2115 n\u271d\nhf : \u2200 (a : \u2115), eval cf (a :: \u2191v) = Part.some [f (a ::\u1d65 v)]\nn : \u2115\nhn : f (n ::\u1d65 v) = 0\nhm : \u2200 {m : \u2115}, m < n \u2192 \u00acf (m ::\u1d65 v) = 0\nw : List \u2115\nthis :\n  w \u2208\n    PFun.fix\n      (fun v =>\n        Part.bind (eval cf v) fun y =>\n          Part.some\n            (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)\n            else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))\n      (n :: \u2191v)\n\u22a2 w \u2208\n    PFun.fix\n      (fun v =>\n        Part.bind (eval cf v) fun y =>\n          Part.some\n            (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)\n            else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))\n      (0 :: \u2191v)", "state_after": "case intro.mpr.intro.intro.intro\nn\u271d\u00b9 : \u2115\nf\u271d : Vector \u2115 n\u271d\u00b9 \u2192. \u2115\nn\u271d : \u2115\nf : Vector \u2115 (n\u271d + 1) \u2192 \u2115\na\u271d : Nat.Partrec' \u2191f\ncf : Code\nv : Vector \u2115 n\u271d\nhf : \u2200 (a : \u2115), eval cf (a :: \u2191v) = Part.some [f (a ::\u1d65 v)]\nn : \u2115\nhm : \u2200 {m : \u2115}, m < n \u2192 \u00acf (m ::\u1d65 v) = 0\nw : List \u2115\nthis :\n  w \u2208\n    PFun.fix\n      (fun v =>\n        Part.bind (eval cf v) fun y =>\n          Part.some\n            (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)\n            else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))\n      (n :: \u2191v)\n\u22a2 w \u2208\n    PFun.fix\n      (fun v =>\n        Part.bind (eval cf v) fun y =>\n          Part.some\n            (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)\n            else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))\n      (0 :: \u2191v)"}, {"tactic": "induction' n with n IH", "annotated_tactic": ["induction' n with n IH", []], "state_before": "case intro.mpr.intro.intro.intro\nn\u271d\u00b9 : \u2115\nf\u271d : Vector \u2115 n\u271d\u00b9 \u2192. \u2115\nn\u271d : \u2115\nf : Vector \u2115 (n\u271d + 1) \u2192 \u2115\na\u271d : Nat.Partrec' \u2191f\ncf : Code\nv : Vector \u2115 n\u271d\nhf : \u2200 (a : \u2115), eval cf (a :: \u2191v) = Part.some [f (a ::\u1d65 v)]\nn : \u2115\nhm : \u2200 {m : \u2115}, m < n \u2192 \u00acf (m ::\u1d65 v) = 0\nw : List \u2115\nthis :\n  w \u2208\n    PFun.fix\n      (fun v =>\n        Part.bind (eval cf v) fun y =>\n          Part.some\n            (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)\n            else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))\n      (n :: \u2191v)\n\u22a2 w \u2208\n    PFun.fix\n      (fun v =>\n        Part.bind (eval cf v) fun y =>\n          Part.some\n            (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)\n            else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))\n      (0 :: \u2191v)", "state_after": "case intro.mpr.intro.intro.intro.zero\nn\u271d\u00b9 : \u2115\nf\u271d : Vector \u2115 n\u271d\u00b9 \u2192. \u2115\nn\u271d : \u2115\nf : Vector \u2115 (n\u271d + 1) \u2192 \u2115\na\u271d : Nat.Partrec' \u2191f\ncf : Code\nv : Vector \u2115 n\u271d\nhf : \u2200 (a : \u2115), eval cf (a :: \u2191v) = Part.some [f (a ::\u1d65 v)]\nn : \u2115\nhm\u271d : \u2200 {m : \u2115}, m < n \u2192 \u00acf (m ::\u1d65 v) = 0\nw : List \u2115\nthis\u271d :\n  w \u2208\n    PFun.fix\n      (fun v =>\n        Part.bind (eval cf v) fun y =>\n          Part.some\n            (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)\n            else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))\n      (n :: \u2191v)\nhm : \u2200 {m : \u2115}, m < Nat.zero \u2192 \u00acf (m ::\u1d65 v) = 0\nthis :\n  w \u2208\n    PFun.fix\n      (fun v =>\n        Part.bind (eval cf v) fun y =>\n          Part.some\n            (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)\n            else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))\n      (Nat.zero :: \u2191v)\n\u22a2 w \u2208\n    PFun.fix\n      (fun v =>\n        Part.bind (eval cf v) fun y =>\n          Part.some\n            (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)\n            else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))\n      (0 :: \u2191v)\n\ncase intro.mpr.intro.intro.intro.succ\nn\u271d\u00b2 : \u2115\nf\u271d : Vector \u2115 n\u271d\u00b2 \u2192. \u2115\nn\u271d\u00b9 : \u2115\nf : Vector \u2115 (n\u271d\u00b9 + 1) \u2192 \u2115\na\u271d : Nat.Partrec' \u2191f\ncf : Code\nv : Vector \u2115 n\u271d\u00b9\nhf : \u2200 (a : \u2115), eval cf (a :: \u2191v) = Part.some [f (a ::\u1d65 v)]\nn\u271d : \u2115\nhm\u271d : \u2200 {m : \u2115}, m < n\u271d \u2192 \u00acf (m ::\u1d65 v) = 0\nw : List \u2115\nthis\u271d :\n  w \u2208\n    PFun.fix\n      (fun v =>\n        Part.bind (eval cf v) fun y =>\n          Part.some\n            (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)\n            else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))\n      (n\u271d :: \u2191v)\nn : \u2115\nIH :\n  (\u2200 {m : \u2115}, m < n \u2192 \u00acf (m ::\u1d65 v) = 0) \u2192\n    w \u2208\n        PFun.fix\n          (fun v =>\n            Part.bind (eval cf v) fun y =>\n              Part.some\n                (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)\n                else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))\n          (n :: \u2191v) \u2192\n      w \u2208\n        PFun.fix\n          (fun v =>\n            Part.bind (eval cf v) fun y =>\n              Part.some\n                (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)\n                else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))\n          (0 :: \u2191v)\nhm : \u2200 {m : \u2115}, m < Nat.succ n \u2192 \u00acf (m ::\u1d65 v) = 0\nthis :\n  w \u2208\n    PFun.fix\n      (fun v =>\n        Part.bind (eval cf v) fun y =>\n          Part.some\n            (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)\n            else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))\n      (Nat.succ n :: \u2191v)\n\u22a2 w \u2208\n    PFun.fix\n      (fun v =>\n        Part.bind (eval cf v) fun y =>\n          Part.some\n            (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)\n            else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))\n      (0 :: \u2191v)"}, {"tactic": "refine' IH (fun {m} h' => hm (Nat.lt_succ_of_lt h'))\n  (PFun.mem_fix_iff.2 (Or.inr \u27e8_, _, this\u27e9))", "annotated_tactic": ["refine' IH (fun {m} h' => hm (<a>Nat.lt_succ_of_lt</a> h'))\n        (<a>PFun.mem_fix_iff</a>.2 (<a>Or.inr</a> \u27e8_, _, this\u27e9))", [{"full_name": "Nat.lt_succ_of_lt", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [226, 9], "def_end_pos": [226, 22]}, {"full_name": "PFun.mem_fix_iff", "def_path": "Mathlib/Data/PFun.lean", "def_pos": [266, 9], "def_end_pos": [266, 20]}, {"full_name": "Or.inr", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [519, 5], "def_end_pos": [519, 8]}]], "state_before": "case intro.mpr.intro.intro.intro.succ\nn\u271d\u00b2 : \u2115\nf\u271d : Vector \u2115 n\u271d\u00b2 \u2192. \u2115\nn\u271d\u00b9 : \u2115\nf : Vector \u2115 (n\u271d\u00b9 + 1) \u2192 \u2115\na\u271d : Nat.Partrec' \u2191f\ncf : Code\nv : Vector \u2115 n\u271d\u00b9\nhf : \u2200 (a : \u2115), eval cf (a :: \u2191v) = Part.some [f (a ::\u1d65 v)]\nn\u271d : \u2115\nhm\u271d : \u2200 {m : \u2115}, m < n\u271d \u2192 \u00acf (m ::\u1d65 v) = 0\nw : List \u2115\nthis\u271d :\n  w \u2208\n    PFun.fix\n      (fun v =>\n        Part.bind (eval cf v) fun y =>\n          Part.some\n            (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)\n            else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))\n      (n\u271d :: \u2191v)\nn : \u2115\nIH :\n  (\u2200 {m : \u2115}, m < n \u2192 \u00acf (m ::\u1d65 v) = 0) \u2192\n    w \u2208\n        PFun.fix\n          (fun v =>\n            Part.bind (eval cf v) fun y =>\n              Part.some\n                (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)\n                else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))\n          (n :: \u2191v) \u2192\n      w \u2208\n        PFun.fix\n          (fun v =>\n            Part.bind (eval cf v) fun y =>\n              Part.some\n                (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)\n                else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))\n          (0 :: \u2191v)\nhm : \u2200 {m : \u2115}, m < Nat.succ n \u2192 \u00acf (m ::\u1d65 v) = 0\nthis :\n  w \u2208\n    PFun.fix\n      (fun v =>\n        Part.bind (eval cf v) fun y =>\n          Part.some\n            (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)\n            else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))\n      (Nat.succ n :: \u2191v)\n\u22a2 w \u2208\n    PFun.fix\n      (fun v =>\n        Part.bind (eval cf v) fun y =>\n          Part.some\n            (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)\n            else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))\n      (0 :: \u2191v)", "state_after": "case intro.mpr.intro.intro.intro.succ\nn\u271d\u00b2 : \u2115\nf\u271d : Vector \u2115 n\u271d\u00b2 \u2192. \u2115\nn\u271d\u00b9 : \u2115\nf : Vector \u2115 (n\u271d\u00b9 + 1) \u2192 \u2115\na\u271d : Nat.Partrec' \u2191f\ncf : Code\nv : Vector \u2115 n\u271d\u00b9\nhf : \u2200 (a : \u2115), eval cf (a :: \u2191v) = Part.some [f (a ::\u1d65 v)]\nn\u271d : \u2115\nhm\u271d : \u2200 {m : \u2115}, m < n\u271d \u2192 \u00acf (m ::\u1d65 v) = 0\nw : List \u2115\nthis\u271d :\n  w \u2208\n    PFun.fix\n      (fun v =>\n        Part.bind (eval cf v) fun y =>\n          Part.some\n            (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)\n            else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))\n      (n\u271d :: \u2191v)\nn : \u2115\nIH :\n  (\u2200 {m : \u2115}, m < n \u2192 \u00acf (m ::\u1d65 v) = 0) \u2192\n    w \u2208\n        PFun.fix\n          (fun v =>\n            Part.bind (eval cf v) fun y =>\n              Part.some\n                (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)\n                else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))\n          (n :: \u2191v) \u2192\n      w \u2208\n        PFun.fix\n          (fun v =>\n            Part.bind (eval cf v) fun y =>\n              Part.some\n                (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)\n                else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))\n          (0 :: \u2191v)\nhm : \u2200 {m : \u2115}, m < Nat.succ n \u2192 \u00acf (m ::\u1d65 v) = 0\nthis :\n  w \u2208\n    PFun.fix\n      (fun v =>\n        Part.bind (eval cf v) fun y =>\n          Part.some\n            (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)\n            else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))\n      (Nat.succ n :: \u2191v)\n\u22a2 Sum.inr (Nat.succ n :: \u2191v) \u2208\n    Part.bind (eval cf (n :: \u2191v)) fun y =>\n      Part.some\n        (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v))\n        else Sum.inr (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v)))"}, {"tactic": "simp only [hf, hm n.lt_succ_self, Part.bind_some, List.headI, eq_self_iff_true, if_false,\n  Part.mem_some_iff, and_self_iff, List.tail_cons]", "annotated_tactic": ["simp only [hf, hm n.lt_succ_self, <a>Part.bind_some</a>, <a>List.headI</a>, <a>eq_self_iff_true</a>, <a>if_false</a>,\n        <a>Part.mem_some_iff</a>, <a>and_self_iff</a>, <a>List.tail_cons</a>]", [{"full_name": "Part.bind_some", "def_path": "Mathlib/Data/Part.lean", "def_pos": [518, 9], "def_end_pos": [518, 18]}, {"full_name": "List.headI", "def_path": "Mathlib/Init/Data/List/Basic.lean", "def_pos": [39, 5], "def_end_pos": [39, 10]}, {"full_name": "eq_self_iff_true", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [86, 9], "def_end_pos": [86, 25]}, {"full_name": "if_false", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [729, 17], "def_end_pos": [729, 25]}, {"full_name": "Part.mem_some_iff", "def_path": "Mathlib/Data/Part.lean", "def_pos": [170, 9], "def_end_pos": [170, 21]}, {"full_name": "and_self_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [155, 9], "def_end_pos": [155, 21]}, {"full_name": "List.tail_cons", "def_path": "lake-packages/std/Std/Data/List/Basic.lean", "def_pos": [316, 17], "def_end_pos": [316, 26]}]], "state_before": "case intro.mpr.intro.intro.intro.succ\nn\u271d\u00b2 : \u2115\nf\u271d : Vector \u2115 n\u271d\u00b2 \u2192. \u2115\nn\u271d\u00b9 : \u2115\nf : Vector \u2115 (n\u271d\u00b9 + 1) \u2192 \u2115\na\u271d : Nat.Partrec' \u2191f\ncf : Code\nv : Vector \u2115 n\u271d\u00b9\nhf : \u2200 (a : \u2115), eval cf (a :: \u2191v) = Part.some [f (a ::\u1d65 v)]\nn\u271d : \u2115\nhm\u271d : \u2200 {m : \u2115}, m < n\u271d \u2192 \u00acf (m ::\u1d65 v) = 0\nw : List \u2115\nthis\u271d :\n  w \u2208\n    PFun.fix\n      (fun v =>\n        Part.bind (eval cf v) fun y =>\n          Part.some\n            (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)\n            else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))\n      (n\u271d :: \u2191v)\nn : \u2115\nIH :\n  (\u2200 {m : \u2115}, m < n \u2192 \u00acf (m ::\u1d65 v) = 0) \u2192\n    w \u2208\n        PFun.fix\n          (fun v =>\n            Part.bind (eval cf v) fun y =>\n              Part.some\n                (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)\n                else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))\n          (n :: \u2191v) \u2192\n      w \u2208\n        PFun.fix\n          (fun v =>\n            Part.bind (eval cf v) fun y =>\n              Part.some\n                (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)\n                else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))\n          (0 :: \u2191v)\nhm : \u2200 {m : \u2115}, m < Nat.succ n \u2192 \u00acf (m ::\u1d65 v) = 0\nthis :\n  w \u2208\n    PFun.fix\n      (fun v =>\n        Part.bind (eval cf v) fun y =>\n          Part.some\n            (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)\n            else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))\n      (Nat.succ n :: \u2191v)\n\u22a2 Sum.inr (Nat.succ n :: \u2191v) \u2208\n    Part.bind (eval cf (n :: \u2191v)) fun y =>\n      Part.some\n        (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v))\n        else Sum.inr (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v)))", "state_after": "no goals"}, {"tactic": "simp [hf, hn]", "annotated_tactic": ["simp [hf, hn]", []], "state_before": "n\u271d\u00b9 : \u2115\nf\u271d : Vector \u2115 n\u271d\u00b9 \u2192. \u2115\nn\u271d : \u2115\nf : Vector \u2115 (n\u271d + 1) \u2192 \u2115\na\u271d : Nat.Partrec' \u2191f\ncf : Code\nv : Vector \u2115 n\u271d\nhf : \u2200 (a : \u2115), eval cf (a :: \u2191v) = Part.some [f (a ::\u1d65 v)]\nn : \u2115\nhn : f (n ::\u1d65 v) = 0\nhm : \u2200 {m : \u2115}, m < n \u2192 \u00acf (m ::\u1d65 v) = 0\n\u22a2 Sum.inl (Nat.succ n :: \u2191v) \u2208\n    Part.bind (eval cf (n :: \u2191v)) fun y =>\n      Part.some\n        (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v))\n        else Sum.inr (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v)))", "state_after": "no goals"}, {"tactic": "exact this", "annotated_tactic": ["exact this", []], "state_before": "case intro.mpr.intro.intro.intro.zero\nn\u271d\u00b9 : \u2115\nf\u271d : Vector \u2115 n\u271d\u00b9 \u2192. \u2115\nn\u271d : \u2115\nf : Vector \u2115 (n\u271d + 1) \u2192 \u2115\na\u271d : Nat.Partrec' \u2191f\ncf : Code\nv : Vector \u2115 n\u271d\nhf : \u2200 (a : \u2115), eval cf (a :: \u2191v) = Part.some [f (a ::\u1d65 v)]\nn : \u2115\nhm\u271d : \u2200 {m : \u2115}, m < n \u2192 \u00acf (m ::\u1d65 v) = 0\nw : List \u2115\nthis\u271d :\n  w \u2208\n    PFun.fix\n      (fun v =>\n        Part.bind (eval cf v) fun y =>\n          Part.some\n            (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)\n            else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))\n      (n :: \u2191v)\nhm : \u2200 {m : \u2115}, m < Nat.zero \u2192 \u00acf (m ::\u1d65 v) = 0\nthis :\n  w \u2208\n    PFun.fix\n      (fun v =>\n        Part.bind (eval cf v) fun y =>\n          Part.some\n            (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)\n            else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))\n      (Nat.zero :: \u2191v)\n\u22a2 w \u2208\n    PFun.fix\n      (fun v =>\n        Part.bind (eval cf v) fun y =>\n          Part.some\n            (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)\n            else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))\n      (0 :: \u2191v)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "full_name": "IntervalIntegrable.comp_add_right", "start": [316, 1], "end": [325, 31], "traced_tactics": [{"tactic": "wlog h : a \u2264 b generalizing a b", "annotated_tactic": ["wlog h : a \u2264 b generalizing a b", []], "state_before": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedRing A\nf g : \u211d \u2192 E\na b : \u211d\n\u03bc : Measure \u211d\nhf : IntervalIntegrable f volume a b\nc : \u211d\n\u22a2 IntervalIntegrable (fun x => f (x + c)) volume (a - c) (b - c)", "state_after": "case inr\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedRing A\nf g : \u211d \u2192 E\na b : \u211d\n\u03bc : Measure \u211d\nhf : IntervalIntegrable f volume a b\nc : \u211d\nthis :\n  \u2200 {a b : \u211d}, IntervalIntegrable f volume a b \u2192 a \u2264 b \u2192 IntervalIntegrable (fun x => f (x + c)) volume (a - c) (b - c)\nh : \u00aca \u2264 b\n\u22a2 IntervalIntegrable (fun x => f (x + c)) volume (a - c) (b - c)\n\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedRing A\nf g : \u211d \u2192 E\na\u271d b\u271d : \u211d\n\u03bc : Measure \u211d\nc a b : \u211d\nhf : IntervalIntegrable f volume a b\nh : a \u2264 b\n\u22a2 IntervalIntegrable (fun x => f (x + c)) volume (a - c) (b - c)"}, {"tactic": "rw [intervalIntegrable_iff'] at hf \u22a2", "annotated_tactic": ["rw [<a>intervalIntegrable_iff'</a>] at hf \u22a2", [{"full_name": "intervalIntegrable_iff'", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [96, 9], "def_end_pos": [96, 32]}]], "state_before": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedRing A\nf g : \u211d \u2192 E\na\u271d b\u271d : \u211d\n\u03bc : Measure \u211d\nc a b : \u211d\nhf : IntervalIntegrable f volume a b\nh : a \u2264 b\n\u22a2 IntervalIntegrable (fun x => f (x + c)) volume (a - c) (b - c)", "state_after": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedRing A\nf g : \u211d \u2192 E\na\u271d b\u271d : \u211d\n\u03bc : Measure \u211d\nc a b : \u211d\nhf : IntegrableOn f [[a, b]]\nh : a \u2264 b\n\u22a2 IntegrableOn (fun x => f (x + c)) [[a - c, b - c]]"}, {"tactic": "have A : MeasurableEmbedding fun x => x + c :=\n  (Homeomorph.addRight c).closedEmbedding.measurableEmbedding", "annotated_tactic": ["have A : <a>MeasurableEmbedding</a> fun x => x + c :=\n    (<a>Homeomorph.addRight</a> c).closedEmbedding.measurableEmbedding", [{"full_name": "MeasurableEmbedding", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [1178, 11], "def_end_pos": [1178, 30]}, {"full_name": "Homeomorph.addRight", "def_path": "Mathlib/Topology/Algebra/Group/Basic.lean", "def_pos": [101, 3], "def_end_pos": [101, 14]}]], "state_before": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedRing A\nf g : \u211d \u2192 E\na\u271d b\u271d : \u211d\n\u03bc : Measure \u211d\nc a b : \u211d\nhf : IntegrableOn f [[a, b]]\nh : a \u2264 b\n\u22a2 IntegrableOn (fun x => f (x + c)) [[a - c, b - c]]", "state_after": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA\u271d : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedRing A\u271d\nf g : \u211d \u2192 E\na\u271d b\u271d : \u211d\n\u03bc : Measure \u211d\nc a b : \u211d\nhf : IntegrableOn f [[a, b]]\nh : a \u2264 b\nA : MeasurableEmbedding fun x => x + c\n\u22a2 IntegrableOn (fun x => f (x + c)) [[a - c, b - c]]"}, {"tactic": "convert (MeasurableEmbedding.integrableOn_map_iff A).mp hf using 1", "annotated_tactic": ["convert (<a>MeasurableEmbedding.integrableOn_map_iff</a> A).<a>mp</a> hf using 1", [{"full_name": "MeasurableEmbedding.integrableOn_map_iff", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [234, 9], "def_end_pos": [234, 56]}, {"full_name": "Iff.mp", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [90, 3], "def_end_pos": [90, 5]}]], "state_before": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA\u271d : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedRing A\u271d\nf g : \u211d \u2192 E\na\u271d b\u271d : \u211d\n\u03bc : Measure \u211d\nc a b : \u211d\nhf : IntegrableOn f [[a, b]]\nh : a \u2264 b\nA : MeasurableEmbedding fun x => x + c\n\u22a2 IntegrableOn (fun x => f (x + c)) [[a - c, b - c]]", "state_after": "case h.e'_6\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA\u271d : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedRing A\u271d\nf g : \u211d \u2192 E\na\u271d b\u271d : \u211d\n\u03bc : Measure \u211d\nc a b : \u211d\nhf : IntegrableOn f [[a, b]]\nh : a \u2264 b\nA : MeasurableEmbedding fun x => x + c\n\u22a2 [[a - c, b - c]] = (fun x => x + c) \u207b\u00b9' [[a, b]]"}, {"tactic": "rw [preimage_add_const_uIcc]", "annotated_tactic": ["rw [<a>preimage_add_const_uIcc</a>]", [{"full_name": "Set.preimage_add_const_uIcc", "def_path": "Mathlib/Data/Set/Pointwise/Interval.lean", "def_pos": [431, 9], "def_end_pos": [431, 32]}]], "state_before": "case h.e'_6\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA\u271d : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedRing A\u271d\nf g : \u211d \u2192 E\na\u271d b\u271d : \u211d\n\u03bc : Measure \u211d\nc a b : \u211d\nhf : IntegrableOn f [[a, b]]\nh : a \u2264 b\nA : MeasurableEmbedding fun x => x + c\n\u22a2 [[a - c, b - c]] = (fun x => x + c) \u207b\u00b9' [[a, b]]", "state_after": "no goals"}, {"tactic": "exact IntervalIntegrable.symm (this hf.symm (le_of_not_le h))", "annotated_tactic": ["exact <a>IntervalIntegrable.symm</a> (this hf.symm (<a>le_of_not_le</a> h))", [{"full_name": "IntervalIntegrable.symm", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [139, 16], "def_end_pos": [139, 20]}, {"full_name": "le_of_not_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [303, 9], "def_end_pos": [303, 21]}]], "state_before": "case inr\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedRing A\nf g : \u211d \u2192 E\na b : \u211d\n\u03bc : Measure \u211d\nhf : IntervalIntegrable f volume a b\nc : \u211d\nthis :\n  \u2200 {a b : \u211d}, IntervalIntegrable f volume a b \u2192 a \u2264 b \u2192 IntervalIntegrable (fun x => f (x + c)) volume (a - c) (b - c)\nh : \u00aca \u2264 b\n\u22a2 IntervalIntegrable (fun x => f (x + c)) volume (a - c) (b - c)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean", "full_name": "MeasureTheory.addHaarMeasure_eq_volume_pi", "start": [120, 1], "end": [124, 101], "traced_tactics": [{"tactic": "convert (addHaarMeasure_unique volume (piIcc01 \u03b9)).symm", "annotated_tactic": ["convert (<a>addHaarMeasure_unique</a> <a>volume</a> (<a>piIcc01</a> \u03b9)).<a>symm</a>", [{"full_name": "MeasureTheory.Measure.addHaarMeasure_unique", "def_path": "Mathlib/MeasureTheory/Measure/Haar/Basic.lean", "def_pos": [688, 3], "def_end_pos": [688, 14]}, {"full_name": "MeasureTheory.MeasureSpace.volume", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [663, 3], "def_end_pos": [663, 9]}, {"full_name": "TopologicalSpace.PositiveCompacts.piIcc01", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean", "def_pos": [64, 5], "def_end_pos": [64, 46]}, {"full_name": "Eq.symm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [310, 9], "def_end_pos": [310, 16]}]], "state_before": "\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\n\u22a2 addHaarMeasure (piIcc01 \u03b9) = volume", "state_after": "case h.e'_2\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\n\u22a2 addHaarMeasure (piIcc01 \u03b9) = \u2191\u2191volume \u2191(piIcc01 \u03b9) \u2022 addHaarMeasure (piIcc01 \u03b9)"}, {"tactic": "simp only [piIcc01, volume_pi_pi fun _ => Icc (0 : \u211d) 1, PositiveCompacts.coe_mk,\n  Compacts.coe_mk, Finset.prod_const_one, ENNReal.ofReal_one, Real.volume_Icc, one_smul, sub_zero]", "annotated_tactic": ["simp only [<a>piIcc01</a>, <a>volume_pi_pi</a> fun _ => <a>Icc</a> (0 : \u211d) 1, <a>PositiveCompacts.coe_mk</a>,\n    <a>Compacts.coe_mk</a>, <a>Finset.prod_const_one</a>, <a>ENNReal.ofReal_one</a>, <a>Real.volume_Icc</a>, <a>one_smul</a>, <a>sub_zero</a>]", [{"full_name": "TopologicalSpace.PositiveCompacts.piIcc01", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean", "def_pos": [64, 5], "def_end_pos": [64, 46]}, {"full_name": "MeasureTheory.volume_pi_pi", "def_path": "Mathlib/MeasureTheory/Constructions/Pi.lean", "def_pos": [676, 9], "def_end_pos": [676, 21]}, {"full_name": "Set.Icc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [59, 5], "def_end_pos": [59, 8]}, {"full_name": "TopologicalSpace.PositiveCompacts.coe_mk", "def_path": "Mathlib/Topology/Sets/Compacts.lean", "def_pos": [352, 9], "def_end_pos": [352, 15]}, {"full_name": "TopologicalSpace.Compacts.coe_mk", "def_path": "Mathlib/Topology/Sets/Compacts.lean", "def_pos": [67, 9], "def_end_pos": [67, 15]}, {"full_name": "Finset.prod_const_one", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [357, 9], "def_end_pos": [357, 23]}, {"full_name": "ENNReal.ofReal_one", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [248, 17], "def_end_pos": [248, 27]}, {"full_name": "Real.volume_Icc", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean", "def_pos": [84, 9], "def_end_pos": [84, 19]}, {"full_name": "one_smul", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [492, 9], "def_end_pos": [492, 17]}, {"full_name": "sub_zero", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [339, 3], "def_end_pos": [339, 14]}]], "state_before": "case h.e'_2\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\n\u22a2 addHaarMeasure (piIcc01 \u03b9) = \u2191\u2191volume \u2191(piIcc01 \u03b9) \u2022 addHaarMeasure (piIcc01 \u03b9)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "full_name": "MeasureTheory.norm_integral_le_lintegral_norm", "start": [964, 1], "end": [971, 24], "traced_tactics": [{"tactic": "by_cases hG : CompleteSpace G", "annotated_tactic": ["by_cases hG : <a>CompleteSpace</a> G", [{"full_name": "CompleteSpace", "def_path": "Mathlib/Topology/UniformSpace/Cauchy.lean", "def_pos": [397, 7], "def_end_pos": [397, 20]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 G\n\u22a2 \u2016\u222b (a : \u03b1), f a \u2202\u03bc\u2016 \u2264 ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016f a\u2016 \u2202\u03bc)", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 G\nhG : CompleteSpace G\n\u22a2 \u2016\u222b (a : \u03b1), f a \u2202\u03bc\u2016 \u2264 ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016f a\u2016 \u2202\u03bc)\n\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 G\nhG : \u00acCompleteSpace G\n\u22a2 \u2016\u222b (a : \u03b1), f a \u2202\u03bc\u2016 \u2264 ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016f a\u2016 \u2202\u03bc)"}, {"tactic": "by_cases hf : Integrable f \u03bc", "annotated_tactic": ["by_cases hf : <a>Integrable</a> f \u03bc", [{"full_name": "MeasureTheory.Integrable", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [442, 5], "def_end_pos": [442, 15]}]], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 G\nhG : CompleteSpace G\n\u22a2 \u2016\u222b (a : \u03b1), f a \u2202\u03bc\u2016 \u2264 ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016f a\u2016 \u2202\u03bc)", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 G\nhG : CompleteSpace G\nhf : Integrable f\n\u22a2 \u2016\u222b (a : \u03b1), f a \u2202\u03bc\u2016 \u2264 ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016f a\u2016 \u2202\u03bc)\n\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 G\nhG : CompleteSpace G\nhf : \u00acIntegrable f\n\u22a2 \u2016\u222b (a : \u03b1), f a \u2202\u03bc\u2016 \u2264 ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016f a\u2016 \u2202\u03bc)"}, {"tactic": "rw [integral_eq f hf, \u2190 Integrable.norm_toL1_eq_lintegral_norm f hf]", "annotated_tactic": ["rw [<a>integral_eq</a> f hf, \u2190 <a>Integrable.norm_toL1_eq_lintegral_norm</a> f hf]", [{"full_name": "MeasureTheory.integral_eq", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [821, 9], "def_end_pos": [821, 20]}, {"full_name": "MeasureTheory.Integrable.norm_toL1_eq_lintegral_norm", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [1457, 9], "def_end_pos": [1457, 36]}]], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 G\nhG : CompleteSpace G\nhf : Integrable f\n\u22a2 \u2016\u222b (a : \u03b1), f a \u2202\u03bc\u2016 \u2264 ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016f a\u2016 \u2202\u03bc)", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup 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\u2192 G\n\u03bc : Measure \u03b1\nm : MeasurableSpace \u03b1\nE : Type u_3\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : AddGroup E\ninst\u271d\u00b9 : MeasurableSingletonClass E\ninst\u271d : MeasurableSub\u2082 E\nf g : \u03b1 \u2192 E\nhf : Measurable f\nhg : Measurable g\nh_set_eq : {x | f x = g x} = {x | (f - g) x = 0}\n\u22a2 MeasurableSet {x | (f - g) x = 0}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "full_name": "ContinuousLinearMap.add_compLp", "start": [1126, 1], "end": [1134, 28], "traced_tactics": [{"tactic": "ext1", "annotated_tactic": ["ext1", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedAddCommGroup G\ng : E \u2192 F\nc : \u211d\u22650\n\ud835\udd5c : Type u_5\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \ud835\udd5c F\nL L' : E \u2192L[\ud835\udd5c] F\nf : { x // x \u2208 Lp E p }\n\u22a2 compLp (L + L') f = compLp L f + compLp L' f", "state_after": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedAddCommGroup G\ng : E \u2192 F\nc : \u211d\u22650\n\ud835\udd5c : Type u_5\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \ud835\udd5c F\nL L' : E \u2192L[\ud835\udd5c] F\nf : { x // x \u2208 Lp E p }\n\u22a2 \u2191\u2191(compLp (L + L') f) =\u1d50[\u03bc] \u2191\u2191(compLp L f + compLp L' f)"}, {"tactic": "refine' (coeFn_compLp' (L + L') f).trans _", "annotated_tactic": ["refine' (<a>coeFn_compLp'</a> (L + L') f).<a>trans</a> _", [{"full_name": "ContinuousLinearMap.coeFn_compLp'", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [1094, 9], "def_end_pos": [1094, 22]}, {"full_name": "Filter.EventuallyEq.trans", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1503, 9], "def_end_pos": [1503, 27]}]], "state_before": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedAddCommGroup G\ng : E \u2192 F\nc : \u211d\u22650\n\ud835\udd5c : Type u_5\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \ud835\udd5c F\nL L' : E \u2192L[\ud835\udd5c] F\nf : { x // x \u2208 Lp E p }\n\u22a2 \u2191\u2191(compLp (L + L') f) =\u1d50[\u03bc] \u2191\u2191(compLp L f + compLp L' f)", "state_after": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedAddCommGroup G\ng : E \u2192 F\nc : \u211d\u22650\n\ud835\udd5c : Type u_5\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \ud835\udd5c F\nL L' : E \u2192L[\ud835\udd5c] F\nf : { x // x \u2208 Lp E p }\n\u22a2 (fun a => \u2191(L + L') (\u2191\u2191f a)) =\u1d50[\u03bc] \u2191\u2191(compLp L f + compLp L' f)"}, {"tactic": "refine' EventuallyEq.trans _ (Lp.coeFn_add _ _).symm", "annotated_tactic": ["refine' <a>EventuallyEq.trans</a> _ (<a>Lp.coeFn_add</a> _ _).<a>symm</a>", [{"full_name": "Filter.EventuallyEq.trans", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1503, 9], "def_end_pos": [1503, 27]}, {"full_name": "MeasureTheory.Lp.coeFn_add", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [232, 9], "def_end_pos": [232, 18]}, {"full_name": "Filter.EventuallyEq.symm", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1498, 9], "def_end_pos": [1498, 26]}]], "state_before": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedAddCommGroup G\ng : E \u2192 F\nc : \u211d\u22650\n\ud835\udd5c : Type u_5\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \ud835\udd5c F\nL L' : E \u2192L[\ud835\udd5c] F\nf : { x // x \u2208 Lp E p }\n\u22a2 (fun a => \u2191(L + L') (\u2191\u2191f a)) =\u1d50[\u03bc] \u2191\u2191(compLp L f + compLp L' f)", "state_after": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedAddCommGroup G\ng : E \u2192 F\nc : \u211d\u22650\n\ud835\udd5c : Type u_5\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \ud835\udd5c F\nL L' : E \u2192L[\ud835\udd5c] F\nf : { x // x \u2208 Lp E p }\n\u22a2 (fun a => \u2191(L + L') (\u2191\u2191f a)) =\u1d50[\u03bc] \u2191\u2191(compLp L f) + \u2191\u2191(compLp L' f)"}, {"tactic": "refine'\n  EventuallyEq.trans _ (EventuallyEq.add (L.coeFn_compLp' f).symm (L'.coeFn_compLp' f).symm)", "annotated_tactic": ["refine'\n    <a>EventuallyEq.trans</a> _ (<a>EventuallyEq.add</a> (L.coeFn_compLp' f).<a>symm</a> (L'.coeFn_compLp' f).<a>symm</a>)", [{"full_name": "Filter.EventuallyEq.trans", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1503, 9], "def_end_pos": [1503, 27]}, {"full_name": "Filter.EventuallyEq.add", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1530, 3], "def_end_pos": [1530, 14]}, {"full_name": "Filter.EventuallyEq.symm", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1498, 9], "def_end_pos": [1498, 26]}, {"full_name": "Filter.EventuallyEq.symm", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1498, 9], "def_end_pos": [1498, 26]}]], "state_before": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedAddCommGroup G\ng : E \u2192 F\nc : \u211d\u22650\n\ud835\udd5c : Type u_5\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \ud835\udd5c F\nL L' : E \u2192L[\ud835\udd5c] F\nf : { x // x \u2208 Lp E p }\n\u22a2 (fun a => \u2191(L + L') (\u2191\u2191f a)) =\u1d50[\u03bc] \u2191\u2191(compLp L f) + \u2191\u2191(compLp L' f)", "state_after": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedAddCommGroup G\ng : E \u2192 F\nc : \u211d\u22650\n\ud835\udd5c : Type u_5\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \ud835\udd5c F\nL L' : E \u2192L[\ud835\udd5c] F\nf : { x // x \u2208 Lp E p }\n\u22a2 (fun a => \u2191(L + L') (\u2191\u2191f a)) =\u1d50[\u03bc] fun x => \u2191L (\u2191\u2191f x) + \u2191L' (\u2191\u2191f x)"}, {"tactic": "refine' eventually_of_forall fun x => _", "annotated_tactic": ["refine' <a>eventually_of_forall</a> fun x => _", [{"full_name": "Filter.eventually_of_forall", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1111, 9], "def_end_pos": [1111, 29]}]], "state_before": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedAddCommGroup G\ng : E \u2192 F\nc : \u211d\u22650\n\ud835\udd5c : Type u_5\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \ud835\udd5c F\nL L' : E \u2192L[\ud835\udd5c] F\nf : { x // x \u2208 Lp E p }\n\u22a2 (fun a => \u2191(L + L') (\u2191\u2191f a)) =\u1d50[\u03bc] fun x => \u2191L (\u2191\u2191f x) + \u2191L' (\u2191\u2191f x)", "state_after": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedAddCommGroup G\ng : E \u2192 F\nc : \u211d\u22650\n\ud835\udd5c : Type u_5\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \ud835\udd5c F\nL L' : E \u2192L[\ud835\udd5c] F\nf : { x // x \u2208 Lp E p }\nx : \u03b1\n\u22a2 (fun a => \u2191(L + L') (\u2191\u2191f a)) x = (fun x => \u2191L (\u2191\u2191f x) + \u2191L' (\u2191\u2191f x)) x"}, {"tactic": "rw [coe_add', Pi.add_def]", "annotated_tactic": ["rw [<a>coe_add'</a>, <a>Pi.add_def</a>]", [{"full_name": "ContinuousLinearMap.coe_add'", "def_path": "Mathlib/Topology/Algebra/Module/Basic.lean", "def_pos": [736, 9], "def_end_pos": [736, 17]}, {"full_name": "Pi.add_def", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [88, 3], "def_end_pos": [88, 14]}]], "state_before": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedAddCommGroup G\ng : E \u2192 F\nc : \u211d\u22650\n\ud835\udd5c : Type u_5\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \ud835\udd5c F\nL L' : E \u2192L[\ud835\udd5c] F\nf : { x // x \u2208 Lp E p }\nx : \u03b1\n\u22a2 (fun a => \u2191(L + L') (\u2191\u2191f a)) x = (fun x => \u2191L (\u2191\u2191f x) + \u2191L' (\u2191\u2191f x)) x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Decomposition/SignedHahn.lean", "full_name": "MeasureTheory.SignedMeasure.restrictNonposSeq_measurableSet", "start": [201, 9], "end": [205, 45], "traced_tactics": [{"tactic": "rw [restrictNonposSeq]", "annotated_tactic": ["rw [<a>restrictNonposSeq</a>]", [{"full_name": "_private.Mathlib.MeasureTheory.Decomposition.SignedHahn.0.MeasureTheory.SignedMeasure.restrictNonposSeq", "def_path": "Mathlib/MeasureTheory/Decomposition/SignedHahn.lean", "def_pos": [163, 13], "def_end_pos": [163, 30]}]], "state_before": "case succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : TopologicalSpace M\ninst\u271d : OrderedAddCommMonoid M\ns : SignedMeasure \u03b1\ni j : Set \u03b1\nn\u271d : \u2115\n\u22a2 MeasurableSet (MeasureTheory.SignedMeasure.restrictNonposSeq s i (Nat.succ n\u271d))", "state_after": "case succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : TopologicalSpace M\ninst\u271d : OrderedAddCommMonoid M\ns : SignedMeasure \u03b1\ni j : Set \u03b1\nn\u271d : \u2115\n\u22a2 MeasurableSet\n    (MeasureTheory.SignedMeasure.someExistsOneDivLT s\n      (i \\\n        \u22c3 k,\n          \u22c3 (H : k \u2264 n\u271d),\n            let_fun this := (_ : k < Nat.succ n\u271d);\n            MeasureTheory.SignedMeasure.restrictNonposSeq s i k))"}, {"tactic": "exact someExistsOneDivLT_measurableSet", "annotated_tactic": ["exact <a>someExistsOneDivLT_measurableSet</a>", [{"full_name": "_private.Mathlib.MeasureTheory.Decomposition.SignedHahn.0.MeasureTheory.SignedMeasure.someExistsOneDivLT_measurableSet", "def_path": "Mathlib/MeasureTheory/Decomposition/SignedHahn.lean", "def_pos": [144, 17], "def_end_pos": [144, 49]}]], "state_before": "case succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : TopologicalSpace M\ninst\u271d : OrderedAddCommMonoid M\ns : SignedMeasure \u03b1\ni j : Set \u03b1\nn\u271d : \u2115\n\u22a2 MeasurableSet\n    (MeasureTheory.SignedMeasure.someExistsOneDivLT s\n      (i \\\n        \u22c3 k,\n          \u22c3 (H : k \u2264 n\u271d),\n            let_fun this := (_ : k < Nat.succ n\u271d);\n            MeasureTheory.SignedMeasure.restrictNonposSeq s i k))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "full_name": "MeasureTheory.L1.SimpleFunc.setToL1S_mono", "start": [852, 1], "end": [858, 51], "traced_tactics": [{"tactic": "rw [\u2190 sub_nonneg] at hfg \u22a2", "annotated_tactic": ["rw [\u2190 <a>sub_nonneg</a>] at hfg \u22a2", [{"full_name": "sub_nonneg", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [720, 30], "def_end_pos": [720, 40]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : NormedAddCommGroup F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : NormedField \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nG'' : Type u_7\nG' : Type u_8\ninst\u271d\u00b3 : NormedLatticeAddCommGroup G'\ninst\u271d\u00b2 : NormedSpace \u211d G'\ninst\u271d\u00b9 : NormedLatticeAddCommGroup G''\ninst\u271d : NormedSpace \u211d G''\nT : Set \u03b1 \u2192 G'' \u2192L[\u211d] G'\nh_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s = 0 \u2192 T s = 0\nh_add : FinMeasAdditive \u03bc T\nhT_nonneg : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 (x : G''), 0 \u2264 x \u2192 0 \u2264 \u2191(T s) x\nf g : { x // x \u2208 simpleFunc G'' 1 \u03bc }\nhfg : f \u2264 g\n\u22a2 setToL1S T f \u2264 setToL1S T g", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : NormedAddCommGroup F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : NormedField \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nG'' : Type u_7\nG' : Type u_8\ninst\u271d\u00b3 : NormedLatticeAddCommGroup G'\ninst\u271d\u00b2 : NormedSpace \u211d G'\ninst\u271d\u00b9 : NormedLatticeAddCommGroup G''\ninst\u271d : NormedSpace \u211d G''\nT : Set \u03b1 \u2192 G'' \u2192L[\u211d] G'\nh_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s = 0 \u2192 T s = 0\nh_add : FinMeasAdditive \u03bc T\nhT_nonneg : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 (x : G''), 0 \u2264 x \u2192 0 \u2264 \u2191(T s) x\nf g : { x // x \u2208 simpleFunc G'' 1 \u03bc }\nhfg\u271d : f \u2264 g\nhfg : 0 \u2264 g - f\n\u22a2 0 \u2264 setToL1S T g - setToL1S T f"}, {"tactic": "rw [\u2190 setToL1S_sub h_zero h_add]", "annotated_tactic": ["rw [\u2190 <a>setToL1S_sub</a> h_zero h_add]", [{"full_name": "MeasureTheory.L1.SimpleFunc.setToL1S_sub", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [772, 9], "def_end_pos": [772, 21]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : NormedAddCommGroup F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : NormedField \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nG'' : Type u_7\nG' : Type u_8\ninst\u271d\u00b3 : NormedLatticeAddCommGroup G'\ninst\u271d\u00b2 : NormedSpace \u211d G'\ninst\u271d\u00b9 : NormedLatticeAddCommGroup G''\ninst\u271d : NormedSpace \u211d G''\nT : Set \u03b1 \u2192 G'' \u2192L[\u211d] G'\nh_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s = 0 \u2192 T s = 0\nh_add : FinMeasAdditive \u03bc T\nhT_nonneg : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 (x : G''), 0 \u2264 x \u2192 0 \u2264 \u2191(T s) x\nf g : { x // x \u2208 simpleFunc G'' 1 \u03bc }\nhfg\u271d : f \u2264 g\nhfg : 0 \u2264 g - f\n\u22a2 0 \u2264 setToL1S T g - setToL1S T f", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : NormedAddCommGroup F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : NormedField \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nG'' : Type u_7\nG' : Type u_8\ninst\u271d\u00b3 : NormedLatticeAddCommGroup G'\ninst\u271d\u00b2 : NormedSpace \u211d G'\ninst\u271d\u00b9 : NormedLatticeAddCommGroup G''\ninst\u271d : NormedSpace \u211d G''\nT : Set \u03b1 \u2192 G'' \u2192L[\u211d] G'\nh_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s = 0 \u2192 T s = 0\nh_add : FinMeasAdditive \u03bc T\nhT_nonneg : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 (x : G''), 0 \u2264 x \u2192 0 \u2264 \u2191(T s) x\nf g : { x // x \u2208 simpleFunc G'' 1 \u03bc }\nhfg\u271d : f \u2264 g\nhfg : 0 \u2264 g - f\n\u22a2 0 \u2264 setToL1S (fun s => T s) (g - f)"}, {"tactic": "exact setToL1S_nonneg h_zero h_add hT_nonneg hfg", "annotated_tactic": ["exact <a>setToL1S_nonneg</a> h_zero h_add hT_nonneg hfg", [{"full_name": "MeasureTheory.L1.SimpleFunc.setToL1S_nonneg", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [839, 9], "def_end_pos": [839, 24]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : NormedAddCommGroup F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : NormedField \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nG'' : Type u_7\nG' : Type u_8\ninst\u271d\u00b3 : NormedLatticeAddCommGroup G'\ninst\u271d\u00b2 : NormedSpace \u211d G'\ninst\u271d\u00b9 : NormedLatticeAddCommGroup G''\ninst\u271d : NormedSpace \u211d G''\nT : Set \u03b1 \u2192 G'' \u2192L[\u211d] G'\nh_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s = 0 \u2192 T s = 0\nh_add : FinMeasAdditive \u03bc T\nhT_nonneg : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 (x : G''), 0 \u2264 x \u2192 0 \u2264 \u2191(T s) x\nf g : { x // x \u2208 simpleFunc G'' 1 \u03bc }\nhfg\u271d : f \u2264 g\nhfg : 0 \u2264 g - f\n\u22a2 0 \u2264 setToL1S (fun s => T s) (g - f)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Variance.lean", "full_name": "ProbabilityTheory.variance_def'", "start": [210, 1], "end": [220, 7], "traced_tactics": [{"tactic": "rw [hX.variance_eq, sub_sq', integral_sub', integral_add']", "annotated_tactic": ["rw [hX.variance_eq, <a>sub_sq'</a>, <a>integral_sub'</a>, <a>integral_add'</a>]", [{"full_name": "sub_sq'", "def_path": "Mathlib/Algebra/GroupPower/Ring.lean", "def_pos": [288, 9], "def_end_pos": [288, 16]}, {"full_name": "MeasureTheory.integral_sub'", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [909, 9], "def_end_pos": [909, 22]}, {"full_name": "MeasureTheory.integral_add'", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [876, 9], "def_end_pos": [876, 22]}]], "state_before": "\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhX : Mem\u2112p X 2\n\u22a2 variance X \u2119 = (\u222b (a : \u03a9), (X ^ 2) a) - (\u222b (a : \u03a9), X a) ^ 2", "state_after": "\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhX : Mem\u2112p X 2\n\u22a2 ((\u222b (a : \u03a9), (X ^ 2) a) + \u222b (a : \u03a9), ((fun x => \u222b (x : \u03a9), X x) ^ 2) a) -\n      \u222b (a : \u03a9), (2 * X * fun x => \u222b (x : \u03a9), X x) a =\n    (\u222b (a : \u03a9), (X ^ 2) a) - (\u222b (a : \u03a9), X a) ^ 2\n\ncase hf\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhX : Mem\u2112p X 2\n\u22a2 Integrable (X ^ 2)\n\ncase hg\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhX : Mem\u2112p X 2\n\u22a2 Integrable ((fun x => \u222b (x : \u03a9), X x) ^ 2)\n\ncase hf\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhX : Mem\u2112p X 2\n\u22a2 Integrable (X ^ 2 + (fun x => \u222b (x : \u03a9), X x) ^ 2)\n\ncase hg\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhX : Mem\u2112p X 2\n\u22a2 Integrable (2 * X * fun x => \u222b (x : \u03a9), X x)"}, {"tactic": "rotate_left", "annotated_tactic": ["rotate_left", []], "state_before": "\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhX : Mem\u2112p X 2\n\u22a2 ((\u222b (a : \u03a9), (X ^ 2) a) + \u222b (a : \u03a9), ((fun x => \u222b (x : \u03a9), X x) ^ 2) a) -\n      \u222b (a : \u03a9), (2 * X * fun x => \u222b (x : \u03a9), X x) a =\n    (\u222b (a : \u03a9), (X ^ 2) a) - (\u222b (a : \u03a9), X a) ^ 2\n\ncase hf\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhX : Mem\u2112p X 2\n\u22a2 Integrable (X ^ 2)\n\ncase hg\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhX : Mem\u2112p X 2\n\u22a2 Integrable ((fun x => \u222b (x : \u03a9), X x) ^ 2)\n\ncase hf\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhX : Mem\u2112p X 2\n\u22a2 Integrable (X ^ 2 + (fun x => \u222b (x : \u03a9), X x) ^ 2)\n\ncase hg\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhX : Mem\u2112p X 2\n\u22a2 Integrable (2 * X * fun x => \u222b (x : \u03a9), X x)", "state_after": "case hf\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhX : Mem\u2112p X 2\n\u22a2 Integrable (X ^ 2)\n\ncase hg\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhX : Mem\u2112p X 2\n\u22a2 Integrable ((fun x => \u222b (x : \u03a9), X x) ^ 2)\n\ncase hf\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhX : Mem\u2112p X 2\n\u22a2 Integrable (X ^ 2 + (fun x => \u222b (x : \u03a9), X x) ^ 2)\n\ncase hg\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhX : Mem\u2112p X 2\n\u22a2 Integrable (2 * X * fun x => \u222b (x : \u03a9), X x)\n\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhX : Mem\u2112p X 2\n\u22a2 ((\u222b (a : \u03a9), (X ^ 2) a) + \u222b (a : \u03a9), ((fun x => \u222b (x : \u03a9), X x) ^ 2) a) -\n      \u222b (a : \u03a9), (2 * X * fun x => \u222b (x : \u03a9), X x) a =\n    (\u222b (a : \u03a9), (X ^ 2) a) - (\u222b (a : \u03a9), X a) ^ 2"}, {"tactic": "simp only [Pi.pow_apply, integral_const, measure_univ, ENNReal.one_toReal, smul_eq_mul, one_mul,\n  Pi.mul_apply, Pi.ofNat_apply, Nat.cast_ofNat, integral_mul_right, integral_mul_left]", "annotated_tactic": ["simp only [<a>Pi.pow_apply</a>, <a>integral_const</a>, <a>measure_univ</a>, <a>ENNReal.one_toReal</a>, <a>smul_eq_mul</a>, <a>one_mul</a>,\n    <a>Pi.mul_apply</a>, <a>Pi.ofNat_apply</a>, <a>Nat.cast_ofNat</a>, <a>integral_mul_right</a>, <a>integral_mul_left</a>]", [{"full_name": "Pi.pow_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [117, 9], "def_end_pos": [117, 18]}, {"full_name": "MeasureTheory.integral_const", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1409, 9], "def_end_pos": [1409, 23]}, {"full_name": "MeasureTheory.IsProbabilityMeasure.measure_univ", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3027, 3], "def_end_pos": [3027, 15]}, {"full_name": "ENNReal.one_toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [230, 17], "def_end_pos": [230, 27]}, {"full_name": "smul_eq_mul", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [93, 9], "def_end_pos": [93, 20]}, {"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [464, 9], "def_end_pos": [464, 16]}, {"full_name": "Pi.mul_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [83, 9], "def_end_pos": [83, 18]}, {"full_name": "Pi.ofNat_apply", "def_path": "Mathlib/Data/Nat/Cast/Basic.lean", "def_pos": [200, 9], "def_end_pos": [200, 20]}, {"full_name": "Nat.cast_ofNat", "def_path": "Mathlib/Data/Nat/Cast/Defs.lean", "def_pos": [66, 28], "def_end_pos": [66, 42]}, {"full_name": "MeasureTheory.integral_mul_right", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [928, 9], "def_end_pos": [928, 27]}, {"full_name": "MeasureTheory.integral_mul_left", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [923, 9], "def_end_pos": [923, 26]}]], "state_before": "\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhX : Mem\u2112p X 2\n\u22a2 ((\u222b (a : \u03a9), (X ^ 2) a) + \u222b (a : \u03a9), ((fun x => \u222b (x : \u03a9), X x) ^ 2) a) -\n      \u222b (a : \u03a9), (2 * X * fun x => \u222b (x : \u03a9), X x) a =\n    (\u222b (a : \u03a9), (X ^ 2) a) - (\u222b (a : \u03a9), X a) ^ 2", "state_after": "\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhX : Mem\u2112p X 2\n\u22a2 (\u222b (a : \u03a9), X a ^ 2) + (\u222b (x : \u03a9), X x) ^ 2 - (2 * \u222b (x : \u03a9), X x) * \u222b (x : \u03a9), X x =\n    (\u222b (a : \u03a9), X a ^ 2) - (\u222b (x : \u03a9), X x) ^ 2"}, {"tactic": "ring", "annotated_tactic": ["ring", []], "state_before": "\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhX : Mem\u2112p X 2\n\u22a2 (\u222b (a : \u03a9), X a ^ 2) + (\u222b (x : \u03a9), X x) ^ 2 - (2 * \u222b (x : \u03a9), X x) * \u222b (x : \u03a9), X x =\n    (\u222b (a : \u03a9), X a ^ 2) - (\u222b (x : \u03a9), X x) ^ 2", "state_after": "no goals"}, {"tactic": "exact hX.integrable_sq", "annotated_tactic": ["exact hX.integrable_sq", []], "state_before": "case hf\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhX : Mem\u2112p X 2\n\u22a2 Integrable (X ^ 2)", "state_after": "no goals"}, {"tactic": "convert @integrable_const \u03a9 \u211d (_) \u2119 _ _ (\ud835\udd3c[X] ^ 2)", "annotated_tactic": ["convert @<a>integrable_const</a> \u03a9 \u211d (_) \u2119 _ _ (\ud835\udd3c[X] ^ 2)", [{"full_name": "MeasureTheory.integrable_const", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [506, 9], "def_end_pos": [506, 25]}]], "state_before": "case hg\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhX : Mem\u2112p X 2\n\u22a2 Integrable ((fun x => \u222b (x : \u03a9), X x) ^ 2)", "state_after": "no goals"}, {"tactic": "apply hX.integrable_sq.add", "annotated_tactic": ["apply hX.integrable_sq.add", []], "state_before": "case hf\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhX : Mem\u2112p X 2\n\u22a2 Integrable (X ^ 2 + (fun x => \u222b (x : \u03a9), X x) ^ 2)", "state_after": "case hf\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhX : Mem\u2112p X 2\n\u22a2 Integrable ((fun x => \u222b (x : \u03a9), X x) ^ 2)"}, {"tactic": "convert @integrable_const \u03a9 \u211d (_) \u2119 _ _ (\ud835\udd3c[X] ^ 2)", "annotated_tactic": ["convert @<a>integrable_const</a> \u03a9 \u211d (_) \u2119 _ _ (\ud835\udd3c[X] ^ 2)", [{"full_name": "MeasureTheory.integrable_const", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [506, 9], "def_end_pos": [506, 25]}]], "state_before": "case hf\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhX : Mem\u2112p X 2\n\u22a2 Integrable ((fun x => \u222b (x : \u03a9), X x) ^ 2)", "state_after": "no goals"}, {"tactic": "exact ((hX.integrable one_le_two).const_mul 2).mul_const' _", "annotated_tactic": ["exact ((hX.integrable <a>one_le_two</a>).<a>const_mul</a> 2).<a>mul_const'</a> _", [{"full_name": "one_le_two", "def_path": "Mathlib/Algebra/Order/Monoid/NatCast.lean", "def_pos": [50, 7], "def_end_pos": [50, 17]}, {"full_name": "MeasureTheory.Integrable.const_mul", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [1128, 9], "def_end_pos": [1128, 29]}, {"full_name": "MeasureTheory.Integrable.mul_const'", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [1143, 9], "def_end_pos": [1143, 30]}]], "state_before": "case hg\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhX : Mem\u2112p X 2\n\u22a2 Integrable (2 * X * fun x => \u222b (x : \u03a9), X x)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/List/Basic.lean", "full_name": "List.pmap_eq_pmapImpl", "start": [774, 18], "end": [781, 28], "traced_tactics": [{"tactic": "funext \u03b1 \u03b2 p f L h'", "annotated_tactic": ["funext \u03b1 \u03b2 p f L h'", []], "state_before": "\u22a2 @pmap = @List.pmapImpl", "state_after": "case h.h.h.h.h.h\n\u03b1 : Type u_2\n\u03b2 : Type u_1\np : \u03b1 \u2192 Prop\nf : (a : \u03b1) \u2192 p a \u2192 \u03b2\nL : List \u03b1\nh' : \u2200 (a : \u03b1), a \u2208 L \u2192 p a\n\u22a2 pmap f L h' = List.pmapImpl f L h'"}, {"tactic": "exact go L fun _ hx => hx", "annotated_tactic": ["exact go L fun _ hx => hx", []], "state_before": "case h.h.h.h.h.h\n\u03b1 : Type u_2\n\u03b2 : Type u_1\np : \u03b1 \u2192 Prop\nf : (a : \u03b1) \u2192 p a \u2192 \u03b2\nL : List \u03b1\nh' : \u2200 (a : \u03b1), a \u2208 L \u2192 p a\n\u22a2 pmap f L h' = List.pmapImpl f L h'", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Sups.lean", "full_name": "Finset.disjSups_subset_sups", "start": [481, 1], "end": [483, 58], "traced_tactics": [{"tactic": "simp_rw [subset_iff, mem_sups, mem_disjSups]", "annotated_tactic": ["simp_rw [<a>subset_iff</a>, <a>mem_sups</a>, <a>mem_disjSups</a>]", [{"full_name": "Finset.subset_iff", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [371, 9], "def_end_pos": [371, 19]}, {"full_name": "Finset.mem_sups", "def_path": "Mathlib/Data/Finset/Sups.lean", "def_pos": [58, 9], "def_end_pos": [58, 17]}, {"full_name": "Finset.mem_disjSups", "def_path": "Mathlib/Data/Finset/Sups.lean", "def_pos": [477, 9], "def_end_pos": [477, 21]}]], "state_before": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u2074 : DecidableEq \u03b1\ninst\u271d\u00b3 : DecidableEq \u03b2\ninst\u271d\u00b2 : SemilatticeSup \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\ninst\u271d : DecidableRel Disjoint\ns s\u2081 s\u2082 t t\u2081 t\u2082 u : Finset \u03b1\na b c : \u03b1\n\u22a2 s \u25cb t \u2286 s \u22bb t", "state_after": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u2074 : DecidableEq \u03b1\ninst\u271d\u00b3 : DecidableEq \u03b2\ninst\u271d\u00b2 : SemilatticeSup \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\ninst\u271d : DecidableRel Disjoint\ns s\u2081 s\u2082 t t\u2081 t\u2082 u : Finset \u03b1\na b c : \u03b1\n\u22a2 \u2200 \u2983x : \u03b1\u2984, (\u2203 a, a \u2208 s \u2227 \u2203 b, b \u2208 t \u2227 Disjoint a b \u2227 a \u2294 b = x) \u2192 \u2203 a, a \u2208 s \u2227 \u2203 b, b \u2208 t \u2227 a \u2294 b = x"}, {"tactic": "exact fun c \u27e8a, b, ha, hb, _, hc\u27e9 => \u27e8a, b, ha, hb, hc\u27e9", "annotated_tactic": ["exact fun c \u27e8a, b, ha, hb, _, hc\u27e9 => \u27e8a, b, ha, hb, hc\u27e9", []], "state_before": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u2074 : DecidableEq \u03b1\ninst\u271d\u00b3 : DecidableEq \u03b2\ninst\u271d\u00b2 : SemilatticeSup \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\ninst\u271d : DecidableRel Disjoint\ns s\u2081 s\u2082 t t\u2081 t\u2082 u : Finset \u03b1\na b c : \u03b1\n\u22a2 \u2200 \u2983x : \u03b1\u2984, (\u2203 a, a \u2208 s \u2227 \u2203 b, b \u2208 t \u2227 Disjoint a b \u2227 a \u2294 b = x) \u2192 \u2203 a, a \u2208 s \u2227 \u2203 b, b \u2208 t \u2227 a \u2294 b = x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "full_name": "MeasureTheory.L1.SimpleFunc.norm_Integral_le_one", "start": [573, 1], "end": [577, 35], "traced_tactics": [{"tactic": "rw [one_mul]", "annotated_tactic": ["rw [<a>one_mul</a>]", [{"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [464, 9], "def_end_pos": [464, 16]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedAddCommGroup F\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2078 : NormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\nF' : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\nE' : Type u_6\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : NormedSpace \u211d E'\ninst\u271d : NormedSpace \ud835\udd5c E'\nf : { x // x \u2208 simpleFunc E 1 \u03bc }\n\u22a2 \u2016\u2191{\n            toAddHom :=\n              { toFun := integral,\n                map_add' := (_ : \u2200 (f g : { x // x \u2208 simpleFunc E 1 \u03bc }), integral (f + g) = integral f + integral g) },\n            map_smul' := (_ : \u2200 (c : \u211d) (f : { x // x \u2208 simpleFunc E 1 \u03bc }), integral (c \u2022 f) = c \u2022 integral f) }\n        f\u2016 \u2264\n    1 * \u2016f\u2016", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedAddCommGroup F\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2078 : NormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\nF' : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\nE' : Type u_6\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : NormedSpace \u211d E'\ninst\u271d : NormedSpace \ud835\udd5c E'\nf : { x // x \u2208 simpleFunc E 1 \u03bc }\n\u22a2 \u2016\u2191{\n            toAddHom :=\n              { toFun := integral,\n                map_add' := (_ : \u2200 (f g : { x // x \u2208 simpleFunc E 1 \u03bc }), integral (f + g) = integral f + integral g) },\n            map_smul' := (_ : \u2200 (c : \u211d) (f : { x // x \u2208 simpleFunc E 1 \u03bc }), integral (c \u2022 f) = c \u2022 integral f) }\n        f\u2016 \u2264\n    \u2016f\u2016"}, {"tactic": "exact norm_integral_le_norm f", "annotated_tactic": ["exact <a>norm_integral_le_norm</a> f", [{"full_name": "MeasureTheory.L1.SimpleFunc.norm_integral_le_norm", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [546, 9], "def_end_pos": [546, 30]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedAddCommGroup F\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2078 : NormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\nF' : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\nE' : Type u_6\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : NormedSpace \u211d E'\ninst\u271d : NormedSpace \ud835\udd5c E'\nf : { x // x \u2208 simpleFunc E 1 \u03bc }\n\u22a2 \u2016\u2191{\n            toAddHom :=\n              { toFun := integral,\n                map_add' := (_ : \u2200 (f g : { x // x \u2208 simpleFunc E 1 \u03bc }), integral (f + g) = integral f + integral g) },\n            map_smul' := (_ : \u2200 (c : \u211d) (f : { x // x \u2208 simpleFunc E 1 \u03bc }), integral (c \u2022 f) = c \u2022 integral f) }\n        f\u2016 \u2264\n    \u2016f\u2016", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "Finset.sdiff_insert_insert_of_mem_of_not_mem", "start": [2355, 1], "end": [2357, 61], "traced_tactics": [{"tactic": "rw [sdiff_insert, insert_erase (mem_sdiff.mpr \u27e8hxs, hxt\u27e9)]", "annotated_tactic": ["rw [<a>sdiff_insert</a>, <a>insert_erase</a> (mem_sdiff.mpr \u27e8hxs, hxt\u27e9)]", [{"full_name": "Finset.sdiff_insert", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2350, 9], "def_end_pos": [2350, 21]}, {"full_name": "Finset.insert_erase", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1957, 9], "def_end_pos": [1957, 21]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d u v : Finset \u03b1\na b : \u03b1\ns t : Finset \u03b1\nx : \u03b1\nhxs : x \u2208 s\nhxt : \u00acx \u2208 t\n\u22a2 insert x (s \\ insert x t) = s \\ t", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Array/Init/Lemmas.lean", "full_name": "Array.foldrM_eq_reverse_foldlM_data.aux", "start": [51, 1], "end": [57, 76], "traced_tactics": [{"tactic": "unfold foldrM.fold", "annotated_tactic": ["unfold <a>foldrM.fold</a>", [{"full_name": "Array.foldrM.fold", "def_path": "lake-packages/lean4/src/lean/Init/Data/Array/Basic.lean", "def_pos": [234, 11], "def_end_pos": [234, 15]}]], "state_before": "m : Type u_1 \u2192 Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_1\ninst\u271d : Monad m\nf : \u03b1 \u2192 \u03b2 \u2192 m \u03b2\narr : Array \u03b1\ninit : \u03b2\ni : Nat\nh : i \u2264 size arr\n\u22a2 List.foldlM (fun x y => f y x) init (List.reverse (List.take i arr.data)) = foldrM.fold f arr 0 i h init", "state_after": "m : Type u_1 \u2192 Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_1\ninst\u271d : Monad m\nf : \u03b1 \u2192 \u03b2 \u2192 m \u03b2\narr : Array \u03b1\ninit : \u03b2\ni : Nat\nh : i \u2264 size arr\n\u22a2 List.foldlM (fun x y => f y x) init (List.reverse (List.take i arr.data)) =\n    if (i == 0) = true then pure init\n    else\n      match i, h with\n      | 0, x => pure init\n      | Nat.succ i, h =>\n        let_fun this := (_ : i < size arr);\n        do\n        let __do_lift \u2190 f arr[i] init\n        foldrM.fold f arr 0 i (_ : i \u2264 size arr) __do_lift"}, {"tactic": "simp [List.foldlM, List.take]", "annotated_tactic": ["simp [<a>List.foldlM</a>, <a>List.take</a>]", [{"full_name": "List.foldlM", "def_path": "lake-packages/lean4/src/lean/Init/Data/List/Control.lean", "def_pos": [94, 15], "def_end_pos": [94, 21]}, {"full_name": "List.take", "def_path": "lake-packages/lean4/src/lean/Init/Data/List/Basic.lean", "def_pos": [494, 5], "def_end_pos": [494, 9]}]], "state_before": "m : Type u_1 \u2192 Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_1\ninst\u271d : Monad m\nf : \u03b1 \u2192 \u03b2 \u2192 m \u03b2\narr : Array \u03b1\ninit : \u03b2\ni : Nat\nh : 0 \u2264 size arr\n\u22a2 List.foldlM (fun x y => f y x) init (List.reverse (List.take 0 arr.data)) =\n    if (0 == 0) = true then pure init\n    else\n      match 0, h with\n      | 0, x => pure init\n      | Nat.succ i, h =>\n        let_fun this := (_ : i < size arr);\n        do\n        let __do_lift \u2190 f arr[i] init\n        foldrM.fold f arr 0 i (_ : i \u2264 size arr) __do_lift", "state_after": "no goals"}, {"tactic": "rw [\u2190 List.take_concat_get _ _ h]", "annotated_tactic": ["rw [\u2190 <a>List.take_concat_get</a> _ _ h]", [{"full_name": "List.take_concat_get", "def_path": "lake-packages/std/Std/Data/List/Init/Lemmas.lean", "def_pos": [171, 9], "def_end_pos": [171, 24]}]], "state_before": "m : Type u_1 \u2192 Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_1\ninst\u271d : Monad m\nf : \u03b1 \u2192 \u03b2 \u2192 m \u03b2\narr : Array \u03b1\ninit : \u03b2\ni\u271d i : Nat\nh : i + 1 \u2264 size arr\n\u22a2 List.foldlM (fun x y => f y x) init (List.reverse (List.take (i + 1) arr.data)) =\n    if (i + 1 == 0) = true then pure init\n    else\n      match i + 1, h with\n      | 0, x => pure init\n      | Nat.succ i, h =>\n        let_fun this := (_ : i < size arr);\n        do\n        let __do_lift \u2190 f arr[i] init\n        foldrM.fold f arr 0 i (_ : i \u2264 size arr) __do_lift", "state_after": "m : Type u_1 \u2192 Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_1\ninst\u271d : Monad m\nf : \u03b1 \u2192 \u03b2 \u2192 m \u03b2\narr : Array \u03b1\ninit : \u03b2\ni\u271d i : Nat\nh : i + 1 \u2264 size arr\n\u22a2 List.foldlM (fun x y => f y x) init (List.reverse (List.concat (List.take i arr.data) arr.data[i])) =\n    if (i + 1 == 0) = true then pure init\n    else\n      match i + 1, h with\n      | 0, x => pure init\n      | Nat.succ i, h =>\n        let_fun this := (_ : i < size arr);\n        do\n        let __do_lift \u2190 f arr[i] init\n        foldrM.fold f arr 0 i (_ : i \u2264 size arr) __do_lift"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "m : Type u_1 \u2192 Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_1\ninst\u271d : Monad m\nf : \u03b1 \u2192 \u03b2 \u2192 m \u03b2\narr : Array \u03b1\ninit : \u03b2\ni\u271d i : Nat\nh : i + 1 \u2264 size arr\n\u22a2 (do\n      let init \u2190 f arr.data[i] init\n      List.foldrM (fun x y => f x y) init (List.take i arr.data)) =\n    if i + 1 = 0 then pure init\n    else do\n      let init \u2190 f arr[i] init\n      List.foldrM (fun x y => f x y) init (List.take i arr.data)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Kernel/CondCdf.lean", "full_name": "ProbabilityTheory.ofReal_condCdfRat_ae_eq", "start": [661, 1], "end": [665, 55], "traced_tactics": [{"tactic": "filter_upwards [condCdfRat_ae_eq \u03c1 r, preCdf_le_one \u03c1] with a ha ha_le_one", "annotated_tactic": ["filter_upwards [<a>condCdfRat_ae_eq</a> \u03c1 r, <a>preCdf_le_one</a> \u03c1] with a ha ha_le_one", [{"full_name": "ProbabilityTheory.condCdfRat_ae_eq", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [656, 9], "def_end_pos": [656, 25]}, {"full_name": "ProbabilityTheory.preCdf_le_one", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [349, 9], "def_end_pos": [349, 22]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nr : \u211a\n\u22a2 (fun a => ENNReal.ofReal (condCdfRat \u03c1 a r)) =\u1d50[Measure.fst \u03c1] preCdf \u03c1 r", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nr : \u211a\na : \u03b1\nha : condCdfRat \u03c1 a r = ENNReal.toReal (preCdf \u03c1 r a)\nha_le_one : \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\n\u22a2 ENNReal.ofReal (condCdfRat \u03c1 a r) = preCdf \u03c1 r a"}, {"tactic": "rw [ha, ENNReal.ofReal_toReal]", "annotated_tactic": ["rw [ha, <a>ENNReal.ofReal_toReal</a>]", [{"full_name": "ENNReal.ofReal_toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [186, 9], "def_end_pos": [186, 22]}]], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nr : \u211a\na : \u03b1\nha : condCdfRat \u03c1 a r = ENNReal.toReal (preCdf \u03c1 r a)\nha_le_one : \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\n\u22a2 ENNReal.ofReal (condCdfRat \u03c1 a r) = preCdf \u03c1 r a", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nr : \u211a\na : \u03b1\nha : condCdfRat \u03c1 a r = ENNReal.toReal (preCdf \u03c1 r a)\nha_le_one : \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\n\u22a2 preCdf \u03c1 r a \u2260 \u22a4"}, {"tactic": "exact ((ha_le_one r).trans_lt ENNReal.one_lt_top).ne", "annotated_tactic": ["exact ((ha_le_one r).<a>trans_lt</a> <a>ENNReal.one_lt_top</a>).<a>ne</a>", [{"full_name": "LE.le.trans_lt", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [124, 7], "def_end_pos": [124, 21]}, {"full_name": "ENNReal.one_lt_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [723, 17], "def_end_pos": [723, 27]}, {"full_name": "LT.lt.ne", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [152, 7], "def_end_pos": [152, 15]}]], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nr : \u211a\na : \u03b1\nha : condCdfRat \u03c1 a r = ENNReal.toReal (preCdf \u03c1 r a)\nha_le_one : \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\n\u22a2 preCdf \u03c1 r a \u2260 \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Covering/LiminfLimsup.lean", "full_name": "blimsup_thickening_mul_ae_eq", "start": [272, 1], "end": [288, 95], "traced_tactics": [{"tactic": "let q : \u2115 \u2192 Prop := fun i => p i \u2227 0 < r i", "annotated_tactic": ["let q : \u2115 \u2192 Prop := fun i => p i \u2227 0 < r i", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nM : \u211d\nhM : 0 < M\nr : \u2115 \u2192 \u211d\nhr : Tendsto r atTop (\ud835\udcdd 0)\n\u22a2 blimsup (fun i => thickening (M * r i) (s i)) atTop p =\u1d50[\u03bc] blimsup (fun i => thickening (r i) (s i)) atTop p", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nM : \u211d\nhM : 0 < M\nr : \u2115 \u2192 \u211d\nhr : Tendsto r atTop (\ud835\udcdd 0)\nq : \u2115 \u2192 Prop := fun i => p i \u2227 0 < r i\n\u22a2 blimsup (fun i => thickening (M * r i) (s i)) atTop p =\u1d50[\u03bc] blimsup (fun i => thickening (r i) (s i)) atTop p"}, {"tactic": "rw [h\u2081, h\u2082]", "annotated_tactic": ["rw [h\u2081, h\u2082]", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nM : \u211d\nhM : 0 < M\nr : \u2115 \u2192 \u211d\nhr : Tendsto r atTop (\ud835\udcdd 0)\nq : \u2115 \u2192 Prop := fun i => p i \u2227 0 < r i\nh\u2081 : blimsup (fun i => thickening (r i) (s i)) atTop p = blimsup (fun i => thickening (r i) (s i)) atTop q\nh\u2082 : blimsup (fun i => thickening (M * r i) (s i)) atTop p = blimsup (fun i => thickening (M * r i) (s i)) atTop q\n\u22a2 blimsup (fun i => thickening (M * r i) (s i)) atTop p =\u1d50[\u03bc] blimsup (fun i => thickening (r i) (s i)) atTop p", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nM : \u211d\nhM : 0 < M\nr : \u2115 \u2192 \u211d\nhr : Tendsto r atTop (\ud835\udcdd 0)\nq : \u2115 \u2192 Prop := fun i => p i \u2227 0 < r i\nh\u2081 : blimsup (fun i => thickening (r i) (s i)) atTop p = blimsup (fun i => thickening (r i) (s i)) atTop q\nh\u2082 : blimsup (fun i => thickening (M * r i) (s i)) atTop p = blimsup (fun i => thickening (M * r i) (s i)) atTop q\n\u22a2 blimsup (fun i => thickening (M * r i) (s i)) atTop q =\u1d50[\u03bc] blimsup (fun i => thickening (r i) (s i)) atTop q"}, {"tactic": "exact blimsup_thickening_mul_ae_eq_aux \u03bc q s hM r hr (eventually_of_forall fun i hi => hi.2)", "annotated_tactic": ["exact <a>blimsup_thickening_mul_ae_eq_aux</a> \u03bc q s hM r hr (<a>eventually_of_forall</a> fun i hi => hi.2)", [{"full_name": "blimsup_thickening_mul_ae_eq_aux", "def_path": "Mathlib/MeasureTheory/Covering/LiminfLimsup.lean", "def_pos": [248, 9], "def_end_pos": [248, 41]}, {"full_name": "Filter.eventually_of_forall", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1111, 9], "def_end_pos": [1111, 29]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nM : \u211d\nhM : 0 < M\nr : \u2115 \u2192 \u211d\nhr : Tendsto r atTop (\ud835\udcdd 0)\nq : \u2115 \u2192 Prop := fun i => p i \u2227 0 < r i\nh\u2081 : blimsup (fun i => thickening (r i) (s i)) atTop p = blimsup (fun i => thickening (r i) (s i)) atTop q\nh\u2082 : blimsup (fun i => thickening (M * r i) (s i)) atTop p = blimsup (fun i => thickening (M * r i) (s i)) atTop q\n\u22a2 blimsup (fun i => thickening (M * r i) (s i)) atTop q =\u1d50[\u03bc] blimsup (fun i => thickening (r i) (s i)) atTop q", "state_after": "no goals"}, {"tactic": "refine' blimsup_congr' (eventually_of_forall fun i h => _)", "annotated_tactic": ["refine' <a>blimsup_congr'</a> (<a>eventually_of_forall</a> fun i h => _)", [{"full_name": "Filter.blimsup_congr'", "def_path": "Mathlib/Order/LiminfLimsup.lean", "def_pos": [809, 9], "def_end_pos": [809, 23]}, {"full_name": "Filter.eventually_of_forall", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1111, 9], "def_end_pos": [1111, 29]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nM : \u211d\nhM : 0 < M\nr : \u2115 \u2192 \u211d\nhr : Tendsto r atTop (\ud835\udcdd 0)\nq : \u2115 \u2192 Prop := fun i => p i \u2227 0 < r i\n\u22a2 blimsup (fun i => thickening (r i) (s i)) atTop p = blimsup (fun i => thickening (r i) (s i)) atTop q", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nM : \u211d\nhM : 0 < M\nr : \u2115 \u2192 \u211d\nhr : Tendsto r atTop (\ud835\udcdd 0)\nq : \u2115 \u2192 Prop := fun i => p i \u2227 0 < r i\ni : \u2115\nh : thickening (r i) (s i) \u2260 \u22a5\n\u22a2 p i \u2194 q i"}, {"tactic": "replace hi : 0 < r i", "annotated_tactic": ["replace hi : 0 < r i", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nM : \u211d\nhM : 0 < M\nr : \u2115 \u2192 \u211d\nhr : Tendsto r atTop (\ud835\udcdd 0)\nq : \u2115 \u2192 Prop := fun i => p i \u2227 0 < r i\ni : \u2115\nh : thickening (r i) (s i) \u2260 \u22a5\n\u22a2 p i \u2194 q i", "state_after": "case hi\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nM : \u211d\nhM : 0 < M\nr : \u2115 \u2192 \u211d\nhr : Tendsto r atTop (\ud835\udcdd 0)\nq : \u2115 \u2192 Prop := fun i => p i \u2227 0 < r i\ni : \u2115\nh : thickening (r i) (s i) \u2260 \u22a5\n\u22a2 0 < r i\n\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nM : \u211d\nhM : 0 < M\nr : \u2115 \u2192 \u211d\nhr : Tendsto r atTop (\ud835\udcdd 0)\nq : \u2115 \u2192 Prop := fun i => p i \u2227 0 < r i\ni : \u2115\nh : thickening (r i) (s i) \u2260 \u22a5\nhi : 0 < r i\n\u22a2 p i \u2194 q i"}, {"tactic": "simp only [hi, iff_self_and, imp_true_iff]", "annotated_tactic": ["simp only [hi, <a>iff_self_and</a>, <a>imp_true_iff</a>]", [{"full_name": "iff_self_and", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [214, 17], "def_end_pos": [214, 29]}, {"full_name": "imp_true_iff", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [116, 9], "def_end_pos": [116, 21]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nM : \u211d\nhM : 0 < M\nr : \u2115 \u2192 \u211d\nhr : Tendsto r atTop (\ud835\udcdd 0)\nq : \u2115 \u2192 Prop := fun i => p i \u2227 0 < r i\ni : \u2115\nh : thickening (r i) (s i) \u2260 \u22a5\nhi : 0 < r i\n\u22a2 p i \u2194 q i", "state_after": "no goals"}, {"tactic": "contrapose! h", "annotated_tactic": ["contrapose! h", []], "state_before": "case hi\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nM : \u211d\nhM : 0 < M\nr : \u2115 \u2192 \u211d\nhr : Tendsto r atTop (\ud835\udcdd 0)\nq : \u2115 \u2192 Prop := fun i => p i \u2227 0 < r i\ni : \u2115\nh : thickening (r i) (s i) \u2260 \u22a5\n\u22a2 0 < r i", "state_after": "case hi\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nM : \u211d\nhM : 0 < M\nr : \u2115 \u2192 \u211d\nhr : Tendsto r atTop (\ud835\udcdd 0)\nq : \u2115 \u2192 Prop := fun i => p i \u2227 0 < r i\ni : \u2115\nh : r i \u2264 0\n\u22a2 thickening (r i) (s i) = \u22a5"}, {"tactic": "apply thickening_of_nonpos h", "annotated_tactic": ["apply <a>thickening_of_nonpos</a> h", [{"full_name": "Metric.thickening_of_nonpos", "def_path": "Mathlib/Topology/MetricSpace/HausdorffDistance.lean", "def_pos": [932, 9], "def_end_pos": [932, 29]}]], "state_before": "case hi\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nM : \u211d\nhM : 0 < M\nr : \u2115 \u2192 \u211d\nhr : Tendsto r atTop (\ud835\udcdd 0)\nq : \u2115 \u2192 Prop := fun i => p i \u2227 0 < r i\ni : \u2115\nh : r i \u2264 0\n\u22a2 thickening (r i) (s i) = \u22a5", "state_after": "no goals"}, {"tactic": "refine' blimsup_congr' (eventually_of_forall fun i h => _)", "annotated_tactic": ["refine' <a>blimsup_congr'</a> (<a>eventually_of_forall</a> fun i h => _)", [{"full_name": "Filter.blimsup_congr'", "def_path": "Mathlib/Order/LiminfLimsup.lean", "def_pos": [809, 9], "def_end_pos": [809, 23]}, {"full_name": "Filter.eventually_of_forall", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1111, 9], "def_end_pos": [1111, 29]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nM : \u211d\nhM : 0 < M\nr : \u2115 \u2192 \u211d\nhr : Tendsto r atTop (\ud835\udcdd 0)\nq : \u2115 \u2192 Prop := fun i => p i \u2227 0 < r i\nh\u2081 : blimsup (fun i => thickening (r i) (s i)) atTop p = blimsup (fun i => thickening (r i) (s i)) atTop q\n\u22a2 blimsup (fun i => thickening (M * r i) (s i)) atTop p = blimsup (fun i => thickening (M * r i) (s i)) atTop q", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nM : \u211d\nhM : 0 < M\nr : \u2115 \u2192 \u211d\nhr : Tendsto r atTop (\ud835\udcdd 0)\nq : \u2115 \u2192 Prop := fun i => p i \u2227 0 < r i\nh\u2081 : blimsup (fun i => thickening (r i) (s i)) atTop p = blimsup (fun i => thickening (r i) (s i)) atTop q\ni : \u2115\nh : thickening (M * r i) (s i) \u2260 \u22a5\n\u22a2 p i \u2194 q i"}, {"tactic": "replace h : 0 < r i", "annotated_tactic": ["replace h : 0 < r i", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nM : \u211d\nhM : 0 < M\nr : \u2115 \u2192 \u211d\nhr : Tendsto r atTop (\ud835\udcdd 0)\nq : \u2115 \u2192 Prop := fun i => p i \u2227 0 < r i\nh\u2081 : blimsup (fun i => thickening (r i) (s i)) atTop p = blimsup (fun i => thickening (r i) (s i)) atTop q\ni : \u2115\nh : thickening (M * r i) (s i) \u2260 \u22a5\n\u22a2 p i \u2194 q i", "state_after": "case h\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nM : \u211d\nhM : 0 < M\nr : \u2115 \u2192 \u211d\nhr : Tendsto r atTop (\ud835\udcdd 0)\nq : \u2115 \u2192 Prop := fun i => p i \u2227 0 < r i\nh\u2081 : blimsup (fun i => thickening (r i) (s i)) atTop p = blimsup (fun i => thickening (r i) (s i)) atTop q\ni : \u2115\nh : thickening (M * r i) (s i) \u2260 \u22a5\n\u22a2 0 < r i\n\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nM : \u211d\nhM : 0 < M\nr : \u2115 \u2192 \u211d\nhr : Tendsto r atTop (\ud835\udcdd 0)\nq : \u2115 \u2192 Prop := fun i => p i \u2227 0 < r i\nh\u2081 : blimsup (fun i => thickening (r i) (s i)) atTop p = blimsup (fun i => thickening (r i) (s i)) atTop q\ni : \u2115\nh : 0 < r i\n\u22a2 p i \u2194 q i"}, {"tactic": "simp only [h, iff_self_and, imp_true_iff]", "annotated_tactic": ["simp only [h, <a>iff_self_and</a>, <a>imp_true_iff</a>]", [{"full_name": "iff_self_and", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [214, 17], "def_end_pos": [214, 29]}, {"full_name": "imp_true_iff", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [116, 9], "def_end_pos": [116, 21]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nM : \u211d\nhM : 0 < M\nr : \u2115 \u2192 \u211d\nhr : Tendsto r atTop (\ud835\udcdd 0)\nq : \u2115 \u2192 Prop := fun i => p i \u2227 0 < r i\nh\u2081 : blimsup (fun i => thickening (r i) (s i)) atTop p = blimsup (fun i => thickening (r i) (s i)) atTop q\ni : \u2115\nh : 0 < r i\n\u22a2 p i \u2194 q i", "state_after": "no goals"}, {"tactic": "rw [\u2190 zero_lt_mul_left hM]", "annotated_tactic": ["rw [\u2190 <a>zero_lt_mul_left</a> hM]", [{"full_name": "zero_lt_mul_left", "def_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "def_pos": [353, 9], "def_end_pos": [353, 25]}]], "state_before": "case h\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nM : \u211d\nhM : 0 < M\nr : \u2115 \u2192 \u211d\nhr : Tendsto r atTop (\ud835\udcdd 0)\nq : \u2115 \u2192 Prop := fun i => p i \u2227 0 < r i\nh\u2081 : blimsup (fun i => thickening (r i) (s i)) atTop p = blimsup (fun i => thickening (r i) (s i)) atTop q\ni : \u2115\nh : thickening (M * r i) (s i) \u2260 \u22a5\n\u22a2 0 < r i", "state_after": "case h\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nM : \u211d\nhM : 0 < M\nr : \u2115 \u2192 \u211d\nhr : Tendsto r atTop (\ud835\udcdd 0)\nq : \u2115 \u2192 Prop := fun i => p i \u2227 0 < r i\nh\u2081 : blimsup (fun i => thickening (r i) (s i)) atTop p = blimsup (fun i => thickening (r i) (s i)) atTop q\ni : \u2115\nh : thickening (M * r i) (s i) \u2260 \u22a5\n\u22a2 0 < M * r i"}, {"tactic": "contrapose! h", "annotated_tactic": ["contrapose! h", []], "state_before": "case h\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nM : \u211d\nhM : 0 < M\nr : \u2115 \u2192 \u211d\nhr : Tendsto r atTop (\ud835\udcdd 0)\nq : \u2115 \u2192 Prop := fun i => p i \u2227 0 < r i\nh\u2081 : blimsup (fun i => thickening (r i) (s i)) atTop p = blimsup (fun i => thickening (r i) (s i)) atTop q\ni : \u2115\nh : thickening (M * r i) (s i) \u2260 \u22a5\n\u22a2 0 < M * r i", "state_after": "case h\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nM : \u211d\nhM : 0 < M\nr : \u2115 \u2192 \u211d\nhr : Tendsto r atTop (\ud835\udcdd 0)\nq : \u2115 \u2192 Prop := fun i => p i \u2227 0 < r i\nh\u2081 : blimsup (fun i => thickening (r i) (s i)) atTop p = blimsup (fun i => thickening (r i) (s i)) atTop q\ni : \u2115\nh : M * r i \u2264 0\n\u22a2 thickening (M * r i) (s i) = \u22a5"}, {"tactic": "apply thickening_of_nonpos h", "annotated_tactic": ["apply <a>thickening_of_nonpos</a> h", [{"full_name": "Metric.thickening_of_nonpos", "def_path": "Mathlib/Topology/MetricSpace/HausdorffDistance.lean", "def_pos": [932, 9], "def_end_pos": [932, 29]}]], "state_before": "case h\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nM : \u211d\nhM : 0 < M\nr : \u2115 \u2192 \u211d\nhr : Tendsto r atTop (\ud835\udcdd 0)\nq : \u2115 \u2192 Prop := fun i => p i \u2227 0 < r i\nh\u2081 : blimsup (fun i => thickening (r i) (s i)) atTop p = blimsup (fun i => thickening (r i) (s i)) atTop q\ni : \u2115\nh : M * r i \u2264 0\n\u22a2 thickening (M * r i) (s i) = \u22a5", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "full_name": "String.prev_of_valid", "start": [296, 1], "end": [299, 37], "traced_tactics": [{"tactic": "simp [prev]", "annotated_tactic": ["simp [<a>prev</a>]", [{"full_name": "String.prev", "def_path": "lake-packages/lean4/src/lean/Init/Data/String/Basic.lean", "def_pos": [101, 5], "def_end_pos": [101, 9]}]], "state_before": "cs : List Char\nc : Char\ncs' : List Char\n\u22a2 prev { data := cs ++ c :: cs' } { byteIdx := utf8Len cs + csize c } = { byteIdx := utf8Len cs }", "state_after": "cs : List Char\nc : Char\ncs' : List Char\n\u22a2 (if { byteIdx := utf8Len cs + csize c } = 0 then 0\n    else utf8PrevAux (cs ++ c :: cs') 0 { byteIdx := utf8Len cs + csize c }) =\n    { byteIdx := utf8Len cs }"}, {"tactic": "refine (if_neg (Pos.ne_of_gt add_csize_pos)).trans ?_", "annotated_tactic": ["refine (<a>if_neg</a> (<a>Pos.ne_of_gt</a> <a>add_csize_pos</a>)).<a>trans</a> ?_", [{"full_name": "if_neg", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [795, 9], "def_end_pos": [795, 15]}, {"full_name": "String.Pos.ne_of_gt", "def_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "def_pos": [136, 9], "def_end_pos": [136, 17]}, {"full_name": "_private.\u00ablake-packages\u00bb.std.Std.Data.String.Lemmas.0.String.add_csize_pos", "def_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "def_pos": [44, 17], "def_end_pos": [44, 30]}, {"full_name": "Eq.trans", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [322, 9], "def_end_pos": [322, 17]}]], "state_before": "cs : List Char\nc : Char\ncs' : List Char\n\u22a2 (if { byteIdx := utf8Len cs + csize c } = 0 then 0\n    else utf8PrevAux (cs ++ c :: cs') 0 { byteIdx := utf8Len cs + csize c }) =\n    { byteIdx := utf8Len cs }", "state_after": "cs : List Char\nc : Char\ncs' : List Char\n\u22a2 utf8PrevAux (cs ++ c :: cs') 0 { byteIdx := utf8Len cs + csize c } = { byteIdx := utf8Len cs }"}, {"tactic": "rw [utf8PrevAux_of_valid] <;> simp", "annotated_tactic": ["rw [<a>utf8PrevAux_of_valid</a>] <;> simp", [{"full_name": "String.utf8PrevAux_of_valid", "def_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "def_pos": [284, 9], "def_end_pos": [284, 29]}]], "state_before": "cs : List Char\nc : Char\ncs' : List Char\n\u22a2 utf8PrevAux (cs ++ c :: cs') 0 { byteIdx := utf8Len cs + csize c } = { byteIdx := utf8Len cs }", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Constructions/Polish.lean", "full_name": "MeasureTheory.borel_eq_borel_of_le", "start": [899, 1], "end": [908, 31], "traced_tactics": [{"tactic": "refine' le_antisymm _ (borel_anti hle)", "annotated_tactic": ["refine' <a>le_antisymm</a> _ (<a>borel_anti</a> hle)", [{"full_name": "le_antisymm", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [188, 9], "def_end_pos": [188, 20]}, {"full_name": "borel_anti", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [63, 9], "def_end_pos": [63, 19]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\nt\u03b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : T2Space \u03b2\ninst\u271d : MeasurableSpace \u03b2\ns : Set \u03b3\nf : \u03b3 \u2192 \u03b2\nt t' : TopologicalSpace \u03b3\nht : PolishSpace \u03b3\nht' : PolishSpace \u03b3\nhle : t \u2264 t'\n\u22a2 borel \u03b3 = borel \u03b3", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\nt\u03b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : T2Space \u03b2\ninst\u271d : MeasurableSpace \u03b2\ns : Set \u03b3\nf : \u03b3 \u2192 \u03b2\nt t' : TopologicalSpace \u03b3\nht : PolishSpace \u03b3\nht' : PolishSpace \u03b3\nhle : t \u2264 t'\n\u22a2 borel \u03b3 \u2264 borel \u03b3"}, {"tactic": "intro s hs", "annotated_tactic": ["intro s hs", []], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\nt\u03b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : T2Space \u03b2\ninst\u271d : MeasurableSpace \u03b2\ns : Set \u03b3\nf : \u03b3 \u2192 \u03b2\nt t' : TopologicalSpace \u03b3\nht : PolishSpace \u03b3\nht' : PolishSpace \u03b3\nhle : t \u2264 t'\n\u22a2 borel \u03b3 \u2264 borel \u03b3", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\nt\u03b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : T2Space \u03b2\ninst\u271d : MeasurableSpace \u03b2\ns\u271d : Set \u03b3\nf : \u03b3 \u2192 \u03b2\nt t' : TopologicalSpace \u03b3\nht : PolishSpace \u03b3\nht' : PolishSpace \u03b3\nhle : t \u2264 t'\ns : Set \u03b3\nhs : MeasurableSet s\n\u22a2 MeasurableSet s"}, {"tactic": "have e := @Continuous.measurableEmbedding\n  _ _ t' _ (@borel _ t') _ (@BorelSpace.mk _ _ (borel \u03b3) rfl)\n  t _ (@borel _ t) (@BorelSpace.mk _ t (@borel _ t) rfl) (continuous_id_of_le hle) injective_id", "annotated_tactic": ["have e := @<a>Continuous.measurableEmbedding</a>\n    _ _ t' _ (@<a>borel</a> _ t') _ (@<a>BorelSpace.mk</a> _ _ (<a>borel</a> \u03b3) <a>rfl</a>)\n    t _ (@<a>borel</a> _ t) (@<a>BorelSpace.mk</a> _ t (@<a>borel</a> _ t) <a>rfl</a>) (<a>continuous_id_of_le</a> hle) <a>injective_id</a>", [{"full_name": "Continuous.measurableEmbedding", "def_path": "Mathlib/MeasureTheory/Constructions/Polish.lean", "def_pos": [860, 9], "def_end_pos": [860, 46]}, {"full_name": "borel", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [59, 5], "def_end_pos": [59, 10]}, {"full_name": "BorelSpace.mk", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [202, 18], "def_end_pos": [202, 70]}, {"full_name": "borel", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [59, 5], "def_end_pos": [59, 10]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}, {"full_name": "borel", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [59, 5], "def_end_pos": [59, 10]}, {"full_name": "BorelSpace.mk", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [202, 18], "def_end_pos": [202, 70]}, {"full_name": "borel", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [59, 5], "def_end_pos": [59, 10]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}, {"full_name": "continuous_id_of_le", "def_path": "Mathlib/Topology/Order.lean", "def_pos": [847, 9], "def_end_pos": [847, 28]}, {"full_name": "Function.injective_id", "def_path": "Mathlib/Init/Function.lean", "def_pos": [194, 9], "def_end_pos": [194, 21]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\nt\u03b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : T2Space \u03b2\ninst\u271d : MeasurableSpace \u03b2\ns\u271d : Set \u03b3\nf : \u03b3 \u2192 \u03b2\nt t' : TopologicalSpace \u03b3\nht : PolishSpace \u03b3\nht' : PolishSpace \u03b3\nhle : t \u2264 t'\ns : Set \u03b3\nhs : MeasurableSet s\n\u22a2 MeasurableSet s", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\nt\u03b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : T2Space \u03b2\ninst\u271d : MeasurableSpace \u03b2\ns\u271d : Set \u03b3\nf : \u03b3 \u2192 \u03b2\nt t' : TopologicalSpace \u03b3\nht : PolishSpace \u03b3\nht' : PolishSpace \u03b3\nhle : t \u2264 t'\ns : Set \u03b3\nhs : MeasurableSet s\ne : MeasurableEmbedding id\n\u22a2 MeasurableSet s"}, {"tactic": "convert e.measurableSet_image.2 hs", "annotated_tactic": ["convert e.measurableSet_image.2 hs", []], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\nt\u03b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : T2Space \u03b2\ninst\u271d : MeasurableSpace \u03b2\ns\u271d : Set \u03b3\nf : \u03b3 \u2192 \u03b2\nt t' : TopologicalSpace \u03b3\nht : PolishSpace \u03b3\nht' : PolishSpace \u03b3\nhle : t \u2264 t'\ns : Set \u03b3\nhs : MeasurableSet s\ne : MeasurableEmbedding id\n\u22a2 MeasurableSet s", "state_after": "case h.e'_3\n\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\nt\u03b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : T2Space \u03b2\ninst\u271d : MeasurableSpace \u03b2\ns\u271d : Set \u03b3\nf : \u03b3 \u2192 \u03b2\nt t' : TopologicalSpace \u03b3\nht : PolishSpace \u03b3\nht' : PolishSpace \u03b3\nhle : t \u2264 t'\ns : Set \u03b3\nhs : MeasurableSet s\ne : MeasurableEmbedding id\n\u22a2 s = id '' s"}, {"tactic": "simp only [id_eq, image_id']", "annotated_tactic": ["simp only [<a>id_eq</a>, <a>image_id'</a>]", [{"full_name": "id_eq", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [284, 17], "def_end_pos": [284, 22]}, {"full_name": "Set.image_id'", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [396, 9], "def_end_pos": [396, 18]}]], "state_before": "case h.e'_3\n\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\nt\u03b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : T2Space \u03b2\ninst\u271d : MeasurableSpace \u03b2\ns\u271d : Set \u03b3\nf : \u03b3 \u2192 \u03b2\nt t' : TopologicalSpace \u03b3\nht : PolishSpace \u03b3\nht' : PolishSpace \u03b3\nhle : t \u2264 t'\ns : Set \u03b3\nhs : MeasurableSet s\ne : MeasurableEmbedding id\n\u22a2 s = id '' s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Vector/Basic.lean", "full_name": "Vector.map_id", "start": [232, 1], "end": [233, 64], "traced_tactics": [{"tactic": "simp only [List.map_id, Vector.toList_map]", "annotated_tactic": ["simp only [<a>List.map_id</a>, <a>Vector.toList_map</a>]", [{"full_name": "List.map_id", "def_path": "lake-packages/std/Std/Data/List/Init/Lemmas.lean", "def_pos": [94, 17], "def_end_pos": [94, 23]}, {"full_name": "Vector.toList_map", "def_path": "Mathlib/Data/Vector/Basic.lean", "def_pos": [99, 9], "def_end_pos": [99, 19]}]], "state_before": "n\u271d : \u2115\n\u03b1 : Type u_1\nn : \u2115\nv : Vector \u03b1 n\n\u22a2 toList (map id v) = toList v", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Num/Lemmas.lean", "full_name": "PosNum.to_nat_inj", "start": [529, 1], "end": [530, 91], "traced_tactics": [{"tactic": "rw [\u2190 PosNum.of_to_nat, \u2190 PosNum.of_to_nat, h]", "annotated_tactic": ["rw [\u2190 <a>PosNum.of_to_nat</a>, \u2190 <a>PosNum.of_to_nat</a>, h]", [{"full_name": "PosNum.of_to_nat", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [524, 9], "def_end_pos": [524, 18]}, {"full_name": "PosNum.of_to_nat", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [524, 9], "def_end_pos": [524, 18]}]], "state_before": "\u03b1 : Type u_1\nm n : PosNum\nh : \u2191m = \u2191n\n\u22a2 pos m = pos n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/IntervalAverage.lean", "full_name": "interval_average_eq_div", "start": [52, 1], "end": [54, 56], "traced_tactics": [{"tactic": "rw [interval_average_eq, smul_eq_mul, div_eq_inv_mul]", "annotated_tactic": ["rw [<a>interval_average_eq</a>, <a>smul_eq_mul</a>, <a>div_eq_inv_mul</a>]", [{"full_name": "interval_average_eq", "def_path": "Mathlib/MeasureTheory/Integral/IntervalAverage.lean", "def_pos": [43, 9], "def_end_pos": [43, 28]}, {"full_name": "smul_eq_mul", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [93, 9], "def_end_pos": [93, 20]}, {"full_name": "div_eq_inv_mul", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [492, 9], "def_end_pos": [492, 23]}]], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf : \u211d \u2192 \u211d\na b : \u211d\n\u22a2 \u2a0d (x : \u211d) in a..b, f x = (\u222b (x : \u211d) in a..b, f x) / (b - a)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Sum/Basic.lean", "full_name": "Sum.getLeft_inl", "start": [83, 9], "end": [83, 88], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Decomposition/Lebesgue.lean", "full_name": "MeasureTheory.Measure.LebesgueDecomposition.iSup_mem_measurableLE", "start": [498, 1], "end": [509, 66], "traced_tactics": [{"tactic": "induction' n with m hm", "annotated_tactic": ["induction' n with m hm", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bc \u03bd\nn : \u2115\n\u22a2 (fun x => \u2a06 k, \u2a06 (_ : k \u2264 n), f k x) \u2208 measurableLE \u03bc \u03bd", "state_after": "case zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bc \u03bd\n\u22a2 (fun x => \u2a06 k, \u2a06 (_ : k \u2264 Nat.zero), f k x) \u2208 measurableLE \u03bc \u03bd\n\ncase succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u271d : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bc \u03bd\nm : \u2115\nhm : (fun x => \u2a06 k, \u2a06 (_ : k \u2264 m), f k x) \u2208 measurableLE \u03bc \u03bd\n\u22a2 (fun x => \u2a06 k, \u2a06 (_ : k \u2264 Nat.succ m), f k x) \u2208 measurableLE \u03bc \u03bd"}, {"tactic": "refine' \u27e8_, _\u27e9", "annotated_tactic": ["refine' \u27e8_, _\u27e9", []], "state_before": "case zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bc \u03bd\n\u22a2 (fun x => \u2a06 k, \u2a06 (_ : k \u2264 Nat.zero), f k x) \u2208 measurableLE \u03bc \u03bd", "state_after": "case zero.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bc \u03bd\n\u22a2 Measurable fun x => \u2a06 k, \u2a06 (_ : k \u2264 Nat.zero), f k x\n\ncase zero.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bc \u03bd\n\u22a2 \u2200 (A : Set \u03b1), MeasurableSet A \u2192 \u222b\u207b (x : \u03b1) in A, (fun x => \u2a06 k, \u2a06 (_ : k \u2264 Nat.zero), f k x) x \u2202\u03bc \u2264 \u2191\u2191\u03bd A"}, {"tactic": "simp [(hf 0).1]", "annotated_tactic": ["simp [(hf 0).1]", []], "state_before": "case zero.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bc \u03bd\n\u22a2 Measurable fun x => \u2a06 k, \u2a06 (_ : k \u2264 Nat.zero), f k x", "state_after": "no goals"}, {"tactic": "intro A hA", "annotated_tactic": ["intro A hA", []], "state_before": "case zero.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bc \u03bd\n\u22a2 \u2200 (A : Set \u03b1), MeasurableSet A \u2192 \u222b\u207b (x : \u03b1) in A, (fun x => \u2a06 k, \u2a06 (_ : k \u2264 Nat.zero), f k x) x \u2202\u03bc \u2264 \u2191\u2191\u03bd A", "state_after": "case zero.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bc \u03bd\nA : Set \u03b1\nhA : MeasurableSet A\n\u22a2 \u222b\u207b (x : \u03b1) in A, (fun x => \u2a06 k, \u2a06 (_ : k \u2264 Nat.zero), f k x) x \u2202\u03bc \u2264 \u2191\u2191\u03bd A"}, {"tactic": "simp [(hf 0).2 A hA]", "annotated_tactic": ["simp [(hf 0).2 A hA]", []], "state_before": "case zero.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bc \u03bd\nA : Set \u03b1\nhA : MeasurableSet A\n\u22a2 \u222b\u207b (x : \u03b1) in A, (fun x => \u2a06 k, \u2a06 (_ : k \u2264 Nat.zero), f k x) x \u2202\u03bc \u2264 \u2191\u2191\u03bd A", "state_after": "no goals"}, {"tactic": "have :\n  (fun a : \u03b1 => \u2a06 (k : \u2115) (_ : k \u2264 m + 1), f k a) = fun a =>\n    f m.succ a \u2294 \u2a06 (k : \u2115) (_ : k \u2264 m), f k a :=\n  funext fun _ => iSup_succ_eq_sup _ _ _", "annotated_tactic": ["have :\n      (fun a : \u03b1 => \u2a06 (k : \u2115) (_ : k \u2264 m + 1), f k a) = fun a =>\n        f m.succ a \u2294 \u2a06 (k : \u2115) (_ : k \u2264 m), f k a :=\n      <a>funext</a> fun _ => <a>iSup_succ_eq_sup</a> _ _ _", [{"full_name": "funext", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [1555, 9], "def_end_pos": [1555, 15]}, {"full_name": "MeasureTheory.Measure.LebesgueDecomposition.iSup_succ_eq_sup", "def_path": "Mathlib/MeasureTheory/Decomposition/Lebesgue.lean", "def_pos": [478, 9], "def_end_pos": [478, 25]}]], "state_before": "case succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u271d : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bc \u03bd\nm : \u2115\nhm : (fun x => \u2a06 k, \u2a06 (_ : k \u2264 m), f k x) \u2208 measurableLE \u03bc \u03bd\n\u22a2 (fun x => \u2a06 k, \u2a06 (_ : k \u2264 Nat.succ m), f k x) \u2208 measurableLE \u03bc \u03bd", "state_after": "case succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u271d : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bc \u03bd\nm : \u2115\nhm : (fun x => \u2a06 k, \u2a06 (_ : k \u2264 m), f k x) \u2208 measurableLE \u03bc \u03bd\nthis : (fun a => \u2a06 k, \u2a06 (_ : k \u2264 m + 1), f k a) = fun a => f (Nat.succ m) a \u2294 \u2a06 k, \u2a06 (_ : k \u2264 m), f k a\n\u22a2 (fun x => \u2a06 k, \u2a06 (_ : k \u2264 Nat.succ m), f k x) \u2208 measurableLE \u03bc \u03bd"}, {"tactic": "refine' \u27e8measurable_iSup fun n => Measurable.iSup_Prop _ (hf n).1, fun A hA => _\u27e9", "annotated_tactic": ["refine' \u27e8<a>measurable_iSup</a> fun n => <a>Measurable.iSup_Prop</a> _ (hf n).1, fun A hA => _\u27e9", [{"full_name": "measurable_iSup", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [1360, 9], "def_end_pos": [1360, 24]}, {"full_name": "Measurable.iSup_Prop", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [1342, 9], "def_end_pos": [1342, 29]}]], "state_before": "case succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u271d : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bc \u03bd\nm : \u2115\nhm : (fun x => \u2a06 k, \u2a06 (_ : k \u2264 m), f k x) \u2208 measurableLE \u03bc \u03bd\nthis : (fun a => \u2a06 k, \u2a06 (_ : k \u2264 m + 1), f k a) = fun a => f (Nat.succ m) a \u2294 \u2a06 k, \u2a06 (_ : k \u2264 m), f k a\n\u22a2 (fun x => \u2a06 k, \u2a06 (_ : k \u2264 Nat.succ m), f k x) \u2208 measurableLE \u03bc \u03bd", "state_after": "case succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u271d : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bc \u03bd\nm : \u2115\nhm : (fun x => \u2a06 k, \u2a06 (_ : k \u2264 m), f k x) \u2208 measurableLE \u03bc \u03bd\nthis : (fun a => \u2a06 k, \u2a06 (_ : k \u2264 m + 1), f k a) = fun a => f (Nat.succ m) a \u2294 \u2a06 k, \u2a06 (_ : k \u2264 m), f k a\nA : Set \u03b1\nhA : MeasurableSet A\n\u22a2 \u222b\u207b (x : \u03b1) in A, (fun x => \u2a06 k, \u2a06 (_ : k \u2264 Nat.succ m), f k x) x \u2202\u03bc \u2264 \u2191\u2191\u03bd A"}, {"tactic": "rw [this]", "annotated_tactic": ["rw [this]", []], "state_before": "case succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u271d : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bc \u03bd\nm : \u2115\nhm : (fun x => \u2a06 k, \u2a06 (_ : k \u2264 m), f k x) \u2208 measurableLE \u03bc \u03bd\nthis : (fun a => \u2a06 k, \u2a06 (_ : k \u2264 m + 1), f k a) = fun a => f (Nat.succ m) a \u2294 \u2a06 k, \u2a06 (_ : k \u2264 m), f k a\nA : Set \u03b1\nhA : MeasurableSet A\n\u22a2 \u222b\u207b (x : \u03b1) in A, (fun x => \u2a06 k, \u2a06 (_ : k \u2264 Nat.succ m), f k x) x \u2202\u03bc \u2264 \u2191\u2191\u03bd A", "state_after": "case succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u271d : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bc \u03bd\nm : \u2115\nhm : (fun x => \u2a06 k, \u2a06 (_ : k \u2264 m), f k x) \u2208 measurableLE \u03bc \u03bd\nthis : (fun a => \u2a06 k, \u2a06 (_ : k \u2264 m + 1), f k a) = fun a => f (Nat.succ m) a \u2294 \u2a06 k, \u2a06 (_ : k \u2264 m), f k a\nA : Set \u03b1\nhA : MeasurableSet A\n\u22a2 \u222b\u207b (a : \u03b1) in A, f (Nat.succ m) a \u2294 \u2a06 k, \u2a06 (_ : k \u2264 m), f k a \u2202\u03bc \u2264 \u2191\u2191\u03bd A"}, {"tactic": "exact (sup_mem_measurableLE (hf m.succ) hm).2 A hA", "annotated_tactic": ["exact (<a>sup_mem_measurableLE</a> (hf m.succ) hm).2 A hA", [{"full_name": "MeasureTheory.Measure.LebesgueDecomposition.sup_mem_measurableLE", "def_path": "Mathlib/MeasureTheory/Decomposition/Lebesgue.lean", "def_pos": [466, 9], "def_end_pos": [466, 29]}]], "state_before": "case succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u271d : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bc \u03bd\nm : \u2115\nhm : (fun x => \u2a06 k, \u2a06 (_ : k \u2264 m), f k x) \u2208 measurableLE \u03bc \u03bd\nthis : (fun a => \u2a06 k, \u2a06 (_ : k \u2264 m + 1), f k a) = fun a => f (Nat.succ m) a \u2294 \u2a06 k, \u2a06 (_ : k \u2264 m), f k a\nA : Set \u03b1\nhA : MeasurableSet A\n\u22a2 \u222b\u207b (a : \u03b1) in A, f (Nat.succ m) a \u2294 \u2a06 k, \u2a06 (_ : k \u2264 m), f k a \u2202\u03bc \u2264 \u2191\u2191\u03bd A", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/Egorov.lean", "full_name": "MeasureTheory.Egorov.notConvergentSeq_antitone", "start": [55, 1], "end": [56, 84], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Lattice.lean", "full_name": "Finset.inf_erase_top", "start": [431, 1], "end": [432, 29], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/ProbabilityMassFunction/Basic.lean", "full_name": "PMF.toOuterMeasure_apply_eq_one_iff", "start": [201, 1], "end": [211, 75], "traced_tactics": [{"tactic": "refine' (p.toOuterMeasure_apply s).symm \u25b8 \u27e8fun h a hap => _, fun h => _\u27e9", "annotated_tactic": ["refine' (p.toOuterMeasure_apply s).<a>symm</a> \u25b8 \u27e8fun h a hap => _, fun h => _\u27e9", [{"full_name": "Eq.symm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [310, 9], "def_end_pos": [310, 16]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\ns t : Set \u03b1\n\u22a2 \u2191(toOuterMeasure p) s = 1 \u2194 support p \u2286 s", "state_after": "case refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\ns t : Set \u03b1\nh : \u2211' (x : \u03b1), Set.indicator s (\u2191p) x = 1\na : \u03b1\nhap : a \u2208 support p\n\u22a2 a \u2208 s\n\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\ns t : Set \u03b1\nh : support p \u2286 s\n\u22a2 \u2211' (x : \u03b1), Set.indicator s (\u2191p) x = 1"}, {"tactic": "refine' by_contra fun hs => ne_of_lt _ (h.trans p.tsum_coe.symm)", "annotated_tactic": ["refine' <a>by_contra</a> fun hs => <a>ne_of_lt</a> _ (h.trans p.tsum_coe.symm)", [{"full_name": "by_contra", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [223, 7], "def_end_pos": [223, 16]}, {"full_name": "ne_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [101, 9], "def_end_pos": [101, 17]}]], "state_before": "case refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\ns t : Set \u03b1\nh : \u2211' (x : \u03b1), Set.indicator s (\u2191p) x = 1\na : \u03b1\nhap : a \u2208 support p\n\u22a2 a \u2208 s", "state_after": "case refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\ns t : Set \u03b1\nh : \u2211' (x : \u03b1), Set.indicator s (\u2191p) x = 1\na : \u03b1\nhap : a \u2208 support p\nhs : \u00aca \u2208 s\n\u22a2 \u2211' (x : \u03b1), Set.indicator s (\u2191p) x < \u2211' (a : \u03b1), \u2191p a"}, {"tactic": "have hs' : s.indicator p a = 0 := Set.indicator_apply_eq_zero.2 fun hs' => False.elim <| hs hs'", "annotated_tactic": ["have hs' : s.indicator p a = 0 := <a>Set.indicator_apply_eq_zero</a>.2 fun hs' => <a>False.elim</a> <| hs hs'", [{"full_name": "Set.indicator_apply_eq_zero", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [109, 3], "def_end_pos": [109, 14]}, {"full_name": "False.elim", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [223, 21], "def_end_pos": [223, 31]}]], "state_before": "case refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\ns t : Set \u03b1\nh : \u2211' (x : \u03b1), Set.indicator s (\u2191p) x = 1\na : \u03b1\nhap : a \u2208 support p\nhs : \u00aca \u2208 s\n\u22a2 \u2211' (x : \u03b1), Set.indicator s (\u2191p) x < \u2211' (a : \u03b1), \u2191p a", "state_after": "case refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\ns t : Set \u03b1\nh : \u2211' (x : \u03b1), Set.indicator s (\u2191p) x = 1\na : \u03b1\nhap : a \u2208 support p\nhs : \u00aca \u2208 s\nhs' : Set.indicator s (\u2191p) a = 0\n\u22a2 \u2211' (x : \u03b1), Set.indicator s (\u2191p) x < \u2211' (a : \u03b1), \u2191p a"}, {"tactic": "have hsa : s.indicator p a < p a := hs'.symm \u25b8 (p.apply_pos_iff a).2 hap", "annotated_tactic": ["have hsa : s.indicator p a < p a := hs'.symm \u25b8 (p.apply_pos_iff a).2 hap", []], "state_before": "case refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\ns t : Set \u03b1\nh : \u2211' (x : \u03b1), Set.indicator s (\u2191p) x = 1\na : \u03b1\nhap : a \u2208 support p\nhs : \u00aca \u2208 s\nhs' : Set.indicator s (\u2191p) a = 0\n\u22a2 \u2211' (x : \u03b1), Set.indicator s (\u2191p) x < \u2211' (a : \u03b1), \u2191p a", "state_after": "case refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\ns t : Set \u03b1\nh : \u2211' (x : \u03b1), Set.indicator s (\u2191p) x = 1\na : \u03b1\nhap : a \u2208 support p\nhs : \u00aca \u2208 s\nhs' : Set.indicator s (\u2191p) a = 0\nhsa : Set.indicator s (\u2191p) a < \u2191p a\n\u22a2 \u2211' (x : \u03b1), Set.indicator s (\u2191p) x < \u2211' (a : \u03b1), \u2191p a"}, {"tactic": "exact ENNReal.tsum_lt_tsum (p.tsum_coe_indicator_ne_top s)\n  (fun x => Set.indicator_apply_le fun _ => le_rfl) hsa", "annotated_tactic": ["exact <a>ENNReal.tsum_lt_tsum</a> (p.tsum_coe_indicator_ne_top s)\n      (fun x => <a>Set.indicator_apply_le</a> fun _ => <a>le_rfl</a>) hsa", [{"full_name": "ENNReal.tsum_lt_tsum", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [1282, 9], "def_end_pos": [1282, 21]}, {"full_name": "Set.indicator_apply_le", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [906, 3], "def_end_pos": [906, 14]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}]], "state_before": "case refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\ns t : Set \u03b1\nh : \u2211' (x : \u03b1), Set.indicator s (\u2191p) x = 1\na : \u03b1\nhap : a \u2208 support p\nhs : \u00aca \u2208 s\nhs' : Set.indicator s (\u2191p) a = 0\nhsa : Set.indicator s (\u2191p) a < \u2191p a\n\u22a2 \u2211' (x : \u03b1), Set.indicator s (\u2191p) x < \u2211' (a : \u03b1), \u2191p a", "state_after": "no goals"}, {"tactic": "suffices : \u2200 (x) (_ : x \u2209 s), p x = 0", "annotated_tactic": ["suffices : \u2200 (x) (_ : x \u2209 s), p x = 0", []], "state_before": "case refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\ns t : Set \u03b1\nh : support p \u2286 s\n\u22a2 \u2211' (x : \u03b1), Set.indicator s (\u2191p) x = 1", "state_after": "case refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\ns t : Set \u03b1\nh : support p \u2286 s\nthis : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 \u2191p x = 0\n\u22a2 \u2211' (x : \u03b1), Set.indicator s (\u2191p) x = 1\n\ncase this\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\ns t : Set \u03b1\nh : support p \u2286 s\n\u22a2 \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 \u2191p x = 0"}, {"tactic": "exact _root_.trans (tsum_congr\n  fun a => (Set.indicator_apply s p a).trans (ite_eq_left_iff.2 <| symm \u2218 this a)) p.tsum_coe", "annotated_tactic": ["exact <a>_root_.trans</a> (<a>tsum_congr</a>\n      fun a => (<a>Set.indicator_apply</a> s p a).<a>trans</a> (<a>ite_eq_left_iff</a>.2 <| <a>symm</a> \u2218 this a)) p.tsum_coe", [{"full_name": "trans", "def_path": "Mathlib/Init/Algebra/Classes.lean", "def_pos": [308, 9], "def_end_pos": [308, 14]}, {"full_name": "tsum_congr", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/Basic.lean", "def_pos": [498, 9], "def_end_pos": [498, 19]}, {"full_name": "Set.indicator_apply", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [59, 3], "def_end_pos": [59, 14]}, {"full_name": "Eq.trans", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [322, 9], "def_end_pos": [322, 17]}, {"full_name": "ite_eq_left_iff", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [1159, 17], "def_end_pos": [1159, 32]}, {"full_name": "symm", "def_path": "Mathlib/Init/Algebra/Classes.lean", "def_pos": [312, 9], "def_end_pos": [312, 13]}]], "state_before": "case refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\ns t : Set \u03b1\nh : support p \u2286 s\nthis : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 \u2191p x = 0\n\u22a2 \u2211' (x : \u03b1), Set.indicator s (\u2191p) x = 1\n\ncase this\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\ns t : Set \u03b1\nh : support p \u2286 s\n\u22a2 \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 \u2191p x = 0", "state_after": "case this\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\ns t : Set \u03b1\nh : support p \u2286 s\n\u22a2 \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 \u2191p x = 0"}, {"tactic": "exact fun a ha => (p.apply_eq_zero_iff a).2 <| Set.not_mem_subset h ha", "annotated_tactic": ["exact fun a ha => (p.apply_eq_zero_iff a).2 <| <a>Set.not_mem_subset</a> h ha", [{"full_name": "Set.not_mem_subset", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [387, 9], "def_end_pos": [387, 23]}]], "state_before": "case this\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\ns t : Set \u03b1\nh : support p \u2286 s\n\u22a2 \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 \u2191p x = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Intervals/Monoid.lean", "full_name": "Set.image_add_const_Ico", "start": [94, 1], "end": [95, 31], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/Basic.lean", "full_name": "MvPolynomial.eval\u2082_mul_monomial", "start": [1002, 1], "end": [1017, 23], "traced_tactics": [{"tactic": "apply MvPolynomial.induction_on p", "annotated_tactic": ["apply <a>MvPolynomial.induction_on</a> p", [{"full_name": "MvPolynomial.induction_on", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [451, 9], "def_end_pos": [451, 21]}]], "state_before": "R : Type u\nS\u2081 : Type v\nS\u2082 : Type w\nS\u2083 : Type x\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : CommSemiring S\u2081\np q : MvPolynomial \u03c3 R\nf : R \u2192+* S\u2081\ng : \u03c3 \u2192 S\u2081\n\u22a2 \u2200 {s : \u03c3 \u2192\u2080 \u2115} {a : R}, eval\u2082 f g (p * \u2191(monomial s) a) = eval\u2082 f g p * \u2191f a * Finsupp.prod s fun n e => g n ^ e", "state_after": "case h_C\nR : Type u\nS\u2081 : Type v\nS\u2082 : Type w\nS\u2083 : Type x\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : CommSemiring S\u2081\np q : MvPolynomial \u03c3 R\nf : R \u2192+* S\u2081\ng : \u03c3 \u2192 S\u2081\n\u22a2 \u2200 (a : R) {s : \u03c3 \u2192\u2080 \u2115} {a_1 : R},\n    eval\u2082 f g (\u2191C a * \u2191(monomial s) a_1) = eval\u2082 f g (\u2191C a) * \u2191f a_1 * Finsupp.prod s fun n e => g n ^ e\n\ncase h_add\nR : Type u\nS\u2081 : Type v\nS\u2082 : Type w\nS\u2083 : Type x\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : CommSemiring S\u2081\np q : MvPolynomial \u03c3 R\nf : R \u2192+* S\u2081\ng : \u03c3 \u2192 S\u2081\n\u22a2 \u2200 (p q : MvPolynomial \u03c3 R),\n    (\u2200 {s : \u03c3 \u2192\u2080 \u2115} {a : R}, eval\u2082 f g (p * \u2191(monomial s) a) = eval\u2082 f g p * \u2191f a * Finsupp.prod s fun n e => g n ^ e) \u2192\n      (\u2200 {s : \u03c3 \u2192\u2080 \u2115} {a : R},\n          eval\u2082 f g (q * \u2191(monomial s) a) = eval\u2082 f g q * \u2191f a * Finsupp.prod s fun n e => g n ^ e) \u2192\n        \u2200 {s : \u03c3 \u2192\u2080 \u2115} {a : R},\n          eval\u2082 f g ((p + q) * \u2191(monomial s) a) = eval\u2082 f g (p + q) * \u2191f a * Finsupp.prod s fun n e => g n ^ e\n\ncase h_X\nR : Type u\nS\u2081 : Type v\nS\u2082 : Type w\nS\u2083 : Type x\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : CommSemiring S\u2081\np q : MvPolynomial \u03c3 R\nf : R \u2192+* S\u2081\ng : \u03c3 \u2192 S\u2081\n\u22a2 \u2200 (p : MvPolynomial \u03c3 R) (n : \u03c3),\n    (\u2200 {s : \u03c3 \u2192\u2080 \u2115} {a : R}, eval\u2082 f g (p * \u2191(monomial s) a) = eval\u2082 f g p * \u2191f a * Finsupp.prod s fun n e => g n ^ e) \u2192\n      \u2200 {s : \u03c3 \u2192\u2080 \u2115} {a : R},\n        eval\u2082 f g (p * X n * \u2191(monomial s) a) = eval\u2082 f g (p * X n) * \u2191f a * Finsupp.prod s fun n e => g n ^ e"}, {"tactic": "intro a' s a", "annotated_tactic": ["intro a' s a", []], "state_before": "case h_C\nR : Type u\nS\u2081 : Type v\nS\u2082 : Type w\nS\u2083 : Type x\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : CommSemiring S\u2081\np q : MvPolynomial \u03c3 R\nf : R \u2192+* S\u2081\ng : \u03c3 \u2192 S\u2081\n\u22a2 \u2200 (a : R) {s : \u03c3 \u2192\u2080 \u2115} {a_1 : R},\n    eval\u2082 f g (\u2191C a * \u2191(monomial s) a_1) = eval\u2082 f g (\u2191C a) * \u2191f a_1 * Finsupp.prod s fun n e => g n ^ e", "state_after": "case h_C\nR : Type u\nS\u2081 : Type v\nS\u2082 : Type w\nS\u2083 : Type x\n\u03c3 : Type u_1\na\u271d a'\u271d a\u2081 a\u2082 : R\ne : \u2115\nn m : \u03c3\ns\u271d : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : CommSemiring S\u2081\np q : MvPolynomial \u03c3 R\nf : R \u2192+* S\u2081\ng : \u03c3 \u2192 S\u2081\na' : R\ns : \u03c3 \u2192\u2080 \u2115\na : R\n\u22a2 eval\u2082 f g (\u2191C a' * \u2191(monomial s) a) = eval\u2082 f g (\u2191C a') * \u2191f a * Finsupp.prod s fun n e => g n ^ e"}, {"tactic": "simp [C_mul_monomial, eval\u2082_monomial, f.map_mul]", "annotated_tactic": ["simp [<a>C_mul_monomial</a>, <a>eval\u2082_monomial</a>, f.map_mul]", [{"full_name": "MvPolynomial.C_mul_monomial", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [221, 9], "def_end_pos": [221, 23]}, {"full_name": "MvPolynomial.eval\u2082_monomial", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [983, 9], "def_end_pos": [983, 23]}]], "state_before": "case h_C\nR : Type u\nS\u2081 : Type v\nS\u2082 : Type w\nS\u2083 : Type x\n\u03c3 : Type u_1\na\u271d a'\u271d a\u2081 a\u2082 : R\ne : \u2115\nn m : \u03c3\ns\u271d : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : CommSemiring S\u2081\np q : MvPolynomial \u03c3 R\nf : R \u2192+* S\u2081\ng : \u03c3 \u2192 S\u2081\na' : R\ns : \u03c3 \u2192\u2080 \u2115\na : R\n\u22a2 eval\u2082 f g (\u2191C a' * \u2191(monomial s) a) = eval\u2082 f g (\u2191C a') * \u2191f a * Finsupp.prod s fun n e => g n ^ e", "state_after": "no goals"}, {"tactic": "intro p q ih_p ih_q", "annotated_tactic": ["intro p q ih_p ih_q", []], "state_before": "case h_add\nR : Type u\nS\u2081 : Type v\nS\u2082 : Type w\nS\u2083 : Type x\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : CommSemiring S\u2081\np q : MvPolynomial \u03c3 R\nf : R \u2192+* S\u2081\ng : \u03c3 \u2192 S\u2081\n\u22a2 \u2200 (p q : MvPolynomial \u03c3 R),\n    (\u2200 {s : \u03c3 \u2192\u2080 \u2115} {a : R}, eval\u2082 f g (p * \u2191(monomial s) a) = eval\u2082 f g p * \u2191f a * Finsupp.prod s fun n e => g n ^ e) \u2192\n      (\u2200 {s : \u03c3 \u2192\u2080 \u2115} {a : R},\n          eval\u2082 f g (q * \u2191(monomial s) a) = eval\u2082 f g q * \u2191f a * Finsupp.prod s fun n e => g n ^ e) \u2192\n        \u2200 {s : \u03c3 \u2192\u2080 \u2115} {a : R},\n          eval\u2082 f g ((p + q) * \u2191(monomial s) a) = eval\u2082 f g (p + q) * \u2191f a * Finsupp.prod s fun n e => g n ^ e", "state_after": "case h_add\nR : Type u\nS\u2081 : Type v\nS\u2082 : Type w\nS\u2083 : Type x\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : CommSemiring S\u2081\np\u271d q\u271d : MvPolynomial \u03c3 R\nf : R \u2192+* S\u2081\ng : \u03c3 \u2192 S\u2081\np q : MvPolynomial \u03c3 R\nih_p : \u2200 {s : \u03c3 \u2192\u2080 \u2115} {a : R}, eval\u2082 f g (p * \u2191(monomial s) a) = eval\u2082 f g p * \u2191f a * Finsupp.prod s fun n e => g n ^ e\nih_q : \u2200 {s : \u03c3 \u2192\u2080 \u2115} {a : R}, eval\u2082 f g (q * \u2191(monomial s) a) = eval\u2082 f g q * \u2191f a * Finsupp.prod s fun n e => g n ^ e\n\u22a2 \u2200 {s : \u03c3 \u2192\u2080 \u2115} {a : R},\n    eval\u2082 f g ((p + q) * \u2191(monomial s) a) = eval\u2082 f g (p + q) * \u2191f a * Finsupp.prod s fun n e => g n ^ e"}, {"tactic": "simp [add_mul, eval\u2082_add, ih_p, ih_q]", "annotated_tactic": ["simp [<a>add_mul</a>, <a>eval\u2082_add</a>, ih_p, ih_q]", [{"full_name": "add_mul", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [91, 7], "def_end_pos": [91, 14]}, {"full_name": "MvPolynomial.eval\u2082_add", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [978, 9], "def_end_pos": [978, 18]}]], "state_before": "case h_add\nR : Type u\nS\u2081 : Type v\nS\u2082 : Type w\nS\u2083 : Type x\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : CommSemiring S\u2081\np\u271d q\u271d : MvPolynomial \u03c3 R\nf : R \u2192+* S\u2081\ng : \u03c3 \u2192 S\u2081\np q : MvPolynomial \u03c3 R\nih_p : \u2200 {s : \u03c3 \u2192\u2080 \u2115} {a : R}, eval\u2082 f g (p * \u2191(monomial s) a) = eval\u2082 f g p * \u2191f a * Finsupp.prod s fun n e => g n ^ e\nih_q : \u2200 {s : \u03c3 \u2192\u2080 \u2115} {a : R}, eval\u2082 f g (q * \u2191(monomial s) a) = eval\u2082 f g q * \u2191f a * Finsupp.prod s fun n e => g n ^ e\n\u22a2 \u2200 {s : \u03c3 \u2192\u2080 \u2115} {a : R},\n    eval\u2082 f g ((p + q) * \u2191(monomial s) a) = eval\u2082 f g (p + q) * \u2191f a * Finsupp.prod s fun n e => g n ^ e", "state_after": "no goals"}, {"tactic": "intro p n ih s a", "annotated_tactic": ["intro p n ih s a", []], "state_before": "case h_X\nR : Type u\nS\u2081 : Type v\nS\u2082 : Type w\nS\u2083 : Type x\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : CommSemiring S\u2081\np q : MvPolynomial \u03c3 R\nf : R \u2192+* S\u2081\ng : \u03c3 \u2192 S\u2081\n\u22a2 \u2200 (p : MvPolynomial \u03c3 R) (n : \u03c3),\n    (\u2200 {s : \u03c3 \u2192\u2080 \u2115} {a : R}, eval\u2082 f g (p * \u2191(monomial s) a) = eval\u2082 f g p * \u2191f a * Finsupp.prod s fun n e => g n ^ e) \u2192\n      \u2200 {s : \u03c3 \u2192\u2080 \u2115} {a : R},\n        eval\u2082 f g (p * X n * \u2191(monomial s) a) = eval\u2082 f g (p * X n) * \u2191f a * Finsupp.prod s fun n e => g n ^ e", "state_after": "case h_X\nR : Type u\nS\u2081 : Type v\nS\u2082 : Type w\nS\u2083 : Type x\n\u03c3 : Type u_1\na\u271d a' a\u2081 a\u2082 : R\ne : \u2115\nn\u271d m : \u03c3\ns\u271d : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : CommSemiring S\u2081\np\u271d q : MvPolynomial \u03c3 R\nf : R \u2192+* S\u2081\ng : \u03c3 \u2192 S\u2081\np : MvPolynomial \u03c3 R\nn : \u03c3\nih : \u2200 {s : \u03c3 \u2192\u2080 \u2115} {a : R}, eval\u2082 f g (p * \u2191(monomial s) a) = eval\u2082 f g p * \u2191f a * Finsupp.prod s fun n e => g n ^ e\ns : \u03c3 \u2192\u2080 \u2115\na : R\n\u22a2 eval\u2082 f g (p * X n * \u2191(monomial s) a) = eval\u2082 f g (p * X n) * \u2191f a * Finsupp.prod s fun n e => g n ^ e"}, {"tactic": "exact\n  calc\n    (p * X n * monomial s a).eval\u2082 f g = (p * monomial (Finsupp.single n 1 + s) a).eval\u2082 f g :=\n      by rw [monomial_single_add, pow_one, mul_assoc]\n    _ = (p * monomial (Finsupp.single n 1) 1).eval\u2082 f g * f a * s.prod fun n e => g n ^ e := by\n      simp [ih, prod_single_index, prod_add_index, pow_one, pow_add, mul_assoc, mul_left_comm,\n        f.map_one]", "annotated_tactic": ["exact\n      calc\n        (p * <a>X</a> n * <a>monomial</a> s a).<a>eval\u2082</a> f g = (p * <a>monomial</a> (<a>Finsupp.single</a> n 1 + s) a).<a>eval\u2082</a> f g :=\n          by rw [<a>monomial_single_add</a>, <a>pow_one</a>, <a>mul_assoc</a>]\n        _ = (p * <a>monomial</a> (<a>Finsupp.single</a> n 1) 1).<a>eval\u2082</a> f g * f a * s.prod fun n e => g n ^ e := by\n          simp [ih, <a>prod_single_index</a>, <a>prod_add_index</a>, <a>pow_one</a>, <a>pow_add</a>, <a>mul_assoc</a>, <a>mul_left_comm</a>,\n            f.map_one]", [{"full_name": "MvPolynomial.X", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [193, 5], "def_end_pos": [193, 6]}, {"full_name": "MvPolynomial.monomial", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [167, 5], "def_end_pos": [167, 13]}, {"full_name": "MvPolynomial.eval\u2082", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [955, 5], "def_end_pos": [955, 10]}, {"full_name": "MvPolynomial.monomial", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [167, 5], "def_end_pos": [167, 13]}, {"full_name": "Finsupp.single", "def_path": "Mathlib/Data/Finsupp/Defs.lean", "def_pos": [289, 5], "def_end_pos": [289, 11]}, {"full_name": "MvPolynomial.eval\u2082", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [955, 5], "def_end_pos": [955, 10]}, {"full_name": "MvPolynomial.monomial_single_add", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [334, 9], "def_end_pos": [334, 28]}, {"full_name": "pow_one", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [97, 9], "def_end_pos": [97, 16]}, {"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [264, 9], "def_end_pos": [264, 18]}, {"full_name": "MvPolynomial.monomial", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [167, 5], "def_end_pos": [167, 13]}, {"full_name": "Finsupp.single", "def_path": "Mathlib/Data/Finsupp/Defs.lean", "def_pos": [289, 5], "def_end_pos": [289, 11]}, {"full_name": "MvPolynomial.eval\u2082", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [955, 5], "def_end_pos": [955, 10]}, {"full_name": "Finsupp.prod_single_index", "def_path": "Mathlib/Algebra/BigOperators/Finsupp.lean", "def_pos": [75, 9], "def_end_pos": [75, 26]}, {"full_name": "Finsupp.prod_add_index", "def_path": "Mathlib/Algebra/BigOperators/Finsupp.lean", "def_pos": [387, 9], "def_end_pos": [387, 23]}, {"full_name": "pow_one", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [97, 9], "def_end_pos": [97, 16]}, {"full_name": "pow_add", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [118, 9], "def_end_pos": [118, 16]}, {"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [264, 9], "def_end_pos": [264, 18]}, {"full_name": "mul_left_comm", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [96, 9], "def_end_pos": [96, 22]}]], "state_before": "case h_X\nR : Type u\nS\u2081 : Type v\nS\u2082 : Type w\nS\u2083 : Type x\n\u03c3 : Type u_1\na\u271d a' a\u2081 a\u2082 : R\ne : \u2115\nn\u271d m : \u03c3\ns\u271d : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : CommSemiring S\u2081\np\u271d q : MvPolynomial \u03c3 R\nf : R \u2192+* S\u2081\ng : \u03c3 \u2192 S\u2081\np : MvPolynomial \u03c3 R\nn : \u03c3\nih : \u2200 {s : \u03c3 \u2192\u2080 \u2115} {a : R}, eval\u2082 f g (p * \u2191(monomial s) a) = eval\u2082 f g p * \u2191f a * Finsupp.prod s fun n e => g n ^ e\ns : \u03c3 \u2192\u2080 \u2115\na : R\n\u22a2 eval\u2082 f g (p * X n * \u2191(monomial s) a) = eval\u2082 f g (p * X n) * \u2191f a * Finsupp.prod s fun n e => g n ^ e", "state_after": "no goals"}, {"tactic": "rw [monomial_single_add, pow_one, mul_assoc]", "annotated_tactic": ["rw [<a>monomial_single_add</a>, <a>pow_one</a>, <a>mul_assoc</a>]", [{"full_name": "MvPolynomial.monomial_single_add", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [334, 9], "def_end_pos": [334, 28]}, {"full_name": "pow_one", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [97, 9], "def_end_pos": [97, 16]}, {"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [264, 9], "def_end_pos": [264, 18]}]], "state_before": "R : Type u\nS\u2081 : Type v\nS\u2082 : Type w\nS\u2083 : Type x\n\u03c3 : Type u_1\na\u271d a' a\u2081 a\u2082 : R\ne : \u2115\nn\u271d m : \u03c3\ns\u271d : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : CommSemiring S\u2081\np\u271d q : MvPolynomial \u03c3 R\nf : R \u2192+* S\u2081\ng : \u03c3 \u2192 S\u2081\np : MvPolynomial \u03c3 R\nn : \u03c3\nih : \u2200 {s : \u03c3 \u2192\u2080 \u2115} {a : R}, eval\u2082 f g (p * \u2191(monomial s) a) = eval\u2082 f g p * \u2191f a * Finsupp.prod s fun n e => g n ^ e\ns : \u03c3 \u2192\u2080 \u2115\na : R\n\u22a2 eval\u2082 f g (p * X n * \u2191(monomial s) a) = eval\u2082 f g (p * \u2191(monomial ((fun\u2080 | n => 1) + s)) a)", "state_after": "no goals"}, {"tactic": "simp [ih, prod_single_index, prod_add_index, pow_one, pow_add, mul_assoc, mul_left_comm,\n  f.map_one]", "annotated_tactic": ["simp [ih, <a>prod_single_index</a>, <a>prod_add_index</a>, <a>pow_one</a>, <a>pow_add</a>, <a>mul_assoc</a>, <a>mul_left_comm</a>,\n            f.map_one]", [{"full_name": "Finsupp.prod_single_index", "def_path": "Mathlib/Algebra/BigOperators/Finsupp.lean", "def_pos": [75, 9], "def_end_pos": [75, 26]}, {"full_name": "Finsupp.prod_add_index", "def_path": "Mathlib/Algebra/BigOperators/Finsupp.lean", "def_pos": [387, 9], "def_end_pos": [387, 23]}, {"full_name": "pow_one", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [97, 9], "def_end_pos": [97, 16]}, {"full_name": "pow_add", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [118, 9], "def_end_pos": [118, 16]}, {"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [264, 9], "def_end_pos": [264, 18]}, {"full_name": "mul_left_comm", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [96, 9], "def_end_pos": [96, 22]}]], "state_before": "R : Type u\nS\u2081 : Type v\nS\u2082 : Type w\nS\u2083 : Type x\n\u03c3 : Type u_1\na\u271d a' a\u2081 a\u2082 : R\ne : \u2115\nn\u271d m : \u03c3\ns\u271d : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : CommSemiring S\u2081\np\u271d q : MvPolynomial \u03c3 R\nf : R \u2192+* S\u2081\ng : \u03c3 \u2192 S\u2081\np : MvPolynomial \u03c3 R\nn : \u03c3\nih : \u2200 {s : \u03c3 \u2192\u2080 \u2115} {a : R}, eval\u2082 f g (p * \u2191(monomial s) a) = eval\u2082 f g p * \u2191f a * Finsupp.prod s fun n e => g n ^ e\ns : \u03c3 \u2192\u2080 \u2115\na : R\n\u22a2 eval\u2082 f g (p * \u2191(monomial ((fun\u2080 | n => 1) + s)) a) =\n    eval\u2082 f g (p * \u2191(monomial fun\u2080 | n => 1) 1) * \u2191f a * Finsupp.prod s fun n e => g n ^ e", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "full_name": "MeasureTheory.Measure.finiteAt_nhdsWithin", "start": [4074, 1], "end": [4076, 34], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "full_name": "Dense.borel_eq_generateFrom_Ico_mem_aux", "start": [651, 1], "end": [685, 62], "traced_tactics": [{"tactic": "set S : Set (Set \u03b1) := { S | \u2203 l \u2208 s, \u2203 u \u2208 s, l < u \u2227 Ico l u = S }", "annotated_tactic": ["set S : <a>Set</a> (<a>Set</a> \u03b1) := { S | \u2203 l \u2208 s, \u2203 u \u2208 s, l < u \u2227 <a>Ico</a> l u = S }", [{"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}, {"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}, {"full_name": "Set.Ico", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [49, 5], "def_end_pos": [49, 8]}]], "state_before": "\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t u : Set \u03b1\u271d\ninst\u271d\u00b2\u2070 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9\u2079 : MeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2078 : OpensMeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2077 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u2076 : MeasurableSpace \u03b2\ninst\u271d\u00b9\u2075 : OpensMeasurableSpace \u03b2\ninst\u271d\u00b9\u2074 : TopologicalSpace \u03b3\ninst\u271d\u00b9\u00b3 : MeasurableSpace \u03b3\ninst\u271d\u00b9\u00b2 : BorelSpace \u03b3\ninst\u271d\u00b9\u00b9 : TopologicalSpace \u03b3\u2082\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b3\u2082\ninst\u271d\u2079 : BorelSpace \u03b3\u2082\ninst\u271d\u2078 : MeasurableSpace \u03b4\n\u03b1' : Type u_6\ninst\u271d\u2077 : TopologicalSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : LinearOrder \u03b1\u271d\ninst\u271d\u2074 : OrderClosedTopology \u03b1\u271d\na b x : \u03b1\u271d\n\u03b1 : Type u_7\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1\nhd : Dense s\nhbot : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s\nhIoo : \u2200 (x y : \u03b1), x < y \u2192 Ioo x y = \u2205 \u2192 y \u2208 s\n\u22a2 borel \u03b1 = MeasurableSpace.generateFrom {S | \u2203 l, l \u2208 s \u2227 \u2203 u, u \u2208 s \u2227 l < u \u2227 Ico l u = S}", "state_after": "\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t u : Set \u03b1\u271d\ninst\u271d\u00b2\u2070 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9\u2079 : MeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2078 : OpensMeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2077 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u2076 : MeasurableSpace \u03b2\ninst\u271d\u00b9\u2075 : OpensMeasurableSpace \u03b2\ninst\u271d\u00b9\u2074 : TopologicalSpace \u03b3\ninst\u271d\u00b9\u00b3 : MeasurableSpace \u03b3\ninst\u271d\u00b9\u00b2 : BorelSpace \u03b3\ninst\u271d\u00b9\u00b9 : TopologicalSpace \u03b3\u2082\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b3\u2082\ninst\u271d\u2079 : BorelSpace \u03b3\u2082\ninst\u271d\u2078 : MeasurableSpace \u03b4\n\u03b1' : Type u_6\ninst\u271d\u2077 : TopologicalSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : LinearOrder \u03b1\u271d\ninst\u271d\u2074 : OrderClosedTopology \u03b1\u271d\na b x : \u03b1\u271d\n\u03b1 : Type u_7\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1\nhd : Dense s\nhbot : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s\nhIoo : \u2200 (x y : \u03b1), x < y \u2192 Ioo x y = \u2205 \u2192 y \u2208 s\nS : Set (Set \u03b1) := {S | \u2203 l, l \u2208 s \u2227 \u2203 u, u \u2208 s \u2227 l < u \u2227 Ico l u = S}\n\u22a2 borel \u03b1 = MeasurableSpace.generateFrom S"}, {"tactic": "refine' le_antisymm _ (generateFrom_Ico_mem_le_borel _ _)", "annotated_tactic": ["refine' <a>le_antisymm</a> _ (<a>generateFrom_Ico_mem_le_borel</a> _ _)", [{"full_name": "le_antisymm", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [188, 9], "def_end_pos": [188, 20]}, {"full_name": "generateFrom_Ico_mem_le_borel", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [641, 9], "def_end_pos": [641, 38]}]], "state_before": "\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t u : Set \u03b1\u271d\ninst\u271d\u00b2\u2070 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9\u2079 : MeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2078 : OpensMeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2077 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u2076 : MeasurableSpace \u03b2\ninst\u271d\u00b9\u2075 : OpensMeasurableSpace \u03b2\ninst\u271d\u00b9\u2074 : TopologicalSpace \u03b3\ninst\u271d\u00b9\u00b3 : MeasurableSpace \u03b3\ninst\u271d\u00b9\u00b2 : BorelSpace \u03b3\ninst\u271d\u00b9\u00b9 : TopologicalSpace \u03b3\u2082\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b3\u2082\ninst\u271d\u2079 : BorelSpace \u03b3\u2082\ninst\u271d\u2078 : MeasurableSpace \u03b4\n\u03b1' : Type u_6\ninst\u271d\u2077 : TopologicalSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : LinearOrder \u03b1\u271d\ninst\u271d\u2074 : OrderClosedTopology \u03b1\u271d\na b x : \u03b1\u271d\n\u03b1 : Type u_7\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1\nhd : Dense s\nhbot : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s\nhIoo : \u2200 (x y : \u03b1), x < y \u2192 Ioo x y = \u2205 \u2192 y \u2208 s\nS : Set (Set \u03b1) := {S | \u2203 l, l \u2208 s \u2227 \u2203 u, u \u2208 s \u2227 l < u \u2227 Ico l u = S}\n\u22a2 borel \u03b1 = MeasurableSpace.generateFrom S", "state_after": "\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t u : Set \u03b1\u271d\ninst\u271d\u00b2\u2070 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9\u2079 : MeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2078 : OpensMeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2077 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u2076 : MeasurableSpace \u03b2\ninst\u271d\u00b9\u2075 : OpensMeasurableSpace \u03b2\ninst\u271d\u00b9\u2074 : TopologicalSpace \u03b3\ninst\u271d\u00b9\u00b3 : MeasurableSpace \u03b3\ninst\u271d\u00b9\u00b2 : BorelSpace \u03b3\ninst\u271d\u00b9\u00b9 : TopologicalSpace \u03b3\u2082\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b3\u2082\ninst\u271d\u2079 : BorelSpace \u03b3\u2082\ninst\u271d\u2078 : MeasurableSpace \u03b4\n\u03b1' : Type u_6\ninst\u271d\u2077 : TopologicalSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : LinearOrder \u03b1\u271d\ninst\u271d\u2074 : OrderClosedTopology \u03b1\u271d\na b x : \u03b1\u271d\n\u03b1 : Type u_7\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1\nhd : Dense s\nhbot : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s\nhIoo : \u2200 (x y : \u03b1), x < y \u2192 Ioo x y = \u2205 \u2192 y \u2208 s\nS : Set (Set \u03b1) := {S | \u2203 l, l \u2208 s \u2227 \u2203 u, u \u2208 s \u2227 l < u \u2227 Ico l u = S}\n\u22a2 borel \u03b1 \u2264 MeasurableSpace.generateFrom S"}, {"tactic": "letI : MeasurableSpace \u03b1 := generateFrom S", "annotated_tactic": ["letI : <a>MeasurableSpace</a> \u03b1 := <a>generateFrom</a> S", [{"full_name": "MeasurableSpace", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [48, 20], "def_end_pos": [48, 35]}, {"full_name": "MeasurableSpace.generateFrom", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [363, 5], "def_end_pos": [363, 17]}]], "state_before": "\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t u : Set \u03b1\u271d\ninst\u271d\u00b2\u2070 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9\u2079 : MeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2078 : OpensMeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2077 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u2076 : MeasurableSpace \u03b2\ninst\u271d\u00b9\u2075 : OpensMeasurableSpace \u03b2\ninst\u271d\u00b9\u2074 : TopologicalSpace \u03b3\ninst\u271d\u00b9\u00b3 : MeasurableSpace \u03b3\ninst\u271d\u00b9\u00b2 : BorelSpace \u03b3\ninst\u271d\u00b9\u00b9 : TopologicalSpace \u03b3\u2082\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b3\u2082\ninst\u271d\u2079 : BorelSpace \u03b3\u2082\ninst\u271d\u2078 : MeasurableSpace \u03b4\n\u03b1' : Type u_6\ninst\u271d\u2077 : TopologicalSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : LinearOrder \u03b1\u271d\ninst\u271d\u2074 : OrderClosedTopology \u03b1\u271d\na b x : \u03b1\u271d\n\u03b1 : Type u_7\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1\nhd : Dense s\nhbot : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s\nhIoo : \u2200 (x y : \u03b1), x < y \u2192 Ioo x y = \u2205 \u2192 y \u2208 s\nS : Set (Set \u03b1) := {S | \u2203 l, l \u2208 s \u2227 \u2203 u, u \u2208 s \u2227 l < u \u2227 Ico l u = S}\n\u22a2 borel \u03b1 \u2264 MeasurableSpace.generateFrom S", "state_after": "\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t u : Set \u03b1\u271d\ninst\u271d\u00b2\u2070 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9\u2079 : MeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2078 : OpensMeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2077 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u2076 : MeasurableSpace \u03b2\ninst\u271d\u00b9\u2075 : OpensMeasurableSpace \u03b2\ninst\u271d\u00b9\u2074 : TopologicalSpace \u03b3\ninst\u271d\u00b9\u00b3 : MeasurableSpace \u03b3\ninst\u271d\u00b9\u00b2 : BorelSpace \u03b3\ninst\u271d\u00b9\u00b9 : TopologicalSpace \u03b3\u2082\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b3\u2082\ninst\u271d\u2079 : BorelSpace \u03b3\u2082\ninst\u271d\u2078 : MeasurableSpace \u03b4\n\u03b1' : Type u_6\ninst\u271d\u2077 : TopologicalSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : LinearOrder \u03b1\u271d\ninst\u271d\u2074 : OrderClosedTopology \u03b1\u271d\na b x : \u03b1\u271d\n\u03b1 : Type u_7\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1\nhd : Dense s\nhbot : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s\nhIoo : \u2200 (x y : \u03b1), x < y \u2192 Ioo x y = \u2205 \u2192 y \u2208 s\nS : Set (Set \u03b1) := {S | \u2203 l, l \u2208 s \u2227 \u2203 u, u \u2208 s \u2227 l < u \u2227 Ico l u = S}\nthis : MeasurableSpace \u03b1 := MeasurableSpace.generateFrom S\n\u22a2 borel \u03b1 \u2264 MeasurableSpace.generateFrom S"}, {"tactic": "rw [borel_eq_generateFrom_Iio]", "annotated_tactic": ["rw [<a>borel_eq_generateFrom_Iio</a>]", [{"full_name": "borel_eq_generateFrom_Iio", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [124, 9], "def_end_pos": [124, 34]}]], "state_before": "\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t u : Set \u03b1\u271d\ninst\u271d\u00b2\u2070 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9\u2079 : MeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2078 : OpensMeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2077 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u2076 : MeasurableSpace \u03b2\ninst\u271d\u00b9\u2075 : OpensMeasurableSpace \u03b2\ninst\u271d\u00b9\u2074 : TopologicalSpace \u03b3\ninst\u271d\u00b9\u00b3 : MeasurableSpace \u03b3\ninst\u271d\u00b9\u00b2 : BorelSpace \u03b3\ninst\u271d\u00b9\u00b9 : TopologicalSpace \u03b3\u2082\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b3\u2082\ninst\u271d\u2079 : BorelSpace \u03b3\u2082\ninst\u271d\u2078 : MeasurableSpace \u03b4\n\u03b1' : Type u_6\ninst\u271d\u2077 : TopologicalSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : LinearOrder \u03b1\u271d\ninst\u271d\u2074 : OrderClosedTopology \u03b1\u271d\na b x : \u03b1\u271d\n\u03b1 : Type u_7\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1\nhd : Dense s\nhbot : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s\nhIoo : \u2200 (x y : \u03b1), x < y \u2192 Ioo x y = \u2205 \u2192 y \u2208 s\nS : Set (Set \u03b1) := {S | \u2203 l, l \u2208 s \u2227 \u2203 u, u \u2208 s \u2227 l < u \u2227 Ico l u = S}\nthis : MeasurableSpace \u03b1 := MeasurableSpace.generateFrom S\n\u22a2 borel \u03b1 \u2264 MeasurableSpace.generateFrom S", "state_after": "\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t u : Set \u03b1\u271d\ninst\u271d\u00b2\u2070 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9\u2079 : MeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2078 : OpensMeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2077 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u2076 : MeasurableSpace \u03b2\ninst\u271d\u00b9\u2075 : OpensMeasurableSpace \u03b2\ninst\u271d\u00b9\u2074 : TopologicalSpace \u03b3\ninst\u271d\u00b9\u00b3 : MeasurableSpace \u03b3\ninst\u271d\u00b9\u00b2 : BorelSpace \u03b3\ninst\u271d\u00b9\u00b9 : TopologicalSpace \u03b3\u2082\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b3\u2082\ninst\u271d\u2079 : BorelSpace \u03b3\u2082\ninst\u271d\u2078 : MeasurableSpace \u03b4\n\u03b1' : Type u_6\ninst\u271d\u2077 : TopologicalSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : LinearOrder \u03b1\u271d\ninst\u271d\u2074 : OrderClosedTopology \u03b1\u271d\na b x : \u03b1\u271d\n\u03b1 : Type u_7\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1\nhd : Dense s\nhbot : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s\nhIoo : \u2200 (x y : \u03b1), x < y \u2192 Ioo x y = \u2205 \u2192 y \u2208 s\nS : Set (Set \u03b1) := {S | \u2203 l, l \u2208 s \u2227 \u2203 u, u \u2208 s \u2227 l < u \u2227 Ico l u = S}\nthis : MeasurableSpace \u03b1 := MeasurableSpace.generateFrom S\n\u22a2 MeasurableSpace.generateFrom (range Iio) \u2264 MeasurableSpace.generateFrom S"}, {"tactic": "refine' generateFrom_le (forall_range_iff.2 fun a => _)", "annotated_tactic": ["refine' <a>generateFrom_le</a> (<a>forall_range_iff</a>.2 fun a => _)", [{"full_name": "MeasurableSpace.generateFrom_le", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [384, 9], "def_end_pos": [384, 24]}, {"full_name": "Set.forall_range_iff", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [684, 9], "def_end_pos": [684, 25]}]], "state_before": "\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t u : Set \u03b1\u271d\ninst\u271d\u00b2\u2070 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9\u2079 : MeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2078 : OpensMeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2077 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u2076 : MeasurableSpace \u03b2\ninst\u271d\u00b9\u2075 : OpensMeasurableSpace \u03b2\ninst\u271d\u00b9\u2074 : TopologicalSpace \u03b3\ninst\u271d\u00b9\u00b3 : MeasurableSpace \u03b3\ninst\u271d\u00b9\u00b2 : BorelSpace \u03b3\ninst\u271d\u00b9\u00b9 : TopologicalSpace \u03b3\u2082\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b3\u2082\ninst\u271d\u2079 : BorelSpace \u03b3\u2082\ninst\u271d\u2078 : MeasurableSpace \u03b4\n\u03b1' : Type u_6\ninst\u271d\u2077 : TopologicalSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : LinearOrder \u03b1\u271d\ninst\u271d\u2074 : OrderClosedTopology \u03b1\u271d\na b x : \u03b1\u271d\n\u03b1 : Type u_7\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1\nhd : Dense s\nhbot : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s\nhIoo : \u2200 (x y : \u03b1), x < y \u2192 Ioo x y = \u2205 \u2192 y \u2208 s\nS : Set (Set \u03b1) := {S | \u2203 l, l \u2208 s \u2227 \u2203 u, u \u2208 s \u2227 l < u \u2227 Ico l u = S}\nthis : MeasurableSpace \u03b1 := MeasurableSpace.generateFrom S\n\u22a2 MeasurableSpace.generateFrom (range Iio) \u2264 MeasurableSpace.generateFrom S", "state_after": "\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t u : Set \u03b1\u271d\ninst\u271d\u00b2\u2070 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9\u2079 : MeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2078 : OpensMeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2077 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u2076 : MeasurableSpace \u03b2\ninst\u271d\u00b9\u2075 : OpensMeasurableSpace \u03b2\ninst\u271d\u00b9\u2074 : TopologicalSpace \u03b3\ninst\u271d\u00b9\u00b3 : MeasurableSpace \u03b3\ninst\u271d\u00b9\u00b2 : BorelSpace \u03b3\ninst\u271d\u00b9\u00b9 : TopologicalSpace \u03b3\u2082\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b3\u2082\ninst\u271d\u2079 : BorelSpace \u03b3\u2082\ninst\u271d\u2078 : MeasurableSpace \u03b4\n\u03b1' : Type u_6\ninst\u271d\u2077 : TopologicalSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : LinearOrder \u03b1\u271d\ninst\u271d\u2074 : OrderClosedTopology \u03b1\u271d\na\u271d b x : \u03b1\u271d\n\u03b1 : Type u_7\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1\nhd : Dense s\nhbot : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s\nhIoo : \u2200 (x y : \u03b1), x < y \u2192 Ioo x y = \u2205 \u2192 y \u2208 s\nS : Set (Set \u03b1) := {S | \u2203 l, l \u2208 s \u2227 \u2203 u, u \u2208 s \u2227 l < u \u2227 Ico l u = S}\nthis : MeasurableSpace \u03b1 := MeasurableSpace.generateFrom S\na : \u03b1\n\u22a2 MeasurableSet (Iio a)"}, {"tactic": "rcases hd.exists_countable_dense_subset_bot_top with \u27e8t, hts, hc, htd, htb, -\u27e9", "annotated_tactic": ["rcases hd.exists_countable_dense_subset_bot_top with \u27e8t, hts, hc, htd, htb, -\u27e9", []], "state_before": "\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t u : Set \u03b1\u271d\ninst\u271d\u00b2\u2070 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9\u2079 : MeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2078 : OpensMeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2077 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u2076 : MeasurableSpace \u03b2\ninst\u271d\u00b9\u2075 : OpensMeasurableSpace \u03b2\ninst\u271d\u00b9\u2074 : TopologicalSpace \u03b3\ninst\u271d\u00b9\u00b3 : MeasurableSpace \u03b3\ninst\u271d\u00b9\u00b2 : BorelSpace \u03b3\ninst\u271d\u00b9\u00b9 : TopologicalSpace \u03b3\u2082\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b3\u2082\ninst\u271d\u2079 : BorelSpace \u03b3\u2082\ninst\u271d\u2078 : MeasurableSpace \u03b4\n\u03b1' : Type u_6\ninst\u271d\u2077 : TopologicalSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : LinearOrder \u03b1\u271d\ninst\u271d\u2074 : OrderClosedTopology \u03b1\u271d\na\u271d b x : \u03b1\u271d\n\u03b1 : Type u_7\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1\nhd : Dense s\nhbot : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s\nhIoo : \u2200 (x y : \u03b1), x < y \u2192 Ioo x y = \u2205 \u2192 y \u2208 s\nS : Set (Set \u03b1) := {S | \u2203 l, l \u2208 s \u2227 \u2203 u, u \u2208 s \u2227 l < u \u2227 Ico l u = S}\nthis : MeasurableSpace \u03b1 := MeasurableSpace.generateFrom S\na : \u03b1\n\u22a2 MeasurableSet (Iio a)", "state_after": "case intro.intro.intro.intro.intro\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\u271d\ninst\u271d\u00b2\u2070 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9\u2079 : MeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2078 : OpensMeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2077 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u2076 : MeasurableSpace \u03b2\ninst\u271d\u00b9\u2075 : OpensMeasurableSpace \u03b2\ninst\u271d\u00b9\u2074 : TopologicalSpace \u03b3\ninst\u271d\u00b9\u00b3 : MeasurableSpace \u03b3\ninst\u271d\u00b9\u00b2 : BorelSpace \u03b3\ninst\u271d\u00b9\u00b9 : TopologicalSpace \u03b3\u2082\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b3\u2082\ninst\u271d\u2079 : BorelSpace \u03b3\u2082\ninst\u271d\u2078 : MeasurableSpace \u03b4\n\u03b1' : Type u_6\ninst\u271d\u2077 : TopologicalSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : LinearOrder \u03b1\u271d\ninst\u271d\u2074 : OrderClosedTopology \u03b1\u271d\na\u271d b x : \u03b1\u271d\n\u03b1 : Type u_7\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1\nhd : Dense s\nhbot : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s\nhIoo : \u2200 (x y : \u03b1), x < y \u2192 Ioo x y = \u2205 \u2192 y \u2208 s\nS : Set (Set \u03b1) := {S | \u2203 l, l \u2208 s \u2227 \u2203 u, u \u2208 s \u2227 l < u \u2227 Ico l u = S}\nthis : MeasurableSpace \u03b1 := MeasurableSpace.generateFrom S\na : \u03b1\nt : Set \u03b1\nhts : t \u2286 s\nhc : Set.Countable t\nhtd : Dense t\nhtb : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s \u2192 x \u2208 t\n\u22a2 MeasurableSet (Iio a)"}, {"tactic": "by_cases ha : \u2200 b < a, (Ioo b a).Nonempty", "annotated_tactic": ["by_cases ha : \u2200 b < a, (<a>Ioo</a> b a).<a>Nonempty</a>", [{"full_name": "Set.Ioo", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [44, 5], "def_end_pos": [44, 8]}, {"full_name": "Set.Nonempty", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [439, 15], "def_end_pos": [439, 23]}]], "state_before": "case intro.intro.intro.intro.intro\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\u271d\ninst\u271d\u00b2\u2070 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9\u2079 : MeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2078 : OpensMeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2077 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u2076 : MeasurableSpace \u03b2\ninst\u271d\u00b9\u2075 : OpensMeasurableSpace \u03b2\ninst\u271d\u00b9\u2074 : TopologicalSpace \u03b3\ninst\u271d\u00b9\u00b3 : MeasurableSpace \u03b3\ninst\u271d\u00b9\u00b2 : BorelSpace \u03b3\ninst\u271d\u00b9\u00b9 : TopologicalSpace \u03b3\u2082\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b3\u2082\ninst\u271d\u2079 : BorelSpace \u03b3\u2082\ninst\u271d\u2078 : MeasurableSpace \u03b4\n\u03b1' : Type u_6\ninst\u271d\u2077 : TopologicalSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : LinearOrder \u03b1\u271d\ninst\u271d\u2074 : OrderClosedTopology \u03b1\u271d\na\u271d b x : \u03b1\u271d\n\u03b1 : Type u_7\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1\nhd : Dense s\nhbot : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s\nhIoo : \u2200 (x y : \u03b1), x < y \u2192 Ioo x y = \u2205 \u2192 y \u2208 s\nS : Set (Set \u03b1) := {S | \u2203 l, l \u2208 s \u2227 \u2203 u, u \u2208 s \u2227 l < u \u2227 Ico l u = S}\nthis : MeasurableSpace \u03b1 := MeasurableSpace.generateFrom S\na : \u03b1\nt : Set \u03b1\nhts : t \u2286 s\nhc : Set.Countable t\nhtd : Dense t\nhtb : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s \u2192 x \u2208 t\n\u22a2 MeasurableSet (Iio a)", "state_after": "case pos\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\u271d\ninst\u271d\u00b2\u2070 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9\u2079 : MeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2078 : OpensMeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2077 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u2076 : MeasurableSpace \u03b2\ninst\u271d\u00b9\u2075 : OpensMeasurableSpace \u03b2\ninst\u271d\u00b9\u2074 : TopologicalSpace \u03b3\ninst\u271d\u00b9\u00b3 : MeasurableSpace \u03b3\ninst\u271d\u00b9\u00b2 : BorelSpace \u03b3\ninst\u271d\u00b9\u00b9 : TopologicalSpace \u03b3\u2082\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b3\u2082\ninst\u271d\u2079 : BorelSpace \u03b3\u2082\ninst\u271d\u2078 : MeasurableSpace \u03b4\n\u03b1' : Type u_6\ninst\u271d\u2077 : TopologicalSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : LinearOrder \u03b1\u271d\ninst\u271d\u2074 : OrderClosedTopology \u03b1\u271d\na\u271d b x : \u03b1\u271d\n\u03b1 : Type u_7\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1\nhd : Dense s\nhbot : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s\nhIoo : \u2200 (x y : \u03b1), x < y \u2192 Ioo x y = \u2205 \u2192 y \u2208 s\nS : Set (Set \u03b1) := {S | \u2203 l, l \u2208 s \u2227 \u2203 u, u \u2208 s \u2227 l < u \u2227 Ico l u = S}\nthis : MeasurableSpace \u03b1 := MeasurableSpace.generateFrom S\na : \u03b1\nt : Set \u03b1\nhts : t \u2286 s\nhc : Set.Countable t\nhtd : Dense t\nhtb : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s \u2192 x \u2208 t\nha : \u2200 (b : \u03b1), b < a \u2192 Set.Nonempty (Ioo b a)\n\u22a2 MeasurableSet (Iio a)\n\ncase neg\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\u271d\ninst\u271d\u00b2\u2070 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9\u2079 : MeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2078 : OpensMeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2077 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u2076 : MeasurableSpace \u03b2\ninst\u271d\u00b9\u2075 : OpensMeasurableSpace \u03b2\ninst\u271d\u00b9\u2074 : TopologicalSpace \u03b3\ninst\u271d\u00b9\u00b3 : MeasurableSpace \u03b3\ninst\u271d\u00b9\u00b2 : BorelSpace \u03b3\ninst\u271d\u00b9\u00b9 : TopologicalSpace \u03b3\u2082\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b3\u2082\ninst\u271d\u2079 : BorelSpace \u03b3\u2082\ninst\u271d\u2078 : MeasurableSpace \u03b4\n\u03b1' : Type u_6\ninst\u271d\u2077 : TopologicalSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : LinearOrder \u03b1\u271d\ninst\u271d\u2074 : OrderClosedTopology \u03b1\u271d\na\u271d b x : \u03b1\u271d\n\u03b1 : Type u_7\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1\nhd : Dense s\nhbot : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s\nhIoo : \u2200 (x y : \u03b1), x < y \u2192 Ioo x y = \u2205 \u2192 y \u2208 s\nS : Set (Set \u03b1) := {S | \u2203 l, l \u2208 s \u2227 \u2203 u, u \u2208 s \u2227 l < u \u2227 Ico l u = S}\nthis : MeasurableSpace \u03b1 := MeasurableSpace.generateFrom S\na : \u03b1\nt : Set \u03b1\nhts : t \u2286 s\nhc : Set.Countable t\nhtd : Dense t\nhtb : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s \u2192 x \u2208 t\nha : \u00ac\u2200 (b : \u03b1), b < a \u2192 Set.Nonempty (Ioo b a)\n\u22a2 MeasurableSet (Iio a)"}, {"tactic": "convert_to MeasurableSet (\u22c3 (l \u2208 t) (u \u2208 t) (_ : l < u) (_ : u \u2264 a), Ico l u)", "annotated_tactic": ["convert_to <a>MeasurableSet</a> (\u22c3 (l \u2208 t) (u \u2208 t) (_ : l < u) (_ : u \u2264 a), <a>Ico</a> l u)", [{"full_name": "MeasurableSet", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [64, 5], "def_end_pos": [64, 18]}, {"full_name": "Set.Ico", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [49, 5], "def_end_pos": [49, 8]}]], "state_before": "case pos\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\u271d\ninst\u271d\u00b2\u2070 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9\u2079 : MeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2078 : OpensMeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2077 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u2076 : MeasurableSpace \u03b2\ninst\u271d\u00b9\u2075 : OpensMeasurableSpace \u03b2\ninst\u271d\u00b9\u2074 : TopologicalSpace \u03b3\ninst\u271d\u00b9\u00b3 : MeasurableSpace \u03b3\ninst\u271d\u00b9\u00b2 : BorelSpace \u03b3\ninst\u271d\u00b9\u00b9 : TopologicalSpace \u03b3\u2082\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b3\u2082\ninst\u271d\u2079 : BorelSpace \u03b3\u2082\ninst\u271d\u2078 : MeasurableSpace \u03b4\n\u03b1' : Type u_6\ninst\u271d\u2077 : TopologicalSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : LinearOrder \u03b1\u271d\ninst\u271d\u2074 : OrderClosedTopology \u03b1\u271d\na\u271d b x : \u03b1\u271d\n\u03b1 : Type u_7\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1\nhd : Dense s\nhbot : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s\nhIoo : \u2200 (x y : \u03b1), x < y \u2192 Ioo x y = \u2205 \u2192 y \u2208 s\nS : Set (Set \u03b1) := {S | \u2203 l, l \u2208 s \u2227 \u2203 u, u \u2208 s \u2227 l < u \u2227 Ico l u = S}\nthis : MeasurableSpace \u03b1 := MeasurableSpace.generateFrom S\na : \u03b1\nt : Set \u03b1\nhts : t \u2286 s\nhc : Set.Countable t\nhtd : Dense t\nhtb : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s \u2192 x \u2208 t\nha : \u2200 (b : \u03b1), b < a \u2192 Set.Nonempty (Ioo b a)\n\u22a2 MeasurableSet (Iio a)", "state_after": "case h.e'_3\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\u271d\ninst\u271d\u00b2\u2070 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9\u2079 : MeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2078 : OpensMeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2077 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u2076 : MeasurableSpace \u03b2\ninst\u271d\u00b9\u2075 : OpensMeasurableSpace \u03b2\ninst\u271d\u00b9\u2074 : TopologicalSpace \u03b3\ninst\u271d\u00b9\u00b3 : MeasurableSpace \u03b3\ninst\u271d\u00b9\u00b2 : BorelSpace \u03b3\ninst\u271d\u00b9\u00b9 : TopologicalSpace \u03b3\u2082\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b3\u2082\ninst\u271d\u2079 : BorelSpace \u03b3\u2082\ninst\u271d\u2078 : MeasurableSpace \u03b4\n\u03b1' : Type u_6\ninst\u271d\u2077 : TopologicalSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : LinearOrder \u03b1\u271d\ninst\u271d\u2074 : OrderClosedTopology \u03b1\u271d\na\u271d b x : \u03b1\u271d\n\u03b1 : Type u_7\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1\nhd : Dense s\nhbot : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s\nhIoo : \u2200 (x y : \u03b1), x < y \u2192 Ioo x y = \u2205 \u2192 y \u2208 s\nS : Set (Set \u03b1) := {S | \u2203 l, l \u2208 s \u2227 \u2203 u, u \u2208 s \u2227 l < u \u2227 Ico l u = S}\nthis : MeasurableSpace \u03b1 := MeasurableSpace.generateFrom S\na : \u03b1\nt : Set \u03b1\nhts : t \u2286 s\nhc : Set.Countable t\nhtd : Dense t\nhtb : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s \u2192 x \u2208 t\nha : \u2200 (b : \u03b1), b < a \u2192 Set.Nonempty (Ioo b a)\n\u22a2 Iio a = \u22c3 l \u2208 t, \u22c3 u \u2208 t, \u22c3 (_ : l < u), \u22c3 (_ : u \u2264 a), Ico l u\n\ncase pos\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\u271d\ninst\u271d\u00b2\u2070 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9\u2079 : MeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2078 : OpensMeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2077 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u2076 : MeasurableSpace \u03b2\ninst\u271d\u00b9\u2075 : OpensMeasurableSpace \u03b2\ninst\u271d\u00b9\u2074 : TopologicalSpace \u03b3\ninst\u271d\u00b9\u00b3 : MeasurableSpace \u03b3\ninst\u271d\u00b9\u00b2 : BorelSpace \u03b3\ninst\u271d\u00b9\u00b9 : TopologicalSpace \u03b3\u2082\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b3\u2082\ninst\u271d\u2079 : BorelSpace \u03b3\u2082\ninst\u271d\u2078 : MeasurableSpace \u03b4\n\u03b1' : Type u_6\ninst\u271d\u2077 : TopologicalSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : LinearOrder \u03b1\u271d\ninst\u271d\u2074 : OrderClosedTopology \u03b1\u271d\na\u271d b x : \u03b1\u271d\n\u03b1 : Type u_7\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1\nhd : Dense s\nhbot : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s\nhIoo : \u2200 (x y : \u03b1), x < y \u2192 Ioo x y = \u2205 \u2192 y \u2208 s\nS : Set (Set \u03b1) := {S | \u2203 l, l \u2208 s \u2227 \u2203 u, u \u2208 s \u2227 l < u \u2227 Ico l u = S}\nthis : MeasurableSpace \u03b1 := MeasurableSpace.generateFrom S\na : \u03b1\nt : Set \u03b1\nhts : t \u2286 s\nhc : Set.Countable t\nhtd : Dense t\nhtb : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s \u2192 x \u2208 t\nha : \u2200 (b : \u03b1), b < a \u2192 Set.Nonempty (Ioo b a)\n\u22a2 MeasurableSet (\u22c3 l \u2208 t, \u22c3 u \u2208 t, \u22c3 (_ : l < u), \u22c3 (_ : u \u2264 a), Ico l u)"}, {"tactic": "ext y", "annotated_tactic": ["ext y", []], "state_before": "case h.e'_3\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\u271d\ninst\u271d\u00b2\u2070 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9\u2079 : MeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2078 : OpensMeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2077 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u2076 : MeasurableSpace \u03b2\ninst\u271d\u00b9\u2075 : OpensMeasurableSpace \u03b2\ninst\u271d\u00b9\u2074 : TopologicalSpace \u03b3\ninst\u271d\u00b9\u00b3 : MeasurableSpace \u03b3\ninst\u271d\u00b9\u00b2 : BorelSpace \u03b3\ninst\u271d\u00b9\u00b9 : TopologicalSpace \u03b3\u2082\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b3\u2082\ninst\u271d\u2079 : BorelSpace \u03b3\u2082\ninst\u271d\u2078 : MeasurableSpace \u03b4\n\u03b1' : Type u_6\ninst\u271d\u2077 : TopologicalSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : LinearOrder \u03b1\u271d\ninst\u271d\u2074 : OrderClosedTopology \u03b1\u271d\na\u271d b x : \u03b1\u271d\n\u03b1 : Type u_7\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1\nhd : Dense s\nhbot : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s\nhIoo : \u2200 (x y : \u03b1), x < y \u2192 Ioo x y = \u2205 \u2192 y \u2208 s\nS : Set (Set \u03b1) := {S | \u2203 l, l \u2208 s \u2227 \u2203 u, u \u2208 s \u2227 l < u \u2227 Ico l u = S}\nthis : MeasurableSpace \u03b1 := MeasurableSpace.generateFrom S\na : \u03b1\nt : Set \u03b1\nhts : t \u2286 s\nhc : Set.Countable t\nhtd : Dense t\nhtb : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s \u2192 x \u2208 t\nha : \u2200 (b : \u03b1), b < a \u2192 Set.Nonempty (Ioo b a)\n\u22a2 Iio a = \u22c3 l \u2208 t, \u22c3 u \u2208 t, \u22c3 (_ : l < u), \u22c3 (_ : u \u2264 a), Ico l u", "state_after": "case h.e'_3.h\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\u271d\ninst\u271d\u00b2\u2070 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9\u2079 : MeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2078 : OpensMeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2077 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u2076 : MeasurableSpace \u03b2\ninst\u271d\u00b9\u2075 : OpensMeasurableSpace \u03b2\ninst\u271d\u00b9\u2074 : TopologicalSpace \u03b3\ninst\u271d\u00b9\u00b3 : MeasurableSpace \u03b3\ninst\u271d\u00b9\u00b2 : BorelSpace \u03b3\ninst\u271d\u00b9\u00b9 : TopologicalSpace \u03b3\u2082\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b3\u2082\ninst\u271d\u2079 : BorelSpace \u03b3\u2082\ninst\u271d\u2078 : MeasurableSpace \u03b4\n\u03b1' : Type u_6\ninst\u271d\u2077 : TopologicalSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : LinearOrder \u03b1\u271d\ninst\u271d\u2074 : OrderClosedTopology \u03b1\u271d\na\u271d b x : \u03b1\u271d\n\u03b1 : Type u_7\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1\nhd : Dense s\nhbot : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s\nhIoo : \u2200 (x y : \u03b1), x < y \u2192 Ioo x y = \u2205 \u2192 y \u2208 s\nS : Set (Set \u03b1) := {S | \u2203 l, l \u2208 s \u2227 \u2203 u, u \u2208 s \u2227 l < u \u2227 Ico l u = S}\nthis : MeasurableSpace \u03b1 := MeasurableSpace.generateFrom S\na : \u03b1\nt : Set \u03b1\nhts : t \u2286 s\nhc : Set.Countable t\nhtd : Dense t\nhtb : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s \u2192 x \u2208 t\nha : \u2200 (b : \u03b1), b < a \u2192 Set.Nonempty (Ioo b a)\ny : \u03b1\n\u22a2 y \u2208 Iio a \u2194 y \u2208 \u22c3 l \u2208 t, \u22c3 u \u2208 t, \u22c3 (_ : l < u), \u22c3 (_ : u \u2264 a), Ico l u"}, {"tactic": "simp only [mem_iUnion, mem_Iio, mem_Ico]", "annotated_tactic": ["simp only [<a>mem_iUnion</a>, <a>mem_Iio</a>, <a>mem_Ico</a>]", [{"full_name": "Set.mem_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [201, 9], "def_end_pos": [201, 19]}, {"full_name": "Set.mem_Iio", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [126, 9], "def_end_pos": [126, 16]}, {"full_name": "Set.mem_Ico", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [121, 9], "def_end_pos": [121, 16]}]], "state_before": "case h.e'_3.h\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\u271d\ninst\u271d\u00b2\u2070 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9\u2079 : MeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2078 : OpensMeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2077 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u2076 : MeasurableSpace \u03b2\ninst\u271d\u00b9\u2075 : OpensMeasurableSpace \u03b2\ninst\u271d\u00b9\u2074 : TopologicalSpace \u03b3\ninst\u271d\u00b9\u00b3 : MeasurableSpace \u03b3\ninst\u271d\u00b9\u00b2 : BorelSpace \u03b3\ninst\u271d\u00b9\u00b9 : TopologicalSpace \u03b3\u2082\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b3\u2082\ninst\u271d\u2079 : BorelSpace \u03b3\u2082\ninst\u271d\u2078 : MeasurableSpace \u03b4\n\u03b1' : Type u_6\ninst\u271d\u2077 : TopologicalSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : LinearOrder \u03b1\u271d\ninst\u271d\u2074 : OrderClosedTopology \u03b1\u271d\na\u271d b x : \u03b1\u271d\n\u03b1 : Type u_7\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1\nhd : Dense s\nhbot : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s\nhIoo : \u2200 (x y : \u03b1), x < y \u2192 Ioo x y = \u2205 \u2192 y \u2208 s\nS : Set (Set \u03b1) := {S | \u2203 l, l \u2208 s \u2227 \u2203 u, u \u2208 s \u2227 l < u \u2227 Ico l u = S}\nthis : MeasurableSpace \u03b1 := MeasurableSpace.generateFrom S\na : \u03b1\nt : Set \u03b1\nhts : t \u2286 s\nhc : Set.Countable t\nhtd : Dense t\nhtb : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s \u2192 x \u2208 t\nha : \u2200 (b : \u03b1), b < a \u2192 Set.Nonempty (Ioo b a)\ny : \u03b1\n\u22a2 y \u2208 Iio a \u2194 y \u2208 \u22c3 l \u2208 t, \u22c3 u \u2208 t, \u22c3 (_ : l < u), \u22c3 (_ : u \u2264 a), Ico l u", "state_after": "case h.e'_3.h\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\u271d\ninst\u271d\u00b2\u2070 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9\u2079 : MeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2078 : OpensMeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2077 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u2076 : MeasurableSpace \u03b2\ninst\u271d\u00b9\u2075 : OpensMeasurableSpace \u03b2\ninst\u271d\u00b9\u2074 : TopologicalSpace \u03b3\ninst\u271d\u00b9\u00b3 : MeasurableSpace \u03b3\ninst\u271d\u00b9\u00b2 : BorelSpace \u03b3\ninst\u271d\u00b9\u00b9 : TopologicalSpace \u03b3\u2082\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b3\u2082\ninst\u271d\u2079 : BorelSpace \u03b3\u2082\ninst\u271d\u2078 : MeasurableSpace \u03b4\n\u03b1' : Type u_6\ninst\u271d\u2077 : TopologicalSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : LinearOrder \u03b1\u271d\ninst\u271d\u2074 : OrderClosedTopology \u03b1\u271d\na\u271d b x : \u03b1\u271d\n\u03b1 : Type u_7\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1\nhd : Dense s\nhbot : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s\nhIoo : \u2200 (x y : \u03b1), x < y \u2192 Ioo x y = \u2205 \u2192 y \u2208 s\nS : Set (Set \u03b1) := {S | \u2203 l, l \u2208 s \u2227 \u2203 u, u \u2208 s \u2227 l < u \u2227 Ico l u = S}\nthis : MeasurableSpace \u03b1 := MeasurableSpace.generateFrom S\na : \u03b1\nt : Set \u03b1\nhts : t \u2286 s\nhc : Set.Countable t\nhtd : Dense t\nhtb : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s \u2192 x \u2208 t\nha : \u2200 (b : \u03b1), b < a \u2192 Set.Nonempty (Ioo b a)\ny : \u03b1\n\u22a2 y < a \u2194 \u2203 i h i_1 h h h, i \u2264 y \u2227 y < i_1"}, {"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "case h.e'_3.h\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\u271d\ninst\u271d\u00b2\u2070 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9\u2079 : MeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2078 : OpensMeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2077 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u2076 : MeasurableSpace \u03b2\ninst\u271d\u00b9\u2075 : OpensMeasurableSpace \u03b2\ninst\u271d\u00b9\u2074 : TopologicalSpace \u03b3\ninst\u271d\u00b9\u00b3 : MeasurableSpace \u03b3\ninst\u271d\u00b9\u00b2 : BorelSpace \u03b3\ninst\u271d\u00b9\u00b9 : TopologicalSpace \u03b3\u2082\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b3\u2082\ninst\u271d\u2079 : BorelSpace \u03b3\u2082\ninst\u271d\u2078 : MeasurableSpace \u03b4\n\u03b1' : Type u_6\ninst\u271d\u2077 : TopologicalSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : LinearOrder \u03b1\u271d\ninst\u271d\u2074 : OrderClosedTopology \u03b1\u271d\na\u271d b x : \u03b1\u271d\n\u03b1 : Type u_7\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1\nhd : Dense s\nhbot : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s\nhIoo : \u2200 (x y : \u03b1), x < y \u2192 Ioo x y = \u2205 \u2192 y \u2208 s\nS : Set (Set \u03b1) := {S | \u2203 l, l \u2208 s \u2227 \u2203 u, u \u2208 s \u2227 l < u \u2227 Ico l u = S}\nthis : MeasurableSpace \u03b1 := MeasurableSpace.generateFrom S\na : \u03b1\nt : Set \u03b1\nhts : t \u2286 s\nhc : Set.Countable t\nhtd : Dense t\nhtb : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s \u2192 x \u2208 t\nha : \u2200 (b : \u03b1), b < a \u2192 Set.Nonempty (Ioo b a)\ny : \u03b1\n\u22a2 y < a \u2194 \u2203 i h i_1 h h h, i \u2264 y \u2227 y < i_1", "state_after": "case h.e'_3.h.mp\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\u271d\ninst\u271d\u00b2\u2070 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9\u2079 : MeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2078 : OpensMeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2077 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u2076 : MeasurableSpace \u03b2\ninst\u271d\u00b9\u2075 : OpensMeasurableSpace \u03b2\ninst\u271d\u00b9\u2074 : TopologicalSpace \u03b3\ninst\u271d\u00b9\u00b3 : MeasurableSpace \u03b3\ninst\u271d\u00b9\u00b2 : BorelSpace \u03b3\ninst\u271d\u00b9\u00b9 : TopologicalSpace \u03b3\u2082\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b3\u2082\ninst\u271d\u2079 : BorelSpace \u03b3\u2082\ninst\u271d\u2078 : MeasurableSpace \u03b4\n\u03b1' : Type u_6\ninst\u271d\u2077 : TopologicalSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : LinearOrder \u03b1\u271d\ninst\u271d\u2074 : OrderClosedTopology \u03b1\u271d\na\u271d b x : \u03b1\u271d\n\u03b1 : Type u_7\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1\nhd : Dense s\nhbot : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s\nhIoo : \u2200 (x y : \u03b1), x < y \u2192 Ioo x y = \u2205 \u2192 y \u2208 s\nS : Set (Set \u03b1) := {S | \u2203 l, l \u2208 s \u2227 \u2203 u, u \u2208 s \u2227 l < u \u2227 Ico l u = S}\nthis : MeasurableSpace \u03b1 := MeasurableSpace.generateFrom S\na : \u03b1\nt : Set \u03b1\nhts : t \u2286 s\nhc : Set.Countable t\nhtd : Dense t\nhtb : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s \u2192 x \u2208 t\nha : \u2200 (b : \u03b1), b < a \u2192 Set.Nonempty (Ioo b a)\ny : \u03b1\n\u22a2 y < a \u2192 \u2203 i h i_1 h h h, i \u2264 y \u2227 y < i_1\n\ncase h.e'_3.h.mpr\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\u271d\ninst\u271d\u00b2\u2070 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9\u2079 : MeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2078 : OpensMeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2077 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u2076 : MeasurableSpace \u03b2\ninst\u271d\u00b9\u2075 : OpensMeasurableSpace \u03b2\ninst\u271d\u00b9\u2074 : TopologicalSpace \u03b3\ninst\u271d\u00b9\u00b3 : MeasurableSpace \u03b3\ninst\u271d\u00b9\u00b2 : BorelSpace \u03b3\ninst\u271d\u00b9\u00b9 : TopologicalSpace \u03b3\u2082\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b3\u2082\ninst\u271d\u2079 : BorelSpace \u03b3\u2082\ninst\u271d\u2078 : MeasurableSpace \u03b4\n\u03b1' : Type u_6\ninst\u271d\u2077 : TopologicalSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : LinearOrder \u03b1\u271d\ninst\u271d\u2074 : OrderClosedTopology \u03b1\u271d\na\u271d b x : \u03b1\u271d\n\u03b1 : Type u_7\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1\nhd : Dense s\nhbot : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s\nhIoo : \u2200 (x y : \u03b1), x < y \u2192 Ioo x y = \u2205 \u2192 y \u2208 s\nS : Set (Set \u03b1) := {S | \u2203 l, l \u2208 s \u2227 \u2203 u, u \u2208 s \u2227 l < u \u2227 Ico l u = S}\nthis : MeasurableSpace \u03b1 := MeasurableSpace.generateFrom S\na : \u03b1\nt : Set \u03b1\nhts : t \u2286 s\nhc : Set.Countable t\nhtd : Dense t\nhtb : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s \u2192 x \u2208 t\nha : \u2200 (b : \u03b1), b < a \u2192 Set.Nonempty (Ioo b a)\ny : \u03b1\n\u22a2 (\u2203 i h i_1 h h h, i \u2264 y \u2227 y < i_1) \u2192 y < a"}, {"tactic": "intro hy", "annotated_tactic": ["intro hy", []], "state_before": "case h.e'_3.h.mp\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\u271d\ninst\u271d\u00b2\u2070 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9\u2079 : MeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2078 : OpensMeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2077 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u2076 : MeasurableSpace \u03b2\ninst\u271d\u00b9\u2075 : OpensMeasurableSpace \u03b2\ninst\u271d\u00b9\u2074 : TopologicalSpace \u03b3\ninst\u271d\u00b9\u00b3 : MeasurableSpace \u03b3\ninst\u271d\u00b9\u00b2 : BorelSpace \u03b3\ninst\u271d\u00b9\u00b9 : TopologicalSpace \u03b3\u2082\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b3\u2082\ninst\u271d\u2079 : BorelSpace \u03b3\u2082\ninst\u271d\u2078 : MeasurableSpace \u03b4\n\u03b1' : Type u_6\ninst\u271d\u2077 : TopologicalSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : LinearOrder \u03b1\u271d\ninst\u271d\u2074 : OrderClosedTopology \u03b1\u271d\na\u271d b x : \u03b1\u271d\n\u03b1 : Type u_7\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1\nhd : Dense s\nhbot : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s\nhIoo : \u2200 (x y : \u03b1), x < y \u2192 Ioo x y = \u2205 \u2192 y \u2208 s\nS : Set (Set \u03b1) := {S | \u2203 l, l \u2208 s \u2227 \u2203 u, u \u2208 s \u2227 l < u \u2227 Ico l u = S}\nthis : MeasurableSpace \u03b1 := MeasurableSpace.generateFrom S\na : \u03b1\nt : Set \u03b1\nhts : t \u2286 s\nhc : Set.Countable t\nhtd : Dense t\nhtb : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s \u2192 x \u2208 t\nha : \u2200 (b : \u03b1), b < a \u2192 Set.Nonempty (Ioo b a)\ny : \u03b1\n\u22a2 y < a \u2192 \u2203 i h i_1 h h h, i \u2264 y \u2227 y < i_1", "state_after": "case h.e'_3.h.mp\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\u271d\ninst\u271d\u00b2\u2070 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9\u2079 : MeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2078 : OpensMeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2077 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u2076 : MeasurableSpace \u03b2\ninst\u271d\u00b9\u2075 : OpensMeasurableSpace \u03b2\ninst\u271d\u00b9\u2074 : TopologicalSpace \u03b3\ninst\u271d\u00b9\u00b3 : MeasurableSpace \u03b3\ninst\u271d\u00b9\u00b2 : BorelSpace \u03b3\ninst\u271d\u00b9\u00b9 : TopologicalSpace \u03b3\u2082\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b3\u2082\ninst\u271d\u2079 : BorelSpace \u03b3\u2082\ninst\u271d\u2078 : MeasurableSpace \u03b4\n\u03b1' : Type u_6\ninst\u271d\u2077 : TopologicalSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : LinearOrder \u03b1\u271d\ninst\u271d\u2074 : OrderClosedTopology \u03b1\u271d\na\u271d b x : \u03b1\u271d\n\u03b1 : Type u_7\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1\nhd : Dense s\nhbot : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s\nhIoo : \u2200 (x y : \u03b1), x < y \u2192 Ioo x y = \u2205 \u2192 y \u2208 s\nS : Set (Set \u03b1) := {S | \u2203 l, l \u2208 s \u2227 \u2203 u, u \u2208 s \u2227 l < u \u2227 Ico l u = S}\nthis : MeasurableSpace \u03b1 := MeasurableSpace.generateFrom S\na : \u03b1\nt : Set \u03b1\nhts : t \u2286 s\nhc : Set.Countable t\nhtd : Dense t\nhtb : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s \u2192 x \u2208 t\nha : \u2200 (b : \u03b1), b < a \u2192 Set.Nonempty (Ioo b a)\ny : \u03b1\nhy : y < a\n\u22a2 \u2203 i h i_1 h h h, i \u2264 y \u2227 y < i_1"}, {"tactic": "rcases htd.exists_le' (fun b hb => htb _ hb (hbot b hb)) y with \u27e8l, hlt, hly\u27e9", "annotated_tactic": ["rcases htd.exists_le' (fun b hb => htb _ hb (hbot b hb)) y with \u27e8l, hlt, hly\u27e9", []], "state_before": "case h.e'_3.h.mp\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\u271d\ninst\u271d\u00b2\u2070 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9\u2079 : MeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2078 : OpensMeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2077 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u2076 : MeasurableSpace \u03b2\ninst\u271d\u00b9\u2075 : OpensMeasurableSpace \u03b2\ninst\u271d\u00b9\u2074 : TopologicalSpace \u03b3\ninst\u271d\u00b9\u00b3 : MeasurableSpace \u03b3\ninst\u271d\u00b9\u00b2 : BorelSpace \u03b3\ninst\u271d\u00b9\u00b9 : TopologicalSpace \u03b3\u2082\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b3\u2082\ninst\u271d\u2079 : BorelSpace \u03b3\u2082\ninst\u271d\u2078 : MeasurableSpace \u03b4\n\u03b1' : Type u_6\ninst\u271d\u2077 : TopologicalSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : LinearOrder \u03b1\u271d\ninst\u271d\u2074 : OrderClosedTopology \u03b1\u271d\na\u271d b x : \u03b1\u271d\n\u03b1 : Type u_7\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1\nhd : Dense s\nhbot : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s\nhIoo : \u2200 (x y : \u03b1), x < y \u2192 Ioo x y = \u2205 \u2192 y \u2208 s\nS : Set (Set \u03b1) := {S | \u2203 l, l \u2208 s \u2227 \u2203 u, u \u2208 s \u2227 l < u \u2227 Ico l u = S}\nthis : MeasurableSpace \u03b1 := MeasurableSpace.generateFrom S\na : \u03b1\nt : Set \u03b1\nhts : t \u2286 s\nhc : Set.Countable t\nhtd : Dense t\nhtb : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s \u2192 x \u2208 t\nha : \u2200 (b : \u03b1), b < a \u2192 Set.Nonempty (Ioo b a)\ny : \u03b1\nhy : y < a\n\u22a2 \u2203 i h i_1 h h h, i \u2264 y \u2227 y < i_1", "state_after": "case h.e'_3.h.mp.intro.intro\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\u271d\ninst\u271d\u00b2\u2070 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9\u2079 : MeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2078 : OpensMeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2077 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u2076 : MeasurableSpace \u03b2\ninst\u271d\u00b9\u2075 : OpensMeasurableSpace \u03b2\ninst\u271d\u00b9\u2074 : TopologicalSpace \u03b3\ninst\u271d\u00b9\u00b3 : MeasurableSpace \u03b3\ninst\u271d\u00b9\u00b2 : BorelSpace \u03b3\ninst\u271d\u00b9\u00b9 : TopologicalSpace \u03b3\u2082\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b3\u2082\ninst\u271d\u2079 : BorelSpace \u03b3\u2082\ninst\u271d\u2078 : MeasurableSpace \u03b4\n\u03b1' : Type u_6\ninst\u271d\u2077 : TopologicalSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : LinearOrder \u03b1\u271d\ninst\u271d\u2074 : OrderClosedTopology \u03b1\u271d\na\u271d b x : \u03b1\u271d\n\u03b1 : Type u_7\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1\nhd : Dense s\nhbot : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s\nhIoo : \u2200 (x y : \u03b1), x < y \u2192 Ioo x y = \u2205 \u2192 y \u2208 s\nS : Set (Set \u03b1) := {S | \u2203 l, l \u2208 s \u2227 \u2203 u, u \u2208 s \u2227 l < u \u2227 Ico l u = S}\nthis : MeasurableSpace \u03b1 := MeasurableSpace.generateFrom S\na : \u03b1\nt : Set \u03b1\nhts : t \u2286 s\nhc : Set.Countable t\nhtd : Dense t\nhtb : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s \u2192 x \u2208 t\nha : \u2200 (b : \u03b1), b < a \u2192 Set.Nonempty (Ioo b a)\ny : \u03b1\nhy : y < a\nl : \u03b1\nhlt : l \u2208 t\nhly : l \u2264 y\n\u22a2 \u2203 i h i_1 h h h, i \u2264 y \u2227 y < i_1"}, {"tactic": "rcases htd.exists_mem_open isOpen_Ioo (ha y hy) with \u27e8u, hut, hyu, hua\u27e9", "annotated_tactic": ["rcases htd.exists_mem_open <a>isOpen_Ioo</a> (ha y hy) with \u27e8u, hut, hyu, hua\u27e9", [{"full_name": "isOpen_Ioo", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [336, 9], "def_end_pos": [336, 19]}]], "state_before": "case h.e'_3.h.mp.intro.intro\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\u271d\ninst\u271d\u00b2\u2070 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9\u2079 : MeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2078 : OpensMeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2077 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u2076 : MeasurableSpace \u03b2\ninst\u271d\u00b9\u2075 : OpensMeasurableSpace \u03b2\ninst\u271d\u00b9\u2074 : TopologicalSpace \u03b3\ninst\u271d\u00b9\u00b3 : MeasurableSpace \u03b3\ninst\u271d\u00b9\u00b2 : BorelSpace \u03b3\ninst\u271d\u00b9\u00b9 : TopologicalSpace \u03b3\u2082\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b3\u2082\ninst\u271d\u2079 : BorelSpace \u03b3\u2082\ninst\u271d\u2078 : MeasurableSpace \u03b4\n\u03b1' : Type u_6\ninst\u271d\u2077 : TopologicalSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : LinearOrder \u03b1\u271d\ninst\u271d\u2074 : OrderClosedTopology \u03b1\u271d\na\u271d b x : \u03b1\u271d\n\u03b1 : Type u_7\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1\nhd : Dense s\nhbot : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s\nhIoo : \u2200 (x y : \u03b1), x < y \u2192 Ioo x y = \u2205 \u2192 y \u2208 s\nS : Set (Set \u03b1) := {S | \u2203 l, l \u2208 s \u2227 \u2203 u, u \u2208 s \u2227 l < u \u2227 Ico l u = S}\nthis : MeasurableSpace \u03b1 := MeasurableSpace.generateFrom S\na : \u03b1\nt : Set \u03b1\nhts : t \u2286 s\nhc : Set.Countable t\nhtd : Dense t\nhtb : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s \u2192 x \u2208 t\nha : \u2200 (b : \u03b1), b < a \u2192 Set.Nonempty (Ioo b a)\ny : \u03b1\nhy : y < a\nl : \u03b1\nhlt : l \u2208 t\nhly : l \u2264 y\n\u22a2 \u2203 i h i_1 h h h, i \u2264 y \u2227 y < i_1", "state_after": "case h.e'_3.h.mp.intro.intro.intro.intro.intro\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u\u271d : Set \u03b1\u271d\ninst\u271d\u00b2\u2070 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9\u2079 : MeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2078 : OpensMeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2077 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u2076 : MeasurableSpace \u03b2\ninst\u271d\u00b9\u2075 : OpensMeasurableSpace \u03b2\ninst\u271d\u00b9\u2074 : TopologicalSpace \u03b3\ninst\u271d\u00b9\u00b3 : MeasurableSpace \u03b3\ninst\u271d\u00b9\u00b2 : BorelSpace \u03b3\ninst\u271d\u00b9\u00b9 : TopologicalSpace \u03b3\u2082\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b3\u2082\ninst\u271d\u2079 : BorelSpace \u03b3\u2082\ninst\u271d\u2078 : MeasurableSpace \u03b4\n\u03b1' : Type u_6\ninst\u271d\u2077 : TopologicalSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : LinearOrder \u03b1\u271d\ninst\u271d\u2074 : OrderClosedTopology \u03b1\u271d\na\u271d b x : \u03b1\u271d\n\u03b1 : Type u_7\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1\nhd : Dense s\nhbot : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s\nhIoo : \u2200 (x y : \u03b1), x < y \u2192 Ioo x y = \u2205 \u2192 y \u2208 s\nS : Set (Set \u03b1) := {S | \u2203 l, l \u2208 s \u2227 \u2203 u, u \u2208 s \u2227 l < u \u2227 Ico l u = S}\nthis : MeasurableSpace \u03b1 := MeasurableSpace.generateFrom S\na : \u03b1\nt : Set \u03b1\nhts : t \u2286 s\nhc : Set.Countable t\nhtd : Dense t\nhtb : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s \u2192 x \u2208 t\nha : \u2200 (b : \u03b1), b < a \u2192 Set.Nonempty (Ioo b a)\ny : \u03b1\nhy : y < a\nl : \u03b1\nhlt : l \u2208 t\nhly : l \u2264 y\nu : \u03b1\nhut : u \u2208 t\nhyu : y < u\nhua : u < a\n\u22a2 \u2203 i h i_1 h h h, i \u2264 y \u2227 y < i_1"}, {"tactic": "exact \u27e8l, hlt, u, hut, hly.trans_lt hyu, hua.le, hly, hyu\u27e9", "annotated_tactic": ["exact \u27e8l, hlt, u, hut, hly.trans_lt hyu, hua.le, hly, hyu\u27e9", []], "state_before": "case h.e'_3.h.mp.intro.intro.intro.intro.intro\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u\u271d : Set \u03b1\u271d\ninst\u271d\u00b2\u2070 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9\u2079 : MeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2078 : OpensMeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2077 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u2076 : MeasurableSpace \u03b2\ninst\u271d\u00b9\u2075 : OpensMeasurableSpace \u03b2\ninst\u271d\u00b9\u2074 : TopologicalSpace \u03b3\ninst\u271d\u00b9\u00b3 : MeasurableSpace \u03b3\ninst\u271d\u00b9\u00b2 : BorelSpace \u03b3\ninst\u271d\u00b9\u00b9 : TopologicalSpace \u03b3\u2082\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b3\u2082\ninst\u271d\u2079 : BorelSpace \u03b3\u2082\ninst\u271d\u2078 : MeasurableSpace \u03b4\n\u03b1' : Type u_6\ninst\u271d\u2077 : TopologicalSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : LinearOrder \u03b1\u271d\ninst\u271d\u2074 : OrderClosedTopology \u03b1\u271d\na\u271d b x : \u03b1\u271d\n\u03b1 : Type u_7\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1\nhd : Dense s\nhbot : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s\nhIoo : \u2200 (x y : \u03b1), x < y \u2192 Ioo x y = \u2205 \u2192 y \u2208 s\nS : Set (Set \u03b1) := {S | \u2203 l, l \u2208 s \u2227 \u2203 u, u \u2208 s \u2227 l < u \u2227 Ico l u = S}\nthis : MeasurableSpace \u03b1 := MeasurableSpace.generateFrom S\na : \u03b1\nt : Set \u03b1\nhts : t \u2286 s\nhc : Set.Countable t\nhtd : Dense t\nhtb : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s \u2192 x \u2208 t\nha : \u2200 (b : \u03b1), b < a \u2192 Set.Nonempty (Ioo b a)\ny : \u03b1\nhy : y < a\nl : \u03b1\nhlt : l \u2208 t\nhly : l \u2264 y\nu : \u03b1\nhut : u \u2208 t\nhyu : y < u\nhua : u < a\n\u22a2 \u2203 i h i_1 h h h, i \u2264 y \u2227 y < i_1", "state_after": "no goals"}, {"tactic": "rintro \u27e8l, -, u, -, -, hua, -, hyu\u27e9", "annotated_tactic": ["rintro \u27e8l, -, u, -, -, hua, -, hyu\u27e9", []], "state_before": "case h.e'_3.h.mpr\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\u271d\ninst\u271d\u00b2\u2070 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9\u2079 : MeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2078 : OpensMeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2077 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u2076 : MeasurableSpace \u03b2\ninst\u271d\u00b9\u2075 : OpensMeasurableSpace \u03b2\ninst\u271d\u00b9\u2074 : TopologicalSpace \u03b3\ninst\u271d\u00b9\u00b3 : MeasurableSpace \u03b3\ninst\u271d\u00b9\u00b2 : BorelSpace \u03b3\ninst\u271d\u00b9\u00b9 : TopologicalSpace \u03b3\u2082\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b3\u2082\ninst\u271d\u2079 : BorelSpace \u03b3\u2082\ninst\u271d\u2078 : MeasurableSpace \u03b4\n\u03b1' : Type u_6\ninst\u271d\u2077 : TopologicalSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : LinearOrder \u03b1\u271d\ninst\u271d\u2074 : OrderClosedTopology \u03b1\u271d\na\u271d b x : \u03b1\u271d\n\u03b1 : Type u_7\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1\nhd : Dense s\nhbot : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s\nhIoo : \u2200 (x y : \u03b1), x < y \u2192 Ioo x y = \u2205 \u2192 y \u2208 s\nS : Set (Set \u03b1) := {S | \u2203 l, l \u2208 s \u2227 \u2203 u, u \u2208 s \u2227 l < u \u2227 Ico l u = S}\nthis : MeasurableSpace \u03b1 := MeasurableSpace.generateFrom S\na : \u03b1\nt : Set \u03b1\nhts : t \u2286 s\nhc : Set.Countable t\nhtd : Dense t\nhtb : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s \u2192 x \u2208 t\nha : \u2200 (b : \u03b1), b < a \u2192 Set.Nonempty (Ioo b a)\ny : \u03b1\n\u22a2 (\u2203 i h i_1 h h h, i \u2264 y \u2227 y < i_1) \u2192 y < a", "state_after": "case h.e'_3.h.mpr.intro.intro.intro.intro.intro.intro.intro\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u\u271d : Set \u03b1\u271d\ninst\u271d\u00b2\u2070 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9\u2079 : MeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2078 : OpensMeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2077 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u2076 : MeasurableSpace \u03b2\ninst\u271d\u00b9\u2075 : OpensMeasurableSpace \u03b2\ninst\u271d\u00b9\u2074 : TopologicalSpace \u03b3\ninst\u271d\u00b9\u00b3 : MeasurableSpace \u03b3\ninst\u271d\u00b9\u00b2 : BorelSpace \u03b3\ninst\u271d\u00b9\u00b9 : TopologicalSpace \u03b3\u2082\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b3\u2082\ninst\u271d\u2079 : BorelSpace \u03b3\u2082\ninst\u271d\u2078 : MeasurableSpace \u03b4\n\u03b1' : Type u_6\ninst\u271d\u2077 : TopologicalSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : LinearOrder \u03b1\u271d\ninst\u271d\u2074 : OrderClosedTopology \u03b1\u271d\na\u271d b x : \u03b1\u271d\n\u03b1 : Type u_7\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1\nhd : Dense s\nhbot : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s\nhIoo : \u2200 (x y : \u03b1), x < y \u2192 Ioo x y = \u2205 \u2192 y \u2208 s\nS : Set (Set \u03b1) := {S | \u2203 l, l \u2208 s \u2227 \u2203 u, u \u2208 s \u2227 l < u \u2227 Ico l u = S}\nthis : MeasurableSpace \u03b1 := MeasurableSpace.generateFrom S\na : \u03b1\nt : Set \u03b1\nhts : t \u2286 s\nhc : Set.Countable t\nhtd : Dense t\nhtb : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s \u2192 x \u2208 t\nha : \u2200 (b : \u03b1), b < a \u2192 Set.Nonempty (Ioo b a)\ny l u : \u03b1\nhua : u \u2264 a\nhyu : y < u\n\u22a2 y < a"}, {"tactic": "exact hyu.trans_le hua", "annotated_tactic": ["exact hyu.trans_le hua", []], "state_before": "case h.e'_3.h.mpr.intro.intro.intro.intro.intro.intro.intro\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u\u271d : Set \u03b1\u271d\ninst\u271d\u00b2\u2070 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9\u2079 : MeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2078 : OpensMeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2077 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u2076 : MeasurableSpace \u03b2\ninst\u271d\u00b9\u2075 : OpensMeasurableSpace \u03b2\ninst\u271d\u00b9\u2074 : TopologicalSpace \u03b3\ninst\u271d\u00b9\u00b3 : MeasurableSpace \u03b3\ninst\u271d\u00b9\u00b2 : BorelSpace \u03b3\ninst\u271d\u00b9\u00b9 : TopologicalSpace \u03b3\u2082\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b3\u2082\ninst\u271d\u2079 : BorelSpace \u03b3\u2082\ninst\u271d\u2078 : MeasurableSpace \u03b4\n\u03b1' : Type u_6\ninst\u271d\u2077 : TopologicalSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : LinearOrder \u03b1\u271d\ninst\u271d\u2074 : OrderClosedTopology \u03b1\u271d\na\u271d b x : \u03b1\u271d\n\u03b1 : Type u_7\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1\nhd : Dense s\nhbot : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s\nhIoo : \u2200 (x y : \u03b1), x < y \u2192 Ioo x y = \u2205 \u2192 y \u2208 s\nS : Set (Set \u03b1) := {S | \u2203 l, l \u2208 s \u2227 \u2203 u, u \u2208 s \u2227 l < u \u2227 Ico l u = S}\nthis : MeasurableSpace \u03b1 := MeasurableSpace.generateFrom S\na : \u03b1\nt : Set \u03b1\nhts : t \u2286 s\nhc : Set.Countable t\nhtd : Dense t\nhtb : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s \u2192 x \u2208 t\nha : \u2200 (b : \u03b1), b < a \u2192 Set.Nonempty (Ioo b a)\ny l u : \u03b1\nhua : u \u2264 a\nhyu : y < u\n\u22a2 y < a", "state_after": "no goals"}, {"tactic": "refine' MeasurableSet.biUnion hc fun a ha => MeasurableSet.biUnion hc fun b hb => _", "annotated_tactic": ["refine' <a>MeasurableSet.biUnion</a> hc fun a ha => <a>MeasurableSet.biUnion</a> hc fun b hb => _", [{"full_name": "MeasurableSet.biUnion", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [130, 19], "def_end_pos": [130, 40]}, {"full_name": "MeasurableSet.biUnion", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [130, 19], "def_end_pos": [130, 40]}]], "state_before": "case pos\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\u271d\ninst\u271d\u00b2\u2070 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9\u2079 : MeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2078 : OpensMeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2077 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u2076 : MeasurableSpace \u03b2\ninst\u271d\u00b9\u2075 : OpensMeasurableSpace \u03b2\ninst\u271d\u00b9\u2074 : TopologicalSpace \u03b3\ninst\u271d\u00b9\u00b3 : MeasurableSpace \u03b3\ninst\u271d\u00b9\u00b2 : BorelSpace \u03b3\ninst\u271d\u00b9\u00b9 : TopologicalSpace \u03b3\u2082\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b3\u2082\ninst\u271d\u2079 : BorelSpace \u03b3\u2082\ninst\u271d\u2078 : MeasurableSpace \u03b4\n\u03b1' : Type u_6\ninst\u271d\u2077 : TopologicalSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : LinearOrder \u03b1\u271d\ninst\u271d\u2074 : OrderClosedTopology \u03b1\u271d\na\u271d b x : \u03b1\u271d\n\u03b1 : Type u_7\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1\nhd : Dense s\nhbot : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s\nhIoo : \u2200 (x y : \u03b1), x < y \u2192 Ioo x y = \u2205 \u2192 y \u2208 s\nS : Set (Set \u03b1) := {S | \u2203 l, l \u2208 s \u2227 \u2203 u, u \u2208 s \u2227 l < u \u2227 Ico l u = S}\nthis : MeasurableSpace \u03b1 := MeasurableSpace.generateFrom S\na : \u03b1\nt : Set \u03b1\nhts : t \u2286 s\nhc : Set.Countable t\nhtd : Dense t\nhtb : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s \u2192 x \u2208 t\nha : \u2200 (b : \u03b1), b < a \u2192 Set.Nonempty (Ioo b a)\n\u22a2 MeasurableSet (\u22c3 l \u2208 t, \u22c3 u \u2208 t, \u22c3 (_ : l < u), \u22c3 (_ : u \u2264 a), Ico l u)", "state_after": "case pos\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\u271d\ninst\u271d\u00b2\u2070 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9\u2079 : MeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2078 : OpensMeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2077 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u2076 : MeasurableSpace \u03b2\ninst\u271d\u00b9\u2075 : OpensMeasurableSpace \u03b2\ninst\u271d\u00b9\u2074 : TopologicalSpace \u03b3\ninst\u271d\u00b9\u00b3 : MeasurableSpace \u03b3\ninst\u271d\u00b9\u00b2 : BorelSpace \u03b3\ninst\u271d\u00b9\u00b9 : TopologicalSpace \u03b3\u2082\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b3\u2082\ninst\u271d\u2079 : BorelSpace \u03b3\u2082\ninst\u271d\u2078 : MeasurableSpace \u03b4\n\u03b1' : Type u_6\ninst\u271d\u2077 : TopologicalSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : LinearOrder \u03b1\u271d\ninst\u271d\u2074 : OrderClosedTopology \u03b1\u271d\na\u271d\u00b9 b\u271d x : \u03b1\u271d\n\u03b1 : Type u_7\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1\nhd : Dense s\nhbot : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s\nhIoo : \u2200 (x y : \u03b1), x < y \u2192 Ioo x y = \u2205 \u2192 y \u2208 s\nS : Set (Set \u03b1) := {S | \u2203 l, l \u2208 s \u2227 \u2203 u, u \u2208 s \u2227 l < u \u2227 Ico l u = S}\nthis : MeasurableSpace \u03b1 := MeasurableSpace.generateFrom S\na\u271d : \u03b1\nt : Set \u03b1\nhts : t \u2286 s\nhc : Set.Countable t\nhtd : Dense t\nhtb : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s \u2192 x \u2208 t\nha\u271d : \u2200 (b : \u03b1), b < a\u271d \u2192 Set.Nonempty (Ioo b a\u271d)\na : \u03b1\nha : a \u2208 t\nb : \u03b1\nhb : b \u2208 t\n\u22a2 MeasurableSet (\u22c3 (_ : a < b), \u22c3 (_ : b \u2264 a\u271d), Ico a b)"}, {"tactic": "refine' MeasurableSet.iUnion fun hab => MeasurableSet.iUnion fun _ => _", "annotated_tactic": ["refine' <a>MeasurableSet.iUnion</a> fun hab => <a>MeasurableSet.iUnion</a> fun _ => _", [{"full_name": "MeasurableSet.iUnion", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [115, 19], "def_end_pos": [115, 39]}, {"full_name": "MeasurableSet.iUnion", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [115, 19], "def_end_pos": [115, 39]}]], "state_before": "case pos\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\u271d\ninst\u271d\u00b2\u2070 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9\u2079 : MeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2078 : OpensMeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2077 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u2076 : MeasurableSpace \u03b2\ninst\u271d\u00b9\u2075 : OpensMeasurableSpace \u03b2\ninst\u271d\u00b9\u2074 : TopologicalSpace \u03b3\ninst\u271d\u00b9\u00b3 : MeasurableSpace \u03b3\ninst\u271d\u00b9\u00b2 : BorelSpace \u03b3\ninst\u271d\u00b9\u00b9 : TopologicalSpace \u03b3\u2082\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b3\u2082\ninst\u271d\u2079 : BorelSpace \u03b3\u2082\ninst\u271d\u2078 : MeasurableSpace \u03b4\n\u03b1' : Type u_6\ninst\u271d\u2077 : TopologicalSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : LinearOrder \u03b1\u271d\ninst\u271d\u2074 : OrderClosedTopology \u03b1\u271d\na\u271d\u00b9 b\u271d x : \u03b1\u271d\n\u03b1 : Type u_7\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1\nhd : Dense s\nhbot : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s\nhIoo : \u2200 (x y : \u03b1), x < y \u2192 Ioo x y = \u2205 \u2192 y \u2208 s\nS : Set (Set \u03b1) := {S | \u2203 l, l \u2208 s \u2227 \u2203 u, u \u2208 s \u2227 l < u \u2227 Ico l u = S}\nthis : MeasurableSpace \u03b1 := MeasurableSpace.generateFrom S\na\u271d : \u03b1\nt : Set \u03b1\nhts : t \u2286 s\nhc : Set.Countable t\nhtd : Dense t\nhtb : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s \u2192 x \u2208 t\nha\u271d : \u2200 (b : \u03b1), b < a\u271d \u2192 Set.Nonempty (Ioo b a\u271d)\na : \u03b1\nha : a \u2208 t\nb : \u03b1\nhb : b \u2208 t\n\u22a2 MeasurableSet (\u22c3 (_ : a < b), \u22c3 (_ : b \u2264 a\u271d), Ico a b)", "state_after": "case pos\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\u271d\ninst\u271d\u00b2\u2070 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9\u2079 : MeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2078 : OpensMeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2077 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u2076 : MeasurableSpace \u03b2\ninst\u271d\u00b9\u2075 : OpensMeasurableSpace \u03b2\ninst\u271d\u00b9\u2074 : TopologicalSpace \u03b3\ninst\u271d\u00b9\u00b3 : MeasurableSpace \u03b3\ninst\u271d\u00b9\u00b2 : BorelSpace \u03b3\ninst\u271d\u00b9\u00b9 : TopologicalSpace \u03b3\u2082\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b3\u2082\ninst\u271d\u2079 : BorelSpace \u03b3\u2082\ninst\u271d\u2078 : MeasurableSpace \u03b4\n\u03b1' : Type u_6\ninst\u271d\u2077 : TopologicalSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : LinearOrder \u03b1\u271d\ninst\u271d\u2074 : OrderClosedTopology \u03b1\u271d\na\u271d\u00b9 b\u271d x : \u03b1\u271d\n\u03b1 : Type u_7\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1\nhd : Dense s\nhbot : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s\nhIoo : \u2200 (x y : \u03b1), x < y \u2192 Ioo x y = \u2205 \u2192 y \u2208 s\nS : Set (Set \u03b1) := {S | \u2203 l, l \u2208 s \u2227 \u2203 u, u \u2208 s \u2227 l < u \u2227 Ico l u = S}\nthis : MeasurableSpace \u03b1 := MeasurableSpace.generateFrom S\na\u271d : \u03b1\nt : Set \u03b1\nhts : t \u2286 s\nhc : Set.Countable t\nhtd : Dense t\nhtb : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s \u2192 x \u2208 t\nha\u271d : \u2200 (b : \u03b1), b < a\u271d \u2192 Set.Nonempty (Ioo b a\u271d)\na : \u03b1\nha : a \u2208 t\nb : \u03b1\nhb : b \u2208 t\nhab : a < b\nx\u271d : b \u2264 a\u271d\n\u22a2 MeasurableSet (Ico a b)"}, {"tactic": "exact .basic _ \u27e8a, hts ha, b, hts hb, hab, mem_singleton _\u27e9", "annotated_tactic": ["exact .basic _ \u27e8a, hts ha, b, hts hb, hab, <a>mem_singleton</a> _\u27e9", [{"full_name": "Set.mem_singleton", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1289, 9], "def_end_pos": [1289, 22]}]], "state_before": "case pos\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\u271d\ninst\u271d\u00b2\u2070 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9\u2079 : MeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2078 : OpensMeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2077 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u2076 : MeasurableSpace \u03b2\ninst\u271d\u00b9\u2075 : OpensMeasurableSpace \u03b2\ninst\u271d\u00b9\u2074 : TopologicalSpace \u03b3\ninst\u271d\u00b9\u00b3 : MeasurableSpace \u03b3\ninst\u271d\u00b9\u00b2 : BorelSpace \u03b3\ninst\u271d\u00b9\u00b9 : TopologicalSpace \u03b3\u2082\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b3\u2082\ninst\u271d\u2079 : BorelSpace \u03b3\u2082\ninst\u271d\u2078 : MeasurableSpace \u03b4\n\u03b1' : Type u_6\ninst\u271d\u2077 : TopologicalSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : LinearOrder \u03b1\u271d\ninst\u271d\u2074 : OrderClosedTopology \u03b1\u271d\na\u271d\u00b9 b\u271d x : \u03b1\u271d\n\u03b1 : Type u_7\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1\nhd : Dense s\nhbot : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s\nhIoo : \u2200 (x y : \u03b1), x < y \u2192 Ioo x y = \u2205 \u2192 y \u2208 s\nS : Set (Set \u03b1) := {S | \u2203 l, l \u2208 s \u2227 \u2203 u, u \u2208 s \u2227 l < u \u2227 Ico l u = S}\nthis : MeasurableSpace \u03b1 := MeasurableSpace.generateFrom S\na\u271d : \u03b1\nt : Set \u03b1\nhts : t \u2286 s\nhc : Set.Countable t\nhtd : Dense t\nhtb : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s \u2192 x \u2208 t\nha\u271d : \u2200 (b : \u03b1), b < a\u271d \u2192 Set.Nonempty (Ioo b a\u271d)\na : \u03b1\nha : a \u2208 t\nb : \u03b1\nhb : b \u2208 t\nhab : a < b\nx\u271d : b \u2264 a\u271d\n\u22a2 MeasurableSet (Ico a b)", "state_after": "no goals"}, {"tactic": "simp only [not_forall, not_nonempty_iff_eq_empty] at ha", "annotated_tactic": ["simp only [<a>not_forall</a>, <a>not_nonempty_iff_eq_empty</a>] at ha", [{"full_name": "Classical.not_forall", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [686, 9], "def_end_pos": [686, 19]}, {"full_name": "Set.not_nonempty_iff_eq_empty", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [605, 9], "def_end_pos": [605, 34]}]], "state_before": "case neg\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\u271d\ninst\u271d\u00b2\u2070 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9\u2079 : MeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2078 : OpensMeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2077 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u2076 : MeasurableSpace \u03b2\ninst\u271d\u00b9\u2075 : OpensMeasurableSpace \u03b2\ninst\u271d\u00b9\u2074 : TopologicalSpace \u03b3\ninst\u271d\u00b9\u00b3 : MeasurableSpace \u03b3\ninst\u271d\u00b9\u00b2 : BorelSpace \u03b3\ninst\u271d\u00b9\u00b9 : TopologicalSpace \u03b3\u2082\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b3\u2082\ninst\u271d\u2079 : BorelSpace \u03b3\u2082\ninst\u271d\u2078 : MeasurableSpace \u03b4\n\u03b1' : Type u_6\ninst\u271d\u2077 : TopologicalSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : LinearOrder \u03b1\u271d\ninst\u271d\u2074 : OrderClosedTopology \u03b1\u271d\na\u271d b x : \u03b1\u271d\n\u03b1 : Type u_7\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1\nhd : Dense s\nhbot : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s\nhIoo : \u2200 (x y : \u03b1), x < y \u2192 Ioo x y = \u2205 \u2192 y \u2208 s\nS : Set (Set \u03b1) := {S | \u2203 l, l \u2208 s \u2227 \u2203 u, u \u2208 s \u2227 l < u \u2227 Ico l u = S}\nthis : MeasurableSpace \u03b1 := MeasurableSpace.generateFrom S\na : \u03b1\nt : Set \u03b1\nhts : t \u2286 s\nhc : Set.Countable t\nhtd : Dense t\nhtb : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s \u2192 x \u2208 t\nha : \u00ac\u2200 (b : \u03b1), b < a \u2192 Set.Nonempty (Ioo b a)\n\u22a2 MeasurableSet (Iio a)", "state_after": "case neg\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\u271d\ninst\u271d\u00b2\u2070 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9\u2079 : MeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2078 : OpensMeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2077 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u2076 : MeasurableSpace \u03b2\ninst\u271d\u00b9\u2075 : OpensMeasurableSpace \u03b2\ninst\u271d\u00b9\u2074 : TopologicalSpace \u03b3\ninst\u271d\u00b9\u00b3 : MeasurableSpace \u03b3\ninst\u271d\u00b9\u00b2 : BorelSpace \u03b3\ninst\u271d\u00b9\u00b9 : TopologicalSpace \u03b3\u2082\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b3\u2082\ninst\u271d\u2079 : BorelSpace \u03b3\u2082\ninst\u271d\u2078 : MeasurableSpace \u03b4\n\u03b1' : Type u_6\ninst\u271d\u2077 : TopologicalSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : LinearOrder \u03b1\u271d\ninst\u271d\u2074 : OrderClosedTopology \u03b1\u271d\na\u271d b x : \u03b1\u271d\n\u03b1 : Type u_7\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1\nhd : Dense s\nhbot : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s\nhIoo : \u2200 (x y : \u03b1), x < y \u2192 Ioo x y = \u2205 \u2192 y \u2208 s\nS : Set (Set \u03b1) := {S | \u2203 l, l \u2208 s \u2227 \u2203 u, u \u2208 s \u2227 l < u \u2227 Ico l u = S}\nthis : MeasurableSpace \u03b1 := MeasurableSpace.generateFrom S\na : \u03b1\nt : Set \u03b1\nhts : t \u2286 s\nhc : Set.Countable t\nhtd : Dense t\nhtb : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s \u2192 x \u2208 t\nha : \u2203 x h, Ioo x a = \u2205\n\u22a2 MeasurableSet (Iio a)"}, {"tactic": "replace ha : a \u2208 s := hIoo ha.choose a ha.choose_spec.fst ha.choose_spec.snd", "annotated_tactic": ["replace ha : a \u2208 s := hIoo ha.choose a ha.choose_spec.fst ha.choose_spec.snd", []], "state_before": "case neg\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\u271d\ninst\u271d\u00b2\u2070 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9\u2079 : MeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2078 : OpensMeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2077 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u2076 : MeasurableSpace \u03b2\ninst\u271d\u00b9\u2075 : OpensMeasurableSpace \u03b2\ninst\u271d\u00b9\u2074 : TopologicalSpace \u03b3\ninst\u271d\u00b9\u00b3 : MeasurableSpace \u03b3\ninst\u271d\u00b9\u00b2 : BorelSpace \u03b3\ninst\u271d\u00b9\u00b9 : TopologicalSpace \u03b3\u2082\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b3\u2082\ninst\u271d\u2079 : BorelSpace \u03b3\u2082\ninst\u271d\u2078 : MeasurableSpace \u03b4\n\u03b1' : Type u_6\ninst\u271d\u2077 : TopologicalSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : LinearOrder \u03b1\u271d\ninst\u271d\u2074 : OrderClosedTopology \u03b1\u271d\na\u271d b x : \u03b1\u271d\n\u03b1 : Type u_7\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1\nhd : Dense s\nhbot : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s\nhIoo : \u2200 (x y : \u03b1), x < y \u2192 Ioo x y = \u2205 \u2192 y \u2208 s\nS : Set (Set \u03b1) := {S | \u2203 l, l \u2208 s \u2227 \u2203 u, u \u2208 s \u2227 l < u \u2227 Ico l u = S}\nthis : MeasurableSpace \u03b1 := MeasurableSpace.generateFrom S\na : \u03b1\nt : Set \u03b1\nhts : t \u2286 s\nhc : Set.Countable t\nhtd : Dense t\nhtb : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s \u2192 x \u2208 t\nha : \u2203 x h, Ioo x a = \u2205\n\u22a2 MeasurableSet (Iio a)", "state_after": "case neg\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\u271d\ninst\u271d\u00b2\u2070 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9\u2079 : MeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2078 : OpensMeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2077 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u2076 : MeasurableSpace \u03b2\ninst\u271d\u00b9\u2075 : OpensMeasurableSpace \u03b2\ninst\u271d\u00b9\u2074 : TopologicalSpace \u03b3\ninst\u271d\u00b9\u00b3 : MeasurableSpace \u03b3\ninst\u271d\u00b9\u00b2 : BorelSpace \u03b3\ninst\u271d\u00b9\u00b9 : TopologicalSpace \u03b3\u2082\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b3\u2082\ninst\u271d\u2079 : BorelSpace \u03b3\u2082\ninst\u271d\u2078 : MeasurableSpace \u03b4\n\u03b1' : Type u_6\ninst\u271d\u2077 : TopologicalSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : LinearOrder \u03b1\u271d\ninst\u271d\u2074 : OrderClosedTopology \u03b1\u271d\na\u271d b x : \u03b1\u271d\n\u03b1 : Type u_7\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1\nhd : Dense s\nhbot : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s\nhIoo : \u2200 (x y : \u03b1), x < y \u2192 Ioo x y = \u2205 \u2192 y \u2208 s\nS : Set (Set \u03b1) := {S | \u2203 l, l \u2208 s \u2227 \u2203 u, u \u2208 s \u2227 l < u \u2227 Ico l u = S}\nthis : MeasurableSpace \u03b1 := MeasurableSpace.generateFrom S\na : \u03b1\nt : Set \u03b1\nhts : t \u2286 s\nhc : Set.Countable t\nhtd : Dense t\nhtb : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s \u2192 x \u2208 t\nha : a \u2208 s\n\u22a2 MeasurableSet (Iio a)"}, {"tactic": "convert_to MeasurableSet (\u22c3 (l \u2208 t) (_ : l < a), Ico l a)", "annotated_tactic": ["convert_to <a>MeasurableSet</a> (\u22c3 (l \u2208 t) (_ : l < a), <a>Ico</a> l a)", [{"full_name": "MeasurableSet", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [64, 5], "def_end_pos": [64, 18]}, {"full_name": "Set.Ico", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [49, 5], "def_end_pos": [49, 8]}]], "state_before": "case neg\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\u271d\ninst\u271d\u00b2\u2070 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9\u2079 : MeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2078 : OpensMeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2077 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u2076 : MeasurableSpace \u03b2\ninst\u271d\u00b9\u2075 : OpensMeasurableSpace \u03b2\ninst\u271d\u00b9\u2074 : TopologicalSpace \u03b3\ninst\u271d\u00b9\u00b3 : MeasurableSpace \u03b3\ninst\u271d\u00b9\u00b2 : BorelSpace \u03b3\ninst\u271d\u00b9\u00b9 : TopologicalSpace \u03b3\u2082\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b3\u2082\ninst\u271d\u2079 : BorelSpace \u03b3\u2082\ninst\u271d\u2078 : MeasurableSpace \u03b4\n\u03b1' : Type u_6\ninst\u271d\u2077 : TopologicalSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : LinearOrder \u03b1\u271d\ninst\u271d\u2074 : OrderClosedTopology \u03b1\u271d\na\u271d b x : \u03b1\u271d\n\u03b1 : Type u_7\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1\nhd : Dense s\nhbot : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s\nhIoo : \u2200 (x y : \u03b1), x < y \u2192 Ioo x y = \u2205 \u2192 y \u2208 s\nS : Set (Set \u03b1) := {S | \u2203 l, l \u2208 s \u2227 \u2203 u, u \u2208 s \u2227 l < u \u2227 Ico l u = S}\nthis : MeasurableSpace \u03b1 := MeasurableSpace.generateFrom S\na : \u03b1\nt : Set \u03b1\nhts : t \u2286 s\nhc : Set.Countable t\nhtd : Dense t\nhtb : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s \u2192 x \u2208 t\nha : a \u2208 s\n\u22a2 MeasurableSet (Iio a)", "state_after": "case h.e'_3\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\u271d\ninst\u271d\u00b2\u2070 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9\u2079 : MeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2078 : OpensMeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2077 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u2076 : MeasurableSpace \u03b2\ninst\u271d\u00b9\u2075 : OpensMeasurableSpace \u03b2\ninst\u271d\u00b9\u2074 : TopologicalSpace \u03b3\ninst\u271d\u00b9\u00b3 : MeasurableSpace \u03b3\ninst\u271d\u00b9\u00b2 : BorelSpace \u03b3\ninst\u271d\u00b9\u00b9 : TopologicalSpace \u03b3\u2082\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b3\u2082\ninst\u271d\u2079 : BorelSpace \u03b3\u2082\ninst\u271d\u2078 : MeasurableSpace \u03b4\n\u03b1' : Type u_6\ninst\u271d\u2077 : TopologicalSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : LinearOrder \u03b1\u271d\ninst\u271d\u2074 : OrderClosedTopology \u03b1\u271d\na\u271d b x : \u03b1\u271d\n\u03b1 : Type u_7\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1\nhd : Dense s\nhbot : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s\nhIoo : \u2200 (x y : \u03b1), x < y \u2192 Ioo x y = \u2205 \u2192 y \u2208 s\nS : Set (Set \u03b1) := {S | \u2203 l, l \u2208 s \u2227 \u2203 u, u \u2208 s \u2227 l < u \u2227 Ico l u = S}\nthis : MeasurableSpace \u03b1 := MeasurableSpace.generateFrom S\na : \u03b1\nt : Set \u03b1\nhts : t \u2286 s\nhc : Set.Countable t\nhtd : Dense t\nhtb : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s \u2192 x \u2208 t\nha : a \u2208 s\n\u22a2 Iio a = \u22c3 l \u2208 t, \u22c3 (_ : l < a), Ico l a\n\ncase neg\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\u271d\ninst\u271d\u00b2\u2070 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9\u2079 : MeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2078 : OpensMeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2077 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u2076 : MeasurableSpace \u03b2\ninst\u271d\u00b9\u2075 : OpensMeasurableSpace \u03b2\ninst\u271d\u00b9\u2074 : TopologicalSpace \u03b3\ninst\u271d\u00b9\u00b3 : MeasurableSpace \u03b3\ninst\u271d\u00b9\u00b2 : BorelSpace \u03b3\ninst\u271d\u00b9\u00b9 : TopologicalSpace \u03b3\u2082\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b3\u2082\ninst\u271d\u2079 : BorelSpace \u03b3\u2082\ninst\u271d\u2078 : MeasurableSpace \u03b4\n\u03b1' : Type u_6\ninst\u271d\u2077 : TopologicalSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : LinearOrder \u03b1\u271d\ninst\u271d\u2074 : OrderClosedTopology \u03b1\u271d\na\u271d b x : \u03b1\u271d\n\u03b1 : Type u_7\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1\nhd : Dense s\nhbot : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s\nhIoo : \u2200 (x y : \u03b1), x < y \u2192 Ioo x y = \u2205 \u2192 y \u2208 s\nS : Set (Set \u03b1) := {S | \u2203 l, l \u2208 s \u2227 \u2203 u, u \u2208 s \u2227 l < u \u2227 Ico l u = S}\nthis : MeasurableSpace \u03b1 := MeasurableSpace.generateFrom S\na : \u03b1\nt : Set \u03b1\nhts : t \u2286 s\nhc : Set.Countable t\nhtd : Dense t\nhtb : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s \u2192 x \u2208 t\nha : a \u2208 s\n\u22a2 MeasurableSet (\u22c3 l \u2208 t, \u22c3 (_ : l < a), Ico l a)"}, {"tactic": "symm", "annotated_tactic": ["symm", []], "state_before": "case h.e'_3\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\u271d\ninst\u271d\u00b2\u2070 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9\u2079 : MeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2078 : OpensMeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2077 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u2076 : MeasurableSpace \u03b2\ninst\u271d\u00b9\u2075 : OpensMeasurableSpace \u03b2\ninst\u271d\u00b9\u2074 : TopologicalSpace \u03b3\ninst\u271d\u00b9\u00b3 : MeasurableSpace \u03b3\ninst\u271d\u00b9\u00b2 : BorelSpace \u03b3\ninst\u271d\u00b9\u00b9 : TopologicalSpace \u03b3\u2082\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b3\u2082\ninst\u271d\u2079 : BorelSpace \u03b3\u2082\ninst\u271d\u2078 : MeasurableSpace \u03b4\n\u03b1' : Type u_6\ninst\u271d\u2077 : TopologicalSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : LinearOrder \u03b1\u271d\ninst\u271d\u2074 : OrderClosedTopology \u03b1\u271d\na\u271d b x : \u03b1\u271d\n\u03b1 : Type u_7\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1\nhd : Dense s\nhbot : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s\nhIoo : \u2200 (x y : \u03b1), x < y \u2192 Ioo x y = \u2205 \u2192 y \u2208 s\nS : Set (Set \u03b1) := {S | \u2203 l, l \u2208 s \u2227 \u2203 u, u \u2208 s \u2227 l < u \u2227 Ico l u = S}\nthis : MeasurableSpace \u03b1 := MeasurableSpace.generateFrom S\na : \u03b1\nt : Set \u03b1\nhts : t \u2286 s\nhc : Set.Countable t\nhtd : Dense t\nhtb : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s \u2192 x \u2208 t\nha : a \u2208 s\n\u22a2 Iio a = \u22c3 l \u2208 t, \u22c3 (_ : l < a), Ico l a", "state_after": "case h.e'_3\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\u271d\ninst\u271d\u00b2\u2070 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9\u2079 : MeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2078 : OpensMeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2077 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u2076 : MeasurableSpace \u03b2\ninst\u271d\u00b9\u2075 : OpensMeasurableSpace \u03b2\ninst\u271d\u00b9\u2074 : TopologicalSpace \u03b3\ninst\u271d\u00b9\u00b3 : MeasurableSpace \u03b3\ninst\u271d\u00b9\u00b2 : BorelSpace \u03b3\ninst\u271d\u00b9\u00b9 : TopologicalSpace \u03b3\u2082\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b3\u2082\ninst\u271d\u2079 : BorelSpace \u03b3\u2082\ninst\u271d\u2078 : MeasurableSpace \u03b4\n\u03b1' : Type u_6\ninst\u271d\u2077 : TopologicalSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : LinearOrder \u03b1\u271d\ninst\u271d\u2074 : OrderClosedTopology \u03b1\u271d\na\u271d b x : \u03b1\u271d\n\u03b1 : Type u_7\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1\nhd : Dense s\nhbot : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s\nhIoo : \u2200 (x y : \u03b1), x < y \u2192 Ioo x y = \u2205 \u2192 y \u2208 s\nS : Set (Set \u03b1) := {S | \u2203 l, l \u2208 s \u2227 \u2203 u, u \u2208 s \u2227 l < u \u2227 Ico l u = S}\nthis : MeasurableSpace \u03b1 := MeasurableSpace.generateFrom S\na : \u03b1\nt : Set \u03b1\nhts : t \u2286 s\nhc : Set.Countable t\nhtd : Dense t\nhtb : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s \u2192 x \u2208 t\nha : a \u2208 s\n\u22a2 \u22c3 l \u2208 t, \u22c3 (_ : l < a), Ico l a = Iio a"}, {"tactic": "simp only [\u2190 Ici_inter_Iio, \u2190 iUnion_inter, inter_eq_right, subset_def, mem_iUnion,\n  mem_Ici, mem_Iio]", "annotated_tactic": ["simp only [\u2190 <a>Ici_inter_Iio</a>, \u2190 <a>iUnion_inter</a>, <a>inter_eq_right</a>, <a>subset_def</a>, <a>mem_iUnion</a>,\n        <a>mem_Ici</a>, <a>mem_Iio</a>]", [{"full_name": "Set.Ici_inter_Iio", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [622, 9], "def_end_pos": [622, 22]}, {"full_name": "Set.iUnion_inter", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [639, 9], "def_end_pos": [639, 21]}, {"full_name": "Set.inter_eq_right", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [984, 15], "def_end_pos": [984, 29]}, {"full_name": "Set.subset_def", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [345, 9], "def_end_pos": [345, 19]}, {"full_name": "Set.mem_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [201, 9], "def_end_pos": [201, 19]}, {"full_name": "Set.mem_Ici", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [146, 9], "def_end_pos": [146, 16]}, {"full_name": "Set.mem_Iio", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [126, 9], "def_end_pos": [126, 16]}]], "state_before": "case h.e'_3\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\u271d\ninst\u271d\u00b2\u2070 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9\u2079 : MeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2078 : OpensMeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2077 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u2076 : MeasurableSpace \u03b2\ninst\u271d\u00b9\u2075 : OpensMeasurableSpace \u03b2\ninst\u271d\u00b9\u2074 : TopologicalSpace \u03b3\ninst\u271d\u00b9\u00b3 : MeasurableSpace \u03b3\ninst\u271d\u00b9\u00b2 : BorelSpace \u03b3\ninst\u271d\u00b9\u00b9 : TopologicalSpace \u03b3\u2082\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b3\u2082\ninst\u271d\u2079 : BorelSpace \u03b3\u2082\ninst\u271d\u2078 : MeasurableSpace \u03b4\n\u03b1' : Type u_6\ninst\u271d\u2077 : TopologicalSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : LinearOrder \u03b1\u271d\ninst\u271d\u2074 : OrderClosedTopology \u03b1\u271d\na\u271d b x : \u03b1\u271d\n\u03b1 : Type u_7\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1\nhd : Dense s\nhbot : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s\nhIoo : \u2200 (x y : \u03b1), x < y \u2192 Ioo x y = \u2205 \u2192 y \u2208 s\nS : Set (Set \u03b1) := {S | \u2203 l, l \u2208 s \u2227 \u2203 u, u \u2208 s \u2227 l < u \u2227 Ico l u = S}\nthis : MeasurableSpace \u03b1 := MeasurableSpace.generateFrom S\na : \u03b1\nt : Set \u03b1\nhts : t \u2286 s\nhc : Set.Countable t\nhtd : Dense t\nhtb : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s \u2192 x \u2208 t\nha : a \u2208 s\n\u22a2 \u22c3 l \u2208 t, \u22c3 (_ : l < a), Ico l a = Iio a", "state_after": "case h.e'_3\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\u271d\ninst\u271d\u00b2\u2070 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9\u2079 : MeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2078 : OpensMeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2077 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u2076 : MeasurableSpace \u03b2\ninst\u271d\u00b9\u2075 : OpensMeasurableSpace \u03b2\ninst\u271d\u00b9\u2074 : TopologicalSpace \u03b3\ninst\u271d\u00b9\u00b3 : MeasurableSpace \u03b3\ninst\u271d\u00b9\u00b2 : BorelSpace \u03b3\ninst\u271d\u00b9\u00b9 : TopologicalSpace \u03b3\u2082\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b3\u2082\ninst\u271d\u2079 : BorelSpace \u03b3\u2082\ninst\u271d\u2078 : MeasurableSpace \u03b4\n\u03b1' : Type u_6\ninst\u271d\u2077 : TopologicalSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : LinearOrder \u03b1\u271d\ninst\u271d\u2074 : OrderClosedTopology \u03b1\u271d\na\u271d b x : \u03b1\u271d\n\u03b1 : Type u_7\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1\nhd : Dense s\nhbot : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s\nhIoo : \u2200 (x y : \u03b1), x < y \u2192 Ioo x y = \u2205 \u2192 y \u2208 s\nS : Set (Set \u03b1) := {S | \u2203 l, l \u2208 s \u2227 \u2203 u, u \u2208 s \u2227 l < u \u2227 Ico l u = S}\nthis : MeasurableSpace \u03b1 := MeasurableSpace.generateFrom S\na : \u03b1\nt : Set \u03b1\nhts : t \u2286 s\nhc : Set.Countable t\nhtd : Dense t\nhtb : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s \u2192 x \u2208 t\nha : a \u2208 s\n\u22a2 \u2200 (x : \u03b1), x < a \u2192 \u2203 i h h, i \u2264 x"}, {"tactic": "intro x hx", "annotated_tactic": ["intro x hx", []], "state_before": "case h.e'_3\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\u271d\ninst\u271d\u00b2\u2070 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9\u2079 : MeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2078 : OpensMeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2077 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u2076 : MeasurableSpace \u03b2\ninst\u271d\u00b9\u2075 : OpensMeasurableSpace \u03b2\ninst\u271d\u00b9\u2074 : TopologicalSpace \u03b3\ninst\u271d\u00b9\u00b3 : MeasurableSpace \u03b3\ninst\u271d\u00b9\u00b2 : BorelSpace \u03b3\ninst\u271d\u00b9\u00b9 : TopologicalSpace \u03b3\u2082\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b3\u2082\ninst\u271d\u2079 : BorelSpace \u03b3\u2082\ninst\u271d\u2078 : MeasurableSpace \u03b4\n\u03b1' : Type u_6\ninst\u271d\u2077 : TopologicalSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : LinearOrder \u03b1\u271d\ninst\u271d\u2074 : OrderClosedTopology \u03b1\u271d\na\u271d b x : \u03b1\u271d\n\u03b1 : Type u_7\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1\nhd : Dense s\nhbot : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s\nhIoo : \u2200 (x y : \u03b1), x < y \u2192 Ioo x y = \u2205 \u2192 y \u2208 s\nS : Set (Set \u03b1) := {S | \u2203 l, l \u2208 s \u2227 \u2203 u, u \u2208 s \u2227 l < u \u2227 Ico l u = S}\nthis : MeasurableSpace \u03b1 := MeasurableSpace.generateFrom S\na : \u03b1\nt : Set \u03b1\nhts : t \u2286 s\nhc : Set.Countable t\nhtd : Dense t\nhtb : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s \u2192 x \u2208 t\nha : a \u2208 s\n\u22a2 \u2200 (x : \u03b1), x < a \u2192 \u2203 i h h, i \u2264 x", "state_after": "case h.e'_3\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\u271d\ninst\u271d\u00b2\u2070 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9\u2079 : MeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2078 : OpensMeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2077 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u2076 : MeasurableSpace \u03b2\ninst\u271d\u00b9\u2075 : OpensMeasurableSpace \u03b2\ninst\u271d\u00b9\u2074 : TopologicalSpace \u03b3\ninst\u271d\u00b9\u00b3 : MeasurableSpace \u03b3\ninst\u271d\u00b9\u00b2 : BorelSpace \u03b3\ninst\u271d\u00b9\u00b9 : TopologicalSpace \u03b3\u2082\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b3\u2082\ninst\u271d\u2079 : BorelSpace \u03b3\u2082\ninst\u271d\u2078 : MeasurableSpace \u03b4\n\u03b1' : Type u_6\ninst\u271d\u2077 : TopologicalSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : LinearOrder \u03b1\u271d\ninst\u271d\u2074 : OrderClosedTopology \u03b1\u271d\na\u271d b x\u271d : \u03b1\u271d\n\u03b1 : Type u_7\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1\nhd : Dense s\nhbot : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s\nhIoo : \u2200 (x y : \u03b1), x < y \u2192 Ioo x y = \u2205 \u2192 y \u2208 s\nS : Set (Set \u03b1) := {S | \u2203 l, l \u2208 s \u2227 \u2203 u, u \u2208 s \u2227 l < u \u2227 Ico l u = S}\nthis : MeasurableSpace \u03b1 := MeasurableSpace.generateFrom S\na : \u03b1\nt : Set \u03b1\nhts : t \u2286 s\nhc : Set.Countable t\nhtd : Dense t\nhtb : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s \u2192 x \u2208 t\nha : a \u2208 s\nx : \u03b1\nhx : x < a\n\u22a2 \u2203 i h h, i \u2264 x"}, {"tactic": "rcases htd.exists_le' (fun b hb => htb _ hb (hbot b hb)) x with \u27e8z, hzt, hzx\u27e9", "annotated_tactic": ["rcases htd.exists_le' (fun b hb => htb _ hb (hbot b hb)) x with \u27e8z, hzt, hzx\u27e9", []], "state_before": "case h.e'_3\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\u271d\ninst\u271d\u00b2\u2070 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9\u2079 : MeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2078 : OpensMeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2077 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u2076 : MeasurableSpace \u03b2\ninst\u271d\u00b9\u2075 : OpensMeasurableSpace \u03b2\ninst\u271d\u00b9\u2074 : TopologicalSpace \u03b3\ninst\u271d\u00b9\u00b3 : MeasurableSpace \u03b3\ninst\u271d\u00b9\u00b2 : BorelSpace \u03b3\ninst\u271d\u00b9\u00b9 : TopologicalSpace \u03b3\u2082\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b3\u2082\ninst\u271d\u2079 : BorelSpace \u03b3\u2082\ninst\u271d\u2078 : MeasurableSpace \u03b4\n\u03b1' : Type u_6\ninst\u271d\u2077 : TopologicalSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : LinearOrder \u03b1\u271d\ninst\u271d\u2074 : OrderClosedTopology \u03b1\u271d\na\u271d b x\u271d : \u03b1\u271d\n\u03b1 : Type u_7\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1\nhd : Dense s\nhbot : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s\nhIoo : \u2200 (x y : \u03b1), x < y \u2192 Ioo x y = \u2205 \u2192 y \u2208 s\nS : Set (Set \u03b1) := {S | \u2203 l, l \u2208 s \u2227 \u2203 u, u \u2208 s \u2227 l < u \u2227 Ico l u = S}\nthis : MeasurableSpace \u03b1 := MeasurableSpace.generateFrom S\na : \u03b1\nt : Set \u03b1\nhts : t \u2286 s\nhc : Set.Countable t\nhtd : Dense t\nhtb : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s \u2192 x \u2208 t\nha : a \u2208 s\nx : \u03b1\nhx : x < a\n\u22a2 \u2203 i h h, i \u2264 x", "state_after": "case h.e'_3.intro.intro\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\u271d\ninst\u271d\u00b2\u2070 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9\u2079 : MeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2078 : OpensMeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2077 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u2076 : MeasurableSpace \u03b2\ninst\u271d\u00b9\u2075 : OpensMeasurableSpace \u03b2\ninst\u271d\u00b9\u2074 : TopologicalSpace \u03b3\ninst\u271d\u00b9\u00b3 : MeasurableSpace \u03b3\ninst\u271d\u00b9\u00b2 : BorelSpace \u03b3\ninst\u271d\u00b9\u00b9 : TopologicalSpace \u03b3\u2082\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b3\u2082\ninst\u271d\u2079 : BorelSpace \u03b3\u2082\ninst\u271d\u2078 : MeasurableSpace \u03b4\n\u03b1' : Type u_6\ninst\u271d\u2077 : TopologicalSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : LinearOrder \u03b1\u271d\ninst\u271d\u2074 : OrderClosedTopology \u03b1\u271d\na\u271d b x\u271d : \u03b1\u271d\n\u03b1 : Type u_7\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1\nhd : Dense s\nhbot : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s\nhIoo : \u2200 (x y : \u03b1), x < y \u2192 Ioo x y = \u2205 \u2192 y \u2208 s\nS : Set (Set \u03b1) := {S | \u2203 l, l \u2208 s \u2227 \u2203 u, u \u2208 s \u2227 l < u \u2227 Ico l u = S}\nthis : MeasurableSpace \u03b1 := MeasurableSpace.generateFrom S\na : \u03b1\nt : Set \u03b1\nhts : t \u2286 s\nhc : Set.Countable t\nhtd : Dense t\nhtb : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s \u2192 x \u2208 t\nha : a \u2208 s\nx : \u03b1\nhx : x < a\nz : \u03b1\nhzt : z \u2208 t\nhzx : z \u2264 x\n\u22a2 \u2203 i h h, i \u2264 x"}, {"tactic": "exact \u27e8z, hzt, hzx.trans_lt hx, hzx\u27e9", "annotated_tactic": ["exact \u27e8z, hzt, hzx.trans_lt hx, hzx\u27e9", []], "state_before": "case h.e'_3.intro.intro\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\u271d\ninst\u271d\u00b2\u2070 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9\u2079 : MeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2078 : OpensMeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2077 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u2076 : MeasurableSpace \u03b2\ninst\u271d\u00b9\u2075 : OpensMeasurableSpace \u03b2\ninst\u271d\u00b9\u2074 : TopologicalSpace \u03b3\ninst\u271d\u00b9\u00b3 : MeasurableSpace \u03b3\ninst\u271d\u00b9\u00b2 : BorelSpace \u03b3\ninst\u271d\u00b9\u00b9 : TopologicalSpace \u03b3\u2082\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b3\u2082\ninst\u271d\u2079 : BorelSpace \u03b3\u2082\ninst\u271d\u2078 : MeasurableSpace \u03b4\n\u03b1' : Type u_6\ninst\u271d\u2077 : TopologicalSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : LinearOrder \u03b1\u271d\ninst\u271d\u2074 : OrderClosedTopology \u03b1\u271d\na\u271d b x\u271d : \u03b1\u271d\n\u03b1 : Type u_7\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1\nhd : Dense s\nhbot : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s\nhIoo : \u2200 (x y : \u03b1), x < y \u2192 Ioo x y = \u2205 \u2192 y \u2208 s\nS : Set (Set \u03b1) := {S | \u2203 l, l \u2208 s \u2227 \u2203 u, u \u2208 s \u2227 l < u \u2227 Ico l u = S}\nthis : MeasurableSpace \u03b1 := MeasurableSpace.generateFrom S\na : \u03b1\nt : Set \u03b1\nhts : t \u2286 s\nhc : Set.Countable t\nhtd : Dense t\nhtb : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s \u2192 x \u2208 t\nha : a \u2208 s\nx : \u03b1\nhx : x < a\nz : \u03b1\nhzt : z \u2208 t\nhzx : z \u2264 x\n\u22a2 \u2203 i h h, i \u2264 x", "state_after": "no goals"}, {"tactic": "refine' .biUnion hc fun x hx => MeasurableSet.iUnion fun hlt => _", "annotated_tactic": ["refine' .biUnion hc fun x hx => <a>MeasurableSet.iUnion</a> fun hlt => _", [{"full_name": "MeasurableSet.iUnion", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [115, 19], "def_end_pos": [115, 39]}]], "state_before": "case neg\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\u271d\ninst\u271d\u00b2\u2070 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9\u2079 : MeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2078 : OpensMeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2077 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u2076 : MeasurableSpace \u03b2\ninst\u271d\u00b9\u2075 : OpensMeasurableSpace \u03b2\ninst\u271d\u00b9\u2074 : TopologicalSpace \u03b3\ninst\u271d\u00b9\u00b3 : MeasurableSpace \u03b3\ninst\u271d\u00b9\u00b2 : BorelSpace \u03b3\ninst\u271d\u00b9\u00b9 : TopologicalSpace \u03b3\u2082\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b3\u2082\ninst\u271d\u2079 : BorelSpace \u03b3\u2082\ninst\u271d\u2078 : MeasurableSpace \u03b4\n\u03b1' : Type u_6\ninst\u271d\u2077 : TopologicalSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : LinearOrder \u03b1\u271d\ninst\u271d\u2074 : OrderClosedTopology \u03b1\u271d\na\u271d b x : \u03b1\u271d\n\u03b1 : Type u_7\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1\nhd : Dense s\nhbot : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s\nhIoo : \u2200 (x y : \u03b1), x < y \u2192 Ioo x y = \u2205 \u2192 y \u2208 s\nS : Set (Set \u03b1) := {S | \u2203 l, l \u2208 s \u2227 \u2203 u, u \u2208 s \u2227 l < u \u2227 Ico l u = S}\nthis : MeasurableSpace \u03b1 := MeasurableSpace.generateFrom S\na : \u03b1\nt : Set \u03b1\nhts : t \u2286 s\nhc : Set.Countable t\nhtd : Dense t\nhtb : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s \u2192 x \u2208 t\nha : a \u2208 s\n\u22a2 MeasurableSet (\u22c3 l \u2208 t, \u22c3 (_ : l < a), Ico l a)", "state_after": "case neg\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\u271d\ninst\u271d\u00b2\u2070 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9\u2079 : MeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2078 : OpensMeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2077 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u2076 : MeasurableSpace \u03b2\ninst\u271d\u00b9\u2075 : OpensMeasurableSpace \u03b2\ninst\u271d\u00b9\u2074 : TopologicalSpace \u03b3\ninst\u271d\u00b9\u00b3 : MeasurableSpace \u03b3\ninst\u271d\u00b9\u00b2 : BorelSpace \u03b3\ninst\u271d\u00b9\u00b9 : TopologicalSpace \u03b3\u2082\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b3\u2082\ninst\u271d\u2079 : BorelSpace \u03b3\u2082\ninst\u271d\u2078 : MeasurableSpace \u03b4\n\u03b1' : Type u_6\ninst\u271d\u2077 : TopologicalSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : LinearOrder \u03b1\u271d\ninst\u271d\u2074 : OrderClosedTopology \u03b1\u271d\na\u271d b x\u271d : \u03b1\u271d\n\u03b1 : Type u_7\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1\nhd : Dense s\nhbot : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s\nhIoo : \u2200 (x y : \u03b1), x < y \u2192 Ioo x y = \u2205 \u2192 y \u2208 s\nS : Set (Set \u03b1) := {S | \u2203 l, l \u2208 s \u2227 \u2203 u, u \u2208 s \u2227 l < u \u2227 Ico l u = S}\nthis : MeasurableSpace \u03b1 := MeasurableSpace.generateFrom S\na : \u03b1\nt : Set \u03b1\nhts : t \u2286 s\nhc : Set.Countable t\nhtd : Dense t\nhtb : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s \u2192 x \u2208 t\nha : a \u2208 s\nx : \u03b1\nhx : x \u2208 t\nhlt : x < a\n\u22a2 MeasurableSet (Ico x a)"}, {"tactic": "exact .basic _ \u27e8x, hts hx, a, ha, hlt, mem_singleton _\u27e9", "annotated_tactic": ["exact .basic _ \u27e8x, hts hx, a, ha, hlt, <a>mem_singleton</a> _\u27e9", [{"full_name": "Set.mem_singleton", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1289, 9], "def_end_pos": [1289, 22]}]], "state_before": "case neg\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\u271d\ninst\u271d\u00b2\u2070 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9\u2079 : MeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2078 : OpensMeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2077 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u2076 : MeasurableSpace \u03b2\ninst\u271d\u00b9\u2075 : OpensMeasurableSpace \u03b2\ninst\u271d\u00b9\u2074 : TopologicalSpace \u03b3\ninst\u271d\u00b9\u00b3 : MeasurableSpace \u03b3\ninst\u271d\u00b9\u00b2 : BorelSpace \u03b3\ninst\u271d\u00b9\u00b9 : TopologicalSpace \u03b3\u2082\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b3\u2082\ninst\u271d\u2079 : BorelSpace \u03b3\u2082\ninst\u271d\u2078 : MeasurableSpace \u03b4\n\u03b1' : Type u_6\ninst\u271d\u2077 : TopologicalSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : LinearOrder \u03b1\u271d\ninst\u271d\u2074 : OrderClosedTopology \u03b1\u271d\na\u271d b x\u271d : \u03b1\u271d\n\u03b1 : Type u_7\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1\nhd : Dense s\nhbot : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s\nhIoo : \u2200 (x y : \u03b1), x < y \u2192 Ioo x y = \u2205 \u2192 y \u2208 s\nS : Set (Set \u03b1) := {S | \u2203 l, l \u2208 s \u2227 \u2203 u, u \u2208 s \u2227 l < u \u2227 Ico l u = S}\nthis : MeasurableSpace \u03b1 := MeasurableSpace.generateFrom S\na : \u03b1\nt : Set \u03b1\nhts : t \u2286 s\nhc : Set.Countable t\nhtd : Dense t\nhtb : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s \u2192 x \u2208 t\nha : a \u2208 s\nx : \u03b1\nhx : x \u2208 t\nhlt : x < a\n\u22a2 MeasurableSet (Ico x a)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Intervals/ProjIcc.lean", "full_name": "Set.IicExtend_self", "start": [257, 1], "end": [258, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Hausdorff.lean", "full_name": "MeasureTheory.OuterMeasure.isometry_comap_mkMetric", "start": [371, 1], "end": [384, 61], "traced_tactics": [{"tactic": "simp only [mkMetric, mkMetric', mkMetric'.pre, inducedOuterMeasure, comap_iSup]", "annotated_tactic": ["simp only [<a>mkMetric</a>, <a>mkMetric'</a>, <a>mkMetric'.pre</a>, <a>inducedOuterMeasure</a>, <a>comap_iSup</a>]", [{"full_name": "MeasureTheory.OuterMeasure.mkMetric", "def_path": "Mathlib/MeasureTheory/Measure/Hausdorff.lean", "def_pos": [264, 5], "def_end_pos": [264, 13]}, {"full_name": "MeasureTheory.OuterMeasure.mkMetric'", "def_path": "Mathlib/MeasureTheory/Measure/Hausdorff.lean", "def_pos": [258, 5], "def_end_pos": [258, 14]}, {"full_name": "MeasureTheory.OuterMeasure.mkMetric'.pre", "def_path": "Mathlib/MeasureTheory/Measure/Hausdorff.lean", "def_pos": [251, 5], "def_end_pos": [251, 18]}, {"full_name": "MeasureTheory.inducedOuterMeasure", "def_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "def_pos": [1436, 5], "def_end_pos": [1436, 24]}, {"full_name": "MeasureTheory.OuterMeasure.comap_iSup", "def_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "def_pos": [550, 9], "def_end_pos": [550, 19]}]], "state_before": "\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nf : X \u2192 Y\nhf : Isometry f\nH : Monotone m \u2228 Surjective f\n\u22a2 \u2191(comap f) (mkMetric m) = mkMetric m", "state_after": "\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nf : X \u2192 Y\nhf : Isometry f\nH : Monotone m \u2228 Surjective f\n\u22a2 \u2a06 i, \u2a06 (_ : i > 0), \u2191(comap f) (boundedBy (extend fun s x => m (diam s))) =\n    \u2a06 r, \u2a06 (_ : r > 0), boundedBy (extend fun s x => m (diam s))"}, {"tactic": "refine' surjective_id.iSup_congr id fun \u03b5 => surjective_id.iSup_congr id fun h\u03b5 => _", "annotated_tactic": ["refine' surjective_id.iSup_congr <a>id</a> fun \u03b5 => surjective_id.iSup_congr <a>id</a> fun h\u03b5 => _", [{"full_name": "id", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [33, 15], "def_end_pos": [33, 17]}, {"full_name": "id", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [33, 15], "def_end_pos": [33, 17]}]], "state_before": "\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nf : X \u2192 Y\nhf : Isometry f\nH : Monotone m \u2228 Surjective f\n\u22a2 \u2a06 i, \u2a06 (_ : i > 0), \u2191(comap f) (boundedBy (extend fun s x => m (diam s))) =\n    \u2a06 r, \u2a06 (_ : r > 0), boundedBy (extend fun s x => m (diam s))", "state_after": "\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nf : X \u2192 Y\nhf : Isometry f\nH : Monotone m \u2228 Surjective f\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : id \u03b5 > 0\n\u22a2 \u2191(comap f) (boundedBy (extend fun s x => m (diam s))) = boundedBy (extend fun s x => m (diam s))"}, {"tactic": "rw [comap_boundedBy _ (H.imp _ id)]", "annotated_tactic": ["rw [<a>comap_boundedBy</a> _ (H.imp _ <a>id</a>)]", [{"full_name": "MeasureTheory.OuterMeasure.comap_boundedBy", "def_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "def_pos": [898, 9], "def_end_pos": [898, 24]}, {"full_name": "id", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [33, 15], "def_end_pos": [33, 17]}]], "state_before": "\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nf : X \u2192 Y\nhf : Isometry f\nH : Monotone m \u2228 Surjective f\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : id \u03b5 > 0\n\u22a2 \u2191(comap f) (boundedBy (extend fun s x => m (diam s))) = boundedBy (extend fun s x => m (diam s))", "state_after": "\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nf : X \u2192 Y\nhf : Isometry f\nH : Monotone m \u2228 Surjective f\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : id \u03b5 > 0\n\u22a2 (boundedBy fun s => extend (fun s x => m (diam s)) (f '' s)) = boundedBy (extend fun s x => m (diam s))\n\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nf : X \u2192 Y\nhf : Isometry f\nH : Monotone m \u2228 Surjective f\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : id \u03b5 > 0\n\u22a2 Monotone m \u2192 Monotone fun s => extend (fun s x => m (diam s)) \u2191s"}, {"tactic": "congr with s : 1", "annotated_tactic": ["congr with s : 1", []], "state_before": "\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nf : X \u2192 Y\nhf : Isometry f\nH : Monotone m \u2228 Surjective f\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : id \u03b5 > 0\n\u22a2 (boundedBy fun s => extend (fun s x => m (diam s)) (f '' s)) = boundedBy (extend fun s x => m (diam s))", "state_after": "case e_m.h\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nf : X \u2192 Y\nhf : Isometry f\nH : Monotone m \u2228 Surjective f\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : id \u03b5 > 0\ns : Set X\n\u22a2 extend (fun s x => m (diam s)) (f '' s) = extend (fun s x => m (diam s)) s"}, {"tactic": "apply extend_congr", "annotated_tactic": ["apply <a>extend_congr</a>", [{"full_name": "MeasureTheory.extend_congr", "def_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "def_pos": [1339, 9], "def_end_pos": [1339, 21]}]], "state_before": "case e_m.h\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nf : X \u2192 Y\nhf : Isometry f\nH : Monotone m \u2228 Surjective f\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : id \u03b5 > 0\ns : Set X\n\u22a2 extend (fun s x => m (diam s)) (f '' s) = extend (fun s x => m (diam s)) s", "state_after": "case e_m.h.hP\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nf : X \u2192 Y\nhf : Isometry f\nH : Monotone m \u2228 Surjective f\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : id \u03b5 > 0\ns : Set X\n\u22a2 diam (f '' s) \u2264 \u03b5 \u2194 diam s \u2264 id \u03b5\n\ncase e_m.h.hm\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nf : X \u2192 Y\nhf : Isometry f\nH : Monotone m \u2228 Surjective f\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : id \u03b5 > 0\ns : Set X\n\u22a2 diam (f '' s) \u2264 \u03b5 \u2192 diam s \u2264 id \u03b5 \u2192 m (diam (f '' s)) = m (diam s)"}, {"tactic": "simp [hf.ediam_image]", "annotated_tactic": ["simp [hf.ediam_image]", []], "state_before": "case e_m.h.hP\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nf : X \u2192 Y\nhf : Isometry f\nH : Monotone m \u2228 Surjective f\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : id \u03b5 > 0\ns : Set X\n\u22a2 diam (f '' s) \u2264 \u03b5 \u2194 diam s \u2264 id \u03b5", "state_after": "no goals"}, {"tactic": "intros", "annotated_tactic": ["intros", []], "state_before": "case e_m.h.hm\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nf : X \u2192 Y\nhf : Isometry f\nH : Monotone m \u2228 Surjective f\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : id \u03b5 > 0\ns : Set X\n\u22a2 diam (f '' s) \u2264 \u03b5 \u2192 diam s \u2264 id \u03b5 \u2192 m (diam (f '' s)) = m (diam s)", "state_after": "case e_m.h.hm\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nf : X \u2192 Y\nhf : Isometry f\nH : Monotone m \u2228 Surjective f\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : id \u03b5 > 0\ns : Set X\nha\u271d : diam (f '' s) \u2264 \u03b5\nhb\u271d : diam s \u2264 id \u03b5\n\u22a2 m (diam (f '' s)) = m (diam s)"}, {"tactic": "simp [hf.injective.subsingleton_image_iff, hf.ediam_image]", "annotated_tactic": ["simp [hf.injective.subsingleton_image_iff, hf.ediam_image]", []], "state_before": "case e_m.h.hm\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nf : X \u2192 Y\nhf : Isometry f\nH : Monotone m \u2228 Surjective f\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : id \u03b5 > 0\ns : Set X\nha\u271d : diam (f '' s) \u2264 \u03b5\nhb\u271d : diam s \u2264 id \u03b5\n\u22a2 m (diam (f '' s)) = m (diam s)", "state_after": "no goals"}, {"tactic": "intro h_mono s t hst", "annotated_tactic": ["intro h_mono s t hst", []], "state_before": "\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nf : X \u2192 Y\nhf : Isometry f\nH : Monotone m \u2228 Surjective f\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : id \u03b5 > 0\n\u22a2 Monotone m \u2192 Monotone fun s => extend (fun s x => m (diam s)) \u2191s", "state_after": "\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nf : X \u2192 Y\nhf : Isometry f\nH : Monotone m \u2228 Surjective f\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : id \u03b5 > 0\nh_mono : Monotone m\ns t : { s // Set.Nonempty s }\nhst : s \u2264 t\n\u22a2 (fun s => extend (fun s x => m (diam s)) \u2191s) s \u2264 (fun s => extend (fun s x => m (diam s)) \u2191s) t"}, {"tactic": "simp only [extend, le_iInf_iff]", "annotated_tactic": ["simp only [<a>extend</a>, <a>le_iInf_iff</a>]", [{"full_name": "MeasureTheory.extend", "def_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "def_pos": [1312, 5], "def_end_pos": [1312, 11]}, {"full_name": "le_iInf_iff", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [969, 9], "def_end_pos": [969, 20]}]], "state_before": "\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nf : X \u2192 Y\nhf : Isometry f\nH : Monotone m \u2228 Surjective f\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : id \u03b5 > 0\nh_mono : Monotone m\ns t : { s // Set.Nonempty s }\nhst : s \u2264 t\n\u22a2 (fun s => extend (fun s x => m (diam s)) \u2191s) s \u2264 (fun s => extend (fun s x => m (diam s)) \u2191s) t", "state_after": "\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nf : X \u2192 Y\nhf : Isometry f\nH : Monotone m \u2228 Surjective f\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : id \u03b5 > 0\nh_mono : Monotone m\ns t : { s // Set.Nonempty s }\nhst : s \u2264 t\n\u22a2 diam \u2191t \u2264 \u03b5 \u2192 \u2a05 (_ : diam \u2191s \u2264 \u03b5), m (diam \u2191s) \u2264 m (diam \u2191t)"}, {"tactic": "intro ht", "annotated_tactic": ["intro ht", []], "state_before": "\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nf : X \u2192 Y\nhf : Isometry f\nH : Monotone m \u2228 Surjective f\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : id \u03b5 > 0\nh_mono : Monotone m\ns t : { s // Set.Nonempty s }\nhst : s \u2264 t\n\u22a2 diam \u2191t \u2264 \u03b5 \u2192 \u2a05 (_ : diam \u2191s \u2264 \u03b5), m (diam \u2191s) \u2264 m (diam \u2191t)", "state_after": "\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nf : X \u2192 Y\nhf : Isometry f\nH : Monotone m \u2228 Surjective f\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : id \u03b5 > 0\nh_mono : Monotone m\ns t : { s // Set.Nonempty s }\nhst : s \u2264 t\nht : diam \u2191t \u2264 \u03b5\n\u22a2 \u2a05 (_ : diam \u2191s \u2264 \u03b5), m (diam \u2191s) \u2264 m (diam \u2191t)"}, {"tactic": "apply le_trans _ (h_mono (diam_mono hst))", "annotated_tactic": ["apply <a>le_trans</a> _ (h_mono (<a>diam_mono</a> hst))", [{"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}, {"full_name": "EMetric.diam_mono", "def_path": "Mathlib/Topology/EMetricSpace/Basic.lean", "def_pos": [948, 9], "def_end_pos": [948, 18]}]], "state_before": "\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nf : X \u2192 Y\nhf : Isometry f\nH : Monotone m \u2228 Surjective f\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : id \u03b5 > 0\nh_mono : Monotone m\ns t : { s // Set.Nonempty s }\nhst : s \u2264 t\nht : diam \u2191t \u2264 \u03b5\n\u22a2 \u2a05 (_ : diam \u2191s \u2264 \u03b5), m (diam \u2191s) \u2264 m (diam \u2191t)", "state_after": "\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nf : X \u2192 Y\nhf : Isometry f\nH : Monotone m \u2228 Surjective f\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : id \u03b5 > 0\nh_mono : Monotone m\ns t : { s // Set.Nonempty s }\nhst : s \u2264 t\nht : diam \u2191t \u2264 \u03b5\n\u22a2 \u2a05 (_ : diam \u2191s \u2264 \u03b5), m (diam \u2191s) \u2264 m (diam ((fun a => \u2191a) s))"}, {"tactic": "simp only [(diam_mono hst).trans ht, le_refl, ciInf_pos]", "annotated_tactic": ["simp only [(<a>diam_mono</a> hst).<a>trans</a> ht, <a>le_refl</a>, <a>ciInf_pos</a>]", [{"full_name": "EMetric.diam_mono", "def_path": "Mathlib/Topology/EMetricSpace/Basic.lean", "def_pos": [948, 9], "def_end_pos": [948, 18]}, {"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [120, 7], "def_end_pos": [120, 18]}, {"full_name": "le_refl", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [50, 9], "def_end_pos": [50, 16]}, {"full_name": "ciInf_pos", "def_path": "Mathlib/Order/ConditionallyCompleteLattice/Basic.lean", "def_pos": [880, 9], "def_end_pos": [880, 18]}]], "state_before": "\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nf : X \u2192 Y\nhf : Isometry f\nH : Monotone m \u2228 Surjective f\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : id \u03b5 > 0\nh_mono : Monotone m\ns t : { s // Set.Nonempty s }\nhst : s \u2264 t\nht : diam \u2191t \u2264 \u03b5\n\u22a2 \u2a05 (_ : diam \u2191s \u2264 \u03b5), m (diam \u2191s) \u2264 m (diam ((fun a => \u2191a) s))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "full_name": "Int.mul_self_lt_mul_self", "start": [1239, 11], "end": [1240, 69], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Haar/Basic.lean", "full_name": "MeasureTheory.Measure.haar.haarContent_apply", "start": [554, 1], "end": [556, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Intervals/OrdConnectedComponent.lean", "full_name": "Set.ordConnectedComponent_eq_empty", "start": [62, 1], "end": [63, 67], "traced_tactics": [{"tactic": "rw [\u2190 not_nonempty_iff_eq_empty, nonempty_ordConnectedComponent]", "annotated_tactic": ["rw [\u2190 <a>not_nonempty_iff_eq_empty</a>, <a>nonempty_ordConnectedComponent</a>]", [{"full_name": "Set.not_nonempty_iff_eq_empty", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [605, 9], "def_end_pos": [605, 34]}, {"full_name": "Set.nonempty_ordConnectedComponent", "def_path": "Mathlib/Data/Set/Intervals/OrdConnectedComponent.lean", "def_pos": [57, 9], "def_end_pos": [57, 39]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\ns t : Set \u03b1\nx y z : \u03b1\n\u22a2 ordConnectedComponent s x = \u2205 \u2194 \u00acx \u2208 s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Int/GCD.lean", "full_name": "Nat.xgcdAux_val", "start": [121, 1], "end": [122, 39], "traced_tactics": [{"tactic": "rw [xgcd, \u2190 xgcdAux_fst x y 1 0 0 1]", "annotated_tactic": ["rw [<a>xgcd</a>, \u2190 <a>xgcdAux_fst</a> x y 1 0 0 1]", [{"full_name": "Nat.xgcd", "def_path": "Mathlib/Data/Int/GCD.lean", "def_pos": [68, 5], "def_end_pos": [68, 9]}, {"full_name": "Nat.xgcdAux_fst", "def_path": "Mathlib/Data/Int/GCD.lean", "def_pos": [115, 9], "def_end_pos": [115, 20]}]], "state_before": "x y : \u2115\n\u22a2 xgcdAux x 1 0 y 0 1 = (gcd x y, xgcd x y)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Function.lean", "full_name": "Set.RightInvOn.congr_right", "start": [1136, 1], "end": [1138, 33], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/ZMod/Basic.lean", "full_name": "ZMod.val_add_val_of_le", "start": [627, 1], "end": [631, 65], "traced_tactics": [{"tactic": "rw [val_add, Nat.add_mod_add_of_le_add_mod, Nat.mod_eq_of_lt (val_lt _),\n  Nat.mod_eq_of_lt (val_lt _)]", "annotated_tactic": ["rw [<a>val_add</a>, <a>Nat.add_mod_add_of_le_add_mod</a>, <a>Nat.mod_eq_of_lt</a> (<a>val_lt</a> _),\n    <a>Nat.mod_eq_of_lt</a> (<a>val_lt</a> _)]", [{"full_name": "ZMod.val_add", "def_path": "Mathlib/Data/ZMod/Basic.lean", "def_pos": [616, 9], "def_end_pos": [616, 16]}, {"full_name": "Nat.add_mod_add_of_le_add_mod", "def_path": "Mathlib/Data/Nat/ModEq.lean", "def_pos": [447, 9], "def_end_pos": [447, 34]}, {"full_name": "Nat.mod_eq_of_lt", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Div.lean", "def_pos": [111, 9], "def_end_pos": [111, 21]}, {"full_name": "ZMod.val_lt", "def_path": "Mathlib/Data/ZMod/Basic.lean", "def_pos": [52, 9], "def_end_pos": [52, 15]}, {"full_name": "Nat.mod_eq_of_lt", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Div.lean", "def_pos": [111, 9], "def_end_pos": [111, 21]}, {"full_name": "ZMod.val_lt", "def_path": "Mathlib/Data/ZMod/Basic.lean", "def_pos": [52, 9], "def_end_pos": [52, 15]}]], "state_before": "n : \u2115\ninst\u271d : NeZero n\na b : ZMod n\nh : n \u2264 val a + val b\n\u22a2 val a + val b = val (a + b) + n", "state_after": "n : \u2115\ninst\u271d : NeZero n\na b : ZMod n\nh : n \u2264 val a + val b\n\u22a2 n \u2264 val a % n + val b % n"}, {"tactic": "rwa [Nat.mod_eq_of_lt (val_lt _), Nat.mod_eq_of_lt (val_lt _)]", "annotated_tactic": ["rwa [<a>Nat.mod_eq_of_lt</a> (<a>val_lt</a> _), <a>Nat.mod_eq_of_lt</a> (<a>val_lt</a> _)]", [{"full_name": "Nat.mod_eq_of_lt", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Div.lean", "def_pos": [111, 9], "def_end_pos": [111, 21]}, {"full_name": "ZMod.val_lt", "def_path": "Mathlib/Data/ZMod/Basic.lean", "def_pos": [52, 9], "def_end_pos": [52, 15]}, {"full_name": "Nat.mod_eq_of_lt", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Div.lean", "def_pos": [111, 9], "def_end_pos": [111, 21]}, {"full_name": "ZMod.val_lt", "def_path": "Mathlib/Data/ZMod/Basic.lean", "def_pos": [52, 9], "def_end_pos": [52, 15]}]], "state_before": "n : \u2115\ninst\u271d : NeZero n\na b : ZMod n\nh : n \u2264 val a + val b\n\u22a2 n \u2264 val a % n + val b % n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/CondCount.lean", "full_name": "ProbabilityTheory.pred_true_of_condCount_eq_one", "start": [118, 1], "end": [126, 96], "traced_tactics": [{"tactic": "have hsf := finite_of_condCount_ne_zero (by rw [h]; exact one_ne_zero)", "annotated_tactic": ["have hsf := <a>finite_of_condCount_ne_zero</a> (by rw [h]; exact <a>one_ne_zero</a>)", [{"full_name": "ProbabilityTheory.finite_of_condCount_ne_zero", "def_path": "Mathlib/Probability/CondCount.lean", "def_pos": [65, 9], "def_end_pos": [65, 36]}, {"full_name": "one_ne_zero", "def_path": "Mathlib/Algebra/NeZero.lean", "def_pos": [55, 15], "def_end_pos": [55, 26]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nh : \u2191\u2191(condCount s) t = 1\n\u22a2 s \u2286 t", "state_after": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nh : \u2191\u2191(condCount s) t = 1\nhsf : Set.Finite s\n\u22a2 s \u2286 t"}, {"tactic": "rw [condCount, cond_apply _ hsf.measurableSet, mul_comm] at h", "annotated_tactic": ["rw [<a>condCount</a>, <a>cond_apply</a> _ hsf.measurableSet, <a>mul_comm</a>] at h", [{"full_name": "ProbabilityTheory.condCount", "def_path": "Mathlib/Probability/CondCount.lean", "def_pos": [54, 5], "def_end_pos": [54, 14]}, {"full_name": "ProbabilityTheory.cond_apply", "def_path": "Mathlib/Probability/ConditionalProbability.lean", "def_pos": [102, 9], "def_end_pos": [102, 19]}, {"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [302, 9], "def_end_pos": [302, 17]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nh : \u2191\u2191(condCount s) t = 1\nhsf : Set.Finite s\n\u22a2 s \u2286 t", "state_after": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nh : \u2191\u2191Measure.count (s \u2229 t) * (\u2191\u2191Measure.count s)\u207b\u00b9 = 1\nhsf : Set.Finite s\n\u22a2 s \u2286 t"}, {"tactic": "replace h := ENNReal.eq_inv_of_mul_eq_one_left h", "annotated_tactic": ["replace h := <a>ENNReal.eq_inv_of_mul_eq_one_left</a> h", [{"full_name": "ENNReal.eq_inv_of_mul_eq_one_left", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1701, 19], "def_end_pos": [1701, 44]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nh : \u2191\u2191Measure.count (s \u2229 t) * (\u2191\u2191Measure.count s)\u207b\u00b9 = 1\nhsf : Set.Finite s\n\u22a2 s \u2286 t", "state_after": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nhsf : Set.Finite s\nh : \u2191\u2191Measure.count (s \u2229 t) = (\u2191\u2191Measure.count s)\u207b\u00b9\u207b\u00b9\n\u22a2 s \u2286 t"}, {"tactic": "rw [inv_inv, Measure.count_apply_finite _ hsf, Measure.count_apply_finite _ (hsf.inter_of_left _),\n  Nat.cast_inj] at h", "annotated_tactic": ["rw [<a>inv_inv</a>, <a>Measure.count_apply_finite</a> _ hsf, <a>Measure.count_apply_finite</a> _ (hsf.inter_of_left _),\n    <a>Nat.cast_inj</a>] at h", [{"full_name": "inv_inv", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [800, 9], "def_end_pos": [800, 16]}, {"full_name": "MeasureTheory.Measure.count_apply_finite", "def_path": "Mathlib/MeasureTheory/Measure/Count.lean", "def_pos": [69, 9], "def_end_pos": [69, 27]}, {"full_name": "MeasureTheory.Measure.count_apply_finite", "def_path": "Mathlib/MeasureTheory/Measure/Count.lean", "def_pos": [69, 9], "def_end_pos": [69, 27]}, {"full_name": "Nat.cast_inj", "def_path": "Mathlib/Algebra/CharZero/Defs.lean", "def_pos": [75, 9], "def_end_pos": [75, 17]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nhsf : Set.Finite s\nh : \u2191\u2191Measure.count (s \u2229 t) = (\u2191\u2191Measure.count s)\u207b\u00b9\u207b\u00b9\n\u22a2 s \u2286 t", "state_after": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nhsf : Set.Finite s\nh : Finset.card (Set.Finite.toFinset (_ : Set.Finite (s \u2229 t))) = Finset.card (Set.Finite.toFinset hsf)\n\u22a2 s \u2286 t"}, {"tactic": "suffices s \u2229 t = s by exact this \u25b8 fun x hx => hx.2", "annotated_tactic": ["suffices s \u2229 t = s by exact this \u25b8 fun x hx => hx.2", []], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nhsf : Set.Finite s\nh : Finset.card (Set.Finite.toFinset (_ : Set.Finite (s \u2229 t))) = Finset.card (Set.Finite.toFinset hsf)\n\u22a2 s \u2286 t", "state_after": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nhsf : Set.Finite s\nh : Finset.card (Set.Finite.toFinset (_ : Set.Finite (s \u2229 t))) = Finset.card (Set.Finite.toFinset hsf)\n\u22a2 s \u2229 t = s"}, {"tactic": "rw [\u2190 @Set.Finite.toFinset_inj _ _ _ (hsf.inter_of_left _) hsf]", "annotated_tactic": ["rw [\u2190 @<a>Set.Finite.toFinset_inj</a> _ _ _ (hsf.inter_of_left _) hsf]", [{"full_name": "Set.Finite.toFinset_inj", "def_path": "Mathlib/Data/Set/Finite.lean", "def_pos": [193, 19], "def_end_pos": [193, 31]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nhsf : Set.Finite s\nh : Finset.card (Set.Finite.toFinset (_ : Set.Finite (s \u2229 t))) = Finset.card (Set.Finite.toFinset hsf)\n\u22a2 s \u2229 t = s", "state_after": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nhsf : Set.Finite s\nh : Finset.card (Set.Finite.toFinset (_ : Set.Finite (s \u2229 t))) = Finset.card (Set.Finite.toFinset hsf)\n\u22a2 Set.Finite.toFinset (_ : Set.Finite (s \u2229 t)) = Set.Finite.toFinset hsf"}, {"tactic": "exact Finset.eq_of_subset_of_card_le (Set.Finite.toFinset_mono <| s.inter_subset_left t) h.ge", "annotated_tactic": ["exact <a>Finset.eq_of_subset_of_card_le</a> (<a>Set.Finite.toFinset_mono</a> <| s.inter_subset_left t) h.ge", [{"full_name": "Finset.eq_of_subset_of_card_le", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [275, 9], "def_end_pos": [275, 32]}, {"full_name": "Set.Finite.toFinset_mono", "def_path": "Mathlib/Data/Set/Finite.lean", "def_pos": [227, 11], "def_end_pos": [227, 24]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nhsf : Set.Finite s\nh : Finset.card (Set.Finite.toFinset (_ : Set.Finite (s \u2229 t))) = Finset.card (Set.Finite.toFinset hsf)\n\u22a2 Set.Finite.toFinset (_ : Set.Finite (s \u2229 t)) = Set.Finite.toFinset hsf", "state_after": "no goals"}, {"tactic": "rw [h]", "annotated_tactic": ["rw [h]", []], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nh : \u2191\u2191(condCount s) t = 1\n\u22a2 \u2191\u2191(condCount ?m.9264) ?m.9265 \u2260 0", "state_after": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nh : \u2191\u2191(condCount s) t = 1\n\u22a2 1 \u2260 0"}, {"tactic": "exact one_ne_zero", "annotated_tactic": ["exact <a>one_ne_zero</a>", [{"full_name": "one_ne_zero", "def_path": "Mathlib/Algebra/NeZero.lean", "def_pos": [55, 15], "def_end_pos": [55, 26]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nh : \u2191\u2191(condCount s) t = 1\n\u22a2 1 \u2260 0", "state_after": "no goals"}, {"tactic": "exact this \u25b8 fun x hx => hx.2", "annotated_tactic": ["exact this \u25b8 fun x hx => hx.2", []], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nhsf : Set.Finite s\nh : Finset.card (Set.Finite.toFinset (_ : Set.Finite (s \u2229 t))) = Finset.card (Set.Finite.toFinset hsf)\nthis : s \u2229 t = s\n\u22a2 s \u2286 t", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Pointwise.lean", "full_name": "Finset.card_le_card_mul_right", "start": [1920, 1], "end": [1921, 52], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "full_name": "MeasureTheory.SimpleFunc.lintegral_map", "start": [1139, 1], "end": [1141, 86], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Pointwise.lean", "full_name": "Finset.mem_prod_list_ofFn", "start": [911, 1], "end": [914, 6], "traced_tactics": [{"tactic": "rw [\u2190 mem_coe, coe_list_prod, List.map_ofFn, Set.mem_prod_list_ofFn]", "annotated_tactic": ["rw [\u2190 <a>mem_coe</a>, <a>coe_list_prod</a>, <a>List.map_ofFn</a>, <a>Set.mem_prod_list_ofFn</a>]", [{"full_name": "Finset.mem_coe", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [208, 9], "def_end_pos": [208, 16]}, {"full_name": "Finset.coe_list_prod", "def_path": "Mathlib/Data/Finset/Pointwise.lean", "def_pos": [905, 9], "def_end_pos": [905, 22]}, {"full_name": "List.map_ofFn", "def_path": "Mathlib/Data/List/OfFn.lean", "def_pos": [84, 9], "def_end_pos": [84, 17]}, {"full_name": "Set.mem_prod_list_ofFn", "def_path": "Mathlib/Data/Set/Pointwise/ListOfFn.lean", "def_pos": [26, 9], "def_end_pos": [26, 27]}]], "state_before": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : Monoid \u03b1\ns\u271d t : Finset \u03b1\na\u271d : \u03b1\nm n : \u2115\na : \u03b1\ns : Fin n \u2192 Finset \u03b1\n\u22a2 a \u2208 List.prod (List.ofFn s) \u2194 \u2203 f, List.prod (List.ofFn fun i => \u2191(f i)) = a", "state_after": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : Monoid \u03b1\ns\u271d t : Finset \u03b1\na\u271d : \u03b1\nm n : \u2115\na : \u03b1\ns : Fin n \u2192 Finset \u03b1\n\u22a2 (\u2203 f, List.prod (List.ofFn fun i => \u2191(f i)) = a) \u2194 \u2203 f, List.prod (List.ofFn fun i => \u2191(f i)) = a"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : Monoid \u03b1\ns\u271d t : Finset \u03b1\na\u271d : \u03b1\nm n : \u2115\na : \u03b1\ns : Fin n \u2192 Finset \u03b1\n\u22a2 (\u2203 f, List.prod (List.ofFn fun i => \u2191(f i)) = a) \u2194 \u2203 f, List.prod (List.ofFn fun i => \u2191(f i)) = a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/FundThmCalculus.lean", "full_name": "intervalIntegral.deriv_integral_of_tendsto_ae_right", "start": [765, 1], "end": [768, 62], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Partrec.lean", "full_name": "Nat.Partrec.ppred", "start": [215, 1], "end": [230, 24], "traced_tactics": [{"tactic": "cases n <;> simp", "annotated_tactic": ["cases n <;> simp", []], "state_before": "this : Primrec\u2082 fun n m => if n = Nat.succ m then 0 else 1\nn : \u2115\n\u22a2 (Nat.rfind fun n_1 =>\n      (fun m => decide (m = 0)) <$> \u2191(unpaired fun n m => if n = Nat.succ m then 0 else 1) (Nat.pair n n_1)) =\n    \u2191(Nat.ppred n)", "state_after": "case zero\nthis : Primrec\u2082 fun n m => if n = Nat.succ m then 0 else 1\n\u22a2 (Nat.rfind fun n => Part.some false) = Part.none\n\ncase succ\nthis : Primrec\u2082 fun n m => if n = Nat.succ m then 0 else 1\nn\u271d : \u2115\n\u22a2 (Nat.rfind fun n => Part.some (decide (\u00acn\u271d = n \u2192 False))) = Part.some n\u271d"}, {"tactic": "exact\n  eq_none_iff.2 fun a \u27e8\u27e8m, h, _\u27e9, _\u27e9 => by\n    simp [show 0 \u2260 m.succ by intro h; injection h] at h", "annotated_tactic": ["exact\n        <a>eq_none_iff</a>.2 fun a \u27e8\u27e8m, h, _\u27e9, _\u27e9 => by\n          simp [show 0 \u2260 m.succ by intro h; injection h] at h", [{"full_name": "Part.eq_none_iff", "def_path": "Mathlib/Data/Part.lean", "def_pos": [178, 9], "def_end_pos": [178, 20]}]], "state_before": "case zero\nthis : Primrec\u2082 fun n m => if n = Nat.succ m then 0 else 1\n\u22a2 (Nat.rfind fun n => Part.some false) = Part.none", "state_after": "no goals"}, {"tactic": "simp [show 0 \u2260 m.succ by intro h; injection h] at h", "annotated_tactic": ["simp [show 0 \u2260 m.succ by intro h; injection h] at h", []], "state_before": "this : Primrec\u2082 fun n m => if n = Nat.succ m then 0 else 1\na : \u2115\nx\u271d : a \u2208 Nat.rfind fun n => Part.some false\nm : \u2115\nh : true \u2208 (fun n => Part.some false) m\nright\u271d : \u2200 (k : \u2115), k < m \u2192 ((fun n => Part.some false) k).Dom\nh\u271d :\n  Part.get (Nat.rfind fun n => Part.some false)\n      (_ : \u2203 n, true \u2208 (fun n => Part.some false) n \u2227 \u2200 (k : \u2115), k < n \u2192 ((fun n => Part.some false) k).Dom) =\n    a\n\u22a2 False", "state_after": "no goals"}, {"tactic": "intro h", "annotated_tactic": ["intro h", []], "state_before": "this : Primrec\u2082 fun n m => if n = Nat.succ m then 0 else 1\na : \u2115\nx\u271d : a \u2208 Nat.rfind fun n => Part.some false\nm : \u2115\nh : true \u2208 (fun n => Part.some false) m\nright\u271d : \u2200 (k : \u2115), k < m \u2192 ((fun n => Part.some false) k).Dom\nh\u271d :\n  Part.get (Nat.rfind fun n => Part.some false)\n      (_ : \u2203 n, true \u2208 (fun n => Part.some false) n \u2227 \u2200 (k : \u2115), k < n \u2192 ((fun n => Part.some false) k).Dom) =\n    a\n\u22a2 0 \u2260 Nat.succ m", "state_after": "this : Primrec\u2082 fun n m => if n = Nat.succ m then 0 else 1\na : \u2115\nx\u271d : a \u2208 Nat.rfind fun n => Part.some false\nm : \u2115\nh\u271d\u00b9 : true \u2208 (fun n => Part.some false) m\nright\u271d : \u2200 (k : \u2115), k < m \u2192 ((fun n => Part.some false) k).Dom\nh\u271d :\n  Part.get (Nat.rfind fun n => Part.some false)\n      (_ : \u2203 n, true \u2208 (fun n => Part.some false) n \u2227 \u2200 (k : \u2115), k < n \u2192 ((fun n => Part.some false) k).Dom) =\n    a\nh : 0 = Nat.succ m\n\u22a2 False"}, {"tactic": "injection h", "annotated_tactic": ["injection h", []], "state_before": "this : Primrec\u2082 fun n m => if n = Nat.succ m then 0 else 1\na : \u2115\nx\u271d : a \u2208 Nat.rfind fun n => Part.some false\nm : \u2115\nh\u271d\u00b9 : true \u2208 (fun n => Part.some false) m\nright\u271d : \u2200 (k : \u2115), k < m \u2192 ((fun n => Part.some false) k).Dom\nh\u271d :\n  Part.get (Nat.rfind fun n => Part.some false)\n      (_ : \u2203 n, true \u2208 (fun n => Part.some false) n \u2227 \u2200 (k : \u2115), k < n \u2192 ((fun n => Part.some false) k).Dom) =\n    a\nh : 0 = Nat.succ m\n\u22a2 False", "state_after": "no goals"}, {"tactic": "refine' eq_some_iff.2 _", "annotated_tactic": ["refine' <a>eq_some_iff</a>.2 _", [{"full_name": "Part.eq_some_iff", "def_path": "Mathlib/Data/Part.lean", "def_pos": [174, 9], "def_end_pos": [174, 20]}]], "state_before": "case succ\nthis : Primrec\u2082 fun n m => if n = Nat.succ m then 0 else 1\nn\u271d : \u2115\n\u22a2 (Nat.rfind fun n => Part.some (decide (\u00acn\u271d = n \u2192 False))) = Part.some n\u271d", "state_after": "case succ\nthis : Primrec\u2082 fun n m => if n = Nat.succ m then 0 else 1\nn\u271d : \u2115\n\u22a2 n\u271d \u2208 Nat.rfind fun n => Part.some (decide (\u00acn\u271d = n \u2192 False))"}, {"tactic": "simp only [mem_rfind, not_true, IsEmpty.forall_iff, decide_True, mem_some_iff,\n  Bool.false_eq_decide_iff, true_and]", "annotated_tactic": ["simp only [<a>mem_rfind</a>, <a>not_true</a>, <a>IsEmpty.forall_iff</a>, <a>decide_True</a>, <a>mem_some_iff</a>,\n        <a>Bool.false_eq_decide_iff</a>, <a>true_and</a>]", [{"full_name": "Nat.mem_rfind", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [102, 9], "def_end_pos": [102, 18]}, {"full_name": "not_true", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [80, 17], "def_end_pos": [80, 25]}, {"full_name": "IsEmpty.forall_iff", "def_path": "Mathlib/Logic/IsEmpty.lean", "def_pos": [121, 9], "def_end_pos": [121, 19]}, {"full_name": "decide_True", "def_path": "lake-packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [158, 17], "def_end_pos": [158, 28]}, {"full_name": "Part.mem_some_iff", "def_path": "Mathlib/Data/Part.lean", "def_pos": [170, 9], "def_end_pos": [170, 21]}, {"full_name": "Bool.false_eq_decide_iff", "def_path": "Mathlib/Data/Bool/Basic.lean", "def_pos": [58, 9], "def_end_pos": [58, 28]}, {"full_name": "true_and", "def_path": "lake-packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [84, 17], "def_end_pos": [84, 25]}]], "state_before": "case succ\nthis : Primrec\u2082 fun n m => if n = Nat.succ m then 0 else 1\nn\u271d : \u2115\n\u22a2 n\u271d \u2208 Nat.rfind fun n => Part.some (decide (\u00acn\u271d = n \u2192 False))", "state_after": "case succ\nthis : Primrec\u2082 fun n m => if n = Nat.succ m then 0 else 1\nn\u271d : \u2115\n\u22a2 \u2200 {m : \u2115}, m < n\u271d \u2192 \u00ac(\u00acn\u271d = m \u2192 False)"}, {"tactic": "intro m h", "annotated_tactic": ["intro m h", []], "state_before": "case succ\nthis : Primrec\u2082 fun n m => if n = Nat.succ m then 0 else 1\nn\u271d : \u2115\n\u22a2 \u2200 {m : \u2115}, m < n\u271d \u2192 \u00ac(\u00acn\u271d = m \u2192 False)", "state_after": "case succ\nthis : Primrec\u2082 fun n m => if n = Nat.succ m then 0 else 1\nn\u271d m : \u2115\nh : m < n\u271d\n\u22a2 \u00ac(\u00acn\u271d = m \u2192 False)"}, {"tactic": "simp [ne_of_gt h]", "annotated_tactic": ["simp [<a>ne_of_gt</a> h]", [{"full_name": "Nat.ne_of_gt", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [136, 9], "def_end_pos": [136, 17]}]], "state_before": "case succ\nthis : Primrec\u2082 fun n m => if n = Nat.succ m then 0 else 1\nn\u271d m : \u2115\nh : m < n\u271d\n\u22a2 \u00ac(\u00acn\u271d = m \u2192 False)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/WF.lean", "full_name": "Acc.rec_eq_recC", "start": [63, 18], "end": [68, 38], "traced_tactics": [{"tactic": "funext \u03b1 r motive intro a t", "annotated_tactic": ["funext \u03b1 r motive intro a t", []], "state_before": "\u22a2 @rec = @Acc.recC", "state_after": "case h.h.h.h.h.h\n\u03b1 : Sort u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nmotive : (a : \u03b1) \u2192 Acc r a \u2192 Sort u_2\nintro :\n  (x : \u03b1) \u2192 (h : \u2200 (y : \u03b1), r y x \u2192 Acc r y) \u2192 ((y : \u03b1) \u2192 (a : r y x) \u2192 motive y (_ : Acc r y)) \u2192 motive x (_ : Acc r x)\na : \u03b1\nt : Acc r a\n\u22a2 rec intro t = Acc.recC intro t"}, {"tactic": "induction t with\n| intro x h ih =>\n  dsimp only [recC_intro intro h]\n  congr; funext y hr; exact ih _ hr", "annotated_tactic": ["induction t with\n  | <a>intro</a> x h ih =>\n    dsimp only [<a>recC_intro</a> intro h]\n    congr; funext y hr; exact ih _ hr", [{"full_name": "Acc.intro", "def_path": "lake-packages/lean4/src/lean/Init/WF.lean", "def_pos": [13, 5], "def_end_pos": [13, 10]}, {"full_name": "_private.\u00ablake-packages\u00bb.std.Std.WF.0.Acc.recC_intro", "def_path": "lake-packages/std/Std/WF.lean", "def_pos": [56, 17], "def_end_pos": [56, 27]}]], "state_before": "case h.h.h.h.h.h\n\u03b1 : Sort u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nmotive : (a : \u03b1) \u2192 Acc r a \u2192 Sort u_2\nintro :\n  (x : \u03b1) \u2192 (h : \u2200 (y : \u03b1), r y x \u2192 Acc r y) \u2192 ((y : \u03b1) \u2192 (a : r y x) \u2192 motive y (_ : Acc r y)) \u2192 motive x (_ : Acc r x)\na : \u03b1\nt : Acc r a\n\u22a2 rec intro t = Acc.recC intro t", "state_after": "no goals"}, {"tactic": "dsimp only [recC_intro intro h]", "annotated_tactic": ["dsimp only [<a>recC_intro</a> intro h]", [{"full_name": "_private.\u00ablake-packages\u00bb.std.Std.WF.0.Acc.recC_intro", "def_path": "lake-packages/std/Std/WF.lean", "def_pos": [56, 17], "def_end_pos": [56, 27]}]], "state_before": "case h.h.h.h.h.h.intro\n\u03b1 : Sort u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nmotive : (a : \u03b1) \u2192 Acc r a \u2192 Sort u_2\nintro :\n  (x : \u03b1) \u2192 (h : \u2200 (y : \u03b1), r y x \u2192 Acc r y) \u2192 ((y : \u03b1) \u2192 (a : r y x) \u2192 motive y (_ : Acc r y)) \u2192 motive x (_ : Acc r x)\na x : \u03b1\nh : \u2200 (y : \u03b1), r y x \u2192 Acc r y\nih : \u2200 (y : \u03b1) (a : r y x), rec intro (_ : Acc ?m.2399 y) = Acc.recC intro (_ : Acc ?m.2399 y)\n\u22a2 rec intro (_ : Acc r x) = Acc.recC intro (_ : Acc r x)", "state_after": "case h.h.h.h.h.h.intro\n\u03b1 : Sort u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nmotive : (a : \u03b1) \u2192 Acc r a \u2192 Sort u_2\nintro :\n  (x : \u03b1) \u2192 (h : \u2200 (y : \u03b1), r y x \u2192 Acc r y) \u2192 ((y : \u03b1) \u2192 (a : r y x) \u2192 motive y (_ : Acc r y)) \u2192 motive x (_ : Acc r x)\na x : \u03b1\nh : \u2200 (y : \u03b1), r y x \u2192 Acc r y\nih : \u2200 (y : \u03b1) (a : r y x), rec intro (_ : Acc ?m.2399 y) = Acc.recC intro (_ : Acc ?m.2399 y)\n\u22a2 (intro x h fun y a => rec intro (_ : Acc ?m.2399 y)) = intro x h fun y hr => Acc.recC intro (_ : Acc ?m.2399 y)"}, {"tactic": "congr", "annotated_tactic": ["congr", []], "state_before": "case h.h.h.h.h.h.intro\n\u03b1 : Sort u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nmotive : (a : \u03b1) \u2192 Acc r a \u2192 Sort u_2\nintro :\n  (x : \u03b1) \u2192 (h : \u2200 (y : \u03b1), r y x \u2192 Acc r y) \u2192 ((y : \u03b1) \u2192 (a : r y x) \u2192 motive y (_ : Acc r y)) \u2192 motive x (_ : Acc r x)\na x : \u03b1\nh : \u2200 (y : \u03b1), r y x \u2192 Acc r y\nih : \u2200 (y : \u03b1) (a : r y x), rec intro (_ : Acc ?m.2399 y) = Acc.recC intro (_ : Acc ?m.2399 y)\n\u22a2 (intro x h fun y a => rec intro (_ : Acc ?m.2399 y)) = intro x h fun y hr => Acc.recC intro (_ : Acc ?m.2399 y)", "state_after": "case h.h.h.h.h.h.intro.e_h_ih\n\u03b1 : Sort u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nmotive : (a : \u03b1) \u2192 Acc r a \u2192 Sort u_2\nintro :\n  (x : \u03b1) \u2192 (h : \u2200 (y : \u03b1), r y x \u2192 Acc r y) \u2192 ((y : \u03b1) \u2192 (a : r y x) \u2192 motive y (_ : Acc r y)) \u2192 motive x (_ : Acc r x)\na x : \u03b1\nh : \u2200 (y : \u03b1), r y x \u2192 Acc r y\nih : \u2200 (y : \u03b1) (a : r y x), rec intro (_ : Acc ?m.2399 y) = Acc.recC intro (_ : Acc ?m.2399 y)\n\u22a2 (fun y a => rec intro (_ : Acc ?m.2399 y)) = fun y hr => Acc.recC intro (_ : Acc ?m.2399 y)"}, {"tactic": "funext y hr", "annotated_tactic": ["funext y hr", []], "state_before": "case h.h.h.h.h.h.intro.e_h_ih\n\u03b1 : Sort u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nmotive : (a : \u03b1) \u2192 Acc r a \u2192 Sort u_2\nintro :\n  (x : \u03b1) \u2192 (h : \u2200 (y : \u03b1), r y x \u2192 Acc r y) \u2192 ((y : \u03b1) \u2192 (a : r y x) \u2192 motive y (_ : Acc r y)) \u2192 motive x (_ : Acc r x)\na x : \u03b1\nh : \u2200 (y : \u03b1), r y x \u2192 Acc r y\nih : \u2200 (y : \u03b1) (a : r y x), rec intro (_ : Acc ?m.2399 y) = Acc.recC intro (_ : Acc ?m.2399 y)\n\u22a2 (fun y a => rec intro (_ : Acc ?m.2399 y)) = fun y hr => Acc.recC intro (_ : Acc ?m.2399 y)", "state_after": "case h.h.h.h.h.h.intro.e_h_ih.h.h\n\u03b1 : Sort u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nmotive : (a : \u03b1) \u2192 Acc r a \u2192 Sort u_2\nintro :\n  (x : \u03b1) \u2192 (h : \u2200 (y : \u03b1), r y x \u2192 Acc r y) \u2192 ((y : \u03b1) \u2192 (a : r y x) \u2192 motive y (_ : Acc r y)) \u2192 motive x (_ : Acc r x)\na x : \u03b1\nh : \u2200 (y : \u03b1), r y x \u2192 Acc r y\nih : \u2200 (y : \u03b1) (a : r y x), rec intro (_ : Acc ?m.2399 y) = Acc.recC intro (_ : Acc ?m.2399 y)\ny : \u03b1\nhr : r y x\n\u22a2 rec intro (_ : Acc ?m.2399 y) = Acc.recC intro (_ : Acc ?m.2399 y)"}, {"tactic": "exact ih _ hr", "annotated_tactic": ["exact ih _ hr", []], "state_before": "case h.h.h.h.h.h.intro.e_h_ih.h.h\n\u03b1 : Sort u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nmotive : (a : \u03b1) \u2192 Acc r a \u2192 Sort u_2\nintro :\n  (x : \u03b1) \u2192 (h : \u2200 (y : \u03b1), r y x \u2192 Acc r y) \u2192 ((y : \u03b1) \u2192 (a : r y x) \u2192 motive y (_ : Acc r y)) \u2192 motive x (_ : Acc r x)\na x : \u03b1\nh : \u2200 (y : \u03b1), r y x \u2192 Acc r y\nih : \u2200 (y : \u03b1) (a : r y x), rec intro (_ : Acc ?m.2399 y) = Acc.recC intro (_ : Acc ?m.2399 y)\ny : \u03b1\nhr : r y x\n\u22a2 rec intro (_ : Acc ?m.2399 y) = Acc.recC intro (_ : Acc ?m.2399 y)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "full_name": "MeasureTheory.integral_map_of_stronglyMeasurable", "start": [1587, 1], "end": [1607, 7], "traced_tactics": [{"tactic": "by_cases hG : CompleteSpace G", "annotated_tactic": ["by_cases hG : <a>CompleteSpace</a> G", [{"full_name": "CompleteSpace", "def_path": "Mathlib/Topology/UniformSpace/Cauchy.lean", "def_pos": [397, 7], "def_end_pos": [397, 20]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\nG : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b2 : Type u_7\ninst\u271d : MeasurableSpace \u03b2\n\u03c6 : \u03b1 \u2192 \u03b2\nh\u03c6 : Measurable \u03c6\nf : \u03b2 \u2192 G\nhfm : StronglyMeasurable f\n\u22a2 \u222b (y : \u03b2), f y \u2202Measure.map \u03c6 \u03bc = \u222b (x : \u03b1), f (\u03c6 x) \u2202\u03bc", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\nG : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b2 : Type u_7\ninst\u271d : MeasurableSpace \u03b2\n\u03c6 : \u03b1 \u2192 \u03b2\nh\u03c6 : Measurable \u03c6\nf : \u03b2 \u2192 G\nhfm : StronglyMeasurable f\nhG : CompleteSpace G\n\u22a2 \u222b (y : \u03b2), f y \u2202Measure.map \u03c6 \u03bc = \u222b (x : \u03b1), f (\u03c6 x) \u2202\u03bc\n\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\nG : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b2 : Type u_7\ninst\u271d : MeasurableSpace \u03b2\n\u03c6 : \u03b1 \u2192 \u03b2\nh\u03c6 : Measurable \u03c6\nf : \u03b2 \u2192 G\nhfm : StronglyMeasurable f\nhG : \u00acCompleteSpace G\n\u22a2 \u222b (y : \u03b2), f y \u2202Measure.map \u03c6 \u03bc = \u222b (x : \u03b1), f (\u03c6 x) \u2202\u03bc"}, {"tactic": "swap", "annotated_tactic": ["swap", []], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\nG : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b2 : Type u_7\ninst\u271d : MeasurableSpace \u03b2\n\u03c6 : \u03b1 \u2192 \u03b2\nh\u03c6 : Measurable \u03c6\nf : \u03b2 \u2192 G\nhfm : StronglyMeasurable f\nhG : CompleteSpace G\n\u22a2 \u222b (y : \u03b2), f y \u2202Measure.map \u03c6 \u03bc = \u222b (x : \u03b1), f (\u03c6 x) \u2202\u03bc\n\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\nG : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b2 : Type u_7\ninst\u271d : MeasurableSpace \u03b2\n\u03c6 : \u03b1 \u2192 \u03b2\nh\u03c6 : Measurable \u03c6\nf : \u03b2 \u2192 G\nhfm : StronglyMeasurable f\nhG : \u00acCompleteSpace G\n\u22a2 \u222b (y : \u03b2), f y \u2202Measure.map \u03c6 \u03bc = \u222b (x : \u03b1), f (\u03c6 x) \u2202\u03bc", "state_after": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\nG : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b2 : Type u_7\ninst\u271d : MeasurableSpace \u03b2\n\u03c6 : \u03b1 \u2192 \u03b2\nh\u03c6 : Measurable \u03c6\nf : \u03b2 \u2192 G\nhfm : StronglyMeasurable f\nhG : \u00acCompleteSpace G\n\u22a2 \u222b (y : \u03b2), f y \u2202Measure.map \u03c6 \u03bc = \u222b (x : \u03b1), f (\u03c6 x) \u2202\u03bc\n\ncase pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\nG : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b2 : Type u_7\ninst\u271d : MeasurableSpace \u03b2\n\u03c6 : \u03b1 \u2192 \u03b2\nh\u03c6 : Measurable \u03c6\nf : \u03b2 \u2192 G\nhfm : StronglyMeasurable f\nhG : CompleteSpace G\n\u22a2 \u222b (y : \u03b2), f y \u2202Measure.map \u03c6 \u03bc = \u222b (x : \u03b1), f (\u03c6 x) \u2202\u03bc"}, {"tactic": "by_cases hfi : Integrable f (Measure.map \u03c6 \u03bc)", "annotated_tactic": ["by_cases hfi : <a>Integrable</a> f (<a>Measure.map</a> \u03c6 \u03bc)", [{"full_name": "MeasureTheory.Integrable", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [442, 5], "def_end_pos": [442, 15]}, {"full_name": "MeasureTheory.Measure.map", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1163, 17], "def_end_pos": [1163, 20]}]], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\nG : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b2 : Type u_7\ninst\u271d : MeasurableSpace \u03b2\n\u03c6 : \u03b1 \u2192 \u03b2\nh\u03c6 : Measurable \u03c6\nf : \u03b2 \u2192 G\nhfm : StronglyMeasurable f\nhG : CompleteSpace G\n\u22a2 \u222b (y : \u03b2), f y \u2202Measure.map \u03c6 \u03bc = \u222b (x : \u03b1), f (\u03c6 x) \u2202\u03bc", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\nG : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b2 : Type u_7\ninst\u271d : MeasurableSpace \u03b2\n\u03c6 : \u03b1 \u2192 \u03b2\nh\u03c6 : Measurable \u03c6\nf : \u03b2 \u2192 G\nhfm : StronglyMeasurable f\nhG : CompleteSpace G\nhfi : Integrable f\n\u22a2 \u222b (y : \u03b2), f y \u2202Measure.map \u03c6 \u03bc = \u222b (x : \u03b1), f (\u03c6 x) \u2202\u03bc\n\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\nG : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b2 : Type u_7\ninst\u271d : MeasurableSpace \u03b2\n\u03c6 : \u03b1 \u2192 \u03b2\nh\u03c6 : Measurable \u03c6\nf : \u03b2 \u2192 G\nhfm : StronglyMeasurable f\nhG : CompleteSpace G\nhfi : \u00acIntegrable f\n\u22a2 \u222b (y : \u03b2), f y \u2202Measure.map \u03c6 \u03bc = \u222b (x : \u03b1), f (\u03c6 x) \u2202\u03bc"}, {"tactic": "swap", "annotated_tactic": ["swap", []], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\nG : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b2 : Type u_7\ninst\u271d : MeasurableSpace \u03b2\n\u03c6 : \u03b1 \u2192 \u03b2\nh\u03c6 : Measurable \u03c6\nf : \u03b2 \u2192 G\nhfm : StronglyMeasurable f\nhG : CompleteSpace G\nhfi : Integrable f\n\u22a2 \u222b (y : \u03b2), f y \u2202Measure.map \u03c6 \u03bc = \u222b (x : \u03b1), f (\u03c6 x) \u2202\u03bc\n\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\nG : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b2 : Type u_7\ninst\u271d : MeasurableSpace \u03b2\n\u03c6 : \u03b1 \u2192 \u03b2\nh\u03c6 : Measurable \u03c6\nf : \u03b2 \u2192 G\nhfm : StronglyMeasurable f\nhG : CompleteSpace G\nhfi : \u00acIntegrable f\n\u22a2 \u222b (y : \u03b2), f y \u2202Measure.map \u03c6 \u03bc = \u222b (x : \u03b1), f (\u03c6 x) \u2202\u03bc", "state_after": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\nG : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b2 : Type u_7\ninst\u271d : MeasurableSpace \u03b2\n\u03c6 : \u03b1 \u2192 \u03b2\nh\u03c6 : Measurable \u03c6\nf : \u03b2 \u2192 G\nhfm : StronglyMeasurable f\nhG : CompleteSpace G\nhfi : \u00acIntegrable f\n\u22a2 \u222b (y : \u03b2), f y \u2202Measure.map \u03c6 \u03bc = \u222b (x : \u03b1), f (\u03c6 x) \u2202\u03bc\n\ncase pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\nG : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b2 : Type u_7\ninst\u271d : MeasurableSpace \u03b2\n\u03c6 : \u03b1 \u2192 \u03b2\nh\u03c6 : Measurable \u03c6\nf : \u03b2 \u2192 G\nhfm : StronglyMeasurable f\nhG : CompleteSpace G\nhfi : Integrable f\n\u22a2 \u222b (y : \u03b2), f y \u2202Measure.map \u03c6 \u03bc = \u222b (x : \u03b1), f (\u03c6 x) \u2202\u03bc"}, {"tactic": "borelize G", "annotated_tactic": ["borelize G", []], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\nG : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b2 : Type u_7\ninst\u271d : MeasurableSpace \u03b2\n\u03c6 : \u03b1 \u2192 \u03b2\nh\u03c6 : Measurable \u03c6\nf : \u03b2 \u2192 G\nhfm : StronglyMeasurable f\nhG : CompleteSpace G\nhfi : Integrable f\n\u22a2 \u222b (y : \u03b2), f y \u2202Measure.map \u03c6 \u03bc = \u222b (x : \u03b1), f (\u03c6 x) \u2202\u03bc", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\nG : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b2 : Type u_7\ninst\u271d : MeasurableSpace \u03b2\n\u03c6 : \u03b1 \u2192 \u03b2\nh\u03c6 : Measurable \u03c6\nf : \u03b2 \u2192 G\nhfm : StronglyMeasurable f\nhG : CompleteSpace G\nhfi : Integrable f\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\n\u22a2 \u222b (y : \u03b2), f y \u2202Measure.map \u03c6 \u03bc = \u222b (x : \u03b1), f (\u03c6 x) \u2202\u03bc"}, {"tactic": "have : SeparableSpace (range f \u222a {0} : Set G) := hfm.separableSpace_range_union_singleton", "annotated_tactic": ["have : <a>SeparableSpace</a> (<a>range</a> f \u222a {0} : <a>Set</a> G) := hfm.separableSpace_range_union_singleton", [{"full_name": "TopologicalSpace.SeparableSpace", "def_path": "Mathlib/Topology/Bases.lean", "def_pos": [313, 17], "def_end_pos": [313, 31]}, {"full_name": "Set.range", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [668, 5], "def_end_pos": [668, 10]}, {"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}]], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\nG : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b2 : Type u_7\ninst\u271d : MeasurableSpace \u03b2\n\u03c6 : \u03b1 \u2192 \u03b2\nh\u03c6 : Measurable \u03c6\nf : \u03b2 \u2192 G\nhfm : StronglyMeasurable f\nhG : CompleteSpace G\nhfi : Integrable f\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\n\u22a2 \u222b (y : \u03b2), f y \u2202Measure.map \u03c6 \u03bc = \u222b (x : \u03b1), f (\u03c6 x) \u2202\u03bc", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\nG : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b2 : Type u_7\ninst\u271d : MeasurableSpace \u03b2\n\u03c6 : \u03b1 \u2192 \u03b2\nh\u03c6 : Measurable \u03c6\nf : \u03b2 \u2192 G\nhfm : StronglyMeasurable f\nhG : CompleteSpace G\nhfi : Integrable f\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nthis : SeparableSpace \u2191(range f \u222a {0})\n\u22a2 \u222b (y : \u03b2), f y \u2202Measure.map \u03c6 \u03bc = \u222b (x : \u03b1), f (\u03c6 x) \u2202\u03bc"}, {"tactic": "refine' tendsto_nhds_unique\n  (tendsto_integral_approxOn_of_measurable_of_range_subset hfm.measurable hfi _ Subset.rfl) _", "annotated_tactic": ["refine' <a>tendsto_nhds_unique</a>\n    (<a>tendsto_integral_approxOn_of_measurable_of_range_subset</a> hfm.measurable hfi _ <a>Subset.rfl</a>) _", [{"full_name": "tendsto_nhds_unique", "def_path": "Mathlib/Topology/Separation.lean", "def_pos": [994, 9], "def_end_pos": [994, 28]}, {"full_name": "MeasureTheory.tendsto_integral_approxOn_of_measurable_of_range_subset", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1440, 9], "def_end_pos": [1440, 64]}, {"full_name": "Set.Subset.rfl", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [357, 9], "def_end_pos": [357, 19]}]], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\nG : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b2 : Type u_7\ninst\u271d : MeasurableSpace \u03b2\n\u03c6 : \u03b1 \u2192 \u03b2\nh\u03c6 : Measurable \u03c6\nf : \u03b2 \u2192 G\nhfm : StronglyMeasurable f\nhG : CompleteSpace G\nhfi : Integrable f\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nthis : SeparableSpace \u2191(range f \u222a {0})\n\u22a2 \u222b (y : \u03b2), f y \u2202Measure.map \u03c6 \u03bc = \u222b (x : \u03b1), f (\u03c6 x) \u2202\u03bc", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\nG : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b2 : Type u_7\ninst\u271d : MeasurableSpace \u03b2\n\u03c6 : \u03b1 \u2192 \u03b2\nh\u03c6 : Measurable \u03c6\nf : \u03b2 \u2192 G\nhfm : StronglyMeasurable f\nhG : CompleteSpace G\nhfi : Integrable f\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nthis : SeparableSpace \u2191(range f \u222a {0})\n\u22a2 Tendsto\n    (fun n =>\n      SimpleFunc.integral (Measure.map \u03c6 \u03bc) (approxOn f (_ : Measurable f) (range f \u222a {0}) 0 (_ : 0 \u2208 range f \u222a {0}) n))\n    atTop (\ud835\udcdd (\u222b (x : \u03b1), f (\u03c6 x) \u2202\u03bc))"}, {"tactic": "convert tendsto_integral_approxOn_of_measurable_of_range_subset (hfm.measurable.comp h\u03c6)\n  ((integrable_map_measure hfm.aestronglyMeasurable h\u03c6.aemeasurable).1 hfi) (range f \u222a {0})\n  (by simp [insert_subset_insert, Set.range_comp_subset_range]) using 1", "annotated_tactic": ["convert <a>tendsto_integral_approxOn_of_measurable_of_range_subset</a> (hfm.measurable.comp h\u03c6)\n    ((<a>integrable_map_measure</a> hfm.aestronglyMeasurable h\u03c6.aemeasurable).1 hfi) (<a>range</a> f \u222a {0})\n    (by simp [<a>insert_subset_insert</a>, <a>Set.range_comp_subset_range</a>]) using 1", [{"full_name": "MeasureTheory.tendsto_integral_approxOn_of_measurable_of_range_subset", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1440, 9], "def_end_pos": [1440, 64]}, {"full_name": "MeasureTheory.integrable_map_measure", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [610, 9], "def_end_pos": [610, 31]}, {"full_name": "Set.range", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [668, 5], "def_end_pos": [668, 10]}, {"full_name": "Set.insert_subset_insert", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1176, 9], "def_end_pos": [1176, 29]}, {"full_name": "Set.range_comp_subset_range", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [763, 9], "def_end_pos": [763, 32]}]], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\nG : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b2 : Type u_7\ninst\u271d : MeasurableSpace \u03b2\n\u03c6 : \u03b1 \u2192 \u03b2\nh\u03c6 : Measurable \u03c6\nf : \u03b2 \u2192 G\nhfm : StronglyMeasurable f\nhG : CompleteSpace G\nhfi : Integrable f\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nthis : SeparableSpace \u2191(range f \u222a {0})\n\u22a2 Tendsto\n    (fun n =>\n      SimpleFunc.integral (Measure.map \u03c6 \u03bc) (approxOn f (_ : Measurable f) (range f \u222a {0}) 0 (_ : 0 \u2208 range f \u222a {0}) n))\n    atTop (\ud835\udcdd (\u222b (x : \u03b1), f (\u03c6 x) \u2202\u03bc))", "state_after": "case h.e'_3\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\nG : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b2 : Type u_7\ninst\u271d : MeasurableSpace \u03b2\n\u03c6 : \u03b1 \u2192 \u03b2\nh\u03c6 : Measurable \u03c6\nf : \u03b2 \u2192 G\nhfm : StronglyMeasurable f\nhG : CompleteSpace G\nhfi : Integrable f\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nthis : SeparableSpace \u2191(range f \u222a {0})\n\u22a2 (fun n =>\n      SimpleFunc.integral (Measure.map \u03c6 \u03bc)\n        (approxOn f (_ : Measurable f) (range f \u222a {0}) 0 (_ : 0 \u2208 range f \u222a {0}) n)) =\n    fun n =>\n    SimpleFunc.integral \u03bc (approxOn (f \u2218 \u03c6) (_ : Measurable (f \u2218 \u03c6)) (range f \u222a {0}) 0 (_ : 0 \u2208 range f \u222a {0}) n)"}, {"tactic": "ext1 i", "annotated_tactic": ["ext1 i", []], "state_before": "case h.e'_3\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\nG : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b2 : Type u_7\ninst\u271d : MeasurableSpace \u03b2\n\u03c6 : \u03b1 \u2192 \u03b2\nh\u03c6 : Measurable \u03c6\nf : \u03b2 \u2192 G\nhfm : StronglyMeasurable f\nhG : CompleteSpace G\nhfi : Integrable f\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nthis : SeparableSpace \u2191(range f \u222a {0})\n\u22a2 (fun n =>\n      SimpleFunc.integral (Measure.map \u03c6 \u03bc)\n        (approxOn f (_ : Measurable f) (range f \u222a {0}) 0 (_ : 0 \u2208 range f \u222a {0}) n)) =\n    fun n =>\n    SimpleFunc.integral \u03bc (approxOn (f \u2218 \u03c6) (_ : Measurable (f \u2218 \u03c6)) (range f \u222a {0}) 0 (_ : 0 \u2208 range f \u222a {0}) n)", "state_after": "case h.e'_3.h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\nG : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b2 : Type u_7\ninst\u271d : MeasurableSpace \u03b2\n\u03c6 : \u03b1 \u2192 \u03b2\nh\u03c6 : Measurable \u03c6\nf : \u03b2 \u2192 G\nhfm : StronglyMeasurable f\nhG : CompleteSpace G\nhfi : Integrable f\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nthis : SeparableSpace \u2191(range f \u222a {0})\ni : \u2115\n\u22a2 SimpleFunc.integral (Measure.map \u03c6 \u03bc) (approxOn f (_ : Measurable f) (range f \u222a {0}) 0 (_ : 0 \u2208 range f \u222a {0}) i) =\n    SimpleFunc.integral \u03bc (approxOn (f \u2218 \u03c6) (_ : Measurable (f \u2218 \u03c6)) (range f \u222a {0}) 0 (_ : 0 \u2208 range f \u222a {0}) i)"}, {"tactic": "simp only [SimpleFunc.approxOn_comp, SimpleFunc.integral_eq, Measure.map_apply, h\u03c6,\n  SimpleFunc.measurableSet_preimage, \u2190 preimage_comp, SimpleFunc.coe_comp]", "annotated_tactic": ["simp only [<a>SimpleFunc.approxOn_comp</a>, <a>SimpleFunc.integral_eq</a>, <a>Measure.map_apply</a>, h\u03c6,\n    <a>SimpleFunc.measurableSet_preimage</a>, \u2190 <a>preimage_comp</a>, <a>SimpleFunc.coe_comp</a>]", [{"full_name": "MeasureTheory.SimpleFunc.approxOn_comp", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDense.lean", "def_pos": [148, 9], "def_end_pos": [148, 22]}, {"full_name": "MeasureTheory.SimpleFunc.integral_eq", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [317, 9], "def_end_pos": [317, 20]}, {"full_name": "MeasureTheory.Measure.map_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1236, 9], "def_end_pos": [1236, 18]}, {"full_name": "MeasureTheory.SimpleFunc.measurableSet_preimage", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [195, 9], "def_end_pos": [195, 31]}, {"full_name": "Set.preimage_comp", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [163, 9], "def_end_pos": [163, 22]}, {"full_name": "MeasureTheory.SimpleFunc.coe_comp", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [337, 9], "def_end_pos": [337, 17]}]], "state_before": "case h.e'_3.h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\nG : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b2 : Type u_7\ninst\u271d : MeasurableSpace \u03b2\n\u03c6 : \u03b1 \u2192 \u03b2\nh\u03c6 : Measurable \u03c6\nf : \u03b2 \u2192 G\nhfm : StronglyMeasurable f\nhG : CompleteSpace G\nhfi : Integrable f\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nthis : SeparableSpace \u2191(range f \u222a {0})\ni : \u2115\n\u22a2 SimpleFunc.integral (Measure.map \u03c6 \u03bc) (approxOn f (_ : Measurable f) (range f \u222a {0}) 0 (_ : 0 \u2208 range f \u222a {0}) i) =\n    SimpleFunc.integral \u03bc (approxOn (f \u2218 \u03c6) (_ : Measurable (f \u2218 \u03c6)) (range f \u222a {0}) 0 (_ : 0 \u2208 range f \u222a {0}) i)", "state_after": "case h.e'_3.h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\nG : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b2 : Type u_7\ninst\u271d : MeasurableSpace \u03b2\n\u03c6 : \u03b1 \u2192 \u03b2\nh\u03c6 : Measurable \u03c6\nf : \u03b2 \u2192 G\nhfm : StronglyMeasurable f\nhG : CompleteSpace G\nhfi : Integrable f\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nthis : SeparableSpace \u2191(range f \u222a {0})\ni : \u2115\n\u22a2 \u2211 x in SimpleFunc.range (approxOn f (_ : Measurable f) (range f \u222a {0}) 0 (_ : 0 \u2208 range f \u222a {0}) i),\n      ENNReal.toReal (\u2191\u2191\u03bc (\u2191(approxOn f (_ : Measurable f) (range f \u222a {0}) 0 (_ : 0 \u2208 range f \u222a {0}) i) \u2218 \u03c6 \u207b\u00b9' {x})) \u2022\n        x =\n    \u2211 x in SimpleFunc.range (approxOn (f \u2218 \u03c6) (_ : Measurable (f \u2218 \u03c6)) (range f \u222a {0}) 0 (_ : 0 \u2208 range f \u222a {0}) i),\n      ENNReal.toReal\n          (\u2191\u2191\u03bc (\u2191(approxOn (f \u2218 \u03c6) (_ : Measurable (f \u2218 \u03c6)) (range f \u222a {0}) 0 (_ : 0 \u2208 range f \u222a {0}) i) \u207b\u00b9' {x})) \u2022\n        x"}, {"tactic": "refine' (Finset.sum_subset (SimpleFunc.range_comp_subset_range _ h\u03c6) fun y _ hy => _).symm", "annotated_tactic": ["refine' (<a>Finset.sum_subset</a> (<a>SimpleFunc.range_comp_subset_range</a> _ h\u03c6) fun y _ hy => _).<a>symm</a>", [{"full_name": "Finset.sum_subset", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [744, 3], "def_end_pos": [744, 14]}, {"full_name": "MeasureTheory.SimpleFunc.range_comp_subset_range", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [342, 9], "def_end_pos": [342, 32]}, {"full_name": "Eq.symm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [310, 9], "def_end_pos": [310, 16]}]], "state_before": "case h.e'_3.h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\nG : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b2 : Type u_7\ninst\u271d : MeasurableSpace \u03b2\n\u03c6 : \u03b1 \u2192 \u03b2\nh\u03c6 : Measurable \u03c6\nf : \u03b2 \u2192 G\nhfm : StronglyMeasurable f\nhG : CompleteSpace G\nhfi : Integrable f\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nthis : SeparableSpace \u2191(range f \u222a {0})\ni : \u2115\n\u22a2 \u2211 x in SimpleFunc.range (approxOn f (_ : Measurable f) (range f \u222a {0}) 0 (_ : 0 \u2208 range f \u222a {0}) i),\n      ENNReal.toReal (\u2191\u2191\u03bc (\u2191(approxOn f (_ : Measurable f) (range f \u222a {0}) 0 (_ : 0 \u2208 range f \u222a {0}) i) \u2218 \u03c6 \u207b\u00b9' {x})) \u2022\n        x =\n    \u2211 x in SimpleFunc.range (approxOn (f \u2218 \u03c6) (_ : Measurable (f \u2218 \u03c6)) (range f \u222a {0}) 0 (_ : 0 \u2208 range f \u222a {0}) i),\n      ENNReal.toReal\n          (\u2191\u2191\u03bc (\u2191(approxOn (f \u2218 \u03c6) (_ : Measurable (f \u2218 \u03c6)) (range f \u222a {0}) 0 (_ : 0 \u2208 range f \u222a {0}) i) \u207b\u00b9' {x})) \u2022\n        x", "state_after": "case h.e'_3.h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\nG : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b2 : Type u_7\ninst\u271d : MeasurableSpace \u03b2\n\u03c6 : \u03b1 \u2192 \u03b2\nh\u03c6 : Measurable \u03c6\nf : \u03b2 \u2192 G\nhfm : StronglyMeasurable f\nhG : CompleteSpace G\nhfi : Integrable f\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nthis : SeparableSpace \u2191(range f \u222a {0})\ni : \u2115\ny : G\nx\u271d : y \u2208 SimpleFunc.range (approxOn f (_ : Measurable f) (range f \u222a {0}) 0 (_ : 0 \u2208 range f \u222a {0}) i)\nhy :\n  \u00acy \u2208\n      SimpleFunc.range\n        (SimpleFunc.comp (approxOn f (_ : Measurable f) (range f \u222a {0}) 0 (_ : 0 \u2208 range f \u222a {0}) i) \u03c6 h\u03c6)\n\u22a2 ENNReal.toReal (\u2191\u2191\u03bc (\u2191(approxOn f (_ : Measurable f) (range f \u222a {0}) 0 (_ : 0 \u2208 range f \u222a {0}) i) \u2218 \u03c6 \u207b\u00b9' {y})) \u2022 y =\n    0"}, {"tactic": "rw [SimpleFunc.mem_range, \u2190 Set.preimage_singleton_eq_empty, SimpleFunc.coe_comp] at hy", "annotated_tactic": ["rw [<a>SimpleFunc.mem_range</a>, \u2190 <a>Set.preimage_singleton_eq_empty</a>, <a>SimpleFunc.coe_comp</a>] at hy", [{"full_name": "MeasureTheory.SimpleFunc.mem_range", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [108, 9], "def_end_pos": [108, 18]}, {"full_name": "Set.preimage_singleton_eq_empty", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [1048, 9], "def_end_pos": [1048, 36]}, {"full_name": "MeasureTheory.SimpleFunc.coe_comp", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [337, 9], "def_end_pos": [337, 17]}]], "state_before": "case h.e'_3.h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\nG : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b2 : Type u_7\ninst\u271d : MeasurableSpace \u03b2\n\u03c6 : \u03b1 \u2192 \u03b2\nh\u03c6 : Measurable \u03c6\nf : \u03b2 \u2192 G\nhfm : StronglyMeasurable f\nhG : CompleteSpace G\nhfi : Integrable f\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nthis : SeparableSpace \u2191(range f \u222a {0})\ni : \u2115\ny : G\nx\u271d : y \u2208 SimpleFunc.range (approxOn f (_ : Measurable f) (range f \u222a {0}) 0 (_ : 0 \u2208 range f \u222a {0}) i)\nhy :\n  \u00acy \u2208\n      SimpleFunc.range\n        (SimpleFunc.comp (approxOn f (_ : Measurable f) (range f \u222a {0}) 0 (_ : 0 \u2208 range f \u222a {0}) i) \u03c6 h\u03c6)\n\u22a2 ENNReal.toReal (\u2191\u2191\u03bc (\u2191(approxOn f (_ : Measurable f) (range f \u222a {0}) 0 (_ : 0 \u2208 range f \u222a {0}) i) \u2218 \u03c6 \u207b\u00b9' {y})) \u2022 y =\n    0", "state_after": "case h.e'_3.h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\nG : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b2 : Type u_7\ninst\u271d : MeasurableSpace \u03b2\n\u03c6 : \u03b1 \u2192 \u03b2\nh\u03c6 : Measurable \u03c6\nf : \u03b2 \u2192 G\nhfm : StronglyMeasurable f\nhG : CompleteSpace G\nhfi : Integrable f\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nthis : SeparableSpace \u2191(range f \u222a {0})\ni : \u2115\ny : G\nx\u271d : y \u2208 SimpleFunc.range (approxOn f (_ : Measurable f) (range f \u222a {0}) 0 (_ : 0 \u2208 range f \u222a {0}) i)\nhy : \u2191(approxOn f (_ : Measurable f) (range f \u222a {0}) 0 (_ : 0 \u2208 range f \u222a {0}) i) \u2218 \u03c6 \u207b\u00b9' {y} = \u2205\n\u22a2 ENNReal.toReal (\u2191\u2191\u03bc (\u2191(approxOn f (_ : Measurable f) (range f \u222a {0}) 0 (_ : 0 \u2208 range f \u222a {0}) i) \u2218 \u03c6 \u207b\u00b9' {y})) \u2022 y =\n    0"}, {"tactic": "rw [hy]", "annotated_tactic": ["rw [hy]", []], "state_before": "case h.e'_3.h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\nG : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b2 : Type u_7\ninst\u271d : MeasurableSpace \u03b2\n\u03c6 : \u03b1 \u2192 \u03b2\nh\u03c6 : Measurable \u03c6\nf : \u03b2 \u2192 G\nhfm : StronglyMeasurable f\nhG : CompleteSpace G\nhfi : Integrable f\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nthis : SeparableSpace \u2191(range f \u222a {0})\ni : \u2115\ny : G\nx\u271d : y \u2208 SimpleFunc.range (approxOn f (_ : Measurable f) (range f \u222a {0}) 0 (_ : 0 \u2208 range f \u222a {0}) i)\nhy : \u2191(approxOn f (_ : Measurable f) (range f \u222a {0}) 0 (_ : 0 \u2208 range f \u222a {0}) i) \u2218 \u03c6 \u207b\u00b9' {y} = \u2205\n\u22a2 ENNReal.toReal (\u2191\u2191\u03bc (\u2191(approxOn f (_ : Measurable f) (range f \u222a {0}) 0 (_ : 0 \u2208 range f \u222a {0}) i) \u2218 \u03c6 \u207b\u00b9' {y})) \u2022 y =\n    0", "state_after": "case h.e'_3.h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\nG : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b2 : Type u_7\ninst\u271d : MeasurableSpace \u03b2\n\u03c6 : \u03b1 \u2192 \u03b2\nh\u03c6 : Measurable \u03c6\nf : \u03b2 \u2192 G\nhfm : StronglyMeasurable f\nhG : CompleteSpace G\nhfi : Integrable f\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nthis : SeparableSpace \u2191(range f \u222a {0})\ni : \u2115\ny : G\nx\u271d : y \u2208 SimpleFunc.range (approxOn f (_ : Measurable f) (range f \u222a {0}) 0 (_ : 0 \u2208 range f \u222a {0}) i)\nhy : \u2191(approxOn f (_ : Measurable f) (range f \u222a {0}) 0 (_ : 0 \u2208 range f \u222a {0}) i) \u2218 \u03c6 \u207b\u00b9' {y} = \u2205\n\u22a2 ENNReal.toReal (\u2191\u2191\u03bc \u2205) \u2022 y = 0"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case h.e'_3.h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\nG : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b2 : Type u_7\ninst\u271d : MeasurableSpace \u03b2\n\u03c6 : \u03b1 \u2192 \u03b2\nh\u03c6 : Measurable \u03c6\nf : \u03b2 \u2192 G\nhfm : StronglyMeasurable f\nhG : CompleteSpace G\nhfi : Integrable f\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nthis : SeparableSpace \u2191(range f \u222a {0})\ni : \u2115\ny : G\nx\u271d : y \u2208 SimpleFunc.range (approxOn f (_ : Measurable f) (range f \u222a {0}) 0 (_ : 0 \u2208 range f \u222a {0}) i)\nhy : \u2191(approxOn f (_ : Measurable f) (range f \u222a {0}) 0 (_ : 0 \u2208 range f \u222a {0}) i) \u2218 \u03c6 \u207b\u00b9' {y} = \u2205\n\u22a2 ENNReal.toReal (\u2191\u2191\u03bc \u2205) \u2022 y = 0", "state_after": "no goals"}, {"tactic": "simp [integral, hG]", "annotated_tactic": ["simp [<a>integral</a>, hG]", [{"full_name": "MeasureTheory.integral", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [791, 17], "def_end_pos": [791, 25]}]], "state_before": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\nG : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b2 : Type u_7\ninst\u271d : MeasurableSpace \u03b2\n\u03c6 : \u03b1 \u2192 \u03b2\nh\u03c6 : Measurable \u03c6\nf : \u03b2 \u2192 G\nhfm : StronglyMeasurable f\nhG : \u00acCompleteSpace G\n\u22a2 \u222b (y : \u03b2), f y \u2202Measure.map \u03c6 \u03bc = \u222b (x : \u03b1), f (\u03c6 x) \u2202\u03bc", "state_after": "no goals"}, {"tactic": "rw [integral_undef hfi, integral_undef]", "annotated_tactic": ["rw [<a>integral_undef</a> hfi, <a>integral_undef</a>]", [{"full_name": "MeasureTheory.integral_undef", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [836, 9], "def_end_pos": [836, 23]}, {"full_name": "MeasureTheory.integral_undef", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [836, 9], "def_end_pos": [836, 23]}]], "state_before": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\nG : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b2 : Type u_7\ninst\u271d : MeasurableSpace \u03b2\n\u03c6 : \u03b1 \u2192 \u03b2\nh\u03c6 : Measurable \u03c6\nf : \u03b2 \u2192 G\nhfm : StronglyMeasurable f\nhG : CompleteSpace G\nhfi : \u00acIntegrable f\n\u22a2 \u222b (y : \u03b2), f y \u2202Measure.map \u03c6 \u03bc = \u222b (x : \u03b1), f (\u03c6 x) \u2202\u03bc", "state_after": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\nG : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b2 : Type u_7\ninst\u271d : MeasurableSpace \u03b2\n\u03c6 : \u03b1 \u2192 \u03b2\nh\u03c6 : Measurable \u03c6\nf : \u03b2 \u2192 G\nhfm : StronglyMeasurable f\nhG : CompleteSpace G\nhfi : \u00acIntegrable f\n\u22a2 \u00acIntegrable fun x => f (\u03c6 x)"}, {"tactic": "exact fun hf\u03c6 => hfi ((integrable_map_measure hfm.aestronglyMeasurable h\u03c6.aemeasurable).2 hf\u03c6)", "annotated_tactic": ["exact fun hf\u03c6 => hfi ((<a>integrable_map_measure</a> hfm.aestronglyMeasurable h\u03c6.aemeasurable).2 hf\u03c6)", [{"full_name": "MeasureTheory.integrable_map_measure", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [610, 9], "def_end_pos": [610, 31]}]], "state_before": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\nG : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b2 : Type u_7\ninst\u271d : MeasurableSpace \u03b2\n\u03c6 : \u03b1 \u2192 \u03b2\nh\u03c6 : Measurable \u03c6\nf : \u03b2 \u2192 G\nhfm : StronglyMeasurable f\nhG : CompleteSpace G\nhfi : \u00acIntegrable f\n\u22a2 \u00acIntegrable fun x => f (\u03c6 x)", "state_after": "no goals"}, {"tactic": "simp [insert_subset_insert, Set.range_comp_subset_range]", "annotated_tactic": ["simp [<a>insert_subset_insert</a>, <a>Set.range_comp_subset_range</a>]", [{"full_name": "Set.insert_subset_insert", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1176, 9], "def_end_pos": [1176, 29]}, {"full_name": "Set.range_comp_subset_range", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [763, 9], "def_end_pos": [763, 32]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\nG : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b2 : Type u_7\ninst\u271d : MeasurableSpace \u03b2\n\u03c6 : \u03b1 \u2192 \u03b2\nh\u03c6 : Measurable \u03c6\nf : \u03b2 \u2192 G\nhfm : StronglyMeasurable f\nhG : CompleteSpace G\nhfi : Integrable f\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nthis : SeparableSpace \u2191(range f \u222a {0})\n\u22a2 range (f \u2218 \u03c6) \u222a {0} \u2286 range f \u222a {0}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/ZMod/Basic.lean", "full_name": "ZMod.ker_int_castAddHom", "start": [560, 1], "end": [564, 35], "traced_tactics": [{"tactic": "ext", "annotated_tactic": ["ext", []], "state_before": "n : \u2115\n\u22a2 AddMonoidHom.ker (Int.castAddHom (ZMod n)) = AddSubgroup.zmultiples \u2191n", "state_after": "case h\nn : \u2115\nx\u271d : \u2124\n\u22a2 x\u271d \u2208 AddMonoidHom.ker (Int.castAddHom (ZMod n)) \u2194 x\u271d \u2208 AddSubgroup.zmultiples \u2191n"}, {"tactic": "rw [Int.mem_zmultiples_iff, AddMonoidHom.mem_ker, Int.coe_castAddHom,\n  int_cast_zmod_eq_zero_iff_dvd]", "annotated_tactic": ["rw [<a>Int.mem_zmultiples_iff</a>, <a>AddMonoidHom.mem_ker</a>, <a>Int.coe_castAddHom</a>,\n    <a>int_cast_zmod_eq_zero_iff_dvd</a>]", [{"full_name": "Int.mem_zmultiples_iff", "def_path": "Mathlib/GroupTheory/Subgroup/ZPowers.lean", "def_pos": [163, 9], "def_end_pos": [163, 31]}, {"full_name": "AddMonoidHom.mem_ker", "def_path": "Mathlib/GroupTheory/Subgroup/Basic.lean", "def_pos": [2822, 3], "def_end_pos": [2822, 14]}, {"full_name": "Int.coe_castAddHom", "def_path": "Mathlib/Data/Int/Cast/Lemmas.lean", "def_pos": [73, 9], "def_end_pos": [73, 23]}, {"full_name": "ZMod.int_cast_zmod_eq_zero_iff_dvd", "def_path": "Mathlib/Data/ZMod/Basic.lean", "def_pos": [477, 9], "def_end_pos": [477, 38]}]], "state_before": "case h\nn : \u2115\nx\u271d : \u2124\n\u22a2 x\u271d \u2208 AddMonoidHom.ker (Int.castAddHom (ZMod n)) \u2194 x\u271d \u2208 AddSubgroup.zmultiples \u2191n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Pointwise.lean", "full_name": "Finset.smul_univ", "start": [2012, 1], "end": [2015, 27], "traced_tactics": [{"tactic": "push_cast", "annotated_tactic": ["push_cast", []], "state_before": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b3 : DecidableEq \u03b2\ninst\u271d\u00b2 : Group \u03b1\ninst\u271d\u00b9 : MulAction \u03b1 \u03b2\ns\u271d t : Finset \u03b2\na : \u03b1\nb : \u03b2\ninst\u271d : Fintype \u03b2\ns : Finset \u03b1\nhs : Finset.Nonempty s\n\u22a2 \u2191(s \u2022 univ) = \u2191univ", "state_after": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b3 : DecidableEq \u03b2\ninst\u271d\u00b2 : Group \u03b1\ninst\u271d\u00b9 : MulAction \u03b1 \u03b2\ns\u271d t : Finset \u03b2\na : \u03b1\nb : \u03b2\ninst\u271d : Fintype \u03b2\ns : Finset \u03b1\nhs : Finset.Nonempty s\n\u22a2 \u2191s \u2022 Set.univ = Set.univ"}, {"tactic": "exact Set.smul_univ hs", "annotated_tactic": ["exact <a>Set.smul_univ</a> hs", [{"full_name": "Set.smul_univ", "def_path": "Mathlib/Data/Set/Pointwise/SMul.lean", "def_pos": [956, 9], "def_end_pos": [956, 18]}]], "state_before": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b3 : DecidableEq \u03b2\ninst\u271d\u00b2 : Group \u03b1\ninst\u271d\u00b9 : MulAction \u03b1 \u03b2\ns\u271d t : Finset \u03b2\na : \u03b1\nb : \u03b2\ninst\u271d : Fintype \u03b2\ns : Finset \u03b1\nhs : Finset.Nonempty s\n\u22a2 \u2191s \u2022 Set.univ = Set.univ", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/LocallyFinite.lean", "full_name": "Finset.Ioo_subset_Iic_self", "start": [461, 1], "end": [462, 48], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/OpenPos.lean", "full_name": "MeasureTheory.Measure.eqOn_Ico_of_ae_eq", "start": [197, 1], "end": [200, 60], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/Jacobian.lean", "full_name": "MeasureTheory.integral_image_eq_integral_abs_det_fderiv_smul", "start": [1207, 1], "end": [1218, 65], "traced_tactics": [{"tactic": "rw [\u2190 restrict_map_withDensity_abs_det_fderiv_eq_addHaar \u03bc hs hf' hf,\n  (measurableEmbedding_of_fderivWithin hs hf' hf).integral_map]", "annotated_tactic": ["rw [\u2190 <a>restrict_map_withDensity_abs_det_fderiv_eq_addHaar</a> \u03bc hs hf' hf,\n    (<a>measurableEmbedding_of_fderivWithin</a> hs hf' hf).<a>integral_map</a>]", [{"full_name": "MeasureTheory.restrict_map_withDensity_abs_det_fderiv_eq_addHaar", "def_path": "Mathlib/MeasureTheory/Function/Jacobian.lean", "def_pos": [1137, 9], "def_end_pos": [1137, 59]}, {"full_name": "MeasureTheory.measurableEmbedding_of_fderivWithin", "def_path": "Mathlib/MeasureTheory/Function/Jacobian.lean", "def_pos": [786, 9], "def_end_pos": [786, 44]}, {"full_name": "MeasurableEmbedding.integral_map", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1623, 9], "def_end_pos": [1623, 48]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\ng : E \u2192 F\n\u22a2 \u222b (x : E) in f '' s, g x \u2202\u03bc = \u222b (x : E) in s, |ContinuousLinearMap.det (f' x)| \u2022 g (f x) \u2202\u03bc", "state_after": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\ng : E \u2192 F\n\u22a2 \u222b (x : \u2191s),\n      g\n        (Set.restrict s f\n          x) \u2202Measure.comap Subtype.val (withDensity \u03bc fun x => ENNReal.ofReal |ContinuousLinearMap.det (f' x)|) =\n    \u222b (x : E) in s, |ContinuousLinearMap.det (f' x)| \u2022 g (f x) \u2202\u03bc"}, {"tactic": "have : \u2200 x : s, g (s.restrict f x) = (g \u2218 f) x := fun x => rfl", "annotated_tactic": ["have : \u2200 x : s, g (s.restrict f x) = (g \u2218 f) x := fun x => <a>rfl</a>", [{"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\ng : E \u2192 F\n\u22a2 \u222b (x : \u2191s),\n      g\n        (Set.restrict s f\n          x) \u2202Measure.comap Subtype.val (withDensity \u03bc fun x => ENNReal.ofReal |ContinuousLinearMap.det (f' x)|) =\n    \u222b (x : E) in s, |ContinuousLinearMap.det (f' x)| \u2022 g (f x) \u2202\u03bc", "state_after": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\ng : E \u2192 F\nthis : \u2200 (x : \u2191s), g (Set.restrict s f x) = (g \u2218 f) \u2191x\n\u22a2 \u222b (x : \u2191s),\n      g\n        (Set.restrict s f\n          x) \u2202Measure.comap Subtype.val (withDensity \u03bc fun x => ENNReal.ofReal |ContinuousLinearMap.det (f' x)|) =\n    \u222b (x : E) in s, |ContinuousLinearMap.det (f' x)| \u2022 g (f x) \u2202\u03bc"}, {"tactic": "simp only [this, ENNReal.ofReal]", "annotated_tactic": ["simp only [this, <a>ENNReal.ofReal</a>]", [{"full_name": "ENNReal.ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [172, 29], "def_end_pos": [172, 35]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\ng : E \u2192 F\nthis : \u2200 (x : \u2191s), g (Set.restrict s f x) = (g \u2218 f) \u2191x\n\u22a2 \u222b (x : \u2191s),\n      g\n        (Set.restrict s f\n          x) \u2202Measure.comap Subtype.val (withDensity \u03bc fun x => ENNReal.ofReal |ContinuousLinearMap.det (f' x)|) =\n    \u222b (x : E) in s, |ContinuousLinearMap.det (f' x)| \u2022 g (f x) \u2202\u03bc", "state_after": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\ng : E \u2192 F\nthis : \u2200 (x : \u2191s), g (Set.restrict s f x) = (g \u2218 f) \u2191x\n\u22a2 \u222b (x : \u2191s),\n      (g \u2218 f) \u2191x \u2202Measure.comap Subtype.val (withDensity \u03bc fun x => \u2191(Real.toNNReal |ContinuousLinearMap.det (f' x)|)) =\n    \u222b (x : E) in s, |ContinuousLinearMap.det (f' x)| \u2022 g (f x) \u2202\u03bc"}, {"tactic": "rw [\u2190 (MeasurableEmbedding.subtype_coe hs).integral_map, map_comap_subtype_coe hs,\n  set_integral_withDensity_eq_set_integral_smul\u2080\n    (aemeasurable_toNNReal_abs_det_fderivWithin \u03bc hs hf') _ hs]", "annotated_tactic": ["rw [\u2190 (<a>MeasurableEmbedding.subtype_coe</a> hs).<a>integral_map</a>, <a>map_comap_subtype_coe</a> hs,\n    <a>set_integral_withDensity_eq_set_integral_smul\u2080</a>\n      (<a>aemeasurable_toNNReal_abs_det_fderivWithin</a> \u03bc hs hf') _ hs]", [{"full_name": "MeasurableEmbedding.subtype_coe", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [1208, 9], "def_end_pos": [1208, 20]}, {"full_name": "MeasurableEmbedding.integral_map", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1623, 9], "def_end_pos": [1623, 48]}, {"full_name": "map_comap_subtype_coe", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [4159, 9], "def_end_pos": [4159, 30]}, {"full_name": "set_integral_withDensity_eq_set_integral_smul\u2080", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [1339, 9], "def_end_pos": [1339, 55]}, {"full_name": "MeasureTheory.aemeasurable_toNNReal_abs_det_fderivWithin", "def_path": "Mathlib/MeasureTheory/Function/Jacobian.lean", "def_pos": [767, 9], "def_end_pos": [767, 51]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\ng : E \u2192 F\nthis : \u2200 (x : \u2191s), g (Set.restrict s f x) = (g \u2218 f) \u2191x\n\u22a2 \u222b (x : \u2191s),\n      (g \u2218 f) \u2191x \u2202Measure.comap Subtype.val (withDensity \u03bc fun x => \u2191(Real.toNNReal |ContinuousLinearMap.det (f' x)|)) =\n    \u222b (x : E) in s, |ContinuousLinearMap.det (f' x)| \u2022 g (f x) \u2202\u03bc", "state_after": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\ng : E \u2192 F\nthis : \u2200 (x : \u2191s), g (Set.restrict s f x) = (g \u2218 f) \u2191x\n\u22a2 \u222b (a : E) in s, Real.toNNReal |ContinuousLinearMap.det (f' a)| \u2022 (g \u2218 f) a \u2202\u03bc =\n    \u222b (x : E) in s, |ContinuousLinearMap.det (f' x)| \u2022 g (f x) \u2202\u03bc"}, {"tactic": "congr with x", "annotated_tactic": ["congr with x", []], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\ng : E \u2192 F\nthis : \u2200 (x : \u2191s), g (Set.restrict s f x) = (g \u2218 f) \u2191x\n\u22a2 \u222b (a : E) in s, Real.toNNReal |ContinuousLinearMap.det (f' a)| \u2022 (g \u2218 f) a \u2202\u03bc =\n    \u222b (x : E) in s, |ContinuousLinearMap.det (f' x)| \u2022 g (f x) \u2202\u03bc", "state_after": "case e_f.h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\ng : E \u2192 F\nthis : \u2200 (x : \u2191s), g (Set.restrict s f x) = (g \u2218 f) \u2191x\nx : E\n\u22a2 Real.toNNReal |ContinuousLinearMap.det (f' x)| \u2022 (g \u2218 f) x = |ContinuousLinearMap.det (f' x)| \u2022 g (f x)"}, {"tactic": "conv_rhs => rw [\u2190 Real.coe_toNNReal _ (abs_nonneg (f' x).det)]", "annotated_tactic": ["conv_rhs => rw [\u2190 <a>Real.coe_toNNReal</a> _ (<a>abs_nonneg</a> (f' x).<a>det</a>)]", [{"full_name": "Real.coe_toNNReal", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [122, 9], "def_end_pos": [122, 33]}, {"full_name": "abs_nonneg", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [169, 9], "def_end_pos": [169, 19]}, {"full_name": "ContinuousLinearMap.det", "def_path": "Mathlib/Topology/Algebra/Module/Determinant.lean", "def_pos": [22, 19], "def_end_pos": [22, 22]}]], "state_before": "case e_f.h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\ng : E \u2192 F\nthis : \u2200 (x : \u2191s), g (Set.restrict s f x) = (g \u2218 f) \u2191x\nx : E\n\u22a2 Real.toNNReal |ContinuousLinearMap.det (f' x)| \u2022 (g \u2218 f) x = |ContinuousLinearMap.det (f' x)| \u2022 g (f x)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/ZMod/Basic.lean", "full_name": "ZMod.valMinAbs_zero", "start": [1044, 1], "end": [1046, 90], "traced_tactics": [{"tactic": "simp only [valMinAbs_def_zero]", "annotated_tactic": ["simp only [<a>valMinAbs_def_zero</a>]", [{"full_name": "ZMod.valMinAbs_def_zero", "def_path": "Mathlib/Data/ZMod/Basic.lean", "def_pos": [959, 9], "def_end_pos": [959, 27]}]], "state_before": "\u22a2 valMinAbs 0 = 0", "state_after": "no goals"}, {"tactic": "simp only [valMinAbs_def_pos, if_true, Int.ofNat_zero, zero_le, val_zero]", "annotated_tactic": ["simp only [<a>valMinAbs_def_pos</a>, <a>if_true</a>, <a>Int.ofNat_zero</a>, <a>zero_le</a>, <a>val_zero</a>]", [{"full_name": "ZMod.valMinAbs_def_pos", "def_path": "Mathlib/Data/ZMod/Basic.lean", "def_pos": [963, 9], "def_end_pos": [963, 26]}, {"full_name": "if_true", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [727, 17], "def_end_pos": [727, 24]}, {"full_name": "Int.ofNat_zero", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [19, 17], "def_end_pos": [19, 27]}, {"full_name": "zero_le", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [217, 30], "def_end_pos": [217, 37]}, {"full_name": "ZMod.val_zero", "def_path": "Mathlib/Data/ZMod/Basic.lean", "def_pos": [63, 9], "def_end_pos": [63, 17]}]], "state_before": "n : \u2115\n\u22a2 valMinAbs 0 = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Kernel/CondCdf.lean", "full_name": "ProbabilityTheory.tendsto_preCdf_atBot_zero", "start": [443, 1], "end": [508, 53], "traced_tactics": [{"tactic": "suffices \u2200\u1d50 a \u2202\u03c1.fst, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd 0) by\n  filter_upwards [this] with a ha\n  have h_eq_neg : (fun r : \u211a => preCdf \u03c1 r a) = fun r : \u211a => preCdf \u03c1 (- -r) a := by\n    simp_rw [neg_neg]\n  rw [h_eq_neg]\n  exact ha.comp tendsto_neg_atBot_atTop", "annotated_tactic": ["suffices \u2200\u1d50 a \u2202\u03c1.fst, <a>Tendsto</a> (fun r => <a>preCdf</a> \u03c1 (-r) a) <a>atTop</a> (\ud835\udcdd 0) by\n    filter_upwards [this] with a ha\n    have h_eq_neg : (fun r : \u211a => <a>preCdf</a> \u03c1 r a) = fun r : \u211a => <a>preCdf</a> \u03c1 (- -r) a := by\n      simp_rw [<a>neg_neg</a>]\n    rw [h_eq_neg]\n    exact ha.comp <a>tendsto_neg_atBot_atTop</a>", [{"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2939, 5], "def_end_pos": [2939, 12]}, {"full_name": "ProbabilityTheory.preCdf", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [294, 19], "def_end_pos": [294, 25]}, {"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}, {"full_name": "ProbabilityTheory.preCdf", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [294, 19], "def_end_pos": [294, 25]}, {"full_name": "ProbabilityTheory.preCdf", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [294, 19], "def_end_pos": [294, 25]}, {"full_name": "neg_neg", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [799, 3], "def_end_pos": [799, 14]}, {"full_name": "Filter.tendsto_neg_atBot_atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [897, 9], "def_end_pos": [897, 32]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atBot (\ud835\udcdd 0)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd 0)"}, {"tactic": "filter_upwards [this] with a ha", "annotated_tactic": ["filter_upwards [this] with a ha", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nthis : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd 0)\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atBot (\ud835\udcdd 0)", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nthis : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd 0)\na : \u03b1\nha : Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd 0)\n\u22a2 Tendsto (fun r => preCdf \u03c1 r a) atBot (\ud835\udcdd 0)"}, {"tactic": "have h_eq_neg : (fun r : \u211a => preCdf \u03c1 r a) = fun r : \u211a => preCdf \u03c1 (- -r) a := by\n  simp_rw [neg_neg]", "annotated_tactic": ["have h_eq_neg : (fun r : \u211a => <a>preCdf</a> \u03c1 r a) = fun r : \u211a => <a>preCdf</a> \u03c1 (- -r) a := by\n      simp_rw [<a>neg_neg</a>]", [{"full_name": "ProbabilityTheory.preCdf", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [294, 19], "def_end_pos": [294, 25]}, {"full_name": "ProbabilityTheory.preCdf", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [294, 19], "def_end_pos": [294, 25]}, {"full_name": "neg_neg", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [799, 3], "def_end_pos": [799, 14]}]], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nthis : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd 0)\na : \u03b1\nha : Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd 0)\n\u22a2 Tendsto (fun r => preCdf \u03c1 r a) atBot (\ud835\udcdd 0)", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nthis : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd 0)\na : \u03b1\nha : Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd 0)\nh_eq_neg : (fun r => preCdf \u03c1 r a) = fun r => preCdf \u03c1 (- -r) a\n\u22a2 Tendsto (fun r => preCdf \u03c1 r a) atBot (\ud835\udcdd 0)"}, {"tactic": "rw [h_eq_neg]", "annotated_tactic": ["rw [h_eq_neg]", []], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nthis : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd 0)\na : \u03b1\nha : Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd 0)\nh_eq_neg : (fun r => preCdf \u03c1 r a) = fun r => preCdf \u03c1 (- -r) a\n\u22a2 Tendsto (fun r => preCdf \u03c1 r a) atBot (\ud835\udcdd 0)", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nthis : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd 0)\na : \u03b1\nha : Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd 0)\nh_eq_neg : (fun r => preCdf \u03c1 r a) = fun r => preCdf \u03c1 (- -r) a\n\u22a2 Tendsto (fun r => preCdf \u03c1 (- -r) a) atBot (\ud835\udcdd 0)"}, {"tactic": "exact ha.comp tendsto_neg_atBot_atTop", "annotated_tactic": ["exact ha.comp <a>tendsto_neg_atBot_atTop</a>", [{"full_name": "Filter.tendsto_neg_atBot_atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [897, 9], "def_end_pos": [897, 32]}]], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nthis : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd 0)\na : \u03b1\nha : Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd 0)\nh_eq_neg : (fun r => preCdf \u03c1 r a) = fun r => preCdf \u03c1 (- -r) a\n\u22a2 Tendsto (fun r => preCdf \u03c1 (- -r) a) atBot (\ud835\udcdd 0)", "state_after": "no goals"}, {"tactic": "simp_rw [neg_neg]", "annotated_tactic": ["simp_rw [<a>neg_neg</a>]", [{"full_name": "neg_neg", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [799, 3], "def_end_pos": [799, 14]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nthis : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd 0)\na : \u03b1\nha : Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd 0)\n\u22a2 (fun r => preCdf \u03c1 r a) = fun r => preCdf \u03c1 (- -r) a", "state_after": "no goals"}, {"tactic": "filter_upwards [monotone_preCdf \u03c1] with a ha", "annotated_tactic": ["filter_upwards [<a>monotone_preCdf</a> \u03c1] with a ha", [{"full_name": "ProbabilityTheory.monotone_preCdf", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [318, 9], "def_end_pos": [318, 24]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\na : \u03b1\nha : Monotone fun r => preCdf \u03c1 r a\n\u22a2 \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)"}, {"tactic": "have h_anti : Antitone fun r => preCdf \u03c1 (-r) a := fun p q hpq => ha (neg_le_neg hpq)", "annotated_tactic": ["have h_anti : <a>Antitone</a> fun r => <a>preCdf</a> \u03c1 (-r) a := fun p q hpq => ha (<a>neg_le_neg</a> hpq)", [{"full_name": "Antitone", "def_path": "Mathlib/Order/Monotone/Basic.lean", "def_pos": [82, 5], "def_end_pos": [82, 13]}, {"full_name": "ProbabilityTheory.preCdf", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [294, 19], "def_end_pos": [294, 25]}, {"full_name": "neg_le_neg", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [1238, 15], "def_end_pos": [1238, 25]}]], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\na : \u03b1\nha : Monotone fun r => preCdf \u03c1 r a\n\u22a2 \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\na : \u03b1\nha : Monotone fun r => preCdf \u03c1 r a\nh_anti : Antitone fun r => preCdf \u03c1 (-r) a\n\u22a2 \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)"}, {"tactic": "have h_tendsto :\n  Tendsto (fun r => preCdf \u03c1 (-r) a) atTop atBot \u2228\n    \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) :=\n  tendsto_of_antitone h_anti", "annotated_tactic": ["have h_tendsto :\n      <a>Tendsto</a> (fun r => <a>preCdf</a> \u03c1 (-r) a) <a>atTop</a> <a>atBot</a> \u2228\n        \u2203 l, <a>Tendsto</a> (fun r => <a>preCdf</a> \u03c1 (-r) a) <a>atTop</a> (\ud835\udcdd l) :=\n      <a>tendsto_of_antitone</a> h_anti", [{"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2939, 5], "def_end_pos": [2939, 12]}, {"full_name": "ProbabilityTheory.preCdf", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [294, 19], "def_end_pos": [294, 25]}, {"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}, {"full_name": "Filter.atBot", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [49, 5], "def_end_pos": [49, 10]}, {"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2939, 5], "def_end_pos": [2939, 12]}, {"full_name": "ProbabilityTheory.preCdf", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [294, 19], "def_end_pos": [294, 25]}, {"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}, {"full_name": "tendsto_of_antitone", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [112, 9], "def_end_pos": [112, 28]}]], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\na : \u03b1\nha : Monotone fun r => preCdf \u03c1 r a\nh_anti : Antitone fun r => preCdf \u03c1 (-r) a\n\u22a2 \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\na : \u03b1\nha : Monotone fun r => preCdf \u03c1 r a\nh_anti : Antitone fun r => preCdf \u03c1 (-r) a\nh_tendsto : Tendsto (fun r => preCdf \u03c1 (-r) a) atTop atBot \u2228 \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\n\u22a2 \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)"}, {"tactic": "cases' h_tendsto with h_bot h_tendsto", "annotated_tactic": ["cases' h_tendsto with h_bot h_tendsto", []], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\na : \u03b1\nha : Monotone fun r => preCdf \u03c1 r a\nh_anti : Antitone fun r => preCdf \u03c1 (-r) a\nh_tendsto : Tendsto (fun r => preCdf \u03c1 (-r) a) atTop atBot \u2228 \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\n\u22a2 \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)", "state_after": "case h.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\na : \u03b1\nha : Monotone fun r => preCdf \u03c1 r a\nh_anti : Antitone fun r => preCdf \u03c1 (-r) a\nh_bot : Tendsto (fun r => preCdf \u03c1 (-r) a) atTop atBot\n\u22a2 \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\n\ncase h.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\na : \u03b1\nha : Monotone fun r => preCdf \u03c1 r a\nh_anti : Antitone fun r => preCdf \u03c1 (-r) a\nh_tendsto : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\n\u22a2 \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)"}, {"tactic": "exact \u27e80, Tendsto.mono_right h_bot atBot_le_nhds_bot\u27e9", "annotated_tactic": ["exact \u27e80, <a>Tendsto.mono_right</a> h_bot <a>atBot_le_nhds_bot</a>\u27e9", [{"full_name": "Filter.Tendsto.mono_right", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [3041, 9], "def_end_pos": [3041, 27]}, {"full_name": "atBot_le_nhds_bot", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [91, 9], "def_end_pos": [91, 26]}]], "state_before": "case h.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\na : \u03b1\nha : Monotone fun r => preCdf \u03c1 r a\nh_anti : Antitone fun r => preCdf \u03c1 (-r) a\nh_bot : Tendsto (fun r => preCdf \u03c1 (-r) a) atTop atBot\n\u22a2 \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)", "state_after": "no goals"}, {"tactic": "exact h_tendsto", "annotated_tactic": ["exact h_tendsto", []], "state_before": "case h.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\na : \u03b1\nha : Monotone fun r => preCdf \u03c1 r a\nh_anti : Antitone fun r => preCdf \u03c1 (-r) a\nh_tendsto : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\n\u22a2 \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)", "state_after": "no goals"}, {"tactic": "let F : \u03b1 \u2192 \u211d\u22650\u221e := fun a =>\n  if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then h.choose else 0", "annotated_tactic": ["let F : \u03b1 \u2192 \u211d\u22650\u221e := fun a =>\n    if h : \u2203 l, <a>Tendsto</a> (fun r => <a>preCdf</a> \u03c1 (-r) a) <a>atTop</a> (\ud835\udcdd l) then h.choose else 0", [{"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2939, 5], "def_end_pos": [2939, 12]}, {"full_name": "ProbabilityTheory.preCdf", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [294, 19], "def_end_pos": [294, 25]}, {"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd 0)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd 0)"}, {"tactic": "have h_tendsto : \u2200\u1d50 a \u2202\u03c1.fst, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a)) := by\n  filter_upwards [h_exists] with a ha\n  simp_rw [dif_pos ha]\n  exact ha.choose_spec", "annotated_tactic": ["have h_tendsto : \u2200\u1d50 a \u2202\u03c1.fst, <a>Tendsto</a> (fun r => <a>preCdf</a> \u03c1 (-r) a) <a>atTop</a> (\ud835\udcdd (F a)) := by\n    filter_upwards [h_exists] with a ha\n    simp_rw [<a>dif_pos</a> ha]\n    exact ha.choose_spec", [{"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2939, 5], "def_end_pos": [2939, 12]}, {"full_name": "ProbabilityTheory.preCdf", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [294, 19], "def_end_pos": [294, 25]}, {"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}, {"full_name": "dif_pos", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [807, 9], "def_end_pos": [807, 16]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd 0)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a))\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd 0)"}, {"tactic": "suffices h_lintegral_eq : \u222b\u207b a, F a \u2202\u03c1.fst = 0", "annotated_tactic": ["suffices h_lintegral_eq : \u222b\u207b a, F a \u2202\u03c1.fst = 0", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a))\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd 0)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a))\nh_lintegral_eq : \u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1 = 0\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd 0)\n\ncase h_lintegral_eq\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a))\n\u22a2 \u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1 = 0"}, {"tactic": "have h_lintegral' : Tendsto (fun r => \u222b\u207b a, preCdf \u03c1 (-r) a \u2202\u03c1.fst) atTop (\ud835\udcdd 0) := by\n  have h_lintegral_eq :\n    (fun r => \u222b\u207b a, preCdf \u03c1 (-r) a \u2202\u03c1.fst) = fun r : \u211a => \u03c1 (univ \u00d7\u02e2 Iic (-r : \u211d)) := by\n    ext1 n\n    rw [\u2190 set_lintegral_univ, set_lintegral_preCdf_fst \u03c1 _ MeasurableSet.univ,\n      Measure.IicSnd_univ]\n    norm_cast\n  rw [h_lintegral_eq]\n  have h_zero_eq_measure_iInter : (0 : \u211d\u22650\u221e) = \u03c1 (\u22c2 r : \u211a, univ \u00d7\u02e2 Iic (-r : \u211d)) := by\n    suffices \u22c2 r : \u211a, Iic (-(r : \u211d)) = \u2205 by rw [\u2190 prod_iInter, this, prod_empty, measure_empty]\n    ext1 x\n    simp only [mem_iInter, mem_Iic, mem_empty_iff_false, iff_false_iff, not_forall, not_le]\n    simp_rw [neg_lt]\n    exact exists_rat_gt _\n  rw [h_zero_eq_measure_iInter]\n  refine'\n    tendsto_measure_iInter (fun n => MeasurableSet.univ.prod measurableSet_Iic)\n      (fun i j hij x => _) \u27e80, measure_ne_top \u03c1 _\u27e9\n  simp only [mem_prod, mem_univ, mem_Iic, true_and_iff]\n  refine' fun hxj => hxj.trans (neg_le_neg _)\n  exact_mod_cast hij", "annotated_tactic": ["have h_lintegral' : <a>Tendsto</a> (fun r => \u222b\u207b a, <a>preCdf</a> \u03c1 (-r) a \u2202\u03c1.fst) <a>atTop</a> (\ud835\udcdd 0) := by\n    have h_lintegral_eq :\n      (fun r => \u222b\u207b a, <a>preCdf</a> \u03c1 (-r) a \u2202\u03c1.fst) = fun r : \u211a => \u03c1 (<a>univ</a> \u00d7\u02e2 <a>Iic</a> (-r : \u211d)) := by\n      ext1 n\n      rw [\u2190 <a>set_lintegral_univ</a>, <a>set_lintegral_preCdf_fst</a> \u03c1 _ <a>MeasurableSet.univ</a>,\n        <a>Measure.IicSnd_univ</a>]\n      norm_cast\n    rw [h_lintegral_eq]\n    have h_zero_eq_measure_iInter : (0 : \u211d\u22650\u221e) = \u03c1 (\u22c2 r : \u211a, <a>univ</a> \u00d7\u02e2 <a>Iic</a> (-r : \u211d)) := by\n      suffices \u22c2 r : \u211a, <a>Iic</a> (-(r : \u211d)) = \u2205 by rw [\u2190 <a>prod_iInter</a>, this, <a>prod_empty</a>, <a>measure_empty</a>]\n      ext1 x\n      simp only [<a>mem_iInter</a>, <a>mem_Iic</a>, <a>mem_empty_iff_false</a>, <a>iff_false_iff</a>, <a>not_forall</a>, <a>not_le</a>]\n      simp_rw [<a>neg_lt</a>]\n      exact <a>exists_rat_gt</a> _\n    rw [h_zero_eq_measure_iInter]\n    refine'\n      <a>tendsto_measure_iInter</a> (fun n => MeasurableSet.univ.prod <a>measurableSet_Iic</a>)\n        (fun i j hij x => _) \u27e80, <a>measure_ne_top</a> \u03c1 _\u27e9\n    simp only [<a>mem_prod</a>, <a>mem_univ</a>, <a>mem_Iic</a>, <a>true_and_iff</a>]\n    refine' fun hxj => hxj.trans (<a>neg_le_neg</a> _)\n    exact_mod_cast hij", [{"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2939, 5], "def_end_pos": [2939, 12]}, {"full_name": "ProbabilityTheory.preCdf", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [294, 19], "def_end_pos": [294, 25]}, {"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}, {"full_name": "ProbabilityTheory.preCdf", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [294, 19], "def_end_pos": [294, 25]}, {"full_name": "Set.univ", "def_path": "Mathlib/Init/Set.lean", "def_pos": [90, 5], "def_end_pos": [90, 9]}, {"full_name": "Set.Iic", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [64, 5], "def_end_pos": [64, 8]}, {"full_name": "MeasureTheory.set_lintegral_univ", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [645, 9], "def_end_pos": [645, 27]}, {"full_name": "ProbabilityTheory.set_lintegral_preCdf_fst", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [307, 9], "def_end_pos": [307, 33]}, {"full_name": "MeasurableSet.univ", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [101, 19], "def_end_pos": [101, 37]}, {"full_name": "MeasureTheory.Measure.IicSnd_univ", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [190, 9], "def_end_pos": [190, 20]}, {"full_name": "Set.univ", "def_path": "Mathlib/Init/Set.lean", "def_pos": [90, 5], "def_end_pos": [90, 9]}, {"full_name": "Set.Iic", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [64, 5], "def_end_pos": [64, 8]}, {"full_name": "Set.Iic", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [64, 5], "def_end_pos": [64, 8]}, {"full_name": "prod_iInter", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [70, 9], "def_end_pos": [70, 20]}, {"full_name": "Set.prod_empty", "def_path": "Mathlib/Data/Set/Prod.lean", "def_pos": [113, 9], "def_end_pos": [113, 19]}, {"full_name": "MeasureTheory.measure_empty", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [185, 9], "def_end_pos": [185, 22]}, {"full_name": "Set.mem_iInter", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [207, 9], "def_end_pos": [207, 19]}, {"full_name": "Set.mem_Iic", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [136, 9], "def_end_pos": [136, 16]}, {"full_name": "Set.mem_empty_iff_false", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [562, 9], "def_end_pos": [562, 28]}, {"full_name": "iff_false_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [201, 9], "def_end_pos": [201, 22]}, {"full_name": "Classical.not_forall", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [686, 9], "def_end_pos": [686, 19]}, {"full_name": "not_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [373, 9], "def_end_pos": [373, 15]}, {"full_name": "neg_lt", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [388, 15], "def_end_pos": [388, 21]}, {"full_name": "exists_rat_gt", "def_path": "Mathlib/Algebra/Order/Archimedean.lean", "def_pos": [253, 9], "def_end_pos": [253, 22]}, {"full_name": "MeasureTheory.tendsto_measure_iInter", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [538, 9], "def_end_pos": [538, 31]}, {"full_name": "measurableSet_Iic", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [515, 9], "def_end_pos": [515, 26]}, {"full_name": "MeasureTheory.measure_ne_top", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2875, 9], "def_end_pos": [2875, 23]}, {"full_name": "Set.mem_prod", "def_path": "Mathlib/Data/Set/Prod.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}, {"full_name": "Set.mem_univ", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [676, 9], "def_end_pos": [676, 17]}, {"full_name": "Set.mem_Iic", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [136, 9], "def_end_pos": [136, 16]}, {"full_name": "true_and_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [147, 9], "def_end_pos": [147, 21]}, {"full_name": "neg_le_neg", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [1238, 15], "def_end_pos": [1238, 25]}]], "state_before": "case h_lintegral_eq\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a))\nh_lintegral : Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd (\u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1))\n\u22a2 \u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1 = 0", "state_after": "case h_lintegral_eq\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a))\nh_lintegral : Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd (\u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1))\nh_lintegral' : Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd 0)\n\u22a2 \u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1 = 0"}, {"tactic": "exact tendsto_nhds_unique h_lintegral h_lintegral'", "annotated_tactic": ["exact <a>tendsto_nhds_unique</a> h_lintegral h_lintegral'", [{"full_name": "tendsto_nhds_unique", "def_path": "Mathlib/Topology/Separation.lean", "def_pos": [994, 9], "def_end_pos": [994, 28]}]], "state_before": "case h_lintegral_eq\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a))\nh_lintegral : Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd (\u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1))\nh_lintegral' : Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd 0)\n\u22a2 \u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1 = 0", "state_after": "no goals"}, {"tactic": "filter_upwards [h_exists] with a ha", "annotated_tactic": ["filter_upwards [h_exists] with a ha", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a))", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0\na : \u03b1\nha : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\n\u22a2 Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a))"}, {"tactic": "simp_rw [dif_pos ha]", "annotated_tactic": ["simp_rw [<a>dif_pos</a> ha]", [{"full_name": "dif_pos", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [807, 9], "def_end_pos": [807, 16]}]], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0\na : \u03b1\nha : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\n\u22a2 Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a))", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0\na : \u03b1\nha : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\n\u22a2 Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (Exists.choose ha))"}, {"tactic": "exact ha.choose_spec", "annotated_tactic": ["exact ha.choose_spec", []], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0\na : \u03b1\nha : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\n\u22a2 Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (Exists.choose ha))", "state_after": "no goals"}, {"tactic": "have hF_ae_meas : AEMeasurable F \u03c1.fst := by\n  refine' aemeasurable_of_tendsto_metrizable_ae _ (fun n => _) h_tendsto\n  exact measurable_preCdf.aemeasurable", "annotated_tactic": ["have hF_ae_meas : <a>AEMeasurable</a> F \u03c1.fst := by\n      refine' <a>aemeasurable_of_tendsto_metrizable_ae</a> _ (fun n => _) h_tendsto\n      exact measurable_preCdf.aemeasurable", [{"full_name": "AEMeasurable", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [708, 5], "def_end_pos": [708, 17]}, {"full_name": "aemeasurable_of_tendsto_metrizable_ae", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Metrizable.lean", "def_pos": [92, 9], "def_end_pos": [92, 46]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a))\nh_lintegral_eq : \u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1 = 0\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd 0)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a))\nh_lintegral_eq : \u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1 = 0\nhF_ae_meas : AEMeasurable F\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd 0)"}, {"tactic": "rw [lintegral_eq_zero_iff' hF_ae_meas] at h_lintegral_eq", "annotated_tactic": ["rw [<a>lintegral_eq_zero_iff'</a> hF_ae_meas] at h_lintegral_eq", [{"full_name": "MeasureTheory.lintegral_eq_zero_iff'", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [888, 9], "def_end_pos": [888, 31]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a))\nh_lintegral_eq : \u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1 = 0\nhF_ae_meas : AEMeasurable F\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd 0)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a))\nh_lintegral_eq : F =\u1d50[Measure.fst \u03c1] 0\nhF_ae_meas : AEMeasurable F\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd 0)"}, {"tactic": "filter_upwards [h_tendsto, h_lintegral_eq] with a ha_tendsto ha_eq", "annotated_tactic": ["filter_upwards [h_tendsto, h_lintegral_eq] with a ha_tendsto ha_eq", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a))\nh_lintegral_eq : F =\u1d50[Measure.fst \u03c1] 0\nhF_ae_meas : AEMeasurable F\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd 0)", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a))\nh_lintegral_eq : F =\u1d50[Measure.fst \u03c1] 0\nhF_ae_meas : AEMeasurable F\na : \u03b1\nha_tendsto : Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a))\nha_eq : F a = OfNat.ofNat 0 a\n\u22a2 Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd 0)"}, {"tactic": "rwa [ha_eq] at ha_tendsto", "annotated_tactic": ["rwa [ha_eq] at ha_tendsto", []], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a))\nh_lintegral_eq : F =\u1d50[Measure.fst \u03c1] 0\nhF_ae_meas : AEMeasurable F\na : \u03b1\nha_tendsto : Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a))\nha_eq : F a = OfNat.ofNat 0 a\n\u22a2 Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd 0)", "state_after": "no goals"}, {"tactic": "refine' aemeasurable_of_tendsto_metrizable_ae _ (fun n => _) h_tendsto", "annotated_tactic": ["refine' <a>aemeasurable_of_tendsto_metrizable_ae</a> _ (fun n => _) h_tendsto", [{"full_name": "aemeasurable_of_tendsto_metrizable_ae", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Metrizable.lean", "def_pos": [92, 9], "def_end_pos": [92, 46]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a))\nh_lintegral_eq : \u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1 = 0\n\u22a2 AEMeasurable F", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a))\nh_lintegral_eq : \u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1 = 0\nn : \u211a\n\u22a2 AEMeasurable fun x => preCdf \u03c1 (-n) x"}, {"tactic": "exact measurable_preCdf.aemeasurable", "annotated_tactic": ["exact measurable_preCdf.aemeasurable", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a))\nh_lintegral_eq : \u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1 = 0\nn : \u211a\n\u22a2 AEMeasurable fun x => preCdf \u03c1 (-n) x", "state_after": "no goals"}, {"tactic": "refine'\n  tendsto_lintegral_filter_of_dominated_convergence (fun _ => 1)\n    (eventually_of_forall fun _ => measurable_preCdf) (eventually_of_forall fun _ => _) _\n    h_tendsto", "annotated_tactic": ["refine'\n      <a>tendsto_lintegral_filter_of_dominated_convergence</a> (fun _ => 1)\n        (<a>eventually_of_forall</a> fun _ => <a>measurable_preCdf</a>) (<a>eventually_of_forall</a> fun _ => _) _\n        h_tendsto", [{"full_name": "MeasureTheory.tendsto_lintegral_filter_of_dominated_convergence", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [1097, 9], "def_end_pos": [1097, 58]}, {"full_name": "Filter.eventually_of_forall", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1111, 9], "def_end_pos": [1111, 29]}, {"full_name": "ProbabilityTheory.measurable_preCdf", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [298, 9], "def_end_pos": [298, 26]}, {"full_name": "Filter.eventually_of_forall", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1111, 9], "def_end_pos": [1111, 29]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a))\n\u22a2 Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd (\u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1))", "state_after": "case refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a))\nx\u271d : \u211a\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, preCdf \u03c1 (-x\u271d) a \u2264 (fun x => 1) a\n\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a))\n\u22a2 \u222b\u207b (a : \u03b1), (fun x => 1) a \u2202Measure.fst \u03c1 \u2260 \u22a4"}, {"tactic": "filter_upwards [preCdf_le_one \u03c1] with a ha using ha _", "annotated_tactic": ["filter_upwards [<a>preCdf_le_one</a> \u03c1] with a ha using ha _", [{"full_name": "ProbabilityTheory.preCdf_le_one", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [349, 9], "def_end_pos": [349, 22]}]], "state_before": "case refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a))\nx\u271d : \u211a\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, preCdf \u03c1 (-x\u271d) a \u2264 (fun x => 1) a", "state_after": "no goals"}, {"tactic": "rw [lintegral_one]", "annotated_tactic": ["rw [<a>lintegral_one</a>]", [{"full_name": "MeasureTheory.lintegral_one", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [149, 9], "def_end_pos": [149, 22]}]], "state_before": "case refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a))\n\u22a2 \u222b\u207b (a : \u03b1), (fun x => 1) a \u2202Measure.fst \u03c1 \u2260 \u22a4", "state_after": "case refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a))\n\u22a2 \u2191\u2191(Measure.fst \u03c1) univ \u2260 \u22a4"}, {"tactic": "exact measure_ne_top _ _", "annotated_tactic": ["exact <a>measure_ne_top</a> _ _", [{"full_name": "MeasureTheory.measure_ne_top", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2875, 9], "def_end_pos": [2875, 23]}]], "state_before": "case refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a))\n\u22a2 \u2191\u2191(Measure.fst \u03c1) univ \u2260 \u22a4", "state_after": "no goals"}, {"tactic": "have h_lintegral_eq :\n  (fun r => \u222b\u207b a, preCdf \u03c1 (-r) a \u2202\u03c1.fst) = fun r : \u211a => \u03c1 (univ \u00d7\u02e2 Iic (-r : \u211d)) := by\n  ext1 n\n  rw [\u2190 set_lintegral_univ, set_lintegral_preCdf_fst \u03c1 _ MeasurableSet.univ,\n    Measure.IicSnd_univ]\n  norm_cast", "annotated_tactic": ["have h_lintegral_eq :\n      (fun r => \u222b\u207b a, <a>preCdf</a> \u03c1 (-r) a \u2202\u03c1.fst) = fun r : \u211a => \u03c1 (<a>univ</a> \u00d7\u02e2 <a>Iic</a> (-r : \u211d)) := by\n      ext1 n\n      rw [\u2190 <a>set_lintegral_univ</a>, <a>set_lintegral_preCdf_fst</a> \u03c1 _ <a>MeasurableSet.univ</a>,\n        <a>Measure.IicSnd_univ</a>]\n      norm_cast", [{"full_name": "ProbabilityTheory.preCdf", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [294, 19], "def_end_pos": [294, 25]}, {"full_name": "Set.univ", "def_path": "Mathlib/Init/Set.lean", "def_pos": [90, 5], "def_end_pos": [90, 9]}, {"full_name": "Set.Iic", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [64, 5], "def_end_pos": [64, 8]}, {"full_name": "MeasureTheory.set_lintegral_univ", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [645, 9], "def_end_pos": [645, 27]}, {"full_name": "ProbabilityTheory.set_lintegral_preCdf_fst", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [307, 9], "def_end_pos": [307, 33]}, {"full_name": "MeasurableSet.univ", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [101, 19], "def_end_pos": [101, 37]}, {"full_name": "MeasureTheory.Measure.IicSnd_univ", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [190, 9], "def_end_pos": [190, 20]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a))\nh_lintegral : Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd (\u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1))\n\u22a2 Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd 0)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a))\nh_lintegral : Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd (\u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1))\nh_lintegral_eq : (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) = fun r => \u2191\u2191\u03c1 (univ \u00d7\u02e2 Iic (-\u2191r))\n\u22a2 Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd 0)"}, {"tactic": "rw [h_lintegral_eq]", "annotated_tactic": ["rw [h_lintegral_eq]", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a))\nh_lintegral : Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd (\u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1))\nh_lintegral_eq : (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) = fun r => \u2191\u2191\u03c1 (univ \u00d7\u02e2 Iic (-\u2191r))\n\u22a2 Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd 0)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a))\nh_lintegral : Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd (\u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1))\nh_lintegral_eq : (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) = fun r => \u2191\u2191\u03c1 (univ \u00d7\u02e2 Iic (-\u2191r))\n\u22a2 Tendsto (fun r => \u2191\u2191\u03c1 (univ \u00d7\u02e2 Iic (-\u2191r))) atTop (\ud835\udcdd 0)"}, {"tactic": "have h_zero_eq_measure_iInter : (0 : \u211d\u22650\u221e) = \u03c1 (\u22c2 r : \u211a, univ \u00d7\u02e2 Iic (-r : \u211d)) := by\n  suffices \u22c2 r : \u211a, Iic (-(r : \u211d)) = \u2205 by rw [\u2190 prod_iInter, this, prod_empty, measure_empty]\n  ext1 x\n  simp only [mem_iInter, mem_Iic, mem_empty_iff_false, iff_false_iff, not_forall, not_le]\n  simp_rw [neg_lt]\n  exact exists_rat_gt _", "annotated_tactic": ["have h_zero_eq_measure_iInter : (0 : \u211d\u22650\u221e) = \u03c1 (\u22c2 r : \u211a, <a>univ</a> \u00d7\u02e2 <a>Iic</a> (-r : \u211d)) := by\n      suffices \u22c2 r : \u211a, <a>Iic</a> (-(r : \u211d)) = \u2205 by rw [\u2190 <a>prod_iInter</a>, this, <a>prod_empty</a>, <a>measure_empty</a>]\n      ext1 x\n      simp only [<a>mem_iInter</a>, <a>mem_Iic</a>, <a>mem_empty_iff_false</a>, <a>iff_false_iff</a>, <a>not_forall</a>, <a>not_le</a>]\n      simp_rw [<a>neg_lt</a>]\n      exact <a>exists_rat_gt</a> _", [{"full_name": "Set.univ", "def_path": "Mathlib/Init/Set.lean", "def_pos": [90, 5], "def_end_pos": [90, 9]}, {"full_name": "Set.Iic", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [64, 5], "def_end_pos": [64, 8]}, {"full_name": "Set.Iic", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [64, 5], "def_end_pos": [64, 8]}, {"full_name": "prod_iInter", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [70, 9], "def_end_pos": [70, 20]}, {"full_name": "Set.prod_empty", "def_path": "Mathlib/Data/Set/Prod.lean", "def_pos": [113, 9], "def_end_pos": [113, 19]}, {"full_name": "MeasureTheory.measure_empty", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [185, 9], "def_end_pos": [185, 22]}, {"full_name": "Set.mem_iInter", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [207, 9], "def_end_pos": [207, 19]}, {"full_name": "Set.mem_Iic", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [136, 9], "def_end_pos": [136, 16]}, {"full_name": "Set.mem_empty_iff_false", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [562, 9], "def_end_pos": [562, 28]}, {"full_name": "iff_false_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [201, 9], "def_end_pos": [201, 22]}, {"full_name": "Classical.not_forall", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [686, 9], "def_end_pos": [686, 19]}, {"full_name": "not_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [373, 9], "def_end_pos": [373, 15]}, {"full_name": "neg_lt", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [388, 15], "def_end_pos": [388, 21]}, {"full_name": "exists_rat_gt", "def_path": "Mathlib/Algebra/Order/Archimedean.lean", "def_pos": [253, 9], "def_end_pos": [253, 22]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a))\nh_lintegral : Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd (\u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1))\nh_lintegral_eq : (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) = fun r => \u2191\u2191\u03c1 (univ \u00d7\u02e2 Iic (-\u2191r))\n\u22a2 Tendsto (fun r => \u2191\u2191\u03c1 (univ \u00d7\u02e2 Iic (-\u2191r))) atTop (\ud835\udcdd 0)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a))\nh_lintegral : Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd (\u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1))\nh_lintegral_eq : (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) = fun r => \u2191\u2191\u03c1 (univ \u00d7\u02e2 Iic (-\u2191r))\nh_zero_eq_measure_iInter : 0 = \u2191\u2191\u03c1 (\u22c2 r, univ \u00d7\u02e2 Iic (-\u2191r))\n\u22a2 Tendsto (fun r => \u2191\u2191\u03c1 (univ \u00d7\u02e2 Iic (-\u2191r))) atTop (\ud835\udcdd 0)"}, {"tactic": "rw [h_zero_eq_measure_iInter]", "annotated_tactic": ["rw [h_zero_eq_measure_iInter]", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a))\nh_lintegral : Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd (\u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1))\nh_lintegral_eq : (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) = fun r => \u2191\u2191\u03c1 (univ \u00d7\u02e2 Iic (-\u2191r))\nh_zero_eq_measure_iInter : 0 = \u2191\u2191\u03c1 (\u22c2 r, univ \u00d7\u02e2 Iic (-\u2191r))\n\u22a2 Tendsto (fun r => \u2191\u2191\u03c1 (univ \u00d7\u02e2 Iic (-\u2191r))) atTop (\ud835\udcdd 0)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a))\nh_lintegral : Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd (\u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1))\nh_lintegral_eq : (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) = fun r => \u2191\u2191\u03c1 (univ \u00d7\u02e2 Iic (-\u2191r))\nh_zero_eq_measure_iInter : 0 = \u2191\u2191\u03c1 (\u22c2 r, univ \u00d7\u02e2 Iic (-\u2191r))\n\u22a2 Tendsto (fun r => \u2191\u2191\u03c1 (univ \u00d7\u02e2 Iic (-\u2191r))) atTop (\ud835\udcdd (\u2191\u2191\u03c1 (\u22c2 r, univ \u00d7\u02e2 Iic (-\u2191r))))"}, {"tactic": "refine'\n  tendsto_measure_iInter (fun n => MeasurableSet.univ.prod measurableSet_Iic)\n    (fun i j hij x => _) \u27e80, measure_ne_top \u03c1 _\u27e9", "annotated_tactic": ["refine'\n      <a>tendsto_measure_iInter</a> (fun n => MeasurableSet.univ.prod <a>measurableSet_Iic</a>)\n        (fun i j hij x => _) \u27e80, <a>measure_ne_top</a> \u03c1 _\u27e9", [{"full_name": "MeasureTheory.tendsto_measure_iInter", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [538, 9], "def_end_pos": [538, 31]}, {"full_name": "measurableSet_Iic", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [515, 9], "def_end_pos": [515, 26]}, {"full_name": "MeasureTheory.measure_ne_top", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2875, 9], "def_end_pos": [2875, 23]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a))\nh_lintegral : Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd (\u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1))\nh_lintegral_eq : (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) = fun r => \u2191\u2191\u03c1 (univ \u00d7\u02e2 Iic (-\u2191r))\nh_zero_eq_measure_iInter : 0 = \u2191\u2191\u03c1 (\u22c2 r, univ \u00d7\u02e2 Iic (-\u2191r))\n\u22a2 Tendsto (fun r => \u2191\u2191\u03c1 (univ \u00d7\u02e2 Iic (-\u2191r))) atTop (\ud835\udcdd (\u2191\u2191\u03c1 (\u22c2 r, univ \u00d7\u02e2 Iic (-\u2191r))))", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a))\nh_lintegral : Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd (\u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1))\nh_lintegral_eq : (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) = fun r => \u2191\u2191\u03c1 (univ \u00d7\u02e2 Iic (-\u2191r))\nh_zero_eq_measure_iInter : 0 = \u2191\u2191\u03c1 (\u22c2 r, univ \u00d7\u02e2 Iic (-\u2191r))\ni j : \u211a\nhij : i \u2264 j\nx : \u03b1 \u00d7 \u211d\n\u22a2 x \u2208 univ \u00d7\u02e2 Iic (-\u2191j) \u2192 x \u2208 univ \u00d7\u02e2 Iic (-\u2191i)"}, {"tactic": "simp only [mem_prod, mem_univ, mem_Iic, true_and_iff]", "annotated_tactic": ["simp only [<a>mem_prod</a>, <a>mem_univ</a>, <a>mem_Iic</a>, <a>true_and_iff</a>]", [{"full_name": "Set.mem_prod", "def_path": "Mathlib/Data/Set/Prod.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}, {"full_name": "Set.mem_univ", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [676, 9], "def_end_pos": [676, 17]}, {"full_name": "Set.mem_Iic", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [136, 9], "def_end_pos": [136, 16]}, {"full_name": "true_and_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [147, 9], "def_end_pos": [147, 21]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a))\nh_lintegral : Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd (\u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1))\nh_lintegral_eq : (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) = fun r => \u2191\u2191\u03c1 (univ \u00d7\u02e2 Iic (-\u2191r))\nh_zero_eq_measure_iInter : 0 = \u2191\u2191\u03c1 (\u22c2 r, univ \u00d7\u02e2 Iic (-\u2191r))\ni j : \u211a\nhij : i \u2264 j\nx : \u03b1 \u00d7 \u211d\n\u22a2 x \u2208 univ \u00d7\u02e2 Iic (-\u2191j) \u2192 x \u2208 univ \u00d7\u02e2 Iic (-\u2191i)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a))\nh_lintegral : Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd (\u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1))\nh_lintegral_eq : (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) = fun r => \u2191\u2191\u03c1 (univ \u00d7\u02e2 Iic (-\u2191r))\nh_zero_eq_measure_iInter : 0 = \u2191\u2191\u03c1 (\u22c2 r, univ \u00d7\u02e2 Iic (-\u2191r))\ni j : \u211a\nhij : i \u2264 j\nx : \u03b1 \u00d7 \u211d\n\u22a2 x.2 \u2264 -\u2191j \u2192 x.2 \u2264 -\u2191i"}, {"tactic": "refine' fun hxj => hxj.trans (neg_le_neg _)", "annotated_tactic": ["refine' fun hxj => hxj.trans (<a>neg_le_neg</a> _)", [{"full_name": "neg_le_neg", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [1238, 15], "def_end_pos": [1238, 25]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a))\nh_lintegral : Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd (\u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1))\nh_lintegral_eq : (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) = fun r => \u2191\u2191\u03c1 (univ \u00d7\u02e2 Iic (-\u2191r))\nh_zero_eq_measure_iInter : 0 = \u2191\u2191\u03c1 (\u22c2 r, univ \u00d7\u02e2 Iic (-\u2191r))\ni j : \u211a\nhij : i \u2264 j\nx : \u03b1 \u00d7 \u211d\n\u22a2 x.2 \u2264 -\u2191j \u2192 x.2 \u2264 -\u2191i", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a))\nh_lintegral : Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd (\u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1))\nh_lintegral_eq : (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) = fun r => \u2191\u2191\u03c1 (univ \u00d7\u02e2 Iic (-\u2191r))\nh_zero_eq_measure_iInter : 0 = \u2191\u2191\u03c1 (\u22c2 r, univ \u00d7\u02e2 Iic (-\u2191r))\ni j : \u211a\nhij : i \u2264 j\nx : \u03b1 \u00d7 \u211d\nhxj : x.2 \u2264 -\u2191j\n\u22a2 \u2191i \u2264 \u2191j"}, {"tactic": "exact_mod_cast hij", "annotated_tactic": ["exact_mod_cast hij", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a))\nh_lintegral : Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd (\u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1))\nh_lintegral_eq : (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) = fun r => \u2191\u2191\u03c1 (univ \u00d7\u02e2 Iic (-\u2191r))\nh_zero_eq_measure_iInter : 0 = \u2191\u2191\u03c1 (\u22c2 r, univ \u00d7\u02e2 Iic (-\u2191r))\ni j : \u211a\nhij : i \u2264 j\nx : \u03b1 \u00d7 \u211d\nhxj : x.2 \u2264 -\u2191j\n\u22a2 \u2191i \u2264 \u2191j", "state_after": "no goals"}, {"tactic": "ext1 n", "annotated_tactic": ["ext1 n", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a))\nh_lintegral : Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd (\u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1))\n\u22a2 (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) = fun r => \u2191\u2191\u03c1 (univ \u00d7\u02e2 Iic (-\u2191r))", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a))\nh_lintegral : Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd (\u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1))\nn : \u211a\n\u22a2 \u222b\u207b (a : \u03b1), preCdf \u03c1 (-n) a \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (univ \u00d7\u02e2 Iic (-\u2191n))"}, {"tactic": "rw [\u2190 set_lintegral_univ, set_lintegral_preCdf_fst \u03c1 _ MeasurableSet.univ,\n  Measure.IicSnd_univ]", "annotated_tactic": ["rw [\u2190 <a>set_lintegral_univ</a>, <a>set_lintegral_preCdf_fst</a> \u03c1 _ <a>MeasurableSet.univ</a>,\n        <a>Measure.IicSnd_univ</a>]", [{"full_name": "MeasureTheory.set_lintegral_univ", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [645, 9], "def_end_pos": [645, 27]}, {"full_name": "ProbabilityTheory.set_lintegral_preCdf_fst", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [307, 9], "def_end_pos": [307, 33]}, {"full_name": "MeasurableSet.univ", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [101, 19], "def_end_pos": [101, 37]}, {"full_name": "MeasureTheory.Measure.IicSnd_univ", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [190, 9], "def_end_pos": [190, 20]}]], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a))\nh_lintegral : Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd (\u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1))\nn : \u211a\n\u22a2 \u222b\u207b (a : \u03b1), preCdf \u03c1 (-n) a \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (univ \u00d7\u02e2 Iic (-\u2191n))", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a))\nh_lintegral : Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd (\u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1))\nn : \u211a\n\u22a2 \u2191\u2191\u03c1 (univ \u00d7\u02e2 Iic \u2191(-n)) = \u2191\u2191\u03c1 (univ \u00d7\u02e2 Iic (-\u2191n))"}, {"tactic": "norm_cast", "annotated_tactic": ["norm_cast", []], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a))\nh_lintegral : Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd (\u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1))\nn : \u211a\n\u22a2 \u2191\u2191\u03c1 (univ \u00d7\u02e2 Iic \u2191(-n)) = \u2191\u2191\u03c1 (univ \u00d7\u02e2 Iic (-\u2191n))", "state_after": "no goals"}, {"tactic": "suffices \u22c2 r : \u211a, Iic (-(r : \u211d)) = \u2205 by rw [\u2190 prod_iInter, this, prod_empty, measure_empty]", "annotated_tactic": ["suffices \u22c2 r : \u211a, <a>Iic</a> (-(r : \u211d)) = \u2205 by rw [\u2190 <a>prod_iInter</a>, this, <a>prod_empty</a>, <a>measure_empty</a>]", [{"full_name": "Set.Iic", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [64, 5], "def_end_pos": [64, 8]}, {"full_name": "prod_iInter", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [70, 9], "def_end_pos": [70, 20]}, {"full_name": "Set.prod_empty", "def_path": "Mathlib/Data/Set/Prod.lean", "def_pos": [113, 9], "def_end_pos": [113, 19]}, {"full_name": "MeasureTheory.measure_empty", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [185, 9], "def_end_pos": [185, 22]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a))\nh_lintegral : Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd (\u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1))\nh_lintegral_eq : (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) = fun r => \u2191\u2191\u03c1 (univ \u00d7\u02e2 Iic (-\u2191r))\n\u22a2 0 = \u2191\u2191\u03c1 (\u22c2 r, univ \u00d7\u02e2 Iic (-\u2191r))", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a))\nh_lintegral : Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd (\u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1))\nh_lintegral_eq : (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) = fun r => \u2191\u2191\u03c1 (univ \u00d7\u02e2 Iic (-\u2191r))\n\u22a2 \u22c2 r, Iic (-\u2191r) = \u2205"}, {"tactic": "ext1 x", "annotated_tactic": ["ext1 x", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a))\nh_lintegral : Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd (\u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1))\nh_lintegral_eq : (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) = fun r => \u2191\u2191\u03c1 (univ \u00d7\u02e2 Iic (-\u2191r))\n\u22a2 \u22c2 r, Iic (-\u2191r) = \u2205", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a))\nh_lintegral : Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd (\u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1))\nh_lintegral_eq : (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) = fun r => \u2191\u2191\u03c1 (univ \u00d7\u02e2 Iic (-\u2191r))\nx : \u211d\n\u22a2 x \u2208 \u22c2 r, Iic (-\u2191r) \u2194 x \u2208 \u2205"}, {"tactic": "simp only [mem_iInter, mem_Iic, mem_empty_iff_false, iff_false_iff, not_forall, not_le]", "annotated_tactic": ["simp only [<a>mem_iInter</a>, <a>mem_Iic</a>, <a>mem_empty_iff_false</a>, <a>iff_false_iff</a>, <a>not_forall</a>, <a>not_le</a>]", [{"full_name": "Set.mem_iInter", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [207, 9], "def_end_pos": [207, 19]}, {"full_name": "Set.mem_Iic", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [136, 9], "def_end_pos": [136, 16]}, {"full_name": "Set.mem_empty_iff_false", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [562, 9], "def_end_pos": [562, 28]}, {"full_name": "iff_false_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [201, 9], "def_end_pos": [201, 22]}, {"full_name": "Classical.not_forall", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [686, 9], "def_end_pos": [686, 19]}, {"full_name": "not_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [373, 9], "def_end_pos": [373, 15]}]], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a))\nh_lintegral : Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd (\u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1))\nh_lintegral_eq : (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) = fun r => \u2191\u2191\u03c1 (univ \u00d7\u02e2 Iic (-\u2191r))\nx : \u211d\n\u22a2 x \u2208 \u22c2 r, Iic (-\u2191r) \u2194 x \u2208 \u2205", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a))\nh_lintegral : Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd (\u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1))\nh_lintegral_eq : (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) = fun r => \u2191\u2191\u03c1 (univ \u00d7\u02e2 Iic (-\u2191r))\nx : \u211d\n\u22a2 \u2203 x_1, -\u2191x_1 < x"}, {"tactic": "simp_rw [neg_lt]", "annotated_tactic": ["simp_rw [<a>neg_lt</a>]", [{"full_name": "neg_lt", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [388, 15], "def_end_pos": [388, 21]}]], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a))\nh_lintegral : Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd (\u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1))\nh_lintegral_eq : (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) = fun r => \u2191\u2191\u03c1 (univ \u00d7\u02e2 Iic (-\u2191r))\nx : \u211d\n\u22a2 \u2203 x_1, -\u2191x_1 < x", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a))\nh_lintegral : Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd (\u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1))\nh_lintegral_eq : (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) = fun r => \u2191\u2191\u03c1 (univ \u00d7\u02e2 Iic (-\u2191r))\nx : \u211d\n\u22a2 \u2203 x_1, -x < \u2191x_1"}, {"tactic": "exact exists_rat_gt _", "annotated_tactic": ["exact <a>exists_rat_gt</a> _", [{"full_name": "exists_rat_gt", "def_path": "Mathlib/Algebra/Order/Archimedean.lean", "def_pos": [253, 9], "def_end_pos": [253, 22]}]], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a))\nh_lintegral : Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd (\u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1))\nh_lintegral_eq : (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) = fun r => \u2191\u2191\u03c1 (univ \u00d7\u02e2 Iic (-\u2191r))\nx : \u211d\n\u22a2 \u2203 x_1, -x < \u2191x_1", "state_after": "no goals"}, {"tactic": "rw [\u2190 prod_iInter, this, prod_empty, measure_empty]", "annotated_tactic": ["rw [\u2190 <a>prod_iInter</a>, this, <a>prod_empty</a>, <a>measure_empty</a>]", [{"full_name": "prod_iInter", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [70, 9], "def_end_pos": [70, 20]}, {"full_name": "Set.prod_empty", "def_path": "Mathlib/Data/Set/Prod.lean", "def_pos": [113, 9], "def_end_pos": [113, 19]}, {"full_name": "MeasureTheory.measure_empty", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [185, 9], "def_end_pos": [185, 22]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a))\nh_lintegral : Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd (\u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1))\nh_lintegral_eq : (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) = fun r => \u2191\u2191\u03c1 (univ \u00d7\u02e2 Iic (-\u2191r))\nthis : \u22c2 r, Iic (-\u2191r) = \u2205\n\u22a2 0 = \u2191\u2191\u03c1 (\u22c2 r, univ \u00d7\u02e2 Iic (-\u2191r))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Kernel/MeasurableIntegral.lean", "full_name": "Measurable.lintegral_kernel_prod_right", "start": [149, 1], "end": [181, 33], "traced_tactics": [{"tactic": "let F : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) \u211d\u22650\u221e := SimpleFunc.eapprox (uncurry f)", "annotated_tactic": ["let F : \u2115 \u2192 <a>SimpleFunc</a> (\u03b1 \u00d7 \u03b2) \u211d\u22650\u221e := <a>SimpleFunc.eapprox</a> (<a>uncurry</a> f)", [{"full_name": "MeasureTheory.SimpleFunc", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [44, 11], "def_end_pos": [44, 21]}, {"full_name": "MeasureTheory.SimpleFunc.eapprox", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [873, 5], "def_end_pos": [873, 12]}, {"full_name": "Function.uncurry", "def_path": "Mathlib/Init/Function.lean", "def_pos": [217, 5], "def_end_pos": [217, 12]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na : \u03b1\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\nhf : Measurable (uncurry f)\n\u22a2 Measurable fun a => \u222b\u207b (b : \u03b2), f a b \u2202\u2191\u03ba a", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na : \u03b1\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\nhf : Measurable (uncurry f)\nF : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) \u211d\u22650\u221e := SimpleFunc.eapprox (uncurry f)\n\u22a2 Measurable fun a => \u222b\u207b (b : \u03b2), f a b \u2202\u2191\u03ba a"}, {"tactic": "have h : \u2200 a, \u2a06 n, F n a = uncurry f a := SimpleFunc.iSup_eapprox_apply (uncurry f) hf", "annotated_tactic": ["have h : \u2200 a, \u2a06 n, F n a = <a>uncurry</a> f a := <a>SimpleFunc.iSup_eapprox_apply</a> (<a>uncurry</a> f) hf", [{"full_name": "Function.uncurry", "def_path": "Mathlib/Init/Function.lean", "def_pos": [217, 5], "def_end_pos": [217, 12]}, {"full_name": "MeasureTheory.SimpleFunc.iSup_eapprox_apply", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [896, 9], "def_end_pos": [896, 27]}, {"full_name": "Function.uncurry", "def_path": "Mathlib/Init/Function.lean", "def_pos": [217, 5], "def_end_pos": [217, 12]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na : \u03b1\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\nhf : Measurable (uncurry f)\nF : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) \u211d\u22650\u221e := SimpleFunc.eapprox (uncurry f)\n\u22a2 Measurable fun a => \u222b\u207b (b : \u03b2), f a b \u2202\u2191\u03ba a", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na : \u03b1\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\nhf : Measurable (uncurry f)\nF : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) \u211d\u22650\u221e := SimpleFunc.eapprox (uncurry f)\nh : \u2200 (a : \u03b1 \u00d7 \u03b2), \u2a06 n, \u2191(F n) a = uncurry f a\n\u22a2 Measurable fun a => \u222b\u207b (b : \u03b2), f a b \u2202\u2191\u03ba a"}, {"tactic": "simp only [Prod.forall, uncurry_apply_pair] at h", "annotated_tactic": ["simp only [<a>Prod.forall</a>, <a>uncurry_apply_pair</a>] at h", [{"full_name": "Prod.forall", "def_path": "Mathlib/Data/Prod/Basic.lean", "def_pos": [36, 9], "def_end_pos": [36, 17]}, {"full_name": "Function.uncurry_apply_pair", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [817, 9], "def_end_pos": [817, 27]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na : \u03b1\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\nhf : Measurable (uncurry f)\nF : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) \u211d\u22650\u221e := SimpleFunc.eapprox (uncurry f)\nh : \u2200 (a : \u03b1 \u00d7 \u03b2), \u2a06 n, \u2191(F n) a = uncurry f a\n\u22a2 Measurable fun a => \u222b\u207b (b : \u03b2), f a b \u2202\u2191\u03ba a", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na : \u03b1\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\nhf : Measurable (uncurry f)\nF : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) \u211d\u22650\u221e := SimpleFunc.eapprox (uncurry f)\nh : \u2200 (a : \u03b1) (b : \u03b2), \u2a06 n, \u2191(SimpleFunc.eapprox (uncurry f) n) (a, b) = f a b\n\u22a2 Measurable fun a => \u222b\u207b (b : \u03b2), f a b \u2202\u2191\u03ba a"}, {"tactic": "simp_rw [\u2190 h]", "annotated_tactic": ["simp_rw [\u2190 h]", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na : \u03b1\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\nhf : Measurable (uncurry f)\nF : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) \u211d\u22650\u221e := SimpleFunc.eapprox (uncurry f)\nh : \u2200 (a : \u03b1) (b : \u03b2), \u2a06 n, \u2191(SimpleFunc.eapprox (uncurry f) n) (a, b) = f a b\n\u22a2 Measurable fun a => \u222b\u207b (b : \u03b2), f a b \u2202\u2191\u03ba a", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na : \u03b1\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\nhf : Measurable (uncurry f)\nF : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) \u211d\u22650\u221e := SimpleFunc.eapprox (uncurry f)\nh : \u2200 (a : \u03b1) (b : \u03b2), \u2a06 n, \u2191(SimpleFunc.eapprox (uncurry f) n) (a, b) = f a b\n\u22a2 Measurable fun a => \u222b\u207b (b : \u03b2), \u2a06 n, \u2191(SimpleFunc.eapprox (uncurry f) n) (a, b) \u2202\u2191\u03ba a"}, {"tactic": "simp_rw [this]", "annotated_tactic": ["simp_rw [this]", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na : \u03b1\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\nhf : Measurable (uncurry f)\nF : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) \u211d\u22650\u221e := SimpleFunc.eapprox (uncurry f)\nh : \u2200 (a : \u03b1) (b : \u03b2), \u2a06 n, \u2191(SimpleFunc.eapprox (uncurry f) n) (a, b) = f a b\nthis : \u2200 (a : \u03b1), \u222b\u207b (b : \u03b2), \u2a06 n, \u2191(F n) (a, b) \u2202\u2191\u03ba a = \u2a06 n, \u222b\u207b (b : \u03b2), \u2191(F n) (a, b) \u2202\u2191\u03ba a\n\u22a2 Measurable fun a => \u222b\u207b (b : \u03b2), \u2a06 n, \u2191(SimpleFunc.eapprox (uncurry f) n) (a, b) \u2202\u2191\u03ba a", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na : \u03b1\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\nhf : Measurable (uncurry f)\nF : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) \u211d\u22650\u221e := SimpleFunc.eapprox (uncurry f)\nh : \u2200 (a : \u03b1) (b : \u03b2), \u2a06 n, \u2191(SimpleFunc.eapprox (uncurry f) n) (a, b) = f a b\nthis : \u2200 (a : \u03b1), \u222b\u207b (b : \u03b2), \u2a06 n, \u2191(F n) (a, b) \u2202\u2191\u03ba a = \u2a06 n, \u222b\u207b (b : \u03b2), \u2191(F n) (a, b) \u2202\u2191\u03ba a\n\u22a2 Measurable fun a => \u2a06 n, \u222b\u207b (b : \u03b2), \u2191(SimpleFunc.eapprox (uncurry f) n) (a, b) \u2202\u2191\u03ba a"}, {"tactic": "refine' measurable_iSup fun n => _", "annotated_tactic": ["refine' <a>measurable_iSup</a> fun n => _", [{"full_name": "measurable_iSup", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [1360, 9], "def_end_pos": [1360, 24]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na : \u03b1\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\nhf : Measurable (uncurry f)\nF : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) \u211d\u22650\u221e := SimpleFunc.eapprox (uncurry f)\nh : \u2200 (a : \u03b1) (b : \u03b2), \u2a06 n, \u2191(SimpleFunc.eapprox (uncurry f) n) (a, b) = f a b\nthis : \u2200 (a : \u03b1), \u222b\u207b (b : \u03b2), \u2a06 n, \u2191(F n) (a, b) \u2202\u2191\u03ba a = \u2a06 n, \u222b\u207b (b : \u03b2), \u2191(F n) (a, b) \u2202\u2191\u03ba a\n\u22a2 Measurable fun a => \u2a06 n, \u222b\u207b (b : \u03b2), \u2191(SimpleFunc.eapprox (uncurry f) n) (a, b) \u2202\u2191\u03ba a", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na : \u03b1\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\nhf : Measurable (uncurry f)\nF : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) \u211d\u22650\u221e := SimpleFunc.eapprox (uncurry f)\nh : \u2200 (a : \u03b1) (b : \u03b2), \u2a06 n, \u2191(SimpleFunc.eapprox (uncurry f) n) (a, b) = f a b\nthis : \u2200 (a : \u03b1), \u222b\u207b (b : \u03b2), \u2a06 n, \u2191(F n) (a, b) \u2202\u2191\u03ba a = \u2a06 n, \u222b\u207b (b : \u03b2), \u2191(F n) (a, b) \u2202\u2191\u03ba a\nn : \u2115\n\u22a2 Measurable fun a => \u222b\u207b (b : \u03b2), \u2191(SimpleFunc.eapprox (uncurry f) n) (a, b) \u2202\u2191\u03ba a"}, {"tactic": "refine' SimpleFunc.induction\n  (P := fun f => Measurable (fun (a : \u03b1) => \u222b\u207b (b : \u03b2), f (a, b) \u2202\u03ba a)) _ _ (F n)", "annotated_tactic": ["refine' <a>SimpleFunc.induction</a>\n    (P := fun f => <a>Measurable</a> (fun (a : \u03b1) => \u222b\u207b (b : \u03b2), f (a, b) \u2202\u03ba a)) _ _ (F n)", [{"full_name": "MeasureTheory.SimpleFunc.induction", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [1266, 19], "def_end_pos": [1266, 28]}, {"full_name": "Measurable", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [535, 5], "def_end_pos": [535, 15]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na : \u03b1\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\nhf : Measurable (uncurry f)\nF : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) \u211d\u22650\u221e := SimpleFunc.eapprox (uncurry f)\nh : \u2200 (a : \u03b1) (b : \u03b2), \u2a06 n, \u2191(SimpleFunc.eapprox (uncurry f) n) (a, b) = f a b\nthis : \u2200 (a : \u03b1), \u222b\u207b (b : \u03b2), \u2a06 n, \u2191(F n) (a, b) \u2202\u2191\u03ba a = \u2a06 n, \u222b\u207b (b : \u03b2), \u2191(F n) (a, b) \u2202\u2191\u03ba a\nn : \u2115\n\u22a2 Measurable fun a => \u222b\u207b (b : \u03b2), \u2191(SimpleFunc.eapprox (uncurry f) n) (a, b) \u2202\u2191\u03ba a", "state_after": "case refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na : \u03b1\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\nhf : Measurable (uncurry f)\nF : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) \u211d\u22650\u221e := SimpleFunc.eapprox (uncurry f)\nh : \u2200 (a : \u03b1) (b : \u03b2), \u2a06 n, \u2191(SimpleFunc.eapprox (uncurry f) n) (a, b) = f a b\nthis : \u2200 (a : \u03b1), \u222b\u207b (b : \u03b2), \u2a06 n, \u2191(F n) (a, b) \u2202\u2191\u03ba a = \u2a06 n, \u222b\u207b (b : \u03b2), \u2191(F n) (a, b) \u2202\u2191\u03ba a\nn : \u2115\n\u22a2 \u2200 (c : \u211d\u22650\u221e) {s : Set (\u03b1 \u00d7 \u03b2)} (hs : MeasurableSet s),\n    (fun f => Measurable fun a => \u222b\u207b (b : \u03b2), \u2191f (a, b) \u2202\u2191\u03ba a)\n      (SimpleFunc.piecewise s hs (SimpleFunc.const (\u03b1 \u00d7 \u03b2) c) (SimpleFunc.const (\u03b1 \u00d7 \u03b2) 0))\n\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na : \u03b1\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\nhf : Measurable (uncurry f)\nF : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) \u211d\u22650\u221e := SimpleFunc.eapprox (uncurry f)\nh : \u2200 (a : \u03b1) (b : \u03b2), \u2a06 n, \u2191(SimpleFunc.eapprox (uncurry f) n) (a, b) = f a b\nthis : \u2200 (a : \u03b1), \u222b\u207b (b : \u03b2), \u2a06 n, \u2191(F n) (a, b) \u2202\u2191\u03ba a = \u2a06 n, \u222b\u207b (b : \u03b2), \u2191(F n) (a, b) \u2202\u2191\u03ba a\nn : \u2115\n\u22a2 \u2200 \u2983f g : SimpleFunc (\u03b1 \u00d7 \u03b2) \u211d\u22650\u221e\u2984,\n    Disjoint (support \u2191f) (support \u2191g) \u2192\n      (fun f => Measurable fun a => \u222b\u207b (b : \u03b2), \u2191f (a, b) \u2202\u2191\u03ba a) f \u2192\n        (fun f => Measurable fun a => \u222b\u207b (b : \u03b2), \u2191f (a, b) \u2202\u2191\u03ba a) g \u2192\n          (fun f => Measurable fun a => \u222b\u207b (b : \u03b2), \u2191f (a, b) \u2202\u2191\u03ba a) (f + g)"}, {"tactic": "intro a", "annotated_tactic": ["intro a", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na : \u03b1\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\nhf : Measurable (uncurry f)\nF : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) \u211d\u22650\u221e := SimpleFunc.eapprox (uncurry f)\nh : \u2200 (a : \u03b1) (b : \u03b2), \u2a06 n, \u2191(SimpleFunc.eapprox (uncurry f) n) (a, b) = f a b\n\u22a2 \u2200 (a : \u03b1), \u222b\u207b (b : \u03b2), \u2a06 n, \u2191(F n) (a, b) \u2202\u2191\u03ba a = \u2a06 n, \u222b\u207b (b : \u03b2), \u2191(F n) (a, b) \u2202\u2191\u03ba a", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na\u271d : \u03b1\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\nhf : Measurable (uncurry f)\nF : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) \u211d\u22650\u221e := SimpleFunc.eapprox (uncurry f)\nh : \u2200 (a : \u03b1) (b : \u03b2), \u2a06 n, \u2191(SimpleFunc.eapprox (uncurry f) n) (a, b) = f a b\na : \u03b1\n\u22a2 \u222b\u207b (b : \u03b2), \u2a06 n, \u2191(F n) (a, b) \u2202\u2191\u03ba a = \u2a06 n, \u222b\u207b (b : \u03b2), \u2191(F n) (a, b) \u2202\u2191\u03ba a"}, {"tactic": "rw [lintegral_iSup]", "annotated_tactic": ["rw [<a>lintegral_iSup</a>]", [{"full_name": "MeasureTheory.lintegral_iSup", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [345, 9], "def_end_pos": [345, 23]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na\u271d : \u03b1\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\nhf : Measurable (uncurry f)\nF : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) \u211d\u22650\u221e := SimpleFunc.eapprox (uncurry f)\nh : \u2200 (a : \u03b1) (b : \u03b2), \u2a06 n, \u2191(SimpleFunc.eapprox (uncurry f) n) (a, b) = f a b\na : \u03b1\n\u22a2 \u222b\u207b (b : \u03b2), \u2a06 n, \u2191(F n) (a, b) \u2202\u2191\u03ba a = \u2a06 n, \u222b\u207b (b : \u03b2), \u2191(F n) (a, b) \u2202\u2191\u03ba a", "state_after": "case hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na\u271d : \u03b1\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\nhf : Measurable (uncurry f)\nF : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) \u211d\u22650\u221e := SimpleFunc.eapprox (uncurry f)\nh : \u2200 (a : \u03b1) (b : \u03b2), \u2a06 n, \u2191(SimpleFunc.eapprox (uncurry f) n) (a, b) = f a b\na : \u03b1\n\u22a2 \u2200 (n : \u2115), Measurable fun b => \u2191(F n) (a, b)\n\ncase h_mono\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na\u271d : \u03b1\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\nhf : Measurable (uncurry f)\nF : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) \u211d\u22650\u221e := SimpleFunc.eapprox (uncurry f)\nh : \u2200 (a : \u03b1) (b : \u03b2), \u2a06 n, \u2191(SimpleFunc.eapprox (uncurry f) n) (a, b) = f a b\na : \u03b1\n\u22a2 Monotone fun n b => \u2191(F n) (a, b)"}, {"tactic": "exact fun n => (F n).measurable.comp measurable_prod_mk_left", "annotated_tactic": ["exact fun n => (F n).measurable.comp <a>measurable_prod_mk_left</a>", [{"full_name": "measurable_prod_mk_left", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [736, 9], "def_end_pos": [736, 32]}]], "state_before": "case hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na\u271d : \u03b1\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\nhf : Measurable (uncurry f)\nF : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) \u211d\u22650\u221e := SimpleFunc.eapprox (uncurry f)\nh : \u2200 (a : \u03b1) (b : \u03b2), \u2a06 n, \u2191(SimpleFunc.eapprox (uncurry f) n) (a, b) = f a b\na : \u03b1\n\u22a2 \u2200 (n : \u2115), Measurable fun b => \u2191(F n) (a, b)", "state_after": "no goals"}, {"tactic": "exact fun i j hij b => SimpleFunc.monotone_eapprox (uncurry f) hij _", "annotated_tactic": ["exact fun i j hij b => <a>SimpleFunc.monotone_eapprox</a> (<a>uncurry</a> f) hij _", [{"full_name": "MeasureTheory.SimpleFunc.monotone_eapprox", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [892, 9], "def_end_pos": [892, 25]}, {"full_name": "Function.uncurry", "def_path": "Mathlib/Init/Function.lean", "def_pos": [217, 5], "def_end_pos": [217, 12]}]], "state_before": "case h_mono\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na\u271d : \u03b1\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\nhf : Measurable (uncurry f)\nF : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) \u211d\u22650\u221e := SimpleFunc.eapprox (uncurry f)\nh : \u2200 (a : \u03b1) (b : \u03b2), \u2a06 n, \u2191(SimpleFunc.eapprox (uncurry f) n) (a, b) = f a b\na : \u03b1\n\u22a2 Monotone fun n b => \u2191(F n) (a, b)", "state_after": "no goals"}, {"tactic": "intro c t ht", "annotated_tactic": ["intro c t ht", []], "state_before": "case refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na : \u03b1\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\nhf : Measurable (uncurry f)\nF : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) \u211d\u22650\u221e := SimpleFunc.eapprox (uncurry f)\nh : \u2200 (a : \u03b1) (b : \u03b2), \u2a06 n, \u2191(SimpleFunc.eapprox (uncurry f) n) (a, b) = f a b\nthis : \u2200 (a : \u03b1), \u222b\u207b (b : \u03b2), \u2a06 n, \u2191(F n) (a, b) \u2202\u2191\u03ba a = \u2a06 n, \u222b\u207b (b : \u03b2), \u2191(F n) (a, b) \u2202\u2191\u03ba a\nn : \u2115\n\u22a2 \u2200 (c : \u211d\u22650\u221e) {s : Set (\u03b1 \u00d7 \u03b2)} (hs : MeasurableSet s),\n    (fun f => Measurable fun a => \u222b\u207b (b : \u03b2), \u2191f (a, b) \u2202\u2191\u03ba a)\n      (SimpleFunc.piecewise s hs (SimpleFunc.const (\u03b1 \u00d7 \u03b2) c) (SimpleFunc.const (\u03b1 \u00d7 \u03b2) 0))", "state_after": "case refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na : \u03b1\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\nhf : Measurable (uncurry f)\nF : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) \u211d\u22650\u221e := SimpleFunc.eapprox (uncurry f)\nh : \u2200 (a : \u03b1) (b : \u03b2), \u2a06 n, \u2191(SimpleFunc.eapprox (uncurry f) n) (a, b) = f a b\nthis : \u2200 (a : \u03b1), \u222b\u207b (b : \u03b2), \u2a06 n, \u2191(F n) (a, b) \u2202\u2191\u03ba a = \u2a06 n, \u222b\u207b (b : \u03b2), \u2191(F n) (a, b) \u2202\u2191\u03ba a\nn : \u2115\nc : \u211d\u22650\u221e\nt : Set (\u03b1 \u00d7 \u03b2)\nht : MeasurableSet t\n\u22a2 Measurable fun a =>\n    \u222b\u207b (b : \u03b2), \u2191(SimpleFunc.piecewise t ht (SimpleFunc.const (\u03b1 \u00d7 \u03b2) c) (SimpleFunc.const (\u03b1 \u00d7 \u03b2) 0)) (a, b) \u2202\u2191\u03ba a"}, {"tactic": "simp only [SimpleFunc.const_zero, SimpleFunc.coe_piecewise, SimpleFunc.coe_const,\n  SimpleFunc.coe_zero, Set.piecewise_eq_indicator]", "annotated_tactic": ["simp only [<a>SimpleFunc.const_zero</a>, <a>SimpleFunc.coe_piecewise</a>, <a>SimpleFunc.coe_const</a>,\n      <a>SimpleFunc.coe_zero</a>, <a>Set.piecewise_eq_indicator</a>]", [{"full_name": "MeasureTheory.SimpleFunc.const_zero", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [457, 3], "def_end_pos": [457, 14]}, {"full_name": "MeasureTheory.SimpleFunc.coe_piecewise", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [230, 9], "def_end_pos": [230, 22]}, {"full_name": "MeasureTheory.SimpleFunc.coe_const", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [158, 9], "def_end_pos": [158, 18]}, {"full_name": "MeasureTheory.SimpleFunc.coe_zero", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [463, 3], "def_end_pos": [463, 14]}, {"full_name": "Set.piecewise_eq_indicator", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [52, 3], "def_end_pos": [52, 14]}]], "state_before": "case refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na : \u03b1\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\nhf : Measurable (uncurry f)\nF : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) \u211d\u22650\u221e := SimpleFunc.eapprox (uncurry f)\nh : \u2200 (a : \u03b1) (b : \u03b2), \u2a06 n, \u2191(SimpleFunc.eapprox (uncurry f) n) (a, b) = f a b\nthis : \u2200 (a : \u03b1), \u222b\u207b (b : \u03b2), \u2a06 n, \u2191(F n) (a, b) \u2202\u2191\u03ba a = \u2a06 n, \u222b\u207b (b : \u03b2), \u2191(F n) (a, b) \u2202\u2191\u03ba a\nn : \u2115\nc : \u211d\u22650\u221e\nt : Set (\u03b1 \u00d7 \u03b2)\nht : MeasurableSet t\n\u22a2 Measurable fun a =>\n    \u222b\u207b (b : \u03b2), \u2191(SimpleFunc.piecewise t ht (SimpleFunc.const (\u03b1 \u00d7 \u03b2) c) (SimpleFunc.const (\u03b1 \u00d7 \u03b2) 0)) (a, b) \u2202\u2191\u03ba a", "state_after": "case refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na : \u03b1\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\nhf : Measurable (uncurry f)\nF : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) \u211d\u22650\u221e := SimpleFunc.eapprox (uncurry f)\nh : \u2200 (a : \u03b1) (b : \u03b2), \u2a06 n, \u2191(SimpleFunc.eapprox (uncurry f) n) (a, b) = f a b\nthis : \u2200 (a : \u03b1), \u222b\u207b (b : \u03b2), \u2a06 n, \u2191(F n) (a, b) \u2202\u2191\u03ba a = \u2a06 n, \u222b\u207b (b : \u03b2), \u2191(F n) (a, b) \u2202\u2191\u03ba a\nn : \u2115\nc : \u211d\u22650\u221e\nt : Set (\u03b1 \u00d7 \u03b2)\nht : MeasurableSet t\n\u22a2 Measurable fun a => \u222b\u207b (b : \u03b2), Set.piecewise t (Function.const (\u03b1 \u00d7 \u03b2) c) 0 (a, b) \u2202\u2191\u03ba a"}, {"tactic": "exact kernel.measurable_lintegral_indicator_const (\u03ba := \u03ba) ht c", "annotated_tactic": ["exact <a>kernel.measurable_lintegral_indicator_const</a> (\u03ba := \u03ba) ht c", [{"full_name": "ProbabilityTheory.kernel.measurable_lintegral_indicator_const", "def_path": "Mathlib/Probability/Kernel/MeasurableIntegral.lean", "def_pos": [134, 9], "def_end_pos": [134, 52]}]], "state_before": "case refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na : \u03b1\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\nhf : Measurable (uncurry f)\nF : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) \u211d\u22650\u221e := SimpleFunc.eapprox (uncurry f)\nh : \u2200 (a : \u03b1) (b : \u03b2), \u2a06 n, \u2191(SimpleFunc.eapprox (uncurry f) n) (a, b) = f a b\nthis : \u2200 (a : \u03b1), \u222b\u207b (b : \u03b2), \u2a06 n, \u2191(F n) (a, b) \u2202\u2191\u03ba a = \u2a06 n, \u222b\u207b (b : \u03b2), \u2191(F n) (a, b) \u2202\u2191\u03ba a\nn : \u2115\nc : \u211d\u22650\u221e\nt : Set (\u03b1 \u00d7 \u03b2)\nht : MeasurableSet t\n\u22a2 Measurable fun a => \u222b\u207b (b : \u03b2), Set.piecewise t (Function.const (\u03b1 \u00d7 \u03b2) c) 0 (a, b) \u2202\u2191\u03ba a", "state_after": "no goals"}, {"tactic": "intro g\u2081 g\u2082 _ hm\u2081 hm\u2082", "annotated_tactic": ["intro g\u2081 g\u2082 _ hm\u2081 hm\u2082", []], "state_before": "case refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na : \u03b1\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\nhf : Measurable (uncurry f)\nF : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) \u211d\u22650\u221e := SimpleFunc.eapprox (uncurry f)\nh : \u2200 (a : \u03b1) (b : \u03b2), \u2a06 n, \u2191(SimpleFunc.eapprox (uncurry f) n) (a, b) = f a b\nthis : \u2200 (a : \u03b1), \u222b\u207b (b : \u03b2), \u2a06 n, \u2191(F n) (a, b) \u2202\u2191\u03ba a = \u2a06 n, \u222b\u207b (b : \u03b2), \u2191(F n) (a, b) \u2202\u2191\u03ba a\nn : \u2115\n\u22a2 \u2200 \u2983f g : SimpleFunc (\u03b1 \u00d7 \u03b2) \u211d\u22650\u221e\u2984,\n    Disjoint (support \u2191f) (support \u2191g) \u2192\n      (fun f => Measurable fun a => \u222b\u207b (b : \u03b2), \u2191f (a, b) \u2202\u2191\u03ba a) f \u2192\n        (fun f => Measurable fun a => \u222b\u207b (b : \u03b2), \u2191f (a, b) \u2202\u2191\u03ba a) g \u2192\n          (fun f => Measurable fun a => \u222b\u207b (b : \u03b2), \u2191f (a, b) \u2202\u2191\u03ba a) (f + g)", "state_after": "case refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na : \u03b1\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\nhf : Measurable (uncurry f)\nF : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) \u211d\u22650\u221e := SimpleFunc.eapprox (uncurry f)\nh : \u2200 (a : \u03b1) (b : \u03b2), \u2a06 n, \u2191(SimpleFunc.eapprox (uncurry f) n) (a, b) = f a b\nthis : \u2200 (a : \u03b1), \u222b\u207b (b : \u03b2), \u2a06 n, \u2191(F n) (a, b) \u2202\u2191\u03ba a = \u2a06 n, \u222b\u207b (b : \u03b2), \u2191(F n) (a, b) \u2202\u2191\u03ba a\nn : \u2115\ng\u2081 g\u2082 : SimpleFunc (\u03b1 \u00d7 \u03b2) \u211d\u22650\u221e\na\u271d : Disjoint (support \u2191g\u2081) (support \u2191g\u2082)\nhm\u2081 : Measurable fun a => \u222b\u207b (b : \u03b2), \u2191g\u2081 (a, b) \u2202\u2191\u03ba a\nhm\u2082 : Measurable fun a => \u222b\u207b (b : \u03b2), \u2191g\u2082 (a, b) \u2202\u2191\u03ba a\n\u22a2 Measurable fun a => \u222b\u207b (b : \u03b2), \u2191(g\u2081 + g\u2082) (a, b) \u2202\u2191\u03ba a"}, {"tactic": "simp only [SimpleFunc.coe_add, Pi.add_apply]", "annotated_tactic": ["simp only [<a>SimpleFunc.coe_add</a>, <a>Pi.add_apply</a>]", [{"full_name": "MeasureTheory.SimpleFunc.coe_add", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [469, 3], "def_end_pos": [469, 14]}, {"full_name": "Pi.add_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [82, 3], "def_end_pos": [82, 14]}]], "state_before": "case refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na : \u03b1\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\nhf : Measurable (uncurry f)\nF : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) \u211d\u22650\u221e := SimpleFunc.eapprox (uncurry f)\nh : \u2200 (a : \u03b1) (b : \u03b2), \u2a06 n, \u2191(SimpleFunc.eapprox (uncurry f) n) (a, b) = f a b\nthis : \u2200 (a : \u03b1), \u222b\u207b (b : \u03b2), \u2a06 n, \u2191(F n) (a, b) \u2202\u2191\u03ba a = \u2a06 n, \u222b\u207b (b : \u03b2), \u2191(F n) (a, b) \u2202\u2191\u03ba a\nn : \u2115\ng\u2081 g\u2082 : SimpleFunc (\u03b1 \u00d7 \u03b2) \u211d\u22650\u221e\na\u271d : Disjoint (support \u2191g\u2081) (support \u2191g\u2082)\nhm\u2081 : Measurable fun a => \u222b\u207b (b : \u03b2), \u2191g\u2081 (a, b) \u2202\u2191\u03ba a\nhm\u2082 : Measurable fun a => \u222b\u207b (b : \u03b2), \u2191g\u2082 (a, b) \u2202\u2191\u03ba a\n\u22a2 Measurable fun a => \u222b\u207b (b : \u03b2), \u2191(g\u2081 + g\u2082) (a, b) \u2202\u2191\u03ba a", "state_after": "case refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na : \u03b1\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\nhf : Measurable (uncurry f)\nF : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) \u211d\u22650\u221e := SimpleFunc.eapprox (uncurry f)\nh : \u2200 (a : \u03b1) (b : \u03b2), \u2a06 n, \u2191(SimpleFunc.eapprox (uncurry f) n) (a, b) = f a b\nthis : \u2200 (a : \u03b1), \u222b\u207b (b : \u03b2), \u2a06 n, \u2191(F n) (a, b) \u2202\u2191\u03ba a = \u2a06 n, \u222b\u207b (b : \u03b2), \u2191(F n) (a, b) \u2202\u2191\u03ba a\nn : \u2115\ng\u2081 g\u2082 : SimpleFunc (\u03b1 \u00d7 \u03b2) \u211d\u22650\u221e\na\u271d : Disjoint (support \u2191g\u2081) (support \u2191g\u2082)\nhm\u2081 : Measurable fun a => \u222b\u207b (b : \u03b2), \u2191g\u2081 (a, b) \u2202\u2191\u03ba a\nhm\u2082 : Measurable fun a => \u222b\u207b (b : \u03b2), \u2191g\u2082 (a, b) \u2202\u2191\u03ba a\n\u22a2 Measurable fun a => \u222b\u207b (b : \u03b2), \u2191g\u2081 (a, b) + \u2191g\u2082 (a, b) \u2202\u2191\u03ba a"}, {"tactic": "have h_add :\n  (fun a => \u222b\u207b b, g\u2081 (a, b) + g\u2082 (a, b) \u2202\u03ba a) =\n    (fun a => \u222b\u207b b, g\u2081 (a, b) \u2202\u03ba a) + fun a => \u222b\u207b b, g\u2082 (a, b) \u2202\u03ba a := by\n  ext1 a\n  rw [Pi.add_apply]\n  erw [lintegral_add_left (g\u2081.measurable.comp measurable_prod_mk_left)]\n  simp_rw [Function.comp_apply]", "annotated_tactic": ["have h_add :\n      (fun a => \u222b\u207b b, g\u2081 (a, b) + g\u2082 (a, b) \u2202\u03ba a) =\n        (fun a => \u222b\u207b b, g\u2081 (a, b) \u2202\u03ba a) + fun a => \u222b\u207b b, g\u2082 (a, b) \u2202\u03ba a := by\n      ext1 a\n      rw [<a>Pi.add_apply</a>]\n      -- Porting note: was `rw` (`Function.comp` reducibility)\n      erw [<a>lintegral_add_left</a> (g\u2081.measurable.comp <a>measurable_prod_mk_left</a>)]\n      simp_rw [<a>Function.comp_apply</a>]", [{"full_name": "Pi.add_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [82, 3], "def_end_pos": [82, 14]}, {"full_name": "MeasureTheory.lintegral_add_left", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [554, 9], "def_end_pos": [554, 27]}, {"full_name": "measurable_prod_mk_left", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [736, 9], "def_end_pos": [736, 32]}, {"full_name": "Function.comp_apply", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [33, 17], "def_end_pos": [33, 36]}]], "state_before": "case refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na : \u03b1\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\nhf : Measurable (uncurry f)\nF : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) \u211d\u22650\u221e := SimpleFunc.eapprox (uncurry f)\nh : \u2200 (a : \u03b1) (b : \u03b2), \u2a06 n, \u2191(SimpleFunc.eapprox (uncurry f) n) (a, b) = f a b\nthis : \u2200 (a : \u03b1), \u222b\u207b (b : \u03b2), \u2a06 n, \u2191(F n) (a, b) \u2202\u2191\u03ba a = \u2a06 n, \u222b\u207b (b : \u03b2), \u2191(F n) (a, b) \u2202\u2191\u03ba a\nn : \u2115\ng\u2081 g\u2082 : SimpleFunc (\u03b1 \u00d7 \u03b2) \u211d\u22650\u221e\na\u271d : Disjoint (support \u2191g\u2081) (support \u2191g\u2082)\nhm\u2081 : Measurable fun a => \u222b\u207b (b : \u03b2), \u2191g\u2081 (a, b) \u2202\u2191\u03ba a\nhm\u2082 : Measurable fun a => \u222b\u207b (b : \u03b2), \u2191g\u2082 (a, b) \u2202\u2191\u03ba a\n\u22a2 Measurable fun a => \u222b\u207b (b : \u03b2), \u2191g\u2081 (a, b) + \u2191g\u2082 (a, b) \u2202\u2191\u03ba a", "state_after": "case refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na : \u03b1\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\nhf : Measurable (uncurry f)\nF : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) \u211d\u22650\u221e := SimpleFunc.eapprox (uncurry f)\nh : \u2200 (a : \u03b1) (b : \u03b2), \u2a06 n, \u2191(SimpleFunc.eapprox (uncurry f) n) (a, b) = f a b\nthis : \u2200 (a : \u03b1), \u222b\u207b (b : \u03b2), \u2a06 n, \u2191(F n) (a, b) \u2202\u2191\u03ba a = \u2a06 n, \u222b\u207b (b : \u03b2), \u2191(F n) (a, b) \u2202\u2191\u03ba a\nn : \u2115\ng\u2081 g\u2082 : SimpleFunc (\u03b1 \u00d7 \u03b2) \u211d\u22650\u221e\na\u271d : Disjoint (support \u2191g\u2081) (support \u2191g\u2082)\nhm\u2081 : Measurable fun a => \u222b\u207b (b : \u03b2), \u2191g\u2081 (a, b) \u2202\u2191\u03ba a\nhm\u2082 : Measurable fun a => \u222b\u207b (b : \u03b2), \u2191g\u2082 (a, b) \u2202\u2191\u03ba a\nh_add :\n  (fun a => \u222b\u207b (b : \u03b2), \u2191g\u2081 (a, b) + \u2191g\u2082 (a, b) \u2202\u2191\u03ba a) =\n    (fun a => \u222b\u207b (b : \u03b2), \u2191g\u2081 (a, b) \u2202\u2191\u03ba a) + fun a => \u222b\u207b (b : \u03b2), \u2191g\u2082 (a, b) \u2202\u2191\u03ba a\n\u22a2 Measurable fun a => \u222b\u207b (b : \u03b2), \u2191g\u2081 (a, b) + \u2191g\u2082 (a, b) \u2202\u2191\u03ba a"}, {"tactic": "rw [h_add]", "annotated_tactic": ["rw [h_add]", []], "state_before": "case refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na : \u03b1\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\nhf : Measurable (uncurry f)\nF : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) \u211d\u22650\u221e := SimpleFunc.eapprox (uncurry f)\nh : \u2200 (a : \u03b1) (b : \u03b2), \u2a06 n, \u2191(SimpleFunc.eapprox (uncurry f) n) (a, b) = f a b\nthis : \u2200 (a : \u03b1), \u222b\u207b (b : \u03b2), \u2a06 n, \u2191(F n) (a, b) \u2202\u2191\u03ba a = \u2a06 n, \u222b\u207b (b : \u03b2), \u2191(F n) (a, b) \u2202\u2191\u03ba a\nn : \u2115\ng\u2081 g\u2082 : SimpleFunc (\u03b1 \u00d7 \u03b2) \u211d\u22650\u221e\na\u271d : Disjoint (support \u2191g\u2081) (support \u2191g\u2082)\nhm\u2081 : Measurable fun a => \u222b\u207b (b : \u03b2), \u2191g\u2081 (a, b) \u2202\u2191\u03ba a\nhm\u2082 : Measurable fun a => \u222b\u207b (b : \u03b2), \u2191g\u2082 (a, b) \u2202\u2191\u03ba a\nh_add :\n  (fun a => \u222b\u207b (b : \u03b2), \u2191g\u2081 (a, b) + \u2191g\u2082 (a, b) \u2202\u2191\u03ba a) =\n    (fun a => \u222b\u207b (b : \u03b2), \u2191g\u2081 (a, b) \u2202\u2191\u03ba a) + fun a => \u222b\u207b (b : \u03b2), \u2191g\u2082 (a, b) \u2202\u2191\u03ba a\n\u22a2 Measurable fun a => \u222b\u207b (b : \u03b2), \u2191g\u2081 (a, b) + \u2191g\u2082 (a, b) \u2202\u2191\u03ba a", "state_after": "case refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na : \u03b1\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\nhf : Measurable (uncurry f)\nF : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) \u211d\u22650\u221e := SimpleFunc.eapprox (uncurry f)\nh : \u2200 (a : \u03b1) (b : \u03b2), \u2a06 n, \u2191(SimpleFunc.eapprox (uncurry f) n) (a, b) = f a b\nthis : \u2200 (a : \u03b1), \u222b\u207b (b : \u03b2), \u2a06 n, \u2191(F n) (a, b) \u2202\u2191\u03ba a = \u2a06 n, \u222b\u207b (b : \u03b2), \u2191(F n) (a, b) \u2202\u2191\u03ba a\nn : \u2115\ng\u2081 g\u2082 : SimpleFunc (\u03b1 \u00d7 \u03b2) \u211d\u22650\u221e\na\u271d : Disjoint (support \u2191g\u2081) (support \u2191g\u2082)\nhm\u2081 : Measurable fun a => \u222b\u207b (b : \u03b2), \u2191g\u2081 (a, b) \u2202\u2191\u03ba a\nhm\u2082 : Measurable fun a => \u222b\u207b (b : \u03b2), \u2191g\u2082 (a, b) \u2202\u2191\u03ba a\nh_add :\n  (fun a => \u222b\u207b (b : \u03b2), \u2191g\u2081 (a, b) + \u2191g\u2082 (a, b) \u2202\u2191\u03ba a) =\n    (fun a => \u222b\u207b (b : \u03b2), \u2191g\u2081 (a, b) \u2202\u2191\u03ba a) + fun a => \u222b\u207b (b : \u03b2), \u2191g\u2082 (a, b) \u2202\u2191\u03ba a\n\u22a2 Measurable ((fun a => \u222b\u207b (b : \u03b2), \u2191g\u2081 (a, b) \u2202\u2191\u03ba a) + fun a => \u222b\u207b (b : \u03b2), \u2191g\u2082 (a, b) \u2202\u2191\u03ba a)"}, {"tactic": "exact Measurable.add hm\u2081 hm\u2082", "annotated_tactic": ["exact <a>Measurable.add</a> hm\u2081 hm\u2082", [{"full_name": "Measurable.add", "def_path": "Mathlib/MeasureTheory/Group/Arithmetic.lean", "def_pos": [140, 3], "def_end_pos": [140, 14]}]], "state_before": "case refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na : \u03b1\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\nhf : Measurable (uncurry f)\nF : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) \u211d\u22650\u221e := SimpleFunc.eapprox (uncurry f)\nh : \u2200 (a : \u03b1) (b : \u03b2), \u2a06 n, \u2191(SimpleFunc.eapprox (uncurry f) n) (a, b) = f a b\nthis : \u2200 (a : \u03b1), \u222b\u207b (b : \u03b2), \u2a06 n, \u2191(F n) (a, b) \u2202\u2191\u03ba a = \u2a06 n, \u222b\u207b (b : \u03b2), \u2191(F n) (a, b) \u2202\u2191\u03ba a\nn : \u2115\ng\u2081 g\u2082 : SimpleFunc (\u03b1 \u00d7 \u03b2) \u211d\u22650\u221e\na\u271d : Disjoint (support \u2191g\u2081) (support \u2191g\u2082)\nhm\u2081 : Measurable fun a => \u222b\u207b (b : \u03b2), \u2191g\u2081 (a, b) \u2202\u2191\u03ba a\nhm\u2082 : Measurable fun a => \u222b\u207b (b : \u03b2), \u2191g\u2082 (a, b) \u2202\u2191\u03ba a\nh_add :\n  (fun a => \u222b\u207b (b : \u03b2), \u2191g\u2081 (a, b) + \u2191g\u2082 (a, b) \u2202\u2191\u03ba a) =\n    (fun a => \u222b\u207b (b : \u03b2), \u2191g\u2081 (a, b) \u2202\u2191\u03ba a) + fun a => \u222b\u207b (b : \u03b2), \u2191g\u2082 (a, b) \u2202\u2191\u03ba a\n\u22a2 Measurable ((fun a => \u222b\u207b (b : \u03b2), \u2191g\u2081 (a, b) \u2202\u2191\u03ba a) + fun a => \u222b\u207b (b : \u03b2), \u2191g\u2082 (a, b) \u2202\u2191\u03ba a)", "state_after": "no goals"}, {"tactic": "ext1 a", "annotated_tactic": ["ext1 a", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na : \u03b1\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\nhf : Measurable (uncurry f)\nF : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) \u211d\u22650\u221e := SimpleFunc.eapprox (uncurry f)\nh : \u2200 (a : \u03b1) (b : \u03b2), \u2a06 n, \u2191(SimpleFunc.eapprox (uncurry f) n) (a, b) = f a b\nthis : \u2200 (a : \u03b1), \u222b\u207b (b : \u03b2), \u2a06 n, \u2191(F n) (a, b) \u2202\u2191\u03ba a = \u2a06 n, \u222b\u207b (b : \u03b2), \u2191(F n) (a, b) \u2202\u2191\u03ba a\nn : \u2115\ng\u2081 g\u2082 : SimpleFunc (\u03b1 \u00d7 \u03b2) \u211d\u22650\u221e\na\u271d : Disjoint (support \u2191g\u2081) (support \u2191g\u2082)\nhm\u2081 : Measurable fun a => \u222b\u207b (b : \u03b2), \u2191g\u2081 (a, b) \u2202\u2191\u03ba a\nhm\u2082 : Measurable fun a => \u222b\u207b (b : \u03b2), \u2191g\u2082 (a, b) \u2202\u2191\u03ba a\n\u22a2 (fun a => \u222b\u207b (b : \u03b2), \u2191g\u2081 (a, b) + \u2191g\u2082 (a, b) \u2202\u2191\u03ba a) =\n    (fun a => \u222b\u207b (b : \u03b2), \u2191g\u2081 (a, b) \u2202\u2191\u03ba a) + fun a => \u222b\u207b (b : \u03b2), \u2191g\u2082 (a, b) \u2202\u2191\u03ba a", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na\u271d\u00b9 : \u03b1\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\nhf : Measurable (uncurry f)\nF : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) \u211d\u22650\u221e := SimpleFunc.eapprox (uncurry f)\nh : \u2200 (a : \u03b1) (b : \u03b2), \u2a06 n, \u2191(SimpleFunc.eapprox (uncurry f) n) (a, b) = f a b\nthis : \u2200 (a : \u03b1), \u222b\u207b (b : \u03b2), \u2a06 n, \u2191(F n) (a, b) \u2202\u2191\u03ba a = \u2a06 n, \u222b\u207b (b : \u03b2), \u2191(F n) (a, b) \u2202\u2191\u03ba a\nn : \u2115\ng\u2081 g\u2082 : SimpleFunc (\u03b1 \u00d7 \u03b2) \u211d\u22650\u221e\na\u271d : Disjoint (support \u2191g\u2081) (support \u2191g\u2082)\nhm\u2081 : Measurable fun a => \u222b\u207b (b : \u03b2), \u2191g\u2081 (a, b) \u2202\u2191\u03ba a\nhm\u2082 : Measurable fun a => \u222b\u207b (b : \u03b2), \u2191g\u2082 (a, b) \u2202\u2191\u03ba a\na : \u03b1\n\u22a2 \u222b\u207b (b : \u03b2), \u2191g\u2081 (a, b) + \u2191g\u2082 (a, b) \u2202\u2191\u03ba a =\n    ((fun a => \u222b\u207b (b : \u03b2), \u2191g\u2081 (a, b) \u2202\u2191\u03ba a) + fun a => \u222b\u207b (b : \u03b2), \u2191g\u2082 (a, b) \u2202\u2191\u03ba a) a"}, {"tactic": "rw [Pi.add_apply]", "annotated_tactic": ["rw [<a>Pi.add_apply</a>]", [{"full_name": "Pi.add_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [82, 3], "def_end_pos": [82, 14]}]], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na\u271d\u00b9 : \u03b1\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\nhf : Measurable (uncurry f)\nF : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) \u211d\u22650\u221e := SimpleFunc.eapprox (uncurry f)\nh : \u2200 (a : \u03b1) (b : \u03b2), \u2a06 n, \u2191(SimpleFunc.eapprox (uncurry f) n) (a, b) = f a b\nthis : \u2200 (a : \u03b1), \u222b\u207b (b : \u03b2), \u2a06 n, \u2191(F n) (a, b) \u2202\u2191\u03ba a = \u2a06 n, \u222b\u207b (b : \u03b2), \u2191(F n) (a, b) \u2202\u2191\u03ba a\nn : \u2115\ng\u2081 g\u2082 : SimpleFunc (\u03b1 \u00d7 \u03b2) \u211d\u22650\u221e\na\u271d : Disjoint (support \u2191g\u2081) (support \u2191g\u2082)\nhm\u2081 : Measurable fun a => \u222b\u207b (b : \u03b2), \u2191g\u2081 (a, b) \u2202\u2191\u03ba a\nhm\u2082 : Measurable fun a => \u222b\u207b (b : \u03b2), \u2191g\u2082 (a, b) \u2202\u2191\u03ba a\na : \u03b1\n\u22a2 \u222b\u207b (b : \u03b2), \u2191g\u2081 (a, b) + \u2191g\u2082 (a, b) \u2202\u2191\u03ba a =\n    ((fun a => \u222b\u207b (b : \u03b2), \u2191g\u2081 (a, b) \u2202\u2191\u03ba a) + fun a => \u222b\u207b (b : \u03b2), \u2191g\u2082 (a, b) \u2202\u2191\u03ba a) a", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na\u271d\u00b9 : \u03b1\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\nhf : Measurable (uncurry f)\nF : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) \u211d\u22650\u221e := SimpleFunc.eapprox (uncurry f)\nh : \u2200 (a : \u03b1) (b : \u03b2), \u2a06 n, \u2191(SimpleFunc.eapprox (uncurry f) n) (a, b) = f a b\nthis : \u2200 (a : \u03b1), \u222b\u207b (b : \u03b2), \u2a06 n, \u2191(F n) (a, b) \u2202\u2191\u03ba a = \u2a06 n, \u222b\u207b (b : \u03b2), \u2191(F n) (a, b) \u2202\u2191\u03ba a\nn : \u2115\ng\u2081 g\u2082 : SimpleFunc (\u03b1 \u00d7 \u03b2) \u211d\u22650\u221e\na\u271d : Disjoint (support \u2191g\u2081) (support \u2191g\u2082)\nhm\u2081 : Measurable fun a => \u222b\u207b (b : \u03b2), \u2191g\u2081 (a, b) \u2202\u2191\u03ba a\nhm\u2082 : Measurable fun a => \u222b\u207b (b : \u03b2), \u2191g\u2082 (a, b) \u2202\u2191\u03ba a\na : \u03b1\n\u22a2 \u222b\u207b (b : \u03b2), \u2191g\u2081 (a, b) + \u2191g\u2082 (a, b) \u2202\u2191\u03ba a = \u222b\u207b (b : \u03b2), \u2191g\u2081 (a, b) \u2202\u2191\u03ba a + \u222b\u207b (b : \u03b2), \u2191g\u2082 (a, b) \u2202\u2191\u03ba a"}, {"tactic": "erw [lintegral_add_left (g\u2081.measurable.comp measurable_prod_mk_left)]", "annotated_tactic": ["erw [<a>lintegral_add_left</a> (g\u2081.measurable.comp <a>measurable_prod_mk_left</a>)]", [{"full_name": "MeasureTheory.lintegral_add_left", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [554, 9], "def_end_pos": [554, 27]}, {"full_name": "measurable_prod_mk_left", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [736, 9], "def_end_pos": [736, 32]}]], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na\u271d\u00b9 : \u03b1\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\nhf : Measurable (uncurry f)\nF : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) \u211d\u22650\u221e := SimpleFunc.eapprox (uncurry f)\nh : \u2200 (a : \u03b1) (b : \u03b2), \u2a06 n, \u2191(SimpleFunc.eapprox (uncurry f) n) (a, b) = f a b\nthis : \u2200 (a : \u03b1), \u222b\u207b (b : \u03b2), \u2a06 n, \u2191(F n) (a, b) \u2202\u2191\u03ba a = \u2a06 n, \u222b\u207b (b : \u03b2), \u2191(F n) (a, b) \u2202\u2191\u03ba a\nn : \u2115\ng\u2081 g\u2082 : SimpleFunc (\u03b1 \u00d7 \u03b2) \u211d\u22650\u221e\na\u271d : Disjoint (support \u2191g\u2081) (support \u2191g\u2082)\nhm\u2081 : Measurable fun a => \u222b\u207b (b : \u03b2), \u2191g\u2081 (a, b) \u2202\u2191\u03ba a\nhm\u2082 : Measurable fun a => \u222b\u207b (b : \u03b2), \u2191g\u2082 (a, b) \u2202\u2191\u03ba a\na : \u03b1\n\u22a2 \u222b\u207b (b : \u03b2), \u2191g\u2081 (a, b) + \u2191g\u2082 (a, b) \u2202\u2191\u03ba a = \u222b\u207b (b : \u03b2), \u2191g\u2081 (a, b) \u2202\u2191\u03ba a + \u222b\u207b (b : \u03b2), \u2191g\u2082 (a, b) \u2202\u2191\u03ba a", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na\u271d\u00b9 : \u03b1\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\nhf : Measurable (uncurry f)\nF : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) \u211d\u22650\u221e := SimpleFunc.eapprox (uncurry f)\nh : \u2200 (a : \u03b1) (b : \u03b2), \u2a06 n, \u2191(SimpleFunc.eapprox (uncurry f) n) (a, b) = f a b\nthis : \u2200 (a : \u03b1), \u222b\u207b (b : \u03b2), \u2a06 n, \u2191(F n) (a, b) \u2202\u2191\u03ba a = \u2a06 n, \u222b\u207b (b : \u03b2), \u2191(F n) (a, b) \u2202\u2191\u03ba a\nn : \u2115\ng\u2081 g\u2082 : SimpleFunc (\u03b1 \u00d7 \u03b2) \u211d\u22650\u221e\na\u271d : Disjoint (support \u2191g\u2081) (support \u2191g\u2082)\nhm\u2081 : Measurable fun a => \u222b\u207b (b : \u03b2), \u2191g\u2081 (a, b) \u2202\u2191\u03ba a\nhm\u2082 : Measurable fun a => \u222b\u207b (b : \u03b2), \u2191g\u2082 (a, b) \u2202\u2191\u03ba a\na : \u03b1\n\u22a2 \u222b\u207b (a_1 : \u03b2), (\u2191g\u2081 \u2218 Prod.mk a) a_1 \u2202\u2191\u03ba a + \u222b\u207b (a_1 : \u03b2), \u2191g\u2082 (a, a_1) \u2202\u2191\u03ba a =\n    \u222b\u207b (b : \u03b2), \u2191g\u2081 (a, b) \u2202\u2191\u03ba a + \u222b\u207b (b : \u03b2), \u2191g\u2082 (a, b) \u2202\u2191\u03ba a"}, {"tactic": "simp_rw [Function.comp_apply]", "annotated_tactic": ["simp_rw [<a>Function.comp_apply</a>]", [{"full_name": "Function.comp_apply", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [33, 17], "def_end_pos": [33, 36]}]], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na\u271d\u00b9 : \u03b1\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\nf : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\nhf : Measurable (uncurry f)\nF : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) \u211d\u22650\u221e := SimpleFunc.eapprox (uncurry f)\nh : \u2200 (a : \u03b1) (b : \u03b2), \u2a06 n, \u2191(SimpleFunc.eapprox (uncurry f) n) (a, b) = f a b\nthis : \u2200 (a : \u03b1), \u222b\u207b (b : \u03b2), \u2a06 n, \u2191(F n) (a, b) \u2202\u2191\u03ba a = \u2a06 n, \u222b\u207b (b : \u03b2), \u2191(F n) (a, b) \u2202\u2191\u03ba a\nn : \u2115\ng\u2081 g\u2082 : SimpleFunc (\u03b1 \u00d7 \u03b2) \u211d\u22650\u221e\na\u271d : Disjoint (support \u2191g\u2081) (support \u2191g\u2082)\nhm\u2081 : Measurable fun a => \u222b\u207b (b : \u03b2), \u2191g\u2081 (a, b) \u2202\u2191\u03ba a\nhm\u2082 : Measurable fun a => \u222b\u207b (b : \u03b2), \u2191g\u2082 (a, b) \u2202\u2191\u03ba a\na : \u03b1\n\u22a2 \u222b\u207b (a_1 : \u03b2), (\u2191g\u2081 \u2218 Prod.mk a) a_1 \u2202\u2191\u03ba a + \u222b\u207b (a_1 : \u03b2), \u2191g\u2082 (a, a_1) \u2202\u2191\u03ba a =\n    \u222b\u207b (b : \u03b2), \u2191g\u2081 (a, b) \u2202\u2191\u03ba a + \u222b\u207b (b : \u03b2), \u2191g\u2082 (a, b) \u2202\u2191\u03ba a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "full_name": "MeasureTheory.norm_set_integral_le_of_norm_le_const_ae", "start": [548, 1], "end": [552, 45], "traced_tactics": [{"tactic": "rw [\u2190 Measure.restrict_apply_univ] at *", "annotated_tactic": ["rw [\u2190 <a>Measure.restrict_apply_univ</a>] at *", [{"full_name": "MeasureTheory.Measure.restrict_apply_univ", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1604, 9], "def_end_pos": [1604, 28]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\nf g : \u03b1 \u2192 E\ns t : Set \u03b1\n\u03bc \u03bd : Measure \u03b1\nl l' : Filter \u03b1\ninst\u271d : NormedSpace \u211d E\nC : \u211d\nhs : \u2191\u2191\u03bc s < \u22a4\nhC : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc s, \u2016f x\u2016 \u2264 C\n\u22a2 \u2016\u222b (x : \u03b1) in s, f x \u2202\u03bc\u2016 \u2264 C * ENNReal.toReal (\u2191\u2191\u03bc s)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\nf g : \u03b1 \u2192 E\ns t : Set \u03b1\n\u03bc \u03bd : Measure \u03b1\nl l' : Filter \u03b1\ninst\u271d : NormedSpace \u211d E\nC : \u211d\nhs : \u2191\u2191(Measure.restrict \u03bc s) univ < \u22a4\nhC : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc s, \u2016f x\u2016 \u2264 C\n\u22a2 \u2016\u222b (x : \u03b1) in s, f x \u2202\u03bc\u2016 \u2264 C * ENNReal.toReal (\u2191\u2191(Measure.restrict \u03bc s) univ)"}, {"tactic": "haveI : IsFiniteMeasure (\u03bc.restrict s) := \u27e8\u2039_\u203a\u27e9", "annotated_tactic": ["haveI : <a>IsFiniteMeasure</a> (\u03bc.restrict s) := \u27e8\u2039_\u203a\u27e9", [{"full_name": "MeasureTheory.IsFiniteMeasure", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2850, 7], "def_end_pos": [2850, 22]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\nf g : \u03b1 \u2192 E\ns t : Set \u03b1\n\u03bc \u03bd : Measure \u03b1\nl l' : Filter \u03b1\ninst\u271d : NormedSpace \u211d E\nC : \u211d\nhs : \u2191\u2191(Measure.restrict \u03bc s) univ < \u22a4\nhC : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc s, \u2016f x\u2016 \u2264 C\n\u22a2 \u2016\u222b (x : \u03b1) in s, f x \u2202\u03bc\u2016 \u2264 C * ENNReal.toReal (\u2191\u2191(Measure.restrict \u03bc s) univ)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\nf g : \u03b1 \u2192 E\ns t : Set \u03b1\n\u03bc \u03bd : Measure \u03b1\nl l' : Filter \u03b1\ninst\u271d : NormedSpace \u211d E\nC : \u211d\nhs : \u2191\u2191(Measure.restrict \u03bc s) univ < \u22a4\nhC : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc s, \u2016f x\u2016 \u2264 C\nthis : IsFiniteMeasure (Measure.restrict \u03bc s)\n\u22a2 \u2016\u222b (x : \u03b1) in s, f x \u2202\u03bc\u2016 \u2264 C * ENNReal.toReal (\u2191\u2191(Measure.restrict \u03bc s) univ)"}, {"tactic": "exact norm_integral_le_of_norm_le_const hC", "annotated_tactic": ["exact <a>norm_integral_le_of_norm_le_const</a> hC", [{"full_name": "MeasureTheory.norm_integral_le_of_norm_le_const", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1422, 9], "def_end_pos": [1422, 42]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\nf g : \u03b1 \u2192 E\ns t : Set \u03b1\n\u03bc \u03bd : Measure \u03b1\nl l' : Filter \u03b1\ninst\u271d : NormedSpace \u211d E\nC : \u211d\nhs : \u2191\u2191(Measure.restrict \u03bc s) univ < \u22a4\nhC : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc s, \u2016f x\u2016 \u2264 C\nthis : IsFiniteMeasure (Measure.restrict \u03bc s)\n\u22a2 \u2016\u222b (x : \u03b1) in s, f x \u2202\u03bc\u2016 \u2264 C * ENNReal.toReal (\u2191\u2191(Measure.restrict \u03bc s) univ)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Doubling.lean", "full_name": "IsUnifLocDoublingMeasure.eventually_measure_le_scaling_constant_mul'", "start": [142, 1], "end": [146, 37], "traced_tactics": [{"tactic": "convert eventually_nhdsWithin_pos_mul_left hK (eventually_measure_le_scaling_constant_mul \u03bc K\u207b\u00b9)", "annotated_tactic": ["convert <a>eventually_nhdsWithin_pos_mul_left</a> hK (<a>eventually_measure_le_scaling_constant_mul</a> \u03bc K\u207b\u00b9)", [{"full_name": "eventually_nhdsWithin_pos_mul_left", "def_path": "Mathlib/Topology/Algebra/Order/Field.lean", "def_pos": [232, 9], "def_end_pos": [232, 43]}, {"full_name": "IsUnifLocDoublingMeasure.eventually_measure_le_scaling_constant_mul", "def_path": "Mathlib/MeasureTheory/Measure/Doubling.lean", "def_pos": [135, 9], "def_end_pos": [135, 51]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b2 : MetricSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\nK : \u211d\nhK : 0 < K\n\u22a2 \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x r) \u2264 \u2191(scalingConstantOf \u03bc K\u207b\u00b9) * \u2191\u2191\u03bc (closedBall x (K * r))", "state_after": "case h.e'_2.h.h.h.e'_3.h.e'_3.h.e'_4\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MetricSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\nK : \u211d\nhK : 0 < K\nx\u271d : \u211d\na\u271d : \u03b1\n\u22a2 x\u271d = K\u207b\u00b9 * (K * x\u271d)"}, {"tactic": "simp [inv_mul_cancel_left\u2080 hK.ne']", "annotated_tactic": ["simp [<a>inv_mul_cancel_left\u2080</a> hK.ne']", [{"full_name": "inv_mul_cancel_left\u2080", "def_path": "Mathlib/Algebra/GroupWithZero/Basic.lean", "def_pos": [243, 9], "def_end_pos": [243, 29]}]], "state_before": "case h.e'_2.h.h.h.e'_3.h.e'_3.h.e'_4\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MetricSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\nK : \u211d\nhK : 0 < K\nx\u271d : \u211d\na\u271d : \u03b1\n\u22a2 x\u271d = K\u207b\u00b9 * (K * x\u271d)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Lattice.lean", "full_name": "Finset.min_mono", "start": [1391, 1], "end": [1392, 14], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Classes/LawfulMonad.lean", "full_name": "SatisfiesM.trivial", "start": [83, 11], "end": [84, 59], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/Monad.lean", "full_name": "MvPolynomial.join\u2082_comp_map", "start": [200, 1], "end": [201, 29], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "full_name": "MeasureTheory.Measure.restrict_finset_biUnion_congr", "start": [1850, 1], "end": [1855, 34], "traced_tactics": [{"tactic": "induction' s using Finset.induction_on with i s _ hs", "annotated_tactic": ["induction' s using <a>Finset.induction_on</a> with i s _ hs", [{"full_name": "Finset.induction_on", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1251, 19], "def_end_pos": [1251, 31]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\ns : Finset \u03b9\nt : \u03b9 \u2192 Set \u03b1\n\u22a2 restrict \u03bc (\u22c3 i \u2208 s, t i) = restrict \u03bd (\u22c3 i \u2208 s, t i) \u2194 \u2200 (i : \u03b9), i \u2208 s \u2192 restrict \u03bc (t i) = restrict \u03bd (t i)", "state_after": "case empty\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t\u271d : Set \u03b1\nt : \u03b9 \u2192 Set \u03b1\n\u22a2 restrict \u03bc (\u22c3 i \u2208 \u2205, t i) = restrict \u03bd (\u22c3 i \u2208 \u2205, t i) \u2194 \u2200 (i : \u03b9), i \u2208 \u2205 \u2192 restrict \u03bc (t i) = restrict \u03bd (t i)\n\ncase insert\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\nt : \u03b9 \u2192 Set \u03b1\ni : \u03b9\ns : Finset \u03b9\na\u271d : \u00aci \u2208 s\nhs : restrict \u03bc (\u22c3 i \u2208 s, t i) = restrict \u03bd (\u22c3 i \u2208 s, t i) \u2194 \u2200 (i : \u03b9), i \u2208 s \u2192 restrict \u03bc (t i) = restrict \u03bd (t i)\n\u22a2 restrict \u03bc (\u22c3 i_1 \u2208 insert i s, t i_1) = restrict \u03bd (\u22c3 i_1 \u2208 insert i s, t i_1) \u2194\n    \u2200 (i_1 : \u03b9), i_1 \u2208 insert i s \u2192 restrict \u03bc (t i_1) = restrict \u03bd (t i_1)"}, {"tactic": "simp only [forall_eq_or_imp, iUnion_iUnion_eq_or_left, Finset.mem_insert]", "annotated_tactic": ["simp only [<a>forall_eq_or_imp</a>, <a>iUnion_iUnion_eq_or_left</a>, <a>Finset.mem_insert</a>]", [{"full_name": "forall_eq_or_imp", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [473, 17], "def_end_pos": [473, 33]}, {"full_name": "Set.iUnion_iUnion_eq_or_left", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [950, 9], "def_end_pos": [950, 33]}, {"full_name": "Finset.mem_insert", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1087, 9], "def_end_pos": [1087, 19]}]], "state_before": "case insert\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\nt : \u03b9 \u2192 Set \u03b1\ni : \u03b9\ns : Finset \u03b9\na\u271d : \u00aci \u2208 s\nhs : restrict \u03bc (\u22c3 i \u2208 s, t i) = restrict \u03bd (\u22c3 i \u2208 s, t i) \u2194 \u2200 (i : \u03b9), i \u2208 s \u2192 restrict \u03bc (t i) = restrict \u03bd (t i)\n\u22a2 restrict \u03bc (\u22c3 i_1 \u2208 insert i s, t i_1) = restrict \u03bd (\u22c3 i_1 \u2208 insert i s, t i_1) \u2194\n    \u2200 (i_1 : \u03b9), i_1 \u2208 insert i s \u2192 restrict \u03bc (t i_1) = restrict \u03bd (t i_1)", "state_after": "case insert\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\nt : \u03b9 \u2192 Set \u03b1\ni : \u03b9\ns : Finset \u03b9\na\u271d : \u00aci \u2208 s\nhs : restrict \u03bc (\u22c3 i \u2208 s, t i) = restrict \u03bd (\u22c3 i \u2208 s, t i) \u2194 \u2200 (i : \u03b9), i \u2208 s \u2192 restrict \u03bc (t i) = restrict \u03bd (t i)\n\u22a2 restrict \u03bc (t i \u222a \u22c3 x \u2208 s, t x) = restrict \u03bd (t i \u222a \u22c3 x \u2208 s, t x) \u2194\n    restrict \u03bc (t i) = restrict \u03bd (t i) \u2227 \u2200 (a : \u03b9), a \u2208 s \u2192 restrict \u03bc (t a) = restrict \u03bd (t a)"}, {"tactic": "rw [restrict_union_congr, \u2190 hs]", "annotated_tactic": ["rw [<a>restrict_union_congr</a>, \u2190 hs]", [{"full_name": "MeasureTheory.Measure.restrict_union_congr", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1824, 9], "def_end_pos": [1824, 29]}]], "state_before": "case insert\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\nt : \u03b9 \u2192 Set \u03b1\ni : \u03b9\ns : Finset \u03b9\na\u271d : \u00aci \u2208 s\nhs : restrict \u03bc (\u22c3 i \u2208 s, t i) = restrict \u03bd (\u22c3 i \u2208 s, t i) \u2194 \u2200 (i : \u03b9), i \u2208 s \u2192 restrict \u03bc (t i) = restrict \u03bd (t i)\n\u22a2 restrict \u03bc (t i \u222a \u22c3 x \u2208 s, t x) = restrict \u03bd (t i \u222a \u22c3 x \u2208 s, t x) \u2194\n    restrict \u03bc (t i) = restrict \u03bd (t i) \u2227 \u2200 (a : \u03b9), a \u2208 s \u2192 restrict \u03bc (t a) = restrict \u03bd (t a)", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case empty\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t\u271d : Set \u03b1\nt : \u03b9 \u2192 Set \u03b1\n\u22a2 restrict \u03bc (\u22c3 i \u2208 \u2205, t i) = restrict \u03bd (\u22c3 i \u2208 \u2205, t i) \u2194 \u2200 (i : \u03b9), i \u2208 \u2205 \u2192 restrict \u03bc (t i) = restrict \u03bd (t i)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/List/Basic.lean", "full_name": "List.range'_eq_range'TR", "start": [1206, 10], "end": [1215, 24], "traced_tactics": [{"tactic": "funext s n step", "annotated_tactic": ["funext s n step", []], "state_before": "\u22a2 range' = range'TR", "state_after": "case h.h.h\ns n : Nat\nstep : optParam Nat 1\n\u22a2 range' s n step = range'TR s n step"}, {"tactic": "exact (go s n 0).symm", "annotated_tactic": ["exact (go s n 0).<a>symm</a>", [{"full_name": "Eq.symm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [310, 9], "def_end_pos": [310, 16]}]], "state_before": "case h.h.h\ns n : Nat\nstep : optParam Nat 1\n\u22a2 range' s n step = range'TR s n step", "state_after": "no goals"}, {"tactic": "simp [range'TR.go]", "annotated_tactic": ["simp [<a>range'TR.go</a>]", [{"full_name": "List.range'TR.go", "def_path": "lake-packages/std/Std/Data/List/Basic.lean", "def_pos": [1202, 3], "def_end_pos": [1202, 5]}]], "state_before": "s\u271d n : Nat\nstep : optParam Nat 1\ns m : Nat\n\u22a2 range'TR.go step 0 (s + step * 0) (range' (s + step * 0) m step) = range' s (0 + m) step", "state_after": "no goals"}, {"tactic": "simp [range'TR.go]", "annotated_tactic": ["simp [<a>range'TR.go</a>]", [{"full_name": "List.range'TR.go", "def_path": "lake-packages/std/Std/Data/List/Basic.lean", "def_pos": [1202, 3], "def_end_pos": [1202, 5]}]], "state_before": "s\u271d n\u271d : Nat\nstep : optParam Nat 1\ns n m : Nat\n\u22a2 range'TR.go step (n + 1) (s + step * (n + 1)) (range' (s + step * (n + 1)) m step) = range' s (n + 1 + m) step", "state_after": "s\u271d n\u271d : Nat\nstep : optParam Nat 1\ns n m : Nat\n\u22a2 range'TR.go step n (s + step * (n + 1) - step) ((s + step * (n + 1) - step) :: range' (s + step * (n + 1)) m step) =\n    range' s (n + 1 + m) step"}, {"tactic": "rw [Nat.mul_succ, \u2190 Nat.add_assoc, Nat.add_sub_cancel, Nat.add_right_comm n]", "annotated_tactic": ["rw [<a>Nat.mul_succ</a>, \u2190 <a>Nat.add_assoc</a>, <a>Nat.add_sub_cancel</a>, <a>Nat.add_right_comm</a> n]", [{"full_name": "Nat.mul_succ", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [162, 9], "def_end_pos": [162, 17]}, {"full_name": "Nat.add_assoc", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [138, 19], "def_end_pos": [138, 28]}, {"full_name": "Nat.add_sub_cancel", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [594, 19], "def_end_pos": [594, 33]}, {"full_name": "Nat.add_right_comm", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [145, 19], "def_end_pos": [145, 33]}]], "state_before": "s\u271d n\u271d : Nat\nstep : optParam Nat 1\ns n m : Nat\n\u22a2 range'TR.go step n (s + step * (n + 1) - step) ((s + step * (n + 1) - step) :: range' (s + step * (n + 1)) m step) =\n    range' s (n + 1 + m) step", "state_after": "s\u271d n\u271d : Nat\nstep : optParam Nat 1\ns n m : Nat\n\u22a2 range'TR.go step n (s + step * n) ((s + step * n) :: range' (s + step * n + step) m step) = range' s (n + m + 1) step"}, {"tactic": "exact go s n (m + 1)", "annotated_tactic": ["exact go s n (m + 1)", []], "state_before": "s\u271d n\u271d : Nat\nstep : optParam Nat 1\ns n m : Nat\n\u22a2 range'TR.go step n (s + step * n) ((s + step * n) :: range' (s + step * n + step) m step) = range' s (n + m + 1) step", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Array/Lemmas.lean", "full_name": "Array.get?_push_eq", "start": [82, 1], "end": [83, 33], "traced_tactics": [{"tactic": "rw [getElem?_pos, get_push_eq]", "annotated_tactic": ["rw [<a>getElem?_pos</a>, <a>get_push_eq</a>]", [{"full_name": "getElem?_pos", "def_path": "lake-packages/std/Std/Data/Array/Lemmas.lean", "def_pos": [23, 9], "def_end_pos": [23, 21]}, {"full_name": "Array.get_push_eq", "def_path": "lake-packages/std/Std/Data/Array/Init/Lemmas.lean", "def_pos": [131, 17], "def_end_pos": [131, 28]}]], "state_before": "\u03b1 : Type u_1\na : Array \u03b1\nx : \u03b1\n\u22a2 (push a x)[size a]? = some x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "full_name": "List.get?_set_eq", "start": [913, 1], "end": [914, 50], "traced_tactics": [{"tactic": "simp only [set_eq_modifyNth, get?_modifyNth_eq]", "annotated_tactic": ["simp only [<a>set_eq_modifyNth</a>, <a>get?_modifyNth_eq</a>]", [{"full_name": "List.set_eq_modifyNth", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [889, 9], "def_end_pos": [889, 25]}, {"full_name": "List.get?_modifyNth_eq", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [873, 17], "def_end_pos": [873, 34]}]], "state_before": "\u03b1 : Type u_1\na : \u03b1\nn : Nat\nl : List \u03b1\n\u22a2 get? (set l n a) n = (fun x => a) <$> get? l n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL2.lean", "full_name": "MeasureTheory.condexpIndSMul_nonneg", "start": [516, 1], "end": [521, 26], "traced_tactics": [{"tactic": "refine' EventuallyLE.trans_eq _ (condexpIndSMul_ae_eq_smul hm hs h\u03bcs x).symm", "annotated_tactic": ["refine' <a>EventuallyLE.trans_eq</a> _ (<a>condexpIndSMul_ae_eq_smul</a> hm hs h\u03bcs x).<a>symm</a>", [{"full_name": "Filter.EventuallyLE.trans_eq", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1693, 9], "def_end_pos": [1693, 30]}, {"full_name": "MeasureTheory.condexpIndSMul_ae_eq_smul", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL2.lean", "def_pos": [419, 9], "def_end_pos": [419, 34]}, {"full_name": "Filter.EventuallyEq.symm", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1498, 9], "def_end_pos": [1498, 26]}]], "state_before": "\u03b1 : Type u_1\nE\u271d : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b2\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b2\u00b2 : NormedAddCommGroup E\u271d\ninst\u271d\u00b2\u00b9 : InnerProductSpace \ud835\udd5c E\u271d\ninst\u271d\u00b2\u2070 : CompleteSpace E\u271d\ninst\u271d\u00b9\u2079 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u2077 : CompleteSpace E'\ninst\u271d\u00b9\u2076 : NormedSpace \u211d E'\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d G'\ninst\u271d\u00b9\u2070 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2079 : IsROrC \ud835\udd5c'\ninst\u271d\u2078 : NormedAddCommGroup E''\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u2076 : CompleteSpace E''\ninst\u271d\u2075 : NormedSpace \u211d E''\ninst\u271d\u2074 : NormedSpace \u211d G\nhm : m \u2264 m0\nE : Type u_10\ninst\u271d\u00b3 : NormedLatticeAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : OrderedSMul \u211d E\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx : E\nhx : 0 \u2264 x\n\u22a2 0 \u2264\u1d50[\u03bc] \u2191\u2191(condexpIndSMul hm hs h\u03bcs x)", "state_after": "\u03b1 : Type u_1\nE\u271d : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b2\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b2\u00b2 : NormedAddCommGroup E\u271d\ninst\u271d\u00b2\u00b9 : InnerProductSpace \ud835\udd5c E\u271d\ninst\u271d\u00b2\u2070 : CompleteSpace E\u271d\ninst\u271d\u00b9\u2079 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u2077 : CompleteSpace E'\ninst\u271d\u00b9\u2076 : NormedSpace \u211d E'\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d G'\ninst\u271d\u00b9\u2070 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2079 : IsROrC \ud835\udd5c'\ninst\u271d\u2078 : NormedAddCommGroup E''\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u2076 : CompleteSpace E''\ninst\u271d\u2075 : NormedSpace \u211d E''\ninst\u271d\u2074 : NormedSpace \u211d G\nhm : m \u2264 m0\nE : Type u_10\ninst\u271d\u00b3 : NormedLatticeAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : OrderedSMul \u211d E\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx : E\nhx : 0 \u2264 x\n\u22a2 0 \u2264\u1d50[\u03bc] fun a => \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)) a \u2022 x"}, {"tactic": "filter_upwards [condexpL2_indicator_nonneg hm hs h\u03bcs] with a ha", "annotated_tactic": ["filter_upwards [<a>condexpL2_indicator_nonneg</a> hm hs h\u03bcs] with a ha", [{"full_name": "MeasureTheory.condexpL2_indicator_nonneg", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL2.lean", "def_pos": [491, 9], "def_end_pos": [491, 35]}]], "state_before": "\u03b1 : Type u_1\nE\u271d : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b2\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b2\u00b2 : NormedAddCommGroup E\u271d\ninst\u271d\u00b2\u00b9 : InnerProductSpace \ud835\udd5c E\u271d\ninst\u271d\u00b2\u2070 : CompleteSpace E\u271d\ninst\u271d\u00b9\u2079 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u2077 : CompleteSpace E'\ninst\u271d\u00b9\u2076 : NormedSpace \u211d E'\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d G'\ninst\u271d\u00b9\u2070 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2079 : IsROrC \ud835\udd5c'\ninst\u271d\u2078 : NormedAddCommGroup E''\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u2076 : CompleteSpace E''\ninst\u271d\u2075 : NormedSpace \u211d E''\ninst\u271d\u2074 : NormedSpace \u211d G\nhm : m \u2264 m0\nE : Type u_10\ninst\u271d\u00b3 : NormedLatticeAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : OrderedSMul \u211d E\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx : E\nhx : 0 \u2264 x\n\u22a2 0 \u2264\u1d50[\u03bc] fun a => \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)) a \u2022 x", "state_after": "case h\n\u03b1 : Type u_1\nE\u271d : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b2\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b2\u00b2 : NormedAddCommGroup E\u271d\ninst\u271d\u00b2\u00b9 : InnerProductSpace \ud835\udd5c E\u271d\ninst\u271d\u00b2\u2070 : CompleteSpace E\u271d\ninst\u271d\u00b9\u2079 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u2077 : CompleteSpace E'\ninst\u271d\u00b9\u2076 : NormedSpace \u211d E'\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d G'\ninst\u271d\u00b9\u2070 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2079 : IsROrC \ud835\udd5c'\ninst\u271d\u2078 : NormedAddCommGroup E''\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u2076 : CompleteSpace E''\ninst\u271d\u2075 : NormedSpace \u211d E''\ninst\u271d\u2074 : NormedSpace \u211d G\nhm : m \u2264 m0\nE : Type u_10\ninst\u271d\u00b3 : NormedLatticeAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : OrderedSMul \u211d E\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx : E\nhx : 0 \u2264 x\na : \u03b1\nha : OfNat.ofNat 0 a \u2264 \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)) a\n\u22a2 OfNat.ofNat 0 a \u2264 \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)) a \u2022 x"}, {"tactic": "exact smul_nonneg ha hx", "annotated_tactic": ["exact <a>smul_nonneg</a> ha hx", [{"full_name": "smul_nonneg", "def_path": "Mathlib/Algebra/Order/SMul.lean", "def_pos": [102, 9], "def_end_pos": [102, 20]}]], "state_before": "case h\n\u03b1 : Type u_1\nE\u271d : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b2\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b2\u00b2 : NormedAddCommGroup E\u271d\ninst\u271d\u00b2\u00b9 : InnerProductSpace \ud835\udd5c E\u271d\ninst\u271d\u00b2\u2070 : CompleteSpace E\u271d\ninst\u271d\u00b9\u2079 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u2077 : CompleteSpace E'\ninst\u271d\u00b9\u2076 : NormedSpace \u211d E'\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d G'\ninst\u271d\u00b9\u2070 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2079 : IsROrC \ud835\udd5c'\ninst\u271d\u2078 : NormedAddCommGroup E''\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u2076 : CompleteSpace E''\ninst\u271d\u2075 : NormedSpace \u211d E''\ninst\u271d\u2074 : NormedSpace \u211d G\nhm : m \u2264 m0\nE : Type u_10\ninst\u271d\u00b3 : NormedLatticeAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : OrderedSMul \u211d E\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx : E\nhx : 0 \u2264 x\na : \u03b1\nha : OfNat.ofNat 0 a \u2264 \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)) a\n\u22a2 OfNat.ofNat 0 a \u2264 \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)) a \u2022 x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/TMToPartrec.lean", "full_name": "Turing.PartrecToTM2.tr_respects", "start": [1706, 1], "end": [1708, 30], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/ZMod/Basic.lean", "full_name": "ZMod.mul_inv_eq_gcd", "start": [694, 1], "end": [708, 80], "traced_tactics": [{"tactic": "cases' n with n", "annotated_tactic": ["cases' n with n", []], "state_before": "n : \u2115\na : ZMod n\n\u22a2 a * a\u207b\u00b9 = \u2191(Nat.gcd (val a) n)", "state_after": "case zero\na : ZMod Nat.zero\n\u22a2 a * a\u207b\u00b9 = \u2191(Nat.gcd (val a) Nat.zero)\n\ncase succ\nn : \u2115\na : ZMod (Nat.succ n)\n\u22a2 a * a\u207b\u00b9 = \u2191(Nat.gcd (val a) (Nat.succ n))"}, {"tactic": "dsimp [ZMod] at a \u22a2", "annotated_tactic": ["dsimp [<a>ZMod</a>] at a \u22a2", [{"full_name": "ZMod", "def_path": "Mathlib/Data/ZMod/Defs.lean", "def_pos": [94, 5], "def_end_pos": [94, 9]}]], "state_before": "case zero\na : ZMod Nat.zero\n\u22a2 a * a\u207b\u00b9 = \u2191(Nat.gcd (val a) Nat.zero)", "state_after": "case zero\na : \u2124\n\u22a2 a * a\u207b\u00b9 = \u2191(Nat.gcd (val a) 0)"}, {"tactic": "calc\n  _ = a * Int.sign a := rfl\n  _ = a.natAbs := by rw [Int.mul_sign]\n  _ = a.natAbs.gcd 0 := by rw [Nat.gcd_zero_right]", "annotated_tactic": ["calc\n      _ = a * <a>Int.sign</a> a := <a>rfl</a>\n      _ = a.natAbs := by rw [<a>Int.mul_sign</a>]\n      _ = a.natAbs.gcd 0 := by rw [<a>Nat.gcd_zero_right</a>]", [{"full_name": "Int.sign", "def_path": "lake-packages/std/Std/Data/Int/Basic.lean", "def_pos": [24, 5], "def_end_pos": [24, 9]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}, {"full_name": "Int.mul_sign", "def_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "def_pos": [840, 9], "def_end_pos": [840, 17]}, {"full_name": "Nat.gcd_zero_right", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Gcd.lean", "def_pos": [30, 17], "def_end_pos": [30, 31]}]], "state_before": "case zero\na : \u2124\n\u22a2 a * a\u207b\u00b9 = \u2191(Nat.gcd (val a) 0)", "state_after": "no goals"}, {"tactic": "rw [Int.mul_sign]", "annotated_tactic": ["rw [<a>Int.mul_sign</a>]", [{"full_name": "Int.mul_sign", "def_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "def_pos": [840, 9], "def_end_pos": [840, 17]}]], "state_before": "a : \u2124\n\u22a2 a * Int.sign a = \u2191(Int.natAbs a)", "state_after": "no goals"}, {"tactic": "rw [Nat.gcd_zero_right]", "annotated_tactic": ["rw [<a>Nat.gcd_zero_right</a>]", [{"full_name": "Nat.gcd_zero_right", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Gcd.lean", "def_pos": [30, 17], "def_end_pos": [30, 31]}]], "state_before": "a : \u2124\n\u22a2 \u2191(Int.natAbs a) = \u2191(Nat.gcd (Int.natAbs a) 0)", "state_after": "no goals"}, {"tactic": "calc\n  a * a\u207b\u00b9 = a * a\u207b\u00b9 + n.succ * Nat.gcdB (val a) n.succ := by\n    rw [nat_cast_self, zero_mul, add_zero]\n  _ = \u2191(\u2191a.val * Nat.gcdA (val a) n.succ + n.succ * Nat.gcdB (val a) n.succ) := by\n    push_cast\n    rw [nat_cast_zmod_val]\n    rfl\n  _ = Nat.gcd a.val n.succ := by rw [\u2190 Nat.gcd_eq_gcd_ab a.val n.succ]; rfl", "annotated_tactic": ["calc\n      a * a\u207b\u00b9 = a * a\u207b\u00b9 + n.succ * <a>Nat.gcdB</a> (<a>val</a> a) n.succ := by\n        rw [<a>nat_cast_self</a>, <a>zero_mul</a>, <a>add_zero</a>]\n      _ = \u2191(\u2191a.val * <a>Nat.gcdA</a> (<a>val</a> a) n.succ + n.succ * <a>Nat.gcdB</a> (<a>val</a> a) n.succ) := by\n        push_cast\n        rw [<a>nat_cast_zmod_val</a>]\n        rfl\n      _ = <a>Nat.gcd</a> a.val n.succ := by rw [\u2190 <a>Nat.gcd_eq_gcd_ab</a> a.val n.succ]; rfl", [{"full_name": "Nat.gcdB", "def_path": "Mathlib/Data/Int/GCD.lean", "def_pos": [78, 5], "def_end_pos": [78, 9]}, {"full_name": "ZMod.val", "def_path": "Mathlib/Data/ZMod/Basic.lean", "def_pos": [47, 5], "def_end_pos": [47, 8]}, {"full_name": "ZMod.nat_cast_self", "def_path": "Mathlib/Data/ZMod/Basic.lean", "def_pos": [132, 9], "def_end_pos": [132, 22]}, {"full_name": "MulZeroClass.zero_mul", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [36, 3], "def_end_pos": [36, 11]}, {"full_name": "add_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [469, 3], "def_end_pos": [469, 14]}, {"full_name": "Nat.gcdA", "def_path": "Mathlib/Data/Int/GCD.lean", "def_pos": [73, 5], "def_end_pos": [73, 9]}, {"full_name": "ZMod.val", "def_path": "Mathlib/Data/ZMod/Basic.lean", "def_pos": [47, 5], "def_end_pos": [47, 8]}, {"full_name": "Nat.gcdB", "def_path": "Mathlib/Data/Int/GCD.lean", "def_pos": [78, 5], "def_end_pos": [78, 9]}, {"full_name": "ZMod.val", "def_path": "Mathlib/Data/ZMod/Basic.lean", "def_pos": [47, 5], "def_end_pos": [47, 8]}, {"full_name": "ZMod.nat_cast_zmod_val", "def_path": "Mathlib/Data/ZMod/Basic.lean", "def_pos": [195, 9], "def_end_pos": [195, 26]}, {"full_name": "Nat.gcd", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Gcd.lean", "def_pos": [17, 5], "def_end_pos": [17, 8]}, {"full_name": "Nat.gcd_eq_gcd_ab", "def_path": "Mathlib/Data/Int/GCD.lean", "def_pos": [152, 9], "def_end_pos": [152, 22]}]], "state_before": "case succ\nn : \u2115\na : ZMod (Nat.succ n)\n\u22a2 a * a\u207b\u00b9 = \u2191(Nat.gcd (val a) (Nat.succ n))", "state_after": "no goals"}, {"tactic": "rw [nat_cast_self, zero_mul, add_zero]", "annotated_tactic": ["rw [<a>nat_cast_self</a>, <a>zero_mul</a>, <a>add_zero</a>]", [{"full_name": "ZMod.nat_cast_self", "def_path": "Mathlib/Data/ZMod/Basic.lean", "def_pos": [132, 9], "def_end_pos": [132, 22]}, {"full_name": "MulZeroClass.zero_mul", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [36, 3], "def_end_pos": [36, 11]}, {"full_name": "add_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [469, 3], "def_end_pos": [469, 14]}]], "state_before": "n : \u2115\na : ZMod (Nat.succ n)\n\u22a2 a * a\u207b\u00b9 = a * a\u207b\u00b9 + \u2191(Nat.succ n) * \u2191(Nat.gcdB (val a) (Nat.succ n))", "state_after": "no goals"}, {"tactic": "push_cast", "annotated_tactic": ["push_cast", []], "state_before": "n : \u2115\na : ZMod (Nat.succ n)\n\u22a2 a * a\u207b\u00b9 + \u2191(Nat.succ n) * \u2191(Nat.gcdB (val a) (Nat.succ n)) =\n    \u2191(\u2191(val a) * Nat.gcdA (val a) (Nat.succ n) + \u2191(Nat.succ n) * Nat.gcdB (val a) (Nat.succ n))", "state_after": "n : \u2115\na : ZMod (Nat.succ n)\n\u22a2 a * a\u207b\u00b9 + (\u2191n + 1) * \u2191(Nat.gcdB (val a) (Nat.succ n)) =\n    \u2191(val a) * \u2191(Nat.gcdA (val a) (Nat.succ n)) + (\u2191n + 1) * \u2191(Nat.gcdB (val a) (Nat.succ n))"}, {"tactic": "rw [nat_cast_zmod_val]", "annotated_tactic": ["rw [<a>nat_cast_zmod_val</a>]", [{"full_name": "ZMod.nat_cast_zmod_val", "def_path": "Mathlib/Data/ZMod/Basic.lean", "def_pos": [195, 9], "def_end_pos": [195, 26]}]], "state_before": "n : \u2115\na : ZMod (Nat.succ n)\n\u22a2 a * a\u207b\u00b9 + (\u2191n + 1) * \u2191(Nat.gcdB (val a) (Nat.succ n)) =\n    \u2191(val a) * \u2191(Nat.gcdA (val a) (Nat.succ n)) + (\u2191n + 1) * \u2191(Nat.gcdB (val a) (Nat.succ n))", "state_after": "n : \u2115\na : ZMod (Nat.succ n)\n\u22a2 a * a\u207b\u00b9 + (\u2191n + 1) * \u2191(Nat.gcdB (val a) (Nat.succ n)) =\n    a * \u2191(Nat.gcdA (val a) (Nat.succ n)) + (\u2191n + 1) * \u2191(Nat.gcdB (val a) (Nat.succ n))"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "n : \u2115\na : ZMod (Nat.succ n)\n\u22a2 a * a\u207b\u00b9 + (\u2191n + 1) * \u2191(Nat.gcdB (val a) (Nat.succ n)) =\n    a * \u2191(Nat.gcdA (val a) (Nat.succ n)) + (\u2191n + 1) * \u2191(Nat.gcdB (val a) (Nat.succ n))", "state_after": "no goals"}, {"tactic": "rw [\u2190 Nat.gcd_eq_gcd_ab a.val n.succ]", "annotated_tactic": ["rw [\u2190 <a>Nat.gcd_eq_gcd_ab</a> a.val n.succ]", [{"full_name": "Nat.gcd_eq_gcd_ab", "def_path": "Mathlib/Data/Int/GCD.lean", "def_pos": [152, 9], "def_end_pos": [152, 22]}]], "state_before": "n : \u2115\na : ZMod (Nat.succ n)\n\u22a2 \u2191(\u2191(val a) * Nat.gcdA (val a) (Nat.succ n) + \u2191(Nat.succ n) * Nat.gcdB (val a) (Nat.succ n)) =\n    \u2191(Nat.gcd (val a) (Nat.succ n))", "state_after": "n : \u2115\na : ZMod (Nat.succ n)\n\u22a2 \u2191\u2191(Nat.gcd (val a) (Nat.succ n)) = \u2191(Nat.gcd (val a) (Nat.succ n))"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "n : \u2115\na : ZMod (Nat.succ n)\n\u22a2 \u2191\u2191(Nat.gcd (val a) (Nat.succ n)) = \u2191(Nat.gcd (val a) (Nat.succ n))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/List.lean", "full_name": "Set.range_list_map_coe", "start": [35, 1], "end": [36, 41], "traced_tactics": [{"tactic": "rw [range_list_map, Subtype.range_coe]", "annotated_tactic": ["rw [<a>range_list_map</a>, <a>Subtype.range_coe</a>]", [{"full_name": "Set.range_list_map", "def_path": "Mathlib/Data/Set/List.lean", "def_pos": [25, 9], "def_end_pos": [25, 23]}, {"full_name": "Subtype.range_coe", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [1409, 9], "def_end_pos": [1409, 18]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nl : List \u03b1\ns : Set \u03b1\n\u22a2 range (map Subtype.val) = {l | \u2200 (x : \u03b1), x \u2208 l \u2192 x \u2208 s}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Array/Lemmas.lean", "full_name": "Array.get?_swap", "start": [122, 1], "end": [124, 55], "traced_tactics": [{"tactic": "simp [swap_def, get?_set, \u2190 getElem_fin_eq_data_get]", "annotated_tactic": ["simp [<a>swap_def</a>, <a>get?_set</a>, \u2190 <a>getElem_fin_eq_data_get</a>]", [{"full_name": "Array.swap_def", "def_path": "lake-packages/std/Std/Data/Array/Lemmas.lean", "def_pos": [115, 9], "def_end_pos": [115, 17]}, {"full_name": "Array.get?_set", "def_path": "lake-packages/std/Std/Data/Array/Lemmas.lean", "def_pos": [102, 9], "def_end_pos": [102, 17]}, {"full_name": "Array.getElem_fin_eq_data_get", "def_path": "lake-packages/std/Std/Data/Array/Lemmas.lean", "def_pos": [42, 9], "def_end_pos": [42, 32]}]], "state_before": "\u03b1 : Type u_1\na : Array \u03b1\ni j : Fin (size a)\nk : Nat\n\u22a2 (swap a i j)[k]? = if j.val = k then some a[i.val] else if i.val = k then some a[j.val] else a[k]?", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Lebesgue/Integral.lean", "full_name": "integral_comp_neg_Ioi", "start": [102, 1], "end": [105, 22], "traced_tactics": [{"tactic": "rw [\u2190 neg_neg c, \u2190 integral_comp_neg_Iic]", "annotated_tactic": ["rw [\u2190 <a>neg_neg</a> c, \u2190 <a>integral_comp_neg_Iic</a>]", [{"full_name": "neg_neg", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [799, 3], "def_end_pos": [799, 14]}, {"full_name": "integral_comp_neg_Iic", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/Integral.lean", "def_pos": [91, 9], "def_end_pos": [91, 30]}]], "state_before": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nc : \u211d\nf : \u211d \u2192 E\n\u22a2 \u222b (x : \u211d) in Ioi c, f (-x) = \u222b (x : \u211d) in Iic (-c), f x", "state_after": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nc : \u211d\nf : \u211d \u2192 E\n\u22a2 \u222b (x : \u211d) in Iic (-c), f (- -x) = \u222b (x : \u211d) in Iic (- - -c), f x"}, {"tactic": "simp only [neg_neg]", "annotated_tactic": ["simp only [<a>neg_neg</a>]", [{"full_name": "neg_neg", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [799, 3], "def_end_pos": [799, 14]}]], "state_before": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nc : \u211d\nf : \u211d \u2192 E\n\u22a2 \u222b (x : \u211d) in Iic (-c), f (- -x) = \u222b (x : \u211d) in Iic (- - -c), f x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Regular.lean", "full_name": "MeasureTheory.Measure.WeaklyRegular.restrict_of_measurableSet", "start": [607, 1], "end": 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(\u2191\u2191\u03bc A < \u22a4)\n\u22a2 WeaklyRegular (restrict \u03bc A)"}, {"tactic": "refine' InnerRegular.weaklyRegular_of_finite (\u03bc.restrict A) fun V V_open => _", "annotated_tactic": ["refine' <a>InnerRegular.weaklyRegular_of_finite</a> (\u03bc.restrict A) fun V V_open => _", [{"full_name": "MeasureTheory.Measure.InnerRegular.weaklyRegular_of_finite", "def_path": "Mathlib/MeasureTheory/Measure/Regular.lean", "def_pos": [374, 9], "def_end_pos": [374, 32]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\ninst\u271d : WeaklyRegular \u03bc\nA : Set \u03b1\nhA : MeasurableSet A\nh'A : \u2191\u2191\u03bc A \u2260 \u22a4\nthis : Fact (\u2191\u2191\u03bc A < \u22a4)\n\u22a2 WeaklyRegular (restrict \u03bc A)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : 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MeasurableSet A\nh'A : \u2191\u2191\u03bc A \u2260 \u22a4\nthis : Fact (\u2191\u2191\u03bc A < \u22a4)\nV : Set \u03b1\nV_open : IsOpen V\n\u22a2 \u2200 (r : \u211d\u22650\u221e), r < \u2191\u2191(restrict \u03bc A) V \u2192 \u2203 K, K \u2286 V \u2227 IsClosed K \u2227 r < \u2191\u2191(restrict \u03bc A) K", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\ninst\u271d : WeaklyRegular \u03bc\nA : Set \u03b1\nhA : MeasurableSet A\nh'A : \u2191\u2191\u03bc A \u2260 \u22a4\nthis : Fact (\u2191\u2191\u03bc A < \u22a4)\nV : Set \u03b1\nV_open : IsOpen V\n\u22a2 \u2200 (r : \u211d\u22650\u221e), r < \u2191\u2191\u03bc (V \u2229 A) \u2192 \u2203 K, K \u2286 V \u2227 IsClosed K \u2227 r < \u2191\u2191\u03bc (K \u2229 A)"}, {"tactic": "intro r hr", "annotated_tactic": ["intro r hr", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\ninst\u271d : WeaklyRegular \u03bc\nA : Set \u03b1\nhA : MeasurableSet A\nh'A : \u2191\u2191\u03bc A \u2260 \u22a4\nthis : Fact (\u2191\u2191\u03bc A < \u22a4)\nV : Set \u03b1\nV_open : IsOpen V\n\u22a2 \u2200 (r : \u211d\u22650\u221e), r < \u2191\u2191\u03bc (V \u2229 A) \u2192 \u2203 K, K \u2286 V \u2227 IsClosed K \u2227 r < \u2191\u2191\u03bc (K \u2229 A)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\ninst\u271d : WeaklyRegular \u03bc\nA : Set \u03b1\nhA : MeasurableSet A\nh'A : \u2191\u2191\u03bc A \u2260 \u22a4\nthis : Fact (\u2191\u2191\u03bc A < \u22a4)\nV : Set \u03b1\nV_open : IsOpen V\nr : \u211d\u22650\u221e\nhr : r < \u2191\u2191\u03bc (V \u2229 A)\n\u22a2 \u2203 K, K \u2286 V \u2227 IsClosed K \u2227 r < \u2191\u2191\u03bc (K \u2229 A)"}, {"tactic": "have : \u03bc (V \u2229 A) \u2260 \u221e := ne_top_of_le_ne_top h'A (measure_mono <| inter_subset_right _ _)", "annotated_tactic": ["have : \u03bc (V \u2229 A) \u2260 \u221e := <a>ne_top_of_le_ne_top</a> h'A (<a>measure_mono</a> <| <a>inter_subset_right</a> _ _)", [{"full_name": "ne_top_of_le_ne_top", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [194, 9], "def_end_pos": [194, 28]}, {"full_name": "MeasureTheory.measure_mono", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [193, 9], "def_end_pos": [193, 21]}, {"full_name": "Set.inter_subset_right", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [969, 9], "def_end_pos": [969, 27]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\ninst\u271d : WeaklyRegular \u03bc\nA : Set \u03b1\nhA : MeasurableSet A\nh'A : \u2191\u2191\u03bc A \u2260 \u22a4\nthis : Fact (\u2191\u2191\u03bc A < \u22a4)\nV : Set \u03b1\nV_open : IsOpen V\nr : \u211d\u22650\u221e\nhr : r < \u2191\u2191\u03bc (V \u2229 A)\n\u22a2 \u2203 K, K \u2286 V \u2227 IsClosed K \u2227 r < \u2191\u2191\u03bc (K \u2229 A)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\ninst\u271d : WeaklyRegular \u03bc\nA : Set \u03b1\nhA : MeasurableSet A\nh'A : \u2191\u2191\u03bc A \u2260 \u22a4\nthis\u271d : Fact (\u2191\u2191\u03bc A < \u22a4)\nV : Set \u03b1\nV_open : IsOpen V\nr : \u211d\u22650\u221e\nhr : r < \u2191\u2191\u03bc (V \u2229 A)\nthis : \u2191\u2191\u03bc (V \u2229 A) \u2260 \u22a4\n\u22a2 \u2203 K, K \u2286 V \u2227 IsClosed K \u2227 r < \u2191\u2191\u03bc (K \u2229 A)"}, {"tactic": "rcases (V_open.measurableSet.inter hA).exists_lt_isClosed_of_ne_top this hr with\n  \u27e8F, hFVA, hFc, hF\u27e9", "annotated_tactic": ["rcases (V_open.measurableSet.inter hA).<a>exists_lt_isClosed_of_ne_top</a> this hr with\n    \u27e8F, hFVA, hFc, hF\u27e9", [{"full_name": "MeasurableSet.exists_lt_isClosed_of_ne_top", "def_path": "Mathlib/MeasureTheory/Measure/Regular.lean", "def_pos": [592, 9], "def_end_pos": [592, 58]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\ninst\u271d : WeaklyRegular \u03bc\nA : Set \u03b1\nhA : MeasurableSet A\nh'A : \u2191\u2191\u03bc A \u2260 \u22a4\nthis\u271d : Fact (\u2191\u2191\u03bc A < \u22a4)\nV : Set \u03b1\nV_open : IsOpen V\nr : \u211d\u22650\u221e\nhr : r < \u2191\u2191\u03bc (V \u2229 A)\nthis : \u2191\u2191\u03bc (V \u2229 A) \u2260 \u22a4\n\u22a2 \u2203 K, K \u2286 V \u2227 IsClosed K \u2227 r < \u2191\u2191\u03bc (K \u2229 A)", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\ninst\u271d : WeaklyRegular \u03bc\nA : Set \u03b1\nhA : MeasurableSet A\nh'A : \u2191\u2191\u03bc A \u2260 \u22a4\nthis\u271d : Fact (\u2191\u2191\u03bc A < \u22a4)\nV : Set \u03b1\nV_open : IsOpen V\nr : \u211d\u22650\u221e\nhr : r < \u2191\u2191\u03bc (V \u2229 A)\nthis : \u2191\u2191\u03bc (V \u2229 A) \u2260 \u22a4\nF : Set \u03b1\nhFVA : F \u2286 V \u2229 A\nhFc : IsClosed F\nhF : r < \u2191\u2191\u03bc F\n\u22a2 \u2203 K, K \u2286 V \u2227 IsClosed K \u2227 r < \u2191\u2191\u03bc (K \u2229 A)"}, {"tactic": "refine' \u27e8F, hFVA.trans (inter_subset_left _ _), hFc, _\u27e9", "annotated_tactic": ["refine' \u27e8F, hFVA.trans (<a>inter_subset_left</a> _ _), hFc, _\u27e9", [{"full_name": "Set.inter_subset_left", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [965, 9], "def_end_pos": [965, 26]}]], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\ninst\u271d : WeaklyRegular \u03bc\nA : Set \u03b1\nhA : MeasurableSet A\nh'A : \u2191\u2191\u03bc A \u2260 \u22a4\nthis\u271d : Fact (\u2191\u2191\u03bc A < \u22a4)\nV : Set \u03b1\nV_open : IsOpen V\nr : \u211d\u22650\u221e\nhr : r < \u2191\u2191\u03bc (V \u2229 A)\nthis : \u2191\u2191\u03bc (V \u2229 A) \u2260 \u22a4\nF : Set \u03b1\nhFVA : F \u2286 V \u2229 A\nhFc : IsClosed F\nhF : r < \u2191\u2191\u03bc F\n\u22a2 \u2203 K, K \u2286 V \u2227 IsClosed K \u2227 r < \u2191\u2191\u03bc (K \u2229 A)", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\ninst\u271d : WeaklyRegular \u03bc\nA : Set \u03b1\nhA : MeasurableSet A\nh'A : \u2191\u2191\u03bc A \u2260 \u22a4\nthis\u271d : Fact (\u2191\u2191\u03bc A < \u22a4)\nV : Set \u03b1\nV_open : IsOpen V\nr : \u211d\u22650\u221e\nhr : r < \u2191\u2191\u03bc (V \u2229 A)\nthis : \u2191\u2191\u03bc (V \u2229 A) \u2260 \u22a4\nF : Set \u03b1\nhFVA : F \u2286 V \u2229 A\nhFc : IsClosed F\nhF : r < \u2191\u2191\u03bc F\n\u22a2 r < \u2191\u2191\u03bc (F \u2229 A)"}, {"tactic": "rwa [inter_eq_self_of_subset_left (hFVA.trans <| inter_subset_right _ _)]", "annotated_tactic": ["rwa [<a>inter_eq_self_of_subset_left</a> (hFVA.trans <| <a>inter_subset_right</a> _ _)]", [{"full_name": "Set.inter_eq_self_of_subset_left", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [987, 9], "def_end_pos": [987, 37]}, {"full_name": "Set.inter_subset_right", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [969, 9], "def_end_pos": [969, 27]}]], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\ninst\u271d : WeaklyRegular \u03bc\nA : Set \u03b1\nhA : MeasurableSet A\nh'A : \u2191\u2191\u03bc A \u2260 \u22a4\nthis\u271d : Fact (\u2191\u2191\u03bc A < \u22a4)\nV : Set \u03b1\nV_open : IsOpen V\nr : \u211d\u22650\u221e\nhr : r < \u2191\u2191\u03bc (V \u2229 A)\nthis : \u2191\u2191\u03bc (V \u2229 A) \u2260 \u22a4\nF : Set \u03b1\nhFVA : F \u2286 V \u2229 A\nhFc : IsClosed F\nhF : r < \u2191\u2191\u03bc F\n\u22a2 r < \u2191\u2191\u03bc (F \u2229 A)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Group/Action.lean", "full_name": "MeasureTheory.measure_pos_iff_nonempty_of_smulInvariant", "start": [279, 1], "end": [282, 60], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "full_name": "MeasureTheory.OuterMeasure.map_iInf", "start": [1230, 1], "end": [1233, 39], "traced_tactics": [{"tactic": "refine' Eq.trans _ (map_comap _ _)", "annotated_tactic": ["refine' <a>Eq.trans</a> _ (<a>map_comap</a> _ _)", [{"full_name": "Eq.trans", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [322, 9], "def_end_pos": [322, 17]}, {"full_name": "MeasureTheory.OuterMeasure.map_comap", "def_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "def_pos": [587, 9], "def_end_pos": [587, 18]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Sort u_2\n\u03b2 : Type u_3\nf : \u03b1 \u2192 \u03b2\nhf : Injective f\nm : \u03b9 \u2192 OuterMeasure \u03b1\n\u22a2 \u2191(map f) (\u2a05 i, m i) = \u2191(restrict (range f)) (\u2a05 i, \u2191(map f) (m i))", "state_after": "\u03b1 : Type u_1\n\u03b9 : Sort u_2\n\u03b2 : Type u_3\nf : \u03b1 \u2192 \u03b2\nhf : Injective f\nm : \u03b9 \u2192 OuterMeasure \u03b1\n\u22a2 \u2191(map f) (\u2a05 i, m i) = \u2191(map f) (\u2191(comap f) (\u2a05 i, \u2191(map f) (m i)))"}, {"tactic": "simp only [comap_iInf, comap_map hf]", "annotated_tactic": ["simp only [<a>comap_iInf</a>, <a>comap_map</a> hf]", [{"full_name": "MeasureTheory.OuterMeasure.comap_iInf", "def_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "def_pos": [1220, 9], "def_end_pos": [1220, 19]}, {"full_name": "MeasureTheory.OuterMeasure.comap_map", "def_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "def_pos": [614, 9], "def_end_pos": [614, 18]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Sort u_2\n\u03b2 : Type u_3\nf : \u03b1 \u2192 \u03b2\nhf : Injective f\nm : \u03b9 \u2192 OuterMeasure \u03b1\n\u22a2 \u2191(map f) (\u2a05 i, m i) = \u2191(map f) (\u2191(comap f) (\u2a05 i, \u2191(map f) (m i)))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Primrec.lean", "full_name": "Primrec.eq", "start": [750, 11], "end": [750, 73], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/PFun.lean", "full_name": "PFun.core_eq", "start": [518, 1], "end": [520, 83], "traced_tactics": [{"tactic": "rw [preimage_eq, Set.union_distrib_right, Set.union_comm (Dom f), Set.compl_union_self,\n  Set.inter_univ, Set.union_eq_self_of_subset_right (f.compl_dom_subset_core s)]", "annotated_tactic": ["rw [<a>preimage_eq</a>, <a>Set.union_distrib_right</a>, <a>Set.union_comm</a> (<a>Dom</a> f), <a>Set.compl_union_self</a>,\n    <a>Set.inter_univ</a>, <a>Set.union_eq_self_of_subset_right</a> (f.compl_dom_subset_core s)]", [{"full_name": "PFun.preimage_eq", "def_path": "Mathlib/Data/PFun.lean", "def_pos": [510, 9], "def_end_pos": [510, 20]}, {"full_name": "Set.union_distrib_right", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1079, 9], "def_end_pos": [1079, 28]}, {"full_name": "Set.union_comm", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [786, 9], "def_end_pos": [786, 19]}, {"full_name": "PFun.Dom", "def_path": "Mathlib/Data/PFun.lean", "def_pos": [75, 5], "def_end_pos": [75, 8]}, {"full_name": "Set.compl_union_self", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1744, 9], "def_end_pos": [1744, 25]}, {"full_name": "Set.inter_univ", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1012, 9], "def_end_pos": [1012, 19]}, {"full_name": "Set.union_eq_self_of_subset_right", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [824, 9], "def_end_pos": [824, 38]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf\u271d f : \u03b1 \u2192. \u03b2\ns : Set \u03b2\n\u22a2 core f s = preimage f s \u222a (Dom f)\u1d9c", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/ZMod/Quotient.lean", "full_name": "AddAction.orbitZmultiplesEquiv_symm_apply'", "start": [167, 1], "end": [173, 57], "traced_tactics": [{"tactic": "rw [AddAction.orbit_zmultiples_equiv_symm_apply, ZMod.coe_int_cast]", "annotated_tactic": ["rw [<a>AddAction.orbit_zmultiples_equiv_symm_apply</a>, <a>ZMod.coe_int_cast</a>]", [{"full_name": "AddAction.orbit_zmultiples_equiv_symm_apply", "def_path": "Mathlib/Data/ZMod/Quotient.lean", "def_pos": [152, 15], "def_end_pos": [152, 48]}, {"full_name": "ZMod.coe_int_cast", "def_path": "Mathlib/Data/ZMod/Basic.lean", "def_pos": [496, 9], "def_end_pos": [496, 21]}]], "state_before": "n : \u2115\nA : Type u_1\nR : Type u_2\ninst\u271d\u2075 : AddGroup A\ninst\u271d\u2074 : Ring R\n\u03b1\u271d : Type u_3\n\u03b2\u271d : Type u_4\ninst\u271d\u00b3 : Group \u03b1\u271d\na\u271d : \u03b1\u271d\ninst\u271d\u00b2 : MulAction \u03b1\u271d \u03b2\u271d\nb\u271d : \u03b2\u271d\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u00b9 : AddGroup \u03b1\na : \u03b1\ninst\u271d : AddAction \u03b1 \u03b2\nb : \u03b2\nk : \u2124\n\u22a2 \u2191(orbitZmultiplesEquiv a b).symm \u2191k =\n    k \u2022 { val := a, property := (_ : a \u2208 zmultiples a) } +\u1d65\n      { val := b, property := (_ : b \u2208 AddAction.orbit { x // x \u2208 zmultiples a } b) }", "state_after": "n : \u2115\nA : Type u_1\nR : Type u_2\ninst\u271d\u2075 : AddGroup A\ninst\u271d\u2074 : Ring R\n\u03b1\u271d : Type u_3\n\u03b2\u271d : Type u_4\ninst\u271d\u00b3 : Group \u03b1\u271d\na\u271d : \u03b1\u271d\ninst\u271d\u00b2 : MulAction \u03b1\u271d \u03b2\u271d\nb\u271d : \u03b2\u271d\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u00b9 : AddGroup \u03b1\na : \u03b1\ninst\u271d : AddAction \u03b1 \u03b2\nb : \u03b2\nk : \u2124\n\u22a2 (k % \u2191(minimalPeriod ((fun x x_1 => x +\u1d65 x_1) a) b)) \u2022 { val := a, property := (_ : a \u2208 zmultiples a) } +\u1d65\n      { val := b, property := (_ : b \u2208 AddAction.orbit { x // x \u2208 zmultiples a } b) } =\n    k \u2022 { val := a, property := (_ : a \u2208 zmultiples a) } +\u1d65\n      { val := b, property := (_ : b \u2208 AddAction.orbit { x // x \u2208 zmultiples a } b) }"}, {"tactic": "exact Subtype.ext (zsmul_vadd_mod_minimalPeriod a b k)", "annotated_tactic": ["exact <a>Subtype.ext</a> (<a>zsmul_vadd_mod_minimalPeriod</a> a b k)", [{"full_name": "Subtype.ext", "def_path": "Mathlib/Data/Subtype.lean", "def_pos": [65, 19], "def_end_pos": [65, 22]}, {"full_name": "AddAction.zsmul_vadd_mod_minimalPeriod", "def_path": "Mathlib/Dynamics/PeriodicPts.lean", "def_pos": [662, 3], "def_end_pos": [662, 14]}]], "state_before": "n : \u2115\nA : Type u_1\nR : Type u_2\ninst\u271d\u2075 : AddGroup A\ninst\u271d\u2074 : Ring R\n\u03b1\u271d : Type u_3\n\u03b2\u271d : Type u_4\ninst\u271d\u00b3 : Group \u03b1\u271d\na\u271d : \u03b1\u271d\ninst\u271d\u00b2 : MulAction \u03b1\u271d \u03b2\u271d\nb\u271d : \u03b2\u271d\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u00b9 : AddGroup \u03b1\na : \u03b1\ninst\u271d : AddAction \u03b1 \u03b2\nb : \u03b2\nk : \u2124\n\u22a2 (k % \u2191(minimalPeriod ((fun x x_1 => x +\u1d65 x_1) a) b)) \u2022 { val := a, property := (_ : a \u2208 zmultiples a) } +\u1d65\n      { val := b, property := (_ : b \u2208 AddAction.orbit { x // x \u2208 zmultiples a } b) } =\n    k \u2022 { val := a, property := (_ : a \u2208 zmultiples a) } +\u1d65\n      { val := b, property := (_ : b \u2208 AddAction.orbit { x // x \u2208 zmultiples a } b) }", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "full_name": "MeasureTheory.SimpleFunc.iSup_eapprox_apply", "start": [896, 1], "end": [907, 47], "traced_tactics": [{"tactic": "rw [eapprox, iSup_approx_apply ennrealRatEmbed f a hf rfl]", "annotated_tactic": ["rw [<a>eapprox</a>, <a>iSup_approx_apply</a> <a>ennrealRatEmbed</a> f a hf <a>rfl</a>]", [{"full_name": "MeasureTheory.SimpleFunc.eapprox", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [873, 5], "def_end_pos": [873, 12]}, {"full_name": "MeasureTheory.SimpleFunc.iSup_approx_apply", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [843, 9], "def_end_pos": [843, 26]}, {"full_name": "MeasureTheory.SimpleFunc.ennrealRatEmbed", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [864, 5], "def_end_pos": [864, 20]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d : MeasurableSpace \u03b1\nK : Type u_5\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\na : \u03b1\n\u22a2 \u2a06 n, \u2191(eapprox f n) a = f a", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d : MeasurableSpace \u03b1\nK : Type u_5\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\na : \u03b1\n\u22a2 \u2a06 k, \u2a06 (_ : ennrealRatEmbed k \u2264 f a), ennrealRatEmbed k = f a"}, {"tactic": "refine' le_antisymm (iSup_le fun i => iSup_le fun hi => hi) (le_of_not_gt _)", "annotated_tactic": ["refine' <a>le_antisymm</a> (<a>iSup_le</a> fun i => <a>iSup_le</a> fun hi => hi) (<a>le_of_not_gt</a> _)", [{"full_name": "le_antisymm", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [188, 9], "def_end_pos": [188, 20]}, {"full_name": "iSup_le", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [875, 9], "def_end_pos": [875, 16]}, {"full_name": "iSup_le", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [875, 9], "def_end_pos": [875, 16]}, {"full_name": "le_of_not_gt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [328, 9], "def_end_pos": [328, 21]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d : MeasurableSpace \u03b1\nK : Type u_5\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\na : \u03b1\n\u22a2 \u2a06 k, \u2a06 (_ : ennrealRatEmbed k \u2264 f a), ennrealRatEmbed k = f a", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d : MeasurableSpace \u03b1\nK : Type u_5\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\na : \u03b1\n\u22a2 \u00acf a > \u2a06 k, \u2a06 (_ : ennrealRatEmbed k \u2264 f a), ennrealRatEmbed k"}, {"tactic": "intro h", "annotated_tactic": ["intro h", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d : MeasurableSpace \u03b1\nK : Type u_5\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\na : \u03b1\n\u22a2 \u00acf a > \u2a06 k, \u2a06 (_ : ennrealRatEmbed k \u2264 f a), ennrealRatEmbed k", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d : MeasurableSpace \u03b1\nK : Type u_5\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\na : \u03b1\nh : f a > \u2a06 k, \u2a06 (_ : ennrealRatEmbed k \u2264 f a), ennrealRatEmbed k\n\u22a2 False"}, {"tactic": "rcases ENNReal.lt_iff_exists_rat_btwn.1 h with \u27e8q, _, lt_q, q_lt\u27e9", "annotated_tactic": ["rcases <a>ENNReal.lt_iff_exists_rat_btwn</a>.1 h with \u27e8q, _, lt_q, q_lt\u27e9", [{"full_name": "ENNReal.lt_iff_exists_rat_btwn", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [835, 9], "def_end_pos": [835, 31]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d : MeasurableSpace \u03b1\nK : Type u_5\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\na : \u03b1\nh : f a > \u2a06 k, \u2a06 (_ : ennrealRatEmbed k \u2264 f a), ennrealRatEmbed k\n\u22a2 False", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d : MeasurableSpace \u03b1\nK : Type u_5\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\na : \u03b1\nh : f a > \u2a06 k, \u2a06 (_ : ennrealRatEmbed k \u2264 f a), ennrealRatEmbed k\nq : \u211a\nleft\u271d : 0 \u2264 q\nlt_q : \u2a06 k, \u2a06 (_ : ennrealRatEmbed k \u2264 f a), ennrealRatEmbed k < \u2191(Real.toNNReal \u2191q)\nq_lt : \u2191(Real.toNNReal \u2191q) < f a\n\u22a2 False"}, {"tactic": "have :\n  (Real.toNNReal q : \u211d\u22650\u221e) \u2264 \u2a06 (k : \u2115) (_ : ennrealRatEmbed k \u2264 f a), ennrealRatEmbed k := by\n  refine' le_iSup_of_le (Encodable.encode q) _\n  rw [ennrealRatEmbed_encode q]\n  exact le_iSup_of_le (le_of_lt q_lt) le_rfl", "annotated_tactic": ["have :\n    (<a>Real.toNNReal</a> q : \u211d\u22650\u221e) \u2264 \u2a06 (k : \u2115) (_ : <a>ennrealRatEmbed</a> k \u2264 f a), <a>ennrealRatEmbed</a> k := by\n    refine' <a>le_iSup_of_le</a> (<a>Encodable.encode</a> q) _\n    rw [<a>ennrealRatEmbed_encode</a> q]\n    exact <a>le_iSup_of_le</a> (<a>le_of_lt</a> q_lt) <a>le_rfl</a>", [{"full_name": "Real.toNNReal", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [118, 19], "def_end_pos": [118, 39]}, {"full_name": "MeasureTheory.SimpleFunc.ennrealRatEmbed", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [864, 5], "def_end_pos": [864, 20]}, {"full_name": "MeasureTheory.SimpleFunc.ennrealRatEmbed", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [864, 5], "def_end_pos": [864, 20]}, {"full_name": "le_iSup_of_le", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [849, 9], "def_end_pos": [849, 22]}, {"full_name": "Encodable.encode", "def_path": "Mathlib/Logic/Encodable/Basic.lean", "def_pos": [47, 3], "def_end_pos": [47, 9]}, {"full_name": "MeasureTheory.SimpleFunc.ennrealRatEmbed_encode", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [868, 9], "def_end_pos": [868, 31]}, {"full_name": "le_iSup_of_le", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [849, 9], "def_end_pos": [849, 22]}, {"full_name": "le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [110, 9], "def_end_pos": [110, 17]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}]], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d : MeasurableSpace \u03b1\nK : Type u_5\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\na : \u03b1\nh : f a > \u2a06 k, \u2a06 (_ : ennrealRatEmbed k \u2264 f a), ennrealRatEmbed k\nq : \u211a\nleft\u271d : 0 \u2264 q\nlt_q : \u2a06 k, \u2a06 (_ : ennrealRatEmbed k \u2264 f a), ennrealRatEmbed k < \u2191(Real.toNNReal \u2191q)\nq_lt : \u2191(Real.toNNReal \u2191q) < f a\n\u22a2 False", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d : MeasurableSpace \u03b1\nK : Type u_5\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\na : \u03b1\nh : f a > \u2a06 k, \u2a06 (_ : ennrealRatEmbed k \u2264 f a), ennrealRatEmbed k\nq : \u211a\nleft\u271d : 0 \u2264 q\nlt_q : \u2a06 k, \u2a06 (_ : ennrealRatEmbed k \u2264 f a), ennrealRatEmbed k < \u2191(Real.toNNReal \u2191q)\nq_lt : \u2191(Real.toNNReal \u2191q) < f a\nthis : \u2191(Real.toNNReal \u2191q) \u2264 \u2a06 k, \u2a06 (_ : ennrealRatEmbed k \u2264 f a), ennrealRatEmbed k\n\u22a2 False"}, {"tactic": "exact lt_irrefl _ (lt_of_le_of_lt this lt_q)", "annotated_tactic": ["exact <a>lt_irrefl</a> _ (<a>lt_of_le_of_lt</a> this lt_q)", [{"full_name": "lt_irrefl", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [79, 9], "def_end_pos": [79, 18]}, {"full_name": "lt_of_le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [122, 9], "def_end_pos": [122, 23]}]], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d : MeasurableSpace \u03b1\nK : Type u_5\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\na : \u03b1\nh : f a > \u2a06 k, \u2a06 (_ : ennrealRatEmbed k \u2264 f a), ennrealRatEmbed k\nq : \u211a\nleft\u271d : 0 \u2264 q\nlt_q : \u2a06 k, \u2a06 (_ : ennrealRatEmbed k \u2264 f a), ennrealRatEmbed k < \u2191(Real.toNNReal \u2191q)\nq_lt : \u2191(Real.toNNReal \u2191q) < f a\nthis : \u2191(Real.toNNReal \u2191q) \u2264 \u2a06 k, \u2a06 (_ : ennrealRatEmbed k \u2264 f a), ennrealRatEmbed k\n\u22a2 False", "state_after": "no goals"}, {"tactic": "refine' le_iSup_of_le (Encodable.encode q) _", "annotated_tactic": ["refine' <a>le_iSup_of_le</a> (<a>Encodable.encode</a> q) _", [{"full_name": "le_iSup_of_le", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [849, 9], "def_end_pos": [849, 22]}, {"full_name": "Encodable.encode", "def_path": "Mathlib/Logic/Encodable/Basic.lean", "def_pos": [47, 3], "def_end_pos": [47, 9]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d : MeasurableSpace \u03b1\nK : Type u_5\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\na : \u03b1\nh : f a > \u2a06 k, \u2a06 (_ : ennrealRatEmbed k \u2264 f a), ennrealRatEmbed k\nq : \u211a\nleft\u271d : 0 \u2264 q\nlt_q : \u2a06 k, \u2a06 (_ : ennrealRatEmbed k \u2264 f a), ennrealRatEmbed k < \u2191(Real.toNNReal \u2191q)\nq_lt : \u2191(Real.toNNReal \u2191q) < f a\n\u22a2 \u2191(Real.toNNReal \u2191q) \u2264 \u2a06 k, \u2a06 (_ : ennrealRatEmbed k \u2264 f a), ennrealRatEmbed k", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d : MeasurableSpace \u03b1\nK : Type u_5\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\na : \u03b1\nh : f a > \u2a06 k, \u2a06 (_ : ennrealRatEmbed k \u2264 f a), ennrealRatEmbed k\nq : \u211a\nleft\u271d : 0 \u2264 q\nlt_q : \u2a06 k, \u2a06 (_ : ennrealRatEmbed k \u2264 f a), ennrealRatEmbed k < \u2191(Real.toNNReal \u2191q)\nq_lt : \u2191(Real.toNNReal \u2191q) < f a\n\u22a2 \u2191(Real.toNNReal \u2191q) \u2264 \u2a06 (_ : ennrealRatEmbed (Encodable.encode q) \u2264 f a), ennrealRatEmbed (Encodable.encode q)"}, {"tactic": "rw [ennrealRatEmbed_encode q]", "annotated_tactic": ["rw [<a>ennrealRatEmbed_encode</a> q]", [{"full_name": "MeasureTheory.SimpleFunc.ennrealRatEmbed_encode", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [868, 9], "def_end_pos": [868, 31]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d : MeasurableSpace \u03b1\nK : Type u_5\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\na : \u03b1\nh : f a > \u2a06 k, \u2a06 (_ : ennrealRatEmbed k \u2264 f a), ennrealRatEmbed k\nq : \u211a\nleft\u271d : 0 \u2264 q\nlt_q : \u2a06 k, \u2a06 (_ : ennrealRatEmbed k \u2264 f a), ennrealRatEmbed k < \u2191(Real.toNNReal \u2191q)\nq_lt : \u2191(Real.toNNReal \u2191q) < f a\n\u22a2 \u2191(Real.toNNReal \u2191q) \u2264 \u2a06 (_ : ennrealRatEmbed (Encodable.encode q) \u2264 f a), ennrealRatEmbed (Encodable.encode q)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d : MeasurableSpace \u03b1\nK : Type u_5\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\na : \u03b1\nh : f a > \u2a06 k, \u2a06 (_ : ennrealRatEmbed k \u2264 f a), ennrealRatEmbed k\nq : \u211a\nleft\u271d : 0 \u2264 q\nlt_q : \u2a06 k, \u2a06 (_ : ennrealRatEmbed k \u2264 f a), ennrealRatEmbed k < \u2191(Real.toNNReal \u2191q)\nq_lt : \u2191(Real.toNNReal \u2191q) < f a\n\u22a2 \u2191(Real.toNNReal \u2191q) \u2264 \u2a06 (_ : \u2191(Real.toNNReal \u2191q) \u2264 f a), \u2191(Real.toNNReal \u2191q)"}, {"tactic": "exact le_iSup_of_le (le_of_lt q_lt) le_rfl", "annotated_tactic": ["exact <a>le_iSup_of_le</a> (<a>le_of_lt</a> q_lt) <a>le_rfl</a>", [{"full_name": "le_iSup_of_le", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [849, 9], "def_end_pos": [849, 22]}, {"full_name": "le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [110, 9], "def_end_pos": [110, 17]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d : MeasurableSpace \u03b1\nK : Type u_5\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\na : \u03b1\nh : f a > \u2a06 k, \u2a06 (_ : ennrealRatEmbed k \u2264 f a), ennrealRatEmbed k\nq : \u211a\nleft\u271d : 0 \u2264 q\nlt_q : \u2a06 k, \u2a06 (_ : ennrealRatEmbed k \u2264 f a), ennrealRatEmbed k < \u2191(Real.toNNReal \u2191q)\nq_lt : \u2191(Real.toNNReal \u2191q) < f a\n\u22a2 \u2191(Real.toNNReal \u2191q) \u2264 \u2a06 (_ : \u2191(Real.toNNReal \u2191q) \u2264 f a), \u2191(Real.toNNReal \u2191q)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Card.lean", "full_name": "Finset.one_lt_card_iff_nontrivial_coe", "start": [585, 1], "end": [586, 96], "traced_tactics": [{"tactic": "rw [\u2190 not_iff_not, not_lt, not_nontrivial_iff_subsingleton, card_le_one_iff_subsingleton_coe]", "annotated_tactic": ["rw [\u2190 <a>not_iff_not</a>, <a>not_lt</a>, <a>not_nontrivial_iff_subsingleton</a>, <a>card_le_one_iff_subsingleton_coe</a>]", [{"full_name": "not_iff_not", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [439, 9], "def_end_pos": [439, 20]}, {"full_name": "not_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [368, 9], "def_end_pos": [368, 15]}, {"full_name": "not_nontrivial_iff_subsingleton", "def_path": "Mathlib/Logic/Nontrivial/Defs.lean", "def_pos": [81, 9], "def_end_pos": [81, 40]}, {"full_name": "Finset.card_le_one_iff_subsingleton_coe", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [556, 9], "def_end_pos": [556, 41]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\n\u22a2 1 < card s \u2194 Nontrivial { x // x \u2208 s }", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Group/FundamentalDomain.lean", "full_name": "MeasureTheory.pairwise_disjoint_fundamentalInterior", "start": [635, 1], "end": [642, 59], "traced_tactics": [{"tactic": "refine' fun a b hab => disjoint_left.2 _", "annotated_tactic": ["refine' fun a b hab => <a>disjoint_left</a>.2 _", [{"full_name": "Set.disjoint_left", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1546, 9], "def_end_pos": [1546, 22]}]], "state_before": "G : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9 : Group G\ninst\u271d : MulAction G \u03b1\ns : Set \u03b1\nx : \u03b1\n\u22a2 Pairwise (Disjoint on fun g => g \u2022 fundamentalInterior G s)", "state_after": "G : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9 : Group G\ninst\u271d : MulAction G \u03b1\ns : Set \u03b1\nx : \u03b1\na b : G\nhab : a \u2260 b\n\u22a2 \u2200 \u2983a_1 : \u03b1\u2984, a_1 \u2208 (fun g => g \u2022 fundamentalInterior G s) a \u2192 \u00aca_1 \u2208 (fun g => g \u2022 fundamentalInterior G s) b"}, {"tactic": "rintro _ \u27e8x, hx, rfl\u27e9 \u27e8y, hy, hxy\u27e9", "annotated_tactic": ["rintro _ \u27e8x, hx, rfl\u27e9 \u27e8y, hy, hxy\u27e9", []], "state_before": "G : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9 : Group G\ninst\u271d : MulAction G \u03b1\ns : Set \u03b1\nx : \u03b1\na b : G\nhab : a \u2260 b\n\u22a2 \u2200 \u2983a_1 : \u03b1\u2984, a_1 \u2208 (fun g => g \u2022 fundamentalInterior G s) a \u2192 \u00aca_1 \u2208 (fun g => g \u2022 fundamentalInterior G s) b", "state_after": "case intro.intro.intro.intro\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9 : Group G\ninst\u271d : MulAction G \u03b1\ns : Set \u03b1\nx\u271d : \u03b1\na b : G\nhab : a \u2260 b\nx : \u03b1\nhx : x \u2208 fundamentalInterior G s\ny : \u03b1\nhy : y \u2208 fundamentalInterior G s\nhxy : (fun x => b \u2022 x) y = (fun x => a \u2022 x) x\n\u22a2 False"}, {"tactic": "rw [mem_fundamentalInterior] at hx hy", "annotated_tactic": ["rw [<a>mem_fundamentalInterior</a>] at hx hy", [{"full_name": "MeasureTheory.mem_fundamentalInterior", "def_path": "Mathlib/MeasureTheory/Group/FundamentalDomain.lean", "def_pos": [567, 9], "def_end_pos": [567, 32]}]], "state_before": "case intro.intro.intro.intro\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9 : Group G\ninst\u271d : MulAction G \u03b1\ns : Set \u03b1\nx\u271d : \u03b1\na b : G\nhab : a \u2260 b\nx : \u03b1\nhx : x \u2208 fundamentalInterior G s\ny : \u03b1\nhy : y \u2208 fundamentalInterior G s\nhxy : (fun x => b \u2022 x) y = (fun x => a \u2022 x) x\n\u22a2 False", "state_after": "case intro.intro.intro.intro\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9 : Group G\ninst\u271d : MulAction G \u03b1\ns : Set \u03b1\nx\u271d : \u03b1\na b : G\nhab : a \u2260 b\nx : \u03b1\nhx : x \u2208 s \u2227 \u2200 (g : G), g \u2260 1 \u2192 \u00acx \u2208 g \u2022 s\ny : \u03b1\nhy : y \u2208 s \u2227 \u2200 (g : G), g \u2260 1 \u2192 \u00acy \u2208 g \u2022 s\nhxy : (fun x => b \u2022 x) y = (fun x => a \u2022 x) x\n\u22a2 False"}, {"tactic": "refine' hx.2 (a\u207b\u00b9 * b) _ _", "annotated_tactic": ["refine' hx.2 (a\u207b\u00b9 * b) _ _", []], "state_before": "case intro.intro.intro.intro\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9 : Group G\ninst\u271d : MulAction G \u03b1\ns : Set \u03b1\nx\u271d : \u03b1\na b : G\nhab : a \u2260 b\nx : \u03b1\nhx : x \u2208 s \u2227 \u2200 (g : G), g \u2260 1 \u2192 \u00acx \u2208 g \u2022 s\ny : \u03b1\nhy : y \u2208 s \u2227 \u2200 (g : G), g \u2260 1 \u2192 \u00acy \u2208 g \u2022 s\nhxy : (fun x => b \u2022 x) y = (fun x => a \u2022 x) x\n\u22a2 False", "state_after": "case intro.intro.intro.intro.refine'_1\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9 : Group G\ninst\u271d : MulAction G \u03b1\ns : Set \u03b1\nx\u271d : \u03b1\na b : G\nhab : a \u2260 b\nx : \u03b1\nhx : x \u2208 s \u2227 \u2200 (g : G), g \u2260 1 \u2192 \u00acx \u2208 g \u2022 s\ny : \u03b1\nhy : y \u2208 s \u2227 \u2200 (g : G), g \u2260 1 \u2192 \u00acy \u2208 g \u2022 s\nhxy : (fun x => b \u2022 x) y = (fun x => a \u2022 x) x\n\u22a2 a\u207b\u00b9 * b \u2260 1\n\ncase intro.intro.intro.intro.refine'_2\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9 : Group G\ninst\u271d : MulAction G \u03b1\ns : Set \u03b1\nx\u271d : \u03b1\na b : G\nhab : a \u2260 b\nx : \u03b1\nhx : x \u2208 s \u2227 \u2200 (g : G), g \u2260 1 \u2192 \u00acx \u2208 g \u2022 s\ny : \u03b1\nhy : y \u2208 s \u2227 \u2200 (g : G), g \u2260 1 \u2192 \u00acy \u2208 g \u2022 s\nhxy : (fun x => b \u2022 x) y = (fun x => a \u2022 x) x\n\u22a2 x \u2208 (a\u207b\u00b9 * b) \u2022 s"}, {"tactic": "rwa [Ne.def, inv_mul_eq_iff_eq_mul, mul_one, eq_comm]", "annotated_tactic": ["rwa [<a>Ne.def</a>, <a>inv_mul_eq_iff_eq_mul</a>, <a>mul_one</a>, <a>eq_comm</a>]", [{"full_name": "Ne.def", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [59, 9], "def_end_pos": [59, 15]}, {"full_name": "inv_mul_eq_iff_eq_mul", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [691, 9], "def_end_pos": [691, 30]}, {"full_name": "mul_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [470, 9], "def_end_pos": [470, 16]}, {"full_name": "eq_comm", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [104, 9], "def_end_pos": [104, 16]}]], "state_before": "case intro.intro.intro.intro.refine'_1\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9 : Group G\ninst\u271d : MulAction G \u03b1\ns : Set \u03b1\nx\u271d : \u03b1\na b : G\nhab : a \u2260 b\nx : \u03b1\nhx : x \u2208 s \u2227 \u2200 (g : G), g \u2260 1 \u2192 \u00acx \u2208 g \u2022 s\ny : \u03b1\nhy : y \u2208 s \u2227 \u2200 (g : G), g \u2260 1 \u2192 \u00acy \u2208 g \u2022 s\nhxy : (fun x => b \u2022 x) y = (fun x => a \u2022 x) x\n\u22a2 a\u207b\u00b9 * b \u2260 1\n\ncase intro.intro.intro.intro.refine'_2\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9 : Group G\ninst\u271d : MulAction G \u03b1\ns : Set \u03b1\nx\u271d : \u03b1\na b : G\nhab : a \u2260 b\nx : \u03b1\nhx : x \u2208 s \u2227 \u2200 (g : G), g \u2260 1 \u2192 \u00acx \u2208 g \u2022 s\ny : \u03b1\nhy : y \u2208 s \u2227 \u2200 (g : G), g \u2260 1 \u2192 \u00acy \u2208 g \u2022 s\nhxy : (fun x => b \u2022 x) y = (fun x => a \u2022 x) x\n\u22a2 x \u2208 (a\u207b\u00b9 * b) \u2022 s", "state_after": "case intro.intro.intro.intro.refine'_2\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9 : Group G\ninst\u271d : MulAction G \u03b1\ns : Set \u03b1\nx\u271d : \u03b1\na b : G\nhab : a \u2260 b\nx : \u03b1\nhx : x \u2208 s \u2227 \u2200 (g : G), g \u2260 1 \u2192 \u00acx \u2208 g \u2022 s\ny : \u03b1\nhy : y \u2208 s \u2227 \u2200 (g : G), g \u2260 1 \u2192 \u00acy \u2208 g \u2022 s\nhxy : (fun x => b \u2022 x) y = (fun x => a \u2022 x) x\n\u22a2 x \u2208 (a\u207b\u00b9 * b) \u2022 s"}, {"tactic": "simpa [mul_smul, \u2190 hxy, mem_inv_smul_set_iff] using hy.1", "annotated_tactic": ["simpa [<a>mul_smul</a>, \u2190 hxy, <a>mem_inv_smul_set_iff</a>] using hy.1", [{"full_name": "MulAction.mul_smul", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [112, 3], "def_end_pos": [112, 11]}, {"full_name": "Set.mem_inv_smul_set_iff", "def_path": "Mathlib/Data/Set/Pointwise/SMul.lean", "def_pos": [893, 9], "def_end_pos": [893, 29]}]], "state_before": "case intro.intro.intro.intro.refine'_2\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9 : Group G\ninst\u271d : MulAction G \u03b1\ns : Set \u03b1\nx\u271d : \u03b1\na b : G\nhab : a \u2260 b\nx : \u03b1\nhx : x \u2208 s \u2227 \u2200 (g : G), g \u2260 1 \u2192 \u00acx \u2208 g \u2022 s\ny : \u03b1\nhy : y \u2208 s \u2227 \u2200 (g : G), g \u2260 1 \u2192 \u00acy \u2208 g \u2022 s\nhxy : (fun x => b \u2022 x) y = (fun x => a \u2022 x) x\n\u22a2 x \u2208 (a\u207b\u00b9 * b) \u2022 s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/NullMeasurable.lean", "full_name": "Set.Finite.nullMeasurableSet_sInter", "start": [396, 1], "end": [398, 51], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Reduce.lean", "full_name": "manyOneReducible_toNat_toNat", "start": [348, 1], "end": [350, 67], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Int/Units.lean", "full_name": "Int.isUnit_ne_iff_eq_neg", "start": [46, 1], "end": [47, 91], "traced_tactics": [{"tactic": "simpa only [Ne, Units.ext_iff] using units_ne_iff_eq_neg (u := hu.unit) (u' := hu'.unit)", "annotated_tactic": ["simpa only [<a>Ne</a>, <a>Units.ext_iff</a>] using <a>units_ne_iff_eq_neg</a> (u := hu.unit) (u' := hu'.unit)", [{"full_name": "Ne", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [560, 18], "def_end_pos": [560, 20]}, {"full_name": "Units.ext_iff", "def_path": "Mathlib/Algebra/Group/Units.lean", "def_pos": [153, 9], "def_end_pos": [153, 16]}, {"full_name": "Int.units_ne_iff_eq_neg", "def_path": "Mathlib/Data/Int/Units.lean", "def_pos": [41, 9], "def_end_pos": [41, 28]}]], "state_before": "u u' : \u2124\nhu : IsUnit u\nhu' : IsUnit u'\n\u22a2 u \u2260 u' \u2194 u = -u'", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/CircleIntegral.lean", "full_name": "range_circleMap", "start": [142, 1], "end": [149, 11], "traced_tactics": [{"tactic": "rw [Complex.range_exp_mul_I, smul_sphere R 0 zero_le_one]", "annotated_tactic": ["rw [<a>Complex.range_exp_mul_I</a>, <a>smul_sphere</a> R 0 <a>zero_le_one</a>]", [{"full_name": "Complex.range_exp_mul_I", "def_path": "Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean", "def_pos": [82, 9], "def_end_pos": [82, 24]}, {"full_name": "smul_sphere", "def_path": "Mathlib/Analysis/NormedSpace/Pointwise.lean", "def_pos": [423, 9], "def_end_pos": [423, 20]}, {"full_name": "zero_le_one", "def_path": "Mathlib/Algebra/Order/ZeroLEOne.lean", "def_pos": [26, 15], "def_end_pos": [26, 26]}]], "state_before": "E : Type u_1\ninst\u271d : NormedAddCommGroup E\nc : \u2102\nR : \u211d\n\u22a2 (c +\u1d65 R \u2022 range fun \u03b8 => cexp (\u2191\u03b8 * I)) = sphere c |R|", "state_after": "E : Type u_1\ninst\u271d : NormedAddCommGroup E\nc : \u2102\nR : \u211d\n\u22a2 c +\u1d65 sphere (R \u2022 0) (\u2016R\u2016 * 1) = sphere c |R|"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "E : Type u_1\ninst\u271d : NormedAddCommGroup E\nc : \u2102\nR : \u211d\n\u22a2 c +\u1d65 sphere (R \u2022 0) (\u2016R\u2016 * 1) = sphere c |R|", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "full_name": "List.aestronglyMeasurable_prod'", "start": [1387, 1], "end": [1392, 28], "traced_tactics": [{"tactic": "induction' l with f l ihl", "annotated_tactic": ["induction' l with f l ihl", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u2075 : Countable \u03b9\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b3\nf g : \u03b1 \u2192 \u03b2\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : TopologicalSpace M\ninst\u271d : ContinuousMul M\nl : List (\u03b1 \u2192 M)\nhl : \u2200 (f : \u03b1 \u2192 M), f \u2208 l \u2192 AEStronglyMeasurable f \u03bc\n\u22a2 AEStronglyMeasurable (List.prod l) \u03bc", "state_after": "case nil\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u2075 : Countable \u03b9\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b3\nf g : \u03b1 \u2192 \u03b2\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : TopologicalSpace M\ninst\u271d : ContinuousMul M\nl : List (\u03b1 \u2192 M)\nhl\u271d : \u2200 (f : \u03b1 \u2192 M), f \u2208 l \u2192 AEStronglyMeasurable f \u03bc\nhl : \u2200 (f : \u03b1 \u2192 M), f \u2208 [] \u2192 AEStronglyMeasurable f \u03bc\n\u22a2 AEStronglyMeasurable (List.prod []) \u03bc\n\ncase cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u2075 : Countable \u03b9\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b3\nf\u271d g : \u03b1 \u2192 \u03b2\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : TopologicalSpace M\ninst\u271d : ContinuousMul M\nl\u271d : List (\u03b1 \u2192 M)\nhl\u271d : \u2200 (f : \u03b1 \u2192 M), f \u2208 l\u271d \u2192 AEStronglyMeasurable f \u03bc\nf : \u03b1 \u2192 M\nl : List (\u03b1 \u2192 M)\nihl : (\u2200 (f : \u03b1 \u2192 M), f \u2208 l \u2192 AEStronglyMeasurable f \u03bc) \u2192 AEStronglyMeasurable (List.prod l) \u03bc\nhl : \u2200 (f_1 : \u03b1 \u2192 M), f_1 \u2208 f :: l \u2192 AEStronglyMeasurable f_1 \u03bc\n\u22a2 AEStronglyMeasurable (List.prod (f :: l)) \u03bc"}, {"tactic": "rw [List.forall_mem_cons] at hl", "annotated_tactic": ["rw [<a>List.forall_mem_cons</a>] at hl", [{"full_name": "List.forall_mem_cons", "def_path": "lake-packages/std/Std/Data/List/Init/Lemmas.lean", "def_pos": [120, 9], "def_end_pos": [120, 24]}]], "state_before": "case cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u2075 : Countable \u03b9\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b3\nf\u271d g : \u03b1 \u2192 \u03b2\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : TopologicalSpace M\ninst\u271d : ContinuousMul M\nl\u271d : List (\u03b1 \u2192 M)\nhl\u271d : \u2200 (f : \u03b1 \u2192 M), f \u2208 l\u271d \u2192 AEStronglyMeasurable f \u03bc\nf : \u03b1 \u2192 M\nl : List (\u03b1 \u2192 M)\nihl : (\u2200 (f : \u03b1 \u2192 M), f \u2208 l \u2192 AEStronglyMeasurable f \u03bc) \u2192 AEStronglyMeasurable (List.prod l) \u03bc\nhl : \u2200 (f_1 : \u03b1 \u2192 M), f_1 \u2208 f :: l \u2192 AEStronglyMeasurable f_1 \u03bc\n\u22a2 AEStronglyMeasurable (List.prod (f :: l)) \u03bc", "state_after": "case cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u2075 : Countable \u03b9\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b3\nf\u271d g : \u03b1 \u2192 \u03b2\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : TopologicalSpace M\ninst\u271d : ContinuousMul M\nl\u271d : List (\u03b1 \u2192 M)\nhl\u271d : \u2200 (f : \u03b1 \u2192 M), f \u2208 l\u271d \u2192 AEStronglyMeasurable f \u03bc\nf : \u03b1 \u2192 M\nl : List (\u03b1 \u2192 M)\nihl : (\u2200 (f : \u03b1 \u2192 M), f \u2208 l \u2192 AEStronglyMeasurable f \u03bc) \u2192 AEStronglyMeasurable (List.prod l) \u03bc\nhl : AEStronglyMeasurable f \u03bc \u2227 \u2200 (x : \u03b1 \u2192 M), x \u2208 l \u2192 AEStronglyMeasurable x \u03bc\n\u22a2 AEStronglyMeasurable (List.prod (f :: l)) \u03bc"}, {"tactic": "rw [List.prod_cons]", "annotated_tactic": ["rw [<a>List.prod_cons</a>]", [{"full_name": "List.prod_cons", "def_path": "Mathlib/Data/List/BigOperators/Basic.lean", "def_pos": [41, 9], "def_end_pos": [41, 18]}]], "state_before": "case cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u2075 : Countable \u03b9\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b3\nf\u271d g : \u03b1 \u2192 \u03b2\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : TopologicalSpace M\ninst\u271d : ContinuousMul M\nl\u271d : List (\u03b1 \u2192 M)\nhl\u271d : \u2200 (f : \u03b1 \u2192 M), f \u2208 l\u271d \u2192 AEStronglyMeasurable f \u03bc\nf : \u03b1 \u2192 M\nl : List (\u03b1 \u2192 M)\nihl : (\u2200 (f : \u03b1 \u2192 M), f \u2208 l \u2192 AEStronglyMeasurable f \u03bc) \u2192 AEStronglyMeasurable (List.prod l) \u03bc\nhl : AEStronglyMeasurable f \u03bc \u2227 \u2200 (x : \u03b1 \u2192 M), x \u2208 l \u2192 AEStronglyMeasurable x \u03bc\n\u22a2 AEStronglyMeasurable (List.prod (f :: l)) \u03bc", "state_after": "case cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u2075 : Countable \u03b9\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b3\nf\u271d g : \u03b1 \u2192 \u03b2\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : TopologicalSpace M\ninst\u271d : ContinuousMul M\nl\u271d : List (\u03b1 \u2192 M)\nhl\u271d : \u2200 (f : \u03b1 \u2192 M), f \u2208 l\u271d \u2192 AEStronglyMeasurable f \u03bc\nf : \u03b1 \u2192 M\nl : List (\u03b1 \u2192 M)\nihl : (\u2200 (f : \u03b1 \u2192 M), f \u2208 l \u2192 AEStronglyMeasurable f \u03bc) \u2192 AEStronglyMeasurable (List.prod l) \u03bc\nhl : AEStronglyMeasurable f \u03bc \u2227 \u2200 (x : \u03b1 \u2192 M), x \u2208 l \u2192 AEStronglyMeasurable x \u03bc\n\u22a2 AEStronglyMeasurable (f * List.prod l) \u03bc"}, {"tactic": "exact hl.1.mul (ihl hl.2)", "annotated_tactic": ["exact hl.1.<a>mul</a> (ihl hl.2)", [{"full_name": "MeasureTheory.AEStronglyMeasurable.mul", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1294, 19], "def_end_pos": [1294, 22]}]], "state_before": "case cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u2075 : Countable \u03b9\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b3\nf\u271d g : \u03b1 \u2192 \u03b2\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : TopologicalSpace M\ninst\u271d : ContinuousMul M\nl\u271d : List (\u03b1 \u2192 M)\nhl\u271d : \u2200 (f : \u03b1 \u2192 M), f \u2208 l\u271d \u2192 AEStronglyMeasurable f \u03bc\nf : \u03b1 \u2192 M\nl : List (\u03b1 \u2192 M)\nihl : (\u2200 (f : \u03b1 \u2192 M), f \u2208 l \u2192 AEStronglyMeasurable f \u03bc) \u2192 AEStronglyMeasurable (List.prod l) \u03bc\nhl : AEStronglyMeasurable f \u03bc \u2227 \u2200 (x : \u03b1 \u2192 M), x \u2208 l \u2192 AEStronglyMeasurable x \u03bc\n\u22a2 AEStronglyMeasurable (f * List.prod l) \u03bc", "state_after": "no goals"}, {"tactic": "exact aestronglyMeasurable_one", "annotated_tactic": ["exact <a>aestronglyMeasurable_one</a>", [{"full_name": "MeasureTheory.aestronglyMeasurable_one", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1161, 9], "def_end_pos": [1161, 33]}]], "state_before": "case nil\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u2075 : Countable \u03b9\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b3\nf g : \u03b1 \u2192 \u03b2\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : TopologicalSpace M\ninst\u271d : ContinuousMul M\nl : List (\u03b1 \u2192 M)\nhl\u271d : \u2200 (f : \u03b1 \u2192 M), f \u2208 l \u2192 AEStronglyMeasurable f \u03bc\nhl : \u2200 (f : \u03b1 \u2192 M), f \u2208 [] \u2192 AEStronglyMeasurable f \u03bc\n\u22a2 AEStronglyMeasurable (List.prod []) \u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Array/Init/Lemmas.lean", "full_name": "Array.toListRev_eq", "start": [101, 9], "end": [102, 77], "traced_tactics": [{"tactic": "rw [toListRev, foldl_eq_foldl_data, \u2190 List.foldr_reverse, List.foldr_self]", "annotated_tactic": ["rw [<a>toListRev</a>, <a>foldl_eq_foldl_data</a>, \u2190 <a>List.foldr_reverse</a>, <a>List.foldr_self</a>]", [{"full_name": "Array.toListRev", "def_path": "lake-packages/std/Std/Data/Array/Init/Lemmas.lean", "def_pos": [99, 15], "def_end_pos": [99, 24]}, {"full_name": "Array.foldl_eq_foldl_data", "def_path": "lake-packages/std/Std/Data/Array/Init/Lemmas.lean", "def_pos": [47, 9], "def_end_pos": [47, 28]}, {"full_name": "List.foldr_reverse", "def_path": "lake-packages/std/Std/Data/List/Init/Lemmas.lean", "def_pos": [218, 17], "def_end_pos": [218, 30]}, {"full_name": "List.foldr_self", "def_path": "lake-packages/std/Std/Data/List/Init/Lemmas.lean", "def_pos": [243, 9], "def_end_pos": [243, 19]}]], "state_before": "\u03b1 : Type u_1\narr : Array \u03b1\n\u22a2 toListRev arr = List.reverse arr.data", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/W/Cardinal.lean", "full_name": "WType.cardinal_mk_le_max_aleph0_of_finite", "start": [57, 1], "end": [81, 96], "traced_tactics": [{"tactic": "intro h", "annotated_tactic": ["intro h", []], "state_before": "\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type u\ninst\u271d : \u2200 (a : \u03b1), Finite (\u03b2 a)\n\u22a2 IsEmpty \u03b1 \u2192 #(WType \u03b2) \u2264 max #\u03b1 \u2135\u2080", "state_after": "\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type u\ninst\u271d : \u2200 (a : \u03b1), Finite (\u03b2 a)\nh : IsEmpty \u03b1\n\u22a2 #(WType \u03b2) \u2264 max #\u03b1 \u2135\u2080"}, {"tactic": "rw [Cardinal.mk_eq_zero (WType \u03b2)]", "annotated_tactic": ["rw [<a>Cardinal.mk_eq_zero</a> (<a>WType</a> \u03b2)]", [{"full_name": "Cardinal.mk_eq_zero", "def_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "def_pos": [381, 9], "def_end_pos": [381, 19]}, {"full_name": "WType", "def_path": "Mathlib/Data/W/Basic.lean", "def_pos": [37, 11], "def_end_pos": [37, 16]}]], "state_before": "\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type u\ninst\u271d : \u2200 (a : \u03b1), Finite (\u03b2 a)\nh : IsEmpty \u03b1\n\u22a2 #(WType \u03b2) \u2264 max #\u03b1 \u2135\u2080", "state_after": "\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type u\ninst\u271d : \u2200 (a : \u03b1), Finite (\u03b2 a)\nh : IsEmpty \u03b1\n\u22a2 0 \u2264 max #\u03b1 \u2135\u2080"}, {"tactic": "exact zero_le _", "annotated_tactic": ["exact <a>zero_le</a> _", [{"full_name": "zero_le", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [217, 30], "def_end_pos": [217, 37]}]], "state_before": "\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type u\ninst\u271d : \u2200 (a : \u03b1), Finite (\u03b2 a)\nh : IsEmpty \u03b1\n\u22a2 0 \u2264 max #\u03b1 \u2135\u2080", "state_after": "no goals"}, {"tactic": "rw [succ_zero]", "annotated_tactic": ["rw [<a>succ_zero</a>]", [{"full_name": "Cardinal.succ_zero", "def_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "def_pos": [1389, 9], "def_end_pos": [1389, 18]}]], "state_before": "\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type u\ninst\u271d : \u2200 (a : \u03b1), Finite (\u03b2 a)\nhn : Nonempty \u03b1\nm : Cardinal.{u} := max #\u03b1 \u2135\u2080\n\u22a2 Order.succ 0 \u2264 \u2a06 a, m ^ #(\u03b2 a)", "state_after": "\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type u\ninst\u271d : \u2200 (a : \u03b1), Finite (\u03b2 a)\nhn : Nonempty \u03b1\nm : Cardinal.{u} := max #\u03b1 \u2135\u2080\n\u22a2 1 \u2264 \u2a06 a, m ^ #(\u03b2 a)"}, {"tactic": "obtain \u27e8a\u27e9 : Nonempty \u03b1 := hn", "annotated_tactic": ["obtain \u27e8a\u27e9 : <a>Nonempty</a> \u03b1 := hn", [{"full_name": "Nonempty", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [686, 17], "def_end_pos": [686, 25]}]], "state_before": "\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type u\ninst\u271d : \u2200 (a : \u03b1), Finite (\u03b2 a)\nhn : Nonempty \u03b1\nm : Cardinal.{u} := max #\u03b1 \u2135\u2080\n\u22a2 1 \u2264 \u2a06 a, m ^ #(\u03b2 a)", "state_after": "case intro\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type u\ninst\u271d : \u2200 (a : \u03b1), Finite (\u03b2 a)\nm : Cardinal.{u} := max #\u03b1 \u2135\u2080\na : \u03b1\n\u22a2 1 \u2264 \u2a06 a, m ^ #(\u03b2 a)"}, {"tactic": "refine' le_trans _ (le_ciSup (bddAbove_range.{u, u} _) a)", "annotated_tactic": ["refine' <a>le_trans</a> _ (<a>le_ciSup</a> (<a>bddAbove_range</a>.{u, u} _) a)", [{"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}, {"full_name": "le_ciSup", "def_path": "Mathlib/Order/ConditionallyCompleteLattice/Basic.lean", "def_pos": [800, 9], "def_end_pos": [800, 17]}, {"full_name": "Cardinal.bddAbove_range", "def_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "def_pos": [945, 9], "def_end_pos": [945, 23]}]], "state_before": "case intro\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type u\ninst\u271d : \u2200 (a : \u03b1), Finite (\u03b2 a)\nm : Cardinal.{u} := max #\u03b1 \u2135\u2080\na : \u03b1\n\u22a2 1 \u2264 \u2a06 a, m ^ #(\u03b2 a)", "state_after": "case intro\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type u\ninst\u271d : \u2200 (a : \u03b1), Finite (\u03b2 a)\nm : Cardinal.{u} := max #\u03b1 \u2135\u2080\na : \u03b1\n\u22a2 1 \u2264 m ^ #(\u03b2 a)"}, {"tactic": "rw [\u2190 power_zero]", "annotated_tactic": ["rw [\u2190 <a>power_zero</a>]", [{"full_name": "Cardinal.power_zero", "def_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "def_pos": [511, 9], "def_end_pos": [511, 19]}]], "state_before": "case intro\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type u\ninst\u271d : \u2200 (a : \u03b1), Finite (\u03b2 a)\nm : Cardinal.{u} := max #\u03b1 \u2135\u2080\na : \u03b1\n\u22a2 1 \u2264 m ^ #(\u03b2 a)", "state_after": "case intro\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type u\ninst\u271d : \u2200 (a : \u03b1), Finite (\u03b2 a)\nm : Cardinal.{u} := max #\u03b1 \u2135\u2080\na : \u03b1\n\u22a2 ?m.2501 ^ 0 \u2264 m ^ #(\u03b2 a)\n\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type u\ninst\u271d : \u2200 (a : \u03b1), Finite (\u03b2 a)\nm : Cardinal.{u} := max #\u03b1 \u2135\u2080\na : \u03b1\n\u22a2 Cardinal.{u}"}, {"tactic": "exact\n  power_le_power_left\n    (pos_iff_ne_zero.1 (aleph0_pos.trans_le (le_max_right _ _))) (zero_le _)", "annotated_tactic": ["exact\n                    <a>power_le_power_left</a>\n                      (<a>pos_iff_ne_zero</a>.1 (aleph0_pos.trans_le (<a>le_max_right</a> _ _))) (<a>zero_le</a> _)", [{"full_name": "Cardinal.power_le_power_left", "def_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "def_pos": [724, 9], "def_end_pos": [724, 28]}, {"full_name": "pos_iff_ne_zero", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [243, 3], "def_end_pos": [243, 14]}, {"full_name": "le_max_right", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [61, 9], "def_end_pos": [61, 21]}, {"full_name": "zero_le", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [217, 30], "def_end_pos": [217, 37]}]], "state_before": "case intro\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type u\ninst\u271d : \u2200 (a : \u03b1), Finite (\u03b2 a)\nm : Cardinal.{u} := max #\u03b1 \u2135\u2080\na : \u03b1\n\u22a2 ?m.2501 ^ 0 \u2264 m ^ #(\u03b2 a)\n\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type u\ninst\u271d : \u2200 (a : \u03b1), Finite (\u03b2 a)\nm : Cardinal.{u} := max #\u03b1 \u2135\u2080\na : \u03b1\n\u22a2 Cardinal.{u}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Fin/Lemmas.lean", "full_name": "Fin.add_one_le_iff", "start": [250, 9], "end": [259, 85], "traced_tactics": [{"tactic": "match n with\n| 0 =>\n  intro (k : Fin 1)\n  exact iff_of_true (Subsingleton.elim (\u03b1 := Fin 1) (k+1) _ \u25b8 Nat.le_refl _) (fin_one_eq_zero ..)\n| n + 1 =>\n  intro (k : Fin (n+2))\n  rw [\u2190 add_one_lt_iff, lt_def, le_def, Nat.lt_iff_le_and_ne, and_iff_left]\n  rw [val_add_one]\n  split <;> simp [*, (Nat.succ_ne_zero _).symm, Nat.ne_of_gt (Nat.lt_succ_self _)]", "annotated_tactic": ["match n with\n  | 0 =>\n    intro (k : <a>Fin</a> 1)\n    exact <a>iff_of_true</a> (<a>Subsingleton.elim</a> (\u03b1 := <a>Fin</a> 1) (k+1) _ \u25b8 <a>Nat.le_refl</a> _) (<a>fin_one_eq_zero</a> ..)\n  | n + 1 =>\n    intro (k : <a>Fin</a> (n+2))\n    rw [\u2190 <a>add_one_lt_iff</a>, <a>lt_def</a>, <a>le_def</a>, <a>Nat.lt_iff_le_and_ne</a>, <a>and_iff_left</a>]\n    rw [<a>val_add_one</a>]\n    split <;> simp [*, (<a>Nat.succ_ne_zero</a> _).<a>symm</a>, <a>Nat.ne_of_gt</a> (<a>Nat.lt_succ_self</a> _)]", [{"full_name": "Fin", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1745, 11], "def_end_pos": [1745, 14]}, {"full_name": "iff_of_true", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [59, 9], "def_end_pos": [59, 20]}, {"full_name": "Subsingleton.elim", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [873, 19], "def_end_pos": [873, 36]}, {"full_name": "Fin", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1745, 11], "def_end_pos": [1745, 14]}, {"full_name": "Nat.le_refl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1605, 19], "def_end_pos": [1605, 30]}, {"full_name": "Fin.fin_one_eq_zero", "def_path": "lake-packages/std/Std/Data/Fin/Lemmas.lean", "def_pos": [174, 9], "def_end_pos": [174, 24]}, {"full_name": "Fin", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1745, 11], "def_end_pos": [1745, 14]}, {"full_name": "Fin.add_one_lt_iff", "def_path": "lake-packages/std/Std/Data/Fin/Lemmas.lean", "def_pos": [243, 17], "def_end_pos": [243, 31]}, {"full_name": "Fin.lt_def", "def_path": "lake-packages/std/Std/Data/Fin/Lemmas.lean", "def_pos": [81, 9], "def_end_pos": [81, 15]}, {"full_name": "Fin.le_def", "def_path": "lake-packages/std/Std/Data/Fin/Lemmas.lean", "def_pos": [79, 9], "def_end_pos": [79, 15]}, {"full_name": "Nat.lt_iff_le_and_ne", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [143, 19], "def_end_pos": [143, 35]}, {"full_name": "and_iff_left", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [204, 9], "def_end_pos": [204, 21]}, {"full_name": "Fin.val_add_one", "def_path": "lake-packages/std/Std/Data/Fin/Lemmas.lean", "def_pos": [189, 9], "def_end_pos": [189, 20]}, {"full_name": "Nat.succ_ne_zero", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [432, 9], "def_end_pos": [432, 21]}, {"full_name": "Ne.symm", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [575, 9], "def_end_pos": [575, 16]}, {"full_name": "Nat.ne_of_gt", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [136, 9], "def_end_pos": [136, 17]}, {"full_name": "Nat.lt_succ_self", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [294, 9], "def_end_pos": [294, 21]}]], "state_before": "n : Nat\n\u22a2 \u2200 {k : Fin (n + 1)}, k + 1 \u2264 k \u2194 k = last n", "state_after": "no goals"}, {"tactic": "intro (k : Fin 1)", "annotated_tactic": ["intro (k : <a>Fin</a> 1)", [{"full_name": "Fin", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1745, 11], "def_end_pos": [1745, 14]}]], "state_before": "n : Nat\n\u22a2 \u2200 {k : Fin (0 + 1)}, k + 1 \u2264 k \u2194 k = last 0", "state_after": "n : Nat\nk : Fin 1\n\u22a2 k + 1 \u2264 k \u2194 k = last 0"}, {"tactic": "exact iff_of_true (Subsingleton.elim (\u03b1 := Fin 1) (k+1) _ \u25b8 Nat.le_refl _) (fin_one_eq_zero ..)", "annotated_tactic": ["exact <a>iff_of_true</a> (<a>Subsingleton.elim</a> (\u03b1 := <a>Fin</a> 1) (k+1) _ \u25b8 <a>Nat.le_refl</a> _) (<a>fin_one_eq_zero</a> ..)", [{"full_name": "iff_of_true", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [59, 9], "def_end_pos": [59, 20]}, {"full_name": "Subsingleton.elim", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [873, 19], "def_end_pos": [873, 36]}, {"full_name": "Fin", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1745, 11], "def_end_pos": [1745, 14]}, {"full_name": "Nat.le_refl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1605, 19], "def_end_pos": [1605, 30]}, {"full_name": "Fin.fin_one_eq_zero", "def_path": "lake-packages/std/Std/Data/Fin/Lemmas.lean", "def_pos": [174, 9], "def_end_pos": [174, 24]}]], "state_before": "n : Nat\nk : Fin 1\n\u22a2 k + 1 \u2264 k \u2194 k = last 0", "state_after": "no goals"}, {"tactic": "intro (k : Fin (n+2))", "annotated_tactic": ["intro (k : <a>Fin</a> (n+2))", [{"full_name": "Fin", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1745, 11], "def_end_pos": [1745, 14]}]], "state_before": "n\u271d n : Nat\n\u22a2 \u2200 {k : Fin (n + 1 + 1)}, k + 1 \u2264 k \u2194 k = last (n + 1)", "state_after": "n\u271d n : Nat\nk : Fin (n + 2)\n\u22a2 k + 1 \u2264 k \u2194 k = last (n + 1)"}, {"tactic": "rw [\u2190 add_one_lt_iff, lt_def, le_def, Nat.lt_iff_le_and_ne, and_iff_left]", "annotated_tactic": ["rw [\u2190 <a>add_one_lt_iff</a>, <a>lt_def</a>, <a>le_def</a>, <a>Nat.lt_iff_le_and_ne</a>, <a>and_iff_left</a>]", [{"full_name": "Fin.add_one_lt_iff", "def_path": "lake-packages/std/Std/Data/Fin/Lemmas.lean", "def_pos": [243, 17], "def_end_pos": [243, 31]}, {"full_name": "Fin.lt_def", "def_path": "lake-packages/std/Std/Data/Fin/Lemmas.lean", "def_pos": [81, 9], "def_end_pos": [81, 15]}, {"full_name": "Fin.le_def", "def_path": "lake-packages/std/Std/Data/Fin/Lemmas.lean", "def_pos": [79, 9], "def_end_pos": [79, 15]}, {"full_name": "Nat.lt_iff_le_and_ne", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [143, 19], "def_end_pos": [143, 35]}, {"full_name": "and_iff_left", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [204, 9], "def_end_pos": [204, 21]}]], "state_before": "n\u271d n : Nat\nk : Fin (n + 2)\n\u22a2 k + 1 \u2264 k \u2194 k = last (n + 1)", "state_after": "n\u271d n : Nat\nk : Fin (n + 2)\n\u22a2 \u2191(k + 1) \u2260 \u2191k"}, {"tactic": "rw [val_add_one]", "annotated_tactic": ["rw [<a>val_add_one</a>]", [{"full_name": "Fin.val_add_one", "def_path": "lake-packages/std/Std/Data/Fin/Lemmas.lean", "def_pos": [189, 9], "def_end_pos": [189, 20]}]], "state_before": "n\u271d n : Nat\nk : Fin (n + 2)\n\u22a2 \u2191(k + 1) \u2260 \u2191k", "state_after": "n\u271d n : Nat\nk : Fin (n + 2)\n\u22a2 (if k = last (n + 1) then 0 else \u2191k + 1) \u2260 \u2191k"}, {"tactic": "split <;> simp [*, (Nat.succ_ne_zero _).symm, Nat.ne_of_gt (Nat.lt_succ_self _)]", "annotated_tactic": ["split <;> simp [*, (<a>Nat.succ_ne_zero</a> _).<a>symm</a>, <a>Nat.ne_of_gt</a> (<a>Nat.lt_succ_self</a> _)]", [{"full_name": "Nat.succ_ne_zero", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [432, 9], "def_end_pos": [432, 21]}, {"full_name": "Ne.symm", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [575, 9], "def_end_pos": [575, 16]}, {"full_name": "Nat.ne_of_gt", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [136, 9], "def_end_pos": [136, 17]}, {"full_name": "Nat.lt_succ_self", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [294, 9], "def_end_pos": [294, 21]}]], "state_before": "n\u271d n : Nat\nk : Fin (n + 2)\n\u22a2 (if k = last (n + 1) then 0 else \u2191k + 1) \u2260 \u2191k", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean", "full_name": "MeasureTheory.lpMeasSubgroupToLpTrim_neg", "start": [409, 1], "end": [419, 45], "traced_tactics": [{"tactic": "ext1", "annotated_tactic": ["ext1", []], "state_before": "\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\n\u03b9 : Type u_6\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 lpMeasSubgroup F m p \u03bc }\n\u22a2 lpMeasSubgroupToLpTrim F p \u03bc hm (-f) = -lpMeasSubgroupToLpTrim F p \u03bc hm f", "state_after": "case h\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\n\u03b9 : Type u_6\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 lpMeasSubgroup F m p \u03bc }\n\u22a2 \u2191\u2191(lpMeasSubgroupToLpTrim F p \u03bc hm (-f)) =\u1d50[Measure.trim \u03bc hm] \u2191\u2191(-lpMeasSubgroupToLpTrim F p \u03bc hm f)"}, {"tactic": "refine' EventuallyEq.trans _ (Lp.coeFn_neg _).symm", "annotated_tactic": ["refine' <a>EventuallyEq.trans</a> _ (<a>Lp.coeFn_neg</a> _).<a>symm</a>", [{"full_name": "Filter.EventuallyEq.trans", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1503, 9], "def_end_pos": [1503, 27]}, {"full_name": "MeasureTheory.Lp.coeFn_neg", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [228, 9], "def_end_pos": [228, 18]}, {"full_name": "Filter.EventuallyEq.symm", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1498, 9], "def_end_pos": [1498, 26]}]], "state_before": "case h\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\n\u03b9 : Type u_6\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 lpMeasSubgroup F m p \u03bc }\n\u22a2 \u2191\u2191(lpMeasSubgroupToLpTrim F p \u03bc hm (-f)) =\u1d50[Measure.trim \u03bc hm] \u2191\u2191(-lpMeasSubgroupToLpTrim F p \u03bc hm f)", "state_after": "case h\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\n\u03b9 : Type u_6\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 lpMeasSubgroup F m p \u03bc }\n\u22a2 \u2191\u2191(lpMeasSubgroupToLpTrim F p \u03bc hm (-f)) =\u1d50[Measure.trim \u03bc hm] -\u2191\u2191(lpMeasSubgroupToLpTrim F p \u03bc hm f)"}, {"tactic": "refine' ae_eq_trim_of_stronglyMeasurable hm (Lp.stronglyMeasurable _) _ _", "annotated_tactic": ["refine' <a>ae_eq_trim_of_stronglyMeasurable</a> hm (<a>Lp.stronglyMeasurable</a> _) _ _", [{"full_name": "MeasureTheory.ae_eq_trim_of_stronglyMeasurable", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1907, 9], "def_end_pos": [1907, 41]}, {"full_name": "MeasureTheory.Lp.stronglyMeasurable", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [207, 19], "def_end_pos": [207, 37]}]], "state_before": "case h\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\n\u03b9 : Type u_6\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 lpMeasSubgroup F m p \u03bc }\n\u22a2 \u2191\u2191(lpMeasSubgroupToLpTrim F p \u03bc hm (-f)) =\u1d50[Measure.trim \u03bc hm] -\u2191\u2191(lpMeasSubgroupToLpTrim F p \u03bc hm f)", "state_after": "case h.refine'_1\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\n\u03b9 : Type u_6\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 lpMeasSubgroup F m p \u03bc }\n\u22a2 StronglyMeasurable (-\u2191\u2191(lpMeasSubgroupToLpTrim F p \u03bc hm f))\n\ncase h.refine'_2\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\n\u03b9 : Type u_6\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 lpMeasSubgroup F m p \u03bc }\n\u22a2 \u2191\u2191(lpMeasSubgroupToLpTrim F p \u03bc hm (-f)) =\u1d50[\u03bc] -\u2191\u2191(lpMeasSubgroupToLpTrim F p \u03bc hm f)"}, {"tactic": "refine' (lpMeasSubgroupToLpTrim_ae_eq hm _).trans _", "annotated_tactic": ["refine' (<a>lpMeasSubgroupToLpTrim_ae_eq</a> hm _).<a>trans</a> _", [{"full_name": "MeasureTheory.lpMeasSubgroupToLpTrim_ae_eq", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean", "def_pos": [346, 9], "def_end_pos": [346, 37]}, {"full_name": "Filter.EventuallyEq.trans", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1503, 9], "def_end_pos": [1503, 27]}]], "state_before": "case h.refine'_2\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\n\u03b9 : Type u_6\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 lpMeasSubgroup F m p \u03bc }\n\u22a2 \u2191\u2191(lpMeasSubgroupToLpTrim F p \u03bc hm (-f)) =\u1d50[\u03bc] -\u2191\u2191(lpMeasSubgroupToLpTrim F p \u03bc hm f)", "state_after": "case h.refine'_2\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\n\u03b9 : Type u_6\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 lpMeasSubgroup F m p \u03bc }\n\u22a2 \u2191\u2191\u2191(-f) =\u1d50[\u03bc] -\u2191\u2191(lpMeasSubgroupToLpTrim F p \u03bc hm f)"}, {"tactic": "refine' EventuallyEq.trans _ (EventuallyEq.neg (lpMeasSubgroupToLpTrim_ae_eq hm f).symm)", "annotated_tactic": ["refine' <a>EventuallyEq.trans</a> _ (<a>EventuallyEq.neg</a> (<a>lpMeasSubgroupToLpTrim_ae_eq</a> hm f).<a>symm</a>)", [{"full_name": "Filter.EventuallyEq.trans", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1503, 9], "def_end_pos": [1503, 27]}, {"full_name": "Filter.EventuallyEq.neg", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1537, 3], "def_end_pos": [1537, 14]}, {"full_name": "MeasureTheory.lpMeasSubgroupToLpTrim_ae_eq", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean", "def_pos": [346, 9], "def_end_pos": [346, 37]}, {"full_name": "Filter.EventuallyEq.symm", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1498, 9], "def_end_pos": [1498, 26]}]], "state_before": "case h.refine'_2\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\n\u03b9 : Type u_6\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 lpMeasSubgroup F m p \u03bc }\n\u22a2 \u2191\u2191\u2191(-f) =\u1d50[\u03bc] -\u2191\u2191(lpMeasSubgroupToLpTrim F p \u03bc hm f)", "state_after": "case h.refine'_2\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\n\u03b9 : Type u_6\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 lpMeasSubgroup F m p \u03bc }\n\u22a2 \u2191\u2191\u2191(-f) =\u1d50[\u03bc] fun x => -\u2191\u2191\u2191f x"}, {"tactic": "refine' (Lp.coeFn_neg _).trans _", "annotated_tactic": ["refine' (<a>Lp.coeFn_neg</a> _).<a>trans</a> _", [{"full_name": "MeasureTheory.Lp.coeFn_neg", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [228, 9], "def_end_pos": [228, 18]}, {"full_name": "Filter.EventuallyEq.trans", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1503, 9], "def_end_pos": [1503, 27]}]], "state_before": "case h.refine'_2\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\n\u03b9 : Type u_6\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 lpMeasSubgroup F m p \u03bc }\n\u22a2 \u2191\u2191\u2191(-f) =\u1d50[\u03bc] fun x => -\u2191\u2191\u2191f x", "state_after": "case h.refine'_2\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\n\u03b9 : Type u_6\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 lpMeasSubgroup F m p \u03bc }\n\u22a2 -\u2191\u2191\u2191f =\u1d50[\u03bc] fun x => -\u2191\u2191\u2191f x"}, {"tactic": "exact eventually_of_forall fun x => by rfl", "annotated_tactic": ["exact <a>eventually_of_forall</a> fun x => by rfl", [{"full_name": "Filter.eventually_of_forall", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1111, 9], "def_end_pos": [1111, 29]}]], "state_before": "case h.refine'_2\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\n\u03b9 : Type u_6\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 lpMeasSubgroup F m p \u03bc }\n\u22a2 -\u2191\u2191\u2191f =\u1d50[\u03bc] fun x => -\u2191\u2191\u2191f x", "state_after": "no goals"}, {"tactic": "exact @StronglyMeasurable.neg _ _ _ m _ _ _ (Lp.stronglyMeasurable _)", "annotated_tactic": ["exact @<a>StronglyMeasurable.neg</a> _ _ _ m _ _ _ (<a>Lp.stronglyMeasurable</a> _)", [{"full_name": "MeasureTheory.StronglyMeasurable.neg", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [429, 3], "def_end_pos": [429, 14]}, {"full_name": "MeasureTheory.Lp.stronglyMeasurable", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [207, 19], "def_end_pos": [207, 37]}]], "state_before": "case h.refine'_1\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\n\u03b9 : Type u_6\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 lpMeasSubgroup F m p \u03bc }\n\u22a2 StronglyMeasurable (-\u2191\u2191(lpMeasSubgroupToLpTrim F p \u03bc hm f))", "state_after": "no goals"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\n\u03b9 : Type u_6\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 lpMeasSubgroup F m p \u03bc }\nx : \u03b1\n\u22a2 (-\u2191\u2191\u2191f) x = (fun x => -\u2191\u2191\u2191f x) x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Haar/Basic.lean", "full_name": "MeasureTheory.Measure.haar.chaar_mono", "start": [455, 1], "end": [464, 62], "traced_tactics": [{"tactic": "let eval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2082 - f K\u2081", "annotated_tactic": ["let eval : (<a>Compacts</a> G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2082 - f K\u2081", [{"full_name": "TopologicalSpace.Compacts", "def_path": "Mathlib/Topology/Sets/Compacts.lean", "def_pos": [36, 11], "def_end_pos": [36, 19]}]], "state_before": "G : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\nh : \u2191K\u2081 \u2286 \u2191K\u2082\n\u22a2 chaar K\u2080 K\u2081 \u2264 chaar K\u2080 K\u2082", "state_after": "G : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\nh : \u2191K\u2081 \u2286 \u2191K\u2082\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2082 - f K\u2081\n\u22a2 chaar K\u2080 K\u2081 \u2264 chaar K\u2080 K\u2082"}, {"tactic": "have : Continuous eval := (continuous_apply K\u2082).sub (continuous_apply K\u2081)", "annotated_tactic": ["have : <a>Continuous</a> eval := (<a>continuous_apply</a> K\u2082).<a>sub</a> (<a>continuous_apply</a> K\u2081)", [{"full_name": "Continuous", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [1591, 11], "def_end_pos": [1591, 21]}, {"full_name": "continuous_apply", "def_path": "Mathlib/Topology/Constructions.lean", "def_pos": [1208, 9], "def_end_pos": [1208, 25]}, {"full_name": "Continuous.sub", "def_path": "Mathlib/Topology/Algebra/Group/Basic.lean", "def_pos": [1104, 36], "def_end_pos": [1104, 39]}, {"full_name": "continuous_apply", "def_path": "Mathlib/Topology/Constructions.lean", "def_pos": [1208, 9], "def_end_pos": [1208, 25]}]], "state_before": "G : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\nh : \u2191K\u2081 \u2286 \u2191K\u2082\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2082 - f K\u2081\n\u22a2 chaar K\u2080 K\u2081 \u2264 chaar K\u2080 K\u2082", "state_after": "G : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\nh : \u2191K\u2081 \u2286 \u2191K\u2082\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2082 - f K\u2081\nthis : Continuous eval\n\u22a2 chaar K\u2080 K\u2081 \u2264 chaar K\u2080 K\u2082"}, {"tactic": "rw [\u2190 sub_nonneg]", "annotated_tactic": ["rw [\u2190 <a>sub_nonneg</a>]", [{"full_name": "sub_nonneg", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [720, 30], "def_end_pos": [720, 40]}]], "state_before": "G : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\nh : \u2191K\u2081 \u2286 \u2191K\u2082\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2082 - f K\u2081\nthis : Continuous eval\n\u22a2 chaar K\u2080 K\u2081 \u2264 chaar K\u2080 K\u2082", "state_after": "G : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\nh : \u2191K\u2081 \u2286 \u2191K\u2082\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2082 - f K\u2081\nthis : Continuous eval\n\u22a2 0 \u2264 chaar K\u2080 K\u2082 - chaar K\u2080 K\u2081"}, {"tactic": "show chaar K\u2080 \u2208 eval \u207b\u00b9' Ici (0 : \u211d)", "annotated_tactic": ["show <a>chaar</a> K\u2080 \u2208 eval \u207b\u00b9' <a>Ici</a> (0 : \u211d)", [{"full_name": "MeasureTheory.Measure.haar.chaar", "def_path": "Mathlib/MeasureTheory/Measure/Haar/Basic.lean", "def_pos": [404, 19], "def_end_pos": [404, 24]}, {"full_name": "Set.Ici", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [74, 5], "def_end_pos": [74, 8]}]], "state_before": "G : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\nh : \u2191K\u2081 \u2286 \u2191K\u2082\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2082 - f K\u2081\nthis : Continuous eval\n\u22a2 0 \u2264 chaar K\u2080 K\u2082 - chaar K\u2080 K\u2081", "state_after": "G : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\nh : \u2191K\u2081 \u2286 \u2191K\u2082\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2082 - f K\u2081\nthis : Continuous eval\n\u22a2 chaar K\u2080 \u2208 eval \u207b\u00b9' Ici 0"}, {"tactic": "apply mem_of_subset_of_mem _ (chaar_mem_clPrehaar K\u2080 \u22a4)", "annotated_tactic": ["apply <a>mem_of_subset_of_mem</a> _ (<a>chaar_mem_clPrehaar</a> K\u2080 \u22a4)", [{"full_name": "Set.mem_of_subset_of_mem", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [383, 9], "def_end_pos": [383, 29]}, {"full_name": "MeasureTheory.Measure.haar.chaar_mem_clPrehaar", "def_path": "Mathlib/MeasureTheory/Measure/Haar/Basic.lean", "def_pos": [416, 9], "def_end_pos": [416, 28]}]], "state_before": "G : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\nh : \u2191K\u2081 \u2286 \u2191K\u2082\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2082 - f K\u2081\nthis : Continuous eval\n\u22a2 chaar K\u2080 \u2208 eval \u207b\u00b9' Ici 0", "state_after": "G : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\nh : \u2191K\u2081 \u2286 \u2191K\u2082\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2082 - f K\u2081\nthis : Continuous eval\n\u22a2 clPrehaar \u2191K\u2080 \u22a4 \u2286 eval \u207b\u00b9' Ici 0"}, {"tactic": "unfold clPrehaar", "annotated_tactic": ["unfold <a>clPrehaar</a>", [{"full_name": "MeasureTheory.Measure.haar.clPrehaar", "def_path": "Mathlib/MeasureTheory/Measure/Haar/Basic.lean", "def_pos": [154, 5], "def_end_pos": [154, 14]}]], "state_before": "G : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\nh : \u2191K\u2081 \u2286 \u2191K\u2082\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2082 - f K\u2081\nthis : Continuous eval\n\u22a2 clPrehaar \u2191K\u2080 \u22a4 \u2286 eval \u207b\u00b9' Ici 0", "state_after": "G : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\nh : \u2191K\u2081 \u2286 \u2191K\u2082\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2082 - f K\u2081\nthis : Continuous eval\n\u22a2 closure (prehaar \u2191K\u2080 '' {U | U \u2286 \u2191\u22a4.toOpens \u2227 IsOpen U \u2227 1 \u2208 U}) \u2286 eval \u207b\u00b9' Ici 0"}, {"tactic": "rw [IsClosed.closure_subset_iff]", "annotated_tactic": ["rw [<a>IsClosed.closure_subset_iff</a>]", [{"full_name": "IsClosed.closure_subset_iff", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [465, 9], "def_end_pos": [465, 36]}]], "state_before": "G : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\nh : \u2191K\u2081 \u2286 \u2191K\u2082\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2082 - f K\u2081\nthis : Continuous eval\n\u22a2 closure (prehaar \u2191K\u2080 '' {U | U \u2286 \u2191\u22a4.toOpens \u2227 IsOpen U \u2227 1 \u2208 U}) \u2286 eval \u207b\u00b9' Ici 0", "state_after": "G : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\nh : \u2191K\u2081 \u2286 \u2191K\u2082\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2082 - f K\u2081\nthis : Continuous eval\n\u22a2 prehaar \u2191K\u2080 '' {U | U \u2286 \u2191\u22a4.toOpens \u2227 IsOpen U \u2227 1 \u2208 U} \u2286 eval \u207b\u00b9' Ici 0\n\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\nh : \u2191K\u2081 \u2286 \u2191K\u2082\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2082 - f K\u2081\nthis : Continuous eval\n\u22a2 IsClosed (eval \u207b\u00b9' Ici 0)"}, {"tactic": "rintro _ \u27e8U, \u27e8_, h2U, h3U\u27e9, rfl\u27e9", "annotated_tactic": ["rintro _ \u27e8U, \u27e8_, h2U, h3U\u27e9, rfl\u27e9", []], "state_before": "G : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\nh : \u2191K\u2081 \u2286 \u2191K\u2082\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2082 - f K\u2081\nthis : Continuous eval\n\u22a2 prehaar \u2191K\u2080 '' {U | U \u2286 \u2191\u22a4.toOpens \u2227 IsOpen U \u2227 1 \u2208 U} \u2286 eval \u207b\u00b9' Ici 0", "state_after": "case intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\nh : \u2191K\u2081 \u2286 \u2191K\u2082\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2082 - f K\u2081\nthis : Continuous eval\nU : Set G\nleft\u271d : U \u2286 \u2191\u22a4.toOpens\nh2U : IsOpen U\nh3U : 1 \u2208 U\n\u22a2 prehaar (\u2191K\u2080) U \u2208 eval \u207b\u00b9' Ici 0"}, {"tactic": "simp only [mem_preimage, mem_Ici, sub_nonneg]", "annotated_tactic": ["simp only [<a>mem_preimage</a>, <a>mem_Ici</a>, <a>sub_nonneg</a>]", [{"full_name": "Set.mem_preimage", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [64, 9], "def_end_pos": [64, 21]}, {"full_name": "Set.mem_Ici", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [146, 9], "def_end_pos": [146, 16]}, {"full_name": "sub_nonneg", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [720, 30], "def_end_pos": [720, 40]}]], "state_before": "case intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\nh : \u2191K\u2081 \u2286 \u2191K\u2082\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2082 - f K\u2081\nthis : Continuous eval\nU : Set G\nleft\u271d : U \u2286 \u2191\u22a4.toOpens\nh2U : IsOpen U\nh3U : 1 \u2208 U\n\u22a2 prehaar (\u2191K\u2080) U \u2208 eval \u207b\u00b9' Ici 0", "state_after": "case intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\nh : \u2191K\u2081 \u2286 \u2191K\u2082\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2082 - f K\u2081\nthis : Continuous eval\nU : Set G\nleft\u271d : U \u2286 \u2191\u22a4.toOpens\nh2U : IsOpen U\nh3U : 1 \u2208 U\n\u22a2 prehaar (\u2191K\u2080) U K\u2081 \u2264 prehaar (\u2191K\u2080) U K\u2082"}, {"tactic": "apply prehaar_mono _ h", "annotated_tactic": ["apply <a>prehaar_mono</a> _ h", [{"full_name": "MeasureTheory.Measure.haar.prehaar_mono", "def_path": "Mathlib/MeasureTheory/Measure/Haar/Basic.lean", "def_pos": [316, 9], "def_end_pos": [316, 21]}]], "state_before": "case intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\nh : \u2191K\u2081 \u2286 \u2191K\u2082\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2082 - f K\u2081\nthis : Continuous eval\nU : Set G\nleft\u271d : U \u2286 \u2191\u22a4.toOpens\nh2U : IsOpen U\nh3U : 1 \u2208 U\n\u22a2 prehaar (\u2191K\u2080) U K\u2081 \u2264 prehaar (\u2191K\u2080) U K\u2082", "state_after": "G : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\nh : \u2191K\u2081 \u2286 \u2191K\u2082\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2082 - f K\u2081\nthis : Continuous eval\nU : Set G\nleft\u271d : U \u2286 \u2191\u22a4.toOpens\nh2U : IsOpen U\nh3U : 1 \u2208 U\n\u22a2 Set.Nonempty (interior U)"}, {"tactic": "rw [h2U.interior_eq]", "annotated_tactic": ["rw [h2U.interior_eq]", []], "state_before": "G : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\nh : \u2191K\u2081 \u2286 \u2191K\u2082\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2082 - f K\u2081\nthis : Continuous eval\nU : Set G\nleft\u271d : U \u2286 \u2191\u22a4.toOpens\nh2U : IsOpen U\nh3U : 1 \u2208 U\n\u22a2 Set.Nonempty (interior U)", "state_after": "G : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\nh : \u2191K\u2081 \u2286 \u2191K\u2082\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2082 - f K\u2081\nthis : Continuous eval\nU : Set G\nleft\u271d : U \u2286 \u2191\u22a4.toOpens\nh2U : IsOpen U\nh3U : 1 \u2208 U\n\u22a2 Set.Nonempty U"}, {"tactic": "exact \u27e81, h3U\u27e9", "annotated_tactic": ["exact \u27e81, h3U\u27e9", []], "state_before": "G : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\nh : \u2191K\u2081 \u2286 \u2191K\u2082\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2082 - f K\u2081\nthis : Continuous eval\nU : Set G\nleft\u271d : U \u2286 \u2191\u22a4.toOpens\nh2U : IsOpen U\nh3U : 1 \u2208 U\n\u22a2 Set.Nonempty U", "state_after": "no goals"}, {"tactic": "apply continuous_iff_isClosed.mp this", "annotated_tactic": ["apply continuous_iff_isClosed.mp this", []], "state_before": "G : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\nh : \u2191K\u2081 \u2286 \u2191K\u2082\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2082 - f K\u2081\nthis : Continuous eval\n\u22a2 IsClosed (eval \u207b\u00b9' Ici 0)", "state_after": "case a\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\nh : \u2191K\u2081 \u2286 \u2191K\u2082\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2082 - f K\u2081\nthis : Continuous eval\n\u22a2 IsClosed (Ici 0)"}, {"tactic": "exact isClosed_Ici", "annotated_tactic": ["exact <a>isClosed_Ici</a>", [{"full_name": "isClosed_Ici", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [159, 9], "def_end_pos": [159, 21]}]], "state_before": "case a\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\nh : \u2191K\u2081 \u2286 \u2191K\u2082\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2082 - f K\u2081\nthis : Continuous eval\n\u22a2 IsClosed (Ici 0)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Array/Lemmas.lean", "full_name": "Array.get!_eq_get?", "start": [67, 9], "end": [68, 22], "traced_tactics": [{"tactic": "simp [get!_eq_getD]", "annotated_tactic": ["simp [<a>get!_eq_getD</a>]", [{"full_name": "Array.get!_eq_getD", "def_path": "lake-packages/std/Std/Data/Array/Lemmas.lean", "def_pos": [65, 9], "def_end_pos": [65, 21]}]], "state_before": "\u03b1 : Type u_1\nn : Nat\ninst\u271d : Inhabited \u03b1\na : Array \u03b1\n\u22a2 get! a n = Option.getD (get? a n) default", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/CondCount.lean", "full_name": "ProbabilityTheory.condCount_inter", "start": [138, 1], "end": [148, 66], "traced_tactics": [{"tactic": "by_cases hst : s \u2229 t = \u2205", "annotated_tactic": ["by_cases hst : s \u2229 t = \u2205", []], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nhs : Set.Finite s\n\u22a2 \u2191\u2191(condCount s) (t \u2229 u) = \u2191\u2191(condCount (s \u2229 t)) u * \u2191\u2191(condCount s) t", "state_after": "case pos\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nhs : Set.Finite s\nhst : s \u2229 t = \u2205\n\u22a2 \u2191\u2191(condCount s) (t \u2229 u) = \u2191\u2191(condCount (s \u2229 t)) u * \u2191\u2191(condCount s) t\n\ncase neg\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nhs : Set.Finite s\nhst : \u00acs \u2229 t = \u2205\n\u22a2 \u2191\u2191(condCount s) (t \u2229 u) = \u2191\u2191(condCount (s \u2229 t)) u * \u2191\u2191(condCount s) t"}, {"tactic": "rw [condCount, condCount, cond_apply _ hs.measurableSet, cond_apply _ hs.measurableSet,\n  cond_apply _ (hs.inter_of_left _).measurableSet, mul_comm _ (Measure.count (s \u2229 t)),\n  \u2190 mul_assoc, mul_comm _ (Measure.count (s \u2229 t)), \u2190 mul_assoc, ENNReal.mul_inv_cancel, one_mul,\n  mul_comm, Set.inter_assoc]", "annotated_tactic": ["rw [<a>condCount</a>, <a>condCount</a>, <a>cond_apply</a> _ hs.measurableSet, <a>cond_apply</a> _ hs.measurableSet,\n    <a>cond_apply</a> _ (hs.inter_of_left _).<a>measurableSet</a>, <a>mul_comm</a> _ (<a>Measure.count</a> (s \u2229 t)),\n    \u2190 <a>mul_assoc</a>, <a>mul_comm</a> _ (<a>Measure.count</a> (s \u2229 t)), \u2190 <a>mul_assoc</a>, <a>ENNReal.mul_inv_cancel</a>, <a>one_mul</a>,\n    <a>mul_comm</a>, <a>Set.inter_assoc</a>]", [{"full_name": "ProbabilityTheory.condCount", "def_path": "Mathlib/Probability/CondCount.lean", "def_pos": [54, 5], "def_end_pos": [54, 14]}, {"full_name": "ProbabilityTheory.condCount", "def_path": "Mathlib/Probability/CondCount.lean", "def_pos": [54, 5], "def_end_pos": [54, 14]}, {"full_name": "ProbabilityTheory.cond_apply", "def_path": "Mathlib/Probability/ConditionalProbability.lean", "def_pos": [102, 9], "def_end_pos": [102, 19]}, {"full_name": "ProbabilityTheory.cond_apply", "def_path": "Mathlib/Probability/ConditionalProbability.lean", "def_pos": [102, 9], "def_end_pos": [102, 19]}, {"full_name": "ProbabilityTheory.cond_apply", "def_path": "Mathlib/Probability/ConditionalProbability.lean", "def_pos": [102, 9], "def_end_pos": [102, 19]}, {"full_name": "Set.Finite.measurableSet", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [305, 9], "def_end_pos": [305, 33]}, {"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [302, 9], "def_end_pos": [302, 17]}, {"full_name": "MeasureTheory.Measure.count", "def_path": "Mathlib/MeasureTheory/Measure/Count.lean", "def_pos": [28, 5], "def_end_pos": [28, 10]}, {"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [264, 9], "def_end_pos": [264, 18]}, {"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [302, 9], "def_end_pos": [302, 17]}, {"full_name": "MeasureTheory.Measure.count", "def_path": "Mathlib/MeasureTheory/Measure/Count.lean", "def_pos": [28, 5], "def_end_pos": [28, 10]}, {"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [264, 9], "def_end_pos": [264, 18]}, {"full_name": "ENNReal.mul_inv_cancel", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1418, 19], "def_end_pos": [1418, 33]}, {"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [464, 9], "def_end_pos": [464, 16]}, {"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [302, 9], "def_end_pos": [302, 17]}, {"full_name": "Set.inter_assoc", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [944, 9], "def_end_pos": [944, 20]}]], "state_before": "case neg\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nhs : Set.Finite s\nhst : \u00acs \u2229 t = \u2205\n\u22a2 \u2191\u2191(condCount s) (t \u2229 u) = \u2191\u2191(condCount (s \u2229 t)) u * \u2191\u2191(condCount s) t", "state_after": "case neg.h0\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nhs : Set.Finite s\nhst : \u00acs \u2229 t = \u2205\n\u22a2 \u2191\u2191Measure.count (s \u2229 t) \u2260 0\n\ncase neg.ht\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nhs : Set.Finite s\nhst : \u00acs \u2229 t = \u2205\n\u22a2 \u2191\u2191Measure.count (s \u2229 t) \u2260 \u22a4"}, {"tactic": "rw [hst, condCount_empty_meas, Measure.coe_zero, Pi.zero_apply, zero_mul,\n  condCount_eq_zero_iff hs, \u2190 Set.inter_assoc, hst, Set.empty_inter]", "annotated_tactic": ["rw [hst, <a>condCount_empty_meas</a>, <a>Measure.coe_zero</a>, <a>Pi.zero_apply</a>, <a>zero_mul</a>,\n      <a>condCount_eq_zero_iff</a> hs, \u2190 <a>Set.inter_assoc</a>, hst, <a>Set.empty_inter</a>]", [{"full_name": "ProbabilityTheory.condCount_empty_meas", "def_path": "Mathlib/Probability/CondCount.lean", "def_pos": [59, 9], "def_end_pos": [59, 29]}, {"full_name": "MeasureTheory.Measure.coe_zero", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [760, 9], "def_end_pos": [760, 17]}, {"full_name": "Pi.zero_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [46, 3], "def_end_pos": [46, 14]}, {"full_name": "MulZeroClass.zero_mul", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [36, 3], "def_end_pos": [36, 11]}, {"full_name": "ProbabilityTheory.condCount_eq_zero_iff", "def_path": "Mathlib/Probability/CondCount.lean", "def_pos": [129, 9], "def_end_pos": [129, 30]}, {"full_name": "Set.inter_assoc", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [944, 9], "def_end_pos": [944, 20]}, {"full_name": "Set.empty_inter", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [936, 9], "def_end_pos": [936, 20]}]], "state_before": "case pos\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nhs : Set.Finite s\nhst : s \u2229 t = \u2205\n\u22a2 \u2191\u2191(condCount s) (t \u2229 u) = \u2191\u2191(condCount (s \u2229 t)) u * \u2191\u2191(condCount s) t", "state_after": "no goals"}, {"tactic": "rwa [\u2190 Measure.count_eq_zero_iff] at hst", "annotated_tactic": ["rwa [\u2190 <a>Measure.count_eq_zero_iff</a>] at hst", [{"full_name": "MeasureTheory.Measure.count_eq_zero_iff", "def_path": "Mathlib/MeasureTheory/Measure/Count.lean", "def_pos": [139, 9], "def_end_pos": [139, 26]}]], "state_before": "case neg.h0\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nhs : Set.Finite s\nhst : \u00acs \u2229 t = \u2205\n\u22a2 \u2191\u2191Measure.count (s \u2229 t) \u2260 0", "state_after": "no goals"}, {"tactic": "exact (Measure.count_apply_lt_top.2 <| hs.inter_of_left _).ne", "annotated_tactic": ["exact (<a>Measure.count_apply_lt_top</a>.2 <| hs.inter_of_left _).<a>ne</a>", [{"full_name": "MeasureTheory.Measure.count_apply_lt_top", "def_path": "Mathlib/MeasureTheory/Measure/Count.lean", "def_pos": [111, 9], "def_end_pos": [111, 27]}, {"full_name": "LT.lt.ne", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [152, 7], "def_end_pos": [152, 15]}]], "state_before": "case neg.ht\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nhs : Set.Finite s\nhst : \u00acs \u2229 t = \u2205\n\u22a2 \u2191\u2191Measure.count (s \u2229 t) \u2260 \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finmap.lean", "full_name": "Finmap.lookup_insert_of_ne", "start": [503, 1], "end": [505, 94], "traced_tactics": [{"tactic": "simp only [insert_toFinmap, lookup_toFinmap, lookup_insert_ne h]", "annotated_tactic": ["simp only [<a>insert_toFinmap</a>, <a>lookup_toFinmap</a>, <a>lookup_insert_ne</a> h]", [{"full_name": "Finmap.insert_toFinmap", "def_path": "Mathlib/Data/Finmap.lean", "def_pos": [480, 9], "def_end_pos": [480, 24]}, {"full_name": "Finmap.lookup_toFinmap", "def_path": "Mathlib/Data/Finmap.lean", "def_pos": [267, 9], "def_end_pos": [267, 24]}, {"full_name": "AList.lookup_insert_ne", "def_path": "Mathlib/Data/List/AList.lean", "def_pos": [315, 9], "def_end_pos": [315, 25]}]], "state_before": "\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\na a' : \u03b1\nb : \u03b2 a\ns\u271d : Finmap \u03b2\nh : a' \u2260 a\ns : AList \u03b2\n\u22a2 lookup a' (insert a b \u27e6s\u27e7) = lookup a' \u27e6s\u27e7", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "full_name": "MeasureTheory.OuterMeasure.boundedBy_zero", "start": [887, 1], "end": [889, 21], "traced_tactics": [{"tactic": "rw [\u2190 coe_bot, eq_bot_iff]", "annotated_tactic": ["rw [\u2190 <a>coe_bot</a>, <a>eq_bot_iff</a>]", [{"full_name": "MeasureTheory.OuterMeasure.coe_bot", "def_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "def_pos": [363, 9], "def_end_pos": [363, 16]}, {"full_name": "eq_bot_iff", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [363, 9], "def_end_pos": [363, 19]}]], "state_before": "\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\n\u22a2 boundedBy 0 = 0", "state_after": "\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\n\u22a2 boundedBy 0 \u2264 \u22a5"}, {"tactic": "apply boundedBy_le", "annotated_tactic": ["apply <a>boundedBy_le</a>", [{"full_name": "MeasureTheory.OuterMeasure.boundedBy_le", "def_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "def_pos": [838, 9], "def_end_pos": [838, 21]}]], "state_before": "\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\n\u22a2 boundedBy 0 \u2264 \u22a5", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "Finset.inter_filter", "start": [2891, 1], "end": [2892, 44], "traced_tactics": [{"tactic": "rw [inter_comm, filter_inter, inter_comm]", "annotated_tactic": ["rw [<a>inter_comm</a>, <a>filter_inter</a>, <a>inter_comm</a>]", [{"full_name": "Finset.inter_comm", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1642, 9], "def_end_pos": [1642, 19]}, {"full_name": "Finset.filter_inter", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2886, 9], "def_end_pos": [2886, 21]}, {"full_name": "Finset.inter_comm", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1642, 9], "def_end_pos": [1642, 19]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np q : \u03b1 \u2192 Prop\ninst\u271d\u00b2 : DecidablePred p\ninst\u271d\u00b9 : DecidablePred q\ns\u271d : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\n\u22a2 s \u2229 filter p t = filter p (s \u2229 t)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Int/GCD.lean", "full_name": "Int.lcm_one_left", "start": [467, 1], "end": [469, 25], "traced_tactics": [{"tactic": "rw [Int.lcm]", "annotated_tactic": ["rw [<a>Int.lcm</a>]", [{"full_name": "Int.lcm", "def_path": "Mathlib/Data/Int/GCD.lean", "def_pos": [235, 5], "def_end_pos": [235, 8]}]], "state_before": "i : \u2124\n\u22a2 lcm 1 i = natAbs i", "state_after": "i : \u2124\n\u22a2 Nat.lcm (natAbs 1) (natAbs i) = natAbs i"}, {"tactic": "apply Nat.lcm_one_left", "annotated_tactic": ["apply <a>Nat.lcm_one_left</a>", [{"full_name": "Nat.lcm_one_left", "def_path": "lake-packages/std/Std/Data/Nat/Gcd.lean", "def_pos": [203, 17], "def_end_pos": [203, 29]}]], "state_before": "i : \u2124\n\u22a2 Nat.lcm (natAbs 1) (natAbs i) = natAbs i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "full_name": "ContinuousLinearMap.smul_compLpL", "start": [1191, 1], "end": [1193, 34], "traced_tactics": [{"tactic": "ext1 f", "annotated_tactic": ["ext1 f", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedAddCommGroup G\ng : E \u2192 F\nc\u271d : \u211d\u22650\n\ud835\udd5c : Type u_5\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\ninst\u271d\u2074 : Fact (1 \u2264 p)\n\ud835\udd5c' : Type u_6\ninst\u271d\u00b3 : NormedRing \ud835\udd5c'\ninst\u271d\u00b2 : Module \ud835\udd5c' F\ninst\u271d\u00b9 : BoundedSMul \ud835\udd5c' F\ninst\u271d : SMulCommClass \ud835\udd5c \ud835\udd5c' F\nc : \ud835\udd5c'\nL : E \u2192L[\ud835\udd5c] F\n\u22a2 compLpL p \u03bc (c \u2022 L) = c \u2022 compLpL p \u03bc L", "state_after": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedAddCommGroup G\ng : E \u2192 F\nc\u271d : \u211d\u22650\n\ud835\udd5c : Type u_5\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\ninst\u271d\u2074 : Fact (1 \u2264 p)\n\ud835\udd5c' : Type u_6\ninst\u271d\u00b3 : NormedRing \ud835\udd5c'\ninst\u271d\u00b2 : Module \ud835\udd5c' F\ninst\u271d\u00b9 : BoundedSMul \ud835\udd5c' F\ninst\u271d : SMulCommClass \ud835\udd5c \ud835\udd5c' F\nc : \ud835\udd5c'\nL : E \u2192L[\ud835\udd5c] F\nf : { x // x \u2208 Lp E p }\n\u22a2 \u2191(compLpL p \u03bc (c \u2022 L)) f = \u2191(c \u2022 compLpL p \u03bc L) f"}, {"tactic": "exact smul_compLp c L f", "annotated_tactic": ["exact <a>smul_compLp</a> c L f", [{"full_name": "ContinuousLinearMap.smul_compLp", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [1137, 9], "def_end_pos": [1137, 20]}]], "state_before": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedAddCommGroup G\ng : E \u2192 F\nc\u271d : \u211d\u22650\n\ud835\udd5c : Type u_5\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\ninst\u271d\u2074 : Fact (1 \u2264 p)\n\ud835\udd5c' : Type u_6\ninst\u271d\u00b3 : NormedRing \ud835\udd5c'\ninst\u271d\u00b2 : Module \ud835\udd5c' F\ninst\u271d\u00b9 : BoundedSMul \ud835\udd5c' F\ninst\u271d : SMulCommClass \ud835\udd5c \ud835\udd5c' F\nc : \ud835\udd5c'\nL : E \u2192L[\ud835\udd5c] F\nf : { x // x \u2208 Lp E p }\n\u22a2 \u2191(compLpL p \u03bc (c \u2022 L)) f = \u2191(c \u2022 compLpL p \u03bc L) f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/OpenPos.lean", "full_name": "MeasureTheory.Measure.eqOn_Ioo_of_ae_eq", "start": [186, 1], "end": [188, 54], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Holor.lean", "full_name": "Holor.slice_sum", "start": [278, 1], "end": [284, 79], "traced_tactics": [{"tactic": "letI := Classical.decEq \u03b2", "annotated_tactic": ["letI := <a>Classical.decEq</a> \u03b2", [{"full_name": "Classical.decEq", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [983, 19], "def_end_pos": [983, 24]}]], "state_before": "\u03b1 : Type\nd : \u2115\nds ds\u2081 ds\u2082 ds\u2083 : List \u2115\ninst\u271d : AddCommMonoid \u03b1\n\u03b2 : Type\ni : \u2115\nhid : i < d\ns : Finset \u03b2\nf : \u03b2 \u2192 Holor \u03b1 (d :: ds)\n\u22a2 \u2211 x in s, slice (f x) i hid = slice (\u2211 x in s, f x) i hid", "state_after": "\u03b1 : Type\nd : \u2115\nds ds\u2081 ds\u2082 ds\u2083 : List \u2115\ninst\u271d : AddCommMonoid \u03b1\n\u03b2 : Type\ni : \u2115\nhid : i < d\ns : Finset \u03b2\nf : \u03b2 \u2192 Holor \u03b1 (d :: ds)\nthis : DecidableEq \u03b2 := Classical.decEq \u03b2\n\u22a2 \u2211 x in s, slice (f x) i hid = slice (\u2211 x in s, f x) i hid"}, {"tactic": "refine' Finset.induction_on s _ _", "annotated_tactic": ["refine' <a>Finset.induction_on</a> s _ _", [{"full_name": "Finset.induction_on", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1251, 19], "def_end_pos": [1251, 31]}]], "state_before": "\u03b1 : Type\nd : \u2115\nds ds\u2081 ds\u2082 ds\u2083 : List \u2115\ninst\u271d : AddCommMonoid \u03b1\n\u03b2 : Type\ni : \u2115\nhid : i < d\ns : Finset \u03b2\nf : \u03b2 \u2192 Holor \u03b1 (d :: ds)\nthis : DecidableEq \u03b2 := Classical.decEq \u03b2\n\u22a2 \u2211 x in s, slice (f x) i hid = slice (\u2211 x in s, f x) i hid", "state_after": "case refine'_1\n\u03b1 : Type\nd : \u2115\nds ds\u2081 ds\u2082 ds\u2083 : List \u2115\ninst\u271d : AddCommMonoid \u03b1\n\u03b2 : Type\ni : \u2115\nhid : i < d\ns : Finset \u03b2\nf : \u03b2 \u2192 Holor \u03b1 (d :: ds)\nthis : DecidableEq \u03b2 := Classical.decEq \u03b2\n\u22a2 \u2211 x in \u2205, slice (f x) i hid = slice (\u2211 x in \u2205, f x) i hid\n\ncase refine'_2\n\u03b1 : Type\nd : \u2115\nds ds\u2081 ds\u2082 ds\u2083 : List \u2115\ninst\u271d : AddCommMonoid \u03b1\n\u03b2 : Type\ni : \u2115\nhid : i < d\ns : Finset \u03b2\nf : \u03b2 \u2192 Holor \u03b1 (d :: ds)\nthis : DecidableEq \u03b2 := Classical.decEq \u03b2\n\u22a2 \u2200 \u2983a : \u03b2\u2984 {s : Finset \u03b2},\n    \u00aca \u2208 s \u2192\n      \u2211 x in s, slice (f x) i hid = slice (\u2211 x in s, f x) i hid \u2192\n        \u2211 x in insert a s, slice (f x) i hid = slice (\u2211 x in insert a s, f x) i hid"}, {"tactic": "simp [slice_zero]", "annotated_tactic": ["simp [<a>slice_zero</a>]", [{"full_name": "Holor.slice_zero", "def_path": "Mathlib/Data/Holor.lean", "def_pos": [274, 9], "def_end_pos": [274, 19]}]], "state_before": "case refine'_1\n\u03b1 : Type\nd : \u2115\nds ds\u2081 ds\u2082 ds\u2083 : List \u2115\ninst\u271d : AddCommMonoid \u03b1\n\u03b2 : Type\ni : \u2115\nhid : i < d\ns : Finset \u03b2\nf : \u03b2 \u2192 Holor \u03b1 (d :: ds)\nthis : DecidableEq \u03b2 := Classical.decEq \u03b2\n\u22a2 \u2211 x in \u2205, slice (f x) i hid = slice (\u2211 x in \u2205, f x) i hid", "state_after": "no goals"}, {"tactic": "intro _ _ h_not_in ih", "annotated_tactic": ["intro _ _ h_not_in ih", []], "state_before": "case refine'_2\n\u03b1 : Type\nd : \u2115\nds ds\u2081 ds\u2082 ds\u2083 : List \u2115\ninst\u271d : AddCommMonoid \u03b1\n\u03b2 : Type\ni : \u2115\nhid : i < d\ns : Finset \u03b2\nf : \u03b2 \u2192 Holor \u03b1 (d :: ds)\nthis : DecidableEq \u03b2 := Classical.decEq \u03b2\n\u22a2 \u2200 \u2983a : \u03b2\u2984 {s : Finset \u03b2},\n    \u00aca \u2208 s \u2192\n      \u2211 x in s, slice (f x) i hid = slice (\u2211 x in s, f x) i hid \u2192\n        \u2211 x in insert a s, slice (f x) i hid = slice (\u2211 x in insert a s, f x) i hid", "state_after": "case refine'_2\n\u03b1 : Type\nd : \u2115\nds ds\u2081 ds\u2082 ds\u2083 : List \u2115\ninst\u271d : AddCommMonoid \u03b1\n\u03b2 : Type\ni : \u2115\nhid : i < d\ns : Finset \u03b2\nf : \u03b2 \u2192 Holor \u03b1 (d :: ds)\nthis : DecidableEq \u03b2 := Classical.decEq \u03b2\na\u271d : \u03b2\ns\u271d : Finset \u03b2\nh_not_in : \u00aca\u271d \u2208 s\u271d\nih : \u2211 x in s\u271d, slice (f x) i hid = slice (\u2211 x in s\u271d, f x) i hid\n\u22a2 \u2211 x in insert a\u271d s\u271d, slice (f x) i hid = slice (\u2211 x in insert a\u271d s\u271d, f x) i hid"}, {"tactic": "rw [Finset.sum_insert h_not_in, ih, slice_add, Finset.sum_insert h_not_in]", "annotated_tactic": ["rw [<a>Finset.sum_insert</a> h_not_in, ih, <a>slice_add</a>, <a>Finset.sum_insert</a> h_not_in]", [{"full_name": "Finset.sum_insert", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [316, 3], "def_end_pos": [316, 14]}, {"full_name": "Holor.slice_add", "def_path": "Mathlib/Data/Holor.lean", "def_pos": [269, 9], "def_end_pos": [269, 18]}, {"full_name": "Finset.sum_insert", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [316, 3], "def_end_pos": [316, 14]}]], "state_before": "case refine'_2\n\u03b1 : Type\nd : \u2115\nds ds\u2081 ds\u2082 ds\u2083 : List \u2115\ninst\u271d : AddCommMonoid \u03b1\n\u03b2 : Type\ni : \u2115\nhid : i < d\ns : Finset \u03b2\nf : \u03b2 \u2192 Holor \u03b1 (d :: ds)\nthis : DecidableEq \u03b2 := Classical.decEq \u03b2\na\u271d : \u03b2\ns\u271d : Finset \u03b2\nh_not_in : \u00aca\u271d \u2208 s\u271d\nih : \u2211 x in s\u271d, slice (f x) i hid = slice (\u2211 x in s\u271d, f x) i hid\n\u22a2 \u2211 x in insert a\u271d s\u271d, slice (f x) i hid = slice (\u2211 x in insert a\u271d s\u271d, f x) i hid", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Constructions/Pi.lean", "full_name": "MeasureTheory.Measure.tprod_tprod", "start": [252, 1], "end": [256, 65], "traced_tactics": [{"tactic": "induction' l with i l ih", "annotated_tactic": ["induction' l with i l ih", []], "state_before": "\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : \u03b9 \u2192 Type u_3\ninst\u271d\u00b3 : Fintype \u03b9\nm : (i : \u03b9) \u2192 OuterMeasure (\u03b1 i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (\u03b1 i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (\u03b1 i)\n\u03b4 : Type u_4\n\u03c0 : \u03b4 \u2192 Type u_5\ninst\u271d\u00b9 : (x : \u03b4) \u2192 MeasurableSpace (\u03c0 x)\nl : List \u03b4\n\u03bc : (i : \u03b4) \u2192 Measure (\u03c0 i)\ninst\u271d : \u2200 (i : \u03b4), SigmaFinite (\u03bc i)\ns : (i : \u03b4) \u2192 Set (\u03c0 i)\n\u22a2 \u2191\u2191(Measure.tprod l \u03bc) (Set.tprod l s) = List.prod (List.map (fun i => \u2191\u2191(\u03bc i) (s i)) l)", "state_after": "case nil\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : \u03b9 \u2192 Type u_3\ninst\u271d\u00b3 : Fintype \u03b9\nm : (i : \u03b9) \u2192 OuterMeasure (\u03b1 i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (\u03b1 i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (\u03b1 i)\n\u03b4 : Type u_4\n\u03c0 : \u03b4 \u2192 Type u_5\ninst\u271d\u00b9 : (x : \u03b4) \u2192 MeasurableSpace (\u03c0 x)\n\u03bc : (i : \u03b4) \u2192 Measure (\u03c0 i)\ninst\u271d : \u2200 (i : \u03b4), SigmaFinite (\u03bc i)\ns : (i : \u03b4) \u2192 Set (\u03c0 i)\n\u22a2 \u2191\u2191(Measure.tprod [] \u03bc) (Set.tprod [] s) = List.prod (List.map (fun i => \u2191\u2191(\u03bc i) (s i)) [])\n\ncase cons\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : \u03b9 \u2192 Type u_3\ninst\u271d\u00b3 : Fintype \u03b9\nm : (i : \u03b9) \u2192 OuterMeasure (\u03b1 i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (\u03b1 i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (\u03b1 i)\n\u03b4 : Type u_4\n\u03c0 : \u03b4 \u2192 Type u_5\ninst\u271d\u00b9 : (x : \u03b4) \u2192 MeasurableSpace (\u03c0 x)\n\u03bc : (i : \u03b4) \u2192 Measure (\u03c0 i)\ninst\u271d : \u2200 (i : \u03b4), SigmaFinite (\u03bc i)\ns : (i : \u03b4) \u2192 Set (\u03c0 i)\ni : \u03b4\nl : List \u03b4\nih : \u2191\u2191(Measure.tprod l \u03bc) (Set.tprod l s) = List.prod (List.map (fun i => \u2191\u2191(\u03bc i) (s i)) l)\n\u22a2 \u2191\u2191(Measure.tprod (i :: l) \u03bc) (Set.tprod (i :: l) s) = List.prod (List.map (fun i => \u2191\u2191(\u03bc i) (s i)) (i :: l))"}, {"tactic": "rw [tprod_cons, Set.tprod, prod_prod, map_cons, prod_cons, ih]", "annotated_tactic": ["rw [<a>tprod_cons</a>, <a>Set.tprod</a>, <a>prod_prod</a>, <a>map_cons</a>, <a>prod_cons</a>, ih]", [{"full_name": "MeasureTheory.Measure.tprod_cons", "def_path": "Mathlib/MeasureTheory/Constructions/Pi.lean", "def_pos": [240, 9], "def_end_pos": [240, 19]}, {"full_name": "Set.tprod", "def_path": "Mathlib/Data/Prod/TProd.lean", "def_pos": [152, 15], "def_end_pos": [152, 20]}, {"full_name": "MeasureTheory.Measure.prod_prod", "def_path": "Mathlib/MeasureTheory/Constructions/Prod/Basic.lean", "def_pos": [316, 9], "def_end_pos": [316, 18]}, {"full_name": "List.map_cons", "def_path": "lake-packages/std/Std/Data/List/Init/Lemmas.lean", "def_pos": [89, 17], "def_end_pos": [89, 25]}, {"full_name": "List.prod_cons", "def_path": "Mathlib/Data/List/BigOperators/Basic.lean", "def_pos": [41, 9], "def_end_pos": [41, 18]}]], "state_before": "case cons\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : \u03b9 \u2192 Type u_3\ninst\u271d\u00b3 : Fintype \u03b9\nm : (i : \u03b9) \u2192 OuterMeasure (\u03b1 i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (\u03b1 i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (\u03b1 i)\n\u03b4 : Type u_4\n\u03c0 : \u03b4 \u2192 Type u_5\ninst\u271d\u00b9 : (x : \u03b4) \u2192 MeasurableSpace (\u03c0 x)\n\u03bc : (i : \u03b4) \u2192 Measure (\u03c0 i)\ninst\u271d : \u2200 (i : \u03b4), SigmaFinite (\u03bc i)\ns : (i : \u03b4) \u2192 Set (\u03c0 i)\ni : \u03b4\nl : List \u03b4\nih : \u2191\u2191(Measure.tprod l \u03bc) (Set.tprod l s) = List.prod (List.map (fun i => \u2191\u2191(\u03bc i) (s i)) l)\n\u22a2 \u2191\u2191(Measure.tprod (i :: l) \u03bc) (Set.tprod (i :: l) s) = List.prod (List.map (fun i => \u2191\u2191(\u03bc i) (s i)) (i :: l))", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case nil\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : \u03b9 \u2192 Type u_3\ninst\u271d\u00b3 : Fintype \u03b9\nm : (i : \u03b9) \u2192 OuterMeasure (\u03b1 i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (\u03b1 i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (\u03b1 i)\n\u03b4 : Type u_4\n\u03c0 : \u03b4 \u2192 Type u_5\ninst\u271d\u00b9 : (x : \u03b4) \u2192 MeasurableSpace (\u03c0 x)\n\u03bc : (i : \u03b4) \u2192 Measure (\u03c0 i)\ninst\u271d : \u2200 (i : \u03b4), SigmaFinite (\u03bc i)\ns : (i : \u03b4) \u2192 Set (\u03c0 i)\n\u22a2 \u2191\u2191(Measure.tprod [] \u03bc) (Set.tprod [] s) = List.prod (List.map (fun i => \u2191\u2191(\u03bc i) (s i)) [])", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Sort.lean", "full_name": "Finset.orderEmbOfFin_singleton", "start": [206, 1], "end": [208, 94], "traced_tactics": [{"tactic": "rw [Subsingleton.elim i \u27e80, zero_lt_one\u27e9, orderEmbOfFin_zero _ zero_lt_one, min'_singleton]", "annotated_tactic": ["rw [<a>Subsingleton.elim</a> i \u27e80, <a>zero_lt_one</a>\u27e9, <a>orderEmbOfFin_zero</a> _ <a>zero_lt_one</a>, <a>min'_singleton</a>]", [{"full_name": "Subsingleton.elim", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [873, 19], "def_end_pos": [873, 36]}, {"full_name": "zero_lt_one", "def_path": "Mathlib/Algebra/Order/ZeroLEOne.lean", "def_pos": [39, 15], "def_end_pos": [39, 26]}, {"full_name": "Finset.orderEmbOfFin_zero", "def_path": "Mathlib/Data/Finset/Sort.lean", "def_pos": [192, 9], "def_end_pos": [192, 27]}, {"full_name": "zero_lt_one", "def_path": "Mathlib/Algebra/Order/ZeroLEOne.lean", "def_pos": [39, 15], "def_end_pos": [39, 26]}, {"full_name": "Finset.min'_singleton", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [1438, 9], "def_end_pos": [1438, 23]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na : \u03b1\ni : Fin 1\n\u22a2 \u2191(orderEmbOfFin {a} (_ : card {a} = 1)) i = a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Intervals/ProjIcc.lean", "full_name": "StrictMono.strictMonoOn_IicExtend", "start": [334, 1], "end": [336, 44], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/Floor.lean", "full_name": "measurable_fract", "start": [47, 1], "end": [50, 93], "traced_tactics": [{"tactic": "intro s hs", "annotated_tactic": ["intro s hs", []], "state_before": "\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : LinearOrderedRing R\ninst\u271d\u2074 : FloorRing R\ninst\u271d\u00b3 : TopologicalSpace R\ninst\u271d\u00b2 : OrderTopology R\ninst\u271d\u00b9 : MeasurableSpace R\ninst\u271d : BorelSpace R\n\u22a2 Measurable Int.fract", "state_after": "\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : LinearOrderedRing R\ninst\u271d\u2074 : FloorRing R\ninst\u271d\u00b3 : TopologicalSpace R\ninst\u271d\u00b2 : OrderTopology R\ninst\u271d\u00b9 : MeasurableSpace R\ninst\u271d : BorelSpace R\ns : Set R\nhs : MeasurableSet s\n\u22a2 MeasurableSet (Int.fract \u207b\u00b9' s)"}, {"tactic": "rw [Int.preimage_fract]", "annotated_tactic": ["rw [<a>Int.preimage_fract</a>]", [{"full_name": "Int.preimage_fract", "def_path": "Mathlib/Algebra/Order/Floor.lean", "def_pos": [1058, 9], "def_end_pos": [1058, 23]}]], "state_before": "\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : LinearOrderedRing R\ninst\u271d\u2074 : FloorRing R\ninst\u271d\u00b3 : TopologicalSpace R\ninst\u271d\u00b2 : OrderTopology R\ninst\u271d\u00b9 : MeasurableSpace R\ninst\u271d : BorelSpace R\ns : Set R\nhs : MeasurableSet s\n\u22a2 MeasurableSet (Int.fract \u207b\u00b9' s)", "state_after": "\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : LinearOrderedRing R\ninst\u271d\u2074 : FloorRing R\ninst\u271d\u00b3 : TopologicalSpace R\ninst\u271d\u00b2 : OrderTopology R\ninst\u271d\u00b9 : MeasurableSpace R\ninst\u271d : BorelSpace R\ns : Set R\nhs : MeasurableSet s\n\u22a2 MeasurableSet (\u22c3 m, (fun x => x - \u2191m) \u207b\u00b9' (s \u2229 Ico 0 1))"}, {"tactic": "exact MeasurableSet.iUnion fun z => measurable_id.sub_const _ (hs.inter measurableSet_Ico)", "annotated_tactic": ["exact <a>MeasurableSet.iUnion</a> fun z => measurable_id.sub_const _ (hs.inter <a>measurableSet_Ico</a>)", [{"full_name": "MeasurableSet.iUnion", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [115, 19], "def_end_pos": [115, 39]}, {"full_name": "measurableSet_Ico", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [594, 9], "def_end_pos": [594, 26]}]], "state_before": "\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : LinearOrderedRing R\ninst\u271d\u2074 : FloorRing R\ninst\u271d\u00b3 : TopologicalSpace R\ninst\u271d\u00b2 : OrderTopology R\ninst\u271d\u00b9 : MeasurableSpace R\ninst\u271d : BorelSpace R\ns : Set R\nhs : MeasurableSet s\n\u22a2 MeasurableSet (\u22c3 m, (fun x => x - \u2191m) \u207b\u00b9' (s \u2229 Ico 0 1))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean", "full_name": "Basis.parallelepiped_basisFun", "start": [74, 1], "end": [79, 24], "traced_tactics": [{"tactic": "refine' Eq.trans _ ((uIcc_of_le _).trans (Set.pi_univ_Icc _ _).symm)", "annotated_tactic": ["refine' <a>Eq.trans</a> _ ((<a>uIcc_of_le</a> _).<a>trans</a> (<a>Set.pi_univ_Icc</a> _ _).<a>symm</a>)", [{"full_name": "Eq.trans", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [322, 9], "def_end_pos": [322, 17]}, {"full_name": "Set.uIcc_of_le", "def_path": "Mathlib/Data/Set/Intervals/UnorderedInterval.lean", "def_pos": [69, 7], "def_end_pos": [69, 17]}, {"full_name": "Eq.trans", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [322, 9], "def_end_pos": [322, 17]}, {"full_name": "Set.pi_univ_Icc", "def_path": "Mathlib/Data/Set/Intervals/Pi.lean", "def_pos": [43, 9], "def_end_pos": [43, 20]}, {"full_name": "Eq.symm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [310, 9], "def_end_pos": [310, 16]}]], "state_before": "\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\n\u22a2 \u2191(parallelepiped (Pi.basisFun \u211d \u03b9)) = \u2191(PositiveCompacts.piIcc01 \u03b9)", "state_after": "case refine'_1\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\n\u22a2 \u2191(parallelepiped (Pi.basisFun \u211d \u03b9)) = uIcc (fun i => 0) fun i => 1\n\ncase refine'_2\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\n\u22a2 (fun i => 0) \u2264 fun i => 1"}, {"tactic": "classical convert parallelepiped_single (\u03b9 := \u03b9) 1", "annotated_tactic": ["classical convert <a>parallelepiped_single</a> (\u03b9 := \u03b9) 1", [{"full_name": "parallelepiped_single", "def_path": "Mathlib/MeasureTheory/Measure/Haar/OfBasis.lean", "def_pos": [139, 9], "def_end_pos": [139, 30]}]], "state_before": "case refine'_1\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\n\u22a2 \u2191(parallelepiped (Pi.basisFun \u211d \u03b9)) = uIcc (fun i => 0) fun i => 1", "state_after": "no goals"}, {"tactic": "convert parallelepiped_single (\u03b9 := \u03b9) 1", "annotated_tactic": ["convert <a>parallelepiped_single</a> (\u03b9 := \u03b9) 1", [{"full_name": "parallelepiped_single", "def_path": "Mathlib/MeasureTheory/Measure/Haar/OfBasis.lean", "def_pos": [139, 9], "def_end_pos": [139, 30]}]], "state_before": "case refine'_1\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\n\u22a2 \u2191(parallelepiped (Pi.basisFun \u211d \u03b9)) = uIcc (fun i => 0) fun i => 1", "state_after": "no goals"}, {"tactic": "exact zero_le_one", "annotated_tactic": ["exact <a>zero_le_one</a>", [{"full_name": "zero_le_one", "def_path": "Mathlib/Algebra/Order/ZeroLEOne.lean", "def_pos": [26, 15], "def_end_pos": [26, 26]}]], "state_before": "case refine'_2\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\n\u22a2 (fun i => 0) \u2264 fun i => 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Countable.lean", "full_name": "Set.exists_seq_iSup_eq_top_iff_countable", "start": [156, 1], "end": [171, 42], "traced_tactics": [{"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d : CompleteLattice \u03b1\np : \u03b1 \u2192 Prop\nh : \u2203 x, p x\n\u22a2 (\u2203 s, (\u2200 (n : \u2115), p (s n)) \u2227 \u2a06 n, s n = \u22a4) \u2194 \u2203 S, Set.Countable S \u2227 (\u2200 (s : \u03b1), s \u2208 S \u2192 p s) \u2227 sSup S = \u22a4", "state_after": "case mp\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d : CompleteLattice \u03b1\np : \u03b1 \u2192 Prop\nh : \u2203 x, p x\n\u22a2 (\u2203 s, (\u2200 (n : \u2115), p (s n)) \u2227 \u2a06 n, s n = \u22a4) \u2192 \u2203 S, Set.Countable S \u2227 (\u2200 (s : \u03b1), s \u2208 S \u2192 p s) \u2227 sSup S = \u22a4\n\ncase mpr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d : CompleteLattice \u03b1\np : \u03b1 \u2192 Prop\nh : \u2203 x, p x\n\u22a2 (\u2203 S, Set.Countable S \u2227 (\u2200 (s : \u03b1), s \u2208 S \u2192 p s) \u2227 sSup S = \u22a4) \u2192 \u2203 s, (\u2200 (n : \u2115), p (s n)) \u2227 \u2a06 n, s n = \u22a4"}, {"tactic": "rintro \u27e8s, hps, hs\u27e9", "annotated_tactic": ["rintro \u27e8s, hps, hs\u27e9", []], "state_before": "case mp\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d : CompleteLattice \u03b1\np : \u03b1 \u2192 Prop\nh : \u2203 x, p x\n\u22a2 (\u2203 s, (\u2200 (n : \u2115), p (s n)) \u2227 \u2a06 n, s n = \u22a4) \u2192 \u2203 S, Set.Countable S \u2227 (\u2200 (s : \u03b1), s \u2208 S \u2192 p s) \u2227 sSup S = \u22a4", "state_after": "case mp.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d : CompleteLattice \u03b1\np : \u03b1 \u2192 Prop\nh : \u2203 x, p x\ns : \u2115 \u2192 \u03b1\nhps : \u2200 (n : \u2115), p (s n)\nhs : \u2a06 n, s n = \u22a4\n\u22a2 \u2203 S, Set.Countable S \u2227 (\u2200 (s : \u03b1), s \u2208 S \u2192 p s) \u2227 sSup S = \u22a4"}, {"tactic": "refine' \u27e8range s, countable_range s, forall_range_iff.2 hps, _\u27e9", "annotated_tactic": ["refine' \u27e8<a>range</a> s, <a>countable_range</a> s, <a>forall_range_iff</a>.2 hps, _\u27e9", [{"full_name": "Set.range", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [668, 5], "def_end_pos": [668, 10]}, {"full_name": "Set.countable_range", "def_path": "Mathlib/Data/Set/Countable.lean", "def_pos": [89, 9], "def_end_pos": [89, 24]}, {"full_name": "Set.forall_range_iff", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [684, 9], "def_end_pos": [684, 25]}]], "state_before": "case mp.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d : CompleteLattice \u03b1\np : \u03b1 \u2192 Prop\nh : \u2203 x, p x\ns : \u2115 \u2192 \u03b1\nhps : \u2200 (n : \u2115), p (s n)\nhs : \u2a06 n, s n = \u22a4\n\u22a2 \u2203 S, Set.Countable S \u2227 (\u2200 (s : \u03b1), s \u2208 S \u2192 p s) \u2227 sSup S = \u22a4", "state_after": "case mp.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d : CompleteLattice \u03b1\np : \u03b1 \u2192 Prop\nh : \u2203 x, p x\ns : \u2115 \u2192 \u03b1\nhps : \u2200 (n : \u2115), p (s n)\nhs : \u2a06 n, s n = \u22a4\n\u22a2 sSup (range s) = \u22a4"}, {"tactic": "rwa [sSup_range]", "annotated_tactic": ["rwa [<a>sSup_range</a>]", [{"full_name": "sSup_range", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [685, 9], "def_end_pos": [685, 19]}]], "state_before": "case mp.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d : CompleteLattice \u03b1\np : \u03b1 \u2192 Prop\nh : \u2203 x, p x\ns : \u2115 \u2192 \u03b1\nhps : \u2200 (n : \u2115), p (s n)\nhs : \u2a06 n, s n = \u22a4\n\u22a2 sSup (range s) = \u22a4", "state_after": "no goals"}, {"tactic": "rintro \u27e8S, hSc, hps, hS\u27e9", "annotated_tactic": ["rintro \u27e8S, hSc, hps, hS\u27e9", []], "state_before": "case mpr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d : CompleteLattice \u03b1\np : \u03b1 \u2192 Prop\nh : \u2203 x, p x\n\u22a2 (\u2203 S, Set.Countable S \u2227 (\u2200 (s : \u03b1), s \u2208 S \u2192 p s) \u2227 sSup S = \u22a4) \u2192 \u2203 s, (\u2200 (n : \u2115), p (s n)) \u2227 \u2a06 n, s n = \u22a4", "state_after": "case mpr.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d : CompleteLattice \u03b1\np : \u03b1 \u2192 Prop\nh : \u2203 x, p x\nS : Set \u03b1\nhSc : Set.Countable S\nhps : \u2200 (s : \u03b1), s \u2208 S \u2192 p s\nhS : sSup S = \u22a4\n\u22a2 \u2203 s, (\u2200 (n : \u2115), p (s n)) \u2227 \u2a06 n, s n = \u22a4"}, {"tactic": "rcases eq_empty_or_nonempty S with (rfl | hne)", "annotated_tactic": ["rcases <a>eq_empty_or_nonempty</a> S with (rfl | hne)", [{"full_name": "Set.eq_empty_or_nonempty", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [635, 9], "def_end_pos": [635, 29]}]], "state_before": "case mpr.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d : CompleteLattice \u03b1\np : \u03b1 \u2192 Prop\nh : \u2203 x, p x\nS : Set \u03b1\nhSc : Set.Countable S\nhps : \u2200 (s : \u03b1), s \u2208 S \u2192 p s\nhS : sSup S = \u22a4\n\u22a2 \u2203 s, (\u2200 (n : \u2115), p (s n)) \u2227 \u2a06 n, s n = \u22a4", "state_after": "case mpr.intro.intro.intro.inl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d : CompleteLattice \u03b1\np : \u03b1 \u2192 Prop\nh : \u2203 x, p x\nhSc : Set.Countable \u2205\nhps : \u2200 (s : \u03b1), s \u2208 \u2205 \u2192 p s\nhS : sSup \u2205 = \u22a4\n\u22a2 \u2203 s, (\u2200 (n : \u2115), p (s n)) \u2227 \u2a06 n, s n = \u22a4\n\ncase mpr.intro.intro.intro.inr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d : CompleteLattice \u03b1\np : \u03b1 \u2192 Prop\nh : \u2203 x, p x\nS : Set \u03b1\nhSc : Set.Countable S\nhps : \u2200 (s : \u03b1), s \u2208 S \u2192 p s\nhS : sSup S = \u22a4\nhne : Set.Nonempty S\n\u22a2 \u2203 s, (\u2200 (n : \u2115), p (s n)) \u2227 \u2a06 n, s n = \u22a4"}, {"tactic": "rw [sSup_empty] at hS", "annotated_tactic": ["rw [<a>sSup_empty</a>] at hS", [{"full_name": "sSup_empty", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [543, 9], "def_end_pos": [543, 19]}]], "state_before": "case mpr.intro.intro.intro.inl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d : CompleteLattice \u03b1\np : \u03b1 \u2192 Prop\nh : \u2203 x, p x\nhSc : Set.Countable \u2205\nhps : \u2200 (s : \u03b1), s \u2208 \u2205 \u2192 p s\nhS : sSup \u2205 = \u22a4\n\u22a2 \u2203 s, (\u2200 (n : \u2115), p (s n)) \u2227 \u2a06 n, s n = \u22a4", "state_after": "case mpr.intro.intro.intro.inl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d : CompleteLattice \u03b1\np : \u03b1 \u2192 Prop\nh : \u2203 x, p x\nhSc : Set.Countable \u2205\nhps : \u2200 (s : \u03b1), s \u2208 \u2205 \u2192 p s\nhS : \u22a5 = \u22a4\n\u22a2 \u2203 s, (\u2200 (n : \u2115), p (s n)) \u2227 \u2a06 n, s n = \u22a4"}, {"tactic": "haveI := subsingleton_of_bot_eq_top hS", "annotated_tactic": ["haveI := <a>subsingleton_of_bot_eq_top</a> hS", [{"full_name": "subsingleton_of_bot_eq_top", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [686, 9], "def_end_pos": [686, 35]}]], "state_before": "case mpr.intro.intro.intro.inl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d : CompleteLattice \u03b1\np : \u03b1 \u2192 Prop\nh : \u2203 x, p x\nhSc : Set.Countable \u2205\nhps : \u2200 (s : \u03b1), s \u2208 \u2205 \u2192 p s\nhS : \u22a5 = \u22a4\n\u22a2 \u2203 s, (\u2200 (n : \u2115), p (s n)) \u2227 \u2a06 n, s n = \u22a4", "state_after": "case mpr.intro.intro.intro.inl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d : CompleteLattice \u03b1\np : \u03b1 \u2192 Prop\nh : \u2203 x, p x\nhSc : Set.Countable \u2205\nhps : \u2200 (s : \u03b1), s \u2208 \u2205 \u2192 p s\nhS : \u22a5 = \u22a4\nthis : Subsingleton \u03b1\n\u22a2 \u2203 s, (\u2200 (n : \u2115), p (s n)) \u2227 \u2a06 n, s n = \u22a4"}, {"tactic": "rcases h with \u27e8x, hx\u27e9", "annotated_tactic": ["rcases h with \u27e8x, hx\u27e9", []], "state_before": "case mpr.intro.intro.intro.inl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d : CompleteLattice \u03b1\np : \u03b1 \u2192 Prop\nh : \u2203 x, p x\nhSc : Set.Countable \u2205\nhps : \u2200 (s : \u03b1), s \u2208 \u2205 \u2192 p s\nhS : \u22a5 = \u22a4\nthis : Subsingleton \u03b1\n\u22a2 \u2203 s, (\u2200 (n : \u2115), p (s n)) \u2227 \u2a06 n, s n = \u22a4", "state_after": "case mpr.intro.intro.intro.inl.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d : CompleteLattice \u03b1\np : \u03b1 \u2192 Prop\nhSc : Set.Countable \u2205\nhps : \u2200 (s : \u03b1), s \u2208 \u2205 \u2192 p s\nhS : \u22a5 = \u22a4\nthis : Subsingleton \u03b1\nx : \u03b1\nhx : p x\n\u22a2 \u2203 s, (\u2200 (n : \u2115), p (s n)) \u2227 \u2a06 n, s n = \u22a4"}, {"tactic": "exact \u27e8fun _ => x, fun _ => hx, Subsingleton.elim _ _\u27e9", "annotated_tactic": ["exact \u27e8fun _ => x, fun _ => hx, <a>Subsingleton.elim</a> _ _\u27e9", [{"full_name": "Subsingleton.elim", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [873, 19], "def_end_pos": [873, 36]}]], "state_before": "case mpr.intro.intro.intro.inl.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d : CompleteLattice \u03b1\np : \u03b1 \u2192 Prop\nhSc : Set.Countable \u2205\nhps : \u2200 (s : \u03b1), s \u2208 \u2205 \u2192 p s\nhS : \u22a5 = \u22a4\nthis : Subsingleton \u03b1\nx : \u03b1\nhx : p x\n\u22a2 \u2203 s, (\u2200 (n : \u2115), p (s n)) \u2227 \u2a06 n, s n = \u22a4", "state_after": "no goals"}, {"tactic": "rcases (Set.countable_iff_exists_surjective hne).1 hSc with \u27e8s, hs\u27e9", "annotated_tactic": ["rcases (<a>Set.countable_iff_exists_surjective</a> hne).1 hSc with \u27e8s, hs\u27e9", [{"full_name": "Set.countable_iff_exists_surjective", "def_path": "Mathlib/Data/Set/Countable.lean", "def_pos": [104, 19], "def_end_pos": [104, 50]}]], "state_before": "case mpr.intro.intro.intro.inr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d : CompleteLattice \u03b1\np : \u03b1 \u2192 Prop\nh : \u2203 x, p x\nS : Set \u03b1\nhSc : Set.Countable S\nhps : \u2200 (s : \u03b1), s \u2208 S \u2192 p s\nhS : sSup S = \u22a4\nhne : Set.Nonempty S\n\u22a2 \u2203 s, (\u2200 (n : \u2115), p (s n)) \u2227 \u2a06 n, s n = \u22a4", "state_after": "case mpr.intro.intro.intro.inr.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d : CompleteLattice \u03b1\np : \u03b1 \u2192 Prop\nh : \u2203 x, p x\nS : Set \u03b1\nhSc : Set.Countable S\nhps : \u2200 (s : \u03b1), s \u2208 S \u2192 p s\nhS : sSup S = \u22a4\nhne : Set.Nonempty S\ns : \u2115 \u2192 \u2191S\nhs : Surjective s\n\u22a2 \u2203 s, (\u2200 (n : \u2115), p (s n)) \u2227 \u2a06 n, s n = \u22a4"}, {"tactic": "refine' \u27e8fun n => s n, fun n => hps _ (s n).coe_prop, _\u27e9", "annotated_tactic": ["refine' \u27e8fun n => s n, fun n => hps _ (s n).<a>coe_prop</a>, _\u27e9", [{"full_name": "Subtype.coe_prop", "def_path": "Mathlib/Data/Subtype.lean", "def_pos": [262, 9], "def_end_pos": [262, 17]}]], "state_before": "case mpr.intro.intro.intro.inr.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d : CompleteLattice \u03b1\np : \u03b1 \u2192 Prop\nh : \u2203 x, p x\nS : Set \u03b1\nhSc : Set.Countable S\nhps : \u2200 (s : \u03b1), s \u2208 S \u2192 p s\nhS : sSup S = \u22a4\nhne : Set.Nonempty S\ns : \u2115 \u2192 \u2191S\nhs : Surjective s\n\u22a2 \u2203 s, (\u2200 (n : \u2115), p (s n)) \u2227 \u2a06 n, s n = \u22a4", "state_after": "case mpr.intro.intro.intro.inr.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d : CompleteLattice \u03b1\np : \u03b1 \u2192 Prop\nh : \u2203 x, p x\nS : Set \u03b1\nhSc : Set.Countable S\nhps : \u2200 (s : \u03b1), s \u2208 S \u2192 p s\nhS : sSup S = \u22a4\nhne : Set.Nonempty S\ns : \u2115 \u2192 \u2191S\nhs : Surjective s\n\u22a2 \u2a06 n, (fun n => \u2191(s n)) n = \u22a4"}, {"tactic": "rwa [hs.iSup_comp, \u2190 sSup_eq_iSup']", "annotated_tactic": ["rwa [hs.iSup_comp, \u2190 <a>sSup_eq_iSup'</a>]", [{"full_name": "sSup_eq_iSup'", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [689, 9], "def_end_pos": [689, 22]}]], "state_before": "case mpr.intro.intro.intro.inr.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d : CompleteLattice \u03b1\np : \u03b1 \u2192 Prop\nh : \u2203 x, p x\nS : Set \u03b1\nhSc : Set.Countable S\nhps : \u2200 (s : \u03b1), s \u2208 S \u2192 p s\nhS : sSup S = \u22a4\nhne : Set.Nonempty S\ns : \u2115 \u2192 \u2191S\nhs : Surjective s\n\u22a2 \u2a06 n, (fun n => \u2191(s n)) n = \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Lattice.lean", "full_name": "Finset.not_mem_of_lt_min", "start": [1387, 1], "end": [1388, 79], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Process/Stopping.lean", "full_name": "MeasureTheory.ProgMeasurable.stoppedProcess", "start": [845, 1], "end": [847, 74], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Finite.lean", "full_name": "Set.iInter_iUnion_of_antitone", "start": [1548, 1], "end": [1551, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "full_name": "Nat.sub_le_iff_le_add", "start": [433, 11], "end": [434, 43], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "full_name": "MeasureTheory.L1.SimpleFunc.setToL1SCLM_mono", "start": [986, 1], "end": [989, 73], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "full_name": "Int.emod_nonneg", "start": [369, 1], "end": [371, 88], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Covering/BesicovitchVectorSpace.lean", "full_name": "Besicovitch.exists_good\u03b4", "start": [207, 1], "end": [279, 72], "traced_tactics": [{"tactic": "by_contra' h", "annotated_tactic": ["by_contra' h", []], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\n\u22a2 \u2203 \u03b4,\n    0 < \u03b4 \u2227\n      \u03b4 < 1 \u2227\n        \u2200 (s : Finset E),\n          (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2192\n            (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2192 Finset.card s \u2264 multiplicity E", "state_after": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nh :\n  \u2200 (\u03b4 : \u211d),\n    0 < \u03b4 \u2192\n      \u03b4 < 1 \u2192\n        \u2203 s,\n          (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227\n            (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s\n\u22a2 False"}, {"tactic": "set N := multiplicity E + 1 with hN", "annotated_tactic": ["set N := <a>multiplicity</a> E + 1 with hN", [{"full_name": "Besicovitch.multiplicity", "def_path": "Mathlib/MeasureTheory/Covering/BesicovitchVectorSpace.lean", "def_pos": [131, 5], "def_end_pos": [131, 17]}]], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nh :\n  \u2200 (\u03b4 : \u211d),\n    0 < \u03b4 \u2192\n      \u03b4 < 1 \u2192\n        \u2203 s,\n          (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227\n            (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s\n\u22a2 False", "state_after": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nh :\n  \u2200 (\u03b4 : \u211d),\n    0 < \u03b4 \u2192\n      \u03b4 < 1 \u2192\n        \u2203 s,\n          (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227\n            (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s\nN : \u2115 := multiplicity E + 1\nhN : N = multiplicity E + 1\n\u22a2 False"}, {"tactic": "choose! F hF using this", "annotated_tactic": ["choose! F hF using this", []], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nh :\n  \u2200 (\u03b4 : \u211d),\n    0 < \u03b4 \u2192\n      \u03b4 < 1 \u2192\n        \u2203 s,\n          (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227\n            (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s\nN : \u2115 := multiplicity E + 1\nhN : N = multiplicity E + 1\nthis : \u2200 (\u03b4 : \u211d), 0 < \u03b4 \u2192 \u2203 f, (\u2200 (i : Fin N), \u2016f i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016f i - f j\u2016\n\u22a2 False", "state_after": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nh :\n  \u2200 (\u03b4 : \u211d),\n    0 < \u03b4 \u2192\n      \u03b4 < 1 \u2192\n        \u2203 s,\n          (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227\n            (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s\nN : \u2115 := multiplicity E + 1\nhN : N = multiplicity E + 1\nF : \u211d \u2192 Fin N \u2192 E\nhF : \u2200 (\u03b4 : \u211d), 0 < \u03b4 \u2192 (\u2200 (i : Fin N), \u2016F \u03b4 i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016F \u03b4 i - F \u03b4 j\u2016\n\u22a2 False"}, {"tactic": "rcases this with \u27e8f, hf, h'f\u27e9", "annotated_tactic": ["rcases this with \u27e8f, hf, h'f\u27e9", []], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nh :\n  \u2200 (\u03b4 : \u211d),\n    0 < \u03b4 \u2192\n      \u03b4 < 1 \u2192\n        \u2203 s,\n          (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227\n            (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s\nN : \u2115 := multiplicity E + 1\nhN : N = multiplicity E + 1\nF : \u211d \u2192 Fin N \u2192 E\nhF : \u2200 (\u03b4 : \u211d), 0 < \u03b4 \u2192 (\u2200 (i : Fin N), \u2016F \u03b4 i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016F \u03b4 i - F \u03b4 j\u2016\nthis : \u2203 f, (\u2200 (i : Fin N), \u2016f i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 \u2264 \u2016f i - f j\u2016\n\u22a2 False", "state_after": "case intro.intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nh :\n  \u2200 (\u03b4 : \u211d),\n    0 < \u03b4 \u2192\n      \u03b4 < 1 \u2192\n        \u2203 s,\n          (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227\n            (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s\nN : \u2115 := multiplicity E + 1\nhN : N = multiplicity E + 1\nF : \u211d \u2192 Fin N \u2192 E\nhF : \u2200 (\u03b4 : \u211d), 0 < \u03b4 \u2192 (\u2200 (i : Fin N), \u2016F \u03b4 i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016F \u03b4 i - F \u03b4 j\u2016\nf : Fin N \u2192 E\nhf : \u2200 (i : Fin N), \u2016f i\u2016 \u2264 2\nh'f : \u2200 (i j : Fin N), i \u2260 j \u2192 1 \u2264 \u2016f i - f j\u2016\n\u22a2 False"}, {"tactic": "have finj : Function.Injective f := by\n  intro i j hij\n  by_contra h\n  have : 1 \u2264 \u2016f i - f j\u2016 := h'f i j h\n  simp only [hij, norm_zero, sub_self] at this\n  exact lt_irrefl _ (this.trans_lt zero_lt_one)", "annotated_tactic": ["have finj : <a>Function.Injective</a> f := by\n    intro i j hij\n    by_contra h\n    have : 1 \u2264 \u2016f i - f j\u2016 := h'f i j h\n    simp only [hij, <a>norm_zero</a>, <a>sub_self</a>] at this\n    exact <a>lt_irrefl</a> _ (this.trans_lt <a>zero_lt_one</a>)", [{"full_name": "Function.Injective", "def_path": "Mathlib/Init/Function.lean", "def_pos": [109, 5], "def_end_pos": [109, 14]}, {"full_name": "norm_zero", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [528, 30], "def_end_pos": [528, 39]}, {"full_name": "sub_self", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [734, 30], "def_end_pos": [734, 38]}, {"full_name": "lt_irrefl", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [79, 9], "def_end_pos": [79, 18]}, {"full_name": "zero_lt_one", "def_path": "Mathlib/Algebra/Order/ZeroLEOne.lean", "def_pos": [39, 15], "def_end_pos": [39, 26]}]], "state_before": "case intro.intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nh :\n  \u2200 (\u03b4 : \u211d),\n    0 < \u03b4 \u2192\n      \u03b4 < 1 \u2192\n        \u2203 s,\n          (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227\n            (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s\nN : \u2115 := multiplicity E + 1\nhN : N = multiplicity E + 1\nF : \u211d \u2192 Fin N \u2192 E\nhF : \u2200 (\u03b4 : \u211d), 0 < \u03b4 \u2192 (\u2200 (i : Fin N), \u2016F \u03b4 i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016F \u03b4 i - F \u03b4 j\u2016\nf : Fin N \u2192 E\nhf : \u2200 (i : Fin N), \u2016f i\u2016 \u2264 2\nh'f : \u2200 (i j : Fin N), i \u2260 j \u2192 1 \u2264 \u2016f i - f j\u2016\n\u22a2 False", "state_after": "case intro.intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nh :\n  \u2200 (\u03b4 : \u211d),\n    0 < \u03b4 \u2192\n      \u03b4 < 1 \u2192\n        \u2203 s,\n          (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227\n            (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s\nN : \u2115 := multiplicity E + 1\nhN : N = multiplicity E + 1\nF : \u211d \u2192 Fin N \u2192 E\nhF : \u2200 (\u03b4 : \u211d), 0 < \u03b4 \u2192 (\u2200 (i : Fin N), \u2016F \u03b4 i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016F \u03b4 i - F \u03b4 j\u2016\nf : Fin N \u2192 E\nhf : \u2200 (i : Fin N), \u2016f i\u2016 \u2264 2\nh'f : \u2200 (i j : Fin N), i \u2260 j \u2192 1 \u2264 \u2016f i - f j\u2016\nfinj : Function.Injective f\n\u22a2 False"}, {"tactic": "let s := Finset.image f Finset.univ", "annotated_tactic": ["let s := <a>Finset.image</a> f <a>Finset.univ</a>", [{"full_name": "Finset.image", "def_path": "Mathlib/Data/Finset/Image.lean", "def_pos": [313, 5], "def_end_pos": [313, 10]}, {"full_name": "Finset.univ", "def_path": "Mathlib/Data/Fintype/Basic.lean", "def_pos": [67, 5], "def_end_pos": [67, 9]}]], "state_before": "case intro.intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nh :\n  \u2200 (\u03b4 : \u211d),\n    0 < \u03b4 \u2192\n      \u03b4 < 1 \u2192\n        \u2203 s,\n          (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227\n            (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s\nN : \u2115 := multiplicity E + 1\nhN : N = multiplicity E + 1\nF : \u211d \u2192 Fin N \u2192 E\nhF : \u2200 (\u03b4 : \u211d), 0 < \u03b4 \u2192 (\u2200 (i : Fin N), \u2016F \u03b4 i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016F \u03b4 i - F \u03b4 j\u2016\nf : Fin N \u2192 E\nhf : \u2200 (i : Fin N), \u2016f i\u2016 \u2264 2\nh'f : \u2200 (i j : Fin N), i \u2260 j \u2192 1 \u2264 \u2016f i - f j\u2016\nfinj : Function.Injective f\n\u22a2 False", "state_after": "case intro.intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nh :\n  \u2200 (\u03b4 : \u211d),\n    0 < \u03b4 \u2192\n      \u03b4 < 1 \u2192\n        \u2203 s,\n          (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227\n            (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s\nN : \u2115 := multiplicity E + 1\nhN : N = multiplicity E + 1\nF : \u211d \u2192 Fin N \u2192 E\nhF : \u2200 (\u03b4 : \u211d), 0 < \u03b4 \u2192 (\u2200 (i : Fin N), \u2016F \u03b4 i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016F \u03b4 i - F \u03b4 j\u2016\nf : Fin N \u2192 E\nhf : \u2200 (i : Fin N), \u2016f i\u2016 \u2264 2\nh'f : \u2200 (i j : Fin N), i \u2260 j \u2192 1 \u2264 \u2016f i - f j\u2016\nfinj : Function.Injective f\ns : Finset E := Finset.image f Finset.univ\n\u22a2 False"}, {"tactic": "have s_card : s.card = N := by rw [Finset.card_image_of_injective _ finj]; exact Finset.card_fin N", "annotated_tactic": ["have s_card : s.card = N := by rw [<a>Finset.card_image_of_injective</a> _ finj]; exact <a>Finset.card_fin</a> N", [{"full_name": "Finset.card_image_of_injective", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [250, 9], "def_end_pos": [250, 32]}, {"full_name": "Finset.card_fin", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [322, 9], "def_end_pos": [322, 24]}]], "state_before": "case intro.intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nh :\n  \u2200 (\u03b4 : \u211d),\n    0 < \u03b4 \u2192\n      \u03b4 < 1 \u2192\n        \u2203 s,\n          (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227\n            (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s\nN : \u2115 := multiplicity E + 1\nhN : N = multiplicity E + 1\nF : \u211d \u2192 Fin N \u2192 E\nhF : \u2200 (\u03b4 : \u211d), 0 < \u03b4 \u2192 (\u2200 (i : Fin N), \u2016F \u03b4 i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016F \u03b4 i - F \u03b4 j\u2016\nf : Fin N \u2192 E\nhf : \u2200 (i : Fin N), \u2016f i\u2016 \u2264 2\nh'f : \u2200 (i j : Fin N), i \u2260 j \u2192 1 \u2264 \u2016f i - f j\u2016\nfinj : Function.Injective f\ns : Finset E := Finset.image f Finset.univ\n\u22a2 False", "state_after": "case intro.intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nh :\n  \u2200 (\u03b4 : \u211d),\n    0 < \u03b4 \u2192\n      \u03b4 < 1 \u2192\n        \u2203 s,\n          (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227\n            (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s\nN : \u2115 := multiplicity E + 1\nhN : N = multiplicity E + 1\nF : \u211d \u2192 Fin N \u2192 E\nhF : \u2200 (\u03b4 : \u211d), 0 < \u03b4 \u2192 (\u2200 (i : Fin N), \u2016F \u03b4 i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016F \u03b4 i - F \u03b4 j\u2016\nf : Fin N \u2192 E\nhf : \u2200 (i : Fin N), \u2016f i\u2016 \u2264 2\nh'f : \u2200 (i j : Fin N), i \u2260 j \u2192 1 \u2264 \u2016f i - f j\u2016\nfinj : Function.Injective f\ns : Finset E := Finset.image f Finset.univ\ns_card : Finset.card s = N\n\u22a2 False"}, {"tactic": "have hs : \u2200 c \u2208 s, \u2016c\u2016 \u2264 2 := by\n  simp only [hf, forall_apply_eq_imp_iff, forall_const, forall_exists_index, Finset.mem_univ,\n    Finset.mem_image, true_and]", "annotated_tactic": ["have hs : \u2200 c \u2208 s, \u2016c\u2016 \u2264 2 := by\n    simp only [hf, <a>forall_apply_eq_imp_iff</a>, <a>forall_const</a>, <a>forall_exists_index</a>, <a>Finset.mem_univ</a>,\n      <a>Finset.mem_image</a>, <a>true_and</a>]", [{"full_name": "forall_apply_eq_imp_iff", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [499, 17], "def_end_pos": [499, 40]}, {"full_name": "forall_const", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [435, 17], "def_end_pos": [435, 29]}, {"full_name": "forall_exists_index", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [356, 17], "def_end_pos": [356, 36]}, {"full_name": "Finset.mem_univ", "def_path": "Mathlib/Data/Fintype/Basic.lean", "def_pos": [72, 9], "def_end_pos": [72, 17]}, {"full_name": "Finset.mem_image", "def_path": "Mathlib/Data/Finset/Image.lean", "def_pos": [330, 9], "def_end_pos": [330, 18]}, {"full_name": "true_and", "def_path": "lake-packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [84, 17], "def_end_pos": [84, 25]}]], "state_before": "case intro.intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nh :\n  \u2200 (\u03b4 : \u211d),\n    0 < \u03b4 \u2192\n      \u03b4 < 1 \u2192\n        \u2203 s,\n          (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227\n            (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s\nN : \u2115 := multiplicity E + 1\nhN : N = multiplicity E + 1\nF : \u211d \u2192 Fin N \u2192 E\nhF : \u2200 (\u03b4 : \u211d), 0 < \u03b4 \u2192 (\u2200 (i : Fin N), \u2016F \u03b4 i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016F \u03b4 i - F \u03b4 j\u2016\nf : Fin N \u2192 E\nhf : \u2200 (i : Fin N), \u2016f i\u2016 \u2264 2\nh'f : \u2200 (i j : Fin N), i \u2260 j \u2192 1 \u2264 \u2016f i - f j\u2016\nfinj : Function.Injective f\ns : Finset E := Finset.image f Finset.univ\ns_card : Finset.card s = N\n\u22a2 False", "state_after": "case intro.intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nh :\n  \u2200 (\u03b4 : \u211d),\n    0 < \u03b4 \u2192\n      \u03b4 < 1 \u2192\n        \u2203 s,\n          (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227\n            (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s\nN : \u2115 := multiplicity E + 1\nhN : N = multiplicity E + 1\nF : \u211d \u2192 Fin N \u2192 E\nhF : \u2200 (\u03b4 : \u211d), 0 < \u03b4 \u2192 (\u2200 (i : Fin N), \u2016F \u03b4 i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016F \u03b4 i - F \u03b4 j\u2016\nf : Fin N \u2192 E\nhf : \u2200 (i : Fin N), \u2016f i\u2016 \u2264 2\nh'f : \u2200 (i j : Fin N), i \u2260 j \u2192 1 \u2264 \u2016f i - f j\u2016\nfinj : Function.Injective f\ns : Finset E := Finset.image f Finset.univ\ns_card : Finset.card s = N\nhs : \u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2\n\u22a2 False"}, {"tactic": "have h's : \u2200 c \u2208 s, \u2200 d \u2208 s, c \u2260 d \u2192 1 \u2264 \u2016c - d\u2016 := by\n  simp only [forall_apply_eq_imp_iff, forall_exists_index, Finset.mem_univ, Finset.mem_image,\n    Ne.def, exists_true_left, forall_apply_eq_imp_iff, forall_true_left, true_and]\n  intro i j hij\n  have : i \u2260 j := fun h => by rw [h] at hij; exact hij rfl\n  exact h'f i j this", "annotated_tactic": ["have h's : \u2200 c \u2208 s, \u2200 d \u2208 s, c \u2260 d \u2192 1 \u2264 \u2016c - d\u2016 := by\n    simp only [<a>forall_apply_eq_imp_iff</a>, <a>forall_exists_index</a>, <a>Finset.mem_univ</a>, <a>Finset.mem_image</a>,\n      <a>Ne.def</a>, <a>exists_true_left</a>, <a>forall_apply_eq_imp_iff</a>, <a>forall_true_left</a>, <a>true_and</a>]\n    intro i j hij\n    have : i \u2260 j := fun h => by rw [h] at hij; exact hij <a>rfl</a>\n    exact h'f i j this", [{"full_name": "forall_apply_eq_imp_iff", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [499, 17], "def_end_pos": [499, 40]}, {"full_name": "forall_exists_index", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [356, 17], "def_end_pos": [356, 36]}, {"full_name": "Finset.mem_univ", "def_path": "Mathlib/Data/Fintype/Basic.lean", "def_pos": [72, 9], "def_end_pos": [72, 17]}, {"full_name": "Finset.mem_image", "def_path": "Mathlib/Data/Finset/Image.lean", "def_pos": [330, 9], "def_end_pos": [330, 18]}, {"full_name": "Ne.def", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [59, 9], "def_end_pos": [59, 15]}, {"full_name": "exists_true_left", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [912, 17], "def_end_pos": [912, 33]}, {"full_name": "forall_apply_eq_imp_iff", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [499, 17], "def_end_pos": [499, 40]}, {"full_name": "forall_true_left", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [931, 17], "def_end_pos": [931, 33]}, {"full_name": "true_and", "def_path": "lake-packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [84, 17], "def_end_pos": [84, 25]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "case intro.intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nh :\n  \u2200 (\u03b4 : \u211d),\n    0 < \u03b4 \u2192\n      \u03b4 < 1 \u2192\n        \u2203 s,\n          (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227\n            (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s\nN : \u2115 := multiplicity E + 1\nhN : N = multiplicity E + 1\nF : \u211d \u2192 Fin N \u2192 E\nhF : \u2200 (\u03b4 : \u211d), 0 < \u03b4 \u2192 (\u2200 (i : Fin N), \u2016F \u03b4 i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016F \u03b4 i - F \u03b4 j\u2016\nf : Fin N \u2192 E\nhf : \u2200 (i : Fin N), \u2016f i\u2016 \u2264 2\nh'f : \u2200 (i j : Fin N), i \u2260 j \u2192 1 \u2264 \u2016f i - f j\u2016\nfinj : Function.Injective f\ns : Finset E := Finset.image f Finset.univ\ns_card : Finset.card s = N\nhs : \u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2\n\u22a2 False", "state_after": "case intro.intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nh :\n  \u2200 (\u03b4 : \u211d),\n    0 < \u03b4 \u2192\n      \u03b4 < 1 \u2192\n        \u2203 s,\n          (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227\n            (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s\nN : \u2115 := multiplicity E + 1\nhN : N = multiplicity E + 1\nF : \u211d \u2192 Fin N \u2192 E\nhF : \u2200 (\u03b4 : \u211d), 0 < \u03b4 \u2192 (\u2200 (i : Fin N), \u2016F \u03b4 i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016F \u03b4 i - F \u03b4 j\u2016\nf : Fin N \u2192 E\nhf : \u2200 (i : Fin N), \u2016f i\u2016 \u2264 2\nh'f : \u2200 (i j : Fin N), i \u2260 j \u2192 1 \u2264 \u2016f i - f j\u2016\nfinj : Function.Injective f\ns : Finset E := Finset.image f Finset.univ\ns_card : Finset.card s = N\nhs : \u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2\nh's : \u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 \u2264 \u2016c - d\u2016\n\u22a2 False"}, {"tactic": "have : s.card \u2264 multiplicity E := card_le_multiplicity hs h's", "annotated_tactic": ["have : s.card \u2264 <a>multiplicity</a> E := <a>card_le_multiplicity</a> hs h's", [{"full_name": "Besicovitch.multiplicity", "def_path": "Mathlib/MeasureTheory/Covering/BesicovitchVectorSpace.lean", "def_pos": [131, 5], "def_end_pos": [131, 17]}, {"full_name": "Besicovitch.card_le_multiplicity", "def_path": "Mathlib/MeasureTheory/Covering/BesicovitchVectorSpace.lean", "def_pos": [193, 9], "def_end_pos": [193, 29]}]], "state_before": "case intro.intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nh :\n  \u2200 (\u03b4 : \u211d),\n    0 < \u03b4 \u2192\n      \u03b4 < 1 \u2192\n        \u2203 s,\n          (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227\n            (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s\nN : \u2115 := multiplicity E + 1\nhN : N = multiplicity E + 1\nF : \u211d \u2192 Fin N \u2192 E\nhF : \u2200 (\u03b4 : \u211d), 0 < \u03b4 \u2192 (\u2200 (i : Fin N), \u2016F \u03b4 i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016F \u03b4 i - F \u03b4 j\u2016\nf : Fin N \u2192 E\nhf : \u2200 (i : Fin N), \u2016f i\u2016 \u2264 2\nh'f : \u2200 (i j : Fin N), i \u2260 j \u2192 1 \u2264 \u2016f i - f j\u2016\nfinj : Function.Injective f\ns : Finset E := Finset.image f Finset.univ\ns_card : Finset.card s = N\nhs : \u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2\nh's : \u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 \u2264 \u2016c - d\u2016\n\u22a2 False", "state_after": "case intro.intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nh :\n  \u2200 (\u03b4 : \u211d),\n    0 < \u03b4 \u2192\n      \u03b4 < 1 \u2192\n        \u2203 s,\n          (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227\n            (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s\nN : \u2115 := multiplicity E + 1\nhN : N = multiplicity E + 1\nF : \u211d \u2192 Fin N \u2192 E\nhF : \u2200 (\u03b4 : \u211d), 0 < \u03b4 \u2192 (\u2200 (i : Fin N), \u2016F \u03b4 i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016F \u03b4 i - F \u03b4 j\u2016\nf : Fin N \u2192 E\nhf : \u2200 (i : Fin N), \u2016f i\u2016 \u2264 2\nh'f : \u2200 (i j : Fin N), i \u2260 j \u2192 1 \u2264 \u2016f i - f j\u2016\nfinj : Function.Injective f\ns : Finset E := Finset.image f Finset.univ\ns_card : Finset.card s = N\nhs : \u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2\nh's : \u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 \u2264 \u2016c - d\u2016\nthis : Finset.card s \u2264 multiplicity E\n\u22a2 False"}, {"tactic": "rw [s_card, hN] at this", "annotated_tactic": ["rw [s_card, hN] at this", []], "state_before": "case intro.intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nh :\n  \u2200 (\u03b4 : \u211d),\n    0 < \u03b4 \u2192\n      \u03b4 < 1 \u2192\n        \u2203 s,\n          (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227\n            (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s\nN : \u2115 := multiplicity E + 1\nhN : N = multiplicity E + 1\nF : \u211d \u2192 Fin N \u2192 E\nhF : \u2200 (\u03b4 : \u211d), 0 < \u03b4 \u2192 (\u2200 (i : Fin N), \u2016F \u03b4 i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016F \u03b4 i - F \u03b4 j\u2016\nf : Fin N \u2192 E\nhf : \u2200 (i : Fin N), \u2016f i\u2016 \u2264 2\nh'f : \u2200 (i j : Fin N), i \u2260 j \u2192 1 \u2264 \u2016f i - f j\u2016\nfinj : Function.Injective f\ns : Finset E := Finset.image f Finset.univ\ns_card : Finset.card s = N\nhs : \u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2\nh's : \u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 \u2264 \u2016c - d\u2016\nthis : Finset.card s \u2264 multiplicity E\n\u22a2 False", "state_after": "case intro.intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nh :\n  \u2200 (\u03b4 : \u211d),\n    0 < \u03b4 \u2192\n      \u03b4 < 1 \u2192\n        \u2203 s,\n          (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227\n            (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s\nN : \u2115 := multiplicity E + 1\nhN : N = multiplicity E + 1\nF : \u211d \u2192 Fin N \u2192 E\nhF : \u2200 (\u03b4 : \u211d), 0 < \u03b4 \u2192 (\u2200 (i : Fin N), \u2016F \u03b4 i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016F \u03b4 i - F \u03b4 j\u2016\nf : Fin N \u2192 E\nhf : \u2200 (i : Fin N), \u2016f i\u2016 \u2264 2\nh'f : \u2200 (i j : Fin N), i \u2260 j \u2192 1 \u2264 \u2016f i - f j\u2016\nfinj : Function.Injective f\ns : Finset E := Finset.image f Finset.univ\ns_card : Finset.card s = N\nhs : \u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2\nh's : \u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 \u2264 \u2016c - d\u2016\nthis : multiplicity E + 1 \u2264 multiplicity E\n\u22a2 False"}, {"tactic": "exact lt_irrefl _ ((Nat.lt_succ_self (multiplicity E)).trans_le this)", "annotated_tactic": ["exact <a>lt_irrefl</a> _ ((<a>Nat.lt_succ_self</a> (<a>multiplicity</a> E)).<a>trans_le</a> this)", [{"full_name": "lt_irrefl", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [79, 9], "def_end_pos": [79, 18]}, {"full_name": "Nat.lt_succ_self", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [294, 9], "def_end_pos": [294, 21]}, {"full_name": "Besicovitch.multiplicity", "def_path": "Mathlib/MeasureTheory/Covering/BesicovitchVectorSpace.lean", "def_pos": [131, 5], "def_end_pos": [131, 17]}, {"full_name": "LT.lt.trans_le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [148, 7], "def_end_pos": [148, 21]}]], "state_before": "case intro.intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nh :\n  \u2200 (\u03b4 : \u211d),\n    0 < \u03b4 \u2192\n      \u03b4 < 1 \u2192\n        \u2203 s,\n          (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227\n            (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s\nN : \u2115 := multiplicity E + 1\nhN : N = multiplicity E + 1\nF : \u211d \u2192 Fin N \u2192 E\nhF : \u2200 (\u03b4 : \u211d), 0 < \u03b4 \u2192 (\u2200 (i : Fin N), \u2016F \u03b4 i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016F \u03b4 i - F \u03b4 j\u2016\nf : Fin N \u2192 E\nhf : \u2200 (i : Fin N), \u2016f i\u2016 \u2264 2\nh'f : \u2200 (i j : Fin N), i \u2260 j \u2192 1 \u2264 \u2016f i - f j\u2016\nfinj : Function.Injective f\ns : Finset E := Finset.image f Finset.univ\ns_card : Finset.card s = N\nhs : \u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2\nh's : \u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 \u2264 \u2016c - d\u2016\nthis : multiplicity E + 1 \u2264 multiplicity E\n\u22a2 False", "state_after": "no goals"}, {"tactic": "intro \u03b4 h\u03b4", "annotated_tactic": ["intro \u03b4 h\u03b4", []], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nh :\n  \u2200 (\u03b4 : \u211d),\n    0 < \u03b4 \u2192\n      \u03b4 < 1 \u2192\n        \u2203 s,\n          (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227\n            (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s\nN : \u2115 := multiplicity E + 1\nhN : N = multiplicity E + 1\n\u22a2 \u2200 (\u03b4 : \u211d), 0 < \u03b4 \u2192 \u2203 f, (\u2200 (i : Fin N), \u2016f i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016f i - f j\u2016", "state_after": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nh :\n  \u2200 (\u03b4 : \u211d),\n    0 < \u03b4 \u2192\n      \u03b4 < 1 \u2192\n        \u2203 s,\n          (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227\n            (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s\nN : \u2115 := multiplicity E + 1\nhN : N = multiplicity E + 1\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\n\u22a2 \u2203 f, (\u2200 (i : Fin N), \u2016f i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016f i - f j\u2016"}, {"tactic": "rcases lt_or_le \u03b4 1 with (h\u03b4' | h\u03b4')", "annotated_tactic": ["rcases <a>lt_or_le</a> \u03b4 1 with (h\u03b4' | h\u03b4')", [{"full_name": "lt_or_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [336, 9], "def_end_pos": [336, 17]}]], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nh :\n  \u2200 (\u03b4 : \u211d),\n    0 < \u03b4 \u2192\n      \u03b4 < 1 \u2192\n        \u2203 s,\n          (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227\n            (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s\nN : \u2115 := multiplicity E + 1\nhN : N = multiplicity E + 1\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\n\u22a2 \u2203 f, (\u2200 (i : Fin N), \u2016f i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016f i - f j\u2016", "state_after": "case inl\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nh :\n  \u2200 (\u03b4 : \u211d),\n    0 < \u03b4 \u2192\n      \u03b4 < 1 \u2192\n        \u2203 s,\n          (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227\n            (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s\nN : \u2115 := multiplicity E + 1\nhN : N = multiplicity E + 1\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nh\u03b4' : \u03b4 < 1\n\u22a2 \u2203 f, (\u2200 (i : Fin N), \u2016f i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016f i - f j\u2016\n\ncase inr\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nh :\n  \u2200 (\u03b4 : \u211d),\n    0 < \u03b4 \u2192\n      \u03b4 < 1 \u2192\n        \u2203 s,\n          (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227\n            (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s\nN : \u2115 := multiplicity E + 1\nhN : N = multiplicity E + 1\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nh\u03b4' : 1 \u2264 \u03b4\n\u22a2 \u2203 f, (\u2200 (i : Fin N), \u2016f i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016f i - f j\u2016"}, {"tactic": "rcases h \u03b4 h\u03b4 h\u03b4' with \u27e8s, hs, h's, s_card\u27e9", "annotated_tactic": ["rcases h \u03b4 h\u03b4 h\u03b4' with \u27e8s, hs, h's, s_card\u27e9", []], "state_before": "case inl\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nh :\n  \u2200 (\u03b4 : \u211d),\n    0 < \u03b4 \u2192\n      \u03b4 < 1 \u2192\n        \u2203 s,\n          (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227\n            (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s\nN : \u2115 := multiplicity E + 1\nhN : N = multiplicity E + 1\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nh\u03b4' : \u03b4 < 1\n\u22a2 \u2203 f, (\u2200 (i : Fin N), \u2016f i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016f i - f j\u2016", "state_after": "case inl.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nh :\n  \u2200 (\u03b4 : \u211d),\n    0 < \u03b4 \u2192\n      \u03b4 < 1 \u2192\n        \u2203 s,\n          (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227\n            (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s\nN : \u2115 := multiplicity E + 1\nhN : N = multiplicity E + 1\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nh\u03b4' : \u03b4 < 1\ns : Finset E\nhs : \u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2\nh's : \u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016\ns_card : multiplicity E < Finset.card s\n\u22a2 \u2203 f, (\u2200 (i : Fin N), \u2016f i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016f i - f j\u2016"}, {"tactic": "obtain \u27e8f, f_inj, hfs\u27e9 : \u2203 f : Fin N \u2192 E, Function.Injective f \u2227 range f \u2286 \u2191s := by\n  have : Fintype.card (Fin N) \u2264 s.card := by simp only [Fintype.card_fin]; exact s_card\n  rcases Function.Embedding.exists_of_card_le_finset this with \u27e8f, hf\u27e9\n  exact \u27e8f, f.injective, hf\u27e9", "annotated_tactic": ["obtain \u27e8f, f_inj, hfs\u27e9 : \u2203 f : <a>Fin</a> N \u2192 E, <a>Function.Injective</a> f \u2227 <a>range</a> f \u2286 \u2191s := by\n        have : <a>Fintype.card</a> (<a>Fin</a> N) \u2264 s.card := by simp only [<a>Fintype.card_fin</a>]; exact s_card\n        rcases <a>Function.Embedding.exists_of_card_le_finset</a> this with \u27e8f, hf\u27e9\n        exact \u27e8f, f.injective, hf\u27e9", [{"full_name": "Fin", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1745, 11], "def_end_pos": [1745, 14]}, {"full_name": "Function.Injective", "def_path": "Mathlib/Init/Function.lean", "def_pos": [109, 5], "def_end_pos": [109, 14]}, {"full_name": "Set.range", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [668, 5], "def_end_pos": [668, 10]}, {"full_name": "Fintype.card", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [61, 5], "def_end_pos": [61, 9]}, {"full_name": "Fin", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1745, 11], "def_end_pos": [1745, 14]}, {"full_name": "Fintype.card_fin", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [308, 9], "def_end_pos": [308, 25]}, {"full_name": "Function.Embedding.exists_of_card_le_finset", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [828, 9], "def_end_pos": [828, 33]}]], "state_before": "case inl.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nh :\n  \u2200 (\u03b4 : \u211d),\n    0 < \u03b4 \u2192\n      \u03b4 < 1 \u2192\n        \u2203 s,\n          (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227\n            (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s\nN : \u2115 := multiplicity E + 1\nhN : N = multiplicity E + 1\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nh\u03b4' : \u03b4 < 1\ns : Finset E\nhs : \u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2\nh's : \u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016\ns_card : multiplicity E < Finset.card s\n\u22a2 \u2203 f, (\u2200 (i : Fin N), \u2016f i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016f i - f j\u2016", "state_after": "case inl.intro.intro.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nh :\n  \u2200 (\u03b4 : \u211d),\n    0 < \u03b4 \u2192\n      \u03b4 < 1 \u2192\n        \u2203 s,\n          (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227\n            (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s\nN : \u2115 := multiplicity E + 1\nhN : N = multiplicity E + 1\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nh\u03b4' : \u03b4 < 1\ns : Finset E\nhs : \u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2\nh's : \u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016\ns_card : multiplicity E < Finset.card s\nf : Fin N \u2192 E\nf_inj : Function.Injective f\nhfs : range f \u2286 \u2191s\n\u22a2 \u2203 f, (\u2200 (i : Fin N), \u2016f i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016f i - f j\u2016"}, {"tactic": "simp only [range_subset_iff, Finset.mem_coe] at hfs", "annotated_tactic": ["simp only [<a>range_subset_iff</a>, <a>Finset.mem_coe</a>] at hfs", [{"full_name": "Set.range_subset_iff", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [749, 9], "def_end_pos": [749, 25]}, {"full_name": "Finset.mem_coe", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [208, 9], "def_end_pos": [208, 16]}]], "state_before": "case inl.intro.intro.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nh :\n  \u2200 (\u03b4 : \u211d),\n    0 < \u03b4 \u2192\n      \u03b4 < 1 \u2192\n        \u2203 s,\n          (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227\n            (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s\nN : \u2115 := multiplicity E + 1\nhN : N = multiplicity E + 1\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nh\u03b4' : \u03b4 < 1\ns : Finset E\nhs : \u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2\nh's : \u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016\ns_card : multiplicity E < Finset.card s\nf : Fin N \u2192 E\nf_inj : Function.Injective f\nhfs : range f \u2286 \u2191s\n\u22a2 \u2203 f, (\u2200 (i : Fin N), \u2016f i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016f i - f j\u2016", "state_after": "case inl.intro.intro.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nh :\n  \u2200 (\u03b4 : \u211d),\n    0 < \u03b4 \u2192\n      \u03b4 < 1 \u2192\n        \u2203 s,\n          (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227\n            (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s\nN : \u2115 := multiplicity E + 1\nhN : N = multiplicity E + 1\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nh\u03b4' : \u03b4 < 1\ns : Finset E\nhs : \u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2\nh's : \u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016\ns_card : multiplicity E < Finset.card s\nf : Fin N \u2192 E\nf_inj : Function.Injective f\nhfs : \u2200 (y : Fin (multiplicity E + 1)), f y \u2208 s\n\u22a2 \u2203 f, (\u2200 (i : Fin N), \u2016f i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016f i - f j\u2016"}, {"tactic": "refine' \u27e8f, fun i => hs _ (hfs i), fun i j hij => h's _ (hfs i) _ (hfs j) (f_inj.ne hij)\u27e9", "annotated_tactic": ["refine' \u27e8f, fun i => hs _ (hfs i), fun i j hij => h's _ (hfs i) _ (hfs j) (f_inj.ne hij)\u27e9", []], "state_before": "case inl.intro.intro.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nh :\n  \u2200 (\u03b4 : \u211d),\n    0 < \u03b4 \u2192\n      \u03b4 < 1 \u2192\n        \u2203 s,\n          (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227\n            (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s\nN : \u2115 := multiplicity E + 1\nhN : N = multiplicity E + 1\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nh\u03b4' : \u03b4 < 1\ns : Finset E\nhs : \u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2\nh's : \u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016\ns_card : multiplicity E < Finset.card s\nf : Fin N \u2192 E\nf_inj : Function.Injective f\nhfs : \u2200 (y : Fin (multiplicity E + 1)), f y \u2208 s\n\u22a2 \u2203 f, (\u2200 (i : Fin N), \u2016f i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016f i - f j\u2016", "state_after": "no goals"}, {"tactic": "have : Fintype.card (Fin N) \u2264 s.card := by simp only [Fintype.card_fin]; exact s_card", "annotated_tactic": ["have : <a>Fintype.card</a> (<a>Fin</a> N) \u2264 s.card := by simp only [<a>Fintype.card_fin</a>]; exact s_card", [{"full_name": "Fintype.card", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [61, 5], "def_end_pos": [61, 9]}, {"full_name": "Fin", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1745, 11], "def_end_pos": [1745, 14]}, {"full_name": "Fintype.card_fin", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [308, 9], "def_end_pos": [308, 25]}]], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nh :\n  \u2200 (\u03b4 : \u211d),\n    0 < \u03b4 \u2192\n      \u03b4 < 1 \u2192\n        \u2203 s,\n          (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227\n            (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s\nN : \u2115 := multiplicity E + 1\nhN : N = multiplicity E + 1\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nh\u03b4' : \u03b4 < 1\ns : Finset E\nhs : \u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2\nh's : \u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016\ns_card : multiplicity E < Finset.card s\n\u22a2 \u2203 f, Function.Injective f \u2227 range f \u2286 \u2191s", "state_after": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nh :\n  \u2200 (\u03b4 : \u211d),\n    0 < \u03b4 \u2192\n      \u03b4 < 1 \u2192\n        \u2203 s,\n          (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227\n            (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s\nN : \u2115 := multiplicity E + 1\nhN : N = multiplicity E + 1\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nh\u03b4' : \u03b4 < 1\ns : Finset E\nhs : \u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2\nh's : \u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016\ns_card : multiplicity E < Finset.card s\nthis : Fintype.card (Fin N) \u2264 Finset.card s\n\u22a2 \u2203 f, Function.Injective f \u2227 range f \u2286 \u2191s"}, {"tactic": "rcases Function.Embedding.exists_of_card_le_finset this with \u27e8f, hf\u27e9", "annotated_tactic": ["rcases <a>Function.Embedding.exists_of_card_le_finset</a> this with \u27e8f, hf\u27e9", [{"full_name": "Function.Embedding.exists_of_card_le_finset", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [828, 9], "def_end_pos": [828, 33]}]], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nh :\n  \u2200 (\u03b4 : \u211d),\n    0 < \u03b4 \u2192\n      \u03b4 < 1 \u2192\n        \u2203 s,\n          (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227\n            (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s\nN : \u2115 := multiplicity E + 1\nhN : N = multiplicity E + 1\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nh\u03b4' : \u03b4 < 1\ns : Finset E\nhs : \u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2\nh's : \u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016\ns_card : multiplicity E < Finset.card s\nthis : Fintype.card (Fin N) \u2264 Finset.card s\n\u22a2 \u2203 f, Function.Injective f \u2227 range f \u2286 \u2191s", "state_after": "case intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nh :\n  \u2200 (\u03b4 : \u211d),\n    0 < \u03b4 \u2192\n      \u03b4 < 1 \u2192\n        \u2203 s,\n          (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227\n            (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s\nN : \u2115 := multiplicity E + 1\nhN : N = multiplicity E + 1\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nh\u03b4' : \u03b4 < 1\ns : Finset E\nhs : \u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2\nh's : \u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016\ns_card : multiplicity E < Finset.card s\nthis : Fintype.card (Fin N) \u2264 Finset.card s\nf : Fin N \u21aa E\nhf : range \u2191f \u2286 \u2191s\n\u22a2 \u2203 f, Function.Injective f \u2227 range f \u2286 \u2191s"}, {"tactic": "exact \u27e8f, f.injective, hf\u27e9", "annotated_tactic": ["exact \u27e8f, f.injective, hf\u27e9", []], "state_before": "case intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nh :\n  \u2200 (\u03b4 : \u211d),\n    0 < \u03b4 \u2192\n      \u03b4 < 1 \u2192\n        \u2203 s,\n          (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227\n            (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s\nN : \u2115 := multiplicity E + 1\nhN : N = multiplicity E + 1\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nh\u03b4' : \u03b4 < 1\ns : Finset E\nhs : \u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2\nh's : \u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016\ns_card : multiplicity E < Finset.card s\nthis : Fintype.card (Fin N) \u2264 Finset.card s\nf : Fin N \u21aa E\nhf : range \u2191f \u2286 \u2191s\n\u22a2 \u2203 f, Function.Injective f \u2227 range f \u2286 \u2191s", "state_after": "no goals"}, {"tactic": "simp only [Fintype.card_fin]", "annotated_tactic": ["simp only [<a>Fintype.card_fin</a>]", [{"full_name": "Fintype.card_fin", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [308, 9], "def_end_pos": [308, 25]}]], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nh :\n  \u2200 (\u03b4 : \u211d),\n    0 < \u03b4 \u2192\n      \u03b4 < 1 \u2192\n        \u2203 s,\n          (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227\n            (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s\nN : \u2115 := multiplicity E + 1\nhN : N = multiplicity E + 1\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nh\u03b4' : \u03b4 < 1\ns : Finset E\nhs : \u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2\nh's : \u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016\ns_card : multiplicity E < Finset.card s\n\u22a2 Fintype.card (Fin N) \u2264 Finset.card s", "state_after": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nh :\n  \u2200 (\u03b4 : \u211d),\n    0 < \u03b4 \u2192\n      \u03b4 < 1 \u2192\n        \u2203 s,\n          (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227\n            (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s\nN : \u2115 := multiplicity E + 1\nhN : N = multiplicity E + 1\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nh\u03b4' : \u03b4 < 1\ns : Finset E\nhs : \u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2\nh's : \u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016\ns_card : multiplicity E < Finset.card s\n\u22a2 multiplicity E + 1 \u2264 Finset.card s"}, {"tactic": "exact s_card", "annotated_tactic": ["exact s_card", []], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nh :\n  \u2200 (\u03b4 : \u211d),\n    0 < \u03b4 \u2192\n      \u03b4 < 1 \u2192\n        \u2203 s,\n          (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227\n            (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s\nN : \u2115 := multiplicity E + 1\nhN : N = multiplicity E + 1\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nh\u03b4' : \u03b4 < 1\ns : Finset E\nhs : \u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2\nh's : \u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016\ns_card : multiplicity E < Finset.card s\n\u22a2 multiplicity E + 1 \u2264 Finset.card s", "state_after": "no goals"}, {"tactic": "exact\n  \u27e8fun _ => 0, fun i => by simp; norm_num, fun i j _ => by\n    simpa only [norm_zero, sub_nonpos, sub_self]\u27e9", "annotated_tactic": ["exact\n        \u27e8fun _ => 0, fun i => by simp; norm_num, fun i j _ => by\n          simpa only [<a>norm_zero</a>, <a>sub_nonpos</a>, <a>sub_self</a>]\u27e9", [{"full_name": "norm_zero", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [528, 30], "def_end_pos": [528, 39]}, {"full_name": "sub_nonpos", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [730, 30], "def_end_pos": [730, 40]}, {"full_name": "sub_self", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [734, 30], "def_end_pos": [734, 38]}]], "state_before": "case inr\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nh :\n  \u2200 (\u03b4 : \u211d),\n    0 < \u03b4 \u2192\n      \u03b4 < 1 \u2192\n        \u2203 s,\n          (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227\n            (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s\nN : \u2115 := multiplicity E + 1\nhN : N = multiplicity E + 1\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nh\u03b4' : 1 \u2264 \u03b4\n\u22a2 \u2203 f, (\u2200 (i : Fin N), \u2016f i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016f i - f j\u2016", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nh :\n  \u2200 (\u03b4 : \u211d),\n    0 < \u03b4 \u2192\n      \u03b4 < 1 \u2192\n        \u2203 s,\n          (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227\n            (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s\nN : \u2115 := multiplicity E + 1\nhN : N = multiplicity E + 1\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nh\u03b4' : 1 \u2264 \u03b4\ni : Fin N\n\u22a2 \u2016(fun x => 0) i\u2016 \u2264 2", "state_after": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nh :\n  \u2200 (\u03b4 : \u211d),\n    0 < \u03b4 \u2192\n      \u03b4 < 1 \u2192\n        \u2203 s,\n          (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227\n            (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s\nN : \u2115 := multiplicity E + 1\nhN : N = multiplicity E + 1\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nh\u03b4' : 1 \u2264 \u03b4\ni : Fin N\n\u22a2 0 \u2264 2"}, {"tactic": "norm_num", "annotated_tactic": ["norm_num", []], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nh :\n  \u2200 (\u03b4 : \u211d),\n    0 < \u03b4 \u2192\n      \u03b4 < 1 \u2192\n        \u2203 s,\n          (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227\n            (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s\nN : \u2115 := multiplicity E + 1\nhN : N = multiplicity E + 1\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nh\u03b4' : 1 \u2264 \u03b4\ni : Fin N\n\u22a2 0 \u2264 2", "state_after": "no goals"}, {"tactic": "simpa only [norm_zero, sub_nonpos, sub_self]", "annotated_tactic": ["simpa only [<a>norm_zero</a>, <a>sub_nonpos</a>, <a>sub_self</a>]", [{"full_name": "norm_zero", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [528, 30], "def_end_pos": [528, 39]}, {"full_name": "sub_nonpos", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [730, 30], "def_end_pos": [730, 40]}, {"full_name": "sub_self", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [734, 30], "def_end_pos": [734, 38]}]], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nh :\n  \u2200 (\u03b4 : \u211d),\n    0 < \u03b4 \u2192\n      \u03b4 < 1 \u2192\n        \u2203 s,\n          (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227\n            (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s\nN : \u2115 := multiplicity E + 1\nhN : N = multiplicity E + 1\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nh\u03b4' : 1 \u2264 \u03b4\ni j : Fin N\nx\u271d : i \u2260 j\n\u22a2 1 - \u03b4 \u2264 \u2016(fun x => 0) i - (fun x => 0) j\u2016", "state_after": "no goals"}, {"tactic": "obtain \u27e8u, _, zero_lt_u, hu\u27e9 :\n  \u2203 u : \u2115 \u2192 \u211d,\n    (\u2200 m n : \u2115, m < n \u2192 u n < u m) \u2227 (\u2200 n : \u2115, 0 < u n) \u2227 Filter.Tendsto u Filter.atTop (\ud835\udcdd 0) :=\n  exists_seq_strictAnti_tendsto (0 : \u211d)", "annotated_tactic": ["obtain \u27e8u, _, zero_lt_u, hu\u27e9 :\n      \u2203 u : \u2115 \u2192 \u211d,\n        (\u2200 m n : \u2115, m < n \u2192 u n < u m) \u2227 (\u2200 n : \u2115, 0 < u n) \u2227 <a>Filter.Tendsto</a> u <a>Filter.atTop</a> (\ud835\udcdd 0) :=\n      <a>exists_seq_strictAnti_tendsto</a> (0 : \u211d)", [{"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2939, 5], "def_end_pos": [2939, 12]}, {"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}, {"full_name": "exists_seq_strictAnti_tendsto", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [2258, 9], "def_end_pos": [2258, 38]}]], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nh :\n  \u2200 (\u03b4 : \u211d),\n    0 < \u03b4 \u2192\n      \u03b4 < 1 \u2192\n        \u2203 s,\n          (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227\n            (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s\nN : \u2115 := multiplicity E + 1\nhN : N = multiplicity E + 1\nF : \u211d \u2192 Fin N \u2192 E\nhF : \u2200 (\u03b4 : \u211d), 0 < \u03b4 \u2192 (\u2200 (i : Fin N), \u2016F \u03b4 i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016F \u03b4 i - F \u03b4 j\u2016\n\u22a2 \u2203 f, (\u2200 (i : Fin N), \u2016f i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 \u2264 \u2016f i - f j\u2016", "state_after": "case intro.intro.intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nh :\n  \u2200 (\u03b4 : \u211d),\n    0 < \u03b4 \u2192\n      \u03b4 < 1 \u2192\n        \u2203 s,\n          (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227\n            (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s\nN : \u2115 := multiplicity E + 1\nhN : N = multiplicity E + 1\nF : \u211d \u2192 Fin N \u2192 E\nhF : \u2200 (\u03b4 : \u211d), 0 < \u03b4 \u2192 (\u2200 (i : Fin N), \u2016F \u03b4 i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016F \u03b4 i - F \u03b4 j\u2016\nu : \u2115 \u2192 \u211d\nleft\u271d : \u2200 (m n : \u2115), m < n \u2192 u n < u m\nzero_lt_u : \u2200 (n : \u2115), 0 < u n\nhu : Tendsto u atTop (\ud835\udcdd 0)\n\u22a2 \u2203 f, (\u2200 (i : Fin N), \u2016f i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 \u2264 \u2016f i - f j\u2016"}, {"tactic": "have A : \u2200 n, F (u n) \u2208 closedBall (0 : Fin N \u2192 E) 2 := by\n  intro n\n  simp only [pi_norm_le_iff_of_nonneg zero_le_two, mem_closedBall, dist_zero_right,\n    (hF (u n) (zero_lt_u n)).left, forall_const]", "annotated_tactic": ["have A : \u2200 n, F (u n) \u2208 <a>closedBall</a> (0 : <a>Fin</a> N \u2192 E) 2 := by\n      intro n\n      simp only [<a>pi_norm_le_iff_of_nonneg</a> <a>zero_le_two</a>, <a>mem_closedBall</a>, <a>dist_zero_right</a>,\n        (hF (u n) (zero_lt_u n)).<a>left</a>, <a>forall_const</a>]", [{"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "Fin", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1745, 11], "def_end_pos": [1745, 14]}, {"full_name": "pi_norm_le_iff_of_nonneg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [2523, 15], "def_end_pos": [2523, 39]}, {"full_name": "zero_le_two", "def_path": "Mathlib/Algebra/Order/Monoid/NatCast.lean", "def_pos": [32, 7], "def_end_pos": [32, 18]}, {"full_name": "Metric.mem_closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [478, 17], "def_end_pos": [478, 31]}, {"full_name": "dist_zero_right", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [395, 3], "def_end_pos": [395, 14]}, {"full_name": "And.left", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [504, 3], "def_end_pos": [504, 7]}, {"full_name": "forall_const", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [435, 17], "def_end_pos": [435, 29]}]], "state_before": "case intro.intro.intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nh :\n  \u2200 (\u03b4 : \u211d),\n    0 < \u03b4 \u2192\n      \u03b4 < 1 \u2192\n        \u2203 s,\n          (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227\n            (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s\nN : \u2115 := multiplicity E + 1\nhN : N = multiplicity E + 1\nF : \u211d \u2192 Fin N \u2192 E\nhF : \u2200 (\u03b4 : \u211d), 0 < \u03b4 \u2192 (\u2200 (i : Fin N), \u2016F \u03b4 i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016F \u03b4 i - F \u03b4 j\u2016\nu : \u2115 \u2192 \u211d\nleft\u271d : \u2200 (m n : \u2115), m < n \u2192 u n < u m\nzero_lt_u : \u2200 (n : \u2115), 0 < u n\nhu : Tendsto u atTop (\ud835\udcdd 0)\n\u22a2 \u2203 f, (\u2200 (i : Fin N), \u2016f i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 \u2264 \u2016f i - f j\u2016", "state_after": "case intro.intro.intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nh :\n  \u2200 (\u03b4 : \u211d),\n    0 < \u03b4 \u2192\n      \u03b4 < 1 \u2192\n        \u2203 s,\n          (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227\n            (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s\nN : \u2115 := multiplicity E + 1\nhN : N = multiplicity E + 1\nF : \u211d \u2192 Fin N \u2192 E\nhF : \u2200 (\u03b4 : \u211d), 0 < \u03b4 \u2192 (\u2200 (i : Fin N), \u2016F \u03b4 i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016F \u03b4 i - F \u03b4 j\u2016\nu : \u2115 \u2192 \u211d\nleft\u271d : \u2200 (m n : \u2115), m < n \u2192 u n < u m\nzero_lt_u : \u2200 (n : \u2115), 0 < u n\nhu : Tendsto u atTop (\ud835\udcdd 0)\nA : \u2200 (n : \u2115), F (u n) \u2208 closedBall 0 2\n\u22a2 \u2203 f, (\u2200 (i : Fin N), \u2016f i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 \u2264 \u2016f i - f j\u2016"}, {"tactic": "obtain \u27e8f, fmem, \u03c6, \u03c6_mono, hf\u27e9 :\n  \u2203 f \u2208 closedBall (0 : Fin N \u2192 E) 2,\n    \u2203 \u03c6 : \u2115 \u2192 \u2115, StrictMono \u03c6 \u2227 Tendsto ((F \u2218 u) \u2218 \u03c6) atTop (\ud835\udcdd f) :=\n  IsCompact.tendsto_subseq (isCompact_closedBall _ _) A", "annotated_tactic": ["obtain \u27e8f, fmem, \u03c6, \u03c6_mono, hf\u27e9 :\n      \u2203 f \u2208 <a>closedBall</a> (0 : <a>Fin</a> N \u2192 E) 2,\n        \u2203 \u03c6 : \u2115 \u2192 \u2115, <a>StrictMono</a> \u03c6 \u2227 <a>Tendsto</a> ((F \u2218 u) \u2218 \u03c6) <a>atTop</a> (\ud835\udcdd f) :=\n      <a>IsCompact.tendsto_subseq</a> (<a>isCompact_closedBall</a> _ _) A", [{"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "Fin", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1745, 11], "def_end_pos": [1745, 14]}, {"full_name": "StrictMono", "def_path": "Mathlib/Order/Monotone/Basic.lean", "def_pos": [97, 5], "def_end_pos": [97, 15]}, {"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2939, 5], "def_end_pos": [2939, 12]}, {"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}, {"full_name": "IsCompact.tendsto_subseq", "def_path": "Mathlib/Topology/Sequences.lean", "def_pos": [297, 9], "def_end_pos": [297, 33]}, {"full_name": "ProperSpace.isCompact_closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [2199, 3], "def_end_pos": [2199, 23]}]], "state_before": "case intro.intro.intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nh :\n  \u2200 (\u03b4 : \u211d),\n    0 < \u03b4 \u2192\n      \u03b4 < 1 \u2192\n        \u2203 s,\n          (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227\n            (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s\nN : \u2115 := multiplicity E + 1\nhN : N = multiplicity E + 1\nF : \u211d \u2192 Fin N \u2192 E\nhF : \u2200 (\u03b4 : \u211d), 0 < \u03b4 \u2192 (\u2200 (i : Fin N), \u2016F \u03b4 i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016F \u03b4 i - F \u03b4 j\u2016\nu : \u2115 \u2192 \u211d\nleft\u271d : \u2200 (m n : \u2115), m < n \u2192 u n < u m\nzero_lt_u : \u2200 (n : \u2115), 0 < u n\nhu : Tendsto u atTop (\ud835\udcdd 0)\nA : \u2200 (n : \u2115), F (u n) \u2208 closedBall 0 2\n\u22a2 \u2203 f, (\u2200 (i : Fin N), \u2016f i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 \u2264 \u2016f i - f j\u2016", "state_after": "case intro.intro.intro.intro.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nh :\n  \u2200 (\u03b4 : \u211d),\n    0 < \u03b4 \u2192\n      \u03b4 < 1 \u2192\n        \u2203 s,\n          (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227\n            (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s\nN : \u2115 := multiplicity E + 1\nhN : N = multiplicity E + 1\nF : \u211d \u2192 Fin N \u2192 E\nhF : \u2200 (\u03b4 : \u211d), 0 < \u03b4 \u2192 (\u2200 (i : Fin N), \u2016F \u03b4 i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016F \u03b4 i - F \u03b4 j\u2016\nu : \u2115 \u2192 \u211d\nleft\u271d : \u2200 (m n : \u2115), m < n \u2192 u n < u m\nzero_lt_u : \u2200 (n : \u2115), 0 < u n\nhu : Tendsto u atTop (\ud835\udcdd 0)\nA : \u2200 (n : \u2115), F (u n) \u2208 closedBall 0 2\nf : Fin N \u2192 E\nfmem : f \u2208 closedBall 0 2\n\u03c6 : \u2115 \u2192 \u2115\n\u03c6_mono : StrictMono \u03c6\nhf : Tendsto ((F \u2218 u) \u2218 \u03c6) atTop (\ud835\udcdd f)\n\u22a2 \u2203 f, (\u2200 (i : Fin N), \u2016f i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 \u2264 \u2016f i - f j\u2016"}, {"tactic": "refine' \u27e8f, fun i => _, fun i j hij => _\u27e9", "annotated_tactic": ["refine' \u27e8f, fun i => _, fun i j hij => _\u27e9", []], "state_before": "case intro.intro.intro.intro.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nh :\n  \u2200 (\u03b4 : \u211d),\n    0 < \u03b4 \u2192\n      \u03b4 < 1 \u2192\n        \u2203 s,\n          (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227\n            (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s\nN : \u2115 := multiplicity E + 1\nhN : N = multiplicity E + 1\nF : \u211d \u2192 Fin N \u2192 E\nhF : \u2200 (\u03b4 : \u211d), 0 < \u03b4 \u2192 (\u2200 (i : Fin N), \u2016F \u03b4 i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016F \u03b4 i - F \u03b4 j\u2016\nu : \u2115 \u2192 \u211d\nleft\u271d : \u2200 (m n : \u2115), m < n \u2192 u n < u m\nzero_lt_u : \u2200 (n : \u2115), 0 < u n\nhu : Tendsto u atTop (\ud835\udcdd 0)\nA : \u2200 (n : \u2115), F (u n) \u2208 closedBall 0 2\nf : Fin N \u2192 E\nfmem : f \u2208 closedBall 0 2\n\u03c6 : \u2115 \u2192 \u2115\n\u03c6_mono : StrictMono \u03c6\nhf : Tendsto ((F \u2218 u) \u2218 \u03c6) atTop (\ud835\udcdd f)\n\u22a2 \u2203 f, (\u2200 (i : Fin N), \u2016f i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 \u2264 \u2016f i - f j\u2016", "state_after": "case intro.intro.intro.intro.intro.intro.intro.refine'_1\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nh :\n  \u2200 (\u03b4 : \u211d),\n    0 < \u03b4 \u2192\n      \u03b4 < 1 \u2192\n        \u2203 s,\n          (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227\n            (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s\nN : \u2115 := multiplicity E + 1\nhN : N = multiplicity E + 1\nF : \u211d \u2192 Fin N \u2192 E\nhF : \u2200 (\u03b4 : \u211d), 0 < \u03b4 \u2192 (\u2200 (i : Fin N), \u2016F \u03b4 i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016F \u03b4 i - F \u03b4 j\u2016\nu : \u2115 \u2192 \u211d\nleft\u271d : \u2200 (m n : \u2115), m < n \u2192 u n < u m\nzero_lt_u : \u2200 (n : \u2115), 0 < u n\nhu : Tendsto u atTop (\ud835\udcdd 0)\nA : \u2200 (n : \u2115), F (u n) \u2208 closedBall 0 2\nf : Fin N \u2192 E\nfmem : f \u2208 closedBall 0 2\n\u03c6 : \u2115 \u2192 \u2115\n\u03c6_mono : StrictMono \u03c6\nhf : Tendsto ((F \u2218 u) \u2218 \u03c6) atTop (\ud835\udcdd f)\ni : Fin N\n\u22a2 \u2016f i\u2016 \u2264 2\n\ncase intro.intro.intro.intro.intro.intro.intro.refine'_2\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nh :\n  \u2200 (\u03b4 : \u211d),\n    0 < \u03b4 \u2192\n      \u03b4 < 1 \u2192\n        \u2203 s,\n          (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227\n            (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s\nN : \u2115 := multiplicity E + 1\nhN : N = multiplicity E + 1\nF : \u211d \u2192 Fin N \u2192 E\nhF : \u2200 (\u03b4 : \u211d), 0 < \u03b4 \u2192 (\u2200 (i : Fin N), \u2016F \u03b4 i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016F \u03b4 i - F \u03b4 j\u2016\nu : \u2115 \u2192 \u211d\nleft\u271d : \u2200 (m n : \u2115), m < n \u2192 u n < u m\nzero_lt_u : \u2200 (n : \u2115), 0 < u n\nhu : Tendsto u atTop (\ud835\udcdd 0)\nA : \u2200 (n : \u2115), F (u n) \u2208 closedBall 0 2\nf : Fin N \u2192 E\nfmem : f \u2208 closedBall 0 2\n\u03c6 : \u2115 \u2192 \u2115\n\u03c6_mono : StrictMono \u03c6\nhf : Tendsto ((F \u2218 u) \u2218 \u03c6) atTop (\ud835\udcdd f)\ni j : Fin N\nhij : i \u2260 j\n\u22a2 1 \u2264 \u2016f i - f j\u2016"}, {"tactic": "intro n", "annotated_tactic": ["intro n", []], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nh :\n  \u2200 (\u03b4 : \u211d),\n    0 < \u03b4 \u2192\n      \u03b4 < 1 \u2192\n        \u2203 s,\n          (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227\n            (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s\nN : \u2115 := multiplicity E + 1\nhN : N = multiplicity E + 1\nF : \u211d \u2192 Fin N \u2192 E\nhF : \u2200 (\u03b4 : \u211d), 0 < \u03b4 \u2192 (\u2200 (i : Fin N), \u2016F \u03b4 i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016F \u03b4 i - F \u03b4 j\u2016\nu : \u2115 \u2192 \u211d\nleft\u271d : \u2200 (m n : \u2115), m < n \u2192 u n < u m\nzero_lt_u : \u2200 (n : \u2115), 0 < u n\nhu : Tendsto u atTop (\ud835\udcdd 0)\n\u22a2 \u2200 (n : \u2115), F (u n) \u2208 closedBall 0 2", "state_after": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nh :\n  \u2200 (\u03b4 : \u211d),\n    0 < \u03b4 \u2192\n      \u03b4 < 1 \u2192\n        \u2203 s,\n          (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227\n            (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s\nN : \u2115 := multiplicity E + 1\nhN : N = multiplicity E + 1\nF : \u211d \u2192 Fin N \u2192 E\nhF : \u2200 (\u03b4 : \u211d), 0 < \u03b4 \u2192 (\u2200 (i : Fin N), \u2016F \u03b4 i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016F \u03b4 i - F \u03b4 j\u2016\nu : \u2115 \u2192 \u211d\nleft\u271d : \u2200 (m n : \u2115), m < n \u2192 u n < u m\nzero_lt_u : \u2200 (n : \u2115), 0 < u n\nhu : Tendsto u atTop (\ud835\udcdd 0)\nn : \u2115\n\u22a2 F (u n) \u2208 closedBall 0 2"}, {"tactic": "simp only [pi_norm_le_iff_of_nonneg zero_le_two, mem_closedBall, dist_zero_right,\n  (hF (u n) (zero_lt_u n)).left, forall_const]", "annotated_tactic": ["simp only [<a>pi_norm_le_iff_of_nonneg</a> <a>zero_le_two</a>, <a>mem_closedBall</a>, <a>dist_zero_right</a>,\n        (hF (u n) (zero_lt_u n)).<a>left</a>, <a>forall_const</a>]", [{"full_name": "pi_norm_le_iff_of_nonneg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [2523, 15], "def_end_pos": [2523, 39]}, {"full_name": "zero_le_two", "def_path": "Mathlib/Algebra/Order/Monoid/NatCast.lean", "def_pos": [32, 7], "def_end_pos": [32, 18]}, {"full_name": "Metric.mem_closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [478, 17], "def_end_pos": [478, 31]}, {"full_name": "dist_zero_right", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [395, 3], "def_end_pos": [395, 14]}, {"full_name": "And.left", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [504, 3], "def_end_pos": [504, 7]}, {"full_name": "forall_const", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [435, 17], "def_end_pos": [435, 29]}]], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nh :\n  \u2200 (\u03b4 : \u211d),\n    0 < \u03b4 \u2192\n      \u03b4 < 1 \u2192\n        \u2203 s,\n          (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227\n            (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s\nN : \u2115 := multiplicity E + 1\nhN : N = multiplicity E + 1\nF : \u211d \u2192 Fin N \u2192 E\nhF : \u2200 (\u03b4 : \u211d), 0 < \u03b4 \u2192 (\u2200 (i : Fin N), \u2016F \u03b4 i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016F \u03b4 i - F \u03b4 j\u2016\nu : \u2115 \u2192 \u211d\nleft\u271d : \u2200 (m n : \u2115), m < n \u2192 u n < u m\nzero_lt_u : \u2200 (n : \u2115), 0 < u n\nhu : Tendsto u atTop (\ud835\udcdd 0)\nn : \u2115\n\u22a2 F (u n) \u2208 closedBall 0 2", "state_after": "no goals"}, {"tactic": "simp only [pi_norm_le_iff_of_nonneg zero_le_two, mem_closedBall, dist_zero_right] at fmem", "annotated_tactic": ["simp only [<a>pi_norm_le_iff_of_nonneg</a> <a>zero_le_two</a>, <a>mem_closedBall</a>, <a>dist_zero_right</a>] at fmem", [{"full_name": "pi_norm_le_iff_of_nonneg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [2523, 15], "def_end_pos": [2523, 39]}, {"full_name": "zero_le_two", "def_path": "Mathlib/Algebra/Order/Monoid/NatCast.lean", "def_pos": [32, 7], "def_end_pos": [32, 18]}, {"full_name": "Metric.mem_closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [478, 17], "def_end_pos": [478, 31]}, {"full_name": "dist_zero_right", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [395, 3], "def_end_pos": [395, 14]}]], "state_before": "case intro.intro.intro.intro.intro.intro.intro.refine'_1\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nh :\n  \u2200 (\u03b4 : \u211d),\n    0 < \u03b4 \u2192\n      \u03b4 < 1 \u2192\n        \u2203 s,\n          (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227\n            (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s\nN : \u2115 := multiplicity E + 1\nhN : N = multiplicity E + 1\nF : \u211d \u2192 Fin N \u2192 E\nhF : \u2200 (\u03b4 : \u211d), 0 < \u03b4 \u2192 (\u2200 (i : Fin N), \u2016F \u03b4 i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016F \u03b4 i - F \u03b4 j\u2016\nu : \u2115 \u2192 \u211d\nleft\u271d : \u2200 (m n : \u2115), m < n \u2192 u n < u m\nzero_lt_u : \u2200 (n : \u2115), 0 < u n\nhu : Tendsto u atTop (\ud835\udcdd 0)\nA : \u2200 (n : \u2115), F (u n) \u2208 closedBall 0 2\nf : Fin N \u2192 E\nfmem : f \u2208 closedBall 0 2\n\u03c6 : \u2115 \u2192 \u2115\n\u03c6_mono : StrictMono \u03c6\nhf : Tendsto ((F \u2218 u) \u2218 \u03c6) atTop (\ud835\udcdd f)\ni : Fin N\n\u22a2 \u2016f i\u2016 \u2264 2", "state_after": "case intro.intro.intro.intro.intro.intro.intro.refine'_1\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nh :\n  \u2200 (\u03b4 : \u211d),\n    0 < \u03b4 \u2192\n      \u03b4 < 1 \u2192\n        \u2203 s,\n          (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227\n            (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s\nN : \u2115 := multiplicity E + 1\nhN : N = multiplicity E + 1\nF : \u211d \u2192 Fin N \u2192 E\nhF : \u2200 (\u03b4 : \u211d), 0 < \u03b4 \u2192 (\u2200 (i : Fin N), \u2016F \u03b4 i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016F \u03b4 i - F \u03b4 j\u2016\nu : \u2115 \u2192 \u211d\nleft\u271d : \u2200 (m n : \u2115), m < n \u2192 u n < u m\nzero_lt_u : \u2200 (n : \u2115), 0 < u n\nhu : Tendsto u atTop (\ud835\udcdd 0)\nA : \u2200 (n : \u2115), F (u n) \u2208 closedBall 0 2\nf : Fin N \u2192 E\n\u03c6 : \u2115 \u2192 \u2115\n\u03c6_mono : StrictMono \u03c6\nhf : Tendsto ((F \u2218 u) \u2218 \u03c6) atTop (\ud835\udcdd f)\ni : Fin N\nfmem : \u2200 (i : Fin N), \u2016f i\u2016 \u2264 2\n\u22a2 \u2016f i\u2016 \u2264 2"}, {"tactic": "exact fmem i", "annotated_tactic": ["exact fmem i", []], "state_before": "case intro.intro.intro.intro.intro.intro.intro.refine'_1\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nh :\n  \u2200 (\u03b4 : \u211d),\n    0 < \u03b4 \u2192\n      \u03b4 < 1 \u2192\n        \u2203 s,\n          (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227\n            (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s\nN : \u2115 := multiplicity E + 1\nhN : N = multiplicity E + 1\nF : \u211d \u2192 Fin N \u2192 E\nhF : \u2200 (\u03b4 : \u211d), 0 < \u03b4 \u2192 (\u2200 (i : Fin N), \u2016F \u03b4 i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016F \u03b4 i - F \u03b4 j\u2016\nu : \u2115 \u2192 \u211d\nleft\u271d : \u2200 (m n : \u2115), m < n \u2192 u n < u m\nzero_lt_u : \u2200 (n : \u2115), 0 < u n\nhu : Tendsto u atTop (\ud835\udcdd 0)\nA : \u2200 (n : \u2115), F (u n) \u2208 closedBall 0 2\nf : Fin N \u2192 E\n\u03c6 : \u2115 \u2192 \u2115\n\u03c6_mono : StrictMono \u03c6\nhf : Tendsto ((F \u2218 u) \u2218 \u03c6) atTop (\ud835\udcdd f)\ni : Fin N\nfmem : \u2200 (i : Fin N), \u2016f i\u2016 \u2264 2\n\u22a2 \u2016f i\u2016 \u2264 2", "state_after": "no goals"}, {"tactic": "have A : Tendsto (fun n => \u2016F (u (\u03c6 n)) i - F (u (\u03c6 n)) j\u2016) atTop (\ud835\udcdd \u2016f i - f j\u2016) :=\n  ((hf.apply i).sub (hf.apply j)).norm", "annotated_tactic": ["have A : <a>Tendsto</a> (fun n => \u2016F (u (\u03c6 n)) i - F (u (\u03c6 n)) j\u2016) <a>atTop</a> (\ud835\udcdd \u2016f i - f j\u2016) :=\n        ((hf.apply i).<a>sub</a> (hf.apply j)).<a>norm</a>", [{"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2939, 5], "def_end_pos": [2939, 12]}, {"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}, {"full_name": "Filter.Tendsto.sub", "def_path": "Mathlib/Topology/Algebra/Group/Basic.lean", "def_pos": [1081, 15], "def_end_pos": [1081, 18]}, {"full_name": "Filter.Tendsto.norm", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [1240, 15], "def_end_pos": [1240, 34]}]], "state_before": "case intro.intro.intro.intro.intro.intro.intro.refine'_2\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nh :\n  \u2200 (\u03b4 : \u211d),\n    0 < \u03b4 \u2192\n      \u03b4 < 1 \u2192\n        \u2203 s,\n          (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227\n            (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s\nN : \u2115 := multiplicity E + 1\nhN : N = multiplicity E + 1\nF : \u211d \u2192 Fin N \u2192 E\nhF : \u2200 (\u03b4 : \u211d), 0 < \u03b4 \u2192 (\u2200 (i : Fin N), \u2016F \u03b4 i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016F \u03b4 i - F \u03b4 j\u2016\nu : \u2115 \u2192 \u211d\nleft\u271d : \u2200 (m n : \u2115), m < n \u2192 u n < u m\nzero_lt_u : \u2200 (n : \u2115), 0 < u n\nhu : Tendsto u atTop (\ud835\udcdd 0)\nA : \u2200 (n : \u2115), F (u n) \u2208 closedBall 0 2\nf : Fin N \u2192 E\nfmem : f \u2208 closedBall 0 2\n\u03c6 : \u2115 \u2192 \u2115\n\u03c6_mono : StrictMono \u03c6\nhf : Tendsto ((F \u2218 u) \u2218 \u03c6) atTop (\ud835\udcdd f)\ni j : Fin N\nhij : i \u2260 j\n\u22a2 1 \u2264 \u2016f i - f j\u2016", "state_after": "case intro.intro.intro.intro.intro.intro.intro.refine'_2\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nh :\n  \u2200 (\u03b4 : \u211d),\n    0 < \u03b4 \u2192\n      \u03b4 < 1 \u2192\n        \u2203 s,\n          (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227\n            (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s\nN : \u2115 := multiplicity E + 1\nhN : N = multiplicity E + 1\nF : \u211d \u2192 Fin N \u2192 E\nhF : \u2200 (\u03b4 : \u211d), 0 < \u03b4 \u2192 (\u2200 (i : Fin N), \u2016F \u03b4 i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016F \u03b4 i - F \u03b4 j\u2016\nu : \u2115 \u2192 \u211d\nleft\u271d : \u2200 (m n : \u2115), m < n \u2192 u n < u m\nzero_lt_u : \u2200 (n : \u2115), 0 < u n\nhu : Tendsto u atTop (\ud835\udcdd 0)\nA\u271d : \u2200 (n : \u2115), F (u n) \u2208 closedBall 0 2\nf : Fin N \u2192 E\nfmem : f \u2208 closedBall 0 2\n\u03c6 : \u2115 \u2192 \u2115\n\u03c6_mono : StrictMono \u03c6\nhf : Tendsto ((F \u2218 u) \u2218 \u03c6) atTop (\ud835\udcdd f)\ni j : Fin N\nhij : i \u2260 j\nA : Tendsto (fun n => \u2016F (u (\u03c6 n)) i - F (u (\u03c6 n)) j\u2016) atTop (\ud835\udcdd \u2016f i - f j\u2016)\n\u22a2 1 \u2264 \u2016f i - f j\u2016"}, {"tactic": "have B : Tendsto (fun n => 1 - u (\u03c6 n)) atTop (\ud835\udcdd (1 - 0)) :=\n  tendsto_const_nhds.sub (hu.comp \u03c6_mono.tendsto_atTop)", "annotated_tactic": ["have B : <a>Tendsto</a> (fun n => 1 - u (\u03c6 n)) <a>atTop</a> (\ud835\udcdd (1 - 0)) :=\n        tendsto_const_nhds.sub (hu.comp \u03c6_mono.tendsto_atTop)", [{"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2939, 5], "def_end_pos": [2939, 12]}, {"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}]], "state_before": "case intro.intro.intro.intro.intro.intro.intro.refine'_2\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nh :\n  \u2200 (\u03b4 : \u211d),\n    0 < \u03b4 \u2192\n      \u03b4 < 1 \u2192\n        \u2203 s,\n          (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227\n            (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s\nN : \u2115 := multiplicity E + 1\nhN : N = multiplicity E + 1\nF : \u211d \u2192 Fin N \u2192 E\nhF : \u2200 (\u03b4 : \u211d), 0 < \u03b4 \u2192 (\u2200 (i : Fin N), \u2016F \u03b4 i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016F \u03b4 i - F \u03b4 j\u2016\nu : \u2115 \u2192 \u211d\nleft\u271d : \u2200 (m n : \u2115), m < n \u2192 u n < u m\nzero_lt_u : \u2200 (n : \u2115), 0 < u n\nhu : Tendsto u atTop (\ud835\udcdd 0)\nA\u271d : \u2200 (n : \u2115), F (u n) \u2208 closedBall 0 2\nf : Fin N \u2192 E\nfmem : f \u2208 closedBall 0 2\n\u03c6 : \u2115 \u2192 \u2115\n\u03c6_mono : StrictMono \u03c6\nhf : Tendsto ((F \u2218 u) \u2218 \u03c6) atTop (\ud835\udcdd f)\ni j : Fin N\nhij : i \u2260 j\nA : Tendsto (fun n => \u2016F (u (\u03c6 n)) i - F (u (\u03c6 n)) j\u2016) atTop (\ud835\udcdd \u2016f i - f j\u2016)\n\u22a2 1 \u2264 \u2016f i - f j\u2016", "state_after": "case intro.intro.intro.intro.intro.intro.intro.refine'_2\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nh :\n  \u2200 (\u03b4 : \u211d),\n    0 < \u03b4 \u2192\n      \u03b4 < 1 \u2192\n        \u2203 s,\n          (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227\n            (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s\nN : \u2115 := multiplicity E + 1\nhN : N = multiplicity E + 1\nF : \u211d \u2192 Fin N \u2192 E\nhF : \u2200 (\u03b4 : \u211d), 0 < \u03b4 \u2192 (\u2200 (i : Fin N), \u2016F \u03b4 i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016F \u03b4 i - F \u03b4 j\u2016\nu : \u2115 \u2192 \u211d\nleft\u271d : \u2200 (m n : \u2115), m < n \u2192 u n < u m\nzero_lt_u : \u2200 (n : \u2115), 0 < u n\nhu : Tendsto u atTop (\ud835\udcdd 0)\nA\u271d : \u2200 (n : \u2115), F (u n) \u2208 closedBall 0 2\nf : Fin N \u2192 E\nfmem : f \u2208 closedBall 0 2\n\u03c6 : \u2115 \u2192 \u2115\n\u03c6_mono : StrictMono \u03c6\nhf : Tendsto ((F \u2218 u) \u2218 \u03c6) atTop (\ud835\udcdd f)\ni j : Fin N\nhij : i \u2260 j\nA : Tendsto (fun n => \u2016F (u (\u03c6 n)) i - F (u (\u03c6 n)) j\u2016) atTop (\ud835\udcdd \u2016f i - f j\u2016)\nB : Tendsto (fun n => 1 - u (\u03c6 n)) atTop (\ud835\udcdd (1 - 0))\n\u22a2 1 \u2264 \u2016f i - f j\u2016"}, {"tactic": "rw [sub_zero] at B", "annotated_tactic": ["rw [<a>sub_zero</a>] at B", [{"full_name": "sub_zero", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [339, 3], "def_end_pos": [339, 14]}]], "state_before": "case intro.intro.intro.intro.intro.intro.intro.refine'_2\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nh :\n  \u2200 (\u03b4 : \u211d),\n    0 < \u03b4 \u2192\n      \u03b4 < 1 \u2192\n        \u2203 s,\n          (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227\n            (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s\nN : \u2115 := multiplicity E + 1\nhN : N = multiplicity E + 1\nF : \u211d \u2192 Fin N \u2192 E\nhF : \u2200 (\u03b4 : \u211d), 0 < \u03b4 \u2192 (\u2200 (i : Fin N), \u2016F \u03b4 i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016F \u03b4 i - F \u03b4 j\u2016\nu : \u2115 \u2192 \u211d\nleft\u271d : \u2200 (m n : \u2115), m < n \u2192 u n < u m\nzero_lt_u : \u2200 (n : \u2115), 0 < u n\nhu : Tendsto u atTop (\ud835\udcdd 0)\nA\u271d : \u2200 (n : \u2115), F (u n) \u2208 closedBall 0 2\nf : Fin N \u2192 E\nfmem : f \u2208 closedBall 0 2\n\u03c6 : \u2115 \u2192 \u2115\n\u03c6_mono : StrictMono \u03c6\nhf : Tendsto ((F \u2218 u) \u2218 \u03c6) atTop (\ud835\udcdd f)\ni j : Fin N\nhij : i \u2260 j\nA : Tendsto (fun n => \u2016F (u (\u03c6 n)) i - F (u (\u03c6 n)) j\u2016) atTop (\ud835\udcdd \u2016f i - f j\u2016)\nB : Tendsto (fun n => 1 - u (\u03c6 n)) atTop (\ud835\udcdd (1 - 0))\n\u22a2 1 \u2264 \u2016f i - f j\u2016", "state_after": "case intro.intro.intro.intro.intro.intro.intro.refine'_2\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nh :\n  \u2200 (\u03b4 : \u211d),\n    0 < \u03b4 \u2192\n      \u03b4 < 1 \u2192\n        \u2203 s,\n          (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227\n            (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s\nN : \u2115 := multiplicity E + 1\nhN : N = multiplicity E + 1\nF : \u211d \u2192 Fin N \u2192 E\nhF : \u2200 (\u03b4 : \u211d), 0 < \u03b4 \u2192 (\u2200 (i : Fin N), \u2016F \u03b4 i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016F \u03b4 i - F \u03b4 j\u2016\nu : \u2115 \u2192 \u211d\nleft\u271d : \u2200 (m n : \u2115), m < n \u2192 u n < u m\nzero_lt_u : \u2200 (n : \u2115), 0 < u n\nhu : Tendsto u atTop (\ud835\udcdd 0)\nA\u271d : \u2200 (n : \u2115), F (u n) \u2208 closedBall 0 2\nf : Fin N \u2192 E\nfmem : f \u2208 closedBall 0 2\n\u03c6 : \u2115 \u2192 \u2115\n\u03c6_mono : StrictMono \u03c6\nhf : Tendsto ((F \u2218 u) \u2218 \u03c6) atTop (\ud835\udcdd f)\ni j : Fin N\nhij : i \u2260 j\nA : Tendsto (fun n => \u2016F (u (\u03c6 n)) i - F (u (\u03c6 n)) j\u2016) atTop (\ud835\udcdd \u2016f i - f j\u2016)\nB : Tendsto (fun n => 1 - u (\u03c6 n)) atTop (\ud835\udcdd 1)\n\u22a2 1 \u2264 \u2016f i - f j\u2016"}, {"tactic": "exact le_of_tendsto_of_tendsto' B A fun n => (hF (u (\u03c6 n)) (zero_lt_u _)).2 i j hij", "annotated_tactic": ["exact <a>le_of_tendsto_of_tendsto'</a> B A fun n => (hF (u (\u03c6 n)) (zero_lt_u _)).2 i j hij", [{"full_name": "le_of_tendsto_of_tendsto'", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [232, 9], "def_end_pos": [232, 34]}]], "state_before": "case intro.intro.intro.intro.intro.intro.intro.refine'_2\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nh :\n  \u2200 (\u03b4 : \u211d),\n    0 < \u03b4 \u2192\n      \u03b4 < 1 \u2192\n        \u2203 s,\n          (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227\n            (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s\nN : \u2115 := multiplicity E + 1\nhN : N = multiplicity E + 1\nF : \u211d \u2192 Fin N \u2192 E\nhF : \u2200 (\u03b4 : \u211d), 0 < \u03b4 \u2192 (\u2200 (i : Fin N), \u2016F \u03b4 i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016F \u03b4 i - F \u03b4 j\u2016\nu : \u2115 \u2192 \u211d\nleft\u271d : \u2200 (m n : \u2115), m < n \u2192 u n < u m\nzero_lt_u : \u2200 (n : \u2115), 0 < u n\nhu : Tendsto u atTop (\ud835\udcdd 0)\nA\u271d : \u2200 (n : \u2115), F (u n) \u2208 closedBall 0 2\nf : Fin N \u2192 E\nfmem : f \u2208 closedBall 0 2\n\u03c6 : \u2115 \u2192 \u2115\n\u03c6_mono : StrictMono \u03c6\nhf : Tendsto ((F \u2218 u) \u2218 \u03c6) atTop (\ud835\udcdd f)\ni j : Fin N\nhij : i \u2260 j\nA : Tendsto (fun n => \u2016F (u (\u03c6 n)) i - F (u (\u03c6 n)) j\u2016) atTop (\ud835\udcdd \u2016f i - f j\u2016)\nB : Tendsto (fun n => 1 - u (\u03c6 n)) atTop (\ud835\udcdd 1)\n\u22a2 1 \u2264 \u2016f i - f j\u2016", "state_after": "no goals"}, {"tactic": "intro i j hij", "annotated_tactic": ["intro i j hij", []], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nh :\n  \u2200 (\u03b4 : \u211d),\n    0 < \u03b4 \u2192\n      \u03b4 < 1 \u2192\n        \u2203 s,\n          (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227\n            (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s\nN : \u2115 := multiplicity E + 1\nhN : N = multiplicity E + 1\nF : \u211d \u2192 Fin N \u2192 E\nhF : \u2200 (\u03b4 : \u211d), 0 < \u03b4 \u2192 (\u2200 (i : Fin N), \u2016F \u03b4 i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016F \u03b4 i - F \u03b4 j\u2016\nf : Fin N \u2192 E\nhf : \u2200 (i : Fin N), \u2016f i\u2016 \u2264 2\nh'f : \u2200 (i j : Fin N), i \u2260 j \u2192 1 \u2264 \u2016f i - f j\u2016\n\u22a2 Function.Injective f", "state_after": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nh :\n  \u2200 (\u03b4 : \u211d),\n    0 < \u03b4 \u2192\n      \u03b4 < 1 \u2192\n        \u2203 s,\n          (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227\n            (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s\nN : \u2115 := multiplicity E + 1\nhN : N = multiplicity E + 1\nF : \u211d \u2192 Fin N \u2192 E\nhF : \u2200 (\u03b4 : \u211d), 0 < \u03b4 \u2192 (\u2200 (i : Fin N), \u2016F \u03b4 i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016F \u03b4 i - F \u03b4 j\u2016\nf : Fin N \u2192 E\nhf : \u2200 (i : Fin N), \u2016f i\u2016 \u2264 2\nh'f : \u2200 (i j : Fin N), i \u2260 j \u2192 1 \u2264 \u2016f i - f j\u2016\ni j : Fin N\nhij : f i = f j\n\u22a2 i = j"}, {"tactic": "by_contra h", "annotated_tactic": ["by_contra h", []], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nh :\n  \u2200 (\u03b4 : \u211d),\n    0 < \u03b4 \u2192\n      \u03b4 < 1 \u2192\n        \u2203 s,\n          (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227\n            (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s\nN : \u2115 := multiplicity E + 1\nhN : N = multiplicity E + 1\nF : \u211d \u2192 Fin N \u2192 E\nhF : \u2200 (\u03b4 : \u211d), 0 < \u03b4 \u2192 (\u2200 (i : Fin N), \u2016F \u03b4 i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016F \u03b4 i - F \u03b4 j\u2016\nf : Fin N \u2192 E\nhf : \u2200 (i : Fin N), \u2016f i\u2016 \u2264 2\nh'f : \u2200 (i j : Fin N), i \u2260 j \u2192 1 \u2264 \u2016f i - f j\u2016\ni j : Fin N\nhij : f i = f j\n\u22a2 i = j", "state_after": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nh\u271d :\n  \u2200 (\u03b4 : \u211d),\n    0 < \u03b4 \u2192\n      \u03b4 < 1 \u2192\n        \u2203 s,\n          (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227\n            (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s\nN : \u2115 := multiplicity E + 1\nhN : N = multiplicity E + 1\nF : \u211d \u2192 Fin N \u2192 E\nhF : \u2200 (\u03b4 : \u211d), 0 < \u03b4 \u2192 (\u2200 (i : Fin N), \u2016F \u03b4 i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016F \u03b4 i - F \u03b4 j\u2016\nf : Fin N \u2192 E\nhf : \u2200 (i : Fin N), \u2016f i\u2016 \u2264 2\nh'f : \u2200 (i j : Fin N), i \u2260 j \u2192 1 \u2264 \u2016f i - f j\u2016\ni j : Fin N\nhij : f i = f j\nh : \u00aci = j\n\u22a2 False"}, {"tactic": "have : 1 \u2264 \u2016f i - f j\u2016 := h'f i j h", "annotated_tactic": ["have : 1 \u2264 \u2016f i - f j\u2016 := h'f i j h", []], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nh\u271d :\n  \u2200 (\u03b4 : \u211d),\n    0 < \u03b4 \u2192\n      \u03b4 < 1 \u2192\n        \u2203 s,\n          (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227\n            (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s\nN : \u2115 := multiplicity E + 1\nhN : N = multiplicity E + 1\nF : \u211d \u2192 Fin N \u2192 E\nhF : \u2200 (\u03b4 : \u211d), 0 < \u03b4 \u2192 (\u2200 (i : Fin N), \u2016F \u03b4 i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016F \u03b4 i - F \u03b4 j\u2016\nf : Fin N \u2192 E\nhf : \u2200 (i : Fin N), \u2016f i\u2016 \u2264 2\nh'f : \u2200 (i j : Fin N), i \u2260 j \u2192 1 \u2264 \u2016f i - f j\u2016\ni j : Fin N\nhij : f i = f j\nh : \u00aci = j\n\u22a2 False", "state_after": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nh\u271d :\n  \u2200 (\u03b4 : \u211d),\n    0 < \u03b4 \u2192\n      \u03b4 < 1 \u2192\n        \u2203 s,\n          (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227\n            (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s\nN : \u2115 := multiplicity E + 1\nhN : N = multiplicity E + 1\nF : \u211d \u2192 Fin N \u2192 E\nhF : \u2200 (\u03b4 : \u211d), 0 < \u03b4 \u2192 (\u2200 (i : Fin N), \u2016F \u03b4 i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016F \u03b4 i - F \u03b4 j\u2016\nf : Fin N \u2192 E\nhf : \u2200 (i : Fin N), \u2016f i\u2016 \u2264 2\nh'f : \u2200 (i j : Fin N), i \u2260 j \u2192 1 \u2264 \u2016f i - f j\u2016\ni j : Fin N\nhij : f i = f j\nh : \u00aci = j\nthis : 1 \u2264 \u2016f i - f j\u2016\n\u22a2 False"}, {"tactic": "simp only [hij, norm_zero, sub_self] at this", "annotated_tactic": ["simp only [hij, <a>norm_zero</a>, <a>sub_self</a>] at this", [{"full_name": "norm_zero", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [528, 30], "def_end_pos": [528, 39]}, {"full_name": "sub_self", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [734, 30], "def_end_pos": [734, 38]}]], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nh\u271d :\n  \u2200 (\u03b4 : \u211d),\n    0 < \u03b4 \u2192\n      \u03b4 < 1 \u2192\n        \u2203 s,\n          (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227\n            (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s\nN : \u2115 := multiplicity E + 1\nhN : N = multiplicity E + 1\nF : \u211d \u2192 Fin N \u2192 E\nhF : \u2200 (\u03b4 : \u211d), 0 < \u03b4 \u2192 (\u2200 (i : Fin N), \u2016F \u03b4 i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016F \u03b4 i - F \u03b4 j\u2016\nf : Fin N \u2192 E\nhf : \u2200 (i : Fin N), \u2016f i\u2016 \u2264 2\nh'f : \u2200 (i j : Fin N), i \u2260 j \u2192 1 \u2264 \u2016f i - f j\u2016\ni j : Fin N\nhij : f i = f j\nh : \u00aci = j\nthis : 1 \u2264 \u2016f i - f j\u2016\n\u22a2 False", "state_after": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nh\u271d :\n  \u2200 (\u03b4 : \u211d),\n    0 < \u03b4 \u2192\n      \u03b4 < 1 \u2192\n        \u2203 s,\n          (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227\n            (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s\nN : \u2115 := multiplicity E + 1\nhN : N = multiplicity E + 1\nF : \u211d \u2192 Fin N \u2192 E\nhF : \u2200 (\u03b4 : \u211d), 0 < \u03b4 \u2192 (\u2200 (i : Fin N), \u2016F \u03b4 i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016F \u03b4 i - F \u03b4 j\u2016\nf : Fin N \u2192 E\nhf : \u2200 (i : Fin N), \u2016f i\u2016 \u2264 2\nh'f : \u2200 (i j : Fin N), i \u2260 j \u2192 1 \u2264 \u2016f i - f j\u2016\ni j : Fin N\nhij : f i = f j\nh : \u00aci = j\nthis : 1 \u2264 0\n\u22a2 False"}, {"tactic": "exact lt_irrefl _ (this.trans_lt zero_lt_one)", "annotated_tactic": ["exact <a>lt_irrefl</a> _ (this.trans_lt <a>zero_lt_one</a>)", [{"full_name": "lt_irrefl", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [79, 9], "def_end_pos": [79, 18]}, {"full_name": "zero_lt_one", "def_path": "Mathlib/Algebra/Order/ZeroLEOne.lean", "def_pos": [39, 15], "def_end_pos": [39, 26]}]], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nh\u271d :\n  \u2200 (\u03b4 : \u211d),\n    0 < \u03b4 \u2192\n      \u03b4 < 1 \u2192\n        \u2203 s,\n          (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227\n            (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s\nN : \u2115 := multiplicity E + 1\nhN : N = multiplicity E + 1\nF : \u211d \u2192 Fin N \u2192 E\nhF : \u2200 (\u03b4 : \u211d), 0 < \u03b4 \u2192 (\u2200 (i : Fin N), \u2016F \u03b4 i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016F \u03b4 i - F \u03b4 j\u2016\nf : Fin N \u2192 E\nhf : \u2200 (i : Fin N), \u2016f i\u2016 \u2264 2\nh'f : \u2200 (i j : Fin N), i \u2260 j \u2192 1 \u2264 \u2016f i - f j\u2016\ni j : Fin N\nhij : f i = f j\nh : \u00aci = j\nthis : 1 \u2264 0\n\u22a2 False", "state_after": "no goals"}, {"tactic": "rw [Finset.card_image_of_injective _ finj]", "annotated_tactic": ["rw [<a>Finset.card_image_of_injective</a> _ finj]", [{"full_name": "Finset.card_image_of_injective", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [250, 9], "def_end_pos": [250, 32]}]], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nh :\n  \u2200 (\u03b4 : \u211d),\n    0 < \u03b4 \u2192\n      \u03b4 < 1 \u2192\n        \u2203 s,\n          (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227\n            (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s\nN : \u2115 := multiplicity E + 1\nhN : N = multiplicity E + 1\nF : \u211d \u2192 Fin N \u2192 E\nhF : \u2200 (\u03b4 : \u211d), 0 < \u03b4 \u2192 (\u2200 (i : Fin N), \u2016F \u03b4 i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016F \u03b4 i - F \u03b4 j\u2016\nf : Fin N \u2192 E\nhf : \u2200 (i : Fin N), \u2016f i\u2016 \u2264 2\nh'f : \u2200 (i j : Fin N), i \u2260 j \u2192 1 \u2264 \u2016f i - f j\u2016\nfinj : Function.Injective f\ns : Finset E := Finset.image f Finset.univ\n\u22a2 Finset.card s = N", "state_after": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nh :\n  \u2200 (\u03b4 : \u211d),\n    0 < \u03b4 \u2192\n      \u03b4 < 1 \u2192\n        \u2203 s,\n          (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227\n            (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s\nN : \u2115 := multiplicity E + 1\nhN : N = multiplicity E + 1\nF : \u211d \u2192 Fin N \u2192 E\nhF : \u2200 (\u03b4 : \u211d), 0 < \u03b4 \u2192 (\u2200 (i : Fin N), \u2016F \u03b4 i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016F \u03b4 i - F \u03b4 j\u2016\nf : Fin N \u2192 E\nhf : \u2200 (i : Fin N), \u2016f i\u2016 \u2264 2\nh'f : \u2200 (i j : Fin N), i \u2260 j \u2192 1 \u2264 \u2016f i - f j\u2016\nfinj : Function.Injective f\ns : Finset E := Finset.image f Finset.univ\n\u22a2 Finset.card Finset.univ = N"}, {"tactic": "exact Finset.card_fin N", "annotated_tactic": ["exact <a>Finset.card_fin</a> N", [{"full_name": "Finset.card_fin", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [322, 9], "def_end_pos": [322, 24]}]], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nh :\n  \u2200 (\u03b4 : \u211d),\n    0 < \u03b4 \u2192\n      \u03b4 < 1 \u2192\n        \u2203 s,\n          (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227\n            (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s\nN : \u2115 := multiplicity E + 1\nhN : N = multiplicity E + 1\nF : \u211d \u2192 Fin N \u2192 E\nhF : \u2200 (\u03b4 : \u211d), 0 < \u03b4 \u2192 (\u2200 (i : Fin N), \u2016F \u03b4 i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016F \u03b4 i - F \u03b4 j\u2016\nf : Fin N \u2192 E\nhf : \u2200 (i : Fin N), \u2016f i\u2016 \u2264 2\nh'f : \u2200 (i j : Fin N), i \u2260 j \u2192 1 \u2264 \u2016f i - f j\u2016\nfinj : Function.Injective f\ns : Finset E := Finset.image f Finset.univ\n\u22a2 Finset.card Finset.univ = N", "state_after": "no goals"}, {"tactic": "simp only [hf, forall_apply_eq_imp_iff, forall_const, forall_exists_index, Finset.mem_univ,\n  Finset.mem_image, true_and]", "annotated_tactic": ["simp only [hf, <a>forall_apply_eq_imp_iff</a>, <a>forall_const</a>, <a>forall_exists_index</a>, <a>Finset.mem_univ</a>,\n      <a>Finset.mem_image</a>, <a>true_and</a>]", [{"full_name": "forall_apply_eq_imp_iff", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [499, 17], "def_end_pos": [499, 40]}, {"full_name": "forall_const", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [435, 17], "def_end_pos": [435, 29]}, {"full_name": "forall_exists_index", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [356, 17], "def_end_pos": [356, 36]}, {"full_name": "Finset.mem_univ", "def_path": "Mathlib/Data/Fintype/Basic.lean", "def_pos": [72, 9], "def_end_pos": [72, 17]}, {"full_name": "Finset.mem_image", "def_path": "Mathlib/Data/Finset/Image.lean", "def_pos": [330, 9], "def_end_pos": [330, 18]}, {"full_name": "true_and", "def_path": "lake-packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [84, 17], "def_end_pos": [84, 25]}]], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nh :\n  \u2200 (\u03b4 : \u211d),\n    0 < \u03b4 \u2192\n      \u03b4 < 1 \u2192\n        \u2203 s,\n          (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227\n            (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s\nN : \u2115 := multiplicity E + 1\nhN : N = multiplicity E + 1\nF : \u211d \u2192 Fin N \u2192 E\nhF : \u2200 (\u03b4 : \u211d), 0 < \u03b4 \u2192 (\u2200 (i : Fin N), \u2016F \u03b4 i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016F \u03b4 i - F \u03b4 j\u2016\nf : Fin N \u2192 E\nhf : \u2200 (i : Fin N), \u2016f i\u2016 \u2264 2\nh'f : \u2200 (i j : Fin N), i \u2260 j \u2192 1 \u2264 \u2016f i - f j\u2016\nfinj : Function.Injective f\ns : Finset E := Finset.image f Finset.univ\ns_card : Finset.card s = N\n\u22a2 \u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2", "state_after": "no goals"}, {"tactic": "simp only [forall_apply_eq_imp_iff, forall_exists_index, Finset.mem_univ, Finset.mem_image,\n  Ne.def, exists_true_left, forall_apply_eq_imp_iff, forall_true_left, true_and]", "annotated_tactic": ["simp only [<a>forall_apply_eq_imp_iff</a>, <a>forall_exists_index</a>, <a>Finset.mem_univ</a>, <a>Finset.mem_image</a>,\n      <a>Ne.def</a>, <a>exists_true_left</a>, <a>forall_apply_eq_imp_iff</a>, <a>forall_true_left</a>, <a>true_and</a>]", [{"full_name": "forall_apply_eq_imp_iff", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [499, 17], "def_end_pos": [499, 40]}, {"full_name": "forall_exists_index", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [356, 17], "def_end_pos": [356, 36]}, {"full_name": "Finset.mem_univ", "def_path": "Mathlib/Data/Fintype/Basic.lean", "def_pos": [72, 9], "def_end_pos": [72, 17]}, {"full_name": "Finset.mem_image", "def_path": "Mathlib/Data/Finset/Image.lean", "def_pos": [330, 9], "def_end_pos": [330, 18]}, {"full_name": "Ne.def", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [59, 9], "def_end_pos": [59, 15]}, {"full_name": "exists_true_left", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [912, 17], "def_end_pos": [912, 33]}, {"full_name": "forall_apply_eq_imp_iff", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [499, 17], "def_end_pos": [499, 40]}, {"full_name": "forall_true_left", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [931, 17], "def_end_pos": [931, 33]}, {"full_name": "true_and", "def_path": "lake-packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [84, 17], "def_end_pos": [84, 25]}]], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nh :\n  \u2200 (\u03b4 : \u211d),\n    0 < \u03b4 \u2192\n      \u03b4 < 1 \u2192\n        \u2203 s,\n          (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227\n            (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s\nN : \u2115 := multiplicity E + 1\nhN : N = multiplicity E + 1\nF : \u211d \u2192 Fin N \u2192 E\nhF : \u2200 (\u03b4 : \u211d), 0 < \u03b4 \u2192 (\u2200 (i : Fin N), \u2016F \u03b4 i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016F \u03b4 i - F \u03b4 j\u2016\nf : Fin N \u2192 E\nhf : \u2200 (i : Fin N), \u2016f i\u2016 \u2264 2\nh'f : \u2200 (i j : Fin N), i \u2260 j \u2192 1 \u2264 \u2016f i - f j\u2016\nfinj : Function.Injective f\ns : Finset E := Finset.image f Finset.univ\ns_card : Finset.card s = N\nhs : \u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2\n\u22a2 \u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 \u2264 \u2016c - d\u2016", "state_after": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nh :\n  \u2200 (\u03b4 : \u211d),\n    0 < \u03b4 \u2192\n      \u03b4 < 1 \u2192\n        \u2203 s,\n          (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227\n            (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s\nN : \u2115 := multiplicity E + 1\nhN : N = multiplicity E + 1\nF : \u211d \u2192 Fin N \u2192 E\nhF : \u2200 (\u03b4 : \u211d), 0 < \u03b4 \u2192 (\u2200 (i : Fin N), \u2016F \u03b4 i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016F \u03b4 i - F \u03b4 j\u2016\nf : Fin N \u2192 E\nhf : \u2200 (i : Fin N), \u2016f i\u2016 \u2264 2\nh'f : \u2200 (i j : Fin N), i \u2260 j \u2192 1 \u2264 \u2016f i - f j\u2016\nfinj : Function.Injective f\ns : Finset E := Finset.image f Finset.univ\ns_card : Finset.card s = N\nhs : \u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2\n\u22a2 \u2200 (a a_1 : Fin (multiplicity E + 1)), \u00acf a = f a_1 \u2192 1 \u2264 \u2016f a - f a_1\u2016"}, {"tactic": "intro i j hij", "annotated_tactic": ["intro i j hij", []], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nh :\n  \u2200 (\u03b4 : \u211d),\n    0 < \u03b4 \u2192\n      \u03b4 < 1 \u2192\n        \u2203 s,\n          (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227\n            (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s\nN : \u2115 := multiplicity E + 1\nhN : N = multiplicity E + 1\nF : \u211d \u2192 Fin N \u2192 E\nhF : \u2200 (\u03b4 : \u211d), 0 < \u03b4 \u2192 (\u2200 (i : Fin N), \u2016F \u03b4 i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016F \u03b4 i - F \u03b4 j\u2016\nf : Fin N \u2192 E\nhf : \u2200 (i : Fin N), \u2016f i\u2016 \u2264 2\nh'f : \u2200 (i j : Fin N), i \u2260 j \u2192 1 \u2264 \u2016f i - f j\u2016\nfinj : Function.Injective f\ns : Finset E := Finset.image f Finset.univ\ns_card : Finset.card s = N\nhs : \u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2\n\u22a2 \u2200 (a a_1 : Fin (multiplicity E + 1)), \u00acf a = f a_1 \u2192 1 \u2264 \u2016f a - f a_1\u2016", "state_after": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nh :\n  \u2200 (\u03b4 : \u211d),\n    0 < \u03b4 \u2192\n      \u03b4 < 1 \u2192\n        \u2203 s,\n          (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227\n            (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s\nN : \u2115 := multiplicity E + 1\nhN : N = multiplicity E + 1\nF : \u211d \u2192 Fin N \u2192 E\nhF : \u2200 (\u03b4 : \u211d), 0 < \u03b4 \u2192 (\u2200 (i : Fin N), \u2016F \u03b4 i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016F \u03b4 i - F \u03b4 j\u2016\nf : Fin N \u2192 E\nhf : \u2200 (i : Fin N), \u2016f i\u2016 \u2264 2\nh'f : \u2200 (i j : Fin N), i \u2260 j \u2192 1 \u2264 \u2016f i - f j\u2016\nfinj : Function.Injective f\ns : Finset E := Finset.image f Finset.univ\ns_card : Finset.card s = N\nhs : \u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2\ni j : Fin (multiplicity E + 1)\nhij : \u00acf i = f j\n\u22a2 1 \u2264 \u2016f i - f j\u2016"}, {"tactic": "have : i \u2260 j := fun h => by rw [h] at hij; exact hij rfl", "annotated_tactic": ["have : i \u2260 j := fun h => by rw [h] at hij; exact hij <a>rfl</a>", [{"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nh :\n  \u2200 (\u03b4 : \u211d),\n    0 < \u03b4 \u2192\n      \u03b4 < 1 \u2192\n        \u2203 s,\n          (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227\n            (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s\nN : \u2115 := multiplicity E + 1\nhN : N = multiplicity E + 1\nF : \u211d \u2192 Fin N \u2192 E\nhF : \u2200 (\u03b4 : \u211d), 0 < \u03b4 \u2192 (\u2200 (i : Fin N), \u2016F \u03b4 i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016F \u03b4 i - F \u03b4 j\u2016\nf : Fin N \u2192 E\nhf : \u2200 (i : Fin N), \u2016f i\u2016 \u2264 2\nh'f : \u2200 (i j : Fin N), i \u2260 j \u2192 1 \u2264 \u2016f i - f j\u2016\nfinj : Function.Injective f\ns : Finset E := Finset.image f Finset.univ\ns_card : Finset.card s = N\nhs : \u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2\ni j : Fin (multiplicity E + 1)\nhij : \u00acf i = f j\n\u22a2 1 \u2264 \u2016f i - f j\u2016", "state_after": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nh :\n  \u2200 (\u03b4 : \u211d),\n    0 < \u03b4 \u2192\n      \u03b4 < 1 \u2192\n        \u2203 s,\n          (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227\n            (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s\nN : \u2115 := multiplicity E + 1\nhN : N = multiplicity E + 1\nF : \u211d \u2192 Fin N \u2192 E\nhF : \u2200 (\u03b4 : \u211d), 0 < \u03b4 \u2192 (\u2200 (i : Fin N), \u2016F \u03b4 i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016F \u03b4 i - F \u03b4 j\u2016\nf : Fin N \u2192 E\nhf : \u2200 (i : Fin N), \u2016f i\u2016 \u2264 2\nh'f : \u2200 (i j : Fin N), i \u2260 j \u2192 1 \u2264 \u2016f i - f j\u2016\nfinj : Function.Injective f\ns : Finset E := Finset.image f Finset.univ\ns_card : Finset.card s = N\nhs : \u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2\ni j : Fin (multiplicity E + 1)\nhij : \u00acf i = f j\nthis : i \u2260 j\n\u22a2 1 \u2264 \u2016f i - f j\u2016"}, {"tactic": "exact h'f i j this", "annotated_tactic": ["exact h'f i j this", []], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nh :\n  \u2200 (\u03b4 : \u211d),\n    0 < \u03b4 \u2192\n      \u03b4 < 1 \u2192\n        \u2203 s,\n          (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227\n            (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s\nN : \u2115 := multiplicity E + 1\nhN : N = multiplicity E + 1\nF : \u211d \u2192 Fin N \u2192 E\nhF : \u2200 (\u03b4 : \u211d), 0 < \u03b4 \u2192 (\u2200 (i : Fin N), \u2016F \u03b4 i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016F \u03b4 i - F \u03b4 j\u2016\nf : Fin N \u2192 E\nhf : \u2200 (i : Fin N), \u2016f i\u2016 \u2264 2\nh'f : \u2200 (i j : Fin N), i \u2260 j \u2192 1 \u2264 \u2016f i - f j\u2016\nfinj : Function.Injective f\ns : Finset E := Finset.image f Finset.univ\ns_card : Finset.card s = N\nhs : \u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2\ni j : Fin (multiplicity E + 1)\nhij : \u00acf i = f j\nthis : i \u2260 j\n\u22a2 1 \u2264 \u2016f i - f j\u2016", "state_after": "no goals"}, {"tactic": "rw [h] at hij", "annotated_tactic": ["rw [h] at hij", []], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nh\u271d :\n  \u2200 (\u03b4 : \u211d),\n    0 < \u03b4 \u2192\n      \u03b4 < 1 \u2192\n        \u2203 s,\n          (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227\n            (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s\nN : \u2115 := multiplicity E + 1\nhN : N = multiplicity E + 1\nF : \u211d \u2192 Fin N \u2192 E\nhF : \u2200 (\u03b4 : \u211d), 0 < \u03b4 \u2192 (\u2200 (i : Fin N), \u2016F \u03b4 i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016F \u03b4 i - F \u03b4 j\u2016\nf : Fin N \u2192 E\nhf : \u2200 (i : Fin N), \u2016f i\u2016 \u2264 2\nh'f : \u2200 (i j : Fin N), i \u2260 j \u2192 1 \u2264 \u2016f i - f j\u2016\nfinj : Function.Injective f\ns : Finset E := Finset.image f Finset.univ\ns_card : Finset.card s = N\nhs : \u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2\ni j : Fin (multiplicity E + 1)\nhij : \u00acf i = f j\nh : i = j\n\u22a2 False", "state_after": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nh\u271d :\n  \u2200 (\u03b4 : \u211d),\n    0 < \u03b4 \u2192\n      \u03b4 < 1 \u2192\n        \u2203 s,\n          (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227\n            (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s\nN : \u2115 := multiplicity E + 1\nhN : N = multiplicity E + 1\nF : \u211d \u2192 Fin N \u2192 E\nhF : \u2200 (\u03b4 : \u211d), 0 < \u03b4 \u2192 (\u2200 (i : Fin N), \u2016F \u03b4 i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016F \u03b4 i - F \u03b4 j\u2016\nf : Fin N \u2192 E\nhf : \u2200 (i : Fin N), \u2016f i\u2016 \u2264 2\nh'f : \u2200 (i j : Fin N), i \u2260 j \u2192 1 \u2264 \u2016f i - f j\u2016\nfinj : Function.Injective f\ns : Finset E := Finset.image f Finset.univ\ns_card : Finset.card s = N\nhs : \u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2\ni j : Fin (multiplicity E + 1)\nhij : \u00acf j = f j\nh : i = j\n\u22a2 False"}, {"tactic": "exact hij rfl", "annotated_tactic": ["exact hij <a>rfl</a>", [{"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nh\u271d :\n  \u2200 (\u03b4 : \u211d),\n    0 < \u03b4 \u2192\n      \u03b4 < 1 \u2192\n        \u2203 s,\n          (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227\n            (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s\nN : \u2115 := multiplicity E + 1\nhN : N = multiplicity E + 1\nF : \u211d \u2192 Fin N \u2192 E\nhF : \u2200 (\u03b4 : \u211d), 0 < \u03b4 \u2192 (\u2200 (i : Fin N), \u2016F \u03b4 i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016F \u03b4 i - F \u03b4 j\u2016\nf : Fin N \u2192 E\nhf : \u2200 (i : Fin N), \u2016f i\u2016 \u2264 2\nh'f : \u2200 (i j : Fin N), i \u2260 j \u2192 1 \u2264 \u2016f i - f j\u2016\nfinj : Function.Injective f\ns : Finset E := Finset.image f Finset.univ\ns_card : Finset.card s = N\nhs : \u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2\ni j : Fin (multiplicity E + 1)\nhij : \u00acf j = f j\nh : i = j\n\u22a2 False", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "full_name": "Substring.Valid.validFor", "start": [973, 1], "end": [983, 73], "traced_tactics": [{"tactic": "simp at *", "annotated_tactic": ["simp at *", []], "state_before": "l mr lm r : List Char\ne :\n  lm ++ r =\n    { str := { data := l ++ mr }, startPos := { byteIdx := utf8ByteSize.go l },\n          stopPos := { byteIdx := utf8ByteSize.go lm } }.str.data\nh :\n  { str := { data := l ++ mr }, startPos := { byteIdx := utf8ByteSize.go l },\n        stopPos := { byteIdx := utf8ByteSize.go lm } }.startPos \u2264\n    { str := { data := l ++ mr }, startPos := { byteIdx := utf8ByteSize.go l },\n        stopPos := { byteIdx := utf8ByteSize.go lm } }.stopPos\n\u22a2 \u2203 l_1 m r,\n    ValidFor l_1 m r\n      { str := { data := l ++ mr }, startPos := { byteIdx := utf8ByteSize.go l },\n        stopPos := { byteIdx := utf8ByteSize.go lm } }", "state_after": "l mr lm r : List Char\ne : lm ++ r = l ++ mr\nh : utf8Len l \u2264 utf8Len lm\n\u22a2 \u2203 l_1 m r,\n    ValidFor l_1 m r\n      { str := { data := l ++ mr }, startPos := { byteIdx := utf8Len l }, stopPos := { byteIdx := utf8Len lm } }"}, {"tactic": "have := (or_iff_right_iff_imp.2 fun h => ?x).1 (List.append_eq_append_iff.1 e)", "annotated_tactic": ["have := (<a>or_iff_right_iff_imp</a>.2 fun h => ?x).1 (<a>List.append_eq_append_iff</a>.1 e)", [{"full_name": "or_iff_right_iff_imp", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [298, 17], "def_end_pos": [298, 37]}, {"full_name": "List.append_eq_append_iff", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [131, 9], "def_end_pos": [131, 29]}]], "state_before": "l mr lm r : List Char\ne : lm ++ r = l ++ mr\nh : utf8Len l \u2264 utf8Len lm\n\u22a2 \u2203 l_1 m r,\n    ValidFor l_1 m r\n      { str := { data := l ++ mr }, startPos := { byteIdx := utf8Len l }, stopPos := { byteIdx := utf8Len lm } }", "state_after": "l mr lm r : List Char\ne : lm ++ r = l ++ mr\nh : utf8Len l \u2264 utf8Len lm\nthis : \u2203 c', lm = l ++ c' \u2227 mr = c' ++ r\n\u22a2 \u2203 l_1 m r,\n    ValidFor l_1 m r\n      { str := { data := l ++ mr }, startPos := { byteIdx := utf8Len l }, stopPos := { byteIdx := utf8Len lm } }\n\ncase x\nl mr lm r : List Char\ne : lm ++ r = l ++ mr\nh\u271d : utf8Len l \u2264 utf8Len lm\nh : \u2203 a', l = lm ++ a' \u2227 r = a' ++ mr\n\u22a2 \u2203 c', lm = l ++ c' \u2227 mr = c' ++ r"}, {"tactic": "case x =>\n  match l, r, h with | _, _, \u27e8m, rfl, rfl\u27e9 => ?_\n  simp at h\n  cases utf8Len_eq_zero.1 <| Nat.le_zero.1 (Nat.le_of_add_le_add_left (c := 0) h)\n  exact \u27e8[], by simp\u27e9", "annotated_tactic": ["case x =>\n      match l, r, h with | _, _, \u27e8m, <a>rfl</a>, <a>rfl</a>\u27e9 => ?_\n      simp at h\n      cases <a>utf8Len_eq_zero</a>.1 <| <a>Nat.le_zero</a>.1 (<a>Nat.le_of_add_le_add_left</a> (c := 0) h)\n      exact \u27e8[], by simp\u27e9", [{"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}, {"full_name": "String.utf8Len_eq_zero", "def_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "def_pos": [73, 17], "def_end_pos": [73, 32]}, {"full_name": "Nat.le_zero", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [206, 9], "def_end_pos": [206, 16]}, {"full_name": "Nat.le_of_add_le_add_left", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [410, 19], "def_end_pos": [410, 40]}]], "state_before": "l mr lm r : List Char\ne : lm ++ r = l ++ mr\nh : utf8Len l \u2264 utf8Len lm\nthis : \u2203 c', lm = l ++ c' \u2227 mr = c' ++ r\n\u22a2 \u2203 l_1 m r,\n    ValidFor l_1 m r\n      { str := { data := l ++ mr }, startPos := { byteIdx := utf8Len l }, stopPos := { byteIdx := utf8Len lm } }\n\ncase x\nl mr lm r : List Char\ne : lm ++ r = l ++ mr\nh\u271d : utf8Len l \u2264 utf8Len lm\nh : \u2203 a', l = lm ++ a' \u2227 r = a' ++ mr\n\u22a2 \u2203 c', lm = l ++ c' \u2227 mr = c' ++ r", "state_after": "l mr lm r : List Char\ne : lm ++ r = l ++ mr\nh : utf8Len l \u2264 utf8Len lm\nthis : \u2203 c', lm = l ++ c' \u2227 mr = c' ++ r\n\u22a2 \u2203 l_1 m r,\n    ValidFor l_1 m r\n      { str := { data := l ++ mr }, startPos := { byteIdx := utf8Len l }, stopPos := { byteIdx := utf8Len lm } }"}, {"tactic": "match lm, mr, this with\n| _, _, \u27e8m, rfl, rfl\u27e9 => exact \u27e8l, m, r, by simpa using ValidFor.mk\u27e9", "annotated_tactic": ["match lm, mr, this with\n    | _, _, \u27e8m, <a>rfl</a>, <a>rfl</a>\u27e9 => exact \u27e8l, m, r, by simpa using <a>ValidFor.mk</a>\u27e9", [{"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}, {"full_name": "Substring.ValidFor.mk", "def_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "def_pos": [802, 5], "def_end_pos": [802, 7]}]], "state_before": "l mr lm r : List Char\ne : lm ++ r = l ++ mr\nh : utf8Len l \u2264 utf8Len lm\nthis : \u2203 c', lm = l ++ c' \u2227 mr = c' ++ r\n\u22a2 \u2203 l_1 m r,\n    ValidFor l_1 m r\n      { str := { data := l ++ mr }, startPos := { byteIdx := utf8Len l }, stopPos := { byteIdx := utf8Len lm } }", "state_after": "no goals"}, {"tactic": "match l, r, h with | _, _, \u27e8m, rfl, rfl\u27e9 => ?_", "annotated_tactic": ["match l, r, h with | _, _, \u27e8m, <a>rfl</a>, <a>rfl</a>\u27e9 => ?_", [{"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "l mr lm r : List Char\ne : lm ++ r = l ++ mr\nh\u271d : utf8Len l \u2264 utf8Len lm\nh : \u2203 a', l = lm ++ a' \u2227 r = a' ++ mr\n\u22a2 \u2203 c', lm = l ++ c' \u2227 mr = c' ++ r", "state_after": "l mr lm r : List Char\nh\u271d : \u2203 a', l = lm ++ a' \u2227 r = a' ++ mr\nm : List Char\ne : lm ++ (m ++ mr) = lm ++ m ++ mr\nh : utf8Len (lm ++ m) \u2264 utf8Len lm\n\u22a2 \u2203 c', lm = lm ++ m ++ c' \u2227 mr = c' ++ (m ++ mr)"}, {"tactic": "simp at h", "annotated_tactic": ["simp at h", []], "state_before": "l mr lm r : List Char\nh\u271d : \u2203 a', l = lm ++ a' \u2227 r = a' ++ mr\nm : List Char\ne : lm ++ (m ++ mr) = lm ++ m ++ mr\nh : utf8Len (lm ++ m) \u2264 utf8Len lm\n\u22a2 \u2203 c', lm = lm ++ m ++ c' \u2227 mr = c' ++ (m ++ mr)", "state_after": "l mr lm r : List Char\nh\u271d : \u2203 a', l = lm ++ a' \u2227 r = a' ++ mr\nm : List Char\ne : lm ++ (m ++ mr) = lm ++ m ++ mr\nh : utf8Len lm + utf8Len m \u2264 utf8Len lm\n\u22a2 \u2203 c', lm = lm ++ m ++ c' \u2227 mr = c' ++ (m ++ mr)"}, {"tactic": "cases utf8Len_eq_zero.1 <| Nat.le_zero.1 (Nat.le_of_add_le_add_left (c := 0) h)", "annotated_tactic": ["cases <a>utf8Len_eq_zero</a>.1 <| <a>Nat.le_zero</a>.1 (<a>Nat.le_of_add_le_add_left</a> (c := 0) h)", [{"full_name": "String.utf8Len_eq_zero", "def_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "def_pos": [73, 17], "def_end_pos": [73, 32]}, {"full_name": "Nat.le_zero", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [206, 9], "def_end_pos": [206, 16]}, {"full_name": "Nat.le_of_add_le_add_left", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [410, 19], "def_end_pos": [410, 40]}]], "state_before": "l mr lm r : List Char\nh\u271d : \u2203 a', l = lm ++ a' \u2227 r = a' ++ mr\nm : List Char\ne : lm ++ (m ++ mr) = lm ++ m ++ mr\nh : utf8Len lm + utf8Len m \u2264 utf8Len lm\n\u22a2 \u2203 c', lm = lm ++ m ++ c' \u2227 mr = c' ++ (m ++ mr)", "state_after": "case refl\nl mr lm r : List Char\nh\u271d : \u2203 a', l = lm ++ a' \u2227 r = a' ++ mr\ne : lm ++ ([] ++ mr) = lm ++ [] ++ mr\nh : utf8Len lm + utf8Len [] \u2264 utf8Len lm\n\u22a2 \u2203 c', lm = lm ++ [] ++ c' \u2227 mr = c' ++ ([] ++ mr)"}, {"tactic": "exact \u27e8[], by simp\u27e9", "annotated_tactic": ["exact \u27e8[], by simp\u27e9", []], "state_before": "case refl\nl mr lm r : List Char\nh\u271d : \u2203 a', l = lm ++ a' \u2227 r = a' ++ mr\ne : lm ++ ([] ++ mr) = lm ++ [] ++ mr\nh : utf8Len lm + utf8Len [] \u2264 utf8Len lm\n\u22a2 \u2203 c', lm = lm ++ [] ++ c' \u2227 mr = c' ++ ([] ++ mr)", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "l mr lm r : List Char\nh\u271d : \u2203 a', l = lm ++ a' \u2227 r = a' ++ mr\ne : lm ++ ([] ++ mr) = lm ++ [] ++ mr\nh : utf8Len lm + utf8Len [] \u2264 utf8Len lm\n\u22a2 lm = lm ++ [] ++ [] \u2227 mr = [] ++ ([] ++ mr)", "state_after": "no goals"}, {"tactic": "exact \u27e8l, m, r, by simpa using ValidFor.mk\u27e9", "annotated_tactic": ["exact \u27e8l, m, r, by simpa using <a>ValidFor.mk</a>\u27e9", [{"full_name": "Substring.ValidFor.mk", "def_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "def_pos": [802, 5], "def_end_pos": [802, 7]}]], "state_before": "l mr lm r : List Char\nthis : \u2203 c', lm = l ++ c' \u2227 mr = c' ++ r\nm : List Char\ne : l ++ m ++ r = l ++ (m ++ r)\nh : utf8Len l \u2264 utf8Len (l ++ m)\n\u22a2 \u2203 l_1 m_1 r_1,\n    ValidFor l_1 m_1 r_1\n      { str := { data := l ++ (m ++ r) }, startPos := { byteIdx := utf8Len l },\n        stopPos := { byteIdx := utf8Len (l ++ m) } }", "state_after": "no goals"}, {"tactic": "simpa using ValidFor.mk", "annotated_tactic": ["simpa using <a>ValidFor.mk</a>", [{"full_name": "Substring.ValidFor.mk", "def_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "def_pos": [802, 5], "def_end_pos": [802, 7]}]], "state_before": "l mr lm r : List Char\nthis : \u2203 c', lm = l ++ c' \u2227 mr = c' ++ r\nm : List Char\ne : l ++ m ++ r = l ++ (m ++ r)\nh : utf8Len l \u2264 utf8Len (l ++ m)\n\u22a2 ValidFor l m r\n    { str := { data := l ++ (m ++ r) }, startPos := { byteIdx := utf8Len l },\n      stopPos := { byteIdx := utf8Len (l ++ m) } }", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/AEMeasurable.lean", "full_name": "aemeasurable_iUnion_iff", "start": [140, 1], "end": [142, 95], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "full_name": "MeasureTheory.Lp.cauchySeq_Lp_iff_cauchySeq_\u2112p", "start": [1409, 1], "end": [1415, 23], "traced_tactics": [{"tactic": "simp_rw [cauchySeq_iff_tendsto_dist_atTop_0, dist_def]", "annotated_tactic": ["simp_rw [<a>cauchySeq_iff_tendsto_dist_atTop_0</a>, <a>dist_def</a>]", [{"full_name": "cauchySeq_iff_tendsto_dist_atTop_0", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [1472, 9], "def_end_pos": [1472, 43]}, {"full_name": "MeasureTheory.Lp.dist_def", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [290, 9], "def_end_pos": [290, 17]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedAddCommGroup G\n\u03b9 : Type u_5\ninst\u271d\u00b9 : Nonempty \u03b9\ninst\u271d : SemilatticeSup \u03b9\nhp : Fact (1 \u2264 p)\nf : \u03b9 \u2192 { x // x \u2208 Lp E p }\n\u22a2 CauchySeq f \u2194 Tendsto (fun n => snorm (\u2191\u2191(f n.1) - \u2191\u2191(f n.2)) p \u03bc) atTop (\ud835\udcdd 0)", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedAddCommGroup G\n\u03b9 : Type u_5\ninst\u271d\u00b9 : Nonempty \u03b9\ninst\u271d : SemilatticeSup \u03b9\nhp : Fact (1 \u2264 p)\nf : \u03b9 \u2192 { x // x \u2208 Lp E p }\n\u22a2 Tendsto (fun n => ENNReal.toReal (snorm (\u2191\u2191(f n.1) - \u2191\u2191(f n.2)) p \u03bc)) atTop (\ud835\udcdd 0) \u2194\n    Tendsto (fun n => snorm (\u2191\u2191(f n.1) - \u2191\u2191(f n.2)) p \u03bc) atTop (\ud835\udcdd 0)"}, {"tactic": "rw [\u2190 ENNReal.zero_toReal, ENNReal.tendsto_toReal_iff (fun n => ?_) ENNReal.zero_ne_top]", "annotated_tactic": ["rw [\u2190 <a>ENNReal.zero_toReal</a>, <a>ENNReal.tendsto_toReal_iff</a> (fun n => ?_) <a>ENNReal.zero_ne_top</a>]", [{"full_name": "ENNReal.zero_toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [242, 17], "def_end_pos": [242, 28]}, {"full_name": "ENNReal.tendsto_toReal_iff", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [1062, 9], "def_end_pos": [1062, 27]}, {"full_name": "ENNReal.zero_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [334, 17], "def_end_pos": [334, 28]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedAddCommGroup G\n\u03b9 : Type u_5\ninst\u271d\u00b9 : Nonempty \u03b9\ninst\u271d : SemilatticeSup \u03b9\nhp : Fact (1 \u2264 p)\nf : \u03b9 \u2192 { x // x \u2208 Lp E p }\n\u22a2 Tendsto (fun n => ENNReal.toReal (snorm (\u2191\u2191(f n.1) - \u2191\u2191(f n.2)) p \u03bc)) atTop (\ud835\udcdd 0) \u2194\n    Tendsto (fun n => snorm (\u2191\u2191(f n.1) - \u2191\u2191(f n.2)) p \u03bc) atTop (\ud835\udcdd 0)", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedAddCommGroup G\n\u03b9 : Type u_5\ninst\u271d\u00b9 : Nonempty \u03b9\ninst\u271d : SemilatticeSup \u03b9\nhp : Fact (1 \u2264 p)\nf : \u03b9 \u2192 { x // x \u2208 Lp E p }\nn : \u03b9 \u00d7 \u03b9\n\u22a2 snorm (\u2191\u2191(f n.1) - \u2191\u2191(f n.2)) p \u03bc \u2260 \u22a4"}, {"tactic": "rw [snorm_congr_ae (Lp.coeFn_sub _ _).symm]", "annotated_tactic": ["rw [<a>snorm_congr_ae</a> (<a>Lp.coeFn_sub</a> _ _).<a>symm</a>]", [{"full_name": "MeasureTheory.snorm_congr_ae", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [549, 9], "def_end_pos": [549, 23]}, {"full_name": "MeasureTheory.Lp.coeFn_sub", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [236, 9], "def_end_pos": [236, 18]}, {"full_name": "Filter.EventuallyEq.symm", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1498, 9], "def_end_pos": [1498, 26]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedAddCommGroup G\n\u03b9 : Type u_5\ninst\u271d\u00b9 : Nonempty \u03b9\ninst\u271d : SemilatticeSup \u03b9\nhp : Fact (1 \u2264 p)\nf : \u03b9 \u2192 { x // x \u2208 Lp E p }\nn : \u03b9 \u00d7 \u03b9\n\u22a2 snorm (\u2191\u2191(f n.1) - \u2191\u2191(f n.2)) p \u03bc \u2260 \u22a4", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedAddCommGroup G\n\u03b9 : Type u_5\ninst\u271d\u00b9 : Nonempty \u03b9\ninst\u271d : SemilatticeSup \u03b9\nhp : Fact (1 \u2264 p)\nf : \u03b9 \u2192 { x // x \u2208 Lp E p }\nn : \u03b9 \u00d7 \u03b9\n\u22a2 snorm (\u2191\u2191(f n.1 - f n.2)) p \u03bc \u2260 \u22a4"}, {"tactic": "exact snorm_ne_top _", "annotated_tactic": ["exact <a>snorm_ne_top</a> _", [{"full_name": "MeasureTheory.Lp.snorm_ne_top", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [202, 9], "def_end_pos": [202, 21]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedAddCommGroup G\n\u03b9 : Type u_5\ninst\u271d\u00b9 : Nonempty \u03b9\ninst\u271d : SemilatticeSup \u03b9\nhp : Fact (1 \u2264 p)\nf : \u03b9 \u2192 { x // x \u2208 Lp E p }\nn : \u03b9 \u00d7 \u03b9\n\u22a2 snorm (\u2191\u2191(f n.1 - f n.2)) p \u03bc \u2260 \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Regular.lean", "full_name": "MeasureTheory.Measure.InnerRegular.measurableSet_of_open", "start": [348, 1], "end": [367, 48], "traced_tactics": [{"tactic": "rintro s \u27e8hs, h\u03bcs\u27e9 r hr", "annotated_tactic": ["rintro s \u27e8hs, h\u03bcs\u27e9 r hr", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03bc : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU s : Set \u03b1\n\u03b5 r : \u211d\u22650\u221e\ninst\u271d : OuterRegular \u03bc\nH : InnerRegular \u03bc p IsOpen\nh0 : p \u2205\nhd : \u2200 \u2983s U : Set \u03b1\u2984, p s \u2192 IsOpen U \u2192 p (s \\ U)\n\u22a2 InnerRegular \u03bc p fun s => MeasurableSet s \u2227 \u2191\u2191\u03bc s \u2260 \u22a4", "state_after": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03bc : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU s\u271d : Set \u03b1\n\u03b5 r\u271d : \u211d\u22650\u221e\ninst\u271d : OuterRegular \u03bc\nH : InnerRegular \u03bc p IsOpen\nh0 : p \u2205\nhd : \u2200 \u2983s U : Set \u03b1\u2984, p s \u2192 IsOpen U \u2192 p (s \\ U)\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nr : \u211d\u22650\u221e\nhr : r < \u2191\u2191\u03bc s\n\u22a2 \u2203 K, K \u2286 s \u2227 p K \u2227 r < \u2191\u2191\u03bc K"}, {"tactic": "obtain \u27e8\u03b5, h\u03b5, h\u03b5s, rfl\u27e9 : \u2203 (\u03b5 : _) (_ : \u03b5 \u2260 0), \u03b5 + \u03b5 \u2264 \u03bc s \u2227 r = \u03bc s - (\u03b5 + \u03b5) := by\n  use (\u03bc s - r) / 2\n  simp [*, hr.le, ENNReal.add_halves, ENNReal.sub_sub_cancel, le_add_right]", "annotated_tactic": ["obtain \u27e8\u03b5, h\u03b5, h\u03b5s, rfl\u27e9 : \u2203 (\u03b5 : _) (_ : \u03b5 \u2260 0), \u03b5 + \u03b5 \u2264 \u03bc s \u2227 r = \u03bc s - (\u03b5 + \u03b5) := by\n    use (\u03bc s - r) / 2\n    simp [*, hr.le, <a>ENNReal.add_halves</a>, <a>ENNReal.sub_sub_cancel</a>, <a>le_add_right</a>]", [{"full_name": "ENNReal.add_halves", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1781, 19], "def_end_pos": [1781, 29]}, {"full_name": "ENNReal.sub_sub_cancel", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1225, 9], "def_end_pos": [1225, 23]}, {"full_name": "le_add_right", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [278, 3], "def_end_pos": [278, 14]}]], "state_before": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03bc : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU s\u271d : Set \u03b1\n\u03b5 r\u271d : \u211d\u22650\u221e\ninst\u271d : OuterRegular \u03bc\nH : InnerRegular \u03bc p IsOpen\nh0 : p \u2205\nhd : \u2200 \u2983s U : Set \u03b1\u2984, p s \u2192 IsOpen U \u2192 p (s \\ U)\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nr : \u211d\u22650\u221e\nhr : r < \u2191\u2191\u03bc s\n\u22a2 \u2203 K, K \u2286 s \u2227 p K \u2227 r < \u2191\u2191\u03bc K", "state_after": "case intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03bc : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU s\u271d : Set \u03b1\n\u03b5\u271d r : \u211d\u22650\u221e\ninst\u271d : OuterRegular \u03bc\nH : InnerRegular \u03bc p IsOpen\nh0 : p \u2205\nhd : \u2200 \u2983s U : Set \u03b1\u2984, p s \u2192 IsOpen U \u2192 p (s \\ U)\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nh\u03b5s : \u03b5 + \u03b5 \u2264 \u2191\u2191\u03bc s\nhr : \u2191\u2191\u03bc s - (\u03b5 + \u03b5) < \u2191\u2191\u03bc s\n\u22a2 \u2203 K, K \u2286 s \u2227 p K \u2227 \u2191\u2191\u03bc s - (\u03b5 + \u03b5) < \u2191\u2191\u03bc K"}, {"tactic": "rcases hs.exists_isOpen_diff_lt h\u03bcs h\u03b5 with \u27e8U, hsU, hUo, hUt, h\u03bcU\u27e9", "annotated_tactic": ["rcases hs.exists_isOpen_diff_lt h\u03bcs h\u03b5 with \u27e8U, hsU, hUo, hUt, h\u03bcU\u27e9", []], "state_before": "case intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03bc : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU s\u271d : Set \u03b1\n\u03b5\u271d r : \u211d\u22650\u221e\ninst\u271d : OuterRegular \u03bc\nH : InnerRegular \u03bc p IsOpen\nh0 : p \u2205\nhd : \u2200 \u2983s U : Set \u03b1\u2984, p s \u2192 IsOpen U \u2192 p (s \\ U)\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nh\u03b5s : \u03b5 + \u03b5 \u2264 \u2191\u2191\u03bc s\nhr : \u2191\u2191\u03bc s - (\u03b5 + \u03b5) < \u2191\u2191\u03bc s\n\u22a2 \u2203 K, K \u2286 s \u2227 p K \u2227 \u2191\u2191\u03bc s - (\u03b5 + \u03b5) < \u2191\u2191\u03bc K", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03bc : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU\u271d s\u271d : Set \u03b1\n\u03b5\u271d r : \u211d\u22650\u221e\ninst\u271d : OuterRegular \u03bc\nH : InnerRegular \u03bc p IsOpen\nh0 : p \u2205\nhd : \u2200 \u2983s U : Set \u03b1\u2984, p s \u2192 IsOpen U \u2192 p (s \\ U)\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nh\u03b5s : \u03b5 + \u03b5 \u2264 \u2191\u2191\u03bc s\nhr : \u2191\u2191\u03bc s - (\u03b5 + \u03b5) < \u2191\u2191\u03bc s\nU : Set \u03b1\nhsU : U \u2287 s\nhUo : IsOpen U\nhUt : \u2191\u2191\u03bc U < \u22a4\nh\u03bcU : \u2191\u2191\u03bc (U \\ s) < \u03b5\n\u22a2 \u2203 K, K \u2286 s \u2227 p K \u2227 \u2191\u2191\u03bc s - (\u03b5 + \u03b5) < \u2191\u2191\u03bc K"}, {"tactic": "rcases (U \\ s).exists_isOpen_lt_of_lt _ h\u03bcU with \u27e8U', hsU', hU'o, h\u03bcU'\u27e9", "annotated_tactic": ["rcases (U \\ s).<a>exists_isOpen_lt_of_lt</a> _ h\u03bcU with \u27e8U', hsU', hU'o, h\u03bcU'\u27e9", [{"full_name": "Set.exists_isOpen_lt_of_lt", "def_path": "Mathlib/MeasureTheory/Measure/Regular.lean", "def_pos": [242, 9], "def_end_pos": [242, 42]}]], "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03bc : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU\u271d s\u271d : Set \u03b1\n\u03b5\u271d r : \u211d\u22650\u221e\ninst\u271d : OuterRegular \u03bc\nH : InnerRegular \u03bc p IsOpen\nh0 : p \u2205\nhd : \u2200 \u2983s U : Set \u03b1\u2984, p s \u2192 IsOpen U \u2192 p (s \\ U)\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nh\u03b5s : \u03b5 + \u03b5 \u2264 \u2191\u2191\u03bc s\nhr : \u2191\u2191\u03bc s - (\u03b5 + \u03b5) < \u2191\u2191\u03bc s\nU : Set \u03b1\nhsU : U \u2287 s\nhUo : IsOpen U\nhUt : \u2191\u2191\u03bc U < \u22a4\nh\u03bcU : \u2191\u2191\u03bc (U \\ s) < \u03b5\n\u22a2 \u2203 K, K \u2286 s \u2227 p K \u2227 \u2191\u2191\u03bc s - (\u03b5 + \u03b5) < \u2191\u2191\u03bc K", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03bc : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU\u271d s\u271d : Set \u03b1\n\u03b5\u271d r : \u211d\u22650\u221e\ninst\u271d : OuterRegular \u03bc\nH : InnerRegular \u03bc p IsOpen\nh0 : p \u2205\nhd : \u2200 \u2983s U : Set \u03b1\u2984, p s \u2192 IsOpen U \u2192 p (s \\ U)\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nh\u03b5s : \u03b5 + \u03b5 \u2264 \u2191\u2191\u03bc s\nhr : \u2191\u2191\u03bc s - (\u03b5 + \u03b5) < \u2191\u2191\u03bc s\nU : Set \u03b1\nhsU : U \u2287 s\nhUo : IsOpen U\nhUt : \u2191\u2191\u03bc U < \u22a4\nh\u03bcU : \u2191\u2191\u03bc (U \\ s) < \u03b5\nU' : Set \u03b1\nhsU' : U' \u2287 U \\ s\nhU'o : IsOpen U'\nh\u03bcU' : \u2191\u2191\u03bc U' < \u03b5\n\u22a2 \u2203 K, K \u2286 s \u2227 p K \u2227 \u2191\u2191\u03bc s - (\u03b5 + \u03b5) < \u2191\u2191\u03bc K"}, {"tactic": "replace hsU' := diff_subset_comm.1 hsU'", "annotated_tactic": ["replace hsU' := <a>diff_subset_comm</a>.1 hsU'", [{"full_name": "Set.diff_subset_comm", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1974, 9], "def_end_pos": [1974, 25]}]], "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03bc : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU\u271d s\u271d : Set \u03b1\n\u03b5\u271d r : \u211d\u22650\u221e\ninst\u271d : OuterRegular \u03bc\nH : InnerRegular \u03bc p IsOpen\nh0 : p \u2205\nhd : \u2200 \u2983s U : Set \u03b1\u2984, p s \u2192 IsOpen U \u2192 p (s \\ U)\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nh\u03b5s : \u03b5 + \u03b5 \u2264 \u2191\u2191\u03bc s\nhr : \u2191\u2191\u03bc s - (\u03b5 + \u03b5) < \u2191\u2191\u03bc s\nU : Set \u03b1\nhsU : U \u2287 s\nhUo : IsOpen U\nhUt : \u2191\u2191\u03bc U < \u22a4\nh\u03bcU : \u2191\u2191\u03bc (U \\ s) < \u03b5\nU' : Set \u03b1\nhsU' : U' \u2287 U \\ s\nhU'o : IsOpen U'\nh\u03bcU' : \u2191\u2191\u03bc U' < \u03b5\n\u22a2 \u2203 K, K \u2286 s \u2227 p K \u2227 \u2191\u2191\u03bc s - (\u03b5 + \u03b5) < \u2191\u2191\u03bc K", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03bc : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU\u271d s\u271d : Set \u03b1\n\u03b5\u271d r : \u211d\u22650\u221e\ninst\u271d : OuterRegular \u03bc\nH : InnerRegular \u03bc p IsOpen\nh0 : p \u2205\nhd : \u2200 \u2983s U : Set \u03b1\u2984, p s \u2192 IsOpen U \u2192 p (s \\ U)\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nh\u03b5s : \u03b5 + \u03b5 \u2264 \u2191\u2191\u03bc s\nhr : \u2191\u2191\u03bc s - (\u03b5 + \u03b5) < \u2191\u2191\u03bc s\nU : Set \u03b1\nhsU : U \u2287 s\nhUo : IsOpen U\nhUt : \u2191\u2191\u03bc U < \u22a4\nh\u03bcU : \u2191\u2191\u03bc (U \\ s) < \u03b5\nU' : Set \u03b1\nhU'o : IsOpen U'\nh\u03bcU' : \u2191\u2191\u03bc U' < \u03b5\nhsU' : U \\ U' \u2286 s\n\u22a2 \u2203 K, K \u2286 s \u2227 p K \u2227 \u2191\u2191\u03bc s - (\u03b5 + \u03b5) < \u2191\u2191\u03bc K"}, {"tactic": "rcases H.exists_subset_lt_add h0 hUo hUt.ne h\u03b5 with \u27e8K, hKU, hKc, hKr\u27e9", "annotated_tactic": ["rcases H.exists_subset_lt_add h0 hUo hUt.ne h\u03b5 with \u27e8K, hKU, hKc, hKr\u27e9", []], "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03bc : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU\u271d s\u271d : Set \u03b1\n\u03b5\u271d r : \u211d\u22650\u221e\ninst\u271d : OuterRegular \u03bc\nH : InnerRegular \u03bc p IsOpen\nh0 : p \u2205\nhd : \u2200 \u2983s U : Set \u03b1\u2984, p s \u2192 IsOpen U \u2192 p (s \\ U)\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nh\u03b5s : \u03b5 + \u03b5 \u2264 \u2191\u2191\u03bc s\nhr : \u2191\u2191\u03bc s - (\u03b5 + \u03b5) < \u2191\u2191\u03bc s\nU : Set \u03b1\nhsU : U \u2287 s\nhUo : IsOpen U\nhUt : \u2191\u2191\u03bc U < \u22a4\nh\u03bcU : \u2191\u2191\u03bc (U \\ s) < \u03b5\nU' : Set \u03b1\nhU'o : IsOpen U'\nh\u03bcU' : \u2191\u2191\u03bc U' < \u03b5\nhsU' : U \\ U' \u2286 s\n\u22a2 \u2203 K, K \u2286 s \u2227 p K \u2227 \u2191\u2191\u03bc s - (\u03b5 + \u03b5) < \u2191\u2191\u03bc K", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03bc : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU\u271d s\u271d : Set \u03b1\n\u03b5\u271d r : \u211d\u22650\u221e\ninst\u271d : OuterRegular \u03bc\nH : InnerRegular \u03bc p IsOpen\nh0 : p \u2205\nhd : \u2200 \u2983s U : Set \u03b1\u2984, p s \u2192 IsOpen U \u2192 p (s \\ U)\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nh\u03b5s : \u03b5 + \u03b5 \u2264 \u2191\u2191\u03bc s\nhr : \u2191\u2191\u03bc s - (\u03b5 + \u03b5) < \u2191\u2191\u03bc s\nU : Set \u03b1\nhsU : U \u2287 s\nhUo : IsOpen U\nhUt : \u2191\u2191\u03bc U < \u22a4\nh\u03bcU : \u2191\u2191\u03bc (U \\ s) < \u03b5\nU' : Set \u03b1\nhU'o : IsOpen U'\nh\u03bcU' : \u2191\u2191\u03bc U' < \u03b5\nhsU' : U \\ U' \u2286 s\nK : Set \u03b1\nhKU : K \u2286 U\nhKc : p K\nhKr : \u2191\u2191\u03bc U < \u2191\u2191\u03bc K + \u03b5\n\u22a2 \u2203 K, K \u2286 s \u2227 p K \u2227 \u2191\u2191\u03bc s - (\u03b5 + \u03b5) < \u2191\u2191\u03bc K"}, {"tactic": "refine' \u27e8K \\ U', fun x hx => hsU' \u27e8hKU hx.1, hx.2\u27e9, hd hKc hU'o, ENNReal.sub_lt_of_lt_add h\u03b5s _\u27e9", "annotated_tactic": ["refine' \u27e8K \\ U', fun x hx => hsU' \u27e8hKU hx.1, hx.2\u27e9, hd hKc hU'o, <a>ENNReal.sub_lt_of_lt_add</a> h\u03b5s _\u27e9", [{"full_name": "ENNReal.sub_lt_of_lt_add", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1205, 19], "def_end_pos": [1205, 35]}]], "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03bc : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU\u271d s\u271d : Set \u03b1\n\u03b5\u271d r : \u211d\u22650\u221e\ninst\u271d : OuterRegular \u03bc\nH : InnerRegular \u03bc p IsOpen\nh0 : p \u2205\nhd : \u2200 \u2983s U : Set \u03b1\u2984, p s \u2192 IsOpen U \u2192 p (s \\ U)\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nh\u03b5s : \u03b5 + \u03b5 \u2264 \u2191\u2191\u03bc s\nhr : \u2191\u2191\u03bc s - (\u03b5 + \u03b5) < \u2191\u2191\u03bc s\nU : Set \u03b1\nhsU : U \u2287 s\nhUo : IsOpen U\nhUt : \u2191\u2191\u03bc U < \u22a4\nh\u03bcU : \u2191\u2191\u03bc (U \\ s) < \u03b5\nU' : Set \u03b1\nhU'o : IsOpen U'\nh\u03bcU' : \u2191\u2191\u03bc U' < \u03b5\nhsU' : U \\ U' \u2286 s\nK : Set \u03b1\nhKU : K \u2286 U\nhKc : p K\nhKr : \u2191\u2191\u03bc U < \u2191\u2191\u03bc K + \u03b5\n\u22a2 \u2203 K, K \u2286 s \u2227 p K \u2227 \u2191\u2191\u03bc s - (\u03b5 + \u03b5) < \u2191\u2191\u03bc K", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03bc : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU\u271d s\u271d : Set \u03b1\n\u03b5\u271d r : \u211d\u22650\u221e\ninst\u271d : OuterRegular \u03bc\nH : InnerRegular \u03bc p IsOpen\nh0 : p \u2205\nhd : \u2200 \u2983s U : Set \u03b1\u2984, p s \u2192 IsOpen U \u2192 p (s \\ U)\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nh\u03b5s : \u03b5 + \u03b5 \u2264 \u2191\u2191\u03bc s\nhr : \u2191\u2191\u03bc s - (\u03b5 + \u03b5) < \u2191\u2191\u03bc s\nU : Set \u03b1\nhsU : U \u2287 s\nhUo : IsOpen U\nhUt : \u2191\u2191\u03bc U < \u22a4\nh\u03bcU : \u2191\u2191\u03bc (U \\ s) < \u03b5\nU' : Set \u03b1\nhU'o : IsOpen U'\nh\u03bcU' : \u2191\u2191\u03bc U' < \u03b5\nhsU' : U \\ U' \u2286 s\nK : Set \u03b1\nhKU : K \u2286 U\nhKc : p K\nhKr : \u2191\u2191\u03bc U < \u2191\u2191\u03bc K + \u03b5\n\u22a2 \u2191\u2191\u03bc s < \u2191\u2191\u03bc (K \\ U') + (\u03b5 + \u03b5)"}, {"tactic": "calc\n  \u03bc s \u2264 \u03bc U := \u03bc.mono hsU\n  _ < \u03bc K + \u03b5 := hKr\n  _ \u2264 \u03bc (K \\ U') + \u03bc U' + \u03b5 := (add_le_add_right (tsub_le_iff_right.1 le_measure_diff) _)\n  _ \u2264 \u03bc (K \\ U') + \u03b5 + \u03b5 := by\n    apply add_le_add_right; apply add_le_add_left\n    exact h\u03bcU'.le\n  _ = \u03bc (K \\ U') + (\u03b5 + \u03b5) := add_assoc _ _ _", "annotated_tactic": ["calc\n    \u03bc s \u2264 \u03bc U := \u03bc.mono hsU\n    _ < \u03bc K + \u03b5 := hKr\n    _ \u2264 \u03bc (K \\ U') + \u03bc U' + \u03b5 := (<a>add_le_add_right</a> (<a>tsub_le_iff_right</a>.1 <a>le_measure_diff</a>) _)\n    _ \u2264 \u03bc (K \\ U') + \u03b5 + \u03b5 := by\n      apply <a>add_le_add_right</a>; apply <a>add_le_add_left</a>\n      exact h\u03bcU'.le\n    _ = \u03bc (K \\ U') + (\u03b5 + \u03b5) := <a>add_assoc</a> _ _ _", [{"full_name": "add_le_add_right", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [66, 15], "def_end_pos": [66, 31]}, {"full_name": "tsub_le_iff_right", "def_path": "Mathlib/Algebra/Order/Sub/Defs.lean", "def_pos": [65, 9], "def_end_pos": [65, 26]}, {"full_name": "MeasureTheory.le_measure_diff", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [256, 9], "def_end_pos": [256, 24]}, {"full_name": "add_le_add_right", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [66, 15], "def_end_pos": [66, 31]}, {"full_name": "add_le_add_left", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [49, 15], "def_end_pos": [49, 30]}, {"full_name": "add_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [263, 3], "def_end_pos": [263, 14]}]], "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03bc : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU\u271d s\u271d : Set \u03b1\n\u03b5\u271d r : \u211d\u22650\u221e\ninst\u271d : OuterRegular \u03bc\nH : InnerRegular \u03bc p IsOpen\nh0 : p \u2205\nhd : \u2200 \u2983s U : Set \u03b1\u2984, p s \u2192 IsOpen U \u2192 p (s \\ U)\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nh\u03b5s : \u03b5 + \u03b5 \u2264 \u2191\u2191\u03bc s\nhr : \u2191\u2191\u03bc s - (\u03b5 + \u03b5) < \u2191\u2191\u03bc s\nU : Set \u03b1\nhsU : U \u2287 s\nhUo : IsOpen U\nhUt : \u2191\u2191\u03bc U < \u22a4\nh\u03bcU : \u2191\u2191\u03bc (U \\ s) < \u03b5\nU' : Set \u03b1\nhU'o : IsOpen U'\nh\u03bcU' : \u2191\u2191\u03bc U' < \u03b5\nhsU' : U \\ U' \u2286 s\nK : Set \u03b1\nhKU : K \u2286 U\nhKc : p K\nhKr : \u2191\u2191\u03bc U < \u2191\u2191\u03bc K + \u03b5\n\u22a2 \u2191\u2191\u03bc s < \u2191\u2191\u03bc (K \\ U') + (\u03b5 + \u03b5)", "state_after": "no goals"}, {"tactic": "use (\u03bc s - r) / 2", "annotated_tactic": ["use (\u03bc s - r) / 2", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03bc : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU s\u271d : Set \u03b1\n\u03b5 r\u271d : \u211d\u22650\u221e\ninst\u271d : OuterRegular \u03bc\nH : InnerRegular \u03bc p IsOpen\nh0 : p \u2205\nhd : \u2200 \u2983s U : Set \u03b1\u2984, p s \u2192 IsOpen U \u2192 p (s \\ U)\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nr : \u211d\u22650\u221e\nhr : r < \u2191\u2191\u03bc s\n\u22a2 \u2203 \u03b5 x, \u03b5 + \u03b5 \u2264 \u2191\u2191\u03bc s \u2227 r = \u2191\u2191\u03bc s - (\u03b5 + \u03b5)", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03bc : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU s\u271d : Set \u03b1\n\u03b5 r\u271d : \u211d\u22650\u221e\ninst\u271d : OuterRegular \u03bc\nH : InnerRegular \u03bc p IsOpen\nh0 : p \u2205\nhd : \u2200 \u2983s U : Set \u03b1\u2984, p s \u2192 IsOpen U \u2192 p (s \\ U)\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nr : \u211d\u22650\u221e\nhr : r < \u2191\u2191\u03bc s\n\u22a2 \u2203 x, (\u2191\u2191\u03bc s - r) / 2 + (\u2191\u2191\u03bc s - r) / 2 \u2264 \u2191\u2191\u03bc s \u2227 r = \u2191\u2191\u03bc s - ((\u2191\u2191\u03bc s - r) / 2 + (\u2191\u2191\u03bc s - r) / 2)"}, {"tactic": "simp [*, hr.le, ENNReal.add_halves, ENNReal.sub_sub_cancel, le_add_right]", "annotated_tactic": ["simp [*, hr.le, <a>ENNReal.add_halves</a>, <a>ENNReal.sub_sub_cancel</a>, <a>le_add_right</a>]", [{"full_name": "ENNReal.add_halves", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1781, 19], "def_end_pos": [1781, 29]}, {"full_name": "ENNReal.sub_sub_cancel", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1225, 9], "def_end_pos": [1225, 23]}, {"full_name": "le_add_right", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [278, 3], "def_end_pos": [278, 14]}]], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03bc : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU s\u271d : Set \u03b1\n\u03b5 r\u271d : \u211d\u22650\u221e\ninst\u271d : OuterRegular \u03bc\nH : InnerRegular \u03bc p IsOpen\nh0 : p \u2205\nhd : \u2200 \u2983s U : Set \u03b1\u2984, p s \u2192 IsOpen U \u2192 p (s \\ U)\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nr : \u211d\u22650\u221e\nhr : r < \u2191\u2191\u03bc s\n\u22a2 \u2203 x, (\u2191\u2191\u03bc s - r) / 2 + (\u2191\u2191\u03bc s - r) / 2 \u2264 \u2191\u2191\u03bc s \u2227 r = \u2191\u2191\u03bc s - ((\u2191\u2191\u03bc s - r) / 2 + (\u2191\u2191\u03bc s - r) / 2)", "state_after": "no goals"}, {"tactic": "apply add_le_add_right", "annotated_tactic": ["apply <a>add_le_add_right</a>", [{"full_name": "add_le_add_right", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [66, 15], "def_end_pos": [66, 31]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03bc : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU\u271d s\u271d : Set \u03b1\n\u03b5\u271d r : \u211d\u22650\u221e\ninst\u271d : OuterRegular \u03bc\nH : InnerRegular \u03bc p IsOpen\nh0 : p \u2205\nhd : \u2200 \u2983s U : Set \u03b1\u2984, p s \u2192 IsOpen U \u2192 p (s \\ U)\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nh\u03b5s : \u03b5 + \u03b5 \u2264 \u2191\u2191\u03bc s\nhr : \u2191\u2191\u03bc s - (\u03b5 + \u03b5) < \u2191\u2191\u03bc s\nU : Set \u03b1\nhsU : U \u2287 s\nhUo : IsOpen U\nhUt : \u2191\u2191\u03bc U < \u22a4\nh\u03bcU : \u2191\u2191\u03bc (U \\ s) < \u03b5\nU' : Set \u03b1\nhU'o : IsOpen U'\nh\u03bcU' : \u2191\u2191\u03bc U' < \u03b5\nhsU' : U \\ U' \u2286 s\nK : Set \u03b1\nhKU : K \u2286 U\nhKc : p K\nhKr : \u2191\u2191\u03bc U < \u2191\u2191\u03bc K + \u03b5\n\u22a2 \u2191\u2191\u03bc (K \\ U') + \u2191\u2191\u03bc U' + \u03b5 \u2264 \u2191\u2191\u03bc (K \\ U') + \u03b5 + \u03b5", "state_after": "case bc\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03bc : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU\u271d s\u271d : Set \u03b1\n\u03b5\u271d r : \u211d\u22650\u221e\ninst\u271d : OuterRegular \u03bc\nH : InnerRegular \u03bc p IsOpen\nh0 : p \u2205\nhd : \u2200 \u2983s U : Set \u03b1\u2984, p s \u2192 IsOpen U \u2192 p (s \\ U)\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nh\u03b5s : \u03b5 + \u03b5 \u2264 \u2191\u2191\u03bc s\nhr : \u2191\u2191\u03bc s - (\u03b5 + \u03b5) < \u2191\u2191\u03bc s\nU : Set \u03b1\nhsU : U \u2287 s\nhUo : IsOpen U\nhUt : \u2191\u2191\u03bc U < \u22a4\nh\u03bcU : \u2191\u2191\u03bc (U \\ s) < \u03b5\nU' : Set \u03b1\nhU'o : IsOpen U'\nh\u03bcU' : \u2191\u2191\u03bc U' < \u03b5\nhsU' : U \\ U' \u2286 s\nK : Set \u03b1\nhKU : K \u2286 U\nhKc : p K\nhKr : \u2191\u2191\u03bc U < \u2191\u2191\u03bc K + \u03b5\n\u22a2 \u2191\u2191\u03bc (K \\ U') + \u2191\u2191\u03bc U' \u2264 \u2191\u2191\u03bc (K \\ U') + \u03b5"}, {"tactic": "apply add_le_add_left", "annotated_tactic": ["apply <a>add_le_add_left</a>", [{"full_name": "add_le_add_left", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [49, 15], "def_end_pos": [49, 30]}]], "state_before": "case bc\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03bc : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU\u271d s\u271d : Set \u03b1\n\u03b5\u271d r : \u211d\u22650\u221e\ninst\u271d : OuterRegular \u03bc\nH : InnerRegular \u03bc p IsOpen\nh0 : p \u2205\nhd : \u2200 \u2983s U : Set \u03b1\u2984, p s \u2192 IsOpen U \u2192 p (s \\ U)\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nh\u03b5s : \u03b5 + \u03b5 \u2264 \u2191\u2191\u03bc s\nhr : \u2191\u2191\u03bc s - (\u03b5 + \u03b5) < \u2191\u2191\u03bc s\nU : Set \u03b1\nhsU : U \u2287 s\nhUo : IsOpen U\nhUt : \u2191\u2191\u03bc U < \u22a4\nh\u03bcU : \u2191\u2191\u03bc (U \\ s) < \u03b5\nU' : Set \u03b1\nhU'o : IsOpen U'\nh\u03bcU' : \u2191\u2191\u03bc U' < \u03b5\nhsU' : U \\ U' \u2286 s\nK : Set \u03b1\nhKU : K \u2286 U\nhKc : p K\nhKr : \u2191\u2191\u03bc U < \u2191\u2191\u03bc K + \u03b5\n\u22a2 \u2191\u2191\u03bc (K \\ U') + \u2191\u2191\u03bc U' \u2264 \u2191\u2191\u03bc (K \\ U') + \u03b5", "state_after": "case bc.bc\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03bc : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU\u271d s\u271d : Set \u03b1\n\u03b5\u271d r : \u211d\u22650\u221e\ninst\u271d : OuterRegular \u03bc\nH : InnerRegular \u03bc p IsOpen\nh0 : p \u2205\nhd : \u2200 \u2983s U : Set \u03b1\u2984, p s \u2192 IsOpen U \u2192 p (s \\ U)\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nh\u03b5s : \u03b5 + \u03b5 \u2264 \u2191\u2191\u03bc s\nhr : \u2191\u2191\u03bc s - (\u03b5 + \u03b5) < \u2191\u2191\u03bc s\nU : Set \u03b1\nhsU : U \u2287 s\nhUo : IsOpen U\nhUt : \u2191\u2191\u03bc U < \u22a4\nh\u03bcU : \u2191\u2191\u03bc (U \\ s) < \u03b5\nU' : Set \u03b1\nhU'o : IsOpen U'\nh\u03bcU' : \u2191\u2191\u03bc U' < \u03b5\nhsU' : U \\ U' \u2286 s\nK : Set \u03b1\nhKU : K \u2286 U\nhKc : p K\nhKr : \u2191\u2191\u03bc U < \u2191\u2191\u03bc K + \u03b5\n\u22a2 \u2191\u2191\u03bc U' \u2264 \u03b5"}, {"tactic": "exact h\u03bcU'.le", "annotated_tactic": ["exact h\u03bcU'.le", []], "state_before": "case bc.bc\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03bc : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU\u271d s\u271d : Set \u03b1\n\u03b5\u271d r : \u211d\u22650\u221e\ninst\u271d : OuterRegular \u03bc\nH : InnerRegular \u03bc p IsOpen\nh0 : p \u2205\nhd : \u2200 \u2983s U : Set \u03b1\u2984, p s \u2192 IsOpen U \u2192 p (s \\ U)\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nh\u03b5s : \u03b5 + \u03b5 \u2264 \u2191\u2191\u03bc s\nhr : \u2191\u2191\u03bc s - (\u03b5 + \u03b5) < \u2191\u2191\u03bc s\nU : Set \u03b1\nhsU : U \u2287 s\nhUo : IsOpen U\nhUt : \u2191\u2191\u03bc U < \u22a4\nh\u03bcU : \u2191\u2191\u03bc (U \\ s) < \u03b5\nU' : Set \u03b1\nhU'o : IsOpen U'\nh\u03bcU' : \u2191\u2191\u03bc U' < \u03b5\nhsU' : U \\ U' \u2286 s\nK : Set \u03b1\nhKU : K \u2286 U\nhKc : p K\nhKr : \u2191\u2191\u03bc U < \u2191\u2191\u03bc K + \u03b5\n\u22a2 \u2191\u2191\u03bc U' \u2264 \u03b5", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Constructions/Pi.lean", "full_name": "MeasureTheory.Measure.univ_pi_Ico_ae_eq_Icc", "start": [535, 1], "end": [537, 48], "traced_tactics": [{"tactic": "rw [\u2190 pi_univ_Icc]", "annotated_tactic": ["rw [\u2190 <a>pi_univ_Icc</a>]", [{"full_name": "Set.pi_univ_Icc", "def_path": "Mathlib/Data/Set/Intervals/Pi.lean", "def_pos": [43, 9], "def_end_pos": [43, 20]}]], "state_before": "\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : \u03b9 \u2192 Type u_3\ninst\u271d\u2074 : Fintype \u03b9\nm : (i : \u03b9) \u2192 OuterMeasure (\u03b1 i)\ninst\u271d\u00b3 : (i : \u03b9) \u2192 MeasurableSpace (\u03b1 i)\n\u03bc : (i : \u03b9) \u2192 Measure (\u03b1 i)\ninst\u271d\u00b2 : \u2200 (i : \u03b9), SigmaFinite (\u03bc i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 PartialOrder (\u03b1 i)\ninst\u271d : \u2200 (i : \u03b9), NoAtoms (\u03bc i)\nf g : (i : \u03b9) \u2192 \u03b1 i\n\u22a2 (Set.pi univ fun i => Ico (f i) (g i)) =\u1da0[ae (Measure.pi \u03bc)] Icc f g", "state_after": "\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : \u03b9 \u2192 Type u_3\ninst\u271d\u2074 : Fintype \u03b9\nm : (i : \u03b9) \u2192 OuterMeasure (\u03b1 i)\ninst\u271d\u00b3 : (i : \u03b9) \u2192 MeasurableSpace (\u03b1 i)\n\u03bc : (i : \u03b9) \u2192 Measure (\u03b1 i)\ninst\u271d\u00b2 : \u2200 (i : \u03b9), SigmaFinite (\u03bc i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 PartialOrder (\u03b1 i)\ninst\u271d : \u2200 (i : \u03b9), NoAtoms (\u03bc i)\nf g : (i : \u03b9) \u2192 \u03b1 i\n\u22a2 (Set.pi univ fun i => Ico (f i) (g i)) =\u1da0[ae (Measure.pi \u03bc)] Set.pi univ fun i => Icc (f i) (g i)"}, {"tactic": "exact pi_Ico_ae_eq_pi_Icc", "annotated_tactic": ["exact <a>pi_Ico_ae_eq_pi_Icc</a>", [{"full_name": "MeasureTheory.Measure.pi_Ico_ae_eq_pi_Icc", "def_path": "Mathlib/MeasureTheory/Constructions/Pi.lean", "def_pos": [530, 9], "def_end_pos": [530, 28]}]], "state_before": "\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : \u03b9 \u2192 Type u_3\ninst\u271d\u2074 : Fintype \u03b9\nm : (i : \u03b9) \u2192 OuterMeasure (\u03b1 i)\ninst\u271d\u00b3 : (i : \u03b9) \u2192 MeasurableSpace (\u03b1 i)\n\u03bc : (i : \u03b9) \u2192 Measure (\u03b1 i)\ninst\u271d\u00b2 : \u2200 (i : \u03b9), SigmaFinite (\u03bc i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 PartialOrder (\u03b1 i)\ninst\u271d : \u2200 (i : \u03b9), NoAtoms (\u03bc i)\nf g : (i : \u03b9) \u2192 \u03b1 i\n\u22a2 (Set.pi univ fun i => Ico (f i) (g i)) =\u1da0[ae (Measure.pi \u03bc)] Set.pi univ fun i => Icc (f i) (g i)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "full_name": "MeasureTheory.Lp.simpleFunc.smul_toSimpleFunc", "start": [670, 1], "end": [675, 25], "traced_tactics": [{"tactic": "filter_upwards [toSimpleFunc_eq_toFun (k \u2022 f), toSimpleFunc_eq_toFun f,\n  Lp.coeFn_smul k (f : Lp E p \u03bc)] with _", "annotated_tactic": ["filter_upwards [<a>toSimpleFunc_eq_toFun</a> (k \u2022 f), <a>toSimpleFunc_eq_toFun</a> f,\n    <a>Lp.coeFn_smul</a> k (f : <a>Lp</a> E p \u03bc)] with _", [{"full_name": "MeasureTheory.Lp.simpleFunc.toSimpleFunc_eq_toFun", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "def_pos": [614, 9], "def_end_pos": [614, 30]}, {"full_name": "MeasureTheory.Lp.simpleFunc.toSimpleFunc_eq_toFun", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "def_pos": [614, 9], "def_end_pos": [614, 30]}, {"full_name": "MeasureTheory.Lp.coeFn_smul", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [498, 9], "def_end_pos": [498, 19]}, {"full_name": "MeasureTheory.Lp", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [98, 5], "def_end_pos": [98, 7]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedRing \ud835\udd5c\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : BoundedSMul \ud835\udd5c E\nk : \ud835\udd5c\nf : { x // x \u2208 simpleFunc E p \u03bc }\n\u22a2 \u2191(toSimpleFunc (k \u2022 f)) =\u1d50[\u03bc] k \u2022 \u2191(toSimpleFunc f)", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedRing \ud835\udd5c\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : BoundedSMul \ud835\udd5c E\nk : \ud835\udd5c\nf : { x // x \u2208 simpleFunc E p \u03bc }\na\u271d : \u03b1\n\u22a2 \u2191(toSimpleFunc (k \u2022 f)) a\u271d = \u2191\u2191\u2191(k \u2022 f) a\u271d \u2192\n    \u2191(toSimpleFunc f) a\u271d = \u2191\u2191\u2191f a\u271d \u2192\n      \u2191\u2191(k \u2022 \u2191f) a\u271d = (k \u2022 \u2191\u2191\u2191f) a\u271d \u2192 \u2191(toSimpleFunc (k \u2022 f)) a\u271d = (k \u2022 \u2191(toSimpleFunc f)) a\u271d"}, {"tactic": "simp only [Pi.smul_apply, coe_smul]", "annotated_tactic": ["simp only [<a>Pi.smul_apply</a>, <a>coe_smul</a>]", [{"full_name": "Pi.smul_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [116, 60], "def_end_pos": [116, 70]}, {"full_name": "MeasureTheory.Lp.simpleFunc.coe_smul", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "def_pos": [498, 9], "def_end_pos": [498, 17]}]], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedRing \ud835\udd5c\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : BoundedSMul \ud835\udd5c E\nk : \ud835\udd5c\nf : { x // x \u2208 simpleFunc E p \u03bc }\na\u271d : \u03b1\n\u22a2 \u2191(toSimpleFunc (k \u2022 f)) a\u271d = \u2191\u2191\u2191(k \u2022 f) a\u271d \u2192\n    \u2191(toSimpleFunc f) a\u271d = \u2191\u2191\u2191f a\u271d \u2192\n      \u2191\u2191(k \u2022 \u2191f) a\u271d = (k \u2022 \u2191\u2191\u2191f) a\u271d \u2192 \u2191(toSimpleFunc (k \u2022 f)) a\u271d = (k \u2022 \u2191(toSimpleFunc f)) a\u271d", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedRing \ud835\udd5c\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : BoundedSMul \ud835\udd5c E\nk : \ud835\udd5c\nf : { x // x \u2208 simpleFunc E p \u03bc }\na\u271d : \u03b1\n\u22a2 \u2191(toSimpleFunc (k \u2022 f)) a\u271d = \u2191\u2191(k \u2022 \u2191f) a\u271d \u2192\n    \u2191(toSimpleFunc f) a\u271d = \u2191\u2191\u2191f a\u271d \u2192 \u2191\u2191(k \u2022 \u2191f) a\u271d = k \u2022 \u2191\u2191\u2191f a\u271d \u2192 \u2191(toSimpleFunc (k \u2022 f)) a\u271d = k \u2022 \u2191(toSimpleFunc f) a\u271d"}, {"tactic": "repeat intro h; rw [h]", "annotated_tactic": ["repeat intro h; rw [h]", []], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedRing \ud835\udd5c\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : BoundedSMul \ud835\udd5c E\nk : \ud835\udd5c\nf : { x // x \u2208 simpleFunc E p \u03bc }\na\u271d : \u03b1\n\u22a2 \u2191(toSimpleFunc (k \u2022 f)) a\u271d = \u2191\u2191(k \u2022 \u2191f) a\u271d \u2192\n    \u2191(toSimpleFunc f) a\u271d = \u2191\u2191\u2191f a\u271d \u2192 \u2191\u2191(k \u2022 \u2191f) a\u271d = k \u2022 \u2191\u2191\u2191f a\u271d \u2192 \u2191(toSimpleFunc (k \u2022 f)) a\u271d = k \u2022 \u2191(toSimpleFunc f) a\u271d", "state_after": "no goals"}, {"tactic": "intro h", "annotated_tactic": ["intro h", []], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedRing \ud835\udd5c\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : BoundedSMul \ud835\udd5c E\nk : \ud835\udd5c\nf : { x // x \u2208 simpleFunc E p \u03bc }\na\u271d : \u03b1\nh\u271d : \u2191(toSimpleFunc (k \u2022 f)) a\u271d = \u2191\u2191(k \u2022 \u2191f) a\u271d\nh : \u2191(toSimpleFunc f) a\u271d = \u2191\u2191\u2191f a\u271d\n\u22a2 \u2191\u2191(k \u2022 \u2191f) a\u271d = k \u2022 \u2191\u2191\u2191f a\u271d \u2192 \u2191\u2191(k \u2022 \u2191f) a\u271d = k \u2022 \u2191\u2191\u2191f a\u271d", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedRing \ud835\udd5c\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : BoundedSMul \ud835\udd5c E\nk : \ud835\udd5c\nf : { x // x \u2208 simpleFunc E p \u03bc }\na\u271d : \u03b1\nh\u271d\u00b9 : \u2191(toSimpleFunc (k \u2022 f)) a\u271d = \u2191\u2191(k \u2022 \u2191f) a\u271d\nh\u271d : \u2191(toSimpleFunc f) a\u271d = \u2191\u2191\u2191f a\u271d\nh : \u2191\u2191(k \u2022 \u2191f) a\u271d = k \u2022 \u2191\u2191\u2191f a\u271d\n\u22a2 \u2191\u2191(k \u2022 \u2191f) a\u271d = k \u2022 \u2191\u2191\u2191f a\u271d"}, {"tactic": "rw [h]", "annotated_tactic": ["rw [h]", []], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedRing \ud835\udd5c\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : BoundedSMul \ud835\udd5c E\nk : \ud835\udd5c\nf : { x // x \u2208 simpleFunc E p \u03bc }\na\u271d : \u03b1\nh\u271d\u00b9 : \u2191(toSimpleFunc (k \u2022 f)) a\u271d = \u2191\u2191(k \u2022 \u2191f) a\u271d\nh\u271d : \u2191(toSimpleFunc f) a\u271d = \u2191\u2191\u2191f a\u271d\nh : \u2191\u2191(k \u2022 \u2191f) a\u271d = k \u2022 \u2191\u2191\u2191f a\u271d\n\u22a2 \u2191\u2191(k \u2022 \u2191f) a\u271d = k \u2022 \u2191\u2191\u2191f a\u271d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Pointwise.lean", "full_name": "Finset.univ_pow", "start": [950, 1], "end": [951, 62], "traced_tactics": [{"tactic": "rw [coe_pow, coe_univ, Set.univ_pow hn]", "annotated_tactic": ["rw [<a>coe_pow</a>, <a>coe_univ</a>, <a>Set.univ_pow</a> hn]", [{"full_name": "Finset.coe_pow", "def_path": "Mathlib/Data/Finset/Pointwise.lean", "def_pos": [857, 9], "def_end_pos": [857, 16]}, {"full_name": "Finset.coe_univ", "def_path": "Mathlib/Data/Fintype/Basic.lean", "def_pos": [89, 9], "def_end_pos": [89, 17]}, {"full_name": "Set.univ_pow", "def_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "def_pos": [1009, 9], "def_end_pos": [1009, 17]}]], "state_before": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b3 : DecidableEq \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b2\ninst\u271d\u00b9 : Monoid \u03b1\ns t : Finset \u03b1\na : \u03b1\nm n : \u2115\ninst\u271d : Fintype \u03b1\nhn : n \u2260 0\n\u22a2 \u2191(univ ^ n) = \u2191univ", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/TuringMachine.lean", "full_name": "Turing.PointedMap.headI_map", "start": [374, 1], "end": [376, 54], "traced_tactics": [{"tactic": "cases l <;> [exact (PointedMap.map_pt f).symm; rfl]", "annotated_tactic": ["cases l <;> [exact (<a>PointedMap.map_pt</a> f).<a>symm</a>; rfl]", [{"full_name": "Turing.PointedMap.map_pt", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [368, 9], "def_end_pos": [368, 26]}, {"full_name": "Eq.symm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [310, 9], "def_end_pos": [310, 16]}]], "state_before": "\u0393 : Type u_1\n\u0393' : Type u_2\ninst\u271d\u00b9 : Inhabited \u0393\ninst\u271d : Inhabited \u0393'\nf : PointedMap \u0393 \u0393'\nl : List \u0393\n\u22a2 List.headI (List.map f.f l) = Turing.PointedMap.f f (List.headI l)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "full_name": "measurable_ereal_toReal", "start": [2202, 1], "end": [2203, 69], "traced_tactics": [{"tactic": "simpa using measurable_id", "annotated_tactic": ["simpa using <a>measurable_id</a>", [{"full_name": "measurable_id", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [550, 9], "def_end_pos": [550, 22]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns t u : Set \u03b1\ninst\u271d : MeasurableSpace \u03b1\n\u22a2 Measurable fun p => EReal.toReal \u2191p", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/ProbabilityMassFunction/Constructions.lean", "full_name": "PMF.filter_apply_ne_zero_iff", "start": [295, 1], "end": [296, 86], "traced_tactics": [{"tactic": "rw [Ne.def, filter_apply_eq_zero_iff, not_or, Classical.not_not, Classical.not_not]", "annotated_tactic": ["rw [<a>Ne.def</a>, <a>filter_apply_eq_zero_iff</a>, <a>not_or</a>, <a>Classical.not_not</a>, <a>Classical.not_not</a>]", [{"full_name": "Ne.def", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [59, 9], "def_end_pos": [59, 15]}, {"full_name": "PMF.filter_apply_eq_zero_iff", "def_path": "Mathlib/Probability/ProbabilityMassFunction/Constructions.lean", "def_pos": [291, 9], "def_end_pos": [291, 33]}, {"full_name": "not_or", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [340, 9], "def_end_pos": [340, 15]}, {"full_name": "Classical.not_not", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [683, 24], "def_end_pos": [683, 31]}, {"full_name": "Classical.not_not", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [683, 24], "def_end_pos": [683, 31]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\ns : Set \u03b1\nh : \u2203 a, a \u2208 s \u2227 a \u2208 support p\na : \u03b1\n\u22a2 \u2191(filter p s h) a \u2260 0 \u2194 a \u2208 s \u2227 a \u2208 support p", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/TMToPartrec.lean", "full_name": "Turing.ToPartrec.cont_eval_fix", "start": [625, 1], "end": [695, 67], "traced_tactics": [{"tactic": "refine' Part.ext fun x => _", "annotated_tactic": ["refine' <a>Part.ext</a> fun x => _", [{"full_name": "Part.ext", "def_path": "Mathlib/Data/Part.lean", "def_pos": [116, 9], "def_end_pos": [116, 12]}]], "state_before": "f : Code\nk : Cont\nv : List \u2115\nfok : Code.Ok f\n\u22a2 eval step (stepNormal f (Cont.fix f k) v) = do\n    let v \u2190 Code.eval (Code.fix f) v\n    eval step (Cfg.ret k v)", "state_after": "f : Code\nk : Cont\nv : List \u2115\nfok : Code.Ok f\nx : Cfg\n\u22a2 x \u2208 eval step (stepNormal f (Cont.fix f k) v) \u2194\n    x \u2208 do\n      let v \u2190 Code.eval (Code.fix f) v\n      eval step (Cfg.ret k v)"}, {"tactic": "simp only [Part.bind_eq_bind, Part.mem_bind_iff]", "annotated_tactic": ["simp only [<a>Part.bind_eq_bind</a>, <a>Part.mem_bind_iff</a>]", [{"full_name": "Part.bind_eq_bind", "def_path": "Mathlib/Data/Part.lean", "def_pos": [614, 9], "def_end_pos": [614, 21]}, {"full_name": "Part.mem_bind_iff", "def_path": "Mathlib/Data/Part.lean", "def_pos": [494, 9], "def_end_pos": [494, 21]}]], "state_before": "f : Code\nk : Cont\nv : List \u2115\nfok : Code.Ok f\nx : Cfg\n\u22a2 x \u2208 eval step (stepNormal f (Cont.fix f k) v) \u2194\n    x \u2208 do\n      let v \u2190 Code.eval (Code.fix f) v\n      eval step (Cfg.ret k v)", "state_after": "f : Code\nk : Cont\nv : List \u2115\nfok : Code.Ok f\nx : Cfg\n\u22a2 x \u2208 eval step (stepNormal f (Cont.fix f k) v) \u2194 \u2203 a, a \u2208 Code.eval (Code.fix f) v \u2227 x \u2208 eval step (Cfg.ret k a)"}, {"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "f : Code\nk : Cont\nv : List \u2115\nfok : Code.Ok f\nx : Cfg\n\u22a2 x \u2208 eval step (stepNormal f (Cont.fix f k) v) \u2194 \u2203 a, a \u2208 Code.eval (Code.fix f) v \u2227 x \u2208 eval step (Cfg.ret k a)", "state_after": "case mp\nf : Code\nk : Cont\nv : List \u2115\nfok : Code.Ok f\nx : Cfg\n\u22a2 x \u2208 eval step (stepNormal f (Cont.fix f k) v) \u2192 \u2203 a, a \u2208 Code.eval (Code.fix f) v \u2227 x \u2208 eval step (Cfg.ret k a)\n\ncase mpr\nf : Code\nk : Cont\nv : List \u2115\nfok : Code.Ok f\nx : Cfg\n\u22a2 (\u2203 a, a \u2208 Code.eval (Code.fix f) v \u2227 x \u2208 eval step (Cfg.ret k a)) \u2192 x \u2208 eval step (stepNormal f (Cont.fix f k) v)"}, {"tactic": "refine' fun c he => evalInduction he fun y h IH => _", "annotated_tactic": ["refine' fun c he => <a>evalInduction</a> he fun y h IH => _", [{"full_name": "Turing.evalInduction", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [824, 5], "def_end_pos": [824, 18]}]], "state_before": "case mp\nf : Code\nk : Cont\nv : List \u2115\nfok : Code.Ok f\nx : Cfg\n\u22a2 \u2200 (c : Cfg),\n    x \u2208 eval step c \u2192\n      \u2200 (v : List \u2115) (c' : Cfg),\n        c = Cfg.then c' (Cont.fix f k) \u2192\n          Reaches step (stepNormal f Cont.halt v) c' \u2192\n            \u2203 v\u2081,\n              v\u2081 \u2208 Code.eval f v \u2227\n                \u2203 v\u2082,\n                  (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227\n                    x \u2208 eval step (Cfg.ret k v\u2082)", "state_after": "case mp\nf : Code\nk : Cont\nv : List \u2115\nfok : Code.Ok f\nx c : Cfg\nhe : x \u2208 eval step c\ny : Cfg\nh : x \u2208 eval step y\nIH :\n  \u2200 (a' : Cfg),\n    step y = some a' \u2192\n      \u2200 (v : List \u2115) (c' : Cfg),\n        a' = Cfg.then c' (Cont.fix f k) \u2192\n          Reaches step (stepNormal f Cont.halt v) c' \u2192\n            \u2203 v\u2081,\n              v\u2081 \u2208 Code.eval f v \u2227\n                \u2203 v\u2082,\n                  (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227\n                    x \u2208 eval step (Cfg.ret k v\u2082)\n\u22a2 \u2200 (v : List \u2115) (c' : Cfg),\n    y = Cfg.then c' (Cont.fix f k) \u2192\n      Reaches step (stepNormal f Cont.halt v) c' \u2192\n        \u2203 v\u2081,\n          v\u2081 \u2208 Code.eval f v \u2227\n            \u2203 v\u2082,\n              (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227\n                x \u2208 eval step (Cfg.ret k v\u2082)"}, {"tactic": "rintro v (\u27e8v'\u27e9 | \u27e8k', v'\u27e9) rfl hr <;> rw [Cfg.then] at h IH <;> simp only [] at h IH", "annotated_tactic": ["rintro v (\u27e8v'\u27e9 | \u27e8k', v'\u27e9) rfl hr <;> rw [<a>Cfg.then</a>] at h IH <;> simp only [] at h IH", [{"full_name": "Turing.ToPartrec.Cfg.then", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [559, 5], "def_end_pos": [559, 13]}]], "state_before": "case mp\nf : Code\nk : Cont\nv : List \u2115\nfok : Code.Ok f\nx c : Cfg\nhe : x \u2208 eval step c\ny : Cfg\nh : x \u2208 eval step y\nIH :\n  \u2200 (a' : Cfg),\n    step y = some a' \u2192\n      \u2200 (v : List \u2115) (c' : Cfg),\n        a' = Cfg.then c' (Cont.fix f k) \u2192\n          Reaches step (stepNormal f Cont.halt v) c' \u2192\n            \u2203 v\u2081,\n              v\u2081 \u2208 Code.eval f v \u2227\n                \u2203 v\u2082,\n                  (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227\n                    x \u2208 eval step (Cfg.ret k v\u2082)\n\u22a2 \u2200 (v : List \u2115) (c' : Cfg),\n    y = Cfg.then c' (Cont.fix f k) \u2192\n      Reaches step (stepNormal f Cont.halt v) c' \u2192\n        \u2203 v\u2081,\n          v\u2081 \u2208 Code.eval f v \u2227\n            \u2203 v\u2082,\n              (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227\n                x \u2208 eval step (Cfg.ret k v\u2082)", "state_after": "case mp.halt\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx c : Cfg\nhe : x \u2208 eval step c\nv v' : List \u2115\nh : x \u2208 eval step (stepRet (Cont.fix f k) v')\nIH :\n  \u2200 (a' : Cfg),\n    step (stepRet (Cont.fix f k) v') = some a' \u2192\n      \u2200 (v : List \u2115) (c' : Cfg),\n        a' = Cfg.then c' (Cont.fix f k) \u2192\n          Reaches step (stepNormal f Cont.halt v) c' \u2192\n            \u2203 v\u2081,\n              v\u2081 \u2208 Code.eval f v \u2227\n                \u2203 v\u2082,\n                  (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227\n                    x \u2208 eval step (Cfg.ret k v\u2082)\nhr : Reaches step (stepNormal f Cont.halt v) (Cfg.halt v')\n\u22a2 \u2203 v\u2081,\n    v\u2081 \u2208 Code.eval f v \u2227\n      \u2203 v\u2082,\n        (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227\n          x \u2208 eval step (Cfg.ret k v\u2082)\n\ncase mp.ret\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx c : Cfg\nhe : x \u2208 eval step c\nv : List \u2115\nk' : Cont\nv' : List \u2115\nh : x \u2208 eval step (Cfg.ret (Cont.then k' (Cont.fix f k)) v')\nIH :\n  \u2200 (a' : Cfg),\n    step (Cfg.ret (Cont.then k' (Cont.fix f k)) v') = some a' \u2192\n      \u2200 (v : List \u2115) (c' : Cfg),\n        a' = Cfg.then c' (Cont.fix f k) \u2192\n          Reaches step (stepNormal f Cont.halt v) c' \u2192\n            \u2203 v\u2081,\n              v\u2081 \u2208 Code.eval f v \u2227\n                \u2203 v\u2082,\n                  (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227\n                    x \u2208 eval step (Cfg.ret k v\u2082)\nhr : Reaches step (stepNormal f Cont.halt v) (Cfg.ret k' v')\n\u22a2 \u2203 v\u2081,\n    v\u2081 \u2208 Code.eval f v \u2227\n      \u2203 v\u2082,\n        (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227\n          x \u2208 eval step (Cfg.ret k v\u2082)"}, {"tactic": "intro h", "annotated_tactic": ["intro h", []], "state_before": "f : Code\nk : Cont\nv : List \u2115\nfok : Code.Ok f\nx : Cfg\nthis :\n  \u2200 (c : Cfg),\n    x \u2208 eval step c \u2192\n      \u2200 (v : List \u2115) (c' : Cfg),\n        c = Cfg.then c' (Cont.fix f k) \u2192\n          Reaches step (stepNormal f Cont.halt v) c' \u2192\n            \u2203 v\u2081,\n              v\u2081 \u2208 Code.eval f v \u2227\n                \u2203 v\u2082,\n                  (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227\n                    x \u2208 eval step (Cfg.ret k v\u2082)\n\u22a2 x \u2208 eval step (stepNormal f (Cont.fix f k) v) \u2192 \u2203 a, a \u2208 Code.eval (Code.fix f) v \u2227 x \u2208 eval step (Cfg.ret k a)", "state_after": "f : Code\nk : Cont\nv : List \u2115\nfok : Code.Ok f\nx : Cfg\nthis :\n  \u2200 (c : Cfg),\n    x \u2208 eval step c \u2192\n      \u2200 (v : List \u2115) (c' : Cfg),\n        c = Cfg.then c' (Cont.fix f k) \u2192\n          Reaches step (stepNormal f Cont.halt v) c' \u2192\n            \u2203 v\u2081,\n              v\u2081 \u2208 Code.eval f v \u2227\n                \u2203 v\u2082,\n                  (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227\n                    x \u2208 eval step (Cfg.ret k v\u2082)\nh : x \u2208 eval step (stepNormal f (Cont.fix f k) v)\n\u22a2 \u2203 a, a \u2208 Code.eval (Code.fix f) v \u2227 x \u2208 eval step (Cfg.ret k a)"}, {"tactic": "obtain \u27e8v\u2081, hv\u2081, v\u2082, hv\u2082, h\u2083\u27e9 :=\n  this _ h _ _ (stepNormal_then _ Cont.halt _ _) ReflTransGen.refl", "annotated_tactic": ["obtain \u27e8v\u2081, hv\u2081, v\u2082, hv\u2082, h\u2083\u27e9 :=\n        this _ h _ _ (<a>stepNormal_then</a> _ <a>Cont.halt</a> _ _) <a>ReflTransGen.refl</a>", [{"full_name": "Turing.ToPartrec.stepNormal_then", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [567, 9], "def_end_pos": [567, 24]}, {"full_name": "Turing.ToPartrec.Cont.halt", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [433, 5], "def_end_pos": [433, 9]}, {"full_name": "Relation.ReflTransGen.refl", "def_path": "Mathlib/Logic/Relation.lean", "def_pos": [223, 5], "def_end_pos": [223, 9]}]], "state_before": "f : Code\nk : Cont\nv : List \u2115\nfok : Code.Ok f\nx : Cfg\nthis :\n  \u2200 (c : Cfg),\n    x \u2208 eval step c \u2192\n      \u2200 (v : List \u2115) (c' : Cfg),\n        c = Cfg.then c' (Cont.fix f k) \u2192\n          Reaches step (stepNormal f Cont.halt v) c' \u2192\n            \u2203 v\u2081,\n              v\u2081 \u2208 Code.eval f v \u2227\n                \u2203 v\u2082,\n                  (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227\n                    x \u2208 eval step (Cfg.ret k v\u2082)\nh : x \u2208 eval step (stepNormal f (Cont.fix f k) v)\n\u22a2 \u2203 a, a \u2208 Code.eval (Code.fix f) v \u2227 x \u2208 eval step (Cfg.ret k a)", "state_after": "case intro.intro.intro.intro\nf : Code\nk : Cont\nv : List \u2115\nfok : Code.Ok f\nx : Cfg\nthis :\n  \u2200 (c : Cfg),\n    x \u2208 eval step c \u2192\n      \u2200 (v : List \u2115) (c' : Cfg),\n        c = Cfg.then c' (Cont.fix f k) \u2192\n          Reaches step (stepNormal f Cont.halt v) c' \u2192\n            \u2203 v\u2081,\n              v\u2081 \u2208 Code.eval f v \u2227\n                \u2203 v\u2082,\n                  (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227\n                    x \u2208 eval step (Cfg.ret k v\u2082)\nh : x \u2208 eval step (stepNormal f (Cont.fix f k) v)\nv\u2081 : List \u2115\nhv\u2081 : v\u2081 \u2208 Code.eval f v\nv\u2082 : List \u2115\nhv\u2082 : v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)\nh\u2083 : x \u2208 eval step (Cfg.ret k v\u2082)\n\u22a2 \u2203 a, a \u2208 Code.eval (Code.fix f) v \u2227 x \u2208 eval step (Cfg.ret k a)"}, {"tactic": "refine' \u27e8v\u2082, PFun.mem_fix_iff.2 _, h\u2083\u27e9", "annotated_tactic": ["refine' \u27e8v\u2082, <a>PFun.mem_fix_iff</a>.2 _, h\u2083\u27e9", [{"full_name": "PFun.mem_fix_iff", "def_path": "Mathlib/Data/PFun.lean", "def_pos": [266, 9], "def_end_pos": [266, 20]}]], "state_before": "case intro.intro.intro.intro\nf : Code\nk : Cont\nv : List \u2115\nfok : Code.Ok f\nx : Cfg\nthis :\n  \u2200 (c : Cfg),\n    x \u2208 eval step c \u2192\n      \u2200 (v : List \u2115) (c' : Cfg),\n        c = Cfg.then c' (Cont.fix f k) \u2192\n          Reaches step (stepNormal f Cont.halt v) c' \u2192\n            \u2203 v\u2081,\n              v\u2081 \u2208 Code.eval f v \u2227\n                \u2203 v\u2082,\n                  (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227\n                    x \u2208 eval step (Cfg.ret k v\u2082)\nh : x \u2208 eval step (stepNormal f (Cont.fix f k) v)\nv\u2081 : List \u2115\nhv\u2081 : v\u2081 \u2208 Code.eval f v\nv\u2082 : List \u2115\nhv\u2082 : v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)\nh\u2083 : x \u2208 eval step (Cfg.ret k v\u2082)\n\u22a2 \u2203 a, a \u2208 Code.eval (Code.fix f) v \u2227 x \u2208 eval step (Cfg.ret k a)", "state_after": "case intro.intro.intro.intro\nf : Code\nk : Cont\nv : List \u2115\nfok : Code.Ok f\nx : Cfg\nthis :\n  \u2200 (c : Cfg),\n    x \u2208 eval step c \u2192\n      \u2200 (v : List \u2115) (c' : Cfg),\n        c = Cfg.then c' (Cont.fix f k) \u2192\n          Reaches step (stepNormal f Cont.halt v) c' \u2192\n            \u2203 v\u2081,\n              v\u2081 \u2208 Code.eval f v \u2227\n                \u2203 v\u2082,\n                  (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227\n                    x \u2208 eval step (Cfg.ret k v\u2082)\nh : x \u2208 eval step (stepNormal f (Cont.fix f k) v)\nv\u2081 : List \u2115\nhv\u2081 : v\u2081 \u2208 Code.eval f v\nv\u2082 : List \u2115\nhv\u2082 : v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)\nh\u2083 : x \u2208 eval step (Cfg.ret k v\u2082)\n\u22a2 Sum.inl v\u2082 \u2208\n      Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v) \u2228\n    \u2203 a',\n      Sum.inr a' \u2208\n          Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v))\n            (Code.eval f v) \u2227\n        v\u2082 \u2208\n          PFun.fix\n            (fun v =>\n              Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v))\n                (Code.eval f v))\n            a'"}, {"tactic": "simp only [Part.eq_some_iff.2 hv\u2081, Part.map_some]", "annotated_tactic": ["simp only [<a>Part.eq_some_iff</a>.2 hv\u2081, <a>Part.map_some</a>]", [{"full_name": "Part.eq_some_iff", "def_path": "Mathlib/Data/Part.lean", "def_pos": [174, 9], "def_end_pos": [174, 20]}, {"full_name": "Part.map_some", "def_path": "Mathlib/Data/Part.lean", "def_pos": [457, 9], "def_end_pos": [457, 17]}]], "state_before": "case intro.intro.intro.intro\nf : Code\nk : Cont\nv : List \u2115\nfok : Code.Ok f\nx : Cfg\nthis :\n  \u2200 (c : Cfg),\n    x \u2208 eval step c \u2192\n      \u2200 (v : List \u2115) (c' : Cfg),\n        c = Cfg.then c' (Cont.fix f k) \u2192\n          Reaches step (stepNormal f Cont.halt v) c' \u2192\n            \u2203 v\u2081,\n              v\u2081 \u2208 Code.eval f v \u2227\n                \u2203 v\u2082,\n                  (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227\n                    x \u2208 eval step (Cfg.ret k v\u2082)\nh : x \u2208 eval step (stepNormal f (Cont.fix f k) v)\nv\u2081 : List \u2115\nhv\u2081 : v\u2081 \u2208 Code.eval f v\nv\u2082 : List \u2115\nhv\u2082 : v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)\nh\u2083 : x \u2208 eval step (Cfg.ret k v\u2082)\n\u22a2 Sum.inl v\u2082 \u2208\n      Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v) \u2228\n    \u2203 a',\n      Sum.inr a' \u2208\n          Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v))\n            (Code.eval f v) \u2227\n        v\u2082 \u2208\n          PFun.fix\n            (fun v =>\n              Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v))\n                (Code.eval f v))\n            a'", "state_after": "case intro.intro.intro.intro\nf : Code\nk : Cont\nv : List \u2115\nfok : Code.Ok f\nx : Cfg\nthis :\n  \u2200 (c : Cfg),\n    x \u2208 eval step c \u2192\n      \u2200 (v : List \u2115) (c' : Cfg),\n        c = Cfg.then c' (Cont.fix f k) \u2192\n          Reaches step (stepNormal f Cont.halt v) c' \u2192\n            \u2203 v\u2081,\n              v\u2081 \u2208 Code.eval f v \u2227\n                \u2203 v\u2082,\n                  (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227\n                    x \u2208 eval step (Cfg.ret k v\u2082)\nh : x \u2208 eval step (stepNormal f (Cont.fix f k) v)\nv\u2081 : List \u2115\nhv\u2081 : v\u2081 \u2208 Code.eval f v\nv\u2082 : List \u2115\nhv\u2082 : v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)\nh\u2083 : x \u2208 eval step (Cfg.ret k v\u2082)\n\u22a2 Sum.inl v\u2082 \u2208 Part.some (if List.headI v\u2081 = 0 then Sum.inl (List.tail v\u2081) else Sum.inr (List.tail v\u2081)) \u2228\n    \u2203 a',\n      Sum.inr a' \u2208 Part.some (if List.headI v\u2081 = 0 then Sum.inl (List.tail v\u2081) else Sum.inr (List.tail v\u2081)) \u2227\n        v\u2082 \u2208\n          PFun.fix\n            (fun v =>\n              Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v))\n                (Code.eval f v))\n            a'"}, {"tactic": "split_ifs at hv\u2082 \u22a2", "annotated_tactic": ["split_ifs at hv\u2082 \u22a2", []], "state_before": "case intro.intro.intro.intro\nf : Code\nk : Cont\nv : List \u2115\nfok : Code.Ok f\nx : Cfg\nthis :\n  \u2200 (c : Cfg),\n    x \u2208 eval step c \u2192\n      \u2200 (v : List \u2115) (c' : Cfg),\n        c = Cfg.then c' (Cont.fix f k) \u2192\n          Reaches step (stepNormal f Cont.halt v) c' \u2192\n            \u2203 v\u2081,\n              v\u2081 \u2208 Code.eval f v \u2227\n                \u2203 v\u2082,\n                  (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227\n                    x \u2208 eval step (Cfg.ret k v\u2082)\nh : x \u2208 eval step (stepNormal f (Cont.fix f k) v)\nv\u2081 : List \u2115\nhv\u2081 : v\u2081 \u2208 Code.eval f v\nv\u2082 : List \u2115\nhv\u2082 : v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)\nh\u2083 : x \u2208 eval step (Cfg.ret k v\u2082)\n\u22a2 Sum.inl v\u2082 \u2208 Part.some (if List.headI v\u2081 = 0 then Sum.inl (List.tail v\u2081) else Sum.inr (List.tail v\u2081)) \u2228\n    \u2203 a',\n      Sum.inr a' \u2208 Part.some (if List.headI v\u2081 = 0 then Sum.inl (List.tail v\u2081) else Sum.inr (List.tail v\u2081)) \u2227\n        v\u2082 \u2208\n          PFun.fix\n            (fun v =>\n              Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v))\n                (Code.eval f v))\n            a'", "state_after": "case pos\nf : Code\nk : Cont\nv : List \u2115\nfok : Code.Ok f\nx : Cfg\nthis :\n  \u2200 (c : Cfg),\n    x \u2208 eval step c \u2192\n      \u2200 (v : List \u2115) (c' : Cfg),\n        c = Cfg.then c' (Cont.fix f k) \u2192\n          Reaches step (stepNormal f Cont.halt v) c' \u2192\n            \u2203 v\u2081,\n              v\u2081 \u2208 Code.eval f v \u2227\n                \u2203 v\u2082,\n                  (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227\n                    x \u2208 eval step (Cfg.ret k v\u2082)\nh : x \u2208 eval step (stepNormal f (Cont.fix f k) v)\nv\u2081 : List \u2115\nhv\u2081 : v\u2081 \u2208 Code.eval f v\nv\u2082 : List \u2115\nh\u2083 : x \u2208 eval step (Cfg.ret k v\u2082)\nh\u271d : List.headI v\u2081 = 0\nhv\u2082 : v\u2082 \u2208 pure (List.tail v\u2081)\n\u22a2 Sum.inl v\u2082 \u2208 Part.some (Sum.inl (List.tail v\u2081)) \u2228\n    \u2203 a',\n      Sum.inr a' \u2208 Part.some (Sum.inl (List.tail v\u2081)) \u2227\n        v\u2082 \u2208\n          PFun.fix\n            (fun v =>\n              Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v))\n                (Code.eval f v))\n            a'\n\ncase neg\nf : Code\nk : Cont\nv : List \u2115\nfok : Code.Ok f\nx : Cfg\nthis :\n  \u2200 (c : Cfg),\n    x \u2208 eval step c \u2192\n      \u2200 (v : List \u2115) (c' : Cfg),\n        c = Cfg.then c' (Cont.fix f k) \u2192\n          Reaches step (stepNormal f Cont.halt v) c' \u2192\n            \u2203 v\u2081,\n              v\u2081 \u2208 Code.eval f v \u2227\n                \u2203 v\u2082,\n                  (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227\n                    x \u2208 eval step (Cfg.ret k v\u2082)\nh : x \u2208 eval step (stepNormal f (Cont.fix f k) v)\nv\u2081 : List \u2115\nhv\u2081 : v\u2081 \u2208 Code.eval f v\nv\u2082 : List \u2115\nh\u2083 : x \u2208 eval step (Cfg.ret k v\u2082)\nh\u271d : \u00acList.headI v\u2081 = 0\nhv\u2082 : v\u2082 \u2208 Code.eval (Code.fix f) (List.tail v\u2081)\n\u22a2 Sum.inl v\u2082 \u2208 Part.some (Sum.inr (List.tail v\u2081)) \u2228\n    \u2203 a',\n      Sum.inr a' \u2208 Part.some (Sum.inr (List.tail v\u2081)) \u2227\n        v\u2082 \u2208\n          PFun.fix\n            (fun v =>\n              Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v))\n                (Code.eval f v))\n            a'"}, {"tactic": "rw [Part.mem_some_iff.1 hv\u2082]", "annotated_tactic": ["rw [<a>Part.mem_some_iff</a>.1 hv\u2082]", [{"full_name": "Part.mem_some_iff", "def_path": "Mathlib/Data/Part.lean", "def_pos": [170, 9], "def_end_pos": [170, 21]}]], "state_before": "case pos\nf : Code\nk : Cont\nv : List \u2115\nfok : Code.Ok f\nx : Cfg\nthis :\n  \u2200 (c : Cfg),\n    x \u2208 eval step c \u2192\n      \u2200 (v : List \u2115) (c' : Cfg),\n        c = Cfg.then c' (Cont.fix f k) \u2192\n          Reaches step (stepNormal f Cont.halt v) c' \u2192\n            \u2203 v\u2081,\n              v\u2081 \u2208 Code.eval f v \u2227\n                \u2203 v\u2082,\n                  (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227\n                    x \u2208 eval step (Cfg.ret k v\u2082)\nh : x \u2208 eval step (stepNormal f (Cont.fix f k) v)\nv\u2081 : List \u2115\nhv\u2081 : v\u2081 \u2208 Code.eval f v\nv\u2082 : List \u2115\nh\u2083 : x \u2208 eval step (Cfg.ret k v\u2082)\nh\u271d : List.headI v\u2081 = 0\nhv\u2082 : v\u2082 \u2208 pure (List.tail v\u2081)\n\u22a2 Sum.inl v\u2082 \u2208 Part.some (Sum.inl (List.tail v\u2081)) \u2228\n    \u2203 a',\n      Sum.inr a' \u2208 Part.some (Sum.inl (List.tail v\u2081)) \u2227\n        v\u2082 \u2208\n          PFun.fix\n            (fun v =>\n              Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v))\n                (Code.eval f v))\n            a'", "state_after": "case pos\nf : Code\nk : Cont\nv : List \u2115\nfok : Code.Ok f\nx : Cfg\nthis :\n  \u2200 (c : Cfg),\n    x \u2208 eval step c \u2192\n      \u2200 (v : List \u2115) (c' : Cfg),\n        c = Cfg.then c' (Cont.fix f k) \u2192\n          Reaches step (stepNormal f Cont.halt v) c' \u2192\n            \u2203 v\u2081,\n              v\u2081 \u2208 Code.eval f v \u2227\n                \u2203 v\u2082,\n                  (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227\n                    x \u2208 eval step (Cfg.ret k v\u2082)\nh : x \u2208 eval step (stepNormal f (Cont.fix f k) v)\nv\u2081 : List \u2115\nhv\u2081 : v\u2081 \u2208 Code.eval f v\nv\u2082 : List \u2115\nh\u2083 : x \u2208 eval step (Cfg.ret k v\u2082)\nh\u271d : List.headI v\u2081 = 0\nhv\u2082 : v\u2082 \u2208 pure (List.tail v\u2081)\n\u22a2 Sum.inl (List.tail v\u2081) \u2208 Part.some (Sum.inl (List.tail v\u2081)) \u2228\n    \u2203 a',\n      Sum.inr a' \u2208 Part.some (Sum.inl (List.tail v\u2081)) \u2227\n        List.tail v\u2081 \u2208\n          PFun.fix\n            (fun v =>\n              Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v))\n                (Code.eval f v))\n            a'"}, {"tactic": "exact Or.inl (Part.mem_some _)", "annotated_tactic": ["exact <a>Or.inl</a> (<a>Part.mem_some</a> _)", [{"full_name": "Or.inl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [517, 5], "def_end_pos": [517, 8]}, {"full_name": "Part.mem_some", "def_path": "Mathlib/Data/Part.lean", "def_pos": [165, 9], "def_end_pos": [165, 17]}]], "state_before": "case pos\nf : Code\nk : Cont\nv : List \u2115\nfok : Code.Ok f\nx : Cfg\nthis :\n  \u2200 (c : Cfg),\n    x \u2208 eval step c \u2192\n      \u2200 (v : List \u2115) (c' : Cfg),\n        c = Cfg.then c' (Cont.fix f k) \u2192\n          Reaches step (stepNormal f Cont.halt v) c' \u2192\n            \u2203 v\u2081,\n              v\u2081 \u2208 Code.eval f v \u2227\n                \u2203 v\u2082,\n                  (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227\n                    x \u2208 eval step (Cfg.ret k v\u2082)\nh : x \u2208 eval step (stepNormal f (Cont.fix f k) v)\nv\u2081 : List \u2115\nhv\u2081 : v\u2081 \u2208 Code.eval f v\nv\u2082 : List \u2115\nh\u2083 : x \u2208 eval step (Cfg.ret k v\u2082)\nh\u271d : List.headI v\u2081 = 0\nhv\u2082 : v\u2082 \u2208 pure (List.tail v\u2081)\n\u22a2 Sum.inl (List.tail v\u2081) \u2208 Part.some (Sum.inl (List.tail v\u2081)) \u2228\n    \u2203 a',\n      Sum.inr a' \u2208 Part.some (Sum.inl (List.tail v\u2081)) \u2227\n        List.tail v\u2081 \u2208\n          PFun.fix\n            (fun v =>\n              Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v))\n                (Code.eval f v))\n            a'", "state_after": "no goals"}, {"tactic": "exact Or.inr \u27e8_, Part.mem_some _, hv\u2082\u27e9", "annotated_tactic": ["exact <a>Or.inr</a> \u27e8_, <a>Part.mem_some</a> _, hv\u2082\u27e9", [{"full_name": "Or.inr", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [519, 5], "def_end_pos": [519, 8]}, {"full_name": "Part.mem_some", "def_path": "Mathlib/Data/Part.lean", "def_pos": [165, 9], "def_end_pos": [165, 17]}]], "state_before": "case neg\nf : Code\nk : Cont\nv : List \u2115\nfok : Code.Ok f\nx : Cfg\nthis :\n  \u2200 (c : Cfg),\n    x \u2208 eval step c \u2192\n      \u2200 (v : List \u2115) (c' : Cfg),\n        c = Cfg.then c' (Cont.fix f k) \u2192\n          Reaches step (stepNormal f Cont.halt v) c' \u2192\n            \u2203 v\u2081,\n              v\u2081 \u2208 Code.eval f v \u2227\n                \u2203 v\u2082,\n                  (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227\n                    x \u2208 eval step (Cfg.ret k v\u2082)\nh : x \u2208 eval step (stepNormal f (Cont.fix f k) v)\nv\u2081 : List \u2115\nhv\u2081 : v\u2081 \u2208 Code.eval f v\nv\u2082 : List \u2115\nh\u2083 : x \u2208 eval step (Cfg.ret k v\u2082)\nh\u271d : \u00acList.headI v\u2081 = 0\nhv\u2082 : v\u2082 \u2208 Code.eval (Code.fix f) (List.tail v\u2081)\n\u22a2 Sum.inl v\u2082 \u2208 Part.some (Sum.inr (List.tail v\u2081)) \u2228\n    \u2203 a',\n      Sum.inr a' \u2208 Part.some (Sum.inr (List.tail v\u2081)) \u2227\n        v\u2082 \u2208\n          PFun.fix\n            (fun v =>\n              Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v))\n                (Code.eval f v))\n            a'", "state_after": "no goals"}, {"tactic": "have := mem_eval.2 \u27e8hr, rfl\u27e9", "annotated_tactic": ["have := <a>mem_eval</a>.2 \u27e8hr, <a>rfl</a>\u27e9", [{"full_name": "Turing.mem_eval", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [830, 9], "def_end_pos": [830, 17]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "case mp.halt\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx c : Cfg\nhe : x \u2208 eval step c\nv v' : List \u2115\nh : x \u2208 eval step (stepRet (Cont.fix f k) v')\nIH :\n  \u2200 (a' : Cfg),\n    step (stepRet (Cont.fix f k) v') = some a' \u2192\n      \u2200 (v : List \u2115) (c' : Cfg),\n        a' = Cfg.then c' (Cont.fix f k) \u2192\n          Reaches step (stepNormal f Cont.halt v) c' \u2192\n            \u2203 v\u2081,\n              v\u2081 \u2208 Code.eval f v \u2227\n                \u2203 v\u2082,\n                  (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227\n                    x \u2208 eval step (Cfg.ret k v\u2082)\nhr : Reaches step (stepNormal f Cont.halt v) (Cfg.halt v')\n\u22a2 \u2203 v\u2081,\n    v\u2081 \u2208 Code.eval f v \u2227\n      \u2203 v\u2082,\n        (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227\n          x \u2208 eval step (Cfg.ret k v\u2082)", "state_after": "case mp.halt\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx c : Cfg\nhe : x \u2208 eval step c\nv v' : List \u2115\nh : x \u2208 eval step (stepRet (Cont.fix f k) v')\nIH :\n  \u2200 (a' : Cfg),\n    step (stepRet (Cont.fix f k) v') = some a' \u2192\n      \u2200 (v : List \u2115) (c' : Cfg),\n        a' = Cfg.then c' (Cont.fix f k) \u2192\n          Reaches step (stepNormal f Cont.halt v) c' \u2192\n            \u2203 v\u2081,\n              v\u2081 \u2208 Code.eval f v \u2227\n                \u2203 v\u2082,\n                  (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227\n                    x \u2208 eval step (Cfg.ret k v\u2082)\nhr : Reaches step (stepNormal f Cont.halt v) (Cfg.halt v')\nthis : Cfg.halt v' \u2208 eval step (stepNormal f Cont.halt v)\n\u22a2 \u2203 v\u2081,\n    v\u2081 \u2208 Code.eval f v \u2227\n      \u2203 v\u2082,\n        (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227\n          x \u2208 eval step (Cfg.ret k v\u2082)"}, {"tactic": "rw [fok, Part.bind_eq_bind, Part.mem_bind_iff] at this", "annotated_tactic": ["rw [fok, <a>Part.bind_eq_bind</a>, <a>Part.mem_bind_iff</a>] at this", [{"full_name": "Part.bind_eq_bind", "def_path": "Mathlib/Data/Part.lean", "def_pos": [614, 9], "def_end_pos": [614, 21]}, {"full_name": "Part.mem_bind_iff", "def_path": "Mathlib/Data/Part.lean", "def_pos": [494, 9], "def_end_pos": [494, 21]}]], "state_before": "case mp.halt\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx c : Cfg\nhe : x \u2208 eval step c\nv v' : List \u2115\nh : x \u2208 eval step (stepRet (Cont.fix f k) v')\nIH :\n  \u2200 (a' : Cfg),\n    step (stepRet (Cont.fix f k) v') = some a' \u2192\n      \u2200 (v : List \u2115) (c' : Cfg),\n        a' = Cfg.then c' (Cont.fix f k) \u2192\n          Reaches step (stepNormal f Cont.halt v) c' \u2192\n            \u2203 v\u2081,\n              v\u2081 \u2208 Code.eval f v \u2227\n                \u2203 v\u2082,\n                  (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227\n                    x \u2208 eval step (Cfg.ret k v\u2082)\nhr : Reaches step (stepNormal f Cont.halt v) (Cfg.halt v')\nthis : Cfg.halt v' \u2208 eval step (stepNormal f Cont.halt v)\n\u22a2 \u2203 v\u2081,\n    v\u2081 \u2208 Code.eval f v \u2227\n      \u2203 v\u2082,\n        (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227\n          x \u2208 eval step (Cfg.ret k v\u2082)", "state_after": "case mp.halt\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx c : Cfg\nhe : x \u2208 eval step c\nv v' : List \u2115\nh : x \u2208 eval step (stepRet (Cont.fix f k) v')\nIH :\n  \u2200 (a' : Cfg),\n    step (stepRet (Cont.fix f k) v') = some a' \u2192\n      \u2200 (v : List \u2115) (c' : Cfg),\n        a' = Cfg.then c' (Cont.fix f k) \u2192\n          Reaches step (stepNormal f Cont.halt v) c' \u2192\n            \u2203 v\u2081,\n              v\u2081 \u2208 Code.eval f v \u2227\n                \u2203 v\u2082,\n                  (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227\n                    x \u2208 eval step (Cfg.ret k v\u2082)\nhr : Reaches step (stepNormal f Cont.halt v) (Cfg.halt v')\nthis : \u2203 a, a \u2208 Code.eval f v \u2227 Cfg.halt v' \u2208 eval step (Cfg.ret Cont.halt a)\n\u22a2 \u2203 v\u2081,\n    v\u2081 \u2208 Code.eval f v \u2227\n      \u2203 v\u2082,\n        (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227\n          x \u2208 eval step (Cfg.ret k v\u2082)"}, {"tactic": "obtain \u27e8v'', h\u2081, h\u2082\u27e9 := this", "annotated_tactic": ["obtain \u27e8v'', h\u2081, h\u2082\u27e9 := this", []], "state_before": "case mp.halt\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx c : Cfg\nhe : x \u2208 eval step c\nv v' : List \u2115\nh : x \u2208 eval step (stepRet (Cont.fix f k) v')\nIH :\n  \u2200 (a' : Cfg),\n    step (stepRet (Cont.fix f k) v') = some a' \u2192\n      \u2200 (v : List \u2115) (c' : Cfg),\n        a' = Cfg.then c' (Cont.fix f k) \u2192\n          Reaches step (stepNormal f Cont.halt v) c' \u2192\n            \u2203 v\u2081,\n              v\u2081 \u2208 Code.eval f v \u2227\n                \u2203 v\u2082,\n                  (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227\n                    x \u2208 eval step (Cfg.ret k v\u2082)\nhr : Reaches step (stepNormal f Cont.halt v) (Cfg.halt v')\nthis : \u2203 a, a \u2208 Code.eval f v \u2227 Cfg.halt v' \u2208 eval step (Cfg.ret Cont.halt a)\n\u22a2 \u2203 v\u2081,\n    v\u2081 \u2208 Code.eval f v \u2227\n      \u2203 v\u2082,\n        (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227\n          x \u2208 eval step (Cfg.ret k v\u2082)", "state_after": "case mp.halt.intro.intro\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx c : Cfg\nhe : x \u2208 eval step c\nv v' : List \u2115\nh : x \u2208 eval step (stepRet (Cont.fix f k) v')\nIH :\n  \u2200 (a' : Cfg),\n    step (stepRet (Cont.fix f k) v') = some a' \u2192\n      \u2200 (v : List \u2115) (c' : Cfg),\n        a' = Cfg.then c' (Cont.fix f k) \u2192\n          Reaches step (stepNormal f Cont.halt v) c' \u2192\n            \u2203 v\u2081,\n              v\u2081 \u2208 Code.eval f v \u2227\n                \u2203 v\u2082,\n                  (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227\n                    x \u2208 eval step (Cfg.ret k v\u2082)\nhr : Reaches step (stepNormal f Cont.halt v) (Cfg.halt v')\nv'' : List \u2115\nh\u2081 : v'' \u2208 Code.eval f v\nh\u2082 : Cfg.halt v' \u2208 eval step (Cfg.ret Cont.halt v'')\n\u22a2 \u2203 v\u2081,\n    v\u2081 \u2208 Code.eval f v \u2227\n      \u2203 v\u2082,\n        (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227\n          x \u2208 eval step (Cfg.ret k v\u2082)"}, {"tactic": "rw [reaches_eval] at h\u2082", "annotated_tactic": ["rw [<a>reaches_eval</a>] at h\u2082", [{"full_name": "Turing.reaches_eval", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [863, 9], "def_end_pos": [863, 21]}]], "state_before": "case mp.halt.intro.intro\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx c : Cfg\nhe : x \u2208 eval step c\nv v' : List \u2115\nh : x \u2208 eval step (stepRet (Cont.fix f k) v')\nIH :\n  \u2200 (a' : Cfg),\n    step (stepRet (Cont.fix f k) v') = some a' \u2192\n      \u2200 (v : List \u2115) (c' : Cfg),\n        a' = Cfg.then c' (Cont.fix f k) \u2192\n          Reaches step (stepNormal f Cont.halt v) c' \u2192\n            \u2203 v\u2081,\n              v\u2081 \u2208 Code.eval f v \u2227\n                \u2203 v\u2082,\n                  (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227\n                    x \u2208 eval step (Cfg.ret k v\u2082)\nhr : Reaches step (stepNormal f Cont.halt v) (Cfg.halt v')\nv'' : List \u2115\nh\u2081 : v'' \u2208 Code.eval f v\nh\u2082 : Cfg.halt v' \u2208 eval step (Cfg.ret Cont.halt v'')\n\u22a2 \u2203 v\u2081,\n    v\u2081 \u2208 Code.eval f v \u2227\n      \u2203 v\u2082,\n        (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227\n          x \u2208 eval step (Cfg.ret k v\u2082)", "state_after": "case mp.halt.intro.intro\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx c : Cfg\nhe : x \u2208 eval step c\nv v' : List \u2115\nh : x \u2208 eval step (stepRet (Cont.fix f k) v')\nIH :\n  \u2200 (a' : Cfg),\n    step (stepRet (Cont.fix f k) v') = some a' \u2192\n      \u2200 (v : List \u2115) (c' : Cfg),\n        a' = Cfg.then c' (Cont.fix f k) \u2192\n          Reaches step (stepNormal f Cont.halt v) c' \u2192\n            \u2203 v\u2081,\n              v\u2081 \u2208 Code.eval f v \u2227\n                \u2203 v\u2082,\n                  (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227\n                    x \u2208 eval step (Cfg.ret k v\u2082)\nhr : Reaches step (stepNormal f Cont.halt v) (Cfg.halt v')\nv'' : List \u2115\nh\u2081 : v'' \u2208 Code.eval f v\nh\u2082\u271d : Cfg.halt v' \u2208 eval step (Cfg.ret Cont.halt v'')\nh\u2082 : Cfg.halt v' \u2208 eval step ?m.194376\n\u22a2 \u2203 v\u2081,\n    v\u2081 \u2208 Code.eval f v \u2227\n      \u2203 v\u2082,\n        (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227\n          x \u2208 eval step (Cfg.ret k v\u2082)\n\ncase mp.halt.intro.intro\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx c : Cfg\nhe : x \u2208 eval step c\nv v' : List \u2115\nh : x \u2208 eval step (stepRet (Cont.fix f k) v')\nIH :\n  \u2200 (a' : Cfg),\n    step (stepRet (Cont.fix f k) v') = some a' \u2192\n      \u2200 (v : List \u2115) (c' : Cfg),\n        a' = Cfg.then c' (Cont.fix f k) \u2192\n          Reaches step (stepNormal f Cont.halt v) c' \u2192\n            \u2203 v\u2081,\n              v\u2081 \u2208 Code.eval f v \u2227\n                \u2203 v\u2082,\n                  (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227\n                    x \u2208 eval step (Cfg.ret k v\u2082)\nhr : Reaches step (stepNormal f Cont.halt v) (Cfg.halt v')\nv'' : List \u2115\nh\u2081 : v'' \u2208 Code.eval f v\nh\u2082 : Cfg.halt v' \u2208 eval step (Cfg.ret Cont.halt v'')\n\u22a2 Reaches step (Cfg.ret Cont.halt v'') ?m.194376\n\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx c : Cfg\nhe : x \u2208 eval step c\nv v' : List \u2115\nh : x \u2208 eval step (stepRet (Cont.fix f k) v')\nIH :\n  \u2200 (a' : Cfg),\n    step (stepRet (Cont.fix f k) v') = some a' \u2192\n      \u2200 (v : List \u2115) (c' : Cfg),\n        a' = Cfg.then c' (Cont.fix f k) \u2192\n          Reaches step (stepNormal f Cont.halt v) c' \u2192\n            \u2203 v\u2081,\n              v\u2081 \u2208 Code.eval f v \u2227\n                \u2203 v\u2082,\n                  (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227\n                    x \u2208 eval step (Cfg.ret k v\u2082)\nhr : Reaches step (stepNormal f Cont.halt v) (Cfg.halt v')\nv'' : List \u2115\nh\u2081 : v'' \u2208 Code.eval f v\nh\u2082 : Cfg.halt v' \u2208 eval step (Cfg.ret Cont.halt v'')\n\u22a2 Cfg"}, {"tactic": "swap", "annotated_tactic": ["swap", []], "state_before": "case mp.halt.intro.intro\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx c : Cfg\nhe : x \u2208 eval step c\nv v' : List \u2115\nh : x \u2208 eval step (stepRet (Cont.fix f k) v')\nIH :\n  \u2200 (a' : Cfg),\n    step (stepRet (Cont.fix f k) v') = some a' \u2192\n      \u2200 (v : List \u2115) (c' : Cfg),\n        a' = Cfg.then c' (Cont.fix f k) \u2192\n          Reaches step (stepNormal f Cont.halt v) c' \u2192\n            \u2203 v\u2081,\n              v\u2081 \u2208 Code.eval f v \u2227\n                \u2203 v\u2082,\n                  (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227\n                    x \u2208 eval step (Cfg.ret k v\u2082)\nhr : Reaches step (stepNormal f Cont.halt v) (Cfg.halt v')\nv'' : List \u2115\nh\u2081 : v'' \u2208 Code.eval f v\nh\u2082\u271d : Cfg.halt v' \u2208 eval step (Cfg.ret Cont.halt v'')\nh\u2082 : Cfg.halt v' \u2208 eval step ?m.194376\n\u22a2 \u2203 v\u2081,\n    v\u2081 \u2208 Code.eval f v \u2227\n      \u2203 v\u2082,\n        (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227\n          x \u2208 eval step (Cfg.ret k v\u2082)\n\ncase mp.halt.intro.intro\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx c : Cfg\nhe : x \u2208 eval step c\nv v' : List \u2115\nh : x \u2208 eval step (stepRet (Cont.fix f k) v')\nIH :\n  \u2200 (a' : Cfg),\n    step (stepRet (Cont.fix f k) v') = some a' \u2192\n      \u2200 (v : List \u2115) (c' : Cfg),\n        a' = Cfg.then c' (Cont.fix f k) \u2192\n          Reaches step (stepNormal f Cont.halt v) c' \u2192\n            \u2203 v\u2081,\n              v\u2081 \u2208 Code.eval f v \u2227\n                \u2203 v\u2082,\n                  (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227\n                    x \u2208 eval step (Cfg.ret k v\u2082)\nhr : Reaches step (stepNormal f Cont.halt v) (Cfg.halt v')\nv'' : List \u2115\nh\u2081 : v'' \u2208 Code.eval f v\nh\u2082 : Cfg.halt v' \u2208 eval step (Cfg.ret Cont.halt v'')\n\u22a2 Reaches step (Cfg.ret Cont.halt v'') ?m.194376\n\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx c : Cfg\nhe : x \u2208 eval step c\nv v' : List \u2115\nh : x \u2208 eval step (stepRet (Cont.fix f k) v')\nIH :\n  \u2200 (a' : Cfg),\n    step (stepRet (Cont.fix f k) v') = some a' \u2192\n      \u2200 (v : List \u2115) (c' : Cfg),\n        a' = Cfg.then c' (Cont.fix f k) \u2192\n          Reaches step (stepNormal f Cont.halt v) c' \u2192\n            \u2203 v\u2081,\n              v\u2081 \u2208 Code.eval f v \u2227\n                \u2203 v\u2082,\n                  (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227\n                    x \u2208 eval step (Cfg.ret k v\u2082)\nhr : Reaches step (stepNormal f Cont.halt v) (Cfg.halt v')\nv'' : List \u2115\nh\u2081 : v'' \u2208 Code.eval f v\nh\u2082 : Cfg.halt v' \u2208 eval step (Cfg.ret Cont.halt v'')\n\u22a2 Cfg", "state_after": "case mp.halt.intro.intro\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx c : Cfg\nhe : x \u2208 eval step c\nv v' : List \u2115\nh : x \u2208 eval step (stepRet (Cont.fix f k) v')\nIH :\n  \u2200 (a' : Cfg),\n    step (stepRet (Cont.fix f k) v') = some a' \u2192\n      \u2200 (v : List \u2115) (c' : Cfg),\n        a' = Cfg.then c' (Cont.fix f k) \u2192\n          Reaches step (stepNormal f Cont.halt v) c' \u2192\n            \u2203 v\u2081,\n              v\u2081 \u2208 Code.eval f v \u2227\n                \u2203 v\u2082,\n                  (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227\n                    x \u2208 eval step (Cfg.ret k v\u2082)\nhr : Reaches step (stepNormal f Cont.halt v) (Cfg.halt v')\nv'' : List \u2115\nh\u2081 : v'' \u2208 Code.eval f v\nh\u2082 : Cfg.halt v' \u2208 eval step (Cfg.ret Cont.halt v'')\n\u22a2 Reaches step (Cfg.ret Cont.halt v'') ?m.194376\n\ncase mp.halt.intro.intro\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx c : Cfg\nhe : x \u2208 eval step c\nv v' : List \u2115\nh : x \u2208 eval step (stepRet (Cont.fix f k) v')\nIH :\n  \u2200 (a' : Cfg),\n    step (stepRet (Cont.fix f k) v') = some a' \u2192\n      \u2200 (v : List \u2115) (c' : Cfg),\n        a' = Cfg.then c' (Cont.fix f k) \u2192\n          Reaches step (stepNormal f Cont.halt v) c' \u2192\n            \u2203 v\u2081,\n              v\u2081 \u2208 Code.eval f v \u2227\n                \u2203 v\u2082,\n                  (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227\n                    x \u2208 eval step (Cfg.ret k v\u2082)\nhr : Reaches step (stepNormal f Cont.halt v) (Cfg.halt v')\nv'' : List \u2115\nh\u2081 : v'' \u2208 Code.eval f v\nh\u2082\u271d : Cfg.halt v' \u2208 eval step (Cfg.ret Cont.halt v'')\nh\u2082 : Cfg.halt v' \u2208 eval step ?m.194376\n\u22a2 \u2203 v\u2081,\n    v\u2081 \u2208 Code.eval f v \u2227\n      \u2203 v\u2082,\n        (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227\n          x \u2208 eval step (Cfg.ret k v\u2082)\n\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx c : Cfg\nhe : x \u2208 eval step c\nv v' : List \u2115\nh : x \u2208 eval step (stepRet (Cont.fix f k) v')\nIH :\n  \u2200 (a' : Cfg),\n    step (stepRet (Cont.fix f k) v') = some a' \u2192\n      \u2200 (v : List \u2115) (c' : Cfg),\n        a' = Cfg.then c' (Cont.fix f k) \u2192\n          Reaches step (stepNormal f Cont.halt v) c' \u2192\n            \u2203 v\u2081,\n              v\u2081 \u2208 Code.eval f v \u2227\n                \u2203 v\u2082,\n                  (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227\n                    x \u2208 eval step (Cfg.ret k v\u2082)\nhr : Reaches step (stepNormal f Cont.halt v) (Cfg.halt v')\nv'' : List \u2115\nh\u2081 : v'' \u2208 Code.eval f v\nh\u2082 : Cfg.halt v' \u2208 eval step (Cfg.ret Cont.halt v'')\n\u22a2 Cfg"}, {"tactic": "exact ReflTransGen.single rfl", "annotated_tactic": ["exact <a>ReflTransGen.single</a> <a>rfl</a>", [{"full_name": "Relation.ReflTransGen.single", "def_path": "Mathlib/Logic/Relation.lean", "def_pos": [276, 9], "def_end_pos": [276, 15]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "case mp.halt.intro.intro\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx c : Cfg\nhe : x \u2208 eval step c\nv v' : List \u2115\nh : x \u2208 eval step (stepRet (Cont.fix f k) v')\nIH :\n  \u2200 (a' : Cfg),\n    step (stepRet (Cont.fix f k) v') = some a' \u2192\n      \u2200 (v : List \u2115) (c' : Cfg),\n        a' = Cfg.then c' (Cont.fix f k) \u2192\n          Reaches step (stepNormal f Cont.halt v) c' \u2192\n            \u2203 v\u2081,\n              v\u2081 \u2208 Code.eval f v \u2227\n                \u2203 v\u2082,\n                  (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227\n                    x \u2208 eval step (Cfg.ret k v\u2082)\nhr : Reaches step (stepNormal f Cont.halt v) (Cfg.halt v')\nv'' : List \u2115\nh\u2081 : v'' \u2208 Code.eval f v\nh\u2082 : Cfg.halt v' \u2208 eval step (Cfg.ret Cont.halt v'')\n\u22a2 Reaches step (Cfg.ret Cont.halt v'') ?m.194376\n\ncase mp.halt.intro.intro\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx c : Cfg\nhe : x \u2208 eval step c\nv v' : List \u2115\nh : x \u2208 eval step (stepRet (Cont.fix f k) v')\nIH :\n  \u2200 (a' : Cfg),\n    step (stepRet (Cont.fix f k) v') = some a' \u2192\n      \u2200 (v : List \u2115) (c' : Cfg),\n        a' = Cfg.then c' (Cont.fix f k) \u2192\n          Reaches step (stepNormal f Cont.halt v) c' \u2192\n            \u2203 v\u2081,\n              v\u2081 \u2208 Code.eval f v \u2227\n                \u2203 v\u2082,\n                  (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227\n                    x \u2208 eval step (Cfg.ret k v\u2082)\nhr : Reaches step (stepNormal f Cont.halt v) (Cfg.halt v')\nv'' : List \u2115\nh\u2081 : v'' \u2208 Code.eval f v\nh\u2082\u271d : Cfg.halt v' \u2208 eval step (Cfg.ret Cont.halt v'')\nh\u2082 : Cfg.halt v' \u2208 eval step ?m.194376\n\u22a2 \u2203 v\u2081,\n    v\u2081 \u2208 Code.eval f v \u2227\n      \u2203 v\u2082,\n        (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227\n          x \u2208 eval step (Cfg.ret k v\u2082)\n\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx c : Cfg\nhe : x \u2208 eval step c\nv v' : List \u2115\nh : x \u2208 eval step (stepRet (Cont.fix f k) v')\nIH :\n  \u2200 (a' : Cfg),\n    step (stepRet (Cont.fix f k) v') = some a' \u2192\n      \u2200 (v : List \u2115) (c' : Cfg),\n        a' = Cfg.then c' (Cont.fix f k) \u2192\n          Reaches step (stepNormal f Cont.halt v) c' \u2192\n            \u2203 v\u2081,\n              v\u2081 \u2208 Code.eval f v \u2227\n                \u2203 v\u2082,\n                  (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227\n                    x \u2208 eval step (Cfg.ret k v\u2082)\nhr : Reaches step (stepNormal f Cont.halt v) (Cfg.halt v')\nv'' : List \u2115\nh\u2081 : v'' \u2208 Code.eval f v\nh\u2082 : Cfg.halt v' \u2208 eval step (Cfg.ret Cont.halt v'')\n\u22a2 Cfg", "state_after": "case mp.halt.intro.intro\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx c : Cfg\nhe : x \u2208 eval step c\nv v' : List \u2115\nh : x \u2208 eval step (stepRet (Cont.fix f k) v')\nIH :\n  \u2200 (a' : Cfg),\n    step (stepRet (Cont.fix f k) v') = some a' \u2192\n      \u2200 (v : List \u2115) (c' : Cfg),\n        a' = Cfg.then c' (Cont.fix f k) \u2192\n          Reaches step (stepNormal f Cont.halt v) c' \u2192\n            \u2203 v\u2081,\n              v\u2081 \u2208 Code.eval f v \u2227\n                \u2203 v\u2082,\n                  (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227\n                    x \u2208 eval step (Cfg.ret k v\u2082)\nhr : Reaches step (stepNormal f Cont.halt v) (Cfg.halt v')\nv'' : List \u2115\nh\u2081 : v'' \u2208 Code.eval f v\nh\u2082\u271d : Cfg.halt v' \u2208 eval step (Cfg.ret Cont.halt v'')\nh\u2082 : Cfg.halt v' \u2208 eval step (stepRet Cont.halt v'')\n\u22a2 \u2203 v\u2081,\n    v\u2081 \u2208 Code.eval f v \u2227\n      \u2203 v\u2082,\n        (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227\n          x \u2208 eval step (Cfg.ret k v\u2082)"}, {"tactic": "cases Part.mem_unique h\u2082 (mem_eval.2 \u27e8ReflTransGen.refl, rfl\u27e9)", "annotated_tactic": ["cases <a>Part.mem_unique</a> h\u2082 (<a>mem_eval</a>.2 \u27e8<a>ReflTransGen.refl</a>, <a>rfl</a>\u27e9)", [{"full_name": "Part.mem_unique", "def_path": "Mathlib/Data/Part.lean", "def_pos": [144, 9], "def_end_pos": [144, 19]}, {"full_name": "Turing.mem_eval", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [830, 9], "def_end_pos": [830, 17]}, {"full_name": "Relation.ReflTransGen.refl", "def_path": "Mathlib/Logic/Relation.lean", "def_pos": [223, 5], "def_end_pos": [223, 9]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "case mp.halt.intro.intro\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx c : Cfg\nhe : x \u2208 eval step c\nv v' : List \u2115\nh : x \u2208 eval step (stepRet (Cont.fix f k) v')\nIH :\n  \u2200 (a' : Cfg),\n    step (stepRet (Cont.fix f k) v') = some a' \u2192\n      \u2200 (v : List \u2115) (c' : Cfg),\n        a' = Cfg.then c' (Cont.fix f k) \u2192\n          Reaches step (stepNormal f Cont.halt v) c' \u2192\n            \u2203 v\u2081,\n              v\u2081 \u2208 Code.eval f v \u2227\n                \u2203 v\u2082,\n                  (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227\n                    x \u2208 eval step (Cfg.ret k v\u2082)\nhr : Reaches step (stepNormal f Cont.halt v) (Cfg.halt v')\nv'' : List \u2115\nh\u2081 : v'' \u2208 Code.eval f v\nh\u2082\u271d : Cfg.halt v' \u2208 eval step (Cfg.ret Cont.halt v'')\nh\u2082 : Cfg.halt v' \u2208 eval step (stepRet Cont.halt v'')\n\u22a2 \u2203 v\u2081,\n    v\u2081 \u2208 Code.eval f v \u2227\n      \u2203 v\u2082,\n        (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227\n          x \u2208 eval step (Cfg.ret k v\u2082)", "state_after": "case mp.halt.intro.intro.refl\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx c : Cfg\nhe : x \u2208 eval step c\nv v' : List \u2115\nh : x \u2208 eval step (stepRet (Cont.fix f k) v')\nIH :\n  \u2200 (a' : Cfg),\n    step (stepRet (Cont.fix f k) v') = some a' \u2192\n      \u2200 (v : List \u2115) (c' : Cfg),\n        a' = Cfg.then c' (Cont.fix f k) \u2192\n          Reaches step (stepNormal f Cont.halt v) c' \u2192\n            \u2203 v\u2081,\n              v\u2081 \u2208 Code.eval f v \u2227\n                \u2203 v\u2082,\n                  (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227\n                    x \u2208 eval step (Cfg.ret k v\u2082)\nhr : Reaches step (stepNormal f Cont.halt v) (Cfg.halt v')\nh\u2081 : v' \u2208 Code.eval f v\nh\u2082\u271d : Cfg.halt v' \u2208 eval step (Cfg.ret Cont.halt v')\nh\u2082 : Cfg.halt v' \u2208 eval step (stepRet Cont.halt v')\n\u22a2 \u2203 v\u2081,\n    v\u2081 \u2208 Code.eval f v \u2227\n      \u2203 v\u2082,\n        (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227\n          x \u2208 eval step (Cfg.ret k v\u2082)"}, {"tactic": "refine' \u27e8v', h\u2081, _\u27e9", "annotated_tactic": ["refine' \u27e8v', h\u2081, _\u27e9", []], "state_before": "case mp.halt.intro.intro.refl\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx c : Cfg\nhe : x \u2208 eval step c\nv v' : List \u2115\nh : x \u2208 eval step (stepRet (Cont.fix f k) v')\nIH :\n  \u2200 (a' : Cfg),\n    step (stepRet (Cont.fix f k) v') = some a' \u2192\n      \u2200 (v : List \u2115) (c' : Cfg),\n        a' = Cfg.then c' (Cont.fix f k) \u2192\n          Reaches step (stepNormal f Cont.halt v) c' \u2192\n            \u2203 v\u2081,\n              v\u2081 \u2208 Code.eval f v \u2227\n                \u2203 v\u2082,\n                  (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227\n                    x \u2208 eval step (Cfg.ret k v\u2082)\nhr : Reaches step (stepNormal f Cont.halt v) (Cfg.halt v')\nh\u2081 : v' \u2208 Code.eval f v\nh\u2082\u271d : Cfg.halt v' \u2208 eval step (Cfg.ret Cont.halt v')\nh\u2082 : Cfg.halt v' \u2208 eval step (stepRet Cont.halt v')\n\u22a2 \u2203 v\u2081,\n    v\u2081 \u2208 Code.eval f v \u2227\n      \u2203 v\u2082,\n        (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227\n          x \u2208 eval step (Cfg.ret k v\u2082)", "state_after": "case mp.halt.intro.intro.refl\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx c : Cfg\nhe : x \u2208 eval step c\nv v' : List \u2115\nh : x \u2208 eval step (stepRet (Cont.fix f k) v')\nIH :\n  \u2200 (a' : Cfg),\n    step (stepRet (Cont.fix f k) v') = some a' \u2192\n      \u2200 (v : List \u2115) (c' : Cfg),\n        a' = Cfg.then c' (Cont.fix f k) \u2192\n          Reaches step (stepNormal f Cont.halt v) c' \u2192\n            \u2203 v\u2081,\n              v\u2081 \u2208 Code.eval f v \u2227\n                \u2203 v\u2082,\n                  (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227\n                    x \u2208 eval step (Cfg.ret k v\u2082)\nhr : Reaches step (stepNormal f Cont.halt v) (Cfg.halt v')\nh\u2081 : v' \u2208 Code.eval f v\nh\u2082\u271d : Cfg.halt v' \u2208 eval step (Cfg.ret Cont.halt v')\nh\u2082 : Cfg.halt v' \u2208 eval step (stepRet Cont.halt v')\n\u22a2 \u2203 v\u2082,\n    (v\u2082 \u2208 if List.headI v' = 0 then pure (List.tail v') else Code.eval (Code.fix f) (List.tail v')) \u2227\n      x \u2208 eval step (Cfg.ret k v\u2082)"}, {"tactic": "rw [stepRet] at h", "annotated_tactic": ["rw [<a>stepRet</a>] at h", [{"full_name": "Turing.ToPartrec.stepRet", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [513, 5], "def_end_pos": [513, 12]}]], "state_before": "case mp.halt.intro.intro.refl\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx c : Cfg\nhe : x \u2208 eval step c\nv v' : List \u2115\nh : x \u2208 eval step (stepRet (Cont.fix f k) v')\nIH :\n  \u2200 (a' : Cfg),\n    step (stepRet (Cont.fix f k) v') = some a' \u2192\n      \u2200 (v : List \u2115) (c' : Cfg),\n        a' = Cfg.then c' (Cont.fix f k) \u2192\n          Reaches step (stepNormal f Cont.halt v) c' \u2192\n            \u2203 v\u2081,\n              v\u2081 \u2208 Code.eval f v \u2227\n                \u2203 v\u2082,\n                  (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227\n                    x \u2208 eval step (Cfg.ret k v\u2082)\nhr : Reaches step (stepNormal f Cont.halt v) (Cfg.halt v')\nh\u2081 : v' \u2208 Code.eval f v\nh\u2082\u271d : Cfg.halt v' \u2208 eval step (Cfg.ret Cont.halt v')\nh\u2082 : Cfg.halt v' \u2208 eval step (stepRet Cont.halt v')\n\u22a2 \u2203 v\u2082,\n    (v\u2082 \u2208 if List.headI v' = 0 then pure (List.tail v') else Code.eval (Code.fix f) (List.tail v')) \u2227\n      x \u2208 eval step (Cfg.ret k v\u2082)", "state_after": "case mp.halt.intro.intro.refl\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx c : Cfg\nhe : x \u2208 eval step c\nv v' : List \u2115\nh : x \u2208 eval step (if List.headI v' = 0 then stepRet k (List.tail v') else stepNormal f (Cont.fix f k) (List.tail v'))\nIH :\n  \u2200 (a' : Cfg),\n    step (stepRet (Cont.fix f k) v') = some a' \u2192\n      \u2200 (v : List \u2115) (c' : Cfg),\n        a' = Cfg.then c' (Cont.fix f k) \u2192\n          Reaches step (stepNormal f Cont.halt v) c' \u2192\n            \u2203 v\u2081,\n              v\u2081 \u2208 Code.eval f v \u2227\n                \u2203 v\u2082,\n                  (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227\n                    x \u2208 eval step (Cfg.ret k v\u2082)\nhr : Reaches step (stepNormal f Cont.halt v) (Cfg.halt v')\nh\u2081 : v' \u2208 Code.eval f v\nh\u2082\u271d : Cfg.halt v' \u2208 eval step (Cfg.ret Cont.halt v')\nh\u2082 : Cfg.halt v' \u2208 eval step (stepRet Cont.halt v')\n\u22a2 \u2203 v\u2082,\n    (v\u2082 \u2208 if List.headI v' = 0 then pure (List.tail v') else Code.eval (Code.fix f) (List.tail v')) \u2227\n      x \u2208 eval step (Cfg.ret k v\u2082)"}, {"tactic": "revert h", "annotated_tactic": ["revert h", []], "state_before": "case mp.halt.intro.intro.refl\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx c : Cfg\nhe : x \u2208 eval step c\nv v' : List \u2115\nh : x \u2208 eval step (if List.headI v' = 0 then stepRet k (List.tail v') else stepNormal f (Cont.fix f k) (List.tail v'))\nIH :\n  \u2200 (a' : Cfg),\n    step (stepRet (Cont.fix f k) v') = some a' \u2192\n      \u2200 (v : List \u2115) (c' : Cfg),\n        a' = Cfg.then c' (Cont.fix f k) \u2192\n          Reaches step (stepNormal f Cont.halt v) c' \u2192\n            \u2203 v\u2081,\n              v\u2081 \u2208 Code.eval f v \u2227\n                \u2203 v\u2082,\n                  (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227\n                    x \u2208 eval step (Cfg.ret k v\u2082)\nhr : Reaches step (stepNormal f Cont.halt v) (Cfg.halt v')\nh\u2081 : v' \u2208 Code.eval f v\nh\u2082\u271d : Cfg.halt v' \u2208 eval step (Cfg.ret Cont.halt v')\nh\u2082 : Cfg.halt v' \u2208 eval step (stepRet Cont.halt v')\n\u22a2 \u2203 v\u2082,\n    (v\u2082 \u2208 if List.headI v' = 0 then pure (List.tail v') else Code.eval (Code.fix f) (List.tail v')) \u2227\n      x \u2208 eval step (Cfg.ret k v\u2082)", "state_after": "case mp.halt.intro.intro.refl\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx c : Cfg\nhe : x \u2208 eval step c\nv v' : List \u2115\nIH :\n  \u2200 (a' : Cfg),\n    step (stepRet (Cont.fix f k) v') = some a' \u2192\n      \u2200 (v : List \u2115) (c' : Cfg),\n        a' = Cfg.then c' (Cont.fix f k) \u2192\n          Reaches step (stepNormal f Cont.halt v) c' \u2192\n            \u2203 v\u2081,\n              v\u2081 \u2208 Code.eval f v \u2227\n                \u2203 v\u2082,\n                  (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227\n                    x \u2208 eval step (Cfg.ret k v\u2082)\nhr : Reaches step (stepNormal f Cont.halt v) (Cfg.halt v')\nh\u2081 : v' \u2208 Code.eval f v\nh\u2082\u271d : Cfg.halt v' \u2208 eval step (Cfg.ret Cont.halt v')\nh\u2082 : Cfg.halt v' \u2208 eval step (stepRet Cont.halt v')\n\u22a2 x \u2208 eval step (if List.headI v' = 0 then stepRet k (List.tail v') else stepNormal f (Cont.fix f k) (List.tail v')) \u2192\n    \u2203 v\u2082,\n      (v\u2082 \u2208 if List.headI v' = 0 then pure (List.tail v') else Code.eval (Code.fix f) (List.tail v')) \u2227\n        x \u2208 eval step (Cfg.ret k v\u2082)"}, {"tactic": "by_cases he : v'.headI = 0 <;> simp only [exists_prop, if_pos, if_false, he] <;> intro h", "annotated_tactic": ["by_cases he : v'.headI = 0 <;> simp only [<a>exists_prop</a>, <a>if_pos</a>, <a>if_false</a>, he] <;> intro h", [{"full_name": "exists_prop", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [485, 17], "def_end_pos": [485, 28]}, {"full_name": "if_pos", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [790, 9], "def_end_pos": [790, 15]}, {"full_name": "if_false", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [729, 17], "def_end_pos": [729, 25]}]], "state_before": "case mp.halt.intro.intro.refl\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx c : Cfg\nhe : x \u2208 eval step c\nv v' : List \u2115\nIH :\n  \u2200 (a' : Cfg),\n    step (stepRet (Cont.fix f k) v') = some a' \u2192\n      \u2200 (v : List \u2115) (c' : Cfg),\n        a' = Cfg.then c' (Cont.fix f k) \u2192\n          Reaches step (stepNormal f Cont.halt v) c' \u2192\n            \u2203 v\u2081,\n              v\u2081 \u2208 Code.eval f v \u2227\n                \u2203 v\u2082,\n                  (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227\n                    x \u2208 eval step (Cfg.ret k v\u2082)\nhr : Reaches step (stepNormal f Cont.halt v) (Cfg.halt v')\nh\u2081 : v' \u2208 Code.eval f v\nh\u2082\u271d : Cfg.halt v' \u2208 eval step (Cfg.ret Cont.halt v')\nh\u2082 : Cfg.halt v' \u2208 eval step (stepRet Cont.halt v')\n\u22a2 x \u2208 eval step (if List.headI v' = 0 then stepRet k (List.tail v') else stepNormal f (Cont.fix f k) (List.tail v')) \u2192\n    \u2203 v\u2082,\n      (v\u2082 \u2208 if List.headI v' = 0 then pure (List.tail v') else Code.eval (Code.fix f) (List.tail v')) \u2227\n        x \u2208 eval step (Cfg.ret k v\u2082)", "state_after": "case pos\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx c : Cfg\nhe\u271d : x \u2208 eval step c\nv v' : List \u2115\nIH :\n  \u2200 (a' : Cfg),\n    step (stepRet (Cont.fix f k) v') = some a' \u2192\n      \u2200 (v : List \u2115) (c' : Cfg),\n        a' = Cfg.then c' (Cont.fix f k) \u2192\n          Reaches step (stepNormal f Cont.halt v) c' \u2192\n            \u2203 v\u2081,\n              v\u2081 \u2208 Code.eval f v \u2227\n                \u2203 v\u2082,\n                  (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227\n                    x \u2208 eval step (Cfg.ret k v\u2082)\nhr : Reaches step (stepNormal f Cont.halt v) (Cfg.halt v')\nh\u2081 : v' \u2208 Code.eval f v\nh\u2082\u271d : Cfg.halt v' \u2208 eval step (Cfg.ret Cont.halt v')\nh\u2082 : Cfg.halt v' \u2208 eval step (stepRet Cont.halt v')\nhe : List.headI v' = 0\nh : x \u2208 eval step (stepRet k (List.tail v'))\n\u22a2 \u2203 v\u2082, v\u2082 \u2208 pure (List.tail v') \u2227 x \u2208 eval step (Cfg.ret k v\u2082)\n\ncase neg\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx c : Cfg\nhe\u271d : x \u2208 eval step c\nv v' : List \u2115\nIH :\n  \u2200 (a' : Cfg),\n    step (stepRet (Cont.fix f k) v') = some a' \u2192\n      \u2200 (v : List \u2115) (c' : Cfg),\n        a' = Cfg.then c' (Cont.fix f k) \u2192\n          Reaches step (stepNormal f Cont.halt v) c' \u2192\n            \u2203 v\u2081,\n              v\u2081 \u2208 Code.eval f v \u2227\n                \u2203 v\u2082,\n                  (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227\n                    x \u2208 eval step (Cfg.ret k v\u2082)\nhr : Reaches step (stepNormal f Cont.halt v) (Cfg.halt v')\nh\u2081 : v' \u2208 Code.eval f v\nh\u2082\u271d : Cfg.halt v' \u2208 eval step (Cfg.ret Cont.halt v')\nh\u2082 : Cfg.halt v' \u2208 eval step (stepRet Cont.halt v')\nhe : \u00acList.headI v' = 0\nh : x \u2208 eval step (stepNormal f (Cont.fix f k) (List.tail v'))\n\u22a2 \u2203 v\u2082, v\u2082 \u2208 Code.eval (Code.fix f) (List.tail v') \u2227 x \u2208 eval step (Cfg.ret k v\u2082)"}, {"tactic": "refine' \u27e8_, Part.mem_some _, _\u27e9", "annotated_tactic": ["refine' \u27e8_, <a>Part.mem_some</a> _, _\u27e9", [{"full_name": "Part.mem_some", "def_path": "Mathlib/Data/Part.lean", "def_pos": [165, 9], "def_end_pos": [165, 17]}]], "state_before": "case pos\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx c : Cfg\nhe\u271d : x \u2208 eval step c\nv v' : List \u2115\nIH :\n  \u2200 (a' : Cfg),\n    step (stepRet (Cont.fix f k) v') = some a' \u2192\n      \u2200 (v : List \u2115) (c' : Cfg),\n        a' = Cfg.then c' (Cont.fix f k) \u2192\n          Reaches step (stepNormal f Cont.halt v) c' \u2192\n            \u2203 v\u2081,\n              v\u2081 \u2208 Code.eval f v \u2227\n                \u2203 v\u2082,\n                  (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227\n                    x \u2208 eval step (Cfg.ret k v\u2082)\nhr : Reaches step (stepNormal f Cont.halt v) (Cfg.halt v')\nh\u2081 : v' \u2208 Code.eval f v\nh\u2082\u271d : Cfg.halt v' \u2208 eval step (Cfg.ret Cont.halt v')\nh\u2082 : Cfg.halt v' \u2208 eval step (stepRet Cont.halt v')\nhe : List.headI v' = 0\nh : x \u2208 eval step (stepRet k (List.tail v'))\n\u22a2 \u2203 v\u2082, v\u2082 \u2208 pure (List.tail v') \u2227 x \u2208 eval step (Cfg.ret k v\u2082)", "state_after": "case pos\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx c : Cfg\nhe\u271d : x \u2208 eval step c\nv v' : List \u2115\nIH :\n  \u2200 (a' : Cfg),\n    step (stepRet (Cont.fix f k) v') = some a' \u2192\n      \u2200 (v : List \u2115) (c' : Cfg),\n        a' = Cfg.then c' (Cont.fix f k) \u2192\n          Reaches step (stepNormal f Cont.halt v) c' \u2192\n            \u2203 v\u2081,\n              v\u2081 \u2208 Code.eval f v \u2227\n                \u2203 v\u2082,\n                  (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227\n                    x \u2208 eval step (Cfg.ret k v\u2082)\nhr : Reaches step (stepNormal f Cont.halt v) (Cfg.halt v')\nh\u2081 : v' \u2208 Code.eval f v\nh\u2082\u271d : Cfg.halt v' \u2208 eval step (Cfg.ret Cont.halt v')\nh\u2082 : Cfg.halt v' \u2208 eval step (stepRet Cont.halt v')\nhe : List.headI v' = 0\nh : x \u2208 eval step (stepRet k (List.tail v'))\n\u22a2 x \u2208 eval step (Cfg.ret k (List.tail v'))"}, {"tactic": "rw [reaches_eval]", "annotated_tactic": ["rw [<a>reaches_eval</a>]", [{"full_name": "Turing.reaches_eval", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [863, 9], "def_end_pos": [863, 21]}]], "state_before": "case pos\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx c : Cfg\nhe\u271d : x \u2208 eval step c\nv v' : List \u2115\nIH :\n  \u2200 (a' : Cfg),\n    step (stepRet (Cont.fix f k) v') = some a' \u2192\n      \u2200 (v : List \u2115) (c' : Cfg),\n        a' = Cfg.then c' (Cont.fix f k) \u2192\n          Reaches step (stepNormal f Cont.halt v) c' \u2192\n            \u2203 v\u2081,\n              v\u2081 \u2208 Code.eval f v \u2227\n                \u2203 v\u2082,\n                  (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227\n                    x \u2208 eval step (Cfg.ret k v\u2082)\nhr : Reaches step (stepNormal f Cont.halt v) (Cfg.halt v')\nh\u2081 : v' \u2208 Code.eval f v\nh\u2082\u271d : Cfg.halt v' \u2208 eval step (Cfg.ret Cont.halt v')\nh\u2082 : Cfg.halt v' \u2208 eval step (stepRet Cont.halt v')\nhe : List.headI v' = 0\nh : x \u2208 eval step (stepRet k (List.tail v'))\n\u22a2 x \u2208 eval step (Cfg.ret k (List.tail v'))", "state_after": "case pos\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx c : Cfg\nhe\u271d : x \u2208 eval step c\nv v' : List \u2115\nIH :\n  \u2200 (a' : Cfg),\n    step (stepRet (Cont.fix f k) v') = some a' \u2192\n      \u2200 (v : List \u2115) (c' : Cfg),\n        a' = Cfg.then c' (Cont.fix f k) \u2192\n          Reaches step (stepNormal f Cont.halt v) c' \u2192\n            \u2203 v\u2081,\n              v\u2081 \u2208 Code.eval f v \u2227\n                \u2203 v\u2082,\n                  (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227\n                    x \u2208 eval step (Cfg.ret k v\u2082)\nhr : Reaches step (stepNormal f Cont.halt v) (Cfg.halt v')\nh\u2081 : v' \u2208 Code.eval f v\nh\u2082\u271d : Cfg.halt v' \u2208 eval step (Cfg.ret Cont.halt v')\nh\u2082 : Cfg.halt v' \u2208 eval step (stepRet Cont.halt v')\nhe : List.headI v' = 0\nh : x \u2208 eval step (stepRet k (List.tail v'))\n\u22a2 x \u2208 eval step ?m.195368\n\ncase pos\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx c : Cfg\nhe\u271d : x \u2208 eval step c\nv v' : List \u2115\nIH :\n  \u2200 (a' : Cfg),\n    step (stepRet (Cont.fix f k) v') = some a' \u2192\n      \u2200 (v : List \u2115) (c' : Cfg),\n        a' = Cfg.then c' (Cont.fix f k) \u2192\n          Reaches step (stepNormal f Cont.halt v) c' \u2192\n            \u2203 v\u2081,\n              v\u2081 \u2208 Code.eval f v \u2227\n                \u2203 v\u2082,\n                  (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227\n                    x \u2208 eval step (Cfg.ret k v\u2082)\nhr : Reaches step (stepNormal f Cont.halt v) (Cfg.halt v')\nh\u2081 : v' \u2208 Code.eval f v\nh\u2082\u271d : Cfg.halt v' \u2208 eval step (Cfg.ret Cont.halt v')\nh\u2082 : Cfg.halt v' \u2208 eval step (stepRet Cont.halt v')\nhe : List.headI v' = 0\nh : x \u2208 eval step (stepRet k (List.tail v'))\n\u22a2 Reaches step (Cfg.ret k (List.tail v')) ?m.195368\n\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx c : Cfg\nhe\u271d : x \u2208 eval step c\nv v' : List \u2115\nIH :\n  \u2200 (a' : Cfg),\n    step (stepRet (Cont.fix f k) v') = some a' \u2192\n      \u2200 (v : List \u2115) (c' : Cfg),\n        a' = Cfg.then c' (Cont.fix f k) \u2192\n          Reaches step (stepNormal f Cont.halt v) c' \u2192\n            \u2203 v\u2081,\n              v\u2081 \u2208 Code.eval f v \u2227\n                \u2203 v\u2082,\n                  (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227\n                    x \u2208 eval step (Cfg.ret k v\u2082)\nhr : Reaches step (stepNormal f Cont.halt v) (Cfg.halt v')\nh\u2081 : v' \u2208 Code.eval f v\nh\u2082\u271d : Cfg.halt v' \u2208 eval step (Cfg.ret Cont.halt v')\nh\u2082 : Cfg.halt v' \u2208 eval step (stepRet Cont.halt v')\nhe : List.headI v' = 0\nh : x \u2208 eval step (stepRet k (List.tail v'))\n\u22a2 Cfg"}, {"tactic": "exact h", "annotated_tactic": ["exact h", []], "state_before": "case pos\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx c : Cfg\nhe\u271d : x \u2208 eval step c\nv v' : List \u2115\nIH :\n  \u2200 (a' : Cfg),\n    step (stepRet (Cont.fix f k) v') = some a' \u2192\n      \u2200 (v : List \u2115) (c' : Cfg),\n        a' = Cfg.then c' (Cont.fix f k) \u2192\n          Reaches step (stepNormal f Cont.halt v) c' \u2192\n            \u2203 v\u2081,\n              v\u2081 \u2208 Code.eval f v \u2227\n                \u2203 v\u2082,\n                  (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227\n                    x \u2208 eval step (Cfg.ret k v\u2082)\nhr : Reaches step (stepNormal f Cont.halt v) (Cfg.halt v')\nh\u2081 : v' \u2208 Code.eval f v\nh\u2082\u271d : Cfg.halt v' \u2208 eval step (Cfg.ret Cont.halt v')\nh\u2082 : Cfg.halt v' \u2208 eval step (stepRet Cont.halt v')\nhe : List.headI v' = 0\nh : x \u2208 eval step (stepRet k (List.tail v'))\n\u22a2 x \u2208 eval step ?m.195368\n\ncase pos\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx c : Cfg\nhe\u271d : x \u2208 eval step c\nv v' : List \u2115\nIH :\n  \u2200 (a' : Cfg),\n    step (stepRet (Cont.fix f k) v') = some a' \u2192\n      \u2200 (v : List \u2115) (c' : Cfg),\n        a' = Cfg.then c' (Cont.fix f k) \u2192\n          Reaches step (stepNormal f Cont.halt v) c' \u2192\n            \u2203 v\u2081,\n              v\u2081 \u2208 Code.eval f v \u2227\n                \u2203 v\u2082,\n                  (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227\n                    x \u2208 eval step (Cfg.ret k v\u2082)\nhr : Reaches step (stepNormal f Cont.halt v) (Cfg.halt v')\nh\u2081 : v' \u2208 Code.eval f v\nh\u2082\u271d : Cfg.halt v' \u2208 eval step (Cfg.ret Cont.halt v')\nh\u2082 : Cfg.halt v' \u2208 eval step (stepRet Cont.halt v')\nhe : List.headI v' = 0\nh : x \u2208 eval step (stepRet k (List.tail v'))\n\u22a2 Reaches step (Cfg.ret k (List.tail v')) ?m.195368\n\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx c : Cfg\nhe\u271d : x \u2208 eval step c\nv v' : List \u2115\nIH :\n  \u2200 (a' : Cfg),\n    step (stepRet (Cont.fix f k) v') = some a' \u2192\n      \u2200 (v : List \u2115) (c' : Cfg),\n        a' = Cfg.then c' (Cont.fix f k) \u2192\n          Reaches step (stepNormal f Cont.halt v) c' \u2192\n            \u2203 v\u2081,\n              v\u2081 \u2208 Code.eval f v \u2227\n                \u2203 v\u2082,\n                  (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227\n                    x \u2208 eval step (Cfg.ret k v\u2082)\nhr : Reaches step (stepNormal f Cont.halt v) (Cfg.halt v')\nh\u2081 : v' \u2208 Code.eval f v\nh\u2082\u271d : Cfg.halt v' \u2208 eval step (Cfg.ret Cont.halt v')\nh\u2082 : Cfg.halt v' \u2208 eval step (stepRet Cont.halt v')\nhe : List.headI v' = 0\nh : x \u2208 eval step (stepRet k (List.tail v'))\n\u22a2 Cfg", "state_after": "case pos\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx c : Cfg\nhe\u271d : x \u2208 eval step c\nv v' : List \u2115\nIH :\n  \u2200 (a' : Cfg),\n    step (stepRet (Cont.fix f k) v') = some a' \u2192\n      \u2200 (v : List \u2115) (c' : Cfg),\n        a' = Cfg.then c' (Cont.fix f k) \u2192\n          Reaches step (stepNormal f Cont.halt v) c' \u2192\n            \u2203 v\u2081,\n              v\u2081 \u2208 Code.eval f v \u2227\n                \u2203 v\u2082,\n                  (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227\n                    x \u2208 eval step (Cfg.ret k v\u2082)\nhr : Reaches step (stepNormal f Cont.halt v) (Cfg.halt v')\nh\u2081 : v' \u2208 Code.eval f v\nh\u2082\u271d : Cfg.halt v' \u2208 eval step (Cfg.ret Cont.halt v')\nh\u2082 : Cfg.halt v' \u2208 eval step (stepRet Cont.halt v')\nhe : List.headI v' = 0\nh : x \u2208 eval step (stepRet k (List.tail v'))\n\u22a2 Reaches step (Cfg.ret k (List.tail v')) (stepRet k (List.tail v'))"}, {"tactic": "exact ReflTransGen.single rfl", "annotated_tactic": ["exact <a>ReflTransGen.single</a> <a>rfl</a>", [{"full_name": "Relation.ReflTransGen.single", "def_path": "Mathlib/Logic/Relation.lean", "def_pos": [276, 9], "def_end_pos": [276, 15]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "case pos\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx c : Cfg\nhe\u271d : x \u2208 eval step c\nv v' : List \u2115\nIH :\n  \u2200 (a' : Cfg),\n    step (stepRet (Cont.fix f k) v') = some a' \u2192\n      \u2200 (v : List \u2115) (c' : Cfg),\n        a' = Cfg.then c' (Cont.fix f k) \u2192\n          Reaches step (stepNormal f Cont.halt v) c' \u2192\n            \u2203 v\u2081,\n              v\u2081 \u2208 Code.eval f v \u2227\n                \u2203 v\u2082,\n                  (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227\n                    x \u2208 eval step (Cfg.ret k v\u2082)\nhr : Reaches step (stepNormal f Cont.halt v) (Cfg.halt v')\nh\u2081 : v' \u2208 Code.eval f v\nh\u2082\u271d : Cfg.halt v' \u2208 eval step (Cfg.ret Cont.halt v')\nh\u2082 : Cfg.halt v' \u2208 eval step (stepRet Cont.halt v')\nhe : List.headI v' = 0\nh : x \u2208 eval step (stepRet k (List.tail v'))\n\u22a2 Reaches step (Cfg.ret k (List.tail v')) (stepRet k (List.tail v'))", "state_after": "no goals"}, {"tactic": "obtain \u27e8k\u2080, v\u2080, e\u2080\u27e9 := stepNormal.is_ret f Cont.halt v'.tail", "annotated_tactic": ["obtain \u27e8k\u2080, v\u2080, e\u2080\u27e9 := <a>stepNormal.is_ret</a> f <a>Cont.halt</a> v'.tail", [{"full_name": "Turing.ToPartrec.stepNormal.is_ret", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [613, 9], "def_end_pos": [613, 26]}, {"full_name": "Turing.ToPartrec.Cont.halt", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [433, 5], "def_end_pos": [433, 9]}]], "state_before": "case neg\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx c : Cfg\nhe\u271d : x \u2208 eval step c\nv v' : List \u2115\nIH :\n  \u2200 (a' : Cfg),\n    step (stepRet (Cont.fix f k) v') = some a' \u2192\n      \u2200 (v : List \u2115) (c' : Cfg),\n        a' = Cfg.then c' (Cont.fix f k) \u2192\n          Reaches step (stepNormal f Cont.halt v) c' \u2192\n            \u2203 v\u2081,\n              v\u2081 \u2208 Code.eval f v \u2227\n                \u2203 v\u2082,\n                  (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227\n                    x \u2208 eval step (Cfg.ret k v\u2082)\nhr : Reaches step (stepNormal f Cont.halt v) (Cfg.halt v')\nh\u2081 : v' \u2208 Code.eval f v\nh\u2082\u271d : Cfg.halt v' \u2208 eval step (Cfg.ret Cont.halt v')\nh\u2082 : Cfg.halt v' \u2208 eval step (stepRet Cont.halt v')\nhe : \u00acList.headI v' = 0\nh : x \u2208 eval step (stepNormal f (Cont.fix f k) (List.tail v'))\n\u22a2 \u2203 v\u2082, v\u2082 \u2208 Code.eval (Code.fix f) (List.tail v') \u2227 x \u2208 eval step (Cfg.ret k v\u2082)", "state_after": "case neg.intro.intro\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx c : Cfg\nhe\u271d : x \u2208 eval step c\nv v' : List \u2115\nIH :\n  \u2200 (a' : Cfg),\n    step (stepRet (Cont.fix f k) v') = some a' \u2192\n      \u2200 (v : List \u2115) (c' : Cfg),\n        a' = Cfg.then c' (Cont.fix f k) \u2192\n          Reaches step (stepNormal f Cont.halt v) c' \u2192\n            \u2203 v\u2081,\n              v\u2081 \u2208 Code.eval f v \u2227\n                \u2203 v\u2082,\n                  (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227\n                    x \u2208 eval step (Cfg.ret k v\u2082)\nhr : Reaches step (stepNormal f Cont.halt v) (Cfg.halt v')\nh\u2081 : v' \u2208 Code.eval f v\nh\u2082\u271d : Cfg.halt v' \u2208 eval step (Cfg.ret Cont.halt v')\nh\u2082 : Cfg.halt v' \u2208 eval step (stepRet Cont.halt v')\nhe : \u00acList.headI v' = 0\nh : x \u2208 eval step (stepNormal f (Cont.fix f k) (List.tail v'))\nk\u2080 : Cont\nv\u2080 : List \u2115\ne\u2080 : stepNormal f Cont.halt (List.tail v') = Cfg.ret k\u2080 v\u2080\n\u22a2 \u2203 v\u2082, v\u2082 \u2208 Code.eval (Code.fix f) (List.tail v') \u2227 x \u2208 eval step (Cfg.ret k v\u2082)"}, {"tactic": "have e\u2081 := stepNormal_then f Cont.halt (Cont.fix f k) v'.tail", "annotated_tactic": ["have e\u2081 := <a>stepNormal_then</a> f <a>Cont.halt</a> (<a>Cont.fix</a> f k) v'.tail", [{"full_name": "Turing.ToPartrec.stepNormal_then", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [567, 9], "def_end_pos": [567, 24]}, {"full_name": "Turing.ToPartrec.Cont.halt", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [433, 5], "def_end_pos": [433, 9]}, {"full_name": "Turing.ToPartrec.Cont.fix", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [437, 5], "def_end_pos": [437, 8]}]], "state_before": "case neg.intro.intro\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx c : Cfg\nhe\u271d : x \u2208 eval step c\nv v' : List \u2115\nIH :\n  \u2200 (a' : Cfg),\n    step (stepRet (Cont.fix f k) v') = some a' \u2192\n      \u2200 (v : List \u2115) (c' : Cfg),\n        a' = Cfg.then c' (Cont.fix f k) \u2192\n          Reaches step (stepNormal f Cont.halt v) c' \u2192\n            \u2203 v\u2081,\n              v\u2081 \u2208 Code.eval f v \u2227\n                \u2203 v\u2082,\n                  (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227\n                    x \u2208 eval step (Cfg.ret k v\u2082)\nhr : Reaches step (stepNormal f Cont.halt v) (Cfg.halt v')\nh\u2081 : v' \u2208 Code.eval f v\nh\u2082\u271d : Cfg.halt v' \u2208 eval step (Cfg.ret Cont.halt v')\nh\u2082 : Cfg.halt v' \u2208 eval step (stepRet Cont.halt v')\nhe : \u00acList.headI v' = 0\nh : x \u2208 eval step (stepNormal f (Cont.fix f k) (List.tail v'))\nk\u2080 : Cont\nv\u2080 : List \u2115\ne\u2080 : stepNormal f Cont.halt (List.tail v') = Cfg.ret k\u2080 v\u2080\n\u22a2 \u2203 v\u2082, v\u2082 \u2208 Code.eval (Code.fix f) (List.tail v') \u2227 x \u2208 eval step (Cfg.ret k v\u2082)", "state_after": "case neg.intro.intro\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx c : Cfg\nhe\u271d : x \u2208 eval step c\nv v' : List \u2115\nIH :\n  \u2200 (a' : Cfg),\n    step (stepRet (Cont.fix f k) v') = some a' \u2192\n      \u2200 (v : List \u2115) (c' : Cfg),\n        a' = Cfg.then c' (Cont.fix f k) \u2192\n          Reaches step (stepNormal f Cont.halt v) c' \u2192\n            \u2203 v\u2081,\n              v\u2081 \u2208 Code.eval f v \u2227\n                \u2203 v\u2082,\n                  (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227\n                    x \u2208 eval step (Cfg.ret k v\u2082)\nhr : Reaches step (stepNormal f Cont.halt v) (Cfg.halt v')\nh\u2081 : v' \u2208 Code.eval f v\nh\u2082\u271d : Cfg.halt v' \u2208 eval step (Cfg.ret Cont.halt v')\nh\u2082 : Cfg.halt v' \u2208 eval step (stepRet Cont.halt v')\nhe : \u00acList.headI v' = 0\nh : x \u2208 eval step (stepNormal f (Cont.fix f k) (List.tail v'))\nk\u2080 : Cont\nv\u2080 : List \u2115\ne\u2080 : stepNormal f Cont.halt (List.tail v') = Cfg.ret k\u2080 v\u2080\ne\u2081 :\n  stepNormal f (Cont.then Cont.halt (Cont.fix f k)) (List.tail v') =\n    Cfg.then (stepNormal f Cont.halt (List.tail v')) (Cont.fix f k)\n\u22a2 \u2203 v\u2082, v\u2082 \u2208 Code.eval (Code.fix f) (List.tail v') \u2227 x \u2208 eval step (Cfg.ret k v\u2082)"}, {"tactic": "rw [e\u2080, Cont.then, Cfg.then] at e\u2081", "annotated_tactic": ["rw [e\u2080, <a>Cont.then</a>, <a>Cfg.then</a>] at e\u2081", [{"full_name": "Turing.ToPartrec.Cont.then", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [541, 5], "def_end_pos": [541, 14]}, {"full_name": "Turing.ToPartrec.Cfg.then", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [559, 5], "def_end_pos": [559, 13]}]], "state_before": "case neg.intro.intro\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx c : Cfg\nhe\u271d : x \u2208 eval step c\nv v' : List \u2115\nIH :\n  \u2200 (a' : Cfg),\n    step (stepRet (Cont.fix f k) v') = some a' \u2192\n      \u2200 (v : List \u2115) (c' : Cfg),\n        a' = Cfg.then c' (Cont.fix f k) \u2192\n          Reaches step (stepNormal f Cont.halt v) c' \u2192\n            \u2203 v\u2081,\n              v\u2081 \u2208 Code.eval f v \u2227\n                \u2203 v\u2082,\n                  (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227\n                    x \u2208 eval step (Cfg.ret k v\u2082)\nhr : Reaches step (stepNormal f Cont.halt v) (Cfg.halt v')\nh\u2081 : v' \u2208 Code.eval f v\nh\u2082\u271d : Cfg.halt v' \u2208 eval step (Cfg.ret Cont.halt v')\nh\u2082 : Cfg.halt v' \u2208 eval step (stepRet Cont.halt v')\nhe : \u00acList.headI v' = 0\nh : x \u2208 eval step (stepNormal f (Cont.fix f k) (List.tail v'))\nk\u2080 : Cont\nv\u2080 : List \u2115\ne\u2080 : stepNormal f Cont.halt (List.tail v') = Cfg.ret k\u2080 v\u2080\ne\u2081 :\n  stepNormal f (Cont.then Cont.halt (Cont.fix f k)) (List.tail v') =\n    Cfg.then (stepNormal f Cont.halt (List.tail v')) (Cont.fix f k)\n\u22a2 \u2203 v\u2082, v\u2082 \u2208 Code.eval (Code.fix f) (List.tail v') \u2227 x \u2208 eval step (Cfg.ret k v\u2082)", "state_after": "case neg.intro.intro\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx c : Cfg\nhe\u271d : x \u2208 eval step c\nv v' : List \u2115\nIH :\n  \u2200 (a' : Cfg),\n    step (stepRet (Cont.fix f k) v') = some a' \u2192\n      \u2200 (v : List \u2115) (c' : Cfg),\n        a' = Cfg.then c' (Cont.fix f k) \u2192\n          Reaches step (stepNormal f Cont.halt v) c' \u2192\n            \u2203 v\u2081,\n              v\u2081 \u2208 Code.eval f v \u2227\n                \u2203 v\u2082,\n                  (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227\n                    x \u2208 eval step (Cfg.ret k v\u2082)\nhr : Reaches step (stepNormal f Cont.halt v) (Cfg.halt v')\nh\u2081 : v' \u2208 Code.eval f v\nh\u2082\u271d : Cfg.halt v' \u2208 eval step (Cfg.ret Cont.halt v')\nh\u2082 : Cfg.halt v' \u2208 eval step (stepRet Cont.halt v')\nhe : \u00acList.headI v' = 0\nh : x \u2208 eval step (stepNormal f (Cont.fix f k) (List.tail v'))\nk\u2080 : Cont\nv\u2080 : List \u2115\ne\u2080 : stepNormal f Cont.halt (List.tail v') = Cfg.ret k\u2080 v\u2080\ne\u2081 :\n  stepNormal f ((fun k' => k') (Cont.fix f k)) (List.tail v') =\n    (match (motive := Cfg \u2192 Cont \u2192 Cfg) Cfg.ret k\u2080 v\u2080 with\n      | Cfg.halt v => fun k' => stepRet k' v\n      | Cfg.ret k v => fun k' => Cfg.ret (Cont.then k k') v)\n      (Cont.fix f k)\n\u22a2 \u2203 v\u2082, v\u2082 \u2208 Code.eval (Code.fix f) (List.tail v') \u2227 x \u2208 eval step (Cfg.ret k v\u2082)"}, {"tactic": "simp only [] at e\u2081", "annotated_tactic": ["simp only [] at e\u2081", []], "state_before": "case neg.intro.intro\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx c : Cfg\nhe\u271d : x \u2208 eval step c\nv v' : List \u2115\nIH :\n  \u2200 (a' : Cfg),\n    step (stepRet (Cont.fix f k) v') = some a' \u2192\n      \u2200 (v : List \u2115) (c' : Cfg),\n        a' = Cfg.then c' (Cont.fix f k) \u2192\n          Reaches step (stepNormal f Cont.halt v) c' \u2192\n            \u2203 v\u2081,\n              v\u2081 \u2208 Code.eval f v \u2227\n                \u2203 v\u2082,\n                  (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227\n                    x \u2208 eval step (Cfg.ret k v\u2082)\nhr : Reaches step (stepNormal f Cont.halt v) (Cfg.halt v')\nh\u2081 : v' \u2208 Code.eval f v\nh\u2082\u271d : Cfg.halt v' \u2208 eval step (Cfg.ret Cont.halt v')\nh\u2082 : Cfg.halt v' \u2208 eval step (stepRet Cont.halt v')\nhe : \u00acList.headI v' = 0\nh : x \u2208 eval step (stepNormal f (Cont.fix f k) (List.tail v'))\nk\u2080 : Cont\nv\u2080 : List \u2115\ne\u2080 : stepNormal f Cont.halt (List.tail v') = Cfg.ret k\u2080 v\u2080\ne\u2081 :\n  stepNormal f ((fun k' => k') (Cont.fix f k)) (List.tail v') =\n    (match (motive := Cfg \u2192 Cont \u2192 Cfg) Cfg.ret k\u2080 v\u2080 with\n      | Cfg.halt v => fun k' => stepRet k' v\n      | Cfg.ret k v => fun k' => Cfg.ret (Cont.then k k') v)\n      (Cont.fix f k)\n\u22a2 \u2203 v\u2082, v\u2082 \u2208 Code.eval (Code.fix f) (List.tail v') \u2227 x \u2208 eval step (Cfg.ret k v\u2082)", "state_after": "case neg.intro.intro\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx c : Cfg\nhe\u271d : x \u2208 eval step c\nv v' : List \u2115\nIH :\n  \u2200 (a' : Cfg),\n    step (stepRet (Cont.fix f k) v') = some a' \u2192\n      \u2200 (v : List \u2115) (c' : Cfg),\n        a' = Cfg.then c' (Cont.fix f k) \u2192\n          Reaches step (stepNormal f Cont.halt v) c' \u2192\n            \u2203 v\u2081,\n              v\u2081 \u2208 Code.eval f v \u2227\n                \u2203 v\u2082,\n                  (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227\n                    x \u2208 eval step (Cfg.ret k v\u2082)\nhr : Reaches step (stepNormal f Cont.halt v) (Cfg.halt v')\nh\u2081 : v' \u2208 Code.eval f v\nh\u2082\u271d : Cfg.halt v' \u2208 eval step (Cfg.ret Cont.halt v')\nh\u2082 : Cfg.halt v' \u2208 eval step (stepRet Cont.halt v')\nhe : \u00acList.headI v' = 0\nh : x \u2208 eval step (stepNormal f (Cont.fix f k) (List.tail v'))\nk\u2080 : Cont\nv\u2080 : List \u2115\ne\u2080 : stepNormal f Cont.halt (List.tail v') = Cfg.ret k\u2080 v\u2080\ne\u2081 : stepNormal f (Cont.fix f k) (List.tail v') = Cfg.ret (Cont.then k\u2080 (Cont.fix f k)) v\u2080\n\u22a2 \u2203 v\u2082, v\u2082 \u2208 Code.eval (Code.fix f) (List.tail v') \u2227 x \u2208 eval step (Cfg.ret k v\u2082)"}, {"tactic": "obtain \u27e8v\u2081, hv\u2081, v\u2082, hv\u2082, h\u2083\u27e9 :=\n  IH (stepRet (k\u2080.then (Cont.fix f k)) v\u2080) (by rw [stepRet, if_neg he, e\u2081]; rfl)\n    v'.tail _ stepRet_then (by apply ReflTransGen.single; rw [e\u2080]; rfl)", "annotated_tactic": ["obtain \u27e8v\u2081, hv\u2081, v\u2082, hv\u2082, h\u2083\u27e9 :=\n          IH (<a>stepRet</a> (k\u2080.then (<a>Cont.fix</a> f k)) v\u2080) (by rw [<a>stepRet</a>, <a>if_neg</a> he, e\u2081]; rfl)\n            v'.tail _ <a>stepRet_then</a> (by apply <a>ReflTransGen.single</a>; rw [e\u2080]; rfl)", [{"full_name": "Turing.ToPartrec.stepRet", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [513, 5], "def_end_pos": [513, 12]}, {"full_name": "Turing.ToPartrec.Cont.fix", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [437, 5], "def_end_pos": [437, 8]}, {"full_name": "Turing.ToPartrec.stepRet", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [513, 5], "def_end_pos": [513, 12]}, {"full_name": "if_neg", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [795, 9], "def_end_pos": [795, 15]}, {"full_name": "Turing.ToPartrec.stepRet_then", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [580, 9], "def_end_pos": [580, 21]}, {"full_name": "Relation.ReflTransGen.single", "def_path": "Mathlib/Logic/Relation.lean", "def_pos": [276, 9], "def_end_pos": [276, 15]}]], "state_before": "case neg.intro.intro\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx c : Cfg\nhe\u271d : x \u2208 eval step c\nv v' : List \u2115\nIH :\n  \u2200 (a' : Cfg),\n    step (stepRet (Cont.fix f k) v') = some a' \u2192\n      \u2200 (v : List \u2115) (c' : Cfg),\n        a' = Cfg.then c' (Cont.fix f k) \u2192\n          Reaches step (stepNormal f Cont.halt v) c' \u2192\n            \u2203 v\u2081,\n              v\u2081 \u2208 Code.eval f v \u2227\n                \u2203 v\u2082,\n                  (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227\n                    x \u2208 eval step (Cfg.ret k v\u2082)\nhr : Reaches step (stepNormal f Cont.halt v) (Cfg.halt v')\nh\u2081 : v' \u2208 Code.eval f v\nh\u2082\u271d : Cfg.halt v' \u2208 eval step (Cfg.ret Cont.halt v')\nh\u2082 : Cfg.halt v' \u2208 eval step (stepRet Cont.halt v')\nhe : \u00acList.headI v' = 0\nh : x \u2208 eval step (stepNormal f (Cont.fix f k) (List.tail v'))\nk\u2080 : Cont\nv\u2080 : List \u2115\ne\u2080 : stepNormal f Cont.halt (List.tail v') = Cfg.ret k\u2080 v\u2080\ne\u2081 : stepNormal f (Cont.fix f k) (List.tail v') = Cfg.ret (Cont.then k\u2080 (Cont.fix f k)) v\u2080\n\u22a2 \u2203 v\u2082, v\u2082 \u2208 Code.eval (Code.fix f) (List.tail v') \u2227 x \u2208 eval step (Cfg.ret k v\u2082)", "state_after": "case neg.intro.intro.intro.intro.intro.intro\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx c : Cfg\nhe\u271d : x \u2208 eval step c\nv v' : List \u2115\nIH :\n  \u2200 (a' : Cfg),\n    step (stepRet (Cont.fix f k) v') = some a' \u2192\n      \u2200 (v : List \u2115) (c' : Cfg),\n        a' = Cfg.then c' (Cont.fix f k) \u2192\n          Reaches step (stepNormal f Cont.halt v) c' \u2192\n            \u2203 v\u2081,\n              v\u2081 \u2208 Code.eval f v \u2227\n                \u2203 v\u2082,\n                  (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227\n                    x \u2208 eval step (Cfg.ret k v\u2082)\nhr : Reaches step (stepNormal f Cont.halt v) (Cfg.halt v')\nh\u2081 : v' \u2208 Code.eval f v\nh\u2082\u271d : Cfg.halt v' \u2208 eval step (Cfg.ret Cont.halt v')\nh\u2082 : Cfg.halt v' \u2208 eval step (stepRet Cont.halt v')\nhe : \u00acList.headI v' = 0\nh : x \u2208 eval step (stepNormal f (Cont.fix f k) (List.tail v'))\nk\u2080 : Cont\nv\u2080 : List \u2115\ne\u2080 : stepNormal f Cont.halt (List.tail v') = Cfg.ret k\u2080 v\u2080\ne\u2081 : stepNormal f (Cont.fix f k) (List.tail v') = Cfg.ret (Cont.then k\u2080 (Cont.fix f k)) v\u2080\nv\u2081 : List \u2115\nhv\u2081 : v\u2081 \u2208 Code.eval f (List.tail v')\nv\u2082 : List \u2115\nhv\u2082 : v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)\nh\u2083 : x \u2208 eval step (Cfg.ret k v\u2082)\n\u22a2 \u2203 v\u2082, v\u2082 \u2208 Code.eval (Code.fix f) (List.tail v') \u2227 x \u2208 eval step (Cfg.ret k v\u2082)"}, {"tactic": "rw [stepRet, if_neg he, e\u2081]", "annotated_tactic": ["rw [<a>stepRet</a>, <a>if_neg</a> he, e\u2081]", [{"full_name": "Turing.ToPartrec.stepRet", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [513, 5], "def_end_pos": [513, 12]}, {"full_name": "if_neg", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [795, 9], "def_end_pos": [795, 15]}]], "state_before": "f : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx c : Cfg\nhe\u271d : x \u2208 eval step c\nv v' : List \u2115\nIH :\n  \u2200 (a' : Cfg),\n    step (stepRet (Cont.fix f k) v') = some a' \u2192\n      \u2200 (v : List \u2115) (c' : Cfg),\n        a' = Cfg.then c' (Cont.fix f k) \u2192\n          Reaches step (stepNormal f Cont.halt v) c' \u2192\n            \u2203 v\u2081,\n              v\u2081 \u2208 Code.eval f v \u2227\n                \u2203 v\u2082,\n                  (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227\n                    x \u2208 eval step (Cfg.ret k v\u2082)\nhr : Reaches step (stepNormal f Cont.halt v) (Cfg.halt v')\nh\u2081 : v' \u2208 Code.eval f v\nh\u2082\u271d : Cfg.halt v' \u2208 eval step (Cfg.ret Cont.halt v')\nh\u2082 : Cfg.halt v' \u2208 eval step (stepRet Cont.halt v')\nhe : \u00acList.headI v' = 0\nh : x \u2208 eval step (stepNormal f (Cont.fix f k) (List.tail v'))\nk\u2080 : Cont\nv\u2080 : List \u2115\ne\u2080 : stepNormal f Cont.halt (List.tail v') = Cfg.ret k\u2080 v\u2080\ne\u2081 : stepNormal f (Cont.fix f k) (List.tail v') = Cfg.ret (Cont.then k\u2080 (Cont.fix f k)) v\u2080\n\u22a2 step (stepRet (Cont.fix f k) v') = some (stepRet (Cont.then k\u2080 (Cont.fix f k)) v\u2080)", "state_after": "f : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx c : Cfg\nhe\u271d : x \u2208 eval step c\nv v' : List \u2115\nIH :\n  \u2200 (a' : Cfg),\n    step (stepRet (Cont.fix f k) v') = some a' \u2192\n      \u2200 (v : List \u2115) (c' : Cfg),\n        a' = Cfg.then c' (Cont.fix f k) \u2192\n          Reaches step (stepNormal f Cont.halt v) c' \u2192\n            \u2203 v\u2081,\n              v\u2081 \u2208 Code.eval f v \u2227\n                \u2203 v\u2082,\n                  (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227\n                    x \u2208 eval step (Cfg.ret k v\u2082)\nhr : Reaches step (stepNormal f Cont.halt v) (Cfg.halt v')\nh\u2081 : v' \u2208 Code.eval f v\nh\u2082\u271d : Cfg.halt v' \u2208 eval step (Cfg.ret Cont.halt v')\nh\u2082 : Cfg.halt v' \u2208 eval step (stepRet Cont.halt v')\nhe : \u00acList.headI v' = 0\nh : x \u2208 eval step (stepNormal f (Cont.fix f k) (List.tail v'))\nk\u2080 : Cont\nv\u2080 : List \u2115\ne\u2080 : stepNormal f Cont.halt (List.tail v') = Cfg.ret k\u2080 v\u2080\ne\u2081 : stepNormal f (Cont.fix f k) (List.tail v') = Cfg.ret (Cont.then k\u2080 (Cont.fix f k)) v\u2080\n\u22a2 step (Cfg.ret (Cont.then k\u2080 (Cont.fix f k)) v\u2080) = some (stepRet (Cont.then k\u2080 (Cont.fix f k)) v\u2080)"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "f : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx c : Cfg\nhe\u271d : x \u2208 eval step c\nv v' : List \u2115\nIH :\n  \u2200 (a' : Cfg),\n    step (stepRet (Cont.fix f k) v') = some a' \u2192\n      \u2200 (v : List \u2115) (c' : Cfg),\n        a' = Cfg.then c' (Cont.fix f k) \u2192\n          Reaches step (stepNormal f Cont.halt v) c' \u2192\n            \u2203 v\u2081,\n              v\u2081 \u2208 Code.eval f v \u2227\n                \u2203 v\u2082,\n                  (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227\n                    x \u2208 eval step (Cfg.ret k v\u2082)\nhr : Reaches step (stepNormal f Cont.halt v) (Cfg.halt v')\nh\u2081 : v' \u2208 Code.eval f v\nh\u2082\u271d : Cfg.halt v' \u2208 eval step (Cfg.ret Cont.halt v')\nh\u2082 : Cfg.halt v' \u2208 eval step (stepRet Cont.halt v')\nhe : \u00acList.headI v' = 0\nh : x \u2208 eval step (stepNormal f (Cont.fix f k) (List.tail v'))\nk\u2080 : Cont\nv\u2080 : List \u2115\ne\u2080 : stepNormal f Cont.halt (List.tail v') = Cfg.ret k\u2080 v\u2080\ne\u2081 : stepNormal f (Cont.fix f k) (List.tail v') = Cfg.ret (Cont.then k\u2080 (Cont.fix f k)) v\u2080\n\u22a2 step (Cfg.ret (Cont.then k\u2080 (Cont.fix f k)) v\u2080) = some (stepRet (Cont.then k\u2080 (Cont.fix f k)) v\u2080)", "state_after": "no goals"}, {"tactic": "apply ReflTransGen.single", "annotated_tactic": ["apply <a>ReflTransGen.single</a>", [{"full_name": "Relation.ReflTransGen.single", "def_path": "Mathlib/Logic/Relation.lean", "def_pos": [276, 9], "def_end_pos": [276, 15]}]], "state_before": "f : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx c : Cfg\nhe\u271d : x \u2208 eval step c\nv v' : List \u2115\nIH :\n  \u2200 (a' : Cfg),\n    step (stepRet (Cont.fix f k) v') = some a' \u2192\n      \u2200 (v : List \u2115) (c' : Cfg),\n        a' = Cfg.then c' (Cont.fix f k) \u2192\n          Reaches step (stepNormal f Cont.halt v) c' \u2192\n            \u2203 v\u2081,\n              v\u2081 \u2208 Code.eval f v \u2227\n                \u2203 v\u2082,\n                  (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227\n                    x \u2208 eval step (Cfg.ret k v\u2082)\nhr : Reaches step (stepNormal f Cont.halt v) (Cfg.halt v')\nh\u2081 : v' \u2208 Code.eval f v\nh\u2082\u271d : Cfg.halt v' \u2208 eval step (Cfg.ret Cont.halt v')\nh\u2082 : Cfg.halt v' \u2208 eval step (stepRet Cont.halt v')\nhe : \u00acList.headI v' = 0\nh : x \u2208 eval step (stepNormal f (Cont.fix f k) (List.tail v'))\nk\u2080 : Cont\nv\u2080 : List \u2115\ne\u2080 : stepNormal f Cont.halt (List.tail v') = Cfg.ret k\u2080 v\u2080\ne\u2081 : stepNormal f (Cont.fix f k) (List.tail v') = Cfg.ret (Cont.then k\u2080 (Cont.fix f k)) v\u2080\n\u22a2 Reaches step (stepNormal f Cont.halt (List.tail v')) (stepRet k\u2080 v\u2080)", "state_after": "case hab\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx c : Cfg\nhe\u271d : x \u2208 eval step c\nv v' : List \u2115\nIH :\n  \u2200 (a' : Cfg),\n    step (stepRet (Cont.fix f k) v') = some a' \u2192\n      \u2200 (v : List \u2115) (c' : Cfg),\n        a' = Cfg.then c' (Cont.fix f k) \u2192\n          Reaches step (stepNormal f Cont.halt v) c' \u2192\n            \u2203 v\u2081,\n              v\u2081 \u2208 Code.eval f v \u2227\n                \u2203 v\u2082,\n                  (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227\n                    x \u2208 eval step (Cfg.ret k v\u2082)\nhr : Reaches step (stepNormal f Cont.halt v) (Cfg.halt v')\nh\u2081 : v' \u2208 Code.eval f v\nh\u2082\u271d : Cfg.halt v' \u2208 eval step (Cfg.ret Cont.halt v')\nh\u2082 : Cfg.halt v' \u2208 eval step (stepRet Cont.halt v')\nhe : \u00acList.headI v' = 0\nh : x \u2208 eval step (stepNormal f (Cont.fix f k) (List.tail v'))\nk\u2080 : Cont\nv\u2080 : List \u2115\ne\u2080 : stepNormal f Cont.halt (List.tail v') = Cfg.ret k\u2080 v\u2080\ne\u2081 : stepNormal f (Cont.fix f k) (List.tail v') = Cfg.ret (Cont.then k\u2080 (Cont.fix f k)) v\u2080\n\u22a2 stepRet k\u2080 v\u2080 \u2208 step (stepNormal f Cont.halt (List.tail v'))"}, {"tactic": "rw [e\u2080]", "annotated_tactic": ["rw [e\u2080]", []], "state_before": "case hab\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx c : Cfg\nhe\u271d : x \u2208 eval step c\nv v' : List \u2115\nIH :\n  \u2200 (a' : Cfg),\n    step (stepRet (Cont.fix f k) v') = some a' \u2192\n      \u2200 (v : List \u2115) (c' : Cfg),\n        a' = Cfg.then c' (Cont.fix f k) \u2192\n          Reaches step (stepNormal f Cont.halt v) c' \u2192\n            \u2203 v\u2081,\n              v\u2081 \u2208 Code.eval f v \u2227\n                \u2203 v\u2082,\n                  (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227\n                    x \u2208 eval step (Cfg.ret k v\u2082)\nhr : Reaches step (stepNormal f Cont.halt v) (Cfg.halt v')\nh\u2081 : v' \u2208 Code.eval f v\nh\u2082\u271d : Cfg.halt v' \u2208 eval step (Cfg.ret Cont.halt v')\nh\u2082 : Cfg.halt v' \u2208 eval step (stepRet Cont.halt v')\nhe : \u00acList.headI v' = 0\nh : x \u2208 eval step (stepNormal f (Cont.fix f k) (List.tail v'))\nk\u2080 : Cont\nv\u2080 : List \u2115\ne\u2080 : stepNormal f Cont.halt (List.tail v') = Cfg.ret k\u2080 v\u2080\ne\u2081 : stepNormal f (Cont.fix f k) (List.tail v') = Cfg.ret (Cont.then k\u2080 (Cont.fix f k)) v\u2080\n\u22a2 stepRet k\u2080 v\u2080 \u2208 step (stepNormal f Cont.halt (List.tail v'))", "state_after": "case hab\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx c : Cfg\nhe\u271d : x \u2208 eval step c\nv v' : List \u2115\nIH :\n  \u2200 (a' : Cfg),\n    step (stepRet (Cont.fix f k) v') = some a' \u2192\n      \u2200 (v : List \u2115) (c' : Cfg),\n        a' = Cfg.then c' (Cont.fix f k) \u2192\n          Reaches step (stepNormal f Cont.halt v) c' \u2192\n            \u2203 v\u2081,\n              v\u2081 \u2208 Code.eval f v \u2227\n                \u2203 v\u2082,\n                  (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227\n                    x \u2208 eval step (Cfg.ret k v\u2082)\nhr : Reaches step (stepNormal f Cont.halt v) (Cfg.halt v')\nh\u2081 : v' \u2208 Code.eval f v\nh\u2082\u271d : Cfg.halt v' \u2208 eval step (Cfg.ret Cont.halt v')\nh\u2082 : Cfg.halt v' \u2208 eval step (stepRet Cont.halt v')\nhe : \u00acList.headI v' = 0\nh : x \u2208 eval step (stepNormal f (Cont.fix f k) (List.tail v'))\nk\u2080 : Cont\nv\u2080 : List \u2115\ne\u2080 : stepNormal f Cont.halt (List.tail v') = Cfg.ret k\u2080 v\u2080\ne\u2081 : stepNormal f (Cont.fix f k) (List.tail v') = Cfg.ret (Cont.then k\u2080 (Cont.fix f k)) v\u2080\n\u22a2 stepRet k\u2080 v\u2080 \u2208 step (Cfg.ret k\u2080 v\u2080)"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case hab\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx c : Cfg\nhe\u271d : x \u2208 eval step c\nv v' : List \u2115\nIH :\n  \u2200 (a' : Cfg),\n    step (stepRet (Cont.fix f k) v') = some a' \u2192\n      \u2200 (v : List \u2115) (c' : Cfg),\n        a' = Cfg.then c' (Cont.fix f k) \u2192\n          Reaches step (stepNormal f Cont.halt v) c' \u2192\n            \u2203 v\u2081,\n              v\u2081 \u2208 Code.eval f v \u2227\n                \u2203 v\u2082,\n                  (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227\n                    x \u2208 eval step (Cfg.ret k v\u2082)\nhr : Reaches step (stepNormal f Cont.halt v) (Cfg.halt v')\nh\u2081 : v' \u2208 Code.eval f v\nh\u2082\u271d : Cfg.halt v' \u2208 eval step (Cfg.ret Cont.halt v')\nh\u2082 : Cfg.halt v' \u2208 eval step (stepRet Cont.halt v')\nhe : \u00acList.headI v' = 0\nh : x \u2208 eval step (stepNormal f (Cont.fix f k) (List.tail v'))\nk\u2080 : Cont\nv\u2080 : List \u2115\ne\u2080 : stepNormal f Cont.halt (List.tail v') = Cfg.ret k\u2080 v\u2080\ne\u2081 : stepNormal f (Cont.fix f k) (List.tail v') = Cfg.ret (Cont.then k\u2080 (Cont.fix f k)) v\u2080\n\u22a2 stepRet k\u2080 v\u2080 \u2208 step (Cfg.ret k\u2080 v\u2080)", "state_after": "no goals"}, {"tactic": "refine' \u27e8_, PFun.mem_fix_iff.2 _, h\u2083\u27e9", "annotated_tactic": ["refine' \u27e8_, <a>PFun.mem_fix_iff</a>.2 _, h\u2083\u27e9", [{"full_name": "PFun.mem_fix_iff", "def_path": "Mathlib/Data/PFun.lean", "def_pos": [266, 9], "def_end_pos": [266, 20]}]], "state_before": "case neg.intro.intro.intro.intro.intro.intro\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx c : Cfg\nhe\u271d : x \u2208 eval step c\nv v' : List \u2115\nIH :\n  \u2200 (a' : Cfg),\n    step (stepRet (Cont.fix f k) v') = some a' \u2192\n      \u2200 (v : List \u2115) (c' : Cfg),\n        a' = Cfg.then c' (Cont.fix f k) \u2192\n          Reaches step (stepNormal f Cont.halt v) c' \u2192\n            \u2203 v\u2081,\n              v\u2081 \u2208 Code.eval f v \u2227\n                \u2203 v\u2082,\n                  (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227\n                    x \u2208 eval step (Cfg.ret k v\u2082)\nhr : Reaches step (stepNormal f Cont.halt v) (Cfg.halt v')\nh\u2081 : v' \u2208 Code.eval f v\nh\u2082\u271d : Cfg.halt v' \u2208 eval step (Cfg.ret Cont.halt v')\nh\u2082 : Cfg.halt v' \u2208 eval step (stepRet Cont.halt v')\nhe : \u00acList.headI v' = 0\nh : x \u2208 eval step (stepNormal f (Cont.fix f k) (List.tail v'))\nk\u2080 : Cont\nv\u2080 : List \u2115\ne\u2080 : stepNormal f Cont.halt (List.tail v') = Cfg.ret k\u2080 v\u2080\ne\u2081 : stepNormal f (Cont.fix f k) (List.tail v') = Cfg.ret (Cont.then k\u2080 (Cont.fix f k)) v\u2080\nv\u2081 : List \u2115\nhv\u2081 : v\u2081 \u2208 Code.eval f (List.tail v')\nv\u2082 : List \u2115\nhv\u2082 : v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)\nh\u2083 : x \u2208 eval step (Cfg.ret k v\u2082)\n\u22a2 \u2203 v\u2082, v\u2082 \u2208 Code.eval (Code.fix f) (List.tail v') \u2227 x \u2208 eval step (Cfg.ret k v\u2082)", "state_after": "case neg.intro.intro.intro.intro.intro.intro\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx c : Cfg\nhe\u271d : x \u2208 eval step c\nv v' : List \u2115\nIH :\n  \u2200 (a' : Cfg),\n    step (stepRet (Cont.fix f k) v') = some a' \u2192\n      \u2200 (v : List \u2115) (c' : Cfg),\n        a' = Cfg.then c' (Cont.fix f k) \u2192\n          Reaches step (stepNormal f Cont.halt v) c' \u2192\n            \u2203 v\u2081,\n              v\u2081 \u2208 Code.eval f v \u2227\n                \u2203 v\u2082,\n                  (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227\n                    x \u2208 eval step (Cfg.ret k v\u2082)\nhr : Reaches step (stepNormal f Cont.halt v) (Cfg.halt v')\nh\u2081 : v' \u2208 Code.eval f v\nh\u2082\u271d : Cfg.halt v' \u2208 eval step (Cfg.ret Cont.halt v')\nh\u2082 : Cfg.halt v' \u2208 eval step (stepRet Cont.halt v')\nhe : \u00acList.headI v' = 0\nh : x \u2208 eval step (stepNormal f (Cont.fix f k) (List.tail v'))\nk\u2080 : Cont\nv\u2080 : List \u2115\ne\u2080 : stepNormal f Cont.halt (List.tail v') = Cfg.ret k\u2080 v\u2080\ne\u2081 : stepNormal f (Cont.fix f k) (List.tail v') = Cfg.ret (Cont.then k\u2080 (Cont.fix f k)) v\u2080\nv\u2081 : List \u2115\nhv\u2081 : v\u2081 \u2208 Code.eval f (List.tail v')\nv\u2082 : List \u2115\nhv\u2082 : v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)\nh\u2083 : x \u2208 eval step (Cfg.ret k v\u2082)\n\u22a2 Sum.inl v\u2082 \u2208\n      Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v))\n        (Code.eval f (List.tail v')) \u2228\n    \u2203 a',\n      Sum.inr a' \u2208\n          Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v))\n            (Code.eval f (List.tail v')) \u2227\n        v\u2082 \u2208\n          PFun.fix\n            (fun v =>\n              Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v))\n                (Code.eval f v))\n            a'"}, {"tactic": "simp only [Part.eq_some_iff.2 hv\u2081, Part.map_some, Part.mem_some_iff]", "annotated_tactic": ["simp only [<a>Part.eq_some_iff</a>.2 hv\u2081, <a>Part.map_some</a>, <a>Part.mem_some_iff</a>]", [{"full_name": "Part.eq_some_iff", "def_path": "Mathlib/Data/Part.lean", "def_pos": [174, 9], "def_end_pos": [174, 20]}, {"full_name": "Part.map_some", "def_path": "Mathlib/Data/Part.lean", "def_pos": [457, 9], "def_end_pos": [457, 17]}, {"full_name": "Part.mem_some_iff", "def_path": "Mathlib/Data/Part.lean", "def_pos": [170, 9], "def_end_pos": [170, 21]}]], "state_before": "case neg.intro.intro.intro.intro.intro.intro\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx c : Cfg\nhe\u271d : x \u2208 eval step c\nv v' : List \u2115\nIH :\n  \u2200 (a' : Cfg),\n    step (stepRet (Cont.fix f k) v') = some a' \u2192\n      \u2200 (v : List \u2115) (c' : Cfg),\n        a' = Cfg.then c' (Cont.fix f k) \u2192\n          Reaches step (stepNormal f Cont.halt v) c' \u2192\n            \u2203 v\u2081,\n              v\u2081 \u2208 Code.eval f v \u2227\n                \u2203 v\u2082,\n                  (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227\n                    x \u2208 eval step (Cfg.ret k v\u2082)\nhr : Reaches step (stepNormal f Cont.halt v) (Cfg.halt v')\nh\u2081 : v' \u2208 Code.eval f v\nh\u2082\u271d : Cfg.halt v' \u2208 eval step (Cfg.ret Cont.halt v')\nh\u2082 : Cfg.halt v' \u2208 eval step (stepRet Cont.halt v')\nhe : \u00acList.headI v' = 0\nh : x \u2208 eval step (stepNormal f (Cont.fix f k) (List.tail v'))\nk\u2080 : Cont\nv\u2080 : List \u2115\ne\u2080 : stepNormal f Cont.halt (List.tail v') = Cfg.ret k\u2080 v\u2080\ne\u2081 : stepNormal f (Cont.fix f k) (List.tail v') = Cfg.ret (Cont.then k\u2080 (Cont.fix f k)) v\u2080\nv\u2081 : List \u2115\nhv\u2081 : v\u2081 \u2208 Code.eval f (List.tail v')\nv\u2082 : List \u2115\nhv\u2082 : v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)\nh\u2083 : x \u2208 eval step (Cfg.ret k v\u2082)\n\u22a2 Sum.inl v\u2082 \u2208\n      Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v))\n        (Code.eval f (List.tail v')) \u2228\n    \u2203 a',\n      Sum.inr a' \u2208\n          Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v))\n            (Code.eval f (List.tail v')) \u2227\n        v\u2082 \u2208\n          PFun.fix\n            (fun v =>\n              Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v))\n                (Code.eval f v))\n            a'", "state_after": "case neg.intro.intro.intro.intro.intro.intro\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx c : Cfg\nhe\u271d : x \u2208 eval step c\nv v' : List \u2115\nIH :\n  \u2200 (a' : Cfg),\n    step (stepRet (Cont.fix f k) v') = some a' \u2192\n      \u2200 (v : List \u2115) (c' : Cfg),\n        a' = Cfg.then c' (Cont.fix f k) \u2192\n          Reaches step (stepNormal f Cont.halt v) c' \u2192\n            \u2203 v\u2081,\n              v\u2081 \u2208 Code.eval f v \u2227\n                \u2203 v\u2082,\n                  (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227\n                    x \u2208 eval step (Cfg.ret k v\u2082)\nhr : Reaches step (stepNormal f Cont.halt v) (Cfg.halt v')\nh\u2081 : v' \u2208 Code.eval f v\nh\u2082\u271d : Cfg.halt v' \u2208 eval step (Cfg.ret Cont.halt v')\nh\u2082 : Cfg.halt v' \u2208 eval step (stepRet Cont.halt v')\nhe : \u00acList.headI v' = 0\nh : x \u2208 eval step (stepNormal f (Cont.fix f k) (List.tail v'))\nk\u2080 : Cont\nv\u2080 : List \u2115\ne\u2080 : stepNormal f Cont.halt (List.tail v') = Cfg.ret k\u2080 v\u2080\ne\u2081 : stepNormal f (Cont.fix f k) (List.tail v') = Cfg.ret (Cont.then k\u2080 (Cont.fix f k)) v\u2080\nv\u2081 : List \u2115\nhv\u2081 : v\u2081 \u2208 Code.eval f (List.tail v')\nv\u2082 : List \u2115\nhv\u2082 : v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)\nh\u2083 : x \u2208 eval step (Cfg.ret k v\u2082)\n\u22a2 (Sum.inl v\u2082 = if List.headI v\u2081 = 0 then Sum.inl (List.tail v\u2081) else Sum.inr (List.tail v\u2081)) \u2228\n    \u2203 a',\n      (Sum.inr a' = if List.headI v\u2081 = 0 then Sum.inl (List.tail v\u2081) else Sum.inr (List.tail v\u2081)) \u2227\n        v\u2082 \u2208\n          PFun.fix\n            (fun v =>\n              Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v))\n                (Code.eval f v))\n            a'"}, {"tactic": "split_ifs at hv\u2082 \u22a2 <;> [exact Or.inl (congr_arg Sum.inl (Part.mem_some_iff.1 hv\u2082));\n  exact Or.inr \u27e8_, rfl, hv\u2082\u27e9]", "annotated_tactic": ["split_ifs at hv\u2082 \u22a2 <;> [exact <a>Or.inl</a> (<a>congr_arg</a> <a>Sum.inl</a> (<a>Part.mem_some_iff</a>.1 hv\u2082));\n            exact <a>Or.inr</a> \u27e8_, <a>rfl</a>, hv\u2082\u27e9]", [{"full_name": "Or.inl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [517, 5], "def_end_pos": [517, 8]}, {"full_name": "congr_arg", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [43, 7], "def_end_pos": [43, 16]}, {"full_name": "Sum.inl", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [104, 5], "def_end_pos": [104, 8]}, {"full_name": "Part.mem_some_iff", "def_path": "Mathlib/Data/Part.lean", "def_pos": [170, 9], "def_end_pos": [170, 21]}, {"full_name": "Or.inr", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [519, 5], "def_end_pos": [519, 8]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "case neg.intro.intro.intro.intro.intro.intro\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx c : Cfg\nhe\u271d : x \u2208 eval step c\nv v' : List \u2115\nIH :\n  \u2200 (a' : Cfg),\n    step (stepRet (Cont.fix f k) v') = some a' \u2192\n      \u2200 (v : List \u2115) (c' : Cfg),\n        a' = Cfg.then c' (Cont.fix f k) \u2192\n          Reaches step (stepNormal f Cont.halt v) c' \u2192\n            \u2203 v\u2081,\n              v\u2081 \u2208 Code.eval f v \u2227\n                \u2203 v\u2082,\n                  (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227\n                    x \u2208 eval step (Cfg.ret k v\u2082)\nhr : Reaches step (stepNormal f Cont.halt v) (Cfg.halt v')\nh\u2081 : v' \u2208 Code.eval f v\nh\u2082\u271d : Cfg.halt v' \u2208 eval step (Cfg.ret Cont.halt v')\nh\u2082 : Cfg.halt v' \u2208 eval step (stepRet Cont.halt v')\nhe : \u00acList.headI v' = 0\nh : x \u2208 eval step (stepNormal f (Cont.fix f k) (List.tail v'))\nk\u2080 : Cont\nv\u2080 : List \u2115\ne\u2080 : stepNormal f Cont.halt (List.tail v') = Cfg.ret k\u2080 v\u2080\ne\u2081 : stepNormal f (Cont.fix f k) (List.tail v') = Cfg.ret (Cont.then k\u2080 (Cont.fix f k)) v\u2080\nv\u2081 : List \u2115\nhv\u2081 : v\u2081 \u2208 Code.eval f (List.tail v')\nv\u2082 : List \u2115\nhv\u2082 : v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)\nh\u2083 : x \u2208 eval step (Cfg.ret k v\u2082)\n\u22a2 (Sum.inl v\u2082 = if List.headI v\u2081 = 0 then Sum.inl (List.tail v\u2081) else Sum.inr (List.tail v\u2081)) \u2228\n    \u2203 a',\n      (Sum.inr a' = if List.headI v\u2081 = 0 then Sum.inl (List.tail v\u2081) else Sum.inr (List.tail v\u2081)) \u2227\n        v\u2082 \u2208\n          PFun.fix\n            (fun v =>\n              Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v))\n                (Code.eval f v))\n            a'", "state_after": "no goals"}, {"tactic": "exact IH _ rfl _ _ stepRet_then (ReflTransGen.tail hr rfl)", "annotated_tactic": ["exact IH _ <a>rfl</a> _ _ <a>stepRet_then</a> (<a>ReflTransGen.tail</a> hr <a>rfl</a>)", [{"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}, {"full_name": "Turing.ToPartrec.stepRet_then", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [580, 9], "def_end_pos": [580, 21]}, {"full_name": "Relation.ReflTransGen.tail", "def_path": "Mathlib/Logic/Relation.lean", "def_pos": [224, 5], "def_end_pos": [224, 9]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "case mp.ret\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx c : Cfg\nhe : x \u2208 eval step c\nv : List \u2115\nk' : Cont\nv' : List \u2115\nh : x \u2208 eval step (Cfg.ret (Cont.then k' (Cont.fix f k)) v')\nIH :\n  \u2200 (a' : Cfg),\n    step (Cfg.ret (Cont.then k' (Cont.fix f k)) v') = some a' \u2192\n      \u2200 (v : List \u2115) (c' : Cfg),\n        a' = Cfg.then c' (Cont.fix f k) \u2192\n          Reaches step (stepNormal f Cont.halt v) c' \u2192\n            \u2203 v\u2081,\n              v\u2081 \u2208 Code.eval f v \u2227\n                \u2203 v\u2082,\n                  (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227\n                    x \u2208 eval step (Cfg.ret k v\u2082)\nhr : Reaches step (stepNormal f Cont.halt v) (Cfg.ret k' v')\n\u22a2 \u2203 v\u2081,\n    v\u2081 \u2208 Code.eval f v \u2227\n      \u2203 v\u2082,\n        (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227\n          x \u2208 eval step (Cfg.ret k v\u2082)", "state_after": "no goals"}, {"tactic": "rintro \u27e8v', he, hr\u27e9", "annotated_tactic": ["rintro \u27e8v', he, hr\u27e9", []], "state_before": "case mpr\nf : Code\nk : Cont\nv : List \u2115\nfok : Code.Ok f\nx : Cfg\n\u22a2 (\u2203 a, a \u2208 Code.eval (Code.fix f) v \u2227 x \u2208 eval step (Cfg.ret k a)) \u2192 x \u2208 eval step (stepNormal f (Cont.fix f k) v)", "state_after": "case mpr.intro.intro\nf : Code\nk : Cont\nv : List \u2115\nfok : Code.Ok f\nx : Cfg\nv' : List \u2115\nhe : v' \u2208 Code.eval (Code.fix f) v\nhr : x \u2208 eval step (Cfg.ret k v')\n\u22a2 x \u2208 eval step (stepNormal f (Cont.fix f k) v)"}, {"tactic": "rw [reaches_eval] at hr", "annotated_tactic": ["rw [<a>reaches_eval</a>] at hr", [{"full_name": "Turing.reaches_eval", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [863, 9], "def_end_pos": [863, 21]}]], "state_before": "case mpr.intro.intro\nf : Code\nk : Cont\nv : List \u2115\nfok : Code.Ok f\nx : Cfg\nv' : List \u2115\nhe : v' \u2208 Code.eval (Code.fix f) v\nhr : x \u2208 eval step (Cfg.ret k v')\n\u22a2 x \u2208 eval step (stepNormal f (Cont.fix f k) v)", "state_after": "case mpr.intro.intro\nf : Code\nk : Cont\nv : List \u2115\nfok : Code.Ok f\nx : Cfg\nv' : List \u2115\nhe : v' \u2208 Code.eval (Code.fix f) v\nhr\u271d : x \u2208 eval step (Cfg.ret k v')\nhr : x \u2208 eval step ?m.197346\n\u22a2 x \u2208 eval step (stepNormal f (Cont.fix f k) v)\n\ncase mpr.intro.intro\nf : Code\nk : Cont\nv : List \u2115\nfok : Code.Ok f\nx : Cfg\nv' : List \u2115\nhe : v' \u2208 Code.eval (Code.fix f) v\nhr : x \u2208 eval step (Cfg.ret k v')\n\u22a2 Reaches step (Cfg.ret k v') ?m.197346\n\nf : Code\nk : Cont\nv : List \u2115\nfok : Code.Ok f\nx : Cfg\nv' : List \u2115\nhe : v' \u2208 Code.eval (Code.fix f) v\nhr : x \u2208 eval step (Cfg.ret k v')\n\u22a2 Cfg"}, {"tactic": "swap", "annotated_tactic": ["swap", []], "state_before": "case mpr.intro.intro\nf : Code\nk : Cont\nv : List \u2115\nfok : Code.Ok f\nx : Cfg\nv' : List \u2115\nhe : v' \u2208 Code.eval (Code.fix f) v\nhr\u271d : x \u2208 eval step (Cfg.ret k v')\nhr : x \u2208 eval step ?m.197346\n\u22a2 x \u2208 eval step (stepNormal f (Cont.fix f k) v)\n\ncase mpr.intro.intro\nf : Code\nk : Cont\nv : List \u2115\nfok : Code.Ok f\nx : Cfg\nv' : List \u2115\nhe : v' \u2208 Code.eval (Code.fix f) v\nhr : x \u2208 eval step (Cfg.ret k v')\n\u22a2 Reaches step (Cfg.ret k v') ?m.197346\n\nf : Code\nk : Cont\nv : List \u2115\nfok : Code.Ok f\nx : Cfg\nv' : List \u2115\nhe : v' \u2208 Code.eval (Code.fix f) v\nhr : x \u2208 eval step (Cfg.ret k v')\n\u22a2 Cfg", "state_after": "case mpr.intro.intro\nf : Code\nk : Cont\nv : List \u2115\nfok : Code.Ok f\nx : Cfg\nv' : List \u2115\nhe : v' \u2208 Code.eval (Code.fix f) v\nhr : x \u2208 eval step (Cfg.ret k v')\n\u22a2 Reaches step (Cfg.ret k v') ?m.197346\n\ncase mpr.intro.intro\nf : Code\nk : Cont\nv : List \u2115\nfok : Code.Ok f\nx : Cfg\nv' : List \u2115\nhe : v' \u2208 Code.eval (Code.fix f) v\nhr\u271d : x \u2208 eval step (Cfg.ret k v')\nhr : x \u2208 eval step ?m.197346\n\u22a2 x \u2208 eval step (stepNormal f (Cont.fix f k) v)\n\nf : Code\nk : Cont\nv : List \u2115\nfok : Code.Ok f\nx : Cfg\nv' : List \u2115\nhe : v' \u2208 Code.eval (Code.fix f) v\nhr : x \u2208 eval step (Cfg.ret k v')\n\u22a2 Cfg"}, {"tactic": "exact ReflTransGen.single rfl", "annotated_tactic": ["exact <a>ReflTransGen.single</a> <a>rfl</a>", [{"full_name": "Relation.ReflTransGen.single", "def_path": "Mathlib/Logic/Relation.lean", "def_pos": [276, 9], "def_end_pos": [276, 15]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "case mpr.intro.intro\nf : Code\nk : Cont\nv : List \u2115\nfok : Code.Ok f\nx : Cfg\nv' : List \u2115\nhe : v' \u2208 Code.eval (Code.fix f) v\nhr : x \u2208 eval step (Cfg.ret k v')\n\u22a2 Reaches step (Cfg.ret k v') ?m.197346\n\ncase mpr.intro.intro\nf : Code\nk : Cont\nv : List \u2115\nfok : Code.Ok f\nx : Cfg\nv' : List \u2115\nhe : v' \u2208 Code.eval (Code.fix f) v\nhr\u271d : x \u2208 eval step (Cfg.ret k v')\nhr : x \u2208 eval step ?m.197346\n\u22a2 x \u2208 eval step (stepNormal f (Cont.fix f k) v)\n\nf : Code\nk : Cont\nv : List \u2115\nfok : Code.Ok f\nx : Cfg\nv' : List \u2115\nhe : v' \u2208 Code.eval (Code.fix f) v\nhr : x \u2208 eval step (Cfg.ret k v')\n\u22a2 Cfg", "state_after": "case mpr.intro.intro\nf : Code\nk : Cont\nv : List \u2115\nfok : Code.Ok f\nx : Cfg\nv' : List \u2115\nhe : v' \u2208 Code.eval (Code.fix f) v\nhr\u271d : x \u2208 eval step (Cfg.ret k v')\nhr : x \u2208 eval step (stepRet k v')\n\u22a2 x \u2208 eval step (stepNormal f (Cont.fix f k) v)"}, {"tactic": "refine' PFun.fixInduction he fun v (he : v' \u2208 f.fix.eval v) IH => _", "annotated_tactic": ["refine' <a>PFun.fixInduction</a> he fun v (he : v' \u2208 f.fix.eval v) IH => _", [{"full_name": "PFun.fixInduction", "def_path": "Mathlib/Data/PFun.lean", "def_pos": [328, 5], "def_end_pos": [328, 17]}]], "state_before": "case mpr.intro.intro\nf : Code\nk : Cont\nv : List \u2115\nfok : Code.Ok f\nx : Cfg\nv' : List \u2115\nhe : v' \u2208 Code.eval (Code.fix f) v\nhr\u271d : x \u2208 eval step (Cfg.ret k v')\nhr : x \u2208 eval step (stepRet k v')\n\u22a2 x \u2208 eval step (stepNormal f (Cont.fix f k) v)", "state_after": "case mpr.intro.intro\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx : Cfg\nv' : List \u2115\nhe\u271d : v' \u2208 Code.eval (Code.fix f) v\u271d\nhr\u271d : x \u2208 eval step (Cfg.ret k v')\nhr : x \u2208 eval step (stepRet k v')\nv : List \u2115\nhe : v' \u2208 Code.eval (Code.fix f) v\nIH :\n  \u2200 (a'' : List \u2115),\n    Sum.inr a'' \u2208\n        Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v) \u2192\n      x \u2208 eval step (stepNormal f (Cont.fix f k) a'')\n\u22a2 x \u2208 eval step (stepNormal f (Cont.fix f k) v)"}, {"tactic": "rw [fok, Part.bind_eq_bind, Part.mem_bind_iff]", "annotated_tactic": ["rw [fok, <a>Part.bind_eq_bind</a>, <a>Part.mem_bind_iff</a>]", [{"full_name": "Part.bind_eq_bind", "def_path": "Mathlib/Data/Part.lean", "def_pos": [614, 9], "def_end_pos": [614, 21]}, {"full_name": "Part.mem_bind_iff", "def_path": "Mathlib/Data/Part.lean", "def_pos": [494, 9], "def_end_pos": [494, 21]}]], "state_before": "case mpr.intro.intro\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx : Cfg\nv' : List \u2115\nhe\u271d : v' \u2208 Code.eval (Code.fix f) v\u271d\nhr\u271d : x \u2208 eval step (Cfg.ret k v')\nhr : x \u2208 eval step (stepRet k v')\nv : List \u2115\nhe : v' \u2208 Code.eval (Code.fix f) v\nIH :\n  \u2200 (a'' : List \u2115),\n    Sum.inr a'' \u2208\n        Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v) \u2192\n      x \u2208 eval step (stepNormal f (Cont.fix f k) a'')\n\u22a2 x \u2208 eval step (stepNormal f (Cont.fix f k) v)", "state_after": "case mpr.intro.intro\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx : Cfg\nv' : List \u2115\nhe\u271d : v' \u2208 Code.eval (Code.fix f) v\u271d\nhr\u271d : x \u2208 eval step (Cfg.ret k v')\nhr : x \u2208 eval step (stepRet k v')\nv : List \u2115\nhe : v' \u2208 Code.eval (Code.fix f) v\nIH :\n  \u2200 (a'' : List \u2115),\n    Sum.inr a'' \u2208\n        Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v) \u2192\n      x \u2208 eval step (stepNormal f (Cont.fix f k) a'')\n\u22a2 \u2203 a, a \u2208 Code.eval f v \u2227 x \u2208 eval step (Cfg.ret (Cont.fix f k) a)"}, {"tactic": "obtain he | \u27e8v'', he\u2081', _\u27e9 := PFun.mem_fix_iff.1 he", "annotated_tactic": ["obtain he | \u27e8v'', he\u2081', _\u27e9 := <a>PFun.mem_fix_iff</a>.1 he", [{"full_name": "PFun.mem_fix_iff", "def_path": "Mathlib/Data/PFun.lean", "def_pos": [266, 9], "def_end_pos": [266, 20]}]], "state_before": "case mpr.intro.intro\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx : Cfg\nv' : List \u2115\nhe\u271d : v' \u2208 Code.eval (Code.fix f) v\u271d\nhr\u271d : x \u2208 eval step (Cfg.ret k v')\nhr : x \u2208 eval step (stepRet k v')\nv : List \u2115\nhe : v' \u2208 Code.eval (Code.fix f) v\nIH :\n  \u2200 (a'' : List \u2115),\n    Sum.inr a'' \u2208\n        Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v) \u2192\n      x \u2208 eval step (stepNormal f (Cont.fix f k) a'')\n\u22a2 \u2203 a, a \u2208 Code.eval f v \u2227 x \u2208 eval step (Cfg.ret (Cont.fix f k) a)", "state_after": "case mpr.intro.intro.inl\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx : Cfg\nv' : List \u2115\nhe\u271d\u00b9 : v' \u2208 Code.eval (Code.fix f) v\u271d\nhr\u271d : x \u2208 eval step (Cfg.ret k v')\nhr : x \u2208 eval step (stepRet k v')\nv : List \u2115\nhe\u271d : v' \u2208 Code.eval (Code.fix f) v\nIH :\n  \u2200 (a'' : List \u2115),\n    Sum.inr a'' \u2208\n        Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v) \u2192\n      x \u2208 eval step (stepNormal f (Cont.fix f k) a'')\nhe :\n  Sum.inl v' \u2208\n    Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v)\n\u22a2 \u2203 a, a \u2208 Code.eval f v \u2227 x \u2208 eval step (Cfg.ret (Cont.fix f k) a)\n\ncase mpr.intro.intro.inr.intro.intro\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx : Cfg\nv' : List \u2115\nhe\u271d : v' \u2208 Code.eval (Code.fix f) v\u271d\nhr\u271d : x \u2208 eval step (Cfg.ret k v')\nhr : x \u2208 eval step (stepRet k v')\nv : List \u2115\nhe : v' \u2208 Code.eval (Code.fix f) v\nIH :\n  \u2200 (a'' : List \u2115),\n    Sum.inr a'' \u2208\n        Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v) \u2192\n      x \u2208 eval step (stepNormal f (Cont.fix f k) a'')\nv'' : List \u2115\nhe\u2081' :\n  Sum.inr v'' \u2208\n    Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v)\nright\u271d :\n  v' \u2208\n    PFun.fix\n      (fun v =>\n        Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v))\n      v''\n\u22a2 \u2203 a, a \u2208 Code.eval f v \u2227 x \u2208 eval step (Cfg.ret (Cont.fix f k) a)"}, {"tactic": "obtain \u27e8v', he\u2081, he\u2082\u27e9 := (Part.mem_map_iff _).1 he", "annotated_tactic": ["obtain \u27e8v', he\u2081, he\u2082\u27e9 := (<a>Part.mem_map_iff</a> _).1 he", [{"full_name": "Part.mem_map_iff", "def_path": "Mathlib/Data/Part.lean", "def_pos": [445, 9], "def_end_pos": [445, 20]}]], "state_before": "case mpr.intro.intro.inl\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx : Cfg\nv' : List \u2115\nhe\u271d\u00b9 : v' \u2208 Code.eval (Code.fix f) v\u271d\nhr\u271d : x \u2208 eval step (Cfg.ret k v')\nhr : x \u2208 eval step (stepRet k v')\nv : List \u2115\nhe\u271d : v' \u2208 Code.eval (Code.fix f) v\nIH :\n  \u2200 (a'' : List \u2115),\n    Sum.inr a'' \u2208\n        Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v) \u2192\n      x \u2208 eval step (stepNormal f (Cont.fix f k) a'')\nhe :\n  Sum.inl v' \u2208\n    Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v)\n\u22a2 \u2203 a, a \u2208 Code.eval f v \u2227 x \u2208 eval step (Cfg.ret (Cont.fix f k) a)", "state_after": "case mpr.intro.intro.inl.intro.intro\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx : Cfg\nv'\u271d : List \u2115\nhe\u271d\u00b9 : v'\u271d \u2208 Code.eval (Code.fix f) v\u271d\nhr\u271d : x \u2208 eval step (Cfg.ret k v'\u271d)\nhr : x \u2208 eval step (stepRet k v'\u271d)\nv : List \u2115\nhe\u271d : v'\u271d \u2208 Code.eval (Code.fix f) v\nIH :\n  \u2200 (a'' : List \u2115),\n    Sum.inr a'' \u2208\n        Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v) \u2192\n      x \u2208 eval step (stepNormal f (Cont.fix f k) a'')\nhe :\n  Sum.inl v'\u271d \u2208\n    Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v)\nv' : List \u2115\nhe\u2081 : v' \u2208 Code.eval f v\nhe\u2082 : (if List.headI v' = 0 then Sum.inl (List.tail v') else Sum.inr (List.tail v')) = Sum.inl v'\u271d\n\u22a2 \u2203 a, a \u2208 Code.eval f v \u2227 x \u2208 eval step (Cfg.ret (Cont.fix f k) a)"}, {"tactic": "split_ifs at he\u2082 with h", "annotated_tactic": ["split_ifs at he\u2082 with h", []], "state_before": "case mpr.intro.intro.inl.intro.intro\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx : Cfg\nv'\u271d : List \u2115\nhe\u271d\u00b9 : v'\u271d \u2208 Code.eval (Code.fix f) v\u271d\nhr\u271d : x \u2208 eval step (Cfg.ret k v'\u271d)\nhr : x \u2208 eval step (stepRet k v'\u271d)\nv : List \u2115\nhe\u271d : v'\u271d \u2208 Code.eval (Code.fix f) v\nIH :\n  \u2200 (a'' : List \u2115),\n    Sum.inr a'' \u2208\n        Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v) \u2192\n      x \u2208 eval step (stepNormal f (Cont.fix f k) a'')\nhe :\n  Sum.inl v'\u271d \u2208\n    Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v)\nv' : List \u2115\nhe\u2081 : v' \u2208 Code.eval f v\nhe\u2082 : (if List.headI v' = 0 then Sum.inl (List.tail v') else Sum.inr (List.tail v')) = Sum.inl v'\u271d\n\u22a2 \u2203 a, a \u2208 Code.eval f v \u2227 x \u2208 eval step (Cfg.ret (Cont.fix f k) a)", "state_after": "case pos\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx : Cfg\nv'\u271d : List \u2115\nhe\u271d\u00b9 : v'\u271d \u2208 Code.eval (Code.fix f) v\u271d\nhr\u271d : x \u2208 eval step (Cfg.ret k v'\u271d)\nhr : x \u2208 eval step (stepRet k v'\u271d)\nv : List \u2115\nhe\u271d : v'\u271d \u2208 Code.eval (Code.fix f) v\nIH :\n  \u2200 (a'' : List \u2115),\n    Sum.inr a'' \u2208\n        Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v) \u2192\n      x \u2208 eval step (stepNormal f (Cont.fix f k) a'')\nhe :\n  Sum.inl v'\u271d \u2208\n    Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v)\nv' : List \u2115\nhe\u2081 : v' \u2208 Code.eval f v\nh : List.headI v' = 0\nhe\u2082 : Sum.inl (List.tail v') = Sum.inl v'\u271d\n\u22a2 \u2203 a, a \u2208 Code.eval f v \u2227 x \u2208 eval step (Cfg.ret (Cont.fix f k) a)"}, {"tactic": "cases he\u2082", "annotated_tactic": ["cases he\u2082", []], "state_before": "case pos\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx : Cfg\nv'\u271d : List \u2115\nhe\u271d\u00b9 : v'\u271d \u2208 Code.eval (Code.fix f) v\u271d\nhr\u271d : x \u2208 eval step (Cfg.ret k v'\u271d)\nhr : x \u2208 eval step (stepRet k v'\u271d)\nv : List \u2115\nhe\u271d : v'\u271d \u2208 Code.eval (Code.fix f) v\nIH :\n  \u2200 (a'' : List \u2115),\n    Sum.inr a'' \u2208\n        Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v) \u2192\n      x \u2208 eval step (stepNormal f (Cont.fix f k) a'')\nhe :\n  Sum.inl v'\u271d \u2208\n    Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v)\nv' : List \u2115\nhe\u2081 : v' \u2208 Code.eval f v\nh : List.headI v' = 0\nhe\u2082 : Sum.inl (List.tail v') = Sum.inl v'\u271d\n\u22a2 \u2203 a, a \u2208 Code.eval f v \u2227 x \u2208 eval step (Cfg.ret (Cont.fix f k) a)", "state_after": "case pos.refl\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx : Cfg\nv : List \u2115\nIH :\n  \u2200 (a'' : List \u2115),\n    Sum.inr a'' \u2208\n        Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v) \u2192\n      x \u2208 eval step (stepNormal f (Cont.fix f k) a'')\nv' : List \u2115\nhe\u2081 : v' \u2208 Code.eval f v\nh : List.headI v' = 0\nhe\u271d\u00b9 : List.tail v' \u2208 Code.eval (Code.fix f) v\u271d\nhr\u271d : x \u2208 eval step (Cfg.ret k (List.tail v'))\nhr : x \u2208 eval step (stepRet k (List.tail v'))\nhe\u271d : List.tail v' \u2208 Code.eval (Code.fix f) v\nhe :\n  Sum.inl (List.tail v') \u2208\n    Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v)\n\u22a2 \u2203 a, a \u2208 Code.eval f v \u2227 x \u2208 eval step (Cfg.ret (Cont.fix f k) a)"}, {"tactic": "refine' \u27e8_, he\u2081, _\u27e9", "annotated_tactic": ["refine' \u27e8_, he\u2081, _\u27e9", []], "state_before": "case pos.refl\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx : Cfg\nv : List \u2115\nIH :\n  \u2200 (a'' : List \u2115),\n    Sum.inr a'' \u2208\n        Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v) \u2192\n      x \u2208 eval step (stepNormal f (Cont.fix f k) a'')\nv' : List \u2115\nhe\u2081 : v' \u2208 Code.eval f v\nh : List.headI v' = 0\nhe\u271d\u00b9 : List.tail v' \u2208 Code.eval (Code.fix f) v\u271d\nhr\u271d : x \u2208 eval step (Cfg.ret k (List.tail v'))\nhr : x \u2208 eval step (stepRet k (List.tail v'))\nhe\u271d : List.tail v' \u2208 Code.eval (Code.fix f) v\nhe :\n  Sum.inl (List.tail v') \u2208\n    Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v)\n\u22a2 \u2203 a, a \u2208 Code.eval f v \u2227 x \u2208 eval step (Cfg.ret (Cont.fix f k) a)", "state_after": "case pos.refl\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx : Cfg\nv : List \u2115\nIH :\n  \u2200 (a'' : List \u2115),\n    Sum.inr a'' \u2208\n        Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v) \u2192\n      x \u2208 eval step (stepNormal f (Cont.fix f k) a'')\nv' : List \u2115\nhe\u2081 : v' \u2208 Code.eval f v\nh : List.headI v' = 0\nhe\u271d\u00b9 : List.tail v' \u2208 Code.eval (Code.fix f) v\u271d\nhr\u271d : x \u2208 eval step (Cfg.ret k (List.tail v'))\nhr : x \u2208 eval step (stepRet k (List.tail v'))\nhe\u271d : List.tail v' \u2208 Code.eval (Code.fix f) v\nhe :\n  Sum.inl (List.tail v') \u2208\n    Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v)\n\u22a2 x \u2208 eval step (Cfg.ret (Cont.fix f k) v')"}, {"tactic": "rw [reaches_eval]", "annotated_tactic": ["rw [<a>reaches_eval</a>]", [{"full_name": "Turing.reaches_eval", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [863, 9], "def_end_pos": [863, 21]}]], "state_before": "case pos.refl\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx : Cfg\nv : List \u2115\nIH :\n  \u2200 (a'' : List \u2115),\n    Sum.inr a'' \u2208\n        Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v) \u2192\n      x \u2208 eval step (stepNormal f (Cont.fix f k) a'')\nv' : List \u2115\nhe\u2081 : v' \u2208 Code.eval f v\nh : List.headI v' = 0\nhe\u271d\u00b9 : List.tail v' \u2208 Code.eval (Code.fix f) v\u271d\nhr\u271d : x \u2208 eval step (Cfg.ret k (List.tail v'))\nhr : x \u2208 eval step (stepRet k (List.tail v'))\nhe\u271d : List.tail v' \u2208 Code.eval (Code.fix f) v\nhe :\n  Sum.inl (List.tail v') \u2208\n    Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v)\n\u22a2 x \u2208 eval step (Cfg.ret (Cont.fix f k) v')", "state_after": "case pos.refl\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx : Cfg\nv : List \u2115\nIH :\n  \u2200 (a'' : List \u2115),\n    Sum.inr a'' \u2208\n        Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v) \u2192\n      x \u2208 eval step (stepNormal f (Cont.fix f k) a'')\nv' : List \u2115\nhe\u2081 : v' \u2208 Code.eval f v\nh : List.headI v' = 0\nhe\u271d\u00b9 : List.tail v' \u2208 Code.eval (Code.fix f) v\u271d\nhr\u271d : x \u2208 eval step (Cfg.ret k (List.tail v'))\nhr : x \u2208 eval step (stepRet k (List.tail v'))\nhe\u271d : List.tail v' \u2208 Code.eval (Code.fix f) v\nhe :\n  Sum.inl (List.tail v') \u2208\n    Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v)\n\u22a2 x \u2208 eval step ?m.198247\n\ncase pos.refl\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx : Cfg\nv : List \u2115\nIH :\n  \u2200 (a'' : List \u2115),\n    Sum.inr a'' \u2208\n        Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v) \u2192\n      x \u2208 eval step (stepNormal f (Cont.fix f k) a'')\nv' : List \u2115\nhe\u2081 : v' \u2208 Code.eval f v\nh : List.headI v' = 0\nhe\u271d\u00b9 : List.tail v' \u2208 Code.eval (Code.fix f) v\u271d\nhr\u271d : x \u2208 eval step (Cfg.ret k (List.tail v'))\nhr : x \u2208 eval step (stepRet k (List.tail v'))\nhe\u271d : List.tail v' \u2208 Code.eval (Code.fix f) v\nhe :\n  Sum.inl (List.tail v') \u2208\n    Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v)\n\u22a2 Reaches step (Cfg.ret (Cont.fix f k) v') ?m.198247\n\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx : Cfg\nv : List \u2115\nIH :\n  \u2200 (a'' : List \u2115),\n    Sum.inr a'' \u2208\n        Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v) \u2192\n      x \u2208 eval step (stepNormal f (Cont.fix f k) a'')\nv' : List \u2115\nhe\u2081 : v' \u2208 Code.eval f v\nh : List.headI v' = 0\nhe\u271d\u00b9 : List.tail v' \u2208 Code.eval (Code.fix f) v\u271d\nhr\u271d : x \u2208 eval step (Cfg.ret k (List.tail v'))\nhr : x \u2208 eval step (stepRet k (List.tail v'))\nhe\u271d : List.tail v' \u2208 Code.eval (Code.fix f) v\nhe :\n  Sum.inl (List.tail v') \u2208\n    Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v)\n\u22a2 Cfg"}, {"tactic": "swap", "annotated_tactic": ["swap", []], "state_before": "case pos.refl\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx : Cfg\nv : List \u2115\nIH :\n  \u2200 (a'' : List \u2115),\n    Sum.inr a'' \u2208\n        Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v) \u2192\n      x \u2208 eval step (stepNormal f (Cont.fix f k) a'')\nv' : List \u2115\nhe\u2081 : v' \u2208 Code.eval f v\nh : List.headI v' = 0\nhe\u271d\u00b9 : List.tail v' \u2208 Code.eval (Code.fix f) v\u271d\nhr\u271d : x \u2208 eval step (Cfg.ret k (List.tail v'))\nhr : x \u2208 eval step (stepRet k (List.tail v'))\nhe\u271d : List.tail v' \u2208 Code.eval (Code.fix f) v\nhe :\n  Sum.inl (List.tail v') \u2208\n    Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v)\n\u22a2 x \u2208 eval step ?m.198247\n\ncase pos.refl\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx : Cfg\nv : List \u2115\nIH :\n  \u2200 (a'' : List \u2115),\n    Sum.inr a'' \u2208\n        Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v) \u2192\n      x \u2208 eval step (stepNormal f (Cont.fix f k) a'')\nv' : List \u2115\nhe\u2081 : v' \u2208 Code.eval f v\nh : List.headI v' = 0\nhe\u271d\u00b9 : List.tail v' \u2208 Code.eval (Code.fix f) v\u271d\nhr\u271d : x \u2208 eval step (Cfg.ret k (List.tail v'))\nhr : x \u2208 eval step (stepRet k (List.tail v'))\nhe\u271d : List.tail v' \u2208 Code.eval (Code.fix f) v\nhe :\n  Sum.inl (List.tail v') \u2208\n    Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v)\n\u22a2 Reaches step (Cfg.ret (Cont.fix f k) v') ?m.198247\n\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx : Cfg\nv : List \u2115\nIH :\n  \u2200 (a'' : List \u2115),\n    Sum.inr a'' \u2208\n        Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v) \u2192\n      x \u2208 eval step (stepNormal f (Cont.fix f k) a'')\nv' : List \u2115\nhe\u2081 : v' \u2208 Code.eval f v\nh : List.headI v' = 0\nhe\u271d\u00b9 : List.tail v' \u2208 Code.eval (Code.fix f) v\u271d\nhr\u271d : x \u2208 eval step (Cfg.ret k (List.tail v'))\nhr : x \u2208 eval step (stepRet k (List.tail v'))\nhe\u271d : List.tail v' \u2208 Code.eval (Code.fix f) v\nhe :\n  Sum.inl (List.tail v') \u2208\n    Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v)\n\u22a2 Cfg", "state_after": "case pos.refl\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx : Cfg\nv : List \u2115\nIH :\n  \u2200 (a'' : List \u2115),\n    Sum.inr a'' \u2208\n        Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v) \u2192\n      x \u2208 eval step (stepNormal f (Cont.fix f k) a'')\nv' : List \u2115\nhe\u2081 : v' \u2208 Code.eval f v\nh : List.headI v' = 0\nhe\u271d\u00b9 : List.tail v' \u2208 Code.eval (Code.fix f) v\u271d\nhr\u271d : x \u2208 eval step (Cfg.ret k (List.tail v'))\nhr : x \u2208 eval step (stepRet k (List.tail v'))\nhe\u271d : List.tail v' \u2208 Code.eval (Code.fix f) v\nhe :\n  Sum.inl (List.tail v') \u2208\n    Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v)\n\u22a2 Reaches step (Cfg.ret (Cont.fix f k) v') ?m.198247\n\ncase pos.refl\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx : Cfg\nv : List \u2115\nIH :\n  \u2200 (a'' : List \u2115),\n    Sum.inr a'' \u2208\n        Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v) \u2192\n      x \u2208 eval step (stepNormal f (Cont.fix f k) a'')\nv' : List \u2115\nhe\u2081 : v' \u2208 Code.eval f v\nh : List.headI v' = 0\nhe\u271d\u00b9 : List.tail v' \u2208 Code.eval (Code.fix f) v\u271d\nhr\u271d : x \u2208 eval step (Cfg.ret k (List.tail v'))\nhr : x \u2208 eval step (stepRet k (List.tail v'))\nhe\u271d : List.tail v' \u2208 Code.eval (Code.fix f) v\nhe :\n  Sum.inl (List.tail v') \u2208\n    Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v)\n\u22a2 x \u2208 eval step ?m.198247\n\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx : Cfg\nv : List \u2115\nIH :\n  \u2200 (a'' : List \u2115),\n    Sum.inr a'' \u2208\n        Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v) \u2192\n      x \u2208 eval step (stepNormal f (Cont.fix f k) a'')\nv' : List \u2115\nhe\u2081 : v' \u2208 Code.eval f v\nh : List.headI v' = 0\nhe\u271d\u00b9 : List.tail v' \u2208 Code.eval (Code.fix f) v\u271d\nhr\u271d : x \u2208 eval step (Cfg.ret k (List.tail v'))\nhr : x \u2208 eval step (stepRet k (List.tail v'))\nhe\u271d : List.tail v' \u2208 Code.eval (Code.fix f) v\nhe :\n  Sum.inl (List.tail v') \u2208\n    Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v)\n\u22a2 Cfg"}, {"tactic": "exact ReflTransGen.single rfl", "annotated_tactic": ["exact <a>ReflTransGen.single</a> <a>rfl</a>", [{"full_name": "Relation.ReflTransGen.single", "def_path": "Mathlib/Logic/Relation.lean", "def_pos": [276, 9], "def_end_pos": [276, 15]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "case pos.refl\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx : Cfg\nv : List \u2115\nIH :\n  \u2200 (a'' : List \u2115),\n    Sum.inr a'' \u2208\n        Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v) \u2192\n      x \u2208 eval step (stepNormal f (Cont.fix f k) a'')\nv' : List \u2115\nhe\u2081 : v' \u2208 Code.eval f v\nh : List.headI v' = 0\nhe\u271d\u00b9 : List.tail v' \u2208 Code.eval (Code.fix f) v\u271d\nhr\u271d : x \u2208 eval step (Cfg.ret k (List.tail v'))\nhr : x \u2208 eval step (stepRet k (List.tail v'))\nhe\u271d : List.tail v' \u2208 Code.eval (Code.fix f) v\nhe :\n  Sum.inl (List.tail v') \u2208\n    Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v)\n\u22a2 Reaches step (Cfg.ret (Cont.fix f k) v') ?m.198247\n\ncase pos.refl\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx : Cfg\nv : List \u2115\nIH :\n  \u2200 (a'' : List \u2115),\n    Sum.inr a'' \u2208\n        Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v) \u2192\n      x \u2208 eval step (stepNormal f (Cont.fix f k) a'')\nv' : List \u2115\nhe\u2081 : v' \u2208 Code.eval f v\nh : List.headI v' = 0\nhe\u271d\u00b9 : List.tail v' \u2208 Code.eval (Code.fix f) v\u271d\nhr\u271d : x \u2208 eval step (Cfg.ret k (List.tail v'))\nhr : x \u2208 eval step (stepRet k (List.tail v'))\nhe\u271d : List.tail v' \u2208 Code.eval (Code.fix f) v\nhe :\n  Sum.inl (List.tail v') \u2208\n    Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v)\n\u22a2 x \u2208 eval step ?m.198247\n\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx : Cfg\nv : List \u2115\nIH :\n  \u2200 (a'' : List \u2115),\n    Sum.inr a'' \u2208\n        Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v) \u2192\n      x \u2208 eval step (stepNormal f (Cont.fix f k) a'')\nv' : List \u2115\nhe\u2081 : v' \u2208 Code.eval f v\nh : List.headI v' = 0\nhe\u271d\u00b9 : List.tail v' \u2208 Code.eval (Code.fix f) v\u271d\nhr\u271d : x \u2208 eval step (Cfg.ret k (List.tail v'))\nhr : x \u2208 eval step (stepRet k (List.tail v'))\nhe\u271d : List.tail v' \u2208 Code.eval (Code.fix f) v\nhe :\n  Sum.inl (List.tail v') \u2208\n    Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v)\n\u22a2 Cfg", "state_after": "case pos.refl\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx : Cfg\nv : List \u2115\nIH :\n  \u2200 (a'' : List \u2115),\n    Sum.inr a'' \u2208\n        Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v) \u2192\n      x \u2208 eval step (stepNormal f (Cont.fix f k) a'')\nv' : List \u2115\nhe\u2081 : v' \u2208 Code.eval f v\nh : List.headI v' = 0\nhe\u271d\u00b9 : List.tail v' \u2208 Code.eval (Code.fix f) v\u271d\nhr\u271d : x \u2208 eval step (Cfg.ret k (List.tail v'))\nhr : x \u2208 eval step (stepRet k (List.tail v'))\nhe\u271d : List.tail v' \u2208 Code.eval (Code.fix f) v\nhe :\n  Sum.inl (List.tail v') \u2208\n    Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v)\n\u22a2 x \u2208 eval step (stepRet (Cont.fix f k) v')"}, {"tactic": "rwa [stepRet, if_pos h]", "annotated_tactic": ["rwa [<a>stepRet</a>, <a>if_pos</a> h]", [{"full_name": "Turing.ToPartrec.stepRet", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [513, 5], "def_end_pos": [513, 12]}, {"full_name": "if_pos", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [790, 9], "def_end_pos": [790, 15]}]], "state_before": "case pos.refl\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx : Cfg\nv : List \u2115\nIH :\n  \u2200 (a'' : List \u2115),\n    Sum.inr a'' \u2208\n        Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v) \u2192\n      x \u2208 eval step (stepNormal f (Cont.fix f k) a'')\nv' : List \u2115\nhe\u2081 : v' \u2208 Code.eval f v\nh : List.headI v' = 0\nhe\u271d\u00b9 : List.tail v' \u2208 Code.eval (Code.fix f) v\u271d\nhr\u271d : x \u2208 eval step (Cfg.ret k (List.tail v'))\nhr : x \u2208 eval step (stepRet k (List.tail v'))\nhe\u271d : List.tail v' \u2208 Code.eval (Code.fix f) v\nhe :\n  Sum.inl (List.tail v') \u2208\n    Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v)\n\u22a2 x \u2208 eval step (stepRet (Cont.fix f k) v')", "state_after": "no goals"}, {"tactic": "obtain \u27e8v\u2081, he\u2081, he\u2082\u27e9 := (Part.mem_map_iff _).1 he\u2081'", "annotated_tactic": ["obtain \u27e8v\u2081, he\u2081, he\u2082\u27e9 := (<a>Part.mem_map_iff</a> _).1 he\u2081'", [{"full_name": "Part.mem_map_iff", "def_path": "Mathlib/Data/Part.lean", "def_pos": [445, 9], "def_end_pos": [445, 20]}]], "state_before": "case mpr.intro.intro.inr.intro.intro\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx : Cfg\nv' : List \u2115\nhe\u271d : v' \u2208 Code.eval (Code.fix f) v\u271d\nhr\u271d : x \u2208 eval step (Cfg.ret k v')\nhr : x \u2208 eval step (stepRet k v')\nv : List \u2115\nhe : v' \u2208 Code.eval (Code.fix f) v\nIH :\n  \u2200 (a'' : List \u2115),\n    Sum.inr a'' \u2208\n        Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v) \u2192\n      x \u2208 eval step (stepNormal f (Cont.fix f k) a'')\nv'' : List \u2115\nhe\u2081' :\n  Sum.inr v'' \u2208\n    Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v)\nright\u271d :\n  v' \u2208\n    PFun.fix\n      (fun v =>\n        Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v))\n      v''\n\u22a2 \u2203 a, a \u2208 Code.eval f v \u2227 x \u2208 eval step (Cfg.ret (Cont.fix f k) a)", "state_after": "case mpr.intro.intro.inr.intro.intro.intro.intro\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx : Cfg\nv' : List \u2115\nhe\u271d : v' \u2208 Code.eval (Code.fix f) v\u271d\nhr\u271d : x \u2208 eval step (Cfg.ret k v')\nhr : x \u2208 eval step (stepRet k v')\nv : List \u2115\nhe : v' \u2208 Code.eval (Code.fix f) v\nIH :\n  \u2200 (a'' : List \u2115),\n    Sum.inr a'' \u2208\n        Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v) \u2192\n      x \u2208 eval step (stepNormal f (Cont.fix f k) a'')\nv'' : List \u2115\nhe\u2081' :\n  Sum.inr v'' \u2208\n    Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v)\nright\u271d :\n  v' \u2208\n    PFun.fix\n      (fun v =>\n        Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v))\n      v''\nv\u2081 : List \u2115\nhe\u2081 : v\u2081 \u2208 Code.eval f v\nhe\u2082 : (if List.headI v\u2081 = 0 then Sum.inl (List.tail v\u2081) else Sum.inr (List.tail v\u2081)) = Sum.inr v''\n\u22a2 \u2203 a, a \u2208 Code.eval f v \u2227 x \u2208 eval step (Cfg.ret (Cont.fix f k) a)"}, {"tactic": "split_ifs at he\u2082 with h", "annotated_tactic": ["split_ifs at he\u2082 with h", []], "state_before": "case mpr.intro.intro.inr.intro.intro.intro.intro\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx : Cfg\nv' : List \u2115\nhe\u271d : v' \u2208 Code.eval (Code.fix f) v\u271d\nhr\u271d : x \u2208 eval step (Cfg.ret k v')\nhr : x \u2208 eval step (stepRet k v')\nv : List \u2115\nhe : v' \u2208 Code.eval (Code.fix f) v\nIH :\n  \u2200 (a'' : List \u2115),\n    Sum.inr a'' \u2208\n        Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v) \u2192\n      x \u2208 eval step (stepNormal f (Cont.fix f k) a'')\nv'' : List \u2115\nhe\u2081' :\n  Sum.inr v'' \u2208\n    Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v)\nright\u271d :\n  v' \u2208\n    PFun.fix\n      (fun v =>\n        Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v))\n      v''\nv\u2081 : List \u2115\nhe\u2081 : v\u2081 \u2208 Code.eval f v\nhe\u2082 : (if List.headI v\u2081 = 0 then Sum.inl (List.tail v\u2081) else Sum.inr (List.tail v\u2081)) = Sum.inr v''\n\u22a2 \u2203 a, a \u2208 Code.eval f v \u2227 x \u2208 eval step (Cfg.ret (Cont.fix f k) a)", "state_after": "case neg\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx : Cfg\nv' : List \u2115\nhe\u271d : v' \u2208 Code.eval (Code.fix f) v\u271d\nhr\u271d : x \u2208 eval step (Cfg.ret k v')\nhr : x \u2208 eval step (stepRet k v')\nv : List \u2115\nhe : v' \u2208 Code.eval (Code.fix f) v\nIH :\n  \u2200 (a'' : List \u2115),\n    Sum.inr a'' \u2208\n        Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v) \u2192\n      x \u2208 eval step (stepNormal f (Cont.fix f k) a'')\nv'' : List \u2115\nhe\u2081' :\n  Sum.inr v'' \u2208\n    Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v)\nright\u271d :\n  v' \u2208\n    PFun.fix\n      (fun v =>\n        Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v))\n      v''\nv\u2081 : List \u2115\nhe\u2081 : v\u2081 \u2208 Code.eval f v\nh : \u00acList.headI v\u2081 = 0\nhe\u2082 : Sum.inr (List.tail v\u2081) = Sum.inr v''\n\u22a2 \u2203 a, a \u2208 Code.eval f v \u2227 x \u2208 eval step (Cfg.ret (Cont.fix f k) a)"}, {"tactic": "cases he\u2082", "annotated_tactic": ["cases he\u2082", []], "state_before": "case neg\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx : Cfg\nv' : List \u2115\nhe\u271d : v' \u2208 Code.eval (Code.fix f) v\u271d\nhr\u271d : x \u2208 eval step (Cfg.ret k v')\nhr : x \u2208 eval step (stepRet k v')\nv : List \u2115\nhe : v' \u2208 Code.eval (Code.fix f) v\nIH :\n  \u2200 (a'' : List \u2115),\n    Sum.inr a'' \u2208\n        Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v) \u2192\n      x \u2208 eval step (stepNormal f (Cont.fix f k) a'')\nv'' : List \u2115\nhe\u2081' :\n  Sum.inr v'' \u2208\n    Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v)\nright\u271d :\n  v' \u2208\n    PFun.fix\n      (fun v =>\n        Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v))\n      v''\nv\u2081 : List \u2115\nhe\u2081 : v\u2081 \u2208 Code.eval f v\nh : \u00acList.headI v\u2081 = 0\nhe\u2082 : Sum.inr (List.tail v\u2081) = Sum.inr v''\n\u22a2 \u2203 a, a \u2208 Code.eval f v \u2227 x \u2208 eval step (Cfg.ret (Cont.fix f k) a)", "state_after": "case neg.refl\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx : Cfg\nv' : List \u2115\nhe\u271d : v' \u2208 Code.eval (Code.fix f) v\u271d\nhr\u271d : x \u2208 eval step (Cfg.ret k v')\nhr : x \u2208 eval step (stepRet k v')\nv : List \u2115\nhe : v' \u2208 Code.eval (Code.fix f) v\nIH :\n  \u2200 (a'' : List \u2115),\n    Sum.inr a'' \u2208\n        Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v) \u2192\n      x \u2208 eval step (stepNormal f (Cont.fix f k) a'')\nv\u2081 : List \u2115\nhe\u2081 : v\u2081 \u2208 Code.eval f v\nh : \u00acList.headI v\u2081 = 0\nhe\u2081' :\n  Sum.inr (List.tail v\u2081) \u2208\n    Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v)\nright\u271d :\n  v' \u2208\n    PFun.fix\n      (fun v =>\n        Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v))\n      (List.tail v\u2081)\n\u22a2 \u2203 a, a \u2208 Code.eval f v \u2227 x \u2208 eval step (Cfg.ret (Cont.fix f k) a)"}, {"tactic": "clear he\u2081'", "annotated_tactic": ["clear he\u2081'", []], "state_before": "case neg.refl\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx : Cfg\nv' : List \u2115\nhe\u271d : v' \u2208 Code.eval (Code.fix f) v\u271d\nhr\u271d : x \u2208 eval step (Cfg.ret k v')\nhr : x \u2208 eval step (stepRet k v')\nv : List \u2115\nhe : v' \u2208 Code.eval (Code.fix f) v\nIH :\n  \u2200 (a'' : List \u2115),\n    Sum.inr a'' \u2208\n        Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v) \u2192\n      x \u2208 eval step (stepNormal f (Cont.fix f k) a'')\nv\u2081 : List \u2115\nhe\u2081 : v\u2081 \u2208 Code.eval f v\nh : \u00acList.headI v\u2081 = 0\nhe\u2081' :\n  Sum.inr (List.tail v\u2081) \u2208\n    Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v)\nright\u271d :\n  v' \u2208\n    PFun.fix\n      (fun v =>\n        Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v))\n      (List.tail v\u2081)\n\u22a2 \u2203 a, a \u2208 Code.eval f v \u2227 x \u2208 eval step (Cfg.ret (Cont.fix f k) a)", "state_after": "case neg.refl\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx : Cfg\nv' : List \u2115\nhe\u271d : v' \u2208 Code.eval (Code.fix f) v\u271d\nhr\u271d : x \u2208 eval step (Cfg.ret k v')\nhr : x \u2208 eval step (stepRet k v')\nv : List \u2115\nhe : v' \u2208 Code.eval (Code.fix f) v\nIH :\n  \u2200 (a'' : List \u2115),\n    Sum.inr a'' \u2208\n        Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v) \u2192\n      x \u2208 eval step (stepNormal f (Cont.fix f k) a'')\nv\u2081 : List \u2115\nhe\u2081 : v\u2081 \u2208 Code.eval f v\nh : \u00acList.headI v\u2081 = 0\nright\u271d :\n  v' \u2208\n    PFun.fix\n      (fun v =>\n        Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v))\n      (List.tail v\u2081)\n\u22a2 \u2203 a, a \u2208 Code.eval f v \u2227 x \u2208 eval step (Cfg.ret (Cont.fix f k) a)"}, {"tactic": "refine' \u27e8_, he\u2081, _\u27e9", "annotated_tactic": ["refine' \u27e8_, he\u2081, _\u27e9", []], "state_before": "case neg.refl\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx : Cfg\nv' : List \u2115\nhe\u271d : v' \u2208 Code.eval (Code.fix f) v\u271d\nhr\u271d : x \u2208 eval step (Cfg.ret k v')\nhr : x \u2208 eval step (stepRet k v')\nv : List \u2115\nhe : v' \u2208 Code.eval (Code.fix f) v\nIH :\n  \u2200 (a'' : List \u2115),\n    Sum.inr a'' \u2208\n        Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v) \u2192\n      x \u2208 eval step (stepNormal f (Cont.fix f k) a'')\nv\u2081 : List \u2115\nhe\u2081 : v\u2081 \u2208 Code.eval f v\nh : \u00acList.headI v\u2081 = 0\nright\u271d :\n  v' \u2208\n    PFun.fix\n      (fun v =>\n        Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v))\n      (List.tail v\u2081)\n\u22a2 \u2203 a, a \u2208 Code.eval f v \u2227 x \u2208 eval step (Cfg.ret (Cont.fix f k) a)", "state_after": "case neg.refl\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx : Cfg\nv' : List \u2115\nhe\u271d : v' \u2208 Code.eval (Code.fix f) v\u271d\nhr\u271d : x \u2208 eval step (Cfg.ret k v')\nhr : x \u2208 eval step (stepRet k v')\nv : List \u2115\nhe : v' \u2208 Code.eval (Code.fix f) v\nIH :\n  \u2200 (a'' : List \u2115),\n    Sum.inr a'' \u2208\n        Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v) \u2192\n      x \u2208 eval step (stepNormal f (Cont.fix f k) a'')\nv\u2081 : List \u2115\nhe\u2081 : v\u2081 \u2208 Code.eval f v\nh : \u00acList.headI v\u2081 = 0\nright\u271d :\n  v' \u2208\n    PFun.fix\n      (fun v =>\n        Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v))\n      (List.tail v\u2081)\n\u22a2 x \u2208 eval step (Cfg.ret (Cont.fix f k) v\u2081)"}, {"tactic": "rw [reaches_eval]", "annotated_tactic": ["rw [<a>reaches_eval</a>]", [{"full_name": "Turing.reaches_eval", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [863, 9], "def_end_pos": [863, 21]}]], "state_before": "case neg.refl\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx : Cfg\nv' : List \u2115\nhe\u271d : v' \u2208 Code.eval (Code.fix f) v\u271d\nhr\u271d : x \u2208 eval step (Cfg.ret k v')\nhr : x \u2208 eval step (stepRet k v')\nv : List \u2115\nhe : v' \u2208 Code.eval (Code.fix f) v\nIH :\n  \u2200 (a'' : List \u2115),\n    Sum.inr a'' \u2208\n        Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v) \u2192\n      x \u2208 eval step (stepNormal f (Cont.fix f k) a'')\nv\u2081 : List \u2115\nhe\u2081 : v\u2081 \u2208 Code.eval f v\nh : \u00acList.headI v\u2081 = 0\nright\u271d :\n  v' \u2208\n    PFun.fix\n      (fun v =>\n        Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v))\n      (List.tail v\u2081)\n\u22a2 x \u2208 eval step (Cfg.ret (Cont.fix f k) v\u2081)", "state_after": "case neg.refl\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx : Cfg\nv' : List \u2115\nhe\u271d : v' \u2208 Code.eval (Code.fix f) v\u271d\nhr\u271d : x \u2208 eval step (Cfg.ret k v')\nhr : x \u2208 eval step (stepRet k v')\nv : List \u2115\nhe : v' \u2208 Code.eval (Code.fix f) v\nIH :\n  \u2200 (a'' : List \u2115),\n    Sum.inr a'' \u2208\n        Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v) \u2192\n      x \u2208 eval step (stepNormal f (Cont.fix f k) a'')\nv\u2081 : List \u2115\nhe\u2081 : v\u2081 \u2208 Code.eval f v\nh : \u00acList.headI v\u2081 = 0\nright\u271d :\n  v' \u2208\n    PFun.fix\n      (fun v =>\n        Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v))\n      (List.tail v\u2081)\n\u22a2 x \u2208 eval step ?m.198933\n\ncase neg.refl\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx : Cfg\nv' : List \u2115\nhe\u271d : v' \u2208 Code.eval (Code.fix f) v\u271d\nhr\u271d : x \u2208 eval step (Cfg.ret k v')\nhr : x \u2208 eval step (stepRet k v')\nv : List \u2115\nhe : v' \u2208 Code.eval (Code.fix f) v\nIH :\n  \u2200 (a'' : List \u2115),\n    Sum.inr a'' \u2208\n        Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v) \u2192\n      x \u2208 eval step (stepNormal f (Cont.fix f k) a'')\nv\u2081 : List \u2115\nhe\u2081 : v\u2081 \u2208 Code.eval f v\nh : \u00acList.headI v\u2081 = 0\nright\u271d :\n  v' \u2208\n    PFun.fix\n      (fun v =>\n        Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v))\n      (List.tail v\u2081)\n\u22a2 Reaches step (Cfg.ret (Cont.fix f k) v\u2081) ?m.198933\n\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx : Cfg\nv' : List \u2115\nhe\u271d : v' \u2208 Code.eval (Code.fix f) v\u271d\nhr\u271d : x \u2208 eval step (Cfg.ret k v')\nhr : x \u2208 eval step (stepRet k v')\nv : List \u2115\nhe : v' \u2208 Code.eval (Code.fix f) v\nIH :\n  \u2200 (a'' : List \u2115),\n    Sum.inr a'' \u2208\n        Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v) \u2192\n      x \u2208 eval step (stepNormal f (Cont.fix f k) a'')\nv\u2081 : List \u2115\nhe\u2081 : v\u2081 \u2208 Code.eval f v\nh : \u00acList.headI v\u2081 = 0\nright\u271d :\n  v' \u2208\n    PFun.fix\n      (fun v =>\n        Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v))\n      (List.tail v\u2081)\n\u22a2 Cfg"}, {"tactic": "swap", "annotated_tactic": ["swap", []], "state_before": "case neg.refl\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx : Cfg\nv' : List \u2115\nhe\u271d : v' \u2208 Code.eval (Code.fix f) v\u271d\nhr\u271d : x \u2208 eval step (Cfg.ret k v')\nhr : x \u2208 eval step (stepRet k v')\nv : List \u2115\nhe : v' \u2208 Code.eval (Code.fix f) v\nIH :\n  \u2200 (a'' : List \u2115),\n    Sum.inr a'' \u2208\n        Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v) \u2192\n      x \u2208 eval step (stepNormal f (Cont.fix f k) a'')\nv\u2081 : List \u2115\nhe\u2081 : v\u2081 \u2208 Code.eval f v\nh : \u00acList.headI v\u2081 = 0\nright\u271d :\n  v' \u2208\n    PFun.fix\n      (fun v =>\n        Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v))\n      (List.tail v\u2081)\n\u22a2 x \u2208 eval step ?m.198933\n\ncase neg.refl\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx : Cfg\nv' : List \u2115\nhe\u271d : v' \u2208 Code.eval (Code.fix f) v\u271d\nhr\u271d : x \u2208 eval step (Cfg.ret k v')\nhr : x \u2208 eval step (stepRet k v')\nv : List \u2115\nhe : v' \u2208 Code.eval (Code.fix f) v\nIH :\n  \u2200 (a'' : List \u2115),\n    Sum.inr a'' \u2208\n        Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v) \u2192\n      x \u2208 eval step (stepNormal f (Cont.fix f k) a'')\nv\u2081 : List \u2115\nhe\u2081 : v\u2081 \u2208 Code.eval f v\nh : \u00acList.headI v\u2081 = 0\nright\u271d :\n  v' \u2208\n    PFun.fix\n      (fun v =>\n        Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v))\n      (List.tail v\u2081)\n\u22a2 Reaches step (Cfg.ret (Cont.fix f k) v\u2081) ?m.198933\n\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx : Cfg\nv' : List \u2115\nhe\u271d : v' \u2208 Code.eval (Code.fix f) v\u271d\nhr\u271d : x \u2208 eval step (Cfg.ret k v')\nhr : x \u2208 eval step (stepRet k v')\nv : List \u2115\nhe : v' \u2208 Code.eval (Code.fix f) v\nIH :\n  \u2200 (a'' : List \u2115),\n    Sum.inr a'' \u2208\n        Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v) \u2192\n      x \u2208 eval step (stepNormal f (Cont.fix f k) a'')\nv\u2081 : List \u2115\nhe\u2081 : v\u2081 \u2208 Code.eval f v\nh : \u00acList.headI v\u2081 = 0\nright\u271d :\n  v' \u2208\n    PFun.fix\n      (fun v =>\n        Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v))\n      (List.tail v\u2081)\n\u22a2 Cfg", "state_after": "case neg.refl\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx : Cfg\nv' : List \u2115\nhe\u271d : v' \u2208 Code.eval (Code.fix f) v\u271d\nhr\u271d : x \u2208 eval step (Cfg.ret k v')\nhr : x \u2208 eval step (stepRet k v')\nv : List \u2115\nhe : v' \u2208 Code.eval (Code.fix f) v\nIH :\n  \u2200 (a'' : List \u2115),\n    Sum.inr a'' \u2208\n        Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v) \u2192\n      x \u2208 eval step (stepNormal f (Cont.fix f k) a'')\nv\u2081 : List \u2115\nhe\u2081 : v\u2081 \u2208 Code.eval f v\nh : \u00acList.headI v\u2081 = 0\nright\u271d :\n  v' \u2208\n    PFun.fix\n      (fun v =>\n        Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v))\n      (List.tail v\u2081)\n\u22a2 Reaches step (Cfg.ret (Cont.fix f k) v\u2081) ?m.198933\n\ncase neg.refl\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx : Cfg\nv' : List \u2115\nhe\u271d : v' \u2208 Code.eval (Code.fix f) v\u271d\nhr\u271d : x \u2208 eval step (Cfg.ret k v')\nhr : x \u2208 eval step (stepRet k v')\nv : List \u2115\nhe : v' \u2208 Code.eval (Code.fix f) v\nIH :\n  \u2200 (a'' : List \u2115),\n    Sum.inr a'' \u2208\n        Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v) \u2192\n      x \u2208 eval step (stepNormal f (Cont.fix f k) a'')\nv\u2081 : List \u2115\nhe\u2081 : v\u2081 \u2208 Code.eval f v\nh : \u00acList.headI v\u2081 = 0\nright\u271d :\n  v' \u2208\n    PFun.fix\n      (fun v =>\n        Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v))\n      (List.tail v\u2081)\n\u22a2 x \u2208 eval step ?m.198933\n\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx : Cfg\nv' : List \u2115\nhe\u271d : v' \u2208 Code.eval (Code.fix f) v\u271d\nhr\u271d : x \u2208 eval step (Cfg.ret k v')\nhr : x \u2208 eval step (stepRet k v')\nv : List \u2115\nhe : v' \u2208 Code.eval (Code.fix f) v\nIH :\n  \u2200 (a'' : List \u2115),\n    Sum.inr a'' \u2208\n        Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v) \u2192\n      x \u2208 eval step (stepNormal f (Cont.fix f k) a'')\nv\u2081 : List \u2115\nhe\u2081 : v\u2081 \u2208 Code.eval f v\nh : \u00acList.headI v\u2081 = 0\nright\u271d :\n  v' \u2208\n    PFun.fix\n      (fun v =>\n        Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v))\n      (List.tail v\u2081)\n\u22a2 Cfg"}, {"tactic": "exact ReflTransGen.single rfl", "annotated_tactic": ["exact <a>ReflTransGen.single</a> <a>rfl</a>", [{"full_name": "Relation.ReflTransGen.single", "def_path": "Mathlib/Logic/Relation.lean", "def_pos": [276, 9], "def_end_pos": [276, 15]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "case neg.refl\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx : Cfg\nv' : List \u2115\nhe\u271d : v' \u2208 Code.eval (Code.fix f) v\u271d\nhr\u271d : x \u2208 eval step (Cfg.ret k v')\nhr : x \u2208 eval step (stepRet k v')\nv : List \u2115\nhe : v' \u2208 Code.eval (Code.fix f) v\nIH :\n  \u2200 (a'' : List \u2115),\n    Sum.inr a'' \u2208\n        Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v) \u2192\n      x \u2208 eval step (stepNormal f (Cont.fix f k) a'')\nv\u2081 : List \u2115\nhe\u2081 : v\u2081 \u2208 Code.eval f v\nh : \u00acList.headI v\u2081 = 0\nright\u271d :\n  v' \u2208\n    PFun.fix\n      (fun v =>\n        Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v))\n      (List.tail v\u2081)\n\u22a2 Reaches step (Cfg.ret (Cont.fix f k) v\u2081) ?m.198933\n\ncase neg.refl\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx : Cfg\nv' : List \u2115\nhe\u271d : v' \u2208 Code.eval (Code.fix f) v\u271d\nhr\u271d : x \u2208 eval step (Cfg.ret k v')\nhr : x \u2208 eval step (stepRet k v')\nv : List \u2115\nhe : v' \u2208 Code.eval (Code.fix f) v\nIH :\n  \u2200 (a'' : List \u2115),\n    Sum.inr a'' \u2208\n        Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v) \u2192\n      x \u2208 eval step (stepNormal f (Cont.fix f k) a'')\nv\u2081 : List \u2115\nhe\u2081 : v\u2081 \u2208 Code.eval f v\nh : \u00acList.headI v\u2081 = 0\nright\u271d :\n  v' \u2208\n    PFun.fix\n      (fun v =>\n        Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v))\n      (List.tail v\u2081)\n\u22a2 x \u2208 eval step ?m.198933\n\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx : Cfg\nv' : List \u2115\nhe\u271d : v' \u2208 Code.eval (Code.fix f) v\u271d\nhr\u271d : x \u2208 eval step (Cfg.ret k v')\nhr : x \u2208 eval step (stepRet k v')\nv : List \u2115\nhe : v' \u2208 Code.eval (Code.fix f) v\nIH :\n  \u2200 (a'' : List \u2115),\n    Sum.inr a'' \u2208\n        Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v) \u2192\n      x \u2208 eval step (stepNormal f (Cont.fix f k) a'')\nv\u2081 : List \u2115\nhe\u2081 : v\u2081 \u2208 Code.eval f v\nh : \u00acList.headI v\u2081 = 0\nright\u271d :\n  v' \u2208\n    PFun.fix\n      (fun v =>\n        Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v))\n      (List.tail v\u2081)\n\u22a2 Cfg", "state_after": "case neg.refl\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx : Cfg\nv' : List \u2115\nhe\u271d : v' \u2208 Code.eval (Code.fix f) v\u271d\nhr\u271d : x \u2208 eval step (Cfg.ret k v')\nhr : x \u2208 eval step (stepRet k v')\nv : List \u2115\nhe : v' \u2208 Code.eval (Code.fix f) v\nIH :\n  \u2200 (a'' : List \u2115),\n    Sum.inr a'' \u2208\n        Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v) \u2192\n      x \u2208 eval step (stepNormal f (Cont.fix f k) a'')\nv\u2081 : List \u2115\nhe\u2081 : v\u2081 \u2208 Code.eval f v\nh : \u00acList.headI v\u2081 = 0\nright\u271d :\n  v' \u2208\n    PFun.fix\n      (fun v =>\n        Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v))\n      (List.tail v\u2081)\n\u22a2 x \u2208 eval step (stepRet (Cont.fix f k) v\u2081)"}, {"tactic": "rw [stepRet, if_neg h]", "annotated_tactic": ["rw [<a>stepRet</a>, <a>if_neg</a> h]", [{"full_name": "Turing.ToPartrec.stepRet", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [513, 5], "def_end_pos": [513, 12]}, {"full_name": "if_neg", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [795, 9], "def_end_pos": [795, 15]}]], "state_before": "case neg.refl\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx : Cfg\nv' : List \u2115\nhe\u271d : v' \u2208 Code.eval (Code.fix f) v\u271d\nhr\u271d : x \u2208 eval step (Cfg.ret k v')\nhr : x \u2208 eval step (stepRet k v')\nv : List \u2115\nhe : v' \u2208 Code.eval (Code.fix f) v\nIH :\n  \u2200 (a'' : List \u2115),\n    Sum.inr a'' \u2208\n        Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v) \u2192\n      x \u2208 eval step (stepNormal f (Cont.fix f k) a'')\nv\u2081 : List \u2115\nhe\u2081 : v\u2081 \u2208 Code.eval f v\nh : \u00acList.headI v\u2081 = 0\nright\u271d :\n  v' \u2208\n    PFun.fix\n      (fun v =>\n        Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v))\n      (List.tail v\u2081)\n\u22a2 x \u2208 eval step (stepRet (Cont.fix f k) v\u2081)", "state_after": "case neg.refl\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx : Cfg\nv' : List \u2115\nhe\u271d : v' \u2208 Code.eval (Code.fix f) v\u271d\nhr\u271d : x \u2208 eval step (Cfg.ret k v')\nhr : x \u2208 eval step (stepRet k v')\nv : List \u2115\nhe : v' \u2208 Code.eval (Code.fix f) v\nIH :\n  \u2200 (a'' : List \u2115),\n    Sum.inr a'' \u2208\n        Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v) \u2192\n      x \u2208 eval step (stepNormal f (Cont.fix f k) a'')\nv\u2081 : List \u2115\nhe\u2081 : v\u2081 \u2208 Code.eval f v\nh : \u00acList.headI v\u2081 = 0\nright\u271d :\n  v' \u2208\n    PFun.fix\n      (fun v =>\n        Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v))\n      (List.tail v\u2081)\n\u22a2 x \u2208 eval step (stepNormal f (Cont.fix f k) (List.tail v\u2081))"}, {"tactic": "exact IH v\u2081.tail ((Part.mem_map_iff _).2 \u27e8_, he\u2081, if_neg h\u27e9)", "annotated_tactic": ["exact IH v\u2081.tail ((<a>Part.mem_map_iff</a> _).2 \u27e8_, he\u2081, <a>if_neg</a> h\u27e9)", [{"full_name": "Part.mem_map_iff", "def_path": "Mathlib/Data/Part.lean", "def_pos": [445, 9], "def_end_pos": [445, 20]}, {"full_name": "if_neg", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [795, 9], "def_end_pos": [795, 15]}]], "state_before": "case neg.refl\nf : Code\nk : Cont\nv\u271d : List \u2115\nfok : Code.Ok f\nx : Cfg\nv' : List \u2115\nhe\u271d : v' \u2208 Code.eval (Code.fix f) v\u271d\nhr\u271d : x \u2208 eval step (Cfg.ret k v')\nhr : x \u2208 eval step (stepRet k v')\nv : List \u2115\nhe : v' \u2208 Code.eval (Code.fix f) v\nIH :\n  \u2200 (a'' : List \u2115),\n    Sum.inr a'' \u2208\n        Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v) \u2192\n      x \u2208 eval step (stepNormal f (Cont.fix f k) a'')\nv\u2081 : List \u2115\nhe\u2081 : v\u2081 \u2208 Code.eval f v\nh : \u00acList.headI v\u2081 = 0\nright\u271d :\n  v' \u2208\n    PFun.fix\n      (fun v =>\n        Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v))\n      (List.tail v\u2081)\n\u22a2 x \u2208 eval step (stepNormal f (Cont.fix f k) (List.tail v\u2081))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Image.lean", "full_name": "Set.mem_compl_image", "start": [389, 1], "end": [391, 41], "traced_tactics": [{"tactic": "simp [\u2190 preimage_compl_eq_image_compl]", "annotated_tactic": ["simp [\u2190 <a>preimage_compl_eq_image_compl</a>]", [{"full_name": "Set.preimage_compl_eq_image_compl", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [382, 9], "def_end_pos": [382, 38]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\nf : \u03b1 \u2192 \u03b2\ns t\u271d : Set \u03b1\ninst\u271d : BooleanAlgebra \u03b1\nt : \u03b1\nS : Set \u03b1\n\u22a2 t \u2208 compl '' S \u2194 t\u1d9c \u2208 S", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Function.lean", "full_name": "Set.InvOn.mono", "start": [1201, 1], "end": [1202, 29], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "full_name": "MeasureTheory.Lp.simpleFunc.denseInducing", "start": [781, 11], "end": [783, 56], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Language.lean", "full_name": "Language.mem_pow", "start": [229, 1], "end": [244, 51], "traced_tactics": [{"tactic": "induction' n with n ihn generalizing x", "annotated_tactic": ["induction' n with n ihn generalizing x", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nl\u271d m : Language \u03b1\na b x\u271d : List \u03b1\nl : Language \u03b1\nx : List \u03b1\nn : \u2115\n\u22a2 x \u2208 l ^ n \u2194 \u2203 S, x = join S \u2227 length S = n \u2227 \u2200 (y : List \u03b1), y \u2208 S \u2192 y \u2208 l", "state_after": "case zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nl\u271d m : Language \u03b1\na b x\u271d\u00b9 : List \u03b1\nl : Language \u03b1\nx\u271d x : List \u03b1\n\u22a2 x \u2208 l ^ Nat.zero \u2194 \u2203 S, x = join S \u2227 length S = Nat.zero \u2227 \u2200 (y : List \u03b1), y \u2208 S \u2192 y \u2208 l\n\ncase succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nl\u271d m : Language \u03b1\na b x\u271d\u00b9 : List \u03b1\nl : Language \u03b1\nx\u271d : List \u03b1\nn : \u2115\nihn : \u2200 {x : List \u03b1}, x \u2208 l ^ n \u2194 \u2203 S, x = join S \u2227 length S = n \u2227 \u2200 (y : List \u03b1), y \u2208 S \u2192 y \u2208 l\nx : List \u03b1\n\u22a2 x \u2208 l ^ Nat.succ n \u2194 \u2203 S, x = join S \u2227 length S = Nat.succ n \u2227 \u2200 (y : List \u03b1), y \u2208 S \u2192 y \u2208 l"}, {"tactic": "simp only [mem_one, pow_zero, length_eq_zero]", "annotated_tactic": ["simp only [<a>mem_one</a>, <a>pow_zero</a>, <a>length_eq_zero</a>]", [{"full_name": "Language.mem_one", "def_path": "Mathlib/Computability/Language.lean", "def_pos": [102, 9], "def_end_pos": [102, 16]}, {"full_name": "pow_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [639, 9], "def_end_pos": [639, 17]}, {"full_name": "List.length_eq_zero", "def_path": "lake-packages/std/Std/Data/List/Init/Lemmas.lean", "def_pos": [50, 9], "def_end_pos": [50, 23]}]], "state_before": "case zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nl\u271d m : Language \u03b1\na b x\u271d\u00b9 : List \u03b1\nl : Language \u03b1\nx\u271d x : List \u03b1\n\u22a2 x \u2208 l ^ Nat.zero \u2194 \u2203 S, x = join S \u2227 length S = Nat.zero \u2227 \u2200 (y : List \u03b1), y \u2208 S \u2192 y \u2208 l", "state_after": "case zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nl\u271d m : Language \u03b1\na b x\u271d\u00b9 : List \u03b1\nl : Language \u03b1\nx\u271d x : List \u03b1\n\u22a2 x = [] \u2194 \u2203 S, x = join S \u2227 S = [] \u2227 \u2200 (y : List \u03b1), y \u2208 S \u2192 y \u2208 l"}, {"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "case zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nl\u271d m : Language \u03b1\na b x\u271d\u00b9 : List \u03b1\nl : Language \u03b1\nx\u271d x : List \u03b1\n\u22a2 x = [] \u2194 \u2203 S, x = join S \u2227 S = [] \u2227 \u2200 (y : List \u03b1), y \u2208 S \u2192 y \u2208 l", "state_after": "case zero.mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nl\u271d m : Language \u03b1\na b x\u271d\u00b9 : List \u03b1\nl : Language \u03b1\nx\u271d x : List \u03b1\n\u22a2 x = [] \u2192 \u2203 S, x = join S \u2227 S = [] \u2227 \u2200 (y : List \u03b1), y \u2208 S \u2192 y \u2208 l\n\ncase zero.mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nl\u271d m : Language \u03b1\na b x\u271d\u00b9 : List \u03b1\nl : Language \u03b1\nx\u271d x : List \u03b1\n\u22a2 (\u2203 S, x = join S \u2227 S = [] \u2227 \u2200 (y : List \u03b1), y \u2208 S \u2192 y \u2208 l) \u2192 x = []"}, {"tactic": "rintro rfl", "annotated_tactic": ["rintro rfl", []], "state_before": "case zero.mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nl\u271d m : Language \u03b1\na b x\u271d\u00b9 : List \u03b1\nl : Language \u03b1\nx\u271d x : List \u03b1\n\u22a2 x = [] \u2192 \u2203 S, x = join S \u2227 S = [] \u2227 \u2200 (y : List \u03b1), y \u2208 S \u2192 y \u2208 l", "state_after": "case zero.mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nl\u271d m : Language \u03b1\na b x\u271d : List \u03b1\nl : Language \u03b1\nx : List \u03b1\n\u22a2 \u2203 S, [] = join S \u2227 S = [] \u2227 \u2200 (y : List \u03b1), y \u2208 S \u2192 y \u2208 l"}, {"tactic": "exact \u27e8[], rfl, rfl, fun _ h \u21a6 by contradiction\u27e9", "annotated_tactic": ["exact \u27e8[], <a>rfl</a>, <a>rfl</a>, fun _ h \u21a6 by contradiction\u27e9", [{"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "case zero.mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nl\u271d m : Language \u03b1\na b x\u271d : List \u03b1\nl : Language \u03b1\nx : List \u03b1\n\u22a2 \u2203 S, [] = join S \u2227 S = [] \u2227 \u2200 (y : List \u03b1), y \u2208 S \u2192 y \u2208 l", "state_after": "no goals"}, {"tactic": "contradiction", "annotated_tactic": ["contradiction", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nl\u271d m : Language \u03b1\na b x\u271d\u00b9 : List \u03b1\nl : Language \u03b1\nx x\u271d : List \u03b1\nh : x\u271d \u2208 []\n\u22a2 x\u271d \u2208 l", "state_after": "no goals"}, {"tactic": "rintro \u27e8_, rfl, rfl, _\u27e9", "annotated_tactic": ["rintro \u27e8_, rfl, rfl, _\u27e9", []], "state_before": "case zero.mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nl\u271d m : Language \u03b1\na b x\u271d\u00b9 : List \u03b1\nl : Language \u03b1\nx\u271d x : List \u03b1\n\u22a2 (\u2203 S, x = join S \u2227 S = [] \u2227 \u2200 (y : List \u03b1), y \u2208 S \u2192 y \u2208 l) \u2192 x = []", "state_after": "case zero.mpr.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nl\u271d m : Language \u03b1\na b x\u271d : List \u03b1\nl : Language \u03b1\nx : List \u03b1\nright\u271d : \u2200 (y : List \u03b1), y \u2208 [] \u2192 y \u2208 l\n\u22a2 join [] = []"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case zero.mpr.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nl\u271d m : Language \u03b1\na b x\u271d : List \u03b1\nl : Language \u03b1\nx : List \u03b1\nright\u271d : \u2200 (y : List \u03b1), y \u2208 [] \u2192 y \u2208 l\n\u22a2 join [] = []", "state_after": "no goals"}, {"tactic": "simp only [pow_succ, mem_mul, ihn]", "annotated_tactic": ["simp only [<a>pow_succ</a>, <a>mem_mul</a>, ihn]", [{"full_name": "pow_succ", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [645, 9], "def_end_pos": [645, 17]}, {"full_name": "Language.mem_mul", "def_path": "Mathlib/Computability/Language.lean", "def_pos": [113, 9], "def_end_pos": [113, 16]}]], "state_before": "case succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nl\u271d m : Language \u03b1\na b x\u271d\u00b9 : List \u03b1\nl : Language \u03b1\nx\u271d : List \u03b1\nn : \u2115\nihn : \u2200 {x : List \u03b1}, x \u2208 l ^ n \u2194 \u2203 S, x = join S \u2227 length S = n \u2227 \u2200 (y : List \u03b1), y \u2208 S \u2192 y \u2208 l\nx : List \u03b1\n\u22a2 x \u2208 l ^ Nat.succ n \u2194 \u2203 S, x = join S \u2227 length S = Nat.succ n \u2227 \u2200 (y : List \u03b1), y \u2208 S \u2192 y \u2208 l", "state_after": "case succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nl\u271d m : Language \u03b1\na b x\u271d\u00b9 : List \u03b1\nl : Language \u03b1\nx\u271d : List \u03b1\nn : \u2115\nihn : \u2200 {x : List \u03b1}, x \u2208 l ^ n \u2194 \u2203 S, x = join S \u2227 length S = n \u2227 \u2200 (y : List \u03b1), y \u2208 S \u2192 y \u2208 l\nx : List \u03b1\n\u22a2 (\u2203 a b, a \u2208 l \u2227 (\u2203 S, b = join S \u2227 length S = n \u2227 \u2200 (y : List \u03b1), y \u2208 S \u2192 y \u2208 l) \u2227 a ++ b = x) \u2194\n    \u2203 S, x = join S \u2227 length S = Nat.succ n \u2227 \u2200 (y : List \u03b1), y \u2208 S \u2192 y \u2208 l"}, {"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "case succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nl\u271d m : Language \u03b1\na b x\u271d\u00b9 : List \u03b1\nl : Language \u03b1\nx\u271d : List \u03b1\nn : \u2115\nihn : \u2200 {x : List \u03b1}, x \u2208 l ^ n \u2194 \u2203 S, x = join S \u2227 length S = n \u2227 \u2200 (y : List \u03b1), y \u2208 S \u2192 y \u2208 l\nx : List \u03b1\n\u22a2 (\u2203 a b, a \u2208 l \u2227 (\u2203 S, b = join S \u2227 length S = n \u2227 \u2200 (y : List \u03b1), y \u2208 S \u2192 y \u2208 l) \u2227 a ++ b = x) \u2194\n    \u2203 S, x = join S \u2227 length S = Nat.succ n \u2227 \u2200 (y : List \u03b1), y \u2208 S \u2192 y \u2208 l", "state_after": "case succ.mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nl\u271d m : Language \u03b1\na b x\u271d\u00b9 : List \u03b1\nl : Language \u03b1\nx\u271d : List \u03b1\nn : \u2115\nihn : \u2200 {x : List \u03b1}, x \u2208 l ^ n \u2194 \u2203 S, x = join S \u2227 length S = n \u2227 \u2200 (y : List \u03b1), y \u2208 S \u2192 y \u2208 l\nx : List \u03b1\n\u22a2 (\u2203 a b, a \u2208 l \u2227 (\u2203 S, b = join S \u2227 length S = n \u2227 \u2200 (y : List \u03b1), y \u2208 S \u2192 y \u2208 l) \u2227 a ++ b = x) \u2192\n    \u2203 S, x = join S \u2227 length S = Nat.succ n \u2227 \u2200 (y : List \u03b1), y \u2208 S \u2192 y \u2208 l\n\ncase succ.mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nl\u271d m : Language \u03b1\na b x\u271d\u00b9 : List \u03b1\nl : Language \u03b1\nx\u271d : List \u03b1\nn : \u2115\nihn : \u2200 {x : List \u03b1}, x \u2208 l ^ n \u2194 \u2203 S, x = join S \u2227 length S = n \u2227 \u2200 (y : List \u03b1), y \u2208 S \u2192 y \u2208 l\nx : List \u03b1\n\u22a2 (\u2203 S, x = join S \u2227 length S = Nat.succ n \u2227 \u2200 (y : List \u03b1), y \u2208 S \u2192 y \u2208 l) \u2192\n    \u2203 a b, a \u2208 l \u2227 (\u2203 S, b = join S \u2227 length S = n \u2227 \u2200 (y : List \u03b1), y \u2208 S \u2192 y \u2208 l) \u2227 a ++ b = x"}, {"tactic": "rintro \u27e8a, b, ha, \u27e8S, rfl, rfl, hS\u27e9, rfl\u27e9", "annotated_tactic": ["rintro \u27e8a, b, ha, \u27e8S, rfl, rfl, hS\u27e9, rfl\u27e9", []], "state_before": "case succ.mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nl\u271d m : Language \u03b1\na b x\u271d\u00b9 : List \u03b1\nl : Language \u03b1\nx\u271d : List \u03b1\nn : \u2115\nihn : \u2200 {x : List \u03b1}, x \u2208 l ^ n \u2194 \u2203 S, x = join S \u2227 length S = n \u2227 \u2200 (y : List \u03b1), y \u2208 S \u2192 y \u2208 l\nx : List \u03b1\n\u22a2 (\u2203 a b, a \u2208 l \u2227 (\u2203 S, b = join S \u2227 length S = n \u2227 \u2200 (y : List \u03b1), y \u2208 S \u2192 y \u2208 l) \u2227 a ++ b = x) \u2192\n    \u2203 S, x = join S \u2227 length S = Nat.succ n \u2227 \u2200 (y : List \u03b1), y \u2208 S \u2192 y \u2208 l", "state_after": "case succ.mp.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nl\u271d m : Language \u03b1\na\u271d b x\u271d : List \u03b1\nl : Language \u03b1\nx a : List \u03b1\nha : a \u2208 l\nS : List (List \u03b1)\nhS : \u2200 (y : List \u03b1), y \u2208 S \u2192 y \u2208 l\nihn : \u2200 {x : List \u03b1}, x \u2208 l ^ length S \u2194 \u2203 S_1, x = join S_1 \u2227 length S_1 = length S \u2227 \u2200 (y : List \u03b1), y \u2208 S_1 \u2192 y \u2208 l\n\u22a2 \u2203 S_1, a ++ join S = join S_1 \u2227 length S_1 = Nat.succ (length S) \u2227 \u2200 (y : List \u03b1), y \u2208 S_1 \u2192 y \u2208 l"}, {"tactic": "exact \u27e8a :: S, rfl, rfl, forall_mem_cons.2 \u27e8ha, hS\u27e9\u27e9", "annotated_tactic": ["exact \u27e8a :: S, <a>rfl</a>, <a>rfl</a>, <a>forall_mem_cons</a>.2 \u27e8ha, hS\u27e9\u27e9", [{"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}, {"full_name": "List.forall_mem_cons", "def_path": "lake-packages/std/Std/Data/List/Init/Lemmas.lean", "def_pos": [120, 9], "def_end_pos": [120, 24]}]], "state_before": "case succ.mp.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nl\u271d m : Language \u03b1\na\u271d b x\u271d : List \u03b1\nl : Language \u03b1\nx a : List \u03b1\nha : a \u2208 l\nS : List (List \u03b1)\nhS : \u2200 (y : List \u03b1), y \u2208 S \u2192 y \u2208 l\nihn : \u2200 {x : List \u03b1}, x \u2208 l ^ length S \u2194 \u2203 S_1, x = join S_1 \u2227 length S_1 = length S \u2227 \u2200 (y : List \u03b1), y \u2208 S_1 \u2192 y \u2208 l\n\u22a2 \u2203 S_1, a ++ join S = join S_1 \u2227 length S_1 = Nat.succ (length S) \u2227 \u2200 (y : List \u03b1), y \u2208 S_1 \u2192 y \u2208 l", "state_after": "no goals"}, {"tactic": "rintro \u27e8_ | \u27e8a, S\u27e9, rfl, hn, hS\u27e9 <;> cases hn", "annotated_tactic": ["rintro \u27e8_ | \u27e8a, S\u27e9, rfl, hn, hS\u27e9 <;> cases hn", []], "state_before": "case succ.mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nl\u271d m : Language \u03b1\na b x\u271d\u00b9 : List \u03b1\nl : Language \u03b1\nx\u271d : List \u03b1\nn : \u2115\nihn : \u2200 {x : List \u03b1}, x \u2208 l ^ n \u2194 \u2203 S, x = join S \u2227 length S = n \u2227 \u2200 (y : List \u03b1), y \u2208 S \u2192 y \u2208 l\nx : List \u03b1\n\u22a2 (\u2203 S, x = join S \u2227 length S = Nat.succ n \u2227 \u2200 (y : List \u03b1), y \u2208 S \u2192 y \u2208 l) \u2192\n    \u2203 a b, a \u2208 l \u2227 (\u2203 S, b = join S \u2227 length S = n \u2227 \u2200 (y : List \u03b1), y \u2208 S \u2192 y \u2208 l) \u2227 a ++ b = x", "state_after": "case succ.mpr.intro.cons.intro.intro.refl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nl\u271d m : Language \u03b1\na\u271d b x\u271d : List \u03b1\nl : Language \u03b1\nx a : List \u03b1\nS : List (List \u03b1)\nhS : \u2200 (y : List \u03b1), y \u2208 a :: S \u2192 y \u2208 l\nihn :\n  \u2200 {x : List \u03b1},\n    x \u2208 l ^ Nat.add (length S) 0 \u2194\n      \u2203 S_1, x = join S_1 \u2227 length S_1 = Nat.add (length S) 0 \u2227 \u2200 (y : List \u03b1), y \u2208 S_1 \u2192 y \u2208 l\n\u22a2 \u2203 a_1 b,\n    a_1 \u2208 l \u2227\n      (\u2203 S_1, b = join S_1 \u2227 length S_1 = Nat.add (length S) 0 \u2227 \u2200 (y : List \u03b1), y \u2208 S_1 \u2192 y \u2208 l) \u2227\n        a_1 ++ b = join (a :: S)"}, {"tactic": "rw [forall_mem_cons] at hS", "annotated_tactic": ["rw [<a>forall_mem_cons</a>] at hS", [{"full_name": "List.forall_mem_cons", "def_path": "lake-packages/std/Std/Data/List/Init/Lemmas.lean", "def_pos": [120, 9], "def_end_pos": [120, 24]}]], "state_before": "case succ.mpr.intro.cons.intro.intro.refl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nl\u271d m : Language \u03b1\na\u271d b x\u271d : List \u03b1\nl : Language \u03b1\nx a : List \u03b1\nS : List (List \u03b1)\nhS : \u2200 (y : List \u03b1), y \u2208 a :: S \u2192 y \u2208 l\nihn :\n  \u2200 {x : List \u03b1},\n    x \u2208 l ^ Nat.add (length S) 0 \u2194\n      \u2203 S_1, x = join S_1 \u2227 length S_1 = Nat.add (length S) 0 \u2227 \u2200 (y : List \u03b1), y \u2208 S_1 \u2192 y \u2208 l\n\u22a2 \u2203 a_1 b,\n    a_1 \u2208 l \u2227\n      (\u2203 S_1, b = join S_1 \u2227 length S_1 = Nat.add (length S) 0 \u2227 \u2200 (y : List \u03b1), y \u2208 S_1 \u2192 y \u2208 l) \u2227\n        a_1 ++ b = join (a :: S)", "state_after": "case succ.mpr.intro.cons.intro.intro.refl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nl\u271d m : Language \u03b1\na\u271d b x\u271d : List \u03b1\nl : Language \u03b1\nx a : List \u03b1\nS : List (List \u03b1)\nhS : a \u2208 l \u2227 \u2200 (x : List \u03b1), x \u2208 S \u2192 x \u2208 l\nihn :\n  \u2200 {x : List \u03b1},\n    x \u2208 l ^ Nat.add (length S) 0 \u2194\n      \u2203 S_1, x = join S_1 \u2227 length S_1 = Nat.add (length S) 0 \u2227 \u2200 (y : List \u03b1), y \u2208 S_1 \u2192 y \u2208 l\n\u22a2 \u2203 a_1 b,\n    a_1 \u2208 l \u2227\n      (\u2203 S_1, b = join S_1 \u2227 length S_1 = Nat.add (length S) 0 \u2227 \u2200 (y : List \u03b1), y \u2208 S_1 \u2192 y \u2208 l) \u2227\n        a_1 ++ b = join (a :: S)"}, {"tactic": "exact \u27e8a, _, hS.1, \u27e8S, rfl, rfl, hS.2\u27e9, rfl\u27e9", "annotated_tactic": ["exact \u27e8a, _, hS.1, \u27e8S, <a>rfl</a>, <a>rfl</a>, hS.2\u27e9, <a>rfl</a>\u27e9", [{"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "case succ.mpr.intro.cons.intro.intro.refl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nl\u271d m : Language \u03b1\na\u271d b x\u271d : List \u03b1\nl : Language \u03b1\nx a : List \u03b1\nS : List (List \u03b1)\nhS : a \u2208 l \u2227 \u2200 (x : List \u03b1), x \u2208 S \u2192 x \u2208 l\nihn :\n  \u2200 {x : List \u03b1},\n    x \u2208 l ^ Nat.add (length S) 0 \u2194\n      \u2203 S_1, x = join S_1 \u2227 length S_1 = Nat.add (length S) 0 \u2227 \u2200 (y : List \u03b1), y \u2208 S_1 \u2192 y \u2208 l\n\u22a2 \u2203 a_1 b,\n    a_1 \u2208 l \u2227\n      (\u2203 S_1, b = join S_1 \u2227 length S_1 = Nat.add (length S) 0 \u2227 \u2200 (y : List \u03b1), y \u2208 S_1 \u2192 y \u2208 l) \u2227\n        a_1 ++ b = join (a :: S)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/VitaliCaratheodory.lean", "full_name": "MeasureTheory.exists_le_lowerSemicontinuous_lintegral_ge", "start": [164, 1], "end": [195, 83], "traced_tactics": [{"tactic": "rcases ENNReal.exists_pos_sum_of_countable' \u03b5pos \u2115 with \u27e8\u03b4, \u03b4pos, h\u03b4\u27e9", "annotated_tactic": ["rcases <a>ENNReal.exists_pos_sum_of_countable'</a> \u03b5pos \u2115 with \u27e8\u03b4, \u03b4pos, h\u03b4\u27e9", [{"full_name": "ENNReal.exists_pos_sum_of_countable'", "def_path": "Mathlib/Analysis/SpecificLimits/Basic.lean", "def_pos": [536, 9], "def_end_pos": [536, 37]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 \u2260 0\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), f x \u2264 g x) \u2227 LowerSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), f x \u2202\u03bc + \u03b5", "state_after": "case intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 \u2260 0\n\u03b4 : \u2115 \u2192 \u211d\u22650\u221e\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\nh\u03b4 : \u2211' (i : \u2115), \u03b4 i < \u03b5\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), f x \u2264 g x) \u2227 LowerSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), f x \u2202\u03bc + \u03b5"}, {"tactic": "have :\n  \u2200 n,\n    \u2203 g : \u03b1 \u2192 \u211d\u22650,\n      (\u2200 x, SimpleFunc.eapproxDiff f n x \u2264 g x) \u2227\n        LowerSemicontinuous g \u2227\n          (\u222b\u207b x, g x \u2202\u03bc) \u2264 (\u222b\u207b x, SimpleFunc.eapproxDiff f n x \u2202\u03bc) + \u03b4 n :=\n  fun n =>\n  SimpleFunc.exists_le_lowerSemicontinuous_lintegral_ge \u03bc (SimpleFunc.eapproxDiff f n)\n    (\u03b4pos n).ne'", "annotated_tactic": ["have :\n    \u2200 n,\n      \u2203 g : \u03b1 \u2192 \u211d\u22650,\n        (\u2200 x, <a>SimpleFunc.eapproxDiff</a> f n x \u2264 g x) \u2227\n          <a>LowerSemicontinuous</a> g \u2227\n            (\u222b\u207b x, g x \u2202\u03bc) \u2264 (\u222b\u207b x, <a>SimpleFunc.eapproxDiff</a> f n x \u2202\u03bc) + \u03b4 n :=\n    fun n =>\n    <a>SimpleFunc.exists_le_lowerSemicontinuous_lintegral_ge</a> \u03bc (<a>SimpleFunc.eapproxDiff</a> f n)\n      (\u03b4pos n).<a>ne'</a>", [{"full_name": "MeasureTheory.SimpleFunc.eapproxDiff", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [917, 5], "def_end_pos": [917, 16]}, {"full_name": "LowerSemicontinuous", "def_path": "Mathlib/Topology/Semicontinuous.lean", "def_pos": [91, 5], "def_end_pos": [91, 24]}, {"full_name": "MeasureTheory.SimpleFunc.eapproxDiff", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [917, 5], "def_end_pos": [917, 16]}, {"full_name": "MeasureTheory.SimpleFunc.exists_le_lowerSemicontinuous_lintegral_ge", "def_path": "Mathlib/MeasureTheory/Integral/VitaliCaratheodory.lean", "def_pos": [93, 9], "def_end_pos": [93, 62]}, {"full_name": "MeasureTheory.SimpleFunc.eapproxDiff", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [917, 5], "def_end_pos": [917, 16]}, {"full_name": "LT.lt.ne'", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [328, 9], "def_end_pos": [328, 12]}]], "state_before": "case intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 \u2260 0\n\u03b4 : \u2115 \u2192 \u211d\u22650\u221e\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\nh\u03b4 : \u2211' (i : \u2115), \u03b4 i < \u03b5\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), f x \u2264 g x) \u2227 LowerSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), f x \u2202\u03bc + \u03b5", "state_after": "case intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 \u2260 0\n\u03b4 : \u2115 \u2192 \u211d\u22650\u221e\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\nh\u03b4 : \u2211' (i : \u2115), \u03b4 i < \u03b5\nthis :\n  \u2200 (n : \u2115),\n    \u2203 g,\n      (\u2200 (x : \u03b1), \u2191(SimpleFunc.eapproxDiff f n) x \u2264 g x) \u2227\n        LowerSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191(SimpleFunc.eapproxDiff f n) x) \u2202\u03bc + \u03b4 n\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), f x \u2264 g x) \u2227 LowerSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), f x \u2202\u03bc + \u03b5"}, {"tactic": "choose g f_le_g gcont hg using this", "annotated_tactic": ["choose g f_le_g gcont hg using this", []], "state_before": "case intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 \u2260 0\n\u03b4 : \u2115 \u2192 \u211d\u22650\u221e\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\nh\u03b4 : \u2211' (i : \u2115), \u03b4 i < \u03b5\nthis :\n  \u2200 (n : \u2115),\n    \u2203 g,\n      (\u2200 (x : \u03b1), \u2191(SimpleFunc.eapproxDiff f n) x \u2264 g x) \u2227\n        LowerSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191(SimpleFunc.eapproxDiff f n) x) \u2202\u03bc + \u03b4 n\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), f x \u2264 g x) \u2227 LowerSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), f x \u2202\u03bc + \u03b5", "state_after": "case intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 \u2260 0\n\u03b4 : \u2115 \u2192 \u211d\u22650\u221e\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\nh\u03b4 : \u2211' (i : \u2115), \u03b4 i < \u03b5\ng : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\nf_le_g : \u2200 (n : \u2115) (x : \u03b1), \u2191(SimpleFunc.eapproxDiff f n) x \u2264 g n x\ngcont : \u2200 (n : \u2115), LowerSemicontinuous (g n)\nhg : \u2200 (n : \u2115), \u222b\u207b (x : \u03b1), \u2191(g n x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191(SimpleFunc.eapproxDiff f n) x) \u2202\u03bc + \u03b4 n\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), f x \u2264 g x) \u2227 LowerSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), f x \u2202\u03bc + \u03b5"}, {"tactic": "refine' \u27e8fun x => \u2211' n, g n x, fun x => _, _, _\u27e9", "annotated_tactic": ["refine' \u27e8fun x => \u2211' n, g n x, fun x => _, _, _\u27e9", []], "state_before": "case intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 \u2260 0\n\u03b4 : \u2115 \u2192 \u211d\u22650\u221e\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\nh\u03b4 : \u2211' (i : \u2115), \u03b4 i < \u03b5\ng : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\nf_le_g : \u2200 (n : \u2115) (x : \u03b1), \u2191(SimpleFunc.eapproxDiff f n) x \u2264 g n x\ngcont : \u2200 (n : \u2115), LowerSemicontinuous (g n)\nhg : \u2200 (n : \u2115), \u222b\u207b (x : \u03b1), \u2191(g n x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191(SimpleFunc.eapproxDiff f n) x) \u2202\u03bc + \u03b4 n\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), f x \u2264 g x) \u2227 LowerSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), f x \u2202\u03bc + \u03b5", "state_after": "case intro.intro.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 \u2260 0\n\u03b4 : \u2115 \u2192 \u211d\u22650\u221e\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\nh\u03b4 : \u2211' (i : \u2115), \u03b4 i < \u03b5\ng : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\nf_le_g : \u2200 (n : \u2115) (x : \u03b1), \u2191(SimpleFunc.eapproxDiff f n) x \u2264 g n x\ngcont : \u2200 (n : \u2115), LowerSemicontinuous (g n)\nhg : \u2200 (n : \u2115), \u222b\u207b (x : \u03b1), \u2191(g n x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191(SimpleFunc.eapproxDiff f n) x) \u2202\u03bc + \u03b4 n\nx : \u03b1\n\u22a2 f x \u2264 (fun x => \u2211' (n : \u2115), \u2191(g n x)) x\n\ncase intro.intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 \u2260 0\n\u03b4 : \u2115 \u2192 \u211d\u22650\u221e\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\nh\u03b4 : \u2211' (i : \u2115), \u03b4 i < \u03b5\ng : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\nf_le_g : \u2200 (n : \u2115) (x : \u03b1), \u2191(SimpleFunc.eapproxDiff f n) x \u2264 g n x\ngcont : \u2200 (n : \u2115), LowerSemicontinuous (g n)\nhg : \u2200 (n : \u2115), \u222b\u207b (x : \u03b1), \u2191(g n x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191(SimpleFunc.eapproxDiff f n) x) \u2202\u03bc + \u03b4 n\n\u22a2 LowerSemicontinuous fun x => \u2211' (n : \u2115), \u2191(g n x)\n\ncase intro.intro.refine'_3\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 \u2260 0\n\u03b4 : \u2115 \u2192 \u211d\u22650\u221e\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\nh\u03b4 : \u2211' (i : \u2115), \u03b4 i < \u03b5\ng : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\nf_le_g : \u2200 (n : \u2115) (x : \u03b1), \u2191(SimpleFunc.eapproxDiff f n) x \u2264 g n x\ngcont : \u2200 (n : \u2115), LowerSemicontinuous (g n)\nhg : \u2200 (n : \u2115), \u222b\u207b (x : \u03b1), \u2191(g n x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191(SimpleFunc.eapproxDiff f n) x) \u2202\u03bc + \u03b4 n\n\u22a2 \u222b\u207b (x : \u03b1), (fun x => \u2211' (n : \u2115), \u2191(g n x)) x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), f x \u2202\u03bc + \u03b5"}, {"tactic": "rw [\u2190 SimpleFunc.tsum_eapproxDiff f hf]", "annotated_tactic": ["rw [\u2190 <a>SimpleFunc.tsum_eapproxDiff</a> f hf]", [{"full_name": "MeasureTheory.SimpleFunc.tsum_eapproxDiff", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [935, 9], "def_end_pos": [935, 25]}]], "state_before": "case intro.intro.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 \u2260 0\n\u03b4 : \u2115 \u2192 \u211d\u22650\u221e\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\nh\u03b4 : \u2211' (i : \u2115), \u03b4 i < \u03b5\ng : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\nf_le_g : \u2200 (n : \u2115) (x : \u03b1), \u2191(SimpleFunc.eapproxDiff f n) x \u2264 g n x\ngcont : \u2200 (n : \u2115), LowerSemicontinuous (g n)\nhg : \u2200 (n : \u2115), \u222b\u207b (x : \u03b1), \u2191(g n x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191(SimpleFunc.eapproxDiff f n) x) \u2202\u03bc + \u03b4 n\nx : \u03b1\n\u22a2 f x \u2264 (fun x => \u2211' (n : \u2115), \u2191(g n x)) x", "state_after": "case intro.intro.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 \u2260 0\n\u03b4 : \u2115 \u2192 \u211d\u22650\u221e\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\nh\u03b4 : \u2211' (i : \u2115), \u03b4 i < \u03b5\ng : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\nf_le_g : \u2200 (n : \u2115) (x : \u03b1), \u2191(SimpleFunc.eapproxDiff f n) x \u2264 g n x\ngcont : \u2200 (n : \u2115), LowerSemicontinuous (g n)\nhg : \u2200 (n : \u2115), \u222b\u207b (x : \u03b1), \u2191(g n x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191(SimpleFunc.eapproxDiff f n) x) \u2202\u03bc + \u03b4 n\nx : \u03b1\n\u22a2 \u2211' (n : \u2115), \u2191(\u2191(SimpleFunc.eapproxDiff f n) x) \u2264 (fun x => \u2211' (n : \u2115), \u2191(g n x)) x"}, {"tactic": "exact ENNReal.tsum_le_tsum fun n => ENNReal.coe_le_coe.2 (f_le_g n x)", "annotated_tactic": ["exact <a>ENNReal.tsum_le_tsum</a> fun n => <a>ENNReal.coe_le_coe</a>.2 (f_le_g n x)", [{"full_name": "ENNReal.tsum_le_tsum", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [827, 19], "def_end_pos": [827, 31]}, {"full_name": "ENNReal.coe_le_coe", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [349, 28], "def_end_pos": [349, 38]}]], "state_before": "case intro.intro.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 \u2260 0\n\u03b4 : \u2115 \u2192 \u211d\u22650\u221e\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\nh\u03b4 : \u2211' (i : \u2115), \u03b4 i < \u03b5\ng : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\nf_le_g : \u2200 (n : \u2115) (x : \u03b1), \u2191(SimpleFunc.eapproxDiff f n) x \u2264 g n x\ngcont : \u2200 (n : \u2115), LowerSemicontinuous (g n)\nhg : \u2200 (n : \u2115), \u222b\u207b (x : \u03b1), \u2191(g n x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191(SimpleFunc.eapproxDiff f n) x) \u2202\u03bc + \u03b4 n\nx : \u03b1\n\u22a2 \u2211' (n : \u2115), \u2191(\u2191(SimpleFunc.eapproxDiff f n) x) \u2264 (fun x => \u2211' (n : \u2115), \u2191(g n x)) x", "state_after": "no goals"}, {"tactic": "refine' lowerSemicontinuous_tsum fun n => _", "annotated_tactic": ["refine' <a>lowerSemicontinuous_tsum</a> fun n => _", [{"full_name": "lowerSemicontinuous_tsum", "def_path": "Mathlib/Topology/Semicontinuous.lean", "def_pos": [653, 9], "def_end_pos": [653, 33]}]], "state_before": "case intro.intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 \u2260 0\n\u03b4 : \u2115 \u2192 \u211d\u22650\u221e\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\nh\u03b4 : \u2211' (i : \u2115), \u03b4 i < \u03b5\ng : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\nf_le_g : \u2200 (n : \u2115) (x : \u03b1), \u2191(SimpleFunc.eapproxDiff f n) x \u2264 g n x\ngcont : \u2200 (n : \u2115), LowerSemicontinuous (g n)\nhg : \u2200 (n : \u2115), \u222b\u207b (x : \u03b1), \u2191(g n x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191(SimpleFunc.eapproxDiff f n) x) \u2202\u03bc + \u03b4 n\n\u22a2 LowerSemicontinuous fun x => \u2211' (n : \u2115), \u2191(g n x)", "state_after": "case intro.intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 \u2260 0\n\u03b4 : \u2115 \u2192 \u211d\u22650\u221e\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\nh\u03b4 : \u2211' (i : \u2115), \u03b4 i < \u03b5\ng : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\nf_le_g : \u2200 (n : \u2115) (x : \u03b1), \u2191(SimpleFunc.eapproxDiff f n) x \u2264 g n x\ngcont : \u2200 (n : \u2115), LowerSemicontinuous (g n)\nhg : \u2200 (n : \u2115), \u222b\u207b (x : \u03b1), \u2191(g n x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191(SimpleFunc.eapproxDiff f n) x) \u2202\u03bc + \u03b4 n\nn : \u2115\n\u22a2 LowerSemicontinuous fun x => \u2191(g n x)"}, {"tactic": "exact\n  ENNReal.continuous_coe.comp_lowerSemicontinuous (gcont n) fun x y hxy =>\n    ENNReal.coe_le_coe.2 hxy", "annotated_tactic": ["exact\n      ENNReal.continuous_coe.comp_lowerSemicontinuous (gcont n) fun x y hxy =>\n        <a>ENNReal.coe_le_coe</a>.2 hxy", [{"full_name": "ENNReal.coe_le_coe", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [349, 28], "def_end_pos": [349, 38]}]], "state_before": "case intro.intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 \u2260 0\n\u03b4 : \u2115 \u2192 \u211d\u22650\u221e\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\nh\u03b4 : \u2211' (i : \u2115), \u03b4 i < \u03b5\ng : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\nf_le_g : \u2200 (n : \u2115) (x : \u03b1), \u2191(SimpleFunc.eapproxDiff f n) x \u2264 g n x\ngcont : \u2200 (n : \u2115), LowerSemicontinuous (g n)\nhg : \u2200 (n : \u2115), \u222b\u207b (x : \u03b1), \u2191(g n x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191(SimpleFunc.eapproxDiff f n) x) \u2202\u03bc + \u03b4 n\nn : \u2115\n\u22a2 LowerSemicontinuous fun x => \u2191(g n x)", "state_after": "no goals"}, {"tactic": "rw [lintegral_tsum fun n => (gcont n).measurable.coe_nnreal_ennreal.aemeasurable]", "annotated_tactic": ["rw [<a>lintegral_tsum</a> fun n => (gcont n).measurable.coe_nnreal_ennreal.aemeasurable]", [{"full_name": "MeasureTheory.lintegral_tsum", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [1184, 9], "def_end_pos": [1184, 23]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 \u2260 0\n\u03b4 : \u2115 \u2192 \u211d\u22650\u221e\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\nh\u03b4 : \u2211' (i : \u2115), \u03b4 i < \u03b5\ng : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\nf_le_g : \u2200 (n : \u2115) (x : \u03b1), \u2191(SimpleFunc.eapproxDiff f n) x \u2264 g n x\ngcont : \u2200 (n : \u2115), LowerSemicontinuous (g n)\nhg : \u2200 (n : \u2115), \u222b\u207b (x : \u03b1), \u2191(g n x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191(SimpleFunc.eapproxDiff f n) x) \u2202\u03bc + \u03b4 n\n\u22a2 \u222b\u207b (x : \u03b1), \u2211' (n : \u2115), \u2191(g n x) \u2202\u03bc = \u2211' (n : \u2115), \u222b\u207b (x : \u03b1), \u2191(g n x) \u2202\u03bc", "state_after": "no goals"}, {"tactic": "refine' add_le_add _ h\u03b4.le", "annotated_tactic": ["refine' <a>add_le_add</a> _ h\u03b4.le", [{"full_name": "add_le_add", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [205, 15], "def_end_pos": [205, 25]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 \u2260 0\n\u03b4 : \u2115 \u2192 \u211d\u22650\u221e\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\nh\u03b4 : \u2211' (i : \u2115), \u03b4 i < \u03b5\ng : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\nf_le_g : \u2200 (n : \u2115) (x : \u03b1), \u2191(SimpleFunc.eapproxDiff f n) x \u2264 g n x\ngcont : \u2200 (n : \u2115), LowerSemicontinuous (g n)\nhg : \u2200 (n : \u2115), \u222b\u207b (x : \u03b1), \u2191(g n x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191(SimpleFunc.eapproxDiff f n) x) \u2202\u03bc + \u03b4 n\n\u22a2 \u2211' (n : \u2115), \u222b\u207b (x : \u03b1), \u2191(\u2191(SimpleFunc.eapproxDiff f n) x) \u2202\u03bc + \u2211' (n : \u2115), \u03b4 n \u2264 \u222b\u207b (x : \u03b1), f x \u2202\u03bc + \u03b5", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 \u2260 0\n\u03b4 : \u2115 \u2192 \u211d\u22650\u221e\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\nh\u03b4 : \u2211' (i : \u2115), \u03b4 i < \u03b5\ng : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\nf_le_g : \u2200 (n : \u2115) (x : \u03b1), \u2191(SimpleFunc.eapproxDiff f n) x \u2264 g n x\ngcont : \u2200 (n : \u2115), LowerSemicontinuous (g n)\nhg : \u2200 (n : \u2115), \u222b\u207b (x : \u03b1), \u2191(g n x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191(SimpleFunc.eapproxDiff f n) x) \u2202\u03bc + \u03b4 n\n\u22a2 \u2211' (n : \u2115), \u222b\u207b (x : \u03b1), \u2191(\u2191(SimpleFunc.eapproxDiff f n) x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), f x \u2202\u03bc"}, {"tactic": "rw [\u2190 lintegral_tsum]", "annotated_tactic": ["rw [\u2190 <a>lintegral_tsum</a>]", [{"full_name": "MeasureTheory.lintegral_tsum", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [1184, 9], "def_end_pos": [1184, 23]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 \u2260 0\n\u03b4 : \u2115 \u2192 \u211d\u22650\u221e\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\nh\u03b4 : \u2211' (i : \u2115), \u03b4 i < \u03b5\ng : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\nf_le_g : \u2200 (n : \u2115) (x : \u03b1), \u2191(SimpleFunc.eapproxDiff f n) x \u2264 g n x\ngcont : \u2200 (n : \u2115), LowerSemicontinuous (g n)\nhg : \u2200 (n : \u2115), \u222b\u207b (x : \u03b1), \u2191(g n x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191(SimpleFunc.eapproxDiff f n) x) \u2202\u03bc + \u03b4 n\n\u22a2 \u2211' (n : \u2115), \u222b\u207b (x : \u03b1), \u2191(\u2191(SimpleFunc.eapproxDiff f n) x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), f x \u2202\u03bc", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 \u2260 0\n\u03b4 : \u2115 \u2192 \u211d\u22650\u221e\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\nh\u03b4 : \u2211' (i : \u2115), \u03b4 i < \u03b5\ng : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\nf_le_g : \u2200 (n : \u2115) (x : \u03b1), \u2191(SimpleFunc.eapproxDiff f n) x \u2264 g n x\ngcont : \u2200 (n : \u2115), LowerSemicontinuous (g n)\nhg : \u2200 (n : \u2115), \u222b\u207b (x : \u03b1), \u2191(g n x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191(SimpleFunc.eapproxDiff f n) x) \u2202\u03bc + \u03b4 n\n\u22a2 \u222b\u207b (a : \u03b1), \u2211' (i : \u2115), \u2191(\u2191(SimpleFunc.eapproxDiff f i) a) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), f x \u2202\u03bc\n\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 \u2260 0\n\u03b4 : \u2115 \u2192 \u211d\u22650\u221e\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\nh\u03b4 : \u2211' (i : \u2115), \u03b4 i < \u03b5\ng : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\nf_le_g : \u2200 (n : \u2115) (x : \u03b1), \u2191(SimpleFunc.eapproxDiff f n) x \u2264 g n x\ngcont : \u2200 (n : \u2115), LowerSemicontinuous (g n)\nhg : \u2200 (n : \u2115), \u222b\u207b (x : \u03b1), \u2191(g n x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191(SimpleFunc.eapproxDiff f n) x) \u2202\u03bc + \u03b4 n\n\u22a2 \u2200 (i : \u2115), AEMeasurable fun x => \u2191(\u2191(SimpleFunc.eapproxDiff f i) x)"}, {"tactic": "simp_rw [SimpleFunc.tsum_eapproxDiff f hf, le_refl]", "annotated_tactic": ["simp_rw [<a>SimpleFunc.tsum_eapproxDiff</a> f hf, <a>le_refl</a>]", [{"full_name": "MeasureTheory.SimpleFunc.tsum_eapproxDiff", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [935, 9], "def_end_pos": [935, 25]}, {"full_name": "le_refl", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [50, 9], "def_end_pos": [50, 16]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 \u2260 0\n\u03b4 : \u2115 \u2192 \u211d\u22650\u221e\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\nh\u03b4 : \u2211' (i : \u2115), \u03b4 i < \u03b5\ng : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\nf_le_g : \u2200 (n : \u2115) (x : \u03b1), \u2191(SimpleFunc.eapproxDiff f n) x \u2264 g n x\ngcont : \u2200 (n : \u2115), LowerSemicontinuous (g n)\nhg : \u2200 (n : \u2115), \u222b\u207b (x : \u03b1), \u2191(g n x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191(SimpleFunc.eapproxDiff f n) x) \u2202\u03bc + \u03b4 n\n\u22a2 \u222b\u207b (a : \u03b1), \u2211' (i : \u2115), \u2191(\u2191(SimpleFunc.eapproxDiff f i) a) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), f x \u2202\u03bc", "state_after": "no goals"}, {"tactic": "intro n", "annotated_tactic": ["intro n", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 \u2260 0\n\u03b4 : \u2115 \u2192 \u211d\u22650\u221e\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\nh\u03b4 : \u2211' (i : \u2115), \u03b4 i < \u03b5\ng : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\nf_le_g : \u2200 (n : \u2115) (x : \u03b1), \u2191(SimpleFunc.eapproxDiff f n) x \u2264 g n x\ngcont : \u2200 (n : \u2115), LowerSemicontinuous (g n)\nhg : \u2200 (n : \u2115), \u222b\u207b (x : \u03b1), \u2191(g n x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191(SimpleFunc.eapproxDiff f n) x) \u2202\u03bc + \u03b4 n\n\u22a2 \u2200 (i : \u2115), AEMeasurable fun x => \u2191(\u2191(SimpleFunc.eapproxDiff f i) x)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 \u2260 0\n\u03b4 : \u2115 \u2192 \u211d\u22650\u221e\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\nh\u03b4 : \u2211' (i : \u2115), \u03b4 i < \u03b5\ng : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\nf_le_g : \u2200 (n : \u2115) (x : \u03b1), \u2191(SimpleFunc.eapproxDiff f n) x \u2264 g n x\ngcont : \u2200 (n : \u2115), LowerSemicontinuous (g n)\nhg : \u2200 (n : \u2115), \u222b\u207b (x : \u03b1), \u2191(g n x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191(SimpleFunc.eapproxDiff f n) x) \u2202\u03bc + \u03b4 n\nn : \u2115\n\u22a2 AEMeasurable fun x => \u2191(\u2191(SimpleFunc.eapproxDiff f n) x)"}, {"tactic": "exact (SimpleFunc.measurable _).coe_nnreal_ennreal.aemeasurable", "annotated_tactic": ["exact (<a>SimpleFunc.measurable</a> _).coe_nnreal_ennreal.aemeasurable", [{"full_name": "MeasureTheory.SimpleFunc.measurable", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [201, 19], "def_end_pos": [201, 29]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 \u2260 0\n\u03b4 : \u2115 \u2192 \u211d\u22650\u221e\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\nh\u03b4 : \u2211' (i : \u2115), \u03b4 i < \u03b5\ng : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\nf_le_g : \u2200 (n : \u2115) (x : \u03b1), \u2191(SimpleFunc.eapproxDiff f n) x \u2264 g n x\ngcont : \u2200 (n : \u2115), LowerSemicontinuous (g n)\nhg : \u2200 (n : \u2115), \u222b\u207b (x : \u03b1), \u2191(g n x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191(SimpleFunc.eapproxDiff f n) x) \u2202\u03bc + \u03b4 n\nn : \u2115\n\u22a2 AEMeasurable fun x => \u2191(\u2191(SimpleFunc.eapproxDiff f n) x)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Independence/Basic.lean", "full_name": "ProbabilityTheory.indepSets_of_indepSets_of_le_right", "start": [258, 1], "end": [261, 56], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/LocallyFinite.lean", "full_name": "Finset.uIcc_toDual", "start": [914, 1], "end": [915, 17], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "full_name": "MeasureTheory.SignedMeasure.toMeasureOfZeroLE_apply", "start": [1349, 1], "end": [1353, 46], "traced_tactics": [{"tactic": "simp_rw [toMeasureOfZeroLE, Measure.ofMeasurable_apply _ hj\u2081, toMeasureOfZeroLE',\n  s.restrict_apply hi\u2081 hj\u2081, Set.inter_comm]", "annotated_tactic": ["simp_rw [<a>toMeasureOfZeroLE</a>, <a>Measure.ofMeasurable_apply</a> _ hj\u2081, <a>toMeasureOfZeroLE'</a>,\n    s.restrict_apply hi\u2081 hj\u2081, <a>Set.inter_comm</a>]", [{"full_name": "MeasureTheory.SignedMeasure.toMeasureOfZeroLE", "def_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "def_pos": [1326, 5], "def_end_pos": [1326, 22]}, {"full_name": "MeasureTheory.Measure.ofMeasurable_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [120, 9], "def_end_pos": [120, 27]}, {"full_name": "MeasureTheory.SignedMeasure.toMeasureOfZeroLE'", "def_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "def_pos": [1319, 5], "def_end_pos": [1319, 23]}, {"full_name": "Set.inter_comm", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [940, 9], "def_end_pos": [940, 19]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\ni j : Set \u03b1\nhi : VectorMeasure.restrict 0 i \u2264 VectorMeasure.restrict s i\nhi\u2081 : MeasurableSet i\nhj\u2081 : MeasurableSet j\n\u22a2 \u2191\u2191(toMeasureOfZeroLE s i hi\u2081 hi) j = \u2191{ val := \u2191s (i \u2229 j), property := (_ : 0 \u2264 \u2191s (i \u2229 j)) }", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Lattice.lean", "full_name": "Finset.ofDual_max'", "start": [1530, 1], "end": [1535, 6], "traced_tactics": [{"tactic": "rw [\u2190 WithTop.coe_eq_coe]", "annotated_tactic": ["rw [\u2190 <a>WithTop.coe_eq_coe</a>]", [{"full_name": "WithTop.coe_eq_coe", "def_path": "Mathlib/Order/WithBot.lean", "def_pos": [728, 9], "def_end_pos": [728, 19]}]], "state_before": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Type u_5\n\u03ba : Type u_6\ninst\u271d : LinearOrder \u03b1\ns\u271d : Finset \u03b1\nH : Finset.Nonempty s\u271d\nx : \u03b1\ns : Finset \u03b1\u1d52\u1d48\nhs : Finset.Nonempty s\n\u22a2 \u2191ofDual (max' s hs) = min' (image (\u2191ofDual) s) (_ : Finset.Nonempty (image (\u2191ofDual) s))", "state_after": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Type u_5\n\u03ba : Type u_6\ninst\u271d : LinearOrder \u03b1\ns\u271d : Finset \u03b1\nH : Finset.Nonempty s\u271d\nx : \u03b1\ns : Finset \u03b1\u1d52\u1d48\nhs : Finset.Nonempty s\n\u22a2 \u2191(\u2191ofDual (max' s hs)) = \u2191(min' (image (\u2191ofDual) s) (_ : Finset.Nonempty (image (\u2191ofDual) s)))"}, {"tactic": "simp only [max'_eq_sup', id_eq, ofDual_sup', Function.comp_apply, coe_inf', min'_eq_inf',\n  inf_image]", "annotated_tactic": ["simp only [<a>max'_eq_sup'</a>, <a>id_eq</a>, <a>ofDual_sup'</a>, <a>Function.comp_apply</a>, <a>coe_inf'</a>, <a>min'_eq_inf'</a>,\n    <a>inf_image</a>]", [{"full_name": "Finset.max'_eq_sup'", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [1472, 9], "def_end_pos": [1472, 21]}, {"full_name": "id_eq", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [284, 17], "def_end_pos": [284, 22]}, {"full_name": "Finset.ofDual_sup'", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [1124, 9], "def_end_pos": [1124, 20]}, {"full_name": "Function.comp_apply", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [33, 17], "def_end_pos": [33, 36]}, {"full_name": "Finset.coe_inf'", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [939, 9], "def_end_pos": [939, 17]}, {"full_name": "Finset.min'_eq_inf'", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [1476, 9], "def_end_pos": [1476, 21]}, {"full_name": "Finset.inf_image", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [332, 9], "def_end_pos": [332, 18]}]], "state_before": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Type u_5\n\u03ba : Type u_6\ninst\u271d : LinearOrder \u03b1\ns\u271d : Finset \u03b1\nH : Finset.Nonempty s\u271d\nx : \u03b1\ns : Finset \u03b1\u1d52\u1d48\nhs : Finset.Nonempty s\n\u22a2 \u2191(\u2191ofDual (max' s hs)) = \u2191(min' (image (\u2191ofDual) s) (_ : Finset.Nonempty (image (\u2191ofDual) s)))", "state_after": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Type u_5\n\u03ba : Type u_6\ninst\u271d : LinearOrder \u03b1\ns\u271d : Finset \u03b1\nH : Finset.Nonempty s\u271d\nx : \u03b1\ns : Finset \u03b1\u1d52\u1d48\nhs : Finset.Nonempty s\n\u22a2 inf s (WithTop.some \u2218 fun x => \u2191ofDual x) = inf s ((WithTop.some \u2218 fun x => x) \u2218 \u2191ofDual)"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Type u_5\n\u03ba : Type u_6\ninst\u271d : LinearOrder \u03b1\ns\u271d : Finset \u03b1\nH : Finset.Nonempty s\u271d\nx : \u03b1\ns : Finset \u03b1\u1d52\u1d48\nhs : Finset.Nonempty s\n\u22a2 inf s (WithTop.some \u2218 fun x => \u2191ofDual x) = inf s ((WithTop.some \u2218 fun x => x) \u2218 \u2191ofDual)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "full_name": "MeasureTheory.L1.setToFun_eq_setToL1", "start": [1281, 1], "end": [1283, 69], "traced_tactics": [{"tactic": "rw [setToFun_eq hT (L1.integrable_coeFn f), Integrable.toL1_coeFn]", "annotated_tactic": ["rw [<a>setToFun_eq</a> hT (<a>L1.integrable_coeFn</a> f), <a>Integrable.toL1_coeFn</a>]", [{"full_name": "MeasureTheory.setToFun_eq", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [1276, 9], "def_end_pos": [1276, 20]}, {"full_name": "MeasureTheory.L1.integrable_coeFn", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [1324, 9], "def_end_pos": [1324, 25]}, {"full_name": "MeasureTheory.Integrable.toL1_coeFn", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [1407, 9], "def_end_pos": [1407, 19]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\nf : { x // x \u2208 Lp E 1 }\n\u22a2 setToFun \u03bc T hT \u2191\u2191f = \u2191(setToL1 hT) f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL2.lean", "full_name": "MeasureTheory.condexpL2_indicator_of_measurable", "start": [125, 1], "end": [136, 31], "traced_tactics": [{"tactic": "rw [condexpL2]", "annotated_tactic": ["rw [<a>condexpL2</a>]", [{"full_name": "MeasureTheory.condexpL2", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL2.lean", "def_pos": [71, 19], "def_end_pos": [71, 28]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nc : E\n\u22a2 \u2191(\u2191(condexpL2 E \ud835\udd5c hm) (indicatorConstLp 2 (_ : MeasurableSet s) h\u03bcs c)) =\n    indicatorConstLp 2 (_ : MeasurableSet s) h\u03bcs c", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nc : E\n\u22a2 \u2191(\u2191(orthogonalProjection (lpMeas E \ud835\udd5c m 2 \u03bc)) (indicatorConstLp 2 (_ : MeasurableSet s) h\u03bcs c)) =\n    indicatorConstLp 2 (_ : MeasurableSet s) h\u03bcs c"}, {"tactic": "haveI : Fact (m \u2264 m0) := \u27e8hm\u27e9", "annotated_tactic": ["haveI : <a>Fact</a> (m \u2264 m0) := \u27e8hm\u27e9", [{"full_name": "Fact", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [115, 7], "def_end_pos": [115, 11]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nc : E\n\u22a2 \u2191(\u2191(orthogonalProjection (lpMeas E \ud835\udd5c m 2 \u03bc)) (indicatorConstLp 2 (_ : MeasurableSet s) h\u03bcs c)) =\n    indicatorConstLp 2 (_ : MeasurableSet s) h\u03bcs c", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nc : E\nthis : Fact (m \u2264 m0)\n\u22a2 \u2191(\u2191(orthogonalProjection (lpMeas E \ud835\udd5c m 2 \u03bc)) (indicatorConstLp 2 (_ : MeasurableSet s) h\u03bcs c)) =\n    indicatorConstLp 2 (_ : MeasurableSet s) h\u03bcs c"}, {"tactic": "have h_mem : indicatorConstLp 2 (hm s hs) h\u03bcs c \u2208 lpMeas E \ud835\udd5c m 2 \u03bc :=\n  mem_lpMeas_indicatorConstLp hm hs h\u03bcs", "annotated_tactic": ["have h_mem : <a>indicatorConstLp</a> 2 (hm s hs) h\u03bcs c \u2208 <a>lpMeas</a> E \ud835\udd5c m 2 \u03bc :=\n    <a>mem_lpMeas_indicatorConstLp</a> hm hs h\u03bcs", [{"full_name": "MeasureTheory.indicatorConstLp", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [744, 5], "def_end_pos": [744, 21]}, {"full_name": "MeasureTheory.lpMeas", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean", "def_pos": [222, 5], "def_end_pos": [222, 11]}, {"full_name": "MeasureTheory.mem_lpMeas_indicatorConstLp", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean", "def_pos": [262, 9], "def_end_pos": [262, 36]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nc : E\nthis : Fact (m \u2264 m0)\n\u22a2 \u2191(\u2191(orthogonalProjection (lpMeas E \ud835\udd5c m 2 \u03bc)) (indicatorConstLp 2 (_ : MeasurableSet s) h\u03bcs c)) =\n    indicatorConstLp 2 (_ : MeasurableSet s) h\u03bcs c", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nc : E\nthis : Fact (m \u2264 m0)\nh_mem : indicatorConstLp 2 (_ : MeasurableSet s) h\u03bcs c \u2208 lpMeas E \ud835\udd5c m 2 \u03bc\n\u22a2 \u2191(\u2191(orthogonalProjection (lpMeas E \ud835\udd5c m 2 \u03bc)) (indicatorConstLp 2 (_ : MeasurableSet s) h\u03bcs c)) =\n    indicatorConstLp 2 (_ : MeasurableSet s) h\u03bcs c"}, {"tactic": "let ind := (\u27e8indicatorConstLp 2 (hm s hs) h\u03bcs c, h_mem\u27e9 : lpMeas E \ud835\udd5c m 2 \u03bc)", "annotated_tactic": ["let ind := (\u27e8<a>indicatorConstLp</a> 2 (hm s hs) h\u03bcs c, h_mem\u27e9 : <a>lpMeas</a> E \ud835\udd5c m 2 \u03bc)", [{"full_name": "MeasureTheory.indicatorConstLp", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [744, 5], "def_end_pos": [744, 21]}, {"full_name": "MeasureTheory.lpMeas", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean", "def_pos": [222, 5], "def_end_pos": [222, 11]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nc : E\nthis : Fact (m \u2264 m0)\nh_mem : indicatorConstLp 2 (_ : MeasurableSet s) h\u03bcs c \u2208 lpMeas E \ud835\udd5c m 2 \u03bc\n\u22a2 \u2191(\u2191(orthogonalProjection (lpMeas E \ud835\udd5c m 2 \u03bc)) (indicatorConstLp 2 (_ : MeasurableSet s) h\u03bcs c)) =\n    indicatorConstLp 2 (_ : MeasurableSet s) h\u03bcs c", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nc : E\nthis : Fact (m \u2264 m0)\nh_mem : indicatorConstLp 2 (_ : MeasurableSet s) h\u03bcs c \u2208 lpMeas E \ud835\udd5c m 2 \u03bc\nind : { x // x \u2208 lpMeas E \ud835\udd5c m 2 \u03bc } := { val := indicatorConstLp 2 (_ : MeasurableSet s) h\u03bcs c, property := h_mem }\n\u22a2 \u2191(\u2191(orthogonalProjection (lpMeas E \ud835\udd5c m 2 \u03bc)) (indicatorConstLp 2 (_ : MeasurableSet s) h\u03bcs c)) =\n    indicatorConstLp 2 (_ : MeasurableSet s) h\u03bcs c"}, {"tactic": "have h_coe_ind : (ind : \u03b1 \u2192\u2082[\u03bc] E) = indicatorConstLp 2 (hm s hs) h\u03bcs c := rfl", "annotated_tactic": ["have h_coe_ind : (ind : \u03b1 \u2192\u2082[\u03bc] E) = <a>indicatorConstLp</a> 2 (hm s hs) h\u03bcs c := <a>rfl</a>", [{"full_name": "MeasureTheory.indicatorConstLp", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [744, 5], "def_end_pos": [744, 21]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nc : E\nthis : Fact (m \u2264 m0)\nh_mem : indicatorConstLp 2 (_ : MeasurableSet s) h\u03bcs c \u2208 lpMeas E \ud835\udd5c m 2 \u03bc\nind : { x // x \u2208 lpMeas E \ud835\udd5c m 2 \u03bc } := { val := indicatorConstLp 2 (_ : MeasurableSet s) h\u03bcs c, property := h_mem }\n\u22a2 \u2191(\u2191(orthogonalProjection (lpMeas E \ud835\udd5c m 2 \u03bc)) (indicatorConstLp 2 (_ : MeasurableSet s) h\u03bcs c)) =\n    indicatorConstLp 2 (_ : MeasurableSet s) h\u03bcs c", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nc : E\nthis : Fact (m \u2264 m0)\nh_mem : indicatorConstLp 2 (_ : MeasurableSet s) h\u03bcs c \u2208 lpMeas E \ud835\udd5c m 2 \u03bc\nind : { x // x \u2208 lpMeas E \ud835\udd5c m 2 \u03bc } := { val := indicatorConstLp 2 (_ : MeasurableSet s) h\u03bcs c, property := h_mem }\nh_coe_ind : \u2191ind = indicatorConstLp 2 (_ : MeasurableSet s) h\u03bcs c\n\u22a2 \u2191(\u2191(orthogonalProjection (lpMeas E \ud835\udd5c m 2 \u03bc)) (indicatorConstLp 2 (_ : MeasurableSet s) h\u03bcs c)) =\n    indicatorConstLp 2 (_ : MeasurableSet s) h\u03bcs c"}, {"tactic": "have h_orth_mem := orthogonalProjection_mem_subspace_eq_self ind", "annotated_tactic": ["have h_orth_mem := <a>orthogonalProjection_mem_subspace_eq_self</a> ind", [{"full_name": "orthogonalProjection_mem_subspace_eq_self", "def_path": "Mathlib/Analysis/InnerProductSpace/Projection.lean", "def_pos": [570, 9], "def_end_pos": [570, 50]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nc : E\nthis : Fact (m \u2264 m0)\nh_mem : indicatorConstLp 2 (_ : MeasurableSet s) h\u03bcs c \u2208 lpMeas E \ud835\udd5c m 2 \u03bc\nind : { x // x \u2208 lpMeas E \ud835\udd5c m 2 \u03bc } := { val := indicatorConstLp 2 (_ : MeasurableSet s) h\u03bcs c, property := h_mem }\nh_coe_ind : \u2191ind = indicatorConstLp 2 (_ : MeasurableSet s) h\u03bcs c\n\u22a2 \u2191(\u2191(orthogonalProjection (lpMeas E \ud835\udd5c m 2 \u03bc)) (indicatorConstLp 2 (_ : MeasurableSet s) h\u03bcs c)) =\n    indicatorConstLp 2 (_ : MeasurableSet s) h\u03bcs c", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nc : E\nthis : Fact (m \u2264 m0)\nh_mem : indicatorConstLp 2 (_ : MeasurableSet s) h\u03bcs c \u2208 lpMeas E \ud835\udd5c m 2 \u03bc\nind : { x // x \u2208 lpMeas E \ud835\udd5c m 2 \u03bc } := { val := indicatorConstLp 2 (_ : MeasurableSet s) h\u03bcs c, property := h_mem }\nh_coe_ind : \u2191ind = indicatorConstLp 2 (_ : MeasurableSet s) h\u03bcs c\nh_orth_mem : \u2191(orthogonalProjection (lpMeas E \ud835\udd5c m 2 \u03bc)) \u2191ind = ind\n\u22a2 \u2191(\u2191(orthogonalProjection (lpMeas E \ud835\udd5c m 2 \u03bc)) (indicatorConstLp 2 (_ : MeasurableSet s) h\u03bcs c)) =\n    indicatorConstLp 2 (_ : MeasurableSet s) h\u03bcs c"}, {"tactic": "rw [\u2190 h_coe_ind, h_orth_mem]", "annotated_tactic": ["rw [\u2190 h_coe_ind, h_orth_mem]", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nc : E\nthis : Fact (m \u2264 m0)\nh_mem : indicatorConstLp 2 (_ : MeasurableSet s) h\u03bcs c \u2208 lpMeas E \ud835\udd5c m 2 \u03bc\nind : { x // x \u2208 lpMeas E \ud835\udd5c m 2 \u03bc } := { val := indicatorConstLp 2 (_ : MeasurableSet s) h\u03bcs c, property := h_mem }\nh_coe_ind : \u2191ind = indicatorConstLp 2 (_ : MeasurableSet s) h\u03bcs c\nh_orth_mem : \u2191(orthogonalProjection (lpMeas E \ud835\udd5c m 2 \u03bc)) \u2191ind = ind\n\u22a2 \u2191(\u2191(orthogonalProjection (lpMeas E \ud835\udd5c m 2 \u03bc)) (indicatorConstLp 2 (_ : MeasurableSet s) h\u03bcs c)) =\n    indicatorConstLp 2 (_ : MeasurableSet s) h\u03bcs c", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Num/Lemmas.lean", "full_name": "ZNum.cast_le", "start": [1398, 1], "end": [1399, 41], "traced_tactics": [{"tactic": "rw [\u2190 not_lt]", "annotated_tactic": ["rw [\u2190 <a>not_lt</a>]", [{"full_name": "not_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [368, 9], "def_end_pos": [368, 15]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : LinearOrderedRing \u03b1\nm n : ZNum\n\u22a2 \u2191m \u2264 \u2191n \u2194 m \u2264 n", "state_after": "\u03b1 : Type u_1\ninst\u271d : LinearOrderedRing \u03b1\nm n : ZNum\n\u22a2 \u00ac\u2191n < \u2191m \u2194 m \u2264 n"}, {"tactic": "exact not_congr cast_lt", "annotated_tactic": ["exact <a>not_congr</a> <a>cast_lt</a>", [{"full_name": "not_congr", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [22, 9], "def_end_pos": [22, 18]}, {"full_name": "ZNum.cast_lt", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [1393, 9], "def_end_pos": [1393, 16]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : LinearOrderedRing \u03b1\nm n : ZNum\n\u22a2 \u00ac\u2191n < \u2191m \u2194 m \u2264 n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/Average.lean", "full_name": "MeasureTheory.laverage_add_measure", "start": [142, 1], "end": [153, 62], "traced_tactics": [{"tactic": "by_cases h\u03bc : IsFiniteMeasure \u03bc", "annotated_tactic": ["by_cases h\u03bc : <a>IsFiniteMeasure</a> \u03bc", [{"full_name": "MeasureTheory.IsFiniteMeasure", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2850, 7], "def_end_pos": [2850, 22]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\n\u22a2 \u2a0d\u207b (x : \u03b1), f x \u2202(\u03bc + \u03bd) =\n    \u2191\u2191\u03bc univ / (\u2191\u2191\u03bc univ + \u2191\u2191\u03bd univ) * \u2a0d\u207b (x : \u03b1), f x \u2202\u03bc + \u2191\u2191\u03bd univ / (\u2191\u2191\u03bc univ + \u2191\u2191\u03bd univ) * \u2a0d\u207b (x : \u03b1), f x \u2202\u03bd", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nh\u03bc : IsFiniteMeasure \u03bc\n\u22a2 \u2a0d\u207b (x : \u03b1), f x \u2202(\u03bc + \u03bd) =\n    \u2191\u2191\u03bc univ / (\u2191\u2191\u03bc univ + \u2191\u2191\u03bd univ) * \u2a0d\u207b (x : \u03b1), f x \u2202\u03bc + \u2191\u2191\u03bd univ / (\u2191\u2191\u03bc univ + \u2191\u2191\u03bd univ) * \u2a0d\u207b (x : \u03b1), f x \u2202\u03bd\n\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nh\u03bc : \u00acIsFiniteMeasure \u03bc\n\u22a2 \u2a0d\u207b (x : \u03b1), f x \u2202(\u03bc + \u03bd) =\n    \u2191\u2191\u03bc univ / (\u2191\u2191\u03bc univ + \u2191\u2191\u03bd univ) * \u2a0d\u207b (x : \u03b1), f x \u2202\u03bc + \u2191\u2191\u03bd univ / (\u2191\u2191\u03bc univ + \u2191\u2191\u03bd univ) * \u2a0d\u207b (x : \u03b1), f x \u2202\u03bd"}, {"tactic": "swap", "annotated_tactic": ["swap", []], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nh\u03bc : IsFiniteMeasure \u03bc\n\u22a2 \u2a0d\u207b (x : \u03b1), f x \u2202(\u03bc + \u03bd) =\n    \u2191\u2191\u03bc univ / (\u2191\u2191\u03bc univ + \u2191\u2191\u03bd univ) * \u2a0d\u207b (x : \u03b1), f x \u2202\u03bc + \u2191\u2191\u03bd univ / (\u2191\u2191\u03bc univ + \u2191\u2191\u03bd univ) * \u2a0d\u207b (x : \u03b1), f x \u2202\u03bd\n\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nh\u03bc : \u00acIsFiniteMeasure \u03bc\n\u22a2 \u2a0d\u207b (x : \u03b1), f x \u2202(\u03bc + \u03bd) =\n    \u2191\u2191\u03bc univ / (\u2191\u2191\u03bc univ + \u2191\u2191\u03bd univ) * \u2a0d\u207b (x : \u03b1), f x \u2202\u03bc + \u2191\u2191\u03bd univ / (\u2191\u2191\u03bc univ + \u2191\u2191\u03bd univ) * \u2a0d\u207b (x : \u03b1), f x \u2202\u03bd", "state_after": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nh\u03bc : \u00acIsFiniteMeasure \u03bc\n\u22a2 \u2a0d\u207b (x : \u03b1), f x \u2202(\u03bc + \u03bd) =\n    \u2191\u2191\u03bc univ / (\u2191\u2191\u03bc univ + \u2191\u2191\u03bd univ) * \u2a0d\u207b (x : \u03b1), f x \u2202\u03bc + \u2191\u2191\u03bd univ / (\u2191\u2191\u03bc univ + \u2191\u2191\u03bd univ) * \u2a0d\u207b (x : \u03b1), f x \u2202\u03bd\n\ncase pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nh\u03bc : IsFiniteMeasure \u03bc\n\u22a2 \u2a0d\u207b (x : \u03b1), f x \u2202(\u03bc + \u03bd) =\n    \u2191\u2191\u03bc univ / (\u2191\u2191\u03bc univ + \u2191\u2191\u03bd univ) * \u2a0d\u207b (x : \u03b1), f x \u2202\u03bc + \u2191\u2191\u03bd univ / (\u2191\u2191\u03bc univ + \u2191\u2191\u03bd univ) * \u2a0d\u207b (x : \u03b1), f x \u2202\u03bd"}, {"tactic": "by_cases h\u03bd : IsFiniteMeasure \u03bd", "annotated_tactic": ["by_cases h\u03bd : <a>IsFiniteMeasure</a> \u03bd", [{"full_name": "MeasureTheory.IsFiniteMeasure", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2850, 7], "def_end_pos": [2850, 22]}]], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nh\u03bc : IsFiniteMeasure \u03bc\n\u22a2 \u2a0d\u207b (x : \u03b1), f x \u2202(\u03bc + \u03bd) =\n    \u2191\u2191\u03bc univ / (\u2191\u2191\u03bc univ + \u2191\u2191\u03bd univ) * \u2a0d\u207b (x : \u03b1), f x \u2202\u03bc + \u2191\u2191\u03bd univ / (\u2191\u2191\u03bc univ + \u2191\u2191\u03bd univ) * \u2a0d\u207b (x : \u03b1), f x \u2202\u03bd", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nh\u03bc : IsFiniteMeasure \u03bc\nh\u03bd : IsFiniteMeasure \u03bd\n\u22a2 \u2a0d\u207b (x : \u03b1), f x \u2202(\u03bc + \u03bd) =\n    \u2191\u2191\u03bc univ / (\u2191\u2191\u03bc univ + \u2191\u2191\u03bd univ) * \u2a0d\u207b (x : \u03b1), f x \u2202\u03bc + \u2191\u2191\u03bd univ / (\u2191\u2191\u03bc univ + \u2191\u2191\u03bd univ) * \u2a0d\u207b (x : \u03b1), f x \u2202\u03bd\n\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nh\u03bc : IsFiniteMeasure \u03bc\nh\u03bd : \u00acIsFiniteMeasure \u03bd\n\u22a2 \u2a0d\u207b (x : \u03b1), f x \u2202(\u03bc + \u03bd) =\n    \u2191\u2191\u03bc univ / (\u2191\u2191\u03bc univ + \u2191\u2191\u03bd univ) * \u2a0d\u207b (x : \u03b1), f x \u2202\u03bc + \u2191\u2191\u03bd univ / (\u2191\u2191\u03bc univ + \u2191\u2191\u03bd univ) * \u2a0d\u207b (x : \u03b1), f x \u2202\u03bd"}, {"tactic": "swap", "annotated_tactic": ["swap", []], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nh\u03bc : IsFiniteMeasure \u03bc\nh\u03bd : IsFiniteMeasure \u03bd\n\u22a2 \u2a0d\u207b (x : \u03b1), f x \u2202(\u03bc + \u03bd) =\n    \u2191\u2191\u03bc univ / (\u2191\u2191\u03bc univ + \u2191\u2191\u03bd univ) * \u2a0d\u207b (x : \u03b1), f x \u2202\u03bc + \u2191\u2191\u03bd univ / (\u2191\u2191\u03bc univ + \u2191\u2191\u03bd univ) * \u2a0d\u207b (x : \u03b1), f x \u2202\u03bd\n\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nh\u03bc : IsFiniteMeasure \u03bc\nh\u03bd : \u00acIsFiniteMeasure \u03bd\n\u22a2 \u2a0d\u207b (x : \u03b1), f x \u2202(\u03bc + \u03bd) =\n    \u2191\u2191\u03bc univ / (\u2191\u2191\u03bc univ + \u2191\u2191\u03bd univ) * \u2a0d\u207b (x : \u03b1), f x \u2202\u03bc + \u2191\u2191\u03bd univ / (\u2191\u2191\u03bc univ + \u2191\u2191\u03bd univ) * \u2a0d\u207b (x : \u03b1), f x \u2202\u03bd", "state_after": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nh\u03bc : IsFiniteMeasure \u03bc\nh\u03bd : \u00acIsFiniteMeasure \u03bd\n\u22a2 \u2a0d\u207b (x : \u03b1), f x \u2202(\u03bc + \u03bd) =\n    \u2191\u2191\u03bc univ / (\u2191\u2191\u03bc univ + \u2191\u2191\u03bd univ) * \u2a0d\u207b (x : \u03b1), f x \u2202\u03bc + \u2191\u2191\u03bd univ / (\u2191\u2191\u03bc univ + \u2191\u2191\u03bd univ) * \u2a0d\u207b (x : \u03b1), f x \u2202\u03bd\n\ncase pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nh\u03bc : IsFiniteMeasure \u03bc\nh\u03bd : IsFiniteMeasure \u03bd\n\u22a2 \u2a0d\u207b (x : \u03b1), f x \u2202(\u03bc + \u03bd) =\n    \u2191\u2191\u03bc univ / (\u2191\u2191\u03bc univ + \u2191\u2191\u03bd univ) * \u2a0d\u207b (x : \u03b1), f x \u2202\u03bc + \u2191\u2191\u03bd univ / (\u2191\u2191\u03bc univ + \u2191\u2191\u03bd univ) * \u2a0d\u207b (x : \u03b1), f x \u2202\u03bd"}, {"tactic": "haveI := h\u03bc", "annotated_tactic": ["haveI := h\u03bc", []], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nh\u03bc : IsFiniteMeasure \u03bc\nh\u03bd : IsFiniteMeasure \u03bd\n\u22a2 \u2a0d\u207b (x : \u03b1), f x \u2202(\u03bc + \u03bd) =\n    \u2191\u2191\u03bc univ / (\u2191\u2191\u03bc univ + \u2191\u2191\u03bd univ) * \u2a0d\u207b (x : \u03b1), f x \u2202\u03bc + \u2191\u2191\u03bd univ / (\u2191\u2191\u03bc univ + \u2191\u2191\u03bd univ) * \u2a0d\u207b (x : \u03b1), f x \u2202\u03bd", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nh\u03bc : IsFiniteMeasure \u03bc\nh\u03bd : IsFiniteMeasure \u03bd\nthis : IsFiniteMeasure \u03bc\n\u22a2 \u2a0d\u207b (x : \u03b1), f x \u2202(\u03bc + \u03bd) =\n    \u2191\u2191\u03bc univ / (\u2191\u2191\u03bc univ + \u2191\u2191\u03bd univ) * \u2a0d\u207b (x : \u03b1), f x \u2202\u03bc + \u2191\u2191\u03bd univ / (\u2191\u2191\u03bc univ + \u2191\u2191\u03bd univ) * \u2a0d\u207b (x : \u03b1), f x \u2202\u03bd"}, {"tactic": "haveI := h\u03bd", "annotated_tactic": ["haveI := h\u03bd", []], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nh\u03bc : IsFiniteMeasure \u03bc\nh\u03bd : IsFiniteMeasure \u03bd\nthis : IsFiniteMeasure \u03bc\n\u22a2 \u2a0d\u207b (x : \u03b1), f x \u2202(\u03bc + \u03bd) =\n    \u2191\u2191\u03bc univ / (\u2191\u2191\u03bc univ + \u2191\u2191\u03bd univ) * \u2a0d\u207b (x : \u03b1), f x \u2202\u03bc + \u2191\u2191\u03bd univ / (\u2191\u2191\u03bc univ + \u2191\u2191\u03bd univ) * \u2a0d\u207b (x : \u03b1), f x \u2202\u03bd", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nh\u03bc : IsFiniteMeasure \u03bc\nh\u03bd : IsFiniteMeasure \u03bd\nthis\u271d : IsFiniteMeasure \u03bc\nthis : IsFiniteMeasure \u03bd\n\u22a2 \u2a0d\u207b (x : \u03b1), f x \u2202(\u03bc + \u03bd) =\n    \u2191\u2191\u03bc univ / (\u2191\u2191\u03bc univ + \u2191\u2191\u03bd univ) * \u2a0d\u207b (x : \u03b1), f x \u2202\u03bc + \u2191\u2191\u03bd univ / (\u2191\u2191\u03bc univ + \u2191\u2191\u03bd univ) * \u2a0d\u207b (x : \u03b1), f x \u2202\u03bd"}, {"tactic": "simp only [\u2190ENNReal.mul_div_right_comm, measure_mul_laverage, \u2190ENNReal.add_div,\n  \u2190lintegral_add_measure, \u2190Measure.add_apply, \u2190laverage_eq]", "annotated_tactic": ["simp only [\u2190<a>ENNReal.mul_div_right_comm</a>, <a>measure_mul_laverage</a>, \u2190<a>ENNReal.add_div</a>,\n    \u2190<a>lintegral_add_measure</a>, \u2190<a>Measure.add_apply</a>, \u2190<a>laverage_eq</a>]", [{"full_name": "ENNReal.mul_div_right_comm", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1441, 19], "def_end_pos": [1441, 37]}, {"full_name": "MeasureTheory.measure_mul_laverage", "def_path": "Mathlib/MeasureTheory/Integral/Average.lean", "def_pos": [100, 9], "def_end_pos": [100, 29]}, {"full_name": "ENNReal.add_div", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1738, 19], "def_end_pos": [1738, 26]}, {"full_name": "MeasureTheory.lintegral_add_measure", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [619, 9], "def_end_pos": [619, 30]}, {"full_name": "MeasureTheory.Measure.add_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [798, 9], "def_end_pos": [798, 18]}, {"full_name": "MeasureTheory.laverage_eq", "def_path": "Mathlib/MeasureTheory/Integral/Average.lean", "def_pos": [91, 9], "def_end_pos": [91, 20]}]], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nh\u03bc : IsFiniteMeasure \u03bc\nh\u03bd : IsFiniteMeasure \u03bd\nthis\u271d : IsFiniteMeasure \u03bc\nthis : IsFiniteMeasure \u03bd\n\u22a2 \u2a0d\u207b (x : \u03b1), f x \u2202(\u03bc + \u03bd) =\n    \u2191\u2191\u03bc univ / (\u2191\u2191\u03bc univ + \u2191\u2191\u03bd univ) * \u2a0d\u207b (x : \u03b1), f x \u2202\u03bc + \u2191\u2191\u03bd univ / (\u2191\u2191\u03bc univ + \u2191\u2191\u03bd univ) * \u2a0d\u207b (x : \u03b1), f x \u2202\u03bd", "state_after": "no goals"}, {"tactic": "rw [not_isFiniteMeasure_iff] at h\u03bc", "annotated_tactic": ["rw [<a>not_isFiniteMeasure_iff</a>] at h\u03bc", [{"full_name": "MeasureTheory.not_isFiniteMeasure_iff", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2855, 9], "def_end_pos": [2855, 32]}]], "state_before": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nh\u03bc : \u00acIsFiniteMeasure \u03bc\n\u22a2 \u2a0d\u207b (x : \u03b1), f x \u2202(\u03bc + \u03bd) =\n    \u2191\u2191\u03bc univ / (\u2191\u2191\u03bc univ + \u2191\u2191\u03bd univ) * \u2a0d\u207b (x : \u03b1), f x \u2202\u03bc + \u2191\u2191\u03bd univ / (\u2191\u2191\u03bc univ + \u2191\u2191\u03bd univ) * \u2a0d\u207b (x : \u03b1), f x \u2202\u03bd", "state_after": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nh\u03bc : \u2191\u2191\u03bc univ = \u22a4\n\u22a2 \u2a0d\u207b (x : \u03b1), f x \u2202(\u03bc + \u03bd) =\n    \u2191\u2191\u03bc univ / (\u2191\u2191\u03bc univ + \u2191\u2191\u03bd univ) * \u2a0d\u207b (x : \u03b1), f x \u2202\u03bc + \u2191\u2191\u03bd univ / (\u2191\u2191\u03bc univ + \u2191\u2191\u03bd univ) * \u2a0d\u207b (x : \u03b1), f x \u2202\u03bd"}, {"tactic": "simp [laverage_eq, h\u03bc]", "annotated_tactic": ["simp [<a>laverage_eq</a>, h\u03bc]", [{"full_name": "MeasureTheory.laverage_eq", "def_path": "Mathlib/MeasureTheory/Integral/Average.lean", "def_pos": [91, 9], "def_end_pos": [91, 20]}]], "state_before": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nh\u03bc : \u2191\u2191\u03bc univ = \u22a4\n\u22a2 \u2a0d\u207b (x : \u03b1), f x \u2202(\u03bc + \u03bd) =\n    \u2191\u2191\u03bc univ / (\u2191\u2191\u03bc univ + \u2191\u2191\u03bd univ) * \u2a0d\u207b (x : \u03b1), f x \u2202\u03bc + \u2191\u2191\u03bd univ / (\u2191\u2191\u03bc univ + \u2191\u2191\u03bd univ) * \u2a0d\u207b (x : \u03b1), f x \u2202\u03bd", "state_after": "no goals"}, {"tactic": "rw [not_isFiniteMeasure_iff] at h\u03bd", "annotated_tactic": ["rw [<a>not_isFiniteMeasure_iff</a>] at h\u03bd", [{"full_name": "MeasureTheory.not_isFiniteMeasure_iff", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2855, 9], "def_end_pos": [2855, 32]}]], "state_before": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nh\u03bc : IsFiniteMeasure \u03bc\nh\u03bd : \u00acIsFiniteMeasure \u03bd\n\u22a2 \u2a0d\u207b (x : \u03b1), f x \u2202(\u03bc + \u03bd) =\n    \u2191\u2191\u03bc univ / (\u2191\u2191\u03bc univ + \u2191\u2191\u03bd univ) * \u2a0d\u207b (x : \u03b1), f x \u2202\u03bc + \u2191\u2191\u03bd univ / (\u2191\u2191\u03bc univ + \u2191\u2191\u03bd univ) * \u2a0d\u207b (x : \u03b1), f x \u2202\u03bd", "state_after": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nh\u03bc : IsFiniteMeasure \u03bc\nh\u03bd : \u2191\u2191\u03bd univ = \u22a4\n\u22a2 \u2a0d\u207b (x : \u03b1), f x \u2202(\u03bc + \u03bd) =\n    \u2191\u2191\u03bc univ / (\u2191\u2191\u03bc univ + \u2191\u2191\u03bd univ) * \u2a0d\u207b (x : \u03b1), f x \u2202\u03bc + \u2191\u2191\u03bd univ / (\u2191\u2191\u03bc univ + \u2191\u2191\u03bd univ) * \u2a0d\u207b (x : \u03b1), f x \u2202\u03bd"}, {"tactic": "simp [laverage_eq, h\u03bd]", "annotated_tactic": ["simp [<a>laverage_eq</a>, h\u03bd]", [{"full_name": "MeasureTheory.laverage_eq", "def_path": "Mathlib/MeasureTheory/Integral/Average.lean", "def_pos": [91, 9], "def_end_pos": [91, 20]}]], "state_before": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nh\u03bc : IsFiniteMeasure \u03bc\nh\u03bd : \u2191\u2191\u03bd univ = \u22a4\n\u22a2 \u2a0d\u207b (x : \u03b1), f x \u2202(\u03bc + \u03bd) =\n    \u2191\u2191\u03bc univ / (\u2191\u2191\u03bc univ + \u2191\u2191\u03bd univ) * \u2a0d\u207b (x : \u03b1), f x \u2202\u03bc + \u2191\u2191\u03bd univ / (\u2191\u2191\u03bc univ + \u2191\u2191\u03bd univ) * \u2a0d\u207b (x : \u03b1), f x \u2202\u03bd", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Array/Init/Lemmas.lean", "full_name": "Array.SatisfiesM_mapM", "start": [155, 1], "end": [171, 85], "traced_tactics": [{"tactic": "rw [mapM_eq_foldlM]", "annotated_tactic": ["rw [<a>mapM_eq_foldlM</a>]", [{"full_name": "Array.mapM_eq_foldlM", "def_path": "lake-packages/std/Std/Data/Array/Init/Lemmas.lean", "def_pos": [143, 9], "def_end_pos": [143, 23]}]], "state_before": "m : Type u_1 \u2192 Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_1\ninst\u271d\u00b9 : Monad m\ninst\u271d : LawfulMonad m\nas : Array \u03b1\nf : \u03b1 \u2192 m \u03b2\nmotive : Nat \u2192 Prop\nh0 : motive 0\np : Fin (size as) \u2192 \u03b2 \u2192 Prop\nhs : \u2200 (i : Fin (size as)), motive i.val \u2192 SatisfiesM (fun x => p i x \u2227 motive (i.val + 1)) (f as[i])\n\u22a2 SatisfiesM (fun arr => motive (size as) \u2227 \u2203 eq, \u2200 (i : Nat) (h : i < size as), p { val := i, isLt := h } arr[i])\n    (mapM f as)", "state_after": "m : Type u_1 \u2192 Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_1\ninst\u271d\u00b9 : Monad m\ninst\u271d : LawfulMonad m\nas : Array \u03b1\nf : \u03b1 \u2192 m \u03b2\nmotive : Nat \u2192 Prop\nh0 : motive 0\np : Fin (size as) \u2192 \u03b2 \u2192 Prop\nhs : \u2200 (i : Fin (size as)), motive i.val \u2192 SatisfiesM (fun x => p i x \u2227 motive (i.val + 1)) (f as[i])\n\u22a2 SatisfiesM (fun arr => motive (size as) \u2227 \u2203 eq, \u2200 (i : Nat) (h : i < size as), p { val := i, isLt := h } arr[i])\n    (foldlM (fun bs a => push bs <$> f a) #[] as 0 (size as))"}, {"tactic": "refine SatisfiesM_foldlM (m := m) (\u03b2 := Array \u03b2)\n  (motive := fun i arr => motive i \u2227 arr.size = i \u2227 \u2200 i h2, p i (arr[i.1]'h2)) ?z ?s\n  |>.imp fun \u27e8h\u2081, eq, h\u2082\u27e9 => \u27e8h\u2081, eq, fun _ _ => h\u2082 ..\u27e9", "annotated_tactic": ["refine <a>SatisfiesM_foldlM</a> (m := m) (\u03b2 := <a>Array</a> \u03b2)\n    (motive := fun i arr => motive i \u2227 arr.size = i \u2227 \u2200 i h2, p i (arr[i.1]'h2)) ?z ?s\n    |>.<a>imp</a> fun \u27e8h\u2081, eq, h\u2082\u27e9 => \u27e8h\u2081, eq, fun _ _ => h\u2082 ..\u27e9", [{"full_name": "Array.SatisfiesM_foldlM", "def_path": "lake-packages/std/Std/Data/Array/Init/Lemmas.lean", "def_pos": [104, 9], "def_end_pos": [104, 26]}, {"full_name": "Array", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2495, 11], "def_end_pos": [2495, 16]}, {"full_name": "SatisfiesM.imp", "def_path": "lake-packages/std/Std/Classes/LawfulMonad.lean", "def_pos": [87, 9], "def_end_pos": [87, 12]}]], "state_before": "m : Type u_1 \u2192 Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_1\ninst\u271d\u00b9 : Monad m\ninst\u271d : LawfulMonad m\nas : Array \u03b1\nf : \u03b1 \u2192 m \u03b2\nmotive : Nat \u2192 Prop\nh0 : motive 0\np : Fin (size as) \u2192 \u03b2 \u2192 Prop\nhs : \u2200 (i : Fin (size as)), motive i.val \u2192 SatisfiesM (fun x => p i x \u2227 motive (i.val + 1)) (f as[i])\n\u22a2 SatisfiesM (fun arr => motive (size as) \u2227 \u2203 eq, \u2200 (i : Nat) (h : i < size as), p { val := i, isLt := h } arr[i])\n    (foldlM (fun bs a => push bs <$> f a) #[] as 0 (size as))", "state_after": "case z\nm : Type u_1 \u2192 Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_1\ninst\u271d\u00b9 : Monad m\ninst\u271d : LawfulMonad m\nas : Array \u03b1\nf : \u03b1 \u2192 m \u03b2\nmotive : Nat \u2192 Prop\nh0 : motive 0\np : Fin (size as) \u2192 \u03b2 \u2192 Prop\nhs : \u2200 (i : Fin (size as)), motive i.val \u2192 SatisfiesM (fun x => p i x \u2227 motive (i.val + 1)) (f as[i])\n\u22a2 (fun i arr => motive i \u2227 size arr = i \u2227 \u2200 (i : Fin (size as)) (h2 : i.val < size arr), p i arr[i.val]) 0 #[]\n\ncase s\nm : Type u_1 \u2192 Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_1\ninst\u271d\u00b9 : Monad m\ninst\u271d : LawfulMonad m\nas : Array \u03b1\nf : \u03b1 \u2192 m \u03b2\nmotive : Nat \u2192 Prop\nh0 : motive 0\np : Fin (size as) \u2192 \u03b2 \u2192 Prop\nhs : \u2200 (i : Fin (size as)), motive i.val \u2192 SatisfiesM (fun x => p i x \u2227 motive (i.val + 1)) (f as[i])\n\u22a2 \u2200 (i : Fin (size as)) (b : Array \u03b2),\n    (fun i arr => motive i \u2227 size arr = i \u2227 \u2200 (i : Fin (size as)) (h2 : i.val < size arr), p i arr[i.val]) i.val b \u2192\n      SatisfiesM\n        ((fun i arr => motive i \u2227 size arr = i \u2227 \u2200 (i : Fin (size as)) (h2 : i.val < size arr), p i arr[i.val])\n          (i.val + 1))\n        (push b <$> f as[i])"}, {"tactic": "case z => exact \u27e8h0, rfl, fun.\u27e9", "annotated_tactic": ["case z => exact \u27e8h0, <a>rfl</a>, fun.\u27e9", [{"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "case z\nm : Type u_1 \u2192 Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_1\ninst\u271d\u00b9 : Monad m\ninst\u271d : LawfulMonad m\nas : Array \u03b1\nf : \u03b1 \u2192 m \u03b2\nmotive : Nat \u2192 Prop\nh0 : motive 0\np : Fin (size as) \u2192 \u03b2 \u2192 Prop\nhs : \u2200 (i : Fin (size as)), motive i.val \u2192 SatisfiesM (fun x => p i x \u2227 motive (i.val + 1)) (f as[i])\n\u22a2 (fun i arr => motive i \u2227 size arr = i \u2227 \u2200 (i : Fin (size as)) (h2 : i.val < size arr), p i arr[i.val]) 0 #[]", "state_after": "no goals"}, {"tactic": "exact \u27e8h0, rfl, fun.\u27e9", "annotated_tactic": ["exact \u27e8h0, <a>rfl</a>, fun.\u27e9", [{"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "m : Type u_1 \u2192 Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_1\ninst\u271d\u00b9 : Monad m\ninst\u271d : LawfulMonad m\nas : Array \u03b1\nf : \u03b1 \u2192 m \u03b2\nmotive : Nat \u2192 Prop\nh0 : motive 0\np : Fin (size as) \u2192 \u03b2 \u2192 Prop\nhs : \u2200 (i : Fin (size as)), motive i.val \u2192 SatisfiesM (fun x => p i x \u2227 motive (i.val + 1)) (f as[i])\n\u22a2 (fun i arr => motive i \u2227 size arr = i \u2227 \u2200 (i : Fin (size as)) (h2 : i.val < size arr), p i arr[i.val]) 0 #[]", "state_after": "no goals"}, {"tactic": "case s =>\nintro \u27e8i, hi\u27e9 arr \u27e8ih\u2081, eq, ih\u2082\u27e9\nrefine (hs _ ih\u2081).map fun \u27e8h\u2081, h\u2082\u27e9 => \u27e8h\u2082, by simp [eq], fun j hj => ?_\u27e9\nsimp [get_push] at hj \u22a2; split; {apply ih\u2082}\ncases j; cases (Nat.le_or_eq_of_le_succ hj).resolve_left \u2039_\u203a; cases eq; exact h\u2081", "annotated_tactic": ["case s =>\n    intro \u27e8i, hi\u27e9 arr \u27e8ih\u2081, eq, ih\u2082\u27e9\n    refine (hs _ ih\u2081).<a>map</a> fun \u27e8h\u2081, h\u2082\u27e9 => \u27e8h\u2082, by simp [eq], fun j hj => ?_\u27e9\n    simp [<a>get_push</a>] at hj \u22a2; split; {apply ih\u2082}\n    cases j; cases (<a>Nat.le_or_eq_of_le_succ</a> hj).<a>resolve_left</a> \u2039_\u203a; cases eq; exact h\u2081", [{"full_name": "SatisfiesM.map", "def_path": "lake-packages/std/Std/Classes/LawfulMonad.lean", "def_pos": [92, 19], "def_end_pos": [92, 22]}, {"full_name": "Array.get_push", "def_path": "lake-packages/std/Std/Data/Array/Init/Lemmas.lean", "def_pos": [135, 9], "def_end_pos": [135, 17]}, {"full_name": "Nat.le_or_eq_of_le_succ", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [332, 9], "def_end_pos": [332, 28]}, {"full_name": "Or.resolve_left", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [268, 9], "def_end_pos": [268, 24]}]], "state_before": "case s\nm : Type u_1 \u2192 Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_1\ninst\u271d\u00b9 : Monad m\ninst\u271d : LawfulMonad m\nas : Array \u03b1\nf : \u03b1 \u2192 m \u03b2\nmotive : Nat \u2192 Prop\nh0 : motive 0\np : Fin (size as) \u2192 \u03b2 \u2192 Prop\nhs : \u2200 (i : Fin (size as)), motive i.val \u2192 SatisfiesM (fun x => p i x \u2227 motive (i.val + 1)) (f as[i])\n\u22a2 \u2200 (i : Fin (size as)) (b : Array \u03b2),\n    (fun i arr => motive i \u2227 size arr = i \u2227 \u2200 (i : Fin (size as)) (h2 : i.val < size arr), p i arr[i.val]) i.val b \u2192\n      SatisfiesM\n        ((fun i arr => motive i \u2227 size arr = i \u2227 \u2200 (i : Fin (size as)) (h2 : i.val < size arr), p i arr[i.val])\n          (i.val + 1))\n        (push b <$> f as[i])", "state_after": "no goals"}, {"tactic": "intro \u27e8i, hi\u27e9 arr \u27e8ih\u2081, eq, ih\u2082\u27e9", "annotated_tactic": ["intro \u27e8i, hi\u27e9 arr \u27e8ih\u2081, eq, ih\u2082\u27e9", []], "state_before": "m : Type u_1 \u2192 Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_1\ninst\u271d\u00b9 : Monad m\ninst\u271d : LawfulMonad m\nas : Array \u03b1\nf : \u03b1 \u2192 m \u03b2\nmotive : Nat \u2192 Prop\nh0 : motive 0\np : Fin (size as) \u2192 \u03b2 \u2192 Prop\nhs : \u2200 (i : Fin (size as)), motive i.val \u2192 SatisfiesM (fun x => p i x \u2227 motive (i.val + 1)) (f as[i])\n\u22a2 \u2200 (i : Fin (size as)) (b : Array \u03b2),\n    (fun i arr => motive i \u2227 size arr = i \u2227 \u2200 (i : Fin (size as)) (h2 : i.val < size arr), p i arr[i.val]) i.val b \u2192\n      SatisfiesM\n        ((fun i arr => motive i \u2227 size arr = i \u2227 \u2200 (i : Fin (size as)) (h2 : i.val < size arr), p i arr[i.val])\n          (i.val + 1))\n        (push b <$> f as[i])", "state_after": "m : Type u_1 \u2192 Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_1\ninst\u271d\u00b9 : Monad m\ninst\u271d : LawfulMonad m\nas : Array \u03b1\nf : \u03b1 \u2192 m \u03b2\nmotive : Nat \u2192 Prop\nh0 : motive 0\np : Fin (size as) \u2192 \u03b2 \u2192 Prop\nhs : \u2200 (i : Fin (size as)), motive i.val \u2192 SatisfiesM (fun x => p i x \u2227 motive (i.val + 1)) (f as[i])\ni : Nat\nhi : i < size as\narr : Array \u03b2\nih\u2081 : motive { val := i, isLt := hi }.val\neq : size arr = { val := i, isLt := hi }.val\nih\u2082 : \u2200 (i : Fin (size as)) (h2 : i.val < size arr), p i arr[i.val]\n\u22a2 SatisfiesM\n    ((fun i arr => motive i \u2227 size arr = i \u2227 \u2200 (i : Fin (size as)) (h2 : i.val < size arr), p i arr[i.val])\n      ({ val := i, isLt := hi }.val + 1))\n    (push arr <$> f as[{ val := i, isLt := hi }])"}, {"tactic": "refine (hs _ ih\u2081).map fun \u27e8h\u2081, h\u2082\u27e9 => \u27e8h\u2082, by simp [eq], fun j hj => ?_\u27e9", "annotated_tactic": ["refine (hs _ ih\u2081).<a>map</a> fun \u27e8h\u2081, h\u2082\u27e9 => \u27e8h\u2082, by simp [eq], fun j hj => ?_\u27e9", [{"full_name": "SatisfiesM.map", "def_path": "lake-packages/std/Std/Classes/LawfulMonad.lean", "def_pos": [92, 19], "def_end_pos": [92, 22]}]], "state_before": "m : Type u_1 \u2192 Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_1\ninst\u271d\u00b9 : Monad m\ninst\u271d : LawfulMonad m\nas : Array \u03b1\nf : \u03b1 \u2192 m \u03b2\nmotive : Nat \u2192 Prop\nh0 : motive 0\np : Fin (size as) \u2192 \u03b2 \u2192 Prop\nhs : \u2200 (i : Fin (size as)), motive i.val \u2192 SatisfiesM (fun x => p i x \u2227 motive (i.val + 1)) (f as[i])\ni : Nat\nhi : i < size as\narr : Array \u03b2\nih\u2081 : motive { val := i, isLt := hi }.val\neq : size arr = { val := i, isLt := hi }.val\nih\u2082 : \u2200 (i : Fin (size as)) (h2 : i.val < size arr), p i arr[i.val]\n\u22a2 SatisfiesM\n    ((fun i arr => motive i \u2227 size arr = i \u2227 \u2200 (i : Fin (size as)) (h2 : i.val < size arr), p i arr[i.val])\n      ({ val := i, isLt := hi }.val + 1))\n    (push arr <$> f as[{ val := i, isLt := hi }])", "state_after": "m : Type u_1 \u2192 Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_1\ninst\u271d\u00b9 : Monad m\ninst\u271d : LawfulMonad m\nas : Array \u03b1\nf : \u03b1 \u2192 m \u03b2\nmotive : Nat \u2192 Prop\nh0 : motive 0\np : Fin (size as) \u2192 \u03b2 \u2192 Prop\nhs : \u2200 (i : Fin (size as)), motive i.val \u2192 SatisfiesM (fun x => p i x \u2227 motive (i.val + 1)) (f as[i])\ni : Nat\nhi : i < size as\narr : Array \u03b2\nih\u2081 : motive { val := i, isLt := hi }.val\neq : size arr = { val := i, isLt := hi }.val\nih\u2082 : \u2200 (i : Fin (size as)) (h2 : i.val < size arr), p i arr[i.val]\na\u271d : \u03b2\nx\u271d : p { val := i, isLt := hi } a\u271d \u2227 motive ({ val := i, isLt := hi }.val + 1)\nh\u2081 : p { val := i, isLt := hi } a\u271d\nh\u2082 : motive ({ val := i, isLt := hi }.val + 1)\nj : Fin (size as)\nhj : j.val < size (push arr a\u271d)\n\u22a2 p j (push arr a\u271d)[j.val]"}, {"tactic": "simp [get_push] at hj \u22a2", "annotated_tactic": ["simp [<a>get_push</a>] at hj \u22a2", [{"full_name": "Array.get_push", "def_path": "lake-packages/std/Std/Data/Array/Init/Lemmas.lean", "def_pos": [135, 9], "def_end_pos": [135, 17]}]], "state_before": "m : Type u_1 \u2192 Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_1\ninst\u271d\u00b9 : Monad m\ninst\u271d : LawfulMonad m\nas : Array \u03b1\nf : \u03b1 \u2192 m \u03b2\nmotive : Nat \u2192 Prop\nh0 : motive 0\np : Fin (size as) \u2192 \u03b2 \u2192 Prop\nhs : \u2200 (i : Fin (size as)), motive i.val \u2192 SatisfiesM (fun x => p i x \u2227 motive (i.val + 1)) (f as[i])\ni : Nat\nhi : i < size as\narr : Array \u03b2\nih\u2081 : motive { val := i, isLt := hi }.val\neq : size arr = { val := i, isLt := hi }.val\nih\u2082 : \u2200 (i : Fin (size as)) (h2 : i.val < size arr), p i arr[i.val]\na\u271d : \u03b2\nx\u271d : p { val := i, isLt := hi } a\u271d \u2227 motive ({ val := i, isLt := hi }.val + 1)\nh\u2081 : p { val := i, isLt := hi } a\u271d\nh\u2082 : motive ({ val := i, isLt := hi }.val + 1)\nj : Fin (size as)\nhj : j.val < size (push arr a\u271d)\n\u22a2 p j (push arr a\u271d)[j.val]", "state_after": "m : Type u_1 \u2192 Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_1\ninst\u271d\u00b9 : Monad m\ninst\u271d : LawfulMonad m\nas : Array \u03b1\nf : \u03b1 \u2192 m \u03b2\nmotive : Nat \u2192 Prop\nh0 : motive 0\np : Fin (size as) \u2192 \u03b2 \u2192 Prop\nhs : \u2200 (i : Fin (size as)), motive i.val \u2192 SatisfiesM (fun x => p i x \u2227 motive (i.val + 1)) (f as[i])\ni : Nat\nhi : i < size as\narr : Array \u03b2\nih\u2081 : motive { val := i, isLt := hi }.val\neq : size arr = { val := i, isLt := hi }.val\nih\u2082 : \u2200 (i : Fin (size as)) (h2 : i.val < size arr), p i arr[i.val]\na\u271d : \u03b2\nx\u271d : p { val := i, isLt := hi } a\u271d \u2227 motive ({ val := i, isLt := hi }.val + 1)\nh\u2081 : p { val := i, isLt := hi } a\u271d\nh\u2082 : motive ({ val := i, isLt := hi }.val + 1)\nj : Fin (size as)\nhj : j.val < size arr + 1\n\u22a2 p j (if h : j.val < size arr then arr[j.val] else a\u271d)"}, {"tactic": "split", "annotated_tactic": ["split", []], "state_before": "m : Type u_1 \u2192 Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_1\ninst\u271d\u00b9 : Monad m\ninst\u271d : LawfulMonad m\nas : Array \u03b1\nf : \u03b1 \u2192 m \u03b2\nmotive : Nat \u2192 Prop\nh0 : motive 0\np : Fin (size as) \u2192 \u03b2 \u2192 Prop\nhs : \u2200 (i : Fin (size as)), motive i.val \u2192 SatisfiesM (fun x => p i x \u2227 motive (i.val + 1)) (f as[i])\ni : Nat\nhi : i < size as\narr : Array \u03b2\nih\u2081 : motive { val := i, isLt := hi }.val\neq : size arr = { val := i, isLt := hi }.val\nih\u2082 : \u2200 (i : Fin (size as)) (h2 : i.val < size arr), p i arr[i.val]\na\u271d : \u03b2\nx\u271d : p { val := i, isLt := hi } a\u271d \u2227 motive ({ val := i, isLt := hi }.val + 1)\nh\u2081 : p { val := i, isLt := hi } a\u271d\nh\u2082 : motive ({ val := i, isLt := hi }.val + 1)\nj : Fin (size as)\nhj : j.val < size arr + 1\n\u22a2 p j (if h : j.val < size arr then arr[j.val] else a\u271d)", "state_after": "case inl\nm : Type u_1 \u2192 Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_1\ninst\u271d\u00b9 : Monad m\ninst\u271d : LawfulMonad m\nas : Array \u03b1\nf : \u03b1 \u2192 m \u03b2\nmotive : Nat \u2192 Prop\nh0 : motive 0\np : Fin (size as) \u2192 \u03b2 \u2192 Prop\nhs : \u2200 (i : Fin (size as)), motive i.val \u2192 SatisfiesM (fun x => p i x \u2227 motive (i.val + 1)) (f as[i])\ni : Nat\nhi : i < size as\narr : Array \u03b2\nih\u2081 : motive { val := i, isLt := hi }.val\neq : size arr = { val := i, isLt := hi }.val\nih\u2082 : \u2200 (i : Fin (size as)) (h2 : i.val < size arr), p i arr[i.val]\na\u271d : \u03b2\nx\u271d : p { val := i, isLt := hi } a\u271d \u2227 motive ({ val := i, isLt := hi }.val + 1)\nh\u2081 : p { val := i, isLt := hi } a\u271d\nh\u2082 : motive ({ val := i, isLt := hi }.val + 1)\nj : Fin (size as)\nhj : j.val < size arr + 1\nh\u271d : j.val < size arr\n\u22a2 p j arr[j.val]\n\ncase inr\nm : Type u_1 \u2192 Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_1\ninst\u271d\u00b9 : Monad m\ninst\u271d : LawfulMonad m\nas : Array \u03b1\nf : \u03b1 \u2192 m \u03b2\nmotive : Nat \u2192 Prop\nh0 : motive 0\np : Fin (size as) \u2192 \u03b2 \u2192 Prop\nhs : \u2200 (i : Fin (size as)), motive i.val \u2192 SatisfiesM (fun x => p i x \u2227 motive (i.val + 1)) (f as[i])\ni : Nat\nhi : i < size as\narr : Array \u03b2\nih\u2081 : motive { val := i, isLt := hi }.val\neq : size arr = { val := i, isLt := hi }.val\nih\u2082 : \u2200 (i : Fin (size as)) (h2 : i.val < size arr), p i arr[i.val]\na\u271d : \u03b2\nx\u271d : p { val := i, isLt := hi } a\u271d \u2227 motive ({ val := i, isLt := hi }.val + 1)\nh\u2081 : p { val := i, isLt := hi } a\u271d\nh\u2082 : motive ({ val := i, isLt := hi }.val + 1)\nj : Fin (size as)\nhj : j.val < size arr + 1\nh\u271d : \u00acj.val < size arr\n\u22a2 p j a\u271d"}, {"tactic": "{apply ih\u2082}", "annotated_tactic": ["{apply ih\u2082}", []], "state_before": "case inl\nm : Type u_1 \u2192 Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_1\ninst\u271d\u00b9 : Monad m\ninst\u271d : LawfulMonad m\nas : Array \u03b1\nf : \u03b1 \u2192 m \u03b2\nmotive : Nat \u2192 Prop\nh0 : motive 0\np : Fin (size as) \u2192 \u03b2 \u2192 Prop\nhs : \u2200 (i : Fin (size as)), motive i.val \u2192 SatisfiesM (fun x => p i x \u2227 motive (i.val + 1)) (f as[i])\ni : Nat\nhi : i < size as\narr : Array \u03b2\nih\u2081 : motive { val := i, isLt := hi }.val\neq : size arr = { val := i, isLt := hi }.val\nih\u2082 : \u2200 (i : Fin (size as)) (h2 : i.val < size arr), p i arr[i.val]\na\u271d : \u03b2\nx\u271d : p { val := i, isLt := hi } a\u271d \u2227 motive ({ val := i, isLt := hi }.val + 1)\nh\u2081 : p { val := i, isLt := hi } a\u271d\nh\u2082 : motive ({ val := i, isLt := hi }.val + 1)\nj : Fin (size as)\nhj : j.val < size arr + 1\nh\u271d : j.val < size arr\n\u22a2 p j arr[j.val]\n\ncase inr\nm : Type u_1 \u2192 Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_1\ninst\u271d\u00b9 : Monad m\ninst\u271d : LawfulMonad m\nas : Array \u03b1\nf : \u03b1 \u2192 m \u03b2\nmotive : Nat \u2192 Prop\nh0 : motive 0\np : Fin (size as) \u2192 \u03b2 \u2192 Prop\nhs : \u2200 (i : Fin (size as)), motive i.val \u2192 SatisfiesM (fun x => p i x \u2227 motive (i.val + 1)) (f as[i])\ni : Nat\nhi : i < size as\narr : Array \u03b2\nih\u2081 : motive { val := i, isLt := hi }.val\neq : size arr = { val := i, isLt := hi }.val\nih\u2082 : \u2200 (i : Fin (size as)) (h2 : i.val < size arr), p i arr[i.val]\na\u271d : \u03b2\nx\u271d : p { val := i, isLt := hi } a\u271d \u2227 motive ({ val := i, isLt := hi }.val + 1)\nh\u2081 : p { val := i, isLt := hi } a\u271d\nh\u2082 : motive ({ val := i, isLt := hi }.val + 1)\nj : Fin (size as)\nhj : j.val < size arr + 1\nh\u271d : \u00acj.val < size arr\n\u22a2 p j a\u271d", "state_after": "case inr\nm : Type u_1 \u2192 Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_1\ninst\u271d\u00b9 : Monad m\ninst\u271d : LawfulMonad m\nas : Array \u03b1\nf : \u03b1 \u2192 m \u03b2\nmotive : Nat \u2192 Prop\nh0 : motive 0\np : Fin (size as) \u2192 \u03b2 \u2192 Prop\nhs : \u2200 (i : Fin (size as)), motive i.val \u2192 SatisfiesM (fun x => p i x \u2227 motive (i.val + 1)) (f as[i])\ni : Nat\nhi : i < size as\narr : Array \u03b2\nih\u2081 : motive { val := i, isLt := hi }.val\neq : size arr = { val := i, isLt := hi }.val\nih\u2082 : \u2200 (i : Fin (size as)) (h2 : i.val < size arr), p i arr[i.val]\na\u271d : \u03b2\nx\u271d : p { val := i, isLt := hi } a\u271d \u2227 motive ({ val := i, isLt := hi }.val + 1)\nh\u2081 : p { val := i, isLt := hi } a\u271d\nh\u2082 : motive ({ val := i, isLt := hi }.val + 1)\nj : Fin (size as)\nhj : j.val < size arr + 1\nh\u271d : \u00acj.val < size arr\n\u22a2 p j a\u271d"}, {"tactic": "cases j", "annotated_tactic": ["cases j", []], "state_before": "case inr\nm : Type u_1 \u2192 Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_1\ninst\u271d\u00b9 : Monad m\ninst\u271d : LawfulMonad m\nas : Array \u03b1\nf : \u03b1 \u2192 m \u03b2\nmotive : Nat \u2192 Prop\nh0 : motive 0\np : Fin (size as) \u2192 \u03b2 \u2192 Prop\nhs : \u2200 (i : Fin (size as)), motive i.val \u2192 SatisfiesM (fun x => p i x \u2227 motive (i.val + 1)) (f as[i])\ni : Nat\nhi : i < size as\narr : Array \u03b2\nih\u2081 : motive { val := i, isLt := hi }.val\neq : size arr = { val := i, isLt := hi }.val\nih\u2082 : \u2200 (i : Fin (size as)) (h2 : i.val < size arr), p i arr[i.val]\na\u271d : \u03b2\nx\u271d : p { val := i, isLt := hi } a\u271d \u2227 motive ({ val := i, isLt := hi }.val + 1)\nh\u2081 : p { val := i, isLt := hi } a\u271d\nh\u2082 : motive ({ val := i, isLt := hi }.val + 1)\nj : Fin (size as)\nhj : j.val < size arr + 1\nh\u271d : \u00acj.val < size arr\n\u22a2 p j a\u271d", "state_after": "case inr.mk\nm : Type u_1 \u2192 Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_1\ninst\u271d\u00b9 : Monad m\ninst\u271d : LawfulMonad m\nas : Array \u03b1\nf : \u03b1 \u2192 m \u03b2\nmotive : Nat \u2192 Prop\nh0 : motive 0\np : Fin (size as) \u2192 \u03b2 \u2192 Prop\nhs : \u2200 (i : Fin (size as)), motive i.val \u2192 SatisfiesM (fun x => p i x \u2227 motive (i.val + 1)) (f as[i])\ni : Nat\nhi : i < size as\narr : Array \u03b2\nih\u2081 : motive { val := i, isLt := hi }.val\neq : size arr = { val := i, isLt := hi }.val\nih\u2082 : \u2200 (i : Fin (size as)) (h2 : i.val < size arr), p i arr[i.val]\na\u271d : \u03b2\nx\u271d : p { val := i, isLt := hi } a\u271d \u2227 motive ({ val := i, isLt := hi }.val + 1)\nh\u2081 : p { val := i, isLt := hi } a\u271d\nh\u2082 : motive ({ val := i, isLt := hi }.val + 1)\nval\u271d : Nat\nisLt\u271d : val\u271d < size as\nhj : { val := val\u271d, isLt := isLt\u271d }.val < size arr + 1\nh\u271d : \u00ac{ val := val\u271d, isLt := isLt\u271d }.val < size arr\n\u22a2 p { val := val\u271d, isLt := isLt\u271d } a\u271d"}, {"tactic": "cases (Nat.le_or_eq_of_le_succ hj).resolve_left \u2039_\u203a", "annotated_tactic": ["cases (<a>Nat.le_or_eq_of_le_succ</a> hj).<a>resolve_left</a> \u2039_\u203a", [{"full_name": "Nat.le_or_eq_of_le_succ", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [332, 9], "def_end_pos": [332, 28]}, {"full_name": "Or.resolve_left", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [268, 9], "def_end_pos": [268, 24]}]], "state_before": "case inr.mk\nm : Type u_1 \u2192 Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_1\ninst\u271d\u00b9 : Monad m\ninst\u271d : LawfulMonad m\nas : Array \u03b1\nf : \u03b1 \u2192 m \u03b2\nmotive : Nat \u2192 Prop\nh0 : motive 0\np : Fin (size as) \u2192 \u03b2 \u2192 Prop\nhs : \u2200 (i : Fin (size as)), motive i.val \u2192 SatisfiesM (fun x => p i x \u2227 motive (i.val + 1)) (f as[i])\ni : Nat\nhi : i < size as\narr : Array \u03b2\nih\u2081 : motive { val := i, isLt := hi }.val\neq : size arr = { val := i, isLt := hi }.val\nih\u2082 : \u2200 (i : Fin (size as)) (h2 : i.val < size arr), p i arr[i.val]\na\u271d : \u03b2\nx\u271d : p { val := i, isLt := hi } a\u271d \u2227 motive ({ val := i, isLt := hi }.val + 1)\nh\u2081 : p { val := i, isLt := hi } a\u271d\nh\u2082 : motive ({ val := i, isLt := hi }.val + 1)\nval\u271d : Nat\nisLt\u271d : val\u271d < size as\nhj : { val := val\u271d, isLt := isLt\u271d }.val < size arr + 1\nh\u271d : \u00ac{ val := val\u271d, isLt := isLt\u271d }.val < size arr\n\u22a2 p { val := val\u271d, isLt := isLt\u271d } a\u271d", "state_after": "case inr.mk.refl\nm : Type u_1 \u2192 Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_1\ninst\u271d\u00b9 : Monad m\ninst\u271d : LawfulMonad m\nas : Array \u03b1\nf : \u03b1 \u2192 m \u03b2\nmotive : Nat \u2192 Prop\nh0 : motive 0\np : Fin (size as) \u2192 \u03b2 \u2192 Prop\nhs : \u2200 (i : Fin (size as)), motive i.val \u2192 SatisfiesM (fun x => p i x \u2227 motive (i.val + 1)) (f as[i])\ni : Nat\nhi : i < size as\narr : Array \u03b2\nih\u2081 : motive { val := i, isLt := hi }.val\neq : size arr = { val := i, isLt := hi }.val\nih\u2082 : \u2200 (i : Fin (size as)) (h2 : i.val < size arr), p i arr[i.val]\na\u271d : \u03b2\nx\u271d : p { val := i, isLt := hi } a\u271d \u2227 motive ({ val := i, isLt := hi }.val + 1)\nh\u2081 : p { val := i, isLt := hi } a\u271d\nh\u2082 : motive ({ val := i, isLt := hi }.val + 1)\nisLt\u271d : List.length arr.data < size as\nhj : { val := List.length arr.data, isLt := isLt\u271d }.val < size arr + 1\nh\u271d : \u00ac{ val := List.length arr.data, isLt := isLt\u271d }.val < size arr\n\u22a2 p { val := List.length arr.data, isLt := isLt\u271d } a\u271d"}, {"tactic": "cases eq", "annotated_tactic": ["cases eq", []], "state_before": "case inr.mk.refl\nm : Type u_1 \u2192 Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_1\ninst\u271d\u00b9 : Monad m\ninst\u271d : LawfulMonad m\nas : Array \u03b1\nf : \u03b1 \u2192 m \u03b2\nmotive : Nat \u2192 Prop\nh0 : motive 0\np : Fin (size as) \u2192 \u03b2 \u2192 Prop\nhs : \u2200 (i : Fin (size as)), motive i.val \u2192 SatisfiesM (fun x => p i x \u2227 motive (i.val + 1)) (f as[i])\ni : Nat\nhi : i < size as\narr : Array \u03b2\nih\u2081 : motive { val := i, isLt := hi }.val\neq : size arr = { val := i, isLt := hi }.val\nih\u2082 : \u2200 (i : Fin (size as)) (h2 : i.val < size arr), p i arr[i.val]\na\u271d : \u03b2\nx\u271d : p { val := i, isLt := hi } a\u271d \u2227 motive ({ val := i, isLt := hi }.val + 1)\nh\u2081 : p { val := i, isLt := hi } a\u271d\nh\u2082 : motive ({ val := i, isLt := hi }.val + 1)\nisLt\u271d : List.length arr.data < size as\nhj : { val := List.length arr.data, isLt := isLt\u271d }.val < size arr + 1\nh\u271d : \u00ac{ val := List.length arr.data, isLt := isLt\u271d }.val < size arr\n\u22a2 p { val := List.length arr.data, isLt := isLt\u271d } a\u271d", "state_after": "case inr.mk.refl.refl\nm : Type u_1 \u2192 Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_1\ninst\u271d\u00b9 : Monad m\ninst\u271d : LawfulMonad m\nas : Array \u03b1\nf : \u03b1 \u2192 m \u03b2\nmotive : Nat \u2192 Prop\nh0 : motive 0\np : Fin (size as) \u2192 \u03b2 \u2192 Prop\nhs : \u2200 (i : Fin (size as)), motive i.val \u2192 SatisfiesM (fun x => p i x \u2227 motive (i.val + 1)) (f as[i])\narr : Array \u03b2\nih\u2082 : \u2200 (i : Fin (size as)) (h2 : i.val < size arr), p i arr[i.val]\na\u271d : \u03b2\nisLt\u271d : List.length arr.data < size as\nhj : { val := List.length arr.data, isLt := isLt\u271d }.val < size arr + 1\nh\u271d : \u00ac{ val := List.length arr.data, isLt := isLt\u271d }.val < size arr\nhi : List.length arr.data < size as\nih\u2081 : motive { val := List.length arr.data, isLt := hi }.val\nx\u271d : p { val := List.length arr.data, isLt := hi } a\u271d \u2227 motive ({ val := List.length arr.data, isLt := hi }.val + 1)\nh\u2081 : p { val := List.length arr.data, isLt := hi } a\u271d\nh\u2082 : motive ({ val := List.length arr.data, isLt := hi }.val + 1)\n\u22a2 p { val := List.length arr.data, isLt := isLt\u271d } a\u271d"}, {"tactic": "exact h\u2081", "annotated_tactic": ["exact h\u2081", []], "state_before": "case inr.mk.refl.refl\nm : Type u_1 \u2192 Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_1\ninst\u271d\u00b9 : Monad m\ninst\u271d : LawfulMonad m\nas : Array \u03b1\nf : \u03b1 \u2192 m \u03b2\nmotive : Nat \u2192 Prop\nh0 : motive 0\np : Fin (size as) \u2192 \u03b2 \u2192 Prop\nhs : \u2200 (i : Fin (size as)), motive i.val \u2192 SatisfiesM (fun x => p i x \u2227 motive (i.val + 1)) (f as[i])\narr : Array \u03b2\nih\u2082 : \u2200 (i : Fin (size as)) (h2 : i.val < size arr), p i arr[i.val]\na\u271d : \u03b2\nisLt\u271d : List.length arr.data < size as\nhj : { val := List.length arr.data, isLt := isLt\u271d }.val < size arr + 1\nh\u271d : \u00ac{ val := List.length arr.data, isLt := isLt\u271d }.val < size arr\nhi : List.length arr.data < size as\nih\u2081 : motive { val := List.length arr.data, isLt := hi }.val\nx\u271d : p { val := List.length arr.data, isLt := hi } a\u271d \u2227 motive ({ val := List.length arr.data, isLt := hi }.val + 1)\nh\u2081 : p { val := List.length arr.data, isLt := hi } a\u271d\nh\u2082 : motive ({ val := List.length arr.data, isLt := hi }.val + 1)\n\u22a2 p { val := List.length arr.data, isLt := isLt\u271d } a\u271d", "state_after": "no goals"}, {"tactic": "simp [eq]", "annotated_tactic": ["simp [eq]", []], "state_before": "m : Type u_1 \u2192 Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_1\ninst\u271d\u00b9 : Monad m\ninst\u271d : LawfulMonad m\nas : Array \u03b1\nf : \u03b1 \u2192 m \u03b2\nmotive : Nat \u2192 Prop\nh0 : motive 0\np : Fin (size as) \u2192 \u03b2 \u2192 Prop\nhs : \u2200 (i : Fin (size as)), motive i.val \u2192 SatisfiesM (fun x => p i x \u2227 motive (i.val + 1)) (f as[i])\ni : Nat\nhi : i < size as\narr : Array \u03b2\nih\u2081 : motive { val := i, isLt := hi }.val\neq : size arr = { val := i, isLt := hi }.val\nih\u2082 : \u2200 (i : Fin (size as)) (h2 : i.val < size arr), p i arr[i.val]\na\u271d : \u03b2\nx\u271d : p { val := i, isLt := hi } a\u271d \u2227 motive ({ val := i, isLt := hi }.val + 1)\nh\u2081 : p { val := i, isLt := hi } a\u271d\nh\u2082 : motive ({ val := i, isLt := hi }.val + 1)\n\u22a2 size (push arr a\u271d) = { val := i, isLt := hi }.val + 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/RegularExpressions.lean", "full_name": "RegularExpression.matches'_map", "start": [441, 1], "end": [455, 28], "traced_tactics": [{"tactic": "rw [eq_comm]", "annotated_tactic": ["rw [<a>eq_comm</a>]", [{"full_name": "eq_comm", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [104, 9], "def_end_pos": [104, 16]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na\u271d b : \u03b1\nf : \u03b1 \u2192 \u03b2\na : \u03b1\n\u22a2 matches' (map f (char a)) = \u2191(Language.map f) (matches' (char a))", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na\u271d b : \u03b1\nf : \u03b1 \u2192 \u03b2\na : \u03b1\n\u22a2 \u2191(Language.map f) (matches' (char a)) = matches' (map f (char a))"}, {"tactic": "exact image_singleton", "annotated_tactic": ["exact <a>image_singleton</a>", [{"full_name": "Set.image_singleton", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [363, 9], "def_end_pos": [363, 24]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na\u271d b : \u03b1\nf : \u03b1 \u2192 \u03b2\na : \u03b1\n\u22a2 \u2191(Language.map f) (matches' (char a)) = matches' (map f (char a))", "state_after": "no goals"}, {"tactic": "simp only [matches'_map, map, matches'_add]", "annotated_tactic": ["simp only [matches'_map, <a>map</a>, <a>matches'_add</a>]", [{"full_name": "RegularExpression.map", "def_path": "Mathlib/Computability/RegularExpressions.lean", "def_pos": [403, 5], "def_end_pos": [403, 8]}, {"full_name": "RegularExpression.matches'_add", "def_path": "Mathlib/Computability/RegularExpressions.lean", "def_pos": [135, 9], "def_end_pos": [135, 21]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\nf : \u03b1 \u2192 \u03b2\nR S : RegularExpression \u03b1\n\u22a2 matches' (map f (R + S)) = \u2191(Language.map f) (matches' (R + S))", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\nf : \u03b1 \u2192 \u03b2\nR S : RegularExpression \u03b1\n\u22a2 \u2191(Language.map f) (matches' R) + \u2191(Language.map f) (matches' S) = \u2191(Language.map f) (matches' R + matches' S)"}, {"tactic": "rw [map_add]", "annotated_tactic": ["rw [<a>map_add</a>]", [{"full_name": "map_add", "def_path": "Mathlib/Algebra/Hom/Group/Defs.lean", "def_pos": [298, 3], "def_end_pos": [298, 14]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\nf : \u03b1 \u2192 \u03b2\nR S : RegularExpression \u03b1\n\u22a2 \u2191(Language.map f) (matches' R) + \u2191(Language.map f) (matches' S) = \u2191(Language.map f) (matches' R + matches' S)", "state_after": "no goals"}, {"tactic": "simp only [matches'_map, map, matches'_mul]", "annotated_tactic": ["simp only [matches'_map, <a>map</a>, <a>matches'_mul</a>]", [{"full_name": "RegularExpression.map", "def_path": "Mathlib/Computability/RegularExpressions.lean", "def_pos": [403, 5], "def_end_pos": [403, 8]}, {"full_name": "RegularExpression.matches'_mul", "def_path": "Mathlib/Computability/RegularExpressions.lean", "def_pos": [140, 9], "def_end_pos": [140, 21]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\nf : \u03b1 \u2192 \u03b2\nR S : RegularExpression \u03b1\n\u22a2 matches' (map f (R * S)) = \u2191(Language.map f) (matches' (R * S))", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\nf : \u03b1 \u2192 \u03b2\nR S : RegularExpression \u03b1\n\u22a2 \u2191(Language.map f) (matches' R) * \u2191(Language.map f) (matches' S) = \u2191(Language.map f) (matches' R * matches' S)"}, {"tactic": "rw [map_mul]", "annotated_tactic": ["rw [<a>map_mul</a>]", [{"full_name": "map_mul", "def_path": "Mathlib/Algebra/Hom/Group/Defs.lean", "def_pos": [299, 9], "def_end_pos": [299, 16]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\nf : \u03b1 \u2192 \u03b2\nR S : RegularExpression \u03b1\n\u22a2 \u2191(Language.map f) (matches' R) * \u2191(Language.map f) (matches' S) = \u2191(Language.map f) (matches' R * matches' S)", "state_after": "no goals"}, {"tactic": "simp_rw [map, matches', matches'_map]", "annotated_tactic": ["simp_rw [<a>map</a>, <a>matches'</a>, matches'_map]", [{"full_name": "RegularExpression.map", "def_path": "Mathlib/Computability/RegularExpressions.lean", "def_pos": [403, 5], "def_end_pos": [403, 8]}, {"full_name": "RegularExpression.matches'", "def_path": "Mathlib/Computability/RegularExpressions.lean", "def_pos": [110, 5], "def_end_pos": [110, 13]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\nf : \u03b1 \u2192 \u03b2\nR : RegularExpression \u03b1\n\u22a2 matches' (map f (star R)) = \u2191(Language.map f) (matches' (star R))", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\nf : \u03b1 \u2192 \u03b2\nR : RegularExpression \u03b1\n\u22a2 (\u2191(Language.map f) (matches' R))\u2217 = \u2191(Language.map f) (matches' R)\u2217"}, {"tactic": "rw [Language.kstar_eq_iSup_pow, Language.kstar_eq_iSup_pow]", "annotated_tactic": ["rw [<a>Language.kstar_eq_iSup_pow</a>, <a>Language.kstar_eq_iSup_pow</a>]", [{"full_name": "Language.kstar_eq_iSup_pow", "def_path": "Mathlib/Computability/Language.lean", "def_pos": [247, 9], "def_end_pos": [247, 26]}, {"full_name": "Language.kstar_eq_iSup_pow", "def_path": "Mathlib/Computability/Language.lean", "def_pos": [247, 9], "def_end_pos": [247, 26]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\nf : \u03b1 \u2192 \u03b2\nR : RegularExpression \u03b1\n\u22a2 (\u2191(Language.map f) (matches' R))\u2217 = \u2191(Language.map f) (matches' R)\u2217", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\nf : \u03b1 \u2192 \u03b2\nR : RegularExpression \u03b1\n\u22a2 \u2a06 i, \u2191(Language.map f) (matches' R) ^ i = \u2191(Language.map f) (\u2a06 i, matches' R ^ i)"}, {"tactic": "simp_rw [\u2190 map_pow]", "annotated_tactic": ["simp_rw [\u2190 <a>map_pow</a>]", [{"full_name": "map_pow", "def_path": "Mathlib/Algebra/Hom/Group/Defs.lean", "def_pos": [435, 9], "def_end_pos": [435, 16]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\nf : \u03b1 \u2192 \u03b2\nR : RegularExpression \u03b1\n\u22a2 \u2a06 i, \u2191(Language.map f) (matches' R) ^ i = \u2191(Language.map f) (\u2a06 i, matches' R ^ i)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\nf : \u03b1 \u2192 \u03b2\nR : RegularExpression \u03b1\n\u22a2 \u2a06 i, \u2191(Language.map f) (matches' R ^ i) = \u2191(Language.map f) (\u2a06 i, matches' R ^ i)"}, {"tactic": "exact image_iUnion.symm", "annotated_tactic": ["exact image_iUnion.symm", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\nf : \u03b1 \u2192 \u03b2\nR : RegularExpression \u03b1\n\u22a2 \u2a06 i, \u2191(Language.map f) (matches' R ^ i) = \u2191(Language.map f) (\u2a06 i, matches' R ^ i)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Constructions/Pi.lean", "full_name": "MeasureTheory.Measure.pi_eq_generateFrom", "start": [360, 1], "end": [373, 59], "traced_tactics": [{"tactic": "apply_assumption", "annotated_tactic": ["apply_assumption", []], "state_before": "\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : \u03b9 \u2192 Type u_3\ninst\u271d\u00b9 : Fintype \u03b9\nm : (i : \u03b9) \u2192 OuterMeasure (\u03b1 i)\ninst\u271d : (i : \u03b9) \u2192 MeasurableSpace (\u03b1 i)\n\u03bc : (i : \u03b9) \u2192 Measure (\u03b1 i)\nC : (i : \u03b9) \u2192 Set (Set (\u03b1 i))\ni : \u03b9\n\u22a2 MeasurableSpace (\u03b1 i)", "state_after": "no goals"}, {"tactic": "have h4C : \u2200 (i) (s : Set (\u03b1 i)), s \u2208 C i \u2192 MeasurableSet s := by\n  intro i s hs; rw [\u2190 hC]; exact measurableSet_generateFrom hs", "annotated_tactic": ["have h4C : \u2200 (i) (s : <a>Set</a> (\u03b1 i)), s \u2208 C i \u2192 <a>MeasurableSet</a> s := by\n    intro i s hs; rw [\u2190 hC]; exact <a>measurableSet_generateFrom</a> hs", [{"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}, {"full_name": "MeasurableSet", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [64, 5], "def_end_pos": [64, 18]}, {"full_name": "MeasurableSpace.measurableSet_generateFrom", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [370, 9], "def_end_pos": [370, 35]}]], "state_before": "\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : \u03b9 \u2192 Type u_3\ninst\u271d\u00b9 : Fintype \u03b9\nm : (i : \u03b9) \u2192 OuterMeasure (\u03b1 i)\ninst\u271d : (i : \u03b9) \u2192 MeasurableSpace (\u03b1 i)\n\u03bc : (i : \u03b9) \u2192 Measure (\u03b1 i)\nC : (i : \u03b9) \u2192 Set (Set (\u03b1 i))\nhC : \u2200 (i : \u03b9), generateFrom (C i) = inst\u271d i\nh2C : \u2200 (i : \u03b9), IsPiSystem (C i)\nh3C : (i : \u03b9) \u2192 FiniteSpanningSetsIn (\u03bc i) (C i)\n\u03bc\u03bd : Measure ((i : \u03b9) \u2192 \u03b1 i)\nh\u2081 : \u2200 (s : (i : \u03b9) \u2192 Set (\u03b1 i)), (\u2200 (i : \u03b9), s i \u2208 C i) \u2192 \u2191\u2191\u03bc\u03bd (Set.pi univ s) = \u220f i : \u03b9, \u2191\u2191(\u03bc i) (s i)\n\u22a2 Measure.pi \u03bc = \u03bc\u03bd", "state_after": "\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : \u03b9 \u2192 Type u_3\ninst\u271d\u00b9 : Fintype \u03b9\nm : (i : \u03b9) \u2192 OuterMeasure (\u03b1 i)\ninst\u271d : (i : \u03b9) \u2192 MeasurableSpace (\u03b1 i)\n\u03bc : (i : \u03b9) \u2192 Measure (\u03b1 i)\nC : (i : \u03b9) \u2192 Set (Set (\u03b1 i))\nhC : \u2200 (i : \u03b9), generateFrom (C i) = inst\u271d i\nh2C : \u2200 (i : \u03b9), IsPiSystem (C i)\nh3C : (i : \u03b9) \u2192 FiniteSpanningSetsIn (\u03bc i) (C i)\n\u03bc\u03bd : Measure ((i : \u03b9) \u2192 \u03b1 i)\nh\u2081 : \u2200 (s : (i : \u03b9) \u2192 Set (\u03b1 i)), (\u2200 (i : \u03b9), s i \u2208 C i) \u2192 \u2191\u2191\u03bc\u03bd (Set.pi univ s) = \u220f i : \u03b9, \u2191\u2191(\u03bc i) (s i)\nh4C : \u2200 (i : \u03b9) (s : Set (\u03b1 i)), s \u2208 C i \u2192 MeasurableSet s\n\u22a2 Measure.pi \u03bc = \u03bc\u03bd"}, {"tactic": "refine'\n  (FiniteSpanningSetsIn.pi h3C).ext\n    (generateFrom_eq_pi hC fun i => (h3C i).isCountablySpanning).symm (IsPiSystem.pi h2C) _", "annotated_tactic": ["refine'\n    (<a>FiniteSpanningSetsIn.pi</a> h3C).<a>ext</a>\n      (<a>generateFrom_eq_pi</a> hC fun i => (h3C i).<a>isCountablySpanning</a>).<a>symm</a> (<a>IsPiSystem.pi</a> h2C) _", [{"full_name": "MeasureTheory.Measure.FiniteSpanningSetsIn.pi", "def_path": "Mathlib/MeasureTheory/Constructions/Pi.lean", "def_pos": [332, 5], "def_end_pos": [332, 28]}, {"full_name": "MeasureTheory.Measure.FiniteSpanningSetsIn.ext", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3652, 19], "def_end_pos": [3652, 22]}, {"full_name": "generateFrom_eq_pi", "def_path": "Mathlib/MeasureTheory/Constructions/Pi.lean", "def_pos": [133, 9], "def_end_pos": [133, 27]}, {"full_name": "MeasureTheory.Measure.FiniteSpanningSetsIn.isCountablySpanning", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3657, 19], "def_end_pos": [3657, 38]}, {"full_name": "Eq.symm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [310, 9], "def_end_pos": [310, 16]}, {"full_name": "IsPiSystem.pi", "def_path": "Mathlib/MeasureTheory/Constructions/Pi.lean", "def_pos": [72, 9], "def_end_pos": [72, 22]}]], "state_before": "\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : \u03b9 \u2192 Type u_3\ninst\u271d\u00b9 : Fintype \u03b9\nm : (i : \u03b9) \u2192 OuterMeasure (\u03b1 i)\ninst\u271d : (i : \u03b9) \u2192 MeasurableSpace (\u03b1 i)\n\u03bc : (i : \u03b9) \u2192 Measure (\u03b1 i)\nC : (i : \u03b9) \u2192 Set (Set (\u03b1 i))\nhC : \u2200 (i : \u03b9), generateFrom (C i) = inst\u271d i\nh2C : \u2200 (i : \u03b9), IsPiSystem (C i)\nh3C : (i : \u03b9) \u2192 FiniteSpanningSetsIn (\u03bc i) (C i)\n\u03bc\u03bd : Measure ((i : \u03b9) \u2192 \u03b1 i)\nh\u2081 : \u2200 (s : (i : \u03b9) \u2192 Set (\u03b1 i)), (\u2200 (i : \u03b9), s i \u2208 C i) \u2192 \u2191\u2191\u03bc\u03bd (Set.pi univ s) = \u220f i : \u03b9, \u2191\u2191(\u03bc i) (s i)\nh4C : \u2200 (i : \u03b9) (s : Set (\u03b1 i)), s \u2208 C i \u2192 MeasurableSet s\n\u22a2 Measure.pi \u03bc = \u03bc\u03bd", "state_after": "\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : \u03b9 \u2192 Type u_3\ninst\u271d\u00b9 : Fintype \u03b9\nm : (i : \u03b9) \u2192 OuterMeasure (\u03b1 i)\ninst\u271d : (i : \u03b9) \u2192 MeasurableSpace (\u03b1 i)\n\u03bc : (i : \u03b9) \u2192 Measure (\u03b1 i)\nC : (i : \u03b9) \u2192 Set (Set (\u03b1 i))\nhC : \u2200 (i : \u03b9), generateFrom (C i) = inst\u271d i\nh2C : \u2200 (i : \u03b9), IsPiSystem (C i)\nh3C : (i : \u03b9) \u2192 FiniteSpanningSetsIn (\u03bc i) (C i)\n\u03bc\u03bd : Measure ((i : \u03b9) \u2192 \u03b1 i)\nh\u2081 : \u2200 (s : (i : \u03b9) \u2192 Set (\u03b1 i)), (\u2200 (i : \u03b9), s i \u2208 C i) \u2192 \u2191\u2191\u03bc\u03bd (Set.pi univ s) = \u220f i : \u03b9, \u2191\u2191(\u03bc i) (s i)\nh4C : \u2200 (i : \u03b9) (s : Set (\u03b1 i)), s \u2208 C i \u2192 MeasurableSet s\n\u22a2 \u2200 (s : Set ((i : \u03b9) \u2192 \u03b1 i)), (s \u2208 Set.pi univ '' Set.pi univ fun i => C i) \u2192 \u2191\u2191(Measure.pi \u03bc) s = \u2191\u2191\u03bc\u03bd s"}, {"tactic": "rintro _ \u27e8s, hs, rfl\u27e9", "annotated_tactic": ["rintro _ \u27e8s, hs, rfl\u27e9", []], "state_before": "\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : \u03b9 \u2192 Type u_3\ninst\u271d\u00b9 : Fintype \u03b9\nm : (i : \u03b9) \u2192 OuterMeasure (\u03b1 i)\ninst\u271d : (i : \u03b9) \u2192 MeasurableSpace (\u03b1 i)\n\u03bc : (i : \u03b9) \u2192 Measure (\u03b1 i)\nC : (i : \u03b9) \u2192 Set (Set (\u03b1 i))\nhC : \u2200 (i : \u03b9), generateFrom (C i) = inst\u271d i\nh2C : \u2200 (i : \u03b9), IsPiSystem (C i)\nh3C : (i : \u03b9) \u2192 FiniteSpanningSetsIn (\u03bc i) (C i)\n\u03bc\u03bd : Measure ((i : \u03b9) \u2192 \u03b1 i)\nh\u2081 : \u2200 (s : (i : \u03b9) \u2192 Set (\u03b1 i)), (\u2200 (i : \u03b9), s i \u2208 C i) \u2192 \u2191\u2191\u03bc\u03bd (Set.pi univ s) = \u220f i : \u03b9, \u2191\u2191(\u03bc i) (s i)\nh4C : \u2200 (i : \u03b9) (s : Set (\u03b1 i)), s \u2208 C i \u2192 MeasurableSet s\n\u22a2 \u2200 (s : Set ((i : \u03b9) \u2192 \u03b1 i)), (s \u2208 Set.pi univ '' Set.pi univ fun i => C i) \u2192 \u2191\u2191(Measure.pi \u03bc) s = \u2191\u2191\u03bc\u03bd s", "state_after": "case intro.intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : \u03b9 \u2192 Type u_3\ninst\u271d\u00b9 : Fintype \u03b9\nm : (i : \u03b9) \u2192 OuterMeasure (\u03b1 i)\ninst\u271d : (i : \u03b9) \u2192 MeasurableSpace (\u03b1 i)\n\u03bc : (i : \u03b9) \u2192 Measure (\u03b1 i)\nC : (i : \u03b9) \u2192 Set (Set (\u03b1 i))\nhC : \u2200 (i : \u03b9), generateFrom (C i) = inst\u271d i\nh2C : \u2200 (i : \u03b9), IsPiSystem (C i)\nh3C : (i : \u03b9) \u2192 FiniteSpanningSetsIn (\u03bc i) (C i)\n\u03bc\u03bd : Measure ((i : \u03b9) \u2192 \u03b1 i)\nh\u2081 : \u2200 (s : (i : \u03b9) \u2192 Set (\u03b1 i)), (\u2200 (i : \u03b9), s i \u2208 C i) \u2192 \u2191\u2191\u03bc\u03bd (Set.pi univ s) = \u220f i : \u03b9, \u2191\u2191(\u03bc i) (s i)\nh4C : \u2200 (i : \u03b9) (s : Set (\u03b1 i)), s \u2208 C i \u2192 MeasurableSet s\ns : (i : \u03b9) \u2192 Set (\u03b1 i)\nhs : s \u2208 Set.pi univ fun i => C i\n\u22a2 \u2191\u2191(Measure.pi \u03bc) (Set.pi univ s) = \u2191\u2191\u03bc\u03bd (Set.pi univ s)"}, {"tactic": "rw [mem_univ_pi] at hs", "annotated_tactic": ["rw [<a>mem_univ_pi</a>] at hs", [{"full_name": "Set.mem_univ_pi", "def_path": "Mathlib/Data/Set/Prod.lean", "def_pos": [675, 9], "def_end_pos": [675, 20]}]], "state_before": "case intro.intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : \u03b9 \u2192 Type u_3\ninst\u271d\u00b9 : Fintype \u03b9\nm : (i : \u03b9) \u2192 OuterMeasure (\u03b1 i)\ninst\u271d : (i : \u03b9) \u2192 MeasurableSpace (\u03b1 i)\n\u03bc : (i : \u03b9) \u2192 Measure (\u03b1 i)\nC : (i : \u03b9) \u2192 Set (Set (\u03b1 i))\nhC : \u2200 (i : \u03b9), generateFrom (C i) = inst\u271d i\nh2C : \u2200 (i : \u03b9), IsPiSystem (C i)\nh3C : (i : \u03b9) \u2192 FiniteSpanningSetsIn (\u03bc i) (C i)\n\u03bc\u03bd : Measure ((i : \u03b9) \u2192 \u03b1 i)\nh\u2081 : \u2200 (s : (i : \u03b9) \u2192 Set (\u03b1 i)), (\u2200 (i : \u03b9), s i \u2208 C i) \u2192 \u2191\u2191\u03bc\u03bd (Set.pi univ s) = \u220f i : \u03b9, \u2191\u2191(\u03bc i) (s i)\nh4C : \u2200 (i : \u03b9) (s : Set (\u03b1 i)), s \u2208 C i \u2192 MeasurableSet s\ns : (i : \u03b9) \u2192 Set (\u03b1 i)\nhs : s \u2208 Set.pi univ fun i => C i\n\u22a2 \u2191\u2191(Measure.pi \u03bc) (Set.pi univ s) = \u2191\u2191\u03bc\u03bd (Set.pi univ s)", "state_after": "case intro.intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : \u03b9 \u2192 Type u_3\ninst\u271d\u00b9 : Fintype \u03b9\nm : (i : \u03b9) \u2192 OuterMeasure (\u03b1 i)\ninst\u271d : (i : \u03b9) \u2192 MeasurableSpace (\u03b1 i)\n\u03bc : (i : \u03b9) \u2192 Measure (\u03b1 i)\nC : (i : \u03b9) \u2192 Set (Set (\u03b1 i))\nhC : \u2200 (i : \u03b9), generateFrom (C i) = inst\u271d i\nh2C : \u2200 (i : \u03b9), IsPiSystem (C i)\nh3C : (i : \u03b9) \u2192 FiniteSpanningSetsIn (\u03bc i) (C i)\n\u03bc\u03bd : Measure ((i : \u03b9) \u2192 \u03b1 i)\nh\u2081 : \u2200 (s : (i : \u03b9) \u2192 Set (\u03b1 i)), (\u2200 (i : \u03b9), s i \u2208 C i) \u2192 \u2191\u2191\u03bc\u03bd (Set.pi univ s) = \u220f i : \u03b9, \u2191\u2191(\u03bc i) (s i)\nh4C : \u2200 (i : \u03b9) (s : Set (\u03b1 i)), s \u2208 C i \u2192 MeasurableSet s\ns : (i : \u03b9) \u2192 Set (\u03b1 i)\nhs : \u2200 (i : \u03b9), s i \u2208 C i\n\u22a2 \u2191\u2191(Measure.pi \u03bc) (Set.pi univ s) = \u2191\u2191\u03bc\u03bd (Set.pi univ s)"}, {"tactic": "haveI := fun i => (h3C i).sigmaFinite", "annotated_tactic": ["haveI := fun i => (h3C i).<a>sigmaFinite</a>", [{"full_name": "MeasureTheory.Measure.FiniteSpanningSetsIn.sigmaFinite", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3646, 19], "def_end_pos": [3646, 30]}]], "state_before": "case intro.intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : \u03b9 \u2192 Type u_3\ninst\u271d\u00b9 : Fintype \u03b9\nm : (i : \u03b9) \u2192 OuterMeasure (\u03b1 i)\ninst\u271d : (i : \u03b9) \u2192 MeasurableSpace (\u03b1 i)\n\u03bc : (i : \u03b9) \u2192 Measure (\u03b1 i)\nC : (i : \u03b9) \u2192 Set (Set (\u03b1 i))\nhC : \u2200 (i : \u03b9), generateFrom (C i) = inst\u271d i\nh2C : \u2200 (i : \u03b9), IsPiSystem (C i)\nh3C : (i : \u03b9) \u2192 FiniteSpanningSetsIn (\u03bc i) (C i)\n\u03bc\u03bd : Measure ((i : \u03b9) \u2192 \u03b1 i)\nh\u2081 : \u2200 (s : (i : \u03b9) \u2192 Set (\u03b1 i)), (\u2200 (i : \u03b9), s i \u2208 C i) \u2192 \u2191\u2191\u03bc\u03bd (Set.pi univ s) = \u220f i : \u03b9, \u2191\u2191(\u03bc i) (s i)\nh4C : \u2200 (i : \u03b9) (s : Set (\u03b1 i)), s \u2208 C i \u2192 MeasurableSet s\ns : (i : \u03b9) \u2192 Set (\u03b1 i)\nhs : \u2200 (i : \u03b9), s i \u2208 C i\n\u22a2 \u2191\u2191(Measure.pi \u03bc) (Set.pi univ s) = \u2191\u2191\u03bc\u03bd (Set.pi univ s)", "state_after": "case intro.intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : \u03b9 \u2192 Type u_3\ninst\u271d\u00b9 : Fintype \u03b9\nm : (i : \u03b9) \u2192 OuterMeasure (\u03b1 i)\ninst\u271d : (i : \u03b9) \u2192 MeasurableSpace (\u03b1 i)\n\u03bc : (i : \u03b9) \u2192 Measure (\u03b1 i)\nC : (i : \u03b9) \u2192 Set (Set (\u03b1 i))\nhC : \u2200 (i : \u03b9), generateFrom (C i) = inst\u271d i\nh2C : \u2200 (i : \u03b9), IsPiSystem (C i)\nh3C : (i : \u03b9) \u2192 FiniteSpanningSetsIn (\u03bc i) (C i)\n\u03bc\u03bd : Measure ((i : \u03b9) \u2192 \u03b1 i)\nh\u2081 : \u2200 (s : (i : \u03b9) \u2192 Set (\u03b1 i)), (\u2200 (i : \u03b9), s i \u2208 C i) \u2192 \u2191\u2191\u03bc\u03bd (Set.pi univ s) = \u220f i : \u03b9, \u2191\u2191(\u03bc i) (s i)\nh4C : \u2200 (i : \u03b9) (s : Set (\u03b1 i)), s \u2208 C i \u2192 MeasurableSet s\ns : (i : \u03b9) \u2192 Set (\u03b1 i)\nhs : \u2200 (i : \u03b9), s i \u2208 C i\nthis : \u2200 (i : \u03b9), SigmaFinite (\u03bc i)\n\u22a2 \u2191\u2191(Measure.pi \u03bc) (Set.pi univ s) = \u2191\u2191\u03bc\u03bd (Set.pi univ s)"}, {"tactic": "simp_rw [h\u2081 s hs, pi_pi_aux \u03bc s fun i => h4C i _ (hs i)]", "annotated_tactic": ["simp_rw [h\u2081 s hs, <a>pi_pi_aux</a> \u03bc s fun i => h4C i _ (hs i)]", [{"full_name": "MeasureTheory.Measure.pi_pi_aux", "def_path": "Mathlib/MeasureTheory/Constructions/Pi.lean", "def_pos": [314, 9], "def_end_pos": [314, 18]}]], "state_before": "case intro.intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : \u03b9 \u2192 Type u_3\ninst\u271d\u00b9 : Fintype \u03b9\nm : (i : \u03b9) \u2192 OuterMeasure (\u03b1 i)\ninst\u271d : (i : \u03b9) \u2192 MeasurableSpace (\u03b1 i)\n\u03bc : (i : \u03b9) \u2192 Measure (\u03b1 i)\nC : (i : \u03b9) \u2192 Set (Set (\u03b1 i))\nhC : \u2200 (i : \u03b9), generateFrom (C i) = inst\u271d i\nh2C : \u2200 (i : \u03b9), IsPiSystem (C i)\nh3C : (i : \u03b9) \u2192 FiniteSpanningSetsIn (\u03bc i) (C i)\n\u03bc\u03bd : Measure ((i : \u03b9) \u2192 \u03b1 i)\nh\u2081 : \u2200 (s : (i : \u03b9) \u2192 Set (\u03b1 i)), (\u2200 (i : \u03b9), s i \u2208 C i) \u2192 \u2191\u2191\u03bc\u03bd (Set.pi univ s) = \u220f i : \u03b9, \u2191\u2191(\u03bc i) (s i)\nh4C : \u2200 (i : \u03b9) (s : Set (\u03b1 i)), s \u2208 C i \u2192 MeasurableSet s\ns : (i : \u03b9) \u2192 Set (\u03b1 i)\nhs : \u2200 (i : \u03b9), s i \u2208 C i\nthis : \u2200 (i : \u03b9), SigmaFinite (\u03bc i)\n\u22a2 \u2191\u2191(Measure.pi \u03bc) (Set.pi univ s) = \u2191\u2191\u03bc\u03bd (Set.pi univ s)", "state_after": "no goals"}, {"tactic": "intro i s hs", "annotated_tactic": ["intro i s hs", []], "state_before": "\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : \u03b9 \u2192 Type u_3\ninst\u271d\u00b9 : Fintype \u03b9\nm : (i : \u03b9) \u2192 OuterMeasure (\u03b1 i)\ninst\u271d : (i : \u03b9) \u2192 MeasurableSpace (\u03b1 i)\n\u03bc : (i : \u03b9) \u2192 Measure (\u03b1 i)\nC : (i : \u03b9) \u2192 Set (Set (\u03b1 i))\nhC : \u2200 (i : \u03b9), generateFrom (C i) = inst\u271d i\nh2C : \u2200 (i : \u03b9), IsPiSystem (C i)\nh3C : (i : \u03b9) \u2192 FiniteSpanningSetsIn (\u03bc i) (C i)\n\u03bc\u03bd : Measure ((i : \u03b9) \u2192 \u03b1 i)\nh\u2081 : \u2200 (s : (i : \u03b9) \u2192 Set (\u03b1 i)), (\u2200 (i : \u03b9), s i \u2208 C i) \u2192 \u2191\u2191\u03bc\u03bd (Set.pi univ s) = \u220f i : \u03b9, \u2191\u2191(\u03bc i) (s i)\n\u22a2 \u2200 (i : \u03b9) (s : Set (\u03b1 i)), s \u2208 C i \u2192 MeasurableSet s", "state_after": "\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : \u03b9 \u2192 Type u_3\ninst\u271d\u00b9 : Fintype \u03b9\nm : (i : \u03b9) \u2192 OuterMeasure (\u03b1 i)\ninst\u271d : (i : \u03b9) \u2192 MeasurableSpace (\u03b1 i)\n\u03bc : (i : \u03b9) \u2192 Measure (\u03b1 i)\nC : (i : \u03b9) \u2192 Set (Set (\u03b1 i))\nhC : \u2200 (i : \u03b9), generateFrom (C i) = inst\u271d i\nh2C : \u2200 (i : \u03b9), IsPiSystem (C i)\nh3C : (i : \u03b9) \u2192 FiniteSpanningSetsIn (\u03bc i) (C i)\n\u03bc\u03bd : Measure ((i : \u03b9) \u2192 \u03b1 i)\nh\u2081 : \u2200 (s : (i : \u03b9) \u2192 Set (\u03b1 i)), (\u2200 (i : \u03b9), s i \u2208 C i) \u2192 \u2191\u2191\u03bc\u03bd (Set.pi univ s) = \u220f i : \u03b9, \u2191\u2191(\u03bc i) (s i)\ni : \u03b9\ns : Set (\u03b1 i)\nhs : s \u2208 C i\n\u22a2 MeasurableSet s"}, {"tactic": "exact measurableSet_generateFrom hs", "annotated_tactic": ["exact <a>measurableSet_generateFrom</a> hs", [{"full_name": "MeasurableSpace.measurableSet_generateFrom", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [370, 9], "def_end_pos": [370, 35]}]], "state_before": "\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : \u03b9 \u2192 Type u_3\ninst\u271d\u00b9 : Fintype \u03b9\nm : (i : \u03b9) \u2192 OuterMeasure (\u03b1 i)\ninst\u271d : (i : \u03b9) \u2192 MeasurableSpace (\u03b1 i)\n\u03bc : (i : \u03b9) \u2192 Measure (\u03b1 i)\nC : (i : \u03b9) \u2192 Set (Set (\u03b1 i))\nhC : \u2200 (i : \u03b9), generateFrom (C i) = inst\u271d i\nh2C : \u2200 (i : \u03b9), IsPiSystem (C i)\nh3C : (i : \u03b9) \u2192 FiniteSpanningSetsIn (\u03bc i) (C i)\n\u03bc\u03bd : Measure ((i : \u03b9) \u2192 \u03b1 i)\nh\u2081 : \u2200 (s : (i : \u03b9) \u2192 Set (\u03b1 i)), (\u2200 (i : \u03b9), s i \u2208 C i) \u2192 \u2191\u2191\u03bc\u03bd (Set.pi univ s) = \u220f i : \u03b9, \u2191\u2191(\u03bc i) (s i)\ni : \u03b9\ns : Set (\u03b1 i)\nhs : s \u2208 C i\n\u22a2 MeasurableSet s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Prod.lean", "full_name": "Finset.product_disjUnion", "start": [276, 1], "end": [278, 42], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/ProbabilityMassFunction/Constructions.lean", "full_name": "PMF.filter_apply_eq_zero_iff", "start": [291, 1], "end": [292, 73], "traced_tactics": [{"tactic": "erw [apply_eq_zero_iff, support_filter, Set.mem_inter_iff, not_and_or]", "annotated_tactic": ["erw [<a>apply_eq_zero_iff</a>, <a>support_filter</a>, <a>Set.mem_inter_iff</a>, <a>not_and_or</a>]", [{"full_name": "PMF.apply_eq_zero_iff", "def_path": "Mathlib/Probability/ProbabilityMassFunction/Basic.lean", "def_pos": [106, 9], "def_end_pos": [106, 26]}, {"full_name": "PMF.support_filter", "def_path": "Mathlib/Probability/ProbabilityMassFunction/Constructions.lean", "def_pos": [287, 9], "def_end_pos": [287, 23]}, {"full_name": "Set.mem_inter_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [909, 9], "def_end_pos": [909, 22]}, {"full_name": "not_and_or", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [473, 9], "def_end_pos": [473, 19]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\ns : Set \u03b1\nh : \u2203 a, a \u2208 s \u2227 a \u2208 support p\na : \u03b1\n\u22a2 \u2191(filter p s h) a = 0 \u2194 \u00aca \u2208 s \u2228 \u00aca \u2208 support p", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/WithDensity.lean", "full_name": "MeasureTheory.withDensity_sum", "start": [83, 1], "end": [86, 93], "traced_tactics": [{"tactic": "ext1 s hs", "annotated_tactic": ["ext1 s hs", []], "state_before": "\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\n\u03b9 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc : \u03b9 \u2192 Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u22a2 withDensity (sum \u03bc) f = sum fun n => withDensity (\u03bc n) f", "state_after": "case h\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\n\u03b9 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc : \u03b9 \u2192 Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(withDensity (sum \u03bc) f) s = \u2191\u2191(sum fun n => withDensity (\u03bc n) f) s"}, {"tactic": "simp_rw [sum_apply _ hs, withDensity_apply f hs, restrict_sum \u03bc hs, lintegral_sum_measure]", "annotated_tactic": ["simp_rw [<a>sum_apply</a> _ hs, <a>withDensity_apply</a> f hs, <a>restrict_sum</a> \u03bc hs, <a>lintegral_sum_measure</a>]", [{"full_name": "MeasureTheory.Measure.sum_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1989, 9], "def_end_pos": [1989, 18]}, {"full_name": "MeasureTheory.withDensity_apply", "def_path": "Mathlib/MeasureTheory/Measure/WithDensity.lean", "def_pos": [39, 9], "def_end_pos": [39, 26]}, {"full_name": "MeasureTheory.Measure.restrict_sum", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2060, 9], "def_end_pos": [2060, 21]}, {"full_name": "MeasureTheory.lintegral_sum_measure", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [596, 9], "def_end_pos": [596, 30]}]], "state_before": "case h\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\n\u03b9 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc : \u03b9 \u2192 Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(withDensity (sum \u03bc) f) s = \u2191\u2191(sum fun n => withDensity (\u03bc n) f) s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/BinomialHeap/Lemmas.lean", "full_name": "Std.BinomialHeap.Imp.Heap.deleteMin_fst", "start": [16, 1], "end": [19, 78], "traced_tactics": [{"tactic": "simp only [deleteMin, findMin_val, Option.map, head?]", "annotated_tactic": ["simp only [<a>deleteMin</a>, <a>findMin_val</a>, <a>Option.map</a>, <a>head?</a>]", [{"full_name": "Std.BinomialHeap.Imp.Heap.deleteMin", "def_path": "lake-packages/std/Std/Data/BinomialHeap/Basic.lean", "def_pos": [193, 5], "def_end_pos": [193, 19]}, {"full_name": "Std.BinomialHeap.Imp.Heap.findMin_val", "def_path": "lake-packages/std/Std/Data/BinomialHeap/Lemmas.lean", "def_pos": [11, 9], "def_end_pos": [11, 25]}, {"full_name": "Option.map", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2167, 25], "def_end_pos": [2167, 35]}, {"full_name": "Std.BinomialHeap.Imp.Heap.head?", "def_path": "lake-packages/std/Std/Data/BinomialHeap/Basic.lean", "def_pos": [161, 15], "def_end_pos": [161, 25]}]], "state_before": "\u03b1 : Type u_1\ns\u271d : Heap \u03b1\nle : \u03b1 \u2192 \u03b1 \u2192 Bool\nr : Nat\na : \u03b1\nc : HeapNode \u03b1\ns : Heap \u03b1\n\u22a2 Option.map (fun x => x.fst) (deleteMin le (cons r a c s)) = head? le (cons r a c s)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Martingale/Upcrossing.lean", "full_name": "MeasureTheory.upperCrossingTime_eq_upperCrossingTime_of_lt", "start": [535, 1], "end": [540, 65], "traced_tactics": [{"tactic": "cases n", "annotated_tactic": ["cases n", []], "state_before": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\nM : \u2115\nhNM : N \u2264 M\nh : upperCrossingTime a b f N n \u03c9 < N\n\u22a2 upperCrossingTime a b f M n \u03c9 = upperCrossingTime a b f N n \u03c9", "state_after": "case zero\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\nM : \u2115\nhNM : N \u2264 M\nh : upperCrossingTime a b f N Nat.zero \u03c9 < N\n\u22a2 upperCrossingTime a b f M Nat.zero \u03c9 = upperCrossingTime a b f N Nat.zero \u03c9\n\ncase succ\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\nM : \u2115\nhNM : N \u2264 M\nn\u271d : \u2115\nh : upperCrossingTime a b f N (Nat.succ n\u271d) \u03c9 < N\n\u22a2 upperCrossingTime a b f M (Nat.succ n\u271d) \u03c9 = upperCrossingTime a b f N (Nat.succ n\u271d) \u03c9"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case zero\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\nM : \u2115\nhNM : N \u2264 M\nh : upperCrossingTime a b f N Nat.zero \u03c9 < N\n\u22a2 upperCrossingTime a b f M Nat.zero \u03c9 = upperCrossingTime a b f N Nat.zero \u03c9", "state_after": "no goals"}, {"tactic": "exact (crossing_eq_crossing_of_upperCrossingTime_lt hNM h).1", "annotated_tactic": ["exact (<a>crossing_eq_crossing_of_upperCrossingTime_lt</a> hNM h).1", [{"full_name": "MeasureTheory.crossing_eq_crossing_of_upperCrossingTime_lt", "def_path": "Mathlib/Probability/Martingale/Upcrossing.lean", "def_pos": [521, 9], "def_end_pos": [521, 53]}]], "state_before": "case succ\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\nM : \u2115\nhNM : N \u2264 M\nn\u271d : \u2115\nh : upperCrossingTime a b f N (Nat.succ n\u271d) \u03c9 < N\n\u22a2 upperCrossingTime a b f M (Nat.succ n\u271d) \u03c9 = upperCrossingTime a b f N (Nat.succ n\u271d) \u03c9", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "full_name": "MeasureTheory.SimpleFunc.eapprox_comp", "start": [910, 1], "end": [912, 38], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/TuringMachine.lean", "full_name": "Turing.BlankRel.equivalence", "start": [165, 1], "end": [166, 59], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/TMToPartrec.lean", "full_name": "Turing.ToPartrec.Code.pred_eval", "start": [211, 1], "end": [212, 38], "traced_tactics": [{"tactic": "simp [pred]", "annotated_tactic": ["simp [<a>pred</a>]", [{"full_name": "Turing.ToPartrec.Code.pred", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [206, 5], "def_end_pos": [206, 9]}]], "state_before": "v : List \u2115\n\u22a2 eval pred v = pure [Nat.pred (List.headI v)]", "state_after": "v : List \u2115\n\u22a2 Nat.rec (Part.some [0]) (fun y x => Part.some [y]) (List.headI v) = Part.some [Nat.pred (List.headI v)]"}, {"tactic": "cases v.headI <;> simp", "annotated_tactic": ["cases v.headI <;> simp", []], "state_before": "v : List \u2115\n\u22a2 Nat.rec (Part.some [0]) (fun y x => Part.some [y]) (List.headI v) = Part.some [Nat.pred (List.headI v)]", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Fin/Lemmas.lean", "full_name": "Fin.pred_mk_succ'", "start": [503, 9], "end": [504, 81], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Pointwise/Finite.lean", "full_name": "Set.Infinite.of_smul_set", "start": [110, 1], "end": [111, 22], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/VitaliCaratheodory.lean", "full_name": "MeasureTheory.SimpleFunc.exists_upperSemicontinuous_le_lintegral_le", "start": [323, 1], "end": [380, 9], "traced_tactics": [{"tactic": "induction' f using MeasureTheory.SimpleFunc.induction with c s hs f\u2081 f\u2082 _ h\u2081 h\u2082 generalizing \u03b5", "annotated_tactic": ["induction' f using <a>MeasureTheory.SimpleFunc.induction</a> with c s hs f\u2081 f\u2082 _ h\u2081 h\u2082 generalizing \u03b5", [{"full_name": "MeasureTheory.SimpleFunc.induction", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [1266, 19], "def_end_pos": [1266, 28]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nint_f : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), g x \u2264 \u2191f x) \u2227 UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5", "state_after": "case h_ind\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nint_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\nint_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), g x \u2264 \u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2227\n      UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5\n\ncase h_add\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nint_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (Function.support \u2191f\u2081) (Function.support \u2191f\u2082)\nh\u2081 :\n  \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2260 \u22a4 \u2192\n    \u2200 {\u03b5 : \u211d\u22650\u221e},\n      \u03b5 \u2260 0 \u2192\n        \u2203 g, (\u2200 (x : \u03b1), g x \u2264 \u2191f\u2081 x) \u2227 UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5\nh\u2082 :\n  \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2260 \u22a4 \u2192\n    \u2200 {\u03b5 : \u211d\u22650\u221e},\n      \u03b5 \u2260 0 \u2192\n        \u2203 g, (\u2200 (x : \u03b1), g x \u2264 \u2191f\u2082 x) \u2227 UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5\nint_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(f\u2081 + f\u2082) x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), g x \u2264 \u2191(f\u2081 + f\u2082) x) \u2227 UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191(f\u2081 + f\u2082) x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5"}, {"tactic": "by_cases hc : c = 0", "annotated_tactic": ["by_cases hc : c = 0", []], "state_before": "case h_ind\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nint_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\nint_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), g x \u2264 \u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2227\n      UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5", "state_after": "case pos\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nint_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\nint_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nhc : c = 0\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), g x \u2264 \u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2227\n      UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5\n\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nint_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\nint_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nhc : \u00acc = 0\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), g x \u2264 \u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2227\n      UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5"}, {"tactic": "have \u03bcs_lt_top : \u03bc s < \u221e := by\n  classical\n  simpa only [hs, hc, lt_top_iff_ne_top, true_and_iff, SimpleFunc.coe_const, or_false_iff,\n    lintegral_const, ENNReal.coe_indicator, Set.univ_inter, ENNReal.coe_ne_top,\n    Measure.restrict_apply MeasurableSet.univ, ENNReal.mul_eq_top, SimpleFunc.const_zero,\n    Function.const_apply, lintegral_indicator, ENNReal.coe_eq_zero, Ne.def, not_false_iff,\n    SimpleFunc.coe_zero, Set.piecewise_eq_indicator, SimpleFunc.coe_piecewise,\n    false_and_iff] using int_f", "annotated_tactic": ["have \u03bcs_lt_top : \u03bc s < \u221e := by\n      classical\n      simpa only [hs, hc, <a>lt_top_iff_ne_top</a>, <a>true_and_iff</a>, <a>SimpleFunc.coe_const</a>, <a>or_false_iff</a>,\n        <a>lintegral_const</a>, <a>ENNReal.coe_indicator</a>, <a>Set.univ_inter</a>, <a>ENNReal.coe_ne_top</a>,\n        <a>Measure.restrict_apply</a> <a>MeasurableSet.univ</a>, <a>ENNReal.mul_eq_top</a>, <a>SimpleFunc.const_zero</a>,\n        <a>Function.const_apply</a>, <a>lintegral_indicator</a>, <a>ENNReal.coe_eq_zero</a>, <a>Ne.def</a>, <a>not_false_iff</a>,\n        <a>SimpleFunc.coe_zero</a>, <a>Set.piecewise_eq_indicator</a>, <a>SimpleFunc.coe_piecewise</a>,\n        <a>false_and_iff</a>] using int_f", [{"full_name": "lt_top_iff_ne_top", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [173, 9], "def_end_pos": [173, 26]}, {"full_name": "true_and_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [147, 9], "def_end_pos": [147, 21]}, {"full_name": "MeasureTheory.SimpleFunc.coe_const", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [158, 9], "def_end_pos": [158, 18]}, {"full_name": "or_false_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [188, 9], "def_end_pos": [188, 21]}, {"full_name": "MeasureTheory.lintegral_const", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [136, 9], "def_end_pos": [136, 24]}, {"full_name": "ENNReal.coe_indicator", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [548, 9], "def_end_pos": [548, 22]}, {"full_name": "Set.univ_inter", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1017, 9], "def_end_pos": [1017, 19]}, {"full_name": "ENNReal.coe_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [302, 17], "def_end_pos": [302, 27]}, {"full_name": "MeasureTheory.Measure.restrict_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1533, 9], "def_end_pos": [1533, 23]}, {"full_name": "MeasurableSet.univ", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [101, 19], "def_end_pos": [101, 37]}, {"full_name": "ENNReal.mul_eq_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [608, 9], "def_end_pos": [608, 19]}, {"full_name": "MeasureTheory.SimpleFunc.const_zero", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [457, 3], "def_end_pos": [457, 14]}, {"full_name": "Function.const_apply", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [31, 17], "def_end_pos": [31, 37]}, {"full_name": "MeasureTheory.lintegral_indicator", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [762, 9], "def_end_pos": [762, 28]}, {"full_name": "ENNReal.coe_eq_zero", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [368, 28], "def_end_pos": [368, 39]}, {"full_name": "Ne.def", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [59, 9], "def_end_pos": [59, 15]}, {"full_name": "not_false_iff", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [82, 9], "def_end_pos": [82, 22]}, {"full_name": "MeasureTheory.SimpleFunc.coe_zero", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [463, 3], "def_end_pos": [463, 14]}, {"full_name": "Set.piecewise_eq_indicator", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [52, 3], "def_end_pos": [52, 14]}, {"full_name": "MeasureTheory.SimpleFunc.coe_piecewise", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [230, 9], "def_end_pos": [230, 22]}, {"full_name": "false_and_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [151, 9], "def_end_pos": [151, 22]}]], "state_before": "case neg\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nint_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\nint_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nhc : \u00acc = 0\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), g x \u2264 \u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2227\n      UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5", "state_after": "case neg\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nint_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\nint_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nhc : \u00acc = 0\n\u03bcs_lt_top : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), g x \u2264 \u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2227\n      UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5"}, {"tactic": "have : (0 : \u211d\u22650\u221e) < \u03b5 / c := ENNReal.div_pos_iff.2 \u27e8\u03b50, ENNReal.coe_ne_top\u27e9", "annotated_tactic": ["have : (0 : \u211d\u22650\u221e) < \u03b5 / c := <a>ENNReal.div_pos_iff</a>.2 \u27e8\u03b50, <a>ENNReal.coe_ne_top</a>\u27e9", [{"full_name": "ENNReal.div_pos_iff", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1793, 17], "def_end_pos": [1793, 28]}, {"full_name": "ENNReal.coe_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [302, 17], "def_end_pos": [302, 27]}]], "state_before": "case neg\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nint_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\nint_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nhc : \u00acc = 0\n\u03bcs_lt_top : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), g x \u2264 \u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2227\n      UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5", "state_after": "case neg\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nint_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\nint_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nhc : \u00acc = 0\n\u03bcs_lt_top : \u2191\u2191\u03bc s < \u22a4\nthis : 0 < \u03b5 / \u2191c\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), g x \u2264 \u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2227\n      UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5"}, {"tactic": "obtain \u27e8F, Fs, F_closed, \u03bcF\u27e9 : \u2203 (F : _), F \u2286 s \u2227 IsClosed F \u2227 \u03bc s < \u03bc F + \u03b5 / c :=\n  hs.exists_isClosed_lt_add \u03bcs_lt_top.ne this.ne'", "annotated_tactic": ["obtain \u27e8F, Fs, F_closed, \u03bcF\u27e9 : \u2203 (F : _), F \u2286 s \u2227 <a>IsClosed</a> F \u2227 \u03bc s < \u03bc F + \u03b5 / c :=\n      hs.exists_isClosed_lt_add \u03bcs_lt_top.ne this.ne'", [{"full_name": "IsClosed", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [196, 7], "def_end_pos": [196, 15]}]], "state_before": "case neg\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nint_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\nint_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nhc : \u00acc = 0\n\u03bcs_lt_top : \u2191\u2191\u03bc s < \u22a4\nthis : 0 < \u03b5 / \u2191c\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), g x \u2264 \u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2227\n      UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5", "state_after": "case neg.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nint_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\nint_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nhc : \u00acc = 0\n\u03bcs_lt_top : \u2191\u2191\u03bc s < \u22a4\nthis : 0 < \u03b5 / \u2191c\nF : Set \u03b1\nFs : F \u2286 s\nF_closed : IsClosed F\n\u03bcF : \u2191\u2191\u03bc s < \u2191\u2191\u03bc F + \u03b5 / \u2191c\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), g x \u2264 \u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2227\n      UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5"}, {"tactic": "refine'\n  \u27e8Set.indicator F fun _ => c, fun x => _, F_closed.upperSemicontinuous_indicator (zero_le _),\n    _\u27e9", "annotated_tactic": ["refine'\n      \u27e8<a>Set.indicator</a> F fun _ => c, fun x => _, F_closed.upperSemicontinuous_indicator (<a>zero_le</a> _),\n        _\u27e9", [{"full_name": "Set.indicator", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [46, 3], "def_end_pos": [46, 14]}, {"full_name": "zero_le", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [217, 30], "def_end_pos": [217, 37]}]], "state_before": "case neg.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nint_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\nint_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nhc : \u00acc = 0\n\u03bcs_lt_top : \u2191\u2191\u03bc s < \u22a4\nthis : 0 < \u03b5 / \u2191c\nF : Set \u03b1\nFs : F \u2286 s\nF_closed : IsClosed F\n\u03bcF : \u2191\u2191\u03bc s < \u2191\u2191\u03bc F + \u03b5 / \u2191c\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), g x \u2264 \u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2227\n      UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5", "state_after": "case neg.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nint_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\nint_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nhc : \u00acc = 0\n\u03bcs_lt_top : \u2191\u2191\u03bc s < \u22a4\nthis : 0 < \u03b5 / \u2191c\nF : Set \u03b1\nFs : F \u2286 s\nF_closed : IsClosed F\n\u03bcF : \u2191\u2191\u03bc s < \u2191\u2191\u03bc F + \u03b5 / \u2191c\nx : \u03b1\n\u22a2 Set.indicator F (fun x => c) x \u2264 \u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x\n\ncase neg.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nint_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\nint_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nhc : \u00acc = 0\n\u03bcs_lt_top : \u2191\u2191\u03bc s < \u22a4\nthis : 0 < \u03b5 / \u2191c\nF : Set \u03b1\nFs : F \u2286 s\nF_closed : IsClosed F\n\u03bcF : \u2191\u2191\u03bc s < \u2191\u2191\u03bc F + \u03b5 / \u2191c\n\u22a2 \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(Set.indicator F (fun x => c) x) \u2202\u03bc + \u03b5"}, {"tactic": "refine' \u27e8fun _ => 0, _, upperSemicontinuous_const, _\u27e9", "annotated_tactic": ["refine' \u27e8fun _ => 0, _, <a>upperSemicontinuous_const</a>, _\u27e9", [{"full_name": "upperSemicontinuous_const", "def_path": "Mathlib/Topology/Semicontinuous.lean", "def_pos": [724, 9], "def_end_pos": [724, 34]}]], "state_before": "case pos\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nint_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\nint_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nhc : c = 0\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), g x \u2264 \u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2227\n      UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5", "state_after": "case pos.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nint_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\nint_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nhc : c = 0\n\u22a2 \u2200 (x : \u03b1), (fun x => 0) x \u2264 \u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x\n\ncase pos.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nint_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\nint_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nhc : c = 0\n\u22a2 \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191((fun x => 0) x) \u2202\u03bc + \u03b5"}, {"tactic": "classical\nsimp only [hc, Set.indicator_zero', Pi.zero_apply, SimpleFunc.const_zero, imp_true_iff,\n  eq_self_iff_true, SimpleFunc.coe_zero, Set.piecewise_eq_indicator,\n  SimpleFunc.coe_piecewise, le_zero_iff]", "annotated_tactic": ["classical\n        simp only [hc, <a>Set.indicator_zero'</a>, <a>Pi.zero_apply</a>, <a>SimpleFunc.const_zero</a>, <a>imp_true_iff</a>,\n          <a>eq_self_iff_true</a>, <a>SimpleFunc.coe_zero</a>, <a>Set.piecewise_eq_indicator</a>,\n          <a>SimpleFunc.coe_piecewise</a>, <a>le_zero_iff</a>]", [{"full_name": "Set.indicator_zero'", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [213, 3], "def_end_pos": [213, 14]}, {"full_name": "Pi.zero_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [46, 3], "def_end_pos": [46, 14]}, {"full_name": "MeasureTheory.SimpleFunc.const_zero", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [457, 3], "def_end_pos": [457, 14]}, {"full_name": "imp_true_iff", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [116, 9], "def_end_pos": [116, 21]}, {"full_name": "eq_self_iff_true", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [86, 9], "def_end_pos": [86, 25]}, {"full_name": "MeasureTheory.SimpleFunc.coe_zero", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [463, 3], "def_end_pos": [463, 14]}, {"full_name": "Set.piecewise_eq_indicator", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [52, 3], "def_end_pos": [52, 14]}, {"full_name": "MeasureTheory.SimpleFunc.coe_piecewise", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [230, 9], "def_end_pos": [230, 22]}, {"full_name": "le_zero_iff", "def_path": "Mathlib/Algebra/Order/WithZero.lean", "def_pos": [102, 9], "def_end_pos": [102, 20]}]], "state_before": "case pos.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nint_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\nint_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nhc : c = 0\n\u22a2 \u2200 (x : \u03b1), (fun x => 0) x \u2264 \u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x", "state_after": "no goals"}, {"tactic": "simp only [hc, Set.indicator_zero', Pi.zero_apply, SimpleFunc.const_zero, imp_true_iff,\n  eq_self_iff_true, SimpleFunc.coe_zero, Set.piecewise_eq_indicator,\n  SimpleFunc.coe_piecewise, le_zero_iff]", "annotated_tactic": ["simp only [hc, <a>Set.indicator_zero'</a>, <a>Pi.zero_apply</a>, <a>SimpleFunc.const_zero</a>, <a>imp_true_iff</a>,\n          <a>eq_self_iff_true</a>, <a>SimpleFunc.coe_zero</a>, <a>Set.piecewise_eq_indicator</a>,\n          <a>SimpleFunc.coe_piecewise</a>, <a>le_zero_iff</a>]", [{"full_name": "Set.indicator_zero'", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [213, 3], "def_end_pos": [213, 14]}, {"full_name": "Pi.zero_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [46, 3], "def_end_pos": [46, 14]}, {"full_name": "MeasureTheory.SimpleFunc.const_zero", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [457, 3], "def_end_pos": [457, 14]}, {"full_name": "imp_true_iff", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [116, 9], "def_end_pos": [116, 21]}, {"full_name": "eq_self_iff_true", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [86, 9], "def_end_pos": [86, 25]}, {"full_name": "MeasureTheory.SimpleFunc.coe_zero", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [463, 3], "def_end_pos": [463, 14]}, {"full_name": "Set.piecewise_eq_indicator", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [52, 3], "def_end_pos": [52, 14]}, {"full_name": "MeasureTheory.SimpleFunc.coe_piecewise", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [230, 9], "def_end_pos": [230, 22]}, {"full_name": "le_zero_iff", "def_path": "Mathlib/Algebra/Order/WithZero.lean", "def_pos": [102, 9], "def_end_pos": [102, 20]}]], "state_before": "case pos.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nint_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\nint_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nhc : c = 0\n\u22a2 \u2200 (x : \u03b1), (fun x => 0) x \u2264 \u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x", "state_after": "no goals"}, {"tactic": "classical\nsimp only [hc, Set.indicator_zero', lintegral_const, zero_mul, Pi.zero_apply,\n  SimpleFunc.const_zero, zero_add, zero_le', SimpleFunc.coe_zero,\n  Set.piecewise_eq_indicator, ENNReal.coe_zero, SimpleFunc.coe_piecewise, zero_le]", "annotated_tactic": ["classical\n        simp only [hc, <a>Set.indicator_zero'</a>, <a>lintegral_const</a>, <a>zero_mul</a>, <a>Pi.zero_apply</a>,\n          <a>SimpleFunc.const_zero</a>, <a>zero_add</a>, <a>zero_le'</a>, <a>SimpleFunc.coe_zero</a>,\n          <a>Set.piecewise_eq_indicator</a>, <a>ENNReal.coe_zero</a>, <a>SimpleFunc.coe_piecewise</a>, <a>zero_le</a>]", [{"full_name": "Set.indicator_zero'", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [213, 3], "def_end_pos": [213, 14]}, {"full_name": "MeasureTheory.lintegral_const", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [136, 9], "def_end_pos": [136, 24]}, {"full_name": "MulZeroClass.zero_mul", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [36, 3], "def_end_pos": [36, 11]}, {"full_name": "Pi.zero_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [46, 3], "def_end_pos": [46, 14]}, {"full_name": "MeasureTheory.SimpleFunc.const_zero", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [457, 3], "def_end_pos": [457, 14]}, {"full_name": "zero_add", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [463, 3], "def_end_pos": [463, 14]}, {"full_name": "zero_le'", "def_path": "Mathlib/Algebra/Order/WithZero.lean", "def_pos": [93, 9], "def_end_pos": [93, 17]}, {"full_name": "MeasureTheory.SimpleFunc.coe_zero", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [463, 3], "def_end_pos": [463, 14]}, {"full_name": "Set.piecewise_eq_indicator", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [52, 3], "def_end_pos": [52, 14]}, {"full_name": "ENNReal.coe_zero", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [215, 28], "def_end_pos": [215, 36]}, {"full_name": "MeasureTheory.SimpleFunc.coe_piecewise", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [230, 9], "def_end_pos": [230, 22]}, {"full_name": "zero_le", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [217, 30], "def_end_pos": [217, 37]}]], "state_before": "case pos.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nint_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\nint_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nhc : c = 0\n\u22a2 \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191((fun x => 0) x) \u2202\u03bc + \u03b5", "state_after": "no goals"}, {"tactic": "simp only [hc, Set.indicator_zero', lintegral_const, zero_mul, Pi.zero_apply,\n  SimpleFunc.const_zero, zero_add, zero_le', SimpleFunc.coe_zero,\n  Set.piecewise_eq_indicator, ENNReal.coe_zero, SimpleFunc.coe_piecewise, zero_le]", "annotated_tactic": ["simp only [hc, <a>Set.indicator_zero'</a>, <a>lintegral_const</a>, <a>zero_mul</a>, <a>Pi.zero_apply</a>,\n          <a>SimpleFunc.const_zero</a>, <a>zero_add</a>, <a>zero_le'</a>, <a>SimpleFunc.coe_zero</a>,\n          <a>Set.piecewise_eq_indicator</a>, <a>ENNReal.coe_zero</a>, <a>SimpleFunc.coe_piecewise</a>, <a>zero_le</a>]", [{"full_name": "Set.indicator_zero'", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [213, 3], "def_end_pos": [213, 14]}, {"full_name": "MeasureTheory.lintegral_const", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [136, 9], "def_end_pos": [136, 24]}, {"full_name": "MulZeroClass.zero_mul", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [36, 3], "def_end_pos": [36, 11]}, {"full_name": "Pi.zero_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [46, 3], "def_end_pos": [46, 14]}, {"full_name": "MeasureTheory.SimpleFunc.const_zero", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [457, 3], "def_end_pos": [457, 14]}, {"full_name": "zero_add", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [463, 3], "def_end_pos": [463, 14]}, {"full_name": "zero_le'", "def_path": "Mathlib/Algebra/Order/WithZero.lean", "def_pos": [93, 9], "def_end_pos": [93, 17]}, {"full_name": "MeasureTheory.SimpleFunc.coe_zero", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [463, 3], "def_end_pos": [463, 14]}, {"full_name": "Set.piecewise_eq_indicator", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [52, 3], "def_end_pos": [52, 14]}, {"full_name": "ENNReal.coe_zero", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [215, 28], "def_end_pos": [215, 36]}, {"full_name": "MeasureTheory.SimpleFunc.coe_piecewise", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [230, 9], "def_end_pos": [230, 22]}, {"full_name": "zero_le", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [217, 30], "def_end_pos": [217, 37]}]], "state_before": "case pos.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nint_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\nint_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nhc : c = 0\n\u22a2 \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191((fun x => 0) x) \u2202\u03bc + \u03b5", "state_after": "no goals"}, {"tactic": "classical\nsimpa only [hs, hc, lt_top_iff_ne_top, true_and_iff, SimpleFunc.coe_const, or_false_iff,\n  lintegral_const, ENNReal.coe_indicator, Set.univ_inter, ENNReal.coe_ne_top,\n  Measure.restrict_apply MeasurableSet.univ, ENNReal.mul_eq_top, SimpleFunc.const_zero,\n  Function.const_apply, lintegral_indicator, ENNReal.coe_eq_zero, Ne.def, not_false_iff,\n  SimpleFunc.coe_zero, Set.piecewise_eq_indicator, SimpleFunc.coe_piecewise,\n  false_and_iff] using int_f", "annotated_tactic": ["classical\n      simpa only [hs, hc, <a>lt_top_iff_ne_top</a>, <a>true_and_iff</a>, <a>SimpleFunc.coe_const</a>, <a>or_false_iff</a>,\n        <a>lintegral_const</a>, <a>ENNReal.coe_indicator</a>, <a>Set.univ_inter</a>, <a>ENNReal.coe_ne_top</a>,\n        <a>Measure.restrict_apply</a> <a>MeasurableSet.univ</a>, <a>ENNReal.mul_eq_top</a>, <a>SimpleFunc.const_zero</a>,\n        <a>Function.const_apply</a>, <a>lintegral_indicator</a>, <a>ENNReal.coe_eq_zero</a>, <a>Ne.def</a>, <a>not_false_iff</a>,\n        <a>SimpleFunc.coe_zero</a>, <a>Set.piecewise_eq_indicator</a>, <a>SimpleFunc.coe_piecewise</a>,\n        <a>false_and_iff</a>] using int_f", [{"full_name": "lt_top_iff_ne_top", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [173, 9], "def_end_pos": [173, 26]}, {"full_name": "true_and_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [147, 9], "def_end_pos": [147, 21]}, {"full_name": "MeasureTheory.SimpleFunc.coe_const", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [158, 9], "def_end_pos": [158, 18]}, {"full_name": "or_false_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [188, 9], "def_end_pos": [188, 21]}, {"full_name": "MeasureTheory.lintegral_const", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [136, 9], "def_end_pos": [136, 24]}, {"full_name": "ENNReal.coe_indicator", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [548, 9], "def_end_pos": [548, 22]}, {"full_name": "Set.univ_inter", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1017, 9], "def_end_pos": [1017, 19]}, {"full_name": "ENNReal.coe_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [302, 17], "def_end_pos": [302, 27]}, {"full_name": "MeasureTheory.Measure.restrict_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1533, 9], "def_end_pos": [1533, 23]}, {"full_name": "MeasurableSet.univ", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [101, 19], "def_end_pos": [101, 37]}, {"full_name": "ENNReal.mul_eq_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [608, 9], "def_end_pos": [608, 19]}, {"full_name": "MeasureTheory.SimpleFunc.const_zero", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [457, 3], "def_end_pos": [457, 14]}, {"full_name": "Function.const_apply", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [31, 17], "def_end_pos": [31, 37]}, {"full_name": "MeasureTheory.lintegral_indicator", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [762, 9], "def_end_pos": [762, 28]}, {"full_name": "ENNReal.coe_eq_zero", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [368, 28], "def_end_pos": [368, 39]}, {"full_name": "Ne.def", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [59, 9], "def_end_pos": [59, 15]}, {"full_name": "not_false_iff", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [82, 9], "def_end_pos": [82, 22]}, {"full_name": "MeasureTheory.SimpleFunc.coe_zero", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [463, 3], "def_end_pos": [463, 14]}, {"full_name": "Set.piecewise_eq_indicator", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [52, 3], "def_end_pos": [52, 14]}, {"full_name": "MeasureTheory.SimpleFunc.coe_piecewise", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [230, 9], "def_end_pos": [230, 22]}, {"full_name": "false_and_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [151, 9], "def_end_pos": [151, 22]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nint_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\nint_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nhc : \u00acc = 0\n\u22a2 \u2191\u2191\u03bc s < \u22a4", "state_after": "no goals"}, {"tactic": "simpa only [hs, hc, lt_top_iff_ne_top, true_and_iff, SimpleFunc.coe_const, or_false_iff,\n  lintegral_const, ENNReal.coe_indicator, Set.univ_inter, ENNReal.coe_ne_top,\n  Measure.restrict_apply MeasurableSet.univ, ENNReal.mul_eq_top, SimpleFunc.const_zero,\n  Function.const_apply, lintegral_indicator, ENNReal.coe_eq_zero, Ne.def, not_false_iff,\n  SimpleFunc.coe_zero, Set.piecewise_eq_indicator, SimpleFunc.coe_piecewise,\n  false_and_iff] using int_f", "annotated_tactic": ["simpa only [hs, hc, <a>lt_top_iff_ne_top</a>, <a>true_and_iff</a>, <a>SimpleFunc.coe_const</a>, <a>or_false_iff</a>,\n        <a>lintegral_const</a>, <a>ENNReal.coe_indicator</a>, <a>Set.univ_inter</a>, <a>ENNReal.coe_ne_top</a>,\n        <a>Measure.restrict_apply</a> <a>MeasurableSet.univ</a>, <a>ENNReal.mul_eq_top</a>, <a>SimpleFunc.const_zero</a>,\n        <a>Function.const_apply</a>, <a>lintegral_indicator</a>, <a>ENNReal.coe_eq_zero</a>, <a>Ne.def</a>, <a>not_false_iff</a>,\n        <a>SimpleFunc.coe_zero</a>, <a>Set.piecewise_eq_indicator</a>, <a>SimpleFunc.coe_piecewise</a>,\n        <a>false_and_iff</a>] using int_f", [{"full_name": "lt_top_iff_ne_top", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [173, 9], "def_end_pos": [173, 26]}, {"full_name": "true_and_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [147, 9], "def_end_pos": [147, 21]}, {"full_name": "MeasureTheory.SimpleFunc.coe_const", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [158, 9], "def_end_pos": [158, 18]}, {"full_name": "or_false_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [188, 9], "def_end_pos": [188, 21]}, {"full_name": "MeasureTheory.lintegral_const", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [136, 9], "def_end_pos": [136, 24]}, {"full_name": "ENNReal.coe_indicator", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [548, 9], "def_end_pos": [548, 22]}, {"full_name": "Set.univ_inter", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1017, 9], "def_end_pos": [1017, 19]}, {"full_name": "ENNReal.coe_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [302, 17], "def_end_pos": [302, 27]}, {"full_name": "MeasureTheory.Measure.restrict_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1533, 9], "def_end_pos": [1533, 23]}, {"full_name": "MeasurableSet.univ", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [101, 19], "def_end_pos": [101, 37]}, {"full_name": "ENNReal.mul_eq_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [608, 9], "def_end_pos": [608, 19]}, {"full_name": "MeasureTheory.SimpleFunc.const_zero", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [457, 3], "def_end_pos": [457, 14]}, {"full_name": "Function.const_apply", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [31, 17], "def_end_pos": [31, 37]}, {"full_name": "MeasureTheory.lintegral_indicator", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [762, 9], "def_end_pos": [762, 28]}, {"full_name": "ENNReal.coe_eq_zero", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [368, 28], "def_end_pos": [368, 39]}, {"full_name": "Ne.def", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [59, 9], "def_end_pos": [59, 15]}, {"full_name": "not_false_iff", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [82, 9], "def_end_pos": [82, 22]}, {"full_name": "MeasureTheory.SimpleFunc.coe_zero", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [463, 3], "def_end_pos": [463, 14]}, {"full_name": "Set.piecewise_eq_indicator", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [52, 3], "def_end_pos": [52, 14]}, {"full_name": "MeasureTheory.SimpleFunc.coe_piecewise", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [230, 9], "def_end_pos": [230, 22]}, {"full_name": "false_and_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [151, 9], "def_end_pos": [151, 22]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nint_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\nint_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nhc : \u00acc = 0\n\u22a2 \u2191\u2191\u03bc s < \u22a4", "state_after": "no goals"}, {"tactic": "simp only [SimpleFunc.coe_const, SimpleFunc.const_zero, SimpleFunc.coe_zero,\n  Set.piecewise_eq_indicator, SimpleFunc.coe_piecewise]", "annotated_tactic": ["simp only [<a>SimpleFunc.coe_const</a>, <a>SimpleFunc.const_zero</a>, <a>SimpleFunc.coe_zero</a>,\n        <a>Set.piecewise_eq_indicator</a>, <a>SimpleFunc.coe_piecewise</a>]", [{"full_name": "MeasureTheory.SimpleFunc.coe_const", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [158, 9], "def_end_pos": [158, 18]}, {"full_name": "MeasureTheory.SimpleFunc.const_zero", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [457, 3], "def_end_pos": [457, 14]}, {"full_name": "MeasureTheory.SimpleFunc.coe_zero", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [463, 3], "def_end_pos": [463, 14]}, {"full_name": "Set.piecewise_eq_indicator", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [52, 3], "def_end_pos": [52, 14]}, {"full_name": "MeasureTheory.SimpleFunc.coe_piecewise", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [230, 9], "def_end_pos": [230, 22]}]], "state_before": "case neg.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nint_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\nint_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nhc : \u00acc = 0\n\u03bcs_lt_top : \u2191\u2191\u03bc s < \u22a4\nthis : 0 < \u03b5 / \u2191c\nF : Set \u03b1\nFs : F \u2286 s\nF_closed : IsClosed F\n\u03bcF : \u2191\u2191\u03bc s < \u2191\u2191\u03bc F + \u03b5 / \u2191c\nx : \u03b1\n\u22a2 Set.indicator F (fun x => c) x \u2264 \u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x", "state_after": "case neg.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nint_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\nint_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nhc : \u00acc = 0\n\u03bcs_lt_top : \u2191\u2191\u03bc s < \u22a4\nthis : 0 < \u03b5 / \u2191c\nF : Set \u03b1\nFs : F \u2286 s\nF_closed : IsClosed F\n\u03bcF : \u2191\u2191\u03bc s < \u2191\u2191\u03bc F + \u03b5 / \u2191c\nx : \u03b1\n\u22a2 Set.indicator F (fun x => c) x \u2264 Set.piecewise s (Function.const \u03b1 c) 0 x"}, {"tactic": "exact Set.indicator_le_indicator_of_subset Fs (fun x => zero_le _) _", "annotated_tactic": ["exact <a>Set.indicator_le_indicator_of_subset</a> Fs (fun x => <a>zero_le</a> _) _", [{"full_name": "Set.indicator_le_indicator_of_subset", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [849, 3], "def_end_pos": [849, 14]}, {"full_name": "zero_le", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [217, 30], "def_end_pos": [217, 37]}]], "state_before": "case neg.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nint_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\nint_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nhc : \u00acc = 0\n\u03bcs_lt_top : \u2191\u2191\u03bc s < \u22a4\nthis : 0 < \u03b5 / \u2191c\nF : Set \u03b1\nFs : F \u2286 s\nF_closed : IsClosed F\n\u03bcF : \u2191\u2191\u03bc s < \u2191\u2191\u03bc F + \u03b5 / \u2191c\nx : \u03b1\n\u22a2 Set.indicator F (fun x => c) x \u2264 Set.piecewise s (Function.const \u03b1 c) 0 x", "state_after": "no goals"}, {"tactic": "suffices (c : \u211d\u22650\u221e) * \u03bc s \u2264 c * \u03bc F + \u03b5 by\n  classical\n  simpa only [hs, F_closed.measurableSet, SimpleFunc.coe_const, Function.const_apply,\n    lintegral_const, ENNReal.coe_indicator, Set.univ_inter, MeasurableSet.univ,\n    SimpleFunc.const_zero, lintegral_indicator, SimpleFunc.coe_zero,\n    Set.piecewise_eq_indicator, SimpleFunc.coe_piecewise, Measure.restrict_apply]", "annotated_tactic": ["suffices (c : \u211d\u22650\u221e) * \u03bc s \u2264 c * \u03bc F + \u03b5 by\n        classical\n        simpa only [hs, F_closed.measurableSet, <a>SimpleFunc.coe_const</a>, <a>Function.const_apply</a>,\n          <a>lintegral_const</a>, <a>ENNReal.coe_indicator</a>, <a>Set.univ_inter</a>, <a>MeasurableSet.univ</a>,\n          <a>SimpleFunc.const_zero</a>, <a>lintegral_indicator</a>, <a>SimpleFunc.coe_zero</a>,\n          <a>Set.piecewise_eq_indicator</a>, <a>SimpleFunc.coe_piecewise</a>, <a>Measure.restrict_apply</a>]", [{"full_name": "MeasureTheory.SimpleFunc.coe_const", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [158, 9], "def_end_pos": [158, 18]}, {"full_name": "Function.const_apply", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [31, 17], "def_end_pos": [31, 37]}, {"full_name": "MeasureTheory.lintegral_const", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [136, 9], "def_end_pos": [136, 24]}, {"full_name": "ENNReal.coe_indicator", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [548, 9], "def_end_pos": [548, 22]}, {"full_name": "Set.univ_inter", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1017, 9], "def_end_pos": [1017, 19]}, {"full_name": "MeasurableSet.univ", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [101, 19], "def_end_pos": [101, 37]}, {"full_name": "MeasureTheory.SimpleFunc.const_zero", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [457, 3], "def_end_pos": [457, 14]}, {"full_name": "MeasureTheory.lintegral_indicator", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [762, 9], "def_end_pos": [762, 28]}, {"full_name": "MeasureTheory.SimpleFunc.coe_zero", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [463, 3], "def_end_pos": [463, 14]}, {"full_name": "Set.piecewise_eq_indicator", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [52, 3], "def_end_pos": [52, 14]}, {"full_name": "MeasureTheory.SimpleFunc.coe_piecewise", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [230, 9], "def_end_pos": [230, 22]}, {"full_name": "MeasureTheory.Measure.restrict_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1533, 9], "def_end_pos": [1533, 23]}]], "state_before": "case neg.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nint_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\nint_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nhc : \u00acc = 0\n\u03bcs_lt_top : \u2191\u2191\u03bc s < \u22a4\nthis : 0 < \u03b5 / \u2191c\nF : Set \u03b1\nFs : F \u2286 s\nF_closed : IsClosed F\n\u03bcF : \u2191\u2191\u03bc s < \u2191\u2191\u03bc F + \u03b5 / \u2191c\n\u22a2 \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(Set.indicator F (fun x => c) x) \u2202\u03bc + \u03b5", "state_after": "case neg.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nint_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\nint_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nhc : \u00acc = 0\n\u03bcs_lt_top : \u2191\u2191\u03bc s < \u22a4\nthis : 0 < \u03b5 / \u2191c\nF : Set \u03b1\nFs : F \u2286 s\nF_closed : IsClosed F\n\u03bcF : \u2191\u2191\u03bc s < \u2191\u2191\u03bc F + \u03b5 / \u2191c\n\u22a2 \u2191c * \u2191\u2191\u03bc s \u2264 \u2191c * \u2191\u2191\u03bc F + \u03b5"}, {"tactic": "calc\n  (c : \u211d\u22650\u221e) * \u03bc s \u2264 c * (\u03bc F + \u03b5 / c) := mul_le_mul_left' \u03bcF.le _\n  _ = c * \u03bc F + \u03b5 := by\n    simp_rw [mul_add]\n    rw [ENNReal.mul_div_cancel' _ ENNReal.coe_ne_top]\n    simpa using hc", "annotated_tactic": ["calc\n        (c : \u211d\u22650\u221e) * \u03bc s \u2264 c * (\u03bc F + \u03b5 / c) := <a>mul_le_mul_left'</a> \u03bcF.le _\n        _ = c * \u03bc F + \u03b5 := by\n          simp_rw [<a>mul_add</a>]\n          rw [<a>ENNReal.mul_div_cancel'</a> _ <a>ENNReal.coe_ne_top</a>]\n          simpa using hc", [{"full_name": "mul_le_mul_left'", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [50, 9], "def_end_pos": [50, 25]}, {"full_name": "mul_add", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [83, 7], "def_end_pos": [83, 14]}, {"full_name": "ENNReal.mul_div_cancel'", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1432, 19], "def_end_pos": [1432, 34]}, {"full_name": "ENNReal.coe_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [302, 17], "def_end_pos": [302, 27]}]], "state_before": "case neg.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nint_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\nint_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nhc : \u00acc = 0\n\u03bcs_lt_top : \u2191\u2191\u03bc s < \u22a4\nthis : 0 < \u03b5 / \u2191c\nF : Set \u03b1\nFs : F \u2286 s\nF_closed : IsClosed F\n\u03bcF : \u2191\u2191\u03bc s < \u2191\u2191\u03bc F + \u03b5 / \u2191c\n\u22a2 \u2191c * \u2191\u2191\u03bc s \u2264 \u2191c * \u2191\u2191\u03bc F + \u03b5", "state_after": "no goals"}, {"tactic": "classical\nsimpa only [hs, F_closed.measurableSet, SimpleFunc.coe_const, Function.const_apply,\n  lintegral_const, ENNReal.coe_indicator, Set.univ_inter, MeasurableSet.univ,\n  SimpleFunc.const_zero, lintegral_indicator, SimpleFunc.coe_zero,\n  Set.piecewise_eq_indicator, SimpleFunc.coe_piecewise, Measure.restrict_apply]", "annotated_tactic": ["classical\n        simpa only [hs, F_closed.measurableSet, <a>SimpleFunc.coe_const</a>, <a>Function.const_apply</a>,\n          <a>lintegral_const</a>, <a>ENNReal.coe_indicator</a>, <a>Set.univ_inter</a>, <a>MeasurableSet.univ</a>,\n          <a>SimpleFunc.const_zero</a>, <a>lintegral_indicator</a>, <a>SimpleFunc.coe_zero</a>,\n          <a>Set.piecewise_eq_indicator</a>, <a>SimpleFunc.coe_piecewise</a>, <a>Measure.restrict_apply</a>]", [{"full_name": "MeasureTheory.SimpleFunc.coe_const", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [158, 9], "def_end_pos": [158, 18]}, {"full_name": "Function.const_apply", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [31, 17], "def_end_pos": [31, 37]}, {"full_name": "MeasureTheory.lintegral_const", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [136, 9], "def_end_pos": [136, 24]}, {"full_name": "ENNReal.coe_indicator", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [548, 9], "def_end_pos": [548, 22]}, {"full_name": "Set.univ_inter", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1017, 9], "def_end_pos": [1017, 19]}, {"full_name": "MeasurableSet.univ", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [101, 19], "def_end_pos": [101, 37]}, {"full_name": "MeasureTheory.SimpleFunc.const_zero", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [457, 3], "def_end_pos": [457, 14]}, {"full_name": "MeasureTheory.lintegral_indicator", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [762, 9], "def_end_pos": [762, 28]}, {"full_name": "MeasureTheory.SimpleFunc.coe_zero", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [463, 3], "def_end_pos": [463, 14]}, {"full_name": "Set.piecewise_eq_indicator", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [52, 3], "def_end_pos": [52, 14]}, {"full_name": "MeasureTheory.SimpleFunc.coe_piecewise", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [230, 9], "def_end_pos": [230, 22]}, {"full_name": "MeasureTheory.Measure.restrict_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1533, 9], "def_end_pos": [1533, 23]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nint_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\nint_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nhc : \u00acc = 0\n\u03bcs_lt_top : \u2191\u2191\u03bc s < \u22a4\nthis\u271d : 0 < \u03b5 / \u2191c\nF : Set \u03b1\nFs : F \u2286 s\nF_closed : IsClosed F\n\u03bcF : \u2191\u2191\u03bc s < \u2191\u2191\u03bc F + \u03b5 / \u2191c\nthis : \u2191c * \u2191\u2191\u03bc s \u2264 \u2191c * \u2191\u2191\u03bc F + \u03b5\n\u22a2 \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(Set.indicator F (fun x => c) x) \u2202\u03bc + \u03b5", "state_after": "no goals"}, {"tactic": "simpa only [hs, F_closed.measurableSet, SimpleFunc.coe_const, Function.const_apply,\n  lintegral_const, ENNReal.coe_indicator, Set.univ_inter, MeasurableSet.univ,\n  SimpleFunc.const_zero, lintegral_indicator, SimpleFunc.coe_zero,\n  Set.piecewise_eq_indicator, SimpleFunc.coe_piecewise, Measure.restrict_apply]", "annotated_tactic": ["simpa only [hs, F_closed.measurableSet, <a>SimpleFunc.coe_const</a>, <a>Function.const_apply</a>,\n          <a>lintegral_const</a>, <a>ENNReal.coe_indicator</a>, <a>Set.univ_inter</a>, <a>MeasurableSet.univ</a>,\n          <a>SimpleFunc.const_zero</a>, <a>lintegral_indicator</a>, <a>SimpleFunc.coe_zero</a>,\n          <a>Set.piecewise_eq_indicator</a>, <a>SimpleFunc.coe_piecewise</a>, <a>Measure.restrict_apply</a>]", [{"full_name": "MeasureTheory.SimpleFunc.coe_const", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [158, 9], "def_end_pos": [158, 18]}, {"full_name": "Function.const_apply", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [31, 17], "def_end_pos": [31, 37]}, {"full_name": "MeasureTheory.lintegral_const", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [136, 9], "def_end_pos": [136, 24]}, {"full_name": "ENNReal.coe_indicator", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [548, 9], "def_end_pos": [548, 22]}, {"full_name": "Set.univ_inter", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1017, 9], "def_end_pos": [1017, 19]}, {"full_name": "MeasurableSet.univ", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [101, 19], "def_end_pos": [101, 37]}, {"full_name": "MeasureTheory.SimpleFunc.const_zero", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [457, 3], "def_end_pos": [457, 14]}, {"full_name": "MeasureTheory.lintegral_indicator", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [762, 9], "def_end_pos": [762, 28]}, {"full_name": "MeasureTheory.SimpleFunc.coe_zero", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [463, 3], "def_end_pos": [463, 14]}, {"full_name": "Set.piecewise_eq_indicator", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [52, 3], "def_end_pos": [52, 14]}, {"full_name": "MeasureTheory.SimpleFunc.coe_piecewise", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [230, 9], "def_end_pos": [230, 22]}, {"full_name": "MeasureTheory.Measure.restrict_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1533, 9], "def_end_pos": [1533, 23]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nint_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\nint_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nhc : \u00acc = 0\n\u03bcs_lt_top : \u2191\u2191\u03bc s < \u22a4\nthis\u271d : 0 < \u03b5 / \u2191c\nF : Set \u03b1\nFs : F \u2286 s\nF_closed : IsClosed F\n\u03bcF : \u2191\u2191\u03bc s < \u2191\u2191\u03bc F + \u03b5 / \u2191c\nthis : \u2191c * \u2191\u2191\u03bc s \u2264 \u2191c * \u2191\u2191\u03bc F + \u03b5\n\u22a2 \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(Set.indicator F (fun x => c) x) \u2202\u03bc + \u03b5", "state_after": "no goals"}, {"tactic": "simp_rw [mul_add]", "annotated_tactic": ["simp_rw [<a>mul_add</a>]", [{"full_name": "mul_add", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [83, 7], "def_end_pos": [83, 14]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nint_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\nint_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nhc : \u00acc = 0\n\u03bcs_lt_top : \u2191\u2191\u03bc s < \u22a4\nthis : 0 < \u03b5 / \u2191c\nF : Set \u03b1\nFs : F \u2286 s\nF_closed : IsClosed F\n\u03bcF : \u2191\u2191\u03bc s < \u2191\u2191\u03bc F + \u03b5 / \u2191c\n\u22a2 \u2191c * (\u2191\u2191\u03bc F + \u03b5 / \u2191c) = \u2191c * \u2191\u2191\u03bc F + \u03b5", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nint_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\nint_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nhc : \u00acc = 0\n\u03bcs_lt_top : \u2191\u2191\u03bc s < \u22a4\nthis : 0 < \u03b5 / \u2191c\nF : Set \u03b1\nFs : F \u2286 s\nF_closed : IsClosed F\n\u03bcF : \u2191\u2191\u03bc s < \u2191\u2191\u03bc F + \u03b5 / \u2191c\n\u22a2 \u2191c * \u2191\u2191\u03bc F + \u2191c * (\u03b5 / \u2191c) = \u2191c * \u2191\u2191\u03bc F + \u03b5"}, {"tactic": "rw [ENNReal.mul_div_cancel' _ ENNReal.coe_ne_top]", "annotated_tactic": ["rw [<a>ENNReal.mul_div_cancel'</a> _ <a>ENNReal.coe_ne_top</a>]", [{"full_name": "ENNReal.mul_div_cancel'", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1432, 19], "def_end_pos": [1432, 34]}, {"full_name": "ENNReal.coe_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [302, 17], "def_end_pos": [302, 27]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nint_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\nint_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nhc : \u00acc = 0\n\u03bcs_lt_top : \u2191\u2191\u03bc s < \u22a4\nthis : 0 < \u03b5 / \u2191c\nF : Set \u03b1\nFs : F \u2286 s\nF_closed : IsClosed F\n\u03bcF : \u2191\u2191\u03bc s < \u2191\u2191\u03bc F + \u03b5 / \u2191c\n\u22a2 \u2191c * \u2191\u2191\u03bc F + \u2191c * (\u03b5 / \u2191c) = \u2191c * \u2191\u2191\u03bc F + \u03b5", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nint_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\nint_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nhc : \u00acc = 0\n\u03bcs_lt_top : \u2191\u2191\u03bc s < \u22a4\nthis : 0 < \u03b5 / \u2191c\nF : Set \u03b1\nFs : F \u2286 s\nF_closed : IsClosed F\n\u03bcF : \u2191\u2191\u03bc s < \u2191\u2191\u03bc F + \u03b5 / \u2191c\n\u22a2 \u2191c \u2260 0"}, {"tactic": "simpa using hc", "annotated_tactic": ["simpa using hc", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nint_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\nint_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nhc : \u00acc = 0\n\u03bcs_lt_top : \u2191\u2191\u03bc s < \u22a4\nthis : 0 < \u03b5 / \u2191c\nF : Set \u03b1\nFs : F \u2286 s\nF_closed : IsClosed F\n\u03bcF : \u2191\u2191\u03bc s < \u2191\u2191\u03bc F + \u03b5 / \u2191c\n\u22a2 \u2191c \u2260 0", "state_after": "no goals"}, {"tactic": "have A : ((\u222b\u207b x : \u03b1, f\u2081 x \u2202\u03bc) + \u222b\u207b x : \u03b1, f\u2082 x \u2202\u03bc) \u2260 \u22a4 := by\n  rwa [\u2190 lintegral_add_left f\u2081.measurable.coe_nnreal_ennreal]", "annotated_tactic": ["have A : ((\u222b\u207b x : \u03b1, f\u2081 x \u2202\u03bc) + \u222b\u207b x : \u03b1, f\u2082 x \u2202\u03bc) \u2260 \u22a4 := by\n      rwa [\u2190 <a>lintegral_add_left</a> f\u2081.measurable.coe_nnreal_ennreal]", [{"full_name": "MeasureTheory.lintegral_add_left", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [554, 9], "def_end_pos": [554, 27]}]], "state_before": "case h_add\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nint_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (Function.support \u2191f\u2081) (Function.support \u2191f\u2082)\nh\u2081 :\n  \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2260 \u22a4 \u2192\n    \u2200 {\u03b5 : \u211d\u22650\u221e},\n      \u03b5 \u2260 0 \u2192\n        \u2203 g, (\u2200 (x : \u03b1), g x \u2264 \u2191f\u2081 x) \u2227 UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5\nh\u2082 :\n  \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2260 \u22a4 \u2192\n    \u2200 {\u03b5 : \u211d\u22650\u221e},\n      \u03b5 \u2260 0 \u2192\n        \u2203 g, (\u2200 (x : \u03b1), g x \u2264 \u2191f\u2082 x) \u2227 UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5\nint_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(f\u2081 + f\u2082) x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), g x \u2264 \u2191(f\u2081 + f\u2082) x) \u2227 UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191(f\u2081 + f\u2082) x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5", "state_after": "case h_add\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nint_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (Function.support \u2191f\u2081) (Function.support \u2191f\u2082)\nh\u2081 :\n  \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2260 \u22a4 \u2192\n    \u2200 {\u03b5 : \u211d\u22650\u221e},\n      \u03b5 \u2260 0 \u2192\n        \u2203 g, (\u2200 (x : \u03b1), g x \u2264 \u2191f\u2081 x) \u2227 UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5\nh\u2082 :\n  \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2260 \u22a4 \u2192\n    \u2200 {\u03b5 : \u211d\u22650\u221e},\n      \u03b5 \u2260 0 \u2192\n        \u2203 g, (\u2200 (x : \u03b1), g x \u2264 \u2191f\u2082 x) \u2227 UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5\nint_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(f\u2081 + f\u2082) x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nA : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2260 \u22a4\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), g x \u2264 \u2191(f\u2081 + f\u2082) x) \u2227 UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191(f\u2081 + f\u2082) x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5"}, {"tactic": "rcases h\u2081 (ENNReal.add_ne_top.1 A).1 (ENNReal.half_pos \u03b50).ne' with\n  \u27e8g\u2081, f\u2081_le_g\u2081, g\u2081cont, g\u2081int\u27e9", "annotated_tactic": ["rcases h\u2081 (<a>ENNReal.add_ne_top</a>.1 A).1 (<a>ENNReal.half_pos</a> \u03b50).<a>ne'</a> with\n      \u27e8g\u2081, f\u2081_le_g\u2081, g\u2081cont, g\u2081int\u27e9", [{"full_name": "ENNReal.add_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [574, 9], "def_end_pos": [574, 19]}, {"full_name": "ENNReal.half_pos", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1796, 19], "def_end_pos": [1796, 27]}, {"full_name": "LT.lt.ne'", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [328, 9], "def_end_pos": [328, 12]}]], "state_before": "case h_add\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nint_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (Function.support \u2191f\u2081) (Function.support \u2191f\u2082)\nh\u2081 :\n  \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2260 \u22a4 \u2192\n    \u2200 {\u03b5 : \u211d\u22650\u221e},\n      \u03b5 \u2260 0 \u2192\n        \u2203 g, (\u2200 (x : \u03b1), g x \u2264 \u2191f\u2081 x) \u2227 UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5\nh\u2082 :\n  \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2260 \u22a4 \u2192\n    \u2200 {\u03b5 : \u211d\u22650\u221e},\n      \u03b5 \u2260 0 \u2192\n        \u2203 g, (\u2200 (x : \u03b1), g x \u2264 \u2191f\u2082 x) \u2227 UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5\nint_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(f\u2081 + f\u2082) x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nA : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2260 \u22a4\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), g x \u2264 \u2191(f\u2081 + f\u2082) x) \u2227 UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191(f\u2081 + f\u2082) x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5", "state_after": "case h_add.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nint_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (Function.support \u2191f\u2081) (Function.support \u2191f\u2082)\nh\u2081 :\n  \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2260 \u22a4 \u2192\n    \u2200 {\u03b5 : \u211d\u22650\u221e},\n      \u03b5 \u2260 0 \u2192\n        \u2203 g, (\u2200 (x : \u03b1), g x \u2264 \u2191f\u2081 x) \u2227 UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5\nh\u2082 :\n  \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2260 \u22a4 \u2192\n    \u2200 {\u03b5 : \u211d\u22650\u221e},\n      \u03b5 \u2260 0 \u2192\n        \u2203 g, (\u2200 (x : \u03b1), g x \u2264 \u2191f\u2082 x) \u2227 UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5\nint_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(f\u2081 + f\u2082) x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nA : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2260 \u22a4\ng\u2081 : \u03b1 \u2192 \u211d\u22650\nf\u2081_le_g\u2081 : \u2200 (x : \u03b1), g\u2081 x \u2264 \u2191f\u2081 x\ng\u2081cont : UpperSemicontinuous g\u2081\ng\u2081int : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g\u2081 x) \u2202\u03bc + \u03b5 / 2\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), g x \u2264 \u2191(f\u2081 + f\u2082) x) \u2227 UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191(f\u2081 + f\u2082) x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5"}, {"tactic": "rcases h\u2082 (ENNReal.add_ne_top.1 A).2 (ENNReal.half_pos \u03b50).ne' with\n  \u27e8g\u2082, f\u2082_le_g\u2082, g\u2082cont, g\u2082int\u27e9", "annotated_tactic": ["rcases h\u2082 (<a>ENNReal.add_ne_top</a>.1 A).2 (<a>ENNReal.half_pos</a> \u03b50).<a>ne'</a> with\n      \u27e8g\u2082, f\u2082_le_g\u2082, g\u2082cont, g\u2082int\u27e9", [{"full_name": "ENNReal.add_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [574, 9], "def_end_pos": [574, 19]}, {"full_name": "ENNReal.half_pos", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1796, 19], "def_end_pos": [1796, 27]}, {"full_name": "LT.lt.ne'", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [328, 9], "def_end_pos": [328, 12]}]], "state_before": "case h_add.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nint_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (Function.support \u2191f\u2081) (Function.support \u2191f\u2082)\nh\u2081 :\n  \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2260 \u22a4 \u2192\n    \u2200 {\u03b5 : \u211d\u22650\u221e},\n      \u03b5 \u2260 0 \u2192\n        \u2203 g, (\u2200 (x : \u03b1), g x \u2264 \u2191f\u2081 x) \u2227 UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5\nh\u2082 :\n  \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2260 \u22a4 \u2192\n    \u2200 {\u03b5 : \u211d\u22650\u221e},\n      \u03b5 \u2260 0 \u2192\n        \u2203 g, (\u2200 (x : \u03b1), g x \u2264 \u2191f\u2082 x) \u2227 UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5\nint_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(f\u2081 + f\u2082) x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nA : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2260 \u22a4\ng\u2081 : \u03b1 \u2192 \u211d\u22650\nf\u2081_le_g\u2081 : \u2200 (x : \u03b1), g\u2081 x \u2264 \u2191f\u2081 x\ng\u2081cont : UpperSemicontinuous g\u2081\ng\u2081int : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g\u2081 x) \u2202\u03bc + \u03b5 / 2\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), g x \u2264 \u2191(f\u2081 + f\u2082) x) \u2227 UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191(f\u2081 + f\u2082) x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5", "state_after": "case h_add.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nint_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (Function.support \u2191f\u2081) (Function.support \u2191f\u2082)\nh\u2081 :\n  \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2260 \u22a4 \u2192\n    \u2200 {\u03b5 : \u211d\u22650\u221e},\n      \u03b5 \u2260 0 \u2192\n        \u2203 g, (\u2200 (x : \u03b1), g x \u2264 \u2191f\u2081 x) \u2227 UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5\nh\u2082 :\n  \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2260 \u22a4 \u2192\n    \u2200 {\u03b5 : \u211d\u22650\u221e},\n      \u03b5 \u2260 0 \u2192\n        \u2203 g, (\u2200 (x : \u03b1), g x \u2264 \u2191f\u2082 x) \u2227 UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5\nint_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(f\u2081 + f\u2082) x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nA : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2260 \u22a4\ng\u2081 : \u03b1 \u2192 \u211d\u22650\nf\u2081_le_g\u2081 : \u2200 (x : \u03b1), g\u2081 x \u2264 \u2191f\u2081 x\ng\u2081cont : UpperSemicontinuous g\u2081\ng\u2081int : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g\u2081 x) \u2202\u03bc + \u03b5 / 2\ng\u2082 : \u03b1 \u2192 \u211d\u22650\nf\u2082_le_g\u2082 : \u2200 (x : \u03b1), g\u2082 x \u2264 \u2191f\u2082 x\ng\u2082cont : UpperSemicontinuous g\u2082\ng\u2082int : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g\u2082 x) \u2202\u03bc + \u03b5 / 2\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), g x \u2264 \u2191(f\u2081 + f\u2082) x) \u2227 UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191(f\u2081 + f\u2082) x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5"}, {"tactic": "refine'\n  \u27e8fun x => g\u2081 x + g\u2082 x, fun x => add_le_add (f\u2081_le_g\u2081 x) (f\u2082_le_g\u2082 x), g\u2081cont.add g\u2082cont, _\u27e9", "annotated_tactic": ["refine'\n      \u27e8fun x => g\u2081 x + g\u2082 x, fun x => <a>add_le_add</a> (f\u2081_le_g\u2081 x) (f\u2082_le_g\u2082 x), g\u2081cont.add g\u2082cont, _\u27e9", [{"full_name": "add_le_add", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [205, 15], "def_end_pos": [205, 25]}]], "state_before": "case h_add.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nint_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (Function.support \u2191f\u2081) (Function.support \u2191f\u2082)\nh\u2081 :\n  \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2260 \u22a4 \u2192\n    \u2200 {\u03b5 : \u211d\u22650\u221e},\n      \u03b5 \u2260 0 \u2192\n        \u2203 g, (\u2200 (x : \u03b1), g x \u2264 \u2191f\u2081 x) \u2227 UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5\nh\u2082 :\n  \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2260 \u22a4 \u2192\n    \u2200 {\u03b5 : \u211d\u22650\u221e},\n      \u03b5 \u2260 0 \u2192\n        \u2203 g, (\u2200 (x : \u03b1), g x \u2264 \u2191f\u2082 x) \u2227 UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5\nint_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(f\u2081 + f\u2082) x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nA : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2260 \u22a4\ng\u2081 : \u03b1 \u2192 \u211d\u22650\nf\u2081_le_g\u2081 : \u2200 (x : \u03b1), g\u2081 x \u2264 \u2191f\u2081 x\ng\u2081cont : UpperSemicontinuous g\u2081\ng\u2081int : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g\u2081 x) \u2202\u03bc + \u03b5 / 2\ng\u2082 : \u03b1 \u2192 \u211d\u22650\nf\u2082_le_g\u2082 : \u2200 (x : \u03b1), g\u2082 x \u2264 \u2191f\u2082 x\ng\u2082cont : UpperSemicontinuous g\u2082\ng\u2082int : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g\u2082 x) \u2202\u03bc + \u03b5 / 2\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), g x \u2264 \u2191(f\u2081 + f\u2082) x) \u2227 UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191(f\u2081 + f\u2082) x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5", "state_after": "case h_add.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nint_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (Function.support \u2191f\u2081) (Function.support \u2191f\u2082)\nh\u2081 :\n  \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2260 \u22a4 \u2192\n    \u2200 {\u03b5 : \u211d\u22650\u221e},\n      \u03b5 \u2260 0 \u2192\n        \u2203 g, (\u2200 (x : \u03b1), g x \u2264 \u2191f\u2081 x) \u2227 UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5\nh\u2082 :\n  \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2260 \u22a4 \u2192\n    \u2200 {\u03b5 : \u211d\u22650\u221e},\n      \u03b5 \u2260 0 \u2192\n        \u2203 g, (\u2200 (x : \u03b1), g x \u2264 \u2191f\u2082 x) \u2227 UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5\nint_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(f\u2081 + f\u2082) x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nA : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2260 \u22a4\ng\u2081 : \u03b1 \u2192 \u211d\u22650\nf\u2081_le_g\u2081 : \u2200 (x : \u03b1), g\u2081 x \u2264 \u2191f\u2081 x\ng\u2081cont : UpperSemicontinuous g\u2081\ng\u2081int : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g\u2081 x) \u2202\u03bc + \u03b5 / 2\ng\u2082 : \u03b1 \u2192 \u211d\u22650\nf\u2082_le_g\u2082 : \u2200 (x : \u03b1), g\u2082 x \u2264 \u2191f\u2082 x\ng\u2082cont : UpperSemicontinuous g\u2082\ng\u2082int : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g\u2082 x) \u2202\u03bc + \u03b5 / 2\n\u22a2 \u222b\u207b (x : \u03b1), \u2191(\u2191(f\u2081 + f\u2082) x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191((fun x => g\u2081 x + g\u2082 x) x) \u2202\u03bc + \u03b5"}, {"tactic": "simp only [SimpleFunc.coe_add, ENNReal.coe_add, Pi.add_apply]", "annotated_tactic": ["simp only [<a>SimpleFunc.coe_add</a>, <a>ENNReal.coe_add</a>, <a>Pi.add_apply</a>]", [{"full_name": "MeasureTheory.SimpleFunc.coe_add", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [469, 3], "def_end_pos": [469, 14]}, {"full_name": "ENNReal.coe_add", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [386, 28], "def_end_pos": [386, 35]}, {"full_name": "Pi.add_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [82, 3], "def_end_pos": [82, 14]}]], "state_before": "case h_add.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nint_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (Function.support \u2191f\u2081) (Function.support \u2191f\u2082)\nh\u2081 :\n  \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2260 \u22a4 \u2192\n    \u2200 {\u03b5 : \u211d\u22650\u221e},\n      \u03b5 \u2260 0 \u2192\n        \u2203 g, (\u2200 (x : \u03b1), g x \u2264 \u2191f\u2081 x) \u2227 UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5\nh\u2082 :\n  \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2260 \u22a4 \u2192\n    \u2200 {\u03b5 : \u211d\u22650\u221e},\n      \u03b5 \u2260 0 \u2192\n        \u2203 g, (\u2200 (x : \u03b1), g x \u2264 \u2191f\u2082 x) \u2227 UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5\nint_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(f\u2081 + f\u2082) x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nA : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2260 \u22a4\ng\u2081 : \u03b1 \u2192 \u211d\u22650\nf\u2081_le_g\u2081 : \u2200 (x : \u03b1), g\u2081 x \u2264 \u2191f\u2081 x\ng\u2081cont : UpperSemicontinuous g\u2081\ng\u2081int : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g\u2081 x) \u2202\u03bc + \u03b5 / 2\ng\u2082 : \u03b1 \u2192 \u211d\u22650\nf\u2082_le_g\u2082 : \u2200 (x : \u03b1), g\u2082 x \u2264 \u2191f\u2082 x\ng\u2082cont : UpperSemicontinuous g\u2082\ng\u2082int : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g\u2082 x) \u2202\u03bc + \u03b5 / 2\n\u22a2 \u222b\u207b (x : \u03b1), \u2191(\u2191(f\u2081 + f\u2082) x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191((fun x => g\u2081 x + g\u2082 x) x) \u2202\u03bc + \u03b5", "state_after": "case h_add.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nint_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (Function.support \u2191f\u2081) (Function.support \u2191f\u2082)\nh\u2081 :\n  \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2260 \u22a4 \u2192\n    \u2200 {\u03b5 : \u211d\u22650\u221e},\n      \u03b5 \u2260 0 \u2192\n        \u2203 g, (\u2200 (x : \u03b1), g x \u2264 \u2191f\u2081 x) \u2227 UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5\nh\u2082 :\n  \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2260 \u22a4 \u2192\n    \u2200 {\u03b5 : \u211d\u22650\u221e},\n      \u03b5 \u2260 0 \u2192\n        \u2203 g, (\u2200 (x : \u03b1), g x \u2264 \u2191f\u2082 x) \u2227 UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5\nint_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(f\u2081 + f\u2082) x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nA : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2260 \u22a4\ng\u2081 : \u03b1 \u2192 \u211d\u22650\nf\u2081_le_g\u2081 : \u2200 (x : \u03b1), g\u2081 x \u2264 \u2191f\u2081 x\ng\u2081cont : UpperSemicontinuous g\u2081\ng\u2081int : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g\u2081 x) \u2202\u03bc + \u03b5 / 2\ng\u2082 : \u03b1 \u2192 \u211d\u22650\nf\u2082_le_g\u2082 : \u2200 (x : \u03b1), g\u2082 x \u2264 \u2191f\u2082 x\ng\u2082cont : UpperSemicontinuous g\u2082\ng\u2082int : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g\u2082 x) \u2202\u03bc + \u03b5 / 2\n\u22a2 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) + \u2191(\u2191f\u2082 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g\u2081 x) + \u2191(g\u2082 x) \u2202\u03bc + \u03b5"}, {"tactic": "rw [lintegral_add_left f\u2081.measurable.coe_nnreal_ennreal,\n  lintegral_add_left g\u2081cont.measurable.coe_nnreal_ennreal]", "annotated_tactic": ["rw [<a>lintegral_add_left</a> f\u2081.measurable.coe_nnreal_ennreal,\n      <a>lintegral_add_left</a> g\u2081cont.measurable.coe_nnreal_ennreal]", [{"full_name": "MeasureTheory.lintegral_add_left", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [554, 9], "def_end_pos": [554, 27]}, {"full_name": "MeasureTheory.lintegral_add_left", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [554, 9], "def_end_pos": [554, 27]}]], "state_before": "case h_add.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nint_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (Function.support \u2191f\u2081) (Function.support \u2191f\u2082)\nh\u2081 :\n  \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2260 \u22a4 \u2192\n    \u2200 {\u03b5 : \u211d\u22650\u221e},\n      \u03b5 \u2260 0 \u2192\n        \u2203 g, (\u2200 (x : \u03b1), g x \u2264 \u2191f\u2081 x) \u2227 UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5\nh\u2082 :\n  \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2260 \u22a4 \u2192\n    \u2200 {\u03b5 : \u211d\u22650\u221e},\n      \u03b5 \u2260 0 \u2192\n        \u2203 g, (\u2200 (x : \u03b1), g x \u2264 \u2191f\u2082 x) \u2227 UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5\nint_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(f\u2081 + f\u2082) x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nA : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2260 \u22a4\ng\u2081 : \u03b1 \u2192 \u211d\u22650\nf\u2081_le_g\u2081 : \u2200 (x : \u03b1), g\u2081 x \u2264 \u2191f\u2081 x\ng\u2081cont : UpperSemicontinuous g\u2081\ng\u2081int : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g\u2081 x) \u2202\u03bc + \u03b5 / 2\ng\u2082 : \u03b1 \u2192 \u211d\u22650\nf\u2082_le_g\u2082 : \u2200 (x : \u03b1), g\u2082 x \u2264 \u2191f\u2082 x\ng\u2082cont : UpperSemicontinuous g\u2082\ng\u2082int : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g\u2082 x) \u2202\u03bc + \u03b5 / 2\n\u22a2 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) + \u2191(\u2191f\u2082 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g\u2081 x) + \u2191(g\u2082 x) \u2202\u03bc + \u03b5", "state_after": "case h_add.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nint_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (Function.support \u2191f\u2081) (Function.support \u2191f\u2082)\nh\u2081 :\n  \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2260 \u22a4 \u2192\n    \u2200 {\u03b5 : \u211d\u22650\u221e},\n      \u03b5 \u2260 0 \u2192\n        \u2203 g, (\u2200 (x : \u03b1), g x \u2264 \u2191f\u2081 x) \u2227 UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5\nh\u2082 :\n  \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2260 \u22a4 \u2192\n    \u2200 {\u03b5 : \u211d\u22650\u221e},\n      \u03b5 \u2260 0 \u2192\n        \u2203 g, (\u2200 (x : \u03b1), g x \u2264 \u2191f\u2082 x) \u2227 UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5\nint_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(f\u2081 + f\u2082) x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nA : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2260 \u22a4\ng\u2081 : \u03b1 \u2192 \u211d\u22650\nf\u2081_le_g\u2081 : \u2200 (x : \u03b1), g\u2081 x \u2264 \u2191f\u2081 x\ng\u2081cont : UpperSemicontinuous g\u2081\ng\u2081int : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g\u2081 x) \u2202\u03bc + \u03b5 / 2\ng\u2082 : \u03b1 \u2192 \u211d\u22650\nf\u2082_le_g\u2082 : \u2200 (x : \u03b1), g\u2082 x \u2264 \u2191f\u2082 x\ng\u2082cont : UpperSemicontinuous g\u2082\ng\u2082int : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g\u2082 x) \u2202\u03bc + \u03b5 / 2\n\u22a2 \u222b\u207b (a : \u03b1), \u2191(\u2191f\u2081 a) \u2202\u03bc + \u222b\u207b (a : \u03b1), \u2191(\u2191f\u2082 a) \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), \u2191(g\u2081 a) \u2202\u03bc + \u222b\u207b (a : \u03b1), \u2191(g\u2082 a) \u2202\u03bc + \u03b5"}, {"tactic": "convert add_le_add g\u2081int g\u2082int using 1", "annotated_tactic": ["convert <a>add_le_add</a> g\u2081int g\u2082int using 1", [{"full_name": "add_le_add", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [205, 15], "def_end_pos": [205, 25]}]], "state_before": "case h_add.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nint_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (Function.support \u2191f\u2081) (Function.support \u2191f\u2082)\nh\u2081 :\n  \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2260 \u22a4 \u2192\n    \u2200 {\u03b5 : \u211d\u22650\u221e},\n      \u03b5 \u2260 0 \u2192\n        \u2203 g, (\u2200 (x : \u03b1), g x \u2264 \u2191f\u2081 x) \u2227 UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5\nh\u2082 :\n  \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2260 \u22a4 \u2192\n    \u2200 {\u03b5 : \u211d\u22650\u221e},\n      \u03b5 \u2260 0 \u2192\n        \u2203 g, (\u2200 (x : \u03b1), g x \u2264 \u2191f\u2082 x) \u2227 UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5\nint_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(f\u2081 + f\u2082) x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nA : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2260 \u22a4\ng\u2081 : \u03b1 \u2192 \u211d\u22650\nf\u2081_le_g\u2081 : \u2200 (x : \u03b1), g\u2081 x \u2264 \u2191f\u2081 x\ng\u2081cont : UpperSemicontinuous g\u2081\ng\u2081int : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g\u2081 x) \u2202\u03bc + \u03b5 / 2\ng\u2082 : \u03b1 \u2192 \u211d\u22650\nf\u2082_le_g\u2082 : \u2200 (x : \u03b1), g\u2082 x \u2264 \u2191f\u2082 x\ng\u2082cont : UpperSemicontinuous g\u2082\ng\u2082int : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g\u2082 x) \u2202\u03bc + \u03b5 / 2\n\u22a2 \u222b\u207b (a : \u03b1), \u2191(\u2191f\u2081 a) \u2202\u03bc + \u222b\u207b (a : \u03b1), \u2191(\u2191f\u2082 a) \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), \u2191(g\u2081 a) \u2202\u03bc + \u222b\u207b (a : \u03b1), \u2191(g\u2082 a) \u2202\u03bc + \u03b5", "state_after": "case h.e'_4\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nint_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (Function.support \u2191f\u2081) (Function.support \u2191f\u2082)\nh\u2081 :\n  \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2260 \u22a4 \u2192\n    \u2200 {\u03b5 : \u211d\u22650\u221e},\n      \u03b5 \u2260 0 \u2192\n        \u2203 g, (\u2200 (x : \u03b1), g x \u2264 \u2191f\u2081 x) \u2227 UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5\nh\u2082 :\n  \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2260 \u22a4 \u2192\n    \u2200 {\u03b5 : \u211d\u22650\u221e},\n      \u03b5 \u2260 0 \u2192\n        \u2203 g, (\u2200 (x : \u03b1), g x \u2264 \u2191f\u2082 x) \u2227 UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5\nint_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(f\u2081 + f\u2082) x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nA : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2260 \u22a4\ng\u2081 : \u03b1 \u2192 \u211d\u22650\nf\u2081_le_g\u2081 : \u2200 (x : \u03b1), g\u2081 x \u2264 \u2191f\u2081 x\ng\u2081cont : UpperSemicontinuous g\u2081\ng\u2081int : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g\u2081 x) \u2202\u03bc + \u03b5 / 2\ng\u2082 : \u03b1 \u2192 \u211d\u22650\nf\u2082_le_g\u2082 : \u2200 (x : \u03b1), g\u2082 x \u2264 \u2191f\u2082 x\ng\u2082cont : UpperSemicontinuous g\u2082\ng\u2082int : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g\u2082 x) \u2202\u03bc + \u03b5 / 2\n\u22a2 \u222b\u207b (a : \u03b1), \u2191(g\u2081 a) \u2202\u03bc + \u222b\u207b (a : \u03b1), \u2191(g\u2082 a) \u2202\u03bc + \u03b5 =\n    \u222b\u207b (x : \u03b1), \u2191(g\u2081 x) \u2202\u03bc + \u03b5 / 2 + (\u222b\u207b (x : \u03b1), \u2191(g\u2082 x) \u2202\u03bc + \u03b5 / 2)"}, {"tactic": "conv_lhs => rw [\u2190 ENNReal.add_halves \u03b5]", "annotated_tactic": ["conv_lhs => rw [\u2190 <a>ENNReal.add_halves</a> \u03b5]", [{"full_name": "ENNReal.add_halves", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1781, 19], "def_end_pos": [1781, 29]}]], "state_before": "case h.e'_4\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nint_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (Function.support \u2191f\u2081) (Function.support \u2191f\u2082)\nh\u2081 :\n  \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2260 \u22a4 \u2192\n    \u2200 {\u03b5 : \u211d\u22650\u221e},\n      \u03b5 \u2260 0 \u2192\n        \u2203 g, (\u2200 (x : \u03b1), g x \u2264 \u2191f\u2081 x) \u2227 UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5\nh\u2082 :\n  \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2260 \u22a4 \u2192\n    \u2200 {\u03b5 : \u211d\u22650\u221e},\n      \u03b5 \u2260 0 \u2192\n        \u2203 g, (\u2200 (x : \u03b1), g x \u2264 \u2191f\u2082 x) \u2227 UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5\nint_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(f\u2081 + f\u2082) x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nA : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2260 \u22a4\ng\u2081 : \u03b1 \u2192 \u211d\u22650\nf\u2081_le_g\u2081 : \u2200 (x : \u03b1), g\u2081 x \u2264 \u2191f\u2081 x\ng\u2081cont : UpperSemicontinuous g\u2081\ng\u2081int : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g\u2081 x) \u2202\u03bc + \u03b5 / 2\ng\u2082 : \u03b1 \u2192 \u211d\u22650\nf\u2082_le_g\u2082 : \u2200 (x : \u03b1), g\u2082 x \u2264 \u2191f\u2082 x\ng\u2082cont : UpperSemicontinuous g\u2082\ng\u2082int : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g\u2082 x) \u2202\u03bc + \u03b5 / 2\n\u22a2 \u222b\u207b (a : \u03b1), \u2191(g\u2081 a) \u2202\u03bc + \u222b\u207b (a : \u03b1), \u2191(g\u2082 a) \u2202\u03bc + \u03b5 =\n    \u222b\u207b (x : \u03b1), \u2191(g\u2081 x) \u2202\u03bc + \u03b5 / 2 + (\u222b\u207b (x : \u03b1), \u2191(g\u2082 x) \u2202\u03bc + \u03b5 / 2)", "state_after": "case h.e'_4\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nint_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (Function.support \u2191f\u2081) (Function.support \u2191f\u2082)\nh\u2081 :\n  \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2260 \u22a4 \u2192\n    \u2200 {\u03b5 : \u211d\u22650\u221e},\n      \u03b5 \u2260 0 \u2192\n        \u2203 g, (\u2200 (x : \u03b1), g x \u2264 \u2191f\u2081 x) \u2227 UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5\nh\u2082 :\n  \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2260 \u22a4 \u2192\n    \u2200 {\u03b5 : \u211d\u22650\u221e},\n      \u03b5 \u2260 0 \u2192\n        \u2203 g, (\u2200 (x : \u03b1), g x \u2264 \u2191f\u2082 x) \u2227 UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5\nint_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(f\u2081 + f\u2082) x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nA : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2260 \u22a4\ng\u2081 : \u03b1 \u2192 \u211d\u22650\nf\u2081_le_g\u2081 : \u2200 (x : \u03b1), g\u2081 x \u2264 \u2191f\u2081 x\ng\u2081cont : UpperSemicontinuous g\u2081\ng\u2081int : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g\u2081 x) \u2202\u03bc + \u03b5 / 2\ng\u2082 : \u03b1 \u2192 \u211d\u22650\nf\u2082_le_g\u2082 : \u2200 (x : \u03b1), g\u2082 x \u2264 \u2191f\u2082 x\ng\u2082cont : UpperSemicontinuous g\u2082\ng\u2082int : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g\u2082 x) \u2202\u03bc + \u03b5 / 2\n\u22a2 \u222b\u207b (a : \u03b1), \u2191(g\u2081 a) \u2202\u03bc + \u222b\u207b (a : \u03b1), \u2191(g\u2082 a) \u2202\u03bc + (\u03b5 / 2 + \u03b5 / 2) =\n    \u222b\u207b (x : \u03b1), \u2191(g\u2081 x) \u2202\u03bc + \u03b5 / 2 + (\u222b\u207b (x : \u03b1), \u2191(g\u2082 x) \u2202\u03bc + \u03b5 / 2)"}, {"tactic": "abel", "annotated_tactic": ["abel", []], "state_before": "case h.e'_4\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nint_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (Function.support \u2191f\u2081) (Function.support \u2191f\u2082)\nh\u2081 :\n  \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2260 \u22a4 \u2192\n    \u2200 {\u03b5 : \u211d\u22650\u221e},\n      \u03b5 \u2260 0 \u2192\n        \u2203 g, (\u2200 (x : \u03b1), g x \u2264 \u2191f\u2081 x) \u2227 UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5\nh\u2082 :\n  \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2260 \u22a4 \u2192\n    \u2200 {\u03b5 : \u211d\u22650\u221e},\n      \u03b5 \u2260 0 \u2192\n        \u2203 g, (\u2200 (x : \u03b1), g x \u2264 \u2191f\u2082 x) \u2227 UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5\nint_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(f\u2081 + f\u2082) x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nA : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2260 \u22a4\ng\u2081 : \u03b1 \u2192 \u211d\u22650\nf\u2081_le_g\u2081 : \u2200 (x : \u03b1), g\u2081 x \u2264 \u2191f\u2081 x\ng\u2081cont : UpperSemicontinuous g\u2081\ng\u2081int : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g\u2081 x) \u2202\u03bc + \u03b5 / 2\ng\u2082 : \u03b1 \u2192 \u211d\u22650\nf\u2082_le_g\u2082 : \u2200 (x : \u03b1), g\u2082 x \u2264 \u2191f\u2082 x\ng\u2082cont : UpperSemicontinuous g\u2082\ng\u2082int : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g\u2082 x) \u2202\u03bc + \u03b5 / 2\n\u22a2 \u222b\u207b (a : \u03b1), \u2191(g\u2081 a) \u2202\u03bc + \u222b\u207b (a : \u03b1), \u2191(g\u2082 a) \u2202\u03bc + (\u03b5 / 2 + \u03b5 / 2) =\n    \u222b\u207b (x : \u03b1), \u2191(g\u2081 x) \u2202\u03bc + \u03b5 / 2 + (\u222b\u207b (x : \u03b1), \u2191(g\u2082 x) \u2202\u03bc + \u03b5 / 2)", "state_after": "no goals"}, {"tactic": "rwa [\u2190 lintegral_add_left f\u2081.measurable.coe_nnreal_ennreal]", "annotated_tactic": ["rwa [\u2190 <a>lintegral_add_left</a> f\u2081.measurable.coe_nnreal_ennreal]", [{"full_name": "MeasureTheory.lintegral_add_left", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [554, 9], "def_end_pos": [554, 27]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nint_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (Function.support \u2191f\u2081) (Function.support \u2191f\u2082)\nh\u2081 :\n  \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2260 \u22a4 \u2192\n    \u2200 {\u03b5 : \u211d\u22650\u221e},\n      \u03b5 \u2260 0 \u2192\n        \u2203 g, (\u2200 (x : \u03b1), g x \u2264 \u2191f\u2081 x) \u2227 UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5\nh\u2082 :\n  \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2260 \u22a4 \u2192\n    \u2200 {\u03b5 : \u211d\u22650\u221e},\n      \u03b5 \u2260 0 \u2192\n        \u2203 g, (\u2200 (x : \u03b1), g x \u2264 \u2191f\u2082 x) \u2227 UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5\nint_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(f\u2081 + f\u2082) x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\n\u22a2 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2260 \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Array/Init/Lemmas.lean", "full_name": "Array.map_data", "start": [185, 9], "end": [190, 11], "traced_tactics": [{"tactic": "rw [map, mapM_eq_foldlM]", "annotated_tactic": ["rw [<a>map</a>, <a>mapM_eq_foldlM</a>]", [{"full_name": "Array.map", "def_path": "lake-packages/lean4/src/lean/Init/Data/Array/Basic.lean", "def_pos": [397, 5], "def_end_pos": [397, 8]}, {"full_name": "Array.mapM_eq_foldlM", "def_path": "lake-packages/std/Std/Data/Array/Init/Lemmas.lean", "def_pos": [143, 9], "def_end_pos": [143, 23]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b2\narr : Array \u03b1\n\u22a2 (map f arr).data = List.map f arr.data", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b2\narr : Array \u03b1\n\u22a2 (Id.run (foldlM (fun bs a => push bs <$> f a) #[] arr 0 (size arr))).data = List.map f arr.data"}, {"tactic": "apply congrArg data (foldl_eq_foldl_data (fun bs a => push bs (f a)) #[] arr) |>.trans", "annotated_tactic": ["apply <a>congrArg</a> <a>data</a> (<a>foldl_eq_foldl_data</a> (fun bs a => <a>push</a> bs (f a)) #[] arr) |>.<a>trans</a>", [{"full_name": "congrArg", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [349, 9], "def_end_pos": [349, 17]}, {"full_name": "Array.data", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2501, 3], "def_end_pos": [2501, 7]}, {"full_name": "Array.foldl_eq_foldl_data", "def_path": "lake-packages/std/Std/Data/Array/Init/Lemmas.lean", "def_pos": [47, 9], "def_end_pos": [47, 28]}, {"full_name": "Array.push", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2541, 5], "def_end_pos": [2541, 15]}, {"full_name": "Eq.trans", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [322, 9], "def_end_pos": [322, 17]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b2\narr : Array \u03b1\n\u22a2 (Id.run (foldlM (fun bs a => push bs <$> f a) #[] arr 0 (size arr))).data = List.map f arr.data", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b2\narr : Array \u03b1\n\u22a2 (List.foldl (fun bs a => push bs (f a)) #[] arr.data).data = List.map f arr.data"}, {"tactic": "have H (l arr) : List.foldl (fun bs a => push bs (f a)) arr l = \u27e8arr.data ++ l.map f\u27e9 := by\n  induction l generalizing arr <;> simp [*]", "annotated_tactic": ["have H (l arr) : <a>List.foldl</a> (fun bs a => <a>push</a> bs (f a)) arr l = \u27e8arr.data ++ l.map f\u27e9 := by\n    induction l generalizing arr <;> simp [*]", [{"full_name": "List.foldl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2212, 5], "def_end_pos": [2212, 15]}, {"full_name": "Array.push", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2541, 5], "def_end_pos": [2541, 15]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b2\narr : Array \u03b1\n\u22a2 (List.foldl (fun bs a => push bs (f a)) #[] arr.data).data = List.map f arr.data", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b2\narr : Array \u03b1\nH : \u2200 (l : List \u03b1) (arr : Array \u03b2), List.foldl (fun bs a => push bs (f a)) arr l = { data := arr.data ++ List.map f l }\n\u22a2 (List.foldl (fun bs a => push bs (f a)) #[] arr.data).data = List.map f arr.data"}, {"tactic": "simp [H]", "annotated_tactic": ["simp [H]", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b2\narr : Array \u03b1\nH : \u2200 (l : List \u03b1) (arr : Array \u03b2), List.foldl (fun bs a => push bs (f a)) arr l = { data := arr.data ++ List.map f l }\n\u22a2 (List.foldl (fun bs a => push bs (f a)) #[] arr.data).data = List.map f arr.data", "state_after": "no goals"}, {"tactic": "induction l generalizing arr <;> simp [*]", "annotated_tactic": ["induction l generalizing arr <;> simp [*]", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b2\narr\u271d : Array \u03b1\nl : List \u03b1\narr : Array \u03b2\n\u22a2 List.foldl (fun bs a => push bs (f a)) arr l = { data := arr.data ++ List.map f l }", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Martingale/Upcrossing.lean", "full_name": "MeasureTheory.crossing_pos_eq", "start": [670, 1], "end": [715, 10], "traced_tactics": [{"tactic": "have hab' : 0 < b - a := sub_pos.2 hab", "annotated_tactic": ["have hab' : 0 < b - a := <a>sub_pos</a>.2 hab", [{"full_name": "sub_pos", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [883, 30], "def_end_pos": [883, 37]}]], "state_before": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\nhab : a < b\n\u22a2 upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N n = upperCrossingTime a b f N n \u2227\n    lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N n = lowerCrossingTime a b f N n", "state_after": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\nhab : a < b\nhab' : 0 < b - a\n\u22a2 upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N n = upperCrossingTime a b f N n \u2227\n    lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N n = lowerCrossingTime a b f N n"}, {"tactic": "have hf' : \u2200 \u03c9 i, (f i \u03c9 - a)\u207a \u2264 0 \u2194 f i \u03c9 \u2264 a := by\n  intro \u03c9 i\n  rw [LatticeOrderedGroup.pos_nonpos_iff, sub_nonpos]", "annotated_tactic": ["have hf' : \u2200 \u03c9 i, (f i \u03c9 - a)\u207a \u2264 0 \u2194 f i \u03c9 \u2264 a := by\n    intro \u03c9 i\n    rw [<a>LatticeOrderedGroup.pos_nonpos_iff</a>, <a>sub_nonpos</a>]", [{"full_name": "LatticeOrderedGroup.pos_nonpos_iff", "def_path": "Mathlib/Algebra/Order/LatticeGroup.lean", "def_pos": [214, 3], "def_end_pos": [214, 14]}, {"full_name": "sub_nonpos", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [730, 30], "def_end_pos": [730, 40]}]], "state_before": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\nhab : a < b\nhab' : 0 < b - a\nhf : \u2200 (\u03c9 : \u03a9) (i : \u2115), b - a \u2264 (f i \u03c9 - a)\u207a \u2194 b \u2264 f i \u03c9\n\u22a2 upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N n = upperCrossingTime a b f N n \u2227\n    lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N n = lowerCrossingTime a b f N n", "state_after": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\nhab : a < b\nhab' : 0 < b - a\nhf : \u2200 (\u03c9 : \u03a9) (i : \u2115), b - a \u2264 (f i \u03c9 - a)\u207a \u2194 b \u2264 f i \u03c9\nhf' : \u2200 (\u03c9 : \u03a9) (i : \u2115), (f i \u03c9 - a)\u207a \u2264 0 \u2194 f i \u03c9 \u2264 a\n\u22a2 upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N n = upperCrossingTime a b f N n \u2227\n    lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N n = lowerCrossingTime a b f N n"}, {"tactic": "induction' n with k ih", "annotated_tactic": ["induction' n with k ih", []], "state_before": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\nhab : a < b\nhab' : 0 < b - a\nhf : \u2200 (\u03c9 : \u03a9) (i : \u2115), b - a \u2264 (f i \u03c9 - a)\u207a \u2194 b \u2264 f i \u03c9\nhf' : \u2200 (\u03c9 : \u03a9) (i : \u2115), (f i \u03c9 - a)\u207a \u2264 0 \u2194 f i \u03c9 \u2264 a\n\u22a2 upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N n = upperCrossingTime a b f N n \u2227\n    lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N n = lowerCrossingTime a b f N n", "state_after": "case zero\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\nhab : a < b\nhab' : 0 < b - a\nhf : \u2200 (\u03c9 : \u03a9) (i : \u2115), b - a \u2264 (f i \u03c9 - a)\u207a \u2194 b \u2264 f i \u03c9\nhf' : \u2200 (\u03c9 : \u03a9) (i : \u2115), (f i \u03c9 - a)\u207a \u2264 0 \u2194 f i \u03c9 \u2264 a\n\u22a2 upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N Nat.zero = upperCrossingTime a b f N Nat.zero \u2227\n    lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N Nat.zero = lowerCrossingTime a b f N Nat.zero\n\ncase succ\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\nhab : a < b\nhab' : 0 < b - a\nhf : \u2200 (\u03c9 : \u03a9) (i : \u2115), b - a \u2264 (f i \u03c9 - a)\u207a \u2194 b \u2264 f i \u03c9\nhf' : \u2200 (\u03c9 : \u03a9) (i : \u2115), (f i \u03c9 - a)\u207a \u2264 0 \u2194 f i \u03c9 \u2264 a\nk : \u2115\nih :\n  upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k = upperCrossingTime a b f N k \u2227\n    lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k = lowerCrossingTime a b f N k\n\u22a2 upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N (Nat.succ k) = upperCrossingTime a b f N (Nat.succ k) \u2227\n    lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N (Nat.succ k) = lowerCrossingTime a b f N (Nat.succ k)"}, {"tactic": "intro i \u03c9", "annotated_tactic": ["intro i \u03c9", []], "state_before": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\nhab : a < b\nhab' : 0 < b - a\n\u22a2 \u2200 (\u03c9 : \u03a9) (i : \u2115), b - a \u2264 (f i \u03c9 - a)\u207a \u2194 b \u2264 f i \u03c9", "state_after": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9\u271d : \u03a9\n\u2131 : Filtration \u2115 m0\nhab : a < b\nhab' : 0 < b - a\ni : \u03a9\n\u03c9 : \u2115\n\u22a2 b - a \u2264 (f \u03c9 i - a)\u207a \u2194 b \u2264 f \u03c9 i"}, {"tactic": "refine' \u27e8fun h => _, fun h => _\u27e9", "annotated_tactic": ["refine' \u27e8fun h => _, fun h => _\u27e9", []], "state_before": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9\u271d : \u03a9\n\u2131 : Filtration \u2115 m0\nhab : a < b\nhab' : 0 < b - a\ni : \u03a9\n\u03c9 : \u2115\n\u22a2 b - a \u2264 (f \u03c9 i - a)\u207a \u2194 b \u2264 f \u03c9 i", "state_after": "case refine'_1\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9\u271d : \u03a9\n\u2131 : Filtration \u2115 m0\nhab : a < b\nhab' : 0 < b - a\ni : \u03a9\n\u03c9 : \u2115\nh : b - a \u2264 (f \u03c9 i - a)\u207a\n\u22a2 b \u2264 f \u03c9 i\n\ncase refine'_2\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9\u271d : \u03a9\n\u2131 : Filtration \u2115 m0\nhab : a < b\nhab' : 0 < b - a\ni : \u03a9\n\u03c9 : \u2115\nh : b \u2264 f \u03c9 i\n\u22a2 b - a \u2264 (f \u03c9 i - a)\u207a"}, {"tactic": "rwa [\u2190 sub_le_sub_iff_right a, \u2190\n  LatticeOrderedGroup.pos_eq_self_of_pos_pos (lt_of_lt_of_le hab' h)]", "annotated_tactic": ["rwa [\u2190 <a>sub_le_sub_iff_right</a> a, \u2190\n        <a>LatticeOrderedGroup.pos_eq_self_of_pos_pos</a> (<a>lt_of_lt_of_le</a> hab' h)]", [{"full_name": "sub_le_sub_iff_right", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [708, 3], "def_end_pos": [708, 14]}, {"full_name": "LatticeOrderedGroup.pos_eq_self_of_pos_pos", "def_path": "Mathlib/Algebra/Order/LatticeGroup.lean", "def_pos": [336, 3], "def_end_pos": [336, 14]}, {"full_name": "lt_of_lt_of_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [115, 9], "def_end_pos": [115, 23]}]], "state_before": "case refine'_1\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9\u271d : \u03a9\n\u2131 : Filtration \u2115 m0\nhab : a < b\nhab' : 0 < b - a\ni : \u03a9\n\u03c9 : \u2115\nh : b - a \u2264 (f \u03c9 i - a)\u207a\n\u22a2 b \u2264 f \u03c9 i", "state_after": "no goals"}, {"tactic": "rw [\u2190 sub_le_sub_iff_right a] at h", "annotated_tactic": ["rw [\u2190 <a>sub_le_sub_iff_right</a> a] at h", [{"full_name": "sub_le_sub_iff_right", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [708, 3], "def_end_pos": [708, 14]}]], "state_before": "case refine'_2\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9\u271d : \u03a9\n\u2131 : Filtration \u2115 m0\nhab : a < b\nhab' : 0 < b - a\ni : \u03a9\n\u03c9 : \u2115\nh : b \u2264 f \u03c9 i\n\u22a2 b - a \u2264 (f \u03c9 i - a)\u207a", "state_after": "case refine'_2\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9\u271d : \u03a9\n\u2131 : Filtration \u2115 m0\nhab : a < b\nhab' : 0 < b - a\ni : \u03a9\n\u03c9 : \u2115\nh : b - a \u2264 f \u03c9 i - a\n\u22a2 b - a \u2264 (f \u03c9 i - a)\u207a"}, {"tactic": "rwa [LatticeOrderedGroup.pos_of_nonneg _ (le_trans hab'.le h)]", "annotated_tactic": ["rwa [<a>LatticeOrderedGroup.pos_of_nonneg</a> _ (<a>le_trans</a> hab'.le h)]", [{"full_name": "LatticeOrderedGroup.pos_of_nonneg", "def_path": "Mathlib/Algebra/Order/LatticeGroup.lean", "def_pos": [299, 3], "def_end_pos": [299, 14]}, {"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}]], "state_before": "case refine'_2\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9\u271d : \u03a9\n\u2131 : Filtration \u2115 m0\nhab : a < b\nhab' : 0 < b - a\ni : \u03a9\n\u03c9 : \u2115\nh : b - a \u2264 f \u03c9 i - a\n\u22a2 b - a \u2264 (f \u03c9 i - a)\u207a", "state_after": "no goals"}, {"tactic": "intro \u03c9 i", "annotated_tactic": ["intro \u03c9 i", []], "state_before": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\nhab : a < b\nhab' : 0 < b - a\nhf : \u2200 (\u03c9 : \u03a9) (i : \u2115), b - a \u2264 (f i \u03c9 - a)\u207a \u2194 b \u2264 f i \u03c9\n\u22a2 \u2200 (\u03c9 : \u03a9) (i : \u2115), (f i \u03c9 - a)\u207a \u2264 0 \u2194 f i \u03c9 \u2264 a", "state_after": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9\u271d : \u03a9\n\u2131 : Filtration \u2115 m0\nhab : a < b\nhab' : 0 < b - a\nhf : \u2200 (\u03c9 : \u03a9) (i : \u2115), b - a \u2264 (f i \u03c9 - a)\u207a \u2194 b \u2264 f i \u03c9\n\u03c9 : \u03a9\ni : \u2115\n\u22a2 (f i \u03c9 - a)\u207a \u2264 0 \u2194 f i \u03c9 \u2264 a"}, {"tactic": "rw [LatticeOrderedGroup.pos_nonpos_iff, sub_nonpos]", "annotated_tactic": ["rw [<a>LatticeOrderedGroup.pos_nonpos_iff</a>, <a>sub_nonpos</a>]", [{"full_name": "LatticeOrderedGroup.pos_nonpos_iff", "def_path": "Mathlib/Algebra/Order/LatticeGroup.lean", "def_pos": [214, 3], "def_end_pos": [214, 14]}, {"full_name": "sub_nonpos", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [730, 30], "def_end_pos": [730, 40]}]], "state_before": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9\u271d : \u03a9\n\u2131 : Filtration \u2115 m0\nhab : a < b\nhab' : 0 < b - a\nhf : \u2200 (\u03c9 : \u03a9) (i : \u2115), b - a \u2264 (f i \u03c9 - a)\u207a \u2194 b \u2264 f i \u03c9\n\u03c9 : \u03a9\ni : \u2115\n\u22a2 (f i \u03c9 - a)\u207a \u2264 0 \u2194 f i \u03c9 \u2264 a", "state_after": "no goals"}, {"tactic": "refine' \u27e8rfl, _\u27e9", "annotated_tactic": ["refine' \u27e8<a>rfl</a>, _\u27e9", [{"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "case zero\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\nhab : a < b\nhab' : 0 < b - a\nhf : \u2200 (\u03c9 : \u03a9) (i : \u2115), b - a \u2264 (f i \u03c9 - a)\u207a \u2194 b \u2264 f i \u03c9\nhf' : \u2200 (\u03c9 : \u03a9) (i : \u2115), (f i \u03c9 - a)\u207a \u2264 0 \u2194 f i \u03c9 \u2264 a\n\u22a2 upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N Nat.zero = upperCrossingTime a b f N Nat.zero \u2227\n    lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N Nat.zero = lowerCrossingTime a b f N Nat.zero", "state_after": "case zero\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\nhab : a < b\nhab' : 0 < b - a\nhf : \u2200 (\u03c9 : \u03a9) (i : \u2115), b - a \u2264 (f i \u03c9 - a)\u207a \u2194 b \u2264 f i \u03c9\nhf' : \u2200 (\u03c9 : \u03a9) (i : \u2115), (f i \u03c9 - a)\u207a \u2264 0 \u2194 f i \u03c9 \u2264 a\n\u22a2 lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N Nat.zero = lowerCrossingTime a b f N Nat.zero"}, {"tactic": "simp only [lowerCrossingTime_zero, hitting, Set.mem_Icc, Set.mem_Iic, Nat.zero_eq]", "annotated_tactic": ["simp only [<a>lowerCrossingTime_zero</a>, <a>hitting</a>, <a>Set.mem_Icc</a>, <a>Set.mem_Iic</a>, <a>Nat.zero_eq</a>]", [{"full_name": "MeasureTheory.lowerCrossingTime_zero", "def_path": "Mathlib/Probability/Martingale/Upcrossing.lean", "def_pos": [164, 9], "def_end_pos": [164, 31]}, {"full_name": "MeasureTheory.hitting", "def_path": "Mathlib/Probability/Process/HittingTime.lean", "def_pos": [51, 19], "def_end_pos": [51, 26]}, {"full_name": "Set.mem_Icc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [131, 9], "def_end_pos": [131, 16]}, {"full_name": "Set.mem_Iic", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [136, 9], "def_end_pos": [136, 16]}, {"full_name": "Nat.zero_eq", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [83, 17], "def_end_pos": [83, 24]}]], "state_before": "case zero\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\nhab : a < b\nhab' : 0 < b - a\nhf : \u2200 (\u03c9 : \u03a9) (i : \u2115), b - a \u2264 (f i \u03c9 - a)\u207a \u2194 b \u2264 f i \u03c9\nhf' : \u2200 (\u03c9 : \u03a9) (i : \u2115), (f i \u03c9 - a)\u207a \u2264 0 \u2194 f i \u03c9 \u2264 a\n\u22a2 lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N Nat.zero = lowerCrossingTime a b f N Nat.zero", "state_after": "case zero\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\nhab : a < b\nhab' : 0 < b - a\nhf : \u2200 (\u03c9 : \u03a9) (i : \u2115), b - a \u2264 (f i \u03c9 - a)\u207a \u2194 b \u2264 f i \u03c9\nhf' : \u2200 (\u03c9 : \u03a9) (i : \u2115), (f i \u03c9 - a)\u207a \u2264 0 \u2194 f i \u03c9 \u2264 a\n\u22a2 (fun x =>\n      if \u2203 j, j \u2208 Set.Icc \u22a5 N \u2227 (f j x - a)\u207a \u2208 Set.Iic 0 then sInf (Set.Icc \u22a5 N \u2229 {i | (f i x - a)\u207a \u2264 0}) else N) =\n    fun x => if \u2203 j, j \u2208 Set.Icc \u22a5 N \u2227 f j x \u2208 Set.Iic a then sInf (Set.Icc \u22a5 N \u2229 {i | f i x \u2264 a}) else N"}, {"tactic": "ext \u03c9", "annotated_tactic": ["ext \u03c9", []], "state_before": "case zero\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\nhab : a < b\nhab' : 0 < b - a\nhf : \u2200 (\u03c9 : \u03a9) (i : \u2115), b - a \u2264 (f i \u03c9 - a)\u207a \u2194 b \u2264 f i \u03c9\nhf' : \u2200 (\u03c9 : \u03a9) (i : \u2115), (f i \u03c9 - a)\u207a \u2264 0 \u2194 f i \u03c9 \u2264 a\n\u22a2 (fun x =>\n      if \u2203 j, j \u2208 Set.Icc \u22a5 N \u2227 (f j x - a)\u207a \u2208 Set.Iic 0 then sInf (Set.Icc \u22a5 N \u2229 {i | (f i x - a)\u207a \u2264 0}) else N) =\n    fun x => if \u2203 j, j \u2208 Set.Icc \u22a5 N \u2227 f j x \u2208 Set.Iic a then sInf (Set.Icc \u22a5 N \u2229 {i | f i x \u2264 a}) else N", "state_after": "case zero.h\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9\u271d : \u03a9\n\u2131 : Filtration \u2115 m0\nhab : a < b\nhab' : 0 < b - a\nhf : \u2200 (\u03c9 : \u03a9) (i : \u2115), b - a \u2264 (f i \u03c9 - a)\u207a \u2194 b \u2264 f i \u03c9\nhf' : \u2200 (\u03c9 : \u03a9) (i : \u2115), (f i \u03c9 - a)\u207a \u2264 0 \u2194 f i \u03c9 \u2264 a\n\u03c9 : \u03a9\n\u22a2 (if \u2203 j, j \u2208 Set.Icc \u22a5 N \u2227 (f j \u03c9 - a)\u207a \u2208 Set.Iic 0 then sInf (Set.Icc \u22a5 N \u2229 {i | (f i \u03c9 - a)\u207a \u2264 0}) else N) =\n    if \u2203 j, j \u2208 Set.Icc \u22a5 N \u2227 f j \u03c9 \u2208 Set.Iic a then sInf (Set.Icc \u22a5 N \u2229 {i | f i \u03c9 \u2264 a}) else N"}, {"tactic": "split_ifs with h\u2081 h\u2082 h\u2082", "annotated_tactic": ["split_ifs with h\u2081 h\u2082 h\u2082", []], "state_before": "case zero.h\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9\u271d : \u03a9\n\u2131 : Filtration \u2115 m0\nhab : a < b\nhab' : 0 < b - a\nhf : \u2200 (\u03c9 : \u03a9) (i : \u2115), b - a \u2264 (f i \u03c9 - a)\u207a \u2194 b \u2264 f i \u03c9\nhf' : \u2200 (\u03c9 : \u03a9) (i : \u2115), (f i \u03c9 - a)\u207a \u2264 0 \u2194 f i \u03c9 \u2264 a\n\u03c9 : \u03a9\n\u22a2 (if \u2203 j, j \u2208 Set.Icc \u22a5 N \u2227 (f j \u03c9 - a)\u207a \u2208 Set.Iic 0 then sInf (Set.Icc \u22a5 N \u2229 {i | (f i \u03c9 - a)\u207a \u2264 0}) else N) =\n    if \u2203 j, j \u2208 Set.Icc \u22a5 N \u2227 f j \u03c9 \u2208 Set.Iic a then sInf (Set.Icc \u22a5 N \u2229 {i | f i \u03c9 \u2264 a}) else N", "state_after": "case pos\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9\u271d : \u03a9\n\u2131 : Filtration \u2115 m0\nhab : a < b\nhab' : 0 < b - a\nhf : \u2200 (\u03c9 : \u03a9) (i : \u2115), b - a \u2264 (f i \u03c9 - a)\u207a \u2194 b \u2264 f i \u03c9\nhf' : \u2200 (\u03c9 : \u03a9) (i : \u2115), (f i \u03c9 - a)\u207a \u2264 0 \u2194 f i \u03c9 \u2264 a\n\u03c9 : \u03a9\nh\u2081 : \u2203 j, j \u2208 Set.Icc \u22a5 N \u2227 (f j \u03c9 - a)\u207a \u2208 Set.Iic 0\nh\u2082 : \u2203 j, j \u2208 Set.Icc \u22a5 N \u2227 f j \u03c9 \u2208 Set.Iic a\n\u22a2 sInf (Set.Icc \u22a5 N \u2229 {i | (f i \u03c9 - a)\u207a \u2264 0}) = sInf (Set.Icc \u22a5 N \u2229 {i | f i \u03c9 \u2264 a})\n\ncase neg\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9\u271d : \u03a9\n\u2131 : Filtration \u2115 m0\nhab : a < b\nhab' : 0 < b - a\nhf : \u2200 (\u03c9 : \u03a9) (i : \u2115), b - a \u2264 (f i \u03c9 - a)\u207a \u2194 b \u2264 f i \u03c9\nhf' : \u2200 (\u03c9 : \u03a9) (i : \u2115), (f i \u03c9 - a)\u207a \u2264 0 \u2194 f i \u03c9 \u2264 a\n\u03c9 : \u03a9\nh\u2081 : \u2203 j, j \u2208 Set.Icc \u22a5 N \u2227 (f j \u03c9 - a)\u207a \u2208 Set.Iic 0\nh\u2082 : \u00ac\u2203 j, j \u2208 Set.Icc \u22a5 N \u2227 f j \u03c9 \u2208 Set.Iic a\n\u22a2 sInf (Set.Icc \u22a5 N \u2229 {i | (f i \u03c9 - a)\u207a \u2264 0}) = N\n\ncase pos\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9\u271d : \u03a9\n\u2131 : Filtration \u2115 m0\nhab : a < b\nhab' : 0 < b - a\nhf : \u2200 (\u03c9 : \u03a9) (i : \u2115), b - a \u2264 (f i \u03c9 - a)\u207a \u2194 b \u2264 f i \u03c9\nhf' : \u2200 (\u03c9 : \u03a9) (i : \u2115), (f i \u03c9 - a)\u207a \u2264 0 \u2194 f i \u03c9 \u2264 a\n\u03c9 : \u03a9\nh\u2081 : \u00ac\u2203 j, j \u2208 Set.Icc \u22a5 N \u2227 (f j \u03c9 - a)\u207a \u2208 Set.Iic 0\nh\u2082 : \u2203 j, j \u2208 Set.Icc \u22a5 N \u2227 f j \u03c9 \u2208 Set.Iic a\n\u22a2 N = sInf (Set.Icc \u22a5 N \u2229 {i | f i \u03c9 \u2264 a})\n\ncase neg\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9\u271d : \u03a9\n\u2131 : Filtration \u2115 m0\nhab : a < b\nhab' : 0 < b - a\nhf : \u2200 (\u03c9 : \u03a9) (i : \u2115), b - a \u2264 (f i \u03c9 - a)\u207a \u2194 b \u2264 f i \u03c9\nhf' : \u2200 (\u03c9 : \u03a9) (i : \u2115), (f i \u03c9 - a)\u207a \u2264 0 \u2194 f i \u03c9 \u2264 a\n\u03c9 : \u03a9\nh\u2081 : \u00ac\u2203 j, j \u2208 Set.Icc \u22a5 N \u2227 (f j \u03c9 - a)\u207a \u2208 Set.Iic 0\nh\u2082 : \u00ac\u2203 j, j \u2208 Set.Icc \u22a5 N \u2227 f j \u03c9 \u2208 Set.Iic a\n\u22a2 N = N"}, {"tactic": "simp_rw [hf']", "annotated_tactic": ["simp_rw [hf']", []], "state_before": "case pos\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9\u271d : \u03a9\n\u2131 : Filtration \u2115 m0\nhab : a < b\nhab' : 0 < b - a\nhf : \u2200 (\u03c9 : \u03a9) (i : \u2115), b - a \u2264 (f i \u03c9 - a)\u207a \u2194 b \u2264 f i \u03c9\nhf' : \u2200 (\u03c9 : \u03a9) (i : \u2115), (f i \u03c9 - a)\u207a \u2264 0 \u2194 f i \u03c9 \u2264 a\n\u03c9 : \u03a9\nh\u2081 : \u2203 j, j \u2208 Set.Icc \u22a5 N \u2227 (f j \u03c9 - a)\u207a \u2208 Set.Iic 0\nh\u2082 : \u2203 j, j \u2208 Set.Icc \u22a5 N \u2227 f j \u03c9 \u2208 Set.Iic a\n\u22a2 sInf (Set.Icc \u22a5 N \u2229 {i | (f i \u03c9 - a)\u207a \u2264 0}) = sInf (Set.Icc \u22a5 N \u2229 {i | f i \u03c9 \u2264 a})", "state_after": "no goals"}, {"tactic": "simp_rw [Set.mem_Iic, \u2190 hf' _ _] at h\u2082", "annotated_tactic": ["simp_rw [<a>Set.mem_Iic</a>, \u2190 hf' _ _] at h\u2082", [{"full_name": "Set.mem_Iic", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [136, 9], "def_end_pos": [136, 16]}]], "state_before": "case neg\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9\u271d : \u03a9\n\u2131 : Filtration \u2115 m0\nhab : a < b\nhab' : 0 < b - a\nhf : \u2200 (\u03c9 : \u03a9) (i : \u2115), b - a \u2264 (f i \u03c9 - a)\u207a \u2194 b \u2264 f i \u03c9\nhf' : \u2200 (\u03c9 : \u03a9) (i : \u2115), (f i \u03c9 - a)\u207a \u2264 0 \u2194 f i \u03c9 \u2264 a\n\u03c9 : \u03a9\nh\u2081 : \u2203 j, j \u2208 Set.Icc \u22a5 N \u2227 (f j \u03c9 - a)\u207a \u2208 Set.Iic 0\nh\u2082 : \u00ac\u2203 j, j \u2208 Set.Icc \u22a5 N \u2227 f j \u03c9 \u2208 Set.Iic a\n\u22a2 sInf (Set.Icc \u22a5 N \u2229 {i | (f i \u03c9 - a)\u207a \u2264 0}) = N", "state_after": "case neg\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9\u271d : \u03a9\n\u2131 : Filtration \u2115 m0\nhab : a < b\nhab' : 0 < b - a\nhf : \u2200 (\u03c9 : \u03a9) (i : \u2115), b - a \u2264 (f i \u03c9 - a)\u207a \u2194 b \u2264 f i \u03c9\nhf' : \u2200 (\u03c9 : \u03a9) (i : \u2115), (f i \u03c9 - a)\u207a \u2264 0 \u2194 f i \u03c9 \u2264 a\n\u03c9 : \u03a9\nh\u2081 : \u2203 j, j \u2208 Set.Icc \u22a5 N \u2227 (f j \u03c9 - a)\u207a \u2208 Set.Iic 0\nh\u2082 : \u00ac\u2203 j, j \u2208 Set.Icc \u22a5 N \u2227 (f j \u03c9 - a)\u207a \u2264 0\n\u22a2 sInf (Set.Icc \u22a5 N \u2229 {i | (f i \u03c9 - a)\u207a \u2264 0}) = N"}, {"tactic": "exact False.elim (h\u2082 h\u2081)", "annotated_tactic": ["exact <a>False.elim</a> (h\u2082 h\u2081)", [{"full_name": "False.elim", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [223, 21], "def_end_pos": [223, 31]}]], "state_before": "case neg\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9\u271d : \u03a9\n\u2131 : Filtration \u2115 m0\nhab : a < b\nhab' : 0 < b - a\nhf : \u2200 (\u03c9 : \u03a9) (i : \u2115), b - a \u2264 (f i \u03c9 - a)\u207a \u2194 b \u2264 f i \u03c9\nhf' : \u2200 (\u03c9 : \u03a9) (i : \u2115), (f i \u03c9 - a)\u207a \u2264 0 \u2194 f i \u03c9 \u2264 a\n\u03c9 : \u03a9\nh\u2081 : \u2203 j, j \u2208 Set.Icc \u22a5 N \u2227 (f j \u03c9 - a)\u207a \u2208 Set.Iic 0\nh\u2082 : \u00ac\u2203 j, j \u2208 Set.Icc \u22a5 N \u2227 (f j \u03c9 - a)\u207a \u2264 0\n\u22a2 sInf (Set.Icc \u22a5 N \u2229 {i | (f i \u03c9 - a)\u207a \u2264 0}) = N", "state_after": "no goals"}, {"tactic": "simp_rw [Set.mem_Iic, hf' _ _] at h\u2081", "annotated_tactic": ["simp_rw [<a>Set.mem_Iic</a>, hf' _ _] at h\u2081", [{"full_name": "Set.mem_Iic", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [136, 9], "def_end_pos": [136, 16]}]], "state_before": "case pos\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9\u271d : \u03a9\n\u2131 : Filtration \u2115 m0\nhab : a < b\nhab' : 0 < b - a\nhf : \u2200 (\u03c9 : \u03a9) (i : \u2115), b - a \u2264 (f i \u03c9 - a)\u207a \u2194 b \u2264 f i \u03c9\nhf' : \u2200 (\u03c9 : \u03a9) (i : \u2115), (f i \u03c9 - a)\u207a \u2264 0 \u2194 f i \u03c9 \u2264 a\n\u03c9 : \u03a9\nh\u2081 : \u00ac\u2203 j, j \u2208 Set.Icc \u22a5 N \u2227 (f j \u03c9 - a)\u207a \u2208 Set.Iic 0\nh\u2082 : \u2203 j, j \u2208 Set.Icc \u22a5 N \u2227 f j \u03c9 \u2208 Set.Iic a\n\u22a2 N = sInf (Set.Icc \u22a5 N \u2229 {i | f i \u03c9 \u2264 a})", "state_after": "case pos\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9\u271d : \u03a9\n\u2131 : Filtration \u2115 m0\nhab : a < b\nhab' : 0 < b - a\nhf : \u2200 (\u03c9 : \u03a9) (i : \u2115), b - a \u2264 (f i \u03c9 - a)\u207a \u2194 b \u2264 f i \u03c9\nhf' : \u2200 (\u03c9 : \u03a9) (i : \u2115), (f i \u03c9 - a)\u207a \u2264 0 \u2194 f i \u03c9 \u2264 a\n\u03c9 : \u03a9\nh\u2082 : \u2203 j, j \u2208 Set.Icc \u22a5 N \u2227 f j \u03c9 \u2208 Set.Iic a\nh\u2081 : \u00ac\u2203 j, j \u2208 Set.Icc \u22a5 N \u2227 f j \u03c9 \u2264 a\n\u22a2 N = sInf (Set.Icc \u22a5 N \u2229 {i | f i \u03c9 \u2264 a})"}, {"tactic": "exact False.elim (h\u2081 h\u2082)", "annotated_tactic": ["exact <a>False.elim</a> (h\u2081 h\u2082)", [{"full_name": "False.elim", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [223, 21], "def_end_pos": [223, 31]}]], "state_before": "case pos\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9\u271d : \u03a9\n\u2131 : Filtration \u2115 m0\nhab : a < b\nhab' : 0 < b - a\nhf : \u2200 (\u03c9 : \u03a9) (i : \u2115), b - a \u2264 (f i \u03c9 - a)\u207a \u2194 b \u2264 f i \u03c9\nhf' : \u2200 (\u03c9 : \u03a9) (i : \u2115), (f i \u03c9 - a)\u207a \u2264 0 \u2194 f i \u03c9 \u2264 a\n\u03c9 : \u03a9\nh\u2082 : \u2203 j, j \u2208 Set.Icc \u22a5 N \u2227 f j \u03c9 \u2208 Set.Iic a\nh\u2081 : \u00ac\u2203 j, j \u2208 Set.Icc \u22a5 N \u2227 f j \u03c9 \u2264 a\n\u22a2 N = sInf (Set.Icc \u22a5 N \u2229 {i | f i \u03c9 \u2264 a})", "state_after": "no goals"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case neg\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9\u271d : \u03a9\n\u2131 : Filtration \u2115 m0\nhab : a < b\nhab' : 0 < b - a\nhf : \u2200 (\u03c9 : \u03a9) (i : \u2115), b - a \u2264 (f i \u03c9 - a)\u207a \u2194 b \u2264 f i \u03c9\nhf' : \u2200 (\u03c9 : \u03a9) (i : \u2115), (f i \u03c9 - a)\u207a \u2264 0 \u2194 f i \u03c9 \u2264 a\n\u03c9 : \u03a9\nh\u2081 : \u00ac\u2203 j, j \u2208 Set.Icc \u22a5 N \u2227 (f j \u03c9 - a)\u207a \u2208 Set.Iic 0\nh\u2082 : \u00ac\u2203 j, j \u2208 Set.Icc \u22a5 N \u2227 f j \u03c9 \u2208 Set.Iic a\n\u22a2 N = N", "state_after": "no goals"}, {"tactic": "refine' \u27e8this, _\u27e9", "annotated_tactic": ["refine' \u27e8this, _\u27e9", []], "state_before": "case succ\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\nhab : a < b\nhab' : 0 < b - a\nhf : \u2200 (\u03c9 : \u03a9) (i : \u2115), b - a \u2264 (f i \u03c9 - a)\u207a \u2194 b \u2264 f i \u03c9\nhf' : \u2200 (\u03c9 : \u03a9) (i : \u2115), (f i \u03c9 - a)\u207a \u2264 0 \u2194 f i \u03c9 \u2264 a\nk : \u2115\nih :\n  upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k = upperCrossingTime a b f N k \u2227\n    lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k = lowerCrossingTime a b f N k\nthis : upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N (k + 1) = upperCrossingTime a b f N (k + 1)\n\u22a2 upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N (Nat.succ k) = upperCrossingTime a b f N (Nat.succ k) \u2227\n    lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N (Nat.succ k) = lowerCrossingTime a b f N (Nat.succ k)", "state_after": "case succ\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\nhab : a < b\nhab' : 0 < b - a\nhf : \u2200 (\u03c9 : \u03a9) (i : \u2115), b - a \u2264 (f i \u03c9 - a)\u207a \u2194 b \u2264 f i \u03c9\nhf' : \u2200 (\u03c9 : \u03a9) (i : \u2115), (f i \u03c9 - a)\u207a \u2264 0 \u2194 f i \u03c9 \u2264 a\nk : \u2115\nih :\n  upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k = upperCrossingTime a b f N k \u2227\n    lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k = lowerCrossingTime a b f N k\nthis : upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N (k + 1) = upperCrossingTime a b f N (k + 1)\n\u22a2 lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N (Nat.succ k) = lowerCrossingTime a b f N (Nat.succ k)"}, {"tactic": "ext \u03c9", "annotated_tactic": ["ext \u03c9", []], "state_before": "case succ\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\nhab : a < b\nhab' : 0 < b - a\nhf : \u2200 (\u03c9 : \u03a9) (i : \u2115), b - a \u2264 (f i \u03c9 - a)\u207a \u2194 b \u2264 f i \u03c9\nhf' : \u2200 (\u03c9 : \u03a9) (i : \u2115), (f i \u03c9 - a)\u207a \u2264 0 \u2194 f i \u03c9 \u2264 a\nk : \u2115\nih :\n  upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k = upperCrossingTime a b f N k \u2227\n    lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k = lowerCrossingTime a b f N k\nthis : upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N (k + 1) = upperCrossingTime a b f N (k + 1)\n\u22a2 lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N (Nat.succ k) = lowerCrossingTime a b f N (Nat.succ k)", "state_after": "case succ.h\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9\u271d : \u03a9\n\u2131 : Filtration \u2115 m0\nhab : a < b\nhab' : 0 < b - a\nhf : \u2200 (\u03c9 : \u03a9) (i : \u2115), b - a \u2264 (f i \u03c9 - a)\u207a \u2194 b \u2264 f i \u03c9\nhf' : \u2200 (\u03c9 : \u03a9) (i : \u2115), (f i \u03c9 - a)\u207a \u2264 0 \u2194 f i \u03c9 \u2264 a\nk : \u2115\nih :\n  upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k = upperCrossingTime a b f N k \u2227\n    lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k = lowerCrossingTime a b f N k\nthis : upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N (k + 1) = upperCrossingTime a b f N (k + 1)\n\u03c9 : \u03a9\n\u22a2 lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N (Nat.succ k) \u03c9 = lowerCrossingTime a b f N (Nat.succ k) \u03c9"}, {"tactic": "simp only [lowerCrossingTime, this, hitting, Set.mem_Iic]", "annotated_tactic": ["simp only [<a>lowerCrossingTime</a>, this, <a>hitting</a>, <a>Set.mem_Iic</a>]", [{"full_name": "MeasureTheory.lowerCrossingTime", "def_path": "Mathlib/Probability/Martingale/Upcrossing.lean", "def_pos": [148, 19], "def_end_pos": [148, 36]}, {"full_name": "MeasureTheory.hitting", "def_path": "Mathlib/Probability/Process/HittingTime.lean", "def_pos": [51, 19], "def_end_pos": [51, 26]}, {"full_name": "Set.mem_Iic", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [136, 9], "def_end_pos": [136, 16]}]], "state_before": "case succ.h\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9\u271d : \u03a9\n\u2131 : Filtration \u2115 m0\nhab : a < b\nhab' : 0 < b - a\nhf : \u2200 (\u03c9 : \u03a9) (i : \u2115), b - a \u2264 (f i \u03c9 - a)\u207a \u2194 b \u2264 f i \u03c9\nhf' : \u2200 (\u03c9 : \u03a9) (i : \u2115), (f i \u03c9 - a)\u207a \u2264 0 \u2194 f i \u03c9 \u2264 a\nk : \u2115\nih :\n  upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k = upperCrossingTime a b f N k \u2227\n    lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k = lowerCrossingTime a b f N k\nthis : upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N (k + 1) = upperCrossingTime a b f N (k + 1)\n\u03c9 : \u03a9\n\u22a2 lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N (Nat.succ k) \u03c9 = lowerCrossingTime a b f N (Nat.succ k) \u03c9", "state_after": "case succ.h\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9\u271d : \u03a9\n\u2131 : Filtration \u2115 m0\nhab : a < b\nhab' : 0 < b - a\nhf : \u2200 (\u03c9 : \u03a9) (i : \u2115), b - a \u2264 (f i \u03c9 - a)\u207a \u2194 b \u2264 f i \u03c9\nhf' : \u2200 (\u03c9 : \u03a9) (i : \u2115), (f i \u03c9 - a)\u207a \u2264 0 \u2194 f i \u03c9 \u2264 a\nk : \u2115\nih :\n  upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k = upperCrossingTime a b f N k \u2227\n    lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k = lowerCrossingTime a b f N k\nthis : upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N (k + 1) = upperCrossingTime a b f N (k + 1)\n\u03c9 : \u03a9\n\u22a2 (if\n        \u2203 j,\n          j \u2208 Set.Icc (upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N (Nat.succ k) \u03c9) N \u2227\n            (f j \u03c9 - a)\u207a \u2208 Set.Iic 0 then\n      sInf (Set.Icc (upperCrossingTime a b f N (k + 1) \u03c9) N \u2229 {i | (f i \u03c9 - a)\u207a \u2264 0})\n    else N) =\n    if \u2203 j, j \u2208 Set.Icc (upperCrossingTime a b f N (Nat.succ k) \u03c9) N \u2227 f j \u03c9 \u2208 Set.Iic a then\n      sInf (Set.Icc (upperCrossingTime a b f N (Nat.succ k) \u03c9) N \u2229 {i | f i \u03c9 \u2264 a})\n    else N"}, {"tactic": "split_ifs with h\u2081 h\u2082 h\u2082", "annotated_tactic": ["split_ifs with h\u2081 h\u2082 h\u2082", []], "state_before": "case succ.h\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9\u271d : \u03a9\n\u2131 : Filtration \u2115 m0\nhab : a < b\nhab' : 0 < b - a\nhf : \u2200 (\u03c9 : \u03a9) (i : \u2115), b - a \u2264 (f i \u03c9 - a)\u207a \u2194 b \u2264 f i \u03c9\nhf' : \u2200 (\u03c9 : \u03a9) (i : \u2115), (f i \u03c9 - a)\u207a \u2264 0 \u2194 f i \u03c9 \u2264 a\nk : \u2115\nih :\n  upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k = upperCrossingTime a b f N k \u2227\n    lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k = lowerCrossingTime a b f N k\nthis : upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N (k + 1) = upperCrossingTime a b f N (k + 1)\n\u03c9 : \u03a9\n\u22a2 (if\n        \u2203 j,\n          j \u2208 Set.Icc (upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N (Nat.succ k) \u03c9) N \u2227\n            (f j \u03c9 - a)\u207a \u2208 Set.Iic 0 then\n      sInf (Set.Icc (upperCrossingTime a b f N (k + 1) \u03c9) N \u2229 {i | (f i \u03c9 - a)\u207a \u2264 0})\n    else N) =\n    if \u2203 j, j \u2208 Set.Icc (upperCrossingTime a b f N (Nat.succ k) \u03c9) N \u2227 f j \u03c9 \u2208 Set.Iic a then\n      sInf (Set.Icc (upperCrossingTime a b f N (Nat.succ k) \u03c9) N \u2229 {i | f i \u03c9 \u2264 a})\n    else N", "state_after": "case pos\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9\u271d : \u03a9\n\u2131 : Filtration \u2115 m0\nhab : a < b\nhab' : 0 < b - a\nhf : \u2200 (\u03c9 : \u03a9) (i : \u2115), b - a \u2264 (f i \u03c9 - a)\u207a \u2194 b \u2264 f i \u03c9\nhf' : \u2200 (\u03c9 : \u03a9) (i : \u2115), (f i \u03c9 - a)\u207a \u2264 0 \u2194 f i \u03c9 \u2264 a\nk : \u2115\nih :\n  upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k = upperCrossingTime a b f N k \u2227\n    lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k = lowerCrossingTime a b f N k\nthis : upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N (k + 1) = upperCrossingTime a b f N (k + 1)\n\u03c9 : \u03a9\nh\u2081 :\n  \u2203 j, j \u2208 Set.Icc (upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N (Nat.succ k) \u03c9) N \u2227 (f j \u03c9 - a)\u207a \u2208 Set.Iic 0\nh\u2082 : \u2203 j, j \u2208 Set.Icc (upperCrossingTime a b f N (Nat.succ k) \u03c9) N \u2227 f j \u03c9 \u2208 Set.Iic a\n\u22a2 sInf (Set.Icc (upperCrossingTime a b f N (k + 1) \u03c9) N \u2229 {i | (f i \u03c9 - a)\u207a \u2264 0}) =\n    sInf (Set.Icc (upperCrossingTime a b f N (Nat.succ k) \u03c9) N \u2229 {i | f i \u03c9 \u2264 a})\n\ncase neg\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9\u271d : \u03a9\n\u2131 : Filtration \u2115 m0\nhab : a < b\nhab' : 0 < b - a\nhf : \u2200 (\u03c9 : \u03a9) (i : \u2115), b - a \u2264 (f i \u03c9 - a)\u207a \u2194 b \u2264 f i \u03c9\nhf' : \u2200 (\u03c9 : \u03a9) (i : \u2115), (f i \u03c9 - a)\u207a \u2264 0 \u2194 f i \u03c9 \u2264 a\nk : \u2115\nih :\n  upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k = upperCrossingTime a b f N k \u2227\n    lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k = lowerCrossingTime a b f N k\nthis : upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N (k + 1) = upperCrossingTime a b f N (k + 1)\n\u03c9 : \u03a9\nh\u2081 :\n  \u2203 j, j \u2208 Set.Icc (upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N (Nat.succ k) \u03c9) N \u2227 (f j \u03c9 - a)\u207a \u2208 Set.Iic 0\nh\u2082 : \u00ac\u2203 j, j \u2208 Set.Icc (upperCrossingTime a b f N (Nat.succ k) \u03c9) N \u2227 f j \u03c9 \u2208 Set.Iic a\n\u22a2 sInf (Set.Icc (upperCrossingTime a b f N (k + 1) \u03c9) N \u2229 {i | (f i \u03c9 - a)\u207a \u2264 0}) = N\n\ncase pos\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9\u271d : \u03a9\n\u2131 : Filtration \u2115 m0\nhab : a < b\nhab' : 0 < b - a\nhf : \u2200 (\u03c9 : \u03a9) (i : \u2115), b - a \u2264 (f i \u03c9 - a)\u207a \u2194 b \u2264 f i \u03c9\nhf' : \u2200 (\u03c9 : \u03a9) (i : \u2115), (f i \u03c9 - a)\u207a \u2264 0 \u2194 f i \u03c9 \u2264 a\nk : \u2115\nih :\n  upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k = upperCrossingTime a b f N k \u2227\n    lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k = lowerCrossingTime a b f N k\nthis : upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N (k + 1) = upperCrossingTime a b f N (k + 1)\n\u03c9 : \u03a9\nh\u2081 :\n  \u00ac\u2203 j,\n      j \u2208 Set.Icc (upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N (Nat.succ k) \u03c9) N \u2227 (f j \u03c9 - a)\u207a \u2208 Set.Iic 0\nh\u2082 : \u2203 j, j \u2208 Set.Icc (upperCrossingTime a b f N (Nat.succ k) \u03c9) N \u2227 f j \u03c9 \u2208 Set.Iic a\n\u22a2 N = sInf (Set.Icc (upperCrossingTime a b f N (Nat.succ k) \u03c9) N \u2229 {i | f i \u03c9 \u2264 a})\n\ncase neg\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9\u271d : \u03a9\n\u2131 : Filtration \u2115 m0\nhab : a < b\nhab' : 0 < b - a\nhf : \u2200 (\u03c9 : \u03a9) (i : \u2115), b - a \u2264 (f i \u03c9 - a)\u207a \u2194 b \u2264 f i \u03c9\nhf' : \u2200 (\u03c9 : \u03a9) (i : \u2115), (f i \u03c9 - a)\u207a \u2264 0 \u2194 f i \u03c9 \u2264 a\nk : \u2115\nih :\n  upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k = upperCrossingTime a b f N k \u2227\n    lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k = lowerCrossingTime a b f N k\nthis : upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N (k + 1) = upperCrossingTime a b f N (k + 1)\n\u03c9 : \u03a9\nh\u2081 :\n  \u00ac\u2203 j,\n      j \u2208 Set.Icc (upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N (Nat.succ k) \u03c9) N \u2227 (f j \u03c9 - a)\u207a \u2208 Set.Iic 0\nh\u2082 : \u00ac\u2203 j, j \u2208 Set.Icc (upperCrossingTime a b f N (Nat.succ k) \u03c9) N \u2227 f j \u03c9 \u2208 Set.Iic a\n\u22a2 N = N"}, {"tactic": "ext \u03c9", "annotated_tactic": ["ext \u03c9", []], "state_before": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\nhab : a < b\nhab' : 0 < b - a\nhf : \u2200 (\u03c9 : \u03a9) (i : \u2115), b - a \u2264 (f i \u03c9 - a)\u207a \u2194 b \u2264 f i \u03c9\nhf' : \u2200 (\u03c9 : \u03a9) (i : \u2115), (f i \u03c9 - a)\u207a \u2264 0 \u2194 f i \u03c9 \u2264 a\nk : \u2115\nih :\n  upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k = upperCrossingTime a b f N k \u2227\n    lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k = lowerCrossingTime a b f N k\n\u22a2 upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N (k + 1) = upperCrossingTime a b f N (k + 1)", "state_after": "case h\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9\u271d : \u03a9\n\u2131 : Filtration \u2115 m0\nhab : a < b\nhab' : 0 < b - a\nhf : \u2200 (\u03c9 : \u03a9) (i : \u2115), b - a \u2264 (f i \u03c9 - a)\u207a \u2194 b \u2264 f i \u03c9\nhf' : \u2200 (\u03c9 : \u03a9) (i : \u2115), (f i \u03c9 - a)\u207a \u2264 0 \u2194 f i \u03c9 \u2264 a\nk : \u2115\nih :\n  upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k = upperCrossingTime a b f N k \u2227\n    lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k = lowerCrossingTime a b f N k\n\u03c9 : \u03a9\n\u22a2 upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N (k + 1) \u03c9 = upperCrossingTime a b f N (k + 1) \u03c9"}, {"tactic": "simp only [upperCrossingTime_succ_eq, \u2190 ih.2, hitting, Set.mem_Ici, tsub_le_iff_right]", "annotated_tactic": ["simp only [<a>upperCrossingTime_succ_eq</a>, \u2190 ih.2, <a>hitting</a>, <a>Set.mem_Ici</a>, <a>tsub_le_iff_right</a>]", [{"full_name": "MeasureTheory.upperCrossingTime_succ_eq", "def_path": "Mathlib/Probability/Martingale/Upcrossing.lean", "def_pos": [173, 9], "def_end_pos": [173, 34]}, {"full_name": "MeasureTheory.hitting", "def_path": "Mathlib/Probability/Process/HittingTime.lean", "def_pos": [51, 19], "def_end_pos": [51, 26]}, {"full_name": "Set.mem_Ici", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [146, 9], "def_end_pos": [146, 16]}, {"full_name": "tsub_le_iff_right", "def_path": "Mathlib/Algebra/Order/Sub/Defs.lean", "def_pos": [65, 9], "def_end_pos": [65, 26]}]], "state_before": "case h\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9\u271d : \u03a9\n\u2131 : Filtration \u2115 m0\nhab : a < b\nhab' : 0 < b - a\nhf : \u2200 (\u03c9 : \u03a9) (i : \u2115), b - a \u2264 (f i \u03c9 - a)\u207a \u2194 b \u2264 f i \u03c9\nhf' : \u2200 (\u03c9 : \u03a9) (i : \u2115), (f i \u03c9 - a)\u207a \u2264 0 \u2194 f i \u03c9 \u2264 a\nk : \u2115\nih :\n  upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k = upperCrossingTime a b f N k \u2227\n    lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k = lowerCrossingTime a b f N k\n\u03c9 : \u03a9\n\u22a2 upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N (k + 1) \u03c9 = upperCrossingTime a b f N (k + 1) \u03c9", "state_after": "case h\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9\u271d : \u03a9\n\u2131 : Filtration \u2115 m0\nhab : a < b\nhab' : 0 < b - a\nhf : \u2200 (\u03c9 : \u03a9) (i : \u2115), b - a \u2264 (f i \u03c9 - a)\u207a \u2194 b \u2264 f i \u03c9\nhf' : \u2200 (\u03c9 : \u03a9) (i : \u2115), (f i \u03c9 - a)\u207a \u2264 0 \u2194 f i \u03c9 \u2264 a\nk : \u2115\nih :\n  upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k = upperCrossingTime a b f N k \u2227\n    lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k = lowerCrossingTime a b f N k\n\u03c9 : \u03a9\n\u22a2 (if\n        \u2203 j,\n          j \u2208 Set.Icc (lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k \u03c9) N \u2227\n            (f j \u03c9 - a)\u207a \u2208 Set.Ici (b - a) then\n      sInf (Set.Icc (lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k \u03c9) N \u2229 {i | b \u2264 (f i \u03c9 - a)\u207a + a})\n    else N) =\n    if \u2203 j, j \u2208 Set.Icc (lowerCrossingTime a b f N k \u03c9) N \u2227 f j \u03c9 \u2208 Set.Ici b then\n      sInf (Set.Icc (lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k \u03c9) N \u2229 {i | b \u2264 f i \u03c9})\n    else N"}, {"tactic": "split_ifs with h\u2081 h\u2082 h\u2082", "annotated_tactic": ["split_ifs with h\u2081 h\u2082 h\u2082", []], "state_before": "case h\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9\u271d : \u03a9\n\u2131 : Filtration \u2115 m0\nhab : a < b\nhab' : 0 < b - a\nhf : \u2200 (\u03c9 : \u03a9) (i : \u2115), b - a \u2264 (f i \u03c9 - a)\u207a \u2194 b \u2264 f i \u03c9\nhf' : \u2200 (\u03c9 : \u03a9) (i : \u2115), (f i \u03c9 - a)\u207a \u2264 0 \u2194 f i \u03c9 \u2264 a\nk : \u2115\nih :\n  upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k = upperCrossingTime a b f N k \u2227\n    lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k = lowerCrossingTime a b f N k\n\u03c9 : \u03a9\n\u22a2 (if\n        \u2203 j,\n          j \u2208 Set.Icc (lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k \u03c9) N \u2227\n            (f j \u03c9 - a)\u207a \u2208 Set.Ici (b - a) then\n      sInf (Set.Icc (lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k \u03c9) N \u2229 {i | b \u2264 (f i \u03c9 - a)\u207a + a})\n    else N) =\n    if \u2203 j, j \u2208 Set.Icc (lowerCrossingTime a b f N k \u03c9) N \u2227 f j \u03c9 \u2208 Set.Ici b then\n      sInf (Set.Icc (lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k \u03c9) N \u2229 {i | b \u2264 f i \u03c9})\n    else N", "state_after": "case pos\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9\u271d : \u03a9\n\u2131 : Filtration \u2115 m0\nhab : a < b\nhab' : 0 < b - a\nhf : \u2200 (\u03c9 : \u03a9) (i : \u2115), b - a \u2264 (f i \u03c9 - a)\u207a \u2194 b \u2264 f i \u03c9\nhf' : \u2200 (\u03c9 : \u03a9) (i : \u2115), (f i \u03c9 - a)\u207a \u2264 0 \u2194 f i \u03c9 \u2264 a\nk : \u2115\nih :\n  upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k = upperCrossingTime a b f N k \u2227\n    lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k = lowerCrossingTime a b f N k\n\u03c9 : \u03a9\nh\u2081 : \u2203 j, j \u2208 Set.Icc (lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k \u03c9) N \u2227 (f j \u03c9 - a)\u207a \u2208 Set.Ici (b - a)\nh\u2082 : \u2203 j, j \u2208 Set.Icc (lowerCrossingTime a b f N k \u03c9) N \u2227 f j \u03c9 \u2208 Set.Ici b\n\u22a2 sInf (Set.Icc (lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k \u03c9) N \u2229 {i | b \u2264 (f i \u03c9 - a)\u207a + a}) =\n    sInf (Set.Icc (lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k \u03c9) N \u2229 {i | b \u2264 f i \u03c9})\n\ncase neg\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9\u271d : \u03a9\n\u2131 : Filtration \u2115 m0\nhab : a < b\nhab' : 0 < b - a\nhf : \u2200 (\u03c9 : \u03a9) (i : \u2115), b - a \u2264 (f i \u03c9 - a)\u207a \u2194 b \u2264 f i \u03c9\nhf' : \u2200 (\u03c9 : \u03a9) (i : \u2115), (f i \u03c9 - a)\u207a \u2264 0 \u2194 f i \u03c9 \u2264 a\nk : \u2115\nih :\n  upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k = upperCrossingTime a b f N k \u2227\n    lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k = lowerCrossingTime a b f N k\n\u03c9 : \u03a9\nh\u2081 : \u2203 j, j \u2208 Set.Icc (lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k \u03c9) N \u2227 (f j \u03c9 - a)\u207a \u2208 Set.Ici (b - a)\nh\u2082 : \u00ac\u2203 j, j \u2208 Set.Icc (lowerCrossingTime a b f N k \u03c9) N \u2227 f j \u03c9 \u2208 Set.Ici b\n\u22a2 sInf (Set.Icc (lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k \u03c9) N \u2229 {i | b \u2264 (f i \u03c9 - a)\u207a + a}) = N\n\ncase pos\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9\u271d : \u03a9\n\u2131 : Filtration \u2115 m0\nhab : a < b\nhab' : 0 < b - a\nhf : \u2200 (\u03c9 : \u03a9) (i : \u2115), b - a \u2264 (f i \u03c9 - a)\u207a \u2194 b \u2264 f i \u03c9\nhf' : \u2200 (\u03c9 : \u03a9) (i : \u2115), (f i \u03c9 - a)\u207a \u2264 0 \u2194 f i \u03c9 \u2264 a\nk : \u2115\nih :\n  upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k = upperCrossingTime a b f N k \u2227\n    lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k = lowerCrossingTime a b f N k\n\u03c9 : \u03a9\nh\u2081 : \u00ac\u2203 j, j \u2208 Set.Icc (lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k \u03c9) N \u2227 (f j \u03c9 - a)\u207a \u2208 Set.Ici (b - a)\nh\u2082 : \u2203 j, j \u2208 Set.Icc (lowerCrossingTime a b f N k \u03c9) N \u2227 f j \u03c9 \u2208 Set.Ici b\n\u22a2 N = sInf (Set.Icc (lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k \u03c9) N \u2229 {i | b \u2264 f i \u03c9})\n\ncase neg\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9\u271d : \u03a9\n\u2131 : Filtration \u2115 m0\nhab : a < b\nhab' : 0 < b - a\nhf : \u2200 (\u03c9 : \u03a9) (i : \u2115), b - a \u2264 (f i \u03c9 - a)\u207a \u2194 b \u2264 f i \u03c9\nhf' : \u2200 (\u03c9 : \u03a9) (i : \u2115), (f i \u03c9 - a)\u207a \u2264 0 \u2194 f i \u03c9 \u2264 a\nk : \u2115\nih :\n  upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k = upperCrossingTime a b f N k \u2227\n    lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k = lowerCrossingTime a b f N k\n\u03c9 : \u03a9\nh\u2081 : \u00ac\u2203 j, j \u2208 Set.Icc (lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k \u03c9) N \u2227 (f j \u03c9 - a)\u207a \u2208 Set.Ici (b - a)\nh\u2082 : \u00ac\u2203 j, j \u2208 Set.Icc (lowerCrossingTime a b f N k \u03c9) N \u2227 f j \u03c9 \u2208 Set.Ici b\n\u22a2 N = N"}, {"tactic": "simp_rw [\u2190 sub_le_iff_le_add, hf \u03c9]", "annotated_tactic": ["simp_rw [\u2190 <a>sub_le_iff_le_add</a>, hf \u03c9]", [{"full_name": "sub_le_iff_le_add", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [750, 3], "def_end_pos": [750, 14]}]], "state_before": "case pos\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9\u271d : \u03a9\n\u2131 : Filtration \u2115 m0\nhab : a < b\nhab' : 0 < b - a\nhf : \u2200 (\u03c9 : \u03a9) (i : \u2115), b - a \u2264 (f i \u03c9 - a)\u207a \u2194 b \u2264 f i \u03c9\nhf' : \u2200 (\u03c9 : \u03a9) (i : \u2115), (f i \u03c9 - a)\u207a \u2264 0 \u2194 f i \u03c9 \u2264 a\nk : \u2115\nih :\n  upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k = upperCrossingTime a b f N k \u2227\n    lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k = lowerCrossingTime a b f N k\n\u03c9 : \u03a9\nh\u2081 : \u2203 j, j \u2208 Set.Icc (lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k \u03c9) N \u2227 (f j \u03c9 - a)\u207a \u2208 Set.Ici (b - a)\nh\u2082 : \u2203 j, j \u2208 Set.Icc (lowerCrossingTime a b f N k \u03c9) N \u2227 f j \u03c9 \u2208 Set.Ici b\n\u22a2 sInf (Set.Icc (lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k \u03c9) N \u2229 {i | b \u2264 (f i \u03c9 - a)\u207a + a}) =\n    sInf (Set.Icc (lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k \u03c9) N \u2229 {i | b \u2264 f i \u03c9})", "state_after": "no goals"}, {"tactic": "refine' False.elim (h\u2082 _)", "annotated_tactic": ["refine' <a>False.elim</a> (h\u2082 _)", [{"full_name": "False.elim", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [223, 21], "def_end_pos": [223, 31]}]], "state_before": "case neg\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9\u271d : \u03a9\n\u2131 : Filtration \u2115 m0\nhab : a < b\nhab' : 0 < b - a\nhf : \u2200 (\u03c9 : \u03a9) (i : \u2115), b - a \u2264 (f i \u03c9 - a)\u207a \u2194 b \u2264 f i \u03c9\nhf' : \u2200 (\u03c9 : \u03a9) (i : \u2115), (f i \u03c9 - a)\u207a \u2264 0 \u2194 f i \u03c9 \u2264 a\nk : \u2115\nih :\n  upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k = upperCrossingTime a b f N k \u2227\n    lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k = lowerCrossingTime a b f N k\n\u03c9 : \u03a9\nh\u2081 : \u2203 j, j \u2208 Set.Icc (lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k \u03c9) N \u2227 (f j \u03c9 - a)\u207a \u2208 Set.Ici (b - a)\nh\u2082 : \u00ac\u2203 j, j \u2208 Set.Icc (lowerCrossingTime a b f N k \u03c9) N \u2227 f j \u03c9 \u2208 Set.Ici b\n\u22a2 sInf (Set.Icc (lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k \u03c9) N \u2229 {i | b \u2264 (f i \u03c9 - a)\u207a + a}) = N", "state_after": "case neg\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9\u271d : \u03a9\n\u2131 : Filtration \u2115 m0\nhab : a < b\nhab' : 0 < b - a\nhf : \u2200 (\u03c9 : \u03a9) (i : \u2115), b - a \u2264 (f i \u03c9 - a)\u207a \u2194 b \u2264 f i \u03c9\nhf' : \u2200 (\u03c9 : \u03a9) (i : \u2115), (f i \u03c9 - a)\u207a \u2264 0 \u2194 f i \u03c9 \u2264 a\nk : \u2115\nih :\n  upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k = upperCrossingTime a b f N k \u2227\n    lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k = lowerCrossingTime a b f N k\n\u03c9 : \u03a9\nh\u2081 : \u2203 j, j \u2208 Set.Icc (lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k \u03c9) N \u2227 (f j \u03c9 - a)\u207a \u2208 Set.Ici (b - a)\nh\u2082 : \u00ac\u2203 j, j \u2208 Set.Icc (lowerCrossingTime a b f N k \u03c9) N \u2227 f j \u03c9 \u2208 Set.Ici b\n\u22a2 \u2203 j, j \u2208 Set.Icc (lowerCrossingTime a b f N k \u03c9) N \u2227 f j \u03c9 \u2208 Set.Ici b"}, {"tactic": "simp_all only [Set.mem_Ici]", "annotated_tactic": ["simp_all only [<a>Set.mem_Ici</a>]", [{"full_name": "Set.mem_Ici", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [146, 9], "def_end_pos": [146, 16]}]], "state_before": "case neg\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9\u271d : \u03a9\n\u2131 : Filtration \u2115 m0\nhab : a < b\nhab' : 0 < b - a\nhf : \u2200 (\u03c9 : \u03a9) (i : \u2115), b - a \u2264 (f i \u03c9 - a)\u207a \u2194 b \u2264 f i \u03c9\nhf' : \u2200 (\u03c9 : \u03a9) (i : \u2115), (f i \u03c9 - a)\u207a \u2264 0 \u2194 f i \u03c9 \u2264 a\nk : \u2115\nih :\n  upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k = upperCrossingTime a b f N k \u2227\n    lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k = lowerCrossingTime a b f N k\n\u03c9 : \u03a9\nh\u2081 : \u2203 j, j \u2208 Set.Icc (lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k \u03c9) N \u2227 (f j \u03c9 - a)\u207a \u2208 Set.Ici (b - a)\nh\u2082 : \u00ac\u2203 j, j \u2208 Set.Icc (lowerCrossingTime a b f N k \u03c9) N \u2227 f j \u03c9 \u2208 Set.Ici b\n\u22a2 \u2203 j, j \u2208 Set.Icc (lowerCrossingTime a b f N k \u03c9) N \u2227 f j \u03c9 \u2208 Set.Ici b", "state_after": "no goals"}, {"tactic": "refine' False.elim (h\u2081 _)", "annotated_tactic": ["refine' <a>False.elim</a> (h\u2081 _)", [{"full_name": "False.elim", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [223, 21], "def_end_pos": [223, 31]}]], "state_before": "case pos\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9\u271d : \u03a9\n\u2131 : Filtration \u2115 m0\nhab : a < b\nhab' : 0 < b - a\nhf : \u2200 (\u03c9 : \u03a9) (i : \u2115), b - a \u2264 (f i \u03c9 - a)\u207a \u2194 b \u2264 f i \u03c9\nhf' : \u2200 (\u03c9 : \u03a9) (i : \u2115), (f i \u03c9 - a)\u207a \u2264 0 \u2194 f i \u03c9 \u2264 a\nk : \u2115\nih :\n  upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k = upperCrossingTime a b f N k \u2227\n    lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k = lowerCrossingTime a b f N k\n\u03c9 : \u03a9\nh\u2081 : \u00ac\u2203 j, j \u2208 Set.Icc (lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k \u03c9) N \u2227 (f j \u03c9 - a)\u207a \u2208 Set.Ici (b - a)\nh\u2082 : \u2203 j, j \u2208 Set.Icc (lowerCrossingTime a b f N k \u03c9) N \u2227 f j \u03c9 \u2208 Set.Ici b\n\u22a2 N = sInf (Set.Icc (lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k \u03c9) N \u2229 {i | b \u2264 f i \u03c9})", "state_after": "case pos\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9\u271d : \u03a9\n\u2131 : Filtration \u2115 m0\nhab : a < b\nhab' : 0 < b - a\nhf : \u2200 (\u03c9 : \u03a9) (i : \u2115), b - a \u2264 (f i \u03c9 - a)\u207a \u2194 b \u2264 f i \u03c9\nhf' : \u2200 (\u03c9 : \u03a9) (i : \u2115), (f i \u03c9 - a)\u207a \u2264 0 \u2194 f i \u03c9 \u2264 a\nk : \u2115\nih :\n  upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k = upperCrossingTime a b f N k \u2227\n    lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k = lowerCrossingTime a b f N k\n\u03c9 : \u03a9\nh\u2081 : \u00ac\u2203 j, j \u2208 Set.Icc (lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k \u03c9) N \u2227 (f j \u03c9 - a)\u207a \u2208 Set.Ici (b - a)\nh\u2082 : \u2203 j, j \u2208 Set.Icc (lowerCrossingTime a b f N k \u03c9) N \u2227 f j \u03c9 \u2208 Set.Ici b\n\u22a2 \u2203 j, j \u2208 Set.Icc (lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k \u03c9) N \u2227 (f j \u03c9 - a)\u207a \u2208 Set.Ici (b - a)"}, {"tactic": "simp_all only [Set.mem_Ici]", "annotated_tactic": ["simp_all only [<a>Set.mem_Ici</a>]", [{"full_name": "Set.mem_Ici", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [146, 9], "def_end_pos": [146, 16]}]], "state_before": "case pos\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9\u271d : \u03a9\n\u2131 : Filtration \u2115 m0\nhab : a < b\nhab' : 0 < b - a\nhf : \u2200 (\u03c9 : \u03a9) (i : \u2115), b - a \u2264 (f i \u03c9 - a)\u207a \u2194 b \u2264 f i \u03c9\nhf' : \u2200 (\u03c9 : \u03a9) (i : \u2115), (f i \u03c9 - a)\u207a \u2264 0 \u2194 f i \u03c9 \u2264 a\nk : \u2115\nih :\n  upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k = upperCrossingTime a b f N k \u2227\n    lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k = lowerCrossingTime a b f N k\n\u03c9 : \u03a9\nh\u2081 : \u00ac\u2203 j, j \u2208 Set.Icc (lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k \u03c9) N \u2227 (f j \u03c9 - a)\u207a \u2208 Set.Ici (b - a)\nh\u2082 : \u2203 j, j \u2208 Set.Icc (lowerCrossingTime a b f N k \u03c9) N \u2227 f j \u03c9 \u2208 Set.Ici b\n\u22a2 \u2203 j, j \u2208 Set.Icc (lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k \u03c9) N \u2227 (f j \u03c9 - a)\u207a \u2208 Set.Ici (b - a)", "state_after": "no goals"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case neg\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9\u271d : \u03a9\n\u2131 : Filtration \u2115 m0\nhab : a < b\nhab' : 0 < b - a\nhf : \u2200 (\u03c9 : \u03a9) (i : \u2115), b - a \u2264 (f i \u03c9 - a)\u207a \u2194 b \u2264 f i \u03c9\nhf' : \u2200 (\u03c9 : \u03a9) (i : \u2115), (f i \u03c9 - a)\u207a \u2264 0 \u2194 f i \u03c9 \u2264 a\nk : \u2115\nih :\n  upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k = upperCrossingTime a b f N k \u2227\n    lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k = lowerCrossingTime a b f N k\n\u03c9 : \u03a9\nh\u2081 : \u00ac\u2203 j, j \u2208 Set.Icc (lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k \u03c9) N \u2227 (f j \u03c9 - a)\u207a \u2208 Set.Ici (b - a)\nh\u2082 : \u00ac\u2203 j, j \u2208 Set.Icc (lowerCrossingTime a b f N k \u03c9) N \u2227 f j \u03c9 \u2208 Set.Ici b\n\u22a2 N = N", "state_after": "no goals"}, {"tactic": "simp_rw [hf' \u03c9]", "annotated_tactic": ["simp_rw [hf' \u03c9]", []], "state_before": "case pos\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9\u271d : \u03a9\n\u2131 : Filtration \u2115 m0\nhab : a < b\nhab' : 0 < b - a\nhf : \u2200 (\u03c9 : \u03a9) (i : \u2115), b - a \u2264 (f i \u03c9 - a)\u207a \u2194 b \u2264 f i \u03c9\nhf' : \u2200 (\u03c9 : \u03a9) (i : \u2115), (f i \u03c9 - a)\u207a \u2264 0 \u2194 f i \u03c9 \u2264 a\nk : \u2115\nih :\n  upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k = upperCrossingTime a b f N k \u2227\n    lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k = lowerCrossingTime a b f N k\nthis : upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N (k + 1) = upperCrossingTime a b f N (k + 1)\n\u03c9 : \u03a9\nh\u2081 :\n  \u2203 j, j \u2208 Set.Icc (upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N (Nat.succ k) \u03c9) N \u2227 (f j \u03c9 - a)\u207a \u2208 Set.Iic 0\nh\u2082 : \u2203 j, j \u2208 Set.Icc (upperCrossingTime a b f N (Nat.succ k) \u03c9) N \u2227 f j \u03c9 \u2208 Set.Iic a\n\u22a2 sInf (Set.Icc (upperCrossingTime a b f N (k + 1) \u03c9) N \u2229 {i | (f i \u03c9 - a)\u207a \u2264 0}) =\n    sInf (Set.Icc (upperCrossingTime a b f N (Nat.succ k) \u03c9) N \u2229 {i | f i \u03c9 \u2264 a})", "state_after": "no goals"}, {"tactic": "refine' False.elim (h\u2082 _)", "annotated_tactic": ["refine' <a>False.elim</a> (h\u2082 _)", [{"full_name": "False.elim", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [223, 21], "def_end_pos": [223, 31]}]], "state_before": "case neg\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9\u271d : \u03a9\n\u2131 : Filtration \u2115 m0\nhab : a < b\nhab' : 0 < b - a\nhf : \u2200 (\u03c9 : \u03a9) (i : \u2115), b - a \u2264 (f i \u03c9 - a)\u207a \u2194 b \u2264 f i \u03c9\nhf' : \u2200 (\u03c9 : \u03a9) (i : \u2115), (f i \u03c9 - a)\u207a \u2264 0 \u2194 f i \u03c9 \u2264 a\nk : \u2115\nih :\n  upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k = upperCrossingTime a b f N k \u2227\n    lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k = lowerCrossingTime a b f N k\nthis : upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N (k + 1) = upperCrossingTime a b f N (k + 1)\n\u03c9 : \u03a9\nh\u2081 :\n  \u2203 j, j \u2208 Set.Icc (upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N (Nat.succ k) \u03c9) N \u2227 (f j \u03c9 - a)\u207a \u2208 Set.Iic 0\nh\u2082 : \u00ac\u2203 j, j \u2208 Set.Icc (upperCrossingTime a b f N (Nat.succ k) \u03c9) N \u2227 f j \u03c9 \u2208 Set.Iic a\n\u22a2 sInf (Set.Icc (upperCrossingTime a b f N (k + 1) \u03c9) N \u2229 {i | (f i \u03c9 - a)\u207a \u2264 0}) = N", "state_after": "case neg\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9\u271d : \u03a9\n\u2131 : Filtration \u2115 m0\nhab : a < b\nhab' : 0 < b - a\nhf : \u2200 (\u03c9 : \u03a9) (i : \u2115), b - a \u2264 (f i \u03c9 - a)\u207a \u2194 b \u2264 f i \u03c9\nhf' : \u2200 (\u03c9 : \u03a9) (i : \u2115), (f i \u03c9 - a)\u207a \u2264 0 \u2194 f i \u03c9 \u2264 a\nk : \u2115\nih :\n  upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k = upperCrossingTime a b f N k \u2227\n    lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k = lowerCrossingTime a b f N k\nthis : upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N (k + 1) = upperCrossingTime a b f N (k + 1)\n\u03c9 : \u03a9\nh\u2081 :\n  \u2203 j, j \u2208 Set.Icc (upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N (Nat.succ k) \u03c9) N \u2227 (f j \u03c9 - a)\u207a \u2208 Set.Iic 0\nh\u2082 : \u00ac\u2203 j, j \u2208 Set.Icc (upperCrossingTime a b f N (Nat.succ k) \u03c9) N \u2227 f j \u03c9 \u2208 Set.Iic a\n\u22a2 \u2203 j, j \u2208 Set.Icc (upperCrossingTime a b f N (Nat.succ k) \u03c9) N \u2227 f j \u03c9 \u2208 Set.Iic a"}, {"tactic": "simp_all only [Set.mem_Iic]", "annotated_tactic": ["simp_all only [<a>Set.mem_Iic</a>]", [{"full_name": "Set.mem_Iic", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [136, 9], "def_end_pos": [136, 16]}]], "state_before": "case neg\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9\u271d : \u03a9\n\u2131 : Filtration \u2115 m0\nhab : a < b\nhab' : 0 < b - a\nhf : \u2200 (\u03c9 : \u03a9) (i : \u2115), b - a \u2264 (f i \u03c9 - a)\u207a \u2194 b \u2264 f i \u03c9\nhf' : \u2200 (\u03c9 : \u03a9) (i : \u2115), (f i \u03c9 - a)\u207a \u2264 0 \u2194 f i \u03c9 \u2264 a\nk : \u2115\nih :\n  upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k = upperCrossingTime a b f N k \u2227\n    lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k = lowerCrossingTime a b f N k\nthis : upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N (k + 1) = upperCrossingTime a b f N (k + 1)\n\u03c9 : \u03a9\nh\u2081 :\n  \u2203 j, j \u2208 Set.Icc (upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N (Nat.succ k) \u03c9) N \u2227 (f j \u03c9 - a)\u207a \u2208 Set.Iic 0\nh\u2082 : \u00ac\u2203 j, j \u2208 Set.Icc (upperCrossingTime a b f N (Nat.succ k) \u03c9) N \u2227 f j \u03c9 \u2208 Set.Iic a\n\u22a2 \u2203 j, j \u2208 Set.Icc (upperCrossingTime a b f N (Nat.succ k) \u03c9) N \u2227 f j \u03c9 \u2208 Set.Iic a", "state_after": "no goals"}, {"tactic": "refine' False.elim (h\u2081 _)", "annotated_tactic": ["refine' <a>False.elim</a> (h\u2081 _)", [{"full_name": "False.elim", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [223, 21], "def_end_pos": [223, 31]}]], "state_before": "case pos\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9\u271d : \u03a9\n\u2131 : Filtration \u2115 m0\nhab : a < b\nhab' : 0 < b - a\nhf : \u2200 (\u03c9 : \u03a9) (i : \u2115), b - a \u2264 (f i \u03c9 - a)\u207a \u2194 b \u2264 f i \u03c9\nhf' : \u2200 (\u03c9 : \u03a9) (i : \u2115), (f i \u03c9 - a)\u207a \u2264 0 \u2194 f i \u03c9 \u2264 a\nk : \u2115\nih :\n  upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k = upperCrossingTime a b f N k \u2227\n    lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k = lowerCrossingTime a b f N k\nthis : upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N (k + 1) = upperCrossingTime a b f N (k + 1)\n\u03c9 : \u03a9\nh\u2081 :\n  \u00ac\u2203 j,\n      j \u2208 Set.Icc (upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N (Nat.succ k) \u03c9) N \u2227 (f j \u03c9 - a)\u207a \u2208 Set.Iic 0\nh\u2082 : \u2203 j, j \u2208 Set.Icc (upperCrossingTime a b f N (Nat.succ k) \u03c9) N \u2227 f j \u03c9 \u2208 Set.Iic a\n\u22a2 N = sInf (Set.Icc (upperCrossingTime a b f N (Nat.succ k) \u03c9) N \u2229 {i | f i \u03c9 \u2264 a})", "state_after": "case pos\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9\u271d : \u03a9\n\u2131 : Filtration \u2115 m0\nhab : a < b\nhab' : 0 < b - a\nhf : \u2200 (\u03c9 : \u03a9) (i : \u2115), b - a \u2264 (f i \u03c9 - a)\u207a \u2194 b \u2264 f i \u03c9\nhf' : \u2200 (\u03c9 : \u03a9) (i : \u2115), (f i \u03c9 - a)\u207a \u2264 0 \u2194 f i \u03c9 \u2264 a\nk : \u2115\nih :\n  upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k = upperCrossingTime a b f N k \u2227\n    lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k = lowerCrossingTime a b f N k\nthis : upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N (k + 1) = upperCrossingTime a b f N (k + 1)\n\u03c9 : \u03a9\nh\u2081 :\n  \u00ac\u2203 j,\n      j \u2208 Set.Icc (upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N (Nat.succ k) \u03c9) N \u2227 (f j \u03c9 - a)\u207a \u2208 Set.Iic 0\nh\u2082 : \u2203 j, j \u2208 Set.Icc (upperCrossingTime a b f N (Nat.succ k) \u03c9) N \u2227 f j \u03c9 \u2208 Set.Iic a\n\u22a2 \u2203 j, j \u2208 Set.Icc (upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N (Nat.succ k) \u03c9) N \u2227 (f j \u03c9 - a)\u207a \u2208 Set.Iic 0"}, {"tactic": "simp_all only [Set.mem_Iic]", "annotated_tactic": ["simp_all only [<a>Set.mem_Iic</a>]", [{"full_name": "Set.mem_Iic", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [136, 9], "def_end_pos": [136, 16]}]], "state_before": "case pos\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9\u271d : \u03a9\n\u2131 : Filtration \u2115 m0\nhab : a < b\nhab' : 0 < b - a\nhf : \u2200 (\u03c9 : \u03a9) (i : \u2115), b - a \u2264 (f i \u03c9 - a)\u207a \u2194 b \u2264 f i \u03c9\nhf' : \u2200 (\u03c9 : \u03a9) (i : \u2115), (f i \u03c9 - a)\u207a \u2264 0 \u2194 f i \u03c9 \u2264 a\nk : \u2115\nih :\n  upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k = upperCrossingTime a b f N k \u2227\n    lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k = lowerCrossingTime a b f N k\nthis : upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N (k + 1) = upperCrossingTime a b f N (k + 1)\n\u03c9 : \u03a9\nh\u2081 :\n  \u00ac\u2203 j,\n      j \u2208 Set.Icc (upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N (Nat.succ k) \u03c9) N \u2227 (f j \u03c9 - a)\u207a \u2208 Set.Iic 0\nh\u2082 : \u2203 j, j \u2208 Set.Icc (upperCrossingTime a b f N (Nat.succ k) \u03c9) N \u2227 f j \u03c9 \u2208 Set.Iic a\n\u22a2 \u2203 j, j \u2208 Set.Icc (upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N (Nat.succ k) \u03c9) N \u2227 (f j \u03c9 - a)\u207a \u2208 Set.Iic 0", "state_after": "no goals"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case neg\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9\u271d : \u03a9\n\u2131 : Filtration \u2115 m0\nhab : a < b\nhab' : 0 < b - a\nhf : \u2200 (\u03c9 : \u03a9) (i : \u2115), b - a \u2264 (f i \u03c9 - a)\u207a \u2194 b \u2264 f i \u03c9\nhf' : \u2200 (\u03c9 : \u03a9) (i : \u2115), (f i \u03c9 - a)\u207a \u2264 0 \u2194 f i \u03c9 \u2264 a\nk : \u2115\nih :\n  upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k = upperCrossingTime a b f N k \u2227\n    lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k = lowerCrossingTime a b f N k\nthis : upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N (k + 1) = upperCrossingTime a b f N (k + 1)\n\u03c9 : \u03a9\nh\u2081 :\n  \u00ac\u2203 j,\n      j \u2208 Set.Icc (upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N (Nat.succ k) \u03c9) N \u2227 (f j \u03c9 - a)\u207a \u2208 Set.Iic 0\nh\u2082 : \u00ac\u2203 j, j \u2208 Set.Icc (upperCrossingTime a b f N (Nat.succ k) \u03c9) N \u2227 f j \u03c9 \u2208 Set.Iic a\n\u22a2 N = N", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/QPF/Multivariate/Constructions/Cofix.lean", "full_name": "MvQPF.Cofix.dest_corec", "start": [184, 1], "end": [190, 71], "traced_tactics": [{"tactic": "conv =>\n  lhs\n  rw [Cofix.dest, Cofix.corec];", "annotated_tactic": ["conv =>\n    lhs\n    rw [<a>Cofix.dest</a>, <a>Cofix.corec</a>];", [{"full_name": "MvQPF.Cofix.dest", "def_path": "Mathlib/Data/QPF/Multivariate/Constructions/Cofix.lean", "def_pos": [138, 5], "def_end_pos": [138, 15]}, {"full_name": "MvQPF.Cofix.corec", "def_path": "Mathlib/Data/QPF/Multivariate/Constructions/Cofix.lean", "def_pos": [133, 5], "def_end_pos": [133, 16]}]], "state_before": "n : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\n\u03b2 : Type u\ng : \u03b2 \u2192 F (\u03b1 ::: \u03b2)\nx : \u03b2\n\u22a2 dest (corec g x) = (TypeVec.id ::: corec g) <$$> g x", "state_after": "n : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\n\u03b2 : Type u\ng : \u03b2 \u2192 F (\u03b1 ::: \u03b2)\nx : \u03b2\n\u22a2 Quot.lift (fun x => (TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) x))\n      (_ :\n        \u2200 (x y : M (P F) \u03b1),\n          Mcongr x y \u2192\n            (fun x => (TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) x)) x =\n              (fun x => (TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) x)) y)\n      (Quot.mk Mcongr (corecF g x)) =\n    (TypeVec.id ::: corec g) <$$> g x"}, {"tactic": "dsimp", "annotated_tactic": ["dsimp", []], "state_before": "n : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\n\u03b2 : Type u\ng : \u03b2 \u2192 F (\u03b1 ::: \u03b2)\nx : \u03b2\n\u22a2 Quot.lift (fun x => (TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) x))\n      (_ :\n        \u2200 (x y : M (P F) \u03b1),\n          Mcongr x y \u2192\n            (fun x => (TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) x)) x =\n              (fun x => (TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) x)) y)\n      (Quot.mk Mcongr (corecF g x)) =\n    (TypeVec.id ::: corec g) <$$> g x", "state_after": "n : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\n\u03b2 : Type u\ng : \u03b2 \u2192 F (\u03b1 ::: \u03b2)\nx : \u03b2\n\u22a2 (TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) (corecF g x)) = (TypeVec.id ::: corec g) <$$> g x"}, {"tactic": "rw [corecF_eq, abs_map, abs_repr, \u2190 comp_map, \u2190 appendFun_comp]", "annotated_tactic": ["rw [<a>corecF_eq</a>, <a>abs_map</a>, <a>abs_repr</a>, \u2190 <a>comp_map</a>, \u2190 <a>appendFun_comp</a>]", [{"full_name": "MvQPF.corecF_eq", "def_path": "Mathlib/Data/QPF/Multivariate/Constructions/Cofix.lean", "def_pos": [64, 9], "def_end_pos": [64, 18]}, {"full_name": "MvQPF.abs_map", "def_path": "Mathlib/Data/QPF/Multivariate/Basic.lean", "def_pos": [90, 3], "def_end_pos": [90, 10]}, {"full_name": "MvQPF.abs_repr", "def_path": "Mathlib/Data/QPF/Multivariate/Basic.lean", "def_pos": [89, 3], "def_end_pos": [89, 11]}, {"full_name": "MvQPF.comp_map", "def_path": "Mathlib/Data/QPF/Multivariate/Basic.lean", "def_pos": [112, 9], "def_end_pos": [112, 17]}, {"full_name": "TypeVec.appendFun_comp", "def_path": "Mathlib/Data/TypeVec.lean", "def_pos": [252, 9], "def_end_pos": [252, 23]}]], "state_before": "n : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\n\u03b2 : Type u\ng : \u03b2 \u2192 F (\u03b1 ::: \u03b2)\nx : \u03b2\n\u22a2 (TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) (corecF g x)) = (TypeVec.id ::: corec g) <$$> g x", "state_after": "n : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\n\u03b2 : Type u\ng : \u03b2 \u2192 F (\u03b1 ::: \u03b2)\nx : \u03b2\n\u22a2 (TypeVec.id \u229a TypeVec.id ::: Quot.mk Mcongr \u2218 corecF g) <$$> g x = (TypeVec.id ::: corec g) <$$> g x"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "n : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\n\u03b2 : Type u\ng : \u03b2 \u2192 F (\u03b1 ::: \u03b2)\nx : \u03b2\n\u22a2 (TypeVec.id \u229a TypeVec.id ::: Quot.mk Mcongr \u2218 corecF g) <$$> g x = (TypeVec.id ::: corec g) <$$> g x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/SpecialFunctions/Basic.lean", "full_name": "Complex.measurable_arg", "start": [114, 1], "end": [121, 22], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Constructions/Prod/Basic.lean", "full_name": "MeasureTheory.Measure.zero_prod", "start": [690, 1], "end": [692, 25], "traced_tactics": [{"tactic": "rw [Measure.prod]", "annotated_tactic": ["rw [<a>Measure.prod</a>]", [{"full_name": "MeasureTheory.Measure.prod", "def_path": "Mathlib/MeasureTheory/Constructions/Prod/Basic.lean", "def_pos": [292, 27], "def_end_pos": [292, 31]}]], "state_before": "\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2'\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd\u271d \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : SigmaFinite \u03bd\u271d\ninst\u271d : SigmaFinite \u03bc\n\u03bd : Measure \u03b2\n\u22a2 Measure.prod 0 \u03bd = 0", "state_after": "\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2'\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd\u271d \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : SigmaFinite \u03bd\u271d\ninst\u271d : SigmaFinite \u03bc\n\u03bd : Measure \u03b2\n\u22a2 (bind 0 fun x => map (Prod.mk x) \u03bd) = 0"}, {"tactic": "exact bind_zero_left _", "annotated_tactic": ["exact <a>bind_zero_left</a> _", [{"full_name": "MeasureTheory.Measure.bind_zero_left", "def_path": "Mathlib/MeasureTheory/Measure/GiryMonad.lean", "def_pos": [157, 9], "def_end_pos": [157, 23]}]], "state_before": "\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2'\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd\u271d \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : SigmaFinite \u03bd\u271d\ninst\u271d : SigmaFinite \u03bc\n\u03bd : Measure \u03b2\n\u22a2 (bind 0 fun x => map (Prod.mk x) \u03bd) = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/VitaliCaratheodory.lean", "full_name": "MeasureTheory.exists_upperSemicontinuous_lt_integral_gt", "start": [536, 1], "end": [555, 36], "traced_tactics": [{"tactic": "rcases exists_lt_lowerSemicontinuous_integral_lt (fun x => -f x) hf.neg \u03b5pos with\n  \u27e8g, g_lt_f, gcont, g_integrable, g_lt_top, gint\u27e9", "annotated_tactic": ["rcases <a>exists_lt_lowerSemicontinuous_integral_lt</a> (fun x => -f x) hf.neg \u03b5pos with\n    \u27e8g, g_lt_f, gcont, g_integrable, g_lt_top, gint\u27e9", [{"full_name": "MeasureTheory.exists_lt_lowerSemicontinuous_integral_lt", "def_path": "Mathlib/MeasureTheory/Integral/VitaliCaratheodory.lean", "def_pos": [457, 9], "def_end_pos": [457, 50]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), g x < \u2191(f x)) \u2227\n      UpperSemicontinuous g \u2227\n        (Integrable fun x => EReal.toReal (g x)) \u2227\n          (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u22a5 < g x) \u2227 \u222b (x : \u03b1), f x \u2202\u03bc < \u222b (x : \u03b1), EReal.toReal (g x) \u2202\u03bc + \u03b5", "state_after": "case intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\ng : \u03b1 \u2192 EReal\ng_lt_f : \u2200 (x : \u03b1), \u2191(-f x) < g x\ngcont : LowerSemicontinuous g\ng_integrable : Integrable fun x => EReal.toReal (g x)\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\ngint : \u222b (x : \u03b1), EReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), -f x \u2202\u03bc + \u03b5\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), g x < \u2191(f x)) \u2227\n      UpperSemicontinuous g \u2227\n        (Integrable fun x => EReal.toReal (g x)) \u2227\n          (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u22a5 < g x) \u2227 \u222b (x : \u03b1), f x \u2202\u03bc < \u222b (x : \u03b1), EReal.toReal (g x) \u2202\u03bc + \u03b5"}, {"tactic": "refine' \u27e8fun x => -g x, _, _, _, _, _\u27e9", "annotated_tactic": ["refine' \u27e8fun x => -g x, _, _, _, _, _\u27e9", []], "state_before": "case intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\ng : \u03b1 \u2192 EReal\ng_lt_f : \u2200 (x : \u03b1), \u2191(-f x) < g x\ngcont : LowerSemicontinuous g\ng_integrable : Integrable fun x => EReal.toReal (g x)\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\ngint : \u222b (x : \u03b1), EReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), -f x \u2202\u03bc + \u03b5\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), g x < \u2191(f x)) \u2227\n      UpperSemicontinuous g \u2227\n        (Integrable fun x => EReal.toReal (g x)) \u2227\n          (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u22a5 < g x) \u2227 \u222b (x : \u03b1), f x \u2202\u03bc < \u222b (x : \u03b1), EReal.toReal (g x) \u2202\u03bc + \u03b5", "state_after": "case intro.intro.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\ng : \u03b1 \u2192 EReal\ng_lt_f : \u2200 (x : \u03b1), \u2191(-f x) < g x\ngcont : LowerSemicontinuous g\ng_integrable : Integrable fun x => EReal.toReal (g x)\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\ngint : \u222b (x : \u03b1), EReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), -f x \u2202\u03bc + \u03b5\n\u22a2 \u2200 (x : \u03b1), (fun x => -g x) x < \u2191(f x)\n\ncase intro.intro.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\ng : \u03b1 \u2192 EReal\ng_lt_f : \u2200 (x : \u03b1), \u2191(-f x) < g x\ngcont : LowerSemicontinuous g\ng_integrable : Integrable fun x => EReal.toReal (g x)\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\ngint : \u222b (x : \u03b1), EReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), -f x \u2202\u03bc + \u03b5\n\u22a2 UpperSemicontinuous fun x => -g x\n\ncase intro.intro.intro.intro.intro.refine'_3\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\ng : \u03b1 \u2192 EReal\ng_lt_f : \u2200 (x : \u03b1), \u2191(-f x) < g x\ngcont : LowerSemicontinuous g\ng_integrable : Integrable fun x => EReal.toReal (g x)\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\ngint : \u222b (x : \u03b1), EReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), -f x \u2202\u03bc + \u03b5\n\u22a2 Integrable fun x => EReal.toReal ((fun x => -g x) x)\n\ncase intro.intro.intro.intro.intro.refine'_4\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\ng : \u03b1 \u2192 EReal\ng_lt_f : \u2200 (x : \u03b1), \u2191(-f x) < g x\ngcont : LowerSemicontinuous g\ng_integrable : Integrable fun x => EReal.toReal (g x)\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\ngint : \u222b (x : \u03b1), EReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), -f x \u2202\u03bc + \u03b5\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u22a5 < (fun x => -g x) x\n\ncase intro.intro.intro.intro.intro.refine'_5\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\ng : \u03b1 \u2192 EReal\ng_lt_f : \u2200 (x : \u03b1), \u2191(-f x) < g x\ngcont : LowerSemicontinuous g\ng_integrable : Integrable fun x => EReal.toReal (g x)\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\ngint : \u222b (x : \u03b1), EReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), -f x \u2202\u03bc + \u03b5\n\u22a2 \u222b (x : \u03b1), f x \u2202\u03bc < \u222b (x : \u03b1), EReal.toReal ((fun x => -g x) x) \u2202\u03bc + \u03b5"}, {"tactic": "exact fun x => EReal.neg_lt_iff_neg_lt.1 (by simpa only [EReal.coe_neg] using g_lt_f x)", "annotated_tactic": ["exact fun x => <a>EReal.neg_lt_iff_neg_lt</a>.1 (by simpa only [<a>EReal.coe_neg</a>] using g_lt_f x)", [{"full_name": "EReal.neg_lt_iff_neg_lt", "def_path": "Mathlib/Data/Real/EReal.lean", "def_pos": [833, 9], "def_end_pos": [833, 26]}, {"full_name": "EReal.coe_neg", "def_path": "Mathlib/Data/Real/EReal.lean", "def_pos": [759, 28], "def_end_pos": [759, 35]}]], "state_before": "case intro.intro.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\ng : \u03b1 \u2192 EReal\ng_lt_f : \u2200 (x : \u03b1), \u2191(-f x) < g x\ngcont : LowerSemicontinuous g\ng_integrable : Integrable fun x => EReal.toReal (g x)\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\ngint : \u222b (x : \u03b1), EReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), -f x \u2202\u03bc + \u03b5\n\u22a2 \u2200 (x : \u03b1), (fun x => -g x) x < \u2191(f x)", "state_after": "no goals"}, {"tactic": "simpa only [EReal.coe_neg] using g_lt_f x", "annotated_tactic": ["simpa only [<a>EReal.coe_neg</a>] using g_lt_f x", [{"full_name": "EReal.coe_neg", "def_path": "Mathlib/Data/Real/EReal.lean", "def_pos": [759, 28], "def_end_pos": [759, 35]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\ng : \u03b1 \u2192 EReal\ng_lt_f : \u2200 (x : \u03b1), \u2191(-f x) < g x\ngcont : LowerSemicontinuous g\ng_integrable : Integrable fun x => EReal.toReal (g x)\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\ngint : \u222b (x : \u03b1), EReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), -f x \u2202\u03bc + \u03b5\nx : \u03b1\n\u22a2 -\u2191(f x) < g x", "state_after": "no goals"}, {"tactic": "exact\n  continuous_neg.comp_lowerSemicontinuous_antitone gcont fun x y hxy =>\n    EReal.neg_le_neg_iff.2 hxy", "annotated_tactic": ["exact\n      continuous_neg.comp_lowerSemicontinuous_antitone gcont fun x y hxy =>\n        <a>EReal.neg_le_neg_iff</a>.2 hxy", [{"full_name": "EReal.neg_le_neg_iff", "def_path": "Mathlib/Data/Real/EReal.lean", "def_pos": [805, 17], "def_end_pos": [805, 31]}]], "state_before": "case intro.intro.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\ng : \u03b1 \u2192 EReal\ng_lt_f : \u2200 (x : \u03b1), \u2191(-f x) < g x\ngcont : LowerSemicontinuous g\ng_integrable : Integrable fun x => EReal.toReal (g x)\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\ngint : \u222b (x : \u03b1), EReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), -f x \u2202\u03bc + \u03b5\n\u22a2 UpperSemicontinuous fun x => -g x", "state_after": "no goals"}, {"tactic": "convert g_integrable.neg", "annotated_tactic": ["convert g_integrable.neg", []], "state_before": "case intro.intro.intro.intro.intro.refine'_3\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\ng : \u03b1 \u2192 EReal\ng_lt_f : \u2200 (x : \u03b1), \u2191(-f x) < g x\ngcont : LowerSemicontinuous g\ng_integrable : Integrable fun x => EReal.toReal (g x)\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\ngint : \u222b (x : \u03b1), EReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), -f x \u2202\u03bc + \u03b5\n\u22a2 Integrable fun x => EReal.toReal ((fun x => -g x) x)", "state_after": "case h.e'_5.h\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\ng : \u03b1 \u2192 EReal\ng_lt_f : \u2200 (x : \u03b1), \u2191(-f x) < g x\ngcont : LowerSemicontinuous g\ng_integrable : Integrable fun x => EReal.toReal (g x)\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\ngint : \u222b (x : \u03b1), EReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), -f x \u2202\u03bc + \u03b5\nx\u271d : \u03b1\n\u22a2 EReal.toReal ((fun x => -g x) x\u271d) = (-fun x => EReal.toReal (g x)) x\u271d"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case h.e'_5.h\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\ng : \u03b1 \u2192 EReal\ng_lt_f : \u2200 (x : \u03b1), \u2191(-f x) < g x\ngcont : LowerSemicontinuous g\ng_integrable : Integrable fun x => EReal.toReal (g x)\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\ngint : \u222b (x : \u03b1), EReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), -f x \u2202\u03bc + \u03b5\nx\u271d : \u03b1\n\u22a2 EReal.toReal ((fun x => -g x) x\u271d) = (-fun x => EReal.toReal (g x)) x\u271d", "state_after": "no goals"}, {"tactic": "simpa [bot_lt_iff_ne_bot, lt_top_iff_ne_top] using g_lt_top", "annotated_tactic": ["simpa [<a>bot_lt_iff_ne_bot</a>, <a>lt_top_iff_ne_top</a>] using g_lt_top", [{"full_name": "bot_lt_iff_ne_bot", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [371, 9], "def_end_pos": [371, 26]}, {"full_name": "lt_top_iff_ne_top", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [173, 9], "def_end_pos": [173, 26]}]], "state_before": "case intro.intro.intro.intro.intro.refine'_4\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\ng : \u03b1 \u2192 EReal\ng_lt_f : \u2200 (x : \u03b1), \u2191(-f x) < g x\ngcont : LowerSemicontinuous g\ng_integrable : Integrable fun x => EReal.toReal (g x)\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\ngint : \u222b (x : \u03b1), EReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), -f x \u2202\u03bc + \u03b5\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u22a5 < (fun x => -g x) x", "state_after": "no goals"}, {"tactic": "simp_rw [integral_neg, lt_neg_add_iff_add_lt] at gint", "annotated_tactic": ["simp_rw [<a>integral_neg</a>, <a>lt_neg_add_iff_add_lt</a>] at gint", [{"full_name": "MeasureTheory.integral_neg", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [890, 9], "def_end_pos": [890, 21]}, {"full_name": "lt_neg_add_iff_add_lt", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [170, 3], "def_end_pos": [170, 14]}]], "state_before": "case intro.intro.intro.intro.intro.refine'_5\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\ng : \u03b1 \u2192 EReal\ng_lt_f : \u2200 (x : \u03b1), \u2191(-f x) < g x\ngcont : LowerSemicontinuous g\ng_integrable : Integrable fun x => EReal.toReal (g x)\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\ngint : \u222b (x : \u03b1), EReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), -f x \u2202\u03bc + \u03b5\n\u22a2 \u222b (x : \u03b1), f x \u2202\u03bc < \u222b (x : \u03b1), EReal.toReal ((fun x => -g x) x) \u2202\u03bc + \u03b5", "state_after": "case intro.intro.intro.intro.intro.refine'_5\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\ng : \u03b1 \u2192 EReal\ng_lt_f : \u2200 (x : \u03b1), \u2191(-f x) < g x\ngcont : LowerSemicontinuous g\ng_integrable : Integrable fun x => EReal.toReal (g x)\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\ngint : \u222b (a : \u03b1), f a \u2202\u03bc + \u222b (x : \u03b1), EReal.toReal (g x) \u2202\u03bc < \u03b5\n\u22a2 \u222b (x : \u03b1), f x \u2202\u03bc < \u222b (x : \u03b1), EReal.toReal ((fun x => -g x) x) \u2202\u03bc + \u03b5"}, {"tactic": "rw [add_comm] at gint", "annotated_tactic": ["rw [<a>add_comm</a>] at gint", [{"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [301, 3], "def_end_pos": [301, 14]}]], "state_before": "case intro.intro.intro.intro.intro.refine'_5\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\ng : \u03b1 \u2192 EReal\ng_lt_f : \u2200 (x : \u03b1), \u2191(-f x) < g x\ngcont : LowerSemicontinuous g\ng_integrable : Integrable fun x => EReal.toReal (g x)\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\ngint : \u222b (a : \u03b1), f a \u2202\u03bc + \u222b (x : \u03b1), EReal.toReal (g x) \u2202\u03bc < \u03b5\n\u22a2 \u222b (x : \u03b1), f x \u2202\u03bc < \u222b (x : \u03b1), EReal.toReal ((fun x => -g x) x) \u2202\u03bc + \u03b5", "state_after": "case intro.intro.intro.intro.intro.refine'_5\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\ng : \u03b1 \u2192 EReal\ng_lt_f : \u2200 (x : \u03b1), \u2191(-f x) < g x\ngcont : LowerSemicontinuous g\ng_integrable : Integrable fun x => EReal.toReal (g x)\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\ngint : \u222b (x : \u03b1), EReal.toReal (g x) \u2202\u03bc + \u222b (a : \u03b1), f a \u2202\u03bc < \u03b5\n\u22a2 \u222b (x : \u03b1), f x \u2202\u03bc < \u222b (x : \u03b1), EReal.toReal ((fun x => -g x) x) \u2202\u03bc + \u03b5"}, {"tactic": "simpa [integral_neg] using gint", "annotated_tactic": ["simpa [<a>integral_neg</a>] using gint", [{"full_name": "MeasureTheory.integral_neg", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [890, 9], "def_end_pos": [890, 21]}]], "state_before": "case intro.intro.intro.intro.intro.refine'_5\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\ng : \u03b1 \u2192 EReal\ng_lt_f : \u2200 (x : \u03b1), \u2191(-f x) < g x\ngcont : LowerSemicontinuous g\ng_integrable : Integrable fun x => EReal.toReal (g x)\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\ngint : \u222b (x : \u03b1), EReal.toReal (g x) \u2202\u03bc + \u222b (a : \u03b1), f a \u2202\u03bc < \u03b5\n\u22a2 \u222b (x : \u03b1), f x \u2202\u03bc < \u222b (x : \u03b1), EReal.toReal ((fun x => -g x) x) \u2202\u03bc + \u03b5", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "full_name": "MeasureTheory.L1.SimpleFunc.integral_eq_lintegral", "start": [525, 8], "end": [527, 82], "traced_tactics": [{"tactic": "rw [integral, SimpleFunc.integral_eq_lintegral (SimpleFunc.integrable f) h_pos]", "annotated_tactic": ["rw [<a>integral</a>, <a>SimpleFunc.integral_eq_lintegral</a> (<a>SimpleFunc.integrable</a> f) h_pos]", [{"full_name": "MeasureTheory.L1.SimpleFunc.integral", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [518, 5], "def_end_pos": [518, 13]}, {"full_name": "MeasureTheory.SimpleFunc.integral_eq_lintegral", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [405, 9], "def_end_pos": [405, 30]}, {"full_name": "MeasureTheory.L1.SimpleFunc.integrable", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "def_pos": [1040, 19], "def_end_pos": [1040, 43]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedAddCommGroup F\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : NormedField \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : SMulCommClass \u211d \ud835\udd5c E\nF' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup F'\ninst\u271d : NormedSpace \u211d F'\nf : { x // x \u2208 simpleFunc \u211d 1 \u03bc }\nh_pos : 0 \u2264\u1d50[\u03bc] \u2191(toSimpleFunc f)\n\u22a2 integral f = ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (\u2191(toSimpleFunc f) a) \u2202\u03bc)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/EssSup.lean", "full_name": "essInf_const", "start": [76, 1], "end": [77, 40], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "full_name": "String.set_of_valid", "start": [257, 1], "end": [259, 51], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/EssSup.lean", "full_name": "ENNReal.essSup_liminf_le", "start": [325, 1], "end": [329, 64], "traced_tactics": [{"tactic": "simp_rw [essSup]", "annotated_tactic": ["simp_rw [<a>essSup</a>]", [{"full_name": "essSup", "def_path": "Mathlib/MeasureTheory/Function/EssSup.lean", "def_pos": [44, 5], "def_end_pos": [44, 11]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf\u271d : \u03b1 \u2192 \u211d\u22650\u221e\n\u03b9 : Type u_3\ninst\u271d\u00b9 : Countable \u03b9\ninst\u271d : LinearOrder \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\n\u22a2 essSup (fun x => liminf (fun n => f n x) atTop) \u03bc \u2264 liminf (fun n => essSup (fun x => f n x) \u03bc) atTop", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf\u271d : \u03b1 \u2192 \u211d\u22650\u221e\n\u03b9 : Type u_3\ninst\u271d\u00b9 : Countable \u03b9\ninst\u271d : LinearOrder \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\n\u22a2 limsup (fun x => liminf (fun n => f n x) atTop) (Measure.ae \u03bc) \u2264\n    liminf (fun n => limsup (fun x => f n x) (Measure.ae \u03bc)) atTop"}, {"tactic": "exact ENNReal.limsup_liminf_le_liminf_limsup fun a b => f b a", "annotated_tactic": ["exact <a>ENNReal.limsup_liminf_le_liminf_limsup</a> fun a b => f b a", [{"full_name": "ENNReal.limsup_liminf_le_liminf_limsup", "def_path": "Mathlib/Order/Filter/ENNReal.lean", "def_pos": [86, 9], "def_end_pos": [86, 39]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf\u271d : \u03b1 \u2192 \u211d\u22650\u221e\n\u03b9 : Type u_3\ninst\u271d\u00b9 : Countable \u03b9\ninst\u271d : LinearOrder \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\n\u22a2 limsup (fun x => liminf (fun n => f n x) atTop) (Measure.ae \u03bc) \u2264\n    liminf (fun n => limsup (fun x => f n x) (Measure.ae \u03bc)) atTop", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Pi.lean", "full_name": "Finset.pi_const_singleton", "start": [122, 1], "end": [124, 29], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Function.lean", "full_name": "Set.piecewise_op\u2082", "start": [1525, 1], "end": [1528, 54], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Num/Lemmas.lean", "full_name": "Num.add_of_nat", "start": [474, 1], "end": [475, 17], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Pointwise/SMul.lean", "full_name": "Set.smul_neg", "start": [1085, 11], "end": [1087, 41], "traced_tactics": [{"tactic": "simp_rw [\u2190 image_neg]", "annotated_tactic": ["simp_rw [\u2190 <a>image_neg</a>]", [{"full_name": "Set.image_neg", "def_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "def_pos": [256, 3], "def_end_pos": [256, 14]}]], "state_before": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : Monoid \u03b1\ninst\u271d\u00b9 : AddGroup \u03b2\ninst\u271d : DistribMulAction \u03b1 \u03b2\na : \u03b1\ns : Set \u03b1\nt : Set \u03b2\n\u22a2 s \u2022 -t = -(s \u2022 t)", "state_after": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : Monoid \u03b1\ninst\u271d\u00b9 : AddGroup \u03b2\ninst\u271d : DistribMulAction \u03b1 \u03b2\na : \u03b1\ns : Set \u03b1\nt : Set \u03b2\n\u22a2 s \u2022 Neg.neg '' t = Neg.neg '' (s \u2022 t)"}, {"tactic": "exact image_image2_right_comm smul_neg", "annotated_tactic": ["exact <a>image_image2_right_comm</a> <a>smul_neg</a>", [{"full_name": "Set.image_image2_right_comm", "def_path": "Mathlib/Data/Set/NAry.lean", "def_pos": [377, 9], "def_end_pos": [377, 32]}, {"full_name": "smul_neg", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [984, 9], "def_end_pos": [984, 17]}]], "state_before": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : Monoid \u03b1\ninst\u271d\u00b9 : AddGroup \u03b2\ninst\u271d : DistribMulAction \u03b1 \u03b2\na : \u03b1\ns : Set \u03b1\nt : Set \u03b2\n\u22a2 s \u2022 Neg.neg '' t = Neg.neg '' (s \u2022 t)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "full_name": "String.next_of_valid", "start": [269, 1], "end": [270, 85], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "full_name": "intervalIntegral.norm_integral_le_of_norm_le_const", "start": [572, 1], "end": [574, 65], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "full_name": "Substring.ValidFor.all", "start": [947, 1], "end": [948, 68], "traced_tactics": [{"tactic": "simp [Substring.all, h.any, List.all_eq_not_any_not]", "annotated_tactic": ["simp [<a>Substring.all</a>, h.any, <a>List.all_eq_not_any_not</a>]", [{"full_name": "Substring.all", "def_path": "lake-packages/lean4/src/lean/Init/Data/String/Basic.lean", "def_pos": [630, 15], "def_end_pos": [630, 18]}, {"full_name": "List.all_eq_not_any_not", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [990, 9], "def_end_pos": [990, 27]}]], "state_before": "l m r : List Char\nf : Char \u2192 Bool\nx\u271d : Substring\nh : ValidFor l m r x\u271d\n\u22a2 Substring.all x\u271d f = List.all m f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "full_name": "MeasureTheory.IntegrableOn.restrict_toMeasurable", "start": [291, 1], "end": [305, 38], "traced_tactics": [{"tactic": "rcases exists_seq_strictAnti_tendsto (0 : \u211d) with \u27e8u, _, u_pos, u_lim\u27e9", "annotated_tactic": ["rcases <a>exists_seq_strictAnti_tendsto</a> (0 : \u211d) with \u27e8u, _, u_pos, u_lim\u27e9", [{"full_name": "exists_seq_strictAnti_tendsto", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [2258, 9], "def_end_pos": [2258, 38]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf g : \u03b1 \u2192 E\ns t : Set \u03b1\n\u03bc \u03bd : Measure \u03b1\nhf : IntegrableOn f s\nh's : \u2200 (x : \u03b1), x \u2208 s \u2192 f x \u2260 0\n\u22a2 Measure.restrict \u03bc (toMeasurable \u03bc s) = Measure.restrict \u03bc s", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf g : \u03b1 \u2192 E\ns t : Set \u03b1\n\u03bc \u03bd : Measure \u03b1\nhf : IntegrableOn f s\nh's : \u2200 (x : \u03b1), x \u2208 s \u2192 f x \u2260 0\nu : \u2115 \u2192 \u211d\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\n\u22a2 Measure.restrict \u03bc (toMeasurable \u03bc s) = Measure.restrict \u03bc s"}, {"tactic": "let v n := toMeasurable (\u03bc.restrict s) { x | u n \u2264 \u2016f x\u2016 }", "annotated_tactic": ["let v n := <a>toMeasurable</a> (\u03bc.restrict s) { x | u n \u2264 \u2016f x\u2016 }", [{"full_name": "MeasureTheory.toMeasurable", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [626, 17], "def_end_pos": [626, 29]}]], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf g : \u03b1 \u2192 E\ns t : Set \u03b1\n\u03bc \u03bd : Measure \u03b1\nhf : IntegrableOn f s\nh's : \u2200 (x : \u03b1), x \u2208 s \u2192 f x \u2260 0\nu : \u2115 \u2192 \u211d\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\n\u22a2 Measure.restrict \u03bc (toMeasurable \u03bc s) = Measure.restrict \u03bc s", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf g : \u03b1 \u2192 E\ns t : Set \u03b1\n\u03bc \u03bd : Measure \u03b1\nhf : IntegrableOn f s\nh's : \u2200 (x : \u03b1), x \u2208 s \u2192 f x \u2260 0\nu : \u2115 \u2192 \u211d\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nv : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable (Measure.restrict \u03bc s) {x | u n \u2264 \u2016f x\u2016}\n\u22a2 Measure.restrict \u03bc (toMeasurable \u03bc s) = Measure.restrict \u03bc s"}, {"tactic": "have A : \u2200 n, \u03bc (s \u2229 v n) \u2260 \u221e := by\n  intro n\n  rw [inter_comm, \u2190 Measure.restrict_apply (measurableSet_toMeasurable _ _),\n    measure_toMeasurable]\n  exact (hf.measure_norm_ge_lt_top (u_pos n)).ne", "annotated_tactic": ["have A : \u2200 n, \u03bc (s \u2229 v n) \u2260 \u221e := by\n    intro n\n    rw [<a>inter_comm</a>, \u2190 <a>Measure.restrict_apply</a> (<a>measurableSet_toMeasurable</a> _ _),\n      <a>measure_toMeasurable</a>]\n    exact (hf.measure_norm_ge_lt_top (u_pos n)).<a>ne</a>", [{"full_name": "Set.inter_comm", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [940, 9], "def_end_pos": [940, 19]}, {"full_name": "MeasureTheory.Measure.restrict_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1533, 9], "def_end_pos": [1533, 23]}, {"full_name": "MeasureTheory.measurableSet_toMeasurable", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [645, 9], "def_end_pos": [645, 35]}, {"full_name": "MeasureTheory.measure_toMeasurable", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [653, 9], "def_end_pos": [653, 29]}, {"full_name": "LT.lt.ne", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [152, 7], "def_end_pos": [152, 15]}]], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf g : \u03b1 \u2192 E\ns t : Set \u03b1\n\u03bc \u03bd : Measure \u03b1\nhf : IntegrableOn f s\nh's : \u2200 (x : \u03b1), x \u2208 s \u2192 f x \u2260 0\nu : \u2115 \u2192 \u211d\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nv : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable (Measure.restrict \u03bc s) {x | u n \u2264 \u2016f x\u2016}\n\u22a2 Measure.restrict \u03bc (toMeasurable \u03bc s) = Measure.restrict \u03bc s", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf g : \u03b1 \u2192 E\ns t : Set \u03b1\n\u03bc \u03bd : Measure \u03b1\nhf : IntegrableOn f s\nh's : \u2200 (x : \u03b1), x \u2208 s \u2192 f x \u2260 0\nu : \u2115 \u2192 \u211d\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nv : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable (Measure.restrict \u03bc s) {x | u n \u2264 \u2016f x\u2016}\nA : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \u2229 v n) \u2260 \u22a4\n\u22a2 Measure.restrict \u03bc (toMeasurable \u03bc s) = Measure.restrict \u03bc s"}, {"tactic": "apply Measure.restrict_toMeasurable_of_cover _ A", "annotated_tactic": ["apply <a>Measure.restrict_toMeasurable_of_cover</a> _ A", [{"full_name": "MeasureTheory.Measure.restrict_toMeasurable_of_cover", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3598, 9], "def_end_pos": [3598, 39]}]], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf g : \u03b1 \u2192 E\ns t : Set \u03b1\n\u03bc \u03bd : Measure \u03b1\nhf : IntegrableOn f s\nh's : \u2200 (x : \u03b1), x \u2208 s \u2192 f x \u2260 0\nu : \u2115 \u2192 \u211d\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nv : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable (Measure.restrict \u03bc s) {x | u n \u2264 \u2016f x\u2016}\nA : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \u2229 v n) \u2260 \u22a4\n\u22a2 Measure.restrict \u03bc (toMeasurable \u03bc s) = Measure.restrict \u03bc s", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf g : \u03b1 \u2192 E\ns t : Set \u03b1\n\u03bc \u03bd : Measure \u03b1\nhf : IntegrableOn f s\nh's : \u2200 (x : \u03b1), x \u2208 s \u2192 f x \u2260 0\nu : \u2115 \u2192 \u211d\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nv : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable (Measure.restrict \u03bc s) {x | u n \u2264 \u2016f x\u2016}\nA : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \u2229 v n) \u2260 \u22a4\n\u22a2 s \u2286 \u22c3 n, v n"}, {"tactic": "intro x hx", "annotated_tactic": ["intro x hx", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf g : \u03b1 \u2192 E\ns t : Set \u03b1\n\u03bc \u03bd : Measure \u03b1\nhf : IntegrableOn f s\nh's : \u2200 (x : \u03b1), x \u2208 s \u2192 f x \u2260 0\nu : \u2115 \u2192 \u211d\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nv : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable (Measure.restrict \u03bc s) {x | u n \u2264 \u2016f x\u2016}\nA : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \u2229 v n) \u2260 \u22a4\n\u22a2 s \u2286 \u22c3 n, v n", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf g : \u03b1 \u2192 E\ns t : Set \u03b1\n\u03bc \u03bd : Measure \u03b1\nhf : IntegrableOn f s\nh's : \u2200 (x : \u03b1), x \u2208 s \u2192 f x \u2260 0\nu : \u2115 \u2192 \u211d\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nv : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable (Measure.restrict \u03bc s) {x | u n \u2264 \u2016f x\u2016}\nA : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \u2229 v n) \u2260 \u22a4\nx : \u03b1\nhx : x \u2208 s\n\u22a2 x \u2208 \u22c3 n, v n"}, {"tactic": "have : 0 < \u2016f x\u2016 := by simp only [h's x hx, norm_pos_iff, Ne.def, not_false_iff]", "annotated_tactic": ["have : 0 < \u2016f x\u2016 := by simp only [h's x hx, <a>norm_pos_iff</a>, <a>Ne.def</a>, <a>not_false_iff</a>]", [{"full_name": "norm_pos_iff", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [2030, 30], "def_end_pos": [2030, 42]}, {"full_name": "Ne.def", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [59, 9], "def_end_pos": [59, 15]}, {"full_name": "not_false_iff", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [82, 9], "def_end_pos": [82, 22]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf g : \u03b1 \u2192 E\ns t : Set \u03b1\n\u03bc \u03bd : Measure \u03b1\nhf : IntegrableOn f s\nh's : \u2200 (x : \u03b1), x \u2208 s \u2192 f x \u2260 0\nu : \u2115 \u2192 \u211d\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nv : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable (Measure.restrict \u03bc s) {x | u n \u2264 \u2016f x\u2016}\nA : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \u2229 v n) \u2260 \u22a4\nx : \u03b1\nhx : x \u2208 s\n\u22a2 x \u2208 \u22c3 n, v n", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf g : \u03b1 \u2192 E\ns t : Set \u03b1\n\u03bc \u03bd : Measure \u03b1\nhf : IntegrableOn f s\nh's : \u2200 (x : \u03b1), x \u2208 s \u2192 f x \u2260 0\nu : \u2115 \u2192 \u211d\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nv : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable (Measure.restrict \u03bc s) {x | u n \u2264 \u2016f x\u2016}\nA : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \u2229 v n) \u2260 \u22a4\nx : \u03b1\nhx : x \u2208 s\nthis : 0 < \u2016f x\u2016\n\u22a2 x \u2208 \u22c3 n, v n"}, {"tactic": "obtain \u27e8n, hn\u27e9 : \u2203 n, u n < \u2016f x\u2016", "annotated_tactic": ["obtain \u27e8n, hn\u27e9 : \u2203 n, u n < \u2016f x\u2016", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf g : \u03b1 \u2192 E\ns t : Set \u03b1\n\u03bc \u03bd : Measure \u03b1\nhf : IntegrableOn f s\nh's : \u2200 (x : \u03b1), x \u2208 s \u2192 f x \u2260 0\nu : \u2115 \u2192 \u211d\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nv : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable (Measure.restrict \u03bc s) {x | u n \u2264 \u2016f x\u2016}\nA : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \u2229 v n) \u2260 \u22a4\nx : \u03b1\nhx : x \u2208 s\nthis : 0 < \u2016f x\u2016\n\u22a2 x \u2208 \u22c3 n, v n", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf g : \u03b1 \u2192 E\ns t : Set \u03b1\n\u03bc \u03bd : Measure \u03b1\nhf : IntegrableOn f s\nh's : \u2200 (x : \u03b1), x \u2208 s \u2192 f x \u2260 0\nu : \u2115 \u2192 \u211d\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nv : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable (Measure.restrict \u03bc s) {x | u n \u2264 \u2016f x\u2016}\nA : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \u2229 v n) \u2260 \u22a4\nx : \u03b1\nhx : x \u2208 s\nthis : 0 < \u2016f x\u2016\n\u22a2 \u2203 n, u n < \u2016f x\u2016\n\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf g : \u03b1 \u2192 E\ns t : Set \u03b1\n\u03bc \u03bd : Measure \u03b1\nhf : IntegrableOn f s\nh's : \u2200 (x : \u03b1), x \u2208 s \u2192 f x \u2260 0\nu : \u2115 \u2192 \u211d\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nv : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable (Measure.restrict \u03bc s) {x | u n \u2264 \u2016f x\u2016}\nA : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \u2229 v n) \u2260 \u22a4\nx : \u03b1\nhx : x \u2208 s\nthis : 0 < \u2016f x\u2016\nn : \u2115\nhn : u n < \u2016f x\u2016\n\u22a2 x \u2208 \u22c3 n, v n"}, {"tactic": "exact ((tendsto_order.1 u_lim).2 _ this).exists", "annotated_tactic": ["exact ((<a>tendsto_order</a>.1 u_lim).2 _ this).<a>exists</a>", [{"full_name": "tendsto_order", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [919, 9], "def_end_pos": [919, 22]}, {"full_name": "Filter.Eventually.exists", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1308, 9], "def_end_pos": [1308, 26]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf g : \u03b1 \u2192 E\ns t : Set \u03b1\n\u03bc \u03bd : Measure \u03b1\nhf : IntegrableOn f s\nh's : \u2200 (x : \u03b1), x \u2208 s \u2192 f x \u2260 0\nu : \u2115 \u2192 \u211d\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nv : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable (Measure.restrict \u03bc s) {x | u n \u2264 \u2016f x\u2016}\nA : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \u2229 v n) \u2260 \u22a4\nx : \u03b1\nhx : x \u2208 s\nthis : 0 < \u2016f x\u2016\n\u22a2 \u2203 n, u n < \u2016f x\u2016\n\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf g : \u03b1 \u2192 E\ns t : Set \u03b1\n\u03bc \u03bd : Measure \u03b1\nhf : IntegrableOn f s\nh's : \u2200 (x : \u03b1), x \u2208 s \u2192 f x \u2260 0\nu : \u2115 \u2192 \u211d\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nv : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable (Measure.restrict \u03bc s) {x | u n \u2264 \u2016f x\u2016}\nA : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \u2229 v n) \u2260 \u22a4\nx : \u03b1\nhx : x \u2208 s\nthis : 0 < \u2016f x\u2016\nn : \u2115\nhn : u n < \u2016f x\u2016\n\u22a2 x \u2208 \u22c3 n, v n", "state_after": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf g : \u03b1 \u2192 E\ns t : Set \u03b1\n\u03bc \u03bd : Measure \u03b1\nhf : IntegrableOn f s\nh's : \u2200 (x : \u03b1), x \u2208 s \u2192 f x \u2260 0\nu : \u2115 \u2192 \u211d\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nv : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable (Measure.restrict \u03bc s) {x | u n \u2264 \u2016f x\u2016}\nA : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \u2229 v n) \u2260 \u22a4\nx : \u03b1\nhx : x \u2208 s\nthis : 0 < \u2016f x\u2016\nn : \u2115\nhn : u n < \u2016f x\u2016\n\u22a2 x \u2208 \u22c3 n, v n"}, {"tactic": "refine' mem_iUnion.2 \u27e8n, _\u27e9", "annotated_tactic": ["refine' <a>mem_iUnion</a>.2 \u27e8n, _\u27e9", [{"full_name": "Set.mem_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [201, 9], "def_end_pos": [201, 19]}]], "state_before": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf g : \u03b1 \u2192 E\ns t : Set \u03b1\n\u03bc \u03bd : Measure \u03b1\nhf : IntegrableOn f s\nh's : \u2200 (x : \u03b1), x \u2208 s \u2192 f x \u2260 0\nu : \u2115 \u2192 \u211d\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nv : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable (Measure.restrict \u03bc s) {x | u n \u2264 \u2016f x\u2016}\nA : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \u2229 v n) \u2260 \u22a4\nx : \u03b1\nhx : x \u2208 s\nthis : 0 < \u2016f x\u2016\nn : \u2115\nhn : u n < \u2016f x\u2016\n\u22a2 x \u2208 \u22c3 n, v n", "state_after": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf g : \u03b1 \u2192 E\ns t : Set \u03b1\n\u03bc \u03bd : Measure \u03b1\nhf : IntegrableOn f s\nh's : \u2200 (x : \u03b1), x \u2208 s \u2192 f x \u2260 0\nu : \u2115 \u2192 \u211d\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nv : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable (Measure.restrict \u03bc s) {x | u n \u2264 \u2016f x\u2016}\nA : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \u2229 v n) \u2260 \u22a4\nx : \u03b1\nhx : x \u2208 s\nthis : 0 < \u2016f x\u2016\nn : \u2115\nhn : u n < \u2016f x\u2016\n\u22a2 x \u2208 v n"}, {"tactic": "exact subset_toMeasurable _ _ hn.le", "annotated_tactic": ["exact <a>subset_toMeasurable</a> _ _ hn.le", [{"full_name": "MeasureTheory.subset_toMeasurable", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [633, 9], "def_end_pos": [633, 28]}]], "state_before": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf g : \u03b1 \u2192 E\ns t : Set \u03b1\n\u03bc \u03bd : Measure \u03b1\nhf : IntegrableOn f s\nh's : \u2200 (x : \u03b1), x \u2208 s \u2192 f x \u2260 0\nu : \u2115 \u2192 \u211d\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nv : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable (Measure.restrict \u03bc s) {x | u n \u2264 \u2016f x\u2016}\nA : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \u2229 v n) \u2260 \u22a4\nx : \u03b1\nhx : x \u2208 s\nthis : 0 < \u2016f x\u2016\nn : \u2115\nhn : u n < \u2016f x\u2016\n\u22a2 x \u2208 v n", "state_after": "no goals"}, {"tactic": "intro n", "annotated_tactic": ["intro n", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf g : \u03b1 \u2192 E\ns t : Set \u03b1\n\u03bc \u03bd : Measure \u03b1\nhf : IntegrableOn f s\nh's : \u2200 (x : \u03b1), x \u2208 s \u2192 f x \u2260 0\nu : \u2115 \u2192 \u211d\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nv : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable (Measure.restrict \u03bc s) {x | u n \u2264 \u2016f x\u2016}\n\u22a2 \u2200 (n : \u2115), \u2191\u2191\u03bc (s \u2229 v n) \u2260 \u22a4", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf g : \u03b1 \u2192 E\ns t : Set \u03b1\n\u03bc \u03bd : Measure \u03b1\nhf : IntegrableOn f s\nh's : \u2200 (x : \u03b1), x \u2208 s \u2192 f x \u2260 0\nu : \u2115 \u2192 \u211d\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nv : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable (Measure.restrict \u03bc s) {x | u n \u2264 \u2016f x\u2016}\nn : \u2115\n\u22a2 \u2191\u2191\u03bc (s \u2229 v n) \u2260 \u22a4"}, {"tactic": "rw [inter_comm, \u2190 Measure.restrict_apply (measurableSet_toMeasurable _ _),\n  measure_toMeasurable]", "annotated_tactic": ["rw [<a>inter_comm</a>, \u2190 <a>Measure.restrict_apply</a> (<a>measurableSet_toMeasurable</a> _ _),\n      <a>measure_toMeasurable</a>]", [{"full_name": "Set.inter_comm", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [940, 9], "def_end_pos": [940, 19]}, {"full_name": "MeasureTheory.Measure.restrict_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1533, 9], "def_end_pos": [1533, 23]}, {"full_name": "MeasureTheory.measurableSet_toMeasurable", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [645, 9], "def_end_pos": [645, 35]}, {"full_name": "MeasureTheory.measure_toMeasurable", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [653, 9], "def_end_pos": [653, 29]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf g : \u03b1 \u2192 E\ns t : Set \u03b1\n\u03bc \u03bd : Measure \u03b1\nhf : IntegrableOn f s\nh's : \u2200 (x : \u03b1), x \u2208 s \u2192 f x \u2260 0\nu : \u2115 \u2192 \u211d\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nv : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable (Measure.restrict \u03bc s) {x | u n \u2264 \u2016f x\u2016}\nn : \u2115\n\u22a2 \u2191\u2191\u03bc (s \u2229 v n) \u2260 \u22a4", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf g : \u03b1 \u2192 E\ns t : Set \u03b1\n\u03bc \u03bd : Measure \u03b1\nhf : IntegrableOn f s\nh's : \u2200 (x : \u03b1), x \u2208 s \u2192 f x \u2260 0\nu : \u2115 \u2192 \u211d\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nv : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable (Measure.restrict \u03bc s) {x | u n \u2264 \u2016f x\u2016}\nn : \u2115\n\u22a2 \u2191\u2191(Measure.restrict \u03bc s) {x | u n \u2264 \u2016f x\u2016} \u2260 \u22a4"}, {"tactic": "exact (hf.measure_norm_ge_lt_top (u_pos n)).ne", "annotated_tactic": ["exact (hf.measure_norm_ge_lt_top (u_pos n)).<a>ne</a>", [{"full_name": "LT.lt.ne", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [152, 7], "def_end_pos": [152, 15]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf g : \u03b1 \u2192 E\ns t : Set \u03b1\n\u03bc \u03bd : Measure \u03b1\nhf : IntegrableOn f s\nh's : \u2200 (x : \u03b1), x \u2208 s \u2192 f x \u2260 0\nu : \u2115 \u2192 \u211d\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nv : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable (Measure.restrict \u03bc s) {x | u n \u2264 \u2016f x\u2016}\nn : \u2115\n\u22a2 \u2191\u2191(Measure.restrict \u03bc s) {x | u n \u2264 \u2016f x\u2016} \u2260 \u22a4", "state_after": "no goals"}, {"tactic": "simp only [h's x hx, norm_pos_iff, Ne.def, not_false_iff]", "annotated_tactic": ["simp only [h's x hx, <a>norm_pos_iff</a>, <a>Ne.def</a>, <a>not_false_iff</a>]", [{"full_name": "norm_pos_iff", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [2030, 30], "def_end_pos": [2030, 42]}, {"full_name": "Ne.def", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [59, 9], "def_end_pos": [59, 15]}, {"full_name": "not_false_iff", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [82, 9], "def_end_pos": [82, 22]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf g : \u03b1 \u2192 E\ns t : Set \u03b1\n\u03bc \u03bd : Measure \u03b1\nhf : IntegrableOn f s\nh's : \u2200 (x : \u03b1), x \u2208 s \u2192 f x \u2260 0\nu : \u2115 \u2192 \u211d\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nv : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable (Measure.restrict \u03bc s) {x | u n \u2264 \u2016f x\u2016}\nA : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \u2229 v n) \u2260 \u22a4\nx : \u03b1\nhx : x \u2208 s\n\u22a2 0 < \u2016f x\u2016", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "full_name": "MeasureTheory.Lp.simpleFunc.denseRange_coeSimpleFuncNonnegToLpNonneg", "start": [852, 1], "end": [904, 8], "traced_tactics": [{"tactic": "borelize G", "annotated_tactic": ["borelize G", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\n\u22a2 g \u2208 closure (Set.range (coeSimpleFuncNonnegToLpNonneg p \u03bc G))", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\n\u22a2 g \u2208 closure (Set.range (coeSimpleFuncNonnegToLpNonneg p \u03bc G))"}, {"tactic": "rw [mem_closure_iff_seq_limit]", "annotated_tactic": ["rw [<a>mem_closure_iff_seq_limit</a>]", [{"full_name": "mem_closure_iff_seq_limit", "def_path": "Mathlib/Topology/Sequences.lean", "def_pos": [131, 9], "def_end_pos": [131, 34]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\n\u22a2 g \u2208 closure (Set.range (coeSimpleFuncNonnegToLpNonneg p \u03bc G))", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\n\u22a2 \u2203 x, (\u2200 (n : \u2115), x n \u2208 Set.range (coeSimpleFuncNonnegToLpNonneg p \u03bc G)) \u2227 Tendsto x atTop (\ud835\udcdd g)"}, {"tactic": "have hg_mem\u2112p : Mem\u2112p (g : \u03b1 \u2192 G) p \u03bc := Lp.mem\u2112p (g : Lp G p \u03bc)", "annotated_tactic": ["have hg_mem\u2112p : <a>Mem\u2112p</a> (g : \u03b1 \u2192 G) p \u03bc := <a>Lp.mem\u2112p</a> (g : <a>Lp</a> G p \u03bc)", [{"full_name": "MeasureTheory.Mem\u2112p", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [108, 5], "def_end_pos": [108, 10]}, {"full_name": "MeasureTheory.Lp.mem\u2112p", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [216, 19], "def_end_pos": [216, 24]}, {"full_name": "MeasureTheory.Lp", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [98, 5], "def_end_pos": [98, 7]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\n\u22a2 \u2203 x, (\u2200 (n : \u2115), x n \u2208 Set.range (coeSimpleFuncNonnegToLpNonneg p \u03bc G)) \u2227 Tendsto x atTop (\ud835\udcdd g)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\n\u22a2 \u2203 x, (\u2200 (n : \u2115), x n \u2208 Set.range (coeSimpleFuncNonnegToLpNonneg p \u03bc G)) \u2227 Tendsto x atTop (\ud835\udcdd g)"}, {"tactic": "have zero_mem : (0 : G) \u2208 (range (g : \u03b1 \u2192 G) \u222a {0} : Set G) \u2229 { y | 0 \u2264 y } := by\n  simp only [union_singleton, mem_inter_iff, mem_insert_iff, eq_self_iff_true, true_or_iff,\n    mem_setOf_eq, le_refl, and_self_iff]", "annotated_tactic": ["have zero_mem : (0 : G) \u2208 (<a>range</a> (g : \u03b1 \u2192 G) \u222a {0} : <a>Set</a> G) \u2229 { y | 0 \u2264 y } := by\n    simp only [<a>union_singleton</a>, <a>mem_inter_iff</a>, <a>mem_insert_iff</a>, <a>eq_self_iff_true</a>, <a>true_or_iff</a>,\n      <a>mem_setOf_eq</a>, <a>le_refl</a>, <a>and_self_iff</a>]", [{"full_name": "Set.range", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [668, 5], "def_end_pos": [668, 10]}, {"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}, {"full_name": "Set.union_singleton", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1347, 9], "def_end_pos": [1347, 24]}, {"full_name": "Set.mem_inter_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [909, 9], "def_end_pos": [909, 22]}, {"full_name": "Set.mem_insert_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1147, 9], "def_end_pos": [1147, 23]}, {"full_name": "eq_self_iff_true", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [86, 9], "def_end_pos": [86, 25]}, {"full_name": "true_or_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [182, 9], "def_end_pos": [182, 20]}, {"full_name": "Set.mem_setOf_eq", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [256, 29], "def_end_pos": [256, 41]}, {"full_name": "le_refl", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [50, 9], "def_end_pos": [50, 16]}, {"full_name": "and_self_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [155, 9], "def_end_pos": [155, 21]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\n\u22a2 \u2203 x, (\u2200 (n : \u2115), x n \u2208 Set.range (coeSimpleFuncNonnegToLpNonneg p \u03bc G)) \u2227 Tendsto x atTop (\ud835\udcdd g)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\nzero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}\n\u22a2 \u2203 x, (\u2200 (n : \u2115), x n \u2208 Set.range (coeSimpleFuncNonnegToLpNonneg p \u03bc G)) \u2227 Tendsto x atTop (\ud835\udcdd g)"}, {"tactic": "have : SeparableSpace ((range (g : \u03b1 \u2192 G) \u222a {0}) \u2229 { y | 0 \u2264 y } : Set G) := by\n  apply IsSeparable.separableSpace\n  apply IsSeparable.mono _ (Set.inter_subset_left _ _)\n  exact\n    (Lp.stronglyMeasurable (g : Lp G p \u03bc)).isSeparable_range.union\n      (finite_singleton _).isSeparable", "annotated_tactic": ["have : <a>SeparableSpace</a> ((<a>range</a> (g : \u03b1 \u2192 G) \u222a {0}) \u2229 { y | 0 \u2264 y } : <a>Set</a> G) := by\n    apply <a>IsSeparable.separableSpace</a>\n    apply <a>IsSeparable.mono</a> _ (<a>Set.inter_subset_left</a> _ _)\n    exact\n      (<a>Lp.stronglyMeasurable</a> (g : <a>Lp</a> G p \u03bc)).isSeparable_range.union\n        (<a>finite_singleton</a> _).<a>isSeparable</a>", [{"full_name": "TopologicalSpace.SeparableSpace", "def_path": "Mathlib/Topology/Bases.lean", "def_pos": [313, 17], "def_end_pos": [313, 31]}, {"full_name": "Set.range", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [668, 5], "def_end_pos": [668, 10]}, {"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}, {"full_name": "TopologicalSpace.IsSeparable.separableSpace", "def_path": "Mathlib/Topology/EMetricSpace/Basic.lean", "def_pos": [827, 9], "def_end_pos": [827, 59]}, {"full_name": "TopologicalSpace.IsSeparable.mono", "def_path": "Mathlib/Topology/Bases.lean", "def_pos": [442, 9], "def_end_pos": [442, 25]}, {"full_name": "Set.inter_subset_left", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [965, 9], "def_end_pos": [965, 26]}, {"full_name": "MeasureTheory.Lp.stronglyMeasurable", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [207, 19], "def_end_pos": [207, 37]}, {"full_name": "MeasureTheory.Lp", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [98, 5], "def_end_pos": [98, 7]}, {"full_name": "Set.finite_singleton", "def_path": "Mathlib/Data/Set/Finite.lean", "def_pos": [848, 9], "def_end_pos": [848, 25]}, {"full_name": "Set.Finite.isSeparable", "def_path": "Mathlib/Topology/Bases.lean", "def_pos": [492, 9], "def_end_pos": [492, 38]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\nzero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}\n\u22a2 \u2203 x, (\u2200 (n : \u2115), x n \u2208 Set.range (coeSimpleFuncNonnegToLpNonneg p \u03bc G)) \u2227 Tendsto x atTop (\ud835\udcdd g)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\nzero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}\nthis : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y})\n\u22a2 \u2203 x, (\u2200 (n : \u2115), x n \u2208 Set.range (coeSimpleFuncNonnegToLpNonneg p \u03bc G)) \u2227 Tendsto x atTop (\ud835\udcdd g)"}, {"tactic": "have g_meas : Measurable (g : \u03b1 \u2192 G) := (Lp.stronglyMeasurable (g : Lp G p \u03bc)).measurable", "annotated_tactic": ["have g_meas : <a>Measurable</a> (g : \u03b1 \u2192 G) := (<a>Lp.stronglyMeasurable</a> (g : <a>Lp</a> G p \u03bc)).<a>measurable</a>", [{"full_name": "Measurable", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [535, 5], "def_end_pos": [535, 15]}, {"full_name": "MeasureTheory.Lp.stronglyMeasurable", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [207, 19], "def_end_pos": [207, 37]}, {"full_name": "MeasureTheory.Lp", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [98, 5], "def_end_pos": [98, 7]}, {"full_name": "MeasureTheory.StronglyMeasurable.measurable", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [341, 19], "def_end_pos": [341, 29]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\nzero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}\nthis : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y})\n\u22a2 \u2203 x, (\u2200 (n : \u2115), x n \u2208 Set.range (coeSimpleFuncNonnegToLpNonneg p \u03bc G)) \u2227 Tendsto x atTop (\ud835\udcdd g)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\nzero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}\nthis : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y})\ng_meas : Measurable \u2191\u2191\u2191g\n\u22a2 \u2203 x, (\u2200 (n : \u2115), x n \u2208 Set.range (coeSimpleFuncNonnegToLpNonneg p \u03bc G)) \u2227 Tendsto x atTop (\ud835\udcdd g)"}, {"tactic": "let x n := SimpleFunc.approxOn g g_meas ((range (g : \u03b1 \u2192 G) \u222a {0}) \u2229 { y | 0 \u2264 y }) 0 zero_mem n", "annotated_tactic": ["let x n := <a>SimpleFunc.approxOn</a> g g_meas ((<a>range</a> (g : \u03b1 \u2192 G) \u222a {0}) \u2229 { y | 0 \u2264 y }) 0 zero_mem n", [{"full_name": "MeasureTheory.SimpleFunc.approxOn", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDense.lean", "def_pos": [127, 19], "def_end_pos": [127, 27]}, {"full_name": "Set.range", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [668, 5], "def_end_pos": [668, 10]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\nzero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}\nthis : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y})\ng_meas : Measurable \u2191\u2191\u2191g\n\u22a2 \u2203 x, (\u2200 (n : \u2115), x n \u2208 Set.range (coeSimpleFuncNonnegToLpNonneg p \u03bc G)) \u2227 Tendsto x atTop (\ud835\udcdd g)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\nzero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}\nthis : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y})\ng_meas : Measurable \u2191\u2191\u2191g\nx : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n\n\u22a2 \u2203 x, (\u2200 (n : \u2115), x n \u2208 Set.range (coeSimpleFuncNonnegToLpNonneg p \u03bc G)) \u2227 Tendsto x atTop (\ud835\udcdd g)"}, {"tactic": "have hx_nonneg : \u2200 n, 0 \u2264 x n := by\n  intro n a\n  change x n a \u2208 { y : G | 0 \u2264 y }\n  have A : (range (g : \u03b1 \u2192 G) \u222a {0} : Set G) \u2229 { y | 0 \u2264 y } \u2286 { y | 0 \u2264 y } :=\n    inter_subset_right _ _\n  apply A\n  exact SimpleFunc.approxOn_mem g_meas _ n a", "annotated_tactic": ["have hx_nonneg : \u2200 n, 0 \u2264 x n := by\n    intro n a\n    change x n a \u2208 { y : G | 0 \u2264 y }\n    have A : (<a>range</a> (g : \u03b1 \u2192 G) \u222a {0} : <a>Set</a> G) \u2229 { y | 0 \u2264 y } \u2286 { y | 0 \u2264 y } :=\n      <a>inter_subset_right</a> _ _\n    apply A\n    exact <a>SimpleFunc.approxOn_mem</a> g_meas _ n a", [{"full_name": "Set.range", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [668, 5], "def_end_pos": [668, 10]}, {"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}, {"full_name": "Set.inter_subset_right", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [969, 9], "def_end_pos": [969, 27]}, {"full_name": "MeasureTheory.SimpleFunc.approxOn_mem", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDense.lean", "def_pos": [139, 9], "def_end_pos": [139, 21]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\nzero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}\nthis : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y})\ng_meas : Measurable \u2191\u2191\u2191g\nx : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n\n\u22a2 \u2203 x, (\u2200 (n : \u2115), x n \u2208 Set.range (coeSimpleFuncNonnegToLpNonneg p \u03bc G)) \u2227 Tendsto x atTop (\ud835\udcdd g)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\nzero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}\nthis : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y})\ng_meas : Measurable \u2191\u2191\u2191g\nx : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n\nhx_nonneg : \u2200 (n : \u2115), 0 \u2264 x n\n\u22a2 \u2203 x, (\u2200 (n : \u2115), x n \u2208 Set.range (coeSimpleFuncNonnegToLpNonneg p \u03bc G)) \u2227 Tendsto x atTop (\ud835\udcdd g)"}, {"tactic": "have hx_mem\u2112p : \u2200 n, Mem\u2112p (x n) p \u03bc :=\n  SimpleFunc.mem\u2112p_approxOn _ hg_mem\u2112p _ \u27e8aestronglyMeasurable_const, by simp\u27e9", "annotated_tactic": ["have hx_mem\u2112p : \u2200 n, <a>Mem\u2112p</a> (x n) p \u03bc :=\n    <a>SimpleFunc.mem\u2112p_approxOn</a> _ hg_mem\u2112p _ \u27e8<a>aestronglyMeasurable_const</a>, by simp\u27e9", [{"full_name": "MeasureTheory.Mem\u2112p", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [108, 5], "def_end_pos": [108, 10]}, {"full_name": "MeasureTheory.SimpleFunc.mem\u2112p_approxOn", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "def_pos": [139, 9], "def_end_pos": [139, 23]}, {"full_name": "MeasureTheory.aestronglyMeasurable_const", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1155, 9], "def_end_pos": [1155, 35]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\nzero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}\nthis : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y})\ng_meas : Measurable \u2191\u2191\u2191g\nx : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n\nhx_nonneg : \u2200 (n : \u2115), 0 \u2264 x n\n\u22a2 \u2203 x, (\u2200 (n : \u2115), x n \u2208 Set.range (coeSimpleFuncNonnegToLpNonneg p \u03bc G)) \u2227 Tendsto x atTop (\ud835\udcdd g)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\nzero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}\nthis : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y})\ng_meas : Measurable \u2191\u2191\u2191g\nx : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n\nhx_nonneg : \u2200 (n : \u2115), 0 \u2264 x n\nhx_mem\u2112p : \u2200 (n : \u2115), Mem\u2112p (\u2191(x n)) p\n\u22a2 \u2203 x, (\u2200 (n : \u2115), x n \u2208 Set.range (coeSimpleFuncNonnegToLpNonneg p \u03bc G)) \u2227 Tendsto x atTop (\ud835\udcdd g)"}, {"tactic": "have h_toLp := fun n => Mem\u2112p.coeFn_toLp (hx_mem\u2112p n)", "annotated_tactic": ["have h_toLp := fun n => <a>Mem\u2112p.coeFn_toLp</a> (hx_mem\u2112p n)", [{"full_name": "MeasureTheory.Mem\u2112p.coeFn_toLp", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [119, 9], "def_end_pos": [119, 19]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\nzero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}\nthis : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y})\ng_meas : Measurable \u2191\u2191\u2191g\nx : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n\nhx_nonneg : \u2200 (n : \u2115), 0 \u2264 x n\nhx_mem\u2112p : \u2200 (n : \u2115), Mem\u2112p (\u2191(x n)) p\n\u22a2 \u2203 x, (\u2200 (n : \u2115), x n \u2208 Set.range (coeSimpleFuncNonnegToLpNonneg p \u03bc G)) \u2227 Tendsto x atTop (\ud835\udcdd g)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\nzero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}\nthis : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y})\ng_meas : Measurable \u2191\u2191\u2191g\nx : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n\nhx_nonneg : \u2200 (n : \u2115), 0 \u2264 x n\nhx_mem\u2112p : \u2200 (n : \u2115), Mem\u2112p (\u2191(x n)) p\nh_toLp : \u2200 (n : \u2115), \u2191\u2191(Mem\u2112p.toLp \u2191(x n) (_ : Mem\u2112p (\u2191(x n)) p)) =\u1d50[\u03bc] \u2191(x n)\n\u22a2 \u2203 x, (\u2200 (n : \u2115), x n \u2208 Set.range (coeSimpleFuncNonnegToLpNonneg p \u03bc G)) \u2227 Tendsto x atTop (\ud835\udcdd g)"}, {"tactic": "have hx_nonneg_Lp : \u2200 n, 0 \u2264 toLp (x n) (hx_mem\u2112p n) := by\n  intro n\n  rw [\u2190 Lp.simpleFunc.coeFn_le, Lp.simpleFunc.toLp_eq_toLp]\n  filter_upwards [Lp.simpleFunc.coeFn_zero p \u03bc G, h_toLp n] with a ha0 ha_toLp\n  rw [ha0, ha_toLp]\n  exact hx_nonneg n a", "annotated_tactic": ["have hx_nonneg_Lp : \u2200 n, 0 \u2264 <a>toLp</a> (x n) (hx_mem\u2112p n) := by\n    intro n\n    rw [\u2190 <a>Lp.simpleFunc.coeFn_le</a>, <a>Lp.simpleFunc.toLp_eq_toLp</a>]\n    filter_upwards [<a>Lp.simpleFunc.coeFn_zero</a> p \u03bc G, h_toLp n] with a ha0 ha_toLp\n    rw [ha0, ha_toLp]\n    exact hx_nonneg n a", [{"full_name": "MeasureTheory.Lp.simpleFunc.toLp", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "def_pos": [539, 5], "def_end_pos": [539, 9]}, {"full_name": "MeasureTheory.Lp.simpleFunc.coeFn_le", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "def_pos": [810, 9], "def_end_pos": [810, 17]}, {"full_name": "MeasureTheory.Lp.simpleFunc.toLp_eq_toLp", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "def_pos": [543, 9], "def_end_pos": [543, 21]}, {"full_name": "MeasureTheory.Lp.simpleFunc.coeFn_zero", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "def_pos": [827, 9], "def_end_pos": [827, 19]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\nzero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}\nthis : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y})\ng_meas : Measurable \u2191\u2191\u2191g\nx : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n\nhx_nonneg : \u2200 (n : \u2115), 0 \u2264 x n\nhx_mem\u2112p : \u2200 (n : \u2115), Mem\u2112p (\u2191(x n)) p\nh_toLp : \u2200 (n : \u2115), \u2191\u2191(Mem\u2112p.toLp \u2191(x n) (_ : Mem\u2112p (\u2191(x n)) p)) =\u1d50[\u03bc] \u2191(x n)\n\u22a2 \u2203 x, (\u2200 (n : \u2115), x n \u2208 Set.range (coeSimpleFuncNonnegToLpNonneg p \u03bc G)) \u2227 Tendsto x atTop (\ud835\udcdd g)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\nzero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}\nthis : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y})\ng_meas : Measurable \u2191\u2191\u2191g\nx : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n\nhx_nonneg : \u2200 (n : \u2115), 0 \u2264 x n\nhx_mem\u2112p : \u2200 (n : \u2115), Mem\u2112p (\u2191(x n)) p\nh_toLp : \u2200 (n : \u2115), \u2191\u2191(Mem\u2112p.toLp \u2191(x n) (_ : Mem\u2112p (\u2191(x n)) p)) =\u1d50[\u03bc] \u2191(x n)\nhx_nonneg_Lp : \u2200 (n : \u2115), 0 \u2264 toLp (x n) (_ : Mem\u2112p (\u2191(x n)) p)\n\u22a2 \u2203 x, (\u2200 (n : \u2115), x n \u2208 Set.range (coeSimpleFuncNonnegToLpNonneg p \u03bc G)) \u2227 Tendsto x atTop (\ud835\udcdd g)"}, {"tactic": "refine'\n  \u27e8fun n =>\n    (coeSimpleFuncNonnegToLpNonneg p \u03bc G) \u27e8toLp (x n) (hx_mem\u2112p n), hx_nonneg_Lp n\u27e9,\n    fun n => mem_range_self _, _\u27e9", "annotated_tactic": ["refine'\n    \u27e8fun n =>\n      (<a>coeSimpleFuncNonnegToLpNonneg</a> p \u03bc G) \u27e8<a>toLp</a> (x n) (hx_mem\u2112p n), hx_nonneg_Lp n\u27e9,\n      fun n => <a>mem_range_self</a> _, _\u27e9", [{"full_name": "MeasureTheory.Lp.simpleFunc.coeSimpleFuncNonnegToLpNonneg", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "def_pos": [848, 5], "def_end_pos": [848, 34]}, {"full_name": "MeasureTheory.Lp.simpleFunc.toLp", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "def_pos": [539, 5], "def_end_pos": [539, 9]}, {"full_name": "Set.mem_range_self", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [680, 9], "def_end_pos": [680, 23]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\nzero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}\nthis : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y})\ng_meas : Measurable \u2191\u2191\u2191g\nx : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n\nhx_nonneg : \u2200 (n : \u2115), 0 \u2264 x n\nhx_mem\u2112p : \u2200 (n : \u2115), Mem\u2112p (\u2191(x n)) p\nh_toLp : \u2200 (n : \u2115), \u2191\u2191(Mem\u2112p.toLp \u2191(x n) (_ : Mem\u2112p (\u2191(x n)) p)) =\u1d50[\u03bc] \u2191(x n)\nhx_nonneg_Lp : \u2200 (n : \u2115), 0 \u2264 toLp (x n) (_ : Mem\u2112p (\u2191(x n)) p)\nhx_tendsto : Tendsto (fun n => snorm (\u2191(x n) - \u2191\u2191\u2191g) p \u03bc) atTop (\ud835\udcdd 0)\n\u22a2 \u2203 x, (\u2200 (n : \u2115), x n \u2208 Set.range (coeSimpleFuncNonnegToLpNonneg p \u03bc G)) \u2227 Tendsto x atTop (\ud835\udcdd g)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\nzero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}\nthis : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y})\ng_meas : Measurable \u2191\u2191\u2191g\nx : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n\nhx_nonneg : \u2200 (n : \u2115), 0 \u2264 x n\nhx_mem\u2112p : \u2200 (n : \u2115), Mem\u2112p (\u2191(x n)) p\nh_toLp : \u2200 (n : \u2115), \u2191\u2191(Mem\u2112p.toLp \u2191(x n) (_ : Mem\u2112p (\u2191(x n)) p)) =\u1d50[\u03bc] \u2191(x n)\nhx_nonneg_Lp : \u2200 (n : \u2115), 0 \u2264 toLp (x n) (_ : Mem\u2112p (\u2191(x n)) p)\nhx_tendsto : Tendsto (fun n => snorm (\u2191(x n) - \u2191\u2191\u2191g) p \u03bc) atTop (\ud835\udcdd 0)\n\u22a2 Tendsto\n    (fun n =>\n      coeSimpleFuncNonnegToLpNonneg p \u03bc G\n        { val := toLp (x n) (_ : Mem\u2112p (\u2191(x n)) p), property := (_ : 0 \u2264 toLp (x n) (_ : Mem\u2112p (\u2191(x n)) p)) })\n    atTop (\ud835\udcdd g)"}, {"tactic": "suffices Tendsto (fun n : \u2115 => (toLp (x n) (hx_mem\u2112p n) : Lp G p \u03bc)) atTop (\ud835\udcdd (g : Lp G p \u03bc)) by\n  rw [tendsto_iff_dist_tendsto_zero] at this \u22a2\n  simp_rw [Subtype.dist_eq]\n  exact this", "annotated_tactic": ["suffices <a>Tendsto</a> (fun n : \u2115 => (<a>toLp</a> (x n) (hx_mem\u2112p n) : <a>Lp</a> G p \u03bc)) <a>atTop</a> (\ud835\udcdd (g : <a>Lp</a> G p \u03bc)) by\n    rw [<a>tendsto_iff_dist_tendsto_zero</a>] at this \u22a2\n    simp_rw [<a>Subtype.dist_eq</a>]\n    exact this", [{"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2939, 5], "def_end_pos": [2939, 12]}, {"full_name": "MeasureTheory.Lp.simpleFunc.toLp", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "def_pos": [539, 5], "def_end_pos": [539, 9]}, {"full_name": "MeasureTheory.Lp", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [98, 5], "def_end_pos": [98, 7]}, {"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}, {"full_name": "MeasureTheory.Lp", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [98, 5], "def_end_pos": [98, 7]}, {"full_name": "tendsto_iff_dist_tendsto_zero", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [1850, 9], "def_end_pos": [1850, 38]}, {"full_name": "Subtype.dist_eq", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [1656, 9], "def_end_pos": [1656, 24]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\nzero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}\nthis : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y})\ng_meas : Measurable \u2191\u2191\u2191g\nx : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n\nhx_nonneg : \u2200 (n : \u2115), 0 \u2264 x n\nhx_mem\u2112p : \u2200 (n : \u2115), Mem\u2112p (\u2191(x n)) p\nh_toLp : \u2200 (n : \u2115), \u2191\u2191(Mem\u2112p.toLp \u2191(x n) (_ : Mem\u2112p (\u2191(x n)) p)) =\u1d50[\u03bc] \u2191(x n)\nhx_nonneg_Lp : \u2200 (n : \u2115), 0 \u2264 toLp (x n) (_ : Mem\u2112p (\u2191(x n)) p)\nhx_tendsto : Tendsto (fun n => snorm (\u2191(x n) - \u2191\u2191\u2191g) p \u03bc) atTop (\ud835\udcdd 0)\n\u22a2 Tendsto\n    (fun n =>\n      coeSimpleFuncNonnegToLpNonneg p \u03bc G\n        { val := toLp (x n) (_ : Mem\u2112p (\u2191(x n)) p), property := (_ : 0 \u2264 toLp (x n) (_ : Mem\u2112p (\u2191(x n)) p)) })\n    atTop (\ud835\udcdd g)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\nzero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}\nthis : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y})\ng_meas : Measurable \u2191\u2191\u2191g\nx : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n\nhx_nonneg : \u2200 (n : \u2115), 0 \u2264 x n\nhx_mem\u2112p : \u2200 (n : \u2115), Mem\u2112p (\u2191(x n)) p\nh_toLp : \u2200 (n : \u2115), \u2191\u2191(Mem\u2112p.toLp \u2191(x n) (_ : Mem\u2112p (\u2191(x n)) p)) =\u1d50[\u03bc] \u2191(x n)\nhx_nonneg_Lp : \u2200 (n : \u2115), 0 \u2264 toLp (x n) (_ : Mem\u2112p (\u2191(x n)) p)\nhx_tendsto : Tendsto (fun n => snorm (\u2191(x n) - \u2191\u2191\u2191g) p \u03bc) atTop (\ud835\udcdd 0)\n\u22a2 Tendsto (fun n => \u2191(toLp (x n) (_ : Mem\u2112p (\u2191(x n)) p))) atTop (\ud835\udcdd \u2191g)"}, {"tactic": "rw [Lp.tendsto_Lp_iff_tendsto_\u2112p']", "annotated_tactic": ["rw [<a>Lp.tendsto_Lp_iff_tendsto_\u2112p'</a>]", [{"full_name": "MeasureTheory.Lp.tendsto_Lp_iff_tendsto_\u2112p'", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [1367, 9], "def_end_pos": [1367, 35]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\nzero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}\nthis : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y})\ng_meas : Measurable \u2191\u2191\u2191g\nx : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n\nhx_nonneg : \u2200 (n : \u2115), 0 \u2264 x n\nhx_mem\u2112p : \u2200 (n : \u2115), Mem\u2112p (\u2191(x n)) p\nh_toLp : \u2200 (n : \u2115), \u2191\u2191(Mem\u2112p.toLp \u2191(x n) (_ : Mem\u2112p (\u2191(x n)) p)) =\u1d50[\u03bc] \u2191(x n)\nhx_nonneg_Lp : \u2200 (n : \u2115), 0 \u2264 toLp (x n) (_ : Mem\u2112p (\u2191(x n)) p)\nhx_tendsto : Tendsto (fun n => snorm (\u2191(x n) - \u2191\u2191\u2191g) p \u03bc) atTop (\ud835\udcdd 0)\n\u22a2 Tendsto (fun n => \u2191(toLp (x n) (_ : Mem\u2112p (\u2191(x n)) p))) atTop (\ud835\udcdd \u2191g)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\nzero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}\nthis : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y})\ng_meas : Measurable \u2191\u2191\u2191g\nx : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n\nhx_nonneg : \u2200 (n : \u2115), 0 \u2264 x n\nhx_mem\u2112p : \u2200 (n : \u2115), Mem\u2112p (\u2191(x n)) p\nh_toLp : \u2200 (n : \u2115), \u2191\u2191(Mem\u2112p.toLp \u2191(x n) (_ : Mem\u2112p (\u2191(x n)) p)) =\u1d50[\u03bc] \u2191(x n)\nhx_nonneg_Lp : \u2200 (n : \u2115), 0 \u2264 toLp (x n) (_ : Mem\u2112p (\u2191(x n)) p)\nhx_tendsto : Tendsto (fun n => snorm (\u2191(x n) - \u2191\u2191\u2191g) p \u03bc) atTop (\ud835\udcdd 0)\n\u22a2 Tendsto (fun n => snorm (\u2191\u2191\u2191(toLp (x n) (_ : Mem\u2112p (\u2191(x n)) p)) - \u2191\u2191\u2191g) p \u03bc) atTop (\ud835\udcdd 0)"}, {"tactic": "refine Filter.Tendsto.congr (fun n => snorm_congr_ae (EventuallyEq.sub ?_ ?_)) hx_tendsto", "annotated_tactic": ["refine <a>Filter.Tendsto.congr</a> (fun n => <a>snorm_congr_ae</a> (<a>EventuallyEq.sub</a> ?_ ?_)) hx_tendsto", [{"full_name": "Filter.Tendsto.congr", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [3019, 9], "def_end_pos": [3019, 22]}, {"full_name": "MeasureTheory.snorm_congr_ae", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [549, 9], "def_end_pos": [549, 23]}, {"full_name": "Filter.EventuallyEq.sub", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1544, 3], "def_end_pos": [1544, 14]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\nzero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}\nthis : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y})\ng_meas : Measurable \u2191\u2191\u2191g\nx : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n\nhx_nonneg : \u2200 (n : \u2115), 0 \u2264 x n\nhx_mem\u2112p : \u2200 (n : \u2115), Mem\u2112p (\u2191(x n)) p\nh_toLp : \u2200 (n : \u2115), \u2191\u2191(Mem\u2112p.toLp \u2191(x n) (_ : Mem\u2112p (\u2191(x n)) p)) =\u1d50[\u03bc] \u2191(x n)\nhx_nonneg_Lp : \u2200 (n : \u2115), 0 \u2264 toLp (x n) (_ : Mem\u2112p (\u2191(x n)) p)\nhx_tendsto : Tendsto (fun n => snorm (\u2191(x n) - \u2191\u2191\u2191g) p \u03bc) atTop (\ud835\udcdd 0)\n\u22a2 Tendsto (fun n => snorm (\u2191\u2191\u2191(toLp (x n) (_ : Mem\u2112p (\u2191(x n)) p)) - \u2191\u2191\u2191g) p \u03bc) atTop (\ud835\udcdd 0)", "state_after": "case refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\nzero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}\nthis : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y})\ng_meas : Measurable \u2191\u2191\u2191g\nx : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n\nhx_nonneg : \u2200 (n : \u2115), 0 \u2264 x n\nhx_mem\u2112p : \u2200 (n : \u2115), Mem\u2112p (\u2191(x n)) p\nh_toLp : \u2200 (n : \u2115), \u2191\u2191(Mem\u2112p.toLp \u2191(x n) (_ : Mem\u2112p (\u2191(x n)) p)) =\u1d50[\u03bc] \u2191(x n)\nhx_nonneg_Lp : \u2200 (n : \u2115), 0 \u2264 toLp (x n) (_ : Mem\u2112p (\u2191(x n)) p)\nhx_tendsto : Tendsto (fun n => snorm (\u2191(x n) - \u2191\u2191\u2191g) p \u03bc) atTop (\ud835\udcdd 0)\nn : \u2115\n\u22a2 (fun x_1 => \u2191(x n) x_1) =\u1d50[\u03bc] fun x_1 => \u2191\u2191\u2191(toLp (x n) (_ : Mem\u2112p (\u2191(x n)) p)) x_1\n\ncase refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\nzero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}\nthis : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y})\ng_meas : Measurable \u2191\u2191\u2191g\nx : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n\nhx_nonneg : \u2200 (n : \u2115), 0 \u2264 x n\nhx_mem\u2112p : \u2200 (n : \u2115), Mem\u2112p (\u2191(x n)) p\nh_toLp : \u2200 (n : \u2115), \u2191\u2191(Mem\u2112p.toLp \u2191(x n) (_ : Mem\u2112p (\u2191(x n)) p)) =\u1d50[\u03bc] \u2191(x n)\nhx_nonneg_Lp : \u2200 (n : \u2115), 0 \u2264 toLp (x n) (_ : Mem\u2112p (\u2191(x n)) p)\nhx_tendsto : Tendsto (fun n => snorm (\u2191(x n) - \u2191\u2191\u2191g) p \u03bc) atTop (\ud835\udcdd 0)\nn : \u2115\n\u22a2 (fun x => \u2191\u2191\u2191g x) =\u1d50[\u03bc] fun x => \u2191\u2191\u2191g x"}, {"tactic": "simp only [union_singleton, mem_inter_iff, mem_insert_iff, eq_self_iff_true, true_or_iff,\n  mem_setOf_eq, le_refl, and_self_iff]", "annotated_tactic": ["simp only [<a>union_singleton</a>, <a>mem_inter_iff</a>, <a>mem_insert_iff</a>, <a>eq_self_iff_true</a>, <a>true_or_iff</a>,\n      <a>mem_setOf_eq</a>, <a>le_refl</a>, <a>and_self_iff</a>]", [{"full_name": "Set.union_singleton", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1347, 9], "def_end_pos": [1347, 24]}, {"full_name": "Set.mem_inter_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [909, 9], "def_end_pos": [909, 22]}, {"full_name": "Set.mem_insert_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1147, 9], "def_end_pos": [1147, 23]}, {"full_name": "eq_self_iff_true", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [86, 9], "def_end_pos": [86, 25]}, {"full_name": "true_or_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [182, 9], "def_end_pos": [182, 20]}, {"full_name": "Set.mem_setOf_eq", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [256, 29], "def_end_pos": [256, 41]}, {"full_name": "le_refl", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [50, 9], "def_end_pos": [50, 16]}, {"full_name": "and_self_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [155, 9], "def_end_pos": [155, 21]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\n\u22a2 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}", "state_after": "no goals"}, {"tactic": "apply IsSeparable.separableSpace", "annotated_tactic": ["apply <a>IsSeparable.separableSpace</a>", [{"full_name": "TopologicalSpace.IsSeparable.separableSpace", "def_path": "Mathlib/Topology/EMetricSpace/Basic.lean", "def_pos": [827, 9], "def_end_pos": [827, 59]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\nzero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}\n\u22a2 SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y})", "state_after": "case hs\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\nzero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}\n\u22a2 IsSeparable ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y})"}, {"tactic": "apply IsSeparable.mono _ (Set.inter_subset_left _ _)", "annotated_tactic": ["apply <a>IsSeparable.mono</a> _ (<a>Set.inter_subset_left</a> _ _)", [{"full_name": "TopologicalSpace.IsSeparable.mono", "def_path": "Mathlib/Topology/Bases.lean", "def_pos": [442, 9], "def_end_pos": [442, 25]}, {"full_name": "Set.inter_subset_left", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [965, 9], "def_end_pos": [965, 26]}]], "state_before": "case hs\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\nzero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}\n\u22a2 IsSeparable ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y})", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\nzero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}\n\u22a2 IsSeparable (Set.range \u2191\u2191\u2191g \u222a {0})"}, {"tactic": "exact\n  (Lp.stronglyMeasurable (g : Lp G p \u03bc)).isSeparable_range.union\n    (finite_singleton _).isSeparable", "annotated_tactic": ["exact\n      (<a>Lp.stronglyMeasurable</a> (g : <a>Lp</a> G p \u03bc)).isSeparable_range.union\n        (<a>finite_singleton</a> _).<a>isSeparable</a>", [{"full_name": "MeasureTheory.Lp.stronglyMeasurable", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [207, 19], "def_end_pos": [207, 37]}, {"full_name": "MeasureTheory.Lp", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [98, 5], "def_end_pos": [98, 7]}, {"full_name": "Set.finite_singleton", "def_path": "Mathlib/Data/Set/Finite.lean", "def_pos": [848, 9], "def_end_pos": [848, 25]}, {"full_name": "Set.Finite.isSeparable", "def_path": "Mathlib/Topology/Bases.lean", "def_pos": [492, 9], "def_end_pos": [492, 38]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\nzero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}\n\u22a2 IsSeparable (Set.range \u2191\u2191\u2191g \u222a {0})", "state_after": "no goals"}, {"tactic": "intro n a", "annotated_tactic": ["intro n a", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\nzero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}\nthis : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y})\ng_meas : Measurable \u2191\u2191\u2191g\nx : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n\n\u22a2 \u2200 (n : \u2115), 0 \u2264 x n", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\nzero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}\nthis : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y})\ng_meas : Measurable \u2191\u2191\u2191g\nx : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n\nn : \u2115\na : \u03b1\n\u22a2 \u21910 a \u2264 \u2191(x n) a"}, {"tactic": "change x n a \u2208 { y : G | 0 \u2264 y }", "annotated_tactic": ["change x n a \u2208 { y : G | 0 \u2264 y }", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\nzero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}\nthis : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y})\ng_meas : Measurable \u2191\u2191\u2191g\nx : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n\nn : \u2115\na : \u03b1\n\u22a2 \u21910 a \u2264 \u2191(x n) a", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\nzero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}\nthis : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y})\ng_meas : Measurable \u2191\u2191\u2191g\nx : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n\nn : \u2115\na : \u03b1\n\u22a2 \u2191(x n) a \u2208 {y | 0 \u2264 y}"}, {"tactic": "have A : (range (g : \u03b1 \u2192 G) \u222a {0} : Set G) \u2229 { y | 0 \u2264 y } \u2286 { y | 0 \u2264 y } :=\n  inter_subset_right _ _", "annotated_tactic": ["have A : (<a>range</a> (g : \u03b1 \u2192 G) \u222a {0} : <a>Set</a> G) \u2229 { y | 0 \u2264 y } \u2286 { y | 0 \u2264 y } :=\n      <a>inter_subset_right</a> _ _", [{"full_name": "Set.range", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [668, 5], "def_end_pos": [668, 10]}, {"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}, {"full_name": "Set.inter_subset_right", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [969, 9], "def_end_pos": [969, 27]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\nzero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}\nthis : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y})\ng_meas : Measurable \u2191\u2191\u2191g\nx : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n\nn : \u2115\na : \u03b1\n\u22a2 \u2191(x n) a \u2208 {y | 0 \u2264 y}", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\nzero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}\nthis : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y})\ng_meas : Measurable \u2191\u2191\u2191g\nx : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n\nn : \u2115\na : \u03b1\nA : (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y} \u2286 {y | 0 \u2264 y}\n\u22a2 \u2191(x n) a \u2208 {y | 0 \u2264 y}"}, {"tactic": "apply A", "annotated_tactic": ["apply A", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\nzero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}\nthis : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y})\ng_meas : Measurable \u2191\u2191\u2191g\nx : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n\nn : \u2115\na : \u03b1\nA : (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y} \u2286 {y | 0 \u2264 y}\n\u22a2 \u2191(x n) a \u2208 {y | 0 \u2264 y}", "state_after": "case a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\nzero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}\nthis : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y})\ng_meas : Measurable \u2191\u2191\u2191g\nx : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n\nn : \u2115\na : \u03b1\nA : (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y} \u2286 {y | 0 \u2264 y}\n\u22a2 \u2191(x n) a \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}"}, {"tactic": "exact SimpleFunc.approxOn_mem g_meas _ n a", "annotated_tactic": ["exact <a>SimpleFunc.approxOn_mem</a> g_meas _ n a", [{"full_name": "MeasureTheory.SimpleFunc.approxOn_mem", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDense.lean", "def_pos": [139, 9], "def_end_pos": [139, 21]}]], "state_before": "case a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\nzero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}\nthis : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y})\ng_meas : Measurable \u2191\u2191\u2191g\nx : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n\nn : \u2115\na : \u03b1\nA : (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y} \u2286 {y | 0 \u2264 y}\n\u22a2 \u2191(x n) a \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\nzero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}\nthis : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y})\ng_meas : Measurable \u2191\u2191\u2191g\nx : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n\nhx_nonneg : \u2200 (n : \u2115), 0 \u2264 x n\n\u22a2 snorm (fun x => 0) p \u03bc < \u22a4", "state_after": "no goals"}, {"tactic": "intro n", "annotated_tactic": ["intro n", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\nzero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}\nthis : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y})\ng_meas : Measurable \u2191\u2191\u2191g\nx : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n\nhx_nonneg : \u2200 (n : \u2115), 0 \u2264 x n\nhx_mem\u2112p : \u2200 (n : \u2115), Mem\u2112p (\u2191(x n)) p\nh_toLp : \u2200 (n : \u2115), \u2191\u2191(Mem\u2112p.toLp \u2191(x n) (_ : Mem\u2112p (\u2191(x n)) p)) =\u1d50[\u03bc] \u2191(x n)\n\u22a2 \u2200 (n : \u2115), 0 \u2264 toLp (x n) (_ : Mem\u2112p (\u2191(x n)) p)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\nzero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}\nthis : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y})\ng_meas : Measurable \u2191\u2191\u2191g\nx : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n\nhx_nonneg : \u2200 (n : \u2115), 0 \u2264 x n\nhx_mem\u2112p : \u2200 (n : \u2115), Mem\u2112p (\u2191(x n)) p\nh_toLp : \u2200 (n : \u2115), \u2191\u2191(Mem\u2112p.toLp \u2191(x n) (_ : Mem\u2112p (\u2191(x n)) p)) =\u1d50[\u03bc] \u2191(x n)\nn : \u2115\n\u22a2 0 \u2264 toLp (x n) (_ : Mem\u2112p (\u2191(x n)) p)"}, {"tactic": "rw [\u2190 Lp.simpleFunc.coeFn_le, Lp.simpleFunc.toLp_eq_toLp]", "annotated_tactic": ["rw [\u2190 <a>Lp.simpleFunc.coeFn_le</a>, <a>Lp.simpleFunc.toLp_eq_toLp</a>]", [{"full_name": "MeasureTheory.Lp.simpleFunc.coeFn_le", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "def_pos": [810, 9], "def_end_pos": [810, 17]}, {"full_name": "MeasureTheory.Lp.simpleFunc.toLp_eq_toLp", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "def_pos": [543, 9], "def_end_pos": [543, 21]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\nzero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}\nthis : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y})\ng_meas : Measurable \u2191\u2191\u2191g\nx : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n\nhx_nonneg : \u2200 (n : \u2115), 0 \u2264 x n\nhx_mem\u2112p : \u2200 (n : \u2115), Mem\u2112p (\u2191(x n)) p\nh_toLp : \u2200 (n : \u2115), \u2191\u2191(Mem\u2112p.toLp \u2191(x n) (_ : Mem\u2112p (\u2191(x n)) p)) =\u1d50[\u03bc] \u2191(x n)\nn : \u2115\n\u22a2 0 \u2264 toLp (x n) (_ : Mem\u2112p (\u2191(x n)) p)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\nzero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}\nthis : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y})\ng_meas : Measurable \u2191\u2191\u2191g\nx : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n\nhx_nonneg : \u2200 (n : \u2115), 0 \u2264 x n\nhx_mem\u2112p : \u2200 (n : \u2115), Mem\u2112p (\u2191(x n)) p\nh_toLp : \u2200 (n : \u2115), \u2191\u2191(Mem\u2112p.toLp \u2191(x n) (_ : Mem\u2112p (\u2191(x n)) p)) =\u1d50[\u03bc] \u2191(x n)\nn : \u2115\n\u22a2 \u2191\u2191\u21910 \u2264\u1d50[\u03bc] \u2191\u2191(Mem\u2112p.toLp \u2191(x n) (_ : Mem\u2112p (\u2191(x n)) p))"}, {"tactic": "filter_upwards [Lp.simpleFunc.coeFn_zero p \u03bc G, h_toLp n] with a ha0 ha_toLp", "annotated_tactic": ["filter_upwards [<a>Lp.simpleFunc.coeFn_zero</a> p \u03bc G, h_toLp n] with a ha0 ha_toLp", [{"full_name": "MeasureTheory.Lp.simpleFunc.coeFn_zero", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "def_pos": [827, 9], "def_end_pos": [827, 19]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\nzero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}\nthis : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y})\ng_meas : Measurable \u2191\u2191\u2191g\nx : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n\nhx_nonneg : \u2200 (n : \u2115), 0 \u2264 x n\nhx_mem\u2112p : \u2200 (n : \u2115), Mem\u2112p (\u2191(x n)) p\nh_toLp : \u2200 (n : \u2115), \u2191\u2191(Mem\u2112p.toLp \u2191(x n) (_ : Mem\u2112p (\u2191(x n)) p)) =\u1d50[\u03bc] \u2191(x n)\nn : \u2115\n\u22a2 \u2191\u2191\u21910 \u2264\u1d50[\u03bc] \u2191\u2191(Mem\u2112p.toLp \u2191(x n) (_ : Mem\u2112p (\u2191(x n)) p))", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\nzero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}\nthis : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y})\ng_meas : Measurable \u2191\u2191\u2191g\nx : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n\nhx_nonneg : \u2200 (n : \u2115), 0 \u2264 x n\nhx_mem\u2112p : \u2200 (n : \u2115), Mem\u2112p (\u2191(x n)) p\nh_toLp : \u2200 (n : \u2115), \u2191\u2191(Mem\u2112p.toLp \u2191(x n) (_ : Mem\u2112p (\u2191(x n)) p)) =\u1d50[\u03bc] \u2191(x n)\nn : \u2115\na : \u03b1\nha0 : \u2191\u2191\u21910 a = OfNat.ofNat 0 a\nha_toLp : \u2191\u2191(Mem\u2112p.toLp \u2191(x n) (_ : Mem\u2112p (\u2191(x n)) p)) a = \u2191(x n) a\n\u22a2 \u2191\u2191\u21910 a \u2264 \u2191\u2191(Mem\u2112p.toLp \u2191(x n) (_ : Mem\u2112p (\u2191(x n)) p)) a"}, {"tactic": "rw [ha0, ha_toLp]", "annotated_tactic": ["rw [ha0, ha_toLp]", []], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\nzero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}\nthis : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y})\ng_meas : Measurable \u2191\u2191\u2191g\nx : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n\nhx_nonneg : \u2200 (n : \u2115), 0 \u2264 x n\nhx_mem\u2112p : \u2200 (n : \u2115), Mem\u2112p (\u2191(x n)) p\nh_toLp : \u2200 (n : \u2115), \u2191\u2191(Mem\u2112p.toLp \u2191(x n) (_ : Mem\u2112p (\u2191(x n)) p)) =\u1d50[\u03bc] \u2191(x n)\nn : \u2115\na : \u03b1\nha0 : \u2191\u2191\u21910 a = OfNat.ofNat 0 a\nha_toLp : \u2191\u2191(Mem\u2112p.toLp \u2191(x n) (_ : Mem\u2112p (\u2191(x n)) p)) a = \u2191(x n) a\n\u22a2 \u2191\u2191\u21910 a \u2264 \u2191\u2191(Mem\u2112p.toLp \u2191(x n) (_ : Mem\u2112p (\u2191(x n)) p)) a", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\nzero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}\nthis : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y})\ng_meas : Measurable \u2191\u2191\u2191g\nx : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n\nhx_nonneg : \u2200 (n : \u2115), 0 \u2264 x n\nhx_mem\u2112p : \u2200 (n : \u2115), Mem\u2112p (\u2191(x n)) p\nh_toLp : \u2200 (n : \u2115), \u2191\u2191(Mem\u2112p.toLp \u2191(x n) (_ : Mem\u2112p (\u2191(x n)) p)) =\u1d50[\u03bc] \u2191(x n)\nn : \u2115\na : \u03b1\nha0 : \u2191\u2191\u21910 a = OfNat.ofNat 0 a\nha_toLp : \u2191\u2191(Mem\u2112p.toLp \u2191(x n) (_ : Mem\u2112p (\u2191(x n)) p)) a = \u2191(x n) a\n\u22a2 OfNat.ofNat 0 a \u2264 \u2191(x n) a"}, {"tactic": "exact hx_nonneg n a", "annotated_tactic": ["exact hx_nonneg n a", []], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\nzero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}\nthis : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y})\ng_meas : Measurable \u2191\u2191\u2191g\nx : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n\nhx_nonneg : \u2200 (n : \u2115), 0 \u2264 x n\nhx_mem\u2112p : \u2200 (n : \u2115), Mem\u2112p (\u2191(x n)) p\nh_toLp : \u2200 (n : \u2115), \u2191\u2191(Mem\u2112p.toLp \u2191(x n) (_ : Mem\u2112p (\u2191(x n)) p)) =\u1d50[\u03bc] \u2191(x n)\nn : \u2115\na : \u03b1\nha0 : \u2191\u2191\u21910 a = OfNat.ofNat 0 a\nha_toLp : \u2191\u2191(Mem\u2112p.toLp \u2191(x n) (_ : Mem\u2112p (\u2191(x n)) p)) a = \u2191(x n) a\n\u22a2 OfNat.ofNat 0 a \u2264 \u2191(x n) a", "state_after": "no goals"}, {"tactic": "apply SimpleFunc.tendsto_approxOn_Lp_snorm g_meas zero_mem hp_ne_top", "annotated_tactic": ["apply <a>SimpleFunc.tendsto_approxOn_Lp_snorm</a> g_meas zero_mem hp_ne_top", [{"full_name": "MeasureTheory.SimpleFunc.tendsto_approxOn_Lp_snorm", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "def_pos": [94, 9], "def_end_pos": [94, 34]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\nzero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}\nthis : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y})\ng_meas : Measurable \u2191\u2191\u2191g\nx : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n\nhx_nonneg : \u2200 (n : \u2115), 0 \u2264 x n\nhx_mem\u2112p : \u2200 (n : \u2115), Mem\u2112p (\u2191(x n)) p\nh_toLp : \u2200 (n : \u2115), \u2191\u2191(Mem\u2112p.toLp \u2191(x n) (_ : Mem\u2112p (\u2191(x n)) p)) =\u1d50[\u03bc] \u2191(x n)\nhx_nonneg_Lp : \u2200 (n : \u2115), 0 \u2264 toLp (x n) (_ : Mem\u2112p (\u2191(x n)) p)\n\u22a2 Tendsto (fun n => snorm (\u2191(x n) - \u2191\u2191\u2191g) p \u03bc) atTop (\ud835\udcdd 0)", "state_after": "case h\u03bc\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\nzero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}\nthis : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y})\ng_meas : Measurable \u2191\u2191\u2191g\nx : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n\nhx_nonneg : \u2200 (n : \u2115), 0 \u2264 x n\nhx_mem\u2112p : \u2200 (n : \u2115), Mem\u2112p (\u2191(x n)) p\nh_toLp : \u2200 (n : \u2115), \u2191\u2191(Mem\u2112p.toLp \u2191(x n) (_ : Mem\u2112p (\u2191(x n)) p)) =\u1d50[\u03bc] \u2191(x n)\nhx_nonneg_Lp : \u2200 (n : \u2115), 0 \u2264 toLp (x n) (_ : Mem\u2112p (\u2191(x n)) p)\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2191\u2191\u2191g x \u2208 closure ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y})\n\ncase hi\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\nzero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}\nthis : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y})\ng_meas : Measurable \u2191\u2191\u2191g\nx : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n\nhx_nonneg : \u2200 (n : \u2115), 0 \u2264 x n\nhx_mem\u2112p : \u2200 (n : \u2115), Mem\u2112p (\u2191(x n)) p\nh_toLp : \u2200 (n : \u2115), \u2191\u2191(Mem\u2112p.toLp \u2191(x n) (_ : Mem\u2112p (\u2191(x n)) p)) =\u1d50[\u03bc] \u2191(x n)\nhx_nonneg_Lp : \u2200 (n : \u2115), 0 \u2264 toLp (x n) (_ : Mem\u2112p (\u2191(x n)) p)\n\u22a2 snorm (fun x => \u2191\u2191\u2191g x - 0) p \u03bc < \u22a4"}, {"tactic": "have hg_nonneg : (0 : \u03b1 \u2192 G) \u2264\u1d50[\u03bc] g := (Lp.coeFn_nonneg _).mpr g.2", "annotated_tactic": ["have hg_nonneg : (0 : \u03b1 \u2192 G) \u2264\u1d50[\u03bc] g := (<a>Lp.coeFn_nonneg</a> _).<a>mpr</a> g.2", [{"full_name": "MeasureTheory.Lp.coeFn_nonneg", "def_path": "Mathlib/MeasureTheory/Function/LpOrder.lean", "def_pos": [46, 9], "def_end_pos": [46, 21]}, {"full_name": "Iff.mpr", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [92, 3], "def_end_pos": [92, 6]}]], "state_before": "case h\u03bc\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\nzero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}\nthis : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y})\ng_meas : Measurable \u2191\u2191\u2191g\nx : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n\nhx_nonneg : \u2200 (n : \u2115), 0 \u2264 x n\nhx_mem\u2112p : \u2200 (n : \u2115), Mem\u2112p (\u2191(x n)) p\nh_toLp : \u2200 (n : \u2115), \u2191\u2191(Mem\u2112p.toLp \u2191(x n) (_ : Mem\u2112p (\u2191(x n)) p)) =\u1d50[\u03bc] \u2191(x n)\nhx_nonneg_Lp : \u2200 (n : \u2115), 0 \u2264 toLp (x n) (_ : Mem\u2112p (\u2191(x n)) p)\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2191\u2191\u2191g x \u2208 closure ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y})", "state_after": "case h\u03bc\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\nzero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}\nthis : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y})\ng_meas : Measurable \u2191\u2191\u2191g\nx : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n\nhx_nonneg : \u2200 (n : \u2115), 0 \u2264 x n\nhx_mem\u2112p : \u2200 (n : \u2115), Mem\u2112p (\u2191(x n)) p\nh_toLp : \u2200 (n : \u2115), \u2191\u2191(Mem\u2112p.toLp \u2191(x n) (_ : Mem\u2112p (\u2191(x n)) p)) =\u1d50[\u03bc] \u2191(x n)\nhx_nonneg_Lp : \u2200 (n : \u2115), 0 \u2264 toLp (x n) (_ : Mem\u2112p (\u2191(x n)) p)\nhg_nonneg : 0 \u2264\u1d50[\u03bc] \u2191\u2191\u2191g\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2191\u2191\u2191g x \u2208 closure ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y})"}, {"tactic": "refine' hg_nonneg.mono fun a ha => subset_closure _", "annotated_tactic": ["refine' hg_nonneg.mono fun a ha => <a>subset_closure</a> _", [{"full_name": "subset_closure", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [435, 9], "def_end_pos": [435, 23]}]], "state_before": "case h\u03bc\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\nzero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}\nthis : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y})\ng_meas : Measurable \u2191\u2191\u2191g\nx : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n\nhx_nonneg : \u2200 (n : \u2115), 0 \u2264 x n\nhx_mem\u2112p : \u2200 (n : \u2115), Mem\u2112p (\u2191(x n)) p\nh_toLp : \u2200 (n : \u2115), \u2191\u2191(Mem\u2112p.toLp \u2191(x n) (_ : Mem\u2112p (\u2191(x n)) p)) =\u1d50[\u03bc] \u2191(x n)\nhx_nonneg_Lp : \u2200 (n : \u2115), 0 \u2264 toLp (x n) (_ : Mem\u2112p (\u2191(x n)) p)\nhg_nonneg : 0 \u2264\u1d50[\u03bc] \u2191\u2191\u2191g\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2191\u2191\u2191g x \u2208 closure ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y})", "state_after": "case h\u03bc\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\nzero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}\nthis : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y})\ng_meas : Measurable \u2191\u2191\u2191g\nx : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n\nhx_nonneg : \u2200 (n : \u2115), 0 \u2264 x n\nhx_mem\u2112p : \u2200 (n : \u2115), Mem\u2112p (\u2191(x n)) p\nh_toLp : \u2200 (n : \u2115), \u2191\u2191(Mem\u2112p.toLp \u2191(x n) (_ : Mem\u2112p (\u2191(x n)) p)) =\u1d50[\u03bc] \u2191(x n)\nhx_nonneg_Lp : \u2200 (n : \u2115), 0 \u2264 toLp (x n) (_ : Mem\u2112p (\u2191(x n)) p)\nhg_nonneg : 0 \u2264\u1d50[\u03bc] \u2191\u2191\u2191g\na : \u03b1\nha : OfNat.ofNat 0 a \u2264 \u2191\u2191\u2191g a\n\u22a2 \u2191\u2191\u2191g a \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}"}, {"tactic": "simpa using ha", "annotated_tactic": ["simpa using ha", []], "state_before": "case h\u03bc\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\nzero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}\nthis : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y})\ng_meas : Measurable \u2191\u2191\u2191g\nx : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n\nhx_nonneg : \u2200 (n : \u2115), 0 \u2264 x n\nhx_mem\u2112p : \u2200 (n : \u2115), Mem\u2112p (\u2191(x n)) p\nh_toLp : \u2200 (n : \u2115), \u2191\u2191(Mem\u2112p.toLp \u2191(x n) (_ : Mem\u2112p (\u2191(x n)) p)) =\u1d50[\u03bc] \u2191(x n)\nhx_nonneg_Lp : \u2200 (n : \u2115), 0 \u2264 toLp (x n) (_ : Mem\u2112p (\u2191(x n)) p)\nhg_nonneg : 0 \u2264\u1d50[\u03bc] \u2191\u2191\u2191g\na : \u03b1\nha : OfNat.ofNat 0 a \u2264 \u2191\u2191\u2191g a\n\u22a2 \u2191\u2191\u2191g a \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}", "state_after": "no goals"}, {"tactic": "simp_rw [sub_zero]", "annotated_tactic": ["simp_rw [<a>sub_zero</a>]", [{"full_name": "sub_zero", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [339, 3], "def_end_pos": [339, 14]}]], "state_before": "case hi\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\nzero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}\nthis : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y})\ng_meas : Measurable \u2191\u2191\u2191g\nx : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n\nhx_nonneg : \u2200 (n : \u2115), 0 \u2264 x n\nhx_mem\u2112p : \u2200 (n : \u2115), Mem\u2112p (\u2191(x n)) p\nh_toLp : \u2200 (n : \u2115), \u2191\u2191(Mem\u2112p.toLp \u2191(x n) (_ : Mem\u2112p (\u2191(x n)) p)) =\u1d50[\u03bc] \u2191(x n)\nhx_nonneg_Lp : \u2200 (n : \u2115), 0 \u2264 toLp (x n) (_ : Mem\u2112p (\u2191(x n)) p)\n\u22a2 snorm (fun x => \u2191\u2191\u2191g x - 0) p \u03bc < \u22a4", "state_after": "case hi\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\nzero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}\nthis : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y})\ng_meas : Measurable \u2191\u2191\u2191g\nx : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n\nhx_nonneg : \u2200 (n : \u2115), 0 \u2264 x n\nhx_mem\u2112p : \u2200 (n : \u2115), Mem\u2112p (\u2191(x n)) p\nh_toLp : \u2200 (n : \u2115), \u2191\u2191(Mem\u2112p.toLp \u2191(x n) (_ : Mem\u2112p (\u2191(x n)) p)) =\u1d50[\u03bc] \u2191(x n)\nhx_nonneg_Lp : \u2200 (n : \u2115), 0 \u2264 toLp (x n) (_ : Mem\u2112p (\u2191(x n)) p)\n\u22a2 snorm (fun x => \u2191\u2191\u2191g x) p \u03bc < \u22a4"}, {"tactic": "exact hg_mem\u2112p.snorm_lt_top", "annotated_tactic": ["exact hg_mem\u2112p.snorm_lt_top", []], "state_before": "case hi\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\nzero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}\nthis : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y})\ng_meas : Measurable \u2191\u2191\u2191g\nx : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n\nhx_nonneg : \u2200 (n : \u2115), 0 \u2264 x n\nhx_mem\u2112p : \u2200 (n : \u2115), Mem\u2112p (\u2191(x n)) p\nh_toLp : \u2200 (n : \u2115), \u2191\u2191(Mem\u2112p.toLp \u2191(x n) (_ : Mem\u2112p (\u2191(x n)) p)) =\u1d50[\u03bc] \u2191(x n)\nhx_nonneg_Lp : \u2200 (n : \u2115), 0 \u2264 toLp (x n) (_ : Mem\u2112p (\u2191(x n)) p)\n\u22a2 snorm (fun x => \u2191\u2191\u2191g x) p \u03bc < \u22a4", "state_after": "no goals"}, {"tactic": "rw [tendsto_iff_dist_tendsto_zero] at this \u22a2", "annotated_tactic": ["rw [<a>tendsto_iff_dist_tendsto_zero</a>] at this \u22a2", [{"full_name": "tendsto_iff_dist_tendsto_zero", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [1850, 9], "def_end_pos": [1850, 38]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b2 : MeasurableSpace G := borel G\nthis\u271d\u00b9 : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\nzero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}\nthis\u271d : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y})\ng_meas : Measurable \u2191\u2191\u2191g\nx : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n\nhx_nonneg : \u2200 (n : \u2115), 0 \u2264 x n\nhx_mem\u2112p : \u2200 (n : \u2115), Mem\u2112p (\u2191(x n)) p\nh_toLp : \u2200 (n : \u2115), \u2191\u2191(Mem\u2112p.toLp \u2191(x n) (_ : Mem\u2112p (\u2191(x n)) p)) =\u1d50[\u03bc] \u2191(x n)\nhx_nonneg_Lp : \u2200 (n : \u2115), 0 \u2264 toLp (x n) (_ : Mem\u2112p (\u2191(x n)) p)\nhx_tendsto : Tendsto (fun n => snorm (\u2191(x n) - \u2191\u2191\u2191g) p \u03bc) atTop (\ud835\udcdd 0)\nthis : Tendsto (fun n => \u2191(toLp (x n) (_ : Mem\u2112p (\u2191(x n)) p))) atTop (\ud835\udcdd \u2191g)\n\u22a2 Tendsto\n    (fun n =>\n      coeSimpleFuncNonnegToLpNonneg p \u03bc G\n        { val := toLp (x n) (_ : Mem\u2112p (\u2191(x n)) p), property := (_ : 0 \u2264 toLp (x n) (_ : Mem\u2112p (\u2191(x n)) p)) })\n    atTop (\ud835\udcdd g)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b2 : MeasurableSpace G := borel G\nthis\u271d\u00b9 : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\nzero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}\nthis\u271d : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y})\ng_meas : Measurable \u2191\u2191\u2191g\nx : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n\nhx_nonneg : \u2200 (n : \u2115), 0 \u2264 x n\nhx_mem\u2112p : \u2200 (n : \u2115), Mem\u2112p (\u2191(x n)) p\nh_toLp : \u2200 (n : \u2115), \u2191\u2191(Mem\u2112p.toLp \u2191(x n) (_ : Mem\u2112p (\u2191(x n)) p)) =\u1d50[\u03bc] \u2191(x n)\nhx_nonneg_Lp : \u2200 (n : \u2115), 0 \u2264 toLp (x n) (_ : Mem\u2112p (\u2191(x n)) p)\nhx_tendsto : Tendsto (fun n => snorm (\u2191(x n) - \u2191\u2191\u2191g) p \u03bc) atTop (\ud835\udcdd 0)\nthis : Tendsto (fun b => dist \u2191(toLp (x b) (_ : Mem\u2112p (\u2191(x b)) p)) \u2191g) atTop (\ud835\udcdd 0)\n\u22a2 Tendsto\n    (fun b =>\n      dist\n        (coeSimpleFuncNonnegToLpNonneg p \u03bc G\n          { val := toLp (x b) (_ : Mem\u2112p (\u2191(x b)) p), property := (_ : 0 \u2264 toLp (x b) (_ : Mem\u2112p (\u2191(x b)) p)) })\n        g)\n    atTop (\ud835\udcdd 0)"}, {"tactic": "simp_rw [Subtype.dist_eq]", "annotated_tactic": ["simp_rw [<a>Subtype.dist_eq</a>]", [{"full_name": "Subtype.dist_eq", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [1656, 9], "def_end_pos": [1656, 24]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b2 : MeasurableSpace G := borel G\nthis\u271d\u00b9 : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\nzero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}\nthis\u271d : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y})\ng_meas : Measurable \u2191\u2191\u2191g\nx : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n\nhx_nonneg : \u2200 (n : \u2115), 0 \u2264 x n\nhx_mem\u2112p : \u2200 (n : \u2115), Mem\u2112p (\u2191(x n)) p\nh_toLp : \u2200 (n : \u2115), \u2191\u2191(Mem\u2112p.toLp \u2191(x n) (_ : Mem\u2112p (\u2191(x n)) p)) =\u1d50[\u03bc] \u2191(x n)\nhx_nonneg_Lp : \u2200 (n : \u2115), 0 \u2264 toLp (x n) (_ : Mem\u2112p (\u2191(x n)) p)\nhx_tendsto : Tendsto (fun n => snorm (\u2191(x n) - \u2191\u2191\u2191g) p \u03bc) atTop (\ud835\udcdd 0)\nthis : Tendsto (fun b => dist \u2191(toLp (x b) (_ : Mem\u2112p (\u2191(x b)) p)) \u2191g) atTop (\ud835\udcdd 0)\n\u22a2 Tendsto\n    (fun b =>\n      dist\n        (coeSimpleFuncNonnegToLpNonneg p \u03bc G\n          { val := toLp (x b) (_ : Mem\u2112p (\u2191(x b)) p), property := (_ : 0 \u2264 toLp (x b) (_ : Mem\u2112p (\u2191(x b)) p)) })\n        g)\n    atTop (\ud835\udcdd 0)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b2 : MeasurableSpace G := borel G\nthis\u271d\u00b9 : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\nzero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}\nthis\u271d : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y})\ng_meas : Measurable \u2191\u2191\u2191g\nx : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n\nhx_nonneg : \u2200 (n : \u2115), 0 \u2264 x n\nhx_mem\u2112p : \u2200 (n : \u2115), Mem\u2112p (\u2191(x n)) p\nh_toLp : \u2200 (n : \u2115), \u2191\u2191(Mem\u2112p.toLp \u2191(x n) (_ : Mem\u2112p (\u2191(x n)) p)) =\u1d50[\u03bc] \u2191(x n)\nhx_nonneg_Lp : \u2200 (n : \u2115), 0 \u2264 toLp (x n) (_ : Mem\u2112p (\u2191(x n)) p)\nhx_tendsto : Tendsto (fun n => snorm (\u2191(x n) - \u2191\u2191\u2191g) p \u03bc) atTop (\ud835\udcdd 0)\nthis : Tendsto (fun b => dist \u2191(toLp (x b) (_ : Mem\u2112p (\u2191(x b)) p)) \u2191g) atTop (\ud835\udcdd 0)\n\u22a2 Tendsto\n    (fun b =>\n      dist\n        \u2191(coeSimpleFuncNonnegToLpNonneg p \u03bc G\n            {\n              val :=\n                toLp (SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem b)\n                  (_ : Mem\u2112p (\u2191(x b)) p),\n              property := (_ : 0 \u2264 toLp (x b) (_ : Mem\u2112p (\u2191(x b)) p)) })\n        \u2191g)\n    atTop (\ud835\udcdd 0)"}, {"tactic": "exact this", "annotated_tactic": ["exact this", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b2 : MeasurableSpace G := borel G\nthis\u271d\u00b9 : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\nzero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}\nthis\u271d : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y})\ng_meas : Measurable \u2191\u2191\u2191g\nx : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n\nhx_nonneg : \u2200 (n : \u2115), 0 \u2264 x n\nhx_mem\u2112p : \u2200 (n : \u2115), Mem\u2112p (\u2191(x n)) p\nh_toLp : \u2200 (n : \u2115), \u2191\u2191(Mem\u2112p.toLp \u2191(x n) (_ : Mem\u2112p (\u2191(x n)) p)) =\u1d50[\u03bc] \u2191(x n)\nhx_nonneg_Lp : \u2200 (n : \u2115), 0 \u2264 toLp (x n) (_ : Mem\u2112p (\u2191(x n)) p)\nhx_tendsto : Tendsto (fun n => snorm (\u2191(x n) - \u2191\u2191\u2191g) p \u03bc) atTop (\ud835\udcdd 0)\nthis : Tendsto (fun b => dist \u2191(toLp (x b) (_ : Mem\u2112p (\u2191(x b)) p)) \u2191g) atTop (\ud835\udcdd 0)\n\u22a2 Tendsto\n    (fun b =>\n      dist\n        \u2191(coeSimpleFuncNonnegToLpNonneg p \u03bc G\n            {\n              val :=\n                toLp (SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem b)\n                  (_ : Mem\u2112p (\u2191(x b)) p),\n              property := (_ : 0 \u2264 toLp (x b) (_ : Mem\u2112p (\u2191(x b)) p)) })\n        \u2191g)\n    atTop (\ud835\udcdd 0)", "state_after": "no goals"}, {"tactic": "symm", "annotated_tactic": ["symm", []], "state_before": "case refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\nzero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}\nthis : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y})\ng_meas : Measurable \u2191\u2191\u2191g\nx : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n\nhx_nonneg : \u2200 (n : \u2115), 0 \u2264 x n\nhx_mem\u2112p : \u2200 (n : \u2115), Mem\u2112p (\u2191(x n)) p\nh_toLp : \u2200 (n : \u2115), \u2191\u2191(Mem\u2112p.toLp \u2191(x n) (_ : Mem\u2112p (\u2191(x n)) p)) =\u1d50[\u03bc] \u2191(x n)\nhx_nonneg_Lp : \u2200 (n : \u2115), 0 \u2264 toLp (x n) (_ : Mem\u2112p (\u2191(x n)) p)\nhx_tendsto : Tendsto (fun n => snorm (\u2191(x n) - \u2191\u2191\u2191g) p \u03bc) atTop (\ud835\udcdd 0)\nn : \u2115\n\u22a2 (fun x_1 => \u2191(x n) x_1) =\u1d50[\u03bc] fun x_1 => \u2191\u2191\u2191(toLp (x n) (_ : Mem\u2112p (\u2191(x n)) p)) x_1", "state_after": "case refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\nzero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}\nthis : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y})\ng_meas : Measurable \u2191\u2191\u2191g\nx : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n\nhx_nonneg : \u2200 (n : \u2115), 0 \u2264 x n\nhx_mem\u2112p : \u2200 (n : \u2115), Mem\u2112p (\u2191(x n)) p\nh_toLp : \u2200 (n : \u2115), \u2191\u2191(Mem\u2112p.toLp \u2191(x n) (_ : Mem\u2112p (\u2191(x n)) p)) =\u1d50[\u03bc] \u2191(x n)\nhx_nonneg_Lp : \u2200 (n : \u2115), 0 \u2264 toLp (x n) (_ : Mem\u2112p (\u2191(x n)) p)\nhx_tendsto : Tendsto (fun n => snorm (\u2191(x n) - \u2191\u2191\u2191g) p \u03bc) atTop (\ud835\udcdd 0)\nn : \u2115\n\u22a2 (fun x_1 => \u2191\u2191\u2191(toLp (x n) (_ : Mem\u2112p (\u2191(x n)) p)) x_1) =\u1d50[\u03bc] fun x_1 => \u2191(x n) x_1"}, {"tactic": "rw [Lp.simpleFunc.toLp_eq_toLp]", "annotated_tactic": ["rw [<a>Lp.simpleFunc.toLp_eq_toLp</a>]", [{"full_name": "MeasureTheory.Lp.simpleFunc.toLp_eq_toLp", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "def_pos": [543, 9], "def_end_pos": [543, 21]}]], "state_before": "case refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\nzero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}\nthis : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y})\ng_meas : Measurable \u2191\u2191\u2191g\nx : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n\nhx_nonneg : \u2200 (n : \u2115), 0 \u2264 x n\nhx_mem\u2112p : \u2200 (n : \u2115), Mem\u2112p (\u2191(x n)) p\nh_toLp : \u2200 (n : \u2115), \u2191\u2191(Mem\u2112p.toLp \u2191(x n) (_ : Mem\u2112p (\u2191(x n)) p)) =\u1d50[\u03bc] \u2191(x n)\nhx_nonneg_Lp : \u2200 (n : \u2115), 0 \u2264 toLp (x n) (_ : Mem\u2112p (\u2191(x n)) p)\nhx_tendsto : Tendsto (fun n => snorm (\u2191(x n) - \u2191\u2191\u2191g) p \u03bc) atTop (\ud835\udcdd 0)\nn : \u2115\n\u22a2 (fun x_1 => \u2191\u2191\u2191(toLp (x n) (_ : Mem\u2112p (\u2191(x n)) p)) x_1) =\u1d50[\u03bc] fun x_1 => \u2191(x n) x_1", "state_after": "case refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\nzero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}\nthis : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y})\ng_meas : Measurable \u2191\u2191\u2191g\nx : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n\nhx_nonneg : \u2200 (n : \u2115), 0 \u2264 x n\nhx_mem\u2112p : \u2200 (n : \u2115), Mem\u2112p (\u2191(x n)) p\nh_toLp : \u2200 (n : \u2115), \u2191\u2191(Mem\u2112p.toLp \u2191(x n) (_ : Mem\u2112p (\u2191(x n)) p)) =\u1d50[\u03bc] \u2191(x n)\nhx_nonneg_Lp : \u2200 (n : \u2115), 0 \u2264 toLp (x n) (_ : Mem\u2112p (\u2191(x n)) p)\nhx_tendsto : Tendsto (fun n => snorm (\u2191(x n) - \u2191\u2191\u2191g) p \u03bc) atTop (\ud835\udcdd 0)\nn : \u2115\n\u22a2 (fun x_1 => \u2191\u2191(Mem\u2112p.toLp \u2191(x n) (_ : Mem\u2112p (\u2191(x n)) p)) x_1) =\u1d50[\u03bc] fun x_1 => \u2191(x n) x_1"}, {"tactic": "exact h_toLp n", "annotated_tactic": ["exact h_toLp n", []], "state_before": "case refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\nzero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}\nthis : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y})\ng_meas : Measurable \u2191\u2191\u2191g\nx : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n\nhx_nonneg : \u2200 (n : \u2115), 0 \u2264 x n\nhx_mem\u2112p : \u2200 (n : \u2115), Mem\u2112p (\u2191(x n)) p\nh_toLp : \u2200 (n : \u2115), \u2191\u2191(Mem\u2112p.toLp \u2191(x n) (_ : Mem\u2112p (\u2191(x n)) p)) =\u1d50[\u03bc] \u2191(x n)\nhx_nonneg_Lp : \u2200 (n : \u2115), 0 \u2264 toLp (x n) (_ : Mem\u2112p (\u2191(x n)) p)\nhx_tendsto : Tendsto (fun n => snorm (\u2191(x n) - \u2191\u2191\u2191g) p \u03bc) atTop (\ud835\udcdd 0)\nn : \u2115\n\u22a2 (fun x_1 => \u2191\u2191(Mem\u2112p.toLp \u2191(x n) (_ : Mem\u2112p (\u2191(x n)) p)) x_1) =\u1d50[\u03bc] fun x_1 => \u2191(x n) x_1", "state_after": "no goals"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\nzero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}\nthis : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y})\ng_meas : Measurable \u2191\u2191\u2191g\nx : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n\nhx_nonneg : \u2200 (n : \u2115), 0 \u2264 x n\nhx_mem\u2112p : \u2200 (n : \u2115), Mem\u2112p (\u2191(x n)) p\nh_toLp : \u2200 (n : \u2115), \u2191\u2191(Mem\u2112p.toLp \u2191(x n) (_ : Mem\u2112p (\u2191(x n)) p)) =\u1d50[\u03bc] \u2191(x n)\nhx_nonneg_Lp : \u2200 (n : \u2115), 0 \u2264 toLp (x n) (_ : Mem\u2112p (\u2191(x n)) p)\nhx_tendsto : Tendsto (fun n => snorm (\u2191(x n) - \u2191\u2191\u2191g) p \u03bc) atTop (\ud835\udcdd 0)\nn : \u2115\n\u22a2 (fun x => \u2191\u2191\u2191g x) =\u1d50[\u03bc] fun x => \u2191\u2191\u2191g x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Pointwise/SMul.lean", "full_name": "Set.vsub_self_mono", "start": [682, 1], "end": [683, 23], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Martingale/BorelCantelli.lean", "full_name": "MeasureTheory.norm_stoppedValue_leastGE_le", "start": [114, 1], "end": [126, 78], "traced_tactics": [{"tactic": "filter_upwards [hbdd] with \u03c9 hbdd\u03c9", "annotated_tactic": ["filter_upwards [hbdd] with \u03c9 hbdd\u03c9", []], "state_before": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nr : \u211d\nR : \u211d\u22650\nhr : 0 \u2264 r\nhf0 : f 0 = 0\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\ni : \u2115\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, stoppedValue f (leastGE f r i) \u03c9 \u2264 r + \u2191R", "state_after": "case h\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\nhr : 0 \u2264 r\nhf0 : f 0 = 0\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\ni : \u2115\n\u03c9 : \u03a9\nhbdd\u03c9 : \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\n\u22a2 stoppedValue f (leastGE f r i) \u03c9 \u2264 r + \u2191R"}, {"tactic": "change f (leastGE f r i \u03c9) \u03c9 \u2264 r + R", "annotated_tactic": ["change f (<a>leastGE</a> f r i \u03c9) \u03c9 \u2264 r + R", [{"full_name": "MeasureTheory.leastGE", "def_path": "Mathlib/Probability/Martingale/BorelCantelli.lean", "def_pos": [50, 19], "def_end_pos": [50, 26]}]], "state_before": "case h\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\nhr : 0 \u2264 r\nhf0 : f 0 = 0\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\ni : \u2115\n\u03c9 : \u03a9\nhbdd\u03c9 : \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\n\u22a2 stoppedValue f (leastGE f r i) \u03c9 \u2264 r + \u2191R", "state_after": "case h\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\nhr : 0 \u2264 r\nhf0 : f 0 = 0\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\ni : \u2115\n\u03c9 : \u03a9\nhbdd\u03c9 : \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\n\u22a2 f (leastGE f r i \u03c9) \u03c9 \u2264 r + \u2191R"}, {"tactic": "by_cases heq : leastGE f r i \u03c9 = 0", "annotated_tactic": ["by_cases heq : <a>leastGE</a> f r i \u03c9 = 0", [{"full_name": "MeasureTheory.leastGE", "def_path": "Mathlib/Probability/Martingale/BorelCantelli.lean", "def_pos": [50, 19], "def_end_pos": [50, 26]}]], "state_before": "case h\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\nhr : 0 \u2264 r\nhf0 : f 0 = 0\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\ni : \u2115\n\u03c9 : \u03a9\nhbdd\u03c9 : \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\n\u22a2 f (leastGE f r i \u03c9) \u03c9 \u2264 r + \u2191R", "state_after": "case pos\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\nhr : 0 \u2264 r\nhf0 : f 0 = 0\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\ni : \u2115\n\u03c9 : \u03a9\nhbdd\u03c9 : \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\nheq : leastGE f r i \u03c9 = 0\n\u22a2 f (leastGE f r i \u03c9) \u03c9 \u2264 r + \u2191R\n\ncase neg\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\nhr : 0 \u2264 r\nhf0 : f 0 = 0\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\ni : \u2115\n\u03c9 : \u03a9\nhbdd\u03c9 : \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\nheq : \u00acleastGE f r i \u03c9 = 0\n\u22a2 f (leastGE f r i \u03c9) \u03c9 \u2264 r + \u2191R"}, {"tactic": "rw [heq, hf0, Pi.zero_apply]", "annotated_tactic": ["rw [heq, hf0, <a>Pi.zero_apply</a>]", [{"full_name": "Pi.zero_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [46, 3], "def_end_pos": [46, 14]}]], "state_before": "case pos\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\nhr : 0 \u2264 r\nhf0 : f 0 = 0\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\ni : \u2115\n\u03c9 : \u03a9\nhbdd\u03c9 : \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\nheq : leastGE f r i \u03c9 = 0\n\u22a2 f (leastGE f r i \u03c9) \u03c9 \u2264 r + \u2191R", "state_after": "case pos\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\nhr : 0 \u2264 r\nhf0 : f 0 = 0\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\ni : \u2115\n\u03c9 : \u03a9\nhbdd\u03c9 : \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\nheq : leastGE f r i \u03c9 = 0\n\u22a2 0 \u2264 r + \u2191R"}, {"tactic": "exact add_nonneg hr R.coe_nonneg", "annotated_tactic": ["exact <a>add_nonneg</a> hr R.coe_nonneg", [{"full_name": "add_nonneg", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [1092, 24], "def_end_pos": [1092, 34]}]], "state_before": "case pos\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\nhr : 0 \u2264 r\nhf0 : f 0 = 0\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\ni : \u2115\n\u03c9 : \u03a9\nhbdd\u03c9 : \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\nheq : leastGE f r i \u03c9 = 0\n\u22a2 0 \u2264 r + \u2191R", "state_after": "no goals"}, {"tactic": "obtain \u27e8k, hk\u27e9 := Nat.exists_eq_succ_of_ne_zero heq", "annotated_tactic": ["obtain \u27e8k, hk\u27e9 := <a>Nat.exists_eq_succ_of_ne_zero</a> heq", [{"full_name": "Nat.exists_eq_succ_of_ne_zero", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [238, 9], "def_end_pos": [238, 34]}]], "state_before": "case neg\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\nhr : 0 \u2264 r\nhf0 : f 0 = 0\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\ni : \u2115\n\u03c9 : \u03a9\nhbdd\u03c9 : \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\nheq : \u00acleastGE f r i \u03c9 = 0\n\u22a2 f (leastGE f r i \u03c9) \u03c9 \u2264 r + \u2191R", "state_after": "case neg.intro\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\nhr : 0 \u2264 r\nhf0 : f 0 = 0\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\ni : \u2115\n\u03c9 : \u03a9\nhbdd\u03c9 : \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\nheq : \u00acleastGE f r i \u03c9 = 0\nk : \u2115\nhk : leastGE f r i \u03c9 = Nat.succ k\n\u22a2 f (leastGE f r i \u03c9) \u03c9 \u2264 r + \u2191R"}, {"tactic": "rw [hk, add_comm, \u2190 sub_le_iff_le_add]", "annotated_tactic": ["rw [hk, <a>add_comm</a>, \u2190 <a>sub_le_iff_le_add</a>]", [{"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [301, 3], "def_end_pos": [301, 14]}, {"full_name": "sub_le_iff_le_add", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [750, 3], "def_end_pos": [750, 14]}]], "state_before": "case neg.intro\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\nhr : 0 \u2264 r\nhf0 : f 0 = 0\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\ni : \u2115\n\u03c9 : \u03a9\nhbdd\u03c9 : \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\nheq : \u00acleastGE f r i \u03c9 = 0\nk : \u2115\nhk : leastGE f r i \u03c9 = Nat.succ k\n\u22a2 f (leastGE f r i \u03c9) \u03c9 \u2264 r + \u2191R", "state_after": "case neg.intro\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\nhr : 0 \u2264 r\nhf0 : f 0 = 0\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\ni : \u2115\n\u03c9 : \u03a9\nhbdd\u03c9 : \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\nheq : \u00acleastGE f r i \u03c9 = 0\nk : \u2115\nhk : leastGE f r i \u03c9 = Nat.succ k\n\u22a2 f (Nat.succ k) \u03c9 - r \u2264 \u2191R"}, {"tactic": "have := not_mem_of_lt_hitting (hk.symm \u25b8 k.lt_succ_self : k < leastGE f r i \u03c9) (zero_le _)", "annotated_tactic": ["have := <a>not_mem_of_lt_hitting</a> (hk.symm \u25b8 k.lt_succ_self : k < <a>leastGE</a> f r i \u03c9) (<a>zero_le</a> _)", [{"full_name": "MeasureTheory.not_mem_of_lt_hitting", "def_path": "Mathlib/Probability/Process/HittingTime.lean", "def_pos": [82, 9], "def_end_pos": [82, 30]}, {"full_name": "MeasureTheory.leastGE", "def_path": "Mathlib/Probability/Martingale/BorelCantelli.lean", "def_pos": [50, 19], "def_end_pos": [50, 26]}, {"full_name": "zero_le", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [217, 30], "def_end_pos": [217, 37]}]], "state_before": "case neg.intro\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\nhr : 0 \u2264 r\nhf0 : f 0 = 0\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\ni : \u2115\n\u03c9 : \u03a9\nhbdd\u03c9 : \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\nheq : \u00acleastGE f r i \u03c9 = 0\nk : \u2115\nhk : leastGE f r i \u03c9 = Nat.succ k\n\u22a2 f (Nat.succ k) \u03c9 - r \u2264 \u2191R", "state_after": "case neg.intro\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\nhr : 0 \u2264 r\nhf0 : f 0 = 0\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\ni : \u2115\n\u03c9 : \u03a9\nhbdd\u03c9 : \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\nheq : \u00acleastGE f r i \u03c9 = 0\nk : \u2115\nhk : leastGE f r i \u03c9 = Nat.succ k\nthis : \u00acf k \u03c9 \u2208 Set.Ici r\n\u22a2 f (Nat.succ k) \u03c9 - r \u2264 \u2191R"}, {"tactic": "simp only [Set.mem_union, Set.mem_Iic, Set.mem_Ici, not_or, not_le] at this", "annotated_tactic": ["simp only [<a>Set.mem_union</a>, <a>Set.mem_Iic</a>, <a>Set.mem_Ici</a>, <a>not_or</a>, <a>not_le</a>] at this", [{"full_name": "Set.mem_union", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [767, 9], "def_end_pos": [767, 18]}, {"full_name": "Set.mem_Iic", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [136, 9], "def_end_pos": [136, 16]}, {"full_name": "Set.mem_Ici", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [146, 9], "def_end_pos": [146, 16]}, {"full_name": "not_or", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [340, 9], "def_end_pos": [340, 15]}, {"full_name": "not_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [373, 9], "def_end_pos": [373, 15]}]], "state_before": "case neg.intro\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\nhr : 0 \u2264 r\nhf0 : f 0 = 0\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\ni : \u2115\n\u03c9 : \u03a9\nhbdd\u03c9 : \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\nheq : \u00acleastGE f r i \u03c9 = 0\nk : \u2115\nhk : leastGE f r i \u03c9 = Nat.succ k\nthis : \u00acf k \u03c9 \u2208 Set.Ici r\n\u22a2 f (Nat.succ k) \u03c9 - r \u2264 \u2191R", "state_after": "case neg.intro\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\nhr : 0 \u2264 r\nhf0 : f 0 = 0\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\ni : \u2115\n\u03c9 : \u03a9\nhbdd\u03c9 : \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\nheq : \u00acleastGE f r i \u03c9 = 0\nk : \u2115\nhk : leastGE f r i \u03c9 = Nat.succ k\nthis : f k \u03c9 < r\n\u22a2 f (Nat.succ k) \u03c9 - r \u2264 \u2191R"}, {"tactic": "exact (sub_lt_sub_left this _).le.trans ((le_abs_self _).trans (hbdd\u03c9 _))", "annotated_tactic": ["exact (<a>sub_lt_sub_left</a> this _).le.trans ((<a>le_abs_self</a> _).<a>trans</a> (hbdd\u03c9 _))", [{"full_name": "sub_lt_sub_left", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [946, 15], "def_end_pos": [946, 30]}, {"full_name": "le_abs_self", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [57, 9], "def_end_pos": [57, 20]}, {"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [120, 7], "def_end_pos": [120, 18]}]], "state_before": "case neg.intro\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\nhr : 0 \u2264 r\nhf0 : f 0 = 0\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\ni : \u2115\n\u03c9 : \u03a9\nhbdd\u03c9 : \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\nheq : \u00acleastGE f r i \u03c9 = 0\nk : \u2115\nhk : leastGE f r i \u03c9 = Nat.succ k\nthis : f k \u03c9 < r\n\u22a2 f (Nat.succ k) \u03c9 - r \u2264 \u2191R", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Prod.lean", "full_name": "Set.univ_pi_subset_univ_pi_iff", "start": [886, 1], "end": [887, 29], "traced_tactics": [{"tactic": "simp [pi_subset_pi_iff]", "annotated_tactic": ["simp [<a>pi_subset_pi_iff</a>]", [{"full_name": "Set.pi_subset_pi_iff", "def_path": "Mathlib/Data/Set/Prod.lean", "def_pos": [877, 9], "def_end_pos": [877, 25]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\ns s\u2081 s\u2082 : Set \u03b9\nt t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\ni : \u03b9\n\u22a2 pi univ t\u2081 \u2286 pi univ t\u2082 \u2194 (\u2200 (i : \u03b9), t\u2081 i \u2286 t\u2082 i) \u2228 \u2203 i, t\u2081 i = \u2205", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/NoncommProd.lean", "full_name": "Finset.noncommProd_mul_distrib", "start": [391, 1], "end": [407, 100], "traced_tactics": [{"tactic": "induction' s using Finset.induction_on with x s hnmem ih", "annotated_tactic": ["induction' s using <a>Finset.induction_on</a> with x s hnmem ih", [{"full_name": "Finset.induction_on", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1251, 19], "def_end_pos": [1251, 31]}]], "state_before": "F : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b2\ninst\u271d : Monoid \u03b3\ns : Finset \u03b1\nf g : \u03b1 \u2192 \u03b2\ncomm_ff : Set.Pairwise \u2191s fun x y => Commute (f x) (f y)\ncomm_gg : Set.Pairwise \u2191s fun x y => Commute (g x) (g y)\ncomm_gf : Set.Pairwise \u2191s fun x y => Commute (g x) (f y)\n\u22a2 noncommProd s (f * g) (_ : Set.Pairwise \u2191s fun x y => Commute ((f * g) x) ((f * g) y)) =\n    noncommProd s f comm_ff * noncommProd s g comm_gg", "state_after": "case empty\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b2\ninst\u271d : Monoid \u03b3\ns : Finset \u03b1\nf g : \u03b1 \u2192 \u03b2\ncomm_ff\u271d : Set.Pairwise \u2191s fun x y => Commute (f x) (f y)\ncomm_gg\u271d : Set.Pairwise \u2191s fun x y => Commute (g x) (g y)\ncomm_gf\u271d : Set.Pairwise \u2191s fun x y => Commute (g x) (f y)\ncomm_ff : Set.Pairwise \u2191\u2205 fun x y => Commute (f x) (f y)\ncomm_gg : Set.Pairwise \u2191\u2205 fun x y => Commute (g x) (g y)\ncomm_gf : Set.Pairwise \u2191\u2205 fun x y => Commute (g x) (f y)\n\u22a2 noncommProd \u2205 (f * g) (_ : Set.Pairwise \u2191\u2205 fun x y => Commute ((f * g) x) ((f * g) y)) =\n    noncommProd \u2205 f comm_ff * noncommProd \u2205 g comm_gg\n\ncase insert\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b2\ninst\u271d : Monoid \u03b3\ns\u271d : Finset \u03b1\nf g : \u03b1 \u2192 \u03b2\ncomm_ff\u271d : Set.Pairwise \u2191s\u271d fun x y => Commute (f x) (f y)\ncomm_gg\u271d : Set.Pairwise \u2191s\u271d fun x y => Commute (g x) (g y)\ncomm_gf\u271d : Set.Pairwise \u2191s\u271d fun x y => Commute (g x) (f y)\nx : \u03b1\ns : Finset \u03b1\nhnmem : \u00acx \u2208 s\nih :\n  \u2200 (comm_ff : Set.Pairwise \u2191s fun x y => Commute (f x) (f y))\n    (comm_gg : Set.Pairwise \u2191s fun x y => Commute (g x) (g y))\n    (comm_gf : Set.Pairwise \u2191s fun x y => Commute (g x) (f y)),\n    noncommProd s (f * g) (_ : Set.Pairwise \u2191s fun x y => Commute ((f * g) x) ((f * g) y)) =\n      noncommProd s f comm_ff * noncommProd s g comm_gg\ncomm_ff : Set.Pairwise \u2191(insert x s) fun x y => Commute (f x) (f y)\ncomm_gg : Set.Pairwise \u2191(insert x s) fun x y => Commute (g x) (g y)\ncomm_gf : Set.Pairwise \u2191(insert x s) fun x y => Commute (g x) (f y)\n\u22a2 noncommProd (insert x s) (f * g) (_ : Set.Pairwise \u2191(insert x s) fun x y => Commute ((f * g) x) ((f * g) y)) =\n    noncommProd (insert x s) f comm_ff * noncommProd (insert x s) g comm_gg"}, {"tactic": "simp only [Finset.noncommProd_insert_of_not_mem _ _ _ _ hnmem]", "annotated_tactic": ["simp only [<a>Finset.noncommProd_insert_of_not_mem</a> _ _ _ _ hnmem]", [{"full_name": "Finset.noncommProd_insert_of_not_mem", "def_path": "Mathlib/Data/Finset/NoncommProd.lean", "def_pos": [276, 9], "def_end_pos": [276, 38]}]], "state_before": "case insert\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b2\ninst\u271d : Monoid \u03b3\ns\u271d : Finset \u03b1\nf g : \u03b1 \u2192 \u03b2\ncomm_ff\u271d : Set.Pairwise \u2191s\u271d fun x y => Commute (f x) (f y)\ncomm_gg\u271d : Set.Pairwise \u2191s\u271d fun x y => Commute (g x) (g y)\ncomm_gf\u271d : Set.Pairwise \u2191s\u271d fun x y => Commute (g x) (f y)\nx : \u03b1\ns : Finset \u03b1\nhnmem : \u00acx \u2208 s\nih :\n  \u2200 (comm_ff : Set.Pairwise \u2191s fun x y => Commute (f x) (f y))\n    (comm_gg : Set.Pairwise \u2191s fun x y => Commute (g x) (g y))\n    (comm_gf : Set.Pairwise \u2191s fun x y => Commute (g x) (f y)),\n    noncommProd s (f * g) (_ : Set.Pairwise \u2191s fun x y => Commute ((f * g) x) ((f * g) y)) =\n      noncommProd s f comm_ff * noncommProd s g comm_gg\ncomm_ff : Set.Pairwise \u2191(insert x s) fun x y => Commute (f x) (f y)\ncomm_gg : Set.Pairwise \u2191(insert x s) fun x y => Commute (g x) (g y)\ncomm_gf : Set.Pairwise \u2191(insert x s) fun x y => Commute (g x) (f y)\n\u22a2 noncommProd (insert x s) (f * g) (_ : Set.Pairwise \u2191(insert x s) fun x y => Commute ((f * g) x) ((f * g) y)) =\n    noncommProd (insert x s) f comm_ff * noncommProd (insert x s) g comm_gg", "state_after": "case insert\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b2\ninst\u271d : Monoid \u03b3\ns\u271d : Finset \u03b1\nf g : \u03b1 \u2192 \u03b2\ncomm_ff\u271d : Set.Pairwise \u2191s\u271d fun x y => Commute (f x) (f y)\ncomm_gg\u271d : Set.Pairwise \u2191s\u271d fun x y => Commute (g x) (g y)\ncomm_gf\u271d : Set.Pairwise \u2191s\u271d fun x y => Commute (g x) (f y)\nx : \u03b1\ns : Finset \u03b1\nhnmem : \u00acx \u2208 s\nih :\n  \u2200 (comm_ff : Set.Pairwise \u2191s fun x y => Commute (f x) (f y))\n    (comm_gg : Set.Pairwise \u2191s fun x y => Commute (g x) (g y))\n    (comm_gf : Set.Pairwise \u2191s fun x y => Commute (g x) (f y)),\n    noncommProd s (f * g) (_ : Set.Pairwise \u2191s fun x y => Commute ((f * g) x) ((f * g) y)) =\n      noncommProd s f comm_ff * noncommProd s g comm_gg\ncomm_ff : Set.Pairwise \u2191(insert x s) fun x y => Commute (f x) (f y)\ncomm_gg : Set.Pairwise \u2191(insert x s) fun x y => Commute (g x) (g y)\ncomm_gf : Set.Pairwise \u2191(insert x s) fun x y => Commute (g x) (f y)\n\u22a2 (f * g) x * noncommProd s (f * g) (_ : Set.Pairwise \u2191s fun a b => Commute ((f * g) a) ((f * g) b)) =\n    f x * noncommProd s f (_ : Set.Pairwise \u2191s fun a b => Commute (f a) (f b)) *\n      (g x * noncommProd s g (_ : Set.Pairwise \u2191s fun a b => Commute (g a) (g b)))"}, {"tactic": "specialize\n  ih (comm_ff.mono fun _ => mem_insert_of_mem) (comm_gg.mono fun _ => mem_insert_of_mem)\n    (comm_gf.mono fun _ => mem_insert_of_mem)", "annotated_tactic": ["specialize\n      ih (comm_ff.mono fun _ => <a>mem_insert_of_mem</a>) (comm_gg.mono fun _ => <a>mem_insert_of_mem</a>)\n        (comm_gf.mono fun _ => <a>mem_insert_of_mem</a>)", [{"full_name": "Finset.mem_insert_of_mem", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1095, 9], "def_end_pos": [1095, 26]}, {"full_name": "Finset.mem_insert_of_mem", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1095, 9], "def_end_pos": [1095, 26]}, {"full_name": "Finset.mem_insert_of_mem", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1095, 9], "def_end_pos": [1095, 26]}]], "state_before": "case insert\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b2\ninst\u271d : Monoid \u03b3\ns\u271d : Finset \u03b1\nf g : \u03b1 \u2192 \u03b2\ncomm_ff\u271d : Set.Pairwise \u2191s\u271d fun x y => Commute (f x) (f y)\ncomm_gg\u271d : Set.Pairwise \u2191s\u271d fun x y => Commute (g x) (g y)\ncomm_gf\u271d : Set.Pairwise \u2191s\u271d fun x y => Commute (g x) (f y)\nx : \u03b1\ns : Finset \u03b1\nhnmem : \u00acx \u2208 s\nih :\n  \u2200 (comm_ff : Set.Pairwise \u2191s fun x y => Commute (f x) (f y))\n    (comm_gg : Set.Pairwise \u2191s fun x y => Commute (g x) (g y))\n    (comm_gf : Set.Pairwise \u2191s fun x y => Commute (g x) (f y)),\n    noncommProd s (f * g) (_ : Set.Pairwise \u2191s fun x y => Commute ((f * g) x) ((f * g) y)) =\n      noncommProd s f comm_ff * noncommProd s g comm_gg\ncomm_ff : Set.Pairwise \u2191(insert x s) fun x y => Commute (f x) (f y)\ncomm_gg : Set.Pairwise \u2191(insert x s) fun x y => Commute (g x) (g y)\ncomm_gf : Set.Pairwise \u2191(insert x s) fun x y => Commute (g x) (f y)\n\u22a2 (f * g) x * noncommProd s (f * g) (_ : Set.Pairwise \u2191s fun a b => Commute ((f * g) a) ((f * g) b)) =\n    f x * noncommProd s f (_ : Set.Pairwise \u2191s fun a b => Commute (f a) (f b)) *\n      (g x * noncommProd s g (_ : Set.Pairwise \u2191s fun a b => Commute (g a) (g b)))", "state_after": "case insert\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b2\ninst\u271d : Monoid \u03b3\ns\u271d : Finset \u03b1\nf g : \u03b1 \u2192 \u03b2\ncomm_ff\u271d : Set.Pairwise \u2191s\u271d fun x y => Commute (f x) (f y)\ncomm_gg\u271d : Set.Pairwise \u2191s\u271d fun x y => Commute (g x) (g y)\ncomm_gf\u271d : Set.Pairwise \u2191s\u271d fun x y => Commute (g x) (f y)\nx : \u03b1\ns : Finset \u03b1\nhnmem : \u00acx \u2208 s\ncomm_ff : Set.Pairwise \u2191(insert x s) fun x y => Commute (f x) (f y)\ncomm_gg : Set.Pairwise \u2191(insert x s) fun x y => Commute (g x) (g y)\ncomm_gf : Set.Pairwise \u2191(insert x s) fun x y => Commute (g x) (f y)\nih :\n  noncommProd s (f * g) (_ : Set.Pairwise \u2191s fun x y => Commute ((f * g) x) ((f * g) y)) =\n    noncommProd s f (_ : Set.Pairwise \u2191s fun x y => Commute (f x) (f y)) *\n      noncommProd s g (_ : Set.Pairwise \u2191s fun x y => Commute (g x) (g y))\n\u22a2 (f * g) x * noncommProd s (f * g) (_ : Set.Pairwise \u2191s fun a b => Commute ((f * g) a) ((f * g) b)) =\n    f x * noncommProd s f (_ : Set.Pairwise \u2191s fun a b => Commute (f a) (f b)) *\n      (g x * noncommProd s g (_ : Set.Pairwise \u2191s fun a b => Commute (g a) (g b)))"}, {"tactic": "rw [ih, Pi.mul_apply]", "annotated_tactic": ["rw [ih, <a>Pi.mul_apply</a>]", [{"full_name": "Pi.mul_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [83, 9], "def_end_pos": [83, 18]}]], "state_before": "case insert\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b2\ninst\u271d : Monoid \u03b3\ns\u271d : Finset \u03b1\nf g : \u03b1 \u2192 \u03b2\ncomm_ff\u271d : Set.Pairwise \u2191s\u271d fun x y => Commute (f x) (f y)\ncomm_gg\u271d : Set.Pairwise \u2191s\u271d fun x y => Commute (g x) (g y)\ncomm_gf\u271d : Set.Pairwise \u2191s\u271d fun x y => Commute (g x) (f y)\nx : \u03b1\ns : Finset \u03b1\nhnmem : \u00acx \u2208 s\ncomm_ff : Set.Pairwise \u2191(insert x s) fun x y => Commute (f x) (f y)\ncomm_gg : Set.Pairwise \u2191(insert x s) fun x y => Commute (g x) (g y)\ncomm_gf : Set.Pairwise \u2191(insert x s) fun x y => Commute (g x) (f y)\nih :\n  noncommProd s (f * g) (_ : Set.Pairwise \u2191s fun x y => Commute ((f * g) x) ((f * g) y)) =\n    noncommProd s f (_ : Set.Pairwise \u2191s fun x y => Commute (f x) (f y)) *\n      noncommProd s g (_ : Set.Pairwise \u2191s fun x y => Commute (g x) (g y))\n\u22a2 (f * g) x * noncommProd s (f * g) (_ : Set.Pairwise \u2191s fun a b => Commute ((f * g) a) ((f * g) b)) =\n    f x * noncommProd s f (_ : Set.Pairwise \u2191s fun a b => Commute (f a) (f b)) *\n      (g x * noncommProd s g (_ : Set.Pairwise \u2191s fun a b => Commute (g a) (g b)))", "state_after": "case insert\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b2\ninst\u271d : Monoid \u03b3\ns\u271d : Finset \u03b1\nf g : \u03b1 \u2192 \u03b2\ncomm_ff\u271d : Set.Pairwise \u2191s\u271d fun x y => Commute (f x) (f y)\ncomm_gg\u271d : Set.Pairwise \u2191s\u271d fun x y => Commute (g x) (g y)\ncomm_gf\u271d : Set.Pairwise \u2191s\u271d fun x y => Commute (g x) (f y)\nx : \u03b1\ns : Finset \u03b1\nhnmem : \u00acx \u2208 s\ncomm_ff : Set.Pairwise \u2191(insert x s) fun x y => Commute (f x) (f y)\ncomm_gg : Set.Pairwise \u2191(insert x s) fun x y => Commute (g x) (g y)\ncomm_gf : Set.Pairwise \u2191(insert x s) fun x y => Commute (g x) (f y)\nih :\n  noncommProd s (f * g) (_ : Set.Pairwise \u2191s fun x y => Commute ((f * g) x) ((f * g) y)) =\n    noncommProd s f (_ : Set.Pairwise \u2191s fun x y => Commute (f x) (f y)) *\n      noncommProd s g (_ : Set.Pairwise \u2191s fun x y => Commute (g x) (g y))\n\u22a2 f x * g x *\n      (noncommProd s f (_ : Set.Pairwise \u2191s fun x y => Commute (f x) (f y)) *\n        noncommProd s g (_ : Set.Pairwise \u2191s fun x y => Commute (g x) (g y))) =\n    f x * noncommProd s f (_ : Set.Pairwise \u2191s fun a b => Commute (f a) (f b)) *\n      (g x * noncommProd s g (_ : Set.Pairwise \u2191s fun a b => Commute (g a) (g b)))"}, {"tactic": "simp only [mul_assoc]", "annotated_tactic": ["simp only [<a>mul_assoc</a>]", [{"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [264, 9], "def_end_pos": [264, 18]}]], "state_before": "case insert\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b2\ninst\u271d : Monoid \u03b3\ns\u271d : Finset \u03b1\nf g : \u03b1 \u2192 \u03b2\ncomm_ff\u271d : Set.Pairwise \u2191s\u271d fun x y => Commute (f x) (f y)\ncomm_gg\u271d : Set.Pairwise \u2191s\u271d fun x y => Commute (g x) (g y)\ncomm_gf\u271d : Set.Pairwise \u2191s\u271d fun x y => Commute (g x) (f y)\nx : \u03b1\ns : Finset \u03b1\nhnmem : \u00acx \u2208 s\ncomm_ff : Set.Pairwise \u2191(insert x s) fun x y => Commute (f x) (f y)\ncomm_gg : Set.Pairwise \u2191(insert x s) fun x y => Commute (g x) (g y)\ncomm_gf : Set.Pairwise \u2191(insert x s) fun x y => Commute (g x) (f y)\nih :\n  noncommProd s (f * g) (_ : Set.Pairwise \u2191s fun x y => Commute ((f * g) x) ((f * g) y)) =\n    noncommProd s f (_ : Set.Pairwise \u2191s fun x y => Commute (f x) (f y)) *\n      noncommProd s g (_ : Set.Pairwise \u2191s fun x y => Commute (g x) (g y))\n\u22a2 f x * g x *\n      (noncommProd s f (_ : Set.Pairwise \u2191s fun x y => Commute (f x) (f y)) *\n        noncommProd s g (_ : Set.Pairwise \u2191s fun x y => Commute (g x) (g y))) =\n    f x * noncommProd s f (_ : Set.Pairwise \u2191s fun a b => Commute (f a) (f b)) *\n      (g x * noncommProd s g (_ : Set.Pairwise \u2191s fun a b => Commute (g a) (g b)))", "state_after": "case insert\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b2\ninst\u271d : Monoid \u03b3\ns\u271d : Finset \u03b1\nf g : \u03b1 \u2192 \u03b2\ncomm_ff\u271d : Set.Pairwise \u2191s\u271d fun x y => Commute (f x) (f y)\ncomm_gg\u271d : Set.Pairwise \u2191s\u271d fun x y => Commute (g x) (g y)\ncomm_gf\u271d : Set.Pairwise \u2191s\u271d fun x y => Commute (g x) (f y)\nx : \u03b1\ns : Finset \u03b1\nhnmem : \u00acx \u2208 s\ncomm_ff : Set.Pairwise \u2191(insert x s) fun x y => Commute (f x) (f y)\ncomm_gg : Set.Pairwise \u2191(insert x s) fun x y => Commute (g x) (g y)\ncomm_gf : Set.Pairwise \u2191(insert x s) fun x y => Commute (g x) (f y)\nih :\n  noncommProd s (f * g) (_ : Set.Pairwise \u2191s fun x y => Commute ((f * g) x) ((f * g) y)) =\n    noncommProd s f (_ : Set.Pairwise \u2191s fun x y => Commute (f x) (f y)) *\n      noncommProd s g (_ : Set.Pairwise \u2191s fun x y => Commute (g x) (g y))\n\u22a2 f x *\n      (g x *\n        (noncommProd s f (_ : Set.Pairwise \u2191s fun x y => Commute (f x) (f y)) *\n          noncommProd s g (_ : Set.Pairwise \u2191s fun x y => Commute (g x) (g y)))) =\n    f x *\n      (noncommProd s f (_ : Set.Pairwise \u2191s fun x y => Commute (f x) (f y)) *\n        (g x * noncommProd s g (_ : Set.Pairwise \u2191s fun x y => Commute (g x) (g y))))"}, {"tactic": "congr 1", "annotated_tactic": ["congr 1", []], "state_before": "case insert\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b2\ninst\u271d : Monoid \u03b3\ns\u271d : Finset \u03b1\nf g : \u03b1 \u2192 \u03b2\ncomm_ff\u271d : Set.Pairwise \u2191s\u271d fun x y => Commute (f x) (f y)\ncomm_gg\u271d : Set.Pairwise \u2191s\u271d fun x y => Commute (g x) (g y)\ncomm_gf\u271d : Set.Pairwise \u2191s\u271d fun x y => Commute (g x) (f y)\nx : \u03b1\ns : Finset \u03b1\nhnmem : \u00acx \u2208 s\ncomm_ff : Set.Pairwise \u2191(insert x s) fun x y => Commute (f x) (f y)\ncomm_gg : Set.Pairwise \u2191(insert x s) fun x y => Commute (g x) (g y)\ncomm_gf : Set.Pairwise \u2191(insert x s) fun x y => Commute (g x) (f y)\nih :\n  noncommProd s (f * g) (_ : Set.Pairwise \u2191s fun x y => Commute ((f * g) x) ((f * g) y)) =\n    noncommProd s f (_ : Set.Pairwise \u2191s fun x y => Commute (f x) (f y)) *\n      noncommProd s g (_ : Set.Pairwise \u2191s fun x y => Commute (g x) (g y))\n\u22a2 f x *\n      (g x *\n        (noncommProd s f (_ : Set.Pairwise \u2191s fun x y => Commute (f x) (f y)) *\n          noncommProd s g (_ : Set.Pairwise \u2191s fun x y => Commute (g x) (g y)))) =\n    f x *\n      (noncommProd s f (_ : Set.Pairwise \u2191s fun x y => Commute (f x) (f y)) *\n        (g x * noncommProd s g (_ : Set.Pairwise \u2191s fun x y => Commute (g x) (g y))))", "state_after": "case insert.e_a\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b2\ninst\u271d : Monoid \u03b3\ns\u271d : Finset \u03b1\nf g : \u03b1 \u2192 \u03b2\ncomm_ff\u271d : Set.Pairwise \u2191s\u271d fun x y => Commute (f x) (f y)\ncomm_gg\u271d : Set.Pairwise \u2191s\u271d fun x y => Commute (g x) (g y)\ncomm_gf\u271d : Set.Pairwise \u2191s\u271d fun x y => Commute (g x) (f y)\nx : \u03b1\ns : Finset \u03b1\nhnmem : \u00acx \u2208 s\ncomm_ff : Set.Pairwise \u2191(insert x s) fun x y => Commute (f x) (f y)\ncomm_gg : Set.Pairwise \u2191(insert x s) fun x y => Commute (g x) (g y)\ncomm_gf : Set.Pairwise \u2191(insert x s) fun x y => Commute (g x) (f y)\nih :\n  noncommProd s (f * g) (_ : Set.Pairwise \u2191s fun x y => Commute ((f * g) x) ((f * g) y)) =\n    noncommProd s f (_ : Set.Pairwise \u2191s fun x y => Commute (f x) (f y)) *\n      noncommProd s g (_ : Set.Pairwise \u2191s fun x y => Commute (g x) (g y))\n\u22a2 g x *\n      (noncommProd s f (_ : Set.Pairwise \u2191s fun x y => Commute (f x) (f y)) *\n        noncommProd s g (_ : Set.Pairwise \u2191s fun x y => Commute (g x) (g y))) =\n    noncommProd s f (_ : Set.Pairwise \u2191s fun x y => Commute (f x) (f y)) *\n      (g x * noncommProd s g (_ : Set.Pairwise \u2191s fun x y => Commute (g x) (g y)))"}, {"tactic": "simp only [\u2190 mul_assoc]", "annotated_tactic": ["simp only [\u2190 <a>mul_assoc</a>]", [{"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [264, 9], "def_end_pos": [264, 18]}]], "state_before": "case insert.e_a\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b2\ninst\u271d : Monoid \u03b3\ns\u271d : Finset \u03b1\nf g : \u03b1 \u2192 \u03b2\ncomm_ff\u271d : Set.Pairwise \u2191s\u271d fun x y => Commute (f x) (f y)\ncomm_gg\u271d : Set.Pairwise \u2191s\u271d fun x y => Commute (g x) (g y)\ncomm_gf\u271d : Set.Pairwise \u2191s\u271d fun x y => Commute (g x) (f y)\nx : \u03b1\ns : Finset \u03b1\nhnmem : \u00acx \u2208 s\ncomm_ff : Set.Pairwise \u2191(insert x s) fun x y => Commute (f x) (f y)\ncomm_gg : Set.Pairwise \u2191(insert x s) fun x y => Commute (g x) (g y)\ncomm_gf : Set.Pairwise \u2191(insert x s) fun x y => Commute (g x) (f y)\nih :\n  noncommProd s (f * g) (_ : Set.Pairwise \u2191s fun x y => Commute ((f * g) x) ((f * g) y)) =\n    noncommProd s f (_ : Set.Pairwise \u2191s fun x y => Commute (f x) (f y)) *\n      noncommProd s g (_ : Set.Pairwise \u2191s fun x y => Commute (g x) (g y))\n\u22a2 g x *\n      (noncommProd s f (_ : Set.Pairwise \u2191s fun x y => Commute (f x) (f y)) *\n        noncommProd s g (_ : Set.Pairwise \u2191s fun x y => Commute (g x) (g y))) =\n    noncommProd s f (_ : Set.Pairwise \u2191s fun x y => Commute (f x) (f y)) *\n      (g x * noncommProd s g (_ : Set.Pairwise \u2191s fun x y => Commute (g x) (g y)))", "state_after": "case insert.e_a\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b2\ninst\u271d : Monoid \u03b3\ns\u271d : Finset \u03b1\nf g : \u03b1 \u2192 \u03b2\ncomm_ff\u271d : Set.Pairwise \u2191s\u271d fun x y => Commute (f x) (f y)\ncomm_gg\u271d : Set.Pairwise \u2191s\u271d fun x y => Commute (g x) (g y)\ncomm_gf\u271d : Set.Pairwise \u2191s\u271d fun x y => Commute (g x) (f y)\nx : \u03b1\ns : Finset \u03b1\nhnmem : \u00acx \u2208 s\ncomm_ff : Set.Pairwise \u2191(insert x s) fun x y => Commute (f x) (f y)\ncomm_gg : Set.Pairwise \u2191(insert x s) fun x y => Commute (g x) (g y)\ncomm_gf : Set.Pairwise \u2191(insert x s) fun x y => Commute (g x) (f y)\nih :\n  noncommProd s (f * g) (_ : Set.Pairwise \u2191s fun x y => Commute ((f * g) x) ((f * g) y)) =\n    noncommProd s f (_ : Set.Pairwise \u2191s fun x y => Commute (f x) (f y)) *\n      noncommProd s g (_ : Set.Pairwise \u2191s fun x y => Commute (g x) (g y))\n\u22a2 g x * noncommProd s f (_ : Set.Pairwise \u2191s fun x y => Commute (f x) (f y)) *\n      noncommProd s g (_ : Set.Pairwise \u2191s fun x y => Commute (g x) (g y)) =\n    noncommProd s f (_ : Set.Pairwise \u2191s fun x y => Commute (f x) (f y)) * g x *\n      noncommProd s g (_ : Set.Pairwise \u2191s fun x y => Commute (g x) (g y))"}, {"tactic": "congr 1", "annotated_tactic": ["congr 1", []], "state_before": "case insert.e_a\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b2\ninst\u271d : Monoid \u03b3\ns\u271d : Finset \u03b1\nf g : \u03b1 \u2192 \u03b2\ncomm_ff\u271d : Set.Pairwise \u2191s\u271d fun x y => Commute (f x) (f y)\ncomm_gg\u271d : Set.Pairwise \u2191s\u271d fun x y => Commute (g x) (g y)\ncomm_gf\u271d : Set.Pairwise \u2191s\u271d fun x y => Commute (g x) (f y)\nx : \u03b1\ns : Finset \u03b1\nhnmem : \u00acx \u2208 s\ncomm_ff : Set.Pairwise \u2191(insert x s) fun x y => Commute (f x) (f y)\ncomm_gg : Set.Pairwise \u2191(insert x s) fun x y => Commute (g x) (g y)\ncomm_gf : Set.Pairwise \u2191(insert x s) fun x y => Commute (g x) (f y)\nih :\n  noncommProd s (f * g) (_ : Set.Pairwise \u2191s fun x y => Commute ((f * g) x) ((f * g) y)) =\n    noncommProd s f (_ : Set.Pairwise \u2191s fun x y => Commute (f x) (f y)) *\n      noncommProd s g (_ : Set.Pairwise \u2191s fun x y => Commute (g x) (g y))\n\u22a2 g x * noncommProd s f (_ : Set.Pairwise \u2191s fun x y => Commute (f x) (f y)) *\n      noncommProd s g (_ : Set.Pairwise \u2191s fun x y => Commute (g x) (g y)) =\n    noncommProd s f (_ : Set.Pairwise \u2191s fun x y => Commute (f x) (f y)) * g x *\n      noncommProd s g (_ : Set.Pairwise \u2191s fun x y => Commute (g x) (g y))", "state_after": "case insert.e_a.e_a\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b2\ninst\u271d : Monoid \u03b3\ns\u271d : Finset \u03b1\nf g : \u03b1 \u2192 \u03b2\ncomm_ff\u271d : Set.Pairwise \u2191s\u271d fun x y => Commute (f x) (f y)\ncomm_gg\u271d : Set.Pairwise \u2191s\u271d fun x y => Commute (g x) (g y)\ncomm_gf\u271d : Set.Pairwise \u2191s\u271d fun x y => Commute (g x) (f y)\nx : \u03b1\ns : Finset \u03b1\nhnmem : \u00acx \u2208 s\ncomm_ff : Set.Pairwise \u2191(insert x s) fun x y => Commute (f x) (f y)\ncomm_gg : Set.Pairwise \u2191(insert x s) fun x y => Commute (g x) (g y)\ncomm_gf : Set.Pairwise \u2191(insert x s) fun x y => Commute (g x) (f y)\nih :\n  noncommProd s (f * g) (_ : Set.Pairwise \u2191s fun x y => Commute ((f * g) x) ((f * g) y)) =\n    noncommProd s f (_ : Set.Pairwise \u2191s fun x y => Commute (f x) (f y)) *\n      noncommProd s g (_ : Set.Pairwise \u2191s fun x y => Commute (g x) (g y))\n\u22a2 g x * noncommProd s f (_ : Set.Pairwise \u2191s fun x y => Commute (f x) (f y)) =\n    noncommProd s f (_ : Set.Pairwise \u2191s fun x y => Commute (f x) (f y)) * g x"}, {"tactic": "refine' noncommProd_commute _ _ _ _ fun y hy => _", "annotated_tactic": ["refine' <a>noncommProd_commute</a> _ _ _ _ fun y hy => _", [{"full_name": "Finset.noncommProd_commute", "def_path": "Mathlib/Data/Finset/NoncommProd.lean", "def_pos": [335, 9], "def_end_pos": [335, 28]}]], "state_before": "case insert.e_a.e_a\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b2\ninst\u271d : Monoid \u03b3\ns\u271d : Finset \u03b1\nf g : \u03b1 \u2192 \u03b2\ncomm_ff\u271d : Set.Pairwise \u2191s\u271d fun x y => Commute (f x) (f y)\ncomm_gg\u271d : Set.Pairwise \u2191s\u271d fun x y => Commute (g x) (g y)\ncomm_gf\u271d : Set.Pairwise \u2191s\u271d fun x y => Commute (g x) (f y)\nx : \u03b1\ns : Finset \u03b1\nhnmem : \u00acx \u2208 s\ncomm_ff : Set.Pairwise \u2191(insert x s) fun x y => Commute (f x) (f y)\ncomm_gg : Set.Pairwise \u2191(insert x s) fun x y => Commute (g x) (g y)\ncomm_gf : Set.Pairwise \u2191(insert x s) fun x y => Commute (g x) (f y)\nih :\n  noncommProd s (f * g) (_ : Set.Pairwise \u2191s fun x y => Commute ((f * g) x) ((f * g) y)) =\n    noncommProd s f (_ : Set.Pairwise \u2191s fun x y => Commute (f x) (f y)) *\n      noncommProd s g (_ : Set.Pairwise \u2191s fun x y => Commute (g x) (g y))\n\u22a2 g x * noncommProd s f (_ : Set.Pairwise \u2191s fun x y => Commute (f x) (f y)) =\n    noncommProd s f (_ : Set.Pairwise \u2191s fun x y => Commute (f x) (f y)) * g x", "state_after": "case insert.e_a.e_a\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b2\ninst\u271d : Monoid \u03b3\ns\u271d : Finset \u03b1\nf g : \u03b1 \u2192 \u03b2\ncomm_ff\u271d : Set.Pairwise \u2191s\u271d fun x y => Commute (f x) (f y)\ncomm_gg\u271d : Set.Pairwise \u2191s\u271d fun x y => Commute (g x) (g y)\ncomm_gf\u271d : Set.Pairwise \u2191s\u271d fun x y => Commute (g x) (f y)\nx : \u03b1\ns : Finset \u03b1\nhnmem : \u00acx \u2208 s\ncomm_ff : Set.Pairwise \u2191(insert x s) fun x y => Commute (f x) (f y)\ncomm_gg : Set.Pairwise \u2191(insert x s) fun x y => Commute (g x) (g y)\ncomm_gf : Set.Pairwise \u2191(insert x s) fun x y => Commute (g x) (f y)\nih :\n  noncommProd s (f * g) (_ : Set.Pairwise \u2191s fun x y => Commute ((f * g) x) ((f * g) y)) =\n    noncommProd s f (_ : Set.Pairwise \u2191s fun x y => Commute (f x) (f y)) *\n      noncommProd s g (_ : Set.Pairwise \u2191s fun x y => Commute (g x) (g y))\ny : \u03b1\nhy : y \u2208 s\n\u22a2 Commute (g x) (f y)"}, {"tactic": "exact comm_gf (mem_insert_self x s) (mem_insert_of_mem hy) (ne_of_mem_of_not_mem hy hnmem).symm", "annotated_tactic": ["exact comm_gf (<a>mem_insert_self</a> x s) (<a>mem_insert_of_mem</a> hy) (<a>ne_of_mem_of_not_mem</a> hy hnmem).<a>symm</a>", [{"full_name": "Finset.mem_insert_self", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1091, 9], "def_end_pos": [1091, 24]}, {"full_name": "Finset.mem_insert_of_mem", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1095, 9], "def_end_pos": [1095, 26]}, {"full_name": "ne_of_mem_of_not_mem", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [719, 9], "def_end_pos": [719, 29]}, {"full_name": "Ne.symm", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [575, 9], "def_end_pos": [575, 16]}]], "state_before": "case insert.e_a.e_a\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b2\ninst\u271d : Monoid \u03b3\ns\u271d : Finset \u03b1\nf g : \u03b1 \u2192 \u03b2\ncomm_ff\u271d : Set.Pairwise \u2191s\u271d fun x y => Commute (f x) (f y)\ncomm_gg\u271d : Set.Pairwise \u2191s\u271d fun x y => Commute (g x) (g y)\ncomm_gf\u271d : Set.Pairwise \u2191s\u271d fun x y => Commute (g x) (f y)\nx : \u03b1\ns : Finset \u03b1\nhnmem : \u00acx \u2208 s\ncomm_ff : Set.Pairwise \u2191(insert x s) fun x y => Commute (f x) (f y)\ncomm_gg : Set.Pairwise \u2191(insert x s) fun x y => Commute (g x) (g y)\ncomm_gf : Set.Pairwise \u2191(insert x s) fun x y => Commute (g x) (f y)\nih :\n  noncommProd s (f * g) (_ : Set.Pairwise \u2191s fun x y => Commute ((f * g) x) ((f * g) y)) =\n    noncommProd s f (_ : Set.Pairwise \u2191s fun x y => Commute (f x) (f y)) *\n      noncommProd s g (_ : Set.Pairwise \u2191s fun x y => Commute (g x) (g y))\ny : \u03b1\nhy : y \u2208 s\n\u22a2 Commute (g x) (f y)", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case empty\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b2\ninst\u271d : Monoid \u03b3\ns : Finset \u03b1\nf g : \u03b1 \u2192 \u03b2\ncomm_ff\u271d : Set.Pairwise \u2191s fun x y => Commute (f x) (f y)\ncomm_gg\u271d : Set.Pairwise \u2191s fun x y => Commute (g x) (g y)\ncomm_gf\u271d : Set.Pairwise \u2191s fun x y => Commute (g x) (f y)\ncomm_ff : Set.Pairwise \u2191\u2205 fun x y => Commute (f x) (f y)\ncomm_gg : Set.Pairwise \u2191\u2205 fun x y => Commute (g x) (g y)\ncomm_gf : Set.Pairwise \u2191\u2205 fun x y => Commute (g x) (f y)\n\u22a2 noncommProd \u2205 (f * g) (_ : Set.Pairwise \u2191\u2205 fun x y => Commute ((f * g) x) ((f * g) y)) =\n    noncommProd \u2205 f comm_ff * noncommProd \u2205 g comm_gg", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Martingale/Basic.lean", "full_name": "MeasureTheory.Submartingale.set_integral_le", "start": [250, 1], "end": [254, 54], "traced_tactics": [{"tactic": "rw [\u2190 neg_le_neg_iff, \u2190 integral_neg, \u2190 integral_neg]", "annotated_tactic": ["rw [\u2190 <a>neg_le_neg_iff</a>, \u2190 <a>integral_neg</a>, \u2190 <a>integral_neg</a>]", [{"full_name": "neg_le_neg_iff", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [342, 3], "def_end_pos": [342, 14]}, {"full_name": "MeasureTheory.integral_neg", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [890, 9], "def_end_pos": [890, 21]}, {"full_name": "MeasureTheory.integral_neg", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [890, 9], "def_end_pos": [890, 21]}]], "state_before": "\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2074 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\nf : \u03b9 \u2192 \u03a9 \u2192 \u211d\nhf : Submartingale f \u2131 \u03bc\ni j : \u03b9\nhij : i \u2264 j\ns : Set \u03a9\nhs : MeasurableSet s\n\u22a2 \u222b (\u03c9 : \u03a9) in s, f i \u03c9 \u2202\u03bc \u2264 \u222b (\u03c9 : \u03a9) in s, f j \u03c9 \u2202\u03bc", "state_after": "\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2074 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\nf : \u03b9 \u2192 \u03a9 \u2192 \u211d\nhf : Submartingale f \u2131 \u03bc\ni j : \u03b9\nhij : i \u2264 j\ns : Set \u03a9\nhs : MeasurableSet s\n\u22a2 \u222b (a : \u03a9) in s, -f j a \u2202\u03bc \u2264 \u222b (a : \u03a9) in s, -f i a \u2202\u03bc"}, {"tactic": "exact Supermartingale.set_integral_le hf.neg hij hs", "annotated_tactic": ["exact <a>Supermartingale.set_integral_le</a> hf.neg hij hs", [{"full_name": "MeasureTheory.Supermartingale.set_integral_le", "def_path": "Mathlib/Probability/Martingale/Basic.lean", "def_pos": [176, 9], "def_end_pos": [176, 24]}]], "state_before": "\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2074 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\nf : \u03b9 \u2192 \u03a9 \u2192 \u211d\nhf : Submartingale f \u2131 \u03bc\ni j : \u03b9\nhij : i \u2264 j\ns : Set \u03a9\nhs : MeasurableSet s\n\u22a2 \u222b (a : \u03a9) in s, -f j a \u2202\u03bc \u2264 \u222b (a : \u03a9) in s, -f i a \u2202\u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/ProbabilityMeasure.lean", "full_name": "MeasureTheory.FiniteMeasure.tendsto_normalize_testAgainstNN_of_tendsto", "start": [438, 1], "end": [461, 34], "traced_tactics": [{"tactic": "have lim_mass := \u03bcs_lim.mass", "annotated_tactic": ["have lim_mass := \u03bcs_lim.mass", []], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b2 : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_2\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\n\u03bcs_lim : Tendsto \u03bcs F (\ud835\udcdd \u03bc)\nnonzero : \u03bc \u2260 0\nf : \u03a9 \u2192\u1d47 \u211d\u22650\n\u22a2 Tendsto (fun i => testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize (\u03bcs i))) f) F\n    (\ud835\udcdd (testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize \u03bc)) f))", "state_after": "\u03a9 : Type u_1\ninst\u271d\u00b2 : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_2\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\n\u03bcs_lim : Tendsto \u03bcs F (\ud835\udcdd \u03bc)\nnonzero : \u03bc \u2260 0\nf : \u03a9 \u2192\u1d47 \u211d\u22650\nlim_mass : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd (mass \u03bc))\n\u22a2 Tendsto (fun i => testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize (\u03bcs i))) f) F\n    (\ud835\udcdd (testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize \u03bc)) f))"}, {"tactic": "have aux : {(0 : \u211d\u22650)}\u1d9c \u2208 \ud835\udcdd \u03bc.mass :=\n  isOpen_compl_singleton.mem_nhds (\u03bc.mass_nonzero_iff.mpr nonzero)", "annotated_tactic": ["have aux : {(0 : \u211d\u22650)}\u1d9c \u2208 \ud835\udcdd \u03bc.mass :=\n    isOpen_compl_singleton.mem_nhds (\u03bc.mass_nonzero_iff.mpr nonzero)", []], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b2 : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_2\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\n\u03bcs_lim : Tendsto \u03bcs F (\ud835\udcdd \u03bc)\nnonzero : \u03bc \u2260 0\nf : \u03a9 \u2192\u1d47 \u211d\u22650\nlim_mass : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd (mass \u03bc))\n\u22a2 Tendsto (fun i => testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize (\u03bcs i))) f) F\n    (\ud835\udcdd (testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize \u03bc)) f))", "state_after": "\u03a9 : Type u_1\ninst\u271d\u00b2 : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_2\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\n\u03bcs_lim : Tendsto \u03bcs F (\ud835\udcdd \u03bc)\nnonzero : \u03bc \u2260 0\nf : \u03a9 \u2192\u1d47 \u211d\u22650\nlim_mass : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd (mass \u03bc))\naux : {0}\u1d9c \u2208 \ud835\udcdd (mass \u03bc)\n\u22a2 Tendsto (fun i => testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize (\u03bcs i))) f) F\n    (\ud835\udcdd (testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize \u03bc)) f))"}, {"tactic": "have eventually_nonzero : \u2200\u1da0 i in F, \u03bcs i \u2260 0 := by\n  simp_rw [\u2190 mass_nonzero_iff]\n  exact lim_mass aux", "annotated_tactic": ["have eventually_nonzero : \u2200\u1da0 i in F, \u03bcs i \u2260 0 := by\n    simp_rw [\u2190 <a>mass_nonzero_iff</a>]\n    exact lim_mass aux", [{"full_name": "MeasureTheory.FiniteMeasure.mass_nonzero_iff", "def_path": "Mathlib/MeasureTheory/Measure/FiniteMeasure.lean", "def_pos": [191, 9], "def_end_pos": [191, 25]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b2 : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_2\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\n\u03bcs_lim : Tendsto \u03bcs F (\ud835\udcdd \u03bc)\nnonzero : \u03bc \u2260 0\nf : \u03a9 \u2192\u1d47 \u211d\u22650\nlim_mass : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd (mass \u03bc))\naux : {0}\u1d9c \u2208 \ud835\udcdd (mass \u03bc)\n\u22a2 Tendsto (fun i => testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize (\u03bcs i))) f) F\n    (\ud835\udcdd (testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize \u03bc)) f))", "state_after": "\u03a9 : Type u_1\ninst\u271d\u00b2 : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_2\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\n\u03bcs_lim : Tendsto \u03bcs F (\ud835\udcdd \u03bc)\nnonzero : \u03bc \u2260 0\nf : \u03a9 \u2192\u1d47 \u211d\u22650\nlim_mass : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd (mass \u03bc))\naux : {0}\u1d9c \u2208 \ud835\udcdd (mass \u03bc)\neventually_nonzero : \u2200\u1da0 (i : \u03b3) in F, \u03bcs i \u2260 0\n\u22a2 Tendsto (fun i => testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize (\u03bcs i))) f) F\n    (\ud835\udcdd (testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize \u03bc)) f))"}, {"tactic": "have eve : \u2200\u1da0 i in F,\n    (\u03bcs i).normalize.toFiniteMeasure.testAgainstNN f =\n      (\u03bcs i).mass\u207b\u00b9 * (\u03bcs i).testAgainstNN f := by\n  filter_upwards [eventually_iff.mp eventually_nonzero]\n  intro i hi\n  apply normalize_testAgainstNN _ hi", "annotated_tactic": ["have eve : \u2200\u1da0 i in F,\n      (\u03bcs i).normalize.toFiniteMeasure.testAgainstNN f =\n        (\u03bcs i).<a>mass</a>\u207b\u00b9 * (\u03bcs i).<a>testAgainstNN</a> f := by\n    filter_upwards [eventually_iff.mp eventually_nonzero]\n    intro i hi\n    apply <a>normalize_testAgainstNN</a> _ hi", [{"full_name": "MeasureTheory.FiniteMeasure.mass", "def_path": "Mathlib/MeasureTheory/Measure/FiniteMeasure.lean", "def_pos": [163, 5], "def_end_pos": [163, 9]}, {"full_name": "MeasureTheory.FiniteMeasure.testAgainstNN", "def_path": "Mathlib/MeasureTheory/Measure/FiniteMeasure.lean", "def_pos": [321, 5], "def_end_pos": [321, 18]}, {"full_name": "MeasureTheory.FiniteMeasure.normalize_testAgainstNN", "def_path": "Mathlib/MeasureTheory/Measure/ProbabilityMeasure.lean", "def_pos": [410, 9], "def_end_pos": [410, 32]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b2 : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_2\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\n\u03bcs_lim : Tendsto \u03bcs F (\ud835\udcdd \u03bc)\nnonzero : \u03bc \u2260 0\nf : \u03a9 \u2192\u1d47 \u211d\u22650\nlim_mass : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd (mass \u03bc))\naux : {0}\u1d9c \u2208 \ud835\udcdd (mass \u03bc)\neventually_nonzero : \u2200\u1da0 (i : \u03b3) in F, \u03bcs i \u2260 0\n\u22a2 Tendsto (fun i => testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize (\u03bcs i))) f) F\n    (\ud835\udcdd (testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize \u03bc)) f))", "state_after": "\u03a9 : Type u_1\ninst\u271d\u00b2 : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_2\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\n\u03bcs_lim : Tendsto \u03bcs F (\ud835\udcdd \u03bc)\nnonzero : \u03bc \u2260 0\nf : \u03a9 \u2192\u1d47 \u211d\u22650\nlim_mass : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd (mass \u03bc))\naux : {0}\u1d9c \u2208 \ud835\udcdd (mass \u03bc)\neventually_nonzero : \u2200\u1da0 (i : \u03b3) in F, \u03bcs i \u2260 0\neve :\n  \u2200\u1da0 (i : \u03b3) in F,\n    testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize (\u03bcs i))) f = (mass (\u03bcs i))\u207b\u00b9 * testAgainstNN (\u03bcs i) f\n\u22a2 Tendsto (fun i => testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize (\u03bcs i))) f) F\n    (\ud835\udcdd (testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize \u03bc)) f))"}, {"tactic": "simp_rw [tendsto_congr' eve, \u03bc.normalize_testAgainstNN nonzero]", "annotated_tactic": ["simp_rw [<a>tendsto_congr'</a> eve, \u03bc.normalize_testAgainstNN nonzero]", [{"full_name": "Filter.tendsto_congr'", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [3005, 9], "def_end_pos": [3005, 23]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b2 : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_2\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\n\u03bcs_lim : Tendsto \u03bcs F (\ud835\udcdd \u03bc)\nnonzero : \u03bc \u2260 0\nf : \u03a9 \u2192\u1d47 \u211d\u22650\nlim_mass : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd (mass \u03bc))\naux : {0}\u1d9c \u2208 \ud835\udcdd (mass \u03bc)\neventually_nonzero : \u2200\u1da0 (i : \u03b3) in F, \u03bcs i \u2260 0\neve :\n  \u2200\u1da0 (i : \u03b3) in F,\n    testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize (\u03bcs i))) f = (mass (\u03bcs i))\u207b\u00b9 * testAgainstNN (\u03bcs i) f\n\u22a2 Tendsto (fun i => testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize (\u03bcs i))) f) F\n    (\ud835\udcdd (testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize \u03bc)) f))", "state_after": "\u03a9 : Type u_1\ninst\u271d\u00b2 : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_2\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\n\u03bcs_lim : Tendsto \u03bcs F (\ud835\udcdd \u03bc)\nnonzero : \u03bc \u2260 0\nf : \u03a9 \u2192\u1d47 \u211d\u22650\nlim_mass : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd (mass \u03bc))\naux : {0}\u1d9c \u2208 \ud835\udcdd (mass \u03bc)\neventually_nonzero : \u2200\u1da0 (i : \u03b3) in F, \u03bcs i \u2260 0\neve :\n  \u2200\u1da0 (i : \u03b3) in F,\n    testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize (\u03bcs i))) f = (mass (\u03bcs i))\u207b\u00b9 * testAgainstNN (\u03bcs i) f\n\u22a2 Tendsto (fun x => (mass (\u03bcs x))\u207b\u00b9 * testAgainstNN (\u03bcs x) f) F (\ud835\udcdd ((mass \u03bc)\u207b\u00b9 * testAgainstNN \u03bc f))"}, {"tactic": "exact tendsto_mul.comp lim_pair", "annotated_tactic": ["exact tendsto_mul.comp lim_pair", []], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b2 : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_2\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\n\u03bcs_lim : Tendsto \u03bcs F (\ud835\udcdd \u03bc)\nnonzero : \u03bc \u2260 0\nf : \u03a9 \u2192\u1d47 \u211d\u22650\nlim_mass : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd (mass \u03bc))\naux : {0}\u1d9c \u2208 \ud835\udcdd (mass \u03bc)\neventually_nonzero : \u2200\u1da0 (i : \u03b3) in F, \u03bcs i \u2260 0\neve :\n  \u2200\u1da0 (i : \u03b3) in F,\n    testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize (\u03bcs i))) f = (mass (\u03bcs i))\u207b\u00b9 * testAgainstNN (\u03bcs i) f\nlim_pair : Tendsto (fun i => ((mass (\u03bcs i))\u207b\u00b9, testAgainstNN (\u03bcs i) f)) F (\ud835\udcdd ((mass \u03bc)\u207b\u00b9, testAgainstNN \u03bc f))\n\u22a2 Tendsto (fun x => (mass (\u03bcs x))\u207b\u00b9 * testAgainstNN (\u03bcs x) f) F (\ud835\udcdd ((mass \u03bc)\u207b\u00b9 * testAgainstNN \u03bc f))", "state_after": "no goals"}, {"tactic": "simp_rw [\u2190 mass_nonzero_iff]", "annotated_tactic": ["simp_rw [\u2190 <a>mass_nonzero_iff</a>]", [{"full_name": "MeasureTheory.FiniteMeasure.mass_nonzero_iff", "def_path": "Mathlib/MeasureTheory/Measure/FiniteMeasure.lean", "def_pos": [191, 9], "def_end_pos": [191, 25]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b2 : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_2\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\n\u03bcs_lim : Tendsto \u03bcs F (\ud835\udcdd \u03bc)\nnonzero : \u03bc \u2260 0\nf : \u03a9 \u2192\u1d47 \u211d\u22650\nlim_mass : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd (mass \u03bc))\naux : {0}\u1d9c \u2208 \ud835\udcdd (mass \u03bc)\n\u22a2 \u2200\u1da0 (i : \u03b3) in F, \u03bcs i \u2260 0", "state_after": "\u03a9 : Type u_1\ninst\u271d\u00b2 : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_2\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\n\u03bcs_lim : Tendsto \u03bcs F (\ud835\udcdd \u03bc)\nnonzero : \u03bc \u2260 0\nf : \u03a9 \u2192\u1d47 \u211d\u22650\nlim_mass : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd (mass \u03bc))\naux : {0}\u1d9c \u2208 \ud835\udcdd (mass \u03bc)\n\u22a2 \u2200\u1da0 (i : \u03b3) in F, mass (\u03bcs i) \u2260 0"}, {"tactic": "exact lim_mass aux", "annotated_tactic": ["exact lim_mass aux", []], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b2 : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_2\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\n\u03bcs_lim : Tendsto \u03bcs F (\ud835\udcdd \u03bc)\nnonzero : \u03bc \u2260 0\nf : \u03a9 \u2192\u1d47 \u211d\u22650\nlim_mass : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd (mass \u03bc))\naux : {0}\u1d9c \u2208 \ud835\udcdd (mass \u03bc)\n\u22a2 \u2200\u1da0 (i : \u03b3) in F, mass (\u03bcs i) \u2260 0", "state_after": "no goals"}, {"tactic": "filter_upwards [eventually_iff.mp eventually_nonzero]", "annotated_tactic": ["filter_upwards [eventually_iff.mp eventually_nonzero]", []], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b2 : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_2\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\n\u03bcs_lim : Tendsto \u03bcs F (\ud835\udcdd \u03bc)\nnonzero : \u03bc \u2260 0\nf : \u03a9 \u2192\u1d47 \u211d\u22650\nlim_mass : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd (mass \u03bc))\naux : {0}\u1d9c \u2208 \ud835\udcdd (mass \u03bc)\neventually_nonzero : \u2200\u1da0 (i : \u03b3) in F, \u03bcs i \u2260 0\n\u22a2 \u2200\u1da0 (i : \u03b3) in F,\n    testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize (\u03bcs i))) f = (mass (\u03bcs i))\u207b\u00b9 * testAgainstNN (\u03bcs i) f", "state_after": "case h\n\u03a9 : Type u_1\ninst\u271d\u00b2 : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_2\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\n\u03bcs_lim : Tendsto \u03bcs F (\ud835\udcdd \u03bc)\nnonzero : \u03bc \u2260 0\nf : \u03a9 \u2192\u1d47 \u211d\u22650\nlim_mass : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd (mass \u03bc))\naux : {0}\u1d9c \u2208 \ud835\udcdd (mass \u03bc)\neventually_nonzero : \u2200\u1da0 (i : \u03b3) in F, \u03bcs i \u2260 0\n\u22a2 \u2200 (a : \u03b3),\n    \u03bcs a \u2260 0 \u2192\n      testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize (\u03bcs a))) f = (mass (\u03bcs a))\u207b\u00b9 * testAgainstNN (\u03bcs a) f"}, {"tactic": "intro i hi", "annotated_tactic": ["intro i hi", []], "state_before": "case h\n\u03a9 : Type u_1\ninst\u271d\u00b2 : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_2\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\n\u03bcs_lim : Tendsto \u03bcs F (\ud835\udcdd \u03bc)\nnonzero : \u03bc \u2260 0\nf : \u03a9 \u2192\u1d47 \u211d\u22650\nlim_mass : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd (mass \u03bc))\naux : {0}\u1d9c \u2208 \ud835\udcdd (mass \u03bc)\neventually_nonzero : \u2200\u1da0 (i : \u03b3) in F, \u03bcs i \u2260 0\n\u22a2 \u2200 (a : \u03b3),\n    \u03bcs a \u2260 0 \u2192\n      testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize (\u03bcs a))) f = (mass (\u03bcs a))\u207b\u00b9 * testAgainstNN (\u03bcs a) f", "state_after": "case h\n\u03a9 : Type u_1\ninst\u271d\u00b2 : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_2\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\n\u03bcs_lim : Tendsto \u03bcs F (\ud835\udcdd \u03bc)\nnonzero : \u03bc \u2260 0\nf : \u03a9 \u2192\u1d47 \u211d\u22650\nlim_mass : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd (mass \u03bc))\naux : {0}\u1d9c \u2208 \ud835\udcdd (mass \u03bc)\neventually_nonzero : \u2200\u1da0 (i : \u03b3) in F, \u03bcs i \u2260 0\ni : \u03b3\nhi : \u03bcs i \u2260 0\n\u22a2 testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize (\u03bcs i))) f = (mass (\u03bcs i))\u207b\u00b9 * testAgainstNN (\u03bcs i) f"}, {"tactic": "apply normalize_testAgainstNN _ hi", "annotated_tactic": ["apply <a>normalize_testAgainstNN</a> _ hi", [{"full_name": "MeasureTheory.FiniteMeasure.normalize_testAgainstNN", "def_path": "Mathlib/MeasureTheory/Measure/ProbabilityMeasure.lean", "def_pos": [410, 9], "def_end_pos": [410, 32]}]], "state_before": "case h\n\u03a9 : Type u_1\ninst\u271d\u00b2 : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_2\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\n\u03bcs_lim : Tendsto \u03bcs F (\ud835\udcdd \u03bc)\nnonzero : \u03bc \u2260 0\nf : \u03a9 \u2192\u1d47 \u211d\u22650\nlim_mass : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd (mass \u03bc))\naux : {0}\u1d9c \u2208 \ud835\udcdd (mass \u03bc)\neventually_nonzero : \u2200\u1da0 (i : \u03b3) in F, \u03bcs i \u2260 0\ni : \u03b3\nhi : \u03bcs i \u2260 0\n\u22a2 testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize (\u03bcs i))) f = (mass (\u03bcs i))\u207b\u00b9 * testAgainstNN (\u03bcs i) f", "state_after": "no goals"}, {"tactic": "refine' (Prod.tendsto_iff _ _).mpr \u27e8_, _\u27e9", "annotated_tactic": ["refine' (<a>Prod.tendsto_iff</a> _ _).<a>mpr</a> \u27e8_, _\u27e9", [{"full_name": "Prod.tendsto_iff", "def_path": "Mathlib/Topology/Constructions.lean", "def_pos": [554, 9], "def_end_pos": [554, 25]}, {"full_name": "Iff.mpr", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [92, 3], "def_end_pos": [92, 6]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b2 : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_2\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\n\u03bcs_lim : Tendsto \u03bcs F (\ud835\udcdd \u03bc)\nnonzero : \u03bc \u2260 0\nf : \u03a9 \u2192\u1d47 \u211d\u22650\nlim_mass : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd (mass \u03bc))\naux : {0}\u1d9c \u2208 \ud835\udcdd (mass \u03bc)\neventually_nonzero : \u2200\u1da0 (i : \u03b3) in F, \u03bcs i \u2260 0\neve :\n  \u2200\u1da0 (i : \u03b3) in F,\n    testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize (\u03bcs i))) f = (mass (\u03bcs i))\u207b\u00b9 * testAgainstNN (\u03bcs i) f\n\u22a2 Tendsto (fun i => ((mass (\u03bcs i))\u207b\u00b9, testAgainstNN (\u03bcs i) f)) F (\ud835\udcdd ((mass \u03bc)\u207b\u00b9, testAgainstNN \u03bc f))", "state_after": "case refine'_1\n\u03a9 : Type u_1\ninst\u271d\u00b2 : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_2\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\n\u03bcs_lim : Tendsto \u03bcs F (\ud835\udcdd \u03bc)\nnonzero : \u03bc \u2260 0\nf : \u03a9 \u2192\u1d47 \u211d\u22650\nlim_mass : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd (mass \u03bc))\naux : {0}\u1d9c \u2208 \ud835\udcdd (mass \u03bc)\neventually_nonzero : \u2200\u1da0 (i : \u03b3) in F, \u03bcs i \u2260 0\neve :\n  \u2200\u1da0 (i : \u03b3) in F,\n    testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize (\u03bcs i))) f = (mass (\u03bcs i))\u207b\u00b9 * testAgainstNN (\u03bcs i) f\n\u22a2 Tendsto (fun n => ((mass (\u03bcs n))\u207b\u00b9, testAgainstNN (\u03bcs n) f).1) F (\ud835\udcdd ((mass \u03bc)\u207b\u00b9, testAgainstNN \u03bc f).1)\n\ncase refine'_2\n\u03a9 : Type u_1\ninst\u271d\u00b2 : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_2\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\n\u03bcs_lim : Tendsto \u03bcs F (\ud835\udcdd \u03bc)\nnonzero : \u03bc \u2260 0\nf : \u03a9 \u2192\u1d47 \u211d\u22650\nlim_mass : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd (mass \u03bc))\naux : {0}\u1d9c \u2208 \ud835\udcdd (mass \u03bc)\neventually_nonzero : \u2200\u1da0 (i : \u03b3) in F, \u03bcs i \u2260 0\neve :\n  \u2200\u1da0 (i : \u03b3) in F,\n    testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize (\u03bcs i))) f = (mass (\u03bcs i))\u207b\u00b9 * testAgainstNN (\u03bcs i) f\n\u22a2 Tendsto (fun n => ((mass (\u03bcs n))\u207b\u00b9, testAgainstNN (\u03bcs n) f).2) F (\ud835\udcdd ((mass \u03bc)\u207b\u00b9, testAgainstNN \u03bc f).2)"}, {"tactic": "exact (continuousOn_inv\u2080.continuousAt aux).tendsto.comp lim_mass", "annotated_tactic": ["exact (continuousOn_inv\u2080.continuousAt aux).tendsto.comp lim_mass", []], "state_before": "case refine'_1\n\u03a9 : Type u_1\ninst\u271d\u00b2 : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_2\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\n\u03bcs_lim : Tendsto \u03bcs F (\ud835\udcdd \u03bc)\nnonzero : \u03bc \u2260 0\nf : \u03a9 \u2192\u1d47 \u211d\u22650\nlim_mass : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd (mass \u03bc))\naux : {0}\u1d9c \u2208 \ud835\udcdd (mass \u03bc)\neventually_nonzero : \u2200\u1da0 (i : \u03b3) in F, \u03bcs i \u2260 0\neve :\n  \u2200\u1da0 (i : \u03b3) in F,\n    testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize (\u03bcs i))) f = (mass (\u03bcs i))\u207b\u00b9 * testAgainstNN (\u03bcs i) f\n\u22a2 Tendsto (fun n => ((mass (\u03bcs n))\u207b\u00b9, testAgainstNN (\u03bcs n) f).1) F (\ud835\udcdd ((mass \u03bc)\u207b\u00b9, testAgainstNN \u03bc f).1)", "state_after": "no goals"}, {"tactic": "exact tendsto_iff_forall_testAgainstNN_tendsto.mp \u03bcs_lim f", "annotated_tactic": ["exact tendsto_iff_forall_testAgainstNN_tendsto.mp \u03bcs_lim f", []], "state_before": "case refine'_2\n\u03a9 : Type u_1\ninst\u271d\u00b2 : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_2\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\n\u03bcs_lim : Tendsto \u03bcs F (\ud835\udcdd \u03bc)\nnonzero : \u03bc \u2260 0\nf : \u03a9 \u2192\u1d47 \u211d\u22650\nlim_mass : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd (mass \u03bc))\naux : {0}\u1d9c \u2208 \ud835\udcdd (mass \u03bc)\neventually_nonzero : \u2200\u1da0 (i : \u03b3) in F, \u03bcs i \u2260 0\neve :\n  \u2200\u1da0 (i : \u03b3) in F,\n    testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize (\u03bcs i))) f = (mass (\u03bcs i))\u207b\u00b9 * testAgainstNN (\u03bcs i) f\n\u22a2 Tendsto (fun n => ((mass (\u03bcs n))\u207b\u00b9, testAgainstNN (\u03bcs n) f).2) F (\ud835\udcdd ((mass \u03bc)\u207b\u00b9, testAgainstNN \u03bc f).2)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/Equiv.lean", "full_name": "MvPolynomial.finSuccEquiv_apply", "start": [332, 1], "end": [336, 42], "traced_tactics": [{"tactic": "rw [\u2190 finSuccEquiv_eq, RingHom.coe_coe]", "annotated_tactic": ["rw [\u2190 <a>finSuccEquiv_eq</a>, <a>RingHom.coe_coe</a>]", [{"full_name": "MvPolynomial.finSuccEquiv_eq", "def_path": "Mathlib/Data/MvPolynomial/Equiv.lean", "def_pos": [320, 9], "def_end_pos": [320, 24]}, {"full_name": "RingHom.coe_coe", "def_path": "Mathlib/Algebra/Hom/Ring/Defs.lean", "def_pos": [454, 9], "def_end_pos": [454, 16]}]], "state_before": "R : Type u\nS\u2081 : Type v\nS\u2082 : Type w\nS\u2083 : Type x\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\nn : \u2115\np : MvPolynomial (Fin (n + 1)) R\n\u22a2 \u2191(finSuccEquiv R n) p =\n    \u2191(eval\u2082Hom (RingHom.comp Polynomial.C C) fun i => Fin.cases Polynomial.X (fun k => \u2191Polynomial.C (X k)) i) p", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Moments.lean", "full_name": "ProbabilityTheory.iIndepFun.integrable_exp_mul_sum", "start": [285, 1], "end": [298, 74], "traced_tactics": [{"tactic": "induction' s using Finset.induction_on with i s hi_notin_s h_rec h_int", "annotated_tactic": ["induction' s using <a>Finset.induction_on</a> with i s hi_notin_s h_rec h_int", [{"full_name": "Finset.induction_on", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1251, 19], "def_end_pos": [1251, 31]}]], "state_before": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\np : \u2115\n\u03bc : Measure \u03a9\nt : \u211d\ninst\u271d : IsProbabilityMeasure \u03bc\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\nh_indep : iIndepFun (fun i => inferInstance) X\nh_meas : \u2200 (i : \u03b9), Measurable (X i)\ns : Finset \u03b9\nh_int : \u2200 (i : \u03b9), i \u2208 s \u2192 Integrable fun \u03c9 => rexp (t * X i \u03c9)\n\u22a2 Integrable fun \u03c9 => rexp (t * Finset.sum s (fun i => X i) \u03c9)", "state_after": "case empty\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\np : \u2115\n\u03bc : Measure \u03a9\nt : \u211d\ninst\u271d : IsProbabilityMeasure \u03bc\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\nh_indep : iIndepFun (fun i => inferInstance) X\nh_meas : \u2200 (i : \u03b9), Measurable (X i)\ns : Finset \u03b9\nh_int\u271d : \u2200 (i : \u03b9), i \u2208 s \u2192 Integrable fun \u03c9 => rexp (t * X i \u03c9)\nh_int : \u2200 (i : \u03b9), i \u2208 \u2205 \u2192 Integrable fun \u03c9 => rexp (t * X i \u03c9)\n\u22a2 Integrable fun \u03c9 => rexp (t * Finset.sum \u2205 (fun i => X i) \u03c9)\n\ncase insert\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\np : \u2115\n\u03bc : Measure \u03a9\nt : \u211d\ninst\u271d : IsProbabilityMeasure \u03bc\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\nh_indep : iIndepFun (fun i => inferInstance) X\nh_meas : \u2200 (i : \u03b9), Measurable (X i)\ns\u271d : Finset \u03b9\nh_int\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Integrable fun \u03c9 => rexp (t * X i \u03c9)\ni : \u03b9\ns : Finset \u03b9\nhi_notin_s : \u00aci \u2208 s\nh_rec :\n  (\u2200 (i : \u03b9), i \u2208 s \u2192 Integrable fun \u03c9 => rexp (t * X i \u03c9)) \u2192\n    Integrable fun \u03c9 => rexp (t * Finset.sum s (fun i => X i) \u03c9)\nh_int : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i s \u2192 Integrable fun \u03c9 => rexp (t * X i_1 \u03c9)\n\u22a2 Integrable fun \u03c9 => rexp (t * Finset.sum (insert i s) (fun i => X i) \u03c9)"}, {"tactic": "simp only [Pi.zero_apply, sum_apply, sum_empty, mul_zero, exp_zero]", "annotated_tactic": ["simp only [<a>Pi.zero_apply</a>, <a>sum_apply</a>, <a>sum_empty</a>, <a>mul_zero</a>, <a>exp_zero</a>]", [{"full_name": "Pi.zero_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [46, 3], "def_end_pos": [46, 14]}, {"full_name": "Finset.sum_apply", "def_path": "Mathlib/Algebra/BigOperators/Pi.lean", "def_pos": [41, 3], "def_end_pos": [41, 14]}, {"full_name": "Finset.sum_empty", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [298, 3], "def_end_pos": [298, 14]}, {"full_name": "MulZeroClass.mul_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [38, 3], "def_end_pos": [38, 11]}, {"full_name": "Real.exp_zero", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [1135, 9], "def_end_pos": [1135, 17]}]], "state_before": "case empty\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\np : \u2115\n\u03bc : Measure \u03a9\nt : \u211d\ninst\u271d : IsProbabilityMeasure \u03bc\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\nh_indep : iIndepFun (fun i => inferInstance) X\nh_meas : \u2200 (i : \u03b9), Measurable (X i)\ns : Finset \u03b9\nh_int\u271d : \u2200 (i : \u03b9), i \u2208 s \u2192 Integrable fun \u03c9 => rexp (t * X i \u03c9)\nh_int : \u2200 (i : \u03b9), i \u2208 \u2205 \u2192 Integrable fun \u03c9 => rexp (t * X i \u03c9)\n\u22a2 Integrable fun \u03c9 => rexp (t * Finset.sum \u2205 (fun i => X i) \u03c9)", "state_after": "case empty\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\np : \u2115\n\u03bc : Measure \u03a9\nt : \u211d\ninst\u271d : IsProbabilityMeasure \u03bc\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\nh_indep : iIndepFun (fun i => inferInstance) X\nh_meas : \u2200 (i : \u03b9), Measurable (X i)\ns : Finset \u03b9\nh_int\u271d : \u2200 (i : \u03b9), i \u2208 s \u2192 Integrable fun \u03c9 => rexp (t * X i \u03c9)\nh_int : \u2200 (i : \u03b9), i \u2208 \u2205 \u2192 Integrable fun \u03c9 => rexp (t * X i \u03c9)\n\u22a2 Integrable fun \u03c9 => 1"}, {"tactic": "exact integrable_const _", "annotated_tactic": ["exact <a>integrable_const</a> _", [{"full_name": "MeasureTheory.integrable_const", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [506, 9], "def_end_pos": [506, 25]}]], "state_before": "case empty\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\np : \u2115\n\u03bc : Measure \u03a9\nt : \u211d\ninst\u271d : IsProbabilityMeasure \u03bc\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\nh_indep : iIndepFun (fun i => inferInstance) X\nh_meas : \u2200 (i : \u03b9), Measurable (X i)\ns : Finset \u03b9\nh_int\u271d : \u2200 (i : \u03b9), i \u2208 s \u2192 Integrable fun \u03c9 => rexp (t * X i \u03c9)\nh_int : \u2200 (i : \u03b9), i \u2208 \u2205 \u2192 Integrable fun \u03c9 => rexp (t * X i \u03c9)\n\u22a2 Integrable fun \u03c9 => 1", "state_after": "no goals"}, {"tactic": "have : \u2200 i : \u03b9, i \u2208 s \u2192 Integrable (fun \u03c9 : \u03a9 => exp (t * X i \u03c9)) \u03bc := fun i hi =>\n  h_int i (mem_insert_of_mem hi)", "annotated_tactic": ["have : \u2200 i : \u03b9, i \u2208 s \u2192 <a>Integrable</a> (fun \u03c9 : \u03a9 => <a>exp</a> (t * X i \u03c9)) \u03bc := fun i hi =>\n      h_int i (<a>mem_insert_of_mem</a> hi)", [{"full_name": "MeasureTheory.Integrable", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [442, 5], "def_end_pos": [442, 15]}, {"full_name": "Real.exp", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [434, 12], "def_end_pos": [434, 15]}, {"full_name": "Finset.mem_insert_of_mem", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1095, 9], "def_end_pos": [1095, 26]}]], "state_before": "case insert\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\np : \u2115\n\u03bc : Measure \u03a9\nt : \u211d\ninst\u271d : IsProbabilityMeasure \u03bc\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\nh_indep : iIndepFun (fun i => inferInstance) X\nh_meas : \u2200 (i : \u03b9), Measurable (X i)\ns\u271d : Finset \u03b9\nh_int\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Integrable fun \u03c9 => rexp (t * X i \u03c9)\ni : \u03b9\ns : Finset \u03b9\nhi_notin_s : \u00aci \u2208 s\nh_rec :\n  (\u2200 (i : \u03b9), i \u2208 s \u2192 Integrable fun \u03c9 => rexp (t * X i \u03c9)) \u2192\n    Integrable fun \u03c9 => rexp (t * Finset.sum s (fun i => X i) \u03c9)\nh_int : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i s \u2192 Integrable fun \u03c9 => rexp (t * X i_1 \u03c9)\n\u22a2 Integrable fun \u03c9 => rexp (t * Finset.sum (insert i s) (fun i => X i) \u03c9)", "state_after": "case insert\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\np : \u2115\n\u03bc : Measure \u03a9\nt : \u211d\ninst\u271d : IsProbabilityMeasure \u03bc\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\nh_indep : iIndepFun (fun i => inferInstance) X\nh_meas : \u2200 (i : \u03b9), Measurable (X i)\ns\u271d : Finset \u03b9\nh_int\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Integrable fun \u03c9 => rexp (t * X i \u03c9)\ni : \u03b9\ns : Finset \u03b9\nhi_notin_s : \u00aci \u2208 s\nh_rec :\n  (\u2200 (i : \u03b9), i \u2208 s \u2192 Integrable fun \u03c9 => rexp (t * X i \u03c9)) \u2192\n    Integrable fun \u03c9 => rexp (t * Finset.sum s (fun i => X i) \u03c9)\nh_int : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i s \u2192 Integrable fun \u03c9 => rexp (t * X i_1 \u03c9)\nthis : \u2200 (i : \u03b9), i \u2208 s \u2192 Integrable fun \u03c9 => rexp (t * X i \u03c9)\n\u22a2 Integrable fun \u03c9 => rexp (t * Finset.sum (insert i s) (fun i => X i) \u03c9)"}, {"tactic": "specialize h_rec this", "annotated_tactic": ["specialize h_rec this", []], "state_before": "case insert\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\np : \u2115\n\u03bc : Measure \u03a9\nt : \u211d\ninst\u271d : IsProbabilityMeasure \u03bc\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\nh_indep : iIndepFun (fun i => inferInstance) X\nh_meas : \u2200 (i : \u03b9), Measurable (X i)\ns\u271d : Finset \u03b9\nh_int\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Integrable fun \u03c9 => rexp (t * X i \u03c9)\ni : \u03b9\ns : Finset \u03b9\nhi_notin_s : \u00aci \u2208 s\nh_rec :\n  (\u2200 (i : \u03b9), i \u2208 s \u2192 Integrable fun \u03c9 => rexp (t * X i \u03c9)) \u2192\n    Integrable fun \u03c9 => rexp (t * Finset.sum s (fun i => X i) \u03c9)\nh_int : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i s \u2192 Integrable fun \u03c9 => rexp (t * X i_1 \u03c9)\nthis : \u2200 (i : \u03b9), i \u2208 s \u2192 Integrable fun \u03c9 => rexp (t * X i \u03c9)\n\u22a2 Integrable fun \u03c9 => rexp (t * Finset.sum (insert i s) (fun i => X i) \u03c9)", "state_after": "case insert\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\np : \u2115\n\u03bc : Measure \u03a9\nt : \u211d\ninst\u271d : IsProbabilityMeasure \u03bc\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\nh_indep : iIndepFun (fun i => inferInstance) X\nh_meas : \u2200 (i : \u03b9), Measurable (X i)\ns\u271d : Finset \u03b9\nh_int\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Integrable fun \u03c9 => rexp (t * X i \u03c9)\ni : \u03b9\ns : Finset \u03b9\nhi_notin_s : \u00aci \u2208 s\nh_int : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i s \u2192 Integrable fun \u03c9 => rexp (t * X i_1 \u03c9)\nthis : \u2200 (i : \u03b9), i \u2208 s \u2192 Integrable fun \u03c9 => rexp (t * X i \u03c9)\nh_rec : Integrable fun \u03c9 => rexp (t * Finset.sum s (fun i => X i) \u03c9)\n\u22a2 Integrable fun \u03c9 => rexp (t * Finset.sum (insert i s) (fun i => X i) \u03c9)"}, {"tactic": "rw [sum_insert hi_notin_s]", "annotated_tactic": ["rw [<a>sum_insert</a> hi_notin_s]", [{"full_name": "Finset.sum_insert", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [316, 3], "def_end_pos": [316, 14]}]], "state_before": "case insert\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\np : \u2115\n\u03bc : Measure \u03a9\nt : \u211d\ninst\u271d : IsProbabilityMeasure \u03bc\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\nh_indep : iIndepFun (fun i => inferInstance) X\nh_meas : \u2200 (i : \u03b9), Measurable (X i)\ns\u271d : Finset \u03b9\nh_int\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Integrable fun \u03c9 => rexp (t * X i \u03c9)\ni : \u03b9\ns : Finset \u03b9\nhi_notin_s : \u00aci \u2208 s\nh_int : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i s \u2192 Integrable fun \u03c9 => rexp (t * X i_1 \u03c9)\nthis : \u2200 (i : \u03b9), i \u2208 s \u2192 Integrable fun \u03c9 => rexp (t * X i \u03c9)\nh_rec : Integrable fun \u03c9 => rexp (t * Finset.sum s (fun i => X i) \u03c9)\n\u22a2 Integrable fun \u03c9 => rexp (t * Finset.sum (insert i s) (fun i => X i) \u03c9)", "state_after": "case insert\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\np : \u2115\n\u03bc : Measure \u03a9\nt : \u211d\ninst\u271d : IsProbabilityMeasure \u03bc\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\nh_indep : iIndepFun (fun i => inferInstance) X\nh_meas : \u2200 (i : \u03b9), Measurable (X i)\ns\u271d : Finset \u03b9\nh_int\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Integrable fun \u03c9 => rexp (t * X i \u03c9)\ni : \u03b9\ns : Finset \u03b9\nhi_notin_s : \u00aci \u2208 s\nh_int : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i s \u2192 Integrable fun \u03c9 => rexp (t * X i_1 \u03c9)\nthis : \u2200 (i : \u03b9), i \u2208 s \u2192 Integrable fun \u03c9 => rexp (t * X i \u03c9)\nh_rec : Integrable fun \u03c9 => rexp (t * Finset.sum s (fun i => X i) \u03c9)\n\u22a2 Integrable fun \u03c9 => rexp (t * (X i + \u2211 x in s, X x) \u03c9)"}, {"tactic": "refine' IndepFun.integrable_exp_mul_add _ (h_int i (mem_insert_self _ _)) h_rec", "annotated_tactic": ["refine' <a>IndepFun.integrable_exp_mul_add</a> _ (h_int i (<a>mem_insert_self</a> _ _)) h_rec", [{"full_name": "ProbabilityTheory.IndepFun.integrable_exp_mul_add", "def_path": "Mathlib/Probability/Moments.lean", "def_pos": [277, 9], "def_end_pos": [277, 40]}, {"full_name": "Finset.mem_insert_self", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1091, 9], "def_end_pos": [1091, 24]}]], "state_before": "case insert\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\np : \u2115\n\u03bc : Measure \u03a9\nt : \u211d\ninst\u271d : IsProbabilityMeasure \u03bc\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\nh_indep : iIndepFun (fun i => inferInstance) X\nh_meas : \u2200 (i : \u03b9), Measurable (X i)\ns\u271d : Finset \u03b9\nh_int\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Integrable fun \u03c9 => rexp (t * X i \u03c9)\ni : \u03b9\ns : Finset \u03b9\nhi_notin_s : \u00aci \u2208 s\nh_int : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i s \u2192 Integrable fun \u03c9 => rexp (t * X i_1 \u03c9)\nthis : \u2200 (i : \u03b9), i \u2208 s \u2192 Integrable fun \u03c9 => rexp (t * X i \u03c9)\nh_rec : Integrable fun \u03c9 => rexp (t * Finset.sum s (fun i => X i) \u03c9)\n\u22a2 Integrable fun \u03c9 => rexp (t * (X i + \u2211 x in s, X x) \u03c9)", "state_after": "case insert\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\np : \u2115\n\u03bc : Measure \u03a9\nt : \u211d\ninst\u271d : IsProbabilityMeasure \u03bc\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\nh_indep : iIndepFun (fun i => inferInstance) X\nh_meas : \u2200 (i : \u03b9), Measurable (X i)\ns\u271d : Finset \u03b9\nh_int\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Integrable fun \u03c9 => rexp (t * X i \u03c9)\ni : \u03b9\ns : Finset \u03b9\nhi_notin_s : \u00aci \u2208 s\nh_int : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i s \u2192 Integrable fun \u03c9 => rexp (t * X i_1 \u03c9)\nthis : \u2200 (i : \u03b9), i \u2208 s \u2192 Integrable fun \u03c9 => rexp (t * X i \u03c9)\nh_rec : Integrable fun \u03c9 => rexp (t * Finset.sum s (fun i => X i) \u03c9)\n\u22a2 IndepFun (X i) (\u2211 x in s, X x)"}, {"tactic": "exact (h_indep.indepFun_finset_sum_of_not_mem h_meas hi_notin_s).symm", "annotated_tactic": ["exact (h_indep.indepFun_finset_sum_of_not_mem h_meas hi_notin_s).<a>symm</a>", [{"full_name": "ProbabilityTheory.IndepFun.symm", "def_path": "Mathlib/Probability/Independence/Basic.lean", "def_pos": [563, 16], "def_end_pos": [563, 29]}]], "state_before": "case insert\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\np : \u2115\n\u03bc : Measure \u03a9\nt : \u211d\ninst\u271d : IsProbabilityMeasure \u03bc\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\nh_indep : iIndepFun (fun i => inferInstance) X\nh_meas : \u2200 (i : \u03b9), Measurable (X i)\ns\u271d : Finset \u03b9\nh_int\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Integrable fun \u03c9 => rexp (t * X i \u03c9)\ni : \u03b9\ns : Finset \u03b9\nhi_notin_s : \u00aci \u2208 s\nh_int : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i s \u2192 Integrable fun \u03c9 => rexp (t * X i_1 \u03c9)\nthis : \u2200 (i : \u03b9), i \u2208 s \u2192 Integrable fun \u03c9 => rexp (t * X i \u03c9)\nh_rec : Integrable fun \u03c9 => rexp (t * Finset.sum s (fun i => X i) \u03c9)\n\u22a2 IndepFun (X i) (\u2211 x in s, X x)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/ZMod/Basic.lean", "full_name": "ZMod.cast_sub'", "start": [391, 1], "end": [392, 23], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Kernel/IntegralCompProd.lean", "full_name": "ProbabilityTheory.hasFiniteIntegral_compProd_iff'", "start": [107, 1], "end": [121, 53], "traced_tactics": [{"tactic": "rw [hasFiniteIntegral_congr h1f.ae_eq_mk,\n  hasFiniteIntegral_compProd_iff h1f.stronglyMeasurable_mk]", "annotated_tactic": ["rw [<a>hasFiniteIntegral_congr</a> h1f.ae_eq_mk,\n    <a>hasFiniteIntegral_compProd_iff</a> h1f.stronglyMeasurable_mk]", [{"full_name": "MeasureTheory.hasFiniteIntegral_congr", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [166, 9], "def_end_pos": [166, 32]}, {"full_name": "ProbabilityTheory.hasFiniteIntegral_compProd_iff", "def_path": "Mathlib/Probability/Kernel/IntegralCompProd.lean", "def_pos": [88, 9], "def_end_pos": [88, 39]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nE : Type u_4\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d\u00b2 : NormedAddCommGroup E\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d : IsSFiniteKernel \u03b7\na : \u03b1\nf : \u03b2 \u00d7 \u03b3 \u2192 E\nh1f : AEStronglyMeasurable f (\u2191(\u03ba \u2297\u2096 \u03b7) a)\n\u22a2 HasFiniteIntegral f \u2194\n    (\u2200\u1d50 (x : \u03b2) \u2202\u2191\u03ba a, HasFiniteIntegral fun y => f (x, y)) \u2227\n      HasFiniteIntegral fun x => \u222b (y : \u03b3), \u2016f (x, y)\u2016 \u2202\u2191\u03b7 (a, x)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nE : Type u_4\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d\u00b2 : NormedAddCommGroup E\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d : IsSFiniteKernel \u03b7\na : \u03b1\nf : \u03b2 \u00d7 \u03b3 \u2192 E\nh1f : AEStronglyMeasurable f (\u2191(\u03ba \u2297\u2096 \u03b7) a)\n\u22a2 ((\u2200\u1d50 (x : \u03b2) \u2202\u2191\u03ba a, HasFiniteIntegral fun y => AEStronglyMeasurable.mk f h1f (x, y)) \u2227\n      HasFiniteIntegral fun x => \u222b (y : \u03b3), \u2016AEStronglyMeasurable.mk f h1f (x, y)\u2016 \u2202\u2191\u03b7 (a, x)) \u2194\n    (\u2200\u1d50 (x : \u03b2) \u2202\u2191\u03ba a, HasFiniteIntegral fun y => f (x, y)) \u2227\n      HasFiniteIntegral fun x => \u222b (y : \u03b3), \u2016f (x, y)\u2016 \u2202\u2191\u03b7 (a, x)"}, {"tactic": "apply and_congr", "annotated_tactic": ["apply <a>and_congr</a>", [{"full_name": "and_congr", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [156, 9], "def_end_pos": [156, 18]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nE : Type u_4\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d\u00b2 : NormedAddCommGroup E\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d : IsSFiniteKernel \u03b7\na : \u03b1\nf : \u03b2 \u00d7 \u03b3 \u2192 E\nh1f : AEStronglyMeasurable f (\u2191(\u03ba \u2297\u2096 \u03b7) a)\n\u22a2 ((\u2200\u1d50 (x : \u03b2) \u2202\u2191\u03ba a, HasFiniteIntegral fun y => AEStronglyMeasurable.mk f h1f (x, y)) \u2227\n      HasFiniteIntegral fun x => \u222b (y : \u03b3), \u2016AEStronglyMeasurable.mk f h1f (x, y)\u2016 \u2202\u2191\u03b7 (a, x)) \u2194\n    (\u2200\u1d50 (x : \u03b2) \u2202\u2191\u03ba a, HasFiniteIntegral fun y => f (x, y)) \u2227\n      HasFiniteIntegral fun x => \u222b (y : \u03b3), \u2016f (x, y)\u2016 \u2202\u2191\u03b7 (a, x)", "state_after": "case h\u2081\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nE : Type u_4\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d\u00b2 : NormedAddCommGroup E\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d : IsSFiniteKernel \u03b7\na : \u03b1\nf : \u03b2 \u00d7 \u03b3 \u2192 E\nh1f : AEStronglyMeasurable f (\u2191(\u03ba \u2297\u2096 \u03b7) a)\n\u22a2 (\u2200\u1d50 (x : \u03b2) \u2202\u2191\u03ba a, HasFiniteIntegral fun y => AEStronglyMeasurable.mk f h1f (x, y)) \u2194\n    \u2200\u1d50 (x : \u03b2) \u2202\u2191\u03ba a, HasFiniteIntegral fun y => f (x, y)\n\ncase h\u2082\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nE : Type u_4\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d\u00b2 : NormedAddCommGroup E\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d : IsSFiniteKernel \u03b7\na : \u03b1\nf : \u03b2 \u00d7 \u03b3 \u2192 E\nh1f : AEStronglyMeasurable f (\u2191(\u03ba \u2297\u2096 \u03b7) a)\n\u22a2 (HasFiniteIntegral fun x => \u222b (y : \u03b3), \u2016AEStronglyMeasurable.mk f h1f (x, y)\u2016 \u2202\u2191\u03b7 (a, x)) \u2194\n    HasFiniteIntegral fun x => \u222b (y : \u03b3), \u2016f (x, y)\u2016 \u2202\u2191\u03b7 (a, x)"}, {"tactic": "apply eventually_congr", "annotated_tactic": ["apply <a>eventually_congr</a>", [{"full_name": "Filter.eventually_congr", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1161, 9], "def_end_pos": [1161, 25]}]], "state_before": "case h\u2081\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nE : Type u_4\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d\u00b2 : NormedAddCommGroup E\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d : IsSFiniteKernel \u03b7\na : \u03b1\nf : \u03b2 \u00d7 \u03b3 \u2192 E\nh1f : AEStronglyMeasurable f (\u2191(\u03ba \u2297\u2096 \u03b7) a)\n\u22a2 (\u2200\u1d50 (x : \u03b2) \u2202\u2191\u03ba a, HasFiniteIntegral fun y => AEStronglyMeasurable.mk f h1f (x, y)) \u2194\n    \u2200\u1d50 (x : \u03b2) \u2202\u2191\u03ba a, HasFiniteIntegral fun y => f (x, y)", "state_after": "case h\u2081.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nE : Type u_4\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d\u00b2 : NormedAddCommGroup E\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d : IsSFiniteKernel \u03b7\na : \u03b1\nf : \u03b2 \u00d7 \u03b3 \u2192 E\nh1f : AEStronglyMeasurable f (\u2191(\u03ba \u2297\u2096 \u03b7) a)\n\u22a2 \u2200\u1d50 (x : \u03b2) \u2202\u2191\u03ba a,\n    (HasFiniteIntegral fun y => AEStronglyMeasurable.mk f h1f (x, y)) \u2194 HasFiniteIntegral fun y => f (x, y)"}, {"tactic": "filter_upwards [ae_ae_of_ae_compProd h1f.ae_eq_mk.symm]", "annotated_tactic": ["filter_upwards [<a>ae_ae_of_ae_compProd</a> h1f.ae_eq_mk.symm]", [{"full_name": "ProbabilityTheory.kernel.ae_ae_of_ae_compProd", "def_path": "Mathlib/Probability/Kernel/Composition.lean", "def_pos": [317, 9], "def_end_pos": [317, 29]}]], "state_before": "case h\u2081.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nE : Type u_4\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d\u00b2 : NormedAddCommGroup E\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d : IsSFiniteKernel \u03b7\na : \u03b1\nf : \u03b2 \u00d7 \u03b3 \u2192 E\nh1f : AEStronglyMeasurable f (\u2191(\u03ba \u2297\u2096 \u03b7) a)\n\u22a2 \u2200\u1d50 (x : \u03b2) \u2202\u2191\u03ba a,\n    (HasFiniteIntegral fun y => AEStronglyMeasurable.mk f h1f (x, y)) \u2194 HasFiniteIntegral fun y => f (x, y)", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nE : Type u_4\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d\u00b2 : NormedAddCommGroup E\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d : IsSFiniteKernel \u03b7\na : \u03b1\nf : \u03b2 \u00d7 \u03b3 \u2192 E\nh1f : AEStronglyMeasurable f (\u2191(\u03ba \u2297\u2096 \u03b7) a)\n\u22a2 \u2200 (a_1 : \u03b2),\n    (\u2200\u1d50 (c : \u03b3) \u2202\u2191\u03b7 (a, a_1), AEStronglyMeasurable.mk f h1f (a_1, c) = f (a_1, c)) \u2192\n      ((HasFiniteIntegral fun y => AEStronglyMeasurable.mk f h1f (a_1, y)) \u2194 HasFiniteIntegral fun y => f (a_1, y))"}, {"tactic": "intro x hx", "annotated_tactic": ["intro x hx", []], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nE : Type u_4\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d\u00b2 : NormedAddCommGroup E\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d : IsSFiniteKernel \u03b7\na : \u03b1\nf : \u03b2 \u00d7 \u03b3 \u2192 E\nh1f : AEStronglyMeasurable f (\u2191(\u03ba \u2297\u2096 \u03b7) a)\n\u22a2 \u2200 (a_1 : \u03b2),\n    (\u2200\u1d50 (c : \u03b3) \u2202\u2191\u03b7 (a, a_1), AEStronglyMeasurable.mk f h1f (a_1, c) = f (a_1, c)) \u2192\n      ((HasFiniteIntegral fun y => AEStronglyMeasurable.mk f h1f (a_1, y)) \u2194 HasFiniteIntegral fun y => f (a_1, y))", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nE : Type u_4\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d\u00b2 : NormedAddCommGroup E\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d : IsSFiniteKernel \u03b7\na : \u03b1\nf : \u03b2 \u00d7 \u03b3 \u2192 E\nh1f : AEStronglyMeasurable f (\u2191(\u03ba \u2297\u2096 \u03b7) a)\nx : \u03b2\nhx : \u2200\u1d50 (c : \u03b3) \u2202\u2191\u03b7 (a, x), AEStronglyMeasurable.mk f h1f (x, c) = f (x, c)\n\u22a2 (HasFiniteIntegral fun y => AEStronglyMeasurable.mk f h1f (x, y)) \u2194 HasFiniteIntegral fun y => f (x, y)"}, {"tactic": "exact hasFiniteIntegral_congr hx", "annotated_tactic": ["exact <a>hasFiniteIntegral_congr</a> hx", [{"full_name": "MeasureTheory.hasFiniteIntegral_congr", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [166, 9], "def_end_pos": [166, 32]}]], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nE : Type u_4\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d\u00b2 : NormedAddCommGroup E\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d : IsSFiniteKernel \u03b7\na : \u03b1\nf : \u03b2 \u00d7 \u03b3 \u2192 E\nh1f : AEStronglyMeasurable f (\u2191(\u03ba \u2297\u2096 \u03b7) a)\nx : \u03b2\nhx : \u2200\u1d50 (c : \u03b3) \u2202\u2191\u03b7 (a, x), AEStronglyMeasurable.mk f h1f (x, c) = f (x, c)\n\u22a2 (HasFiniteIntegral fun y => AEStronglyMeasurable.mk f h1f (x, y)) \u2194 HasFiniteIntegral fun y => f (x, y)", "state_after": "no goals"}, {"tactic": "apply hasFiniteIntegral_congr", "annotated_tactic": ["apply <a>hasFiniteIntegral_congr</a>", [{"full_name": "MeasureTheory.hasFiniteIntegral_congr", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [166, 9], "def_end_pos": [166, 32]}]], "state_before": "case h\u2082\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nE : Type u_4\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d\u00b2 : NormedAddCommGroup E\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d : IsSFiniteKernel \u03b7\na : \u03b1\nf : \u03b2 \u00d7 \u03b3 \u2192 E\nh1f : AEStronglyMeasurable f (\u2191(\u03ba \u2297\u2096 \u03b7) a)\n\u22a2 (HasFiniteIntegral fun x => \u222b (y : \u03b3), \u2016AEStronglyMeasurable.mk f h1f (x, y)\u2016 \u2202\u2191\u03b7 (a, x)) \u2194\n    HasFiniteIntegral fun x => \u222b (y : \u03b3), \u2016f (x, y)\u2016 \u2202\u2191\u03b7 (a, x)", "state_after": "case h\u2082.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nE : Type u_4\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d\u00b2 : NormedAddCommGroup E\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d : IsSFiniteKernel \u03b7\na : \u03b1\nf : \u03b2 \u00d7 \u03b3 \u2192 E\nh1f : AEStronglyMeasurable f (\u2191(\u03ba \u2297\u2096 \u03b7) a)\n\u22a2 (fun x => \u222b (y : \u03b3), \u2016AEStronglyMeasurable.mk f h1f (x, y)\u2016 \u2202\u2191\u03b7 (a, x)) =\u1d50[\u2191\u03ba a] fun x =>\n    \u222b (y : \u03b3), \u2016f (x, y)\u2016 \u2202\u2191\u03b7 (a, x)"}, {"tactic": "filter_upwards [ae_ae_of_ae_compProd h1f.ae_eq_mk.symm] with _ hx using\n  integral_congr_ae (EventuallyEq.fun_comp hx _)", "annotated_tactic": ["filter_upwards [<a>ae_ae_of_ae_compProd</a> h1f.ae_eq_mk.symm] with _ hx using\n      <a>integral_congr_ae</a> (<a>EventuallyEq.fun_comp</a> hx _)", [{"full_name": "ProbabilityTheory.kernel.ae_ae_of_ae_compProd", "def_path": "Mathlib/Probability/Kernel/Composition.lean", "def_pos": [317, 9], "def_end_pos": [317, 29]}, {"full_name": "MeasureTheory.integral_congr_ae", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [938, 9], "def_end_pos": [938, 26]}, {"full_name": "Filter.EventuallyEq.fun_comp", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1520, 9], "def_end_pos": [1520, 30]}]], "state_before": "case h\u2082.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nE : Type u_4\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d\u00b2 : NormedAddCommGroup E\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d : IsSFiniteKernel \u03b7\na : \u03b1\nf : \u03b2 \u00d7 \u03b3 \u2192 E\nh1f : AEStronglyMeasurable f (\u2191(\u03ba \u2297\u2096 \u03b7) a)\n\u22a2 (fun x => \u222b (y : \u03b3), \u2016AEStronglyMeasurable.mk f h1f (x, y)\u2016 \u2202\u2191\u03b7 (a, x)) =\u1d50[\u2191\u03ba a] fun x =>\n    \u222b (y : \u03b3), \u2016f (x, y)\u2016 \u2202\u2191\u03b7 (a, x)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/PEquiv.lean", "full_name": "PEquiv.trans_symm_eq_iff_forall_isSome", "start": [285, 1], "end": [287, 67], "traced_tactics": [{"tactic": "rw [self_trans_symm, ofSet_eq_refl, Set.eq_univ_iff_forall]", "annotated_tactic": ["rw [<a>self_trans_symm</a>, <a>ofSet_eq_refl</a>, <a>Set.eq_univ_iff_forall</a>]", [{"full_name": "PEquiv.self_trans_symm", "def_path": "Mathlib/Data/PEquiv.lean", "def_pos": [270, 9], "def_end_pos": [270, 24]}, {"full_name": "PEquiv.ofSet_eq_refl", "def_path": "Mathlib/Data/PEquiv.lean", "def_pos": [255, 9], "def_end_pos": [255, 22]}, {"full_name": "Set.eq_univ_iff_forall", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [701, 9], "def_end_pos": [701, 27]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nf : \u03b1 \u2243. \u03b2\n\u22a2 PEquiv.trans f (PEquiv.symm f) = PEquiv.refl \u03b1 \u2194 \u2200 (a : \u03b1), isSome (\u2191f a) = true", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nf : \u03b1 \u2243. \u03b2\n\u22a2 (\u2200 (x : \u03b1), x \u2208 {a | isSome (\u2191f a) = true}) \u2194 \u2200 (a : \u03b1), isSome (\u2191f a) = true"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nf : \u03b1 \u2243. \u03b2\n\u22a2 (\u2200 (x : \u03b1), x \u2208 {a | isSome (\u2191f a) = true}) \u2194 \u2200 (a : \u03b1), isSome (\u2191f a) = true", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Density.lean", "full_name": "MeasureTheory.pdf.IsUniform.hasPDF", "start": [320, 1], "end": [333, 48], "traced_tactics": [{"tactic": "intro hpdf", "annotated_tactic": ["intro hpdf", []], "state_before": "\u03a9 : Type u_1\nE : Type u_2\ninst\u271d : MeasurableSpace E\nm\u271d : MeasurableSpace \u03a9\n\u2119\u271d : Measure \u03a9\n\u03bc\u271d : Measure E\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 E\n\u2119 : Measure \u03a9\n\u03bc : Measure E\ns : Set E\nhns : \u2191\u2191\u03bc s \u2260 0\nhnt : \u2191\u2191\u03bc s \u2260 \u22a4\nhu : IsUniform X s \u2119\n\u22a2 pdf X \u2119 \u2260 0", "state_after": "\u03a9 : Type u_1\nE : Type u_2\ninst\u271d : MeasurableSpace E\nm\u271d : MeasurableSpace \u03a9\n\u2119\u271d : Measure \u03a9\n\u03bc\u271d : Measure E\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 E\n\u2119 : Measure \u03a9\n\u03bc : Measure E\ns : Set E\nhns : \u2191\u2191\u03bc s \u2260 0\nhnt : \u2191\u2191\u03bc s \u2260 \u22a4\nhu : IsUniform X s \u2119\nhpdf : pdf X \u2119 = 0\n\u22a2 False"}, {"tactic": "simp only [IsUniform, hpdf] at hu", "annotated_tactic": ["simp only [<a>IsUniform</a>, hpdf] at hu", [{"full_name": "MeasureTheory.pdf.IsUniform", "def_path": "Mathlib/Probability/Density.lean", "def_pos": [313, 5], "def_end_pos": [313, 14]}]], "state_before": "\u03a9 : Type u_1\nE : Type u_2\ninst\u271d : MeasurableSpace E\nm\u271d : MeasurableSpace \u03a9\n\u2119\u271d : Measure \u03a9\n\u03bc\u271d : Measure E\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 E\n\u2119 : Measure \u03a9\n\u03bc : Measure E\ns : Set E\nhns : \u2191\u2191\u03bc s \u2260 0\nhnt : \u2191\u2191\u03bc s \u2260 \u22a4\nhu : IsUniform X s \u2119\nhpdf : pdf X \u2119 = 0\n\u22a2 False", "state_after": "\u03a9 : Type u_1\nE : Type u_2\ninst\u271d : MeasurableSpace E\nm\u271d : MeasurableSpace \u03a9\n\u2119\u271d : Measure \u03a9\n\u03bc\u271d : Measure E\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 E\n\u2119 : Measure \u03a9\n\u03bc : Measure E\ns : Set E\nhns : \u2191\u2191\u03bc s \u2260 0\nhnt : \u2191\u2191\u03bc s \u2260 \u22a4\nhpdf : pdf X \u2119 = 0\nhu : 0 =\u1da0[ae \u03bc] Set.indicator s ((\u2191\u2191\u03bc s)\u207b\u00b9 \u2022 1)\n\u22a2 False"}, {"tactic": "suffices \u03bc (s \u2229 Function.support ((\u03bc s)\u207b\u00b9 \u2022 (1 : E \u2192 \u211d\u22650\u221e))) = 0 by\n  have heq : Function.support ((\u03bc s)\u207b\u00b9 \u2022 (1 : E \u2192 \u211d\u22650\u221e)) = Set.univ := by\n    ext x\n    rw [Function.mem_support]\n    simp [hnt]\n  rw [heq, Set.inter_univ] at this\n  exact hns this", "annotated_tactic": ["suffices \u03bc (s \u2229 <a>Function.support</a> ((\u03bc s)\u207b\u00b9 \u2022 (1 : E \u2192 \u211d\u22650\u221e))) = 0 by\n        have heq : <a>Function.support</a> ((\u03bc s)\u207b\u00b9 \u2022 (1 : E \u2192 \u211d\u22650\u221e)) = <a>Set.univ</a> := by\n          ext x\n          rw [<a>Function.mem_support</a>]\n          simp [hnt]\n        rw [heq, <a>Set.inter_univ</a>] at this\n        exact hns this", [{"full_name": "Function.support", "def_path": "Mathlib/Algebra/Support.lean", "def_pos": [37, 5], "def_end_pos": [37, 12]}, {"full_name": "Function.support", "def_path": "Mathlib/Algebra/Support.lean", "def_pos": [37, 5], "def_end_pos": [37, 12]}, {"full_name": "Set.univ", "def_path": "Mathlib/Init/Set.lean", "def_pos": [90, 5], "def_end_pos": [90, 9]}, {"full_name": "Function.mem_support", "def_path": "Mathlib/Algebra/Support.lean", "def_pos": [65, 3], "def_end_pos": [65, 14]}, {"full_name": "Set.inter_univ", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1012, 9], "def_end_pos": [1012, 19]}]], "state_before": "\u03a9 : Type u_1\nE : Type u_2\ninst\u271d : MeasurableSpace E\nm\u271d : MeasurableSpace \u03a9\n\u2119\u271d : Measure \u03a9\n\u03bc\u271d : Measure E\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 E\n\u2119 : Measure \u03a9\n\u03bc : Measure E\ns : Set E\nhns : \u2191\u2191\u03bc s \u2260 0\nhnt : \u2191\u2191\u03bc s \u2260 \u22a4\nhpdf : pdf X \u2119 = 0\nhu : 0 =\u1da0[ae \u03bc] Set.indicator s ((\u2191\u2191\u03bc s)\u207b\u00b9 \u2022 1)\n\u22a2 False", "state_after": "\u03a9 : Type u_1\nE : Type u_2\ninst\u271d : MeasurableSpace E\nm\u271d : MeasurableSpace \u03a9\n\u2119\u271d : Measure \u03a9\n\u03bc\u271d : Measure E\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 E\n\u2119 : Measure \u03a9\n\u03bc : Measure E\ns : Set E\nhns : \u2191\u2191\u03bc s \u2260 0\nhnt : \u2191\u2191\u03bc s \u2260 \u22a4\nhpdf : pdf X \u2119 = 0\nhu : 0 =\u1da0[ae \u03bc] Set.indicator s ((\u2191\u2191\u03bc s)\u207b\u00b9 \u2022 1)\n\u22a2 \u2191\u2191\u03bc (s \u2229 Function.support ((\u2191\u2191\u03bc s)\u207b\u00b9 \u2022 1)) = 0"}, {"tactic": "exact Set.indicator_ae_eq_zero.1 hu.symm", "annotated_tactic": ["exact <a>Set.indicator_ae_eq_zero</a>.1 hu.symm", [{"full_name": "Set.indicator_ae_eq_zero", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [587, 3], "def_end_pos": [587, 14]}]], "state_before": "\u03a9 : Type u_1\nE : Type u_2\ninst\u271d : MeasurableSpace E\nm\u271d : MeasurableSpace \u03a9\n\u2119\u271d : Measure \u03a9\n\u03bc\u271d : Measure E\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 E\n\u2119 : Measure \u03a9\n\u03bc : Measure E\ns : Set E\nhns : \u2191\u2191\u03bc s \u2260 0\nhnt : \u2191\u2191\u03bc s \u2260 \u22a4\nhpdf : pdf X \u2119 = 0\nhu : 0 =\u1da0[ae \u03bc] Set.indicator s ((\u2191\u2191\u03bc s)\u207b\u00b9 \u2022 1)\n\u22a2 \u2191\u2191\u03bc (s \u2229 Function.support ((\u2191\u2191\u03bc s)\u207b\u00b9 \u2022 1)) = 0", "state_after": "no goals"}, {"tactic": "have heq : Function.support ((\u03bc s)\u207b\u00b9 \u2022 (1 : E \u2192 \u211d\u22650\u221e)) = Set.univ := by\n  ext x\n  rw [Function.mem_support]\n  simp [hnt]", "annotated_tactic": ["have heq : <a>Function.support</a> ((\u03bc s)\u207b\u00b9 \u2022 (1 : E \u2192 \u211d\u22650\u221e)) = <a>Set.univ</a> := by\n          ext x\n          rw [<a>Function.mem_support</a>]\n          simp [hnt]", [{"full_name": "Function.support", "def_path": "Mathlib/Algebra/Support.lean", "def_pos": [37, 5], "def_end_pos": [37, 12]}, {"full_name": "Set.univ", "def_path": "Mathlib/Init/Set.lean", "def_pos": [90, 5], "def_end_pos": [90, 9]}, {"full_name": "Function.mem_support", "def_path": "Mathlib/Algebra/Support.lean", "def_pos": [65, 3], "def_end_pos": [65, 14]}]], "state_before": "\u03a9 : Type u_1\nE : Type u_2\ninst\u271d : MeasurableSpace E\nm\u271d : MeasurableSpace \u03a9\n\u2119\u271d : Measure \u03a9\n\u03bc\u271d : Measure E\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 E\n\u2119 : Measure \u03a9\n\u03bc : Measure E\ns : Set E\nhns : \u2191\u2191\u03bc s \u2260 0\nhnt : \u2191\u2191\u03bc s \u2260 \u22a4\nhpdf : pdf X \u2119 = 0\nhu : 0 =\u1da0[ae \u03bc] Set.indicator s ((\u2191\u2191\u03bc s)\u207b\u00b9 \u2022 1)\nthis : \u2191\u2191\u03bc (s \u2229 Function.support ((\u2191\u2191\u03bc s)\u207b\u00b9 \u2022 1)) = 0\n\u22a2 False", "state_after": "\u03a9 : Type u_1\nE : Type u_2\ninst\u271d : MeasurableSpace E\nm\u271d : MeasurableSpace \u03a9\n\u2119\u271d : Measure \u03a9\n\u03bc\u271d : Measure E\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 E\n\u2119 : Measure \u03a9\n\u03bc : Measure E\ns : Set E\nhns : \u2191\u2191\u03bc s \u2260 0\nhnt : \u2191\u2191\u03bc s \u2260 \u22a4\nhpdf : pdf X \u2119 = 0\nhu : 0 =\u1da0[ae \u03bc] Set.indicator s ((\u2191\u2191\u03bc s)\u207b\u00b9 \u2022 1)\nthis : \u2191\u2191\u03bc (s \u2229 Function.support ((\u2191\u2191\u03bc s)\u207b\u00b9 \u2022 1)) = 0\nheq : Function.support ((\u2191\u2191\u03bc s)\u207b\u00b9 \u2022 1) = Set.univ\n\u22a2 False"}, {"tactic": "rw [heq, Set.inter_univ] at this", "annotated_tactic": ["rw [heq, <a>Set.inter_univ</a>] at this", [{"full_name": "Set.inter_univ", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1012, 9], "def_end_pos": [1012, 19]}]], "state_before": "\u03a9 : Type u_1\nE : Type u_2\ninst\u271d : MeasurableSpace E\nm\u271d : MeasurableSpace \u03a9\n\u2119\u271d : Measure \u03a9\n\u03bc\u271d : Measure E\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 E\n\u2119 : Measure \u03a9\n\u03bc : Measure E\ns : Set E\nhns : \u2191\u2191\u03bc s \u2260 0\nhnt : \u2191\u2191\u03bc s \u2260 \u22a4\nhpdf : pdf X \u2119 = 0\nhu : 0 =\u1da0[ae \u03bc] Set.indicator s ((\u2191\u2191\u03bc s)\u207b\u00b9 \u2022 1)\nthis : \u2191\u2191\u03bc (s \u2229 Function.support ((\u2191\u2191\u03bc s)\u207b\u00b9 \u2022 1)) = 0\nheq : Function.support ((\u2191\u2191\u03bc s)\u207b\u00b9 \u2022 1) = Set.univ\n\u22a2 False", "state_after": "\u03a9 : Type u_1\nE : Type u_2\ninst\u271d : MeasurableSpace E\nm\u271d : MeasurableSpace \u03a9\n\u2119\u271d : Measure \u03a9\n\u03bc\u271d : Measure E\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 E\n\u2119 : Measure \u03a9\n\u03bc : Measure E\ns : Set E\nhns : \u2191\u2191\u03bc s \u2260 0\nhnt : \u2191\u2191\u03bc s \u2260 \u22a4\nhpdf : pdf X \u2119 = 0\nhu : 0 =\u1da0[ae \u03bc] Set.indicator s ((\u2191\u2191\u03bc s)\u207b\u00b9 \u2022 1)\nthis : \u2191\u2191\u03bc s = 0\nheq : Function.support ((\u2191\u2191\u03bc s)\u207b\u00b9 \u2022 1) = Set.univ\n\u22a2 False"}, {"tactic": "exact hns this", "annotated_tactic": ["exact hns this", []], "state_before": "\u03a9 : Type u_1\nE : Type u_2\ninst\u271d : MeasurableSpace E\nm\u271d : MeasurableSpace \u03a9\n\u2119\u271d : Measure \u03a9\n\u03bc\u271d : Measure E\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 E\n\u2119 : Measure \u03a9\n\u03bc : Measure E\ns : Set E\nhns : \u2191\u2191\u03bc s \u2260 0\nhnt : \u2191\u2191\u03bc s \u2260 \u22a4\nhpdf : pdf X \u2119 = 0\nhu : 0 =\u1da0[ae \u03bc] Set.indicator s ((\u2191\u2191\u03bc s)\u207b\u00b9 \u2022 1)\nthis : \u2191\u2191\u03bc s = 0\nheq : Function.support ((\u2191\u2191\u03bc s)\u207b\u00b9 \u2022 1) = Set.univ\n\u22a2 False", "state_after": "no goals"}, {"tactic": "ext x", "annotated_tactic": ["ext x", []], "state_before": "\u03a9 : Type u_1\nE : Type u_2\ninst\u271d : MeasurableSpace E\nm\u271d : MeasurableSpace \u03a9\n\u2119\u271d : Measure \u03a9\n\u03bc\u271d : Measure E\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 E\n\u2119 : Measure \u03a9\n\u03bc : Measure E\ns : Set E\nhns : \u2191\u2191\u03bc s \u2260 0\nhnt : \u2191\u2191\u03bc s \u2260 \u22a4\nhpdf : pdf X \u2119 = 0\nhu : 0 =\u1da0[ae \u03bc] Set.indicator s ((\u2191\u2191\u03bc s)\u207b\u00b9 \u2022 1)\nthis : \u2191\u2191\u03bc (s \u2229 Function.support ((\u2191\u2191\u03bc s)\u207b\u00b9 \u2022 1)) = 0\n\u22a2 Function.support ((\u2191\u2191\u03bc s)\u207b\u00b9 \u2022 1) = Set.univ", "state_after": "case h\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d : MeasurableSpace E\nm\u271d : MeasurableSpace \u03a9\n\u2119\u271d : Measure \u03a9\n\u03bc\u271d : Measure E\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 E\n\u2119 : Measure \u03a9\n\u03bc : Measure E\ns : Set E\nhns : \u2191\u2191\u03bc s \u2260 0\nhnt : \u2191\u2191\u03bc s \u2260 \u22a4\nhpdf : pdf X \u2119 = 0\nhu : 0 =\u1da0[ae \u03bc] Set.indicator s ((\u2191\u2191\u03bc s)\u207b\u00b9 \u2022 1)\nthis : \u2191\u2191\u03bc (s \u2229 Function.support ((\u2191\u2191\u03bc s)\u207b\u00b9 \u2022 1)) = 0\nx : E\n\u22a2 x \u2208 Function.support ((\u2191\u2191\u03bc s)\u207b\u00b9 \u2022 1) \u2194 x \u2208 Set.univ"}, {"tactic": "rw [Function.mem_support]", "annotated_tactic": ["rw [<a>Function.mem_support</a>]", [{"full_name": "Function.mem_support", "def_path": "Mathlib/Algebra/Support.lean", "def_pos": [65, 3], "def_end_pos": [65, 14]}]], "state_before": "case h\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d : MeasurableSpace E\nm\u271d : MeasurableSpace \u03a9\n\u2119\u271d : Measure \u03a9\n\u03bc\u271d : Measure E\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 E\n\u2119 : Measure \u03a9\n\u03bc : Measure E\ns : Set E\nhns : \u2191\u2191\u03bc s \u2260 0\nhnt : \u2191\u2191\u03bc s \u2260 \u22a4\nhpdf : pdf X \u2119 = 0\nhu : 0 =\u1da0[ae \u03bc] Set.indicator s ((\u2191\u2191\u03bc s)\u207b\u00b9 \u2022 1)\nthis : \u2191\u2191\u03bc (s \u2229 Function.support ((\u2191\u2191\u03bc s)\u207b\u00b9 \u2022 1)) = 0\nx : E\n\u22a2 x \u2208 Function.support ((\u2191\u2191\u03bc s)\u207b\u00b9 \u2022 1) \u2194 x \u2208 Set.univ", "state_after": "case h\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d : MeasurableSpace E\nm\u271d : MeasurableSpace \u03a9\n\u2119\u271d : Measure \u03a9\n\u03bc\u271d : Measure E\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 E\n\u2119 : Measure \u03a9\n\u03bc : Measure E\ns : Set E\nhns : \u2191\u2191\u03bc s \u2260 0\nhnt : \u2191\u2191\u03bc s \u2260 \u22a4\nhpdf : pdf X \u2119 = 0\nhu : 0 =\u1da0[ae \u03bc] Set.indicator s ((\u2191\u2191\u03bc s)\u207b\u00b9 \u2022 1)\nthis : \u2191\u2191\u03bc (s \u2229 Function.support ((\u2191\u2191\u03bc s)\u207b\u00b9 \u2022 1)) = 0\nx : E\n\u22a2 ((\u2191\u2191\u03bc s)\u207b\u00b9 \u2022 1) x \u2260 0 \u2194 x \u2208 Set.univ"}, {"tactic": "simp [hnt]", "annotated_tactic": ["simp [hnt]", []], "state_before": "case h\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d : MeasurableSpace E\nm\u271d : MeasurableSpace \u03a9\n\u2119\u271d : Measure \u03a9\n\u03bc\u271d : Measure E\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 E\n\u2119 : Measure \u03a9\n\u03bc : Measure E\ns : Set E\nhns : \u2191\u2191\u03bc s \u2260 0\nhnt : \u2191\u2191\u03bc s \u2260 \u22a4\nhpdf : pdf X \u2119 = 0\nhu : 0 =\u1da0[ae \u03bc] Set.indicator s ((\u2191\u2191\u03bc s)\u207b\u00b9 \u2022 1)\nthis : \u2191\u2191\u03bc (s \u2229 Function.support ((\u2191\u2191\u03bc s)\u207b\u00b9 \u2022 1)) = 0\nx : E\n\u22a2 ((\u2191\u2191\u03bc s)\u207b\u00b9 \u2022 1) x \u2260 0 \u2194 x \u2208 Set.univ", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Intervals/Disjoint.lean", "full_name": "iUnion_Ici_eq_Ioi_iInf", "start": [234, 1], "end": [237, 38], "traced_tactics": [{"tactic": "simp only [\u2190 IsGLB.biUnion_Ici_eq_Ioi (@isGLB_iInf _ _ _ f) no_least_elem, mem_range,\n  iUnion_exists, iUnion_iUnion_eq']", "annotated_tactic": ["simp only [\u2190 <a>IsGLB.biUnion_Ici_eq_Ioi</a> (@<a>isGLB_iInf</a> _ _ _ f) no_least_elem, <a>mem_range</a>,\n    <a>iUnion_exists</a>, <a>iUnion_iUnion_eq'</a>]", [{"full_name": "IsGLB.biUnion_Ici_eq_Ioi", "def_path": "Mathlib/Data/Set/Intervals/Disjoint.lean", "def_pos": [208, 9], "def_end_pos": [208, 33]}, {"full_name": "isGLB_iInf", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [837, 9], "def_end_pos": [837, 19]}, {"full_name": "Set.mem_range", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [673, 9], "def_end_pos": [673, 18]}, {"full_name": "Set.iUnion_exists", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [804, 9], "def_end_pos": [804, 22]}, {"full_name": "Set.iUnion_iUnion_eq'", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [1822, 9], "def_end_pos": [1822, 26]}]], "state_before": "\u03b9 : Sort u\n\u03b1 : Type v\n\u03b2 : Type w\ninst\u271d\u00b9 : LinearOrder \u03b1\ns : Set \u03b1\na : \u03b1\nf\u271d : \u03b9 \u2192 \u03b1\nR : Type u_1\ninst\u271d : CompleteLinearOrder R\nf : \u03b9 \u2192 R\nno_least_elem : \u00ac\u2a05 i, f i \u2208 range f\n\u22a2 \u22c3 i, Ici (f i) = Ioi (\u2a05 i, f i)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "full_name": "MeasureTheory.Mem\u2112p.mem\u2112p_of_exponent_le", "start": [1186, 1], "end": [1212, 98], "traced_tactics": [{"tactic": "cases' hfq with hfq_m hfq_lt_top", "annotated_tactic": ["cases' hfq with hfq_m hfq_lt_top", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\np q : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 E\nhfq : Mem\u2112p f q\nhpq : p \u2264 q\n\u22a2 Mem\u2112p f p", "state_after": "case intro\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\np q : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 E\nhpq : p \u2264 q\nhfq_m : AEStronglyMeasurable f \u03bc\nhfq_lt_top : snorm f q \u03bc < \u22a4\n\u22a2 Mem\u2112p f p"}, {"tactic": "by_cases hp0 : p = 0", "annotated_tactic": ["by_cases hp0 : p = 0", []], "state_before": "case intro\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\np q : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 E\nhpq : p \u2264 q\nhfq_m : AEStronglyMeasurable f \u03bc\nhfq_lt_top : snorm f q \u03bc < \u22a4\n\u22a2 Mem\u2112p f p", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\np q : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 E\nhpq : p \u2264 q\nhfq_m : AEStronglyMeasurable f \u03bc\nhfq_lt_top : snorm f q \u03bc < \u22a4\nhp0 : p = 0\n\u22a2 Mem\u2112p f p\n\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\np q : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 E\nhpq : p \u2264 q\nhfq_m : AEStronglyMeasurable f \u03bc\nhfq_lt_top : snorm f q \u03bc < \u22a4\nhp0 : \u00acp = 0\n\u22a2 Mem\u2112p f p"}, {"tactic": "rw [\u2190 Ne.def] at hp0", "annotated_tactic": ["rw [\u2190 <a>Ne.def</a>] at hp0", [{"full_name": "Ne.def", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [59, 9], "def_end_pos": [59, 15]}]], "state_before": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\np q : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 E\nhpq : p \u2264 q\nhfq_m : AEStronglyMeasurable f \u03bc\nhfq_lt_top : snorm f q \u03bc < \u22a4\nhp0 : \u00acp = 0\n\u22a2 Mem\u2112p f p", "state_after": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\np q : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 E\nhpq : p \u2264 q\nhfq_m : AEStronglyMeasurable f \u03bc\nhfq_lt_top : snorm f q \u03bc < \u22a4\nhp0 : p \u2260 0\n\u22a2 Mem\u2112p f p"}, {"tactic": "refine' \u27e8hfq_m, _\u27e9", "annotated_tactic": ["refine' \u27e8hfq_m, _\u27e9", []], "state_before": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\np q : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 E\nhpq : p \u2264 q\nhfq_m : AEStronglyMeasurable f \u03bc\nhfq_lt_top : snorm f q \u03bc < \u22a4\nhp0 : p \u2260 0\n\u22a2 Mem\u2112p f p", "state_after": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\np q : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 E\nhpq : p \u2264 q\nhfq_m : AEStronglyMeasurable f \u03bc\nhfq_lt_top : snorm f q \u03bc < \u22a4\nhp0 : p \u2260 0\n\u22a2 snorm f p \u03bc < \u22a4"}, {"tactic": "by_cases hp_top : p = \u221e", "annotated_tactic": ["by_cases hp_top : p = \u221e", []], "state_before": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\np q : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 E\nhpq : p \u2264 q\nhfq_m : AEStronglyMeasurable f \u03bc\nhfq_lt_top : snorm f q \u03bc < \u22a4\nhp0 : p \u2260 0\n\u22a2 snorm f p \u03bc < \u22a4", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\np q : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 E\nhpq : p \u2264 q\nhfq_m : AEStronglyMeasurable f \u03bc\nhfq_lt_top : snorm f q \u03bc < \u22a4\nhp0 : p \u2260 0\nhp_top : p = \u22a4\n\u22a2 snorm f p \u03bc < \u22a4\n\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\np q : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 E\nhpq : p \u2264 q\nhfq_m : AEStronglyMeasurable f \u03bc\nhfq_lt_top : snorm f q \u03bc < \u22a4\nhp0 : p \u2260 0\nhp_top : \u00acp = \u22a4\n\u22a2 snorm f p \u03bc < \u22a4"}, {"tactic": "have hp_pos : 0 < p.toReal := ENNReal.toReal_pos hp0 hp_top", "annotated_tactic": ["have hp_pos : 0 < p.toReal := <a>ENNReal.toReal_pos</a> hp0 hp_top", [{"full_name": "ENNReal.toReal_pos", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2131, 9], "def_end_pos": [2131, 19]}]], "state_before": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\np q : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 E\nhpq : p \u2264 q\nhfq_m : AEStronglyMeasurable f \u03bc\nhfq_lt_top : snorm f q \u03bc < \u22a4\nhp0 : p \u2260 0\nhp_top : \u00acp = \u22a4\n\u22a2 snorm f p \u03bc < \u22a4", "state_after": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\np q : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 E\nhpq : p \u2264 q\nhfq_m : AEStronglyMeasurable f \u03bc\nhfq_lt_top : snorm f q \u03bc < \u22a4\nhp0 : p \u2260 0\nhp_top : \u00acp = \u22a4\nhp_pos : 0 < ENNReal.toReal p\n\u22a2 snorm f p \u03bc < \u22a4"}, {"tactic": "by_cases hq_top : q = \u221e", "annotated_tactic": ["by_cases hq_top : q = \u221e", []], "state_before": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\np q : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 E\nhpq : p \u2264 q\nhfq_m : AEStronglyMeasurable f \u03bc\nhfq_lt_top : snorm f q \u03bc < \u22a4\nhp0 : p \u2260 0\nhp_top : \u00acp = \u22a4\nhp_pos : 0 < ENNReal.toReal p\n\u22a2 snorm f p \u03bc < \u22a4", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\np q : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 E\nhpq : p \u2264 q\nhfq_m : AEStronglyMeasurable f \u03bc\nhfq_lt_top : snorm f q \u03bc < \u22a4\nhp0 : p \u2260 0\nhp_top : \u00acp = \u22a4\nhp_pos : 0 < ENNReal.toReal p\nhq_top : q = \u22a4\n\u22a2 snorm f p \u03bc < \u22a4\n\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\np q : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 E\nhpq : p \u2264 q\nhfq_m : AEStronglyMeasurable f \u03bc\nhfq_lt_top : snorm f q \u03bc < \u22a4\nhp0 : p \u2260 0\nhp_top : \u00acp = \u22a4\nhp_pos : 0 < ENNReal.toReal p\nhq_top : \u00acq = \u22a4\n\u22a2 snorm f p \u03bc < \u22a4"}, {"tactic": "have hq0 : q \u2260 0 := by\n  by_contra hq_eq_zero\n  have hp_eq_zero : p = 0 := le_antisymm (by rwa [hq_eq_zero] at hpq) (zero_le _)\n  rw [hp_eq_zero, ENNReal.zero_toReal] at hp_pos\n  exact (lt_irrefl _) hp_pos", "annotated_tactic": ["have hq0 : q \u2260 0 := by\n    by_contra hq_eq_zero\n    have hp_eq_zero : p = 0 := <a>le_antisymm</a> (by rwa [hq_eq_zero] at hpq) (<a>zero_le</a> _)\n    rw [hp_eq_zero, <a>ENNReal.zero_toReal</a>] at hp_pos\n    exact (<a>lt_irrefl</a> _) hp_pos", [{"full_name": "le_antisymm", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [188, 9], "def_end_pos": [188, 20]}, {"full_name": "zero_le", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [217, 30], "def_end_pos": [217, 37]}, {"full_name": "ENNReal.zero_toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [242, 17], "def_end_pos": [242, 28]}, {"full_name": "lt_irrefl", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [79, 9], "def_end_pos": [79, 18]}]], "state_before": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\np q : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 E\nhpq : p \u2264 q\nhfq_m : AEStronglyMeasurable f \u03bc\nhfq_lt_top : snorm f q \u03bc < \u22a4\nhp0 : p \u2260 0\nhp_top : \u00acp = \u22a4\nhp_pos : 0 < ENNReal.toReal p\nhq_top : \u00acq = \u22a4\n\u22a2 snorm f p \u03bc < \u22a4", "state_after": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\np q : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 E\nhpq : p \u2264 q\nhfq_m : AEStronglyMeasurable f \u03bc\nhfq_lt_top : snorm f q \u03bc < \u22a4\nhp0 : p \u2260 0\nhp_top : \u00acp = \u22a4\nhp_pos : 0 < ENNReal.toReal p\nhq_top : \u00acq = \u22a4\nhq0 : q \u2260 0\n\u22a2 snorm f p \u03bc < \u22a4"}, {"tactic": "have hpq_real : p.toReal \u2264 q.toReal := by rwa [ENNReal.toReal_le_toReal hp_top hq_top]", "annotated_tactic": ["have hpq_real : p.toReal \u2264 q.toReal := by rwa [<a>ENNReal.toReal_le_toReal</a> hp_top hq_top]", [{"full_name": "ENNReal.toReal_le_toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2036, 9], "def_end_pos": [2036, 25]}]], "state_before": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\np q : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 E\nhpq : p \u2264 q\nhfq_m : AEStronglyMeasurable f \u03bc\nhfq_lt_top : snorm f q \u03bc < \u22a4\nhp0 : p \u2260 0\nhp_top : \u00acp = \u22a4\nhp_pos : 0 < ENNReal.toReal p\nhq_top : \u00acq = \u22a4\nhq0 : q \u2260 0\n\u22a2 snorm f p \u03bc < \u22a4", "state_after": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\np q : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 E\nhpq : p \u2264 q\nhfq_m : AEStronglyMeasurable f \u03bc\nhfq_lt_top : snorm f q \u03bc < \u22a4\nhp0 : p \u2260 0\nhp_top : \u00acp = \u22a4\nhp_pos : 0 < ENNReal.toReal p\nhq_top : \u00acq = \u22a4\nhq0 : q \u2260 0\nhpq_real : ENNReal.toReal p \u2264 ENNReal.toReal q\n\u22a2 snorm f p \u03bc < \u22a4"}, {"tactic": "rw [snorm_eq_snorm' hp0 hp_top]", "annotated_tactic": ["rw [<a>snorm_eq_snorm'</a> hp0 hp_top]", [{"full_name": "MeasureTheory.snorm_eq_snorm'", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [88, 9], "def_end_pos": [88, 24]}]], "state_before": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\np q : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 E\nhpq : p \u2264 q\nhfq_m : AEStronglyMeasurable f \u03bc\nhfq_lt_top : snorm f q \u03bc < \u22a4\nhp0 : p \u2260 0\nhp_top : \u00acp = \u22a4\nhp_pos : 0 < ENNReal.toReal p\nhq_top : \u00acq = \u22a4\nhq0 : q \u2260 0\nhpq_real : ENNReal.toReal p \u2264 ENNReal.toReal q\n\u22a2 snorm f p \u03bc < \u22a4", "state_after": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\np q : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 E\nhpq : p \u2264 q\nhfq_m : AEStronglyMeasurable f \u03bc\nhfq_lt_top : snorm f q \u03bc < \u22a4\nhp0 : p \u2260 0\nhp_top : \u00acp = \u22a4\nhp_pos : 0 < ENNReal.toReal p\nhq_top : \u00acq = \u22a4\nhq0 : q \u2260 0\nhpq_real : ENNReal.toReal p \u2264 ENNReal.toReal q\n\u22a2 snorm' f (ENNReal.toReal p) \u03bc < \u22a4"}, {"tactic": "rw [snorm_eq_snorm' hq0 hq_top] at hfq_lt_top", "annotated_tactic": ["rw [<a>snorm_eq_snorm'</a> hq0 hq_top] at hfq_lt_top", [{"full_name": "MeasureTheory.snorm_eq_snorm'", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [88, 9], "def_end_pos": [88, 24]}]], "state_before": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\np q : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 E\nhpq : p \u2264 q\nhfq_m : AEStronglyMeasurable f \u03bc\nhfq_lt_top : snorm f q \u03bc < \u22a4\nhp0 : p \u2260 0\nhp_top : \u00acp = \u22a4\nhp_pos : 0 < ENNReal.toReal p\nhq_top : \u00acq = \u22a4\nhq0 : q \u2260 0\nhpq_real : ENNReal.toReal p \u2264 ENNReal.toReal q\n\u22a2 snorm' f (ENNReal.toReal p) \u03bc < \u22a4", "state_after": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\np q : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 E\nhpq : p \u2264 q\nhfq_m : AEStronglyMeasurable f \u03bc\nhfq_lt_top : snorm' f (ENNReal.toReal q) \u03bc < \u22a4\nhp0 : p \u2260 0\nhp_top : \u00acp = \u22a4\nhp_pos : 0 < ENNReal.toReal p\nhq_top : \u00acq = \u22a4\nhq0 : q \u2260 0\nhpq_real : ENNReal.toReal p \u2264 ENNReal.toReal q\n\u22a2 snorm' f (ENNReal.toReal p) \u03bc < \u22a4"}, {"tactic": "exact snorm'_lt_top_of_snorm'_lt_top_of_exponent_le hfq_m hfq_lt_top (le_of_lt hp_pos) hpq_real", "annotated_tactic": ["exact <a>snorm'_lt_top_of_snorm'_lt_top_of_exponent_le</a> hfq_m hfq_lt_top (<a>le_of_lt</a> hp_pos) hpq_real", [{"full_name": "MeasureTheory.snorm'_lt_top_of_snorm'_lt_top_of_exponent_le", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [1123, 9], "def_end_pos": [1123, 54]}, {"full_name": "le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [110, 9], "def_end_pos": [110, 17]}]], "state_before": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\np q : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 E\nhpq : p \u2264 q\nhfq_m : AEStronglyMeasurable f \u03bc\nhfq_lt_top : snorm' f (ENNReal.toReal q) \u03bc < \u22a4\nhp0 : p \u2260 0\nhp_top : \u00acp = \u22a4\nhp_pos : 0 < ENNReal.toReal p\nhq_top : \u00acq = \u22a4\nhq0 : q \u2260 0\nhpq_real : ENNReal.toReal p \u2264 ENNReal.toReal q\n\u22a2 snorm' f (ENNReal.toReal p) \u03bc < \u22a4", "state_after": "no goals"}, {"tactic": "rwa [hp0, mem\u2112p_zero_iff_aestronglyMeasurable]", "annotated_tactic": ["rwa [hp0, <a>mem\u2112p_zero_iff_aestronglyMeasurable</a>]", [{"full_name": "MeasureTheory.mem\u2112p_zero_iff_aestronglyMeasurable", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [179, 9], "def_end_pos": [179, 44]}]], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\np q : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 E\nhpq : p \u2264 q\nhfq_m : AEStronglyMeasurable f \u03bc\nhfq_lt_top : snorm f q \u03bc < \u22a4\nhp0 : p = 0\n\u22a2 Mem\u2112p f p", "state_after": "no goals"}, {"tactic": "have hq_top : q = \u221e := by rwa [hp_top, top_le_iff] at hpq", "annotated_tactic": ["have hq_top : q = \u221e := by rwa [hp_top, <a>top_le_iff</a>] at hpq", [{"full_name": "top_le_iff", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [157, 9], "def_end_pos": [157, 19]}]], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\np q : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 E\nhpq : p \u2264 q\nhfq_m : AEStronglyMeasurable f \u03bc\nhfq_lt_top : snorm f q \u03bc < \u22a4\nhp0 : p \u2260 0\nhp_top : p = \u22a4\n\u22a2 snorm f p \u03bc < \u22a4", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\np q : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 E\nhpq : p \u2264 q\nhfq_m : AEStronglyMeasurable f \u03bc\nhfq_lt_top : snorm f q \u03bc < \u22a4\nhp0 : p \u2260 0\nhp_top : p = \u22a4\nhq_top : q = \u22a4\n\u22a2 snorm f p \u03bc < \u22a4"}, {"tactic": "rw [hp_top]", "annotated_tactic": ["rw [hp_top]", []], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\np q : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 E\nhpq : p \u2264 q\nhfq_m : AEStronglyMeasurable f \u03bc\nhfq_lt_top : snorm f q \u03bc < \u22a4\nhp0 : p \u2260 0\nhp_top : p = \u22a4\nhq_top : q = \u22a4\n\u22a2 snorm f p \u03bc < \u22a4", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\np q : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 E\nhpq : p \u2264 q\nhfq_m : AEStronglyMeasurable f \u03bc\nhfq_lt_top : snorm f q \u03bc < \u22a4\nhp0 : p \u2260 0\nhp_top : p = \u22a4\nhq_top : q = \u22a4\n\u22a2 snorm f \u22a4 \u03bc < \u22a4"}, {"tactic": "rwa [hq_top] at hfq_lt_top", "annotated_tactic": ["rwa [hq_top] at hfq_lt_top", []], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\np q : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 E\nhpq : p \u2264 q\nhfq_m : AEStronglyMeasurable f \u03bc\nhfq_lt_top : snorm f q \u03bc < \u22a4\nhp0 : p \u2260 0\nhp_top : p = \u22a4\nhq_top : q = \u22a4\n\u22a2 snorm f \u22a4 \u03bc < \u22a4", "state_after": "no goals"}, {"tactic": "rwa [hp_top, top_le_iff] at hpq", "annotated_tactic": ["rwa [hp_top, <a>top_le_iff</a>] at hpq", [{"full_name": "top_le_iff", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [157, 9], "def_end_pos": [157, 19]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\np q : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 E\nhpq : p \u2264 q\nhfq_m : AEStronglyMeasurable f \u03bc\nhfq_lt_top : snorm f q \u03bc < \u22a4\nhp0 : p \u2260 0\nhp_top : p = \u22a4\n\u22a2 q = \u22a4", "state_after": "no goals"}, {"tactic": "rw [snorm_eq_snorm' hp0 hp_top]", "annotated_tactic": ["rw [<a>snorm_eq_snorm'</a> hp0 hp_top]", [{"full_name": "MeasureTheory.snorm_eq_snorm'", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [88, 9], "def_end_pos": [88, 24]}]], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\np q : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 E\nhpq : p \u2264 q\nhfq_m : AEStronglyMeasurable f \u03bc\nhfq_lt_top : snorm f q \u03bc < \u22a4\nhp0 : p \u2260 0\nhp_top : \u00acp = \u22a4\nhp_pos : 0 < ENNReal.toReal p\nhq_top : q = \u22a4\n\u22a2 snorm f p \u03bc < \u22a4", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\np q : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 E\nhpq : p \u2264 q\nhfq_m : AEStronglyMeasurable f \u03bc\nhfq_lt_top : snorm f q \u03bc < \u22a4\nhp0 : p \u2260 0\nhp_top : \u00acp = \u22a4\nhp_pos : 0 < ENNReal.toReal p\nhq_top : q = \u22a4\n\u22a2 snorm' f (ENNReal.toReal p) \u03bc < \u22a4"}, {"tactic": "rw [hq_top, snorm_exponent_top] at hfq_lt_top", "annotated_tactic": ["rw [hq_top, <a>snorm_exponent_top</a>] at hfq_lt_top", [{"full_name": "MeasureTheory.snorm_exponent_top", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [103, 9], "def_end_pos": [103, 27]}]], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\np q : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 E\nhpq : p \u2264 q\nhfq_m : AEStronglyMeasurable f \u03bc\nhfq_lt_top : snorm f q \u03bc < \u22a4\nhp0 : p \u2260 0\nhp_top : \u00acp = \u22a4\nhp_pos : 0 < ENNReal.toReal p\nhq_top : q = \u22a4\n\u22a2 snorm' f (ENNReal.toReal p) \u03bc < \u22a4", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\np q : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 E\nhpq : p \u2264 q\nhfq_m : AEStronglyMeasurable f \u03bc\nhfq_lt_top : snormEssSup f \u03bc < \u22a4\nhp0 : p \u2260 0\nhp_top : \u00acp = \u22a4\nhp_pos : 0 < ENNReal.toReal p\nhq_top : q = \u22a4\n\u22a2 snorm' f (ENNReal.toReal p) \u03bc < \u22a4"}, {"tactic": "refine' lt_of_le_of_lt (snorm'_le_snormEssSup_mul_rpow_measure_univ hp_pos) _", "annotated_tactic": ["refine' <a>lt_of_le_of_lt</a> (<a>snorm'_le_snormEssSup_mul_rpow_measure_univ</a> hp_pos) _", [{"full_name": "lt_of_le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [122, 9], "def_end_pos": [122, 23]}, {"full_name": "MeasureTheory.snorm'_le_snormEssSup_mul_rpow_measure_univ", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [1066, 9], "def_end_pos": [1066, 52]}]], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\np q : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 E\nhpq : p \u2264 q\nhfq_m : AEStronglyMeasurable f \u03bc\nhfq_lt_top : snormEssSup f \u03bc < \u22a4\nhp0 : p \u2260 0\nhp_top : \u00acp = \u22a4\nhp_pos : 0 < ENNReal.toReal p\nhq_top : q = \u22a4\n\u22a2 snorm' f (ENNReal.toReal p) \u03bc < \u22a4", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\np q : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 E\nhpq : p \u2264 q\nhfq_m : AEStronglyMeasurable f \u03bc\nhfq_lt_top : snormEssSup f \u03bc < \u22a4\nhp0 : p \u2260 0\nhp_top : \u00acp = \u22a4\nhp_pos : 0 < ENNReal.toReal p\nhq_top : q = \u22a4\n\u22a2 snormEssSup f \u03bc * \u2191\u2191\u03bc Set.univ ^ (1 / ENNReal.toReal p) < \u22a4"}, {"tactic": "refine' ENNReal.mul_lt_top hfq_lt_top.ne _", "annotated_tactic": ["refine' <a>ENNReal.mul_lt_top</a> hfq_lt_top.ne _", [{"full_name": "ENNReal.mul_lt_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [612, 9], "def_end_pos": [612, 19]}]], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\np q : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 E\nhpq : p \u2264 q\nhfq_m : AEStronglyMeasurable f \u03bc\nhfq_lt_top : snormEssSup f \u03bc < \u22a4\nhp0 : p \u2260 0\nhp_top : \u00acp = \u22a4\nhp_pos : 0 < ENNReal.toReal p\nhq_top : q = \u22a4\n\u22a2 snormEssSup f \u03bc * \u2191\u2191\u03bc Set.univ ^ (1 / ENNReal.toReal p) < \u22a4", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\np q : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 E\nhpq : p \u2264 q\nhfq_m : AEStronglyMeasurable f \u03bc\nhfq_lt_top : snormEssSup f \u03bc < \u22a4\nhp0 : p \u2260 0\nhp_top : \u00acp = \u22a4\nhp_pos : 0 < ENNReal.toReal p\nhq_top : q = \u22a4\n\u22a2 \u2191\u2191\u03bc Set.univ ^ (1 / ENNReal.toReal p) \u2260 \u22a4"}, {"tactic": "exact (ENNReal.rpow_lt_top_of_nonneg (by simp [hp_pos.le]) (measure_ne_top \u03bc Set.univ)).ne", "annotated_tactic": ["exact (<a>ENNReal.rpow_lt_top_of_nonneg</a> (by simp [hp_pos.le]) (<a>measure_ne_top</a> \u03bc <a>Set.univ</a>)).<a>ne</a>", [{"full_name": "ENNReal.rpow_lt_top_of_nonneg", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [503, 9], "def_end_pos": [503, 30]}, {"full_name": "MeasureTheory.measure_ne_top", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2875, 9], "def_end_pos": [2875, 23]}, {"full_name": "Set.univ", "def_path": "Mathlib/Init/Set.lean", "def_pos": [90, 5], "def_end_pos": [90, 9]}, {"full_name": "LT.lt.ne", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [152, 7], "def_end_pos": [152, 15]}]], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\np q : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 E\nhpq : p \u2264 q\nhfq_m : AEStronglyMeasurable f \u03bc\nhfq_lt_top : snormEssSup f \u03bc < \u22a4\nhp0 : p \u2260 0\nhp_top : \u00acp = \u22a4\nhp_pos : 0 < ENNReal.toReal p\nhq_top : q = \u22a4\n\u22a2 \u2191\u2191\u03bc Set.univ ^ (1 / ENNReal.toReal p) \u2260 \u22a4", "state_after": "no goals"}, {"tactic": "simp [hp_pos.le]", "annotated_tactic": ["simp [hp_pos.le]", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\np q : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 E\nhpq : p \u2264 q\nhfq_m : AEStronglyMeasurable f \u03bc\nhfq_lt_top : snormEssSup f \u03bc < \u22a4\nhp0 : p \u2260 0\nhp_top : \u00acp = \u22a4\nhp_pos : 0 < ENNReal.toReal p\nhq_top : q = \u22a4\n\u22a2 0 \u2264 1 / ENNReal.toReal p", "state_after": "no goals"}, {"tactic": "by_contra hq_eq_zero", "annotated_tactic": ["by_contra hq_eq_zero", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\np q : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 E\nhpq : p \u2264 q\nhfq_m : AEStronglyMeasurable f \u03bc\nhfq_lt_top : snorm f q \u03bc < \u22a4\nhp0 : p \u2260 0\nhp_top : \u00acp = \u22a4\nhp_pos : 0 < ENNReal.toReal p\nhq_top : \u00acq = \u22a4\n\u22a2 q \u2260 0", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\np q : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 E\nhpq : p \u2264 q\nhfq_m : AEStronglyMeasurable f \u03bc\nhfq_lt_top : snorm f q \u03bc < \u22a4\nhp0 : p \u2260 0\nhp_top : \u00acp = \u22a4\nhp_pos : 0 < ENNReal.toReal p\nhq_top : \u00acq = \u22a4\nhq_eq_zero : q = 0\n\u22a2 False"}, {"tactic": "have hp_eq_zero : p = 0 := le_antisymm (by rwa [hq_eq_zero] at hpq) (zero_le _)", "annotated_tactic": ["have hp_eq_zero : p = 0 := <a>le_antisymm</a> (by rwa [hq_eq_zero] at hpq) (<a>zero_le</a> _)", [{"full_name": "le_antisymm", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [188, 9], "def_end_pos": [188, 20]}, {"full_name": "zero_le", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [217, 30], "def_end_pos": [217, 37]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\np q : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 E\nhpq : p \u2264 q\nhfq_m : AEStronglyMeasurable f \u03bc\nhfq_lt_top : snorm f q \u03bc < \u22a4\nhp0 : p \u2260 0\nhp_top : \u00acp = \u22a4\nhp_pos : 0 < ENNReal.toReal p\nhq_top : \u00acq = \u22a4\nhq_eq_zero : q = 0\n\u22a2 False", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\np q : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 E\nhpq : p \u2264 q\nhfq_m : AEStronglyMeasurable f \u03bc\nhfq_lt_top : snorm f q \u03bc < \u22a4\nhp0 : p \u2260 0\nhp_top : \u00acp = \u22a4\nhp_pos : 0 < ENNReal.toReal p\nhq_top : \u00acq = \u22a4\nhq_eq_zero : q = 0\nhp_eq_zero : p = 0\n\u22a2 False"}, {"tactic": "rw [hp_eq_zero, ENNReal.zero_toReal] at hp_pos", "annotated_tactic": ["rw [hp_eq_zero, <a>ENNReal.zero_toReal</a>] at hp_pos", [{"full_name": "ENNReal.zero_toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [242, 17], "def_end_pos": [242, 28]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\np q : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 E\nhpq : p \u2264 q\nhfq_m : AEStronglyMeasurable f \u03bc\nhfq_lt_top : snorm f q \u03bc < \u22a4\nhp0 : p \u2260 0\nhp_top : \u00acp = \u22a4\nhp_pos : 0 < ENNReal.toReal p\nhq_top : \u00acq = \u22a4\nhq_eq_zero : q = 0\nhp_eq_zero : p = 0\n\u22a2 False", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\np q : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 E\nhpq : p \u2264 q\nhfq_m : AEStronglyMeasurable f \u03bc\nhfq_lt_top : snorm f q \u03bc < \u22a4\nhp0 : p \u2260 0\nhp_top : \u00acp = \u22a4\nhp_pos : 0 < 0\nhq_top : \u00acq = \u22a4\nhq_eq_zero : q = 0\nhp_eq_zero : p = 0\n\u22a2 False"}, {"tactic": "exact (lt_irrefl _) hp_pos", "annotated_tactic": ["exact (<a>lt_irrefl</a> _) hp_pos", [{"full_name": "lt_irrefl", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [79, 9], "def_end_pos": [79, 18]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\np q : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 E\nhpq : p \u2264 q\nhfq_m : AEStronglyMeasurable f \u03bc\nhfq_lt_top : snorm f q \u03bc < \u22a4\nhp0 : p \u2260 0\nhp_top : \u00acp = \u22a4\nhp_pos : 0 < 0\nhq_top : \u00acq = \u22a4\nhq_eq_zero : q = 0\nhp_eq_zero : p = 0\n\u22a2 False", "state_after": "no goals"}, {"tactic": "rwa [hq_eq_zero] at hpq", "annotated_tactic": ["rwa [hq_eq_zero] at hpq", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\np q : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 E\nhpq : p \u2264 q\nhfq_m : AEStronglyMeasurable f \u03bc\nhfq_lt_top : snorm f q \u03bc < \u22a4\nhp0 : p \u2260 0\nhp_top : \u00acp = \u22a4\nhp_pos : 0 < ENNReal.toReal p\nhq_top : \u00acq = \u22a4\nhq_eq_zero : q = 0\n\u22a2 p \u2264 0", "state_after": "no goals"}, {"tactic": "rwa [ENNReal.toReal_le_toReal hp_top hq_top]", "annotated_tactic": ["rwa [<a>ENNReal.toReal_le_toReal</a> hp_top hq_top]", [{"full_name": "ENNReal.toReal_le_toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2036, 9], "def_end_pos": [2036, 25]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\np q : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 E\nhpq : p \u2264 q\nhfq_m : AEStronglyMeasurable f \u03bc\nhfq_lt_top : snorm f q \u03bc < \u22a4\nhp0 : p \u2260 0\nhp_top : \u00acp = \u22a4\nhp_pos : 0 < ENNReal.toReal p\nhq_top : \u00acq = \u22a4\nhq0 : q \u2260 0\n\u22a2 ENNReal.toReal p \u2264 ENNReal.toReal q", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/Average.lean", "full_name": "MeasureTheory.average_union", "start": [343, 1], "end": [349, 100], "traced_tactics": [{"tactic": "haveI := Fact.mk hs\u03bc.lt_top", "annotated_tactic": ["haveI := <a>Fact.mk</a> hs\u03bc.lt_top", [{"full_name": "Fact.mk", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [115, 12], "def_end_pos": [115, 22]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns\u271d t\u271d : Set \u03b1\nf\u271d g f : \u03b1 \u2192 E\ns t : Set \u03b1\nhd : AEDisjoint \u03bc s t\nht : NullMeasurableSet t\nhs\u03bc : \u2191\u2191\u03bc s \u2260 \u22a4\nht\u03bc : \u2191\u2191\u03bc t \u2260 \u22a4\nhfs : IntegrableOn f s\nhft : IntegrableOn f t\n\u22a2 \u2a0d (x : \u03b1) in s \u222a t, f x \u2202\u03bc =\n    (ENNReal.toReal (\u2191\u2191\u03bc s) / (ENNReal.toReal (\u2191\u2191\u03bc s) + ENNReal.toReal (\u2191\u2191\u03bc t))) \u2022 \u2a0d (x : \u03b1) in s, f x \u2202\u03bc +\n      (ENNReal.toReal (\u2191\u2191\u03bc t) / (ENNReal.toReal (\u2191\u2191\u03bc s) + ENNReal.toReal (\u2191\u2191\u03bc t))) \u2022 \u2a0d (x : \u03b1) in t, f x \u2202\u03bc", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns\u271d t\u271d : Set \u03b1\nf\u271d g f : \u03b1 \u2192 E\ns t : Set \u03b1\nhd : AEDisjoint \u03bc s t\nht : NullMeasurableSet t\nhs\u03bc : \u2191\u2191\u03bc s \u2260 \u22a4\nht\u03bc : \u2191\u2191\u03bc t \u2260 \u22a4\nhfs : IntegrableOn f s\nhft : IntegrableOn f t\nthis : Fact (\u2191\u2191\u03bc s < \u22a4)\n\u22a2 \u2a0d (x : \u03b1) in s \u222a t, f x \u2202\u03bc =\n    (ENNReal.toReal (\u2191\u2191\u03bc s) / (ENNReal.toReal (\u2191\u2191\u03bc s) + ENNReal.toReal (\u2191\u2191\u03bc t))) \u2022 \u2a0d (x : \u03b1) in s, f x \u2202\u03bc +\n      (ENNReal.toReal (\u2191\u2191\u03bc t) / (ENNReal.toReal (\u2191\u2191\u03bc s) + ENNReal.toReal (\u2191\u2191\u03bc t))) \u2022 \u2a0d (x : \u03b1) in t, f x \u2202\u03bc"}, {"tactic": "haveI := Fact.mk ht\u03bc.lt_top", "annotated_tactic": ["haveI := <a>Fact.mk</a> ht\u03bc.lt_top", [{"full_name": "Fact.mk", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [115, 12], "def_end_pos": [115, 22]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns\u271d t\u271d : Set \u03b1\nf\u271d g f : \u03b1 \u2192 E\ns t : Set \u03b1\nhd : AEDisjoint \u03bc s t\nht : NullMeasurableSet t\nhs\u03bc : \u2191\u2191\u03bc s \u2260 \u22a4\nht\u03bc : \u2191\u2191\u03bc t \u2260 \u22a4\nhfs : IntegrableOn f s\nhft : IntegrableOn f t\nthis : Fact (\u2191\u2191\u03bc s < \u22a4)\n\u22a2 \u2a0d (x : \u03b1) in s \u222a t, f x \u2202\u03bc =\n    (ENNReal.toReal (\u2191\u2191\u03bc s) / (ENNReal.toReal (\u2191\u2191\u03bc s) + ENNReal.toReal (\u2191\u2191\u03bc t))) \u2022 \u2a0d (x : \u03b1) in s, f x \u2202\u03bc +\n      (ENNReal.toReal (\u2191\u2191\u03bc t) / (ENNReal.toReal (\u2191\u2191\u03bc s) + ENNReal.toReal (\u2191\u2191\u03bc t))) \u2022 \u2a0d (x : \u03b1) in t, f x \u2202\u03bc", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns\u271d t\u271d : Set \u03b1\nf\u271d g f : \u03b1 \u2192 E\ns t : Set \u03b1\nhd : AEDisjoint \u03bc s t\nht : NullMeasurableSet t\nhs\u03bc : \u2191\u2191\u03bc s \u2260 \u22a4\nht\u03bc : \u2191\u2191\u03bc t \u2260 \u22a4\nhfs : IntegrableOn f s\nhft : IntegrableOn f t\nthis\u271d : Fact (\u2191\u2191\u03bc s < \u22a4)\nthis : Fact (\u2191\u2191\u03bc t < \u22a4)\n\u22a2 \u2a0d (x : \u03b1) in s \u222a t, f x \u2202\u03bc =\n    (ENNReal.toReal (\u2191\u2191\u03bc s) / (ENNReal.toReal (\u2191\u2191\u03bc s) + ENNReal.toReal (\u2191\u2191\u03bc t))) \u2022 \u2a0d (x : \u03b1) in s, f x \u2202\u03bc +\n      (ENNReal.toReal (\u2191\u2191\u03bc t) / (ENNReal.toReal (\u2191\u2191\u03bc s) + ENNReal.toReal (\u2191\u2191\u03bc t))) \u2022 \u2a0d (x : \u03b1) in t, f x \u2202\u03bc"}, {"tactic": "rw [restrict_union\u2080 hd ht, average_add_measure hfs hft, restrict_apply_univ, restrict_apply_univ]", "annotated_tactic": ["rw [<a>restrict_union\u2080</a> hd ht, <a>average_add_measure</a> hfs hft, <a>restrict_apply_univ</a>, <a>restrict_apply_univ</a>]", [{"full_name": "MeasureTheory.Measure.restrict_union\u2080", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1733, 9], "def_end_pos": [1733, 24]}, {"full_name": "MeasureTheory.average_add_measure", "def_path": "Mathlib/MeasureTheory/Integral/Average.lean", "def_pos": [322, 9], "def_end_pos": [322, 28]}, {"full_name": "MeasureTheory.Measure.restrict_apply_univ", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1604, 9], "def_end_pos": [1604, 28]}, {"full_name": "MeasureTheory.Measure.restrict_apply_univ", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1604, 9], "def_end_pos": [1604, 28]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns\u271d t\u271d : Set \u03b1\nf\u271d g f : \u03b1 \u2192 E\ns t : Set \u03b1\nhd : AEDisjoint \u03bc s t\nht : NullMeasurableSet t\nhs\u03bc : \u2191\u2191\u03bc s \u2260 \u22a4\nht\u03bc : \u2191\u2191\u03bc t \u2260 \u22a4\nhfs : IntegrableOn f s\nhft : IntegrableOn f t\nthis\u271d : Fact (\u2191\u2191\u03bc s < \u22a4)\nthis : Fact (\u2191\u2191\u03bc t < \u22a4)\n\u22a2 \u2a0d (x : \u03b1) in s \u222a t, f x \u2202\u03bc =\n    (ENNReal.toReal (\u2191\u2191\u03bc s) / (ENNReal.toReal (\u2191\u2191\u03bc s) + ENNReal.toReal (\u2191\u2191\u03bc t))) \u2022 \u2a0d (x : \u03b1) in s, f x \u2202\u03bc +\n      (ENNReal.toReal (\u2191\u2191\u03bc t) / (ENNReal.toReal (\u2191\u2191\u03bc s) + ENNReal.toReal (\u2191\u2191\u03bc t))) \u2022 \u2a0d (x : \u03b1) in t, f x \u2202\u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Portmanteau.lean", "full_name": "MeasureTheory.measure_of_cont_bdd_of_tendsto_indicator", "start": [312, 1], "end": [323, 99], "traced_tactics": [{"tactic": "have fs_lim' :\n  \u2200 \u03c9, Tendsto (fun n : \u2115 => (fs n \u03c9 : \u211d\u22650)) atTop (\ud835\udcdd (indicator E (fun _ => (1 : \u211d\u22650)) \u03c9)) := by\n  rw [tendsto_pi_nhds] at fs_lim\n  exact fun \u03c9 => fs_lim \u03c9", "annotated_tactic": ["have fs_lim' :\n    \u2200 \u03c9, <a>Tendsto</a> (fun n : \u2115 => (fs n \u03c9 : \u211d\u22650)) <a>atTop</a> (\ud835\udcdd (<a>indicator</a> E (fun _ => (1 : \u211d\u22650)) \u03c9)) := by\n    rw [<a>tendsto_pi_nhds</a>] at fs_lim\n    exact fun \u03c9 => fs_lim \u03c9", [{"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2939, 5], "def_end_pos": [2939, 12]}, {"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}, {"full_name": "Set.indicator", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [46, 3], "def_end_pos": [46, 14]}, {"full_name": "tendsto_pi_nhds", "def_path": "Mathlib/Topology/Constructions.lean", "def_pos": [1231, 9], "def_end_pos": [1231, 24]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b3 : MeasurableSpace \u03a9\ninst\u271d\u00b2 : TopologicalSpace \u03a9\ninst\u271d\u00b9 : OpensMeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d : IsFiniteMeasure \u03bc\nc : \u211d\u22650\nE : Set \u03a9\nE_mble : MeasurableSet E\nfs : \u2115 \u2192 \u03a9 \u2192\u1d47 \u211d\u22650\nfs_bdd : \u2200 (n : \u2115) (\u03c9 : \u03a9), \u2191(fs n) \u03c9 \u2264 c\nfs_lim : Tendsto (fun n => \u2191(fs n)) atTop (\ud835\udcdd (indicator E fun x => 1))\n\u22a2 Tendsto (fun n => \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191(fs n) \u03c9) \u2202\u03bc) atTop (\ud835\udcdd (\u2191\u2191\u03bc E))", "state_after": "\u03a9 : Type u_1\ninst\u271d\u00b3 : MeasurableSpace \u03a9\ninst\u271d\u00b2 : TopologicalSpace \u03a9\ninst\u271d\u00b9 : OpensMeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d : IsFiniteMeasure \u03bc\nc : \u211d\u22650\nE : Set \u03a9\nE_mble : MeasurableSet E\nfs : \u2115 \u2192 \u03a9 \u2192\u1d47 \u211d\u22650\nfs_bdd : \u2200 (n : \u2115) (\u03c9 : \u03a9), \u2191(fs n) \u03c9 \u2264 c\nfs_lim : Tendsto (fun n => \u2191(fs n)) atTop (\ud835\udcdd (indicator E fun x => 1))\nfs_lim' : \u2200 (\u03c9 : \u03a9), Tendsto (fun n => \u2191(fs n) \u03c9) atTop (\ud835\udcdd (indicator E (fun x => 1) \u03c9))\n\u22a2 Tendsto (fun n => \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191(fs n) \u03c9) \u2202\u03bc) atTop (\ud835\udcdd (\u2191\u2191\u03bc E))"}, {"tactic": "apply measure_of_cont_bdd_of_tendsto_filter_indicator \u03bc E_mble fs\n  (eventually_of_forall fun n => eventually_of_forall (fs_bdd n)) (eventually_of_forall fs_lim')", "annotated_tactic": ["apply <a>measure_of_cont_bdd_of_tendsto_filter_indicator</a> \u03bc E_mble fs\n    (<a>eventually_of_forall</a> fun n => <a>eventually_of_forall</a> (fs_bdd n)) (<a>eventually_of_forall</a> fs_lim')", [{"full_name": "MeasureTheory.measure_of_cont_bdd_of_tendsto_filter_indicator", "def_path": "Mathlib/MeasureTheory/Measure/Portmanteau.lean", "def_pos": [291, 9], "def_end_pos": [291, 56]}, {"full_name": "Filter.eventually_of_forall", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1111, 9], "def_end_pos": [1111, 29]}, {"full_name": "Filter.eventually_of_forall", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1111, 9], "def_end_pos": [1111, 29]}, {"full_name": "Filter.eventually_of_forall", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1111, 9], "def_end_pos": [1111, 29]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b3 : MeasurableSpace \u03a9\ninst\u271d\u00b2 : TopologicalSpace \u03a9\ninst\u271d\u00b9 : OpensMeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d : IsFiniteMeasure \u03bc\nc : \u211d\u22650\nE : Set \u03a9\nE_mble : MeasurableSet E\nfs : \u2115 \u2192 \u03a9 \u2192\u1d47 \u211d\u22650\nfs_bdd : \u2200 (n : \u2115) (\u03c9 : \u03a9), \u2191(fs n) \u03c9 \u2264 c\nfs_lim : Tendsto (fun n => \u2191(fs n)) atTop (\ud835\udcdd (indicator E fun x => 1))\nfs_lim' : \u2200 (\u03c9 : \u03a9), Tendsto (fun n => \u2191(fs n) \u03c9) atTop (\ud835\udcdd (indicator E (fun x => 1) \u03c9))\n\u22a2 Tendsto (fun n => \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191(fs n) \u03c9) \u2202\u03bc) atTop (\ud835\udcdd (\u2191\u2191\u03bc E))", "state_after": "no goals"}, {"tactic": "rw [tendsto_pi_nhds] at fs_lim", "annotated_tactic": ["rw [<a>tendsto_pi_nhds</a>] at fs_lim", [{"full_name": "tendsto_pi_nhds", "def_path": "Mathlib/Topology/Constructions.lean", "def_pos": [1231, 9], "def_end_pos": [1231, 24]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b3 : MeasurableSpace \u03a9\ninst\u271d\u00b2 : TopologicalSpace \u03a9\ninst\u271d\u00b9 : OpensMeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d : IsFiniteMeasure \u03bc\nc : \u211d\u22650\nE : Set \u03a9\nE_mble : MeasurableSet E\nfs : \u2115 \u2192 \u03a9 \u2192\u1d47 \u211d\u22650\nfs_bdd : \u2200 (n : \u2115) (\u03c9 : \u03a9), \u2191(fs n) \u03c9 \u2264 c\nfs_lim : Tendsto (fun n => \u2191(fs n)) atTop (\ud835\udcdd (indicator E fun x => 1))\n\u22a2 \u2200 (\u03c9 : \u03a9), Tendsto (fun n => \u2191(fs n) \u03c9) atTop (\ud835\udcdd (indicator E (fun x => 1) \u03c9))", "state_after": "\u03a9 : Type u_1\ninst\u271d\u00b3 : MeasurableSpace \u03a9\ninst\u271d\u00b2 : TopologicalSpace \u03a9\ninst\u271d\u00b9 : OpensMeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d : IsFiniteMeasure \u03bc\nc : \u211d\u22650\nE : Set \u03a9\nE_mble : MeasurableSet E\nfs : \u2115 \u2192 \u03a9 \u2192\u1d47 \u211d\u22650\nfs_bdd : \u2200 (n : \u2115) (\u03c9 : \u03a9), \u2191(fs n) \u03c9 \u2264 c\nfs_lim : \u2200 (x : \u03a9), Tendsto (fun i => \u2191(fs i) x) atTop (\ud835\udcdd (indicator E (fun x => 1) x))\n\u22a2 \u2200 (\u03c9 : \u03a9), Tendsto (fun n => \u2191(fs n) \u03c9) atTop (\ud835\udcdd (indicator E (fun x => 1) \u03c9))"}, {"tactic": "exact fun \u03c9 => fs_lim \u03c9", "annotated_tactic": ["exact fun \u03c9 => fs_lim \u03c9", []], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b3 : MeasurableSpace \u03a9\ninst\u271d\u00b2 : TopologicalSpace \u03a9\ninst\u271d\u00b9 : OpensMeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d : IsFiniteMeasure \u03bc\nc : \u211d\u22650\nE : Set \u03a9\nE_mble : MeasurableSet E\nfs : \u2115 \u2192 \u03a9 \u2192\u1d47 \u211d\u22650\nfs_bdd : \u2200 (n : \u2115) (\u03c9 : \u03a9), \u2191(fs n) \u03c9 \u2264 c\nfs_lim : \u2200 (x : \u03a9), Tendsto (fun i => \u2191(fs i) x) atTop (\ud835\udcdd (indicator E (fun x => 1) x))\n\u22a2 \u2200 (\u03c9 : \u03a9), Tendsto (fun n => \u2191(fs n) \u03c9) atTop (\ud835\udcdd (indicator E (fun x => 1) \u03c9))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Decomposition/Lebesgue.lean", "full_name": "MeasureTheory.SignedMeasure.singularPart_totalVariation", "start": [864, 1], "end": [875, 30], "traced_tactics": [{"tactic": "have :\n  (s.singularPart \u03bc).toJordanDecomposition =\n    \u27e8s.toJordanDecomposition.posPart.singularPart \u03bc,\n      s.toJordanDecomposition.negPart.singularPart \u03bc, singularPart_mutuallySingular s \u03bc\u27e9 := by\n  refine' JordanDecomposition.toSignedMeasure_injective _\n  rw [toSignedMeasure_toJordanDecomposition]\n  rfl", "annotated_tactic": ["have :\n    (s.singularPart \u03bc).<a>toJordanDecomposition</a> =\n      \u27e8s.toJordanDecomposition.posPart.singularPart \u03bc,\n        s.toJordanDecomposition.negPart.singularPart \u03bc, <a>singularPart_mutuallySingular</a> s \u03bc\u27e9 := by\n    refine' <a>JordanDecomposition.toSignedMeasure_injective</a> _\n    rw [<a>toSignedMeasure_toJordanDecomposition</a>]\n    rfl", [{"full_name": "MeasureTheory.SignedMeasure.toJordanDecomposition", "def_path": "Mathlib/MeasureTheory/Decomposition/Jordan.lean", "def_pos": [228, 5], "def_end_pos": [228, 26]}, {"full_name": "MeasureTheory.SignedMeasure.singularPart_mutuallySingular", "def_path": "Mathlib/MeasureTheory/Decomposition/Lebesgue.lean", "def_pos": [846, 9], "def_end_pos": [846, 38]}, {"full_name": "MeasureTheory.JordanDecomposition.toSignedMeasure_injective", "def_path": "Mathlib/MeasureTheory/Decomposition/Jordan.lean", "def_pos": [372, 9], "def_end_pos": [372, 34]}, {"full_name": "MeasureTheory.SignedMeasure.toSignedMeasure_toJordanDecomposition", "def_path": "Mathlib/MeasureTheory/Decomposition/Jordan.lean", "def_pos": [260, 9], "def_end_pos": [260, 46]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\ns : SignedMeasure \u03b1\n\u03bc : Measure \u03b1\n\u22a2 totalVariation (singularPart s \u03bc) =\n    Measure.singularPart (toJordanDecomposition s).posPart \u03bc + Measure.singularPart (toJordanDecomposition s).negPart \u03bc", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\ns : SignedMeasure \u03b1\n\u03bc : Measure \u03b1\nthis :\n  toJordanDecomposition (singularPart s \u03bc) =\n    JordanDecomposition.mk (Measure.singularPart (toJordanDecomposition s).posPart \u03bc)\n      (Measure.singularPart (toJordanDecomposition s).negPart \u03bc)\n      (_ :\n        Measure.singularPart (toJordanDecomposition s).posPart \u03bc \u27c2\u2098\n          Measure.singularPart (toJordanDecomposition s).negPart \u03bc)\n\u22a2 totalVariation (singularPart s \u03bc) =\n    Measure.singularPart (toJordanDecomposition s).posPart \u03bc + Measure.singularPart (toJordanDecomposition s).negPart \u03bc"}, {"tactic": "refine' JordanDecomposition.toSignedMeasure_injective _", "annotated_tactic": ["refine' <a>JordanDecomposition.toSignedMeasure_injective</a> _", [{"full_name": "MeasureTheory.JordanDecomposition.toSignedMeasure_injective", "def_path": "Mathlib/MeasureTheory/Decomposition/Jordan.lean", "def_pos": [372, 9], "def_end_pos": [372, 34]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\ns : SignedMeasure \u03b1\n\u03bc : Measure \u03b1\n\u22a2 toJordanDecomposition (singularPart s \u03bc) =\n    JordanDecomposition.mk (Measure.singularPart (toJordanDecomposition s).posPart \u03bc)\n      (Measure.singularPart (toJordanDecomposition s).negPart \u03bc)\n      (_ :\n        Measure.singularPart (toJordanDecomposition s).posPart \u03bc \u27c2\u2098\n          Measure.singularPart (toJordanDecomposition s).negPart \u03bc)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\ns : SignedMeasure \u03b1\n\u03bc : Measure \u03b1\n\u22a2 JordanDecomposition.toSignedMeasure (toJordanDecomposition (singularPart s \u03bc)) =\n    JordanDecomposition.toSignedMeasure\n      (JordanDecomposition.mk (Measure.singularPart (toJordanDecomposition s).posPart \u03bc)\n        (Measure.singularPart (toJordanDecomposition s).negPart \u03bc)\n        (_ :\n          Measure.singularPart (toJordanDecomposition s).posPart \u03bc \u27c2\u2098\n            Measure.singularPart (toJordanDecomposition s).negPart \u03bc))"}, {"tactic": "rw [toSignedMeasure_toJordanDecomposition]", "annotated_tactic": ["rw [<a>toSignedMeasure_toJordanDecomposition</a>]", [{"full_name": "MeasureTheory.SignedMeasure.toSignedMeasure_toJordanDecomposition", "def_path": "Mathlib/MeasureTheory/Decomposition/Jordan.lean", "def_pos": [260, 9], "def_end_pos": [260, 46]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\ns : SignedMeasure \u03b1\n\u03bc : Measure \u03b1\n\u22a2 JordanDecomposition.toSignedMeasure (toJordanDecomposition (singularPart s \u03bc)) =\n    JordanDecomposition.toSignedMeasure\n      (JordanDecomposition.mk (Measure.singularPart (toJordanDecomposition s).posPart \u03bc)\n        (Measure.singularPart (toJordanDecomposition s).negPart \u03bc)\n        (_ :\n          Measure.singularPart (toJordanDecomposition s).posPart \u03bc \u27c2\u2098\n            Measure.singularPart (toJordanDecomposition s).negPart \u03bc))", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\ns : SignedMeasure \u03b1\n\u03bc : Measure \u03b1\n\u22a2 singularPart s \u03bc =\n    JordanDecomposition.toSignedMeasure\n      (JordanDecomposition.mk (Measure.singularPart (toJordanDecomposition s).posPart \u03bc)\n        (Measure.singularPart (toJordanDecomposition s).negPart \u03bc)\n        (_ :\n          Measure.singularPart (toJordanDecomposition s).posPart \u03bc \u27c2\u2098\n            Measure.singularPart (toJordanDecomposition s).negPart \u03bc))"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\ns : SignedMeasure \u03b1\n\u03bc : Measure \u03b1\n\u22a2 singularPart s \u03bc =\n    JordanDecomposition.toSignedMeasure\n      (JordanDecomposition.mk (Measure.singularPart (toJordanDecomposition s).posPart \u03bc)\n        (Measure.singularPart (toJordanDecomposition s).negPart \u03bc)\n        (_ :\n          Measure.singularPart (toJordanDecomposition s).posPart \u03bc \u27c2\u2098\n            Measure.singularPart (toJordanDecomposition s).negPart \u03bc))", "state_after": "no goals"}, {"tactic": "rw [totalVariation, this]", "annotated_tactic": ["rw [<a>totalVariation</a>, this]", [{"full_name": "MeasureTheory.SignedMeasure.totalVariation", "def_path": "Mathlib/MeasureTheory/Decomposition/Jordan.lean", "def_pos": [494, 5], "def_end_pos": [494, 19]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\ns : SignedMeasure \u03b1\n\u03bc : Measure \u03b1\nthis :\n  toJordanDecomposition (singularPart s \u03bc) =\n    JordanDecomposition.mk (Measure.singularPart (toJordanDecomposition s).posPart \u03bc)\n      (Measure.singularPart (toJordanDecomposition s).negPart \u03bc)\n      (_ :\n        Measure.singularPart (toJordanDecomposition s).posPart \u03bc \u27c2\u2098\n          Measure.singularPart (toJordanDecomposition s).negPart \u03bc)\n\u22a2 totalVariation (singularPart s \u03bc) =\n    Measure.singularPart (toJordanDecomposition s).posPart \u03bc + Measure.singularPart (toJordanDecomposition s).negPart \u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/MeanInequalities.lean", "full_name": "ENNReal.lintegral_Lp_add_le_of_le_one", "start": [377, 1], "end": [391, 64], "traced_tactics": [{"tactic": "rcases eq_or_lt_of_le hp0 with (rfl | hp)", "annotated_tactic": ["rcases <a>eq_or_lt_of_le</a> hp0 with (rfl | hp)", [{"full_name": "eq_or_lt_of_le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [414, 9], "def_end_pos": [414, 23]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhp0 : 0 \u2264 p\nhp1 : p \u2264 1\n\u22a2 (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2264\n    2 ^ (1 / p - 1) * ((\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p))", "state_after": "case inl\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhp0 : 0 \u2264 0\nhp1 : 0 \u2264 1\n\u22a2 (\u222b\u207b (a : \u03b1), (f + g) a ^ 0 \u2202\u03bc) ^ (1 / 0) \u2264\n    2 ^ (1 / 0 - 1) * ((\u222b\u207b (a : \u03b1), f a ^ 0 \u2202\u03bc) ^ (1 / 0) + (\u222b\u207b (a : \u03b1), g a ^ 0 \u2202\u03bc) ^ (1 / 0))\n\ncase inr\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhp0 : 0 \u2264 p\nhp1 : p \u2264 1\nhp : 0 < p\n\u22a2 (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2264\n    2 ^ (1 / p - 1) * ((\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p))"}, {"tactic": "calc\n  (\u222b\u207b a, (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2264 ((\u222b\u207b a, f a ^ p \u2202\u03bc) + \u222b\u207b a, g a ^ p \u2202\u03bc) ^ (1 / p) := by\n    apply rpow_le_rpow _ (div_nonneg zero_le_one hp0)\n    rw [\u2190 lintegral_add_left' (hf.pow_const p)]\n    exact lintegral_mono fun a => rpow_add_le_add_rpow _ _ hp0 hp1\n  _ \u2264 (2 : \u211d\u22650\u221e) ^ (1 / p - 1) * ((\u222b\u207b a, f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b a, g a ^ p \u2202\u03bc) ^ (1 / p)) :=\n    rpow_add_le_mul_rpow_add_rpow _ _ ((one_le_div hp).2 hp1)", "annotated_tactic": ["calc\n    (\u222b\u207b a, (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2264 ((\u222b\u207b a, f a ^ p \u2202\u03bc) + \u222b\u207b a, g a ^ p \u2202\u03bc) ^ (1 / p) := by\n      apply <a>rpow_le_rpow</a> _ (<a>div_nonneg</a> <a>zero_le_one</a> hp0)\n      rw [\u2190 <a>lintegral_add_left'</a> (hf.pow_const p)]\n      exact <a>lintegral_mono</a> fun a => <a>rpow_add_le_add_rpow</a> _ _ hp0 hp1\n    _ \u2264 (2 : \u211d\u22650\u221e) ^ (1 / p - 1) * ((\u222b\u207b a, f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b a, g a ^ p \u2202\u03bc) ^ (1 / p)) :=\n      <a>rpow_add_le_mul_rpow_add_rpow</a> _ _ ((<a>one_le_div</a> hp).2 hp1)", [{"full_name": "ENNReal.rpow_le_rpow", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [642, 9], "def_end_pos": [642, 21]}, {"full_name": "div_nonneg", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [94, 9], "def_end_pos": [94, 19]}, {"full_name": "zero_le_one", "def_path": "Mathlib/Algebra/Order/ZeroLEOne.lean", "def_pos": [26, 15], "def_end_pos": [26, 26]}, {"full_name": "MeasureTheory.lintegral_add_left'", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [566, 9], "def_end_pos": [566, 28]}, {"full_name": "MeasureTheory.lintegral_mono", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [99, 9], "def_end_pos": [99, 23]}, {"full_name": "ENNReal.rpow_add_le_add_rpow", "def_path": "Mathlib/Analysis/MeanInequalitiesPow.lean", "def_pos": [338, 9], "def_end_pos": [338, 29]}, {"full_name": "ENNReal.rpow_add_le_mul_rpow_add_rpow", "def_path": "Mathlib/Analysis/MeanInequalitiesPow.lean", "def_pos": [292, 9], "def_end_pos": [292, 38]}, {"full_name": "one_le_div", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [422, 9], "def_end_pos": [422, 19]}]], "state_before": "case inr\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhp0 : 0 \u2264 p\nhp1 : p \u2264 1\nhp : 0 < p\n\u22a2 (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2264\n    2 ^ (1 / p - 1) * ((\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p))", "state_after": "no goals"}, {"tactic": "simp only [Pi.add_apply, rpow_zero, lintegral_one, _root_.div_zero, zero_sub]", "annotated_tactic": ["simp only [<a>Pi.add_apply</a>, <a>rpow_zero</a>, <a>lintegral_one</a>, <a>_root_.div_zero</a>, <a>zero_sub</a>]", [{"full_name": "Pi.add_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [82, 3], "def_end_pos": [82, 14]}, {"full_name": "ENNReal.rpow_zero", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [381, 9], "def_end_pos": [381, 18]}, {"full_name": "MeasureTheory.lintegral_one", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [149, 9], "def_end_pos": [149, 22]}, {"full_name": "div_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Basic.lean", "def_pos": [295, 9], "def_end_pos": [295, 17]}, {"full_name": "zero_sub", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [317, 3], "def_end_pos": [317, 14]}]], "state_before": "case inl\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhp0 : 0 \u2264 0\nhp1 : 0 \u2264 1\n\u22a2 (\u222b\u207b (a : \u03b1), (f + g) a ^ 0 \u2202\u03bc) ^ (1 / 0) \u2264\n    2 ^ (1 / 0 - 1) * ((\u222b\u207b (a : \u03b1), f a ^ 0 \u2202\u03bc) ^ (1 / 0) + (\u222b\u207b (a : \u03b1), g a ^ 0 \u2202\u03bc) ^ (1 / 0))", "state_after": "case inl\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhp0 : 0 \u2264 0\nhp1 : 0 \u2264 1\n\u22a2 1 \u2264 2 ^ (-1) * (1 + 1)"}, {"tactic": "norm_num", "annotated_tactic": ["norm_num", []], "state_before": "case inl\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhp0 : 0 \u2264 0\nhp1 : 0 \u2264 1\n\u22a2 1 \u2264 2 ^ (-1) * (1 + 1)", "state_after": "case inl\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhp0 : 0 \u2264 0\nhp1 : 0 \u2264 1\n\u22a2 1 \u2264 2 ^ (-1) * 2"}, {"tactic": "rw [rpow_neg, rpow_one, ENNReal.inv_mul_cancel two_ne_zero two_ne_top]", "annotated_tactic": ["rw [<a>rpow_neg</a>, <a>rpow_one</a>, <a>ENNReal.inv_mul_cancel</a> <a>two_ne_zero</a> <a>two_ne_top</a>]", [{"full_name": "ENNReal.rpow_neg", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [514, 9], "def_end_pos": [514, 17]}, {"full_name": "ENNReal.rpow_one", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [450, 9], "def_end_pos": [450, 17]}, {"full_name": "ENNReal.inv_mul_cancel", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1424, 19], "def_end_pos": [1424, 33]}, {"full_name": "two_ne_zero", "def_path": "Mathlib/Algebra/NeZero.lean", "def_pos": [62, 7], "def_end_pos": [62, 18]}, {"full_name": "ENNReal.two_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [431, 9], "def_end_pos": [431, 19]}]], "state_before": "case inl\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhp0 : 0 \u2264 0\nhp1 : 0 \u2264 1\n\u22a2 1 \u2264 2 ^ (-1) * 2", "state_after": "no goals"}, {"tactic": "apply rpow_le_rpow _ (div_nonneg zero_le_one hp0)", "annotated_tactic": ["apply <a>rpow_le_rpow</a> _ (<a>div_nonneg</a> <a>zero_le_one</a> hp0)", [{"full_name": "ENNReal.rpow_le_rpow", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [642, 9], "def_end_pos": [642, 21]}, {"full_name": "div_nonneg", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [94, 9], "def_end_pos": [94, 19]}, {"full_name": "zero_le_one", "def_path": "Mathlib/Algebra/Order/ZeroLEOne.lean", "def_pos": [26, 15], "def_end_pos": [26, 26]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhp0 : 0 \u2264 p\nhp1 : p \u2264 1\nhp : 0 < p\n\u22a2 (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc + \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)", "state_after": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhp0 : 0 \u2264 p\nhp1 : p \u2264 1\nhp : 0 < p\n\u22a2 \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc + \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc"}, {"tactic": "rw [\u2190 lintegral_add_left' (hf.pow_const p)]", "annotated_tactic": ["rw [\u2190 <a>lintegral_add_left'</a> (hf.pow_const p)]", [{"full_name": "MeasureTheory.lintegral_add_left'", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [566, 9], "def_end_pos": [566, 28]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhp0 : 0 \u2264 p\nhp1 : p \u2264 1\nhp : 0 < p\n\u22a2 \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc + \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc", "state_after": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhp0 : 0 \u2264 p\nhp1 : p \u2264 1\nhp : 0 < p\n\u22a2 \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), f a ^ p + g a ^ p \u2202\u03bc"}, {"tactic": "exact lintegral_mono fun a => rpow_add_le_add_rpow _ _ hp0 hp1", "annotated_tactic": ["exact <a>lintegral_mono</a> fun a => <a>rpow_add_le_add_rpow</a> _ _ hp0 hp1", [{"full_name": "MeasureTheory.lintegral_mono", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [99, 9], "def_end_pos": [99, 23]}, {"full_name": "ENNReal.rpow_add_le_add_rpow", "def_path": "Mathlib/Analysis/MeanInequalitiesPow.lean", "def_pos": [338, 9], "def_end_pos": [338, 29]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhp0 : 0 \u2264 p\nhp1 : p \u2264 1\nhp : 0 < p\n\u22a2 \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), f a ^ p + g a ^ p \u2202\u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/FiniteMeasure.lean", "full_name": "MeasureTheory.FiniteMeasure.coeFn_smul_apply", "start": [276, 1], "end": [278, 33], "traced_tactics": [{"tactic": "rw [coeFn_smul, Pi.smul_apply]", "annotated_tactic": ["rw [<a>coeFn_smul</a>, <a>Pi.smul_apply</a>]", [{"full_name": "MeasureTheory.FiniteMeasure.coeFn_smul", "def_path": "Mathlib/MeasureTheory/Measure/FiniteMeasure.lean", "def_pos": [253, 9], "def_end_pos": [253, 19]}, {"full_name": "Pi.smul_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [116, 60], "def_end_pos": [116, 70]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u2075 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2074 : SMul R \u211d\u22650\ninst\u271d\u00b3 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b9 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d : IsScalarTower R \u211d\u22650 \u211d\u22650\nc : R\n\u03bc : FiniteMeasure \u03a9\ns : Set \u03a9\n\u22a2 (fun s => ENNReal.toNNReal (\u2191\u2191\u2191(c \u2022 \u03bc) s)) s = c \u2022 (fun s => ENNReal.toNNReal (\u2191\u2191\u2191\u03bc s)) s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean", "full_name": "Real.volume_preserving_transvectionStruct", "start": [379, 1], "end": [418, 35], "traced_tactics": [{"tactic": "let p : \u03b9 \u2192 Prop := fun i => i \u2260 t.i", "annotated_tactic": ["let p : \u03b9 \u2192 Prop := fun i => i \u2260 t.i", []], "state_before": "\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nt : TransvectionStruct \u03b9 \u211d\n\u22a2 MeasurePreserving \u2191(\u2191toLin' (TransvectionStruct.toMatrix t))", "state_after": "\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nt : TransvectionStruct \u03b9 \u211d\np : \u03b9 \u2192 Prop := fun i => i \u2260 t.i\n\u22a2 MeasurePreserving \u2191(\u2191toLin' (TransvectionStruct.toMatrix t))"}, {"tactic": "let \u03b1 : Type _ := { x // p x }", "annotated_tactic": ["let \u03b1 : Type _ := { x // p x }", []], "state_before": "\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nt : TransvectionStruct \u03b9 \u211d\np : \u03b9 \u2192 Prop := fun i => i \u2260 t.i\n\u22a2 MeasurePreserving \u2191(\u2191toLin' (TransvectionStruct.toMatrix t))", "state_after": "\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nt : TransvectionStruct \u03b9 \u211d\np : \u03b9 \u2192 Prop := fun i => i \u2260 t.i\n\u03b1 : Type u_1 := { x // p x }\n\u22a2 MeasurePreserving \u2191(\u2191toLin' (TransvectionStruct.toMatrix t))"}, {"tactic": "let \u03b2 : Type _ := { x // \u00acp x }", "annotated_tactic": ["let \u03b2 : Type _ := { x // \u00acp x }", []], "state_before": "\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nt : TransvectionStruct \u03b9 \u211d\np : \u03b9 \u2192 Prop := fun i => i \u2260 t.i\n\u03b1 : Type u_1 := { x // p x }\n\u22a2 MeasurePreserving \u2191(\u2191toLin' (TransvectionStruct.toMatrix t))", "state_after": "\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nt : TransvectionStruct \u03b9 \u211d\np : \u03b9 \u2192 Prop := fun i => i \u2260 t.i\n\u03b1 : Type u_1 := { x // p x }\n\u03b2 : Type u_1 := { x // \u00acp x }\n\u22a2 MeasurePreserving \u2191(\u2191toLin' (TransvectionStruct.toMatrix t))"}, {"tactic": "let g : (\u03b1 \u2192 \u211d) \u2192 (\u03b2 \u2192 \u211d) \u2192 \u03b2 \u2192 \u211d := fun a b => (fun _ => t.c * a \u27e8t.j, t.hij.symm\u27e9) + b", "annotated_tactic": ["let g : (\u03b1 \u2192 \u211d) \u2192 (\u03b2 \u2192 \u211d) \u2192 \u03b2 \u2192 \u211d := fun a b => (fun _ => t.c * a \u27e8t.j, t.hij.symm\u27e9) + b", []], "state_before": "\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nt : TransvectionStruct \u03b9 \u211d\np : \u03b9 \u2192 Prop := fun i => i \u2260 t.i\n\u03b1 : Type u_1 := { x // p x }\n\u03b2 : Type u_1 := { x // \u00acp x }\n\u22a2 MeasurePreserving \u2191(\u2191toLin' (TransvectionStruct.toMatrix t))", "state_after": "\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nt : TransvectionStruct \u03b9 \u211d\np : \u03b9 \u2192 Prop := fun i => i \u2260 t.i\n\u03b1 : Type u_1 := { x // p x }\n\u03b2 : Type u_1 := { x // \u00acp x }\ng : (\u03b1 \u2192 \u211d) \u2192 (\u03b2 \u2192 \u211d) \u2192 \u03b2 \u2192 \u211d := fun a b => (fun x => t.c * a { val := t.j, property := (_ : t.j \u2260 t.i) }) + b\n\u22a2 MeasurePreserving \u2191(\u2191toLin' (TransvectionStruct.toMatrix t))"}, {"tactic": "let F : (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) \u2192 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := fun p => (id p.1, g p.1 p.2)", "annotated_tactic": ["let F : (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) \u2192 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := fun p => (<a>id</a> p.1, g p.1 p.2)", [{"full_name": "id", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [33, 15], "def_end_pos": [33, 17]}]], "state_before": "\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nt : TransvectionStruct \u03b9 \u211d\np : \u03b9 \u2192 Prop := fun i => i \u2260 t.i\n\u03b1 : Type u_1 := { x // p x }\n\u03b2 : Type u_1 := { x // \u00acp x }\ng : (\u03b1 \u2192 \u211d) \u2192 (\u03b2 \u2192 \u211d) \u2192 \u03b2 \u2192 \u211d := fun a b => (fun x => t.c * a { val := t.j, property := (_ : t.j \u2260 t.i) }) + b\n\u22a2 MeasurePreserving \u2191(\u2191toLin' (TransvectionStruct.toMatrix t))", "state_after": "\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nt : TransvectionStruct \u03b9 \u211d\np : \u03b9 \u2192 Prop := fun i => i \u2260 t.i\n\u03b1 : Type u_1 := { x // p x }\n\u03b2 : Type u_1 := { x // \u00acp x }\ng : (\u03b1 \u2192 \u211d) \u2192 (\u03b2 \u2192 \u211d) \u2192 \u03b2 \u2192 \u211d := fun a b => (fun x => t.c * a { val := t.j, property := (_ : t.j \u2260 t.i) }) + b\nF : (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) \u2192 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := fun p => (id p.1, g p.1 p.2)\n\u22a2 MeasurePreserving \u2191(\u2191toLin' (TransvectionStruct.toMatrix t))"}, {"tactic": "let e : (\u03b9 \u2192 \u211d) \u2243\u1d50 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := MeasurableEquiv.piEquivPiSubtypeProd (fun _ : \u03b9 => \u211d) p", "annotated_tactic": ["let e : (\u03b9 \u2192 \u211d) \u2243\u1d50 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := <a>MeasurableEquiv.piEquivPiSubtypeProd</a> (fun _ : \u03b9 => \u211d) p", [{"full_name": "MeasurableEquiv.piEquivPiSubtypeProd", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [1670, 5], "def_end_pos": [1670, 25]}]], "state_before": "\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nt : TransvectionStruct \u03b9 \u211d\np : \u03b9 \u2192 Prop := fun i => i \u2260 t.i\n\u03b1 : Type u_1 := { x // p x }\n\u03b2 : Type u_1 := { x // \u00acp x }\ng : (\u03b1 \u2192 \u211d) \u2192 (\u03b2 \u2192 \u211d) \u2192 \u03b2 \u2192 \u211d := fun a b => (fun x => t.c * a { val := t.j, property := (_ : t.j \u2260 t.i) }) + b\nF : (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) \u2192 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := fun p => (id p.1, g p.1 p.2)\n\u22a2 MeasurePreserving \u2191(\u2191toLin' (TransvectionStruct.toMatrix t))", "state_after": "\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nt : TransvectionStruct \u03b9 \u211d\np : \u03b9 \u2192 Prop := fun i => i \u2260 t.i\n\u03b1 : Type u_1 := { x // p x }\n\u03b2 : Type u_1 := { x // \u00acp x }\ng : (\u03b1 \u2192 \u211d) \u2192 (\u03b2 \u2192 \u211d) \u2192 \u03b2 \u2192 \u211d := fun a b => (fun x => t.c * a { val := t.j, property := (_ : t.j \u2260 t.i) }) + b\nF : (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) \u2192 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := fun p => (id p.1, g p.1 p.2)\ne : (\u03b9 \u2192 \u211d) \u2243\u1d50 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := MeasurableEquiv.piEquivPiSubtypeProd (fun x => \u211d) p\n\u22a2 MeasurePreserving \u2191(\u2191toLin' (TransvectionStruct.toMatrix t))"}, {"tactic": "rw [this]", "annotated_tactic": ["rw [this]", []], "state_before": "\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nt : TransvectionStruct \u03b9 \u211d\np : \u03b9 \u2192 Prop := fun i => i \u2260 t.i\n\u03b1 : Type u_1 := { x // p x }\n\u03b2 : Type u_1 := { x // \u00acp x }\ng : (\u03b1 \u2192 \u211d) \u2192 (\u03b2 \u2192 \u211d) \u2192 \u03b2 \u2192 \u211d := fun a b => (fun x => t.c * a { val := t.j, property := (_ : t.j \u2260 t.i) }) + b\nF : (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) \u2192 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := fun p => (id p.1, g p.1 p.2)\ne : (\u03b9 \u2192 \u211d) \u2243\u1d50 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := MeasurableEquiv.piEquivPiSubtypeProd (fun x => \u211d) p\nthis : \u2191(\u2191toLin' (TransvectionStruct.toMatrix t)) = \u2191(MeasurableEquiv.symm e) \u2218 F \u2218 \u2191e\n\u22a2 MeasurePreserving \u2191(\u2191toLin' (TransvectionStruct.toMatrix t))", "state_after": "\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nt : TransvectionStruct \u03b9 \u211d\np : \u03b9 \u2192 Prop := fun i => i \u2260 t.i\n\u03b1 : Type u_1 := { x // p x }\n\u03b2 : Type u_1 := { x // \u00acp x }\ng : (\u03b1 \u2192 \u211d) \u2192 (\u03b2 \u2192 \u211d) \u2192 \u03b2 \u2192 \u211d := fun a b => (fun x => t.c * a { val := t.j, property := (_ : t.j \u2260 t.i) }) + b\nF : (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) \u2192 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := fun p => (id p.1, g p.1 p.2)\ne : (\u03b9 \u2192 \u211d) \u2243\u1d50 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := MeasurableEquiv.piEquivPiSubtypeProd (fun x => \u211d) p\nthis : \u2191(\u2191toLin' (TransvectionStruct.toMatrix t)) = \u2191(MeasurableEquiv.symm e) \u2218 F \u2218 \u2191e\n\u22a2 MeasurePreserving (\u2191(MeasurableEquiv.symm e) \u2218 F \u2218 \u2191e)"}, {"tactic": "have A : MeasurePreserving e := by\n  convert volume_preserving_piEquivPiSubtypeProd (fun _ : \u03b9 => \u211d) p", "annotated_tactic": ["have A : <a>MeasurePreserving</a> e := by\n    convert <a>volume_preserving_piEquivPiSubtypeProd</a> (fun _ : \u03b9 => \u211d) p", [{"full_name": "MeasureTheory.MeasurePreserving", "def_path": "Mathlib/Dynamics/Ergodic/MeasurePreserving.lean", "def_pos": [42, 11], "def_end_pos": [42, 28]}, {"full_name": "MeasureTheory.volume_preserving_piEquivPiSubtypeProd", "def_path": "Mathlib/MeasureTheory/Constructions/Pi.lean", "def_pos": [747, 9], "def_end_pos": [747, 47]}]], "state_before": "\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nt : TransvectionStruct \u03b9 \u211d\np : \u03b9 \u2192 Prop := fun i => i \u2260 t.i\n\u03b1 : Type u_1 := { x // p x }\n\u03b2 : Type u_1 := { x // \u00acp x }\ng : (\u03b1 \u2192 \u211d) \u2192 (\u03b2 \u2192 \u211d) \u2192 \u03b2 \u2192 \u211d := fun a b => (fun x => t.c * a { val := t.j, property := (_ : t.j \u2260 t.i) }) + b\nF : (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) \u2192 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := fun p => (id p.1, g p.1 p.2)\ne : (\u03b9 \u2192 \u211d) \u2243\u1d50 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := MeasurableEquiv.piEquivPiSubtypeProd (fun x => \u211d) p\nthis : \u2191(\u2191toLin' (TransvectionStruct.toMatrix t)) = \u2191(MeasurableEquiv.symm e) \u2218 F \u2218 \u2191e\n\u22a2 MeasurePreserving (\u2191(MeasurableEquiv.symm e) \u2218 F \u2218 \u2191e)", "state_after": "\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nt : TransvectionStruct \u03b9 \u211d\np : \u03b9 \u2192 Prop := fun i => i \u2260 t.i\n\u03b1 : Type u_1 := { x // p x }\n\u03b2 : Type u_1 := { x // \u00acp x }\ng : (\u03b1 \u2192 \u211d) \u2192 (\u03b2 \u2192 \u211d) \u2192 \u03b2 \u2192 \u211d := fun a b => (fun x => t.c * a { val := t.j, property := (_ : t.j \u2260 t.i) }) + b\nF : (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) \u2192 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := fun p => (id p.1, g p.1 p.2)\ne : (\u03b9 \u2192 \u211d) \u2243\u1d50 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := MeasurableEquiv.piEquivPiSubtypeProd (fun x => \u211d) p\nthis : \u2191(\u2191toLin' (TransvectionStruct.toMatrix t)) = \u2191(MeasurableEquiv.symm e) \u2218 F \u2218 \u2191e\nA : MeasurePreserving \u2191e\n\u22a2 MeasurePreserving (\u2191(MeasurableEquiv.symm e) \u2218 F \u2218 \u2191e)"}, {"tactic": "have B : MeasurePreserving F :=\n  haveI g_meas : Measurable (Function.uncurry g) := by\n    have : Measurable fun c : \u03b1 \u2192 \u211d => c \u27e8t.j, t.hij.symm\u27e9 :=\n      measurable_pi_apply \u27e8t.j, t.hij.symm\u27e9\n    refine Measurable.add ?_ measurable_snd\n    refine measurable_pi_lambda _ fun _ => Measurable.const_mul ?_ _\n    exact this.comp measurable_fst\n  (MeasurePreserving.id _).skew_product g_meas\n    (eventually_of_forall fun a => map_add_left_eq_self\n      (Measure.pi fun _ => (stdOrthonormalBasis \u211d \u211d).toBasis.addHaar) _)", "annotated_tactic": ["have B : <a>MeasurePreserving</a> F :=\n    haveI g_meas : <a>Measurable</a> (<a>Function.uncurry</a> g) := by\n      have : <a>Measurable</a> fun c : \u03b1 \u2192 \u211d => c \u27e8t.j, t.hij.symm\u27e9 :=\n        <a>measurable_pi_apply</a> \u27e8t.j, t.hij.symm\u27e9\n      refine <a>Measurable.add</a> ?_ <a>measurable_snd</a>\n      refine <a>measurable_pi_lambda</a> _ fun _ => <a>Measurable.const_mul</a> ?_ _\n      exact this.comp <a>measurable_fst</a>\n    (<a>MeasurePreserving.id</a> _).<a>skew_product</a> g_meas\n      (<a>eventually_of_forall</a> fun a => <a>map_add_left_eq_self</a>\n        (<a>Measure.pi</a> fun _ => (<a>stdOrthonormalBasis</a> \u211d \u211d).toBasis.addHaar) _)", [{"full_name": "MeasureTheory.MeasurePreserving", "def_path": "Mathlib/Dynamics/Ergodic/MeasurePreserving.lean", "def_pos": [42, 11], "def_end_pos": [42, 28]}, {"full_name": "Measurable", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [535, 5], "def_end_pos": [535, 15]}, {"full_name": "Function.uncurry", "def_path": "Mathlib/Init/Function.lean", "def_pos": [217, 5], "def_end_pos": [217, 12]}, {"full_name": "Measurable", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [535, 5], "def_end_pos": [535, 15]}, {"full_name": "measurable_pi_apply", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [896, 9], "def_end_pos": [896, 28]}, {"full_name": "Measurable.add", "def_path": "Mathlib/MeasureTheory/Group/Arithmetic.lean", "def_pos": [140, 3], "def_end_pos": [140, 14]}, {"full_name": "measurable_snd", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [698, 9], "def_end_pos": [698, 23]}, {"full_name": "measurable_pi_lambda", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [907, 9], "def_end_pos": [907, 29]}, {"full_name": "Measurable.const_mul", "def_path": "Mathlib/MeasureTheory/Group/Arithmetic.lean", "def_pos": [106, 9], "def_end_pos": [106, 29]}, {"full_name": "measurable_fst", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [692, 9], "def_end_pos": [692, 23]}, {"full_name": "MeasureTheory.MeasurePreserving.id", "def_path": "Mathlib/Dynamics/Ergodic/MeasurePreserving.lean", "def_pos": [57, 19], "def_end_pos": [57, 21]}, {"full_name": "MeasureTheory.MeasurePreserving.skew_product", "def_path": "Mathlib/MeasureTheory/Constructions/Prod/Basic.lean", "def_pos": [718, 9], "def_end_pos": [718, 21]}, {"full_name": "Filter.eventually_of_forall", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1111, 9], "def_end_pos": [1111, 29]}, {"full_name": "MeasureTheory.map_add_left_eq_self", "def_path": "Mathlib/MeasureTheory/Group/Measure.lean", "def_pos": [83, 3], "def_end_pos": [83, 14]}, {"full_name": "MeasureTheory.Measure.pi", "def_path": "Mathlib/MeasureTheory/Constructions/Pi.lean", "def_pos": [303, 27], "def_end_pos": [303, 29]}, {"full_name": "stdOrthonormalBasis", "def_path": "Mathlib/Analysis/InnerProductSpace/PiL2.lean", "def_pos": [833, 17], "def_end_pos": [833, 36]}]], "state_before": "\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nt : TransvectionStruct \u03b9 \u211d\np : \u03b9 \u2192 Prop := fun i => i \u2260 t.i\n\u03b1 : Type u_1 := { x // p x }\n\u03b2 : Type u_1 := { x // \u00acp x }\ng : (\u03b1 \u2192 \u211d) \u2192 (\u03b2 \u2192 \u211d) \u2192 \u03b2 \u2192 \u211d := fun a b => (fun x => t.c * a { val := t.j, property := (_ : t.j \u2260 t.i) }) + b\nF : (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) \u2192 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := fun p => (id p.1, g p.1 p.2)\ne : (\u03b9 \u2192 \u211d) \u2243\u1d50 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := MeasurableEquiv.piEquivPiSubtypeProd (fun x => \u211d) p\nthis : \u2191(\u2191toLin' (TransvectionStruct.toMatrix t)) = \u2191(MeasurableEquiv.symm e) \u2218 F \u2218 \u2191e\nA : MeasurePreserving \u2191e\n\u22a2 MeasurePreserving (\u2191(MeasurableEquiv.symm e) \u2218 F \u2218 \u2191e)", "state_after": "\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nt : TransvectionStruct \u03b9 \u211d\np : \u03b9 \u2192 Prop := fun i => i \u2260 t.i\n\u03b1 : Type u_1 := { x // p x }\n\u03b2 : Type u_1 := { x // \u00acp x }\ng : (\u03b1 \u2192 \u211d) \u2192 (\u03b2 \u2192 \u211d) \u2192 \u03b2 \u2192 \u211d := fun a b => (fun x => t.c * a { val := t.j, property := (_ : t.j \u2260 t.i) }) + b\nF : (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) \u2192 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := fun p => (id p.1, g p.1 p.2)\ne : (\u03b9 \u2192 \u211d) \u2243\u1d50 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := MeasurableEquiv.piEquivPiSubtypeProd (fun x => \u211d) p\nthis : \u2191(\u2191toLin' (TransvectionStruct.toMatrix t)) = \u2191(MeasurableEquiv.symm e) \u2218 F \u2218 \u2191e\nA : MeasurePreserving \u2191e\nB : MeasurePreserving F\n\u22a2 MeasurePreserving (\u2191(MeasurableEquiv.symm e) \u2218 F \u2218 \u2191e)"}, {"tactic": "exact ((A.symm e).comp B).comp A", "annotated_tactic": ["exact ((A.symm e).<a>comp</a> B).<a>comp</a> A", [{"full_name": "MeasureTheory.MeasurePreserving.comp", "def_path": "Mathlib/Dynamics/Ergodic/MeasurePreserving.lean", "def_pos": [102, 19], "def_end_pos": [102, 23]}, {"full_name": "MeasureTheory.MeasurePreserving.comp", "def_path": "Mathlib/Dynamics/Ergodic/MeasurePreserving.lean", "def_pos": [102, 19], "def_end_pos": [102, 23]}]], "state_before": "\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nt : TransvectionStruct \u03b9 \u211d\np : \u03b9 \u2192 Prop := fun i => i \u2260 t.i\n\u03b1 : Type u_1 := { x // p x }\n\u03b2 : Type u_1 := { x // \u00acp x }\ng : (\u03b1 \u2192 \u211d) \u2192 (\u03b2 \u2192 \u211d) \u2192 \u03b2 \u2192 \u211d := fun a b => (fun x => t.c * a { val := t.j, property := (_ : t.j \u2260 t.i) }) + b\nF : (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) \u2192 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := fun p => (id p.1, g p.1 p.2)\ne : (\u03b9 \u2192 \u211d) \u2243\u1d50 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := MeasurableEquiv.piEquivPiSubtypeProd (fun x => \u211d) p\nthis : \u2191(\u2191toLin' (TransvectionStruct.toMatrix t)) = \u2191(MeasurableEquiv.symm e) \u2218 F \u2218 \u2191e\nA : MeasurePreserving \u2191e\nB : MeasurePreserving F\n\u22a2 MeasurePreserving (\u2191(MeasurableEquiv.symm e) \u2218 F \u2218 \u2191e)", "state_after": "no goals"}, {"tactic": "ext f k", "annotated_tactic": ["ext f k", []], "state_before": "case mk\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nt_i t_j : \u03b9\nt_hij : t_i \u2260 t_j\nt_c : \u211d\np : \u03b9 \u2192 Prop := fun i => i \u2260 { i := t_i, j := t_j, hij := t_hij, c := t_c }.i\n\u03b1 : Type u_1 := { x // p x }\n\u03b2 : Type u_1 := { x // \u00acp x }\ng : (\u03b1 \u2192 \u211d) \u2192 (\u03b2 \u2192 \u211d) \u2192 \u03b2 \u2192 \u211d :=\n  fun a b =>\n    (fun x =>\n        { i := t_i, j := t_j, hij := t_hij, c := t_c }.c *\n          a\n            { val := { i := t_i, j := t_j, hij := t_hij, c := t_c }.j,\n              property :=\n                (_ :\n                  { i := t_i, j := t_j, hij := t_hij, c := t_c }.j \u2260\n                    { i := t_i, j := t_j, hij := t_hij, c := t_c }.i) }) +\n      b\nF : (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) \u2192 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := fun p => (id p.1, g p.1 p.2)\ne : (\u03b9 \u2192 \u211d) \u2243\u1d50 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := MeasurableEquiv.piEquivPiSubtypeProd (fun x => \u211d) p\n\u22a2 \u2191(\u2191toLin' (TransvectionStruct.toMatrix { i := t_i, j := t_j, hij := t_hij, c := t_c })) =\n    \u2191(MeasurableEquiv.symm e) \u2218 F \u2218 \u2191e", "state_after": "case mk.h.h\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nt_i t_j : \u03b9\nt_hij : t_i \u2260 t_j\nt_c : \u211d\np : \u03b9 \u2192 Prop := fun i => i \u2260 { i := t_i, j := t_j, hij := t_hij, c := t_c }.i\n\u03b1 : Type u_1 := { x // p x }\n\u03b2 : Type u_1 := { x // \u00acp x }\ng : (\u03b1 \u2192 \u211d) \u2192 (\u03b2 \u2192 \u211d) \u2192 \u03b2 \u2192 \u211d :=\n  fun a b =>\n    (fun x =>\n        { i := t_i, j := t_j, hij := t_hij, c := t_c }.c *\n          a\n            { val := { i := t_i, j := t_j, hij := t_hij, c := t_c }.j,\n              property :=\n                (_ :\n                  { i := t_i, j := t_j, hij := t_hij, c := t_c }.j \u2260\n                    { i := t_i, j := t_j, hij := t_hij, c := t_c }.i) }) +\n      b\nF : (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) \u2192 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := fun p => (id p.1, g p.1 p.2)\ne : (\u03b9 \u2192 \u211d) \u2243\u1d50 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := MeasurableEquiv.piEquivPiSubtypeProd (fun x => \u211d) p\nf : \u03b9 \u2192 \u211d\nk : \u03b9\n\u22a2 \u2191(\u2191toLin' (TransvectionStruct.toMatrix { i := t_i, j := t_j, hij := t_hij, c := t_c })) f k =\n    (\u2191(MeasurableEquiv.symm e) \u2218 F \u2218 \u2191e) f k"}, {"tactic": "simp only [LinearEquiv.map_smul, dite_eq_ite, LinearMap.id_coe, ite_not,\n  Algebra.id.smul_eq_mul, one_mul, dotProduct, stdBasisMatrix,\n  MeasurableEquiv.piEquivPiSubtypeProd_symm_apply, id.def, transvection, Pi.add_apply,\n  zero_mul, LinearMap.smul_apply, Function.comp_apply,\n  MeasurableEquiv.piEquivPiSubtypeProd_apply, Matrix.TransvectionStruct.toMatrix_mk,\n  Matrix.mulVec, LinearEquiv.map_add, ite_mul, Matrix.toLin'_apply, Pi.smul_apply,\n  Subtype.coe_mk, LinearMap.add_apply, Finset.sum_congr, Matrix.toLin'_one]", "annotated_tactic": ["simp only [<a>LinearEquiv.map_smul</a>, <a>dite_eq_ite</a>, <a>LinearMap.id_coe</a>, <a>ite_not</a>,\n      <a>Algebra.id.smul_eq_mul</a>, <a>one_mul</a>, <a>dotProduct</a>, <a>stdBasisMatrix</a>,\n      <a>MeasurableEquiv.piEquivPiSubtypeProd_symm_apply</a>, <a>id.def</a>, <a>transvection</a>, <a>Pi.add_apply</a>,\n      <a>zero_mul</a>, <a>LinearMap.smul_apply</a>, <a>Function.comp_apply</a>,\n      <a>MeasurableEquiv.piEquivPiSubtypeProd_apply</a>, <a>Matrix.TransvectionStruct.toMatrix_mk</a>,\n      <a>Matrix.mulVec</a>, <a>LinearEquiv.map_add</a>, <a>ite_mul</a>, <a>Matrix.toLin'_apply</a>, <a>Pi.smul_apply</a>,\n      <a>Subtype.coe_mk</a>, <a>LinearMap.add_apply</a>, <a>Finset.sum_congr</a>, <a>Matrix.toLin'_one</a>]", [{"full_name": "LinearEquiv.map_smul", "def_path": "Mathlib/Algebra/Module/Equiv.lean", "def_pos": [502, 9], "def_end_pos": [502, 17]}, {"full_name": "dite_eq_ite", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [1217, 17], "def_end_pos": [1217, 28]}, {"full_name": "LinearMap.id_coe", "def_path": "Mathlib/Algebra/Module/LinearMap.lean", "def_pos": [272, 9], "def_end_pos": [272, 15]}, {"full_name": "ite_not", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [749, 17], "def_end_pos": [749, 24]}, {"full_name": "Algebra.id.smul_eq_mul", "def_path": "Mathlib/Algebra/Algebra/Basic.lean", "def_pos": [453, 9], "def_end_pos": [453, 20]}, {"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [464, 9], "def_end_pos": [464, 16]}, {"full_name": "Matrix.dotProduct", "def_path": "Mathlib/Data/Matrix/Basic.lean", "def_pos": [705, 5], "def_end_pos": [705, 15]}, {"full_name": "Matrix.stdBasisMatrix", "def_path": "Mathlib/Data/Matrix/Basis.lean", "def_pos": [36, 5], "def_end_pos": [36, 19]}, {"full_name": "MeasurableEquiv.piEquivPiSubtypeProd_symm_apply", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [1669, 3], "def_end_pos": [1669, 47]}, {"full_name": "id.def", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [527, 9], "def_end_pos": [527, 15]}, {"full_name": "Matrix.transvection", "def_path": "Mathlib/LinearAlgebra/Matrix/Transvection.lean", "def_pos": [84, 5], "def_end_pos": [84, 17]}, {"full_name": "Pi.add_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [82, 3], "def_end_pos": [82, 14]}, {"full_name": "MulZeroClass.zero_mul", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [36, 3], "def_end_pos": [36, 11]}, {"full_name": "LinearMap.smul_apply", "def_path": "Mathlib/Algebra/Module/LinearMap.lean", "def_pos": [841, 9], "def_end_pos": [841, 19]}, {"full_name": "Function.comp_apply", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [33, 17], "def_end_pos": [33, 36]}, {"full_name": "MeasurableEquiv.piEquivPiSubtypeProd_apply", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [1669, 3], "def_end_pos": [1669, 47]}, {"full_name": "Matrix.TransvectionStruct.toMatrix_mk", "def_path": "Mathlib/LinearAlgebra/Matrix/Transvection.lean", "def_pos": [174, 9], "def_end_pos": [174, 20]}, {"full_name": "Matrix.mulVec", "def_path": "Mathlib/Data/Matrix/Basic.lean", "def_pos": [1672, 5], "def_end_pos": [1672, 11]}, {"full_name": "LinearEquiv.map_add", "def_path": "Mathlib/Algebra/Module/Equiv.lean", "def_pos": [490, 19], "def_end_pos": [490, 26]}, {"full_name": "ite_mul", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [201, 9], "def_end_pos": [201, 16]}, {"full_name": "Matrix.toLin'_apply", "def_path": "Mathlib/LinearAlgebra/Matrix/ToLin.lean", "def_pos": [349, 9], "def_end_pos": [349, 28]}, {"full_name": "Pi.smul_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [116, 60], "def_end_pos": [116, 70]}, {"full_name": "Subtype.coe_mk", "def_path": "Mathlib/Data/Subtype.lean", "def_pos": [99, 9], "def_end_pos": [99, 15]}, {"full_name": "LinearMap.add_apply", "def_path": "Mathlib/Algebra/Module/LinearMap.lean", "def_pos": [912, 9], "def_end_pos": [912, 18]}, {"full_name": "Finset.sum_congr", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [376, 3], "def_end_pos": [376, 14]}, {"full_name": "Matrix.toLin'_one", "def_path": "Mathlib/LinearAlgebra/Matrix/ToLin.lean", "def_pos": [354, 9], "def_end_pos": [354, 26]}]], "state_before": "case mk.h.h\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nt_i t_j : \u03b9\nt_hij : t_i \u2260 t_j\nt_c : \u211d\np : \u03b9 \u2192 Prop := fun i => i \u2260 { i := t_i, j := t_j, hij := t_hij, c := t_c }.i\n\u03b1 : Type u_1 := { x // p x }\n\u03b2 : Type u_1 := { x // \u00acp x }\ng : (\u03b1 \u2192 \u211d) \u2192 (\u03b2 \u2192 \u211d) \u2192 \u03b2 \u2192 \u211d :=\n  fun a b =>\n    (fun x =>\n        { i := t_i, j := t_j, hij := t_hij, c := t_c }.c *\n          a\n            { val := { i := t_i, j := t_j, hij := t_hij, c := t_c }.j,\n              property :=\n                (_ :\n                  { i := t_i, j := t_j, hij := t_hij, c := t_c }.j \u2260\n                    { i := t_i, j := t_j, hij := t_hij, c := t_c }.i) }) +\n      b\nF : (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) \u2192 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := fun p => (id p.1, g p.1 p.2)\ne : (\u03b9 \u2192 \u211d) \u2243\u1d50 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := MeasurableEquiv.piEquivPiSubtypeProd (fun x => \u211d) p\nf : \u03b9 \u2192 \u211d\nk : \u03b9\n\u22a2 \u2191(\u2191toLin' (TransvectionStruct.toMatrix { i := t_i, j := t_j, hij := t_hij, c := t_c })) f k =\n    (\u2191(MeasurableEquiv.symm e) \u2218 F \u2218 \u2191e) f k", "state_after": "case mk.h.h\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nt_i t_j : \u03b9\nt_hij : t_i \u2260 t_j\nt_c : \u211d\np : \u03b9 \u2192 Prop := fun i => i \u2260 { i := t_i, j := t_j, hij := t_hij, c := t_c }.i\n\u03b1 : Type u_1 := { x // p x }\n\u03b2 : Type u_1 := { x // \u00acp x }\ng : (\u03b1 \u2192 \u211d) \u2192 (\u03b2 \u2192 \u211d) \u2192 \u03b2 \u2192 \u211d :=\n  fun a b =>\n    (fun x =>\n        { i := t_i, j := t_j, hij := t_hij, c := t_c }.c *\n          a\n            { val := { i := t_i, j := t_j, hij := t_hij, c := t_c }.j,\n              property :=\n                (_ :\n                  { i := t_i, j := t_j, hij := t_hij, c := t_c }.j \u2260\n                    { i := t_i, j := t_j, hij := t_hij, c := t_c }.i) }) +\n      b\nF : (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) \u2192 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := fun p => (id p.1, g p.1 p.2)\ne : (\u03b9 \u2192 \u211d) \u2243\u1d50 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := MeasurableEquiv.piEquivPiSubtypeProd (fun x => \u211d) p\nf : \u03b9 \u2192 \u211d\nk : \u03b9\n\u22a2 (f k + \u2211 x : \u03b9, if t_i = k \u2227 t_j = x then t_c * f x else 0) = if k = t_i then t_c * f t_j + f k else f k"}, {"tactic": "by_cases h : t_i = k", "annotated_tactic": ["by_cases h : t_i = k", []], "state_before": "case mk.h.h\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nt_i t_j : \u03b9\nt_hij : t_i \u2260 t_j\nt_c : \u211d\np : \u03b9 \u2192 Prop := fun i => i \u2260 { i := t_i, j := t_j, hij := t_hij, c := t_c }.i\n\u03b1 : Type u_1 := { x // p x }\n\u03b2 : Type u_1 := { x // \u00acp x }\ng : (\u03b1 \u2192 \u211d) \u2192 (\u03b2 \u2192 \u211d) \u2192 \u03b2 \u2192 \u211d :=\n  fun a b =>\n    (fun x =>\n        { i := t_i, j := t_j, hij := t_hij, c := t_c }.c *\n          a\n            { val := { i := t_i, j := t_j, hij := t_hij, c := t_c }.j,\n              property :=\n                (_ :\n                  { i := t_i, j := t_j, hij := t_hij, c := t_c }.j \u2260\n                    { i := t_i, j := t_j, hij := t_hij, c := t_c }.i) }) +\n      b\nF : (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) \u2192 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := fun p => (id p.1, g p.1 p.2)\ne : (\u03b9 \u2192 \u211d) \u2243\u1d50 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := MeasurableEquiv.piEquivPiSubtypeProd (fun x => \u211d) p\nf : \u03b9 \u2192 \u211d\nk : \u03b9\n\u22a2 (f k + \u2211 x : \u03b9, if t_i = k \u2227 t_j = x then t_c * f x else 0) = if k = t_i then t_c * f t_j + f k else f k", "state_after": "case pos\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nt_i t_j : \u03b9\nt_hij : t_i \u2260 t_j\nt_c : \u211d\np : \u03b9 \u2192 Prop := fun i => i \u2260 { i := t_i, j := t_j, hij := t_hij, c := t_c }.i\n\u03b1 : Type u_1 := { x // p x }\n\u03b2 : Type u_1 := { x // \u00acp x }\ng : (\u03b1 \u2192 \u211d) \u2192 (\u03b2 \u2192 \u211d) \u2192 \u03b2 \u2192 \u211d :=\n  fun a b =>\n    (fun x =>\n        { i := t_i, j := t_j, hij := t_hij, c := t_c }.c *\n          a\n            { val := { i := t_i, j := t_j, hij := t_hij, c := t_c }.j,\n              property :=\n                (_ :\n                  { i := t_i, j := t_j, hij := t_hij, c := t_c }.j \u2260\n                    { i := t_i, j := t_j, hij := t_hij, c := t_c }.i) }) +\n      b\nF : (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) \u2192 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := fun p => (id p.1, g p.1 p.2)\ne : (\u03b9 \u2192 \u211d) \u2243\u1d50 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := MeasurableEquiv.piEquivPiSubtypeProd (fun x => \u211d) p\nf : \u03b9 \u2192 \u211d\nk : \u03b9\nh : t_i = k\n\u22a2 (f k + \u2211 x : \u03b9, if t_i = k \u2227 t_j = x then t_c * f x else 0) = if k = t_i then t_c * f t_j + f k else f k\n\ncase neg\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nt_i t_j : \u03b9\nt_hij : t_i \u2260 t_j\nt_c : \u211d\np : \u03b9 \u2192 Prop := fun i => i \u2260 { i := t_i, j := t_j, hij := t_hij, c := t_c }.i\n\u03b1 : Type u_1 := { x // p x }\n\u03b2 : Type u_1 := { x // \u00acp x }\ng : (\u03b1 \u2192 \u211d) \u2192 (\u03b2 \u2192 \u211d) \u2192 \u03b2 \u2192 \u211d :=\n  fun a b =>\n    (fun x =>\n        { i := t_i, j := t_j, hij := t_hij, c := t_c }.c *\n          a\n            { val := { i := t_i, j := t_j, hij := t_hij, c := t_c }.j,\n              property :=\n                (_ :\n                  { i := t_i, j := t_j, hij := t_hij, c := t_c }.j \u2260\n                    { i := t_i, j := t_j, hij := t_hij, c := t_c }.i) }) +\n      b\nF : (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) \u2192 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := fun p => (id p.1, g p.1 p.2)\ne : (\u03b9 \u2192 \u211d) \u2243\u1d50 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := MeasurableEquiv.piEquivPiSubtypeProd (fun x => \u211d) p\nf : \u03b9 \u2192 \u211d\nk : \u03b9\nh : \u00act_i = k\n\u22a2 (f k + \u2211 x : \u03b9, if t_i = k \u2227 t_j = x then t_c * f x else 0) = if k = t_i then t_c * f t_j + f k else f k"}, {"tactic": "simp only [h, true_and_iff, Finset.mem_univ, if_true, eq_self_iff_true, Finset.sum_ite_eq,\n  one_apply, boole_mul, add_comm]", "annotated_tactic": ["simp only [h, <a>true_and_iff</a>, <a>Finset.mem_univ</a>, <a>if_true</a>, <a>eq_self_iff_true</a>, <a>Finset.sum_ite_eq</a>,\n        <a>one_apply</a>, <a>boole_mul</a>, <a>add_comm</a>]", [{"full_name": "true_and_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [147, 9], "def_end_pos": [147, 21]}, {"full_name": "Finset.mem_univ", "def_path": "Mathlib/Data/Fintype/Basic.lean", "def_pos": [72, 9], "def_end_pos": [72, 17]}, {"full_name": "if_true", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [727, 17], "def_end_pos": [727, 24]}, {"full_name": "eq_self_iff_true", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [86, 9], "def_end_pos": [86, 25]}, {"full_name": "Finset.sum_ite_eq", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [1084, 3], "def_end_pos": [1084, 14]}, {"full_name": "Matrix.one_apply", "def_path": "Mathlib/Data/Matrix/Basic.lean", "def_pos": [531, 9], "def_end_pos": [531, 18]}, {"full_name": "boole_mul", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [222, 9], "def_end_pos": [222, 18]}, {"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [301, 3], "def_end_pos": [301, 14]}]], "state_before": "case pos\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nt_i t_j : \u03b9\nt_hij : t_i \u2260 t_j\nt_c : \u211d\np : \u03b9 \u2192 Prop := fun i => i \u2260 { i := t_i, j := t_j, hij := t_hij, c := t_c }.i\n\u03b1 : Type u_1 := { x // p x }\n\u03b2 : Type u_1 := { x // \u00acp x }\ng : (\u03b1 \u2192 \u211d) \u2192 (\u03b2 \u2192 \u211d) \u2192 \u03b2 \u2192 \u211d :=\n  fun a b =>\n    (fun x =>\n        { i := t_i, j := t_j, hij := t_hij, c := t_c }.c *\n          a\n            { val := { i := t_i, j := t_j, hij := t_hij, c := t_c }.j,\n              property :=\n                (_ :\n                  { i := t_i, j := t_j, hij := t_hij, c := t_c }.j \u2260\n                    { i := t_i, j := t_j, hij := t_hij, c := t_c }.i) }) +\n      b\nF : (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) \u2192 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := fun p => (id p.1, g p.1 p.2)\ne : (\u03b9 \u2192 \u211d) \u2243\u1d50 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := MeasurableEquiv.piEquivPiSubtypeProd (fun x => \u211d) p\nf : \u03b9 \u2192 \u211d\nk : \u03b9\nh : t_i = k\n\u22a2 (f k + \u2211 x : \u03b9, if t_i = k \u2227 t_j = x then t_c * f x else 0) = if k = t_i then t_c * f t_j + f k else f k", "state_after": "no goals"}, {"tactic": "simp only [h, Ne.symm h, add_zero, if_false, Finset.sum_const_zero, false_and_iff,\n  mul_zero]", "annotated_tactic": ["simp only [h, <a>Ne.symm</a> h, <a>add_zero</a>, <a>if_false</a>, <a>Finset.sum_const_zero</a>, <a>false_and_iff</a>,\n        <a>mul_zero</a>]", [{"full_name": "Ne.symm", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [575, 9], "def_end_pos": [575, 16]}, {"full_name": "add_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [469, 3], "def_end_pos": [469, 14]}, {"full_name": "if_false", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [729, 17], "def_end_pos": [729, 25]}, {"full_name": "Finset.sum_const_zero", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [356, 3], "def_end_pos": [356, 14]}, {"full_name": "false_and_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [151, 9], "def_end_pos": [151, 22]}, {"full_name": "MulZeroClass.mul_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [38, 3], "def_end_pos": [38, 11]}]], "state_before": "case neg\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nt_i t_j : \u03b9\nt_hij : t_i \u2260 t_j\nt_c : \u211d\np : \u03b9 \u2192 Prop := fun i => i \u2260 { i := t_i, j := t_j, hij := t_hij, c := t_c }.i\n\u03b1 : Type u_1 := { x // p x }\n\u03b2 : Type u_1 := { x // \u00acp x }\ng : (\u03b1 \u2192 \u211d) \u2192 (\u03b2 \u2192 \u211d) \u2192 \u03b2 \u2192 \u211d :=\n  fun a b =>\n    (fun x =>\n        { i := t_i, j := t_j, hij := t_hij, c := t_c }.c *\n          a\n            { val := { i := t_i, j := t_j, hij := t_hij, c := t_c }.j,\n              property :=\n                (_ :\n                  { i := t_i, j := t_j, hij := t_hij, c := t_c }.j \u2260\n                    { i := t_i, j := t_j, hij := t_hij, c := t_c }.i) }) +\n      b\nF : (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) \u2192 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := fun p => (id p.1, g p.1 p.2)\ne : (\u03b9 \u2192 \u211d) \u2243\u1d50 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := MeasurableEquiv.piEquivPiSubtypeProd (fun x => \u211d) p\nf : \u03b9 \u2192 \u211d\nk : \u03b9\nh : \u00act_i = k\n\u22a2 (f k + \u2211 x : \u03b9, if t_i = k \u2227 t_j = x then t_c * f x else 0) = if k = t_i then t_c * f t_j + f k else f k", "state_after": "no goals"}, {"tactic": "convert volume_preserving_piEquivPiSubtypeProd (fun _ : \u03b9 => \u211d) p", "annotated_tactic": ["convert <a>volume_preserving_piEquivPiSubtypeProd</a> (fun _ : \u03b9 => \u211d) p", [{"full_name": "MeasureTheory.volume_preserving_piEquivPiSubtypeProd", "def_path": "Mathlib/MeasureTheory/Constructions/Pi.lean", "def_pos": [747, 9], "def_end_pos": [747, 47]}]], "state_before": "\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nt : TransvectionStruct \u03b9 \u211d\np : \u03b9 \u2192 Prop := fun i => i \u2260 t.i\n\u03b1 : Type u_1 := { x // p x }\n\u03b2 : Type u_1 := { x // \u00acp x }\ng : (\u03b1 \u2192 \u211d) \u2192 (\u03b2 \u2192 \u211d) \u2192 \u03b2 \u2192 \u211d := fun a b => (fun x => t.c * a { val := t.j, property := (_ : t.j \u2260 t.i) }) + b\nF : (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) \u2192 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := fun p => (id p.1, g p.1 p.2)\ne : (\u03b9 \u2192 \u211d) \u2243\u1d50 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := MeasurableEquiv.piEquivPiSubtypeProd (fun x => \u211d) p\nthis : \u2191(\u2191toLin' (TransvectionStruct.toMatrix t)) = \u2191(MeasurableEquiv.symm e) \u2218 F \u2218 \u2191e\n\u22a2 MeasurePreserving \u2191e", "state_after": "no goals"}, {"tactic": "have : Measurable fun c : \u03b1 \u2192 \u211d => c \u27e8t.j, t.hij.symm\u27e9 :=\n  measurable_pi_apply \u27e8t.j, t.hij.symm\u27e9", "annotated_tactic": ["have : <a>Measurable</a> fun c : \u03b1 \u2192 \u211d => c \u27e8t.j, t.hij.symm\u27e9 :=\n        <a>measurable_pi_apply</a> \u27e8t.j, t.hij.symm\u27e9", [{"full_name": "Measurable", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [535, 5], "def_end_pos": [535, 15]}, {"full_name": "measurable_pi_apply", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [896, 9], "def_end_pos": [896, 28]}]], "state_before": "\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nt : TransvectionStruct \u03b9 \u211d\np : \u03b9 \u2192 Prop := fun i => i \u2260 t.i\n\u03b1 : Type u_1 := { x // p x }\n\u03b2 : Type u_1 := { x // \u00acp x }\ng : (\u03b1 \u2192 \u211d) \u2192 (\u03b2 \u2192 \u211d) \u2192 \u03b2 \u2192 \u211d := fun a b => (fun x => t.c * a { val := t.j, property := (_ : t.j \u2260 t.i) }) + b\nF : (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) \u2192 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := fun p => (id p.1, g p.1 p.2)\ne : (\u03b9 \u2192 \u211d) \u2243\u1d50 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := MeasurableEquiv.piEquivPiSubtypeProd (fun x => \u211d) p\nthis : \u2191(\u2191toLin' (TransvectionStruct.toMatrix t)) = \u2191(MeasurableEquiv.symm e) \u2218 F \u2218 \u2191e\nA : MeasurePreserving \u2191e\n\u22a2 Measurable (Function.uncurry g)", "state_after": "\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nt : TransvectionStruct \u03b9 \u211d\np : \u03b9 \u2192 Prop := fun i => i \u2260 t.i\n\u03b1 : Type u_1 := { x // p x }\n\u03b2 : Type u_1 := { x // \u00acp x }\ng : (\u03b1 \u2192 \u211d) \u2192 (\u03b2 \u2192 \u211d) \u2192 \u03b2 \u2192 \u211d := fun a b => (fun x => t.c * a { val := t.j, property := (_ : t.j \u2260 t.i) }) + b\nF : (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) \u2192 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := fun p => (id p.1, g p.1 p.2)\ne : (\u03b9 \u2192 \u211d) \u2243\u1d50 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := MeasurableEquiv.piEquivPiSubtypeProd (fun x => \u211d) p\nthis\u271d : \u2191(\u2191toLin' (TransvectionStruct.toMatrix t)) = \u2191(MeasurableEquiv.symm e) \u2218 F \u2218 \u2191e\nA : MeasurePreserving \u2191e\nthis : Measurable fun c => c { val := t.j, property := (_ : t.j \u2260 t.i) }\n\u22a2 Measurable (Function.uncurry g)"}, {"tactic": "refine Measurable.add ?_ measurable_snd", "annotated_tactic": ["refine <a>Measurable.add</a> ?_ <a>measurable_snd</a>", [{"full_name": "Measurable.add", "def_path": "Mathlib/MeasureTheory/Group/Arithmetic.lean", "def_pos": [140, 3], "def_end_pos": [140, 14]}, {"full_name": "measurable_snd", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [698, 9], "def_end_pos": [698, 23]}]], "state_before": "\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nt : TransvectionStruct \u03b9 \u211d\np : \u03b9 \u2192 Prop := fun i => i \u2260 t.i\n\u03b1 : Type u_1 := { x // p x }\n\u03b2 : Type u_1 := { x // \u00acp x }\ng : (\u03b1 \u2192 \u211d) \u2192 (\u03b2 \u2192 \u211d) \u2192 \u03b2 \u2192 \u211d := fun a b => (fun x => t.c * a { val := t.j, property := (_ : t.j \u2260 t.i) }) + b\nF : (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) \u2192 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := fun p => (id p.1, g p.1 p.2)\ne : (\u03b9 \u2192 \u211d) \u2243\u1d50 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := MeasurableEquiv.piEquivPiSubtypeProd (fun x => \u211d) p\nthis\u271d : \u2191(\u2191toLin' (TransvectionStruct.toMatrix t)) = \u2191(MeasurableEquiv.symm e) \u2218 F \u2218 \u2191e\nA : MeasurePreserving \u2191e\nthis : Measurable fun c => c { val := t.j, property := (_ : t.j \u2260 t.i) }\n\u22a2 Measurable (Function.uncurry g)", "state_after": "\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nt : TransvectionStruct \u03b9 \u211d\np : \u03b9 \u2192 Prop := fun i => i \u2260 t.i\n\u03b1 : Type u_1 := { x // p x }\n\u03b2 : Type u_1 := { x // \u00acp x }\ng : (\u03b1 \u2192 \u211d) \u2192 (\u03b2 \u2192 \u211d) \u2192 \u03b2 \u2192 \u211d := fun a b => (fun x => t.c * a { val := t.j, property := (_ : t.j \u2260 t.i) }) + b\nF : (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) \u2192 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := fun p => (id p.1, g p.1 p.2)\ne : (\u03b9 \u2192 \u211d) \u2243\u1d50 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := MeasurableEquiv.piEquivPiSubtypeProd (fun x => \u211d) p\nthis\u271d : \u2191(\u2191toLin' (TransvectionStruct.toMatrix t)) = \u2191(MeasurableEquiv.symm e) \u2218 F \u2218 \u2191e\nA : MeasurePreserving \u2191e\nthis : Measurable fun c => c { val := t.j, property := (_ : t.j \u2260 t.i) }\n\u22a2 Measurable fun a x => t.c * a.1 { val := t.j, property := (_ : t.j \u2260 t.i) }"}, {"tactic": "refine measurable_pi_lambda _ fun _ => Measurable.const_mul ?_ _", "annotated_tactic": ["refine <a>measurable_pi_lambda</a> _ fun _ => <a>Measurable.const_mul</a> ?_ _", [{"full_name": "measurable_pi_lambda", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [907, 9], "def_end_pos": [907, 29]}, {"full_name": "Measurable.const_mul", "def_path": "Mathlib/MeasureTheory/Group/Arithmetic.lean", "def_pos": [106, 9], "def_end_pos": [106, 29]}]], "state_before": "\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nt : TransvectionStruct \u03b9 \u211d\np : \u03b9 \u2192 Prop := fun i => i \u2260 t.i\n\u03b1 : Type u_1 := { x // p x }\n\u03b2 : Type u_1 := { x // \u00acp x }\ng : (\u03b1 \u2192 \u211d) \u2192 (\u03b2 \u2192 \u211d) \u2192 \u03b2 \u2192 \u211d := fun a b => (fun x => t.c * a { val := t.j, property := (_ : t.j \u2260 t.i) }) + b\nF : (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) \u2192 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := fun p => (id p.1, g p.1 p.2)\ne : (\u03b9 \u2192 \u211d) \u2243\u1d50 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := MeasurableEquiv.piEquivPiSubtypeProd (fun x => \u211d) p\nthis\u271d : \u2191(\u2191toLin' (TransvectionStruct.toMatrix t)) = \u2191(MeasurableEquiv.symm e) \u2218 F \u2218 \u2191e\nA : MeasurePreserving \u2191e\nthis : Measurable fun c => c { val := t.j, property := (_ : t.j \u2260 t.i) }\n\u22a2 Measurable fun a x => t.c * a.1 { val := t.j, property := (_ : t.j \u2260 t.i) }", "state_after": "\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nt : TransvectionStruct \u03b9 \u211d\np : \u03b9 \u2192 Prop := fun i => i \u2260 t.i\n\u03b1 : Type u_1 := { x // p x }\n\u03b2 : Type u_1 := { x // \u00acp x }\ng : (\u03b1 \u2192 \u211d) \u2192 (\u03b2 \u2192 \u211d) \u2192 \u03b2 \u2192 \u211d := fun a b => (fun x => t.c * a { val := t.j, property := (_ : t.j \u2260 t.i) }) + b\nF : (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) \u2192 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := fun p => (id p.1, g p.1 p.2)\ne : (\u03b9 \u2192 \u211d) \u2243\u1d50 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := MeasurableEquiv.piEquivPiSubtypeProd (fun x => \u211d) p\nthis\u271d : \u2191(\u2191toLin' (TransvectionStruct.toMatrix t)) = \u2191(MeasurableEquiv.symm e) \u2218 F \u2218 \u2191e\nA : MeasurePreserving \u2191e\nthis : Measurable fun c => c { val := t.j, property := (_ : t.j \u2260 t.i) }\nx\u271d : \u03b2\n\u22a2 Measurable fun c => c.1 { val := t.j, property := (_ : t.j \u2260 t.i) }"}, {"tactic": "exact this.comp measurable_fst", "annotated_tactic": ["exact this.comp <a>measurable_fst</a>", [{"full_name": "measurable_fst", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [692, 9], "def_end_pos": [692, 23]}]], "state_before": "\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nt : TransvectionStruct \u03b9 \u211d\np : \u03b9 \u2192 Prop := fun i => i \u2260 t.i\n\u03b1 : Type u_1 := { x // p x }\n\u03b2 : Type u_1 := { x // \u00acp x }\ng : (\u03b1 \u2192 \u211d) \u2192 (\u03b2 \u2192 \u211d) \u2192 \u03b2 \u2192 \u211d := fun a b => (fun x => t.c * a { val := t.j, property := (_ : t.j \u2260 t.i) }) + b\nF : (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) \u2192 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := fun p => (id p.1, g p.1 p.2)\ne : (\u03b9 \u2192 \u211d) \u2243\u1d50 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := MeasurableEquiv.piEquivPiSubtypeProd (fun x => \u211d) p\nthis\u271d : \u2191(\u2191toLin' (TransvectionStruct.toMatrix t)) = \u2191(MeasurableEquiv.symm e) \u2218 F \u2218 \u2191e\nA : MeasurePreserving \u2191e\nthis : Measurable fun c => c { val := t.j, property := (_ : t.j \u2260 t.i) }\nx\u271d : \u03b2\n\u22a2 Measurable fun c => c.1 { val := t.j, property := (_ : t.j \u2260 t.i) }", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Fin/Lemmas.lean", "full_name": "USize.mod_toNat", "start": [744, 9], "end": [745, 17], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Card.lean", "full_name": "Set.exists_subset_or_subset_of_two_mul_lt_ncard", "start": [980, 1], "end": [988, 43], "traced_tactics": [{"tactic": "classical\nhave hu := finite_of_ncard_ne_zero ((Nat.zero_le _).trans_lt hst).ne.symm\nrw [ncard_eq_toFinset_card _ hu,\n  Finite.toFinset_union (hu.subset (subset_union_left _ _))\n    (hu.subset (subset_union_right _ _))] at hst\nobtain \u27e8r', hnr', hr'\u27e9 := Finset.exists_subset_or_subset_of_two_mul_lt_card hst\nexact \u27e8r', by simpa, by simpa using hr'\u27e9", "annotated_tactic": ["classical\n  have hu := <a>finite_of_ncard_ne_zero</a> ((<a>Nat.zero_le</a> _).<a>trans_lt</a> hst).ne.symm\n  rw [<a>ncard_eq_toFinset_card</a> _ hu,\n    <a>Finite.toFinset_union</a> (hu.subset (<a>subset_union_left</a> _ _))\n      (hu.subset (<a>subset_union_right</a> _ _))] at hst\n  obtain \u27e8r', hnr', hr'\u27e9 := <a>Finset.exists_subset_or_subset_of_two_mul_lt_card</a> hst\n  exact \u27e8r', by simpa, by simpa using hr'\u27e9", [{"full_name": "Set.finite_of_ncard_ne_zero", "def_path": "Mathlib/Data/Set/Card.lean", "def_pos": [546, 9], "def_end_pos": [546, 32]}, {"full_name": "Nat.zero_le", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1578, 9], "def_end_pos": [1578, 20]}, {"full_name": "LE.le.trans_lt", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [124, 7], "def_end_pos": [124, 21]}, {"full_name": "Set.ncard_eq_toFinset_card", "def_path": "Mathlib/Data/Set/Card.lean", "def_pos": [489, 9], "def_end_pos": [489, 31]}, {"full_name": "Set.Finite.toFinset_union", "def_path": "Mathlib/Data/Set/Finite.lean", "def_pos": [258, 19], "def_end_pos": [258, 33]}, {"full_name": "Set.subset_union_left", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [829, 9], "def_end_pos": [829, 26]}, {"full_name": "Set.subset_union_right", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [833, 9], "def_end_pos": [833, 27]}, {"full_name": "Finset.exists_subset_or_subset_of_two_mul_lt_card", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [508, 9], "def_end_pos": [508, 51]}]], "state_before": "\u03b1 : Type u_1\ns t : Set \u03b1\nn : \u2115\nhst : 2 * n < ncard (s \u222a t)\n\u22a2 \u2203 r, n < ncard r \u2227 (r \u2286 s \u2228 r \u2286 t)", "state_after": "no goals"}, {"tactic": "have hu := finite_of_ncard_ne_zero ((Nat.zero_le _).trans_lt hst).ne.symm", "annotated_tactic": ["have hu := <a>finite_of_ncard_ne_zero</a> ((<a>Nat.zero_le</a> _).<a>trans_lt</a> hst).ne.symm", [{"full_name": "Set.finite_of_ncard_ne_zero", "def_path": "Mathlib/Data/Set/Card.lean", "def_pos": [546, 9], "def_end_pos": [546, 32]}, {"full_name": "Nat.zero_le", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1578, 9], "def_end_pos": [1578, 20]}, {"full_name": "LE.le.trans_lt", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [124, 7], "def_end_pos": [124, 21]}]], "state_before": "\u03b1 : Type u_1\ns t : Set \u03b1\nn : \u2115\nhst : 2 * n < ncard (s \u222a t)\n\u22a2 \u2203 r, n < ncard r \u2227 (r \u2286 s \u2228 r \u2286 t)", "state_after": "\u03b1 : Type u_1\ns t : Set \u03b1\nn : \u2115\nhst : 2 * n < ncard (s \u222a t)\nhu : Set.Finite (s \u222a t)\n\u22a2 \u2203 r, n < ncard r \u2227 (r \u2286 s \u2228 r \u2286 t)"}, {"tactic": "rw [ncard_eq_toFinset_card _ hu,\n  Finite.toFinset_union (hu.subset (subset_union_left _ _))\n    (hu.subset (subset_union_right _ _))] at hst", "annotated_tactic": ["rw [<a>ncard_eq_toFinset_card</a> _ hu,\n    <a>Finite.toFinset_union</a> (hu.subset (<a>subset_union_left</a> _ _))\n      (hu.subset (<a>subset_union_right</a> _ _))] at hst", [{"full_name": "Set.ncard_eq_toFinset_card", "def_path": "Mathlib/Data/Set/Card.lean", "def_pos": [489, 9], "def_end_pos": [489, 31]}, {"full_name": "Set.Finite.toFinset_union", "def_path": "Mathlib/Data/Set/Finite.lean", "def_pos": [258, 19], "def_end_pos": [258, 33]}, {"full_name": "Set.subset_union_left", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [829, 9], "def_end_pos": [829, 26]}, {"full_name": "Set.subset_union_right", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [833, 9], "def_end_pos": [833, 27]}]], "state_before": "\u03b1 : Type u_1\ns t : Set \u03b1\nn : \u2115\nhst : 2 * n < ncard (s \u222a t)\nhu : Set.Finite (s \u222a t)\n\u22a2 \u2203 r, n < ncard r \u2227 (r \u2286 s \u2228 r \u2286 t)", "state_after": "\u03b1 : Type u_1\ns t : Set \u03b1\nn : \u2115\nhu : Set.Finite (s \u222a t)\nhst : 2 * n < Finset.card (Finite.toFinset (_ : Set.Finite s) \u222a Finite.toFinset (_ : Set.Finite t))\n\u22a2 \u2203 r, n < ncard r \u2227 (r \u2286 s \u2228 r \u2286 t)"}, {"tactic": "obtain \u27e8r', hnr', hr'\u27e9 := Finset.exists_subset_or_subset_of_two_mul_lt_card hst", "annotated_tactic": ["obtain \u27e8r', hnr', hr'\u27e9 := <a>Finset.exists_subset_or_subset_of_two_mul_lt_card</a> hst", [{"full_name": "Finset.exists_subset_or_subset_of_two_mul_lt_card", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [508, 9], "def_end_pos": [508, 51]}]], "state_before": "\u03b1 : Type u_1\ns t : Set \u03b1\nn : \u2115\nhu : Set.Finite (s \u222a t)\nhst : 2 * n < Finset.card (Finite.toFinset (_ : Set.Finite s) \u222a Finite.toFinset (_ : Set.Finite t))\n\u22a2 \u2203 r, n < ncard r \u2227 (r \u2286 s \u2228 r \u2286 t)", "state_after": "case intro.intro\n\u03b1 : Type u_1\ns t : Set \u03b1\nn : \u2115\nhu : Set.Finite (s \u222a t)\nhst : 2 * n < Finset.card (Finite.toFinset (_ : Set.Finite s) \u222a Finite.toFinset (_ : Set.Finite t))\nr' : Finset \u03b1\nhnr' : n < Finset.card r'\nhr' : r' \u2286 Finite.toFinset (_ : Set.Finite s) \u2228 r' \u2286 Finite.toFinset (_ : Set.Finite t)\n\u22a2 \u2203 r, n < ncard r \u2227 (r \u2286 s \u2228 r \u2286 t)"}, {"tactic": "exact \u27e8r', by simpa, by simpa using hr'\u27e9", "annotated_tactic": ["exact \u27e8r', by simpa, by simpa using hr'\u27e9", []], "state_before": "case intro.intro\n\u03b1 : Type u_1\ns t : Set \u03b1\nn : \u2115\nhu : Set.Finite (s \u222a t)\nhst : 2 * n < Finset.card (Finite.toFinset (_ : Set.Finite s) \u222a Finite.toFinset (_ : Set.Finite t))\nr' : Finset \u03b1\nhnr' : n < Finset.card r'\nhr' : r' \u2286 Finite.toFinset (_ : Set.Finite s) \u2228 r' \u2286 Finite.toFinset (_ : Set.Finite t)\n\u22a2 \u2203 r, n < ncard r \u2227 (r \u2286 s \u2228 r \u2286 t)", "state_after": "no goals"}, {"tactic": "simpa", "annotated_tactic": ["simpa", []], "state_before": "\u03b1 : Type u_1\ns t : Set \u03b1\nn : \u2115\nhu : Set.Finite (s \u222a t)\nhst : 2 * n < Finset.card (Finite.toFinset (_ : Set.Finite s) \u222a Finite.toFinset (_ : Set.Finite t))\nr' : Finset \u03b1\nhnr' : n < Finset.card r'\nhr' : r' \u2286 Finite.toFinset (_ : Set.Finite s) \u2228 r' \u2286 Finite.toFinset (_ : Set.Finite t)\n\u22a2 n < ncard \u2191r'", "state_after": "no goals"}, {"tactic": "simpa using hr'", "annotated_tactic": ["simpa using hr'", []], "state_before": "\u03b1 : Type u_1\ns t : Set \u03b1\nn : \u2115\nhu : Set.Finite (s \u222a t)\nhst : 2 * n < Finset.card (Finite.toFinset (_ : Set.Finite s) \u222a Finite.toFinset (_ : Set.Finite t))\nr' : Finset \u03b1\nhnr' : n < Finset.card r'\nhr' : r' \u2286 Finite.toFinset (_ : Set.Finite s) \u2228 r' \u2286 Finite.toFinset (_ : Set.Finite t)\n\u22a2 \u2191r' \u2286 s \u2228 \u2191r' \u2286 t", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/DFA.lean", "full_name": "DFA.evalFrom_split", "start": [101, 1], "end": [134, 50], "traced_tactics": [{"tactic": "obtain \u27e8n, m, hneq, heq\u27e9 :=\n  Fintype.exists_ne_map_eq_of_card_lt\n    (fun n : Fin (Fintype.card \u03c3 + 1) => M.evalFrom s (x.take n)) (by norm_num)", "annotated_tactic": ["obtain \u27e8n, m, hneq, heq\u27e9 :=\n    <a>Fintype.exists_ne_map_eq_of_card_lt</a>\n      (fun n : <a>Fin</a> (<a>Fintype.card</a> \u03c3 + 1) => M.evalFrom s (x.take n)) (by norm_num)", [{"full_name": "Fintype.exists_ne_map_eq_of_card_lt", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [518, 9], "def_end_pos": [518, 36]}, {"full_name": "Fin", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1745, 11], "def_end_pos": [1745, 14]}, {"full_name": "Fintype.card", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [61, 5], "def_end_pos": [61, 9]}]], "state_before": "\u03b1 : Type u\n\u03c3 : Type v\nM : DFA \u03b1 \u03c3\ninst\u271d : Fintype \u03c3\nx : List \u03b1\ns t : \u03c3\nhlen : Fintype.card \u03c3 \u2264 List.length x\nhx : evalFrom M s x = t\n\u22a2 \u2203 q a b c,\n    x = a ++ b ++ c \u2227\n      List.length a + List.length b \u2264 Fintype.card \u03c3 \u2227\n        b \u2260 [] \u2227 evalFrom M s a = q \u2227 evalFrom M q b = q \u2227 evalFrom M q c = t", "state_after": "case intro.intro.intro\n\u03b1 : Type u\n\u03c3 : Type v\nM : DFA \u03b1 \u03c3\ninst\u271d : Fintype \u03c3\nx : List \u03b1\ns t : \u03c3\nhlen : Fintype.card \u03c3 \u2264 List.length x\nhx : evalFrom M s x = t\nn m : Fin (Fintype.card \u03c3 + 1)\nhneq : n \u2260 m\nheq : evalFrom M s (List.take (\u2191n) x) = evalFrom M s (List.take (\u2191m) x)\n\u22a2 \u2203 q a b c,\n    x = a ++ b ++ c \u2227\n      List.length a + List.length b \u2264 Fintype.card \u03c3 \u2227\n        b \u2260 [] \u2227 evalFrom M s a = q \u2227 evalFrom M q b = q \u2227 evalFrom M q c = t"}, {"tactic": "wlog hle : (n : \u2115) \u2264 m", "annotated_tactic": ["wlog hle : (n : \u2115) \u2264 m", []], "state_before": "case intro.intro.intro\n\u03b1 : Type u\n\u03c3 : Type v\nM : DFA \u03b1 \u03c3\ninst\u271d : Fintype \u03c3\nx : List \u03b1\ns t : \u03c3\nhlen : Fintype.card \u03c3 \u2264 List.length x\nhx : evalFrom M s x = t\nn m : Fin (Fintype.card \u03c3 + 1)\nhneq : n \u2260 m\nheq : evalFrom M s (List.take (\u2191n) x) = evalFrom M s (List.take (\u2191m) x)\n\u22a2 \u2203 q a b c,\n    x = a ++ b ++ c \u2227\n      List.length a + List.length b \u2264 Fintype.card \u03c3 \u2227\n        b \u2260 [] \u2227 evalFrom M s a = q \u2227 evalFrom M q b = q \u2227 evalFrom M q c = t", "state_after": "case intro.intro.intro.inr\n\u03b1 : Type u\n\u03c3 : Type v\nM : DFA \u03b1 \u03c3\ninst\u271d : Fintype \u03c3\nx : List \u03b1\ns t : \u03c3\nhlen : Fintype.card \u03c3 \u2264 List.length x\nhx : evalFrom M s x = t\nn m : Fin (Fintype.card \u03c3 + 1)\nhneq : n \u2260 m\nheq : evalFrom M s (List.take (\u2191n) x) = evalFrom M s (List.take (\u2191m) x)\nthis :\n  \u2200 {\u03b1 : Type u} {\u03c3 : Type v} (M : DFA \u03b1 \u03c3) [inst : Fintype \u03c3] {x : List \u03b1} {s t : \u03c3},\n    Fintype.card \u03c3 \u2264 List.length x \u2192\n      evalFrom M s x = t \u2192\n        \u2200 (n m : Fin (Fintype.card \u03c3 + 1)),\n          n \u2260 m \u2192\n            evalFrom M s (List.take (\u2191n) x) = evalFrom M s (List.take (\u2191m) x) \u2192\n              \u2191n \u2264 \u2191m \u2192\n                \u2203 q a b c,\n                  x = a ++ b ++ c \u2227\n                    List.length a + List.length b \u2264 Fintype.card \u03c3 \u2227\n                      b \u2260 [] \u2227 evalFrom M s a = q \u2227 evalFrom M q b = q \u2227 evalFrom M q c = t\nhle : \u00ac\u2191n \u2264 \u2191m\n\u22a2 \u2203 q a b c,\n    x = a ++ b ++ c \u2227\n      List.length a + List.length b \u2264 Fintype.card \u03c3 \u2227\n        b \u2260 [] \u2227 evalFrom M s a = q \u2227 evalFrom M q b = q \u2227 evalFrom M q c = t\n\n\u03b1\u271d : Type u\n\u03c3\u271d : Type v\nM\u271d : DFA \u03b1\u271d \u03c3\u271d\n\u03b1 : Type u\n\u03c3 : Type v\nM : DFA \u03b1 \u03c3\ninst\u271d : Fintype \u03c3\nx : List \u03b1\ns t : \u03c3\nhlen : Fintype.card \u03c3 \u2264 List.length x\nhx : evalFrom M s x = t\nn m : Fin (Fintype.card \u03c3 + 1)\nhneq : n \u2260 m\nheq : evalFrom M s (List.take (\u2191n) x) = evalFrom M s (List.take (\u2191m) x)\nhle : \u2191n \u2264 \u2191m\n\u22a2 \u2203 q a b c,\n    x = a ++ b ++ c \u2227\n      List.length a + List.length b \u2264 Fintype.card \u03c3 \u2227\n        b \u2260 [] \u2227 evalFrom M s a = q \u2227 evalFrom M q b = q \u2227 evalFrom M q c = t"}, {"tactic": "have hm : (m : \u2115) \u2264 Fintype.card \u03c3 := Fin.is_le m", "annotated_tactic": ["have hm : (m : \u2115) \u2264 <a>Fintype.card</a> \u03c3 := <a>Fin.is_le</a> m", [{"full_name": "Fintype.card", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [61, 5], "def_end_pos": [61, 9]}, {"full_name": "Fin.is_le", "def_path": "lake-packages/std/Std/Data/Fin/Lemmas.lean", "def_pos": [91, 9], "def_end_pos": [91, 14]}]], "state_before": "\u03b1\u271d : Type u\n\u03c3\u271d : Type v\nM\u271d : DFA \u03b1\u271d \u03c3\u271d\n\u03b1 : Type u\n\u03c3 : Type v\nM : DFA \u03b1 \u03c3\ninst\u271d : Fintype \u03c3\nx : List \u03b1\ns t : \u03c3\nhlen : Fintype.card \u03c3 \u2264 List.length x\nhx : evalFrom M s x = t\nn m : Fin (Fintype.card \u03c3 + 1)\nhneq : n \u2260 m\nheq : evalFrom M s (List.take (\u2191n) x) = evalFrom M s (List.take (\u2191m) x)\nhle : \u2191n \u2264 \u2191m\n\u22a2 \u2203 q a b c,\n    x = a ++ b ++ c \u2227\n      List.length a + List.length b \u2264 Fintype.card \u03c3 \u2227\n        b \u2260 [] \u2227 evalFrom M s a = q \u2227 evalFrom M q b = q \u2227 evalFrom M q c = t", "state_after": "\u03b1\u271d : Type u\n\u03c3\u271d : Type v\nM\u271d : DFA \u03b1\u271d \u03c3\u271d\n\u03b1 : Type u\n\u03c3 : Type v\nM : DFA \u03b1 \u03c3\ninst\u271d : Fintype \u03c3\nx : List \u03b1\ns t : \u03c3\nhlen : Fintype.card \u03c3 \u2264 List.length x\nhx : evalFrom M s x = t\nn m : Fin (Fintype.card \u03c3 + 1)\nhneq : n \u2260 m\nheq : evalFrom M s (List.take (\u2191n) x) = evalFrom M s (List.take (\u2191m) x)\nhle : \u2191n \u2264 \u2191m\nhm : \u2191m \u2264 Fintype.card \u03c3\n\u22a2 \u2203 q a b c,\n    x = a ++ b ++ c \u2227\n      List.length a + List.length b \u2264 Fintype.card \u03c3 \u2227\n        b \u2260 [] \u2227 evalFrom M s a = q \u2227 evalFrom M q b = q \u2227 evalFrom M q c = t"}, {"tactic": "refine'\n  \u27e8M.evalFrom s ((x.take m).take n), (x.take m).take n, (x.take m).drop n, x.drop m, _, _, _, by\n    rfl, _\u27e9", "annotated_tactic": ["refine'\n    \u27e8M.evalFrom s ((x.take m).<a>take</a> n), (x.take m).<a>take</a> n, (x.take m).<a>drop</a> n, x.drop m, _, _, _, by\n      rfl, _\u27e9", [{"full_name": "List.take", "def_path": "lake-packages/lean4/src/lean/Init/Data/List/Basic.lean", "def_pos": [494, 5], "def_end_pos": [494, 9]}, {"full_name": "List.take", "def_path": "lake-packages/lean4/src/lean/Init/Data/List/Basic.lean", "def_pos": [494, 5], "def_end_pos": [494, 9]}, {"full_name": "List.drop", "def_path": "lake-packages/lean4/src/lean/Init/Data/List/Basic.lean", "def_pos": [475, 5], "def_end_pos": [475, 9]}]], "state_before": "\u03b1\u271d : Type u\n\u03c3\u271d : Type v\nM\u271d : DFA \u03b1\u271d \u03c3\u271d\n\u03b1 : Type u\n\u03c3 : Type v\nM : DFA \u03b1 \u03c3\ninst\u271d : Fintype \u03c3\nx : List \u03b1\ns t : \u03c3\nhlen : Fintype.card \u03c3 \u2264 List.length x\nhx : evalFrom M s x = t\nn m : Fin (Fintype.card \u03c3 + 1)\nhneq : n \u2260 m\nheq : evalFrom M s (List.take (\u2191n) x) = evalFrom M s (List.take (\u2191m) x)\nhle : \u2191n \u2264 \u2191m\nhm : \u2191m \u2264 Fintype.card \u03c3\n\u22a2 \u2203 q a b c,\n    x = a ++ b ++ c \u2227\n      List.length a + List.length b \u2264 Fintype.card \u03c3 \u2227\n        b \u2260 [] \u2227 evalFrom M s a = q \u2227 evalFrom M q b = q \u2227 evalFrom M q c = t", "state_after": "case refine'_1\n\u03b1\u271d : Type u\n\u03c3\u271d : Type v\nM\u271d : DFA \u03b1\u271d \u03c3\u271d\n\u03b1 : Type u\n\u03c3 : Type v\nM : DFA \u03b1 \u03c3\ninst\u271d : Fintype \u03c3\nx : List \u03b1\ns t : \u03c3\nhlen : Fintype.card \u03c3 \u2264 List.length x\nhx : evalFrom M s x = t\nn m : Fin (Fintype.card \u03c3 + 1)\nhneq : n \u2260 m\nheq : evalFrom M s (List.take (\u2191n) x) = evalFrom M s (List.take (\u2191m) x)\nhle : \u2191n \u2264 \u2191m\nhm : \u2191m \u2264 Fintype.card \u03c3\n\u22a2 x = List.take (\u2191n) (List.take (\u2191m) x) ++ List.drop (\u2191n) (List.take (\u2191m) x) ++ List.drop (\u2191m) x\n\ncase refine'_2\n\u03b1\u271d : Type u\n\u03c3\u271d : Type v\nM\u271d : DFA \u03b1\u271d \u03c3\u271d\n\u03b1 : Type u\n\u03c3 : Type v\nM : DFA \u03b1 \u03c3\ninst\u271d : Fintype \u03c3\nx : List \u03b1\ns t : \u03c3\nhlen : Fintype.card \u03c3 \u2264 List.length x\nhx : evalFrom M s x = t\nn m : Fin (Fintype.card \u03c3 + 1)\nhneq : n \u2260 m\nheq : evalFrom M s (List.take (\u2191n) x) = evalFrom M s (List.take (\u2191m) x)\nhle : \u2191n \u2264 \u2191m\nhm : \u2191m \u2264 Fintype.card \u03c3\n\u22a2 List.length (List.take (\u2191n) (List.take (\u2191m) x)) + List.length (List.drop (\u2191n) (List.take (\u2191m) x)) \u2264 Fintype.card \u03c3\n\ncase refine'_3\n\u03b1\u271d : Type u\n\u03c3\u271d : Type v\nM\u271d : DFA \u03b1\u271d \u03c3\u271d\n\u03b1 : Type u\n\u03c3 : Type v\nM : DFA \u03b1 \u03c3\ninst\u271d : Fintype \u03c3\nx : List \u03b1\ns t : \u03c3\nhlen : Fintype.card \u03c3 \u2264 List.length x\nhx : evalFrom M s x = t\nn m : Fin (Fintype.card \u03c3 + 1)\nhneq : n \u2260 m\nheq : evalFrom M s (List.take (\u2191n) x) = evalFrom M s (List.take (\u2191m) x)\nhle : \u2191n \u2264 \u2191m\nhm : \u2191m \u2264 Fintype.card \u03c3\n\u22a2 List.drop (\u2191n) (List.take (\u2191m) x) \u2260 []\n\ncase refine'_4\n\u03b1\u271d : Type u\n\u03c3\u271d : Type v\nM\u271d : DFA \u03b1\u271d \u03c3\u271d\n\u03b1 : Type u\n\u03c3 : Type v\nM : DFA \u03b1 \u03c3\ninst\u271d : Fintype \u03c3\nx : List \u03b1\ns t : \u03c3\nhlen : Fintype.card \u03c3 \u2264 List.length x\nhx : evalFrom M s x = t\nn m : Fin (Fintype.card \u03c3 + 1)\nhneq : n \u2260 m\nheq : evalFrom M s (List.take (\u2191n) x) = evalFrom M s (List.take (\u2191m) x)\nhle : \u2191n \u2264 \u2191m\nhm : \u2191m \u2264 Fintype.card \u03c3\n\u22a2 evalFrom M (evalFrom M s (List.take (\u2191n) (List.take (\u2191m) x))) (List.drop (\u2191n) (List.take (\u2191m) x)) =\n      evalFrom M s (List.take (\u2191n) (List.take (\u2191m) x)) \u2227\n    evalFrom M (evalFrom M s (List.take (\u2191n) (List.take (\u2191m) x))) (List.drop (\u2191m) x) = t"}, {"tactic": "have hq : M.evalFrom (M.evalFrom s ((x.take m).take n)) ((x.take m).drop n) =\n    M.evalFrom s ((x.take m).take n) := by\n  rw [List.take_take, min_eq_left hle, \u2190 evalFrom_of_append, heq, \u2190 min_eq_left hle, \u2190\n    List.take_take, min_eq_left hle, List.take_append_drop]", "annotated_tactic": ["have hq : M.evalFrom (M.evalFrom s ((x.take m).<a>take</a> n)) ((x.take m).<a>drop</a> n) =\n      M.evalFrom s ((x.take m).<a>take</a> n) := by\n    rw [<a>List.take_take</a>, <a>min_eq_left</a> hle, \u2190 <a>evalFrom_of_append</a>, heq, \u2190 <a>min_eq_left</a> hle, \u2190\n      <a>List.take_take</a>, <a>min_eq_left</a> hle, <a>List.take_append_drop</a>]", [{"full_name": "List.take", "def_path": "lake-packages/lean4/src/lean/Init/Data/List/Basic.lean", "def_pos": [494, 5], "def_end_pos": [494, 9]}, {"full_name": "List.drop", "def_path": "lake-packages/lean4/src/lean/Init/Data/List/Basic.lean", "def_pos": [475, 5], "def_end_pos": [475, 9]}, {"full_name": "List.take", "def_path": "lake-packages/lean4/src/lean/Init/Data/List/Basic.lean", "def_pos": [494, 5], "def_end_pos": [494, 9]}, {"full_name": "List.take_take", "def_path": "Mathlib/Data/List/Basic.lean", "def_pos": [1886, 9], "def_end_pos": [1886, 18]}, {"full_name": "min_eq_left", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [100, 9], "def_end_pos": [100, 20]}, {"full_name": "DFA.evalFrom_of_append", "def_path": "Mathlib/Computability/DFA.lean", "def_pos": [89, 9], "def_end_pos": [89, 27]}, {"full_name": "min_eq_left", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [100, 9], "def_end_pos": [100, 20]}, {"full_name": "List.take_take", "def_path": "Mathlib/Data/List/Basic.lean", "def_pos": [1886, 9], "def_end_pos": [1886, 18]}, {"full_name": "min_eq_left", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [100, 9], "def_end_pos": [100, 20]}, {"full_name": "List.take_append_drop", "def_path": "lake-packages/std/Std/Data/List/Init/Lemmas.lean", "def_pos": [138, 17], "def_end_pos": [138, 33]}]], "state_before": "case refine'_4\n\u03b1\u271d : Type u\n\u03c3\u271d : Type v\nM\u271d : DFA \u03b1\u271d \u03c3\u271d\n\u03b1 : Type u\n\u03c3 : Type v\nM : DFA \u03b1 \u03c3\ninst\u271d : Fintype \u03c3\nx : List \u03b1\ns t : \u03c3\nhlen : Fintype.card \u03c3 \u2264 List.length x\nhx : evalFrom M s x = t\nn m : Fin (Fintype.card \u03c3 + 1)\nhneq : n \u2260 m\nheq : evalFrom M s (List.take (\u2191n) x) = evalFrom M s (List.take (\u2191m) x)\nhle : \u2191n \u2264 \u2191m\nhm : \u2191m \u2264 Fintype.card \u03c3\n\u22a2 evalFrom M (evalFrom M s (List.take (\u2191n) (List.take (\u2191m) x))) (List.drop (\u2191n) (List.take (\u2191m) x)) =\n      evalFrom M s (List.take (\u2191n) (List.take (\u2191m) x)) \u2227\n    evalFrom M (evalFrom M s (List.take (\u2191n) (List.take (\u2191m) x))) (List.drop (\u2191m) x) = t", "state_after": "case refine'_4\n\u03b1\u271d : Type u\n\u03c3\u271d : Type v\nM\u271d : DFA \u03b1\u271d \u03c3\u271d\n\u03b1 : Type u\n\u03c3 : Type v\nM : DFA \u03b1 \u03c3\ninst\u271d : Fintype \u03c3\nx : List \u03b1\ns t : \u03c3\nhlen : Fintype.card \u03c3 \u2264 List.length x\nhx : evalFrom M s x = t\nn m : Fin (Fintype.card \u03c3 + 1)\nhneq : n \u2260 m\nheq : evalFrom M s (List.take (\u2191n) x) = evalFrom M s (List.take (\u2191m) x)\nhle : \u2191n \u2264 \u2191m\nhm : \u2191m \u2264 Fintype.card \u03c3\nhq :\n  evalFrom M (evalFrom M s (List.take (\u2191n) (List.take (\u2191m) x))) (List.drop (\u2191n) (List.take (\u2191m) x)) =\n    evalFrom M s (List.take (\u2191n) (List.take (\u2191m) x))\n\u22a2 evalFrom M (evalFrom M s (List.take (\u2191n) (List.take (\u2191m) x))) (List.drop (\u2191n) (List.take (\u2191m) x)) =\n      evalFrom M s (List.take (\u2191n) (List.take (\u2191m) x)) \u2227\n    evalFrom M (evalFrom M s (List.take (\u2191n) (List.take (\u2191m) x))) (List.drop (\u2191m) x) = t"}, {"tactic": "use hq", "annotated_tactic": ["use hq", []], "state_before": "case refine'_4\n\u03b1\u271d : Type u\n\u03c3\u271d : Type v\nM\u271d : DFA \u03b1\u271d \u03c3\u271d\n\u03b1 : Type u\n\u03c3 : Type v\nM : DFA \u03b1 \u03c3\ninst\u271d : Fintype \u03c3\nx : List \u03b1\ns t : \u03c3\nhlen : Fintype.card \u03c3 \u2264 List.length x\nhx : evalFrom M s x = t\nn m : Fin (Fintype.card \u03c3 + 1)\nhneq : n \u2260 m\nheq : evalFrom M s (List.take (\u2191n) x) = evalFrom M s (List.take (\u2191m) x)\nhle : \u2191n \u2264 \u2191m\nhm : \u2191m \u2264 Fintype.card \u03c3\nhq :\n  evalFrom M (evalFrom M s (List.take (\u2191n) (List.take (\u2191m) x))) (List.drop (\u2191n) (List.take (\u2191m) x)) =\n    evalFrom M s (List.take (\u2191n) (List.take (\u2191m) x))\n\u22a2 evalFrom M (evalFrom M s (List.take (\u2191n) (List.take (\u2191m) x))) (List.drop (\u2191n) (List.take (\u2191m) x)) =\n      evalFrom M s (List.take (\u2191n) (List.take (\u2191m) x)) \u2227\n    evalFrom M (evalFrom M s (List.take (\u2191n) (List.take (\u2191m) x))) (List.drop (\u2191m) x) = t", "state_after": "case right\n\u03b1\u271d : Type u\n\u03c3\u271d : Type v\nM\u271d : DFA \u03b1\u271d \u03c3\u271d\n\u03b1 : Type u\n\u03c3 : Type v\nM : DFA \u03b1 \u03c3\ninst\u271d : Fintype \u03c3\nx : List \u03b1\ns t : \u03c3\nhlen : Fintype.card \u03c3 \u2264 List.length x\nhx : evalFrom M s x = t\nn m : Fin (Fintype.card \u03c3 + 1)\nhneq : n \u2260 m\nheq : evalFrom M s (List.take (\u2191n) x) = evalFrom M s (List.take (\u2191m) x)\nhle : \u2191n \u2264 \u2191m\nhm : \u2191m \u2264 Fintype.card \u03c3\nhq :\n  evalFrom M (evalFrom M s (List.take (\u2191n) (List.take (\u2191m) x))) (List.drop (\u2191n) (List.take (\u2191m) x)) =\n    evalFrom M s (List.take (\u2191n) (List.take (\u2191m) x))\n\u22a2 evalFrom M (evalFrom M s (List.take (\u2191n) (List.take (\u2191m) x))) (List.drop (\u2191m) x) = t"}, {"tactic": "rwa [\u2190 hq, \u2190 evalFrom_of_append, \u2190 evalFrom_of_append, \u2190 List.append_assoc,\n  List.take_append_drop, List.take_append_drop]", "annotated_tactic": ["rwa [\u2190 hq, \u2190 <a>evalFrom_of_append</a>, \u2190 <a>evalFrom_of_append</a>, \u2190 <a>List.append_assoc</a>,\n    <a>List.take_append_drop</a>, <a>List.take_append_drop</a>]", [{"full_name": "DFA.evalFrom_of_append", "def_path": "Mathlib/Computability/DFA.lean", "def_pos": [89, 9], "def_end_pos": [89, 27]}, {"full_name": "DFA.evalFrom_of_append", "def_path": "Mathlib/Computability/DFA.lean", "def_pos": [89, 9], "def_end_pos": [89, 27]}, {"full_name": "List.append_assoc", "def_path": "lake-packages/lean4/src/lean/Init/Data/List/Basic.lean", "def_pos": [103, 9], "def_end_pos": [103, 21]}, {"full_name": "List.take_append_drop", "def_path": "lake-packages/std/Std/Data/List/Init/Lemmas.lean", "def_pos": [138, 17], "def_end_pos": [138, 33]}, {"full_name": "List.take_append_drop", "def_path": "lake-packages/std/Std/Data/List/Init/Lemmas.lean", "def_pos": [138, 17], "def_end_pos": [138, 33]}]], "state_before": "case right\n\u03b1\u271d : Type u\n\u03c3\u271d : Type v\nM\u271d : DFA \u03b1\u271d \u03c3\u271d\n\u03b1 : Type u\n\u03c3 : Type v\nM : DFA \u03b1 \u03c3\ninst\u271d : Fintype \u03c3\nx : List \u03b1\ns t : \u03c3\nhlen : Fintype.card \u03c3 \u2264 List.length x\nhx : evalFrom M s x = t\nn m : Fin (Fintype.card \u03c3 + 1)\nhneq : n \u2260 m\nheq : evalFrom M s (List.take (\u2191n) x) = evalFrom M s (List.take (\u2191m) x)\nhle : \u2191n \u2264 \u2191m\nhm : \u2191m \u2264 Fintype.card \u03c3\nhq :\n  evalFrom M (evalFrom M s (List.take (\u2191n) (List.take (\u2191m) x))) (List.drop (\u2191n) (List.take (\u2191m) x)) =\n    evalFrom M s (List.take (\u2191n) (List.take (\u2191m) x))\n\u22a2 evalFrom M (evalFrom M s (List.take (\u2191n) (List.take (\u2191m) x))) (List.drop (\u2191m) x) = t", "state_after": "no goals"}, {"tactic": "norm_num", "annotated_tactic": ["norm_num", []], "state_before": "\u03b1 : Type u\n\u03c3 : Type v\nM : DFA \u03b1 \u03c3\ninst\u271d : Fintype \u03c3\nx : List \u03b1\ns t : \u03c3\nhlen : Fintype.card \u03c3 \u2264 List.length x\nhx : evalFrom M s x = t\n\u22a2 Fintype.card \u03c3 < Fintype.card (Fin (Fintype.card \u03c3 + 1))", "state_after": "no goals"}, {"tactic": "exact this _ hlen hx _ _ hneq.symm heq.symm (le_of_not_le hle)", "annotated_tactic": ["exact this _ hlen hx _ _ hneq.symm heq.symm (<a>le_of_not_le</a> hle)", [{"full_name": "le_of_not_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [303, 9], "def_end_pos": [303, 21]}]], "state_before": "case intro.intro.intro.inr\n\u03b1 : Type u\n\u03c3 : Type v\nM : DFA \u03b1 \u03c3\ninst\u271d : Fintype \u03c3\nx : List \u03b1\ns t : \u03c3\nhlen : Fintype.card \u03c3 \u2264 List.length x\nhx : evalFrom M s x = t\nn m : Fin (Fintype.card \u03c3 + 1)\nhneq : n \u2260 m\nheq : evalFrom M s (List.take (\u2191n) x) = evalFrom M s (List.take (\u2191m) x)\nthis :\n  \u2200 {\u03b1 : Type u} {\u03c3 : Type v} (M : DFA \u03b1 \u03c3) [inst : Fintype \u03c3] {x : List \u03b1} {s t : \u03c3},\n    Fintype.card \u03c3 \u2264 List.length x \u2192\n      evalFrom M s x = t \u2192\n        \u2200 (n m : Fin (Fintype.card \u03c3 + 1)),\n          n \u2260 m \u2192\n            evalFrom M s (List.take (\u2191n) x) = evalFrom M s (List.take (\u2191m) x) \u2192\n              \u2191n \u2264 \u2191m \u2192\n                \u2203 q a b c,\n                  x = a ++ b ++ c \u2227\n                    List.length a + List.length b \u2264 Fintype.card \u03c3 \u2227\n                      b \u2260 [] \u2227 evalFrom M s a = q \u2227 evalFrom M q b = q \u2227 evalFrom M q c = t\nhle : \u00ac\u2191n \u2264 \u2191m\n\u22a2 \u2203 q a b c,\n    x = a ++ b ++ c \u2227\n      List.length a + List.length b \u2264 Fintype.card \u03c3 \u2227\n        b \u2260 [] \u2227 evalFrom M s a = q \u2227 evalFrom M q b = q \u2227 evalFrom M q c = t", "state_after": "no goals"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\u03b1\u271d : Type u\n\u03c3\u271d : Type v\nM\u271d : DFA \u03b1\u271d \u03c3\u271d\n\u03b1 : Type u\n\u03c3 : Type v\nM : DFA \u03b1 \u03c3\ninst\u271d : Fintype \u03c3\nx : List \u03b1\ns t : \u03c3\nhlen : Fintype.card \u03c3 \u2264 List.length x\nhx : evalFrom M s x = t\nn m : Fin (Fintype.card \u03c3 + 1)\nhneq : n \u2260 m\nheq : evalFrom M s (List.take (\u2191n) x) = evalFrom M s (List.take (\u2191m) x)\nhle : \u2191n \u2264 \u2191m\nhm : \u2191m \u2264 Fintype.card \u03c3\n\u22a2 evalFrom M s (List.take (\u2191n) (List.take (\u2191m) x)) = evalFrom M s (List.take (\u2191n) (List.take (\u2191m) x))", "state_after": "no goals"}, {"tactic": "rw [List.take_append_drop, List.take_append_drop]", "annotated_tactic": ["rw [<a>List.take_append_drop</a>, <a>List.take_append_drop</a>]", [{"full_name": "List.take_append_drop", "def_path": "lake-packages/std/Std/Data/List/Init/Lemmas.lean", "def_pos": [138, 17], "def_end_pos": [138, 33]}, {"full_name": "List.take_append_drop", "def_path": "lake-packages/std/Std/Data/List/Init/Lemmas.lean", "def_pos": [138, 17], "def_end_pos": [138, 33]}]], "state_before": "case refine'_1\n\u03b1\u271d : Type u\n\u03c3\u271d : Type v\nM\u271d : DFA \u03b1\u271d \u03c3\u271d\n\u03b1 : Type u\n\u03c3 : Type v\nM : DFA \u03b1 \u03c3\ninst\u271d : Fintype \u03c3\nx : List \u03b1\ns t : \u03c3\nhlen : Fintype.card \u03c3 \u2264 List.length x\nhx : evalFrom M s x = t\nn m : Fin (Fintype.card \u03c3 + 1)\nhneq : n \u2260 m\nheq : evalFrom M s (List.take (\u2191n) x) = evalFrom M s (List.take (\u2191m) x)\nhle : \u2191n \u2264 \u2191m\nhm : \u2191m \u2264 Fintype.card \u03c3\n\u22a2 x = List.take (\u2191n) (List.take (\u2191m) x) ++ List.drop (\u2191n) (List.take (\u2191m) x) ++ List.drop (\u2191m) x", "state_after": "no goals"}, {"tactic": "simp only [List.length_drop, List.length_take]", "annotated_tactic": ["simp only [<a>List.length_drop</a>, <a>List.length_take</a>]", [{"full_name": "List.length_drop", "def_path": "lake-packages/std/Std/Data/List/Init/Lemmas.lean", "def_pos": [143, 17], "def_end_pos": [143, 28]}, {"full_name": "List.length_take", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [786, 17], "def_end_pos": [786, 28]}]], "state_before": "case refine'_2\n\u03b1\u271d : Type u\n\u03c3\u271d : Type v\nM\u271d : DFA \u03b1\u271d \u03c3\u271d\n\u03b1 : Type u\n\u03c3 : Type v\nM : DFA \u03b1 \u03c3\ninst\u271d : Fintype \u03c3\nx : List \u03b1\ns t : \u03c3\nhlen : Fintype.card \u03c3 \u2264 List.length x\nhx : evalFrom M s x = t\nn m : Fin (Fintype.card \u03c3 + 1)\nhneq : n \u2260 m\nheq : evalFrom M s (List.take (\u2191n) x) = evalFrom M s (List.take (\u2191m) x)\nhle : \u2191n \u2264 \u2191m\nhm : \u2191m \u2264 Fintype.card \u03c3\n\u22a2 List.length (List.take (\u2191n) (List.take (\u2191m) x)) + List.length (List.drop (\u2191n) (List.take (\u2191m) x)) \u2264 Fintype.card \u03c3", "state_after": "case refine'_2\n\u03b1\u271d : Type u\n\u03c3\u271d : Type v\nM\u271d : DFA \u03b1\u271d \u03c3\u271d\n\u03b1 : Type u\n\u03c3 : Type v\nM : DFA \u03b1 \u03c3\ninst\u271d : Fintype \u03c3\nx : List \u03b1\ns t : \u03c3\nhlen : Fintype.card \u03c3 \u2264 List.length x\nhx : evalFrom M s x = t\nn m : Fin (Fintype.card \u03c3 + 1)\nhneq : n \u2260 m\nheq : evalFrom M s (List.take (\u2191n) x) = evalFrom M s (List.take (\u2191m) x)\nhle : \u2191n \u2264 \u2191m\nhm : \u2191m \u2264 Fintype.card \u03c3\n\u22a2 min (\u2191n) (min (\u2191m) (List.length x)) + (min (\u2191m) (List.length x) - \u2191n) \u2264 Fintype.card \u03c3"}, {"tactic": "rw [min_eq_left (hm.trans hlen), min_eq_left hle, add_tsub_cancel_of_le hle]", "annotated_tactic": ["rw [<a>min_eq_left</a> (hm.trans hlen), <a>min_eq_left</a> hle, <a>add_tsub_cancel_of_le</a> hle]", [{"full_name": "min_eq_left", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [100, 9], "def_end_pos": [100, 20]}, {"full_name": "min_eq_left", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [100, 9], "def_end_pos": [100, 20]}, {"full_name": "add_tsub_cancel_of_le", "def_path": "Mathlib/Algebra/Order/Sub/Canonical.lean", "def_pos": [24, 9], "def_end_pos": [24, 30]}]], "state_before": "case refine'_2\n\u03b1\u271d : Type u\n\u03c3\u271d : Type v\nM\u271d : DFA \u03b1\u271d \u03c3\u271d\n\u03b1 : Type u\n\u03c3 : Type v\nM : DFA \u03b1 \u03c3\ninst\u271d : Fintype \u03c3\nx : List \u03b1\ns t : \u03c3\nhlen : Fintype.card \u03c3 \u2264 List.length x\nhx : evalFrom M s x = t\nn m : Fin (Fintype.card \u03c3 + 1)\nhneq : n \u2260 m\nheq : evalFrom M s (List.take (\u2191n) x) = evalFrom M s (List.take (\u2191m) x)\nhle : \u2191n \u2264 \u2191m\nhm : \u2191m \u2264 Fintype.card \u03c3\n\u22a2 min (\u2191n) (min (\u2191m) (List.length x)) + (min (\u2191m) (List.length x) - \u2191n) \u2264 Fintype.card \u03c3", "state_after": "case refine'_2\n\u03b1\u271d : Type u\n\u03c3\u271d : Type v\nM\u271d : DFA \u03b1\u271d \u03c3\u271d\n\u03b1 : Type u\n\u03c3 : Type v\nM : DFA \u03b1 \u03c3\ninst\u271d : Fintype \u03c3\nx : List \u03b1\ns t : \u03c3\nhlen : Fintype.card \u03c3 \u2264 List.length x\nhx : evalFrom M s x = t\nn m : Fin (Fintype.card \u03c3 + 1)\nhneq : n \u2260 m\nheq : evalFrom M s (List.take (\u2191n) x) = evalFrom M s (List.take (\u2191m) x)\nhle : \u2191n \u2264 \u2191m\nhm : \u2191m \u2264 Fintype.card \u03c3\n\u22a2 \u2191m \u2264 Fintype.card \u03c3"}, {"tactic": "exact hm", "annotated_tactic": ["exact hm", []], "state_before": "case refine'_2\n\u03b1\u271d : Type u\n\u03c3\u271d : Type v\nM\u271d : DFA \u03b1\u271d \u03c3\u271d\n\u03b1 : Type u\n\u03c3 : Type v\nM : DFA \u03b1 \u03c3\ninst\u271d : Fintype \u03c3\nx : List \u03b1\ns t : \u03c3\nhlen : Fintype.card \u03c3 \u2264 List.length x\nhx : evalFrom M s x = t\nn m : Fin (Fintype.card \u03c3 + 1)\nhneq : n \u2260 m\nheq : evalFrom M s (List.take (\u2191n) x) = evalFrom M s (List.take (\u2191m) x)\nhle : \u2191n \u2264 \u2191m\nhm : \u2191m \u2264 Fintype.card \u03c3\n\u22a2 \u2191m \u2264 Fintype.card \u03c3", "state_after": "no goals"}, {"tactic": "intro h", "annotated_tactic": ["intro h", []], "state_before": "case refine'_3\n\u03b1\u271d : Type u\n\u03c3\u271d : Type v\nM\u271d : DFA \u03b1\u271d \u03c3\u271d\n\u03b1 : Type u\n\u03c3 : Type v\nM : DFA \u03b1 \u03c3\ninst\u271d : Fintype \u03c3\nx : List \u03b1\ns t : \u03c3\nhlen : Fintype.card \u03c3 \u2264 List.length x\nhx : evalFrom M s x = t\nn m : Fin (Fintype.card \u03c3 + 1)\nhneq : n \u2260 m\nheq : evalFrom M s (List.take (\u2191n) x) = evalFrom M s (List.take (\u2191m) x)\nhle : \u2191n \u2264 \u2191m\nhm : \u2191m \u2264 Fintype.card \u03c3\n\u22a2 List.drop (\u2191n) (List.take (\u2191m) x) \u2260 []", "state_after": "case refine'_3\n\u03b1\u271d : Type u\n\u03c3\u271d : Type v\nM\u271d : DFA \u03b1\u271d \u03c3\u271d\n\u03b1 : Type u\n\u03c3 : Type v\nM : DFA \u03b1 \u03c3\ninst\u271d : Fintype \u03c3\nx : List \u03b1\ns t : \u03c3\nhlen : Fintype.card \u03c3 \u2264 List.length x\nhx : evalFrom M s x = t\nn m : Fin (Fintype.card \u03c3 + 1)\nhneq : n \u2260 m\nheq : evalFrom M s (List.take (\u2191n) x) = evalFrom M s (List.take (\u2191m) x)\nhle : \u2191n \u2264 \u2191m\nhm : \u2191m \u2264 Fintype.card \u03c3\nh : List.drop (\u2191n) (List.take (\u2191m) x) = []\n\u22a2 False"}, {"tactic": "have hlen' := congr_arg List.length h", "annotated_tactic": ["have hlen' := <a>congr_arg</a> <a>List.length</a> h", [{"full_name": "congr_arg", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [43, 7], "def_end_pos": [43, 16]}, {"full_name": "List.length", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2232, 5], "def_end_pos": [2232, 16]}]], "state_before": "case refine'_3\n\u03b1\u271d : Type u\n\u03c3\u271d : Type v\nM\u271d : DFA \u03b1\u271d \u03c3\u271d\n\u03b1 : Type u\n\u03c3 : Type v\nM : DFA \u03b1 \u03c3\ninst\u271d : Fintype \u03c3\nx : List \u03b1\ns t : \u03c3\nhlen : Fintype.card \u03c3 \u2264 List.length x\nhx : evalFrom M s x = t\nn m : Fin (Fintype.card \u03c3 + 1)\nhneq : n \u2260 m\nheq : evalFrom M s (List.take (\u2191n) x) = evalFrom M s (List.take (\u2191m) x)\nhle : \u2191n \u2264 \u2191m\nhm : \u2191m \u2264 Fintype.card \u03c3\nh : List.drop (\u2191n) (List.take (\u2191m) x) = []\n\u22a2 False", "state_after": "case refine'_3\n\u03b1\u271d : Type u\n\u03c3\u271d : Type v\nM\u271d : DFA \u03b1\u271d \u03c3\u271d\n\u03b1 : Type u\n\u03c3 : Type v\nM : DFA \u03b1 \u03c3\ninst\u271d : Fintype \u03c3\nx : List \u03b1\ns t : \u03c3\nhlen : Fintype.card \u03c3 \u2264 List.length x\nhx : evalFrom M s x = t\nn m : Fin (Fintype.card \u03c3 + 1)\nhneq : n \u2260 m\nheq : evalFrom M s (List.take (\u2191n) x) = evalFrom M s (List.take (\u2191m) x)\nhle : \u2191n \u2264 \u2191m\nhm : \u2191m \u2264 Fintype.card \u03c3\nh : List.drop (\u2191n) (List.take (\u2191m) x) = []\nhlen' : List.length (List.drop (\u2191n) (List.take (\u2191m) x)) = List.length []\n\u22a2 False"}, {"tactic": "simp only [List.length_drop, List.length, List.length_take] at hlen'", "annotated_tactic": ["simp only [<a>List.length_drop</a>, <a>List.length</a>, <a>List.length_take</a>] at hlen'", [{"full_name": "List.length_drop", "def_path": "lake-packages/std/Std/Data/List/Init/Lemmas.lean", "def_pos": [143, 17], "def_end_pos": [143, 28]}, {"full_name": "List.length", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2232, 5], "def_end_pos": [2232, 16]}, {"full_name": "List.length_take", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [786, 17], "def_end_pos": [786, 28]}]], "state_before": "case refine'_3\n\u03b1\u271d : Type u\n\u03c3\u271d : Type v\nM\u271d : DFA \u03b1\u271d \u03c3\u271d\n\u03b1 : Type u\n\u03c3 : Type v\nM : DFA \u03b1 \u03c3\ninst\u271d : Fintype \u03c3\nx : List \u03b1\ns t : \u03c3\nhlen : Fintype.card \u03c3 \u2264 List.length x\nhx : evalFrom M s x = t\nn m : Fin (Fintype.card \u03c3 + 1)\nhneq : n \u2260 m\nheq : evalFrom M s (List.take (\u2191n) x) = evalFrom M s (List.take (\u2191m) x)\nhle : \u2191n \u2264 \u2191m\nhm : \u2191m \u2264 Fintype.card \u03c3\nh : List.drop (\u2191n) (List.take (\u2191m) x) = []\nhlen' : List.length (List.drop (\u2191n) (List.take (\u2191m) x)) = List.length []\n\u22a2 False", "state_after": "case refine'_3\n\u03b1\u271d : Type u\n\u03c3\u271d : Type v\nM\u271d : DFA \u03b1\u271d \u03c3\u271d\n\u03b1 : Type u\n\u03c3 : Type v\nM : DFA \u03b1 \u03c3\ninst\u271d : Fintype \u03c3\nx : List \u03b1\ns t : \u03c3\nhlen : Fintype.card \u03c3 \u2264 List.length x\nhx : evalFrom M s x = t\nn m : Fin (Fintype.card \u03c3 + 1)\nhneq : n \u2260 m\nheq : evalFrom M s (List.take (\u2191n) x) = evalFrom M s (List.take (\u2191m) x)\nhle : \u2191n \u2264 \u2191m\nhm : \u2191m \u2264 Fintype.card \u03c3\nh : List.drop (\u2191n) (List.take (\u2191m) x) = []\nhlen' : min (\u2191m) (List.length x) - \u2191n = 0\n\u22a2 False"}, {"tactic": "rw [min_eq_left, tsub_eq_zero_iff_le] at hlen'", "annotated_tactic": ["rw [<a>min_eq_left</a>, <a>tsub_eq_zero_iff_le</a>] at hlen'", [{"full_name": "min_eq_left", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [100, 9], "def_end_pos": [100, 20]}, {"full_name": "tsub_eq_zero_iff_le", "def_path": "Mathlib/Algebra/Order/Sub/Canonical.lean", "def_pos": [324, 9], "def_end_pos": [324, 28]}]], "state_before": "case refine'_3\n\u03b1\u271d : Type u\n\u03c3\u271d : Type v\nM\u271d : DFA \u03b1\u271d \u03c3\u271d\n\u03b1 : Type u\n\u03c3 : Type v\nM : DFA \u03b1 \u03c3\ninst\u271d : Fintype \u03c3\nx : List \u03b1\ns t : \u03c3\nhlen : Fintype.card \u03c3 \u2264 List.length x\nhx : evalFrom M s x = t\nn m : Fin (Fintype.card \u03c3 + 1)\nhneq : n \u2260 m\nheq : evalFrom M s (List.take (\u2191n) x) = evalFrom M s (List.take (\u2191m) x)\nhle : \u2191n \u2264 \u2191m\nhm : \u2191m \u2264 Fintype.card \u03c3\nh : List.drop (\u2191n) (List.take (\u2191m) x) = []\nhlen' : min (\u2191m) (List.length x) - \u2191n = 0\n\u22a2 False", "state_after": "case refine'_3\n\u03b1\u271d : Type u\n\u03c3\u271d : Type v\nM\u271d : DFA \u03b1\u271d \u03c3\u271d\n\u03b1 : Type u\n\u03c3 : Type v\nM : DFA \u03b1 \u03c3\ninst\u271d : Fintype \u03c3\nx : List \u03b1\ns t : \u03c3\nhlen : Fintype.card \u03c3 \u2264 List.length x\nhx : evalFrom M s x = t\nn m : Fin (Fintype.card \u03c3 + 1)\nhneq : n \u2260 m\nheq : evalFrom M s (List.take (\u2191n) x) = evalFrom M s (List.take (\u2191m) x)\nhle : \u2191n \u2264 \u2191m\nhm : \u2191m \u2264 Fintype.card \u03c3\nh : List.drop (\u2191n) (List.take (\u2191m) x) = []\nhlen' : \u2191m \u2264 \u2191n\n\u22a2 False\n\ncase refine'_3\n\u03b1\u271d : Type u\n\u03c3\u271d : Type v\nM\u271d : DFA \u03b1\u271d \u03c3\u271d\n\u03b1 : Type u\n\u03c3 : Type v\nM : DFA \u03b1 \u03c3\ninst\u271d : Fintype \u03c3\nx : List \u03b1\ns t : \u03c3\nhlen : Fintype.card \u03c3 \u2264 List.length x\nhx : evalFrom M s x = t\nn m : Fin (Fintype.card \u03c3 + 1)\nhneq : n \u2260 m\nheq : evalFrom M s (List.take (\u2191n) x) = evalFrom M s (List.take (\u2191m) x)\nhle : \u2191n \u2264 \u2191m\nhm : \u2191m \u2264 Fintype.card \u03c3\nh : List.drop (\u2191n) (List.take (\u2191m) x) = []\nhlen' : min (\u2191m) (List.length x) - \u2191n = 0\n\u22a2 \u2191m \u2264 List.length x"}, {"tactic": "exact hm.trans hlen", "annotated_tactic": ["exact hm.trans hlen", []], "state_before": "case refine'_3\n\u03b1\u271d : Type u\n\u03c3\u271d : Type v\nM\u271d : DFA \u03b1\u271d \u03c3\u271d\n\u03b1 : Type u\n\u03c3 : Type v\nM : DFA \u03b1 \u03c3\ninst\u271d : Fintype \u03c3\nx : List \u03b1\ns t : \u03c3\nhlen : Fintype.card \u03c3 \u2264 List.length x\nhx : evalFrom M s x = t\nn m : Fin (Fintype.card \u03c3 + 1)\nhneq : n \u2260 m\nheq : evalFrom M s (List.take (\u2191n) x) = evalFrom M s (List.take (\u2191m) x)\nhle : \u2191n \u2264 \u2191m\nhm : \u2191m \u2264 Fintype.card \u03c3\nh : List.drop (\u2191n) (List.take (\u2191m) x) = []\nhlen' : min (\u2191m) (List.length x) - \u2191n = 0\n\u22a2 \u2191m \u2264 List.length x", "state_after": "no goals"}, {"tactic": "apply hneq", "annotated_tactic": ["apply hneq", []], "state_before": "case refine'_3\n\u03b1\u271d : Type u\n\u03c3\u271d : Type v\nM\u271d : DFA \u03b1\u271d \u03c3\u271d\n\u03b1 : Type u\n\u03c3 : Type v\nM : DFA \u03b1 \u03c3\ninst\u271d : Fintype \u03c3\nx : List \u03b1\ns t : \u03c3\nhlen : Fintype.card \u03c3 \u2264 List.length x\nhx : evalFrom M s x = t\nn m : Fin (Fintype.card \u03c3 + 1)\nhneq : n \u2260 m\nheq : evalFrom M s (List.take (\u2191n) x) = evalFrom M s (List.take (\u2191m) x)\nhle : \u2191n \u2264 \u2191m\nhm : \u2191m \u2264 Fintype.card \u03c3\nh : List.drop (\u2191n) (List.take (\u2191m) x) = []\nhlen' : \u2191m \u2264 \u2191n\n\u22a2 False", "state_after": "case refine'_3\n\u03b1\u271d : Type u\n\u03c3\u271d : Type v\nM\u271d : DFA \u03b1\u271d \u03c3\u271d\n\u03b1 : Type u\n\u03c3 : Type v\nM : DFA \u03b1 \u03c3\ninst\u271d : Fintype \u03c3\nx : List \u03b1\ns t : \u03c3\nhlen : Fintype.card \u03c3 \u2264 List.length x\nhx : evalFrom M s x = t\nn m : Fin (Fintype.card \u03c3 + 1)\nhneq : n \u2260 m\nheq : evalFrom M s (List.take (\u2191n) x) = evalFrom M s (List.take (\u2191m) x)\nhle : \u2191n \u2264 \u2191m\nhm : \u2191m \u2264 Fintype.card \u03c3\nh : List.drop (\u2191n) (List.take (\u2191m) x) = []\nhlen' : \u2191m \u2264 \u2191n\n\u22a2 n = m"}, {"tactic": "apply le_antisymm", "annotated_tactic": ["apply <a>le_antisymm</a>", [{"full_name": "le_antisymm", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [188, 9], "def_end_pos": [188, 20]}]], "state_before": "case refine'_3\n\u03b1\u271d : Type u\n\u03c3\u271d : Type v\nM\u271d : DFA \u03b1\u271d \u03c3\u271d\n\u03b1 : Type u\n\u03c3 : Type v\nM : DFA \u03b1 \u03c3\ninst\u271d : Fintype \u03c3\nx : List \u03b1\ns t : \u03c3\nhlen : Fintype.card \u03c3 \u2264 List.length x\nhx : evalFrom M s x = t\nn m : Fin (Fintype.card \u03c3 + 1)\nhneq : n \u2260 m\nheq : evalFrom M s (List.take (\u2191n) x) = evalFrom M s (List.take (\u2191m) x)\nhle : \u2191n \u2264 \u2191m\nhm : \u2191m \u2264 Fintype.card \u03c3\nh : List.drop (\u2191n) (List.take (\u2191m) x) = []\nhlen' : \u2191m \u2264 \u2191n\n\u22a2 n = m", "state_after": "case refine'_3.a\n\u03b1\u271d : Type u\n\u03c3\u271d : Type v\nM\u271d : DFA \u03b1\u271d \u03c3\u271d\n\u03b1 : Type u\n\u03c3 : Type v\nM : DFA \u03b1 \u03c3\ninst\u271d : Fintype \u03c3\nx : List \u03b1\ns t : \u03c3\nhlen : Fintype.card \u03c3 \u2264 List.length x\nhx : evalFrom M s x = t\nn m : Fin (Fintype.card \u03c3 + 1)\nhneq : n \u2260 m\nheq : evalFrom M s (List.take (\u2191n) x) = evalFrom M s (List.take (\u2191m) x)\nhle : \u2191n \u2264 \u2191m\nhm : \u2191m \u2264 Fintype.card \u03c3\nh : List.drop (\u2191n) (List.take (\u2191m) x) = []\nhlen' : \u2191m \u2264 \u2191n\n\u22a2 n \u2264 m\n\ncase refine'_3.a\n\u03b1\u271d : Type u\n\u03c3\u271d : Type v\nM\u271d : DFA \u03b1\u271d \u03c3\u271d\n\u03b1 : Type u\n\u03c3 : Type v\nM : DFA \u03b1 \u03c3\ninst\u271d : Fintype \u03c3\nx : List \u03b1\ns t : \u03c3\nhlen : Fintype.card \u03c3 \u2264 List.length x\nhx : evalFrom M s x = t\nn m : Fin (Fintype.card \u03c3 + 1)\nhneq : n \u2260 m\nheq : evalFrom M s (List.take (\u2191n) x) = evalFrom M s (List.take (\u2191m) x)\nhle : \u2191n \u2264 \u2191m\nhm : \u2191m \u2264 Fintype.card \u03c3\nh : List.drop (\u2191n) (List.take (\u2191m) x) = []\nhlen' : \u2191m \u2264 \u2191n\n\u22a2 m \u2264 n"}, {"tactic": "assumption'", "annotated_tactic": ["assumption'", []], "state_before": "case refine'_3.a\n\u03b1\u271d : Type u\n\u03c3\u271d : Type v\nM\u271d : DFA \u03b1\u271d \u03c3\u271d\n\u03b1 : Type u\n\u03c3 : Type v\nM : DFA \u03b1 \u03c3\ninst\u271d : Fintype \u03c3\nx : List \u03b1\ns t : \u03c3\nhlen : Fintype.card \u03c3 \u2264 List.length x\nhx : evalFrom M s x = t\nn m : Fin (Fintype.card \u03c3 + 1)\nhneq : n \u2260 m\nheq : evalFrom M s (List.take (\u2191n) x) = evalFrom M s (List.take (\u2191m) x)\nhle : \u2191n \u2264 \u2191m\nhm : \u2191m \u2264 Fintype.card \u03c3\nh : List.drop (\u2191n) (List.take (\u2191m) x) = []\nhlen' : \u2191m \u2264 \u2191n\n\u22a2 n \u2264 m\n\ncase refine'_3.a\n\u03b1\u271d : Type u\n\u03c3\u271d : Type v\nM\u271d : DFA \u03b1\u271d \u03c3\u271d\n\u03b1 : Type u\n\u03c3 : Type v\nM : DFA \u03b1 \u03c3\ninst\u271d : Fintype \u03c3\nx : List \u03b1\ns t : \u03c3\nhlen : Fintype.card \u03c3 \u2264 List.length x\nhx : evalFrom M s x = t\nn m : Fin (Fintype.card \u03c3 + 1)\nhneq : n \u2260 m\nheq : evalFrom M s (List.take (\u2191n) x) = evalFrom M s (List.take (\u2191m) x)\nhle : \u2191n \u2264 \u2191m\nhm : \u2191m \u2264 Fintype.card \u03c3\nh : List.drop (\u2191n) (List.take (\u2191m) x) = []\nhlen' : \u2191m \u2264 \u2191n\n\u22a2 m \u2264 n", "state_after": "no goals"}, {"tactic": "rw [List.take_take, min_eq_left hle, \u2190 evalFrom_of_append, heq, \u2190 min_eq_left hle, \u2190\n  List.take_take, min_eq_left hle, List.take_append_drop]", "annotated_tactic": ["rw [<a>List.take_take</a>, <a>min_eq_left</a> hle, \u2190 <a>evalFrom_of_append</a>, heq, \u2190 <a>min_eq_left</a> hle, \u2190\n      <a>List.take_take</a>, <a>min_eq_left</a> hle, <a>List.take_append_drop</a>]", [{"full_name": "List.take_take", "def_path": "Mathlib/Data/List/Basic.lean", "def_pos": [1886, 9], "def_end_pos": [1886, 18]}, {"full_name": "min_eq_left", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [100, 9], "def_end_pos": [100, 20]}, {"full_name": "DFA.evalFrom_of_append", "def_path": "Mathlib/Computability/DFA.lean", "def_pos": [89, 9], "def_end_pos": [89, 27]}, {"full_name": "min_eq_left", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [100, 9], "def_end_pos": [100, 20]}, {"full_name": "List.take_take", "def_path": "Mathlib/Data/List/Basic.lean", "def_pos": [1886, 9], "def_end_pos": [1886, 18]}, {"full_name": "min_eq_left", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [100, 9], "def_end_pos": [100, 20]}, {"full_name": "List.take_append_drop", "def_path": "lake-packages/std/Std/Data/List/Init/Lemmas.lean", "def_pos": [138, 17], "def_end_pos": [138, 33]}]], "state_before": "\u03b1\u271d : Type u\n\u03c3\u271d : Type v\nM\u271d : DFA \u03b1\u271d \u03c3\u271d\n\u03b1 : Type u\n\u03c3 : Type v\nM : DFA \u03b1 \u03c3\ninst\u271d : Fintype \u03c3\nx : List \u03b1\ns t : \u03c3\nhlen : Fintype.card \u03c3 \u2264 List.length x\nhx : evalFrom M s x = t\nn m : Fin (Fintype.card \u03c3 + 1)\nhneq : n \u2260 m\nheq : evalFrom M s (List.take (\u2191n) x) = evalFrom M s (List.take (\u2191m) x)\nhle : \u2191n \u2264 \u2191m\nhm : \u2191m \u2264 Fintype.card \u03c3\n\u22a2 evalFrom M (evalFrom M s (List.take (\u2191n) (List.take (\u2191m) x))) (List.drop (\u2191n) (List.take (\u2191m) x)) =\n    evalFrom M s (List.take (\u2191n) (List.take (\u2191m) x))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Vector/Basic.lean", "full_name": "Vector.ofFn_get", "start": [155, 1], "end": [159, 49], "traced_tactics": [{"tactic": "rcases v with \u27e8l, rfl\u27e9", "annotated_tactic": ["rcases v with \u27e8l, rfl\u27e9", []], "state_before": "n : \u2115\n\u03b1 : Type u_1\nv : Vector \u03b1 n\n\u22a2 ofFn (get v) = v", "state_after": "case mk\n\u03b1 : Type u_1\nl : List \u03b1\n\u22a2 ofFn (get { val := l, property := (_ : List.length l = List.length l) }) =\n    { val := l, property := (_ : List.length l = List.length l) }"}, {"tactic": "apply toList_injective", "annotated_tactic": ["apply <a>toList_injective</a>", [{"full_name": "Vector.toList_injective", "def_path": "Mathlib/Data/Vector/Basic.lean", "def_pos": [39, 9], "def_end_pos": [39, 25]}]], "state_before": "case mk\n\u03b1 : Type u_1\nl : List \u03b1\n\u22a2 ofFn (get { val := l, property := (_ : List.length l = List.length l) }) =\n    { val := l, property := (_ : List.length l = List.length l) }", "state_after": "case mk.a\n\u03b1 : Type u_1\nl : List \u03b1\n\u22a2 toList (ofFn (get { val := l, property := (_ : List.length l = List.length l) })) =\n    toList { val := l, property := (_ : List.length l = List.length l) }"}, {"tactic": "dsimp", "annotated_tactic": ["dsimp", []], "state_before": "case mk.a\n\u03b1 : Type u_1\nl : List \u03b1\n\u22a2 toList (ofFn (get { val := l, property := (_ : List.length l = List.length l) })) =\n    toList { val := l, property := (_ : List.length l = List.length l) }", "state_after": "case mk.a\n\u03b1 : Type u_1\nl : List \u03b1\n\u22a2 toList (ofFn (get { val := l, property := (_ : List.length l = List.length l) })) = l"}, {"tactic": "simpa only [toList_ofFn] using List.ofFn_get _", "annotated_tactic": ["simpa only [<a>toList_ofFn</a>] using <a>List.ofFn_get</a> _", [{"full_name": "Vector.toList_ofFn", "def_path": "Mathlib/Data/Vector/Basic.lean", "def_pos": [78, 9], "def_end_pos": [78, 20]}, {"full_name": "List.ofFn_get", "def_path": "Mathlib/Data/List/OfFn.lean", "def_pos": [184, 9], "def_end_pos": [184, 17]}]], "state_before": "case mk.a\n\u03b1 : Type u_1\nl : List \u03b1\n\u22a2 toList (ofFn (get { val := l, property := (_ : List.length l = List.length l) })) = l", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Nat/Gcd.lean", "full_name": "Nat.gcd_gcd_self_right_right", "start": [165, 9], "end": [166, 45], "traced_tactics": [{"tactic": "rw [gcd_comm n m, gcd_gcd_self_right_left]", "annotated_tactic": ["rw [<a>gcd_comm</a> n m, <a>gcd_gcd_self_right_left</a>]", [{"full_name": "Nat.gcd_comm", "def_path": "lake-packages/std/Std/Data/Nat/Gcd.lean", "def_pos": [59, 9], "def_end_pos": [59, 17]}, {"full_name": "Nat.gcd_gcd_self_right_left", "def_path": "lake-packages/std/Std/Data/Nat/Gcd.lean", "def_pos": [162, 17], "def_end_pos": [162, 40]}]], "state_before": "m n : Nat\n\u22a2 gcd m (gcd n m) = gcd n m", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/Average.lean", "full_name": "MeasureTheory.exists_not_mem_null_integral_le", "start": [597, 1], "end": [600, 74], "traced_tactics": [{"tactic": "simpa only [average_eq_integral] using\n  exists_not_mem_null_average_le (IsProbabilityMeasure.ne_zero \u03bc) hf hN", "annotated_tactic": ["simpa only [<a>average_eq_integral</a>] using\n    <a>exists_not_mem_null_average_le</a> (<a>IsProbabilityMeasure.ne_zero</a> \u03bc) hf hN", [{"full_name": "MeasureTheory.average_eq_integral", "def_path": "Mathlib/MeasureTheory/Integral/Average.lean", "def_pos": [285, 9], "def_end_pos": [285, 28]}, {"full_name": "MeasureTheory.exists_not_mem_null_average_le", "def_path": "Mathlib/MeasureTheory/Integral/Average.lean", "def_pos": [552, 9], "def_end_pos": [552, 39]}, {"full_name": "MeasureTheory.IsProbabilityMeasure.ne_zero", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3040, 9], "def_end_pos": [3040, 37]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t N : Set \u03b1\nf : \u03b1 \u2192 \u211d\ninst\u271d : IsProbabilityMeasure \u03bc\nhf : Integrable f\nhN : \u2191\u2191\u03bc N = 0\n\u22a2 \u2203 x, \u00acx \u2208 N \u2227 \u222b (a : \u03b1), f a \u2202\u03bc \u2264 f x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Vector/Basic.lean", "full_name": "Vector.reverse_get_zero", "start": [300, 1], "end": [307, 9], "traced_tactics": [{"tactic": "rw [\u2190 get_zero, last_def, get_eq_get, get_eq_get]", "annotated_tactic": ["rw [\u2190 <a>get_zero</a>, <a>last_def</a>, <a>get_eq_get</a>, <a>get_eq_get</a>]", [{"full_name": "Vector.get_zero", "def_path": "Mathlib/Data/Vector/Basic.lean", "def_pos": [264, 9], "def_end_pos": [264, 17]}, {"full_name": "Vector.last_def", "def_path": "Mathlib/Data/Vector/Basic.lean", "def_pos": [295, 9], "def_end_pos": [295, 17]}, {"full_name": "Vector.get_eq_get", "def_path": "Mathlib/Data/Vector/Basic.lean", "def_pos": [116, 9], "def_end_pos": [116, 19]}, {"full_name": "Vector.get_eq_get", "def_path": "Mathlib/Data/Vector/Basic.lean", "def_pos": [116, 9], "def_end_pos": [116, 19]}]], "state_before": "n : \u2115\n\u03b1 : Type u_1\nv : Vector \u03b1 (n + 1)\n\u22a2 head (reverse v) = last v", "state_after": "n : \u2115\n\u03b1 : Type u_1\nv : Vector \u03b1 (n + 1)\n\u22a2 List.get (toList (reverse v)) (Fin.cast (_ : Nat.succ n = List.length (toList (reverse v))) 0) =\n    List.get (toList v) (Fin.cast (_ : n + 1 = List.length (toList v)) (Fin.last n))"}, {"tactic": "simp_rw [toList_reverse]", "annotated_tactic": ["simp_rw [<a>toList_reverse</a>]", [{"full_name": "Vector.toList_reverse", "def_path": "Mathlib/Data/Vector/Basic.lean", "def_pos": [253, 9], "def_end_pos": [253, 23]}]], "state_before": "n : \u2115\n\u03b1 : Type u_1\nv : Vector \u03b1 (n + 1)\n\u22a2 List.get (toList (reverse v)) (Fin.cast (_ : Nat.succ n = List.length (toList (reverse v))) 0) =\n    List.get (toList v) (Fin.cast (_ : n + 1 = List.length (toList v)) (Fin.last n))", "state_after": "n : \u2115\n\u03b1 : Type u_1\nv : Vector \u03b1 (n + 1)\n\u22a2 List.get (List.reverse (toList v)) (Fin.cast (_ : Nat.succ n = List.length (toList (reverse v))) 0) =\n    List.get (toList v) (Fin.cast (_ : n + 1 = List.length (toList v)) (Fin.last n))"}, {"tactic": "rw [\u2190 Option.some_inj, Fin.cast, Fin.cast, \u2190 List.get?_eq_get, \u2190 List.get?_eq_get,\n  List.get?_reverse]", "annotated_tactic": ["rw [\u2190 <a>Option.some_inj</a>, <a>Fin.cast</a>, <a>Fin.cast</a>, \u2190 <a>List.get?_eq_get</a>, \u2190 <a>List.get?_eq_get</a>,\n    <a>List.get?_reverse</a>]", [{"full_name": "Option.some_inj", "def_path": "lake-packages/std/Std/Data/Option/Basic.lean", "def_pos": [27, 9], "def_end_pos": [27, 17]}, {"full_name": "Fin.cast", "def_path": "lake-packages/std/Std/Data/Fin/Basic.lean", "def_pos": [23, 15], "def_end_pos": [23, 19]}, {"full_name": "Fin.cast", "def_path": "lake-packages/std/Std/Data/Fin/Basic.lean", "def_pos": [23, 15], "def_end_pos": [23, 19]}, {"full_name": "List.get?_eq_get", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [581, 9], "def_end_pos": [581, 20]}, {"full_name": "List.get?_eq_get", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [581, 9], "def_end_pos": [581, 20]}, {"full_name": "List.get?_reverse", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [762, 9], "def_end_pos": [762, 21]}]], "state_before": "n : \u2115\n\u03b1 : Type u_1\nv : Vector \u03b1 (n + 1)\n\u22a2 List.get (List.reverse (toList v)) (Fin.cast (_ : Nat.succ n = List.length (toList (reverse v))) 0) =\n    List.get (toList v) (Fin.cast (_ : n + 1 = List.length (toList v)) (Fin.last n))", "state_after": "n : \u2115\n\u03b1 : Type u_1\nv : Vector \u03b1 (n + 1)\n\u22a2 List.get? (toList v) (List.length (toList v) - 1 - \u21910) = List.get? (toList v) \u2191(Fin.last n)\n\ncase h\nn : \u2115\n\u03b1 : Type u_1\nv : Vector \u03b1 (n + 1)\n\u22a2 \u21910 < List.length (toList v)"}, {"tactic": "congr", "annotated_tactic": ["congr", []], "state_before": "n : \u2115\n\u03b1 : Type u_1\nv : Vector \u03b1 (n + 1)\n\u22a2 List.get? (toList v) (List.length (toList v) - 1 - \u21910) = List.get? (toList v) \u2191(Fin.last n)", "state_after": "case e_a\nn : \u2115\n\u03b1 : Type u_1\nv : Vector \u03b1 (n + 1)\n\u22a2 List.length (toList v) - 1 - \u21910 = \u2191(Fin.last n)"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case e_a\nn : \u2115\n\u03b1 : Type u_1\nv : Vector \u03b1 (n + 1)\n\u22a2 List.length (toList v) - 1 - \u21910 = \u2191(Fin.last n)", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case h\nn : \u2115\n\u03b1 : Type u_1\nv : Vector \u03b1 (n + 1)\n\u22a2 \u21910 < List.length (toList v)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/ProbabilityMassFunction/Monad.lean", "full_name": "PMF.bind_pure", "start": [139, 1], "end": [142, 38], "traced_tactics": [{"tactic": "rw [pure_apply_of_ne _ _ hy.symm, mul_zero]", "annotated_tactic": ["rw [<a>pure_apply_of_ne</a> _ _ hy.symm, <a>mul_zero</a>]", [{"full_name": "PMF.pure_apply_of_ne", "def_path": "Mathlib/Probability/ProbabilityMassFunction/Monad.lean", "def_pos": [61, 9], "def_end_pos": [61, 25]}, {"full_name": "MulZeroClass.mul_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [38, 3], "def_end_pos": [38, 11]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\nf : \u03b1 \u2192 PMF \u03b2\ng : \u03b2 \u2192 PMF \u03b3\nx y : \u03b1\nhy : y \u2260 x\n\u22a2 \u2191p y * \u2191(pure y) x = 0", "state_after": "no goals"}, {"tactic": "rw [pure_apply_self, mul_one]", "annotated_tactic": ["rw [<a>pure_apply_self</a>, <a>mul_one</a>]", [{"full_name": "PMF.pure_apply_self", "def_path": "Mathlib/Probability/ProbabilityMassFunction/Monad.lean", "def_pos": [57, 9], "def_end_pos": [57, 24]}, {"full_name": "mul_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [470, 9], "def_end_pos": [470, 16]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\nf : \u03b1 \u2192 PMF \u03b2\ng : \u03b2 \u2192 PMF \u03b3\nx : \u03b1\n\u22a2 \u2191p x * \u2191(pure x) x = \u2191p x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Sum.lean", "full_name": "Finset.empty_disjSum", "start": [40, 1], "end": [41, 39], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/StrongLaw.lean", "full_name": "ProbabilityTheory.sum_variance_truncation_le", "start": [339, 1], "end": [400, 80], "traced_tactics": [{"tactic": "set Y := fun n : \u2115 => truncation X n", "annotated_tactic": ["set Y := fun n : \u2115 => <a>truncation</a> X n", [{"full_name": "ProbabilityTheory.truncation", "def_path": "Mathlib/Probability/StrongLaw.lean", "def_pos": [78, 5], "def_end_pos": [78, 15]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\n\u22a2 \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * \u222b (a : \u03a9), (truncation X \u2191j ^ 2) a \u2264 2 * \u222b (a : \u03a9), X a", "state_after": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u22a2 \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * \u222b (a : \u03a9), (truncation X \u2191j ^ 2) a \u2264 2 * \u222b (a : \u03a9), X a"}, {"tactic": "let \u03c1 : Measure \u211d := Measure.map X \u2119", "annotated_tactic": ["let \u03c1 : <a>Measure</a> \u211d := <a>Measure.map</a> X \u2119", [{"full_name": "MeasureTheory.Measure", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [74, 11], "def_end_pos": [74, 18]}, {"full_name": "MeasureTheory.Measure.map", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1163, 17], "def_end_pos": [1163, 20]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u22a2 \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * \u222b (a : \u03a9), (truncation X \u2191j ^ 2) a \u2264 2 * \u222b (a : \u03a9), X a", "state_after": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\n\u22a2 \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * \u222b (a : \u03a9), (truncation X \u2191j ^ 2) a \u2264 2 * \u222b (a : \u03a9), X a"}, {"tactic": "have Y2 : \u2200 n, \ud835\udd3c[Y n ^ 2] = \u222b x in (0)..n, x ^ 2 \u2202\u03c1 := by\n  intro n\n  change \ud835\udd3c[fun x => Y n x ^ 2] = _\n  rw [moment_truncation_eq_intervalIntegral_of_nonneg hint.1 two_ne_zero hnonneg]", "annotated_tactic": ["have Y2 : \u2200 n, \ud835\udd3c[Y n ^ 2] = \u222b x in (0)..n, x ^ 2 \u2202\u03c1 := by\n    intro n\n    change \ud835\udd3c[fun x => Y n x ^ 2] = _\n    rw [<a>moment_truncation_eq_intervalIntegral_of_nonneg</a> hint.1 <a>two_ne_zero</a> hnonneg]", [{"full_name": "ProbabilityTheory.moment_truncation_eq_intervalIntegral_of_nonneg", "def_path": "Mathlib/Probability/StrongLaw.lean", "def_pos": [153, 9], "def_end_pos": [153, 56]}, {"full_name": "two_ne_zero", "def_path": "Mathlib/Algebra/NeZero.lean", "def_pos": [62, 7], "def_end_pos": [62, 18]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\n\u22a2 \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * \u222b (a : \u03a9), (truncation X \u2191j ^ 2) a \u2264 2 * \u222b (a : \u03a9), X a", "state_after": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\n\u22a2 \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * \u222b (a : \u03a9), (truncation X \u2191j ^ 2) a \u2264 2 * \u222b (a : \u03a9), X a"}, {"tactic": "intro n", "annotated_tactic": ["intro n", []], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\n\u22a2 \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1", "state_after": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nn : \u2115\n\u22a2 \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1"}, {"tactic": "change \ud835\udd3c[fun x => Y n x ^ 2] = _", "annotated_tactic": ["change \ud835\udd3c[fun x => Y n x ^ 2] = _", []], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nn : \u2115\n\u22a2 \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1", "state_after": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nn : \u2115\n\u22a2 \u222b (a : \u03a9), (fun x => Y n x ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1"}, {"tactic": "rw [moment_truncation_eq_intervalIntegral_of_nonneg hint.1 two_ne_zero hnonneg]", "annotated_tactic": ["rw [<a>moment_truncation_eq_intervalIntegral_of_nonneg</a> hint.1 <a>two_ne_zero</a> hnonneg]", [{"full_name": "ProbabilityTheory.moment_truncation_eq_intervalIntegral_of_nonneg", "def_path": "Mathlib/Probability/StrongLaw.lean", "def_pos": [153, 9], "def_end_pos": [153, 56]}, {"full_name": "two_ne_zero", "def_path": "Mathlib/Algebra/NeZero.lean", "def_pos": [62, 7], "def_end_pos": [62, 18]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nn : \u2115\n\u22a2 \u222b (a : \u03a9), (fun x => Y n x ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1", "state_after": "no goals"}, {"tactic": "simp_rw [Y2]", "annotated_tactic": ["simp_rw [Y2]", []], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\n\u22a2 \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * \u222b (a : \u03a9), (Y j ^ 2) a = \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * \u222b (x : \u211d) in 0 ..\u2191j, x ^ 2 \u2202\u03c1", "state_after": "no goals"}, {"tactic": "congr 1 with j", "annotated_tactic": ["congr 1 with j", []], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\n\u22a2 \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * \u222b (x : \u211d) in 0 ..\u2191j, x ^ 2 \u2202\u03c1 =\n    \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * \u2211 k in range j, \u222b (x : \u211d) in \u2191k..\u2191(k + 1), x ^ 2 \u2202\u03c1", "state_after": "case e_f.h\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\nj : \u2115\n\u22a2 (\u2191j ^ 2)\u207b\u00b9 * \u222b (x : \u211d) in 0 ..\u2191j, x ^ 2 \u2202\u03c1 = (\u2191j ^ 2)\u207b\u00b9 * \u2211 k in range j, \u222b (x : \u211d) in \u2191k..\u2191(k + 1), x ^ 2 \u2202\u03c1"}, {"tactic": "congr 1", "annotated_tactic": ["congr 1", []], "state_before": "case e_f.h\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\nj : \u2115\n\u22a2 (\u2191j ^ 2)\u207b\u00b9 * \u222b (x : \u211d) in 0 ..\u2191j, x ^ 2 \u2202\u03c1 = (\u2191j ^ 2)\u207b\u00b9 * \u2211 k in range j, \u222b (x : \u211d) in \u2191k..\u2191(k + 1), x ^ 2 \u2202\u03c1", "state_after": "case e_f.h.e_a\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\nj : \u2115\n\u22a2 \u222b (x : \u211d) in 0 ..\u2191j, x ^ 2 \u2202\u03c1 = \u2211 k in range j, \u222b (x : \u211d) in \u2191k..\u2191(k + 1), x ^ 2 \u2202\u03c1"}, {"tactic": "rw [intervalIntegral.sum_integral_adjacent_intervals]", "annotated_tactic": ["rw [<a>intervalIntegral.sum_integral_adjacent_intervals</a>]", [{"full_name": "intervalIntegral.sum_integral_adjacent_intervals", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [922, 9], "def_end_pos": [922, 40]}]], "state_before": "case e_f.h.e_a\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\nj : \u2115\n\u22a2 \u222b (x : \u211d) in 0 ..\u2191j, x ^ 2 \u2202\u03c1 = \u2211 k in range j, \u222b (x : \u211d) in \u2191k..\u2191(k + 1), x ^ 2 \u2202\u03c1", "state_after": "case e_f.h.e_a\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\nj : \u2115\n\u22a2 \u222b (x : \u211d) in 0 ..\u2191j, x ^ 2 \u2202\u03c1 = \u222b (x : \u211d) in \u21910 ..\u2191j, x ^ 2 \u2202\u03c1\n\ncase e_f.h.e_a\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\nj : \u2115\n\u22a2 \u2200 (k : \u2115), k < j \u2192 IntervalIntegrable (fun x => x ^ 2) \u03c1 \u2191k \u2191(k + 1)"}, {"tactic": "intro k _", "annotated_tactic": ["intro k _", []], "state_before": "case e_f.h.e_a\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\nj : \u2115\n\u22a2 \u2200 (k : \u2115), k < j \u2192 IntervalIntegrable (fun x => x ^ 2) \u03c1 \u2191k \u2191(k + 1)", "state_after": "case e_f.h.e_a\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\nj k : \u2115\na\u271d : k < j\n\u22a2 IntervalIntegrable (fun x => x ^ 2) \u03c1 \u2191k \u2191(k + 1)"}, {"tactic": "exact (continuous_id.pow _).intervalIntegrable _ _", "annotated_tactic": ["exact (continuous_id.pow _).<a>intervalIntegrable</a> _ _", [{"full_name": "Continuous.intervalIntegrable", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [366, 9], "def_end_pos": [366, 38]}]], "state_before": "case e_f.h.e_a\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\nj k : \u2115\na\u271d : k < j\n\u22a2 IntervalIntegrable (fun x => x ^ 2) \u03c1 \u2191k \u2191(k + 1)", "state_after": "no goals"}, {"tactic": "norm_cast", "annotated_tactic": ["norm_cast", []], "state_before": "case e_f.h.e_a\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\nj : \u2115\n\u22a2 \u222b (x : \u211d) in 0 ..\u2191j, x ^ 2 \u2202\u03c1 = \u222b (x : \u211d) in \u21910 ..\u2191j, x ^ 2 \u2202\u03c1", "state_after": "no goals"}, {"tactic": "simp_rw [mul_sum, sum_mul, sum_sigma']", "annotated_tactic": ["simp_rw [<a>mul_sum</a>, <a>sum_mul</a>, <a>sum_sigma'</a>]", [{"full_name": "Finset.mul_sum", "def_path": "Mathlib/Algebra/BigOperators/Ring.lean", "def_pos": [55, 9], "def_end_pos": [55, 16]}, {"full_name": "Finset.sum_mul", "def_path": "Mathlib/Algebra/BigOperators/Ring.lean", "def_pos": [51, 9], "def_end_pos": [51, 16]}, {"full_name": "Finset.sum_sigma'", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [531, 3], "def_end_pos": [531, 14]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\n\u22a2 \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * \u2211 k in range j, \u222b (x : \u211d) in \u2191k..\u2191(k + 1), x ^ 2 \u2202\u03c1 =\n    \u2211 k in range K, (\u2211 j in Ioo k K, (\u2191j ^ 2)\u207b\u00b9) * \u222b (x : \u211d) in \u2191k..\u2191(k + 1), x ^ 2 \u2202\u03c1", "state_after": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\n\u22a2 \u2211 x in Finset.sigma (range K) fun a => range a,\n      (\u2191x.fst ^ 2)\u207b\u00b9 * \u222b (x : \u211d) in \u2191x.snd..\u2191(x.snd + 1), x ^ 2 \u2202Measure.map X \u2119 =\n    \u2211 x in Finset.sigma (range K) fun a => Ioo a K,\n      (\u2191x.snd ^ 2)\u207b\u00b9 * \u222b (x : \u211d) in \u2191x.fst..\u2191(x.fst + 1), x ^ 2 \u2202Measure.map X \u2119"}, {"tactic": "refine' sum_bij' (fun (p : \u03a3 _ : \u2115, \u2115) _ => (\u27e8p.2, p.1\u27e9 : \u03a3 _ : \u2115, \u2115)) _ (fun a _ => rfl)\n  (fun (p : \u03a3 _ : \u2115, \u2115) _ => (\u27e8p.2, p.1\u27e9 : \u03a3 _ : \u2115, \u2115)) _ _ _", "annotated_tactic": ["refine' <a>sum_bij'</a> (fun (p : \u03a3 _ : \u2115, \u2115) _ => (\u27e8p.2, p.1\u27e9 : \u03a3 _ : \u2115, \u2115)) _ (fun a _ => <a>rfl</a>)\n        (fun (p : \u03a3 _ : \u2115, \u2115) _ => (\u27e8p.2, p.1\u27e9 : \u03a3 _ : \u2115, \u2115)) _ _ _", [{"full_name": "Finset.sum_bij'", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [560, 3], "def_end_pos": [560, 14]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\n\u22a2 \u2211 x in Finset.sigma (range K) fun a => range a,\n      (\u2191x.fst ^ 2)\u207b\u00b9 * \u222b (x : \u211d) in \u2191x.snd..\u2191(x.snd + 1), x ^ 2 \u2202Measure.map X \u2119 =\n    \u2211 x in Finset.sigma (range K) fun a => Ioo a K,\n      (\u2191x.snd ^ 2)\u207b\u00b9 * \u222b (x : \u211d) in \u2191x.fst..\u2191(x.fst + 1), x ^ 2 \u2202Measure.map X \u2119", "state_after": "case refine'_1\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\n\u22a2 \u2200 (a : (_ : \u2115) \u00d7 \u2115) (ha : a \u2208 Finset.sigma (range K) fun a => range a),\n    (fun p x => { fst := p.snd, snd := p.fst }) a ha \u2208 Finset.sigma (range K) fun a => Ioo a K\n\ncase refine'_2\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\n\u22a2 \u2200 (a : (_ : \u2115) \u00d7 \u2115) (ha : a \u2208 Finset.sigma (range K) fun a => Ioo a K),\n    (fun p x => { fst := p.snd, snd := p.fst }) a ha \u2208 Finset.sigma (range K) fun a => range a\n\ncase refine'_3\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\n\u22a2 \u2200 (a : (_ : \u2115) \u00d7 \u2115) (ha : a \u2208 Finset.sigma (range K) fun a => range a),\n    (fun p x => { fst := p.snd, snd := p.fst }) ((fun p x => { fst := p.snd, snd := p.fst }) a ha)\n        (_ : (fun p x => { fst := p.snd, snd := p.fst }) a ha \u2208 ?m.184176) =\n      a\n\ncase refine'_4\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\n\u22a2 \u2200 (a : (_ : \u2115) \u00d7 \u2115) (ha : a \u2208 Finset.sigma (range K) fun a => Ioo a K),\n    (fun p x => { fst := p.snd, snd := p.fst }) ((fun p x => { fst := p.snd, snd := p.fst }) a ha)\n        (_ : (fun p x => { fst := p.snd, snd := p.fst }) a ha \u2208 ?m.184175) =\n      a"}, {"tactic": "rintro \u27e8i, j\u27e9 hij", "annotated_tactic": ["rintro \u27e8i, j\u27e9 hij", []], "state_before": "case refine'_1\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\n\u22a2 \u2200 (a : (_ : \u2115) \u00d7 \u2115) (ha : a \u2208 Finset.sigma (range K) fun a => range a),\n    (fun p x => { fst := p.snd, snd := p.fst }) a ha \u2208 Finset.sigma (range K) fun a => Ioo a K", "state_after": "case refine'_1.mk\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\ni j : \u2115\nhij : { fst := i, snd := j } \u2208 Finset.sigma (range K) fun a => range a\n\u22a2 (fun p x => { fst := p.snd, snd := p.fst }) { fst := i, snd := j } hij \u2208 Finset.sigma (range K) fun a => Ioo a K"}, {"tactic": "simp only [mem_sigma, mem_range, mem_filter] at hij", "annotated_tactic": ["simp only [<a>mem_sigma</a>, <a>mem_range</a>, <a>mem_filter</a>] at hij", [{"full_name": "Finset.mem_sigma", "def_path": "Mathlib/Data/Finset/Sigma.lean", "def_pos": [49, 9], "def_end_pos": [49, 18]}, {"full_name": "Finset.mem_range", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3037, 9], "def_end_pos": [3037, 18]}, {"full_name": "Finset.mem_filter", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2708, 9], "def_end_pos": [2708, 19]}]], "state_before": "case refine'_1.mk\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\ni j : \u2115\nhij : { fst := i, snd := j } \u2208 Finset.sigma (range K) fun a => range a\n\u22a2 (fun p x => { fst := p.snd, snd := p.fst }) { fst := i, snd := j } hij \u2208 Finset.sigma (range K) fun a => Ioo a K", "state_after": "case refine'_1.mk\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\ni j : \u2115\nhij\u271d : { fst := i, snd := j } \u2208 Finset.sigma (range K) fun a => range a\nhij : i < K \u2227 j < i\n\u22a2 (fun p x => { fst := p.snd, snd := p.fst }) { fst := i, snd := j } hij\u271d \u2208 Finset.sigma (range K) fun a => Ioo a K"}, {"tactic": "simp [hij, mem_sigma, mem_range, and_self_iff, hij.2.trans hij.1]", "annotated_tactic": ["simp [hij, <a>mem_sigma</a>, <a>mem_range</a>, <a>and_self_iff</a>, hij.2.<a>trans</a> hij.1]", [{"full_name": "Finset.mem_sigma", "def_path": "Mathlib/Data/Finset/Sigma.lean", "def_pos": [49, 9], "def_end_pos": [49, 18]}, {"full_name": "Finset.mem_range", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3037, 9], "def_end_pos": [3037, 18]}, {"full_name": "and_self_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [155, 9], "def_end_pos": [155, 21]}, {"full_name": "LT.lt.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [144, 7], "def_end_pos": [144, 18]}]], "state_before": "case refine'_1.mk\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\ni j : \u2115\nhij\u271d : { fst := i, snd := j } \u2208 Finset.sigma (range K) fun a => range a\nhij : i < K \u2227 j < i\n\u22a2 (fun p x => { fst := p.snd, snd := p.fst }) { fst := i, snd := j } hij\u271d \u2208 Finset.sigma (range K) fun a => Ioo a K", "state_after": "no goals"}, {"tactic": "rintro \u27e8i, j\u27e9 hij", "annotated_tactic": ["rintro \u27e8i, j\u27e9 hij", []], "state_before": "case refine'_2\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\n\u22a2 \u2200 (a : (_ : \u2115) \u00d7 \u2115) (ha : a \u2208 Finset.sigma (range K) fun a => Ioo a K),\n    (fun p x => { fst := p.snd, snd := p.fst }) a ha \u2208 Finset.sigma (range K) fun a => range a", "state_after": "case refine'_2.mk\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\ni j : \u2115\nhij : { fst := i, snd := j } \u2208 Finset.sigma (range K) fun a => Ioo a K\n\u22a2 (fun p x => { fst := p.snd, snd := p.fst }) { fst := i, snd := j } hij \u2208 Finset.sigma (range K) fun a => range a"}, {"tactic": "simp only [mem_sigma, mem_range, mem_Ioo] at hij", "annotated_tactic": ["simp only [<a>mem_sigma</a>, <a>mem_range</a>, <a>mem_Ioo</a>] at hij", [{"full_name": "Finset.mem_sigma", "def_path": "Mathlib/Data/Finset/Sigma.lean", "def_pos": [49, 9], "def_end_pos": [49, 18]}, {"full_name": "Finset.mem_range", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3037, 9], "def_end_pos": [3037, 18]}, {"full_name": "Finset.mem_Ioo", "def_path": "Mathlib/Order/LocallyFinite.lean", "def_pos": [341, 9], "def_end_pos": [341, 16]}]], "state_before": "case refine'_2.mk\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\ni j : \u2115\nhij : { fst := i, snd := j } \u2208 Finset.sigma (range K) fun a => Ioo a K\n\u22a2 (fun p x => { fst := p.snd, snd := p.fst }) { fst := i, snd := j } hij \u2208 Finset.sigma (range K) fun a => range a", "state_after": "case refine'_2.mk\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\ni j : \u2115\nhij\u271d : { fst := i, snd := j } \u2208 Finset.sigma (range K) fun a => Ioo a K\nhij : i < K \u2227 i < j \u2227 j < K\n\u22a2 (fun p x => { fst := p.snd, snd := p.fst }) { fst := i, snd := j } hij\u271d \u2208 Finset.sigma (range K) fun a => range a"}, {"tactic": "simp only [hij, mem_sigma, mem_range, and_self_iff]", "annotated_tactic": ["simp only [hij, <a>mem_sigma</a>, <a>mem_range</a>, <a>and_self_iff</a>]", [{"full_name": "Finset.mem_sigma", "def_path": "Mathlib/Data/Finset/Sigma.lean", "def_pos": [49, 9], "def_end_pos": [49, 18]}, {"full_name": "Finset.mem_range", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3037, 9], "def_end_pos": [3037, 18]}, {"full_name": "and_self_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [155, 9], "def_end_pos": [155, 21]}]], "state_before": "case refine'_2.mk\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\ni j : \u2115\nhij\u271d : { fst := i, snd := j } \u2208 Finset.sigma (range K) fun a => Ioo a K\nhij : i < K \u2227 i < j \u2227 j < K\n\u22a2 (fun p x => { fst := p.snd, snd := p.fst }) { fst := i, snd := j } hij\u271d \u2208 Finset.sigma (range K) fun a => range a", "state_after": "no goals"}, {"tactic": "rintro \u27e8i, j\u27e9 hij", "annotated_tactic": ["rintro \u27e8i, j\u27e9 hij", []], "state_before": "case refine'_3\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\n\u22a2 \u2200 (a : (_ : \u2115) \u00d7 \u2115) (ha : a \u2208 Finset.sigma (range K) fun a => range a),\n    (fun p x => { fst := p.snd, snd := p.fst }) ((fun p x => { fst := p.snd, snd := p.fst }) a ha)\n        (_ : (fun p x => { fst := p.snd, snd := p.fst }) a ha \u2208 Finset.sigma (range K) fun a => Ioo a K) =\n      a", "state_after": "case refine'_3.mk\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\ni j : \u2115\nhij : { fst := i, snd := j } \u2208 Finset.sigma (range K) fun a => range a\n\u22a2 (fun p x => { fst := p.snd, snd := p.fst }) ((fun p x => { fst := p.snd, snd := p.fst }) { fst := i, snd := j } hij)\n      (_ :\n        (fun p x => { fst := p.snd, snd := p.fst }) { fst := i, snd := j } hij \u2208\n          Finset.sigma (range K) fun a => Ioo a K) =\n    { fst := i, snd := j }"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case refine'_3.mk\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\ni j : \u2115\nhij : { fst := i, snd := j } \u2208 Finset.sigma (range K) fun a => range a\n\u22a2 (fun p x => { fst := p.snd, snd := p.fst }) ((fun p x => { fst := p.snd, snd := p.fst }) { fst := i, snd := j } hij)\n      (_ :\n        (fun p x => { fst := p.snd, snd := p.fst }) { fst := i, snd := j } hij \u2208\n          Finset.sigma (range K) fun a => Ioo a K) =\n    { fst := i, snd := j }", "state_after": "no goals"}, {"tactic": "rintro \u27e8i, j\u27e9 hij", "annotated_tactic": ["rintro \u27e8i, j\u27e9 hij", []], "state_before": "case refine'_4\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\n\u22a2 \u2200 (a : (_ : \u2115) \u00d7 \u2115) (ha : a \u2208 Finset.sigma (range K) fun a => Ioo a K),\n    (fun p x => { fst := p.snd, snd := p.fst }) ((fun p x => { fst := p.snd, snd := p.fst }) a ha)\n        (_ : (fun p x => { fst := p.snd, snd := p.fst }) a ha \u2208 Finset.sigma (range K) fun a => range a) =\n      a", "state_after": "case refine'_4.mk\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\ni j : \u2115\nhij : { fst := i, snd := j } \u2208 Finset.sigma (range K) fun a => Ioo a K\n\u22a2 (fun p x => { fst := p.snd, snd := p.fst }) ((fun p x => { fst := p.snd, snd := p.fst }) { fst := i, snd := j } hij)\n      (_ :\n        (fun p x => { fst := p.snd, snd := p.fst }) { fst := i, snd := j } hij \u2208\n          Finset.sigma (range K) fun a => range a) =\n    { fst := i, snd := j }"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case refine'_4.mk\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\ni j : \u2115\nhij : { fst := i, snd := j } \u2208 Finset.sigma (range K) fun a => Ioo a K\n\u22a2 (fun p x => { fst := p.snd, snd := p.fst }) ((fun p x => { fst := p.snd, snd := p.fst }) { fst := i, snd := j } hij)\n      (_ :\n        (fun p x => { fst := p.snd, snd := p.fst }) { fst := i, snd := j } hij \u2208\n          Finset.sigma (range K) fun a => range a) =\n    { fst := i, snd := j }", "state_after": "no goals"}, {"tactic": "apply sum_le_sum fun k _ => ?_", "annotated_tactic": ["apply <a>sum_le_sum</a> fun k _ => ?_", [{"full_name": "Finset.sum_le_sum", "def_path": "Mathlib/Algebra/BigOperators/Order.lean", "def_pos": [111, 15], "def_end_pos": [111, 25]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\n\u22a2 \u2211 k in range K, (\u2211 j in Ioo k K, (\u2191j ^ 2)\u207b\u00b9) * \u222b (x : \u211d) in \u2191k..\u2191(k + 1), x ^ 2 \u2202\u03c1 \u2264\n    \u2211 k in range K, 2 / (\u2191k + 1) * \u222b (x : \u211d) in \u2191k..\u2191(k + 1), x ^ 2 \u2202\u03c1", "state_after": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\nk : \u2115\nx\u271d : k \u2208 range K\n\u22a2 (\u2211 j in Ioo k K, (\u2191j ^ 2)\u207b\u00b9) * \u222b (x : \u211d) in \u2191k..\u2191(k + 1), x ^ 2 \u2202\u03c1 \u2264\n    2 / (\u2191k + 1) * \u222b (x : \u211d) in \u2191k..\u2191(k + 1), x ^ 2 \u2202\u03c1"}, {"tactic": "refine' mul_le_mul_of_nonneg_right (sum_Ioo_inv_sq_le _ _) _", "annotated_tactic": ["refine' <a>mul_le_mul_of_nonneg_right</a> (<a>sum_Ioo_inv_sq_le</a> _ _) _", [{"full_name": "mul_le_mul_of_nonneg_right", "def_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "def_pos": [156, 9], "def_end_pos": [156, 35]}, {"full_name": "sum_Ioo_inv_sq_le", "def_path": "Mathlib/Analysis/PSeries.lean", "def_pos": [316, 9], "def_end_pos": [316, 26]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\nk : \u2115\nx\u271d : k \u2208 range K\n\u22a2 (\u2211 j in Ioo k K, (\u2191j ^ 2)\u207b\u00b9) * \u222b (x : \u211d) in \u2191k..\u2191(k + 1), x ^ 2 \u2202\u03c1 \u2264\n    2 / (\u2191k + 1) * \u222b (x : \u211d) in \u2191k..\u2191(k + 1), x ^ 2 \u2202\u03c1", "state_after": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\nk : \u2115\nx\u271d : k \u2208 range K\n\u22a2 0 \u2264 \u222b (x : \u211d) in \u2191k..\u2191(k + 1), x ^ 2 \u2202\u03c1"}, {"tactic": "refine' intervalIntegral.integral_nonneg_of_forall _ fun u => sq_nonneg _", "annotated_tactic": ["refine' <a>intervalIntegral.integral_nonneg_of_forall</a> _ fun u => <a>sq_nonneg</a> _", [{"full_name": "intervalIntegral.integral_nonneg_of_forall", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [1364, 9], "def_end_pos": [1364, 34]}, {"full_name": "sq_nonneg", "def_path": "Mathlib/Algebra/GroupPower/Order.lean", "def_pos": [645, 9], "def_end_pos": [645, 18]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\nk : \u2115\nx\u271d : k \u2208 range K\n\u22a2 0 \u2264 \u222b (x : \u211d) in \u2191k..\u2191(k + 1), x ^ 2 \u2202\u03c1", "state_after": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\nk : \u2115\nx\u271d : k \u2208 range K\n\u22a2 \u2191k \u2264 \u2191(k + 1)"}, {"tactic": "simp only [Nat.cast_add, Nat.cast_one, le_add_iff_nonneg_right, zero_le_one]", "annotated_tactic": ["simp only [<a>Nat.cast_add</a>, <a>Nat.cast_one</a>, <a>le_add_iff_nonneg_right</a>, <a>zero_le_one</a>]", [{"full_name": "Nat.cast_add", "def_path": "Mathlib/Data/Nat/Cast/Defs.lean", "def_pos": [146, 9], "def_end_pos": [146, 17]}, {"full_name": "Nat.cast_one", "def_path": "Mathlib/Data/Nat/Cast/Defs.lean", "def_pos": [141, 9], "def_end_pos": [141, 17]}, {"full_name": "le_add_iff_nonneg_right", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [457, 30], "def_end_pos": [457, 53]}, {"full_name": "zero_le_one", "def_path": "Mathlib/Algebra/Order/ZeroLEOne.lean", "def_pos": [26, 15], "def_end_pos": [26, 26]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\nk : \u2115\nx\u271d : k \u2208 range K\n\u22a2 \u2191k \u2264 \u2191(k + 1)", "state_after": "no goals"}, {"tactic": "apply sum_le_sum fun k _ => ?_", "annotated_tactic": ["apply <a>sum_le_sum</a> fun k _ => ?_", [{"full_name": "Finset.sum_le_sum", "def_path": "Mathlib/Algebra/BigOperators/Order.lean", "def_pos": [111, 15], "def_end_pos": [111, 25]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\n\u22a2 \u2211 k in range K, 2 / (\u2191k + 1) * \u222b (x : \u211d) in \u2191k..\u2191(k + 1), x ^ 2 \u2202\u03c1 \u2264\n    \u2211 k in range K, \u222b (x : \u211d) in \u2191k..\u2191(k + 1), 2 * x \u2202\u03c1", "state_after": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\nk : \u2115\nx\u271d : k \u2208 range K\n\u22a2 2 / (\u2191k + 1) * \u222b (x : \u211d) in \u2191k..\u2191(k + 1), x ^ 2 \u2202\u03c1 \u2264 \u222b (x : \u211d) in \u2191k..\u2191(k + 1), 2 * x \u2202\u03c1"}, {"tactic": "have Ik : (k : \u211d) \u2264 (k + 1 : \u2115) := by simp", "annotated_tactic": ["have Ik : (k : \u211d) \u2264 (k + 1 : \u2115) := by simp", []], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\nk : \u2115\nx\u271d : k \u2208 range K\n\u22a2 2 / (\u2191k + 1) * \u222b (x : \u211d) in \u2191k..\u2191(k + 1), x ^ 2 \u2202\u03c1 \u2264 \u222b (x : \u211d) in \u2191k..\u2191(k + 1), 2 * x \u2202\u03c1", "state_after": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\nk : \u2115\nx\u271d : k \u2208 range K\nIk : \u2191k \u2264 \u2191(k + 1)\n\u22a2 2 / (\u2191k + 1) * \u222b (x : \u211d) in \u2191k..\u2191(k + 1), x ^ 2 \u2202\u03c1 \u2264 \u222b (x : \u211d) in \u2191k..\u2191(k + 1), 2 * x \u2202\u03c1"}, {"tactic": "rw [\u2190 intervalIntegral.integral_const_mul, intervalIntegral.integral_of_le Ik,\n  intervalIntegral.integral_of_le Ik]", "annotated_tactic": ["rw [\u2190 <a>intervalIntegral.integral_const_mul</a>, <a>intervalIntegral.integral_of_le</a> Ik,\n        <a>intervalIntegral.integral_of_le</a> Ik]", [{"full_name": "intervalIntegral.integral_const_mul", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [615, 9], "def_end_pos": [615, 27]}, {"full_name": "intervalIntegral.integral_of_le", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [465, 9], "def_end_pos": [465, 23]}, {"full_name": "intervalIntegral.integral_of_le", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [465, 9], "def_end_pos": [465, 23]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\nk : \u2115\nx\u271d : k \u2208 range K\nIk : \u2191k \u2264 \u2191(k + 1)\n\u22a2 2 / (\u2191k + 1) * \u222b (x : \u211d) in \u2191k..\u2191(k + 1), x ^ 2 \u2202\u03c1 \u2264 \u222b (x : \u211d) in \u2191k..\u2191(k + 1), 2 * x \u2202\u03c1", "state_after": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\nk : \u2115\nx\u271d : k \u2208 range K\nIk : \u2191k \u2264 \u2191(k + 1)\n\u22a2 \u222b (x : \u211d) in Set.Ioc \u2191k \u2191(k + 1), 2 / (\u2191k + 1) * x ^ 2 \u2202\u03c1 \u2264 \u222b (x : \u211d) in Set.Ioc \u2191k \u2191(k + 1), 2 * x \u2202\u03c1"}, {"tactic": "refine' set_integral_mono_on _ _ measurableSet_Ioc fun x hx => _", "annotated_tactic": ["refine' <a>set_integral_mono_on</a> _ _ <a>measurableSet_Ioc</a> fun x hx => _", [{"full_name": "MeasureTheory.set_integral_mono_on", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [721, 9], "def_end_pos": [721, 29]}, {"full_name": "measurableSet_Ioc", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [589, 9], "def_end_pos": [589, 26]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\nk : \u2115\nx\u271d : k \u2208 range K\nIk : \u2191k \u2264 \u2191(k + 1)\n\u22a2 \u222b (x : \u211d) in Set.Ioc \u2191k \u2191(k + 1), 2 / (\u2191k + 1) * x ^ 2 \u2202\u03c1 \u2264 \u222b (x : \u211d) in Set.Ioc \u2191k \u2191(k + 1), 2 * x \u2202\u03c1", "state_after": "case refine'_1\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\nk : \u2115\nx\u271d : k \u2208 range K\nIk : \u2191k \u2264 \u2191(k + 1)\n\u22a2 IntegrableOn (fun x => 2 / (\u2191k + 1) * x ^ 2) (Set.Ioc \u2191k \u2191(k + 1))\n\ncase refine'_2\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\nk : \u2115\nx\u271d : k \u2208 range K\nIk : \u2191k \u2264 \u2191(k + 1)\n\u22a2 IntegrableOn (fun x => 2 * x) (Set.Ioc \u2191k \u2191(k + 1))\n\ncase refine'_3\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\nk : \u2115\nx\u271d : k \u2208 range K\nIk : \u2191k \u2264 \u2191(k + 1)\nx : \u211d\nhx : x \u2208 Set.Ioc \u2191k \u2191(k + 1)\n\u22a2 2 / (\u2191k + 1) * x ^ 2 \u2264 2 * x"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\nk : \u2115\nx\u271d : k \u2208 range K\n\u22a2 \u2191k \u2264 \u2191(k + 1)", "state_after": "no goals"}, {"tactic": "apply Continuous.integrableOn_Ioc", "annotated_tactic": ["apply <a>Continuous.integrableOn_Ioc</a>", [{"full_name": "Continuous.integrableOn_Ioc", "def_path": "Mathlib/MeasureTheory/Function/LocallyIntegrable.lean", "def_pos": [411, 9], "def_end_pos": [411, 36]}]], "state_before": "case refine'_1\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\nk : \u2115\nx\u271d : k \u2208 range K\nIk : \u2191k \u2264 \u2191(k + 1)\n\u22a2 IntegrableOn (fun x => 2 / (\u2191k + 1) * x ^ 2) (Set.Ioc \u2191k \u2191(k + 1))", "state_after": "case refine'_1.hf\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\nk : \u2115\nx\u271d : k \u2208 range K\nIk : \u2191k \u2264 \u2191(k + 1)\n\u22a2 Continuous fun x => 2 / (\u2191k + 1) * x ^ 2"}, {"tactic": "exact continuous_const.mul (continuous_pow 2)", "annotated_tactic": ["exact continuous_const.mul (<a>continuous_pow</a> 2)", [{"full_name": "continuous_pow", "def_path": "Mathlib/Topology/Algebra/Monoid.lean", "def_pos": [558, 9], "def_end_pos": [558, 23]}]], "state_before": "case refine'_1.hf\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\nk : \u2115\nx\u271d : k \u2208 range K\nIk : \u2191k \u2264 \u2191(k + 1)\n\u22a2 Continuous fun x => 2 / (\u2191k + 1) * x ^ 2", "state_after": "no goals"}, {"tactic": "apply Continuous.integrableOn_Ioc", "annotated_tactic": ["apply <a>Continuous.integrableOn_Ioc</a>", [{"full_name": "Continuous.integrableOn_Ioc", "def_path": "Mathlib/MeasureTheory/Function/LocallyIntegrable.lean", "def_pos": [411, 9], "def_end_pos": [411, 36]}]], "state_before": "case refine'_2\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\nk : \u2115\nx\u271d : k \u2208 range K\nIk : \u2191k \u2264 \u2191(k + 1)\n\u22a2 IntegrableOn (fun x => 2 * x) (Set.Ioc \u2191k \u2191(k + 1))", "state_after": "case refine'_2.hf\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\nk : \u2115\nx\u271d : k \u2208 range K\nIk : \u2191k \u2264 \u2191(k + 1)\n\u22a2 Continuous fun x => 2 * x"}, {"tactic": "exact continuous_const.mul continuous_id'", "annotated_tactic": ["exact continuous_const.mul <a>continuous_id'</a>", [{"full_name": "continuous_id'", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [1671, 9], "def_end_pos": [1671, 23]}]], "state_before": "case refine'_2.hf\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\nk : \u2115\nx\u271d : k \u2208 range K\nIk : \u2191k \u2264 \u2191(k + 1)\n\u22a2 Continuous fun x => 2 * x", "state_after": "no goals"}, {"tactic": "calc\n  \u21912 / (\u2191k + \u21911) * x ^ 2 = x / (k + 1) * (2 * x) := by ring\n  _ \u2264 1 * (2 * x) :=\n    (mul_le_mul_of_nonneg_right (by\n      convert (div_le_one _).2 hx.2; norm_cast\n      simp only [Nat.cast_add, Nat.cast_one]\n      linarith only [show (0 : \u211d) \u2264 k from Nat.cast_nonneg k])\n      (mul_nonneg zero_le_two ((Nat.cast_nonneg k).trans hx.1.le)))\n  _ = 2 * x := by rw [one_mul]", "annotated_tactic": ["calc\n          \u21912 / (\u2191k + \u21911) * x ^ 2 = x / (k + 1) * (2 * x) := by ring\n          _ \u2264 1 * (2 * x) :=\n            (<a>mul_le_mul_of_nonneg_right</a> (by\n              convert (<a>div_le_one</a> _).2 hx.2; norm_cast\n              simp only [<a>Nat.cast_add</a>, <a>Nat.cast_one</a>]\n              linarith only [show (0 : \u211d) \u2264 k from <a>Nat.cast_nonneg</a> k])\n              (<a>mul_nonneg</a> <a>zero_le_two</a> ((<a>Nat.cast_nonneg</a> k).<a>trans</a> hx.1.<a>le</a>)))\n          _ = 2 * x := by rw [<a>one_mul</a>]", [{"full_name": "mul_le_mul_of_nonneg_right", "def_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "def_pos": [156, 9], "def_end_pos": [156, 35]}, {"full_name": "div_le_one", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [425, 9], "def_end_pos": [425, 19]}, {"full_name": "Nat.cast_add", "def_path": "Mathlib/Data/Nat/Cast/Defs.lean", "def_pos": [146, 9], "def_end_pos": [146, 17]}, {"full_name": "Nat.cast_one", "def_path": "Mathlib/Data/Nat/Cast/Defs.lean", "def_pos": [141, 9], "def_end_pos": [141, 17]}, {"full_name": "Nat.cast_nonneg", "def_path": "Mathlib/Data/Nat/Cast/Order.lean", "def_pos": [44, 9], "def_end_pos": [44, 20]}, {"full_name": "mul_nonneg", "def_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "def_pos": [380, 7], "def_end_pos": [380, 17]}, {"full_name": "zero_le_two", "def_path": "Mathlib/Algebra/Order/Monoid/NatCast.lean", "def_pos": [32, 7], "def_end_pos": [32, 18]}, {"full_name": "Nat.cast_nonneg", "def_path": "Mathlib/Data/Nat/Cast/Order.lean", "def_pos": [44, 9], "def_end_pos": [44, 20]}, {"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [120, 7], "def_end_pos": [120, 18]}, {"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [142, 7], "def_end_pos": [142, 15]}, {"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [464, 9], "def_end_pos": [464, 16]}]], "state_before": "case refine'_3\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\nk : \u2115\nx\u271d : k \u2208 range K\nIk : \u2191k \u2264 \u2191(k + 1)\nx : \u211d\nhx : x \u2208 Set.Ioc \u2191k \u2191(k + 1)\n\u22a2 2 / (\u2191k + 1) * x ^ 2 \u2264 2 * x", "state_after": "no goals"}, {"tactic": "ring", "annotated_tactic": ["ring", []], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\nk : \u2115\nx\u271d : k \u2208 range K\nIk : \u2191k \u2264 \u2191(k + 1)\nx : \u211d\nhx : x \u2208 Set.Ioc \u2191k \u2191(k + 1)\n\u22a2 2 / (\u2191k + 1) * x ^ 2 = x / (\u2191k + 1) * (2 * x)", "state_after": "no goals"}, {"tactic": "convert (div_le_one _).2 hx.2", "annotated_tactic": ["convert (<a>div_le_one</a> _).2 hx.2", [{"full_name": "div_le_one", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [425, 9], "def_end_pos": [425, 19]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\nk : \u2115\nx\u271d : k \u2208 range K\nIk : \u2191k \u2264 \u2191(k + 1)\nx : \u211d\nhx : x \u2208 Set.Ioc \u2191k \u2191(k + 1)\n\u22a2 x / (\u2191k + 1) \u2264 1", "state_after": "case h.e'_3.h.e'_6\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\nk : \u2115\nx\u271d : k \u2208 range K\nIk : \u2191k \u2264 \u2191(k + 1)\nx : \u211d\nhx : x \u2208 Set.Ioc \u2191k \u2191(k + 1)\n\u22a2 \u2191k + 1 = \u2191(k + 1)\n\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\nk : \u2115\nx\u271d : k \u2208 range K\nIk : \u2191k \u2264 \u2191(k + 1)\nx : \u211d\nhx : x \u2208 Set.Ioc \u2191k \u2191(k + 1)\n\u22a2 0 < \u2191(k + 1)"}, {"tactic": "norm_cast", "annotated_tactic": ["norm_cast", []], "state_before": "case h.e'_3.h.e'_6\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\nk : \u2115\nx\u271d : k \u2208 range K\nIk : \u2191k \u2264 \u2191(k + 1)\nx : \u211d\nhx : x \u2208 Set.Ioc \u2191k \u2191(k + 1)\n\u22a2 \u2191k + 1 = \u2191(k + 1)\n\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\nk : \u2115\nx\u271d : k \u2208 range K\nIk : \u2191k \u2264 \u2191(k + 1)\nx : \u211d\nhx : x \u2208 Set.Ioc \u2191k \u2191(k + 1)\n\u22a2 0 < \u2191(k + 1)", "state_after": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\nk : \u2115\nx\u271d : k \u2208 range K\nIk : \u2191k \u2264 \u2191(k + 1)\nx : \u211d\nhx : x \u2208 Set.Ioc \u2191k \u2191(k + 1)\n\u22a2 0 < \u2191(k + 1)"}, {"tactic": "simp only [Nat.cast_add, Nat.cast_one]", "annotated_tactic": ["simp only [<a>Nat.cast_add</a>, <a>Nat.cast_one</a>]", [{"full_name": "Nat.cast_add", "def_path": "Mathlib/Data/Nat/Cast/Defs.lean", "def_pos": [146, 9], "def_end_pos": [146, 17]}, {"full_name": "Nat.cast_one", "def_path": "Mathlib/Data/Nat/Cast/Defs.lean", "def_pos": [141, 9], "def_end_pos": [141, 17]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\nk : \u2115\nx\u271d : k \u2208 range K\nIk : \u2191k \u2264 \u2191(k + 1)\nx : \u211d\nhx : x \u2208 Set.Ioc \u2191k \u2191(k + 1)\n\u22a2 0 < \u2191(k + 1)", "state_after": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\nk : \u2115\nx\u271d : k \u2208 range K\nIk : \u2191k \u2264 \u2191(k + 1)\nx : \u211d\nhx : x \u2208 Set.Ioc \u2191k \u2191(k + 1)\n\u22a2 0 < \u2191k + 1"}, {"tactic": "linarith only [show (0 : \u211d) \u2264 k from Nat.cast_nonneg k]", "annotated_tactic": ["linarith only [show (0 : \u211d) \u2264 k from <a>Nat.cast_nonneg</a> k]", [{"full_name": "Nat.cast_nonneg", "def_path": "Mathlib/Data/Nat/Cast/Order.lean", "def_pos": [44, 9], "def_end_pos": [44, 20]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\nk : \u2115\nx\u271d : k \u2208 range K\nIk : \u2191k \u2264 \u2191(k + 1)\nx : \u211d\nhx : x \u2208 Set.Ioc \u2191k \u2191(k + 1)\n\u22a2 0 < \u2191k + 1", "state_after": "no goals"}, {"tactic": "rw [one_mul]", "annotated_tactic": ["rw [<a>one_mul</a>]", [{"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [464, 9], "def_end_pos": [464, 16]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\nk : \u2115\nx\u271d : k \u2208 range K\nIk : \u2191k \u2264 \u2191(k + 1)\nx : \u211d\nhx : x \u2208 Set.Ioc \u2191k \u2191(k + 1)\n\u22a2 1 * (2 * x) = 2 * x", "state_after": "no goals"}, {"tactic": "rw [intervalIntegral.sum_integral_adjacent_intervals fun k _ => ?_]", "annotated_tactic": ["rw [<a>intervalIntegral.sum_integral_adjacent_intervals</a> fun k _ => ?_]", [{"full_name": "intervalIntegral.sum_integral_adjacent_intervals", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [922, 9], "def_end_pos": [922, 40]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\n\u22a2 \u2211 k in range K, \u222b (x : \u211d) in \u2191k..\u2191(k + 1), 2 * x \u2202\u03c1 = 2 * \u222b (x : \u211d) in 0 ..\u2191K, x \u2202\u03c1", "state_after": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\n\u22a2 \u222b (x : \u211d) in \u21910 ..\u2191K, 2 * x \u2202\u03c1 = 2 * \u222b (x : \u211d) in 0 ..\u2191K, x \u2202\u03c1\n\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\nk : \u2115\nx\u271d : k < K\n\u22a2 IntervalIntegrable (fun x => 2 * x) \u03c1 \u2191k \u2191(k + 1)"}, {"tactic": "swap", "annotated_tactic": ["swap", []], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\n\u22a2 \u222b (x : \u211d) in \u21910 ..\u2191K, 2 * x \u2202\u03c1 = 2 * \u222b (x : \u211d) in 0 ..\u2191K, x \u2202\u03c1\n\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\nk : \u2115\nx\u271d : k < K\n\u22a2 IntervalIntegrable (fun x => 2 * x) \u03c1 \u2191k \u2191(k + 1)", "state_after": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\nk : \u2115\nx\u271d : k < K\n\u22a2 IntervalIntegrable (fun x => 2 * x) \u03c1 \u2191k \u2191(k + 1)\n\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\n\u22a2 \u222b (x : \u211d) in \u21910 ..\u2191K, 2 * x \u2202\u03c1 = 2 * \u222b (x : \u211d) in 0 ..\u2191K, x \u2202\u03c1"}, {"tactic": "rw [intervalIntegral.integral_const_mul]", "annotated_tactic": ["rw [<a>intervalIntegral.integral_const_mul</a>]", [{"full_name": "intervalIntegral.integral_const_mul", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [615, 9], "def_end_pos": [615, 27]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\n\u22a2 \u222b (x : \u211d) in \u21910 ..\u2191K, 2 * x \u2202\u03c1 = 2 * \u222b (x : \u211d) in 0 ..\u2191K, x \u2202\u03c1", "state_after": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\n\u22a2 2 * \u222b (x : \u211d) in \u21910 ..\u2191K, x \u2202\u03c1 = 2 * \u222b (x : \u211d) in 0 ..\u2191K, x \u2202\u03c1"}, {"tactic": "norm_cast", "annotated_tactic": ["norm_cast", []], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\n\u22a2 2 * \u222b (x : \u211d) in \u21910 ..\u2191K, x \u2202\u03c1 = 2 * \u222b (x : \u211d) in 0 ..\u2191K, x \u2202\u03c1", "state_after": "no goals"}, {"tactic": "exact (continuous_const.mul continuous_id').intervalIntegrable _ _", "annotated_tactic": ["exact (continuous_const.mul <a>continuous_id'</a>).<a>intervalIntegrable</a> _ _", [{"full_name": "continuous_id'", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [1671, 9], "def_end_pos": [1671, 23]}, {"full_name": "Continuous.intervalIntegrable", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [366, 9], "def_end_pos": [366, 38]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\nk : \u2115\nx\u271d : k < K\n\u22a2 IntervalIntegrable (fun x => 2 * x) \u03c1 \u2191k \u2191(k + 1)", "state_after": "no goals"}, {"tactic": "rw [\u2190 integral_truncation_eq_intervalIntegral_of_nonneg hint.1 hnonneg]", "annotated_tactic": ["rw [\u2190 <a>integral_truncation_eq_intervalIntegral_of_nonneg</a> hint.1 hnonneg]", [{"full_name": "ProbabilityTheory.integral_truncation_eq_intervalIntegral_of_nonneg", "def_path": "Mathlib/Probability/StrongLaw.lean", "def_pos": [185, 9], "def_end_pos": [185, 58]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\n\u22a2 \u222b (x : \u211d) in 0 ..\u2191K, x \u2202\u03c1 \u2264 \u222b (a : \u03a9), X a", "state_after": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\n\u22a2 \u222b (x : \u03a9), truncation X (\u2191K) x \u2264 \u222b (a : \u03a9), X a"}, {"tactic": "exact integral_truncation_le_integral_of_nonneg hint hnonneg", "annotated_tactic": ["exact <a>integral_truncation_le_integral_of_nonneg</a> hint hnonneg", [{"full_name": "ProbabilityTheory.integral_truncation_le_integral_of_nonneg", "def_path": "Mathlib/Probability/StrongLaw.lean", "def_pos": [190, 9], "def_end_pos": [190, 50]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\n\u22a2 \u222b (x : \u03a9), truncation X (\u2191K) x \u2264 \u222b (a : \u03a9), X a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Moments.lean", "full_name": "ProbabilityTheory.centralMoment_one", "start": [81, 1], "end": [91, 29], "traced_tactics": [{"tactic": "by_cases h_int : Integrable X \u03bc", "annotated_tactic": ["by_cases h_int : <a>Integrable</a> X \u03bc", [{"full_name": "MeasureTheory.Integrable", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [442, 5], "def_end_pos": [442, 15]}]], "state_before": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u211d\np : \u2115\n\u03bc : Measure \u03a9\ninst\u271d : IsProbabilityMeasure \u03bc\n\u22a2 centralMoment X 1 \u03bc = 0", "state_after": "case pos\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u211d\np : \u2115\n\u03bc : Measure \u03a9\ninst\u271d : IsProbabilityMeasure \u03bc\nh_int : Integrable X\n\u22a2 centralMoment X 1 \u03bc = 0\n\ncase neg\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u211d\np : \u2115\n\u03bc : Measure \u03a9\ninst\u271d : IsProbabilityMeasure \u03bc\nh_int : \u00acIntegrable X\n\u22a2 centralMoment X 1 \u03bc = 0"}, {"tactic": "rw [centralMoment_one' h_int]", "annotated_tactic": ["rw [<a>centralMoment_one'</a> h_int]", [{"full_name": "ProbabilityTheory.centralMoment_one'", "def_path": "Mathlib/Probability/Moments.lean", "def_pos": [73, 9], "def_end_pos": [73, 27]}]], "state_before": "case pos\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u211d\np : \u2115\n\u03bc : Measure \u03a9\ninst\u271d : IsProbabilityMeasure \u03bc\nh_int : Integrable X\n\u22a2 centralMoment X 1 \u03bc = 0", "state_after": "case pos\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u211d\np : \u2115\n\u03bc : Measure \u03a9\ninst\u271d : IsProbabilityMeasure \u03bc\nh_int : Integrable X\n\u22a2 (1 - ENNReal.toReal (\u2191\u2191\u03bc Set.univ)) * \u222b (x : \u03a9), X x \u2202\u03bc = 0"}, {"tactic": "simp only [measure_univ, ENNReal.one_toReal, sub_self, zero_mul]", "annotated_tactic": ["simp only [<a>measure_univ</a>, <a>ENNReal.one_toReal</a>, <a>sub_self</a>, <a>zero_mul</a>]", [{"full_name": "MeasureTheory.IsProbabilityMeasure.measure_univ", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3027, 3], "def_end_pos": [3027, 15]}, {"full_name": "ENNReal.one_toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [230, 17], "def_end_pos": [230, 27]}, {"full_name": "sub_self", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [734, 30], "def_end_pos": [734, 38]}, {"full_name": "MulZeroClass.zero_mul", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [36, 3], "def_end_pos": [36, 11]}]], "state_before": "case pos\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u211d\np : \u2115\n\u03bc : Measure \u03a9\ninst\u271d : IsProbabilityMeasure \u03bc\nh_int : Integrable X\n\u22a2 (1 - ENNReal.toReal (\u2191\u2191\u03bc Set.univ)) * \u222b (x : \u03a9), X x \u2202\u03bc = 0", "state_after": "no goals"}, {"tactic": "simp only [centralMoment, Pi.sub_apply, pow_one]", "annotated_tactic": ["simp only [<a>centralMoment</a>, <a>Pi.sub_apply</a>, <a>pow_one</a>]", [{"full_name": "ProbabilityTheory.centralMoment", "def_path": "Mathlib/Probability/Moments.lean", "def_pos": [56, 5], "def_end_pos": [56, 18]}, {"full_name": "Pi.sub_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [200, 3], "def_end_pos": [200, 14]}, {"full_name": "pow_one", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [97, 9], "def_end_pos": [97, 16]}]], "state_before": "case neg\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u211d\np : \u2115\n\u03bc : Measure \u03a9\ninst\u271d : IsProbabilityMeasure \u03bc\nh_int : \u00acIntegrable X\n\u22a2 centralMoment X 1 \u03bc = 0", "state_after": "case neg\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u211d\np : \u2115\n\u03bc : Measure \u03a9\ninst\u271d : IsProbabilityMeasure \u03bc\nh_int : \u00acIntegrable X\n\u22a2 \u222b (x : \u03a9), X x - \u222b (x : \u03a9), X x \u2202\u03bc \u2202\u03bc = 0"}, {"tactic": "have : \u00acIntegrable (fun x => X x - integral \u03bc X) \u03bc := by\n  refine' fun h_sub => h_int _\n  have h_add : X = (fun x => X x - integral \u03bc X) + fun _ => integral \u03bc X := by ext1 x; simp\n  rw [h_add]\n  exact h_sub.add (integrable_const _)", "annotated_tactic": ["have : \u00ac<a>Integrable</a> (fun x => X x - <a>integral</a> \u03bc X) \u03bc := by\n      refine' fun h_sub => h_int _\n      have h_add : X = (fun x => X x - <a>integral</a> \u03bc X) + fun _ => <a>integral</a> \u03bc X := by ext1 x; simp\n      rw [h_add]\n      exact h_sub.add (<a>integrable_const</a> _)", [{"full_name": "MeasureTheory.Integrable", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [442, 5], "def_end_pos": [442, 15]}, {"full_name": "MeasureTheory.integral", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [791, 17], "def_end_pos": [791, 25]}, {"full_name": "MeasureTheory.integral", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [791, 17], "def_end_pos": [791, 25]}, {"full_name": "MeasureTheory.integral", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [791, 17], "def_end_pos": [791, 25]}, {"full_name": "MeasureTheory.integrable_const", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [506, 9], "def_end_pos": [506, 25]}]], "state_before": "case neg\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u211d\np : \u2115\n\u03bc : Measure \u03a9\ninst\u271d : IsProbabilityMeasure \u03bc\nh_int : \u00acIntegrable X\n\u22a2 \u222b (x : \u03a9), X x - \u222b (x : \u03a9), X x \u2202\u03bc \u2202\u03bc = 0", "state_after": "case neg\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u211d\np : \u2115\n\u03bc : Measure \u03a9\ninst\u271d : IsProbabilityMeasure \u03bc\nh_int : \u00acIntegrable X\nthis : \u00acIntegrable fun x => X x - integral \u03bc X\n\u22a2 \u222b (x : \u03a9), X x - \u222b (x : \u03a9), X x \u2202\u03bc \u2202\u03bc = 0"}, {"tactic": "rw [integral_undef this]", "annotated_tactic": ["rw [<a>integral_undef</a> this]", [{"full_name": "MeasureTheory.integral_undef", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [836, 9], "def_end_pos": [836, 23]}]], "state_before": "case neg\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u211d\np : \u2115\n\u03bc : Measure \u03a9\ninst\u271d : IsProbabilityMeasure \u03bc\nh_int : \u00acIntegrable X\nthis : \u00acIntegrable fun x => X x - integral \u03bc X\n\u22a2 \u222b (x : \u03a9), X x - \u222b (x : \u03a9), X x \u2202\u03bc \u2202\u03bc = 0", "state_after": "no goals"}, {"tactic": "refine' fun h_sub => h_int _", "annotated_tactic": ["refine' fun h_sub => h_int _", []], "state_before": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u211d\np : \u2115\n\u03bc : Measure \u03a9\ninst\u271d : IsProbabilityMeasure \u03bc\nh_int : \u00acIntegrable X\n\u22a2 \u00acIntegrable fun x => X x - integral \u03bc X", "state_after": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u211d\np : \u2115\n\u03bc : Measure \u03a9\ninst\u271d : IsProbabilityMeasure \u03bc\nh_int : \u00acIntegrable X\nh_sub : Integrable fun x => X x - integral \u03bc X\n\u22a2 Integrable X"}, {"tactic": "have h_add : X = (fun x => X x - integral \u03bc X) + fun _ => integral \u03bc X := by ext1 x; simp", "annotated_tactic": ["have h_add : X = (fun x => X x - <a>integral</a> \u03bc X) + fun _ => <a>integral</a> \u03bc X := by ext1 x; simp", [{"full_name": "MeasureTheory.integral", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [791, 17], "def_end_pos": [791, 25]}, {"full_name": "MeasureTheory.integral", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [791, 17], "def_end_pos": [791, 25]}]], "state_before": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u211d\np : \u2115\n\u03bc : Measure \u03a9\ninst\u271d : IsProbabilityMeasure \u03bc\nh_int : \u00acIntegrable X\nh_sub : Integrable fun x => X x - integral \u03bc X\n\u22a2 Integrable X", "state_after": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u211d\np : \u2115\n\u03bc : Measure \u03a9\ninst\u271d : IsProbabilityMeasure \u03bc\nh_int : \u00acIntegrable X\nh_sub : Integrable fun x => X x - integral \u03bc X\nh_add : X = (fun x => X x - integral \u03bc X) + fun x => integral \u03bc X\n\u22a2 Integrable X"}, {"tactic": "rw [h_add]", "annotated_tactic": ["rw [h_add]", []], "state_before": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u211d\np : \u2115\n\u03bc : Measure \u03a9\ninst\u271d : IsProbabilityMeasure \u03bc\nh_int : \u00acIntegrable X\nh_sub : Integrable fun x => X x - integral \u03bc X\nh_add : X = (fun x => X x - integral \u03bc X) + fun x => integral \u03bc X\n\u22a2 Integrable X", "state_after": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u211d\np : \u2115\n\u03bc : Measure \u03a9\ninst\u271d : IsProbabilityMeasure \u03bc\nh_int : \u00acIntegrable X\nh_sub : Integrable fun x => X x - integral \u03bc X\nh_add : X = (fun x => X x - integral \u03bc X) + fun x => integral \u03bc X\n\u22a2 Integrable ((fun x => X x - integral \u03bc X) + fun x => integral \u03bc X)"}, {"tactic": "exact h_sub.add (integrable_const _)", "annotated_tactic": ["exact h_sub.add (<a>integrable_const</a> _)", [{"full_name": "MeasureTheory.integrable_const", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [506, 9], "def_end_pos": [506, 25]}]], "state_before": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u211d\np : \u2115\n\u03bc : Measure \u03a9\ninst\u271d : IsProbabilityMeasure \u03bc\nh_int : \u00acIntegrable X\nh_sub : Integrable fun x => X x - integral \u03bc X\nh_add : X = (fun x => X x - integral \u03bc X) + fun x => integral \u03bc X\n\u22a2 Integrable ((fun x => X x - integral \u03bc X) + fun x => integral \u03bc X)", "state_after": "no goals"}, {"tactic": "ext1 x", "annotated_tactic": ["ext1 x", []], "state_before": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u211d\np : \u2115\n\u03bc : Measure \u03a9\ninst\u271d : IsProbabilityMeasure \u03bc\nh_int : \u00acIntegrable X\nh_sub : Integrable fun x => X x - integral \u03bc X\n\u22a2 X = (fun x => X x - integral \u03bc X) + fun x => integral \u03bc X", "state_after": "case h\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u211d\np : \u2115\n\u03bc : Measure \u03a9\ninst\u271d : IsProbabilityMeasure \u03bc\nh_int : \u00acIntegrable X\nh_sub : Integrable fun x => X x - integral \u03bc X\nx : \u03a9\n\u22a2 X x = ((fun x => X x - integral \u03bc X) + fun x => integral \u03bc X) x"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case h\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u211d\np : \u2115\n\u03bc : Measure \u03a9\ninst\u271d : IsProbabilityMeasure \u03bc\nh_int : \u00acIntegrable X\nh_sub : Integrable fun x => X x - integral \u03bc X\nx : \u03a9\n\u22a2 X x = ((fun x => X x - integral \u03bc X) + fun x => integral \u03bc X) x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/IdentDistrib.lean", "full_name": "ProbabilityTheory.Mem\u2112p.uniformIntegrable_of_identDistrib", "start": [348, 1], "end": [360, 92], "traced_tactics": [{"tactic": "have hfmeas : \u2200 i, AEStronglyMeasurable (f i) \u03bc := fun i =>\n  (hf i).aestronglyMeasurable_iff.2 h\u2112p.1", "annotated_tactic": ["have hfmeas : \u2200 i, <a>AEStronglyMeasurable</a> (f i) \u03bc := fun i =>\n    (hf i).<a>aestronglyMeasurable_iff</a>.2 h\u2112p.1", [{"full_name": "MeasureTheory.AEStronglyMeasurable", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [93, 5], "def_end_pos": [93, 25]}, {"full_name": "ProbabilityTheory.IdentDistrib.aestronglyMeasurable_iff", "def_path": "Mathlib/Probability/IdentDistrib.lean", "def_pos": [171, 9], "def_end_pos": [171, 33]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b3\ninst\u271d\u2074 : MeasurableSpace \u03b4\nE : Type u_5\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\n\u03b9 : Type u_6\nf : \u03b9 \u2192 \u03b1 \u2192 E\nj : \u03b9\np : \u211d\u22650\u221e\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nh\u2112p : Mem\u2112p (f j) p\nhf : \u2200 (i : \u03b9), IdentDistrib (f i) (f j)\n\u22a2 UniformIntegrable f p \u03bc", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b3\ninst\u271d\u2074 : MeasurableSpace \u03b4\nE : Type u_5\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\n\u03b9 : Type u_6\nf : \u03b9 \u2192 \u03b1 \u2192 E\nj : \u03b9\np : \u211d\u22650\u221e\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nh\u2112p : Mem\u2112p (f j) p\nhf : \u2200 (i : \u03b9), IdentDistrib (f i) (f j)\nhfmeas : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\n\u22a2 UniformIntegrable f p \u03bc"}, {"tactic": "set g : \u03b9 \u2192 \u03b1 \u2192 E := fun i => (hfmeas i).choose", "annotated_tactic": ["set g : \u03b9 \u2192 \u03b1 \u2192 E := fun i => (hfmeas i).<a>choose</a>", [{"full_name": "Exists.choose", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [442, 32], "def_end_pos": [442, 45]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b3\ninst\u271d\u2074 : MeasurableSpace \u03b4\nE : Type u_5\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\n\u03b9 : Type u_6\nf : \u03b9 \u2192 \u03b1 \u2192 E\nj : \u03b9\np : \u211d\u22650\u221e\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nh\u2112p : Mem\u2112p (f j) p\nhf : \u2200 (i : \u03b9), IdentDistrib (f i) (f j)\nhfmeas : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\n\u22a2 UniformIntegrable f p \u03bc", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b3\ninst\u271d\u2074 : MeasurableSpace \u03b4\nE : Type u_5\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\n\u03b9 : Type u_6\nf : \u03b9 \u2192 \u03b1 \u2192 E\nj : \u03b9\np : \u211d\u22650\u221e\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nh\u2112p : Mem\u2112p (f j) p\nhf : \u2200 (i : \u03b9), IdentDistrib (f i) (f j)\nhfmeas : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\ng : \u03b9 \u2192 \u03b1 \u2192 E := fun i => Exists.choose (_ : AEStronglyMeasurable (f i) \u03bc)\n\u22a2 UniformIntegrable f p \u03bc"}, {"tactic": "have hgmeas : \u2200 i, StronglyMeasurable (g i) := fun i => (Exists.choose_spec <| hfmeas i).1", "annotated_tactic": ["have hgmeas : \u2200 i, <a>StronglyMeasurable</a> (g i) := fun i => (<a>Exists.choose_spec</a> <| hfmeas i).1", [{"full_name": "MeasureTheory.StronglyMeasurable", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [78, 5], "def_end_pos": [78, 23]}, {"full_name": "Exists.choose_spec", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [445, 9], "def_end_pos": [445, 27]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b3\ninst\u271d\u2074 : MeasurableSpace \u03b4\nE : Type u_5\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\n\u03b9 : Type u_6\nf : \u03b9 \u2192 \u03b1 \u2192 E\nj : \u03b9\np : \u211d\u22650\u221e\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nh\u2112p : Mem\u2112p (f j) p\nhf : \u2200 (i : \u03b9), IdentDistrib (f i) (f j)\nhfmeas : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\ng : \u03b9 \u2192 \u03b1 \u2192 E := fun i => Exists.choose (_ : AEStronglyMeasurable (f i) \u03bc)\n\u22a2 UniformIntegrable f p \u03bc", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b3\ninst\u271d\u2074 : MeasurableSpace \u03b4\nE : Type u_5\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\n\u03b9 : Type u_6\nf : \u03b9 \u2192 \u03b1 \u2192 E\nj : \u03b9\np : \u211d\u22650\u221e\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nh\u2112p : Mem\u2112p (f j) p\nhf : \u2200 (i : \u03b9), IdentDistrib (f i) (f j)\nhfmeas : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\ng : \u03b9 \u2192 \u03b1 \u2192 E := fun i => Exists.choose (_ : AEStronglyMeasurable (f i) \u03bc)\nhgmeas : \u2200 (i : \u03b9), StronglyMeasurable (g i)\n\u22a2 UniformIntegrable f p \u03bc"}, {"tactic": "have hgeq : \u2200 i, g i =\u1d50[\u03bc] f i := fun i => (Exists.choose_spec <| hfmeas i).2.symm", "annotated_tactic": ["have hgeq : \u2200 i, g i =\u1d50[\u03bc] f i := fun i => (<a>Exists.choose_spec</a> <| hfmeas i).2.<a>symm</a>", [{"full_name": "Exists.choose_spec", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [445, 9], "def_end_pos": [445, 27]}, {"full_name": "Filter.EventuallyEq.symm", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1498, 9], "def_end_pos": [1498, 26]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b3\ninst\u271d\u2074 : MeasurableSpace \u03b4\nE : Type u_5\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\n\u03b9 : Type u_6\nf : \u03b9 \u2192 \u03b1 \u2192 E\nj : \u03b9\np : \u211d\u22650\u221e\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nh\u2112p : Mem\u2112p (f j) p\nhf : \u2200 (i : \u03b9), IdentDistrib (f i) (f j)\nhfmeas : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\ng : \u03b9 \u2192 \u03b1 \u2192 E := fun i => Exists.choose (_ : AEStronglyMeasurable (f i) \u03bc)\nhgmeas : \u2200 (i : \u03b9), StronglyMeasurable (g i)\n\u22a2 UniformIntegrable f p \u03bc", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b3\ninst\u271d\u2074 : MeasurableSpace \u03b4\nE : Type u_5\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\n\u03b9 : Type u_6\nf : \u03b9 \u2192 \u03b1 \u2192 E\nj : \u03b9\np : \u211d\u22650\u221e\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nh\u2112p : Mem\u2112p (f j) p\nhf : \u2200 (i : \u03b9), IdentDistrib (f i) (f j)\nhfmeas : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\ng : \u03b9 \u2192 \u03b1 \u2192 E := fun i => Exists.choose (_ : AEStronglyMeasurable (f i) \u03bc)\nhgmeas : \u2200 (i : \u03b9), StronglyMeasurable (g i)\nhgeq : \u2200 (i : \u03b9), g i =\u1d50[\u03bc] f i\n\u22a2 UniformIntegrable f p \u03bc"}, {"tactic": "have hg\u2112p : Mem\u2112p (g j) p \u03bc := h\u2112p.ae_eq (hgeq j).symm", "annotated_tactic": ["have hg\u2112p : <a>Mem\u2112p</a> (g j) p \u03bc := h\u2112p.ae_eq (hgeq j).<a>symm</a>", [{"full_name": "MeasureTheory.Mem\u2112p", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [108, 5], "def_end_pos": [108, 10]}, {"full_name": "Filter.EventuallyEq.symm", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1498, 9], "def_end_pos": [1498, 26]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b3\ninst\u271d\u2074 : MeasurableSpace \u03b4\nE : Type u_5\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\n\u03b9 : Type u_6\nf : \u03b9 \u2192 \u03b1 \u2192 E\nj : \u03b9\np : \u211d\u22650\u221e\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nh\u2112p : Mem\u2112p (f j) p\nhf : \u2200 (i : \u03b9), IdentDistrib (f i) (f j)\nhfmeas : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\ng : \u03b9 \u2192 \u03b1 \u2192 E := fun i => Exists.choose (_ : AEStronglyMeasurable (f i) \u03bc)\nhgmeas : \u2200 (i : \u03b9), StronglyMeasurable (g i)\nhgeq : \u2200 (i : \u03b9), g i =\u1d50[\u03bc] f i\n\u22a2 UniformIntegrable f p \u03bc", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b3\ninst\u271d\u2074 : MeasurableSpace \u03b4\nE : Type u_5\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\n\u03b9 : Type u_6\nf : \u03b9 \u2192 \u03b1 \u2192 E\nj : \u03b9\np : \u211d\u22650\u221e\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nh\u2112p : Mem\u2112p (f j) p\nhf : \u2200 (i : \u03b9), IdentDistrib (f i) (f j)\nhfmeas : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\ng : \u03b9 \u2192 \u03b1 \u2192 E := fun i => Exists.choose (_ : AEStronglyMeasurable (f i) \u03bc)\nhgmeas : \u2200 (i : \u03b9), StronglyMeasurable (g i)\nhgeq : \u2200 (i : \u03b9), g i =\u1d50[\u03bc] f i\nhg\u2112p : Mem\u2112p (g j) p\n\u22a2 UniformIntegrable f p \u03bc"}, {"tactic": "exact UniformIntegrable.ae_eq\n  (Mem\u2112p.uniformIntegrable_of_identDistrib_aux hp hp' hg\u2112p hgmeas fun i =>\n    (IdentDistrib.of_ae_eq (hgmeas i).aemeasurable (hgeq i)).trans\n      ((hf i).trans <| IdentDistrib.of_ae_eq (hfmeas j).aemeasurable (hgeq j).symm)) hgeq", "annotated_tactic": ["exact <a>UniformIntegrable.ae_eq</a>\n    (<a>Mem\u2112p.uniformIntegrable_of_identDistrib_aux</a> hp hp' hg\u2112p hgmeas fun i =>\n      (<a>IdentDistrib.of_ae_eq</a> (hgmeas i).<a>aemeasurable</a> (hgeq i)).<a>trans</a>\n        ((hf i).<a>trans</a> <| <a>IdentDistrib.of_ae_eq</a> (hfmeas j).<a>aemeasurable</a> (hgeq j).<a>symm</a>)) hgeq", [{"full_name": "MeasureTheory.UniformIntegrable.ae_eq", "def_path": "Mathlib/MeasureTheory/Function/UniformIntegrable.lean", "def_pos": [747, 9], "def_end_pos": [747, 32]}, {"full_name": "ProbabilityTheory.Mem\u2112p.uniformIntegrable_of_identDistrib_aux", "def_path": "Mathlib/Probability/IdentDistrib.lean", "def_pos": [322, 9], "def_end_pos": [322, 52]}, {"full_name": "ProbabilityTheory.IdentDistrib.of_ae_eq", "def_path": "Mathlib/Probability/IdentDistrib.lean", "def_pos": [116, 19], "def_end_pos": [116, 27]}, {"full_name": "MeasureTheory.StronglyMeasurable.aemeasurable", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [349, 19], "def_end_pos": [349, 31]}, {"full_name": "ProbabilityTheory.IdentDistrib.trans", "def_path": "Mathlib/Probability/IdentDistrib.lean", "def_pos": [94, 19], "def_end_pos": [94, 24]}, {"full_name": "ProbabilityTheory.IdentDistrib.trans", "def_path": "Mathlib/Probability/IdentDistrib.lean", "def_pos": [94, 19], "def_end_pos": [94, 24]}, {"full_name": "ProbabilityTheory.IdentDistrib.of_ae_eq", "def_path": "Mathlib/Probability/IdentDistrib.lean", "def_pos": [116, 19], "def_end_pos": [116, 27]}, {"full_name": "MeasureTheory.AEStronglyMeasurable.aemeasurable", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1220, 19], "def_end_pos": [1220, 31]}, {"full_name": "Filter.EventuallyEq.symm", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1498, 9], "def_end_pos": [1498, 26]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b3\ninst\u271d\u2074 : MeasurableSpace \u03b4\nE : Type u_5\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\n\u03b9 : Type u_6\nf : \u03b9 \u2192 \u03b1 \u2192 E\nj : \u03b9\np : \u211d\u22650\u221e\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nh\u2112p : Mem\u2112p (f j) p\nhf : \u2200 (i : \u03b9), IdentDistrib (f i) (f j)\nhfmeas : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\ng : \u03b9 \u2192 \u03b1 \u2192 E := fun i => Exists.choose (_ : AEStronglyMeasurable (f i) \u03bc)\nhgmeas : \u2200 (i : \u03b9), StronglyMeasurable (g i)\nhgeq : \u2200 (i : \u03b9), g i =\u1d50[\u03bc] f i\nhg\u2112p : Mem\u2112p (g j) p\n\u22a2 UniformIntegrable f p \u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Constructions/Pi.lean", "full_name": "MeasureTheory.volume_measurePreserving_sumPiEquivProdPi", "start": [789, 1], "end": [792, 54], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/RBMap/Lemmas.lean", "full_name": "Std.RBNode.Ordered.lowerBound?_exists", "start": [286, 1], "end": [299, 21], "traced_tactics": [{"tactic": "refine \u27e8fun \u27e8x, hx\u27e9 => \u27e8_, lowerBound?_mem hx, lowerBound?_le hx\u27e9, fun H => ?_\u27e9", "annotated_tactic": ["refine \u27e8fun \u27e8x, hx\u27e9 => \u27e8_, <a>lowerBound?_mem</a> hx, <a>lowerBound?_le</a> hx\u27e9, fun H => ?_\u27e9", [{"full_name": "Std.RBNode.lowerBound?_mem", "def_path": "lake-packages/std/Std/Data/RBMap/Lemmas.lean", "def_pos": [280, 9], "def_end_pos": [280, 24]}, {"full_name": "Std.RBNode.lowerBound?_le", "def_path": "lake-packages/std/Std/Data/RBMap/Lemmas.lean", "def_pos": [260, 9], "def_end_pos": [260, 23]}]], "state_before": "\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\ncut : \u03b1 \u2192 Ordering\nt : RBNode \u03b1\ninst\u271d\u00b9 : TransCmp cmp\ninst\u271d : IsCut cmp cut\nh : Ordered cmp t\n\u22a2 (\u2203 x, lowerBound? cut t none = some x) \u2194 \u2203 x, x \u2208 t \u2227 cut x \u2260 Ordering.lt", "state_after": "\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\ncut : \u03b1 \u2192 Ordering\nt : RBNode \u03b1\ninst\u271d\u00b9 : TransCmp cmp\ninst\u271d : IsCut cmp cut\nh : Ordered cmp t\nH : \u2203 x, x \u2208 t \u2227 cut x \u2260 Ordering.lt\n\u22a2 \u2203 x, lowerBound? cut t none = some x"}, {"tactic": "obtain \u27e8x, hx, e\u27e9 := H", "annotated_tactic": ["obtain \u27e8x, hx, e\u27e9 := H", []], "state_before": "\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\ncut : \u03b1 \u2192 Ordering\nt : RBNode \u03b1\ninst\u271d\u00b9 : TransCmp cmp\ninst\u271d : IsCut cmp cut\nh : Ordered cmp t\nH : \u2203 x, x \u2208 t \u2227 cut x \u2260 Ordering.lt\n\u22a2 \u2203 x, lowerBound? cut t none = some x", "state_after": "case intro.intro\n\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\ncut : \u03b1 \u2192 Ordering\nt : RBNode \u03b1\ninst\u271d\u00b9 : TransCmp cmp\ninst\u271d : IsCut cmp cut\nh : Ordered cmp t\nx : \u03b1\nhx : x \u2208 t\ne : cut x \u2260 Ordering.lt\n\u22a2 \u2203 x, lowerBound? cut t none = some x"}, {"tactic": "cases hx", "annotated_tactic": ["cases hx", []], "state_before": "case intro.intro.nil\n\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\ncut : \u03b1 \u2192 Ordering\ninst\u271d\u00b9 : TransCmp cmp\ninst\u271d : IsCut cmp cut\nh : Ordered cmp nil\nx : \u03b1\nhx : x \u2208 nil\ne : cut x \u2260 Ordering.lt\n\u22a2 \u2203 x, lowerBound? cut nil none = some x", "state_after": "no goals"}, {"tactic": "simp [lowerBound?]", "annotated_tactic": ["simp [<a>lowerBound?</a>]", [{"full_name": "Std.RBNode.lowerBound?", "def_path": "lake-packages/std/Std/Data/RBMap/Basic.lean", "def_pos": [403, 19], "def_end_pos": [403, 30]}]], "state_before": "case intro.intro.node\n\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\ncut : \u03b1 \u2192 Ordering\ninst\u271d\u00b9 : TransCmp cmp\ninst\u271d : IsCut cmp cut\nc\u271d : RBColor\nl\u271d : RBNode \u03b1\nv\u271d : \u03b1\nr\u271d : RBNode \u03b1\nihl : Ordered cmp l\u271d \u2192 \u2200 (x : \u03b1), x \u2208 l\u271d \u2192 cut x \u2260 Ordering.lt \u2192 \u2203 x, lowerBound? cut l\u271d none = some x\nr_ih\u271d : Ordered cmp r\u271d \u2192 \u2200 (x : \u03b1), x \u2208 r\u271d \u2192 cut x \u2260 Ordering.lt \u2192 \u2203 x, lowerBound? cut r\u271d none = some x\nh : Ordered cmp (node c\u271d l\u271d v\u271d r\u271d)\nx : \u03b1\nhx : x \u2208 node c\u271d l\u271d v\u271d r\u271d\ne : cut x \u2260 Ordering.lt\n\u22a2 \u2203 x, lowerBound? cut (node c\u271d l\u271d v\u271d r\u271d) none = some x", "state_after": "case intro.intro.node\n\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\ncut : \u03b1 \u2192 Ordering\ninst\u271d\u00b9 : TransCmp cmp\ninst\u271d : IsCut cmp cut\nc\u271d : RBColor\nl\u271d : RBNode \u03b1\nv\u271d : \u03b1\nr\u271d : RBNode \u03b1\nihl : Ordered cmp l\u271d \u2192 \u2200 (x : \u03b1), x \u2208 l\u271d \u2192 cut x \u2260 Ordering.lt \u2192 \u2203 x, lowerBound? cut l\u271d none = some x\nr_ih\u271d : Ordered cmp r\u271d \u2192 \u2200 (x : \u03b1), x \u2208 r\u271d \u2192 cut x \u2260 Ordering.lt \u2192 \u2203 x, lowerBound? cut r\u271d none = some x\nh : Ordered cmp (node c\u271d l\u271d v\u271d r\u271d)\nx : \u03b1\nhx : x \u2208 node c\u271d l\u271d v\u271d r\u271d\ne : cut x \u2260 Ordering.lt\n\u22a2 \u2203 x,\n    (match cut v\u271d with\n      | Ordering.lt => lowerBound? cut l\u271d none\n      | Ordering.gt => lowerBound? cut r\u271d (some v\u271d)\n      | Ordering.eq => some v\u271d) =\n      some x"}, {"tactic": "split", "annotated_tactic": ["split", []], "state_before": "case intro.intro.node\n\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\ncut : \u03b1 \u2192 Ordering\ninst\u271d\u00b9 : TransCmp cmp\ninst\u271d : IsCut cmp cut\nc\u271d : RBColor\nl\u271d : RBNode \u03b1\nv\u271d : \u03b1\nr\u271d : RBNode \u03b1\nihl : Ordered cmp l\u271d \u2192 \u2200 (x : \u03b1), x \u2208 l\u271d \u2192 cut x \u2260 Ordering.lt \u2192 \u2203 x, lowerBound? cut l\u271d none = some x\nr_ih\u271d : Ordered cmp r\u271d \u2192 \u2200 (x : \u03b1), x \u2208 r\u271d \u2192 cut x \u2260 Ordering.lt \u2192 \u2203 x, lowerBound? cut r\u271d none = some x\nh : Ordered cmp (node c\u271d l\u271d v\u271d r\u271d)\nx : \u03b1\nhx : x \u2208 node c\u271d l\u271d v\u271d r\u271d\ne : cut x \u2260 Ordering.lt\n\u22a2 \u2203 x,\n    (match cut v\u271d with\n      | Ordering.lt => lowerBound? cut l\u271d none\n      | Ordering.gt => lowerBound? cut r\u271d (some v\u271d)\n      | Ordering.eq => some v\u271d) =\n      some x", "state_after": "case intro.intro.node.h_1\n\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\ncut : \u03b1 \u2192 Ordering\ninst\u271d\u00b9 : TransCmp cmp\ninst\u271d : IsCut cmp cut\nc\u271d : RBColor\nl\u271d : RBNode \u03b1\nv\u271d : \u03b1\nr\u271d : RBNode \u03b1\nihl : Ordered cmp l\u271d \u2192 \u2200 (x : \u03b1), x \u2208 l\u271d \u2192 cut x \u2260 Ordering.lt \u2192 \u2203 x, lowerBound? cut l\u271d none = some x\nr_ih\u271d : Ordered cmp r\u271d \u2192 \u2200 (x : \u03b1), x \u2208 r\u271d \u2192 cut x \u2260 Ordering.lt \u2192 \u2203 x, lowerBound? cut r\u271d none = some x\nh : Ordered cmp (node c\u271d l\u271d v\u271d r\u271d)\nx : \u03b1\nhx : x \u2208 node c\u271d l\u271d v\u271d r\u271d\ne : cut x \u2260 Ordering.lt\nx\u271d : Ordering\nheq\u271d : cut v\u271d = Ordering.lt\n\u22a2 \u2203 x, lowerBound? cut l\u271d none = some x\n\ncase intro.intro.node.h_2\n\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\ncut : \u03b1 \u2192 Ordering\ninst\u271d\u00b9 : TransCmp cmp\ninst\u271d : IsCut cmp cut\nc\u271d : RBColor\nl\u271d : RBNode \u03b1\nv\u271d : \u03b1\nr\u271d : RBNode \u03b1\nihl : Ordered cmp l\u271d \u2192 \u2200 (x : \u03b1), x \u2208 l\u271d \u2192 cut x \u2260 Ordering.lt \u2192 \u2203 x, lowerBound? cut l\u271d none = some x\nr_ih\u271d : Ordered cmp r\u271d \u2192 \u2200 (x : \u03b1), x \u2208 r\u271d \u2192 cut x \u2260 Ordering.lt \u2192 \u2203 x, lowerBound? cut r\u271d none = some x\nh : Ordered cmp (node c\u271d l\u271d v\u271d r\u271d)\nx : \u03b1\nhx : x \u2208 node c\u271d l\u271d v\u271d r\u271d\ne : cut x \u2260 Ordering.lt\nx\u271d : Ordering\nheq\u271d : cut v\u271d = Ordering.gt\n\u22a2 \u2203 x, lowerBound? cut r\u271d (some v\u271d) = some x\n\ncase intro.intro.node.h_3\n\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\ncut : \u03b1 \u2192 Ordering\ninst\u271d\u00b9 : TransCmp cmp\ninst\u271d : IsCut cmp cut\nc\u271d : RBColor\nl\u271d : RBNode \u03b1\nv\u271d : \u03b1\nr\u271d : RBNode \u03b1\nihl : Ordered cmp l\u271d \u2192 \u2200 (x : \u03b1), x \u2208 l\u271d \u2192 cut x \u2260 Ordering.lt \u2192 \u2203 x, lowerBound? cut l\u271d none = some x\nr_ih\u271d : Ordered cmp r\u271d \u2192 \u2200 (x : \u03b1), x \u2208 r\u271d \u2192 cut x \u2260 Ordering.lt \u2192 \u2203 x, lowerBound? cut r\u271d none = some x\nh : Ordered cmp (node c\u271d l\u271d v\u271d r\u271d)\nx : \u03b1\nhx : x \u2208 node c\u271d l\u271d v\u271d r\u271d\ne : cut x \u2260 Ordering.lt\nx\u271d : Ordering\nheq\u271d : cut v\u271d = Ordering.eq\n\u22a2 \u2203 x, some v\u271d = some x"}, {"tactic": "rcases hx with rfl | hx | hx", "annotated_tactic": ["rcases hx with rfl | hx | hx", []], "state_before": "case intro.intro.node.h_1\n\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\ncut : \u03b1 \u2192 Ordering\ninst\u271d\u00b9 : TransCmp cmp\ninst\u271d : IsCut cmp cut\nc\u271d : RBColor\nl\u271d : RBNode \u03b1\nv\u271d : \u03b1\nr\u271d : RBNode \u03b1\nihl : Ordered cmp l\u271d \u2192 \u2200 (x : \u03b1), x \u2208 l\u271d \u2192 cut x \u2260 Ordering.lt \u2192 \u2203 x, lowerBound? cut l\u271d none = some x\nr_ih\u271d : Ordered cmp r\u271d \u2192 \u2200 (x : \u03b1), x \u2208 r\u271d \u2192 cut x \u2260 Ordering.lt \u2192 \u2203 x, lowerBound? cut r\u271d none = some x\nh : Ordered cmp (node c\u271d l\u271d v\u271d r\u271d)\nx : \u03b1\nhx : x \u2208 node c\u271d l\u271d v\u271d r\u271d\ne : cut x \u2260 Ordering.lt\nx\u271d : Ordering\nheq\u271d : cut v\u271d = Ordering.lt\n\u22a2 \u2203 x, lowerBound? cut l\u271d none = some x", "state_after": "case intro.intro.node.h_1.inl\n\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\ncut : \u03b1 \u2192 Ordering\ninst\u271d\u00b9 : TransCmp cmp\ninst\u271d : IsCut cmp cut\nc\u271d : RBColor\nl\u271d r\u271d : RBNode \u03b1\nihl : Ordered cmp l\u271d \u2192 \u2200 (x : \u03b1), x \u2208 l\u271d \u2192 cut x \u2260 Ordering.lt \u2192 \u2203 x, lowerBound? cut l\u271d none = some x\nr_ih\u271d : Ordered cmp r\u271d \u2192 \u2200 (x : \u03b1), x \u2208 r\u271d \u2192 cut x \u2260 Ordering.lt \u2192 \u2203 x, lowerBound? cut r\u271d none = some x\nx : \u03b1\ne : cut x \u2260 Ordering.lt\nx\u271d : Ordering\nh : Ordered cmp (node c\u271d l\u271d x r\u271d)\nheq\u271d : cut x = Ordering.lt\n\u22a2 \u2203 x, lowerBound? cut l\u271d none = some x\n\ncase intro.intro.node.h_1.inr.inl\n\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\ncut : \u03b1 \u2192 Ordering\ninst\u271d\u00b9 : TransCmp cmp\ninst\u271d : IsCut cmp cut\nc\u271d : RBColor\nl\u271d : RBNode \u03b1\nv\u271d : \u03b1\nr\u271d : RBNode \u03b1\nihl : Ordered cmp l\u271d \u2192 \u2200 (x : \u03b1), x \u2208 l\u271d \u2192 cut x \u2260 Ordering.lt \u2192 \u2203 x, lowerBound? cut l\u271d none = some x\nr_ih\u271d : Ordered cmp r\u271d \u2192 \u2200 (x : \u03b1), x \u2208 r\u271d \u2192 cut x \u2260 Ordering.lt \u2192 \u2203 x, lowerBound? cut r\u271d none = some x\nh : Ordered cmp (node c\u271d l\u271d v\u271d r\u271d)\nx : \u03b1\ne : cut x \u2260 Ordering.lt\nx\u271d : Ordering\nheq\u271d : cut v\u271d = Ordering.lt\nhx : Any (fun x_1 => x = x_1) l\u271d\n\u22a2 \u2203 x, lowerBound? cut l\u271d none = some x\n\ncase intro.intro.node.h_1.inr.inr\n\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\ncut : \u03b1 \u2192 Ordering\ninst\u271d\u00b9 : TransCmp cmp\ninst\u271d : IsCut cmp cut\nc\u271d : RBColor\nl\u271d : RBNode \u03b1\nv\u271d : \u03b1\nr\u271d : RBNode \u03b1\nihl : Ordered cmp l\u271d \u2192 \u2200 (x : \u03b1), x \u2208 l\u271d \u2192 cut x \u2260 Ordering.lt \u2192 \u2203 x, lowerBound? cut l\u271d none = some x\nr_ih\u271d : Ordered cmp r\u271d \u2192 \u2200 (x : \u03b1), x \u2208 r\u271d \u2192 cut x \u2260 Ordering.lt \u2192 \u2203 x, lowerBound? cut r\u271d none = some x\nh : Ordered cmp (node c\u271d l\u271d v\u271d r\u271d)\nx : \u03b1\ne : cut x \u2260 Ordering.lt\nx\u271d : Ordering\nheq\u271d : cut v\u271d = Ordering.lt\nhx : Any (fun x_1 => x = x_1) r\u271d\n\u22a2 \u2203 x, lowerBound? cut l\u271d none = some x"}, {"tactic": "contradiction", "annotated_tactic": ["contradiction", []], "state_before": "case intro.intro.node.h_1.inl\n\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\ncut : \u03b1 \u2192 Ordering\ninst\u271d\u00b9 : TransCmp cmp\ninst\u271d : IsCut cmp cut\nc\u271d : RBColor\nl\u271d r\u271d : RBNode \u03b1\nihl : Ordered cmp l\u271d \u2192 \u2200 (x : \u03b1), x \u2208 l\u271d \u2192 cut x \u2260 Ordering.lt \u2192 \u2203 x, lowerBound? cut l\u271d none = some x\nr_ih\u271d : Ordered cmp r\u271d \u2192 \u2200 (x : \u03b1), x \u2208 r\u271d \u2192 cut x \u2260 Ordering.lt \u2192 \u2203 x, lowerBound? cut r\u271d none = some x\nx : \u03b1\ne : cut x \u2260 Ordering.lt\nx\u271d : Ordering\nh : Ordered cmp (node c\u271d l\u271d x r\u271d)\nheq\u271d : cut x = Ordering.lt\n\u22a2 \u2203 x, lowerBound? cut l\u271d none = some x", "state_after": "no goals"}, {"tactic": "exact ihl h.2.2.1 _ hx e", "annotated_tactic": ["exact ihl h.2.2.1 _ hx e", []], "state_before": "case intro.intro.node.h_1.inr.inl\n\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\ncut : \u03b1 \u2192 Ordering\ninst\u271d\u00b9 : TransCmp cmp\ninst\u271d : IsCut cmp cut\nc\u271d : RBColor\nl\u271d : RBNode \u03b1\nv\u271d : \u03b1\nr\u271d : RBNode \u03b1\nihl : Ordered cmp l\u271d \u2192 \u2200 (x : \u03b1), x \u2208 l\u271d \u2192 cut x \u2260 Ordering.lt \u2192 \u2203 x, lowerBound? cut l\u271d none = some x\nr_ih\u271d : Ordered cmp r\u271d \u2192 \u2200 (x : \u03b1), x \u2208 r\u271d \u2192 cut x \u2260 Ordering.lt \u2192 \u2203 x, lowerBound? cut r\u271d none = some x\nh : Ordered cmp (node c\u271d l\u271d v\u271d r\u271d)\nx : \u03b1\ne : cut x \u2260 Ordering.lt\nx\u271d : Ordering\nheq\u271d : cut v\u271d = Ordering.lt\nhx : Any (fun x_1 => x = x_1) l\u271d\n\u22a2 \u2203 x, lowerBound? cut l\u271d none = some x", "state_after": "no goals"}, {"tactic": "next hv => cases e <| IsCut.lt_trans (All_def.1 h.2.1 _ hx).1 hv", "annotated_tactic": ["next hv => cases e <| <a>IsCut.lt_trans</a> (<a>All_def</a>.1 h.2.1 _ hx).1 hv", [{"full_name": "Std.RBNode.IsCut.lt_trans", "def_path": "lake-packages/std/Std/Data/RBMap/Lemmas.lean", "def_pos": [152, 9], "def_end_pos": [152, 23]}, {"full_name": "Std.RBNode.All_def", "def_path": "lake-packages/std/Std/Data/RBMap/Lemmas.lean", "def_pos": [100, 9], "def_end_pos": [100, 16]}]], "state_before": "case intro.intro.node.h_1.inr.inr\n\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\ncut : \u03b1 \u2192 Ordering\ninst\u271d\u00b9 : TransCmp cmp\ninst\u271d : IsCut cmp cut\nc\u271d : RBColor\nl\u271d : RBNode \u03b1\nv\u271d : \u03b1\nr\u271d : RBNode \u03b1\nihl : Ordered cmp l\u271d \u2192 \u2200 (x : \u03b1), x \u2208 l\u271d \u2192 cut x \u2260 Ordering.lt \u2192 \u2203 x, lowerBound? cut l\u271d none = some x\nr_ih\u271d : Ordered cmp r\u271d \u2192 \u2200 (x : \u03b1), x \u2208 r\u271d \u2192 cut x \u2260 Ordering.lt \u2192 \u2203 x, lowerBound? cut r\u271d none = some x\nh : Ordered cmp (node c\u271d l\u271d v\u271d r\u271d)\nx : \u03b1\ne : cut x \u2260 Ordering.lt\nx\u271d : Ordering\nheq\u271d : cut v\u271d = Ordering.lt\nhx : Any (fun x_1 => x = x_1) r\u271d\n\u22a2 \u2203 x, lowerBound? cut l\u271d none = some x", "state_after": "no goals"}, {"tactic": "cases e <| IsCut.lt_trans (All_def.1 h.2.1 _ hx).1 hv", "annotated_tactic": ["cases e <| <a>IsCut.lt_trans</a> (<a>All_def</a>.1 h.2.1 _ hx).1 hv", [{"full_name": "Std.RBNode.IsCut.lt_trans", "def_path": "lake-packages/std/Std/Data/RBMap/Lemmas.lean", "def_pos": [152, 9], "def_end_pos": [152, 23]}, {"full_name": "Std.RBNode.All_def", "def_path": "lake-packages/std/Std/Data/RBMap/Lemmas.lean", "def_pos": [100, 9], "def_end_pos": [100, 16]}]], "state_before": "\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\ncut : \u03b1 \u2192 Ordering\ninst\u271d\u00b9 : TransCmp cmp\ninst\u271d : IsCut cmp cut\nc\u271d : RBColor\nl\u271d : RBNode \u03b1\nv\u271d : \u03b1\nr\u271d : RBNode \u03b1\nihl : Ordered cmp l\u271d \u2192 \u2200 (x : \u03b1), x \u2208 l\u271d \u2192 cut x \u2260 Ordering.lt \u2192 \u2203 x, lowerBound? cut l\u271d none = some x\nr_ih\u271d : Ordered cmp r\u271d \u2192 \u2200 (x : \u03b1), x \u2208 r\u271d \u2192 cut x \u2260 Ordering.lt \u2192 \u2203 x, lowerBound? cut r\u271d none = some x\nh : Ordered cmp (node c\u271d l\u271d v\u271d r\u271d)\nx : \u03b1\ne : cut x \u2260 Ordering.lt\nx\u271d : Ordering\nhv : cut v\u271d = Ordering.lt\nhx : Any (fun x_1 => x = x_1) r\u271d\n\u22a2 \u2203 x, lowerBound? cut l\u271d none = some x", "state_after": "no goals"}, {"tactic": "exact lowerBound?_of_some", "annotated_tactic": ["exact <a>lowerBound?_of_some</a>", [{"full_name": "Std.RBNode.lowerBound?_of_some", "def_path": "lake-packages/std/Std/Data/RBMap/Lemmas.lean", "def_pos": [283, 9], "def_end_pos": [283, 28]}]], "state_before": "case intro.intro.node.h_2\n\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\ncut : \u03b1 \u2192 Ordering\ninst\u271d\u00b9 : TransCmp cmp\ninst\u271d : IsCut cmp cut\nc\u271d : RBColor\nl\u271d : RBNode \u03b1\nv\u271d : \u03b1\nr\u271d : RBNode \u03b1\nihl : Ordered cmp l\u271d \u2192 \u2200 (x : \u03b1), x \u2208 l\u271d \u2192 cut x \u2260 Ordering.lt \u2192 \u2203 x, lowerBound? cut l\u271d none = some x\nr_ih\u271d : Ordered cmp r\u271d \u2192 \u2200 (x : \u03b1), x \u2208 r\u271d \u2192 cut x \u2260 Ordering.lt \u2192 \u2203 x, lowerBound? cut r\u271d none = some x\nh : Ordered cmp (node c\u271d l\u271d v\u271d r\u271d)\nx : \u03b1\nhx : x \u2208 node c\u271d l\u271d v\u271d r\u271d\ne : cut x \u2260 Ordering.lt\nx\u271d : Ordering\nheq\u271d : cut v\u271d = Ordering.gt\n\u22a2 \u2203 x, lowerBound? cut r\u271d (some v\u271d) = some x", "state_after": "no goals"}, {"tactic": "exact \u27e8_, rfl\u27e9", "annotated_tactic": ["exact \u27e8_, <a>rfl</a>\u27e9", [{"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "case intro.intro.node.h_3\n\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\ncut : \u03b1 \u2192 Ordering\ninst\u271d\u00b9 : TransCmp cmp\ninst\u271d : IsCut cmp cut\nc\u271d : RBColor\nl\u271d : RBNode \u03b1\nv\u271d : \u03b1\nr\u271d : RBNode \u03b1\nihl : Ordered cmp l\u271d \u2192 \u2200 (x : \u03b1), x \u2208 l\u271d \u2192 cut x \u2260 Ordering.lt \u2192 \u2203 x, lowerBound? cut l\u271d none = some x\nr_ih\u271d : Ordered cmp r\u271d \u2192 \u2200 (x : \u03b1), x \u2208 r\u271d \u2192 cut x \u2260 Ordering.lt \u2192 \u2203 x, lowerBound? cut r\u271d none = some x\nh : Ordered cmp (node c\u271d l\u271d v\u271d r\u271d)\nx : \u03b1\nhx : x \u2208 node c\u271d l\u271d v\u271d r\u271d\ne : cut x \u2260 Ordering.lt\nx\u271d : Ordering\nheq\u271d : cut v\u271d = Ordering.eq\n\u22a2 \u2203 x, some v\u271d = some x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/LocallyFinite.lean", "full_name": "Finset.Ioc_subset_Ici_self", "start": [424, 1], "end": [425, 48], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Pointwise/SMul.lean", "full_name": "Set.mul_subset_iff_left", "start": [433, 1], "end": [434, 25], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/NoncommProd.lean", "full_name": "Finset.noncommProd_union_of_disjoint", "start": [357, 1], "end": [370, 81], "traced_tactics": [{"tactic": "obtain \u27e8sl, sl', rfl\u27e9 := exists_list_nodup_eq s", "annotated_tactic": ["obtain \u27e8sl, sl', rfl\u27e9 := <a>exists_list_nodup_eq</a> s", [{"full_name": "Finset.exists_list_nodup_eq", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3475, 9], "def_end_pos": [3475, 29]}]], "state_before": "F : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b2 : Monoid \u03b2\ninst\u271d\u00b9 : Monoid \u03b3\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\nh : Disjoint s t\nf : \u03b1 \u2192 \u03b2\ncomm : Set.Pairwise {x | x \u2208 s \u222a t} fun a b => Commute (f a) (f b)\n\u22a2 noncommProd (s \u222a t) f comm =\n    noncommProd s f (_ : Set.Pairwise \u2191s fun a b => Commute (f a) (f b)) *\n      noncommProd t f (_ : Set.Pairwise \u2191t fun a b => Commute (f a) (f b))", "state_after": "case intro.intro\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b2 : Monoid \u03b2\ninst\u271d\u00b9 : Monoid \u03b3\ninst\u271d : DecidableEq \u03b1\nt : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nsl : List \u03b1\nsl' : List.Nodup sl\nh : Disjoint (List.toFinset sl) t\ncomm : Set.Pairwise {x | x \u2208 List.toFinset sl \u222a t} fun a b => Commute (f a) (f b)\n\u22a2 noncommProd (List.toFinset sl \u222a t) f comm =\n    noncommProd (List.toFinset sl) f (_ : Set.Pairwise \u2191(List.toFinset sl) fun a b => Commute (f a) (f b)) *\n      noncommProd t f (_ : Set.Pairwise \u2191t fun a b => Commute (f a) (f b))"}, {"tactic": "obtain \u27e8tl, tl', rfl\u27e9 := exists_list_nodup_eq t", "annotated_tactic": ["obtain \u27e8tl, tl', rfl\u27e9 := <a>exists_list_nodup_eq</a> t", [{"full_name": "Finset.exists_list_nodup_eq", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3475, 9], "def_end_pos": [3475, 29]}]], "state_before": "case intro.intro\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b2 : Monoid \u03b2\ninst\u271d\u00b9 : Monoid \u03b3\ninst\u271d : DecidableEq \u03b1\nt : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nsl : List \u03b1\nsl' : List.Nodup sl\nh : Disjoint (List.toFinset sl) t\ncomm : Set.Pairwise {x | x \u2208 List.toFinset sl \u222a t} fun a b => Commute (f a) (f b)\n\u22a2 noncommProd (List.toFinset sl \u222a t) f comm =\n    noncommProd (List.toFinset sl) f (_ : Set.Pairwise \u2191(List.toFinset sl) fun a b => Commute (f a) (f b)) *\n      noncommProd t f (_ : Set.Pairwise \u2191t fun a b => Commute (f a) (f b))", "state_after": "case intro.intro.intro.intro\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b2 : Monoid \u03b2\ninst\u271d\u00b9 : Monoid \u03b3\ninst\u271d : DecidableEq \u03b1\nf : \u03b1 \u2192 \u03b2\nsl : List \u03b1\nsl' : List.Nodup sl\ntl : List \u03b1\ntl' : List.Nodup tl\nh : Disjoint (List.toFinset sl) (List.toFinset tl)\ncomm : Set.Pairwise {x | x \u2208 List.toFinset sl \u222a List.toFinset tl} fun a b => Commute (f a) (f b)\n\u22a2 noncommProd (List.toFinset sl \u222a List.toFinset tl) f comm =\n    noncommProd (List.toFinset sl) f (_ : Set.Pairwise \u2191(List.toFinset sl) fun a b => Commute (f a) (f b)) *\n      noncommProd (List.toFinset tl) f (_ : Set.Pairwise \u2191(List.toFinset tl) fun a b => Commute (f a) (f b))"}, {"tactic": "rw [List.disjoint_toFinset_iff_disjoint] at h", "annotated_tactic": ["rw [<a>List.disjoint_toFinset_iff_disjoint</a>] at h", [{"full_name": "List.disjoint_toFinset_iff_disjoint", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3876, 9], "def_end_pos": [3876, 39]}]], "state_before": "case intro.intro.intro.intro\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b2 : Monoid \u03b2\ninst\u271d\u00b9 : Monoid \u03b3\ninst\u271d : DecidableEq \u03b1\nf : \u03b1 \u2192 \u03b2\nsl : List \u03b1\nsl' : List.Nodup sl\ntl : List \u03b1\ntl' : List.Nodup tl\nh : Disjoint (List.toFinset sl) (List.toFinset tl)\ncomm : Set.Pairwise {x | x \u2208 List.toFinset sl \u222a List.toFinset tl} fun a b => Commute (f a) (f b)\n\u22a2 noncommProd (List.toFinset sl \u222a List.toFinset tl) f comm =\n    noncommProd (List.toFinset sl) f (_ : Set.Pairwise \u2191(List.toFinset sl) fun a b => Commute (f a) (f b)) *\n      noncommProd (List.toFinset tl) f (_ : Set.Pairwise \u2191(List.toFinset tl) fun a b => Commute (f a) (f b))", "state_after": "case intro.intro.intro.intro\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b2 : Monoid \u03b2\ninst\u271d\u00b9 : Monoid \u03b3\ninst\u271d : DecidableEq \u03b1\nf : \u03b1 \u2192 \u03b2\nsl : List \u03b1\nsl' : List.Nodup sl\ntl : List \u03b1\ntl' : List.Nodup tl\nh : List.Disjoint sl tl\ncomm : Set.Pairwise {x | x \u2208 List.toFinset sl \u222a List.toFinset tl} fun a b => Commute (f a) (f b)\n\u22a2 noncommProd (List.toFinset sl \u222a List.toFinset tl) f comm =\n    noncommProd (List.toFinset sl) f (_ : Set.Pairwise \u2191(List.toFinset sl) fun a b => Commute (f a) (f b)) *\n      noncommProd (List.toFinset tl) f (_ : Set.Pairwise \u2191(List.toFinset tl) fun a b => Commute (f a) (f b))"}, {"tactic": "calc noncommProd (List.toFinset sl \u222a List.toFinset tl) f comm\n   = noncommProd \u27e8\u2191(sl ++ tl), Multiset.coe_nodup.2 (sl'.append tl' h)\u27e9 f\n       (by convert comm; simp [Set.ext_iff]) := noncommProd_congr (by ext; simp) (by simp) _\n _ = noncommProd (List.toFinset sl) f (comm.mono <| coe_subset.2 <| subset_union_left _ _) *\n       noncommProd (List.toFinset tl) f (comm.mono <| coe_subset.2 <| subset_union_right _ _) :=\n  by simp [noncommProd, List.dedup_eq_self.2 sl', List.dedup_eq_self.2 tl', h]", "annotated_tactic": ["calc <a>noncommProd</a> (<a>List.toFinset</a> sl \u222a <a>List.toFinset</a> tl) f comm\n     = <a>noncommProd</a> \u27e8\u2191(sl ++ tl), <a>Multiset.coe_nodup</a>.2 (sl'.append tl' h)\u27e9 f\n         (by convert comm; simp [<a>Set.ext_iff</a>]) := <a>noncommProd_congr</a> (by ext; simp) (by simp) _\n   _ = <a>noncommProd</a> (<a>List.toFinset</a> sl) f (comm.mono <| <a>coe_subset</a>.2 <| <a>subset_union_left</a> _ _) *\n         <a>noncommProd</a> (<a>List.toFinset</a> tl) f (comm.mono <| <a>coe_subset</a>.2 <| <a>subset_union_right</a> _ _) :=\n    by simp [<a>noncommProd</a>, <a>List.dedup_eq_self</a>.2 sl', <a>List.dedup_eq_self</a>.2 tl', h]", [{"full_name": "Finset.noncommProd", "def_path": "Mathlib/Data/Finset/NoncommProd.lean", "def_pos": [242, 5], "def_end_pos": [242, 16]}, {"full_name": "List.toFinset", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3300, 5], "def_end_pos": [3300, 13]}, {"full_name": "List.toFinset", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3300, 5], "def_end_pos": [3300, 13]}, {"full_name": "Finset.noncommProd", "def_path": "Mathlib/Data/Finset/NoncommProd.lean", "def_pos": [242, 5], "def_end_pos": [242, 16]}, {"full_name": "Multiset.coe_nodup", "def_path": "Mathlib/Data/Multiset/Nodup.lean", "def_pos": [31, 9], "def_end_pos": [31, 18]}, {"full_name": "Set.ext_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [241, 9], "def_end_pos": [241, 16]}, {"full_name": "Finset.noncommProd_congr", "def_path": "Mathlib/Data/Finset/NoncommProd.lean", "def_pos": [249, 9], "def_end_pos": [249, 26]}, {"full_name": "Finset.noncommProd", "def_path": "Mathlib/Data/Finset/NoncommProd.lean", "def_pos": [242, 5], "def_end_pos": [242, 16]}, {"full_name": "List.toFinset", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3300, 5], "def_end_pos": [3300, 13]}, {"full_name": "Finset.coe_subset", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [376, 9], "def_end_pos": [376, 19]}, {"full_name": "Finset.subset_union_left", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1407, 9], "def_end_pos": [1407, 26]}, {"full_name": "Finset.noncommProd", "def_path": "Mathlib/Data/Finset/NoncommProd.lean", "def_pos": [242, 5], "def_end_pos": [242, 16]}, {"full_name": "List.toFinset", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3300, 5], "def_end_pos": [3300, 13]}, {"full_name": "Finset.coe_subset", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [376, 9], "def_end_pos": [376, 19]}, {"full_name": "Finset.subset_union_right", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1410, 9], "def_end_pos": [1410, 27]}, {"full_name": "Finset.noncommProd", "def_path": "Mathlib/Data/Finset/NoncommProd.lean", "def_pos": [242, 5], "def_end_pos": [242, 16]}, {"full_name": "List.dedup_eq_self", "def_path": "Mathlib/Data/List/Dedup.lean", "def_pos": [90, 9], "def_end_pos": [90, 22]}, {"full_name": "List.dedup_eq_self", "def_path": "Mathlib/Data/List/Dedup.lean", "def_pos": [90, 9], "def_end_pos": [90, 22]}]], "state_before": "case intro.intro.intro.intro\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b2 : Monoid \u03b2\ninst\u271d\u00b9 : Monoid \u03b3\ninst\u271d : DecidableEq \u03b1\nf : \u03b1 \u2192 \u03b2\nsl : List \u03b1\nsl' : List.Nodup sl\ntl : List \u03b1\ntl' : List.Nodup tl\nh : List.Disjoint sl tl\ncomm : Set.Pairwise {x | x \u2208 List.toFinset sl \u222a List.toFinset tl} fun a b => Commute (f a) (f b)\n\u22a2 noncommProd (List.toFinset sl \u222a List.toFinset tl) f comm =\n    noncommProd (List.toFinset sl) f (_ : Set.Pairwise \u2191(List.toFinset sl) fun a b => Commute (f a) (f b)) *\n      noncommProd (List.toFinset tl) f (_ : Set.Pairwise \u2191(List.toFinset tl) fun a b => Commute (f a) (f b))", "state_after": "no goals"}, {"tactic": "convert comm", "annotated_tactic": ["convert comm", []], "state_before": "F : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b2 : Monoid \u03b2\ninst\u271d\u00b9 : Monoid \u03b3\ninst\u271d : DecidableEq \u03b1\nf : \u03b1 \u2192 \u03b2\nsl : List \u03b1\nsl' : List.Nodup sl\ntl : List \u03b1\ntl' : List.Nodup tl\nh : List.Disjoint sl tl\ncomm : Set.Pairwise {x | x \u2208 List.toFinset sl \u222a List.toFinset tl} fun a b => Commute (f a) (f b)\n\u22a2 Set.Pairwise \u2191{ val := \u2191(sl ++ tl), nodup := (_ : Multiset.Nodup \u2191(sl ++ tl)) } fun a b => Commute (f a) (f b)", "state_after": "case h.e'_2\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b2 : Monoid \u03b2\ninst\u271d\u00b9 : Monoid \u03b3\ninst\u271d : DecidableEq \u03b1\nf : \u03b1 \u2192 \u03b2\nsl : List \u03b1\nsl' : List.Nodup sl\ntl : List \u03b1\ntl' : List.Nodup tl\nh : List.Disjoint sl tl\ncomm : Set.Pairwise {x | x \u2208 List.toFinset sl \u222a List.toFinset tl} fun a b => Commute (f a) (f b)\n\u22a2 \u2191{ val := \u2191(sl ++ tl), nodup := (_ : Multiset.Nodup \u2191(sl ++ tl)) } = {x | x \u2208 List.toFinset sl \u222a List.toFinset tl}"}, {"tactic": "simp [Set.ext_iff]", "annotated_tactic": ["simp [<a>Set.ext_iff</a>]", [{"full_name": "Set.ext_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [241, 9], "def_end_pos": [241, 16]}]], "state_before": "case h.e'_2\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b2 : Monoid \u03b2\ninst\u271d\u00b9 : Monoid \u03b3\ninst\u271d : DecidableEq \u03b1\nf : \u03b1 \u2192 \u03b2\nsl : List \u03b1\nsl' : List.Nodup sl\ntl : List \u03b1\ntl' : List.Nodup tl\nh : List.Disjoint sl tl\ncomm : Set.Pairwise {x | x \u2208 List.toFinset sl \u222a List.toFinset tl} fun a b => Commute (f a) (f b)\n\u22a2 \u2191{ val := \u2191(sl ++ tl), nodup := (_ : Multiset.Nodup \u2191(sl ++ tl)) } = {x | x \u2208 List.toFinset sl \u222a List.toFinset tl}", "state_after": "no goals"}, {"tactic": "ext", "annotated_tactic": ["ext", []], "state_before": "F : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b2 : Monoid \u03b2\ninst\u271d\u00b9 : Monoid \u03b3\ninst\u271d : DecidableEq \u03b1\nf : \u03b1 \u2192 \u03b2\nsl : List \u03b1\nsl' : List.Nodup sl\ntl : List \u03b1\ntl' : List.Nodup tl\nh : List.Disjoint sl tl\ncomm : Set.Pairwise {x | x \u2208 List.toFinset sl \u222a List.toFinset tl} fun a b => Commute (f a) (f b)\n\u22a2 List.toFinset sl \u222a List.toFinset tl = { val := \u2191(sl ++ tl), nodup := (_ : Multiset.Nodup \u2191(sl ++ tl)) }", "state_after": "case a\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b2 : Monoid \u03b2\ninst\u271d\u00b9 : Monoid \u03b3\ninst\u271d : DecidableEq \u03b1\nf : \u03b1 \u2192 \u03b2\nsl : List \u03b1\nsl' : List.Nodup sl\ntl : List \u03b1\ntl' : List.Nodup tl\nh : List.Disjoint sl tl\ncomm : Set.Pairwise {x | x \u2208 List.toFinset sl \u222a List.toFinset tl} fun a b => Commute (f a) (f b)\na\u271d : \u03b1\n\u22a2 a\u271d \u2208 List.toFinset sl \u222a List.toFinset tl \u2194 a\u271d \u2208 { val := \u2191(sl ++ tl), nodup := (_ : Multiset.Nodup \u2191(sl ++ tl)) }"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case a\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b2 : Monoid \u03b2\ninst\u271d\u00b9 : Monoid \u03b3\ninst\u271d : DecidableEq \u03b1\nf : \u03b1 \u2192 \u03b2\nsl : List \u03b1\nsl' : List.Nodup sl\ntl : List \u03b1\ntl' : List.Nodup tl\nh : List.Disjoint sl tl\ncomm : Set.Pairwise {x | x \u2208 List.toFinset sl \u222a List.toFinset tl} fun a b => Commute (f a) (f b)\na\u271d : \u03b1\n\u22a2 a\u271d \u2208 List.toFinset sl \u222a List.toFinset tl \u2194 a\u271d \u2208 { val := \u2191(sl ++ tl), nodup := (_ : Multiset.Nodup \u2191(sl ++ tl)) }", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "F : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b2 : Monoid \u03b2\ninst\u271d\u00b9 : Monoid \u03b3\ninst\u271d : DecidableEq \u03b1\nf : \u03b1 \u2192 \u03b2\nsl : List \u03b1\nsl' : List.Nodup sl\ntl : List \u03b1\ntl' : List.Nodup tl\nh : List.Disjoint sl tl\ncomm : Set.Pairwise {x | x \u2208 List.toFinset sl \u222a List.toFinset tl} fun a b => Commute (f a) (f b)\n\u22a2 \u2200 (x : \u03b1), x \u2208 { val := \u2191(sl ++ tl), nodup := (_ : Multiset.Nodup \u2191(sl ++ tl)) } \u2192 f x = f x", "state_after": "no goals"}, {"tactic": "simp [noncommProd, List.dedup_eq_self.2 sl', List.dedup_eq_self.2 tl', h]", "annotated_tactic": ["simp [<a>noncommProd</a>, <a>List.dedup_eq_self</a>.2 sl', <a>List.dedup_eq_self</a>.2 tl', h]", [{"full_name": "Finset.noncommProd", "def_path": "Mathlib/Data/Finset/NoncommProd.lean", "def_pos": [242, 5], "def_end_pos": [242, 16]}, {"full_name": "List.dedup_eq_self", "def_path": "Mathlib/Data/List/Dedup.lean", "def_pos": [90, 9], "def_end_pos": [90, 22]}, {"full_name": "List.dedup_eq_self", "def_path": "Mathlib/Data/List/Dedup.lean", "def_pos": [90, 9], "def_end_pos": [90, 22]}]], "state_before": "F : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b2 : Monoid \u03b2\ninst\u271d\u00b9 : Monoid \u03b3\ninst\u271d : DecidableEq \u03b1\nf : \u03b1 \u2192 \u03b2\nsl : List \u03b1\nsl' : List.Nodup sl\ntl : List \u03b1\ntl' : List.Nodup tl\nh : List.Disjoint sl tl\ncomm : Set.Pairwise {x | x \u2208 List.toFinset sl \u222a List.toFinset tl} fun a b => Commute (f a) (f b)\n\u22a2 noncommProd { val := \u2191(sl ++ tl), nodup := (_ : Multiset.Nodup \u2191(sl ++ tl)) } f\n      (_ :\n        Set.Pairwise \u2191{ val := \u2191(sl ++ tl), nodup := (_ : Multiset.Nodup \u2191(sl ++ tl)) } fun a b =>\n          Commute (f a) (f b)) =\n    noncommProd (List.toFinset sl) f (_ : Set.Pairwise \u2191(List.toFinset sl) fun a b => Commute (f a) (f b)) *\n      noncommProd (List.toFinset tl) f (_ : Set.Pairwise \u2191(List.toFinset tl) fun a b => Commute (f a) (f b))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Pointwise/BigOperators.lean", "full_name": "Set.mem_finset_prod", "start": [60, 1], "end": [81, 68], "traced_tactics": [{"tactic": "induction' t using Finset.induction_on with i is hi ih generalizing a", "annotated_tactic": ["induction' t using <a>Finset.induction_on</a> with i is hi ih generalizing a", [{"full_name": "Finset.induction_on", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1251, 19], "def_end_pos": [1251, 31]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nF : Type u_4\ninst\u271d\u00b2 : CommMonoid \u03b1\ninst\u271d\u00b9 : CommMonoid \u03b2\ninst\u271d : MonoidHomClass F \u03b1 \u03b2\nt : Finset \u03b9\nf : \u03b9 \u2192 Set \u03b1\na : \u03b1\n\u22a2 a \u2208 \u220f i in t, f i \u2194 \u2203 g x, \u220f i in t, g i = a", "state_after": "case empty\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nF : Type u_4\ninst\u271d\u00b2 : CommMonoid \u03b1\ninst\u271d\u00b9 : CommMonoid \u03b2\ninst\u271d : MonoidHomClass F \u03b1 \u03b2\nf : \u03b9 \u2192 Set \u03b1\na\u271d a : \u03b1\n\u22a2 a \u2208 \u220f i in \u2205, f i \u2194 \u2203 g x, \u220f i in \u2205, g i = a\n\ncase insert\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nF : Type u_4\ninst\u271d\u00b2 : CommMonoid \u03b1\ninst\u271d\u00b9 : CommMonoid \u03b2\ninst\u271d : MonoidHomClass F \u03b1 \u03b2\nf : \u03b9 \u2192 Set \u03b1\na\u271d : \u03b1\ni : \u03b9\nis : Finset \u03b9\nhi : \u00aci \u2208 is\nih : \u2200 (a : \u03b1), a \u2208 \u220f i in is, f i \u2194 \u2203 g x, \u220f i in is, g i = a\na : \u03b1\n\u22a2 a \u2208 \u220f i in insert i is, f i \u2194 \u2203 g x, \u220f i in insert i is, g i = a"}, {"tactic": "rw [Finset.prod_insert hi, Set.mem_mul]", "annotated_tactic": ["rw [<a>Finset.prod_insert</a> hi, <a>Set.mem_mul</a>]", [{"full_name": "Finset.prod_insert", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [317, 9], "def_end_pos": [317, 20]}, {"full_name": "Set.mem_mul", "def_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "def_pos": [336, 9], "def_end_pos": [336, 16]}]], "state_before": "case insert\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nF : Type u_4\ninst\u271d\u00b2 : CommMonoid \u03b1\ninst\u271d\u00b9 : CommMonoid \u03b2\ninst\u271d : MonoidHomClass F \u03b1 \u03b2\nf : \u03b9 \u2192 Set \u03b1\na\u271d : \u03b1\ni : \u03b9\nis : Finset \u03b9\nhi : \u00aci \u2208 is\nih : \u2200 (a : \u03b1), a \u2208 \u220f i in is, f i \u2194 \u2203 g x, \u220f i in is, g i = a\na : \u03b1\n\u22a2 a \u2208 \u220f i in insert i is, f i \u2194 \u2203 g x, \u220f i in insert i is, g i = a", "state_after": "case insert\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nF : Type u_4\ninst\u271d\u00b2 : CommMonoid \u03b1\ninst\u271d\u00b9 : CommMonoid \u03b2\ninst\u271d : MonoidHomClass F \u03b1 \u03b2\nf : \u03b9 \u2192 Set \u03b1\na\u271d : \u03b1\ni : \u03b9\nis : Finset \u03b9\nhi : \u00aci \u2208 is\nih : \u2200 (a : \u03b1), a \u2208 \u220f i in is, f i \u2194 \u2203 g x, \u220f i in is, g i = a\na : \u03b1\n\u22a2 (\u2203 x y, x \u2208 f i \u2227 y \u2208 \u220f x in is, f x \u2227 x * y = a) \u2194 \u2203 g x, \u220f i in insert i is, g i = a"}, {"tactic": "simp_rw [Finset.prod_insert hi]", "annotated_tactic": ["simp_rw [<a>Finset.prod_insert</a> hi]", [{"full_name": "Finset.prod_insert", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [317, 9], "def_end_pos": [317, 20]}]], "state_before": "case insert\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nF : Type u_4\ninst\u271d\u00b2 : CommMonoid \u03b1\ninst\u271d\u00b9 : CommMonoid \u03b2\ninst\u271d : MonoidHomClass F \u03b1 \u03b2\nf : \u03b9 \u2192 Set \u03b1\na\u271d : \u03b1\ni : \u03b9\nis : Finset \u03b9\nhi : \u00aci \u2208 is\nih : \u2200 (a : \u03b1), a \u2208 \u220f i in is, f i \u2194 \u2203 g x, \u220f i in is, g i = a\na : \u03b1\n\u22a2 (\u2203 x y, x \u2208 f i \u2227 y \u2208 \u220f x in is, f x \u2227 x * y = a) \u2194 \u2203 g x, \u220f i in insert i is, g i = a", "state_after": "case insert\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nF : Type u_4\ninst\u271d\u00b2 : CommMonoid \u03b1\ninst\u271d\u00b9 : CommMonoid \u03b2\ninst\u271d : MonoidHomClass F \u03b1 \u03b2\nf : \u03b9 \u2192 Set \u03b1\na\u271d : \u03b1\ni : \u03b9\nis : Finset \u03b9\nhi : \u00aci \u2208 is\nih : \u2200 (a : \u03b1), a \u2208 \u220f i in is, f i \u2194 \u2203 g x, \u220f i in is, g i = a\na : \u03b1\n\u22a2 (\u2203 x y, x \u2208 f i \u2227 y \u2208 \u220f x in is, f x \u2227 x * y = a) \u2194 \u2203 g h, g i * \u220f i in is, g i = a"}, {"tactic": "simp_rw [ih]", "annotated_tactic": ["simp_rw [ih]", []], "state_before": "case insert\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nF : Type u_4\ninst\u271d\u00b2 : CommMonoid \u03b1\ninst\u271d\u00b9 : CommMonoid \u03b2\ninst\u271d : MonoidHomClass F \u03b1 \u03b2\nf : \u03b9 \u2192 Set \u03b1\na\u271d : \u03b1\ni : \u03b9\nis : Finset \u03b9\nhi : \u00aci \u2208 is\nih : \u2200 (a : \u03b1), a \u2208 \u220f i in is, f i \u2194 \u2203 g x, \u220f i in is, g i = a\na : \u03b1\n\u22a2 (\u2203 x y, x \u2208 f i \u2227 y \u2208 \u220f x in is, f x \u2227 x * y = a) \u2194 \u2203 g h, g i * \u220f i in is, g i = a", "state_after": "case insert\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nF : Type u_4\ninst\u271d\u00b2 : CommMonoid \u03b1\ninst\u271d\u00b9 : CommMonoid \u03b2\ninst\u271d : MonoidHomClass F \u03b1 \u03b2\nf : \u03b9 \u2192 Set \u03b1\na\u271d : \u03b1\ni : \u03b9\nis : Finset \u03b9\nhi : \u00aci \u2208 is\nih : \u2200 (a : \u03b1), a \u2208 \u220f i in is, f i \u2194 \u2203 g x, \u220f i in is, g i = a\na : \u03b1\n\u22a2 (\u2203 x y, x \u2208 f i \u2227 (\u2203 g x, \u220f i in is, g i = y) \u2227 x * y = a) \u2194 \u2203 g h, g i * \u220f i in is, g i = a"}, {"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "case insert\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nF : Type u_4\ninst\u271d\u00b2 : CommMonoid \u03b1\ninst\u271d\u00b9 : CommMonoid \u03b2\ninst\u271d : MonoidHomClass F \u03b1 \u03b2\nf : \u03b9 \u2192 Set \u03b1\na\u271d : \u03b1\ni : \u03b9\nis : Finset \u03b9\nhi : \u00aci \u2208 is\nih : \u2200 (a : \u03b1), a \u2208 \u220f i in is, f i \u2194 \u2203 g x, \u220f i in is, g i = a\na : \u03b1\n\u22a2 (\u2203 x y, x \u2208 f i \u2227 (\u2203 g x, \u220f i in is, g i = y) \u2227 x * y = a) \u2194 \u2203 g h, g i * \u220f i in is, g i = a", "state_after": "case insert.mp\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nF : Type u_4\ninst\u271d\u00b2 : CommMonoid \u03b1\ninst\u271d\u00b9 : CommMonoid \u03b2\ninst\u271d : MonoidHomClass F \u03b1 \u03b2\nf : \u03b9 \u2192 Set \u03b1\na\u271d : \u03b1\ni : \u03b9\nis : Finset \u03b9\nhi : \u00aci \u2208 is\nih : \u2200 (a : \u03b1), a \u2208 \u220f i in is, f i \u2194 \u2203 g x, \u220f i in is, g i = a\na : \u03b1\n\u22a2 (\u2203 x y, x \u2208 f i \u2227 (\u2203 g x, \u220f i in is, g i = y) \u2227 x * y = a) \u2192 \u2203 g h, g i * \u220f i in is, g i = a\n\ncase insert.mpr\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nF : Type u_4\ninst\u271d\u00b2 : CommMonoid \u03b1\ninst\u271d\u00b9 : CommMonoid \u03b2\ninst\u271d : MonoidHomClass F \u03b1 \u03b2\nf : \u03b9 \u2192 Set \u03b1\na\u271d : \u03b1\ni : \u03b9\nis : Finset \u03b9\nhi : \u00aci \u2208 is\nih : \u2200 (a : \u03b1), a \u2208 \u220f i in is, f i \u2194 \u2203 g x, \u220f i in is, g i = a\na : \u03b1\n\u22a2 (\u2203 g h, g i * \u220f i in is, g i = a) \u2192 \u2203 x y, x \u2208 f i \u2227 (\u2203 g x, \u220f i in is, g i = y) \u2227 x * y = a"}, {"tactic": "simp_rw [Finset.prod_empty, Set.mem_one]", "annotated_tactic": ["simp_rw [<a>Finset.prod_empty</a>, <a>Set.mem_one</a>]", [{"full_name": "Finset.prod_empty", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [299, 9], "def_end_pos": [299, 19]}, {"full_name": "Set.mem_one", "def_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "def_pos": [100, 9], "def_end_pos": [100, 16]}]], "state_before": "case empty\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nF : Type u_4\ninst\u271d\u00b2 : CommMonoid \u03b1\ninst\u271d\u00b9 : CommMonoid \u03b2\ninst\u271d : MonoidHomClass F \u03b1 \u03b2\nf : \u03b9 \u2192 Set \u03b1\na\u271d a : \u03b1\n\u22a2 a \u2208 \u220f i in \u2205, f i \u2194 \u2203 g x, \u220f i in \u2205, g i = a", "state_after": "case empty\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nF : Type u_4\ninst\u271d\u00b2 : CommMonoid \u03b1\ninst\u271d\u00b9 : CommMonoid \u03b2\ninst\u271d : MonoidHomClass F \u03b1 \u03b2\nf : \u03b9 \u2192 Set \u03b1\na\u271d a : \u03b1\n\u22a2 a = 1 \u2194 \u2203 g h, 1 = a"}, {"tactic": "exact \u27e8fun h \u21a6 \u27e8fun _ \u21a6 a, fun hi \u21a6 False.elim (Finset.not_mem_empty _ hi), h.symm\u27e9,\n  fun \u27e8_, _, hf\u27e9 \u21a6 hf.symm\u27e9", "annotated_tactic": ["exact \u27e8fun h \u21a6 \u27e8fun _ \u21a6 a, fun hi \u21a6 <a>False.elim</a> (<a>Finset.not_mem_empty</a> _ hi), h.symm\u27e9,\n        fun \u27e8_, _, hf\u27e9 \u21a6 hf.symm\u27e9", [{"full_name": "False.elim", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [223, 21], "def_end_pos": [223, 31]}, {"full_name": "Finset.not_mem_empty", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [548, 9], "def_end_pos": [548, 22]}]], "state_before": "case empty\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nF : Type u_4\ninst\u271d\u00b2 : CommMonoid \u03b1\ninst\u271d\u00b9 : CommMonoid \u03b2\ninst\u271d : MonoidHomClass F \u03b1 \u03b2\nf : \u03b9 \u2192 Set \u03b1\na\u271d a : \u03b1\n\u22a2 a = 1 \u2194 \u2203 g h, 1 = a", "state_after": "no goals"}, {"tactic": "rintro \u27e8x, y, hx, \u27e8g, hg, rfl\u27e9, rfl\u27e9", "annotated_tactic": ["rintro \u27e8x, y, hx, \u27e8g, hg, rfl\u27e9, rfl\u27e9", []], "state_before": "case insert.mp\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nF : Type u_4\ninst\u271d\u00b2 : CommMonoid \u03b1\ninst\u271d\u00b9 : CommMonoid \u03b2\ninst\u271d : MonoidHomClass F \u03b1 \u03b2\nf : \u03b9 \u2192 Set \u03b1\na\u271d : \u03b1\ni : \u03b9\nis : Finset \u03b9\nhi : \u00aci \u2208 is\nih : \u2200 (a : \u03b1), a \u2208 \u220f i in is, f i \u2194 \u2203 g x, \u220f i in is, g i = a\na : \u03b1\n\u22a2 (\u2203 x y, x \u2208 f i \u2227 (\u2203 g x, \u220f i in is, g i = y) \u2227 x * y = a) \u2192 \u2203 g h, g i * \u220f i in is, g i = a", "state_after": "case insert.mp.intro.intro.intro.intro.intro.intro\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nF : Type u_4\ninst\u271d\u00b2 : CommMonoid \u03b1\ninst\u271d\u00b9 : CommMonoid \u03b2\ninst\u271d : MonoidHomClass F \u03b1 \u03b2\nf : \u03b9 \u2192 Set \u03b1\na : \u03b1\ni : \u03b9\nis : Finset \u03b9\nhi : \u00aci \u2208 is\nih : \u2200 (a : \u03b1), a \u2208 \u220f i in is, f i \u2194 \u2203 g x, \u220f i in is, g i = a\nx : \u03b1\nhx : x \u2208 f i\ng : \u03b9 \u2192 \u03b1\nhg : \u2200 {i : \u03b9}, i \u2208 is \u2192 g i \u2208 f i\n\u22a2 \u2203 g_1 h, g_1 i * \u220f i in is, g_1 i = x * \u220f i in is, g i"}, {"tactic": "refine \u27e8Function.update g i x, ?_, ?_\u27e9", "annotated_tactic": ["refine \u27e8<a>Function.update</a> g i x, ?_, ?_\u27e9", [{"full_name": "Function.update", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [550, 5], "def_end_pos": [550, 11]}]], "state_before": "case insert.mp.intro.intro.intro.intro.intro.intro\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nF : Type u_4\ninst\u271d\u00b2 : CommMonoid \u03b1\ninst\u271d\u00b9 : CommMonoid \u03b2\ninst\u271d : MonoidHomClass F \u03b1 \u03b2\nf : \u03b9 \u2192 Set \u03b1\na : \u03b1\ni : \u03b9\nis : Finset \u03b9\nhi : \u00aci \u2208 is\nih : \u2200 (a : \u03b1), a \u2208 \u220f i in is, f i \u2194 \u2203 g x, \u220f i in is, g i = a\nx : \u03b1\nhx : x \u2208 f i\ng : \u03b9 \u2192 \u03b1\nhg : \u2200 {i : \u03b9}, i \u2208 is \u2192 g i \u2208 f i\n\u22a2 \u2203 g_1 h, g_1 i * \u220f i in is, g_1 i = x * \u220f i in is, g i", "state_after": "case insert.mp.intro.intro.intro.intro.intro.intro.refine_1\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nF : Type u_4\ninst\u271d\u00b2 : CommMonoid \u03b1\ninst\u271d\u00b9 : CommMonoid \u03b2\ninst\u271d : MonoidHomClass F \u03b1 \u03b2\nf : \u03b9 \u2192 Set \u03b1\na : \u03b1\ni : \u03b9\nis : Finset \u03b9\nhi : \u00aci \u2208 is\nih : \u2200 (a : \u03b1), a \u2208 \u220f i in is, f i \u2194 \u2203 g x, \u220f i in is, g i = a\nx : \u03b1\nhx : x \u2208 f i\ng : \u03b9 \u2192 \u03b1\nhg : \u2200 {i : \u03b9}, i \u2208 is \u2192 g i \u2208 f i\n\u22a2 \u2200 {i_1 : \u03b9}, i_1 \u2208 insert i is \u2192 update g i x i_1 \u2208 f i_1\n\ncase insert.mp.intro.intro.intro.intro.intro.intro.refine_2\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nF : Type u_4\ninst\u271d\u00b2 : CommMonoid \u03b1\ninst\u271d\u00b9 : CommMonoid \u03b2\ninst\u271d : MonoidHomClass F \u03b1 \u03b2\nf : \u03b9 \u2192 Set \u03b1\na : \u03b1\ni : \u03b9\nis : Finset \u03b9\nhi : \u00aci \u2208 is\nih : \u2200 (a : \u03b1), a \u2208 \u220f i in is, f i \u2194 \u2203 g x, \u220f i in is, g i = a\nx : \u03b1\nhx : x \u2208 f i\ng : \u03b9 \u2192 \u03b1\nhg : \u2200 {i : \u03b9}, i \u2208 is \u2192 g i \u2208 f i\n\u22a2 update g i x i * \u220f i_1 in is, update g i x i_1 = x * \u220f i in is, g i"}, {"tactic": "intro j hj", "annotated_tactic": ["intro j hj", []], "state_before": "case insert.mp.intro.intro.intro.intro.intro.intro.refine_1\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nF : Type u_4\ninst\u271d\u00b2 : CommMonoid \u03b1\ninst\u271d\u00b9 : CommMonoid \u03b2\ninst\u271d : MonoidHomClass F \u03b1 \u03b2\nf : \u03b9 \u2192 Set \u03b1\na : \u03b1\ni : \u03b9\nis : Finset \u03b9\nhi : \u00aci \u2208 is\nih : \u2200 (a : \u03b1), a \u2208 \u220f i in is, f i \u2194 \u2203 g x, \u220f i in is, g i = a\nx : \u03b1\nhx : x \u2208 f i\ng : \u03b9 \u2192 \u03b1\nhg : \u2200 {i : \u03b9}, i \u2208 is \u2192 g i \u2208 f i\n\u22a2 \u2200 {i_1 : \u03b9}, i_1 \u2208 insert i is \u2192 update g i x i_1 \u2208 f i_1", "state_after": "case insert.mp.intro.intro.intro.intro.intro.intro.refine_1\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nF : Type u_4\ninst\u271d\u00b2 : CommMonoid \u03b1\ninst\u271d\u00b9 : CommMonoid \u03b2\ninst\u271d : MonoidHomClass F \u03b1 \u03b2\nf : \u03b9 \u2192 Set \u03b1\na : \u03b1\ni : \u03b9\nis : Finset \u03b9\nhi : \u00aci \u2208 is\nih : \u2200 (a : \u03b1), a \u2208 \u220f i in is, f i \u2194 \u2203 g x, \u220f i in is, g i = a\nx : \u03b1\nhx : x \u2208 f i\ng : \u03b9 \u2192 \u03b1\nhg : \u2200 {i : \u03b9}, i \u2208 is \u2192 g i \u2208 f i\nj : \u03b9\nhj : j \u2208 insert i is\n\u22a2 update g i x j \u2208 f j"}, {"tactic": "obtain rfl | hj := Finset.mem_insert.mp hj", "annotated_tactic": ["obtain rfl | hj := Finset.mem_insert.mp hj", []], "state_before": "case insert.mp.intro.intro.intro.intro.intro.intro.refine_1\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nF : Type u_4\ninst\u271d\u00b2 : CommMonoid \u03b1\ninst\u271d\u00b9 : CommMonoid \u03b2\ninst\u271d : MonoidHomClass F \u03b1 \u03b2\nf : \u03b9 \u2192 Set \u03b1\na : \u03b1\ni : \u03b9\nis : Finset \u03b9\nhi : \u00aci \u2208 is\nih : \u2200 (a : \u03b1), a \u2208 \u220f i in is, f i \u2194 \u2203 g x, \u220f i in is, g i = a\nx : \u03b1\nhx : x \u2208 f i\ng : \u03b9 \u2192 \u03b1\nhg : \u2200 {i : \u03b9}, i \u2208 is \u2192 g i \u2208 f i\nj : \u03b9\nhj : j \u2208 insert i is\n\u22a2 update g i x j \u2208 f j", "state_after": "case insert.mp.intro.intro.intro.intro.intro.intro.refine_1.inl\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nF : Type u_4\ninst\u271d\u00b2 : CommMonoid \u03b1\ninst\u271d\u00b9 : CommMonoid \u03b2\ninst\u271d : MonoidHomClass F \u03b1 \u03b2\nf : \u03b9 \u2192 Set \u03b1\na : \u03b1\nis : Finset \u03b9\nih : \u2200 (a : \u03b1), a \u2208 \u220f i in is, f i \u2194 \u2203 g x, \u220f i in is, g i = a\nx : \u03b1\ng : \u03b9 \u2192 \u03b1\nhg : \u2200 {i : \u03b9}, i \u2208 is \u2192 g i \u2208 f i\nj : \u03b9\nhi : \u00acj \u2208 is\nhx : x \u2208 f j\nhj : j \u2208 insert j is\n\u22a2 update g j x j \u2208 f j\n\ncase insert.mp.intro.intro.intro.intro.intro.intro.refine_1.inr\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nF : Type u_4\ninst\u271d\u00b2 : CommMonoid \u03b1\ninst\u271d\u00b9 : CommMonoid \u03b2\ninst\u271d : MonoidHomClass F \u03b1 \u03b2\nf : \u03b9 \u2192 Set \u03b1\na : \u03b1\ni : \u03b9\nis : Finset \u03b9\nhi : \u00aci \u2208 is\nih : \u2200 (a : \u03b1), a \u2208 \u220f i in is, f i \u2194 \u2203 g x, \u220f i in is, g i = a\nx : \u03b1\nhx : x \u2208 f i\ng : \u03b9 \u2192 \u03b1\nhg : \u2200 {i : \u03b9}, i \u2208 is \u2192 g i \u2208 f i\nj : \u03b9\nhj\u271d : j \u2208 insert i is\nhj : j \u2208 is\n\u22a2 update g i x j \u2208 f j"}, {"tactic": "rwa [Function.update_same]", "annotated_tactic": ["rwa [<a>Function.update_same</a>]", [{"full_name": "Function.update_same", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [555, 9], "def_end_pos": [555, 20]}]], "state_before": "case insert.mp.intro.intro.intro.intro.intro.intro.refine_1.inl\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nF : Type u_4\ninst\u271d\u00b2 : CommMonoid \u03b1\ninst\u271d\u00b9 : CommMonoid \u03b2\ninst\u271d : MonoidHomClass F \u03b1 \u03b2\nf : \u03b9 \u2192 Set \u03b1\na : \u03b1\nis : Finset \u03b9\nih : \u2200 (a : \u03b1), a \u2208 \u220f i in is, f i \u2194 \u2203 g x, \u220f i in is, g i = a\nx : \u03b1\ng : \u03b9 \u2192 \u03b1\nhg : \u2200 {i : \u03b9}, i \u2208 is \u2192 g i \u2208 f i\nj : \u03b9\nhi : \u00acj \u2208 is\nhx : x \u2208 f j\nhj : j \u2208 insert j is\n\u22a2 update g j x j \u2208 f j", "state_after": "no goals"}, {"tactic": "rw [update_noteq (ne_of_mem_of_not_mem hj hi)]", "annotated_tactic": ["rw [<a>update_noteq</a> (<a>ne_of_mem_of_not_mem</a> hj hi)]", [{"full_name": "Function.update_noteq", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [560, 9], "def_end_pos": [560, 21]}, {"full_name": "ne_of_mem_of_not_mem", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [719, 9], "def_end_pos": [719, 29]}]], "state_before": "case insert.mp.intro.intro.intro.intro.intro.intro.refine_1.inr\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nF : Type u_4\ninst\u271d\u00b2 : CommMonoid \u03b1\ninst\u271d\u00b9 : CommMonoid \u03b2\ninst\u271d : MonoidHomClass F \u03b1 \u03b2\nf : \u03b9 \u2192 Set \u03b1\na : \u03b1\ni : \u03b9\nis : Finset \u03b9\nhi : \u00aci \u2208 is\nih : \u2200 (a : \u03b1), a \u2208 \u220f i in is, f i \u2194 \u2203 g x, \u220f i in is, g i = a\nx : \u03b1\nhx : x \u2208 f i\ng : \u03b9 \u2192 \u03b1\nhg : \u2200 {i : \u03b9}, i \u2208 is \u2192 g i \u2208 f i\nj : \u03b9\nhj\u271d : j \u2208 insert i is\nhj : j \u2208 is\n\u22a2 update g i x j \u2208 f j", "state_after": "case insert.mp.intro.intro.intro.intro.intro.intro.refine_1.inr\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nF : Type u_4\ninst\u271d\u00b2 : CommMonoid \u03b1\ninst\u271d\u00b9 : CommMonoid \u03b2\ninst\u271d : MonoidHomClass F \u03b1 \u03b2\nf : \u03b9 \u2192 Set \u03b1\na : \u03b1\ni : \u03b9\nis : Finset \u03b9\nhi : \u00aci \u2208 is\nih : \u2200 (a : \u03b1), a \u2208 \u220f i in is, f i \u2194 \u2203 g x, \u220f i in is, g i = a\nx : \u03b1\nhx : x \u2208 f i\ng : \u03b9 \u2192 \u03b1\nhg : \u2200 {i : \u03b9}, i \u2208 is \u2192 g i \u2208 f i\nj : \u03b9\nhj\u271d : j \u2208 insert i is\nhj : j \u2208 is\n\u22a2 g j \u2208 f j"}, {"tactic": "exact hg hj", "annotated_tactic": ["exact hg hj", []], "state_before": "case insert.mp.intro.intro.intro.intro.intro.intro.refine_1.inr\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nF : Type u_4\ninst\u271d\u00b2 : CommMonoid \u03b1\ninst\u271d\u00b9 : CommMonoid \u03b2\ninst\u271d : MonoidHomClass F \u03b1 \u03b2\nf : \u03b9 \u2192 Set \u03b1\na : \u03b1\ni : \u03b9\nis : Finset \u03b9\nhi : \u00aci \u2208 is\nih : \u2200 (a : \u03b1), a \u2208 \u220f i in is, f i \u2194 \u2203 g x, \u220f i in is, g i = a\nx : \u03b1\nhx : x \u2208 f i\ng : \u03b9 \u2192 \u03b1\nhg : \u2200 {i : \u03b9}, i \u2208 is \u2192 g i \u2208 f i\nj : \u03b9\nhj\u271d : j \u2208 insert i is\nhj : j \u2208 is\n\u22a2 g j \u2208 f j", "state_after": "no goals"}, {"tactic": "rw [Finset.prod_update_of_not_mem hi, Function.update_same]", "annotated_tactic": ["rw [<a>Finset.prod_update_of_not_mem</a> hi, <a>Function.update_same</a>]", [{"full_name": "Finset.prod_update_of_not_mem", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [1616, 9], "def_end_pos": [1616, 31]}, {"full_name": "Function.update_same", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [555, 9], "def_end_pos": [555, 20]}]], "state_before": "case insert.mp.intro.intro.intro.intro.intro.intro.refine_2\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nF : Type u_4\ninst\u271d\u00b2 : CommMonoid \u03b1\ninst\u271d\u00b9 : CommMonoid \u03b2\ninst\u271d : MonoidHomClass F \u03b1 \u03b2\nf : \u03b9 \u2192 Set \u03b1\na : \u03b1\ni : \u03b9\nis : Finset \u03b9\nhi : \u00aci \u2208 is\nih : \u2200 (a : \u03b1), a \u2208 \u220f i in is, f i \u2194 \u2203 g x, \u220f i in is, g i = a\nx : \u03b1\nhx : x \u2208 f i\ng : \u03b9 \u2192 \u03b1\nhg : \u2200 {i : \u03b9}, i \u2208 is \u2192 g i \u2208 f i\n\u22a2 update g i x i * \u220f i_1 in is, update g i x i_1 = x * \u220f i in is, g i", "state_after": "no goals"}, {"tactic": "rintro \u27e8g, hg, rfl\u27e9", "annotated_tactic": ["rintro \u27e8g, hg, rfl\u27e9", []], "state_before": "case insert.mpr\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nF : Type u_4\ninst\u271d\u00b2 : CommMonoid \u03b1\ninst\u271d\u00b9 : CommMonoid \u03b2\ninst\u271d : MonoidHomClass F \u03b1 \u03b2\nf : \u03b9 \u2192 Set \u03b1\na\u271d : \u03b1\ni : \u03b9\nis : Finset \u03b9\nhi : \u00aci \u2208 is\nih : \u2200 (a : \u03b1), a \u2208 \u220f i in is, f i \u2194 \u2203 g x, \u220f i in is, g i = a\na : \u03b1\n\u22a2 (\u2203 g h, g i * \u220f i in is, g i = a) \u2192 \u2203 x y, x \u2208 f i \u2227 (\u2203 g x, \u220f i in is, g i = y) \u2227 x * y = a", "state_after": "case insert.mpr.intro.intro\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nF : Type u_4\ninst\u271d\u00b2 : CommMonoid \u03b1\ninst\u271d\u00b9 : CommMonoid \u03b2\ninst\u271d : MonoidHomClass F \u03b1 \u03b2\nf : \u03b9 \u2192 Set \u03b1\na : \u03b1\ni : \u03b9\nis : Finset \u03b9\nhi : \u00aci \u2208 is\nih : \u2200 (a : \u03b1), a \u2208 \u220f i in is, f i \u2194 \u2203 g x, \u220f i in is, g i = a\ng : \u03b9 \u2192 \u03b1\nhg : \u2200 {i_1 : \u03b9}, i_1 \u2208 insert i is \u2192 g i_1 \u2208 f i_1\n\u22a2 \u2203 x y, x \u2208 f i \u2227 (\u2203 g x, \u220f i in is, g i = y) \u2227 x * y = g i * \u220f i in is, g i"}, {"tactic": "exact \u27e8g i, is.prod g, hg (is.mem_insert_self _),\n  \u27e8\u27e8g, fun hi \u21a6 hg (Finset.mem_insert_of_mem hi), rfl\u27e9, rfl\u27e9\u27e9", "annotated_tactic": ["exact \u27e8g i, is.prod g, hg (is.mem_insert_self _),\n        \u27e8\u27e8g, fun hi \u21a6 hg (<a>Finset.mem_insert_of_mem</a> hi), <a>rfl</a>\u27e9, <a>rfl</a>\u27e9\u27e9", [{"full_name": "Finset.mem_insert_of_mem", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1095, 9], "def_end_pos": [1095, 26]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "case insert.mpr.intro.intro\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nF : Type u_4\ninst\u271d\u00b2 : CommMonoid \u03b1\ninst\u271d\u00b9 : CommMonoid \u03b2\ninst\u271d : MonoidHomClass F \u03b1 \u03b2\nf : \u03b9 \u2192 Set \u03b1\na : \u03b1\ni : \u03b9\nis : Finset \u03b9\nhi : \u00aci \u2208 is\nih : \u2200 (a : \u03b1), a \u2208 \u220f i in is, f i \u2194 \u2203 g x, \u220f i in is, g i = a\ng : \u03b9 \u2192 \u03b1\nhg : \u2200 {i_1 : \u03b9}, i_1 \u2208 insert i is \u2192 g i_1 \u2208 f i_1\n\u22a2 \u2203 x y, x \u2208 f i \u2227 (\u2203 g x, \u220f i in is, g i = y) \u2227 x * y = g i * \u220f i in is, g i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Martingale/BorelCantelli.lean", "full_name": "MeasureTheory.BorelCantelli.martingalePart_process_ae_eq", "start": [289, 1], "end": [294, 53], "traced_tactics": [{"tactic": "simp only [martingalePart_eq_sum, process_zero, zero_add]", "annotated_tactic": ["simp only [<a>martingalePart_eq_sum</a>, <a>process_zero</a>, <a>zero_add</a>]", [{"full_name": "MeasureTheory.martingalePart_eq_sum", "def_path": "Mathlib/Probability/Martingale/Centering.lean", "def_pos": [75, 9], "def_end_pos": [75, 30]}, {"full_name": "MeasureTheory.BorelCantelli.process_zero", "def_path": "Mathlib/Probability/Martingale/BorelCantelli.lean", "def_pos": [281, 9], "def_end_pos": [281, 21]}, {"full_name": "zero_add", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [463, 3], "def_end_pos": [463, 14]}]], "state_before": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc\u271d : Measure \u03a9\n\u2131\u271d : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nr : \u211d\nR : \u211d\u22650\ns\u271d : \u2115 \u2192 Set \u03a9\n\u2131 : Filtration \u2115 m0\n\u03bc : Measure \u03a9\ns : \u2115 \u2192 Set \u03a9\nn : \u2115\n\u22a2 martingalePart (process s) \u2131 \u03bc n =\n    \u2211 k in Finset.range n, (Set.indicator (s (k + 1)) 1 - \u03bc[Set.indicator (s (k + 1)) 1|\u2191\u2131 k])", "state_after": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc\u271d : Measure \u03a9\n\u2131\u271d : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nr : \u211d\nR : \u211d\u22650\ns\u271d : \u2115 \u2192 Set \u03a9\n\u2131 : Filtration \u2115 m0\n\u03bc : Measure \u03a9\ns : \u2115 \u2192 Set \u03a9\nn : \u2115\n\u22a2 \u2211 i in Finset.range n, (process s (i + 1) - process s i - \u03bc[process s (i + 1) - process s i|\u2191\u2131 i]) =\n    \u2211 k in Finset.range n, (Set.indicator (s (k + 1)) 1 - \u03bc[Set.indicator (s (k + 1)) 1|\u2191\u2131 k])"}, {"tactic": "refine' Finset.sum_congr rfl fun k _ => _", "annotated_tactic": ["refine' <a>Finset.sum_congr</a> <a>rfl</a> fun k _ => _", [{"full_name": "Finset.sum_congr", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [376, 3], "def_end_pos": [376, 14]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc\u271d : Measure \u03a9\n\u2131\u271d : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nr : \u211d\nR : \u211d\u22650\ns\u271d : \u2115 \u2192 Set \u03a9\n\u2131 : Filtration \u2115 m0\n\u03bc : Measure \u03a9\ns : \u2115 \u2192 Set \u03a9\nn : \u2115\n\u22a2 \u2211 i in Finset.range n, (process s (i + 1) - process s i - \u03bc[process s (i + 1) - process s i|\u2191\u2131 i]) =\n    \u2211 k in Finset.range n, (Set.indicator (s (k + 1)) 1 - \u03bc[Set.indicator (s (k + 1)) 1|\u2191\u2131 k])", "state_after": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc\u271d : Measure \u03a9\n\u2131\u271d : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nr : \u211d\nR : \u211d\u22650\ns\u271d : \u2115 \u2192 Set \u03a9\n\u2131 : Filtration \u2115 m0\n\u03bc : Measure \u03a9\ns : \u2115 \u2192 Set \u03a9\nn k : \u2115\nx\u271d : k \u2208 Finset.range n\n\u22a2 process s (k + 1) - process s k - \u03bc[process s (k + 1) - process s k|\u2191\u2131 k] =\n    Set.indicator (s (k + 1)) 1 - \u03bc[Set.indicator (s (k + 1)) 1|\u2191\u2131 k]"}, {"tactic": "simp only [process, Finset.sum_range_succ_sub_sum]", "annotated_tactic": ["simp only [<a>process</a>, <a>Finset.sum_range_succ_sub_sum</a>]", [{"full_name": "MeasureTheory.BorelCantelli.process", "def_path": "Mathlib/Probability/Martingale/BorelCantelli.lean", "def_pos": [275, 19], "def_end_pos": [275, 26]}, {"full_name": "Finset.sum_range_succ_sub_sum", "def_path": "Mathlib/Algebra/BigOperators/Intervals.lean", "def_pos": [229, 3], "def_end_pos": [229, 14]}]], "state_before": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc\u271d : Measure \u03a9\n\u2131\u271d : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nr : \u211d\nR : \u211d\u22650\ns\u271d : \u2115 \u2192 Set \u03a9\n\u2131 : Filtration \u2115 m0\n\u03bc : Measure \u03a9\ns : \u2115 \u2192 Set \u03a9\nn k : \u2115\nx\u271d : k \u2208 Finset.range n\n\u22a2 process s (k + 1) - process s k - \u03bc[process s (k + 1) - process s k|\u2191\u2131 k] =\n    Set.indicator (s (k + 1)) 1 - \u03bc[Set.indicator (s (k + 1)) 1|\u2191\u2131 k]", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "full_name": "MeasurableSpace.ext", "start": [256, 1], "end": [258, 59], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Num/Lemmas.lean", "full_name": "PosNum.size_to_nat", "start": [566, 1], "end": [570, 81], "traced_tactics": [{"tactic": "rw [size, succ_to_nat, size_to_nat n, cast_bit0, Nat.size_bit0 <| ne_of_gt <| to_nat_pos n]", "annotated_tactic": ["rw [<a>size</a>, <a>succ_to_nat</a>, size_to_nat n, <a>cast_bit0</a>, <a>Nat.size_bit0</a> <| <a>ne_of_gt</a> <| <a>to_nat_pos</a> n]", [{"full_name": "PosNum.size", "def_path": "Mathlib/Data/Num/Basic.lean", "def_pos": [126, 5], "def_end_pos": [126, 9]}, {"full_name": "PosNum.succ_to_nat", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [73, 9], "def_end_pos": [73, 20]}, {"full_name": "PosNum.cast_bit0", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [47, 9], "def_end_pos": [47, 18]}, {"full_name": "Nat.size_bit0", "def_path": "Mathlib/Data/Nat/Size.lean", "def_pos": [71, 9], "def_end_pos": [71, 18]}, {"full_name": "ne_of_gt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [104, 9], "def_end_pos": [104, 17]}, {"full_name": "PosNum.to_nat_pos", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [132, 9], "def_end_pos": [132, 19]}]], "state_before": "\u03b1 : Type u_1\nn : PosNum\n\u22a2 \u2191(size (bit0 n)) = Nat.size \u2191(bit0 n)", "state_after": "no goals"}, {"tactic": "rw [size, succ_to_nat, size_to_nat n, cast_bit1, Nat.size_bit1]", "annotated_tactic": ["rw [<a>size</a>, <a>succ_to_nat</a>, size_to_nat n, <a>cast_bit1</a>, <a>Nat.size_bit1</a>]", [{"full_name": "PosNum.size", "def_path": "Mathlib/Data/Num/Basic.lean", "def_pos": [126, 5], "def_end_pos": [126, 9]}, {"full_name": "PosNum.succ_to_nat", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [73, 9], "def_end_pos": [73, 20]}, {"full_name": "PosNum.cast_bit1", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [52, 9], "def_end_pos": [52, 18]}, {"full_name": "Nat.size_bit1", "def_path": "Mathlib/Data/Nat/Size.lean", "def_pos": [76, 9], "def_end_pos": [76, 18]}]], "state_before": "\u03b1 : Type u_1\nn : PosNum\n\u22a2 \u2191(size (bit1 n)) = Nat.size \u2191(bit1 n)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/FiniteMeasure.lean", "full_name": "MeasureTheory.FiniteMeasure.tendsto_iff_forall_toWeakDualBCNN_tendsto", "start": [487, 1], "end": [491, 93], "traced_tactics": [{"tactic": "rw [tendsto_iff_weak_star_tendsto, tendsto_iff_forall_eval_tendsto_topDualPairing]", "annotated_tactic": ["rw [<a>tendsto_iff_weak_star_tendsto</a>, <a>tendsto_iff_forall_eval_tendsto_topDualPairing</a>]", [{"full_name": "MeasureTheory.FiniteMeasure.tendsto_iff_weak_star_tendsto", "def_path": "Mathlib/MeasureTheory/Measure/FiniteMeasure.lean", "def_pos": [481, 9], "def_end_pos": [481, 38]}, {"full_name": "tendsto_iff_forall_eval_tendsto_topDualPairing", "def_path": "Mathlib/Topology/Algebra/Module/WeakDual.lean", "def_pos": [339, 9], "def_end_pos": [339, 55]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2075 : SMul R \u211d\u22650\ninst\u271d\u2074 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_3\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\n\u03bc : FiniteMeasure \u03a9\n\u22a2 Tendsto \u03bcs F (\ud835\udcdd \u03bc) \u2194 \u2200 (f : \u03a9 \u2192\u1d47 \u211d\u22650), Tendsto (fun i => \u2191(toWeakDualBCNN (\u03bcs i)) f) F (\ud835\udcdd (\u2191(toWeakDualBCNN \u03bc) f))", "state_after": "\u03a9 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2075 : SMul R \u211d\u22650\ninst\u271d\u2074 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_3\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\n\u03bc : FiniteMeasure \u03a9\n\u22a2 (\u2200 (y : \u03a9 \u2192\u1d47 \u211d\u22650),\n      Tendsto (fun i => \u2191(\u2191(topDualPairing \u211d\u22650 (\u03a9 \u2192\u1d47 \u211d\u22650)) (toWeakDualBCNN (\u03bcs i))) y) F\n        (\ud835\udcdd (\u2191(\u2191(topDualPairing \u211d\u22650 (\u03a9 \u2192\u1d47 \u211d\u22650)) (toWeakDualBCNN \u03bc)) y))) \u2194\n    \u2200 (f : \u03a9 \u2192\u1d47 \u211d\u22650), Tendsto (fun i => \u2191(toWeakDualBCNN (\u03bcs i)) f) F (\ud835\udcdd (\u2191(toWeakDualBCNN \u03bc) f))"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\u03a9 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2075 : SMul R \u211d\u22650\ninst\u271d\u2074 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_3\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\n\u03bc : FiniteMeasure \u03a9\n\u22a2 (\u2200 (y : \u03a9 \u2192\u1d47 \u211d\u22650),\n      Tendsto (fun i => \u2191(\u2191(topDualPairing \u211d\u22650 (\u03a9 \u2192\u1d47 \u211d\u22650)) (toWeakDualBCNN (\u03bcs i))) y) F\n        (\ud835\udcdd (\u2191(\u2191(topDualPairing \u211d\u22650 (\u03a9 \u2192\u1d47 \u211d\u22650)) (toWeakDualBCNN \u03bc)) y))) \u2194\n    \u2200 (f : \u03a9 \u2192\u1d47 \u211d\u22650), Tendsto (fun i => \u2191(toWeakDualBCNN (\u03bcs i)) f) F (\ud835\udcdd (\u2191(toWeakDualBCNN \u03bc) f))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "full_name": "Int.lt_div_add_one_mul_self", "start": [548, 1], "end": [550, 72], "traced_tactics": [{"tactic": "rw [Int.add_mul, Int.one_mul, Int.mul_comm]", "annotated_tactic": ["rw [<a>Int.add_mul</a>, <a>Int.one_mul</a>, <a>Int.mul_comm</a>]", [{"full_name": "Int.add_mul", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [459, 19], "def_end_pos": [459, 26]}, {"full_name": "Int.one_mul", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [521, 27], "def_end_pos": [521, 34]}, {"full_name": "Int.mul_comm", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [380, 19], "def_end_pos": [380, 27]}]], "state_before": "a b : Int\nH : 0 < b\n\u22a2 a < (div a b + 1) * b", "state_after": "a b : Int\nH : 0 < b\n\u22a2 a < b * div a b + b"}, {"tactic": "exact Int.lt_add_of_sub_left_lt <| Int.mod_def .. \u25b8 mod_lt_of_pos _ H", "annotated_tactic": ["exact <a>Int.lt_add_of_sub_left_lt</a> <| <a>Int.mod_def</a> .. \u25b8 <a>mod_lt_of_pos</a> _ H", [{"full_name": "Int.lt_add_of_sub_left_lt", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [1074, 19], "def_end_pos": [1074, 40]}, {"full_name": "Int.mod_def", "def_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "def_pos": [322, 9], "def_end_pos": [322, 16]}, {"full_name": "Int.mod_lt_of_pos", "def_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "def_pos": [379, 9], "def_end_pos": [379, 22]}]], "state_before": "a b : Int\nH : 0 < b\n\u22a2 a < b * div a b + b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Halting.lean", "full_name": "Nat.Partrec'.vec_iff", "start": [430, 1], "end": [432, 44], "traced_tactics": [{"tactic": "simpa only [ofFn_get] using vector_ofFn fun i => to_part (h i)", "annotated_tactic": ["simpa only [<a>ofFn_get</a>] using <a>vector_ofFn</a> fun i => <a>to_part</a> (h i)", [{"full_name": "Vector.ofFn_get", "def_path": "Mathlib/Data/Vector/Basic.lean", "def_pos": [155, 9], "def_end_pos": [155, 17]}, {"full_name": "Computable.vector_ofFn", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [751, 9], "def_end_pos": [751, 20]}, {"full_name": "Nat.Partrec'.to_part", "def_path": "Mathlib/Computability/Halting.lean", "def_pos": [295, 9], "def_end_pos": [295, 16]}]], "state_before": "m n : \u2115\nf : Vector \u2115 m \u2192 Vector \u2115 n\nh : Vec f\n\u22a2 Computable f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Num/Lemmas.lean", "full_name": "ZNum.dvd_to_int", "start": [1562, 1], "end": [1563, 86], "traced_tactics": [{"tactic": "rw [\u2190 of_to_int n, e]", "annotated_tactic": ["rw [\u2190 <a>of_to_int</a> n, e]", [{"full_name": "ZNum.of_to_int", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [1540, 9], "def_end_pos": [1540, 18]}]], "state_before": "\u03b1 : Type u_1\nm n : ZNum\nx\u271d : \u2191m \u2223 \u2191n\nk : \u2124\ne : \u2191n = \u2191m * k\n\u22a2 n = m * \u2191k", "state_after": "\u03b1 : Type u_1\nm n : ZNum\nx\u271d : \u2191m \u2223 \u2191n\nk : \u2124\ne : \u2191n = \u2191m * k\n\u22a2 \u2191(\u2191m * k) = m * \u2191k"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03b1 : Type u_1\nm n : ZNum\nx\u271d : \u2191m \u2223 \u2191n\nk : \u2124\ne : \u2191n = \u2191m * k\n\u22a2 \u2191(\u2191m * k) = m * \u2191k", "state_after": "no goals"}, {"tactic": "simp [e]", "annotated_tactic": ["simp [e]", []], "state_before": "\u03b1 : Type u_1\nm n : ZNum\nx\u271d : m \u2223 n\nk : ZNum\ne : n = m * k\n\u22a2 \u2191n = \u2191m * \u2191k", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "full_name": "Embedding.aestronglyMeasurable_comp_iff", "start": [1633, 1], "end": [1652, 53], "traced_tactics": [{"tactic": "letI := pseudoMetrizableSpacePseudoMetric \u03b3", "annotated_tactic": ["letI := <a>pseudoMetrizableSpacePseudoMetric</a> \u03b3", [{"full_name": "TopologicalSpace.pseudoMetrizableSpacePseudoMetric", "def_path": "Mathlib/Topology/MetricSpace/Metrizable.lean", "def_pos": [48, 19], "def_end_pos": [48, 52]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u2074 : Countable \u03b9\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : PseudoMetrizableSpace \u03b2\ninst\u271d : PseudoMetrizableSpace \u03b3\ng : \u03b2 \u2192 \u03b3\nf : \u03b1 \u2192 \u03b2\nhg : _root_.Embedding g\n\u22a2 AEStronglyMeasurable (fun x => g (f x)) \u03bc \u2194 AEStronglyMeasurable f \u03bc", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u2074 : Countable \u03b9\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : PseudoMetrizableSpace \u03b2\ninst\u271d : PseudoMetrizableSpace \u03b3\ng : \u03b2 \u2192 \u03b3\nf : \u03b1 \u2192 \u03b2\nhg : _root_.Embedding g\nthis : PseudoMetricSpace \u03b3 := pseudoMetrizableSpacePseudoMetric \u03b3\n\u22a2 AEStronglyMeasurable (fun x => g (f x)) \u03bc \u2194 AEStronglyMeasurable f \u03bc"}, {"tactic": "borelize \u03b2 \u03b3", "annotated_tactic": ["borelize \u03b2 \u03b3", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u2074 : Countable \u03b9\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : PseudoMetrizableSpace \u03b2\ninst\u271d : PseudoMetrizableSpace \u03b3\ng : \u03b2 \u2192 \u03b3\nf : \u03b1 \u2192 \u03b2\nhg : _root_.Embedding g\nthis : PseudoMetricSpace \u03b3 := pseudoMetrizableSpacePseudoMetric \u03b3\n\u22a2 AEStronglyMeasurable (fun x => g (f x)) \u03bc \u2194 AEStronglyMeasurable f \u03bc", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u2074 : Countable \u03b9\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : PseudoMetrizableSpace \u03b2\ninst\u271d : PseudoMetrizableSpace \u03b3\ng : \u03b2 \u2192 \u03b3\nf : \u03b1 \u2192 \u03b2\nhg : _root_.Embedding g\nthis : PseudoMetricSpace \u03b3 := pseudoMetrizableSpacePseudoMetric \u03b3\nthis\u271d\u00b3 : MeasurableSpace \u03b2 := borel \u03b2\nthis\u271d\u00b2 : BorelSpace \u03b2\nthis\u271d\u00b9 : MeasurableSpace \u03b3 := borel \u03b3\nthis\u271d : BorelSpace \u03b3\n\u22a2 AEStronglyMeasurable (fun x => g (f x)) \u03bc \u2194 AEStronglyMeasurable f \u03bc"}, {"tactic": "refine'\n  \u27e8fun H => aestronglyMeasurable_iff_aemeasurable_separable.2 \u27e8_, _\u27e9, fun H =>\n    hg.continuous.comp_aestronglyMeasurable H\u27e9", "annotated_tactic": ["refine'\n    \u27e8fun H => <a>aestronglyMeasurable_iff_aemeasurable_separable</a>.2 \u27e8_, _\u27e9, fun H =>\n      hg.continuous.comp_aestronglyMeasurable H\u27e9", [{"full_name": "aestronglyMeasurable_iff_aemeasurable_separable", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1599, 9], "def_end_pos": [1599, 63]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u2074 : Countable \u03b9\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : PseudoMetrizableSpace \u03b2\ninst\u271d : PseudoMetrizableSpace \u03b3\ng : \u03b2 \u2192 \u03b3\nf : \u03b1 \u2192 \u03b2\nhg : _root_.Embedding g\nthis : PseudoMetricSpace \u03b3 := pseudoMetrizableSpacePseudoMetric \u03b3\nthis\u271d\u00b3 : MeasurableSpace \u03b2 := borel \u03b2\nthis\u271d\u00b2 : BorelSpace \u03b2\nthis\u271d\u00b9 : MeasurableSpace \u03b3 := borel \u03b3\nthis\u271d : BorelSpace \u03b3\n\u22a2 AEStronglyMeasurable (fun x => g (f x)) \u03bc \u2194 AEStronglyMeasurable f \u03bc", "state_after": "case refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u2074 : Countable \u03b9\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : PseudoMetrizableSpace \u03b2\ninst\u271d : PseudoMetrizableSpace \u03b3\ng : \u03b2 \u2192 \u03b3\nf : \u03b1 \u2192 \u03b2\nhg : _root_.Embedding g\nthis : PseudoMetricSpace \u03b3 := pseudoMetrizableSpacePseudoMetric \u03b3\nthis\u271d\u00b3 : MeasurableSpace \u03b2 := borel \u03b2\nthis\u271d\u00b2 : BorelSpace \u03b2\nthis\u271d\u00b9 : MeasurableSpace \u03b3 := borel \u03b3\nthis\u271d : BorelSpace \u03b3\nH : AEStronglyMeasurable (fun x => g (f x)) \u03bc\n\u22a2 AEMeasurable f\n\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u2074 : Countable \u03b9\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : PseudoMetrizableSpace \u03b2\ninst\u271d : PseudoMetrizableSpace \u03b3\ng : \u03b2 \u2192 \u03b3\nf : \u03b1 \u2192 \u03b2\nhg : _root_.Embedding g\nthis : PseudoMetricSpace \u03b3 := pseudoMetrizableSpacePseudoMetric \u03b3\nthis\u271d\u00b3 : MeasurableSpace \u03b2 := borel \u03b2\nthis\u271d\u00b2 : BorelSpace \u03b2\nthis\u271d\u00b9 : MeasurableSpace \u03b3 := borel \u03b3\nthis\u271d : BorelSpace \u03b3\nH : AEStronglyMeasurable (fun x => g (f x)) \u03bc\n\u22a2 \u2203 t, IsSeparable t \u2227 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t"}, {"tactic": "let G : \u03b2 \u2192 range g := codRestrict g (range g) mem_range_self", "annotated_tactic": ["let G : \u03b2 \u2192 <a>range</a> g := <a>codRestrict</a> g (<a>range</a> g) <a>mem_range_self</a>", [{"full_name": "Set.range", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [668, 5], "def_end_pos": [668, 10]}, {"full_name": "Set.codRestrict", "def_path": "Mathlib/Data/Set/Function.lean", "def_pos": [148, 5], "def_end_pos": [148, 16]}, {"full_name": "Set.range", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [668, 5], "def_end_pos": [668, 10]}, {"full_name": "Set.mem_range_self", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [680, 9], "def_end_pos": [680, 23]}]], "state_before": "case refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u2074 : Countable \u03b9\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : PseudoMetrizableSpace \u03b2\ninst\u271d : PseudoMetrizableSpace \u03b3\ng : \u03b2 \u2192 \u03b3\nf : \u03b1 \u2192 \u03b2\nhg : _root_.Embedding g\nthis : PseudoMetricSpace \u03b3 := pseudoMetrizableSpacePseudoMetric \u03b3\nthis\u271d\u00b3 : MeasurableSpace \u03b2 := borel \u03b2\nthis\u271d\u00b2 : BorelSpace \u03b2\nthis\u271d\u00b9 : MeasurableSpace \u03b3 := borel \u03b3\nthis\u271d : BorelSpace \u03b3\nH : AEStronglyMeasurable (fun x => g (f x)) \u03bc\n\u22a2 AEMeasurable f", "state_after": "case refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u2074 : Countable \u03b9\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : PseudoMetrizableSpace \u03b2\ninst\u271d : PseudoMetrizableSpace \u03b3\ng : \u03b2 \u2192 \u03b3\nf : \u03b1 \u2192 \u03b2\nhg : _root_.Embedding g\nthis : PseudoMetricSpace \u03b3 := pseudoMetrizableSpacePseudoMetric \u03b3\nthis\u271d\u00b3 : MeasurableSpace \u03b2 := borel \u03b2\nthis\u271d\u00b2 : BorelSpace \u03b2\nthis\u271d\u00b9 : MeasurableSpace \u03b3 := borel \u03b3\nthis\u271d : BorelSpace \u03b3\nH : AEStronglyMeasurable (fun x => g (f x)) \u03bc\nG : \u03b2 \u2192 \u2191(range g) := codRestrict g (range g) (_ : \u2200 (i : \u03b2), g i \u2208 range fun x => g x)\n\u22a2 AEMeasurable f"}, {"tactic": "have hG : ClosedEmbedding G :=\n  { hg.codRestrict _ _ with\n    closed_range := by\n      convert isClosed_univ (\u03b1 := \u21a5(range g))\n      apply eq_univ_of_forall\n      rintro \u27e8-, \u27e8x, rfl\u27e9\u27e9\n      exact mem_range_self x }", "annotated_tactic": ["have hG : <a>ClosedEmbedding</a> G :=\n      { hg.codRestrict _ _ with\n        closed_range := by\n          convert <a>isClosed_univ</a> (\u03b1 := \u21a5(<a>range</a> g))\n          apply <a>eq_univ_of_forall</a>\n          rintro \u27e8-, \u27e8x, rfl\u27e9\u27e9\n          exact <a>mem_range_self</a> x }", [{"full_name": "ClosedEmbedding", "def_path": "Mathlib/Topology/Maps.lean", "def_pos": [671, 11], "def_end_pos": [671, 26]}, {"full_name": "isClosed_univ", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [215, 17], "def_end_pos": [215, 30]}, {"full_name": "Set.range", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [668, 5], "def_end_pos": [668, 10]}, {"full_name": "Set.eq_univ_of_forall", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [705, 9], "def_end_pos": [705, 26]}, {"full_name": "Set.mem_range_self", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [680, 9], "def_end_pos": [680, 23]}]], "state_before": "case refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u2074 : Countable \u03b9\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : PseudoMetrizableSpace \u03b2\ninst\u271d : PseudoMetrizableSpace \u03b3\ng : \u03b2 \u2192 \u03b3\nf : \u03b1 \u2192 \u03b2\nhg : _root_.Embedding g\nthis : PseudoMetricSpace \u03b3 := pseudoMetrizableSpacePseudoMetric \u03b3\nthis\u271d\u00b3 : MeasurableSpace \u03b2 := borel \u03b2\nthis\u271d\u00b2 : BorelSpace \u03b2\nthis\u271d\u00b9 : MeasurableSpace \u03b3 := borel \u03b3\nthis\u271d : BorelSpace \u03b3\nH : AEStronglyMeasurable (fun x => g (f x)) \u03bc\nG : \u03b2 \u2192 \u2191(range g) := codRestrict g (range g) (_ : \u2200 (i : \u03b2), g i \u2208 range fun x => g x)\n\u22a2 AEMeasurable f", "state_after": "case refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u2074 : Countable \u03b9\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : PseudoMetrizableSpace \u03b2\ninst\u271d : PseudoMetrizableSpace \u03b3\ng : \u03b2 \u2192 \u03b3\nf : \u03b1 \u2192 \u03b2\nhg : _root_.Embedding g\nthis : PseudoMetricSpace \u03b3 := pseudoMetrizableSpacePseudoMetric \u03b3\nthis\u271d\u00b3 : MeasurableSpace \u03b2 := borel \u03b2\nthis\u271d\u00b2 : BorelSpace \u03b2\nthis\u271d\u00b9 : MeasurableSpace \u03b3 := borel \u03b3\nthis\u271d : BorelSpace \u03b3\nH : AEStronglyMeasurable (fun x => g (f x)) \u03bc\nG : \u03b2 \u2192 \u2191(range g) := codRestrict g (range g) (_ : \u2200 (i : \u03b2), g i \u2208 range fun x => g x)\nhG : ClosedEmbedding G\n\u22a2 AEMeasurable f"}, {"tactic": "have : AEMeasurable (G \u2218 f) \u03bc := AEMeasurable.subtype_mk H.aemeasurable", "annotated_tactic": ["have : <a>AEMeasurable</a> (G \u2218 f) \u03bc := <a>AEMeasurable.subtype_mk</a> H.aemeasurable", [{"full_name": "AEMeasurable", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [708, 5], "def_end_pos": [708, 17]}, {"full_name": "AEMeasurable.subtype_mk", "def_path": "Mathlib/MeasureTheory/Measure/AEMeasurable.lean", "def_pos": [227, 9], "def_end_pos": [227, 19]}]], "state_before": "case refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u2074 : Countable \u03b9\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : PseudoMetrizableSpace \u03b2\ninst\u271d : PseudoMetrizableSpace \u03b3\ng : \u03b2 \u2192 \u03b3\nf : \u03b1 \u2192 \u03b2\nhg : _root_.Embedding g\nthis : PseudoMetricSpace \u03b3 := pseudoMetrizableSpacePseudoMetric \u03b3\nthis\u271d\u00b3 : MeasurableSpace \u03b2 := borel \u03b2\nthis\u271d\u00b2 : BorelSpace \u03b2\nthis\u271d\u00b9 : MeasurableSpace \u03b3 := borel \u03b3\nthis\u271d : BorelSpace \u03b3\nH : AEStronglyMeasurable (fun x => g (f x)) \u03bc\nG : \u03b2 \u2192 \u2191(range g) := codRestrict g (range g) (_ : \u2200 (i : \u03b2), g i \u2208 range fun x => g x)\nhG : ClosedEmbedding G\n\u22a2 AEMeasurable f", "state_after": "case refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u2074 : Countable \u03b9\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : PseudoMetrizableSpace \u03b2\ninst\u271d : PseudoMetrizableSpace \u03b3\ng : \u03b2 \u2192 \u03b3\nf : \u03b1 \u2192 \u03b2\nhg : _root_.Embedding g\nthis\u271d\u2074 : PseudoMetricSpace \u03b3 := pseudoMetrizableSpacePseudoMetric \u03b3\nthis\u271d\u00b3 : MeasurableSpace \u03b2 := borel \u03b2\nthis\u271d\u00b2 : BorelSpace \u03b2\nthis\u271d\u00b9 : MeasurableSpace \u03b3 := borel \u03b3\nthis\u271d : BorelSpace \u03b3\nH : AEStronglyMeasurable (fun x => g (f x)) \u03bc\nG : \u03b2 \u2192 \u2191(range g) := codRestrict g (range g) (_ : \u2200 (i : \u03b2), g i \u2208 range fun x => g x)\nhG : ClosedEmbedding G\nthis : AEMeasurable (G \u2218 f)\n\u22a2 AEMeasurable f"}, {"tactic": "exact hG.measurableEmbedding.aemeasurable_comp_iff.1 this", "annotated_tactic": ["exact hG.measurableEmbedding.aemeasurable_comp_iff.1 this", []], "state_before": "case refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u2074 : Countable \u03b9\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : PseudoMetrizableSpace \u03b2\ninst\u271d : PseudoMetrizableSpace \u03b3\ng : \u03b2 \u2192 \u03b3\nf : \u03b1 \u2192 \u03b2\nhg : _root_.Embedding g\nthis\u271d\u2074 : PseudoMetricSpace \u03b3 := pseudoMetrizableSpacePseudoMetric \u03b3\nthis\u271d\u00b3 : MeasurableSpace \u03b2 := borel \u03b2\nthis\u271d\u00b2 : BorelSpace \u03b2\nthis\u271d\u00b9 : MeasurableSpace \u03b3 := borel \u03b3\nthis\u271d : BorelSpace \u03b3\nH : AEStronglyMeasurable (fun x => g (f x)) \u03bc\nG : \u03b2 \u2192 \u2191(range g) := codRestrict g (range g) (_ : \u2200 (i : \u03b2), g i \u2208 range fun x => g x)\nhG : ClosedEmbedding G\nthis : AEMeasurable (G \u2218 f)\n\u22a2 AEMeasurable f", "state_after": "no goals"}, {"tactic": "convert isClosed_univ (\u03b1 := \u21a5(range g))", "annotated_tactic": ["convert <a>isClosed_univ</a> (\u03b1 := \u21a5(<a>range</a> g))", [{"full_name": "isClosed_univ", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [215, 17], "def_end_pos": [215, 30]}, {"full_name": "Set.range", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [668, 5], "def_end_pos": [668, 10]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u2074 : Countable \u03b9\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : PseudoMetrizableSpace \u03b2\ninst\u271d : PseudoMetrizableSpace \u03b3\ng : \u03b2 \u2192 \u03b3\nf : \u03b1 \u2192 \u03b2\nhg : _root_.Embedding g\nthis : PseudoMetricSpace \u03b3 := pseudoMetrizableSpacePseudoMetric \u03b3\nthis\u271d\u00b3 : MeasurableSpace \u03b2 := borel \u03b2\nthis\u271d\u00b2 : BorelSpace \u03b2\nthis\u271d\u00b9 : MeasurableSpace \u03b3 := borel \u03b3\nthis\u271d : BorelSpace \u03b3\nH : AEStronglyMeasurable (fun x => g (f x)) \u03bc\nG : \u03b2 \u2192 \u2191(range g) := codRestrict g (range g) (_ : \u2200 (i : \u03b2), g i \u2208 range fun x => g x)\nsrc\u271d : _root_.Embedding (codRestrict g (range g) (_ : \u2200 (i : \u03b2), g i \u2208 range fun x => g x)) :=\n  Embedding.codRestrict hg (range g) mem_range_self\n\u22a2 IsClosed (range G)", "state_after": "case h.e'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u2074 : Countable \u03b9\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : PseudoMetrizableSpace \u03b2\ninst\u271d : PseudoMetrizableSpace \u03b3\ng : \u03b2 \u2192 \u03b3\nf : \u03b1 \u2192 \u03b2\nhg : _root_.Embedding g\nthis : PseudoMetricSpace \u03b3 := pseudoMetrizableSpacePseudoMetric \u03b3\nthis\u271d\u00b3 : MeasurableSpace \u03b2 := borel \u03b2\nthis\u271d\u00b2 : BorelSpace \u03b2\nthis\u271d\u00b9 : MeasurableSpace \u03b3 := borel \u03b3\nthis\u271d : BorelSpace \u03b3\nH : AEStronglyMeasurable (fun x => g (f x)) \u03bc\nG : \u03b2 \u2192 \u2191(range g) := codRestrict g (range g) (_ : \u2200 (i : \u03b2), g i \u2208 range fun x => g x)\nsrc\u271d : _root_.Embedding (codRestrict g (range g) (_ : \u2200 (i : \u03b2), g i \u2208 range fun x => g x)) :=\n  Embedding.codRestrict hg (range g) mem_range_self\n\u22a2 range G = univ"}, {"tactic": "apply eq_univ_of_forall", "annotated_tactic": ["apply <a>eq_univ_of_forall</a>", [{"full_name": "Set.eq_univ_of_forall", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [705, 9], "def_end_pos": [705, 26]}]], "state_before": "case h.e'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u2074 : Countable \u03b9\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : PseudoMetrizableSpace \u03b2\ninst\u271d : PseudoMetrizableSpace \u03b3\ng : \u03b2 \u2192 \u03b3\nf : \u03b1 \u2192 \u03b2\nhg : _root_.Embedding g\nthis : PseudoMetricSpace \u03b3 := pseudoMetrizableSpacePseudoMetric \u03b3\nthis\u271d\u00b3 : MeasurableSpace \u03b2 := borel \u03b2\nthis\u271d\u00b2 : BorelSpace \u03b2\nthis\u271d\u00b9 : MeasurableSpace \u03b3 := borel \u03b3\nthis\u271d : BorelSpace \u03b3\nH : AEStronglyMeasurable (fun x => g (f x)) \u03bc\nG : \u03b2 \u2192 \u2191(range g) := codRestrict g (range g) (_ : \u2200 (i : \u03b2), g i \u2208 range fun x => g x)\nsrc\u271d : _root_.Embedding (codRestrict g (range g) (_ : \u2200 (i : \u03b2), g i \u2208 range fun x => g x)) :=\n  Embedding.codRestrict hg (range g) mem_range_self\n\u22a2 range G = univ", "state_after": "case h.e'_3.a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u2074 : Countable \u03b9\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : PseudoMetrizableSpace \u03b2\ninst\u271d : PseudoMetrizableSpace \u03b3\ng : \u03b2 \u2192 \u03b3\nf : \u03b1 \u2192 \u03b2\nhg : _root_.Embedding g\nthis : PseudoMetricSpace \u03b3 := pseudoMetrizableSpacePseudoMetric \u03b3\nthis\u271d\u00b3 : MeasurableSpace \u03b2 := borel \u03b2\nthis\u271d\u00b2 : BorelSpace \u03b2\nthis\u271d\u00b9 : MeasurableSpace \u03b3 := borel \u03b3\nthis\u271d : BorelSpace \u03b3\nH : AEStronglyMeasurable (fun x => g (f x)) \u03bc\nG : \u03b2 \u2192 \u2191(range g) := codRestrict g (range g) (_ : \u2200 (i : \u03b2), g i \u2208 range fun x => g x)\nsrc\u271d : _root_.Embedding (codRestrict g (range g) (_ : \u2200 (i : \u03b2), g i \u2208 range fun x => g x)) :=\n  Embedding.codRestrict hg (range g) mem_range_self\n\u22a2 \u2200 (x : \u2191(range g)), x \u2208 range G"}, {"tactic": "rintro \u27e8-, \u27e8x, rfl\u27e9\u27e9", "annotated_tactic": ["rintro \u27e8-, \u27e8x, rfl\u27e9\u27e9", []], "state_before": "case h.e'_3.a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u2074 : Countable \u03b9\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : PseudoMetrizableSpace \u03b2\ninst\u271d : PseudoMetrizableSpace \u03b3\ng : \u03b2 \u2192 \u03b3\nf : \u03b1 \u2192 \u03b2\nhg : _root_.Embedding g\nthis : PseudoMetricSpace \u03b3 := pseudoMetrizableSpacePseudoMetric \u03b3\nthis\u271d\u00b3 : MeasurableSpace \u03b2 := borel \u03b2\nthis\u271d\u00b2 : BorelSpace \u03b2\nthis\u271d\u00b9 : MeasurableSpace \u03b3 := borel \u03b3\nthis\u271d : BorelSpace \u03b3\nH : AEStronglyMeasurable (fun x => g (f x)) \u03bc\nG : \u03b2 \u2192 \u2191(range g) := codRestrict g (range g) (_ : \u2200 (i : \u03b2), g i \u2208 range fun x => g x)\nsrc\u271d : _root_.Embedding (codRestrict g (range g) (_ : \u2200 (i : \u03b2), g i \u2208 range fun x => g x)) :=\n  Embedding.codRestrict hg (range g) mem_range_self\n\u22a2 \u2200 (x : \u2191(range g)), x \u2208 range G", "state_after": "case h.e'_3.a.mk.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u2074 : Countable \u03b9\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : PseudoMetrizableSpace \u03b2\ninst\u271d : PseudoMetrizableSpace \u03b3\ng : \u03b2 \u2192 \u03b3\nf : \u03b1 \u2192 \u03b2\nhg : _root_.Embedding g\nthis : PseudoMetricSpace \u03b3 := pseudoMetrizableSpacePseudoMetric \u03b3\nthis\u271d\u00b3 : MeasurableSpace \u03b2 := borel \u03b2\nthis\u271d\u00b2 : BorelSpace \u03b2\nthis\u271d\u00b9 : MeasurableSpace \u03b3 := borel \u03b3\nthis\u271d : BorelSpace \u03b3\nH : AEStronglyMeasurable (fun x => g (f x)) \u03bc\nG : \u03b2 \u2192 \u2191(range g) := codRestrict g (range g) (_ : \u2200 (i : \u03b2), g i \u2208 range fun x => g x)\nsrc\u271d : _root_.Embedding (codRestrict g (range g) (_ : \u2200 (i : \u03b2), g i \u2208 range fun x => g x)) :=\n  Embedding.codRestrict hg (range g) mem_range_self\nx : \u03b2\n\u22a2 { val := g x, property := (_ : \u2203 y, g y = g x) } \u2208 range G"}, {"tactic": "exact mem_range_self x", "annotated_tactic": ["exact <a>mem_range_self</a> x", [{"full_name": "Set.mem_range_self", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [680, 9], "def_end_pos": [680, 23]}]], "state_before": "case h.e'_3.a.mk.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u2074 : Countable \u03b9\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : PseudoMetrizableSpace \u03b2\ninst\u271d : PseudoMetrizableSpace \u03b3\ng : \u03b2 \u2192 \u03b3\nf : \u03b1 \u2192 \u03b2\nhg : _root_.Embedding g\nthis : PseudoMetricSpace \u03b3 := pseudoMetrizableSpacePseudoMetric \u03b3\nthis\u271d\u00b3 : MeasurableSpace \u03b2 := borel \u03b2\nthis\u271d\u00b2 : BorelSpace \u03b2\nthis\u271d\u00b9 : MeasurableSpace \u03b3 := borel \u03b3\nthis\u271d : BorelSpace \u03b3\nH : AEStronglyMeasurable (fun x => g (f x)) \u03bc\nG : \u03b2 \u2192 \u2191(range g) := codRestrict g (range g) (_ : \u2200 (i : \u03b2), g i \u2208 range fun x => g x)\nsrc\u271d : _root_.Embedding (codRestrict g (range g) (_ : \u2200 (i : \u03b2), g i \u2208 range fun x => g x)) :=\n  Embedding.codRestrict hg (range g) mem_range_self\nx : \u03b2\n\u22a2 { val := g x, property := (_ : \u2203 y, g y = g x) } \u2208 range G", "state_after": "no goals"}, {"tactic": "rcases (aestronglyMeasurable_iff_aemeasurable_separable.1 H).2 with \u27e8t, ht, h't\u27e9", "annotated_tactic": ["rcases (<a>aestronglyMeasurable_iff_aemeasurable_separable</a>.1 H).2 with \u27e8t, ht, h't\u27e9", [{"full_name": "aestronglyMeasurable_iff_aemeasurable_separable", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1599, 9], "def_end_pos": [1599, 63]}]], "state_before": "case refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u2074 : Countable \u03b9\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : PseudoMetrizableSpace \u03b2\ninst\u271d : PseudoMetrizableSpace \u03b3\ng : \u03b2 \u2192 \u03b3\nf : \u03b1 \u2192 \u03b2\nhg : _root_.Embedding g\nthis : PseudoMetricSpace \u03b3 := pseudoMetrizableSpacePseudoMetric \u03b3\nthis\u271d\u00b3 : MeasurableSpace \u03b2 := borel \u03b2\nthis\u271d\u00b2 : BorelSpace \u03b2\nthis\u271d\u00b9 : MeasurableSpace \u03b3 := borel \u03b3\nthis\u271d : BorelSpace \u03b3\nH : AEStronglyMeasurable (fun x => g (f x)) \u03bc\n\u22a2 \u2203 t, IsSeparable t \u2227 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t", "state_after": "case refine'_2.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u2074 : Countable \u03b9\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : PseudoMetrizableSpace \u03b2\ninst\u271d : PseudoMetrizableSpace \u03b3\ng : \u03b2 \u2192 \u03b3\nf : \u03b1 \u2192 \u03b2\nhg : _root_.Embedding g\nthis : PseudoMetricSpace \u03b3 := pseudoMetrizableSpacePseudoMetric \u03b3\nthis\u271d\u00b3 : MeasurableSpace \u03b2 := borel \u03b2\nthis\u271d\u00b2 : BorelSpace \u03b2\nthis\u271d\u00b9 : MeasurableSpace \u03b3 := borel \u03b3\nthis\u271d : BorelSpace \u03b3\nH : AEStronglyMeasurable (fun x => g (f x)) \u03bc\nt : Set \u03b3\nht : IsSeparable t\nh't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g (f x) \u2208 t\n\u22a2 \u2203 t, IsSeparable t \u2227 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t"}, {"tactic": "exact \u27e8g \u207b\u00b9' t, hg.isSeparable_preimage ht, h't\u27e9", "annotated_tactic": ["exact \u27e8g \u207b\u00b9' t, hg.isSeparable_preimage ht, h't\u27e9", []], "state_before": "case refine'_2.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u2074 : Countable \u03b9\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : PseudoMetrizableSpace \u03b2\ninst\u271d : PseudoMetrizableSpace \u03b3\ng : \u03b2 \u2192 \u03b3\nf : \u03b1 \u2192 \u03b2\nhg : _root_.Embedding g\nthis : PseudoMetricSpace \u03b3 := pseudoMetrizableSpacePseudoMetric \u03b3\nthis\u271d\u00b3 : MeasurableSpace \u03b2 := borel \u03b2\nthis\u271d\u00b2 : BorelSpace \u03b2\nthis\u271d\u00b9 : MeasurableSpace \u03b3 := borel \u03b3\nthis\u271d : BorelSpace \u03b3\nH : AEStronglyMeasurable (fun x => g (f x)) \u03bc\nt : Set \u03b3\nht : IsSeparable t\nh't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g (f x) \u2208 t\n\u22a2 \u2203 t, IsSeparable t \u2227 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "full_name": "MeasureTheory.integral_coe_le_of_lintegral_coe_le", "start": [1244, 1], "end": [1248, 38], "traced_tactics": [{"tactic": "by_cases hf : Integrable (fun a => (f a : \u211d)) \u03bc", "annotated_tactic": ["by_cases hf : <a>Integrable</a> (fun a => (f a : \u211d)) \u03bc", [{"full_name": "MeasureTheory.Integrable", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [442, 5], "def_end_pos": [442, 15]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nf : \u03b1 \u2192 \u211d\u22650\nb : \u211d\u22650\nh : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2264 \u2191b\n\u22a2 \u222b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2264 \u2191b", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nf : \u03b1 \u2192 \u211d\u22650\nb : \u211d\u22650\nh : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2264 \u2191b\nhf : Integrable fun a => \u2191(f a)\n\u22a2 \u222b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2264 \u2191b\n\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nf : \u03b1 \u2192 \u211d\u22650\nb : \u211d\u22650\nh : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2264 \u2191b\nhf : \u00acIntegrable fun a => \u2191(f a)\n\u22a2 \u222b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2264 \u2191b"}, {"tactic": "exact (lintegral_coe_le_coe_iff_integral_le hf).1 h", "annotated_tactic": ["exact (<a>lintegral_coe_le_coe_iff_integral_le</a> hf).1 h", [{"full_name": "MeasureTheory.lintegral_coe_le_coe_iff_integral_le", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1238, 9], "def_end_pos": [1238, 45]}]], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nf : \u03b1 \u2192 \u211d\u22650\nb : \u211d\u22650\nh : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2264 \u2191b\nhf : Integrable fun a => \u2191(f a)\n\u22a2 \u222b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2264 \u2191b", "state_after": "no goals"}, {"tactic": "rw [integral_undef hf]", "annotated_tactic": ["rw [<a>integral_undef</a> hf]", [{"full_name": "MeasureTheory.integral_undef", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [836, 9], "def_end_pos": [836, 23]}]], "state_before": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nf : \u03b1 \u2192 \u211d\u22650\nb : \u211d\u22650\nh : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2264 \u2191b\nhf : \u00acIntegrable fun a => \u2191(f a)\n\u22a2 \u222b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2264 \u2191b", "state_after": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nf : \u03b1 \u2192 \u211d\u22650\nb : \u211d\u22650\nh : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2264 \u2191b\nhf : \u00acIntegrable fun a => \u2191(f a)\n\u22a2 0 \u2264 \u2191b"}, {"tactic": "exact b.2", "annotated_tactic": ["exact b.2", []], "state_before": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nf : \u03b1 \u2192 \u211d\u22650\nb : \u211d\u22650\nh : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2264 \u2191b\nhf : \u00acIntegrable fun a => \u2191(f a)\n\u22a2 0 \u2264 \u2191b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Num/Lemmas.lean", "full_name": "Num.dvd_to_nat", "start": [510, 1], "end": [511, 98], "traced_tactics": [{"tactic": "rw [\u2190 of_to_nat n, e]", "annotated_tactic": ["rw [\u2190 <a>of_to_nat</a> n, e]", [{"full_name": "Num.of_to_nat", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [505, 9], "def_end_pos": [505, 18]}]], "state_before": "m n : Num\nx\u271d : \u2191m \u2223 \u2191n\nk : \u2115\ne : \u2191n = \u2191m * k\n\u22a2 n = m * \u2191k", "state_after": "m n : Num\nx\u271d : \u2191m \u2223 \u2191n\nk : \u2115\ne : \u2191n = \u2191m * k\n\u22a2 \u2191(\u2191m * k) = m * \u2191k"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "m n : Num\nx\u271d : \u2191m \u2223 \u2191n\nk : \u2115\ne : \u2191n = \u2191m * k\n\u22a2 \u2191(\u2191m * k) = m * \u2191k", "state_after": "no goals"}, {"tactic": "simp [e, mul_to_nat]", "annotated_tactic": ["simp [e, <a>mul_to_nat</a>]", [{"full_name": "Num.mul_to_nat", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [315, 9], "def_end_pos": [315, 19]}]], "state_before": "m n : Num\nx\u271d : m \u2223 n\nk : Num\ne : n = m * k\n\u22a2 \u2191n = \u2191m * \u2191k", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/RBMap/WF.lean", "full_name": "Std.RBNode.WF.out", "start": [466, 1], "end": [470, 71], "traced_tactics": [{"tactic": "induction h with\n| mk o h => exact \u27e8o, _, _, h\u27e9\n| insert _ ih => have \u27e8o, _, _, h\u27e9 := ih; exact \u27e8o.insert, h.insert\u27e9\n| erase _ ih => have \u27e8o, _, _, h\u27e9 := ih; exact \u27e8o.erase, _, h.erase\u27e9", "annotated_tactic": ["induction h with\n  | <a>mk</a> o h => exact \u27e8o, _, _, h\u27e9\n  | <a>insert</a> _ ih => have \u27e8o, _, _, h\u27e9 := ih; exact \u27e8o.insert, h.insert\u27e9\n  | <a>erase</a> _ ih => have \u27e8o, _, _, h\u27e9 := ih; exact \u27e8o.erase, _, h.erase\u27e9", [{"full_name": "Std.RBNode.WF.mk", "def_path": "lake-packages/std/Std/Data/RBMap/Basic.lean", "def_pos": [588, 5], "def_end_pos": [588, 7]}, {"full_name": "Std.RBNode.WF.insert", "def_path": "lake-packages/std/Std/Data/RBMap/Basic.lean", "def_pos": [591, 5], "def_end_pos": [591, 11]}, {"full_name": "Std.RBNode.WF.erase", "def_path": "lake-packages/std/Std/Data/RBMap/Basic.lean", "def_pos": [594, 5], "def_end_pos": [594, 10]}]], "state_before": "\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nt : RBNode \u03b1\nh : WF cmp t\n\u22a2 Ordered cmp t \u2227 \u2203 c n, Balanced t c n", "state_after": "no goals"}, {"tactic": "exact \u27e8o, _, _, h\u27e9", "annotated_tactic": ["exact \u27e8o, _, _, h\u27e9", []], "state_before": "case mk\n\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nt t\u271d : RBNode \u03b1\nc\u271d : RBColor\nn\u271d : Nat\no : Ordered cmp t\u271d\nh : Balanced t\u271d c\u271d n\u271d\n\u22a2 Ordered cmp t\u271d \u2227 \u2203 c n, Balanced t\u271d c n", "state_after": "no goals"}, {"tactic": "have \u27e8o, _, _, h\u27e9 := ih", "annotated_tactic": ["have \u27e8o, _, _, h\u27e9 := ih", []], "state_before": "case insert\n\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nt t\u271d : RBNode \u03b1\na\u271d\u00b9 : \u03b1\na\u271d : WF cmp t\u271d\nih : Ordered cmp t\u271d \u2227 \u2203 c n, Balanced t\u271d c n\n\u22a2 Ordered cmp (RBNode.insert cmp t\u271d a\u271d\u00b9) \u2227 \u2203 c n, Balanced (RBNode.insert cmp t\u271d a\u271d\u00b9) c n", "state_after": "case insert\n\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nt t\u271d : RBNode \u03b1\na\u271d\u00b9 : \u03b1\na\u271d : WF cmp t\u271d\nih : Ordered cmp t\u271d \u2227 \u2203 c n, Balanced t\u271d c n\no : Ordered cmp t\u271d\nw\u271d\u00b9 : RBColor\nw\u271d : Nat\nh : Balanced t\u271d w\u271d\u00b9 w\u271d\n\u22a2 Ordered cmp (RBNode.insert cmp t\u271d a\u271d\u00b9) \u2227 \u2203 c n, Balanced (RBNode.insert cmp t\u271d a\u271d\u00b9) c n"}, {"tactic": "exact \u27e8o.insert, h.insert\u27e9", "annotated_tactic": ["exact \u27e8o.insert, h.insert\u27e9", []], "state_before": "case insert\n\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nt t\u271d : RBNode \u03b1\na\u271d\u00b9 : \u03b1\na\u271d : WF cmp t\u271d\nih : Ordered cmp t\u271d \u2227 \u2203 c n, Balanced t\u271d c n\no : Ordered cmp t\u271d\nw\u271d\u00b9 : RBColor\nw\u271d : Nat\nh : Balanced t\u271d w\u271d\u00b9 w\u271d\n\u22a2 Ordered cmp (RBNode.insert cmp t\u271d a\u271d\u00b9) \u2227 \u2203 c n, Balanced (RBNode.insert cmp t\u271d a\u271d\u00b9) c n", "state_after": "no goals"}, {"tactic": "have \u27e8o, _, _, h\u27e9 := ih", "annotated_tactic": ["have \u27e8o, _, _, h\u27e9 := ih", []], "state_before": "case erase\n\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nt t\u271d : RBNode \u03b1\ncut\u271d : \u03b1 \u2192 Ordering\na\u271d : WF cmp t\u271d\nih : Ordered cmp t\u271d \u2227 \u2203 c n, Balanced t\u271d c n\n\u22a2 Ordered cmp (RBNode.erase cut\u271d t\u271d) \u2227 \u2203 c n, Balanced (RBNode.erase cut\u271d t\u271d) c n", "state_after": "case erase\n\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nt t\u271d : RBNode \u03b1\ncut\u271d : \u03b1 \u2192 Ordering\na\u271d : WF cmp t\u271d\nih : Ordered cmp t\u271d \u2227 \u2203 c n, Balanced t\u271d c n\no : Ordered cmp t\u271d\nw\u271d\u00b9 : RBColor\nw\u271d : Nat\nh : Balanced t\u271d w\u271d\u00b9 w\u271d\n\u22a2 Ordered cmp (RBNode.erase cut\u271d t\u271d) \u2227 \u2203 c n, Balanced (RBNode.erase cut\u271d t\u271d) c n"}, {"tactic": "exact \u27e8o.erase, _, h.erase\u27e9", "annotated_tactic": ["exact \u27e8o.erase, _, h.erase\u27e9", []], "state_before": "case erase\n\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nt t\u271d : RBNode \u03b1\ncut\u271d : \u03b1 \u2192 Ordering\na\u271d : WF cmp t\u271d\nih : Ordered cmp t\u271d \u2227 \u2203 c n, Balanced t\u271d c n\no : Ordered cmp t\u271d\nw\u271d\u00b9 : RBColor\nw\u271d : Nat\nh : Balanced t\u271d w\u271d\u00b9 w\u271d\n\u22a2 Ordered cmp (RBNode.erase cut\u271d t\u271d) \u2227 \u2203 c n, Balanced (RBNode.erase cut\u271d t\u271d) c n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean", "full_name": "MeasureTheory.condexpL1Clm_of_aestronglyMeasurable'", "start": [507, 1], "end": [509, 53], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "full_name": "MeasureTheory.lintegral_sub_le", "start": [948, 1], "end": [950, 40], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "full_name": "Int.eq_one_of_mul_eq_one_right", "start": [854, 1], "end": [855, 35], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "full_name": "intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm", "start": [1062, 1], "end": [1065, 60], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Constructions/Prod/Integral.lean", "full_name": "measurableSet_integrable", "start": [64, 1], "end": [67, 87], "traced_tactics": [{"tactic": "simp_rw [Integrable, hf.of_uncurry_left.aestronglyMeasurable, true_and_iff]", "annotated_tactic": ["simp_rw [<a>Integrable</a>, hf.of_uncurry_left.aestronglyMeasurable, <a>true_and_iff</a>]", [{"full_name": "MeasureTheory.Integrable", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [442, 5], "def_end_pos": [442, 15]}, {"full_name": "true_and_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [147, 9], "def_end_pos": [147, 21]}]], "state_before": "\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\n\u22a2 MeasurableSet {x | Integrable (f x)}", "state_after": "\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\n\u22a2 MeasurableSet {x | HasFiniteIntegral (f x)}"}, {"tactic": "exact measurableSet_lt (Measurable.lintegral_prod_right hf.ennnorm) measurable_const", "annotated_tactic": ["exact <a>measurableSet_lt</a> (<a>Measurable.lintegral_prod_right</a> hf.ennnorm) <a>measurable_const</a>", [{"full_name": "measurableSet_lt", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [616, 9], "def_end_pos": [616, 25]}, {"full_name": "Measurable.lintegral_prod_right", "def_path": "Mathlib/MeasureTheory/Constructions/Prod/Basic.lean", "def_pos": [263, 9], "def_end_pos": [263, 40]}, {"full_name": "measurable_const", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [570, 9], "def_end_pos": [570, 25]}]], "state_before": "\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\n\u22a2 MeasurableSet {x | HasFiniteIntegral (f x)}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/CircleIntegral.lean", "full_name": "circleIntegral.integral_sub_inv_of_mem_ball", "start": [645, 1], "end": [661, 30], "traced_tactics": [{"tactic": "have hR : 0 < R := dist_nonneg.trans_lt hw", "annotated_tactic": ["have hR : 0 < R := dist_nonneg.trans_lt hw", []], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nc w : \u2102\nR : \u211d\nhw : w \u2208 ball c R\n\u22a2 (\u222e (z : \u2102) in C(c, R), (z - w)\u207b\u00b9) = 2 * \u2191\u03c0 * I", "state_after": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nc w : \u2102\nR : \u211d\nhw : w \u2208 ball c R\nhR : 0 < R\n\u22a2 (\u222e (z : \u2102) in C(c, R), (z - w)\u207b\u00b9) = 2 * \u2191\u03c0 * I"}, {"tactic": "suffices H : HasSum (fun n : \u2115 => \u222e z in C(c, R), ((w - c) / (z - c)) ^ n * (z - c)\u207b\u00b9) (2 * \u03c0 * I)", "annotated_tactic": ["suffices H : <a>HasSum</a> (fun n : \u2115 => \u222e z in C(c, R), ((w - c) / (z - c)) ^ n * (z - c)\u207b\u00b9) (2 * \u03c0 * <a>I</a>)", [{"full_name": "HasSum", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/Basic.lean", "def_pos": [57, 5], "def_end_pos": [57, 11]}, {"full_name": "Complex.I", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [293, 5], "def_end_pos": [293, 6]}]], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nc w : \u2102\nR : \u211d\nhw : w \u2208 ball c R\nhR : 0 < R\n\u22a2 (\u222e (z : \u2102) in C(c, R), (z - w)\u207b\u00b9) = 2 * \u2191\u03c0 * I", "state_after": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nc w : \u2102\nR : \u211d\nhw : w \u2208 ball c R\nhR : 0 < R\nH : HasSum (fun n => \u222e (z : \u2102) in C(c, R), ((w - c) / (z - c)) ^ n * (z - c)\u207b\u00b9) (2 * \u2191\u03c0 * I)\n\u22a2 (\u222e (z : \u2102) in C(c, R), (z - w)\u207b\u00b9) = 2 * \u2191\u03c0 * I\n\ncase H\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nc w : \u2102\nR : \u211d\nhw : w \u2208 ball c R\nhR : 0 < R\n\u22a2 HasSum (fun n => \u222e (z : \u2102) in C(c, R), ((w - c) / (z - c)) ^ n * (z - c)\u207b\u00b9) (2 * \u2191\u03c0 * I)"}, {"tactic": "have H : \u2200 n : \u2115, n \u2260 0 \u2192 (\u222e z in C(c, R), (z - c) ^ (-n - 1 : \u2124)) = 0 := by\n  refine' fun n hn => integral_sub_zpow_of_ne _ _ _ _; simpa", "annotated_tactic": ["have H : \u2200 n : \u2115, n \u2260 0 \u2192 (\u222e z in C(c, R), (z - c) ^ (-n - 1 : \u2124)) = 0 := by\n    refine' fun n hn => <a>integral_sub_zpow_of_ne</a> _ _ _ _; simpa", [{"full_name": "circleIntegral.integral_sub_zpow_of_ne", "def_path": "Mathlib/MeasureTheory/Integral/CircleIntegral.lean", "def_pos": [499, 9], "def_end_pos": [499, 32]}]], "state_before": "case H\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nc w : \u2102\nR : \u211d\nhw : w \u2208 ball c R\nhR : 0 < R\n\u22a2 HasSum (fun n => \u222e (z : \u2102) in C(c, R), ((w - c) / (z - c)) ^ n * (z - c)\u207b\u00b9) (2 * \u2191\u03c0 * I)", "state_after": "case H\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nc w : \u2102\nR : \u211d\nhw : w \u2208 ball c R\nhR : 0 < R\nH : \u2200 (n : \u2115), n \u2260 0 \u2192 (\u222e (z : \u2102) in C(c, R), (z - c) ^ (-\u2191n - 1)) = 0\n\u22a2 HasSum (fun n => \u222e (z : \u2102) in C(c, R), ((w - c) / (z - c)) ^ n * (z - c)\u207b\u00b9) (2 * \u2191\u03c0 * I)"}, {"tactic": "have : (\u222e z in C(c, R), ((w - c) / (z - c)) ^ 0 * (z - c)\u207b\u00b9) = 2 * \u03c0 * I := by simp [hR.ne']", "annotated_tactic": ["have : (\u222e z in C(c, R), ((w - c) / (z - c)) ^ 0 * (z - c)\u207b\u00b9) = 2 * \u03c0 * <a>I</a> := by simp [hR.ne']", [{"full_name": "Complex.I", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [293, 5], "def_end_pos": [293, 6]}]], "state_before": "case H\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nc w : \u2102\nR : \u211d\nhw : w \u2208 ball c R\nhR : 0 < R\nH : \u2200 (n : \u2115), n \u2260 0 \u2192 (\u222e (z : \u2102) in C(c, R), (z - c) ^ (-\u2191n - 1)) = 0\n\u22a2 HasSum (fun n => \u222e (z : \u2102) in C(c, R), ((w - c) / (z - c)) ^ n * (z - c)\u207b\u00b9) (2 * \u2191\u03c0 * I)", "state_after": "case H\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nc w : \u2102\nR : \u211d\nhw : w \u2208 ball c R\nhR : 0 < R\nH : \u2200 (n : \u2115), n \u2260 0 \u2192 (\u222e (z : \u2102) in C(c, R), (z - c) ^ (-\u2191n - 1)) = 0\nthis : (\u222e (z : \u2102) in C(c, R), ((w - c) / (z - c)) ^ 0 * (z - c)\u207b\u00b9) = 2 * \u2191\u03c0 * I\n\u22a2 HasSum (fun n => \u222e (z : \u2102) in C(c, R), ((w - c) / (z - c)) ^ n * (z - c)\u207b\u00b9) (2 * \u2191\u03c0 * I)"}, {"tactic": "refine' this \u25b8 hasSum_single _ fun n hn => _", "annotated_tactic": ["refine' this \u25b8 <a>hasSum_single</a> _ fun n hn => _", [{"full_name": "hasSum_single", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/Basic.lean", "def_pos": [213, 9], "def_end_pos": [213, 22]}]], "state_before": "case H\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nc w : \u2102\nR : \u211d\nhw : w \u2208 ball c R\nhR : 0 < R\nH : \u2200 (n : \u2115), n \u2260 0 \u2192 (\u222e (z : \u2102) in C(c, R), (z - c) ^ (-\u2191n - 1)) = 0\nthis : (\u222e (z : \u2102) in C(c, R), ((w - c) / (z - c)) ^ 0 * (z - c)\u207b\u00b9) = 2 * \u2191\u03c0 * I\n\u22a2 HasSum (fun n => \u222e (z : \u2102) in C(c, R), ((w - c) / (z - c)) ^ n * (z - c)\u207b\u00b9) (2 * \u2191\u03c0 * I)", "state_after": "case H\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nc w : \u2102\nR : \u211d\nhw : w \u2208 ball c R\nhR : 0 < R\nH : \u2200 (n : \u2115), n \u2260 0 \u2192 (\u222e (z : \u2102) in C(c, R), (z - c) ^ (-\u2191n - 1)) = 0\nthis : (\u222e (z : \u2102) in C(c, R), ((w - c) / (z - c)) ^ 0 * (z - c)\u207b\u00b9) = 2 * \u2191\u03c0 * I\nn : \u2115\nhn : n \u2260 0\n\u22a2 (\u222e (z : \u2102) in C(c, R), ((w - c) / (z - c)) ^ n * (z - c)\u207b\u00b9) = 0"}, {"tactic": "simp only [div_eq_mul_inv, mul_pow, integral_const_mul, mul_assoc]", "annotated_tactic": ["simp only [<a>div_eq_mul_inv</a>, <a>mul_pow</a>, <a>integral_const_mul</a>, <a>mul_assoc</a>]", [{"full_name": "div_eq_mul_inv", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [977, 9], "def_end_pos": [977, 23]}, {"full_name": "mul_pow", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [257, 9], "def_end_pos": [257, 16]}, {"full_name": "circleIntegral.integral_const_mul", "def_path": "Mathlib/MeasureTheory/Integral/CircleIntegral.lean", "def_pos": [454, 9], "def_end_pos": [454, 27]}, {"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [264, 9], "def_end_pos": [264, 18]}]], "state_before": "case H\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nc w : \u2102\nR : \u211d\nhw : w \u2208 ball c R\nhR : 0 < R\nH : \u2200 (n : \u2115), n \u2260 0 \u2192 (\u222e (z : \u2102) in C(c, R), (z - c) ^ (-\u2191n - 1)) = 0\nthis : (\u222e (z : \u2102) in C(c, R), ((w - c) / (z - c)) ^ 0 * (z - c)\u207b\u00b9) = 2 * \u2191\u03c0 * I\nn : \u2115\nhn : n \u2260 0\n\u22a2 (\u222e (z : \u2102) in C(c, R), ((w - c) / (z - c)) ^ n * (z - c)\u207b\u00b9) = 0", "state_after": "case H\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nc w : \u2102\nR : \u211d\nhw : w \u2208 ball c R\nhR : 0 < R\nH : \u2200 (n : \u2115), n \u2260 0 \u2192 (\u222e (z : \u2102) in C(c, R), (z - c) ^ (-\u2191n - 1)) = 0\nthis : (\u222e (z : \u2102) in C(c, R), ((w - c) / (z - c)) ^ 0 * (z - c)\u207b\u00b9) = 2 * \u2191\u03c0 * I\nn : \u2115\nhn : n \u2260 0\n\u22a2 ((w - c) ^ n * \u222e (z : \u2102) in C(c, R), (z - c)\u207b\u00b9 ^ n * (z - c)\u207b\u00b9) = 0"}, {"tactic": "rw [(integral_congr hR.le fun z hz => _).trans (H n hn), mul_zero]", "annotated_tactic": ["rw [(<a>integral_congr</a> hR.le fun z hz => _).<a>trans</a> (H n hn), <a>mul_zero</a>]", [{"full_name": "circleIntegral.integral_congr", "def_path": "Mathlib/MeasureTheory/Integral/CircleIntegral.lean", "def_pos": [363, 9], "def_end_pos": [363, 23]}, {"full_name": "Eq.trans", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [322, 9], "def_end_pos": [322, 17]}, {"full_name": "MulZeroClass.mul_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [38, 3], "def_end_pos": [38, 11]}]], "state_before": "case H\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nc w : \u2102\nR : \u211d\nhw : w \u2208 ball c R\nhR : 0 < R\nH : \u2200 (n : \u2115), n \u2260 0 \u2192 (\u222e (z : \u2102) in C(c, R), (z - c) ^ (-\u2191n - 1)) = 0\nthis : (\u222e (z : \u2102) in C(c, R), ((w - c) / (z - c)) ^ 0 * (z - c)\u207b\u00b9) = 2 * \u2191\u03c0 * I\nn : \u2115\nhn : n \u2260 0\n\u22a2 ((w - c) ^ n * \u222e (z : \u2102) in C(c, R), (z - c)\u207b\u00b9 ^ n * (z - c)\u207b\u00b9) = 0", "state_after": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nc w : \u2102\nR : \u211d\nhw : w \u2208 ball c R\nhR : 0 < R\nH : \u2200 (n : \u2115), n \u2260 0 \u2192 (\u222e (z : \u2102) in C(c, R), (z - c) ^ (-\u2191n - 1)) = 0\nthis : (\u222e (z : \u2102) in C(c, R), ((w - c) / (z - c)) ^ 0 * (z - c)\u207b\u00b9) = 2 * \u2191\u03c0 * I\nn : \u2115\nhn : n \u2260 0\n\u22a2 \u2200 (z : \u2102), z \u2208 sphere c R \u2192 (z - c)\u207b\u00b9 ^ n * (z - c)\u207b\u00b9 = (z - c) ^ (-\u2191n - 1)"}, {"tactic": "intro z _", "annotated_tactic": ["intro z _", []], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nc w : \u2102\nR : \u211d\nhw : w \u2208 ball c R\nhR : 0 < R\nH : \u2200 (n : \u2115), n \u2260 0 \u2192 (\u222e (z : \u2102) in C(c, R), (z - c) ^ (-\u2191n - 1)) = 0\nthis : (\u222e (z : \u2102) in C(c, R), ((w - c) / (z - c)) ^ 0 * (z - c)\u207b\u00b9) = 2 * \u2191\u03c0 * I\nn : \u2115\nhn : n \u2260 0\n\u22a2 \u2200 (z : \u2102), z \u2208 sphere c R \u2192 (z - c)\u207b\u00b9 ^ n * (z - c)\u207b\u00b9 = (z - c) ^ (-\u2191n - 1)", "state_after": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nc w : \u2102\nR : \u211d\nhw : w \u2208 ball c R\nhR : 0 < R\nH : \u2200 (n : \u2115), n \u2260 0 \u2192 (\u222e (z : \u2102) in C(c, R), (z - c) ^ (-\u2191n - 1)) = 0\nthis : (\u222e (z : \u2102) in C(c, R), ((w - c) / (z - c)) ^ 0 * (z - c)\u207b\u00b9) = 2 * \u2191\u03c0 * I\nn : \u2115\nhn : n \u2260 0\nz : \u2102\nhz\u271d : z \u2208 sphere c R\n\u22a2 (z - c)\u207b\u00b9 ^ n * (z - c)\u207b\u00b9 = (z - c) ^ (-\u2191n - 1)"}, {"tactic": "rw [\u2190 pow_succ', \u2190 zpow_ofNat, inv_zpow, \u2190 zpow_neg, Int.ofNat_succ, neg_add,\n  sub_eq_add_neg _ (1 : \u2124)]", "annotated_tactic": ["rw [\u2190 <a>pow_succ'</a>, \u2190 <a>zpow_ofNat</a>, <a>inv_zpow</a>, \u2190 <a>zpow_neg</a>, <a>Int.ofNat_succ</a>, <a>neg_add</a>,\n    <a>sub_eq_add_neg</a> _ (1 : \u2124)]", [{"full_name": "pow_succ'", "def_path": "Mathlib/Algebra/Group/Commute/Defs.lean", "def_pos": [213, 9], "def_end_pos": [213, 25]}, {"full_name": "zpow_ofNat", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [948, 9], "def_end_pos": [948, 19]}, {"full_name": "inv_zpow", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [350, 9], "def_end_pos": [350, 17]}, {"full_name": "zpow_neg", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [332, 9], "def_end_pos": [332, 17]}, {"full_name": "Int.ofNat_succ", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [37, 9], "def_end_pos": [37, 19]}, {"full_name": "neg_add", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [481, 15], "def_end_pos": [481, 22]}, {"full_name": "sub_eq_add_neg", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [975, 3], "def_end_pos": [975, 14]}]], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nc w : \u2102\nR : \u211d\nhw : w \u2208 ball c R\nhR : 0 < R\nH : \u2200 (n : \u2115), n \u2260 0 \u2192 (\u222e (z : \u2102) in C(c, R), (z - c) ^ (-\u2191n - 1)) = 0\nthis : (\u222e (z : \u2102) in C(c, R), ((w - c) / (z - c)) ^ 0 * (z - c)\u207b\u00b9) = 2 * \u2191\u03c0 * I\nn : \u2115\nhn : n \u2260 0\nz : \u2102\nhz\u271d : z \u2208 sphere c R\n\u22a2 (z - c)\u207b\u00b9 ^ n * (z - c)\u207b\u00b9 = (z - c) ^ (-\u2191n - 1)", "state_after": "no goals"}, {"tactic": "have A : CircleIntegrable (fun _ => (1 : \u2102)) c R := continuousOn_const.circleIntegrable'", "annotated_tactic": ["have A : <a>CircleIntegrable</a> (fun _ => (1 : \u2102)) c R := continuousOn_const.circleIntegrable'", [{"full_name": "CircleIntegrable", "def_path": "Mathlib/MeasureTheory/Integral/CircleIntegral.lean", "def_pos": [233, 5], "def_end_pos": [233, 21]}]], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nc w : \u2102\nR : \u211d\nhw : w \u2208 ball c R\nhR : 0 < R\nH : HasSum (fun n => \u222e (z : \u2102) in C(c, R), ((w - c) / (z - c)) ^ n * (z - c)\u207b\u00b9) (2 * \u2191\u03c0 * I)\n\u22a2 (\u222e (z : \u2102) in C(c, R), (z - w)\u207b\u00b9) = 2 * \u2191\u03c0 * I", "state_after": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nc w : \u2102\nR : \u211d\nhw : w \u2208 ball c R\nhR : 0 < R\nH : HasSum (fun n => \u222e (z : \u2102) in C(c, R), ((w - c) / (z - c)) ^ n * (z - c)\u207b\u00b9) (2 * \u2191\u03c0 * I)\nA : CircleIntegrable (fun x => 1) c R\n\u22a2 (\u222e (z : \u2102) in C(c, R), (z - w)\u207b\u00b9) = 2 * \u2191\u03c0 * I"}, {"tactic": "refine' (H.unique _).symm", "annotated_tactic": ["refine' (H.unique _).<a>symm</a>", [{"full_name": "Eq.symm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [310, 9], "def_end_pos": [310, 16]}]], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nc w : \u2102\nR : \u211d\nhw : w \u2208 ball c R\nhR : 0 < R\nH : HasSum (fun n => \u222e (z : \u2102) in C(c, R), ((w - c) / (z - c)) ^ n * (z - c)\u207b\u00b9) (2 * \u2191\u03c0 * I)\nA : CircleIntegrable (fun x => 1) c R\n\u22a2 (\u222e (z : \u2102) in C(c, R), (z - w)\u207b\u00b9) = 2 * \u2191\u03c0 * I", "state_after": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nc w : \u2102\nR : \u211d\nhw : w \u2208 ball c R\nhR : 0 < R\nH : HasSum (fun n => \u222e (z : \u2102) in C(c, R), ((w - c) / (z - c)) ^ n * (z - c)\u207b\u00b9) (2 * \u2191\u03c0 * I)\nA : CircleIntegrable (fun x => 1) c R\n\u22a2 HasSum (fun n => \u222e (z : \u2102) in C(c, R), ((w - c) / (z - c)) ^ n * (z - c)\u207b\u00b9) (\u222e (z : \u2102) in C(c, R), (z - w)\u207b\u00b9)"}, {"tactic": "simpa only [smul_eq_mul, mul_one, add_sub_cancel'_right] using\n  hasSum_two_pi_I_cauchyPowerSeries_integral A hw", "annotated_tactic": ["simpa only [<a>smul_eq_mul</a>, <a>mul_one</a>, <a>add_sub_cancel'_right</a>] using\n      <a>hasSum_two_pi_I_cauchyPowerSeries_integral</a> A hw", [{"full_name": "smul_eq_mul", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [93, 9], "def_end_pos": [93, 20]}, {"full_name": "mul_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [470, 9], "def_end_pos": [470, 16]}, {"full_name": "add_sub_cancel'_right", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [953, 3], "def_end_pos": [953, 14]}, {"full_name": "hasSum_two_pi_I_cauchyPowerSeries_integral", "def_path": "Mathlib/MeasureTheory/Integral/CircleIntegral.lean", "def_pos": [582, 9], "def_end_pos": [582, 51]}]], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nc w : \u2102\nR : \u211d\nhw : w \u2208 ball c R\nhR : 0 < R\nH : HasSum (fun n => \u222e (z : \u2102) in C(c, R), ((w - c) / (z - c)) ^ n * (z - c)\u207b\u00b9) (2 * \u2191\u03c0 * I)\nA : CircleIntegrable (fun x => 1) c R\n\u22a2 HasSum (fun n => \u222e (z : \u2102) in C(c, R), ((w - c) / (z - c)) ^ n * (z - c)\u207b\u00b9) (\u222e (z : \u2102) in C(c, R), (z - w)\u207b\u00b9)", "state_after": "no goals"}, {"tactic": "refine' fun n hn => integral_sub_zpow_of_ne _ _ _ _", "annotated_tactic": ["refine' fun n hn => <a>integral_sub_zpow_of_ne</a> _ _ _ _", [{"full_name": "circleIntegral.integral_sub_zpow_of_ne", "def_path": "Mathlib/MeasureTheory/Integral/CircleIntegral.lean", "def_pos": [499, 9], "def_end_pos": [499, 32]}]], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nc w : \u2102\nR : \u211d\nhw : w \u2208 ball c R\nhR : 0 < R\n\u22a2 \u2200 (n : \u2115), n \u2260 0 \u2192 (\u222e (z : \u2102) in C(c, R), (z - c) ^ (-\u2191n - 1)) = 0", "state_after": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nc w : \u2102\nR : \u211d\nhw : w \u2208 ball c R\nhR : 0 < R\nn : \u2115\nhn : n \u2260 0\n\u22a2 -\u2191n - 1 \u2260 -1"}, {"tactic": "simpa", "annotated_tactic": ["simpa", []], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nc w : \u2102\nR : \u211d\nhw : w \u2208 ball c R\nhR : 0 < R\nn : \u2115\nhn : n \u2260 0\n\u22a2 -\u2191n - 1 \u2260 -1", "state_after": "no goals"}, {"tactic": "simp [hR.ne']", "annotated_tactic": ["simp [hR.ne']", []], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nc w : \u2102\nR : \u211d\nhw : w \u2208 ball c R\nhR : 0 < R\nH : \u2200 (n : \u2115), n \u2260 0 \u2192 (\u222e (z : \u2102) in C(c, R), (z - c) ^ (-\u2191n - 1)) = 0\n\u22a2 (\u222e (z : \u2102) in C(c, R), ((w - c) / (z - c)) ^ 0 * (z - c)\u207b\u00b9) = 2 * \u2191\u03c0 * I", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Constructions/Pi.lean", "full_name": "MeasureTheory.Measure.univ_pi_Ioi_ae_eq_Ici", "start": [500, 1], "end": [502, 48], "traced_tactics": [{"tactic": "rw [\u2190 pi_univ_Ici]", "annotated_tactic": ["rw [\u2190 <a>pi_univ_Ici</a>]", [{"full_name": "Set.pi_univ_Ici", "def_path": "Mathlib/Data/Set/Intervals/Pi.lean", "def_pos": [33, 9], "def_end_pos": [33, 20]}]], "state_before": "\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : \u03b9 \u2192 Type u_3\ninst\u271d\u2074 : Fintype \u03b9\nm : (i : \u03b9) \u2192 OuterMeasure (\u03b1 i)\ninst\u271d\u00b3 : (i : \u03b9) \u2192 MeasurableSpace (\u03b1 i)\n\u03bc : (i : \u03b9) \u2192 Measure (\u03b1 i)\ninst\u271d\u00b2 : \u2200 (i : \u03b9), SigmaFinite (\u03bc i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 PartialOrder (\u03b1 i)\ninst\u271d : \u2200 (i : \u03b9), NoAtoms (\u03bc i)\nf : (i : \u03b9) \u2192 \u03b1 i\n\u22a2 (Set.pi univ fun i => Ioi (f i)) =\u1da0[ae (Measure.pi \u03bc)] Ici f", "state_after": "\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : \u03b9 \u2192 Type u_3\ninst\u271d\u2074 : Fintype \u03b9\nm : (i : \u03b9) \u2192 OuterMeasure (\u03b1 i)\ninst\u271d\u00b3 : (i : \u03b9) \u2192 MeasurableSpace (\u03b1 i)\n\u03bc : (i : \u03b9) \u2192 Measure (\u03b1 i)\ninst\u271d\u00b2 : \u2200 (i : \u03b9), SigmaFinite (\u03bc i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 PartialOrder (\u03b1 i)\ninst\u271d : \u2200 (i : \u03b9), NoAtoms (\u03bc i)\nf : (i : \u03b9) \u2192 \u03b1 i\n\u22a2 (Set.pi univ fun i => Ioi (f i)) =\u1da0[ae (Measure.pi \u03bc)] Set.pi univ fun i => Ici (f i)"}, {"tactic": "exact pi_Ioi_ae_eq_pi_Ici", "annotated_tactic": ["exact <a>pi_Ioi_ae_eq_pi_Ici</a>", [{"full_name": "MeasureTheory.Measure.pi_Ioi_ae_eq_pi_Ici", "def_path": "Mathlib/MeasureTheory/Constructions/Pi.lean", "def_pos": [490, 9], "def_end_pos": [490, 28]}]], "state_before": "\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : \u03b9 \u2192 Type u_3\ninst\u271d\u2074 : Fintype \u03b9\nm : (i : \u03b9) \u2192 OuterMeasure (\u03b1 i)\ninst\u271d\u00b3 : (i : \u03b9) \u2192 MeasurableSpace (\u03b1 i)\n\u03bc : (i : \u03b9) \u2192 Measure (\u03b1 i)\ninst\u271d\u00b2 : \u2200 (i : \u03b9), SigmaFinite (\u03bc i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 PartialOrder (\u03b1 i)\ninst\u271d : \u2200 (i : \u03b9), NoAtoms (\u03bc i)\nf : (i : \u03b9) \u2192 \u03b1 i\n\u22a2 (Set.pi univ fun i => Ioi (f i)) =\u1da0[ae (Measure.pi \u03bc)] Set.pi univ fun i => Ici (f i)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Setoid/Partition.lean", "full_name": "Setoid.exists_of_mem_partition", "start": [235, 1], "end": [238, 38], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "Finset.sdiff_eq_self_of_disjoint", "start": [2407, 1], "end": [2408, 33], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Unique.lean", "full_name": "MeasureTheory.lintegral_nnnorm_le_of_forall_fin_meas_integral_eq", "start": [203, 1], "end": [211, 49], "traced_tactics": [{"tactic": "rw [\u2190 ofReal_integral_norm_eq_lintegral_nnnorm hfi, \u2190\n  ofReal_integral_norm_eq_lintegral_nnnorm hgi, ENNReal.ofReal_le_ofReal_iff]", "annotated_tactic": ["rw [\u2190 <a>ofReal_integral_norm_eq_lintegral_nnnorm</a> hfi, \u2190\n    <a>ofReal_integral_norm_eq_lintegral_nnnorm</a> hgi, <a>ENNReal.ofReal_le_ofReal_iff</a>]", [{"full_name": "MeasureTheory.ofReal_integral_norm_eq_lintegral_nnnorm", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1188, 9], "def_end_pos": [1188, 49]}, {"full_name": "MeasureTheory.ofReal_integral_norm_eq_lintegral_nnnorm", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1188, 9], "def_end_pos": [1188, 49]}, {"full_name": "ENNReal.ofReal_le_ofReal_iff", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2145, 9], "def_end_pos": [2145, 29]}]], "state_before": "\u03b1 : Type u_1\nE' : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2075 : CompleteSpace E'\ninst\u271d\u2074 : NormedSpace \u211d E'\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\ns : Set \u03b1\nhm : m \u2264 m0\nf g : \u03b1 \u2192 \u211d\nhf : StronglyMeasurable f\nhfi : IntegrableOn f s\nhg : StronglyMeasurable g\nhgi : IntegrableOn g s\nhgf : \u2200 (t : Set \u03b1), MeasurableSet t \u2192 \u2191\u2191\u03bc t < \u22a4 \u2192 \u222b (x : \u03b1) in t, g x \u2202\u03bc = \u222b (x : \u03b1) in t, f x \u2202\u03bc\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\n\u22a2 \u222b\u207b (x : \u03b1) in s, \u2191\u2016g x\u2016\u208a \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, \u2191\u2016f x\u2016\u208a \u2202\u03bc", "state_after": "\u03b1 : Type u_1\nE' : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2075 : CompleteSpace E'\ninst\u271d\u2074 : NormedSpace \u211d E'\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\ns : Set \u03b1\nhm : m \u2264 m0\nf g : \u03b1 \u2192 \u211d\nhf : StronglyMeasurable f\nhfi : IntegrableOn f s\nhg : StronglyMeasurable g\nhgi : IntegrableOn g s\nhgf : \u2200 (t : Set \u03b1), MeasurableSet t \u2192 \u2191\u2191\u03bc t < \u22a4 \u2192 \u222b (x : \u03b1) in t, g x \u2202\u03bc = \u222b (x : \u03b1) in t, f x \u2202\u03bc\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\n\u22a2 \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc \u2264 \u222b (x : \u03b1) in s, \u2016f x\u2016 \u2202\u03bc\n\n\u03b1 : Type u_1\nE' : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2075 : CompleteSpace E'\ninst\u271d\u2074 : NormedSpace \u211d E'\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\ns : Set \u03b1\nhm : m \u2264 m0\nf g : \u03b1 \u2192 \u211d\nhf : StronglyMeasurable f\nhfi : IntegrableOn f s\nhg : StronglyMeasurable g\nhgi : IntegrableOn g s\nhgf : \u2200 (t : Set \u03b1), MeasurableSet t \u2192 \u2191\u2191\u03bc t < \u22a4 \u2192 \u222b (x : \u03b1) in t, g x \u2202\u03bc = \u222b (x : \u03b1) in t, f x \u2202\u03bc\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\n\u22a2 0 \u2264 \u222b (x : \u03b1) in s, \u2016f x\u2016 \u2202\u03bc"}, {"tactic": "exact integral_norm_le_of_forall_fin_meas_integral_eq hm hf hfi hg hgi hgf hs h\u03bcs", "annotated_tactic": ["exact <a>integral_norm_le_of_forall_fin_meas_integral_eq</a> hm hf hfi hg hgi hgf hs h\u03bcs", [{"full_name": "MeasureTheory.integral_norm_le_of_forall_fin_meas_integral_eq", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Unique.lean", "def_pos": [170, 9], "def_end_pos": [170, 56]}]], "state_before": "\u03b1 : Type u_1\nE' : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2075 : CompleteSpace E'\ninst\u271d\u2074 : NormedSpace \u211d E'\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\ns : Set \u03b1\nhm : m \u2264 m0\nf g : \u03b1 \u2192 \u211d\nhf : StronglyMeasurable f\nhfi : IntegrableOn f s\nhg : StronglyMeasurable g\nhgi : IntegrableOn g s\nhgf : \u2200 (t : Set \u03b1), MeasurableSet t \u2192 \u2191\u2191\u03bc t < \u22a4 \u2192 \u222b (x : \u03b1) in t, g x \u2202\u03bc = \u222b (x : \u03b1) in t, f x \u2202\u03bc\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\n\u22a2 \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc \u2264 \u222b (x : \u03b1) in s, \u2016f x\u2016 \u2202\u03bc", "state_after": "no goals"}, {"tactic": "exact integral_nonneg fun x => norm_nonneg _", "annotated_tactic": ["exact <a>integral_nonneg</a> fun x => <a>norm_nonneg</a> _", [{"full_name": "MeasureTheory.integral_nonneg", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1251, 9], "def_end_pos": [1251, 24]}, {"full_name": "norm_nonneg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [500, 30], "def_end_pos": [500, 41]}]], "state_before": "\u03b1 : Type u_1\nE' : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2075 : CompleteSpace E'\ninst\u271d\u2074 : NormedSpace \u211d E'\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\ns : Set \u03b1\nhm : m \u2264 m0\nf g : \u03b1 \u2192 \u211d\nhf : StronglyMeasurable f\nhfi : IntegrableOn f s\nhg : StronglyMeasurable g\nhgi : IntegrableOn g s\nhgf : \u2200 (t : Set \u03b1), MeasurableSet t \u2192 \u2191\u2191\u03bc t < \u22a4 \u2192 \u222b (x : \u03b1) in t, g x \u2202\u03bc = \u222b (x : \u03b1) in t, f x \u2202\u03bc\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\n\u22a2 0 \u2264 \u222b (x : \u03b1) in s, \u2016f x\u2016 \u2202\u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/NAry.lean", "full_name": "Set.image3_mono", "start": [241, 1], "end": [243, 70], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/TuringMachine.lean", "full_name": "Turing.TM0.Machine.map_respects", "start": [1173, 1], "end": [1182, 8], "traced_tactics": [{"tactic": "intro c _ \u27e8cs, rfl\u27e9", "annotated_tactic": ["intro c _ \u27e8cs, <a>rfl</a>\u27e9", [{"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u0393' : Type u_2\ninst\u271d\u00b2 : Inhabited \u0393'\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u039b' : Type u_4\ninst\u271d : Inhabited \u039b'\nM : Machine \u0393 \u039b\nf\u2081 : PointedMap \u0393 \u0393'\nf\u2082 : PointedMap \u0393' \u0393\ng\u2081\u271d : \u039b \u2192 \u039b'\ng\u2082\u271d : \u039b' \u2192 \u039b\ng\u2081 : PointedMap \u039b \u039b'\ng\u2082 : \u039b' \u2192 \u039b\nS : Set \u039b\nss : Supports M S\nf\u2082\u2081 : Function.RightInverse f\u2081.f f\u2082.f\ng\u2082\u2081 : \u2200 (q : \u039b), q \u2208 S \u2192 g\u2082 (PointedMap.f g\u2081 q) = q\n\u22a2 Respects (step M) (step (map M f\u2081 f\u2082 g\u2081.f g\u2082)) fun a b => a.q \u2208 S \u2227 Cfg.map f\u2081 g\u2081.f a = b", "state_after": "\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u0393' : Type u_2\ninst\u271d\u00b2 : Inhabited \u0393'\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u039b' : Type u_4\ninst\u271d : Inhabited \u039b'\nM : Machine \u0393 \u039b\nf\u2081 : PointedMap \u0393 \u0393'\nf\u2082 : PointedMap \u0393' \u0393\ng\u2081\u271d : \u039b \u2192 \u039b'\ng\u2082\u271d : \u039b' \u2192 \u039b\ng\u2081 : PointedMap \u039b \u039b'\ng\u2082 : \u039b' \u2192 \u039b\nS : Set \u039b\nss : Supports M S\nf\u2082\u2081 : Function.RightInverse f\u2081.f f\u2082.f\ng\u2082\u2081 : \u2200 (q : \u039b), q \u2208 S \u2192 g\u2082 (PointedMap.f g\u2081 q) = q\nc : Cfg \u0393 \u039b\na\u2082\u271d : Cfg \u0393' \u039b'\ncs : c.q \u2208 S\n\u22a2 match step M c with\n  | some b\u2081 =>\n    \u2203 b\u2082,\n      (fun a b => a.q \u2208 S \u2227 Cfg.map f\u2081 g\u2081.f a = b) b\u2081 b\u2082 \u2227 Reaches\u2081 (step (map M f\u2081 f\u2082 g\u2081.f g\u2082)) (Cfg.map f\u2081 g\u2081.f c) b\u2082\n  | none => step (map M f\u2081 f\u2082 g\u2081.f g\u2082) (Cfg.map f\u2081 g\u2081.f c) = none"}, {"tactic": "cases e : step M c", "annotated_tactic": ["cases e : <a>step</a> M c", [{"full_name": "Turing.TM0.step", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1076, 5], "def_end_pos": [1076, 9]}]], "state_before": "\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u0393' : Type u_2\ninst\u271d\u00b2 : Inhabited \u0393'\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u039b' : Type u_4\ninst\u271d : Inhabited \u039b'\nM : Machine \u0393 \u039b\nf\u2081 : PointedMap \u0393 \u0393'\nf\u2082 : PointedMap \u0393' \u0393\ng\u2081\u271d : \u039b \u2192 \u039b'\ng\u2082\u271d : \u039b' \u2192 \u039b\ng\u2081 : PointedMap \u039b \u039b'\ng\u2082 : \u039b' \u2192 \u039b\nS : Set \u039b\nss : Supports M S\nf\u2082\u2081 : Function.RightInverse f\u2081.f f\u2082.f\ng\u2082\u2081 : \u2200 (q : \u039b), q \u2208 S \u2192 g\u2082 (PointedMap.f g\u2081 q) = q\nc : Cfg \u0393 \u039b\na\u2082\u271d : Cfg \u0393' \u039b'\ncs : c.q \u2208 S\n\u22a2 match step M c with\n  | some b\u2081 =>\n    \u2203 b\u2082,\n      (fun a b => a.q \u2208 S \u2227 Cfg.map f\u2081 g\u2081.f a = b) b\u2081 b\u2082 \u2227 Reaches\u2081 (step (map M f\u2081 f\u2082 g\u2081.f g\u2082)) (Cfg.map f\u2081 g\u2081.f c) b\u2082\n  | none => step (map M f\u2081 f\u2082 g\u2081.f g\u2082) (Cfg.map f\u2081 g\u2081.f c) = none", "state_after": "case none\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u0393' : Type u_2\ninst\u271d\u00b2 : Inhabited \u0393'\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u039b' : Type u_4\ninst\u271d : Inhabited \u039b'\nM : Machine \u0393 \u039b\nf\u2081 : PointedMap \u0393 \u0393'\nf\u2082 : PointedMap \u0393' \u0393\ng\u2081\u271d : \u039b \u2192 \u039b'\ng\u2082\u271d : \u039b' \u2192 \u039b\ng\u2081 : PointedMap \u039b \u039b'\ng\u2082 : \u039b' \u2192 \u039b\nS : Set \u039b\nss : Supports M S\nf\u2082\u2081 : Function.RightInverse f\u2081.f f\u2082.f\ng\u2082\u2081 : \u2200 (q : \u039b), q \u2208 S \u2192 g\u2082 (PointedMap.f g\u2081 q) = q\nc : Cfg \u0393 \u039b\na\u2082\u271d : Cfg \u0393' \u039b'\ncs : c.q \u2208 S\ne : step M c = none\n\u22a2 match none with\n  | some b\u2081 =>\n    \u2203 b\u2082,\n      (fun a b => a.q \u2208 S \u2227 Cfg.map f\u2081 g\u2081.f a = b) b\u2081 b\u2082 \u2227 Reaches\u2081 (step (map M f\u2081 f\u2082 g\u2081.f g\u2082)) (Cfg.map f\u2081 g\u2081.f c) b\u2082\n  | none => step (map M f\u2081 f\u2082 g\u2081.f g\u2082) (Cfg.map f\u2081 g\u2081.f c) = none\n\ncase some\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u0393' : Type u_2\ninst\u271d\u00b2 : Inhabited \u0393'\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u039b' : Type u_4\ninst\u271d : Inhabited \u039b'\nM : Machine \u0393 \u039b\nf\u2081 : PointedMap \u0393 \u0393'\nf\u2082 : PointedMap \u0393' \u0393\ng\u2081\u271d : \u039b \u2192 \u039b'\ng\u2082\u271d : \u039b' \u2192 \u039b\ng\u2081 : PointedMap \u039b \u039b'\ng\u2082 : \u039b' \u2192 \u039b\nS : Set \u039b\nss : Supports M S\nf\u2082\u2081 : Function.RightInverse f\u2081.f f\u2082.f\ng\u2082\u2081 : \u2200 (q : \u039b), q \u2208 S \u2192 g\u2082 (PointedMap.f g\u2081 q) = q\nc : Cfg \u0393 \u039b\na\u2082\u271d : Cfg \u0393' \u039b'\ncs : c.q \u2208 S\nval\u271d : Cfg \u0393 \u039b\ne : step M c = some val\u271d\n\u22a2 match some val\u271d with\n  | some b\u2081 =>\n    \u2203 b\u2082,\n      (fun a b => a.q \u2208 S \u2227 Cfg.map f\u2081 g\u2081.f a = b) b\u2081 b\u2082 \u2227 Reaches\u2081 (step (map M f\u2081 f\u2082 g\u2081.f g\u2082)) (Cfg.map f\u2081 g\u2081.f c) b\u2082\n  | none => step (map M f\u2081 f\u2082 g\u2081.f g\u2082) (Cfg.map f\u2081 g\u2081.f c) = none"}, {"tactic": "rw [\u2190 M.map_step f\u2081 f\u2082 g\u2081 g\u2082 f\u2082\u2081 g\u2082\u2081 _ cs, e]", "annotated_tactic": ["rw [\u2190 M.map_step f\u2081 f\u2082 g\u2081 g\u2082 f\u2082\u2081 g\u2082\u2081 _ cs, e]", []], "state_before": "case none\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u0393' : Type u_2\ninst\u271d\u00b2 : Inhabited \u0393'\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u039b' : Type u_4\ninst\u271d : Inhabited \u039b'\nM : Machine \u0393 \u039b\nf\u2081 : PointedMap \u0393 \u0393'\nf\u2082 : PointedMap \u0393' \u0393\ng\u2081\u271d : \u039b \u2192 \u039b'\ng\u2082\u271d : \u039b' \u2192 \u039b\ng\u2081 : PointedMap \u039b \u039b'\ng\u2082 : \u039b' \u2192 \u039b\nS : Set \u039b\nss : Supports M S\nf\u2082\u2081 : Function.RightInverse f\u2081.f f\u2082.f\ng\u2082\u2081 : \u2200 (q : \u039b), q \u2208 S \u2192 g\u2082 (PointedMap.f g\u2081 q) = q\nc : Cfg \u0393 \u039b\na\u2082\u271d : Cfg \u0393' \u039b'\ncs : c.q \u2208 S\ne : step M c = none\n\u22a2 match none with\n  | some b\u2081 =>\n    \u2203 b\u2082,\n      (fun a b => a.q \u2208 S \u2227 Cfg.map f\u2081 g\u2081.f a = b) b\u2081 b\u2082 \u2227 Reaches\u2081 (step (map M f\u2081 f\u2082 g\u2081.f g\u2082)) (Cfg.map f\u2081 g\u2081.f c) b\u2082\n  | none => step (map M f\u2081 f\u2082 g\u2081.f g\u2082) (Cfg.map f\u2081 g\u2081.f c) = none", "state_after": "case none\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u0393' : Type u_2\ninst\u271d\u00b2 : Inhabited \u0393'\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u039b' : Type u_4\ninst\u271d : Inhabited \u039b'\nM : Machine \u0393 \u039b\nf\u2081 : PointedMap \u0393 \u0393'\nf\u2082 : PointedMap \u0393' \u0393\ng\u2081\u271d : \u039b \u2192 \u039b'\ng\u2082\u271d : \u039b' \u2192 \u039b\ng\u2081 : PointedMap \u039b \u039b'\ng\u2082 : \u039b' \u2192 \u039b\nS : Set \u039b\nss : Supports M S\nf\u2082\u2081 : Function.RightInverse f\u2081.f f\u2082.f\ng\u2082\u2081 : \u2200 (q : \u039b), q \u2208 S \u2192 g\u2082 (PointedMap.f g\u2081 q) = q\nc : Cfg \u0393 \u039b\na\u2082\u271d : Cfg \u0393' \u039b'\ncs : c.q \u2208 S\ne : step M c = none\n\u22a2 match none with\n  | some b\u2081 =>\n    \u2203 b\u2082,\n      (fun a b => a.q \u2208 S \u2227 Cfg.map f\u2081 g\u2081.f a = b) b\u2081 b\u2082 \u2227 Reaches\u2081 (step (map M f\u2081 f\u2082 g\u2081.f g\u2082)) (Cfg.map f\u2081 g\u2081.f c) b\u2082\n  | none => Option.map (Cfg.map f\u2081 g\u2081.f) none = none"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case none\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u0393' : Type u_2\ninst\u271d\u00b2 : Inhabited \u0393'\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u039b' : Type u_4\ninst\u271d : Inhabited \u039b'\nM : Machine \u0393 \u039b\nf\u2081 : PointedMap \u0393 \u0393'\nf\u2082 : PointedMap \u0393' \u0393\ng\u2081\u271d : \u039b \u2192 \u039b'\ng\u2082\u271d : \u039b' \u2192 \u039b\ng\u2081 : PointedMap \u039b \u039b'\ng\u2082 : \u039b' \u2192 \u039b\nS : Set \u039b\nss : Supports M S\nf\u2082\u2081 : Function.RightInverse f\u2081.f f\u2082.f\ng\u2082\u2081 : \u2200 (q : \u039b), q \u2208 S \u2192 g\u2082 (PointedMap.f g\u2081 q) = q\nc : Cfg \u0393 \u039b\na\u2082\u271d : Cfg \u0393' \u039b'\ncs : c.q \u2208 S\ne : step M c = none\n\u22a2 match none with\n  | some b\u2081 =>\n    \u2203 b\u2082,\n      (fun a b => a.q \u2208 S \u2227 Cfg.map f\u2081 g\u2081.f a = b) b\u2081 b\u2082 \u2227 Reaches\u2081 (step (map M f\u2081 f\u2082 g\u2081.f g\u2082)) (Cfg.map f\u2081 g\u2081.f c) b\u2082\n  | none => Option.map (Cfg.map f\u2081 g\u2081.f) none = none", "state_after": "no goals"}, {"tactic": "refine' \u27e8_, \u27e8step_supports M ss e cs, rfl\u27e9, TransGen.single _\u27e9", "annotated_tactic": ["refine' \u27e8_, \u27e8<a>step_supports</a> M ss e cs, <a>rfl</a>\u27e9, <a>TransGen.single</a> _\u27e9", [{"full_name": "Turing.TM0.step_supports", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1110, 9], "def_end_pos": [1110, 22]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}, {"full_name": "Relation.TransGen.single", "def_path": "Mathlib/Logic/Relation.lean", "def_pos": [242, 5], "def_end_pos": [242, 11]}]], "state_before": "case some\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u0393' : Type u_2\ninst\u271d\u00b2 : Inhabited \u0393'\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u039b' : Type u_4\ninst\u271d : Inhabited \u039b'\nM : Machine \u0393 \u039b\nf\u2081 : PointedMap \u0393 \u0393'\nf\u2082 : PointedMap \u0393' \u0393\ng\u2081\u271d : \u039b \u2192 \u039b'\ng\u2082\u271d : \u039b' \u2192 \u039b\ng\u2081 : PointedMap \u039b \u039b'\ng\u2082 : \u039b' \u2192 \u039b\nS : Set \u039b\nss : Supports M S\nf\u2082\u2081 : Function.RightInverse f\u2081.f f\u2082.f\ng\u2082\u2081 : \u2200 (q : \u039b), q \u2208 S \u2192 g\u2082 (PointedMap.f g\u2081 q) = q\nc : Cfg \u0393 \u039b\na\u2082\u271d : Cfg \u0393' \u039b'\ncs : c.q \u2208 S\nval\u271d : Cfg \u0393 \u039b\ne : step M c = some val\u271d\n\u22a2 match some val\u271d with\n  | some b\u2081 =>\n    \u2203 b\u2082,\n      (fun a b => a.q \u2208 S \u2227 Cfg.map f\u2081 g\u2081.f a = b) b\u2081 b\u2082 \u2227 Reaches\u2081 (step (map M f\u2081 f\u2082 g\u2081.f g\u2082)) (Cfg.map f\u2081 g\u2081.f c) b\u2082\n  | none => step (map M f\u2081 f\u2082 g\u2081.f g\u2082) (Cfg.map f\u2081 g\u2081.f c) = none", "state_after": "case some\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u0393' : Type u_2\ninst\u271d\u00b2 : Inhabited \u0393'\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u039b' : Type u_4\ninst\u271d : Inhabited \u039b'\nM : Machine \u0393 \u039b\nf\u2081 : PointedMap \u0393 \u0393'\nf\u2082 : PointedMap \u0393' \u0393\ng\u2081\u271d : \u039b \u2192 \u039b'\ng\u2082\u271d : \u039b' \u2192 \u039b\ng\u2081 : PointedMap \u039b \u039b'\ng\u2082 : \u039b' \u2192 \u039b\nS : Set \u039b\nss : Supports M S\nf\u2082\u2081 : Function.RightInverse f\u2081.f f\u2082.f\ng\u2082\u2081 : \u2200 (q : \u039b), q \u2208 S \u2192 g\u2082 (PointedMap.f g\u2081 q) = q\nc : Cfg \u0393 \u039b\na\u2082\u271d : Cfg \u0393' \u039b'\ncs : c.q \u2208 S\nval\u271d : Cfg \u0393 \u039b\ne : step M c = some val\u271d\n\u22a2 Cfg.map f\u2081 g\u2081.f val\u271d \u2208 step (map M f\u2081 f\u2082 g\u2081.f g\u2082) (Cfg.map f\u2081 g\u2081.f c)"}, {"tactic": "rw [\u2190 M.map_step f\u2081 f\u2082 g\u2081 g\u2082 f\u2082\u2081 g\u2082\u2081 _ cs, e]", "annotated_tactic": ["rw [\u2190 M.map_step f\u2081 f\u2082 g\u2081 g\u2082 f\u2082\u2081 g\u2082\u2081 _ cs, e]", []], "state_before": "case some\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u0393' : Type u_2\ninst\u271d\u00b2 : Inhabited \u0393'\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u039b' : Type u_4\ninst\u271d : Inhabited \u039b'\nM : Machine \u0393 \u039b\nf\u2081 : PointedMap \u0393 \u0393'\nf\u2082 : PointedMap \u0393' \u0393\ng\u2081\u271d : \u039b \u2192 \u039b'\ng\u2082\u271d : \u039b' \u2192 \u039b\ng\u2081 : PointedMap \u039b \u039b'\ng\u2082 : \u039b' \u2192 \u039b\nS : Set \u039b\nss : Supports M S\nf\u2082\u2081 : Function.RightInverse f\u2081.f f\u2082.f\ng\u2082\u2081 : \u2200 (q : \u039b), q \u2208 S \u2192 g\u2082 (PointedMap.f g\u2081 q) = q\nc : Cfg \u0393 \u039b\na\u2082\u271d : Cfg \u0393' \u039b'\ncs : c.q \u2208 S\nval\u271d : Cfg \u0393 \u039b\ne : step M c = some val\u271d\n\u22a2 Cfg.map f\u2081 g\u2081.f val\u271d \u2208 step (map M f\u2081 f\u2082 g\u2081.f g\u2082) (Cfg.map f\u2081 g\u2081.f c)", "state_after": "case some\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u0393' : Type u_2\ninst\u271d\u00b2 : Inhabited \u0393'\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u039b' : Type u_4\ninst\u271d : Inhabited \u039b'\nM : Machine \u0393 \u039b\nf\u2081 : PointedMap \u0393 \u0393'\nf\u2082 : PointedMap \u0393' \u0393\ng\u2081\u271d : \u039b \u2192 \u039b'\ng\u2082\u271d : \u039b' \u2192 \u039b\ng\u2081 : PointedMap \u039b \u039b'\ng\u2082 : \u039b' \u2192 \u039b\nS : Set \u039b\nss : Supports M S\nf\u2082\u2081 : Function.RightInverse f\u2081.f f\u2082.f\ng\u2082\u2081 : \u2200 (q : \u039b), q \u2208 S \u2192 g\u2082 (PointedMap.f g\u2081 q) = q\nc : Cfg \u0393 \u039b\na\u2082\u271d : Cfg \u0393' \u039b'\ncs : c.q \u2208 S\nval\u271d : Cfg \u0393 \u039b\ne : step M c = some val\u271d\n\u22a2 Cfg.map f\u2081 g\u2081.f val\u271d \u2208 Option.map (Cfg.map f\u2081 g\u2081.f) (some val\u271d)"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case some\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u0393' : Type u_2\ninst\u271d\u00b2 : Inhabited \u0393'\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u039b' : Type u_4\ninst\u271d : Inhabited \u039b'\nM : Machine \u0393 \u039b\nf\u2081 : PointedMap \u0393 \u0393'\nf\u2082 : PointedMap \u0393' \u0393\ng\u2081\u271d : \u039b \u2192 \u039b'\ng\u2082\u271d : \u039b' \u2192 \u039b\ng\u2081 : PointedMap \u039b \u039b'\ng\u2082 : \u039b' \u2192 \u039b\nS : Set \u039b\nss : Supports M S\nf\u2082\u2081 : Function.RightInverse f\u2081.f f\u2082.f\ng\u2082\u2081 : \u2200 (q : \u039b), q \u2208 S \u2192 g\u2082 (PointedMap.f g\u2081 q) = q\nc : Cfg \u0393 \u039b\na\u2082\u271d : Cfg \u0393' \u039b'\ncs : c.q \u2208 S\nval\u271d : Cfg \u0393 \u039b\ne : step M c = some val\u271d\n\u22a2 Cfg.map f\u2081 g\u2081.f val\u271d \u2208 Option.map (Cfg.map f\u2081 g\u2081.f) (some val\u271d)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/TuringMachine.lean", "full_name": "Turing.TM1to0.tr_eval", "start": [1516, 1], "end": [1520, 21], "traced_tactics": [{"tactic": "rw [Part.map_eq_map, Part.map_map, TM1.eval]", "annotated_tactic": ["rw [<a>Part.map_eq_map</a>, <a>Part.map_map</a>, <a>TM1.eval</a>]", [{"full_name": "Part.map_eq_map", "def_path": "Mathlib/Data/Part.lean", "def_pos": [609, 9], "def_end_pos": [609, 19]}, {"full_name": "Part.map_map", "def_path": "Mathlib/Data/Part.lean", "def_pos": [559, 9], "def_end_pos": [559, 16]}, {"full_name": "Turing.TM1.eval", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1406, 5], "def_end_pos": [1406, 9]}]], "state_before": "\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\nl : List \u0393\n\u22a2 Part.map (fun c => Tape.right\u2080 c.Tape)\n      ((fun a => trCfg M a) <$> eval (TM1.step M) { l := some default, var := default, Tape := Tape.mk\u2081 l }) =\n    TM1.eval M l", "state_after": "\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\nl : List \u0393\n\u22a2 Part.map ((fun c => Tape.right\u2080 c.Tape) \u2218 fun a => trCfg M a)\n      (eval (TM1.step M) { l := some default, var := default, Tape := Tape.mk\u2081 l }) =\n    Part.map (fun c => Tape.right\u2080 c.Tape) (eval (TM1.step M) (TM1.init l))"}, {"tactic": "congr with \u27e8\u27e9", "annotated_tactic": ["congr with \u27e8\u27e9", []], "state_before": "\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\nl : List \u0393\n\u22a2 Part.map ((fun c => Tape.right\u2080 c.Tape) \u2218 fun a => trCfg M a)\n      (eval (TM1.step M) { l := some default, var := default, Tape := Tape.mk\u2081 l }) =\n    Part.map (fun c => Tape.right\u2080 c.Tape) (eval (TM1.step M) (TM1.init l))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Primrec.lean", "full_name": "Primrec.nat_casesOn", "start": [594, 1], "end": [596, 35], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Constructions/Pi.lean", "full_name": "IsCountablySpanning.pi", "start": [90, 1], "end": [98, 36], "traced_tactics": [{"tactic": "choose s h1s h2s using hC", "annotated_tactic": ["choose s h1s h2s using hC", []], "state_before": "\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : \u03b9 \u2192 Type u_3\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : Finite \u03b9'\nC : (i : \u03b9) \u2192 Set (Set (\u03b1 i))\nhC : \u2200 (i : \u03b9), IsCountablySpanning (C i)\n\u22a2 IsCountablySpanning (Set.pi univ '' Set.pi univ C)", "state_after": "\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : \u03b9 \u2192 Type u_3\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : Finite \u03b9'\nC : (i : \u03b9) \u2192 Set (Set (\u03b1 i))\ns : (i : \u03b9) \u2192 \u2115 \u2192 Set (\u03b1 i)\nh1s : \u2200 (i : \u03b9) (n : \u2115), s i n \u2208 C i\nh2s : \u2200 (i : \u03b9), \u22c3 n, s i n = univ\n\u22a2 IsCountablySpanning (Set.pi univ '' Set.pi univ C)"}, {"tactic": "cases nonempty_encodable (\u03b9 \u2192 \u2115)", "annotated_tactic": ["cases <a>nonempty_encodable</a> (\u03b9 \u2192 \u2115)", [{"full_name": "nonempty_encodable", "def_path": "Mathlib/Logic/Encodable/Basic.lean", "def_pos": [472, 9], "def_end_pos": [472, 27]}]], "state_before": "\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : \u03b9 \u2192 Type u_3\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : Finite \u03b9'\nC : (i : \u03b9) \u2192 Set (Set (\u03b1 i))\ns : (i : \u03b9) \u2192 \u2115 \u2192 Set (\u03b1 i)\nh1s : \u2200 (i : \u03b9) (n : \u2115), s i n \u2208 C i\nh2s : \u2200 (i : \u03b9), \u22c3 n, s i n = univ\n\u22a2 IsCountablySpanning (Set.pi univ '' Set.pi univ C)", "state_after": "case intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : \u03b9 \u2192 Type u_3\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : Finite \u03b9'\nC : (i : \u03b9) \u2192 Set (Set (\u03b1 i))\ns : (i : \u03b9) \u2192 \u2115 \u2192 Set (\u03b1 i)\nh1s : \u2200 (i : \u03b9) (n : \u2115), s i n \u2208 C i\nh2s : \u2200 (i : \u03b9), \u22c3 n, s i n = univ\nval\u271d : Encodable (\u03b9 \u2192 \u2115)\n\u22a2 IsCountablySpanning (Set.pi univ '' Set.pi univ C)"}, {"tactic": "let e : \u2115 \u2192 \u03b9 \u2192 \u2115 := fun n => (@decode (\u03b9 \u2192 \u2115) _ n).iget", "annotated_tactic": ["let e : \u2115 \u2192 \u03b9 \u2192 \u2115 := fun n => (@<a>decode</a> (\u03b9 \u2192 \u2115) _ n).<a>iget</a>", [{"full_name": "Encodable.decode", "def_path": "Mathlib/Logic/Encodable/Basic.lean", "def_pos": [51, 3], "def_end_pos": [51, 9]}, {"full_name": "Option.iget", "def_path": "Mathlib/Data/Option/Defs.lean", "def_pos": [106, 5], "def_end_pos": [106, 9]}]], "state_before": "case intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : \u03b9 \u2192 Type u_3\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : Finite \u03b9'\nC : (i : \u03b9) \u2192 Set (Set (\u03b1 i))\ns : (i : \u03b9) \u2192 \u2115 \u2192 Set (\u03b1 i)\nh1s : \u2200 (i : \u03b9) (n : \u2115), s i n \u2208 C i\nh2s : \u2200 (i : \u03b9), \u22c3 n, s i n = univ\nval\u271d : Encodable (\u03b9 \u2192 \u2115)\n\u22a2 IsCountablySpanning (Set.pi univ '' Set.pi univ C)", "state_after": "case intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : \u03b9 \u2192 Type u_3\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : Finite \u03b9'\nC : (i : \u03b9) \u2192 Set (Set (\u03b1 i))\ns : (i : \u03b9) \u2192 \u2115 \u2192 Set (\u03b1 i)\nh1s : \u2200 (i : \u03b9) (n : \u2115), s i n \u2208 C i\nh2s : \u2200 (i : \u03b9), \u22c3 n, s i n = univ\nval\u271d : Encodable (\u03b9 \u2192 \u2115)\ne : \u2115 \u2192 \u03b9 \u2192 \u2115 := fun n => Option.iget (decode n)\n\u22a2 IsCountablySpanning (Set.pi univ '' Set.pi univ C)"}, {"tactic": "refine' \u27e8fun n => Set.pi univ fun i => s i (e n i), fun n =>\n  mem_image_of_mem _ fun i _ => h1s i _, _\u27e9", "annotated_tactic": ["refine' \u27e8fun n => <a>Set.pi</a> <a>univ</a> fun i => s i (e n i), fun n =>\n    <a>mem_image_of_mem</a> _ fun i _ => h1s i _, _\u27e9", [{"full_name": "Set.pi", "def_path": "Mathlib/Data/Set/Prod.lean", "def_pos": [665, 5], "def_end_pos": [665, 7]}, {"full_name": "Set.univ", "def_path": "Mathlib/Init/Set.lean", "def_pos": [90, 5], "def_end_pos": [90, 9]}, {"full_name": "Set.mem_image_of_mem", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [240, 9], "def_end_pos": [240, 25]}]], "state_before": "case intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : \u03b9 \u2192 Type u_3\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : Finite \u03b9'\nC : (i : \u03b9) \u2192 Set (Set (\u03b1 i))\ns : (i : \u03b9) \u2192 \u2115 \u2192 Set (\u03b1 i)\nh1s : \u2200 (i : \u03b9) (n : \u2115), s i n \u2208 C i\nh2s : \u2200 (i : \u03b9), \u22c3 n, s i n = univ\nval\u271d : Encodable (\u03b9 \u2192 \u2115)\ne : \u2115 \u2192 \u03b9 \u2192 \u2115 := fun n => Option.iget (decode n)\n\u22a2 IsCountablySpanning (Set.pi univ '' Set.pi univ C)", "state_after": "case intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : \u03b9 \u2192 Type u_3\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : Finite \u03b9'\nC : (i : \u03b9) \u2192 Set (Set (\u03b1 i))\ns : (i : \u03b9) \u2192 \u2115 \u2192 Set (\u03b1 i)\nh1s : \u2200 (i : \u03b9) (n : \u2115), s i n \u2208 C i\nh2s : \u2200 (i : \u03b9), \u22c3 n, s i n = univ\nval\u271d : Encodable (\u03b9 \u2192 \u2115)\ne : \u2115 \u2192 \u03b9 \u2192 \u2115 := fun n => Option.iget (decode n)\n\u22a2 \u22c3 n, (fun n => Set.pi univ fun i => s i (e n i)) n = univ"}, {"tactic": "simp_rw [(surjective_decode_iget (\u03b9 \u2192 \u2115)).iUnion_comp fun x => Set.pi univ fun i => s i (x i),\n  iUnion_univ_pi s, h2s, pi_univ]", "annotated_tactic": ["simp_rw [(<a>surjective_decode_iget</a> (\u03b9 \u2192 \u2115)).<a>iUnion_comp</a> fun x => <a>Set.pi</a> <a>univ</a> fun i => s i (x i),\n    <a>iUnion_univ_pi</a> s, h2s, <a>pi_univ</a>]", [{"full_name": "Encodable.surjective_decode_iget", "def_path": "Mathlib/Logic/Encodable/Basic.lean", "def_pos": [78, 9], "def_end_pos": [78, 31]}, {"full_name": "Function.Surjective.iUnion_comp", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [2194, 9], "def_end_pos": [2194, 20]}, {"full_name": "Set.pi", "def_path": "Mathlib/Data/Set/Prod.lean", "def_pos": [665, 5], "def_end_pos": [665, 7]}, {"full_name": "Set.univ", "def_path": "Mathlib/Init/Set.lean", "def_pos": [90, 5], "def_end_pos": [90, 9]}, {"full_name": "Set.iUnion_univ_pi", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [2180, 9], "def_end_pos": [2180, 23]}, {"full_name": "Set.pi_univ", "def_path": "Mathlib/Data/Set/Prod.lean", "def_pos": [689, 9], "def_end_pos": [689, 16]}]], "state_before": "case intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : \u03b9 \u2192 Type u_3\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : Finite \u03b9'\nC : (i : \u03b9) \u2192 Set (Set (\u03b1 i))\ns : (i : \u03b9) \u2192 \u2115 \u2192 Set (\u03b1 i)\nh1s : \u2200 (i : \u03b9) (n : \u2115), s i n \u2208 C i\nh2s : \u2200 (i : \u03b9), \u22c3 n, s i n = univ\nval\u271d : Encodable (\u03b9 \u2192 \u2115)\ne : \u2115 \u2192 \u03b9 \u2192 \u2115 := fun n => Option.iget (decode n)\n\u22a2 \u22c3 n, (fun n => Set.pi univ fun i => s i (e n i)) n = univ", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "full_name": "intervalIntegral.integral_const_of_cdf", "start": [969, 1], "end": [973, 64], "traced_tactics": [{"tactic": "simp only [sub_smul, \u2190 set_integral_const]", "annotated_tactic": ["simp only [<a>sub_smul</a>, \u2190 <a>set_integral_const</a>]", [{"full_name": "sub_smul", "def_path": "Mathlib/Algebra/Module/Basic.lean", "def_pos": [299, 9], "def_end_pos": [299, 17]}, {"full_name": "MeasureTheory.set_integral_const", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [474, 9], "def_end_pos": [474, 27]}]], "state_before": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : NormedSpace \u211d E\na b c\u271d d : \u211d\nf g : \u211d \u2192 E\n\u03bc : Measure \u211d\ninst\u271d : IsFiniteMeasure \u03bc\nc : E\n\u22a2 \u222b (x : \u211d) in a..b, c \u2202\u03bc = (ENNReal.toReal (\u2191\u2191\u03bc (Iic b)) - ENNReal.toReal (\u2191\u2191\u03bc (Iic a))) \u2022 c", "state_after": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : NormedSpace \u211d E\na b c\u271d d : \u211d\nf g : \u211d \u2192 E\n\u03bc : Measure \u211d\ninst\u271d : IsFiniteMeasure \u03bc\nc : E\n\u22a2 \u222b (x : \u211d) in a..b, c \u2202\u03bc = \u222b (x : \u211d) in Iic b, c \u2202\u03bc - \u222b (x : \u211d) in Iic a, c \u2202\u03bc"}, {"tactic": "refine' (integral_Iic_sub_Iic _ _).symm <;>\n  simp only [integrableOn_const, measure_lt_top, or_true_iff]", "annotated_tactic": ["refine' (<a>integral_Iic_sub_Iic</a> _ _).<a>symm</a> <;>\n    simp only [<a>integrableOn_const</a>, <a>measure_lt_top</a>, <a>or_true_iff</a>]", [{"full_name": "intervalIntegral.integral_Iic_sub_Iic", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [959, 9], "def_end_pos": [959, 29]}, {"full_name": "Eq.symm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [310, 9], "def_end_pos": [310, 16]}, {"full_name": "MeasureTheory.integrableOn_const", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [119, 9], "def_end_pos": [119, 27]}, {"full_name": "MeasureTheory.measure_lt_top", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2866, 9], "def_end_pos": [2866, 23]}, {"full_name": "or_true_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [184, 9], "def_end_pos": [184, 20]}]], "state_before": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : NormedSpace \u211d E\na b c\u271d d : \u211d\nf g : \u211d \u2192 E\n\u03bc : Measure \u211d\ninst\u271d : IsFiniteMeasure \u03bc\nc : E\n\u22a2 \u222b (x : \u211d) in a..b, c \u2202\u03bc = \u222b (x : \u211d) in Iic b, c \u2202\u03bc - \u222b (x : \u211d) in Iic a, c \u2202\u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "full_name": "MeasureTheory.eventually_mem_spanningSets", "start": [3378, 1], "end": [3380, 88], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Quot.lean", "full_name": "Trunc.out_eq", "start": [582, 1], "end": [583, 15], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "full_name": "MeasureTheory.tendsto_set_integral_of_monotone", "start": [212, 1], "end": [232, 75], "traced_tactics": [{"tactic": "have hfi' : (\u222b\u207b x in \u22c3 n, s n, \u2016f x\u2016\u208a \u2202\u03bc) < \u221e := hfi.2", "annotated_tactic": ["have hfi' : (\u222b\u207b x in \u22c3 n, s n, \u2016f x\u2016\u208a \u2202\u03bc) < \u221e := hfi.2", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\nf g : \u03b1 \u2192 E\ns\u271d t : Set \u03b1\n\u03bc \u03bd : Measure \u03b1\nl l' : Filter \u03b1\ninst\u271d\u00b2 : NormedSpace \u211d E\n\u03b9 : Type u_5\ninst\u271d\u00b9 : Countable \u03b9\ninst\u271d : SemilatticeSup \u03b9\ns : \u03b9 \u2192 Set \u03b1\nhsm : \u2200 (i : \u03b9), MeasurableSet (s i)\nh_mono : Monotone s\nhfi : IntegrableOn f (\u22c3 n, s n)\n\u22a2 Tendsto (fun i => \u222b (a : \u03b1) in s i, f a \u2202\u03bc) atTop (\ud835\udcdd (\u222b (a : \u03b1) in \u22c3 n, s n, f a \u2202\u03bc))", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\nf g : \u03b1 \u2192 E\ns\u271d t : Set \u03b1\n\u03bc \u03bd : Measure \u03b1\nl l' : Filter \u03b1\ninst\u271d\u00b2 : NormedSpace \u211d E\n\u03b9 : Type u_5\ninst\u271d\u00b9 : Countable \u03b9\ninst\u271d : SemilatticeSup \u03b9\ns : \u03b9 \u2192 Set \u03b1\nhsm : \u2200 (i : \u03b9), MeasurableSet (s i)\nh_mono : Monotone s\nhfi : IntegrableOn f (\u22c3 n, s n)\nhfi' : \u222b\u207b (x : \u03b1) in \u22c3 n, s n, \u2191\u2016f x\u2016\u208a \u2202\u03bc < \u22a4\n\u22a2 Tendsto (fun i => \u222b (a : \u03b1) in s i, f a \u2202\u03bc) atTop (\ud835\udcdd (\u222b (a : \u03b1) in \u22c3 n, s n, f a \u2202\u03bc))"}, {"tactic": "set S := \u22c3 i, s i", "annotated_tactic": ["set S := \u22c3 i, s i", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\nf g : \u03b1 \u2192 E\ns\u271d t : Set \u03b1\n\u03bc \u03bd : Measure \u03b1\nl l' : Filter \u03b1\ninst\u271d\u00b2 : NormedSpace \u211d E\n\u03b9 : Type u_5\ninst\u271d\u00b9 : Countable \u03b9\ninst\u271d : SemilatticeSup \u03b9\ns : \u03b9 \u2192 Set \u03b1\nhsm : \u2200 (i : \u03b9), MeasurableSet (s i)\nh_mono : Monotone s\nhfi : IntegrableOn f (\u22c3 n, s n)\nhfi' : \u222b\u207b (x : \u03b1) in \u22c3 n, s n, \u2191\u2016f x\u2016\u208a \u2202\u03bc < \u22a4\n\u22a2 Tendsto (fun i => \u222b (a : \u03b1) in s i, f a \u2202\u03bc) atTop (\ud835\udcdd (\u222b (a : \u03b1) in \u22c3 n, s n, f a \u2202\u03bc))", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\nf g : \u03b1 \u2192 E\ns\u271d t : Set \u03b1\n\u03bc \u03bd : Measure \u03b1\nl l' : Filter \u03b1\ninst\u271d\u00b2 : NormedSpace \u211d E\n\u03b9 : Type u_5\ninst\u271d\u00b9 : Countable \u03b9\ninst\u271d : SemilatticeSup \u03b9\ns : \u03b9 \u2192 Set \u03b1\nhsm : \u2200 (i : \u03b9), MeasurableSet (s i)\nh_mono : Monotone s\nS : Set \u03b1 := \u22c3 i, s i\nhfi : IntegrableOn f S\nhfi' : \u222b\u207b (x : \u03b1) in S, \u2191\u2016f x\u2016\u208a \u2202\u03bc < \u22a4\n\u22a2 Tendsto (fun i => \u222b (a : \u03b1) in s i, f a \u2202\u03bc) atTop (\ud835\udcdd (\u222b (a : \u03b1) in S, f a \u2202\u03bc))"}, {"tactic": "have hSm : MeasurableSet S := MeasurableSet.iUnion hsm", "annotated_tactic": ["have hSm : <a>MeasurableSet</a> S := <a>MeasurableSet.iUnion</a> hsm", [{"full_name": "MeasurableSet", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [64, 5], "def_end_pos": [64, 18]}, {"full_name": "MeasurableSet.iUnion", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [115, 19], "def_end_pos": [115, 39]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\nf g : \u03b1 \u2192 E\ns\u271d t : Set \u03b1\n\u03bc \u03bd : Measure \u03b1\nl l' : Filter \u03b1\ninst\u271d\u00b2 : NormedSpace \u211d E\n\u03b9 : Type u_5\ninst\u271d\u00b9 : Countable \u03b9\ninst\u271d : SemilatticeSup \u03b9\ns : \u03b9 \u2192 Set \u03b1\nhsm : \u2200 (i : \u03b9), MeasurableSet (s i)\nh_mono : Monotone s\nS : Set \u03b1 := \u22c3 i, s i\nhfi : IntegrableOn f S\nhfi' : \u222b\u207b (x : \u03b1) in S, \u2191\u2016f x\u2016\u208a \u2202\u03bc < \u22a4\n\u22a2 Tendsto (fun i => \u222b (a : \u03b1) in s i, f a \u2202\u03bc) atTop (\ud835\udcdd (\u222b (a : \u03b1) in S, f a \u2202\u03bc))", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\nf g : \u03b1 \u2192 E\ns\u271d t : Set \u03b1\n\u03bc \u03bd : Measure \u03b1\nl l' : Filter \u03b1\ninst\u271d\u00b2 : NormedSpace \u211d E\n\u03b9 : Type u_5\ninst\u271d\u00b9 : Countable \u03b9\ninst\u271d : SemilatticeSup \u03b9\ns : \u03b9 \u2192 Set \u03b1\nhsm : \u2200 (i : \u03b9), MeasurableSet (s i)\nh_mono : Monotone s\nS : Set \u03b1 := \u22c3 i, s i\nhfi : IntegrableOn f S\nhfi' : \u222b\u207b (x : \u03b1) in S, \u2191\u2016f x\u2016\u208a \u2202\u03bc < \u22a4\nhSm : MeasurableSet S\n\u22a2 Tendsto (fun i => \u222b (a : \u03b1) in s i, f a \u2202\u03bc) atTop (\ud835\udcdd (\u222b (a : \u03b1) in S, f a \u2202\u03bc))"}, {"tactic": "have hsub : \u2200 {i}, s i \u2286 S := @(subset_iUnion s)", "annotated_tactic": ["have hsub : \u2200 {i}, s i \u2286 S := @(<a>subset_iUnion</a> s)", [{"full_name": "Set.subset_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [431, 9], "def_end_pos": [431, 22]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\nf g : \u03b1 \u2192 E\ns\u271d t : Set \u03b1\n\u03bc \u03bd : Measure \u03b1\nl l' : Filter \u03b1\ninst\u271d\u00b2 : NormedSpace \u211d E\n\u03b9 : Type u_5\ninst\u271d\u00b9 : Countable \u03b9\ninst\u271d : SemilatticeSup \u03b9\ns : \u03b9 \u2192 Set \u03b1\nhsm : \u2200 (i : \u03b9), MeasurableSet (s i)\nh_mono : Monotone s\nS : Set \u03b1 := \u22c3 i, s i\nhfi : IntegrableOn f S\nhfi' : \u222b\u207b (x : \u03b1) in S, \u2191\u2016f x\u2016\u208a \u2202\u03bc < \u22a4\nhSm : MeasurableSet S\n\u22a2 Tendsto (fun i => \u222b (a : \u03b1) in s i, f a \u2202\u03bc) atTop (\ud835\udcdd (\u222b (a : \u03b1) in S, f a \u2202\u03bc))", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\nf g : \u03b1 \u2192 E\ns\u271d t : Set \u03b1\n\u03bc \u03bd : Measure \u03b1\nl l' : Filter \u03b1\ninst\u271d\u00b2 : NormedSpace \u211d E\n\u03b9 : Type u_5\ninst\u271d\u00b9 : Countable \u03b9\ninst\u271d : SemilatticeSup \u03b9\ns : \u03b9 \u2192 Set \u03b1\nhsm : \u2200 (i : \u03b9), MeasurableSet (s i)\nh_mono : Monotone s\nS : Set \u03b1 := \u22c3 i, s i\nhfi : IntegrableOn f S\nhfi' : \u222b\u207b (x : \u03b1) in S, \u2191\u2016f x\u2016\u208a \u2202\u03bc < \u22a4\nhSm : MeasurableSet S\nhsub : \u2200 {i : \u03b9}, s i \u2286 S\n\u22a2 Tendsto (fun i => \u222b (a : \u03b1) in s i, f a \u2202\u03bc) atTop (\ud835\udcdd (\u222b (a : \u03b1) in S, f a \u2202\u03bc))"}, {"tactic": "rw [\u2190 withDensity_apply _ hSm] at hfi'", "annotated_tactic": ["rw [\u2190 <a>withDensity_apply</a> _ hSm] at hfi'", [{"full_name": "MeasureTheory.withDensity_apply", "def_path": "Mathlib/MeasureTheory/Measure/WithDensity.lean", "def_pos": [39, 9], "def_end_pos": [39, 26]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\nf g : \u03b1 \u2192 E\ns\u271d t : Set \u03b1\n\u03bc \u03bd : Measure \u03b1\nl l' : Filter \u03b1\ninst\u271d\u00b2 : NormedSpace \u211d E\n\u03b9 : Type u_5\ninst\u271d\u00b9 : Countable \u03b9\ninst\u271d : SemilatticeSup \u03b9\ns : \u03b9 \u2192 Set \u03b1\nhsm : \u2200 (i : \u03b9), MeasurableSet (s i)\nh_mono : Monotone s\nS : Set \u03b1 := \u22c3 i, s i\nhfi : IntegrableOn f S\nhfi' : \u222b\u207b (x : \u03b1) in S, \u2191\u2016f x\u2016\u208a \u2202\u03bc < \u22a4\nhSm : MeasurableSet S\nhsub : \u2200 {i : \u03b9}, s i \u2286 S\n\u22a2 Tendsto (fun i => \u222b (a : \u03b1) in s i, f a \u2202\u03bc) atTop (\ud835\udcdd (\u222b (a : \u03b1) in S, f a \u2202\u03bc))", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\nf g : \u03b1 \u2192 E\ns\u271d t : Set \u03b1\n\u03bc \u03bd : Measure \u03b1\nl l' : Filter \u03b1\ninst\u271d\u00b2 : NormedSpace \u211d E\n\u03b9 : Type u_5\ninst\u271d\u00b9 : Countable \u03b9\ninst\u271d : SemilatticeSup \u03b9\ns : \u03b9 \u2192 Set \u03b1\nhsm : \u2200 (i : \u03b9), MeasurableSet (s i)\nh_mono : Monotone s\nS : Set \u03b1 := \u22c3 i, s i\nhfi : IntegrableOn f S\nhfi' : \u2191\u2191(Measure.withDensity \u03bc fun x => \u2191\u2016f x\u2016\u208a) S < \u22a4\nhSm : MeasurableSet S\nhsub : \u2200 {i : \u03b9}, s i \u2286 S\n\u22a2 Tendsto (fun i => \u222b (a : \u03b1) in s i, f a \u2202\u03bc) atTop (\ud835\udcdd (\u222b (a : \u03b1) in S, f a \u2202\u03bc))"}, {"tactic": "set \u03bd := \u03bc.withDensity fun x => \u2016f x\u2016\u208a with h\u03bd", "annotated_tactic": ["set \u03bd := \u03bc.withDensity fun x => \u2016f x\u2016\u208a with h\u03bd", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\nf g : \u03b1 \u2192 E\ns\u271d t : Set \u03b1\n\u03bc \u03bd : Measure \u03b1\nl l' : Filter \u03b1\ninst\u271d\u00b2 : NormedSpace \u211d E\n\u03b9 : Type u_5\ninst\u271d\u00b9 : Countable \u03b9\ninst\u271d : SemilatticeSup \u03b9\ns : \u03b9 \u2192 Set \u03b1\nhsm : \u2200 (i : \u03b9), MeasurableSet (s i)\nh_mono : Monotone s\nS : Set \u03b1 := \u22c3 i, s i\nhfi : IntegrableOn f S\nhfi' : \u2191\u2191(Measure.withDensity \u03bc fun x => \u2191\u2016f x\u2016\u208a) S < \u22a4\nhSm : MeasurableSet S\nhsub : \u2200 {i : \u03b9}, s i \u2286 S\n\u22a2 Tendsto (fun i => \u222b (a : \u03b1) in s i, f a \u2202\u03bc) atTop (\ud835\udcdd (\u222b (a : \u03b1) in S, f a \u2202\u03bc))", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\nf g : \u03b1 \u2192 E\ns\u271d t : Set \u03b1\n\u03bc \u03bd\u271d : Measure \u03b1\nl l' : Filter \u03b1\ninst\u271d\u00b2 : NormedSpace \u211d E\n\u03b9 : Type u_5\ninst\u271d\u00b9 : Countable \u03b9\ninst\u271d : SemilatticeSup \u03b9\ns : \u03b9 \u2192 Set \u03b1\nhsm : \u2200 (i : \u03b9), MeasurableSet (s i)\nh_mono : Monotone s\nS : Set \u03b1 := \u22c3 i, s i\nhfi : IntegrableOn f S\nhSm : MeasurableSet S\nhsub : \u2200 {i : \u03b9}, s i \u2286 S\n\u03bd : Measure \u03b1 := Measure.withDensity \u03bc fun x => \u2191\u2016f x\u2016\u208a\nhfi' : \u2191\u2191\u03bd S < \u22a4\nh\u03bd : \u03bd = Measure.withDensity \u03bc fun x => \u2191\u2016f x\u2016\u208a\n\u22a2 Tendsto (fun i => \u222b (a : \u03b1) in s i, f a \u2202\u03bc) atTop (\ud835\udcdd (\u222b (a : \u03b1) in S, f a \u2202\u03bc))"}, {"tactic": "refine' Metric.nhds_basis_closedBall.tendsto_right_iff.2 fun \u03b5 \u03b50 => _", "annotated_tactic": ["refine' Metric.nhds_basis_closedBall.tendsto_right_iff.2 fun \u03b5 \u03b50 => _", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\nf g : \u03b1 \u2192 E\ns\u271d t : Set \u03b1\n\u03bc \u03bd\u271d : Measure \u03b1\nl l' : Filter \u03b1\ninst\u271d\u00b2 : NormedSpace \u211d E\n\u03b9 : Type u_5\ninst\u271d\u00b9 : Countable \u03b9\ninst\u271d : SemilatticeSup \u03b9\ns : \u03b9 \u2192 Set \u03b1\nhsm : \u2200 (i : \u03b9), MeasurableSet (s i)\nh_mono : Monotone s\nS : Set \u03b1 := \u22c3 i, s i\nhfi : IntegrableOn f S\nhSm : MeasurableSet S\nhsub : \u2200 {i : \u03b9}, s i \u2286 S\n\u03bd : Measure \u03b1 := Measure.withDensity \u03bc fun x => \u2191\u2016f x\u2016\u208a\nhfi' : \u2191\u2191\u03bd S < \u22a4\nh\u03bd : \u03bd = Measure.withDensity \u03bc fun x => \u2191\u2016f x\u2016\u208a\n\u22a2 Tendsto (fun i => \u222b (a : \u03b1) in s i, f a \u2202\u03bc) atTop (\ud835\udcdd (\u222b (a : \u03b1) in S, f a \u2202\u03bc))", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\nf g : \u03b1 \u2192 E\ns\u271d t : Set \u03b1\n\u03bc \u03bd\u271d : Measure \u03b1\nl l' : Filter \u03b1\ninst\u271d\u00b2 : NormedSpace \u211d E\n\u03b9 : Type u_5\ninst\u271d\u00b9 : Countable \u03b9\ninst\u271d : SemilatticeSup \u03b9\ns : \u03b9 \u2192 Set \u03b1\nhsm : \u2200 (i : \u03b9), MeasurableSet (s i)\nh_mono : Monotone s\nS : Set \u03b1 := \u22c3 i, s i\nhfi : IntegrableOn f S\nhSm : MeasurableSet S\nhsub : \u2200 {i : \u03b9}, s i \u2286 S\n\u03bd : Measure \u03b1 := Measure.withDensity \u03bc fun x => \u2191\u2016f x\u2016\u208a\nhfi' : \u2191\u2191\u03bd S < \u22a4\nh\u03bd : \u03bd = Measure.withDensity \u03bc fun x => \u2191\u2016f x\u2016\u208a\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\n\u22a2 \u2200\u1da0 (x : \u03b9) in atTop, \u222b (a : \u03b1) in s x, f a \u2202\u03bc \u2208 Metric.closedBall (\u222b (a : \u03b1) in S, f a \u2202\u03bc) \u03b5"}, {"tactic": "lift \u03b5 to \u211d\u22650 using \u03b50.le", "annotated_tactic": ["lift \u03b5 to \u211d\u22650 using \u03b50.le", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\nf g : \u03b1 \u2192 E\ns\u271d t : Set \u03b1\n\u03bc \u03bd\u271d : Measure \u03b1\nl l' : Filter \u03b1\ninst\u271d\u00b2 : NormedSpace \u211d E\n\u03b9 : Type u_5\ninst\u271d\u00b9 : Countable \u03b9\ninst\u271d : SemilatticeSup \u03b9\ns : \u03b9 \u2192 Set \u03b1\nhsm : \u2200 (i : \u03b9), MeasurableSet (s i)\nh_mono : Monotone s\nS : Set \u03b1 := \u22c3 i, s i\nhfi : IntegrableOn f S\nhSm : MeasurableSet S\nhsub : \u2200 {i : \u03b9}, s i \u2286 S\n\u03bd : Measure \u03b1 := Measure.withDensity \u03bc fun x => \u2191\u2016f x\u2016\u208a\nhfi' : \u2191\u2191\u03bd S < \u22a4\nh\u03bd : \u03bd = Measure.withDensity \u03bc fun x => \u2191\u2016f x\u2016\u208a\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\n\u22a2 \u2200\u1da0 (x : \u03b9) in atTop, \u222b (a : \u03b1) in s x, f a \u2202\u03bc \u2208 Metric.closedBall (\u222b (a : \u03b1) in S, f a \u2202\u03bc) \u03b5", "state_after": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\nf g : \u03b1 \u2192 E\ns\u271d t : Set \u03b1\n\u03bc \u03bd\u271d : Measure \u03b1\nl l' : Filter \u03b1\ninst\u271d\u00b2 : NormedSpace \u211d E\n\u03b9 : Type u_5\ninst\u271d\u00b9 : Countable \u03b9\ninst\u271d : SemilatticeSup \u03b9\ns : \u03b9 \u2192 Set \u03b1\nhsm : \u2200 (i : \u03b9), MeasurableSet (s i)\nh_mono : Monotone s\nS : Set \u03b1 := \u22c3 i, s i\nhfi : IntegrableOn f S\nhSm : MeasurableSet S\nhsub : \u2200 {i : \u03b9}, s i \u2286 S\n\u03bd : Measure \u03b1 := Measure.withDensity \u03bc fun x => \u2191\u2016f x\u2016\u208a\nhfi' : \u2191\u2191\u03bd S < \u22a4\nh\u03bd : \u03bd = Measure.withDensity \u03bc fun x => \u2191\u2016f x\u2016\u208a\n\u03b5 : \u211d\u22650\n\u03b50 : 0 < \u2191\u03b5\n\u22a2 \u2200\u1da0 (x : \u03b9) in atTop, \u222b (a : \u03b1) in s x, f a \u2202\u03bc \u2208 Metric.closedBall (\u222b (a : \u03b1) in S, f a \u2202\u03bc) \u2191\u03b5"}, {"tactic": "have : \u2200\u1da0 i in atTop, \u03bd (s i) \u2208 Icc (\u03bd S - \u03b5) (\u03bd S + \u03b5) :=\n  tendsto_measure_iUnion h_mono (ENNReal.Icc_mem_nhds hfi'.ne (ENNReal.coe_pos.2 \u03b50).ne')", "annotated_tactic": ["have : \u2200\u1da0 i in <a>atTop</a>, \u03bd (s i) \u2208 <a>Icc</a> (\u03bd S - \u03b5) (\u03bd S + \u03b5) :=\n    <a>tendsto_measure_iUnion</a> h_mono (<a>ENNReal.Icc_mem_nhds</a> hfi'.ne (<a>ENNReal.coe_pos</a>.2 \u03b50).<a>ne'</a>)", [{"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}, {"full_name": "Set.Icc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [59, 5], "def_end_pos": [59, 8]}, {"full_name": "MeasureTheory.tendsto_measure_iUnion", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [530, 9], "def_end_pos": [530, 31]}, {"full_name": "ENNReal.Icc_mem_nhds", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [253, 9], "def_end_pos": [253, 21]}, {"full_name": "ENNReal.coe_pos", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [380, 28], "def_end_pos": [380, 35]}, {"full_name": "LT.lt.ne'", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [328, 9], "def_end_pos": [328, 12]}]], "state_before": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\nf g : \u03b1 \u2192 E\ns\u271d t : Set \u03b1\n\u03bc \u03bd\u271d : Measure \u03b1\nl l' : Filter \u03b1\ninst\u271d\u00b2 : NormedSpace \u211d E\n\u03b9 : Type u_5\ninst\u271d\u00b9 : Countable \u03b9\ninst\u271d : SemilatticeSup \u03b9\ns : \u03b9 \u2192 Set \u03b1\nhsm : \u2200 (i : \u03b9), MeasurableSet (s i)\nh_mono : Monotone s\nS : Set \u03b1 := \u22c3 i, s i\nhfi : IntegrableOn f S\nhSm : MeasurableSet S\nhsub : \u2200 {i : \u03b9}, s i \u2286 S\n\u03bd : Measure \u03b1 := Measure.withDensity \u03bc fun x => \u2191\u2016f x\u2016\u208a\nhfi' : \u2191\u2191\u03bd S < \u22a4\nh\u03bd : \u03bd = Measure.withDensity \u03bc fun x => \u2191\u2016f x\u2016\u208a\n\u03b5 : \u211d\u22650\n\u03b50 : 0 < \u2191\u03b5\n\u22a2 \u2200\u1da0 (x : \u03b9) in atTop, \u222b (a : \u03b1) in s x, f a \u2202\u03bc \u2208 Metric.closedBall (\u222b (a : \u03b1) in S, f a \u2202\u03bc) \u2191\u03b5", "state_after": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\nf g : \u03b1 \u2192 E\ns\u271d t : Set \u03b1\n\u03bc \u03bd\u271d : Measure \u03b1\nl l' : Filter \u03b1\ninst\u271d\u00b2 : NormedSpace \u211d E\n\u03b9 : Type u_5\ninst\u271d\u00b9 : Countable \u03b9\ninst\u271d : SemilatticeSup \u03b9\ns : \u03b9 \u2192 Set \u03b1\nhsm : \u2200 (i : \u03b9), MeasurableSet (s i)\nh_mono : Monotone s\nS : Set \u03b1 := \u22c3 i, s i\nhfi : IntegrableOn f S\nhSm : MeasurableSet S\nhsub : \u2200 {i : \u03b9}, s i \u2286 S\n\u03bd : Measure \u03b1 := Measure.withDensity \u03bc fun x => \u2191\u2016f x\u2016\u208a\nhfi' : \u2191\u2191\u03bd S < \u22a4\nh\u03bd : \u03bd = Measure.withDensity \u03bc fun x => \u2191\u2016f x\u2016\u208a\n\u03b5 : \u211d\u22650\n\u03b50 : 0 < \u2191\u03b5\nthis : \u2200\u1da0 (i : \u03b9) in atTop, \u2191\u2191\u03bd (s i) \u2208 Icc (\u2191\u2191\u03bd S - \u2191\u03b5) (\u2191\u2191\u03bd S + \u2191\u03b5)\n\u22a2 \u2200\u1da0 (x : \u03b9) in atTop, \u222b (a : \u03b1) in s x, f a \u2202\u03bc \u2208 Metric.closedBall (\u222b (a : \u03b1) in S, f a \u2202\u03bc) \u2191\u03b5"}, {"tactic": "refine' this.mono fun i hi => _", "annotated_tactic": ["refine' this.mono fun i hi => _", []], "state_before": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\nf g : \u03b1 \u2192 E\ns\u271d t : Set \u03b1\n\u03bc \u03bd\u271d : Measure \u03b1\nl l' : Filter \u03b1\ninst\u271d\u00b2 : NormedSpace \u211d E\n\u03b9 : Type u_5\ninst\u271d\u00b9 : Countable \u03b9\ninst\u271d : SemilatticeSup \u03b9\ns : \u03b9 \u2192 Set \u03b1\nhsm : \u2200 (i : \u03b9), MeasurableSet (s i)\nh_mono : Monotone s\nS : Set \u03b1 := \u22c3 i, s i\nhfi : IntegrableOn f S\nhSm : MeasurableSet S\nhsub : \u2200 {i : \u03b9}, s i \u2286 S\n\u03bd : Measure \u03b1 := Measure.withDensity \u03bc fun x => \u2191\u2016f x\u2016\u208a\nhfi' : \u2191\u2191\u03bd S < \u22a4\nh\u03bd : \u03bd = Measure.withDensity \u03bc fun x => \u2191\u2016f x\u2016\u208a\n\u03b5 : \u211d\u22650\n\u03b50 : 0 < \u2191\u03b5\nthis : \u2200\u1da0 (i : \u03b9) in atTop, \u2191\u2191\u03bd (s i) \u2208 Icc (\u2191\u2191\u03bd S - \u2191\u03b5) (\u2191\u2191\u03bd S + \u2191\u03b5)\n\u22a2 \u2200\u1da0 (x : \u03b9) in atTop, \u222b (a : \u03b1) in s x, f a \u2202\u03bc \u2208 Metric.closedBall (\u222b (a : \u03b1) in S, f a \u2202\u03bc) \u2191\u03b5", "state_after": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\nf g : \u03b1 \u2192 E\ns\u271d t : Set \u03b1\n\u03bc \u03bd\u271d : Measure \u03b1\nl l' : Filter \u03b1\ninst\u271d\u00b2 : NormedSpace \u211d E\n\u03b9 : Type u_5\ninst\u271d\u00b9 : Countable \u03b9\ninst\u271d : SemilatticeSup \u03b9\ns : \u03b9 \u2192 Set \u03b1\nhsm : \u2200 (i : \u03b9), MeasurableSet (s i)\nh_mono : Monotone s\nS : Set \u03b1 := \u22c3 i, s i\nhfi : IntegrableOn f S\nhSm : MeasurableSet S\nhsub : \u2200 {i : \u03b9}, s i \u2286 S\n\u03bd : Measure \u03b1 := Measure.withDensity \u03bc fun x => \u2191\u2016f x\u2016\u208a\nhfi' : \u2191\u2191\u03bd S < \u22a4\nh\u03bd : \u03bd = Measure.withDensity \u03bc fun x => \u2191\u2016f x\u2016\u208a\n\u03b5 : \u211d\u22650\n\u03b50 : 0 < \u2191\u03b5\nthis : \u2200\u1da0 (i : \u03b9) in atTop, \u2191\u2191\u03bd (s i) \u2208 Icc (\u2191\u2191\u03bd S - \u2191\u03b5) (\u2191\u2191\u03bd S + \u2191\u03b5)\ni : \u03b9\nhi : \u2191\u2191\u03bd (s i) \u2208 Icc (\u2191\u2191\u03bd S - \u2191\u03b5) (\u2191\u2191\u03bd S + \u2191\u03b5)\n\u22a2 \u222b (a : \u03b1) in s i, f a \u2202\u03bc \u2208 Metric.closedBall (\u222b (a : \u03b1) in S, f a \u2202\u03bc) \u2191\u03b5"}, {"tactic": "rw [mem_closedBall_iff_norm', \u2190 integral_diff (hsm i) hfi hsub, \u2190 coe_nnnorm, NNReal.coe_le_coe, \u2190\n  ENNReal.coe_le_coe]", "annotated_tactic": ["rw [<a>mem_closedBall_iff_norm'</a>, \u2190 <a>integral_diff</a> (hsm i) hfi hsub, \u2190 <a>coe_nnnorm</a>, <a>NNReal.coe_le_coe</a>, \u2190\n    <a>ENNReal.coe_le_coe</a>]", [{"full_name": "mem_closedBall_iff_norm'", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [658, 15], "def_end_pos": [658, 39]}, {"full_name": "MeasureTheory.integral_diff", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [107, 9], "def_end_pos": [107, 22]}, {"full_name": "coe_nnnorm", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [905, 41], "def_end_pos": [905, 51]}, {"full_name": "NNReal.coe_le_coe", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [366, 19], "def_end_pos": [366, 29]}, {"full_name": "ENNReal.coe_le_coe", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [349, 28], "def_end_pos": [349, 38]}]], "state_before": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\nf g : \u03b1 \u2192 E\ns\u271d t : Set \u03b1\n\u03bc \u03bd\u271d : Measure \u03b1\nl l' : Filter \u03b1\ninst\u271d\u00b2 : NormedSpace \u211d E\n\u03b9 : Type u_5\ninst\u271d\u00b9 : Countable \u03b9\ninst\u271d : SemilatticeSup \u03b9\ns : \u03b9 \u2192 Set \u03b1\nhsm : \u2200 (i : \u03b9), MeasurableSet (s i)\nh_mono : Monotone s\nS : Set \u03b1 := \u22c3 i, s i\nhfi : IntegrableOn f S\nhSm : MeasurableSet S\nhsub : \u2200 {i : \u03b9}, s i \u2286 S\n\u03bd : Measure \u03b1 := Measure.withDensity \u03bc fun x => \u2191\u2016f x\u2016\u208a\nhfi' : \u2191\u2191\u03bd S < \u22a4\nh\u03bd : \u03bd = Measure.withDensity \u03bc fun x => \u2191\u2016f x\u2016\u208a\n\u03b5 : \u211d\u22650\n\u03b50 : 0 < \u2191\u03b5\nthis : \u2200\u1da0 (i : \u03b9) in atTop, \u2191\u2191\u03bd (s i) \u2208 Icc (\u2191\u2191\u03bd S - \u2191\u03b5) (\u2191\u2191\u03bd S + \u2191\u03b5)\ni : \u03b9\nhi : \u2191\u2191\u03bd (s i) \u2208 Icc (\u2191\u2191\u03bd S - \u2191\u03b5) (\u2191\u2191\u03bd S + \u2191\u03b5)\n\u22a2 \u222b (a : \u03b1) in s i, f a \u2202\u03bc \u2208 Metric.closedBall (\u222b (a : \u03b1) in S, f a \u2202\u03bc) \u2191\u03b5", "state_after": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\nf g : \u03b1 \u2192 E\ns\u271d t : Set \u03b1\n\u03bc \u03bd\u271d : Measure \u03b1\nl l' : Filter \u03b1\ninst\u271d\u00b2 : NormedSpace \u211d E\n\u03b9 : Type u_5\ninst\u271d\u00b9 : Countable \u03b9\ninst\u271d : SemilatticeSup \u03b9\ns : \u03b9 \u2192 Set \u03b1\nhsm : \u2200 (i : \u03b9), MeasurableSet (s i)\nh_mono : Monotone s\nS : Set \u03b1 := \u22c3 i, s i\nhfi : IntegrableOn f S\nhSm : MeasurableSet S\nhsub : \u2200 {i : \u03b9}, s i \u2286 S\n\u03bd : Measure \u03b1 := Measure.withDensity \u03bc fun x => \u2191\u2016f x\u2016\u208a\nhfi' : \u2191\u2191\u03bd S < \u22a4\nh\u03bd : \u03bd = Measure.withDensity \u03bc fun x => \u2191\u2016f x\u2016\u208a\n\u03b5 : \u211d\u22650\n\u03b50 : 0 < \u2191\u03b5\nthis : \u2200\u1da0 (i : \u03b9) in atTop, \u2191\u2191\u03bd (s i) \u2208 Icc (\u2191\u2191\u03bd S - \u2191\u03b5) (\u2191\u2191\u03bd S + \u2191\u03b5)\ni : \u03b9\nhi : \u2191\u2191\u03bd (s i) \u2208 Icc (\u2191\u2191\u03bd S - \u2191\u03b5) (\u2191\u2191\u03bd S + \u2191\u03b5)\n\u22a2 \u2191\u2016\u222b (x : \u03b1) in S \\ s i, f x \u2202\u03bc\u2016\u208a \u2264 \u2191\u03b5"}, {"tactic": "refine' (ennnorm_integral_le_lintegral_ennnorm _).trans _", "annotated_tactic": ["refine' (<a>ennnorm_integral_le_lintegral_ennnorm</a> _).<a>trans</a> _", [{"full_name": "MeasureTheory.ennnorm_integral_le_lintegral_ennnorm", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [974, 9], "def_end_pos": [974, 46]}, {"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [120, 7], "def_end_pos": [120, 18]}]], "state_before": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\nf g : \u03b1 \u2192 E\ns\u271d t : Set \u03b1\n\u03bc \u03bd\u271d : Measure \u03b1\nl l' : Filter \u03b1\ninst\u271d\u00b2 : NormedSpace \u211d E\n\u03b9 : Type u_5\ninst\u271d\u00b9 : Countable \u03b9\ninst\u271d : SemilatticeSup \u03b9\ns : \u03b9 \u2192 Set \u03b1\nhsm : \u2200 (i : \u03b9), MeasurableSet (s i)\nh_mono : Monotone s\nS : Set \u03b1 := \u22c3 i, s i\nhfi : IntegrableOn f S\nhSm : MeasurableSet S\nhsub : \u2200 {i : \u03b9}, s i \u2286 S\n\u03bd : Measure \u03b1 := Measure.withDensity \u03bc fun x => \u2191\u2016f x\u2016\u208a\nhfi' : \u2191\u2191\u03bd S < \u22a4\nh\u03bd : \u03bd = Measure.withDensity \u03bc fun x => \u2191\u2016f x\u2016\u208a\n\u03b5 : \u211d\u22650\n\u03b50 : 0 < \u2191\u03b5\nthis : \u2200\u1da0 (i : \u03b9) in atTop, \u2191\u2191\u03bd (s i) \u2208 Icc (\u2191\u2191\u03bd S - \u2191\u03b5) (\u2191\u2191\u03bd S + \u2191\u03b5)\ni : \u03b9\nhi : \u2191\u2191\u03bd (s i) \u2208 Icc (\u2191\u2191\u03bd S - \u2191\u03b5) (\u2191\u2191\u03bd S + \u2191\u03b5)\n\u22a2 \u2191\u2016\u222b (x : \u03b1) in S \\ s i, f x \u2202\u03bc\u2016\u208a \u2264 \u2191\u03b5", "state_after": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\nf g : \u03b1 \u2192 E\ns\u271d t : Set \u03b1\n\u03bc \u03bd\u271d : Measure \u03b1\nl l' : Filter \u03b1\ninst\u271d\u00b2 : NormedSpace \u211d E\n\u03b9 : Type u_5\ninst\u271d\u00b9 : Countable \u03b9\ninst\u271d : SemilatticeSup \u03b9\ns : \u03b9 \u2192 Set \u03b1\nhsm : \u2200 (i : \u03b9), MeasurableSet (s i)\nh_mono : Monotone s\nS : Set \u03b1 := \u22c3 i, s i\nhfi : IntegrableOn f S\nhSm : MeasurableSet S\nhsub : \u2200 {i : \u03b9}, s i \u2286 S\n\u03bd : Measure \u03b1 := Measure.withDensity \u03bc fun x => \u2191\u2016f x\u2016\u208a\nhfi' : \u2191\u2191\u03bd S < \u22a4\nh\u03bd : \u03bd = Measure.withDensity \u03bc fun x => \u2191\u2016f x\u2016\u208a\n\u03b5 : \u211d\u22650\n\u03b50 : 0 < \u2191\u03b5\nthis : \u2200\u1da0 (i : \u03b9) in atTop, \u2191\u2191\u03bd (s i) \u2208 Icc (\u2191\u2191\u03bd S - \u2191\u03b5) (\u2191\u2191\u03bd S + \u2191\u03b5)\ni : \u03b9\nhi : \u2191\u2191\u03bd (s i) \u2208 Icc (\u2191\u2191\u03bd S - \u2191\u03b5) (\u2191\u2191\u03bd S + \u2191\u03b5)\n\u22a2 \u222b\u207b (a : \u03b1) in S \\ s i, \u2191\u2016f a\u2016\u208a \u2202\u03bc \u2264 \u2191\u03b5"}, {"tactic": "rw [\u2190 withDensity_apply _ (hSm.diff (hsm _)), \u2190 h\u03bd, measure_diff hsub (hsm _)]", "annotated_tactic": ["rw [\u2190 <a>withDensity_apply</a> _ (hSm.diff (hsm _)), \u2190 h\u03bd, <a>measure_diff</a> hsub (hsm _)]", [{"full_name": "MeasureTheory.withDensity_apply", "def_path": "Mathlib/MeasureTheory/Measure/WithDensity.lean", "def_pos": [39, 9], "def_end_pos": [39, 26]}, {"full_name": "MeasureTheory.measure_diff", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [252, 9], "def_end_pos": [252, 21]}]], "state_before": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\nf g : \u03b1 \u2192 E\ns\u271d t : Set \u03b1\n\u03bc \u03bd\u271d : Measure \u03b1\nl l' : Filter \u03b1\ninst\u271d\u00b2 : NormedSpace \u211d E\n\u03b9 : Type u_5\ninst\u271d\u00b9 : Countable \u03b9\ninst\u271d : SemilatticeSup \u03b9\ns : \u03b9 \u2192 Set \u03b1\nhsm : \u2200 (i : \u03b9), MeasurableSet (s i)\nh_mono : Monotone s\nS : Set \u03b1 := \u22c3 i, s i\nhfi : IntegrableOn f S\nhSm : MeasurableSet S\nhsub : \u2200 {i : \u03b9}, s i \u2286 S\n\u03bd : Measure \u03b1 := Measure.withDensity \u03bc fun x => \u2191\u2016f x\u2016\u208a\nhfi' : \u2191\u2191\u03bd S < \u22a4\nh\u03bd : \u03bd = Measure.withDensity \u03bc fun x => \u2191\u2016f x\u2016\u208a\n\u03b5 : \u211d\u22650\n\u03b50 : 0 < \u2191\u03b5\nthis : \u2200\u1da0 (i : \u03b9) in atTop, \u2191\u2191\u03bd (s i) \u2208 Icc (\u2191\u2191\u03bd S - \u2191\u03b5) (\u2191\u2191\u03bd S + \u2191\u03b5)\ni : \u03b9\nhi : \u2191\u2191\u03bd (s i) \u2208 Icc (\u2191\u2191\u03bd S - \u2191\u03b5) (\u2191\u2191\u03bd S + \u2191\u03b5)\n\u22a2 \u222b\u207b (a : \u03b1) in S \\ s i, \u2191\u2016f a\u2016\u208a \u2202\u03bc \u2264 \u2191\u03b5", "state_after": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\nf g : \u03b1 \u2192 E\ns\u271d t : Set \u03b1\n\u03bc \u03bd\u271d : Measure \u03b1\nl l' : Filter \u03b1\ninst\u271d\u00b2 : NormedSpace \u211d E\n\u03b9 : Type u_5\ninst\u271d\u00b9 : Countable \u03b9\ninst\u271d : SemilatticeSup \u03b9\ns : \u03b9 \u2192 Set \u03b1\nhsm : \u2200 (i : \u03b9), MeasurableSet (s i)\nh_mono : Monotone s\nS : Set \u03b1 := \u22c3 i, s i\nhfi : IntegrableOn f S\nhSm : MeasurableSet S\nhsub : \u2200 {i : \u03b9}, s i \u2286 S\n\u03bd : Measure \u03b1 := Measure.withDensity \u03bc fun x => \u2191\u2016f x\u2016\u208a\nhfi' : \u2191\u2191\u03bd S < \u22a4\nh\u03bd : \u03bd = Measure.withDensity \u03bc fun x => \u2191\u2016f x\u2016\u208a\n\u03b5 : \u211d\u22650\n\u03b50 : 0 < \u2191\u03b5\nthis : \u2200\u1da0 (i : \u03b9) in atTop, \u2191\u2191\u03bd (s i) \u2208 Icc (\u2191\u2191\u03bd S - \u2191\u03b5) (\u2191\u2191\u03bd S + \u2191\u03b5)\ni : \u03b9\nhi : \u2191\u2191\u03bd (s i) \u2208 Icc (\u2191\u2191\u03bd S - \u2191\u03b5) (\u2191\u2191\u03bd S + \u2191\u03b5)\n\u22a2 \u2191\u2191\u03bd S - \u2191\u2191\u03bd (s i) \u2264 \u2191\u03b5\n\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\nf g : \u03b1 \u2192 E\ns\u271d t : Set \u03b1\n\u03bc \u03bd\u271d : Measure \u03b1\nl l' : Filter \u03b1\ninst\u271d\u00b2 : NormedSpace \u211d E\n\u03b9 : Type u_5\ninst\u271d\u00b9 : Countable \u03b9\ninst\u271d : SemilatticeSup \u03b9\ns : \u03b9 \u2192 Set \u03b1\nhsm : \u2200 (i : \u03b9), MeasurableSet (s i)\nh_mono : Monotone s\nS : Set \u03b1 := \u22c3 i, s i\nhfi : IntegrableOn f S\nhSm : MeasurableSet S\nhsub : \u2200 {i : \u03b9}, s i \u2286 S\n\u03bd : Measure \u03b1 := Measure.withDensity \u03bc fun x => \u2191\u2016f x\u2016\u208a\nhfi' : \u2191\u2191\u03bd S < \u22a4\nh\u03bd : \u03bd = Measure.withDensity \u03bc fun x => \u2191\u2016f x\u2016\u208a\n\u03b5 : \u211d\u22650\n\u03b50 : 0 < \u2191\u03b5\nthis : \u2200\u1da0 (i : \u03b9) in atTop, \u2191\u2191\u03bd (s i) \u2208 Icc (\u2191\u2191\u03bd S - \u2191\u03b5) (\u2191\u2191\u03bd S + \u2191\u03b5)\ni : \u03b9\nhi : \u2191\u2191\u03bd (s i) \u2208 Icc (\u2191\u2191\u03bd S - \u2191\u03b5) (\u2191\u2191\u03bd S + \u2191\u03b5)\n\u22a2 \u2191\u2191\u03bd (s i) \u2260 \u22a4"}, {"tactic": "exacts [tsub_le_iff_tsub_le.mp hi.1,\n  (hi.2.trans_lt <| ENNReal.add_lt_top.2 \u27e8hfi', ENNReal.coe_lt_top\u27e9).ne]", "annotated_tactic": ["exacts [tsub_le_iff_tsub_le.mp hi.1,\n    (hi.2.<a>trans_lt</a> <| <a>ENNReal.add_lt_top</a>.2 \u27e8hfi', <a>ENNReal.coe_lt_top</a>\u27e9).<a>ne</a>]", [{"full_name": "LE.le.trans_lt", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [124, 7], "def_end_pos": [124, 21]}, {"full_name": "ENNReal.add_lt_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [561, 17], "def_end_pos": [561, 27]}, {"full_name": "ENNReal.coe_lt_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [308, 17], "def_end_pos": [308, 27]}, {"full_name": "LT.lt.ne", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [152, 7], "def_end_pos": [152, 15]}]], "state_before": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\nf g : \u03b1 \u2192 E\ns\u271d t : Set \u03b1\n\u03bc \u03bd\u271d : Measure \u03b1\nl l' : Filter \u03b1\ninst\u271d\u00b2 : NormedSpace \u211d E\n\u03b9 : Type u_5\ninst\u271d\u00b9 : Countable \u03b9\ninst\u271d : SemilatticeSup \u03b9\ns : \u03b9 \u2192 Set \u03b1\nhsm : \u2200 (i : \u03b9), MeasurableSet (s i)\nh_mono : Monotone s\nS : Set \u03b1 := \u22c3 i, s i\nhfi : IntegrableOn f S\nhSm : MeasurableSet S\nhsub : \u2200 {i : \u03b9}, s i \u2286 S\n\u03bd : Measure \u03b1 := Measure.withDensity \u03bc fun x => \u2191\u2016f x\u2016\u208a\nhfi' : \u2191\u2191\u03bd S < \u22a4\nh\u03bd : \u03bd = Measure.withDensity \u03bc fun x => \u2191\u2016f x\u2016\u208a\n\u03b5 : \u211d\u22650\n\u03b50 : 0 < \u2191\u03b5\nthis : \u2200\u1da0 (i : \u03b9) in atTop, \u2191\u2191\u03bd (s i) \u2208 Icc (\u2191\u2191\u03bd S - \u2191\u03b5) (\u2191\u2191\u03bd S + \u2191\u03b5)\ni : \u03b9\nhi : \u2191\u2191\u03bd (s i) \u2208 Icc (\u2191\u2191\u03bd S - \u2191\u03b5) (\u2191\u2191\u03bd S + \u2191\u03b5)\n\u22a2 \u2191\u2191\u03bd S - \u2191\u2191\u03bd (s i) \u2264 \u2191\u03b5\n\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\nf g : \u03b1 \u2192 E\ns\u271d t : Set \u03b1\n\u03bc \u03bd\u271d : Measure \u03b1\nl l' : Filter \u03b1\ninst\u271d\u00b2 : NormedSpace \u211d E\n\u03b9 : Type u_5\ninst\u271d\u00b9 : Countable \u03b9\ninst\u271d : SemilatticeSup \u03b9\ns : \u03b9 \u2192 Set \u03b1\nhsm : \u2200 (i : \u03b9), MeasurableSet (s i)\nh_mono : Monotone s\nS : Set \u03b1 := \u22c3 i, s i\nhfi : IntegrableOn f S\nhSm : MeasurableSet S\nhsub : \u2200 {i : \u03b9}, s i \u2286 S\n\u03bd : Measure \u03b1 := Measure.withDensity \u03bc fun x => \u2191\u2016f x\u2016\u208a\nhfi' : \u2191\u2191\u03bd S < \u22a4\nh\u03bd : \u03bd = Measure.withDensity \u03bc fun x => \u2191\u2016f x\u2016\u208a\n\u03b5 : \u211d\u22650\n\u03b50 : 0 < \u2191\u03b5\nthis : \u2200\u1da0 (i : \u03b9) in atTop, \u2191\u2191\u03bd (s i) \u2208 Icc (\u2191\u2191\u03bd S - \u2191\u03b5) (\u2191\u2191\u03bd S + \u2191\u03b5)\ni : \u03b9\nhi : \u2191\u2191\u03bd (s i) \u2208 Icc (\u2191\u2191\u03bd S - \u2191\u03b5) (\u2191\u2191\u03bd S + \u2191\u03b5)\n\u22a2 \u2191\u2191\u03bd (s i) \u2260 \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/QPF/Univariate/Basic.lean", "full_name": "QPF.Cofix.dest_corec", "start": [421, 1], "end": [427, 53], "traced_tactics": [{"tactic": "conv =>\n  lhs\n  rw [Cofix.dest, Cofix.corec];", "annotated_tactic": ["conv =>\n    lhs\n    rw [<a>Cofix.dest</a>, <a>Cofix.corec</a>];", [{"full_name": "QPF.Cofix.dest", "def_path": "Mathlib/Data/QPF/Univariate/Basic.lean", "def_pos": [407, 5], "def_end_pos": [407, 15]}, {"full_name": "QPF.Cofix.corec", "def_path": "Mathlib/Data/QPF/Univariate/Basic.lean", "def_pos": [402, 5], "def_end_pos": [402, 16]}]], "state_before": "F : Type u \u2192 Type u\ninst\u271d : Functor F\nq : QPF F\n\u03b1 : Type u\ng : \u03b1 \u2192 F \u03b1\nx : \u03b1\n\u22a2 dest (corec g x) = corec g <$> g x", "state_after": "F : Type u \u2192 Type u\ninst\u271d : Functor F\nq : QPF F\n\u03b1 : Type u\ng : \u03b1 \u2192 F \u03b1\nx : \u03b1\n\u22a2 Quot.lift (fun x => Quot.mk Mcongr <$> abs (PFunctor.M.dest x))\n      (_ :\n        \u2200 (x y : PFunctor.M (P F)),\n          Mcongr x y \u2192\n            (fun x => Quot.mk Mcongr <$> abs (PFunctor.M.dest x)) x =\n              (fun x => Quot.mk Mcongr <$> abs (PFunctor.M.dest x)) y)\n      (Quot.mk Mcongr (corecF g x)) =\n    corec g <$> g x"}, {"tactic": "dsimp", "annotated_tactic": ["dsimp", []], "state_before": "F : Type u \u2192 Type u\ninst\u271d : Functor F\nq : QPF F\n\u03b1 : Type u\ng : \u03b1 \u2192 F \u03b1\nx : \u03b1\n\u22a2 Quot.lift (fun x => Quot.mk Mcongr <$> abs (PFunctor.M.dest x))\n      (_ :\n        \u2200 (x y : PFunctor.M (P F)),\n          Mcongr x y \u2192\n            (fun x => Quot.mk Mcongr <$> abs (PFunctor.M.dest x)) x =\n              (fun x => Quot.mk Mcongr <$> abs (PFunctor.M.dest x)) y)\n      (Quot.mk Mcongr (corecF g x)) =\n    corec g <$> g x", "state_after": "F : Type u \u2192 Type u\ninst\u271d : Functor F\nq : QPF F\n\u03b1 : Type u\ng : \u03b1 \u2192 F \u03b1\nx : \u03b1\n\u22a2 Quot.mk Mcongr <$> abs (PFunctor.M.dest (corecF g x)) = corec g <$> g x"}, {"tactic": "rw [corecF_eq, abs_map, abs_repr, \u2190 comp_map]", "annotated_tactic": ["rw [<a>corecF_eq</a>, <a>abs_map</a>, <a>abs_repr</a>, \u2190 <a>comp_map</a>]", [{"full_name": "QPF.corecF_eq", "def_path": "Mathlib/Data/QPF/Univariate/Basic.lean", "def_pos": [377, 9], "def_end_pos": [377, 18]}, {"full_name": "QPF.abs_map", "def_path": "Mathlib/Data/QPF/Univariate/Basic.lean", "def_pos": [56, 3], "def_end_pos": [56, 10]}, {"full_name": "QPF.abs_repr", "def_path": "Mathlib/Data/QPF/Univariate/Basic.lean", "def_pos": [55, 3], "def_end_pos": [55, 11]}, {"full_name": "QPF.comp_map", "def_path": "Mathlib/Data/QPF/Univariate/Basic.lean", "def_pos": [78, 9], "def_end_pos": [78, 17]}]], "state_before": "F : Type u \u2192 Type u\ninst\u271d : Functor F\nq : QPF F\n\u03b1 : Type u\ng : \u03b1 \u2192 F \u03b1\nx : \u03b1\n\u22a2 Quot.mk Mcongr <$> abs (PFunctor.M.dest (corecF g x)) = corec g <$> g x", "state_after": "F : Type u \u2192 Type u\ninst\u271d : Functor F\nq : QPF F\n\u03b1 : Type u\ng : \u03b1 \u2192 F \u03b1\nx : \u03b1\n\u22a2 (Quot.mk Mcongr \u2218 corecF g) <$> g x = corec g <$> g x"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "F : Type u \u2192 Type u\ninst\u271d : Functor F\nq : QPF F\n\u03b1 : Type u\ng : \u03b1 \u2192 F \u03b1\nx : \u03b1\n\u22a2 (Quot.mk Mcongr \u2218 corecF g) <$> g x = corec g <$> g x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "full_name": "MeasureTheory.OuterMeasure.ofFunction_caratheodory", "start": [1062, 1], "end": [1077, 45], "traced_tactics": [{"tactic": "apply (isCaratheodory_iff_le _).mpr", "annotated_tactic": ["apply (<a>isCaratheodory_iff_le</a> _).<a>mpr</a>", [{"full_name": "MeasureTheory.OuterMeasure.isCaratheodory_iff_le", "def_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "def_pos": [1047, 9], "def_end_pos": [1047, 30]}, {"full_name": "Iff.mpr", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [92, 3], "def_end_pos": [92, 6]}]], "state_before": "\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nh\u2080 : m \u2205 = 0\nhs : \u2200 (t : Set \u03b1), m (t \u2229 s) + m (t \\ s) \u2264 m t\n\u22a2 MeasurableSet s", "state_after": "\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nh\u2080 : m \u2205 = 0\nhs : \u2200 (t : Set \u03b1), m (t \u2229 s) + m (t \\ s) \u2264 m t\n\u22a2 \u2200 (t : Set \u03b1),\n    \u2191(OuterMeasure.ofFunction m h\u2080) (t \u2229 s) + \u2191(OuterMeasure.ofFunction m h\u2080) (t \\ s) \u2264\n      \u2191(OuterMeasure.ofFunction m h\u2080) t"}, {"tactic": "refine' fun t => le_iInf fun f => le_iInf fun hf => _", "annotated_tactic": ["refine' fun t => <a>le_iInf</a> fun f => <a>le_iInf</a> fun hf => _", [{"full_name": "le_iInf", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [879, 9], "def_end_pos": [879, 16]}, {"full_name": "le_iInf", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [879, 9], "def_end_pos": [879, 16]}]], "state_before": "\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nh\u2080 : m \u2205 = 0\nhs : \u2200 (t : Set \u03b1), m (t \u2229 s) + m (t \\ s) \u2264 m t\n\u22a2 \u2200 (t : Set \u03b1),\n    \u2191(OuterMeasure.ofFunction m h\u2080) (t \u2229 s) + \u2191(OuterMeasure.ofFunction m h\u2080) (t \\ s) \u2264\n      \u2191(OuterMeasure.ofFunction m h\u2080) t", "state_after": "\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nh\u2080 : m \u2205 = 0\nhs : \u2200 (t : Set \u03b1), m (t \u2229 s) + m (t \\ s) \u2264 m t\nt : Set \u03b1\nf : \u2115 \u2192 Set \u03b1\nhf : t \u2286 \u22c3 i, f i\n\u22a2 \u2191(OuterMeasure.ofFunction m h\u2080) (t \u2229 s) + \u2191(OuterMeasure.ofFunction m h\u2080) (t \\ s) \u2264 \u2211' (i : \u2115), m (f i)"}, {"tactic": "refine'\n  le_trans\n    (add_le_add ((iInf_le_of_le fun i => f i \u2229 s) <| iInf_le _ _)\n      ((iInf_le_of_le fun i => f i \\ s) <| iInf_le _ _))\n    _", "annotated_tactic": ["refine'\n    <a>le_trans</a>\n      (<a>add_le_add</a> ((<a>iInf_le_of_le</a> fun i => f i \u2229 s) <| <a>iInf_le</a> _ _)\n        ((<a>iInf_le_of_le</a> fun i => f i \\ s) <| <a>iInf_le</a> _ _))\n      _", [{"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}, {"full_name": "add_le_add", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [205, 15], "def_end_pos": [205, 25]}, {"full_name": "iInf_le_of_le", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [853, 9], "def_end_pos": [853, 22]}, {"full_name": "iInf_le", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [814, 9], "def_end_pos": [814, 16]}, {"full_name": "iInf_le_of_le", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [853, 9], "def_end_pos": [853, 22]}, {"full_name": "iInf_le", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [814, 9], "def_end_pos": [814, 16]}]], "state_before": "\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nh\u2080 : m \u2205 = 0\nhs : \u2200 (t : Set \u03b1), m (t \u2229 s) + m (t \\ s) \u2264 m t\nt : Set \u03b1\nf : \u2115 \u2192 Set \u03b1\nhf : t \u2286 \u22c3 i, f i\n\u22a2 \u2191(OuterMeasure.ofFunction m h\u2080) (t \u2229 s) + \u2191(OuterMeasure.ofFunction m h\u2080) (t \\ s) \u2264 \u2211' (i : \u2115), m (f i)", "state_after": "case refine'_1\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nh\u2080 : m \u2205 = 0\nhs : \u2200 (t : Set \u03b1), m (t \u2229 s) + m (t \\ s) \u2264 m t\nt : Set \u03b1\nf : \u2115 \u2192 Set \u03b1\nhf : t \u2286 \u22c3 i, f i\n\u22a2 t \u2229 s \u2286 \u22c3 i, (fun i => f i \u2229 s) i\n\ncase refine'_2\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nh\u2080 : m \u2205 = 0\nhs : \u2200 (t : Set \u03b1), m (t \u2229 s) + m (t \\ s) \u2264 m t\nt : Set \u03b1\nf : \u2115 \u2192 Set \u03b1\nhf : t \u2286 \u22c3 i, f i\n\u22a2 t \\ s \u2286 \u22c3 i, (fun i => f i \\ s) i\n\ncase refine'_3\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nh\u2080 : m \u2205 = 0\nhs : \u2200 (t : Set \u03b1), m (t \u2229 s) + m (t \\ s) \u2264 m t\nt : Set \u03b1\nf : \u2115 \u2192 Set \u03b1\nhf : t \u2286 \u22c3 i, f i\n\u22a2 \u2211' (i : \u2115), m ((fun i => f i \u2229 s) i) + \u2211' (i : \u2115), m ((fun i => f i \\ s) i) \u2264 \u2211' (i : \u2115), m (f i)"}, {"tactic": "rw [\u2190 iUnion_inter]", "annotated_tactic": ["rw [\u2190 <a>iUnion_inter</a>]", [{"full_name": "Set.iUnion_inter", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [639, 9], "def_end_pos": [639, 21]}]], "state_before": "case refine'_1\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nh\u2080 : m \u2205 = 0\nhs : \u2200 (t : Set \u03b1), m (t \u2229 s) + m (t \\ s) \u2264 m t\nt : Set \u03b1\nf : \u2115 \u2192 Set \u03b1\nhf : t \u2286 \u22c3 i, f i\n\u22a2 t \u2229 s \u2286 \u22c3 i, (fun i => f i \u2229 s) i", "state_after": "case refine'_1\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nh\u2080 : m \u2205 = 0\nhs : \u2200 (t : Set \u03b1), m (t \u2229 s) + m (t \\ s) \u2264 m t\nt : Set \u03b1\nf : \u2115 \u2192 Set \u03b1\nhf : t \u2286 \u22c3 i, f i\n\u22a2 t \u2229 s \u2286 (\u22c3 i, f i) \u2229 s"}, {"tactic": "exact inter_subset_inter_left _ hf", "annotated_tactic": ["exact <a>inter_subset_inter_left</a> _ hf", [{"full_name": "Set.inter_subset_inter_left", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1027, 9], "def_end_pos": [1027, 32]}]], "state_before": "case refine'_1\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nh\u2080 : m \u2205 = 0\nhs : \u2200 (t : Set \u03b1), m (t \u2229 s) + m (t \\ s) \u2264 m t\nt : Set \u03b1\nf : \u2115 \u2192 Set \u03b1\nhf : t \u2286 \u22c3 i, f i\n\u22a2 t \u2229 s \u2286 (\u22c3 i, f i) \u2229 s", "state_after": "no goals"}, {"tactic": "rw [\u2190 iUnion_diff]", "annotated_tactic": ["rw [\u2190 <a>iUnion_diff</a>]", [{"full_name": "Set.iUnion_diff", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [678, 9], "def_end_pos": [678, 20]}]], "state_before": "case refine'_2\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nh\u2080 : m \u2205 = 0\nhs : \u2200 (t : Set \u03b1), m (t \u2229 s) + m (t \\ s) \u2264 m t\nt : Set \u03b1\nf : \u2115 \u2192 Set \u03b1\nhf : t \u2286 \u22c3 i, f i\n\u22a2 t \\ s \u2286 \u22c3 i, (fun i => f i \\ s) i", "state_after": "case refine'_2\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nh\u2080 : m \u2205 = 0\nhs : \u2200 (t : Set \u03b1), m (t \u2229 s) + m (t \\ s) \u2264 m t\nt : Set \u03b1\nf : \u2115 \u2192 Set \u03b1\nhf : t \u2286 \u22c3 i, f i\n\u22a2 t \\ s \u2286 (\u22c3 i, f i) \\ s"}, {"tactic": "exact diff_subset_diff_left hf", "annotated_tactic": ["exact <a>diff_subset_diff_left</a> hf", [{"full_name": "Set.diff_subset_diff_left", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1908, 9], "def_end_pos": [1908, 30]}]], "state_before": "case refine'_2\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nh\u2080 : m \u2205 = 0\nhs : \u2200 (t : Set \u03b1), m (t \u2229 s) + m (t \\ s) \u2264 m t\nt : Set \u03b1\nf : \u2115 \u2192 Set \u03b1\nhf : t \u2286 \u22c3 i, f i\n\u22a2 t \\ s \u2286 (\u22c3 i, f i) \\ s", "state_after": "no goals"}, {"tactic": "rw [\u2190 ENNReal.tsum_add]", "annotated_tactic": ["rw [\u2190 <a>ENNReal.tsum_add</a>]", [{"full_name": "ENNReal.tsum_add", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [823, 19], "def_end_pos": [823, 27]}]], "state_before": "case refine'_3\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nh\u2080 : m \u2205 = 0\nhs : \u2200 (t : Set \u03b1), m (t \u2229 s) + m (t \\ s) \u2264 m t\nt : Set \u03b1\nf : \u2115 \u2192 Set \u03b1\nhf : t \u2286 \u22c3 i, f i\n\u22a2 \u2211' (i : \u2115), m ((fun i => f i \u2229 s) i) + \u2211' (i : \u2115), m ((fun i => f i \\ s) i) \u2264 \u2211' (i : \u2115), m (f i)", "state_after": "case refine'_3\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nh\u2080 : m \u2205 = 0\nhs : \u2200 (t : Set \u03b1), m (t \u2229 s) + m (t \\ s) \u2264 m t\nt : Set \u03b1\nf : \u2115 \u2192 Set \u03b1\nhf : t \u2286 \u22c3 i, f i\n\u22a2 \u2211' (a : \u2115), (m ((fun i => f i \u2229 s) a) + m ((fun i => f i \\ s) a)) \u2264 \u2211' (i : \u2115), m (f i)"}, {"tactic": "exact ENNReal.tsum_le_tsum fun i => hs _", "annotated_tactic": ["exact <a>ENNReal.tsum_le_tsum</a> fun i => hs _", [{"full_name": "ENNReal.tsum_le_tsum", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [827, 19], "def_end_pos": [827, 31]}]], "state_before": "case refine'_3\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nh\u2080 : m \u2205 = 0\nhs : \u2200 (t : Set \u03b1), m (t \u2229 s) + m (t \\ s) \u2264 m t\nt : Set \u03b1\nf : \u2115 \u2192 Set \u03b1\nhf : t \u2286 \u22c3 i, f i\n\u22a2 \u2211' (a : \u2115), (m ((fun i => f i \u2229 s) a) + m ((fun i => f i \\ s) a)) \u2264 \u2211' (i : \u2115), m (f i)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finmap.lean", "full_name": "Finmap.liftOn_toFinmap", "start": [118, 1], "end": [120, 6], "traced_tactics": [{"tactic": "cases s", "annotated_tactic": ["cases s", []], "state_before": "\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\n\u03b3 : Type u_1\ns : AList \u03b2\nf : AList \u03b2 \u2192 \u03b3\nH : \u2200 (a b : AList \u03b2), a.entries ~ b.entries \u2192 f a = f b\n\u22a2 liftOn \u27e6s\u27e7 f H = f s", "state_after": "case mk\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\n\u03b3 : Type u_1\nf : AList \u03b2 \u2192 \u03b3\nH : \u2200 (a b : AList \u03b2), a.entries ~ b.entries \u2192 f a = f b\nentries\u271d : List (Sigma \u03b2)\nnodupKeys\u271d : NodupKeys entries\u271d\n\u22a2 liftOn \u27e6{ entries := entries\u271d, nodupKeys := nodupKeys\u271d }\u27e7 f H = f { entries := entries\u271d, nodupKeys := nodupKeys\u271d }"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case mk\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\n\u03b3 : Type u_1\nf : AList \u03b2 \u2192 \u03b3\nH : \u2200 (a b : AList \u03b2), a.entries ~ b.entries \u2192 f a = f b\nentries\u271d : List (Sigma \u03b2)\nnodupKeys\u271d : NodupKeys entries\u271d\n\u22a2 liftOn \u27e6{ entries := entries\u271d, nodupKeys := nodupKeys\u271d }\u27e7 f H = f { entries := entries\u271d, nodupKeys := nodupKeys\u271d }", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean", "full_name": "MeasureTheory.norm_condexpInd_le", "start": [311, 1], "end": [312, 88], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/ProbabilityMassFunction/Monad.lean", "full_name": "PMF.mem_support_bindOnSupport_iff", "start": [238, 1], "end": [240, 70], "traced_tactics": [{"tactic": "simp only [support_bindOnSupport, Set.mem_setOf_eq, Set.mem_iUnion]", "annotated_tactic": ["simp only [<a>support_bindOnSupport</a>, <a>Set.mem_setOf_eq</a>, <a>Set.mem_iUnion</a>]", [{"full_name": "PMF.support_bindOnSupport", "def_path": "Mathlib/Probability/ProbabilityMassFunction/Monad.lean", "def_pos": [224, 9], "def_end_pos": [224, 30]}, {"full_name": "Set.mem_setOf_eq", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [256, 29], "def_end_pos": [256, 41]}, {"full_name": "Set.mem_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [201, 9], "def_end_pos": [201, 19]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\nf : (a : \u03b1) \u2192 a \u2208 support p \u2192 PMF \u03b2\nb : \u03b2\n\u22a2 b \u2208 support (bindOnSupport p f) \u2194 \u2203 a h, b \u2208 support (f a h)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Intervals/WithBotTop.lean", "full_name": "WithBot.image_coe_Iic", "start": [201, 1], "end": [202, 95], "traced_tactics": [{"tactic": "rw [\u2190 preimage_coe_Iic, image_preimage_eq_inter_range, range_coe, inter_comm, Ioi_inter_Iic]", "annotated_tactic": ["rw [\u2190 <a>preimage_coe_Iic</a>, <a>image_preimage_eq_inter_range</a>, <a>range_coe</a>, <a>inter_comm</a>, <a>Ioi_inter_Iic</a>]", [{"full_name": "WithBot.preimage_coe_Iic", "def_path": "Mathlib/Data/Set/Intervals/WithBotTop.lean", "def_pos": [162, 9], "def_end_pos": [162, 25]}, {"full_name": "Set.image_preimage_eq_inter_range", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [796, 9], "def_end_pos": [796, 38]}, {"full_name": "WithBot.range_coe", "def_path": "Mathlib/Data/Set/Intervals/WithBotTop.lean", "def_pos": [142, 9], "def_end_pos": [142, 18]}, {"full_name": "Set.inter_comm", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [940, 9], "def_end_pos": [940, 19]}, {"full_name": "Set.Ioi_inter_Iic", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [626, 9], "def_end_pos": [626, 22]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : PartialOrder \u03b1\na b : \u03b1\n\u22a2 some '' Iic a = Ioc \u22a5 \u2191a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Int/GCD.lean", "full_name": "Int.gcd_eq_right", "start": [366, 1], "end": [366, 99], "traced_tactics": [{"tactic": "rw [gcd_comm, gcd_eq_left H]", "annotated_tactic": ["rw [<a>gcd_comm</a>, <a>gcd_eq_left</a> H]", [{"full_name": "Int.gcd_comm", "def_path": "Mathlib/Data/Int/GCD.lean", "def_pos": [264, 9], "def_end_pos": [264, 17]}, {"full_name": "Int.gcd_eq_left", "def_path": "Mathlib/Data/Int/GCD.lean", "def_pos": [362, 9], "def_end_pos": [362, 20]}]], "state_before": "i j : \u2124\nH : j \u2223 i\n\u22a2 gcd i j = natAbs j", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "full_name": "MeasureTheory.NullMeasurableSet.mono_ac", "start": [2424, 1], "end": [2426, 60], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Hausdorff.lean", "full_name": "MeasureTheory.hausdorffMeasure_pi_real", "start": [950, 1], "end": [1039, 65], "traced_tactics": [{"tactic": "refine'\n  (pi_eq_generateFrom (fun _ => Real.borel_eq_generateFrom_Ioo_rat.symm)\n      (fun _ => Real.isPiSystem_Ioo_rat) (fun _ => Real.finiteSpanningSetsInIooRat _) _).symm", "annotated_tactic": ["refine'\n    (<a>pi_eq_generateFrom</a> (fun _ => Real.borel_eq_generateFrom_Ioo_rat.symm)\n        (fun _ => <a>Real.isPiSystem_Ioo_rat</a>) (fun _ => <a>Real.finiteSpanningSetsInIooRat</a> _) _).<a>symm</a>", [{"full_name": "MeasureTheory.Measure.pi_eq_generateFrom", "def_path": "Mathlib/MeasureTheory/Constructions/Pi.lean", "def_pos": [360, 9], "def_end_pos": [360, 27]}, {"full_name": "Real.isPiSystem_Ioo_rat", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [1897, 9], "def_end_pos": [1897, 27]}, {"full_name": "Real.finiteSpanningSetsInIooRat", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [1926, 5], "def_end_pos": [1926, 31]}, {"full_name": "Eq.symm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [310, 9], "def_end_pos": [310, 16]}]], "state_before": "\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\n\u22a2 \u03bcH[\u2191(Fintype.card \u03b9)] = volume", "state_after": "\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\n\u22a2 \u2200 (s : \u03b9 \u2192 Set \u211d),\n    (\u2200 (i : \u03b9), s i \u2208 \u22c3 a, \u22c3 b, \u22c3 (_ : a < b), {Ioo \u2191a \u2191b}) \u2192\n      \u2191\u2191\u03bcH[\u2191(Fintype.card \u03b9)] (Set.pi univ s) = \u220f i : \u03b9, \u2191\u2191volume (s i)"}, {"tactic": "simp only [mem_iUnion, mem_singleton_iff]", "annotated_tactic": ["simp only [<a>mem_iUnion</a>, <a>mem_singleton_iff</a>]", [{"full_name": "Set.mem_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [201, 9], "def_end_pos": [201, 19]}, {"full_name": "Set.mem_singleton_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1273, 9], "def_end_pos": [1273, 26]}]], "state_before": "\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\n\u22a2 \u2200 (s : \u03b9 \u2192 Set \u211d),\n    (\u2200 (i : \u03b9), s i \u2208 \u22c3 a, \u22c3 b, \u22c3 (_ : a < b), {Ioo \u2191a \u2191b}) \u2192\n      \u2191\u2191\u03bcH[\u2191(Fintype.card \u03b9)] (Set.pi univ s) = \u220f i : \u03b9, \u2191\u2191volume (s i)", "state_after": "\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\n\u22a2 \u2200 (s : \u03b9 \u2192 Set \u211d),\n    (\u2200 (i : \u03b9), \u2203 i_1 i_2 h, s i = Ioo \u2191i_1 \u2191i_2) \u2192 \u2191\u2191\u03bcH[\u2191(Fintype.card \u03b9)] (Set.pi univ s) = \u220f i : \u03b9, \u2191\u2191volume (s i)"}, {"tactic": "intro s hs", "annotated_tactic": ["intro s hs", []], "state_before": "\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\n\u22a2 \u2200 (s : \u03b9 \u2192 Set \u211d),\n    (\u2200 (i : \u03b9), \u2203 i_1 i_2 h, s i = Ioo \u2191i_1 \u2191i_2) \u2192 \u2191\u2191\u03bcH[\u2191(Fintype.card \u03b9)] (Set.pi univ s) = \u220f i : \u03b9, \u2191\u2191volume (s i)", "state_after": "\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\ns : \u03b9 \u2192 Set \u211d\nhs : \u2200 (i : \u03b9), \u2203 i_1 i_2 h, s i = Ioo \u2191i_1 \u2191i_2\n\u22a2 \u2191\u2191\u03bcH[\u2191(Fintype.card \u03b9)] (Set.pi univ s) = \u220f i : \u03b9, \u2191\u2191volume (s i)"}, {"tactic": "choose a b H using hs", "annotated_tactic": ["choose a b H using hs", []], "state_before": "\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\ns : \u03b9 \u2192 Set \u211d\nhs : \u2200 (i : \u03b9), \u2203 i_1 i_2 h, s i = Ioo \u2191i_1 \u2191i_2\n\u22a2 \u2191\u2191\u03bcH[\u2191(Fintype.card \u03b9)] (Set.pi univ s) = \u220f i : \u03b9, \u2191\u2191volume (s i)", "state_after": "\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\ns : \u03b9 \u2192 Set \u211d\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), \u2203 h, s i = Ioo \u2191(a i) \u2191(b i)\n\u22a2 \u2191\u2191\u03bcH[\u2191(Fintype.card \u03b9)] (Set.pi univ s) = \u220f i : \u03b9, \u2191\u2191volume (s i)"}, {"tactic": "obtain rfl : s = fun i => Ioo (\u03b1 := \u211d) (a i) (b i)", "annotated_tactic": ["obtain rfl : s = fun i => <a>Ioo</a> (\u03b1 := \u211d) (a i) (b i)", [{"full_name": "Set.Ioo", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [44, 5], "def_end_pos": [44, 8]}]], "state_before": "\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\ns : \u03b9 \u2192 Set \u211d\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), \u2203 h, s i = Ioo \u2191(a i) \u2191(b i)\n\u22a2 \u2191\u2191\u03bcH[\u2191(Fintype.card \u03b9)] (Set.pi univ s) = \u220f i : \u03b9, \u2191\u2191volume (s i)", "state_after": "\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\ns : \u03b9 \u2192 Set \u211d\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), \u2203 h, s i = Ioo \u2191(a i) \u2191(b i)\n\u22a2 s = fun i => Ioo \u2191(a i) \u2191(b i)\n\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), \u2203 h, (fun i => Ioo \u2191(a i) \u2191(b i)) i = Ioo \u2191(a i) \u2191(b i)\n\u22a2 \u2191\u2191\u03bcH[\u2191(Fintype.card \u03b9)] (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) = \u220f i : \u03b9, \u2191\u2191volume ((fun i => Ioo \u2191(a i) \u2191(b i)) i)"}, {"tactic": "exact funext fun i => (H i).2", "annotated_tactic": ["exact <a>funext</a> fun i => (H i).2", [{"full_name": "funext", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [1555, 9], "def_end_pos": [1555, 15]}]], "state_before": "\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\ns : \u03b9 \u2192 Set \u211d\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), \u2203 h, s i = Ioo \u2191(a i) \u2191(b i)\n\u22a2 s = fun i => Ioo \u2191(a i) \u2191(b i)\n\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), \u2203 h, (fun i => Ioo \u2191(a i) \u2191(b i)) i = Ioo \u2191(a i) \u2191(b i)\n\u22a2 \u2191\u2191\u03bcH[\u2191(Fintype.card \u03b9)] (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) = \u220f i : \u03b9, \u2191\u2191volume ((fun i => Ioo \u2191(a i) \u2191(b i)) i)", "state_after": "\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), \u2203 h, (fun i => Ioo \u2191(a i) \u2191(b i)) i = Ioo \u2191(a i) \u2191(b i)\n\u22a2 \u2191\u2191\u03bcH[\u2191(Fintype.card \u03b9)] (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) = \u220f i : \u03b9, \u2191\u2191volume ((fun i => Ioo \u2191(a i) \u2191(b i)) i)"}, {"tactic": "replace H := fun i => (H i).1", "annotated_tactic": ["replace H := fun i => (H i).1", []], "state_before": "\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), \u2203 h, (fun i => Ioo \u2191(a i) \u2191(b i)) i = Ioo \u2191(a i) \u2191(b i)\n\u22a2 \u2191\u2191\u03bcH[\u2191(Fintype.card \u03b9)] (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) = \u220f i : \u03b9, \u2191\u2191volume ((fun i => Ioo \u2191(a i) \u2191(b i)) i)", "state_after": "\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\n\u22a2 \u2191\u2191\u03bcH[\u2191(Fintype.card \u03b9)] (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) = \u220f i : \u03b9, \u2191\u2191volume ((fun i => Ioo \u2191(a i) \u2191(b i)) i)"}, {"tactic": "apply le_antisymm _", "annotated_tactic": ["apply <a>le_antisymm</a> _", [{"full_name": "le_antisymm", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [188, 9], "def_end_pos": [188, 20]}]], "state_before": "\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\n\u22a2 \u2191\u2191\u03bcH[\u2191(Fintype.card \u03b9)] (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) = \u220f i : \u03b9, \u2191\u2191volume ((fun i => Ioo \u2191(a i) \u2191(b i)) i)", "state_after": "\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\n\u22a2 \u220f i : \u03b9, \u2191\u2191volume ((fun i => Ioo \u2191(a i) \u2191(b i)) i) \u2264 \u2191\u2191\u03bcH[\u2191(Fintype.card \u03b9)] (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i))\n\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\n\u22a2 \u2191\u2191\u03bcH[\u2191(Fintype.card \u03b9)] (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) \u2264 \u220f i : \u03b9, \u2191\u2191volume ((fun i => Ioo \u2191(a i) \u2191(b i)) i)"}, {"tactic": "have I : \u2200 i, 0 \u2264 (b i : \u211d) - a i := fun i => by\n  simpa only [sub_nonneg, Rat.cast_le] using (H i).le", "annotated_tactic": ["have I : \u2200 i, 0 \u2264 (b i : \u211d) - a i := fun i => by\n    simpa only [<a>sub_nonneg</a>, <a>Rat.cast_le</a>] using (H i).<a>le</a>", [{"full_name": "sub_nonneg", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [720, 30], "def_end_pos": [720, 40]}, {"full_name": "Rat.cast_le", "def_path": "Mathlib/Data/Rat/Cast/Order.lean", "def_pos": [54, 9], "def_end_pos": [54, 16]}, {"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [142, 7], "def_end_pos": [142, 15]}]], "state_before": "\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\n\u22a2 \u2191\u2191\u03bcH[\u2191(Fintype.card \u03b9)] (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) \u2264 \u220f i : \u03b9, \u2191\u2191volume ((fun i => Ioo \u2191(a i) \u2191(b i)) i)", "state_after": "\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u22a2 \u2191\u2191\u03bcH[\u2191(Fintype.card \u03b9)] (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) \u2264 \u220f i : \u03b9, \u2191\u2191volume ((fun i => Ioo \u2191(a i) \u2191(b i)) i)"}, {"tactic": "let \u03b3 := fun n : \u2115 => \u2200 i : \u03b9, Fin \u2308((b i : \u211d) - a i) * n\u2309\u208a", "annotated_tactic": ["let \u03b3 := fun n : \u2115 => \u2200 i : \u03b9, <a>Fin</a> \u2308((b i : \u211d) - a i) * n\u2309\u208a", [{"full_name": "Fin", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1745, 11], "def_end_pos": [1745, 14]}]], "state_before": "\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u22a2 \u2191\u2191\u03bcH[\u2191(Fintype.card \u03b9)] (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) \u2264 \u220f i : \u03b9, \u2191\u2191volume ((fun i => Ioo \u2191(a i) \u2191(b i)) i)", "state_after": "\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\n\u22a2 \u2191\u2191\u03bcH[\u2191(Fintype.card \u03b9)] (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) \u2264 \u220f i : \u03b9, \u2191\u2191volume ((fun i => Ioo \u2191(a i) \u2191(b i)) i)"}, {"tactic": "let t : \u2200 n : \u2115, \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) := fun n f =>\n  Set.pi univ fun i => Icc (a i + f i / n) (a i + (f i + 1) / n)", "annotated_tactic": ["let t : \u2200 n : \u2115, \u03b3 n \u2192 <a>Set</a> (\u03b9 \u2192 \u211d) := fun n f =>\n    <a>Set.pi</a> <a>univ</a> fun i => <a>Icc</a> (a i + f i / n) (a i + (f i + 1) / n)", [{"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}, {"full_name": "Set.pi", "def_path": "Mathlib/Data/Set/Prod.lean", "def_pos": [665, 5], "def_end_pos": [665, 7]}, {"full_name": "Set.univ", "def_path": "Mathlib/Init/Set.lean", "def_pos": [90, 5], "def_end_pos": [90, 9]}, {"full_name": "Set.Icc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [59, 5], "def_end_pos": [59, 8]}]], "state_before": "\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\n\u22a2 \u2191\u2191\u03bcH[\u2191(Fintype.card \u03b9)] (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) \u2264 \u220f i : \u03b9, \u2191\u2191volume ((fun i => Ioo \u2191(a i) \u2191(b i)) i)", "state_after": "\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\n\u22a2 \u2191\u2191\u03bcH[\u2191(Fintype.card \u03b9)] (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) \u2264 \u220f i : \u03b9, \u2191\u2191volume ((fun i => Ioo \u2191(a i) \u2191(b i)) i)"}, {"tactic": "have A : Tendsto (fun n : \u2115 => 1 / (n : \u211d\u22650\u221e)) atTop (\ud835\udcdd 0) := by\n  simp only [one_div, ENNReal.tendsto_inv_nat_nhds_zero]", "annotated_tactic": ["have A : <a>Tendsto</a> (fun n : \u2115 => 1 / (n : \u211d\u22650\u221e)) <a>atTop</a> (\ud835\udcdd 0) := by\n    simp only [<a>one_div</a>, <a>ENNReal.tendsto_inv_nat_nhds_zero</a>]", [{"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2939, 5], "def_end_pos": [2939, 12]}, {"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}, {"full_name": "one_div", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [318, 9], "def_end_pos": [318, 16]}, {"full_name": "ENNReal.tendsto_inv_nat_nhds_zero", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [556, 19], "def_end_pos": [556, 44]}]], "state_before": "\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\n\u22a2 \u2191\u2191\u03bcH[\u2191(Fintype.card \u03b9)] (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) \u2264 \u220f i : \u03b9, \u2191\u2191volume ((fun i => Ioo \u2191(a i) \u2191(b i)) i)", "state_after": "\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\nA : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0)\n\u22a2 \u2191\u2191\u03bcH[\u2191(Fintype.card \u03b9)] (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) \u2264 \u220f i : \u03b9, \u2191\u2191volume ((fun i => Ioo \u2191(a i) \u2191(b i)) i)"}, {"tactic": "have B : \u2200\u1da0 n in atTop, \u2200 i : \u03b3 n, diam (t n i) \u2264 1 / n := by\n  refine' eventually_atTop.2 \u27e81, fun n hn => _\u27e9\n  intro f\n  refine' diam_pi_le_of_le fun b => _\n  simp only [Real.ediam_Icc, add_div, ENNReal.ofReal_div_of_pos (Nat.cast_pos.mpr hn), le_refl,\n    add_sub_add_left_eq_sub, add_sub_cancel', ENNReal.ofReal_one, ENNReal.ofReal_coe_nat]", "annotated_tactic": ["have B : \u2200\u1da0 n in <a>atTop</a>, \u2200 i : \u03b3 n, <a>diam</a> (t n i) \u2264 1 / n := by\n    refine' <a>eventually_atTop</a>.2 \u27e81, fun n hn => _\u27e9\n    intro f\n    refine' <a>diam_pi_le_of_le</a> fun b => _\n    simp only [<a>Real.ediam_Icc</a>, <a>add_div</a>, <a>ENNReal.ofReal_div_of_pos</a> (Nat.cast_pos.mpr hn), <a>le_refl</a>,\n      <a>add_sub_add_left_eq_sub</a>, <a>add_sub_cancel'</a>, <a>ENNReal.ofReal_one</a>, <a>ENNReal.ofReal_coe_nat</a>]", [{"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}, {"full_name": "EMetric.diam", "def_path": "Mathlib/Topology/EMetricSpace/Basic.lean", "def_pos": [881, 19], "def_end_pos": [881, 23]}, {"full_name": "Filter.eventually_atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [178, 9], "def_end_pos": [178, 25]}, {"full_name": "EMetric.diam_pi_le_of_le", "def_path": "Mathlib/Topology/EMetricSpace/Basic.lean", "def_pos": [992, 9], "def_end_pos": [992, 25]}, {"full_name": "Real.ediam_Icc", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [1546, 9], "def_end_pos": [1546, 18]}, {"full_name": "add_div", "def_path": "Mathlib/Algebra/Field/Basic.lean", "def_pos": [29, 9], "def_end_pos": [29, 16]}, {"full_name": "ENNReal.ofReal_div_of_pos", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2248, 9], "def_end_pos": [2248, 26]}, {"full_name": "le_refl", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [50, 9], "def_end_pos": [50, 16]}, {"full_name": "add_sub_add_left_eq_sub", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [894, 3], "def_end_pos": [894, 14]}, {"full_name": "add_sub_cancel'", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [948, 30], "def_end_pos": [948, 45]}, {"full_name": "ENNReal.ofReal_one", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [248, 17], "def_end_pos": [248, 27]}, {"full_name": "ENNReal.ofReal_coe_nat", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [710, 17], "def_end_pos": [710, 31]}]], "state_before": "\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\nA : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0)\n\u22a2 \u2191\u2191\u03bcH[\u2191(Fintype.card \u03b9)] (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) \u2264 \u220f i : \u03b9, \u2191\u2191volume ((fun i => Ioo \u2191(a i) \u2191(b i)) i)", "state_after": "\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\nA : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0)\nB : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n\n\u22a2 \u2191\u2191\u03bcH[\u2191(Fintype.card \u03b9)] (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) \u2264 \u220f i : \u03b9, \u2191\u2191volume ((fun i => Ioo \u2191(a i) \u2191(b i)) i)"}, {"tactic": "have Hle : volume \u2264 (\u03bcH[Fintype.card \u03b9] : Measure (\u03b9 \u2192 \u211d)) := by\n  refine' le_hausdorffMeasure _ _ \u221e ENNReal.coe_lt_top fun s _ => _\n  rw [ENNReal.rpow_nat_cast]\n  exact Real.volume_pi_le_diam_pow s", "annotated_tactic": ["have Hle : <a>volume</a> \u2264 (\u03bcH[<a>Fintype.card</a> \u03b9] : <a>Measure</a> (\u03b9 \u2192 \u211d)) := by\n      refine' <a>le_hausdorffMeasure</a> _ _ \u221e <a>ENNReal.coe_lt_top</a> fun s _ => _\n      rw [<a>ENNReal.rpow_nat_cast</a>]\n      exact <a>Real.volume_pi_le_diam_pow</a> s", [{"full_name": "MeasureTheory.MeasureSpace.volume", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [663, 3], "def_end_pos": [663, 9]}, {"full_name": "Fintype.card", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [61, 5], "def_end_pos": [61, 9]}, {"full_name": "MeasureTheory.Measure", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [74, 11], "def_end_pos": [74, 18]}, {"full_name": "MeasureTheory.Measure.le_hausdorffMeasure", "def_path": "Mathlib/MeasureTheory/Measure/Hausdorff.lean", "def_pos": [585, 9], "def_end_pos": [585, 28]}, {"full_name": "ENNReal.coe_lt_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [308, 17], "def_end_pos": [308, 27]}, {"full_name": "ENNReal.rpow_nat_cast", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [548, 9], "def_end_pos": [548, 22]}, {"full_name": "Real.volume_pi_le_diam_pow", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean", "def_pos": [281, 9], "def_end_pos": [281, 30]}]], "state_before": "\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\n\u22a2 \u220f i : \u03b9, \u2191\u2191volume ((fun i => Ioo \u2191(a i) \u2191(b i)) i) \u2264 \u2191\u2191\u03bcH[\u2191(Fintype.card \u03b9)] (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i))", "state_after": "\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nHle : volume \u2264 \u03bcH[\u2191(Fintype.card \u03b9)]\n\u22a2 \u220f i : \u03b9, \u2191\u2191volume ((fun i => Ioo \u2191(a i) \u2191(b i)) i) \u2264 \u2191\u2191\u03bcH[\u2191(Fintype.card \u03b9)] (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i))"}, {"tactic": "rw [\u2190 volume_pi_pi fun i => Ioo (a i : \u211d) (b i)]", "annotated_tactic": ["rw [\u2190 <a>volume_pi_pi</a> fun i => <a>Ioo</a> (a i : \u211d) (b i)]", [{"full_name": "MeasureTheory.volume_pi_pi", "def_path": "Mathlib/MeasureTheory/Constructions/Pi.lean", "def_pos": [676, 9], "def_end_pos": [676, 21]}, {"full_name": "Set.Ioo", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [44, 5], "def_end_pos": [44, 8]}]], "state_before": "\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nHle : volume \u2264 \u03bcH[\u2191(Fintype.card \u03b9)]\n\u22a2 \u220f i : \u03b9, \u2191\u2191volume ((fun i => Ioo \u2191(a i) \u2191(b i)) i) \u2264 \u2191\u2191\u03bcH[\u2191(Fintype.card \u03b9)] (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i))", "state_after": "\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nHle : volume \u2264 \u03bcH[\u2191(Fintype.card \u03b9)]\n\u22a2 \u2191\u2191volume (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) \u2264 \u2191\u2191\u03bcH[\u2191(Fintype.card \u03b9)] (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i))"}, {"tactic": "exact Measure.le_iff'.1 Hle _", "annotated_tactic": ["exact <a>Measure.le_iff'</a>.1 Hle _", [{"full_name": "MeasureTheory.Measure.le_iff'", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [980, 9], "def_end_pos": [980, 16]}]], "state_before": "\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nHle : volume \u2264 \u03bcH[\u2191(Fintype.card \u03b9)]\n\u22a2 \u2191\u2191volume (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) \u2264 \u2191\u2191\u03bcH[\u2191(Fintype.card \u03b9)] (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i))", "state_after": "no goals"}, {"tactic": "refine' le_hausdorffMeasure _ _ \u221e ENNReal.coe_lt_top fun s _ => _", "annotated_tactic": ["refine' <a>le_hausdorffMeasure</a> _ _ \u221e <a>ENNReal.coe_lt_top</a> fun s _ => _", [{"full_name": "MeasureTheory.Measure.le_hausdorffMeasure", "def_path": "Mathlib/MeasureTheory/Measure/Hausdorff.lean", "def_pos": [585, 9], "def_end_pos": [585, 28]}, {"full_name": "ENNReal.coe_lt_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [308, 17], "def_end_pos": [308, 27]}]], "state_before": "\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\n\u22a2 volume \u2264 \u03bcH[\u2191(Fintype.card \u03b9)]", "state_after": "\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\ns : Set (\u03b9 \u2192 \u211d)\nx\u271d : diam s \u2264 \u22a4\n\u22a2 \u2191\u2191volume s \u2264 diam s ^ \u2191(Fintype.card \u03b9)"}, {"tactic": "rw [ENNReal.rpow_nat_cast]", "annotated_tactic": ["rw [<a>ENNReal.rpow_nat_cast</a>]", [{"full_name": "ENNReal.rpow_nat_cast", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [548, 9], "def_end_pos": [548, 22]}]], "state_before": "\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\ns : Set (\u03b9 \u2192 \u211d)\nx\u271d : diam s \u2264 \u22a4\n\u22a2 \u2191\u2191volume s \u2264 diam s ^ \u2191(Fintype.card \u03b9)", "state_after": "\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\ns : Set (\u03b9 \u2192 \u211d)\nx\u271d : diam s \u2264 \u22a4\n\u22a2 \u2191\u2191volume s \u2264 diam s ^ Fintype.card \u03b9"}, {"tactic": "exact Real.volume_pi_le_diam_pow s", "annotated_tactic": ["exact <a>Real.volume_pi_le_diam_pow</a> s", [{"full_name": "Real.volume_pi_le_diam_pow", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean", "def_pos": [281, 9], "def_end_pos": [281, 30]}]], "state_before": "\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\ns : Set (\u03b9 \u2192 \u211d)\nx\u271d : diam s \u2264 \u22a4\n\u22a2 \u2191\u2191volume s \u2264 diam s ^ Fintype.card \u03b9", "state_after": "no goals"}, {"tactic": "simpa only [sub_nonneg, Rat.cast_le] using (H i).le", "annotated_tactic": ["simpa only [<a>sub_nonneg</a>, <a>Rat.cast_le</a>] using (H i).<a>le</a>", [{"full_name": "sub_nonneg", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [720, 30], "def_end_pos": [720, 40]}, {"full_name": "Rat.cast_le", "def_path": "Mathlib/Data/Rat/Cast/Order.lean", "def_pos": [54, 9], "def_end_pos": [54, 16]}, {"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [142, 7], "def_end_pos": [142, 15]}]], "state_before": "\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\ni : \u03b9\n\u22a2 0 \u2264 \u2191(b i) - \u2191(a i)", "state_after": "no goals"}, {"tactic": "simp only [one_div, ENNReal.tendsto_inv_nat_nhds_zero]", "annotated_tactic": ["simp only [<a>one_div</a>, <a>ENNReal.tendsto_inv_nat_nhds_zero</a>]", [{"full_name": "one_div", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [318, 9], "def_end_pos": [318, 16]}, {"full_name": "ENNReal.tendsto_inv_nat_nhds_zero", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [556, 19], "def_end_pos": [556, 44]}]], "state_before": "\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\n\u22a2 Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0)", "state_after": "no goals"}, {"tactic": "refine' eventually_atTop.2 \u27e81, fun n hn => _\u27e9", "annotated_tactic": ["refine' <a>eventually_atTop</a>.2 \u27e81, fun n hn => _\u27e9", [{"full_name": "Filter.eventually_atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [178, 9], "def_end_pos": [178, 25]}]], "state_before": "\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\nA : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0)\n\u22a2 \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n", "state_after": "\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\nA : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0)\nn : \u2115\nhn : n \u2265 1\n\u22a2 \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n"}, {"tactic": "intro f", "annotated_tactic": ["intro f", []], "state_before": "\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\nA : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0)\nn : \u2115\nhn : n \u2265 1\n\u22a2 \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n", "state_after": "\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\nA : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0)\nn : \u2115\nhn : n \u2265 1\nf : \u03b3 n\n\u22a2 diam (t n f) \u2264 1 / \u2191n"}, {"tactic": "refine' diam_pi_le_of_le fun b => _", "annotated_tactic": ["refine' <a>diam_pi_le_of_le</a> fun b => _", [{"full_name": "EMetric.diam_pi_le_of_le", "def_path": "Mathlib/Topology/EMetricSpace/Basic.lean", "def_pos": [992, 9], "def_end_pos": [992, 25]}]], "state_before": "\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\nA : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0)\nn : \u2115\nhn : n \u2265 1\nf : \u03b3 n\n\u22a2 diam (t n f) \u2264 1 / \u2191n", "state_after": "\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b\u271d : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b\u271d i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b\u271d i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b\u271d i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\nA : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0)\nn : \u2115\nhn : n \u2265 1\nf : \u03b3 n\nb : \u03b9\n\u22a2 diam (Icc (\u2191(a b) + \u2191\u2191(f b) / \u2191n) (\u2191(a b) + (\u2191\u2191(f b) + 1) / \u2191n)) \u2264 1 / \u2191n"}, {"tactic": "simp only [Real.ediam_Icc, add_div, ENNReal.ofReal_div_of_pos (Nat.cast_pos.mpr hn), le_refl,\n  add_sub_add_left_eq_sub, add_sub_cancel', ENNReal.ofReal_one, ENNReal.ofReal_coe_nat]", "annotated_tactic": ["simp only [<a>Real.ediam_Icc</a>, <a>add_div</a>, <a>ENNReal.ofReal_div_of_pos</a> (Nat.cast_pos.mpr hn), <a>le_refl</a>,\n      <a>add_sub_add_left_eq_sub</a>, <a>add_sub_cancel'</a>, <a>ENNReal.ofReal_one</a>, <a>ENNReal.ofReal_coe_nat</a>]", [{"full_name": "Real.ediam_Icc", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [1546, 9], "def_end_pos": [1546, 18]}, {"full_name": "add_div", "def_path": "Mathlib/Algebra/Field/Basic.lean", "def_pos": [29, 9], "def_end_pos": [29, 16]}, {"full_name": "ENNReal.ofReal_div_of_pos", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2248, 9], "def_end_pos": [2248, 26]}, {"full_name": "le_refl", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [50, 9], "def_end_pos": [50, 16]}, {"full_name": "add_sub_add_left_eq_sub", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [894, 3], "def_end_pos": [894, 14]}, {"full_name": "add_sub_cancel'", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [948, 30], "def_end_pos": [948, 45]}, {"full_name": "ENNReal.ofReal_one", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [248, 17], "def_end_pos": [248, 27]}, {"full_name": "ENNReal.ofReal_coe_nat", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [710, 17], "def_end_pos": [710, 31]}]], "state_before": "\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b\u271d : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b\u271d i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b\u271d i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b\u271d i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\nA : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0)\nn : \u2115\nhn : n \u2265 1\nf : \u03b3 n\nb : \u03b9\n\u22a2 diam (Icc (\u2191(a b) + \u2191\u2191(f b) / \u2191n) (\u2191(a b) + (\u2191\u2191(f b) + 1) / \u2191n)) \u2264 1 / \u2191n", "state_after": "no goals"}, {"tactic": "refine' eventually_atTop.2 \u27e81, fun n hn => _\u27e9", "annotated_tactic": ["refine' <a>eventually_atTop</a>.2 \u27e81, fun n hn => _\u27e9", [{"full_name": "Filter.eventually_atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [178, 9], "def_end_pos": [178, 25]}]], "state_before": "\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\nA : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0)\nB : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n\n\u22a2 \u2200\u1da0 (n : \u2115) in atTop, (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) \u2286 \u22c3 i, t n i", "state_after": "\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\nA : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0)\nB : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n\nn : \u2115\nhn : n \u2265 1\n\u22a2 (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) \u2286 \u22c3 i, t n i"}, {"tactic": "have npos : (0 : \u211d) < n := Nat.cast_pos.2 hn", "annotated_tactic": ["have npos : (0 : \u211d) < n := <a>Nat.cast_pos</a>.2 hn", [{"full_name": "Nat.cast_pos", "def_path": "Mathlib/Data/Nat/Cast/Order.lean", "def_pos": [72, 9], "def_end_pos": [72, 17]}]], "state_before": "\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\nA : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0)\nB : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n\nn : \u2115\nhn : n \u2265 1\n\u22a2 (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) \u2286 \u22c3 i, t n i", "state_after": "\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\nA : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0)\nB : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n\nn : \u2115\nhn : n \u2265 1\nnpos : 0 < \u2191n\n\u22a2 (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) \u2286 \u22c3 i, t n i"}, {"tactic": "intro x hx", "annotated_tactic": ["intro x hx", []], "state_before": "\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\nA : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0)\nB : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n\nn : \u2115\nhn : n \u2265 1\nnpos : 0 < \u2191n\n\u22a2 (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) \u2286 \u22c3 i, t n i", "state_after": "\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\nA : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0)\nB : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n\nn : \u2115\nhn : n \u2265 1\nnpos : 0 < \u2191n\nx : \u03b9 \u2192 \u211d\nhx : x \u2208 Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)\n\u22a2 x \u2208 \u22c3 i, t n i"}, {"tactic": "simp only [mem_Ioo, mem_univ_pi] at hx", "annotated_tactic": ["simp only [<a>mem_Ioo</a>, <a>mem_univ_pi</a>] at hx", [{"full_name": "Set.mem_Ioo", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [116, 9], "def_end_pos": [116, 16]}, {"full_name": "Set.mem_univ_pi", "def_path": "Mathlib/Data/Set/Prod.lean", "def_pos": [675, 9], "def_end_pos": [675, 20]}]], "state_before": "\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\nA : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0)\nB : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n\nn : \u2115\nhn : n \u2265 1\nnpos : 0 < \u2191n\nx : \u03b9 \u2192 \u211d\nhx : x \u2208 Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)\n\u22a2 x \u2208 \u22c3 i, t n i", "state_after": "\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\nA : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0)\nB : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n\nn : \u2115\nhn : n \u2265 1\nnpos : 0 < \u2191n\nx : \u03b9 \u2192 \u211d\nhx : \u2200 (i : \u03b9), \u2191(a i) < x i \u2227 x i < \u2191(b i)\n\u22a2 x \u2208 \u22c3 i, t n i"}, {"tactic": "simp only [mem_iUnion, mem_Ioo, mem_univ_pi]", "annotated_tactic": ["simp only [<a>mem_iUnion</a>, <a>mem_Ioo</a>, <a>mem_univ_pi</a>]", [{"full_name": "Set.mem_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [201, 9], "def_end_pos": [201, 19]}, {"full_name": "Set.mem_Ioo", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [116, 9], "def_end_pos": [116, 16]}, {"full_name": "Set.mem_univ_pi", "def_path": "Mathlib/Data/Set/Prod.lean", "def_pos": [675, 9], "def_end_pos": [675, 20]}]], "state_before": "\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\nA : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0)\nB : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n\nn : \u2115\nhn : n \u2265 1\nnpos : 0 < \u2191n\nx : \u03b9 \u2192 \u211d\nhx : \u2200 (i : \u03b9), \u2191(a i) < x i \u2227 x i < \u2191(b i)\n\u22a2 x \u2208 \u22c3 i, t n i", "state_after": "\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\nA : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0)\nB : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n\nn : \u2115\nhn : n \u2265 1\nnpos : 0 < \u2191n\nx : \u03b9 \u2192 \u211d\nhx : \u2200 (i : \u03b9), \u2191(a i) < x i \u2227 x i < \u2191(b i)\n\u22a2 \u2203 i, \u2200 (i_1 : \u03b9), x i_1 \u2208 Icc (\u2191(a i_1) + \u2191\u2191(i i_1) / \u2191n) (\u2191(a i_1) + (\u2191\u2191(i i_1) + 1) / \u2191n)"}, {"tactic": "refine' \u27e8f, fun i => \u27e8_, _\u27e9\u27e9", "annotated_tactic": ["refine' \u27e8f, fun i => \u27e8_, _\u27e9\u27e9", []], "state_before": "\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\nA : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0)\nB : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n\nn : \u2115\nhn : n \u2265 1\nnpos : 0 < \u2191n\nx : \u03b9 \u2192 \u211d\nhx : \u2200 (i : \u03b9), \u2191(a i) < x i \u2227 x i < \u2191(b i)\nf : \u03b3 n := fun i => { val := \u230a(x i - \u2191(a i)) * \u2191n\u230b\u208a, isLt := (_ : \u230a(x i - \u2191(a i)) * \u2191n\u230b\u208a < \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a) }\n\u22a2 \u2203 i, \u2200 (i_1 : \u03b9), x i_1 \u2208 Icc (\u2191(a i_1) + \u2191\u2191(i i_1) / \u2191n) (\u2191(a i_1) + (\u2191\u2191(i i_1) + 1) / \u2191n)", "state_after": "case refine'_1\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\nA : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0)\nB : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n\nn : \u2115\nhn : n \u2265 1\nnpos : 0 < \u2191n\nx : \u03b9 \u2192 \u211d\nhx : \u2200 (i : \u03b9), \u2191(a i) < x i \u2227 x i < \u2191(b i)\nf : \u03b3 n := fun i => { val := \u230a(x i - \u2191(a i)) * \u2191n\u230b\u208a, isLt := (_ : \u230a(x i - \u2191(a i)) * \u2191n\u230b\u208a < \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a) }\ni : \u03b9\n\u22a2 \u2191(a i) + \u2191\u2191(f i) / \u2191n \u2264 x i\n\ncase refine'_2\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\nA : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0)\nB : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n\nn : \u2115\nhn : n \u2265 1\nnpos : 0 < \u2191n\nx : \u03b9 \u2192 \u211d\nhx : \u2200 (i : \u03b9), \u2191(a i) < x i \u2227 x i < \u2191(b i)\nf : \u03b3 n := fun i => { val := \u230a(x i - \u2191(a i)) * \u2191n\u230b\u208a, isLt := (_ : \u230a(x i - \u2191(a i)) * \u2191n\u230b\u208a < \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a) }\ni : \u03b9\n\u22a2 x i \u2264 \u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n"}, {"tactic": "apply Nat.floor_lt_ceil_of_lt_of_pos", "annotated_tactic": ["apply <a>Nat.floor_lt_ceil_of_lt_of_pos</a>", [{"full_name": "Nat.floor_lt_ceil_of_lt_of_pos", "def_path": "Mathlib/Algebra/Order/Floor.lean", "def_pos": [351, 9], "def_end_pos": [351, 35]}]], "state_before": "\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\nA : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0)\nB : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n\nn : \u2115\nhn : n \u2265 1\nnpos : 0 < \u2191n\nx : \u03b9 \u2192 \u211d\nhx : \u2200 (i : \u03b9), \u2191(a i) < x i \u2227 x i < \u2191(b i)\ni : \u03b9\n\u22a2 \u230a(x i - \u2191(a i)) * \u2191n\u230b\u208a < \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a", "state_after": "case h\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\nA : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0)\nB : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n\nn : \u2115\nhn : n \u2265 1\nnpos : 0 < \u2191n\nx : \u03b9 \u2192 \u211d\nhx : \u2200 (i : \u03b9), \u2191(a i) < x i \u2227 x i < \u2191(b i)\ni : \u03b9\n\u22a2 (x i - \u2191(a i)) * \u2191n < (\u2191(b i) - \u2191(a i)) * \u2191n\n\ncase h'\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\nA : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0)\nB : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n\nn : \u2115\nhn : n \u2265 1\nnpos : 0 < \u2191n\nx : \u03b9 \u2192 \u211d\nhx : \u2200 (i : \u03b9), \u2191(a i) < x i \u2227 x i < \u2191(b i)\ni : \u03b9\n\u22a2 0 < (\u2191(b i) - \u2191(a i)) * \u2191n"}, {"tactic": "refine' (mul_lt_mul_right npos).2 _", "annotated_tactic": ["refine' (<a>mul_lt_mul_right</a> npos).2 _", [{"full_name": "mul_lt_mul_right", "def_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "def_pos": [203, 9], "def_end_pos": [203, 25]}]], "state_before": "case h\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\nA : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0)\nB : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n\nn : \u2115\nhn : n \u2265 1\nnpos : 0 < \u2191n\nx : \u03b9 \u2192 \u211d\nhx : \u2200 (i : \u03b9), \u2191(a i) < x i \u2227 x i < \u2191(b i)\ni : \u03b9\n\u22a2 (x i - \u2191(a i)) * \u2191n < (\u2191(b i) - \u2191(a i)) * \u2191n", "state_after": "case h\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\nA : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0)\nB : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n\nn : \u2115\nhn : n \u2265 1\nnpos : 0 < \u2191n\nx : \u03b9 \u2192 \u211d\nhx : \u2200 (i : \u03b9), \u2191(a i) < x i \u2227 x i < \u2191(b i)\ni : \u03b9\n\u22a2 x i - \u2191(a i) < \u2191(b i) - \u2191(a i)"}, {"tactic": "simp only [(hx i).right, sub_lt_sub_iff_right]", "annotated_tactic": ["simp only [(hx i).<a>right</a>, <a>sub_lt_sub_iff_right</a>]", [{"full_name": "And.right", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [507, 3], "def_end_pos": [507, 8]}, {"full_name": "sub_lt_sub_iff_right", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [871, 3], "def_end_pos": [871, 14]}]], "state_before": "case h\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\nA : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0)\nB : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n\nn : \u2115\nhn : n \u2265 1\nnpos : 0 < \u2191n\nx : \u03b9 \u2192 \u211d\nhx : \u2200 (i : \u03b9), \u2191(a i) < x i \u2227 x i < \u2191(b i)\ni : \u03b9\n\u22a2 x i - \u2191(a i) < \u2191(b i) - \u2191(a i)", "state_after": "no goals"}, {"tactic": "refine' mul_pos _ npos", "annotated_tactic": ["refine' <a>mul_pos</a> _ npos", [{"full_name": "mul_pos", "def_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "def_pos": [345, 7], "def_end_pos": [345, 14]}]], "state_before": "case h'\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\nA : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0)\nB : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n\nn : \u2115\nhn : n \u2265 1\nnpos : 0 < \u2191n\nx : \u03b9 \u2192 \u211d\nhx : \u2200 (i : \u03b9), \u2191(a i) < x i \u2227 x i < \u2191(b i)\ni : \u03b9\n\u22a2 0 < (\u2191(b i) - \u2191(a i)) * \u2191n", "state_after": "case h'\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\nA : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0)\nB : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n\nn : \u2115\nhn : n \u2265 1\nnpos : 0 < \u2191n\nx : \u03b9 \u2192 \u211d\nhx : \u2200 (i : \u03b9), \u2191(a i) < x i \u2227 x i < \u2191(b i)\ni : \u03b9\n\u22a2 0 < \u2191(b i) - \u2191(a i)"}, {"tactic": "simpa only [Rat.cast_lt, sub_pos] using H i", "annotated_tactic": ["simpa only [<a>Rat.cast_lt</a>, <a>sub_pos</a>] using H i", [{"full_name": "Rat.cast_lt", "def_path": "Mathlib/Data/Rat/Cast/Order.lean", "def_pos": [59, 9], "def_end_pos": [59, 16]}, {"full_name": "sub_pos", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [883, 30], "def_end_pos": [883, 37]}]], "state_before": "case h'\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\nA : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0)\nB : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n\nn : \u2115\nhn : n \u2265 1\nnpos : 0 < \u2191n\nx : \u03b9 \u2192 \u211d\nhx : \u2200 (i : \u03b9), \u2191(a i) < x i \u2227 x i < \u2191(b i)\ni : \u03b9\n\u22a2 0 < \u2191(b i) - \u2191(a i)", "state_after": "no goals"}, {"tactic": "calc\n  (a i : \u211d) + \u230a(x i - a i) * n\u230b\u208a / n \u2264 (a i : \u211d) + (x i - a i) * n / n := by\n    refine' add_le_add le_rfl ((div_le_div_right npos).2 _)\n    exact Nat.floor_le (mul_nonneg (sub_nonneg.2 (hx i).1.le) npos.le)\n  _ = x i := by field_simp [npos.ne']", "annotated_tactic": ["calc\n        (a i : \u211d) + \u230a(x i - a i) * n\u230b\u208a / n \u2264 (a i : \u211d) + (x i - a i) * n / n := by\n          refine' <a>add_le_add</a> <a>le_rfl</a> ((<a>div_le_div_right</a> npos).2 _)\n          exact <a>Nat.floor_le</a> (<a>mul_nonneg</a> (<a>sub_nonneg</a>.2 (hx i).1.<a>le</a>) npos.le)\n        _ = x i := by field_simp [npos.ne']", [{"full_name": "add_le_add", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [205, 15], "def_end_pos": [205, 25]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}, {"full_name": "div_le_div_right", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [363, 9], "def_end_pos": [363, 25]}, {"full_name": "Nat.floor_le", "def_path": "Mathlib/Algebra/Order/Floor.lean", "def_pos": [153, 9], "def_end_pos": [153, 17]}, {"full_name": "mul_nonneg", "def_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "def_pos": [380, 7], "def_end_pos": [380, 17]}, {"full_name": "sub_nonneg", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [720, 30], "def_end_pos": [720, 40]}, {"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [142, 7], "def_end_pos": [142, 15]}]], "state_before": "case refine'_1\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\nA : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0)\nB : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n\nn : \u2115\nhn : n \u2265 1\nnpos : 0 < \u2191n\nx : \u03b9 \u2192 \u211d\nhx : \u2200 (i : \u03b9), \u2191(a i) < x i \u2227 x i < \u2191(b i)\nf : \u03b3 n := fun i => { val := \u230a(x i - \u2191(a i)) * \u2191n\u230b\u208a, isLt := (_ : \u230a(x i - \u2191(a i)) * \u2191n\u230b\u208a < \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a) }\ni : \u03b9\n\u22a2 \u2191(a i) + \u2191\u2191(f i) / \u2191n \u2264 x i", "state_after": "no goals"}, {"tactic": "refine' add_le_add le_rfl ((div_le_div_right npos).2 _)", "annotated_tactic": ["refine' <a>add_le_add</a> <a>le_rfl</a> ((<a>div_le_div_right</a> npos).2 _)", [{"full_name": "add_le_add", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [205, 15], "def_end_pos": [205, 25]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}, {"full_name": "div_le_div_right", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [363, 9], "def_end_pos": [363, 25]}]], "state_before": "\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\nA : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0)\nB : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n\nn : \u2115\nhn : n \u2265 1\nnpos : 0 < \u2191n\nx : \u03b9 \u2192 \u211d\nhx : \u2200 (i : \u03b9), \u2191(a i) < x i \u2227 x i < \u2191(b i)\nf : \u03b3 n := fun i => { val := \u230a(x i - \u2191(a i)) * \u2191n\u230b\u208a, isLt := (_ : \u230a(x i - \u2191(a i)) * \u2191n\u230b\u208a < \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a) }\ni : \u03b9\n\u22a2 \u2191(a i) + \u2191\u230a(x i - \u2191(a i)) * \u2191n\u230b\u208a / \u2191n \u2264 \u2191(a i) + (x i - \u2191(a i)) * \u2191n / \u2191n", "state_after": "\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\nA : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0)\nB : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n\nn : \u2115\nhn : n \u2265 1\nnpos : 0 < \u2191n\nx : \u03b9 \u2192 \u211d\nhx : \u2200 (i : \u03b9), \u2191(a i) < x i \u2227 x i < \u2191(b i)\nf : \u03b3 n := fun i => { val := \u230a(x i - \u2191(a i)) * \u2191n\u230b\u208a, isLt := (_ : \u230a(x i - \u2191(a i)) * \u2191n\u230b\u208a < \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a) }\ni : \u03b9\n\u22a2 \u2191\u230a(x i - \u2191(a i)) * \u2191n\u230b\u208a \u2264 (x i - \u2191(a i)) * \u2191n"}, {"tactic": "exact Nat.floor_le (mul_nonneg (sub_nonneg.2 (hx i).1.le) npos.le)", "annotated_tactic": ["exact <a>Nat.floor_le</a> (<a>mul_nonneg</a> (<a>sub_nonneg</a>.2 (hx i).1.<a>le</a>) npos.le)", [{"full_name": "Nat.floor_le", "def_path": "Mathlib/Algebra/Order/Floor.lean", "def_pos": [153, 9], "def_end_pos": [153, 17]}, {"full_name": "mul_nonneg", "def_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "def_pos": [380, 7], "def_end_pos": [380, 17]}, {"full_name": "sub_nonneg", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [720, 30], "def_end_pos": [720, 40]}, {"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [142, 7], "def_end_pos": [142, 15]}]], "state_before": "\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\nA : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0)\nB : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n\nn : \u2115\nhn : n \u2265 1\nnpos : 0 < \u2191n\nx : \u03b9 \u2192 \u211d\nhx : \u2200 (i : \u03b9), \u2191(a i) < x i \u2227 x i < \u2191(b i)\nf : \u03b3 n := fun i => { val := \u230a(x i - \u2191(a i)) * \u2191n\u230b\u208a, isLt := (_ : \u230a(x i - \u2191(a i)) * \u2191n\u230b\u208a < \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a) }\ni : \u03b9\n\u22a2 \u2191\u230a(x i - \u2191(a i)) * \u2191n\u230b\u208a \u2264 (x i - \u2191(a i)) * \u2191n", "state_after": "no goals"}, {"tactic": "field_simp [npos.ne']", "annotated_tactic": ["field_simp [npos.ne']", []], "state_before": "\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\nA : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0)\nB : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n\nn : \u2115\nhn : n \u2265 1\nnpos : 0 < \u2191n\nx : \u03b9 \u2192 \u211d\nhx : \u2200 (i : \u03b9), \u2191(a i) < x i \u2227 x i < \u2191(b i)\nf : \u03b3 n := fun i => { val := \u230a(x i - \u2191(a i)) * \u2191n\u230b\u208a, isLt := (_ : \u230a(x i - \u2191(a i)) * \u2191n\u230b\u208a < \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a) }\ni : \u03b9\n\u22a2 \u2191(a i) + (x i - \u2191(a i)) * \u2191n / \u2191n = x i", "state_after": "no goals"}, {"tactic": "calc\n  x i = (a i : \u211d) + (x i - a i) * n / n := by field_simp [npos.ne']\n  _ \u2264 (a i : \u211d) + (\u230a(x i - a i) * n\u230b\u208a + 1) / n :=\n    add_le_add le_rfl ((div_le_div_right npos).2 (Nat.lt_floor_add_one _).le)", "annotated_tactic": ["calc\n        x i = (a i : \u211d) + (x i - a i) * n / n := by field_simp [npos.ne']\n        _ \u2264 (a i : \u211d) + (\u230a(x i - a i) * n\u230b\u208a + 1) / n :=\n          <a>add_le_add</a> <a>le_rfl</a> ((<a>div_le_div_right</a> npos).2 (<a>Nat.lt_floor_add_one</a> _).<a>le</a>)", [{"full_name": "add_le_add", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [205, 15], "def_end_pos": [205, 25]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}, {"full_name": "div_le_div_right", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [363, 9], "def_end_pos": [363, 25]}, {"full_name": "Nat.lt_floor_add_one", "def_path": "Mathlib/Algebra/Order/Floor.lean", "def_pos": [161, 9], "def_end_pos": [161, 25]}, {"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [142, 7], "def_end_pos": [142, 15]}]], "state_before": "case refine'_2\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\nA : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0)\nB : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n\nn : \u2115\nhn : n \u2265 1\nnpos : 0 < \u2191n\nx : \u03b9 \u2192 \u211d\nhx : \u2200 (i : \u03b9), \u2191(a i) < x i \u2227 x i < \u2191(b i)\nf : \u03b3 n := fun i => { val := \u230a(x i - \u2191(a i)) * \u2191n\u230b\u208a, isLt := (_ : \u230a(x i - \u2191(a i)) * \u2191n\u230b\u208a < \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a) }\ni : \u03b9\n\u22a2 x i \u2264 \u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n", "state_after": "no goals"}, {"tactic": "field_simp [npos.ne']", "annotated_tactic": ["field_simp [npos.ne']", []], "state_before": "\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\nA : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0)\nB : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n\nn : \u2115\nhn : n \u2265 1\nnpos : 0 < \u2191n\nx : \u03b9 \u2192 \u211d\nhx : \u2200 (i : \u03b9), \u2191(a i) < x i \u2227 x i < \u2191(b i)\nf : \u03b3 n := fun i => { val := \u230a(x i - \u2191(a i)) * \u2191n\u230b\u208a, isLt := (_ : \u230a(x i - \u2191(a i)) * \u2191n\u230b\u208a < \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a) }\ni : \u03b9\n\u22a2 x i = \u2191(a i) + (x i - \u2191(a i)) * \u2191n / \u2191n", "state_after": "no goals"}, {"tactic": "refine' liminf_le_liminf _ _", "annotated_tactic": ["refine' <a>liminf_le_liminf</a> _ _", [{"full_name": "Filter.liminf_le_liminf", "def_path": "Mathlib/Order/LiminfLimsup.lean", "def_pos": [586, 9], "def_end_pos": [586, 25]}]], "state_before": "\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\nA : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0)\nB : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n\nC : \u2200\u1da0 (n : \u2115) in atTop, (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) \u2286 \u22c3 i, t n i\n\u22a2 liminf (fun n => \u2211 i : \u03b3 n, diam (t n i) ^ \u2191(Fintype.card \u03b9)) atTop \u2264\n    liminf (fun n => \u2211 i : \u03b3 n, (1 / \u2191n) ^ Fintype.card \u03b9) atTop", "state_after": "case refine'_1\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\nA : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0)\nB : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n\nC : \u2200\u1da0 (n : \u2115) in atTop, (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) \u2286 \u22c3 i, t n i\n\u22a2 \u2200\u1da0 (a_1 : \u2115) in atTop, \u2211 i : \u03b3 a_1, diam (t a_1 i) ^ \u2191(Fintype.card \u03b9) \u2264 \u2211 i : \u03b3 a_1, (1 / \u2191a_1) ^ Fintype.card \u03b9\n\ncase refine'_2\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\nA : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0)\nB : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n\nC : \u2200\u1da0 (n : \u2115) in atTop, (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) \u2286 \u22c3 i, t n i\n\u22a2 IsBoundedUnder (fun x x_1 => x \u2265 x_1) atTop fun n => \u2211 i : \u03b3 n, diam (t n i) ^ \u2191(Fintype.card \u03b9)"}, {"tactic": "filter_upwards [B] with _ hn", "annotated_tactic": ["filter_upwards [B] with _ hn", []], "state_before": "case refine'_1\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\nA : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0)\nB : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n\nC : \u2200\u1da0 (n : \u2115) in atTop, (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) \u2286 \u22c3 i, t n i\n\u22a2 \u2200\u1da0 (a_1 : \u2115) in atTop, \u2211 i : \u03b3 a_1, diam (t a_1 i) ^ \u2191(Fintype.card \u03b9) \u2264 \u2211 i : \u03b3 a_1, (1 / \u2191a_1) ^ Fintype.card \u03b9", "state_after": "case h\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\nA : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0)\nB : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n\nC : \u2200\u1da0 (n : \u2115) in atTop, (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) \u2286 \u22c3 i, t n i\na\u271d : \u2115\nhn : \u2200 (i : \u03b3 a\u271d), diam (t a\u271d i) \u2264 1 / \u2191a\u271d\n\u22a2 \u2211 i : \u03b3 a\u271d, diam (t a\u271d i) ^ \u2191(Fintype.card \u03b9) \u2264 \u2211 i : \u03b3 a\u271d, (1 / \u2191a\u271d) ^ Fintype.card \u03b9"}, {"tactic": "apply Finset.sum_le_sum fun i _ => _", "annotated_tactic": ["apply <a>Finset.sum_le_sum</a> fun i _ => _", [{"full_name": "Finset.sum_le_sum", "def_path": "Mathlib/Algebra/BigOperators/Order.lean", "def_pos": [111, 15], "def_end_pos": [111, 25]}]], "state_before": "case h\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\nA : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0)\nB : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n\nC : \u2200\u1da0 (n : \u2115) in atTop, (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) \u2286 \u22c3 i, t n i\na\u271d : \u2115\nhn : \u2200 (i : \u03b3 a\u271d), diam (t a\u271d i) \u2264 1 / \u2191a\u271d\n\u22a2 \u2211 i : \u03b3 a\u271d, diam (t a\u271d i) ^ \u2191(Fintype.card \u03b9) \u2264 \u2211 i : \u03b3 a\u271d, (1 / \u2191a\u271d) ^ Fintype.card \u03b9", "state_after": "\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\nA : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0)\nB : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n\nC : \u2200\u1da0 (n : \u2115) in atTop, (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) \u2286 \u22c3 i, t n i\na\u271d : \u2115\nhn : \u2200 (i : \u03b3 a\u271d), diam (t a\u271d i) \u2264 1 / \u2191a\u271d\n\u22a2 \u2200 (i : \u03b3 a\u271d), i \u2208 Finset.univ \u2192 diam (t a\u271d i) ^ \u2191(Fintype.card \u03b9) \u2264 (1 / \u2191a\u271d) ^ Fintype.card \u03b9"}, {"tactic": "simp only [ENNReal.rpow_nat_cast]", "annotated_tactic": ["simp only [<a>ENNReal.rpow_nat_cast</a>]", [{"full_name": "ENNReal.rpow_nat_cast", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [548, 9], "def_end_pos": [548, 22]}]], "state_before": "\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\nA : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0)\nB : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n\nC : \u2200\u1da0 (n : \u2115) in atTop, (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) \u2286 \u22c3 i, t n i\na\u271d : \u2115\nhn : \u2200 (i : \u03b3 a\u271d), diam (t a\u271d i) \u2264 1 / \u2191a\u271d\n\u22a2 \u2200 (i : \u03b3 a\u271d), i \u2208 Finset.univ \u2192 diam (t a\u271d i) ^ \u2191(Fintype.card \u03b9) \u2264 (1 / \u2191a\u271d) ^ Fintype.card \u03b9", "state_after": "\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\nA : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0)\nB : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n\nC : \u2200\u1da0 (n : \u2115) in atTop, (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) \u2286 \u22c3 i, t n i\na\u271d : \u2115\nhn : \u2200 (i : \u03b3 a\u271d), diam (t a\u271d i) \u2264 1 / \u2191a\u271d\n\u22a2 \u2200 (i : \u03b3 a\u271d),\n    i \u2208 Finset.univ \u2192\n      diam (Set.pi univ fun i_1 => Icc (\u2191(a i_1) + \u2191\u2191(i i_1) / \u2191a\u271d) (\u2191(a i_1) + (\u2191\u2191(i i_1) + 1) / \u2191a\u271d)) ^\n          Fintype.card \u03b9 \u2264\n        (1 / \u2191a\u271d) ^ Fintype.card \u03b9"}, {"tactic": "intros i _", "annotated_tactic": ["intros i _", []], "state_before": "\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\nA : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0)\nB : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n\nC : \u2200\u1da0 (n : \u2115) in atTop, (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) \u2286 \u22c3 i, t n i\na\u271d : \u2115\nhn : \u2200 (i : \u03b3 a\u271d), diam (t a\u271d i) \u2264 1 / \u2191a\u271d\n\u22a2 \u2200 (i : \u03b3 a\u271d),\n    i \u2208 Finset.univ \u2192\n      diam (Set.pi univ fun i_1 => Icc (\u2191(a i_1) + \u2191\u2191(i i_1) / \u2191a\u271d) (\u2191(a i_1) + (\u2191\u2191(i i_1) + 1) / \u2191a\u271d)) ^\n          Fintype.card \u03b9 \u2264\n        (1 / \u2191a\u271d) ^ Fintype.card \u03b9", "state_after": "\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\nA : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0)\nB : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n\nC : \u2200\u1da0 (n : \u2115) in atTop, (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) \u2286 \u22c3 i, t n i\na\u271d : \u2115\nhn : \u2200 (i : \u03b3 a\u271d), diam (t a\u271d i) \u2264 1 / \u2191a\u271d\ni : \u03b3 a\u271d\nx\u271d : i \u2208 Finset.univ\n\u22a2 diam (Set.pi univ fun i_1 => Icc (\u2191(a i_1) + \u2191\u2191(i i_1) / \u2191a\u271d) (\u2191(a i_1) + (\u2191\u2191(i i_1) + 1) / \u2191a\u271d)) ^ Fintype.card \u03b9 \u2264\n    (1 / \u2191a\u271d) ^ Fintype.card \u03b9"}, {"tactic": "exact pow_le_pow_of_le_left' (hn i) _", "annotated_tactic": ["exact <a>pow_le_pow_of_le_left'</a> (hn i) _", [{"full_name": "pow_le_pow_of_le_left'", "def_path": "Mathlib/Algebra/GroupPower/Order.lean", "def_pos": [41, 9], "def_end_pos": [41, 31]}]], "state_before": "\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\nA : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0)\nB : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n\nC : \u2200\u1da0 (n : \u2115) in atTop, (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) \u2286 \u22c3 i, t n i\na\u271d : \u2115\nhn : \u2200 (i : \u03b3 a\u271d), diam (t a\u271d i) \u2264 1 / \u2191a\u271d\ni : \u03b3 a\u271d\nx\u271d : i \u2208 Finset.univ\n\u22a2 diam (Set.pi univ fun i_1 => Icc (\u2191(a i_1) + \u2191\u2191(i i_1) / \u2191a\u271d) (\u2191(a i_1) + (\u2191\u2191(i i_1) + 1) / \u2191a\u271d)) ^ Fintype.card \u03b9 \u2264\n    (1 / \u2191a\u271d) ^ Fintype.card \u03b9", "state_after": "no goals"}, {"tactic": "isBoundedDefault", "annotated_tactic": ["isBoundedDefault", []], "state_before": "case refine'_2\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\nA : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0)\nB : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n\nC : \u2200\u1da0 (n : \u2115) in atTop, (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) \u2286 \u22c3 i, t n i\n\u22a2 IsBoundedUnder (fun x x_1 => x \u2265 x_1) atTop fun n => \u2211 i : \u03b3 n, diam (t n i) ^ \u2191(Fintype.card \u03b9)", "state_after": "no goals"}, {"tactic": "simp only [Finset.card_univ, Nat.cast_prod, one_mul, Fintype.card_fin, Finset.sum_const,\n  nsmul_eq_mul, Fintype.card_pi, div_eq_mul_inv, Finset.prod_mul_distrib, Finset.prod_const]", "annotated_tactic": ["simp only [<a>Finset.card_univ</a>, <a>Nat.cast_prod</a>, <a>one_mul</a>, <a>Fintype.card_fin</a>, <a>Finset.sum_const</a>,\n        <a>nsmul_eq_mul</a>, <a>Fintype.card_pi</a>, <a>div_eq_mul_inv</a>, <a>Finset.prod_mul_distrib</a>, <a>Finset.prod_const</a>]", [{"full_name": "Finset.card_univ", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [247, 9], "def_end_pos": [247, 25]}, {"full_name": "Nat.cast_prod", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [2224, 9], "def_end_pos": [2224, 18]}, {"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [464, 9], "def_end_pos": [464, 16]}, {"full_name": "Fintype.card_fin", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [308, 9], "def_end_pos": [308, 25]}, {"full_name": "Finset.sum_const", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [1440, 3], "def_end_pos": [1440, 14]}, {"full_name": "nsmul_eq_mul", "def_path": "Mathlib/Algebra/GroupPower/Lemmas.lean", "def_pos": [509, 9], "def_end_pos": [509, 21]}, {"full_name": "Fintype.card_pi", "def_path": "Mathlib/Data/Fintype/BigOperators.lean", "def_pos": [140, 9], "def_end_pos": [140, 24]}, {"full_name": "div_eq_mul_inv", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [977, 9], "def_end_pos": [977, 23]}, {"full_name": "Finset.prod_mul_distrib", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [658, 9], "def_end_pos": [658, 25]}, {"full_name": "Finset.prod_const", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [1441, 9], "def_end_pos": [1441, 19]}]], "state_before": "\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\nA : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0)\nB : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n\nC : \u2200\u1da0 (n : \u2115) in atTop, (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) \u2286 \u22c3 i, t n i\n\u22a2 liminf (fun n => \u2211 i : \u03b3 n, (1 / \u2191n) ^ Fintype.card \u03b9) atTop =\n    liminf (fun n => \u220f i : \u03b9, \u2191\u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a / \u2191n) atTop", "state_after": "no goals"}, {"tactic": "simp only [Real.volume_Ioo]", "annotated_tactic": ["simp only [<a>Real.volume_Ioo</a>]", [{"full_name": "Real.volume_Ioo", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean", "def_pos": [88, 9], "def_end_pos": [88, 19]}]], "state_before": "\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\nA : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0)\nB : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n\nC : \u2200\u1da0 (n : \u2115) in atTop, (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) \u2286 \u22c3 i, t n i\n\u22a2 liminf (fun n => \u220f i : \u03b9, \u2191\u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a / \u2191n) atTop = \u220f i : \u03b9, \u2191\u2191volume (Ioo \u2191(a i) \u2191(b i))", "state_after": "\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\nA : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0)\nB : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n\nC : \u2200\u1da0 (n : \u2115) in atTop, (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) \u2286 \u22c3 i, t n i\n\u22a2 liminf (fun n => \u220f i : \u03b9, \u2191\u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a / \u2191n) atTop = \u220f x : \u03b9, ENNReal.ofReal (\u2191(b x) - \u2191(a x))"}, {"tactic": "apply Tendsto.liminf_eq", "annotated_tactic": ["apply <a>Tendsto.liminf_eq</a>", [{"full_name": "Filter.Tendsto.liminf_eq", "def_path": "Mathlib/Topology/Algebra/Order/LiminfLimsup.lean", "def_pos": [158, 9], "def_end_pos": [158, 33]}]], "state_before": "\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\nA : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0)\nB : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n\nC : \u2200\u1da0 (n : \u2115) in atTop, (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) \u2286 \u22c3 i, t n i\n\u22a2 liminf (fun n => \u220f i : \u03b9, \u2191\u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a / \u2191n) atTop = \u220f x : \u03b9, ENNReal.ofReal (\u2191(b x) - \u2191(a x))", "state_after": "case h\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\nA : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0)\nB : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n\nC : \u2200\u1da0 (n : \u2115) in atTop, (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) \u2286 \u22c3 i, t n i\n\u22a2 Tendsto (fun n => \u220f i : \u03b9, \u2191\u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a / \u2191n) atTop (\ud835\udcdd (\u220f x : \u03b9, ENNReal.ofReal (\u2191(b x) - \u2191(a x))))"}, {"tactic": "refine' ENNReal.tendsto_finset_prod_of_ne_top _ (fun i _ => _) fun i _ => _", "annotated_tactic": ["refine' <a>ENNReal.tendsto_finset_prod_of_ne_top</a> _ (fun i _ => _) fun i _ => _", [{"full_name": "ENNReal.tendsto_finset_prod_of_ne_top", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [384, 9], "def_end_pos": [384, 38]}]], "state_before": "case h\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\nA : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0)\nB : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n\nC : \u2200\u1da0 (n : \u2115) in atTop, (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) \u2286 \u22c3 i, t n i\n\u22a2 Tendsto (fun n => \u220f i : \u03b9, \u2191\u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a / \u2191n) atTop (\ud835\udcdd (\u220f x : \u03b9, ENNReal.ofReal (\u2191(b x) - \u2191(a x))))", "state_after": "case h.refine'_1\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\nA : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0)\nB : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n\nC : \u2200\u1da0 (n : \u2115) in atTop, (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) \u2286 \u22c3 i, t n i\ni : \u03b9\nx\u271d : i \u2208 Finset.univ\n\u22a2 Tendsto (fun n => \u2191\u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a / \u2191n) atTop (\ud835\udcdd (ENNReal.ofReal (\u2191(b i) - \u2191(a i))))\n\ncase h.refine'_2\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\nA : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0)\nB : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n\nC : \u2200\u1da0 (n : \u2115) in atTop, (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) \u2286 \u22c3 i, t n i\ni : \u03b9\nx\u271d : i \u2208 Finset.univ\n\u22a2 ENNReal.ofReal (\u2191(b i) - \u2191(a i)) \u2260 \u22a4"}, {"tactic": "apply\n  Tendsto.congr' _\n    ((ENNReal.continuous_ofReal.tendsto _).comp\n      ((tendsto_nat_ceil_mul_div_atTop (I i)).comp tendsto_nat_cast_atTop_atTop))", "annotated_tactic": ["apply\n          <a>Tendsto.congr'</a> _\n            ((ENNReal.continuous_ofReal.tendsto _).<a>comp</a>\n              ((<a>tendsto_nat_ceil_mul_div_atTop</a> (I i)).<a>comp</a> <a>tendsto_nat_cast_atTop_atTop</a>))", [{"full_name": "Filter.Tendsto.congr'", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [3009, 9], "def_end_pos": [3009, 23]}, {"full_name": "Filter.Tendsto.comp", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [3032, 9], "def_end_pos": [3032, 21]}, {"full_name": "tendsto_nat_ceil_mul_div_atTop", "def_path": "Mathlib/Analysis/SpecificLimits/Basic.lean", "def_pos": [623, 9], "def_end_pos": [623, 39]}, {"full_name": "Filter.Tendsto.comp", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [3032, 9], "def_end_pos": [3032, 21]}, {"full_name": "tendsto_nat_cast_atTop_atTop", "def_path": "Mathlib/Order/Filter/Archimedean.lean", "def_pos": [37, 9], "def_end_pos": [37, 37]}]], "state_before": "case h.refine'_1\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\nA : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0)\nB : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n\nC : \u2200\u1da0 (n : \u2115) in atTop, (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) \u2286 \u22c3 i, t n i\ni : \u03b9\nx\u271d : i \u2208 Finset.univ\n\u22a2 Tendsto (fun n => \u2191\u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a / \u2191n) atTop (\ud835\udcdd (ENNReal.ofReal (\u2191(b i) - \u2191(a i))))", "state_after": "\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\nA : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0)\nB : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n\nC : \u2200\u1da0 (n : \u2115) in atTop, (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) \u2286 \u22c3 i, t n i\ni : \u03b9\nx\u271d : i \u2208 Finset.univ\n\u22a2 ENNReal.ofReal \u2218 (fun x => \u2191\u2308(\u2191(b i) - \u2191(a i)) * x\u2309\u208a / x) \u2218 Nat.cast =\u1da0[atTop] fun n =>\n    \u2191\u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a / \u2191n"}, {"tactic": "apply eventually_atTop.2 \u27e81, fun n hn => _\u27e9", "annotated_tactic": ["apply <a>eventually_atTop</a>.2 \u27e81, fun n hn => _\u27e9", [{"full_name": "Filter.eventually_atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [178, 9], "def_end_pos": [178, 25]}]], "state_before": "\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\nA : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0)\nB : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n\nC : \u2200\u1da0 (n : \u2115) in atTop, (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) \u2286 \u22c3 i, t n i\ni : \u03b9\nx\u271d : i \u2208 Finset.univ\n\u22a2 ENNReal.ofReal \u2218 (fun x => \u2191\u2308(\u2191(b i) - \u2191(a i)) * x\u2309\u208a / x) \u2218 Nat.cast =\u1da0[atTop] fun n =>\n    \u2191\u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a / \u2191n", "state_after": "\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\nA : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0)\nB : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n\nC : \u2200\u1da0 (n : \u2115) in atTop, (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) \u2286 \u22c3 i, t n i\ni : \u03b9\nx\u271d : i \u2208 Finset.univ\n\u22a2 \u2200 (n : \u2115),\n    n \u2265 1 \u2192\n      (ENNReal.ofReal \u2218 (fun x => \u2191\u2308(\u2191(b i) - \u2191(a i)) * x\u2309\u208a / x) \u2218 Nat.cast) n =\n        (fun n => \u2191\u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a / \u2191n) n"}, {"tactic": "intros n hn", "annotated_tactic": ["intros n hn", []], "state_before": "\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\nA : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0)\nB : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n\nC : \u2200\u1da0 (n : \u2115) in atTop, (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) \u2286 \u22c3 i, t n i\ni : \u03b9\nx\u271d : i \u2208 Finset.univ\n\u22a2 \u2200 (n : \u2115),\n    n \u2265 1 \u2192\n      (ENNReal.ofReal \u2218 (fun x => \u2191\u2308(\u2191(b i) - \u2191(a i)) * x\u2309\u208a / x) \u2218 Nat.cast) n =\n        (fun n => \u2191\u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a / \u2191n) n", "state_after": "\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\nA : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0)\nB : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n\nC : \u2200\u1da0 (n : \u2115) in atTop, (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) \u2286 \u22c3 i, t n i\ni : \u03b9\nx\u271d : i \u2208 Finset.univ\nn : \u2115\nhn : n \u2265 1\n\u22a2 (ENNReal.ofReal \u2218 (fun x => \u2191\u2308(\u2191(b i) - \u2191(a i)) * x\u2309\u208a / x) \u2218 Nat.cast) n =\n    (fun n => \u2191\u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a / \u2191n) n"}, {"tactic": "simp only [ENNReal.ofReal_div_of_pos (Nat.cast_pos.mpr hn), comp_apply,\n  ENNReal.ofReal_coe_nat]", "annotated_tactic": ["simp only [<a>ENNReal.ofReal_div_of_pos</a> (Nat.cast_pos.mpr hn), <a>comp_apply</a>,\n          <a>ENNReal.ofReal_coe_nat</a>]", [{"full_name": "ENNReal.ofReal_div_of_pos", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2248, 9], "def_end_pos": [2248, 26]}, {"full_name": "Function.comp_apply", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [33, 17], "def_end_pos": [33, 36]}, {"full_name": "ENNReal.ofReal_coe_nat", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [710, 17], "def_end_pos": [710, 31]}]], "state_before": "\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\nA : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0)\nB : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n\nC : \u2200\u1da0 (n : \u2115) in atTop, (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) \u2286 \u22c3 i, t n i\ni : \u03b9\nx\u271d : i \u2208 Finset.univ\nn : \u2115\nhn : n \u2265 1\n\u22a2 (ENNReal.ofReal \u2218 (fun x => \u2191\u2308(\u2191(b i) - \u2191(a i)) * x\u2309\u208a / x) \u2218 Nat.cast) n =\n    (fun n => \u2191\u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a / \u2191n) n", "state_after": "no goals"}, {"tactic": "simp only [ENNReal.ofReal_ne_top, Ne.def, not_false_iff]", "annotated_tactic": ["simp only [<a>ENNReal.ofReal_ne_top</a>, <a>Ne.def</a>, <a>not_false_iff</a>]", [{"full_name": "ENNReal.ofReal_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [311, 17], "def_end_pos": [311, 30]}, {"full_name": "Ne.def", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [59, 9], "def_end_pos": [59, 15]}, {"full_name": "not_false_iff", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [82, 9], "def_end_pos": [82, 22]}]], "state_before": "case h.refine'_2\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\nA : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0)\nB : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n\nC : \u2200\u1da0 (n : \u2115) in atTop, (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) \u2286 \u22c3 i, t n i\ni : \u03b9\nx\u271d : i \u2208 Finset.univ\n\u22a2 ENNReal.ofReal (\u2191(b i) - \u2191(a i)) \u2260 \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/Derivation.lean", "full_name": "MvPolynomial.mkDerivation_X", "start": [140, 1], "end": [141, 22], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "full_name": "MeasureTheory.integral_map", "start": [1610, 1], "end": [1620, 80], "traced_tactics": [{"tactic": "congr 1", "annotated_tactic": ["congr 1", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\nG : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nf\u271d g\u271d : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b2 : Type u_7\ninst\u271d : MeasurableSpace \u03b2\n\u03c6 : \u03b1 \u2192 \u03b2\nh\u03c6 : AEMeasurable \u03c6\nf : \u03b2 \u2192 G\nhfm : AEStronglyMeasurable f (Measure.map \u03c6 \u03bc)\ng : \u03b2 \u2192 G := AEStronglyMeasurable.mk f hfm\n\u22a2 \u222b (y : \u03b2), g y \u2202Measure.map \u03c6 \u03bc = \u222b (y : \u03b2), g y \u2202Measure.map (AEMeasurable.mk \u03c6 h\u03c6) \u03bc", "state_after": "case e_\u03bc\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\nG : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nf\u271d g\u271d : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b2 : Type u_7\ninst\u271d : MeasurableSpace \u03b2\n\u03c6 : \u03b1 \u2192 \u03b2\nh\u03c6 : AEMeasurable \u03c6\nf : \u03b2 \u2192 G\nhfm : AEStronglyMeasurable f (Measure.map \u03c6 \u03bc)\ng : \u03b2 \u2192 G := AEStronglyMeasurable.mk f hfm\n\u22a2 Measure.map \u03c6 \u03bc = Measure.map (AEMeasurable.mk \u03c6 h\u03c6) \u03bc"}, {"tactic": "exact Measure.map_congr h\u03c6.ae_eq_mk", "annotated_tactic": ["exact <a>Measure.map_congr</a> h\u03c6.ae_eq_mk", [{"full_name": "MeasureTheory.Measure.map_congr", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1191, 9], "def_end_pos": [1191, 18]}]], "state_before": "case e_\u03bc\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\nG : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nf\u271d g\u271d : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b2 : Type u_7\ninst\u271d : MeasurableSpace \u03b2\n\u03c6 : \u03b1 \u2192 \u03b2\nh\u03c6 : AEMeasurable \u03c6\nf : \u03b2 \u2192 G\nhfm : AEStronglyMeasurable f (Measure.map \u03c6 \u03bc)\ng : \u03b2 \u2192 G := AEStronglyMeasurable.mk f hfm\n\u22a2 Measure.map \u03c6 \u03bc = Measure.map (AEMeasurable.mk \u03c6 h\u03c6) \u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Hausdorff.lean", "full_name": "MeasureTheory.hausdorffMeasure_lineMap_image", "start": [1152, 1], "end": [1158, 30], "traced_tactics": [{"tactic": "borelize E", "annotated_tactic": ["borelize E", []], "state_before": "\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9\u00b9 : EMetricSpace X\ninst\u271d\u00b9\u2070 : EMetricSpace Y\ninst\u271d\u2079 : MeasurableSpace X\ninst\u271d\u2078 : BorelSpace X\ninst\u271d\u2077 : MeasurableSpace Y\ninst\u271d\u2076 : BorelSpace Y\n\ud835\udd5c : Type u_4\nE : Type u_5\nP : Type u_6\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace P\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor E P\ninst\u271d : BorelSpace P\nx y : P\ns : Set \u211d\n\u22a2 \u2191\u2191\u03bcH[1] (\u2191(IsometryEquiv.vaddConst x) '' ((fun x_1 => x_1 \u2022 (y -\u1d65 x)) '' s)) = nndist x y \u2022 \u2191\u2191\u03bcH[1] s", "state_after": "\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9\u00b9 : EMetricSpace X\ninst\u271d\u00b9\u2070 : EMetricSpace Y\ninst\u271d\u2079 : MeasurableSpace X\ninst\u271d\u2078 : BorelSpace X\ninst\u271d\u2077 : MeasurableSpace Y\ninst\u271d\u2076 : BorelSpace Y\n\ud835\udd5c : Type u_4\nE : Type u_5\nP : Type u_6\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace P\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor E P\ninst\u271d : BorelSpace P\nx y : P\ns : Set \u211d\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\n\u22a2 \u2191\u2191\u03bcH[1] (\u2191(IsometryEquiv.vaddConst x) '' ((fun x_1 => x_1 \u2022 (y -\u1d65 x)) '' s)) = nndist x y \u2022 \u2191\u2191\u03bcH[1] s"}, {"tactic": "rw [IsometryEquiv.hausdorffMeasure_image, hausdorffMeasure_smul_right_image,\n  nndist_eq_nnnorm_vsub' E]", "annotated_tactic": ["rw [<a>IsometryEquiv.hausdorffMeasure_image</a>, <a>hausdorffMeasure_smul_right_image</a>,\n    <a>nndist_eq_nnnorm_vsub'</a> E]", [{"full_name": "IsometryEquiv.hausdorffMeasure_image", "def_path": "Mathlib/MeasureTheory/Measure/Hausdorff.lean", "def_pos": [906, 9], "def_end_pos": [906, 31]}, {"full_name": "MeasureTheory.hausdorffMeasure_smul_right_image", "def_path": "Mathlib/MeasureTheory/Measure/Hausdorff.lean", "def_pos": [1079, 9], "def_end_pos": [1079, 42]}, {"full_name": "nndist_eq_nnnorm_vsub'", "def_path": "Mathlib/Analysis/Normed/Group/AddTorsor.lean", "def_pos": [89, 9], "def_end_pos": [89, 31]}]], "state_before": "\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9\u00b9 : EMetricSpace X\ninst\u271d\u00b9\u2070 : EMetricSpace Y\ninst\u271d\u2079 : MeasurableSpace X\ninst\u271d\u2078 : BorelSpace X\ninst\u271d\u2077 : MeasurableSpace Y\ninst\u271d\u2076 : BorelSpace Y\n\ud835\udd5c : Type u_4\nE : Type u_5\nP : Type u_6\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace P\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor E P\ninst\u271d : BorelSpace P\nx y : P\ns : Set \u211d\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\n\u22a2 \u2191\u2191\u03bcH[1] (\u2191(IsometryEquiv.vaddConst x) '' ((fun x_1 => x_1 \u2022 (y -\u1d65 x)) '' s)) = nndist x y \u2022 \u2191\u2191\u03bcH[1] s", "state_after": "no goals"}, {"tactic": "simpa only [Set.image_image]", "annotated_tactic": ["simpa only [<a>Set.image_image</a>]", [{"full_name": "Set.image_image", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [299, 9], "def_end_pos": [299, 20]}]], "state_before": "\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9\u00b9 : EMetricSpace X\ninst\u271d\u00b9\u2070 : EMetricSpace Y\ninst\u271d\u2079 : MeasurableSpace X\ninst\u271d\u2078 : BorelSpace X\ninst\u271d\u2077 : MeasurableSpace Y\ninst\u271d\u2076 : BorelSpace Y\n\ud835\udd5c : Type u_4\nE : Type u_5\nP : Type u_6\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace P\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor E P\ninst\u271d : BorelSpace P\nx y : P\ns : Set \u211d\nthis : \u2191\u2191\u03bcH[1] (\u2191(IsometryEquiv.vaddConst x) '' ((fun x_1 => x_1 \u2022 (y -\u1d65 x)) '' s)) = nndist x y \u2022 \u2191\u2191\u03bcH[1] s\n\u22a2 \u2191\u2191\u03bcH[1] (\u2191(AffineMap.lineMap x y) '' s) = nndist x y \u2022 \u2191\u2191\u03bcH[1] s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Prod/TProd.lean", "full_name": "List.TProd.elim_mk", "start": [106, 1], "end": [112, 48], "traced_tactics": [{"tactic": "by_cases hji : j = i", "annotated_tactic": ["by_cases hji : j = i", []], "state_before": "\u03b9 : Type u\n\u03b1 : \u03b9 \u2192 Type v\ni\u271d j\u271d : \u03b9\nl : List \u03b9\nf\u271d : (i : \u03b9) \u2192 \u03b1 i\ninst\u271d : DecidableEq \u03b9\ni : \u03b9\nis : List \u03b9\nf : (i : \u03b9) \u2192 \u03b1 i\nj : \u03b9\nhj : j \u2208 i :: is\n\u22a2 TProd.elim (TProd.mk (i :: is) f) hj = f j", "state_after": "case pos\n\u03b9 : Type u\n\u03b1 : \u03b9 \u2192 Type v\ni\u271d j\u271d : \u03b9\nl : List \u03b9\nf\u271d : (i : \u03b9) \u2192 \u03b1 i\ninst\u271d : DecidableEq \u03b9\ni : \u03b9\nis : List \u03b9\nf : (i : \u03b9) \u2192 \u03b1 i\nj : \u03b9\nhj : j \u2208 i :: is\nhji : j = i\n\u22a2 TProd.elim (TProd.mk (i :: is) f) hj = f j\n\ncase neg\n\u03b9 : Type u\n\u03b1 : \u03b9 \u2192 Type v\ni\u271d j\u271d : \u03b9\nl : List \u03b9\nf\u271d : (i : \u03b9) \u2192 \u03b1 i\ninst\u271d : DecidableEq \u03b9\ni : \u03b9\nis : List \u03b9\nf : (i : \u03b9) \u2192 \u03b1 i\nj : \u03b9\nhj : j \u2208 i :: is\nhji : \u00acj = i\n\u22a2 TProd.elim (TProd.mk (i :: is) f) hj = f j"}, {"tactic": "subst hji", "annotated_tactic": ["subst hji", []], "state_before": "case pos\n\u03b9 : Type u\n\u03b1 : \u03b9 \u2192 Type v\ni\u271d j\u271d : \u03b9\nl : List \u03b9\nf\u271d : (i : \u03b9) \u2192 \u03b1 i\ninst\u271d : DecidableEq \u03b9\ni : \u03b9\nis : List \u03b9\nf : (i : \u03b9) \u2192 \u03b1 i\nj : \u03b9\nhj : j \u2208 i :: is\nhji : j = i\n\u22a2 TProd.elim (TProd.mk (i :: is) f) hj = f j", "state_after": "case pos\n\u03b9 : Type u\n\u03b1 : \u03b9 \u2192 Type v\ni j\u271d : \u03b9\nl : List \u03b9\nf\u271d : (i : \u03b9) \u2192 \u03b1 i\ninst\u271d : DecidableEq \u03b9\nis : List \u03b9\nf : (i : \u03b9) \u2192 \u03b1 i\nj : \u03b9\nhj : j \u2208 j :: is\n\u22a2 TProd.elim (TProd.mk (j :: is) f) hj = f j"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case pos\n\u03b9 : Type u\n\u03b1 : \u03b9 \u2192 Type v\ni j\u271d : \u03b9\nl : List \u03b9\nf\u271d : (i : \u03b9) \u2192 \u03b1 i\ninst\u271d : DecidableEq \u03b9\nis : List \u03b9\nf : (i : \u03b9) \u2192 \u03b1 i\nj : \u03b9\nhj : j \u2208 j :: is\n\u22a2 TProd.elim (TProd.mk (j :: is) f) hj = f j", "state_after": "no goals"}, {"tactic": "rw [TProd.elim_of_ne _ hji, snd_mk, elim_mk is]", "annotated_tactic": ["rw [<a>TProd.elim_of_ne</a> _ hji, <a>snd_mk</a>, elim_mk is]", [{"full_name": "List.TProd.elim_of_ne", "def_path": "Mathlib/Data/Prod/TProd.lean", "def_pos": [94, 9], "def_end_pos": [94, 19]}, {"full_name": "List.TProd.snd_mk", "def_path": "Mathlib/Data/Prod/TProd.lean", "def_pos": [72, 9], "def_end_pos": [72, 15]}]], "state_before": "case neg\n\u03b9 : Type u\n\u03b1 : \u03b9 \u2192 Type v\ni\u271d j\u271d : \u03b9\nl : List \u03b9\nf\u271d : (i : \u03b9) \u2192 \u03b1 i\ninst\u271d : DecidableEq \u03b9\ni : \u03b9\nis : List \u03b9\nf : (i : \u03b9) \u2192 \u03b1 i\nj : \u03b9\nhj : j \u2208 i :: is\nhji : \u00acj = i\n\u22a2 TProd.elim (TProd.mk (i :: is) f) hj = f j", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "full_name": "String.atEnd_of_valid", "start": [317, 1], "end": [320, 61], "traced_tactics": [{"tactic": "rw [atEnd_iff]", "annotated_tactic": ["rw [<a>atEnd_iff</a>]", [{"full_name": "String.atEnd_iff", "def_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "def_pos": [274, 17], "def_end_pos": [274, 26]}]], "state_before": "cs cs' : List Char\n\u22a2 atEnd { data := cs ++ cs' } { byteIdx := utf8Len cs } = true \u2194 cs' = []", "state_after": "cs cs' : List Char\n\u22a2 endPos { data := cs ++ cs' } \u2264 { byteIdx := utf8Len cs } \u2194 cs' = []"}, {"tactic": "cases cs' <;> simp [Nat.lt_add_of_pos_right add_csize_pos]", "annotated_tactic": ["cases cs' <;> simp [<a>Nat.lt_add_of_pos_right</a> <a>add_csize_pos</a>]", [{"full_name": "Nat.lt_add_of_pos_right", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [318, 19], "def_end_pos": [318, 38]}, {"full_name": "_private.\u00ablake-packages\u00bb.std.Std.Data.String.Lemmas.0.String.add_csize_pos", "def_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "def_pos": [44, 17], "def_end_pos": [44, 30]}]], "state_before": "cs cs' : List Char\n\u22a2 endPos { data := cs ++ cs' } \u2264 { byteIdx := utf8Len cs } \u2194 cs' = []", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/LazyList/Basic.lean", "full_name": "LazyList.append_nil", "start": [143, 1], "end": [147, 18], "traced_tactics": [{"tactic": "induction' xs using LazyList.rec with _ _ _ _ ih", "annotated_tactic": ["induction' xs using <a>LazyList.rec</a> with _ _ _ _ ih", [{"full_name": "LazyList.rec", "def_path": "Mathlib/Data/LazyList.lean", "def_pos": [26, 11], "def_end_pos": [26, 19]}]], "state_before": "\u03b1 : Type u_1\nxs : LazyList \u03b1\n\u22a2 append xs (Thunk.pure nil) = xs", "state_after": "case nil\n\u03b1 : Type u_1\n\u22a2 append nil (Thunk.pure nil) = nil\n\ncase cons\n\u03b1 : Type u_1\nxs : LazyList \u03b1\nhd\u271d : \u03b1\ntl\u271d : Thunk (LazyList \u03b1)\ntl_ih\u271d : ?m.62213 tl\u271d\n\u22a2 append (cons hd\u271d tl\u271d) (Thunk.pure nil) = cons hd\u271d tl\u271d\n\ncase mk\n\u03b1 : Type u_1\nxs : LazyList \u03b1\nfn\u271d : Unit \u2192 LazyList \u03b1\nih : \u2200 (a : Unit), append (fn\u271d a) (Thunk.pure nil) = fn\u271d a\n\u22a2 ?m.62213 { fn := fn\u271d }"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case nil\n\u03b1 : Type u_1\n\u22a2 append nil (Thunk.pure nil) = nil", "state_after": "no goals"}, {"tactic": "simpa only [append, cons.injEq, true_and]", "annotated_tactic": ["simpa only [<a>append</a>, cons.injEq, <a>true_and</a>]", [{"full_name": "LazyList.append", "def_path": "Mathlib/Data/LazyList.lean", "def_pos": [74, 5], "def_end_pos": [74, 11]}, {"full_name": "true_and", "def_path": "lake-packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [84, 17], "def_end_pos": [84, 25]}]], "state_before": "case cons\n\u03b1 : Type u_1\nxs : LazyList \u03b1\nhd\u271d : \u03b1\ntl\u271d : Thunk (LazyList \u03b1)\ntl_ih\u271d : ?m.62213 tl\u271d\n\u22a2 append (cons hd\u271d tl\u271d) (Thunk.pure nil) = cons hd\u271d tl\u271d", "state_after": "no goals"}, {"tactic": "ext", "annotated_tactic": ["ext", []], "state_before": "case mk\n\u03b1 : Type u_1\nxs : LazyList \u03b1\nfn\u271d : Unit \u2192 LazyList \u03b1\nih : \u2200 (a : Unit), append (fn\u271d a) (Thunk.pure nil) = fn\u271d a\n\u22a2 { fn := fun x => append (Thunk.get { fn := fn\u271d }) (Thunk.pure nil) } = { fn := fn\u271d }", "state_after": "case mk.eq\n\u03b1 : Type u_1\nxs : LazyList \u03b1\nfn\u271d : Unit \u2192 LazyList \u03b1\nih : \u2200 (a : Unit), append (fn\u271d a) (Thunk.pure nil) = fn\u271d a\n\u22a2 Thunk.get { fn := fun x => append (Thunk.get { fn := fn\u271d }) (Thunk.pure nil) } = Thunk.get { fn := fn\u271d }"}, {"tactic": "apply ih", "annotated_tactic": ["apply ih", []], "state_before": "case mk.eq\n\u03b1 : Type u_1\nxs : LazyList \u03b1\nfn\u271d : Unit \u2192 LazyList \u03b1\nih : \u2200 (a : Unit), append (fn\u271d a) (Thunk.pure nil) = fn\u271d a\n\u22a2 Thunk.get { fn := fun x => append (Thunk.get { fn := fn\u271d }) (Thunk.pure nil) } = Thunk.get { fn := fn\u271d }", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/Layercake.lean", "full_name": "MeasureTheory.lintegral_rpow_eq_lintegral_meas_lt_mul", "start": [553, 1], "end": [562, 10], "traced_tactics": [{"tactic": "rw [lintegral_rpow_eq_lintegral_meas_le_mul \u03bc f_nn f_mble p_pos]", "annotated_tactic": ["rw [<a>lintegral_rpow_eq_lintegral_meas_le_mul</a> \u03bc f_nn f_mble p_pos]", [{"full_name": "MeasureTheory.lintegral_rpow_eq_lintegral_meas_le_mul", "def_path": "Mathlib/MeasureTheory/Integral/Layercake.lean", "def_pos": [469, 9], "def_end_pos": [469, 48]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSingletonClass \u03b2\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nf_mble : AEMeasurable f\np : \u211d\np_pos : 0 < p\n\u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9 ^ p) \u2202\u03bc =\n    ENNReal.ofReal p * \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a} * ENNReal.ofReal (t ^ (p - 1))", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSingletonClass \u03b2\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nf_mble : AEMeasurable f\np : \u211d\np_pos : 0 < p\n\u22a2 ENNReal.ofReal p * \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (t ^ (p - 1)) =\n    ENNReal.ofReal p * \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a} * ENNReal.ofReal (t ^ (p - 1))"}, {"tactic": "apply congr_arg fun z => ENNReal.ofReal p * z", "annotated_tactic": ["apply <a>congr_arg</a> fun z => <a>ENNReal.ofReal</a> p * z", [{"full_name": "congr_arg", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [43, 7], "def_end_pos": [43, 16]}, {"full_name": "ENNReal.ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [172, 29], "def_end_pos": [172, 35]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSingletonClass \u03b2\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nf_mble : AEMeasurable f\np : \u211d\np_pos : 0 < p\n\u22a2 ENNReal.ofReal p * \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (t ^ (p - 1)) =\n    ENNReal.ofReal p * \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a} * ENNReal.ofReal (t ^ (p - 1))", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSingletonClass \u03b2\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nf_mble : AEMeasurable f\np : \u211d\np_pos : 0 < p\n\u22a2 \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (t ^ (p - 1)) =\n    \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a} * ENNReal.ofReal (t ^ (p - 1))"}, {"tactic": "apply lintegral_congr_ae", "annotated_tactic": ["apply <a>lintegral_congr_ae</a>", [{"full_name": "MeasureTheory.lintegral_congr_ae", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [304, 9], "def_end_pos": [304, 27]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSingletonClass \u03b2\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nf_mble : AEMeasurable f\np : \u211d\np_pos : 0 < p\n\u22a2 \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (t ^ (p - 1)) =\n    \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a} * ENNReal.ofReal (t ^ (p - 1))", "state_after": "case h\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSingletonClass \u03b2\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nf_mble : AEMeasurable f\np : \u211d\np_pos : 0 < p\n\u22a2 (fun a => \u2191\u2191\u03bc {a_1 | a \u2264 f a_1} * ENNReal.ofReal (a ^ (p - 1))) =\u1da0[ae (Measure.restrict volume (Ioi 0))] fun a =>\n    \u2191\u2191\u03bc {a_1 | a < f a_1} * ENNReal.ofReal (a ^ (p - 1))"}, {"tactic": "filter_upwards [meas_le_ae_eq_meas_lt \u03bc (volume.restrict (Ioi 0)) f]\n  with t ht", "annotated_tactic": ["filter_upwards [<a>meas_le_ae_eq_meas_lt</a> \u03bc (volume.restrict (<a>Ioi</a> 0)) f]\n    with t ht", [{"full_name": "MeasureTheory.meas_le_ae_eq_meas_lt", "def_path": "Mathlib/MeasureTheory/Integral/Layercake.lean", "def_pos": [84, 9], "def_end_pos": [84, 30]}, {"full_name": "Set.Ioi", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [79, 5], "def_end_pos": [79, 8]}]], "state_before": "case h\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSingletonClass \u03b2\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nf_mble : AEMeasurable f\np : \u211d\np_pos : 0 < p\n\u22a2 (fun a => \u2191\u2191\u03bc {a_1 | a \u2264 f a_1} * ENNReal.ofReal (a ^ (p - 1))) =\u1da0[ae (Measure.restrict volume (Ioi 0))] fun a =>\n    \u2191\u2191\u03bc {a_1 | a < f a_1} * ENNReal.ofReal (a ^ (p - 1))", "state_after": "case h\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSingletonClass \u03b2\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nf_mble : AEMeasurable f\np : \u211d\np_pos : 0 < p\nt : \u211d\nht : \u2191\u2191\u03bc {a | t \u2264 f a} = \u2191\u2191\u03bc {a | t < f a}\n\u22a2 \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (t ^ (p - 1)) = \u2191\u2191\u03bc {a | t < f a} * ENNReal.ofReal (t ^ (p - 1))"}, {"tactic": "rw [ht]", "annotated_tactic": ["rw [ht]", []], "state_before": "case h\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSingletonClass \u03b2\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nf_mble : AEMeasurable f\np : \u211d\np_pos : 0 < p\nt : \u211d\nht : \u2191\u2191\u03bc {a | t \u2264 f a} = \u2191\u2191\u03bc {a | t < f a}\n\u22a2 \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (t ^ (p - 1)) = \u2191\u2191\u03bc {a | t < f a} * ENNReal.ofReal (t ^ (p - 1))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "full_name": "Measurable.eq_mp", "start": [942, 1], "end": [944, 33], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Basic.lean", "full_name": "Set.nontrivial_of_nontrivial_coe", "start": [2619, 1], "end": [2620, 55], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/PFun.lean", "full_name": "PFun.lift_graph", "start": [201, 1], "end": [202, 47], "traced_tactics": [{"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf : \u03b1 \u2192 \u03b2\na : \u03b1\nb : \u03b2\n\u22a2 (\u2203 x, f a = b) \u2194 f a = b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Process/Stopping.lean", "full_name": "MeasureTheory.Adapted.stronglyMeasurable_stoppedProcess", "start": [1040, 1], "end": [1043, 83], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "full_name": "MeasureTheory.L1.integral_of_fun_eq_integral", "start": [947, 1], "end": [952, 38], "traced_tactics": [{"tactic": "by_cases hG : CompleteSpace G", "annotated_tactic": ["by_cases hG : <a>CompleteSpace</a> G", [{"full_name": "CompleteSpace", "def_path": "Mathlib/Topology/UniformSpace/Cauchy.lean", "def_pos": [397, 7], "def_end_pos": [397, 20]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 G\nhf : Integrable f\n\u22a2 \u222b (a : \u03b1), \u2191\u2191(Integrable.toL1 f hf) a \u2202\u03bc = \u222b (a : \u03b1), f a \u2202\u03bc", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 G\nhf : Integrable f\nhG : CompleteSpace G\n\u22a2 \u222b (a : \u03b1), \u2191\u2191(Integrable.toL1 f hf) a \u2202\u03bc = \u222b (a : \u03b1), f a \u2202\u03bc\n\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 G\nhf : Integrable f\nhG : \u00acCompleteSpace G\n\u22a2 \u222b (a : \u03b1), \u2191\u2191(Integrable.toL1 f hf) a \u2202\u03bc = \u222b (a : \u03b1), f a \u2202\u03bc"}, {"tactic": "simp only [MeasureTheory.integral, hG, L1.integral]", "annotated_tactic": ["simp only [<a>MeasureTheory.integral</a>, hG, <a>L1.integral</a>]", [{"full_name": "MeasureTheory.integral", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [791, 17], "def_end_pos": [791, 25]}, {"full_name": "MeasureTheory.L1.integral", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [666, 17], "def_end_pos": [666, 25]}]], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 G\nhf : Integrable f\nhG : CompleteSpace G\n\u22a2 \u222b (a : \u03b1), \u2191\u2191(Integrable.toL1 f hf) a \u2202\u03bc = \u222b (a : \u03b1), f a \u2202\u03bc", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 G\nhf : Integrable f\nhG : CompleteSpace G\n\u22a2 (if h : True then\n      if hf_1 : Integrable fun a => \u2191\u2191(Integrable.toL1 f hf) a then\n        \u2191integralCLM (Integrable.toL1 (fun a => \u2191\u2191(Integrable.toL1 f hf) a) hf_1)\n      else 0\n    else 0) =\n    if h : True then if hf : Integrable fun a => f a then \u2191integralCLM (Integrable.toL1 (fun a => f a) hf) else 0 else 0"}, {"tactic": "exact setToFun_toL1 (dominatedFinMeasAdditive_weightedSMul \u03bc) hf", "annotated_tactic": ["exact <a>setToFun_toL1</a> (<a>dominatedFinMeasAdditive_weightedSMul</a> \u03bc) hf", [{"full_name": "MeasureTheory.setToFun_toL1", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [1438, 9], "def_end_pos": [1438, 22]}, {"full_name": "MeasureTheory.dominatedFinMeasAdditive_weightedSMul", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [243, 9], "def_end_pos": [243, 46]}]], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 G\nhf : Integrable f\nhG : CompleteSpace G\n\u22a2 (if h : True then\n      if hf_1 : Integrable fun a => \u2191\u2191(Integrable.toL1 f hf) a then\n        \u2191integralCLM (Integrable.toL1 (fun a => \u2191\u2191(Integrable.toL1 f hf) a) hf_1)\n      else 0\n    else 0) =\n    if h : True then if hf : Integrable fun a => f a then \u2191integralCLM (Integrable.toL1 (fun a => f a) hf) else 0 else 0", "state_after": "no goals"}, {"tactic": "simp [MeasureTheory.integral, hG]", "annotated_tactic": ["simp [<a>MeasureTheory.integral</a>, hG]", [{"full_name": "MeasureTheory.integral", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [791, 17], "def_end_pos": [791, 25]}]], "state_before": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 G\nhf : Integrable f\nhG : \u00acCompleteSpace G\n\u22a2 \u222b (a : \u03b1), \u2191\u2191(Integrable.toL1 f hf) a \u2202\u03bc = \u222b (a : \u03b1), f a \u2202\u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "full_name": "MeasureTheory.integral_add_measure", "start": [1462, 1], "end": [1480, 34], "traced_tactics": [{"tactic": "by_cases hG : CompleteSpace G", "annotated_tactic": ["by_cases hG : <a>CompleteSpace</a> G", [{"full_name": "CompleteSpace", "def_path": "Mathlib/Topology/UniformSpace/Cauchy.lean", "def_pos": [397, 7], "def_end_pos": [397, 20]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\n\u03bd : Measure \u03b1\nf : \u03b1 \u2192 G\nh\u03bc : Integrable f\nh\u03bd : Integrable f\n\u22a2 \u222b (x : \u03b1), f x \u2202(\u03bc + \u03bd) = \u222b (x : \u03b1), f x \u2202\u03bc + \u222b (x : \u03b1), f x \u2202\u03bd", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\n\u03bd : Measure \u03b1\nf : \u03b1 \u2192 G\nh\u03bc : Integrable f\nh\u03bd : Integrable f\nhG : CompleteSpace G\n\u22a2 \u222b (x : \u03b1), f x \u2202(\u03bc + \u03bd) = \u222b (x : \u03b1), f x \u2202\u03bc + \u222b (x : \u03b1), f x \u2202\u03bd\n\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\n\u03bd : Measure \u03b1\nf : \u03b1 \u2192 G\nh\u03bc : Integrable f\nh\u03bd : Integrable f\nhG : \u00acCompleteSpace G\n\u22a2 \u222b (x : \u03b1), f x \u2202(\u03bc + \u03bd) = \u222b (x : \u03b1), f x \u2202\u03bc + \u222b (x : \u03b1), f x \u2202\u03bd"}, {"tactic": "swap", "annotated_tactic": ["swap", []], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\n\u03bd : Measure \u03b1\nf : \u03b1 \u2192 G\nh\u03bc : Integrable f\nh\u03bd : Integrable f\nhG : CompleteSpace G\n\u22a2 \u222b (x : \u03b1), f x \u2202(\u03bc + \u03bd) = \u222b (x : \u03b1), f x \u2202\u03bc + \u222b (x : \u03b1), f x \u2202\u03bd\n\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\n\u03bd : Measure \u03b1\nf : \u03b1 \u2192 G\nh\u03bc : Integrable f\nh\u03bd : Integrable f\nhG : \u00acCompleteSpace G\n\u22a2 \u222b (x : \u03b1), f x \u2202(\u03bc + \u03bd) = \u222b (x : \u03b1), f x \u2202\u03bc + \u222b (x : \u03b1), f x \u2202\u03bd", "state_after": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\n\u03bd : Measure \u03b1\nf : \u03b1 \u2192 G\nh\u03bc : Integrable f\nh\u03bd : Integrable f\nhG : \u00acCompleteSpace G\n\u22a2 \u222b (x : \u03b1), f x \u2202(\u03bc + \u03bd) = \u222b (x : \u03b1), f x \u2202\u03bc + \u222b (x : \u03b1), f x \u2202\u03bd\n\ncase pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\n\u03bd : Measure \u03b1\nf : \u03b1 \u2192 G\nh\u03bc : Integrable f\nh\u03bd : Integrable f\nhG : CompleteSpace G\n\u22a2 \u222b (x : \u03b1), f x \u2202(\u03bc + \u03bd) = \u222b (x : \u03b1), f x \u2202\u03bc + \u222b (x : \u03b1), f x \u2202\u03bd"}, {"tactic": "have hfi := h\u03bc.add_measure h\u03bd", "annotated_tactic": ["have hfi := h\u03bc.add_measure h\u03bd", []], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\n\u03bd : Measure \u03b1\nf : \u03b1 \u2192 G\nh\u03bc : Integrable f\nh\u03bd : Integrable f\nhG : CompleteSpace G\n\u22a2 \u222b (x : \u03b1), f x \u2202(\u03bc + \u03bd) = \u222b (x : \u03b1), f x \u2202\u03bc + \u222b (x : \u03b1), f x \u2202\u03bd", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\n\u03bd : Measure \u03b1\nf : \u03b1 \u2192 G\nh\u03bc : Integrable f\nh\u03bd : Integrable f\nhG : CompleteSpace G\nhfi : Integrable f\n\u22a2 \u222b (x : \u03b1), f x \u2202(\u03bc + \u03bd) = \u222b (x : \u03b1), f x \u2202\u03bc + \u222b (x : \u03b1), f x \u2202\u03bd"}, {"tactic": "simp_rw [integral_eq_setToFun]", "annotated_tactic": ["simp_rw [<a>integral_eq_setToFun</a>]", [{"full_name": "MeasureTheory.integral_eq_setToFun", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [825, 9], "def_end_pos": [825, 29]}]], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\n\u03bd : Measure \u03b1\nf : \u03b1 \u2192 G\nh\u03bc : Integrable f\nh\u03bd : Integrable f\nhG : CompleteSpace G\nhfi : Integrable f\n\u22a2 \u222b (x : \u03b1), f x \u2202(\u03bc + \u03bd) = \u222b (x : \u03b1), f x \u2202\u03bc + \u222b (x : \u03b1), f x \u2202\u03bd", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\n\u03bd : Measure \u03b1\nf : \u03b1 \u2192 G\nh\u03bc : Integrable f\nh\u03bd : Integrable f\nhG : CompleteSpace G\nhfi : Integrable f\n\u22a2 (setToFun (\u03bc + \u03bd) (weightedSMul (\u03bc + \u03bd)) (_ : DominatedFinMeasAdditive (\u03bc + \u03bd) (weightedSMul (\u03bc + \u03bd)) 1) fun a =>\n      f a) =\n    (setToFun \u03bc (weightedSMul \u03bc) (_ : DominatedFinMeasAdditive \u03bc (weightedSMul \u03bc) 1) fun a => f a) +\n      setToFun \u03bd (weightedSMul \u03bd) (_ : DominatedFinMeasAdditive \u03bd (weightedSMul \u03bd) 1) fun a => f a"}, {"tactic": "have h\u03bc_dfma : DominatedFinMeasAdditive (\u03bc + \u03bd) (weightedSMul \u03bc : Set \u03b1 \u2192 G \u2192L[\u211d] G) 1 :=\n  DominatedFinMeasAdditive.add_measure_right \u03bc \u03bd (dominatedFinMeasAdditive_weightedSMul \u03bc)\n    zero_le_one", "annotated_tactic": ["have h\u03bc_dfma : <a>DominatedFinMeasAdditive</a> (\u03bc + \u03bd) (<a>weightedSMul</a> \u03bc : <a>Set</a> \u03b1 \u2192 G \u2192L[\u211d] G) 1 :=\n    <a>DominatedFinMeasAdditive.add_measure_right</a> \u03bc \u03bd (<a>dominatedFinMeasAdditive_weightedSMul</a> \u03bc)\n      <a>zero_le_one</a>", [{"full_name": "MeasureTheory.DominatedFinMeasAdditive", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [187, 5], "def_end_pos": [187, 29]}, {"full_name": "MeasureTheory.weightedSMul", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [166, 5], "def_end_pos": [166, 17]}, {"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}, {"full_name": "MeasureTheory.DominatedFinMeasAdditive.add_measure_right", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [243, 9], "def_end_pos": [243, 26]}, {"full_name": "MeasureTheory.dominatedFinMeasAdditive_weightedSMul", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [243, 9], "def_end_pos": [243, 46]}, {"full_name": "zero_le_one", "def_path": "Mathlib/Algebra/Order/ZeroLEOne.lean", "def_pos": [26, 15], "def_end_pos": [26, 26]}]], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\n\u03bd : Measure \u03b1\nf : \u03b1 \u2192 G\nh\u03bc : Integrable f\nh\u03bd : Integrable f\nhG : CompleteSpace G\nhfi : Integrable f\n\u22a2 (setToFun (\u03bc + \u03bd) (weightedSMul (\u03bc + \u03bd)) (_ : DominatedFinMeasAdditive (\u03bc + \u03bd) (weightedSMul (\u03bc + \u03bd)) 1) fun a =>\n      f a) =\n    (setToFun \u03bc (weightedSMul \u03bc) (_ : DominatedFinMeasAdditive \u03bc (weightedSMul \u03bc) 1) fun a => f a) +\n      setToFun \u03bd (weightedSMul \u03bd) (_ : DominatedFinMeasAdditive \u03bd (weightedSMul \u03bd) 1) fun a => f a", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\n\u03bd : Measure \u03b1\nf : \u03b1 \u2192 G\nh\u03bc : Integrable f\nh\u03bd : Integrable f\nhG : CompleteSpace G\nhfi : Integrable f\nh\u03bc_dfma : DominatedFinMeasAdditive (\u03bc + \u03bd) (weightedSMul \u03bc) 1\n\u22a2 (setToFun (\u03bc + \u03bd) (weightedSMul (\u03bc + \u03bd)) (_ : DominatedFinMeasAdditive (\u03bc + \u03bd) (weightedSMul (\u03bc + \u03bd)) 1) fun a =>\n      f a) =\n    (setToFun \u03bc (weightedSMul \u03bc) (_ : DominatedFinMeasAdditive \u03bc (weightedSMul \u03bc) 1) fun a => f a) +\n      setToFun \u03bd (weightedSMul \u03bd) (_ : DominatedFinMeasAdditive \u03bd (weightedSMul \u03bd) 1) fun a => f a"}, {"tactic": "have h\u03bd_dfma : DominatedFinMeasAdditive (\u03bc + \u03bd) (weightedSMul \u03bd : Set \u03b1 \u2192 G \u2192L[\u211d] G) 1 :=\n  DominatedFinMeasAdditive.add_measure_left \u03bc \u03bd (dominatedFinMeasAdditive_weightedSMul \u03bd)\n    zero_le_one", "annotated_tactic": ["have h\u03bd_dfma : <a>DominatedFinMeasAdditive</a> (\u03bc + \u03bd) (<a>weightedSMul</a> \u03bd : <a>Set</a> \u03b1 \u2192 G \u2192L[\u211d] G) 1 :=\n    <a>DominatedFinMeasAdditive.add_measure_left</a> \u03bc \u03bd (<a>dominatedFinMeasAdditive_weightedSMul</a> \u03bd)\n      <a>zero_le_one</a>", [{"full_name": "MeasureTheory.DominatedFinMeasAdditive", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [187, 5], "def_end_pos": [187, 29]}, {"full_name": "MeasureTheory.weightedSMul", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [166, 5], "def_end_pos": [166, 17]}, {"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}, {"full_name": "MeasureTheory.DominatedFinMeasAdditive.add_measure_left", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [248, 9], "def_end_pos": [248, 25]}, {"full_name": "MeasureTheory.dominatedFinMeasAdditive_weightedSMul", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [243, 9], "def_end_pos": [243, 46]}, {"full_name": "zero_le_one", "def_path": "Mathlib/Algebra/Order/ZeroLEOne.lean", "def_pos": [26, 15], "def_end_pos": [26, 26]}]], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\n\u03bd : Measure \u03b1\nf : \u03b1 \u2192 G\nh\u03bc : Integrable f\nh\u03bd : Integrable f\nhG : CompleteSpace G\nhfi : Integrable f\nh\u03bc_dfma : DominatedFinMeasAdditive (\u03bc + \u03bd) (weightedSMul \u03bc) 1\n\u22a2 (setToFun (\u03bc + \u03bd) (weightedSMul (\u03bc + \u03bd)) (_ : DominatedFinMeasAdditive (\u03bc + \u03bd) (weightedSMul (\u03bc + \u03bd)) 1) fun a =>\n      f a) =\n    (setToFun \u03bc (weightedSMul \u03bc) (_ : DominatedFinMeasAdditive \u03bc (weightedSMul \u03bc) 1) fun a => f a) +\n      setToFun \u03bd (weightedSMul \u03bd) (_ : DominatedFinMeasAdditive \u03bd (weightedSMul \u03bd) 1) fun a => f a", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\n\u03bd : Measure \u03b1\nf : \u03b1 \u2192 G\nh\u03bc : Integrable f\nh\u03bd : Integrable f\nhG : CompleteSpace G\nhfi : Integrable f\nh\u03bc_dfma : DominatedFinMeasAdditive (\u03bc + \u03bd) (weightedSMul \u03bc) 1\nh\u03bd_dfma : DominatedFinMeasAdditive (\u03bc + \u03bd) (weightedSMul \u03bd) 1\n\u22a2 (setToFun (\u03bc + \u03bd) (weightedSMul (\u03bc + \u03bd)) (_ : DominatedFinMeasAdditive (\u03bc + \u03bd) (weightedSMul (\u03bc + \u03bd)) 1) fun a =>\n      f a) =\n    (setToFun \u03bc (weightedSMul \u03bc) (_ : DominatedFinMeasAdditive \u03bc (weightedSMul \u03bc) 1) fun a => f a) +\n      setToFun \u03bd (weightedSMul \u03bd) (_ : DominatedFinMeasAdditive \u03bd (weightedSMul \u03bd) 1) fun a => f a"}, {"tactic": "rw [\u2190 setToFun_congr_measure_of_add_right h\u03bc_dfma\n      (dominatedFinMeasAdditive_weightedSMul \u03bc) f hfi,\n  \u2190 setToFun_congr_measure_of_add_left h\u03bd_dfma (dominatedFinMeasAdditive_weightedSMul \u03bd) f hfi]", "annotated_tactic": ["rw [\u2190 <a>setToFun_congr_measure_of_add_right</a> h\u03bc_dfma\n        (<a>dominatedFinMeasAdditive_weightedSMul</a> \u03bc) f hfi,\n    \u2190 <a>setToFun_congr_measure_of_add_left</a> h\u03bd_dfma (<a>dominatedFinMeasAdditive_weightedSMul</a> \u03bd) f hfi]", [{"full_name": "MeasureTheory.setToFun_congr_measure_of_add_right", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [1646, 9], "def_end_pos": [1646, 44]}, {"full_name": "MeasureTheory.dominatedFinMeasAdditive_weightedSMul", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [243, 9], "def_end_pos": [243, 46]}, {"full_name": "MeasureTheory.setToFun_congr_measure_of_add_left", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [1656, 9], "def_end_pos": [1656, 43]}, {"full_name": "MeasureTheory.dominatedFinMeasAdditive_weightedSMul", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [243, 9], "def_end_pos": [243, 46]}]], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\n\u03bd : Measure \u03b1\nf : \u03b1 \u2192 G\nh\u03bc : Integrable f\nh\u03bd : Integrable f\nhG : CompleteSpace G\nhfi : Integrable f\nh\u03bc_dfma : DominatedFinMeasAdditive (\u03bc + \u03bd) (weightedSMul \u03bc) 1\nh\u03bd_dfma : DominatedFinMeasAdditive (\u03bc + \u03bd) (weightedSMul \u03bd) 1\n\u22a2 (setToFun (\u03bc + \u03bd) (weightedSMul (\u03bc + \u03bd)) (_ : DominatedFinMeasAdditive (\u03bc + \u03bd) (weightedSMul (\u03bc + \u03bd)) 1) fun a =>\n      f a) =\n    (setToFun \u03bc (weightedSMul \u03bc) (_ : DominatedFinMeasAdditive \u03bc (weightedSMul \u03bc) 1) fun a => f a) +\n      setToFun \u03bd (weightedSMul \u03bd) (_ : DominatedFinMeasAdditive \u03bd (weightedSMul \u03bd) 1) fun a => f a", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\n\u03bd : Measure \u03b1\nf : \u03b1 \u2192 G\nh\u03bc : Integrable f\nh\u03bd : Integrable f\nhG : CompleteSpace G\nhfi : Integrable f\nh\u03bc_dfma : DominatedFinMeasAdditive (\u03bc + \u03bd) (weightedSMul \u03bc) 1\nh\u03bd_dfma : DominatedFinMeasAdditive (\u03bc + \u03bd) (weightedSMul \u03bd) 1\n\u22a2 (setToFun (\u03bc + \u03bd) (weightedSMul (\u03bc + \u03bd)) (_ : DominatedFinMeasAdditive (\u03bc + \u03bd) (weightedSMul (\u03bc + \u03bd)) 1) fun a =>\n      f a) =\n    setToFun (\u03bc + \u03bd) (weightedSMul \u03bc) h\u03bc_dfma f + setToFun (\u03bc + \u03bd) (weightedSMul \u03bd) h\u03bd_dfma f"}, {"tactic": "refine' setToFun_add_left' _ _ _ (fun s _ h\u03bc\u03bds => _) f", "annotated_tactic": ["refine' <a>setToFun_add_left'</a> _ _ _ (fun s _ h\u03bc\u03bds => _) f", [{"full_name": "MeasureTheory.setToFun_add_left'", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [1320, 9], "def_end_pos": [1320, 27]}]], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\n\u03bd : Measure \u03b1\nf : \u03b1 \u2192 G\nh\u03bc : Integrable f\nh\u03bd : Integrable f\nhG : CompleteSpace G\nhfi : Integrable f\nh\u03bc_dfma : DominatedFinMeasAdditive (\u03bc + \u03bd) (weightedSMul \u03bc) 1\nh\u03bd_dfma : DominatedFinMeasAdditive (\u03bc + \u03bd) (weightedSMul \u03bd) 1\n\u22a2 (setToFun (\u03bc + \u03bd) (weightedSMul (\u03bc + \u03bd)) (_ : DominatedFinMeasAdditive (\u03bc + \u03bd) (weightedSMul (\u03bc + \u03bd)) 1) fun a =>\n      f a) =\n    setToFun (\u03bc + \u03bd) (weightedSMul \u03bc) h\u03bc_dfma f + setToFun (\u03bc + \u03bd) (weightedSMul \u03bd) h\u03bd_dfma f", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\n\u03bd : Measure \u03b1\nf : \u03b1 \u2192 G\nh\u03bc : Integrable f\nh\u03bd : Integrable f\nhG : CompleteSpace G\nhfi : Integrable f\nh\u03bc_dfma : DominatedFinMeasAdditive (\u03bc + \u03bd) (weightedSMul \u03bc) 1\nh\u03bd_dfma : DominatedFinMeasAdditive (\u03bc + \u03bd) (weightedSMul \u03bd) 1\ns : Set \u03b1\nx\u271d : MeasurableSet s\nh\u03bc\u03bds : \u2191\u2191(\u03bc + \u03bd) s < \u22a4\n\u22a2 weightedSMul (\u03bc + \u03bd) s = weightedSMul \u03bc s + weightedSMul \u03bd s"}, {"tactic": "rw [Measure.coe_add, Pi.add_apply, add_lt_top] at h\u03bc\u03bds", "annotated_tactic": ["rw [<a>Measure.coe_add</a>, <a>Pi.add_apply</a>, <a>add_lt_top</a>] at h\u03bc\u03bds", [{"full_name": "MeasureTheory.Measure.coe_add", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [794, 9], "def_end_pos": [794, 16]}, {"full_name": "Pi.add_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [82, 3], "def_end_pos": [82, 14]}, {"full_name": "ENNReal.add_lt_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [561, 17], "def_end_pos": [561, 27]}]], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\n\u03bd : Measure \u03b1\nf : \u03b1 \u2192 G\nh\u03bc : Integrable f\nh\u03bd : Integrable f\nhG : CompleteSpace G\nhfi : Integrable f\nh\u03bc_dfma : DominatedFinMeasAdditive (\u03bc + \u03bd) (weightedSMul \u03bc) 1\nh\u03bd_dfma : DominatedFinMeasAdditive (\u03bc + \u03bd) (weightedSMul \u03bd) 1\ns : Set \u03b1\nx\u271d : MeasurableSet s\nh\u03bc\u03bds : \u2191\u2191(\u03bc + \u03bd) s < \u22a4\n\u22a2 weightedSMul (\u03bc + \u03bd) s = weightedSMul \u03bc s + weightedSMul \u03bd s", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\n\u03bd : Measure \u03b1\nf : \u03b1 \u2192 G\nh\u03bc : Integrable f\nh\u03bd : Integrable f\nhG : CompleteSpace G\nhfi : Integrable f\nh\u03bc_dfma : DominatedFinMeasAdditive (\u03bc + \u03bd) (weightedSMul \u03bc) 1\nh\u03bd_dfma : DominatedFinMeasAdditive (\u03bc + \u03bd) (weightedSMul \u03bd) 1\ns : Set \u03b1\nx\u271d : MeasurableSet s\nh\u03bc\u03bds : \u2191\u2191\u03bc s < \u22a4 \u2227 \u2191\u2191\u03bd s < \u22a4\n\u22a2 weightedSMul (\u03bc + \u03bd) s = weightedSMul \u03bc s + weightedSMul \u03bd s"}, {"tactic": "rw [weightedSMul, weightedSMul, weightedSMul, \u2190 add_smul, Measure.coe_add, Pi.add_apply,\ntoReal_add h\u03bc\u03bds.1.ne h\u03bc\u03bds.2.ne]", "annotated_tactic": ["rw [<a>weightedSMul</a>, <a>weightedSMul</a>, <a>weightedSMul</a>, \u2190 <a>add_smul</a>, <a>Measure.coe_add</a>, <a>Pi.add_apply</a>,\n  <a>toReal_add</a> h\u03bc\u03bds.1.<a>ne</a> h\u03bc\u03bds.2.<a>ne</a>]", [{"full_name": "MeasureTheory.weightedSMul", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [166, 5], "def_end_pos": [166, 17]}, {"full_name": "MeasureTheory.weightedSMul", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [166, 5], "def_end_pos": [166, 17]}, {"full_name": "MeasureTheory.weightedSMul", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [166, 5], "def_end_pos": [166, 17]}, {"full_name": "add_smul", "def_path": "Mathlib/Algebra/Module/Basic.lean", "def_pos": [91, 9], "def_end_pos": [91, 17]}, {"full_name": "MeasureTheory.Measure.coe_add", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [794, 9], "def_end_pos": [794, 16]}, {"full_name": "Pi.add_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [82, 3], "def_end_pos": [82, 14]}, {"full_name": "ENNReal.toReal_add", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1997, 9], "def_end_pos": [1997, 19]}, {"full_name": "LT.lt.ne", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [152, 7], "def_end_pos": [152, 15]}, {"full_name": "LT.lt.ne", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [152, 7], "def_end_pos": [152, 15]}]], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\n\u03bd : Measure \u03b1\nf : \u03b1 \u2192 G\nh\u03bc : Integrable f\nh\u03bd : Integrable f\nhG : CompleteSpace G\nhfi : Integrable f\nh\u03bc_dfma : DominatedFinMeasAdditive (\u03bc + \u03bd) (weightedSMul \u03bc) 1\nh\u03bd_dfma : DominatedFinMeasAdditive (\u03bc + \u03bd) (weightedSMul \u03bd) 1\ns : Set \u03b1\nx\u271d : MeasurableSet s\nh\u03bc\u03bds : \u2191\u2191\u03bc s < \u22a4 \u2227 \u2191\u2191\u03bd s < \u22a4\n\u22a2 weightedSMul (\u03bc + \u03bd) s = weightedSMul \u03bc s + weightedSMul \u03bd s", "state_after": "no goals"}, {"tactic": "simp [integral, hG]", "annotated_tactic": ["simp [<a>integral</a>, hG]", [{"full_name": "MeasureTheory.integral", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [791, 17], "def_end_pos": [791, 25]}]], "state_before": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\n\u03bd : Measure \u03b1\nf : \u03b1 \u2192 G\nh\u03bc : Integrable f\nh\u03bd : Integrable f\nhG : \u00acCompleteSpace G\n\u22a2 \u222b (x : \u03b1), f x \u2202(\u03bc + \u03bd) = \u222b (x : \u03b1), f x \u2202\u03bc + \u222b (x : \u03b1), f x \u2202\u03bd", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Decomposition/Jordan.lean", "full_name": "MeasureTheory.JordanDecomposition.toSignedMeasure_zero", "start": [173, 1], "end": [177, 43], "traced_tactics": [{"tactic": "ext1 i hi", "annotated_tactic": ["ext1 i hi", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj : JordanDecomposition \u03b1\n\u22a2 toSignedMeasure 0 = 0", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj : JordanDecomposition \u03b1\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 \u2191(toSignedMeasure 0) i = \u21910 i"}, {"tactic": "rw [toSignedMeasure, toSignedMeasure_sub_apply hi, zero_posPart, zero_negPart, sub_self,\n  VectorMeasure.coe_zero, Pi.zero_apply]", "annotated_tactic": ["rw [<a>toSignedMeasure</a>, <a>toSignedMeasure_sub_apply</a> hi, <a>zero_posPart</a>, <a>zero_negPart</a>, <a>sub_self</a>,\n    <a>VectorMeasure.coe_zero</a>, <a>Pi.zero_apply</a>]", [{"full_name": "MeasureTheory.JordanDecomposition.toSignedMeasure", "def_path": "Mathlib/MeasureTheory/Decomposition/Jordan.lean", "def_pos": [169, 5], "def_end_pos": [169, 20]}, {"full_name": "MeasureTheory.Measure.toSignedMeasure_sub_apply", "def_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "def_pos": [514, 9], "def_end_pos": [514, 34]}, {"full_name": "MeasureTheory.JordanDecomposition.zero_posPart", "def_path": "Mathlib/MeasureTheory/Decomposition/Jordan.lean", "def_pos": [100, 9], "def_end_pos": [100, 21]}, {"full_name": "MeasureTheory.JordanDecomposition.zero_negPart", "def_path": "Mathlib/MeasureTheory/Decomposition/Jordan.lean", "def_pos": [105, 9], "def_end_pos": [105, 21]}, {"full_name": "sub_self", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [734, 30], "def_end_pos": [734, 38]}, {"full_name": "MeasureTheory.VectorMeasure.coe_zero", "def_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "def_pos": [290, 9], "def_end_pos": [290, 17]}, {"full_name": "Pi.zero_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [46, 3], "def_end_pos": [46, 14]}]], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj : JordanDecomposition \u03b1\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 \u2191(toSignedMeasure 0) i = \u21910 i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "full_name": "String.extract.go\u2081_cons_addChar", "start": [430, 1], "end": [433, 29], "traced_tactics": [{"tactic": "simp [go\u2081, Pos.ext_iff, Nat.ne_of_lt add_csize_pos]", "annotated_tactic": ["simp [<a>go\u2081</a>, <a>Pos.ext_iff</a>, <a>Nat.ne_of_lt</a> <a>add_csize_pos</a>]", [{"full_name": "String.extract.go\u2081", "def_path": "lake-packages/lean4/src/lean/Init/Data/String/Basic.lean", "def_pos": [223, 3], "def_end_pos": [223, 6]}, {"full_name": "String.Pos.ext_iff", "def_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "def_pos": [108, 9], "def_end_pos": [108, 16]}, {"full_name": "Nat.ne_of_lt", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [329, 9], "def_end_pos": [329, 17]}, {"full_name": "_private.\u00ablake-packages\u00bb.std.Std.Data.String.Lemmas.0.String.add_csize_pos", "def_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "def_pos": [44, 17], "def_end_pos": [44, 30]}]], "state_before": "c : Char\ncs : List Char\nb e : Pos\n\u22a2 go\u2081 (c :: cs) 0 (b + c) (e + c) = go\u2081 cs 0 b e", "state_after": "c : Char\ncs : List Char\nb e : Pos\n\u22a2 go\u2081 cs (0 + c) (b + c) (e + c) = go\u2081 cs 0 b e"}, {"tactic": "apply go\u2081_add_right_cancel", "annotated_tactic": ["apply <a>go\u2081_add_right_cancel</a>", [{"full_name": "String.extract.go\u2081_add_right_cancel", "def_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "def_pos": [417, 9], "def_end_pos": [417, 37]}]], "state_before": "c : Char\ncs : List Char\nb e : Pos\n\u22a2 go\u2081 cs (0 + c) (b + c) (e + c) = go\u2081 cs 0 b e", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/AEMeasurable.lean", "full_name": "AEMeasurable.mono_measure", "start": [52, 1], "end": [53, 36], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/TuringMachine.lean", "full_name": "Turing.TM2to1.addBottom_modifyNth", "start": [2378, 1], "end": [2383, 57], "traced_tactics": [{"tactic": "cases n <;>\n  simp only [addBottom, ListBlank.head_cons, ListBlank.modifyNth, ListBlank.tail_cons]", "annotated_tactic": ["cases n <;>\n    simp only [<a>addBottom</a>, <a>ListBlank.head_cons</a>, <a>ListBlank.modifyNth</a>, <a>ListBlank.tail_cons</a>]", [{"full_name": "Turing.TM2to1.addBottom", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [2366, 5], "def_end_pos": [2366, 14]}, {"full_name": "Turing.ListBlank.head_cons", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [253, 9], "def_end_pos": [253, 28]}, {"full_name": "Turing.ListBlank.modifyNth", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [332, 5], "def_end_pos": [332, 24]}, {"full_name": "Turing.ListBlank.tail_cons", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [258, 9], "def_end_pos": [258, 28]}]], "state_before": "K : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nf : ((k : K) \u2192 Option (\u0393 k)) \u2192 (k : K) \u2192 Option (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nn : \u2115\n\u22a2 ListBlank.modifyNth (fun a => (a.1, f a.2)) n (addBottom L) = addBottom (ListBlank.modifyNth f n L)", "state_after": "case succ\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nf : ((k : K) \u2192 Option (\u0393 k)) \u2192 (k : K) \u2192 Option (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nn\u271d : \u2115\n\u22a2 ListBlank.cons (true, ListBlank.head L)\n      (ListBlank.modifyNth (fun a => (a.1, f a.2)) n\u271d\n        (ListBlank.map { f := Prod.mk false, map_pt' := (_ : (false, default) = (false, default)) }\n          (ListBlank.tail L))) =\n    ListBlank.cons (true, ListBlank.head L)\n      (ListBlank.map { f := Prod.mk false, map_pt' := (_ : (false, default) = (false, default)) }\n        (ListBlank.modifyNth f n\u271d (ListBlank.tail L)))"}, {"tactic": "congr", "annotated_tactic": ["congr", []], "state_before": "case succ\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nf : ((k : K) \u2192 Option (\u0393 k)) \u2192 (k : K) \u2192 Option (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nn\u271d : \u2115\n\u22a2 ListBlank.cons (true, ListBlank.head L)\n      (ListBlank.modifyNth (fun a => (a.1, f a.2)) n\u271d\n        (ListBlank.map { f := Prod.mk false, map_pt' := (_ : (false, default) = (false, default)) }\n          (ListBlank.tail L))) =\n    ListBlank.cons (true, ListBlank.head L)\n      (ListBlank.map { f := Prod.mk false, map_pt' := (_ : (false, default) = (false, default)) }\n        (ListBlank.modifyNth f n\u271d (ListBlank.tail L)))", "state_after": "case succ.e_l\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nf : ((k : K) \u2192 Option (\u0393 k)) \u2192 (k : K) \u2192 Option (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nn\u271d : \u2115\n\u22a2 ListBlank.modifyNth (fun a => (a.1, f a.2)) n\u271d\n      (ListBlank.map { f := Prod.mk false, map_pt' := (_ : (false, default) = (false, default)) } (ListBlank.tail L)) =\n    ListBlank.map { f := Prod.mk false, map_pt' := (_ : (false, default) = (false, default)) }\n      (ListBlank.modifyNth f n\u271d (ListBlank.tail L))"}, {"tactic": "symm", "annotated_tactic": ["symm", []], "state_before": "case succ.e_l\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nf : ((k : K) \u2192 Option (\u0393 k)) \u2192 (k : K) \u2192 Option (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nn\u271d : \u2115\n\u22a2 ListBlank.modifyNth (fun a => (a.1, f a.2)) n\u271d\n      (ListBlank.map { f := Prod.mk false, map_pt' := (_ : (false, default) = (false, default)) } (ListBlank.tail L)) =\n    ListBlank.map { f := Prod.mk false, map_pt' := (_ : (false, default) = (false, default)) }\n      (ListBlank.modifyNth f n\u271d (ListBlank.tail L))", "state_after": "case succ.e_l\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nf : ((k : K) \u2192 Option (\u0393 k)) \u2192 (k : K) \u2192 Option (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nn\u271d : \u2115\n\u22a2 ListBlank.map { f := Prod.mk false, map_pt' := (_ : (false, default) = (false, default)) }\n      (ListBlank.modifyNth f n\u271d (ListBlank.tail L)) =\n    ListBlank.modifyNth (fun a => (a.1, f a.2)) n\u271d\n      (ListBlank.map { f := Prod.mk false, map_pt' := (_ : (false, default) = (false, default)) } (ListBlank.tail L))"}, {"tactic": "apply ListBlank.map_modifyNth", "annotated_tactic": ["apply <a>ListBlank.map_modifyNth</a>", [{"full_name": "Turing.ListBlank.map_modifyNth", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [438, 9], "def_end_pos": [438, 32]}]], "state_before": "case succ.e_l\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nf : ((k : K) \u2192 Option (\u0393 k)) \u2192 (k : K) \u2192 Option (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nn\u271d : \u2115\n\u22a2 ListBlank.map { f := Prod.mk false, map_pt' := (_ : (false, default) = (false, default)) }\n      (ListBlank.modifyNth f n\u271d (ListBlank.tail L)) =\n    ListBlank.modifyNth (fun a => (a.1, f a.2)) n\u271d\n      (ListBlank.map { f := Prod.mk false, map_pt' := (_ : (false, default) = (false, default)) } (ListBlank.tail L))", "state_after": "case succ.e_l.H\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nf : ((k : K) \u2192 Option (\u0393 k)) \u2192 (k : K) \u2192 Option (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nn\u271d : \u2115\n\u22a2 \u2200 (x : (k : K) \u2192 Option (\u0393 k)),\n    PointedMap.f { f := Prod.mk false, map_pt' := (_ : (false, default) = (false, default)) } (f x) =\n      ((PointedMap.f { f := Prod.mk false, map_pt' := (_ : (false, default) = (false, default)) } x).1,\n        f (PointedMap.f { f := Prod.mk false, map_pt' := (_ : (false, default) = (false, default)) } x).2)"}, {"tactic": "intro", "annotated_tactic": ["intro", []], "state_before": "case succ.e_l.H\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nf : ((k : K) \u2192 Option (\u0393 k)) \u2192 (k : K) \u2192 Option (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nn\u271d : \u2115\n\u22a2 \u2200 (x : (k : K) \u2192 Option (\u0393 k)),\n    PointedMap.f { f := Prod.mk false, map_pt' := (_ : (false, default) = (false, default)) } (f x) =\n      ((PointedMap.f { f := Prod.mk false, map_pt' := (_ : (false, default) = (false, default)) } x).1,\n        f (PointedMap.f { f := Prod.mk false, map_pt' := (_ : (false, default) = (false, default)) } x).2)", "state_after": "case succ.e_l.H\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nf : ((k : K) \u2192 Option (\u0393 k)) \u2192 (k : K) \u2192 Option (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nn\u271d : \u2115\nx\u271d : (k : K) \u2192 Option (\u0393 k)\n\u22a2 PointedMap.f { f := Prod.mk false, map_pt' := (_ : (false, default) = (false, default)) } (f x\u271d) =\n    ((PointedMap.f { f := Prod.mk false, map_pt' := (_ : (false, default) = (false, default)) } x\u271d).1,\n      f (PointedMap.f { f := Prod.mk false, map_pt' := (_ : (false, default) = (false, default)) } x\u271d).2)"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case succ.e_l.H\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nf : ((k : K) \u2192 Option (\u0393 k)) \u2192 (k : K) \u2192 Option (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nn\u271d : \u2115\nx\u271d : (k : K) \u2192 Option (\u0393 k)\n\u22a2 PointedMap.f { f := Prod.mk false, map_pt' := (_ : (false, default) = (false, default)) } (f x\u271d) =\n    ((PointedMap.f { f := Prod.mk false, map_pt' := (_ : (false, default) = (false, default)) } x\u271d).1,\n      f (PointedMap.f { f := Prod.mk false, map_pt' := (_ : (false, default) = (false, default)) } x\u271d).2)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "full_name": "MeasureTheory.L1.setToL1_eq_setToL1SCLM", "start": [1024, 1], "end": [1027, 47], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "full_name": "Int.ne_of_lt", "start": [629, 11], "end": [630, 35], "traced_tactics": [{"tactic": "cases e", "annotated_tactic": ["cases e", []], "state_before": "a b : Int\nh : a < b\ne : a = b\n\u22a2 False", "state_after": "case refl\na : Int\nh : a < a\n\u22a2 False"}, {"tactic": "exact Int.lt_irrefl _ h", "annotated_tactic": ["exact <a>Int.lt_irrefl</a> _ h", [{"full_name": "Int.lt_irrefl", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [622, 19], "def_end_pos": [622, 28]}]], "state_before": "case refl\na : Int\nh : a < a\n\u22a2 False", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "full_name": "MeasureTheory.L1.SimpleFunc.setToL1S_congr_left", "start": [710, 1], "end": [713, 101], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Primrec.lean", "full_name": "Primrec.vector_ofFn'", "start": [1330, 1], "end": [1331, 11], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Int/Bitwise.lean", "full_name": "Int.bit_val", "start": [130, 1], "end": [133, 17], "traced_tactics": [{"tactic": "cases b", "annotated_tactic": ["cases b", []], "state_before": "b : Bool\nn : \u2124\n\u22a2 bit b n = 2 * n + bif b then 1 else 0", "state_after": "case false\nn : \u2124\n\u22a2 bit false n = 2 * n + bif false then 1 else 0\n\ncase true\nn : \u2124\n\u22a2 bit true n = 2 * n + bif true then 1 else 0"}, {"tactic": "apply (bit0_val n).trans (add_zero _).symm", "annotated_tactic": ["apply (<a>bit0_val</a> n).<a>trans</a> (<a>add_zero</a> _).<a>symm</a>", [{"full_name": "Int.bit0_val", "def_path": "Mathlib/Data/Int/Bitwise.lean", "def_pos": [121, 9], "def_end_pos": [121, 17]}, {"full_name": "Eq.trans", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [322, 9], "def_end_pos": [322, 17]}, {"full_name": "add_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [469, 3], "def_end_pos": [469, 14]}, {"full_name": "Eq.symm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [310, 9], "def_end_pos": [310, 16]}]], "state_before": "case false\nn : \u2124\n\u22a2 bit false n = 2 * n + bif false then 1 else 0\n\ncase true\nn : \u2124\n\u22a2 bit true n = 2 * n + bif true then 1 else 0", "state_after": "case true\nn : \u2124\n\u22a2 bit true n = 2 * n + bif true then 1 else 0"}, {"tactic": "apply bit1_val", "annotated_tactic": ["apply <a>bit1_val</a>", [{"full_name": "Int.bit1_val", "def_path": "Mathlib/Data/Int/Bitwise.lean", "def_pos": [126, 9], "def_end_pos": [126, 17]}]], "state_before": "case true\nn : \u2124\n\u22a2 bit true n = 2 * n + bif true then 1 else 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "List.toFinset_cons", "start": [3334, 1], "end": [3335, 98], "traced_tactics": [{"tactic": "by_cases h : a \u2208 l <;> simp [Finset.insert_val', Multiset.dedup_cons, h]", "annotated_tactic": ["by_cases h : a \u2208 l <;> simp [<a>Finset.insert_val'</a>, <a>Multiset.dedup_cons</a>, h]", [{"full_name": "Finset.insert_val'", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1078, 9], "def_end_pos": [1078, 20]}, {"full_name": "Multiset.dedup_cons", "def_path": "Mathlib/Data/Multiset/FinsetOps.lean", "def_pos": [83, 9], "def_end_pos": [83, 19]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\nl l' : List \u03b1\na : \u03b1\n\u22a2 (toFinset (a :: l)).val = (insert a (toFinset l)).val", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Hausdorff.lean", "full_name": "MeasureTheory.hausdorffMeasure_affineSegment", "start": [1163, 1], "end": [1166, 32], "traced_tactics": [{"tactic": "rw [affineSegment, hausdorffMeasure_lineMap_image, hausdorffMeasure_real, Real.volume_Icc,\n  sub_zero, ENNReal.ofReal_one, \u2190 Algebra.algebraMap_eq_smul_one]", "annotated_tactic": ["rw [<a>affineSegment</a>, <a>hausdorffMeasure_lineMap_image</a>, <a>hausdorffMeasure_real</a>, <a>Real.volume_Icc</a>,\n    <a>sub_zero</a>, <a>ENNReal.ofReal_one</a>, \u2190 <a>Algebra.algebraMap_eq_smul_one</a>]", [{"full_name": "affineSegment", "def_path": "Mathlib/Analysis/Convex/Between.lean", "def_pos": [44, 5], "def_end_pos": [44, 18]}, {"full_name": "MeasureTheory.hausdorffMeasure_lineMap_image", "def_path": "Mathlib/MeasureTheory/Measure/Hausdorff.lean", "def_pos": [1152, 9], "def_end_pos": [1152, 39]}, {"full_name": "MeasureTheory.hausdorffMeasure_real", "def_path": "Mathlib/MeasureTheory/Measure/Hausdorff.lean", "def_pos": [1059, 9], "def_end_pos": [1059, 30]}, {"full_name": "Real.volume_Icc", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean", "def_pos": [84, 9], "def_end_pos": [84, 19]}, {"full_name": "sub_zero", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [339, 3], "def_end_pos": [339, 14]}, {"full_name": "ENNReal.ofReal_one", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [248, 17], "def_end_pos": [248, 27]}, {"full_name": "Algebra.algebraMap_eq_smul_one", "def_path": "Mathlib/Algebra/Algebra/Basic.lean", "def_pos": [351, 9], "def_end_pos": [351, 31]}]], "state_before": "\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9\u00b9 : EMetricSpace X\ninst\u271d\u00b9\u2070 : EMetricSpace Y\ninst\u271d\u2079 : MeasurableSpace X\ninst\u271d\u2078 : BorelSpace X\ninst\u271d\u2077 : MeasurableSpace Y\ninst\u271d\u2076 : BorelSpace Y\n\ud835\udd5c : Type u_4\nE : Type u_5\nP : Type u_6\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace P\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor E P\ninst\u271d : BorelSpace P\nx y : P\n\u22a2 \u2191\u2191\u03bcH[1] (affineSegment \u211d x y) = edist x y", "state_after": "\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9\u00b9 : EMetricSpace X\ninst\u271d\u00b9\u2070 : EMetricSpace Y\ninst\u271d\u2079 : MeasurableSpace X\ninst\u271d\u2078 : BorelSpace X\ninst\u271d\u2077 : MeasurableSpace Y\ninst\u271d\u2076 : BorelSpace Y\n\ud835\udd5c : Type u_4\nE : Type u_5\nP : Type u_6\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace P\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor E P\ninst\u271d : BorelSpace P\nx y : P\n\u22a2 \u2191(algebraMap \u211d\u22650 \u211d\u22650\u221e) (nndist x y) = edist x y"}, {"tactic": "exact (edist_nndist _ _).symm", "annotated_tactic": ["exact (<a>edist_nndist</a> _ _).<a>symm</a>", [{"full_name": "edist_nndist", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [314, 9], "def_end_pos": [314, 21]}, {"full_name": "Eq.symm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [310, 9], "def_end_pos": [310, 16]}]], "state_before": "\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9\u00b9 : EMetricSpace X\ninst\u271d\u00b9\u2070 : EMetricSpace Y\ninst\u271d\u2079 : MeasurableSpace X\ninst\u271d\u2078 : BorelSpace X\ninst\u271d\u2077 : MeasurableSpace Y\ninst\u271d\u2076 : BorelSpace Y\n\ud835\udd5c : Type u_4\nE : Type u_5\nP : Type u_6\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace P\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor E P\ninst\u271d : BorelSpace P\nx y : P\n\u22a2 \u2191(algebraMap \u211d\u22650 \u211d\u22650\u221e) (nndist x y) = edist x y", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Kernel/CondCdf.lean", "full_name": "ProbabilityTheory.set_integral_condCdf", "start": [948, 1], "end": [955, 61], "traced_tactics": [{"tactic": "have h := set_lintegral_condCdf \u03c1 x hs", "annotated_tactic": ["have h := <a>set_lintegral_condCdf</a> \u03c1 x hs", [{"full_name": "ProbabilityTheory.set_lintegral_condCdf", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [874, 9], "def_end_pos": [874, 30]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u222b (a : \u03b1) in s, \u2191(condCdf \u03c1 a) x \u2202Measure.fst \u03c1 = ENNReal.toReal (\u2191\u2191\u03c1 (s \u00d7\u02e2 Iic x))", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\ns : Set \u03b1\nhs : MeasurableSet s\nh : \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic x)\n\u22a2 \u222b (a : \u03b1) in s, \u2191(condCdf \u03c1 a) x \u2202Measure.fst \u03c1 = ENNReal.toReal (\u2191\u2191\u03c1 (s \u00d7\u02e2 Iic x))"}, {"tactic": "rw [\u2190 ofReal_integral_eq_lintegral_ofReal] at h", "annotated_tactic": ["rw [\u2190 <a>ofReal_integral_eq_lintegral_ofReal</a>] at h", [{"full_name": "MeasureTheory.ofReal_integral_eq_lintegral_ofReal", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1219, 9], "def_end_pos": [1219, 44]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\ns : Set \u03b1\nhs : MeasurableSet s\nh : \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic x)\n\u22a2 \u222b (a : \u03b1) in s, \u2191(condCdf \u03c1 a) x \u2202Measure.fst \u03c1 = ENNReal.toReal (\u2191\u2191\u03c1 (s \u00d7\u02e2 Iic x))", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\ns : Set \u03b1\nhs : MeasurableSet s\nh : ENNReal.ofReal (\u222b (x_1 : \u03b1) in s, \u2191(condCdf \u03c1 x_1) x \u2202Measure.fst \u03c1) = \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic x)\n\u22a2 \u222b (a : \u03b1) in s, \u2191(condCdf \u03c1 a) x \u2202Measure.fst \u03c1 = ENNReal.toReal (\u2191\u2191\u03c1 (s \u00d7\u02e2 Iic x))\n\ncase hfi\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\ns : Set \u03b1\nhs : MeasurableSet s\nh : \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic x)\n\u22a2 Integrable fun a => \u2191(condCdf \u03c1 a) x\n\ncase f_nn\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\ns : Set \u03b1\nhs : MeasurableSet s\nh : \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic x)\n\u22a2 0 \u2264\u1d50[Measure.restrict (Measure.fst \u03c1) s] fun a => \u2191(condCdf \u03c1 a) x"}, {"tactic": "rw [\u2190 h, ENNReal.toReal_ofReal]", "annotated_tactic": ["rw [\u2190 h, <a>ENNReal.toReal_ofReal</a>]", [{"full_name": "ENNReal.toReal_ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [191, 9], "def_end_pos": [191, 22]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\ns : Set \u03b1\nhs : MeasurableSet s\nh : ENNReal.ofReal (\u222b (x_1 : \u03b1) in s, \u2191(condCdf \u03c1 x_1) x \u2202Measure.fst \u03c1) = \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic x)\n\u22a2 \u222b (a : \u03b1) in s, \u2191(condCdf \u03c1 a) x \u2202Measure.fst \u03c1 = ENNReal.toReal (\u2191\u2191\u03c1 (s \u00d7\u02e2 Iic x))", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\ns : Set \u03b1\nhs : MeasurableSet s\nh : ENNReal.ofReal (\u222b (x_1 : \u03b1) in s, \u2191(condCdf \u03c1 x_1) x \u2202Measure.fst \u03c1) = \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic x)\n\u22a2 0 \u2264 \u222b (x_1 : \u03b1) in s, \u2191(condCdf \u03c1 x_1) x \u2202Measure.fst \u03c1"}, {"tactic": "exact integral_nonneg fun _ => condCdf_nonneg _ _ _", "annotated_tactic": ["exact <a>integral_nonneg</a> fun _ => <a>condCdf_nonneg</a> _ _ _", [{"full_name": "MeasureTheory.integral_nonneg", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1251, 9], "def_end_pos": [1251, 24]}, {"full_name": "ProbabilityTheory.condCdf_nonneg", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [789, 9], "def_end_pos": [789, 23]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\ns : Set \u03b1\nhs : MeasurableSet s\nh : ENNReal.ofReal (\u222b (x_1 : \u03b1) in s, \u2191(condCdf \u03c1 x_1) x \u2202Measure.fst \u03c1) = \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic x)\n\u22a2 0 \u2264 \u222b (x_1 : \u03b1) in s, \u2191(condCdf \u03c1 x_1) x \u2202Measure.fst \u03c1", "state_after": "no goals"}, {"tactic": "exact (integrable_condCdf _ _).integrableOn", "annotated_tactic": ["exact (<a>integrable_condCdf</a> _ _).<a>integrableOn</a>", [{"full_name": "ProbabilityTheory.integrable_condCdf", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [931, 9], "def_end_pos": [931, 27]}, {"full_name": "MeasureTheory.Integrable.integrableOn", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [163, 9], "def_end_pos": [163, 32]}]], "state_before": "case hfi\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\ns : Set \u03b1\nhs : MeasurableSet s\nh : \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic x)\n\u22a2 Integrable fun a => \u2191(condCdf \u03c1 a) x", "state_after": "no goals"}, {"tactic": "exact eventually_of_forall fun _ => condCdf_nonneg _ _ _", "annotated_tactic": ["exact <a>eventually_of_forall</a> fun _ => <a>condCdf_nonneg</a> _ _ _", [{"full_name": "Filter.eventually_of_forall", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1111, 9], "def_end_pos": [1111, 29]}, {"full_name": "ProbabilityTheory.condCdf_nonneg", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [789, 9], "def_end_pos": [789, 23]}]], "state_before": "case f_nn\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\ns : Set \u03b1\nhs : MeasurableSet s\nh : \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic x)\n\u22a2 0 \u2264\u1d50[Measure.restrict (Measure.fst \u03c1) s] fun a => \u2191(condCdf \u03c1 a) x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "full_name": "List.Sublist.reverse", "start": [430, 1], "end": [433, 85], "traced_tactics": [{"tactic": "rw [reverse_cons]", "annotated_tactic": ["rw [<a>reverse_cons</a>]", [{"full_name": "List.reverse_cons", "def_path": "lake-packages/lean4/src/lean/Init/Data/List/Basic.lean", "def_pos": [174, 17], "def_end_pos": [174, 29]}]], "state_before": "\u03b1\u271d : Type u_1\nl\u2081\u271d l\u2082 l\u2081 l\u2082\u271d : List \u03b1\u271d\na\u271d : \u03b1\u271d\nh : l\u2081 <+ l\u2082\u271d\n\u22a2 List.reverse l\u2081 <+ List.reverse (a\u271d :: l\u2082\u271d)", "state_after": "\u03b1\u271d : Type u_1\nl\u2081\u271d l\u2082 l\u2081 l\u2082\u271d : List \u03b1\u271d\na\u271d : \u03b1\u271d\nh : l\u2081 <+ l\u2082\u271d\n\u22a2 List.reverse l\u2081 <+ List.reverse l\u2082\u271d ++ [a\u271d]"}, {"tactic": "exact sublist_append_of_sublist_left h.reverse", "annotated_tactic": ["exact <a>sublist_append_of_sublist_left</a> h.reverse", [{"full_name": "List.sublist_append_of_sublist_left", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [399, 9], "def_end_pos": [399, 39]}]], "state_before": "\u03b1\u271d : Type u_1\nl\u2081\u271d l\u2082 l\u2081 l\u2082\u271d : List \u03b1\u271d\na\u271d : \u03b1\u271d\nh : l\u2081 <+ l\u2082\u271d\n\u22a2 List.reverse l\u2081 <+ List.reverse l\u2082\u271d ++ [a\u271d]", "state_after": "no goals"}, {"tactic": "rw [reverse_cons, reverse_cons]", "annotated_tactic": ["rw [<a>reverse_cons</a>, <a>reverse_cons</a>]", [{"full_name": "List.reverse_cons", "def_path": "lake-packages/lean4/src/lean/Init/Data/List/Basic.lean", "def_pos": [174, 17], "def_end_pos": [174, 29]}, {"full_name": "List.reverse_cons", "def_path": "lake-packages/lean4/src/lean/Init/Data/List/Basic.lean", "def_pos": [174, 17], "def_end_pos": [174, 29]}]], "state_before": "\u03b1\u271d : Type u_1\nl\u2081 l\u2082 l\u2081\u271d l\u2082\u271d : List \u03b1\u271d\na\u271d : \u03b1\u271d\nh : l\u2081\u271d <+ l\u2082\u271d\n\u22a2 List.reverse (a\u271d :: l\u2081\u271d) <+ List.reverse (a\u271d :: l\u2082\u271d)", "state_after": "\u03b1\u271d : Type u_1\nl\u2081 l\u2082 l\u2081\u271d l\u2082\u271d : List \u03b1\u271d\na\u271d : \u03b1\u271d\nh : l\u2081\u271d <+ l\u2082\u271d\n\u22a2 List.reverse l\u2081\u271d ++ [a\u271d] <+ List.reverse l\u2082\u271d ++ [a\u271d]"}, {"tactic": "exact h.reverse.append_right _", "annotated_tactic": ["exact h.reverse.append_right _", []], "state_before": "\u03b1\u271d : Type u_1\nl\u2081 l\u2082 l\u2081\u271d l\u2082\u271d : List \u03b1\u271d\na\u271d : \u03b1\u271d\nh : l\u2081\u271d <+ l\u2082\u271d\n\u22a2 List.reverse l\u2081\u271d ++ [a\u271d] <+ List.reverse l\u2082\u271d ++ [a\u271d]", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "full_name": "integral_re_add_im", "start": [1219, 1], "end": [1222, 75], "traced_tactics": [{"tactic": "rw [\u2190 integral_ofReal, \u2190 integral_ofReal, integral_coe_re_add_coe_im hf]", "annotated_tactic": ["rw [\u2190 <a>integral_ofReal</a>, \u2190 <a>integral_ofReal</a>, <a>integral_coe_re_add_coe_im</a> hf]", [{"full_name": "integral_ofReal", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [1191, 9], "def_end_pos": [1191, 24]}, {"full_name": "integral_ofReal", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [1191, 9], "def_end_pos": [1191, 24]}, {"full_name": "integral_coe_re_add_coe_im", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [1209, 9], "def_end_pos": [1209, 35]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2075 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \ud835\udd5c\nhf : Integrable f\n\u22a2 \u2191(\u222b (x : \u03b1), \u2191IsROrC.re (f x) \u2202\u03bc) + \u2191(\u222b (x : \u03b1), \u2191IsROrC.im (f x) \u2202\u03bc) * IsROrC.I = \u222b (x : \u03b1), f x \u2202\u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/PiSystem.lean", "full_name": "mem_generatePiSystem_iUnion_elim'", "start": [327, 1], "end": [358, 16], "traced_tactics": [{"tactic": "rcases @mem_generatePiSystem_iUnion_elim \u03b1 (Subtype s) (g \u2218 Subtype.val)\n    (fun b => h_pi b.val b.property) t this with\n  \u27e8T, \u27e8f, \u27e8rfl, h_t'\u27e9\u27e9\u27e9", "annotated_tactic": ["rcases @<a>mem_generatePiSystem_iUnion_elim</a> \u03b1 (<a>Subtype</a> s) (g \u2218 <a>Subtype.val</a>)\n      (fun b => h_pi b.val b.property) t this with\n    \u27e8T, \u27e8f, \u27e8rfl, h_t'\u27e9\u27e9\u27e9", [{"full_name": "mem_generatePiSystem_iUnion_elim", "def_path": "Mathlib/MeasureTheory/PiSystem.lean", "def_pos": [293, 9], "def_end_pos": [293, 41]}, {"full_name": "Subtype", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [560, 11], "def_end_pos": [560, 18]}, {"full_name": "Subtype.val", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [564, 3], "def_end_pos": [564, 6]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ng : \u03b2 \u2192 Set (Set \u03b1)\ns : Set \u03b2\nh_pi : \u2200 (b : \u03b2), b \u2208 s \u2192 IsPiSystem (g b)\nt : Set \u03b1\nh_t : t \u2208 generatePiSystem (\u22c3 b \u2208 s, g b)\nthis : t \u2208 generatePiSystem (\u22c3 b, (g \u2218 Subtype.val) b)\n\u22a2 \u2203 T f, \u2191T \u2286 s \u2227 t = \u22c2 b \u2208 T, f b \u2227 \u2200 (b : \u03b2), b \u2208 T \u2192 f b \u2208 g b", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ng : \u03b2 \u2192 Set (Set \u03b1)\ns : Set \u03b2\nh_pi : \u2200 (b : \u03b2), b \u2208 s \u2192 IsPiSystem (g b)\nT : Finset (Subtype s)\nf : Subtype s \u2192 Set \u03b1\nh_t' : \u2200 (b : Subtype s), b \u2208 T \u2192 f b \u2208 (g \u2218 Subtype.val) b\nh_t : \u22c2 b \u2208 T, f b \u2208 generatePiSystem (\u22c3 b \u2208 s, g b)\nthis : \u22c2 b \u2208 T, f b \u2208 generatePiSystem (\u22c3 b, (g \u2218 Subtype.val) b)\n\u22a2 \u2203 T_1 f_1, \u2191T_1 \u2286 s \u2227 \u22c2 b \u2208 T, f b = \u22c2 b \u2208 T_1, f_1 b \u2227 \u2200 (b : \u03b2), b \u2208 T_1 \u2192 f_1 b \u2208 g b"}, {"tactic": "refine'\n  \u27e8T.image (fun x : s => (x : \u03b2)),\n    Function.extend (fun x : s => (x : \u03b2)) f fun _ : \u03b2 => (\u2205 : Set \u03b1), by simp, _, _\u27e9", "annotated_tactic": ["refine'\n    \u27e8T.image (fun x : s => (x : \u03b2)),\n      <a>Function.extend</a> (fun x : s => (x : \u03b2)) f fun _ : \u03b2 => (\u2205 : <a>Set</a> \u03b1), by simp, _, _\u27e9", [{"full_name": "Function.extend", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [711, 5], "def_end_pos": [711, 11]}, {"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}]], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ng : \u03b2 \u2192 Set (Set \u03b1)\ns : Set \u03b2\nh_pi : \u2200 (b : \u03b2), b \u2208 s \u2192 IsPiSystem (g b)\nT : Finset (Subtype s)\nf : Subtype s \u2192 Set \u03b1\nh_t' : \u2200 (b : Subtype s), b \u2208 T \u2192 f b \u2208 (g \u2218 Subtype.val) b\nh_t : \u22c2 b \u2208 T, f b \u2208 generatePiSystem (\u22c3 b \u2208 s, g b)\nthis : \u22c2 b \u2208 T, f b \u2208 generatePiSystem (\u22c3 b, (g \u2218 Subtype.val) b)\n\u22a2 \u2203 T_1 f_1, \u2191T_1 \u2286 s \u2227 \u22c2 b \u2208 T, f b = \u22c2 b \u2208 T_1, f_1 b \u2227 \u2200 (b : \u03b2), b \u2208 T_1 \u2192 f_1 b \u2208 g b", "state_after": "case intro.intro.intro.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ng : \u03b2 \u2192 Set (Set \u03b1)\ns : Set \u03b2\nh_pi : \u2200 (b : \u03b2), b \u2208 s \u2192 IsPiSystem (g b)\nT : Finset (Subtype s)\nf : Subtype s \u2192 Set \u03b1\nh_t' : \u2200 (b : Subtype s), b \u2208 T \u2192 f b \u2208 (g \u2218 Subtype.val) b\nh_t : \u22c2 b \u2208 T, f b \u2208 generatePiSystem (\u22c3 b \u2208 s, g b)\nthis : \u22c2 b \u2208 T, f b \u2208 generatePiSystem (\u22c3 b, (g \u2218 Subtype.val) b)\n\u22a2 \u22c2 b \u2208 T, f b = \u22c2 b \u2208 Finset.image (fun x => \u2191x) T, Function.extend (fun x => \u2191x) f (fun x => \u2205) b\n\ncase intro.intro.intro.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ng : \u03b2 \u2192 Set (Set \u03b1)\ns : Set \u03b2\nh_pi : \u2200 (b : \u03b2), b \u2208 s \u2192 IsPiSystem (g b)\nT : Finset (Subtype s)\nf : Subtype s \u2192 Set \u03b1\nh_t' : \u2200 (b : Subtype s), b \u2208 T \u2192 f b \u2208 (g \u2218 Subtype.val) b\nh_t : \u22c2 b \u2208 T, f b \u2208 generatePiSystem (\u22c3 b \u2208 s, g b)\nthis : \u22c2 b \u2208 T, f b \u2208 generatePiSystem (\u22c3 b, (g \u2218 Subtype.val) b)\n\u22a2 \u2200 (b : \u03b2), b \u2208 Finset.image (fun x => \u2191x) T \u2192 Function.extend (fun x => \u2191x) f (fun x => \u2205) b \u2208 g b"}, {"tactic": "suffices h1 : \u22c3 b : Subtype s, (g \u2218 Subtype.val) b = \u22c3 b \u2208 s, g b", "annotated_tactic": ["suffices h1 : \u22c3 b : <a>Subtype</a> s, (g \u2218 <a>Subtype.val</a>) b = \u22c3 b \u2208 s, g b", [{"full_name": "Subtype", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [560, 11], "def_end_pos": [560, 18]}, {"full_name": "Subtype.val", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [564, 3], "def_end_pos": [564, 6]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ng : \u03b2 \u2192 Set (Set \u03b1)\ns : Set \u03b2\nh_pi : \u2200 (b : \u03b2), b \u2208 s \u2192 IsPiSystem (g b)\nt : Set \u03b1\nh_t : t \u2208 generatePiSystem (\u22c3 b \u2208 s, g b)\n\u22a2 t \u2208 generatePiSystem (\u22c3 b, (g \u2218 Subtype.val) b)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ng : \u03b2 \u2192 Set (Set \u03b1)\ns : Set \u03b2\nh_pi : \u2200 (b : \u03b2), b \u2208 s \u2192 IsPiSystem (g b)\nt : Set \u03b1\nh_t : t \u2208 generatePiSystem (\u22c3 b \u2208 s, g b)\nh1 : \u22c3 b, (g \u2218 Subtype.val) b = \u22c3 b \u2208 s, g b\n\u22a2 t \u2208 generatePiSystem (\u22c3 b, (g \u2218 Subtype.val) b)\n\ncase h1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ng : \u03b2 \u2192 Set (Set \u03b1)\ns : Set \u03b2\nh_pi : \u2200 (b : \u03b2), b \u2208 s \u2192 IsPiSystem (g b)\nt : Set \u03b1\nh_t : t \u2208 generatePiSystem (\u22c3 b \u2208 s, g b)\n\u22a2 \u22c3 b, (g \u2218 Subtype.val) b = \u22c3 b \u2208 s, g b"}, {"tactic": "ext x", "annotated_tactic": ["ext x", []], "state_before": "case h1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ng : \u03b2 \u2192 Set (Set \u03b1)\ns : Set \u03b2\nh_pi : \u2200 (b : \u03b2), b \u2208 s \u2192 IsPiSystem (g b)\nt : Set \u03b1\nh_t : t \u2208 generatePiSystem (\u22c3 b \u2208 s, g b)\n\u22a2 \u22c3 b, (g \u2218 Subtype.val) b = \u22c3 b \u2208 s, g b", "state_after": "case h1.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ng : \u03b2 \u2192 Set (Set \u03b1)\ns : Set \u03b2\nh_pi : \u2200 (b : \u03b2), b \u2208 s \u2192 IsPiSystem (g b)\nt : Set \u03b1\nh_t : t \u2208 generatePiSystem (\u22c3 b \u2208 s, g b)\nx : Set \u03b1\n\u22a2 x \u2208 \u22c3 b, (g \u2218 Subtype.val) b \u2194 x \u2208 \u22c3 b \u2208 s, g b"}, {"tactic": "simp only [exists_prop, Set.mem_iUnion, Function.comp_apply, Subtype.exists, Subtype.coe_mk]", "annotated_tactic": ["simp only [<a>exists_prop</a>, <a>Set.mem_iUnion</a>, <a>Function.comp_apply</a>, <a>Subtype.exists</a>, <a>Subtype.coe_mk</a>]", [{"full_name": "exists_prop", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [485, 17], "def_end_pos": [485, 28]}, {"full_name": "Set.mem_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [201, 9], "def_end_pos": [201, 19]}, {"full_name": "Function.comp_apply", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [33, 17], "def_end_pos": [33, 36]}, {"full_name": "Subtype.exists", "def_path": "Mathlib/Data/Subtype.lean", "def_pos": [54, 19], "def_end_pos": [54, 27]}, {"full_name": "Subtype.coe_mk", "def_path": "Mathlib/Data/Subtype.lean", "def_pos": [99, 9], "def_end_pos": [99, 15]}]], "state_before": "case h1.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ng : \u03b2 \u2192 Set (Set \u03b1)\ns : Set \u03b2\nh_pi : \u2200 (b : \u03b2), b \u2208 s \u2192 IsPiSystem (g b)\nt : Set \u03b1\nh_t : t \u2208 generatePiSystem (\u22c3 b \u2208 s, g b)\nx : Set \u03b1\n\u22a2 x \u2208 \u22c3 b, (g \u2218 Subtype.val) b \u2194 x \u2208 \u22c3 b \u2208 s, g b", "state_after": "case h1.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ng : \u03b2 \u2192 Set (Set \u03b1)\ns : Set \u03b2\nh_pi : \u2200 (b : \u03b2), b \u2208 s \u2192 IsPiSystem (g b)\nt : Set \u03b1\nh_t : t \u2208 generatePiSystem (\u22c3 b \u2208 s, g b)\nx : Set \u03b1\n\u22a2 (\u2203 a, s a \u2227 x \u2208 g a) \u2194 \u2203 i, i \u2208 s \u2227 x \u2208 g i"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case h1.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ng : \u03b2 \u2192 Set (Set \u03b1)\ns : Set \u03b2\nh_pi : \u2200 (b : \u03b2), b \u2208 s \u2192 IsPiSystem (g b)\nt : Set \u03b1\nh_t : t \u2208 generatePiSystem (\u22c3 b \u2208 s, g b)\nx : Set \u03b1\n\u22a2 (\u2203 a, s a \u2227 x \u2208 g a) \u2194 \u2203 i, i \u2208 s \u2227 x \u2208 g i", "state_after": "no goals"}, {"tactic": "rwa [h1]", "annotated_tactic": ["rwa [h1]", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ng : \u03b2 \u2192 Set (Set \u03b1)\ns : Set \u03b2\nh_pi : \u2200 (b : \u03b2), b \u2208 s \u2192 IsPiSystem (g b)\nt : Set \u03b1\nh_t : t \u2208 generatePiSystem (\u22c3 b \u2208 s, g b)\nh1 : \u22c3 b, (g \u2218 Subtype.val) b = \u22c3 b \u2208 s, g b\n\u22a2 t \u2208 generatePiSystem (\u22c3 b, (g \u2218 Subtype.val) b)", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ng : \u03b2 \u2192 Set (Set \u03b1)\ns : Set \u03b2\nh_pi : \u2200 (b : \u03b2), b \u2208 s \u2192 IsPiSystem (g b)\nT : Finset (Subtype s)\nf : Subtype s \u2192 Set \u03b1\nh_t' : \u2200 (b : Subtype s), b \u2208 T \u2192 f b \u2208 (g \u2218 Subtype.val) b\nh_t : \u22c2 b \u2208 T, f b \u2208 generatePiSystem (\u22c3 b \u2208 s, g b)\nthis : \u22c2 b \u2208 T, f b \u2208 generatePiSystem (\u22c3 b, (g \u2218 Subtype.val) b)\n\u22a2 \u2191(Finset.image (fun x => \u2191x) T) \u2286 s", "state_after": "no goals"}, {"tactic": "ext a", "annotated_tactic": ["ext a", []], "state_before": "case intro.intro.intro.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ng : \u03b2 \u2192 Set (Set \u03b1)\ns : Set \u03b2\nh_pi : \u2200 (b : \u03b2), b \u2208 s \u2192 IsPiSystem (g b)\nT : Finset (Subtype s)\nf : Subtype s \u2192 Set \u03b1\nh_t' : \u2200 (b : Subtype s), b \u2208 T \u2192 f b \u2208 (g \u2218 Subtype.val) b\nh_t : \u22c2 b \u2208 T, f b \u2208 generatePiSystem (\u22c3 b \u2208 s, g b)\nthis : \u22c2 b \u2208 T, f b \u2208 generatePiSystem (\u22c3 b, (g \u2218 Subtype.val) b)\n\u22a2 \u22c2 b \u2208 T, f b = \u22c2 b \u2208 Finset.image (fun x => \u2191x) T, Function.extend (fun x => \u2191x) f (fun x => \u2205) b", "state_after": "case intro.intro.intro.refine'_1.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ng : \u03b2 \u2192 Set (Set \u03b1)\ns : Set \u03b2\nh_pi : \u2200 (b : \u03b2), b \u2208 s \u2192 IsPiSystem (g b)\nT : Finset (Subtype s)\nf : Subtype s \u2192 Set \u03b1\nh_t' : \u2200 (b : Subtype s), b \u2208 T \u2192 f b \u2208 (g \u2218 Subtype.val) b\nh_t : \u22c2 b \u2208 T, f b \u2208 generatePiSystem (\u22c3 b \u2208 s, g b)\nthis : \u22c2 b \u2208 T, f b \u2208 generatePiSystem (\u22c3 b, (g \u2218 Subtype.val) b)\na : \u03b1\n\u22a2 a \u2208 \u22c2 b \u2208 T, f b \u2194 a \u2208 \u22c2 b \u2208 Finset.image (fun x => \u2191x) T, Function.extend (fun x => \u2191x) f (fun x => \u2205) b"}, {"tactic": "simp (config := { proj := false }) only\n  [Set.mem_iInter, Subtype.forall, Finset.set_biInter_finset_image]", "annotated_tactic": ["simp (config := { proj := <a>false</a> }) only\n          [<a>Set.mem_iInter</a>, <a>Subtype.forall</a>, <a>Finset.set_biInter_finset_image</a>]", [{"full_name": "Bool.false", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [547, 5], "def_end_pos": [547, 10]}, {"full_name": "Set.mem_iInter", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [207, 9], "def_end_pos": [207, 19]}, {"full_name": "Subtype.forall", "def_path": "Mathlib/Data/Subtype.lean", "def_pos": [43, 19], "def_end_pos": [43, 27]}, {"full_name": "Finset.set_biInter_finset_image", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [2151, 9], "def_end_pos": [2151, 33]}]], "state_before": "case intro.intro.intro.refine'_1.h.mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ng : \u03b2 \u2192 Set (Set \u03b1)\ns : Set \u03b2\nh_pi : \u2200 (b : \u03b2), b \u2208 s \u2192 IsPiSystem (g b)\nT : Finset (Subtype s)\nf : Subtype s \u2192 Set \u03b1\nh_t' : \u2200 (b : Subtype s), b \u2208 T \u2192 f b \u2208 (g \u2218 Subtype.val) b\nh_t : \u22c2 b \u2208 T, f b \u2208 generatePiSystem (\u22c3 b \u2208 s, g b)\nthis : \u22c2 b \u2208 T, f b \u2208 generatePiSystem (\u22c3 b, (g \u2218 Subtype.val) b)\na : \u03b1\n\u22a2 a \u2208 \u22c2 b \u2208 Finset.image (fun x => \u2191x) T, Function.extend (fun x => \u2191x) f (fun x => \u2205) b \u2192 a \u2208 \u22c2 b \u2208 T, f b", "state_after": "case intro.intro.intro.refine'_1.h.mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ng : \u03b2 \u2192 Set (Set \u03b1)\ns : Set \u03b2\nh_pi : \u2200 (b : \u03b2), b \u2208 s \u2192 IsPiSystem (g b)\nT : Finset (Subtype s)\nf : Subtype s \u2192 Set \u03b1\nh_t' : \u2200 (b : Subtype s), b \u2208 T \u2192 f b \u2208 (g \u2218 Subtype.val) b\nh_t : \u22c2 b \u2208 T, f b \u2208 generatePiSystem (\u22c3 b \u2208 s, g b)\nthis : \u22c2 b \u2208 T, f b \u2208 generatePiSystem (\u22c3 b, (g \u2218 Subtype.val) b)\na : \u03b1\n\u22a2 (\u2200 (a_1 : \u03b2) (b : a_1 \u2208 s),\n      { val := a_1, property := b } \u2208 T \u2192\n        a \u2208 Function.extend (fun x => \u2191x) f (fun x => \u2205) \u2191{ val := a_1, property := b }) \u2192\n    \u2200 (a_2 : \u03b2) (b : s a_2), { val := a_2, property := b } \u2208 T \u2192 a \u2208 f { val := a_2, property := b }"}, {"tactic": "intro h1 b h_b h_b_in_T", "annotated_tactic": ["intro h1 b h_b h_b_in_T", []], "state_before": "case intro.intro.intro.refine'_1.h.mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ng : \u03b2 \u2192 Set (Set \u03b1)\ns : Set \u03b2\nh_pi : \u2200 (b : \u03b2), b \u2208 s \u2192 IsPiSystem (g b)\nT : Finset (Subtype s)\nf : Subtype s \u2192 Set \u03b1\nh_t' : \u2200 (b : Subtype s), b \u2208 T \u2192 f b \u2208 (g \u2218 Subtype.val) b\nh_t : \u22c2 b \u2208 T, f b \u2208 generatePiSystem (\u22c3 b \u2208 s, g b)\nthis : \u22c2 b \u2208 T, f b \u2208 generatePiSystem (\u22c3 b, (g \u2218 Subtype.val) b)\na : \u03b1\n\u22a2 (\u2200 (a_1 : \u03b2) (b : a_1 \u2208 s),\n      { val := a_1, property := b } \u2208 T \u2192\n        a \u2208 Function.extend (fun x => \u2191x) f (fun x => \u2205) \u2191{ val := a_1, property := b }) \u2192\n    \u2200 (a_2 : \u03b2) (b : s a_2), { val := a_2, property := b } \u2208 T \u2192 a \u2208 f { val := a_2, property := b }", "state_after": "case intro.intro.intro.refine'_1.h.mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ng : \u03b2 \u2192 Set (Set \u03b1)\ns : Set \u03b2\nh_pi : \u2200 (b : \u03b2), b \u2208 s \u2192 IsPiSystem (g b)\nT : Finset (Subtype s)\nf : Subtype s \u2192 Set \u03b1\nh_t' : \u2200 (b : Subtype s), b \u2208 T \u2192 f b \u2208 (g \u2218 Subtype.val) b\nh_t : \u22c2 b \u2208 T, f b \u2208 generatePiSystem (\u22c3 b \u2208 s, g b)\nthis : \u22c2 b \u2208 T, f b \u2208 generatePiSystem (\u22c3 b, (g \u2218 Subtype.val) b)\na : \u03b1\nh1 :\n  \u2200 (a_1 : \u03b2) (b : a_1 \u2208 s),\n    { val := a_1, property := b } \u2208 T \u2192 a \u2208 Function.extend (fun x => \u2191x) f (fun x => \u2205) \u2191{ val := a_1, property := b }\nb : \u03b2\nh_b : s b\nh_b_in_T : { val := b, property := h_b } \u2208 T\n\u22a2 a \u2208 f { val := b, property := h_b }"}, {"tactic": "have h2 := h1 b h_b h_b_in_T", "annotated_tactic": ["have h2 := h1 b h_b h_b_in_T", []], "state_before": "case intro.intro.intro.refine'_1.h.mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ng : \u03b2 \u2192 Set (Set \u03b1)\ns : Set \u03b2\nh_pi : \u2200 (b : \u03b2), b \u2208 s \u2192 IsPiSystem (g b)\nT : Finset (Subtype s)\nf : Subtype s \u2192 Set \u03b1\nh_t' : \u2200 (b : Subtype s), b \u2208 T \u2192 f b \u2208 (g \u2218 Subtype.val) b\nh_t : \u22c2 b \u2208 T, f b \u2208 generatePiSystem (\u22c3 b \u2208 s, g b)\nthis : \u22c2 b \u2208 T, f b \u2208 generatePiSystem (\u22c3 b, (g \u2218 Subtype.val) b)\na : \u03b1\nh1 :\n  \u2200 (a_1 : \u03b2) (b : a_1 \u2208 s),\n    { val := a_1, property := b } \u2208 T \u2192 a \u2208 Function.extend (fun x => \u2191x) f (fun x => \u2205) \u2191{ val := a_1, property := b }\nb : \u03b2\nh_b : s b\nh_b_in_T : { val := b, property := h_b } \u2208 T\n\u22a2 a \u2208 f { val := b, property := h_b }", "state_after": "case intro.intro.intro.refine'_1.h.mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ng : \u03b2 \u2192 Set (Set \u03b1)\ns : Set \u03b2\nh_pi : \u2200 (b : \u03b2), b \u2208 s \u2192 IsPiSystem (g b)\nT : Finset (Subtype s)\nf : Subtype s \u2192 Set \u03b1\nh_t' : \u2200 (b : Subtype s), b \u2208 T \u2192 f b \u2208 (g \u2218 Subtype.val) b\nh_t : \u22c2 b \u2208 T, f b \u2208 generatePiSystem (\u22c3 b \u2208 s, g b)\nthis : \u22c2 b \u2208 T, f b \u2208 generatePiSystem (\u22c3 b, (g \u2218 Subtype.val) b)\na : \u03b1\nh1 :\n  \u2200 (a_1 : \u03b2) (b : a_1 \u2208 s),\n    { val := a_1, property := b } \u2208 T \u2192 a \u2208 Function.extend (fun x => \u2191x) f (fun x => \u2205) \u2191{ val := a_1, property := b }\nb : \u03b2\nh_b : s b\nh_b_in_T : { val := b, property := h_b } \u2208 T\nh2 : a \u2208 Function.extend (fun x => \u2191x) f (fun x => \u2205) \u2191{ val := b, property := h_b }\n\u22a2 a \u2208 f { val := b, property := h_b }"}, {"tactic": "revert h2", "annotated_tactic": ["revert h2", []], "state_before": "case intro.intro.intro.refine'_1.h.mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ng : \u03b2 \u2192 Set (Set \u03b1)\ns : Set \u03b2\nh_pi : \u2200 (b : \u03b2), b \u2208 s \u2192 IsPiSystem (g b)\nT : Finset (Subtype s)\nf : Subtype s \u2192 Set \u03b1\nh_t' : \u2200 (b : Subtype s), b \u2208 T \u2192 f b \u2208 (g \u2218 Subtype.val) b\nh_t : \u22c2 b \u2208 T, f b \u2208 generatePiSystem (\u22c3 b \u2208 s, g b)\nthis : \u22c2 b \u2208 T, f b \u2208 generatePiSystem (\u22c3 b, (g \u2218 Subtype.val) b)\na : \u03b1\nh1 :\n  \u2200 (a_1 : \u03b2) (b : a_1 \u2208 s),\n    { val := a_1, property := b } \u2208 T \u2192 a \u2208 Function.extend (fun x => \u2191x) f (fun x => \u2205) \u2191{ val := a_1, property := b }\nb : \u03b2\nh_b : s b\nh_b_in_T : { val := b, property := h_b } \u2208 T\nh2 : a \u2208 Function.extend (fun x => \u2191x) f (fun x => \u2205) \u2191{ val := b, property := h_b }\n\u22a2 a \u2208 f { val := b, property := h_b }", "state_after": "case intro.intro.intro.refine'_1.h.mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ng : \u03b2 \u2192 Set (Set \u03b1)\ns : Set \u03b2\nh_pi : \u2200 (b : \u03b2), b \u2208 s \u2192 IsPiSystem (g b)\nT : Finset (Subtype s)\nf : Subtype s \u2192 Set \u03b1\nh_t' : \u2200 (b : Subtype s), b \u2208 T \u2192 f b \u2208 (g \u2218 Subtype.val) b\nh_t : \u22c2 b \u2208 T, f b \u2208 generatePiSystem (\u22c3 b \u2208 s, g b)\nthis : \u22c2 b \u2208 T, f b \u2208 generatePiSystem (\u22c3 b, (g \u2218 Subtype.val) b)\na : \u03b1\nh1 :\n  \u2200 (a_1 : \u03b2) (b : a_1 \u2208 s),\n    { val := a_1, property := b } \u2208 T \u2192 a \u2208 Function.extend (fun x => \u2191x) f (fun x => \u2205) \u2191{ val := a_1, property := b }\nb : \u03b2\nh_b : s b\nh_b_in_T : { val := b, property := h_b } \u2208 T\n\u22a2 a \u2208 Function.extend (fun x => \u2191x) f (fun x => \u2205) \u2191{ val := b, property := h_b } \u2192 a \u2208 f { val := b, property := h_b }"}, {"tactic": "rw [Subtype.val_injective.extend_apply]", "annotated_tactic": ["rw [Subtype.val_injective.extend_apply]", []], "state_before": "case intro.intro.intro.refine'_1.h.mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ng : \u03b2 \u2192 Set (Set \u03b1)\ns : Set \u03b2\nh_pi : \u2200 (b : \u03b2), b \u2208 s \u2192 IsPiSystem (g b)\nT : Finset (Subtype s)\nf : Subtype s \u2192 Set \u03b1\nh_t' : \u2200 (b : Subtype s), b \u2208 T \u2192 f b \u2208 (g \u2218 Subtype.val) b\nh_t : \u22c2 b \u2208 T, f b \u2208 generatePiSystem (\u22c3 b \u2208 s, g b)\nthis : \u22c2 b \u2208 T, f b \u2208 generatePiSystem (\u22c3 b, (g \u2218 Subtype.val) b)\na : \u03b1\nh1 :\n  \u2200 (a_1 : \u03b2) (b : a_1 \u2208 s),\n    { val := a_1, property := b } \u2208 T \u2192 a \u2208 Function.extend (fun x => \u2191x) f (fun x => \u2205) \u2191{ val := a_1, property := b }\nb : \u03b2\nh_b : s b\nh_b_in_T : { val := b, property := h_b } \u2208 T\n\u22a2 a \u2208 Function.extend (fun x => \u2191x) f (fun x => \u2205) \u2191{ val := b, property := h_b } \u2192 a \u2208 f { val := b, property := h_b }", "state_after": "case intro.intro.intro.refine'_1.h.mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ng : \u03b2 \u2192 Set (Set \u03b1)\ns : Set \u03b2\nh_pi : \u2200 (b : \u03b2), b \u2208 s \u2192 IsPiSystem (g b)\nT : Finset (Subtype s)\nf : Subtype s \u2192 Set \u03b1\nh_t' : \u2200 (b : Subtype s), b \u2208 T \u2192 f b \u2208 (g \u2218 Subtype.val) b\nh_t : \u22c2 b \u2208 T, f b \u2208 generatePiSystem (\u22c3 b \u2208 s, g b)\nthis : \u22c2 b \u2208 T, f b \u2208 generatePiSystem (\u22c3 b, (g \u2218 Subtype.val) b)\na : \u03b1\nh1 :\n  \u2200 (a_1 : \u03b2) (b : a_1 \u2208 s),\n    { val := a_1, property := b } \u2208 T \u2192 a \u2208 Function.extend (fun x => \u2191x) f (fun x => \u2205) \u2191{ val := a_1, property := b }\nb : \u03b2\nh_b : s b\nh_b_in_T : { val := b, property := h_b } \u2208 T\n\u22a2 a \u2208 f { val := b, property := h_b } \u2192 a \u2208 f { val := b, property := h_b }"}, {"tactic": "apply id", "annotated_tactic": ["apply <a>id</a>", [{"full_name": "id", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [33, 15], "def_end_pos": [33, 17]}]], "state_before": "case intro.intro.intro.refine'_1.h.mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ng : \u03b2 \u2192 Set (Set \u03b1)\ns : Set \u03b2\nh_pi : \u2200 (b : \u03b2), b \u2208 s \u2192 IsPiSystem (g b)\nT : Finset (Subtype s)\nf : Subtype s \u2192 Set \u03b1\nh_t' : \u2200 (b : Subtype s), b \u2208 T \u2192 f b \u2208 (g \u2218 Subtype.val) b\nh_t : \u22c2 b \u2208 T, f b \u2208 generatePiSystem (\u22c3 b \u2208 s, g b)\nthis : \u22c2 b \u2208 T, f b \u2208 generatePiSystem (\u22c3 b, (g \u2218 Subtype.val) b)\na : \u03b1\nh1 :\n  \u2200 (a_1 : \u03b2) (b : a_1 \u2208 s),\n    { val := a_1, property := b } \u2208 T \u2192 a \u2208 Function.extend (fun x => \u2191x) f (fun x => \u2205) \u2191{ val := a_1, property := b }\nb : \u03b2\nh_b : s b\nh_b_in_T : { val := b, property := h_b } \u2208 T\n\u22a2 a \u2208 f { val := b, property := h_b } \u2192 a \u2208 f { val := b, property := h_b }", "state_after": "no goals"}, {"tactic": "intros b h_b", "annotated_tactic": ["intros b h_b", []], "state_before": "case intro.intro.intro.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ng : \u03b2 \u2192 Set (Set \u03b1)\ns : Set \u03b2\nh_pi : \u2200 (b : \u03b2), b \u2208 s \u2192 IsPiSystem (g b)\nT : Finset (Subtype s)\nf : Subtype s \u2192 Set \u03b1\nh_t' : \u2200 (b : Subtype s), b \u2208 T \u2192 f b \u2208 (g \u2218 Subtype.val) b\nh_t : \u22c2 b \u2208 T, f b \u2208 generatePiSystem (\u22c3 b \u2208 s, g b)\nthis : \u22c2 b \u2208 T, f b \u2208 generatePiSystem (\u22c3 b, (g \u2218 Subtype.val) b)\n\u22a2 \u2200 (b : \u03b2), b \u2208 Finset.image (fun x => \u2191x) T \u2192 Function.extend (fun x => \u2191x) f (fun x => \u2205) b \u2208 g b", "state_after": "case intro.intro.intro.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ng : \u03b2 \u2192 Set (Set \u03b1)\ns : Set \u03b2\nh_pi : \u2200 (b : \u03b2), b \u2208 s \u2192 IsPiSystem (g b)\nT : Finset (Subtype s)\nf : Subtype s \u2192 Set \u03b1\nh_t' : \u2200 (b : Subtype s), b \u2208 T \u2192 f b \u2208 (g \u2218 Subtype.val) b\nh_t : \u22c2 b \u2208 T, f b \u2208 generatePiSystem (\u22c3 b \u2208 s, g b)\nthis : \u22c2 b \u2208 T, f b \u2208 generatePiSystem (\u22c3 b, (g \u2218 Subtype.val) b)\nb : \u03b2\nh_b : b \u2208 Finset.image (fun x => \u2191x) T\n\u22a2 Function.extend (fun x => \u2191x) f (fun x => \u2205) b \u2208 g b"}, {"tactic": "simp_rw [Finset.mem_image, Subtype.exists, exists_and_right, exists_eq_right]\n  at h_b", "annotated_tactic": ["simp_rw [<a>Finset.mem_image</a>, <a>Subtype.exists</a>, <a>exists_and_right</a>, <a>exists_eq_right</a>]\n      at h_b", [{"full_name": "Finset.mem_image", "def_path": "Mathlib/Data/Finset/Image.lean", "def_pos": [330, 9], "def_end_pos": [330, 18]}, {"full_name": "Subtype.exists", "def_path": "Mathlib/Data/Subtype.lean", "def_pos": [54, 19], "def_end_pos": [54, 27]}, {"full_name": "exists_and_right", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [468, 17], "def_end_pos": [468, 33]}, {"full_name": "exists_eq_right", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [462, 17], "def_end_pos": [462, 32]}]], "state_before": "case intro.intro.intro.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ng : \u03b2 \u2192 Set (Set \u03b1)\ns : Set \u03b2\nh_pi : \u2200 (b : \u03b2), b \u2208 s \u2192 IsPiSystem (g b)\nT : Finset (Subtype s)\nf : Subtype s \u2192 Set \u03b1\nh_t' : \u2200 (b : Subtype s), b \u2208 T \u2192 f b \u2208 (g \u2218 Subtype.val) b\nh_t : \u22c2 b \u2208 T, f b \u2208 generatePiSystem (\u22c3 b \u2208 s, g b)\nthis : \u22c2 b \u2208 T, f b \u2208 generatePiSystem (\u22c3 b, (g \u2218 Subtype.val) b)\nb : \u03b2\nh_b : b \u2208 Finset.image (fun x => \u2191x) T\n\u22a2 Function.extend (fun x => \u2191x) f (fun x => \u2205) b \u2208 g b", "state_after": "case intro.intro.intro.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ng : \u03b2 \u2192 Set (Set \u03b1)\ns : Set \u03b2\nh_pi : \u2200 (b : \u03b2), b \u2208 s \u2192 IsPiSystem (g b)\nT : Finset (Subtype s)\nf : Subtype s \u2192 Set \u03b1\nh_t' : \u2200 (b : Subtype s), b \u2208 T \u2192 f b \u2208 (g \u2218 Subtype.val) b\nh_t : \u22c2 b \u2208 T, f b \u2208 generatePiSystem (\u22c3 b \u2208 s, g b)\nthis : \u22c2 b \u2208 T, f b \u2208 generatePiSystem (\u22c3 b, (g \u2218 Subtype.val) b)\nb : \u03b2\nh_b : \u2203 x, { val := b, property := (_ : b \u2208 s) } \u2208 T\n\u22a2 Function.extend (fun x => \u2191x) f (fun x => \u2205) b \u2208 g b"}, {"tactic": "cases' h_b with h_b_w h_b_h", "annotated_tactic": ["cases' h_b with h_b_w h_b_h", []], "state_before": "case intro.intro.intro.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ng : \u03b2 \u2192 Set (Set \u03b1)\ns : Set \u03b2\nh_pi : \u2200 (b : \u03b2), b \u2208 s \u2192 IsPiSystem (g b)\nT : Finset (Subtype s)\nf : Subtype s \u2192 Set \u03b1\nh_t' : \u2200 (b : Subtype s), b \u2208 T \u2192 f b \u2208 (g \u2218 Subtype.val) b\nh_t : \u22c2 b \u2208 T, f b \u2208 generatePiSystem (\u22c3 b \u2208 s, g b)\nthis : \u22c2 b \u2208 T, f b \u2208 generatePiSystem (\u22c3 b, (g \u2218 Subtype.val) b)\nb : \u03b2\nh_b : \u2203 x, { val := b, property := (_ : b \u2208 s) } \u2208 T\n\u22a2 Function.extend (fun x => \u2191x) f (fun x => \u2205) b \u2208 g b", "state_after": "case intro.intro.intro.refine'_2.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ng : \u03b2 \u2192 Set (Set \u03b1)\ns : Set \u03b2\nh_pi : \u2200 (b : \u03b2), b \u2208 s \u2192 IsPiSystem (g b)\nT : Finset (Subtype s)\nf : Subtype s \u2192 Set \u03b1\nh_t' : \u2200 (b : Subtype s), b \u2208 T \u2192 f b \u2208 (g \u2218 Subtype.val) b\nh_t : \u22c2 b \u2208 T, f b \u2208 generatePiSystem (\u22c3 b \u2208 s, g b)\nthis : \u22c2 b \u2208 T, f b \u2208 generatePiSystem (\u22c3 b, (g \u2218 Subtype.val) b)\nb : \u03b2\nh_b_w : b \u2208 s\nh_b_h : { val := b, property := (_ : b \u2208 s) } \u2208 T\n\u22a2 Function.extend (fun x => \u2191x) f (fun x => \u2205) b \u2208 g b"}, {"tactic": "have h_b_alt : b = (Subtype.mk b h_b_w).val := rfl", "annotated_tactic": ["have h_b_alt : b = (<a>Subtype.mk</a> b h_b_w).<a>val</a> := <a>rfl</a>", [{"full_name": "Subtype.mk", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [560, 19], "def_end_pos": [560, 46]}, {"full_name": "Subtype.val", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [564, 3], "def_end_pos": [564, 6]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "case intro.intro.intro.refine'_2.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ng : \u03b2 \u2192 Set (Set \u03b1)\ns : Set \u03b2\nh_pi : \u2200 (b : \u03b2), b \u2208 s \u2192 IsPiSystem (g b)\nT : Finset (Subtype s)\nf : Subtype s \u2192 Set \u03b1\nh_t' : \u2200 (b : Subtype s), b \u2208 T \u2192 f b \u2208 (g \u2218 Subtype.val) b\nh_t : \u22c2 b \u2208 T, f b \u2208 generatePiSystem (\u22c3 b \u2208 s, g b)\nthis : \u22c2 b \u2208 T, f b \u2208 generatePiSystem (\u22c3 b, (g \u2218 Subtype.val) b)\nb : \u03b2\nh_b_w : b \u2208 s\nh_b_h : { val := b, property := (_ : b \u2208 s) } \u2208 T\n\u22a2 Function.extend (fun x => \u2191x) f (fun x => \u2205) b \u2208 g b", "state_after": "case intro.intro.intro.refine'_2.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ng : \u03b2 \u2192 Set (Set \u03b1)\ns : Set \u03b2\nh_pi : \u2200 (b : \u03b2), b \u2208 s \u2192 IsPiSystem (g b)\nT : Finset (Subtype s)\nf : Subtype s \u2192 Set \u03b1\nh_t' : \u2200 (b : Subtype s), b \u2208 T \u2192 f b \u2208 (g \u2218 Subtype.val) b\nh_t : \u22c2 b \u2208 T, f b \u2208 generatePiSystem (\u22c3 b \u2208 s, g b)\nthis : \u22c2 b \u2208 T, f b \u2208 generatePiSystem (\u22c3 b, (g \u2218 Subtype.val) b)\nb : \u03b2\nh_b_w : b \u2208 s\nh_b_h : { val := b, property := (_ : b \u2208 s) } \u2208 T\nh_b_alt : b = \u2191{ val := b, property := h_b_w }\n\u22a2 Function.extend (fun x => \u2191x) f (fun x => \u2205) b \u2208 g b"}, {"tactic": "rw [h_b_alt, Subtype.val_injective.extend_apply]", "annotated_tactic": ["rw [h_b_alt, Subtype.val_injective.extend_apply]", []], "state_before": "case intro.intro.intro.refine'_2.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ng : \u03b2 \u2192 Set (Set \u03b1)\ns : Set \u03b2\nh_pi : \u2200 (b : \u03b2), b \u2208 s \u2192 IsPiSystem (g b)\nT : Finset (Subtype s)\nf : Subtype s \u2192 Set \u03b1\nh_t' : \u2200 (b : Subtype s), b \u2208 T \u2192 f b \u2208 (g \u2218 Subtype.val) b\nh_t : \u22c2 b \u2208 T, f b \u2208 generatePiSystem (\u22c3 b \u2208 s, g b)\nthis : \u22c2 b \u2208 T, f b \u2208 generatePiSystem (\u22c3 b, (g \u2218 Subtype.val) b)\nb : \u03b2\nh_b_w : b \u2208 s\nh_b_h : { val := b, property := (_ : b \u2208 s) } \u2208 T\nh_b_alt : b = \u2191{ val := b, property := h_b_w }\n\u22a2 Function.extend (fun x => \u2191x) f (fun x => \u2205) b \u2208 g b", "state_after": "case intro.intro.intro.refine'_2.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ng : \u03b2 \u2192 Set (Set \u03b1)\ns : Set \u03b2\nh_pi : \u2200 (b : \u03b2), b \u2208 s \u2192 IsPiSystem (g b)\nT : Finset (Subtype s)\nf : Subtype s \u2192 Set \u03b1\nh_t' : \u2200 (b : Subtype s), b \u2208 T \u2192 f b \u2208 (g \u2218 Subtype.val) b\nh_t : \u22c2 b \u2208 T, f b \u2208 generatePiSystem (\u22c3 b \u2208 s, g b)\nthis : \u22c2 b \u2208 T, f b \u2208 generatePiSystem (\u22c3 b, (g \u2218 Subtype.val) b)\nb : \u03b2\nh_b_w : b \u2208 s\nh_b_h : { val := b, property := (_ : b \u2208 s) } \u2208 T\nh_b_alt : b = \u2191{ val := b, property := h_b_w }\n\u22a2 f { val := b, property := h_b_w } \u2208 g \u2191{ val := b, property := h_b_w }"}, {"tactic": "apply h_t'", "annotated_tactic": ["apply h_t'", []], "state_before": "case intro.intro.intro.refine'_2.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ng : \u03b2 \u2192 Set (Set \u03b1)\ns : Set \u03b2\nh_pi : \u2200 (b : \u03b2), b \u2208 s \u2192 IsPiSystem (g b)\nT : Finset (Subtype s)\nf : Subtype s \u2192 Set \u03b1\nh_t' : \u2200 (b : Subtype s), b \u2208 T \u2192 f b \u2208 (g \u2218 Subtype.val) b\nh_t : \u22c2 b \u2208 T, f b \u2208 generatePiSystem (\u22c3 b \u2208 s, g b)\nthis : \u22c2 b \u2208 T, f b \u2208 generatePiSystem (\u22c3 b, (g \u2218 Subtype.val) b)\nb : \u03b2\nh_b_w : b \u2208 s\nh_b_h : { val := b, property := (_ : b \u2208 s) } \u2208 T\nh_b_alt : b = \u2191{ val := b, property := h_b_w }\n\u22a2 f { val := b, property := h_b_w } \u2208 g \u2191{ val := b, property := h_b_w }", "state_after": "case intro.intro.intro.refine'_2.intro.a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ng : \u03b2 \u2192 Set (Set \u03b1)\ns : Set \u03b2\nh_pi : \u2200 (b : \u03b2), b \u2208 s \u2192 IsPiSystem (g b)\nT : Finset (Subtype s)\nf : Subtype s \u2192 Set \u03b1\nh_t' : \u2200 (b : Subtype s), b \u2208 T \u2192 f b \u2208 (g \u2218 Subtype.val) b\nh_t : \u22c2 b \u2208 T, f b \u2208 generatePiSystem (\u22c3 b \u2208 s, g b)\nthis : \u22c2 b \u2208 T, f b \u2208 generatePiSystem (\u22c3 b, (g \u2218 Subtype.val) b)\nb : \u03b2\nh_b_w : b \u2208 s\nh_b_h : { val := b, property := (_ : b \u2208 s) } \u2208 T\nh_b_alt : b = \u2191{ val := b, property := h_b_w }\n\u22a2 { val := b, property := h_b_w } \u2208 T"}, {"tactic": "apply h_b_h", "annotated_tactic": ["apply h_b_h", []], "state_before": "case intro.intro.intro.refine'_2.intro.a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ng : \u03b2 \u2192 Set (Set \u03b1)\ns : Set \u03b2\nh_pi : \u2200 (b : \u03b2), b \u2208 s \u2192 IsPiSystem (g b)\nT : Finset (Subtype s)\nf : Subtype s \u2192 Set \u03b1\nh_t' : \u2200 (b : Subtype s), b \u2208 T \u2192 f b \u2208 (g \u2218 Subtype.val) b\nh_t : \u22c2 b \u2208 T, f b \u2208 generatePiSystem (\u22c3 b \u2208 s, g b)\nthis : \u22c2 b \u2208 T, f b \u2208 generatePiSystem (\u22c3 b, (g \u2218 Subtype.val) b)\nb : \u03b2\nh_b_w : b \u2208 s\nh_b_h : { val := b, property := (_ : b \u2208 s) } \u2208 T\nh_b_alt : b = \u2191{ val := b, property := h_b_w }\n\u22a2 { val := b, property := h_b_w } \u2208 T", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Stieltjes.lean", "full_name": "StieltjesFunction.length_Ioc", "start": [159, 1], "end": [167, 71], "traced_tactics": [{"tactic": "refine'\n  le_antisymm (iInf_le_of_le a <| iInf\u2082_le b Subset.rfl)\n    (le_iInf fun a' => le_iInf fun b' => le_iInf fun h => ENNReal.coe_le_coe.2 _)", "annotated_tactic": ["refine'\n    <a>le_antisymm</a> (<a>iInf_le_of_le</a> a <| <a>iInf\u2082_le</a> b <a>Subset.rfl</a>)\n      (<a>le_iInf</a> fun a' => <a>le_iInf</a> fun b' => <a>le_iInf</a> fun h => <a>ENNReal.coe_le_coe</a>.2 _)", [{"full_name": "le_antisymm", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [188, 9], "def_end_pos": [188, 20]}, {"full_name": "iInf_le_of_le", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [853, 9], "def_end_pos": [853, 22]}, {"full_name": "iInf\u2082_le", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [861, 9], "def_end_pos": [861, 17]}, {"full_name": "Set.Subset.rfl", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [357, 9], "def_end_pos": [357, 19]}, {"full_name": "le_iInf", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [879, 9], "def_end_pos": [879, 16]}, {"full_name": "le_iInf", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [879, 9], "def_end_pos": [879, 16]}, {"full_name": "le_iInf", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [879, 9], "def_end_pos": [879, 16]}, {"full_name": "ENNReal.coe_le_coe", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [349, 28], "def_end_pos": [349, 38]}]], "state_before": "f : StieltjesFunction\na b : \u211d\n\u22a2 length f (Ioc a b) = ofReal (\u2191f b - \u2191f a)", "state_after": "f : StieltjesFunction\na b a' b' : \u211d\nh : Ioc a b \u2286 Ioc a' b'\n\u22a2 Real.toNNReal (\u2191f b - \u2191f a) \u2264 Real.toNNReal (\u2191f b' - \u2191f a')"}, {"tactic": "cases' le_or_lt b a with ab ab", "annotated_tactic": ["cases' <a>le_or_lt</a> b a with ab ab", [{"full_name": "le_or_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [340, 9], "def_end_pos": [340, 17]}]], "state_before": "f : StieltjesFunction\na b a' b' : \u211d\nh : Ioc a b \u2286 Ioc a' b'\n\u22a2 Real.toNNReal (\u2191f b - \u2191f a) \u2264 Real.toNNReal (\u2191f b' - \u2191f a')", "state_after": "case inl\nf : StieltjesFunction\na b a' b' : \u211d\nh : Ioc a b \u2286 Ioc a' b'\nab : b \u2264 a\n\u22a2 Real.toNNReal (\u2191f b - \u2191f a) \u2264 Real.toNNReal (\u2191f b' - \u2191f a')\n\ncase inr\nf : StieltjesFunction\na b a' b' : \u211d\nh : Ioc a b \u2286 Ioc a' b'\nab : a < b\n\u22a2 Real.toNNReal (\u2191f b - \u2191f a) \u2264 Real.toNNReal (\u2191f b' - \u2191f a')"}, {"tactic": "cases' (Ioc_subset_Ioc_iff ab).1 h with h\u2081 h\u2082", "annotated_tactic": ["cases' (<a>Ioc_subset_Ioc_iff</a> ab).1 h with h\u2081 h\u2082", [{"full_name": "Set.Ioc_subset_Ioc_iff", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [1152, 9], "def_end_pos": [1152, 27]}]], "state_before": "case inr\nf : StieltjesFunction\na b a' b' : \u211d\nh : Ioc a b \u2286 Ioc a' b'\nab : a < b\n\u22a2 Real.toNNReal (\u2191f b - \u2191f a) \u2264 Real.toNNReal (\u2191f b' - \u2191f a')", "state_after": "case inr.intro\nf : StieltjesFunction\na b a' b' : \u211d\nh : Ioc a b \u2286 Ioc a' b'\nab : a < b\nh\u2081 : b \u2264 b'\nh\u2082 : a' \u2264 a\n\u22a2 Real.toNNReal (\u2191f b - \u2191f a) \u2264 Real.toNNReal (\u2191f b' - \u2191f a')"}, {"tactic": "exact Real.toNNReal_le_toNNReal (sub_le_sub (f.mono h\u2081) (f.mono h\u2082))", "annotated_tactic": ["exact <a>Real.toNNReal_le_toNNReal</a> (<a>sub_le_sub</a> (f.mono h\u2081) (f.mono h\u2082))", [{"full_name": "Real.toNNReal_le_toNNReal", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [677, 9], "def_end_pos": [677, 29]}, {"full_name": "sub_le_sub", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [850, 15], "def_end_pos": [850, 25]}]], "state_before": "case inr.intro\nf : StieltjesFunction\na b a' b' : \u211d\nh : Ioc a b \u2286 Ioc a' b'\nab : a < b\nh\u2081 : b \u2264 b'\nh\u2082 : a' \u2264 a\n\u22a2 Real.toNNReal (\u2191f b - \u2191f a) \u2264 Real.toNNReal (\u2191f b' - \u2191f a')", "state_after": "no goals"}, {"tactic": "rw [Real.toNNReal_of_nonpos (sub_nonpos.2 (f.mono ab))]", "annotated_tactic": ["rw [<a>Real.toNNReal_of_nonpos</a> (<a>sub_nonpos</a>.2 (f.mono ab))]", [{"full_name": "Real.toNNReal_of_nonpos", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [637, 9], "def_end_pos": [637, 27]}, {"full_name": "sub_nonpos", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [730, 30], "def_end_pos": [730, 40]}]], "state_before": "case inl\nf : StieltjesFunction\na b a' b' : \u211d\nh : Ioc a b \u2286 Ioc a' b'\nab : b \u2264 a\n\u22a2 Real.toNNReal (\u2191f b - \u2191f a) \u2264 Real.toNNReal (\u2191f b' - \u2191f a')", "state_after": "case inl\nf : StieltjesFunction\na b a' b' : \u211d\nh : Ioc a b \u2286 Ioc a' b'\nab : b \u2264 a\n\u22a2 0 \u2264 Real.toNNReal (\u2191f b' - \u2191f a')"}, {"tactic": "apply zero_le", "annotated_tactic": ["apply <a>zero_le</a>", [{"full_name": "zero_le", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [217, 30], "def_end_pos": [217, 37]}]], "state_before": "case inl\nf : StieltjesFunction\na b a' b' : \u211d\nh : Ioc a b \u2286 Ioc a' b'\nab : b \u2264 a\n\u22a2 0 \u2264 Real.toNNReal (\u2191f b' - \u2191f a')", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean", "full_name": "MeasureTheory.condexp_ae_eq_condexpL1", "start": [136, 1], "end": [148, 32], "traced_tactics": [{"tactic": "rw [condexp_of_sigmaFinite hm]", "annotated_tactic": ["rw [<a>condexp_of_sigmaFinite</a> hm]", [{"full_name": "MeasureTheory.condexp_of_sigmaFinite", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean", "def_pos": [113, 9], "def_end_pos": [113, 31]}]], "state_before": "\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g : \u03b1 \u2192 F'\ns : Set \u03b1\nhm : m \u2264 m0\nh\u03bcm : SigmaFinite (Measure.trim \u03bc hm)\nf : \u03b1 \u2192 F'\n\u22a2 \u03bc[f|m] =\u1d50[\u03bc] \u2191\u2191(condexpL1 hm \u03bc f)", "state_after": "\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g : \u03b1 \u2192 F'\ns : Set \u03b1\nhm : m \u2264 m0\nh\u03bcm : SigmaFinite (Measure.trim \u03bc hm)\nf : \u03b1 \u2192 F'\n\u22a2 (if Integrable f then\n      if StronglyMeasurable f then f\n      else AEStronglyMeasurable'.mk \u2191\u2191(condexpL1 hm \u03bc f) (_ : AEStronglyMeasurable' m (\u2191\u2191(condexpL1 hm \u03bc f)) \u03bc)\n    else 0) =\u1d50[\u03bc]\n    \u2191\u2191(condexpL1 hm \u03bc f)"}, {"tactic": "by_cases hfi : Integrable f \u03bc", "annotated_tactic": ["by_cases hfi : <a>Integrable</a> f \u03bc", [{"full_name": "MeasureTheory.Integrable", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [442, 5], "def_end_pos": [442, 15]}]], "state_before": "\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g : \u03b1 \u2192 F'\ns : Set \u03b1\nhm : m \u2264 m0\nh\u03bcm : SigmaFinite (Measure.trim \u03bc hm)\nf : \u03b1 \u2192 F'\n\u22a2 (if Integrable f then\n      if StronglyMeasurable f then f\n      else AEStronglyMeasurable'.mk \u2191\u2191(condexpL1 hm \u03bc f) (_ : AEStronglyMeasurable' m (\u2191\u2191(condexpL1 hm \u03bc f)) \u03bc)\n    else 0) =\u1d50[\u03bc]\n    \u2191\u2191(condexpL1 hm \u03bc f)", "state_after": "case pos\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g : \u03b1 \u2192 F'\ns : Set \u03b1\nhm : m \u2264 m0\nh\u03bcm : SigmaFinite (Measure.trim \u03bc hm)\nf : \u03b1 \u2192 F'\nhfi : Integrable f\n\u22a2 (if Integrable f then\n      if StronglyMeasurable f then f\n      else AEStronglyMeasurable'.mk \u2191\u2191(condexpL1 hm \u03bc f) (_ : AEStronglyMeasurable' m (\u2191\u2191(condexpL1 hm \u03bc f)) \u03bc)\n    else 0) =\u1d50[\u03bc]\n    \u2191\u2191(condexpL1 hm \u03bc f)\n\ncase neg\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g : \u03b1 \u2192 F'\ns : Set \u03b1\nhm : m \u2264 m0\nh\u03bcm : SigmaFinite (Measure.trim \u03bc hm)\nf : \u03b1 \u2192 F'\nhfi : \u00acIntegrable f\n\u22a2 (if Integrable f then\n      if StronglyMeasurable f then f\n      else AEStronglyMeasurable'.mk \u2191\u2191(condexpL1 hm \u03bc f) (_ : AEStronglyMeasurable' m (\u2191\u2191(condexpL1 hm \u03bc f)) \u03bc)\n    else 0) =\u1d50[\u03bc]\n    \u2191\u2191(condexpL1 hm \u03bc f)"}, {"tactic": "rw [if_neg hfi, condexpL1_undef hfi]", "annotated_tactic": ["rw [<a>if_neg</a> hfi, <a>condexpL1_undef</a> hfi]", [{"full_name": "if_neg", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [795, 9], "def_end_pos": [795, 15]}, {"full_name": "MeasureTheory.condexpL1_undef", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean", "def_pos": [518, 9], "def_end_pos": [518, 24]}]], "state_before": "case neg\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g : \u03b1 \u2192 F'\ns : Set \u03b1\nhm : m \u2264 m0\nh\u03bcm : SigmaFinite (Measure.trim \u03bc hm)\nf : \u03b1 \u2192 F'\nhfi : \u00acIntegrable f\n\u22a2 (if Integrable f then\n      if StronglyMeasurable f then f\n      else AEStronglyMeasurable'.mk \u2191\u2191(condexpL1 hm \u03bc f) (_ : AEStronglyMeasurable' m (\u2191\u2191(condexpL1 hm \u03bc f)) \u03bc)\n    else 0) =\u1d50[\u03bc]\n    \u2191\u2191(condexpL1 hm \u03bc f)", "state_after": "case neg\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g : \u03b1 \u2192 F'\ns : Set \u03b1\nhm : m \u2264 m0\nh\u03bcm : SigmaFinite (Measure.trim \u03bc hm)\nf : \u03b1 \u2192 F'\nhfi : \u00acIntegrable f\n\u22a2 0 =\u1d50[\u03bc] \u2191\u21910"}, {"tactic": "exact (coeFn_zero _ _ _).symm", "annotated_tactic": ["exact (<a>coeFn_zero</a> _ _ _).<a>symm</a>", [{"full_name": "MeasureTheory.Lp.coeFn_zero", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [222, 9], "def_end_pos": [222, 19]}, {"full_name": "Filter.EventuallyEq.symm", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1498, 9], "def_end_pos": [1498, 26]}]], "state_before": "case neg\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g : \u03b1 \u2192 F'\ns : Set \u03b1\nhm : m \u2264 m0\nh\u03bcm : SigmaFinite (Measure.trim \u03bc hm)\nf : \u03b1 \u2192 F'\nhfi : \u00acIntegrable f\n\u22a2 0 =\u1d50[\u03bc] \u2191\u21910", "state_after": "no goals"}, {"tactic": "rw [if_pos hfi]", "annotated_tactic": ["rw [<a>if_pos</a> hfi]", [{"full_name": "if_pos", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [790, 9], "def_end_pos": [790, 15]}]], "state_before": "case pos\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g : \u03b1 \u2192 F'\ns : Set \u03b1\nhm : m \u2264 m0\nh\u03bcm : SigmaFinite (Measure.trim \u03bc hm)\nf : \u03b1 \u2192 F'\nhfi : Integrable f\n\u22a2 (if Integrable f then\n      if StronglyMeasurable f then f\n      else AEStronglyMeasurable'.mk \u2191\u2191(condexpL1 hm \u03bc f) (_ : AEStronglyMeasurable' m (\u2191\u2191(condexpL1 hm \u03bc f)) \u03bc)\n    else 0) =\u1d50[\u03bc]\n    \u2191\u2191(condexpL1 hm \u03bc f)", "state_after": "case pos\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g : \u03b1 \u2192 F'\ns : Set \u03b1\nhm : m \u2264 m0\nh\u03bcm : SigmaFinite (Measure.trim \u03bc hm)\nf : \u03b1 \u2192 F'\nhfi : Integrable f\n\u22a2 (if StronglyMeasurable f then f\n    else AEStronglyMeasurable'.mk \u2191\u2191(condexpL1 hm \u03bc f) (_ : AEStronglyMeasurable' m (\u2191\u2191(condexpL1 hm \u03bc f)) \u03bc)) =\u1d50[\u03bc]\n    \u2191\u2191(condexpL1 hm \u03bc f)"}, {"tactic": "by_cases hfm : StronglyMeasurable[m] f", "annotated_tactic": ["by_cases hfm : StronglyMeasurable[m] f", []], "state_before": "case pos\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g : \u03b1 \u2192 F'\ns : Set \u03b1\nhm : m \u2264 m0\nh\u03bcm : SigmaFinite (Measure.trim \u03bc hm)\nf : \u03b1 \u2192 F'\nhfi : Integrable f\n\u22a2 (if StronglyMeasurable f then f\n    else AEStronglyMeasurable'.mk \u2191\u2191(condexpL1 hm \u03bc f) (_ : AEStronglyMeasurable' m (\u2191\u2191(condexpL1 hm \u03bc f)) \u03bc)) =\u1d50[\u03bc]\n    \u2191\u2191(condexpL1 hm \u03bc f)", "state_after": "case pos\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g : \u03b1 \u2192 F'\ns : Set \u03b1\nhm : m \u2264 m0\nh\u03bcm : SigmaFinite (Measure.trim \u03bc hm)\nf : \u03b1 \u2192 F'\nhfi : Integrable f\nhfm : StronglyMeasurable f\n\u22a2 (if StronglyMeasurable f then f\n    else AEStronglyMeasurable'.mk \u2191\u2191(condexpL1 hm \u03bc f) (_ : AEStronglyMeasurable' m (\u2191\u2191(condexpL1 hm \u03bc f)) \u03bc)) =\u1d50[\u03bc]\n    \u2191\u2191(condexpL1 hm \u03bc f)\n\ncase neg\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g : \u03b1 \u2192 F'\ns : Set \u03b1\nhm : m \u2264 m0\nh\u03bcm : SigmaFinite (Measure.trim \u03bc hm)\nf : \u03b1 \u2192 F'\nhfi : Integrable f\nhfm : \u00acStronglyMeasurable f\n\u22a2 (if StronglyMeasurable f then f\n    else AEStronglyMeasurable'.mk \u2191\u2191(condexpL1 hm \u03bc f) (_ : AEStronglyMeasurable' m (\u2191\u2191(condexpL1 hm \u03bc f)) \u03bc)) =\u1d50[\u03bc]\n    \u2191\u2191(condexpL1 hm \u03bc f)"}, {"tactic": "rw [if_pos hfm]", "annotated_tactic": ["rw [<a>if_pos</a> hfm]", [{"full_name": "if_pos", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [790, 9], "def_end_pos": [790, 15]}]], "state_before": "case pos\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g : \u03b1 \u2192 F'\ns : Set \u03b1\nhm : m \u2264 m0\nh\u03bcm : SigmaFinite (Measure.trim \u03bc hm)\nf : \u03b1 \u2192 F'\nhfi : Integrable f\nhfm : StronglyMeasurable f\n\u22a2 (if StronglyMeasurable f then f\n    else AEStronglyMeasurable'.mk \u2191\u2191(condexpL1 hm \u03bc f) (_ : AEStronglyMeasurable' m (\u2191\u2191(condexpL1 hm \u03bc f)) \u03bc)) =\u1d50[\u03bc]\n    \u2191\u2191(condexpL1 hm \u03bc f)", "state_after": "case pos\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g : \u03b1 \u2192 F'\ns : Set \u03b1\nhm : m \u2264 m0\nh\u03bcm : SigmaFinite (Measure.trim \u03bc hm)\nf : \u03b1 \u2192 F'\nhfi : Integrable f\nhfm : StronglyMeasurable f\n\u22a2 f =\u1d50[\u03bc] \u2191\u2191(condexpL1 hm \u03bc f)"}, {"tactic": "exact (condexpL1_of_aestronglyMeasurable' (StronglyMeasurable.aeStronglyMeasurable' hfm)\n  hfi).symm", "annotated_tactic": ["exact (<a>condexpL1_of_aestronglyMeasurable'</a> (<a>StronglyMeasurable.aeStronglyMeasurable'</a> hfm)\n        hfi).<a>symm</a>", [{"full_name": "MeasureTheory.condexpL1_of_aestronglyMeasurable'", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean", "def_pos": [584, 9], "def_end_pos": [584, 43]}, {"full_name": "MeasureTheory.StronglyMeasurable.aeStronglyMeasurable'", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean", "def_pos": [138, 9], "def_end_pos": [138, 49]}, {"full_name": "Filter.EventuallyEq.symm", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1498, 9], "def_end_pos": [1498, 26]}]], "state_before": "case pos\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g : \u03b1 \u2192 F'\ns : Set \u03b1\nhm : m \u2264 m0\nh\u03bcm : SigmaFinite (Measure.trim \u03bc hm)\nf : \u03b1 \u2192 F'\nhfi : Integrable f\nhfm : StronglyMeasurable f\n\u22a2 f =\u1d50[\u03bc] \u2191\u2191(condexpL1 hm \u03bc f)", "state_after": "no goals"}, {"tactic": "rw [if_neg hfm]", "annotated_tactic": ["rw [<a>if_neg</a> hfm]", [{"full_name": "if_neg", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [795, 9], "def_end_pos": [795, 15]}]], "state_before": "case neg\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g : \u03b1 \u2192 F'\ns : Set \u03b1\nhm : m \u2264 m0\nh\u03bcm : SigmaFinite (Measure.trim \u03bc hm)\nf : \u03b1 \u2192 F'\nhfi : Integrable f\nhfm : \u00acStronglyMeasurable f\n\u22a2 (if StronglyMeasurable f then f\n    else AEStronglyMeasurable'.mk \u2191\u2191(condexpL1 hm \u03bc f) (_ : AEStronglyMeasurable' m (\u2191\u2191(condexpL1 hm \u03bc f)) \u03bc)) =\u1d50[\u03bc]\n    \u2191\u2191(condexpL1 hm \u03bc f)", "state_after": "case neg\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g : \u03b1 \u2192 F'\ns : Set \u03b1\nhm : m \u2264 m0\nh\u03bcm : SigmaFinite (Measure.trim \u03bc hm)\nf : \u03b1 \u2192 F'\nhfi : Integrable f\nhfm : \u00acStronglyMeasurable f\n\u22a2 AEStronglyMeasurable'.mk \u2191\u2191(condexpL1 hm \u03bc f) (_ : AEStronglyMeasurable' m (\u2191\u2191(condexpL1 hm \u03bc f)) \u03bc) =\u1d50[\u03bc]\n    \u2191\u2191(condexpL1 hm \u03bc f)"}, {"tactic": "exact (AEStronglyMeasurable'.ae_eq_mk aestronglyMeasurable'_condexpL1).symm", "annotated_tactic": ["exact (<a>AEStronglyMeasurable'.ae_eq_mk</a> <a>aestronglyMeasurable'_condexpL1</a>).<a>symm</a>", [{"full_name": "MeasureTheory.AEStronglyMeasurable'.ae_eq_mk", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean", "def_pos": [119, 9], "def_end_pos": [119, 17]}, {"full_name": "MeasureTheory.aestronglyMeasurable'_condexpL1", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean", "def_pos": [536, 9], "def_end_pos": [536, 40]}, {"full_name": "Filter.EventuallyEq.symm", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1498, 9], "def_end_pos": [1498, 26]}]], "state_before": "case neg\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g : \u03b1 \u2192 F'\ns : Set \u03b1\nhm : m \u2264 m0\nh\u03bcm : SigmaFinite (Measure.trim \u03bc hm)\nf : \u03b1 \u2192 F'\nhfi : Integrable f\nhfm : \u00acStronglyMeasurable f\n\u22a2 AEStronglyMeasurable'.mk \u2191\u2191(condexpL1 hm \u03bc f) (_ : AEStronglyMeasurable' m (\u2191\u2191(condexpL1 hm \u03bc f)) \u03bc) =\u1d50[\u03bc]\n    \u2191\u2191(condexpL1 hm \u03bc f)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Group/Measure.lean", "full_name": "MeasureTheory.Measure.IsHaarMeasure.smul", "start": [768, 1], "end": [770, 57], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Basic.lean", "full_name": "Set.mem_ite_empty_right", "start": [2243, 1], "end": [2245, 58], "traced_tactics": [{"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\na b : \u03b1\ns s\u2081 s\u2082 t\u271d t\u2081 t\u2082 u : Set \u03b1\np : Prop\ninst\u271d : Decidable p\nt : Set \u03b1\nx : \u03b1\n\u22a2 (\u2203 h, x \u2208 t) \u2194 p \u2227 x \u2208 t", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Num/Lemmas.lean", "full_name": "Num.cast_succ", "start": [723, 1], "end": [724, 15], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/WithDensity.lean", "full_name": "MeasureTheory.set_lintegral_withDensity_eq_set_lintegral_mul_non_measurable\u2080", "start": [397, 1], "end": [401, 96], "traced_tactics": [{"tactic": "rw [restrict_withDensity hs, lintegral_withDensity_eq_lintegral_mul_non_measurable\u2080 _ hf h'f]", "annotated_tactic": ["rw [<a>restrict_withDensity</a> hs, <a>lintegral_withDensity_eq_lintegral_mul_non_measurable\u2080</a> _ hf h'f]", [{"full_name": "MeasureTheory.restrict_withDensity", "def_path": "Mathlib/MeasureTheory/Measure/WithDensity.lean", "def_pos": [176, 9], "def_end_pos": [176, 29]}, {"full_name": "MeasureTheory.lintegral_withDensity_eq_lintegral_mul_non_measurable\u2080", "def_path": "Mathlib/MeasureTheory/Measure/WithDensity.lean", "def_pos": [378, 9], "def_end_pos": [378, 63]}]], "state_before": "\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhf : AEMeasurable f\ng : \u03b1 \u2192 \u211d\u22650\u221e\nhs : MeasurableSet s\nh'f : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc s, f x < \u22a4\n\u22a2 \u222b\u207b (a : \u03b1) in s, g a \u2202withDensity \u03bc f = \u222b\u207b (a : \u03b1) in s, (f * g) a \u2202\u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "full_name": "MeasureTheory.exists_pos_preimage_ball", "start": [3927, 1], "end": [3929, 100], "traced_tactics": [{"tactic": "rw [\u2190 preimage_iUnion, Metric.iUnion_ball_nat, preimage_univ]", "annotated_tactic": ["rw [\u2190 <a>preimage_iUnion</a>, <a>Metric.iUnion_ball_nat</a>, <a>preimage_univ</a>]", [{"full_name": "Set.preimage_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [1854, 9], "def_end_pos": [1854, 24]}, {"full_name": "Metric.iUnion_ball_nat", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [464, 9], "def_end_pos": [464, 24]}, {"full_name": "Set.preimage_univ", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [78, 9], "def_end_pos": [78, 22]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d : PseudoMetricSpace \u03b4\nf : \u03b1 \u2192 \u03b4\nx : \u03b4\nh\u03bc : \u03bc \u2260 0\n\u22a2 \u22c3 i, f \u207b\u00b9' Metric.ball x \u2191i = univ", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/ProbabilityMassFunction/Basic.lean", "full_name": "PMF.toOuterMeasure_apply_eq_zero_iff", "start": [196, 1], "end": [198, 62], "traced_tactics": [{"tactic": "rw [toOuterMeasure_apply, ENNReal.tsum_eq_zero]", "annotated_tactic": ["rw [<a>toOuterMeasure_apply</a>, <a>ENNReal.tsum_eq_zero</a>]", [{"full_name": "PMF.toOuterMeasure_apply", "def_path": "Mathlib/Probability/ProbabilityMassFunction/Basic.lean", "def_pos": [161, 9], "def_end_pos": [161, 29]}, {"full_name": "ENNReal.tsum_eq_zero", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [862, 19], "def_end_pos": [862, 31]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\ns t : Set \u03b1\n\u22a2 \u2191(toOuterMeasure p) s = 0 \u2194 Disjoint (support p) s", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\ns t : Set \u03b1\n\u22a2 (\u2200 (i : \u03b1), Set.indicator s (\u2191p) i = 0) \u2194 Disjoint (support p) s"}, {"tactic": "exact Function.funext_iff.symm.trans Set.indicator_eq_zero'", "annotated_tactic": ["exact Function.funext_iff.symm.trans <a>Set.indicator_eq_zero'</a>", [{"full_name": "Set.indicator_eq_zero'", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [123, 3], "def_end_pos": [123, 14]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\ns t : Set \u03b1\n\u22a2 (\u2200 (i : \u03b1), Set.indicator s (\u2191p) i = 0) \u2194 Disjoint (support p) s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "full_name": "MeasureTheory.snorm_le_snorm_mul_snorm_of_nnnorm", "start": [1415, 1], "end": [1451, 79], "traced_tactics": [{"tactic": "by_cases hp_zero : p = 0", "annotated_tactic": ["by_cases hp_zero : p = 0", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q r : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhpqr : 1 / p = 1 / q + 1 / r\n\u22a2 snorm (fun x => b (f x) (g x)) p \u03bc \u2264 snorm f q \u03bc * snorm g r \u03bc", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q r : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhpqr : 1 / p = 1 / q + 1 / r\nhp_zero : p = 0\n\u22a2 snorm (fun x => b (f x) (g x)) p \u03bc \u2264 snorm f q \u03bc * snorm g r \u03bc\n\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q r : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhpqr : 1 / p = 1 / q + 1 / r\nhp_zero : \u00acp = 0\n\u22a2 snorm (fun x => b (f x) (g x)) p \u03bc \u2264 snorm f q \u03bc * snorm g r \u03bc"}, {"tactic": "have hq_ne_zero : q \u2260 0 := by\n  intro hq_zero\n  simp only [hq_zero, hp_zero, one_div, ENNReal.inv_zero, top_add, ENNReal.inv_eq_top] at hpqr", "annotated_tactic": ["have hq_ne_zero : q \u2260 0 := by\n    intro hq_zero\n    simp only [hq_zero, hp_zero, <a>one_div</a>, <a>ENNReal.inv_zero</a>, <a>top_add</a>, <a>ENNReal.inv_eq_top</a>] at hpqr", [{"full_name": "one_div", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [318, 9], "def_end_pos": [318, 16]}, {"full_name": "ENNReal.inv_zero", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1370, 17], "def_end_pos": [1370, 25]}, {"full_name": "top_add", "def_path": "Mathlib/Algebra/Order/Monoid/Defs.lean", "def_pos": [105, 9], "def_end_pos": [105, 16]}, {"full_name": "ENNReal.inv_eq_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1449, 17], "def_end_pos": [1449, 27]}]], "state_before": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q r : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhpqr : 1 / p = 1 / q + 1 / r\nhp_zero : \u00acp = 0\n\u22a2 snorm (fun x => b (f x) (g x)) p \u03bc \u2264 snorm f q \u03bc * snorm g r \u03bc", "state_after": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q r : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhpqr : 1 / p = 1 / q + 1 / r\nhp_zero : \u00acp = 0\nhq_ne_zero : q \u2260 0\n\u22a2 snorm (fun x => b (f x) (g x)) p \u03bc \u2264 snorm f q \u03bc * snorm g r \u03bc"}, {"tactic": "have hr_ne_zero : r \u2260 0 := by\n  intro hr_zero\n  simp only [hr_zero, hp_zero, one_div, ENNReal.inv_zero, add_top, ENNReal.inv_eq_top] at hpqr", "annotated_tactic": ["have hr_ne_zero : r \u2260 0 := by\n    intro hr_zero\n    simp only [hr_zero, hp_zero, <a>one_div</a>, <a>ENNReal.inv_zero</a>, <a>add_top</a>, <a>ENNReal.inv_eq_top</a>] at hpqr", [{"full_name": "one_div", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [318, 9], "def_end_pos": [318, 16]}, {"full_name": "ENNReal.inv_zero", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1370, 17], "def_end_pos": [1370, 25]}, {"full_name": "add_top", "def_path": "Mathlib/Algebra/Order/Monoid/Defs.lean", "def_pos": [110, 9], "def_end_pos": [110, 16]}, {"full_name": "ENNReal.inv_eq_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1449, 17], "def_end_pos": [1449, 27]}]], "state_before": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q r : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhpqr : 1 / p = 1 / q + 1 / r\nhp_zero : \u00acp = 0\nhq_ne_zero : q \u2260 0\n\u22a2 snorm (fun x => b (f x) (g x)) p \u03bc \u2264 snorm f q \u03bc * snorm g r \u03bc", "state_after": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q r : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhpqr : 1 / p = 1 / q + 1 / r\nhp_zero : \u00acp = 0\nhq_ne_zero : q \u2260 0\nhr_ne_zero : r \u2260 0\n\u22a2 snorm (fun x => b (f x) (g x)) p \u03bc \u2264 snorm f q \u03bc * snorm g r \u03bc"}, {"tactic": "by_cases hq_top : q = \u221e", "annotated_tactic": ["by_cases hq_top : q = \u221e", []], "state_before": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q r : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhpqr : 1 / p = 1 / q + 1 / r\nhp_zero : \u00acp = 0\nhq_ne_zero : q \u2260 0\nhr_ne_zero : r \u2260 0\n\u22a2 snorm (fun x => b (f x) (g x)) p \u03bc \u2264 snorm f q \u03bc * snorm g r \u03bc", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q r : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhpqr : 1 / p = 1 / q + 1 / r\nhp_zero : \u00acp = 0\nhq_ne_zero : q \u2260 0\nhr_ne_zero : r \u2260 0\nhq_top : q = \u22a4\n\u22a2 snorm (fun x => b (f x) (g x)) p \u03bc \u2264 snorm f q \u03bc * snorm g r \u03bc\n\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q r : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhpqr : 1 / p = 1 / q + 1 / r\nhp_zero : \u00acp = 0\nhq_ne_zero : q \u2260 0\nhr_ne_zero : r \u2260 0\nhq_top : \u00acq = \u22a4\n\u22a2 snorm (fun x => b (f x) (g x)) p \u03bc \u2264 snorm f q \u03bc * snorm g r \u03bc"}, {"tactic": "by_cases hr_top : r = \u221e", "annotated_tactic": ["by_cases hr_top : r = \u221e", []], "state_before": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q r : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhpqr : 1 / p = 1 / q + 1 / r\nhp_zero : \u00acp = 0\nhq_ne_zero : q \u2260 0\nhr_ne_zero : r \u2260 0\nhq_top : \u00acq = \u22a4\n\u22a2 snorm (fun x => b (f x) (g x)) p \u03bc \u2264 snorm f q \u03bc * snorm g r \u03bc", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q r : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhpqr : 1 / p = 1 / q + 1 / r\nhp_zero : \u00acp = 0\nhq_ne_zero : q \u2260 0\nhr_ne_zero : r \u2260 0\nhq_top : \u00acq = \u22a4\nhr_top : r = \u22a4\n\u22a2 snorm (fun x => b (f x) (g x)) p \u03bc \u2264 snorm f q \u03bc * snorm g r \u03bc\n\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q r : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhpqr : 1 / p = 1 / q + 1 / r\nhp_zero : \u00acp = 0\nhq_ne_zero : q \u2260 0\nhr_ne_zero : r \u2260 0\nhq_top : \u00acq = \u22a4\nhr_top : \u00acr = \u22a4\n\u22a2 snorm (fun x => b (f x) (g x)) p \u03bc \u2264 snorm f q \u03bc * snorm g r \u03bc"}, {"tactic": "rw [snorm_eq_snorm' hp_zero (hpq.trans_le le_top).ne, snorm_eq_snorm' hq_ne_zero hq_top,\n  snorm_eq_snorm' hr_ne_zero hr_top]", "annotated_tactic": ["rw [<a>snorm_eq_snorm'</a> hp_zero (hpq.trans_le <a>le_top</a>).<a>ne</a>, <a>snorm_eq_snorm'</a> hq_ne_zero hq_top,\n    <a>snorm_eq_snorm'</a> hr_ne_zero hr_top]", [{"full_name": "MeasureTheory.snorm_eq_snorm'", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [88, 9], "def_end_pos": [88, 24]}, {"full_name": "le_top", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [98, 9], "def_end_pos": [98, 15]}, {"full_name": "LT.lt.ne", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [152, 7], "def_end_pos": [152, 15]}, {"full_name": "MeasureTheory.snorm_eq_snorm'", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [88, 9], "def_end_pos": [88, 24]}, {"full_name": "MeasureTheory.snorm_eq_snorm'", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [88, 9], "def_end_pos": [88, 24]}]], "state_before": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q r : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhpqr : 1 / p = 1 / q + 1 / r\nhp_zero : \u00acp = 0\nhq_ne_zero : q \u2260 0\nhr_ne_zero : r \u2260 0\nhq_top : \u00acq = \u22a4\nhr_top : \u00acr = \u22a4\nhpq : p < q\n\u22a2 snorm (fun x => b (f x) (g x)) p \u03bc \u2264 snorm f q \u03bc * snorm g r \u03bc", "state_after": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q r : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhpqr : 1 / p = 1 / q + 1 / r\nhp_zero : \u00acp = 0\nhq_ne_zero : q \u2260 0\nhr_ne_zero : r \u2260 0\nhq_top : \u00acq = \u22a4\nhr_top : \u00acr = \u22a4\nhpq : p < q\n\u22a2 snorm' (fun x => b (f x) (g x)) (ENNReal.toReal p) \u03bc \u2264 snorm' f (ENNReal.toReal q) \u03bc * snorm' g (ENNReal.toReal r) \u03bc"}, {"tactic": "refine' snorm'_le_snorm'_mul_snorm' hf hg _ h _ _ _", "annotated_tactic": ["refine' <a>snorm'_le_snorm'_mul_snorm'</a> hf hg _ h _ _ _", [{"full_name": "MeasureTheory.snorm'_le_snorm'_mul_snorm'", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [1340, 9], "def_end_pos": [1340, 36]}]], "state_before": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q r : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhpqr : 1 / p = 1 / q + 1 / r\nhp_zero : \u00acp = 0\nhq_ne_zero : q \u2260 0\nhr_ne_zero : r \u2260 0\nhq_top : \u00acq = \u22a4\nhr_top : \u00acr = \u22a4\nhpq : p < q\n\u22a2 snorm' (fun x => b (f x) (g x)) (ENNReal.toReal p) \u03bc \u2264 snorm' f (ENNReal.toReal q) \u03bc * snorm' g (ENNReal.toReal r) \u03bc", "state_after": "case neg.refine'_1\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q r : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhpqr : 1 / p = 1 / q + 1 / r\nhp_zero : \u00acp = 0\nhq_ne_zero : q \u2260 0\nhr_ne_zero : r \u2260 0\nhq_top : \u00acq = \u22a4\nhr_top : \u00acr = \u22a4\nhpq : p < q\n\u22a2 0 < ENNReal.toReal p\n\ncase neg.refine'_2\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q r : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhpqr : 1 / p = 1 / q + 1 / r\nhp_zero : \u00acp = 0\nhq_ne_zero : q \u2260 0\nhr_ne_zero : r \u2260 0\nhq_top : \u00acq = \u22a4\nhr_top : \u00acr = \u22a4\nhpq : p < q\n\u22a2 ENNReal.toReal p < ENNReal.toReal q\n\ncase neg.refine'_3\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q r : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhpqr : 1 / p = 1 / q + 1 / r\nhp_zero : \u00acp = 0\nhq_ne_zero : q \u2260 0\nhr_ne_zero : r \u2260 0\nhq_top : \u00acq = \u22a4\nhr_top : \u00acr = \u22a4\nhpq : p < q\n\u22a2 1 / ENNReal.toReal p = 1 / ENNReal.toReal q + 1 / ENNReal.toReal r"}, {"tactic": "rw [\u2190 ENNReal.one_toReal, \u2190 ENNReal.toReal_div, \u2190 ENNReal.toReal_div, \u2190 ENNReal.toReal_div, hpqr,\n  ENNReal.toReal_add]", "annotated_tactic": ["rw [\u2190 <a>ENNReal.one_toReal</a>, \u2190 <a>ENNReal.toReal_div</a>, \u2190 <a>ENNReal.toReal_div</a>, \u2190 <a>ENNReal.toReal_div</a>, hpqr,\n    <a>ENNReal.toReal_add</a>]", [{"full_name": "ENNReal.one_toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [230, 17], "def_end_pos": [230, 27]}, {"full_name": "ENNReal.toReal_div", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2385, 9], "def_end_pos": [2385, 19]}, {"full_name": "ENNReal.toReal_div", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2385, 9], "def_end_pos": [2385, 19]}, {"full_name": "ENNReal.toReal_div", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2385, 9], "def_end_pos": [2385, 19]}, {"full_name": "ENNReal.toReal_add", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1997, 9], "def_end_pos": [1997, 19]}]], "state_before": "case neg.refine'_3\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q r : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhpqr : 1 / p = 1 / q + 1 / r\nhp_zero : \u00acp = 0\nhq_ne_zero : q \u2260 0\nhr_ne_zero : r \u2260 0\nhq_top : \u00acq = \u22a4\nhr_top : \u00acr = \u22a4\nhpq : p < q\n\u22a2 1 / ENNReal.toReal p = 1 / ENNReal.toReal q + 1 / ENNReal.toReal r", "state_after": "case neg.refine'_3.ha\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q r : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhpqr : 1 / p = 1 / q + 1 / r\nhp_zero : \u00acp = 0\nhq_ne_zero : q \u2260 0\nhr_ne_zero : r \u2260 0\nhq_top : \u00acq = \u22a4\nhr_top : \u00acr = \u22a4\nhpq : p < q\n\u22a2 1 / q \u2260 \u22a4\n\ncase neg.refine'_3.hb\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q r : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhpqr : 1 / p = 1 / q + 1 / r\nhp_zero : \u00acp = 0\nhq_ne_zero : q \u2260 0\nhr_ne_zero : r \u2260 0\nhq_top : \u00acq = \u22a4\nhr_top : \u00acr = \u22a4\nhpq : p < q\n\u22a2 1 / r \u2260 \u22a4"}, {"tactic": "simp [hp_zero]", "annotated_tactic": ["simp [hp_zero]", []], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q r : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhpqr : 1 / p = 1 / q + 1 / r\nhp_zero : p = 0\n\u22a2 snorm (fun x => b (f x) (g x)) p \u03bc \u2264 snorm f q \u03bc * snorm g r \u03bc", "state_after": "no goals"}, {"tactic": "intro hq_zero", "annotated_tactic": ["intro hq_zero", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q r : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhpqr : 1 / p = 1 / q + 1 / r\nhp_zero : \u00acp = 0\n\u22a2 q \u2260 0", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q r : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhpqr : 1 / p = 1 / q + 1 / r\nhp_zero : \u00acp = 0\nhq_zero : q = 0\n\u22a2 False"}, {"tactic": "simp only [hq_zero, hp_zero, one_div, ENNReal.inv_zero, top_add, ENNReal.inv_eq_top] at hpqr", "annotated_tactic": ["simp only [hq_zero, hp_zero, <a>one_div</a>, <a>ENNReal.inv_zero</a>, <a>top_add</a>, <a>ENNReal.inv_eq_top</a>] at hpqr", [{"full_name": "one_div", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [318, 9], "def_end_pos": [318, 16]}, {"full_name": "ENNReal.inv_zero", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1370, 17], "def_end_pos": [1370, 25]}, {"full_name": "top_add", "def_path": "Mathlib/Algebra/Order/Monoid/Defs.lean", "def_pos": [105, 9], "def_end_pos": [105, 16]}, {"full_name": "ENNReal.inv_eq_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1449, 17], "def_end_pos": [1449, 27]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q r : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhpqr : 1 / p = 1 / q + 1 / r\nhp_zero : \u00acp = 0\nhq_zero : q = 0\n\u22a2 False", "state_after": "no goals"}, {"tactic": "intro hr_zero", "annotated_tactic": ["intro hr_zero", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q r : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhpqr : 1 / p = 1 / q + 1 / r\nhp_zero : \u00acp = 0\nhq_ne_zero : q \u2260 0\n\u22a2 r \u2260 0", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q r : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhpqr : 1 / p = 1 / q + 1 / r\nhp_zero : \u00acp = 0\nhq_ne_zero : q \u2260 0\nhr_zero : r = 0\n\u22a2 False"}, {"tactic": "simp only [hr_zero, hp_zero, one_div, ENNReal.inv_zero, add_top, ENNReal.inv_eq_top] at hpqr", "annotated_tactic": ["simp only [hr_zero, hp_zero, <a>one_div</a>, <a>ENNReal.inv_zero</a>, <a>add_top</a>, <a>ENNReal.inv_eq_top</a>] at hpqr", [{"full_name": "one_div", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [318, 9], "def_end_pos": [318, 16]}, {"full_name": "ENNReal.inv_zero", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1370, 17], "def_end_pos": [1370, 25]}, {"full_name": "add_top", "def_path": "Mathlib/Algebra/Order/Monoid/Defs.lean", "def_pos": [110, 9], "def_end_pos": [110, 16]}, {"full_name": "ENNReal.inv_eq_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1449, 17], "def_end_pos": [1449, 27]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q r : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhpqr : 1 / p = 1 / q + 1 / r\nhp_zero : \u00acp = 0\nhq_ne_zero : q \u2260 0\nhr_zero : r = 0\n\u22a2 False", "state_after": "no goals"}, {"tactic": "have hpr : p = r := by\n  simpa only [hq_top, one_div, ENNReal.inv_top, zero_add, inv_inj] using hpqr", "annotated_tactic": ["have hpr : p = r := by\n      simpa only [hq_top, <a>one_div</a>, <a>ENNReal.inv_top</a>, <a>zero_add</a>, <a>inv_inj</a>] using hpqr", [{"full_name": "one_div", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [318, 9], "def_end_pos": [318, 16]}, {"full_name": "ENNReal.inv_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1374, 17], "def_end_pos": [1374, 24]}, {"full_name": "zero_add", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [463, 3], "def_end_pos": [463, 14]}, {"full_name": "inv_inj", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [257, 9], "def_end_pos": [257, 16]}]], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q r : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhpqr : 1 / p = 1 / q + 1 / r\nhp_zero : \u00acp = 0\nhq_ne_zero : q \u2260 0\nhr_ne_zero : r \u2260 0\nhq_top : q = \u22a4\n\u22a2 snorm (fun x => b (f x) (g x)) p \u03bc \u2264 snorm f q \u03bc * snorm g r \u03bc", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q r : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhpqr : 1 / p = 1 / q + 1 / r\nhp_zero : \u00acp = 0\nhq_ne_zero : q \u2260 0\nhr_ne_zero : r \u2260 0\nhq_top : q = \u22a4\nhpr : p = r\n\u22a2 snorm (fun x => b (f x) (g x)) p \u03bc \u2264 snorm f q \u03bc * snorm g r \u03bc"}, {"tactic": "rw [\u2190 hpr, hq_top]", "annotated_tactic": ["rw [\u2190 hpr, hq_top]", []], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q r : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhpqr : 1 / p = 1 / q + 1 / r\nhp_zero : \u00acp = 0\nhq_ne_zero : q \u2260 0\nhr_ne_zero : r \u2260 0\nhq_top : q = \u22a4\nhpr : p = r\n\u22a2 snorm (fun x => b (f x) (g x)) p \u03bc \u2264 snorm f q \u03bc * snorm g r \u03bc", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q r : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhpqr : 1 / p = 1 / q + 1 / r\nhp_zero : \u00acp = 0\nhq_ne_zero : q \u2260 0\nhr_ne_zero : r \u2260 0\nhq_top : q = \u22a4\nhpr : p = r\n\u22a2 snorm (fun x => b (f x) (g x)) p \u03bc \u2264 snorm f \u22a4 \u03bc * snorm g p \u03bc"}, {"tactic": "exact snorm_le_snorm_top_mul_snorm p f hg b h", "annotated_tactic": ["exact <a>snorm_le_snorm_top_mul_snorm</a> p f hg b h", [{"full_name": "MeasureTheory.snorm_le_snorm_top_mul_snorm", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [1357, 9], "def_end_pos": [1357, 37]}]], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q r : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhpqr : 1 / p = 1 / q + 1 / r\nhp_zero : \u00acp = 0\nhq_ne_zero : q \u2260 0\nhr_ne_zero : r \u2260 0\nhq_top : q = \u22a4\nhpr : p = r\n\u22a2 snorm (fun x => b (f x) (g x)) p \u03bc \u2264 snorm f \u22a4 \u03bc * snorm g p \u03bc", "state_after": "no goals"}, {"tactic": "simpa only [hq_top, one_div, ENNReal.inv_top, zero_add, inv_inj] using hpqr", "annotated_tactic": ["simpa only [hq_top, <a>one_div</a>, <a>ENNReal.inv_top</a>, <a>zero_add</a>, <a>inv_inj</a>] using hpqr", [{"full_name": "one_div", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [318, 9], "def_end_pos": [318, 16]}, {"full_name": "ENNReal.inv_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1374, 17], "def_end_pos": [1374, 24]}, {"full_name": "zero_add", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [463, 3], "def_end_pos": [463, 14]}, {"full_name": "inv_inj", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [257, 9], "def_end_pos": [257, 16]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q r : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhpqr : 1 / p = 1 / q + 1 / r\nhp_zero : \u00acp = 0\nhq_ne_zero : q \u2260 0\nhr_ne_zero : r \u2260 0\nhq_top : q = \u22a4\n\u22a2 p = r", "state_after": "no goals"}, {"tactic": "have hpq : p = q := by\n  simpa only [hr_top, one_div, ENNReal.inv_top, add_zero, inv_inj] using hpqr", "annotated_tactic": ["have hpq : p = q := by\n      simpa only [hr_top, <a>one_div</a>, <a>ENNReal.inv_top</a>, <a>add_zero</a>, <a>inv_inj</a>] using hpqr", [{"full_name": "one_div", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [318, 9], "def_end_pos": [318, 16]}, {"full_name": "ENNReal.inv_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1374, 17], "def_end_pos": [1374, 24]}, {"full_name": "add_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [469, 3], "def_end_pos": [469, 14]}, {"full_name": "inv_inj", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [257, 9], "def_end_pos": [257, 16]}]], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q r : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhpqr : 1 / p = 1 / q + 1 / r\nhp_zero : \u00acp = 0\nhq_ne_zero : q \u2260 0\nhr_ne_zero : r \u2260 0\nhq_top : \u00acq = \u22a4\nhr_top : r = \u22a4\n\u22a2 snorm (fun x => b (f x) (g x)) p \u03bc \u2264 snorm f q \u03bc * snorm g r \u03bc", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q r : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhpqr : 1 / p = 1 / q + 1 / r\nhp_zero : \u00acp = 0\nhq_ne_zero : q \u2260 0\nhr_ne_zero : r \u2260 0\nhq_top : \u00acq = \u22a4\nhr_top : r = \u22a4\nhpq : p = q\n\u22a2 snorm (fun x => b (f x) (g x)) p \u03bc \u2264 snorm f q \u03bc * snorm g r \u03bc"}, {"tactic": "rw [\u2190 hpq, hr_top]", "annotated_tactic": ["rw [\u2190 hpq, hr_top]", []], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q r : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhpqr : 1 / p = 1 / q + 1 / r\nhp_zero : \u00acp = 0\nhq_ne_zero : q \u2260 0\nhr_ne_zero : r \u2260 0\nhq_top : \u00acq = \u22a4\nhr_top : r = \u22a4\nhpq : p = q\n\u22a2 snorm (fun x => b (f x) (g x)) p \u03bc \u2264 snorm f q \u03bc * snorm g r \u03bc", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q r : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhpqr : 1 / p = 1 / q + 1 / r\nhp_zero : \u00acp = 0\nhq_ne_zero : q \u2260 0\nhr_ne_zero : r \u2260 0\nhq_top : \u00acq = \u22a4\nhr_top : r = \u22a4\nhpq : p = q\n\u22a2 snorm (fun x => b (f x) (g x)) p \u03bc \u2264 snorm f p \u03bc * snorm g \u22a4 \u03bc"}, {"tactic": "exact snorm_le_snorm_mul_snorm_top p hf g b h", "annotated_tactic": ["exact <a>snorm_le_snorm_mul_snorm_top</a> p hf g b h", [{"full_name": "MeasureTheory.snorm_le_snorm_mul_snorm_top", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [1401, 9], "def_end_pos": [1401, 37]}]], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q r : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhpqr : 1 / p = 1 / q + 1 / r\nhp_zero : \u00acp = 0\nhq_ne_zero : q \u2260 0\nhr_ne_zero : r \u2260 0\nhq_top : \u00acq = \u22a4\nhr_top : r = \u22a4\nhpq : p = q\n\u22a2 snorm (fun x => b (f x) (g x)) p \u03bc \u2264 snorm f p \u03bc * snorm g \u22a4 \u03bc", "state_after": "no goals"}, {"tactic": "simpa only [hr_top, one_div, ENNReal.inv_top, add_zero, inv_inj] using hpqr", "annotated_tactic": ["simpa only [hr_top, <a>one_div</a>, <a>ENNReal.inv_top</a>, <a>add_zero</a>, <a>inv_inj</a>] using hpqr", [{"full_name": "one_div", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [318, 9], "def_end_pos": [318, 16]}, {"full_name": "ENNReal.inv_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1374, 17], "def_end_pos": [1374, 24]}, {"full_name": "add_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [469, 3], "def_end_pos": [469, 14]}, {"full_name": "inv_inj", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [257, 9], "def_end_pos": [257, 16]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q r : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhpqr : 1 / p = 1 / q + 1 / r\nhp_zero : \u00acp = 0\nhq_ne_zero : q \u2260 0\nhr_ne_zero : r \u2260 0\nhq_top : \u00acq = \u22a4\nhr_top : r = \u22a4\n\u22a2 p = q", "state_after": "no goals"}, {"tactic": "suffices 1 / q < 1 / p by rwa [one_div, one_div, ENNReal.inv_lt_inv] at this", "annotated_tactic": ["suffices 1 / q < 1 / p by rwa [<a>one_div</a>, <a>one_div</a>, <a>ENNReal.inv_lt_inv</a>] at this", [{"full_name": "one_div", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [318, 9], "def_end_pos": [318, 16]}, {"full_name": "one_div", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [318, 9], "def_end_pos": [318, 16]}, {"full_name": "ENNReal.inv_lt_inv", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1528, 19], "def_end_pos": [1528, 29]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q r : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhpqr : 1 / p = 1 / q + 1 / r\nhp_zero : \u00acp = 0\nhq_ne_zero : q \u2260 0\nhr_ne_zero : r \u2260 0\nhq_top : \u00acq = \u22a4\nhr_top : \u00acr = \u22a4\n\u22a2 p < q", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q r : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhpqr : 1 / p = 1 / q + 1 / r\nhp_zero : \u00acp = 0\nhq_ne_zero : q \u2260 0\nhr_ne_zero : r \u2260 0\nhq_top : \u00acq = \u22a4\nhr_top : \u00acr = \u22a4\n\u22a2 1 / q < 1 / p"}, {"tactic": "rw [hpqr]", "annotated_tactic": ["rw [hpqr]", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q r : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhpqr : 1 / p = 1 / q + 1 / r\nhp_zero : \u00acp = 0\nhq_ne_zero : q \u2260 0\nhr_ne_zero : r \u2260 0\nhq_top : \u00acq = \u22a4\nhr_top : \u00acr = \u22a4\n\u22a2 1 / q < 1 / p", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q r : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhpqr : 1 / p = 1 / q + 1 / r\nhp_zero : \u00acp = 0\nhq_ne_zero : q \u2260 0\nhr_ne_zero : r \u2260 0\nhq_top : \u00acq = \u22a4\nhr_top : \u00acr = \u22a4\n\u22a2 1 / q < 1 / q + 1 / r"}, {"tactic": "refine' ENNReal.lt_add_right _ _", "annotated_tactic": ["refine' <a>ENNReal.lt_add_right</a> _ _", [{"full_name": "ENNReal.lt_add_right", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [829, 9], "def_end_pos": [829, 21]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q r : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhpqr : 1 / p = 1 / q + 1 / r\nhp_zero : \u00acp = 0\nhq_ne_zero : q \u2260 0\nhr_ne_zero : r \u2260 0\nhq_top : \u00acq = \u22a4\nhr_top : \u00acr = \u22a4\n\u22a2 1 / q < 1 / q + 1 / r", "state_after": "case refine'_1\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q r : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhpqr : 1 / p = 1 / q + 1 / r\nhp_zero : \u00acp = 0\nhq_ne_zero : q \u2260 0\nhr_ne_zero : r \u2260 0\nhq_top : \u00acq = \u22a4\nhr_top : \u00acr = \u22a4\n\u22a2 1 / q \u2260 \u22a4\n\ncase refine'_2\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q r : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhpqr : 1 / p = 1 / q + 1 / r\nhp_zero : \u00acp = 0\nhq_ne_zero : q \u2260 0\nhr_ne_zero : r \u2260 0\nhq_top : \u00acq = \u22a4\nhr_top : \u00acr = \u22a4\n\u22a2 1 / r \u2260 0"}, {"tactic": "rwa [one_div, one_div, ENNReal.inv_lt_inv] at this", "annotated_tactic": ["rwa [<a>one_div</a>, <a>one_div</a>, <a>ENNReal.inv_lt_inv</a>] at this", [{"full_name": "one_div", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [318, 9], "def_end_pos": [318, 16]}, {"full_name": "one_div", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [318, 9], "def_end_pos": [318, 16]}, {"full_name": "ENNReal.inv_lt_inv", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1528, 19], "def_end_pos": [1528, 29]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q r : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhpqr : 1 / p = 1 / q + 1 / r\nhp_zero : \u00acp = 0\nhq_ne_zero : q \u2260 0\nhr_ne_zero : r \u2260 0\nhq_top : \u00acq = \u22a4\nhr_top : \u00acr = \u22a4\nthis : 1 / q < 1 / p\n\u22a2 p < q", "state_after": "no goals"}, {"tactic": "simp only [hq_ne_zero, one_div, Ne.def, ENNReal.inv_eq_top, not_false_iff]", "annotated_tactic": ["simp only [hq_ne_zero, <a>one_div</a>, <a>Ne.def</a>, <a>ENNReal.inv_eq_top</a>, <a>not_false_iff</a>]", [{"full_name": "one_div", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [318, 9], "def_end_pos": [318, 16]}, {"full_name": "Ne.def", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [59, 9], "def_end_pos": [59, 15]}, {"full_name": "ENNReal.inv_eq_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1449, 17], "def_end_pos": [1449, 27]}, {"full_name": "not_false_iff", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [82, 9], "def_end_pos": [82, 22]}]], "state_before": "case refine'_1\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q r : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhpqr : 1 / p = 1 / q + 1 / r\nhp_zero : \u00acp = 0\nhq_ne_zero : q \u2260 0\nhr_ne_zero : r \u2260 0\nhq_top : \u00acq = \u22a4\nhr_top : \u00acr = \u22a4\n\u22a2 1 / q \u2260 \u22a4", "state_after": "no goals"}, {"tactic": "simp only [hr_top, one_div, Ne.def, ENNReal.inv_eq_zero, not_false_iff]", "annotated_tactic": ["simp only [hr_top, <a>one_div</a>, <a>Ne.def</a>, <a>ENNReal.inv_eq_zero</a>, <a>not_false_iff</a>]", [{"full_name": "one_div", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [318, 9], "def_end_pos": [318, 16]}, {"full_name": "Ne.def", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [59, 9], "def_end_pos": [59, 15]}, {"full_name": "ENNReal.inv_eq_zero", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1465, 19], "def_end_pos": [1465, 30]}, {"full_name": "not_false_iff", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [82, 9], "def_end_pos": [82, 22]}]], "state_before": "case refine'_2\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q r : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhpqr : 1 / p = 1 / q + 1 / r\nhp_zero : \u00acp = 0\nhq_ne_zero : q \u2260 0\nhr_ne_zero : r \u2260 0\nhq_top : \u00acq = \u22a4\nhr_top : \u00acr = \u22a4\n\u22a2 1 / r \u2260 0", "state_after": "no goals"}, {"tactic": "exact ENNReal.toReal_pos hp_zero (hpq.trans_le le_top).ne", "annotated_tactic": ["exact <a>ENNReal.toReal_pos</a> hp_zero (hpq.trans_le <a>le_top</a>).<a>ne</a>", [{"full_name": "ENNReal.toReal_pos", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2131, 9], "def_end_pos": [2131, 19]}, {"full_name": "le_top", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [98, 9], "def_end_pos": [98, 15]}, {"full_name": "LT.lt.ne", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [152, 7], "def_end_pos": [152, 15]}]], "state_before": "case neg.refine'_1\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q r : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhpqr : 1 / p = 1 / q + 1 / r\nhp_zero : \u00acp = 0\nhq_ne_zero : q \u2260 0\nhr_ne_zero : r \u2260 0\nhq_top : \u00acq = \u22a4\nhr_top : \u00acr = \u22a4\nhpq : p < q\n\u22a2 0 < ENNReal.toReal p", "state_after": "no goals"}, {"tactic": "exact ENNReal.toReal_strict_mono hq_top hpq", "annotated_tactic": ["exact <a>ENNReal.toReal_strict_mono</a> hq_top hpq", [{"full_name": "ENNReal.toReal_strict_mono", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2059, 9], "def_end_pos": [2059, 27]}]], "state_before": "case neg.refine'_2\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q r : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhpqr : 1 / p = 1 / q + 1 / r\nhp_zero : \u00acp = 0\nhq_ne_zero : q \u2260 0\nhr_ne_zero : r \u2260 0\nhq_top : \u00acq = \u22a4\nhr_top : \u00acr = \u22a4\nhpq : p < q\n\u22a2 ENNReal.toReal p < ENNReal.toReal q", "state_after": "no goals"}, {"tactic": "simp only [hq_ne_zero, one_div, Ne.def, ENNReal.inv_eq_top, not_false_iff]", "annotated_tactic": ["simp only [hq_ne_zero, <a>one_div</a>, <a>Ne.def</a>, <a>ENNReal.inv_eq_top</a>, <a>not_false_iff</a>]", [{"full_name": "one_div", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [318, 9], "def_end_pos": [318, 16]}, {"full_name": "Ne.def", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [59, 9], "def_end_pos": [59, 15]}, {"full_name": "ENNReal.inv_eq_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1449, 17], "def_end_pos": [1449, 27]}, {"full_name": "not_false_iff", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [82, 9], "def_end_pos": [82, 22]}]], "state_before": "case neg.refine'_3.ha\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q r : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhpqr : 1 / p = 1 / q + 1 / r\nhp_zero : \u00acp = 0\nhq_ne_zero : q \u2260 0\nhr_ne_zero : r \u2260 0\nhq_top : \u00acq = \u22a4\nhr_top : \u00acr = \u22a4\nhpq : p < q\n\u22a2 1 / q \u2260 \u22a4", "state_after": "no goals"}, {"tactic": "simp only [hr_ne_zero, one_div, Ne.def, ENNReal.inv_eq_top, not_false_iff]", "annotated_tactic": ["simp only [hr_ne_zero, <a>one_div</a>, <a>Ne.def</a>, <a>ENNReal.inv_eq_top</a>, <a>not_false_iff</a>]", [{"full_name": "one_div", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [318, 9], "def_end_pos": [318, 16]}, {"full_name": "Ne.def", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [59, 9], "def_end_pos": [59, 15]}, {"full_name": "ENNReal.inv_eq_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1449, 17], "def_end_pos": [1449, 27]}, {"full_name": "not_false_iff", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [82, 9], "def_end_pos": [82, 22]}]], "state_before": "case neg.refine'_3.hb\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q r : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhpqr : 1 / p = 1 / q + 1 / r\nhp_zero : \u00acp = 0\nhq_ne_zero : q \u2260 0\nhr_ne_zero : r \u2260 0\nhq_top : \u00acq = \u22a4\nhr_top : \u00acr = \u22a4\nhpq : p < q\n\u22a2 1 / r \u2260 \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean", "full_name": "Real.volume_emetric_closedBall", "start": [127, 1], "end": [132, 59], "traced_tactics": [{"tactic": "rcases eq_or_ne r \u221e with (rfl | hr)", "annotated_tactic": ["rcases <a>eq_or_ne</a> r \u221e with (rfl | hr)", [{"full_name": "eq_or_ne", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [209, 9], "def_end_pos": [209, 17]}]], "state_before": "\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\na : \u211d\nr : \u211d\u22650\u221e\n\u22a2 \u2191\u2191volume (EMetric.closedBall a r) = 2 * r", "state_after": "case inl\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\na : \u211d\n\u22a2 \u2191\u2191volume (EMetric.closedBall a \u22a4) = 2 * \u22a4\n\ncase inr\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\na : \u211d\nr : \u211d\u22650\u221e\nhr : r \u2260 \u22a4\n\u22a2 \u2191\u2191volume (EMetric.closedBall a r) = 2 * r"}, {"tactic": "rw [EMetric.closedBall_top, volume_univ, two_mul, _root_.top_add]", "annotated_tactic": ["rw [<a>EMetric.closedBall_top</a>, <a>volume_univ</a>, <a>two_mul</a>, <a>_root_.top_add</a>]", [{"full_name": "EMetric.closedBall_top", "def_path": "Mathlib/Topology/EMetricSpace/Basic.lean", "def_pos": [550, 9], "def_end_pos": [550, 23]}, {"full_name": "Real.volume_univ", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean", "def_pos": [100, 9], "def_end_pos": [100, 20]}, {"full_name": "two_mul", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [177, 9], "def_end_pos": [177, 16]}, {"full_name": "top_add", "def_path": "Mathlib/Algebra/Order/Monoid/Defs.lean", "def_pos": [105, 9], "def_end_pos": [105, 16]}]], "state_before": "case inl\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\na : \u211d\n\u22a2 \u2191\u2191volume (EMetric.closedBall a \u22a4) = 2 * \u22a4", "state_after": "no goals"}, {"tactic": "lift r to \u211d\u22650 using hr", "annotated_tactic": ["lift r to \u211d\u22650 using hr", []], "state_before": "case inr\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\na : \u211d\nr : \u211d\u22650\u221e\nhr : r \u2260 \u22a4\n\u22a2 \u2191\u2191volume (EMetric.closedBall a r) = 2 * r", "state_after": "case inr.intro\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\na : \u211d\nr : \u211d\u22650\n\u22a2 \u2191\u2191volume (EMetric.closedBall a \u2191r) = 2 * \u2191r"}, {"tactic": "rw [Metric.emetric_closedBall_nnreal, volume_closedBall, two_mul, \u2190 NNReal.coe_add,\n  ENNReal.ofReal_coe_nnreal, ENNReal.coe_add, two_mul]", "annotated_tactic": ["rw [<a>Metric.emetric_closedBall_nnreal</a>, <a>volume_closedBall</a>, <a>two_mul</a>, \u2190 <a>NNReal.coe_add</a>,\n      <a>ENNReal.ofReal_coe_nnreal</a>, <a>ENNReal.coe_add</a>, <a>two_mul</a>]", [{"full_name": "Metric.emetric_closedBall_nnreal", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [1227, 9], "def_end_pos": [1227, 41]}, {"full_name": "Real.volume_closedBall", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean", "def_pos": [113, 9], "def_end_pos": [113, 26]}, {"full_name": "two_mul", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [177, 9], "def_end_pos": [177, 16]}, {"full_name": "NNReal.coe_add", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [181, 19], "def_end_pos": [181, 26]}, {"full_name": "ENNReal.ofReal_coe_nnreal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [212, 17], "def_end_pos": [212, 34]}, {"full_name": "ENNReal.coe_add", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [386, 28], "def_end_pos": [386, 35]}, {"full_name": "two_mul", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [177, 9], "def_end_pos": [177, 16]}]], "state_before": "case inr.intro\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\na : \u211d\nr : \u211d\u22650\n\u22a2 \u2191\u2191volume (EMetric.closedBall a \u2191r) = 2 * \u2191r", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Num/Lemmas.lean", "full_name": "Num.castNum_shiftRight", "start": [962, 1], "end": [984, 68], "traced_tactics": [{"tactic": "cases' m with m <;> dsimp only [\u2190shiftr_eq_shiftRight, shiftr]", "annotated_tactic": ["cases' m with m <;> dsimp only [\u2190<a>shiftr_eq_shiftRight</a>, <a>shiftr</a>]", [{"full_name": "Num.shiftr_eq_shiftRight", "def_path": "Mathlib/Data/Num/Bitwise.lean", "def_pos": [209, 15], "def_end_pos": [209, 35]}, {"full_name": "Num.shiftr", "def_path": "Mathlib/Data/Num/Bitwise.lean", "def_pos": [202, 5], "def_end_pos": [202, 11]}]], "state_before": "\u03b1 : Type u_1\nm : Num\nn : \u2115\n\u22a2 \u2191(m >>> n) = \u2191m >>> n", "state_after": "case zero\n\u03b1 : Type u_1\nn : \u2115\n\u22a2 \u21910 = \u2191zero >>> n\n\ncase pos\n\u03b1 : Type u_1\nn : \u2115\nm : PosNum\n\u22a2 \u2191(m >>> n) = \u2191(pos m) >>> n"}, {"tactic": "induction' n with n IH generalizing m", "annotated_tactic": ["induction' n with n IH generalizing m", []], "state_before": "case pos\n\u03b1 : Type u_1\nn : \u2115\nm : PosNum\n\u22a2 \u2191(m >>> n) = \u2191(pos m) >>> n", "state_after": "case pos.zero\n\u03b1 : Type u_1\nm\u271d m : PosNum\n\u22a2 \u2191(m >>> Nat.zero) = \u2191(pos m) >>> Nat.zero\n\ncase pos.succ\n\u03b1 : Type u_1\nm\u271d : PosNum\nn : \u2115\nIH : \u2200 (m : PosNum), \u2191(m >>> n) = \u2191(pos m) >>> n\nm : PosNum\n\u22a2 \u2191(m >>> Nat.succ n) = \u2191(pos m) >>> Nat.succ n"}, {"tactic": "cases' m with m m <;> dsimp only [PosNum.shiftr, \u2190PosNum.shiftr_eq_shiftRight]", "annotated_tactic": ["cases' m with m m <;> dsimp only [<a>PosNum.shiftr</a>, \u2190<a>PosNum.shiftr_eq_shiftRight</a>]", [{"full_name": "PosNum.shiftr", "def_path": "Mathlib/Data/Num/Bitwise.lean", "def_pos": [129, 5], "def_end_pos": [129, 11]}, {"full_name": "PosNum.shiftr_eq_shiftRight", "def_path": "Mathlib/Data/Num/Bitwise.lean", "def_pos": [138, 15], "def_end_pos": [138, 35]}]], "state_before": "case pos.succ\n\u03b1 : Type u_1\nm\u271d : PosNum\nn : \u2115\nIH : \u2200 (m : PosNum), \u2191(m >>> n) = \u2191(pos m) >>> n\nm : PosNum\n\u22a2 \u2191(m >>> Nat.succ n) = \u2191(pos m) >>> Nat.succ n", "state_after": "case pos.succ.one\n\u03b1 : Type u_1\nm : PosNum\nn : \u2115\nIH : \u2200 (m : PosNum), \u2191(m >>> n) = \u2191(pos m) >>> n\n\u22a2 \u21910 = \u2191(pos one) >>> Nat.succ n\n\ncase pos.succ.bit1\n\u03b1 : Type u_1\nm\u271d : PosNum\nn : \u2115\nIH : \u2200 (m : PosNum), \u2191(m >>> n) = \u2191(pos m) >>> n\nm : PosNum\n\u22a2 \u2191(PosNum.shiftr m n) = \u2191(pos (PosNum.bit1 m)) >>> Nat.succ n\n\ncase pos.succ.bit0\n\u03b1 : Type u_1\nm\u271d : PosNum\nn : \u2115\nIH : \u2200 (m : PosNum), \u2191(m >>> n) = \u2191(pos m) >>> n\nm : PosNum\n\u22a2 \u2191(PosNum.shiftr m n) = \u2191(pos (PosNum.bit0 m)) >>> Nat.succ n"}, {"tactic": "symm", "annotated_tactic": ["symm", []], "state_before": "case zero\n\u03b1 : Type u_1\nn : \u2115\n\u22a2 \u21910 = \u2191zero >>> n", "state_after": "case zero\n\u03b1 : Type u_1\nn : \u2115\n\u22a2 \u2191zero >>> n = \u21910"}, {"tactic": "apply Nat.zero_shiftRight", "annotated_tactic": ["apply <a>Nat.zero_shiftRight</a>", [{"full_name": "Nat.zero_shiftRight", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [981, 17], "def_end_pos": [981, 32]}]], "state_before": "case zero\n\u03b1 : Type u_1\nn : \u2115\n\u22a2 \u2191zero >>> n = \u21910", "state_after": "no goals"}, {"tactic": "cases m <;> rfl", "annotated_tactic": ["cases m <;> rfl", []], "state_before": "case pos.zero\n\u03b1 : Type u_1\nm\u271d m : PosNum\n\u22a2 \u2191(m >>> Nat.zero) = \u2191(pos m) >>> Nat.zero", "state_after": "no goals"}, {"tactic": "rw [Nat.shiftRight_eq_div_pow]", "annotated_tactic": ["rw [<a>Nat.shiftRight_eq_div_pow</a>]", [{"full_name": "Nat.shiftRight_eq_div_pow", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [989, 9], "def_end_pos": [989, 30]}]], "state_before": "case pos.succ.one\n\u03b1 : Type u_1\nm : PosNum\nn : \u2115\nIH : \u2200 (m : PosNum), \u2191(m >>> n) = \u2191(pos m) >>> n\n\u22a2 \u21910 = \u2191(pos one) >>> Nat.succ n", "state_after": "case pos.succ.one\n\u03b1 : Type u_1\nm : PosNum\nn : \u2115\nIH : \u2200 (m : PosNum), \u2191(m >>> n) = \u2191(pos m) >>> n\n\u22a2 \u21910 = \u2191(pos one) / 2 ^ Nat.succ n"}, {"tactic": "symm", "annotated_tactic": ["symm", []], "state_before": "case pos.succ.one\n\u03b1 : Type u_1\nm : PosNum\nn : \u2115\nIH : \u2200 (m : PosNum), \u2191(m >>> n) = \u2191(pos m) >>> n\n\u22a2 \u21910 = \u2191(pos one) / 2 ^ Nat.succ n", "state_after": "case pos.succ.one\n\u03b1 : Type u_1\nm : PosNum\nn : \u2115\nIH : \u2200 (m : PosNum), \u2191(m >>> n) = \u2191(pos m) >>> n\n\u22a2 \u2191(pos one) / 2 ^ Nat.succ n = \u21910"}, {"tactic": "apply Nat.div_eq_of_lt", "annotated_tactic": ["apply <a>Nat.div_eq_of_lt</a>", [{"full_name": "Nat.div_eq_of_lt", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [590, 9], "def_end_pos": [590, 21]}]], "state_before": "case pos.succ.one\n\u03b1 : Type u_1\nm : PosNum\nn : \u2115\nIH : \u2200 (m : PosNum), \u2191(m >>> n) = \u2191(pos m) >>> n\n\u22a2 \u2191(pos one) / 2 ^ Nat.succ n = \u21910", "state_after": "case pos.succ.one.h\u2080\n\u03b1 : Type u_1\nm : PosNum\nn : \u2115\nIH : \u2200 (m : PosNum), \u2191(m >>> n) = \u2191(pos m) >>> n\n\u22a2 \u2191(pos one) < 2 ^ Nat.succ n"}, {"tactic": "simp [@Nat.pow_lt_pow_of_lt_right 2 (by decide) 0 (n + 1) (Nat.succ_pos _)]", "annotated_tactic": ["simp [@<a>Nat.pow_lt_pow_of_lt_right</a> 2 (by decide) 0 (n + 1) (<a>Nat.succ_pos</a> _)]", [{"full_name": "Nat.pow_lt_pow_of_lt_right", "def_path": "Mathlib/Data/Nat/Pow.lean", "def_pos": [31, 9], "def_end_pos": [31, 31]}, {"full_name": "Nat.succ_pos", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1608, 9], "def_end_pos": [1608, 21]}]], "state_before": "case pos.succ.one.h\u2080\n\u03b1 : Type u_1\nm : PosNum\nn : \u2115\nIH : \u2200 (m : PosNum), \u2191(m >>> n) = \u2191(pos m) >>> n\n\u22a2 \u2191(pos one) < 2 ^ Nat.succ n", "state_after": "no goals"}, {"tactic": "decide", "annotated_tactic": ["decide", []], "state_before": "\u03b1 : Type u_1\nm : PosNum\nn : \u2115\nIH : \u2200 (m : PosNum), \u2191(m >>> n) = \u2191(pos m) >>> n\n\u22a2 1 < 2", "state_after": "no goals"}, {"tactic": "trans", "annotated_tactic": ["trans", []], "state_before": "case pos.succ.bit1\n\u03b1 : Type u_1\nm\u271d : PosNum\nn : \u2115\nIH : \u2200 (m : PosNum), \u2191(m >>> n) = \u2191(pos m) >>> n\nm : PosNum\n\u22a2 \u2191(PosNum.shiftr m n) = \u2191(pos (PosNum.bit1 m)) >>> Nat.succ n", "state_after": "\u03b1 : Type u_1\nm\u271d : PosNum\nn : \u2115\nIH : \u2200 (m : PosNum), \u2191(m >>> n) = \u2191(pos m) >>> n\nm : PosNum\n\u22a2 \u2191(PosNum.shiftr m n) = ?m.631939\n\n\u03b1 : Type u_1\nm\u271d : PosNum\nn : \u2115\nIH : \u2200 (m : PosNum), \u2191(m >>> n) = \u2191(pos m) >>> n\nm : PosNum\n\u22a2 ?m.631939 = \u2191(pos (PosNum.bit1 m)) >>> Nat.succ n\n\n\u03b1 : Type u_1\nm\u271d : PosNum\nn : \u2115\nIH : \u2200 (m : PosNum), \u2191(m >>> n) = \u2191(pos m) >>> n\nm : PosNum\n\u22a2 \u2115"}, {"tactic": "apply IH", "annotated_tactic": ["apply IH", []], "state_before": "\u03b1 : Type u_1\nm\u271d : PosNum\nn : \u2115\nIH : \u2200 (m : PosNum), \u2191(m >>> n) = \u2191(pos m) >>> n\nm : PosNum\n\u22a2 \u2191(PosNum.shiftr m n) = ?m.631939\n\n\u03b1 : Type u_1\nm\u271d : PosNum\nn : \u2115\nIH : \u2200 (m : PosNum), \u2191(m >>> n) = \u2191(pos m) >>> n\nm : PosNum\n\u22a2 ?m.631939 = \u2191(pos (PosNum.bit1 m)) >>> Nat.succ n\n\n\u03b1 : Type u_1\nm\u271d : PosNum\nn : \u2115\nIH : \u2200 (m : PosNum), \u2191(m >>> n) = \u2191(pos m) >>> n\nm : PosNum\n\u22a2 \u2115", "state_after": "\u03b1 : Type u_1\nm\u271d : PosNum\nn : \u2115\nIH : \u2200 (m : PosNum), \u2191(m >>> n) = \u2191(pos m) >>> n\nm : PosNum\n\u22a2 \u2191(pos m) >>> n = \u2191(pos (PosNum.bit1 m)) >>> Nat.succ n"}, {"tactic": "change Nat.shiftRight m n = Nat.shiftRight (_root_.bit1 m) (n + 1)", "annotated_tactic": ["change <a>Nat.shiftRight</a> m n = <a>Nat.shiftRight</a> (<a>_root_.bit1</a> m) (n + 1)", [{"full_name": "Nat.shiftRight", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Bitwise.lean", "def_pos": [44, 5], "def_end_pos": [44, 15]}, {"full_name": "Nat.shiftRight", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Bitwise.lean", "def_pos": [44, 5], "def_end_pos": [44, 15]}, {"full_name": "bit1", "def_path": "Mathlib/Init/ZeroOne.lean", "def_pos": [39, 34], "def_end_pos": [39, 38]}]], "state_before": "\u03b1 : Type u_1\nm\u271d : PosNum\nn : \u2115\nIH : \u2200 (m : PosNum), \u2191(m >>> n) = \u2191(pos m) >>> n\nm : PosNum\n\u22a2 \u2191(pos m) >>> n = \u2191(pos (PosNum.bit1 m)) >>> Nat.succ n", "state_after": "\u03b1 : Type u_1\nm\u271d : PosNum\nn : \u2115\nIH : \u2200 (m : PosNum), \u2191(m >>> n) = \u2191(pos m) >>> n\nm : PosNum\n\u22a2 Nat.shiftRight (\u2191m) n = Nat.shiftRight (_root_.bit1 \u2191m) (n + 1)"}, {"tactic": "rw [add_comm n 1, @Nat.shiftRight_eq _ (1 + n), Nat.shiftRight_add]", "annotated_tactic": ["rw [<a>add_comm</a> n 1, @<a>Nat.shiftRight_eq</a> _ (1 + n), <a>Nat.shiftRight_add</a>]", [{"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [301, 3], "def_end_pos": [301, 14]}, {"full_name": "Nat.shiftRight_eq", "def_path": "Mathlib/Init/Data/Nat/Bitwise.lean", "def_pos": [204, 7], "def_end_pos": [204, 20]}, {"full_name": "Nat.shiftRight_add", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [985, 9], "def_end_pos": [985, 23]}]], "state_before": "\u03b1 : Type u_1\nm\u271d : PosNum\nn : \u2115\nIH : \u2200 (m : PosNum), \u2191(m >>> n) = \u2191(pos m) >>> n\nm : PosNum\n\u22a2 Nat.shiftRight (\u2191m) n = Nat.shiftRight (_root_.bit1 \u2191m) (n + 1)", "state_after": "\u03b1 : Type u_1\nm\u271d : PosNum\nn : \u2115\nIH : \u2200 (m : PosNum), \u2191(m >>> n) = \u2191(pos m) >>> n\nm : PosNum\n\u22a2 Nat.shiftRight (\u2191m) n = _root_.bit1 \u2191m >>> 1 >>> n"}, {"tactic": "apply congr_arg fun x => Nat.shiftRight x n", "annotated_tactic": ["apply <a>congr_arg</a> fun x => <a>Nat.shiftRight</a> x n", [{"full_name": "congr_arg", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [43, 7], "def_end_pos": [43, 16]}, {"full_name": "Nat.shiftRight", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Bitwise.lean", "def_pos": [44, 5], "def_end_pos": [44, 15]}]], "state_before": "\u03b1 : Type u_1\nm\u271d : PosNum\nn : \u2115\nIH : \u2200 (m : PosNum), \u2191(m >>> n) = \u2191(pos m) >>> n\nm : PosNum\n\u22a2 Nat.shiftRight (\u2191m) n = _root_.bit1 \u2191m >>> 1 >>> n", "state_after": "\u03b1 : Type u_1\nm\u271d : PosNum\nn : \u2115\nIH : \u2200 (m : PosNum), \u2191(m >>> n) = \u2191(pos m) >>> n\nm : PosNum\n\u22a2 \u2191m = _root_.bit1 \u2191m >>> 1"}, {"tactic": "simp [Nat.shiftRight_succ, Nat.shiftRight_zero, \u2190 Nat.div2_val]", "annotated_tactic": ["simp [<a>Nat.shiftRight_succ</a>, <a>Nat.shiftRight_zero</a>, \u2190 <a>Nat.div2_val</a>]", [{"full_name": "Nat.shiftRight_succ", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [979, 17], "def_end_pos": [979, 32]}, {"full_name": "Nat.shiftRight_zero", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [977, 17], "def_end_pos": [977, 32]}, {"full_name": "Nat.div2_val", "def_path": "Mathlib/Init/Data/Nat/Bitwise.lean", "def_pos": [139, 9], "def_end_pos": [139, 17]}]], "state_before": "\u03b1 : Type u_1\nm\u271d : PosNum\nn : \u2115\nIH : \u2200 (m : PosNum), \u2191(m >>> n) = \u2191(pos m) >>> n\nm : PosNum\n\u22a2 \u2191m = _root_.bit1 \u2191m >>> 1", "state_after": "no goals"}, {"tactic": "trans", "annotated_tactic": ["trans", []], "state_before": "case pos.succ.bit0\n\u03b1 : Type u_1\nm\u271d : PosNum\nn : \u2115\nIH : \u2200 (m : PosNum), \u2191(m >>> n) = \u2191(pos m) >>> n\nm : PosNum\n\u22a2 \u2191(PosNum.shiftr m n) = \u2191(pos (PosNum.bit0 m)) >>> Nat.succ n", "state_after": "\u03b1 : Type u_1\nm\u271d : PosNum\nn : \u2115\nIH : \u2200 (m : PosNum), \u2191(m >>> n) = \u2191(pos m) >>> n\nm : PosNum\n\u22a2 \u2191(PosNum.shiftr m n) = ?m.633935\n\n\u03b1 : Type u_1\nm\u271d : PosNum\nn : \u2115\nIH : \u2200 (m : PosNum), \u2191(m >>> n) = \u2191(pos m) >>> n\nm : PosNum\n\u22a2 ?m.633935 = \u2191(pos (PosNum.bit0 m)) >>> Nat.succ n\n\n\u03b1 : Type u_1\nm\u271d : PosNum\nn : \u2115\nIH : \u2200 (m : PosNum), \u2191(m >>> n) = \u2191(pos m) >>> n\nm : PosNum\n\u22a2 \u2115"}, {"tactic": "apply IH", "annotated_tactic": ["apply IH", []], "state_before": "\u03b1 : Type u_1\nm\u271d : PosNum\nn : \u2115\nIH : \u2200 (m : PosNum), \u2191(m >>> n) = \u2191(pos m) >>> n\nm : PosNum\n\u22a2 \u2191(PosNum.shiftr m n) = ?m.633935\n\n\u03b1 : Type u_1\nm\u271d : PosNum\nn : \u2115\nIH : \u2200 (m : PosNum), \u2191(m >>> n) = \u2191(pos m) >>> n\nm : PosNum\n\u22a2 ?m.633935 = \u2191(pos (PosNum.bit0 m)) >>> Nat.succ n\n\n\u03b1 : Type u_1\nm\u271d : PosNum\nn : \u2115\nIH : \u2200 (m : PosNum), \u2191(m >>> n) = \u2191(pos m) >>> n\nm : PosNum\n\u22a2 \u2115", "state_after": "\u03b1 : Type u_1\nm\u271d : PosNum\nn : \u2115\nIH : \u2200 (m : PosNum), \u2191(m >>> n) = \u2191(pos m) >>> n\nm : PosNum\n\u22a2 \u2191(pos m) >>> n = \u2191(pos (PosNum.bit0 m)) >>> Nat.succ n"}, {"tactic": "change Nat.shiftRight m n = Nat.shiftRight (_root_.bit0 m) (n + 1)", "annotated_tactic": ["change <a>Nat.shiftRight</a> m n = <a>Nat.shiftRight</a> (<a>_root_.bit0</a> m) (n + 1)", [{"full_name": "Nat.shiftRight", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Bitwise.lean", "def_pos": [44, 5], "def_end_pos": [44, 15]}, {"full_name": "Nat.shiftRight", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Bitwise.lean", "def_pos": [44, 5], "def_end_pos": [44, 15]}, {"full_name": "bit0", "def_path": "Mathlib/Init/ZeroOne.lean", "def_pos": [36, 34], "def_end_pos": [36, 38]}]], "state_before": "\u03b1 : Type u_1\nm\u271d : PosNum\nn : \u2115\nIH : \u2200 (m : PosNum), \u2191(m >>> n) = \u2191(pos m) >>> n\nm : PosNum\n\u22a2 \u2191(pos m) >>> n = \u2191(pos (PosNum.bit0 m)) >>> Nat.succ n", "state_after": "\u03b1 : Type u_1\nm\u271d : PosNum\nn : \u2115\nIH : \u2200 (m : PosNum), \u2191(m >>> n) = \u2191(pos m) >>> n\nm : PosNum\n\u22a2 Nat.shiftRight (\u2191m) n = Nat.shiftRight (_root_.bit0 \u2191m) (n + 1)"}, {"tactic": "rw [add_comm n 1,  @Nat.shiftRight_eq _ (1 + n), Nat.shiftRight_add]", "annotated_tactic": ["rw [<a>add_comm</a> n 1,  @<a>Nat.shiftRight_eq</a> _ (1 + n), <a>Nat.shiftRight_add</a>]", [{"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [301, 3], "def_end_pos": [301, 14]}, {"full_name": "Nat.shiftRight_eq", "def_path": "Mathlib/Init/Data/Nat/Bitwise.lean", "def_pos": [204, 7], "def_end_pos": [204, 20]}, {"full_name": "Nat.shiftRight_add", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [985, 9], "def_end_pos": [985, 23]}]], "state_before": "\u03b1 : Type u_1\nm\u271d : PosNum\nn : \u2115\nIH : \u2200 (m : PosNum), \u2191(m >>> n) = \u2191(pos m) >>> n\nm : PosNum\n\u22a2 Nat.shiftRight (\u2191m) n = Nat.shiftRight (_root_.bit0 \u2191m) (n + 1)", "state_after": "\u03b1 : Type u_1\nm\u271d : PosNum\nn : \u2115\nIH : \u2200 (m : PosNum), \u2191(m >>> n) = \u2191(pos m) >>> n\nm : PosNum\n\u22a2 Nat.shiftRight (\u2191m) n = _root_.bit0 \u2191m >>> 1 >>> n"}, {"tactic": "apply congr_arg fun x => Nat.shiftRight x n", "annotated_tactic": ["apply <a>congr_arg</a> fun x => <a>Nat.shiftRight</a> x n", [{"full_name": "congr_arg", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [43, 7], "def_end_pos": [43, 16]}, {"full_name": "Nat.shiftRight", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Bitwise.lean", "def_pos": [44, 5], "def_end_pos": [44, 15]}]], "state_before": "\u03b1 : Type u_1\nm\u271d : PosNum\nn : \u2115\nIH : \u2200 (m : PosNum), \u2191(m >>> n) = \u2191(pos m) >>> n\nm : PosNum\n\u22a2 Nat.shiftRight (\u2191m) n = _root_.bit0 \u2191m >>> 1 >>> n", "state_after": "\u03b1 : Type u_1\nm\u271d : PosNum\nn : \u2115\nIH : \u2200 (m : PosNum), \u2191(m >>> n) = \u2191(pos m) >>> n\nm : PosNum\n\u22a2 \u2191m = _root_.bit0 \u2191m >>> 1"}, {"tactic": "simp [Nat.shiftRight_succ, Nat.shiftRight_zero, \u2190 Nat.div2_val]", "annotated_tactic": ["simp [<a>Nat.shiftRight_succ</a>, <a>Nat.shiftRight_zero</a>, \u2190 <a>Nat.div2_val</a>]", [{"full_name": "Nat.shiftRight_succ", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [979, 17], "def_end_pos": [979, 32]}, {"full_name": "Nat.shiftRight_zero", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [977, 17], "def_end_pos": [977, 32]}, {"full_name": "Nat.div2_val", "def_path": "Mathlib/Init/Data/Nat/Bitwise.lean", "def_pos": [139, 9], "def_end_pos": [139, 17]}]], "state_before": "\u03b1 : Type u_1\nm\u271d : PosNum\nn : \u2115\nIH : \u2200 (m : PosNum), \u2191(m >>> n) = \u2191(pos m) >>> n\nm : PosNum\n\u22a2 \u2191m = _root_.bit0 \u2191m >>> 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "full_name": "measurable_biSup", "start": [1412, 1], "end": [1424, 51], "traced_tactics": [{"tactic": "haveI : Encodable s := hs.toEncodable", "annotated_tactic": ["haveI : <a>Encodable</a> s := hs.toEncodable", [{"full_name": "Encodable", "def_path": "Mathlib/Logic/Encodable/Basic.lean", "def_pos": [45, 7], "def_end_pos": [45, 16]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9\u271d : Sort y\ns\u271d t u : Set \u03b1\ninst\u271d\u00b9\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u2070 : BorelSpace \u03b1\ninst\u271d\u2079 : TopologicalSpace \u03b2\ninst\u271d\u2078 : MeasurableSpace \u03b2\ninst\u271d\u2077 : BorelSpace \u03b2\ninst\u271d\u2076 : TopologicalSpace \u03b3\ninst\u271d\u2075 : MeasurableSpace \u03b3\ninst\u271d\u2074 : BorelSpace \u03b3\ninst\u271d\u00b3 : MeasurableSpace \u03b4\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : SecondCountableTopology \u03b1\n\u03b9 : Type u_6\ns : Set \u03b9\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nhs : Set.Countable s\nhf : \u2200 (i : \u03b9), i \u2208 s \u2192 Measurable (f i)\n\u22a2 Measurable fun b => \u2a06 i \u2208 s, f i b", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9\u271d : Sort y\ns\u271d t u : Set \u03b1\ninst\u271d\u00b9\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u2070 : BorelSpace \u03b1\ninst\u271d\u2079 : TopologicalSpace \u03b2\ninst\u271d\u2078 : MeasurableSpace \u03b2\ninst\u271d\u2077 : BorelSpace \u03b2\ninst\u271d\u2076 : TopologicalSpace \u03b3\ninst\u271d\u2075 : MeasurableSpace \u03b3\ninst\u271d\u2074 : BorelSpace \u03b3\ninst\u271d\u00b3 : MeasurableSpace \u03b4\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : SecondCountableTopology \u03b1\n\u03b9 : Type u_6\ns : Set \u03b9\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nhs : Set.Countable s\nhf : \u2200 (i : \u03b9), i \u2208 s \u2192 Measurable (f i)\nthis : Encodable \u2191s\n\u22a2 Measurable fun b => \u2a06 i \u2208 s, f i b"}, {"tactic": "by_cases H : \u2200 i, i \u2208 s", "annotated_tactic": ["by_cases H : \u2200 i, i \u2208 s", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9\u271d : Sort y\ns\u271d t u : Set \u03b1\ninst\u271d\u00b9\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u2070 : BorelSpace \u03b1\ninst\u271d\u2079 : TopologicalSpace \u03b2\ninst\u271d\u2078 : MeasurableSpace \u03b2\ninst\u271d\u2077 : BorelSpace \u03b2\ninst\u271d\u2076 : TopologicalSpace \u03b3\ninst\u271d\u2075 : MeasurableSpace \u03b3\ninst\u271d\u2074 : BorelSpace \u03b3\ninst\u271d\u00b3 : MeasurableSpace \u03b4\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : SecondCountableTopology \u03b1\n\u03b9 : Type u_6\ns : Set \u03b9\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nhs : Set.Countable s\nhf : \u2200 (i : \u03b9), i \u2208 s \u2192 Measurable (f i)\nthis : Encodable \u2191s\n\u22a2 Measurable fun b => \u2a06 i \u2208 s, f i b", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9\u271d : Sort y\ns\u271d t u : Set \u03b1\ninst\u271d\u00b9\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u2070 : BorelSpace \u03b1\ninst\u271d\u2079 : TopologicalSpace \u03b2\ninst\u271d\u2078 : MeasurableSpace \u03b2\ninst\u271d\u2077 : BorelSpace \u03b2\ninst\u271d\u2076 : TopologicalSpace \u03b3\ninst\u271d\u2075 : MeasurableSpace \u03b3\ninst\u271d\u2074 : BorelSpace \u03b3\ninst\u271d\u00b3 : MeasurableSpace \u03b4\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : SecondCountableTopology \u03b1\n\u03b9 : Type u_6\ns : Set \u03b9\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nhs : Set.Countable s\nhf : \u2200 (i : \u03b9), i \u2208 s \u2192 Measurable (f i)\nthis : Encodable \u2191s\nH : \u2200 (i : \u03b9), i \u2208 s\n\u22a2 Measurable fun b => \u2a06 i \u2208 s, f i b\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9\u271d : Sort y\ns\u271d t u : Set \u03b1\ninst\u271d\u00b9\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u2070 : BorelSpace \u03b1\ninst\u271d\u2079 : TopologicalSpace \u03b2\ninst\u271d\u2078 : MeasurableSpace \u03b2\ninst\u271d\u2077 : BorelSpace \u03b2\ninst\u271d\u2076 : TopologicalSpace \u03b3\ninst\u271d\u2075 : MeasurableSpace \u03b3\ninst\u271d\u2074 : BorelSpace \u03b3\ninst\u271d\u00b3 : MeasurableSpace \u03b4\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : SecondCountableTopology \u03b1\n\u03b9 : Type u_6\ns : Set \u03b9\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nhs : Set.Countable s\nhf : \u2200 (i : \u03b9), i \u2208 s \u2192 Measurable (f i)\nthis : Encodable \u2191s\nH : \u00ac\u2200 (i : \u03b9), i \u2208 s\n\u22a2 Measurable fun b => \u2a06 i \u2208 s, f i b"}, {"tactic": "have : \u2200 b, \u2a06 i \u2208 s, f i b = \u2a06 (i : s), f i b :=\n  fun b \u21a6 cbiSup_eq_of_forall (f := fun i \u21a6 f i b) H", "annotated_tactic": ["have : \u2200 b, \u2a06 i \u2208 s, f i b = \u2a06 (i : s), f i b :=\n      fun b \u21a6 <a>cbiSup_eq_of_forall</a> (f := fun i \u21a6 f i b) H", [{"full_name": "cbiSup_eq_of_forall", "def_path": "Mathlib/Order/ConditionallyCompleteLattice/Basic.lean", "def_pos": [902, 9], "def_end_pos": [902, 28]}]], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9\u271d : Sort y\ns\u271d t u : Set \u03b1\ninst\u271d\u00b9\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u2070 : BorelSpace \u03b1\ninst\u271d\u2079 : TopologicalSpace \u03b2\ninst\u271d\u2078 : MeasurableSpace \u03b2\ninst\u271d\u2077 : BorelSpace \u03b2\ninst\u271d\u2076 : TopologicalSpace \u03b3\ninst\u271d\u2075 : MeasurableSpace \u03b3\ninst\u271d\u2074 : BorelSpace \u03b3\ninst\u271d\u00b3 : MeasurableSpace \u03b4\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : SecondCountableTopology \u03b1\n\u03b9 : Type u_6\ns : Set \u03b9\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nhs : Set.Countable s\nhf : \u2200 (i : \u03b9), i \u2208 s \u2192 Measurable (f i)\nthis : Encodable \u2191s\nH : \u2200 (i : \u03b9), i \u2208 s\n\u22a2 Measurable fun b => \u2a06 i \u2208 s, f i b", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9\u271d : Sort y\ns\u271d t u : Set \u03b1\ninst\u271d\u00b9\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u2070 : BorelSpace \u03b1\ninst\u271d\u2079 : TopologicalSpace \u03b2\ninst\u271d\u2078 : MeasurableSpace \u03b2\ninst\u271d\u2077 : BorelSpace \u03b2\ninst\u271d\u2076 : TopologicalSpace \u03b3\ninst\u271d\u2075 : MeasurableSpace \u03b3\ninst\u271d\u2074 : BorelSpace \u03b3\ninst\u271d\u00b3 : MeasurableSpace \u03b4\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : SecondCountableTopology \u03b1\n\u03b9 : Type u_6\ns : Set \u03b9\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nhs : Set.Countable s\nhf : \u2200 (i : \u03b9), i \u2208 s \u2192 Measurable (f i)\nthis\u271d : Encodable \u2191s\nH : \u2200 (i : \u03b9), i \u2208 s\nthis : \u2200 (b : \u03b4), \u2a06 i \u2208 s, f i b = \u2a06 i, f (\u2191i) b\n\u22a2 Measurable fun b => \u2a06 i \u2208 s, f i b"}, {"tactic": "simp only [this]", "annotated_tactic": ["simp only [this]", []], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9\u271d : Sort y\ns\u271d t u : Set \u03b1\ninst\u271d\u00b9\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u2070 : BorelSpace \u03b1\ninst\u271d\u2079 : TopologicalSpace \u03b2\ninst\u271d\u2078 : MeasurableSpace \u03b2\ninst\u271d\u2077 : BorelSpace \u03b2\ninst\u271d\u2076 : TopologicalSpace \u03b3\ninst\u271d\u2075 : MeasurableSpace \u03b3\ninst\u271d\u2074 : BorelSpace \u03b3\ninst\u271d\u00b3 : MeasurableSpace \u03b4\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : SecondCountableTopology \u03b1\n\u03b9 : Type u_6\ns : Set \u03b9\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nhs : Set.Countable s\nhf : \u2200 (i : \u03b9), i \u2208 s \u2192 Measurable (f i)\nthis\u271d : Encodable \u2191s\nH : \u2200 (i : \u03b9), i \u2208 s\nthis : \u2200 (b : \u03b4), \u2a06 i \u2208 s, f i b = \u2a06 i, f (\u2191i) b\n\u22a2 Measurable fun b => \u2a06 i \u2208 s, f i b", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9\u271d : Sort y\ns\u271d t u : Set \u03b1\ninst\u271d\u00b9\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u2070 : BorelSpace \u03b1\ninst\u271d\u2079 : TopologicalSpace \u03b2\ninst\u271d\u2078 : MeasurableSpace \u03b2\ninst\u271d\u2077 : BorelSpace \u03b2\ninst\u271d\u2076 : TopologicalSpace \u03b3\ninst\u271d\u2075 : MeasurableSpace \u03b3\ninst\u271d\u2074 : BorelSpace \u03b3\ninst\u271d\u00b3 : MeasurableSpace \u03b4\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : SecondCountableTopology \u03b1\n\u03b9 : Type u_6\ns : Set \u03b9\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nhs : Set.Countable s\nhf : \u2200 (i : \u03b9), i \u2208 s \u2192 Measurable (f i)\nthis\u271d : Encodable \u2191s\nH : \u2200 (i : \u03b9), i \u2208 s\nthis : \u2200 (b : \u03b4), \u2a06 i \u2208 s, f i b = \u2a06 i, f (\u2191i) b\n\u22a2 Measurable fun b => \u2a06 i, f (\u2191i) b"}, {"tactic": "exact measurable_iSup (fun (i : s) \u21a6 hf i i.2)", "annotated_tactic": ["exact <a>measurable_iSup</a> (fun (i : s) \u21a6 hf i i.2)", [{"full_name": "measurable_iSup", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [1360, 9], "def_end_pos": [1360, 24]}]], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9\u271d : Sort y\ns\u271d t u : Set \u03b1\ninst\u271d\u00b9\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u2070 : BorelSpace \u03b1\ninst\u271d\u2079 : TopologicalSpace \u03b2\ninst\u271d\u2078 : MeasurableSpace \u03b2\ninst\u271d\u2077 : BorelSpace \u03b2\ninst\u271d\u2076 : TopologicalSpace \u03b3\ninst\u271d\u2075 : MeasurableSpace \u03b3\ninst\u271d\u2074 : BorelSpace \u03b3\ninst\u271d\u00b3 : MeasurableSpace \u03b4\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : SecondCountableTopology \u03b1\n\u03b9 : Type u_6\ns : Set \u03b9\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nhs : Set.Countable s\nhf : \u2200 (i : \u03b9), i \u2208 s \u2192 Measurable (f i)\nthis\u271d : Encodable \u2191s\nH : \u2200 (i : \u03b9), i \u2208 s\nthis : \u2200 (b : \u03b4), \u2a06 i \u2208 s, f i b = \u2a06 i, f (\u2191i) b\n\u22a2 Measurable fun b => \u2a06 i, f (\u2191i) b", "state_after": "no goals"}, {"tactic": "have : \u2200 b, \u2a06 i \u2208 s, f i b = (\u2a06 (i : s), f i b) \u2294 sSup \u2205 :=\n  fun b \u21a6 cbiSup_eq_of_not_forall (f := fun i \u21a6 f i b) H", "annotated_tactic": ["have : \u2200 b, \u2a06 i \u2208 s, f i b = (\u2a06 (i : s), f i b) \u2294 <a>sSup</a> \u2205 :=\n      fun b \u21a6 <a>cbiSup_eq_of_not_forall</a> (f := fun i \u21a6 f i b) H", [{"full_name": "SupSet.sSup", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [55, 3], "def_end_pos": [55, 7]}, {"full_name": "cbiSup_eq_of_not_forall", "def_path": "Mathlib/Order/ConditionallyCompleteLattice/Basic.lean", "def_pos": [1089, 9], "def_end_pos": [1089, 32]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9\u271d : Sort y\ns\u271d t u : Set \u03b1\ninst\u271d\u00b9\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u2070 : BorelSpace \u03b1\ninst\u271d\u2079 : TopologicalSpace \u03b2\ninst\u271d\u2078 : MeasurableSpace \u03b2\ninst\u271d\u2077 : BorelSpace \u03b2\ninst\u271d\u2076 : TopologicalSpace \u03b3\ninst\u271d\u2075 : MeasurableSpace \u03b3\ninst\u271d\u2074 : BorelSpace \u03b3\ninst\u271d\u00b3 : MeasurableSpace \u03b4\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : SecondCountableTopology \u03b1\n\u03b9 : Type u_6\ns : Set \u03b9\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nhs : Set.Countable s\nhf : \u2200 (i : \u03b9), i \u2208 s \u2192 Measurable (f i)\nthis : Encodable \u2191s\nH : \u00ac\u2200 (i : \u03b9), i \u2208 s\n\u22a2 Measurable fun b => \u2a06 i \u2208 s, f i b", "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9\u271d : Sort y\ns\u271d t u : Set \u03b1\ninst\u271d\u00b9\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u2070 : BorelSpace \u03b1\ninst\u271d\u2079 : TopologicalSpace \u03b2\ninst\u271d\u2078 : MeasurableSpace \u03b2\ninst\u271d\u2077 : BorelSpace \u03b2\ninst\u271d\u2076 : TopologicalSpace \u03b3\ninst\u271d\u2075 : MeasurableSpace \u03b3\ninst\u271d\u2074 : BorelSpace \u03b3\ninst\u271d\u00b3 : MeasurableSpace \u03b4\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : SecondCountableTopology \u03b1\n\u03b9 : Type u_6\ns : Set \u03b9\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nhs : Set.Countable s\nhf : \u2200 (i : \u03b9), i \u2208 s \u2192 Measurable (f i)\nthis\u271d : Encodable \u2191s\nH : \u00ac\u2200 (i : \u03b9), i \u2208 s\nthis : \u2200 (b : \u03b4), \u2a06 i \u2208 s, f i b = (\u2a06 i, f (\u2191i) b) \u2294 sSup \u2205\n\u22a2 Measurable fun b => \u2a06 i \u2208 s, f i b"}, {"tactic": "simp only [this]", "annotated_tactic": ["simp only [this]", []], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9\u271d : Sort y\ns\u271d t u : Set \u03b1\ninst\u271d\u00b9\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u2070 : BorelSpace \u03b1\ninst\u271d\u2079 : TopologicalSpace \u03b2\ninst\u271d\u2078 : MeasurableSpace \u03b2\ninst\u271d\u2077 : BorelSpace \u03b2\ninst\u271d\u2076 : TopologicalSpace \u03b3\ninst\u271d\u2075 : MeasurableSpace \u03b3\ninst\u271d\u2074 : BorelSpace \u03b3\ninst\u271d\u00b3 : MeasurableSpace \u03b4\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : SecondCountableTopology \u03b1\n\u03b9 : Type u_6\ns : Set \u03b9\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nhs : Set.Countable s\nhf : \u2200 (i : \u03b9), i \u2208 s \u2192 Measurable (f i)\nthis\u271d : Encodable \u2191s\nH : \u00ac\u2200 (i : \u03b9), i \u2208 s\nthis : \u2200 (b : \u03b4), \u2a06 i \u2208 s, f i b = (\u2a06 i, f (\u2191i) b) \u2294 sSup \u2205\n\u22a2 Measurable fun b => \u2a06 i \u2208 s, f i b", "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9\u271d : Sort y\ns\u271d t u : Set \u03b1\ninst\u271d\u00b9\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u2070 : BorelSpace \u03b1\ninst\u271d\u2079 : TopologicalSpace \u03b2\ninst\u271d\u2078 : MeasurableSpace \u03b2\ninst\u271d\u2077 : BorelSpace \u03b2\ninst\u271d\u2076 : TopologicalSpace \u03b3\ninst\u271d\u2075 : MeasurableSpace \u03b3\ninst\u271d\u2074 : BorelSpace \u03b3\ninst\u271d\u00b3 : MeasurableSpace \u03b4\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : SecondCountableTopology \u03b1\n\u03b9 : Type u_6\ns : Set \u03b9\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nhs : Set.Countable s\nhf : \u2200 (i : \u03b9), i \u2208 s \u2192 Measurable (f i)\nthis\u271d : Encodable \u2191s\nH : \u00ac\u2200 (i : \u03b9), i \u2208 s\nthis : \u2200 (b : \u03b4), \u2a06 i \u2208 s, f i b = (\u2a06 i, f (\u2191i) b) \u2294 sSup \u2205\n\u22a2 Measurable fun b => (\u2a06 i, f (\u2191i) b) \u2294 sSup \u2205"}, {"tactic": "apply Measurable.sup _ measurable_const", "annotated_tactic": ["apply <a>Measurable.sup</a> _ <a>measurable_const</a>", [{"full_name": "Measurable.sup", "def_path": "Mathlib/MeasureTheory/Lattice.lean", "def_pos": [145, 9], "def_end_pos": [145, 23]}, {"full_name": "measurable_const", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [570, 9], "def_end_pos": [570, 25]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9\u271d : Sort y\ns\u271d t u : Set \u03b1\ninst\u271d\u00b9\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u2070 : BorelSpace \u03b1\ninst\u271d\u2079 : TopologicalSpace \u03b2\ninst\u271d\u2078 : MeasurableSpace \u03b2\ninst\u271d\u2077 : BorelSpace \u03b2\ninst\u271d\u2076 : TopologicalSpace \u03b3\ninst\u271d\u2075 : MeasurableSpace \u03b3\ninst\u271d\u2074 : BorelSpace \u03b3\ninst\u271d\u00b3 : MeasurableSpace \u03b4\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : SecondCountableTopology \u03b1\n\u03b9 : Type u_6\ns : Set \u03b9\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nhs : Set.Countable s\nhf : \u2200 (i : \u03b9), i \u2208 s \u2192 Measurable (f i)\nthis\u271d : Encodable \u2191s\nH : \u00ac\u2200 (i : \u03b9), i \u2208 s\nthis : \u2200 (b : \u03b4), \u2a06 i \u2208 s, f i b = (\u2a06 i, f (\u2191i) b) \u2294 sSup \u2205\n\u22a2 Measurable fun b => (\u2a06 i, f (\u2191i) b) \u2294 sSup \u2205", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9\u271d : Sort y\ns\u271d t u : Set \u03b1\ninst\u271d\u00b9\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u2070 : BorelSpace \u03b1\ninst\u271d\u2079 : TopologicalSpace \u03b2\ninst\u271d\u2078 : MeasurableSpace \u03b2\ninst\u271d\u2077 : BorelSpace \u03b2\ninst\u271d\u2076 : TopologicalSpace \u03b3\ninst\u271d\u2075 : MeasurableSpace \u03b3\ninst\u271d\u2074 : BorelSpace \u03b3\ninst\u271d\u00b3 : MeasurableSpace \u03b4\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : SecondCountableTopology \u03b1\n\u03b9 : Type u_6\ns : Set \u03b9\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nhs : Set.Countable s\nhf : \u2200 (i : \u03b9), i \u2208 s \u2192 Measurable (f i)\nthis\u271d : Encodable \u2191s\nH : \u00ac\u2200 (i : \u03b9), i \u2208 s\nthis : \u2200 (b : \u03b4), \u2a06 i \u2208 s, f i b = (\u2a06 i, f (\u2191i) b) \u2294 sSup \u2205\n\u22a2 Measurable fun a => \u2a06 i, f (\u2191i) a"}, {"tactic": "exact measurable_iSup (fun (i : s) \u21a6 hf i i.2)", "annotated_tactic": ["exact <a>measurable_iSup</a> (fun (i : s) \u21a6 hf i i.2)", [{"full_name": "measurable_iSup", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [1360, 9], "def_end_pos": [1360, 24]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9\u271d : Sort y\ns\u271d t u : Set \u03b1\ninst\u271d\u00b9\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u2070 : BorelSpace \u03b1\ninst\u271d\u2079 : TopologicalSpace \u03b2\ninst\u271d\u2078 : MeasurableSpace \u03b2\ninst\u271d\u2077 : BorelSpace \u03b2\ninst\u271d\u2076 : TopologicalSpace \u03b3\ninst\u271d\u2075 : MeasurableSpace \u03b3\ninst\u271d\u2074 : BorelSpace \u03b3\ninst\u271d\u00b3 : MeasurableSpace \u03b4\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : SecondCountableTopology \u03b1\n\u03b9 : Type u_6\ns : Set \u03b9\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nhs : Set.Countable s\nhf : \u2200 (i : \u03b9), i \u2208 s \u2192 Measurable (f i)\nthis\u271d : Encodable \u2191s\nH : \u00ac\u2200 (i : \u03b9), i \u2208 s\nthis : \u2200 (b : \u03b4), \u2a06 i \u2208 s, f i b = (\u2a06 i, f (\u2191i) b) \u2294 sSup \u2205\n\u22a2 Measurable fun a => \u2a06 i, f (\u2191i) a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Process/Stopping.lean", "full_name": "MeasureTheory.stoppedProcess_eq", "start": [1092, 1], "end": [1096, 40], "traced_tactics": [{"tactic": "rw [stoppedProcess_eq'' n]", "annotated_tactic": ["rw [<a>stoppedProcess_eq''</a> n]", [{"full_name": "MeasureTheory.stoppedProcess_eq''", "def_path": "Mathlib/Probability/Process/Stopping.lean", "def_pos": [938, 9], "def_end_pos": [938, 28]}]], "state_before": "\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\nf : Filtration \u2115 m\nu : \u2115 \u2192 \u03a9 \u2192 \u03b2\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : AddCommMonoid \u03b2\nn : \u2115\n\u22a2 stoppedProcess u \u03c4 n = Set.indicator {a | n \u2264 \u03c4 a} (u n) + \u2211 i in Finset.range n, Set.indicator {\u03c9 | \u03c4 \u03c9 = i} (u i)", "state_after": "\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\nf : Filtration \u2115 m\nu : \u2115 \u2192 \u03a9 \u2192 \u03b2\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : AddCommMonoid \u03b2\nn : \u2115\n\u22a2 Set.indicator {a | n \u2264 \u03c4 a} (u n) + \u2211 i in Finset.Iio n, Set.indicator {\u03c9 | \u03c4 \u03c9 = i} (u i) =\n    Set.indicator {a | n \u2264 \u03c4 a} (u n) + \u2211 i in Finset.range n, Set.indicator {\u03c9 | \u03c4 \u03c9 = i} (u i)"}, {"tactic": "congr with i", "annotated_tactic": ["congr with i", []], "state_before": "\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\nf : Filtration \u2115 m\nu : \u2115 \u2192 \u03a9 \u2192 \u03b2\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : AddCommMonoid \u03b2\nn : \u2115\n\u22a2 Set.indicator {a | n \u2264 \u03c4 a} (u n) + \u2211 i in Finset.Iio n, Set.indicator {\u03c9 | \u03c4 \u03c9 = i} (u i) =\n    Set.indicator {a | n \u2264 \u03c4 a} (u n) + \u2211 i in Finset.range n, Set.indicator {\u03c9 | \u03c4 \u03c9 = i} (u i)", "state_after": "case e_a.e_s.a\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\nf : Filtration \u2115 m\nu : \u2115 \u2192 \u03a9 \u2192 \u03b2\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : AddCommMonoid \u03b2\nn i : \u2115\n\u22a2 i \u2208 Finset.Iio n \u2194 i \u2208 Finset.range n"}, {"tactic": "rw [Finset.mem_Iio, Finset.mem_range]", "annotated_tactic": ["rw [<a>Finset.mem_Iio</a>, <a>Finset.mem_range</a>]", [{"full_name": "Finset.mem_Iio", "def_path": "Mathlib/Order/LocallyFinite.lean", "def_pos": [423, 9], "def_end_pos": [423, 16]}, {"full_name": "Finset.mem_range", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3037, 9], "def_end_pos": [3037, 18]}]], "state_before": "case e_a.e_s.a\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\nf : Filtration \u2115 m\nu : \u2115 \u2192 \u03a9 \u2192 \u03b2\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : AddCommMonoid \u03b2\nn i : \u2115\n\u22a2 i \u2208 Finset.Iio n \u2194 i \u2208 Finset.range n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Covering/Vitali.lean", "full_name": "Vitali.exists_disjoint_subfamily_covering_enlargment", "start": [59, 1], "end": [156, 34], "traced_tactics": [{"tactic": "let T : Set (Set \u03b9) := { u | u \u2286 t \u2227 u.PairwiseDisjoint B \u2227\n  \u2200 a \u2208 t, \u2200 b \u2208 u, (B a \u2229 B b).Nonempty \u2192 \u2203 c \u2208 u, (B a \u2229 B c).Nonempty \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c }", "annotated_tactic": ["let T : <a>Set</a> (<a>Set</a> \u03b9) := { u | u \u2286 t \u2227 u.PairwiseDisjoint B \u2227\n    \u2200 a \u2208 t, \u2200 b \u2208 u, (B a \u2229 B b).<a>Nonempty</a> \u2192 \u2203 c \u2208 u, (B a \u2229 B c).<a>Nonempty</a> \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c }", [{"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}, {"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}, {"full_name": "Set.Nonempty", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [439, 15], "def_end_pos": [439, 23]}, {"full_name": "Set.Nonempty", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [439, 15], "def_end_pos": [439, 23]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\nB : \u03b9 \u2192 Set \u03b1\nt : Set \u03b9\n\u03b4 : \u03b9 \u2192 \u211d\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\n\u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a\nR : \u211d\n\u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R\nhne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a)\n\u22a2 \u2203 u x, PairwiseDisjoint u B \u2227 \u2200 (a : \u03b9), a \u2208 t \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 b", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\nB : \u03b9 \u2192 Set \u03b1\nt : Set \u03b9\n\u03b4 : \u03b9 \u2192 \u211d\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\n\u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a\nR : \u211d\n\u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R\nhne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a)\nT : Set (Set \u03b9) :=\n  {u |\n    u \u2286 t \u2227\n      PairwiseDisjoint u B \u2227\n        \u2200 (a : \u03b9),\n          a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c}\n\u22a2 \u2203 u x, PairwiseDisjoint u B \u2227 \u2200 (a : \u03b9), a \u2208 t \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 b"}, {"tactic": "obtain \u27e8u, uT, hu\u27e9 : \u2203 u \u2208 T, \u2200 v \u2208 T, u \u2286 v \u2192 v = u := by\n  refine' zorn_subset _ fun U UT hU => _\n  refine' \u27e8\u22c3\u2080 U, _, fun s hs => subset_sUnion_of_mem hs\u27e9\n  simp only [Set.sUnion_subset_iff, and_imp, exists_prop, forall_exists_index, mem_sUnion,\n    Set.mem_setOf_eq]\n  refine'\n    \u27e8fun u hu => (UT hu).1, (pairwiseDisjoint_sUnion hU.directedOn).2 fun u hu => (UT hu).2.1,\n      fun a hat b u uU hbu hab => _\u27e9\n  obtain \u27e8c, cu, ac, hc\u27e9 : \u2203 c, c \u2208 u \u2227 (B a \u2229 B c).Nonempty \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c :=\n    (UT uU).2.2 a hat b hbu hab\n  exact \u27e8c, \u27e8u, uU, cu\u27e9, ac, hc\u27e9", "annotated_tactic": ["obtain \u27e8u, uT, hu\u27e9 : \u2203 u \u2208 T, \u2200 v \u2208 T, u \u2286 v \u2192 v = u := by\n    refine' <a>zorn_subset</a> _ fun U UT hU => _\n    refine' \u27e8\u22c3\u2080 U, _, fun s hs => <a>subset_sUnion_of_mem</a> hs\u27e9\n    simp only [<a>Set.sUnion_subset_iff</a>, <a>and_imp</a>, <a>exists_prop</a>, <a>forall_exists_index</a>, <a>mem_sUnion</a>,\n      <a>Set.mem_setOf_eq</a>]\n    refine'\n      \u27e8fun u hu => (UT hu).1, (<a>pairwiseDisjoint_sUnion</a> hU.directedOn).2 fun u hu => (UT hu).2.1,\n        fun a hat b u uU hbu hab => _\u27e9\n    obtain \u27e8c, cu, ac, hc\u27e9 : \u2203 c, c \u2208 u \u2227 (B a \u2229 B c).<a>Nonempty</a> \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c :=\n      (UT uU).2.2 a hat b hbu hab\n    exact \u27e8c, \u27e8u, uU, cu\u27e9, ac, hc\u27e9", [{"full_name": "zorn_subset", "def_path": "Mathlib/Order/Zorn.lean", "def_pos": [186, 9], "def_end_pos": [186, 20]}, {"full_name": "Set.subset_sUnion_of_mem", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [1157, 9], "def_end_pos": [1157, 29]}, {"full_name": "Set.sUnion_subset_iff", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [1171, 9], "def_end_pos": [1171, 26]}, {"full_name": "and_imp", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [313, 17], "def_end_pos": [313, 24]}, {"full_name": "exists_prop", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [485, 17], "def_end_pos": [485, 28]}, {"full_name": "forall_exists_index", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [356, 17], "def_end_pos": [356, 36]}, {"full_name": "Set.mem_sUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [98, 9], "def_end_pos": [98, 19]}, {"full_name": "Set.mem_setOf_eq", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [256, 29], "def_end_pos": [256, 41]}, {"full_name": "Set.pairwiseDisjoint_sUnion", "def_path": "Mathlib/Data/Set/Pairwise/Lattice.lean", "def_pos": [59, 9], "def_end_pos": [59, 32]}, {"full_name": "Set.Nonempty", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [439, 15], "def_end_pos": [439, 23]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\nB : \u03b9 \u2192 Set \u03b1\nt : Set \u03b9\n\u03b4 : \u03b9 \u2192 \u211d\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\n\u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a\nR : \u211d\n\u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R\nhne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a)\nT : Set (Set \u03b9) :=\n  {u |\n    u \u2286 t \u2227\n      PairwiseDisjoint u B \u2227\n        \u2200 (a : \u03b9),\n          a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c}\n\u22a2 \u2203 u x, PairwiseDisjoint u B \u2227 \u2200 (a : \u03b9), a \u2208 t \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 b", "state_after": "case intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\nB : \u03b9 \u2192 Set \u03b1\nt : Set \u03b9\n\u03b4 : \u03b9 \u2192 \u211d\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\n\u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a\nR : \u211d\n\u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R\nhne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a)\nT : Set (Set \u03b9) :=\n  {u |\n    u \u2286 t \u2227\n      PairwiseDisjoint u B \u2227\n        \u2200 (a : \u03b9),\n          a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c}\nu : Set \u03b9\nuT : u \u2208 T\nhu : \u2200 (v : Set \u03b9), v \u2208 T \u2192 u \u2286 v \u2192 v = u\n\u22a2 \u2203 u x, PairwiseDisjoint u B \u2227 \u2200 (a : \u03b9), a \u2208 t \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 b"}, {"tactic": "refine' \u27e8u, uT.1, uT.2.1, fun a hat => _\u27e9", "annotated_tactic": ["refine' \u27e8u, uT.1, uT.2.1, fun a hat => _\u27e9", []], "state_before": "case intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\nB : \u03b9 \u2192 Set \u03b1\nt : Set \u03b9\n\u03b4 : \u03b9 \u2192 \u211d\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\n\u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a\nR : \u211d\n\u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R\nhne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a)\nT : Set (Set \u03b9) :=\n  {u |\n    u \u2286 t \u2227\n      PairwiseDisjoint u B \u2227\n        \u2200 (a : \u03b9),\n          a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c}\nu : Set \u03b9\nuT : u \u2208 T\nhu : \u2200 (v : Set \u03b9), v \u2208 T \u2192 u \u2286 v \u2192 v = u\n\u22a2 \u2203 u x, PairwiseDisjoint u B \u2227 \u2200 (a : \u03b9), a \u2208 t \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 b", "state_after": "case intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\nB : \u03b9 \u2192 Set \u03b1\nt : Set \u03b9\n\u03b4 : \u03b9 \u2192 \u211d\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\n\u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a\nR : \u211d\n\u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R\nhne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a)\nT : Set (Set \u03b9) :=\n  {u |\n    u \u2286 t \u2227\n      PairwiseDisjoint u B \u2227\n        \u2200 (a : \u03b9),\n          a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c}\nu : Set \u03b9\nuT : u \u2208 T\nhu : \u2200 (v : Set \u03b9), v \u2208 T \u2192 u \u2286 v \u2192 v = u\na : \u03b9\nhat : a \u2208 t\n\u22a2 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 b"}, {"tactic": "contrapose! hu", "annotated_tactic": ["contrapose! hu", []], "state_before": "case intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\nB : \u03b9 \u2192 Set \u03b1\nt : Set \u03b9\n\u03b4 : \u03b9 \u2192 \u211d\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\n\u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a\nR : \u211d\n\u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R\nhne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a)\nT : Set (Set \u03b9) :=\n  {u |\n    u \u2286 t \u2227\n      PairwiseDisjoint u B \u2227\n        \u2200 (a : \u03b9),\n          a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c}\nu : Set \u03b9\nuT : u \u2208 T\nhu : \u2200 (v : Set \u03b9), v \u2208 T \u2192 u \u2286 v \u2192 v = u\na : \u03b9\nhat : a \u2208 t\n\u22a2 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 b", "state_after": "case intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\nB : \u03b9 \u2192 Set \u03b1\nt : Set \u03b9\n\u03b4 : \u03b9 \u2192 \u211d\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\n\u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a\nR : \u211d\n\u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R\nhne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a)\nT : Set (Set \u03b9) :=\n  {u |\n    u \u2286 t \u2227\n      PairwiseDisjoint u B \u2227\n        \u2200 (a : \u03b9),\n          a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c}\nu : Set \u03b9\nuT : u \u2208 T\na : \u03b9\nhat : a \u2208 t\nhu : \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u03c4 * \u03b4 b < \u03b4 a\n\u22a2 \u2203 v, v \u2208 T \u2227 u \u2286 v \u2227 v \u2260 u"}, {"tactic": "have a_disj : \u2200 c \u2208 u, Disjoint (B a) (B c) := by\n  intro c hc\n  by_contra h\n  rw [not_disjoint_iff_nonempty_inter] at h\n  obtain \u27e8d, du, ad, hd\u27e9 : \u2203 d, d \u2208 u \u2227 (B a \u2229 B d).Nonempty \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 d :=\n    uT.2.2 a hat c hc h\n  exact lt_irrefl _ ((hu d du ad).trans_le hd)", "annotated_tactic": ["have a_disj : \u2200 c \u2208 u, <a>Disjoint</a> (B a) (B c) := by\n    intro c hc\n    by_contra h\n    rw [<a>not_disjoint_iff_nonempty_inter</a>] at h\n    obtain \u27e8d, du, ad, hd\u27e9 : \u2203 d, d \u2208 u \u2227 (B a \u2229 B d).<a>Nonempty</a> \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 d :=\n      uT.2.2 a hat c hc h\n    exact <a>lt_irrefl</a> _ ((hu d du ad).<a>trans_le</a> hd)", [{"full_name": "Disjoint", "def_path": "Mathlib/Order/Disjoint.lean", "def_pos": [41, 5], "def_end_pos": [41, 13]}, {"full_name": "Set.not_disjoint_iff_nonempty_inter", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1557, 7], "def_end_pos": [1557, 38]}, {"full_name": "Set.Nonempty", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [439, 15], "def_end_pos": [439, 23]}, {"full_name": "lt_irrefl", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [79, 9], "def_end_pos": [79, 18]}, {"full_name": "LT.lt.trans_le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [148, 7], "def_end_pos": [148, 21]}]], "state_before": "case intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\nB : \u03b9 \u2192 Set \u03b1\nt : Set \u03b9\n\u03b4 : \u03b9 \u2192 \u211d\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\n\u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a\nR : \u211d\n\u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R\nhne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a)\nT : Set (Set \u03b9) :=\n  {u |\n    u \u2286 t \u2227\n      PairwiseDisjoint u B \u2227\n        \u2200 (a : \u03b9),\n          a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c}\nu : Set \u03b9\nuT : u \u2208 T\na : \u03b9\nhat : a \u2208 t\nhu : \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u03c4 * \u03b4 b < \u03b4 a\n\u22a2 \u2203 v, v \u2208 T \u2227 u \u2286 v \u2227 v \u2260 u", "state_after": "case intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\nB : \u03b9 \u2192 Set \u03b1\nt : Set \u03b9\n\u03b4 : \u03b9 \u2192 \u211d\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\n\u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a\nR : \u211d\n\u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R\nhne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a)\nT : Set (Set \u03b9) :=\n  {u |\n    u \u2286 t \u2227\n      PairwiseDisjoint u B \u2227\n        \u2200 (a : \u03b9),\n          a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c}\nu : Set \u03b9\nuT : u \u2208 T\na : \u03b9\nhat : a \u2208 t\nhu : \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u03c4 * \u03b4 b < \u03b4 a\na_disj : \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a) (B c)\n\u22a2 \u2203 v, v \u2208 T \u2227 u \u2286 v \u2227 v \u2260 u"}, {"tactic": "let A := { a' | a' \u2208 t \u2227 \u2200 c \u2208 u, Disjoint (B a') (B c) }", "annotated_tactic": ["let A := { a' | a' \u2208 t \u2227 \u2200 c \u2208 u, <a>Disjoint</a> (B a') (B c) }", [{"full_name": "Disjoint", "def_path": "Mathlib/Order/Disjoint.lean", "def_pos": [41, 5], "def_end_pos": [41, 13]}]], "state_before": "case intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\nB : \u03b9 \u2192 Set \u03b1\nt : Set \u03b9\n\u03b4 : \u03b9 \u2192 \u211d\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\n\u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a\nR : \u211d\n\u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R\nhne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a)\nT : Set (Set \u03b9) :=\n  {u |\n    u \u2286 t \u2227\n      PairwiseDisjoint u B \u2227\n        \u2200 (a : \u03b9),\n          a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c}\nu : Set \u03b9\nuT : u \u2208 T\na : \u03b9\nhat : a \u2208 t\nhu : \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u03c4 * \u03b4 b < \u03b4 a\na_disj : \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a) (B c)\n\u22a2 \u2203 v, v \u2208 T \u2227 u \u2286 v \u2227 v \u2260 u", "state_after": "case intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\nB : \u03b9 \u2192 Set \u03b1\nt : Set \u03b9\n\u03b4 : \u03b9 \u2192 \u211d\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\n\u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a\nR : \u211d\n\u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R\nhne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a)\nT : Set (Set \u03b9) :=\n  {u |\n    u \u2286 t \u2227\n      PairwiseDisjoint u B \u2227\n        \u2200 (a : \u03b9),\n          a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c}\nu : Set \u03b9\nuT : u \u2208 T\na : \u03b9\nhat : a \u2208 t\nhu : \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u03c4 * \u03b4 b < \u03b4 a\na_disj : \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a) (B c)\nA : Set \u03b9 := {a' | a' \u2208 t \u2227 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a') (B c)}\n\u22a2 \u2203 v, v \u2208 T \u2227 u \u2286 v \u2227 v \u2260 u"}, {"tactic": "have Anonempty : A.Nonempty := \u27e8a, hat, a_disj\u27e9", "annotated_tactic": ["have Anonempty : A.Nonempty := \u27e8a, hat, a_disj\u27e9", []], "state_before": "case intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\nB : \u03b9 \u2192 Set \u03b1\nt : Set \u03b9\n\u03b4 : \u03b9 \u2192 \u211d\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\n\u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a\nR : \u211d\n\u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R\nhne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a)\nT : Set (Set \u03b9) :=\n  {u |\n    u \u2286 t \u2227\n      PairwiseDisjoint u B \u2227\n        \u2200 (a : \u03b9),\n          a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c}\nu : Set \u03b9\nuT : u \u2208 T\na : \u03b9\nhat : a \u2208 t\nhu : \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u03c4 * \u03b4 b < \u03b4 a\na_disj : \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a) (B c)\nA : Set \u03b9 := {a' | a' \u2208 t \u2227 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a') (B c)}\n\u22a2 \u2203 v, v \u2208 T \u2227 u \u2286 v \u2227 v \u2260 u", "state_after": "case intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\nB : \u03b9 \u2192 Set \u03b1\nt : Set \u03b9\n\u03b4 : \u03b9 \u2192 \u211d\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\n\u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a\nR : \u211d\n\u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R\nhne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a)\nT : Set (Set \u03b9) :=\n  {u |\n    u \u2286 t \u2227\n      PairwiseDisjoint u B \u2227\n        \u2200 (a : \u03b9),\n          a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c}\nu : Set \u03b9\nuT : u \u2208 T\na : \u03b9\nhat : a \u2208 t\nhu : \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u03c4 * \u03b4 b < \u03b4 a\na_disj : \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a) (B c)\nA : Set \u03b9 := {a' | a' \u2208 t \u2227 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a') (B c)}\nAnonempty : Set.Nonempty A\n\u22a2 \u2203 v, v \u2208 T \u2227 u \u2286 v \u2227 v \u2260 u"}, {"tactic": "let m := sSup (\u03b4 '' A)", "annotated_tactic": ["let m := <a>sSup</a> (\u03b4 '' A)", [{"full_name": "SupSet.sSup", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [55, 3], "def_end_pos": [55, 7]}]], "state_before": "case intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\nB : \u03b9 \u2192 Set \u03b1\nt : Set \u03b9\n\u03b4 : \u03b9 \u2192 \u211d\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\n\u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a\nR : \u211d\n\u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R\nhne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a)\nT : Set (Set \u03b9) :=\n  {u |\n    u \u2286 t \u2227\n      PairwiseDisjoint u B \u2227\n        \u2200 (a : \u03b9),\n          a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c}\nu : Set \u03b9\nuT : u \u2208 T\na : \u03b9\nhat : a \u2208 t\nhu : \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u03c4 * \u03b4 b < \u03b4 a\na_disj : \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a) (B c)\nA : Set \u03b9 := {a' | a' \u2208 t \u2227 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a') (B c)}\nAnonempty : Set.Nonempty A\n\u22a2 \u2203 v, v \u2208 T \u2227 u \u2286 v \u2227 v \u2260 u", "state_after": "case intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\nB : \u03b9 \u2192 Set \u03b1\nt : Set \u03b9\n\u03b4 : \u03b9 \u2192 \u211d\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\n\u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a\nR : \u211d\n\u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R\nhne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a)\nT : Set (Set \u03b9) :=\n  {u |\n    u \u2286 t \u2227\n      PairwiseDisjoint u B \u2227\n        \u2200 (a : \u03b9),\n          a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c}\nu : Set \u03b9\nuT : u \u2208 T\na : \u03b9\nhat : a \u2208 t\nhu : \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u03c4 * \u03b4 b < \u03b4 a\na_disj : \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a) (B c)\nA : Set \u03b9 := {a' | a' \u2208 t \u2227 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a') (B c)}\nAnonempty : Set.Nonempty A\nm : \u211d := sSup (\u03b4 '' A)\n\u22a2 \u2203 v, v \u2208 T \u2227 u \u2286 v \u2227 v \u2260 u"}, {"tactic": "have bddA : BddAbove (\u03b4 '' A) := by\n  refine' \u27e8R, fun x xA => _\u27e9\n  rcases (mem_image _ _ _).1 xA with \u27e8a', ha', rfl\u27e9\n  exact \u03b4le a' ha'.1", "annotated_tactic": ["have bddA : <a>BddAbove</a> (\u03b4 '' A) := by\n    refine' \u27e8R, fun x xA => _\u27e9\n    rcases (<a>mem_image</a> _ _ _).1 xA with \u27e8a', ha', rfl\u27e9\n    exact \u03b4le a' ha'.1", [{"full_name": "BddAbove", "def_path": "Mathlib/Order/Bounds/Basic.lean", "def_pos": [56, 5], "def_end_pos": [56, 13]}, {"full_name": "Set.mem_image", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [231, 9], "def_end_pos": [231, 18]}]], "state_before": "case intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\nB : \u03b9 \u2192 Set \u03b1\nt : Set \u03b9\n\u03b4 : \u03b9 \u2192 \u211d\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\n\u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a\nR : \u211d\n\u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R\nhne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a)\nT : Set (Set \u03b9) :=\n  {u |\n    u \u2286 t \u2227\n      PairwiseDisjoint u B \u2227\n        \u2200 (a : \u03b9),\n          a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c}\nu : Set \u03b9\nuT : u \u2208 T\na : \u03b9\nhat : a \u2208 t\nhu : \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u03c4 * \u03b4 b < \u03b4 a\na_disj : \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a) (B c)\nA : Set \u03b9 := {a' | a' \u2208 t \u2227 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a') (B c)}\nAnonempty : Set.Nonempty A\nm : \u211d := sSup (\u03b4 '' A)\n\u22a2 \u2203 v, v \u2208 T \u2227 u \u2286 v \u2227 v \u2260 u", "state_after": "case intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\nB : \u03b9 \u2192 Set \u03b1\nt : Set \u03b9\n\u03b4 : \u03b9 \u2192 \u211d\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\n\u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a\nR : \u211d\n\u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R\nhne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a)\nT : Set (Set \u03b9) :=\n  {u |\n    u \u2286 t \u2227\n      PairwiseDisjoint u B \u2227\n        \u2200 (a : \u03b9),\n          a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c}\nu : Set \u03b9\nuT : u \u2208 T\na : \u03b9\nhat : a \u2208 t\nhu : \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u03c4 * \u03b4 b < \u03b4 a\na_disj : \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a) (B c)\nA : Set \u03b9 := {a' | a' \u2208 t \u2227 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a') (B c)}\nAnonempty : Set.Nonempty A\nm : \u211d := sSup (\u03b4 '' A)\nbddA : BddAbove (\u03b4 '' A)\n\u22a2 \u2203 v, v \u2208 T \u2227 u \u2286 v \u2227 v \u2260 u"}, {"tactic": "clear hat hu a_disj a", "annotated_tactic": ["clear hat hu a_disj a", []], "state_before": "case intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\nB : \u03b9 \u2192 Set \u03b1\nt : Set \u03b9\n\u03b4 : \u03b9 \u2192 \u211d\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\n\u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a\nR : \u211d\n\u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R\nhne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a)\nT : Set (Set \u03b9) :=\n  {u |\n    u \u2286 t \u2227\n      PairwiseDisjoint u B \u2227\n        \u2200 (a : \u03b9),\n          a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c}\nu : Set \u03b9\nuT : u \u2208 T\na : \u03b9\nhat : a \u2208 t\nhu : \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u03c4 * \u03b4 b < \u03b4 a\na_disj : \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a) (B c)\nA : Set \u03b9 := {a' | a' \u2208 t \u2227 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a') (B c)}\nAnonempty : Set.Nonempty A\nm : \u211d := sSup (\u03b4 '' A)\nbddA : BddAbove (\u03b4 '' A)\na' : \u03b9\na'A : a' \u2208 A\nha' : m / \u03c4 \u2264 \u03b4 a'\n\u22a2 \u2203 v, v \u2208 T \u2227 u \u2286 v \u2227 v \u2260 u", "state_after": "case intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\nB : \u03b9 \u2192 Set \u03b1\nt : Set \u03b9\n\u03b4 : \u03b9 \u2192 \u211d\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\n\u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a\nR : \u211d\n\u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R\nhne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a)\nT : Set (Set \u03b9) :=\n  {u |\n    u \u2286 t \u2227\n      PairwiseDisjoint u B \u2227\n        \u2200 (a : \u03b9),\n          a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c}\nu : Set \u03b9\nuT : u \u2208 T\nA : Set \u03b9 := {a' | a' \u2208 t \u2227 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a') (B c)}\nAnonempty : Set.Nonempty A\nm : \u211d := sSup (\u03b4 '' A)\nbddA : BddAbove (\u03b4 '' A)\na' : \u03b9\na'A : a' \u2208 A\nha' : m / \u03c4 \u2264 \u03b4 a'\n\u22a2 \u2203 v, v \u2208 T \u2227 u \u2286 v \u2227 v \u2260 u"}, {"tactic": "have a'_ne_u : a' \u2209 u := fun H => (hne _ a'A.1).ne_empty (disjoint_self.1 (a'A.2 _ H))", "annotated_tactic": ["have a'_ne_u : a' \u2209 u := fun H => (hne _ a'A.1).<a>ne_empty</a> (<a>disjoint_self</a>.1 (a'A.2 _ H))", [{"full_name": "Set.Nonempty.ne_empty", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [622, 8], "def_end_pos": [622, 25]}, {"full_name": "disjoint_self", "def_path": "Mathlib/Order/Disjoint.lean", "def_pos": [79, 9], "def_end_pos": [79, 22]}]], "state_before": "case intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\nB : \u03b9 \u2192 Set \u03b1\nt : Set \u03b9\n\u03b4 : \u03b9 \u2192 \u211d\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\n\u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a\nR : \u211d\n\u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R\nhne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a)\nT : Set (Set \u03b9) :=\n  {u |\n    u \u2286 t \u2227\n      PairwiseDisjoint u B \u2227\n        \u2200 (a : \u03b9),\n          a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c}\nu : Set \u03b9\nuT : u \u2208 T\nA : Set \u03b9 := {a' | a' \u2208 t \u2227 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a') (B c)}\nAnonempty : Set.Nonempty A\nm : \u211d := sSup (\u03b4 '' A)\nbddA : BddAbove (\u03b4 '' A)\na' : \u03b9\na'A : a' \u2208 A\nha' : m / \u03c4 \u2264 \u03b4 a'\n\u22a2 \u2203 v, v \u2208 T \u2227 u \u2286 v \u2227 v \u2260 u", "state_after": "case intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\nB : \u03b9 \u2192 Set \u03b1\nt : Set \u03b9\n\u03b4 : \u03b9 \u2192 \u211d\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\n\u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a\nR : \u211d\n\u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R\nhne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a)\nT : Set (Set \u03b9) :=\n  {u |\n    u \u2286 t \u2227\n      PairwiseDisjoint u B \u2227\n        \u2200 (a : \u03b9),\n          a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c}\nu : Set \u03b9\nuT : u \u2208 T\nA : Set \u03b9 := {a' | a' \u2208 t \u2227 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a') (B c)}\nAnonempty : Set.Nonempty A\nm : \u211d := sSup (\u03b4 '' A)\nbddA : BddAbove (\u03b4 '' A)\na' : \u03b9\na'A : a' \u2208 A\nha' : m / \u03c4 \u2264 \u03b4 a'\na'_ne_u : \u00aca' \u2208 u\n\u22a2 \u2203 v, v \u2208 T \u2227 u \u2286 v \u2227 v \u2260 u"}, {"tactic": "refine' \u27e8insert a' u, \u27e8_, _, _\u27e9, subset_insert _ _, (ne_insert_of_not_mem _ a'_ne_u).symm\u27e9", "annotated_tactic": ["refine' \u27e8<a>insert</a> a' u, \u27e8_, _, _\u27e9, <a>subset_insert</a> _ _, (<a>ne_insert_of_not_mem</a> _ a'_ne_u).<a>symm</a>\u27e9", [{"full_name": "Insert.insert", "def_path": "lake-packages/std/Std/Classes/SetNotation.lean", "def_pos": [69, 3], "def_end_pos": [69, 9]}, {"full_name": "Set.subset_insert", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1123, 9], "def_end_pos": [1123, 22]}, {"full_name": "Set.ne_insert_of_not_mem", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1156, 9], "def_end_pos": [1156, 29]}, {"full_name": "Ne.symm", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [575, 9], "def_end_pos": [575, 16]}]], "state_before": "case intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\nB : \u03b9 \u2192 Set \u03b1\nt : Set \u03b9\n\u03b4 : \u03b9 \u2192 \u211d\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\n\u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a\nR : \u211d\n\u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R\nhne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a)\nT : Set (Set \u03b9) :=\n  {u |\n    u \u2286 t \u2227\n      PairwiseDisjoint u B \u2227\n        \u2200 (a : \u03b9),\n          a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c}\nu : Set \u03b9\nuT : u \u2208 T\nA : Set \u03b9 := {a' | a' \u2208 t \u2227 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a') (B c)}\nAnonempty : Set.Nonempty A\nm : \u211d := sSup (\u03b4 '' A)\nbddA : BddAbove (\u03b4 '' A)\na' : \u03b9\na'A : a' \u2208 A\nha' : m / \u03c4 \u2264 \u03b4 a'\na'_ne_u : \u00aca' \u2208 u\n\u22a2 \u2203 v, v \u2208 T \u2227 u \u2286 v \u2227 v \u2260 u", "state_after": "case intro.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\n\u03b9 : Type u_2\nB : \u03b9 \u2192 Set \u03b1\nt : Set \u03b9\n\u03b4 : \u03b9 \u2192 \u211d\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\n\u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a\nR : \u211d\n\u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R\nhne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a)\nT : Set (Set \u03b9) :=\n  {u |\n    u \u2286 t \u2227\n      PairwiseDisjoint u B \u2227\n        \u2200 (a : \u03b9),\n          a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c}\nu : Set \u03b9\nuT : u \u2208 T\nA : Set \u03b9 := {a' | a' \u2208 t \u2227 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a') (B c)}\nAnonempty : Set.Nonempty A\nm : \u211d := sSup (\u03b4 '' A)\nbddA : BddAbove (\u03b4 '' A)\na' : \u03b9\na'A : a' \u2208 A\nha' : m / \u03c4 \u2264 \u03b4 a'\na'_ne_u : \u00aca' \u2208 u\n\u22a2 insert a' u \u2286 t\n\ncase intro.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\n\u03b9 : Type u_2\nB : \u03b9 \u2192 Set \u03b1\nt : Set \u03b9\n\u03b4 : \u03b9 \u2192 \u211d\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\n\u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a\nR : \u211d\n\u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R\nhne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a)\nT : Set (Set \u03b9) :=\n  {u |\n    u \u2286 t \u2227\n      PairwiseDisjoint u B \u2227\n        \u2200 (a : \u03b9),\n          a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c}\nu : Set \u03b9\nuT : u \u2208 T\nA : Set \u03b9 := {a' | a' \u2208 t \u2227 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a') (B c)}\nAnonempty : Set.Nonempty A\nm : \u211d := sSup (\u03b4 '' A)\nbddA : BddAbove (\u03b4 '' A)\na' : \u03b9\na'A : a' \u2208 A\nha' : m / \u03c4 \u2264 \u03b4 a'\na'_ne_u : \u00aca' \u2208 u\n\u22a2 PairwiseDisjoint (insert a' u) B\n\ncase intro.intro.intro.intro.refine'_3\n\u03b1 : Type u_1\n\u03b9 : Type u_2\nB : \u03b9 \u2192 Set \u03b1\nt : Set \u03b9\n\u03b4 : \u03b9 \u2192 \u211d\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\n\u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a\nR : \u211d\n\u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R\nhne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a)\nT : Set (Set \u03b9) :=\n  {u |\n    u \u2286 t \u2227\n      PairwiseDisjoint u B \u2227\n        \u2200 (a : \u03b9),\n          a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c}\nu : Set \u03b9\nuT : u \u2208 T\nA : Set \u03b9 := {a' | a' \u2208 t \u2227 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a') (B c)}\nAnonempty : Set.Nonempty A\nm : \u211d := sSup (\u03b4 '' A)\nbddA : BddAbove (\u03b4 '' A)\na' : \u03b9\na'A : a' \u2208 A\nha' : m / \u03c4 \u2264 \u03b4 a'\na'_ne_u : \u00aca' \u2208 u\n\u22a2 \u2200 (a : \u03b9),\n    a \u2208 t \u2192\n      \u2200 (b : \u03b9),\n        b \u2208 insert a' u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 insert a' u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c"}, {"tactic": "refine' zorn_subset _ fun U UT hU => _", "annotated_tactic": ["refine' <a>zorn_subset</a> _ fun U UT hU => _", [{"full_name": "zorn_subset", "def_path": "Mathlib/Order/Zorn.lean", "def_pos": [186, 9], "def_end_pos": [186, 20]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\nB : \u03b9 \u2192 Set \u03b1\nt : Set \u03b9\n\u03b4 : \u03b9 \u2192 \u211d\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\n\u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a\nR : \u211d\n\u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R\nhne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a)\nT : Set (Set \u03b9) :=\n  {u |\n    u \u2286 t \u2227\n      PairwiseDisjoint u B \u2227\n        \u2200 (a : \u03b9),\n          a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c}\n\u22a2 \u2203 u, u \u2208 T \u2227 \u2200 (v : Set \u03b9), v \u2208 T \u2192 u \u2286 v \u2192 v = u", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\nB : \u03b9 \u2192 Set \u03b1\nt : Set \u03b9\n\u03b4 : \u03b9 \u2192 \u211d\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\n\u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a\nR : \u211d\n\u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R\nhne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a)\nT : Set (Set \u03b9) :=\n  {u |\n    u \u2286 t \u2227\n      PairwiseDisjoint u B \u2227\n        \u2200 (a : \u03b9),\n          a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c}\nU : Set (Set \u03b9)\nUT : U \u2286 T\nhU : IsChain (fun x x_1 => x \u2286 x_1) U\n\u22a2 \u2203 ub, ub \u2208 T \u2227 \u2200 (s : Set \u03b9), s \u2208 U \u2192 s \u2286 ub"}, {"tactic": "refine' \u27e8\u22c3\u2080 U, _, fun s hs => subset_sUnion_of_mem hs\u27e9", "annotated_tactic": ["refine' \u27e8\u22c3\u2080 U, _, fun s hs => <a>subset_sUnion_of_mem</a> hs\u27e9", [{"full_name": "Set.subset_sUnion_of_mem", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [1157, 9], "def_end_pos": [1157, 29]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\nB : \u03b9 \u2192 Set \u03b1\nt : Set \u03b9\n\u03b4 : \u03b9 \u2192 \u211d\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\n\u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a\nR : \u211d\n\u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R\nhne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a)\nT : Set (Set \u03b9) :=\n  {u |\n    u \u2286 t \u2227\n      PairwiseDisjoint u B \u2227\n        \u2200 (a : \u03b9),\n          a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c}\nU : Set (Set \u03b9)\nUT : U \u2286 T\nhU : IsChain (fun x x_1 => x \u2286 x_1) U\n\u22a2 \u2203 ub, ub \u2208 T \u2227 \u2200 (s : Set \u03b9), s \u2208 U \u2192 s \u2286 ub", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\nB : \u03b9 \u2192 Set \u03b1\nt : Set \u03b9\n\u03b4 : \u03b9 \u2192 \u211d\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\n\u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a\nR : \u211d\n\u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R\nhne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a)\nT : Set (Set \u03b9) :=\n  {u |\n    u \u2286 t \u2227\n      PairwiseDisjoint u B \u2227\n        \u2200 (a : \u03b9),\n          a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c}\nU : Set (Set \u03b9)\nUT : U \u2286 T\nhU : IsChain (fun x x_1 => x \u2286 x_1) U\n\u22a2 \u22c3\u2080 U \u2208 T"}, {"tactic": "simp only [Set.sUnion_subset_iff, and_imp, exists_prop, forall_exists_index, mem_sUnion,\n  Set.mem_setOf_eq]", "annotated_tactic": ["simp only [<a>Set.sUnion_subset_iff</a>, <a>and_imp</a>, <a>exists_prop</a>, <a>forall_exists_index</a>, <a>mem_sUnion</a>,\n      <a>Set.mem_setOf_eq</a>]", [{"full_name": "Set.sUnion_subset_iff", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [1171, 9], "def_end_pos": [1171, 26]}, {"full_name": "and_imp", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [313, 17], "def_end_pos": [313, 24]}, {"full_name": "exists_prop", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [485, 17], "def_end_pos": [485, 28]}, {"full_name": "forall_exists_index", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [356, 17], "def_end_pos": [356, 36]}, {"full_name": "Set.mem_sUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [98, 9], "def_end_pos": [98, 19]}, {"full_name": "Set.mem_setOf_eq", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [256, 29], "def_end_pos": [256, 41]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\nB : \u03b9 \u2192 Set \u03b1\nt : Set \u03b9\n\u03b4 : \u03b9 \u2192 \u211d\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\n\u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a\nR : \u211d\n\u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R\nhne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a)\nT : Set (Set \u03b9) :=\n  {u |\n    u \u2286 t \u2227\n      PairwiseDisjoint u B \u2227\n        \u2200 (a : \u03b9),\n          a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c}\nU : Set (Set \u03b9)\nUT : U \u2286 T\nhU : IsChain (fun x x_1 => x \u2286 x_1) U\n\u22a2 \u22c3\u2080 U \u2208 T", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\nB : \u03b9 \u2192 Set \u03b1\nt : Set \u03b9\n\u03b4 : \u03b9 \u2192 \u211d\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\n\u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a\nR : \u211d\n\u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R\nhne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a)\nT : Set (Set \u03b9) :=\n  {u |\n    u \u2286 t \u2227\n      PairwiseDisjoint u B \u2227\n        \u2200 (a : \u03b9),\n          a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c}\nU : Set (Set \u03b9)\nUT : U \u2286 T\nhU : IsChain (fun x x_1 => x \u2286 x_1) U\n\u22a2 (\u2200 (t' : Set \u03b9), t' \u2208 U \u2192 t' \u2286 t) \u2227\n    PairwiseDisjoint (\u22c3\u2080 U) B \u2227\n      \u2200 (a : \u03b9),\n        a \u2208 t \u2192\n          \u2200 (b : \u03b9) (x : Set \u03b9),\n            x \u2208 U \u2192\n              b \u2208 x \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, (\u2203 t, t \u2208 U \u2227 c \u2208 t) \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c"}, {"tactic": "refine'\n  \u27e8fun u hu => (UT hu).1, (pairwiseDisjoint_sUnion hU.directedOn).2 fun u hu => (UT hu).2.1,\n    fun a hat b u uU hbu hab => _\u27e9", "annotated_tactic": ["refine'\n      \u27e8fun u hu => (UT hu).1, (<a>pairwiseDisjoint_sUnion</a> hU.directedOn).2 fun u hu => (UT hu).2.1,\n        fun a hat b u uU hbu hab => _\u27e9", [{"full_name": "Set.pairwiseDisjoint_sUnion", "def_path": "Mathlib/Data/Set/Pairwise/Lattice.lean", "def_pos": [59, 9], "def_end_pos": [59, 32]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\nB : \u03b9 \u2192 Set \u03b1\nt : Set \u03b9\n\u03b4 : \u03b9 \u2192 \u211d\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\n\u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a\nR : \u211d\n\u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R\nhne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a)\nT : Set (Set \u03b9) :=\n  {u |\n    u \u2286 t \u2227\n      PairwiseDisjoint u B \u2227\n        \u2200 (a : \u03b9),\n          a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c}\nU : Set (Set \u03b9)\nUT : U \u2286 T\nhU : IsChain (fun x x_1 => x \u2286 x_1) U\n\u22a2 (\u2200 (t' : Set \u03b9), t' \u2208 U \u2192 t' \u2286 t) \u2227\n    PairwiseDisjoint (\u22c3\u2080 U) B \u2227\n      \u2200 (a : \u03b9),\n        a \u2208 t \u2192\n          \u2200 (b : \u03b9) (x : Set \u03b9),\n            x \u2208 U \u2192\n              b \u2208 x \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, (\u2203 t, t \u2208 U \u2227 c \u2208 t) \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\nB : \u03b9 \u2192 Set \u03b1\nt : Set \u03b9\n\u03b4 : \u03b9 \u2192 \u211d\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\n\u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a\nR : \u211d\n\u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R\nhne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a)\nT : Set (Set \u03b9) :=\n  {u |\n    u \u2286 t \u2227\n      PairwiseDisjoint u B \u2227\n        \u2200 (a : \u03b9),\n          a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c}\nU : Set (Set \u03b9)\nUT : U \u2286 T\nhU : IsChain (fun x x_1 => x \u2286 x_1) U\na : \u03b9\nhat : a \u2208 t\nb : \u03b9\nu : Set \u03b9\nuU : u \u2208 U\nhbu : b \u2208 u\nhab : Set.Nonempty (B a \u2229 B b)\n\u22a2 \u2203 c, (\u2203 t, t \u2208 U \u2227 c \u2208 t) \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c"}, {"tactic": "obtain \u27e8c, cu, ac, hc\u27e9 : \u2203 c, c \u2208 u \u2227 (B a \u2229 B c).Nonempty \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c :=\n  (UT uU).2.2 a hat b hbu hab", "annotated_tactic": ["obtain \u27e8c, cu, ac, hc\u27e9 : \u2203 c, c \u2208 u \u2227 (B a \u2229 B c).<a>Nonempty</a> \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c :=\n      (UT uU).2.2 a hat b hbu hab", [{"full_name": "Set.Nonempty", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [439, 15], "def_end_pos": [439, 23]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\nB : \u03b9 \u2192 Set \u03b1\nt : Set \u03b9\n\u03b4 : \u03b9 \u2192 \u211d\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\n\u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a\nR : \u211d\n\u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R\nhne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a)\nT : Set (Set \u03b9) :=\n  {u |\n    u \u2286 t \u2227\n      PairwiseDisjoint u B \u2227\n        \u2200 (a : \u03b9),\n          a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c}\nU : Set (Set \u03b9)\nUT : U \u2286 T\nhU : IsChain (fun x x_1 => x \u2286 x_1) U\na : \u03b9\nhat : a \u2208 t\nb : \u03b9\nu : Set \u03b9\nuU : u \u2208 U\nhbu : b \u2208 u\nhab : Set.Nonempty (B a \u2229 B b)\n\u22a2 \u2203 c, (\u2203 t, t \u2208 U \u2227 c \u2208 t) \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\nB : \u03b9 \u2192 Set \u03b1\nt : Set \u03b9\n\u03b4 : \u03b9 \u2192 \u211d\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\n\u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a\nR : \u211d\n\u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R\nhne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a)\nT : Set (Set \u03b9) :=\n  {u |\n    u \u2286 t \u2227\n      PairwiseDisjoint u B \u2227\n        \u2200 (a : \u03b9),\n          a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c}\nU : Set (Set \u03b9)\nUT : U \u2286 T\nhU : IsChain (fun x x_1 => x \u2286 x_1) U\na : \u03b9\nhat : a \u2208 t\nb : \u03b9\nu : Set \u03b9\nuU : u \u2208 U\nhbu : b \u2208 u\nhab : Set.Nonempty (B a \u2229 B b)\nc : \u03b9\ncu : c \u2208 u\nac : Set.Nonempty (B a \u2229 B c)\nhc : \u03b4 a \u2264 \u03c4 * \u03b4 c\n\u22a2 \u2203 c, (\u2203 t, t \u2208 U \u2227 c \u2208 t) \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c"}, {"tactic": "exact \u27e8c, \u27e8u, uU, cu\u27e9, ac, hc\u27e9", "annotated_tactic": ["exact \u27e8c, \u27e8u, uU, cu\u27e9, ac, hc\u27e9", []], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\nB : \u03b9 \u2192 Set \u03b1\nt : Set \u03b9\n\u03b4 : \u03b9 \u2192 \u211d\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\n\u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a\nR : \u211d\n\u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R\nhne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a)\nT : Set (Set \u03b9) :=\n  {u |\n    u \u2286 t \u2227\n      PairwiseDisjoint u B \u2227\n        \u2200 (a : \u03b9),\n          a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c}\nU : Set (Set \u03b9)\nUT : U \u2286 T\nhU : IsChain (fun x x_1 => x \u2286 x_1) U\na : \u03b9\nhat : a \u2208 t\nb : \u03b9\nu : Set \u03b9\nuU : u \u2208 U\nhbu : b \u2208 u\nhab : Set.Nonempty (B a \u2229 B b)\nc : \u03b9\ncu : c \u2208 u\nac : Set.Nonempty (B a \u2229 B c)\nhc : \u03b4 a \u2264 \u03c4 * \u03b4 c\n\u22a2 \u2203 c, (\u2203 t, t \u2208 U \u2227 c \u2208 t) \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c", "state_after": "no goals"}, {"tactic": "intro c hc", "annotated_tactic": ["intro c hc", []], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\nB : \u03b9 \u2192 Set \u03b1\nt : Set \u03b9\n\u03b4 : \u03b9 \u2192 \u211d\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\n\u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a\nR : \u211d\n\u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R\nhne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a)\nT : Set (Set \u03b9) :=\n  {u |\n    u \u2286 t \u2227\n      PairwiseDisjoint u B \u2227\n        \u2200 (a : \u03b9),\n          a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c}\nu : Set \u03b9\nuT : u \u2208 T\na : \u03b9\nhat : a \u2208 t\nhu : \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u03c4 * \u03b4 b < \u03b4 a\n\u22a2 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a) (B c)", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\nB : \u03b9 \u2192 Set \u03b1\nt : Set \u03b9\n\u03b4 : \u03b9 \u2192 \u211d\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\n\u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a\nR : \u211d\n\u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R\nhne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a)\nT : Set (Set \u03b9) :=\n  {u |\n    u \u2286 t \u2227\n      PairwiseDisjoint u B \u2227\n        \u2200 (a : \u03b9),\n          a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c}\nu : Set \u03b9\nuT : u \u2208 T\na : \u03b9\nhat : a \u2208 t\nhu : \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u03c4 * \u03b4 b < \u03b4 a\nc : \u03b9\nhc : c \u2208 u\n\u22a2 Disjoint (B a) (B c)"}, {"tactic": "by_contra h", "annotated_tactic": ["by_contra h", []], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\nB : \u03b9 \u2192 Set \u03b1\nt : Set \u03b9\n\u03b4 : \u03b9 \u2192 \u211d\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\n\u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a\nR : \u211d\n\u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R\nhne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a)\nT : Set (Set \u03b9) :=\n  {u |\n    u \u2286 t \u2227\n      PairwiseDisjoint u B \u2227\n        \u2200 (a : \u03b9),\n          a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c}\nu : Set \u03b9\nuT : u \u2208 T\na : \u03b9\nhat : a \u2208 t\nhu : \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u03c4 * \u03b4 b < \u03b4 a\nc : \u03b9\nhc : c \u2208 u\n\u22a2 Disjoint (B a) (B c)", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\nB : \u03b9 \u2192 Set \u03b1\nt : Set \u03b9\n\u03b4 : \u03b9 \u2192 \u211d\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\n\u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a\nR : \u211d\n\u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R\nhne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a)\nT : Set (Set \u03b9) :=\n  {u |\n    u \u2286 t \u2227\n      PairwiseDisjoint u B \u2227\n        \u2200 (a : \u03b9),\n          a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c}\nu : Set \u03b9\nuT : u \u2208 T\na : \u03b9\nhat : a \u2208 t\nhu : \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u03c4 * \u03b4 b < \u03b4 a\nc : \u03b9\nhc : c \u2208 u\nh : \u00acDisjoint (B a) (B c)\n\u22a2 False"}, {"tactic": "rw [not_disjoint_iff_nonempty_inter] at h", "annotated_tactic": ["rw [<a>not_disjoint_iff_nonempty_inter</a>] at h", [{"full_name": "Set.not_disjoint_iff_nonempty_inter", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1557, 7], "def_end_pos": [1557, 38]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\nB : \u03b9 \u2192 Set \u03b1\nt : Set \u03b9\n\u03b4 : \u03b9 \u2192 \u211d\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\n\u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a\nR : \u211d\n\u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R\nhne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a)\nT : Set (Set \u03b9) :=\n  {u |\n    u \u2286 t \u2227\n      PairwiseDisjoint u B \u2227\n        \u2200 (a : \u03b9),\n          a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c}\nu : Set \u03b9\nuT : u \u2208 T\na : \u03b9\nhat : a \u2208 t\nhu : \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u03c4 * \u03b4 b < \u03b4 a\nc : \u03b9\nhc : c \u2208 u\nh : \u00acDisjoint (B a) (B c)\n\u22a2 False", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\nB : \u03b9 \u2192 Set \u03b1\nt : Set \u03b9\n\u03b4 : \u03b9 \u2192 \u211d\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\n\u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a\nR : \u211d\n\u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R\nhne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a)\nT : Set (Set \u03b9) :=\n  {u |\n    u \u2286 t \u2227\n      PairwiseDisjoint u B \u2227\n        \u2200 (a : \u03b9),\n          a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c}\nu : Set \u03b9\nuT : u \u2208 T\na : \u03b9\nhat : a \u2208 t\nhu : \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u03c4 * \u03b4 b < \u03b4 a\nc : \u03b9\nhc : c \u2208 u\nh : Set.Nonempty (B a \u2229 B c)\n\u22a2 False"}, {"tactic": "obtain \u27e8d, du, ad, hd\u27e9 : \u2203 d, d \u2208 u \u2227 (B a \u2229 B d).Nonempty \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 d :=\n  uT.2.2 a hat c hc h", "annotated_tactic": ["obtain \u27e8d, du, ad, hd\u27e9 : \u2203 d, d \u2208 u \u2227 (B a \u2229 B d).<a>Nonempty</a> \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 d :=\n      uT.2.2 a hat c hc h", [{"full_name": "Set.Nonempty", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [439, 15], "def_end_pos": [439, 23]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\nB : \u03b9 \u2192 Set \u03b1\nt : Set \u03b9\n\u03b4 : \u03b9 \u2192 \u211d\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\n\u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a\nR : \u211d\n\u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R\nhne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a)\nT : Set (Set \u03b9) :=\n  {u |\n    u \u2286 t \u2227\n      PairwiseDisjoint u B \u2227\n        \u2200 (a : \u03b9),\n          a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c}\nu : Set \u03b9\nuT : u \u2208 T\na : \u03b9\nhat : a \u2208 t\nhu : \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u03c4 * \u03b4 b < \u03b4 a\nc : \u03b9\nhc : c \u2208 u\nh : Set.Nonempty (B a \u2229 B c)\n\u22a2 False", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\nB : \u03b9 \u2192 Set \u03b1\nt : Set \u03b9\n\u03b4 : \u03b9 \u2192 \u211d\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\n\u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a\nR : \u211d\n\u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R\nhne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a)\nT : Set (Set \u03b9) :=\n  {u |\n    u \u2286 t \u2227\n      PairwiseDisjoint u B \u2227\n        \u2200 (a : \u03b9),\n          a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c}\nu : Set \u03b9\nuT : u \u2208 T\na : \u03b9\nhat : a \u2208 t\nhu : \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u03c4 * \u03b4 b < \u03b4 a\nc : \u03b9\nhc : c \u2208 u\nh : Set.Nonempty (B a \u2229 B c)\nd : \u03b9\ndu : d \u2208 u\nad : Set.Nonempty (B a \u2229 B d)\nhd : \u03b4 a \u2264 \u03c4 * \u03b4 d\n\u22a2 False"}, {"tactic": "exact lt_irrefl _ ((hu d du ad).trans_le hd)", "annotated_tactic": ["exact <a>lt_irrefl</a> _ ((hu d du ad).<a>trans_le</a> hd)", [{"full_name": "lt_irrefl", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [79, 9], "def_end_pos": [79, 18]}, {"full_name": "LT.lt.trans_le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [148, 7], "def_end_pos": [148, 21]}]], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\nB : \u03b9 \u2192 Set \u03b1\nt : Set \u03b9\n\u03b4 : \u03b9 \u2192 \u211d\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\n\u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a\nR : \u211d\n\u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R\nhne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a)\nT : Set (Set \u03b9) :=\n  {u |\n    u \u2286 t \u2227\n      PairwiseDisjoint u B \u2227\n        \u2200 (a : \u03b9),\n          a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c}\nu : Set \u03b9\nuT : u \u2208 T\na : \u03b9\nhat : a \u2208 t\nhu : \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u03c4 * \u03b4 b < \u03b4 a\nc : \u03b9\nhc : c \u2208 u\nh : Set.Nonempty (B a \u2229 B c)\nd : \u03b9\ndu : d \u2208 u\nad : Set.Nonempty (B a \u2229 B d)\nhd : \u03b4 a \u2264 \u03c4 * \u03b4 d\n\u22a2 False", "state_after": "no goals"}, {"tactic": "refine' \u27e8R, fun x xA => _\u27e9", "annotated_tactic": ["refine' \u27e8R, fun x xA => _\u27e9", []], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\nB : \u03b9 \u2192 Set \u03b1\nt : Set \u03b9\n\u03b4 : \u03b9 \u2192 \u211d\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\n\u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a\nR : \u211d\n\u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R\nhne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a)\nT : Set (Set \u03b9) :=\n  {u |\n    u \u2286 t \u2227\n      PairwiseDisjoint u B \u2227\n        \u2200 (a : \u03b9),\n          a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c}\nu : Set \u03b9\nuT : u \u2208 T\na : \u03b9\nhat : a \u2208 t\nhu : \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u03c4 * \u03b4 b < \u03b4 a\na_disj : \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a) (B c)\nA : Set \u03b9 := {a' | a' \u2208 t \u2227 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a') (B c)}\nAnonempty : Set.Nonempty A\nm : \u211d := sSup (\u03b4 '' A)\n\u22a2 BddAbove (\u03b4 '' A)", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\nB : \u03b9 \u2192 Set \u03b1\nt : Set \u03b9\n\u03b4 : \u03b9 \u2192 \u211d\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\n\u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a\nR : \u211d\n\u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R\nhne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a)\nT : Set (Set \u03b9) :=\n  {u |\n    u \u2286 t \u2227\n      PairwiseDisjoint u B \u2227\n        \u2200 (a : \u03b9),\n          a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c}\nu : Set \u03b9\nuT : u \u2208 T\na : \u03b9\nhat : a \u2208 t\nhu : \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u03c4 * \u03b4 b < \u03b4 a\na_disj : \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a) (B c)\nA : Set \u03b9 := {a' | a' \u2208 t \u2227 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a') (B c)}\nAnonempty : Set.Nonempty A\nm : \u211d := sSup (\u03b4 '' A)\nx : \u211d\nxA : x \u2208 \u03b4 '' A\n\u22a2 x \u2264 R"}, {"tactic": "rcases (mem_image _ _ _).1 xA with \u27e8a', ha', rfl\u27e9", "annotated_tactic": ["rcases (<a>mem_image</a> _ _ _).1 xA with \u27e8a', ha', rfl\u27e9", [{"full_name": "Set.mem_image", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [231, 9], "def_end_pos": [231, 18]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\nB : \u03b9 \u2192 Set \u03b1\nt : Set \u03b9\n\u03b4 : \u03b9 \u2192 \u211d\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\n\u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a\nR : \u211d\n\u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R\nhne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a)\nT : Set (Set \u03b9) :=\n  {u |\n    u \u2286 t \u2227\n      PairwiseDisjoint u B \u2227\n        \u2200 (a : \u03b9),\n          a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c}\nu : Set \u03b9\nuT : u \u2208 T\na : \u03b9\nhat : a \u2208 t\nhu : \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u03c4 * \u03b4 b < \u03b4 a\na_disj : \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a) (B c)\nA : Set \u03b9 := {a' | a' \u2208 t \u2227 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a') (B c)}\nAnonempty : Set.Nonempty A\nm : \u211d := sSup (\u03b4 '' A)\nx : \u211d\nxA : x \u2208 \u03b4 '' A\n\u22a2 x \u2264 R", "state_after": "case intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\nB : \u03b9 \u2192 Set \u03b1\nt : Set \u03b9\n\u03b4 : \u03b9 \u2192 \u211d\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\n\u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a\nR : \u211d\n\u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R\nhne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a)\nT : Set (Set \u03b9) :=\n  {u |\n    u \u2286 t \u2227\n      PairwiseDisjoint u B \u2227\n        \u2200 (a : \u03b9),\n          a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c}\nu : Set \u03b9\nuT : u \u2208 T\na : \u03b9\nhat : a \u2208 t\nhu : \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u03c4 * \u03b4 b < \u03b4 a\na_disj : \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a) (B c)\nA : Set \u03b9 := {a' | a' \u2208 t \u2227 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a') (B c)}\nAnonempty : Set.Nonempty A\nm : \u211d := sSup (\u03b4 '' A)\na' : \u03b9\nha' : a' \u2208 A\nxA : \u03b4 a' \u2208 \u03b4 '' A\n\u22a2 \u03b4 a' \u2264 R"}, {"tactic": "exact \u03b4le a' ha'.1", "annotated_tactic": ["exact \u03b4le a' ha'.1", []], "state_before": "case intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\nB : \u03b9 \u2192 Set \u03b1\nt : Set \u03b9\n\u03b4 : \u03b9 \u2192 \u211d\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\n\u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a\nR : \u211d\n\u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R\nhne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a)\nT : Set (Set \u03b9) :=\n  {u |\n    u \u2286 t \u2227\n      PairwiseDisjoint u B \u2227\n        \u2200 (a : \u03b9),\n          a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c}\nu : Set \u03b9\nuT : u \u2208 T\na : \u03b9\nhat : a \u2208 t\nhu : \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u03c4 * \u03b4 b < \u03b4 a\na_disj : \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a) (B c)\nA : Set \u03b9 := {a' | a' \u2208 t \u2227 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a') (B c)}\nAnonempty : Set.Nonempty A\nm : \u211d := sSup (\u03b4 '' A)\na' : \u03b9\nha' : a' \u2208 A\nxA : \u03b4 a' \u2208 \u03b4 '' A\n\u22a2 \u03b4 a' \u2264 R", "state_after": "no goals"}, {"tactic": "have : 0 \u2264 m := (\u03b4nonneg a hat).trans (le_csSup bddA (mem_image_of_mem _ \u27e8hat, a_disj\u27e9))", "annotated_tactic": ["have : 0 \u2264 m := (\u03b4nonneg a hat).<a>trans</a> (<a>le_csSup</a> bddA (<a>mem_image_of_mem</a> _ \u27e8hat, a_disj\u27e9))", [{"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [120, 7], "def_end_pos": [120, 18]}, {"full_name": "le_csSup", "def_path": "Mathlib/Order/ConditionallyCompleteLattice/Basic.lean", "def_pos": [457, 9], "def_end_pos": [457, 17]}, {"full_name": "Set.mem_image_of_mem", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [240, 9], "def_end_pos": [240, 25]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\nB : \u03b9 \u2192 Set \u03b1\nt : Set \u03b9\n\u03b4 : \u03b9 \u2192 \u211d\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\n\u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a\nR : \u211d\n\u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R\nhne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a)\nT : Set (Set \u03b9) :=\n  {u |\n    u \u2286 t \u2227\n      PairwiseDisjoint u B \u2227\n        \u2200 (a : \u03b9),\n          a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c}\nu : Set \u03b9\nuT : u \u2208 T\na : \u03b9\nhat : a \u2208 t\nhu : \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u03c4 * \u03b4 b < \u03b4 a\na_disj : \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a) (B c)\nA : Set \u03b9 := {a' | a' \u2208 t \u2227 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a') (B c)}\nAnonempty : Set.Nonempty A\nm : \u211d := sSup (\u03b4 '' A)\nbddA : BddAbove (\u03b4 '' A)\n\u22a2 \u2203 a', a' \u2208 A \u2227 m / \u03c4 \u2264 \u03b4 a'", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\nB : \u03b9 \u2192 Set \u03b1\nt : Set \u03b9\n\u03b4 : \u03b9 \u2192 \u211d\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\n\u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a\nR : \u211d\n\u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R\nhne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a)\nT : Set (Set \u03b9) :=\n  {u |\n    u \u2286 t \u2227\n      PairwiseDisjoint u B \u2227\n        \u2200 (a : \u03b9),\n          a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c}\nu : Set \u03b9\nuT : u \u2208 T\na : \u03b9\nhat : a \u2208 t\nhu : \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u03c4 * \u03b4 b < \u03b4 a\na_disj : \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a) (B c)\nA : Set \u03b9 := {a' | a' \u2208 t \u2227 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a') (B c)}\nAnonempty : Set.Nonempty A\nm : \u211d := sSup (\u03b4 '' A)\nbddA : BddAbove (\u03b4 '' A)\nthis : 0 \u2264 m\n\u22a2 \u2203 a', a' \u2208 A \u2227 m / \u03c4 \u2264 \u03b4 a'"}, {"tactic": "rcases eq_or_lt_of_le this with (mzero | mpos)", "annotated_tactic": ["rcases <a>eq_or_lt_of_le</a> this with (mzero | mpos)", [{"full_name": "eq_or_lt_of_le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [414, 9], "def_end_pos": [414, 23]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\nB : \u03b9 \u2192 Set \u03b1\nt : Set \u03b9\n\u03b4 : \u03b9 \u2192 \u211d\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\n\u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a\nR : \u211d\n\u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R\nhne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a)\nT : Set (Set \u03b9) :=\n  {u |\n    u \u2286 t \u2227\n      PairwiseDisjoint u B \u2227\n        \u2200 (a : \u03b9),\n          a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c}\nu : Set \u03b9\nuT : u \u2208 T\na : \u03b9\nhat : a \u2208 t\nhu : \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u03c4 * \u03b4 b < \u03b4 a\na_disj : \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a) (B c)\nA : Set \u03b9 := {a' | a' \u2208 t \u2227 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a') (B c)}\nAnonempty : Set.Nonempty A\nm : \u211d := sSup (\u03b4 '' A)\nbddA : BddAbove (\u03b4 '' A)\nthis : 0 \u2264 m\n\u22a2 \u2203 a', a' \u2208 A \u2227 m / \u03c4 \u2264 \u03b4 a'", "state_after": "case inl\n\u03b1 : Type u_1\n\u03b9 : Type u_2\nB : \u03b9 \u2192 Set \u03b1\nt : Set \u03b9\n\u03b4 : \u03b9 \u2192 \u211d\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\n\u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a\nR : \u211d\n\u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R\nhne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a)\nT : Set (Set \u03b9) :=\n  {u |\n    u \u2286 t \u2227\n      PairwiseDisjoint u B \u2227\n        \u2200 (a : \u03b9),\n          a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c}\nu : Set \u03b9\nuT : u \u2208 T\na : \u03b9\nhat : a \u2208 t\nhu : \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u03c4 * \u03b4 b < \u03b4 a\na_disj : \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a) (B c)\nA : Set \u03b9 := {a' | a' \u2208 t \u2227 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a') (B c)}\nAnonempty : Set.Nonempty A\nm : \u211d := sSup (\u03b4 '' A)\nbddA : BddAbove (\u03b4 '' A)\nthis : 0 \u2264 m\nmzero : 0 = m\n\u22a2 \u2203 a', a' \u2208 A \u2227 m / \u03c4 \u2264 \u03b4 a'\n\ncase inr\n\u03b1 : Type u_1\n\u03b9 : Type u_2\nB : \u03b9 \u2192 Set \u03b1\nt : Set \u03b9\n\u03b4 : \u03b9 \u2192 \u211d\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\n\u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a\nR : \u211d\n\u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R\nhne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a)\nT : Set (Set \u03b9) :=\n  {u |\n    u \u2286 t \u2227\n      PairwiseDisjoint u B \u2227\n        \u2200 (a : \u03b9),\n          a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c}\nu : Set \u03b9\nuT : u \u2208 T\na : \u03b9\nhat : a \u2208 t\nhu : \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u03c4 * \u03b4 b < \u03b4 a\na_disj : \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a) (B c)\nA : Set \u03b9 := {a' | a' \u2208 t \u2227 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a') (B c)}\nAnonempty : Set.Nonempty A\nm : \u211d := sSup (\u03b4 '' A)\nbddA : BddAbove (\u03b4 '' A)\nthis : 0 \u2264 m\nmpos : 0 < m\n\u22a2 \u2203 a', a' \u2208 A \u2227 m / \u03c4 \u2264 \u03b4 a'"}, {"tactic": "refine' \u27e8a, \u27e8hat, a_disj\u27e9, _\u27e9", "annotated_tactic": ["refine' \u27e8a, \u27e8hat, a_disj\u27e9, _\u27e9", []], "state_before": "case inl\n\u03b1 : Type u_1\n\u03b9 : Type u_2\nB : \u03b9 \u2192 Set \u03b1\nt : Set \u03b9\n\u03b4 : \u03b9 \u2192 \u211d\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\n\u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a\nR : \u211d\n\u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R\nhne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a)\nT : Set (Set \u03b9) :=\n  {u |\n    u \u2286 t \u2227\n      PairwiseDisjoint u B \u2227\n        \u2200 (a : \u03b9),\n          a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c}\nu : Set \u03b9\nuT : u \u2208 T\na : \u03b9\nhat : a \u2208 t\nhu : \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u03c4 * \u03b4 b < \u03b4 a\na_disj : \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a) (B c)\nA : Set \u03b9 := {a' | a' \u2208 t \u2227 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a') (B c)}\nAnonempty : Set.Nonempty A\nm : \u211d := sSup (\u03b4 '' A)\nbddA : BddAbove (\u03b4 '' A)\nthis : 0 \u2264 m\nmzero : 0 = m\n\u22a2 \u2203 a', a' \u2208 A \u2227 m / \u03c4 \u2264 \u03b4 a'", "state_after": "case inl\n\u03b1 : Type u_1\n\u03b9 : Type u_2\nB : \u03b9 \u2192 Set \u03b1\nt : Set \u03b9\n\u03b4 : \u03b9 \u2192 \u211d\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\n\u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a\nR : \u211d\n\u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R\nhne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a)\nT : Set (Set \u03b9) :=\n  {u |\n    u \u2286 t \u2227\n      PairwiseDisjoint u B \u2227\n        \u2200 (a : \u03b9),\n          a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c}\nu : Set \u03b9\nuT : u \u2208 T\na : \u03b9\nhat : a \u2208 t\nhu : \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u03c4 * \u03b4 b < \u03b4 a\na_disj : \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a) (B c)\nA : Set \u03b9 := {a' | a' \u2208 t \u2227 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a') (B c)}\nAnonempty : Set.Nonempty A\nm : \u211d := sSup (\u03b4 '' A)\nbddA : BddAbove (\u03b4 '' A)\nthis : 0 \u2264 m\nmzero : 0 = m\n\u22a2 m / \u03c4 \u2264 \u03b4 a"}, {"tactic": "simpa only [\u2190 mzero, zero_div] using \u03b4nonneg a hat", "annotated_tactic": ["simpa only [\u2190 mzero, <a>zero_div</a>] using \u03b4nonneg a hat", [{"full_name": "zero_div", "def_path": "Mathlib/Algebra/GroupWithZero/Basic.lean", "def_pos": [291, 9], "def_end_pos": [291, 17]}]], "state_before": "case inl\n\u03b1 : Type u_1\n\u03b9 : Type u_2\nB : \u03b9 \u2192 Set \u03b1\nt : Set \u03b9\n\u03b4 : \u03b9 \u2192 \u211d\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\n\u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a\nR : \u211d\n\u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R\nhne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a)\nT : Set (Set \u03b9) :=\n  {u |\n    u \u2286 t \u2227\n      PairwiseDisjoint u B \u2227\n        \u2200 (a : \u03b9),\n          a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c}\nu : Set \u03b9\nuT : u \u2208 T\na : \u03b9\nhat : a \u2208 t\nhu : \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u03c4 * \u03b4 b < \u03b4 a\na_disj : \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a) (B c)\nA : Set \u03b9 := {a' | a' \u2208 t \u2227 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a') (B c)}\nAnonempty : Set.Nonempty A\nm : \u211d := sSup (\u03b4 '' A)\nbddA : BddAbove (\u03b4 '' A)\nthis : 0 \u2264 m\nmzero : 0 = m\n\u22a2 m / \u03c4 \u2264 \u03b4 a", "state_after": "no goals"}, {"tactic": "have I : m / \u03c4 < m := by\n  rw [div_lt_iff (zero_lt_one.trans h\u03c4)]\n  conv_lhs => rw [\u2190 mul_one m]\n  exact (mul_lt_mul_left mpos).2 h\u03c4", "annotated_tactic": ["have I : m / \u03c4 < m := by\n        rw [<a>div_lt_iff</a> (zero_lt_one.trans h\u03c4)]\n        conv_lhs => rw [\u2190 <a>mul_one</a> m]\n        exact (<a>mul_lt_mul_left</a> mpos).2 h\u03c4", [{"full_name": "div_lt_iff", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [169, 9], "def_end_pos": [169, 19]}, {"full_name": "mul_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [470, 9], "def_end_pos": [470, 16]}, {"full_name": "mul_lt_mul_left", "def_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "def_pos": [197, 9], "def_end_pos": [197, 24]}]], "state_before": "case inr\n\u03b1 : Type u_1\n\u03b9 : Type u_2\nB : \u03b9 \u2192 Set \u03b1\nt : Set \u03b9\n\u03b4 : \u03b9 \u2192 \u211d\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\n\u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a\nR : \u211d\n\u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R\nhne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a)\nT : Set (Set \u03b9) :=\n  {u |\n    u \u2286 t \u2227\n      PairwiseDisjoint u B \u2227\n        \u2200 (a : \u03b9),\n          a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c}\nu : Set \u03b9\nuT : u \u2208 T\na : \u03b9\nhat : a \u2208 t\nhu : \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u03c4 * \u03b4 b < \u03b4 a\na_disj : \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a) (B c)\nA : Set \u03b9 := {a' | a' \u2208 t \u2227 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a') (B c)}\nAnonempty : Set.Nonempty A\nm : \u211d := sSup (\u03b4 '' A)\nbddA : BddAbove (\u03b4 '' A)\nthis : 0 \u2264 m\nmpos : 0 < m\n\u22a2 \u2203 a', a' \u2208 A \u2227 m / \u03c4 \u2264 \u03b4 a'", "state_after": "case inr\n\u03b1 : Type u_1\n\u03b9 : Type u_2\nB : \u03b9 \u2192 Set \u03b1\nt : Set \u03b9\n\u03b4 : \u03b9 \u2192 \u211d\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\n\u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a\nR : \u211d\n\u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R\nhne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a)\nT : Set (Set \u03b9) :=\n  {u |\n    u \u2286 t \u2227\n      PairwiseDisjoint u B \u2227\n        \u2200 (a : \u03b9),\n          a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c}\nu : Set \u03b9\nuT : u \u2208 T\na : \u03b9\nhat : a \u2208 t\nhu : \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u03c4 * \u03b4 b < \u03b4 a\na_disj : \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a) (B c)\nA : Set \u03b9 := {a' | a' \u2208 t \u2227 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a') (B c)}\nAnonempty : Set.Nonempty A\nm : \u211d := sSup (\u03b4 '' A)\nbddA : BddAbove (\u03b4 '' A)\nthis : 0 \u2264 m\nmpos : 0 < m\nI : m / \u03c4 < m\n\u22a2 \u2203 a', a' \u2208 A \u2227 m / \u03c4 \u2264 \u03b4 a'"}, {"tactic": "rcases exists_lt_of_lt_csSup (nonempty_image_iff.2 Anonempty) I with \u27e8x, xA, hx\u27e9", "annotated_tactic": ["rcases <a>exists_lt_of_lt_csSup</a> (<a>nonempty_image_iff</a>.2 Anonempty) I with \u27e8x, xA, hx\u27e9", [{"full_name": "exists_lt_of_lt_csSup", "def_path": "Mathlib/Order/ConditionallyCompleteLattice/Basic.lean", "def_pos": [999, 9], "def_end_pos": [999, 30]}, {"full_name": "Set.nonempty_image_iff", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [481, 9], "def_end_pos": [481, 27]}]], "state_before": "case inr\n\u03b1 : Type u_1\n\u03b9 : Type u_2\nB : \u03b9 \u2192 Set \u03b1\nt : Set \u03b9\n\u03b4 : \u03b9 \u2192 \u211d\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\n\u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a\nR : \u211d\n\u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R\nhne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a)\nT : Set (Set \u03b9) :=\n  {u |\n    u \u2286 t \u2227\n      PairwiseDisjoint u B \u2227\n        \u2200 (a : \u03b9),\n          a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c}\nu : Set \u03b9\nuT : u \u2208 T\na : \u03b9\nhat : a \u2208 t\nhu : \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u03c4 * \u03b4 b < \u03b4 a\na_disj : \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a) (B c)\nA : Set \u03b9 := {a' | a' \u2208 t \u2227 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a') (B c)}\nAnonempty : Set.Nonempty A\nm : \u211d := sSup (\u03b4 '' A)\nbddA : BddAbove (\u03b4 '' A)\nthis : 0 \u2264 m\nmpos : 0 < m\nI : m / \u03c4 < m\n\u22a2 \u2203 a', a' \u2208 A \u2227 m / \u03c4 \u2264 \u03b4 a'", "state_after": "case inr.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\nB : \u03b9 \u2192 Set \u03b1\nt : Set \u03b9\n\u03b4 : \u03b9 \u2192 \u211d\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\n\u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a\nR : \u211d\n\u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R\nhne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a)\nT : Set (Set \u03b9) :=\n  {u |\n    u \u2286 t \u2227\n      PairwiseDisjoint u B \u2227\n        \u2200 (a : \u03b9),\n          a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c}\nu : Set \u03b9\nuT : u \u2208 T\na : \u03b9\nhat : a \u2208 t\nhu : \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u03c4 * \u03b4 b < \u03b4 a\na_disj : \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a) (B c)\nA : Set \u03b9 := {a' | a' \u2208 t \u2227 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a') (B c)}\nAnonempty : Set.Nonempty A\nm : \u211d := sSup (\u03b4 '' A)\nbddA : BddAbove (\u03b4 '' A)\nthis : 0 \u2264 m\nmpos : 0 < m\nI : m / \u03c4 < m\nx : \u211d\nxA : x \u2208 \u03b4 '' A\nhx : m / \u03c4 < x\n\u22a2 \u2203 a', a' \u2208 A \u2227 m / \u03c4 \u2264 \u03b4 a'"}, {"tactic": "rcases (mem_image _ _ _).1 xA with \u27e8a', ha', rfl\u27e9", "annotated_tactic": ["rcases (<a>mem_image</a> _ _ _).1 xA with \u27e8a', ha', rfl\u27e9", [{"full_name": "Set.mem_image", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [231, 9], "def_end_pos": [231, 18]}]], "state_before": "case inr.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\nB : \u03b9 \u2192 Set \u03b1\nt : Set \u03b9\n\u03b4 : \u03b9 \u2192 \u211d\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\n\u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a\nR : \u211d\n\u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R\nhne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a)\nT : Set (Set \u03b9) :=\n  {u |\n    u \u2286 t \u2227\n      PairwiseDisjoint u B \u2227\n        \u2200 (a : \u03b9),\n          a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c}\nu : Set \u03b9\nuT : u \u2208 T\na : \u03b9\nhat : a \u2208 t\nhu : \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u03c4 * \u03b4 b < \u03b4 a\na_disj : \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a) (B c)\nA : Set \u03b9 := {a' | a' \u2208 t \u2227 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a') (B c)}\nAnonempty : Set.Nonempty A\nm : \u211d := sSup (\u03b4 '' A)\nbddA : BddAbove (\u03b4 '' A)\nthis : 0 \u2264 m\nmpos : 0 < m\nI : m / \u03c4 < m\nx : \u211d\nxA : x \u2208 \u03b4 '' A\nhx : m / \u03c4 < x\n\u22a2 \u2203 a', a' \u2208 A \u2227 m / \u03c4 \u2264 \u03b4 a'", "state_after": "case inr.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\nB : \u03b9 \u2192 Set \u03b1\nt : Set \u03b9\n\u03b4 : \u03b9 \u2192 \u211d\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\n\u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a\nR : \u211d\n\u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R\nhne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a)\nT : Set (Set \u03b9) :=\n  {u |\n    u \u2286 t \u2227\n      PairwiseDisjoint u B \u2227\n        \u2200 (a : \u03b9),\n          a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c}\nu : Set \u03b9\nuT : u \u2208 T\na : \u03b9\nhat : a \u2208 t\nhu : \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u03c4 * \u03b4 b < \u03b4 a\na_disj : \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a) (B c)\nA : Set \u03b9 := {a' | a' \u2208 t \u2227 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a') (B c)}\nAnonempty : Set.Nonempty A\nm : \u211d := sSup (\u03b4 '' A)\nbddA : BddAbove (\u03b4 '' A)\nthis : 0 \u2264 m\nmpos : 0 < m\nI : m / \u03c4 < m\na' : \u03b9\nha' : a' \u2208 A\nxA : \u03b4 a' \u2208 \u03b4 '' A\nhx : m / \u03c4 < \u03b4 a'\n\u22a2 \u2203 a', a' \u2208 A \u2227 m / \u03c4 \u2264 \u03b4 a'"}, {"tactic": "exact \u27e8a', ha', hx.le\u27e9", "annotated_tactic": ["exact \u27e8a', ha', hx.le\u27e9", []], "state_before": "case inr.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\nB : \u03b9 \u2192 Set \u03b1\nt : Set \u03b9\n\u03b4 : \u03b9 \u2192 \u211d\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\n\u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a\nR : \u211d\n\u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R\nhne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a)\nT : Set (Set \u03b9) :=\n  {u |\n    u \u2286 t \u2227\n      PairwiseDisjoint u B \u2227\n        \u2200 (a : \u03b9),\n          a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c}\nu : Set \u03b9\nuT : u \u2208 T\na : \u03b9\nhat : a \u2208 t\nhu : \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u03c4 * \u03b4 b < \u03b4 a\na_disj : \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a) (B c)\nA : Set \u03b9 := {a' | a' \u2208 t \u2227 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a') (B c)}\nAnonempty : Set.Nonempty A\nm : \u211d := sSup (\u03b4 '' A)\nbddA : BddAbove (\u03b4 '' A)\nthis : 0 \u2264 m\nmpos : 0 < m\nI : m / \u03c4 < m\na' : \u03b9\nha' : a' \u2208 A\nxA : \u03b4 a' \u2208 \u03b4 '' A\nhx : m / \u03c4 < \u03b4 a'\n\u22a2 \u2203 a', a' \u2208 A \u2227 m / \u03c4 \u2264 \u03b4 a'", "state_after": "no goals"}, {"tactic": "rw [div_lt_iff (zero_lt_one.trans h\u03c4)]", "annotated_tactic": ["rw [<a>div_lt_iff</a> (zero_lt_one.trans h\u03c4)]", [{"full_name": "div_lt_iff", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [169, 9], "def_end_pos": [169, 19]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\nB : \u03b9 \u2192 Set \u03b1\nt : Set \u03b9\n\u03b4 : \u03b9 \u2192 \u211d\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\n\u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a\nR : \u211d\n\u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R\nhne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a)\nT : Set (Set \u03b9) :=\n  {u |\n    u \u2286 t \u2227\n      PairwiseDisjoint u B \u2227\n        \u2200 (a : \u03b9),\n          a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c}\nu : Set \u03b9\nuT : u \u2208 T\na : \u03b9\nhat : a \u2208 t\nhu : \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u03c4 * \u03b4 b < \u03b4 a\na_disj : \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a) (B c)\nA : Set \u03b9 := {a' | a' \u2208 t \u2227 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a') (B c)}\nAnonempty : Set.Nonempty A\nm : \u211d := sSup (\u03b4 '' A)\nbddA : BddAbove (\u03b4 '' A)\nthis : 0 \u2264 m\nmpos : 0 < m\n\u22a2 m / \u03c4 < m", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\nB : \u03b9 \u2192 Set \u03b1\nt : Set \u03b9\n\u03b4 : \u03b9 \u2192 \u211d\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\n\u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a\nR : \u211d\n\u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R\nhne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a)\nT : Set (Set \u03b9) :=\n  {u |\n    u \u2286 t \u2227\n      PairwiseDisjoint u B \u2227\n        \u2200 (a : \u03b9),\n          a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c}\nu : Set \u03b9\nuT : u \u2208 T\na : \u03b9\nhat : a \u2208 t\nhu : \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u03c4 * \u03b4 b < \u03b4 a\na_disj : \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a) (B c)\nA : Set \u03b9 := {a' | a' \u2208 t \u2227 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a') (B c)}\nAnonempty : Set.Nonempty A\nm : \u211d := sSup (\u03b4 '' A)\nbddA : BddAbove (\u03b4 '' A)\nthis : 0 \u2264 m\nmpos : 0 < m\n\u22a2 m < m * \u03c4"}, {"tactic": "conv_lhs => rw [\u2190 mul_one m]", "annotated_tactic": ["conv_lhs => rw [\u2190 <a>mul_one</a> m]", [{"full_name": "mul_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [470, 9], "def_end_pos": [470, 16]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\nB : \u03b9 \u2192 Set \u03b1\nt : Set \u03b9\n\u03b4 : \u03b9 \u2192 \u211d\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\n\u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a\nR : \u211d\n\u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R\nhne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a)\nT : Set (Set \u03b9) :=\n  {u |\n    u \u2286 t \u2227\n      PairwiseDisjoint u B \u2227\n        \u2200 (a : \u03b9),\n          a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c}\nu : Set \u03b9\nuT : u \u2208 T\na : \u03b9\nhat : a \u2208 t\nhu : \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u03c4 * \u03b4 b < \u03b4 a\na_disj : \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a) (B c)\nA : Set \u03b9 := {a' | a' \u2208 t \u2227 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a') (B c)}\nAnonempty : Set.Nonempty A\nm : \u211d := sSup (\u03b4 '' A)\nbddA : BddAbove (\u03b4 '' A)\nthis : 0 \u2264 m\nmpos : 0 < m\n\u22a2 m < m * \u03c4", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\nB : \u03b9 \u2192 Set \u03b1\nt : Set \u03b9\n\u03b4 : \u03b9 \u2192 \u211d\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\n\u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a\nR : \u211d\n\u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R\nhne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a)\nT : Set (Set \u03b9) :=\n  {u |\n    u \u2286 t \u2227\n      PairwiseDisjoint u B \u2227\n        \u2200 (a : \u03b9),\n          a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c}\nu : Set \u03b9\nuT : u \u2208 T\na : \u03b9\nhat : a \u2208 t\nhu : \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u03c4 * \u03b4 b < \u03b4 a\na_disj : \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a) (B c)\nA : Set \u03b9 := {a' | a' \u2208 t \u2227 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a') (B c)}\nAnonempty : Set.Nonempty A\nm : \u211d := sSup (\u03b4 '' A)\nbddA : BddAbove (\u03b4 '' A)\nthis : 0 \u2264 m\nmpos : 0 < m\n\u22a2 m * 1 < m * \u03c4"}, {"tactic": "exact (mul_lt_mul_left mpos).2 h\u03c4", "annotated_tactic": ["exact (<a>mul_lt_mul_left</a> mpos).2 h\u03c4", [{"full_name": "mul_lt_mul_left", "def_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "def_pos": [197, 9], "def_end_pos": [197, 24]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\nB : \u03b9 \u2192 Set \u03b1\nt : Set \u03b9\n\u03b4 : \u03b9 \u2192 \u211d\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\n\u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a\nR : \u211d\n\u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R\nhne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a)\nT : Set (Set \u03b9) :=\n  {u |\n    u \u2286 t \u2227\n      PairwiseDisjoint u B \u2227\n        \u2200 (a : \u03b9),\n          a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c}\nu : Set \u03b9\nuT : u \u2208 T\na : \u03b9\nhat : a \u2208 t\nhu : \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u03c4 * \u03b4 b < \u03b4 a\na_disj : \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a) (B c)\nA : Set \u03b9 := {a' | a' \u2208 t \u2227 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a') (B c)}\nAnonempty : Set.Nonempty A\nm : \u211d := sSup (\u03b4 '' A)\nbddA : BddAbove (\u03b4 '' A)\nthis : 0 \u2264 m\nmpos : 0 < m\n\u22a2 m * 1 < m * \u03c4", "state_after": "no goals"}, {"tactic": "rw [insert_subset_iff]", "annotated_tactic": ["rw [<a>insert_subset_iff</a>]", [{"full_name": "Set.insert_subset_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1169, 9], "def_end_pos": [1169, 26]}]], "state_before": "case intro.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\n\u03b9 : Type u_2\nB : \u03b9 \u2192 Set \u03b1\nt : Set \u03b9\n\u03b4 : \u03b9 \u2192 \u211d\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\n\u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a\nR : \u211d\n\u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R\nhne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a)\nT : Set (Set \u03b9) :=\n  {u |\n    u \u2286 t \u2227\n      PairwiseDisjoint u B \u2227\n        \u2200 (a : \u03b9),\n          a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c}\nu : Set \u03b9\nuT : u \u2208 T\nA : Set \u03b9 := {a' | a' \u2208 t \u2227 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a') (B c)}\nAnonempty : Set.Nonempty A\nm : \u211d := sSup (\u03b4 '' A)\nbddA : BddAbove (\u03b4 '' A)\na' : \u03b9\na'A : a' \u2208 A\nha' : m / \u03c4 \u2264 \u03b4 a'\na'_ne_u : \u00aca' \u2208 u\n\u22a2 insert a' u \u2286 t", "state_after": "case intro.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\n\u03b9 : Type u_2\nB : \u03b9 \u2192 Set \u03b1\nt : Set \u03b9\n\u03b4 : \u03b9 \u2192 \u211d\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\n\u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a\nR : \u211d\n\u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R\nhne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a)\nT : Set (Set \u03b9) :=\n  {u |\n    u \u2286 t \u2227\n      PairwiseDisjoint u B \u2227\n        \u2200 (a : \u03b9),\n          a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c}\nu : Set \u03b9\nuT : u \u2208 T\nA : Set \u03b9 := {a' | a' \u2208 t \u2227 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a') (B c)}\nAnonempty : Set.Nonempty A\nm : \u211d := sSup (\u03b4 '' A)\nbddA : BddAbove (\u03b4 '' A)\na' : \u03b9\na'A : a' \u2208 A\nha' : m / \u03c4 \u2264 \u03b4 a'\na'_ne_u : \u00aca' \u2208 u\n\u22a2 a' \u2208 t \u2227 u \u2286 t"}, {"tactic": "exact \u27e8a'A.1, uT.1\u27e9", "annotated_tactic": ["exact \u27e8a'A.1, uT.1\u27e9", []], "state_before": "case intro.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\n\u03b9 : Type u_2\nB : \u03b9 \u2192 Set \u03b1\nt : Set \u03b9\n\u03b4 : \u03b9 \u2192 \u211d\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\n\u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a\nR : \u211d\n\u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R\nhne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a)\nT : Set (Set \u03b9) :=\n  {u |\n    u \u2286 t \u2227\n      PairwiseDisjoint u B \u2227\n        \u2200 (a : \u03b9),\n          a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c}\nu : Set \u03b9\nuT : u \u2208 T\nA : Set \u03b9 := {a' | a' \u2208 t \u2227 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a') (B c)}\nAnonempty : Set.Nonempty A\nm : \u211d := sSup (\u03b4 '' A)\nbddA : BddAbove (\u03b4 '' A)\na' : \u03b9\na'A : a' \u2208 A\nha' : m / \u03c4 \u2264 \u03b4 a'\na'_ne_u : \u00aca' \u2208 u\n\u22a2 a' \u2208 t \u2227 u \u2286 t", "state_after": "no goals"}, {"tactic": "exact uT.2.1.insert fun b bu _ => a'A.2 b bu", "annotated_tactic": ["exact uT.2.1.<a>insert</a> fun b bu _ => a'A.2 b bu", [{"full_name": "Set.PairwiseDisjoint.insert", "def_path": "Mathlib/Data/Set/Pairwise/Basic.lean", "def_pos": [279, 19], "def_end_pos": [279, 42]}]], "state_before": "case intro.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\n\u03b9 : Type u_2\nB : \u03b9 \u2192 Set \u03b1\nt : Set \u03b9\n\u03b4 : \u03b9 \u2192 \u211d\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\n\u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a\nR : \u211d\n\u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R\nhne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a)\nT : Set (Set \u03b9) :=\n  {u |\n    u \u2286 t \u2227\n      PairwiseDisjoint u B \u2227\n        \u2200 (a : \u03b9),\n          a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c}\nu : Set \u03b9\nuT : u \u2208 T\nA : Set \u03b9 := {a' | a' \u2208 t \u2227 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a') (B c)}\nAnonempty : Set.Nonempty A\nm : \u211d := sSup (\u03b4 '' A)\nbddA : BddAbove (\u03b4 '' A)\na' : \u03b9\na'A : a' \u2208 A\nha' : m / \u03c4 \u2264 \u03b4 a'\na'_ne_u : \u00aca' \u2208 u\n\u22a2 PairwiseDisjoint (insert a' u) B", "state_after": "no goals"}, {"tactic": "intro c ct b ba'u hcb", "annotated_tactic": ["intro c ct b ba'u hcb", []], "state_before": "case intro.intro.intro.intro.refine'_3\n\u03b1 : Type u_1\n\u03b9 : Type u_2\nB : \u03b9 \u2192 Set \u03b1\nt : Set \u03b9\n\u03b4 : \u03b9 \u2192 \u211d\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\n\u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a\nR : \u211d\n\u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R\nhne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a)\nT : Set (Set \u03b9) :=\n  {u |\n    u \u2286 t \u2227\n      PairwiseDisjoint u B \u2227\n        \u2200 (a : \u03b9),\n          a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c}\nu : Set \u03b9\nuT : u \u2208 T\nA : Set \u03b9 := {a' | a' \u2208 t \u2227 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a') (B c)}\nAnonempty : Set.Nonempty A\nm : \u211d := sSup (\u03b4 '' A)\nbddA : BddAbove (\u03b4 '' A)\na' : \u03b9\na'A : a' \u2208 A\nha' : m / \u03c4 \u2264 \u03b4 a'\na'_ne_u : \u00aca' \u2208 u\n\u22a2 \u2200 (a : \u03b9),\n    a \u2208 t \u2192\n      \u2200 (b : \u03b9),\n        b \u2208 insert a' u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 insert a' u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c", "state_after": "case intro.intro.intro.intro.refine'_3\n\u03b1 : Type u_1\n\u03b9 : Type u_2\nB : \u03b9 \u2192 Set \u03b1\nt : Set \u03b9\n\u03b4 : \u03b9 \u2192 \u211d\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\n\u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a\nR : \u211d\n\u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R\nhne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a)\nT : Set (Set \u03b9) :=\n  {u |\n    u \u2286 t \u2227\n      PairwiseDisjoint u B \u2227\n        \u2200 (a : \u03b9),\n          a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c}\nu : Set \u03b9\nuT : u \u2208 T\nA : Set \u03b9 := {a' | a' \u2208 t \u2227 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a') (B c)}\nAnonempty : Set.Nonempty A\nm : \u211d := sSup (\u03b4 '' A)\nbddA : BddAbove (\u03b4 '' A)\na' : \u03b9\na'A : a' \u2208 A\nha' : m / \u03c4 \u2264 \u03b4 a'\na'_ne_u : \u00aca' \u2208 u\nc : \u03b9\nct : c \u2208 t\nb : \u03b9\nba'u : b \u2208 insert a' u\nhcb : Set.Nonempty (B c \u2229 B b)\n\u22a2 \u2203 c_1, c_1 \u2208 insert a' u \u2227 Set.Nonempty (B c \u2229 B c_1) \u2227 \u03b4 c \u2264 \u03c4 * \u03b4 c_1"}, {"tactic": "by_cases H : \u2203 d \u2208 u, (B c \u2229 B d).Nonempty", "annotated_tactic": ["by_cases H : \u2203 d \u2208 u, (B c \u2229 B d).<a>Nonempty</a>", [{"full_name": "Set.Nonempty", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [439, 15], "def_end_pos": [439, 23]}]], "state_before": "case intro.intro.intro.intro.refine'_3\n\u03b1 : Type u_1\n\u03b9 : Type u_2\nB : \u03b9 \u2192 Set \u03b1\nt : Set \u03b9\n\u03b4 : \u03b9 \u2192 \u211d\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\n\u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a\nR : \u211d\n\u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R\nhne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a)\nT : Set (Set \u03b9) :=\n  {u |\n    u \u2286 t \u2227\n      PairwiseDisjoint u B \u2227\n        \u2200 (a : \u03b9),\n          a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c}\nu : Set \u03b9\nuT : u \u2208 T\nA : Set \u03b9 := {a' | a' \u2208 t \u2227 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a') (B c)}\nAnonempty : Set.Nonempty A\nm : \u211d := sSup (\u03b4 '' A)\nbddA : BddAbove (\u03b4 '' A)\na' : \u03b9\na'A : a' \u2208 A\nha' : m / \u03c4 \u2264 \u03b4 a'\na'_ne_u : \u00aca' \u2208 u\nc : \u03b9\nct : c \u2208 t\nb : \u03b9\nba'u : b \u2208 insert a' u\nhcb : Set.Nonempty (B c \u2229 B b)\n\u22a2 \u2203 c_1, c_1 \u2208 insert a' u \u2227 Set.Nonempty (B c \u2229 B c_1) \u2227 \u03b4 c \u2264 \u03c4 * \u03b4 c_1", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b9 : Type u_2\nB : \u03b9 \u2192 Set \u03b1\nt : Set \u03b9\n\u03b4 : \u03b9 \u2192 \u211d\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\n\u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a\nR : \u211d\n\u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R\nhne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a)\nT : Set (Set \u03b9) :=\n  {u |\n    u \u2286 t \u2227\n      PairwiseDisjoint u B \u2227\n        \u2200 (a : \u03b9),\n          a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c}\nu : Set \u03b9\nuT : u \u2208 T\nA : Set \u03b9 := {a' | a' \u2208 t \u2227 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a') (B c)}\nAnonempty : Set.Nonempty A\nm : \u211d := sSup (\u03b4 '' A)\nbddA : BddAbove (\u03b4 '' A)\na' : \u03b9\na'A : a' \u2208 A\nha' : m / \u03c4 \u2264 \u03b4 a'\na'_ne_u : \u00aca' \u2208 u\nc : \u03b9\nct : c \u2208 t\nb : \u03b9\nba'u : b \u2208 insert a' u\nhcb : Set.Nonempty (B c \u2229 B b)\nH : \u2203 d, d \u2208 u \u2227 Set.Nonempty (B c \u2229 B d)\n\u22a2 \u2203 c_1, c_1 \u2208 insert a' u \u2227 Set.Nonempty (B c \u2229 B c_1) \u2227 \u03b4 c \u2264 \u03c4 * \u03b4 c_1\n\ncase neg\n\u03b1 : Type u_1\n\u03b9 : Type u_2\nB : \u03b9 \u2192 Set \u03b1\nt : Set \u03b9\n\u03b4 : \u03b9 \u2192 \u211d\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\n\u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a\nR : \u211d\n\u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R\nhne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a)\nT : Set (Set \u03b9) :=\n  {u |\n    u \u2286 t \u2227\n      PairwiseDisjoint u B \u2227\n        \u2200 (a : \u03b9),\n          a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c}\nu : Set \u03b9\nuT : u \u2208 T\nA : Set \u03b9 := {a' | a' \u2208 t \u2227 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a') (B c)}\nAnonempty : Set.Nonempty A\nm : \u211d := sSup (\u03b4 '' A)\nbddA : BddAbove (\u03b4 '' A)\na' : \u03b9\na'A : a' \u2208 A\nha' : m / \u03c4 \u2264 \u03b4 a'\na'_ne_u : \u00aca' \u2208 u\nc : \u03b9\nct : c \u2208 t\nb : \u03b9\nba'u : b \u2208 insert a' u\nhcb : Set.Nonempty (B c \u2229 B b)\nH : \u00ac\u2203 d, d \u2208 u \u2227 Set.Nonempty (B c \u2229 B d)\n\u22a2 \u2203 c_1, c_1 \u2208 insert a' u \u2227 Set.Nonempty (B c \u2229 B c_1) \u2227 \u03b4 c \u2264 \u03c4 * \u03b4 c_1"}, {"tactic": "rcases H with \u27e8d, du, hd\u27e9", "annotated_tactic": ["rcases H with \u27e8d, du, hd\u27e9", []], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b9 : Type u_2\nB : \u03b9 \u2192 Set \u03b1\nt : Set \u03b9\n\u03b4 : \u03b9 \u2192 \u211d\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\n\u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a\nR : \u211d\n\u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R\nhne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a)\nT : Set (Set \u03b9) :=\n  {u |\n    u \u2286 t \u2227\n      PairwiseDisjoint u B \u2227\n        \u2200 (a : \u03b9),\n          a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c}\nu : Set \u03b9\nuT : u \u2208 T\nA : Set \u03b9 := {a' | a' \u2208 t \u2227 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a') (B c)}\nAnonempty : Set.Nonempty A\nm : \u211d := sSup (\u03b4 '' A)\nbddA : BddAbove (\u03b4 '' A)\na' : \u03b9\na'A : a' \u2208 A\nha' : m / \u03c4 \u2264 \u03b4 a'\na'_ne_u : \u00aca' \u2208 u\nc : \u03b9\nct : c \u2208 t\nb : \u03b9\nba'u : b \u2208 insert a' u\nhcb : Set.Nonempty (B c \u2229 B b)\nH : \u2203 d, d \u2208 u \u2227 Set.Nonempty (B c \u2229 B d)\n\u22a2 \u2203 c_1, c_1 \u2208 insert a' u \u2227 Set.Nonempty (B c \u2229 B c_1) \u2227 \u03b4 c \u2264 \u03c4 * \u03b4 c_1", "state_after": "case pos.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\nB : \u03b9 \u2192 Set \u03b1\nt : Set \u03b9\n\u03b4 : \u03b9 \u2192 \u211d\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\n\u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a\nR : \u211d\n\u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R\nhne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a)\nT : Set (Set \u03b9) :=\n  {u |\n    u \u2286 t \u2227\n      PairwiseDisjoint u B \u2227\n        \u2200 (a : \u03b9),\n          a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c}\nu : Set \u03b9\nuT : u \u2208 T\nA : Set \u03b9 := {a' | a' \u2208 t \u2227 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a') (B c)}\nAnonempty : Set.Nonempty A\nm : \u211d := sSup (\u03b4 '' A)\nbddA : BddAbove (\u03b4 '' A)\na' : \u03b9\na'A : a' \u2208 A\nha' : m / \u03c4 \u2264 \u03b4 a'\na'_ne_u : \u00aca' \u2208 u\nc : \u03b9\nct : c \u2208 t\nb : \u03b9\nba'u : b \u2208 insert a' u\nhcb : Set.Nonempty (B c \u2229 B b)\nd : \u03b9\ndu : d \u2208 u\nhd : Set.Nonempty (B c \u2229 B d)\n\u22a2 \u2203 c_1, c_1 \u2208 insert a' u \u2227 Set.Nonempty (B c \u2229 B c_1) \u2227 \u03b4 c \u2264 \u03c4 * \u03b4 c_1"}, {"tactic": "rcases uT.2.2 c ct d du hd with \u27e8d', d'u, hd'\u27e9", "annotated_tactic": ["rcases uT.2.2 c ct d du hd with \u27e8d', d'u, hd'\u27e9", []], "state_before": "case pos.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\nB : \u03b9 \u2192 Set \u03b1\nt : Set \u03b9\n\u03b4 : \u03b9 \u2192 \u211d\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\n\u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a\nR : \u211d\n\u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R\nhne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a)\nT : Set (Set \u03b9) :=\n  {u |\n    u \u2286 t \u2227\n      PairwiseDisjoint u B \u2227\n        \u2200 (a : \u03b9),\n          a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c}\nu : Set \u03b9\nuT : u \u2208 T\nA : Set \u03b9 := {a' | a' \u2208 t \u2227 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a') (B c)}\nAnonempty : Set.Nonempty A\nm : \u211d := sSup (\u03b4 '' A)\nbddA : BddAbove (\u03b4 '' A)\na' : \u03b9\na'A : a' \u2208 A\nha' : m / \u03c4 \u2264 \u03b4 a'\na'_ne_u : \u00aca' \u2208 u\nc : \u03b9\nct : c \u2208 t\nb : \u03b9\nba'u : b \u2208 insert a' u\nhcb : Set.Nonempty (B c \u2229 B b)\nd : \u03b9\ndu : d \u2208 u\nhd : Set.Nonempty (B c \u2229 B d)\n\u22a2 \u2203 c_1, c_1 \u2208 insert a' u \u2227 Set.Nonempty (B c \u2229 B c_1) \u2227 \u03b4 c \u2264 \u03c4 * \u03b4 c_1", "state_after": "case pos.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\nB : \u03b9 \u2192 Set \u03b1\nt : Set \u03b9\n\u03b4 : \u03b9 \u2192 \u211d\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\n\u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a\nR : \u211d\n\u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R\nhne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a)\nT : Set (Set \u03b9) :=\n  {u |\n    u \u2286 t \u2227\n      PairwiseDisjoint u B \u2227\n        \u2200 (a : \u03b9),\n          a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c}\nu : Set \u03b9\nuT : u \u2208 T\nA : Set \u03b9 := {a' | a' \u2208 t \u2227 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a') (B c)}\nAnonempty : Set.Nonempty A\nm : \u211d := sSup (\u03b4 '' A)\nbddA : BddAbove (\u03b4 '' A)\na' : \u03b9\na'A : a' \u2208 A\nha' : m / \u03c4 \u2264 \u03b4 a'\na'_ne_u : \u00aca' \u2208 u\nc : \u03b9\nct : c \u2208 t\nb : \u03b9\nba'u : b \u2208 insert a' u\nhcb : Set.Nonempty (B c \u2229 B b)\nd : \u03b9\ndu : d \u2208 u\nhd : Set.Nonempty (B c \u2229 B d)\nd' : \u03b9\nd'u : d' \u2208 u\nhd' : Set.Nonempty (B c \u2229 B d') \u2227 \u03b4 c \u2264 \u03c4 * \u03b4 d'\n\u22a2 \u2203 c_1, c_1 \u2208 insert a' u \u2227 Set.Nonempty (B c \u2229 B c_1) \u2227 \u03b4 c \u2264 \u03c4 * \u03b4 c_1"}, {"tactic": "exact \u27e8d', mem_insert_of_mem _ d'u, hd'\u27e9", "annotated_tactic": ["exact \u27e8d', <a>mem_insert_of_mem</a> _ d'u, hd'\u27e9", [{"full_name": "Set.mem_insert_of_mem", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1130, 9], "def_end_pos": [1130, 26]}]], "state_before": "case pos.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\nB : \u03b9 \u2192 Set \u03b1\nt : Set \u03b9\n\u03b4 : \u03b9 \u2192 \u211d\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\n\u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a\nR : \u211d\n\u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R\nhne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a)\nT : Set (Set \u03b9) :=\n  {u |\n    u \u2286 t \u2227\n      PairwiseDisjoint u B \u2227\n        \u2200 (a : \u03b9),\n          a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c}\nu : Set \u03b9\nuT : u \u2208 T\nA : Set \u03b9 := {a' | a' \u2208 t \u2227 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a') (B c)}\nAnonempty : Set.Nonempty A\nm : \u211d := sSup (\u03b4 '' A)\nbddA : BddAbove (\u03b4 '' A)\na' : \u03b9\na'A : a' \u2208 A\nha' : m / \u03c4 \u2264 \u03b4 a'\na'_ne_u : \u00aca' \u2208 u\nc : \u03b9\nct : c \u2208 t\nb : \u03b9\nba'u : b \u2208 insert a' u\nhcb : Set.Nonempty (B c \u2229 B b)\nd : \u03b9\ndu : d \u2208 u\nhd : Set.Nonempty (B c \u2229 B d)\nd' : \u03b9\nd'u : d' \u2208 u\nhd' : Set.Nonempty (B c \u2229 B d') \u2227 \u03b4 c \u2264 \u03c4 * \u03b4 d'\n\u22a2 \u2203 c_1, c_1 \u2208 insert a' u \u2227 Set.Nonempty (B c \u2229 B c_1) \u2227 \u03b4 c \u2264 \u03c4 * \u03b4 c_1", "state_after": "no goals"}, {"tactic": "push_neg at H", "annotated_tactic": ["push_neg at H", []], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b9 : Type u_2\nB : \u03b9 \u2192 Set \u03b1\nt : Set \u03b9\n\u03b4 : \u03b9 \u2192 \u211d\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\n\u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a\nR : \u211d\n\u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R\nhne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a)\nT : Set (Set \u03b9) :=\n  {u |\n    u \u2286 t \u2227\n      PairwiseDisjoint u B \u2227\n        \u2200 (a : \u03b9),\n          a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c}\nu : Set \u03b9\nuT : u \u2208 T\nA : Set \u03b9 := {a' | a' \u2208 t \u2227 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a') (B c)}\nAnonempty : Set.Nonempty A\nm : \u211d := sSup (\u03b4 '' A)\nbddA : BddAbove (\u03b4 '' A)\na' : \u03b9\na'A : a' \u2208 A\nha' : m / \u03c4 \u2264 \u03b4 a'\na'_ne_u : \u00aca' \u2208 u\nc : \u03b9\nct : c \u2208 t\nb : \u03b9\nba'u : b \u2208 insert a' u\nhcb : Set.Nonempty (B c \u2229 B b)\nH : \u00ac\u2203 d, d \u2208 u \u2227 Set.Nonempty (B c \u2229 B d)\n\u22a2 \u2203 c_1, c_1 \u2208 insert a' u \u2227 Set.Nonempty (B c \u2229 B c_1) \u2227 \u03b4 c \u2264 \u03c4 * \u03b4 c_1", "state_after": "case neg\n\u03b1 : Type u_1\n\u03b9 : Type u_2\nB : \u03b9 \u2192 Set \u03b1\nt : Set \u03b9\n\u03b4 : \u03b9 \u2192 \u211d\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\n\u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a\nR : \u211d\n\u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R\nhne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a)\nT : Set (Set \u03b9) :=\n  {u |\n    u \u2286 t \u2227\n      PairwiseDisjoint u B \u2227\n        \u2200 (a : \u03b9),\n          a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c}\nu : Set \u03b9\nuT : u \u2208 T\nA : Set \u03b9 := {a' | a' \u2208 t \u2227 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a') (B c)}\nAnonempty : Set.Nonempty A\nm : \u211d := sSup (\u03b4 '' A)\nbddA : BddAbove (\u03b4 '' A)\na' : \u03b9\na'A : a' \u2208 A\nha' : m / \u03c4 \u2264 \u03b4 a'\na'_ne_u : \u00aca' \u2208 u\nc : \u03b9\nct : c \u2208 t\nb : \u03b9\nba'u : b \u2208 insert a' u\nhcb : Set.Nonempty (B c \u2229 B b)\nH : \u2200 (d : \u03b9), d \u2208 u \u2192 \u00acSet.Nonempty (B c \u2229 B d)\n\u22a2 \u2203 c_1, c_1 \u2208 insert a' u \u2227 Set.Nonempty (B c \u2229 B c_1) \u2227 \u03b4 c \u2264 \u03c4 * \u03b4 c_1"}, {"tactic": "simp only [\u2190 not_disjoint_iff_nonempty_inter, Classical.not_not] at H", "annotated_tactic": ["simp only [\u2190 <a>not_disjoint_iff_nonempty_inter</a>, <a>Classical.not_not</a>] at H", [{"full_name": "Set.not_disjoint_iff_nonempty_inter", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1557, 7], "def_end_pos": [1557, 38]}, {"full_name": "Classical.not_not", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [683, 24], "def_end_pos": [683, 31]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b9 : Type u_2\nB : \u03b9 \u2192 Set \u03b1\nt : Set \u03b9\n\u03b4 : \u03b9 \u2192 \u211d\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\n\u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a\nR : \u211d\n\u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R\nhne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a)\nT : Set (Set \u03b9) :=\n  {u |\n    u \u2286 t \u2227\n      PairwiseDisjoint u B \u2227\n        \u2200 (a : \u03b9),\n          a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c}\nu : Set \u03b9\nuT : u \u2208 T\nA : Set \u03b9 := {a' | a' \u2208 t \u2227 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a') (B c)}\nAnonempty : Set.Nonempty A\nm : \u211d := sSup (\u03b4 '' A)\nbddA : BddAbove (\u03b4 '' A)\na' : \u03b9\na'A : a' \u2208 A\nha' : m / \u03c4 \u2264 \u03b4 a'\na'_ne_u : \u00aca' \u2208 u\nc : \u03b9\nct : c \u2208 t\nb : \u03b9\nba'u : b \u2208 insert a' u\nhcb : Set.Nonempty (B c \u2229 B b)\nH : \u2200 (d : \u03b9), d \u2208 u \u2192 \u00acSet.Nonempty (B c \u2229 B d)\n\u22a2 \u2203 c_1, c_1 \u2208 insert a' u \u2227 Set.Nonempty (B c \u2229 B c_1) \u2227 \u03b4 c \u2264 \u03c4 * \u03b4 c_1", "state_after": "case neg\n\u03b1 : Type u_1\n\u03b9 : Type u_2\nB : \u03b9 \u2192 Set \u03b1\nt : Set \u03b9\n\u03b4 : \u03b9 \u2192 \u211d\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\n\u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a\nR : \u211d\n\u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R\nhne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a)\nT : Set (Set \u03b9) :=\n  {u |\n    u \u2286 t \u2227\n      PairwiseDisjoint u B \u2227\n        \u2200 (a : \u03b9),\n          a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c}\nu : Set \u03b9\nuT : u \u2208 T\nA : Set \u03b9 := {a' | a' \u2208 t \u2227 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a') (B c)}\nAnonempty : Set.Nonempty A\nm : \u211d := sSup (\u03b4 '' A)\nbddA : BddAbove (\u03b4 '' A)\na' : \u03b9\na'A : a' \u2208 A\nha' : m / \u03c4 \u2264 \u03b4 a'\na'_ne_u : \u00aca' \u2208 u\nc : \u03b9\nct : c \u2208 t\nb : \u03b9\nba'u : b \u2208 insert a' u\nhcb : Set.Nonempty (B c \u2229 B b)\nH : \u2200 (d : \u03b9), d \u2208 u \u2192 Disjoint (B c) (B d)\n\u22a2 \u2203 c_1, c_1 \u2208 insert a' u \u2227 Set.Nonempty (B c \u2229 B c_1) \u2227 \u03b4 c \u2264 \u03c4 * \u03b4 c_1"}, {"tactic": "rcases mem_insert_iff.1 ba'u with (rfl | H')", "annotated_tactic": ["rcases <a>mem_insert_iff</a>.1 ba'u with (rfl | H')", [{"full_name": "Set.mem_insert_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1147, 9], "def_end_pos": [1147, 23]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b9 : Type u_2\nB : \u03b9 \u2192 Set \u03b1\nt : Set \u03b9\n\u03b4 : \u03b9 \u2192 \u211d\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\n\u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a\nR : \u211d\n\u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R\nhne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a)\nT : Set (Set \u03b9) :=\n  {u |\n    u \u2286 t \u2227\n      PairwiseDisjoint u B \u2227\n        \u2200 (a : \u03b9),\n          a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c}\nu : Set \u03b9\nuT : u \u2208 T\nA : Set \u03b9 := {a' | a' \u2208 t \u2227 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a') (B c)}\nAnonempty : Set.Nonempty A\nm : \u211d := sSup (\u03b4 '' A)\nbddA : BddAbove (\u03b4 '' A)\na' : \u03b9\na'A : a' \u2208 A\nha' : m / \u03c4 \u2264 \u03b4 a'\na'_ne_u : \u00aca' \u2208 u\nc : \u03b9\nct : c \u2208 t\nb : \u03b9\nba'u : b \u2208 insert a' u\nhcb : Set.Nonempty (B c \u2229 B b)\nH : \u2200 (d : \u03b9), d \u2208 u \u2192 Disjoint (B c) (B d)\n\u22a2 \u2203 c_1, c_1 \u2208 insert a' u \u2227 Set.Nonempty (B c \u2229 B c_1) \u2227 \u03b4 c \u2264 \u03c4 * \u03b4 c_1", "state_after": "case neg.inl\n\u03b1 : Type u_1\n\u03b9 : Type u_2\nB : \u03b9 \u2192 Set \u03b1\nt : Set \u03b9\n\u03b4 : \u03b9 \u2192 \u211d\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\n\u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a\nR : \u211d\n\u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R\nhne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a)\nT : Set (Set \u03b9) :=\n  {u |\n    u \u2286 t \u2227\n      PairwiseDisjoint u B \u2227\n        \u2200 (a : \u03b9),\n          a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c}\nu : Set \u03b9\nuT : u \u2208 T\nA : Set \u03b9 := {a' | a' \u2208 t \u2227 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a') (B c)}\nAnonempty : Set.Nonempty A\nm : \u211d := sSup (\u03b4 '' A)\nbddA : BddAbove (\u03b4 '' A)\nc : \u03b9\nct : c \u2208 t\nb : \u03b9\nhcb : Set.Nonempty (B c \u2229 B b)\nH : \u2200 (d : \u03b9), d \u2208 u \u2192 Disjoint (B c) (B d)\na'A : b \u2208 A\nha' : m / \u03c4 \u2264 \u03b4 b\na'_ne_u : \u00acb \u2208 u\nba'u : b \u2208 insert b u\n\u22a2 \u2203 c_1, c_1 \u2208 insert b u \u2227 Set.Nonempty (B c \u2229 B c_1) \u2227 \u03b4 c \u2264 \u03c4 * \u03b4 c_1\n\ncase neg.inr\n\u03b1 : Type u_1\n\u03b9 : Type u_2\nB : \u03b9 \u2192 Set \u03b1\nt : Set \u03b9\n\u03b4 : \u03b9 \u2192 \u211d\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\n\u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a\nR : \u211d\n\u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R\nhne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a)\nT : Set (Set \u03b9) :=\n  {u |\n    u \u2286 t \u2227\n      PairwiseDisjoint u B \u2227\n        \u2200 (a : \u03b9),\n          a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c}\nu : Set \u03b9\nuT : u \u2208 T\nA : Set \u03b9 := {a' | a' \u2208 t \u2227 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a') (B c)}\nAnonempty : Set.Nonempty A\nm : \u211d := sSup (\u03b4 '' A)\nbddA : BddAbove (\u03b4 '' A)\na' : \u03b9\na'A : a' \u2208 A\nha' : m / \u03c4 \u2264 \u03b4 a'\na'_ne_u : \u00aca' \u2208 u\nc : \u03b9\nct : c \u2208 t\nb : \u03b9\nba'u : b \u2208 insert a' u\nhcb : Set.Nonempty (B c \u2229 B b)\nH : \u2200 (d : \u03b9), d \u2208 u \u2192 Disjoint (B c) (B d)\nH' : b \u2208 u\n\u22a2 \u2203 c_1, c_1 \u2208 insert a' u \u2227 Set.Nonempty (B c \u2229 B c_1) \u2227 \u03b4 c \u2264 \u03c4 * \u03b4 c_1"}, {"tactic": "refine' \u27e8b, mem_insert _ _, hcb, _\u27e9", "annotated_tactic": ["refine' \u27e8b, <a>mem_insert</a> _ _, hcb, _\u27e9", [{"full_name": "Set.mem_insert", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1126, 9], "def_end_pos": [1126, 19]}]], "state_before": "case neg.inl\n\u03b1 : Type u_1\n\u03b9 : Type u_2\nB : \u03b9 \u2192 Set \u03b1\nt : Set \u03b9\n\u03b4 : \u03b9 \u2192 \u211d\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\n\u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a\nR : \u211d\n\u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R\nhne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a)\nT : Set (Set \u03b9) :=\n  {u |\n    u \u2286 t \u2227\n      PairwiseDisjoint u B \u2227\n        \u2200 (a : \u03b9),\n          a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c}\nu : Set \u03b9\nuT : u \u2208 T\nA : Set \u03b9 := {a' | a' \u2208 t \u2227 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a') (B c)}\nAnonempty : Set.Nonempty A\nm : \u211d := sSup (\u03b4 '' A)\nbddA : BddAbove (\u03b4 '' A)\nc : \u03b9\nct : c \u2208 t\nb : \u03b9\nhcb : Set.Nonempty (B c \u2229 B b)\nH : \u2200 (d : \u03b9), d \u2208 u \u2192 Disjoint (B c) (B d)\na'A : b \u2208 A\nha' : m / \u03c4 \u2264 \u03b4 b\na'_ne_u : \u00acb \u2208 u\nba'u : b \u2208 insert b u\n\u22a2 \u2203 c_1, c_1 \u2208 insert b u \u2227 Set.Nonempty (B c \u2229 B c_1) \u2227 \u03b4 c \u2264 \u03c4 * \u03b4 c_1", "state_after": "case neg.inl\n\u03b1 : Type u_1\n\u03b9 : Type u_2\nB : \u03b9 \u2192 Set \u03b1\nt : Set \u03b9\n\u03b4 : \u03b9 \u2192 \u211d\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\n\u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a\nR : \u211d\n\u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R\nhne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a)\nT : Set (Set \u03b9) :=\n  {u |\n    u \u2286 t \u2227\n      PairwiseDisjoint u B \u2227\n        \u2200 (a : \u03b9),\n          a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c}\nu : Set \u03b9\nuT : u \u2208 T\nA : Set \u03b9 := {a' | a' \u2208 t \u2227 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a') (B c)}\nAnonempty : Set.Nonempty A\nm : \u211d := sSup (\u03b4 '' A)\nbddA : BddAbove (\u03b4 '' A)\nc : \u03b9\nct : c \u2208 t\nb : \u03b9\nhcb : Set.Nonempty (B c \u2229 B b)\nH : \u2200 (d : \u03b9), d \u2208 u \u2192 Disjoint (B c) (B d)\na'A : b \u2208 A\nha' : m / \u03c4 \u2264 \u03b4 b\na'_ne_u : \u00acb \u2208 u\nba'u : b \u2208 insert b u\n\u22a2 \u03b4 c \u2264 \u03c4 * \u03b4 b"}, {"tactic": "calc\n  \u03b4 c \u2264 m := le_csSup bddA (mem_image_of_mem _ \u27e8ct, H\u27e9)\n  _ = \u03c4 * (m / \u03c4) := by\n    field_simp [(zero_lt_one.trans h\u03c4).ne']\n    ring\n  _ \u2264 \u03c4 * \u03b4 b := mul_le_mul_of_nonneg_left ha' (zero_le_one.trans h\u03c4.le)", "annotated_tactic": ["calc\n          \u03b4 c \u2264 m := <a>le_csSup</a> bddA (<a>mem_image_of_mem</a> _ \u27e8ct, H\u27e9)\n          _ = \u03c4 * (m / \u03c4) := by\n            field_simp [(zero_lt_one.trans h\u03c4).<a>ne'</a>]\n            ring\n          _ \u2264 \u03c4 * \u03b4 b := <a>mul_le_mul_of_nonneg_left</a> ha' (zero_le_one.trans h\u03c4.le)", [{"full_name": "le_csSup", "def_path": "Mathlib/Order/ConditionallyCompleteLattice/Basic.lean", "def_pos": [457, 9], "def_end_pos": [457, 17]}, {"full_name": "Set.mem_image_of_mem", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [240, 9], "def_end_pos": [240, 25]}, {"full_name": "LT.lt.ne'", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [328, 9], "def_end_pos": [328, 12]}, {"full_name": "mul_le_mul_of_nonneg_left", "def_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "def_pos": [152, 9], "def_end_pos": [152, 34]}]], "state_before": "case neg.inl\n\u03b1 : Type u_1\n\u03b9 : Type u_2\nB : \u03b9 \u2192 Set \u03b1\nt : Set \u03b9\n\u03b4 : \u03b9 \u2192 \u211d\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\n\u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a\nR : \u211d\n\u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R\nhne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a)\nT : Set (Set \u03b9) :=\n  {u |\n    u \u2286 t \u2227\n      PairwiseDisjoint u B \u2227\n        \u2200 (a : \u03b9),\n          a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c}\nu : Set \u03b9\nuT : u \u2208 T\nA : Set \u03b9 := {a' | a' \u2208 t \u2227 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a') (B c)}\nAnonempty : Set.Nonempty A\nm : \u211d := sSup (\u03b4 '' A)\nbddA : BddAbove (\u03b4 '' A)\nc : \u03b9\nct : c \u2208 t\nb : \u03b9\nhcb : Set.Nonempty (B c \u2229 B b)\nH : \u2200 (d : \u03b9), d \u2208 u \u2192 Disjoint (B c) (B d)\na'A : b \u2208 A\nha' : m / \u03c4 \u2264 \u03b4 b\na'_ne_u : \u00acb \u2208 u\nba'u : b \u2208 insert b u\n\u22a2 \u03b4 c \u2264 \u03c4 * \u03b4 b", "state_after": "no goals"}, {"tactic": "field_simp [(zero_lt_one.trans h\u03c4).ne']", "annotated_tactic": ["field_simp [(zero_lt_one.trans h\u03c4).<a>ne'</a>]", [{"full_name": "LT.lt.ne'", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [328, 9], "def_end_pos": [328, 12]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\nB : \u03b9 \u2192 Set \u03b1\nt : Set \u03b9\n\u03b4 : \u03b9 \u2192 \u211d\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\n\u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a\nR : \u211d\n\u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R\nhne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a)\nT : Set (Set \u03b9) :=\n  {u |\n    u \u2286 t \u2227\n      PairwiseDisjoint u B \u2227\n        \u2200 (a : \u03b9),\n          a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c}\nu : Set \u03b9\nuT : u \u2208 T\nA : Set \u03b9 := {a' | a' \u2208 t \u2227 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a') (B c)}\nAnonempty : Set.Nonempty A\nm : \u211d := sSup (\u03b4 '' A)\nbddA : BddAbove (\u03b4 '' A)\nc : \u03b9\nct : c \u2208 t\nb : \u03b9\nhcb : Set.Nonempty (B c \u2229 B b)\nH : \u2200 (d : \u03b9), d \u2208 u \u2192 Disjoint (B c) (B d)\na'A : b \u2208 A\nha' : m / \u03c4 \u2264 \u03b4 b\na'_ne_u : \u00acb \u2208 u\nba'u : b \u2208 insert b u\n\u22a2 m = \u03c4 * (m / \u03c4)", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\nB : \u03b9 \u2192 Set \u03b1\nt : Set \u03b9\n\u03b4 : \u03b9 \u2192 \u211d\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\n\u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a\nR : \u211d\n\u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R\nhne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a)\nT : Set (Set \u03b9) :=\n  {u |\n    u \u2286 t \u2227\n      PairwiseDisjoint u B \u2227\n        \u2200 (a : \u03b9),\n          a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c}\nu : Set \u03b9\nuT : u \u2208 T\nA : Set \u03b9 := {a' | a' \u2208 t \u2227 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a') (B c)}\nAnonempty : Set.Nonempty A\nm : \u211d := sSup (\u03b4 '' A)\nbddA : BddAbove (\u03b4 '' A)\nc : \u03b9\nct : c \u2208 t\nb : \u03b9\nhcb : Set.Nonempty (B c \u2229 B b)\nH : \u2200 (d : \u03b9), d \u2208 u \u2192 Disjoint (B c) (B d)\na'A : b \u2208 A\nha' : m / \u03c4 \u2264 \u03b4 b\na'_ne_u : \u00acb \u2208 u\nba'u : b \u2208 insert b u\n\u22a2 sSup (\u03b4 '' {a' | a' \u2208 t \u2227 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a') (B c)}) * \u03c4 =\n    \u03c4 * sSup (\u03b4 '' {a' | a' \u2208 t \u2227 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a') (B c)})"}, {"tactic": "ring", "annotated_tactic": ["ring", []], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\nB : \u03b9 \u2192 Set \u03b1\nt : Set \u03b9\n\u03b4 : \u03b9 \u2192 \u211d\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\n\u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a\nR : \u211d\n\u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R\nhne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a)\nT : Set (Set \u03b9) :=\n  {u |\n    u \u2286 t \u2227\n      PairwiseDisjoint u B \u2227\n        \u2200 (a : \u03b9),\n          a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c}\nu : Set \u03b9\nuT : u \u2208 T\nA : Set \u03b9 := {a' | a' \u2208 t \u2227 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a') (B c)}\nAnonempty : Set.Nonempty A\nm : \u211d := sSup (\u03b4 '' A)\nbddA : BddAbove (\u03b4 '' A)\nc : \u03b9\nct : c \u2208 t\nb : \u03b9\nhcb : Set.Nonempty (B c \u2229 B b)\nH : \u2200 (d : \u03b9), d \u2208 u \u2192 Disjoint (B c) (B d)\na'A : b \u2208 A\nha' : m / \u03c4 \u2264 \u03b4 b\na'_ne_u : \u00acb \u2208 u\nba'u : b \u2208 insert b u\n\u22a2 sSup (\u03b4 '' {a' | a' \u2208 t \u2227 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a') (B c)}) * \u03c4 =\n    \u03c4 * sSup (\u03b4 '' {a' | a' \u2208 t \u2227 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a') (B c)})", "state_after": "no goals"}, {"tactic": "rw [\u2190 not_disjoint_iff_nonempty_inter] at hcb", "annotated_tactic": ["rw [\u2190 <a>not_disjoint_iff_nonempty_inter</a>] at hcb", [{"full_name": "Set.not_disjoint_iff_nonempty_inter", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1557, 7], "def_end_pos": [1557, 38]}]], "state_before": "case neg.inr\n\u03b1 : Type u_1\n\u03b9 : Type u_2\nB : \u03b9 \u2192 Set \u03b1\nt : Set \u03b9\n\u03b4 : \u03b9 \u2192 \u211d\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\n\u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a\nR : \u211d\n\u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R\nhne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a)\nT : Set (Set \u03b9) :=\n  {u |\n    u \u2286 t \u2227\n      PairwiseDisjoint u B \u2227\n        \u2200 (a : \u03b9),\n          a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c}\nu : Set \u03b9\nuT : u \u2208 T\nA : Set \u03b9 := {a' | a' \u2208 t \u2227 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a') (B c)}\nAnonempty : Set.Nonempty A\nm : \u211d := sSup (\u03b4 '' A)\nbddA : BddAbove (\u03b4 '' A)\na' : \u03b9\na'A : a' \u2208 A\nha' : m / \u03c4 \u2264 \u03b4 a'\na'_ne_u : \u00aca' \u2208 u\nc : \u03b9\nct : c \u2208 t\nb : \u03b9\nba'u : b \u2208 insert a' u\nhcb : Set.Nonempty (B c \u2229 B b)\nH : \u2200 (d : \u03b9), d \u2208 u \u2192 Disjoint (B c) (B d)\nH' : b \u2208 u\n\u22a2 \u2203 c_1, c_1 \u2208 insert a' u \u2227 Set.Nonempty (B c \u2229 B c_1) \u2227 \u03b4 c \u2264 \u03c4 * \u03b4 c_1", "state_after": "case neg.inr\n\u03b1 : Type u_1\n\u03b9 : Type u_2\nB : \u03b9 \u2192 Set \u03b1\nt : Set \u03b9\n\u03b4 : \u03b9 \u2192 \u211d\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\n\u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a\nR : \u211d\n\u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R\nhne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a)\nT : Set (Set \u03b9) :=\n  {u |\n    u \u2286 t \u2227\n      PairwiseDisjoint u B \u2227\n        \u2200 (a : \u03b9),\n          a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c}\nu : Set \u03b9\nuT : u \u2208 T\nA : Set \u03b9 := {a' | a' \u2208 t \u2227 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a') (B c)}\nAnonempty : Set.Nonempty A\nm : \u211d := sSup (\u03b4 '' A)\nbddA : BddAbove (\u03b4 '' A)\na' : \u03b9\na'A : a' \u2208 A\nha' : m / \u03c4 \u2264 \u03b4 a'\na'_ne_u : \u00aca' \u2208 u\nc : \u03b9\nct : c \u2208 t\nb : \u03b9\nba'u : b \u2208 insert a' u\nhcb : \u00acDisjoint (B c) (B b)\nH : \u2200 (d : \u03b9), d \u2208 u \u2192 Disjoint (B c) (B d)\nH' : b \u2208 u\n\u22a2 \u2203 c_1, c_1 \u2208 insert a' u \u2227 Set.Nonempty (B c \u2229 B c_1) \u2227 \u03b4 c \u2264 \u03c4 * \u03b4 c_1"}, {"tactic": "exact (hcb (H _ H')).elim", "annotated_tactic": ["exact (hcb (H _ H')).<a>elim</a>", [{"full_name": "False.elim", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [223, 21], "def_end_pos": [223, 31]}]], "state_before": "case neg.inr\n\u03b1 : Type u_1\n\u03b9 : Type u_2\nB : \u03b9 \u2192 Set \u03b1\nt : Set \u03b9\n\u03b4 : \u03b9 \u2192 \u211d\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\n\u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a\nR : \u211d\n\u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R\nhne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a)\nT : Set (Set \u03b9) :=\n  {u |\n    u \u2286 t \u2227\n      PairwiseDisjoint u B \u2227\n        \u2200 (a : \u03b9),\n          a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c}\nu : Set \u03b9\nuT : u \u2208 T\nA : Set \u03b9 := {a' | a' \u2208 t \u2227 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a') (B c)}\nAnonempty : Set.Nonempty A\nm : \u211d := sSup (\u03b4 '' A)\nbddA : BddAbove (\u03b4 '' A)\na' : \u03b9\na'A : a' \u2208 A\nha' : m / \u03c4 \u2264 \u03b4 a'\na'_ne_u : \u00aca' \u2208 u\nc : \u03b9\nct : c \u2208 t\nb : \u03b9\nba'u : b \u2208 insert a' u\nhcb : \u00acDisjoint (B c) (B b)\nH : \u2200 (d : \u03b9), d \u2208 u \u2192 Disjoint (B c) (B d)\nH' : b \u2208 u\n\u22a2 \u2203 c_1, c_1 \u2208 insert a' u \u2227 Set.Nonempty (B c \u2229 B c_1) \u2227 \u03b4 c \u2264 \u03c4 * \u03b4 c_1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/RBMap/Lemmas.lean", "full_name": "Std.RBNode.Balanced.depth_bound", "start": [83, 1], "end": [85, 76], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "full_name": "MeasureTheory.setToFun_add", "start": [1365, 1], "end": [1368, 29], "traced_tactics": [{"tactic": "rw [setToFun_eq hT (hf.add hg), setToFun_eq hT hf, setToFun_eq hT hg, Integrable.toL1_add,\n  (L1.setToL1 hT).map_add]", "annotated_tactic": ["rw [<a>setToFun_eq</a> hT (hf.add hg), <a>setToFun_eq</a> hT hf, <a>setToFun_eq</a> hT hg, <a>Integrable.toL1_add</a>,\n    (<a>L1.setToL1</a> hT).<a>map_add</a>]", [{"full_name": "MeasureTheory.setToFun_eq", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [1276, 9], "def_end_pos": [1276, 20]}, {"full_name": "MeasureTheory.setToFun_eq", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [1276, 9], "def_end_pos": [1276, 20]}, {"full_name": "MeasureTheory.setToFun_eq", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [1276, 9], "def_end_pos": [1276, 20]}, {"full_name": "MeasureTheory.Integrable.toL1_add", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [1432, 9], "def_end_pos": [1432, 17]}, {"full_name": "MeasureTheory.L1.setToL1", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [1019, 5], "def_end_pos": [1019, 12]}, {"full_name": "ContinuousLinearMap.map_add", "def_path": 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"869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/List/Basic.lean", "full_name": "List.fillNones_eq_fillNonesTR", "start": [1516, 10], "end": [1523, 25], "traced_tactics": [{"tactic": "funext \u03b1 as as'", "annotated_tactic": ["funext \u03b1 as as'", []], "state_before": "\u22a2 @fillNones = @fillNonesTR", "state_after": "case h.h.h\n\u03b1 : Type u_1\nas : List (Option \u03b1)\nas' : List \u03b1\n\u22a2 fillNones as as' = fillNonesTR as as'"}, {"tactic": "simp [fillNonesTR]", "annotated_tactic": ["simp [<a>fillNonesTR</a>]", [{"full_name": "List.fillNonesTR", "def_path": "lake-packages/std/Std/Data/List/Basic.lean", "def_pos": [1508, 15], "def_end_pos": [1508, 26]}]], "state_before": "case h.h.h\n\u03b1 : Type u_1\nas : List (Option \u03b1)\nas' : List \u03b1\n\u22a2 fillNones as as' = fillNonesTR as as'", "state_after": "case h.h.h\n\u03b1 : Type u_1\nas : List (Option \u03b1)\nas' : List \u03b1\n\u22a2 fillNones as as' = fillNonesTR.go as as' 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NormedSpace \u211d E\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : MeasurableSpace E\ninst\u271d\u2074 : MeasurableSpace F\ninst\u271d\u00b3 : BorelSpace E\ninst\u271d\u00b2 : BorelSpace F\ninst\u271d\u00b9 : SecondCountableTopology F\ninst\u271d : SigmaCompactSpace F\nb : Basis \u03b9 \u211d E\nf : E \u2243L[\u211d] F\nthis : IsAddHaarMeasure (Measure.map (\u2191f) (addHaar b))\n\u22a2 \u2191\u2191(addHaar b) (\u2191f \u207b\u00b9' _root_.parallelepiped (\u2191f.toLinearEquiv \u2218 \u2191b)) = 1"}, {"tactic": "erw [\u2190 image_parallelepiped, f.toEquiv.preimage_image, addHaar_self]", "annotated_tactic": ["erw [\u2190 <a>image_parallelepiped</a>, f.toEquiv.preimage_image, <a>addHaar_self</a>]", [{"full_name": "image_parallelepiped", "def_path": "Mathlib/MeasureTheory/Measure/Haar/OfBasis.lean", "def_pos": [53, 9], "def_end_pos": [53, 29]}, {"full_name": "Basis.addHaar_self", "def_path": "Mathlib/MeasureTheory/Measure/Haar/OfBasis.lean", "def_pos": [247, 9], "def_end_pos": [247, 27]}]], "state_before": "\u03b9 : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b9\u2070 : Fintype \u03b9\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : MeasurableSpace E\ninst\u271d\u2074 : MeasurableSpace F\ninst\u271d\u00b3 : BorelSpace E\ninst\u271d\u00b2 : BorelSpace F\ninst\u271d\u00b9 : SecondCountableTopology F\ninst\u271d : SigmaCompactSpace F\nb : Basis \u03b9 \u211d E\nf : E \u2243L[\u211d] F\nthis : IsAddHaarMeasure (Measure.map (\u2191f) (addHaar b))\n\u22a2 \u2191\u2191(addHaar b) (\u2191f \u207b\u00b9' _root_.parallelepiped (\u2191f.toLinearEquiv \u2218 \u2191b)) = 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "full_name": "MeasureTheory.Measure.ext_of_Ico_finite", "start": [739, 1], "end": [747, 16], "traced_tactics": [{"tactic": "refine'\n  ext_of_generate_finite _ (BorelSpace.measurable_eq.trans (borel_eq_generateFrom_Ico \u03b1))\n    (isPiSystem_Ico (id : \u03b1 \u2192 \u03b1) id) _ h\u03bc\u03bd", "annotated_tactic": ["refine'\n    <a>ext_of_generate_finite</a> _ (BorelSpace.measurable_eq.trans (<a>borel_eq_generateFrom_Ico</a> \u03b1))\n      (<a>isPiSystem_Ico</a> (<a>id</a> : \u03b1 \u2192 \u03b1) <a>id</a>) _ h\u03bc\u03bd", [{"full_name": "MeasureTheory.ext_of_generate_finite", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3986, 9], "def_end_pos": [3986, 31]}, {"full_name": "borel_eq_generateFrom_Ico", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [696, 9], "def_end_pos": [696, 34]}, {"full_name": "isPiSystem_Ico", "def_path": "Mathlib/MeasureTheory/PiSystem.lean", "def_pos": [216, 9], "def_end_pos": [216, 23]}, {"full_name": "id", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [33, 15], "def_end_pos": [33, 17]}, {"full_name": "id", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [33, 15], "def_end_pos": [33, 17]}]], "state_before": "\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns t u : Set \u03b1\u271d\ninst\u271d\u00b2\u00b2 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b2\u00b9 : MeasurableSpace \u03b1\u271d\ninst\u271d\u00b2\u2070 : OpensMeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2079 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u2078 : MeasurableSpace \u03b2\ninst\u271d\u00b9\u2077 : OpensMeasurableSpace \u03b2\ninst\u271d\u00b9\u2076 : TopologicalSpace \u03b3\ninst\u271d\u00b9\u2075 : MeasurableSpace \u03b3\ninst\u271d\u00b9\u2074 : BorelSpace \u03b3\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b3\u2082\ninst\u271d\u00b9\u00b2 : MeasurableSpace \u03b3\u2082\ninst\u271d\u00b9\u00b9 : BorelSpace \u03b3\u2082\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b4\n\u03b1' : Type u_6\ninst\u271d\u2079 : TopologicalSpace \u03b1'\ninst\u271d\u2078 : MeasurableSpace \u03b1'\ninst\u271d\u2077 : LinearOrder \u03b1\u271d\ninst\u271d\u2076 : OrderClosedTopology \u03b1\u271d\na b x : \u03b1\u271d\n\u03b1 : Type u_7\ninst\u271d\u2075 : TopologicalSpace \u03b1\nm : MeasurableSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : LinearOrder \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nh\u03bc\u03bd : \u2191\u2191\u03bc univ = \u2191\u2191\u03bd univ\nh : \u2200 \u2983a b : \u03b1\u2984, a < b \u2192 \u2191\u2191\u03bc (Ico a b) = \u2191\u2191\u03bd (Ico a b)\n\u22a2 \u03bc = \u03bd", "state_after": "\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns t u : Set \u03b1\u271d\ninst\u271d\u00b2\u00b2 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b2\u00b9 : MeasurableSpace \u03b1\u271d\ninst\u271d\u00b2\u2070 : OpensMeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2079 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u2078 : MeasurableSpace \u03b2\ninst\u271d\u00b9\u2077 : OpensMeasurableSpace \u03b2\ninst\u271d\u00b9\u2076 : TopologicalSpace \u03b3\ninst\u271d\u00b9\u2075 : MeasurableSpace \u03b3\ninst\u271d\u00b9\u2074 : BorelSpace \u03b3\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b3\u2082\ninst\u271d\u00b9\u00b2 : MeasurableSpace \u03b3\u2082\ninst\u271d\u00b9\u00b9 : BorelSpace \u03b3\u2082\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b4\n\u03b1' : Type u_6\ninst\u271d\u2079 : TopologicalSpace \u03b1'\ninst\u271d\u2078 : MeasurableSpace \u03b1'\ninst\u271d\u2077 : LinearOrder \u03b1\u271d\ninst\u271d\u2076 : OrderClosedTopology \u03b1\u271d\na b x : \u03b1\u271d\n\u03b1 : Type u_7\ninst\u271d\u2075 : TopologicalSpace \u03b1\nm : MeasurableSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : LinearOrder \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nh\u03bc\u03bd : \u2191\u2191\u03bc univ = \u2191\u2191\u03bd univ\nh : \u2200 \u2983a b : \u03b1\u2984, a < b \u2192 \u2191\u2191\u03bc (Ico a b) = \u2191\u2191\u03bd (Ico a b)\n\u22a2 \u2200 (s : Set \u03b1), s \u2208 {S | \u2203 l u, l < u \u2227 Ico l u = S} \u2192 \u2191\u2191\u03bc s = \u2191\u2191\u03bd s"}, {"tactic": "rintro - \u27e8a, b, hlt, rfl\u27e9", "annotated_tactic": ["rintro - \u27e8a, b, hlt, rfl\u27e9", []], "state_before": "\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns t u : Set \u03b1\u271d\ninst\u271d\u00b2\u00b2 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b2\u00b9 : MeasurableSpace \u03b1\u271d\ninst\u271d\u00b2\u2070 : OpensMeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2079 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u2078 : MeasurableSpace \u03b2\ninst\u271d\u00b9\u2077 : OpensMeasurableSpace \u03b2\ninst\u271d\u00b9\u2076 : TopologicalSpace \u03b3\ninst\u271d\u00b9\u2075 : MeasurableSpace \u03b3\ninst\u271d\u00b9\u2074 : BorelSpace \u03b3\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b3\u2082\ninst\u271d\u00b9\u00b2 : MeasurableSpace \u03b3\u2082\ninst\u271d\u00b9\u00b9 : BorelSpace \u03b3\u2082\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b4\n\u03b1' : Type u_6\ninst\u271d\u2079 : TopologicalSpace \u03b1'\ninst\u271d\u2078 : MeasurableSpace \u03b1'\ninst\u271d\u2077 : LinearOrder \u03b1\u271d\ninst\u271d\u2076 : OrderClosedTopology \u03b1\u271d\na b x : \u03b1\u271d\n\u03b1 : Type u_7\ninst\u271d\u2075 : TopologicalSpace \u03b1\nm : MeasurableSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : LinearOrder \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nh\u03bc\u03bd : \u2191\u2191\u03bc univ = \u2191\u2191\u03bd univ\nh : \u2200 \u2983a b : \u03b1\u2984, a < b \u2192 \u2191\u2191\u03bc (Ico a b) = \u2191\u2191\u03bd (Ico a b)\n\u22a2 \u2200 (s : Set \u03b1), s \u2208 {S | \u2203 l u, l < u \u2227 Ico l u = S} \u2192 \u2191\u2191\u03bc s = \u2191\u2191\u03bd s", "state_after": "case intro.intro.intro\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns t u : Set \u03b1\u271d\ninst\u271d\u00b2\u00b2 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b2\u00b9 : MeasurableSpace \u03b1\u271d\ninst\u271d\u00b2\u2070 : OpensMeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2079 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u2078 : MeasurableSpace \u03b2\ninst\u271d\u00b9\u2077 : OpensMeasurableSpace \u03b2\ninst\u271d\u00b9\u2076 : TopologicalSpace \u03b3\ninst\u271d\u00b9\u2075 : MeasurableSpace \u03b3\ninst\u271d\u00b9\u2074 : BorelSpace \u03b3\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b3\u2082\ninst\u271d\u00b9\u00b2 : MeasurableSpace \u03b3\u2082\ninst\u271d\u00b9\u00b9 : BorelSpace \u03b3\u2082\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b4\n\u03b1' : Type u_6\ninst\u271d\u2079 : TopologicalSpace \u03b1'\ninst\u271d\u2078 : MeasurableSpace \u03b1'\ninst\u271d\u2077 : LinearOrder \u03b1\u271d\ninst\u271d\u2076 : OrderClosedTopology \u03b1\u271d\na\u271d b\u271d x : \u03b1\u271d\n\u03b1 : Type u_7\ninst\u271d\u2075 : TopologicalSpace \u03b1\nm : MeasurableSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : LinearOrder \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nh\u03bc\u03bd : \u2191\u2191\u03bc univ = \u2191\u2191\u03bd univ\nh : \u2200 \u2983a b : \u03b1\u2984, a < b \u2192 \u2191\u2191\u03bc (Ico a b) = \u2191\u2191\u03bd (Ico a b)\na b : \u03b1\nhlt : a < b\n\u22a2 \u2191\u2191\u03bc (Ico a b) = \u2191\u2191\u03bd (Ico a b)"}, {"tactic": "exact h hlt", "annotated_tactic": ["exact h hlt", []], "state_before": "case intro.intro.intro\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns t u : Set \u03b1\u271d\ninst\u271d\u00b2\u00b2 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b2\u00b9 : MeasurableSpace \u03b1\u271d\ninst\u271d\u00b2\u2070 : OpensMeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2079 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u2078 : MeasurableSpace \u03b2\ninst\u271d\u00b9\u2077 : OpensMeasurableSpace \u03b2\ninst\u271d\u00b9\u2076 : TopologicalSpace \u03b3\ninst\u271d\u00b9\u2075 : MeasurableSpace \u03b3\ninst\u271d\u00b9\u2074 : BorelSpace \u03b3\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b3\u2082\ninst\u271d\u00b9\u00b2 : MeasurableSpace \u03b3\u2082\ninst\u271d\u00b9\u00b9 : BorelSpace \u03b3\u2082\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b4\n\u03b1' : Type u_6\ninst\u271d\u2079 : TopologicalSpace \u03b1'\ninst\u271d\u2078 : MeasurableSpace \u03b1'\ninst\u271d\u2077 : LinearOrder \u03b1\u271d\ninst\u271d\u2076 : OrderClosedTopology \u03b1\u271d\na\u271d b\u271d x : \u03b1\u271d\n\u03b1 : Type u_7\ninst\u271d\u2075 : TopologicalSpace \u03b1\nm : MeasurableSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : LinearOrder \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nh\u03bc\u03bd : \u2191\u2191\u03bc univ = \u2191\u2191\u03bd univ\nh : \u2200 \u2983a b : \u03b1\u2984, a < b \u2192 \u2191\u2191\u03bc (Ico a b) = \u2191\u2191\u03bd (Ico a b)\na b : \u03b1\nhlt : a < b\n\u22a2 \u2191\u2191\u03bc (Ico a b) = \u2191\u2191\u03bd (Ico a b)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Card.lean", "full_name": "Set.encard_le_coe_iff", "start": [148, 1], "end": [150, 53], "traced_tactics": [{"tactic": "rwa [ENat.le_coe_iff] at h", "annotated_tactic": ["rwa [<a>ENat.le_coe_iff</a>] at h", [{"full_name": "ENat.le_coe_iff", "def_path": "Mathlib/Data/ENat/Basic.lean", "def_pos": [209, 9], "def_end_pos": [209, 19]}]], "state_before": "\u03b1 : Type u_1\ns t : Set \u03b1\nk : \u2115\nh : encard s \u2264 \u2191k\n\u22a2 \u2203 n\u2080, encard s = \u2191n\u2080 \u2227 n\u2080 \u2264 k", "state_after": "no goals"}, {"tactic": "rwa [hs, Nat.cast_le]", "annotated_tactic": ["rwa [hs, <a>Nat.cast_le</a>]", [{"full_name": "Nat.cast_le", "def_path": "Mathlib/Data/Nat/Cast/Order.lean", "def_pos": [91, 9], "def_end_pos": [91, 16]}]], "state_before": "\u03b1 : Type u_1\ns t : Set \u03b1\nk : \u2115\nx\u271d : Set.Finite s \u2227 \u2203 n\u2080, encard s = \u2191n\u2080 \u2227 n\u2080 \u2264 k\nleft\u271d : Set.Finite s\nn\u2080 : \u2115\nhs : encard s = \u2191n\u2080\nhle : n\u2080 \u2264 k\n\u22a2 encard s \u2264 \u2191k", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/Jacobian.lean", "full_name": "MeasureTheory.lintegral_abs_det_fderiv_le_addHaar_image_aux2", "start": [1039, 1], "end": [1054, 72], "traced_tactics": [{"tactic": "have :\n  Tendsto (fun \u03b5 : \u211d\u22650 => \u03bc (f '' s) + 2 * \u03b5 * \u03bc s) (\ud835\udcdd[>] 0)\n    (\ud835\udcdd (\u03bc (f '' s) + 2 * (0 : \u211d\u22650) * \u03bc s)) := by\n  apply Tendsto.mono_left _ nhdsWithin_le_nhds\n  refine' tendsto_const_nhds.add _\n  refine' ENNReal.Tendsto.mul_const _ (Or.inr h's)\n  exact ENNReal.Tendsto.const_mul (ENNReal.tendsto_coe.2 tendsto_id) (Or.inr ENNReal.coe_ne_top)", "annotated_tactic": ["have :\n    <a>Tendsto</a> (fun \u03b5 : \u211d\u22650 => \u03bc (f '' s) + 2 * \u03b5 * \u03bc s) (\ud835\udcdd[>] 0)\n      (\ud835\udcdd (\u03bc (f '' s) + 2 * (0 : \u211d\u22650) * \u03bc s)) := by\n    apply <a>Tendsto.mono_left</a> _ <a>nhdsWithin_le_nhds</a>\n    refine' tendsto_const_nhds.add _\n    refine' <a>ENNReal.Tendsto.mul_const</a> _ (<a>Or.inr</a> h's)\n    exact <a>ENNReal.Tendsto.const_mul</a> (<a>ENNReal.tendsto_coe</a>.2 <a>tendsto_id</a>) (<a>Or.inr</a> <a>ENNReal.coe_ne_top</a>)", [{"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2939, 5], "def_end_pos": [2939, 12]}, {"full_name": "Filter.Tendsto.mono_left", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [3036, 9], "def_end_pos": [3036, 26]}, {"full_name": "nhdsWithin_le_nhds", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [204, 9], "def_end_pos": [204, 27]}, {"full_name": "ENNReal.Tendsto.mul_const", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [379, 19], "def_end_pos": [379, 36]}, {"full_name": "Or.inr", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [519, 5], "def_end_pos": [519, 8]}, {"full_name": "ENNReal.Tendsto.const_mul", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [373, 19], "def_end_pos": [373, 36]}, {"full_name": "ENNReal.tendsto_coe", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [71, 9], "def_end_pos": [71, 20]}, {"full_name": "Filter.tendsto_id", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [3028, 9], "def_end_pos": [3028, 19]}, {"full_name": "Or.inr", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [519, 5], "def_end_pos": [519, 8]}, {"full_name": "ENNReal.coe_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [302, 17], "def_end_pos": [302, 27]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nh's : \u2191\u2191\u03bc s \u2260 \u22a4\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u22a2 \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc \u2264 \u2191\u2191\u03bc (f '' s)", "state_after": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nh's : \u2191\u2191\u03bc s \u2260 \u22a4\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nthis : Tendsto (fun \u03b5 => \u2191\u2191\u03bc (f '' s) + 2 * \u2191\u03b5 * \u2191\u2191\u03bc s) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (f '' s) + 2 * \u21910 * \u2191\u2191\u03bc s))\n\u22a2 \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc \u2264 \u2191\u2191\u03bc (f '' s)"}, {"tactic": "simp only [add_zero, zero_mul, mul_zero, ENNReal.coe_zero] at this", "annotated_tactic": ["simp only [<a>add_zero</a>, <a>zero_mul</a>, <a>mul_zero</a>, <a>ENNReal.coe_zero</a>] at this", [{"full_name": "add_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [469, 3], "def_end_pos": [469, 14]}, {"full_name": "MulZeroClass.zero_mul", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [36, 3], "def_end_pos": [36, 11]}, {"full_name": "MulZeroClass.mul_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [38, 3], "def_end_pos": [38, 11]}, {"full_name": "ENNReal.coe_zero", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [215, 28], "def_end_pos": [215, 36]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nh's : \u2191\u2191\u03bc s \u2260 \u22a4\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nthis : Tendsto (fun \u03b5 => \u2191\u2191\u03bc (f '' s) + 2 * \u2191\u03b5 * \u2191\u2191\u03bc s) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (f '' s) + 2 * \u21910 * \u2191\u2191\u03bc s))\n\u22a2 \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc \u2264 \u2191\u2191\u03bc (f '' s)", "state_after": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nh's : \u2191\u2191\u03bc s \u2260 \u22a4\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nthis : Tendsto (fun \u03b5 => \u2191\u2191\u03bc (f '' s) + 2 * \u2191\u03b5 * \u2191\u2191\u03bc s) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (f '' s)))\n\u22a2 \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc \u2264 \u2191\u2191\u03bc (f '' s)"}, {"tactic": "apply ge_of_tendsto this", "annotated_tactic": ["apply <a>ge_of_tendsto</a> this", [{"full_name": "ge_of_tendsto", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [168, 9], "def_end_pos": [168, 22]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nh's : \u2191\u2191\u03bc s \u2260 \u22a4\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nthis : Tendsto (fun \u03b5 => \u2191\u2191\u03bc (f '' s) + 2 * \u2191\u03b5 * \u2191\u2191\u03bc s) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (f '' s)))\n\u22a2 \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc \u2264 \u2191\u2191\u03bc (f '' s)", "state_after": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nh's : \u2191\u2191\u03bc s \u2260 \u22a4\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nthis : Tendsto (fun \u03b5 => \u2191\u2191\u03bc (f '' s) + 2 * \u2191\u03b5 * \u2191\u2191\u03bc s) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (f '' s)))\n\u22a2 \u2200\u1da0 (c : \u211d\u22650) in \ud835\udcdd[Ioi 0] 0,\n    \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc \u2264 \u2191\u2191\u03bc (f '' s) + 2 * \u2191c * \u2191\u2191\u03bc s"}, {"tactic": "filter_upwards [self_mem_nhdsWithin]", "annotated_tactic": ["filter_upwards [<a>self_mem_nhdsWithin</a>]", [{"full_name": "self_mem_nhdsWithin", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [151, 9], "def_end_pos": [151, 28]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nh's : \u2191\u2191\u03bc s \u2260 \u22a4\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nthis : Tendsto (fun \u03b5 => \u2191\u2191\u03bc (f '' s) + 2 * \u2191\u03b5 * \u2191\u2191\u03bc s) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (f '' s)))\n\u22a2 \u2200\u1da0 (c : \u211d\u22650) in \ud835\udcdd[Ioi 0] 0,\n    \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc \u2264 \u2191\u2191\u03bc (f '' s) + 2 * \u2191c * \u2191\u2191\u03bc s", "state_after": "case h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nh's : \u2191\u2191\u03bc s \u2260 \u22a4\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nthis : Tendsto (fun \u03b5 => \u2191\u2191\u03bc (f '' s) + 2 * \u2191\u03b5 * \u2191\u2191\u03bc s) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (f '' s)))\n\u22a2 \u2200 (a : \u211d\u22650),\n    a \u2208 Ioi 0 \u2192 \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc \u2264 \u2191\u2191\u03bc (f '' s) + 2 * \u2191a * \u2191\u2191\u03bc s"}, {"tactic": "rintro \u03b5 (\u03b5pos : 0 < \u03b5)", "annotated_tactic": ["rintro \u03b5 (\u03b5pos : 0 < \u03b5)", []], "state_before": "case h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nh's : \u2191\u2191\u03bc s \u2260 \u22a4\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nthis : Tendsto (fun \u03b5 => \u2191\u2191\u03bc (f '' s) + 2 * \u2191\u03b5 * \u2191\u2191\u03bc s) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (f '' s)))\n\u22a2 \u2200 (a : \u211d\u22650),\n    a \u2208 Ioi 0 \u2192 \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc \u2264 \u2191\u2191\u03bc (f '' s) + 2 * \u2191a * \u2191\u2191\u03bc s", "state_after": "case h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nh's : \u2191\u2191\u03bc s \u2260 \u22a4\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nthis : Tendsto (fun \u03b5 => \u2191\u2191\u03bc (f '' s) + 2 * \u2191\u03b5 * \u2191\u2191\u03bc s) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (f '' s)))\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\n\u22a2 \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc \u2264 \u2191\u2191\u03bc (f '' s) + 2 * \u2191\u03b5 * \u2191\u2191\u03bc s"}, {"tactic": "exact lintegral_abs_det_fderiv_le_addHaar_image_aux1 \u03bc hs hf' hf \u03b5pos", "annotated_tactic": ["exact <a>lintegral_abs_det_fderiv_le_addHaar_image_aux1</a> \u03bc hs hf' hf \u03b5pos", [{"full_name": "MeasureTheory.lintegral_abs_det_fderiv_le_addHaar_image_aux1", "def_path": "Mathlib/MeasureTheory/Function/Jacobian.lean", "def_pos": [934, 9], "def_end_pos": [934, 55]}]], "state_before": "case h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nh's : \u2191\u2191\u03bc s \u2260 \u22a4\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nthis : Tendsto (fun \u03b5 => \u2191\u2191\u03bc (f '' s) + 2 * \u2191\u03b5 * \u2191\u2191\u03bc s) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (f '' s)))\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\n\u22a2 \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc \u2264 \u2191\u2191\u03bc (f '' s) + 2 * \u2191\u03b5 * \u2191\u2191\u03bc s", "state_after": "no goals"}, {"tactic": "apply Tendsto.mono_left _ nhdsWithin_le_nhds", "annotated_tactic": ["apply <a>Tendsto.mono_left</a> _ <a>nhdsWithin_le_nhds</a>", [{"full_name": "Filter.Tendsto.mono_left", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [3036, 9], "def_end_pos": [3036, 26]}, {"full_name": "nhdsWithin_le_nhds", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [204, 9], "def_end_pos": [204, 27]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nh's : \u2191\u2191\u03bc s \u2260 \u22a4\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u22a2 Tendsto (fun \u03b5 => \u2191\u2191\u03bc (f '' s) + 2 * \u2191\u03b5 * \u2191\u2191\u03bc s) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (f '' s) + 2 * \u21910 * \u2191\u2191\u03bc s))", "state_after": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nh's : \u2191\u2191\u03bc s \u2260 \u22a4\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u22a2 Tendsto (fun \u03b5 => \u2191\u2191\u03bc (f '' s) + 2 * \u2191\u03b5 * \u2191\u2191\u03bc s) (\ud835\udcdd 0) (\ud835\udcdd (\u2191\u2191\u03bc (f '' s) + 2 * \u21910 * \u2191\u2191\u03bc s))"}, {"tactic": "refine' tendsto_const_nhds.add _", "annotated_tactic": ["refine' tendsto_const_nhds.add _", []], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nh's : \u2191\u2191\u03bc s \u2260 \u22a4\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u22a2 Tendsto (fun \u03b5 => \u2191\u2191\u03bc (f '' s) + 2 * \u2191\u03b5 * \u2191\u2191\u03bc s) (\ud835\udcdd 0) (\ud835\udcdd (\u2191\u2191\u03bc (f '' s) + 2 * \u21910 * \u2191\u2191\u03bc s))", "state_after": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nh's : \u2191\u2191\u03bc s \u2260 \u22a4\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u22a2 Tendsto (fun \u03b5 => 2 * \u2191\u03b5 * \u2191\u2191\u03bc s) (\ud835\udcdd 0) (\ud835\udcdd (2 * \u21910 * \u2191\u2191\u03bc s))"}, {"tactic": "refine' ENNReal.Tendsto.mul_const _ (Or.inr h's)", "annotated_tactic": ["refine' <a>ENNReal.Tendsto.mul_const</a> _ (<a>Or.inr</a> h's)", [{"full_name": "ENNReal.Tendsto.mul_const", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [379, 19], "def_end_pos": [379, 36]}, {"full_name": "Or.inr", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [519, 5], "def_end_pos": [519, 8]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nh's : \u2191\u2191\u03bc s \u2260 \u22a4\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u22a2 Tendsto (fun \u03b5 => 2 * \u2191\u03b5 * \u2191\u2191\u03bc s) (\ud835\udcdd 0) (\ud835\udcdd (2 * \u21910 * \u2191\u2191\u03bc s))", "state_after": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nh's : \u2191\u2191\u03bc s \u2260 \u22a4\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u22a2 Tendsto (fun \u03b5 => 2 * \u2191\u03b5) (\ud835\udcdd 0) (\ud835\udcdd (2 * \u21910))"}, {"tactic": "exact ENNReal.Tendsto.const_mul (ENNReal.tendsto_coe.2 tendsto_id) (Or.inr ENNReal.coe_ne_top)", "annotated_tactic": ["exact <a>ENNReal.Tendsto.const_mul</a> (<a>ENNReal.tendsto_coe</a>.2 <a>tendsto_id</a>) (<a>Or.inr</a> <a>ENNReal.coe_ne_top</a>)", [{"full_name": "ENNReal.Tendsto.const_mul", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [373, 19], "def_end_pos": [373, 36]}, {"full_name": "ENNReal.tendsto_coe", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [71, 9], "def_end_pos": [71, 20]}, {"full_name": "Filter.tendsto_id", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [3028, 9], "def_end_pos": [3028, 19]}, {"full_name": "Or.inr", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [519, 5], "def_end_pos": [519, 8]}, {"full_name": "ENNReal.coe_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [302, 17], "def_end_pos": [302, 27]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nh's : \u2191\u2191\u03bc s \u2260 \u22a4\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u22a2 Tendsto (fun \u03b5 => 2 * \u2191\u03b5) (\ud835\udcdd 0) (\ud835\udcdd (2 * \u21910))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Array/Init/Lemmas.lean", "full_name": "Array.appendList_data", "start": [214, 9], "end": [217, 44], "traced_tactics": [{"tactic": "rw [\u2190 appendList_eq_append]", "annotated_tactic": ["rw [\u2190 <a>appendList_eq_append</a>]", [{"full_name": "Array.appendList_eq_append", "def_path": "lake-packages/std/Std/Data/Array/Init/Lemmas.lean", "def_pos": [211, 17], "def_end_pos": [211, 37]}]], "state_before": "\u03b1 : Type u_1\narr : Array \u03b1\nl : List \u03b1\n\u22a2 (arr ++ l).data = arr.data ++ l", "state_after": "\u03b1 : Type u_1\narr : Array \u03b1\nl : List \u03b1\n\u22a2 (Array.appendList arr l).data = arr.data ++ l"}, {"tactic": "unfold Array.appendList", "annotated_tactic": ["unfold <a>Array.appendList</a>", [{"full_name": "Array.appendList", "def_path": "lake-packages/lean4/src/lean/Init/Data/Array/Basic.lean", "def_pos": [491, 15], "def_end_pos": [491, 25]}]], "state_before": "\u03b1 : Type u_1\narr : Array \u03b1\nl : List \u03b1\n\u22a2 (Array.appendList arr l).data = arr.data ++ l", "state_after": "\u03b1 : Type u_1\narr : Array \u03b1\nl : List \u03b1\n\u22a2 (List.foldl (fun r v => push r v) arr l).data = arr.data ++ l"}, {"tactic": "induction l generalizing arr <;> simp [*]", "annotated_tactic": ["induction l generalizing arr <;> simp [*]", []], "state_before": "\u03b1 : Type u_1\narr : Array \u03b1\nl : List \u03b1\n\u22a2 (List.foldl (fun r v => push r v) arr l).data = arr.data ++ l", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Pointwise.lean", "full_name": "Finset.pow_subset_pow", "start": [885, 1], "end": [890, 52], "traced_tactics": [{"tactic": "simp [pow_zero]", "annotated_tactic": ["simp [<a>pow_zero</a>]", [{"full_name": "pow_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [639, 9], "def_end_pos": [639, 17]}]], "state_before": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : Monoid \u03b1\ns t : Finset \u03b1\na : \u03b1\nm n : \u2115\nhst : s \u2286 t\n\u22a2 s ^ 0 \u2286 t ^ 0", "state_after": "no goals"}, {"tactic": "rw [pow_succ]", "annotated_tactic": ["rw [<a>pow_succ</a>]", [{"full_name": "pow_succ", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [645, 9], "def_end_pos": [645, 17]}]], "state_before": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : Monoid \u03b1\ns t : Finset \u03b1\na : \u03b1\nm n\u271d : \u2115\nhst : s \u2286 t\nn : \u2115\n\u22a2 s ^ (n + 1) \u2286 t ^ (n + 1)", "state_after": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : Monoid \u03b1\ns t : Finset \u03b1\na : \u03b1\nm n\u271d : \u2115\nhst : s \u2286 t\nn : \u2115\n\u22a2 s * s ^ n \u2286 t ^ (n + 1)"}, {"tactic": "exact mul_subset_mul hst (pow_subset_pow hst n)", "annotated_tactic": ["exact <a>mul_subset_mul</a> hst (pow_subset_pow hst n)", [{"full_name": "Finset.mul_subset_mul", "def_path": "Mathlib/Data/Finset/Pointwise.lean", "def_pos": [412, 9], "def_end_pos": [412, 23]}]], "state_before": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : Monoid \u03b1\ns t : Finset \u03b1\na : \u03b1\nm n\u271d : \u2115\nhst : s \u2286 t\nn : \u2115\n\u22a2 s * s ^ n \u2286 t ^ (n + 1)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "full_name": "MeasureTheory.setToFun_add_left'", "start": [1320, 1], "end": [1326, 44], "traced_tactics": [{"tactic": "by_cases hf : Integrable f \u03bc", "annotated_tactic": ["by_cases hf : <a>Integrable</a> f \u03bc", [{"full_name": "MeasureTheory.Integrable", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [442, 5], "def_end_pos": [442, 15]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc T' C'\nhT'' : DominatedFinMeasAdditive \u03bc T'' C''\nh_add : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T'' s = T s + T' s\nf : \u03b1 \u2192 E\n\u22a2 setToFun \u03bc T'' hT'' f = setToFun \u03bc T hT f + setToFun \u03bc T' hT' f", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc T' C'\nhT'' : DominatedFinMeasAdditive \u03bc T'' C''\nh_add : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T'' s = T s + T' s\nf : \u03b1 \u2192 E\nhf : Integrable f\n\u22a2 setToFun \u03bc T'' hT'' f = setToFun \u03bc T hT f + setToFun \u03bc T' hT' f\n\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc T' C'\nhT'' : DominatedFinMeasAdditive \u03bc T'' C''\nh_add : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T'' s = T s + T' s\nf : \u03b1 \u2192 E\nhf : \u00acIntegrable f\n\u22a2 setToFun \u03bc T'' hT'' f = setToFun \u03bc T hT f + setToFun \u03bc T' hT' f"}, {"tactic": "simp_rw [setToFun_eq _ hf, L1.setToL1_add_left' hT hT' hT'' h_add]", "annotated_tactic": ["simp_rw [<a>setToFun_eq</a> _ hf, <a>L1.setToL1_add_left'</a> hT hT' hT'' h_add]", [{"full_name": "MeasureTheory.setToFun_eq", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [1276, 9], "def_end_pos": [1276, 20]}, {"full_name": "MeasureTheory.L1.setToL1_add_left'", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [1091, 9], "def_end_pos": [1091, 26]}]], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc T' C'\nhT'' : DominatedFinMeasAdditive \u03bc T'' C''\nh_add : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T'' s = T s + T' s\nf : \u03b1 \u2192 E\nhf : Integrable f\n\u22a2 setToFun \u03bc T'' hT'' f = setToFun \u03bc T hT f + setToFun \u03bc T' hT' f", "state_after": "no goals"}, {"tactic": "simp_rw [setToFun_undef _ hf, add_zero]", "annotated_tactic": ["simp_rw [<a>setToFun_undef</a> _ hf, <a>add_zero</a>]", [{"full_name": "MeasureTheory.setToFun_undef", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [1286, 9], "def_end_pos": [1286, 23]}, {"full_name": "add_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [469, 3], "def_end_pos": [469, 14]}]], "state_before": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc T' C'\nhT'' : DominatedFinMeasAdditive \u03bc T'' C''\nh_add : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T'' s = T s + T' s\nf : \u03b1 \u2192 E\nhf : \u00acIntegrable f\n\u22a2 setToFun \u03bc T'' hT'' f = setToFun \u03bc T hT f + setToFun \u03bc T' hT' f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "Finset.erase_injOn", "start": [2051, 1], "end": [2051, 96], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/LocallyIntegrable.lean", "full_name": "MeasureTheory.LocallyIntegrable.exists_nat_integrableOn", "start": [253, 1], "end": [258, 37], "traced_tactics": [{"tactic": "rcases (hf.locallyIntegrableOn univ).exists_nat_integrableOn with \u27e8u, u_open, u_union, hu\u27e9", "annotated_tactic": ["rcases (hf.locallyIntegrableOn <a>univ</a>).<a>exists_nat_integrableOn</a> with \u27e8u, u_open, u_union, hu\u27e9", [{"full_name": "Set.univ", "def_path": "Mathlib/Init/Set.lean", "def_pos": [90, 5], "def_end_pos": [90, 9]}, {"full_name": "MeasureTheory.LocallyIntegrableOn.exists_nat_integrableOn", "def_path": "Mathlib/MeasureTheory/Function/LocallyIntegrable.lean", "def_pos": [100, 9], "def_end_pos": [100, 52]}]], "state_before": "X : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormedAddCommGroup E\nf g : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : SecondCountableTopology X\nhf : LocallyIntegrable f\n\u22a2 \u2203 u, (\u2200 (n : \u2115), IsOpen (u n)) \u2227 \u22c3 n, u n = univ \u2227 \u2200 (n : \u2115), IntegrableOn f (u n)", "state_after": "case intro.intro.intro\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormedAddCommGroup E\nf g : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : SecondCountableTopology X\nhf : LocallyIntegrable f\nu : \u2115 \u2192 Set X\nu_open : \u2200 (n : \u2115), IsOpen (u n)\nu_union : univ \u2286 \u22c3 n, u n\nhu : \u2200 (n : \u2115), IntegrableOn f (u n \u2229 univ)\n\u22a2 \u2203 u, (\u2200 (n : \u2115), IsOpen (u n)) \u2227 \u22c3 n, u n = univ \u2227 \u2200 (n : \u2115), IntegrableOn f (u n)"}, {"tactic": "refine' \u27e8u, u_open, eq_univ_of_univ_subset u_union, fun n \u21a6 _\u27e9", "annotated_tactic": ["refine' \u27e8u, u_open, <a>eq_univ_of_univ_subset</a> u_union, fun n \u21a6 _\u27e9", [{"full_name": "Set.eq_univ_of_univ_subset", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [698, 8], "def_end_pos": [698, 30]}]], "state_before": "case intro.intro.intro\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormedAddCommGroup E\nf g : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : SecondCountableTopology X\nhf : LocallyIntegrable f\nu : \u2115 \u2192 Set X\nu_open : \u2200 (n : \u2115), IsOpen (u n)\nu_union : univ \u2286 \u22c3 n, u n\nhu : \u2200 (n : \u2115), IntegrableOn f (u n \u2229 univ)\n\u22a2 \u2203 u, (\u2200 (n : \u2115), IsOpen (u n)) \u2227 \u22c3 n, u n = univ \u2227 \u2200 (n : \u2115), IntegrableOn f (u n)", "state_after": "case intro.intro.intro\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormedAddCommGroup E\nf g : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : SecondCountableTopology X\nhf : LocallyIntegrable f\nu : \u2115 \u2192 Set X\nu_open : \u2200 (n : \u2115), IsOpen (u n)\nu_union : univ \u2286 \u22c3 n, u n\nhu : \u2200 (n : \u2115), IntegrableOn f (u n \u2229 univ)\nn : \u2115\n\u22a2 IntegrableOn f (u n)"}, {"tactic": "simpa only [inter_univ] using hu n", "annotated_tactic": ["simpa only [<a>inter_univ</a>] using hu n", [{"full_name": "Set.inter_univ", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1012, 9], "def_end_pos": [1012, 19]}]], "state_before": "case intro.intro.intro\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormedAddCommGroup E\nf g : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : SecondCountableTopology X\nhf : LocallyIntegrable f\nu : \u2115 \u2192 Set X\nu_open : \u2200 (n : \u2115), IsOpen (u n)\nu_union : univ \u2286 \u22c3 n, u n\nhu : \u2200 (n : \u2115), IntegrableOn f (u n \u2229 univ)\nn : \u2115\n\u22a2 IntegrableOn f (u n)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "full_name": "MeasureTheory.integral_union_eq_left_of_forall", "start": [326, 1], "end": [328, 63], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "full_name": "List.exists_or_eq_self_of_eraseP", "start": [1079, 1], "end": [1086, 46], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Pointwise.lean", "full_name": "Finset.isUnit_iff", "start": [1015, 1], "end": [1023, 20], "traced_tactics": [{"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : DivisionMonoid \u03b1\ns t : Finset \u03b1\n\u22a2 IsUnit s \u2194 \u2203 a, s = {a} \u2227 IsUnit a", "state_after": "case mp\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : DivisionMonoid \u03b1\ns t : Finset \u03b1\n\u22a2 IsUnit s \u2192 \u2203 a, s = {a} \u2227 IsUnit a\n\ncase mpr\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : DivisionMonoid \u03b1\ns t : Finset \u03b1\n\u22a2 (\u2203 a, s = {a} \u2227 IsUnit a) \u2192 IsUnit s"}, {"tactic": "rintro \u27e8u, rfl\u27e9", "annotated_tactic": ["rintro \u27e8u, rfl\u27e9", []], "state_before": "case mp\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : DivisionMonoid \u03b1\ns t : Finset \u03b1\n\u22a2 IsUnit s \u2192 \u2203 a, s = {a} \u2227 IsUnit a", "state_after": "case mp.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : DivisionMonoid \u03b1\nt : Finset \u03b1\nu : (Finset \u03b1)\u02e3\n\u22a2 \u2203 a, \u2191u = {a} \u2227 IsUnit a"}, {"tactic": "obtain \u27e8a, b, ha, hb, h\u27e9 := Finset.mul_eq_one_iff.1 u.mul_inv", "annotated_tactic": ["obtain \u27e8a, b, ha, hb, h\u27e9 := <a>Finset.mul_eq_one_iff</a>.1 u.mul_inv", [{"full_name": "Finset.mul_eq_one_iff", "def_path": "Mathlib/Data/Finset/Pointwise.lean", "def_pos": [1001, 19], "def_end_pos": [1001, 33]}]], "state_before": "case mp.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : DivisionMonoid \u03b1\nt : Finset \u03b1\nu : (Finset \u03b1)\u02e3\n\u22a2 \u2203 a, \u2191u = {a} \u2227 IsUnit a", "state_after": "case mp.intro.intro.intro.intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : DivisionMonoid \u03b1\nt : Finset \u03b1\nu : (Finset \u03b1)\u02e3\na b : \u03b1\nha : \u2191u = {a}\nhb : \u2191u\u207b\u00b9 = {b}\nh : a * b = 1\n\u22a2 \u2203 a, \u2191u = {a} \u2227 IsUnit a"}, {"tactic": "refine' \u27e8a, ha, \u27e8a, b, h, singleton_injective _\u27e9, rfl\u27e9", "annotated_tactic": ["refine' \u27e8a, ha, \u27e8a, b, h, <a>singleton_injective</a> _\u27e9, <a>rfl</a>\u27e9", [{"full_name": "Finset.singleton_injective", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [701, 9], "def_end_pos": [701, 28]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "case mp.intro.intro.intro.intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : DivisionMonoid \u03b1\nt : Finset \u03b1\nu : (Finset \u03b1)\u02e3\na b : \u03b1\nha : \u2191u = {a}\nhb : \u2191u\u207b\u00b9 = {b}\nh : a * b = 1\n\u22a2 \u2203 a, \u2191u = {a} \u2227 IsUnit a", "state_after": "case mp.intro.intro.intro.intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : DivisionMonoid \u03b1\nt : Finset \u03b1\nu : (Finset \u03b1)\u02e3\na b : \u03b1\nha : \u2191u = {a}\nhb : \u2191u\u207b\u00b9 = {b}\nh : a * b = 1\n\u22a2 {b * a} = {1}"}, {"tactic": "rw [\u2190 singleton_mul_singleton, \u2190 ha, \u2190 hb]", "annotated_tactic": ["rw [\u2190 <a>singleton_mul_singleton</a>, \u2190 ha, \u2190 hb]", [{"full_name": "Finset.singleton_mul_singleton", "def_path": "Mathlib/Data/Finset/Pointwise.lean", "def_pos": [406, 9], "def_end_pos": [406, 32]}]], "state_before": "case mp.intro.intro.intro.intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : DivisionMonoid \u03b1\nt : Finset \u03b1\nu : (Finset \u03b1)\u02e3\na b : \u03b1\nha : \u2191u = {a}\nhb : \u2191u\u207b\u00b9 = {b}\nh : a * b = 1\n\u22a2 {b * a} = {1}", "state_after": "case mp.intro.intro.intro.intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : DivisionMonoid \u03b1\nt : Finset \u03b1\nu : (Finset \u03b1)\u02e3\na b : \u03b1\nha : \u2191u = {a}\nhb : \u2191u\u207b\u00b9 = {b}\nh : a * b = 1\n\u22a2 \u2191u\u207b\u00b9 * \u2191u = {1}"}, {"tactic": "exact u.inv_mul", "annotated_tactic": ["exact u.inv_mul", []], "state_before": "case mp.intro.intro.intro.intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : DivisionMonoid \u03b1\nt : Finset \u03b1\nu : (Finset \u03b1)\u02e3\na b : \u03b1\nha : \u2191u = {a}\nhb : \u2191u\u207b\u00b9 = {b}\nh : a * b = 1\n\u22a2 \u2191u\u207b\u00b9 * \u2191u = {1}", "state_after": "no goals"}, {"tactic": "rintro \u27e8a, rfl, ha\u27e9", "annotated_tactic": ["rintro \u27e8a, rfl, ha\u27e9", []], "state_before": "case mpr\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : DivisionMonoid \u03b1\ns t : Finset \u03b1\n\u22a2 (\u2203 a, s = {a} \u2227 IsUnit a) \u2192 IsUnit s", "state_after": "case mpr.intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : DivisionMonoid \u03b1\nt : Finset \u03b1\na : \u03b1\nha : IsUnit a\n\u22a2 IsUnit {a}"}, {"tactic": "exact ha.finset", "annotated_tactic": ["exact ha.finset", []], "state_before": "case mpr.intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : DivisionMonoid \u03b1\nt : Finset \u03b1\na : \u03b1\nha : IsUnit a\n\u22a2 IsUnit {a}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/PEquiv.lean", "full_name": "PEquiv.trans_eq_none", "start": [157, 1], "end": [161, 20], "traced_tactics": [{"tactic": "simp only [eq_none_iff_forall_not_mem, mem_trans, imp_iff_not_or.symm]", "annotated_tactic": ["simp only [<a>eq_none_iff_forall_not_mem</a>, <a>mem_trans</a>, imp_iff_not_or.symm]", [{"full_name": "Option.eq_none_iff_forall_not_mem", "def_path": "lake-packages/std/Std/Data/Option/Lemmas.lean", "def_pos": [48, 9], "def_end_pos": [48, 35]}, {"full_name": "PEquiv.mem_trans", "def_path": "Mathlib/Data/PEquiv.lean", "def_pos": [147, 9], "def_end_pos": [147, 18]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nf : \u03b1 \u2243. \u03b2\ng : \u03b2 \u2243. \u03b3\na : \u03b1\n\u22a2 \u2191(PEquiv.trans f g) a = none \u2194 \u2200 (b : \u03b2) (c : \u03b3), \u00acb \u2208 \u2191f a \u2228 \u00acc \u2208 \u2191g b", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nf : \u03b1 \u2243. \u03b2\ng : \u03b2 \u2243. \u03b3\na : \u03b1\n\u22a2 (\u2200 (a_1 : \u03b3), \u00ac\u2203 b, b \u2208 \u2191f a \u2227 a_1 \u2208 \u2191g b) \u2194 \u2200 (b : \u03b2) (c : \u03b3), b \u2208 \u2191f a \u2192 \u00acc \u2208 \u2191g b"}, {"tactic": "push_neg", "annotated_tactic": ["push_neg", []], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nf : \u03b1 \u2243. \u03b2\ng : \u03b2 \u2243. \u03b3\na : \u03b1\n\u22a2 (\u2200 (a_1 : \u03b3), \u00ac\u2203 b, b \u2208 \u2191f a \u2227 a_1 \u2208 \u2191g b) \u2194 \u2200 (b : \u03b2) (c : \u03b3), b \u2208 \u2191f a \u2192 \u00acc \u2208 \u2191g b", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nf : \u03b1 \u2243. \u03b2\ng : \u03b2 \u2243. \u03b3\na : \u03b1\n\u22a2 (\u2200 (a_1 : \u03b3) (b : \u03b2), b \u2208 \u2191f a \u2192 \u00aca_1 \u2208 \u2191g b) \u2194 \u2200 (b : \u03b2) (c : \u03b3), b \u2208 \u2191f a \u2192 \u00acc \u2208 \u2191g b"}, {"tactic": "exact forall_swap", "annotated_tactic": ["exact <a>forall_swap</a>", [{"full_name": "forall_swap", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [667, 9], "def_end_pos": [667, 20]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nf : \u03b1 \u2243. \u03b2\ng : \u03b2 \u2243. \u03b3\na : \u03b1\n\u22a2 (\u2200 (a_1 : \u03b3) (b : \u03b2), b \u2208 \u2191f a \u2192 \u00aca_1 \u2208 \u2191g b) \u2194 \u2200 (b : \u03b2) (c : \u03b3), b \u2208 \u2191f a \u2192 \u00acc \u2208 \u2191g b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Image.lean", "full_name": "Set.disjoint_image_of_injective", "start": [1608, 1], "end": [1610, 95], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Portmanteau.lean", "full_name": "MeasureTheory.FiniteMeasure.limsup_measure_closed_le_of_tendsto", "start": [345, 1], "end": [377, 53], "traced_tactics": [{"tactic": "rcases L.eq_or_neBot with rfl | hne", "annotated_tactic": ["rcases L.eq_or_neBot with rfl | hne", []], "state_before": "\u03a9\u271d : Type u_1\ninst\u271d\u00b3 : MeasurableSpace \u03a9\u271d\n\u03a9 : Type u_2\n\u03b9 : Type u_3\nL : Filter \u03b9\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : PseudoEMetricSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\n\u03bcs : \u03b9 \u2192 FiniteMeasure \u03a9\n\u03bcs_lim : Tendsto \u03bcs L (\ud835\udcdd \u03bc)\nF : Set \u03a9\nF_closed : IsClosed F\n\u22a2 limsup (fun i => \u2191\u2191\u2191(\u03bcs i) F) L \u2264 \u2191\u2191\u2191\u03bc F", "state_after": "case inl\n\u03a9\u271d : Type u_1\ninst\u271d\u00b3 : MeasurableSpace \u03a9\u271d\n\u03a9 : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : PseudoEMetricSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\n\u03bcs : \u03b9 \u2192 FiniteMeasure \u03a9\nF : Set \u03a9\nF_closed : IsClosed F\n\u03bcs_lim : Tendsto \u03bcs \u22a5 (\ud835\udcdd \u03bc)\n\u22a2 limsup (fun i => \u2191\u2191\u2191(\u03bcs i) F) \u22a5 \u2264 \u2191\u2191\u2191\u03bc F\n\ncase inr\n\u03a9\u271d : Type u_1\ninst\u271d\u00b3 : MeasurableSpace \u03a9\u271d\n\u03a9 : Type u_2\n\u03b9 : Type u_3\nL : Filter \u03b9\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : PseudoEMetricSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\n\u03bcs : \u03b9 \u2192 FiniteMeasure \u03a9\n\u03bcs_lim : Tendsto \u03bcs L (\ud835\udcdd \u03bc)\nF : Set \u03a9\nF_closed : IsClosed F\nhne : NeBot L\n\u22a2 limsup (fun i => \u2191\u2191\u2191(\u03bcs i) F) L \u2264 \u2191\u2191\u2191\u03bc F"}, {"tactic": "apply ENNReal.le_of_forall_pos_le_add", "annotated_tactic": ["apply <a>ENNReal.le_of_forall_pos_le_add</a>", [{"full_name": "ENNReal.le_of_forall_pos_le_add", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [867, 9], "def_end_pos": [867, 32]}]], "state_before": "case inr\n\u03a9\u271d : Type u_1\ninst\u271d\u00b3 : MeasurableSpace \u03a9\u271d\n\u03a9 : Type u_2\n\u03b9 : Type u_3\nL : Filter \u03b9\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : PseudoEMetricSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\n\u03bcs : \u03b9 \u2192 FiniteMeasure \u03a9\n\u03bcs_lim : Tendsto \u03bcs L (\ud835\udcdd \u03bc)\nF : Set \u03a9\nF_closed : IsClosed F\nhne : NeBot L\n\u22a2 limsup (fun i => \u2191\u2191\u2191(\u03bcs i) F) L \u2264 \u2191\u2191\u2191\u03bc F", "state_after": "case inr.h\n\u03a9\u271d : Type u_1\ninst\u271d\u00b3 : MeasurableSpace \u03a9\u271d\n\u03a9 : Type u_2\n\u03b9 : Type u_3\nL : Filter \u03b9\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : PseudoEMetricSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\n\u03bcs : \u03b9 \u2192 FiniteMeasure \u03a9\n\u03bcs_lim : Tendsto \u03bcs L (\ud835\udcdd \u03bc)\nF : Set \u03a9\nF_closed : IsClosed F\nhne : NeBot L\n\u22a2 \u2200 (\u03b5 : \u211d\u22650), 0 < \u03b5 \u2192 \u2191\u2191\u2191\u03bc F < \u22a4 \u2192 limsup (fun i => \u2191\u2191\u2191(\u03bcs i) F) L \u2264 \u2191\u2191\u2191\u03bc F + \u2191\u03b5"}, {"tactic": "intro \u03b5 \u03b5_pos _", "annotated_tactic": ["intro \u03b5 \u03b5_pos _", []], "state_before": "case inr.h\n\u03a9\u271d : Type u_1\ninst\u271d\u00b3 : MeasurableSpace \u03a9\u271d\n\u03a9 : Type u_2\n\u03b9 : Type u_3\nL : Filter \u03b9\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : PseudoEMetricSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\n\u03bcs : \u03b9 \u2192 FiniteMeasure \u03a9\n\u03bcs_lim : Tendsto \u03bcs L (\ud835\udcdd \u03bc)\nF : Set \u03a9\nF_closed : IsClosed F\nhne : NeBot L\n\u22a2 \u2200 (\u03b5 : \u211d\u22650), 0 < \u03b5 \u2192 \u2191\u2191\u2191\u03bc F < \u22a4 \u2192 limsup (fun i => \u2191\u2191\u2191(\u03bcs i) F) L \u2264 \u2191\u2191\u2191\u03bc F + \u2191\u03b5", "state_after": "case inr.h\n\u03a9\u271d : Type u_1\ninst\u271d\u00b3 : MeasurableSpace \u03a9\u271d\n\u03a9 : Type u_2\n\u03b9 : Type u_3\nL : Filter \u03b9\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : PseudoEMetricSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\n\u03bcs : \u03b9 \u2192 FiniteMeasure \u03a9\n\u03bcs_lim : Tendsto \u03bcs L (\ud835\udcdd \u03bc)\nF : Set \u03a9\nF_closed : IsClosed F\nhne : NeBot L\n\u03b5 : \u211d\u22650\n\u03b5_pos : 0 < \u03b5\na\u271d : \u2191\u2191\u2191\u03bc F < \u22a4\n\u22a2 limsup (fun i => \u2191\u2191\u2191(\u03bcs i) F) L \u2264 \u2191\u2191\u2191\u03bc F + \u2191\u03b5"}, {"tactic": "let \u03b4s := fun n : \u2115 => (1 : \u211d) / (n + 1)", "annotated_tactic": ["let \u03b4s := fun n : \u2115 => (1 : \u211d) / (n + 1)", []], "state_before": "case inr.h\n\u03a9\u271d : Type u_1\ninst\u271d\u00b3 : MeasurableSpace \u03a9\u271d\n\u03a9 : Type u_2\n\u03b9 : Type u_3\nL : Filter \u03b9\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : PseudoEMetricSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\n\u03bcs : \u03b9 \u2192 FiniteMeasure \u03a9\n\u03bcs_lim : Tendsto \u03bcs L (\ud835\udcdd \u03bc)\nF : Set \u03a9\nF_closed : IsClosed F\nhne : NeBot L\n\u03b5 : \u211d\u22650\n\u03b5_pos : 0 < \u03b5\na\u271d : \u2191\u2191\u2191\u03bc F < \u22a4\n\u22a2 limsup (fun i => \u2191\u2191\u2191(\u03bcs i) F) L \u2264 \u2191\u2191\u2191\u03bc F + \u2191\u03b5", "state_after": "case inr.h\n\u03a9\u271d : Type u_1\ninst\u271d\u00b3 : MeasurableSpace \u03a9\u271d\n\u03a9 : Type u_2\n\u03b9 : Type u_3\nL : Filter \u03b9\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : PseudoEMetricSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\n\u03bcs : \u03b9 \u2192 FiniteMeasure \u03a9\n\u03bcs_lim : Tendsto \u03bcs L (\ud835\udcdd \u03bc)\nF : Set \u03a9\nF_closed : IsClosed F\nhne : NeBot L\n\u03b5 : \u211d\u22650\n\u03b5_pos : 0 < \u03b5\na\u271d : \u2191\u2191\u2191\u03bc F < \u22a4\n\u03b4s : \u2115 \u2192 \u211d := fun n => 1 / (\u2191n + 1)\n\u22a2 limsup (fun i => \u2191\u2191\u2191(\u03bcs i) F) L \u2264 \u2191\u2191\u2191\u03bc F + \u2191\u03b5"}, {"tactic": "have \u03b4s_pos : \u2200 n, 0 < \u03b4s n := fun n => Nat.one_div_pos_of_nat", "annotated_tactic": ["have \u03b4s_pos : \u2200 n, 0 < \u03b4s n := fun n => <a>Nat.one_div_pos_of_nat</a>", [{"full_name": "Nat.one_div_pos_of_nat", "def_path": "Mathlib/Data/Nat/Cast/Field.lean", "def_pos": [65, 9], "def_end_pos": [65, 27]}]], "state_before": "case inr.h\n\u03a9\u271d : Type u_1\ninst\u271d\u00b3 : MeasurableSpace \u03a9\u271d\n\u03a9 : Type u_2\n\u03b9 : Type u_3\nL : Filter \u03b9\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : PseudoEMetricSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\n\u03bcs : \u03b9 \u2192 FiniteMeasure \u03a9\n\u03bcs_lim : Tendsto \u03bcs L (\ud835\udcdd \u03bc)\nF : Set \u03a9\nF_closed : IsClosed F\nhne : NeBot L\n\u03b5 : \u211d\u22650\n\u03b5_pos : 0 < \u03b5\na\u271d : \u2191\u2191\u2191\u03bc F < \u22a4\n\u03b4s : \u2115 \u2192 \u211d := fun n => 1 / (\u2191n + 1)\n\u22a2 limsup (fun i => \u2191\u2191\u2191(\u03bcs i) F) L \u2264 \u2191\u2191\u2191\u03bc F + \u2191\u03b5", "state_after": "case inr.h\n\u03a9\u271d : Type u_1\ninst\u271d\u00b3 : MeasurableSpace \u03a9\u271d\n\u03a9 : Type u_2\n\u03b9 : Type u_3\nL : Filter \u03b9\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : PseudoEMetricSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\n\u03bcs : \u03b9 \u2192 FiniteMeasure \u03a9\n\u03bcs_lim : Tendsto \u03bcs L (\ud835\udcdd \u03bc)\nF : Set \u03a9\nF_closed : IsClosed F\nhne : NeBot L\n\u03b5 : \u211d\u22650\n\u03b5_pos : 0 < \u03b5\na\u271d : \u2191\u2191\u2191\u03bc F < \u22a4\n\u03b4s : \u2115 \u2192 \u211d := fun n => 1 / (\u2191n + 1)\n\u03b4s_pos : \u2200 (n : \u2115), 0 < \u03b4s n\n\u22a2 limsup (fun i => \u2191\u2191\u2191(\u03bcs i) F) L \u2264 \u2191\u2191\u2191\u03bc F + \u2191\u03b5"}, {"tactic": "have \u03b4s_lim : Tendsto \u03b4s atTop (\ud835\udcdd 0) := tendsto_one_div_add_atTop_nhds_0_nat", "annotated_tactic": ["have \u03b4s_lim : <a>Tendsto</a> \u03b4s <a>atTop</a> (\ud835\udcdd 0) := <a>tendsto_one_div_add_atTop_nhds_0_nat</a>", [{"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2939, 5], "def_end_pos": [2939, 12]}, {"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}, {"full_name": "tendsto_one_div_add_atTop_nhds_0_nat", "def_path": "Mathlib/Analysis/SpecificLimits/Basic.lean", "def_pos": [50, 9], "def_end_pos": [50, 45]}]], "state_before": "case inr.h\n\u03a9\u271d : Type u_1\ninst\u271d\u00b3 : MeasurableSpace \u03a9\u271d\n\u03a9 : Type u_2\n\u03b9 : Type u_3\nL : Filter \u03b9\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : PseudoEMetricSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\n\u03bcs : \u03b9 \u2192 FiniteMeasure \u03a9\n\u03bcs_lim : Tendsto \u03bcs L (\ud835\udcdd \u03bc)\nF : Set \u03a9\nF_closed : IsClosed F\nhne : NeBot L\n\u03b5 : \u211d\u22650\n\u03b5_pos : 0 < \u03b5\na\u271d : \u2191\u2191\u2191\u03bc F < \u22a4\n\u03b4s : \u2115 \u2192 \u211d := fun n => 1 / (\u2191n + 1)\n\u03b4s_pos : \u2200 (n : \u2115), 0 < \u03b4s n\n\u22a2 limsup (fun i => \u2191\u2191\u2191(\u03bcs i) F) L \u2264 \u2191\u2191\u2191\u03bc F + \u2191\u03b5", "state_after": "case inr.h\n\u03a9\u271d : Type u_1\ninst\u271d\u00b3 : MeasurableSpace \u03a9\u271d\n\u03a9 : Type u_2\n\u03b9 : Type u_3\nL : Filter \u03b9\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : PseudoEMetricSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\n\u03bcs : \u03b9 \u2192 FiniteMeasure \u03a9\n\u03bcs_lim : Tendsto \u03bcs L (\ud835\udcdd \u03bc)\nF : Set \u03a9\nF_closed : IsClosed F\nhne : NeBot L\n\u03b5 : \u211d\u22650\n\u03b5_pos : 0 < \u03b5\na\u271d : \u2191\u2191\u2191\u03bc F < \u22a4\n\u03b4s : \u2115 \u2192 \u211d := fun n => 1 / (\u2191n + 1)\n\u03b4s_pos : \u2200 (n : \u2115), 0 < \u03b4s n\n\u03b4s_lim : Tendsto \u03b4s atTop (\ud835\udcdd 0)\n\u22a2 limsup (fun i => \u2191\u2191\u2191(\u03bcs i) F) L \u2264 \u2191\u2191\u2191\u03bc F + \u2191\u03b5"}, {"tactic": "have key\u2081 :=\n  tendsto_lintegral_thickenedIndicator_of_isClosed (\u03bc : Measure \u03a9) F_closed \u03b4s_pos \u03b4s_lim", "annotated_tactic": ["have key\u2081 :=\n    <a>tendsto_lintegral_thickenedIndicator_of_isClosed</a> (\u03bc : <a>Measure</a> \u03a9) F_closed \u03b4s_pos \u03b4s_lim", [{"full_name": "MeasureTheory.tendsto_lintegral_thickenedIndicator_of_isClosed", "def_path": "Mathlib/MeasureTheory/Measure/Portmanteau.lean", "def_pos": [329, 9], "def_end_pos": [329, 57]}, {"full_name": "MeasureTheory.Measure", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [74, 11], "def_end_pos": [74, 18]}]], "state_before": "case inr.h\n\u03a9\u271d : Type u_1\ninst\u271d\u00b3 : MeasurableSpace \u03a9\u271d\n\u03a9 : Type u_2\n\u03b9 : Type u_3\nL : Filter \u03b9\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : PseudoEMetricSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\n\u03bcs : \u03b9 \u2192 FiniteMeasure \u03a9\n\u03bcs_lim : Tendsto \u03bcs L (\ud835\udcdd \u03bc)\nF : Set \u03a9\nF_closed : IsClosed F\nhne : NeBot L\n\u03b5 : \u211d\u22650\n\u03b5_pos : 0 < \u03b5\na\u271d : \u2191\u2191\u2191\u03bc F < \u22a4\n\u03b4s : \u2115 \u2192 \u211d := fun n => 1 / (\u2191n + 1)\n\u03b4s_pos : \u2200 (n : \u2115), 0 < \u03b4s n\n\u03b4s_lim : Tendsto \u03b4s atTop (\ud835\udcdd 0)\n\u22a2 limsup (fun i => \u2191\u2191\u2191(\u03bcs i) F) L \u2264 \u2191\u2191\u2191\u03bc F + \u2191\u03b5", "state_after": "case inr.h\n\u03a9\u271d : Type u_1\ninst\u271d\u00b3 : MeasurableSpace \u03a9\u271d\n\u03a9 : Type u_2\n\u03b9 : Type u_3\nL : Filter \u03b9\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : PseudoEMetricSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\n\u03bcs : \u03b9 \u2192 FiniteMeasure \u03a9\n\u03bcs_lim : Tendsto \u03bcs L (\ud835\udcdd \u03bc)\nF : Set \u03a9\nF_closed : IsClosed F\nhne : NeBot L\n\u03b5 : \u211d\u22650\n\u03b5_pos : 0 < \u03b5\na\u271d : \u2191\u2191\u2191\u03bc F < \u22a4\n\u03b4s : \u2115 \u2192 \u211d := fun n => 1 / (\u2191n + 1)\n\u03b4s_pos : \u2200 (n : \u2115), 0 < \u03b4s n\n\u03b4s_lim : Tendsto \u03b4s atTop (\ud835\udcdd 0)\nkey\u2081 : Tendsto (fun n => \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s n) F) \u03c9) \u2202\u2191\u03bc) atTop (\ud835\udcdd (\u2191\u2191\u2191\u03bc F))\n\u22a2 limsup (fun i => \u2191\u2191\u2191(\u03bcs i) F) L \u2264 \u2191\u2191\u2191\u03bc F + \u2191\u03b5"}, {"tactic": "have room\u2081 : (\u03bc : Measure \u03a9) F < (\u03bc : Measure \u03a9) F + \u03b5 / 2 := by\n  apply\n    ENNReal.lt_add_right (measure_lt_top (\u03bc : Measure \u03a9) F).ne\n      (ENNReal.div_pos_iff.mpr \u27e8(ENNReal.coe_pos.mpr \u03b5_pos).ne.symm, ENNReal.two_ne_top\u27e9).ne.symm", "annotated_tactic": ["have room\u2081 : (\u03bc : <a>Measure</a> \u03a9) F < (\u03bc : <a>Measure</a> \u03a9) F + \u03b5 / 2 := by\n    apply\n      <a>ENNReal.lt_add_right</a> (<a>measure_lt_top</a> (\u03bc : <a>Measure</a> \u03a9) F).<a>ne</a>\n        (ENNReal.div_pos_iff.mpr \u27e8(ENNReal.coe_pos.mpr \u03b5_pos).ne.symm, <a>ENNReal.two_ne_top</a>\u27e9).ne.symm", [{"full_name": "MeasureTheory.Measure", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [74, 11], "def_end_pos": [74, 18]}, {"full_name": "MeasureTheory.Measure", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [74, 11], "def_end_pos": [74, 18]}, {"full_name": "ENNReal.lt_add_right", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [829, 9], "def_end_pos": [829, 21]}, {"full_name": "MeasureTheory.measure_lt_top", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2866, 9], "def_end_pos": [2866, 23]}, {"full_name": "MeasureTheory.Measure", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [74, 11], "def_end_pos": [74, 18]}, {"full_name": "LT.lt.ne", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [152, 7], "def_end_pos": [152, 15]}, {"full_name": "ENNReal.two_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [431, 9], "def_end_pos": [431, 19]}]], "state_before": "case inr.h\n\u03a9\u271d : Type u_1\ninst\u271d\u00b3 : MeasurableSpace \u03a9\u271d\n\u03a9 : Type u_2\n\u03b9 : Type u_3\nL : Filter \u03b9\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : PseudoEMetricSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\n\u03bcs : \u03b9 \u2192 FiniteMeasure \u03a9\n\u03bcs_lim : Tendsto \u03bcs L (\ud835\udcdd \u03bc)\nF : Set \u03a9\nF_closed : IsClosed F\nhne : NeBot L\n\u03b5 : \u211d\u22650\n\u03b5_pos : 0 < \u03b5\na\u271d : \u2191\u2191\u2191\u03bc F < \u22a4\n\u03b4s : \u2115 \u2192 \u211d := fun n => 1 / (\u2191n + 1)\n\u03b4s_pos : \u2200 (n : \u2115), 0 < \u03b4s n\n\u03b4s_lim : Tendsto \u03b4s atTop (\ud835\udcdd 0)\nkey\u2081 : Tendsto (fun n => \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s n) F) \u03c9) \u2202\u2191\u03bc) atTop (\ud835\udcdd (\u2191\u2191\u2191\u03bc F))\n\u22a2 limsup (fun i => \u2191\u2191\u2191(\u03bcs i) F) L \u2264 \u2191\u2191\u2191\u03bc F + \u2191\u03b5", "state_after": "case inr.h\n\u03a9\u271d : Type u_1\ninst\u271d\u00b3 : MeasurableSpace \u03a9\u271d\n\u03a9 : Type u_2\n\u03b9 : Type u_3\nL : Filter \u03b9\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : PseudoEMetricSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\n\u03bcs : \u03b9 \u2192 FiniteMeasure \u03a9\n\u03bcs_lim : Tendsto \u03bcs L (\ud835\udcdd \u03bc)\nF : Set \u03a9\nF_closed : IsClosed F\nhne : NeBot L\n\u03b5 : \u211d\u22650\n\u03b5_pos : 0 < \u03b5\na\u271d : \u2191\u2191\u2191\u03bc F < \u22a4\n\u03b4s : \u2115 \u2192 \u211d := fun n => 1 / (\u2191n + 1)\n\u03b4s_pos : \u2200 (n : \u2115), 0 < \u03b4s n\n\u03b4s_lim : Tendsto \u03b4s atTop (\ud835\udcdd 0)\nkey\u2081 : Tendsto (fun n => \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s n) F) \u03c9) \u2202\u2191\u03bc) atTop (\ud835\udcdd (\u2191\u2191\u2191\u03bc F))\nroom\u2081 : \u2191\u2191\u2191\u03bc F < \u2191\u2191\u2191\u03bc F + \u2191\u03b5 / 2\n\u22a2 limsup (fun i => \u2191\u2191\u2191(\u03bcs i) F) L \u2264 \u2191\u2191\u2191\u03bc F + \u2191\u03b5"}, {"tactic": "rcases eventually_atTop.mp (eventually_lt_of_tendsto_lt room\u2081 key\u2081) with \u27e8M, hM\u27e9", "annotated_tactic": ["rcases eventually_atTop.mp (<a>eventually_lt_of_tendsto_lt</a> room\u2081 key\u2081) with \u27e8M, hM\u27e9", [{"full_name": "eventually_lt_of_tendsto_lt", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [391, 9], "def_end_pos": [391, 36]}]], "state_before": "case inr.h\n\u03a9\u271d : Type u_1\ninst\u271d\u00b3 : MeasurableSpace \u03a9\u271d\n\u03a9 : Type u_2\n\u03b9 : Type u_3\nL : Filter \u03b9\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : PseudoEMetricSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\n\u03bcs : \u03b9 \u2192 FiniteMeasure \u03a9\n\u03bcs_lim : Tendsto \u03bcs L (\ud835\udcdd \u03bc)\nF : Set \u03a9\nF_closed : IsClosed F\nhne : NeBot L\n\u03b5 : \u211d\u22650\n\u03b5_pos : 0 < \u03b5\na\u271d : \u2191\u2191\u2191\u03bc F < \u22a4\n\u03b4s : \u2115 \u2192 \u211d := fun n => 1 / (\u2191n + 1)\n\u03b4s_pos : \u2200 (n : \u2115), 0 < \u03b4s n\n\u03b4s_lim : Tendsto \u03b4s atTop (\ud835\udcdd 0)\nkey\u2081 : Tendsto (fun n => \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s n) F) \u03c9) \u2202\u2191\u03bc) atTop (\ud835\udcdd (\u2191\u2191\u2191\u03bc F))\nroom\u2081 : \u2191\u2191\u2191\u03bc F < \u2191\u2191\u2191\u03bc F + \u2191\u03b5 / 2\n\u22a2 limsup (fun i => \u2191\u2191\u2191(\u03bcs i) F) L \u2264 \u2191\u2191\u2191\u03bc F + \u2191\u03b5", "state_after": "case inr.h.intro\n\u03a9\u271d : Type u_1\ninst\u271d\u00b3 : MeasurableSpace \u03a9\u271d\n\u03a9 : Type u_2\n\u03b9 : Type u_3\nL : Filter \u03b9\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : PseudoEMetricSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\n\u03bcs : \u03b9 \u2192 FiniteMeasure \u03a9\n\u03bcs_lim : Tendsto \u03bcs L (\ud835\udcdd \u03bc)\nF : Set \u03a9\nF_closed : IsClosed F\nhne : NeBot L\n\u03b5 : \u211d\u22650\n\u03b5_pos : 0 < \u03b5\na\u271d : \u2191\u2191\u2191\u03bc F < \u22a4\n\u03b4s : \u2115 \u2192 \u211d := fun n => 1 / (\u2191n + 1)\n\u03b4s_pos : \u2200 (n : \u2115), 0 < \u03b4s n\n\u03b4s_lim : Tendsto \u03b4s atTop (\ud835\udcdd 0)\nkey\u2081 : Tendsto (fun n => \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s n) F) \u03c9) \u2202\u2191\u03bc) atTop (\ud835\udcdd (\u2191\u2191\u2191\u03bc F))\nroom\u2081 : \u2191\u2191\u2191\u03bc F < \u2191\u2191\u2191\u03bc F + \u2191\u03b5 / 2\nM : \u2115\nhM : \u2200 (b : \u2115), b \u2265 M \u2192 \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s b) F) \u03c9) \u2202\u2191\u03bc < \u2191\u2191\u2191\u03bc F + \u2191\u03b5 / 2\n\u22a2 limsup (fun i => \u2191\u2191\u2191(\u03bcs i) F) L \u2264 \u2191\u2191\u2191\u03bc F + \u2191\u03b5"}, {"tactic": "have key\u2082 :=\n  FiniteMeasure.tendsto_iff_forall_lintegral_tendsto.mp \u03bcs_lim (thickenedIndicator (\u03b4s_pos M) F)", "annotated_tactic": ["have key\u2082 :=\n    FiniteMeasure.tendsto_iff_forall_lintegral_tendsto.mp \u03bcs_lim (<a>thickenedIndicator</a> (\u03b4s_pos M) F)", [{"full_name": "thickenedIndicator", "def_path": "Mathlib/Topology/MetricSpace/ThickenedIndicator.lean", "def_pos": [163, 5], "def_end_pos": [163, 23]}]], "state_before": "case inr.h.intro\n\u03a9\u271d : Type u_1\ninst\u271d\u00b3 : MeasurableSpace \u03a9\u271d\n\u03a9 : Type u_2\n\u03b9 : Type u_3\nL : Filter \u03b9\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : PseudoEMetricSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\n\u03bcs : \u03b9 \u2192 FiniteMeasure \u03a9\n\u03bcs_lim : Tendsto \u03bcs L (\ud835\udcdd \u03bc)\nF : Set \u03a9\nF_closed : IsClosed F\nhne : NeBot L\n\u03b5 : \u211d\u22650\n\u03b5_pos : 0 < \u03b5\na\u271d : \u2191\u2191\u2191\u03bc F < \u22a4\n\u03b4s : \u2115 \u2192 \u211d := fun n => 1 / (\u2191n + 1)\n\u03b4s_pos : \u2200 (n : \u2115), 0 < \u03b4s n\n\u03b4s_lim : Tendsto \u03b4s atTop (\ud835\udcdd 0)\nkey\u2081 : Tendsto (fun n => \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s n) F) \u03c9) \u2202\u2191\u03bc) atTop (\ud835\udcdd (\u2191\u2191\u2191\u03bc F))\nroom\u2081 : \u2191\u2191\u2191\u03bc F < \u2191\u2191\u2191\u03bc F + \u2191\u03b5 / 2\nM : \u2115\nhM : \u2200 (b : \u2115), b \u2265 M \u2192 \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s b) F) \u03c9) \u2202\u2191\u03bc < \u2191\u2191\u2191\u03bc F + \u2191\u03b5 / 2\n\u22a2 limsup (fun i => \u2191\u2191\u2191(\u03bcs i) F) L \u2264 \u2191\u2191\u2191\u03bc F + \u2191\u03b5", "state_after": "case inr.h.intro\n\u03a9\u271d : Type u_1\ninst\u271d\u00b3 : MeasurableSpace \u03a9\u271d\n\u03a9 : Type u_2\n\u03b9 : Type u_3\nL : Filter \u03b9\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : PseudoEMetricSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\n\u03bcs : \u03b9 \u2192 FiniteMeasure \u03a9\n\u03bcs_lim : Tendsto \u03bcs L (\ud835\udcdd \u03bc)\nF : Set \u03a9\nF_closed : IsClosed F\nhne : NeBot L\n\u03b5 : \u211d\u22650\n\u03b5_pos : 0 < \u03b5\na\u271d : \u2191\u2191\u2191\u03bc F < \u22a4\n\u03b4s : \u2115 \u2192 \u211d := fun n => 1 / (\u2191n + 1)\n\u03b4s_pos : \u2200 (n : \u2115), 0 < \u03b4s n\n\u03b4s_lim : Tendsto \u03b4s atTop (\ud835\udcdd 0)\nkey\u2081 : Tendsto (fun n => \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s n) F) \u03c9) \u2202\u2191\u03bc) atTop (\ud835\udcdd (\u2191\u2191\u2191\u03bc F))\nroom\u2081 : \u2191\u2191\u2191\u03bc F < \u2191\u2191\u2191\u03bc F + \u2191\u03b5 / 2\nM : \u2115\nhM : \u2200 (b : \u2115), b \u2265 M \u2192 \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s b) F) \u03c9) \u2202\u2191\u03bc < \u2191\u2191\u2191\u03bc F + \u2191\u03b5 / 2\nkey\u2082 :\n  Tendsto (fun i => \u222b\u207b (x : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) x) \u2202\u2191(\u03bcs i)) L\n    (\ud835\udcdd (\u222b\u207b (x : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) x) \u2202\u2191\u03bc))\n\u22a2 limsup (fun i => \u2191\u2191\u2191(\u03bcs i) F) L \u2264 \u2191\u2191\u2191\u03bc F + \u2191\u03b5"}, {"tactic": "have room\u2082 :\n  (lintegral (\u03bc : Measure \u03a9) fun a => thickenedIndicator (\u03b4s_pos M) F a) <\n    (lintegral (\u03bc : Measure \u03a9) fun a => thickenedIndicator (\u03b4s_pos M) F a) + \u03b5 / 2 := by\n  apply ENNReal.lt_add_right (ne_of_lt ?_)\n      (ENNReal.div_pos_iff.mpr \u27e8(ENNReal.coe_pos.mpr \u03b5_pos).ne.symm, ENNReal.two_ne_top\u27e9).ne.symm\n  apply BoundedContinuousFunction.lintegral_lt_top_of_nnreal", "annotated_tactic": ["have room\u2082 :\n    (<a>lintegral</a> (\u03bc : <a>Measure</a> \u03a9) fun a => <a>thickenedIndicator</a> (\u03b4s_pos M) F a) <\n      (<a>lintegral</a> (\u03bc : <a>Measure</a> \u03a9) fun a => <a>thickenedIndicator</a> (\u03b4s_pos M) F a) + \u03b5 / 2 := by\n    apply <a>ENNReal.lt_add_right</a> (<a>ne_of_lt</a> ?_)\n        (ENNReal.div_pos_iff.mpr \u27e8(ENNReal.coe_pos.mpr \u03b5_pos).ne.symm, <a>ENNReal.two_ne_top</a>\u27e9).ne.symm\n    apply <a>BoundedContinuousFunction.lintegral_lt_top_of_nnreal</a>", [{"full_name": "MeasureTheory.lintegral", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [60, 17], "def_end_pos": [60, 26]}, {"full_name": "MeasureTheory.Measure", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [74, 11], "def_end_pos": [74, 18]}, {"full_name": "thickenedIndicator", "def_path": "Mathlib/Topology/MetricSpace/ThickenedIndicator.lean", "def_pos": [163, 5], "def_end_pos": [163, 23]}, {"full_name": "MeasureTheory.lintegral", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [60, 17], "def_end_pos": [60, 26]}, {"full_name": "MeasureTheory.Measure", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [74, 11], "def_end_pos": [74, 18]}, {"full_name": "thickenedIndicator", "def_path": "Mathlib/Topology/MetricSpace/ThickenedIndicator.lean", "def_pos": [163, 5], "def_end_pos": [163, 23]}, {"full_name": "ENNReal.lt_add_right", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [829, 9], "def_end_pos": [829, 21]}, {"full_name": "ne_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [101, 9], "def_end_pos": [101, 17]}, {"full_name": "ENNReal.two_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [431, 9], "def_end_pos": [431, 19]}, {"full_name": "BoundedContinuousFunction.lintegral_lt_top_of_nnreal", "def_path": "Mathlib/MeasureTheory/Integral/BoundedContinuousFunction.lean", "def_pos": [34, 9], "def_end_pos": [34, 35]}]], "state_before": "case inr.h.intro\n\u03a9\u271d : Type u_1\ninst\u271d\u00b3 : MeasurableSpace \u03a9\u271d\n\u03a9 : Type u_2\n\u03b9 : Type u_3\nL : Filter \u03b9\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : PseudoEMetricSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\n\u03bcs : \u03b9 \u2192 FiniteMeasure \u03a9\n\u03bcs_lim : Tendsto \u03bcs L (\ud835\udcdd \u03bc)\nF : Set \u03a9\nF_closed : IsClosed F\nhne : NeBot L\n\u03b5 : \u211d\u22650\n\u03b5_pos : 0 < \u03b5\na\u271d : \u2191\u2191\u2191\u03bc F < \u22a4\n\u03b4s : \u2115 \u2192 \u211d := fun n => 1 / (\u2191n + 1)\n\u03b4s_pos : \u2200 (n : \u2115), 0 < \u03b4s n\n\u03b4s_lim : Tendsto \u03b4s atTop (\ud835\udcdd 0)\nkey\u2081 : Tendsto (fun n => \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s n) F) \u03c9) \u2202\u2191\u03bc) atTop (\ud835\udcdd (\u2191\u2191\u2191\u03bc F))\nroom\u2081 : \u2191\u2191\u2191\u03bc F < \u2191\u2191\u2191\u03bc F + \u2191\u03b5 / 2\nM : \u2115\nhM : \u2200 (b : \u2115), b \u2265 M \u2192 \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s b) F) \u03c9) \u2202\u2191\u03bc < \u2191\u2191\u2191\u03bc F + \u2191\u03b5 / 2\nkey\u2082 :\n  Tendsto (fun i => \u222b\u207b (x : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) x) \u2202\u2191(\u03bcs i)) L\n    (\ud835\udcdd (\u222b\u207b (x : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) x) \u2202\u2191\u03bc))\n\u22a2 limsup (fun i => \u2191\u2191\u2191(\u03bcs i) F) L \u2264 \u2191\u2191\u2191\u03bc F + \u2191\u03b5", "state_after": "case inr.h.intro\n\u03a9\u271d : Type u_1\ninst\u271d\u00b3 : MeasurableSpace \u03a9\u271d\n\u03a9 : Type u_2\n\u03b9 : Type u_3\nL : Filter \u03b9\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : PseudoEMetricSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\n\u03bcs : \u03b9 \u2192 FiniteMeasure \u03a9\n\u03bcs_lim : Tendsto \u03bcs L (\ud835\udcdd \u03bc)\nF : Set \u03a9\nF_closed : IsClosed F\nhne : NeBot L\n\u03b5 : \u211d\u22650\n\u03b5_pos : 0 < \u03b5\na\u271d : \u2191\u2191\u2191\u03bc F < \u22a4\n\u03b4s : \u2115 \u2192 \u211d := fun n => 1 / (\u2191n + 1)\n\u03b4s_pos : \u2200 (n : \u2115), 0 < \u03b4s n\n\u03b4s_lim : Tendsto \u03b4s atTop (\ud835\udcdd 0)\nkey\u2081 : Tendsto (fun n => \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s n) F) \u03c9) \u2202\u2191\u03bc) atTop (\ud835\udcdd (\u2191\u2191\u2191\u03bc F))\nroom\u2081 : \u2191\u2191\u2191\u03bc F < \u2191\u2191\u2191\u03bc F + \u2191\u03b5 / 2\nM : \u2115\nhM : \u2200 (b : \u2115), b \u2265 M \u2192 \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s b) F) \u03c9) \u2202\u2191\u03bc < \u2191\u2191\u2191\u03bc F + \u2191\u03b5 / 2\nkey\u2082 :\n  Tendsto (fun i => \u222b\u207b (x : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) x) \u2202\u2191(\u03bcs i)) L\n    (\ud835\udcdd (\u222b\u207b (x : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) x) \u2202\u2191\u03bc))\nroom\u2082 :\n  \u222b\u207b (a : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) a) \u2202\u2191\u03bc <\n    \u222b\u207b (a : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) a) \u2202\u2191\u03bc + \u2191\u03b5 / 2\n\u22a2 limsup (fun i => \u2191\u2191\u2191(\u03bcs i) F) L \u2264 \u2191\u2191\u2191\u03bc F + \u2191\u03b5"}, {"tactic": "have ev_near := Eventually.mono (eventually_lt_of_tendsto_lt room\u2082 key\u2082) fun n => le_of_lt", "annotated_tactic": ["have ev_near := <a>Eventually.mono</a> (<a>eventually_lt_of_tendsto_lt</a> room\u2082 key\u2082) fun n => <a>le_of_lt</a>", [{"full_name": "Filter.Eventually.mono", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1140, 9], "def_end_pos": [1140, 24]}, {"full_name": "eventually_lt_of_tendsto_lt", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [391, 9], "def_end_pos": [391, 36]}, {"full_name": "le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [110, 9], "def_end_pos": [110, 17]}]], "state_before": "case inr.h.intro\n\u03a9\u271d : Type u_1\ninst\u271d\u00b3 : MeasurableSpace \u03a9\u271d\n\u03a9 : Type u_2\n\u03b9 : Type u_3\nL : Filter \u03b9\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : PseudoEMetricSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\n\u03bcs : \u03b9 \u2192 FiniteMeasure \u03a9\n\u03bcs_lim : Tendsto \u03bcs L (\ud835\udcdd \u03bc)\nF : Set \u03a9\nF_closed : IsClosed F\nhne : NeBot L\n\u03b5 : \u211d\u22650\n\u03b5_pos : 0 < \u03b5\na\u271d : \u2191\u2191\u2191\u03bc F < \u22a4\n\u03b4s : \u2115 \u2192 \u211d := fun n => 1 / (\u2191n + 1)\n\u03b4s_pos : \u2200 (n : \u2115), 0 < \u03b4s n\n\u03b4s_lim : Tendsto \u03b4s atTop (\ud835\udcdd 0)\nkey\u2081 : Tendsto (fun n => \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s n) F) \u03c9) \u2202\u2191\u03bc) atTop (\ud835\udcdd (\u2191\u2191\u2191\u03bc F))\nroom\u2081 : \u2191\u2191\u2191\u03bc F < \u2191\u2191\u2191\u03bc F + \u2191\u03b5 / 2\nM : \u2115\nhM : \u2200 (b : \u2115), b \u2265 M \u2192 \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s b) F) \u03c9) \u2202\u2191\u03bc < \u2191\u2191\u2191\u03bc F + \u2191\u03b5 / 2\nkey\u2082 :\n  Tendsto (fun i => \u222b\u207b (x : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) x) \u2202\u2191(\u03bcs i)) L\n    (\ud835\udcdd (\u222b\u207b (x : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) x) \u2202\u2191\u03bc))\nroom\u2082 :\n  \u222b\u207b (a : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) a) \u2202\u2191\u03bc <\n    \u222b\u207b (a : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) a) \u2202\u2191\u03bc + \u2191\u03b5 / 2\n\u22a2 limsup (fun i => \u2191\u2191\u2191(\u03bcs i) F) L \u2264 \u2191\u2191\u2191\u03bc F + \u2191\u03b5", "state_after": "case inr.h.intro\n\u03a9\u271d : Type u_1\ninst\u271d\u00b3 : MeasurableSpace \u03a9\u271d\n\u03a9 : Type u_2\n\u03b9 : Type u_3\nL : Filter \u03b9\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : PseudoEMetricSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\n\u03bcs : \u03b9 \u2192 FiniteMeasure \u03a9\n\u03bcs_lim : Tendsto \u03bcs L (\ud835\udcdd \u03bc)\nF : Set \u03a9\nF_closed : IsClosed F\nhne : NeBot L\n\u03b5 : \u211d\u22650\n\u03b5_pos : 0 < \u03b5\na\u271d : \u2191\u2191\u2191\u03bc F < \u22a4\n\u03b4s : \u2115 \u2192 \u211d := fun n => 1 / (\u2191n + 1)\n\u03b4s_pos : \u2200 (n : \u2115), 0 < \u03b4s n\n\u03b4s_lim : Tendsto \u03b4s atTop (\ud835\udcdd 0)\nkey\u2081 : Tendsto (fun n => \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s n) F) \u03c9) \u2202\u2191\u03bc) atTop (\ud835\udcdd (\u2191\u2191\u2191\u03bc F))\nroom\u2081 : \u2191\u2191\u2191\u03bc F < \u2191\u2191\u2191\u03bc F + \u2191\u03b5 / 2\nM : \u2115\nhM : \u2200 (b : \u2115), b \u2265 M \u2192 \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s b) F) \u03c9) \u2202\u2191\u03bc < \u2191\u2191\u2191\u03bc F + \u2191\u03b5 / 2\nkey\u2082 :\n  Tendsto (fun i => \u222b\u207b (x : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) x) \u2202\u2191(\u03bcs i)) L\n    (\ud835\udcdd (\u222b\u207b (x : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) x) \u2202\u2191\u03bc))\nroom\u2082 :\n  \u222b\u207b (a : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) a) \u2202\u2191\u03bc <\n    \u222b\u207b (a : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) a) \u2202\u2191\u03bc + \u2191\u03b5 / 2\nev_near :\n  \u2200\u1da0 (x : \u03b9) in L,\n    \u222b\u207b (x : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) x) \u2202\u2191(\u03bcs x) \u2264\n      \u222b\u207b (a : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) a) \u2202\u2191\u03bc + \u2191\u03b5 / 2\n\u22a2 limsup (fun i => \u2191\u2191\u2191(\u03bcs i) F) L \u2264 \u2191\u2191\u2191\u03bc F + \u2191\u03b5"}, {"tactic": "have ev_near' := Eventually.mono ev_near fun n => le_trans\n  (measure_le_lintegral_thickenedIndicator (\u03bcs n : Measure \u03a9) F_closed.measurableSet (\u03b4s_pos M))", "annotated_tactic": ["have ev_near' := <a>Eventually.mono</a> ev_near fun n => <a>le_trans</a>\n    (<a>measure_le_lintegral_thickenedIndicator</a> (\u03bcs n : <a>Measure</a> \u03a9) F_closed.measurableSet (\u03b4s_pos M))", [{"full_name": "Filter.Eventually.mono", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1140, 9], "def_end_pos": [1140, 24]}, {"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}, {"full_name": "measure_le_lintegral_thickenedIndicator", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [1360, 9], "def_end_pos": [1360, 48]}, {"full_name": "MeasureTheory.Measure", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [74, 11], "def_end_pos": [74, 18]}]], "state_before": "case inr.h.intro\n\u03a9\u271d : Type u_1\ninst\u271d\u00b3 : MeasurableSpace \u03a9\u271d\n\u03a9 : Type u_2\n\u03b9 : Type u_3\nL : Filter \u03b9\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : PseudoEMetricSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\n\u03bcs : \u03b9 \u2192 FiniteMeasure \u03a9\n\u03bcs_lim : Tendsto \u03bcs L (\ud835\udcdd \u03bc)\nF : Set \u03a9\nF_closed : IsClosed F\nhne : NeBot L\n\u03b5 : \u211d\u22650\n\u03b5_pos : 0 < \u03b5\na\u271d : \u2191\u2191\u2191\u03bc F < \u22a4\n\u03b4s : \u2115 \u2192 \u211d := fun n => 1 / (\u2191n + 1)\n\u03b4s_pos : \u2200 (n : \u2115), 0 < \u03b4s n\n\u03b4s_lim : Tendsto \u03b4s atTop (\ud835\udcdd 0)\nkey\u2081 : Tendsto (fun n => \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s n) F) \u03c9) \u2202\u2191\u03bc) atTop (\ud835\udcdd (\u2191\u2191\u2191\u03bc F))\nroom\u2081 : \u2191\u2191\u2191\u03bc F < \u2191\u2191\u2191\u03bc F + \u2191\u03b5 / 2\nM : \u2115\nhM : \u2200 (b : \u2115), b \u2265 M \u2192 \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s b) F) \u03c9) \u2202\u2191\u03bc < \u2191\u2191\u2191\u03bc F + \u2191\u03b5 / 2\nkey\u2082 :\n  Tendsto (fun i => \u222b\u207b (x : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) x) \u2202\u2191(\u03bcs i)) L\n    (\ud835\udcdd (\u222b\u207b (x : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) x) \u2202\u2191\u03bc))\nroom\u2082 :\n  \u222b\u207b (a : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) a) \u2202\u2191\u03bc <\n    \u222b\u207b (a : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) a) \u2202\u2191\u03bc + \u2191\u03b5 / 2\nev_near :\n  \u2200\u1da0 (x : \u03b9) in L,\n    \u222b\u207b (x : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) x) \u2202\u2191(\u03bcs x) \u2264\n      \u222b\u207b (a : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) a) \u2202\u2191\u03bc + \u2191\u03b5 / 2\n\u22a2 limsup (fun i => \u2191\u2191\u2191(\u03bcs i) F) L \u2264 \u2191\u2191\u2191\u03bc F + \u2191\u03b5", "state_after": "case inr.h.intro\n\u03a9\u271d : Type u_1\ninst\u271d\u00b3 : MeasurableSpace \u03a9\u271d\n\u03a9 : Type u_2\n\u03b9 : Type u_3\nL : Filter \u03b9\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : PseudoEMetricSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\n\u03bcs : \u03b9 \u2192 FiniteMeasure \u03a9\n\u03bcs_lim : Tendsto \u03bcs L (\ud835\udcdd \u03bc)\nF : Set \u03a9\nF_closed : IsClosed F\nhne : NeBot L\n\u03b5 : \u211d\u22650\n\u03b5_pos : 0 < \u03b5\na\u271d : \u2191\u2191\u2191\u03bc F < \u22a4\n\u03b4s : \u2115 \u2192 \u211d := fun n => 1 / (\u2191n + 1)\n\u03b4s_pos : \u2200 (n : \u2115), 0 < \u03b4s n\n\u03b4s_lim : Tendsto \u03b4s atTop (\ud835\udcdd 0)\nkey\u2081 : Tendsto (fun n => \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s n) F) \u03c9) \u2202\u2191\u03bc) atTop (\ud835\udcdd (\u2191\u2191\u2191\u03bc F))\nroom\u2081 : \u2191\u2191\u2191\u03bc F < \u2191\u2191\u2191\u03bc F + \u2191\u03b5 / 2\nM : \u2115\nhM : \u2200 (b : \u2115), b \u2265 M \u2192 \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s b) F) \u03c9) \u2202\u2191\u03bc < \u2191\u2191\u2191\u03bc F + \u2191\u03b5 / 2\nkey\u2082 :\n  Tendsto (fun i => \u222b\u207b (x : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) x) \u2202\u2191(\u03bcs i)) L\n    (\ud835\udcdd (\u222b\u207b (x : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) x) \u2202\u2191\u03bc))\nroom\u2082 :\n  \u222b\u207b (a : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) a) \u2202\u2191\u03bc <\n    \u222b\u207b (a : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) a) \u2202\u2191\u03bc + \u2191\u03b5 / 2\nev_near :\n  \u2200\u1da0 (x : \u03b9) in L,\n    \u222b\u207b (x : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) x) \u2202\u2191(\u03bcs x) \u2264\n      \u222b\u207b (a : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) a) \u2202\u2191\u03bc + \u2191\u03b5 / 2\nev_near' : \u2200\u1da0 (x : \u03b9) in L, \u2191\u2191\u2191(\u03bcs x) F \u2264 \u222b\u207b (a : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) a) \u2202\u2191\u03bc + \u2191\u03b5 / 2\n\u22a2 limsup (fun i => \u2191\u2191\u2191(\u03bcs i) F) L \u2264 \u2191\u2191\u2191\u03bc F + \u2191\u03b5"}, {"tactic": "apply (Filter.limsup_le_limsup ev_near').trans", "annotated_tactic": ["apply (<a>Filter.limsup_le_limsup</a> ev_near').<a>trans</a>", [{"full_name": "Filter.limsup_le_limsup", "def_path": "Mathlib/Order/LiminfLimsup.lean", "def_pos": [578, 9], "def_end_pos": [578, 25]}, {"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [120, 7], "def_end_pos": [120, 18]}]], "state_before": "case inr.h.intro\n\u03a9\u271d : Type u_1\ninst\u271d\u00b3 : MeasurableSpace \u03a9\u271d\n\u03a9 : Type u_2\n\u03b9 : Type u_3\nL : Filter \u03b9\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : PseudoEMetricSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\n\u03bcs : \u03b9 \u2192 FiniteMeasure \u03a9\n\u03bcs_lim : Tendsto \u03bcs L (\ud835\udcdd \u03bc)\nF : Set \u03a9\nF_closed : IsClosed F\nhne : NeBot L\n\u03b5 : \u211d\u22650\n\u03b5_pos : 0 < \u03b5\na\u271d : \u2191\u2191\u2191\u03bc F < \u22a4\n\u03b4s : \u2115 \u2192 \u211d := fun n => 1 / (\u2191n + 1)\n\u03b4s_pos : \u2200 (n : \u2115), 0 < \u03b4s n\n\u03b4s_lim : Tendsto \u03b4s atTop (\ud835\udcdd 0)\nkey\u2081 : Tendsto (fun n => \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s n) F) \u03c9) \u2202\u2191\u03bc) atTop (\ud835\udcdd (\u2191\u2191\u2191\u03bc F))\nroom\u2081 : \u2191\u2191\u2191\u03bc F < \u2191\u2191\u2191\u03bc F + \u2191\u03b5 / 2\nM : \u2115\nhM : \u2200 (b : \u2115), b \u2265 M \u2192 \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s b) F) \u03c9) \u2202\u2191\u03bc < \u2191\u2191\u2191\u03bc F + \u2191\u03b5 / 2\nkey\u2082 :\n  Tendsto (fun i => \u222b\u207b (x : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) x) \u2202\u2191(\u03bcs i)) L\n    (\ud835\udcdd (\u222b\u207b (x : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) x) \u2202\u2191\u03bc))\nroom\u2082 :\n  \u222b\u207b (a : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) a) \u2202\u2191\u03bc <\n    \u222b\u207b (a : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) a) \u2202\u2191\u03bc + \u2191\u03b5 / 2\nev_near :\n  \u2200\u1da0 (x : \u03b9) in L,\n    \u222b\u207b (x : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) x) \u2202\u2191(\u03bcs x) \u2264\n      \u222b\u207b (a : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) a) \u2202\u2191\u03bc + \u2191\u03b5 / 2\nev_near' : \u2200\u1da0 (x : \u03b9) in L, \u2191\u2191\u2191(\u03bcs x) F \u2264 \u222b\u207b (a : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) a) \u2202\u2191\u03bc + \u2191\u03b5 / 2\n\u22a2 limsup (fun i => \u2191\u2191\u2191(\u03bcs i) F) L \u2264 \u2191\u2191\u2191\u03bc F + \u2191\u03b5", "state_after": "case inr.h.intro\n\u03a9\u271d : Type u_1\ninst\u271d\u00b3 : MeasurableSpace \u03a9\u271d\n\u03a9 : Type u_2\n\u03b9 : Type u_3\nL : Filter \u03b9\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : PseudoEMetricSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\n\u03bcs : \u03b9 \u2192 FiniteMeasure \u03a9\n\u03bcs_lim : Tendsto \u03bcs L (\ud835\udcdd \u03bc)\nF : Set \u03a9\nF_closed : IsClosed F\nhne : NeBot L\n\u03b5 : \u211d\u22650\n\u03b5_pos : 0 < \u03b5\na\u271d : \u2191\u2191\u2191\u03bc F < \u22a4\n\u03b4s : \u2115 \u2192 \u211d := fun n => 1 / (\u2191n + 1)\n\u03b4s_pos : \u2200 (n : \u2115), 0 < \u03b4s n\n\u03b4s_lim : Tendsto \u03b4s atTop (\ud835\udcdd 0)\nkey\u2081 : Tendsto (fun n => \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s n) F) \u03c9) \u2202\u2191\u03bc) atTop (\ud835\udcdd (\u2191\u2191\u2191\u03bc F))\nroom\u2081 : \u2191\u2191\u2191\u03bc F < \u2191\u2191\u2191\u03bc F + \u2191\u03b5 / 2\nM : \u2115\nhM : \u2200 (b : \u2115), b \u2265 M \u2192 \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s b) F) \u03c9) \u2202\u2191\u03bc < \u2191\u2191\u2191\u03bc F + \u2191\u03b5 / 2\nkey\u2082 :\n  Tendsto (fun i => \u222b\u207b (x : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) x) \u2202\u2191(\u03bcs i)) L\n    (\ud835\udcdd (\u222b\u207b (x : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) x) \u2202\u2191\u03bc))\nroom\u2082 :\n  \u222b\u207b (a : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) a) \u2202\u2191\u03bc <\n    \u222b\u207b (a : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) a) \u2202\u2191\u03bc + \u2191\u03b5 / 2\nev_near :\n  \u2200\u1da0 (x : \u03b9) in L,\n    \u222b\u207b (x : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) x) \u2202\u2191(\u03bcs x) \u2264\n      \u222b\u207b (a : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) a) \u2202\u2191\u03bc + \u2191\u03b5 / 2\nev_near' : \u2200\u1da0 (x : \u03b9) in L, \u2191\u2191\u2191(\u03bcs x) F \u2264 \u222b\u207b (a : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) a) \u2202\u2191\u03bc + \u2191\u03b5 / 2\n\u22a2 limsup (fun x => \u222b\u207b (a : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) a) \u2202\u2191\u03bc + \u2191\u03b5 / 2) L \u2264 \u2191\u2191\u2191\u03bc F + \u2191\u03b5"}, {"tactic": "rw [limsup_const]", "annotated_tactic": ["rw [<a>limsup_const</a>]", [{"full_name": "Filter.limsup_const", "def_path": "Mathlib/Order/LiminfLimsup.lean", "def_pos": [664, 9], "def_end_pos": [664, 21]}]], "state_before": "case inr.h.intro\n\u03a9\u271d : Type u_1\ninst\u271d\u00b3 : MeasurableSpace \u03a9\u271d\n\u03a9 : Type u_2\n\u03b9 : Type u_3\nL : Filter \u03b9\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : PseudoEMetricSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\n\u03bcs : \u03b9 \u2192 FiniteMeasure \u03a9\n\u03bcs_lim : Tendsto \u03bcs L (\ud835\udcdd \u03bc)\nF : Set \u03a9\nF_closed : IsClosed F\nhne : NeBot L\n\u03b5 : \u211d\u22650\n\u03b5_pos : 0 < \u03b5\na\u271d : \u2191\u2191\u2191\u03bc F < \u22a4\n\u03b4s : \u2115 \u2192 \u211d := fun n => 1 / (\u2191n + 1)\n\u03b4s_pos : \u2200 (n : \u2115), 0 < \u03b4s n\n\u03b4s_lim : Tendsto \u03b4s atTop (\ud835\udcdd 0)\nkey\u2081 : Tendsto (fun n => \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s n) F) \u03c9) \u2202\u2191\u03bc) atTop (\ud835\udcdd (\u2191\u2191\u2191\u03bc F))\nroom\u2081 : \u2191\u2191\u2191\u03bc F < \u2191\u2191\u2191\u03bc F + \u2191\u03b5 / 2\nM : \u2115\nhM : \u2200 (b : \u2115), b \u2265 M \u2192 \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s b) F) \u03c9) \u2202\u2191\u03bc < \u2191\u2191\u2191\u03bc F + \u2191\u03b5 / 2\nkey\u2082 :\n  Tendsto (fun i => \u222b\u207b (x : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) x) \u2202\u2191(\u03bcs i)) L\n    (\ud835\udcdd (\u222b\u207b (x : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) x) \u2202\u2191\u03bc))\nroom\u2082 :\n  \u222b\u207b (a : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) a) \u2202\u2191\u03bc <\n    \u222b\u207b (a : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) a) \u2202\u2191\u03bc + \u2191\u03b5 / 2\nev_near :\n  \u2200\u1da0 (x : \u03b9) in L,\n    \u222b\u207b (x : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) x) \u2202\u2191(\u03bcs x) \u2264\n      \u222b\u207b (a : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) a) \u2202\u2191\u03bc + \u2191\u03b5 / 2\nev_near' : \u2200\u1da0 (x : \u03b9) in L, \u2191\u2191\u2191(\u03bcs x) F \u2264 \u222b\u207b (a : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) a) \u2202\u2191\u03bc + \u2191\u03b5 / 2\n\u22a2 limsup (fun x => \u222b\u207b (a : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) a) \u2202\u2191\u03bc + \u2191\u03b5 / 2) L \u2264 \u2191\u2191\u2191\u03bc F + \u2191\u03b5", "state_after": "case inr.h.intro\n\u03a9\u271d : Type u_1\ninst\u271d\u00b3 : MeasurableSpace \u03a9\u271d\n\u03a9 : Type u_2\n\u03b9 : Type u_3\nL : Filter \u03b9\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : PseudoEMetricSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\n\u03bcs : \u03b9 \u2192 FiniteMeasure \u03a9\n\u03bcs_lim : Tendsto \u03bcs L (\ud835\udcdd \u03bc)\nF : Set \u03a9\nF_closed : IsClosed F\nhne : NeBot L\n\u03b5 : \u211d\u22650\n\u03b5_pos : 0 < \u03b5\na\u271d : \u2191\u2191\u2191\u03bc F < \u22a4\n\u03b4s : \u2115 \u2192 \u211d := fun n => 1 / (\u2191n + 1)\n\u03b4s_pos : \u2200 (n : \u2115), 0 < \u03b4s n\n\u03b4s_lim : Tendsto \u03b4s atTop (\ud835\udcdd 0)\nkey\u2081 : Tendsto (fun n => \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s n) F) \u03c9) \u2202\u2191\u03bc) atTop (\ud835\udcdd (\u2191\u2191\u2191\u03bc F))\nroom\u2081 : \u2191\u2191\u2191\u03bc F < \u2191\u2191\u2191\u03bc F + \u2191\u03b5 / 2\nM : \u2115\nhM : \u2200 (b : \u2115), b \u2265 M \u2192 \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s b) F) \u03c9) \u2202\u2191\u03bc < \u2191\u2191\u2191\u03bc F + \u2191\u03b5 / 2\nkey\u2082 :\n  Tendsto (fun i => \u222b\u207b (x : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) x) \u2202\u2191(\u03bcs i)) L\n    (\ud835\udcdd (\u222b\u207b (x : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) x) \u2202\u2191\u03bc))\nroom\u2082 :\n  \u222b\u207b (a : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) a) \u2202\u2191\u03bc <\n    \u222b\u207b (a : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) a) \u2202\u2191\u03bc + \u2191\u03b5 / 2\nev_near :\n  \u2200\u1da0 (x : \u03b9) in L,\n    \u222b\u207b (x : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) x) \u2202\u2191(\u03bcs x) \u2264\n      \u222b\u207b (a : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) a) \u2202\u2191\u03bc + \u2191\u03b5 / 2\nev_near' : \u2200\u1da0 (x : \u03b9) in L, \u2191\u2191\u2191(\u03bcs x) F \u2264 \u222b\u207b (a : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) a) \u2202\u2191\u03bc + \u2191\u03b5 / 2\n\u22a2 \u222b\u207b (a : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) a) \u2202\u2191\u03bc + \u2191\u03b5 / 2 \u2264 \u2191\u2191\u2191\u03bc F + \u2191\u03b5"}, {"tactic": "apply le_trans (add_le_add (hM M rfl.le).le (le_refl (\u03b5 / 2 : \u211d\u22650\u221e)))", "annotated_tactic": ["apply <a>le_trans</a> (<a>add_le_add</a> (hM M rfl.le).<a>le</a> (<a>le_refl</a> (\u03b5 / 2 : \u211d\u22650\u221e)))", [{"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}, {"full_name": "add_le_add", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [205, 15], "def_end_pos": [205, 25]}, {"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [142, 7], "def_end_pos": [142, 15]}, {"full_name": "le_refl", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [50, 9], "def_end_pos": [50, 16]}]], "state_before": "case inr.h.intro\n\u03a9\u271d : Type u_1\ninst\u271d\u00b3 : MeasurableSpace \u03a9\u271d\n\u03a9 : Type u_2\n\u03b9 : Type u_3\nL : Filter \u03b9\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : PseudoEMetricSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\n\u03bcs : \u03b9 \u2192 FiniteMeasure \u03a9\n\u03bcs_lim : Tendsto \u03bcs L (\ud835\udcdd \u03bc)\nF : Set \u03a9\nF_closed : IsClosed F\nhne : NeBot L\n\u03b5 : \u211d\u22650\n\u03b5_pos : 0 < \u03b5\na\u271d : \u2191\u2191\u2191\u03bc F < \u22a4\n\u03b4s : \u2115 \u2192 \u211d := fun n => 1 / (\u2191n + 1)\n\u03b4s_pos : \u2200 (n : \u2115), 0 < \u03b4s n\n\u03b4s_lim : Tendsto \u03b4s atTop (\ud835\udcdd 0)\nkey\u2081 : Tendsto (fun n => \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s n) F) \u03c9) \u2202\u2191\u03bc) atTop (\ud835\udcdd (\u2191\u2191\u2191\u03bc F))\nroom\u2081 : \u2191\u2191\u2191\u03bc F < \u2191\u2191\u2191\u03bc F + \u2191\u03b5 / 2\nM : \u2115\nhM : \u2200 (b : \u2115), b \u2265 M \u2192 \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s b) F) \u03c9) \u2202\u2191\u03bc < \u2191\u2191\u2191\u03bc F + \u2191\u03b5 / 2\nkey\u2082 :\n  Tendsto (fun i => \u222b\u207b (x : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) x) \u2202\u2191(\u03bcs i)) L\n    (\ud835\udcdd (\u222b\u207b (x : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) x) \u2202\u2191\u03bc))\nroom\u2082 :\n  \u222b\u207b (a : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) a) \u2202\u2191\u03bc <\n    \u222b\u207b (a : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) a) \u2202\u2191\u03bc + \u2191\u03b5 / 2\nev_near :\n  \u2200\u1da0 (x : \u03b9) in L,\n    \u222b\u207b (x : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) x) \u2202\u2191(\u03bcs x) \u2264\n      \u222b\u207b (a : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) a) \u2202\u2191\u03bc + \u2191\u03b5 / 2\nev_near' : \u2200\u1da0 (x : \u03b9) in L, \u2191\u2191\u2191(\u03bcs x) F \u2264 \u222b\u207b (a : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) a) \u2202\u2191\u03bc + \u2191\u03b5 / 2\n\u22a2 \u222b\u207b (a : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) a) \u2202\u2191\u03bc + \u2191\u03b5 / 2 \u2264 \u2191\u2191\u2191\u03bc F + \u2191\u03b5", "state_after": "case inr.h.intro\n\u03a9\u271d : Type u_1\ninst\u271d\u00b3 : MeasurableSpace \u03a9\u271d\n\u03a9 : Type u_2\n\u03b9 : Type u_3\nL : Filter \u03b9\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : PseudoEMetricSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\n\u03bcs : \u03b9 \u2192 FiniteMeasure \u03a9\n\u03bcs_lim : Tendsto \u03bcs L (\ud835\udcdd \u03bc)\nF : Set \u03a9\nF_closed : IsClosed F\nhne : NeBot L\n\u03b5 : \u211d\u22650\n\u03b5_pos : 0 < \u03b5\na\u271d : \u2191\u2191\u2191\u03bc F < \u22a4\n\u03b4s : \u2115 \u2192 \u211d := fun n => 1 / (\u2191n + 1)\n\u03b4s_pos : \u2200 (n : \u2115), 0 < \u03b4s n\n\u03b4s_lim : Tendsto \u03b4s atTop (\ud835\udcdd 0)\nkey\u2081 : Tendsto (fun n => \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s n) F) \u03c9) \u2202\u2191\u03bc) atTop (\ud835\udcdd (\u2191\u2191\u2191\u03bc F))\nroom\u2081 : \u2191\u2191\u2191\u03bc F < \u2191\u2191\u2191\u03bc F + \u2191\u03b5 / 2\nM : \u2115\nhM : \u2200 (b : \u2115), b \u2265 M \u2192 \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s b) F) \u03c9) \u2202\u2191\u03bc < \u2191\u2191\u2191\u03bc F + \u2191\u03b5 / 2\nkey\u2082 :\n  Tendsto (fun i => \u222b\u207b (x : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) x) \u2202\u2191(\u03bcs i)) L\n    (\ud835\udcdd (\u222b\u207b (x : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) x) \u2202\u2191\u03bc))\nroom\u2082 :\n  \u222b\u207b (a : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) a) \u2202\u2191\u03bc <\n    \u222b\u207b (a : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) a) \u2202\u2191\u03bc + \u2191\u03b5 / 2\nev_near :\n  \u2200\u1da0 (x : \u03b9) in L,\n    \u222b\u207b (x : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) x) \u2202\u2191(\u03bcs x) \u2264\n      \u222b\u207b (a : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) a) \u2202\u2191\u03bc + \u2191\u03b5 / 2\nev_near' : \u2200\u1da0 (x : \u03b9) in L, \u2191\u2191\u2191(\u03bcs x) F \u2264 \u222b\u207b (a : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) a) \u2202\u2191\u03bc + \u2191\u03b5 / 2\n\u22a2 \u2191\u2191\u2191\u03bc F + \u2191\u03b5 / 2 + \u2191\u03b5 / 2 \u2264 \u2191\u2191\u2191\u03bc F + \u2191\u03b5"}, {"tactic": "simp only [add_assoc, ENNReal.add_halves, le_refl]", "annotated_tactic": ["simp only [<a>add_assoc</a>, <a>ENNReal.add_halves</a>, <a>le_refl</a>]", [{"full_name": "add_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [263, 3], "def_end_pos": [263, 14]}, {"full_name": "ENNReal.add_halves", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1781, 19], "def_end_pos": [1781, 29]}, {"full_name": "le_refl", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [50, 9], "def_end_pos": [50, 16]}]], "state_before": "case inr.h.intro\n\u03a9\u271d : Type u_1\ninst\u271d\u00b3 : MeasurableSpace \u03a9\u271d\n\u03a9 : Type u_2\n\u03b9 : Type u_3\nL : Filter \u03b9\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : PseudoEMetricSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\n\u03bcs : \u03b9 \u2192 FiniteMeasure \u03a9\n\u03bcs_lim : Tendsto \u03bcs L (\ud835\udcdd \u03bc)\nF : Set \u03a9\nF_closed : IsClosed F\nhne : NeBot L\n\u03b5 : \u211d\u22650\n\u03b5_pos : 0 < \u03b5\na\u271d : \u2191\u2191\u2191\u03bc F < \u22a4\n\u03b4s : \u2115 \u2192 \u211d := fun n => 1 / (\u2191n + 1)\n\u03b4s_pos : \u2200 (n : \u2115), 0 < \u03b4s n\n\u03b4s_lim : Tendsto \u03b4s atTop (\ud835\udcdd 0)\nkey\u2081 : Tendsto (fun n => \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s n) F) \u03c9) \u2202\u2191\u03bc) atTop (\ud835\udcdd (\u2191\u2191\u2191\u03bc F))\nroom\u2081 : \u2191\u2191\u2191\u03bc F < \u2191\u2191\u2191\u03bc F + \u2191\u03b5 / 2\nM : \u2115\nhM : \u2200 (b : \u2115), b \u2265 M \u2192 \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s b) F) \u03c9) \u2202\u2191\u03bc < \u2191\u2191\u2191\u03bc F + \u2191\u03b5 / 2\nkey\u2082 :\n  Tendsto (fun i => \u222b\u207b (x : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) x) \u2202\u2191(\u03bcs i)) L\n    (\ud835\udcdd (\u222b\u207b (x : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) x) \u2202\u2191\u03bc))\nroom\u2082 :\n  \u222b\u207b (a : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) a) \u2202\u2191\u03bc <\n    \u222b\u207b (a : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) a) \u2202\u2191\u03bc + \u2191\u03b5 / 2\nev_near :\n  \u2200\u1da0 (x : \u03b9) in L,\n    \u222b\u207b (x : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) x) \u2202\u2191(\u03bcs x) \u2264\n      \u222b\u207b (a : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) a) \u2202\u2191\u03bc + \u2191\u03b5 / 2\nev_near' : \u2200\u1da0 (x : \u03b9) in L, \u2191\u2191\u2191(\u03bcs x) F \u2264 \u222b\u207b (a : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) a) \u2202\u2191\u03bc + \u2191\u03b5 / 2\n\u22a2 \u2191\u2191\u2191\u03bc F + \u2191\u03b5 / 2 + \u2191\u03b5 / 2 \u2264 \u2191\u2191\u2191\u03bc F + \u2191\u03b5", "state_after": "no goals"}, {"tactic": "simp only [limsup_bot, bot_le]", "annotated_tactic": ["simp only [<a>limsup_bot</a>, <a>bot_le</a>]", [{"full_name": "Filter.limsup_bot", "def_path": "Mathlib/Order/LiminfLimsup.lean", "def_pos": [714, 17], "def_end_pos": [714, 27]}, {"full_name": "bot_le", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [256, 9], "def_end_pos": [256, 15]}]], "state_before": "case inl\n\u03a9\u271d : Type u_1\ninst\u271d\u00b3 : MeasurableSpace \u03a9\u271d\n\u03a9 : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : PseudoEMetricSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\n\u03bcs : \u03b9 \u2192 FiniteMeasure \u03a9\nF : Set \u03a9\nF_closed : IsClosed F\n\u03bcs_lim : Tendsto \u03bcs \u22a5 (\ud835\udcdd \u03bc)\n\u22a2 limsup (fun i => \u2191\u2191\u2191(\u03bcs i) F) \u22a5 \u2264 \u2191\u2191\u2191\u03bc F", "state_after": "no goals"}, {"tactic": "apply\n  ENNReal.lt_add_right (measure_lt_top (\u03bc : Measure \u03a9) F).ne\n    (ENNReal.div_pos_iff.mpr \u27e8(ENNReal.coe_pos.mpr \u03b5_pos).ne.symm, ENNReal.two_ne_top\u27e9).ne.symm", "annotated_tactic": ["apply\n      <a>ENNReal.lt_add_right</a> (<a>measure_lt_top</a> (\u03bc : <a>Measure</a> \u03a9) F).<a>ne</a>\n        (ENNReal.div_pos_iff.mpr \u27e8(ENNReal.coe_pos.mpr \u03b5_pos).ne.symm, <a>ENNReal.two_ne_top</a>\u27e9).ne.symm", [{"full_name": "ENNReal.lt_add_right", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [829, 9], "def_end_pos": [829, 21]}, {"full_name": "MeasureTheory.measure_lt_top", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2866, 9], "def_end_pos": [2866, 23]}, {"full_name": "MeasureTheory.Measure", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [74, 11], "def_end_pos": [74, 18]}, {"full_name": "LT.lt.ne", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [152, 7], "def_end_pos": [152, 15]}, {"full_name": "ENNReal.two_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [431, 9], "def_end_pos": [431, 19]}]], "state_before": "\u03a9\u271d : Type u_1\ninst\u271d\u00b3 : MeasurableSpace \u03a9\u271d\n\u03a9 : Type u_2\n\u03b9 : Type u_3\nL : Filter \u03b9\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : PseudoEMetricSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\n\u03bcs : \u03b9 \u2192 FiniteMeasure \u03a9\n\u03bcs_lim : Tendsto \u03bcs L (\ud835\udcdd \u03bc)\nF : Set \u03a9\nF_closed : IsClosed F\nhne : NeBot L\n\u03b5 : \u211d\u22650\n\u03b5_pos : 0 < \u03b5\na\u271d : \u2191\u2191\u2191\u03bc F < \u22a4\n\u03b4s : \u2115 \u2192 \u211d := fun n => 1 / (\u2191n + 1)\n\u03b4s_pos : \u2200 (n : \u2115), 0 < \u03b4s n\n\u03b4s_lim : Tendsto \u03b4s atTop (\ud835\udcdd 0)\nkey\u2081 : Tendsto (fun n => \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s n) F) \u03c9) \u2202\u2191\u03bc) atTop (\ud835\udcdd (\u2191\u2191\u2191\u03bc F))\n\u22a2 \u2191\u2191\u2191\u03bc F < \u2191\u2191\u2191\u03bc F + \u2191\u03b5 / 2", "state_after": "no goals"}, {"tactic": "apply ENNReal.lt_add_right (ne_of_lt ?_)\n    (ENNReal.div_pos_iff.mpr \u27e8(ENNReal.coe_pos.mpr \u03b5_pos).ne.symm, ENNReal.two_ne_top\u27e9).ne.symm", "annotated_tactic": ["apply <a>ENNReal.lt_add_right</a> (<a>ne_of_lt</a> ?_)\n        (ENNReal.div_pos_iff.mpr \u27e8(ENNReal.coe_pos.mpr \u03b5_pos).ne.symm, <a>ENNReal.two_ne_top</a>\u27e9).ne.symm", [{"full_name": "ENNReal.lt_add_right", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [829, 9], "def_end_pos": [829, 21]}, {"full_name": "ne_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [101, 9], "def_end_pos": [101, 17]}, {"full_name": "ENNReal.two_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [431, 9], "def_end_pos": [431, 19]}]], "state_before": "\u03a9\u271d : Type u_1\ninst\u271d\u00b3 : MeasurableSpace \u03a9\u271d\n\u03a9 : Type u_2\n\u03b9 : Type u_3\nL : Filter \u03b9\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : PseudoEMetricSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\n\u03bcs : \u03b9 \u2192 FiniteMeasure \u03a9\n\u03bcs_lim : Tendsto \u03bcs L (\ud835\udcdd \u03bc)\nF : Set \u03a9\nF_closed : IsClosed F\nhne : NeBot L\n\u03b5 : \u211d\u22650\n\u03b5_pos : 0 < \u03b5\na\u271d : \u2191\u2191\u2191\u03bc F < \u22a4\n\u03b4s : \u2115 \u2192 \u211d := fun n => 1 / (\u2191n + 1)\n\u03b4s_pos : \u2200 (n : \u2115), 0 < \u03b4s n\n\u03b4s_lim : Tendsto \u03b4s atTop (\ud835\udcdd 0)\nkey\u2081 : Tendsto (fun n => \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s n) F) \u03c9) \u2202\u2191\u03bc) atTop (\ud835\udcdd (\u2191\u2191\u2191\u03bc F))\nroom\u2081 : \u2191\u2191\u2191\u03bc F < \u2191\u2191\u2191\u03bc F + \u2191\u03b5 / 2\nM : \u2115\nhM : \u2200 (b : \u2115), b \u2265 M \u2192 \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s b) F) \u03c9) \u2202\u2191\u03bc < \u2191\u2191\u2191\u03bc F + \u2191\u03b5 / 2\nkey\u2082 :\n  Tendsto (fun i => \u222b\u207b (x : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) x) \u2202\u2191(\u03bcs i)) L\n    (\ud835\udcdd (\u222b\u207b (x : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) x) \u2202\u2191\u03bc))\n\u22a2 \u222b\u207b (a : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) a) \u2202\u2191\u03bc <\n    \u222b\u207b (a : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) a) \u2202\u2191\u03bc + \u2191\u03b5 / 2", "state_after": "\u03a9\u271d : Type u_1\ninst\u271d\u00b3 : MeasurableSpace \u03a9\u271d\n\u03a9 : Type u_2\n\u03b9 : Type u_3\nL : Filter \u03b9\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : PseudoEMetricSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\n\u03bcs : \u03b9 \u2192 FiniteMeasure \u03a9\n\u03bcs_lim : Tendsto \u03bcs L (\ud835\udcdd \u03bc)\nF : Set \u03a9\nF_closed : IsClosed F\nhne : NeBot L\n\u03b5 : \u211d\u22650\n\u03b5_pos : 0 < \u03b5\na\u271d : \u2191\u2191\u2191\u03bc F < \u22a4\n\u03b4s : \u2115 \u2192 \u211d := fun n => 1 / (\u2191n + 1)\n\u03b4s_pos : \u2200 (n : \u2115), 0 < \u03b4s n\n\u03b4s_lim : Tendsto \u03b4s atTop (\ud835\udcdd 0)\nkey\u2081 : Tendsto (fun n => \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s n) F) \u03c9) \u2202\u2191\u03bc) atTop (\ud835\udcdd (\u2191\u2191\u2191\u03bc F))\nroom\u2081 : \u2191\u2191\u2191\u03bc F < \u2191\u2191\u2191\u03bc F + \u2191\u03b5 / 2\nM : \u2115\nhM : \u2200 (b : \u2115), b \u2265 M \u2192 \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s b) F) \u03c9) \u2202\u2191\u03bc < \u2191\u2191\u2191\u03bc F + \u2191\u03b5 / 2\nkey\u2082 :\n  Tendsto (fun i => \u222b\u207b (x : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) x) \u2202\u2191(\u03bcs i)) L\n    (\ud835\udcdd (\u222b\u207b (x : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) x) \u2202\u2191\u03bc))\n\u22a2 \u222b\u207b (a : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) a) \u2202\u2191\u03bc < \u22a4"}, {"tactic": "apply BoundedContinuousFunction.lintegral_lt_top_of_nnreal", "annotated_tactic": ["apply <a>BoundedContinuousFunction.lintegral_lt_top_of_nnreal</a>", [{"full_name": "BoundedContinuousFunction.lintegral_lt_top_of_nnreal", "def_path": "Mathlib/MeasureTheory/Integral/BoundedContinuousFunction.lean", "def_pos": [34, 9], "def_end_pos": [34, 35]}]], "state_before": "\u03a9\u271d : Type u_1\ninst\u271d\u00b3 : MeasurableSpace \u03a9\u271d\n\u03a9 : Type u_2\n\u03b9 : Type u_3\nL : Filter \u03b9\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : PseudoEMetricSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\n\u03bcs : \u03b9 \u2192 FiniteMeasure \u03a9\n\u03bcs_lim : Tendsto \u03bcs L (\ud835\udcdd \u03bc)\nF : Set \u03a9\nF_closed : IsClosed F\nhne : NeBot L\n\u03b5 : \u211d\u22650\n\u03b5_pos : 0 < \u03b5\na\u271d : \u2191\u2191\u2191\u03bc F < \u22a4\n\u03b4s : \u2115 \u2192 \u211d := fun n => 1 / (\u2191n + 1)\n\u03b4s_pos : \u2200 (n : \u2115), 0 < \u03b4s n\n\u03b4s_lim : Tendsto \u03b4s atTop (\ud835\udcdd 0)\nkey\u2081 : Tendsto (fun n => \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s n) F) \u03c9) \u2202\u2191\u03bc) atTop (\ud835\udcdd (\u2191\u2191\u2191\u03bc F))\nroom\u2081 : \u2191\u2191\u2191\u03bc F < \u2191\u2191\u2191\u03bc F + \u2191\u03b5 / 2\nM : \u2115\nhM : \u2200 (b : \u2115), b \u2265 M \u2192 \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s b) F) \u03c9) \u2202\u2191\u03bc < \u2191\u2191\u2191\u03bc F + \u2191\u03b5 / 2\nkey\u2082 :\n  Tendsto (fun i => \u222b\u207b (x : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) x) \u2202\u2191(\u03bcs i)) L\n    (\ud835\udcdd (\u222b\u207b (x : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) x) \u2202\u2191\u03bc))\n\u22a2 \u222b\u207b (a : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) a) \u2202\u2191\u03bc < \u22a4", "state_after": "no goals"}]}, 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{"full_name": "Nat.cast_id", "def_path": "Mathlib/Data/Nat/Cast/Basic.lean", "def_pos": [167, 9], "def_end_pos": [167, 20]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : Zero \u03b1\ns : Finset \u03b9\nf\u271d : \u03b9 \u2192\u2080 \u03b1\nf : \u03b9 \u2192\u2080 Finset \u03b1\n\u22a2 \u220f i in f.support, card (\u2191f i) = prod f fun i => \u2191(card (\u2191f i))", "state_after": "no goals"}, {"tactic": "simp only [Pi.nat_apply, Nat.cast_id]", "annotated_tactic": ["simp only [<a>Pi.nat_apply</a>, <a>Nat.cast_id</a>]", [{"full_name": "Pi.nat_apply", "def_path": "Mathlib/Data/Nat/Cast/Basic.lean", "def_pos": [190, 9], "def_end_pos": [190, 18]}, {"full_name": "Nat.cast_id", "def_path": "Mathlib/Data/Nat/Cast/Basic.lean", "def_pos": [167, 9], "def_end_pos": [167, 20]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : Zero \u03b1\ns : Finset \u03b9\nf\u271d : \u03b9 \u2192\u2080 \u03b1\nf : \u03b9 \u2192\u2080 Finset \u03b1\ni : \u03b9\nx\u271d : i \u2208 f.support\n\u22a2 card (\u2191f i) = (fun i => \u2191(card (\u2191f i))) i (\u2191f i)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Setoid/Basic.lean", "full_name": "Setoid.sSup_def", "start": [238, 1], "end": [241, 52], "traced_tactics": [{"tactic": "rw [sSup_eq_eqvGen, sSup_image]", "annotated_tactic": ["rw [<a>sSup_eq_eqvGen</a>, <a>sSup_image</a>]", [{"full_name": "Setoid.sSup_eq_eqvGen", "def_path": "Mathlib/Data/Setoid/Basic.lean", "def_pos": [227, 9], "def_end_pos": [227, 23]}, {"full_name": "sSup_image", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [1452, 9], "def_end_pos": [1452, 19]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ns : Set (Setoid \u03b1)\n\u22a2 sSup s = EqvGen.Setoid (sSup (Rel '' s))", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ns : Set (Setoid \u03b1)\n\u22a2 (EqvGen.Setoid fun x y => \u2203 r, r \u2208 s \u2227 Rel r x y) = EqvGen.Setoid (\u2a06 a \u2208 s, Rel a)"}, {"tactic": "congr with (x y)", "annotated_tactic": ["congr with (x y)", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ns : Set (Setoid \u03b1)\n\u22a2 (EqvGen.Setoid fun x y => \u2203 r, r \u2208 s \u2227 Rel r x y) = EqvGen.Setoid (\u2a06 a \u2208 s, Rel a)", "state_after": "case e_r.h.h.a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns : Set (Setoid \u03b1)\nx y : \u03b1\n\u22a2 (\u2203 r, r \u2208 s \u2227 Rel r x y) \u2194 iSup (fun a => \u2a06 (_ : a \u2208 s), Rel a) x y"}, {"tactic": "simp only [iSup_apply, iSup_Prop_eq, exists_prop]", "annotated_tactic": ["simp only [<a>iSup_apply</a>, <a>iSup_Prop_eq</a>, <a>exists_prop</a>]", [{"full_name": "iSup_apply", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [1844, 9], "def_end_pos": [1844, 19]}, {"full_name": "iSup_Prop_eq", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [1806, 9], "def_end_pos": [1806, 21]}, {"full_name": "exists_prop", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [485, 17], "def_end_pos": [485, 28]}]], "state_before": "case e_r.h.h.a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns : Set (Setoid \u03b1)\nx y : \u03b1\n\u22a2 (\u2203 r, r \u2208 s \u2227 Rel r x y) \u2194 iSup (fun a => \u2a06 (_ : a \u2208 s), Rel a) x y", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "full_name": "String.str_eq", "start": [24, 9], "end": [24, 43], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "full_name": "MeasureTheory.Lp.snorm_lim_le_liminf_snorm", "start": [1350, 1], "end": [1361, 52], "traced_tactics": [{"tactic": "obtain rfl|hp0 := eq_or_ne p 0", "annotated_tactic": ["obtain rfl|hp0 := <a>eq_or_ne</a> p 0", [{"full_name": "eq_or_ne", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [209, 9], "def_end_pos": [209, 17]}]], "state_before": "\u03b1 : Type u_1\nE\u271d : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\u271d\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\nE : Type u_5\ninst\u271d : NormedAddCommGroup E\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nf_lim : \u03b1 \u2192 E\nh_lim : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (f_lim x))\n\u22a2 snorm f_lim p \u03bc \u2264 liminf (fun n => snorm (f n) p \u03bc) atTop", "state_after": "case inl\n\u03b1 : Type u_1\nE\u271d : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\u271d\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\nE : Type u_5\ninst\u271d : NormedAddCommGroup E\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nf_lim : \u03b1 \u2192 E\nh_lim : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (f_lim x))\n\u22a2 snorm f_lim 0 \u03bc \u2264 liminf (fun n => snorm (f n) 0 \u03bc) atTop\n\ncase inr\n\u03b1 : Type u_1\nE\u271d : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\u271d\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\nE : Type u_5\ninst\u271d : NormedAddCommGroup E\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nf_lim : \u03b1 \u2192 E\nh_lim : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (f_lim x))\nhp0 : p \u2260 0\n\u22a2 snorm f_lim p \u03bc \u2264 liminf (fun n => snorm (f n) p \u03bc) atTop"}, {"tactic": "by_cases hp_top : p = \u221e", "annotated_tactic": ["by_cases hp_top : p = \u221e", []], "state_before": "case inr\n\u03b1 : Type u_1\nE\u271d : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\u271d\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\nE : Type u_5\ninst\u271d : NormedAddCommGroup E\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nf_lim : \u03b1 \u2192 E\nh_lim : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (f_lim x))\nhp0 : p \u2260 0\n\u22a2 snorm f_lim p \u03bc \u2264 liminf (fun n => snorm (f n) p \u03bc) atTop", "state_after": "case pos\n\u03b1 : Type u_1\nE\u271d : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\u271d\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\nE : Type u_5\ninst\u271d : NormedAddCommGroup E\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nf_lim : \u03b1 \u2192 E\nh_lim : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (f_lim x))\nhp0 : p \u2260 0\nhp_top : p = \u22a4\n\u22a2 snorm f_lim p \u03bc \u2264 liminf (fun n => snorm (f n) p \u03bc) atTop\n\ncase neg\n\u03b1 : Type u_1\nE\u271d : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\u271d\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\nE : Type u_5\ninst\u271d : NormedAddCommGroup E\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nf_lim : \u03b1 \u2192 E\nh_lim : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (f_lim x))\nhp0 : p \u2260 0\nhp_top : \u00acp = \u22a4\n\u22a2 snorm f_lim p \u03bc \u2264 liminf (fun n => snorm (f n) p \u03bc) atTop"}, {"tactic": "simp_rw [snorm_eq_snorm' hp0 hp_top]", "annotated_tactic": ["simp_rw [<a>snorm_eq_snorm'</a> hp0 hp_top]", [{"full_name": "MeasureTheory.snorm_eq_snorm'", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [88, 9], "def_end_pos": [88, 24]}]], "state_before": "case neg\n\u03b1 : Type u_1\nE\u271d : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\u271d\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\nE : Type u_5\ninst\u271d : NormedAddCommGroup E\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nf_lim : \u03b1 \u2192 E\nh_lim : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (f_lim x))\nhp0 : p \u2260 0\nhp_top : \u00acp = \u22a4\n\u22a2 snorm f_lim p \u03bc \u2264 liminf (fun n => snorm (f n) p \u03bc) atTop", "state_after": "case neg\n\u03b1 : Type u_1\nE\u271d : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\u271d\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\nE : Type u_5\ninst\u271d : NormedAddCommGroup E\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nf_lim : \u03b1 \u2192 E\nh_lim : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (f_lim x))\nhp0 : p \u2260 0\nhp_top : \u00acp = \u22a4\n\u22a2 snorm' f_lim (ENNReal.toReal p) \u03bc \u2264 liminf (fun n => snorm' (f n) (ENNReal.toReal p) \u03bc) atTop"}, {"tactic": "have hp_pos : 0 < p.toReal := ENNReal.toReal_pos hp0 hp_top", "annotated_tactic": ["have hp_pos : 0 < p.toReal := <a>ENNReal.toReal_pos</a> hp0 hp_top", [{"full_name": "ENNReal.toReal_pos", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2131, 9], "def_end_pos": [2131, 19]}]], "state_before": "case neg\n\u03b1 : Type u_1\nE\u271d : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\u271d\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\nE : Type u_5\ninst\u271d : NormedAddCommGroup E\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nf_lim : \u03b1 \u2192 E\nh_lim : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (f_lim x))\nhp0 : p \u2260 0\nhp_top : \u00acp = \u22a4\n\u22a2 snorm' f_lim (ENNReal.toReal p) \u03bc \u2264 liminf (fun n => snorm' (f n) (ENNReal.toReal p) \u03bc) atTop", "state_after": "case neg\n\u03b1 : Type u_1\nE\u271d : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\u271d\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\nE : Type u_5\ninst\u271d : NormedAddCommGroup E\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nf_lim : \u03b1 \u2192 E\nh_lim : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (f_lim x))\nhp0 : p \u2260 0\nhp_top : \u00acp = \u22a4\nhp_pos : 0 < ENNReal.toReal p\n\u22a2 snorm' f_lim (ENNReal.toReal p) \u03bc \u2264 liminf (fun n => snorm' (f n) (ENNReal.toReal p) \u03bc) atTop"}, {"tactic": "exact snorm'_lim_le_liminf_snorm' hp_pos hf h_lim", "annotated_tactic": ["exact <a>snorm'_lim_le_liminf_snorm'</a> hp_pos hf h_lim", [{"full_name": "MeasureTheory.Lp.snorm'_lim_le_liminf_snorm'", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [1313, 9], "def_end_pos": [1313, 36]}]], "state_before": "case neg\n\u03b1 : Type u_1\nE\u271d : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\u271d\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\nE : Type u_5\ninst\u271d : NormedAddCommGroup E\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nf_lim : \u03b1 \u2192 E\nh_lim : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (f_lim x))\nhp0 : p \u2260 0\nhp_top : \u00acp = \u22a4\nhp_pos : 0 < ENNReal.toReal p\n\u22a2 snorm' f_lim (ENNReal.toReal p) \u03bc \u2264 liminf (fun n => snorm' (f n) (ENNReal.toReal p) \u03bc) atTop", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case inl\n\u03b1 : Type u_1\nE\u271d : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\u271d\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\nE : Type u_5\ninst\u271d : NormedAddCommGroup E\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nf_lim : \u03b1 \u2192 E\nh_lim : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (f_lim x))\n\u22a2 snorm f_lim 0 \u03bc \u2264 liminf (fun n => snorm (f n) 0 \u03bc) atTop", "state_after": "no goals"}, {"tactic": "simp_rw [hp_top]", "annotated_tactic": ["simp_rw [hp_top]", []], "state_before": "case pos\n\u03b1 : Type u_1\nE\u271d : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\u271d\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\nE : Type u_5\ninst\u271d : NormedAddCommGroup E\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nf_lim : \u03b1 \u2192 E\nh_lim : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (f_lim x))\nhp0 : p \u2260 0\nhp_top : p = \u22a4\n\u22a2 snorm f_lim p \u03bc \u2264 liminf (fun n => snorm (f n) p \u03bc) atTop", "state_after": "case pos\n\u03b1 : Type u_1\nE\u271d : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\u271d\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\nE : Type u_5\ninst\u271d : NormedAddCommGroup E\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nf_lim : \u03b1 \u2192 E\nh_lim : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (f_lim x))\nhp0 : p \u2260 0\nhp_top : p = \u22a4\n\u22a2 snorm f_lim \u22a4 \u03bc \u2264 liminf (fun n => snorm (f n) \u22a4 \u03bc) atTop"}, {"tactic": "exact snorm_exponent_top_lim_le_liminf_snorm_exponent_top h_lim", "annotated_tactic": ["exact <a>snorm_exponent_top_lim_le_liminf_snorm_exponent_top</a> h_lim", [{"full_name": "MeasureTheory.Lp.snorm_exponent_top_lim_le_liminf_snorm_exponent_top", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [1341, 9], "def_end_pos": [1341, 60]}]], "state_before": "case pos\n\u03b1 : Type u_1\nE\u271d : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\u271d\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\nE : Type u_5\ninst\u271d : NormedAddCommGroup E\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nf_lim : \u03b1 \u2192 E\nh_lim : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (f_lim x))\nhp0 : p \u2260 0\nhp_top : p = \u22a4\n\u22a2 snorm f_lim \u22a4 \u03bc \u2264 liminf (fun n => snorm (f n) \u22a4 \u03bc) atTop", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/List/Init/Lemmas.lean", "full_name": "List.mapM_cons", "start": [265, 9], "end": [266, 91], "traced_tactics": [{"tactic": "simp [\u2190 mapM'_eq_mapM, mapM']", "annotated_tactic": ["simp [\u2190 <a>mapM'_eq_mapM</a>, <a>mapM'</a>]", [{"full_name": "List.mapM'_eq_mapM", "def_path": "lake-packages/std/Std/Data/List/Init/Lemmas.lean", "def_pos": [257, 9], "def_end_pos": [257, 22]}, {"full_name": "List.mapM'", "def_path": "lake-packages/std/Std/Data/List/Init/Lemmas.lean", "def_pos": [248, 5], "def_end_pos": [248, 10]}]], "state_before": "m : Type u_1 \u2192 Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_1\na : \u03b1\nl : List \u03b1\ninst\u271d\u00b9 : Monad m\ninst\u271d : LawfulMonad m\nf : \u03b1 \u2192 m \u03b2\n\u22a2 mapM f (a :: l) = do\n    let __do_lift \u2190 f a\n    let __do_lift_1 \u2190 mapM f l\n    pure (__do_lift :: __do_lift_1)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Kernel/Basic.lean", "full_name": "ProbabilityTheory.kernel.finset_sum_apply'", "start": [107, 1], "end": [108, 98], "traced_tactics": [{"tactic": "rw [finset_sum_apply, Measure.finset_sum_apply]", "annotated_tactic": ["rw [<a>finset_sum_apply</a>, <a>Measure.finset_sum_apply</a>]", [{"full_name": "ProbabilityTheory.kernel.finset_sum_apply", "def_path": "Mathlib/Probability/Kernel/Basic.lean", "def_pos": [103, 9], "def_end_pos": [103, 25]}, {"full_name": "MeasureTheory.Measure.finset_sum_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [885, 9], "def_end_pos": [885, 25]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nI : Finset \u03b9\n\u03ba : \u03b9 \u2192 { x // x \u2208 kernel \u03b1 \u03b2 }\na : \u03b1\ns : Set \u03b2\n\u22a2 \u2191\u2191(\u2191(\u2211 i in I, \u03ba i) a) s = \u2211 i in I, \u2191\u2191(\u2191(\u03ba i) a) s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/RBMap/Lemmas.lean", "full_name": "Std.RBNode.mem_insert_self", "start": [561, 1], "end": [565, 87], "traced_tactics": [{"tactic": "rw [\u2190 mem_toList, List.mem_iff_append]", "annotated_tactic": ["rw [\u2190 <a>mem_toList</a>, <a>List.mem_iff_append</a>]", [{"full_name": "Std.RBNode.mem_toList", "def_path": "lake-packages/std/Std/Data/RBMap/Lemmas.lean", "def_pos": [370, 17], "def_end_pos": [370, 27]}, {"full_name": "List.mem_iff_append", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [152, 9], "def_end_pos": [152, 23]}]], "state_before": "\u03b1 : Type u_1\nc : RBColor\nn : Nat\nv : \u03b1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nt : RBNode \u03b1\nht : Balanced t c n\n\u22a2 v \u2208 insert cmp t v", "state_after": "\u03b1 : Type u_1\nc : RBColor\nn : Nat\nv : \u03b1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nt : RBNode \u03b1\nht : Balanced t c n\n\u22a2 \u2203 s t_1, toList (insert cmp t v) = s ++ v :: t_1"}, {"tactic": "exact match e : zoom (cmp v) t with\n| (nil, p) => let \u27e8_, _, _, h\u27e9 := exists_insert_toList_zoom_nil ht e; \u27e8_, _, h\u27e9\n| (node .., p) => let \u27e8_, _, _, h\u27e9 := exists_insert_toList_zoom_node ht e; \u27e8_, _, h\u27e9", "annotated_tactic": ["exact match e : <a>zoom</a> (cmp v) t with\n  | (<a>nil</a>, p) => let \u27e8_, _, _, h\u27e9 := <a>exists_insert_toList_zoom_nil</a> ht e; \u27e8_, _, h\u27e9\n  | (<a>node</a> .., p) => let \u27e8_, _, _, h\u27e9 := <a>exists_insert_toList_zoom_node</a> ht e; \u27e8_, _, h\u27e9", [{"full_name": "Std.RBNode.zoom", "def_path": "lake-packages/std/Std/Data/RBMap/Basic.lean", "def_pos": [451, 19], "def_end_pos": [451, 23]}, {"full_name": "Std.RBNode.nil", "def_path": "lake-packages/std/Std/Data/RBMap/Basic.lean", "def_pos": [46, 5], "def_end_pos": [46, 8]}, {"full_name": "Std.RBNode.exists_insert_toList_zoom_nil", "def_path": "lake-packages/std/Std/Data/RBMap/Lemmas.lean", "def_pos": [546, 9], "def_end_pos": [546, 38]}, {"full_name": "Std.RBNode.node", "def_path": "lake-packages/std/Std/Data/RBMap/Basic.lean", "def_pos": [50, 5], "def_end_pos": [50, 9]}, {"full_name": "Std.RBNode.exists_insert_toList_zoom_node", "def_path": "lake-packages/std/Std/Data/RBMap/Lemmas.lean", "def_pos": [555, 9], "def_end_pos": [555, 39]}]], "state_before": "\u03b1 : Type u_1\nc : RBColor\nn : Nat\nv : \u03b1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nt : RBNode \u03b1\nht : Balanced t c n\n\u22a2 \u2203 s t_1, toList (insert cmp t v) = s ++ v :: t_1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/LpOrder.lean", "full_name": "MeasureTheory.Mem\u2112p.inf", "start": [73, 1], "end": [76, 71], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Group/Measure.lean", "full_name": "MeasureTheory.map_mul_left_eq_self", "start": [84, 1], "end": [86, 44], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Intervals/WithBotTop.lean", "full_name": "WithBot.image_coe_Ioi", "start": [205, 1], "end": [207, 58], "traced_tactics": [{"tactic": "rw [\u2190 preimage_coe_Ioi, image_preimage_eq_inter_range, range_coe,\n  inter_eq_self_of_subset_left (Ioi_subset_Ioi bot_le)]", "annotated_tactic": ["rw [\u2190 <a>preimage_coe_Ioi</a>, <a>image_preimage_eq_inter_range</a>, <a>range_coe</a>,\n    <a>inter_eq_self_of_subset_left</a> (<a>Ioi_subset_Ioi</a> <a>bot_le</a>)]", [{"full_name": "WithBot.preimage_coe_Ioi", "def_path": "Mathlib/Data/Set/Intervals/WithBotTop.lean", "def_pos": [147, 9], "def_end_pos": [147, 25]}, {"full_name": "Set.image_preimage_eq_inter_range", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [796, 9], "def_end_pos": [796, 38]}, {"full_name": "WithBot.range_coe", "def_path": "Mathlib/Data/Set/Intervals/WithBotTop.lean", "def_pos": [142, 9], "def_end_pos": [142, 18]}, {"full_name": "Set.inter_eq_self_of_subset_left", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [987, 9], "def_end_pos": [987, 37]}, {"full_name": "Set.Ioi_subset_Ioi", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [598, 9], "def_end_pos": [598, 23]}, {"full_name": "bot_le", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [256, 9], "def_end_pos": [256, 15]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : PartialOrder \u03b1\na b : \u03b1\n\u22a2 some '' Ioi a = Ioi \u2191a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Rat/Lemmas.lean", "full_name": "Rat.mkRat_mul_mkRat", "start": [279, 1], "end": [282, 101], "traced_tactics": [{"tactic": "if z\u2081 : d\u2081 = 0 then simp [z\u2081] else if z\u2082 : d\u2082 = 0 then simp [z\u2082] else\nrw [\u2190 normalize_eq_mkRat z\u2081, \u2190 normalize_eq_mkRat z\u2082, normalize_mul_normalize, normalize_eq_mkRat]", "annotated_tactic": ["if z\u2081 : d\u2081 = 0 then simp [z\u2081] else if z\u2082 : d\u2082 = 0 then simp [z\u2082] else\n  rw [\u2190 <a>normalize_eq_mkRat</a> z\u2081, \u2190 <a>normalize_eq_mkRat</a> z\u2082, <a>normalize_mul_normalize</a>, <a>normalize_eq_mkRat</a>]", [{"full_name": "Rat.normalize_eq_mkRat", "def_path": "lake-packages/std/Std/Data/Rat/Lemmas.lean", "def_pos": [87, 9], "def_end_pos": [87, 27]}, {"full_name": "Rat.normalize_eq_mkRat", "def_path": "lake-packages/std/Std/Data/Rat/Lemmas.lean", "def_pos": [87, 9], "def_end_pos": [87, 27]}, {"full_name": "Rat.normalize_mul_normalize", "def_path": "lake-packages/std/Std/Data/Rat/Lemmas.lean", "def_pos": [270, 9], "def_end_pos": [270, 32]}, {"full_name": "Rat.normalize_eq_mkRat", "def_path": "lake-packages/std/Std/Data/Rat/Lemmas.lean", "def_pos": [87, 9], "def_end_pos": [87, 27]}]], "state_before": "n\u2081 n\u2082 : Int\nd\u2081 d\u2082 : Nat\n\u22a2 mkRat n\u2081 d\u2081 * mkRat n\u2082 d\u2082 = mkRat (n\u2081 * n\u2082) (d\u2081 * d\u2082)", "state_after": "no goals"}, {"tactic": "simp [z\u2081]", "annotated_tactic": ["simp [z\u2081]", []], "state_before": "n\u2081 n\u2082 : Int\nd\u2081 d\u2082 : Nat\nz\u2081 : d\u2081 = 0\n\u22a2 mkRat n\u2081 d\u2081 * mkRat n\u2082 d\u2082 = mkRat (n\u2081 * n\u2082) (d\u2081 * d\u2082)", "state_after": "no goals"}, {"tactic": "if z\u2082 : d\u2082 = 0 then simp [z\u2082] else\nrw [\u2190 normalize_eq_mkRat z\u2081, \u2190 normalize_eq_mkRat z\u2082, normalize_mul_normalize, normalize_eq_mkRat]", "annotated_tactic": ["if z\u2082 : d\u2082 = 0 then simp [z\u2082] else\n  rw [\u2190 <a>normalize_eq_mkRat</a> z\u2081, \u2190 <a>normalize_eq_mkRat</a> z\u2082, <a>normalize_mul_normalize</a>, <a>normalize_eq_mkRat</a>]", [{"full_name": "Rat.normalize_eq_mkRat", "def_path": "lake-packages/std/Std/Data/Rat/Lemmas.lean", "def_pos": [87, 9], "def_end_pos": [87, 27]}, {"full_name": "Rat.normalize_eq_mkRat", "def_path": "lake-packages/std/Std/Data/Rat/Lemmas.lean", "def_pos": [87, 9], "def_end_pos": [87, 27]}, {"full_name": "Rat.normalize_mul_normalize", "def_path": "lake-packages/std/Std/Data/Rat/Lemmas.lean", "def_pos": [270, 9], "def_end_pos": [270, 32]}, {"full_name": "Rat.normalize_eq_mkRat", "def_path": "lake-packages/std/Std/Data/Rat/Lemmas.lean", "def_pos": [87, 9], "def_end_pos": [87, 27]}]], "state_before": "n\u2081 n\u2082 : Int\nd\u2081 d\u2082 : Nat\nz\u2081 : \u00acd\u2081 = 0\n\u22a2 mkRat n\u2081 d\u2081 * mkRat n\u2082 d\u2082 = mkRat (n\u2081 * n\u2082) (d\u2081 * d\u2082)", "state_after": "no goals"}, {"tactic": "simp [z\u2082]", "annotated_tactic": ["simp [z\u2082]", []], "state_before": "n\u2081 n\u2082 : Int\nd\u2081 d\u2082 : Nat\nz\u2081 : \u00acd\u2081 = 0\nz\u2082 : d\u2082 = 0\n\u22a2 mkRat n\u2081 d\u2081 * mkRat n\u2082 d\u2082 = mkRat (n\u2081 * n\u2082) (d\u2081 * d\u2082)", "state_after": "no goals"}, {"tactic": "rw [\u2190 normalize_eq_mkRat z\u2081, \u2190 normalize_eq_mkRat z\u2082, normalize_mul_normalize, normalize_eq_mkRat]", "annotated_tactic": ["rw [\u2190 <a>normalize_eq_mkRat</a> z\u2081, \u2190 <a>normalize_eq_mkRat</a> z\u2082, <a>normalize_mul_normalize</a>, <a>normalize_eq_mkRat</a>]", [{"full_name": "Rat.normalize_eq_mkRat", "def_path": "lake-packages/std/Std/Data/Rat/Lemmas.lean", "def_pos": [87, 9], "def_end_pos": [87, 27]}, {"full_name": "Rat.normalize_eq_mkRat", "def_path": "lake-packages/std/Std/Data/Rat/Lemmas.lean", "def_pos": [87, 9], "def_end_pos": [87, 27]}, {"full_name": "Rat.normalize_mul_normalize", "def_path": "lake-packages/std/Std/Data/Rat/Lemmas.lean", "def_pos": [270, 9], "def_end_pos": [270, 32]}, {"full_name": "Rat.normalize_eq_mkRat", "def_path": "lake-packages/std/Std/Data/Rat/Lemmas.lean", "def_pos": [87, 9], "def_end_pos": [87, 27]}]], "state_before": "n\u2081 n\u2082 : Int\nd\u2081 d\u2082 : Nat\nz\u2081 : \u00acd\u2081 = 0\nz\u2082 : \u00acd\u2082 = 0\n\u22a2 mkRat n\u2081 d\u2081 * mkRat n\u2082 d\u2082 = mkRat (n\u2081 * n\u2082) (d\u2081 * d\u2082)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Martingale/Convergence.lean", "full_name": "MeasureTheory.tendsto_of_uncrossing_lt_top", "start": [142, 1], "end": [153, 100], "traced_tactics": [{"tactic": "rw [isBoundedUnder_le_abs] at h", "annotated_tactic": ["rw [<a>isBoundedUnder_le_abs</a>] at h", [{"full_name": "Filter.isBoundedUnder_le_abs", "def_path": "Mathlib/Order/LiminfLimsup.lean", "def_pos": [372, 9], "def_end_pos": [372, 30]}]], "state_before": "case pos\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nR : \u211d\u22650\nhf\u2081 : liminf (fun n => \u2191\u2016f n \u03c9\u2016\u208a) atTop < \u22a4\nhf\u2082 : \u2200 (a b : \u211a), a < b \u2192 upcrossings (\u2191a) (\u2191b) f \u03c9 < \u22a4\nh : IsBoundedUnder (fun x x_1 => x \u2264 x_1) atTop fun n => |f n \u03c9|\n\u22a2 \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c)", "state_after": "case pos\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nR : \u211d\u22650\nhf\u2081 : liminf (fun n => \u2191\u2016f n \u03c9\u2016\u208a) atTop < \u22a4\nhf\u2082 : \u2200 (a b : \u211a), a < b \u2192 upcrossings (\u2191a) (\u2191b) f \u03c9 < \u22a4\nh :\n  (IsBoundedUnder (fun x x_1 => x \u2264 x_1) atTop fun n => f n \u03c9) \u2227\n    IsBoundedUnder (fun x x_1 => x \u2265 x_1) atTop fun n => f n \u03c9\n\u22a2 \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c)"}, {"tactic": "refine' tendsto_of_no_upcrossings Rat.denseRange_cast _ h.1 h.2", "annotated_tactic": ["refine' <a>tendsto_of_no_upcrossings</a> <a>Rat.denseRange_cast</a> _ h.1 h.2", [{"full_name": "tendsto_of_no_upcrossings", "def_path": "Mathlib/Topology/Algebra/Order/LiminfLimsup.lean", "def_pos": [187, 9], "def_end_pos": [187, 34]}, {"full_name": "Rat.denseRange_cast", "def_path": "Mathlib/Topology/Algebra/Order/Archimedean.lean", "def_pos": [29, 9], "def_end_pos": [29, 28]}]], "state_before": "case pos\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nR : \u211d\u22650\nhf\u2081 : liminf (fun n => \u2191\u2016f n \u03c9\u2016\u208a) atTop < \u22a4\nhf\u2082 : \u2200 (a b : \u211a), a < b \u2192 upcrossings (\u2191a) (\u2191b) f \u03c9 < \u22a4\nh :\n  (IsBoundedUnder (fun x x_1 => x \u2264 x_1) atTop fun n => f n \u03c9) \u2227\n    IsBoundedUnder (fun x x_1 => x \u2265 x_1) atTop fun n => f n \u03c9\n\u22a2 \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c)", "state_after": "case pos\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nR : \u211d\u22650\nhf\u2081 : liminf (fun n => \u2191\u2016f n \u03c9\u2016\u208a) atTop < \u22a4\nhf\u2082 : \u2200 (a b : \u211a), a < b \u2192 upcrossings (\u2191a) (\u2191b) f \u03c9 < \u22a4\nh :\n  (IsBoundedUnder (fun x x_1 => x \u2264 x_1) atTop fun n => f n \u03c9) \u2227\n    IsBoundedUnder (fun x x_1 => x \u2265 x_1) atTop fun n => f n \u03c9\n\u22a2 \u2200 (a : \u211d),\n    a \u2208 Set.range Rat.cast \u2192\n      \u2200 (b : \u211d), b \u2208 Set.range Rat.cast \u2192 a < b \u2192 \u00ac((\u2203\u1da0 (n : \u2115) in atTop, f n \u03c9 < a) \u2227 \u2203\u1da0 (n : \u2115) in atTop, b < f n \u03c9)"}, {"tactic": "intro a ha b hb hab", "annotated_tactic": ["intro a ha b hb hab", []], "state_before": "case pos\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nR : \u211d\u22650\nhf\u2081 : liminf (fun n => \u2191\u2016f n \u03c9\u2016\u208a) atTop < \u22a4\nhf\u2082 : \u2200 (a b : \u211a), a < b \u2192 upcrossings (\u2191a) (\u2191b) f \u03c9 < \u22a4\nh :\n  (IsBoundedUnder (fun x x_1 => x \u2264 x_1) atTop fun n => f n \u03c9) \u2227\n    IsBoundedUnder (fun x x_1 => x \u2265 x_1) atTop fun n => f n \u03c9\n\u22a2 \u2200 (a : \u211d),\n    a \u2208 Set.range Rat.cast \u2192\n      \u2200 (b : \u211d), b \u2208 Set.range Rat.cast \u2192 a < b \u2192 \u00ac((\u2203\u1da0 (n : \u2115) in atTop, f n \u03c9 < a) \u2227 \u2203\u1da0 (n : \u2115) in atTop, b < f n \u03c9)", "state_after": "case pos\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na\u271d b\u271d : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nR : \u211d\u22650\nhf\u2081 : liminf (fun n => \u2191\u2016f n \u03c9\u2016\u208a) atTop < \u22a4\nhf\u2082 : \u2200 (a b : \u211a), a < b \u2192 upcrossings (\u2191a) (\u2191b) f \u03c9 < \u22a4\nh :\n  (IsBoundedUnder (fun x x_1 => x \u2264 x_1) atTop fun n => f n \u03c9) \u2227\n    IsBoundedUnder (fun x x_1 => x \u2265 x_1) atTop fun n => f n \u03c9\na : \u211d\nha : a \u2208 Set.range Rat.cast\nb : \u211d\nhb : b \u2208 Set.range Rat.cast\nhab : a < b\n\u22a2 \u00ac((\u2203\u1da0 (n : \u2115) in atTop, f n \u03c9 < a) \u2227 \u2203\u1da0 (n : \u2115) in atTop, b < f n \u03c9)"}, {"tactic": "obtain \u27e8\u27e8a, rfl\u27e9, \u27e8b, rfl\u27e9\u27e9 := ha, hb", "annotated_tactic": ["obtain \u27e8\u27e8a, rfl\u27e9, \u27e8b, rfl\u27e9\u27e9 := ha, hb", []], "state_before": "case pos\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na\u271d b\u271d : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nR : \u211d\u22650\nhf\u2081 : liminf (fun n => \u2191\u2016f n \u03c9\u2016\u208a) atTop < \u22a4\nhf\u2082 : \u2200 (a b : \u211a), a < b \u2192 upcrossings (\u2191a) (\u2191b) f \u03c9 < \u22a4\nh :\n  (IsBoundedUnder (fun x x_1 => x \u2264 x_1) atTop fun n => f n \u03c9) \u2227\n    IsBoundedUnder (fun x x_1 => x \u2265 x_1) atTop fun n => f n \u03c9\na : \u211d\nha : a \u2208 Set.range Rat.cast\nb : \u211d\nhb : b \u2208 Set.range Rat.cast\nhab : a < b\n\u22a2 \u00ac((\u2203\u1da0 (n : \u2115) in atTop, f n \u03c9 < a) \u2227 \u2203\u1da0 (n : \u2115) in atTop, b < f n \u03c9)", "state_after": "case pos.intro.intro\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na\u271d b\u271d : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nR : \u211d\u22650\nhf\u2081 : liminf (fun n => \u2191\u2016f n \u03c9\u2016\u208a) atTop < \u22a4\nhf\u2082 : \u2200 (a b : \u211a), a < b \u2192 upcrossings (\u2191a) (\u2191b) f \u03c9 < \u22a4\nh :\n  (IsBoundedUnder (fun x x_1 => x \u2264 x_1) atTop fun n => f n \u03c9) \u2227\n    IsBoundedUnder (fun x x_1 => x \u2265 x_1) atTop fun n => f n \u03c9\na b : \u211a\nhab : \u2191a < \u2191b\n\u22a2 \u00ac((\u2203\u1da0 (n : \u2115) in atTop, f n \u03c9 < \u2191a) \u2227 \u2203\u1da0 (n : \u2115) in atTop, \u2191b < f n \u03c9)"}, {"tactic": "exact not_frequently_of_upcrossings_lt_top hab (hf\u2082 a b (Rat.cast_lt.1 hab)).ne", "annotated_tactic": ["exact <a>not_frequently_of_upcrossings_lt_top</a> hab (hf\u2082 a b (<a>Rat.cast_lt</a>.1 hab)).<a>ne</a>", [{"full_name": "MeasureTheory.not_frequently_of_upcrossings_lt_top", "def_path": "Mathlib/Probability/Martingale/Convergence.lean", "def_pos": [111, 9], "def_end_pos": [111, 45]}, {"full_name": "Rat.cast_lt", "def_path": "Mathlib/Data/Rat/Cast/Order.lean", "def_pos": [59, 9], "def_end_pos": [59, 16]}, {"full_name": "LT.lt.ne", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [152, 7], "def_end_pos": [152, 15]}]], "state_before": "case pos.intro.intro\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na\u271d b\u271d : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nR : \u211d\u22650\nhf\u2081 : liminf (fun n => \u2191\u2016f n \u03c9\u2016\u208a) atTop < \u22a4\nhf\u2082 : \u2200 (a b : \u211a), a < b \u2192 upcrossings (\u2191a) (\u2191b) f \u03c9 < \u22a4\nh :\n  (IsBoundedUnder (fun x x_1 => x \u2264 x_1) atTop fun n => f n \u03c9) \u2227\n    IsBoundedUnder (fun x x_1 => x \u2265 x_1) atTop fun n => f n \u03c9\na b : \u211a\nhab : \u2191a < \u2191b\n\u22a2 \u00ac((\u2203\u1da0 (n : \u2115) in atTop, f n \u03c9 < \u2191a) \u2227 \u2203\u1da0 (n : \u2115) in atTop, \u2191b < f n \u03c9)", "state_after": "no goals"}, {"tactic": "obtain \u27e8a, b, hab, h\u2081, h\u2082\u27e9 := ENNReal.exists_upcrossings_of_not_bounded_under hf\u2081.ne h", "annotated_tactic": ["obtain \u27e8a, b, hab, h\u2081, h\u2082\u27e9 := <a>ENNReal.exists_upcrossings_of_not_bounded_under</a> hf\u2081.ne h", [{"full_name": "ENNReal.exists_upcrossings_of_not_bounded_under", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [726, 9], "def_end_pos": [726, 48]}]], "state_before": "case neg\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nR : \u211d\u22650\nhf\u2081 : liminf (fun n => \u2191\u2016f n \u03c9\u2016\u208a) atTop < \u22a4\nhf\u2082 : \u2200 (a b : \u211a), a < b \u2192 upcrossings (\u2191a) (\u2191b) f \u03c9 < \u22a4\nh : \u00acIsBoundedUnder (fun x x_1 => x \u2264 x_1) atTop fun n => |f n \u03c9|\n\u22a2 \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c)", "state_after": "case neg.intro.intro.intro.intro\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na\u271d b\u271d : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nR : \u211d\u22650\nhf\u2081 : liminf (fun n => \u2191\u2016f n \u03c9\u2016\u208a) atTop < \u22a4\nhf\u2082 : \u2200 (a b : \u211a), a < b \u2192 upcrossings (\u2191a) (\u2191b) f \u03c9 < \u22a4\nh : \u00acIsBoundedUnder (fun x x_1 => x \u2264 x_1) atTop fun n => |f n \u03c9|\na b : \u211a\nhab : a < b\nh\u2081 : \u2203\u1da0 (i : \u2115) in atTop, f i \u03c9 < \u2191a\nh\u2082 : \u2203\u1da0 (i : \u2115) in atTop, \u2191b < f i \u03c9\n\u22a2 \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c)"}, {"tactic": "exact\n  False.elim ((hf\u2082 a b hab).ne (upcrossings_eq_top_of_frequently_lt (Rat.cast_lt.2 hab) h\u2081 h\u2082))", "annotated_tactic": ["exact\n      <a>False.elim</a> ((hf\u2082 a b hab).<a>ne</a> (<a>upcrossings_eq_top_of_frequently_lt</a> (<a>Rat.cast_lt</a>.2 hab) h\u2081 h\u2082))", [{"full_name": "False.elim", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [223, 21], "def_end_pos": [223, 31]}, {"full_name": "LT.lt.ne", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [152, 7], "def_end_pos": [152, 15]}, {"full_name": "MeasureTheory.upcrossings_eq_top_of_frequently_lt", "def_path": "Mathlib/Probability/Martingale/Convergence.lean", "def_pos": [132, 9], "def_end_pos": [132, 44]}, {"full_name": "Rat.cast_lt", "def_path": "Mathlib/Data/Rat/Cast/Order.lean", "def_pos": [59, 9], "def_end_pos": [59, 16]}]], "state_before": "case neg.intro.intro.intro.intro\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na\u271d b\u271d : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nR : \u211d\u22650\nhf\u2081 : liminf (fun n => \u2191\u2016f n \u03c9\u2016\u208a) atTop < \u22a4\nhf\u2082 : \u2200 (a b : \u211a), a < b \u2192 upcrossings (\u2191a) (\u2191b) f \u03c9 < \u22a4\nh : \u00acIsBoundedUnder (fun x x_1 => x \u2264 x_1) atTop fun n => |f n \u03c9|\na b : \u211a\nhab : a < b\nh\u2081 : \u2203\u1da0 (i : \u2115) in atTop, f i \u03c9 < \u2191a\nh\u2082 : \u2203\u1da0 (i : \u2115) in atTop, \u2191b < f i \u03c9\n\u22a2 \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "full_name": "MeasureTheory.snorm_indicator_const'", "start": [615, 1], "end": [619, 45], "traced_tactics": [{"tactic": "by_cases hp_top : p = \u221e", "annotated_tactic": ["by_cases hp_top : p = \u221e", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nc\u271d : E\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ns : Set \u03b1\nc : G\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 0\nhp : p \u2260 0\n\u22a2 snorm (Set.indicator s fun x => c) p \u03bc = \u2191\u2016c\u2016\u208a * \u2191\u2191\u03bc s ^ (1 / ENNReal.toReal p)", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nc\u271d : E\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ns : Set \u03b1\nc : G\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 0\nhp : p \u2260 0\nhp_top : p = \u22a4\n\u22a2 snorm (Set.indicator s fun x => c) p \u03bc = \u2191\u2016c\u2016\u208a * \u2191\u2191\u03bc s ^ (1 / ENNReal.toReal p)\n\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nc\u271d : E\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ns : Set \u03b1\nc : G\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 0\nhp : p \u2260 0\nhp_top : \u00acp = \u22a4\n\u22a2 snorm (Set.indicator s fun x => c) p \u03bc = \u2191\u2016c\u2016\u208a * \u2191\u2191\u03bc s ^ (1 / ENNReal.toReal p)"}, {"tactic": "simp [hp_top, snormEssSup_indicator_const_eq s c h\u03bcs]", "annotated_tactic": ["simp [hp_top, <a>snormEssSup_indicator_const_eq</a> s c h\u03bcs]", [{"full_name": "MeasureTheory.snormEssSup_indicator_const_eq", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [576, 9], "def_end_pos": [576, 39]}]], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nc\u271d : E\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ns : Set \u03b1\nc : G\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 0\nhp : p \u2260 0\nhp_top : p = \u22a4\n\u22a2 snorm (Set.indicator s fun x => c) p \u03bc = \u2191\u2016c\u2016\u208a * \u2191\u2191\u03bc s ^ (1 / ENNReal.toReal p)", "state_after": "no goals"}, {"tactic": "exact snorm_indicator_const hs hp hp_top", "annotated_tactic": ["exact <a>snorm_indicator_const</a> hs hp hp_top", [{"full_name": "MeasureTheory.snorm_indicator_const", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [610, 9], "def_end_pos": [610, 30]}]], "state_before": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nc\u271d : E\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ns : Set \u03b1\nc : G\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 0\nhp : p \u2260 0\nhp_top : \u00acp = \u22a4\n\u22a2 snorm (Set.indicator s fun x => c) p \u03bc = \u2191\u2016c\u2016\u208a * \u2191\u2191\u03bc s ^ (1 / ENNReal.toReal p)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Decomposition/Lebesgue.lean", "full_name": "MeasureTheory.Measure.eq_rnDeriv", "start": [358, 1], "end": [365, 66], "traced_tactics": [{"tactic": "refine' ae_eq_of_forall_set_lintegral_eq_of_sigmaFinite hf (measurable_rnDeriv \u03bc \u03bd) _", "annotated_tactic": ["refine' <a>ae_eq_of_forall_set_lintegral_eq_of_sigmaFinite</a> hf (<a>measurable_rnDeriv</a> \u03bc \u03bd) _", [{"full_name": "MeasureTheory.ae_eq_of_forall_set_lintegral_eq_of_sigmaFinite", "def_path": "Mathlib/MeasureTheory/Function/AEEqOfIntegral.lean", "def_pos": [225, 9], "def_end_pos": [225, 56]}, {"full_name": "MeasureTheory.Measure.measurable_rnDeriv", "def_path": "Mathlib/MeasureTheory/Decomposition/Lebesgue.lean", "def_pos": [124, 9], "def_end_pos": [124, 27]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : SigmaFinite \u03bd\ns : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhs : s \u27c2\u2098 \u03bd\nhadd : \u03bc = s + withDensity \u03bd f\n\u22a2 f =\u1da0[ae \u03bd] rnDeriv \u03bc \u03bd", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : SigmaFinite \u03bd\ns : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhs : s \u27c2\u2098 \u03bd\nhadd : \u03bc = s + withDensity \u03bd f\n\u22a2 \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bd s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bd = \u222b\u207b (x : \u03b1) in s, rnDeriv \u03bc \u03bd x \u2202\u03bd"}, {"tactic": "intro a ha _", "annotated_tactic": ["intro a ha _", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : SigmaFinite \u03bd\ns : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhs : s \u27c2\u2098 \u03bd\nhadd : \u03bc = s + withDensity \u03bd f\n\u22a2 \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bd s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bd = \u222b\u207b (x : \u03b1) in s, rnDeriv \u03bc \u03bd x \u2202\u03bd", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : SigmaFinite \u03bd\ns : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhs : s \u27c2\u2098 \u03bd\nhadd : \u03bc = s + withDensity \u03bd f\na : Set \u03b1\nha : MeasurableSet a\na\u271d : \u2191\u2191\u03bd a < \u22a4\n\u22a2 \u222b\u207b (x : \u03b1) in a, f x \u2202\u03bd = \u222b\u207b (x : \u03b1) in a, rnDeriv \u03bc \u03bd x \u2202\u03bd"}, {"tactic": "calc\n  \u222b\u207b x : \u03b1 in a, f x \u2202\u03bd = \u03bd.withDensity f a := (withDensity_apply f ha).symm\n  _ = \u03bd.withDensity (\u03bc.rnDeriv \u03bd) a := by rw [eq_withDensity_rnDeriv hf hs hadd]\n  _ = \u222b\u207b x : \u03b1 in a, \u03bc.rnDeriv \u03bd x \u2202\u03bd := withDensity_apply _ ha", "annotated_tactic": ["calc\n    \u222b\u207b x : \u03b1 in a, f x \u2202\u03bd = \u03bd.withDensity f a := (<a>withDensity_apply</a> f ha).<a>symm</a>\n    _ = \u03bd.withDensity (\u03bc.rnDeriv \u03bd) a := by rw [<a>eq_withDensity_rnDeriv</a> hf hs hadd]\n    _ = \u222b\u207b x : \u03b1 in a, \u03bc.rnDeriv \u03bd x \u2202\u03bd := <a>withDensity_apply</a> _ ha", [{"full_name": "MeasureTheory.withDensity_apply", "def_path": "Mathlib/MeasureTheory/Measure/WithDensity.lean", "def_pos": [39, 9], "def_end_pos": [39, 26]}, {"full_name": "Eq.symm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [310, 9], "def_end_pos": [310, 16]}, {"full_name": "MeasureTheory.Measure.eq_withDensity_rnDeriv", "def_path": "Mathlib/MeasureTheory/Decomposition/Lebesgue.lean", "def_pos": [312, 9], "def_end_pos": [312, 31]}, {"full_name": "MeasureTheory.withDensity_apply", "def_path": "Mathlib/MeasureTheory/Measure/WithDensity.lean", "def_pos": [39, 9], "def_end_pos": [39, 26]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : SigmaFinite \u03bd\ns : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhs : s \u27c2\u2098 \u03bd\nhadd : \u03bc = s + withDensity \u03bd f\na : Set \u03b1\nha : MeasurableSet a\na\u271d : \u2191\u2191\u03bd a < \u22a4\n\u22a2 \u222b\u207b (x : \u03b1) in a, f x \u2202\u03bd = \u222b\u207b (x : \u03b1) in a, rnDeriv \u03bc \u03bd x \u2202\u03bd", "state_after": "no goals"}, {"tactic": "rw [eq_withDensity_rnDeriv hf hs hadd]", "annotated_tactic": ["rw [<a>eq_withDensity_rnDeriv</a> hf hs hadd]", [{"full_name": "MeasureTheory.Measure.eq_withDensity_rnDeriv", "def_path": "Mathlib/MeasureTheory/Decomposition/Lebesgue.lean", "def_pos": [312, 9], "def_end_pos": [312, 31]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : SigmaFinite \u03bd\ns : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhs : s \u27c2\u2098 \u03bd\nhadd : \u03bc = s + withDensity \u03bd f\na : Set \u03b1\nha : MeasurableSet a\na\u271d : \u2191\u2191\u03bd a < \u22a4\n\u22a2 \u2191\u2191(withDensity \u03bd f) a = \u2191\u2191(withDensity \u03bd (rnDeriv \u03bc \u03bd)) a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/Basic.lean", "full_name": "MvPolynomial.support_X_mul", "start": [745, 1], "end": [747, 54], "traced_tactics": [{"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "R : Type u\nS\u2081 : Type v\nS\u2082 : Type w\nS\u2083 : Type x\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\nn m : \u03c3\ns\u271d : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : CommSemiring S\u2081\np\u271d q : MvPolynomial \u03c3 R\ns : \u03c3\np : MvPolynomial \u03c3 R\n\u22a2 \u2200 (y : R), 1 * y = 0 \u2194 y = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/Variables.lean", "full_name": "MvPolynomial.le_degrees_add", "start": [193, 1], "end": [213, 63], "traced_tactics": [{"tactic": "apply Finset.sup_le", "annotated_tactic": ["apply <a>Finset.sup_le</a>", [{"full_name": "Finset.sup_le", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [105, 11], "def_end_pos": [105, 17]}]], "state_before": "R : Type u\nS : Type v\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nr : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\np\u271d q\u271d p q : MvPolynomial \u03c3 R\nh : Multiset.Disjoint (degrees p) (degrees q)\n\u22a2 degrees p \u2264 degrees (p + q)", "state_after": "case a\nR : Type u\nS : Type v\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nr : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\np\u271d q\u271d p q : MvPolynomial \u03c3 R\nh : Multiset.Disjoint (degrees p) (degrees q)\n\u22a2 \u2200 (b : \u03c3 \u2192\u2080 \u2115), b \u2208 support p \u2192 \u2191toMultiset b \u2264 degrees (p + q)"}, {"tactic": "intro d hd", "annotated_tactic": ["intro d hd", []], "state_before": "case a\nR : Type u\nS : Type v\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nr : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\np\u271d q\u271d p q : MvPolynomial \u03c3 R\nh : Multiset.Disjoint (degrees p) (degrees q)\n\u22a2 \u2200 (b : \u03c3 \u2192\u2080 \u2115), b \u2208 support p \u2192 \u2191toMultiset b \u2264 degrees (p + q)", "state_after": "case a\nR : Type u\nS : Type v\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nr : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\np\u271d q\u271d p q : MvPolynomial \u03c3 R\nh : Multiset.Disjoint (degrees p) (degrees q)\nd : \u03c3 \u2192\u2080 \u2115\nhd : d \u2208 support p\n\u22a2 \u2191toMultiset d \u2264 degrees (p + q)"}, {"tactic": "rw [Multiset.disjoint_iff_ne] at h", "annotated_tactic": ["rw [<a>Multiset.disjoint_iff_ne</a>] at h", [{"full_name": "Multiset.disjoint_iff_ne", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [2919, 9], "def_end_pos": [2919, 24]}]], "state_before": "case a\nR : Type u\nS : Type v\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nr : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\np\u271d q\u271d p q : MvPolynomial \u03c3 R\nh : Multiset.Disjoint (degrees p) (degrees q)\nd : \u03c3 \u2192\u2080 \u2115\nhd : d \u2208 support p\n\u22a2 \u2191toMultiset d \u2264 degrees (p + q)", "state_after": "case a\nR : Type u\nS : Type v\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nr : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\np\u271d q\u271d p q : MvPolynomial \u03c3 R\nh : \u2200 (a : \u03c3), a \u2208 degrees p \u2192 \u2200 (b : \u03c3), b \u2208 degrees q \u2192 a \u2260 b\nd : \u03c3 \u2192\u2080 \u2115\nhd : d \u2208 support p\n\u22a2 \u2191toMultiset d \u2264 degrees (p + q)"}, {"tactic": "rw [Multiset.le_iff_count]", "annotated_tactic": ["rw [<a>Multiset.le_iff_count</a>]", [{"full_name": "Multiset.le_iff_count", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [2526, 9], "def_end_pos": [2526, 21]}]], "state_before": "case a\nR : Type u\nS : Type v\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nr : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\np\u271d q\u271d p q : MvPolynomial \u03c3 R\nh : \u2200 (a : \u03c3), a \u2208 degrees p \u2192 \u2200 (b : \u03c3), b \u2208 degrees q \u2192 a \u2260 b\nd : \u03c3 \u2192\u2080 \u2115\nhd : d \u2208 support p\n\u22a2 \u2191toMultiset d \u2264 degrees (p + q)", "state_after": "case a\nR : Type u\nS : Type v\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nr : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\np\u271d q\u271d p q : MvPolynomial \u03c3 R\nh : \u2200 (a : \u03c3), a \u2208 degrees p \u2192 \u2200 (b : \u03c3), b \u2208 degrees q \u2192 a \u2260 b\nd : \u03c3 \u2192\u2080 \u2115\nhd : d \u2208 support p\n\u22a2 \u2200 (a : \u03c3), Multiset.count a (\u2191toMultiset d) \u2264 Multiset.count a (degrees (p + q))"}, {"tactic": "intro i", "annotated_tactic": ["intro i", []], "state_before": "case a\nR : Type u\nS : Type v\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nr : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\np\u271d q\u271d p q : MvPolynomial \u03c3 R\nh : \u2200 (a : \u03c3), a \u2208 degrees p \u2192 \u2200 (b : \u03c3), b \u2208 degrees q \u2192 a \u2260 b\nd : \u03c3 \u2192\u2080 \u2115\nhd : d \u2208 support p\n\u22a2 \u2200 (a : \u03c3), Multiset.count a (\u2191toMultiset d) \u2264 Multiset.count a (degrees (p + q))", "state_after": "case a\nR : Type u\nS : Type v\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nr : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\np\u271d q\u271d p q : MvPolynomial \u03c3 R\nh : \u2200 (a : \u03c3), a \u2208 degrees p \u2192 \u2200 (b : \u03c3), b \u2208 degrees q \u2192 a \u2260 b\nd : \u03c3 \u2192\u2080 \u2115\nhd : d \u2208 support p\ni : \u03c3\n\u22a2 Multiset.count i (\u2191toMultiset d) \u2264 Multiset.count i (degrees (p + q))"}, {"tactic": "rw [degrees, Multiset.count_finset_sup]", "annotated_tactic": ["rw [<a>degrees</a>, <a>Multiset.count_finset_sup</a>]", [{"full_name": "MvPolynomial.degrees", "def_path": "Mathlib/Data/MvPolynomial/Variables.lean", "def_pos": [87, 5], "def_end_pos": [87, 12]}, {"full_name": "Multiset.count_finset_sup", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [1834, 9], "def_end_pos": [1834, 25]}]], "state_before": "case a\nR : Type u\nS : Type v\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nr : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\np\u271d q\u271d p q : MvPolynomial \u03c3 R\nh : \u2200 (a : \u03c3), a \u2208 degrees p \u2192 \u2200 (b : \u03c3), b \u2208 degrees q \u2192 a \u2260 b\nd : \u03c3 \u2192\u2080 \u2115\nhd : d \u2208 support p\ni : \u03c3\n\u22a2 Multiset.count i (\u2191toMultiset d) \u2264 Multiset.count i (degrees (p + q))", "state_after": "case a\nR : Type u\nS : Type v\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nr : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\np\u271d q\u271d p q : MvPolynomial \u03c3 R\nh : \u2200 (a : \u03c3), a \u2208 degrees p \u2192 \u2200 (b : \u03c3), b \u2208 degrees q \u2192 a \u2260 b\nd : \u03c3 \u2192\u2080 \u2115\nhd : d \u2208 support p\ni : \u03c3\n\u22a2 Multiset.count i (\u2191toMultiset d) \u2264 Finset.sup (support (p + q)) fun a => Multiset.count i (\u2191toMultiset a)"}, {"tactic": "simp only [Finsupp.count_toMultiset]", "annotated_tactic": ["simp only [<a>Finsupp.count_toMultiset</a>]", [{"full_name": "Finsupp.count_toMultiset", "def_path": "Mathlib/Data/Finsupp/Multiset.lean", "def_pos": [106, 9], "def_end_pos": [106, 25]}]], "state_before": "case a\nR : Type u\nS : Type v\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nr : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\np\u271d q\u271d p q : MvPolynomial \u03c3 R\nh : \u2200 (a : \u03c3), a \u2208 degrees p \u2192 \u2200 (b : \u03c3), b \u2208 degrees q \u2192 a \u2260 b\nd : \u03c3 \u2192\u2080 \u2115\nhd : d \u2208 support p\ni : \u03c3\n\u22a2 Multiset.count i (\u2191toMultiset d) \u2264 Finset.sup (support (p + q)) fun a => Multiset.count i (\u2191toMultiset a)", "state_after": "case a\nR : Type u\nS : Type v\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nr : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\np\u271d q\u271d p q : MvPolynomial \u03c3 R\nh : \u2200 (a : \u03c3), a \u2208 degrees p \u2192 \u2200 (b : \u03c3), b \u2208 degrees q \u2192 a \u2260 b\nd : \u03c3 \u2192\u2080 \u2115\nhd : d \u2208 support p\ni : \u03c3\n\u22a2 \u2191d i \u2264 Finset.sup (support (p + q)) fun a => \u2191a i"}, {"tactic": "by_cases h0 : d = 0", "annotated_tactic": ["by_cases h0 : d = 0", []], "state_before": "case a\nR : Type u\nS : Type v\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nr : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\np\u271d q\u271d p q : MvPolynomial \u03c3 R\nh : \u2200 (a : \u03c3), a \u2208 degrees p \u2192 \u2200 (b : \u03c3), b \u2208 degrees q \u2192 a \u2260 b\nd : \u03c3 \u2192\u2080 \u2115\nhd : d \u2208 support p\ni : \u03c3\n\u22a2 \u2191d i \u2264 Finset.sup (support (p + q)) fun a => \u2191a i", "state_after": "case pos\nR : Type u\nS : Type v\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nr : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\np\u271d q\u271d p q : MvPolynomial \u03c3 R\nh : \u2200 (a : \u03c3), a \u2208 degrees p \u2192 \u2200 (b : \u03c3), b \u2208 degrees q \u2192 a \u2260 b\nd : \u03c3 \u2192\u2080 \u2115\nhd : d \u2208 support p\ni : \u03c3\nh0 : d = 0\n\u22a2 \u2191d i \u2264 Finset.sup (support (p + q)) fun a => \u2191a i\n\ncase neg\nR : Type u\nS : Type v\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nr : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\np\u271d q\u271d p q : MvPolynomial \u03c3 R\nh : \u2200 (a : \u03c3), a \u2208 degrees p \u2192 \u2200 (b : \u03c3), b \u2208 degrees q \u2192 a \u2260 b\nd : \u03c3 \u2192\u2080 \u2115\nhd : d \u2208 support p\ni : \u03c3\nh0 : \u00acd = 0\n\u22a2 \u2191d i \u2264 Finset.sup (support (p + q)) fun a => \u2191a i"}, {"tactic": "simp only [h0, zero_le, Finsupp.zero_apply]", "annotated_tactic": ["simp only [h0, <a>zero_le</a>, <a>Finsupp.zero_apply</a>]", [{"full_name": "zero_le", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [217, 30], "def_end_pos": [217, 37]}, {"full_name": "Finsupp.zero_apply", "def_path": "Mathlib/Data/Finsupp/Defs.lean", "def_pos": [172, 9], "def_end_pos": [172, 19]}]], "state_before": "case pos\nR : Type u\nS : Type v\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nr : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\np\u271d q\u271d p q : MvPolynomial \u03c3 R\nh : \u2200 (a : \u03c3), a \u2208 degrees p \u2192 \u2200 (b : \u03c3), b \u2208 degrees q \u2192 a \u2260 b\nd : \u03c3 \u2192\u2080 \u2115\nhd : d \u2208 support p\ni : \u03c3\nh0 : d = 0\n\u22a2 \u2191d i \u2264 Finset.sup (support (p + q)) fun a => \u2191a i", "state_after": "no goals"}, {"tactic": "refine' @Finset.le_sup _ _ _ _ (p + q).support (fun a => a i) d _", "annotated_tactic": ["refine' @<a>Finset.le_sup</a> _ _ _ _ (p + q).<a>support</a> (fun a => a i) d _", [{"full_name": "Finset.le_sup", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [114, 9], "def_end_pos": [114, 15]}, {"full_name": "MvPolynomial.support", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [526, 5], "def_end_pos": [526, 12]}]], "state_before": "case neg\nR : Type u\nS : Type v\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nr : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\np\u271d q\u271d p q : MvPolynomial \u03c3 R\nh : \u2200 (a : \u03c3), a \u2208 degrees p \u2192 \u2200 (b : \u03c3), b \u2208 degrees q \u2192 a \u2260 b\nd : \u03c3 \u2192\u2080 \u2115\nhd : d \u2208 support p\ni : \u03c3\nh0 : \u00acd = 0\n\u22a2 \u2191d i \u2264 Finset.sup (support (p + q)) fun a => \u2191a i", "state_after": "case neg\nR : Type u\nS : Type v\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nr : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\np\u271d q\u271d p q : MvPolynomial \u03c3 R\nh : \u2200 (a : \u03c3), a \u2208 degrees p \u2192 \u2200 (b : \u03c3), b \u2208 degrees q \u2192 a \u2260 b\nd : \u03c3 \u2192\u2080 \u2115\nhd : d \u2208 support p\ni : \u03c3\nh0 : \u00acd = 0\n\u22a2 d \u2208 support (p + q)"}, {"tactic": "rw [mem_support_iff, coeff_add]", "annotated_tactic": ["rw [<a>mem_support_iff</a>, <a>coeff_add</a>]", [{"full_name": "MvPolynomial.mem_support_iff", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [587, 9], "def_end_pos": [587, 24]}, {"full_name": "MvPolynomial.coeff_add", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [614, 9], "def_end_pos": [614, 18]}]], "state_before": "case neg\nR : Type u\nS : Type v\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nr : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\np\u271d q\u271d p q : MvPolynomial \u03c3 R\nh : \u2200 (a : \u03c3), a \u2208 degrees p \u2192 \u2200 (b : \u03c3), b \u2208 degrees q \u2192 a \u2260 b\nd : \u03c3 \u2192\u2080 \u2115\nhd : d \u2208 support p\ni : \u03c3\nh0 : \u00acd = 0\n\u22a2 d \u2208 support (p + q)", "state_after": "case neg\nR : Type u\nS : Type v\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nr : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\np\u271d q\u271d p q : MvPolynomial \u03c3 R\nh : \u2200 (a : \u03c3), a \u2208 degrees p \u2192 \u2200 (b : \u03c3), b \u2208 degrees q \u2192 a \u2260 b\nd : \u03c3 \u2192\u2080 \u2115\nhd : d \u2208 support p\ni : \u03c3\nh0 : \u00acd = 0\n\u22a2 coeff d p + coeff d q \u2260 0"}, {"tactic": "suffices q.coeff d = 0 by rwa [this, add_zero, coeff, \u2190 Finsupp.mem_support_iff]", "annotated_tactic": ["suffices q.coeff d = 0 by rwa [this, <a>add_zero</a>, <a>coeff</a>, \u2190 <a>Finsupp.mem_support_iff</a>]", [{"full_name": "add_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [469, 3], "def_end_pos": [469, 14]}, {"full_name": "MvPolynomial.coeff", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [580, 5], "def_end_pos": [580, 10]}, {"full_name": "Finsupp.mem_support_iff", "def_path": "Mathlib/Data/Finsupp/Defs.lean", "def_pos": [186, 9], "def_end_pos": [186, 24]}]], "state_before": "case neg\nR : Type u\nS : Type v\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nr : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\np\u271d q\u271d p q : MvPolynomial \u03c3 R\nh : \u2200 (a : \u03c3), a \u2208 degrees p \u2192 \u2200 (b : \u03c3), b \u2208 degrees q \u2192 a \u2260 b\nd : \u03c3 \u2192\u2080 \u2115\nhd : d \u2208 support p\ni : \u03c3\nh0 : \u00acd = 0\n\u22a2 coeff d p + coeff d q \u2260 0", "state_after": "case neg\nR : Type u\nS : Type v\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nr : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\np\u271d q\u271d p q : MvPolynomial \u03c3 R\nh : \u2200 (a : \u03c3), a \u2208 degrees p \u2192 \u2200 (b : \u03c3), b \u2208 degrees q \u2192 a \u2260 b\nd : \u03c3 \u2192\u2080 \u2115\nhd : d \u2208 support p\ni : \u03c3\nh0 : \u00acd = 0\n\u22a2 coeff d q = 0"}, {"tactic": "rw [\u2190 Finsupp.support_eq_empty, \u2190 Ne.def, \u2190 Finset.nonempty_iff_ne_empty] at h0", "annotated_tactic": ["rw [\u2190 <a>Finsupp.support_eq_empty</a>, \u2190 <a>Ne.def</a>, \u2190 <a>Finset.nonempty_iff_ne_empty</a>] at h0", [{"full_name": "Finsupp.support_eq_empty", "def_path": "Mathlib/Data/Finsupp/Defs.lean", "def_pos": [214, 9], "def_end_pos": [214, 25]}, {"full_name": "Ne.def", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [59, 9], "def_end_pos": [59, 15]}, {"full_name": "Finset.nonempty_iff_ne_empty", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [604, 9], "def_end_pos": [604, 30]}]], "state_before": "case neg\nR : Type u\nS : Type v\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nr : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\np\u271d q\u271d p q : MvPolynomial \u03c3 R\nh : \u2200 (a : \u03c3), a \u2208 degrees p \u2192 \u2200 (b : \u03c3), b \u2208 degrees q \u2192 a \u2260 b\nd : \u03c3 \u2192\u2080 \u2115\nhd : d \u2208 support p\ni : \u03c3\nh0 : \u00acd = 0\n\u22a2 coeff d q = 0", "state_after": "case neg\nR : Type u\nS : Type v\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nr : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\np\u271d q\u271d p q : MvPolynomial \u03c3 R\nh : \u2200 (a : \u03c3), a \u2208 degrees p \u2192 \u2200 (b : \u03c3), b \u2208 degrees q \u2192 a \u2260 b\nd : \u03c3 \u2192\u2080 \u2115\nhd : d \u2208 support p\ni : \u03c3\nh0 : Finset.Nonempty d.support\n\u22a2 coeff d q = 0"}, {"tactic": "obtain \u27e8j, hj\u27e9 := h0", "annotated_tactic": ["obtain \u27e8j, hj\u27e9 := h0", []], "state_before": "case neg\nR : Type u\nS : Type v\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nr : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\np\u271d q\u271d p q : MvPolynomial \u03c3 R\nh : \u2200 (a : \u03c3), a \u2208 degrees p \u2192 \u2200 (b : \u03c3), b \u2208 degrees q \u2192 a \u2260 b\nd : \u03c3 \u2192\u2080 \u2115\nhd : d \u2208 support p\ni : \u03c3\nh0 : Finset.Nonempty d.support\n\u22a2 coeff d q = 0", "state_after": "case neg.intro\nR : Type u\nS : Type v\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nr : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\np\u271d q\u271d p q : MvPolynomial \u03c3 R\nh : \u2200 (a : \u03c3), a \u2208 degrees p \u2192 \u2200 (b : \u03c3), b \u2208 degrees q \u2192 a \u2260 b\nd : \u03c3 \u2192\u2080 \u2115\nhd : d \u2208 support p\ni j : \u03c3\nhj : j \u2208 d.support\n\u22a2 coeff d q = 0"}, {"tactic": "contrapose! h", "annotated_tactic": ["contrapose! h", []], "state_before": "case neg.intro\nR : Type u\nS : Type v\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nr : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\np\u271d q\u271d p q : MvPolynomial \u03c3 R\nh : \u2200 (a : \u03c3), a \u2208 degrees p \u2192 \u2200 (b : \u03c3), b \u2208 degrees q \u2192 a \u2260 b\nd : \u03c3 \u2192\u2080 \u2115\nhd : d \u2208 support p\ni j : \u03c3\nhj : j \u2208 d.support\n\u22a2 coeff d q = 0", "state_after": "case neg.intro\nR : Type u\nS : Type v\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nr : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\np\u271d q\u271d p q : MvPolynomial \u03c3 R\nd : \u03c3 \u2192\u2080 \u2115\nhd : d \u2208 support p\ni j : \u03c3\nhj : j \u2208 d.support\nh : coeff d q \u2260 0\n\u22a2 \u2203 a, a \u2208 degrees p \u2227 \u2203 b, b \u2208 degrees q \u2227 a = b"}, {"tactic": "rw [mem_support_iff] at hd", "annotated_tactic": ["rw [<a>mem_support_iff</a>] at hd", [{"full_name": "MvPolynomial.mem_support_iff", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [587, 9], "def_end_pos": [587, 24]}]], "state_before": "case neg.intro\nR : Type u\nS : Type v\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nr : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\np\u271d q\u271d p q : MvPolynomial \u03c3 R\nd : \u03c3 \u2192\u2080 \u2115\nhd : d \u2208 support p\ni j : \u03c3\nhj : j \u2208 d.support\nh : coeff d q \u2260 0\n\u22a2 \u2203 a, a \u2208 degrees p \u2227 \u2203 b, b \u2208 degrees q \u2227 a = b", "state_after": "case neg.intro\nR : Type u\nS : Type v\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nr : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\np\u271d q\u271d p q : MvPolynomial \u03c3 R\nd : \u03c3 \u2192\u2080 \u2115\nhd : coeff d p \u2260 0\ni j : \u03c3\nhj : j \u2208 d.support\nh : coeff d q \u2260 0\n\u22a2 \u2203 a, a \u2208 degrees p \u2227 \u2203 b, b \u2208 degrees q \u2227 a = b"}, {"tactic": "refine' \u27e8j, _, j, _, rfl\u27e9", "annotated_tactic": ["refine' \u27e8j, _, j, _, <a>rfl</a>\u27e9", [{"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "case neg.intro\nR : Type u\nS : Type v\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nr : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\np\u271d q\u271d p q : MvPolynomial \u03c3 R\nd : \u03c3 \u2192\u2080 \u2115\nhd : coeff d p \u2260 0\ni j : \u03c3\nhj : j \u2208 d.support\nh : coeff d q \u2260 0\n\u22a2 \u2203 a, a \u2208 degrees p \u2227 \u2203 b, b \u2208 degrees q \u2227 a = b", "state_after": "case neg.intro.refine'_1\nR : Type u\nS : Type v\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nr : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\np\u271d q\u271d p q : MvPolynomial \u03c3 R\nd : \u03c3 \u2192\u2080 \u2115\nhd : coeff d p \u2260 0\ni j : \u03c3\nhj : j \u2208 d.support\nh : coeff d q \u2260 0\n\u22a2 j \u2208 degrees p\n\ncase neg.intro.refine'_2\nR : Type u\nS : Type v\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nr : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\np\u271d q\u271d p q : MvPolynomial \u03c3 R\nd : \u03c3 \u2192\u2080 \u2115\nhd : coeff d p \u2260 0\ni j : \u03c3\nhj : j \u2208 d.support\nh : coeff d q \u2260 0\n\u22a2 j \u2208 degrees q"}, {"tactic": "all_goals rw [mem_degrees]; refine' \u27e8d, _, hj\u27e9; assumption", "annotated_tactic": ["all_goals rw [<a>mem_degrees</a>]; refine' \u27e8d, _, hj\u27e9; assumption", [{"full_name": "MvPolynomial.mem_degrees", "def_path": "Mathlib/Data/MvPolynomial/Variables.lean", "def_pos": [187, 9], "def_end_pos": [187, 20]}]], "state_before": "case neg.intro.refine'_1\nR : Type u\nS : Type v\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nr : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\np\u271d q\u271d p q : MvPolynomial \u03c3 R\nd : \u03c3 \u2192\u2080 \u2115\nhd : coeff d p \u2260 0\ni j : \u03c3\nhj : j \u2208 d.support\nh : coeff d q \u2260 0\n\u22a2 j \u2208 degrees p\n\ncase neg.intro.refine'_2\nR : Type u\nS : Type v\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nr : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\np\u271d q\u271d p q : MvPolynomial \u03c3 R\nd : \u03c3 \u2192\u2080 \u2115\nhd : coeff d p \u2260 0\ni j : \u03c3\nhj : j \u2208 d.support\nh : coeff d q \u2260 0\n\u22a2 j \u2208 degrees q", "state_after": "no goals"}, {"tactic": "rwa [this, add_zero, coeff, \u2190 Finsupp.mem_support_iff]", "annotated_tactic": ["rwa [this, <a>add_zero</a>, <a>coeff</a>, \u2190 <a>Finsupp.mem_support_iff</a>]", [{"full_name": "add_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [469, 3], "def_end_pos": [469, 14]}, {"full_name": "MvPolynomial.coeff", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [580, 5], "def_end_pos": [580, 10]}, {"full_name": "Finsupp.mem_support_iff", "def_path": "Mathlib/Data/Finsupp/Defs.lean", "def_pos": [186, 9], "def_end_pos": [186, 24]}]], "state_before": "R : Type u\nS : Type v\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nr : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\np\u271d q\u271d p q : MvPolynomial \u03c3 R\nh : \u2200 (a : \u03c3), a \u2208 degrees p \u2192 \u2200 (b : \u03c3), b \u2208 degrees q \u2192 a \u2260 b\nd : \u03c3 \u2192\u2080 \u2115\nhd : d \u2208 support p\ni : \u03c3\nh0 : \u00acd = 0\nthis : coeff d q = 0\n\u22a2 coeff d p + coeff d q \u2260 0", "state_after": "no goals"}, {"tactic": "rw [mem_degrees]", "annotated_tactic": ["rw [<a>mem_degrees</a>]", [{"full_name": "MvPolynomial.mem_degrees", "def_path": "Mathlib/Data/MvPolynomial/Variables.lean", "def_pos": [187, 9], "def_end_pos": [187, 20]}]], "state_before": "case neg.intro.refine'_2\nR : Type u\nS : Type v\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nr : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\np\u271d q\u271d p q : MvPolynomial \u03c3 R\nd : \u03c3 \u2192\u2080 \u2115\nhd : coeff d p \u2260 0\ni j : \u03c3\nhj : j \u2208 d.support\nh : coeff d q \u2260 0\n\u22a2 j \u2208 degrees q", "state_after": "case neg.intro.refine'_2\nR : Type u\nS : Type v\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nr : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\np\u271d q\u271d p q : MvPolynomial \u03c3 R\nd : \u03c3 \u2192\u2080 \u2115\nhd : coeff d p \u2260 0\ni j : \u03c3\nhj : j \u2208 d.support\nh : coeff d q \u2260 0\n\u22a2 \u2203 d, coeff d q \u2260 0 \u2227 j \u2208 d.support"}, {"tactic": "refine' \u27e8d, _, hj\u27e9", "annotated_tactic": ["refine' \u27e8d, _, hj\u27e9", []], "state_before": "case neg.intro.refine'_2\nR : Type u\nS : Type v\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nr : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\np\u271d q\u271d p q : MvPolynomial \u03c3 R\nd : \u03c3 \u2192\u2080 \u2115\nhd : coeff d p \u2260 0\ni j : \u03c3\nhj : j \u2208 d.support\nh : coeff d q \u2260 0\n\u22a2 \u2203 d, coeff d q \u2260 0 \u2227 j \u2208 d.support", "state_after": "case neg.intro.refine'_2\nR : Type u\nS : Type v\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nr : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\np\u271d q\u271d p q : MvPolynomial \u03c3 R\nd : \u03c3 \u2192\u2080 \u2115\nhd : coeff d p \u2260 0\ni j : \u03c3\nhj : j \u2208 d.support\nh : coeff d q \u2260 0\n\u22a2 coeff d q \u2260 0"}, {"tactic": "assumption", "annotated_tactic": ["assumption", []], "state_before": "case neg.intro.refine'_2\nR : Type u\nS : Type v\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nr : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\np\u271d q\u271d p q : MvPolynomial \u03c3 R\nd : \u03c3 \u2192\u2080 \u2115\nhd : coeff d p \u2260 0\ni j : \u03c3\nhj : j \u2208 d.support\nh : coeff d q \u2260 0\n\u22a2 coeff d q \u2260 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "full_name": "QuotientGroup.measurable_from_quotient", "start": [557, 8], "end": [559, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "full_name": "MeasureTheory.Measure.ext_of_generateFrom_of_cover", "start": [1930, 1], "end": [1946, 45], "traced_tactics": [{"tactic": "refine' ext_of_sUnion_eq_univ hc hU fun t ht => _", "annotated_tactic": ["refine' <a>ext_of_sUnion_eq_univ</a> hc hU fun t ht => _", [{"full_name": "MeasureTheory.Measure.ext_of_sUnion_eq_univ", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1927, 11], "def_end_pos": [1927, 32]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nS T : Set (Set \u03b1)\nh_gen : m0 = generateFrom S\nhc : Set.Countable T\nh_inter : IsPiSystem S\nhU : \u22c3\u2080 T = univ\nhtop : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2191\u2191\u03bc t \u2260 \u22a4\nST_eq : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2200 (s : Set \u03b1), s \u2208 S \u2192 \u2191\u2191\u03bc (s \u2229 t) = \u2191\u2191\u03bd (s \u2229 t)\nT_eq : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2191\u2191\u03bc t = \u2191\u2191\u03bd t\n\u22a2 \u03bc = \u03bd", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t\u271d : Set \u03b1\nS T : Set (Set \u03b1)\nh_gen : m0 = generateFrom S\nhc : Set.Countable T\nh_inter : IsPiSystem S\nhU : \u22c3\u2080 T = univ\nhtop : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2191\u2191\u03bc t \u2260 \u22a4\nST_eq : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2200 (s : Set \u03b1), s \u2208 S \u2192 \u2191\u2191\u03bc (s \u2229 t) = \u2191\u2191\u03bd (s \u2229 t)\nT_eq : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2191\u2191\u03bc t = \u2191\u2191\u03bd t\nt : Set \u03b1\nht : t \u2208 T\n\u22a2 restrict \u03bc t = restrict \u03bd t"}, {"tactic": "ext1 u hu", "annotated_tactic": ["ext1 u hu", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t\u271d : Set \u03b1\nS T : Set (Set \u03b1)\nh_gen : m0 = generateFrom S\nhc : Set.Countable T\nh_inter : IsPiSystem S\nhU : \u22c3\u2080 T = univ\nhtop : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2191\u2191\u03bc t \u2260 \u22a4\nST_eq : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2200 (s : Set \u03b1), s \u2208 S \u2192 \u2191\u2191\u03bc (s \u2229 t) = \u2191\u2191\u03bd (s \u2229 t)\nT_eq : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2191\u2191\u03bc t = \u2191\u2191\u03bd t\nt : Set \u03b1\nht : t \u2208 T\n\u22a2 restrict \u03bc t = restrict \u03bd t", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t\u271d : Set \u03b1\nS T : Set (Set \u03b1)\nh_gen : m0 = generateFrom S\nhc : Set.Countable T\nh_inter : IsPiSystem S\nhU : \u22c3\u2080 T = univ\nhtop : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2191\u2191\u03bc t \u2260 \u22a4\nST_eq : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2200 (s : Set \u03b1), s \u2208 S \u2192 \u2191\u2191\u03bc (s \u2229 t) = \u2191\u2191\u03bd (s \u2229 t)\nT_eq : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2191\u2191\u03bc t = \u2191\u2191\u03bd t\nt : Set \u03b1\nht : t \u2208 T\nu : Set \u03b1\nhu : MeasurableSet u\n\u22a2 \u2191\u2191(restrict \u03bc t) u = \u2191\u2191(restrict \u03bd t) u"}, {"tactic": "simp only [restrict_apply hu]", "annotated_tactic": ["simp only [<a>restrict_apply</a> hu]", [{"full_name": "MeasureTheory.Measure.restrict_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1533, 9], "def_end_pos": [1533, 23]}]], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t\u271d : Set \u03b1\nS T : Set (Set \u03b1)\nh_gen : m0 = generateFrom S\nhc : Set.Countable T\nh_inter : IsPiSystem S\nhU : \u22c3\u2080 T = univ\nhtop : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2191\u2191\u03bc t \u2260 \u22a4\nST_eq : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2200 (s : Set \u03b1), s \u2208 S \u2192 \u2191\u2191\u03bc (s \u2229 t) = \u2191\u2191\u03bd (s \u2229 t)\nT_eq : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2191\u2191\u03bc t = \u2191\u2191\u03bd t\nt : Set \u03b1\nht : t \u2208 T\nu : Set \u03b1\nhu : MeasurableSet u\n\u22a2 \u2191\u2191(restrict \u03bc t) u = \u2191\u2191(restrict \u03bd t) u", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t\u271d : Set \u03b1\nS T : Set (Set \u03b1)\nh_gen : m0 = generateFrom S\nhc : Set.Countable T\nh_inter : IsPiSystem S\nhU : \u22c3\u2080 T = univ\nhtop : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2191\u2191\u03bc t \u2260 \u22a4\nST_eq : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2200 (s : Set \u03b1), s \u2208 S \u2192 \u2191\u2191\u03bc (s \u2229 t) = \u2191\u2191\u03bd (s \u2229 t)\nT_eq : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2191\u2191\u03bc t = \u2191\u2191\u03bd t\nt : Set \u03b1\nht : t \u2208 T\nu : Set \u03b1\nhu : MeasurableSet u\n\u22a2 \u2191\u2191\u03bc (u \u2229 t) = \u2191\u2191\u03bd (u \u2229 t)"}, {"tactic": "refine' induction_on_inter h_gen h_inter _ (ST_eq t ht) _ _ hu", "annotated_tactic": ["refine' <a>induction_on_inter</a> h_gen h_inter _ (ST_eq t ht) _ _ hu", [{"full_name": "MeasurableSpace.induction_on_inter", "def_path": "Mathlib/MeasureTheory/PiSystem.lean", "def_pos": [745, 9], "def_end_pos": [745, 27]}]], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t\u271d : Set \u03b1\nS T : Set (Set \u03b1)\nh_gen : m0 = generateFrom S\nhc : Set.Countable T\nh_inter : IsPiSystem S\nhU : \u22c3\u2080 T = univ\nhtop : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2191\u2191\u03bc t \u2260 \u22a4\nST_eq : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2200 (s : Set \u03b1), s \u2208 S \u2192 \u2191\u2191\u03bc (s \u2229 t) = \u2191\u2191\u03bd (s \u2229 t)\nT_eq : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2191\u2191\u03bc t = \u2191\u2191\u03bd t\nt : Set \u03b1\nht : t \u2208 T\nu : Set \u03b1\nhu : MeasurableSet u\n\u22a2 \u2191\u2191\u03bc (u \u2229 t) = \u2191\u2191\u03bd (u \u2229 t)", "state_after": "case h.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t\u271d : Set \u03b1\nS T : Set (Set \u03b1)\nh_gen : m0 = generateFrom S\nhc : Set.Countable T\nh_inter : IsPiSystem S\nhU : \u22c3\u2080 T = univ\nhtop : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2191\u2191\u03bc t \u2260 \u22a4\nST_eq : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2200 (s : Set \u03b1), s \u2208 S \u2192 \u2191\u2191\u03bc (s \u2229 t) = \u2191\u2191\u03bd (s \u2229 t)\nT_eq : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2191\u2191\u03bc t = \u2191\u2191\u03bd t\nt : Set \u03b1\nht : t \u2208 T\nu : Set \u03b1\nhu : MeasurableSet u\n\u22a2 \u2191\u2191\u03bc (\u2205 \u2229 t) = \u2191\u2191\u03bd (\u2205 \u2229 t)\n\ncase h.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t\u271d : Set \u03b1\nS T : Set (Set \u03b1)\nh_gen : m0 = generateFrom S\nhc : Set.Countable T\nh_inter : IsPiSystem S\nhU : \u22c3\u2080 T = univ\nhtop : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2191\u2191\u03bc t \u2260 \u22a4\nST_eq : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2200 (s : Set \u03b1), s \u2208 S \u2192 \u2191\u2191\u03bc (s \u2229 t) = \u2191\u2191\u03bd (s \u2229 t)\nT_eq : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2191\u2191\u03bc t = \u2191\u2191\u03bd t\nt : Set \u03b1\nht : t \u2208 T\nu : Set \u03b1\nhu : MeasurableSet u\n\u22a2 \u2200 (t_1 : Set \u03b1), MeasurableSet t_1 \u2192 \u2191\u2191\u03bc (t_1 \u2229 t) = \u2191\u2191\u03bd (t_1 \u2229 t) \u2192 \u2191\u2191\u03bc (t_1\u1d9c \u2229 t) = \u2191\u2191\u03bd (t_1\u1d9c \u2229 t)\n\ncase h.refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t\u271d : Set \u03b1\nS T : Set (Set \u03b1)\nh_gen : m0 = generateFrom S\nhc : Set.Countable T\nh_inter : IsPiSystem S\nhU : \u22c3\u2080 T = univ\nhtop : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2191\u2191\u03bc t \u2260 \u22a4\nST_eq : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2200 (s : Set \u03b1), s \u2208 S \u2192 \u2191\u2191\u03bc (s \u2229 t) = \u2191\u2191\u03bd (s \u2229 t)\nT_eq : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2191\u2191\u03bc t = \u2191\u2191\u03bd t\nt : Set \u03b1\nht : t \u2208 T\nu : Set \u03b1\nhu : MeasurableSet u\n\u22a2 \u2200 (f : \u2115 \u2192 Set \u03b1),\n    Pairwise (Disjoint on f) \u2192\n      (\u2200 (i : \u2115), MeasurableSet (f i)) \u2192\n        (\u2200 (i : \u2115), \u2191\u2191\u03bc (f i \u2229 t) = \u2191\u2191\u03bd (f i \u2229 t)) \u2192 \u2191\u2191\u03bc ((\u22c3 i, f i) \u2229 t) = \u2191\u2191\u03bd ((\u22c3 i, f i) \u2229 t)"}, {"tactic": "simp only [Set.empty_inter, measure_empty]", "annotated_tactic": ["simp only [<a>Set.empty_inter</a>, <a>measure_empty</a>]", [{"full_name": "Set.empty_inter", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [936, 9], "def_end_pos": [936, 20]}, {"full_name": "MeasureTheory.measure_empty", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [185, 9], "def_end_pos": [185, 22]}]], "state_before": "case h.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t\u271d : Set \u03b1\nS T : Set (Set \u03b1)\nh_gen : m0 = generateFrom S\nhc : Set.Countable T\nh_inter : IsPiSystem S\nhU : \u22c3\u2080 T = univ\nhtop : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2191\u2191\u03bc t \u2260 \u22a4\nST_eq : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2200 (s : Set \u03b1), s \u2208 S \u2192 \u2191\u2191\u03bc (s \u2229 t) = \u2191\u2191\u03bd (s \u2229 t)\nT_eq : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2191\u2191\u03bc t = \u2191\u2191\u03bd t\nt : Set \u03b1\nht : t \u2208 T\nu : Set \u03b1\nhu : MeasurableSet u\n\u22a2 \u2191\u2191\u03bc (\u2205 \u2229 t) = \u2191\u2191\u03bd (\u2205 \u2229 t)", "state_after": "no goals"}, {"tactic": "intro v hv hvt", "annotated_tactic": ["intro v hv hvt", []], "state_before": "case h.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t\u271d : Set \u03b1\nS T : Set (Set \u03b1)\nh_gen : m0 = generateFrom S\nhc : Set.Countable T\nh_inter : IsPiSystem S\nhU : \u22c3\u2080 T = univ\nhtop : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2191\u2191\u03bc t \u2260 \u22a4\nST_eq : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2200 (s : Set \u03b1), s \u2208 S \u2192 \u2191\u2191\u03bc (s \u2229 t) = \u2191\u2191\u03bd (s \u2229 t)\nT_eq : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2191\u2191\u03bc t = \u2191\u2191\u03bd t\nt : Set \u03b1\nht : t \u2208 T\nu : Set \u03b1\nhu : MeasurableSet u\n\u22a2 \u2200 (t_1 : Set \u03b1), MeasurableSet t_1 \u2192 \u2191\u2191\u03bc (t_1 \u2229 t) = \u2191\u2191\u03bd (t_1 \u2229 t) \u2192 \u2191\u2191\u03bc (t_1\u1d9c \u2229 t) = \u2191\u2191\u03bd (t_1\u1d9c \u2229 t)", "state_after": "case h.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t\u271d : Set \u03b1\nS T : Set (Set \u03b1)\nh_gen : m0 = generateFrom S\nhc : Set.Countable T\nh_inter : IsPiSystem S\nhU : \u22c3\u2080 T = univ\nhtop : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2191\u2191\u03bc t \u2260 \u22a4\nST_eq : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2200 (s : Set \u03b1), s \u2208 S \u2192 \u2191\u2191\u03bc (s \u2229 t) = \u2191\u2191\u03bd (s \u2229 t)\nT_eq : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2191\u2191\u03bc t = \u2191\u2191\u03bd t\nt : Set \u03b1\nht : t \u2208 T\nu : Set \u03b1\nhu : MeasurableSet u\nv : Set \u03b1\nhv : MeasurableSet v\nhvt : \u2191\u2191\u03bc (v \u2229 t) = \u2191\u2191\u03bd (v \u2229 t)\n\u22a2 \u2191\u2191\u03bc (v\u1d9c \u2229 t) = \u2191\u2191\u03bd (v\u1d9c \u2229 t)"}, {"tactic": "have := T_eq t ht", "annotated_tactic": ["have := T_eq t ht", []], "state_before": "case h.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t\u271d : Set \u03b1\nS T : Set (Set \u03b1)\nh_gen : m0 = generateFrom S\nhc : Set.Countable T\nh_inter : IsPiSystem S\nhU : \u22c3\u2080 T = univ\nhtop : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2191\u2191\u03bc t \u2260 \u22a4\nST_eq : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2200 (s : Set \u03b1), s \u2208 S \u2192 \u2191\u2191\u03bc (s \u2229 t) = \u2191\u2191\u03bd (s \u2229 t)\nT_eq : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2191\u2191\u03bc t = \u2191\u2191\u03bd t\nt : Set \u03b1\nht : t \u2208 T\nu : Set \u03b1\nhu : MeasurableSet u\nv : Set \u03b1\nhv : MeasurableSet v\nhvt : \u2191\u2191\u03bc (v \u2229 t) = \u2191\u2191\u03bd (v \u2229 t)\n\u22a2 \u2191\u2191\u03bc (v\u1d9c \u2229 t) = \u2191\u2191\u03bd (v\u1d9c \u2229 t)", "state_after": "case h.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t\u271d : Set \u03b1\nS T : Set (Set \u03b1)\nh_gen : m0 = generateFrom S\nhc : Set.Countable T\nh_inter : IsPiSystem S\nhU : \u22c3\u2080 T = univ\nhtop : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2191\u2191\u03bc t \u2260 \u22a4\nST_eq : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2200 (s : Set \u03b1), s \u2208 S \u2192 \u2191\u2191\u03bc (s \u2229 t) = \u2191\u2191\u03bd (s \u2229 t)\nT_eq : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2191\u2191\u03bc t = \u2191\u2191\u03bd t\nt : Set \u03b1\nht : t \u2208 T\nu : Set \u03b1\nhu : MeasurableSet u\nv : Set \u03b1\nhv : MeasurableSet v\nhvt : \u2191\u2191\u03bc (v \u2229 t) = \u2191\u2191\u03bd (v \u2229 t)\nthis : \u2191\u2191\u03bc t = \u2191\u2191\u03bd t\n\u22a2 \u2191\u2191\u03bc (v\u1d9c \u2229 t) = \u2191\u2191\u03bd (v\u1d9c \u2229 t)"}, {"tactic": "rw [Set.inter_comm] at hvt \u22a2", "annotated_tactic": ["rw [<a>Set.inter_comm</a>] at hvt \u22a2", [{"full_name": "Set.inter_comm", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [940, 9], "def_end_pos": [940, 19]}]], "state_before": "case h.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t\u271d : Set \u03b1\nS T : Set (Set \u03b1)\nh_gen : m0 = generateFrom S\nhc : Set.Countable T\nh_inter : IsPiSystem S\nhU : \u22c3\u2080 T = univ\nhtop : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2191\u2191\u03bc t \u2260 \u22a4\nST_eq : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2200 (s : Set \u03b1), s \u2208 S \u2192 \u2191\u2191\u03bc (s \u2229 t) = \u2191\u2191\u03bd (s \u2229 t)\nT_eq : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2191\u2191\u03bc t = \u2191\u2191\u03bd t\nt : Set \u03b1\nht : t \u2208 T\nu : Set \u03b1\nhu : MeasurableSet u\nv : Set \u03b1\nhv : MeasurableSet v\nhvt : \u2191\u2191\u03bc (v \u2229 t) = \u2191\u2191\u03bd (v \u2229 t)\nthis : \u2191\u2191\u03bc t = \u2191\u2191\u03bd t\n\u22a2 \u2191\u2191\u03bc (v\u1d9c \u2229 t) = \u2191\u2191\u03bd (v\u1d9c \u2229 t)", "state_after": "case h.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t\u271d : Set \u03b1\nS T : Set (Set \u03b1)\nh_gen : m0 = generateFrom S\nhc : Set.Countable T\nh_inter : IsPiSystem S\nhU : \u22c3\u2080 T = univ\nhtop : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2191\u2191\u03bc t \u2260 \u22a4\nST_eq : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2200 (s : Set \u03b1), s \u2208 S \u2192 \u2191\u2191\u03bc (s \u2229 t) = \u2191\u2191\u03bd (s \u2229 t)\nT_eq : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2191\u2191\u03bc t = \u2191\u2191\u03bd t\nt : Set \u03b1\nht : t \u2208 T\nu : Set \u03b1\nhu : MeasurableSet u\nv : Set \u03b1\nhv : MeasurableSet v\nhvt : \u2191\u2191\u03bc (t \u2229 v) = \u2191\u2191\u03bd (t \u2229 v)\nthis : \u2191\u2191\u03bc t = \u2191\u2191\u03bd t\n\u22a2 \u2191\u2191\u03bc (t \u2229 v\u1d9c) = \u2191\u2191\u03bd (t \u2229 v\u1d9c)"}, {"tactic": "rwa [\u2190 measure_inter_add_diff t hv, \u2190 measure_inter_add_diff t hv, \u2190 hvt,\n  ENNReal.add_right_inj] at this", "annotated_tactic": ["rwa [\u2190 <a>measure_inter_add_diff</a> t hv, \u2190 <a>measure_inter_add_diff</a> t hv, \u2190 hvt,\n      <a>ENNReal.add_right_inj</a>] at this", [{"full_name": "MeasureTheory.measure_inter_add_diff", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [132, 9], "def_end_pos": [132, 31]}, {"full_name": "MeasureTheory.measure_inter_add_diff", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [132, 9], "def_end_pos": [132, 31]}, {"full_name": "ENNReal.add_right_inj", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1120, 9], "def_end_pos": [1120, 22]}]], "state_before": "case h.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t\u271d : Set \u03b1\nS T : Set (Set \u03b1)\nh_gen : m0 = generateFrom S\nhc : Set.Countable T\nh_inter : IsPiSystem S\nhU : \u22c3\u2080 T = univ\nhtop : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2191\u2191\u03bc t \u2260 \u22a4\nST_eq : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2200 (s : Set \u03b1), s \u2208 S \u2192 \u2191\u2191\u03bc (s \u2229 t) = \u2191\u2191\u03bd (s \u2229 t)\nT_eq : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2191\u2191\u03bc t = \u2191\u2191\u03bd t\nt : Set \u03b1\nht : t \u2208 T\nu : Set \u03b1\nhu : MeasurableSet u\nv : Set \u03b1\nhv : MeasurableSet v\nhvt : \u2191\u2191\u03bc (t \u2229 v) = \u2191\u2191\u03bd (t \u2229 v)\nthis : \u2191\u2191\u03bc t = \u2191\u2191\u03bd t\n\u22a2 \u2191\u2191\u03bc (t \u2229 v\u1d9c) = \u2191\u2191\u03bd (t \u2229 v\u1d9c)", "state_after": "case h.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t\u271d : Set \u03b1\nS T : Set (Set \u03b1)\nh_gen : m0 = generateFrom S\nhc : Set.Countable T\nh_inter : IsPiSystem S\nhU : \u22c3\u2080 T = univ\nhtop : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2191\u2191\u03bc t \u2260 \u22a4\nST_eq : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2200 (s : Set \u03b1), s \u2208 S \u2192 \u2191\u2191\u03bc (s \u2229 t) = \u2191\u2191\u03bd (s \u2229 t)\nT_eq : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2191\u2191\u03bc t = \u2191\u2191\u03bd t\nt : Set \u03b1\nht : t \u2208 T\nu : Set \u03b1\nhu : MeasurableSet u\nv : Set \u03b1\nhv : MeasurableSet v\nhvt : \u2191\u2191\u03bc (t \u2229 v) = \u2191\u2191\u03bd (t \u2229 v)\nthis : \u2191\u2191\u03bc (t \u2229 v) + \u2191\u2191\u03bc (t \\ v) = \u2191\u2191\u03bc (t \u2229 v) + \u2191\u2191\u03bd (t \\ v)\n\u22a2 \u2191\u2191\u03bc (t \u2229 v) \u2260 \u22a4"}, {"tactic": "exact ne_top_of_le_ne_top (htop t ht) (measure_mono <| Set.inter_subset_left _ _)", "annotated_tactic": ["exact <a>ne_top_of_le_ne_top</a> (htop t ht) (<a>measure_mono</a> <| <a>Set.inter_subset_left</a> _ _)", [{"full_name": "ne_top_of_le_ne_top", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [194, 9], "def_end_pos": [194, 28]}, {"full_name": "MeasureTheory.measure_mono", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [193, 9], "def_end_pos": [193, 21]}, {"full_name": "Set.inter_subset_left", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [965, 9], "def_end_pos": [965, 26]}]], "state_before": "case h.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t\u271d : Set \u03b1\nS T : Set (Set \u03b1)\nh_gen : m0 = generateFrom S\nhc : Set.Countable T\nh_inter : IsPiSystem S\nhU : \u22c3\u2080 T = univ\nhtop : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2191\u2191\u03bc t \u2260 \u22a4\nST_eq : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2200 (s : Set \u03b1), s \u2208 S \u2192 \u2191\u2191\u03bc (s \u2229 t) = \u2191\u2191\u03bd (s \u2229 t)\nT_eq : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2191\u2191\u03bc t = \u2191\u2191\u03bd t\nt : Set \u03b1\nht : t \u2208 T\nu : Set \u03b1\nhu : MeasurableSet u\nv : Set \u03b1\nhv : MeasurableSet v\nhvt : \u2191\u2191\u03bc (t \u2229 v) = \u2191\u2191\u03bd (t \u2229 v)\nthis : \u2191\u2191\u03bc (t \u2229 v) + \u2191\u2191\u03bc (t \\ v) = \u2191\u2191\u03bc (t \u2229 v) + \u2191\u2191\u03bd (t \\ v)\n\u22a2 \u2191\u2191\u03bc (t \u2229 v) \u2260 \u22a4", "state_after": "no goals"}, {"tactic": "intro f hfd hfm h_eq", "annotated_tactic": ["intro f hfd hfm h_eq", []], "state_before": "case h.refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t\u271d : Set \u03b1\nS T : Set (Set \u03b1)\nh_gen : m0 = generateFrom S\nhc : Set.Countable T\nh_inter : IsPiSystem S\nhU : \u22c3\u2080 T = univ\nhtop : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2191\u2191\u03bc t \u2260 \u22a4\nST_eq : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2200 (s : Set \u03b1), s \u2208 S \u2192 \u2191\u2191\u03bc (s \u2229 t) = \u2191\u2191\u03bd (s \u2229 t)\nT_eq : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2191\u2191\u03bc t = \u2191\u2191\u03bd t\nt : Set \u03b1\nht : t \u2208 T\nu : Set \u03b1\nhu : MeasurableSet u\n\u22a2 \u2200 (f : \u2115 \u2192 Set \u03b1),\n    Pairwise (Disjoint on f) \u2192\n      (\u2200 (i : \u2115), MeasurableSet (f i)) \u2192\n        (\u2200 (i : \u2115), \u2191\u2191\u03bc (f i \u2229 t) = \u2191\u2191\u03bd (f i \u2229 t)) \u2192 \u2191\u2191\u03bc ((\u22c3 i, f i) \u2229 t) = \u2191\u2191\u03bd ((\u22c3 i, f i) \u2229 t)", "state_after": "case h.refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t\u271d : Set \u03b1\nS T : Set (Set \u03b1)\nh_gen : m0 = generateFrom S\nhc : Set.Countable T\nh_inter : IsPiSystem S\nhU : \u22c3\u2080 T = univ\nhtop : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2191\u2191\u03bc t \u2260 \u22a4\nST_eq : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2200 (s : Set \u03b1), s \u2208 S \u2192 \u2191\u2191\u03bc (s \u2229 t) = \u2191\u2191\u03bd (s \u2229 t)\nT_eq : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2191\u2191\u03bc t = \u2191\u2191\u03bd t\nt : Set \u03b1\nht : t \u2208 T\nu : Set \u03b1\nhu : MeasurableSet u\nf : \u2115 \u2192 Set \u03b1\nhfd : Pairwise (Disjoint on f)\nhfm : \u2200 (i : \u2115), MeasurableSet (f i)\nh_eq : \u2200 (i : \u2115), \u2191\u2191\u03bc (f i \u2229 t) = \u2191\u2191\u03bd (f i \u2229 t)\n\u22a2 \u2191\u2191\u03bc ((\u22c3 i, f i) \u2229 t) = \u2191\u2191\u03bd ((\u22c3 i, f i) \u2229 t)"}, {"tactic": "simp only [\u2190 restrict_apply (hfm _), \u2190 restrict_apply (MeasurableSet.iUnion hfm)] at h_eq \u22a2", "annotated_tactic": ["simp only [\u2190 <a>restrict_apply</a> (hfm _), \u2190 <a>restrict_apply</a> (<a>MeasurableSet.iUnion</a> hfm)] at h_eq \u22a2", [{"full_name": "MeasureTheory.Measure.restrict_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1533, 9], "def_end_pos": [1533, 23]}, {"full_name": "MeasureTheory.Measure.restrict_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1533, 9], "def_end_pos": [1533, 23]}, {"full_name": "MeasurableSet.iUnion", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [115, 19], "def_end_pos": [115, 39]}]], "state_before": "case h.refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t\u271d : Set \u03b1\nS T : Set (Set \u03b1)\nh_gen : m0 = generateFrom S\nhc : Set.Countable T\nh_inter : IsPiSystem S\nhU : \u22c3\u2080 T = univ\nhtop : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2191\u2191\u03bc t \u2260 \u22a4\nST_eq : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2200 (s : Set \u03b1), s \u2208 S \u2192 \u2191\u2191\u03bc (s \u2229 t) = \u2191\u2191\u03bd (s \u2229 t)\nT_eq : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2191\u2191\u03bc t = \u2191\u2191\u03bd t\nt : Set \u03b1\nht : t \u2208 T\nu : Set \u03b1\nhu : MeasurableSet u\nf : \u2115 \u2192 Set \u03b1\nhfd : Pairwise (Disjoint on f)\nhfm : \u2200 (i : \u2115), MeasurableSet (f i)\nh_eq : \u2200 (i : \u2115), \u2191\u2191\u03bc (f i \u2229 t) = \u2191\u2191\u03bd (f i \u2229 t)\n\u22a2 \u2191\u2191\u03bc ((\u22c3 i, f i) \u2229 t) = \u2191\u2191\u03bd ((\u22c3 i, f i) \u2229 t)", "state_after": "case h.refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t\u271d : Set \u03b1\nS T : Set (Set \u03b1)\nh_gen : m0 = generateFrom S\nhc : Set.Countable T\nh_inter : IsPiSystem S\nhU : \u22c3\u2080 T = univ\nhtop : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2191\u2191\u03bc t \u2260 \u22a4\nST_eq : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2200 (s : Set \u03b1), s \u2208 S \u2192 \u2191\u2191\u03bc (s \u2229 t) = \u2191\u2191\u03bd (s \u2229 t)\nT_eq : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2191\u2191\u03bc t = \u2191\u2191\u03bd t\nt : Set \u03b1\nht : t \u2208 T\nu : Set \u03b1\nhu : MeasurableSet u\nf : \u2115 \u2192 Set \u03b1\nhfd : Pairwise (Disjoint on f)\nhfm : \u2200 (i : \u2115), MeasurableSet (f i)\nh_eq : \u2200 (i : \u2115), \u2191\u2191(restrict \u03bc t) (f i) = \u2191\u2191(restrict \u03bd t) (f i)\n\u22a2 \u2191\u2191(restrict \u03bc t) (\u22c3 i, f i) = \u2191\u2191(restrict \u03bd t) (\u22c3 i, f i)"}, {"tactic": "simp only [measure_iUnion hfd hfm, h_eq]", "annotated_tactic": ["simp only [<a>measure_iUnion</a> hfd hfm, h_eq]", [{"full_name": "MeasureTheory.measure_iUnion", "def_path": "Mathlib/MeasureTheory/Measure/NullMeasurable.lean", "def_pos": [272, 9], "def_end_pos": [272, 23]}]], "state_before": "case h.refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t\u271d : Set \u03b1\nS T : Set (Set \u03b1)\nh_gen : m0 = generateFrom S\nhc : Set.Countable T\nh_inter : IsPiSystem S\nhU : \u22c3\u2080 T = univ\nhtop : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2191\u2191\u03bc t \u2260 \u22a4\nST_eq : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2200 (s : Set \u03b1), s \u2208 S \u2192 \u2191\u2191\u03bc (s \u2229 t) = \u2191\u2191\u03bd (s \u2229 t)\nT_eq : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2191\u2191\u03bc t = \u2191\u2191\u03bd t\nt : Set \u03b1\nht : t \u2208 T\nu : Set \u03b1\nhu : MeasurableSet u\nf : \u2115 \u2192 Set \u03b1\nhfd : Pairwise (Disjoint on f)\nhfm : \u2200 (i : \u2115), MeasurableSet (f i)\nh_eq : \u2200 (i : \u2115), \u2191\u2191(restrict \u03bc t) (f i) = \u2191\u2191(restrict \u03bd t) (f i)\n\u22a2 \u2191\u2191(restrict \u03bc t) (\u22c3 i, f i) = \u2191\u2191(restrict \u03bd t) (\u22c3 i, f i)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/Division.lean", "full_name": "MvPolynomial.X_divMonomial", "start": [168, 1], "end": [169, 44], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Rel.lean", "full_name": "Rel.preimage_univ", "start": [206, 1], "end": [206, 95], "traced_tactics": [{"tactic": "rw [preimage, image_univ, codom_inv]", "annotated_tactic": ["rw [<a>preimage</a>, <a>image_univ</a>, <a>codom_inv</a>]", [{"full_name": "Rel.preimage", "def_path": "Mathlib/Data/Rel.lean", "def_pos": [174, 5], "def_end_pos": [174, 13]}, {"full_name": "Rel.image_univ", "def_path": "Mathlib/Data/Rel.lean", "def_pos": [168, 9], "def_end_pos": [168, 19]}, {"full_name": "Rel.codom_inv", "def_path": "Mathlib/Data/Rel.lean", "def_pos": [80, 9], "def_end_pos": [80, 18]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : Rel \u03b1 \u03b2\n\u22a2 preimage r Set.univ = dom r", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Function.lean", "full_name": "Set.mapsTo_union", "start": [486, 1], "end": [490, 28], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/Division.lean", "full_name": "MvPolynomial.X_dvd_iff_modMonomial_eq_zero", "start": [210, 1], "end": [212, 43], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Image.lean", "full_name": "Subtype.forall_set_subtype", "start": [1478, 1], "end": [1480, 44], "traced_tactics": [{"tactic": "rw [\u2190 forall_subset_range_iff, range_coe]", "annotated_tactic": ["rw [\u2190 <a>forall_subset_range_iff</a>, <a>range_coe</a>]", [{"full_name": "Set.forall_subset_range_iff", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [832, 9], "def_end_pos": [832, 32]}, {"full_name": "Subtype.range_coe", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [1409, 9], "def_end_pos": [1409, 18]}]], "state_before": "\u03b1 : Type u_1\nt : Set \u03b1\np : Set \u03b1 \u2192 Prop\n\u22a2 (\u2200 (s : Set \u2191t), p (val '' s)) \u2194 \u2200 (s : Set \u03b1), s \u2286 t \u2192 p s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "full_name": "MeasureTheory.isFiniteMeasure_withDensity_ofReal", "start": [280, 1], "end": [283, 37], "traced_tactics": [{"tactic": "refine' isFiniteMeasure_withDensity ((lintegral_mono fun x => _).trans_lt hfi).ne", "annotated_tactic": ["refine' <a>isFiniteMeasure_withDensity</a> ((<a>lintegral_mono</a> fun x => _).<a>trans_lt</a> hfi).<a>ne</a>", [{"full_name": "MeasureTheory.isFiniteMeasure_withDensity", "def_path": "Mathlib/MeasureTheory/Measure/WithDensity.lean", "def_pos": [111, 9], "def_end_pos": [111, 36]}, {"full_name": "MeasureTheory.lintegral_mono", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [99, 9], "def_end_pos": [99, 23]}, {"full_name": "LE.le.trans_lt", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [124, 7], "def_end_pos": [124, 21]}, {"full_name": "LT.lt.ne", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [152, 7], "def_end_pos": [152, 15]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : NormedAddCommGroup \u03b3\nf : \u03b1 \u2192 \u211d\nhfi : HasFiniteIntegral f\n\u22a2 IsFiniteMeasure (Measure.withDensity \u03bc fun x => ENNReal.ofReal (f x))", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : NormedAddCommGroup \u03b3\nf : \u03b1 \u2192 \u211d\nhfi : HasFiniteIntegral f\nx : \u03b1\n\u22a2 ENNReal.ofReal (f x) \u2264 \u2191\u2016f x\u2016\u208a"}, {"tactic": "exact Real.ofReal_le_ennnorm (f x)", "annotated_tactic": ["exact <a>Real.ofReal_le_ennnorm</a> (f x)", [{"full_name": "Real.ofReal_le_ennnorm", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [1823, 9], "def_end_pos": [1823, 26]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : NormedAddCommGroup \u03b3\nf : \u03b1 \u2192 \u211d\nhfi : HasFiniteIntegral f\nx : \u03b1\n\u22a2 ENNReal.ofReal (f x) \u2264 \u2191\u2016f x\u2016\u208a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "full_name": "MeasureTheory.stronglyMeasurable_of_fintype", "start": [138, 1], "end": [141, 65], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Moments.lean", "full_name": "ProbabilityTheory.cgf_const", "start": [151, 1], "end": [152, 38], "traced_tactics": [{"tactic": "simp only [cgf, mgf_const, log_exp]", "annotated_tactic": ["simp only [<a>cgf</a>, <a>mgf_const</a>, <a>log_exp</a>]", [{"full_name": "ProbabilityTheory.cgf", "def_path": "Mathlib/Probability/Moments.lean", "def_pos": [108, 5], "def_end_pos": [108, 8]}, {"full_name": "ProbabilityTheory.mgf_const", "def_path": "Mathlib/Probability/Moments.lean", "def_pos": [136, 9], "def_end_pos": [136, 18]}, {"full_name": "Real.log_exp", "def_path": "Mathlib/Analysis/SpecialFunctions/Log/Basic.lean", "def_pos": [80, 9], "def_end_pos": [80, 16]}]], "state_before": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u211d\np : \u2115\n\u03bc : Measure \u03a9\nt : \u211d\ninst\u271d : IsProbabilityMeasure \u03bc\nc : \u211d\n\u22a2 cgf (fun x => c) \u03bc t = t * c", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "full_name": "String.get_cons_addChar", "start": [226, 1], "end": [228, 79], "traced_tactics": [{"tactic": "simp [get, utf8GetAux, Pos.zero_ne_addChar, utf8GetAux_addChar_right_cancel]", "annotated_tactic": ["simp [<a>get</a>, <a>utf8GetAux</a>, <a>Pos.zero_ne_addChar</a>, <a>utf8GetAux_addChar_right_cancel</a>]", [{"full_name": "String.get", "def_path": "lake-packages/lean4/src/lean/Init/Data/String/Basic.lean", "def_pos": [57, 5], "def_end_pos": [57, 8]}, {"full_name": "String.utf8GetAux", "def_path": "lake-packages/lean4/src/lean/Init/Data/String/Basic.lean", "def_pos": [47, 5], "def_end_pos": [47, 15]}, {"full_name": "_private.\u00ablake-packages\u00bb.std.Std.Data.String.Lemmas.0.String.Pos.zero_ne_addChar", "def_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "def_pos": [146, 17], "def_end_pos": [146, 32]}, {"full_name": "String.utf8GetAux_addChar_right_cancel", "def_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "def_pos": [209, 9], "def_end_pos": [209, 40]}]], "state_before": "c : Char\ncs : List Char\ni : Pos\n\u22a2 get { data := c :: cs } (i + c) = get { data := cs } i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "full_name": "MeasureTheory.set_integral_eq_integral_of_forall_compl_eq_zero", "start": [401, 1], "end": [403, 72], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Lattice.lean", "full_name": "Finset.measurable_range_sup'", "start": [248, 1], "end": [252, 27], "traced_tactics": [{"tactic": "simp_rw [\u2190 Nat.lt_succ_iff] at hf", "annotated_tactic": ["simp_rw [\u2190 <a>Nat.lt_succ_iff</a>] at hf", [{"full_name": "Nat.lt_succ_iff", "def_path": "Mathlib/Data/Nat/Basic.lean", "def_pos": [207, 9], "def_end_pos": [207, 20]}]], "state_before": "M : Type u_1\ninst\u271d\u00b3 : MeasurableSpace M\n\u03b1 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g : \u03b1 \u2192 M\n\u03b4 : Type u_3\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : SemilatticeSup \u03b1\ninst\u271d : MeasurableSup\u2082 \u03b1\nf : \u2115 \u2192 \u03b4 \u2192 \u03b1\nn : \u2115\nhf : \u2200 (k : \u2115), k \u2264 n \u2192 Measurable (f k)\n\u22a2 Measurable (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) f)", "state_after": "M : Type u_1\ninst\u271d\u00b3 : MeasurableSpace M\n\u03b1 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g : \u03b1 \u2192 M\n\u03b4 : Type u_3\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : SemilatticeSup \u03b1\ninst\u271d : MeasurableSup\u2082 \u03b1\nf : \u2115 \u2192 \u03b4 \u2192 \u03b1\nn : \u2115\nhf : \u2200 (k : \u2115), k < Nat.succ n \u2192 Measurable (f k)\n\u22a2 Measurable (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) f)"}, {"tactic": "refine' Finset.measurable_sup' _ _", "annotated_tactic": ["refine' <a>Finset.measurable_sup'</a> _ _", [{"full_name": "Finset.measurable_sup'", "def_path": "Mathlib/MeasureTheory/Lattice.lean", "def_pos": [242, 9], "def_end_pos": [242, 31]}]], "state_before": "M : Type u_1\ninst\u271d\u00b3 : MeasurableSpace M\n\u03b1 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g : \u03b1 \u2192 M\n\u03b4 : Type u_3\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : SemilatticeSup \u03b1\ninst\u271d : MeasurableSup\u2082 \u03b1\nf : \u2115 \u2192 \u03b4 \u2192 \u03b1\nn : \u2115\nhf : \u2200 (k : \u2115), k < Nat.succ n \u2192 Measurable (f k)\n\u22a2 Measurable (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) f)", "state_after": "M : Type u_1\ninst\u271d\u00b3 : MeasurableSpace M\n\u03b1 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g : \u03b1 \u2192 M\n\u03b4 : Type u_3\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : SemilatticeSup \u03b1\ninst\u271d : MeasurableSup\u2082 \u03b1\nf : \u2115 \u2192 \u03b4 \u2192 \u03b1\nn : \u2115\nhf : \u2200 (k : \u2115), k < Nat.succ n \u2192 Measurable (f k)\n\u22a2 \u2200 (n_1 : \u2115), n_1 \u2208 range (n + 1) \u2192 Measurable (f n_1)"}, {"tactic": "simpa [Finset.mem_range]", "annotated_tactic": ["simpa [<a>Finset.mem_range</a>]", [{"full_name": "Finset.mem_range", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3037, 9], "def_end_pos": [3037, 18]}]], "state_before": "M : Type u_1\ninst\u271d\u00b3 : MeasurableSpace M\n\u03b1 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g : \u03b1 \u2192 M\n\u03b4 : Type u_3\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : SemilatticeSup \u03b1\ninst\u271d : MeasurableSup\u2082 \u03b1\nf : \u2115 \u2192 \u03b4 \u2192 \u03b1\nn : \u2115\nhf : \u2200 (k : \u2115), k < Nat.succ n \u2192 Measurable (f k)\n\u22a2 \u2200 (n_1 : \u2115), n_1 \u2208 range (n + 1) \u2192 Measurable (f n_1)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "full_name": "MeasureTheory.edist_indicatorConstLp_eq_nnnorm", "start": [791, 1], "end": [795, 77], "traced_tactics": [{"tactic": "unfold indicatorConstLp", "annotated_tactic": ["unfold <a>indicatorConstLp</a>", [{"full_name": "MeasureTheory.indicatorConstLp", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [744, 5], "def_end_pos": [744, 21]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nc : E\nt : Set \u03b1\nht : MeasurableSet t\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\n\u22a2 edist (indicatorConstLp p hs h\u03bcs c) (indicatorConstLp p ht h\u03bct c) =\n    \u2191\u2016indicatorConstLp p (_ : MeasurableSet (s \u2206 t)) (_ : \u2191\u2191\u03bc (s \u2206 t) \u2260 \u22a4) c\u2016\u208a", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nc : E\nt : Set \u03b1\nht : MeasurableSet t\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\n\u22a2 edist (Mem\u2112p.toLp (indicator s fun x => c) (_ : Mem\u2112p (indicator s fun x => c) p))\n      (Mem\u2112p.toLp (indicator t fun x => c) (_ : Mem\u2112p (indicator t fun x => c) p)) =\n    \u2191\u2016Mem\u2112p.toLp (indicator (s \u2206 t) fun x => c) (_ : Mem\u2112p (indicator (s \u2206 t) fun x => c) p)\u2016\u208a"}, {"tactic": "rw [Lp.edist_toLp_toLp, snorm_indicator_sub_indicator, Lp.coe_nnnorm_toLp]", "annotated_tactic": ["rw [<a>Lp.edist_toLp_toLp</a>, <a>snorm_indicator_sub_indicator</a>, <a>Lp.coe_nnnorm_toLp</a>]", [{"full_name": "MeasureTheory.Lp.edist_toLp_toLp", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [308, 9], "def_end_pos": [308, 24]}, {"full_name": "MeasureTheory.snorm_indicator_sub_indicator", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [491, 9], "def_end_pos": [491, 38]}, {"full_name": "MeasureTheory.Lp.coe_nnnorm_toLp", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [287, 9], "def_end_pos": [287, 24]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nc : E\nt : Set \u03b1\nht : MeasurableSet t\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\n\u22a2 edist (Mem\u2112p.toLp (indicator s fun x => c) (_ : Mem\u2112p (indicator s fun x => c) p))\n      (Mem\u2112p.toLp (indicator t fun x => c) (_ : Mem\u2112p (indicator t fun x => c) p)) =\n    \u2191\u2016Mem\u2112p.toLp (indicator (s \u2206 t) fun x => c) (_ : Mem\u2112p (indicator (s \u2206 t) fun x => c) p)\u2016\u208a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Decomposition/Jordan.lean", "full_name": "MeasureTheory.JordanDecomposition.toSignedMeasure_injective", "start": [372, 1], "end": [422, 85], "traced_tactics": [{"tactic": "intro j\u2081 j\u2082 hj", "annotated_tactic": ["intro j\u2081 j\u2082 hj", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\n\u22a2 Injective toSignedMeasure", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj\u2081 j\u2082 : JordanDecomposition \u03b1\nhj : toSignedMeasure j\u2081 = toSignedMeasure j\u2082\n\u22a2 j\u2081 = j\u2082"}, {"tactic": "obtain \u27e8S, hS\u2081, hS\u2082, hS\u2083, hS\u2084, hS\u2085\u27e9 := j\u2081.exists_compl_positive_negative", "annotated_tactic": ["obtain \u27e8S, hS\u2081, hS\u2082, hS\u2083, hS\u2084, hS\u2085\u27e9 := j\u2081.exists_compl_positive_negative", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj\u2081 j\u2082 : JordanDecomposition \u03b1\nhj : toSignedMeasure j\u2081 = toSignedMeasure j\u2082\n\u22a2 j\u2081 = j\u2082", "state_after": "case intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj\u2081 j\u2082 : JordanDecomposition \u03b1\nhj : toSignedMeasure j\u2081 = toSignedMeasure j\u2082\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) S \u2264 VectorMeasure.restrict 0 S\nhS\u2083 : VectorMeasure.restrict 0 S\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) S\u1d9c\nhS\u2084 : \u2191\u2191j\u2081.posPart S = 0\nhS\u2085 : \u2191\u2191j\u2081.negPart S\u1d9c = 0\n\u22a2 j\u2081 = j\u2082"}, {"tactic": "obtain \u27e8T, hT\u2081, hT\u2082, hT\u2083, hT\u2084, hT\u2085\u27e9 := j\u2082.exists_compl_positive_negative", "annotated_tactic": ["obtain \u27e8T, hT\u2081, hT\u2082, hT\u2083, hT\u2084, hT\u2085\u27e9 := j\u2082.exists_compl_positive_negative", []], "state_before": "case intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj\u2081 j\u2082 : JordanDecomposition \u03b1\nhj : toSignedMeasure j\u2081 = toSignedMeasure j\u2082\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) S \u2264 VectorMeasure.restrict 0 S\nhS\u2083 : VectorMeasure.restrict 0 S\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) S\u1d9c\nhS\u2084 : \u2191\u2191j\u2081.posPart S = 0\nhS\u2085 : \u2191\u2191j\u2081.negPart S\u1d9c = 0\n\u22a2 j\u2081 = j\u2082", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj\u2081 j\u2082 : JordanDecomposition \u03b1\nhj : toSignedMeasure j\u2081 = toSignedMeasure j\u2082\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) S \u2264 VectorMeasure.restrict 0 S\nhS\u2083 : VectorMeasure.restrict 0 S\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) S\u1d9c\nhS\u2084 : \u2191\u2191j\u2081.posPart S = 0\nhS\u2085 : \u2191\u2191j\u2081.negPart S\u1d9c = 0\nT : Set \u03b1\nhT\u2081 : MeasurableSet T\nhT\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2082) T \u2264 VectorMeasure.restrict 0 T\nhT\u2083 : VectorMeasure.restrict 0 T\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2082) T\u1d9c\nhT\u2084 : \u2191\u2191j\u2082.posPart T = 0\nhT\u2085 : \u2191\u2191j\u2082.negPart T\u1d9c = 0\n\u22a2 j\u2081 = j\u2082"}, {"tactic": "rw [\u2190 hj] at hT\u2082 hT\u2083", "annotated_tactic": ["rw [\u2190 hj] at hT\u2082 hT\u2083", []], "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj\u2081 j\u2082 : JordanDecomposition \u03b1\nhj : toSignedMeasure j\u2081 = toSignedMeasure j\u2082\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) S \u2264 VectorMeasure.restrict 0 S\nhS\u2083 : VectorMeasure.restrict 0 S\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) S\u1d9c\nhS\u2084 : \u2191\u2191j\u2081.posPart S = 0\nhS\u2085 : \u2191\u2191j\u2081.negPart S\u1d9c = 0\nT : Set \u03b1\nhT\u2081 : MeasurableSet T\nhT\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2082) T \u2264 VectorMeasure.restrict 0 T\nhT\u2083 : VectorMeasure.restrict 0 T\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2082) T\u1d9c\nhT\u2084 : \u2191\u2191j\u2082.posPart T = 0\nhT\u2085 : \u2191\u2191j\u2082.negPart T\u1d9c = 0\n\u22a2 j\u2081 = j\u2082", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj\u2081 j\u2082 : JordanDecomposition \u03b1\nhj : toSignedMeasure j\u2081 = toSignedMeasure j\u2082\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) S \u2264 VectorMeasure.restrict 0 S\nhS\u2083 : VectorMeasure.restrict 0 S\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) S\u1d9c\nhS\u2084 : \u2191\u2191j\u2081.posPart S = 0\nhS\u2085 : \u2191\u2191j\u2081.negPart S\u1d9c = 0\nT : Set \u03b1\nhT\u2081 : MeasurableSet T\nhT\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) T \u2264 VectorMeasure.restrict 0 T\nhT\u2083 : VectorMeasure.restrict 0 T\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) T\u1d9c\nhT\u2084 : \u2191\u2191j\u2082.posPart T = 0\nhT\u2085 : \u2191\u2191j\u2082.negPart T\u1d9c = 0\n\u22a2 j\u2081 = j\u2082"}, {"tactic": "obtain \u27e8hST\u2081, -\u27e9 :=\n  of_symmDiff_compl_positive_negative hS\u2081.compl hT\u2081.compl \u27e8hS\u2083, (compl_compl S).symm \u25b8 hS\u2082\u27e9\n    \u27e8hT\u2083, (compl_compl T).symm \u25b8 hT\u2082\u27e9", "annotated_tactic": ["obtain \u27e8hST\u2081, -\u27e9 :=\n    <a>of_symmDiff_compl_positive_negative</a> hS\u2081.compl hT\u2081.compl \u27e8hS\u2083, (<a>compl_compl</a> S).<a>symm</a> \u25b8 hS\u2082\u27e9\n      \u27e8hT\u2083, (<a>compl_compl</a> T).<a>symm</a> \u25b8 hT\u2082\u27e9", [{"full_name": "MeasureTheory.SignedMeasure.of_symmDiff_compl_positive_negative", "def_path": "Mathlib/MeasureTheory/Decomposition/SignedHahn.lean", "def_pos": [420, 9], "def_end_pos": [420, 44]}, {"full_name": "compl_compl", "def_path": "Mathlib/Order/BooleanAlgebra.lean", "def_pos": [634, 9], "def_end_pos": [634, 20]}, {"full_name": "Eq.symm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [310, 9], "def_end_pos": [310, 16]}, {"full_name": "compl_compl", "def_path": "Mathlib/Order/BooleanAlgebra.lean", "def_pos": [634, 9], "def_end_pos": [634, 20]}, {"full_name": "Eq.symm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [310, 9], "def_end_pos": [310, 16]}]], "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj\u2081 j\u2082 : JordanDecomposition \u03b1\nhj : toSignedMeasure j\u2081 = toSignedMeasure j\u2082\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) S \u2264 VectorMeasure.restrict 0 S\nhS\u2083 : VectorMeasure.restrict 0 S\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) S\u1d9c\nhS\u2084 : \u2191\u2191j\u2081.posPart S = 0\nhS\u2085 : \u2191\u2191j\u2081.negPart S\u1d9c = 0\nT : Set \u03b1\nhT\u2081 : MeasurableSet T\nhT\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) T \u2264 VectorMeasure.restrict 0 T\nhT\u2083 : VectorMeasure.restrict 0 T\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) T\u1d9c\nhT\u2084 : \u2191\u2191j\u2082.posPart T = 0\nhT\u2085 : \u2191\u2191j\u2082.negPart T\u1d9c = 0\n\u22a2 j\u2081 = j\u2082", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj\u2081 j\u2082 : JordanDecomposition \u03b1\nhj : toSignedMeasure j\u2081 = toSignedMeasure j\u2082\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) S \u2264 VectorMeasure.restrict 0 S\nhS\u2083 : VectorMeasure.restrict 0 S\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) S\u1d9c\nhS\u2084 : \u2191\u2191j\u2081.posPart S = 0\nhS\u2085 : \u2191\u2191j\u2081.negPart S\u1d9c = 0\nT : Set \u03b1\nhT\u2081 : MeasurableSet T\nhT\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) T \u2264 VectorMeasure.restrict 0 T\nhT\u2083 : VectorMeasure.restrict 0 T\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) T\u1d9c\nhT\u2084 : \u2191\u2191j\u2082.posPart T = 0\nhT\u2085 : \u2191\u2191j\u2082.negPart T\u1d9c = 0\nhST\u2081 : \u2191(toSignedMeasure j\u2081) (S\u1d9c \u2206 T\u1d9c) = 0\n\u22a2 j\u2081 = j\u2082"}, {"tactic": "refine' eq_of_posPart_eq_posPart _ hj", "annotated_tactic": ["refine' <a>eq_of_posPart_eq_posPart</a> _ hj", [{"full_name": "_private.Mathlib.MeasureTheory.Decomposition.Jordan.0.MeasureTheory.JordanDecomposition.eq_of_posPart_eq_posPart", "def_path": "Mathlib/MeasureTheory/Decomposition/Jordan.lean", "def_pos": [361, 17], "def_end_pos": [361, 41]}]], "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj\u2081 j\u2082 : JordanDecomposition \u03b1\nhj : toSignedMeasure j\u2081 = toSignedMeasure j\u2082\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) S \u2264 VectorMeasure.restrict 0 S\nhS\u2083 : VectorMeasure.restrict 0 S\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) S\u1d9c\nhS\u2084 : \u2191\u2191j\u2081.posPart S = 0\nhS\u2085 : \u2191\u2191j\u2081.negPart S\u1d9c = 0\nT : Set \u03b1\nhT\u2081 : MeasurableSet T\nhT\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) T \u2264 VectorMeasure.restrict 0 T\nhT\u2083 : VectorMeasure.restrict 0 T\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) T\u1d9c\nhT\u2084 : \u2191\u2191j\u2082.posPart T = 0\nhT\u2085 : \u2191\u2191j\u2082.negPart T\u1d9c = 0\nhST\u2081 : \u2191(toSignedMeasure j\u2081) (S\u1d9c \u2206 T\u1d9c) = 0\n\u22a2 j\u2081 = j\u2082", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj\u2081 j\u2082 : JordanDecomposition \u03b1\nhj : toSignedMeasure j\u2081 = toSignedMeasure j\u2082\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) S \u2264 VectorMeasure.restrict 0 S\nhS\u2083 : VectorMeasure.restrict 0 S\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) S\u1d9c\nhS\u2084 : \u2191\u2191j\u2081.posPart S = 0\nhS\u2085 : \u2191\u2191j\u2081.negPart S\u1d9c = 0\nT : Set \u03b1\nhT\u2081 : MeasurableSet T\nhT\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) T \u2264 VectorMeasure.restrict 0 T\nhT\u2083 : VectorMeasure.restrict 0 T\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) T\u1d9c\nhT\u2084 : \u2191\u2191j\u2082.posPart T = 0\nhT\u2085 : \u2191\u2191j\u2082.negPart T\u1d9c = 0\nhST\u2081 : \u2191(toSignedMeasure j\u2081) (S\u1d9c \u2206 T\u1d9c) = 0\n\u22a2 j\u2081.posPart = j\u2082.posPart"}, {"tactic": "ext1 i hi", "annotated_tactic": ["ext1 i hi", []], "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj\u2081 j\u2082 : JordanDecomposition \u03b1\nhj : toSignedMeasure j\u2081 = toSignedMeasure j\u2082\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) S \u2264 VectorMeasure.restrict 0 S\nhS\u2083 : VectorMeasure.restrict 0 S\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) S\u1d9c\nhS\u2084 : \u2191\u2191j\u2081.posPart S = 0\nhS\u2085 : \u2191\u2191j\u2081.negPart S\u1d9c = 0\nT : Set \u03b1\nhT\u2081 : MeasurableSet T\nhT\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) T \u2264 VectorMeasure.restrict 0 T\nhT\u2083 : VectorMeasure.restrict 0 T\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) T\u1d9c\nhT\u2084 : \u2191\u2191j\u2082.posPart T = 0\nhT\u2085 : \u2191\u2191j\u2082.negPart T\u1d9c = 0\nhST\u2081 : \u2191(toSignedMeasure j\u2081) (S\u1d9c \u2206 T\u1d9c) = 0\n\u22a2 j\u2081.posPart = j\u2082.posPart", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj\u2081 j\u2082 : JordanDecomposition \u03b1\nhj : toSignedMeasure j\u2081 = toSignedMeasure j\u2082\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) S \u2264 VectorMeasure.restrict 0 S\nhS\u2083 : VectorMeasure.restrict 0 S\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) S\u1d9c\nhS\u2084 : \u2191\u2191j\u2081.posPart S = 0\nhS\u2085 : \u2191\u2191j\u2081.negPart S\u1d9c = 0\nT : Set \u03b1\nhT\u2081 : MeasurableSet T\nhT\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) T \u2264 VectorMeasure.restrict 0 T\nhT\u2083 : VectorMeasure.restrict 0 T\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) T\u1d9c\nhT\u2084 : \u2191\u2191j\u2082.posPart T = 0\nhT\u2085 : \u2191\u2191j\u2082.negPart T\u1d9c = 0\nhST\u2081 : \u2191(toSignedMeasure j\u2081) (S\u1d9c \u2206 T\u1d9c) = 0\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 \u2191\u2191j\u2081.posPart i = \u2191\u2191j\u2082.posPart i"}, {"tactic": "rw [\u2190 ENNReal.toReal_eq_toReal (measure_ne_top _ _) (measure_ne_top _ _), h\u03bc\u2081, h\u03bc\u2082, \u2190 hj]", "annotated_tactic": ["rw [\u2190 <a>ENNReal.toReal_eq_toReal</a> (<a>measure_ne_top</a> _ _) (<a>measure_ne_top</a> _ _), h\u03bc\u2081, h\u03bc\u2082, \u2190 hj]", [{"full_name": "ENNReal.toReal_eq_toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2327, 9], "def_end_pos": [2327, 25]}, {"full_name": "MeasureTheory.measure_ne_top", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2875, 9], "def_end_pos": [2875, 23]}, {"full_name": "MeasureTheory.measure_ne_top", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2875, 9], "def_end_pos": [2875, 23]}]], "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj\u2081 j\u2082 : JordanDecomposition \u03b1\nhj : toSignedMeasure j\u2081 = toSignedMeasure j\u2082\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) S \u2264 VectorMeasure.restrict 0 S\nhS\u2083 : VectorMeasure.restrict 0 S\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) S\u1d9c\nhS\u2084 : \u2191\u2191j\u2081.posPart S = 0\nhS\u2085 : \u2191\u2191j\u2081.negPart S\u1d9c = 0\nT : Set \u03b1\nhT\u2081 : MeasurableSet T\nhT\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) T \u2264 VectorMeasure.restrict 0 T\nhT\u2083 : VectorMeasure.restrict 0 T\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) T\u1d9c\nhT\u2084 : \u2191\u2191j\u2082.posPart T = 0\nhT\u2085 : \u2191\u2191j\u2082.negPart T\u1d9c = 0\nhST\u2081 : \u2191(toSignedMeasure j\u2081) (S\u1d9c \u2206 T\u1d9c) = 0\ni : Set \u03b1\nhi : MeasurableSet i\nh\u03bc\u2081 : ENNReal.toReal (\u2191\u2191j\u2081.posPart i) = \u2191(toSignedMeasure j\u2081) (i \u2229 S\u1d9c)\nh\u03bc\u2082 : ENNReal.toReal (\u2191\u2191j\u2082.posPart i) = \u2191(toSignedMeasure j\u2082) (i \u2229 T\u1d9c)\n\u22a2 \u2191\u2191j\u2081.posPart i = \u2191\u2191j\u2082.posPart i", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj\u2081 j\u2082 : JordanDecomposition \u03b1\nhj : toSignedMeasure j\u2081 = toSignedMeasure j\u2082\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) S \u2264 VectorMeasure.restrict 0 S\nhS\u2083 : VectorMeasure.restrict 0 S\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) S\u1d9c\nhS\u2084 : \u2191\u2191j\u2081.posPart S = 0\nhS\u2085 : \u2191\u2191j\u2081.negPart S\u1d9c = 0\nT : Set \u03b1\nhT\u2081 : MeasurableSet T\nhT\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) T \u2264 VectorMeasure.restrict 0 T\nhT\u2083 : VectorMeasure.restrict 0 T\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) T\u1d9c\nhT\u2084 : \u2191\u2191j\u2082.posPart T = 0\nhT\u2085 : \u2191\u2191j\u2082.negPart T\u1d9c = 0\nhST\u2081 : \u2191(toSignedMeasure j\u2081) (S\u1d9c \u2206 T\u1d9c) = 0\ni : Set \u03b1\nhi : MeasurableSet i\nh\u03bc\u2081 : ENNReal.toReal (\u2191\u2191j\u2081.posPart i) = \u2191(toSignedMeasure j\u2081) (i \u2229 S\u1d9c)\nh\u03bc\u2082 : ENNReal.toReal (\u2191\u2191j\u2082.posPart i) = \u2191(toSignedMeasure j\u2082) (i \u2229 T\u1d9c)\n\u22a2 \u2191(toSignedMeasure j\u2081) (i \u2229 S\u1d9c) = \u2191(toSignedMeasure j\u2081) (i \u2229 T\u1d9c)"}, {"tactic": "exact of_inter_eq_of_symmDiff_eq_zero_positive hS\u2081.compl hT\u2081.compl hi hS\u2083 hT\u2083 hST\u2081", "annotated_tactic": ["exact <a>of_inter_eq_of_symmDiff_eq_zero_positive</a> hS\u2081.compl hT\u2081.compl hi hS\u2083 hT\u2083 hST\u2081", [{"full_name": "MeasureTheory.SignedMeasure.of_inter_eq_of_symmDiff_eq_zero_positive", "def_path": "Mathlib/MeasureTheory/Decomposition/Jordan.lean", "def_pos": [326, 9], "def_end_pos": [326, 49]}]], "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj\u2081 j\u2082 : JordanDecomposition \u03b1\nhj : toSignedMeasure j\u2081 = toSignedMeasure j\u2082\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) S \u2264 VectorMeasure.restrict 0 S\nhS\u2083 : VectorMeasure.restrict 0 S\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) S\u1d9c\nhS\u2084 : \u2191\u2191j\u2081.posPart S = 0\nhS\u2085 : \u2191\u2191j\u2081.negPart S\u1d9c = 0\nT : Set \u03b1\nhT\u2081 : MeasurableSet T\nhT\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) T \u2264 VectorMeasure.restrict 0 T\nhT\u2083 : VectorMeasure.restrict 0 T\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) T\u1d9c\nhT\u2084 : \u2191\u2191j\u2082.posPart T = 0\nhT\u2085 : \u2191\u2191j\u2082.negPart T\u1d9c = 0\nhST\u2081 : \u2191(toSignedMeasure j\u2081) (S\u1d9c \u2206 T\u1d9c) = 0\ni : Set \u03b1\nhi : MeasurableSet i\nh\u03bc\u2081 : ENNReal.toReal (\u2191\u2191j\u2081.posPart i) = \u2191(toSignedMeasure j\u2081) (i \u2229 S\u1d9c)\nh\u03bc\u2082 : ENNReal.toReal (\u2191\u2191j\u2082.posPart i) = \u2191(toSignedMeasure j\u2082) (i \u2229 T\u1d9c)\n\u22a2 \u2191(toSignedMeasure j\u2081) (i \u2229 S\u1d9c) = \u2191(toSignedMeasure j\u2081) (i \u2229 T\u1d9c)", "state_after": "no goals"}, {"tactic": "rw [toSignedMeasure, toSignedMeasure_sub_apply (hi.inter hS\u2081.compl),\n  show j\u2081.negPart (i \u2229 S\u1d9c) = 0 from\n    nonpos_iff_eq_zero.1 (hS\u2085 \u25b8 measure_mono (Set.inter_subset_right _ _)),\n  ENNReal.zero_toReal, sub_zero]", "annotated_tactic": ["rw [<a>toSignedMeasure</a>, <a>toSignedMeasure_sub_apply</a> (hi.inter hS\u2081.compl),\n      show j\u2081.negPart (i \u2229 S\u1d9c) = 0 from\n        <a>nonpos_iff_eq_zero</a>.1 (hS\u2085 \u25b8 <a>measure_mono</a> (<a>Set.inter_subset_right</a> _ _)),\n      <a>ENNReal.zero_toReal</a>, <a>sub_zero</a>]", [{"full_name": "MeasureTheory.JordanDecomposition.toSignedMeasure", "def_path": "Mathlib/MeasureTheory/Decomposition/Jordan.lean", "def_pos": [169, 5], "def_end_pos": [169, 20]}, {"full_name": "MeasureTheory.Measure.toSignedMeasure_sub_apply", "def_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "def_pos": [514, 9], "def_end_pos": [514, 34]}, {"full_name": "nonpos_iff_eq_zero", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [237, 3], "def_end_pos": [237, 14]}, {"full_name": "MeasureTheory.measure_mono", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [193, 9], "def_end_pos": [193, 21]}, {"full_name": "Set.inter_subset_right", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [969, 9], "def_end_pos": [969, 27]}, {"full_name": "ENNReal.zero_toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [242, 17], "def_end_pos": [242, 28]}, {"full_name": "sub_zero", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [339, 3], "def_end_pos": [339, 14]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj\u2081 j\u2082 : JordanDecomposition \u03b1\nhj : toSignedMeasure j\u2081 = toSignedMeasure j\u2082\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) S \u2264 VectorMeasure.restrict 0 S\nhS\u2083 : VectorMeasure.restrict 0 S\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) S\u1d9c\nhS\u2084 : \u2191\u2191j\u2081.posPart S = 0\nhS\u2085 : \u2191\u2191j\u2081.negPart S\u1d9c = 0\nT : Set \u03b1\nhT\u2081 : MeasurableSet T\nhT\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) T \u2264 VectorMeasure.restrict 0 T\nhT\u2083 : VectorMeasure.restrict 0 T\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) T\u1d9c\nhT\u2084 : \u2191\u2191j\u2082.posPart T = 0\nhT\u2085 : \u2191\u2191j\u2082.negPart T\u1d9c = 0\nhST\u2081 : \u2191(toSignedMeasure j\u2081) (S\u1d9c \u2206 T\u1d9c) = 0\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 ENNReal.toReal (\u2191\u2191j\u2081.posPart i) = \u2191(toSignedMeasure j\u2081) (i \u2229 S\u1d9c)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj\u2081 j\u2082 : JordanDecomposition \u03b1\nhj : toSignedMeasure j\u2081 = toSignedMeasure j\u2082\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) S \u2264 VectorMeasure.restrict 0 S\nhS\u2083 : VectorMeasure.restrict 0 S\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) S\u1d9c\nhS\u2084 : \u2191\u2191j\u2081.posPart S = 0\nhS\u2085 : \u2191\u2191j\u2081.negPart S\u1d9c = 0\nT : Set \u03b1\nhT\u2081 : MeasurableSet T\nhT\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) T \u2264 VectorMeasure.restrict 0 T\nhT\u2083 : VectorMeasure.restrict 0 T\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) T\u1d9c\nhT\u2084 : \u2191\u2191j\u2082.posPart T = 0\nhT\u2085 : \u2191\u2191j\u2082.negPart T\u1d9c = 0\nhST\u2081 : \u2191(toSignedMeasure j\u2081) (S\u1d9c \u2206 T\u1d9c) = 0\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 ENNReal.toReal (\u2191\u2191j\u2081.posPart i) = ENNReal.toReal (\u2191\u2191j\u2081.posPart (i \u2229 S\u1d9c))"}, {"tactic": "conv_lhs => rw [\u2190 Set.inter_union_compl i S]", "annotated_tactic": ["conv_lhs => rw [\u2190 <a>Set.inter_union_compl</a> i S]", [{"full_name": "Set.inter_union_compl", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1900, 9], "def_end_pos": [1900, 26]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj\u2081 j\u2082 : JordanDecomposition \u03b1\nhj : toSignedMeasure j\u2081 = toSignedMeasure j\u2082\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) S \u2264 VectorMeasure.restrict 0 S\nhS\u2083 : VectorMeasure.restrict 0 S\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) S\u1d9c\nhS\u2084 : \u2191\u2191j\u2081.posPart S = 0\nhS\u2085 : \u2191\u2191j\u2081.negPart S\u1d9c = 0\nT : Set \u03b1\nhT\u2081 : MeasurableSet T\nhT\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) T \u2264 VectorMeasure.restrict 0 T\nhT\u2083 : VectorMeasure.restrict 0 T\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) T\u1d9c\nhT\u2084 : \u2191\u2191j\u2082.posPart T = 0\nhT\u2085 : \u2191\u2191j\u2082.negPart T\u1d9c = 0\nhST\u2081 : \u2191(toSignedMeasure j\u2081) (S\u1d9c \u2206 T\u1d9c) = 0\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 ENNReal.toReal (\u2191\u2191j\u2081.posPart i) = ENNReal.toReal (\u2191\u2191j\u2081.posPart (i \u2229 S\u1d9c))", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj\u2081 j\u2082 : JordanDecomposition \u03b1\nhj : toSignedMeasure j\u2081 = toSignedMeasure j\u2082\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) S \u2264 VectorMeasure.restrict 0 S\nhS\u2083 : VectorMeasure.restrict 0 S\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) S\u1d9c\nhS\u2084 : \u2191\u2191j\u2081.posPart S = 0\nhS\u2085 : \u2191\u2191j\u2081.negPart S\u1d9c = 0\nT : Set \u03b1\nhT\u2081 : MeasurableSet T\nhT\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) T \u2264 VectorMeasure.restrict 0 T\nhT\u2083 : VectorMeasure.restrict 0 T\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) T\u1d9c\nhT\u2084 : \u2191\u2191j\u2082.posPart T = 0\nhT\u2085 : \u2191\u2191j\u2082.negPart T\u1d9c = 0\nhST\u2081 : \u2191(toSignedMeasure j\u2081) (S\u1d9c \u2206 T\u1d9c) = 0\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 ENNReal.toReal (\u2191\u2191j\u2081.posPart (i \u2229 S \u222a i \u2229 S\u1d9c)) = ENNReal.toReal (\u2191\u2191j\u2081.posPart (i \u2229 S\u1d9c))"}, {"tactic": "rw [measure_union,\n  show j\u2081.posPart (i \u2229 S) = 0 from\n    nonpos_iff_eq_zero.1 (hS\u2084 \u25b8 measure_mono (Set.inter_subset_right _ _)),\n  zero_add]", "annotated_tactic": ["rw [<a>measure_union</a>,\n      show j\u2081.posPart (i \u2229 S) = 0 from\n        <a>nonpos_iff_eq_zero</a>.1 (hS\u2084 \u25b8 <a>measure_mono</a> (<a>Set.inter_subset_right</a> _ _)),\n      <a>zero_add</a>]", [{"full_name": "MeasureTheory.measure_union", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [124, 9], "def_end_pos": [124, 22]}, {"full_name": "nonpos_iff_eq_zero", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [237, 3], "def_end_pos": [237, 14]}, {"full_name": "MeasureTheory.measure_mono", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [193, 9], "def_end_pos": [193, 21]}, {"full_name": "Set.inter_subset_right", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [969, 9], "def_end_pos": [969, 27]}, {"full_name": "zero_add", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [463, 3], "def_end_pos": [463, 14]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj\u2081 j\u2082 : JordanDecomposition \u03b1\nhj : toSignedMeasure j\u2081 = toSignedMeasure j\u2082\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) S \u2264 VectorMeasure.restrict 0 S\nhS\u2083 : VectorMeasure.restrict 0 S\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) S\u1d9c\nhS\u2084 : \u2191\u2191j\u2081.posPart S = 0\nhS\u2085 : \u2191\u2191j\u2081.negPart S\u1d9c = 0\nT : Set \u03b1\nhT\u2081 : MeasurableSet T\nhT\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) T \u2264 VectorMeasure.restrict 0 T\nhT\u2083 : VectorMeasure.restrict 0 T\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) T\u1d9c\nhT\u2084 : \u2191\u2191j\u2082.posPart T = 0\nhT\u2085 : \u2191\u2191j\u2082.negPart T\u1d9c = 0\nhST\u2081 : \u2191(toSignedMeasure j\u2081) (S\u1d9c \u2206 T\u1d9c) = 0\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 ENNReal.toReal (\u2191\u2191j\u2081.posPart (i \u2229 S \u222a i \u2229 S\u1d9c)) = ENNReal.toReal (\u2191\u2191j\u2081.posPart (i \u2229 S\u1d9c))", "state_after": "case hd\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj\u2081 j\u2082 : JordanDecomposition \u03b1\nhj : toSignedMeasure j\u2081 = toSignedMeasure j\u2082\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) S \u2264 VectorMeasure.restrict 0 S\nhS\u2083 : VectorMeasure.restrict 0 S\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) S\u1d9c\nhS\u2084 : \u2191\u2191j\u2081.posPart S = 0\nhS\u2085 : \u2191\u2191j\u2081.negPart S\u1d9c = 0\nT : Set \u03b1\nhT\u2081 : MeasurableSet T\nhT\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) T \u2264 VectorMeasure.restrict 0 T\nhT\u2083 : VectorMeasure.restrict 0 T\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) T\u1d9c\nhT\u2084 : \u2191\u2191j\u2082.posPart T = 0\nhT\u2085 : \u2191\u2191j\u2082.negPart T\u1d9c = 0\nhST\u2081 : \u2191(toSignedMeasure j\u2081) (S\u1d9c \u2206 T\u1d9c) = 0\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 Disjoint (i \u2229 S) (i \u2229 S\u1d9c)\n\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj\u2081 j\u2082 : JordanDecomposition \u03b1\nhj : toSignedMeasure j\u2081 = toSignedMeasure j\u2082\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) S \u2264 VectorMeasure.restrict 0 S\nhS\u2083 : VectorMeasure.restrict 0 S\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) S\u1d9c\nhS\u2084 : \u2191\u2191j\u2081.posPart S = 0\nhS\u2085 : \u2191\u2191j\u2081.negPart S\u1d9c = 0\nT : Set \u03b1\nhT\u2081 : MeasurableSet T\nhT\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) T \u2264 VectorMeasure.restrict 0 T\nhT\u2083 : VectorMeasure.restrict 0 T\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) T\u1d9c\nhT\u2084 : \u2191\u2191j\u2082.posPart T = 0\nhT\u2085 : \u2191\u2191j\u2082.negPart T\u1d9c = 0\nhST\u2081 : \u2191(toSignedMeasure j\u2081) (S\u1d9c \u2206 T\u1d9c) = 0\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 MeasurableSet (i \u2229 S\u1d9c)"}, {"tactic": "refine'\n  Set.disjoint_of_subset_left (Set.inter_subset_right _ _)\n    (Set.disjoint_of_subset_right (Set.inter_subset_right _ _) disjoint_compl_right)", "annotated_tactic": ["refine'\n        <a>Set.disjoint_of_subset_left</a> (<a>Set.inter_subset_right</a> _ _)\n          (<a>Set.disjoint_of_subset_right</a> (<a>Set.inter_subset_right</a> _ _) <a>disjoint_compl_right</a>)", [{"full_name": "Set.disjoint_of_subset_left", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1574, 7], "def_end_pos": [1574, 30]}, {"full_name": "Set.inter_subset_right", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [969, 9], "def_end_pos": [969, 27]}, {"full_name": "Set.disjoint_of_subset_right", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1576, 7], "def_end_pos": [1576, 31]}, {"full_name": "Set.inter_subset_right", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [969, 9], "def_end_pos": [969, 27]}, {"full_name": "disjoint_compl_right", "def_path": "Mathlib/Order/Heyting/Basic.lean", "def_pos": [844, 9], "def_end_pos": [844, 29]}]], "state_before": "case hd\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj\u2081 j\u2082 : JordanDecomposition \u03b1\nhj : toSignedMeasure j\u2081 = toSignedMeasure j\u2082\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) S \u2264 VectorMeasure.restrict 0 S\nhS\u2083 : VectorMeasure.restrict 0 S\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) S\u1d9c\nhS\u2084 : \u2191\u2191j\u2081.posPart S = 0\nhS\u2085 : \u2191\u2191j\u2081.negPart S\u1d9c = 0\nT : Set \u03b1\nhT\u2081 : MeasurableSet T\nhT\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) T \u2264 VectorMeasure.restrict 0 T\nhT\u2083 : VectorMeasure.restrict 0 T\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) T\u1d9c\nhT\u2084 : \u2191\u2191j\u2082.posPart T = 0\nhT\u2085 : \u2191\u2191j\u2082.negPart T\u1d9c = 0\nhST\u2081 : \u2191(toSignedMeasure j\u2081) (S\u1d9c \u2206 T\u1d9c) = 0\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 Disjoint (i \u2229 S) (i \u2229 S\u1d9c)", "state_after": "no goals"}, {"tactic": "exact hi.inter hS\u2081.compl", "annotated_tactic": ["exact hi.inter hS\u2081.compl", []], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj\u2081 j\u2082 : JordanDecomposition \u03b1\nhj : toSignedMeasure j\u2081 = toSignedMeasure j\u2082\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) S \u2264 VectorMeasure.restrict 0 S\nhS\u2083 : VectorMeasure.restrict 0 S\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) S\u1d9c\nhS\u2084 : \u2191\u2191j\u2081.posPart S = 0\nhS\u2085 : \u2191\u2191j\u2081.negPart S\u1d9c = 0\nT : Set \u03b1\nhT\u2081 : MeasurableSet T\nhT\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) T \u2264 VectorMeasure.restrict 0 T\nhT\u2083 : VectorMeasure.restrict 0 T\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) T\u1d9c\nhT\u2084 : \u2191\u2191j\u2082.posPart T = 0\nhT\u2085 : \u2191\u2191j\u2082.negPart T\u1d9c = 0\nhST\u2081 : \u2191(toSignedMeasure j\u2081) (S\u1d9c \u2206 T\u1d9c) = 0\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 MeasurableSet (i \u2229 S\u1d9c)", "state_after": "no goals"}, {"tactic": "rw [toSignedMeasure, toSignedMeasure_sub_apply (hi.inter hT\u2081.compl),\n  show j\u2082.negPart (i \u2229 T\u1d9c) = 0 from\n    nonpos_iff_eq_zero.1 (hT\u2085 \u25b8 measure_mono (Set.inter_subset_right _ _)),\n  ENNReal.zero_toReal, sub_zero]", "annotated_tactic": ["rw [<a>toSignedMeasure</a>, <a>toSignedMeasure_sub_apply</a> (hi.inter hT\u2081.compl),\n      show j\u2082.negPart (i \u2229 T\u1d9c) = 0 from\n        <a>nonpos_iff_eq_zero</a>.1 (hT\u2085 \u25b8 <a>measure_mono</a> (<a>Set.inter_subset_right</a> _ _)),\n      <a>ENNReal.zero_toReal</a>, <a>sub_zero</a>]", [{"full_name": "MeasureTheory.JordanDecomposition.toSignedMeasure", "def_path": "Mathlib/MeasureTheory/Decomposition/Jordan.lean", "def_pos": [169, 5], "def_end_pos": [169, 20]}, {"full_name": "MeasureTheory.Measure.toSignedMeasure_sub_apply", "def_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "def_pos": [514, 9], "def_end_pos": [514, 34]}, {"full_name": "nonpos_iff_eq_zero", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [237, 3], "def_end_pos": [237, 14]}, {"full_name": "MeasureTheory.measure_mono", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [193, 9], "def_end_pos": [193, 21]}, {"full_name": "Set.inter_subset_right", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [969, 9], "def_end_pos": [969, 27]}, {"full_name": "ENNReal.zero_toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [242, 17], "def_end_pos": [242, 28]}, {"full_name": "sub_zero", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [339, 3], "def_end_pos": [339, 14]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj\u2081 j\u2082 : JordanDecomposition \u03b1\nhj : toSignedMeasure j\u2081 = toSignedMeasure j\u2082\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) S \u2264 VectorMeasure.restrict 0 S\nhS\u2083 : VectorMeasure.restrict 0 S\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) S\u1d9c\nhS\u2084 : \u2191\u2191j\u2081.posPart S = 0\nhS\u2085 : \u2191\u2191j\u2081.negPart S\u1d9c = 0\nT : Set \u03b1\nhT\u2081 : MeasurableSet T\nhT\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) T \u2264 VectorMeasure.restrict 0 T\nhT\u2083 : VectorMeasure.restrict 0 T\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) T\u1d9c\nhT\u2084 : \u2191\u2191j\u2082.posPart T = 0\nhT\u2085 : \u2191\u2191j\u2082.negPart T\u1d9c = 0\nhST\u2081 : \u2191(toSignedMeasure j\u2081) (S\u1d9c \u2206 T\u1d9c) = 0\ni : Set \u03b1\nhi : MeasurableSet i\nh\u03bc\u2081 : ENNReal.toReal (\u2191\u2191j\u2081.posPart i) = \u2191(toSignedMeasure j\u2081) (i \u2229 S\u1d9c)\n\u22a2 ENNReal.toReal (\u2191\u2191j\u2082.posPart i) = \u2191(toSignedMeasure j\u2082) (i \u2229 T\u1d9c)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj\u2081 j\u2082 : JordanDecomposition \u03b1\nhj : toSignedMeasure j\u2081 = toSignedMeasure j\u2082\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) S \u2264 VectorMeasure.restrict 0 S\nhS\u2083 : VectorMeasure.restrict 0 S\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) S\u1d9c\nhS\u2084 : \u2191\u2191j\u2081.posPart S = 0\nhS\u2085 : \u2191\u2191j\u2081.negPart S\u1d9c = 0\nT : Set \u03b1\nhT\u2081 : MeasurableSet T\nhT\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) T \u2264 VectorMeasure.restrict 0 T\nhT\u2083 : VectorMeasure.restrict 0 T\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) T\u1d9c\nhT\u2084 : \u2191\u2191j\u2082.posPart T = 0\nhT\u2085 : \u2191\u2191j\u2082.negPart T\u1d9c = 0\nhST\u2081 : \u2191(toSignedMeasure j\u2081) (S\u1d9c \u2206 T\u1d9c) = 0\ni : Set \u03b1\nhi : MeasurableSet i\nh\u03bc\u2081 : ENNReal.toReal (\u2191\u2191j\u2081.posPart i) = \u2191(toSignedMeasure j\u2081) (i \u2229 S\u1d9c)\n\u22a2 ENNReal.toReal (\u2191\u2191j\u2082.posPart i) = ENNReal.toReal (\u2191\u2191j\u2082.posPart (i \u2229 T\u1d9c))"}, {"tactic": "conv_lhs => rw [\u2190 Set.inter_union_compl i T]", "annotated_tactic": ["conv_lhs => rw [\u2190 <a>Set.inter_union_compl</a> i T]", [{"full_name": "Set.inter_union_compl", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1900, 9], "def_end_pos": [1900, 26]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj\u2081 j\u2082 : JordanDecomposition \u03b1\nhj : toSignedMeasure j\u2081 = toSignedMeasure j\u2082\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) S \u2264 VectorMeasure.restrict 0 S\nhS\u2083 : VectorMeasure.restrict 0 S\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) S\u1d9c\nhS\u2084 : \u2191\u2191j\u2081.posPart S = 0\nhS\u2085 : \u2191\u2191j\u2081.negPart S\u1d9c = 0\nT : Set \u03b1\nhT\u2081 : MeasurableSet T\nhT\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) T \u2264 VectorMeasure.restrict 0 T\nhT\u2083 : VectorMeasure.restrict 0 T\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) T\u1d9c\nhT\u2084 : \u2191\u2191j\u2082.posPart T = 0\nhT\u2085 : \u2191\u2191j\u2082.negPart T\u1d9c = 0\nhST\u2081 : \u2191(toSignedMeasure j\u2081) (S\u1d9c \u2206 T\u1d9c) = 0\ni : Set \u03b1\nhi : MeasurableSet i\nh\u03bc\u2081 : ENNReal.toReal (\u2191\u2191j\u2081.posPart i) = \u2191(toSignedMeasure j\u2081) (i \u2229 S\u1d9c)\n\u22a2 ENNReal.toReal (\u2191\u2191j\u2082.posPart i) = ENNReal.toReal (\u2191\u2191j\u2082.posPart (i \u2229 T\u1d9c))", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj\u2081 j\u2082 : JordanDecomposition \u03b1\nhj : toSignedMeasure j\u2081 = toSignedMeasure j\u2082\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) S \u2264 VectorMeasure.restrict 0 S\nhS\u2083 : VectorMeasure.restrict 0 S\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) S\u1d9c\nhS\u2084 : \u2191\u2191j\u2081.posPart S = 0\nhS\u2085 : \u2191\u2191j\u2081.negPart S\u1d9c = 0\nT : Set \u03b1\nhT\u2081 : MeasurableSet T\nhT\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) T \u2264 VectorMeasure.restrict 0 T\nhT\u2083 : VectorMeasure.restrict 0 T\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) T\u1d9c\nhT\u2084 : \u2191\u2191j\u2082.posPart T = 0\nhT\u2085 : \u2191\u2191j\u2082.negPart T\u1d9c = 0\nhST\u2081 : \u2191(toSignedMeasure j\u2081) (S\u1d9c \u2206 T\u1d9c) = 0\ni : Set \u03b1\nhi : MeasurableSet i\nh\u03bc\u2081 : ENNReal.toReal (\u2191\u2191j\u2081.posPart i) = \u2191(toSignedMeasure j\u2081) (i \u2229 S\u1d9c)\n\u22a2 ENNReal.toReal (\u2191\u2191j\u2082.posPart (i \u2229 T \u222a i \u2229 T\u1d9c)) = ENNReal.toReal (\u2191\u2191j\u2082.posPart (i \u2229 T\u1d9c))"}, {"tactic": "rw [measure_union,\n  show j\u2082.posPart (i \u2229 T) = 0 from\n    nonpos_iff_eq_zero.1 (hT\u2084 \u25b8 measure_mono (Set.inter_subset_right _ _)),\n  zero_add]", "annotated_tactic": ["rw [<a>measure_union</a>,\n      show j\u2082.posPart (i \u2229 T) = 0 from\n        <a>nonpos_iff_eq_zero</a>.1 (hT\u2084 \u25b8 <a>measure_mono</a> (<a>Set.inter_subset_right</a> _ _)),\n      <a>zero_add</a>]", [{"full_name": "MeasureTheory.measure_union", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [124, 9], "def_end_pos": [124, 22]}, {"full_name": "nonpos_iff_eq_zero", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [237, 3], "def_end_pos": [237, 14]}, {"full_name": "MeasureTheory.measure_mono", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [193, 9], "def_end_pos": [193, 21]}, {"full_name": "Set.inter_subset_right", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [969, 9], "def_end_pos": [969, 27]}, {"full_name": "zero_add", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [463, 3], "def_end_pos": [463, 14]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj\u2081 j\u2082 : JordanDecomposition \u03b1\nhj : toSignedMeasure j\u2081 = toSignedMeasure j\u2082\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) S \u2264 VectorMeasure.restrict 0 S\nhS\u2083 : VectorMeasure.restrict 0 S\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) S\u1d9c\nhS\u2084 : \u2191\u2191j\u2081.posPart S = 0\nhS\u2085 : \u2191\u2191j\u2081.negPart S\u1d9c = 0\nT : Set \u03b1\nhT\u2081 : MeasurableSet T\nhT\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) T \u2264 VectorMeasure.restrict 0 T\nhT\u2083 : VectorMeasure.restrict 0 T\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) T\u1d9c\nhT\u2084 : \u2191\u2191j\u2082.posPart T = 0\nhT\u2085 : \u2191\u2191j\u2082.negPart T\u1d9c = 0\nhST\u2081 : \u2191(toSignedMeasure j\u2081) (S\u1d9c \u2206 T\u1d9c) = 0\ni : Set \u03b1\nhi : MeasurableSet i\nh\u03bc\u2081 : ENNReal.toReal (\u2191\u2191j\u2081.posPart i) = \u2191(toSignedMeasure j\u2081) (i \u2229 S\u1d9c)\n\u22a2 ENNReal.toReal (\u2191\u2191j\u2082.posPart (i \u2229 T \u222a i \u2229 T\u1d9c)) = ENNReal.toReal (\u2191\u2191j\u2082.posPart (i \u2229 T\u1d9c))", "state_after": "case hd\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj\u2081 j\u2082 : JordanDecomposition \u03b1\nhj : toSignedMeasure j\u2081 = toSignedMeasure j\u2082\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) S \u2264 VectorMeasure.restrict 0 S\nhS\u2083 : VectorMeasure.restrict 0 S\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) S\u1d9c\nhS\u2084 : \u2191\u2191j\u2081.posPart S = 0\nhS\u2085 : \u2191\u2191j\u2081.negPart S\u1d9c = 0\nT : Set \u03b1\nhT\u2081 : MeasurableSet T\nhT\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) T \u2264 VectorMeasure.restrict 0 T\nhT\u2083 : VectorMeasure.restrict 0 T\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) T\u1d9c\nhT\u2084 : \u2191\u2191j\u2082.posPart T = 0\nhT\u2085 : \u2191\u2191j\u2082.negPart T\u1d9c = 0\nhST\u2081 : \u2191(toSignedMeasure j\u2081) (S\u1d9c \u2206 T\u1d9c) = 0\ni : Set \u03b1\nhi : MeasurableSet i\nh\u03bc\u2081 : ENNReal.toReal (\u2191\u2191j\u2081.posPart i) = \u2191(toSignedMeasure j\u2081) (i \u2229 S\u1d9c)\n\u22a2 Disjoint (i \u2229 T) (i \u2229 T\u1d9c)\n\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj\u2081 j\u2082 : JordanDecomposition \u03b1\nhj : toSignedMeasure j\u2081 = toSignedMeasure j\u2082\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) S \u2264 VectorMeasure.restrict 0 S\nhS\u2083 : VectorMeasure.restrict 0 S\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) S\u1d9c\nhS\u2084 : \u2191\u2191j\u2081.posPart S = 0\nhS\u2085 : \u2191\u2191j\u2081.negPart S\u1d9c = 0\nT : Set \u03b1\nhT\u2081 : MeasurableSet T\nhT\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) T \u2264 VectorMeasure.restrict 0 T\nhT\u2083 : VectorMeasure.restrict 0 T\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) T\u1d9c\nhT\u2084 : \u2191\u2191j\u2082.posPart T = 0\nhT\u2085 : \u2191\u2191j\u2082.negPart T\u1d9c = 0\nhST\u2081 : \u2191(toSignedMeasure j\u2081) (S\u1d9c \u2206 T\u1d9c) = 0\ni : Set \u03b1\nhi : MeasurableSet i\nh\u03bc\u2081 : ENNReal.toReal (\u2191\u2191j\u2081.posPart i) = \u2191(toSignedMeasure j\u2081) (i \u2229 S\u1d9c)\n\u22a2 MeasurableSet (i \u2229 T\u1d9c)"}, {"tactic": "exact\n  Set.disjoint_of_subset_left (Set.inter_subset_right _ _)\n    (Set.disjoint_of_subset_right (Set.inter_subset_right _ _) disjoint_compl_right)", "annotated_tactic": ["exact\n        <a>Set.disjoint_of_subset_left</a> (<a>Set.inter_subset_right</a> _ _)\n          (<a>Set.disjoint_of_subset_right</a> (<a>Set.inter_subset_right</a> _ _) <a>disjoint_compl_right</a>)", [{"full_name": "Set.disjoint_of_subset_left", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1574, 7], "def_end_pos": [1574, 30]}, {"full_name": "Set.inter_subset_right", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [969, 9], "def_end_pos": [969, 27]}, {"full_name": "Set.disjoint_of_subset_right", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1576, 7], "def_end_pos": [1576, 31]}, {"full_name": "Set.inter_subset_right", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [969, 9], "def_end_pos": [969, 27]}, {"full_name": "disjoint_compl_right", "def_path": "Mathlib/Order/Heyting/Basic.lean", "def_pos": [844, 9], "def_end_pos": [844, 29]}]], "state_before": "case hd\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj\u2081 j\u2082 : JordanDecomposition \u03b1\nhj : toSignedMeasure j\u2081 = toSignedMeasure j\u2082\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) S \u2264 VectorMeasure.restrict 0 S\nhS\u2083 : VectorMeasure.restrict 0 S\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) S\u1d9c\nhS\u2084 : \u2191\u2191j\u2081.posPart S = 0\nhS\u2085 : \u2191\u2191j\u2081.negPart S\u1d9c = 0\nT : Set \u03b1\nhT\u2081 : MeasurableSet T\nhT\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) T \u2264 VectorMeasure.restrict 0 T\nhT\u2083 : VectorMeasure.restrict 0 T\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) T\u1d9c\nhT\u2084 : \u2191\u2191j\u2082.posPart T = 0\nhT\u2085 : \u2191\u2191j\u2082.negPart T\u1d9c = 0\nhST\u2081 : \u2191(toSignedMeasure j\u2081) (S\u1d9c \u2206 T\u1d9c) = 0\ni : Set \u03b1\nhi : MeasurableSet i\nh\u03bc\u2081 : ENNReal.toReal (\u2191\u2191j\u2081.posPart i) = \u2191(toSignedMeasure j\u2081) (i \u2229 S\u1d9c)\n\u22a2 Disjoint (i \u2229 T) (i \u2229 T\u1d9c)", "state_after": "no goals"}, {"tactic": "exact hi.inter hT\u2081.compl", "annotated_tactic": ["exact hi.inter hT\u2081.compl", []], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj\u2081 j\u2082 : JordanDecomposition \u03b1\nhj : toSignedMeasure j\u2081 = toSignedMeasure j\u2082\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) S \u2264 VectorMeasure.restrict 0 S\nhS\u2083 : VectorMeasure.restrict 0 S\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) S\u1d9c\nhS\u2084 : \u2191\u2191j\u2081.posPart S = 0\nhS\u2085 : \u2191\u2191j\u2081.negPart S\u1d9c = 0\nT : Set \u03b1\nhT\u2081 : MeasurableSet T\nhT\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) T \u2264 VectorMeasure.restrict 0 T\nhT\u2083 : VectorMeasure.restrict 0 T\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) T\u1d9c\nhT\u2084 : \u2191\u2191j\u2082.posPart T = 0\nhT\u2085 : \u2191\u2191j\u2082.negPart T\u1d9c = 0\nhST\u2081 : \u2191(toSignedMeasure j\u2081) (S\u1d9c \u2206 T\u1d9c) = 0\ni : Set \u03b1\nhi : MeasurableSet i\nh\u03bc\u2081 : ENNReal.toReal (\u2191\u2191j\u2081.posPart i) = \u2191(toSignedMeasure j\u2081) (i \u2229 S\u1d9c)\n\u22a2 MeasurableSet (i \u2229 T\u1d9c)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "full_name": "MeasureTheory.union_ae_eq_univ_of_ae_eq_univ_right", "start": [530, 1], "end": [532, 18], "traced_tactics": [{"tactic": "convert ae_eq_set_union (ae_eq_refl s) h", "annotated_tactic": ["convert <a>ae_eq_set_union</a> (<a>ae_eq_refl</a> s) h", [{"full_name": "MeasureTheory.ae_eq_set_union", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [520, 9], "def_end_pos": [520, 24]}, {"full_name": "MeasureTheory.ae_eq_refl", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [436, 9], "def_end_pos": [436, 19]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\ninst\u271d : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns s\u2081 s\u2082 t : Set \u03b1\nh : t =\u1d50[\u03bc] univ\n\u22a2 s \u222a t =\u1d50[\u03bc] univ", "state_after": "case h.e'_5\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\ninst\u271d : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns s\u2081 s\u2082 t : Set \u03b1\nh : t =\u1d50[\u03bc] univ\n\u22a2 univ = s \u222a univ"}, {"tactic": "rw [union_univ]", "annotated_tactic": ["rw [<a>union_univ</a>]", [{"full_name": "Set.union_univ", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [892, 9], "def_end_pos": [892, 19]}]], "state_before": "case h.e'_5\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\ninst\u271d : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns s\u2081 s\u2082 t : Set \u03b1\nh : t =\u1d50[\u03bc] univ\n\u22a2 univ = s \u222a univ", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "full_name": "MeasureTheory.mem_\u21121_toReal_of_lintegral_ne_top", "start": [1036, 1], "end": [1041, 56], "traced_tactics": [{"tactic": "rw [Mem\u2112p, snorm_one_eq_lintegral_nnnorm]", "annotated_tactic": ["rw [<a>Mem\u2112p</a>, <a>snorm_one_eq_lintegral_nnnorm</a>]", [{"full_name": "MeasureTheory.Mem\u2112p", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [108, 5], "def_end_pos": [108, 10]}, {"full_name": "MeasureTheory.snorm_one_eq_lintegral_nnnorm", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [97, 9], "def_end_pos": [97, 38]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : NormedAddCommGroup \u03b3\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhfm : AEMeasurable f\nhfi : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\n\u22a2 Mem\u2112p (fun x => ENNReal.toReal (f x)) 1", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : NormedAddCommGroup \u03b3\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhfm : AEMeasurable f\nhfi : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\n\u22a2 AEStronglyMeasurable (fun x => ENNReal.toReal (f x)) \u03bc \u2227 \u222b\u207b (x : \u03b1), \u2191\u2016ENNReal.toReal (f x)\u2016\u208a \u2202\u03bc < \u22a4"}, {"tactic": "exact\n  \u27e8(AEMeasurable.ennreal_toReal hfm).aestronglyMeasurable,\n    hasFiniteIntegral_toReal_of_lintegral_ne_top hfi\u27e9", "annotated_tactic": ["exact\n    \u27e8(<a>AEMeasurable.ennreal_toReal</a> hfm).<a>aestronglyMeasurable</a>,\n      <a>hasFiniteIntegral_toReal_of_lintegral_ne_top</a> hfi\u27e9", [{"full_name": "AEMeasurable.ennreal_toReal", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [2129, 9], "def_end_pos": [2129, 36]}, {"full_name": "AEMeasurable.aestronglyMeasurable", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1451, 9], "def_end_pos": [1451, 49]}, {"full_name": "MeasureTheory.hasFiniteIntegral_toReal_of_lintegral_ne_top", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [266, 9], "def_end_pos": [266, 53]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : NormedAddCommGroup \u03b3\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhfm : AEMeasurable f\nhfi : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\n\u22a2 AEStronglyMeasurable (fun x => ENNReal.toReal (f x)) \u03bc \u2227 \u222b\u207b (x : \u03b1), \u2191\u2016ENNReal.toReal (f x)\u2016\u208a \u2202\u03bc < \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Intervals/ProjIcc.lean", "full_name": "Monotone.IciExtend", "start": [317, 11], "end": [318, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "full_name": "MeasureTheory.SimpleFunc.integral_eq_integral", "start": [1396, 1], "end": [1400, 81], "traced_tactics": [{"tactic": "rw [MeasureTheory.integral_eq f hfi, \u2190 L1.SimpleFunc.toLp_one_eq_toL1,\n  L1.SimpleFunc.integral_L1_eq_integral, L1.SimpleFunc.integral_eq_integral]", "annotated_tactic": ["rw [<a>MeasureTheory.integral_eq</a> f hfi, \u2190 <a>L1.SimpleFunc.toLp_one_eq_toL1</a>,\n    <a>L1.SimpleFunc.integral_L1_eq_integral</a>, <a>L1.SimpleFunc.integral_eq_integral</a>]", [{"full_name": "MeasureTheory.integral_eq", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [821, 9], "def_end_pos": [821, 20]}, {"full_name": "MeasureTheory.L1.SimpleFunc.toLp_one_eq_toL1", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "def_pos": [1035, 9], "def_end_pos": [1035, 39]}, {"full_name": "MeasureTheory.L1.SimpleFunc.integral_L1_eq_integral", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [680, 9], "def_end_pos": [680, 43]}, {"full_name": "MeasureTheory.L1.SimpleFunc.integral_eq_integral", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [522, 9], "def_end_pos": [522, 29]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nf : \u03b1 \u2192\u209b E\nhfi : Integrable \u2191f\n\u22a2 integral \u03bc f = \u222b (x : \u03b1), \u2191f x \u2202\u03bc", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nf : \u03b1 \u2192\u209b E\nhfi : Integrable \u2191f\n\u22a2 integral \u03bc f = integral \u03bc (Lp.simpleFunc.toSimpleFunc (Lp.simpleFunc.toLp f (_ : Mem\u2112p (\u2191f) 1)))"}, {"tactic": "exact SimpleFunc.integral_congr hfi (Lp.simpleFunc.toSimpleFunc_toLp _ _).symm", "annotated_tactic": ["exact <a>SimpleFunc.integral_congr</a> hfi (<a>Lp.simpleFunc.toSimpleFunc_toLp</a> _ _).<a>symm</a>", [{"full_name": "MeasureTheory.SimpleFunc.integral_congr", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [398, 9], "def_end_pos": [398, 23]}, {"full_name": "MeasureTheory.Lp.simpleFunc.toSimpleFunc_toLp", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "def_pos": [631, 9], "def_end_pos": [631, 26]}, {"full_name": "Filter.EventuallyEq.symm", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1498, 9], "def_end_pos": [1498, 26]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nf : \u03b1 \u2192\u209b E\nhfi : Integrable \u2191f\n\u22a2 integral \u03bc f = integral \u03bc (Lp.simpleFunc.toSimpleFunc (Lp.simpleFunc.toLp f (_ : Mem\u2112p (\u2191f) 1)))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/PEquiv.lean", "full_name": "PEquiv.ofSet_eq_refl", "start": [255, 1], "end": [261, 56], "traced_tactics": [{"tactic": "rw [Set.eq_univ_iff_forall]", "annotated_tactic": ["rw [<a>Set.eq_univ_iff_forall</a>]", [{"full_name": "Set.eq_univ_iff_forall", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [701, 9], "def_end_pos": [701, 27]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\ns\u271d : Set \u03b1\ninst\u271d\u00b9 : DecidablePred fun x => x \u2208 s\u271d\ns : Set \u03b1\ninst\u271d : DecidablePred fun x => x \u2208 s\nh : ofSet s = PEquiv.refl \u03b1\n\u22a2 s = Set.univ", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\ns\u271d : Set \u03b1\ninst\u271d\u00b9 : DecidablePred fun x => x \u2208 s\u271d\ns : Set \u03b1\ninst\u271d : DecidablePred fun x => x \u2208 s\nh : ofSet s = PEquiv.refl \u03b1\n\u22a2 \u2200 (x : \u03b1), x \u2208 s"}, {"tactic": "intro", "annotated_tactic": ["intro", []], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\ns\u271d : Set \u03b1\ninst\u271d\u00b9 : DecidablePred fun x => x \u2208 s\u271d\ns : Set \u03b1\ninst\u271d : DecidablePred fun x => x \u2208 s\nh : ofSet s = PEquiv.refl \u03b1\n\u22a2 \u2200 (x : \u03b1), x \u2208 s", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\ns\u271d : Set \u03b1\ninst\u271d\u00b9 : DecidablePred fun x => x \u2208 s\u271d\ns : Set \u03b1\ninst\u271d : DecidablePred fun x => x \u2208 s\nh : ofSet s = PEquiv.refl \u03b1\nx\u271d : \u03b1\n\u22a2 x\u271d \u2208 s"}, {"tactic": "rw [\u2190 mem_ofSet_self_iff, h]", "annotated_tactic": ["rw [\u2190 <a>mem_ofSet_self_iff</a>, h]", [{"full_name": "PEquiv.mem_ofSet_self_iff", "def_path": "Mathlib/Data/PEquiv.lean", "def_pos": [217, 9], "def_end_pos": [217, 27]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\ns\u271d : Set \u03b1\ninst\u271d\u00b9 : DecidablePred fun x => x \u2208 s\u271d\ns : Set \u03b1\ninst\u271d : DecidablePred fun x => x \u2208 s\nh : ofSet s = PEquiv.refl \u03b1\nx\u271d : \u03b1\n\u22a2 x\u271d \u2208 s", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\ns\u271d : Set \u03b1\ninst\u271d\u00b9 : DecidablePred fun x => x \u2208 s\u271d\ns : Set \u03b1\ninst\u271d : DecidablePred fun x => x \u2208 s\nh : ofSet s = PEquiv.refl \u03b1\nx\u271d : \u03b1\n\u22a2 x\u271d \u2208 \u2191(PEquiv.refl \u03b1) x\u271d"}, {"tactic": "exact rfl", "annotated_tactic": ["exact <a>rfl</a>", [{"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\ns\u271d : Set \u03b1\ninst\u271d\u00b9 : DecidablePred fun x => x \u2208 s\u271d\ns : Set \u03b1\ninst\u271d : DecidablePred fun x => x \u2208 s\nh : ofSet s = PEquiv.refl \u03b1\nx\u271d : \u03b1\n\u22a2 x\u271d \u2208 \u2191(PEquiv.refl \u03b1) x\u271d", "state_after": "no goals"}, {"tactic": "simp only [\u2190 ofSet_univ, h]", "annotated_tactic": ["simp only [\u2190 <a>ofSet_univ</a>, h]", [{"full_name": "PEquiv.ofSet_univ", "def_path": "Mathlib/Data/PEquiv.lean", "def_pos": [250, 9], "def_end_pos": [250, 19]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\ns\u271d : Set \u03b1\ninst\u271d\u00b9 : DecidablePred fun x => x \u2208 s\u271d\ns : Set \u03b1\ninst\u271d : DecidablePred fun x => x \u2208 s\nh : s = Set.univ\n\u22a2 ofSet s = PEquiv.refl \u03b1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "full_name": "Int.mul_div_cancel", "start": [200, 19], "end": [211, 66], "traced_tactics": [{"tactic": "rw [\u2190 ofNat_mul, \u2190 ofNat_div,\n  Nat.mul_div_cancel _ <| Nat.pos_of_ne_zero <| Int.ofNat_ne_zero.1 H]", "annotated_tactic": ["rw [\u2190 <a>ofNat_mul</a>, \u2190 <a>ofNat_div</a>,\n      <a>Nat.mul_div_cancel</a> _ <| <a>Nat.pos_of_ne_zero</a> <| <a>Int.ofNat_ne_zero</a>.1 H]", [{"full_name": "Int.ofNat_mul", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [36, 9], "def_end_pos": [36, 18]}, {"full_name": "Int.ofNat_div", "def_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "def_pos": [19, 9], "def_end_pos": [19, 18]}, {"full_name": "Nat.mul_div_cancel", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [641, 19], "def_end_pos": [641, 33]}, {"full_name": "Nat.pos_of_ne_zero", "def_path": "lake-packages/std/Std/Data/Nat/Init/Lemmas.lean", "def_pos": [25, 19], "def_end_pos": [25, 33]}, {"full_name": "Int.ofNat_ne_zero", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [70, 9], "def_end_pos": [70, 22]}]], "state_before": "a b : Int\nH\u271d : b \u2260 0\na\u271d b\u271d : Nat\nH : \u2191b\u271d \u2260 0\n\u22a2 div (\u2191a\u271d * \u2191b\u271d) \u2191b\u271d = \u2191a\u271d", "state_after": "no goals"}, {"tactic": "rw [Int.mul_neg, Int.neg_div, Int.div_neg, Int.neg_neg,\n  this (Int.neg_ne_zero.1 H)]", "annotated_tactic": ["rw [<a>Int.mul_neg</a>, <a>Int.neg_div</a>, <a>Int.div_neg</a>, <a>Int.neg_neg</a>,\n      this (<a>Int.neg_ne_zero</a>.1 H)]", [{"full_name": "Int.mul_neg", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [471, 33], "def_end_pos": [471, 40]}, {"full_name": "Int.neg_div", "def_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "def_pos": [84, 27], "def_end_pos": [84, 34]}, {"full_name": "Int.div_neg", "def_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "def_pos": [66, 27], "def_end_pos": [66, 34]}, {"full_name": "Int.neg_neg", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [82, 27], "def_end_pos": [82, 34]}, {"full_name": "Int.neg_ne_zero", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [92, 19], "def_end_pos": [92, 30]}]], "state_before": "a\u271d b\u271d : Int\nthis : \u2200 {a b : Nat}, \u2191b \u2260 0 \u2192 div (\u2191a * \u2191b) \u2191b = \u2191a\na b : Nat\nH : -\u2191b \u2260 0\n\u22a2 div (\u2191a * -\u2191b) (-\u2191b) = \u2191a", "state_after": "no goals"}, {"tactic": "rw [Int.neg_mul, Int.neg_div, this H]", "annotated_tactic": ["rw [<a>Int.neg_mul</a>, <a>Int.neg_div</a>, this H]", [{"full_name": "Int.neg_mul", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [468, 33], "def_end_pos": [468, 40]}, {"full_name": "Int.neg_div", "def_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "def_pos": [84, 27], "def_end_pos": [84, 34]}]], "state_before": "a\u271d b\u271d : Int\nthis : \u2200 {a b : Nat}, \u2191b \u2260 0 \u2192 div (\u2191a * \u2191b) \u2191b = \u2191a\na b : Nat\nH : \u2191b \u2260 0\n\u22a2 div (-\u2191a * \u2191b) \u2191b = -\u2191a", "state_after": "no goals"}, {"tactic": "rw [Int.neg_mul_neg, Int.div_neg, this (Int.neg_ne_zero.1 H)]", "annotated_tactic": ["rw [<a>Int.neg_mul_neg</a>, <a>Int.div_neg</a>, this (<a>Int.neg_ne_zero</a>.1 H)]", [{"full_name": "Int.neg_mul_neg", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [474, 19], "def_end_pos": [474, 30]}, {"full_name": "Int.div_neg", "def_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "def_pos": [66, 27], "def_end_pos": [66, 34]}, {"full_name": "Int.neg_ne_zero", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [92, 19], "def_end_pos": [92, 30]}]], "state_before": "a\u271d b\u271d : Int\nthis : \u2200 {a b : Nat}, \u2191b \u2260 0 \u2192 div (\u2191a * \u2191b) \u2191b = \u2191a\na b : Nat\nH : -\u2191b \u2260 0\n\u22a2 div (-\u2191a * -\u2191b) (-\u2191b) = -\u2191a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/HashMap/Basic.lean", "full_name": "Std.HashMap.Imp.Buckets.update_size", "start": [42, 9], "end": [43, 67], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Option/Lemmas.lean", "full_name": "Option.map_eq_none", "start": [134, 1], "end": [134, 64], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/Expand.lean", "full_name": "MvPolynomial.expand_one_apply", "start": [55, 1], "end": [57, 81], "traced_tactics": [{"tactic": "simp only [expand, pow_one, eval\u2082Hom_eq_bind\u2082, bind\u2082_C_left, RingHom.toMonoidHom_eq_coe,\n  RingHom.coe_monoidHom_id, AlgHom.coe_mk, RingHom.coe_mk, MonoidHom.id_apply]", "annotated_tactic": ["simp only [<a>expand</a>, <a>pow_one</a>, <a>eval\u2082Hom_eq_bind\u2082</a>, <a>bind\u2082_C_left</a>, <a>RingHom.toMonoidHom_eq_coe</a>,\n    <a>RingHom.coe_monoidHom_id</a>, <a>AlgHom.coe_mk</a>, <a>RingHom.coe_mk</a>, <a>MonoidHom.id_apply</a>]", [{"full_name": "MvPolynomial.expand", "def_path": "Mathlib/Data/MvPolynomial/Expand.lean", "def_pos": [32, 19], "def_end_pos": [32, 25]}, {"full_name": "pow_one", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [97, 9], "def_end_pos": [97, 16]}, {"full_name": "MvPolynomial.eval\u2082Hom_eq_bind\u2082", "def_path": "Mathlib/Data/MvPolynomial/Monad.lean", "def_pos": [117, 9], "def_end_pos": [117, 26]}, {"full_name": "MvPolynomial.bind\u2082_C_left", "def_path": "Mathlib/Data/MvPolynomial/Monad.lean", "def_pos": [184, 9], "def_end_pos": [184, 21]}, {"full_name": "RingHom.toMonoidHom_eq_coe", "def_path": "Mathlib/Algebra/Hom/Ring/Defs.lean", "def_pos": [468, 9], "def_end_pos": [468, 27]}, {"full_name": "RingHom.coe_monoidHom_id", "def_path": "Mathlib/Algebra/Hom/Ring/Defs.lean", "def_pos": [650, 9], "def_end_pos": [650, 25]}, {"full_name": "AlgHom.coe_mk", "def_path": "Mathlib/Algebra/Algebra/Hom.lean", "def_pos": [157, 9], "def_end_pos": [157, 15]}, {"full_name": "RingHom.coe_mk", "def_path": "Mathlib/Algebra/Hom/Ring/Defs.lean", "def_pos": [449, 9], "def_end_pos": [449, 15]}, {"full_name": "MonoidHom.id_apply", "def_path": "Mathlib/Algebra/Hom/Group/Defs.lean", "def_pos": [1000, 24], "def_end_pos": [1000, 29]}]], "state_before": "\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : CommSemiring S\nf : MvPolynomial \u03c3 R\n\u22a2 \u2191(expand 1) f = f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Intervals/WithBotTop.lean", "full_name": "WithTop.image_coe_Ici", "start": [93, 1], "end": [94, 83], "traced_tactics": [{"tactic": "rw [\u2190 preimage_coe_Ici, image_preimage_eq_inter_range, range_coe, Ici_inter_Iio]", "annotated_tactic": ["rw [\u2190 <a>preimage_coe_Ici</a>, <a>image_preimage_eq_inter_range</a>, <a>range_coe</a>, <a>Ici_inter_Iio</a>]", [{"full_name": "WithTop.preimage_coe_Ici", "def_path": "Mathlib/Data/Set/Intervals/WithBotTop.lean", "def_pos": [44, 9], "def_end_pos": [44, 25]}, {"full_name": "Set.image_preimage_eq_inter_range", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [796, 9], "def_end_pos": [796, 38]}, {"full_name": "WithTop.range_coe", "def_path": "Mathlib/Data/Set/Intervals/WithBotTop.lean", "def_pos": [32, 9], "def_end_pos": [32, 18]}, {"full_name": "Set.Ici_inter_Iio", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [622, 9], "def_end_pos": [622, 22]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : PartialOrder \u03b1\na b : \u03b1\n\u22a2 some '' Ici a = Ico \u2191a \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "full_name": "MeasureTheory.OuterMeasure.iUnion_eq_of_caratheodory", "start": [1052, 11], "end": [1055, 18], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/Rename.lean", "full_name": "MvPolynomial.eval_rename_prod_mk", "start": [220, 1], "end": [222, 44], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/ProbabilityMassFunction/Basic.lean", "full_name": "PMF.toOuterMeasure_caratheodory", "start": [166, 1], "end": [170, 98], "traced_tactics": [{"tactic": "refine' eq_top_iff.2 <| le_trans (le_sInf fun x hx => _) (le_sum_caratheodory _)", "annotated_tactic": ["refine' <a>eq_top_iff</a>.2 <| <a>le_trans</a> (<a>le_sInf</a> fun x hx => _) (<a>le_sum_caratheodory</a> _)", [{"full_name": "eq_top_iff", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [165, 9], "def_end_pos": [165, 19]}, {"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}, {"full_name": "le_sInf", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [269, 9], "def_end_pos": [269, 16]}, {"full_name": "MeasureTheory.OuterMeasure.le_sum_caratheodory", "def_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "def_pos": [1107, 9], "def_end_pos": [1107, 28]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\ns t : Set \u03b1\n\u22a2 OuterMeasure.caratheodory (toOuterMeasure p) = \u22a4", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\ns t : Set \u03b1\nx : MeasurableSpace \u03b1\nhx : x \u2208 Set.range fun i => OuterMeasure.caratheodory (\u2191p i \u2022 dirac i)\n\u22a2 \u22a4 \u2264 x"}, {"tactic": "have \u27e8y, hy\u27e9 := hx", "annotated_tactic": ["have \u27e8y, hy\u27e9 := hx", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\ns t : Set \u03b1\nx : MeasurableSpace \u03b1\nhx : x \u2208 Set.range fun i => OuterMeasure.caratheodory (\u2191p i \u2022 dirac i)\n\u22a2 \u22a4 \u2264 x", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\ns t : Set \u03b1\nx : MeasurableSpace \u03b1\nhx : x \u2208 Set.range fun i => OuterMeasure.caratheodory (\u2191p i \u2022 dirac i)\ny : \u03b1\nhy : (fun i => OuterMeasure.caratheodory (\u2191p i \u2022 dirac i)) y = x\n\u22a2 \u22a4 \u2264 x"}, {"tactic": "exact\n  ((le_of_eq (dirac_caratheodory y).symm).trans (le_smul_caratheodory _ _)).trans (le_of_eq hy)", "annotated_tactic": ["exact\n    ((<a>le_of_eq</a> (<a>dirac_caratheodory</a> y).<a>symm</a>).<a>trans</a> (<a>le_smul_caratheodory</a> _ _)).<a>trans</a> (<a>le_of_eq</a> hy)", [{"full_name": "le_of_eq", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [72, 9], "def_end_pos": [72, 17]}, {"full_name": "MeasureTheory.OuterMeasure.dirac_caratheodory", "def_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "def_pos": [1120, 9], "def_end_pos": [1120, 27]}, {"full_name": "Eq.symm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [310, 9], "def_end_pos": [310, 16]}, {"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [120, 7], "def_end_pos": [120, 18]}, {"full_name": "MeasureTheory.OuterMeasure.le_smul_caratheodory", "def_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "def_pos": [1112, 9], "def_end_pos": [1112, 29]}, {"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [120, 7], "def_end_pos": [120, 18]}, {"full_name": "le_of_eq", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [72, 9], "def_end_pos": [72, 17]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\ns t : Set \u03b1\nx : MeasurableSpace \u03b1\nhx : x \u2208 Set.range fun i => OuterMeasure.caratheodory (\u2191p i \u2022 dirac i)\ny : \u03b1\nhy : (fun i => OuterMeasure.caratheodory (\u2191p i \u2022 dirac i)) y = x\n\u22a2 \u22a4 \u2264 x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/BoolIndicator.lean", "full_name": "Set.preimage_boolIndicator", "start": [54, 1], "end": [58, 42], "traced_tactics": [{"tactic": "simp only [preimage_boolIndicator_eq_union]", "annotated_tactic": ["simp only [<a>preimage_boolIndicator_eq_union</a>]", [{"full_name": "Set.preimage_boolIndicator_eq_union", "def_path": "Mathlib/Data/Set/BoolIndicator.lean", "def_pos": [47, 9], "def_end_pos": [47, 40]}]], "state_before": "\u03b1 : Type u_1\ns : Set \u03b1\nt : Set Bool\n\u22a2 boolIndicator s \u207b\u00b9' t = univ \u2228 boolIndicator s \u207b\u00b9' t = s \u2228 boolIndicator s \u207b\u00b9' t = s\u1d9c \u2228 boolIndicator s \u207b\u00b9' t = \u2205", "state_after": "\u03b1 : Type u_1\ns : Set \u03b1\nt : Set Bool\n\u22a2 ((if true \u2208 t then s else \u2205) \u222a if false \u2208 t then s\u1d9c else \u2205) = univ \u2228\n    ((if true \u2208 t then s else \u2205) \u222a if false \u2208 t then s\u1d9c else \u2205) = s \u2228\n      ((if true \u2208 t then s else \u2205) \u222a if false \u2208 t then s\u1d9c else \u2205) = s\u1d9c \u2228\n        ((if true \u2208 t then s else \u2205) \u222a if false \u2208 t then s\u1d9c else \u2205) = \u2205"}, {"tactic": "split_ifs <;> simp [s.union_compl_self]", "annotated_tactic": ["split_ifs <;> simp [s.union_compl_self]", []], "state_before": "\u03b1 : Type u_1\ns : Set \u03b1\nt : Set Bool\n\u22a2 ((if true \u2208 t then s else \u2205) \u222a if false \u2208 t then s\u1d9c else \u2205) = univ \u2228\n    ((if true \u2208 t then s else \u2205) \u222a if false \u2208 t then s\u1d9c else \u2205) = s \u2228\n      ((if true \u2208 t then s else \u2205) \u222a if false \u2208 t then s\u1d9c else \u2205) = s\u1d9c \u2228\n        ((if true \u2208 t then s else \u2205) \u222a if false \u2208 t then s\u1d9c else \u2205) = \u2205", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/LocallyIntegrable.lean", "full_name": "MeasureTheory.locallyIntegrable_zero", "start": [279, 1], "end": [280, 44], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Constructions/Polish.lean", "full_name": "MeasureTheory.measurableSet_range_of_continuous_injective", "start": [656, 1], "end": [804, 60], "traced_tactics": [{"tactic": "letI := upgradePolishSpace \u03b3", "annotated_tactic": ["letI := <a>upgradePolishSpace</a> \u03b3", [{"full_name": "upgradePolishSpace", "def_path": "Mathlib/Topology/MetricSpace/Polish.lean", "def_pos": [96, 5], "def_end_pos": [96, 23]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\n\u22a2 MeasurableSet (range f)", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\n\u22a2 MeasurableSet (range f)"}, {"tactic": "obtain \u27e8b, b_count, b_nonempty, hb\u27e9 :\n  \u2203 b : Set (Set \u03b3), b.Countable \u2227 \u2205 \u2209 b \u2227 IsTopologicalBasis b := exists_countable_basis \u03b3", "annotated_tactic": ["obtain \u27e8b, b_count, b_nonempty, hb\u27e9 :\n    \u2203 b : <a>Set</a> (<a>Set</a> \u03b3), b.Countable \u2227 \u2205 \u2209 b \u2227 <a>IsTopologicalBasis</a> b := <a>exists_countable_basis</a> \u03b3", [{"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}, {"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}, {"full_name": "TopologicalSpace.IsTopologicalBasis", "def_path": "Mathlib/Topology/Bases.lean", "def_pos": [71, 11], "def_end_pos": [71, 29]}, {"full_name": "TopologicalSpace.exists_countable_basis", "def_path": "Mathlib/Topology/Bases.lean", "def_pos": [683, 9], "def_end_pos": [683, 31]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\n\u22a2 MeasurableSet (range f)", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\n\u22a2 MeasurableSet (range f)"}, {"tactic": "haveI : Encodable b := b_count.toEncodable", "annotated_tactic": ["haveI : <a>Encodable</a> b := b_count.toEncodable", [{"full_name": "Encodable", "def_path": "Mathlib/Logic/Encodable/Basic.lean", "def_pos": [45, 7], "def_end_pos": [45, 16]}]], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\n\u22a2 MeasurableSet (range f)", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\n\u22a2 MeasurableSet (range f)"}, {"tactic": "let A := { p : b \u00d7 b // Disjoint (p.1 : Set \u03b3) p.2 }", "annotated_tactic": ["let A := { p : b \u00d7 b // <a>Disjoint</a> (p.1 : <a>Set</a> \u03b3) p.2 }", [{"full_name": "Disjoint", "def_path": "Mathlib/Order/Disjoint.lean", "def_pos": [41, 5], "def_end_pos": [41, 13]}, {"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}]], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\n\u22a2 MeasurableSet (range f)", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\n\u22a2 MeasurableSet (range f)"}, {"tactic": "have : \u2200 p : A, \u2203 q : Set \u03b2,\n    f '' (p.1.1 : Set \u03b3) \u2286 q \u2227 Disjoint (f '' (p.1.2 : Set \u03b3)) q \u2227 MeasurableSet q := by\n  intro p\n  apply\n    AnalyticSet.measurablySeparable ((hb.isOpen p.1.1.2).analyticSet_image f_cont)\n      ((hb.isOpen p.1.2.2).analyticSet_image f_cont)\n  exact Disjoint.image p.2 (f_inj.injOn univ) (subset_univ _) (subset_univ _)", "annotated_tactic": ["have : \u2200 p : A, \u2203 q : <a>Set</a> \u03b2,\n      f '' (p.1.1 : <a>Set</a> \u03b3) \u2286 q \u2227 <a>Disjoint</a> (f '' (p.1.2 : <a>Set</a> \u03b3)) q \u2227 <a>MeasurableSet</a> q := by\n    intro p\n    apply\n      <a>AnalyticSet.measurablySeparable</a> ((hb.isOpen p.1.1.2).<a>analyticSet_image</a> f_cont)\n        ((hb.isOpen p.1.2.2).<a>analyticSet_image</a> f_cont)\n    exact <a>Disjoint.image</a> p.2 (f_inj.injOn <a>univ</a>) (<a>subset_univ</a> _) (<a>subset_univ</a> _)", [{"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}, {"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}, {"full_name": "Disjoint", "def_path": "Mathlib/Order/Disjoint.lean", "def_pos": [41, 5], "def_end_pos": [41, 13]}, {"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}, {"full_name": "MeasurableSet", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [64, 5], "def_end_pos": [64, 18]}, {"full_name": "MeasureTheory.AnalyticSet.measurablySeparable", "def_path": "Mathlib/MeasureTheory/Constructions/Polish.lean", "def_pos": [511, 9], "def_end_pos": [511, 40]}, {"full_name": "IsOpen.analyticSet_image", "def_path": "Mathlib/MeasureTheory/Constructions/Polish.lean", "def_pos": [186, 9], "def_end_pos": [186, 40]}, {"full_name": "IsOpen.analyticSet_image", "def_path": "Mathlib/MeasureTheory/Constructions/Polish.lean", "def_pos": [186, 9], "def_end_pos": [186, 40]}, {"full_name": "Disjoint.image", "def_path": "Mathlib/Data/Set/Function.lean", "def_pos": [757, 9], "def_end_pos": [757, 30]}, {"full_name": "Set.univ", "def_path": "Mathlib/Init/Set.lean", "def_pos": [90, 5], "def_end_pos": [90, 9]}, {"full_name": "Set.subset_univ", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [691, 9], "def_end_pos": [691, 20]}, {"full_name": "Set.subset_univ", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [691, 9], "def_end_pos": [691, 20]}]], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\n\u22a2 MeasurableSet (range f)", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis\u271d : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nthis : \u2200 (p : A), \u2203 q, f '' \u2191(\u2191p).1 \u2286 q \u2227 Disjoint (f '' \u2191(\u2191p).2) q \u2227 MeasurableSet q\n\u22a2 MeasurableSet (range f)"}, {"tactic": "choose q hq1 hq2 q_meas using this", "annotated_tactic": ["choose q hq1 hq2 q_meas using this", []], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis\u271d : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nthis : \u2200 (p : A), \u2203 q, f '' \u2191(\u2191p).1 \u2286 q \u2227 Disjoint (f '' \u2191(\u2191p).2) q \u2227 MeasurableSet q\n\u22a2 MeasurableSet (range f)", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\n\u22a2 MeasurableSet (range f)"}, {"tactic": "let E : b \u2192 Set \u03b2 := fun s =>\n  closure (f '' s) \u2229 \u22c2 (t : b) (ht : Disjoint s.1 t.1), q \u27e8(s, t), ht\u27e9 \\ q \u27e8(t, s), ht.symm\u27e9", "annotated_tactic": ["let E : b \u2192 <a>Set</a> \u03b2 := fun s =>\n    <a>closure</a> (f '' s) \u2229 \u22c2 (t : b) (ht : <a>Disjoint</a> s.1 t.1), q \u27e8(s, t), ht\u27e9 \\ q \u27e8(t, s), ht.symm\u27e9", [{"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}, {"full_name": "closure", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [422, 5], "def_end_pos": [422, 12]}, {"full_name": "Disjoint", "def_path": "Mathlib/Order/Disjoint.lean", "def_pos": [41, 5], "def_end_pos": [41, 13]}]], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\n\u22a2 MeasurableSet (range f)", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\n\u22a2 MeasurableSet (range f)"}, {"tactic": "obtain \u27e8u, u_anti, u_pos, u_lim\u27e9 :\n  \u2203 u : \u2115 \u2192 \u211d, StrictAnti u \u2227 (\u2200 n : \u2115, 0 < u n) \u2227 Tendsto u atTop (\ud835\udcdd 0) :=\n  exists_seq_strictAnti_tendsto (0 : \u211d)", "annotated_tactic": ["obtain \u27e8u, u_anti, u_pos, u_lim\u27e9 :\n    \u2203 u : \u2115 \u2192 \u211d, <a>StrictAnti</a> u \u2227 (\u2200 n : \u2115, 0 < u n) \u2227 <a>Tendsto</a> u <a>atTop</a> (\ud835\udcdd 0) :=\n    <a>exists_seq_strictAnti_tendsto</a> (0 : \u211d)", [{"full_name": "StrictAnti", "def_path": "Mathlib/Order/Monotone/Basic.lean", "def_pos": [102, 5], "def_end_pos": [102, 15]}, {"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2939, 5], "def_end_pos": [2939, 12]}, {"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}, {"full_name": "exists_seq_strictAnti_tendsto", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [2258, 9], "def_end_pos": [2258, 38]}]], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\n\u22a2 MeasurableSet (range f)", "state_after": "case intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\n\u22a2 MeasurableSet (range f)"}, {"tactic": "let F : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 (s : b) (_ : IsBounded s.1 \u2227 diam s.1 \u2264 u n), E s", "annotated_tactic": ["let F : \u2115 \u2192 <a>Set</a> \u03b2 := fun n => \u22c3 (s : b) (_ : <a>IsBounded</a> s.1 \u2227 <a>diam</a> s.1 \u2264 u n), E s", [{"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}, {"full_name": "Bornology.IsBounded", "def_path": "Mathlib/Topology/Bornology/Basic.lean", "def_pos": [140, 5], "def_end_pos": [140, 14]}, {"full_name": "Metric.diam", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [2651, 19], "def_end_pos": [2651, 23]}]], "state_before": "case intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\n\u22a2 MeasurableSet (range f)", "state_after": "case intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\n\u22a2 MeasurableSet (range f)"}, {"tactic": "suffices range f = \u22c2 n, F n by\n  have E_meas : \u2200 s : b, MeasurableSet (E s) := by\n    intro b\n    refine' isClosed_closure.measurableSet.inter _\n    refine' MeasurableSet.iInter fun s => _\n    exact MeasurableSet.iInter fun hs => (q_meas _).diff (q_meas _)\n  have F_meas : \u2200 n, MeasurableSet (F n) := by\n    intro n\n    refine' MeasurableSet.iUnion fun s => _\n    exact MeasurableSet.iUnion fun _ => E_meas _\n  rw [this]\n  exact MeasurableSet.iInter fun n => F_meas n", "annotated_tactic": ["suffices <a>range</a> f = \u22c2 n, F n by\n    have E_meas : \u2200 s : b, <a>MeasurableSet</a> (E s) := by\n      intro b\n      refine' isClosed_closure.measurableSet.inter _\n      refine' <a>MeasurableSet.iInter</a> fun s => _\n      exact <a>MeasurableSet.iInter</a> fun hs => (q_meas _).<a>diff</a> (q_meas _)\n    have F_meas : \u2200 n, <a>MeasurableSet</a> (F n) := by\n      intro n\n      refine' <a>MeasurableSet.iUnion</a> fun s => _\n      exact <a>MeasurableSet.iUnion</a> fun _ => E_meas _\n    rw [this]\n    exact <a>MeasurableSet.iInter</a> fun n => F_meas n", [{"full_name": "Set.range", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [668, 5], "def_end_pos": [668, 10]}, {"full_name": "MeasurableSet", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [64, 5], "def_end_pos": [64, 18]}, {"full_name": "MeasurableSet.iInter", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [159, 9], "def_end_pos": [159, 29]}, {"full_name": "MeasurableSet.iInter", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [159, 9], "def_end_pos": [159, 29]}, {"full_name": "MeasurableSet.diff", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [205, 19], "def_end_pos": [205, 37]}, {"full_name": "MeasurableSet", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [64, 5], "def_end_pos": [64, 18]}, {"full_name": "MeasurableSet.iUnion", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [115, 19], "def_end_pos": [115, 39]}, {"full_name": "MeasurableSet.iUnion", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [115, 19], "def_end_pos": [115, 39]}, {"full_name": "MeasurableSet.iInter", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [159, 9], "def_end_pos": [159, 29]}]], "state_before": "case intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\n\u22a2 MeasurableSet (range f)", "state_after": "case intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\n\u22a2 range f = \u22c2 n, F n"}, {"tactic": "apply Subset.antisymm", "annotated_tactic": ["apply <a>Subset.antisymm</a>", [{"full_name": "Set.Subset.antisymm", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [370, 9], "def_end_pos": [370, 24]}]], "state_before": "case intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\n\u22a2 range f = \u22c2 n, F n", "state_after": "case intro.intro.intro.intro.intro.intro.h\u2081\n\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\n\u22a2 range f \u2286 \u22c2 n, F n\n\ncase intro.intro.intro.intro.intro.intro.h\u2082\n\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\n\u22a2 \u22c2 n, F n \u2286 range f"}, {"tactic": "intro p", "annotated_tactic": ["intro p", []], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\n\u22a2 \u2200 (p : A), \u2203 q, f '' \u2191(\u2191p).1 \u2286 q \u2227 Disjoint (f '' \u2191(\u2191p).2) q \u2227 MeasurableSet q", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\np : A\n\u22a2 \u2203 q, f '' \u2191(\u2191p).1 \u2286 q \u2227 Disjoint (f '' \u2191(\u2191p).2) q \u2227 MeasurableSet q"}, {"tactic": "apply\n  AnalyticSet.measurablySeparable ((hb.isOpen p.1.1.2).analyticSet_image f_cont)\n    ((hb.isOpen p.1.2.2).analyticSet_image f_cont)", "annotated_tactic": ["apply\n      <a>AnalyticSet.measurablySeparable</a> ((hb.isOpen p.1.1.2).<a>analyticSet_image</a> f_cont)\n        ((hb.isOpen p.1.2.2).<a>analyticSet_image</a> f_cont)", [{"full_name": "MeasureTheory.AnalyticSet.measurablySeparable", "def_path": "Mathlib/MeasureTheory/Constructions/Polish.lean", "def_pos": [511, 9], "def_end_pos": [511, 40]}, {"full_name": "IsOpen.analyticSet_image", "def_path": "Mathlib/MeasureTheory/Constructions/Polish.lean", "def_pos": [186, 9], "def_end_pos": [186, 40]}, {"full_name": "IsOpen.analyticSet_image", "def_path": "Mathlib/MeasureTheory/Constructions/Polish.lean", "def_pos": [186, 9], "def_end_pos": [186, 40]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\np : A\n\u22a2 \u2203 q, f '' \u2191(\u2191p).1 \u2286 q \u2227 Disjoint (f '' \u2191(\u2191p).2) q \u2227 MeasurableSet q", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\np : A\n\u22a2 Disjoint (f '' \u2191(\u2191p).1) (f '' \u2191(\u2191p).2)"}, {"tactic": "exact Disjoint.image p.2 (f_inj.injOn univ) (subset_univ _) (subset_univ _)", "annotated_tactic": ["exact <a>Disjoint.image</a> p.2 (f_inj.injOn <a>univ</a>) (<a>subset_univ</a> _) (<a>subset_univ</a> _)", [{"full_name": "Disjoint.image", "def_path": "Mathlib/Data/Set/Function.lean", "def_pos": [757, 9], "def_end_pos": [757, 30]}, {"full_name": "Set.univ", "def_path": "Mathlib/Init/Set.lean", "def_pos": [90, 5], "def_end_pos": [90, 9]}, {"full_name": "Set.subset_univ", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [691, 9], "def_end_pos": [691, 20]}, {"full_name": "Set.subset_univ", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [691, 9], "def_end_pos": [691, 20]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\np : A\n\u22a2 Disjoint (f '' \u2191(\u2191p).1) (f '' \u2191(\u2191p).2)", "state_after": "no goals"}, {"tactic": "have E_meas : \u2200 s : b, MeasurableSet (E s) := by\n  intro b\n  refine' isClosed_closure.measurableSet.inter _\n  refine' MeasurableSet.iInter fun s => _\n  exact MeasurableSet.iInter fun hs => (q_meas _).diff (q_meas _)", "annotated_tactic": ["have E_meas : \u2200 s : b, <a>MeasurableSet</a> (E s) := by\n      intro b\n      refine' isClosed_closure.measurableSet.inter _\n      refine' <a>MeasurableSet.iInter</a> fun s => _\n      exact <a>MeasurableSet.iInter</a> fun hs => (q_meas _).<a>diff</a> (q_meas _)", [{"full_name": "MeasurableSet", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [64, 5], "def_end_pos": [64, 18]}, {"full_name": "MeasurableSet.iInter", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [159, 9], "def_end_pos": [159, 29]}, {"full_name": "MeasurableSet.iInter", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [159, 9], "def_end_pos": [159, 29]}, {"full_name": "MeasurableSet.diff", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [205, 19], "def_end_pos": [205, 37]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis\u271d : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nthis : range f = \u22c2 n, F n\n\u22a2 MeasurableSet (range f)", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis\u271d : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nthis : range f = \u22c2 n, F n\nE_meas : \u2200 (s : \u2191b), MeasurableSet (E s)\n\u22a2 MeasurableSet (range f)"}, {"tactic": "have F_meas : \u2200 n, MeasurableSet (F n) := by\n  intro n\n  refine' MeasurableSet.iUnion fun s => _\n  exact MeasurableSet.iUnion fun _ => E_meas _", "annotated_tactic": ["have F_meas : \u2200 n, <a>MeasurableSet</a> (F n) := by\n      intro n\n      refine' <a>MeasurableSet.iUnion</a> fun s => _\n      exact <a>MeasurableSet.iUnion</a> fun _ => E_meas _", [{"full_name": "MeasurableSet", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [64, 5], "def_end_pos": [64, 18]}, {"full_name": "MeasurableSet.iUnion", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [115, 19], "def_end_pos": [115, 39]}, {"full_name": "MeasurableSet.iUnion", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [115, 19], "def_end_pos": [115, 39]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis\u271d : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nthis : range f = \u22c2 n, F n\nE_meas : \u2200 (s : \u2191b), MeasurableSet (E s)\n\u22a2 MeasurableSet (range f)", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis\u271d : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nthis : range f = \u22c2 n, F n\nE_meas : \u2200 (s : \u2191b), MeasurableSet (E s)\nF_meas : \u2200 (n : \u2115), MeasurableSet (F n)\n\u22a2 MeasurableSet (range f)"}, {"tactic": "rw [this]", "annotated_tactic": ["rw [this]", []], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis\u271d : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nthis : range f = \u22c2 n, F n\nE_meas : \u2200 (s : \u2191b), MeasurableSet (E s)\nF_meas : \u2200 (n : \u2115), MeasurableSet (F n)\n\u22a2 MeasurableSet (range f)", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis\u271d : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nthis : range f = \u22c2 n, F n\nE_meas : \u2200 (s : \u2191b), MeasurableSet (E s)\nF_meas : \u2200 (n : \u2115), MeasurableSet (F n)\n\u22a2 MeasurableSet (\u22c2 n, F n)"}, {"tactic": "exact MeasurableSet.iInter fun n => F_meas n", "annotated_tactic": ["exact <a>MeasurableSet.iInter</a> fun n => F_meas n", [{"full_name": "MeasurableSet.iInter", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [159, 9], "def_end_pos": [159, 29]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis\u271d : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nthis : range f = \u22c2 n, F n\nE_meas : \u2200 (s : \u2191b), MeasurableSet (E s)\nF_meas : \u2200 (n : \u2115), MeasurableSet (F n)\n\u22a2 MeasurableSet (\u22c2 n, F n)", "state_after": "no goals"}, {"tactic": "intro b", "annotated_tactic": ["intro b", []], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis\u271d : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nthis : range f = \u22c2 n, F n\n\u22a2 \u2200 (s : \u2191b), MeasurableSet (E s)", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb\u271d : Set (Set \u03b3)\nb_count : Set.Countable b\u271d\nb_nonempty : \u00ac\u2205 \u2208 b\u271d\nhb : IsTopologicalBasis b\u271d\nthis\u271d : Encodable \u2191b\u271d\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b\u271d \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nthis : range f = \u22c2 n, F n\nb : \u2191b\u271d\n\u22a2 MeasurableSet (E b)"}, {"tactic": "refine' isClosed_closure.measurableSet.inter _", "annotated_tactic": ["refine' isClosed_closure.measurableSet.inter _", []], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb\u271d : Set (Set \u03b3)\nb_count : Set.Countable b\u271d\nb_nonempty : \u00ac\u2205 \u2208 b\u271d\nhb : IsTopologicalBasis b\u271d\nthis\u271d : Encodable \u2191b\u271d\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b\u271d \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nthis : range f = \u22c2 n, F n\nb : \u2191b\u271d\n\u22a2 MeasurableSet (E b)", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb\u271d : Set (Set \u03b3)\nb_count : Set.Countable b\u271d\nb_nonempty : \u00ac\u2205 \u2208 b\u271d\nhb : IsTopologicalBasis b\u271d\nthis\u271d : Encodable \u2191b\u271d\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b\u271d \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nthis : range f = \u22c2 n, F n\nb : \u2191b\u271d\n\u22a2 MeasurableSet\n    (\u22c2 t,\n      \u22c2 (ht : Disjoint \u2191b \u2191t),\n        q { val := (b, t), property := ht } \\ q { val := (t, b), property := (_ : Disjoint \u2191t \u2191b) })"}, {"tactic": "refine' MeasurableSet.iInter fun s => _", "annotated_tactic": ["refine' <a>MeasurableSet.iInter</a> fun s => _", [{"full_name": "MeasurableSet.iInter", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [159, 9], "def_end_pos": [159, 29]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb\u271d : Set (Set \u03b3)\nb_count : Set.Countable b\u271d\nb_nonempty : \u00ac\u2205 \u2208 b\u271d\nhb : IsTopologicalBasis b\u271d\nthis\u271d : Encodable \u2191b\u271d\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b\u271d \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nthis : range f = \u22c2 n, F n\nb : \u2191b\u271d\n\u22a2 MeasurableSet\n    (\u22c2 t,\n      \u22c2 (ht : Disjoint \u2191b \u2191t),\n        q { val := (b, t), property := ht } \\ q { val := (t, b), property := (_ : Disjoint \u2191t \u2191b) })", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb\u271d : Set (Set \u03b3)\nb_count : Set.Countable b\u271d\nb_nonempty : \u00ac\u2205 \u2208 b\u271d\nhb : IsTopologicalBasis b\u271d\nthis\u271d : Encodable \u2191b\u271d\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b\u271d \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nthis : range f = \u22c2 n, F n\nb s : \u2191b\u271d\n\u22a2 MeasurableSet\n    (\u22c2 (ht : Disjoint \u2191b \u2191s),\n      q { val := (b, s), property := ht } \\ q { val := (s, b), property := (_ : Disjoint \u2191s \u2191b) })"}, {"tactic": "exact MeasurableSet.iInter fun hs => (q_meas _).diff (q_meas _)", "annotated_tactic": ["exact <a>MeasurableSet.iInter</a> fun hs => (q_meas _).<a>diff</a> (q_meas _)", [{"full_name": "MeasurableSet.iInter", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [159, 9], "def_end_pos": [159, 29]}, {"full_name": "MeasurableSet.diff", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [205, 19], "def_end_pos": [205, 37]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb\u271d : Set (Set \u03b3)\nb_count : Set.Countable b\u271d\nb_nonempty : \u00ac\u2205 \u2208 b\u271d\nhb : IsTopologicalBasis b\u271d\nthis\u271d : Encodable \u2191b\u271d\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b\u271d \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nthis : range f = \u22c2 n, F n\nb s : \u2191b\u271d\n\u22a2 MeasurableSet\n    (\u22c2 (ht : Disjoint \u2191b \u2191s),\n      q { val := (b, s), property := ht } \\ q { val := (s, b), property := (_ : Disjoint \u2191s \u2191b) })", "state_after": "no goals"}, {"tactic": "intro n", "annotated_tactic": ["intro n", []], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis\u271d : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nthis : range f = \u22c2 n, F n\nE_meas : \u2200 (s : \u2191b), MeasurableSet (E s)\n\u22a2 \u2200 (n : \u2115), MeasurableSet (F n)", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis\u271d : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nthis : range f = \u22c2 n, F n\nE_meas : \u2200 (s : \u2191b), MeasurableSet (E s)\nn : \u2115\n\u22a2 MeasurableSet (F n)"}, {"tactic": "refine' MeasurableSet.iUnion fun s => _", "annotated_tactic": ["refine' <a>MeasurableSet.iUnion</a> fun s => _", [{"full_name": "MeasurableSet.iUnion", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [115, 19], "def_end_pos": [115, 39]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis\u271d : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nthis : range f = \u22c2 n, F n\nE_meas : \u2200 (s : \u2191b), MeasurableSet (E s)\nn : \u2115\n\u22a2 MeasurableSet (F n)", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis\u271d : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nthis : range f = \u22c2 n, F n\nE_meas : \u2200 (s : \u2191b), MeasurableSet (E s)\nn : \u2115\ns : \u2191b\n\u22a2 MeasurableSet (\u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s)"}, {"tactic": "exact MeasurableSet.iUnion fun _ => E_meas _", "annotated_tactic": ["exact <a>MeasurableSet.iUnion</a> fun _ => E_meas _", [{"full_name": "MeasurableSet.iUnion", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [115, 19], "def_end_pos": [115, 39]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis\u271d : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nthis : range f = \u22c2 n, F n\nE_meas : \u2200 (s : \u2191b), MeasurableSet (E s)\nn : \u2115\ns : \u2191b\n\u22a2 MeasurableSet (\u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s)", "state_after": "no goals"}, {"tactic": "rintro x \u27e8y, rfl\u27e9", "annotated_tactic": ["rintro x \u27e8y, rfl\u27e9", []], "state_before": "case intro.intro.intro.intro.intro.intro.h\u2081\n\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\n\u22a2 range f \u2286 \u22c2 n, F n", "state_after": "case intro.intro.intro.intro.intro.intro.h\u2081.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\ny : \u03b3\n\u22a2 f y \u2208 \u22c2 n, F n"}, {"tactic": "refine mem_iInter.2 fun n => ?_", "annotated_tactic": ["refine <a>mem_iInter</a>.2 fun n => ?_", [{"full_name": "Set.mem_iInter", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [207, 9], "def_end_pos": [207, 19]}]], "state_before": "case intro.intro.intro.intro.intro.intro.h\u2081.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\ny : \u03b3\n\u22a2 f y \u2208 \u22c2 n, F n", "state_after": "case intro.intro.intro.intro.intro.intro.h\u2081.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\ny : \u03b3\nn : \u2115\n\u22a2 f y \u2208 F n"}, {"tactic": "obtain \u27e8s, sb, ys, hs\u27e9 : \u2203 (s : Set \u03b3), s \u2208 b \u2227 y \u2208 s \u2227 s \u2286 ball y (u n / 2) := by\n  apply hb.mem_nhds_iff.1\n  exact ball_mem_nhds _ (half_pos (u_pos n))", "annotated_tactic": ["obtain \u27e8s, sb, ys, hs\u27e9 : \u2203 (s : <a>Set</a> \u03b3), s \u2208 b \u2227 y \u2208 s \u2227 s \u2286 <a>ball</a> y (u n / 2) := by\n      apply hb.mem_nhds_iff.1\n      exact <a>ball_mem_nhds</a> _ (<a>half_pos</a> (u_pos n))", [{"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}, {"full_name": "Metric.ball", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [409, 5], "def_end_pos": [409, 9]}, {"full_name": "Metric.ball_mem_nhds", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [1021, 9], "def_end_pos": [1021, 22]}, {"full_name": "half_pos", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [504, 9], "def_end_pos": [504, 17]}]], "state_before": "case intro.intro.intro.intro.intro.intro.h\u2081.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\ny : \u03b3\nn : \u2115\n\u22a2 f y \u2208 F n", "state_after": "case intro.intro.intro.intro.intro.intro.h\u2081.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\ny : \u03b3\nn : \u2115\ns : Set \u03b3\nsb : s \u2208 b\nys : y \u2208 s\nhs : s \u2286 ball y (u n / 2)\n\u22a2 f y \u2208 F n"}, {"tactic": "have diam_s : diam s \u2264 u n := by\n  apply (diam_mono hs isBounded_ball).trans\n  convert diam_ball (x := y) (half_pos (u_pos n)).le\n  ring", "annotated_tactic": ["have diam_s : <a>diam</a> s \u2264 u n := by\n      apply (<a>diam_mono</a> hs <a>isBounded_ball</a>).<a>trans</a>\n      convert <a>diam_ball</a> (x := y) (<a>half_pos</a> (u_pos n)).<a>le</a>\n      ring", [{"full_name": "Metric.diam", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [2651, 19], "def_end_pos": [2651, 23]}, {"full_name": "Metric.diam_mono", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [2765, 9], "def_end_pos": [2765, 18]}, {"full_name": "Metric.isBounded_ball", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [2371, 9], "def_end_pos": [2371, 23]}, {"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [120, 7], "def_end_pos": [120, 18]}, {"full_name": "Metric.diam_ball", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [2803, 9], "def_end_pos": [2803, 18]}, {"full_name": "half_pos", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [504, 9], "def_end_pos": [504, 17]}, {"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [142, 7], "def_end_pos": [142, 15]}]], "state_before": "case intro.intro.intro.intro.intro.intro.h\u2081.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\ny : \u03b3\nn : \u2115\ns : Set \u03b3\nsb : s \u2208 b\nys : y \u2208 s\nhs : s \u2286 ball y (u n / 2)\n\u22a2 f y \u2208 F n", "state_after": "case intro.intro.intro.intro.intro.intro.h\u2081.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\ny : \u03b3\nn : \u2115\ns : Set \u03b3\nsb : s \u2208 b\nys : y \u2208 s\nhs : s \u2286 ball y (u n / 2)\ndiam_s : diam s \u2264 u n\n\u22a2 f y \u2208 F n"}, {"tactic": "refine' mem_iUnion.2 \u27e8\u27e8s, sb\u27e9, _\u27e9", "annotated_tactic": ["refine' <a>mem_iUnion</a>.2 \u27e8\u27e8s, sb\u27e9, _\u27e9", [{"full_name": "Set.mem_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [201, 9], "def_end_pos": [201, 19]}]], "state_before": "case intro.intro.intro.intro.intro.intro.h\u2081.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\ny : \u03b3\nn : \u2115\ns : Set \u03b3\nsb : s \u2208 b\nys : y \u2208 s\nhs : s \u2286 ball y (u n / 2)\ndiam_s : diam s \u2264 u n\n\u22a2 f y \u2208 F n", "state_after": "case intro.intro.intro.intro.intro.intro.h\u2081.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\ny : \u03b3\nn : \u2115\ns : Set \u03b3\nsb : s \u2208 b\nys : y \u2208 s\nhs : s \u2286 ball y (u n / 2)\ndiam_s : diam s \u2264 u n\n\u22a2 f y \u2208\n    \u22c3 (_ : Bornology.IsBounded \u2191{ val := s, property := sb } \u2227 diam \u2191{ val := s, property := sb } \u2264 u n),\n      E { val := s, property := sb }"}, {"tactic": "refine' mem_iUnion.2 \u27e8\u27e8isBounded_ball.subset hs, diam_s\u27e9, _\u27e9", "annotated_tactic": ["refine' <a>mem_iUnion</a>.2 \u27e8\u27e8isBounded_ball.subset hs, diam_s\u27e9, _\u27e9", [{"full_name": "Set.mem_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [201, 9], "def_end_pos": [201, 19]}]], "state_before": "case intro.intro.intro.intro.intro.intro.h\u2081.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\ny : \u03b3\nn : \u2115\ns : Set \u03b3\nsb : s \u2208 b\nys : y \u2208 s\nhs : s \u2286 ball y (u n / 2)\ndiam_s : diam s \u2264 u n\n\u22a2 f y \u2208\n    \u22c3 (_ : Bornology.IsBounded \u2191{ val := s, property := sb } \u2227 diam \u2191{ val := s, property := sb } \u2264 u n),\n      E { val := s, property := sb }", "state_after": "case intro.intro.intro.intro.intro.intro.h\u2081.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\ny : \u03b3\nn : \u2115\ns : Set \u03b3\nsb : s \u2208 b\nys : y \u2208 s\nhs : s \u2286 ball y (u n / 2)\ndiam_s : diam s \u2264 u n\n\u22a2 f y \u2208 E { val := s, property := sb }"}, {"tactic": "apply mem_inter (subset_closure (mem_image_of_mem _ ys))", "annotated_tactic": ["apply <a>mem_inter</a> (<a>subset_closure</a> (<a>mem_image_of_mem</a> _ ys))", [{"full_name": "Set.mem_inter", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [913, 9], "def_end_pos": [913, 18]}, {"full_name": "subset_closure", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [435, 9], "def_end_pos": [435, 23]}, {"full_name": "Set.mem_image_of_mem", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [240, 9], "def_end_pos": [240, 25]}]], "state_before": "case intro.intro.intro.intro.intro.intro.h\u2081.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\ny : \u03b3\nn : \u2115\ns : Set \u03b3\nsb : s \u2208 b\nys : y \u2208 s\nhs : s \u2286 ball y (u n / 2)\ndiam_s : diam s \u2264 u n\n\u22a2 f y \u2208 E { val := s, property := sb }", "state_after": "case intro.intro.intro.intro.intro.intro.h\u2081.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\ny : \u03b3\nn : \u2115\ns : Set \u03b3\nsb : s \u2208 b\nys : y \u2208 s\nhs : s \u2286 ball y (u n / 2)\ndiam_s : diam s \u2264 u n\n\u22a2 f y \u2208\n    \u22c2 t,\n      \u22c2 (ht : Disjoint \u2191{ val := s, property := sb } \u2191t),\n        q { val := ({ val := s, property := sb }, t), property := ht } \\\n          q { val := (t, { val := s, property := sb }), property := (_ : Disjoint \u2191t \u2191{ val := s, property := sb }) }"}, {"tactic": "refine' mem_iInter.2 fun t => mem_iInter.2 fun ht => \u27e8_, _\u27e9", "annotated_tactic": ["refine' <a>mem_iInter</a>.2 fun t => <a>mem_iInter</a>.2 fun ht => \u27e8_, _\u27e9", [{"full_name": "Set.mem_iInter", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [207, 9], "def_end_pos": [207, 19]}, {"full_name": "Set.mem_iInter", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [207, 9], "def_end_pos": [207, 19]}]], "state_before": "case intro.intro.intro.intro.intro.intro.h\u2081.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\ny : \u03b3\nn : \u2115\ns : Set \u03b3\nsb : s \u2208 b\nys : y \u2208 s\nhs : s \u2286 ball y (u n / 2)\ndiam_s : diam s \u2264 u n\n\u22a2 f y \u2208\n    \u22c2 t,\n      \u22c2 (ht : Disjoint \u2191{ val := s, property := sb } \u2191t),\n        q { val := ({ val := s, property := sb }, t), property := ht } \\\n          q { val := (t, { val := s, property := sb }), property := (_ : Disjoint \u2191t \u2191{ val := s, property := sb }) }", "state_after": "case intro.intro.intro.intro.intro.intro.h\u2081.intro.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\ny : \u03b3\nn : \u2115\ns : Set \u03b3\nsb : s \u2208 b\nys : y \u2208 s\nhs : s \u2286 ball y (u n / 2)\ndiam_s : diam s \u2264 u n\nt : \u2191b\nht : Disjoint \u2191{ val := s, property := sb } \u2191t\n\u22a2 f y \u2208 q { val := ({ val := s, property := sb }, t), property := ht }\n\ncase intro.intro.intro.intro.intro.intro.h\u2081.intro.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\ny : \u03b3\nn : \u2115\ns : Set \u03b3\nsb : s \u2208 b\nys : y \u2208 s\nhs : s \u2286 ball y (u n / 2)\ndiam_s : diam s \u2264 u n\nt : \u2191b\nht : Disjoint \u2191{ val := s, property := sb } \u2191t\n\u22a2 \u00acf y \u2208 q { val := (t, { val := s, property := sb }), property := (_ : Disjoint \u2191t \u2191{ val := s, property := sb }) }"}, {"tactic": "apply hb.mem_nhds_iff.1", "annotated_tactic": ["apply hb.mem_nhds_iff.1", []], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\ny : \u03b3\nn : \u2115\n\u22a2 \u2203 s, s \u2208 b \u2227 y \u2208 s \u2227 s \u2286 ball y (u n / 2)", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\ny : \u03b3\nn : \u2115\n\u22a2 ball y (u n / 2) \u2208 \ud835\udcdd y"}, {"tactic": "exact ball_mem_nhds _ (half_pos (u_pos n))", "annotated_tactic": ["exact <a>ball_mem_nhds</a> _ (<a>half_pos</a> (u_pos n))", [{"full_name": "Metric.ball_mem_nhds", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [1021, 9], "def_end_pos": [1021, 22]}, {"full_name": "half_pos", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [504, 9], "def_end_pos": [504, 17]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\ny : \u03b3\nn : \u2115\n\u22a2 ball y (u n / 2) \u2208 \ud835\udcdd y", "state_after": "no goals"}, {"tactic": "apply (diam_mono hs isBounded_ball).trans", "annotated_tactic": ["apply (<a>diam_mono</a> hs <a>isBounded_ball</a>).<a>trans</a>", [{"full_name": "Metric.diam_mono", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [2765, 9], "def_end_pos": [2765, 18]}, {"full_name": "Metric.isBounded_ball", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [2371, 9], "def_end_pos": [2371, 23]}, {"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [120, 7], "def_end_pos": [120, 18]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\ny : \u03b3\nn : \u2115\ns : Set \u03b3\nsb : s \u2208 b\nys : y \u2208 s\nhs : s \u2286 ball y (u n / 2)\n\u22a2 diam s \u2264 u n", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\ny : \u03b3\nn : \u2115\ns : Set \u03b3\nsb : s \u2208 b\nys : y \u2208 s\nhs : s \u2286 ball y (u n / 2)\n\u22a2 diam (ball y (u n / 2)) \u2264 u n"}, {"tactic": "convert diam_ball (x := y) (half_pos (u_pos n)).le", "annotated_tactic": ["convert <a>diam_ball</a> (x := y) (<a>half_pos</a> (u_pos n)).<a>le</a>", [{"full_name": "Metric.diam_ball", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [2803, 9], "def_end_pos": [2803, 18]}, {"full_name": "half_pos", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [504, 9], "def_end_pos": [504, 17]}, {"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [142, 7], "def_end_pos": [142, 15]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\ny : \u03b3\nn : \u2115\ns : Set \u03b3\nsb : s \u2208 b\nys : y \u2208 s\nhs : s \u2286 ball y (u n / 2)\n\u22a2 diam (ball y (u n / 2)) \u2264 u n", "state_after": "case h.e'_4\n\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\ny : \u03b3\nn : \u2115\ns : Set \u03b3\nsb : s \u2208 b\nys : y \u2208 s\nhs : s \u2286 ball y (u n / 2)\n\u22a2 u n = 2 * (u n / 2)"}, {"tactic": "ring", "annotated_tactic": ["ring", []], "state_before": "case h.e'_4\n\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\ny : \u03b3\nn : \u2115\ns : Set \u03b3\nsb : s \u2208 b\nys : y \u2208 s\nhs : s \u2286 ball y (u n / 2)\n\u22a2 u n = 2 * (u n / 2)", "state_after": "no goals"}, {"tactic": "apply hq1", "annotated_tactic": ["apply hq1", []], "state_before": "case intro.intro.intro.intro.intro.intro.h\u2081.intro.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\ny : \u03b3\nn : \u2115\ns : Set \u03b3\nsb : s \u2208 b\nys : y \u2208 s\nhs : s \u2286 ball y (u n / 2)\ndiam_s : diam s \u2264 u n\nt : \u2191b\nht : Disjoint \u2191{ val := s, property := sb } \u2191t\n\u22a2 f y \u2208 q { val := ({ val := s, property := sb }, t), property := ht }", "state_after": "case intro.intro.intro.intro.intro.intro.h\u2081.intro.intro.intro.intro.refine'_1.a\n\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\ny : \u03b3\nn : \u2115\ns : Set \u03b3\nsb : s \u2208 b\nys : y \u2208 s\nhs : s \u2286 ball y (u n / 2)\ndiam_s : diam s \u2264 u n\nt : \u2191b\nht : Disjoint \u2191{ val := s, property := sb } \u2191t\n\u22a2 f y \u2208 f '' \u2191(\u2191{ val := ({ val := s, property := sb }, t), property := ht }).1"}, {"tactic": "exact mem_image_of_mem _ ys", "annotated_tactic": ["exact <a>mem_image_of_mem</a> _ ys", [{"full_name": "Set.mem_image_of_mem", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [240, 9], "def_end_pos": [240, 25]}]], "state_before": "case intro.intro.intro.intro.intro.intro.h\u2081.intro.intro.intro.intro.refine'_1.a\n\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\ny : \u03b3\nn : \u2115\ns : Set \u03b3\nsb : s \u2208 b\nys : y \u2208 s\nhs : s \u2286 ball y (u n / 2)\ndiam_s : diam s \u2264 u n\nt : \u2191b\nht : Disjoint \u2191{ val := s, property := sb } \u2191t\n\u22a2 f y \u2208 f '' \u2191(\u2191{ val := ({ val := s, property := sb }, t), property := ht }).1", "state_after": "no goals"}, {"tactic": "apply disjoint_left.1 (hq2 \u27e8(t, \u27e8s, sb\u27e9), ht.symm\u27e9)", "annotated_tactic": ["apply <a>disjoint_left</a>.1 (hq2 \u27e8(t, \u27e8s, sb\u27e9), ht.symm\u27e9)", [{"full_name": "Set.disjoint_left", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1546, 9], "def_end_pos": [1546, 22]}]], "state_before": "case intro.intro.intro.intro.intro.intro.h\u2081.intro.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\ny : \u03b3\nn : \u2115\ns : Set \u03b3\nsb : s \u2208 b\nys : y \u2208 s\nhs : s \u2286 ball y (u n / 2)\ndiam_s : diam s \u2264 u n\nt : \u2191b\nht : Disjoint \u2191{ val := s, property := sb } \u2191t\n\u22a2 \u00acf y \u2208 q { val := (t, { val := s, property := sb }), property := (_ : Disjoint \u2191t \u2191{ val := s, property := sb }) }", "state_after": "case intro.intro.intro.intro.intro.intro.h\u2081.intro.intro.intro.intro.refine'_2.a\n\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\ny : \u03b3\nn : \u2115\ns : Set \u03b3\nsb : s \u2208 b\nys : y \u2208 s\nhs : s \u2286 ball y (u n / 2)\ndiam_s : diam s \u2264 u n\nt : \u2191b\nht : Disjoint \u2191{ val := s, property := sb } \u2191t\n\u22a2 f y \u2208\n    f '' \u2191(\u2191{ val := (t, { val := s, property := sb }), property := (_ : Disjoint \u2191t \u2191{ val := s, property := sb }) }).2"}, {"tactic": "exact mem_image_of_mem _ ys", "annotated_tactic": ["exact <a>mem_image_of_mem</a> _ ys", [{"full_name": "Set.mem_image_of_mem", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [240, 9], "def_end_pos": [240, 25]}]], "state_before": "case intro.intro.intro.intro.intro.intro.h\u2081.intro.intro.intro.intro.refine'_2.a\n\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\ny : \u03b3\nn : \u2115\ns : Set \u03b3\nsb : s \u2208 b\nys : y \u2208 s\nhs : s \u2286 ball y (u n / 2)\ndiam_s : diam s \u2264 u n\nt : \u2191b\nht : Disjoint \u2191{ val := s, property := sb } \u2191t\n\u22a2 f y \u2208\n    f '' \u2191(\u2191{ val := (t, { val := s, property := sb }), property := (_ : Disjoint \u2191t \u2191{ val := s, property := sb }) }).2", "state_after": "no goals"}, {"tactic": "intro x hx", "annotated_tactic": ["intro x hx", []], "state_before": "case intro.intro.intro.intro.intro.intro.h\u2082\n\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\n\u22a2 \u22c2 n, F n \u2286 range f", "state_after": "case intro.intro.intro.intro.intro.intro.h\u2082\n\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 n, F n\n\u22a2 x \u2208 range f"}, {"tactic": "have C1 : \u2200 n, \u2203 (s : b) (_ : IsBounded s.1 \u2227 diam s.1 \u2264 u n), x \u2208 E s := fun n => by\n  simpa only [mem_iUnion] using mem_iInter.1 hx n", "annotated_tactic": ["have C1 : \u2200 n, \u2203 (s : b) (_ : <a>IsBounded</a> s.1 \u2227 <a>diam</a> s.1 \u2264 u n), x \u2208 E s := fun n => by\n      simpa only [<a>mem_iUnion</a>] using <a>mem_iInter</a>.1 hx n", [{"full_name": "Bornology.IsBounded", "def_path": "Mathlib/Topology/Bornology/Basic.lean", "def_pos": [140, 5], "def_end_pos": [140, 14]}, {"full_name": "Metric.diam", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [2651, 19], "def_end_pos": [2651, 23]}, {"full_name": "Set.mem_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [201, 9], "def_end_pos": [201, 19]}, {"full_name": "Set.mem_iInter", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [207, 9], "def_end_pos": [207, 19]}]], "state_before": "case intro.intro.intro.intro.intro.intro.h\u2082\n\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 n, F n\n\u22a2 x \u2208 range f", "state_after": "case intro.intro.intro.intro.intro.intro.h\u2082\n\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 n, F n\nC1 : \u2200 (n : \u2115), \u2203 s x_1, x \u2208 E s\n\u22a2 x \u2208 range f"}, {"tactic": "choose s hs hxs using C1", "annotated_tactic": ["choose s hs hxs using C1", []], "state_before": "case intro.intro.intro.intro.intro.intro.h\u2082\n\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 n, F n\nC1 : \u2200 (n : \u2115), \u2203 s x_1, x \u2208 E s\n\u22a2 x \u2208 range f", "state_after": "case intro.intro.intro.intro.intro.intro.h\u2082\n\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 n, F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\n\u22a2 x \u2208 range f"}, {"tactic": "have C2 : \u2200 n, (s n).1.Nonempty := by\n  intro n\n  rw [nonempty_iff_ne_empty]\n  intro hn\n  have := (s n).2\n  rw [hn] at this\n  exact b_nonempty this", "annotated_tactic": ["have C2 : \u2200 n, (s n).1.<a>Nonempty</a> := by\n      intro n\n      rw [<a>nonempty_iff_ne_empty</a>]\n      intro hn\n      have := (s n).2\n      rw [hn] at this\n      exact b_nonempty this", [{"full_name": "Set.Nonempty", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [439, 15], "def_end_pos": [439, 23]}, {"full_name": "Set.nonempty_iff_ne_empty", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [610, 9], "def_end_pos": [610, 30]}]], "state_before": "case intro.intro.intro.intro.intro.intro.h\u2082\n\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 n, F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\n\u22a2 x \u2208 range f", "state_after": "case intro.intro.intro.intro.intro.intro.h\u2082\n\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 n, F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\nC2 : \u2200 (n : \u2115), Set.Nonempty \u2191(s n)\n\u22a2 x \u2208 range f"}, {"tactic": "choose y hy using C2", "annotated_tactic": ["choose y hy using C2", []], "state_before": "case intro.intro.intro.intro.intro.intro.h\u2082\n\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 n, F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\nC2 : \u2200 (n : \u2115), Set.Nonempty \u2191(s n)\n\u22a2 x \u2208 range f", "state_after": "case intro.intro.intro.intro.intro.intro.h\u2082\n\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 n, F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\ny : \u2115 \u2192 \u03b3\nhy : \u2200 (n : \u2115), y n \u2208 \u2191(s n)\n\u22a2 x \u2208 range f"}, {"tactic": "have I : \u2200 m n, ((s m).1 \u2229 (s n).1).Nonempty := by\n  intro m n\n  rw [\u2190 not_disjoint_iff_nonempty_inter]\n  by_contra' h\n  have A : x \u2208 q \u27e8(s m, s n), h\u27e9 \\ q \u27e8(s n, s m), h.symm\u27e9 :=\n    haveI := mem_iInter.1 (hxs m).2 (s n)\n    (mem_iInter.1 this h : _)\n  have B : x \u2208 q \u27e8(s n, s m), h.symm\u27e9 \\ q \u27e8(s m, s n), h\u27e9 :=\n    haveI := mem_iInter.1 (hxs n).2 (s m)\n    (mem_iInter.1 this h.symm : _)\n  exact A.2 B.1", "annotated_tactic": ["have I : \u2200 m n, ((s m).1 \u2229 (s n).1).<a>Nonempty</a> := by\n      intro m n\n      rw [\u2190 <a>not_disjoint_iff_nonempty_inter</a>]\n      by_contra' h\n      have A : x \u2208 q \u27e8(s m, s n), h\u27e9 \\ q \u27e8(s n, s m), h.symm\u27e9 :=\n        haveI := <a>mem_iInter</a>.1 (hxs m).2 (s n)\n        (<a>mem_iInter</a>.1 this h : _)\n      have B : x \u2208 q \u27e8(s n, s m), h.symm\u27e9 \\ q \u27e8(s m, s n), h\u27e9 :=\n        haveI := <a>mem_iInter</a>.1 (hxs n).2 (s m)\n        (<a>mem_iInter</a>.1 this h.symm : _)\n      exact A.2 B.1", [{"full_name": "Set.Nonempty", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [439, 15], "def_end_pos": [439, 23]}, {"full_name": "Set.not_disjoint_iff_nonempty_inter", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1557, 7], "def_end_pos": [1557, 38]}, {"full_name": "Set.mem_iInter", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [207, 9], "def_end_pos": [207, 19]}, {"full_name": "Set.mem_iInter", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [207, 9], "def_end_pos": [207, 19]}, {"full_name": "Set.mem_iInter", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [207, 9], "def_end_pos": [207, 19]}, {"full_name": "Set.mem_iInter", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [207, 9], "def_end_pos": [207, 19]}]], "state_before": "case intro.intro.intro.intro.intro.intro.h\u2082\n\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 n, F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\ny : \u2115 \u2192 \u03b3\nhy : \u2200 (n : \u2115), y n \u2208 \u2191(s n)\n\u22a2 x \u2208 range f", "state_after": "case intro.intro.intro.intro.intro.intro.h\u2082\n\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 n, F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\ny : \u2115 \u2192 \u03b3\nhy : \u2200 (n : \u2115), y n \u2208 \u2191(s n)\nI : \u2200 (m n : \u2115), Set.Nonempty (\u2191(s m) \u2229 \u2191(s n))\n\u22a2 x \u2208 range f"}, {"tactic": "have cauchy_y : CauchySeq y := by\n  have : Tendsto (fun n => 2 * u n) atTop (\ud835\udcdd 0) := by\n    simpa only [mul_zero] using u_lim.const_mul 2\n  refine cauchySeq_of_le_tendsto_0' (fun n => 2 * u n) (fun m n hmn => ?_) this\n  rcases I m n with \u27e8z, zsm, zsn\u27e9\n  calc\n    dist (y m) (y n) \u2264 dist (y m) z + dist z (y n) := dist_triangle _ _ _\n    _ \u2264 u m + u n :=\n      (add_le_add ((dist_le_diam_of_mem (hs m).1 (hy m) zsm).trans (hs m).2)\n        ((dist_le_diam_of_mem (hs n).1 zsn (hy n)).trans (hs n).2))\n    _ \u2264 2 * u m := by linarith [u_anti.antitone hmn]", "annotated_tactic": ["have cauchy_y : <a>CauchySeq</a> y := by\n      have : <a>Tendsto</a> (fun n => 2 * u n) <a>atTop</a> (\ud835\udcdd 0) := by\n        simpa only [<a>mul_zero</a>] using u_lim.const_mul 2\n      refine <a>cauchySeq_of_le_tendsto_0'</a> (fun n => 2 * u n) (fun m n hmn => ?_) this\n      rcases I m n with \u27e8z, zsm, zsn\u27e9\n      calc\n        <a>dist</a> (y m) (y n) \u2264 <a>dist</a> (y m) z + <a>dist</a> z (y n) := <a>dist_triangle</a> _ _ _\n        _ \u2264 u m + u n :=\n          (<a>add_le_add</a> ((<a>dist_le_diam_of_mem</a> (hs m).1 (hy m) zsm).<a>trans</a> (hs m).2)\n            ((<a>dist_le_diam_of_mem</a> (hs n).1 zsn (hy n)).<a>trans</a> (hs n).2))\n        _ \u2264 2 * u m := by linarith [u_anti.antitone hmn]", [{"full_name": "CauchySeq", "def_path": "Mathlib/Topology/UniformSpace/Cauchy.lean", "def_pos": [196, 5], "def_end_pos": [196, 14]}, {"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2939, 5], "def_end_pos": [2939, 12]}, {"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}, {"full_name": "MulZeroClass.mul_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [38, 3], "def_end_pos": [38, 11]}, {"full_name": "cauchySeq_of_le_tendsto_0'", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [1553, 9], "def_end_pos": [1553, 35]}, {"full_name": "Dist.dist", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [94, 3], "def_end_pos": [94, 7]}, {"full_name": "Dist.dist", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [94, 3], "def_end_pos": [94, 7]}, {"full_name": "Dist.dist", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [94, 3], "def_end_pos": [94, 7]}, {"full_name": "dist_triangle", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [196, 9], "def_end_pos": [196, 22]}, {"full_name": "add_le_add", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [205, 15], "def_end_pos": [205, 25]}, {"full_name": "Metric.dist_le_diam_of_mem", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [2751, 9], "def_end_pos": [2751, 28]}, {"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [120, 7], "def_end_pos": [120, 18]}, {"full_name": "Metric.dist_le_diam_of_mem", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [2751, 9], "def_end_pos": [2751, 28]}, {"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [120, 7], "def_end_pos": [120, 18]}]], "state_before": "case intro.intro.intro.intro.intro.intro.h\u2082\n\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 n, F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\ny : \u2115 \u2192 \u03b3\nhy : \u2200 (n : \u2115), y n \u2208 \u2191(s n)\nI : \u2200 (m n : \u2115), Set.Nonempty (\u2191(s m) \u2229 \u2191(s n))\n\u22a2 x \u2208 range f", "state_after": "case intro.intro.intro.intro.intro.intro.h\u2082\n\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 n, F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\ny : \u2115 \u2192 \u03b3\nhy : \u2200 (n : \u2115), y n \u2208 \u2191(s n)\nI : \u2200 (m n : \u2115), Set.Nonempty (\u2191(s m) \u2229 \u2191(s n))\ncauchy_y : CauchySeq y\n\u22a2 x \u2208 range f"}, {"tactic": "haveI : Nonempty \u03b3 := \u27e8y 0\u27e9", "annotated_tactic": ["haveI : <a>Nonempty</a> \u03b3 := \u27e8y 0\u27e9", [{"full_name": "Nonempty", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [686, 17], "def_end_pos": [686, 25]}]], "state_before": "case intro.intro.intro.intro.intro.intro.h\u2082\n\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 n, F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\ny : \u2115 \u2192 \u03b3\nhy : \u2200 (n : \u2115), y n \u2208 \u2191(s n)\nI : \u2200 (m n : \u2115), Set.Nonempty (\u2191(s m) \u2229 \u2191(s n))\ncauchy_y : CauchySeq y\n\u22a2 x \u2208 range f", "state_after": "case intro.intro.intro.intro.intro.intro.h\u2082\n\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis\u271d : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 n, F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\ny : \u2115 \u2192 \u03b3\nhy : \u2200 (n : \u2115), y n \u2208 \u2191(s n)\nI : \u2200 (m n : \u2115), Set.Nonempty (\u2191(s m) \u2229 \u2191(s n))\ncauchy_y : CauchySeq y\nthis : Nonempty \u03b3\n\u22a2 x \u2208 range f"}, {"tactic": "let z := limUnder atTop y", "annotated_tactic": ["let z := <a>limUnder</a> <a>atTop</a> y", [{"full_name": "limUnder", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [1550, 19], "def_end_pos": [1550, 27]}, {"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}]], "state_before": "case intro.intro.intro.intro.intro.intro.h\u2082\n\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis\u271d : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 n, F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\ny : \u2115 \u2192 \u03b3\nhy : \u2200 (n : \u2115), y n \u2208 \u2191(s n)\nI : \u2200 (m n : \u2115), Set.Nonempty (\u2191(s m) \u2229 \u2191(s n))\ncauchy_y : CauchySeq y\nthis : Nonempty \u03b3\n\u22a2 x \u2208 range f", "state_after": "case intro.intro.intro.intro.intro.intro.h\u2082\n\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis\u271d : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 n, F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\ny : \u2115 \u2192 \u03b3\nhy : \u2200 (n : \u2115), y n \u2208 \u2191(s n)\nI : \u2200 (m n : \u2115), Set.Nonempty (\u2191(s m) \u2229 \u2191(s n))\ncauchy_y : CauchySeq y\nthis : Nonempty \u03b3\nz : \u03b3 := limUnder atTop y\n\u22a2 x \u2208 range f"}, {"tactic": "have y_lim : Tendsto y atTop (\ud835\udcdd z) := cauchy_y.tendsto_limUnder", "annotated_tactic": ["have y_lim : <a>Tendsto</a> y <a>atTop</a> (\ud835\udcdd z) := cauchy_y.tendsto_limUnder", [{"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2939, 5], "def_end_pos": [2939, 12]}, {"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}]], "state_before": "case intro.intro.intro.intro.intro.intro.h\u2082\n\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis\u271d : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 n, F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\ny : \u2115 \u2192 \u03b3\nhy : \u2200 (n : \u2115), y n \u2208 \u2191(s n)\nI : \u2200 (m n : \u2115), Set.Nonempty (\u2191(s m) \u2229 \u2191(s n))\ncauchy_y : CauchySeq y\nthis : Nonempty \u03b3\nz : \u03b3 := limUnder atTop y\n\u22a2 x \u2208 range f", "state_after": "case intro.intro.intro.intro.intro.intro.h\u2082\n\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis\u271d : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 n, F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\ny : \u2115 \u2192 \u03b3\nhy : \u2200 (n : \u2115), y n \u2208 \u2191(s n)\nI : \u2200 (m n : \u2115), Set.Nonempty (\u2191(s m) \u2229 \u2191(s n))\ncauchy_y : CauchySeq y\nthis : Nonempty \u03b3\nz : \u03b3 := limUnder atTop y\ny_lim : Tendsto y atTop (\ud835\udcdd z)\n\u22a2 x \u2208 range f"}, {"tactic": "suffices f z = x by\n  rw [\u2190 this]\n  exact mem_range_self _", "annotated_tactic": ["suffices f z = x by\n      rw [\u2190 this]\n      exact <a>mem_range_self</a> _", [{"full_name": "Set.mem_range_self", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [680, 9], "def_end_pos": [680, 23]}]], "state_before": "case intro.intro.intro.intro.intro.intro.h\u2082\n\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis\u271d : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 n, F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\ny : \u2115 \u2192 \u03b3\nhy : \u2200 (n : \u2115), y n \u2208 \u2191(s n)\nI : \u2200 (m n : \u2115), Set.Nonempty (\u2191(s m) \u2229 \u2191(s n))\ncauchy_y : CauchySeq y\nthis : Nonempty \u03b3\nz : \u03b3 := limUnder atTop y\ny_lim : Tendsto y atTop (\ud835\udcdd z)\n\u22a2 x \u2208 range f", "state_after": "case intro.intro.intro.intro.intro.intro.h\u2082\n\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis\u271d : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 n, F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\ny : \u2115 \u2192 \u03b3\nhy : \u2200 (n : \u2115), y n \u2208 \u2191(s n)\nI : \u2200 (m n : \u2115), Set.Nonempty (\u2191(s m) \u2229 \u2191(s n))\ncauchy_y : CauchySeq y\nthis : Nonempty \u03b3\nz : \u03b3 := limUnder atTop y\ny_lim : Tendsto y atTop (\ud835\udcdd z)\n\u22a2 f z = x"}, {"tactic": "by_contra' hne", "annotated_tactic": ["by_contra' hne", []], "state_before": "case intro.intro.intro.intro.intro.intro.h\u2082\n\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis\u271d : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 n, F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\ny : \u2115 \u2192 \u03b3\nhy : \u2200 (n : \u2115), y n \u2208 \u2191(s n)\nI : \u2200 (m n : \u2115), Set.Nonempty (\u2191(s m) \u2229 \u2191(s n))\ncauchy_y : CauchySeq y\nthis : Nonempty \u03b3\nz : \u03b3 := limUnder atTop y\ny_lim : Tendsto y atTop (\ud835\udcdd z)\n\u22a2 f z = x", "state_after": "case intro.intro.intro.intro.intro.intro.h\u2082\n\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis\u271d : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 n, F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\ny : \u2115 \u2192 \u03b3\nhy : \u2200 (n : \u2115), y n \u2208 \u2191(s n)\nI : \u2200 (m n : \u2115), Set.Nonempty (\u2191(s m) \u2229 \u2191(s n))\ncauchy_y : CauchySeq y\nthis : Nonempty \u03b3\nz : \u03b3 := limUnder atTop y\ny_lim : Tendsto y atTop (\ud835\udcdd z)\nhne : f z \u2260 x\n\u22a2 False"}, {"tactic": "obtain \u27e8v, w, v_open, w_open, fzv, xw, hvw\u27e9 := t2_separation hne", "annotated_tactic": ["obtain \u27e8v, w, v_open, w_open, fzv, xw, hvw\u27e9 := <a>t2_separation</a> hne", [{"full_name": "t2_separation", "def_path": "Mathlib/Topology/Separation.lean", "def_pos": [906, 9], "def_end_pos": [906, 22]}]], "state_before": "case intro.intro.intro.intro.intro.intro.h\u2082\n\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis\u271d : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 n, F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\ny : \u2115 \u2192 \u03b3\nhy : \u2200 (n : \u2115), y n \u2208 \u2191(s n)\nI : \u2200 (m n : \u2115), Set.Nonempty (\u2191(s m) \u2229 \u2191(s n))\ncauchy_y : CauchySeq y\nthis : Nonempty \u03b3\nz : \u03b3 := limUnder atTop y\ny_lim : Tendsto y atTop (\ud835\udcdd z)\nhne : f z \u2260 x\n\u22a2 False", "state_after": "case intro.intro.intro.intro.intro.intro.h\u2082.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis\u271d : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 n, F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\ny : \u2115 \u2192 \u03b3\nhy : \u2200 (n : \u2115), y n \u2208 \u2191(s n)\nI : \u2200 (m n : \u2115), Set.Nonempty (\u2191(s m) \u2229 \u2191(s n))\ncauchy_y : CauchySeq y\nthis : Nonempty \u03b3\nz : \u03b3 := limUnder atTop y\ny_lim : Tendsto y atTop (\ud835\udcdd z)\nhne : f z \u2260 x\nv w : Set \u03b2\nv_open : IsOpen v\nw_open : IsOpen w\nfzv : f z \u2208 v\nxw : x \u2208 w\nhvw : Disjoint v w\n\u22a2 False"}, {"tactic": "obtain \u27e8\u03b4, \u03b4pos, h\u03b4\u27e9 : \u2203 \u03b4 > (0 : \u211d), ball z \u03b4 \u2286 f \u207b\u00b9' v := by\n  apply Metric.mem_nhds_iff.1\n  exact f_cont.continuousAt.preimage_mem_nhds (v_open.mem_nhds fzv)", "annotated_tactic": ["obtain \u27e8\u03b4, \u03b4pos, h\u03b4\u27e9 : \u2203 \u03b4 > (0 : \u211d), <a>ball</a> z \u03b4 \u2286 f \u207b\u00b9' v := by\n      apply <a>Metric.mem_nhds_iff</a>.1\n      exact f_cont.continuousAt.preimage_mem_nhds (v_open.mem_nhds fzv)", [{"full_name": "Metric.ball", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [409, 5], "def_end_pos": [409, 9]}, {"full_name": "Metric.mem_nhds_iff", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [954, 9], "def_end_pos": [954, 21]}]], "state_before": "case intro.intro.intro.intro.intro.intro.h\u2082.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis\u271d : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 n, F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\ny : \u2115 \u2192 \u03b3\nhy : \u2200 (n : \u2115), y n \u2208 \u2191(s n)\nI : \u2200 (m n : \u2115), Set.Nonempty (\u2191(s m) \u2229 \u2191(s n))\ncauchy_y : CauchySeq y\nthis : Nonempty \u03b3\nz : \u03b3 := limUnder atTop y\ny_lim : Tendsto y atTop (\ud835\udcdd z)\nhne : f z \u2260 x\nv w : Set \u03b2\nv_open : IsOpen v\nw_open : IsOpen w\nfzv : f z \u2208 v\nxw : x \u2208 w\nhvw : Disjoint v w\n\u22a2 False", "state_after": "case intro.intro.intro.intro.intro.intro.h\u2082.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis\u271d : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 n, F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\ny : \u2115 \u2192 \u03b3\nhy : \u2200 (n : \u2115), y n \u2208 \u2191(s n)\nI : \u2200 (m n : \u2115), Set.Nonempty (\u2191(s m) \u2229 \u2191(s n))\ncauchy_y : CauchySeq y\nthis : Nonempty \u03b3\nz : \u03b3 := limUnder atTop y\ny_lim : Tendsto y atTop (\ud835\udcdd z)\nhne : f z \u2260 x\nv w : Set \u03b2\nv_open : IsOpen v\nw_open : IsOpen w\nfzv : f z \u2208 v\nxw : x \u2208 w\nhvw : Disjoint v w\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\nh\u03b4 : ball z \u03b4 \u2286 f \u207b\u00b9' v\n\u22a2 False"}, {"tactic": "obtain \u27e8n, hn\u27e9 : \u2203 n, u n + dist (y n) z < \u03b4 :=\n  haveI : Tendsto (fun n => u n + dist (y n) z) atTop (\ud835\udcdd 0) := by\n    simpa only [add_zero] using u_lim.add (tendsto_iff_dist_tendsto_zero.1 y_lim)\n  ((tendsto_order.1 this).2 _ \u03b4pos).exists", "annotated_tactic": ["obtain \u27e8n, hn\u27e9 : \u2203 n, u n + <a>dist</a> (y n) z < \u03b4 :=\n      haveI : <a>Tendsto</a> (fun n => u n + <a>dist</a> (y n) z) <a>atTop</a> (\ud835\udcdd 0) := by\n        simpa only [<a>add_zero</a>] using u_lim.add (<a>tendsto_iff_dist_tendsto_zero</a>.1 y_lim)\n      ((<a>tendsto_order</a>.1 this).2 _ \u03b4pos).<a>exists</a>", [{"full_name": "Dist.dist", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [94, 3], "def_end_pos": [94, 7]}, {"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2939, 5], "def_end_pos": [2939, 12]}, {"full_name": "Dist.dist", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [94, 3], "def_end_pos": [94, 7]}, {"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}, {"full_name": "add_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [469, 3], "def_end_pos": [469, 14]}, {"full_name": "tendsto_iff_dist_tendsto_zero", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [1850, 9], "def_end_pos": [1850, 38]}, {"full_name": "tendsto_order", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [919, 9], "def_end_pos": [919, 22]}, {"full_name": "Filter.Eventually.exists", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1308, 9], "def_end_pos": [1308, 26]}]], "state_before": "case intro.intro.intro.intro.intro.intro.h\u2082.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis\u271d : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 n, F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\ny : \u2115 \u2192 \u03b3\nhy : \u2200 (n : \u2115), y n \u2208 \u2191(s n)\nI : \u2200 (m n : \u2115), Set.Nonempty (\u2191(s m) \u2229 \u2191(s n))\ncauchy_y : CauchySeq y\nthis : Nonempty \u03b3\nz : \u03b3 := limUnder atTop y\ny_lim : Tendsto y atTop (\ud835\udcdd z)\nhne : f z \u2260 x\nv w : Set \u03b2\nv_open : IsOpen v\nw_open : IsOpen w\nfzv : f z \u2208 v\nxw : x \u2208 w\nhvw : Disjoint v w\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\nh\u03b4 : ball z \u03b4 \u2286 f \u207b\u00b9' v\n\u22a2 False", "state_after": "case intro.intro.intro.intro.intro.intro.h\u2082.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis\u271d : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 n, F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\ny : \u2115 \u2192 \u03b3\nhy : \u2200 (n : \u2115), y n \u2208 \u2191(s n)\nI : \u2200 (m n : \u2115), Set.Nonempty (\u2191(s m) \u2229 \u2191(s n))\ncauchy_y : CauchySeq y\nthis : Nonempty \u03b3\nz : \u03b3 := limUnder atTop y\ny_lim : Tendsto y atTop (\ud835\udcdd z)\nhne : f z \u2260 x\nv w : Set \u03b2\nv_open : IsOpen v\nw_open : IsOpen w\nfzv : f z \u2208 v\nxw : x \u2208 w\nhvw : Disjoint v w\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\nh\u03b4 : ball z \u03b4 \u2286 f \u207b\u00b9' v\nn : \u2115\nhn : u n + dist (y n) z < \u03b4\n\u22a2 False"}, {"tactic": "have fsnv : f '' s n \u2286 v := by\n  rw [image_subset_iff]\n  apply Subset.trans _ h\u03b4\n  intro a ha\n  calc\n    dist a z \u2264 dist a (y n) + dist (y n) z := dist_triangle _ _ _\n    _ \u2264 u n + dist (y n) z :=\n      (add_le_add_right ((dist_le_diam_of_mem (hs n).1 ha (hy n)).trans (hs n).2) _)\n    _ < \u03b4 := hn", "annotated_tactic": ["have fsnv : f '' s n \u2286 v := by\n      rw [<a>image_subset_iff</a>]\n      apply <a>Subset.trans</a> _ h\u03b4\n      intro a ha\n      calc\n        <a>dist</a> a z \u2264 <a>dist</a> a (y n) + <a>dist</a> (y n) z := <a>dist_triangle</a> _ _ _\n        _ \u2264 u n + <a>dist</a> (y n) z :=\n          (<a>add_le_add_right</a> ((<a>dist_le_diam_of_mem</a> (hs n).1 ha (hy n)).<a>trans</a> (hs n).2) _)\n        _ < \u03b4 := hn", [{"full_name": "Set.image_subset_iff", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [497, 9], "def_end_pos": [497, 25]}, {"full_name": "Set.Subset.trans", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [362, 9], "def_end_pos": [362, 21]}, {"full_name": "Dist.dist", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [94, 3], "def_end_pos": [94, 7]}, {"full_name": "Dist.dist", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [94, 3], "def_end_pos": [94, 7]}, {"full_name": "Dist.dist", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [94, 3], "def_end_pos": [94, 7]}, {"full_name": "dist_triangle", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [196, 9], "def_end_pos": [196, 22]}, {"full_name": "Dist.dist", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [94, 3], "def_end_pos": [94, 7]}, {"full_name": "add_le_add_right", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [66, 15], "def_end_pos": [66, 31]}, {"full_name": "Metric.dist_le_diam_of_mem", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [2751, 9], "def_end_pos": [2751, 28]}, {"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [120, 7], "def_end_pos": [120, 18]}]], "state_before": "case intro.intro.intro.intro.intro.intro.h\u2082.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis\u271d : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 n, F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\ny : \u2115 \u2192 \u03b3\nhy : \u2200 (n : \u2115), y n \u2208 \u2191(s n)\nI : \u2200 (m n : \u2115), Set.Nonempty (\u2191(s m) \u2229 \u2191(s n))\ncauchy_y : CauchySeq y\nthis : Nonempty \u03b3\nz : \u03b3 := limUnder atTop y\ny_lim : Tendsto y atTop (\ud835\udcdd z)\nhne : f z \u2260 x\nv w : Set \u03b2\nv_open : IsOpen v\nw_open : IsOpen w\nfzv : f z \u2208 v\nxw : x \u2208 w\nhvw : Disjoint v w\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\nh\u03b4 : ball z \u03b4 \u2286 f \u207b\u00b9' v\nn : \u2115\nhn : u n + dist (y n) z < \u03b4\n\u22a2 False", "state_after": "case intro.intro.intro.intro.intro.intro.h\u2082.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis\u271d : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 n, F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\ny : \u2115 \u2192 \u03b3\nhy : \u2200 (n : \u2115), y n \u2208 \u2191(s n)\nI : \u2200 (m n : \u2115), Set.Nonempty (\u2191(s m) \u2229 \u2191(s n))\ncauchy_y : CauchySeq y\nthis : Nonempty \u03b3\nz : \u03b3 := limUnder atTop y\ny_lim : Tendsto y atTop (\ud835\udcdd z)\nhne : f z \u2260 x\nv w : Set \u03b2\nv_open : IsOpen v\nw_open : IsOpen w\nfzv : f z \u2208 v\nxw : x \u2208 w\nhvw : Disjoint v w\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\nh\u03b4 : ball z \u03b4 \u2286 f \u207b\u00b9' v\nn : \u2115\nhn : u n + dist (y n) z < \u03b4\nfsnv : f '' \u2191(s n) \u2286 v\n\u22a2 False"}, {"tactic": "have : x \u2208 closure v := closure_mono fsnv (hxs n).1", "annotated_tactic": ["have : x \u2208 <a>closure</a> v := <a>closure_mono</a> fsnv (hxs n).1", [{"full_name": "closure", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [422, 5], "def_end_pos": [422, 12]}, {"full_name": "closure_mono", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [475, 9], "def_end_pos": [475, 21]}]], "state_before": "case intro.intro.intro.intro.intro.intro.h\u2082.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis\u271d : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 n, F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\ny : \u2115 \u2192 \u03b3\nhy : \u2200 (n : \u2115), y n \u2208 \u2191(s n)\nI : \u2200 (m n : \u2115), Set.Nonempty (\u2191(s m) \u2229 \u2191(s n))\ncauchy_y : CauchySeq y\nthis : Nonempty \u03b3\nz : \u03b3 := limUnder atTop y\ny_lim : Tendsto y atTop (\ud835\udcdd z)\nhne : f z \u2260 x\nv w : Set \u03b2\nv_open : IsOpen v\nw_open : IsOpen w\nfzv : f z \u2208 v\nxw : x \u2208 w\nhvw : Disjoint v w\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\nh\u03b4 : ball z \u03b4 \u2286 f \u207b\u00b9' v\nn : \u2115\nhn : u n + dist (y n) z < \u03b4\nfsnv : f '' \u2191(s n) \u2286 v\n\u22a2 False", "state_after": "case intro.intro.intro.intro.intro.intro.h\u2082.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d\u00b2 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis\u271d\u00b9 : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 n, F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\ny : \u2115 \u2192 \u03b3\nhy : \u2200 (n : \u2115), y n \u2208 \u2191(s n)\nI : \u2200 (m n : \u2115), Set.Nonempty (\u2191(s m) \u2229 \u2191(s n))\ncauchy_y : CauchySeq y\nthis\u271d : Nonempty \u03b3\nz : \u03b3 := limUnder atTop y\ny_lim : Tendsto y atTop (\ud835\udcdd z)\nhne : f z \u2260 x\nv w : Set \u03b2\nv_open : IsOpen v\nw_open : IsOpen w\nfzv : f z \u2208 v\nxw : x \u2208 w\nhvw : Disjoint v w\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\nh\u03b4 : ball z \u03b4 \u2286 f \u207b\u00b9' v\nn : \u2115\nhn : u n + dist (y n) z < \u03b4\nfsnv : f '' \u2191(s n) \u2286 v\nthis : x \u2208 closure v\n\u22a2 False"}, {"tactic": "exact disjoint_left.1 (hvw.closure_left w_open) this xw", "annotated_tactic": ["exact <a>disjoint_left</a>.1 (hvw.closure_left w_open) this xw", [{"full_name": "Set.disjoint_left", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1546, 9], "def_end_pos": [1546, 22]}]], "state_before": "case intro.intro.intro.intro.intro.intro.h\u2082.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d\u00b2 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis\u271d\u00b9 : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 n, F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\ny : \u2115 \u2192 \u03b3\nhy : \u2200 (n : \u2115), y n \u2208 \u2191(s n)\nI : \u2200 (m n : \u2115), Set.Nonempty (\u2191(s m) \u2229 \u2191(s n))\ncauchy_y : CauchySeq y\nthis\u271d : Nonempty \u03b3\nz : \u03b3 := limUnder atTop y\ny_lim : Tendsto y atTop (\ud835\udcdd z)\nhne : f z \u2260 x\nv w : Set \u03b2\nv_open : IsOpen v\nw_open : IsOpen w\nfzv : f z \u2208 v\nxw : x \u2208 w\nhvw : Disjoint v w\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\nh\u03b4 : ball z \u03b4 \u2286 f \u207b\u00b9' v\nn : \u2115\nhn : u n + dist (y n) z < \u03b4\nfsnv : f '' \u2191(s n) \u2286 v\nthis : x \u2208 closure v\n\u22a2 False", "state_after": "no goals"}, {"tactic": "simpa only [mem_iUnion] using mem_iInter.1 hx n", "annotated_tactic": ["simpa only [<a>mem_iUnion</a>] using <a>mem_iInter</a>.1 hx n", [{"full_name": "Set.mem_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [201, 9], "def_end_pos": [201, 19]}, {"full_name": "Set.mem_iInter", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [207, 9], "def_end_pos": [207, 19]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 n, F n\nn : \u2115\n\u22a2 \u2203 s x_1, x \u2208 E s", "state_after": "no goals"}, {"tactic": "intro n", "annotated_tactic": ["intro n", []], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 n, F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\n\u22a2 \u2200 (n : \u2115), Set.Nonempty \u2191(s n)", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 n, F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\nn : \u2115\n\u22a2 Set.Nonempty \u2191(s n)"}, {"tactic": "rw [nonempty_iff_ne_empty]", "annotated_tactic": ["rw [<a>nonempty_iff_ne_empty</a>]", [{"full_name": "Set.nonempty_iff_ne_empty", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [610, 9], "def_end_pos": [610, 30]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 n, F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\nn : \u2115\n\u22a2 Set.Nonempty \u2191(s n)", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 n, F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\nn : \u2115\n\u22a2 \u2191(s n) \u2260 \u2205"}, {"tactic": "intro hn", "annotated_tactic": ["intro hn", []], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 n, F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\nn : \u2115\n\u22a2 \u2191(s n) \u2260 \u2205", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 n, F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\nn : \u2115\nhn : \u2191(s n) = \u2205\n\u22a2 False"}, {"tactic": "have := (s n).2", "annotated_tactic": ["have := (s n).2", []], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 n, F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\nn : \u2115\nhn : \u2191(s n) = \u2205\n\u22a2 False", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis\u271d : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 n, F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\nn : \u2115\nhn : \u2191(s n) = \u2205\nthis : \u2191(s n) \u2208 b\n\u22a2 False"}, {"tactic": "rw [hn] at this", "annotated_tactic": ["rw [hn] at this", []], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis\u271d : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 n, F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\nn : \u2115\nhn : \u2191(s n) = \u2205\nthis : \u2191(s n) \u2208 b\n\u22a2 False", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis\u271d : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 n, F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\nn : \u2115\nhn : \u2191(s n) = \u2205\nthis : \u2205 \u2208 b\n\u22a2 False"}, {"tactic": "exact b_nonempty this", "annotated_tactic": ["exact b_nonempty this", []], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis\u271d : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 n, F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\nn : \u2115\nhn : \u2191(s n) = \u2205\nthis : \u2205 \u2208 b\n\u22a2 False", "state_after": "no goals"}, {"tactic": "intro m n", "annotated_tactic": ["intro m n", []], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 n, F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\ny : \u2115 \u2192 \u03b3\nhy : \u2200 (n : \u2115), y n \u2208 \u2191(s n)\n\u22a2 \u2200 (m n : \u2115), Set.Nonempty (\u2191(s m) \u2229 \u2191(s n))", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 n, F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\ny : \u2115 \u2192 \u03b3\nhy : \u2200 (n : \u2115), y n \u2208 \u2191(s n)\nm n : \u2115\n\u22a2 Set.Nonempty (\u2191(s m) \u2229 \u2191(s n))"}, {"tactic": "rw [\u2190 not_disjoint_iff_nonempty_inter]", "annotated_tactic": ["rw [\u2190 <a>not_disjoint_iff_nonempty_inter</a>]", [{"full_name": "Set.not_disjoint_iff_nonempty_inter", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1557, 7], "def_end_pos": [1557, 38]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 n, F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\ny : \u2115 \u2192 \u03b3\nhy : \u2200 (n : \u2115), y n \u2208 \u2191(s n)\nm n : \u2115\n\u22a2 Set.Nonempty (\u2191(s m) \u2229 \u2191(s n))", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 n, F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\ny : \u2115 \u2192 \u03b3\nhy : \u2200 (n : \u2115), y n \u2208 \u2191(s n)\nm n : \u2115\n\u22a2 \u00acDisjoint \u2191(s m) \u2191(s n)"}, {"tactic": "by_contra' h", "annotated_tactic": ["by_contra' h", []], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 n, F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\ny : \u2115 \u2192 \u03b3\nhy : \u2200 (n : \u2115), y n \u2208 \u2191(s n)\nm n : \u2115\n\u22a2 \u00acDisjoint \u2191(s m) \u2191(s n)", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 n, F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\ny : \u2115 \u2192 \u03b3\nhy : \u2200 (n : \u2115), y n \u2208 \u2191(s n)\nm n : \u2115\nh : Disjoint \u2191(s m) \u2191(s n)\n\u22a2 False"}, {"tactic": "have A : x \u2208 q \u27e8(s m, s n), h\u27e9 \\ q \u27e8(s n, s m), h.symm\u27e9 :=\n  haveI := mem_iInter.1 (hxs m).2 (s n)\n  (mem_iInter.1 this h : _)", "annotated_tactic": ["have A : x \u2208 q \u27e8(s m, s n), h\u27e9 \\ q \u27e8(s n, s m), h.symm\u27e9 :=\n        haveI := <a>mem_iInter</a>.1 (hxs m).2 (s n)\n        (<a>mem_iInter</a>.1 this h : _)", [{"full_name": "Set.mem_iInter", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [207, 9], "def_end_pos": [207, 19]}, {"full_name": "Set.mem_iInter", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [207, 9], "def_end_pos": [207, 19]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 n, F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\ny : \u2115 \u2192 \u03b3\nhy : \u2200 (n : \u2115), y n \u2208 \u2191(s n)\nm n : \u2115\nh : Disjoint \u2191(s m) \u2191(s n)\n\u22a2 False", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA\u271d : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A\u271d \u2192 Set \u03b2\nhq1 : \u2200 (p : A\u271d), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A\u271d), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A\u271d), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 n, F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\ny : \u2115 \u2192 \u03b3\nhy : \u2200 (n : \u2115), y n \u2208 \u2191(s n)\nm n : \u2115\nh : Disjoint \u2191(s m) \u2191(s n)\nA : x \u2208 q { val := (s m, s n), property := h } \\ q { val := (s n, s m), property := (_ : Disjoint \u2191(s n) \u2191(s m)) }\n\u22a2 False"}, {"tactic": "have B : x \u2208 q \u27e8(s n, s m), h.symm\u27e9 \\ q \u27e8(s m, s n), h\u27e9 :=\n  haveI := mem_iInter.1 (hxs n).2 (s m)\n  (mem_iInter.1 this h.symm : _)", "annotated_tactic": ["have B : x \u2208 q \u27e8(s n, s m), h.symm\u27e9 \\ q \u27e8(s m, s n), h\u27e9 :=\n        haveI := <a>mem_iInter</a>.1 (hxs n).2 (s m)\n        (<a>mem_iInter</a>.1 this h.symm : _)", [{"full_name": "Set.mem_iInter", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [207, 9], "def_end_pos": [207, 19]}, {"full_name": "Set.mem_iInter", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [207, 9], "def_end_pos": [207, 19]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA\u271d : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A\u271d \u2192 Set \u03b2\nhq1 : \u2200 (p : A\u271d), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A\u271d), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A\u271d), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 n, F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\ny : \u2115 \u2192 \u03b3\nhy : \u2200 (n : \u2115), y n \u2208 \u2191(s n)\nm n : \u2115\nh : Disjoint \u2191(s m) \u2191(s n)\nA : x \u2208 q { val := (s m, s n), property := h } \\ q { val := (s n, s m), property := (_ : Disjoint \u2191(s n) \u2191(s m)) }\n\u22a2 False", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA\u271d : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A\u271d \u2192 Set \u03b2\nhq1 : \u2200 (p : A\u271d), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A\u271d), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A\u271d), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 n, F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\ny : \u2115 \u2192 \u03b3\nhy : \u2200 (n : \u2115), y n \u2208 \u2191(s n)\nm n : \u2115\nh : Disjoint \u2191(s m) \u2191(s n)\nA : x \u2208 q { val := (s m, s n), property := h } \\ q { val := (s n, s m), property := (_ : Disjoint \u2191(s n) \u2191(s m)) }\nB : x \u2208 q { val := (s n, s m), property := (_ : Disjoint \u2191(s n) \u2191(s m)) } \\ q { val := (s m, s n), property := h }\n\u22a2 False"}, {"tactic": "exact A.2 B.1", "annotated_tactic": ["exact A.2 B.1", []], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA\u271d : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A\u271d \u2192 Set \u03b2\nhq1 : \u2200 (p : A\u271d), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A\u271d), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A\u271d), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 n, F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\ny : \u2115 \u2192 \u03b3\nhy : \u2200 (n : \u2115), y n \u2208 \u2191(s n)\nm n : \u2115\nh : Disjoint \u2191(s m) \u2191(s n)\nA : x \u2208 q { val := (s m, s n), property := h } \\ q { val := (s n, s m), property := (_ : Disjoint \u2191(s n) \u2191(s m)) }\nB : x \u2208 q { val := (s n, s m), property := (_ : Disjoint \u2191(s n) \u2191(s m)) } \\ q { val := (s m, s n), property := h }\n\u22a2 False", "state_after": "no goals"}, {"tactic": "have : Tendsto (fun n => 2 * u n) atTop (\ud835\udcdd 0) := by\n  simpa only [mul_zero] using u_lim.const_mul 2", "annotated_tactic": ["have : <a>Tendsto</a> (fun n => 2 * u n) <a>atTop</a> (\ud835\udcdd 0) := by\n        simpa only [<a>mul_zero</a>] using u_lim.const_mul 2", [{"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2939, 5], "def_end_pos": [2939, 12]}, {"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}, {"full_name": "MulZeroClass.mul_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [38, 3], "def_end_pos": [38, 11]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 n, F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\ny : \u2115 \u2192 \u03b3\nhy : \u2200 (n : \u2115), y n \u2208 \u2191(s n)\nI : \u2200 (m n : \u2115), Set.Nonempty (\u2191(s m) \u2229 \u2191(s n))\n\u22a2 CauchySeq y", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis\u271d : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 n, F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\ny : \u2115 \u2192 \u03b3\nhy : \u2200 (n : \u2115), y n \u2208 \u2191(s n)\nI : \u2200 (m n : \u2115), Set.Nonempty (\u2191(s m) \u2229 \u2191(s n))\nthis : Tendsto (fun n => 2 * u n) atTop (\ud835\udcdd 0)\n\u22a2 CauchySeq y"}, {"tactic": "refine cauchySeq_of_le_tendsto_0' (fun n => 2 * u n) (fun m n hmn => ?_) this", "annotated_tactic": ["refine <a>cauchySeq_of_le_tendsto_0'</a> (fun n => 2 * u n) (fun m n hmn => ?_) this", [{"full_name": "cauchySeq_of_le_tendsto_0'", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [1553, 9], "def_end_pos": [1553, 35]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis\u271d : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 n, F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\ny : \u2115 \u2192 \u03b3\nhy : \u2200 (n : \u2115), y n \u2208 \u2191(s n)\nI : \u2200 (m n : \u2115), Set.Nonempty (\u2191(s m) \u2229 \u2191(s n))\nthis : Tendsto (fun n => 2 * u n) atTop (\ud835\udcdd 0)\n\u22a2 CauchySeq y", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis\u271d : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 n, F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\ny : \u2115 \u2192 \u03b3\nhy : \u2200 (n : \u2115), y n \u2208 \u2191(s n)\nI : \u2200 (m n : \u2115), Set.Nonempty (\u2191(s m) \u2229 \u2191(s n))\nthis : Tendsto (fun n => 2 * u n) atTop (\ud835\udcdd 0)\nm n : \u2115\nhmn : m \u2264 n\n\u22a2 dist (y m) (y n) \u2264 (fun n => 2 * u n) m"}, {"tactic": "rcases I m n with \u27e8z, zsm, zsn\u27e9", "annotated_tactic": ["rcases I m n with \u27e8z, zsm, zsn\u27e9", []], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis\u271d : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 n, F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\ny : \u2115 \u2192 \u03b3\nhy : \u2200 (n : \u2115), y n \u2208 \u2191(s n)\nI : \u2200 (m n : \u2115), Set.Nonempty (\u2191(s m) \u2229 \u2191(s n))\nthis : Tendsto (fun n => 2 * u n) atTop (\ud835\udcdd 0)\nm n : \u2115\nhmn : m \u2264 n\n\u22a2 dist (y m) (y n) \u2264 (fun n => 2 * u n) m", "state_after": "case intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis\u271d : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 n, F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\ny : \u2115 \u2192 \u03b3\nhy : \u2200 (n : \u2115), y n \u2208 \u2191(s n)\nI : \u2200 (m n : \u2115), Set.Nonempty (\u2191(s m) \u2229 \u2191(s n))\nthis : Tendsto (fun n => 2 * u n) atTop (\ud835\udcdd 0)\nm n : \u2115\nhmn : m \u2264 n\nz : \u03b3\nzsm : z \u2208 \u2191(s m)\nzsn : z \u2208 \u2191(s n)\n\u22a2 dist (y m) (y n) \u2264 (fun n => 2 * u n) m"}, {"tactic": "calc\n  dist (y m) (y n) \u2264 dist (y m) z + dist z (y n) := dist_triangle _ _ _\n  _ \u2264 u m + u n :=\n    (add_le_add ((dist_le_diam_of_mem (hs m).1 (hy m) zsm).trans (hs m).2)\n      ((dist_le_diam_of_mem (hs n).1 zsn (hy n)).trans (hs n).2))\n  _ \u2264 2 * u m := by linarith [u_anti.antitone hmn]", "annotated_tactic": ["calc\n        <a>dist</a> (y m) (y n) \u2264 <a>dist</a> (y m) z + <a>dist</a> z (y n) := <a>dist_triangle</a> _ _ _\n        _ \u2264 u m + u n :=\n          (<a>add_le_add</a> ((<a>dist_le_diam_of_mem</a> (hs m).1 (hy m) zsm).<a>trans</a> (hs m).2)\n            ((<a>dist_le_diam_of_mem</a> (hs n).1 zsn (hy n)).<a>trans</a> (hs n).2))\n        _ \u2264 2 * u m := by linarith [u_anti.antitone hmn]", [{"full_name": "Dist.dist", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [94, 3], "def_end_pos": [94, 7]}, {"full_name": "Dist.dist", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [94, 3], "def_end_pos": [94, 7]}, {"full_name": "Dist.dist", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [94, 3], "def_end_pos": [94, 7]}, {"full_name": "dist_triangle", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [196, 9], "def_end_pos": [196, 22]}, {"full_name": "add_le_add", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [205, 15], "def_end_pos": [205, 25]}, {"full_name": "Metric.dist_le_diam_of_mem", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [2751, 9], "def_end_pos": [2751, 28]}, {"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [120, 7], "def_end_pos": [120, 18]}, {"full_name": "Metric.dist_le_diam_of_mem", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [2751, 9], "def_end_pos": [2751, 28]}, {"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [120, 7], "def_end_pos": [120, 18]}]], "state_before": "case intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis\u271d : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 n, F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\ny : \u2115 \u2192 \u03b3\nhy : \u2200 (n : \u2115), y n \u2208 \u2191(s n)\nI : \u2200 (m n : \u2115), Set.Nonempty (\u2191(s m) \u2229 \u2191(s n))\nthis : Tendsto (fun n => 2 * u n) atTop (\ud835\udcdd 0)\nm n : \u2115\nhmn : m \u2264 n\nz : \u03b3\nzsm : z \u2208 \u2191(s m)\nzsn : z \u2208 \u2191(s n)\n\u22a2 dist (y m) (y n) \u2264 (fun n => 2 * u n) m", "state_after": "no goals"}, {"tactic": "simpa only [mul_zero] using u_lim.const_mul 2", "annotated_tactic": ["simpa only [<a>mul_zero</a>] using u_lim.const_mul 2", [{"full_name": "MulZeroClass.mul_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [38, 3], "def_end_pos": [38, 11]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 n, F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\ny : \u2115 \u2192 \u03b3\nhy : \u2200 (n : \u2115), y n \u2208 \u2191(s n)\nI : \u2200 (m n : \u2115), Set.Nonempty (\u2191(s m) \u2229 \u2191(s n))\n\u22a2 Tendsto (fun n => 2 * u n) atTop (\ud835\udcdd 0)", "state_after": "no goals"}, {"tactic": "linarith [u_anti.antitone hmn]", "annotated_tactic": ["linarith [u_anti.antitone hmn]", []], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis\u271d : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 n, F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\ny : \u2115 \u2192 \u03b3\nhy : \u2200 (n : \u2115), y n \u2208 \u2191(s n)\nI : \u2200 (m n : \u2115), Set.Nonempty (\u2191(s m) \u2229 \u2191(s n))\nthis : Tendsto (fun n => 2 * u n) atTop (\ud835\udcdd 0)\nm n : \u2115\nhmn : m \u2264 n\nz : \u03b3\nzsm : z \u2208 \u2191(s m)\nzsn : z \u2208 \u2191(s n)\n\u22a2 u m + u n \u2264 2 * u m", "state_after": "no goals"}, {"tactic": "rw [\u2190 this]", "annotated_tactic": ["rw [\u2190 this]", []], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d\u00b2 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis\u271d\u00b9 : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 n, F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\ny : \u2115 \u2192 \u03b3\nhy : \u2200 (n : \u2115), y n \u2208 \u2191(s n)\nI : \u2200 (m n : \u2115), Set.Nonempty (\u2191(s m) \u2229 \u2191(s n))\ncauchy_y : CauchySeq y\nthis\u271d : Nonempty \u03b3\nz : \u03b3 := limUnder atTop y\ny_lim : Tendsto y atTop (\ud835\udcdd z)\nthis : f z = x\n\u22a2 x \u2208 range f", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d\u00b2 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis\u271d\u00b9 : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 n, F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\ny : \u2115 \u2192 \u03b3\nhy : \u2200 (n : \u2115), y n \u2208 \u2191(s n)\nI : \u2200 (m n : \u2115), Set.Nonempty (\u2191(s m) \u2229 \u2191(s n))\ncauchy_y : CauchySeq y\nthis\u271d : Nonempty \u03b3\nz : \u03b3 := limUnder atTop y\ny_lim : Tendsto y atTop (\ud835\udcdd z)\nthis : f z = x\n\u22a2 f z \u2208 range f"}, {"tactic": "exact mem_range_self _", "annotated_tactic": ["exact <a>mem_range_self</a> _", [{"full_name": "Set.mem_range_self", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [680, 9], "def_end_pos": [680, 23]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d\u00b2 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis\u271d\u00b9 : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 n, F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\ny : \u2115 \u2192 \u03b3\nhy : \u2200 (n : \u2115), y n \u2208 \u2191(s n)\nI : \u2200 (m n : \u2115), Set.Nonempty (\u2191(s m) \u2229 \u2191(s n))\ncauchy_y : CauchySeq y\nthis\u271d : Nonempty \u03b3\nz : \u03b3 := limUnder atTop y\ny_lim : Tendsto y atTop (\ud835\udcdd z)\nthis : f z = x\n\u22a2 f z \u2208 range f", "state_after": "no goals"}, {"tactic": "apply Metric.mem_nhds_iff.1", "annotated_tactic": ["apply <a>Metric.mem_nhds_iff</a>.1", [{"full_name": "Metric.mem_nhds_iff", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [954, 9], "def_end_pos": [954, 21]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis\u271d : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 n, F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\ny : \u2115 \u2192 \u03b3\nhy : \u2200 (n : \u2115), y n \u2208 \u2191(s n)\nI : \u2200 (m n : \u2115), Set.Nonempty (\u2191(s m) \u2229 \u2191(s n))\ncauchy_y : CauchySeq y\nthis : Nonempty \u03b3\nz : \u03b3 := limUnder atTop y\ny_lim : Tendsto y atTop (\ud835\udcdd z)\nhne : f z \u2260 x\nv w : Set \u03b2\nv_open : IsOpen v\nw_open : IsOpen w\nfzv : f z \u2208 v\nxw : x \u2208 w\nhvw : Disjoint v w\n\u22a2 \u2203 \u03b4, \u03b4 > 0 \u2227 ball z \u03b4 \u2286 f \u207b\u00b9' v", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis\u271d : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 n, F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\ny : \u2115 \u2192 \u03b3\nhy : \u2200 (n : \u2115), y n \u2208 \u2191(s n)\nI : \u2200 (m n : \u2115), Set.Nonempty (\u2191(s m) \u2229 \u2191(s n))\ncauchy_y : CauchySeq y\nthis : Nonempty \u03b3\nz : \u03b3 := limUnder atTop y\ny_lim : Tendsto y atTop (\ud835\udcdd z)\nhne : f z \u2260 x\nv w : Set \u03b2\nv_open : IsOpen v\nw_open : IsOpen w\nfzv : f z \u2208 v\nxw : x \u2208 w\nhvw : Disjoint v w\n\u22a2 f \u207b\u00b9' v \u2208 \ud835\udcdd z"}, {"tactic": "exact f_cont.continuousAt.preimage_mem_nhds (v_open.mem_nhds fzv)", "annotated_tactic": ["exact f_cont.continuousAt.preimage_mem_nhds (v_open.mem_nhds fzv)", []], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis\u271d : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 n, F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\ny : \u2115 \u2192 \u03b3\nhy : \u2200 (n : \u2115), y n \u2208 \u2191(s n)\nI : \u2200 (m n : \u2115), Set.Nonempty (\u2191(s m) \u2229 \u2191(s n))\ncauchy_y : CauchySeq y\nthis : Nonempty \u03b3\nz : \u03b3 := limUnder atTop y\ny_lim : Tendsto y atTop (\ud835\udcdd z)\nhne : f z \u2260 x\nv w : Set \u03b2\nv_open : IsOpen v\nw_open : IsOpen w\nfzv : f z \u2208 v\nxw : x \u2208 w\nhvw : Disjoint v w\n\u22a2 f \u207b\u00b9' v \u2208 \ud835\udcdd z", "state_after": "no goals"}, {"tactic": "simpa only [add_zero] using u_lim.add (tendsto_iff_dist_tendsto_zero.1 y_lim)", "annotated_tactic": ["simpa only [<a>add_zero</a>] using u_lim.add (<a>tendsto_iff_dist_tendsto_zero</a>.1 y_lim)", [{"full_name": "add_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [469, 3], "def_end_pos": [469, 14]}, {"full_name": "tendsto_iff_dist_tendsto_zero", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [1850, 9], "def_end_pos": [1850, 38]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis\u271d : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 n, F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\ny : \u2115 \u2192 \u03b3\nhy : \u2200 (n : \u2115), y n \u2208 \u2191(s n)\nI : \u2200 (m n : \u2115), Set.Nonempty (\u2191(s m) \u2229 \u2191(s n))\ncauchy_y : CauchySeq y\nthis : Nonempty \u03b3\nz : \u03b3 := limUnder atTop y\ny_lim : Tendsto y atTop (\ud835\udcdd z)\nhne : f z \u2260 x\nv w : Set \u03b2\nv_open : IsOpen v\nw_open : IsOpen w\nfzv : f z \u2208 v\nxw : x \u2208 w\nhvw : Disjoint v w\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\nh\u03b4 : ball z \u03b4 \u2286 f \u207b\u00b9' v\n\u22a2 Tendsto (fun n => u n + dist (y n) z) atTop (\ud835\udcdd 0)", "state_after": "no goals"}, {"tactic": "rw [image_subset_iff]", "annotated_tactic": ["rw [<a>image_subset_iff</a>]", [{"full_name": "Set.image_subset_iff", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [497, 9], "def_end_pos": [497, 25]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis\u271d : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 n, F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\ny : \u2115 \u2192 \u03b3\nhy : \u2200 (n : \u2115), y n \u2208 \u2191(s n)\nI : \u2200 (m n : \u2115), Set.Nonempty (\u2191(s m) \u2229 \u2191(s n))\ncauchy_y : CauchySeq y\nthis : Nonempty \u03b3\nz : \u03b3 := limUnder atTop y\ny_lim : Tendsto y atTop (\ud835\udcdd z)\nhne : f z \u2260 x\nv w : Set \u03b2\nv_open : IsOpen v\nw_open : IsOpen w\nfzv : f z \u2208 v\nxw : x \u2208 w\nhvw : Disjoint v w\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\nh\u03b4 : ball z \u03b4 \u2286 f \u207b\u00b9' v\nn : \u2115\nhn : u n + dist (y n) z < \u03b4\n\u22a2 f '' \u2191(s n) \u2286 v", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis\u271d : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 n, F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\ny : \u2115 \u2192 \u03b3\nhy : \u2200 (n : \u2115), y n \u2208 \u2191(s n)\nI : \u2200 (m n : \u2115), Set.Nonempty (\u2191(s m) \u2229 \u2191(s n))\ncauchy_y : CauchySeq y\nthis : Nonempty \u03b3\nz : \u03b3 := limUnder atTop y\ny_lim : Tendsto y atTop (\ud835\udcdd z)\nhne : f z \u2260 x\nv w : Set \u03b2\nv_open : IsOpen v\nw_open : IsOpen w\nfzv : f z \u2208 v\nxw : x \u2208 w\nhvw : Disjoint v w\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\nh\u03b4 : ball z \u03b4 \u2286 f \u207b\u00b9' v\nn : \u2115\nhn : u n + dist (y n) z < \u03b4\n\u22a2 \u2191(s n) \u2286 f \u207b\u00b9' v"}, {"tactic": "apply Subset.trans _ h\u03b4", "annotated_tactic": ["apply <a>Subset.trans</a> _ h\u03b4", [{"full_name": "Set.Subset.trans", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [362, 9], "def_end_pos": [362, 21]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis\u271d : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 n, F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\ny : \u2115 \u2192 \u03b3\nhy : \u2200 (n : \u2115), y n \u2208 \u2191(s n)\nI : \u2200 (m n : \u2115), Set.Nonempty (\u2191(s m) \u2229 \u2191(s n))\ncauchy_y : CauchySeq y\nthis : Nonempty \u03b3\nz : \u03b3 := limUnder atTop y\ny_lim : Tendsto y atTop (\ud835\udcdd z)\nhne : f z \u2260 x\nv w : Set \u03b2\nv_open : IsOpen v\nw_open : IsOpen w\nfzv : f z \u2208 v\nxw : x \u2208 w\nhvw : Disjoint v w\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\nh\u03b4 : ball z \u03b4 \u2286 f \u207b\u00b9' v\nn : \u2115\nhn : u n + dist (y n) z < \u03b4\n\u22a2 \u2191(s n) \u2286 f \u207b\u00b9' v", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis\u271d : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 n, F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\ny : \u2115 \u2192 \u03b3\nhy : \u2200 (n : \u2115), y n \u2208 \u2191(s n)\nI : \u2200 (m n : \u2115), Set.Nonempty (\u2191(s m) \u2229 \u2191(s n))\ncauchy_y : CauchySeq y\nthis : Nonempty \u03b3\nz : \u03b3 := limUnder atTop y\ny_lim : Tendsto y atTop (\ud835\udcdd z)\nhne : f z \u2260 x\nv w : Set \u03b2\nv_open : IsOpen v\nw_open : IsOpen w\nfzv : f z \u2208 v\nxw : x \u2208 w\nhvw : Disjoint v w\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\nh\u03b4 : ball z \u03b4 \u2286 f \u207b\u00b9' v\nn : \u2115\nhn : u n + dist (y n) z < \u03b4\n\u22a2 \u2191(s n) \u2286 ball z \u03b4"}, {"tactic": "intro a ha", "annotated_tactic": ["intro a ha", []], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis\u271d : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 n, F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\ny : \u2115 \u2192 \u03b3\nhy : \u2200 (n : \u2115), y n \u2208 \u2191(s n)\nI : \u2200 (m n : \u2115), Set.Nonempty (\u2191(s m) \u2229 \u2191(s n))\ncauchy_y : CauchySeq y\nthis : Nonempty \u03b3\nz : \u03b3 := limUnder atTop y\ny_lim : Tendsto y atTop (\ud835\udcdd z)\nhne : f z \u2260 x\nv w : Set \u03b2\nv_open : IsOpen v\nw_open : IsOpen w\nfzv : f z \u2208 v\nxw : x \u2208 w\nhvw : Disjoint v w\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\nh\u03b4 : ball z \u03b4 \u2286 f \u207b\u00b9' v\nn : \u2115\nhn : u n + dist (y n) z < \u03b4\n\u22a2 \u2191(s n) \u2286 ball z \u03b4", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis\u271d : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 n, F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\ny : \u2115 \u2192 \u03b3\nhy : \u2200 (n : \u2115), y n \u2208 \u2191(s n)\nI : \u2200 (m n : \u2115), Set.Nonempty (\u2191(s m) \u2229 \u2191(s n))\ncauchy_y : CauchySeq y\nthis : Nonempty \u03b3\nz : \u03b3 := limUnder atTop y\ny_lim : Tendsto y atTop (\ud835\udcdd z)\nhne : f z \u2260 x\nv w : Set \u03b2\nv_open : IsOpen v\nw_open : IsOpen w\nfzv : f z \u2208 v\nxw : x \u2208 w\nhvw : Disjoint v w\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\nh\u03b4 : ball z \u03b4 \u2286 f \u207b\u00b9' v\nn : \u2115\nhn : u n + dist (y n) z < \u03b4\na : \u03b3\nha : a \u2208 \u2191(s n)\n\u22a2 a \u2208 ball z \u03b4"}, {"tactic": "calc\n  dist a z \u2264 dist a (y n) + dist (y n) z := dist_triangle _ _ _\n  _ \u2264 u n + dist (y n) z :=\n    (add_le_add_right ((dist_le_diam_of_mem (hs n).1 ha (hy n)).trans (hs n).2) _)\n  _ < \u03b4 := hn", "annotated_tactic": ["calc\n        <a>dist</a> a z \u2264 <a>dist</a> a (y n) + <a>dist</a> (y n) z := <a>dist_triangle</a> _ _ _\n        _ \u2264 u n + <a>dist</a> (y n) z :=\n          (<a>add_le_add_right</a> ((<a>dist_le_diam_of_mem</a> (hs n).1 ha (hy n)).<a>trans</a> (hs n).2) _)\n        _ < \u03b4 := hn", [{"full_name": "Dist.dist", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [94, 3], "def_end_pos": [94, 7]}, {"full_name": "Dist.dist", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [94, 3], "def_end_pos": [94, 7]}, {"full_name": "Dist.dist", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [94, 3], "def_end_pos": [94, 7]}, {"full_name": "dist_triangle", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [196, 9], "def_end_pos": [196, 22]}, {"full_name": "Dist.dist", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [94, 3], "def_end_pos": [94, 7]}, {"full_name": "add_le_add_right", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [66, 15], "def_end_pos": [66, 31]}, {"full_name": "Metric.dist_le_diam_of_mem", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [2751, 9], "def_end_pos": [2751, 28]}, {"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [120, 7], "def_end_pos": [120, 18]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis\u271d : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 t,\n        \u22c2 (ht : Disjoint \u2191s \u2191t),\n          q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 n, F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\ny : \u2115 \u2192 \u03b3\nhy : \u2200 (n : \u2115), y n \u2208 \u2191(s n)\nI : \u2200 (m n : \u2115), Set.Nonempty (\u2191(s m) \u2229 \u2191(s n))\ncauchy_y : CauchySeq y\nthis : Nonempty \u03b3\nz : \u03b3 := limUnder atTop y\ny_lim : Tendsto y atTop (\ud835\udcdd z)\nhne : f z \u2260 x\nv w : Set \u03b2\nv_open : IsOpen v\nw_open : IsOpen w\nfzv : f z \u2208 v\nxw : x \u2208 w\nhvw : Disjoint v w\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\nh\u03b4 : ball z \u03b4 \u2286 f \u207b\u00b9' v\nn : \u2115\nhn : u n + dist (y n) z < \u03b4\na : \u03b3\nha : a \u2208 \u2191(s n)\n\u22a2 a \u2208 ball z \u03b4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/LocallyIntegrable.lean", "full_name": "MeasureTheory.LocallyIntegrable.integrable_smul_right_of_hasCompactSupport", "start": [350, 1], "end": [365, 50], "traced_tactics": [{"tactic": "let K := tsupport g", "annotated_tactic": ["let K := <a>tsupport</a> g", [{"full_name": "tsupport", "def_path": "Mathlib/Topology/Support.lean", "def_pos": [42, 3], "def_end_pos": [42, 14]}]], "state_before": "X : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2077 : MeasurableSpace X\ninst\u271d\u2076 : TopologicalSpace X\ninst\u271d\u2075 : MeasurableSpace Y\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : NormedAddCommGroup E\nf\u271d g\u271d : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : OpensMeasurableSpace X\ninst\u271d : T2Space X\nf : X \u2192 \u211d\nhf : LocallyIntegrable f\ng : X \u2192 E\nhg : Continuous g\nh'g : HasCompactSupport g\n\u22a2 Integrable fun x => f x \u2022 g x", "state_after": "X : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2077 : MeasurableSpace X\ninst\u271d\u2076 : TopologicalSpace X\ninst\u271d\u2075 : MeasurableSpace Y\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : NormedAddCommGroup E\nf\u271d g\u271d : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : OpensMeasurableSpace X\ninst\u271d : T2Space X\nf : X \u2192 \u211d\nhf : LocallyIntegrable f\ng : X \u2192 E\nhg : Continuous g\nh'g : HasCompactSupport g\nK : Set X := tsupport g\n\u22a2 Integrable fun x => f x \u2022 g x"}, {"tactic": "have hK : IsCompact K := h'g", "annotated_tactic": ["have hK : <a>IsCompact</a> K := h'g", [{"full_name": "IsCompact", "def_path": "Mathlib/Topology/Compactness/Compact.lean", "def_pos": [40, 5], "def_end_pos": [40, 14]}]], "state_before": "X : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2077 : MeasurableSpace X\ninst\u271d\u2076 : TopologicalSpace X\ninst\u271d\u2075 : MeasurableSpace Y\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : NormedAddCommGroup E\nf\u271d g\u271d : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : OpensMeasurableSpace X\ninst\u271d : T2Space X\nf : X \u2192 \u211d\nhf : LocallyIntegrable f\ng : X \u2192 E\nhg : Continuous g\nh'g : HasCompactSupport g\nK : Set X := tsupport g\n\u22a2 Integrable fun x => f x \u2022 g x", "state_after": "X : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2077 : MeasurableSpace X\ninst\u271d\u2076 : TopologicalSpace X\ninst\u271d\u2075 : MeasurableSpace Y\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : NormedAddCommGroup E\nf\u271d g\u271d : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : OpensMeasurableSpace X\ninst\u271d : T2Space X\nf : X \u2192 \u211d\nhf : LocallyIntegrable f\ng : X \u2192 E\nhg : Continuous g\nh'g : HasCompactSupport g\nK : Set X := tsupport g\nhK : IsCompact K\n\u22a2 Integrable fun x => f x \u2022 g x"}, {"tactic": "have : K.indicator (fun x \u21a6 f x \u2022 g x) = (fun x \u21a6 f x \u2022 g x) := by\n  apply indicator_eq_self.2\n  apply support_subset_iff'.2\n  intros x hx\n  simp [image_eq_zero_of_nmem_tsupport hx]", "annotated_tactic": ["have : K.indicator (fun x \u21a6 f x \u2022 g x) = (fun x \u21a6 f x \u2022 g x) := by\n    apply <a>indicator_eq_self</a>.2\n    apply <a>support_subset_iff'</a>.2\n    intros x hx\n    simp [<a>image_eq_zero_of_nmem_tsupport</a> hx]", [{"full_name": "Set.indicator_eq_self", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [95, 3], "def_end_pos": [95, 14]}, {"full_name": "Function.support_subset_iff'", "def_path": "Mathlib/Algebra/Support.lean", "def_pos": [77, 3], "def_end_pos": [77, 14]}, {"full_name": "image_eq_zero_of_nmem_tsupport", "def_path": "Mathlib/Topology/Support.lean", "def_pos": [66, 3], "def_end_pos": [66, 14]}]], "state_before": "X : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2077 : MeasurableSpace X\ninst\u271d\u2076 : TopologicalSpace X\ninst\u271d\u2075 : MeasurableSpace Y\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : NormedAddCommGroup E\nf\u271d g\u271d : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : OpensMeasurableSpace X\ninst\u271d : T2Space X\nf : X \u2192 \u211d\nhf : LocallyIntegrable f\ng : X \u2192 E\nhg : Continuous g\nh'g : HasCompactSupport g\nK : Set X := tsupport g\nhK : IsCompact K\n\u22a2 Integrable fun x => f x \u2022 g x", "state_after": "X : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2077 : MeasurableSpace X\ninst\u271d\u2076 : TopologicalSpace X\ninst\u271d\u2075 : MeasurableSpace Y\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : NormedAddCommGroup E\nf\u271d g\u271d : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : OpensMeasurableSpace X\ninst\u271d : T2Space X\nf : X \u2192 \u211d\nhf : LocallyIntegrable f\ng : X \u2192 E\nhg : Continuous g\nh'g : HasCompactSupport g\nK : Set X := tsupport g\nhK : IsCompact K\nthis : (Set.indicator K fun x => f x \u2022 g x) = fun x => f x \u2022 g x\n\u22a2 Integrable fun x => f x \u2022 g x"}, {"tactic": "rw [\u2190 this, indicator_smul_left]", "annotated_tactic": ["rw [\u2190 this, <a>indicator_smul_left</a>]", [{"full_name": "Set.indicator_smul_left", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [515, 9], "def_end_pos": [515, 28]}]], "state_before": "X : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2077 : MeasurableSpace X\ninst\u271d\u2076 : TopologicalSpace X\ninst\u271d\u2075 : MeasurableSpace Y\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : NormedAddCommGroup E\nf\u271d g\u271d : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : OpensMeasurableSpace X\ninst\u271d : T2Space X\nf : X \u2192 \u211d\nhf : LocallyIntegrable f\ng : X \u2192 E\nhg : Continuous g\nh'g : HasCompactSupport g\nK : Set X := tsupport g\nhK : IsCompact K\nthis : (Set.indicator K fun x => f x \u2022 g x) = fun x => f x \u2022 g x\n\u22a2 Integrable fun x => f x \u2022 g x", "state_after": "X : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2077 : MeasurableSpace X\ninst\u271d\u2076 : TopologicalSpace X\ninst\u271d\u2075 : MeasurableSpace Y\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : NormedAddCommGroup E\nf\u271d g\u271d : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : OpensMeasurableSpace X\ninst\u271d : T2Space X\nf : X \u2192 \u211d\nhf : LocallyIntegrable f\ng : X \u2192 E\nhg : Continuous g\nh'g : HasCompactSupport g\nK : Set X := tsupport g\nhK : IsCompact K\nthis : (Set.indicator K fun x => f x \u2022 g x) = fun x => f x \u2022 g x\n\u22a2 Integrable fun x => Set.indicator K (fun x => f x) x \u2022 g x"}, {"tactic": "apply Integrable.smul_of_top_left", "annotated_tactic": ["apply <a>Integrable.smul_of_top_left</a>", [{"full_name": "MeasureTheory.Integrable.smul_of_top_left", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [1094, 9], "def_end_pos": [1094, 36]}]], "state_before": "X : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2077 : MeasurableSpace X\ninst\u271d\u2076 : TopologicalSpace X\ninst\u271d\u2075 : MeasurableSpace Y\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : NormedAddCommGroup E\nf\u271d g\u271d : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : OpensMeasurableSpace X\ninst\u271d : T2Space X\nf : X \u2192 \u211d\nhf : LocallyIntegrable f\ng : X \u2192 E\nhg : Continuous g\nh'g : HasCompactSupport g\nK : Set X := tsupport g\nhK : IsCompact K\nthis : (Set.indicator K fun x => f x \u2022 g x) = fun x => f x \u2022 g x\n\u22a2 Integrable fun x => Set.indicator K (fun x => f x) x \u2022 g x", "state_after": "case h\u03c6\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2077 : MeasurableSpace X\ninst\u271d\u2076 : TopologicalSpace X\ninst\u271d\u2075 : MeasurableSpace Y\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : NormedAddCommGroup E\nf\u271d g\u271d : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : OpensMeasurableSpace X\ninst\u271d : T2Space X\nf : X \u2192 \u211d\nhf : LocallyIntegrable f\ng : X \u2192 E\nhg : Continuous g\nh'g : HasCompactSupport g\nK : Set X := tsupport g\nhK : IsCompact K\nthis : (Set.indicator K fun x => f x \u2022 g x) = fun x => f x \u2022 g x\n\u22a2 Integrable fun x => Set.indicator K (fun x => f x) x\n\ncase hf\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2077 : MeasurableSpace X\ninst\u271d\u2076 : TopologicalSpace X\ninst\u271d\u2075 : MeasurableSpace Y\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : NormedAddCommGroup E\nf\u271d g\u271d : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : OpensMeasurableSpace X\ninst\u271d : T2Space X\nf : X \u2192 \u211d\nhf : LocallyIntegrable f\ng : X \u2192 E\nhg : Continuous g\nh'g : HasCompactSupport g\nK : Set X := tsupport g\nhK : IsCompact K\nthis : (Set.indicator K fun x => f x \u2022 g x) = fun x => f x \u2022 g x\n\u22a2 Mem\u2112p (fun x => g x) \u22a4"}, {"tactic": "apply indicator_eq_self.2", "annotated_tactic": ["apply <a>indicator_eq_self</a>.2", [{"full_name": "Set.indicator_eq_self", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [95, 3], "def_end_pos": [95, 14]}]], "state_before": "X : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2077 : MeasurableSpace X\ninst\u271d\u2076 : TopologicalSpace X\ninst\u271d\u2075 : MeasurableSpace Y\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : NormedAddCommGroup E\nf\u271d g\u271d : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : OpensMeasurableSpace X\ninst\u271d : T2Space X\nf : X \u2192 \u211d\nhf : LocallyIntegrable f\ng : X \u2192 E\nhg : Continuous g\nh'g : HasCompactSupport g\nK : Set X := tsupport g\nhK : IsCompact K\n\u22a2 (Set.indicator K fun x => f x \u2022 g x) = fun x => f x \u2022 g x", "state_after": "X : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2077 : MeasurableSpace X\ninst\u271d\u2076 : TopologicalSpace X\ninst\u271d\u2075 : MeasurableSpace Y\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : NormedAddCommGroup E\nf\u271d g\u271d : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : OpensMeasurableSpace X\ninst\u271d : T2Space X\nf : X \u2192 \u211d\nhf : LocallyIntegrable f\ng : X \u2192 E\nhg : Continuous g\nh'g : HasCompactSupport g\nK : Set X := tsupport g\nhK : IsCompact K\n\u22a2 (support fun x => f x \u2022 g x) \u2286 K"}, {"tactic": "apply support_subset_iff'.2", "annotated_tactic": ["apply <a>support_subset_iff'</a>.2", [{"full_name": "Function.support_subset_iff'", "def_path": "Mathlib/Algebra/Support.lean", "def_pos": [77, 3], "def_end_pos": [77, 14]}]], "state_before": "X : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2077 : MeasurableSpace X\ninst\u271d\u2076 : TopologicalSpace X\ninst\u271d\u2075 : MeasurableSpace Y\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : NormedAddCommGroup E\nf\u271d g\u271d : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : OpensMeasurableSpace X\ninst\u271d : T2Space X\nf : X \u2192 \u211d\nhf : LocallyIntegrable f\ng : X \u2192 E\nhg : Continuous g\nh'g : HasCompactSupport g\nK : Set X := tsupport g\nhK : IsCompact K\n\u22a2 (support fun x => f x \u2022 g x) \u2286 K", "state_after": "X : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2077 : MeasurableSpace X\ninst\u271d\u2076 : TopologicalSpace X\ninst\u271d\u2075 : MeasurableSpace Y\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : NormedAddCommGroup E\nf\u271d g\u271d : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : OpensMeasurableSpace X\ninst\u271d : T2Space X\nf : X \u2192 \u211d\nhf : LocallyIntegrable f\ng : X \u2192 E\nhg : Continuous g\nh'g : HasCompactSupport g\nK : Set X := tsupport g\nhK : IsCompact K\n\u22a2 \u2200 (x : X), \u00acx \u2208 K \u2192 f x \u2022 g x = 0"}, {"tactic": "intros x hx", "annotated_tactic": ["intros x hx", []], "state_before": "X : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2077 : MeasurableSpace X\ninst\u271d\u2076 : TopologicalSpace X\ninst\u271d\u2075 : MeasurableSpace Y\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : NormedAddCommGroup E\nf\u271d g\u271d : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : OpensMeasurableSpace X\ninst\u271d : T2Space X\nf : X \u2192 \u211d\nhf : LocallyIntegrable f\ng : X \u2192 E\nhg : Continuous g\nh'g : HasCompactSupport g\nK : Set X := tsupport g\nhK : IsCompact K\n\u22a2 \u2200 (x : X), \u00acx \u2208 K \u2192 f x \u2022 g x = 0", "state_after": "X : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2077 : MeasurableSpace X\ninst\u271d\u2076 : TopologicalSpace X\ninst\u271d\u2075 : MeasurableSpace Y\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : NormedAddCommGroup E\nf\u271d g\u271d : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : OpensMeasurableSpace X\ninst\u271d : T2Space X\nf : X \u2192 \u211d\nhf : LocallyIntegrable f\ng : X \u2192 E\nhg : Continuous g\nh'g : HasCompactSupport g\nK : Set X := tsupport g\nhK : IsCompact K\nx : X\nhx : \u00acx \u2208 K\n\u22a2 f x \u2022 g x = 0"}, {"tactic": "simp [image_eq_zero_of_nmem_tsupport hx]", "annotated_tactic": ["simp [<a>image_eq_zero_of_nmem_tsupport</a> hx]", [{"full_name": "image_eq_zero_of_nmem_tsupport", "def_path": "Mathlib/Topology/Support.lean", "def_pos": [66, 3], "def_end_pos": [66, 14]}]], "state_before": "X : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2077 : MeasurableSpace X\ninst\u271d\u2076 : TopologicalSpace X\ninst\u271d\u2075 : MeasurableSpace Y\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : NormedAddCommGroup E\nf\u271d g\u271d : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : OpensMeasurableSpace X\ninst\u271d : T2Space X\nf : X \u2192 \u211d\nhf : LocallyIntegrable f\ng : X \u2192 E\nhg : Continuous g\nh'g : HasCompactSupport g\nK : Set X := tsupport g\nhK : IsCompact K\nx : X\nhx : \u00acx \u2208 K\n\u22a2 f x \u2022 g x = 0", "state_after": "no goals"}, {"tactic": "rw [integrable_indicator_iff hK.measurableSet]", "annotated_tactic": ["rw [<a>integrable_indicator_iff</a> hK.measurableSet]", [{"full_name": "MeasureTheory.integrable_indicator_iff", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [257, 9], "def_end_pos": [257, 33]}]], "state_before": "case h\u03c6\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2077 : MeasurableSpace X\ninst\u271d\u2076 : TopologicalSpace X\ninst\u271d\u2075 : MeasurableSpace Y\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : NormedAddCommGroup E\nf\u271d g\u271d : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : OpensMeasurableSpace X\ninst\u271d : T2Space X\nf : X \u2192 \u211d\nhf : LocallyIntegrable f\ng : X \u2192 E\nhg : Continuous g\nh'g : HasCompactSupport g\nK : Set X := tsupport g\nhK : IsCompact K\nthis : (Set.indicator K fun x => f x \u2022 g x) = fun x => f x \u2022 g x\n\u22a2 Integrable fun x => Set.indicator K (fun x => f x) x", "state_after": "case h\u03c6\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2077 : MeasurableSpace X\ninst\u271d\u2076 : TopologicalSpace X\ninst\u271d\u2075 : MeasurableSpace Y\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : NormedAddCommGroup E\nf\u271d g\u271d : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : OpensMeasurableSpace X\ninst\u271d : T2Space X\nf : X \u2192 \u211d\nhf : LocallyIntegrable f\ng : X \u2192 E\nhg : Continuous g\nh'g : HasCompactSupport g\nK : Set X := tsupport g\nhK : IsCompact K\nthis : (Set.indicator K fun x => f x \u2022 g x) = fun x => f x \u2022 g x\n\u22a2 IntegrableOn (fun x => f x) K"}, {"tactic": "exact hf.integrableOn_isCompact hK", "annotated_tactic": ["exact hf.integrableOn_isCompact hK", []], "state_before": "case h\u03c6\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2077 : MeasurableSpace X\ninst\u271d\u2076 : TopologicalSpace X\ninst\u271d\u2075 : MeasurableSpace Y\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : NormedAddCommGroup E\nf\u271d g\u271d : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : OpensMeasurableSpace X\ninst\u271d : T2Space X\nf : X \u2192 \u211d\nhf : LocallyIntegrable f\ng : X \u2192 E\nhg : Continuous g\nh'g : HasCompactSupport g\nK : Set X := tsupport g\nhK : IsCompact K\nthis : (Set.indicator K fun x => f x \u2022 g x) = fun x => f x \u2022 g x\n\u22a2 IntegrableOn (fun x => f x) K", "state_after": "no goals"}, {"tactic": "exact hg.mem\u2112p_top_of_hasCompactSupport h'g \u03bc", "annotated_tactic": ["exact hg.mem\u2112p_top_of_hasCompactSupport h'g \u03bc", []], "state_before": "case hf\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2077 : MeasurableSpace X\ninst\u271d\u2076 : TopologicalSpace X\ninst\u271d\u2075 : MeasurableSpace Y\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : NormedAddCommGroup E\nf\u271d g\u271d : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : OpensMeasurableSpace X\ninst\u271d : T2Space X\nf : X \u2192 \u211d\nhf : LocallyIntegrable f\ng : X \u2192 E\nhg : Continuous g\nh'g : HasCompactSupport g\nK : Set X := tsupport g\nhK : IsCompact K\nthis : (Set.indicator K fun x => f x \u2022 g x) = fun x => f x \u2022 g x\n\u22a2 Mem\u2112p (fun x => g x) \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Num/Lemmas.lean", "full_name": "Num.castNum_ldiff", "start": [938, 1], "end": [939, 86], "traced_tactics": [{"tactic": "apply castNum_eq_bitwise PosNum.ldiff <;> intros <;> (try cases_type* Bool) <;> rfl", "annotated_tactic": ["apply <a>castNum_eq_bitwise</a> <a>PosNum.ldiff</a> <;> intros <;> (try cases_type* <a>Bool</a>) <;> rfl", [{"full_name": "Num.castNum_eq_bitwise", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [886, 9], "def_end_pos": [886, 27]}, {"full_name": "PosNum.ldiff", "def_path": "Mathlib/Data/Num/Bitwise.lean", "def_pos": [65, 5], "def_end_pos": [65, 10]}, {"full_name": "Bool", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [545, 11], "def_end_pos": [545, 15]}]], "state_before": "\u03b1 : Type u_1\n\u22a2 \u2200 (m n : Num), \u2191(ldiff m n) = Nat.ldiff \u2191m \u2191n", "state_after": "no goals"}, {"tactic": "try cases_type* Bool", "annotated_tactic": ["try cases_type* <a>Bool</a>", [{"full_name": "Bool", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [545, 11], "def_end_pos": [545, 15]}]], "state_before": "case pbb\n\u03b1 : Type u_1\na\u271d b\u271d : Bool\nm\u271d n\u271d : PosNum\n\u22a2 PosNum.ldiff (PosNum.bit a\u271d m\u271d) (PosNum.bit b\u271d n\u271d) = bit (a\u271d && !b\u271d) (PosNum.ldiff m\u271d n\u271d)", "state_after": "case pbb.false.false\n\u03b1 : Type u_1\nm\u271d n\u271d : PosNum\n\u22a2 PosNum.ldiff (PosNum.bit false m\u271d) (PosNum.bit false n\u271d) = bit (false && !false) 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\u2192 \u2200 (y : \u03c3), y \u2208 g a \u2192 x = y\nk : \u03b1 \u2192. \u03c3\nhk : Partrec k\nK : \u2200 (a : \u03b1), (\u2200 (x : \u03c3), x \u2208 k a \u2192 x \u2208 f a \u2228 x \u2208 g a) \u2227 ((k a).Dom \u2194 (f a).Dom \u2228 (g a).Dom)\na : \u03b1\nx : \u03c3\nh : x \u2208 f a \u2228 x \u2208 g a\nthis : (k a).Dom\n\u22a2 Part.get (k a) this = x"}, {"tactic": "cases' h with h h <;> cases' (K _).1 _ \u27e8this, rfl\u27e9 with h' h'", "annotated_tactic": ["cases' h with h h <;> cases' (K _).1 _ \u27e8this, <a>rfl</a>\u27e9 with h' h'", [{"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf g : \u03b1 \u2192. \u03c3\nhf : Partrec f\nhg : Partrec g\nH : \u2200 (a : \u03b1) (x : \u03c3), x \u2208 f a \u2192 \u2200 (y : \u03c3), y \u2208 g a \u2192 x = y\nk : \u03b1 \u2192. \u03c3\nhk : Partrec k\nK : \u2200 (a : \u03b1), (\u2200 (x : \u03c3), x \u2208 k a \u2192 x \u2208 f a \u2228 x \u2208 g a) \u2227 ((k a).Dom \u2194 (f a).Dom \u2228 (g a).Dom)\na : \u03b1\nx : \u03c3\nh : x \u2208 f a \u2228 x \u2208 g a\nthis : (k a).Dom\n\u22a2 Part.get (k a) this = x", "state_after": "case inl.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf g : \u03b1 \u2192. \u03c3\nhf : Partrec f\nhg : Partrec g\nH : \u2200 (a : \u03b1) (x : \u03c3), x \u2208 f a \u2192 \u2200 (y : \u03c3), y \u2208 g a \u2192 x = y\nk : \u03b1 \u2192. \u03c3\nhk : Partrec k\nK : \u2200 (a : \u03b1), (\u2200 (x : \u03c3), x \u2208 k a \u2192 x \u2208 f a \u2228 x \u2208 g a) \u2227 ((k a).Dom \u2194 (f a).Dom \u2228 (g a).Dom)\na : \u03b1\nx : \u03c3\nthis : (k a).Dom\nh : x \u2208 f a\nh' : Part.get (k a) this \u2208 f a\n\u22a2 Part.get (k a) this = x\n\ncase inl.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf g : \u03b1 \u2192. \u03c3\nhf : Partrec f\nhg : Partrec g\nH : \u2200 (a : \u03b1) (x : \u03c3), x \u2208 f a \u2192 \u2200 (y : \u03c3), y \u2208 g a \u2192 x = y\nk : \u03b1 \u2192. \u03c3\nhk : Partrec k\nK : \u2200 (a : \u03b1), (\u2200 (x : \u03c3), x \u2208 k a \u2192 x \u2208 f a \u2228 x \u2208 g a) \u2227 ((k a).Dom \u2194 (f a).Dom \u2228 (g a).Dom)\na : \u03b1\nx : \u03c3\nthis : (k a).Dom\nh : x \u2208 f a\nh' : Part.get (k a) this \u2208 g a\n\u22a2 Part.get (k a) this = x\n\ncase inr.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf g : \u03b1 \u2192. \u03c3\nhf : Partrec f\nhg : Partrec g\nH : \u2200 (a : \u03b1) (x : \u03c3), x \u2208 f a \u2192 \u2200 (y : \u03c3), y \u2208 g a \u2192 x = y\nk : \u03b1 \u2192. \u03c3\nhk : Partrec k\nK : \u2200 (a : \u03b1), (\u2200 (x : \u03c3), x \u2208 k a \u2192 x \u2208 f a \u2228 x \u2208 g a) \u2227 ((k a).Dom \u2194 (f a).Dom \u2228 (g a).Dom)\na : \u03b1\nx : \u03c3\nthis : (k a).Dom\nh : x \u2208 g a\nh' : Part.get (k a) this \u2208 f a\n\u22a2 Part.get (k a) this = x\n\ncase inr.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf g : \u03b1 \u2192. \u03c3\nhf : Partrec f\nhg : Partrec g\nH : \u2200 (a : \u03b1) (x : \u03c3), x \u2208 f a \u2192 \u2200 (y : \u03c3), y \u2208 g a \u2192 x = y\nk : \u03b1 \u2192. \u03c3\nhk : Partrec k\nK : \u2200 (a : \u03b1), (\u2200 (x : \u03c3), x \u2208 k a \u2192 x \u2208 f a \u2228 x \u2208 g a) \u2227 ((k a).Dom \u2194 (f a).Dom \u2228 (g a).Dom)\na : \u03b1\nx : \u03c3\nthis : (k a).Dom\nh : x \u2208 g a\nh' : Part.get (k a) this \u2208 g a\n\u22a2 Part.get (k a) this = x"}, {"tactic": "exact mem_unique h' h", "annotated_tactic": ["exact <a>mem_unique</a> h' h", [{"full_name": "Part.mem_unique", "def_path": "Mathlib/Data/Part.lean", "def_pos": [144, 9], "def_end_pos": [144, 19]}]], "state_before": "case inl.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf g : \u03b1 \u2192. \u03c3\nhf : Partrec f\nhg : Partrec g\nH : \u2200 (a : \u03b1) (x : \u03c3), x \u2208 f a \u2192 \u2200 (y : \u03c3), y \u2208 g a \u2192 x = y\nk : \u03b1 \u2192. \u03c3\nhk : Partrec k\nK : \u2200 (a : \u03b1), (\u2200 (x : \u03c3), x \u2208 k a \u2192 x \u2208 f a \u2228 x \u2208 g a) \u2227 ((k a).Dom \u2194 (f a).Dom \u2228 (g a).Dom)\na : \u03b1\nx : \u03c3\nthis : (k a).Dom\nh : x \u2208 f a\nh' : Part.get (k a) this \u2208 f a\n\u22a2 Part.get (k a) this = x", "state_after": "no goals"}, {"tactic": "exact (H _ _ h _ h').symm", "annotated_tactic": ["exact (H _ _ h _ h').<a>symm</a>", [{"full_name": "Eq.symm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [310, 9], "def_end_pos": [310, 16]}]], "state_before": "case inl.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf g : \u03b1 \u2192. \u03c3\nhf : Partrec f\nhg : Partrec g\nH : \u2200 (a : \u03b1) (x : \u03c3), x \u2208 f a \u2192 \u2200 (y : \u03c3), y \u2208 g a \u2192 x = y\nk : \u03b1 \u2192. \u03c3\nhk : Partrec k\nK : \u2200 (a : \u03b1), (\u2200 (x : \u03c3), x \u2208 k a \u2192 x \u2208 f a \u2228 x \u2208 g a) \u2227 ((k a).Dom \u2194 (f a).Dom \u2228 (g a).Dom)\na : \u03b1\nx : \u03c3\nthis : (k a).Dom\nh : x \u2208 f a\nh' : Part.get (k a) this \u2208 g a\n\u22a2 Part.get (k a) this = x", "state_after": "no goals"}, {"tactic": "exact H _ _ h' _ h", "annotated_tactic": ["exact H _ _ h' _ h", []], "state_before": "case inr.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf g : \u03b1 \u2192. \u03c3\nhf : Partrec f\nhg : Partrec g\nH : \u2200 (a : \u03b1) (x : \u03c3), x \u2208 f a \u2192 \u2200 (y : \u03c3), y \u2208 g a \u2192 x = y\nk : \u03b1 \u2192. \u03c3\nhk : Partrec k\nK : \u2200 (a : \u03b1), (\u2200 (x : \u03c3), x \u2208 k a \u2192 x \u2208 f a \u2228 x \u2208 g a) \u2227 ((k a).Dom \u2194 (f a).Dom \u2228 (g a).Dom)\na : \u03b1\nx : \u03c3\nthis : (k a).Dom\nh : x \u2208 g a\nh' : Part.get (k a) this \u2208 f a\n\u22a2 Part.get (k a) this = x", "state_after": "no goals"}, {"tactic": "exact mem_unique h' h", "annotated_tactic": ["exact <a>mem_unique</a> h' h", [{"full_name": "Part.mem_unique", "def_path": "Mathlib/Data/Part.lean", "def_pos": [144, 9], "def_end_pos": [144, 19]}]], "state_before": "case inr.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf g : \u03b1 \u2192. \u03c3\nhf : Partrec f\nhg : Partrec g\nH : \u2200 (a : \u03b1) (x : \u03c3), x \u2208 f a \u2192 \u2200 (y : \u03c3), y \u2208 g a \u2192 x = y\nk : \u03b1 \u2192. \u03c3\nhk : Partrec k\nK : \u2200 (a : \u03b1), (\u2200 (x : \u03c3), x \u2208 k a \u2192 x \u2208 f a \u2228 x \u2208 g a) \u2227 ((k a).Dom \u2194 (f a).Dom \u2228 (g a).Dom)\na : \u03b1\nx : \u03c3\nthis : (k a).Dom\nh : x \u2208 g a\nh' : Part.get (k a) this \u2208 g a\n\u22a2 Part.get (k a) this = x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "full_name": "List.disjoint_of_disjoint_cons_right", "start": [1540, 1], "end": [1541, 43], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/PEquiv.lean", "full_name": "PEquiv.ofSet_eq_some_iff", "start": [234, 1], "end": [236, 16], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Lattice.lean", "full_name": "Finset.inf_apply", "start": [1091, 11], "end": [1094, 49], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Covering/Besicovitch.lean", "full_name": "Besicovitch.exists_disjoint_closedBall_covering_ae_of_finiteMeasure_aux", "start": [686, 1], "end": [818, 30], "traced_tactics": [{"tactic": "rcases HasBesicovitchCovering.no_satelliteConfig (\u03b1 := \u03b1) with \u27e8N, \u03c4, h\u03c4, hN\u27e9", "annotated_tactic": ["rcases <a>HasBesicovitchCovering.no_satelliteConfig</a> (\u03b1 := \u03b1) with \u27e8N, \u03c4, h\u03c4, hN\u27e9", [{"full_name": "HasBesicovitchCovering.no_satelliteConfig", "def_path": "Mathlib/MeasureTheory/Covering/Besicovitch.lean", "def_pos": [147, 3], "def_end_pos": [147, 21]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\n\u22a2 \u2203 t,\n    Set.Countable t \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227\n        (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1) \u2227\n          \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2) = 0 \u2227 PairwiseDisjoint t fun p => closedBall p.1 p.2", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\n\u22a2 \u2203 t,\n    Set.Countable t \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227\n        (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1) \u2227\n          \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2) = 0 \u2227 PairwiseDisjoint t fun p => closedBall p.1 p.2"}, {"tactic": "let P : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop := fun t =>\n  ((t : Set (\u03b1 \u00d7 \u211d)).PairwiseDisjoint fun p => closedBall p.1 p.2) \u2227\n    (\u2200 p : \u03b1 \u00d7 \u211d, p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 p : \u03b1 \u00d7 \u211d, p \u2208 t \u2192 p.2 \u2208 f p.1", "annotated_tactic": ["let P : <a>Finset</a> (\u03b1 \u00d7 \u211d) \u2192 Prop := fun t =>\n    ((t : <a>Set</a> (\u03b1 \u00d7 \u211d)).<a>PairwiseDisjoint</a> fun p => <a>closedBall</a> p.1 p.2) \u2227\n      (\u2200 p : \u03b1 \u00d7 \u211d, p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 p : \u03b1 \u00d7 \u211d, p \u2208 t \u2192 p.2 \u2208 f p.1", [{"full_name": "Finset", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [138, 11], "def_end_pos": [138, 17]}, {"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}, {"full_name": "Set.PairwiseDisjoint", "def_path": "Mathlib/Data/Set/Pairwise/Basic.lean", "def_pos": [242, 5], "def_end_pos": [242, 21]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}]], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\n\u22a2 \u2203 t,\n    Set.Countable t \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227\n        (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1) \u2227\n          \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2) = 0 \u2227 PairwiseDisjoint t fun p => closedBall p.1 p.2", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\n\u22a2 \u2203 t,\n    Set.Countable t \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227\n        (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1) \u2227\n          \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2) = 0 \u2227 PairwiseDisjoint t fun p => closedBall p.1 p.2"}, {"tactic": "choose! F hF using this", "annotated_tactic": ["choose! F hF using this", []], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nthis :\n  \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n    P t \u2192\n      \u2203 u, t \u2286 u \u2227 P u \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2)\n\u22a2 \u2203 t,\n    Set.Countable t \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227\n        (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1) \u2227\n          \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2) = 0 \u2227 PairwiseDisjoint t fun p => closedBall p.1 p.2", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n  \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n    P t \u2192\n      t \u2286 F t \u2227\n        P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2)\n\u22a2 \u2203 t,\n    Set.Countable t \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227\n        (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1) \u2227\n          \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2) = 0 \u2227 PairwiseDisjoint t fun p => closedBall p.1 p.2"}, {"tactic": "let u n := F^[n] \u2205", "annotated_tactic": ["let u n := F^[n] \u2205", []], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n  \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n    P t \u2192\n      t \u2286 F t \u2227\n        P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2)\n\u22a2 \u2203 t,\n    Set.Countable t \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227\n        (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1) \u2227\n          \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2) = 0 \u2227 PairwiseDisjoint t fun p => closedBall p.1 p.2", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n  \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n    P t \u2192\n      t \u2286 F t \u2227\n        P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\n\u22a2 \u2203 t,\n    Set.Countable t \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227\n        (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1) \u2227\n          \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2) = 0 \u2227 PairwiseDisjoint t fun p => closedBall p.1 p.2"}, {"tactic": "have u_succ : \u2200 n : \u2115, u n.succ = F (u n) := fun n => by\n  simp only [Function.comp_apply, Function.iterate_succ']", "annotated_tactic": ["have u_succ : \u2200 n : \u2115, u n.succ = F (u n) := fun n => by\n    simp only [<a>Function.comp_apply</a>, <a>Function.iterate_succ'</a>]", [{"full_name": "Function.comp_apply", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [33, 17], "def_end_pos": [33, 36]}, {"full_name": "Function.iterate_succ'", "def_path": "Mathlib/Logic/Function/Iterate.lean", "def_pos": [186, 9], "def_end_pos": [186, 22]}]], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n  \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n    P t \u2192\n      t \u2286 F t \u2227\n        P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\n\u22a2 \u2203 t,\n    Set.Countable t \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227\n        (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1) \u2227\n          \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2) = 0 \u2227 PairwiseDisjoint t fun p => closedBall p.1 p.2", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n  \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n    P t \u2192\n      t \u2286 F t \u2227\n        P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\n\u22a2 \u2203 t,\n    Set.Countable t \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227\n        (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1) \u2227\n          \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2) = 0 \u2227 PairwiseDisjoint t fun p => closedBall p.1 p.2"}, {"tactic": "refine' \u27e8\u22c3 n, u n, countable_iUnion fun n => (u n).countable_toSet, _, _, _, _\u27e9", "annotated_tactic": ["refine' \u27e8\u22c3 n, u n, <a>countable_iUnion</a> fun n => (u n).<a>countable_toSet</a>, _, _, _, _\u27e9", [{"full_name": "Set.countable_iUnion", "def_path": "Mathlib/Data/Set/Countable.lean", "def_pos": [185, 9], "def_end_pos": [185, 25]}, {"full_name": "Finset.countable_toSet", "def_path": "Mathlib/Data/Set/Countable.lean", "def_pos": [314, 9], "def_end_pos": [314, 31]}]], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n  \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n    P t \u2192\n      t \u2286 F t \u2227\n        P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\nPu : \u2200 (n : \u2115), P (u n)\n\u22a2 \u2203 t,\n    Set.Countable t \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227\n        (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1) \u2227\n          \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2) = 0 \u2227 PairwiseDisjoint t fun p => closedBall p.1 p.2", "state_after": "case intro.intro.intro.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n  \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n    P t \u2192\n      t \u2286 F t \u2227\n        P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\nPu : \u2200 (n : \u2115), P (u n)\n\u22a2 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 \u22c3 n, \u2191(u n) \u2192 p.1 \u2208 s\n\ncase intro.intro.intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n  \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n    P t \u2192\n      t \u2286 F t \u2227\n        P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\nPu : \u2200 (n : \u2115), P (u n)\n\u22a2 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 \u22c3 n, \u2191(u n) \u2192 p.2 \u2208 f p.1\n\ncase intro.intro.intro.refine'_3\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n  \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n    P t \u2192\n      t \u2286 F t \u2227\n        P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\nPu : \u2200 (n : \u2115), P (u n)\n\u22a2 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 \u22c3 n, \u2191(u n), closedBall p.1 p.2) = 0\n\ncase intro.intro.intro.refine'_4\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n  \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n    P t \u2192\n      t \u2286 F t \u2227\n        P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\nPu : \u2200 (n : \u2115), P (u n)\n\u22a2 PairwiseDisjoint (\u22c3 n, \u2191(u n)) fun p => closedBall p.1 p.2"}, {"tactic": "intro t ht", "annotated_tactic": ["intro t ht", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\n\u22a2 \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n    P t \u2192\n      \u2203 u, t \u2286 u \u2227 P u \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\n\u22a2 \u2203 u, t \u2286 u \u2227 P u \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2)"}, {"tactic": "set B := \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.1 p.2 with hB", "annotated_tactic": ["set B := \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), <a>closedBall</a> p.1 p.2 with hB", [{"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\n\u22a2 \u2203 u, t \u2286 u \u2227 P u \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2\nhB : B = \u22c3 p \u2208 t, closedBall p.1 p.2\n\u22a2 \u2203 u, t \u2286 u \u2227 P u \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ B)"}, {"tactic": "have B_closed : IsClosed B := isClosed_biUnion_finset fun i _ => isClosed_ball", "annotated_tactic": ["have B_closed : <a>IsClosed</a> B := <a>isClosed_biUnion_finset</a> fun i _ => <a>isClosed_ball</a>", [{"full_name": "IsClosed", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [196, 7], "def_end_pos": [196, 15]}, {"full_name": "isClosed_biUnion_finset", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [264, 7], "def_end_pos": [264, 30]}, {"full_name": "Metric.isClosed_ball", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [1890, 9], "def_end_pos": [1890, 22]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2\nhB : B = \u22c3 p \u2208 t, closedBall p.1 p.2\n\u22a2 \u2203 u, t \u2286 u \u2227 P u \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ B)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2\nhB : B = \u22c3 p \u2208 t, closedBall p.1 p.2\nB_closed : IsClosed B\n\u22a2 \u2203 u, t \u2286 u \u2227 P u \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ B)"}, {"tactic": "set s' := s \\ B", "annotated_tactic": ["set s' := s \\ B", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2\nhB : B = \u22c3 p \u2208 t, closedBall p.1 p.2\nB_closed : IsClosed B\n\u22a2 \u2203 u, t \u2286 u \u2227 P u \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ B)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2\nhB : B = \u22c3 p \u2208 t, closedBall p.1 p.2\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\n\u22a2 \u2203 u, t \u2286 u \u2227 P u \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s'"}, {"tactic": "choose! r hr using this", "annotated_tactic": ["choose! r hr using this", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2\nhB : B = \u22c3 p \u2208 t, closedBall p.1 p.2\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nthis : \u2200 (x : \u03b1), x \u2208 s' \u2192 \u2203 r, r \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x r)\n\u22a2 \u2203 u, t \u2286 u \u2227 P u \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s'", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2\nhB : B = \u22c3 p \u2208 t, closedBall p.1 p.2\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 s' \u2192 r x \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x (r x))\n\u22a2 \u2203 u, t \u2286 u \u2227 P u \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s'"}, {"tactic": "obtain \u27e8v, vs', h\u03bcv, hv\u27e9 :\n  \u2203 v : Finset \u03b1,\n    \u2191v \u2286 s' \u2227\n      \u03bc (s' \\ \u22c3 x \u2208 v, closedBall x (r x)) \u2264 N / (N + 1) * \u03bc s' \u2227\n        (v : Set \u03b1).PairwiseDisjoint fun x : \u03b1 => closedBall x (r x) :=\n  haveI rI : \u2200 x \u2208 s', r x \u2208 Ioo (0 : \u211d) 1 := fun x hx => (hr x hx).1.2\n  exist_finset_disjoint_balls_large_measure \u03bc h\u03c4 hN s' r (fun x hx => (rI x hx).1) fun x hx =>\n    (rI x hx).2.le", "annotated_tactic": ["obtain \u27e8v, vs', h\u03bcv, hv\u27e9 :\n      \u2203 v : <a>Finset</a> \u03b1,\n        \u2191v \u2286 s' \u2227\n          \u03bc (s' \\ \u22c3 x \u2208 v, <a>closedBall</a> x (r x)) \u2264 N / (N + 1) * \u03bc s' \u2227\n            (v : <a>Set</a> \u03b1).<a>PairwiseDisjoint</a> fun x : \u03b1 => <a>closedBall</a> x (r x) :=\n      haveI rI : \u2200 x \u2208 s', r x \u2208 <a>Ioo</a> (0 : \u211d) 1 := fun x hx => (hr x hx).1.2\n      <a>exist_finset_disjoint_balls_large_measure</a> \u03bc h\u03c4 hN s' r (fun x hx => (rI x hx).1) fun x hx =>\n        (rI x hx).2.<a>le</a>", [{"full_name": "Finset", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [138, 11], "def_end_pos": [138, 17]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}, {"full_name": "Set.PairwiseDisjoint", "def_path": "Mathlib/Data/Set/Pairwise/Basic.lean", "def_pos": [242, 5], "def_end_pos": [242, 21]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "Set.Ioo", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [44, 5], "def_end_pos": [44, 8]}, {"full_name": "Besicovitch.exist_finset_disjoint_balls_large_measure", "def_path": "Mathlib/MeasureTheory/Covering/Besicovitch.lean", "def_pos": [546, 9], "def_end_pos": [546, 50]}, {"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [142, 7], "def_end_pos": [142, 15]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2\nhB : B = \u22c3 p \u2208 t, closedBall p.1 p.2\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 s' \u2192 r x \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x (r x))\n\u22a2 \u2203 u, t \u2286 u \u2227 P u \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s'", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2\nhB : B = \u22c3 p \u2208 t, closedBall p.1 p.2\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 s' \u2192 r x \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x (r x))\nv : Finset \u03b1\nvs' : \u2191v \u2286 s'\nh\u03bcv : \u2191\u2191\u03bc (s' \\ \u22c3 x \u2208 v, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s'\nhv : PairwiseDisjoint \u2191v fun x => closedBall x (r x)\n\u22a2 \u2203 u, t \u2286 u \u2227 P u \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s'"}, {"tactic": "refine' \u27e8t \u222a Finset.image (fun x => (x, r x)) v, Finset.subset_union_left _ _, \u27e8_, _, _\u27e9, _\u27e9", "annotated_tactic": ["refine' \u27e8t \u222a <a>Finset.image</a> (fun x => (x, r x)) v, <a>Finset.subset_union_left</a> _ _, \u27e8_, _, _\u27e9, _\u27e9", [{"full_name": "Finset.image", "def_path": "Mathlib/Data/Finset/Image.lean", "def_pos": [313, 5], "def_end_pos": [313, 10]}, {"full_name": "Finset.subset_union_left", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1407, 9], "def_end_pos": [1407, 26]}]], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2\nhB : B = \u22c3 p \u2208 t, closedBall p.1 p.2\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 s' \u2192 r x \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x (r x))\nv : Finset \u03b1\nvs' : \u2191v \u2286 s'\nh\u03bcv : \u2191\u2191\u03bc (s' \\ \u22c3 x \u2208 v, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s'\nhv : PairwiseDisjoint \u2191v fun x => closedBall x (r x)\n\u22a2 \u2203 u, t \u2286 u \u2227 P u \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s'", "state_after": "case intro.intro.intro.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2\nhB : B = \u22c3 p \u2208 t, closedBall p.1 p.2\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 s' \u2192 r x \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x (r x))\nv : Finset \u03b1\nvs' : \u2191v \u2286 s'\nh\u03bcv : \u2191\u2191\u03bc (s' \\ \u22c3 x \u2208 v, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s'\nhv : PairwiseDisjoint \u2191v fun x => closedBall x (r x)\n\u22a2 PairwiseDisjoint \u2191(t \u222a Finset.image (fun x => (x, r x)) v) fun p => closedBall p.1 p.2\n\ncase intro.intro.intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2\nhB : B = \u22c3 p \u2208 t, closedBall p.1 p.2\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 s' \u2192 r x \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x (r x))\nv : Finset \u03b1\nvs' : \u2191v \u2286 s'\nh\u03bcv : \u2191\u2191\u03bc (s' \\ \u22c3 x \u2208 v, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s'\nhv : PairwiseDisjoint \u2191v fun x => closedBall x (r x)\n\u22a2 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u222a Finset.image (fun x => (x, r x)) v \u2192 p.1 \u2208 s\n\ncase intro.intro.intro.refine'_3\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2\nhB : B = \u22c3 p \u2208 t, closedBall p.1 p.2\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 s' \u2192 r x \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x (r x))\nv : Finset \u03b1\nvs' : \u2191v \u2286 s'\nh\u03bcv : \u2191\u2191\u03bc (s' \\ \u22c3 x \u2208 v, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s'\nhv : PairwiseDisjoint \u2191v fun x => closedBall x (r x)\n\u22a2 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u222a Finset.image (fun x => (x, r x)) v \u2192 p.2 \u2208 f p.1\n\ncase intro.intro.intro.refine'_4\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2\nhB : B = \u22c3 p \u2208 t, closedBall p.1 p.2\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 s' \u2192 r x \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x (r x))\nv : Finset \u03b1\nvs' : \u2191v \u2286 s'\nh\u03bcv : \u2191\u2191\u03bc (s' \\ \u22c3 x \u2208 v, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s'\nhv : PairwiseDisjoint \u2191v fun x => closedBall x (r x)\n\u22a2 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t \u222a Finset.image (fun x => (x, r x)) v, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s'"}, {"tactic": "intro x hx", "annotated_tactic": ["intro x hx", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2\nhB : B = \u22c3 p \u2208 t, closedBall p.1 p.2\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\n\u22a2 \u2200 (x : \u03b1), x \u2208 s' \u2192 \u2203 r, r \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x r)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2\nhB : B = \u22c3 p \u2208 t, closedBall p.1 p.2\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nx : \u03b1\nhx : x \u2208 s'\n\u22a2 \u2203 r, r \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x r)"}, {"tactic": "have xs : x \u2208 s := ((mem_diff x).1 hx).1", "annotated_tactic": ["have xs : x \u2208 s := ((<a>mem_diff</a> x).1 hx).1", [{"full_name": "Set.mem_diff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1819, 9], "def_end_pos": [1819, 17]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2\nhB : B = \u22c3 p \u2208 t, closedBall p.1 p.2\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nx : \u03b1\nhx : x \u2208 s'\n\u22a2 \u2203 r, r \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x r)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2\nhB : B = \u22c3 p \u2208 t, closedBall p.1 p.2\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nx : \u03b1\nhx : x \u2208 s'\nxs : x \u2208 s\n\u22a2 \u2203 r, r \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x r)"}, {"tactic": "rcases eq_empty_or_nonempty B with (hB | hB)", "annotated_tactic": ["rcases <a>eq_empty_or_nonempty</a> B with (hB | hB)", [{"full_name": "Set.eq_empty_or_nonempty", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [635, 9], "def_end_pos": [635, 29]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2\nhB : B = \u22c3 p \u2208 t, closedBall p.1 p.2\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nx : \u03b1\nhx : x \u2208 s'\nxs : x \u2208 s\n\u22a2 \u2203 r, r \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x r)", "state_after": "case inl\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2\nhB\u271d : B = \u22c3 p \u2208 t, closedBall p.1 p.2\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nx : \u03b1\nhx : x \u2208 s'\nxs : x \u2208 s\nhB : B = \u2205\n\u22a2 \u2203 r, r \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x r)\n\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2\nhB\u271d : B = \u22c3 p \u2208 t, closedBall p.1 p.2\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nx : \u03b1\nhx : x \u2208 s'\nxs : x \u2208 s\nhB : Set.Nonempty B\n\u22a2 \u2203 r, r \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x r)"}, {"tactic": "rcases hf x xs 1 zero_lt_one with \u27e8r, hr, h'r\u27e9", "annotated_tactic": ["rcases hf x xs 1 <a>zero_lt_one</a> with \u27e8r, hr, h'r\u27e9", [{"full_name": "zero_lt_one", "def_path": "Mathlib/Algebra/Order/ZeroLEOne.lean", "def_pos": [39, 15], "def_end_pos": [39, 26]}]], "state_before": "case inl\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2\nhB\u271d : B = \u22c3 p \u2208 t, closedBall p.1 p.2\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nx : \u03b1\nhx : x \u2208 s'\nxs : x \u2208 s\nhB : B = \u2205\n\u22a2 \u2203 r, r \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x r)", "state_after": "case inl.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2\nhB\u271d : B = \u22c3 p \u2208 t, closedBall p.1 p.2\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nx : \u03b1\nhx : x \u2208 s'\nxs : x \u2208 s\nhB : B = \u2205\nr : \u211d\nhr : r \u2208 f x\nh'r : r \u2208 Ioo 0 1\n\u22a2 \u2203 r, r \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x r)"}, {"tactic": "exact \u27e8r, \u27e8hr, h'r\u27e9, by simp only [hB, empty_disjoint]\u27e9", "annotated_tactic": ["exact \u27e8r, \u27e8hr, h'r\u27e9, by simp only [hB, <a>empty_disjoint</a>]\u27e9", [{"full_name": "Set.empty_disjoint", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1593, 15], "def_end_pos": [1593, 29]}]], "state_before": "case inl.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2\nhB\u271d : B = \u22c3 p \u2208 t, closedBall p.1 p.2\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nx : \u03b1\nhx : x \u2208 s'\nxs : x \u2208 s\nhB : B = \u2205\nr : \u211d\nhr : r \u2208 f x\nh'r : r \u2208 Ioo 0 1\n\u22a2 \u2203 r, r \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x r)", "state_after": "no goals"}, {"tactic": "simp only [hB, empty_disjoint]", "annotated_tactic": ["simp only [hB, <a>empty_disjoint</a>]", [{"full_name": "Set.empty_disjoint", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1593, 15], "def_end_pos": [1593, 29]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2\nhB\u271d : B = \u22c3 p \u2208 t, closedBall p.1 p.2\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nx : \u03b1\nhx : x \u2208 s'\nxs : x \u2208 s\nhB : B = \u2205\nr : \u211d\nhr : r \u2208 f x\nh'r : r \u2208 Ioo 0 1\n\u22a2 Disjoint B (closedBall x r)", "state_after": "no goals"}, {"tactic": "let r := infDist x B", "annotated_tactic": ["let r := <a>infDist</a> x B", [{"full_name": "Metric.infDist", "def_path": "Mathlib/Topology/MetricSpace/HausdorffDistance.lean", "def_pos": [465, 5], "def_end_pos": [465, 12]}]], "state_before": "case inr\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2\nhB\u271d : B = \u22c3 p \u2208 t, closedBall p.1 p.2\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nx : \u03b1\nhx : x \u2208 s'\nxs : x \u2208 s\nhB : Set.Nonempty B\n\u22a2 \u2203 r, r \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x r)", "state_after": "case inr\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2\nhB\u271d : B = \u22c3 p \u2208 t, closedBall p.1 p.2\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nx : \u03b1\nhx : x \u2208 s'\nxs : x \u2208 s\nhB : Set.Nonempty B\nr : \u211d := infDist x B\n\u22a2 \u2203 r, r \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x r)"}, {"tactic": "have : 0 < min r 1 :=\n  lt_min ((B_closed.not_mem_iff_infDist_pos hB).1 ((mem_diff x).1 hx).2) zero_lt_one", "annotated_tactic": ["have : 0 < <a>min</a> r 1 :=\n          <a>lt_min</a> ((B_closed.not_mem_iff_infDist_pos hB).1 ((<a>mem_diff</a> x).1 hx).2) <a>zero_lt_one</a>", [{"full_name": "Min.min", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1103, 3], "def_end_pos": [1103, 6]}, {"full_name": "lt_min", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [159, 9], "def_end_pos": [159, 15]}, {"full_name": "Set.mem_diff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1819, 9], "def_end_pos": [1819, 17]}, {"full_name": "zero_lt_one", "def_path": "Mathlib/Algebra/Order/ZeroLEOne.lean", "def_pos": [39, 15], "def_end_pos": [39, 26]}]], "state_before": "case inr\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2\nhB\u271d : B = \u22c3 p \u2208 t, closedBall p.1 p.2\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nx : \u03b1\nhx : x \u2208 s'\nxs : x \u2208 s\nhB : Set.Nonempty B\nr : \u211d := infDist x B\n\u22a2 \u2203 r, r \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x r)", "state_after": "case inr\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2\nhB\u271d : B = \u22c3 p \u2208 t, closedBall p.1 p.2\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nx : \u03b1\nhx : x \u2208 s'\nxs : x \u2208 s\nhB : Set.Nonempty B\nr : \u211d := infDist x B\nthis : 0 < min r 1\n\u22a2 \u2203 r, r \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x r)"}, {"tactic": "rcases hf x xs _ this with \u27e8r, hr, h'r\u27e9", "annotated_tactic": ["rcases hf x xs _ this with \u27e8r, hr, h'r\u27e9", []], "state_before": "case inr\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2\nhB\u271d : B = \u22c3 p \u2208 t, closedBall p.1 p.2\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nx : \u03b1\nhx : x \u2208 s'\nxs : x \u2208 s\nhB : Set.Nonempty B\nr : \u211d := infDist x B\nthis : 0 < min r 1\n\u22a2 \u2203 r, r \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x r)", "state_after": "case inr.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2\nhB\u271d : B = \u22c3 p \u2208 t, closedBall p.1 p.2\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nx : \u03b1\nhx : x \u2208 s'\nxs : x \u2208 s\nhB : Set.Nonempty B\nr\u271d : \u211d := infDist x B\nthis : 0 < min r\u271d 1\nr : \u211d\nhr : r \u2208 f x\nh'r : r \u2208 Ioo 0 (min r\u271d 1)\n\u22a2 \u2203 r, r \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x r)"}, {"tactic": "refine' \u27e8r, \u27e8hr, \u27e8h'r.1, h'r.2.trans_le (min_le_right _ _)\u27e9\u27e9, _\u27e9", "annotated_tactic": ["refine' \u27e8r, \u27e8hr, \u27e8h'r.1, h'r.2.<a>trans_le</a> (<a>min_le_right</a> _ _)\u27e9\u27e9, _\u27e9", [{"full_name": "LT.lt.trans_le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [148, 7], "def_end_pos": [148, 21]}, {"full_name": "min_le_right", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [40, 9], "def_end_pos": [40, 21]}]], "state_before": "case inr.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2\nhB\u271d : B = \u22c3 p \u2208 t, closedBall p.1 p.2\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nx : \u03b1\nhx : x \u2208 s'\nxs : x \u2208 s\nhB : Set.Nonempty B\nr\u271d : \u211d := infDist x B\nthis : 0 < min r\u271d 1\nr : \u211d\nhr : r \u2208 f x\nh'r : r \u2208 Ioo 0 (min r\u271d 1)\n\u22a2 \u2203 r, r \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x r)", "state_after": "case inr.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2\nhB\u271d : B = \u22c3 p \u2208 t, closedBall p.1 p.2\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nx : \u03b1\nhx : x \u2208 s'\nxs : x \u2208 s\nhB : Set.Nonempty B\nr\u271d : \u211d := infDist x B\nthis : 0 < min r\u271d 1\nr : \u211d\nhr : r \u2208 f x\nh'r : r \u2208 Ioo 0 (min r\u271d 1)\n\u22a2 Disjoint B (closedBall x r)"}, {"tactic": "rw [disjoint_comm]", "annotated_tactic": ["rw [<a>disjoint_comm</a>]", [{"full_name": "disjoint_comm", "def_path": "Mathlib/Order/Disjoint.lean", "def_pos": [45, 9], "def_end_pos": [45, 22]}]], "state_before": "case inr.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2\nhB\u271d : B = \u22c3 p \u2208 t, closedBall p.1 p.2\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nx : \u03b1\nhx : x \u2208 s'\nxs : x \u2208 s\nhB : Set.Nonempty B\nr\u271d : \u211d := infDist x B\nthis : 0 < min r\u271d 1\nr : \u211d\nhr : r \u2208 f x\nh'r : r \u2208 Ioo 0 (min r\u271d 1)\n\u22a2 Disjoint B (closedBall x r)", "state_after": "case inr.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2\nhB\u271d : B = \u22c3 p \u2208 t, closedBall p.1 p.2\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nx : \u03b1\nhx : x \u2208 s'\nxs : x \u2208 s\nhB : Set.Nonempty B\nr\u271d : \u211d := infDist x B\nthis : 0 < min r\u271d 1\nr : \u211d\nhr : r \u2208 f x\nh'r : r \u2208 Ioo 0 (min r\u271d 1)\n\u22a2 Disjoint (closedBall x r) B"}, {"tactic": "exact disjoint_closedBall_of_lt_infDist (h'r.2.trans_le (min_le_left _ _))", "annotated_tactic": ["exact <a>disjoint_closedBall_of_lt_infDist</a> (h'r.2.<a>trans_le</a> (<a>min_le_left</a> _ _))", [{"full_name": "Metric.disjoint_closedBall_of_lt_infDist", "def_path": "Mathlib/Topology/MetricSpace/HausdorffDistance.lean", "def_pos": [546, 9], "def_end_pos": [546, 42]}, {"full_name": "LT.lt.trans_le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [148, 7], "def_end_pos": [148, 21]}, {"full_name": "min_le_left", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [33, 9], "def_end_pos": [33, 20]}]], "state_before": "case inr.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2\nhB\u271d : B = \u22c3 p \u2208 t, closedBall p.1 p.2\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nx : \u03b1\nhx : x \u2208 s'\nxs : x \u2208 s\nhB : Set.Nonempty B\nr\u271d : \u211d := infDist x B\nthis : 0 < min r\u271d 1\nr : \u211d\nhr : r \u2208 f x\nh'r : r \u2208 Ioo 0 (min r\u271d 1)\n\u22a2 Disjoint (closedBall x r) B", "state_after": "no goals"}, {"tactic": "simp only [Finset.coe_union, pairwiseDisjoint_union, ht.1, true_and_iff, Finset.coe_image]", "annotated_tactic": ["simp only [<a>Finset.coe_union</a>, <a>pairwiseDisjoint_union</a>, ht.1, <a>true_and_iff</a>, <a>Finset.coe_image</a>]", [{"full_name": "Finset.coe_union", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1399, 9], "def_end_pos": [1399, 18]}, {"full_name": "Set.pairwiseDisjoint_union", "def_path": "Mathlib/Data/Set/Pairwise/Basic.lean", "def_pos": [306, 9], "def_end_pos": [306, 31]}, {"full_name": "true_and_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [147, 9], "def_end_pos": [147, 21]}, {"full_name": "Finset.coe_image", "def_path": "Mathlib/Data/Finset/Image.lean", "def_pos": [392, 9], "def_end_pos": [392, 18]}]], "state_before": "case intro.intro.intro.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2\nhB : B = \u22c3 p \u2208 t, closedBall p.1 p.2\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 s' \u2192 r x \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x (r x))\nv : Finset \u03b1\nvs' : \u2191v \u2286 s'\nh\u03bcv : \u2191\u2191\u03bc (s' \\ \u22c3 x \u2208 v, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s'\nhv : PairwiseDisjoint \u2191v fun x => closedBall x (r x)\n\u22a2 PairwiseDisjoint \u2191(t \u222a Finset.image (fun x => (x, r x)) v) fun p => closedBall p.1 p.2", "state_after": "case intro.intro.intro.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2\nhB : B = \u22c3 p \u2208 t, closedBall p.1 p.2\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 s' \u2192 r x \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x (r x))\nv : Finset \u03b1\nvs' : \u2191v \u2286 s'\nh\u03bcv : \u2191\u2191\u03bc (s' \\ \u22c3 x \u2208 v, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s'\nhv : PairwiseDisjoint \u2191v fun x => closedBall x (r x)\n\u22a2 (PairwiseDisjoint ((fun x => (x, r x)) '' \u2191v) fun p => closedBall p.1 p.2) \u2227\n    \u2200 \u2983i : \u03b1 \u00d7 \u211d\u2984,\n      i \u2208 \u2191t \u2192 \u2200 \u2983j : \u03b1 \u00d7 \u211d\u2984, j \u2208 (fun x => (x, r x)) '' \u2191v \u2192 i \u2260 j \u2192 Disjoint (closedBall i.1 i.2) (closedBall j.1 j.2)"}, {"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "case intro.intro.intro.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2\nhB : B = \u22c3 p \u2208 t, closedBall p.1 p.2\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 s' \u2192 r x \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x (r x))\nv : Finset \u03b1\nvs' : \u2191v \u2286 s'\nh\u03bcv : \u2191\u2191\u03bc (s' \\ \u22c3 x \u2208 v, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s'\nhv : PairwiseDisjoint \u2191v fun x => closedBall x (r x)\n\u22a2 (PairwiseDisjoint ((fun x => (x, r x)) '' \u2191v) fun p => closedBall p.1 p.2) \u2227\n    \u2200 \u2983i : \u03b1 \u00d7 \u211d\u2984,\n      i \u2208 \u2191t \u2192 \u2200 \u2983j : \u03b1 \u00d7 \u211d\u2984, j \u2208 (fun x => (x, r x)) '' \u2191v \u2192 i \u2260 j \u2192 Disjoint (closedBall i.1 i.2) (closedBall j.1 j.2)", "state_after": "case intro.intro.intro.refine'_1.left\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2\nhB : B = \u22c3 p \u2208 t, closedBall p.1 p.2\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 s' \u2192 r x \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x (r x))\nv : Finset \u03b1\nvs' : \u2191v \u2286 s'\nh\u03bcv : \u2191\u2191\u03bc (s' \\ \u22c3 x \u2208 v, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s'\nhv : PairwiseDisjoint \u2191v fun x => closedBall x (r x)\n\u22a2 PairwiseDisjoint ((fun x => (x, r x)) '' \u2191v) fun p => closedBall p.1 p.2\n\ncase intro.intro.intro.refine'_1.right\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2\nhB : B = \u22c3 p \u2208 t, closedBall p.1 p.2\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 s' \u2192 r x \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x (r x))\nv : Finset \u03b1\nvs' : \u2191v \u2286 s'\nh\u03bcv : \u2191\u2191\u03bc (s' \\ \u22c3 x \u2208 v, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s'\nhv : PairwiseDisjoint \u2191v fun x => closedBall x (r x)\n\u22a2 \u2200 \u2983i : \u03b1 \u00d7 \u211d\u2984,\n    i \u2208 \u2191t \u2192 \u2200 \u2983j : \u03b1 \u00d7 \u211d\u2984, j \u2208 (fun x => (x, r x)) '' \u2191v \u2192 i \u2260 j \u2192 Disjoint (closedBall i.1 i.2) (closedBall j.1 j.2)"}, {"tactic": "intro p hp q hq hpq", "annotated_tactic": ["intro p hp q hq hpq", []], "state_before": "case intro.intro.intro.refine'_1.left\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2\nhB : B = \u22c3 p \u2208 t, closedBall p.1 p.2\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 s' \u2192 r x \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x (r x))\nv : Finset \u03b1\nvs' : \u2191v \u2286 s'\nh\u03bcv : \u2191\u2191\u03bc (s' \\ \u22c3 x \u2208 v, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s'\nhv : PairwiseDisjoint \u2191v fun x => closedBall x (r x)\n\u22a2 PairwiseDisjoint ((fun x => (x, r x)) '' \u2191v) fun p => closedBall p.1 p.2", "state_after": "case intro.intro.intro.refine'_1.left\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2\nhB : B = \u22c3 p \u2208 t, closedBall p.1 p.2\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 s' \u2192 r x \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x (r x))\nv : Finset \u03b1\nvs' : \u2191v \u2286 s'\nh\u03bcv : \u2191\u2191\u03bc (s' \\ \u22c3 x \u2208 v, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s'\nhv : PairwiseDisjoint \u2191v fun x => closedBall x (r x)\np : \u03b1 \u00d7 \u211d\nhp : p \u2208 (fun x => (x, r x)) '' \u2191v\nq : \u03b1 \u00d7 \u211d\nhq : q \u2208 (fun x => (x, r x)) '' \u2191v\nhpq : p \u2260 q\n\u22a2 (Disjoint on fun p => closedBall p.1 p.2) p q"}, {"tactic": "rcases (mem_image _ _ _).1 hp with \u27e8p', p'v, rfl\u27e9", "annotated_tactic": ["rcases (<a>mem_image</a> _ _ _).1 hp with \u27e8p', p'v, rfl\u27e9", [{"full_name": "Set.mem_image", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [231, 9], "def_end_pos": [231, 18]}]], "state_before": "case intro.intro.intro.refine'_1.left\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2\nhB : B = \u22c3 p \u2208 t, closedBall p.1 p.2\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 s' \u2192 r x \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x (r x))\nv : Finset \u03b1\nvs' : \u2191v \u2286 s'\nh\u03bcv : \u2191\u2191\u03bc (s' \\ \u22c3 x \u2208 v, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s'\nhv : PairwiseDisjoint \u2191v fun x => closedBall x (r x)\np : \u03b1 \u00d7 \u211d\nhp : p \u2208 (fun x => (x, r x)) '' \u2191v\nq : \u03b1 \u00d7 \u211d\nhq : q \u2208 (fun x => (x, r x)) '' \u2191v\nhpq : p \u2260 q\n\u22a2 (Disjoint on fun p => closedBall p.1 p.2) p q", "state_after": "case intro.intro.intro.refine'_1.left.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2\nhB : B = \u22c3 p \u2208 t, closedBall p.1 p.2\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 s' \u2192 r x \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x (r x))\nv : Finset \u03b1\nvs' : \u2191v \u2286 s'\nh\u03bcv : \u2191\u2191\u03bc (s' \\ \u22c3 x \u2208 v, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s'\nhv : PairwiseDisjoint \u2191v fun x => closedBall x (r x)\nq : \u03b1 \u00d7 \u211d\nhq : q \u2208 (fun x => (x, r x)) '' \u2191v\np' : \u03b1\np'v : p' \u2208 \u2191v\nhp : (p', r p') \u2208 (fun x => (x, r x)) '' \u2191v\nhpq : (p', r p') \u2260 q\n\u22a2 (Disjoint on fun p => closedBall p.1 p.2) (p', r p') q"}, {"tactic": "rcases (mem_image _ _ _).1 hq with \u27e8q', q'v, rfl\u27e9", "annotated_tactic": ["rcases (<a>mem_image</a> _ _ _).1 hq with \u27e8q', q'v, rfl\u27e9", [{"full_name": "Set.mem_image", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [231, 9], "def_end_pos": [231, 18]}]], "state_before": "case intro.intro.intro.refine'_1.left.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2\nhB : B = \u22c3 p \u2208 t, closedBall p.1 p.2\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 s' \u2192 r x \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x (r x))\nv : Finset \u03b1\nvs' : \u2191v \u2286 s'\nh\u03bcv : \u2191\u2191\u03bc (s' \\ \u22c3 x \u2208 v, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s'\nhv : PairwiseDisjoint \u2191v fun x => closedBall x (r x)\nq : \u03b1 \u00d7 \u211d\nhq : q \u2208 (fun x => (x, r x)) '' \u2191v\np' : \u03b1\np'v : p' \u2208 \u2191v\nhp : (p', r p') \u2208 (fun x => (x, r x)) '' \u2191v\nhpq : (p', r p') \u2260 q\n\u22a2 (Disjoint on fun p => closedBall p.1 p.2) (p', r p') q", "state_after": "case intro.intro.intro.refine'_1.left.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2\nhB : B = \u22c3 p \u2208 t, closedBall p.1 p.2\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 s' \u2192 r x \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x (r x))\nv : Finset \u03b1\nvs' : \u2191v \u2286 s'\nh\u03bcv : \u2191\u2191\u03bc (s' \\ \u22c3 x \u2208 v, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s'\nhv : PairwiseDisjoint \u2191v fun x => closedBall x (r x)\np' : \u03b1\np'v : p' \u2208 \u2191v\nhp : (p', r p') \u2208 (fun x => (x, r x)) '' \u2191v\nq' : \u03b1\nq'v : q' \u2208 \u2191v\nhq : (q', r q') \u2208 (fun x => (x, r x)) '' \u2191v\nhpq : (p', r p') \u2260 (q', r q')\n\u22a2 (Disjoint on fun p => closedBall p.1 p.2) (p', r p') (q', r q')"}, {"tactic": "refine' hv p'v q'v fun hp'q' => _", "annotated_tactic": ["refine' hv p'v q'v fun hp'q' => _", []], "state_before": "case intro.intro.intro.refine'_1.left.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2\nhB : B = \u22c3 p \u2208 t, closedBall p.1 p.2\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 s' \u2192 r x \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x (r x))\nv : Finset \u03b1\nvs' : \u2191v \u2286 s'\nh\u03bcv : \u2191\u2191\u03bc (s' \\ \u22c3 x \u2208 v, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s'\nhv : PairwiseDisjoint \u2191v fun x => closedBall x (r x)\np' : \u03b1\np'v : p' \u2208 \u2191v\nhp : (p', r p') \u2208 (fun x => (x, r x)) '' \u2191v\nq' : \u03b1\nq'v : q' \u2208 \u2191v\nhq : (q', r q') \u2208 (fun x => (x, r x)) '' \u2191v\nhpq : (p', r p') \u2260 (q', r q')\n\u22a2 (Disjoint on fun p => closedBall p.1 p.2) (p', r p') (q', r q')", "state_after": "case intro.intro.intro.refine'_1.left.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2\nhB : B = \u22c3 p \u2208 t, closedBall p.1 p.2\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 s' \u2192 r x \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x (r x))\nv : Finset \u03b1\nvs' : \u2191v \u2286 s'\nh\u03bcv : \u2191\u2191\u03bc (s' \\ \u22c3 x \u2208 v, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s'\nhv : PairwiseDisjoint \u2191v fun x => closedBall x (r x)\np' : \u03b1\np'v : p' \u2208 \u2191v\nhp : (p', r p') \u2208 (fun x => (x, r x)) '' \u2191v\nq' : \u03b1\nq'v : q' \u2208 \u2191v\nhq : (q', r q') \u2208 (fun x => (x, r x)) '' \u2191v\nhpq : (p', r p') \u2260 (q', r q')\nhp'q' : p' = (q', r q').1\n\u22a2 False"}, {"tactic": "rw [hp'q'] at hpq", "annotated_tactic": ["rw [hp'q'] at hpq", []], "state_before": "case intro.intro.intro.refine'_1.left.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2\nhB : B = \u22c3 p \u2208 t, closedBall p.1 p.2\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 s' \u2192 r x \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x (r x))\nv : Finset \u03b1\nvs' : \u2191v \u2286 s'\nh\u03bcv : \u2191\u2191\u03bc (s' \\ \u22c3 x \u2208 v, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s'\nhv : PairwiseDisjoint \u2191v fun x => closedBall x (r x)\np' : \u03b1\np'v : p' \u2208 \u2191v\nhp : (p', r p') \u2208 (fun x => (x, r x)) '' \u2191v\nq' : \u03b1\nq'v : q' \u2208 \u2191v\nhq : (q', r q') \u2208 (fun x => (x, r x)) '' \u2191v\nhpq : (p', r p') \u2260 (q', r q')\nhp'q' : p' = (q', r q').1\n\u22a2 False", "state_after": "case intro.intro.intro.refine'_1.left.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2\nhB : B = \u22c3 p \u2208 t, closedBall p.1 p.2\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 s' \u2192 r x \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x (r x))\nv : Finset \u03b1\nvs' : \u2191v \u2286 s'\nh\u03bcv : \u2191\u2191\u03bc (s' \\ \u22c3 x \u2208 v, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s'\nhv : PairwiseDisjoint \u2191v fun x => closedBall x (r x)\np' : \u03b1\np'v : p' \u2208 \u2191v\nhp : (p', r p') \u2208 (fun x => (x, r x)) '' \u2191v\nq' : \u03b1\nq'v : q' \u2208 \u2191v\nhq : (q', r q') \u2208 (fun x => (x, r x)) '' \u2191v\nhpq : ((q', r q').1, r (q', r q').1) \u2260 (q', r q')\nhp'q' : p' = (q', r q').1\n\u22a2 False"}, {"tactic": "exact hpq rfl", "annotated_tactic": ["exact hpq <a>rfl</a>", [{"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "case intro.intro.intro.refine'_1.left.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2\nhB : B = \u22c3 p \u2208 t, closedBall p.1 p.2\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 s' \u2192 r x \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x (r x))\nv : Finset \u03b1\nvs' : \u2191v \u2286 s'\nh\u03bcv : \u2191\u2191\u03bc (s' \\ \u22c3 x \u2208 v, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s'\nhv : PairwiseDisjoint \u2191v fun x => closedBall x (r x)\np' : \u03b1\np'v : p' \u2208 \u2191v\nhp : (p', r p') \u2208 (fun x => (x, r x)) '' \u2191v\nq' : \u03b1\nq'v : q' \u2208 \u2191v\nhq : (q', r q') \u2208 (fun x => (x, r x)) '' \u2191v\nhpq : ((q', r q').1, r (q', r q').1) \u2260 (q', r q')\nhp'q' : p' = (q', r q').1\n\u22a2 False", "state_after": "no goals"}, {"tactic": "intro p hp q hq hpq", "annotated_tactic": ["intro p hp q hq hpq", []], "state_before": "case intro.intro.intro.refine'_1.right\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2\nhB : B = \u22c3 p \u2208 t, closedBall p.1 p.2\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 s' \u2192 r x \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x (r x))\nv : Finset \u03b1\nvs' : \u2191v \u2286 s'\nh\u03bcv : \u2191\u2191\u03bc (s' \\ \u22c3 x \u2208 v, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s'\nhv : PairwiseDisjoint \u2191v fun x => closedBall x (r x)\n\u22a2 \u2200 \u2983i : \u03b1 \u00d7 \u211d\u2984,\n    i \u2208 \u2191t \u2192 \u2200 \u2983j : \u03b1 \u00d7 \u211d\u2984, j \u2208 (fun x => (x, r x)) '' \u2191v \u2192 i \u2260 j \u2192 Disjoint (closedBall i.1 i.2) (closedBall j.1 j.2)", "state_after": "case intro.intro.intro.refine'_1.right\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2\nhB : B = \u22c3 p \u2208 t, closedBall p.1 p.2\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 s' \u2192 r x \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x (r x))\nv : Finset \u03b1\nvs' : \u2191v \u2286 s'\nh\u03bcv : \u2191\u2191\u03bc (s' \\ \u22c3 x \u2208 v, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s'\nhv : PairwiseDisjoint \u2191v fun x => closedBall x (r x)\np : \u03b1 \u00d7 \u211d\nhp : p \u2208 \u2191t\nq : \u03b1 \u00d7 \u211d\nhq : q \u2208 (fun x => (x, r x)) '' \u2191v\nhpq : p \u2260 q\n\u22a2 Disjoint (closedBall p.1 p.2) (closedBall q.1 q.2)"}, {"tactic": "rcases (mem_image _ _ _).1 hq with \u27e8q', q'v, rfl\u27e9", "annotated_tactic": ["rcases (<a>mem_image</a> _ _ _).1 hq with \u27e8q', q'v, rfl\u27e9", [{"full_name": "Set.mem_image", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [231, 9], "def_end_pos": [231, 18]}]], "state_before": "case intro.intro.intro.refine'_1.right\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2\nhB : B = \u22c3 p \u2208 t, closedBall p.1 p.2\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 s' \u2192 r x \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x (r x))\nv : Finset \u03b1\nvs' : \u2191v \u2286 s'\nh\u03bcv : \u2191\u2191\u03bc (s' \\ \u22c3 x \u2208 v, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s'\nhv : PairwiseDisjoint \u2191v fun x => closedBall x (r x)\np : \u03b1 \u00d7 \u211d\nhp : p \u2208 \u2191t\nq : \u03b1 \u00d7 \u211d\nhq : q \u2208 (fun x => (x, r x)) '' \u2191v\nhpq : p \u2260 q\n\u22a2 Disjoint (closedBall p.1 p.2) (closedBall q.1 q.2)", "state_after": "case intro.intro.intro.refine'_1.right.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2\nhB : B = \u22c3 p \u2208 t, closedBall p.1 p.2\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 s' \u2192 r x \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x (r x))\nv : Finset \u03b1\nvs' : \u2191v \u2286 s'\nh\u03bcv : \u2191\u2191\u03bc (s' \\ \u22c3 x \u2208 v, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s'\nhv : PairwiseDisjoint \u2191v fun x => closedBall x (r x)\np : \u03b1 \u00d7 \u211d\nhp : p \u2208 \u2191t\nq' : \u03b1\nq'v : q' \u2208 \u2191v\nhq : (q', r q') \u2208 (fun x => (x, r x)) '' \u2191v\nhpq : p \u2260 (q', r q')\n\u22a2 Disjoint (closedBall p.1 p.2) (closedBall (q', r q').1 (q', r q').2)"}, {"tactic": "apply disjoint_of_subset_left _ (hr q' (vs' q'v)).2", "annotated_tactic": ["apply <a>disjoint_of_subset_left</a> _ (hr q' (vs' q'v)).2", [{"full_name": "Set.disjoint_of_subset_left", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1574, 7], "def_end_pos": [1574, 30]}]], "state_before": "case intro.intro.intro.refine'_1.right.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2\nhB : B = \u22c3 p \u2208 t, closedBall p.1 p.2\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 s' \u2192 r x \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x (r x))\nv : Finset \u03b1\nvs' : \u2191v \u2286 s'\nh\u03bcv : \u2191\u2191\u03bc (s' \\ \u22c3 x \u2208 v, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s'\nhv : PairwiseDisjoint \u2191v fun x => closedBall x (r x)\np : \u03b1 \u00d7 \u211d\nhp : p \u2208 \u2191t\nq' : \u03b1\nq'v : q' \u2208 \u2191v\nhq : (q', r q') \u2208 (fun x => (x, r x)) '' \u2191v\nhpq : p \u2260 (q', r q')\n\u22a2 Disjoint (closedBall p.1 p.2) (closedBall (q', r q').1 (q', r q').2)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2\nhB : B = \u22c3 p \u2208 t, closedBall p.1 p.2\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 s' \u2192 r x \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x (r x))\nv : Finset \u03b1\nvs' : \u2191v \u2286 s'\nh\u03bcv : \u2191\u2191\u03bc (s' \\ \u22c3 x \u2208 v, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s'\nhv : PairwiseDisjoint \u2191v fun x => closedBall x (r x)\np : \u03b1 \u00d7 \u211d\nhp : p \u2208 \u2191t\nq' : \u03b1\nq'v : q' \u2208 \u2191v\nhq : (q', r q') \u2208 (fun x => (x, r x)) '' \u2191v\nhpq : p \u2260 (q', r q')\n\u22a2 closedBall p.1 p.2 \u2286 B"}, {"tactic": "rw [hB, \u2190 Finset.set_biUnion_coe]", "annotated_tactic": ["rw [hB, \u2190 <a>Finset.set_biUnion_coe</a>]", [{"full_name": "Finset.set_biUnion_coe", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [2087, 9], "def_end_pos": [2087, 24]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2\nhB : B = \u22c3 p \u2208 t, closedBall p.1 p.2\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 s' \u2192 r x \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x (r x))\nv : Finset \u03b1\nvs' : \u2191v \u2286 s'\nh\u03bcv : \u2191\u2191\u03bc (s' \\ \u22c3 x \u2208 v, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s'\nhv : PairwiseDisjoint \u2191v fun x => closedBall x (r x)\np : \u03b1 \u00d7 \u211d\nhp : p \u2208 \u2191t\nq' : \u03b1\nq'v : q' \u2208 \u2191v\nhq : (q', r q') \u2208 (fun x => (x, r x)) '' \u2191v\nhpq : p \u2260 (q', r q')\n\u22a2 closedBall p.1 p.2 \u2286 B", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2\nhB : B = \u22c3 p \u2208 t, closedBall p.1 p.2\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 s' \u2192 r x \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x (r x))\nv : Finset \u03b1\nvs' : \u2191v \u2286 s'\nh\u03bcv : \u2191\u2191\u03bc (s' \\ \u22c3 x \u2208 v, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s'\nhv : PairwiseDisjoint \u2191v fun x => closedBall x (r x)\np : \u03b1 \u00d7 \u211d\nhp : p \u2208 \u2191t\nq' : \u03b1\nq'v : q' \u2208 \u2191v\nhq : (q', r q') \u2208 (fun x => (x, r x)) '' \u2191v\nhpq : p \u2260 (q', r q')\n\u22a2 closedBall p.1 p.2 \u2286 \u22c3 x \u2208 \u2191t, closedBall x.1 x.2"}, {"tactic": "exact subset_biUnion_of_mem (u := fun x : \u03b1 \u00d7 \u211d => closedBall x.1 x.2) hp", "annotated_tactic": ["exact <a>subset_biUnion_of_mem</a> (u := fun x : \u03b1 \u00d7 \u211d => <a>closedBall</a> x.1 x.2) hp", [{"full_name": "Set.subset_biUnion_of_mem", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [978, 9], "def_end_pos": [978, 30]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2\nhB : B = \u22c3 p \u2208 t, closedBall p.1 p.2\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 s' \u2192 r x \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x (r x))\nv : Finset \u03b1\nvs' : \u2191v \u2286 s'\nh\u03bcv : \u2191\u2191\u03bc (s' \\ \u22c3 x \u2208 v, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s'\nhv : PairwiseDisjoint \u2191v fun x => closedBall x (r x)\np : \u03b1 \u00d7 \u211d\nhp : p \u2208 \u2191t\nq' : \u03b1\nq'v : q' \u2208 \u2191v\nhq : (q', r q') \u2208 (fun x => (x, r x)) '' \u2191v\nhpq : p \u2260 (q', r q')\n\u22a2 closedBall p.1 p.2 \u2286 \u22c3 x \u2208 \u2191t, closedBall x.1 x.2", "state_after": "no goals"}, {"tactic": "intro p hp", "annotated_tactic": ["intro p hp", []], "state_before": "case intro.intro.intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2\nhB : B = \u22c3 p \u2208 t, closedBall p.1 p.2\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 s' \u2192 r x \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x (r x))\nv : Finset \u03b1\nvs' : \u2191v \u2286 s'\nh\u03bcv : \u2191\u2191\u03bc (s' \\ \u22c3 x \u2208 v, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s'\nhv : PairwiseDisjoint \u2191v fun x => closedBall x (r x)\n\u22a2 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u222a Finset.image (fun x => (x, r x)) v \u2192 p.1 \u2208 s", "state_after": "case intro.intro.intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2\nhB : B = \u22c3 p \u2208 t, closedBall p.1 p.2\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 s' \u2192 r x \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x (r x))\nv : Finset \u03b1\nvs' : \u2191v \u2286 s'\nh\u03bcv : \u2191\u2191\u03bc (s' \\ \u22c3 x \u2208 v, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s'\nhv : PairwiseDisjoint \u2191v fun x => closedBall x (r x)\np : \u03b1 \u00d7 \u211d\nhp : p \u2208 t \u222a Finset.image (fun x => (x, r x)) v\n\u22a2 p.1 \u2208 s"}, {"tactic": "rcases Finset.mem_union.1 hp with (h'p | h'p)", "annotated_tactic": ["rcases <a>Finset.mem_union</a>.1 hp with (h'p | h'p)", [{"full_name": "Finset.mem_union", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1373, 9], "def_end_pos": [1373, 18]}]], "state_before": "case intro.intro.intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2\nhB : B = \u22c3 p \u2208 t, closedBall p.1 p.2\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 s' \u2192 r x \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x (r x))\nv : Finset \u03b1\nvs' : \u2191v \u2286 s'\nh\u03bcv : \u2191\u2191\u03bc (s' \\ \u22c3 x \u2208 v, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s'\nhv : PairwiseDisjoint \u2191v fun x => closedBall x (r x)\np : \u03b1 \u00d7 \u211d\nhp : p \u2208 t \u222a Finset.image (fun x => (x, r x)) v\n\u22a2 p.1 \u2208 s", "state_after": "case intro.intro.intro.refine'_2.inl\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2\nhB : B = \u22c3 p \u2208 t, closedBall p.1 p.2\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 s' \u2192 r x \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x (r x))\nv : Finset \u03b1\nvs' : \u2191v \u2286 s'\nh\u03bcv : \u2191\u2191\u03bc (s' \\ \u22c3 x \u2208 v, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s'\nhv : PairwiseDisjoint \u2191v fun x => closedBall x (r x)\np : \u03b1 \u00d7 \u211d\nhp : p \u2208 t \u222a Finset.image (fun x => (x, r x)) v\nh'p : p \u2208 t\n\u22a2 p.1 \u2208 s\n\ncase intro.intro.intro.refine'_2.inr\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2\nhB : B = \u22c3 p \u2208 t, closedBall p.1 p.2\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 s' \u2192 r x \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x (r x))\nv : Finset \u03b1\nvs' : \u2191v \u2286 s'\nh\u03bcv : \u2191\u2191\u03bc (s' \\ \u22c3 x \u2208 v, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s'\nhv : PairwiseDisjoint \u2191v fun x => closedBall x (r x)\np : \u03b1 \u00d7 \u211d\nhp : p \u2208 t \u222a Finset.image (fun x => (x, r x)) v\nh'p : p \u2208 Finset.image (fun x => (x, r x)) v\n\u22a2 p.1 \u2208 s"}, {"tactic": "exact ht.2.1 p h'p", "annotated_tactic": ["exact ht.2.1 p h'p", []], "state_before": "case intro.intro.intro.refine'_2.inl\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2\nhB : B = \u22c3 p \u2208 t, closedBall p.1 p.2\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 s' \u2192 r x \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x (r x))\nv : Finset \u03b1\nvs' : \u2191v \u2286 s'\nh\u03bcv : \u2191\u2191\u03bc (s' \\ \u22c3 x \u2208 v, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s'\nhv : PairwiseDisjoint \u2191v fun x => closedBall x (r x)\np : \u03b1 \u00d7 \u211d\nhp : p \u2208 t \u222a Finset.image (fun x => (x, r x)) v\nh'p : p \u2208 t\n\u22a2 p.1 \u2208 s", "state_after": "no goals"}, {"tactic": "rcases Finset.mem_image.1 h'p with \u27e8p', p'v, rfl\u27e9", "annotated_tactic": ["rcases <a>Finset.mem_image</a>.1 h'p with \u27e8p', p'v, rfl\u27e9", [{"full_name": "Finset.mem_image", "def_path": "Mathlib/Data/Finset/Image.lean", "def_pos": [330, 9], "def_end_pos": [330, 18]}]], "state_before": "case intro.intro.intro.refine'_2.inr\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2\nhB : B = \u22c3 p \u2208 t, closedBall p.1 p.2\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 s' \u2192 r x \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x (r x))\nv : Finset \u03b1\nvs' : \u2191v \u2286 s'\nh\u03bcv : \u2191\u2191\u03bc (s' \\ \u22c3 x \u2208 v, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s'\nhv : PairwiseDisjoint \u2191v fun x => closedBall x (r x)\np : \u03b1 \u00d7 \u211d\nhp : p \u2208 t \u222a Finset.image (fun x => (x, r x)) v\nh'p : p \u2208 Finset.image (fun x => (x, r x)) v\n\u22a2 p.1 \u2208 s", "state_after": "case intro.intro.intro.refine'_2.inr.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2\nhB : B = \u22c3 p \u2208 t, closedBall p.1 p.2\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 s' \u2192 r x \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x (r x))\nv : Finset \u03b1\nvs' : \u2191v \u2286 s'\nh\u03bcv : \u2191\u2191\u03bc (s' \\ \u22c3 x \u2208 v, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s'\nhv : PairwiseDisjoint \u2191v fun x => closedBall x (r x)\np' : \u03b1\np'v : p' \u2208 v\nhp : (p', r p') \u2208 t \u222a Finset.image (fun x => (x, r x)) v\nh'p : (p', r p') \u2208 Finset.image (fun x => (x, r x)) v\n\u22a2 (p', r p').1 \u2208 s"}, {"tactic": "exact ((mem_diff _).1 (vs' (Finset.mem_coe.2 p'v))).1", "annotated_tactic": ["exact ((<a>mem_diff</a> _).1 (vs' (<a>Finset.mem_coe</a>.2 p'v))).1", [{"full_name": "Set.mem_diff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1819, 9], "def_end_pos": [1819, 17]}, {"full_name": "Finset.mem_coe", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [208, 9], "def_end_pos": [208, 16]}]], "state_before": "case intro.intro.intro.refine'_2.inr.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2\nhB : B = \u22c3 p \u2208 t, closedBall p.1 p.2\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 s' \u2192 r x \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x (r x))\nv : Finset \u03b1\nvs' : \u2191v \u2286 s'\nh\u03bcv : \u2191\u2191\u03bc (s' \\ \u22c3 x \u2208 v, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s'\nhv : PairwiseDisjoint \u2191v fun x => closedBall x (r x)\np' : \u03b1\np'v : p' \u2208 v\nhp : (p', r p') \u2208 t \u222a Finset.image (fun x => (x, r x)) v\nh'p : (p', r p') \u2208 Finset.image (fun x => (x, r x)) v\n\u22a2 (p', r p').1 \u2208 s", "state_after": "no goals"}, {"tactic": "intro p hp", "annotated_tactic": ["intro p hp", []], "state_before": "case intro.intro.intro.refine'_3\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2\nhB : B = \u22c3 p \u2208 t, closedBall p.1 p.2\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 s' \u2192 r x \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x (r x))\nv : Finset \u03b1\nvs' : \u2191v \u2286 s'\nh\u03bcv : \u2191\u2191\u03bc (s' \\ \u22c3 x \u2208 v, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s'\nhv : PairwiseDisjoint \u2191v fun x => closedBall x (r x)\n\u22a2 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u222a Finset.image (fun x => (x, r x)) v \u2192 p.2 \u2208 f p.1", "state_after": "case intro.intro.intro.refine'_3\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2\nhB : B = \u22c3 p \u2208 t, closedBall p.1 p.2\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 s' \u2192 r x \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x (r x))\nv : Finset \u03b1\nvs' : \u2191v \u2286 s'\nh\u03bcv : \u2191\u2191\u03bc (s' \\ \u22c3 x \u2208 v, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s'\nhv : PairwiseDisjoint \u2191v fun x => closedBall x (r x)\np : \u03b1 \u00d7 \u211d\nhp : p \u2208 t \u222a Finset.image (fun x => (x, r x)) v\n\u22a2 p.2 \u2208 f p.1"}, {"tactic": "rcases Finset.mem_union.1 hp with (h'p | h'p)", "annotated_tactic": ["rcases <a>Finset.mem_union</a>.1 hp with (h'p | h'p)", [{"full_name": "Finset.mem_union", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1373, 9], "def_end_pos": [1373, 18]}]], "state_before": "case intro.intro.intro.refine'_3\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2\nhB : B = \u22c3 p \u2208 t, closedBall p.1 p.2\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 s' \u2192 r x \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x (r x))\nv : Finset \u03b1\nvs' : \u2191v \u2286 s'\nh\u03bcv : \u2191\u2191\u03bc (s' \\ \u22c3 x \u2208 v, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s'\nhv : PairwiseDisjoint \u2191v fun x => closedBall x (r x)\np : \u03b1 \u00d7 \u211d\nhp : p \u2208 t \u222a Finset.image (fun x => (x, r x)) v\n\u22a2 p.2 \u2208 f p.1", "state_after": "case intro.intro.intro.refine'_3.inl\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2\nhB : B = \u22c3 p \u2208 t, closedBall p.1 p.2\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 s' \u2192 r x \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x (r x))\nv : Finset \u03b1\nvs' : \u2191v \u2286 s'\nh\u03bcv : \u2191\u2191\u03bc (s' \\ \u22c3 x \u2208 v, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s'\nhv : PairwiseDisjoint \u2191v fun x => closedBall x (r x)\np : \u03b1 \u00d7 \u211d\nhp : p \u2208 t \u222a Finset.image (fun x => (x, r x)) v\nh'p : p \u2208 t\n\u22a2 p.2 \u2208 f p.1\n\ncase intro.intro.intro.refine'_3.inr\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2\nhB : B = \u22c3 p \u2208 t, closedBall p.1 p.2\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 s' \u2192 r x \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x (r x))\nv : Finset \u03b1\nvs' : \u2191v \u2286 s'\nh\u03bcv : \u2191\u2191\u03bc (s' \\ \u22c3 x \u2208 v, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s'\nhv : PairwiseDisjoint \u2191v fun x => closedBall x (r x)\np : \u03b1 \u00d7 \u211d\nhp : p \u2208 t \u222a Finset.image (fun x => (x, r x)) v\nh'p : p \u2208 Finset.image (fun x => (x, r x)) v\n\u22a2 p.2 \u2208 f p.1"}, {"tactic": "exact ht.2.2 p h'p", "annotated_tactic": ["exact ht.2.2 p h'p", []], "state_before": "case intro.intro.intro.refine'_3.inl\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2\nhB : B = \u22c3 p \u2208 t, closedBall p.1 p.2\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 s' \u2192 r x \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x (r x))\nv : Finset \u03b1\nvs' : \u2191v \u2286 s'\nh\u03bcv : \u2191\u2191\u03bc (s' \\ \u22c3 x \u2208 v, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s'\nhv : PairwiseDisjoint \u2191v fun x => closedBall x (r x)\np : \u03b1 \u00d7 \u211d\nhp : p \u2208 t \u222a Finset.image (fun x => (x, r x)) v\nh'p : p \u2208 t\n\u22a2 p.2 \u2208 f p.1", "state_after": "no goals"}, {"tactic": "rcases Finset.mem_image.1 h'p with \u27e8p', p'v, rfl\u27e9", "annotated_tactic": ["rcases <a>Finset.mem_image</a>.1 h'p with \u27e8p', p'v, rfl\u27e9", [{"full_name": "Finset.mem_image", "def_path": "Mathlib/Data/Finset/Image.lean", "def_pos": [330, 9], "def_end_pos": [330, 18]}]], "state_before": "case intro.intro.intro.refine'_3.inr\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2\nhB : B = \u22c3 p \u2208 t, closedBall p.1 p.2\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 s' \u2192 r x \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x (r x))\nv : Finset \u03b1\nvs' : \u2191v \u2286 s'\nh\u03bcv : \u2191\u2191\u03bc (s' \\ \u22c3 x \u2208 v, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s'\nhv : PairwiseDisjoint \u2191v fun x => closedBall x (r x)\np : \u03b1 \u00d7 \u211d\nhp : p \u2208 t \u222a Finset.image (fun x => (x, r x)) v\nh'p : p \u2208 Finset.image (fun x => (x, r x)) v\n\u22a2 p.2 \u2208 f p.1", "state_after": "case intro.intro.intro.refine'_3.inr.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2\nhB : B = \u22c3 p \u2208 t, closedBall p.1 p.2\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 s' \u2192 r x \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x (r x))\nv : Finset \u03b1\nvs' : \u2191v \u2286 s'\nh\u03bcv : \u2191\u2191\u03bc (s' \\ \u22c3 x \u2208 v, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s'\nhv : PairwiseDisjoint \u2191v fun x => closedBall x (r x)\np' : \u03b1\np'v : p' \u2208 v\nhp : (p', r p') \u2208 t \u222a Finset.image (fun x => (x, r x)) v\nh'p : (p', r p') \u2208 Finset.image (fun x => (x, r x)) v\n\u22a2 (p', r p').2 \u2208 f (p', r p').1"}, {"tactic": "exact (hr p' (vs' p'v)).1.1", "annotated_tactic": ["exact (hr p' (vs' p'v)).1.1", []], "state_before": "case intro.intro.intro.refine'_3.inr.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2\nhB : B = \u22c3 p \u2208 t, closedBall p.1 p.2\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 s' \u2192 r x \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x (r x))\nv : Finset \u03b1\nvs' : \u2191v \u2286 s'\nh\u03bcv : \u2191\u2191\u03bc (s' \\ \u22c3 x \u2208 v, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s'\nhv : PairwiseDisjoint \u2191v fun x => closedBall x (r x)\np' : \u03b1\np'v : p' \u2208 v\nhp : (p', r p') \u2208 t \u222a Finset.image (fun x => (x, r x)) v\nh'p : (p', r p') \u2208 Finset.image (fun x => (x, r x)) v\n\u22a2 (p', r p').2 \u2208 f (p', r p').1", "state_after": "no goals"}, {"tactic": "convert h\u03bcv using 2", "annotated_tactic": ["convert h\u03bcv using 2", []], "state_before": "case intro.intro.intro.refine'_4\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2\nhB : B = \u22c3 p \u2208 t, closedBall p.1 p.2\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 s' \u2192 r x \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x (r x))\nv : Finset \u03b1\nvs' : \u2191v \u2286 s'\nh\u03bcv : \u2191\u2191\u03bc (s' \\ \u22c3 x \u2208 v, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s'\nhv : PairwiseDisjoint \u2191v fun x => closedBall x (r x)\n\u22a2 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t \u222a Finset.image (fun x => (x, r x)) v, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s'", "state_after": "case h.e'_3.h.e'_3\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2\nhB : B = \u22c3 p \u2208 t, closedBall p.1 p.2\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 s' \u2192 r x \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x (r x))\nv : Finset \u03b1\nvs' : \u2191v \u2286 s'\nh\u03bcv : \u2191\u2191\u03bc (s' \\ \u22c3 x \u2208 v, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s'\nhv : PairwiseDisjoint \u2191v fun x => closedBall x (r x)\n\u22a2 s \\ \u22c3 p \u2208 t \u222a Finset.image (fun x => (x, r x)) v, closedBall p.1 p.2 = s' \\ \u22c3 x \u2208 v, closedBall x (r x)"}, {"tactic": "rw [Finset.set_biUnion_union, \u2190 diff_diff, Finset.set_biUnion_finset_image]", "annotated_tactic": ["rw [<a>Finset.set_biUnion_union</a>, \u2190 <a>diff_diff</a>, <a>Finset.set_biUnion_finset_image</a>]", [{"full_name": "Finset.set_biUnion_union", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [2126, 9], "def_end_pos": [2126, 26]}, {"full_name": "Set.diff_diff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1939, 9], "def_end_pos": [1939, 18]}, {"full_name": "Finset.set_biUnion_finset_image", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [2146, 9], "def_end_pos": [2146, 33]}]], "state_before": "case h.e'_3.h.e'_3\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2\nhB : B = \u22c3 p \u2208 t, closedBall p.1 p.2\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 s' \u2192 r x \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x (r x))\nv : Finset \u03b1\nvs' : \u2191v \u2286 s'\nh\u03bcv : \u2191\u2191\u03bc (s' \\ \u22c3 x \u2208 v, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s'\nhv : PairwiseDisjoint \u2191v fun x => closedBall x (r x)\n\u22a2 s \\ \u22c3 p \u2208 t \u222a Finset.image (fun x => (x, r x)) v, closedBall p.1 p.2 = s' \\ \u22c3 x \u2208 v, closedBall x (r x)", "state_after": "no goals"}, {"tactic": "simp only [Function.comp_apply, Function.iterate_succ']", "annotated_tactic": ["simp only [<a>Function.comp_apply</a>, <a>Function.iterate_succ'</a>]", [{"full_name": "Function.comp_apply", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [33, 17], "def_end_pos": [33, 36]}, {"full_name": "Function.iterate_succ'", "def_path": "Mathlib/Logic/Function/Iterate.lean", "def_pos": [186, 9], "def_end_pos": [186, 22]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n  \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n    P t \u2192\n      t \u2286 F t \u2227\n        P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nn : \u2115\n\u22a2 u (Nat.succ n) = F (u n)", "state_after": "no goals"}, {"tactic": "intro n", "annotated_tactic": ["intro n", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n  \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n    P t \u2192\n      t \u2286 F t \u2227\n        P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\n\u22a2 \u2200 (n : \u2115), P (u n)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n  \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n    P t \u2192\n      t \u2286 F t \u2227\n        P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\nn : \u2115\n\u22a2 P (u n)"}, {"tactic": "induction' n with n IH", "annotated_tactic": ["induction' n with n IH", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n  \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n    P t \u2192\n      t \u2286 F t \u2227\n        P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\nn : \u2115\n\u22a2 P (u n)", "state_after": "case zero\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n  \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n    P t \u2192\n      t \u2286 F t \u2227\n        P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\n\u22a2 P (u Nat.zero)\n\ncase succ\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n  \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n    P t \u2192\n      t \u2286 F t \u2227\n        P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\nn : \u2115\nIH : P (u n)\n\u22a2 P (u (Nat.succ n))"}, {"tactic": "simp only [Prod.forall, id.def, Function.iterate_zero, Nat.zero_eq]", "annotated_tactic": ["simp only [<a>Prod.forall</a>, <a>id.def</a>, <a>Function.iterate_zero</a>, <a>Nat.zero_eq</a>]", [{"full_name": "Prod.forall", "def_path": "Mathlib/Data/Prod/Basic.lean", "def_pos": [36, 9], "def_end_pos": [36, 17]}, {"full_name": "id.def", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [527, 9], "def_end_pos": [527, 15]}, {"full_name": "Function.iterate_zero", "def_path": "Mathlib/Logic/Function/Iterate.lean", "def_pos": [53, 9], "def_end_pos": [53, 21]}, {"full_name": "Nat.zero_eq", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [83, 17], "def_end_pos": [83, 24]}]], "state_before": "case zero\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n  \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n    P t \u2192\n      t \u2286 F t \u2227\n        P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\n\u22a2 P (u Nat.zero)", "state_after": "case zero\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n  \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n    P t \u2192\n      t \u2286 F t \u2227\n        P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\n\u22a2 (PairwiseDisjoint \u2191\u2205 fun p => closedBall p.1 p.2) \u2227\n    (\u2200 (a : \u03b1) (b : \u211d), (a, b) \u2208 \u2205 \u2192 a \u2208 s) \u2227 \u2200 (a : \u03b1) (b : \u211d), (a, b) \u2208 \u2205 \u2192 b \u2208 f a"}, {"tactic": "simp only [Finset.not_mem_empty, IsEmpty.forall_iff, Finset.coe_empty, forall\u2082_true_iff,\n  and_self_iff, pairwiseDisjoint_empty]", "annotated_tactic": ["simp only [<a>Finset.not_mem_empty</a>, <a>IsEmpty.forall_iff</a>, <a>Finset.coe_empty</a>, <a>forall\u2082_true_iff</a>,\n        <a>and_self_iff</a>, <a>pairwiseDisjoint_empty</a>]", [{"full_name": "Finset.not_mem_empty", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [548, 9], "def_end_pos": [548, 22]}, {"full_name": "IsEmpty.forall_iff", "def_path": "Mathlib/Logic/IsEmpty.lean", "def_pos": [121, 9], "def_end_pos": [121, 19]}, {"full_name": "Finset.coe_empty", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [618, 9], "def_end_pos": [618, 18]}, {"full_name": "forall\u2082_true_iff", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [731, 9], "def_end_pos": [731, 25]}, {"full_name": "and_self_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [155, 9], "def_end_pos": [155, 21]}, {"full_name": "Set.pairwiseDisjoint_empty", "def_path": "Mathlib/Data/Set/Pairwise/Basic.lean", "def_pos": [259, 9], "def_end_pos": [259, 31]}]], "state_before": "case zero\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n  \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n    P t \u2192\n      t \u2286 F t \u2227\n        P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\n\u22a2 (PairwiseDisjoint \u2191\u2205 fun p => closedBall p.1 p.2) \u2227\n    (\u2200 (a : \u03b1) (b : \u211d), (a, b) \u2208 \u2205 \u2192 a \u2208 s) \u2227 \u2200 (a : \u03b1) (b : \u211d), (a, b) \u2208 \u2205 \u2192 b \u2208 f a", "state_after": "no goals"}, {"tactic": "rw [u_succ]", "annotated_tactic": ["rw [u_succ]", []], "state_before": "case succ\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n  \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n    P t \u2192\n      t \u2286 F t \u2227\n        P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\nn : \u2115\nIH : P (u n)\n\u22a2 P (u (Nat.succ n))", "state_after": "case succ\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n  \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n    P t \u2192\n      t \u2286 F t \u2227\n        P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\nn : \u2115\nIH : P (u n)\n\u22a2 P (F (u n))"}, {"tactic": "exact (hF (u n) IH).2.1", "annotated_tactic": ["exact (hF (u n) IH).2.1", []], "state_before": "case succ\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n  \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n    P t \u2192\n      t \u2286 F t \u2227\n        P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\nn : \u2115\nIH : P (u n)\n\u22a2 P (F (u n))", "state_after": "no goals"}, {"tactic": "intro p hp", "annotated_tactic": ["intro p hp", []], "state_before": "case intro.intro.intro.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n  \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n    P t \u2192\n      t \u2286 F t \u2227\n        P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\nPu : \u2200 (n : \u2115), P (u n)\n\u22a2 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 \u22c3 n, \u2191(u n) \u2192 p.1 \u2208 s", "state_after": "case intro.intro.intro.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n  \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n    P t \u2192\n      t \u2286 F t \u2227\n        P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\nPu : \u2200 (n : \u2115), P (u n)\np : \u03b1 \u00d7 \u211d\nhp : p \u2208 \u22c3 n, \u2191(u n)\n\u22a2 p.1 \u2208 s"}, {"tactic": "rcases mem_iUnion.1 hp with \u27e8n, hn\u27e9", "annotated_tactic": ["rcases <a>mem_iUnion</a>.1 hp with \u27e8n, hn\u27e9", [{"full_name": "Set.mem_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [201, 9], "def_end_pos": [201, 19]}]], "state_before": "case intro.intro.intro.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n  \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n    P t \u2192\n      t \u2286 F t \u2227\n        P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\nPu : \u2200 (n : \u2115), P (u n)\np : \u03b1 \u00d7 \u211d\nhp : p \u2208 \u22c3 n, \u2191(u n)\n\u22a2 p.1 \u2208 s", "state_after": "case intro.intro.intro.refine'_1.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n  \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n    P t \u2192\n      t \u2286 F t \u2227\n        P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\nPu : \u2200 (n : \u2115), P (u n)\np : \u03b1 \u00d7 \u211d\nhp : p \u2208 \u22c3 n, \u2191(u n)\nn : \u2115\nhn : p \u2208 \u2191(u n)\n\u22a2 p.1 \u2208 s"}, {"tactic": "exact (Pu n).2.1 p (Finset.mem_coe.1 hn)", "annotated_tactic": ["exact (Pu n).2.1 p (<a>Finset.mem_coe</a>.1 hn)", [{"full_name": "Finset.mem_coe", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [208, 9], "def_end_pos": [208, 16]}]], "state_before": "case intro.intro.intro.refine'_1.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n  \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n    P t \u2192\n      t \u2286 F t \u2227\n        P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\nPu : \u2200 (n : \u2115), P (u n)\np : \u03b1 \u00d7 \u211d\nhp : p \u2208 \u22c3 n, \u2191(u n)\nn : \u2115\nhn : p \u2208 \u2191(u n)\n\u22a2 p.1 \u2208 s", "state_after": "no goals"}, {"tactic": "intro p hp", "annotated_tactic": ["intro p hp", []], "state_before": "case intro.intro.intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n  \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n    P t \u2192\n      t \u2286 F t \u2227\n        P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\nPu : \u2200 (n : \u2115), P (u n)\n\u22a2 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 \u22c3 n, \u2191(u n) \u2192 p.2 \u2208 f p.1", "state_after": "case intro.intro.intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n  \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n    P t \u2192\n      t \u2286 F t \u2227\n        P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\nPu : \u2200 (n : \u2115), P (u n)\np : \u03b1 \u00d7 \u211d\nhp : p \u2208 \u22c3 n, \u2191(u n)\n\u22a2 p.2 \u2208 f p.1"}, {"tactic": "rcases mem_iUnion.1 hp with \u27e8n, hn\u27e9", "annotated_tactic": ["rcases <a>mem_iUnion</a>.1 hp with \u27e8n, hn\u27e9", [{"full_name": "Set.mem_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [201, 9], "def_end_pos": [201, 19]}]], "state_before": "case intro.intro.intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n  \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n    P t \u2192\n      t \u2286 F t \u2227\n        P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\nPu : \u2200 (n : \u2115), P (u n)\np : \u03b1 \u00d7 \u211d\nhp : p \u2208 \u22c3 n, \u2191(u n)\n\u22a2 p.2 \u2208 f p.1", "state_after": "case intro.intro.intro.refine'_2.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n  \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n    P t \u2192\n      t \u2286 F t \u2227\n        P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\nPu : \u2200 (n : \u2115), P (u n)\np : \u03b1 \u00d7 \u211d\nhp : p \u2208 \u22c3 n, \u2191(u n)\nn : \u2115\nhn : p \u2208 \u2191(u n)\n\u22a2 p.2 \u2208 f p.1"}, {"tactic": "exact (Pu n).2.2 p (Finset.mem_coe.1 hn)", "annotated_tactic": ["exact (Pu n).2.2 p (<a>Finset.mem_coe</a>.1 hn)", [{"full_name": "Finset.mem_coe", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [208, 9], "def_end_pos": [208, 16]}]], "state_before": "case intro.intro.intro.refine'_2.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n  \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n    P t \u2192\n      t \u2286 F t \u2227\n        P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\nPu : \u2200 (n : \u2115), P (u n)\np : \u03b1 \u00d7 \u211d\nhp : p \u2208 \u22c3 n, \u2191(u n)\nn : \u2115\nhn : p \u2208 \u2191(u n)\n\u22a2 p.2 \u2208 f p.1", "state_after": "no goals"}, {"tactic": "have A :\n  \u2200 n,\n    \u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 \u22c3 n : \u2115, (u n : Set (\u03b1 \u00d7 \u211d))), closedBall p.fst p.snd) \u2264\n      \u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 u n), closedBall p.fst p.snd) := by\n  intro n\n  apply measure_mono\n  apply diff_subset_diff (Subset.refl _)\n  exact biUnion_subset_biUnion_left (subset_iUnion (fun i => (u i : Set (\u03b1 \u00d7 \u211d))) n)", "annotated_tactic": ["have A :\n      \u2200 n,\n        \u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 \u22c3 n : \u2115, (u n : <a>Set</a> (\u03b1 \u00d7 \u211d))), <a>closedBall</a> p.fst p.snd) \u2264\n          \u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 u n), <a>closedBall</a> p.fst p.snd) := by\n      intro n\n      apply <a>measure_mono</a>\n      apply <a>diff_subset_diff</a> (<a>Subset.refl</a> _)\n      exact <a>biUnion_subset_biUnion_left</a> (<a>subset_iUnion</a> (fun i => (u i : <a>Set</a> (\u03b1 \u00d7 \u211d))) n)", [{"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "MeasureTheory.measure_mono", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [193, 9], "def_end_pos": [193, 21]}, {"full_name": "Set.diff_subset_diff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1904, 9], "def_end_pos": [1904, 25]}, {"full_name": "Set.Subset.refl", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [354, 9], "def_end_pos": [354, 20]}, {"full_name": "Set.biUnion_subset_biUnion_left", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [990, 9], "def_end_pos": [990, 36]}, {"full_name": "Set.subset_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [431, 9], "def_end_pos": [431, 22]}, {"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}]], "state_before": "case intro.intro.intro.refine'_3\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n  \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n    P t \u2192\n      t \u2286 F t \u2227\n        P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\nPu : \u2200 (n : \u2115), P (u n)\n\u22a2 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 \u22c3 n, \u2191(u n), closedBall p.1 p.2) = 0", "state_after": "case intro.intro.intro.refine'_3\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n  \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n    P t \u2192\n      t \u2286 F t \u2227\n        P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\nPu : \u2200 (n : \u2115), P (u n)\nA : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 \u22c3 n, \u2191(u n), closedBall p.1 p.2) \u2264 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2)\n\u22a2 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 \u22c3 n, \u2191(u n), closedBall p.1 p.2) = 0"}, {"tactic": "rw [zero_mul] at C", "annotated_tactic": ["rw [<a>zero_mul</a>] at C", [{"full_name": "MulZeroClass.zero_mul", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [36, 3], "def_end_pos": [36, 11]}]], "state_before": "case intro.intro.intro.refine'_3\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n  \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n    P t \u2192\n      t \u2286 F t \u2227\n        P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\nPu : \u2200 (n : \u2115), P (u n)\nA : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 \u22c3 n, \u2191(u n), closedBall p.1 p.2) \u2264 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2)\nB : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2) \u2264 (\u2191N / (\u2191N + 1)) ^ n * \u2191\u2191\u03bc s\nC : Tendsto (fun n => (\u2191N / (\u2191N + 1)) ^ n * \u2191\u2191\u03bc s) atTop (\ud835\udcdd (0 * \u2191\u2191\u03bc s))\n\u22a2 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 \u22c3 n, \u2191(u n), closedBall p.1 p.2) = 0", "state_after": "case intro.intro.intro.refine'_3\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n  \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n    P t \u2192\n      t \u2286 F t \u2227\n        P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\nPu : \u2200 (n : \u2115), P (u n)\nA : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 \u22c3 n, \u2191(u n), closedBall p.1 p.2) \u2264 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2)\nB : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2) \u2264 (\u2191N / (\u2191N + 1)) ^ n * \u2191\u2191\u03bc s\nC : Tendsto (fun n => (\u2191N / (\u2191N + 1)) ^ n * \u2191\u2191\u03bc s) atTop (\ud835\udcdd 0)\n\u22a2 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 \u22c3 n, \u2191(u n), closedBall p.1 p.2) = 0"}, {"tactic": "apply le_bot_iff.1", "annotated_tactic": ["apply <a>le_bot_iff</a>.1", [{"full_name": "le_bot_iff", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [355, 9], "def_end_pos": [355, 19]}]], "state_before": "case intro.intro.intro.refine'_3\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n  \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n    P t \u2192\n      t \u2286 F t \u2227\n        P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\nPu : \u2200 (n : \u2115), P (u n)\nA : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 \u22c3 n, \u2191(u n), closedBall p.1 p.2) \u2264 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2)\nB : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2) \u2264 (\u2191N / (\u2191N + 1)) ^ n * \u2191\u2191\u03bc s\nC : Tendsto (fun n => (\u2191N / (\u2191N + 1)) ^ n * \u2191\u2191\u03bc s) atTop (\ud835\udcdd 0)\n\u22a2 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 \u22c3 n, \u2191(u n), closedBall p.1 p.2) = 0", "state_after": "case intro.intro.intro.refine'_3\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n  \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n    P t \u2192\n      t \u2286 F t \u2227\n        P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\nPu : \u2200 (n : \u2115), P (u n)\nA : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 \u22c3 n, \u2191(u n), closedBall p.1 p.2) \u2264 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2)\nB : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2) \u2264 (\u2191N / (\u2191N + 1)) ^ n * \u2191\u2191\u03bc s\nC : Tendsto (fun n => (\u2191N / (\u2191N + 1)) ^ n * \u2191\u2191\u03bc s) atTop (\ud835\udcdd 0)\n\u22a2 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 \u22c3 n, \u2191(u n), closedBall p.1 p.2) \u2264 \u22a5"}, {"tactic": "exact le_of_tendsto_of_tendsto' tendsto_const_nhds C fun n => (A n).trans (B n)", "annotated_tactic": ["exact <a>le_of_tendsto_of_tendsto'</a> <a>tendsto_const_nhds</a> C fun n => (A n).<a>trans</a> (B n)", [{"full_name": "le_of_tendsto_of_tendsto'", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [232, 9], "def_end_pos": [232, 34]}, {"full_name": "tendsto_const_nhds", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [1049, 9], "def_end_pos": [1049, 27]}, {"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [120, 7], "def_end_pos": [120, 18]}]], "state_before": "case intro.intro.intro.refine'_3\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n  \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n    P t \u2192\n      t \u2286 F t \u2227\n        P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\nPu : \u2200 (n : \u2115), P (u n)\nA : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 \u22c3 n, \u2191(u n), closedBall p.1 p.2) \u2264 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2)\nB : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2) \u2264 (\u2191N / (\u2191N + 1)) ^ n * \u2191\u2191\u03bc s\nC : Tendsto (fun n => (\u2191N / (\u2191N + 1)) ^ n * \u2191\u2191\u03bc s) atTop (\ud835\udcdd 0)\n\u22a2 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 \u22c3 n, \u2191(u n), closedBall p.1 p.2) \u2264 \u22a5", "state_after": "no goals"}, {"tactic": "intro n", "annotated_tactic": ["intro n", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n  \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n    P t \u2192\n      t \u2286 F t \u2227\n        P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\nPu : \u2200 (n : \u2115), P (u n)\n\u22a2 \u2200 (n : \u2115), \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 \u22c3 n, \u2191(u n), closedBall p.1 p.2) \u2264 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n  \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n    P t \u2192\n      t \u2286 F t \u2227\n        P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\nPu : \u2200 (n : \u2115), P (u n)\nn : \u2115\n\u22a2 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 \u22c3 n, \u2191(u n), closedBall p.1 p.2) \u2264 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2)"}, {"tactic": "apply measure_mono", "annotated_tactic": ["apply <a>measure_mono</a>", [{"full_name": "MeasureTheory.measure_mono", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [193, 9], "def_end_pos": [193, 21]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n  \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n    P t \u2192\n      t \u2286 F t \u2227\n        P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\nPu : \u2200 (n : \u2115), P (u n)\nn : \u2115\n\u22a2 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 \u22c3 n, \u2191(u n), closedBall p.1 p.2) \u2264 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2)", "state_after": "case h\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n  \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n    P t \u2192\n      t \u2286 F t \u2227\n        P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\nPu : \u2200 (n : \u2115), P (u n)\nn : \u2115\n\u22a2 s \\ \u22c3 p \u2208 \u22c3 n, \u2191(u n), closedBall p.1 p.2 \u2286 s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2"}, {"tactic": "apply diff_subset_diff (Subset.refl _)", "annotated_tactic": ["apply <a>diff_subset_diff</a> (<a>Subset.refl</a> _)", [{"full_name": "Set.diff_subset_diff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1904, 9], "def_end_pos": [1904, 25]}, {"full_name": "Set.Subset.refl", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [354, 9], "def_end_pos": [354, 20]}]], "state_before": "case h\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n  \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n    P t \u2192\n      t \u2286 F t \u2227\n        P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\nPu : \u2200 (n : \u2115), P (u n)\nn : \u2115\n\u22a2 s \\ \u22c3 p \u2208 \u22c3 n, \u2191(u n), closedBall p.1 p.2 \u2286 s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2", "state_after": "case h\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n  \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n    P t \u2192\n      t \u2286 F t \u2227\n        P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\nPu : \u2200 (n : \u2115), P (u n)\nn : \u2115\n\u22a2 \u22c3 p \u2208 u n, closedBall p.1 p.2 \u2286 \u22c3 p \u2208 \u22c3 n, \u2191(u n), closedBall p.1 p.2"}, {"tactic": "exact biUnion_subset_biUnion_left (subset_iUnion (fun i => (u i : Set (\u03b1 \u00d7 \u211d))) n)", "annotated_tactic": ["exact <a>biUnion_subset_biUnion_left</a> (<a>subset_iUnion</a> (fun i => (u i : <a>Set</a> (\u03b1 \u00d7 \u211d))) n)", [{"full_name": "Set.biUnion_subset_biUnion_left", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [990, 9], "def_end_pos": [990, 36]}, {"full_name": "Set.subset_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [431, 9], "def_end_pos": [431, 22]}, {"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}]], "state_before": "case h\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n  \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n    P t \u2192\n      t \u2286 F t \u2227\n        P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\nPu : \u2200 (n : \u2115), P (u n)\nn : \u2115\n\u22a2 \u22c3 p \u2208 u n, closedBall p.1 p.2 \u2286 \u22c3 p \u2208 \u22c3 n, \u2191(u n), closedBall p.1 p.2", "state_after": "no goals"}, {"tactic": "intro n", "annotated_tactic": ["intro n", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n  \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n    P t \u2192\n      t \u2286 F t \u2227\n        P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\nPu : \u2200 (n : \u2115), P (u n)\nA : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 \u22c3 n, \u2191(u n), closedBall p.1 p.2) \u2264 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2)\n\u22a2 \u2200 (n : \u2115), \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2) \u2264 (\u2191N / (\u2191N + 1)) ^ n * \u2191\u2191\u03bc s", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n  \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n    P t \u2192\n      t \u2286 F t \u2227\n        P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\nPu : \u2200 (n : \u2115), P (u n)\nA : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 \u22c3 n, \u2191(u n), closedBall p.1 p.2) \u2264 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2)\nn : \u2115\n\u22a2 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2) \u2264 (\u2191N / (\u2191N + 1)) ^ n * \u2191\u2191\u03bc s"}, {"tactic": "induction' n with n IH", "annotated_tactic": ["induction' n with n IH", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n  \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n    P t \u2192\n      t \u2286 F t \u2227\n        P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\nPu : \u2200 (n : \u2115), P (u n)\nA : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 \u22c3 n, \u2191(u n), closedBall p.1 p.2) \u2264 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2)\nn : \u2115\n\u22a2 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2) \u2264 (\u2191N / (\u2191N + 1)) ^ n * \u2191\u2191\u03bc s", "state_after": "case zero\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n  \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n    P t \u2192\n      t \u2286 F t \u2227\n        P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\nPu : \u2200 (n : \u2115), P (u n)\nA : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 \u22c3 n, \u2191(u n), closedBall p.1 p.2) \u2264 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2)\n\u22a2 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u Nat.zero, closedBall p.1 p.2) \u2264 (\u2191N / (\u2191N + 1)) ^ Nat.zero * \u2191\u2191\u03bc s\n\ncase succ\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n  \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n    P t \u2192\n      t \u2286 F t \u2227\n        P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\nPu : \u2200 (n : \u2115), P (u n)\nA : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 \u22c3 n, \u2191(u n), closedBall p.1 p.2) \u2264 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2)\nn : \u2115\nIH : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2) \u2264 (\u2191N / (\u2191N + 1)) ^ n * \u2191\u2191\u03bc s\n\u22a2 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u (Nat.succ n), closedBall p.1 p.2) \u2264 (\u2191N / (\u2191N + 1)) ^ Nat.succ n * \u2191\u2191\u03bc s"}, {"tactic": "calc\n  \u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 u n.succ), closedBall p.fst p.snd) \u2264\n      N / (N + 1) * \u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 u n), closedBall p.fst p.snd) := by\n    rw [u_succ]; exact (hF (u n) (Pu n)).2.2\n  _ \u2264 (N / (N + 1) : \u211d\u22650\u221e) ^ n.succ * \u03bc s := by\n    rw [pow_succ, mul_assoc]; exact mul_le_mul_left' IH _", "annotated_tactic": ["calc\n        \u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 u n.succ), <a>closedBall</a> p.fst p.snd) \u2264\n            N / (N + 1) * \u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 u n), <a>closedBall</a> p.fst p.snd) := by\n          rw [u_succ]; exact (hF (u n) (Pu n)).2.2\n        _ \u2264 (N / (N + 1) : \u211d\u22650\u221e) ^ n.succ * \u03bc s := by\n          rw [<a>pow_succ</a>, <a>mul_assoc</a>]; exact <a>mul_le_mul_left'</a> IH _", [{"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "pow_succ", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [645, 9], "def_end_pos": [645, 17]}, {"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [264, 9], "def_end_pos": [264, 18]}, {"full_name": "mul_le_mul_left'", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [50, 9], "def_end_pos": [50, 25]}]], "state_before": "case succ\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n  \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n    P t \u2192\n      t \u2286 F t \u2227\n        P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\nPu : \u2200 (n : \u2115), P (u n)\nA : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 \u22c3 n, \u2191(u n), closedBall p.1 p.2) \u2264 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2)\nn : \u2115\nIH : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2) \u2264 (\u2191N / (\u2191N + 1)) ^ n * \u2191\u2191\u03bc s\n\u22a2 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u (Nat.succ n), closedBall p.1 p.2) \u2264 (\u2191N / (\u2191N + 1)) ^ Nat.succ n * \u2191\u2191\u03bc s", "state_after": "no goals"}, {"tactic": "simp only [le_refl, diff_empty, one_mul, iUnion_false, iUnion_empty, pow_zero, Nat.zero_eq,\n  Function.iterate_zero, id.def, Finset.not_mem_empty]", "annotated_tactic": ["simp only [<a>le_refl</a>, <a>diff_empty</a>, <a>one_mul</a>, <a>iUnion_false</a>, <a>iUnion_empty</a>, <a>pow_zero</a>, <a>Nat.zero_eq</a>,\n          <a>Function.iterate_zero</a>, <a>id.def</a>, <a>Finset.not_mem_empty</a>]", [{"full_name": "le_refl", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [50, 9], "def_end_pos": [50, 16]}, {"full_name": "Set.diff_empty", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1930, 9], "def_end_pos": [1930, 19]}, {"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [464, 9], "def_end_pos": [464, 16]}, {"full_name": "Set.iUnion_false", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [783, 9], "def_end_pos": [783, 21]}, {"full_name": "Set.iUnion_empty", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [810, 9], "def_end_pos": [810, 21]}, {"full_name": "pow_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [639, 9], "def_end_pos": [639, 17]}, {"full_name": "Nat.zero_eq", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [83, 17], "def_end_pos": [83, 24]}, {"full_name": "Function.iterate_zero", "def_path": "Mathlib/Logic/Function/Iterate.lean", "def_pos": [53, 9], "def_end_pos": [53, 21]}, {"full_name": "id.def", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [527, 9], "def_end_pos": [527, 15]}, {"full_name": "Finset.not_mem_empty", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [548, 9], "def_end_pos": [548, 22]}]], "state_before": "case zero\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n  \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n    P t \u2192\n      t \u2286 F t \u2227\n        P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\nPu : \u2200 (n : \u2115), P (u n)\nA : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 \u22c3 n, \u2191(u n), closedBall p.1 p.2) \u2264 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2)\n\u22a2 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u Nat.zero, closedBall p.1 p.2) \u2264 (\u2191N / (\u2191N + 1)) ^ Nat.zero * \u2191\u2191\u03bc s", "state_after": "no goals"}, {"tactic": "rw [u_succ]", "annotated_tactic": ["rw [u_succ]", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n  \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n    P t \u2192\n      t \u2286 F t \u2227\n        P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\nPu : \u2200 (n : \u2115), P (u n)\nA : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 \u22c3 n, \u2191(u n), closedBall p.1 p.2) \u2264 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2)\nn : \u2115\nIH : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2) \u2264 (\u2191N / (\u2191N + 1)) ^ n * \u2191\u2191\u03bc s\n\u22a2 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u (Nat.succ n), closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n  \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n    P t \u2192\n      t \u2286 F t \u2227\n        P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\nPu : \u2200 (n : \u2115), P (u n)\nA : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 \u22c3 n, \u2191(u n), closedBall p.1 p.2) \u2264 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2)\nn : \u2115\nIH : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2) \u2264 (\u2191N / (\u2191N + 1)) ^ n * \u2191\u2191\u03bc s\n\u22a2 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F (u n), closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2)"}, {"tactic": "exact (hF (u n) (Pu n)).2.2", "annotated_tactic": ["exact (hF (u n) (Pu n)).2.2", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n  \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n    P t \u2192\n      t \u2286 F t \u2227\n        P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\nPu : \u2200 (n : \u2115), P (u n)\nA : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 \u22c3 n, \u2191(u n), closedBall p.1 p.2) \u2264 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2)\nn : \u2115\nIH : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2) \u2264 (\u2191N / (\u2191N + 1)) ^ n * \u2191\u2191\u03bc s\n\u22a2 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F (u n), closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2)", "state_after": "no goals"}, {"tactic": "rw [pow_succ, mul_assoc]", "annotated_tactic": ["rw [<a>pow_succ</a>, <a>mul_assoc</a>]", [{"full_name": "pow_succ", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [645, 9], "def_end_pos": [645, 17]}, {"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [264, 9], "def_end_pos": [264, 18]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n  \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n    P t \u2192\n      t \u2286 F t \u2227\n        P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\nPu : \u2200 (n : \u2115), P (u n)\nA : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 \u22c3 n, \u2191(u n), closedBall p.1 p.2) \u2264 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2)\nn : \u2115\nIH : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2) \u2264 (\u2191N / (\u2191N + 1)) ^ n * \u2191\u2191\u03bc s\n\u22a2 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2) \u2264 (\u2191N / (\u2191N + 1)) ^ Nat.succ n * \u2191\u2191\u03bc s", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n  \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n    P t \u2192\n      t \u2286 F t \u2227\n        P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\nPu : \u2200 (n : \u2115), P (u n)\nA : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 \u22c3 n, \u2191(u n), closedBall p.1 p.2) \u2264 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2)\nn : \u2115\nIH : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2) \u2264 (\u2191N / (\u2191N + 1)) ^ n * \u2191\u2191\u03bc s\n\u22a2 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * ((\u2191N / (\u2191N + 1)) ^ n * \u2191\u2191\u03bc s)"}, {"tactic": "exact mul_le_mul_left' IH _", "annotated_tactic": ["exact <a>mul_le_mul_left'</a> IH _", [{"full_name": "mul_le_mul_left'", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [50, 9], "def_end_pos": [50, 25]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n  \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n    P t \u2192\n      t \u2286 F t \u2227\n        P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\nPu : \u2200 (n : \u2115), P (u n)\nA : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 \u22c3 n, \u2191(u n), closedBall p.1 p.2) \u2264 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2)\nn : \u2115\nIH : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2) \u2264 (\u2191N / (\u2191N + 1)) ^ n * \u2191\u2191\u03bc s\n\u22a2 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * ((\u2191N / (\u2191N + 1)) ^ n * \u2191\u2191\u03bc s)", "state_after": "no goals"}, {"tactic": "apply ENNReal.Tendsto.mul_const _ (Or.inr (measure_lt_top \u03bc s).ne)", "annotated_tactic": ["apply <a>ENNReal.Tendsto.mul_const</a> _ (<a>Or.inr</a> (<a>measure_lt_top</a> \u03bc s).<a>ne</a>)", [{"full_name": "ENNReal.Tendsto.mul_const", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [379, 19], "def_end_pos": [379, 36]}, {"full_name": "Or.inr", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [519, 5], "def_end_pos": [519, 8]}, {"full_name": "MeasureTheory.measure_lt_top", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2866, 9], "def_end_pos": [2866, 23]}, {"full_name": "LT.lt.ne", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [152, 7], "def_end_pos": [152, 15]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n  \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n    P t \u2192\n      t \u2286 F t \u2227\n        P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\nPu : \u2200 (n : \u2115), P (u n)\nA : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 \u22c3 n, \u2191(u n), closedBall p.1 p.2) \u2264 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2)\nB : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2) \u2264 (\u2191N / (\u2191N + 1)) ^ n * \u2191\u2191\u03bc s\n\u22a2 Tendsto (fun n => (\u2191N / (\u2191N + 1)) ^ n * \u2191\u2191\u03bc s) atTop (\ud835\udcdd (0 * \u2191\u2191\u03bc s))", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n  \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n    P t \u2192\n      t \u2286 F t \u2227\n        P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\nPu : \u2200 (n : \u2115), P (u n)\nA : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 \u22c3 n, \u2191(u n), closedBall p.1 p.2) \u2264 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2)\nB : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2) \u2264 (\u2191N / (\u2191N + 1)) ^ n * \u2191\u2191\u03bc s\n\u22a2 Tendsto (fun x => (\u2191N / (\u2191N + 1)) ^ x) atTop (\ud835\udcdd 0)"}, {"tactic": "apply ENNReal.tendsto_pow_atTop_nhds_0_of_lt_1", "annotated_tactic": ["apply <a>ENNReal.tendsto_pow_atTop_nhds_0_of_lt_1</a>", [{"full_name": "ENNReal.tendsto_pow_atTop_nhds_0_of_lt_1", "def_path": "Mathlib/Analysis/SpecificLimits/Basic.lean", "def_pos": [193, 9], "def_end_pos": [193, 49]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n  \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n    P t \u2192\n      t \u2286 F t \u2227\n        P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\nPu : \u2200 (n : \u2115), P (u n)\nA : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 \u22c3 n, \u2191(u n), closedBall p.1 p.2) \u2264 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2)\nB : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2) \u2264 (\u2191N / (\u2191N + 1)) ^ n * \u2191\u2191\u03bc s\n\u22a2 Tendsto (fun x => (\u2191N / (\u2191N + 1)) ^ x) atTop (\ud835\udcdd 0)", "state_after": "case hr\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n  \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n    P t \u2192\n      t \u2286 F t \u2227\n        P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\nPu : \u2200 (n : \u2115), P (u n)\nA : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 \u22c3 n, \u2191(u n), closedBall p.1 p.2) \u2264 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2)\nB : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2) \u2264 (\u2191N / (\u2191N + 1)) ^ n * \u2191\u2191\u03bc s\n\u22a2 \u2191N / (\u2191N + 1) < 1"}, {"tactic": "rw [ENNReal.div_lt_iff, one_mul]", "annotated_tactic": ["rw [<a>ENNReal.div_lt_iff</a>, <a>one_mul</a>]", [{"full_name": "ENNReal.div_lt_iff", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1658, 19], "def_end_pos": [1658, 29]}, {"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [464, 9], "def_end_pos": [464, 16]}]], "state_before": "case hr\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n  \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n    P t \u2192\n      t \u2286 F t \u2227\n        P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\nPu : \u2200 (n : \u2115), P (u n)\nA : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 \u22c3 n, \u2191(u n), closedBall p.1 p.2) \u2264 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2)\nB : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2) \u2264 (\u2191N / (\u2191N + 1)) ^ n * \u2191\u2191\u03bc s\n\u22a2 \u2191N / (\u2191N + 1) < 1", "state_after": "case hr\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n  \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n    P t \u2192\n      t \u2286 F t \u2227\n        P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\nPu : \u2200 (n : \u2115), P (u n)\nA : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 \u22c3 n, \u2191(u n), closedBall p.1 p.2) \u2264 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2)\nB : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2) \u2264 (\u2191N / (\u2191N + 1)) ^ n * \u2191\u2191\u03bc s\n\u22a2 \u2191N < \u2191N + 1\n\ncase hr.h0\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n  \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n    P t \u2192\n      t \u2286 F t \u2227\n        P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\nPu : \u2200 (n : \u2115), P (u n)\nA : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 \u22c3 n, \u2191(u n), closedBall p.1 p.2) \u2264 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2)\nB : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2) \u2264 (\u2191N / (\u2191N + 1)) ^ n * \u2191\u2191\u03bc s\n\u22a2 \u2191N + 1 \u2260 0 \u2228 \u2191N \u2260 0\n\ncase hr.ht\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n  \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n    P t \u2192\n      t \u2286 F t \u2227\n        P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\nPu : \u2200 (n : \u2115), P (u n)\nA : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 \u22c3 n, \u2191(u n), closedBall p.1 p.2) \u2264 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2)\nB : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2) \u2264 (\u2191N / (\u2191N + 1)) ^ n * \u2191\u2191\u03bc s\n\u22a2 \u2191N + 1 \u2260 \u22a4 \u2228 \u2191N \u2260 \u22a4"}, {"tactic": "conv_lhs => rw [\u2190 add_zero (N : \u211d\u22650\u221e)]", "annotated_tactic": ["conv_lhs => rw [\u2190 <a>add_zero</a> (N : \u211d\u22650\u221e)]", [{"full_name": "add_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [469, 3], "def_end_pos": [469, 14]}]], "state_before": "case hr\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n  \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n    P t \u2192\n      t \u2286 F t \u2227\n        P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\nPu : \u2200 (n : \u2115), P (u n)\nA : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 \u22c3 n, \u2191(u n), closedBall p.1 p.2) \u2264 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2)\nB : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2) \u2264 (\u2191N / (\u2191N + 1)) ^ n * \u2191\u2191\u03bc s\n\u22a2 \u2191N < \u2191N + 1", "state_after": "case hr\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n  \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n    P t \u2192\n      t \u2286 F t \u2227\n        P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\nPu : \u2200 (n : \u2115), P (u n)\nA : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 \u22c3 n, \u2191(u n), closedBall p.1 p.2) \u2264 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2)\nB : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2) \u2264 (\u2191N / (\u2191N + 1)) ^ n * \u2191\u2191\u03bc s\n\u22a2 \u2191N + 0 < \u2191N + 1"}, {"tactic": "exact ENNReal.add_lt_add_left (ENNReal.nat_ne_top N) zero_lt_one", "annotated_tactic": ["exact <a>ENNReal.add_lt_add_left</a> (<a>ENNReal.nat_ne_top</a> N) <a>zero_lt_one</a>", [{"full_name": "ENNReal.add_lt_add_left", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [793, 29], "def_end_pos": [793, 44]}, {"full_name": "ENNReal.nat_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [717, 17], "def_end_pos": [717, 27]}, {"full_name": "zero_lt_one", "def_path": "Mathlib/Algebra/Order/ZeroLEOne.lean", "def_pos": [39, 15], "def_end_pos": [39, 26]}]], "state_before": "case hr\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n  \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n    P t \u2192\n      t \u2286 F t \u2227\n        P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\nPu : \u2200 (n : \u2115), P (u n)\nA : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 \u22c3 n, \u2191(u n), closedBall p.1 p.2) \u2264 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2)\nB : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2) \u2264 (\u2191N / (\u2191N + 1)) ^ n * \u2191\u2191\u03bc s\n\u22a2 \u2191N + 0 < \u2191N + 1", "state_after": "no goals"}, {"tactic": "simp only [true_or_iff, add_eq_zero_iff, Ne.def, not_false_iff, one_ne_zero, and_false_iff]", "annotated_tactic": ["simp only [<a>true_or_iff</a>, <a>add_eq_zero_iff</a>, <a>Ne.def</a>, <a>not_false_iff</a>, <a>one_ne_zero</a>, <a>and_false_iff</a>]", [{"full_name": "true_or_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [182, 9], "def_end_pos": [182, 20]}, {"full_name": "add_eq_zero_iff", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [231, 3], "def_end_pos": [231, 14]}, {"full_name": "Ne.def", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [59, 9], "def_end_pos": [59, 15]}, {"full_name": "not_false_iff", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [82, 9], "def_end_pos": [82, 22]}, {"full_name": "one_ne_zero", "def_path": "Mathlib/Algebra/NeZero.lean", "def_pos": [55, 15], "def_end_pos": [55, 26]}, {"full_name": "and_false_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [149, 9], "def_end_pos": [149, 22]}]], "state_before": "case hr.h0\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n  \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n    P t \u2192\n      t \u2286 F t \u2227\n        P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\nPu : \u2200 (n : \u2115), P (u n)\nA : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 \u22c3 n, \u2191(u n), closedBall p.1 p.2) \u2264 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2)\nB : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2) \u2264 (\u2191N / (\u2191N + 1)) ^ n * \u2191\u2191\u03bc s\n\u22a2 \u2191N + 1 \u2260 0 \u2228 \u2191N \u2260 0", "state_after": "no goals"}, {"tactic": "simp only [ENNReal.nat_ne_top, Ne.def, not_false_iff, or_true_iff]", "annotated_tactic": ["simp only [<a>ENNReal.nat_ne_top</a>, <a>Ne.def</a>, <a>not_false_iff</a>, <a>or_true_iff</a>]", [{"full_name": "ENNReal.nat_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [717, 17], "def_end_pos": [717, 27]}, {"full_name": "Ne.def", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [59, 9], "def_end_pos": [59, 15]}, {"full_name": "not_false_iff", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [82, 9], "def_end_pos": [82, 22]}, {"full_name": "or_true_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [184, 9], "def_end_pos": [184, 20]}]], "state_before": "case hr.ht\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n  \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n    P t \u2192\n      t \u2286 F t \u2227\n        P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\nPu : \u2200 (n : \u2115), P (u n)\nA : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 \u22c3 n, \u2191(u n), closedBall p.1 p.2) \u2264 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2)\nB : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2) \u2264 (\u2191N / (\u2191N + 1)) ^ n * \u2191\u2191\u03bc s\n\u22a2 \u2191N + 1 \u2260 \u22a4 \u2228 \u2191N \u2260 \u22a4", "state_after": "no goals"}, {"tactic": "refine' (pairwiseDisjoint_iUnion _).2 fun n => (Pu n).1", "annotated_tactic": ["refine' (<a>pairwiseDisjoint_iUnion</a> _).2 fun n => (Pu n).1", [{"full_name": "Set.pairwiseDisjoint_iUnion", "def_path": "Mathlib/Data/Set/Pairwise/Lattice.lean", "def_pos": [54, 9], "def_end_pos": [54, 32]}]], "state_before": "case intro.intro.intro.refine'_4\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n  \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n    P t \u2192\n      t \u2286 F t \u2227\n        P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\nPu : \u2200 (n : \u2115), P (u n)\n\u22a2 PairwiseDisjoint (\u22c3 n, \u2191(u n)) fun p => closedBall p.1 p.2", "state_after": "case intro.intro.intro.refine'_4\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n  \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n    P t \u2192\n      t \u2286 F t \u2227\n        P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\nPu : \u2200 (n : \u2115), P (u n)\n\u22a2 Directed (fun x x_1 => x \u2286 x_1) fun n => \u2191(u n)"}, {"tactic": "apply (monotone_nat_of_le_succ fun n => ?_).directed_le", "annotated_tactic": ["apply (<a>monotone_nat_of_le_succ</a> fun n => ?_).<a>directed_le</a>", [{"full_name": "monotone_nat_of_le_succ", "def_path": "Mathlib/Order/Monotone/Basic.lean", "def_pos": [1025, 9], "def_end_pos": [1025, 32]}, {"full_name": "Monotone.directed_le", "def_path": "Mathlib/Order/Directed.lean", "def_pos": [112, 9], "def_end_pos": [112, 29]}]], "state_before": "case intro.intro.intro.refine'_4\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n  \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n    P t \u2192\n      t \u2286 F t \u2227\n        P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\nPu : \u2200 (n : \u2115), P (u n)\n\u22a2 Directed (fun x x_1 => x \u2286 x_1) fun n => \u2191(u n)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n  \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n    P t \u2192\n      t \u2286 F t \u2227\n        P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\nPu : \u2200 (n : \u2115), P (u n)\nn : \u2115\n\u22a2 \u2191(u n) \u2264 \u2191(u (n + 1))"}, {"tactic": "rw [\u2190 Nat.succ_eq_add_one, u_succ]", "annotated_tactic": ["rw [\u2190 <a>Nat.succ_eq_add_one</a>, u_succ]", [{"full_name": "Nat.succ_eq_add_one", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [128, 9], "def_end_pos": [128, 24]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n  \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n    P t \u2192\n      t \u2286 F t \u2227\n        P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\nPu : \u2200 (n : \u2115), P (u n)\nn : \u2115\n\u22a2 \u2191(u n) \u2264 \u2191(u (n + 1))", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n  \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n    P t \u2192\n      t \u2286 F t \u2227\n        P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\nPu : \u2200 (n : \u2115), P (u n)\nn : \u2115\n\u22a2 \u2191(u n) \u2264 \u2191(F (u n))"}, {"tactic": "exact (hF (u n) (Pu n)).1", "annotated_tactic": ["exact (hF (u n) (Pu n)).1", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n  fun t =>\n    (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227\n      (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n  \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n    P t \u2192\n      t \u2286 F t \u2227\n        P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\nPu : \u2200 (n : \u2115), P (u n)\nn : \u2115\n\u22a2 \u2191(u n) \u2264 \u2191(F (u n))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Lebesgue/Integral.lean", "full_name": "Real.integrable_of_summable_norm_Icc", "start": [55, 1], "end": [75, 32], "traced_tactics": [{"tactic": "refine'\n  @integrable_of_summable_norm_restrict \u211d \u2124 E _ volume _ _ _ _ _ _ _ _\n    (summable_of_nonneg_of_le\n      (fun n : \u2124 => mul_nonneg (norm_nonneg\n          (f.restrict (\u27e8Icc (n : \u211d) ((n : \u211d) + 1), isCompact_Icc\u27e9 : Compacts \u211d)))\n          ENNReal.toReal_nonneg)\n      (fun n => _) hf) _", "annotated_tactic": ["refine'\n    @<a>integrable_of_summable_norm_restrict</a> \u211d \u2124 E _ <a>volume</a> _ _ _ _ _ _ _ _\n      (<a>summable_of_nonneg_of_le</a>\n        (fun n : \u2124 => <a>mul_nonneg</a> (<a>norm_nonneg</a>\n            (f.restrict (\u27e8<a>Icc</a> (n : \u211d) ((n : \u211d) + 1), <a>isCompact_Icc</a>\u27e9 : <a>Compacts</a> \u211d)))\n            <a>ENNReal.toReal_nonneg</a>)\n        (fun n => _) hf) _", [{"full_name": "MeasureTheory.integrable_of_summable_norm_restrict", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [855, 9], "def_end_pos": [855, 45]}, {"full_name": "MeasureTheory.MeasureSpace.volume", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [663, 3], "def_end_pos": [663, 9]}, {"full_name": "summable_of_nonneg_of_le", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [1297, 9], "def_end_pos": [1297, 33]}, {"full_name": "mul_nonneg", "def_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "def_pos": [380, 7], "def_end_pos": [380, 17]}, {"full_name": "norm_nonneg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [500, 30], "def_end_pos": [500, 41]}, {"full_name": "Set.Icc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [59, 5], "def_end_pos": [59, 8]}, {"full_name": "CompactIccSpace.isCompact_Icc", "def_path": "Mathlib/Topology/Algebra/Order/Compact.lean", "def_pos": [53, 3], "def_end_pos": [53, 16]}, {"full_name": "TopologicalSpace.Compacts", "def_path": "Mathlib/Topology/Sets/Compacts.lean", "def_pos": [36, 11], "def_end_pos": [36, 19]}, {"full_name": "ENNReal.toReal_nonneg", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [221, 17], "def_end_pos": [221, 30]}]], "state_before": "E : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : C(\u211d, E)\nhf : Summable fun n => \u2016ContinuousMap.restrict (Icc 0 1) (comp f (ContinuousMap.addRight \u2191n))\u2016\n\u22a2 Integrable \u2191f", "state_after": "case refine'_1\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : C(\u211d, E)\nhf : Summable fun n => \u2016ContinuousMap.restrict (Icc 0 1) (comp f (ContinuousMap.addRight \u2191n))\u2016\n\u22a2 \u2124 \u2192 Compacts \u211d\n\ncase refine'_2\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : C(\u211d, E)\nhf : Summable fun n => \u2016ContinuousMap.restrict (Icc 0 1) (comp f (ContinuousMap.addRight \u2191n))\u2016\nn : \u2124\n\u22a2 \u2016ContinuousMap.restrict (\u2191(?refine'_1 n)) f\u2016 * ENNReal.toReal (\u2191\u2191volume \u2191(?refine'_1 n)) \u2264\n    \u2016ContinuousMap.restrict (Icc 0 1) (comp f (ContinuousMap.addRight \u2191n))\u2016\n\ncase refine'_3\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : C(\u211d, E)\nhf : Summable fun n => \u2016ContinuousMap.restrict (Icc 0 1) (comp f (ContinuousMap.addRight \u2191n))\u2016\n\u22a2 \u22c3 i, \u2191(?refine'_1 i) = univ"}, {"tactic": "intro n", "annotated_tactic": ["intro n", []], "state_before": "case refine'_1\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : C(\u211d, E)\nhf : Summable fun n => \u2016ContinuousMap.restrict (Icc 0 1) (comp f (ContinuousMap.addRight \u2191n))\u2016\n\u22a2 \u2124 \u2192 Compacts \u211d", "state_after": "case refine'_1\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : C(\u211d, E)\nhf : Summable fun n => \u2016ContinuousMap.restrict (Icc 0 1) (comp f (ContinuousMap.addRight \u2191n))\u2016\nn : \u2124\n\u22a2 Compacts \u211d"}, {"tactic": "exact \u27e8Icc (n : \u211d) ((n : \u211d) + 1), isCompact_Icc\u27e9", "annotated_tactic": ["exact \u27e8<a>Icc</a> (n : \u211d) ((n : \u211d) + 1), <a>isCompact_Icc</a>\u27e9", [{"full_name": "Set.Icc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [59, 5], "def_end_pos": [59, 8]}, {"full_name": "CompactIccSpace.isCompact_Icc", "def_path": "Mathlib/Topology/Algebra/Order/Compact.lean", "def_pos": [53, 3], "def_end_pos": [53, 16]}]], "state_before": "case refine'_1\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : C(\u211d, E)\nhf : Summable fun n => \u2016ContinuousMap.restrict (Icc 0 1) (comp f (ContinuousMap.addRight \u2191n))\u2016\nn : \u2124\n\u22a2 Compacts \u211d", "state_after": "no goals"}, {"tactic": "simp only [Compacts.coe_mk, Real.volume_Icc, add_sub_cancel', ENNReal.toReal_ofReal zero_le_one,\n  mul_one, norm_le _ (norm_nonneg _)]", "annotated_tactic": ["simp only [<a>Compacts.coe_mk</a>, <a>Real.volume_Icc</a>, <a>add_sub_cancel'</a>, <a>ENNReal.toReal_ofReal</a> <a>zero_le_one</a>,\n      <a>mul_one</a>, <a>norm_le</a> _ (<a>norm_nonneg</a> _)]", [{"full_name": "TopologicalSpace.Compacts.coe_mk", "def_path": "Mathlib/Topology/Sets/Compacts.lean", "def_pos": [67, 9], "def_end_pos": [67, 15]}, {"full_name": "Real.volume_Icc", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean", "def_pos": [84, 9], "def_end_pos": [84, 19]}, {"full_name": "add_sub_cancel'", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [948, 30], "def_end_pos": [948, 45]}, {"full_name": "ENNReal.toReal_ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [191, 9], "def_end_pos": [191, 22]}, {"full_name": "zero_le_one", "def_path": "Mathlib/Algebra/Order/ZeroLEOne.lean", "def_pos": [26, 15], "def_end_pos": [26, 26]}, {"full_name": "mul_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [470, 9], "def_end_pos": [470, 16]}, {"full_name": "ContinuousMap.norm_le", "def_path": "Mathlib/Topology/ContinuousFunction/Compact.lean", "def_pos": [212, 9], "def_end_pos": [212, 16]}, {"full_name": "norm_nonneg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [500, 30], "def_end_pos": [500, 41]}]], "state_before": "case refine'_2\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : C(\u211d, E)\nhf : Summable fun n => \u2016ContinuousMap.restrict (Icc 0 1) (comp f (ContinuousMap.addRight \u2191n))\u2016\nn : \u2124\n\u22a2 \u2016ContinuousMap.restrict (\u2191{ carrier := Icc (\u2191n) (\u2191n + 1), isCompact' := (_ : IsCompact (Icc (\u2191n) (\u2191n + 1))) }) f\u2016 *\n      ENNReal.toReal (\u2191\u2191volume \u2191{ carrier := Icc (\u2191n) (\u2191n + 1), isCompact' := (_ : IsCompact (Icc (\u2191n) (\u2191n + 1))) }) \u2264\n    \u2016ContinuousMap.restrict (Icc 0 1) (comp f (ContinuousMap.addRight \u2191n))\u2016", "state_after": "case refine'_2\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : C(\u211d, E)\nhf : Summable fun n => \u2016ContinuousMap.restrict (Icc 0 1) (comp f (ContinuousMap.addRight \u2191n))\u2016\nn : \u2124\n\u22a2 \u2200 (x : \u2191(Icc (\u2191n) (\u2191n + 1))),\n    \u2016\u2191(ContinuousMap.restrict (Icc (\u2191n) (\u2191n + 1)) f) x\u2016 \u2264\n      \u2016ContinuousMap.restrict (Icc 0 1) (comp f (ContinuousMap.addRight \u2191n))\u2016"}, {"tactic": "intro x", "annotated_tactic": ["intro x", []], "state_before": "case refine'_2\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : C(\u211d, E)\nhf : Summable fun n => \u2016ContinuousMap.restrict (Icc 0 1) (comp f (ContinuousMap.addRight \u2191n))\u2016\nn : \u2124\n\u22a2 \u2200 (x : \u2191(Icc (\u2191n) (\u2191n + 1))),\n    \u2016\u2191(ContinuousMap.restrict (Icc (\u2191n) (\u2191n + 1)) f) x\u2016 \u2264\n      \u2016ContinuousMap.restrict (Icc 0 1) (comp f (ContinuousMap.addRight \u2191n))\u2016", "state_after": "case refine'_2\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : C(\u211d, E)\nhf : Summable fun n => \u2016ContinuousMap.restrict (Icc 0 1) (comp f (ContinuousMap.addRight \u2191n))\u2016\nn : \u2124\nx : \u2191(Icc (\u2191n) (\u2191n + 1))\n\u22a2 \u2016\u2191(ContinuousMap.restrict (Icc (\u2191n) (\u2191n + 1)) f) x\u2016 \u2264\n    \u2016ContinuousMap.restrict (Icc 0 1) (comp f (ContinuousMap.addRight \u2191n))\u2016"}, {"tactic": "have := ((f.comp <| ContinuousMap.addRight n).restrict (Icc 0 1)).norm_coe_le_norm\n    \u27e8x - n, \u27e8sub_nonneg.mpr x.2.1, sub_le_iff_le_add'.mpr x.2.2\u27e9\u27e9", "annotated_tactic": ["have := ((f.comp <| <a>ContinuousMap.addRight</a> n).<a>restrict</a> (<a>Icc</a> 0 1)).<a>norm_coe_le_norm</a>\n        \u27e8x - n, \u27e8sub_nonneg.mpr x.2.1, sub_le_iff_le_add'.mpr x.2.2\u27e9\u27e9", [{"full_name": "ContinuousMap.addRight", "def_path": "Mathlib/Topology/Algebra/Monoid.lean", "def_pos": [871, 3], "def_end_pos": [871, 14]}, {"full_name": "ContinuousMap.restrict", "def_path": "Mathlib/Topology/ContinuousFunction/Basic.lean", "def_pos": [371, 5], "def_end_pos": [371, 13]}, {"full_name": "Set.Icc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [59, 5], "def_end_pos": [59, 8]}, {"full_name": "ContinuousMap.norm_coe_le_norm", "def_path": "Mathlib/Topology/ContinuousFunction/Compact.lean", "def_pos": [202, 9], "def_end_pos": [202, 25]}]], "state_before": "case refine'_2\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : C(\u211d, E)\nhf : Summable fun n => \u2016ContinuousMap.restrict (Icc 0 1) (comp f (ContinuousMap.addRight \u2191n))\u2016\nn : \u2124\nx : \u2191(Icc (\u2191n) (\u2191n + 1))\n\u22a2 \u2016\u2191(ContinuousMap.restrict (Icc (\u2191n) (\u2191n + 1)) f) x\u2016 \u2264\n    \u2016ContinuousMap.restrict (Icc 0 1) (comp f (ContinuousMap.addRight \u2191n))\u2016", "state_after": "case refine'_2\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : C(\u211d, E)\nhf : Summable fun n => \u2016ContinuousMap.restrict (Icc 0 1) (comp f (ContinuousMap.addRight \u2191n))\u2016\nn : \u2124\nx : \u2191(Icc (\u2191n) (\u2191n + 1))\nthis :\n  \u2016\u2191(ContinuousMap.restrict (Icc 0 1) (comp f (ContinuousMap.addRight \u2191n)))\n        { val := \u2191x - \u2191n, property := (_ : 0 \u2264 \u2191x - \u2191n \u2227 \u2191x - \u2191n \u2264 1) }\u2016 \u2264\n    \u2016ContinuousMap.restrict (Icc 0 1) (comp f (ContinuousMap.addRight \u2191n))\u2016\n\u22a2 \u2016\u2191(ContinuousMap.restrict (Icc (\u2191n) (\u2191n + 1)) f) x\u2016 \u2264\n    \u2016ContinuousMap.restrict (Icc 0 1) (comp f (ContinuousMap.addRight \u2191n))\u2016"}, {"tactic": "simpa only [ContinuousMap.restrict_apply, comp_apply, coe_addRight, Subtype.coe_mk,\n  sub_add_cancel] using this", "annotated_tactic": ["simpa only [<a>ContinuousMap.restrict_apply</a>, <a>comp_apply</a>, <a>coe_addRight</a>, <a>Subtype.coe_mk</a>,\n      <a>sub_add_cancel</a>] using this", [{"full_name": "ContinuousMap.restrict_apply", "def_path": "Mathlib/Topology/ContinuousFunction/Basic.lean", "def_pos": [381, 9], "def_end_pos": [381, 23]}, {"full_name": "ContinuousMap.comp_apply", "def_path": "Mathlib/Topology/ContinuousFunction/Basic.lean", "def_pos": [243, 9], "def_end_pos": [243, 19]}, {"full_name": "ContinuousMap.coe_addRight", "def_path": "Mathlib/Topology/Algebra/Monoid.lean", "def_pos": [877, 3], "def_end_pos": [877, 14]}, {"full_name": "Subtype.coe_mk", "def_path": "Mathlib/Data/Subtype.lean", "def_pos": [99, 9], "def_end_pos": [99, 15]}, {"full_name": "sub_add_cancel", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [728, 30], "def_end_pos": [728, 44]}]], "state_before": "case refine'_2\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : C(\u211d, E)\nhf : Summable fun n => \u2016ContinuousMap.restrict (Icc 0 1) (comp f (ContinuousMap.addRight \u2191n))\u2016\nn : \u2124\nx : \u2191(Icc (\u2191n) (\u2191n + 1))\nthis :\n  \u2016\u2191(ContinuousMap.restrict (Icc 0 1) (comp f (ContinuousMap.addRight \u2191n)))\n        { val := \u2191x - \u2191n, property := (_ : 0 \u2264 \u2191x - \u2191n \u2227 \u2191x - \u2191n \u2264 1) }\u2016 \u2264\n    \u2016ContinuousMap.restrict (Icc 0 1) (comp f (ContinuousMap.addRight \u2191n))\u2016\n\u22a2 \u2016\u2191(ContinuousMap.restrict (Icc (\u2191n) (\u2191n + 1)) f) x\u2016 \u2264\n    \u2016ContinuousMap.restrict (Icc 0 1) (comp f (ContinuousMap.addRight \u2191n))\u2016", "state_after": "no goals"}, {"tactic": "exact iUnion_Icc_int_cast \u211d", "annotated_tactic": ["exact <a>iUnion_Icc_int_cast</a> \u211d", [{"full_name": "iUnion_Icc_int_cast", "def_path": "Mathlib/Algebra/Order/ToIntervalMod.lean", "def_pos": [1114, 9], "def_end_pos": [1114, 28]}]], "state_before": "case refine'_3\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : C(\u211d, E)\nhf : Summable fun n => \u2016ContinuousMap.restrict (Icc 0 1) (comp f (ContinuousMap.addRight \u2191n))\u2016\n\u22a2 \u22c3 i, \u2191{ carrier := Icc (\u2191i) (\u2191i + 1), isCompact' := (_ : IsCompact (Icc (\u2191i) (\u2191i + 1))) } = univ", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/Monad.lean", "full_name": "MvPolynomial.bind\u2082_monomial", "start": [344, 1], "end": [347, 42], "traced_tactics": [{"tactic": "simp only [monomial_eq, RingHom.map_mul, bind\u2082_C_right, Finsupp.prod, map_prod,\n  map_pow, bind\u2082_X_right, C_1, one_mul]", "annotated_tactic": ["simp only [<a>monomial_eq</a>, <a>RingHom.map_mul</a>, <a>bind\u2082_C_right</a>, <a>Finsupp.prod</a>, <a>map_prod</a>,\n    <a>map_pow</a>, <a>bind\u2082_X_right</a>, <a>C_1</a>, <a>one_mul</a>]", [{"full_name": "MvPolynomial.monomial_eq", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [394, 9], "def_end_pos": [394, 20]}, {"full_name": "RingHom.map_mul", "def_path": "Mathlib/Algebra/Hom/Ring/Defs.lean", "def_pos": [569, 19], "def_end_pos": [569, 26]}, {"full_name": "MvPolynomial.bind\u2082_C_right", "def_path": "Mathlib/Data/MvPolynomial/Monad.lean", "def_pos": [178, 9], "def_end_pos": [178, 22]}, {"full_name": "Finsupp.prod", "def_path": "Mathlib/Algebra/BigOperators/Finsupp.lean", "def_pos": [52, 5], "def_end_pos": [52, 9]}, {"full_name": "map_prod", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [205, 9], "def_end_pos": [205, 17]}, {"full_name": "map_pow", "def_path": "Mathlib/Algebra/Hom/Group/Defs.lean", "def_pos": [435, 9], "def_end_pos": [435, 16]}, {"full_name": "MvPolynomial.bind\u2082_X_right", "def_path": "Mathlib/Data/MvPolynomial/Monad.lean", "def_pos": [157, 9], "def_end_pos": [157, 22]}, {"full_name": "MvPolynomial.C_1", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [217, 9], "def_end_pos": [217, 12]}, {"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [464, 9], "def_end_pos": [464, 16]}]], "state_before": "\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : CommSemiring T\nf\u271d : \u03c3 \u2192 MvPolynomial \u03c4 R\nf : R \u2192+* MvPolynomial \u03c3 S\nd : \u03c3 \u2192\u2080 \u2115\nr : R\n\u22a2 \u2191(bind\u2082 f) (\u2191(monomial d) r) = \u2191f r * \u2191(monomial d) 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Int/Bitwise.lean", "full_name": "Int.shiftRight_neg", "start": [372, 1], "end": [372, 92], "traced_tactics": [{"tactic": "rw [\u2190 shiftLeft_neg, neg_neg]", "annotated_tactic": ["rw [\u2190 <a>shiftLeft_neg</a>, <a>neg_neg</a>]", [{"full_name": "Int.shiftLeft_neg", "def_path": "Mathlib/Data/Int/Bitwise.lean", "def_pos": [367, 9], "def_end_pos": [367, 22]}, {"full_name": "neg_neg", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [799, 3], "def_end_pos": [799, 14]}]], "state_before": "m n : \u2124\n\u22a2 m >>> (-n) = m <<< n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Int/Units.lean", "full_name": "Int.neg_units_ne_self", "start": [39, 1], "end": [39, 74], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/ZMod/Basic.lean", "full_name": "ZMod.valMinAbs_neg_of_ne_half", "start": [1069, 1], "end": [1078, 42], "traced_tactics": [{"tactic": "cases' eq_zero_or_neZero n with h h", "annotated_tactic": ["cases' <a>eq_zero_or_neZero</a> n with h h", [{"full_name": "eq_zero_or_neZero", "def_path": "Mathlib/Algebra/NeZero.lean", "def_pos": [45, 9], "def_end_pos": [45, 26]}]], "state_before": "n : \u2115\na : ZMod n\nha : 2 * val a \u2260 n\n\u22a2 valMinAbs (-a) = -valMinAbs a", "state_after": "case inl\nn : \u2115\na : ZMod n\nha : 2 * val a \u2260 n\nh : n = 0\n\u22a2 valMinAbs (-a) = -valMinAbs a\n\ncase inr\nn : \u2115\na : ZMod n\nha : 2 * val a \u2260 n\nh : NeZero n\n\u22a2 valMinAbs (-a) = -valMinAbs a"}, {"tactic": "refine' (valMinAbs_spec _ _).2 \u27e8_, _, _\u27e9", "annotated_tactic": ["refine' (<a>valMinAbs_spec</a> _ _).2 \u27e8_, _, _\u27e9", [{"full_name": "ZMod.valMinAbs_spec", "def_path": "Mathlib/Data/ZMod/Basic.lean", "def_pos": [1019, 9], "def_end_pos": [1019, 23]}]], "state_before": "case inr\nn : \u2115\na : ZMod n\nha : 2 * val a \u2260 n\nh : NeZero n\n\u22a2 valMinAbs (-a) = -valMinAbs a", "state_after": "case inr.refine'_1\nn : \u2115\na : ZMod n\nha : 2 * val a \u2260 n\nh : NeZero n\n\u22a2 -a = \u2191(-valMinAbs a)\n\ncase inr.refine'_2\nn : \u2115\na : ZMod n\nha : 2 * val a \u2260 n\nh : NeZero n\n\u22a2 -\u2191n < -valMinAbs a * 2\n\ncase inr.refine'_3\nn : \u2115\na : ZMod n\nha : 2 * val a \u2260 n\nh : NeZero n\n\u22a2 -valMinAbs a * 2 \u2264 \u2191n"}, {"tactic": "subst h", "annotated_tactic": ["subst h", []], "state_before": "case inl\nn : \u2115\na : ZMod n\nha : 2 * val a \u2260 n\nh : n = 0\n\u22a2 valMinAbs (-a) = -valMinAbs a", "state_after": "case inl\na : ZMod 0\nha : 2 * val a \u2260 0\n\u22a2 valMinAbs (-a) = -valMinAbs a"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case inl\na : ZMod 0\nha : 2 * val a \u2260 0\n\u22a2 valMinAbs (-a) = -valMinAbs a", "state_after": "no goals"}, {"tactic": "rw [Int.cast_neg, coe_valMinAbs]", "annotated_tactic": ["rw [<a>Int.cast_neg</a>, <a>coe_valMinAbs</a>]", [{"full_name": "Int.cast_neg", "def_path": "Mathlib/Data/Int/Cast/Basic.lean", "def_pos": [83, 9], "def_end_pos": [83, 17]}, {"full_name": "ZMod.coe_valMinAbs", "def_path": "Mathlib/Data/ZMod/Basic.lean", "def_pos": [971, 9], "def_end_pos": [971, 22]}]], "state_before": "case inr.refine'_1\nn : \u2115\na : ZMod n\nha : 2 * val a \u2260 n\nh : NeZero n\n\u22a2 -a = \u2191(-valMinAbs a)", "state_after": "no goals"}, {"tactic": "rw [neg_mul, neg_lt_neg_iff]", "annotated_tactic": ["rw [<a>neg_mul</a>, <a>neg_lt_neg_iff</a>]", [{"full_name": "neg_mul", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [289, 9], "def_end_pos": [289, 16]}, {"full_name": "neg_lt_neg_iff", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [381, 3], "def_end_pos": [381, 14]}]], "state_before": "case inr.refine'_2\nn : \u2115\na : ZMod n\nha : 2 * val a \u2260 n\nh : NeZero n\n\u22a2 -\u2191n < -valMinAbs a * 2", "state_after": "case inr.refine'_2\nn : \u2115\na : ZMod n\nha : 2 * val a \u2260 n\nh : NeZero n\n\u22a2 valMinAbs a * 2 < \u2191n"}, {"tactic": "exact a.valMinAbs_mem_Ioc.2.lt_of_ne (mt a.valMinAbs_mul_two_eq_iff.1 ha)", "annotated_tactic": ["exact a.valMinAbs_mem_Ioc.2.<a>lt_of_ne</a> (<a>mt</a> a.valMinAbs_mul_two_eq_iff.1 ha)", [{"full_name": "LE.le.lt_of_ne", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [132, 7], "def_end_pos": [132, 21]}, {"full_name": "mt", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [516, 9], "def_end_pos": [516, 11]}]], "state_before": "case inr.refine'_2\nn : \u2115\na : ZMod n\nha : 2 * val a \u2260 n\nh : NeZero n\n\u22a2 valMinAbs a * 2 < \u2191n", "state_after": "no goals"}, {"tactic": "linarith only [a.valMinAbs_mem_Ioc.1]", "annotated_tactic": ["linarith only [a.valMinAbs_mem_Ioc.1]", []], "state_before": "case inr.refine'_3\nn : \u2115\na : ZMod n\nha : 2 * val a \u2260 n\nh : NeZero n\n\u22a2 -valMinAbs a * 2 \u2264 \u2191n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Intervals/Pi.lean", "full_name": "Set.image_mulSingle_uIcc_left", "start": [316, 1], "end": [318, 31], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Haar/Basic.lean", "full_name": "MeasureTheory.Measure.MeasurePreserving.zpow", "start": [877, 1], "end": [880, 40], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Prod.lean", "full_name": "Set.pi_eq_empty_iff'", "start": [757, 1], "end": [757, 86], "traced_tactics": [{"tactic": "simp [pi_eq_empty_iff]", "annotated_tactic": ["simp [<a>pi_eq_empty_iff</a>]", [{"full_name": "Set.pi_eq_empty_iff", "def_path": "Mathlib/Data/Set/Prod.lean", "def_pos": [727, 9], "def_end_pos": [727, 24]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\ns s\u2081 s\u2082 : Set \u03b9\nt t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\ni : \u03b9\ninst\u271d : \u2200 (i : \u03b9), Nonempty (\u03b1 i)\n\u22a2 pi s t = \u2205 \u2194 \u2203 i, i \u2208 s \u2227 t i = \u2205", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/FundThmCalculus.lean", "full_name": "intervalIntegral.integral_deriv_comp_mul_deriv", "start": [1514, 1], "end": [1518, 70], "traced_tactics": [{"tactic": "simpa [mul_comm] using integral_deriv_comp_smul_deriv hf hg hf' hg'", "annotated_tactic": ["simpa [<a>mul_comm</a>] using <a>integral_deriv_comp_smul_deriv</a> hf hg hf' hg'", [{"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [302, 9], "def_end_pos": [302, 17]}, {"full_name": "intervalIntegral.integral_deriv_comp_smul_deriv", "def_path": "Mathlib/MeasureTheory/Integral/FundThmCalculus.lean", "def_pos": [1445, 9], "def_end_pos": [1445, 39]}]], "state_before": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf\u271d\u00b9 : \u211d \u2192 E\ng'\u271d g\u271d \u03c6 : \u211d \u2192 \u211d\nf\u271d f'\u271d : \u211d \u2192 E\na b : \u211d\nf f' g g' : \u211d \u2192 \u211d\nhf : \u2200 (x : \u211d), x \u2208 [[a, b]] \u2192 HasDerivAt f (f' x) x\nhg : \u2200 (x : \u211d), x \u2208 [[a, b]] \u2192 HasDerivAt g (g' (f x)) (f x)\nhf' : ContinuousOn f' [[a, b]]\nhg' : Continuous g'\n\u22a2 \u222b (x : \u211d) in a..b, (g' \u2218 f) x * f' x = (g \u2218 f) b - (g \u2218 f) a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Pointwise/Interval.lean", "full_name": "Set.preimage_const_mul_Ico_of_neg", "start": [674, 1], "end": [676, 66], "traced_tactics": [{"tactic": "simpa only [mul_comm] using preimage_mul_const_Ico_of_neg a b h", "annotated_tactic": ["simpa only [<a>mul_comm</a>] using <a>preimage_mul_const_Ico_of_neg</a> a b h", [{"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [302, 9], "def_end_pos": [302, 17]}, {"full_name": "Set.preimage_mul_const_Ico_of_neg", "def_path": "Mathlib/Data/Set/Pointwise/Interval.lean", "def_pos": [587, 9], "def_end_pos": [587, 38]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na\u271d a b c : \u03b1\nh : c < 0\n\u22a2 (fun x x_1 => x * x_1) c \u207b\u00b9' Ico a b = Ioc (b / c) (a / c)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/UniformIntegrable.lean", "full_name": "MeasureTheory.uniformIntegrable_average", "start": [920, 1], "end": [959, 54], "traced_tactics": [{"tactic": "obtain \u27e8hf\u2081, hf\u2082, hf\u2083\u27e9 := hf", "annotated_tactic": ["obtain \u27e8hf\u2081, hf\u2082, hf\u2083\u27e9 := hf", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nhp : 1 \u2264 p\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf : UniformIntegrable f p \u03bc\n\u22a2 UniformIntegrable (fun n => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in Finset.range n, f i) p \u03bc", "state_after": "case intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nhp : 1 \u2264 p\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf\u2081 : \u2200 (i : \u2115), AEStronglyMeasurable (f i) \u03bc\nhf\u2082 : UnifIntegrable f p \u03bc\nhf\u2083 : \u2203 C, \u2200 (i : \u2115), snorm (f i) p \u03bc \u2264 \u2191C\n\u22a2 UniformIntegrable (fun n => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in Finset.range n, f i) p \u03bc"}, {"tactic": "refine' \u27e8fun n => _, fun \u03b5 h\u03b5 => _, _\u27e9", "annotated_tactic": ["refine' \u27e8fun n => _, fun \u03b5 h\u03b5 => _, _\u27e9", []], "state_before": "case intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nhp : 1 \u2264 p\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf\u2081 : \u2200 (i : \u2115), AEStronglyMeasurable (f i) \u03bc\nhf\u2082 : UnifIntegrable f p \u03bc\nhf\u2083 : \u2203 C, \u2200 (i : \u2115), snorm (f i) p \u03bc \u2264 \u2191C\n\u22a2 UniformIntegrable (fun n => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in Finset.range n, f i) p \u03bc", "state_after": "case intro.intro.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nhp : 1 \u2264 p\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf\u2081 : \u2200 (i : \u2115), AEStronglyMeasurable (f i) \u03bc\nhf\u2082 : UnifIntegrable f p \u03bc\nhf\u2083 : \u2203 C, \u2200 (i : \u2115), snorm (f i) p \u03bc \u2264 \u2191C\nn : \u2115\n\u22a2 AEStronglyMeasurable ((fun n => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in Finset.range n, f i) n) \u03bc\n\ncase intro.intro.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nhp : 1 \u2264 p\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf\u2081 : \u2200 (i : \u2115), AEStronglyMeasurable (f i) \u03bc\nhf\u2082 : UnifIntegrable f p \u03bc\nhf\u2083 : \u2203 C, \u2200 (i : \u2115), snorm (f i) p \u03bc \u2264 \u2191C\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u22a2 \u2203 \u03b4 x,\n    \u2200 (i : \u2115) (s : Set \u03b1),\n      MeasurableSet s \u2192\n        \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192\n          snorm (indicator s ((fun n => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in Finset.range n, f i) i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\n\ncase intro.intro.refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nhp : 1 \u2264 p\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf\u2081 : \u2200 (i : \u2115), AEStronglyMeasurable (f i) \u03bc\nhf\u2082 : UnifIntegrable f p \u03bc\nhf\u2083 : \u2203 C, \u2200 (i : \u2115), snorm (f i) p \u03bc \u2264 \u2191C\n\u22a2 \u2203 C, \u2200 (i : \u2115), snorm ((fun n => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in Finset.range n, f i) i) p \u03bc \u2264 \u2191C"}, {"tactic": "exact (Finset.aestronglyMeasurable_sum' _ fun i _ => hf\u2081 i).const_smul _", "annotated_tactic": ["exact (<a>Finset.aestronglyMeasurable_sum'</a> _ fun i _ => hf\u2081 i).<a>const_smul</a> _", [{"full_name": "Finset.aestronglyMeasurable_sum'", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1426, 3], "def_end_pos": [1426, 14]}, {"full_name": "MeasureTheory.AEStronglyMeasurable.const_smul", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1339, 19], "def_end_pos": [1339, 29]}]], "state_before": "case intro.intro.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nhp : 1 \u2264 p\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf\u2081 : \u2200 (i : \u2115), AEStronglyMeasurable (f i) \u03bc\nhf\u2082 : UnifIntegrable f p \u03bc\nhf\u2083 : \u2203 C, \u2200 (i : \u2115), snorm (f i) p \u03bc \u2264 \u2191C\nn : \u2115\n\u22a2 AEStronglyMeasurable ((fun n => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in Finset.range n, f i) n) \u03bc", "state_after": "no goals"}, {"tactic": "obtain \u27e8\u03b4, h\u03b4\u2081, h\u03b4\u2082\u27e9 := hf\u2082 h\u03b5", "annotated_tactic": ["obtain \u27e8\u03b4, h\u03b4\u2081, h\u03b4\u2082\u27e9 := hf\u2082 h\u03b5", []], "state_before": "case intro.intro.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nhp : 1 \u2264 p\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf\u2081 : \u2200 (i : \u2115), AEStronglyMeasurable (f i) \u03bc\nhf\u2082 : UnifIntegrable f p \u03bc\nhf\u2083 : \u2203 C, \u2200 (i : \u2115), snorm (f i) p \u03bc \u2264 \u2191C\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u22a2 \u2203 \u03b4 x,\n    \u2200 (i : \u2115) (s : Set \u03b1),\n      MeasurableSet s \u2192\n        \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192\n          snorm (indicator s ((fun n => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in Finset.range n, f i) i)) p \u03bc \u2264 ENNReal.ofReal \u03b5", "state_after": "case intro.intro.refine'_2.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nhp : 1 \u2264 p\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf\u2081 : \u2200 (i : \u2115), AEStronglyMeasurable (f i) \u03bc\nhf\u2082 : UnifIntegrable f p \u03bc\nhf\u2083 : \u2203 C, \u2200 (i : \u2115), snorm (f i) p \u03bc \u2264 \u2191C\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u03b4 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\nh\u03b4\u2082 :\n  \u2200 (i : \u2115) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\n\u22a2 \u2203 \u03b4 x,\n    \u2200 (i : \u2115) (s : Set \u03b1),\n      MeasurableSet s \u2192\n        \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192\n          snorm (indicator s ((fun n => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in Finset.range n, f i) i)) p \u03bc \u2264 ENNReal.ofReal \u03b5"}, {"tactic": "refine' \u27e8\u03b4, h\u03b4\u2081, fun n s hs hle => _\u27e9", "annotated_tactic": ["refine' \u27e8\u03b4, h\u03b4\u2081, fun n s hs hle => _\u27e9", []], "state_before": "case intro.intro.refine'_2.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nhp : 1 \u2264 p\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf\u2081 : \u2200 (i : \u2115), AEStronglyMeasurable (f i) \u03bc\nhf\u2082 : UnifIntegrable f p \u03bc\nhf\u2083 : \u2203 C, \u2200 (i : \u2115), snorm (f i) p \u03bc \u2264 \u2191C\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u03b4 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\nh\u03b4\u2082 :\n  \u2200 (i : \u2115) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\n\u22a2 \u2203 \u03b4 x,\n    \u2200 (i : \u2115) (s : Set \u03b1),\n      MeasurableSet s \u2192\n        \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192\n          snorm (indicator s ((fun n => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in Finset.range n, f i) i)) p \u03bc \u2264 ENNReal.ofReal \u03b5", "state_after": "case intro.intro.refine'_2.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nhp : 1 \u2264 p\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf\u2081 : \u2200 (i : \u2115), AEStronglyMeasurable (f i) \u03bc\nhf\u2082 : UnifIntegrable f p \u03bc\nhf\u2083 : \u2203 C, \u2200 (i : \u2115), snorm (f i) p \u03bc \u2264 \u2191C\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u03b4 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\nh\u03b4\u2082 :\n  \u2200 (i : \u2115) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nn : \u2115\ns : Set \u03b1\nhs : MeasurableSet s\nhle : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\n\u22a2 snorm (indicator s ((fun n => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in Finset.range n, f i) n)) p \u03bc \u2264 ENNReal.ofReal \u03b5"}, {"tactic": "simp_rw [Finset.smul_sum, Set.indicator_finset_sum]", "annotated_tactic": ["simp_rw [<a>Finset.smul_sum</a>, <a>Set.indicator_finset_sum</a>]", [{"full_name": "Finset.smul_sum", "def_path": "Mathlib/GroupTheory/GroupAction/BigOperators.lean", "def_pos": [52, 9], "def_end_pos": [52, 24]}, {"full_name": "Set.indicator_finset_sum", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [661, 3], "def_end_pos": [661, 14]}]], "state_before": "case intro.intro.refine'_2.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nhp : 1 \u2264 p\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf\u2081 : \u2200 (i : \u2115), AEStronglyMeasurable (f i) \u03bc\nhf\u2082 : UnifIntegrable f p \u03bc\nhf\u2083 : \u2203 C, \u2200 (i : \u2115), snorm (f i) p \u03bc \u2264 \u2191C\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u03b4 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\nh\u03b4\u2082 :\n  \u2200 (i : \u2115) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nn : \u2115\ns : Set \u03b1\nhs : MeasurableSet s\nhle : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\n\u22a2 snorm (indicator s ((fun n => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in Finset.range n, f i) n)) p \u03bc \u2264 ENNReal.ofReal \u03b5", "state_after": "case intro.intro.refine'_2.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nhp : 1 \u2264 p\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf\u2081 : \u2200 (i : \u2115), AEStronglyMeasurable (f i) \u03bc\nhf\u2082 : UnifIntegrable f p \u03bc\nhf\u2083 : \u2203 C, \u2200 (i : \u2115), snorm (f i) p \u03bc \u2264 \u2191C\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u03b4 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\nh\u03b4\u2082 :\n  \u2200 (i : \u2115) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nn : \u2115\ns : Set \u03b1\nhs : MeasurableSet s\nhle : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\n\u22a2 snorm (\u2211 i in Finset.range n, indicator s ((\u2191n)\u207b\u00b9 \u2022 f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5"}, {"tactic": "refine' le_trans (snorm_sum_le (fun i _ => ((hf\u2081 i).const_smul _).indicator hs) hp) _", "annotated_tactic": ["refine' <a>le_trans</a> (<a>snorm_sum_le</a> (fun i _ => ((hf\u2081 i).<a>const_smul</a> _).<a>indicator</a> hs) hp) _", [{"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}, {"full_name": "MeasureTheory.snorm_sum_le", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [1226, 9], "def_end_pos": [1226, 21]}, {"full_name": "MeasureTheory.AEStronglyMeasurable.const_smul", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1339, 19], "def_end_pos": [1339, 29]}, {"full_name": "MeasureTheory.AEStronglyMeasurable.indicator", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1524, 19], "def_end_pos": [1524, 28]}]], "state_before": "case intro.intro.refine'_2.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nhp : 1 \u2264 p\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf\u2081 : \u2200 (i : \u2115), AEStronglyMeasurable (f i) \u03bc\nhf\u2082 : UnifIntegrable f p \u03bc\nhf\u2083 : \u2203 C, \u2200 (i : \u2115), snorm (f i) p \u03bc \u2264 \u2191C\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u03b4 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\nh\u03b4\u2082 :\n  \u2200 (i : \u2115) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nn : \u2115\ns : Set \u03b1\nhs : MeasurableSet s\nhle : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\n\u22a2 snorm (\u2211 i in Finset.range n, indicator s ((\u2191n)\u207b\u00b9 \u2022 f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5", "state_after": "case intro.intro.refine'_2.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nhp : 1 \u2264 p\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf\u2081 : \u2200 (i : \u2115), AEStronglyMeasurable (f i) \u03bc\nhf\u2082 : UnifIntegrable f p \u03bc\nhf\u2083 : \u2203 C, \u2200 (i : \u2115), snorm (f i) p \u03bc \u2264 \u2191C\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u03b4 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\nh\u03b4\u2082 :\n  \u2200 (i : \u2115) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nn : \u2115\ns : Set \u03b1\nhs : MeasurableSet s\nhle : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\n\u22a2 \u2211 i in Finset.range n, snorm (indicator s ((\u2191n)\u207b\u00b9 \u2022 f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5"}, {"tactic": "have : \u2200 i, s.indicator ((n : \u211d) \u207b\u00b9 \u2022 f i) = (\u2191n : \u211d)\u207b\u00b9 \u2022 s.indicator (f i) :=\n  fun i \u21a6 indicator_const_smul _ _ _", "annotated_tactic": ["have : \u2200 i, s.indicator ((n : \u211d) \u207b\u00b9 \u2022 f i) = (\u2191n : \u211d)\u207b\u00b9 \u2022 s.indicator (f i) :=\n      fun i \u21a6 <a>indicator_const_smul</a> _ _ _", [{"full_name": "Set.indicator_const_smul", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [498, 9], "def_end_pos": [498, 29]}]], "state_before": "case intro.intro.refine'_2.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nhp : 1 \u2264 p\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf\u2081 : \u2200 (i : \u2115), AEStronglyMeasurable (f i) \u03bc\nhf\u2082 : UnifIntegrable f p \u03bc\nhf\u2083 : \u2203 C, \u2200 (i : \u2115), snorm (f i) p \u03bc \u2264 \u2191C\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u03b4 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\nh\u03b4\u2082 :\n  \u2200 (i : \u2115) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nn : \u2115\ns : Set \u03b1\nhs : MeasurableSet s\nhle : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\n\u22a2 \u2211 i in Finset.range n, snorm (indicator s ((\u2191n)\u207b\u00b9 \u2022 f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5", "state_after": "case intro.intro.refine'_2.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nhp : 1 \u2264 p\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf\u2081 : \u2200 (i : \u2115), AEStronglyMeasurable (f i) \u03bc\nhf\u2082 : UnifIntegrable f p \u03bc\nhf\u2083 : \u2203 C, \u2200 (i : \u2115), snorm (f i) p \u03bc \u2264 \u2191C\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u03b4 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\nh\u03b4\u2082 :\n  \u2200 (i : \u2115) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nn : \u2115\ns : Set \u03b1\nhs : MeasurableSet s\nhle : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\nthis : \u2200 (i : \u2115), indicator s ((\u2191n)\u207b\u00b9 \u2022 f i) = (\u2191n)\u207b\u00b9 \u2022 indicator s (f i)\n\u22a2 \u2211 i in Finset.range n, snorm (indicator s ((\u2191n)\u207b\u00b9 \u2022 f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5"}, {"tactic": "simp_rw [this, snorm_const_smul, \u2190 Finset.mul_sum, nnnorm_inv, Real.nnnorm_coe_nat]", "annotated_tactic": ["simp_rw [this, <a>snorm_const_smul</a>, \u2190 <a>Finset.mul_sum</a>, <a>nnnorm_inv</a>, <a>Real.nnnorm_coe_nat</a>]", [{"full_name": "MeasureTheory.snorm_const_smul", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [1578, 9], "def_end_pos": [1578, 25]}, {"full_name": "Finset.mul_sum", "def_path": "Mathlib/Algebra/BigOperators/Ring.lean", "def_pos": [55, 9], "def_end_pos": [55, 16]}, {"full_name": "nnnorm_inv", "def_path": "Mathlib/Analysis/Normed/Field/Basic.lean", "def_pos": [577, 9], "def_end_pos": [577, 19]}, {"full_name": "Real.nnnorm_coe_nat", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [1786, 9], "def_end_pos": [1786, 23]}]], "state_before": "case intro.intro.refine'_2.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nhp : 1 \u2264 p\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf\u2081 : \u2200 (i : \u2115), AEStronglyMeasurable (f i) \u03bc\nhf\u2082 : UnifIntegrable f p \u03bc\nhf\u2083 : \u2203 C, \u2200 (i : \u2115), snorm (f i) p \u03bc \u2264 \u2191C\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u03b4 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\nh\u03b4\u2082 :\n  \u2200 (i : \u2115) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nn : \u2115\ns : Set \u03b1\nhs : MeasurableSet s\nhle : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\nthis : \u2200 (i : \u2115), indicator s ((\u2191n)\u207b\u00b9 \u2022 f i) = (\u2191n)\u207b\u00b9 \u2022 indicator s (f i)\n\u22a2 \u2211 i in Finset.range n, snorm (indicator s ((\u2191n)\u207b\u00b9 \u2022 f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5", "state_after": "case intro.intro.refine'_2.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nhp : 1 \u2264 p\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf\u2081 : \u2200 (i : \u2115), AEStronglyMeasurable (f i) \u03bc\nhf\u2082 : UnifIntegrable f p \u03bc\nhf\u2083 : \u2203 C, \u2200 (i : \u2115), snorm (f i) p \u03bc \u2264 \u2191C\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u03b4 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\nh\u03b4\u2082 :\n  \u2200 (i : \u2115) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nn : \u2115\ns : Set \u03b1\nhs : MeasurableSet s\nhle : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\nthis : \u2200 (i : \u2115), indicator s ((\u2191n)\u207b\u00b9 \u2022 f i) = (\u2191n)\u207b\u00b9 \u2022 indicator s (f i)\n\u22a2 \u2191(\u2191n)\u207b\u00b9 * \u2211 x in Finset.range n, snorm (indicator s (f x)) p \u03bc \u2264 ENNReal.ofReal \u03b5"}, {"tactic": "by_cases hn : (\u2191(\u2191n : \u211d\u22650)\u207b\u00b9 : \u211d\u22650\u221e) = 0", "annotated_tactic": ["by_cases hn : (\u2191(\u2191n : \u211d\u22650)\u207b\u00b9 : \u211d\u22650\u221e) = 0", []], "state_before": "case intro.intro.refine'_2.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nhp : 1 \u2264 p\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf\u2081 : \u2200 (i : \u2115), AEStronglyMeasurable (f i) \u03bc\nhf\u2082 : UnifIntegrable f p \u03bc\nhf\u2083 : \u2203 C, \u2200 (i : \u2115), snorm (f i) p \u03bc \u2264 \u2191C\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u03b4 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\nh\u03b4\u2082 :\n  \u2200 (i : \u2115) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nn : \u2115\ns : Set \u03b1\nhs : MeasurableSet s\nhle : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\nthis : \u2200 (i : \u2115), indicator s ((\u2191n)\u207b\u00b9 \u2022 f i) = (\u2191n)\u207b\u00b9 \u2022 indicator s (f i)\n\u22a2 \u2191(\u2191n)\u207b\u00b9 * \u2211 x in Finset.range n, snorm (indicator s (f x)) p \u03bc \u2264 ENNReal.ofReal \u03b5", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nhp : 1 \u2264 p\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf\u2081 : \u2200 (i : \u2115), AEStronglyMeasurable (f i) \u03bc\nhf\u2082 : UnifIntegrable f p \u03bc\nhf\u2083 : \u2203 C, \u2200 (i : \u2115), snorm (f i) p \u03bc \u2264 \u2191C\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u03b4 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\nh\u03b4\u2082 :\n  \u2200 (i : \u2115) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nn : \u2115\ns : Set \u03b1\nhs : MeasurableSet s\nhle : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\nthis : \u2200 (i : \u2115), indicator s ((\u2191n)\u207b\u00b9 \u2022 f i) = (\u2191n)\u207b\u00b9 \u2022 indicator s (f i)\nhn : \u2191(\u2191n)\u207b\u00b9 = 0\n\u22a2 \u2191(\u2191n)\u207b\u00b9 * \u2211 x in Finset.range n, snorm (indicator s (f x)) p \u03bc \u2264 ENNReal.ofReal \u03b5\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nhp : 1 \u2264 p\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf\u2081 : \u2200 (i : \u2115), AEStronglyMeasurable (f i) \u03bc\nhf\u2082 : UnifIntegrable f p \u03bc\nhf\u2083 : \u2203 C, \u2200 (i : \u2115), snorm (f i) p \u03bc \u2264 \u2191C\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u03b4 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\nh\u03b4\u2082 :\n  \u2200 (i : \u2115) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nn : \u2115\ns : Set \u03b1\nhs : MeasurableSet s\nhle : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\nthis : \u2200 (i : \u2115), indicator s ((\u2191n)\u207b\u00b9 \u2022 f i) = (\u2191n)\u207b\u00b9 \u2022 indicator s (f i)\nhn : \u00ac\u2191(\u2191n)\u207b\u00b9 = 0\n\u22a2 \u2191(\u2191n)\u207b\u00b9 * \u2211 x in Finset.range n, snorm (indicator s (f x)) p \u03bc \u2264 ENNReal.ofReal \u03b5"}, {"tactic": "refine' le_trans _ (_ : \u2191(\u2191n : \u211d\u22650)\u207b\u00b9 * n \u2022 ENNReal.ofReal \u03b5 \u2264 ENNReal.ofReal \u03b5)", "annotated_tactic": ["refine' <a>le_trans</a> _ (_ : \u2191(\u2191n : \u211d\u22650)\u207b\u00b9 * n \u2022 <a>ENNReal.ofReal</a> \u03b5 \u2264 <a>ENNReal.ofReal</a> \u03b5)", [{"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}, {"full_name": "ENNReal.ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [172, 29], "def_end_pos": [172, 35]}, {"full_name": "ENNReal.ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [172, 29], "def_end_pos": [172, 35]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nhp : 1 \u2264 p\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf\u2081 : \u2200 (i : \u2115), AEStronglyMeasurable (f i) \u03bc\nhf\u2082 : UnifIntegrable f p \u03bc\nhf\u2083 : \u2203 C, \u2200 (i : \u2115), snorm (f i) p \u03bc \u2264 \u2191C\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u03b4 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\nh\u03b4\u2082 :\n  \u2200 (i : \u2115) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nn : \u2115\ns : Set \u03b1\nhs : MeasurableSet s\nhle : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\nthis : \u2200 (i : \u2115), indicator s ((\u2191n)\u207b\u00b9 \u2022 f i) = (\u2191n)\u207b\u00b9 \u2022 indicator s (f i)\nhn : \u00ac\u2191(\u2191n)\u207b\u00b9 = 0\n\u22a2 \u2191(\u2191n)\u207b\u00b9 * \u2211 x in Finset.range n, snorm (indicator s (f x)) p \u03bc \u2264 ENNReal.ofReal \u03b5", "state_after": "case neg.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nhp : 1 \u2264 p\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf\u2081 : \u2200 (i : \u2115), AEStronglyMeasurable (f i) \u03bc\nhf\u2082 : UnifIntegrable f p \u03bc\nhf\u2083 : \u2203 C, \u2200 (i : \u2115), snorm (f i) p \u03bc \u2264 \u2191C\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u03b4 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\nh\u03b4\u2082 :\n  \u2200 (i : \u2115) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nn : \u2115\ns : Set \u03b1\nhs : MeasurableSet s\nhle : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\nthis : \u2200 (i : \u2115), indicator s ((\u2191n)\u207b\u00b9 \u2022 f i) = (\u2191n)\u207b\u00b9 \u2022 indicator s (f i)\nhn : \u00ac\u2191(\u2191n)\u207b\u00b9 = 0\n\u22a2 \u2191(\u2191n)\u207b\u00b9 * \u2211 x in Finset.range n, snorm (indicator s (f x)) p \u03bc \u2264 \u2191(\u2191n)\u207b\u00b9 * n \u2022 ENNReal.ofReal \u03b5\n\ncase neg.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nhp : 1 \u2264 p\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf\u2081 : \u2200 (i : \u2115), AEStronglyMeasurable (f i) \u03bc\nhf\u2082 : UnifIntegrable f p \u03bc\nhf\u2083 : \u2203 C, \u2200 (i : \u2115), snorm (f i) p \u03bc \u2264 \u2191C\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u03b4 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\nh\u03b4\u2082 :\n  \u2200 (i : \u2115) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nn : \u2115\ns : Set \u03b1\nhs : MeasurableSet s\nhle : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\nthis : \u2200 (i : \u2115), indicator s ((\u2191n)\u207b\u00b9 \u2022 f i) = (\u2191n)\u207b\u00b9 \u2022 indicator s (f i)\nhn : \u00ac\u2191(\u2191n)\u207b\u00b9 = 0\n\u22a2 \u2191(\u2191n)\u207b\u00b9 * n \u2022 ENNReal.ofReal \u03b5 \u2264 ENNReal.ofReal \u03b5"}, {"tactic": "simp only [hn, zero_mul, zero_le]", "annotated_tactic": ["simp only [hn, <a>zero_mul</a>, <a>zero_le</a>]", [{"full_name": "MulZeroClass.zero_mul", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [36, 3], "def_end_pos": [36, 11]}, {"full_name": "zero_le", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [217, 30], "def_end_pos": [217, 37]}]], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nhp : 1 \u2264 p\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf\u2081 : \u2200 (i : \u2115), AEStronglyMeasurable (f i) \u03bc\nhf\u2082 : UnifIntegrable f p \u03bc\nhf\u2083 : \u2203 C, \u2200 (i : \u2115), snorm (f i) p \u03bc \u2264 \u2191C\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u03b4 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\nh\u03b4\u2082 :\n  \u2200 (i : \u2115) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nn : \u2115\ns : Set \u03b1\nhs : MeasurableSet s\nhle : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\nthis : \u2200 (i : \u2115), indicator s ((\u2191n)\u207b\u00b9 \u2022 f i) = (\u2191n)\u207b\u00b9 \u2022 indicator s (f i)\nhn : \u2191(\u2191n)\u207b\u00b9 = 0\n\u22a2 \u2191(\u2191n)\u207b\u00b9 * \u2211 x in Finset.range n, snorm (indicator s (f x)) p \u03bc \u2264 ENNReal.ofReal \u03b5", "state_after": "no goals"}, {"tactic": "refine' (ENNReal.mul_le_mul_left hn ENNReal.coe_ne_top).2 _", "annotated_tactic": ["refine' (<a>ENNReal.mul_le_mul_left</a> hn <a>ENNReal.coe_ne_top</a>).2 _", [{"full_name": "ENNReal.mul_le_mul_left", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1065, 9], "def_end_pos": [1065, 24]}, {"full_name": "ENNReal.coe_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [302, 17], "def_end_pos": [302, 27]}]], "state_before": "case neg.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nhp : 1 \u2264 p\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf\u2081 : \u2200 (i : \u2115), AEStronglyMeasurable (f i) \u03bc\nhf\u2082 : UnifIntegrable f p \u03bc\nhf\u2083 : \u2203 C, \u2200 (i : \u2115), snorm (f i) p \u03bc \u2264 \u2191C\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u03b4 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\nh\u03b4\u2082 :\n  \u2200 (i : \u2115) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nn : \u2115\ns : Set \u03b1\nhs : MeasurableSet s\nhle : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\nthis : \u2200 (i : \u2115), indicator s ((\u2191n)\u207b\u00b9 \u2022 f i) = (\u2191n)\u207b\u00b9 \u2022 indicator s (f i)\nhn : \u00ac\u2191(\u2191n)\u207b\u00b9 = 0\n\u22a2 \u2191(\u2191n)\u207b\u00b9 * \u2211 x in Finset.range n, snorm (indicator s (f x)) p \u03bc \u2264 \u2191(\u2191n)\u207b\u00b9 * n \u2022 ENNReal.ofReal \u03b5", "state_after": "case neg.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nhp : 1 \u2264 p\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf\u2081 : \u2200 (i : \u2115), AEStronglyMeasurable (f i) \u03bc\nhf\u2082 : UnifIntegrable f p \u03bc\nhf\u2083 : \u2203 C, \u2200 (i : \u2115), snorm (f i) p \u03bc \u2264 \u2191C\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u03b4 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\nh\u03b4\u2082 :\n  \u2200 (i : \u2115) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nn : \u2115\ns : Set \u03b1\nhs : MeasurableSet s\nhle : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\nthis : \u2200 (i : \u2115), indicator s ((\u2191n)\u207b\u00b9 \u2022 f i) = (\u2191n)\u207b\u00b9 \u2022 indicator s (f i)\nhn : \u00ac\u2191(\u2191n)\u207b\u00b9 = 0\n\u22a2 \u2211 x in Finset.range n, snorm (indicator s (f x)) p \u03bc \u2264 n \u2022 ENNReal.ofReal \u03b5"}, {"tactic": "conv_rhs => rw [\u2190 Finset.card_range n]", "annotated_tactic": ["conv_rhs => rw [\u2190 <a>Finset.card_range</a> n]", [{"full_name": "Finset.card_range", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [177, 9], "def_end_pos": [177, 19]}]], "state_before": "case neg.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nhp : 1 \u2264 p\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf\u2081 : \u2200 (i : \u2115), AEStronglyMeasurable (f i) \u03bc\nhf\u2082 : UnifIntegrable f p \u03bc\nhf\u2083 : \u2203 C, \u2200 (i : \u2115), snorm (f i) p \u03bc \u2264 \u2191C\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u03b4 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\nh\u03b4\u2082 :\n  \u2200 (i : \u2115) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nn : \u2115\ns : Set \u03b1\nhs : MeasurableSet s\nhle : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\nthis : \u2200 (i : \u2115), indicator s ((\u2191n)\u207b\u00b9 \u2022 f i) = (\u2191n)\u207b\u00b9 \u2022 indicator s (f i)\nhn : \u00ac\u2191(\u2191n)\u207b\u00b9 = 0\n\u22a2 \u2211 x in Finset.range n, snorm (indicator s (f x)) p \u03bc \u2264 n \u2022 ENNReal.ofReal \u03b5", "state_after": "case neg.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nhp : 1 \u2264 p\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf\u2081 : \u2200 (i : \u2115), AEStronglyMeasurable (f i) \u03bc\nhf\u2082 : UnifIntegrable f p \u03bc\nhf\u2083 : \u2203 C, \u2200 (i : \u2115), snorm (f i) p \u03bc \u2264 \u2191C\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u03b4 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\nh\u03b4\u2082 :\n  \u2200 (i : \u2115) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nn : \u2115\ns : Set \u03b1\nhs : MeasurableSet s\nhle : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\nthis : \u2200 (i : \u2115), indicator s ((\u2191n)\u207b\u00b9 \u2022 f i) = (\u2191n)\u207b\u00b9 \u2022 indicator s (f i)\nhn : \u00ac\u2191(\u2191n)\u207b\u00b9 = 0\n\u22a2 \u2211 x in Finset.range n, snorm (indicator s (f x)) p \u03bc \u2264 Finset.card (Finset.range n) \u2022 ENNReal.ofReal \u03b5"}, {"tactic": "exact Finset.sum_le_card_nsmul _ _ _ fun i _ => h\u03b4\u2082 _ _ hs hle", "annotated_tactic": ["exact <a>Finset.sum_le_card_nsmul</a> _ _ _ fun i _ => h\u03b4\u2082 _ _ hs hle", [{"full_name": "Finset.sum_le_card_nsmul", "def_path": "Mathlib/Algebra/BigOperators/Order.lean", "def_pos": [211, 15], "def_end_pos": [211, 32]}]], "state_before": "case neg.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nhp : 1 \u2264 p\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf\u2081 : \u2200 (i : \u2115), AEStronglyMeasurable (f i) \u03bc\nhf\u2082 : UnifIntegrable f p \u03bc\nhf\u2083 : \u2203 C, \u2200 (i : \u2115), snorm (f i) p \u03bc \u2264 \u2191C\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u03b4 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\nh\u03b4\u2082 :\n  \u2200 (i : \u2115) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nn : \u2115\ns : Set \u03b1\nhs : MeasurableSet s\nhle : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\nthis : \u2200 (i : \u2115), indicator s ((\u2191n)\u207b\u00b9 \u2022 f i) = (\u2191n)\u207b\u00b9 \u2022 indicator s (f i)\nhn : \u00ac\u2191(\u2191n)\u207b\u00b9 = 0\n\u22a2 \u2211 x in Finset.range n, snorm (indicator s (f x)) p \u03bc \u2264 Finset.card (Finset.range n) \u2022 ENNReal.ofReal \u03b5", "state_after": "no goals"}, {"tactic": "simp only [ENNReal.coe_eq_zero, inv_eq_zero, Nat.cast_eq_zero] at hn", "annotated_tactic": ["simp only [<a>ENNReal.coe_eq_zero</a>, <a>inv_eq_zero</a>, <a>Nat.cast_eq_zero</a>] at hn", [{"full_name": "ENNReal.coe_eq_zero", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [368, 28], "def_end_pos": [368, 39]}, {"full_name": "inv_eq_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Basic.lean", "def_pos": [355, 9], "def_end_pos": [355, 20]}, {"full_name": "Nat.cast_eq_zero", "def_path": "Mathlib/Algebra/CharZero/Defs.lean", "def_pos": [80, 9], "def_end_pos": [80, 21]}]], "state_before": "case neg.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nhp : 1 \u2264 p\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf\u2081 : \u2200 (i : \u2115), AEStronglyMeasurable (f i) \u03bc\nhf\u2082 : UnifIntegrable f p \u03bc\nhf\u2083 : \u2203 C, \u2200 (i : \u2115), snorm (f i) p \u03bc \u2264 \u2191C\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u03b4 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\nh\u03b4\u2082 :\n  \u2200 (i : \u2115) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nn : \u2115\ns : Set \u03b1\nhs : MeasurableSet s\nhle : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\nthis : \u2200 (i : \u2115), indicator s ((\u2191n)\u207b\u00b9 \u2022 f i) = (\u2191n)\u207b\u00b9 \u2022 indicator s (f i)\nhn : \u00ac\u2191(\u2191n)\u207b\u00b9 = 0\n\u22a2 \u2191(\u2191n)\u207b\u00b9 * n \u2022 ENNReal.ofReal \u03b5 \u2264 ENNReal.ofReal \u03b5", "state_after": "case neg.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nhp : 1 \u2264 p\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf\u2081 : \u2200 (i : \u2115), AEStronglyMeasurable (f i) \u03bc\nhf\u2082 : UnifIntegrable f p \u03bc\nhf\u2083 : \u2203 C, \u2200 (i : \u2115), snorm (f i) p \u03bc \u2264 \u2191C\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u03b4 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\nh\u03b4\u2082 :\n  \u2200 (i : \u2115) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nn : \u2115\ns : Set \u03b1\nhs : MeasurableSet s\nhle : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\nthis : \u2200 (i : \u2115), indicator s ((\u2191n)\u207b\u00b9 \u2022 f i) = (\u2191n)\u207b\u00b9 \u2022 indicator s (f i)\nhn : \u00acn = 0\n\u22a2 \u2191(\u2191n)\u207b\u00b9 * n \u2022 ENNReal.ofReal \u03b5 \u2264 ENNReal.ofReal \u03b5"}, {"tactic": "rw [nsmul_eq_mul, \u2190 mul_assoc, ENNReal.coe_inv, ENNReal.coe_nat,\n  ENNReal.inv_mul_cancel _ (ENNReal.nat_ne_top _), one_mul]", "annotated_tactic": ["rw [<a>nsmul_eq_mul</a>, \u2190 <a>mul_assoc</a>, <a>ENNReal.coe_inv</a>, <a>ENNReal.coe_nat</a>,\n        <a>ENNReal.inv_mul_cancel</a> _ (<a>ENNReal.nat_ne_top</a> _), <a>one_mul</a>]", [{"full_name": "nsmul_eq_mul", "def_path": "Mathlib/Algebra/GroupPower/Lemmas.lean", "def_pos": [509, 9], "def_end_pos": [509, 21]}, {"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [264, 9], "def_end_pos": [264, 18]}, {"full_name": "ENNReal.coe_inv", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1387, 9], "def_end_pos": [1387, 16]}, {"full_name": "ENNReal.coe_nat", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [707, 9], "def_end_pos": [707, 16]}, {"full_name": "ENNReal.inv_mul_cancel", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1424, 19], "def_end_pos": [1424, 33]}, {"full_name": "ENNReal.nat_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [717, 17], "def_end_pos": [717, 27]}, {"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [464, 9], "def_end_pos": [464, 16]}]], "state_before": "case neg.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nhp : 1 \u2264 p\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf\u2081 : \u2200 (i : \u2115), AEStronglyMeasurable (f i) \u03bc\nhf\u2082 : UnifIntegrable f p \u03bc\nhf\u2083 : \u2203 C, \u2200 (i : \u2115), snorm (f i) p \u03bc \u2264 \u2191C\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u03b4 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\nh\u03b4\u2082 :\n  \u2200 (i : \u2115) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nn : \u2115\ns : Set \u03b1\nhs : MeasurableSet s\nhle : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\nthis : \u2200 (i : \u2115), indicator s ((\u2191n)\u207b\u00b9 \u2022 f i) = (\u2191n)\u207b\u00b9 \u2022 indicator s (f i)\nhn : \u00acn = 0\n\u22a2 \u2191(\u2191n)\u207b\u00b9 * n \u2022 ENNReal.ofReal \u03b5 \u2264 ENNReal.ofReal \u03b5", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nhp : 1 \u2264 p\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf\u2081 : \u2200 (i : \u2115), AEStronglyMeasurable (f i) \u03bc\nhf\u2082 : UnifIntegrable f p \u03bc\nhf\u2083 : \u2203 C, \u2200 (i : \u2115), snorm (f i) p \u03bc \u2264 \u2191C\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u03b4 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\nh\u03b4\u2082 :\n  \u2200 (i : \u2115) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nn : \u2115\ns : Set \u03b1\nhs : MeasurableSet s\nhle : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\nthis : \u2200 (i : \u2115), indicator s ((\u2191n)\u207b\u00b9 \u2022 f i) = (\u2191n)\u207b\u00b9 \u2022 indicator s (f i)\nhn : \u00acn = 0\n\u22a2 \u2191n \u2260 0\n\ncase neg.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nhp : 1 \u2264 p\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf\u2081 : \u2200 (i : \u2115), AEStronglyMeasurable (f i) \u03bc\nhf\u2082 : UnifIntegrable f p \u03bc\nhf\u2083 : \u2203 C, \u2200 (i : \u2115), snorm (f i) p \u03bc \u2264 \u2191C\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u03b4 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\nh\u03b4\u2082 :\n  \u2200 (i : \u2115) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nn : \u2115\ns : Set \u03b1\nhs : MeasurableSet s\nhle : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\nthis : \u2200 (i : \u2115), indicator s ((\u2191n)\u207b\u00b9 \u2022 f i) = (\u2191n)\u207b\u00b9 \u2022 indicator s (f i)\nhn : \u00acn = 0\n\u22a2 \u2191n \u2260 0"}, {"tactic": "all_goals simpa only [Ne.def, Nat.cast_eq_zero]", "annotated_tactic": ["all_goals simpa only [<a>Ne.def</a>, <a>Nat.cast_eq_zero</a>]", [{"full_name": "Ne.def", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [59, 9], "def_end_pos": [59, 15]}, {"full_name": "Nat.cast_eq_zero", "def_path": "Mathlib/Algebra/CharZero/Defs.lean", "def_pos": [80, 9], "def_end_pos": [80, 21]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nhp : 1 \u2264 p\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf\u2081 : \u2200 (i : \u2115), AEStronglyMeasurable (f i) \u03bc\nhf\u2082 : UnifIntegrable f p \u03bc\nhf\u2083 : \u2203 C, \u2200 (i : \u2115), snorm (f i) p \u03bc \u2264 \u2191C\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u03b4 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\nh\u03b4\u2082 :\n  \u2200 (i : \u2115) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nn : \u2115\ns : Set \u03b1\nhs : MeasurableSet s\nhle : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\nthis : \u2200 (i : \u2115), indicator s ((\u2191n)\u207b\u00b9 \u2022 f i) = (\u2191n)\u207b\u00b9 \u2022 indicator s (f i)\nhn : \u00acn = 0\n\u22a2 \u2191n \u2260 0\n\ncase neg.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nhp : 1 \u2264 p\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf\u2081 : \u2200 (i : \u2115), AEStronglyMeasurable (f i) \u03bc\nhf\u2082 : UnifIntegrable f p \u03bc\nhf\u2083 : \u2203 C, \u2200 (i : \u2115), snorm (f i) p \u03bc \u2264 \u2191C\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u03b4 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\nh\u03b4\u2082 :\n  \u2200 (i : \u2115) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nn : \u2115\ns : Set \u03b1\nhs : MeasurableSet s\nhle : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\nthis : \u2200 (i : \u2115), indicator s ((\u2191n)\u207b\u00b9 \u2022 f i) = (\u2191n)\u207b\u00b9 \u2022 indicator s (f i)\nhn : \u00acn = 0\n\u22a2 \u2191n \u2260 0", "state_after": "no goals"}, {"tactic": "simpa only [Ne.def, Nat.cast_eq_zero]", "annotated_tactic": ["simpa only [<a>Ne.def</a>, <a>Nat.cast_eq_zero</a>]", [{"full_name": "Ne.def", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [59, 9], "def_end_pos": [59, 15]}, {"full_name": "Nat.cast_eq_zero", "def_path": "Mathlib/Algebra/CharZero/Defs.lean", "def_pos": [80, 9], "def_end_pos": [80, 21]}]], "state_before": "case neg.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nhp : 1 \u2264 p\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf\u2081 : \u2200 (i : \u2115), AEStronglyMeasurable (f i) \u03bc\nhf\u2082 : UnifIntegrable f p \u03bc\nhf\u2083 : \u2203 C, \u2200 (i : \u2115), snorm (f i) p \u03bc \u2264 \u2191C\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u03b4 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\nh\u03b4\u2082 :\n  \u2200 (i : \u2115) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nn : \u2115\ns : Set \u03b1\nhs : MeasurableSet s\nhle : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\nthis : \u2200 (i : \u2115), indicator s ((\u2191n)\u207b\u00b9 \u2022 f i) = (\u2191n)\u207b\u00b9 \u2022 indicator s (f i)\nhn : \u00acn = 0\n\u22a2 \u2191n \u2260 0", "state_after": "no goals"}, {"tactic": "obtain \u27e8C, hC\u27e9 := hf\u2083", "annotated_tactic": ["obtain \u27e8C, hC\u27e9 := hf\u2083", []], "state_before": "case intro.intro.refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nhp : 1 \u2264 p\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf\u2081 : \u2200 (i : \u2115), AEStronglyMeasurable (f i) \u03bc\nhf\u2082 : UnifIntegrable f p \u03bc\nhf\u2083 : \u2203 C, \u2200 (i : \u2115), snorm (f i) p \u03bc \u2264 \u2191C\n\u22a2 \u2203 C, \u2200 (i : \u2115), snorm ((fun n => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in Finset.range n, f i) i) p \u03bc \u2264 \u2191C", "state_after": "case intro.intro.refine'_3.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nhp : 1 \u2264 p\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf\u2081 : \u2200 (i : \u2115), AEStronglyMeasurable (f i) \u03bc\nhf\u2082 : UnifIntegrable f p \u03bc\nC : \u211d\u22650\nhC : \u2200 (i : \u2115), snorm (f i) p \u03bc \u2264 \u2191C\n\u22a2 \u2203 C, \u2200 (i : \u2115), snorm ((fun n => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in Finset.range n, f i) i) p \u03bc \u2264 \u2191C"}, {"tactic": "simp_rw [Finset.smul_sum]", "annotated_tactic": ["simp_rw [<a>Finset.smul_sum</a>]", [{"full_name": "Finset.smul_sum", "def_path": "Mathlib/GroupTheory/GroupAction/BigOperators.lean", "def_pos": [52, 9], "def_end_pos": [52, 24]}]], "state_before": "case intro.intro.refine'_3.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nhp : 1 \u2264 p\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf\u2081 : \u2200 (i : \u2115), AEStronglyMeasurable (f i) \u03bc\nhf\u2082 : UnifIntegrable f p \u03bc\nC : \u211d\u22650\nhC : \u2200 (i : \u2115), snorm (f i) p \u03bc \u2264 \u2191C\n\u22a2 \u2203 C, \u2200 (i : \u2115), snorm ((fun n => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in Finset.range n, f i) i) p \u03bc \u2264 \u2191C", "state_after": "case intro.intro.refine'_3.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nhp : 1 \u2264 p\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf\u2081 : \u2200 (i : \u2115), AEStronglyMeasurable (f i) \u03bc\nhf\u2082 : UnifIntegrable f p \u03bc\nC : \u211d\u22650\nhC : \u2200 (i : \u2115), snorm (f i) p \u03bc \u2264 \u2191C\n\u22a2 \u2203 C, \u2200 (i : \u2115), snorm (\u2211 x in Finset.range i, (\u2191i)\u207b\u00b9 \u2022 f x) p \u03bc \u2264 \u2191C"}, {"tactic": "refine' \u27e8C, fun n => (snorm_sum_le (fun i _ => (hf\u2081 i).const_smul _) hp).trans _\u27e9", "annotated_tactic": ["refine' \u27e8C, fun n => (<a>snorm_sum_le</a> (fun i _ => (hf\u2081 i).<a>const_smul</a> _) hp).<a>trans</a> _\u27e9", [{"full_name": "MeasureTheory.snorm_sum_le", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [1226, 9], "def_end_pos": [1226, 21]}, {"full_name": "MeasureTheory.AEStronglyMeasurable.const_smul", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1339, 19], "def_end_pos": [1339, 29]}, {"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [120, 7], "def_end_pos": [120, 18]}]], "state_before": "case intro.intro.refine'_3.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nhp : 1 \u2264 p\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf\u2081 : \u2200 (i : \u2115), AEStronglyMeasurable (f i) \u03bc\nhf\u2082 : UnifIntegrable f p \u03bc\nC : \u211d\u22650\nhC : \u2200 (i : \u2115), snorm (f i) p \u03bc \u2264 \u2191C\n\u22a2 \u2203 C, \u2200 (i : \u2115), snorm (\u2211 x in Finset.range i, (\u2191i)\u207b\u00b9 \u2022 f x) p \u03bc \u2264 \u2191C", "state_after": "case intro.intro.refine'_3.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nhp : 1 \u2264 p\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf\u2081 : \u2200 (i : \u2115), AEStronglyMeasurable (f i) \u03bc\nhf\u2082 : UnifIntegrable f p \u03bc\nC : \u211d\u22650\nhC : \u2200 (i : \u2115), snorm (f i) p \u03bc \u2264 \u2191C\nn : \u2115\n\u22a2 \u2211 i in Finset.range n, snorm ((\u2191n)\u207b\u00b9 \u2022 f i) p \u03bc \u2264 \u2191C"}, {"tactic": "simp_rw [snorm_const_smul, \u2190 Finset.mul_sum, nnnorm_inv, Real.nnnorm_coe_nat]", "annotated_tactic": ["simp_rw [<a>snorm_const_smul</a>, \u2190 <a>Finset.mul_sum</a>, <a>nnnorm_inv</a>, <a>Real.nnnorm_coe_nat</a>]", [{"full_name": "MeasureTheory.snorm_const_smul", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [1578, 9], "def_end_pos": [1578, 25]}, {"full_name": "Finset.mul_sum", "def_path": "Mathlib/Algebra/BigOperators/Ring.lean", "def_pos": [55, 9], "def_end_pos": [55, 16]}, {"full_name": "nnnorm_inv", "def_path": "Mathlib/Analysis/Normed/Field/Basic.lean", "def_pos": [577, 9], "def_end_pos": [577, 19]}, {"full_name": "Real.nnnorm_coe_nat", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [1786, 9], "def_end_pos": [1786, 23]}]], "state_before": "case intro.intro.refine'_3.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nhp : 1 \u2264 p\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf\u2081 : \u2200 (i : \u2115), AEStronglyMeasurable (f i) \u03bc\nhf\u2082 : UnifIntegrable f p \u03bc\nC : \u211d\u22650\nhC : \u2200 (i : \u2115), snorm (f i) p \u03bc \u2264 \u2191C\nn : \u2115\n\u22a2 \u2211 i in Finset.range n, snorm ((\u2191n)\u207b\u00b9 \u2022 f i) p \u03bc \u2264 \u2191C", "state_after": "case intro.intro.refine'_3.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nhp : 1 \u2264 p\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf\u2081 : \u2200 (i : \u2115), AEStronglyMeasurable (f i) \u03bc\nhf\u2082 : UnifIntegrable f p \u03bc\nC : \u211d\u22650\nhC : \u2200 (i : \u2115), snorm (f i) p \u03bc \u2264 \u2191C\nn : \u2115\n\u22a2 \u2191(\u2191n)\u207b\u00b9 * \u2211 x in Finset.range n, snorm (f x) p \u03bc \u2264 \u2191C"}, {"tactic": "by_cases hn : (\u2191(\u2191n : \u211d\u22650)\u207b\u00b9 : \u211d\u22650\u221e) = 0", "annotated_tactic": ["by_cases hn : (\u2191(\u2191n : \u211d\u22650)\u207b\u00b9 : \u211d\u22650\u221e) = 0", []], "state_before": "case intro.intro.refine'_3.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nhp : 1 \u2264 p\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf\u2081 : \u2200 (i : \u2115), AEStronglyMeasurable (f i) \u03bc\nhf\u2082 : UnifIntegrable f p \u03bc\nC : \u211d\u22650\nhC : \u2200 (i : \u2115), snorm (f i) p \u03bc \u2264 \u2191C\nn : \u2115\n\u22a2 \u2191(\u2191n)\u207b\u00b9 * \u2211 x in Finset.range n, snorm (f x) p \u03bc \u2264 \u2191C", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nhp : 1 \u2264 p\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf\u2081 : \u2200 (i : \u2115), AEStronglyMeasurable (f i) \u03bc\nhf\u2082 : UnifIntegrable f p \u03bc\nC : \u211d\u22650\nhC : \u2200 (i : \u2115), snorm (f i) p \u03bc \u2264 \u2191C\nn : \u2115\nhn : \u2191(\u2191n)\u207b\u00b9 = 0\n\u22a2 \u2191(\u2191n)\u207b\u00b9 * \u2211 x in Finset.range n, snorm (f x) p \u03bc \u2264 \u2191C\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nhp : 1 \u2264 p\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf\u2081 : \u2200 (i : \u2115), AEStronglyMeasurable (f i) \u03bc\nhf\u2082 : UnifIntegrable f p \u03bc\nC : \u211d\u22650\nhC : \u2200 (i : \u2115), snorm (f i) p \u03bc \u2264 \u2191C\nn : \u2115\nhn : \u00ac\u2191(\u2191n)\u207b\u00b9 = 0\n\u22a2 \u2191(\u2191n)\u207b\u00b9 * \u2211 x in Finset.range n, snorm (f x) p \u03bc \u2264 \u2191C"}, {"tactic": "refine' le_trans _ (_ : \u2191(\u2191n : \u211d\u22650)\u207b\u00b9 * (n \u2022 C : \u211d\u22650\u221e) \u2264 C)", "annotated_tactic": ["refine' <a>le_trans</a> _ (_ : \u2191(\u2191n : \u211d\u22650)\u207b\u00b9 * (n \u2022 C : \u211d\u22650\u221e) \u2264 C)", [{"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nhp : 1 \u2264 p\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf\u2081 : \u2200 (i : \u2115), AEStronglyMeasurable (f i) \u03bc\nhf\u2082 : UnifIntegrable f p \u03bc\nC : \u211d\u22650\nhC : \u2200 (i : \u2115), snorm (f i) p \u03bc \u2264 \u2191C\nn : \u2115\nhn : \u00ac\u2191(\u2191n)\u207b\u00b9 = 0\n\u22a2 \u2191(\u2191n)\u207b\u00b9 * \u2211 x in Finset.range n, snorm (f x) p \u03bc \u2264 \u2191C", "state_after": "case neg.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nhp : 1 \u2264 p\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf\u2081 : \u2200 (i : \u2115), AEStronglyMeasurable (f i) \u03bc\nhf\u2082 : UnifIntegrable f p \u03bc\nC : \u211d\u22650\nhC : \u2200 (i : \u2115), snorm (f i) p \u03bc \u2264 \u2191C\nn : \u2115\nhn : \u00ac\u2191(\u2191n)\u207b\u00b9 = 0\n\u22a2 \u2191(\u2191n)\u207b\u00b9 * \u2211 x in Finset.range n, snorm (f x) p \u03bc \u2264 \u2191(\u2191n)\u207b\u00b9 * \u2191(n \u2022 C)\n\ncase neg.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nhp : 1 \u2264 p\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf\u2081 : \u2200 (i : \u2115), AEStronglyMeasurable (f i) \u03bc\nhf\u2082 : UnifIntegrable f p \u03bc\nC : \u211d\u22650\nhC : \u2200 (i : \u2115), snorm (f i) p \u03bc \u2264 \u2191C\nn : \u2115\nhn : \u00ac\u2191(\u2191n)\u207b\u00b9 = 0\n\u22a2 \u2191(\u2191n)\u207b\u00b9 * \u2191(n \u2022 C) \u2264 \u2191C"}, {"tactic": "simp only [hn, zero_mul, zero_le]", "annotated_tactic": ["simp only [hn, <a>zero_mul</a>, <a>zero_le</a>]", [{"full_name": "MulZeroClass.zero_mul", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [36, 3], "def_end_pos": [36, 11]}, {"full_name": "zero_le", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [217, 30], "def_end_pos": [217, 37]}]], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nhp : 1 \u2264 p\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf\u2081 : \u2200 (i : \u2115), AEStronglyMeasurable (f i) \u03bc\nhf\u2082 : UnifIntegrable f p \u03bc\nC : \u211d\u22650\nhC : \u2200 (i : \u2115), snorm (f i) p \u03bc \u2264 \u2191C\nn : \u2115\nhn : \u2191(\u2191n)\u207b\u00b9 = 0\n\u22a2 \u2191(\u2191n)\u207b\u00b9 * \u2211 x in Finset.range n, snorm (f x) p \u03bc \u2264 \u2191C", "state_after": "no goals"}, {"tactic": "refine' (ENNReal.mul_le_mul_left hn ENNReal.coe_ne_top).2 _", "annotated_tactic": ["refine' (<a>ENNReal.mul_le_mul_left</a> hn <a>ENNReal.coe_ne_top</a>).2 _", [{"full_name": "ENNReal.mul_le_mul_left", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1065, 9], "def_end_pos": [1065, 24]}, {"full_name": "ENNReal.coe_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [302, 17], "def_end_pos": [302, 27]}]], "state_before": "case neg.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nhp : 1 \u2264 p\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf\u2081 : \u2200 (i : \u2115), AEStronglyMeasurable (f i) \u03bc\nhf\u2082 : UnifIntegrable f p \u03bc\nC : \u211d\u22650\nhC : \u2200 (i : \u2115), snorm (f i) p \u03bc \u2264 \u2191C\nn : \u2115\nhn : \u00ac\u2191(\u2191n)\u207b\u00b9 = 0\n\u22a2 \u2191(\u2191n)\u207b\u00b9 * \u2211 x in Finset.range n, snorm (f x) p \u03bc \u2264 \u2191(\u2191n)\u207b\u00b9 * \u2191(n \u2022 C)", "state_after": "case neg.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nhp : 1 \u2264 p\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf\u2081 : \u2200 (i : \u2115), AEStronglyMeasurable (f i) \u03bc\nhf\u2082 : UnifIntegrable f p \u03bc\nC : \u211d\u22650\nhC : \u2200 (i : \u2115), snorm (f i) p \u03bc \u2264 \u2191C\nn : \u2115\nhn : \u00ac\u2191(\u2191n)\u207b\u00b9 = 0\n\u22a2 \u2211 x in Finset.range n, snorm (f x) p \u03bc \u2264 \u2191(n \u2022 C)"}, {"tactic": "conv_rhs => rw [\u2190 Finset.card_range n]", "annotated_tactic": ["conv_rhs => rw [\u2190 <a>Finset.card_range</a> n]", [{"full_name": "Finset.card_range", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [177, 9], "def_end_pos": [177, 19]}]], "state_before": "case neg.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nhp : 1 \u2264 p\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf\u2081 : \u2200 (i : \u2115), AEStronglyMeasurable (f i) \u03bc\nhf\u2082 : UnifIntegrable f p \u03bc\nC : \u211d\u22650\nhC : \u2200 (i : \u2115), snorm (f i) p \u03bc \u2264 \u2191C\nn : \u2115\nhn : \u00ac\u2191(\u2191n)\u207b\u00b9 = 0\n\u22a2 \u2211 x in Finset.range n, snorm (f x) p \u03bc \u2264 \u2191(n \u2022 C)", "state_after": "case neg.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nhp : 1 \u2264 p\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf\u2081 : \u2200 (i : \u2115), AEStronglyMeasurable (f i) \u03bc\nhf\u2082 : UnifIntegrable f p \u03bc\nC : \u211d\u22650\nhC : \u2200 (i : \u2115), snorm (f i) p \u03bc \u2264 \u2191C\nn : \u2115\nhn : \u00ac\u2191(\u2191n)\u207b\u00b9 = 0\n\u22a2 \u2211 x in Finset.range n, snorm (f x) p \u03bc \u2264 \u2191(Finset.card (Finset.range n) \u2022 C)"}, {"tactic": "convert Finset.sum_le_card_nsmul _ _ _ fun i _ => hC i", "annotated_tactic": ["convert <a>Finset.sum_le_card_nsmul</a> _ _ _ fun i _ => hC i", [{"full_name": "Finset.sum_le_card_nsmul", "def_path": "Mathlib/Algebra/BigOperators/Order.lean", "def_pos": [211, 15], "def_end_pos": [211, 32]}]], "state_before": "case neg.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nhp : 1 \u2264 p\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf\u2081 : \u2200 (i : \u2115), AEStronglyMeasurable (f i) \u03bc\nhf\u2082 : UnifIntegrable f p \u03bc\nC : \u211d\u22650\nhC : \u2200 (i : \u2115), snorm (f i) p \u03bc \u2264 \u2191C\nn : \u2115\nhn : \u00ac\u2191(\u2191n)\u207b\u00b9 = 0\n\u22a2 \u2211 x in Finset.range n, snorm (f x) p \u03bc \u2264 \u2191(Finset.card (Finset.range n) \u2022 C)", "state_after": "case h.e'_4\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nhp : 1 \u2264 p\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf\u2081 : \u2200 (i : \u2115), AEStronglyMeasurable (f i) \u03bc\nhf\u2082 : UnifIntegrable f p \u03bc\nC : \u211d\u22650\nhC : \u2200 (i : \u2115), snorm (f i) p \u03bc \u2264 \u2191C\nn : \u2115\nhn : \u00ac\u2191(\u2191n)\u207b\u00b9 = 0\n\u22a2 \u2191(Finset.card (Finset.range n) \u2022 C) = Finset.card (Finset.range n) \u2022 \u2191C"}, {"tactic": "rw [ENNReal.coe_smul]", "annotated_tactic": ["rw [<a>ENNReal.coe_smul</a>]", [{"full_name": "ENNReal.coe_smul", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [539, 9], "def_end_pos": [539, 17]}]], "state_before": "case h.e'_4\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nhp : 1 \u2264 p\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf\u2081 : \u2200 (i : \u2115), AEStronglyMeasurable (f i) \u03bc\nhf\u2082 : UnifIntegrable f p \u03bc\nC : \u211d\u22650\nhC : \u2200 (i : \u2115), snorm (f i) p \u03bc \u2264 \u2191C\nn : \u2115\nhn : \u00ac\u2191(\u2191n)\u207b\u00b9 = 0\n\u22a2 \u2191(Finset.card (Finset.range n) \u2022 C) = Finset.card (Finset.range n) \u2022 \u2191C", "state_after": "no goals"}, {"tactic": "simp only [ENNReal.coe_eq_zero, inv_eq_zero, Nat.cast_eq_zero] at hn", "annotated_tactic": ["simp only [<a>ENNReal.coe_eq_zero</a>, <a>inv_eq_zero</a>, <a>Nat.cast_eq_zero</a>] at hn", [{"full_name": "ENNReal.coe_eq_zero", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [368, 28], "def_end_pos": [368, 39]}, {"full_name": "inv_eq_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Basic.lean", "def_pos": [355, 9], "def_end_pos": [355, 20]}, {"full_name": "Nat.cast_eq_zero", "def_path": "Mathlib/Algebra/CharZero/Defs.lean", "def_pos": [80, 9], "def_end_pos": [80, 21]}]], "state_before": "case neg.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nhp : 1 \u2264 p\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf\u2081 : \u2200 (i : \u2115), AEStronglyMeasurable (f i) \u03bc\nhf\u2082 : UnifIntegrable f p \u03bc\nC : \u211d\u22650\nhC : \u2200 (i : \u2115), snorm (f i) p \u03bc \u2264 \u2191C\nn : \u2115\nhn : \u00ac\u2191(\u2191n)\u207b\u00b9 = 0\n\u22a2 \u2191(\u2191n)\u207b\u00b9 * \u2191(n \u2022 C) \u2264 \u2191C", "state_after": "case neg.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nhp : 1 \u2264 p\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf\u2081 : \u2200 (i : \u2115), AEStronglyMeasurable (f i) \u03bc\nhf\u2082 : UnifIntegrable f p \u03bc\nC : \u211d\u22650\nhC : \u2200 (i : \u2115), snorm (f i) p \u03bc \u2264 \u2191C\nn : \u2115\nhn : \u00acn = 0\n\u22a2 \u2191(\u2191n)\u207b\u00b9 * \u2191(n \u2022 C) \u2264 \u2191C"}, {"tactic": "rw [ENNReal.coe_smul, nsmul_eq_mul, \u2190 mul_assoc, ENNReal.coe_inv, ENNReal.coe_nat,\n  ENNReal.inv_mul_cancel _ (ENNReal.nat_ne_top _), one_mul]", "annotated_tactic": ["rw [<a>ENNReal.coe_smul</a>, <a>nsmul_eq_mul</a>, \u2190 <a>mul_assoc</a>, <a>ENNReal.coe_inv</a>, <a>ENNReal.coe_nat</a>,\n        <a>ENNReal.inv_mul_cancel</a> _ (<a>ENNReal.nat_ne_top</a> _), <a>one_mul</a>]", [{"full_name": "ENNReal.coe_smul", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [539, 9], "def_end_pos": [539, 17]}, {"full_name": "nsmul_eq_mul", "def_path": "Mathlib/Algebra/GroupPower/Lemmas.lean", "def_pos": [509, 9], "def_end_pos": [509, 21]}, {"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [264, 9], "def_end_pos": [264, 18]}, {"full_name": "ENNReal.coe_inv", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1387, 9], "def_end_pos": [1387, 16]}, {"full_name": "ENNReal.coe_nat", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [707, 9], "def_end_pos": [707, 16]}, {"full_name": "ENNReal.inv_mul_cancel", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1424, 19], "def_end_pos": [1424, 33]}, {"full_name": "ENNReal.nat_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [717, 17], "def_end_pos": [717, 27]}, {"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [464, 9], "def_end_pos": [464, 16]}]], "state_before": "case neg.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nhp : 1 \u2264 p\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf\u2081 : \u2200 (i : \u2115), AEStronglyMeasurable (f i) \u03bc\nhf\u2082 : UnifIntegrable f p \u03bc\nC : \u211d\u22650\nhC : \u2200 (i : \u2115), snorm (f i) p \u03bc \u2264 \u2191C\nn : \u2115\nhn : \u00acn = 0\n\u22a2 \u2191(\u2191n)\u207b\u00b9 * \u2191(n \u2022 C) \u2264 \u2191C", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nhp : 1 \u2264 p\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf\u2081 : \u2200 (i : \u2115), AEStronglyMeasurable (f i) \u03bc\nhf\u2082 : UnifIntegrable f p \u03bc\nC : \u211d\u22650\nhC : \u2200 (i : \u2115), snorm (f i) p \u03bc \u2264 \u2191C\nn : \u2115\nhn : \u00acn = 0\n\u22a2 \u2191n \u2260 0\n\ncase neg.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nhp : 1 \u2264 p\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf\u2081 : \u2200 (i : \u2115), AEStronglyMeasurable (f i) \u03bc\nhf\u2082 : UnifIntegrable f p \u03bc\nC : \u211d\u22650\nhC : \u2200 (i : \u2115), snorm (f i) p \u03bc \u2264 \u2191C\nn : \u2115\nhn : \u00acn = 0\n\u22a2 \u2191n \u2260 0"}, {"tactic": "all_goals simpa only [Ne.def, Nat.cast_eq_zero]", "annotated_tactic": ["all_goals simpa only [<a>Ne.def</a>, <a>Nat.cast_eq_zero</a>]", [{"full_name": "Ne.def", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [59, 9], "def_end_pos": [59, 15]}, {"full_name": "Nat.cast_eq_zero", "def_path": "Mathlib/Algebra/CharZero/Defs.lean", "def_pos": [80, 9], "def_end_pos": [80, 21]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nhp : 1 \u2264 p\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf\u2081 : \u2200 (i : \u2115), AEStronglyMeasurable (f i) \u03bc\nhf\u2082 : UnifIntegrable f p \u03bc\nC : \u211d\u22650\nhC : \u2200 (i : \u2115), snorm (f i) p \u03bc \u2264 \u2191C\nn : \u2115\nhn : \u00acn = 0\n\u22a2 \u2191n \u2260 0\n\ncase neg.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nhp : 1 \u2264 p\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf\u2081 : \u2200 (i : \u2115), AEStronglyMeasurable (f i) \u03bc\nhf\u2082 : UnifIntegrable f p \u03bc\nC : \u211d\u22650\nhC : \u2200 (i : \u2115), snorm (f i) p \u03bc \u2264 \u2191C\nn : \u2115\nhn : \u00acn = 0\n\u22a2 \u2191n \u2260 0", "state_after": "no goals"}, {"tactic": "simpa only [Ne.def, Nat.cast_eq_zero]", "annotated_tactic": ["simpa only [<a>Ne.def</a>, <a>Nat.cast_eq_zero</a>]", [{"full_name": "Ne.def", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [59, 9], "def_end_pos": [59, 15]}, {"full_name": "Nat.cast_eq_zero", "def_path": "Mathlib/Algebra/CharZero/Defs.lean", "def_pos": [80, 9], "def_end_pos": [80, 21]}]], "state_before": "case neg.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nhp : 1 \u2264 p\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf\u2081 : \u2200 (i : \u2115), AEStronglyMeasurable (f i) \u03bc\nhf\u2082 : UnifIntegrable f p \u03bc\nC : \u211d\u22650\nhC : \u2200 (i : \u2115), snorm (f i) p \u03bc \u2264 \u2191C\nn : \u2115\nhn : \u00acn = 0\n\u22a2 \u2191n \u2260 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/Rename.lean", "full_name": "MvPolynomial.coeff_rename_ne_zero", "start": [319, 1], "end": [322, 37], "traced_tactics": [{"tactic": "contrapose! h", "annotated_tactic": ["contrapose! h", []], "state_before": "\u03c3 : Type u_1\n\u03c4 : Type u_2\n\u03b1 : Type u_3\nR : Type u_4\nS : Type u_5\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : CommSemiring S\nf : \u03c3 \u2192 \u03c4\n\u03c6 : MvPolynomial \u03c3 R\nd : \u03c4 \u2192\u2080 \u2115\nh : coeff d (\u2191(rename f) \u03c6) \u2260 0\n\u22a2 \u2203 u, Finsupp.mapDomain f u = d \u2227 coeff u \u03c6 \u2260 0", "state_after": "\u03c3 : Type u_1\n\u03c4 : Type u_2\n\u03b1 : Type u_3\nR : Type u_4\nS : Type u_5\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : CommSemiring S\nf : \u03c3 \u2192 \u03c4\n\u03c6 : MvPolynomial \u03c3 R\nd : \u03c4 \u2192\u2080 \u2115\nh : \u2200 (u : \u03c3 \u2192\u2080 \u2115), Finsupp.mapDomain f u = d \u2192 coeff u \u03c6 = 0\n\u22a2 coeff d (\u2191(rename f) \u03c6) = 0"}, {"tactic": "apply coeff_rename_eq_zero _ _ _ h", "annotated_tactic": ["apply <a>coeff_rename_eq_zero</a> _ _ _ h", [{"full_name": "MvPolynomial.coeff_rename_eq_zero", "def_path": "Mathlib/Data/MvPolynomial/Rename.lean", "def_pos": [306, 9], "def_end_pos": [306, 29]}]], "state_before": "\u03c3 : Type u_1\n\u03c4 : Type u_2\n\u03b1 : Type u_3\nR : Type u_4\nS : Type u_5\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : CommSemiring S\nf : \u03c3 \u2192 \u03c4\n\u03c6 : MvPolynomial \u03c3 R\nd : \u03c4 \u2192\u2080 \u2115\nh : \u2200 (u : \u03c3 \u2192\u2080 \u2115), Finsupp.mapDomain f u = d \u2192 coeff u \u03c6 = 0\n\u22a2 coeff d (\u2191(rename f) \u03c6) = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/DList/Defs.lean", "full_name": "Std.DList.toList_append", "start": [75, 1], "end": [77, 70], "traced_tactics": [{"tactic": "cases' l\u2081 with _ l\u2081_invariant", "annotated_tactic": ["cases' l\u2081 with _ l\u2081_invariant", []], "state_before": "\u03b1 : Type u\nl\u2081 l\u2082 : DList \u03b1\n\u22a2 toList (append l\u2081 l\u2082) = toList l\u2081 ++ toList l\u2082", "state_after": "case mk\n\u03b1 : Type u\nl\u2082 : DList \u03b1\napply\u271d : List \u03b1 \u2192 List \u03b1\nl\u2081_invariant : \u2200 (l : List \u03b1), apply\u271d l = apply\u271d [] ++ l\n\u22a2 toList (append { apply := apply\u271d, invariant := l\u2081_invariant } l\u2082) =\n    toList { apply := apply\u271d, invariant := l\u2081_invariant } ++ toList l\u2082"}, {"tactic": "cases' l\u2082", "annotated_tactic": ["cases' l\u2082", []], "state_before": "case mk\n\u03b1 : Type u\nl\u2082 : DList \u03b1\napply\u271d : List \u03b1 \u2192 List \u03b1\nl\u2081_invariant : \u2200 (l : List \u03b1), apply\u271d l = apply\u271d [] ++ l\n\u22a2 toList (append { apply := apply\u271d, invariant := l\u2081_invariant } l\u2082) =\n    toList { apply := apply\u271d, invariant := l\u2081_invariant } ++ toList l\u2082", "state_after": "case mk.mk\n\u03b1 : Type u\napply\u271d\u00b9 : List \u03b1 \u2192 List \u03b1\nl\u2081_invariant : \u2200 (l : List \u03b1), apply\u271d\u00b9 l = apply\u271d\u00b9 [] ++ l\napply\u271d : List \u03b1 \u2192 List \u03b1\ninvariant\u271d : \u2200 (l : List \u03b1), apply\u271d l = apply\u271d [] ++ l\n\u22a2 toList (append { apply := apply\u271d\u00b9, invariant := l\u2081_invariant } { apply := apply\u271d, invariant := invariant\u271d }) =\n    toList { apply := apply\u271d\u00b9, invariant := l\u2081_invariant } ++ toList { apply := apply\u271d, invariant := invariant\u271d }"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case mk.mk\n\u03b1 : Type u\napply\u271d\u00b9 : List \u03b1 \u2192 List \u03b1\nl\u2081_invariant : \u2200 (l : List \u03b1), apply\u271d\u00b9 l = apply\u271d\u00b9 [] ++ l\napply\u271d : List \u03b1 \u2192 List \u03b1\ninvariant\u271d : \u2200 (l : List \u03b1), apply\u271d l = apply\u271d [] ++ l\n\u22a2 toList (append { apply := apply\u271d\u00b9, invariant := l\u2081_invariant } { apply := apply\u271d, invariant := invariant\u271d }) =\n    toList { apply := apply\u271d\u00b9, invariant := l\u2081_invariant } ++ toList { apply := apply\u271d, invariant := invariant\u271d }", "state_after": "case mk.mk\n\u03b1 : Type u\napply\u271d\u00b9 : List \u03b1 \u2192 List \u03b1\nl\u2081_invariant : \u2200 (l : List \u03b1), apply\u271d\u00b9 l = apply\u271d\u00b9 [] ++ l\napply\u271d : List \u03b1 \u2192 List \u03b1\ninvariant\u271d : \u2200 (l : List \u03b1), apply\u271d l = apply\u271d [] ++ l\n\u22a2 apply\u271d\u00b9 (apply\u271d []) = apply\u271d\u00b9 [] ++ apply\u271d []"}, {"tactic": "rw [l\u2081_invariant]", "annotated_tactic": ["rw [l\u2081_invariant]", []], "state_before": "case mk.mk\n\u03b1 : Type u\napply\u271d\u00b9 : List \u03b1 \u2192 List \u03b1\nl\u2081_invariant : \u2200 (l : List \u03b1), apply\u271d\u00b9 l = apply\u271d\u00b9 [] ++ l\napply\u271d : List \u03b1 \u2192 List \u03b1\ninvariant\u271d : \u2200 (l : List \u03b1), apply\u271d l = apply\u271d [] ++ l\n\u22a2 apply\u271d\u00b9 (apply\u271d []) = apply\u271d\u00b9 [] ++ apply\u271d []", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "full_name": "AEMeasurable.edist", "start": [1831, 1], "end": [1833, 46], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/LocallyIntegrable.lean", "full_name": "MeasureTheory.LocallyIntegrable.smul", "start": [316, 11], "end": [318, 57], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Basic.lean", "full_name": "Set.nontrivial_iff_exists_lt", "start": [2541, 1], "end": [2543, 50], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Finite.lean", "full_name": "Set.Finite.toFinset_insert'", "start": [1074, 1], "end": [1076, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "full_name": "Int.neg_add_le_right_of_le_add", "start": [1013, 11], "end": [1015, 40], "traced_tactics": [{"tactic": "rw [Int.add_comm] at h", "annotated_tactic": ["rw [<a>Int.add_comm</a>] at h", [{"full_name": "Int.add_comm", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [218, 19], "def_end_pos": [218, 27]}]], "state_before": "a b c : Int\nh : a \u2264 b + c\n\u22a2 -c + a \u2264 b", "state_after": "a b c : Int\nh : a \u2264 c + b\n\u22a2 -c + a \u2264 b"}, {"tactic": "exact Int.neg_add_le_left_of_le_add h", "annotated_tactic": ["exact <a>Int.neg_add_le_left_of_le_add</a> h", [{"full_name": "Int.neg_add_le_left_of_le_add", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [1005, 19], "def_end_pos": [1005, 44]}]], "state_before": "a b c : Int\nh : a \u2264 c + b\n\u22a2 -c + a \u2264 b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Regular.lean", "full_name": "Set.exists_isOpen_lt_of_lt", "start": [242, 1], "end": [247, 58], "traced_tactics": [{"tactic": "rcases OuterRegular.outerRegular (measurableSet_toMeasurable \u03bc A) r\n    (by rwa [measure_toMeasurable]) with\n  \u27e8U, hAU, hUo, hU\u27e9", "annotated_tactic": ["rcases <a>OuterRegular.outerRegular</a> (<a>measurableSet_toMeasurable</a> \u03bc A) r\n      (by rwa [<a>measure_toMeasurable</a>]) with\n    \u27e8U, hAU, hUo, hU\u27e9", [{"full_name": "MeasureTheory.Measure.OuterRegular.outerRegular", "def_path": "Mathlib/MeasureTheory/Measure/Regular.lean", "def_pos": [203, 13], "def_end_pos": [203, 25]}, {"full_name": "MeasureTheory.measurableSet_toMeasurable", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [645, 9], "def_end_pos": [645, 35]}, {"full_name": "MeasureTheory.measure_toMeasurable", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [653, 9], "def_end_pos": [653, 29]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : OuterRegular \u03bc\nA : Set \u03b1\nr : \u211d\u22650\u221e\nhr : \u2191\u2191\u03bc A < r\n\u22a2 \u2203 U, U \u2287 A \u2227 IsOpen U \u2227 \u2191\u2191\u03bc U < r", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : OuterRegular \u03bc\nA : Set \u03b1\nr : \u211d\u22650\u221e\nhr : \u2191\u2191\u03bc A < r\nU : Set \u03b1\nhAU : U \u2287 toMeasurable \u03bc A\nhUo : IsOpen U\nhU : \u2191\u2191\u03bc U < r\n\u22a2 \u2203 U, U \u2287 A \u2227 IsOpen U \u2227 \u2191\u2191\u03bc U < r"}, {"tactic": "exact \u27e8U, (subset_toMeasurable _ _).trans hAU, hUo, hU\u27e9", "annotated_tactic": ["exact \u27e8U, (<a>subset_toMeasurable</a> _ _).<a>trans</a> hAU, hUo, hU\u27e9", [{"full_name": "MeasureTheory.subset_toMeasurable", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [633, 9], "def_end_pos": [633, 28]}, {"full_name": "HasSubset.Subset.trans", "def_path": "Mathlib/Order/RelClasses.lean", "def_pos": [664, 7], "def_end_pos": [664, 29]}]], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : OuterRegular \u03bc\nA : Set \u03b1\nr : \u211d\u22650\u221e\nhr : \u2191\u2191\u03bc A < r\nU : Set \u03b1\nhAU : U \u2287 toMeasurable \u03bc A\nhUo : IsOpen U\nhU : \u2191\u2191\u03bc U < r\n\u22a2 \u2203 U, U \u2287 A \u2227 IsOpen U \u2227 \u2191\u2191\u03bc U < r", "state_after": "no goals"}, {"tactic": "rwa [measure_toMeasurable]", "annotated_tactic": ["rwa [<a>measure_toMeasurable</a>]", [{"full_name": "MeasureTheory.measure_toMeasurable", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [653, 9], "def_end_pos": [653, 29]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : OuterRegular \u03bc\nA : Set \u03b1\nr : \u211d\u22650\u221e\nhr : \u2191\u2191\u03bc A < r\n\u22a2 r > \u2191\u2191?m.8585 (toMeasurable \u03bc A)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Kernel/CondCdf.lean", "full_name": "ProbabilityTheory.set_lintegral_condCdf_rat", "start": [865, 1], "end": [871, 28], "traced_tactics": [{"tactic": "have : \u2200\u1d50 a \u2202\u03c1.fst, a \u2208 s \u2192 ENNReal.ofReal (condCdf \u03c1 a r) = preCdf \u03c1 r a := by\n  filter_upwards [ofReal_condCdf_ae_eq \u03c1 r] with a ha using fun _ => ha", "annotated_tactic": ["have : \u2200\u1d50 a \u2202\u03c1.fst, a \u2208 s \u2192 <a>ENNReal.ofReal</a> (<a>condCdf</a> \u03c1 a r) = <a>preCdf</a> \u03c1 r a := by\n    filter_upwards [<a>ofReal_condCdf_ae_eq</a> \u03c1 r] with a ha using fun _ => ha", [{"full_name": "ENNReal.ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [172, 29], "def_end_pos": [172, 35]}, {"full_name": "ProbabilityTheory.condCdf", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [777, 19], "def_end_pos": [777, 26]}, {"full_name": "ProbabilityTheory.preCdf", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [294, 19], "def_end_pos": [294, 25]}, {"full_name": "ProbabilityTheory.ofReal_condCdf_ae_eq", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [846, 9], "def_end_pos": [846, 29]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nr : \u211a\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) \u2191r) \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191r)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nr : \u211a\ns : Set \u03b1\nhs : MeasurableSet s\nthis : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 s \u2192 ENNReal.ofReal (\u2191(condCdf \u03c1 a) \u2191r) = preCdf \u03c1 r a\n\u22a2 \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) \u2191r) \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191r)"}, {"tactic": "rw [set_lintegral_congr_fun hs this, set_lintegral_preCdf_fst \u03c1 r hs]", "annotated_tactic": ["rw [<a>set_lintegral_congr_fun</a> hs this, <a>set_lintegral_preCdf_fst</a> \u03c1 r hs]", [{"full_name": "MeasureTheory.set_lintegral_congr_fun", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [316, 9], "def_end_pos": [316, 32]}, {"full_name": "ProbabilityTheory.set_lintegral_preCdf_fst", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [307, 9], "def_end_pos": [307, 33]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nr : \u211a\ns : Set \u03b1\nhs : MeasurableSet s\nthis : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 s \u2192 ENNReal.ofReal (\u2191(condCdf \u03c1 a) \u2191r) = preCdf \u03c1 r a\n\u22a2 \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) \u2191r) \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191r)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nr : \u211a\ns : Set \u03b1\nhs : MeasurableSet s\nthis : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 s \u2192 ENNReal.ofReal (\u2191(condCdf \u03c1 a) \u2191r) = preCdf \u03c1 r a\n\u22a2 \u2191\u2191(Measure.IicSnd \u03c1 \u2191r) s = \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191r)"}, {"tactic": "exact \u03c1.IicSnd_apply r hs", "annotated_tactic": ["exact \u03c1.IicSnd_apply r hs", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nr : \u211a\ns : Set \u03b1\nhs : MeasurableSet s\nthis : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 s \u2192 ENNReal.ofReal (\u2191(condCdf \u03c1 a) \u2191r) = preCdf \u03c1 r a\n\u22a2 \u2191\u2191(Measure.IicSnd \u03c1 \u2191r) s = \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191r)", "state_after": "no goals"}, {"tactic": "filter_upwards [ofReal_condCdf_ae_eq \u03c1 r] with a ha using fun _ => ha", "annotated_tactic": ["filter_upwards [<a>ofReal_condCdf_ae_eq</a> \u03c1 r] with a ha using fun _ => ha", [{"full_name": "ProbabilityTheory.ofReal_condCdf_ae_eq", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [846, 9], "def_end_pos": [846, 29]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nr : \u211a\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 s \u2192 ENNReal.ofReal (\u2191(condCdf \u03c1 a) \u2191r) = preCdf \u03c1 r a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "full_name": "List.mem_range", "start": [2087, 1], "end": [2088, 81], "traced_tactics": [{"tactic": "simp only [range_eq_range', mem_range'_1, Nat.zero_le, true_and, Nat.zero_add]", "annotated_tactic": ["simp only [<a>range_eq_range'</a>, <a>mem_range'_1</a>, <a>Nat.zero_le</a>, <a>true_and</a>, <a>Nat.zero_add</a>]", [{"full_name": "List.range_eq_range'", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [2064, 9], "def_end_pos": [2064, 24]}, {"full_name": "List.mem_range'_1", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [1997, 17], "def_end_pos": [1997, 29]}, {"full_name": "Nat.zero_le", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1578, 9], "def_end_pos": [1578, 20]}, {"full_name": "true_and", "def_path": "lake-packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [84, 17], "def_end_pos": [84, 25]}, {"full_name": "Nat.zero_add", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [114, 27], "def_end_pos": [114, 35]}]], "state_before": "m n : Nat\n\u22a2 m \u2208 range n \u2194 m < n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Countable.lean", "full_name": "Set.countable_pi", "start": [273, 1], "end": [275, 56], "traced_tactics": [{"tactic": "simpa only [\u2190 mem_univ_pi] using countable_univ_pi hs", "annotated_tactic": ["simpa only [\u2190 <a>mem_univ_pi</a>] using <a>countable_univ_pi</a> hs", [{"full_name": "Set.mem_univ_pi", "def_path": "Mathlib/Data/Set/Prod.lean", "def_pos": [675, 9], "def_end_pos": [675, 20]}, {"full_name": "Set.countable_univ_pi", "def_path": "Mathlib/Data/Set/Countable.lean", "def_pos": [267, 9], "def_end_pos": [267, 26]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\n\u03c0 : \u03b1 \u2192 Type u_1\ninst\u271d : Finite \u03b1\ns : (a : \u03b1) \u2192 Set (\u03c0 a)\nhs : \u2200 (a : \u03b1), Set.Countable (s a)\n\u22a2 Set.Countable {f | \u2200 (a : \u03b1), f a \u2208 s a}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Count.lean", "full_name": "MeasureTheory.Measure.count_apply_infinite", "start": [74, 1], "end": [81, 35], "traced_tactics": [{"tactic": "refine' top_unique (le_of_tendsto' ENNReal.tendsto_nat_nhds_top fun n => _)", "annotated_tactic": ["refine' <a>top_unique</a> (<a>le_of_tendsto'</a> <a>ENNReal.tendsto_nat_nhds_top</a> fun n => _)", [{"full_name": "top_unique", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [161, 9], "def_end_pos": [161, 19]}, {"full_name": "le_of_tendsto'", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [145, 9], "def_end_pos": [145, 23]}, {"full_name": "ENNReal.tendsto_nat_nhds_top", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [178, 9], "def_end_pos": [178, 29]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.8763\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\ns : Set \u03b1\nhs : Set.Infinite s\n\u22a2 \u2191\u2191count s = \u22a4", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type ?u.8763\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\ns : Set \u03b1\nhs : Set.Infinite s\nn : \u2115\n\u22a2 \u2191n \u2264 \u2191\u2191count s"}, {"tactic": "rcases hs.exists_subset_card_eq n with \u27e8t, ht, rfl\u27e9", "annotated_tactic": ["rcases hs.exists_subset_card_eq n with \u27e8t, ht, rfl\u27e9", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.8763\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\ns : Set \u03b1\nhs : Set.Infinite s\nn : \u2115\n\u22a2 \u2191n \u2264 \u2191\u2191count s", "state_after": "case intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type ?u.8763\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\ns : Set \u03b1\nhs : Set.Infinite s\nt : Finset \u03b1\nht : \u2191t \u2286 s\n\u22a2 \u2191(Finset.card t) \u2264 \u2191\u2191count s"}, {"tactic": "calc\n  (t.card : \u211d\u22650\u221e) = \u2211 i in t, 1 := by simp\n  _ = \u2211' i : (t : Set \u03b1), 1 := (t.tsum_subtype 1).symm\n  _ \u2264 count (t : Set \u03b1) := le_count_apply\n  _ \u2264 count s := measure_mono ht", "annotated_tactic": ["calc\n    (t.card : \u211d\u22650\u221e) = \u2211 i in t, 1 := by simp\n    _ = \u2211' i : (t : <a>Set</a> \u03b1), 1 := (t.tsum_subtype 1).<a>symm</a>\n    _ \u2264 <a>count</a> (t : <a>Set</a> \u03b1) := <a>le_count_apply</a>\n    _ \u2264 <a>count</a> s := <a>measure_mono</a> ht", [{"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}, {"full_name": "Eq.symm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [310, 9], "def_end_pos": [310, 16]}, {"full_name": "MeasureTheory.Measure.count", "def_path": "Mathlib/MeasureTheory/Measure/Count.lean", "def_pos": [28, 5], "def_end_pos": [28, 10]}, {"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}, {"full_name": "MeasureTheory.Measure.le_count_apply", "def_path": "Mathlib/MeasureTheory/Measure/Count.lean", "def_pos": [32, 9], "def_end_pos": [32, 23]}, {"full_name": "MeasureTheory.Measure.count", "def_path": "Mathlib/MeasureTheory/Measure/Count.lean", "def_pos": [28, 5], "def_end_pos": [28, 10]}, {"full_name": "MeasureTheory.measure_mono", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [193, 9], "def_end_pos": [193, 21]}]], "state_before": "case intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type ?u.8763\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\ns : Set \u03b1\nhs : Set.Infinite s\nt : Finset \u03b1\nht : \u2191t \u2286 s\n\u22a2 \u2191(Finset.card t) \u2264 \u2191\u2191count s", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.8763\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\ns : Set \u03b1\nhs : Set.Infinite s\nt : Finset \u03b1\nht : \u2191t \u2286 s\n\u22a2 \u2191(Finset.card t) = \u2211 i in t, 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "full_name": "MeasureTheory.exists_pos_set_lintegral_lt_of_measure_lt", "start": [467, 1], "end": [494, 44], "traced_tactics": [{"tactic": "rcases exists_between h\u03b5.bot_lt with \u27e8\u03b5\u2082, h\u03b5\u20820 : 0 < \u03b5\u2082, h\u03b5\u2082\u03b5\u27e9", "annotated_tactic": ["rcases <a>exists_between</a> h\u03b5.bot_lt with \u27e8\u03b5\u2082, h\u03b5\u20820 : 0 < \u03b5\u2082, h\u03b5\u2082\u03b5\u27e9", [{"full_name": "exists_between", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [1294, 9], "def_end_pos": [1294, 23]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u22a2 \u2203 \u03b4, \u03b4 > 0 \u2227 \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s < \u03b4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc < \u03b5", "state_after": "case intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03b5\u2082 : \u211d\u22650\u221e\nh\u03b5\u20820 : 0 < \u03b5\u2082\nh\u03b5\u2082\u03b5 : \u03b5\u2082 < \u03b5\n\u22a2 \u2203 \u03b4, \u03b4 > 0 \u2227 \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s < \u03b4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc < \u03b5"}, {"tactic": "rcases exists_between h\u03b5\u20820 with \u27e8\u03b5\u2081, h\u03b5\u20810, h\u03b5\u2081\u2082\u27e9", "annotated_tactic": ["rcases <a>exists_between</a> h\u03b5\u20820 with \u27e8\u03b5\u2081, h\u03b5\u20810, h\u03b5\u2081\u2082\u27e9", [{"full_name": "exists_between", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [1294, 9], "def_end_pos": [1294, 23]}]], "state_before": "case intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03b5\u2082 : \u211d\u22650\u221e\nh\u03b5\u20820 : 0 < \u03b5\u2082\nh\u03b5\u2082\u03b5 : \u03b5\u2082 < \u03b5\n\u22a2 \u2203 \u03b4, \u03b4 > 0 \u2227 \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s < \u03b4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc < \u03b5", "state_after": "case intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03b5\u2082 : \u211d\u22650\u221e\nh\u03b5\u20820 : 0 < \u03b5\u2082\nh\u03b5\u2082\u03b5 : \u03b5\u2082 < \u03b5\n\u03b5\u2081 : \u211d\u22650\u221e\nh\u03b5\u20810 : 0 < \u03b5\u2081\nh\u03b5\u2081\u2082 : \u03b5\u2081 < \u03b5\u2082\n\u22a2 \u2203 \u03b4, \u03b4 > 0 \u2227 \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s < \u03b4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc < \u03b5"}, {"tactic": "rcases exists_simpleFunc_forall_lintegral_sub_lt_of_pos h h\u03b5\u20810.ne' with \u27e8\u03c6, _, h\u03c6\u27e9", "annotated_tactic": ["rcases <a>exists_simpleFunc_forall_lintegral_sub_lt_of_pos</a> h h\u03b5\u20810.ne' with \u27e8\u03c6, _, h\u03c6\u27e9", [{"full_name": "MeasureTheory.exists_simpleFunc_forall_lintegral_sub_lt_of_pos", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [222, 9], "def_end_pos": [222, 57]}]], "state_before": "case intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03b5\u2082 : \u211d\u22650\u221e\nh\u03b5\u20820 : 0 < \u03b5\u2082\nh\u03b5\u2082\u03b5 : \u03b5\u2082 < \u03b5\n\u03b5\u2081 : \u211d\u22650\u221e\nh\u03b5\u20810 : 0 < \u03b5\u2081\nh\u03b5\u2081\u2082 : \u03b5\u2081 < \u03b5\u2082\n\u22a2 \u2203 \u03b4, \u03b4 > 0 \u2227 \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s < \u03b4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc < \u03b5", "state_after": "case intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03b5\u2082 : \u211d\u22650\u221e\nh\u03b5\u20820 : 0 < \u03b5\u2082\nh\u03b5\u2082\u03b5 : \u03b5\u2082 < \u03b5\n\u03b5\u2081 : \u211d\u22650\u221e\nh\u03b5\u20810 : 0 < \u03b5\u2081\nh\u03b5\u2081\u2082 : \u03b5\u2081 < \u03b5\u2082\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\nleft\u271d : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x\nh\u03c6 : \u2200 (\u03c8 : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (\u03c8 - \u03c6)) \u03bc < \u03b5\u2081\n\u22a2 \u2203 \u03b4, \u03b4 > 0 \u2227 \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s < \u03b4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc < \u03b5"}, {"tactic": "rcases \u03c6.exists_forall_le with \u27e8C, hC\u27e9", "annotated_tactic": ["rcases \u03c6.exists_forall_le with \u27e8C, hC\u27e9", []], "state_before": "case intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03b5\u2082 : \u211d\u22650\u221e\nh\u03b5\u20820 : 0 < \u03b5\u2082\nh\u03b5\u2082\u03b5 : \u03b5\u2082 < \u03b5\n\u03b5\u2081 : \u211d\u22650\u221e\nh\u03b5\u20810 : 0 < \u03b5\u2081\nh\u03b5\u2081\u2082 : \u03b5\u2081 < \u03b5\u2082\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\nleft\u271d : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x\nh\u03c6 : \u2200 (\u03c8 : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (\u03c8 - \u03c6)) \u03bc < \u03b5\u2081\n\u22a2 \u2203 \u03b4, \u03b4 > 0 \u2227 \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s < \u03b4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc < \u03b5", "state_after": "case intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03b5\u2082 : \u211d\u22650\u221e\nh\u03b5\u20820 : 0 < \u03b5\u2082\nh\u03b5\u2082\u03b5 : \u03b5\u2082 < \u03b5\n\u03b5\u2081 : \u211d\u22650\u221e\nh\u03b5\u20810 : 0 < \u03b5\u2081\nh\u03b5\u2081\u2082 : \u03b5\u2081 < \u03b5\u2082\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\nleft\u271d : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x\nh\u03c6 : \u2200 (\u03c8 : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (\u03c8 - \u03c6)) \u03bc < \u03b5\u2081\nC : \u211d\u22650\nhC : \u2200 (x : \u03b1), \u2191\u03c6 x \u2264 C\n\u22a2 \u2203 \u03b4, \u03b4 > 0 \u2227 \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s < \u03b4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc < \u03b5"}, {"tactic": "use (\u03b5\u2082 - \u03b5\u2081) / C, ENNReal.div_pos_iff.2 \u27e8(tsub_pos_iff_lt.2 h\u03b5\u2081\u2082).ne', ENNReal.coe_ne_top\u27e9", "annotated_tactic": ["use (\u03b5\u2082 - \u03b5\u2081) / C, <a>ENNReal.div_pos_iff</a>.2 \u27e8(<a>tsub_pos_iff_lt</a>.2 h\u03b5\u2081\u2082).<a>ne'</a>, <a>ENNReal.coe_ne_top</a>\u27e9", [{"full_name": "ENNReal.div_pos_iff", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1793, 17], "def_end_pos": [1793, 28]}, {"full_name": "tsub_pos_iff_lt", "def_path": "Mathlib/Algebra/Order/Sub/Canonical.lean", "def_pos": [420, 9], "def_end_pos": [420, 24]}, {"full_name": "LT.lt.ne'", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [328, 9], "def_end_pos": [328, 12]}, {"full_name": "ENNReal.coe_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [302, 17], "def_end_pos": [302, 27]}]], "state_before": "case intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03b5\u2082 : \u211d\u22650\u221e\nh\u03b5\u20820 : 0 < \u03b5\u2082\nh\u03b5\u2082\u03b5 : \u03b5\u2082 < \u03b5\n\u03b5\u2081 : \u211d\u22650\u221e\nh\u03b5\u20810 : 0 < \u03b5\u2081\nh\u03b5\u2081\u2082 : \u03b5\u2081 < \u03b5\u2082\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\nleft\u271d : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x\nh\u03c6 : \u2200 (\u03c8 : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (\u03c8 - \u03c6)) \u03bc < \u03b5\u2081\nC : \u211d\u22650\nhC : \u2200 (x : \u03b1), \u2191\u03c6 x \u2264 C\n\u22a2 \u2203 \u03b4, \u03b4 > 0 \u2227 \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s < \u03b4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc < \u03b5", "state_after": "case right\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03b5\u2082 : \u211d\u22650\u221e\nh\u03b5\u20820 : 0 < \u03b5\u2082\nh\u03b5\u2082\u03b5 : \u03b5\u2082 < \u03b5\n\u03b5\u2081 : \u211d\u22650\u221e\nh\u03b5\u20810 : 0 < \u03b5\u2081\nh\u03b5\u2081\u2082 : \u03b5\u2081 < \u03b5\u2082\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\nleft\u271d : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x\nh\u03c6 : \u2200 (\u03c8 : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (\u03c8 - \u03c6)) \u03bc < \u03b5\u2081\nC : \u211d\u22650\nhC : \u2200 (x : \u03b1), \u2191\u03c6 x \u2264 C\n\u22a2 \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s < (\u03b5\u2082 - \u03b5\u2081) / \u2191C \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc < \u03b5"}, {"tactic": "refine' fun s hs => lt_of_le_of_lt _ h\u03b5\u2082\u03b5", "annotated_tactic": ["refine' fun s hs => <a>lt_of_le_of_lt</a> _ h\u03b5\u2082\u03b5", [{"full_name": "lt_of_le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [122, 9], "def_end_pos": [122, 23]}]], "state_before": "case right\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03b5\u2082 : \u211d\u22650\u221e\nh\u03b5\u20820 : 0 < \u03b5\u2082\nh\u03b5\u2082\u03b5 : \u03b5\u2082 < \u03b5\n\u03b5\u2081 : \u211d\u22650\u221e\nh\u03b5\u20810 : 0 < \u03b5\u2081\nh\u03b5\u2081\u2082 : \u03b5\u2081 < \u03b5\u2082\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\nleft\u271d : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x\nh\u03c6 : \u2200 (\u03c8 : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (\u03c8 - \u03c6)) \u03bc < \u03b5\u2081\nC : \u211d\u22650\nhC : \u2200 (x : \u03b1), \u2191\u03c6 x \u2264 C\n\u22a2 \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s < (\u03b5\u2082 - \u03b5\u2081) / \u2191C \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc < \u03b5", "state_after": "case right\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03b5\u2082 : \u211d\u22650\u221e\nh\u03b5\u20820 : 0 < \u03b5\u2082\nh\u03b5\u2082\u03b5 : \u03b5\u2082 < \u03b5\n\u03b5\u2081 : \u211d\u22650\u221e\nh\u03b5\u20810 : 0 < \u03b5\u2081\nh\u03b5\u2081\u2082 : \u03b5\u2081 < \u03b5\u2082\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\nleft\u271d : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x\nh\u03c6 : \u2200 (\u03c8 : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (\u03c8 - \u03c6)) \u03bc < \u03b5\u2081\nC : \u211d\u22650\nhC : \u2200 (x : \u03b1), \u2191\u03c6 x \u2264 C\ns : Set \u03b1\nhs : \u2191\u2191\u03bc s < (\u03b5\u2082 - \u03b5\u2081) / \u2191C\n\u22a2 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u03b5\u2082"}, {"tactic": "simp only [lintegral_eq_nnreal, iSup_le_iff]", "annotated_tactic": ["simp only [<a>lintegral_eq_nnreal</a>, <a>iSup_le_iff</a>]", [{"full_name": "MeasureTheory.lintegral_eq_nnreal", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [197, 9], "def_end_pos": [197, 28]}, {"full_name": "iSup_le_iff", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [964, 9], "def_end_pos": [964, 20]}]], "state_before": "case right\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03b5\u2082 : \u211d\u22650\u221e\nh\u03b5\u20820 : 0 < \u03b5\u2082\nh\u03b5\u2082\u03b5 : \u03b5\u2082 < \u03b5\n\u03b5\u2081 : \u211d\u22650\u221e\nh\u03b5\u20810 : 0 < \u03b5\u2081\nh\u03b5\u2081\u2082 : \u03b5\u2081 < \u03b5\u2082\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\nleft\u271d : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x\nh\u03c6 : \u2200 (\u03c8 : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (\u03c8 - \u03c6)) \u03bc < \u03b5\u2081\nC : \u211d\u22650\nhC : \u2200 (x : \u03b1), \u2191\u03c6 x \u2264 C\ns : Set \u03b1\nhs : \u2191\u2191\u03bc s < (\u03b5\u2082 - \u03b5\u2081) / \u2191C\n\u22a2 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u03b5\u2082", "state_after": "case right\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03b5\u2082 : \u211d\u22650\u221e\nh\u03b5\u20820 : 0 < \u03b5\u2082\nh\u03b5\u2082\u03b5 : \u03b5\u2082 < \u03b5\n\u03b5\u2081 : \u211d\u22650\u221e\nh\u03b5\u20810 : 0 < \u03b5\u2081\nh\u03b5\u2081\u2082 : \u03b5\u2081 < \u03b5\u2082\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\nleft\u271d : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x\nh\u03c6 : \u2200 (\u03c8 : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (\u03c8 - \u03c6)) \u03bc < \u03b5\u2081\nC : \u211d\u22650\nhC : \u2200 (x : \u03b1), \u2191\u03c6 x \u2264 C\ns : Set \u03b1\nhs : \u2191\u2191\u03bc s < (\u03b5\u2082 - \u03b5\u2081) / \u2191C\n\u22a2 \u2200 (i : \u03b1 \u2192\u209b \u211d\u22650),\n    (\u2200 (x : \u03b1), \u2191(\u2191i x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some i) (Measure.restrict \u03bc s) \u2264 \u03b5\u2082"}, {"tactic": "intro \u03c8 h\u03c8", "annotated_tactic": ["intro \u03c8 h\u03c8", []], "state_before": "case right\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03b5\u2082 : \u211d\u22650\u221e\nh\u03b5\u20820 : 0 < \u03b5\u2082\nh\u03b5\u2082\u03b5 : \u03b5\u2082 < \u03b5\n\u03b5\u2081 : \u211d\u22650\u221e\nh\u03b5\u20810 : 0 < \u03b5\u2081\nh\u03b5\u2081\u2082 : \u03b5\u2081 < \u03b5\u2082\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\nleft\u271d : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x\nh\u03c6 : \u2200 (\u03c8 : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (\u03c8 - \u03c6)) \u03bc < \u03b5\u2081\nC : \u211d\u22650\nhC : \u2200 (x : \u03b1), \u2191\u03c6 x \u2264 C\ns : Set \u03b1\nhs : \u2191\u2191\u03bc s < (\u03b5\u2082 - \u03b5\u2081) / \u2191C\n\u22a2 \u2200 (i : \u03b1 \u2192\u209b \u211d\u22650),\n    (\u2200 (x : \u03b1), \u2191(\u2191i x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some i) (Measure.restrict \u03bc s) \u2264 \u03b5\u2082", "state_after": "case right\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03b5\u2082 : \u211d\u22650\u221e\nh\u03b5\u20820 : 0 < \u03b5\u2082\nh\u03b5\u2082\u03b5 : \u03b5\u2082 < \u03b5\n\u03b5\u2081 : \u211d\u22650\u221e\nh\u03b5\u20810 : 0 < \u03b5\u2081\nh\u03b5\u2081\u2082 : \u03b5\u2081 < \u03b5\u2082\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\nleft\u271d : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x\nh\u03c6 : \u2200 (\u03c8 : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (\u03c8 - \u03c6)) \u03bc < \u03b5\u2081\nC : \u211d\u22650\nhC : \u2200 (x : \u03b1), \u2191\u03c6 x \u2264 C\ns : Set \u03b1\nhs : \u2191\u2191\u03bc s < (\u03b5\u2082 - \u03b5\u2081) / \u2191C\n\u03c8 : \u03b1 \u2192\u209b \u211d\u22650\nh\u03c8 : \u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x\n\u22a2 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c8) (Measure.restrict \u03bc s) \u2264 \u03b5\u2082"}, {"tactic": "calc\n  (map (\u2191) \u03c8).lintegral (\u03bc.restrict s) \u2264\n      (map (\u2191) \u03c6).lintegral (\u03bc.restrict s) + (map (\u2191) (\u03c8 - \u03c6)).lintegral (\u03bc.restrict s) := by\n    rw [\u2190 SimpleFunc.add_lintegral, \u2190 SimpleFunc.map_add @ENNReal.coe_add]\n    refine' SimpleFunc.lintegral_mono (fun x => _) le_rfl\n    simp only [add_tsub_eq_max, le_max_right, coe_map, Function.comp_apply, SimpleFunc.coe_add,\n      SimpleFunc.coe_sub, Pi.add_apply, Pi.sub_apply, WithTop.coe_max (\u03c6 x) (\u03c8 x), ENNReal.some]\n  _ \u2264 (map (\u2191) \u03c6).lintegral (\u03bc.restrict s) + \u03b5\u2081 := by\n    refine' add_le_add le_rfl (le_trans _ (h\u03c6 _ h\u03c8).le)\n    exact SimpleFunc.lintegral_mono le_rfl Measure.restrict_le_self\n  _ \u2264 (SimpleFunc.const \u03b1 (C : \u211d\u22650\u221e)).lintegral (\u03bc.restrict s) + \u03b5\u2081 :=\n    (add_le_add (SimpleFunc.lintegral_mono (fun x => by exact coe_le_coe.2 (hC x)) le_rfl) le_rfl)\n  _ = C * \u03bc s + \u03b5\u2081 := by\n    simp only [\u2190 SimpleFunc.lintegral_eq_lintegral, coe_const, lintegral_const,\n      Measure.restrict_apply, MeasurableSet.univ, univ_inter, Function.const]\n  _ \u2264 C * ((\u03b5\u2082 - \u03b5\u2081) / C) + \u03b5\u2081 := by gcongr\n  _ \u2264 \u03b5\u2082 - \u03b5\u2081 + \u03b5\u2081 := by gcongr; apply mul_div_le\n  _ = \u03b5\u2082 := tsub_add_cancel_of_le h\u03b5\u2081\u2082.le", "annotated_tactic": ["calc\n    (<a>map</a> (\u2191) \u03c8).<a>lintegral</a> (\u03bc.restrict s) \u2264\n        (<a>map</a> (\u2191) \u03c6).<a>lintegral</a> (\u03bc.restrict s) + (<a>map</a> (\u2191) (\u03c8 - \u03c6)).<a>lintegral</a> (\u03bc.restrict s) := by\n      rw [\u2190 <a>SimpleFunc.add_lintegral</a>, \u2190 <a>SimpleFunc.map_add</a> @<a>ENNReal.coe_add</a>]\n      refine' <a>SimpleFunc.lintegral_mono</a> (fun x => _) <a>le_rfl</a>\n      simp only [<a>add_tsub_eq_max</a>, <a>le_max_right</a>, <a>coe_map</a>, <a>Function.comp_apply</a>, <a>SimpleFunc.coe_add</a>,\n        <a>SimpleFunc.coe_sub</a>, <a>Pi.add_apply</a>, <a>Pi.sub_apply</a>, <a>WithTop.coe_max</a> (\u03c6 x) (\u03c8 x), <a>ENNReal.some</a>]\n    _ \u2264 (<a>map</a> (\u2191) \u03c6).<a>lintegral</a> (\u03bc.restrict s) + \u03b5\u2081 := by\n      refine' <a>add_le_add</a> <a>le_rfl</a> (<a>le_trans</a> _ (h\u03c6 _ h\u03c8).<a>le</a>)\n      exact <a>SimpleFunc.lintegral_mono</a> <a>le_rfl</a> <a>Measure.restrict_le_self</a>\n    _ \u2264 (<a>SimpleFunc.const</a> \u03b1 (C : \u211d\u22650\u221e)).<a>lintegral</a> (\u03bc.restrict s) + \u03b5\u2081 :=\n      (<a>add_le_add</a> (<a>SimpleFunc.lintegral_mono</a> (fun x => by exact <a>coe_le_coe</a>.2 (hC x)) <a>le_rfl</a>) <a>le_rfl</a>)\n    _ = C * \u03bc s + \u03b5\u2081 := by\n      simp only [\u2190 <a>SimpleFunc.lintegral_eq_lintegral</a>, <a>coe_const</a>, <a>lintegral_const</a>,\n        <a>Measure.restrict_apply</a>, <a>MeasurableSet.univ</a>, <a>univ_inter</a>, <a>Function.const</a>]\n    _ \u2264 C * ((\u03b5\u2082 - \u03b5\u2081) / C) + \u03b5\u2081 := by gcongr\n    _ \u2264 \u03b5\u2082 - \u03b5\u2081 + \u03b5\u2081 := by gcongr; apply <a>mul_div_le</a>\n    _ = \u03b5\u2082 := <a>tsub_add_cancel_of_le</a> h\u03b5\u2081\u2082.le", [{"full_name": "MeasureTheory.SimpleFunc.map", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [290, 5], "def_end_pos": [290, 8]}, {"full_name": "MeasureTheory.SimpleFunc.lintegral", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [950, 5], "def_end_pos": [950, 14]}, {"full_name": "MeasureTheory.SimpleFunc.map", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [290, 5], "def_end_pos": [290, 8]}, {"full_name": "MeasureTheory.SimpleFunc.lintegral", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [950, 5], "def_end_pos": [950, 14]}, {"full_name": "MeasureTheory.SimpleFunc.map", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [290, 5], "def_end_pos": [290, 8]}, {"full_name": "MeasureTheory.SimpleFunc.lintegral", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [950, 5], "def_end_pos": [950, 14]}, {"full_name": "MeasureTheory.SimpleFunc.add_lintegral", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [990, 9], "def_end_pos": [990, 22]}, {"full_name": "MeasureTheory.SimpleFunc.map_add", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [565, 3], "def_end_pos": [565, 14]}, {"full_name": "ENNReal.coe_add", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [386, 28], "def_end_pos": [386, 35]}, {"full_name": "MeasureTheory.SimpleFunc.lintegral_mono", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [1105, 9], "def_end_pos": [1105, 23]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}, {"full_name": "add_tsub_eq_max", "def_path": "Mathlib/Algebra/Order/Sub/Canonical.lean", "def_pos": [498, 9], "def_end_pos": [498, 24]}, {"full_name": "le_max_right", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [61, 9], "def_end_pos": [61, 21]}, {"full_name": "MeasureTheory.SimpleFunc.coe_map", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [303, 9], "def_end_pos": [303, 16]}, {"full_name": "Function.comp_apply", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [33, 17], "def_end_pos": [33, 36]}, {"full_name": "MeasureTheory.SimpleFunc.coe_add", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [469, 3], "def_end_pos": [469, 14]}, {"full_name": "MeasureTheory.SimpleFunc.coe_sub", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [481, 3], "def_end_pos": [481, 14]}, {"full_name": "Pi.add_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [82, 3], "def_end_pos": [82, 14]}, {"full_name": "Pi.sub_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [200, 3], "def_end_pos": [200, 14]}, {"full_name": "WithTop.coe_max", "def_path": "Mathlib/Order/WithBot.lean", "def_pos": [1293, 9], "def_end_pos": [1293, 16]}, {"full_name": "ENNReal.some", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [139, 27], "def_end_pos": [139, 31]}, {"full_name": "MeasureTheory.SimpleFunc.map", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [290, 5], "def_end_pos": [290, 8]}, {"full_name": "MeasureTheory.SimpleFunc.lintegral", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [950, 5], "def_end_pos": [950, 14]}, {"full_name": "add_le_add", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [205, 15], "def_end_pos": [205, 25]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}, {"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}, {"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [142, 7], "def_end_pos": [142, 15]}, {"full_name": "MeasureTheory.SimpleFunc.lintegral_mono", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [1105, 9], "def_end_pos": [1105, 23]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}, {"full_name": "MeasureTheory.Measure.restrict_le_self", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1578, 9], "def_end_pos": [1578, 25]}, {"full_name": "MeasureTheory.SimpleFunc.const", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [145, 5], "def_end_pos": [145, 10]}, {"full_name": "MeasureTheory.SimpleFunc.lintegral", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [950, 5], "def_end_pos": [950, 14]}, {"full_name": "add_le_add", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [205, 15], "def_end_pos": [205, 25]}, {"full_name": "MeasureTheory.SimpleFunc.lintegral_mono", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [1105, 9], "def_end_pos": [1105, 23]}, {"full_name": "ENNReal.coe_le_coe", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [349, 28], "def_end_pos": [349, 38]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}, {"full_name": "MeasureTheory.SimpleFunc.lintegral_eq_lintegral", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [80, 9], "def_end_pos": [80, 42]}, {"full_name": "MeasureTheory.SimpleFunc.coe_const", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [158, 9], "def_end_pos": [158, 18]}, {"full_name": "MeasureTheory.lintegral_const", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [136, 9], "def_end_pos": [136, 24]}, {"full_name": "MeasureTheory.Measure.restrict_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1533, 9], "def_end_pos": [1533, 23]}, {"full_name": "MeasurableSet.univ", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [101, 19], "def_end_pos": [101, 37]}, {"full_name": "Set.univ_inter", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1017, 9], "def_end_pos": [1017, 19]}, {"full_name": "Function.const", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [66, 15], "def_end_pos": [66, 29]}, {"full_name": "ENNReal.mul_div_le", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1750, 9], "def_end_pos": [1750, 19]}, {"full_name": "tsub_add_cancel_of_le", "def_path": "Mathlib/Algebra/Order/Sub/Canonical.lean", "def_pos": [30, 9], "def_end_pos": [30, 30]}]], "state_before": "case right\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03b5\u2082 : \u211d\u22650\u221e\nh\u03b5\u20820 : 0 < \u03b5\u2082\nh\u03b5\u2082\u03b5 : \u03b5\u2082 < \u03b5\n\u03b5\u2081 : \u211d\u22650\u221e\nh\u03b5\u20810 : 0 < \u03b5\u2081\nh\u03b5\u2081\u2082 : \u03b5\u2081 < \u03b5\u2082\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\nleft\u271d : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x\nh\u03c6 : \u2200 (\u03c8 : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (\u03c8 - \u03c6)) \u03bc < \u03b5\u2081\nC : \u211d\u22650\nhC : \u2200 (x : \u03b1), \u2191\u03c6 x \u2264 C\ns : Set \u03b1\nhs : \u2191\u2191\u03bc s < (\u03b5\u2082 - \u03b5\u2081) / \u2191C\n\u03c8 : \u03b1 \u2192\u209b \u211d\u22650\nh\u03c8 : \u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x\n\u22a2 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c8) (Measure.restrict \u03bc s) \u2264 \u03b5\u2082", "state_after": "no goals"}, {"tactic": "rw [\u2190 SimpleFunc.add_lintegral, \u2190 SimpleFunc.map_add @ENNReal.coe_add]", "annotated_tactic": ["rw [\u2190 <a>SimpleFunc.add_lintegral</a>, \u2190 <a>SimpleFunc.map_add</a> @<a>ENNReal.coe_add</a>]", [{"full_name": "MeasureTheory.SimpleFunc.add_lintegral", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [990, 9], "def_end_pos": [990, 22]}, {"full_name": "MeasureTheory.SimpleFunc.map_add", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [565, 3], "def_end_pos": [565, 14]}, {"full_name": "ENNReal.coe_add", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [386, 28], "def_end_pos": [386, 35]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03b5\u2082 : \u211d\u22650\u221e\nh\u03b5\u20820 : 0 < \u03b5\u2082\nh\u03b5\u2082\u03b5 : \u03b5\u2082 < \u03b5\n\u03b5\u2081 : \u211d\u22650\u221e\nh\u03b5\u20810 : 0 < \u03b5\u2081\nh\u03b5\u2081\u2082 : \u03b5\u2081 < \u03b5\u2082\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\nleft\u271d : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x\nh\u03c6 : \u2200 (\u03c8 : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (\u03c8 - \u03c6)) \u03bc < \u03b5\u2081\nC : \u211d\u22650\nhC : \u2200 (x : \u03b1), \u2191\u03c6 x \u2264 C\ns : Set \u03b1\nhs : \u2191\u2191\u03bc s < (\u03b5\u2082 - \u03b5\u2081) / \u2191C\n\u03c8 : \u03b1 \u2192\u209b \u211d\u22650\nh\u03c8 : \u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x\n\u22a2 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c8) (Measure.restrict \u03bc s) \u2264\n    SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) (Measure.restrict \u03bc s) +\n      SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (\u03c8 - \u03c6)) (Measure.restrict \u03bc s)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03b5\u2082 : \u211d\u22650\u221e\nh\u03b5\u20820 : 0 < \u03b5\u2082\nh\u03b5\u2082\u03b5 : \u03b5\u2082 < \u03b5\n\u03b5\u2081 : \u211d\u22650\u221e\nh\u03b5\u20810 : 0 < \u03b5\u2081\nh\u03b5\u2081\u2082 : \u03b5\u2081 < \u03b5\u2082\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\nleft\u271d : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x\nh\u03c6 : \u2200 (\u03c8 : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (\u03c8 - \u03c6)) \u03bc < \u03b5\u2081\nC : \u211d\u22650\nhC : \u2200 (x : \u03b1), \u2191\u03c6 x \u2264 C\ns : Set \u03b1\nhs : \u2191\u2191\u03bc s < (\u03b5\u2082 - \u03b5\u2081) / \u2191C\n\u03c8 : \u03b1 \u2192\u209b \u211d\u22650\nh\u03c8 : \u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x\n\u22a2 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c8) (Measure.restrict \u03bc s) \u2264\n    SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (\u03c6 + (\u03c8 - \u03c6))) (Measure.restrict \u03bc s)"}, {"tactic": "refine' SimpleFunc.lintegral_mono (fun x => _) le_rfl", "annotated_tactic": ["refine' <a>SimpleFunc.lintegral_mono</a> (fun x => _) <a>le_rfl</a>", [{"full_name": "MeasureTheory.SimpleFunc.lintegral_mono", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [1105, 9], "def_end_pos": [1105, 23]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03b5\u2082 : \u211d\u22650\u221e\nh\u03b5\u20820 : 0 < \u03b5\u2082\nh\u03b5\u2082\u03b5 : \u03b5\u2082 < \u03b5\n\u03b5\u2081 : \u211d\u22650\u221e\nh\u03b5\u20810 : 0 < \u03b5\u2081\nh\u03b5\u2081\u2082 : \u03b5\u2081 < \u03b5\u2082\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\nleft\u271d : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x\nh\u03c6 : \u2200 (\u03c8 : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (\u03c8 - \u03c6)) \u03bc < \u03b5\u2081\nC : \u211d\u22650\nhC : \u2200 (x : \u03b1), \u2191\u03c6 x \u2264 C\ns : Set \u03b1\nhs : \u2191\u2191\u03bc s < (\u03b5\u2082 - \u03b5\u2081) / \u2191C\n\u03c8 : \u03b1 \u2192\u209b \u211d\u22650\nh\u03c8 : \u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x\n\u22a2 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c8) (Measure.restrict \u03bc s) \u2264\n    SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (\u03c6 + (\u03c8 - \u03c6))) (Measure.restrict \u03bc s)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03b5\u2082 : \u211d\u22650\u221e\nh\u03b5\u20820 : 0 < \u03b5\u2082\nh\u03b5\u2082\u03b5 : \u03b5\u2082 < \u03b5\n\u03b5\u2081 : \u211d\u22650\u221e\nh\u03b5\u20810 : 0 < \u03b5\u2081\nh\u03b5\u2081\u2082 : \u03b5\u2081 < \u03b5\u2082\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\nleft\u271d : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x\nh\u03c6 : \u2200 (\u03c8 : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (\u03c8 - \u03c6)) \u03bc < \u03b5\u2081\nC : \u211d\u22650\nhC : \u2200 (x : \u03b1), \u2191\u03c6 x \u2264 C\ns : Set \u03b1\nhs : \u2191\u2191\u03bc s < (\u03b5\u2082 - \u03b5\u2081) / \u2191C\n\u03c8 : \u03b1 \u2192\u209b \u211d\u22650\nh\u03c8 : \u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x\nx : \u03b1\n\u22a2 \u2191(SimpleFunc.map ENNReal.some \u03c8) x \u2264 \u2191(SimpleFunc.map ENNReal.some (\u03c6 + (\u03c8 - \u03c6))) x"}, {"tactic": "simp only [add_tsub_eq_max, le_max_right, coe_map, Function.comp_apply, SimpleFunc.coe_add,\n  SimpleFunc.coe_sub, Pi.add_apply, Pi.sub_apply, WithTop.coe_max (\u03c6 x) (\u03c8 x), ENNReal.some]", "annotated_tactic": ["simp only [<a>add_tsub_eq_max</a>, <a>le_max_right</a>, <a>coe_map</a>, <a>Function.comp_apply</a>, <a>SimpleFunc.coe_add</a>,\n        <a>SimpleFunc.coe_sub</a>, <a>Pi.add_apply</a>, <a>Pi.sub_apply</a>, <a>WithTop.coe_max</a> (\u03c6 x) (\u03c8 x), <a>ENNReal.some</a>]", [{"full_name": "add_tsub_eq_max", "def_path": "Mathlib/Algebra/Order/Sub/Canonical.lean", "def_pos": [498, 9], "def_end_pos": [498, 24]}, {"full_name": "le_max_right", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [61, 9], "def_end_pos": [61, 21]}, {"full_name": "MeasureTheory.SimpleFunc.coe_map", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [303, 9], "def_end_pos": [303, 16]}, {"full_name": "Function.comp_apply", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [33, 17], "def_end_pos": [33, 36]}, {"full_name": "MeasureTheory.SimpleFunc.coe_add", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [469, 3], "def_end_pos": [469, 14]}, {"full_name": "MeasureTheory.SimpleFunc.coe_sub", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [481, 3], "def_end_pos": [481, 14]}, {"full_name": "Pi.add_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [82, 3], "def_end_pos": [82, 14]}, {"full_name": "Pi.sub_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [200, 3], "def_end_pos": [200, 14]}, {"full_name": "WithTop.coe_max", "def_path": "Mathlib/Order/WithBot.lean", "def_pos": [1293, 9], "def_end_pos": [1293, 16]}, {"full_name": "ENNReal.some", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [139, 27], "def_end_pos": [139, 31]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03b5\u2082 : \u211d\u22650\u221e\nh\u03b5\u20820 : 0 < \u03b5\u2082\nh\u03b5\u2082\u03b5 : \u03b5\u2082 < \u03b5\n\u03b5\u2081 : \u211d\u22650\u221e\nh\u03b5\u20810 : 0 < \u03b5\u2081\nh\u03b5\u2081\u2082 : \u03b5\u2081 < \u03b5\u2082\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\nleft\u271d : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x\nh\u03c6 : \u2200 (\u03c8 : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (\u03c8 - \u03c6)) \u03bc < \u03b5\u2081\nC : \u211d\u22650\nhC : \u2200 (x : \u03b1), \u2191\u03c6 x \u2264 C\ns : Set \u03b1\nhs : \u2191\u2191\u03bc s < (\u03b5\u2082 - \u03b5\u2081) / \u2191C\n\u03c8 : \u03b1 \u2192\u209b \u211d\u22650\nh\u03c8 : \u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x\nx : \u03b1\n\u22a2 \u2191(SimpleFunc.map ENNReal.some \u03c8) x \u2264 \u2191(SimpleFunc.map ENNReal.some (\u03c6 + (\u03c8 - \u03c6))) x", "state_after": "no goals"}, {"tactic": "refine' add_le_add le_rfl (le_trans _ (h\u03c6 _ h\u03c8).le)", "annotated_tactic": ["refine' <a>add_le_add</a> <a>le_rfl</a> (<a>le_trans</a> _ (h\u03c6 _ h\u03c8).<a>le</a>)", [{"full_name": "add_le_add", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [205, 15], "def_end_pos": [205, 25]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}, {"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}, {"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [142, 7], "def_end_pos": [142, 15]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03b5\u2082 : \u211d\u22650\u221e\nh\u03b5\u20820 : 0 < \u03b5\u2082\nh\u03b5\u2082\u03b5 : \u03b5\u2082 < \u03b5\n\u03b5\u2081 : \u211d\u22650\u221e\nh\u03b5\u20810 : 0 < \u03b5\u2081\nh\u03b5\u2081\u2082 : \u03b5\u2081 < \u03b5\u2082\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\nleft\u271d : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x\nh\u03c6 : \u2200 (\u03c8 : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (\u03c8 - \u03c6)) \u03bc < \u03b5\u2081\nC : \u211d\u22650\nhC : \u2200 (x : \u03b1), \u2191\u03c6 x \u2264 C\ns : Set \u03b1\nhs : \u2191\u2191\u03bc s < (\u03b5\u2082 - \u03b5\u2081) / \u2191C\n\u03c8 : \u03b1 \u2192\u209b \u211d\u22650\nh\u03c8 : \u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x\n\u22a2 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) (Measure.restrict \u03bc s) +\n      SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (\u03c8 - \u03c6)) (Measure.restrict \u03bc s) \u2264\n    SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) (Measure.restrict \u03bc s) + \u03b5\u2081", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03b5\u2082 : \u211d\u22650\u221e\nh\u03b5\u20820 : 0 < \u03b5\u2082\nh\u03b5\u2082\u03b5 : \u03b5\u2082 < \u03b5\n\u03b5\u2081 : \u211d\u22650\u221e\nh\u03b5\u20810 : 0 < \u03b5\u2081\nh\u03b5\u2081\u2082 : \u03b5\u2081 < \u03b5\u2082\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\nleft\u271d : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x\nh\u03c6 : \u2200 (\u03c8 : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (\u03c8 - \u03c6)) \u03bc < \u03b5\u2081\nC : \u211d\u22650\nhC : \u2200 (x : \u03b1), \u2191\u03c6 x \u2264 C\ns : Set \u03b1\nhs : \u2191\u2191\u03bc s < (\u03b5\u2082 - \u03b5\u2081) / \u2191C\n\u03c8 : \u03b1 \u2192\u209b \u211d\u22650\nh\u03c8 : \u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x\n\u22a2 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (\u03c8 - \u03c6)) (Measure.restrict \u03bc s) \u2264\n    SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (\u03c8 - \u03c6)) \u03bc"}, {"tactic": "exact SimpleFunc.lintegral_mono le_rfl Measure.restrict_le_self", "annotated_tactic": ["exact <a>SimpleFunc.lintegral_mono</a> <a>le_rfl</a> <a>Measure.restrict_le_self</a>", [{"full_name": "MeasureTheory.SimpleFunc.lintegral_mono", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [1105, 9], "def_end_pos": [1105, 23]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}, {"full_name": "MeasureTheory.Measure.restrict_le_self", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1578, 9], "def_end_pos": [1578, 25]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03b5\u2082 : \u211d\u22650\u221e\nh\u03b5\u20820 : 0 < \u03b5\u2082\nh\u03b5\u2082\u03b5 : \u03b5\u2082 < \u03b5\n\u03b5\u2081 : \u211d\u22650\u221e\nh\u03b5\u20810 : 0 < \u03b5\u2081\nh\u03b5\u2081\u2082 : \u03b5\u2081 < \u03b5\u2082\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\nleft\u271d : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x\nh\u03c6 : \u2200 (\u03c8 : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (\u03c8 - \u03c6)) \u03bc < \u03b5\u2081\nC : \u211d\u22650\nhC : \u2200 (x : \u03b1), \u2191\u03c6 x \u2264 C\ns : Set \u03b1\nhs : \u2191\u2191\u03bc s < (\u03b5\u2082 - \u03b5\u2081) / \u2191C\n\u03c8 : \u03b1 \u2192\u209b \u211d\u22650\nh\u03c8 : \u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x\n\u22a2 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (\u03c8 - \u03c6)) (Measure.restrict \u03bc s) \u2264\n    SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (\u03c8 - \u03c6)) \u03bc", "state_after": "no goals"}, {"tactic": "exact coe_le_coe.2 (hC x)", "annotated_tactic": ["exact <a>coe_le_coe</a>.2 (hC x)", [{"full_name": "ENNReal.coe_le_coe", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [349, 28], "def_end_pos": [349, 38]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03b5\u2082 : \u211d\u22650\u221e\nh\u03b5\u20820 : 0 < \u03b5\u2082\nh\u03b5\u2082\u03b5 : \u03b5\u2082 < \u03b5\n\u03b5\u2081 : \u211d\u22650\u221e\nh\u03b5\u20810 : 0 < \u03b5\u2081\nh\u03b5\u2081\u2082 : \u03b5\u2081 < \u03b5\u2082\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\nleft\u271d : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x\nh\u03c6 : \u2200 (\u03c8 : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (\u03c8 - \u03c6)) \u03bc < \u03b5\u2081\nC : \u211d\u22650\nhC : \u2200 (x : \u03b1), \u2191\u03c6 x \u2264 C\ns : Set \u03b1\nhs : \u2191\u2191\u03bc s < (\u03b5\u2082 - \u03b5\u2081) / \u2191C\n\u03c8 : \u03b1 \u2192\u209b \u211d\u22650\nh\u03c8 : \u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x\nx : \u03b1\n\u22a2 \u2191(SimpleFunc.map ENNReal.some \u03c6) x \u2264 \u2191(const \u03b1 \u2191C) x", "state_after": "no goals"}, {"tactic": "simp only [\u2190 SimpleFunc.lintegral_eq_lintegral, coe_const, lintegral_const,\n  Measure.restrict_apply, MeasurableSet.univ, univ_inter, Function.const]", "annotated_tactic": ["simp only [\u2190 <a>SimpleFunc.lintegral_eq_lintegral</a>, <a>coe_const</a>, <a>lintegral_const</a>,\n        <a>Measure.restrict_apply</a>, <a>MeasurableSet.univ</a>, <a>univ_inter</a>, <a>Function.const</a>]", [{"full_name": "MeasureTheory.SimpleFunc.lintegral_eq_lintegral", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [80, 9], "def_end_pos": [80, 42]}, {"full_name": "MeasureTheory.SimpleFunc.coe_const", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [158, 9], "def_end_pos": [158, 18]}, {"full_name": "MeasureTheory.lintegral_const", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [136, 9], "def_end_pos": [136, 24]}, {"full_name": "MeasureTheory.Measure.restrict_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1533, 9], "def_end_pos": [1533, 23]}, {"full_name": "MeasurableSet.univ", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [101, 19], "def_end_pos": [101, 37]}, {"full_name": "Set.univ_inter", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1017, 9], "def_end_pos": [1017, 19]}, {"full_name": "Function.const", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [66, 15], "def_end_pos": [66, 29]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03b5\u2082 : \u211d\u22650\u221e\nh\u03b5\u20820 : 0 < \u03b5\u2082\nh\u03b5\u2082\u03b5 : \u03b5\u2082 < \u03b5\n\u03b5\u2081 : \u211d\u22650\u221e\nh\u03b5\u20810 : 0 < \u03b5\u2081\nh\u03b5\u2081\u2082 : \u03b5\u2081 < \u03b5\u2082\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\nleft\u271d : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x\nh\u03c6 : \u2200 (\u03c8 : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (\u03c8 - \u03c6)) \u03bc < \u03b5\u2081\nC : \u211d\u22650\nhC : \u2200 (x : \u03b1), \u2191\u03c6 x \u2264 C\ns : Set \u03b1\nhs : \u2191\u2191\u03bc s < (\u03b5\u2082 - \u03b5\u2081) / \u2191C\n\u03c8 : \u03b1 \u2192\u209b \u211d\u22650\nh\u03c8 : \u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x\n\u22a2 SimpleFunc.lintegral (const \u03b1 \u2191C) (Measure.restrict \u03bc s) + \u03b5\u2081 = \u2191C * \u2191\u2191\u03bc s + \u03b5\u2081", "state_after": "no goals"}, {"tactic": "gcongr", "annotated_tactic": ["gcongr", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03b5\u2082 : \u211d\u22650\u221e\nh\u03b5\u20820 : 0 < \u03b5\u2082\nh\u03b5\u2082\u03b5 : \u03b5\u2082 < \u03b5\n\u03b5\u2081 : \u211d\u22650\u221e\nh\u03b5\u20810 : 0 < \u03b5\u2081\nh\u03b5\u2081\u2082 : \u03b5\u2081 < \u03b5\u2082\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\nleft\u271d : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x\nh\u03c6 : \u2200 (\u03c8 : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (\u03c8 - \u03c6)) \u03bc < \u03b5\u2081\nC : \u211d\u22650\nhC : \u2200 (x : \u03b1), \u2191\u03c6 x \u2264 C\ns : Set \u03b1\nhs : \u2191\u2191\u03bc s < (\u03b5\u2082 - \u03b5\u2081) / \u2191C\n\u03c8 : \u03b1 \u2192\u209b \u211d\u22650\nh\u03c8 : \u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x\n\u22a2 \u2191C * \u2191\u2191\u03bc s + \u03b5\u2081 \u2264 \u2191C * ((\u03b5\u2082 - \u03b5\u2081) / \u2191C) + \u03b5\u2081", "state_after": "no goals"}, {"tactic": "gcongr", "annotated_tactic": ["gcongr", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03b5\u2082 : \u211d\u22650\u221e\nh\u03b5\u20820 : 0 < \u03b5\u2082\nh\u03b5\u2082\u03b5 : \u03b5\u2082 < \u03b5\n\u03b5\u2081 : \u211d\u22650\u221e\nh\u03b5\u20810 : 0 < \u03b5\u2081\nh\u03b5\u2081\u2082 : \u03b5\u2081 < \u03b5\u2082\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\nleft\u271d : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x\nh\u03c6 : \u2200 (\u03c8 : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (\u03c8 - \u03c6)) \u03bc < \u03b5\u2081\nC : \u211d\u22650\nhC : \u2200 (x : \u03b1), \u2191\u03c6 x \u2264 C\ns : Set \u03b1\nhs : \u2191\u2191\u03bc s < (\u03b5\u2082 - \u03b5\u2081) / \u2191C\n\u03c8 : \u03b1 \u2192\u209b \u211d\u22650\nh\u03c8 : \u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x\n\u22a2 \u2191C * ((\u03b5\u2082 - \u03b5\u2081) / \u2191C) + \u03b5\u2081 \u2264 \u03b5\u2082 - \u03b5\u2081 + \u03b5\u2081", "state_after": "case bc\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03b5\u2082 : \u211d\u22650\u221e\nh\u03b5\u20820 : 0 < \u03b5\u2082\nh\u03b5\u2082\u03b5 : \u03b5\u2082 < \u03b5\n\u03b5\u2081 : \u211d\u22650\u221e\nh\u03b5\u20810 : 0 < \u03b5\u2081\nh\u03b5\u2081\u2082 : \u03b5\u2081 < \u03b5\u2082\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\nleft\u271d : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x\nh\u03c6 : \u2200 (\u03c8 : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (\u03c8 - \u03c6)) \u03bc < \u03b5\u2081\nC : \u211d\u22650\nhC : \u2200 (x : \u03b1), \u2191\u03c6 x \u2264 C\ns : Set \u03b1\nhs : \u2191\u2191\u03bc s < (\u03b5\u2082 - \u03b5\u2081) / \u2191C\n\u03c8 : \u03b1 \u2192\u209b \u211d\u22650\nh\u03c8 : \u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x\n\u22a2 \u2191C * ((\u03b5\u2082 - \u03b5\u2081) / \u2191C) \u2264 \u03b5\u2082 - \u03b5\u2081"}, {"tactic": "apply mul_div_le", "annotated_tactic": ["apply <a>mul_div_le</a>", [{"full_name": "ENNReal.mul_div_le", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1750, 9], "def_end_pos": [1750, 19]}]], "state_before": "case bc\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03b5\u2082 : \u211d\u22650\u221e\nh\u03b5\u20820 : 0 < \u03b5\u2082\nh\u03b5\u2082\u03b5 : \u03b5\u2082 < \u03b5\n\u03b5\u2081 : \u211d\u22650\u221e\nh\u03b5\u20810 : 0 < \u03b5\u2081\nh\u03b5\u2081\u2082 : \u03b5\u2081 < \u03b5\u2082\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\nleft\u271d : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x\nh\u03c6 : \u2200 (\u03c8 : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (\u03c8 - \u03c6)) \u03bc < \u03b5\u2081\nC : \u211d\u22650\nhC : \u2200 (x : \u03b1), \u2191\u03c6 x \u2264 C\ns : Set \u03b1\nhs : \u2191\u2191\u03bc s < (\u03b5\u2082 - \u03b5\u2081) / \u2191C\n\u03c8 : \u03b1 \u2192\u209b \u211d\u22650\nh\u03c8 : \u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x\n\u22a2 \u2191C * ((\u03b5\u2082 - \u03b5\u2081) / \u2191C) \u2264 \u03b5\u2082 - \u03b5\u2081", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "full_name": "MeasureTheory.OuterMeasure.le_sum_caratheodory", "start": [1107, 1], "end": [1109, 79], "traced_tactics": [{"tactic": "simp [fun i => MeasurableSpace.measurableSet_iInf.1 h i t, ENNReal.tsum_add]", "annotated_tactic": ["simp [fun i => <a>MeasurableSpace.measurableSet_iInf</a>.1 h i t, <a>ENNReal.tsum_add</a>]", [{"full_name": "MeasurableSpace.measurableSet_iInf", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [495, 9], "def_end_pos": [495, 27]}, {"full_name": "ENNReal.tsum_add", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [823, 19], "def_end_pos": [823, 27]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\nm : \u03b9 \u2192 OuterMeasure \u03b1\ns : Set \u03b1\nh : MeasurableSet s\nt : Set \u03b1\n\u22a2 \u2191(sum m) t = \u2191(sum m) (t \u2229 s) + \u2191(sum m) (t \\ s)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/QPF/Multivariate/Constructions/Fix.lean", "full_name": "MvQPF.recF_eq_of_wEquiv", "start": [92, 1], "end": [104, 57], "traced_tactics": [{"tactic": "apply q.P.w_cases _ x", "annotated_tactic": ["apply q.P.w_cases _ x", []], "state_before": "n : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\n\u03b2 : Type u\nu : F (\u03b1 ::: \u03b2) \u2192 \u03b2\nx y : MvPFunctor.W (P F) \u03b1\n\u22a2 WEquiv x y \u2192 recF u x = recF u y", "state_after": "n : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\n\u03b2 : Type u\nu : F (\u03b1 ::: \u03b2) \u2192 \u03b2\nx y : MvPFunctor.W (P F) \u03b1\n\u22a2 \u2200 (a : (P F).A) (f' : MvPFunctor.B (MvPFunctor.drop (P F)) a \u27f9 \u03b1)\n    (f : PFunctor.B (MvPFunctor.last (P F)) a \u2192 MvPFunctor.W (P F) \u03b1),\n    WEquiv (MvPFunctor.wMk (P F) a f' f) y \u2192 recF u (MvPFunctor.wMk (P F) a f' f) = recF u y"}, {"tactic": "intro a\u2080 f'\u2080 f\u2080", "annotated_tactic": ["intro a\u2080 f'\u2080 f\u2080", []], "state_before": "n : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\n\u03b2 : Type u\nu : F (\u03b1 ::: \u03b2) \u2192 \u03b2\nx y : MvPFunctor.W (P F) \u03b1\n\u22a2 \u2200 (a : (P F).A) (f' : MvPFunctor.B (MvPFunctor.drop (P F)) a \u27f9 \u03b1)\n    (f : PFunctor.B (MvPFunctor.last (P F)) a \u2192 MvPFunctor.W (P F) \u03b1),\n    WEquiv (MvPFunctor.wMk (P F) a f' f) y \u2192 recF u (MvPFunctor.wMk (P F) a f' f) = recF u y", "state_after": "n : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\n\u03b2 : Type u\nu : F (\u03b1 ::: \u03b2) \u2192 \u03b2\nx y : MvPFunctor.W (P F) \u03b1\na\u2080 : (P F).A\nf'\u2080 : MvPFunctor.B (MvPFunctor.drop (P F)) a\u2080 \u27f9 \u03b1\nf\u2080 : PFunctor.B (MvPFunctor.last (P F)) a\u2080 \u2192 MvPFunctor.W (P F) \u03b1\n\u22a2 WEquiv (MvPFunctor.wMk (P F) a\u2080 f'\u2080 f\u2080) y \u2192 recF u (MvPFunctor.wMk (P F) a\u2080 f'\u2080 f\u2080) = recF u y"}, {"tactic": "apply q.P.w_cases _ y", "annotated_tactic": ["apply q.P.w_cases _ y", []], "state_before": "n : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\n\u03b2 : Type u\nu : F (\u03b1 ::: \u03b2) \u2192 \u03b2\nx y : MvPFunctor.W (P F) \u03b1\na\u2080 : (P F).A\nf'\u2080 : MvPFunctor.B (MvPFunctor.drop (P F)) a\u2080 \u27f9 \u03b1\nf\u2080 : PFunctor.B (MvPFunctor.last (P F)) a\u2080 \u2192 MvPFunctor.W (P F) \u03b1\n\u22a2 WEquiv (MvPFunctor.wMk (P F) a\u2080 f'\u2080 f\u2080) y \u2192 recF u (MvPFunctor.wMk (P F) a\u2080 f'\u2080 f\u2080) = recF u y", "state_after": "n : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\n\u03b2 : Type u\nu : F (\u03b1 ::: \u03b2) \u2192 \u03b2\nx y : MvPFunctor.W (P F) \u03b1\na\u2080 : (P F).A\nf'\u2080 : MvPFunctor.B (MvPFunctor.drop (P F)) a\u2080 \u27f9 \u03b1\nf\u2080 : PFunctor.B (MvPFunctor.last (P F)) a\u2080 \u2192 MvPFunctor.W (P F) \u03b1\n\u22a2 \u2200 (a : (P F).A) (f' : MvPFunctor.B (MvPFunctor.drop (P F)) a \u27f9 \u03b1)\n    (f : PFunctor.B (MvPFunctor.last (P F)) a \u2192 MvPFunctor.W (P F) \u03b1),\n    WEquiv (MvPFunctor.wMk (P F) a\u2080 f'\u2080 f\u2080) (MvPFunctor.wMk (P F) a f' f) \u2192\n      recF u (MvPFunctor.wMk (P F) a\u2080 f'\u2080 f\u2080) = recF u (MvPFunctor.wMk (P F) a f' f)"}, {"tactic": "intro a\u2081 f'\u2081 f\u2081", "annotated_tactic": ["intro a\u2081 f'\u2081 f\u2081", []], "state_before": "n : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\n\u03b2 : Type u\nu : F (\u03b1 ::: \u03b2) \u2192 \u03b2\nx y : MvPFunctor.W (P F) \u03b1\na\u2080 : (P F).A\nf'\u2080 : MvPFunctor.B (MvPFunctor.drop (P F)) a\u2080 \u27f9 \u03b1\nf\u2080 : PFunctor.B (MvPFunctor.last (P F)) a\u2080 \u2192 MvPFunctor.W (P F) \u03b1\n\u22a2 \u2200 (a : (P F).A) (f' : MvPFunctor.B (MvPFunctor.drop (P F)) a \u27f9 \u03b1)\n    (f : PFunctor.B (MvPFunctor.last (P F)) a \u2192 MvPFunctor.W (P F) \u03b1),\n    WEquiv (MvPFunctor.wMk (P F) a\u2080 f'\u2080 f\u2080) (MvPFunctor.wMk (P F) a f' f) \u2192\n      recF u (MvPFunctor.wMk (P F) a\u2080 f'\u2080 f\u2080) = recF u (MvPFunctor.wMk (P F) a f' f)", "state_after": "n : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\n\u03b2 : Type u\nu : F (\u03b1 ::: \u03b2) \u2192 \u03b2\nx y : MvPFunctor.W (P F) \u03b1\na\u2080 : (P F).A\nf'\u2080 : MvPFunctor.B (MvPFunctor.drop (P F)) a\u2080 \u27f9 \u03b1\nf\u2080 : PFunctor.B (MvPFunctor.last (P F)) a\u2080 \u2192 MvPFunctor.W (P F) \u03b1\na\u2081 : (P F).A\nf'\u2081 : MvPFunctor.B (MvPFunctor.drop (P F)) a\u2081 \u27f9 \u03b1\nf\u2081 : PFunctor.B (MvPFunctor.last (P F)) a\u2081 \u2192 MvPFunctor.W (P F) \u03b1\n\u22a2 WEquiv (MvPFunctor.wMk (P F) a\u2080 f'\u2080 f\u2080) (MvPFunctor.wMk (P F) a\u2081 f'\u2081 f\u2081) \u2192\n    recF u (MvPFunctor.wMk (P F) a\u2080 f'\u2080 f\u2080) = recF u (MvPFunctor.wMk (P F) a\u2081 f'\u2081 f\u2081)"}, {"tactic": "intro h", "annotated_tactic": ["intro h", []], "state_before": "n : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\n\u03b2 : Type u\nu : F (\u03b1 ::: \u03b2) \u2192 \u03b2\nx y : MvPFunctor.W (P F) \u03b1\na\u2080 : (P F).A\nf'\u2080 : MvPFunctor.B (MvPFunctor.drop (P F)) a\u2080 \u27f9 \u03b1\nf\u2080 : PFunctor.B (MvPFunctor.last (P F)) a\u2080 \u2192 MvPFunctor.W (P F) \u03b1\na\u2081 : (P F).A\nf'\u2081 : MvPFunctor.B (MvPFunctor.drop (P F)) a\u2081 \u27f9 \u03b1\nf\u2081 : PFunctor.B (MvPFunctor.last (P F)) a\u2081 \u2192 MvPFunctor.W (P F) \u03b1\n\u22a2 WEquiv (MvPFunctor.wMk (P F) a\u2080 f'\u2080 f\u2080) (MvPFunctor.wMk (P F) a\u2081 f'\u2081 f\u2081) \u2192\n    recF u (MvPFunctor.wMk (P F) a\u2080 f'\u2080 f\u2080) = recF u (MvPFunctor.wMk (P F) a\u2081 f'\u2081 f\u2081)", "state_after": "n : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\n\u03b2 : Type u\nu : F (\u03b1 ::: \u03b2) \u2192 \u03b2\nx y : MvPFunctor.W (P F) \u03b1\na\u2080 : (P F).A\nf'\u2080 : MvPFunctor.B (MvPFunctor.drop (P F)) a\u2080 \u27f9 \u03b1\nf\u2080 : PFunctor.B (MvPFunctor.last (P F)) a\u2080 \u2192 MvPFunctor.W (P F) \u03b1\na\u2081 : (P F).A\nf'\u2081 : MvPFunctor.B (MvPFunctor.drop (P F)) a\u2081 \u27f9 \u03b1\nf\u2081 : PFunctor.B (MvPFunctor.last (P F)) a\u2081 \u2192 MvPFunctor.W (P F) \u03b1\nh : WEquiv (MvPFunctor.wMk (P F) a\u2080 f'\u2080 f\u2080) (MvPFunctor.wMk (P F) a\u2081 f'\u2081 f\u2081)\n\u22a2 recF u (MvPFunctor.wMk (P F) a\u2080 f'\u2080 f\u2080) = recF u (MvPFunctor.wMk (P F) a\u2081 f'\u2081 f\u2081)"}, {"tactic": "refine' @WEquiv.recOn _ _ _ _ _ (\u03bb a a' _ => recF u a = recF u a') _ _ h _ _ _", "annotated_tactic": ["refine' @<a>WEquiv.recOn</a> _ _ _ _ _ (\u03bb a a' _ => <a>recF</a> u a = <a>recF</a> u a') _ _ h _ _ _", [{"full_name": "MvQPF.WEquiv.recOn", "def_path": "Mathlib/Data/QPF/Multivariate/Constructions/Fix.lean", "def_pos": [81, 11], "def_end_pos": [81, 17]}, {"full_name": "MvQPF.recF", "def_path": "Mathlib/Data/QPF/Multivariate/Constructions/Fix.lean", "def_pos": [59, 5], "def_end_pos": [59, 9]}, {"full_name": "MvQPF.recF", "def_path": "Mathlib/Data/QPF/Multivariate/Constructions/Fix.lean", "def_pos": [59, 5], "def_end_pos": [59, 9]}]], "state_before": "n : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\n\u03b2 : Type u\nu : F (\u03b1 ::: \u03b2) \u2192 \u03b2\nx y : MvPFunctor.W (P F) \u03b1\na\u2080 : (P F).A\nf'\u2080 : MvPFunctor.B (MvPFunctor.drop (P F)) a\u2080 \u27f9 \u03b1\nf\u2080 : PFunctor.B (MvPFunctor.last (P F)) a\u2080 \u2192 MvPFunctor.W (P F) \u03b1\na\u2081 : (P F).A\nf'\u2081 : MvPFunctor.B (MvPFunctor.drop (P F)) a\u2081 \u27f9 \u03b1\nf\u2081 : PFunctor.B (MvPFunctor.last (P F)) a\u2081 \u2192 MvPFunctor.W (P F) \u03b1\nh : WEquiv (MvPFunctor.wMk (P F) a\u2080 f'\u2080 f\u2080) (MvPFunctor.wMk (P F) a\u2081 f'\u2081 f\u2081)\n\u22a2 recF u (MvPFunctor.wMk (P F) a\u2080 f'\u2080 f\u2080) = recF u (MvPFunctor.wMk (P F) a\u2081 f'\u2081 f\u2081)", "state_after": "case refine'_1\nn : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\n\u03b2 : Type u\nu : F (\u03b1 ::: \u03b2) \u2192 \u03b2\nx y : MvPFunctor.W (P F) \u03b1\na\u2080 : (P F).A\nf'\u2080 : MvPFunctor.B (MvPFunctor.drop (P F)) a\u2080 \u27f9 \u03b1\nf\u2080 : PFunctor.B (MvPFunctor.last (P F)) a\u2080 \u2192 MvPFunctor.W (P F) \u03b1\na\u2081 : (P F).A\nf'\u2081 : MvPFunctor.B (MvPFunctor.drop (P F)) a\u2081 \u27f9 \u03b1\nf\u2081 : PFunctor.B (MvPFunctor.last (P F)) a\u2081 \u2192 MvPFunctor.W (P F) \u03b1\nh : WEquiv (MvPFunctor.wMk (P F) a\u2080 f'\u2080 f\u2080) (MvPFunctor.wMk (P F) a\u2081 f'\u2081 f\u2081)\n\u22a2 \u2200 (a : (P F).A) (f' : MvPFunctor.B (MvPFunctor.drop (P F)) a \u27f9 \u03b1)\n    (f\u2080 f\u2081 : PFunctor.B (MvPFunctor.last (P F)) a \u2192 MvPFunctor.W (P F) \u03b1)\n    (a_1 : \u2200 (x : PFunctor.B (MvPFunctor.last (P F)) a), WEquiv (f\u2080 x) (f\u2081 x)),\n    (\u2200 (x : PFunctor.B (MvPFunctor.last (P F)) a),\n        (fun a a' x => recF u a = recF u a') (f\u2080 x) (f\u2081 x) (_ : WEquiv (f\u2080 x) (f\u2081 x))) \u2192\n      (fun a a' x => recF u a = recF u a') (MvPFunctor.wMk (P F) a f' f\u2080) (MvPFunctor.wMk (P F) a f' f\u2081)\n        (_ : WEquiv (MvPFunctor.wMk (P F) a f' f\u2080) (MvPFunctor.wMk (P F) a f' f\u2081))\n\ncase refine'_2\nn : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\n\u03b2 : Type u\nu : F (\u03b1 ::: \u03b2) \u2192 \u03b2\nx y : MvPFunctor.W (P F) \u03b1\na\u2080 : (P F).A\nf'\u2080 : MvPFunctor.B (MvPFunctor.drop (P F)) a\u2080 \u27f9 \u03b1\nf\u2080 : PFunctor.B (MvPFunctor.last (P F)) a\u2080 \u2192 MvPFunctor.W (P F) \u03b1\na\u2081 : (P F).A\nf'\u2081 : MvPFunctor.B (MvPFunctor.drop (P F)) a\u2081 \u27f9 \u03b1\nf\u2081 : PFunctor.B (MvPFunctor.last (P F)) a\u2081 \u2192 MvPFunctor.W (P F) \u03b1\nh : WEquiv (MvPFunctor.wMk (P F) a\u2080 f'\u2080 f\u2080) (MvPFunctor.wMk (P F) a\u2081 f'\u2081 f\u2081)\n\u22a2 \u2200 (a\u2080 : (P F).A) (f'\u2080 : MvPFunctor.B (MvPFunctor.drop (P F)) a\u2080 \u27f9 \u03b1)\n    (f\u2080 : PFunctor.B (MvPFunctor.last (P F)) a\u2080 \u2192 MvPFunctor.W (P F) \u03b1) (a\u2081 : (P F).A)\n    (f'\u2081 : MvPFunctor.B (MvPFunctor.drop (P F)) a\u2081 \u27f9 \u03b1)\n    (f\u2081 : PFunctor.B (MvPFunctor.last (P F)) a\u2081 \u2192 MvPFunctor.W (P F) \u03b1)\n    (a :\n      abs { fst := a\u2080, snd := MvPFunctor.appendContents (P F) f'\u2080 f\u2080 } =\n        abs { fst := a\u2081, snd := MvPFunctor.appendContents (P F) f'\u2081 f\u2081 }),\n    (fun a a' x => recF u a = recF u a') (MvPFunctor.wMk (P F) a\u2080 f'\u2080 f\u2080) (MvPFunctor.wMk (P F) a\u2081 f'\u2081 f\u2081)\n      (_ : WEquiv (MvPFunctor.wMk (P F) a\u2080 f'\u2080 f\u2080) (MvPFunctor.wMk (P F) a\u2081 f'\u2081 f\u2081))\n\ncase refine'_3\nn : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\n\u03b2 : Type u\nu : F (\u03b1 ::: \u03b2) \u2192 \u03b2\nx y : MvPFunctor.W (P F) \u03b1\na\u2080 : (P F).A\nf'\u2080 : MvPFunctor.B (MvPFunctor.drop (P F)) a\u2080 \u27f9 \u03b1\nf\u2080 : PFunctor.B (MvPFunctor.last (P F)) a\u2080 \u2192 MvPFunctor.W (P F) \u03b1\na\u2081 : (P F).A\nf'\u2081 : MvPFunctor.B (MvPFunctor.drop (P F)) a\u2081 \u27f9 \u03b1\nf\u2081 : PFunctor.B (MvPFunctor.last (P F)) a\u2081 \u2192 MvPFunctor.W (P F) \u03b1\nh : WEquiv (MvPFunctor.wMk (P F) a\u2080 f'\u2080 f\u2080) (MvPFunctor.wMk (P F) a\u2081 f'\u2081 f\u2081)\n\u22a2 \u2200 (u_1 v w : MvPFunctor.W (P F) \u03b1) (a : WEquiv u_1 v) (a_1 : WEquiv v w),\n    (fun a a' x => recF u a = recF u a') u_1 v a \u2192\n      (fun a a' x => recF u a = recF u a') v w a_1 \u2192 (fun a a' x => recF u a = recF u a') u_1 w (_ : WEquiv u_1 w)"}, {"tactic": "intros a f' f\u2080 f\u2081 _h ih", "annotated_tactic": ["intros a f' f\u2080 f\u2081 _h ih", []], "state_before": "case refine'_1\nn : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\n\u03b2 : Type u\nu : F (\u03b1 ::: \u03b2) \u2192 \u03b2\nx y : MvPFunctor.W (P F) \u03b1\na\u2080 : (P F).A\nf'\u2080 : MvPFunctor.B (MvPFunctor.drop (P F)) a\u2080 \u27f9 \u03b1\nf\u2080 : PFunctor.B (MvPFunctor.last (P F)) a\u2080 \u2192 MvPFunctor.W (P F) \u03b1\na\u2081 : (P F).A\nf'\u2081 : MvPFunctor.B (MvPFunctor.drop (P F)) a\u2081 \u27f9 \u03b1\nf\u2081 : PFunctor.B (MvPFunctor.last (P F)) a\u2081 \u2192 MvPFunctor.W (P F) \u03b1\nh : WEquiv (MvPFunctor.wMk (P F) a\u2080 f'\u2080 f\u2080) (MvPFunctor.wMk (P F) a\u2081 f'\u2081 f\u2081)\n\u22a2 \u2200 (a : (P F).A) (f' : MvPFunctor.B (MvPFunctor.drop (P F)) a \u27f9 \u03b1)\n    (f\u2080 f\u2081 : PFunctor.B (MvPFunctor.last (P F)) a \u2192 MvPFunctor.W (P F) \u03b1)\n    (a_1 : \u2200 (x : PFunctor.B (MvPFunctor.last (P F)) a), WEquiv (f\u2080 x) (f\u2081 x)),\n    (\u2200 (x : PFunctor.B (MvPFunctor.last (P F)) a),\n        (fun a a' x => recF u a = recF u a') (f\u2080 x) (f\u2081 x) (_ : WEquiv (f\u2080 x) (f\u2081 x))) \u2192\n      (fun a a' x => recF u a = recF u a') (MvPFunctor.wMk (P F) a f' f\u2080) (MvPFunctor.wMk (P F) a f' f\u2081)\n        (_ : WEquiv (MvPFunctor.wMk (P F) a f' f\u2080) (MvPFunctor.wMk (P F) a f' f\u2081))", "state_after": "case refine'_1\nn : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\n\u03b2 : Type u\nu : F (\u03b1 ::: \u03b2) \u2192 \u03b2\nx y : MvPFunctor.W (P F) \u03b1\na\u2080 : (P F).A\nf'\u2080 : MvPFunctor.B (MvPFunctor.drop (P F)) a\u2080 \u27f9 \u03b1\nf\u2080\u271d : PFunctor.B (MvPFunctor.last (P F)) a\u2080 \u2192 MvPFunctor.W (P F) \u03b1\na\u2081 : (P F).A\nf'\u2081 : MvPFunctor.B (MvPFunctor.drop (P F)) a\u2081 \u27f9 \u03b1\nf\u2081\u271d : PFunctor.B (MvPFunctor.last (P F)) a\u2081 \u2192 MvPFunctor.W (P F) \u03b1\nh : WEquiv (MvPFunctor.wMk (P F) a\u2080 f'\u2080 f\u2080\u271d) (MvPFunctor.wMk (P F) a\u2081 f'\u2081 f\u2081\u271d)\na : (P F).A\nf' : MvPFunctor.B (MvPFunctor.drop (P F)) a \u27f9 \u03b1\nf\u2080 f\u2081 : PFunctor.B (MvPFunctor.last (P F)) a \u2192 MvPFunctor.W (P F) \u03b1\n_h : \u2200 (x : PFunctor.B (MvPFunctor.last (P F)) a), WEquiv (f\u2080 x) (f\u2081 x)\nih :\n  \u2200 (x : PFunctor.B (MvPFunctor.last (P F)) a),\n    (fun a a' x => recF u a = recF u a') (f\u2080 x) (f\u2081 x) (_ : WEquiv (f\u2080 x) (f\u2081 x))\n\u22a2 recF u (MvPFunctor.wMk (P F) a f' f\u2080) = recF u (MvPFunctor.wMk (P F) a f' f\u2081)"}, {"tactic": "simp only [recF_eq, Function.comp]", "annotated_tactic": ["simp only [<a>recF_eq</a>, <a>Function.comp</a>]", [{"full_name": "MvQPF.recF_eq", "def_path": "Mathlib/Data/QPF/Multivariate/Constructions/Fix.lean", "def_pos": [64, 9], "def_end_pos": [64, 16]}, {"full_name": "Function.comp", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [52, 15], "def_end_pos": [52, 28]}]], "state_before": "case refine'_1\nn : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\n\u03b2 : Type u\nu : F (\u03b1 ::: \u03b2) \u2192 \u03b2\nx y : MvPFunctor.W (P F) \u03b1\na\u2080 : (P F).A\nf'\u2080 : MvPFunctor.B (MvPFunctor.drop (P F)) a\u2080 \u27f9 \u03b1\nf\u2080\u271d : PFunctor.B (MvPFunctor.last (P F)) a\u2080 \u2192 MvPFunctor.W (P F) \u03b1\na\u2081 : (P F).A\nf'\u2081 : MvPFunctor.B (MvPFunctor.drop (P F)) a\u2081 \u27f9 \u03b1\nf\u2081\u271d : PFunctor.B (MvPFunctor.last (P F)) a\u2081 \u2192 MvPFunctor.W (P F) \u03b1\nh : WEquiv (MvPFunctor.wMk (P F) a\u2080 f'\u2080 f\u2080\u271d) (MvPFunctor.wMk (P F) a\u2081 f'\u2081 f\u2081\u271d)\na : (P F).A\nf' : MvPFunctor.B (MvPFunctor.drop (P F)) a \u27f9 \u03b1\nf\u2080 f\u2081 : PFunctor.B (MvPFunctor.last (P F)) a \u2192 MvPFunctor.W (P F) \u03b1\n_h : \u2200 (x : PFunctor.B (MvPFunctor.last (P F)) a), WEquiv (f\u2080 x) (f\u2081 x)\nih :\n  \u2200 (x : PFunctor.B (MvPFunctor.last (P F)) a),\n    (fun a a' x => recF u a = recF u a') (f\u2080 x) (f\u2081 x) (_ : WEquiv (f\u2080 x) (f\u2081 x))\n\u22a2 recF u (MvPFunctor.wMk (P F) a f' f\u2080) = recF u (MvPFunctor.wMk (P F) a f' f\u2081)", "state_after": "case refine'_1\nn : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\n\u03b2 : Type u\nu : F (\u03b1 ::: \u03b2) \u2192 \u03b2\nx y : MvPFunctor.W (P F) \u03b1\na\u2080 : (P F).A\nf'\u2080 : MvPFunctor.B (MvPFunctor.drop (P F)) a\u2080 \u27f9 \u03b1\nf\u2080\u271d : PFunctor.B (MvPFunctor.last (P F)) a\u2080 \u2192 MvPFunctor.W (P F) \u03b1\na\u2081 : (P F).A\nf'\u2081 : MvPFunctor.B (MvPFunctor.drop (P F)) a\u2081 \u27f9 \u03b1\nf\u2081\u271d : PFunctor.B (MvPFunctor.last (P F)) a\u2081 \u2192 MvPFunctor.W (P F) \u03b1\nh : WEquiv (MvPFunctor.wMk (P F) a\u2080 f'\u2080 f\u2080\u271d) (MvPFunctor.wMk (P F) a\u2081 f'\u2081 f\u2081\u271d)\na : (P F).A\nf' : MvPFunctor.B (MvPFunctor.drop (P F)) a \u27f9 \u03b1\nf\u2080 f\u2081 : PFunctor.B (MvPFunctor.last (P F)) a \u2192 MvPFunctor.W (P F) \u03b1\n_h : \u2200 (x : PFunctor.B (MvPFunctor.last (P F)) a), WEquiv (f\u2080 x) (f\u2081 x)\nih :\n  \u2200 (x : PFunctor.B (MvPFunctor.last (P F)) a),\n    (fun a a' x => recF u a = recF u a') (f\u2080 x) (f\u2081 x) (_ : WEquiv (f\u2080 x) (f\u2081 x))\n\u22a2 u (abs { fst := a, snd := splitFun f' fun x => recF u (f\u2080 x) }) =\n    u (abs { fst := a, snd := splitFun f' fun x => recF u (f\u2081 x) })"}, {"tactic": "congr", "annotated_tactic": ["congr", []], "state_before": "case refine'_1\nn : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\n\u03b2 : Type u\nu : F (\u03b1 ::: \u03b2) \u2192 \u03b2\nx y : MvPFunctor.W (P F) \u03b1\na\u2080 : (P F).A\nf'\u2080 : MvPFunctor.B (MvPFunctor.drop (P F)) a\u2080 \u27f9 \u03b1\nf\u2080\u271d : PFunctor.B (MvPFunctor.last (P F)) a\u2080 \u2192 MvPFunctor.W (P F) \u03b1\na\u2081 : (P F).A\nf'\u2081 : MvPFunctor.B (MvPFunctor.drop (P F)) a\u2081 \u27f9 \u03b1\nf\u2081\u271d : PFunctor.B (MvPFunctor.last (P F)) a\u2081 \u2192 MvPFunctor.W (P F) \u03b1\nh : WEquiv (MvPFunctor.wMk (P F) a\u2080 f'\u2080 f\u2080\u271d) (MvPFunctor.wMk (P F) a\u2081 f'\u2081 f\u2081\u271d)\na : (P F).A\nf' : MvPFunctor.B (MvPFunctor.drop (P F)) a \u27f9 \u03b1\nf\u2080 f\u2081 : PFunctor.B (MvPFunctor.last (P F)) a \u2192 MvPFunctor.W (P F) \u03b1\n_h : \u2200 (x : PFunctor.B (MvPFunctor.last (P F)) a), WEquiv (f\u2080 x) (f\u2081 x)\nih :\n  \u2200 (x : PFunctor.B (MvPFunctor.last (P F)) a),\n    (fun a a' x => recF u a = recF u a') (f\u2080 x) (f\u2081 x) (_ : WEquiv (f\u2080 x) (f\u2081 x))\n\u22a2 u (abs { fst := a, snd := splitFun f' fun x => recF u (f\u2080 x) }) =\n    u (abs { fst := a, snd := splitFun f' fun x => recF u (f\u2081 x) })", "state_after": "case refine'_1.e_a.e_a.e_snd\nn : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\n\u03b2 : Type u\nu : F (\u03b1 ::: \u03b2) \u2192 \u03b2\nx y : MvPFunctor.W (P F) \u03b1\na\u2080 : (P F).A\nf'\u2080 : MvPFunctor.B (MvPFunctor.drop (P F)) a\u2080 \u27f9 \u03b1\nf\u2080\u271d : PFunctor.B (MvPFunctor.last (P F)) a\u2080 \u2192 MvPFunctor.W (P F) \u03b1\na\u2081 : (P F).A\nf'\u2081 : MvPFunctor.B (MvPFunctor.drop (P F)) a\u2081 \u27f9 \u03b1\nf\u2081\u271d : PFunctor.B (MvPFunctor.last (P F)) a\u2081 \u2192 MvPFunctor.W (P F) \u03b1\nh : WEquiv (MvPFunctor.wMk (P F) a\u2080 f'\u2080 f\u2080\u271d) (MvPFunctor.wMk (P F) a\u2081 f'\u2081 f\u2081\u271d)\na : (P F).A\nf' : MvPFunctor.B (MvPFunctor.drop (P F)) a \u27f9 \u03b1\nf\u2080 f\u2081 : PFunctor.B (MvPFunctor.last (P F)) a \u2192 MvPFunctor.W (P F) \u03b1\n_h : \u2200 (x : PFunctor.B (MvPFunctor.last (P F)) a), WEquiv (f\u2080 x) (f\u2081 x)\nih :\n  \u2200 (x : PFunctor.B (MvPFunctor.last (P F)) a),\n    (fun a a' x => recF u a = recF u a') (f\u2080 x) (f\u2081 x) (_ : WEquiv (f\u2080 x) (f\u2081 x))\n\u22a2 (splitFun f' fun x => recF u (f\u2080 x)) = splitFun f' fun x => recF u (f\u2081 x)"}, {"tactic": "funext", "annotated_tactic": ["funext", []], "state_before": "case refine'_1.e_a.e_a.e_snd\nn : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\n\u03b2 : Type u\nu : F (\u03b1 ::: \u03b2) \u2192 \u03b2\nx y : MvPFunctor.W (P F) \u03b1\na\u2080 : (P F).A\nf'\u2080 : MvPFunctor.B (MvPFunctor.drop (P F)) a\u2080 \u27f9 \u03b1\nf\u2080\u271d : PFunctor.B (MvPFunctor.last (P F)) a\u2080 \u2192 MvPFunctor.W (P F) \u03b1\na\u2081 : (P F).A\nf'\u2081 : MvPFunctor.B (MvPFunctor.drop (P F)) a\u2081 \u27f9 \u03b1\nf\u2081\u271d : PFunctor.B (MvPFunctor.last (P F)) a\u2081 \u2192 MvPFunctor.W (P F) \u03b1\nh : WEquiv (MvPFunctor.wMk (P F) a\u2080 f'\u2080 f\u2080\u271d) (MvPFunctor.wMk (P F) a\u2081 f'\u2081 f\u2081\u271d)\na : (P F).A\nf' : MvPFunctor.B (MvPFunctor.drop (P F)) a \u27f9 \u03b1\nf\u2080 f\u2081 : PFunctor.B (MvPFunctor.last (P F)) a \u2192 MvPFunctor.W (P F) \u03b1\n_h : \u2200 (x : PFunctor.B (MvPFunctor.last (P F)) a), WEquiv (f\u2080 x) (f\u2081 x)\nih :\n  \u2200 (x : PFunctor.B (MvPFunctor.last (P F)) a),\n    (fun a a' x => recF u a = recF u a') (f\u2080 x) (f\u2081 x) (_ : WEquiv (f\u2080 x) (f\u2081 x))\n\u22a2 (splitFun f' fun x => recF u (f\u2080 x)) = splitFun f' fun x => recF u (f\u2081 x)", "state_after": "case refine'_1.e_a.e_a.e_snd.h.h\nn : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\n\u03b2 : Type u\nu : F (\u03b1 ::: \u03b2) \u2192 \u03b2\nx y : MvPFunctor.W (P F) \u03b1\na\u2080 : (P F).A\nf'\u2080 : MvPFunctor.B (MvPFunctor.drop (P F)) a\u2080 \u27f9 \u03b1\nf\u2080\u271d : PFunctor.B (MvPFunctor.last (P F)) a\u2080 \u2192 MvPFunctor.W (P F) \u03b1\na\u2081 : (P F).A\nf'\u2081 : MvPFunctor.B (MvPFunctor.drop (P F)) a\u2081 \u27f9 \u03b1\nf\u2081\u271d : PFunctor.B (MvPFunctor.last (P F)) a\u2081 \u2192 MvPFunctor.W (P F) \u03b1\nh : WEquiv (MvPFunctor.wMk (P F) a\u2080 f'\u2080 f\u2080\u271d) (MvPFunctor.wMk (P F) a\u2081 f'\u2081 f\u2081\u271d)\na : (P F).A\nf' : MvPFunctor.B (MvPFunctor.drop (P F)) a \u27f9 \u03b1\nf\u2080 f\u2081 : PFunctor.B (MvPFunctor.last (P F)) a \u2192 MvPFunctor.W (P F) \u03b1\n_h : \u2200 (x : PFunctor.B (MvPFunctor.last (P F)) a), WEquiv (f\u2080 x) (f\u2081 x)\nih :\n  \u2200 (x : PFunctor.B (MvPFunctor.last (P F)) a),\n    (fun a a' x => recF u a = recF u a') (f\u2080 x) (f\u2081 x) (_ : WEquiv (f\u2080 x) (f\u2081 x))\nx\u271d\u00b9 : Fin2 (n + 1)\nx\u271d : MvPFunctor.B (P F) a x\u271d\u00b9\n\u22a2 splitFun f' (fun x => recF u (f\u2080 x)) x\u271d\u00b9 x\u271d = splitFun f' (fun x => recF u (f\u2081 x)) x\u271d\u00b9 x\u271d"}, {"tactic": "congr", "annotated_tactic": ["congr", []], "state_before": "case refine'_1.e_a.e_a.e_snd.h.h\nn : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\n\u03b2 : Type u\nu : F (\u03b1 ::: \u03b2) \u2192 \u03b2\nx y : MvPFunctor.W (P F) \u03b1\na\u2080 : (P F).A\nf'\u2080 : MvPFunctor.B (MvPFunctor.drop (P F)) a\u2080 \u27f9 \u03b1\nf\u2080\u271d : PFunctor.B (MvPFunctor.last (P F)) a\u2080 \u2192 MvPFunctor.W (P F) \u03b1\na\u2081 : (P F).A\nf'\u2081 : MvPFunctor.B (MvPFunctor.drop (P F)) a\u2081 \u27f9 \u03b1\nf\u2081\u271d : PFunctor.B (MvPFunctor.last (P F)) a\u2081 \u2192 MvPFunctor.W (P F) \u03b1\nh : WEquiv (MvPFunctor.wMk (P F) a\u2080 f'\u2080 f\u2080\u271d) (MvPFunctor.wMk (P F) a\u2081 f'\u2081 f\u2081\u271d)\na : (P F).A\nf' : MvPFunctor.B (MvPFunctor.drop (P F)) a \u27f9 \u03b1\nf\u2080 f\u2081 : PFunctor.B (MvPFunctor.last (P F)) a \u2192 MvPFunctor.W (P F) \u03b1\n_h : \u2200 (x : PFunctor.B (MvPFunctor.last (P F)) a), WEquiv (f\u2080 x) (f\u2081 x)\nih :\n  \u2200 (x : PFunctor.B (MvPFunctor.last (P F)) a),\n    (fun a a' x => recF u a = recF u a') (f\u2080 x) (f\u2081 x) (_ : WEquiv (f\u2080 x) (f\u2081 x))\nx\u271d\u00b9 : Fin2 (n + 1)\nx\u271d : MvPFunctor.B (P F) a x\u271d\u00b9\n\u22a2 splitFun f' (fun x => recF u (f\u2080 x)) x\u271d\u00b9 x\u271d = splitFun f' (fun x => recF u (f\u2081 x)) x\u271d\u00b9 x\u271d", "state_after": "case refine'_1.e_a.e_a.e_snd.h.h.e_g\nn : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\n\u03b2 : Type u\nu : F (\u03b1 ::: \u03b2) \u2192 \u03b2\nx y : MvPFunctor.W (P F) \u03b1\na\u2080 : (P F).A\nf'\u2080 : MvPFunctor.B (MvPFunctor.drop (P F)) a\u2080 \u27f9 \u03b1\nf\u2080\u271d : PFunctor.B (MvPFunctor.last (P F)) a\u2080 \u2192 MvPFunctor.W (P F) \u03b1\na\u2081 : (P F).A\nf'\u2081 : MvPFunctor.B (MvPFunctor.drop (P F)) a\u2081 \u27f9 \u03b1\nf\u2081\u271d : PFunctor.B (MvPFunctor.last (P F)) a\u2081 \u2192 MvPFunctor.W (P F) \u03b1\nh : WEquiv (MvPFunctor.wMk (P F) a\u2080 f'\u2080 f\u2080\u271d) (MvPFunctor.wMk (P F) a\u2081 f'\u2081 f\u2081\u271d)\na : (P F).A\nf' : MvPFunctor.B (MvPFunctor.drop (P F)) a \u27f9 \u03b1\nf\u2080 f\u2081 : PFunctor.B (MvPFunctor.last (P F)) a \u2192 MvPFunctor.W (P F) \u03b1\n_h : \u2200 (x : PFunctor.B (MvPFunctor.last (P F)) a), WEquiv (f\u2080 x) (f\u2081 x)\nih :\n  \u2200 (x : PFunctor.B (MvPFunctor.last (P F)) a),\n    (fun a a' x => recF u a = recF u a') (f\u2080 x) (f\u2081 x) (_ : WEquiv (f\u2080 x) (f\u2081 x))\nx\u271d\u00b9 : Fin2 (n + 1)\nx\u271d : MvPFunctor.B (P F) a x\u271d\u00b9\n\u22a2 (fun x => recF u (f\u2080 x)) = fun x => recF u (f\u2081 x)"}, {"tactic": "funext", "annotated_tactic": ["funext", []], "state_before": "case refine'_1.e_a.e_a.e_snd.h.h.e_g\nn : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\n\u03b2 : Type u\nu : F (\u03b1 ::: \u03b2) \u2192 \u03b2\nx y : MvPFunctor.W (P F) \u03b1\na\u2080 : (P F).A\nf'\u2080 : MvPFunctor.B (MvPFunctor.drop (P F)) a\u2080 \u27f9 \u03b1\nf\u2080\u271d : PFunctor.B (MvPFunctor.last (P F)) a\u2080 \u2192 MvPFunctor.W (P F) \u03b1\na\u2081 : (P F).A\nf'\u2081 : MvPFunctor.B (MvPFunctor.drop (P F)) a\u2081 \u27f9 \u03b1\nf\u2081\u271d : PFunctor.B (MvPFunctor.last (P F)) a\u2081 \u2192 MvPFunctor.W (P F) \u03b1\nh : WEquiv (MvPFunctor.wMk (P F) a\u2080 f'\u2080 f\u2080\u271d) (MvPFunctor.wMk (P F) a\u2081 f'\u2081 f\u2081\u271d)\na : (P F).A\nf' : MvPFunctor.B (MvPFunctor.drop (P F)) a \u27f9 \u03b1\nf\u2080 f\u2081 : PFunctor.B (MvPFunctor.last (P F)) a \u2192 MvPFunctor.W (P F) \u03b1\n_h : \u2200 (x : PFunctor.B (MvPFunctor.last (P F)) a), WEquiv (f\u2080 x) (f\u2081 x)\nih :\n  \u2200 (x : PFunctor.B (MvPFunctor.last (P F)) a),\n    (fun a a' x => recF u a = recF u a') (f\u2080 x) (f\u2081 x) (_ : WEquiv (f\u2080 x) (f\u2081 x))\nx\u271d\u00b9 : Fin2 (n + 1)\nx\u271d : MvPFunctor.B (P F) a x\u271d\u00b9\n\u22a2 (fun x => recF u (f\u2080 x)) = fun x => recF u (f\u2081 x)", "state_after": "case refine'_1.e_a.e_a.e_snd.h.h.e_g.h\nn : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\n\u03b2 : Type u\nu : F (\u03b1 ::: \u03b2) \u2192 \u03b2\nx y : MvPFunctor.W (P F) \u03b1\na\u2080 : (P F).A\nf'\u2080 : MvPFunctor.B (MvPFunctor.drop (P F)) a\u2080 \u27f9 \u03b1\nf\u2080\u271d : PFunctor.B (MvPFunctor.last (P F)) a\u2080 \u2192 MvPFunctor.W (P F) \u03b1\na\u2081 : (P F).A\nf'\u2081 : MvPFunctor.B (MvPFunctor.drop (P F)) a\u2081 \u27f9 \u03b1\nf\u2081\u271d : PFunctor.B (MvPFunctor.last (P F)) a\u2081 \u2192 MvPFunctor.W (P F) \u03b1\nh : WEquiv (MvPFunctor.wMk (P F) a\u2080 f'\u2080 f\u2080\u271d) (MvPFunctor.wMk (P F) a\u2081 f'\u2081 f\u2081\u271d)\na : (P F).A\nf' : MvPFunctor.B (MvPFunctor.drop (P F)) a \u27f9 \u03b1\nf\u2080 f\u2081 : PFunctor.B (MvPFunctor.last (P F)) a \u2192 MvPFunctor.W (P F) \u03b1\n_h : \u2200 (x : PFunctor.B (MvPFunctor.last (P F)) a), WEquiv (f\u2080 x) (f\u2081 x)\nih :\n  \u2200 (x : PFunctor.B (MvPFunctor.last (P F)) a),\n    (fun a a' x => recF u a = recF u a') (f\u2080 x) (f\u2081 x) (_ : WEquiv (f\u2080 x) (f\u2081 x))\nx\u271d\u00b2 : Fin2 (n + 1)\nx\u271d\u00b9 : MvPFunctor.B (P F) a x\u271d\u00b2\nx\u271d : last (MvPFunctor.B (P F) a)\n\u22a2 recF u (f\u2080 x\u271d) = recF u (f\u2081 x\u271d)"}, {"tactic": "apply ih", "annotated_tactic": ["apply ih", []], "state_before": "case refine'_1.e_a.e_a.e_snd.h.h.e_g.h\nn : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\n\u03b2 : Type u\nu : F (\u03b1 ::: \u03b2) \u2192 \u03b2\nx y : MvPFunctor.W (P F) \u03b1\na\u2080 : (P F).A\nf'\u2080 : MvPFunctor.B (MvPFunctor.drop (P F)) a\u2080 \u27f9 \u03b1\nf\u2080\u271d : PFunctor.B (MvPFunctor.last (P F)) a\u2080 \u2192 MvPFunctor.W (P F) \u03b1\na\u2081 : (P F).A\nf'\u2081 : MvPFunctor.B (MvPFunctor.drop (P F)) a\u2081 \u27f9 \u03b1\nf\u2081\u271d : PFunctor.B (MvPFunctor.last (P F)) a\u2081 \u2192 MvPFunctor.W (P F) \u03b1\nh : WEquiv (MvPFunctor.wMk (P F) a\u2080 f'\u2080 f\u2080\u271d) (MvPFunctor.wMk (P F) a\u2081 f'\u2081 f\u2081\u271d)\na : (P F).A\nf' : MvPFunctor.B (MvPFunctor.drop (P F)) a \u27f9 \u03b1\nf\u2080 f\u2081 : PFunctor.B (MvPFunctor.last (P F)) a \u2192 MvPFunctor.W (P F) \u03b1\n_h : \u2200 (x : PFunctor.B (MvPFunctor.last (P F)) a), WEquiv (f\u2080 x) (f\u2081 x)\nih :\n  \u2200 (x : PFunctor.B (MvPFunctor.last (P F)) a),\n    (fun a a' x => recF u a = recF u a') (f\u2080 x) (f\u2081 x) (_ : WEquiv (f\u2080 x) (f\u2081 x))\nx\u271d\u00b2 : Fin2 (n + 1)\nx\u271d\u00b9 : MvPFunctor.B (P F) a x\u271d\u00b2\nx\u271d : last (MvPFunctor.B (P F) a)\n\u22a2 recF u (f\u2080 x\u271d) = recF u (f\u2081 x\u271d)", "state_after": "no goals"}, {"tactic": "intros a\u2080 f'\u2080 f\u2080 a\u2081 f'\u2081 f\u2081 h", "annotated_tactic": ["intros a\u2080 f'\u2080 f\u2080 a\u2081 f'\u2081 f\u2081 h", []], "state_before": "case refine'_2\nn : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\n\u03b2 : Type u\nu : F (\u03b1 ::: \u03b2) \u2192 \u03b2\nx y : MvPFunctor.W (P F) \u03b1\na\u2080 : (P F).A\nf'\u2080 : MvPFunctor.B (MvPFunctor.drop (P F)) a\u2080 \u27f9 \u03b1\nf\u2080 : PFunctor.B (MvPFunctor.last (P F)) a\u2080 \u2192 MvPFunctor.W (P F) \u03b1\na\u2081 : (P F).A\nf'\u2081 : MvPFunctor.B (MvPFunctor.drop (P F)) a\u2081 \u27f9 \u03b1\nf\u2081 : PFunctor.B (MvPFunctor.last (P F)) a\u2081 \u2192 MvPFunctor.W (P F) \u03b1\nh : WEquiv (MvPFunctor.wMk (P F) a\u2080 f'\u2080 f\u2080) (MvPFunctor.wMk (P F) a\u2081 f'\u2081 f\u2081)\n\u22a2 \u2200 (a\u2080 : (P F).A) (f'\u2080 : MvPFunctor.B (MvPFunctor.drop (P F)) a\u2080 \u27f9 \u03b1)\n    (f\u2080 : PFunctor.B (MvPFunctor.last (P F)) a\u2080 \u2192 MvPFunctor.W (P F) \u03b1) (a\u2081 : (P F).A)\n    (f'\u2081 : MvPFunctor.B (MvPFunctor.drop (P F)) a\u2081 \u27f9 \u03b1)\n    (f\u2081 : PFunctor.B (MvPFunctor.last (P F)) a\u2081 \u2192 MvPFunctor.W (P F) \u03b1)\n    (a :\n      abs { fst := a\u2080, snd := MvPFunctor.appendContents (P F) f'\u2080 f\u2080 } =\n        abs { fst := a\u2081, snd := MvPFunctor.appendContents (P F) f'\u2081 f\u2081 }),\n    (fun a a' x => recF u a = recF u a') (MvPFunctor.wMk (P F) a\u2080 f'\u2080 f\u2080) (MvPFunctor.wMk (P F) a\u2081 f'\u2081 f\u2081)\n      (_ : WEquiv (MvPFunctor.wMk (P F) a\u2080 f'\u2080 f\u2080) (MvPFunctor.wMk (P F) a\u2081 f'\u2081 f\u2081))", "state_after": "case refine'_2\nn : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\n\u03b2 : Type u\nu : F (\u03b1 ::: \u03b2) \u2192 \u03b2\nx y : MvPFunctor.W (P F) \u03b1\na\u2080\u271d : (P F).A\nf'\u2080\u271d : MvPFunctor.B (MvPFunctor.drop (P F)) a\u2080\u271d \u27f9 \u03b1\nf\u2080\u271d : PFunctor.B (MvPFunctor.last (P F)) a\u2080\u271d \u2192 MvPFunctor.W (P F) \u03b1\na\u2081\u271d : (P F).A\nf'\u2081\u271d : MvPFunctor.B (MvPFunctor.drop (P F)) a\u2081\u271d \u27f9 \u03b1\nf\u2081\u271d : PFunctor.B (MvPFunctor.last (P F)) a\u2081\u271d \u2192 MvPFunctor.W (P F) \u03b1\nh\u271d : WEquiv (MvPFunctor.wMk (P F) a\u2080\u271d f'\u2080\u271d f\u2080\u271d) (MvPFunctor.wMk (P F) a\u2081\u271d f'\u2081\u271d f\u2081\u271d)\na\u2080 : (P F).A\nf'\u2080 : MvPFunctor.B (MvPFunctor.drop (P F)) a\u2080 \u27f9 \u03b1\nf\u2080 : PFunctor.B (MvPFunctor.last (P F)) a\u2080 \u2192 MvPFunctor.W (P F) \u03b1\na\u2081 : (P F).A\nf'\u2081 : MvPFunctor.B (MvPFunctor.drop (P F)) a\u2081 \u27f9 \u03b1\nf\u2081 : PFunctor.B (MvPFunctor.last (P F)) a\u2081 \u2192 MvPFunctor.W (P F) \u03b1\nh :\n  abs { fst := a\u2080, snd := MvPFunctor.appendContents (P F) f'\u2080 f\u2080 } =\n    abs { fst := a\u2081, snd := MvPFunctor.appendContents (P F) f'\u2081 f\u2081 }\n\u22a2 recF u (MvPFunctor.wMk (P F) a\u2080 f'\u2080 f\u2080) = recF u (MvPFunctor.wMk (P F) a\u2081 f'\u2081 f\u2081)"}, {"tactic": "simp only [recF_eq', abs_map, MvPFunctor.wDest'_wMk, h]", "annotated_tactic": ["simp only [<a>recF_eq'</a>, <a>abs_map</a>, <a>MvPFunctor.wDest'_wMk</a>, h]", [{"full_name": "MvQPF.recF_eq'", "def_path": "Mathlib/Data/QPF/Multivariate/Constructions/Fix.lean", "def_pos": [71, 9], "def_end_pos": [71, 17]}, {"full_name": "MvQPF.abs_map", "def_path": "Mathlib/Data/QPF/Multivariate/Basic.lean", "def_pos": [90, 3], "def_end_pos": [90, 10]}, {"full_name": "MvPFunctor.wDest'_wMk", "def_path": "Mathlib/Data/PFunctor/Multivariate/W.lean", "def_pos": [306, 9], "def_end_pos": [306, 19]}]], "state_before": "case refine'_2\nn : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\n\u03b2 : Type u\nu : F (\u03b1 ::: \u03b2) \u2192 \u03b2\nx y : MvPFunctor.W (P F) \u03b1\na\u2080\u271d : (P F).A\nf'\u2080\u271d : MvPFunctor.B (MvPFunctor.drop (P F)) a\u2080\u271d \u27f9 \u03b1\nf\u2080\u271d : PFunctor.B (MvPFunctor.last (P F)) a\u2080\u271d \u2192 MvPFunctor.W (P F) \u03b1\na\u2081\u271d : (P F).A\nf'\u2081\u271d : MvPFunctor.B (MvPFunctor.drop (P F)) a\u2081\u271d \u27f9 \u03b1\nf\u2081\u271d : PFunctor.B (MvPFunctor.last (P F)) a\u2081\u271d \u2192 MvPFunctor.W (P F) \u03b1\nh\u271d : WEquiv (MvPFunctor.wMk (P F) a\u2080\u271d f'\u2080\u271d f\u2080\u271d) (MvPFunctor.wMk (P F) a\u2081\u271d f'\u2081\u271d f\u2081\u271d)\na\u2080 : (P F).A\nf'\u2080 : MvPFunctor.B (MvPFunctor.drop (P F)) a\u2080 \u27f9 \u03b1\nf\u2080 : PFunctor.B (MvPFunctor.last (P F)) a\u2080 \u2192 MvPFunctor.W (P F) \u03b1\na\u2081 : (P F).A\nf'\u2081 : MvPFunctor.B (MvPFunctor.drop (P F)) a\u2081 \u27f9 \u03b1\nf\u2081 : PFunctor.B (MvPFunctor.last (P F)) a\u2081 \u2192 MvPFunctor.W (P F) \u03b1\nh :\n  abs { fst := a\u2080, snd := MvPFunctor.appendContents (P F) f'\u2080 f\u2080 } =\n    abs { fst := a\u2081, snd := MvPFunctor.appendContents (P F) f'\u2081 f\u2081 }\n\u22a2 recF u (MvPFunctor.wMk (P F) a\u2080 f'\u2080 f\u2080) = recF u (MvPFunctor.wMk (P F) a\u2081 f'\u2081 f\u2081)", "state_after": "no goals"}, {"tactic": "intros x y z _e\u2081 _e\u2082 ih\u2081 ih\u2082", "annotated_tactic": ["intros x y z _e\u2081 _e\u2082 ih\u2081 ih\u2082", []], "state_before": "case refine'_3\nn : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\n\u03b2 : Type u\nu : F (\u03b1 ::: \u03b2) \u2192 \u03b2\nx y : MvPFunctor.W (P F) \u03b1\na\u2080 : (P F).A\nf'\u2080 : MvPFunctor.B (MvPFunctor.drop (P F)) a\u2080 \u27f9 \u03b1\nf\u2080 : PFunctor.B (MvPFunctor.last (P F)) a\u2080 \u2192 MvPFunctor.W (P F) \u03b1\na\u2081 : (P F).A\nf'\u2081 : MvPFunctor.B (MvPFunctor.drop (P F)) a\u2081 \u27f9 \u03b1\nf\u2081 : PFunctor.B (MvPFunctor.last (P F)) a\u2081 \u2192 MvPFunctor.W (P F) \u03b1\nh : WEquiv (MvPFunctor.wMk (P F) a\u2080 f'\u2080 f\u2080) (MvPFunctor.wMk (P F) a\u2081 f'\u2081 f\u2081)\n\u22a2 \u2200 (u_1 v w : MvPFunctor.W (P F) \u03b1) (a : WEquiv u_1 v) (a_1 : WEquiv v w),\n    (fun a a' x => recF u a = recF u a') u_1 v a \u2192\n      (fun a a' x => recF u a = recF u a') v w a_1 \u2192 (fun a a' x => recF u a = recF u a') u_1 w (_ : WEquiv u_1 w)", "state_after": "case refine'_3\nn : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\n\u03b2 : Type u\nu : F (\u03b1 ::: \u03b2) \u2192 \u03b2\nx\u271d y\u271d : MvPFunctor.W (P F) \u03b1\na\u2080 : (P F).A\nf'\u2080 : MvPFunctor.B (MvPFunctor.drop (P F)) a\u2080 \u27f9 \u03b1\nf\u2080 : PFunctor.B (MvPFunctor.last (P F)) a\u2080 \u2192 MvPFunctor.W (P F) \u03b1\na\u2081 : (P F).A\nf'\u2081 : MvPFunctor.B (MvPFunctor.drop (P F)) a\u2081 \u27f9 \u03b1\nf\u2081 : PFunctor.B (MvPFunctor.last (P F)) a\u2081 \u2192 MvPFunctor.W (P F) \u03b1\nh : WEquiv (MvPFunctor.wMk (P F) a\u2080 f'\u2080 f\u2080) (MvPFunctor.wMk (P F) a\u2081 f'\u2081 f\u2081)\nx y z : MvPFunctor.W (P F) \u03b1\n_e\u2081 : WEquiv x y\n_e\u2082 : WEquiv y z\nih\u2081 : recF u x = recF u y\nih\u2082 : recF u y = recF u z\n\u22a2 recF u x = recF u z"}, {"tactic": "exact Eq.trans ih\u2081 ih\u2082", "annotated_tactic": ["exact <a>Eq.trans</a> ih\u2081 ih\u2082", [{"full_name": "Eq.trans", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [322, 9], "def_end_pos": [322, 17]}]], "state_before": "case refine'_3\nn : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\n\u03b2 : Type u\nu : F (\u03b1 ::: \u03b2) \u2192 \u03b2\nx\u271d y\u271d : MvPFunctor.W (P F) \u03b1\na\u2080 : (P F).A\nf'\u2080 : MvPFunctor.B (MvPFunctor.drop (P F)) a\u2080 \u27f9 \u03b1\nf\u2080 : PFunctor.B (MvPFunctor.last (P F)) a\u2080 \u2192 MvPFunctor.W (P F) \u03b1\na\u2081 : (P F).A\nf'\u2081 : MvPFunctor.B (MvPFunctor.drop (P F)) a\u2081 \u27f9 \u03b1\nf\u2081 : PFunctor.B (MvPFunctor.last (P F)) a\u2081 \u2192 MvPFunctor.W (P F) \u03b1\nh : WEquiv (MvPFunctor.wMk (P F) a\u2080 f'\u2080 f\u2080) (MvPFunctor.wMk (P F) a\u2081 f'\u2081 f\u2081)\nx y z : MvPFunctor.W (P F) \u03b1\n_e\u2081 : WEquiv x y\n_e\u2082 : WEquiv y z\nih\u2081 : recF u x = recF u y\nih\u2082 : recF u y = recF u z\n\u22a2 recF u x = recF u z", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/CondCount.lean", "full_name": "ProbabilityTheory.condCount_add_compl_eq", "start": [193, 1], "end": [202, 33], "traced_tactics": [{"tactic": "have : condCount s t = (condCount (s \u2229 u) t * condCount (s \u2229 u \u222a s \u2229 u\u1d9c) (s \u2229 u) +\n    condCount (s \u2229 u\u1d9c) t * condCount (s \u2229 u \u222a s \u2229 u\u1d9c) (s \u2229 u\u1d9c)) := by\n  rw [condCount_disjoint_union (hs.inter_of_left _) (hs.inter_of_left _)\n    (disjoint_compl_right.mono inf_le_right inf_le_right), Set.inter_union_compl]", "annotated_tactic": ["have : <a>condCount</a> s t = (<a>condCount</a> (s \u2229 u) t * <a>condCount</a> (s \u2229 u \u222a s \u2229 u\u1d9c) (s \u2229 u) +\n      <a>condCount</a> (s \u2229 u\u1d9c) t * <a>condCount</a> (s \u2229 u \u222a s \u2229 u\u1d9c) (s \u2229 u\u1d9c)) := by\n    rw [<a>condCount_disjoint_union</a> (hs.inter_of_left _) (hs.inter_of_left _)\n      (disjoint_compl_right.mono <a>inf_le_right</a> <a>inf_le_right</a>), <a>Set.inter_union_compl</a>]", [{"full_name": "ProbabilityTheory.condCount", "def_path": "Mathlib/Probability/CondCount.lean", "def_pos": [54, 5], "def_end_pos": [54, 14]}, {"full_name": "ProbabilityTheory.condCount", "def_path": "Mathlib/Probability/CondCount.lean", "def_pos": [54, 5], "def_end_pos": [54, 14]}, {"full_name": "ProbabilityTheory.condCount", "def_path": "Mathlib/Probability/CondCount.lean", "def_pos": [54, 5], "def_end_pos": [54, 14]}, {"full_name": "ProbabilityTheory.condCount", "def_path": "Mathlib/Probability/CondCount.lean", "def_pos": [54, 5], "def_end_pos": [54, 14]}, {"full_name": "ProbabilityTheory.condCount", "def_path": "Mathlib/Probability/CondCount.lean", "def_pos": [54, 5], "def_end_pos": [54, 14]}, {"full_name": "ProbabilityTheory.condCount_disjoint_union", "def_path": "Mathlib/Probability/CondCount.lean", "def_pos": [170, 9], "def_end_pos": [170, 33]}, {"full_name": "inf_le_right", "def_path": "Mathlib/Order/Lattice.lean", "def_pos": [399, 9], "def_end_pos": [399, 21]}, {"full_name": "inf_le_right", "def_path": "Mathlib/Order/Lattice.lean", "def_pos": [399, 9], "def_end_pos": [399, 21]}, {"full_name": "Set.inter_union_compl", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1900, 9], "def_end_pos": [1900, 26]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t\u271d u\u271d u t : Set \u03a9\nhs : Set.Finite s\n\u22a2 \u2191\u2191(condCount (s \u2229 u)) t * \u2191\u2191(condCount s) u + \u2191\u2191(condCount (s \u2229 u\u1d9c)) t * \u2191\u2191(condCount s) u\u1d9c = \u2191\u2191(condCount s) t", "state_after": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t\u271d u\u271d u t : Set \u03a9\nhs : Set.Finite s\nthis :\n  \u2191\u2191(condCount s) t =\n    \u2191\u2191(condCount (s \u2229 u)) t * \u2191\u2191(condCount (s \u2229 u \u222a s \u2229 u\u1d9c)) (s \u2229 u) +\n      \u2191\u2191(condCount (s \u2229 u\u1d9c)) t * \u2191\u2191(condCount (s \u2229 u \u222a s \u2229 u\u1d9c)) (s \u2229 u\u1d9c)\n\u22a2 \u2191\u2191(condCount (s \u2229 u)) t * \u2191\u2191(condCount s) u + \u2191\u2191(condCount (s \u2229 u\u1d9c)) t * \u2191\u2191(condCount s) u\u1d9c = \u2191\u2191(condCount s) t"}, {"tactic": "rw [this]", "annotated_tactic": ["rw [this]", []], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t\u271d u\u271d u t : Set \u03a9\nhs : Set.Finite s\nthis :\n  \u2191\u2191(condCount s) t =\n    \u2191\u2191(condCount (s \u2229 u)) t * \u2191\u2191(condCount (s \u2229 u \u222a s \u2229 u\u1d9c)) (s \u2229 u) +\n      \u2191\u2191(condCount (s \u2229 u\u1d9c)) t * \u2191\u2191(condCount (s \u2229 u \u222a s \u2229 u\u1d9c)) (s \u2229 u\u1d9c)\n\u22a2 \u2191\u2191(condCount (s \u2229 u)) t * \u2191\u2191(condCount s) u + \u2191\u2191(condCount (s \u2229 u\u1d9c)) t * \u2191\u2191(condCount s) u\u1d9c = \u2191\u2191(condCount s) t", "state_after": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t\u271d u\u271d u t : Set \u03a9\nhs : Set.Finite s\nthis :\n  \u2191\u2191(condCount s) t =\n    \u2191\u2191(condCount (s \u2229 u)) t * \u2191\u2191(condCount (s \u2229 u \u222a s \u2229 u\u1d9c)) (s \u2229 u) +\n      \u2191\u2191(condCount (s \u2229 u\u1d9c)) t * \u2191\u2191(condCount (s \u2229 u \u222a s \u2229 u\u1d9c)) (s \u2229 u\u1d9c)\n\u22a2 \u2191\u2191(condCount (s \u2229 u)) t * \u2191\u2191(condCount s) u + \u2191\u2191(condCount (s \u2229 u\u1d9c)) t * \u2191\u2191(condCount s) u\u1d9c =\n    \u2191\u2191(condCount (s \u2229 u)) t * \u2191\u2191(condCount (s \u2229 u \u222a s \u2229 u\u1d9c)) (s \u2229 u) +\n      \u2191\u2191(condCount (s \u2229 u\u1d9c)) t * \u2191\u2191(condCount (s \u2229 u \u222a s \u2229 u\u1d9c)) (s \u2229 u\u1d9c)"}, {"tactic": "simp [condCount_inter_self hs]", "annotated_tactic": ["simp [<a>condCount_inter_self</a> hs]", [{"full_name": "ProbabilityTheory.condCount_inter_self", "def_path": "Mathlib/Probability/CondCount.lean", "def_pos": [100, 9], "def_end_pos": [100, 29]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t\u271d u\u271d u t : Set \u03a9\nhs : Set.Finite s\nthis :\n  \u2191\u2191(condCount s) t =\n    \u2191\u2191(condCount (s \u2229 u)) t * \u2191\u2191(condCount (s \u2229 u \u222a s \u2229 u\u1d9c)) (s \u2229 u) +\n      \u2191\u2191(condCount (s \u2229 u\u1d9c)) t * \u2191\u2191(condCount (s \u2229 u \u222a s \u2229 u\u1d9c)) (s \u2229 u\u1d9c)\n\u22a2 \u2191\u2191(condCount (s \u2229 u)) t * \u2191\u2191(condCount s) u + \u2191\u2191(condCount (s \u2229 u\u1d9c)) t * \u2191\u2191(condCount s) u\u1d9c =\n    \u2191\u2191(condCount (s \u2229 u)) t * \u2191\u2191(condCount (s \u2229 u \u222a s \u2229 u\u1d9c)) (s \u2229 u) +\n      \u2191\u2191(condCount (s \u2229 u\u1d9c)) t * \u2191\u2191(condCount (s \u2229 u \u222a s \u2229 u\u1d9c)) (s \u2229 u\u1d9c)", "state_after": "no goals"}, {"tactic": "rw [condCount_disjoint_union (hs.inter_of_left _) (hs.inter_of_left _)\n  (disjoint_compl_right.mono inf_le_right inf_le_right), Set.inter_union_compl]", "annotated_tactic": ["rw [<a>condCount_disjoint_union</a> (hs.inter_of_left _) (hs.inter_of_left _)\n      (disjoint_compl_right.mono <a>inf_le_right</a> <a>inf_le_right</a>), <a>Set.inter_union_compl</a>]", [{"full_name": "ProbabilityTheory.condCount_disjoint_union", "def_path": "Mathlib/Probability/CondCount.lean", "def_pos": [170, 9], "def_end_pos": [170, 33]}, {"full_name": "inf_le_right", "def_path": "Mathlib/Order/Lattice.lean", "def_pos": [399, 9], "def_end_pos": [399, 21]}, {"full_name": "inf_le_right", "def_path": "Mathlib/Order/Lattice.lean", "def_pos": [399, 9], "def_end_pos": [399, 21]}, {"full_name": "Set.inter_union_compl", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1900, 9], "def_end_pos": [1900, 26]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t\u271d u\u271d u t : Set \u03a9\nhs : Set.Finite s\n\u22a2 \u2191\u2191(condCount s) t =\n    \u2191\u2191(condCount (s \u2229 u)) t * \u2191\u2191(condCount (s \u2229 u \u222a s \u2229 u\u1d9c)) (s \u2229 u) +\n      \u2191\u2191(condCount (s \u2229 u\u1d9c)) t * \u2191\u2191(condCount (s \u2229 u \u222a s \u2229 u\u1d9c)) (s \u2229 u\u1d9c)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "full_name": "MeasureTheory.Measure.finiteAt_nhds", "start": [3815, 1], "end": [3817, 40], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "full_name": "MeasureTheory.VectorMeasure.MutuallySingular.smul_left", "start": [1230, 1], "end": [1232, 29], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Process/Stopping.lean", "full_name": "MeasureTheory.IsStoppingTime.measurableSet_eq_stopping_time_of_countable", "start": [733, 1], "end": [754, 70], "traced_tactics": [{"tactic": "rw [h\u03c4.measurableSet]", "annotated_tactic": ["rw [h\u03c4.measurableSet]", []], "state_before": "\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2077 : LinearOrder \u03b9\nf : Filtration \u03b9 m\n\u03c4 \u03c0 : \u03a9 \u2192 \u03b9\ninst\u271d\u2076 : Countable \u03b9\ninst\u271d\u2075 : TopologicalSpace \u03b9\ninst\u271d\u2074 : MeasurableSpace \u03b9\ninst\u271d\u00b3 : BorelSpace \u03b9\ninst\u271d\u00b2 : OrderTopology \u03b9\ninst\u271d\u00b9 : MeasurableSingletonClass \u03b9\ninst\u271d : SecondCountableTopology \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\nh\u03c0 : IsStoppingTime f \u03c0\n\u22a2 MeasurableSet {\u03c9 | \u03c4 \u03c9 = \u03c0 \u03c9}", "state_after": "\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2077 : LinearOrder \u03b9\nf : Filtration \u03b9 m\n\u03c4 \u03c0 : \u03a9 \u2192 \u03b9\ninst\u271d\u2076 : Countable \u03b9\ninst\u271d\u2075 : TopologicalSpace \u03b9\ninst\u271d\u2074 : MeasurableSpace \u03b9\ninst\u271d\u00b3 : BorelSpace \u03b9\ninst\u271d\u00b2 : OrderTopology \u03b9\ninst\u271d\u00b9 : MeasurableSingletonClass \u03b9\ninst\u271d : SecondCountableTopology \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\nh\u03c0 : IsStoppingTime f \u03c0\n\u22a2 \u2200 (i : \u03b9), MeasurableSet ({\u03c9 | \u03c4 \u03c9 = \u03c0 \u03c9} \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 i})"}, {"tactic": "intro j", "annotated_tactic": ["intro j", []], "state_before": "\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2077 : LinearOrder \u03b9\nf : Filtration \u03b9 m\n\u03c4 \u03c0 : \u03a9 \u2192 \u03b9\ninst\u271d\u2076 : Countable \u03b9\ninst\u271d\u2075 : TopologicalSpace \u03b9\ninst\u271d\u2074 : MeasurableSpace \u03b9\ninst\u271d\u00b3 : BorelSpace \u03b9\ninst\u271d\u00b2 : OrderTopology \u03b9\ninst\u271d\u00b9 : MeasurableSingletonClass \u03b9\ninst\u271d : SecondCountableTopology \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\nh\u03c0 : IsStoppingTime f \u03c0\n\u22a2 \u2200 (i : \u03b9), MeasurableSet ({\u03c9 | \u03c4 \u03c9 = \u03c0 \u03c9} \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 i})", "state_after": "\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2077 : LinearOrder \u03b9\nf : Filtration \u03b9 m\n\u03c4 \u03c0 : \u03a9 \u2192 \u03b9\ninst\u271d\u2076 : Countable \u03b9\ninst\u271d\u2075 : TopologicalSpace \u03b9\ninst\u271d\u2074 : MeasurableSpace \u03b9\ninst\u271d\u00b3 : BorelSpace \u03b9\ninst\u271d\u00b2 : OrderTopology \u03b9\ninst\u271d\u00b9 : MeasurableSingletonClass \u03b9\ninst\u271d : SecondCountableTopology \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\nh\u03c0 : IsStoppingTime f \u03c0\nj : \u03b9\n\u22a2 MeasurableSet ({\u03c9 | \u03c4 \u03c9 = \u03c0 \u03c9} \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 j})"}, {"tactic": "rw [this]", "annotated_tactic": ["rw [this]", []], "state_before": "\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2077 : LinearOrder \u03b9\nf : Filtration \u03b9 m\n\u03c4 \u03c0 : \u03a9 \u2192 \u03b9\ninst\u271d\u2076 : Countable \u03b9\ninst\u271d\u2075 : TopologicalSpace \u03b9\ninst\u271d\u2074 : MeasurableSpace \u03b9\ninst\u271d\u00b3 : BorelSpace \u03b9\ninst\u271d\u00b2 : OrderTopology \u03b9\ninst\u271d\u00b9 : MeasurableSingletonClass \u03b9\ninst\u271d : SecondCountableTopology \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\nh\u03c0 : IsStoppingTime f \u03c0\nj : \u03b9\nthis : {\u03c9 | \u03c4 \u03c9 = \u03c0 \u03c9} \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 j} = {\u03c9 | min (\u03c4 \u03c9) j = min (\u03c0 \u03c9) j} \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 j} \u2229 {\u03c9 | \u03c0 \u03c9 \u2264 j}\n\u22a2 MeasurableSet ({\u03c9 | \u03c4 \u03c9 = \u03c0 \u03c9} \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 j})", "state_after": "\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2077 : LinearOrder \u03b9\nf : Filtration \u03b9 m\n\u03c4 \u03c0 : \u03a9 \u2192 \u03b9\ninst\u271d\u2076 : Countable \u03b9\ninst\u271d\u2075 : TopologicalSpace \u03b9\ninst\u271d\u2074 : MeasurableSpace \u03b9\ninst\u271d\u00b3 : BorelSpace \u03b9\ninst\u271d\u00b2 : OrderTopology \u03b9\ninst\u271d\u00b9 : MeasurableSingletonClass \u03b9\ninst\u271d : SecondCountableTopology \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\nh\u03c0 : IsStoppingTime f \u03c0\nj : \u03b9\nthis : {\u03c9 | \u03c4 \u03c9 = \u03c0 \u03c9} \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 j} = {\u03c9 | min (\u03c4 \u03c9) j = min (\u03c0 \u03c9) j} \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 j} \u2229 {\u03c9 | \u03c0 \u03c9 \u2264 j}\n\u22a2 MeasurableSet ({\u03c9 | min (\u03c4 \u03c9) j = min (\u03c0 \u03c9) j} \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 j} \u2229 {\u03c9 | \u03c0 \u03c9 \u2264 j})"}, {"tactic": "refine'\n  MeasurableSet.inter (MeasurableSet.inter _ (h\u03c4.measurableSet_le j)) (h\u03c0.measurableSet_le j)", "annotated_tactic": ["refine'\n    <a>MeasurableSet.inter</a> (<a>MeasurableSet.inter</a> _ (h\u03c4.measurableSet_le j)) (h\u03c0.measurableSet_le j)", [{"full_name": "MeasurableSet.inter", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [198, 19], "def_end_pos": [198, 38]}, {"full_name": "MeasurableSet.inter", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [198, 19], "def_end_pos": [198, 38]}]], "state_before": "\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2077 : LinearOrder \u03b9\nf : Filtration \u03b9 m\n\u03c4 \u03c0 : \u03a9 \u2192 \u03b9\ninst\u271d\u2076 : Countable \u03b9\ninst\u271d\u2075 : TopologicalSpace \u03b9\ninst\u271d\u2074 : MeasurableSpace \u03b9\ninst\u271d\u00b3 : BorelSpace \u03b9\ninst\u271d\u00b2 : OrderTopology \u03b9\ninst\u271d\u00b9 : MeasurableSingletonClass \u03b9\ninst\u271d : SecondCountableTopology \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\nh\u03c0 : IsStoppingTime f \u03c0\nj : \u03b9\nthis : {\u03c9 | \u03c4 \u03c9 = \u03c0 \u03c9} \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 j} = {\u03c9 | min (\u03c4 \u03c9) j = min (\u03c0 \u03c9) j} \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 j} \u2229 {\u03c9 | \u03c0 \u03c9 \u2264 j}\n\u22a2 MeasurableSet ({\u03c9 | min (\u03c4 \u03c9) j = min (\u03c0 \u03c9) j} \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 j} \u2229 {\u03c9 | \u03c0 \u03c9 \u2264 j})", "state_after": "\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2077 : LinearOrder \u03b9\nf : Filtration \u03b9 m\n\u03c4 \u03c0 : \u03a9 \u2192 \u03b9\ninst\u271d\u2076 : Countable \u03b9\ninst\u271d\u2075 : TopologicalSpace \u03b9\ninst\u271d\u2074 : MeasurableSpace \u03b9\ninst\u271d\u00b3 : BorelSpace \u03b9\ninst\u271d\u00b2 : OrderTopology \u03b9\ninst\u271d\u00b9 : MeasurableSingletonClass \u03b9\ninst\u271d : SecondCountableTopology \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\nh\u03c0 : IsStoppingTime f \u03c0\nj : \u03b9\nthis : {\u03c9 | \u03c4 \u03c9 = \u03c0 \u03c9} \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 j} = {\u03c9 | min (\u03c4 \u03c9) j = min (\u03c0 \u03c9) j} \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 j} \u2229 {\u03c9 | \u03c0 \u03c9 \u2264 j}\n\u22a2 MeasurableSet {\u03c9 | min (\u03c4 \u03c9) j = min (\u03c0 \u03c9) j}"}, {"tactic": "apply measurableSet_eq_fun_of_countable", "annotated_tactic": ["apply <a>measurableSet_eq_fun_of_countable</a>", [{"full_name": "measurableSet_eq_fun_of_countable", "def_path": "Mathlib/MeasureTheory/Group/Arithmetic.lean", "def_pos": [396, 9], "def_end_pos": [396, 42]}]], "state_before": "\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2077 : LinearOrder \u03b9\nf : Filtration \u03b9 m\n\u03c4 \u03c0 : \u03a9 \u2192 \u03b9\ninst\u271d\u2076 : Countable \u03b9\ninst\u271d\u2075 : TopologicalSpace \u03b9\ninst\u271d\u2074 : MeasurableSpace \u03b9\ninst\u271d\u00b3 : BorelSpace \u03b9\ninst\u271d\u00b2 : OrderTopology \u03b9\ninst\u271d\u00b9 : MeasurableSingletonClass \u03b9\ninst\u271d : SecondCountableTopology \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\nh\u03c0 : IsStoppingTime f \u03c0\nj : \u03b9\nthis : {\u03c9 | \u03c4 \u03c9 = \u03c0 \u03c9} \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 j} = {\u03c9 | min (\u03c4 \u03c9) j = min (\u03c0 \u03c9) j} \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 j} \u2229 {\u03c9 | \u03c0 \u03c9 \u2264 j}\n\u22a2 MeasurableSet {\u03c9 | min (\u03c4 \u03c9) j = min (\u03c0 \u03c9) j}", "state_after": "case hf\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2077 : LinearOrder \u03b9\nf : Filtration \u03b9 m\n\u03c4 \u03c0 : \u03a9 \u2192 \u03b9\ninst\u271d\u2076 : Countable \u03b9\ninst\u271d\u2075 : TopologicalSpace \u03b9\ninst\u271d\u2074 : MeasurableSpace \u03b9\ninst\u271d\u00b3 : BorelSpace \u03b9\ninst\u271d\u00b2 : OrderTopology \u03b9\ninst\u271d\u00b9 : MeasurableSingletonClass \u03b9\ninst\u271d : SecondCountableTopology \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\nh\u03c0 : IsStoppingTime f \u03c0\nj : \u03b9\nthis : {\u03c9 | \u03c4 \u03c9 = \u03c0 \u03c9} \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 j} = {\u03c9 | min (\u03c4 \u03c9) j = min (\u03c0 \u03c9) j} \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 j} \u2229 {\u03c9 | \u03c0 \u03c9 \u2264 j}\n\u22a2 Measurable fun x => min (\u03c4 x) j\n\ncase hg\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2077 : LinearOrder \u03b9\nf : Filtration \u03b9 m\n\u03c4 \u03c0 : \u03a9 \u2192 \u03b9\ninst\u271d\u2076 : Countable \u03b9\ninst\u271d\u2075 : TopologicalSpace \u03b9\ninst\u271d\u2074 : MeasurableSpace \u03b9\ninst\u271d\u00b3 : BorelSpace \u03b9\ninst\u271d\u00b2 : OrderTopology \u03b9\ninst\u271d\u00b9 : MeasurableSingletonClass \u03b9\ninst\u271d : SecondCountableTopology \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\nh\u03c0 : IsStoppingTime f \u03c0\nj : \u03b9\nthis : {\u03c9 | \u03c4 \u03c9 = \u03c0 \u03c9} \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 j} = {\u03c9 | min (\u03c4 \u03c9) j = min (\u03c0 \u03c9) j} \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 j} \u2229 {\u03c9 | \u03c0 \u03c9 \u2264 j}\n\u22a2 Measurable fun x => min (\u03c0 x) j"}, {"tactic": "ext1 \u03c9", "annotated_tactic": ["ext1 \u03c9", []], "state_before": "\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2077 : LinearOrder \u03b9\nf : Filtration \u03b9 m\n\u03c4 \u03c0 : \u03a9 \u2192 \u03b9\ninst\u271d\u2076 : Countable \u03b9\ninst\u271d\u2075 : TopologicalSpace \u03b9\ninst\u271d\u2074 : MeasurableSpace \u03b9\ninst\u271d\u00b3 : BorelSpace \u03b9\ninst\u271d\u00b2 : OrderTopology \u03b9\ninst\u271d\u00b9 : MeasurableSingletonClass \u03b9\ninst\u271d : SecondCountableTopology \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\nh\u03c0 : IsStoppingTime f \u03c0\nj : \u03b9\n\u22a2 {\u03c9 | \u03c4 \u03c9 = \u03c0 \u03c9} \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 j} = {\u03c9 | min (\u03c4 \u03c9) j = min (\u03c0 \u03c9) j} \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 j} \u2229 {\u03c9 | \u03c0 \u03c9 \u2264 j}", "state_after": "case h\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2077 : LinearOrder \u03b9\nf : Filtration \u03b9 m\n\u03c4 \u03c0 : \u03a9 \u2192 \u03b9\ninst\u271d\u2076 : Countable \u03b9\ninst\u271d\u2075 : TopologicalSpace \u03b9\ninst\u271d\u2074 : MeasurableSpace \u03b9\ninst\u271d\u00b3 : BorelSpace \u03b9\ninst\u271d\u00b2 : OrderTopology \u03b9\ninst\u271d\u00b9 : MeasurableSingletonClass \u03b9\ninst\u271d : SecondCountableTopology \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\nh\u03c0 : IsStoppingTime f \u03c0\nj : \u03b9\n\u03c9 : \u03a9\n\u22a2 \u03c9 \u2208 {\u03c9 | \u03c4 \u03c9 = \u03c0 \u03c9} \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 j} \u2194 \u03c9 \u2208 {\u03c9 | min (\u03c4 \u03c9) j = min (\u03c0 \u03c9) j} \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 j} \u2229 {\u03c9 | \u03c0 \u03c9 \u2264 j}"}, {"tactic": "simp only [Set.mem_inter_iff, Set.mem_setOf_eq]", "annotated_tactic": ["simp only [<a>Set.mem_inter_iff</a>, <a>Set.mem_setOf_eq</a>]", [{"full_name": "Set.mem_inter_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [909, 9], "def_end_pos": [909, 22]}, {"full_name": "Set.mem_setOf_eq", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [256, 29], "def_end_pos": [256, 41]}]], "state_before": "case h\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2077 : LinearOrder \u03b9\nf : Filtration \u03b9 m\n\u03c4 \u03c0 : \u03a9 \u2192 \u03b9\ninst\u271d\u2076 : Countable \u03b9\ninst\u271d\u2075 : TopologicalSpace \u03b9\ninst\u271d\u2074 : MeasurableSpace \u03b9\ninst\u271d\u00b3 : BorelSpace \u03b9\ninst\u271d\u00b2 : OrderTopology \u03b9\ninst\u271d\u00b9 : MeasurableSingletonClass \u03b9\ninst\u271d : SecondCountableTopology \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\nh\u03c0 : IsStoppingTime f \u03c0\nj : \u03b9\n\u03c9 : \u03a9\n\u22a2 \u03c9 \u2208 {\u03c9 | \u03c4 \u03c9 = \u03c0 \u03c9} \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 j} \u2194 \u03c9 \u2208 {\u03c9 | min (\u03c4 \u03c9) j = min (\u03c0 \u03c9) j} \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 j} \u2229 {\u03c9 | \u03c0 \u03c9 \u2264 j}", "state_after": "case h\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2077 : LinearOrder \u03b9\nf : Filtration \u03b9 m\n\u03c4 \u03c0 : \u03a9 \u2192 \u03b9\ninst\u271d\u2076 : Countable \u03b9\ninst\u271d\u2075 : TopologicalSpace \u03b9\ninst\u271d\u2074 : MeasurableSpace \u03b9\ninst\u271d\u00b3 : BorelSpace \u03b9\ninst\u271d\u00b2 : OrderTopology \u03b9\ninst\u271d\u00b9 : MeasurableSingletonClass \u03b9\ninst\u271d : SecondCountableTopology \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\nh\u03c0 : IsStoppingTime f \u03c0\nj : \u03b9\n\u03c9 : \u03a9\n\u22a2 \u03c4 \u03c9 = \u03c0 \u03c9 \u2227 \u03c4 \u03c9 \u2264 j \u2194 (min (\u03c4 \u03c9) j = min (\u03c0 \u03c9) j \u2227 \u03c4 \u03c9 \u2264 j) \u2227 \u03c0 \u03c9 \u2264 j"}, {"tactic": "refine' \u27e8fun h => \u27e8\u27e8_, h.2\u27e9, _\u27e9, fun h => \u27e8_, h.1.2\u27e9\u27e9", "annotated_tactic": ["refine' \u27e8fun h => \u27e8\u27e8_, h.2\u27e9, _\u27e9, fun h => \u27e8_, h.1.2\u27e9\u27e9", []], "state_before": "case h\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2077 : LinearOrder \u03b9\nf : Filtration \u03b9 m\n\u03c4 \u03c0 : \u03a9 \u2192 \u03b9\ninst\u271d\u2076 : Countable \u03b9\ninst\u271d\u2075 : TopologicalSpace \u03b9\ninst\u271d\u2074 : MeasurableSpace \u03b9\ninst\u271d\u00b3 : BorelSpace \u03b9\ninst\u271d\u00b2 : OrderTopology \u03b9\ninst\u271d\u00b9 : MeasurableSingletonClass \u03b9\ninst\u271d : SecondCountableTopology \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\nh\u03c0 : IsStoppingTime f \u03c0\nj : \u03b9\n\u03c9 : \u03a9\n\u22a2 \u03c4 \u03c9 = \u03c0 \u03c9 \u2227 \u03c4 \u03c9 \u2264 j \u2194 (min (\u03c4 \u03c9) j = min (\u03c0 \u03c9) j \u2227 \u03c4 \u03c9 \u2264 j) \u2227 \u03c0 \u03c9 \u2264 j", "state_after": "case h.refine'_1\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2077 : LinearOrder \u03b9\nf : Filtration \u03b9 m\n\u03c4 \u03c0 : \u03a9 \u2192 \u03b9\ninst\u271d\u2076 : Countable \u03b9\ninst\u271d\u2075 : TopologicalSpace \u03b9\ninst\u271d\u2074 : MeasurableSpace \u03b9\ninst\u271d\u00b3 : BorelSpace \u03b9\ninst\u271d\u00b2 : OrderTopology \u03b9\ninst\u271d\u00b9 : MeasurableSingletonClass \u03b9\ninst\u271d : SecondCountableTopology \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\nh\u03c0 : IsStoppingTime f \u03c0\nj : \u03b9\n\u03c9 : \u03a9\nh : \u03c4 \u03c9 = \u03c0 \u03c9 \u2227 \u03c4 \u03c9 \u2264 j\n\u22a2 min (\u03c4 \u03c9) j = min (\u03c0 \u03c9) j\n\ncase h.refine'_2\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2077 : LinearOrder \u03b9\nf : Filtration \u03b9 m\n\u03c4 \u03c0 : \u03a9 \u2192 \u03b9\ninst\u271d\u2076 : Countable \u03b9\ninst\u271d\u2075 : TopologicalSpace \u03b9\ninst\u271d\u2074 : MeasurableSpace \u03b9\ninst\u271d\u00b3 : BorelSpace \u03b9\ninst\u271d\u00b2 : OrderTopology \u03b9\ninst\u271d\u00b9 : MeasurableSingletonClass \u03b9\ninst\u271d : SecondCountableTopology \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\nh\u03c0 : IsStoppingTime f \u03c0\nj : \u03b9\n\u03c9 : \u03a9\nh : \u03c4 \u03c9 = \u03c0 \u03c9 \u2227 \u03c4 \u03c9 \u2264 j\n\u22a2 \u03c0 \u03c9 \u2264 j\n\ncase h.refine'_3\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2077 : LinearOrder \u03b9\nf : Filtration \u03b9 m\n\u03c4 \u03c0 : \u03a9 \u2192 \u03b9\ninst\u271d\u2076 : Countable \u03b9\ninst\u271d\u2075 : TopologicalSpace \u03b9\ninst\u271d\u2074 : MeasurableSpace \u03b9\ninst\u271d\u00b3 : BorelSpace \u03b9\ninst\u271d\u00b2 : OrderTopology \u03b9\ninst\u271d\u00b9 : MeasurableSingletonClass \u03b9\ninst\u271d : SecondCountableTopology \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\nh\u03c0 : IsStoppingTime f \u03c0\nj : \u03b9\n\u03c9 : \u03a9\nh : (min (\u03c4 \u03c9) j = min (\u03c0 \u03c9) j \u2227 \u03c4 \u03c9 \u2264 j) \u2227 \u03c0 \u03c9 \u2264 j\n\u22a2 \u03c4 \u03c9 = \u03c0 \u03c9"}, {"tactic": "rw [h.1]", "annotated_tactic": ["rw [h.1]", []], "state_before": "case h.refine'_1\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2077 : LinearOrder \u03b9\nf : Filtration \u03b9 m\n\u03c4 \u03c0 : \u03a9 \u2192 \u03b9\ninst\u271d\u2076 : Countable \u03b9\ninst\u271d\u2075 : TopologicalSpace \u03b9\ninst\u271d\u2074 : MeasurableSpace \u03b9\ninst\u271d\u00b3 : BorelSpace \u03b9\ninst\u271d\u00b2 : OrderTopology \u03b9\ninst\u271d\u00b9 : MeasurableSingletonClass \u03b9\ninst\u271d : SecondCountableTopology \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\nh\u03c0 : IsStoppingTime f \u03c0\nj : \u03b9\n\u03c9 : \u03a9\nh : \u03c4 \u03c9 = \u03c0 \u03c9 \u2227 \u03c4 \u03c9 \u2264 j\n\u22a2 min (\u03c4 \u03c9) j = min (\u03c0 \u03c9) j", "state_after": "no goals"}, {"tactic": "rw [\u2190 h.1]", "annotated_tactic": ["rw [\u2190 h.1]", []], "state_before": "case h.refine'_2\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2077 : LinearOrder \u03b9\nf : Filtration \u03b9 m\n\u03c4 \u03c0 : \u03a9 \u2192 \u03b9\ninst\u271d\u2076 : Countable \u03b9\ninst\u271d\u2075 : TopologicalSpace \u03b9\ninst\u271d\u2074 : MeasurableSpace \u03b9\ninst\u271d\u00b3 : BorelSpace \u03b9\ninst\u271d\u00b2 : OrderTopology \u03b9\ninst\u271d\u00b9 : MeasurableSingletonClass \u03b9\ninst\u271d : SecondCountableTopology \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\nh\u03c0 : IsStoppingTime f \u03c0\nj : \u03b9\n\u03c9 : \u03a9\nh : \u03c4 \u03c9 = \u03c0 \u03c9 \u2227 \u03c4 \u03c9 \u2264 j\n\u22a2 \u03c0 \u03c9 \u2264 j", "state_after": "case h.refine'_2\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2077 : LinearOrder \u03b9\nf : Filtration \u03b9 m\n\u03c4 \u03c0 : \u03a9 \u2192 \u03b9\ninst\u271d\u2076 : Countable \u03b9\ninst\u271d\u2075 : TopologicalSpace \u03b9\ninst\u271d\u2074 : MeasurableSpace \u03b9\ninst\u271d\u00b3 : BorelSpace \u03b9\ninst\u271d\u00b2 : OrderTopology \u03b9\ninst\u271d\u00b9 : MeasurableSingletonClass \u03b9\ninst\u271d : SecondCountableTopology \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\nh\u03c0 : IsStoppingTime f \u03c0\nj : \u03b9\n\u03c9 : \u03a9\nh : \u03c4 \u03c9 = \u03c0 \u03c9 \u2227 \u03c4 \u03c9 \u2264 j\n\u22a2 \u03c4 \u03c9 \u2264 j"}, {"tactic": "exact h.2", "annotated_tactic": ["exact h.2", []], "state_before": "case h.refine'_2\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2077 : LinearOrder \u03b9\nf : Filtration \u03b9 m\n\u03c4 \u03c0 : \u03a9 \u2192 \u03b9\ninst\u271d\u2076 : Countable \u03b9\ninst\u271d\u2075 : TopologicalSpace \u03b9\ninst\u271d\u2074 : MeasurableSpace \u03b9\ninst\u271d\u00b3 : BorelSpace \u03b9\ninst\u271d\u00b2 : OrderTopology \u03b9\ninst\u271d\u00b9 : MeasurableSingletonClass \u03b9\ninst\u271d : SecondCountableTopology \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\nh\u03c0 : IsStoppingTime f \u03c0\nj : \u03b9\n\u03c9 : \u03a9\nh : \u03c4 \u03c9 = \u03c0 \u03c9 \u2227 \u03c4 \u03c9 \u2264 j\n\u22a2 \u03c4 \u03c9 \u2264 j", "state_after": "no goals"}, {"tactic": "cases' h with h' h\u03c0_le", "annotated_tactic": ["cases' h with h' h\u03c0_le", []], "state_before": "case h.refine'_3\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2077 : LinearOrder \u03b9\nf : Filtration \u03b9 m\n\u03c4 \u03c0 : \u03a9 \u2192 \u03b9\ninst\u271d\u2076 : Countable \u03b9\ninst\u271d\u2075 : TopologicalSpace \u03b9\ninst\u271d\u2074 : MeasurableSpace \u03b9\ninst\u271d\u00b3 : BorelSpace \u03b9\ninst\u271d\u00b2 : OrderTopology \u03b9\ninst\u271d\u00b9 : MeasurableSingletonClass \u03b9\ninst\u271d : SecondCountableTopology \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\nh\u03c0 : IsStoppingTime f \u03c0\nj : \u03b9\n\u03c9 : \u03a9\nh : (min (\u03c4 \u03c9) j = min (\u03c0 \u03c9) j \u2227 \u03c4 \u03c9 \u2264 j) \u2227 \u03c0 \u03c9 \u2264 j\n\u22a2 \u03c4 \u03c9 = \u03c0 \u03c9", "state_after": "case h.refine'_3.intro\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2077 : LinearOrder \u03b9\nf : Filtration \u03b9 m\n\u03c4 \u03c0 : \u03a9 \u2192 \u03b9\ninst\u271d\u2076 : Countable \u03b9\ninst\u271d\u2075 : TopologicalSpace \u03b9\ninst\u271d\u2074 : MeasurableSpace \u03b9\ninst\u271d\u00b3 : BorelSpace \u03b9\ninst\u271d\u00b2 : OrderTopology \u03b9\ninst\u271d\u00b9 : MeasurableSingletonClass \u03b9\ninst\u271d : SecondCountableTopology \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\nh\u03c0 : IsStoppingTime f \u03c0\nj : \u03b9\n\u03c9 : \u03a9\nh' : min (\u03c4 \u03c9) j = min (\u03c0 \u03c9) j \u2227 \u03c4 \u03c9 \u2264 j\nh\u03c0_le : \u03c0 \u03c9 \u2264 j\n\u22a2 \u03c4 \u03c9 = \u03c0 \u03c9"}, {"tactic": "cases' h' with h_eq h\u03c4_le", "annotated_tactic": ["cases' h' with h_eq h\u03c4_le", []], "state_before": "case h.refine'_3.intro\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2077 : LinearOrder \u03b9\nf : Filtration \u03b9 m\n\u03c4 \u03c0 : \u03a9 \u2192 \u03b9\ninst\u271d\u2076 : Countable \u03b9\ninst\u271d\u2075 : TopologicalSpace \u03b9\ninst\u271d\u2074 : MeasurableSpace \u03b9\ninst\u271d\u00b3 : BorelSpace \u03b9\ninst\u271d\u00b2 : OrderTopology \u03b9\ninst\u271d\u00b9 : MeasurableSingletonClass \u03b9\ninst\u271d : SecondCountableTopology \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\nh\u03c0 : IsStoppingTime f \u03c0\nj : \u03b9\n\u03c9 : \u03a9\nh' : min (\u03c4 \u03c9) j = min (\u03c0 \u03c9) j \u2227 \u03c4 \u03c9 \u2264 j\nh\u03c0_le : \u03c0 \u03c9 \u2264 j\n\u22a2 \u03c4 \u03c9 = \u03c0 \u03c9", "state_after": "case h.refine'_3.intro.intro\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2077 : LinearOrder \u03b9\nf : Filtration \u03b9 m\n\u03c4 \u03c0 : \u03a9 \u2192 \u03b9\ninst\u271d\u2076 : Countable \u03b9\ninst\u271d\u2075 : TopologicalSpace \u03b9\ninst\u271d\u2074 : MeasurableSpace \u03b9\ninst\u271d\u00b3 : BorelSpace \u03b9\ninst\u271d\u00b2 : OrderTopology \u03b9\ninst\u271d\u00b9 : MeasurableSingletonClass \u03b9\ninst\u271d : SecondCountableTopology \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\nh\u03c0 : IsStoppingTime f \u03c0\nj : \u03b9\n\u03c9 : \u03a9\nh\u03c0_le : \u03c0 \u03c9 \u2264 j\nh_eq : min (\u03c4 \u03c9) j = min (\u03c0 \u03c9) j\nh\u03c4_le : \u03c4 \u03c9 \u2264 j\n\u22a2 \u03c4 \u03c9 = \u03c0 \u03c9"}, {"tactic": "rwa [min_eq_left h\u03c4_le, min_eq_left h\u03c0_le] at h_eq", "annotated_tactic": ["rwa [<a>min_eq_left</a> h\u03c4_le, <a>min_eq_left</a> h\u03c0_le] at h_eq", [{"full_name": "min_eq_left", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [100, 9], "def_end_pos": [100, 20]}, {"full_name": "min_eq_left", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [100, 9], "def_end_pos": [100, 20]}]], "state_before": "case h.refine'_3.intro.intro\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2077 : LinearOrder \u03b9\nf : Filtration \u03b9 m\n\u03c4 \u03c0 : \u03a9 \u2192 \u03b9\ninst\u271d\u2076 : Countable \u03b9\ninst\u271d\u2075 : TopologicalSpace \u03b9\ninst\u271d\u2074 : MeasurableSpace \u03b9\ninst\u271d\u00b3 : BorelSpace \u03b9\ninst\u271d\u00b2 : OrderTopology \u03b9\ninst\u271d\u00b9 : MeasurableSingletonClass \u03b9\ninst\u271d : SecondCountableTopology \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\nh\u03c0 : IsStoppingTime f \u03c0\nj : \u03b9\n\u03c9 : \u03a9\nh\u03c0_le : \u03c0 \u03c9 \u2264 j\nh_eq : min (\u03c4 \u03c9) j = min (\u03c0 \u03c9) j\nh\u03c4_le : \u03c4 \u03c9 \u2264 j\n\u22a2 \u03c4 \u03c9 = \u03c0 \u03c9", "state_after": "no goals"}, {"tactic": "exact (h\u03c4.min_const j).measurable_of_le fun _ => min_le_right _ _", "annotated_tactic": ["exact (h\u03c4.min_const j).<a>measurable_of_le</a> fun _ => <a>min_le_right</a> _ _", [{"full_name": "MeasureTheory.IsStoppingTime.measurable_of_le", "def_path": "Mathlib/Probability/Process/Stopping.lean", "def_pos": [585, 19], "def_end_pos": [585, 35]}, {"full_name": "min_le_right", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [40, 9], "def_end_pos": [40, 21]}]], "state_before": "case hf\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2077 : LinearOrder \u03b9\nf : Filtration \u03b9 m\n\u03c4 \u03c0 : \u03a9 \u2192 \u03b9\ninst\u271d\u2076 : Countable \u03b9\ninst\u271d\u2075 : TopologicalSpace \u03b9\ninst\u271d\u2074 : MeasurableSpace \u03b9\ninst\u271d\u00b3 : BorelSpace \u03b9\ninst\u271d\u00b2 : OrderTopology \u03b9\ninst\u271d\u00b9 : MeasurableSingletonClass \u03b9\ninst\u271d : SecondCountableTopology \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\nh\u03c0 : IsStoppingTime f \u03c0\nj : \u03b9\nthis : {\u03c9 | \u03c4 \u03c9 = \u03c0 \u03c9} \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 j} = {\u03c9 | min (\u03c4 \u03c9) j = min (\u03c0 \u03c9) j} \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 j} \u2229 {\u03c9 | \u03c0 \u03c9 \u2264 j}\n\u22a2 Measurable fun x => min (\u03c4 x) j", "state_after": "no goals"}, {"tactic": "exact (h\u03c0.min_const j).measurable_of_le fun _ => min_le_right _ _", "annotated_tactic": ["exact (h\u03c0.min_const j).<a>measurable_of_le</a> fun _ => <a>min_le_right</a> _ _", [{"full_name": "MeasureTheory.IsStoppingTime.measurable_of_le", "def_path": "Mathlib/Probability/Process/Stopping.lean", "def_pos": [585, 19], "def_end_pos": [585, 35]}, {"full_name": "min_le_right", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [40, 9], "def_end_pos": [40, 21]}]], "state_before": "case hg\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2077 : LinearOrder \u03b9\nf : Filtration \u03b9 m\n\u03c4 \u03c0 : \u03a9 \u2192 \u03b9\ninst\u271d\u2076 : Countable \u03b9\ninst\u271d\u2075 : TopologicalSpace \u03b9\ninst\u271d\u2074 : MeasurableSpace \u03b9\ninst\u271d\u00b3 : BorelSpace \u03b9\ninst\u271d\u00b2 : OrderTopology \u03b9\ninst\u271d\u00b9 : MeasurableSingletonClass \u03b9\ninst\u271d : SecondCountableTopology \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\nh\u03c0 : IsStoppingTime f \u03c0\nj : \u03b9\nthis : {\u03c9 | \u03c4 \u03c9 = \u03c0 \u03c9} \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 j} = {\u03c9 | min (\u03c4 \u03c9) j = min (\u03c0 \u03c9) j} \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 j} \u2229 {\u03c9 | \u03c0 \u03c9 \u2264 j}\n\u22a2 Measurable fun x => min (\u03c0 x) j", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Decomposition/Jordan.lean", "full_name": "MeasureTheory.SignedMeasure.toJordanDecomposition_neg", "start": [455, 1], "end": [458, 29], "traced_tactics": [{"tactic": "apply toSignedMeasure_injective", "annotated_tactic": ["apply <a>toSignedMeasure_injective</a>", [{"full_name": "MeasureTheory.JordanDecomposition.toSignedMeasure_injective", "def_path": "Mathlib/MeasureTheory/Decomposition/Jordan.lean", "def_pos": [372, 9], "def_end_pos": [372, 34]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u22a2 toJordanDecomposition (-s) = -toJordanDecomposition s", "state_after": "case a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u22a2 toSignedMeasure (toJordanDecomposition (-s)) = toSignedMeasure (-toJordanDecomposition s)"}, {"tactic": "simp [toSignedMeasure_neg]", "annotated_tactic": ["simp [<a>toSignedMeasure_neg</a>]", [{"full_name": "MeasureTheory.JordanDecomposition.toSignedMeasure_neg", "def_path": "Mathlib/MeasureTheory/Decomposition/Jordan.lean", "def_pos": [180, 9], "def_end_pos": [180, 28]}]], "state_before": "case a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u22a2 toSignedMeasure (toJordanDecomposition (-s)) = toSignedMeasure (-toJordanDecomposition s)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Intervals/SurjOn.lean", "full_name": "surjOn_Iic_of_monotone_surjective", "start": [83, 1], "end": [85, 68], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Independence/Kernel.lean", "full_name": "ProbabilityTheory.kernel.iIndepSets.iIndep", "start": [558, 1], "end": [587, 42], "traced_tactics": [{"tactic": "intro s f", "annotated_tactic": ["intro s f", []], "state_before": "\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\n_m\u03a9 : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\ninst\u271d : IsMarkovKernel \u03ba\nm : \u03b9 \u2192 MeasurableSpace \u03a9\nh_le : \u2200 (i : \u03b9), m i \u2264 _m\u03a9\n\u03c0 : \u03b9 \u2192 Set (Set \u03a9)\nh_pi : \u2200 (n : \u03b9), IsPiSystem (\u03c0 n)\nh_generate : \u2200 (i : \u03b9), m i = generateFrom (\u03c0 i)\nh_ind : iIndepSets \u03c0 \u03ba\n\u22a2 kernel.iIndep m \u03ba", "state_after": "\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\n_m\u03a9 : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\ninst\u271d : IsMarkovKernel \u03ba\nm : \u03b9 \u2192 MeasurableSpace \u03a9\nh_le : \u2200 (i : \u03b9), m i \u2264 _m\u03a9\n\u03c0 : \u03b9 \u2192 Set (Set \u03a9)\nh_pi : \u2200 (n : \u03b9), IsPiSystem (\u03c0 n)\nh_generate : \u2200 (i : \u03b9), m i = generateFrom (\u03c0 i)\nh_ind : iIndepSets \u03c0 \u03ba\ns : Finset \u03b9\nf : \u03b9 \u2192 Set \u03a9\n\u22a2 (\u2200 (i : \u03b9), i \u2208 s \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i) \u2192\n    \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (\u22c2 i \u2208 s, f i) = \u220f i in s, \u2191\u2191(\u2191\u03ba a) (f i)"}, {"tactic": "refine Finset.induction ?_ ?_ s", "annotated_tactic": ["refine <a>Finset.induction</a> ?_ ?_ s", [{"full_name": "Finset.induction", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1240, 19], "def_end_pos": [1240, 28]}]], "state_before": "\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\n_m\u03a9 : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\ninst\u271d : IsMarkovKernel \u03ba\nm : \u03b9 \u2192 MeasurableSpace \u03a9\nh_le : \u2200 (i : \u03b9), m i \u2264 _m\u03a9\n\u03c0 : \u03b9 \u2192 Set (Set \u03a9)\nh_pi : \u2200 (n : \u03b9), IsPiSystem (\u03c0 n)\nh_generate : \u2200 (i : \u03b9), m i = generateFrom (\u03c0 i)\nh_ind : iIndepSets \u03c0 \u03ba\ns : Finset \u03b9\nf : \u03b9 \u2192 Set \u03a9\n\u22a2 (\u2200 (i : \u03b9), i \u2208 s \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i) \u2192\n    \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (\u22c2 i \u2208 s, f i) = \u220f i in s, \u2191\u2191(\u2191\u03ba a) (f i)", "state_after": "case refine_1\n\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\n_m\u03a9 : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\ninst\u271d : IsMarkovKernel \u03ba\nm : \u03b9 \u2192 MeasurableSpace \u03a9\nh_le : \u2200 (i : \u03b9), m i \u2264 _m\u03a9\n\u03c0 : \u03b9 \u2192 Set (Set \u03a9)\nh_pi : \u2200 (n : \u03b9), IsPiSystem (\u03c0 n)\nh_generate : \u2200 (i : \u03b9), m i = generateFrom (\u03c0 i)\nh_ind : iIndepSets \u03c0 \u03ba\ns : Finset \u03b9\nf : \u03b9 \u2192 Set \u03a9\n\u22a2 (\u2200 (i : \u03b9), i \u2208 \u2205 \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i) \u2192\n    \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (\u22c2 i \u2208 \u2205, f i) = \u220f i in \u2205, \u2191\u2191(\u2191\u03ba a) (f i)\n\ncase refine_2\n\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\n_m\u03a9 : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\ninst\u271d : IsMarkovKernel \u03ba\nm : \u03b9 \u2192 MeasurableSpace \u03a9\nh_le : \u2200 (i : \u03b9), m i \u2264 _m\u03a9\n\u03c0 : \u03b9 \u2192 Set (Set \u03a9)\nh_pi : \u2200 (n : \u03b9), IsPiSystem (\u03c0 n)\nh_generate : \u2200 (i : \u03b9), m i = generateFrom (\u03c0 i)\nh_ind : iIndepSets \u03c0 \u03ba\ns : Finset \u03b9\nf : \u03b9 \u2192 Set \u03a9\n\u22a2 \u2200 \u2983a : \u03b9\u2984 {s : Finset \u03b9},\n    \u00aca \u2208 s \u2192\n      ((\u2200 (i : \u03b9), i \u2208 s \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i) \u2192\n          \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (\u22c2 i \u2208 s, f i) = \u220f i in s, \u2191\u2191(\u2191\u03ba a) (f i)) \u2192\n        (\u2200 (i : \u03b9), i \u2208 insert a s \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i) \u2192\n          \u2200\u1d50 (a_3 : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a_3) (\u22c2 i \u2208 insert a s, f i) = \u220f i in insert a s, \u2191\u2191(\u2191\u03ba a_3) (f i)"}, {"tactic": "simp only [Finset.not_mem_empty, Set.mem_setOf_eq, IsEmpty.forall_iff, implies_true,\n  Set.iInter_of_empty, Set.iInter_univ, measure_univ, Finset.prod_empty,\n  Filter.eventually_true, forall_true_left]", "annotated_tactic": ["simp only [<a>Finset.not_mem_empty</a>, <a>Set.mem_setOf_eq</a>, <a>IsEmpty.forall_iff</a>, <a>implies_true</a>,\n      <a>Set.iInter_of_empty</a>, <a>Set.iInter_univ</a>, <a>measure_univ</a>, <a>Finset.prod_empty</a>,\n      <a>Filter.eventually_true</a>, <a>forall_true_left</a>]", [{"full_name": "Finset.not_mem_empty", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [548, 9], "def_end_pos": [548, 22]}, {"full_name": "Set.mem_setOf_eq", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [256, 29], "def_end_pos": [256, 41]}, {"full_name": "IsEmpty.forall_iff", "def_path": "Mathlib/Logic/IsEmpty.lean", "def_pos": [121, 9], "def_end_pos": [121, 19]}, {"full_name": "implies_true", "def_path": "lake-packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [98, 17], "def_end_pos": [98, 29]}, {"full_name": "Set.iInter_of_empty", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [1474, 9], "def_end_pos": [1474, 24]}, {"full_name": "Set.iInter_univ", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [815, 9], "def_end_pos": [815, 20]}, {"full_name": "MeasureTheory.IsProbabilityMeasure.measure_univ", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3027, 3], "def_end_pos": [3027, 15]}, {"full_name": "Finset.prod_empty", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [299, 9], "def_end_pos": [299, 19]}, {"full_name": "Filter.eventually_true", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1108, 17], "def_end_pos": [1108, 32]}, {"full_name": "forall_true_left", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [931, 17], "def_end_pos": [931, 33]}]], "state_before": "case refine_1\n\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\n_m\u03a9 : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\ninst\u271d : IsMarkovKernel \u03ba\nm : \u03b9 \u2192 MeasurableSpace \u03a9\nh_le : \u2200 (i : \u03b9), m i \u2264 _m\u03a9\n\u03c0 : \u03b9 \u2192 Set (Set \u03a9)\nh_pi : \u2200 (n : \u03b9), IsPiSystem (\u03c0 n)\nh_generate : \u2200 (i : \u03b9), m i = generateFrom (\u03c0 i)\nh_ind : iIndepSets \u03c0 \u03ba\ns : Finset \u03b9\nf : \u03b9 \u2192 Set \u03a9\n\u22a2 (\u2200 (i : \u03b9), i \u2208 \u2205 \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i) \u2192\n    \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (\u22c2 i \u2208 \u2205, f i) = \u220f i in \u2205, \u2191\u2191(\u2191\u03ba a) (f i)", "state_after": "no goals"}, {"tactic": "intro a S ha_notin_S h_rec hf_m", "annotated_tactic": ["intro a S ha_notin_S h_rec hf_m", []], "state_before": "case refine_2\n\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\n_m\u03a9 : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\ninst\u271d : IsMarkovKernel \u03ba\nm : \u03b9 \u2192 MeasurableSpace \u03a9\nh_le : \u2200 (i : \u03b9), m i \u2264 _m\u03a9\n\u03c0 : \u03b9 \u2192 Set (Set \u03a9)\nh_pi : \u2200 (n : \u03b9), IsPiSystem (\u03c0 n)\nh_generate : \u2200 (i : \u03b9), m i = generateFrom (\u03c0 i)\nh_ind : iIndepSets \u03c0 \u03ba\ns : Finset \u03b9\nf : \u03b9 \u2192 Set \u03a9\n\u22a2 \u2200 \u2983a : \u03b9\u2984 {s : Finset \u03b9},\n    \u00aca \u2208 s \u2192\n      ((\u2200 (i : \u03b9), i \u2208 s \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i) \u2192\n          \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (\u22c2 i \u2208 s, f i) = \u220f i in s, \u2191\u2191(\u2191\u03ba a) (f i)) \u2192\n        (\u2200 (i : \u03b9), i \u2208 insert a s \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i) \u2192\n          \u2200\u1d50 (a_3 : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a_3) (\u22c2 i \u2208 insert a s, f i) = \u220f i in insert a s, \u2191\u2191(\u2191\u03ba a_3) (f i)", "state_after": "case refine_2\n\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\n_m\u03a9 : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\ninst\u271d : IsMarkovKernel \u03ba\nm : \u03b9 \u2192 MeasurableSpace \u03a9\nh_le : \u2200 (i : \u03b9), m i \u2264 _m\u03a9\n\u03c0 : \u03b9 \u2192 Set (Set \u03a9)\nh_pi : \u2200 (n : \u03b9), IsPiSystem (\u03c0 n)\nh_generate : \u2200 (i : \u03b9), m i = generateFrom (\u03c0 i)\nh_ind : iIndepSets \u03c0 \u03ba\ns : Finset \u03b9\nf : \u03b9 \u2192 Set \u03a9\na : \u03b9\nS : Finset \u03b9\nha_notin_S : \u00aca \u2208 S\nh_rec :\n  (\u2200 (i : \u03b9), i \u2208 S \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i) \u2192\n    \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (\u22c2 i \u2208 S, f i) = \u220f i in S, \u2191\u2191(\u2191\u03ba a) (f i)\nhf_m : \u2200 (i : \u03b9), i \u2208 insert a S \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i\n\u22a2 \u2200\u1d50 (a_1 : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a_1) (\u22c2 i \u2208 insert a S, f i) = \u220f i in insert a S, \u2191\u2191(\u2191\u03ba a_1) (f i)"}, {"tactic": "have hf_m_S : \u2200 x \u2208 S, MeasurableSet[m x] (f x) := fun x hx => hf_m x (by simp [hx])", "annotated_tactic": ["have hf_m_S : \u2200 x \u2208 S, MeasurableSet[m x] (f x) := fun x hx => hf_m x (by simp [hx])", []], "state_before": "case refine_2\n\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\n_m\u03a9 : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\ninst\u271d : IsMarkovKernel \u03ba\nm : \u03b9 \u2192 MeasurableSpace \u03a9\nh_le : \u2200 (i : \u03b9), m i \u2264 _m\u03a9\n\u03c0 : \u03b9 \u2192 Set (Set \u03a9)\nh_pi : \u2200 (n : \u03b9), IsPiSystem (\u03c0 n)\nh_generate : \u2200 (i : \u03b9), m i = generateFrom (\u03c0 i)\nh_ind : iIndepSets \u03c0 \u03ba\ns : Finset \u03b9\nf : \u03b9 \u2192 Set \u03a9\na : \u03b9\nS : Finset \u03b9\nha_notin_S : \u00aca \u2208 S\nh_rec :\n  (\u2200 (i : \u03b9), i \u2208 S \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i) \u2192\n    \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (\u22c2 i \u2208 S, f i) = \u220f i in S, \u2191\u2191(\u2191\u03ba a) (f i)\nhf_m : \u2200 (i : \u03b9), i \u2208 insert a S \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i\n\u22a2 \u2200\u1d50 (a_1 : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a_1) (\u22c2 i \u2208 insert a S, f i) = \u220f i in insert a S, \u2191\u2191(\u2191\u03ba a_1) (f i)", "state_after": "case refine_2\n\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\n_m\u03a9 : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\ninst\u271d : IsMarkovKernel \u03ba\nm : \u03b9 \u2192 MeasurableSpace \u03a9\nh_le : \u2200 (i : \u03b9), m i \u2264 _m\u03a9\n\u03c0 : \u03b9 \u2192 Set (Set \u03a9)\nh_pi : \u2200 (n : \u03b9), IsPiSystem (\u03c0 n)\nh_generate : \u2200 (i : \u03b9), m i = generateFrom (\u03c0 i)\nh_ind : iIndepSets \u03c0 \u03ba\ns : Finset \u03b9\nf : \u03b9 \u2192 Set \u03a9\na : \u03b9\nS : Finset \u03b9\nha_notin_S : \u00aca \u2208 S\nh_rec :\n  (\u2200 (i : \u03b9), i \u2208 S \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i) \u2192\n    \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (\u22c2 i \u2208 S, f i) = \u220f i in S, \u2191\u2191(\u2191\u03ba a) (f i)\nhf_m : \u2200 (i : \u03b9), i \u2208 insert a S \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i\nhf_m_S : \u2200 (x : \u03b9), x \u2208 S \u2192 MeasurableSet (f x)\n\u22a2 \u2200\u1d50 (a_1 : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a_1) (\u22c2 i \u2208 insert a S, f i) = \u220f i in insert a S, \u2191\u2191(\u2191\u03ba a_1) (f i)"}, {"tactic": "let p := piiUnionInter \u03c0 S", "annotated_tactic": ["let p := <a>piiUnionInter</a> \u03c0 S", [{"full_name": "piiUnionInter", "def_path": "Mathlib/MeasureTheory/PiSystem.lean", "def_pos": [371, 5], "def_end_pos": [371, 18]}]], "state_before": "case refine_2\n\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\n_m\u03a9 : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\ninst\u271d : IsMarkovKernel \u03ba\nm : \u03b9 \u2192 MeasurableSpace \u03a9\nh_le : \u2200 (i : \u03b9), m i \u2264 _m\u03a9\n\u03c0 : \u03b9 \u2192 Set (Set \u03a9)\nh_pi : \u2200 (n : \u03b9), IsPiSystem (\u03c0 n)\nh_generate : \u2200 (i : \u03b9), m i = generateFrom (\u03c0 i)\nh_ind : iIndepSets \u03c0 \u03ba\ns : Finset \u03b9\nf : \u03b9 \u2192 Set \u03a9\na : \u03b9\nS : Finset \u03b9\nha_notin_S : \u00aca \u2208 S\nh_rec :\n  (\u2200 (i : \u03b9), i \u2208 S \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i) \u2192\n    \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (\u22c2 i \u2208 S, f i) = \u220f i in S, \u2191\u2191(\u2191\u03ba a) (f i)\nhf_m : \u2200 (i : \u03b9), i \u2208 insert a S \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i\nhf_m_S : \u2200 (x : \u03b9), x \u2208 S \u2192 MeasurableSet (f x)\n\u22a2 \u2200\u1d50 (a_1 : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a_1) (\u22c2 i \u2208 insert a S, f i) = \u220f i in insert a S, \u2191\u2191(\u2191\u03ba a_1) (f i)", "state_after": "case refine_2\n\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\n_m\u03a9 : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\ninst\u271d : IsMarkovKernel \u03ba\nm : \u03b9 \u2192 MeasurableSpace \u03a9\nh_le : \u2200 (i : \u03b9), m i \u2264 _m\u03a9\n\u03c0 : \u03b9 \u2192 Set (Set \u03a9)\nh_pi : \u2200 (n : \u03b9), IsPiSystem (\u03c0 n)\nh_generate : \u2200 (i : \u03b9), m i = generateFrom (\u03c0 i)\nh_ind : iIndepSets \u03c0 \u03ba\ns : Finset \u03b9\nf : \u03b9 \u2192 Set \u03a9\na : \u03b9\nS : Finset \u03b9\nha_notin_S : \u00aca \u2208 S\nh_rec :\n  (\u2200 (i : \u03b9), i \u2208 S \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i) \u2192\n    \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (\u22c2 i \u2208 S, f i) = \u220f i in S, \u2191\u2191(\u2191\u03ba a) (f i)\nhf_m : \u2200 (i : \u03b9), i \u2208 insert a S \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i\nhf_m_S : \u2200 (x : \u03b9), x \u2208 S \u2192 MeasurableSet (f x)\np : Set (Set \u03a9) := piiUnionInter \u03c0 \u2191S\n\u22a2 \u2200\u1d50 (a_1 : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a_1) (\u22c2 i \u2208 insert a S, f i) = \u220f i in insert a S, \u2191\u2191(\u2191\u03ba a_1) (f i)"}, {"tactic": "set m_p := generateFrom p with hS_eq_generate", "annotated_tactic": ["set m_p := <a>generateFrom</a> p with hS_eq_generate", [{"full_name": "MeasurableSpace.generateFrom", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [363, 5], "def_end_pos": [363, 17]}]], "state_before": "case refine_2\n\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\n_m\u03a9 : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\ninst\u271d : IsMarkovKernel \u03ba\nm : \u03b9 \u2192 MeasurableSpace \u03a9\nh_le : \u2200 (i : \u03b9), m i \u2264 _m\u03a9\n\u03c0 : \u03b9 \u2192 Set (Set \u03a9)\nh_pi : \u2200 (n : \u03b9), IsPiSystem (\u03c0 n)\nh_generate : \u2200 (i : \u03b9), m i = generateFrom (\u03c0 i)\nh_ind : iIndepSets \u03c0 \u03ba\ns : Finset \u03b9\nf : \u03b9 \u2192 Set \u03a9\na : \u03b9\nS : Finset \u03b9\nha_notin_S : \u00aca \u2208 S\nh_rec :\n  (\u2200 (i : \u03b9), i \u2208 S \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i) \u2192\n    \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (\u22c2 i \u2208 S, f i) = \u220f i in S, \u2191\u2191(\u2191\u03ba a) (f i)\nhf_m : \u2200 (i : \u03b9), i \u2208 insert a S \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i\nhf_m_S : \u2200 (x : \u03b9), x \u2208 S \u2192 MeasurableSet (f x)\np : Set (Set \u03a9) := piiUnionInter \u03c0 \u2191S\n\u22a2 \u2200\u1d50 (a_1 : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a_1) (\u22c2 i \u2208 insert a S, f i) = \u220f i in insert a S, \u2191\u2191(\u2191\u03ba a_1) (f i)", "state_after": "case refine_2\n\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\n_m\u03a9 : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\ninst\u271d : IsMarkovKernel \u03ba\nm : \u03b9 \u2192 MeasurableSpace \u03a9\nh_le : \u2200 (i : \u03b9), m i \u2264 _m\u03a9\n\u03c0 : \u03b9 \u2192 Set (Set \u03a9)\nh_pi : \u2200 (n : \u03b9), IsPiSystem (\u03c0 n)\nh_generate : \u2200 (i : \u03b9), m i = generateFrom (\u03c0 i)\nh_ind : iIndepSets \u03c0 \u03ba\ns : Finset \u03b9\nf : \u03b9 \u2192 Set \u03a9\na : \u03b9\nS : Finset \u03b9\nha_notin_S : \u00aca \u2208 S\nh_rec :\n  (\u2200 (i : \u03b9), i \u2208 S \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i) \u2192\n    \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (\u22c2 i \u2208 S, f i) = \u220f i in S, \u2191\u2191(\u2191\u03ba a) (f i)\nhf_m : \u2200 (i : \u03b9), i \u2208 insert a S \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i\nhf_m_S : \u2200 (x : \u03b9), x \u2208 S \u2192 MeasurableSet (f x)\np : Set (Set \u03a9) := piiUnionInter \u03c0 \u2191S\nm_p : MeasurableSpace \u03a9 := generateFrom p\nhS_eq_generate : m_p = generateFrom p\n\u22a2 \u2200\u1d50 (a_1 : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a_1) (\u22c2 i \u2208 insert a S, f i) = \u220f i in insert a S, \u2191\u2191(\u2191\u03ba a_1) (f i)"}, {"tactic": "have h_indep : Indep m_p (m a) \u03ba \u03bc := by\n  have hp : IsPiSystem p := isPiSystem_piiUnionInter \u03c0 h_pi S\n  have h_le' : \u2200 i, generateFrom (\u03c0 i) \u2264 _m\u03a9 := fun i \u21a6 (h_generate i).symm.trans_le (h_le i)\n  have hm_p : m_p \u2264 _m\u03a9 := generateFrom_piiUnionInter_le \u03c0 h_le' S\n  exact IndepSets.indep hm_p (h_le a) hp (h_pi a) hS_eq_generate (h_generate a)\n    (iIndepSets.piiUnionInter_of_not_mem h_ind ha_notin_S)", "annotated_tactic": ["have h_indep : <a>Indep</a> m_p (m a) \u03ba \u03bc := by\n      have hp : <a>IsPiSystem</a> p := <a>isPiSystem_piiUnionInter</a> \u03c0 h_pi S\n      have h_le' : \u2200 i, <a>generateFrom</a> (\u03c0 i) \u2264 _m\u03a9 := fun i \u21a6 (h_generate i).symm.trans_le (h_le i)\n      have hm_p : m_p \u2264 _m\u03a9 := <a>generateFrom_piiUnionInter_le</a> \u03c0 h_le' S\n      exact <a>IndepSets.indep</a> hm_p (h_le a) hp (h_pi a) hS_eq_generate (h_generate a)\n        (<a>iIndepSets.piiUnionInter_of_not_mem</a> h_ind ha_notin_S)", [{"full_name": "ProbabilityTheory.kernel.Indep", "def_path": "Mathlib/Probability/Independence/Kernel.lean", "def_pos": [82, 5], "def_end_pos": [82, 10]}, {"full_name": "IsPiSystem", "def_path": "Mathlib/MeasureTheory/PiSystem.lean", "def_pos": [66, 5], "def_end_pos": [66, 15]}, {"full_name": "isPiSystem_piiUnionInter", "def_path": "Mathlib/MeasureTheory/PiSystem.lean", "def_pos": [437, 9], "def_end_pos": [437, 33]}, {"full_name": "MeasurableSpace.generateFrom", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [363, 5], "def_end_pos": [363, 17]}, {"full_name": "generateFrom_piiUnionInter_le", "def_path": "Mathlib/MeasureTheory/PiSystem.lean", "def_pos": [486, 9], "def_end_pos": [486, 38]}, {"full_name": "ProbabilityTheory.kernel.IndepSets.indep", "def_path": "Mathlib/Probability/Independence/Kernel.lean", "def_pos": [353, 9], "def_end_pos": [353, 24]}, {"full_name": "ProbabilityTheory.kernel.iIndepSets.piiUnionInter_of_not_mem", "def_path": "Mathlib/Probability/Independence/Kernel.lean", "def_pos": [520, 9], "def_end_pos": [520, 44]}]], "state_before": "case refine_2\n\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\n_m\u03a9 : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\ninst\u271d : IsMarkovKernel \u03ba\nm : \u03b9 \u2192 MeasurableSpace \u03a9\nh_le : \u2200 (i : \u03b9), m i \u2264 _m\u03a9\n\u03c0 : \u03b9 \u2192 Set (Set \u03a9)\nh_pi : \u2200 (n : \u03b9), IsPiSystem (\u03c0 n)\nh_generate : \u2200 (i : \u03b9), m i = generateFrom (\u03c0 i)\nh_ind : iIndepSets \u03c0 \u03ba\ns : Finset \u03b9\nf : \u03b9 \u2192 Set \u03a9\na : \u03b9\nS : Finset \u03b9\nha_notin_S : \u00aca \u2208 S\nh_rec :\n  (\u2200 (i : \u03b9), i \u2208 S \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i) \u2192\n    \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (\u22c2 i \u2208 S, f i) = \u220f i in S, \u2191\u2191(\u2191\u03ba a) (f i)\nhf_m : \u2200 (i : \u03b9), i \u2208 insert a S \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i\nhf_m_S : \u2200 (x : \u03b9), x \u2208 S \u2192 MeasurableSet (f x)\np : Set (Set \u03a9) := piiUnionInter \u03c0 \u2191S\nm_p : MeasurableSpace \u03a9 := generateFrom p\nhS_eq_generate : m_p = generateFrom p\n\u22a2 \u2200\u1d50 (a_1 : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a_1) (\u22c2 i \u2208 insert a S, f i) = \u220f i in insert a S, \u2191\u2191(\u2191\u03ba a_1) (f i)", "state_after": "case refine_2\n\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\n_m\u03a9 : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\ninst\u271d : IsMarkovKernel \u03ba\nm : \u03b9 \u2192 MeasurableSpace \u03a9\nh_le : \u2200 (i : \u03b9), m i \u2264 _m\u03a9\n\u03c0 : \u03b9 \u2192 Set (Set \u03a9)\nh_pi : \u2200 (n : \u03b9), IsPiSystem (\u03c0 n)\nh_generate : \u2200 (i : \u03b9), m i = generateFrom (\u03c0 i)\nh_ind : iIndepSets \u03c0 \u03ba\ns : Finset \u03b9\nf : \u03b9 \u2192 Set \u03a9\na : \u03b9\nS : Finset \u03b9\nha_notin_S : \u00aca \u2208 S\nh_rec :\n  (\u2200 (i : \u03b9), i \u2208 S \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i) \u2192\n    \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (\u22c2 i \u2208 S, f i) = \u220f i in S, \u2191\u2191(\u2191\u03ba a) (f i)\nhf_m : \u2200 (i : \u03b9), i \u2208 insert a S \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i\nhf_m_S : \u2200 (x : \u03b9), x \u2208 S \u2192 MeasurableSet (f x)\np : Set (Set \u03a9) := piiUnionInter \u03c0 \u2191S\nm_p : MeasurableSpace \u03a9 := generateFrom p\nhS_eq_generate : m_p = generateFrom p\nh_indep : Indep m_p (m a) \u03ba\n\u22a2 \u2200\u1d50 (a_1 : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a_1) (\u22c2 i \u2208 insert a S, f i) = \u220f i in insert a S, \u2191\u2191(\u2191\u03ba a_1) (f i)"}, {"tactic": "have h := h_indep.symm (f a) (\u22c2 n \u2208 S, f n) (hf_m a (Finset.mem_insert_self a S)) ?_", "annotated_tactic": ["have h := h_indep.symm (f a) (\u22c2 n \u2208 S, f n) (hf_m a (<a>Finset.mem_insert_self</a> a S)) ?_", [{"full_name": "Finset.mem_insert_self", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1091, 9], "def_end_pos": [1091, 24]}]], "state_before": "case refine_2\n\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\n_m\u03a9 : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\ninst\u271d : IsMarkovKernel \u03ba\nm : \u03b9 \u2192 MeasurableSpace \u03a9\nh_le : \u2200 (i : \u03b9), m i \u2264 _m\u03a9\n\u03c0 : \u03b9 \u2192 Set (Set \u03a9)\nh_pi : \u2200 (n : \u03b9), IsPiSystem (\u03c0 n)\nh_generate : \u2200 (i : \u03b9), m i = generateFrom (\u03c0 i)\nh_ind : iIndepSets \u03c0 \u03ba\ns : Finset \u03b9\nf : \u03b9 \u2192 Set \u03a9\na : \u03b9\nS : Finset \u03b9\nha_notin_S : \u00aca \u2208 S\nh_rec :\n  (\u2200 (i : \u03b9), i \u2208 S \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i) \u2192\n    \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (\u22c2 i \u2208 S, f i) = \u220f i in S, \u2191\u2191(\u2191\u03ba a) (f i)\nhf_m : \u2200 (i : \u03b9), i \u2208 insert a S \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i\nhf_m_S : \u2200 (x : \u03b9), x \u2208 S \u2192 MeasurableSet (f x)\np : Set (Set \u03a9) := piiUnionInter \u03c0 \u2191S\nm_p : MeasurableSpace \u03a9 := generateFrom p\nhS_eq_generate : m_p = generateFrom p\nh_indep : Indep m_p (m a) \u03ba\n\u22a2 \u2200\u1d50 (a_1 : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a_1) (\u22c2 i \u2208 insert a S, f i) = \u220f i in insert a S, \u2191\u2191(\u2191\u03ba a_1) (f i)", "state_after": "case refine_2.refine_2\n\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\n_m\u03a9 : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\ninst\u271d : IsMarkovKernel \u03ba\nm : \u03b9 \u2192 MeasurableSpace \u03a9\nh_le : \u2200 (i : \u03b9), m i \u2264 _m\u03a9\n\u03c0 : \u03b9 \u2192 Set (Set \u03a9)\nh_pi : \u2200 (n : \u03b9), IsPiSystem (\u03c0 n)\nh_generate : \u2200 (i : \u03b9), m i = generateFrom (\u03c0 i)\nh_ind : iIndepSets \u03c0 \u03ba\ns : Finset \u03b9\nf : \u03b9 \u2192 Set \u03a9\na : \u03b9\nS : Finset \u03b9\nha_notin_S : \u00aca \u2208 S\nh_rec :\n  (\u2200 (i : \u03b9), i \u2208 S \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i) \u2192\n    \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (\u22c2 i \u2208 S, f i) = \u220f i in S, \u2191\u2191(\u2191\u03ba a) (f i)\nhf_m : \u2200 (i : \u03b9), i \u2208 insert a S \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i\nhf_m_S : \u2200 (x : \u03b9), x \u2208 S \u2192 MeasurableSet (f x)\np : Set (Set \u03a9) := piiUnionInter \u03c0 \u2191S\nm_p : MeasurableSpace \u03a9 := generateFrom p\nhS_eq_generate : m_p = generateFrom p\nh_indep : Indep m_p (m a) \u03ba\nh : \u2200\u1d50 (a_1 : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a_1) (f a \u2229 \u22c2 n \u2208 S, f n) = \u2191\u2191(\u2191\u03ba a_1) (f a) * \u2191\u2191(\u2191\u03ba a_1) (\u22c2 n \u2208 S, f n)\n\u22a2 \u2200\u1d50 (a_1 : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a_1) (\u22c2 i \u2208 insert a S, f i) = \u220f i in insert a S, \u2191\u2191(\u2191\u03ba a_1) (f i)\n\ncase refine_2.refine_1\n\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\n_m\u03a9 : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\ninst\u271d : IsMarkovKernel \u03ba\nm : \u03b9 \u2192 MeasurableSpace \u03a9\nh_le : \u2200 (i : \u03b9), m i \u2264 _m\u03a9\n\u03c0 : \u03b9 \u2192 Set (Set \u03a9)\nh_pi : \u2200 (n : \u03b9), IsPiSystem (\u03c0 n)\nh_generate : \u2200 (i : \u03b9), m i = generateFrom (\u03c0 i)\nh_ind : iIndepSets \u03c0 \u03ba\ns : Finset \u03b9\nf : \u03b9 \u2192 Set \u03a9\na : \u03b9\nS : Finset \u03b9\nha_notin_S : \u00aca \u2208 S\nh_rec :\n  (\u2200 (i : \u03b9), i \u2208 S \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i) \u2192\n    \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (\u22c2 i \u2208 S, f i) = \u220f i in S, \u2191\u2191(\u2191\u03ba a) (f i)\nhf_m : \u2200 (i : \u03b9), i \u2208 insert a S \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i\nhf_m_S : \u2200 (x : \u03b9), x \u2208 S \u2192 MeasurableSet (f x)\np : Set (Set \u03a9) := piiUnionInter \u03c0 \u2191S\nm_p : MeasurableSpace \u03a9 := generateFrom p\nhS_eq_generate : m_p = generateFrom p\nh_indep : Indep m_p (m a) \u03ba\n\u22a2 \u22c2 n \u2208 S, f n \u2208 {s | MeasurableSet s}"}, {"tactic": "simp [hx]", "annotated_tactic": ["simp [hx]", []], "state_before": "\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\n_m\u03a9 : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\ninst\u271d : IsMarkovKernel \u03ba\nm : \u03b9 \u2192 MeasurableSpace \u03a9\nh_le : \u2200 (i : \u03b9), m i \u2264 _m\u03a9\n\u03c0 : \u03b9 \u2192 Set (Set \u03a9)\nh_pi : \u2200 (n : \u03b9), IsPiSystem (\u03c0 n)\nh_generate : \u2200 (i : \u03b9), m i = generateFrom (\u03c0 i)\nh_ind : iIndepSets \u03c0 \u03ba\ns : Finset \u03b9\nf : \u03b9 \u2192 Set \u03a9\na : \u03b9\nS : Finset \u03b9\nha_notin_S : \u00aca \u2208 S\nh_rec :\n  (\u2200 (i : \u03b9), i \u2208 S \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i) \u2192\n    \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (\u22c2 i \u2208 S, f i) = \u220f i in S, \u2191\u2191(\u2191\u03ba a) (f i)\nhf_m : \u2200 (i : \u03b9), i \u2208 insert a S \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i\nx : \u03b9\nhx : x \u2208 S\n\u22a2 x \u2208 insert a S", "state_after": "no goals"}, {"tactic": "have hp : IsPiSystem p := isPiSystem_piiUnionInter \u03c0 h_pi S", "annotated_tactic": ["have hp : <a>IsPiSystem</a> p := <a>isPiSystem_piiUnionInter</a> \u03c0 h_pi S", [{"full_name": "IsPiSystem", "def_path": "Mathlib/MeasureTheory/PiSystem.lean", "def_pos": [66, 5], "def_end_pos": [66, 15]}, {"full_name": "isPiSystem_piiUnionInter", "def_path": "Mathlib/MeasureTheory/PiSystem.lean", "def_pos": [437, 9], "def_end_pos": [437, 33]}]], "state_before": "\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\n_m\u03a9 : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\ninst\u271d : IsMarkovKernel \u03ba\nm : \u03b9 \u2192 MeasurableSpace \u03a9\nh_le : \u2200 (i : \u03b9), m i \u2264 _m\u03a9\n\u03c0 : \u03b9 \u2192 Set (Set \u03a9)\nh_pi : \u2200 (n : \u03b9), IsPiSystem (\u03c0 n)\nh_generate : \u2200 (i : \u03b9), m i = generateFrom (\u03c0 i)\nh_ind : iIndepSets \u03c0 \u03ba\ns : Finset \u03b9\nf : \u03b9 \u2192 Set \u03a9\na : \u03b9\nS : Finset \u03b9\nha_notin_S : \u00aca \u2208 S\nh_rec :\n  (\u2200 (i : \u03b9), i \u2208 S \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i) \u2192\n    \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (\u22c2 i \u2208 S, f i) = \u220f i in S, \u2191\u2191(\u2191\u03ba a) (f i)\nhf_m : \u2200 (i : \u03b9), i \u2208 insert a S \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i\nhf_m_S : \u2200 (x : \u03b9), x \u2208 S \u2192 MeasurableSet (f x)\np : Set (Set \u03a9) := piiUnionInter \u03c0 \u2191S\nm_p : MeasurableSpace \u03a9 := generateFrom p\nhS_eq_generate : m_p = generateFrom p\n\u22a2 Indep m_p (m a) \u03ba", "state_after": "\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\n_m\u03a9 : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\ninst\u271d : IsMarkovKernel \u03ba\nm : \u03b9 \u2192 MeasurableSpace \u03a9\nh_le : \u2200 (i : \u03b9), m i \u2264 _m\u03a9\n\u03c0 : \u03b9 \u2192 Set (Set \u03a9)\nh_pi : \u2200 (n : \u03b9), IsPiSystem (\u03c0 n)\nh_generate : \u2200 (i : \u03b9), m i = generateFrom (\u03c0 i)\nh_ind : iIndepSets \u03c0 \u03ba\ns : Finset \u03b9\nf : \u03b9 \u2192 Set \u03a9\na : \u03b9\nS : Finset \u03b9\nha_notin_S : \u00aca \u2208 S\nh_rec :\n  (\u2200 (i : \u03b9), i \u2208 S \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i) \u2192\n    \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (\u22c2 i \u2208 S, f i) = \u220f i in S, \u2191\u2191(\u2191\u03ba a) (f i)\nhf_m : \u2200 (i : \u03b9), i \u2208 insert a S \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i\nhf_m_S : \u2200 (x : \u03b9), x \u2208 S \u2192 MeasurableSet (f x)\np : Set (Set \u03a9) := piiUnionInter \u03c0 \u2191S\nm_p : MeasurableSpace \u03a9 := generateFrom p\nhS_eq_generate : m_p = generateFrom p\nhp : IsPiSystem p\n\u22a2 Indep m_p (m a) \u03ba"}, {"tactic": "have h_le' : \u2200 i, generateFrom (\u03c0 i) \u2264 _m\u03a9 := fun i \u21a6 (h_generate i).symm.trans_le (h_le i)", "annotated_tactic": ["have h_le' : \u2200 i, <a>generateFrom</a> (\u03c0 i) \u2264 _m\u03a9 := fun i \u21a6 (h_generate i).symm.trans_le (h_le i)", [{"full_name": "MeasurableSpace.generateFrom", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [363, 5], "def_end_pos": [363, 17]}]], "state_before": "\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\n_m\u03a9 : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\ninst\u271d : IsMarkovKernel \u03ba\nm : \u03b9 \u2192 MeasurableSpace \u03a9\nh_le : \u2200 (i : \u03b9), m i \u2264 _m\u03a9\n\u03c0 : \u03b9 \u2192 Set (Set \u03a9)\nh_pi : \u2200 (n : \u03b9), IsPiSystem (\u03c0 n)\nh_generate : \u2200 (i : \u03b9), m i = generateFrom (\u03c0 i)\nh_ind : iIndepSets \u03c0 \u03ba\ns : Finset \u03b9\nf : \u03b9 \u2192 Set \u03a9\na : \u03b9\nS : Finset \u03b9\nha_notin_S : \u00aca \u2208 S\nh_rec :\n  (\u2200 (i : \u03b9), i \u2208 S \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i) \u2192\n    \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (\u22c2 i \u2208 S, f i) = \u220f i in S, \u2191\u2191(\u2191\u03ba a) (f i)\nhf_m : \u2200 (i : \u03b9), i \u2208 insert a S \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i\nhf_m_S : \u2200 (x : \u03b9), x \u2208 S \u2192 MeasurableSet (f x)\np : Set (Set \u03a9) := piiUnionInter \u03c0 \u2191S\nm_p : MeasurableSpace \u03a9 := generateFrom p\nhS_eq_generate : m_p = generateFrom p\nhp : IsPiSystem p\n\u22a2 Indep m_p (m a) \u03ba", "state_after": "\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\n_m\u03a9 : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\ninst\u271d : IsMarkovKernel \u03ba\nm : \u03b9 \u2192 MeasurableSpace \u03a9\nh_le : \u2200 (i : \u03b9), m i \u2264 _m\u03a9\n\u03c0 : \u03b9 \u2192 Set (Set \u03a9)\nh_pi : \u2200 (n : \u03b9), IsPiSystem (\u03c0 n)\nh_generate : \u2200 (i : \u03b9), m i = generateFrom (\u03c0 i)\nh_ind : iIndepSets \u03c0 \u03ba\ns : Finset \u03b9\nf : \u03b9 \u2192 Set \u03a9\na : \u03b9\nS : Finset \u03b9\nha_notin_S : \u00aca \u2208 S\nh_rec :\n  (\u2200 (i : \u03b9), i \u2208 S \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i) \u2192\n    \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (\u22c2 i \u2208 S, f i) = \u220f i in S, \u2191\u2191(\u2191\u03ba a) (f i)\nhf_m : \u2200 (i : \u03b9), i \u2208 insert a S \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i\nhf_m_S : \u2200 (x : \u03b9), x \u2208 S \u2192 MeasurableSet (f x)\np : Set (Set \u03a9) := piiUnionInter \u03c0 \u2191S\nm_p : MeasurableSpace \u03a9 := generateFrom p\nhS_eq_generate : m_p = generateFrom p\nhp : IsPiSystem p\nh_le' : \u2200 (i : \u03b9), generateFrom (\u03c0 i) \u2264 _m\u03a9\n\u22a2 Indep m_p (m a) \u03ba"}, {"tactic": "have hm_p : m_p \u2264 _m\u03a9 := generateFrom_piiUnionInter_le \u03c0 h_le' S", "annotated_tactic": ["have hm_p : m_p \u2264 _m\u03a9 := <a>generateFrom_piiUnionInter_le</a> \u03c0 h_le' S", [{"full_name": "generateFrom_piiUnionInter_le", "def_path": "Mathlib/MeasureTheory/PiSystem.lean", "def_pos": [486, 9], "def_end_pos": [486, 38]}]], "state_before": "\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\n_m\u03a9 : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\ninst\u271d : IsMarkovKernel \u03ba\nm : \u03b9 \u2192 MeasurableSpace \u03a9\nh_le : \u2200 (i : \u03b9), m i \u2264 _m\u03a9\n\u03c0 : \u03b9 \u2192 Set (Set \u03a9)\nh_pi : \u2200 (n : \u03b9), IsPiSystem (\u03c0 n)\nh_generate : \u2200 (i : \u03b9), m i = generateFrom (\u03c0 i)\nh_ind : iIndepSets \u03c0 \u03ba\ns : Finset \u03b9\nf : \u03b9 \u2192 Set \u03a9\na : \u03b9\nS : Finset \u03b9\nha_notin_S : \u00aca \u2208 S\nh_rec :\n  (\u2200 (i : \u03b9), i \u2208 S \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i) \u2192\n    \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (\u22c2 i \u2208 S, f i) = \u220f i in S, \u2191\u2191(\u2191\u03ba a) (f i)\nhf_m : \u2200 (i : \u03b9), i \u2208 insert a S \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i\nhf_m_S : \u2200 (x : \u03b9), x \u2208 S \u2192 MeasurableSet (f x)\np : Set (Set \u03a9) := piiUnionInter \u03c0 \u2191S\nm_p : MeasurableSpace \u03a9 := generateFrom p\nhS_eq_generate : m_p = generateFrom p\nhp : IsPiSystem p\nh_le' : \u2200 (i : \u03b9), generateFrom (\u03c0 i) \u2264 _m\u03a9\n\u22a2 Indep m_p (m a) \u03ba", "state_after": "\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\n_m\u03a9 : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\ninst\u271d : IsMarkovKernel \u03ba\nm : \u03b9 \u2192 MeasurableSpace \u03a9\nh_le : \u2200 (i : \u03b9), m i \u2264 _m\u03a9\n\u03c0 : \u03b9 \u2192 Set (Set \u03a9)\nh_pi : \u2200 (n : \u03b9), IsPiSystem (\u03c0 n)\nh_generate : \u2200 (i : \u03b9), m i = generateFrom (\u03c0 i)\nh_ind : iIndepSets \u03c0 \u03ba\ns : Finset \u03b9\nf : \u03b9 \u2192 Set \u03a9\na : \u03b9\nS : Finset \u03b9\nha_notin_S : \u00aca \u2208 S\nh_rec :\n  (\u2200 (i : \u03b9), i \u2208 S \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i) \u2192\n    \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (\u22c2 i \u2208 S, f i) = \u220f i in S, \u2191\u2191(\u2191\u03ba a) (f i)\nhf_m : \u2200 (i : \u03b9), i \u2208 insert a S \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i\nhf_m_S : \u2200 (x : \u03b9), x \u2208 S \u2192 MeasurableSet (f x)\np : Set (Set \u03a9) := piiUnionInter \u03c0 \u2191S\nm_p : MeasurableSpace \u03a9 := generateFrom p\nhS_eq_generate : m_p = generateFrom p\nhp : IsPiSystem p\nh_le' : \u2200 (i : \u03b9), generateFrom (\u03c0 i) \u2264 _m\u03a9\nhm_p : m_p \u2264 _m\u03a9\n\u22a2 Indep m_p (m a) \u03ba"}, {"tactic": "exact IndepSets.indep hm_p (h_le a) hp (h_pi a) hS_eq_generate (h_generate a)\n  (iIndepSets.piiUnionInter_of_not_mem h_ind ha_notin_S)", "annotated_tactic": ["exact <a>IndepSets.indep</a> hm_p (h_le a) hp (h_pi a) hS_eq_generate (h_generate a)\n        (<a>iIndepSets.piiUnionInter_of_not_mem</a> h_ind ha_notin_S)", [{"full_name": "ProbabilityTheory.kernel.IndepSets.indep", "def_path": "Mathlib/Probability/Independence/Kernel.lean", "def_pos": [353, 9], "def_end_pos": [353, 24]}, {"full_name": "ProbabilityTheory.kernel.iIndepSets.piiUnionInter_of_not_mem", "def_path": "Mathlib/Probability/Independence/Kernel.lean", "def_pos": [520, 9], "def_end_pos": [520, 44]}]], "state_before": "\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\n_m\u03a9 : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\ninst\u271d : IsMarkovKernel \u03ba\nm : \u03b9 \u2192 MeasurableSpace \u03a9\nh_le : \u2200 (i : \u03b9), m i \u2264 _m\u03a9\n\u03c0 : \u03b9 \u2192 Set (Set \u03a9)\nh_pi : \u2200 (n : \u03b9), IsPiSystem (\u03c0 n)\nh_generate : \u2200 (i : \u03b9), m i = generateFrom (\u03c0 i)\nh_ind : iIndepSets \u03c0 \u03ba\ns : Finset \u03b9\nf : \u03b9 \u2192 Set \u03a9\na : \u03b9\nS : Finset \u03b9\nha_notin_S : \u00aca \u2208 S\nh_rec :\n  (\u2200 (i : \u03b9), i \u2208 S \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i) \u2192\n    \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (\u22c2 i \u2208 S, f i) = \u220f i in S, \u2191\u2191(\u2191\u03ba a) (f i)\nhf_m : \u2200 (i : \u03b9), i \u2208 insert a S \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i\nhf_m_S : \u2200 (x : \u03b9), x \u2208 S \u2192 MeasurableSet (f x)\np : Set (Set \u03a9) := piiUnionInter \u03c0 \u2191S\nm_p : MeasurableSpace \u03a9 := generateFrom p\nhS_eq_generate : m_p = generateFrom p\nhp : IsPiSystem p\nh_le' : \u2200 (i : \u03b9), generateFrom (\u03c0 i) \u2264 _m\u03a9\nhm_p : m_p \u2264 _m\u03a9\n\u22a2 Indep m_p (m a) \u03ba", "state_after": "no goals"}, {"tactic": "filter_upwards [h_rec hf_m_S, h] with a' ha' h'", "annotated_tactic": ["filter_upwards [h_rec hf_m_S, h] with a' ha' h'", []], "state_before": "case refine_2.refine_2\n\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\n_m\u03a9 : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\ninst\u271d : IsMarkovKernel \u03ba\nm : \u03b9 \u2192 MeasurableSpace \u03a9\nh_le : \u2200 (i : \u03b9), m i \u2264 _m\u03a9\n\u03c0 : \u03b9 \u2192 Set (Set \u03a9)\nh_pi : \u2200 (n : \u03b9), IsPiSystem (\u03c0 n)\nh_generate : \u2200 (i : \u03b9), m i = generateFrom (\u03c0 i)\nh_ind : iIndepSets \u03c0 \u03ba\ns : Finset \u03b9\nf : \u03b9 \u2192 Set \u03a9\na : \u03b9\nS : Finset \u03b9\nha_notin_S : \u00aca \u2208 S\nh_rec :\n  (\u2200 (i : \u03b9), i \u2208 S \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i) \u2192\n    \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (\u22c2 i \u2208 S, f i) = \u220f i in S, \u2191\u2191(\u2191\u03ba a) (f i)\nhf_m : \u2200 (i : \u03b9), i \u2208 insert a S \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i\nhf_m_S : \u2200 (x : \u03b9), x \u2208 S \u2192 MeasurableSet (f x)\np : Set (Set \u03a9) := piiUnionInter \u03c0 \u2191S\nm_p : MeasurableSpace \u03a9 := generateFrom p\nhS_eq_generate : m_p = generateFrom p\nh_indep : Indep m_p (m a) \u03ba\nh : \u2200\u1d50 (a_1 : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a_1) (f a \u2229 \u22c2 n \u2208 S, f n) = \u2191\u2191(\u2191\u03ba a_1) (f a) * \u2191\u2191(\u2191\u03ba a_1) (\u22c2 n \u2208 S, f n)\n\u22a2 \u2200\u1d50 (a_1 : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a_1) (\u22c2 i \u2208 insert a S, f i) = \u220f i in insert a S, \u2191\u2191(\u2191\u03ba a_1) (f i)", "state_after": "case h\n\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\n_m\u03a9 : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\ninst\u271d : IsMarkovKernel \u03ba\nm : \u03b9 \u2192 MeasurableSpace \u03a9\nh_le : \u2200 (i : \u03b9), m i \u2264 _m\u03a9\n\u03c0 : \u03b9 \u2192 Set (Set \u03a9)\nh_pi : \u2200 (n : \u03b9), IsPiSystem (\u03c0 n)\nh_generate : \u2200 (i : \u03b9), m i = generateFrom (\u03c0 i)\nh_ind : iIndepSets \u03c0 \u03ba\ns : Finset \u03b9\nf : \u03b9 \u2192 Set \u03a9\na : \u03b9\nS : Finset \u03b9\nha_notin_S : \u00aca \u2208 S\nh_rec :\n  (\u2200 (i : \u03b9), i \u2208 S \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i) \u2192\n    \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (\u22c2 i \u2208 S, f i) = \u220f i in S, \u2191\u2191(\u2191\u03ba a) (f i)\nhf_m : \u2200 (i : \u03b9), i \u2208 insert a S \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i\nhf_m_S : \u2200 (x : \u03b9), x \u2208 S \u2192 MeasurableSet (f x)\np : Set (Set \u03a9) := piiUnionInter \u03c0 \u2191S\nm_p : MeasurableSpace \u03a9 := generateFrom p\nhS_eq_generate : m_p = generateFrom p\nh_indep : Indep m_p (m a) \u03ba\nh : \u2200\u1d50 (a_1 : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a_1) (f a \u2229 \u22c2 n \u2208 S, f n) = \u2191\u2191(\u2191\u03ba a_1) (f a) * \u2191\u2191(\u2191\u03ba a_1) (\u22c2 n \u2208 S, f n)\na' : \u03b1\nha' : \u2191\u2191(\u2191\u03ba a') (\u22c2 i \u2208 S, f i) = \u220f i in S, \u2191\u2191(\u2191\u03ba a') (f i)\nh' : \u2191\u2191(\u2191\u03ba a') (f a \u2229 \u22c2 n \u2208 S, f n) = \u2191\u2191(\u2191\u03ba a') (f a) * \u2191\u2191(\u2191\u03ba a') (\u22c2 n \u2208 S, f n)\n\u22a2 \u2191\u2191(\u2191\u03ba a') (\u22c2 i \u2208 insert a S, f i) = \u220f i in insert a S, \u2191\u2191(\u2191\u03ba a') (f i)"}, {"tactic": "rwa [Finset.set_biInter_insert, Finset.prod_insert ha_notin_S, \u2190 ha']", "annotated_tactic": ["rwa [<a>Finset.set_biInter_insert</a>, <a>Finset.prod_insert</a> ha_notin_S, \u2190 ha']", [{"full_name": "Finset.set_biInter_insert", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [2141, 9], "def_end_pos": [2141, 27]}, {"full_name": "Finset.prod_insert", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [317, 9], "def_end_pos": [317, 20]}]], "state_before": "case h\n\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\n_m\u03a9 : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\ninst\u271d : IsMarkovKernel \u03ba\nm : \u03b9 \u2192 MeasurableSpace \u03a9\nh_le : \u2200 (i : \u03b9), m i \u2264 _m\u03a9\n\u03c0 : \u03b9 \u2192 Set (Set \u03a9)\nh_pi : \u2200 (n : \u03b9), IsPiSystem (\u03c0 n)\nh_generate : \u2200 (i : \u03b9), m i = generateFrom (\u03c0 i)\nh_ind : iIndepSets \u03c0 \u03ba\ns : Finset \u03b9\nf : \u03b9 \u2192 Set \u03a9\na : \u03b9\nS : Finset \u03b9\nha_notin_S : \u00aca \u2208 S\nh_rec :\n  (\u2200 (i : \u03b9), i \u2208 S \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i) \u2192\n    \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (\u22c2 i \u2208 S, f i) = \u220f i in S, \u2191\u2191(\u2191\u03ba a) (f i)\nhf_m : \u2200 (i : \u03b9), i \u2208 insert a S \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i\nhf_m_S : \u2200 (x : \u03b9), x \u2208 S \u2192 MeasurableSet (f x)\np : Set (Set \u03a9) := piiUnionInter \u03c0 \u2191S\nm_p : MeasurableSpace \u03a9 := generateFrom p\nhS_eq_generate : m_p = generateFrom p\nh_indep : Indep m_p (m a) \u03ba\nh : \u2200\u1d50 (a_1 : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a_1) (f a \u2229 \u22c2 n \u2208 S, f n) = \u2191\u2191(\u2191\u03ba a_1) (f a) * \u2191\u2191(\u2191\u03ba a_1) (\u22c2 n \u2208 S, f n)\na' : \u03b1\nha' : \u2191\u2191(\u2191\u03ba a') (\u22c2 i \u2208 S, f i) = \u220f i in S, \u2191\u2191(\u2191\u03ba a') (f i)\nh' : \u2191\u2191(\u2191\u03ba a') (f a \u2229 \u22c2 n \u2208 S, f n) = \u2191\u2191(\u2191\u03ba a') (f a) * \u2191\u2191(\u2191\u03ba a') (\u22c2 n \u2208 S, f n)\n\u22a2 \u2191\u2191(\u2191\u03ba a') (\u22c2 i \u2208 insert a S, f i) = \u220f i in insert a S, \u2191\u2191(\u2191\u03ba a') (f i)", "state_after": "no goals"}, {"tactic": "have h_le_p : \u2200 i \u2208 S, m i \u2264 m_p := by\n  intros n hn\n  rw [hS_eq_generate, h_generate n]\n  refine le_generateFrom_piiUnionInter (S : Set \u03b9) hn", "annotated_tactic": ["have h_le_p : \u2200 i \u2208 S, m i \u2264 m_p := by\n        intros n hn\n        rw [hS_eq_generate, h_generate n]\n        refine <a>le_generateFrom_piiUnionInter</a> (S : <a>Set</a> \u03b9) hn", [{"full_name": "le_generateFrom_piiUnionInter", "def_path": "Mathlib/MeasureTheory/PiSystem.lean", "def_pos": [511, 9], "def_end_pos": [511, 38]}, {"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}]], "state_before": "case refine_2.refine_1\n\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\n_m\u03a9 : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\ninst\u271d : IsMarkovKernel \u03ba\nm : \u03b9 \u2192 MeasurableSpace \u03a9\nh_le : \u2200 (i : \u03b9), m i \u2264 _m\u03a9\n\u03c0 : \u03b9 \u2192 Set (Set \u03a9)\nh_pi : \u2200 (n : \u03b9), IsPiSystem (\u03c0 n)\nh_generate : \u2200 (i : \u03b9), m i = generateFrom (\u03c0 i)\nh_ind : iIndepSets \u03c0 \u03ba\ns : Finset \u03b9\nf : \u03b9 \u2192 Set \u03a9\na : \u03b9\nS : Finset \u03b9\nha_notin_S : \u00aca \u2208 S\nh_rec :\n  (\u2200 (i : \u03b9), i \u2208 S \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i) \u2192\n    \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (\u22c2 i \u2208 S, f i) = \u220f i in S, \u2191\u2191(\u2191\u03ba a) (f i)\nhf_m : \u2200 (i : \u03b9), i \u2208 insert a S \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i\nhf_m_S : \u2200 (x : \u03b9), x \u2208 S \u2192 MeasurableSet (f x)\np : Set (Set \u03a9) := piiUnionInter \u03c0 \u2191S\nm_p : MeasurableSpace \u03a9 := generateFrom p\nhS_eq_generate : m_p = generateFrom p\nh_indep : Indep m_p (m a) \u03ba\n\u22a2 \u22c2 n \u2208 S, f n \u2208 {s | MeasurableSet s}", "state_after": "case refine_2.refine_1\n\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\n_m\u03a9 : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\ninst\u271d : IsMarkovKernel \u03ba\nm : \u03b9 \u2192 MeasurableSpace \u03a9\nh_le : \u2200 (i : \u03b9), m i \u2264 _m\u03a9\n\u03c0 : \u03b9 \u2192 Set (Set \u03a9)\nh_pi : \u2200 (n : \u03b9), IsPiSystem (\u03c0 n)\nh_generate : \u2200 (i : \u03b9), m i = generateFrom (\u03c0 i)\nh_ind : iIndepSets \u03c0 \u03ba\ns : Finset \u03b9\nf : \u03b9 \u2192 Set \u03a9\na : \u03b9\nS : Finset \u03b9\nha_notin_S : \u00aca \u2208 S\nh_rec :\n  (\u2200 (i : \u03b9), i \u2208 S \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i) \u2192\n    \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (\u22c2 i \u2208 S, f i) = \u220f i in S, \u2191\u2191(\u2191\u03ba a) (f i)\nhf_m : \u2200 (i : \u03b9), i \u2208 insert a S \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i\nhf_m_S : \u2200 (x : \u03b9), x \u2208 S \u2192 MeasurableSet (f x)\np : Set (Set \u03a9) := piiUnionInter \u03c0 \u2191S\nm_p : MeasurableSpace \u03a9 := generateFrom p\nhS_eq_generate : m_p = generateFrom p\nh_indep : Indep m_p (m a) \u03ba\nh_le_p : \u2200 (i : \u03b9), i \u2208 S \u2192 m i \u2264 m_p\n\u22a2 \u22c2 n \u2208 S, f n \u2208 {s | MeasurableSet s}"}, {"tactic": "have h_S_f : \u2200 i \u2208 S, MeasurableSet[m_p] (f i) :=\n  fun i hi \u21a6 (h_le_p i hi) (f i) (hf_m_S i hi)", "annotated_tactic": ["have h_S_f : \u2200 i \u2208 S, MeasurableSet[m_p] (f i) :=\n        fun i hi \u21a6 (h_le_p i hi) (f i) (hf_m_S i hi)", []], "state_before": "case refine_2.refine_1\n\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\n_m\u03a9 : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\ninst\u271d : IsMarkovKernel \u03ba\nm : \u03b9 \u2192 MeasurableSpace \u03a9\nh_le : \u2200 (i : \u03b9), m i \u2264 _m\u03a9\n\u03c0 : \u03b9 \u2192 Set (Set \u03a9)\nh_pi : \u2200 (n : \u03b9), IsPiSystem (\u03c0 n)\nh_generate : \u2200 (i : \u03b9), m i = generateFrom (\u03c0 i)\nh_ind : iIndepSets \u03c0 \u03ba\ns : Finset \u03b9\nf : \u03b9 \u2192 Set \u03a9\na : \u03b9\nS : Finset \u03b9\nha_notin_S : \u00aca \u2208 S\nh_rec :\n  (\u2200 (i : \u03b9), i \u2208 S \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i) \u2192\n    \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (\u22c2 i \u2208 S, f i) = \u220f i in S, \u2191\u2191(\u2191\u03ba a) (f i)\nhf_m : \u2200 (i : \u03b9), i \u2208 insert a S \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i\nhf_m_S : \u2200 (x : \u03b9), x \u2208 S \u2192 MeasurableSet (f x)\np : Set (Set \u03a9) := piiUnionInter \u03c0 \u2191S\nm_p : MeasurableSpace \u03a9 := generateFrom p\nhS_eq_generate : m_p = generateFrom p\nh_indep : Indep m_p (m a) \u03ba\nh_le_p : \u2200 (i : \u03b9), i \u2208 S \u2192 m i \u2264 m_p\n\u22a2 \u22c2 n \u2208 S, f n \u2208 {s | MeasurableSet s}", "state_after": "case refine_2.refine_1\n\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\n_m\u03a9 : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\ninst\u271d : IsMarkovKernel \u03ba\nm : \u03b9 \u2192 MeasurableSpace \u03a9\nh_le : \u2200 (i : \u03b9), m i \u2264 _m\u03a9\n\u03c0 : \u03b9 \u2192 Set (Set \u03a9)\nh_pi : \u2200 (n : \u03b9), IsPiSystem (\u03c0 n)\nh_generate : \u2200 (i : \u03b9), m i = generateFrom (\u03c0 i)\nh_ind : iIndepSets \u03c0 \u03ba\ns : Finset \u03b9\nf : \u03b9 \u2192 Set \u03a9\na : \u03b9\nS : Finset \u03b9\nha_notin_S : \u00aca \u2208 S\nh_rec :\n  (\u2200 (i : \u03b9), i \u2208 S \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i) \u2192\n    \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (\u22c2 i \u2208 S, f i) = \u220f i in S, \u2191\u2191(\u2191\u03ba a) (f i)\nhf_m : \u2200 (i : \u03b9), i \u2208 insert a S \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i\nhf_m_S : \u2200 (x : \u03b9), x \u2208 S \u2192 MeasurableSet (f x)\np : Set (Set \u03a9) := piiUnionInter \u03c0 \u2191S\nm_p : MeasurableSpace \u03a9 := generateFrom p\nhS_eq_generate : m_p = generateFrom p\nh_indep : Indep m_p (m a) \u03ba\nh_le_p : \u2200 (i : \u03b9), i \u2208 S \u2192 m i \u2264 m_p\nh_S_f : \u2200 (i : \u03b9), i \u2208 S \u2192 MeasurableSet (f i)\n\u22a2 \u22c2 n \u2208 S, f n \u2208 {s | MeasurableSet s}"}, {"tactic": "exact S.measurableSet_biInter h_S_f", "annotated_tactic": ["exact S.measurableSet_biInter h_S_f", []], "state_before": "case refine_2.refine_1\n\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\n_m\u03a9 : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\ninst\u271d : IsMarkovKernel \u03ba\nm : \u03b9 \u2192 MeasurableSpace \u03a9\nh_le : \u2200 (i : \u03b9), m i \u2264 _m\u03a9\n\u03c0 : \u03b9 \u2192 Set (Set \u03a9)\nh_pi : \u2200 (n : \u03b9), IsPiSystem (\u03c0 n)\nh_generate : \u2200 (i : \u03b9), m i = generateFrom (\u03c0 i)\nh_ind : iIndepSets \u03c0 \u03ba\ns : Finset \u03b9\nf : \u03b9 \u2192 Set \u03a9\na : \u03b9\nS : Finset \u03b9\nha_notin_S : \u00aca \u2208 S\nh_rec :\n  (\u2200 (i : \u03b9), i \u2208 S \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i) \u2192\n    \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (\u22c2 i \u2208 S, f i) = \u220f i in S, \u2191\u2191(\u2191\u03ba a) (f i)\nhf_m : \u2200 (i : \u03b9), i \u2208 insert a S \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i\nhf_m_S : \u2200 (x : \u03b9), x \u2208 S \u2192 MeasurableSet (f x)\np : Set (Set \u03a9) := piiUnionInter \u03c0 \u2191S\nm_p : MeasurableSpace \u03a9 := generateFrom p\nhS_eq_generate : m_p = generateFrom p\nh_indep : Indep m_p (m a) \u03ba\nh_le_p : \u2200 (i : \u03b9), i \u2208 S \u2192 m i \u2264 m_p\nh_S_f : \u2200 (i : \u03b9), i \u2208 S \u2192 MeasurableSet (f i)\n\u22a2 \u22c2 n \u2208 S, f n \u2208 {s | MeasurableSet s}", "state_after": "no goals"}, {"tactic": "intros n hn", "annotated_tactic": ["intros n hn", []], "state_before": "\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\n_m\u03a9 : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\ninst\u271d : IsMarkovKernel \u03ba\nm : \u03b9 \u2192 MeasurableSpace \u03a9\nh_le : \u2200 (i : \u03b9), m i \u2264 _m\u03a9\n\u03c0 : \u03b9 \u2192 Set (Set \u03a9)\nh_pi : \u2200 (n : \u03b9), IsPiSystem (\u03c0 n)\nh_generate : \u2200 (i : \u03b9), m i = generateFrom (\u03c0 i)\nh_ind : iIndepSets \u03c0 \u03ba\ns : Finset \u03b9\nf : \u03b9 \u2192 Set \u03a9\na : \u03b9\nS : Finset \u03b9\nha_notin_S : \u00aca \u2208 S\nh_rec :\n  (\u2200 (i : \u03b9), i \u2208 S \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i) \u2192\n    \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (\u22c2 i \u2208 S, f i) = \u220f i in S, \u2191\u2191(\u2191\u03ba a) (f i)\nhf_m : \u2200 (i : \u03b9), i \u2208 insert a S \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i\nhf_m_S : \u2200 (x : \u03b9), x \u2208 S \u2192 MeasurableSet (f x)\np : Set (Set \u03a9) := piiUnionInter \u03c0 \u2191S\nm_p : MeasurableSpace \u03a9 := generateFrom p\nhS_eq_generate : m_p = generateFrom p\nh_indep : Indep m_p (m a) \u03ba\n\u22a2 \u2200 (i : \u03b9), i \u2208 S \u2192 m i \u2264 m_p", "state_after": "\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\n_m\u03a9 : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\ninst\u271d : IsMarkovKernel \u03ba\nm : \u03b9 \u2192 MeasurableSpace \u03a9\nh_le : \u2200 (i : \u03b9), m i \u2264 _m\u03a9\n\u03c0 : \u03b9 \u2192 Set (Set \u03a9)\nh_pi : \u2200 (n : \u03b9), IsPiSystem (\u03c0 n)\nh_generate : \u2200 (i : \u03b9), m i = generateFrom (\u03c0 i)\nh_ind : iIndepSets \u03c0 \u03ba\ns : Finset \u03b9\nf : \u03b9 \u2192 Set \u03a9\na : \u03b9\nS : Finset \u03b9\nha_notin_S : \u00aca \u2208 S\nh_rec :\n  (\u2200 (i : \u03b9), i \u2208 S \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i) \u2192\n    \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (\u22c2 i \u2208 S, f i) = \u220f i in S, \u2191\u2191(\u2191\u03ba a) (f i)\nhf_m : \u2200 (i : \u03b9), i \u2208 insert a S \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i\nhf_m_S : \u2200 (x : \u03b9), x \u2208 S \u2192 MeasurableSet (f x)\np : Set (Set \u03a9) := piiUnionInter \u03c0 \u2191S\nm_p : MeasurableSpace \u03a9 := generateFrom p\nhS_eq_generate : m_p = generateFrom p\nh_indep : Indep m_p (m a) \u03ba\nn : \u03b9\nhn : n \u2208 S\n\u22a2 m n \u2264 m_p"}, {"tactic": "rw [hS_eq_generate, h_generate n]", "annotated_tactic": ["rw [hS_eq_generate, h_generate n]", []], "state_before": "\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\n_m\u03a9 : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\ninst\u271d : IsMarkovKernel \u03ba\nm : \u03b9 \u2192 MeasurableSpace \u03a9\nh_le : \u2200 (i : \u03b9), m i \u2264 _m\u03a9\n\u03c0 : \u03b9 \u2192 Set (Set \u03a9)\nh_pi : \u2200 (n : \u03b9), IsPiSystem (\u03c0 n)\nh_generate : \u2200 (i : \u03b9), m i = generateFrom (\u03c0 i)\nh_ind : iIndepSets \u03c0 \u03ba\ns : Finset \u03b9\nf : \u03b9 \u2192 Set \u03a9\na : \u03b9\nS : Finset \u03b9\nha_notin_S : \u00aca \u2208 S\nh_rec :\n  (\u2200 (i : \u03b9), i \u2208 S \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i) \u2192\n    \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (\u22c2 i \u2208 S, f i) = \u220f i in S, \u2191\u2191(\u2191\u03ba a) (f i)\nhf_m : \u2200 (i : \u03b9), i \u2208 insert a S \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i\nhf_m_S : \u2200 (x : \u03b9), x \u2208 S \u2192 MeasurableSet (f x)\np : Set (Set \u03a9) := piiUnionInter \u03c0 \u2191S\nm_p : MeasurableSpace \u03a9 := generateFrom p\nhS_eq_generate : m_p = generateFrom p\nh_indep : Indep m_p (m a) \u03ba\nn : \u03b9\nhn : n \u2208 S\n\u22a2 m n \u2264 m_p", "state_after": "\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\n_m\u03a9 : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\ninst\u271d : IsMarkovKernel \u03ba\nm : \u03b9 \u2192 MeasurableSpace \u03a9\nh_le : \u2200 (i : \u03b9), m i \u2264 _m\u03a9\n\u03c0 : \u03b9 \u2192 Set (Set \u03a9)\nh_pi : \u2200 (n : \u03b9), IsPiSystem (\u03c0 n)\nh_generate : \u2200 (i : \u03b9), m i = generateFrom (\u03c0 i)\nh_ind : iIndepSets \u03c0 \u03ba\ns : Finset \u03b9\nf : \u03b9 \u2192 Set \u03a9\na : \u03b9\nS : Finset \u03b9\nha_notin_S : \u00aca \u2208 S\nh_rec :\n  (\u2200 (i : \u03b9), i \u2208 S \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i) \u2192\n    \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (\u22c2 i \u2208 S, f i) = \u220f i in S, \u2191\u2191(\u2191\u03ba a) (f i)\nhf_m : \u2200 (i : \u03b9), i \u2208 insert a S \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i\nhf_m_S : \u2200 (x : \u03b9), x \u2208 S \u2192 MeasurableSet (f x)\np : Set (Set \u03a9) := piiUnionInter \u03c0 \u2191S\nm_p : MeasurableSpace \u03a9 := generateFrom p\nhS_eq_generate : m_p = generateFrom p\nh_indep : Indep m_p (m a) \u03ba\nn : \u03b9\nhn : n \u2208 S\n\u22a2 generateFrom (\u03c0 n) \u2264 generateFrom p"}, {"tactic": "refine le_generateFrom_piiUnionInter (S : Set \u03b9) hn", "annotated_tactic": ["refine <a>le_generateFrom_piiUnionInter</a> (S : <a>Set</a> \u03b9) hn", [{"full_name": "le_generateFrom_piiUnionInter", "def_path": "Mathlib/MeasureTheory/PiSystem.lean", "def_pos": [511, 9], "def_end_pos": [511, 38]}, {"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}]], "state_before": "\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\n_m\u03a9 : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\ninst\u271d : IsMarkovKernel \u03ba\nm : \u03b9 \u2192 MeasurableSpace \u03a9\nh_le : \u2200 (i : \u03b9), m i \u2264 _m\u03a9\n\u03c0 : \u03b9 \u2192 Set (Set \u03a9)\nh_pi : \u2200 (n : \u03b9), IsPiSystem (\u03c0 n)\nh_generate : \u2200 (i : \u03b9), m i = generateFrom (\u03c0 i)\nh_ind : iIndepSets \u03c0 \u03ba\ns : Finset \u03b9\nf : \u03b9 \u2192 Set \u03a9\na : \u03b9\nS : Finset \u03b9\nha_notin_S : \u00aca \u2208 S\nh_rec :\n  (\u2200 (i : \u03b9), i \u2208 S \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i) \u2192\n    \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (\u22c2 i \u2208 S, f i) = \u220f i in S, \u2191\u2191(\u2191\u03ba a) (f i)\nhf_m : \u2200 (i : \u03b9), i \u2208 insert a S \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i\nhf_m_S : \u2200 (x : \u03b9), x \u2208 S \u2192 MeasurableSet (f x)\np : Set (Set \u03a9) := piiUnionInter \u03c0 \u2191S\nm_p : MeasurableSpace \u03a9 := generateFrom p\nhS_eq_generate : m_p = generateFrom p\nh_indep : Indep m_p (m a) \u03ba\nn : \u03b9\nhn : n \u2208 S\n\u22a2 generateFrom (\u03c0 n) \u2264 generateFrom p", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/TuringMachine.lean", "full_name": "Turing.TM0to1.tr_respects", "start": [2024, 1], "end": [2045, 10], "traced_tactics": [{"tactic": "cases' e : M q T.1 with val", "annotated_tactic": ["cases' e : M q T.1 with val", []], "state_before": "\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d : Inhabited \u039b\nM : TM0.Machine \u0393 \u039b\nx\u271d : Cfg\u2080\nq : \u039b\nT : Tape \u0393\n\u22a2 FRespects (TM1.step (tr M)) (fun a => trCfg M a) (trCfg M { q := q, Tape := T }) (TM0.step M { q := q, Tape := T })", "state_after": "case none\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d : Inhabited \u039b\nM : TM0.Machine \u0393 \u039b\nx\u271d : Cfg\u2080\nq : \u039b\nT : Tape \u0393\ne : M q T.head = none\n\u22a2 FRespects (TM1.step (tr M)) (fun a => trCfg M a) (trCfg M { q := q, Tape := T }) (TM0.step M { q := q, Tape := T })\n\ncase some\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d : Inhabited \u039b\nM : TM0.Machine \u0393 \u039b\nx\u271d : Cfg\u2080\nq : \u039b\nT : Tape \u0393\nval : \u039b \u00d7 TM0.Stmt \u0393\ne : M q T.head = some val\n\u22a2 FRespects (TM1.step (tr M)) (fun a => trCfg M a) (trCfg M { q := q, Tape := T }) (TM0.step M { q := q, Tape := T })"}, {"tactic": "cases' val with q' s", "annotated_tactic": ["cases' val with q' s", []], "state_before": "case some\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d : Inhabited \u039b\nM : TM0.Machine \u0393 \u039b\nx\u271d : Cfg\u2080\nq : \u039b\nT : Tape \u0393\nval : \u039b \u00d7 TM0.Stmt \u0393\ne : M q T.head = some val\n\u22a2 FRespects (TM1.step (tr M)) (fun a => trCfg M a) (trCfg M { q := q, Tape := T }) (TM0.step M { q := q, Tape := T })", "state_after": "case some.mk\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d : Inhabited \u039b\nM : TM0.Machine \u0393 \u039b\nx\u271d : Cfg\u2080\nq : \u039b\nT : Tape \u0393\nq' : \u039b\ns : TM0.Stmt \u0393\ne : M q T.head = some (q', s)\n\u22a2 FRespects (TM1.step (tr M)) (fun a => trCfg M a) (trCfg M { q := q, Tape := T }) (TM0.step M { q := q, Tape := T })"}, {"tactic": "simp only [FRespects, TM0.step, trCfg, e, Option.isSome, cond, Option.map_some']", "annotated_tactic": ["simp only [<a>FRespects</a>, <a>TM0.step</a>, <a>trCfg</a>, e, <a>Option.isSome</a>, <a>cond</a>, <a>Option.map_some'</a>]", [{"full_name": "Turing.FRespects", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [957, 5], "def_end_pos": [957, 14]}, {"full_name": "Turing.TM0.step", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1076, 5], "def_end_pos": [1076, 9]}, {"full_name": "Turing.TM0to1.trCfg", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [2020, 5], "def_end_pos": [2020, 10]}, {"full_name": "Option.isSome", "def_path": "lake-packages/lean4/src/lean/Init/Data/Option/Basic.lean", "def_pos": [21, 15], "def_end_pos": [21, 21]}, {"full_name": "cond", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [971, 21], "def_end_pos": [971, 25]}, {"full_name": "Option.map_some'", "def_path": "lake-packages/std/Std/Data/Option/Init/Lemmas.lean", "def_pos": [20, 17], "def_end_pos": [20, 26]}]], "state_before": "case some.mk\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d : Inhabited \u039b\nM : TM0.Machine \u0393 \u039b\nx\u271d : Cfg\u2080\nq : \u039b\nT : Tape \u0393\nq' : \u039b\ns : TM0.Stmt \u0393\ne : M q T.head = some (q', s)\n\u22a2 FRespects (TM1.step (tr M)) (fun a => trCfg M a) (trCfg M { q := q, Tape := T }) (TM0.step M { q := q, Tape := T })", "state_after": "case some.mk\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d : Inhabited \u039b\nM : TM0.Machine \u0393 \u039b\nx\u271d : Cfg\u2080\nq : \u039b\nT : Tape \u0393\nq' : \u039b\ns : TM0.Stmt \u0393\ne : M q T.head = some (q', s)\n\u22a2 Reaches\u2081 (TM1.step (tr M)) { l := some (\u039b'.normal q), var := (), Tape := T }\n    {\n      l :=\n        match\n          match\n            M q'\n              (match s with\n                | TM0.Stmt.move d => Tape.move d T\n                | TM0.Stmt.write a => Tape.write a T).head with\n          | some val => true\n          | none => false with\n        | true => some (\u039b'.normal q')\n        | false => none,\n      var := (),\n      Tape :=\n        match s with\n        | TM0.Stmt.move d => Tape.move d T\n        | TM0.Stmt.write a => Tape.write a T }"}, {"tactic": "revert e", "annotated_tactic": ["revert e", []], "state_before": "case some.mk\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d : Inhabited \u039b\nM : TM0.Machine \u0393 \u039b\nx\u271d : Cfg\u2080\nq : \u039b\nT : Tape \u0393\nq' : \u039b\ns : TM0.Stmt \u0393\ne : M q T.head = some (q', s)\n\u22a2 Reaches\u2081 (TM1.step (tr M)) { l := some (\u039b'.normal q), var := (), Tape := T }\n    {\n      l :=\n        match\n          match\n            M q'\n              (match s with\n                | TM0.Stmt.move d => Tape.move d T\n                | TM0.Stmt.write a => Tape.write a T).head with\n          | some val => true\n          | none => false with\n        | true => some (\u039b'.normal q')\n        | false => none,\n      var := (),\n      Tape :=\n        match s with\n        | TM0.Stmt.move d => Tape.move d T\n        | TM0.Stmt.write a => Tape.write a T }", "state_after": "case some.mk\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d : Inhabited \u039b\nM : TM0.Machine \u0393 \u039b\nx\u271d : Cfg\u2080\nq : \u039b\nT : Tape \u0393\nq' : \u039b\ns : TM0.Stmt \u0393\n\u22a2 M q T.head = some (q', s) \u2192\n    Reaches\u2081 (TM1.step (tr M)) { l := some (\u039b'.normal q), var := (), Tape := T }\n      {\n        l :=\n          match\n            match\n              M q'\n                (match s with\n                  | TM0.Stmt.move d => Tape.move d T\n                  | TM0.Stmt.write a => Tape.write a T).head with\n            | some val => true\n            | none => false with\n          | true => some (\u039b'.normal q')\n          | false => none,\n        var := (),\n        Tape :=\n          match s with\n          | TM0.Stmt.move d => Tape.move d T\n          | TM0.Stmt.write a => Tape.write a T }"}, {"tactic": "have : TM1.step (tr M) \u27e8some (\u039b'.act s q'), (), T\u27e9 = some \u27e8some (\u039b'.normal q'), (), match s with\n    | TM0.Stmt.move d => T.move d\n    | TM0.Stmt.write a => T.write a\u27e9 := by\n  cases' s with d a <;> rfl", "annotated_tactic": ["have : <a>TM1.step</a> (<a>tr</a> M) \u27e8<a>some</a> (<a>\u039b'.act</a> s q'), (), T\u27e9 = <a>some</a> \u27e8<a>some</a> (<a>\u039b'.normal</a> q'), (), match s with\n        | <a>TM0.Stmt.move</a> d => T.move d\n        | <a>TM0.Stmt.write</a> a => T.write a\u27e9 := by\n      cases' s with d a <;> rfl", [{"full_name": "Turing.TM1.step", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1291, 5], "def_end_pos": [1291, 9]}, {"full_name": "Turing.TM0to1.tr", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [2009, 5], "def_end_pos": [2009, 7]}, {"full_name": "Option.some", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2143, 5], "def_end_pos": [2143, 9]}, {"full_name": "Turing.TM0to1.\u039b'.act", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1990, 5], "def_end_pos": [1990, 8]}, {"full_name": "Option.some", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2143, 5], "def_end_pos": [2143, 9]}, {"full_name": "Option.some", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2143, 5], "def_end_pos": [2143, 9]}, {"full_name": "Turing.TM0to1.\u039b'.normal", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1989, 5], "def_end_pos": [1989, 11]}, {"full_name": "Turing.TM0.Stmt.move", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1027, 5], "def_end_pos": [1027, 9]}, {"full_name": "Turing.TM0.Stmt.write", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1028, 5], "def_end_pos": [1028, 10]}]], "state_before": "case some.mk\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d : Inhabited \u039b\nM : TM0.Machine \u0393 \u039b\nx\u271d : Cfg\u2080\nq : \u039b\nT : Tape \u0393\nq' : \u039b\ns : TM0.Stmt \u0393\n\u22a2 M q T.head = some (q', s) \u2192\n    Reaches\u2081 (TM1.step (tr M)) { l := some (\u039b'.normal q), var := (), Tape := T }\n      {\n        l :=\n          match\n            match\n              M q'\n                (match s with\n                  | TM0.Stmt.move d => Tape.move d T\n                  | TM0.Stmt.write a => Tape.write a T).head with\n            | some val => true\n            | none => false with\n          | true => some (\u039b'.normal q')\n          | false => none,\n        var := (),\n        Tape :=\n          match s with\n          | TM0.Stmt.move d => Tape.move d T\n          | TM0.Stmt.write a => Tape.write a T }", "state_after": "case some.mk\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d : Inhabited \u039b\nM : TM0.Machine \u0393 \u039b\nx\u271d : Cfg\u2080\nq : \u039b\nT : Tape \u0393\nq' : \u039b\ns : TM0.Stmt \u0393\nthis :\n  TM1.step (tr M) { l := some (\u039b'.act s q'), var := (), Tape := T } =\n    some\n      { l := some (\u039b'.normal q'), var := (),\n        Tape :=\n          match s with\n          | TM0.Stmt.move d => Tape.move d T\n          | TM0.Stmt.write a => Tape.write a T }\n\u22a2 M q T.head = some (q', s) \u2192\n    Reaches\u2081 (TM1.step (tr M)) { l := some (\u039b'.normal q), var := (), Tape := T }\n      {\n        l :=\n          match\n            match\n              M q'\n                (match s with\n                  | TM0.Stmt.move d => Tape.move d T\n                  | TM0.Stmt.write a => Tape.write a T).head with\n            | some val => true\n            | none => false with\n          | true => some (\u039b'.normal q')\n          | false => none,\n        var := (),\n        Tape :=\n          match s with\n          | TM0.Stmt.move d => Tape.move d T\n          | TM0.Stmt.write a => Tape.write a T }"}, {"tactic": "intro e", "annotated_tactic": ["intro e", []], "state_before": "case some.mk\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d : Inhabited \u039b\nM : TM0.Machine \u0393 \u039b\nx\u271d : Cfg\u2080\nq : \u039b\nT : Tape \u0393\nq' : \u039b\ns : TM0.Stmt \u0393\nthis :\n  TM1.step (tr M) { l := some (\u039b'.act s q'), var := (), Tape := T } =\n    some\n      { l := some (\u039b'.normal q'), var := (),\n        Tape :=\n          match s with\n          | TM0.Stmt.move d => Tape.move d T\n          | TM0.Stmt.write a => Tape.write a T }\n\u22a2 M q T.head = some (q', s) \u2192\n    Reaches\u2081 (TM1.step (tr M)) { l := some (\u039b'.normal q), var := (), Tape := T }\n      {\n        l :=\n          match\n            match\n              M q'\n                (match s with\n                  | TM0.Stmt.move d => Tape.move d T\n                  | TM0.Stmt.write a => Tape.write a T).head with\n            | some val => true\n            | none => false with\n          | true => some (\u039b'.normal q')\n          | false => none,\n        var := (),\n        Tape :=\n          match s with\n          | TM0.Stmt.move d => Tape.move d T\n          | TM0.Stmt.write a => Tape.write a T }", "state_after": "case some.mk\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d : Inhabited \u039b\nM : TM0.Machine \u0393 \u039b\nx\u271d : Cfg\u2080\nq : \u039b\nT : Tape \u0393\nq' : \u039b\ns : TM0.Stmt \u0393\nthis :\n  TM1.step (tr M) { l := some (\u039b'.act s q'), var := (), Tape := T } =\n    some\n      { l := some (\u039b'.normal q'), var := (),\n        Tape :=\n          match s with\n          | TM0.Stmt.move d => Tape.move d T\n          | TM0.Stmt.write a => Tape.write a T }\ne : M q T.head = some (q', s)\n\u22a2 Reaches\u2081 (TM1.step (tr M)) { l := some (\u039b'.normal q), var := (), Tape := T }\n    {\n      l :=\n        match\n          match\n            M q'\n              (match s with\n                | TM0.Stmt.move d => Tape.move d T\n                | TM0.Stmt.write a => Tape.write a T).head with\n          | some val => true\n          | none => false with\n        | true => some (\u039b'.normal q')\n        | false => none,\n      var := (),\n      Tape :=\n        match s with\n        | TM0.Stmt.move d => Tape.move d T\n        | TM0.Stmt.write a => Tape.write a T }"}, {"tactic": "refine' TransGen.head _ (TransGen.head' this _)", "annotated_tactic": ["refine' <a>TransGen.head</a> _ (<a>TransGen.head'</a> this _)", [{"full_name": "Relation.TransGen.head", "def_path": "Mathlib/Logic/Relation.lean", "def_pos": [385, 9], "def_end_pos": [385, 13]}, {"full_name": "Relation.TransGen.head'", "def_path": "Mathlib/Logic/Relation.lean", "def_pos": [375, 9], "def_end_pos": [375, 14]}]], "state_before": "case some.mk\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d : Inhabited \u039b\nM : TM0.Machine \u0393 \u039b\nx\u271d : Cfg\u2080\nq : \u039b\nT : Tape \u0393\nq' : \u039b\ns : TM0.Stmt \u0393\nthis :\n  TM1.step (tr M) { l := some (\u039b'.act s q'), var := (), Tape := T } =\n    some\n      { l := some (\u039b'.normal q'), var := (),\n        Tape :=\n          match s with\n          | TM0.Stmt.move d => Tape.move d T\n          | TM0.Stmt.write a => Tape.write a T }\ne : M q T.head = some (q', s)\n\u22a2 Reaches\u2081 (TM1.step (tr M)) { l := some (\u039b'.normal q), var := (), Tape := T }\n    {\n      l :=\n        match\n          match\n            M q'\n              (match s with\n                | TM0.Stmt.move d => Tape.move d T\n                | TM0.Stmt.write a => Tape.write a T).head with\n          | some val => true\n          | none => false with\n        | true => some (\u039b'.normal q')\n        | false => none,\n      var := (),\n      Tape :=\n        match s with\n        | TM0.Stmt.move d => Tape.move d T\n        | TM0.Stmt.write a => Tape.write a T }", "state_after": "case some.mk.refine'_1\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d : Inhabited \u039b\nM : TM0.Machine \u0393 \u039b\nx\u271d : Cfg\u2080\nq : \u039b\nT : Tape \u0393\nq' : \u039b\ns : TM0.Stmt \u0393\nthis :\n  TM1.step (tr M) { l := some (\u039b'.act s q'), var := (), Tape := T } =\n    some\n      { l := some (\u039b'.normal q'), var := (),\n        Tape :=\n          match s with\n          | TM0.Stmt.move d => Tape.move d T\n          | TM0.Stmt.write a => Tape.write a T }\ne : M q T.head = some (q', s)\n\u22a2 { l := some (\u039b'.act s q'), var := (), Tape := T } \u2208 TM1.step (tr M) { l := some (\u039b'.normal q), var := (), Tape := T }\n\ncase some.mk.refine'_2\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d : Inhabited \u039b\nM : TM0.Machine \u0393 \u039b\nx\u271d : Cfg\u2080\nq : \u039b\nT : Tape \u0393\nq' : \u039b\ns : TM0.Stmt \u0393\nthis :\n  TM1.step (tr M) { l := some (\u039b'.act s q'), var := (), Tape := T } =\n    some\n      { l := some (\u039b'.normal q'), var := (),\n        Tape :=\n          match s with\n          | TM0.Stmt.move d => Tape.move d T\n          | TM0.Stmt.write a => Tape.write a T }\ne : M q T.head = some (q', s)\n\u22a2 ReflTransGen (fun a b => b \u2208 TM1.step (tr M) a)\n    { l := some (\u039b'.normal q'), var := (),\n      Tape :=\n        match s with\n        | TM0.Stmt.move d => Tape.move d T\n        | TM0.Stmt.write a => Tape.write a T }\n    {\n      l :=\n        match\n          match\n            M q'\n              (match s with\n                | TM0.Stmt.move d => Tape.move d T\n                | TM0.Stmt.write a => Tape.write a T).head with\n          | some val => true\n          | none => false with\n        | true => some (\u039b'.normal q')\n        | false => none,\n      var := (),\n      Tape :=\n        match s with\n        | TM0.Stmt.move d => Tape.move d T\n        | TM0.Stmt.write a => Tape.write a T }"}, {"tactic": "cases e' : M q' _", "annotated_tactic": ["cases e' : M q' _", []], "state_before": "case some.mk.refine'_2\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d : Inhabited \u039b\nM : TM0.Machine \u0393 \u039b\nx\u271d : Cfg\u2080\nq : \u039b\nT : Tape \u0393\nq' : \u039b\ns : TM0.Stmt \u0393\nthis :\n  TM1.step (tr M) { l := some (\u039b'.act s q'), var := (), Tape := T } =\n    some\n      { l := some (\u039b'.normal q'), var := (),\n        Tape :=\n          match s with\n          | TM0.Stmt.move d => Tape.move d T\n          | TM0.Stmt.write a => Tape.write a T }\ne : M q T.head = some (q', s)\n\u22a2 ReflTransGen (fun a b => b \u2208 TM1.step (tr M) a)\n    { l := some (\u039b'.normal q'), var := (),\n      Tape :=\n        match s with\n        | TM0.Stmt.move d => Tape.move d T\n        | TM0.Stmt.write a => Tape.write a T }\n    {\n      l :=\n        match\n          match\n            M q'\n              (match s with\n                | TM0.Stmt.move d => Tape.move d T\n                | TM0.Stmt.write a => Tape.write a T).head with\n          | some val => true\n          | none => false with\n        | true => some (\u039b'.normal q')\n        | false => none,\n      var := (),\n      Tape :=\n        match s with\n        | TM0.Stmt.move d => Tape.move d T\n        | TM0.Stmt.write a => Tape.write a T }", "state_after": "case some.mk.refine'_2.none\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d : Inhabited \u039b\nM : TM0.Machine \u0393 \u039b\nx\u271d : Cfg\u2080\nq : \u039b\nT : Tape \u0393\nq' : \u039b\ns : TM0.Stmt \u0393\nthis :\n  TM1.step (tr M) { l := some (\u039b'.act s q'), var := (), Tape := T } =\n    some\n      { l := some (\u039b'.normal q'), var := (),\n        Tape :=\n          match s with\n          | TM0.Stmt.move d => Tape.move d T\n          | TM0.Stmt.write a => Tape.write a T }\ne : M q T.head = some (q', s)\ne' :\n  M q'\n      (match s with\n        | TM0.Stmt.move d => Tape.move d T\n        | TM0.Stmt.write a => Tape.write a T).head =\n    none\n\u22a2 ReflTransGen (fun a b => b \u2208 TM1.step (tr M) a)\n    { l := some (\u039b'.normal q'), var := (),\n      Tape :=\n        match s with\n        | TM0.Stmt.move d => Tape.move d T\n        | TM0.Stmt.write a => Tape.write a T }\n    {\n      l :=\n        match\n          match none with\n          | some val => true\n          | none => false with\n        | true => some (\u039b'.normal q')\n        | false => none,\n      var := (),\n      Tape :=\n        match s with\n        | TM0.Stmt.move d => Tape.move d T\n        | TM0.Stmt.write a => Tape.write a T }\n\ncase some.mk.refine'_2.some\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d : Inhabited \u039b\nM : TM0.Machine \u0393 \u039b\nx\u271d : Cfg\u2080\nq : \u039b\nT : Tape \u0393\nq' : \u039b\ns : TM0.Stmt \u0393\nthis :\n  TM1.step (tr M) { l := some (\u039b'.act s q'), var := (), Tape := T } =\n    some\n      { l := some (\u039b'.normal q'), var := (),\n        Tape :=\n          match s with\n          | TM0.Stmt.move d => Tape.move d T\n          | TM0.Stmt.write a => Tape.write a T }\ne : M q T.head = some (q', s)\nval\u271d : \u039b \u00d7 TM0.Stmt \u0393\ne' :\n  M q'\n      (match s with\n        | TM0.Stmt.move d => Tape.move d T\n        | TM0.Stmt.write a => Tape.write a T).head =\n    some val\u271d\n\u22a2 ReflTransGen (fun a b => b \u2208 TM1.step (tr M) a)\n    { l := some (\u039b'.normal q'), var := (),\n      Tape :=\n        match s with\n        | TM0.Stmt.move d => Tape.move d T\n        | TM0.Stmt.write a => Tape.write a T }\n    {\n      l :=\n        match\n          match some val\u271d with\n          | some val => true\n          | none => false with\n        | true => some (\u039b'.normal q')\n        | false => none,\n      var := (),\n      Tape :=\n        match s with\n        | TM0.Stmt.move d => Tape.move d T\n        | TM0.Stmt.write a => Tape.write a T }"}, {"tactic": "simp only [TM0.step, trCfg, e]", "annotated_tactic": ["simp only [<a>TM0.step</a>, <a>trCfg</a>, e]", [{"full_name": "Turing.TM0.step", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1076, 5], "def_end_pos": [1076, 9]}, {"full_name": "Turing.TM0to1.trCfg", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [2020, 5], "def_end_pos": [2020, 10]}]], "state_before": "case none\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d : Inhabited \u039b\nM : TM0.Machine \u0393 \u039b\nx\u271d : Cfg\u2080\nq : \u039b\nT : Tape \u0393\ne : M q T.head = none\n\u22a2 FRespects (TM1.step (tr M)) (fun a => trCfg M a) (trCfg M { q := q, Tape := T }) (TM0.step M { q := q, Tape := T })", "state_after": "case none\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d : Inhabited \u039b\nM : TM0.Machine \u0393 \u039b\nx\u271d : Cfg\u2080\nq : \u039b\nT : Tape \u0393\ne : M q T.head = none\n\u22a2 FRespects (TM1.step (tr M))\n    (fun a =>\n      { l := bif Option.isSome (M a.q a.Tape.head) then some (\u039b'.normal a.q) else none, var := (), Tape := a.Tape })\n    { l := bif Option.isSome none then some (\u039b'.normal q) else none, var := (), Tape := T }\n    (Option.map\n      (fun x =>\n        { q := x.1,\n          Tape :=\n            match x.2 with\n            | TM0.Stmt.move d => Tape.move d T\n            | TM0.Stmt.write a => Tape.write a T })\n      none)"}, {"tactic": "exact Eq.refl none", "annotated_tactic": ["exact <a>Eq.refl</a> <a>none</a>", [{"full_name": "Eq.refl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [266, 5], "def_end_pos": [266, 9]}, {"full_name": "Option.none", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2141, 5], "def_end_pos": [2141, 9]}]], "state_before": "case none\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d : Inhabited \u039b\nM : TM0.Machine \u0393 \u039b\nx\u271d : Cfg\u2080\nq : \u039b\nT : Tape \u0393\ne : M q T.head = none\n\u22a2 FRespects (TM1.step (tr M))\n    (fun a =>\n      { l := bif Option.isSome (M a.q a.Tape.head) then some (\u039b'.normal a.q) else none, var := (), Tape := a.Tape })\n    { l := bif Option.isSome none then some (\u039b'.normal q) else none, var := (), Tape := T }\n    (Option.map\n      (fun x =>\n        { q := x.1,\n          Tape :=\n            match x.2 with\n            | TM0.Stmt.move d => Tape.move d T\n            | TM0.Stmt.write a => Tape.write a T })\n      none)", "state_after": "no goals"}, {"tactic": "cases' s with d a <;> rfl", "annotated_tactic": ["cases' s with d a <;> rfl", []], "state_before": "\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d : Inhabited \u039b\nM : TM0.Machine \u0393 \u039b\nx\u271d : Cfg\u2080\nq : \u039b\nT : Tape \u0393\nq' : \u039b\ns : TM0.Stmt \u0393\n\u22a2 TM1.step (tr M) { l := some (\u039b'.act s q'), var := (), Tape := T } =\n    some\n      { l := some (\u039b'.normal q'), var := (),\n        Tape :=\n          match s with\n          | TM0.Stmt.move d => Tape.move d T\n          | TM0.Stmt.write a => Tape.write a T }", "state_after": "no goals"}, {"tactic": "simp only [TM1.step, TM1.stepAux]", "annotated_tactic": ["simp only [<a>TM1.step</a>, <a>TM1.stepAux</a>]", [{"full_name": "Turing.TM1.step", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1291, 5], "def_end_pos": [1291, 9]}, {"full_name": "Turing.TM1.stepAux", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1281, 5], "def_end_pos": [1281, 12]}]], "state_before": "case some.mk.refine'_1\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d : Inhabited \u039b\nM : TM0.Machine \u0393 \u039b\nx\u271d : Cfg\u2080\nq : \u039b\nT : Tape \u0393\nq' : \u039b\ns : TM0.Stmt \u0393\nthis :\n  TM1.step (tr M) { l := some (\u039b'.act s q'), var := (), Tape := T } =\n    some\n      { l := some (\u039b'.normal q'), var := (),\n        Tape :=\n          match s with\n          | TM0.Stmt.move d => Tape.move d T\n          | TM0.Stmt.write a => Tape.write a T }\ne : M q T.head = some (q', s)\n\u22a2 { l := some (\u039b'.act s q'), var := (), Tape := T } \u2208 TM1.step (tr M) { l := some (\u039b'.normal q), var := (), Tape := T }", "state_after": "case some.mk.refine'_1\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d : Inhabited \u039b\nM : TM0.Machine \u0393 \u039b\nx\u271d : Cfg\u2080\nq : \u039b\nT : Tape \u0393\nq' : \u039b\ns : TM0.Stmt \u0393\nthis :\n  TM1.step (tr M) { l := some (\u039b'.act s q'), var := (), Tape := T } =\n    some\n      { l := some (\u039b'.normal q'), var := (),\n        Tape :=\n          match s with\n          | TM0.Stmt.move d => Tape.move d T\n          | TM0.Stmt.write a => Tape.write a T }\ne : M q T.head = some (q', s)\n\u22a2 { l := some (\u039b'.act s q'), var := (), Tape := T } \u2208\n    some\n      (bif Option.isNone (M q T.head) then { l := none, var := (), Tape := T }\n      else\n        {\n          l :=\n            some\n              (match M q T.head with\n              | none => default\n              | some (q', s) => \u039b'.act s q'),\n          var := (), Tape := T })"}, {"tactic": "rw [e]", "annotated_tactic": ["rw [e]", []], "state_before": "case some.mk.refine'_1\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d : Inhabited \u039b\nM : TM0.Machine \u0393 \u039b\nx\u271d : Cfg\u2080\nq : \u039b\nT : Tape \u0393\nq' : \u039b\ns : TM0.Stmt \u0393\nthis :\n  TM1.step (tr M) { l := some (\u039b'.act s q'), var := (), Tape := T } =\n    some\n      { l := some (\u039b'.normal q'), var := (),\n        Tape :=\n          match s with\n          | TM0.Stmt.move d => Tape.move d T\n          | TM0.Stmt.write a => Tape.write a T }\ne : M q T.head = some (q', s)\n\u22a2 { l := some (\u039b'.act s q'), var := (), Tape := T } \u2208\n    some\n      (bif Option.isNone (M q T.head) then { l := none, var := (), Tape := T }\n      else\n        {\n          l :=\n            some\n              (match M q T.head with\n              | none => default\n              | some (q', s) => \u039b'.act s q'),\n          var := (), Tape := T })", "state_after": "case some.mk.refine'_1\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d : Inhabited \u039b\nM : TM0.Machine \u0393 \u039b\nx\u271d : Cfg\u2080\nq : \u039b\nT : Tape \u0393\nq' : \u039b\ns : TM0.Stmt \u0393\nthis :\n  TM1.step (tr M) { l := some (\u039b'.act s q'), var := (), Tape := T } =\n    some\n      { l := some (\u039b'.normal q'), var := (),\n        Tape :=\n          match s with\n          | TM0.Stmt.move d => Tape.move d T\n          | TM0.Stmt.write a => Tape.write a T }\ne : M q T.head = some (q', s)\n\u22a2 { l := some (\u039b'.act s q'), var := (), Tape := T } \u2208\n    some\n      (bif Option.isNone (some (q', s)) then { l := none, var := (), Tape := T }\n      else\n        {\n          l :=\n            some\n              (match some (q', s) with\n              | none => default\n              | some (q', s) => \u039b'.act s q'),\n          var := (), Tape := T })"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case some.mk.refine'_1\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d : Inhabited \u039b\nM : TM0.Machine \u0393 \u039b\nx\u271d : Cfg\u2080\nq : \u039b\nT : Tape \u0393\nq' : \u039b\ns : TM0.Stmt \u0393\nthis :\n  TM1.step (tr M) { l := some (\u039b'.act s q'), var := (), Tape := T } =\n    some\n      { l := some (\u039b'.normal q'), var := (),\n        Tape :=\n          match s with\n          | TM0.Stmt.move d => Tape.move d T\n          | TM0.Stmt.write a => Tape.write a T }\ne : M q T.head = some (q', s)\n\u22a2 { l := some (\u039b'.act s q'), var := (), Tape := T } \u2208\n    some\n      (bif Option.isNone (some (q', s)) then { l := none, var := (), Tape := T }\n      else\n        {\n          l :=\n            some\n              (match some (q', s) with\n              | none => default\n              | some (q', s) => \u039b'.act s q'),\n          var := (), Tape := T })", "state_after": "no goals"}, {"tactic": "apply ReflTransGen.single", "annotated_tactic": ["apply <a>ReflTransGen.single</a>", [{"full_name": "Relation.ReflTransGen.single", "def_path": "Mathlib/Logic/Relation.lean", "def_pos": [276, 9], "def_end_pos": [276, 15]}]], "state_before": "case some.mk.refine'_2.none\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d : Inhabited \u039b\nM : TM0.Machine \u0393 \u039b\nx\u271d : Cfg\u2080\nq : \u039b\nT : Tape \u0393\nq' : \u039b\ns : TM0.Stmt \u0393\nthis :\n  TM1.step (tr M) { l := some (\u039b'.act s q'), var := (), Tape := T } =\n    some\n      { l := some (\u039b'.normal q'), var := (),\n        Tape :=\n          match s with\n          | TM0.Stmt.move d => Tape.move d T\n          | TM0.Stmt.write a => Tape.write a T }\ne : M q T.head = some (q', s)\ne' :\n  M q'\n      (match s with\n        | TM0.Stmt.move d => Tape.move d T\n        | TM0.Stmt.write a => Tape.write a T).head =\n    none\n\u22a2 ReflTransGen (fun a b => b \u2208 TM1.step (tr M) a)\n    { l := some (\u039b'.normal q'), var := (),\n      Tape :=\n        match s with\n        | TM0.Stmt.move d => Tape.move d T\n        | TM0.Stmt.write a => Tape.write a T }\n    {\n      l :=\n        match\n          match none with\n          | some val => true\n          | none => false with\n        | true => some (\u039b'.normal q')\n        | false => none,\n      var := (),\n      Tape :=\n        match s with\n        | TM0.Stmt.move d => Tape.move d T\n        | TM0.Stmt.write a => Tape.write a T }", "state_after": "case some.mk.refine'_2.none.hab\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d : Inhabited \u039b\nM : TM0.Machine \u0393 \u039b\nx\u271d : Cfg\u2080\nq : \u039b\nT : Tape \u0393\nq' : \u039b\ns : TM0.Stmt \u0393\nthis :\n  TM1.step (tr M) { l := some (\u039b'.act s q'), var := (), Tape := T } =\n    some\n      { l := some (\u039b'.normal q'), var := (),\n        Tape :=\n          match s with\n          | TM0.Stmt.move d => Tape.move d T\n          | TM0.Stmt.write a => Tape.write a T }\ne : M q T.head = some (q', s)\ne' :\n  M q'\n      (match s with\n        | TM0.Stmt.move d => Tape.move d T\n        | TM0.Stmt.write a => Tape.write a T).head =\n    none\n\u22a2 {\n      l :=\n        match\n          match none with\n          | some val => true\n          | none => false with\n        | true => some (\u039b'.normal q')\n        | false => none,\n      var := (),\n      Tape :=\n        match s with\n        | TM0.Stmt.move d => Tape.move d T\n        | TM0.Stmt.write a => Tape.write a T } \u2208\n    TM1.step (tr M)\n      { l := some (\u039b'.normal q'), var := (),\n        Tape :=\n          match s with\n          | TM0.Stmt.move d => Tape.move d T\n          | TM0.Stmt.write a => Tape.write a T }"}, {"tactic": "simp only [TM1.step, TM1.stepAux]", "annotated_tactic": ["simp only [<a>TM1.step</a>, <a>TM1.stepAux</a>]", [{"full_name": "Turing.TM1.step", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1291, 5], "def_end_pos": [1291, 9]}, {"full_name": "Turing.TM1.stepAux", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1281, 5], "def_end_pos": [1281, 12]}]], "state_before": "case some.mk.refine'_2.none.hab\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d : Inhabited \u039b\nM : TM0.Machine \u0393 \u039b\nx\u271d : Cfg\u2080\nq : \u039b\nT : Tape \u0393\nq' : \u039b\ns : TM0.Stmt \u0393\nthis :\n  TM1.step (tr M) { l := some (\u039b'.act s q'), var := (), Tape := T } =\n    some\n      { l := some (\u039b'.normal q'), var := (),\n        Tape :=\n          match s with\n          | TM0.Stmt.move d => Tape.move d T\n          | TM0.Stmt.write a => Tape.write a T }\ne : M q T.head = some (q', s)\ne' :\n  M q'\n      (match s with\n        | TM0.Stmt.move d => Tape.move d T\n        | TM0.Stmt.write a => Tape.write a T).head =\n    none\n\u22a2 {\n      l :=\n        match\n          match none with\n          | some val => true\n          | none => false with\n        | true => some (\u039b'.normal q')\n        | false => none,\n      var := (),\n      Tape :=\n        match s with\n        | TM0.Stmt.move d => Tape.move d T\n        | TM0.Stmt.write a => Tape.write a T } \u2208\n    TM1.step (tr M)\n      { l := some (\u039b'.normal q'), var := (),\n        Tape :=\n          match s with\n          | TM0.Stmt.move d => Tape.move d T\n          | TM0.Stmt.write a => Tape.write a T }", "state_after": "case some.mk.refine'_2.none.hab\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d : Inhabited \u039b\nM : TM0.Machine \u0393 \u039b\nx\u271d : Cfg\u2080\nq : \u039b\nT : Tape \u0393\nq' : \u039b\ns : TM0.Stmt \u0393\nthis :\n  TM1.step (tr M) { l := some (\u039b'.act s q'), var := (), Tape := T } =\n    some\n      { l := some (\u039b'.normal q'), var := (),\n        Tape :=\n          match s with\n          | TM0.Stmt.move d => Tape.move d T\n          | TM0.Stmt.write a => Tape.write a T }\ne : M q T.head = some (q', s)\ne' :\n  M q'\n      (match s with\n        | TM0.Stmt.move d => Tape.move d T\n        | TM0.Stmt.write a => Tape.write a T).head =\n    none\n\u22a2 { l := none, var := (),\n      Tape :=\n        match s with\n        | TM0.Stmt.move d => Tape.move d T\n        | TM0.Stmt.write a => Tape.write a T } \u2208\n    some\n      (bif\n          Option.isNone\n            (M q'\n              (match s with\n                | TM0.Stmt.move d => Tape.move d T\n                | TM0.Stmt.write a => Tape.write a T).head) then\n        { l := none, var := (),\n          Tape :=\n            match s with\n            | TM0.Stmt.move d => Tape.move d T\n            | TM0.Stmt.write a => Tape.write a T }\n      else\n        {\n          l :=\n            some\n              (match\n                M q'\n                  (match s with\n                    | TM0.Stmt.move d => Tape.move d T\n                    | TM0.Stmt.write a => Tape.write a T).head with\n              | none => default\n              | some (q', s) => \u039b'.act s q'),\n          var := (),\n          Tape :=\n            match s with\n            | TM0.Stmt.move d => Tape.move d T\n            | TM0.Stmt.write a => Tape.write a T })"}, {"tactic": "rw [e']", "annotated_tactic": ["rw [e']", []], "state_before": "case some.mk.refine'_2.none.hab\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d : Inhabited \u039b\nM : TM0.Machine \u0393 \u039b\nx\u271d : Cfg\u2080\nq : \u039b\nT : Tape \u0393\nq' : \u039b\ns : TM0.Stmt \u0393\nthis :\n  TM1.step (tr M) { l := some (\u039b'.act s q'), var := (), Tape := T } =\n    some\n      { l := some (\u039b'.normal q'), var := (),\n        Tape :=\n          match s with\n          | TM0.Stmt.move d => Tape.move d T\n          | TM0.Stmt.write a => Tape.write a T }\ne : M q T.head = some (q', s)\ne' :\n  M q'\n      (match s with\n        | TM0.Stmt.move d => Tape.move d T\n        | TM0.Stmt.write a => Tape.write a T).head =\n    none\n\u22a2 { l := none, var := (),\n      Tape :=\n        match s with\n        | TM0.Stmt.move d => Tape.move d T\n        | TM0.Stmt.write a => Tape.write a T } \u2208\n    some\n      (bif\n          Option.isNone\n            (M q'\n              (match s with\n                | TM0.Stmt.move d => Tape.move d T\n                | TM0.Stmt.write a => Tape.write a T).head) then\n        { l := none, var := (),\n          Tape :=\n            match s with\n            | TM0.Stmt.move d => Tape.move d T\n            | TM0.Stmt.write a => Tape.write a T }\n      else\n        {\n          l :=\n            some\n              (match\n                M q'\n                  (match s with\n                    | TM0.Stmt.move d => Tape.move d T\n                    | TM0.Stmt.write a => Tape.write a T).head with\n              | none => default\n              | some (q', s) => \u039b'.act s q'),\n          var := (),\n          Tape :=\n            match s with\n            | TM0.Stmt.move d => Tape.move d T\n            | TM0.Stmt.write a => Tape.write a T })", "state_after": "case some.mk.refine'_2.none.hab\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d : Inhabited \u039b\nM : TM0.Machine \u0393 \u039b\nx\u271d : Cfg\u2080\nq : \u039b\nT : Tape \u0393\nq' : \u039b\ns : TM0.Stmt \u0393\nthis :\n  TM1.step (tr M) { l := some (\u039b'.act s q'), var := (), Tape := T } =\n    some\n      { l := some (\u039b'.normal q'), var := (),\n        Tape :=\n          match s with\n          | TM0.Stmt.move d => Tape.move d T\n          | TM0.Stmt.write a => Tape.write a T }\ne : M q T.head = some (q', s)\ne' :\n  M q'\n      (match s with\n        | TM0.Stmt.move d => Tape.move d T\n        | TM0.Stmt.write a => Tape.write a T).head =\n    none\n\u22a2 { l := none, var := (),\n      Tape :=\n        match s with\n        | TM0.Stmt.move d => Tape.move d T\n        | TM0.Stmt.write a => Tape.write a T } \u2208\n    some\n      (bif Option.isNone none then\n        { l := none, var := (),\n          Tape :=\n            match s with\n            | TM0.Stmt.move d => Tape.move d T\n            | TM0.Stmt.write a => Tape.write a T }\n      else\n        {\n          l :=\n            some\n              (match none with\n              | none => default\n              | some (q', s) => \u039b'.act s q'),\n          var := (),\n          Tape :=\n            match s with\n            | TM0.Stmt.move d => Tape.move d T\n            | TM0.Stmt.write a => Tape.write a T })"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case some.mk.refine'_2.none.hab\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d : Inhabited \u039b\nM : TM0.Machine \u0393 \u039b\nx\u271d : Cfg\u2080\nq : \u039b\nT : Tape \u0393\nq' : \u039b\ns : TM0.Stmt \u0393\nthis :\n  TM1.step (tr M) { l := some (\u039b'.act s q'), var := (), Tape := T } =\n    some\n      { l := some (\u039b'.normal q'), var := (),\n        Tape :=\n          match s with\n          | TM0.Stmt.move d => Tape.move d T\n          | TM0.Stmt.write a => Tape.write a T }\ne : M q T.head = some (q', s)\ne' :\n  M q'\n      (match s with\n        | TM0.Stmt.move d => Tape.move d T\n        | TM0.Stmt.write a => Tape.write a T).head =\n    none\n\u22a2 { l := none, var := (),\n      Tape :=\n        match s with\n        | TM0.Stmt.move d => Tape.move d T\n        | TM0.Stmt.write a => Tape.write a T } \u2208\n    some\n      (bif Option.isNone none then\n        { l := none, var := (),\n          Tape :=\n            match s with\n            | TM0.Stmt.move d => Tape.move d T\n            | TM0.Stmt.write a => Tape.write a T }\n      else\n        {\n          l :=\n            some\n              (match none with\n              | none => default\n              | some (q', s) => \u039b'.act s q'),\n          var := (),\n          Tape :=\n            match s with\n            | TM0.Stmt.move d => Tape.move d T\n            | TM0.Stmt.write a => Tape.write a T })", "state_after": "no goals"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case some.mk.refine'_2.some\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d : Inhabited \u039b\nM : TM0.Machine \u0393 \u039b\nx\u271d : Cfg\u2080\nq : \u039b\nT : Tape \u0393\nq' : \u039b\ns : TM0.Stmt \u0393\nthis :\n  TM1.step (tr M) { l := some (\u039b'.act s q'), var := (), Tape := T } =\n    some\n      { l := some (\u039b'.normal q'), var := (),\n        Tape :=\n          match s with\n          | TM0.Stmt.move d => Tape.move d T\n          | TM0.Stmt.write a => Tape.write a T }\ne : M q T.head = some (q', s)\nval\u271d : \u039b \u00d7 TM0.Stmt \u0393\ne' :\n  M q'\n      (match s with\n        | TM0.Stmt.move d => Tape.move d T\n        | TM0.Stmt.write a => Tape.write a T).head =\n    some val\u271d\n\u22a2 ReflTransGen (fun a b => b \u2208 TM1.step (tr M) a)\n    { l := some (\u039b'.normal q'), var := (),\n      Tape :=\n        match s with\n        | TM0.Stmt.move d => Tape.move d T\n        | TM0.Stmt.write a => Tape.write a T }\n    {\n      l :=\n        match\n          match some val\u271d with\n          | some val => true\n          | none => false with\n        | true => some (\u039b'.normal q')\n        | false => none,\n      var := (),\n      Tape :=\n        match s with\n        | TM0.Stmt.move d => Tape.move d T\n        | TM0.Stmt.write a => Tape.write a T }", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Regular.lean", "full_name": "IsOpen.exists_lt_isCompact", "start": [472, 1], "end": [474, 31], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/List/Basic.lean", "full_name": "List.scanl_eq_scanlTR", "start": [658, 10], "end": [659, 48], "traced_tactics": [{"tactic": "funext \u03b1 f n l", "annotated_tactic": ["funext \u03b1 f n l", []], "state_before": "\u22a2 @scanl = @scanlTR", "state_after": "case h.h.h.h\n\u03b1 : Type u_2\nf : Type u_1\nn : \u03b1 \u2192 f \u2192 \u03b1\nl : \u03b1\n\u22a2 scanl n l = scanlTR n l"}, {"tactic": "simp [scanlTR, scanlTR_go_eq]", "annotated_tactic": ["simp [<a>scanlTR</a>, <a>scanlTR_go_eq</a>]", [{"full_name": "List.scanlTR", "def_path": "lake-packages/std/Std/Data/List/Basic.lean", "def_pos": [648, 15], "def_end_pos": [648, 22]}, {"full_name": "List.scanlTR_go_eq", "def_path": "lake-packages/std/Std/Data/List/Basic.lean", "def_pos": [654, 9], "def_end_pos": [654, 22]}]], "state_before": "case h.h.h.h\n\u03b1 : Type u_2\nf : Type u_1\nn : \u03b1 \u2192 f \u2192 \u03b1\nl : \u03b1\n\u22a2 scanl n l = scanlTR n l", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/Basic.lean", "full_name": "MvPolynomial.coeff_X", "start": [697, 1], "end": [698, 38], "traced_tactics": [{"tactic": "classical rw [coeff_X', if_pos rfl]", "annotated_tactic": ["classical rw [<a>coeff_X'</a>, <a>if_pos</a> <a>rfl</a>]", [{"full_name": "MvPolynomial.coeff_X'", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [691, 9], "def_end_pos": [691, 17]}, {"full_name": "if_pos", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [790, 9], "def_end_pos": [790, 15]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "R : Type u\nS\u2081 : Type v\nS\u2082 : Type w\nS\u2083 : Type x\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : CommSemiring S\u2081\np q : MvPolynomial \u03c3 R\ni : \u03c3\n\u22a2 coeff (fun\u2080 | i => 1) (X i) = 1", "state_after": "no goals"}, {"tactic": "rw [coeff_X', if_pos rfl]", "annotated_tactic": ["rw [<a>coeff_X'</a>, <a>if_pos</a> <a>rfl</a>]", [{"full_name": "MvPolynomial.coeff_X'", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [691, 9], "def_end_pos": [691, 17]}, {"full_name": "if_pos", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [790, 9], "def_end_pos": [790, 15]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "R : Type u\nS\u2081 : Type v\nS\u2082 : Type w\nS\u2083 : Type x\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : CommSemiring S\u2081\np q : MvPolynomial \u03c3 R\ni : \u03c3\n\u22a2 coeff (fun\u2080 | i => 1) (X i) = 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "full_name": "MeasureTheory.SimpleFunc.measurableSet_preimage", "start": [195, 1], "end": [196, 78], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Finite.lean", "full_name": "Set.Finite.subset", "start": [771, 1], "end": [774, 17], "traced_tactics": [{"tactic": "cases hs", "annotated_tactic": ["cases hs", []], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\n\u03b3 : Type x\ns : Set \u03b1\nhs : Set.Finite s\nt : Set \u03b1\nht : t \u2286 s\n\u22a2 Set.Finite t", "state_after": "case intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\n\u03b3 : Type x\ns t : Set \u03b1\nht : t \u2286 s\na\u271d : Fintype \u2191s\n\u22a2 Set.Finite t"}, {"tactic": "haveI := Finite.Set.subset _ ht", "annotated_tactic": ["haveI := <a>Finite.Set.subset</a> _ ht", [{"full_name": "Finite.Set.subset", "def_path": "Mathlib/Data/Set/Finite.lean", "def_pos": [613, 19], "def_end_pos": [613, 25]}]], "state_before": "case intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\n\u03b3 : Type x\ns t : Set \u03b1\nht : t \u2286 s\na\u271d : Fintype \u2191s\n\u22a2 Set.Finite t", "state_after": "case intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\n\u03b3 : Type x\ns t : Set \u03b1\nht : t \u2286 s\na\u271d : Fintype \u2191s\nthis : Finite \u2191t\n\u22a2 Set.Finite t"}, {"tactic": "apply toFinite", "annotated_tactic": ["apply <a>toFinite</a>", [{"full_name": "Set.toFinite", "def_path": "Mathlib/Data/Set/Finite.lean", "def_pos": [82, 9], "def_end_pos": [82, 17]}]], "state_before": "case intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\n\u03b3 : Type x\ns t : Set \u03b1\nht : t \u2286 s\na\u271d : Fintype \u2191s\nthis : Finite \u2191t\n\u22a2 Set.Finite t", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Num/Lemmas.lean", "full_name": "ZNum.of_int_cast", "start": [1552, 1], "end": [1553, 32], "traced_tactics": [{"tactic": "rw [\u2190 cast_to_int, to_of_int]", "annotated_tactic": ["rw [\u2190 <a>cast_to_int</a>, <a>to_of_int</a>]", [{"full_name": "ZNum.cast_to_int", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [1097, 9], "def_end_pos": [1097, 20]}, {"full_name": "ZNum.to_of_int", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [1543, 9], "def_end_pos": [1543, 18]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : AddGroupWithOne \u03b1\nn : \u2124\n\u22a2 \u2191\u2191n = \u2191n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Haar/Basic.lean", "full_name": "MeasureTheory.Measure.haar.is_left_invariant_chaar", "start": [518, 1], "end": [528, 68], "traced_tactics": [{"tactic": "let eval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f (K.map _ <| continuous_mul_left g) - f K", "annotated_tactic": ["let eval : (<a>Compacts</a> G \u2192 \u211d) \u2192 \u211d := fun f => f (K.map _ <| <a>continuous_mul_left</a> g) - f K", [{"full_name": "TopologicalSpace.Compacts", "def_path": "Mathlib/Topology/Sets/Compacts.lean", "def_pos": [36, 11], "def_end_pos": [36, 19]}, {"full_name": "continuous_mul_left", "def_path": "Mathlib/Topology/Algebra/Monoid.lean", "def_pos": [94, 9], "def_end_pos": [94, 28]}]], "state_before": "G : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\ng : G\nK : Compacts G\n\u22a2 chaar K\u2080 (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) = chaar K\u2080 K", "state_after": "G : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\ng : G\nK : Compacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) - f K\n\u22a2 chaar K\u2080 (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) = chaar K\u2080 K"}, {"tactic": "have : Continuous eval := (continuous_apply (K.map _ _)).sub (continuous_apply K)", "annotated_tactic": ["have : <a>Continuous</a> eval := (<a>continuous_apply</a> (K.map _ _)).<a>sub</a> (<a>continuous_apply</a> K)", [{"full_name": "Continuous", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [1591, 11], "def_end_pos": [1591, 21]}, {"full_name": "continuous_apply", "def_path": "Mathlib/Topology/Constructions.lean", "def_pos": [1208, 9], "def_end_pos": [1208, 25]}, {"full_name": "Continuous.sub", "def_path": "Mathlib/Topology/Algebra/Group/Basic.lean", "def_pos": [1104, 36], "def_end_pos": [1104, 39]}, {"full_name": "continuous_apply", "def_path": "Mathlib/Topology/Constructions.lean", "def_pos": [1208, 9], "def_end_pos": [1208, 25]}]], "state_before": "G : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\ng : G\nK : Compacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) - f K\n\u22a2 chaar K\u2080 (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) = chaar K\u2080 K", "state_after": "G : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\ng : G\nK : Compacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) - f K\nthis : Continuous eval\n\u22a2 chaar K\u2080 (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) = chaar K\u2080 K"}, {"tactic": "rw [\u2190 sub_eq_zero]", "annotated_tactic": ["rw [\u2190 <a>sub_eq_zero</a>]", [{"full_name": "sub_eq_zero", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [801, 3], "def_end_pos": [801, 14]}]], "state_before": "G : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\ng : G\nK : Compacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) - f K\nthis : Continuous eval\n\u22a2 chaar K\u2080 (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) = chaar K\u2080 K", "state_after": "G : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\ng : G\nK : Compacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) - f K\nthis : Continuous eval\n\u22a2 chaar K\u2080 (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) - chaar K\u2080 K = 0"}, {"tactic": "show chaar K\u2080 \u2208 eval \u207b\u00b9' {(0 : \u211d)}", "annotated_tactic": ["show <a>chaar</a> K\u2080 \u2208 eval \u207b\u00b9' {(0 : \u211d)}", [{"full_name": "MeasureTheory.Measure.haar.chaar", "def_path": "Mathlib/MeasureTheory/Measure/Haar/Basic.lean", "def_pos": [404, 19], "def_end_pos": [404, 24]}]], "state_before": "G : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\ng : G\nK : Compacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) - f K\nthis : Continuous eval\n\u22a2 chaar K\u2080 (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) - chaar K\u2080 K = 0", "state_after": "G : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\ng : G\nK : Compacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) - f K\nthis : Continuous eval\n\u22a2 chaar K\u2080 \u2208 eval \u207b\u00b9' {0}"}, {"tactic": "apply mem_of_subset_of_mem _ (chaar_mem_clPrehaar K\u2080 \u22a4)", "annotated_tactic": ["apply <a>mem_of_subset_of_mem</a> _ (<a>chaar_mem_clPrehaar</a> K\u2080 \u22a4)", [{"full_name": "Set.mem_of_subset_of_mem", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [383, 9], "def_end_pos": [383, 29]}, {"full_name": "MeasureTheory.Measure.haar.chaar_mem_clPrehaar", "def_path": "Mathlib/MeasureTheory/Measure/Haar/Basic.lean", "def_pos": [416, 9], "def_end_pos": [416, 28]}]], "state_before": "G : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\ng : G\nK : Compacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) - f K\nthis : Continuous eval\n\u22a2 chaar K\u2080 \u2208 eval \u207b\u00b9' {0}", "state_after": "G : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\ng : G\nK : Compacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) - f K\nthis : Continuous eval\n\u22a2 clPrehaar \u2191K\u2080 \u22a4 \u2286 eval \u207b\u00b9' {0}"}, {"tactic": "unfold clPrehaar", "annotated_tactic": ["unfold <a>clPrehaar</a>", [{"full_name": "MeasureTheory.Measure.haar.clPrehaar", "def_path": "Mathlib/MeasureTheory/Measure/Haar/Basic.lean", "def_pos": [154, 5], "def_end_pos": [154, 14]}]], "state_before": "G : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\ng : G\nK : Compacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) - f K\nthis : Continuous eval\n\u22a2 clPrehaar \u2191K\u2080 \u22a4 \u2286 eval \u207b\u00b9' {0}", "state_after": "G : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\ng : G\nK : Compacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) - f K\nthis : Continuous eval\n\u22a2 closure (prehaar \u2191K\u2080 '' {U | U \u2286 \u2191\u22a4.toOpens \u2227 IsOpen U \u2227 1 \u2208 U}) \u2286 eval \u207b\u00b9' {0}"}, {"tactic": "rw [IsClosed.closure_subset_iff]", "annotated_tactic": ["rw [<a>IsClosed.closure_subset_iff</a>]", [{"full_name": "IsClosed.closure_subset_iff", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [465, 9], "def_end_pos": [465, 36]}]], "state_before": "G : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\ng : G\nK : Compacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) - f K\nthis : Continuous eval\n\u22a2 closure (prehaar \u2191K\u2080 '' {U | U \u2286 \u2191\u22a4.toOpens \u2227 IsOpen U \u2227 1 \u2208 U}) \u2286 eval \u207b\u00b9' {0}", "state_after": "G : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\ng : G\nK : Compacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) - f K\nthis : Continuous eval\n\u22a2 prehaar \u2191K\u2080 '' {U | U \u2286 \u2191\u22a4.toOpens \u2227 IsOpen U \u2227 1 \u2208 U} \u2286 eval \u207b\u00b9' {0}\n\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\ng : G\nK : Compacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) - f K\nthis : Continuous eval\n\u22a2 IsClosed (eval \u207b\u00b9' {0})"}, {"tactic": "rintro _ \u27e8U, \u27e8_, h2U, h3U\u27e9, rfl\u27e9", "annotated_tactic": ["rintro _ \u27e8U, \u27e8_, h2U, h3U\u27e9, rfl\u27e9", []], "state_before": "G : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\ng : G\nK : Compacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) - f K\nthis : Continuous eval\n\u22a2 prehaar \u2191K\u2080 '' {U | U \u2286 \u2191\u22a4.toOpens \u2227 IsOpen U \u2227 1 \u2208 U} \u2286 eval \u207b\u00b9' {0}", "state_after": "case intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\ng : G\nK : Compacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) - f K\nthis : Continuous eval\nU : Set G\nleft\u271d : U \u2286 \u2191\u22a4.toOpens\nh2U : IsOpen U\nh3U : 1 \u2208 U\n\u22a2 prehaar (\u2191K\u2080) U \u2208 eval \u207b\u00b9' {0}"}, {"tactic": "simp only [mem_singleton_iff, mem_preimage, sub_eq_zero]", "annotated_tactic": ["simp only [<a>mem_singleton_iff</a>, <a>mem_preimage</a>, <a>sub_eq_zero</a>]", [{"full_name": "Set.mem_singleton_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1273, 9], "def_end_pos": [1273, 26]}, {"full_name": "Set.mem_preimage", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [64, 9], "def_end_pos": [64, 21]}, {"full_name": "sub_eq_zero", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [801, 3], "def_end_pos": [801, 14]}]], "state_before": "case intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\ng : G\nK : Compacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) - f K\nthis : Continuous eval\nU : Set G\nleft\u271d : U \u2286 \u2191\u22a4.toOpens\nh2U : IsOpen U\nh3U : 1 \u2208 U\n\u22a2 prehaar (\u2191K\u2080) U \u2208 eval \u207b\u00b9' {0}", "state_after": "case intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\ng : G\nK : Compacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) - f K\nthis : Continuous eval\nU : Set G\nleft\u271d : U \u2286 \u2191\u22a4.toOpens\nh2U : IsOpen U\nh3U : 1 \u2208 U\n\u22a2 prehaar (\u2191K\u2080) U (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) = prehaar (\u2191K\u2080) U K"}, {"tactic": "apply is_left_invariant_prehaar", "annotated_tactic": ["apply <a>is_left_invariant_prehaar</a>", [{"full_name": "MeasureTheory.Measure.haar.is_left_invariant_prehaar", "def_path": "Mathlib/MeasureTheory/Measure/Haar/Basic.lean", "def_pos": [352, 9], "def_end_pos": [352, 34]}]], "state_before": "case intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\ng : G\nK : Compacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) - f K\nthis : Continuous eval\nU : Set G\nleft\u271d : U \u2286 \u2191\u22a4.toOpens\nh2U : IsOpen U\nh3U : 1 \u2208 U\n\u22a2 prehaar (\u2191K\u2080) U (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) = prehaar (\u2191K\u2080) U K", "state_after": "case intro.intro.intro.intro.hU\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\ng : G\nK : Compacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) - f K\nthis : Continuous eval\nU : Set G\nleft\u271d : U \u2286 \u2191\u22a4.toOpens\nh2U : IsOpen U\nh3U : 1 \u2208 U\n\u22a2 Set.Nonempty (interior U)"}, {"tactic": "rw [h2U.interior_eq]", "annotated_tactic": ["rw [h2U.interior_eq]", []], "state_before": "case intro.intro.intro.intro.hU\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\ng : G\nK : Compacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) - f K\nthis : Continuous eval\nU : Set G\nleft\u271d : U \u2286 \u2191\u22a4.toOpens\nh2U : IsOpen U\nh3U : 1 \u2208 U\n\u22a2 Set.Nonempty (interior U)", "state_after": "case intro.intro.intro.intro.hU\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\ng : G\nK : Compacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) - f K\nthis : Continuous eval\nU : Set G\nleft\u271d : U \u2286 \u2191\u22a4.toOpens\nh2U : IsOpen U\nh3U : 1 \u2208 U\n\u22a2 Set.Nonempty U"}, {"tactic": "exact \u27e81, h3U\u27e9", "annotated_tactic": ["exact \u27e81, h3U\u27e9", []], "state_before": "case intro.intro.intro.intro.hU\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\ng : G\nK : Compacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) - f K\nthis : Continuous eval\nU : Set G\nleft\u271d : U \u2286 \u2191\u22a4.toOpens\nh2U : IsOpen U\nh3U : 1 \u2208 U\n\u22a2 Set.Nonempty U", "state_after": "no goals"}, {"tactic": "apply continuous_iff_isClosed.mp this", "annotated_tactic": ["apply continuous_iff_isClosed.mp this", []], "state_before": "G : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\ng : G\nK : Compacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) - f K\nthis : Continuous eval\n\u22a2 IsClosed (eval \u207b\u00b9' {0})", "state_after": "case a\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\ng : G\nK : Compacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) - f K\nthis : Continuous eval\n\u22a2 IsClosed {0}"}, {"tactic": "exact isClosed_singleton", "annotated_tactic": ["exact <a>isClosed_singleton</a>", [{"full_name": "isClosed_singleton", "def_path": "Mathlib/Topology/Separation.lean", "def_pos": [384, 9], "def_end_pos": [384, 27]}]], "state_before": "case a\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\ng : G\nK : Compacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) - f K\nthis : Continuous eval\n\u22a2 IsClosed {0}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Process/Stopping.lean", "full_name": "MeasureTheory.IsStoppingTime.measurableSet_inter_le_iff", "start": [655, 1], "end": [666, 14], "traced_tactics": [{"tactic": "constructor <;> intro h", "annotated_tactic": ["constructor <;> intro h", []], "state_before": "\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : LinearOrder \u03b9\nf : Filtration \u03b9 m\n\u03c4 \u03c0 : \u03a9 \u2192 \u03b9\ninst\u271d\u2074 : TopologicalSpace \u03b9\ninst\u271d\u00b3 : SecondCountableTopology \u03b9\ninst\u271d\u00b2 : OrderTopology \u03b9\ninst\u271d\u00b9 : MeasurableSpace \u03b9\ninst\u271d : BorelSpace \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\nh\u03c0 : IsStoppingTime f \u03c0\ns : Set \u03a9\n\u22a2 MeasurableSet (s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c0 \u03c9}) \u2194 MeasurableSet (s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c0 \u03c9})", "state_after": "case mp\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : LinearOrder \u03b9\nf : Filtration \u03b9 m\n\u03c4 \u03c0 : \u03a9 \u2192 \u03b9\ninst\u271d\u2074 : TopologicalSpace \u03b9\ninst\u271d\u00b3 : SecondCountableTopology \u03b9\ninst\u271d\u00b2 : OrderTopology \u03b9\ninst\u271d\u00b9 : MeasurableSpace \u03b9\ninst\u271d : BorelSpace \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\nh\u03c0 : IsStoppingTime f \u03c0\ns : Set \u03a9\nh : MeasurableSet (s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c0 \u03c9})\n\u22a2 MeasurableSet (s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c0 \u03c9})\n\ncase mpr\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : LinearOrder \u03b9\nf : Filtration \u03b9 m\n\u03c4 \u03c0 : \u03a9 \u2192 \u03b9\ninst\u271d\u2074 : TopologicalSpace \u03b9\ninst\u271d\u00b3 : SecondCountableTopology \u03b9\ninst\u271d\u00b2 : OrderTopology \u03b9\ninst\u271d\u00b9 : MeasurableSpace \u03b9\ninst\u271d : BorelSpace \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\nh\u03c0 : IsStoppingTime f \u03c0\ns : Set \u03a9\nh : MeasurableSet (s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c0 \u03c9})\n\u22a2 MeasurableSet (s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c0 \u03c9})"}, {"tactic": "have : s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c0 \u03c9} = s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c0 \u03c9} \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c0 \u03c9} := by\n  rw [Set.inter_assoc, Set.inter_self]", "annotated_tactic": ["have : s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c0 \u03c9} = s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c0 \u03c9} \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c0 \u03c9} := by\n      rw [<a>Set.inter_assoc</a>, <a>Set.inter_self</a>]", [{"full_name": "Set.inter_assoc", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [944, 9], "def_end_pos": [944, 20]}, {"full_name": "Set.inter_self", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [926, 9], "def_end_pos": [926, 19]}]], "state_before": "case mp\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : LinearOrder \u03b9\nf : Filtration \u03b9 m\n\u03c4 \u03c0 : \u03a9 \u2192 \u03b9\ninst\u271d\u2074 : TopologicalSpace \u03b9\ninst\u271d\u00b3 : SecondCountableTopology \u03b9\ninst\u271d\u00b2 : OrderTopology \u03b9\ninst\u271d\u00b9 : MeasurableSpace \u03b9\ninst\u271d : BorelSpace \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\nh\u03c0 : IsStoppingTime f \u03c0\ns : Set \u03a9\nh : MeasurableSet (s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c0 \u03c9})\n\u22a2 MeasurableSet (s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c0 \u03c9})", "state_after": "case mp\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : LinearOrder \u03b9\nf : Filtration \u03b9 m\n\u03c4 \u03c0 : \u03a9 \u2192 \u03b9\ninst\u271d\u2074 : TopologicalSpace \u03b9\ninst\u271d\u00b3 : SecondCountableTopology \u03b9\ninst\u271d\u00b2 : OrderTopology \u03b9\ninst\u271d\u00b9 : MeasurableSpace \u03b9\ninst\u271d : BorelSpace \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\nh\u03c0 : IsStoppingTime f \u03c0\ns : Set \u03a9\nh : MeasurableSet (s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c0 \u03c9})\nthis : s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c0 \u03c9} = s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c0 \u03c9} \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c0 \u03c9}\n\u22a2 MeasurableSet (s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c0 \u03c9})"}, {"tactic": "rw [this]", "annotated_tactic": ["rw [this]", []], "state_before": "case mp\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : LinearOrder \u03b9\nf : Filtration \u03b9 m\n\u03c4 \u03c0 : \u03a9 \u2192 \u03b9\ninst\u271d\u2074 : TopologicalSpace \u03b9\ninst\u271d\u00b3 : SecondCountableTopology \u03b9\ninst\u271d\u00b2 : OrderTopology \u03b9\ninst\u271d\u00b9 : MeasurableSpace \u03b9\ninst\u271d : BorelSpace \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\nh\u03c0 : IsStoppingTime f \u03c0\ns : Set \u03a9\nh : MeasurableSet (s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c0 \u03c9})\nthis : s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c0 \u03c9} = s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c0 \u03c9} \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c0 \u03c9}\n\u22a2 MeasurableSet (s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c0 \u03c9})", "state_after": "case mp\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : LinearOrder \u03b9\nf : Filtration \u03b9 m\n\u03c4 \u03c0 : \u03a9 \u2192 \u03b9\ninst\u271d\u2074 : TopologicalSpace \u03b9\ninst\u271d\u00b3 : SecondCountableTopology \u03b9\ninst\u271d\u00b2 : OrderTopology \u03b9\ninst\u271d\u00b9 : MeasurableSpace \u03b9\ninst\u271d : BorelSpace \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\nh\u03c0 : IsStoppingTime f \u03c0\ns : Set \u03a9\nh : MeasurableSet (s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c0 \u03c9})\nthis : s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c0 \u03c9} = s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c0 \u03c9} \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c0 \u03c9}\n\u22a2 MeasurableSet (s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c0 \u03c9} \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c0 \u03c9})"}, {"tactic": "exact measurableSet_inter_le _ h\u03c0 _ h", "annotated_tactic": ["exact <a>measurableSet_inter_le</a> _ h\u03c0 _ h", [{"full_name": "MeasureTheory.IsStoppingTime.measurableSet_inter_le", "def_path": "Mathlib/Probability/Process/Stopping.lean", "def_pos": [623, 9], "def_end_pos": [623, 31]}]], "state_before": "case mp\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : LinearOrder \u03b9\nf : Filtration \u03b9 m\n\u03c4 \u03c0 : \u03a9 \u2192 \u03b9\ninst\u271d\u2074 : TopologicalSpace \u03b9\ninst\u271d\u00b3 : SecondCountableTopology \u03b9\ninst\u271d\u00b2 : OrderTopology \u03b9\ninst\u271d\u00b9 : MeasurableSpace \u03b9\ninst\u271d : BorelSpace \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\nh\u03c0 : IsStoppingTime f \u03c0\ns : Set \u03a9\nh : MeasurableSet (s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c0 \u03c9})\nthis : s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c0 \u03c9} = s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c0 \u03c9} \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c0 \u03c9}\n\u22a2 MeasurableSet (s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c0 \u03c9} \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c0 \u03c9})", "state_after": "no goals"}, {"tactic": "rw [Set.inter_assoc, Set.inter_self]", "annotated_tactic": ["rw [<a>Set.inter_assoc</a>, <a>Set.inter_self</a>]", [{"full_name": "Set.inter_assoc", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [944, 9], "def_end_pos": [944, 20]}, {"full_name": "Set.inter_self", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [926, 9], "def_end_pos": [926, 19]}]], "state_before": "\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : LinearOrder \u03b9\nf : Filtration \u03b9 m\n\u03c4 \u03c0 : \u03a9 \u2192 \u03b9\ninst\u271d\u2074 : TopologicalSpace \u03b9\ninst\u271d\u00b3 : SecondCountableTopology \u03b9\ninst\u271d\u00b2 : OrderTopology \u03b9\ninst\u271d\u00b9 : MeasurableSpace \u03b9\ninst\u271d : BorelSpace \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\nh\u03c0 : IsStoppingTime f \u03c0\ns : Set \u03a9\nh : MeasurableSet (s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c0 \u03c9})\n\u22a2 s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c0 \u03c9} = s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c0 \u03c9} \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c0 \u03c9}", "state_after": "no goals"}, {"tactic": "rw [measurableSet_min_iff h\u03c4 h\u03c0] at h", "annotated_tactic": ["rw [<a>measurableSet_min_iff</a> h\u03c4 h\u03c0] at h", [{"full_name": "MeasureTheory.IsStoppingTime.measurableSet_min_iff", "def_path": "Mathlib/Probability/Process/Stopping.lean", "def_pos": [606, 9], "def_end_pos": [606, 30]}]], "state_before": "case mpr\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : LinearOrder \u03b9\nf : Filtration \u03b9 m\n\u03c4 \u03c0 : \u03a9 \u2192 \u03b9\ninst\u271d\u2074 : TopologicalSpace \u03b9\ninst\u271d\u00b3 : SecondCountableTopology \u03b9\ninst\u271d\u00b2 : OrderTopology \u03b9\ninst\u271d\u00b9 : MeasurableSpace \u03b9\ninst\u271d : BorelSpace \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\nh\u03c0 : IsStoppingTime f \u03c0\ns : Set \u03a9\nh : MeasurableSet (s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c0 \u03c9})\n\u22a2 MeasurableSet (s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c0 \u03c9})", "state_after": "case mpr\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : LinearOrder \u03b9\nf : Filtration \u03b9 m\n\u03c4 \u03c0 : \u03a9 \u2192 \u03b9\ninst\u271d\u2074 : TopologicalSpace \u03b9\ninst\u271d\u00b3 : SecondCountableTopology \u03b9\ninst\u271d\u00b2 : OrderTopology \u03b9\ninst\u271d\u00b9 : MeasurableSpace \u03b9\ninst\u271d : BorelSpace \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\nh\u03c0 : IsStoppingTime f \u03c0\ns : Set \u03a9\nh : MeasurableSet (s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c0 \u03c9}) \u2227 MeasurableSet (s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c0 \u03c9})\n\u22a2 MeasurableSet (s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c0 \u03c9})"}, {"tactic": "exact h.1", "annotated_tactic": ["exact h.1", []], "state_before": "case mpr\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : LinearOrder \u03b9\nf : Filtration \u03b9 m\n\u03c4 \u03c0 : \u03a9 \u2192 \u03b9\ninst\u271d\u2074 : TopologicalSpace \u03b9\ninst\u271d\u00b3 : SecondCountableTopology \u03b9\ninst\u271d\u00b2 : OrderTopology \u03b9\ninst\u271d\u00b9 : MeasurableSpace \u03b9\ninst\u271d : BorelSpace \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\nh\u03c0 : IsStoppingTime f \u03c0\ns : Set \u03a9\nh : MeasurableSet (s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c0 \u03c9}) \u2227 MeasurableSet (s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c0 \u03c9})\n\u22a2 MeasurableSet (s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c0 \u03c9})", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "full_name": "Int.div_nonpos", "start": [104, 11], "end": [105, 95], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "full_name": "MeasureTheory.lintegral_eq_nnreal", "start": [197, 1], "end": [219, 27], "traced_tactics": [{"tactic": "rw [lintegral]", "annotated_tactic": ["rw [<a>lintegral</a>]", [{"full_name": "MeasureTheory.lintegral", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [60, 17], "def_end_pos": [60, 26]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : Measure \u03b1\n\u22a2 \u222b\u207b (a : \u03b1), f a \u2202\u03bc = \u2a06 \u03c6, \u2a06 (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : Measure \u03b1\n\u22a2 \u2a06 g, \u2a06 (_ : \u2191g \u2264 fun a => f a), SimpleFunc.lintegral g \u03bc =\n    \u2a06 \u03c6, \u2a06 (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc"}, {"tactic": "refine'\n  le_antisymm (iSup\u2082_le fun \u03c6 h\u03c6 => _) (iSup_mono' fun \u03c6 => \u27e8\u03c6.map ((\u2191) : \u211d\u22650 \u2192 \u211d\u22650\u221e), le_rfl\u27e9)", "annotated_tactic": ["refine'\n    <a>le_antisymm</a> (<a>iSup\u2082_le</a> fun \u03c6 h\u03c6 => _) (<a>iSup_mono'</a> fun \u03c6 => \u27e8\u03c6.map ((\u2191) : \u211d\u22650 \u2192 \u211d\u22650\u221e), <a>le_rfl</a>\u27e9)", [{"full_name": "le_antisymm", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [188, 9], "def_end_pos": [188, 20]}, {"full_name": "iSup\u2082_le", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [883, 9], "def_end_pos": [883, 17]}, {"full_name": "iSup_mono'", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [919, 9], "def_end_pos": [919, 19]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : Measure \u03b1\n\u22a2 \u2a06 g, \u2a06 (_ : \u2191g \u2264 fun a => f a), SimpleFunc.lintegral g \u03bc =\n    \u2a06 \u03c6, \u2a06 (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : Measure \u03b1\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\u221e\nh\u03c6 : \u2191\u03c6 \u2264 fun a => f a\n\u22a2 SimpleFunc.lintegral \u03c6 \u03bc \u2264\n    \u2a06 \u03c6, \u2a06 (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc"}, {"tactic": "by_cases h : \u2200\u1d50 a \u2202\u03bc, \u03c6 a \u2260 \u221e", "annotated_tactic": ["by_cases h : \u2200\u1d50 a \u2202\u03bc, \u03c6 a \u2260 \u221e", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : Measure \u03b1\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\u221e\nh\u03c6 : \u2191\u03c6 \u2264 fun a => f a\n\u22a2 SimpleFunc.lintegral \u03c6 \u03bc \u2264\n    \u2a06 \u03c6, \u2a06 (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : Measure \u03b1\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\u221e\nh\u03c6 : \u2191\u03c6 \u2264 fun a => f a\nh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u03c6 a \u2260 \u22a4\n\u22a2 SimpleFunc.lintegral \u03c6 \u03bc \u2264\n    \u2a06 \u03c6, \u2a06 (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : Measure \u03b1\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\u221e\nh\u03c6 : \u2191\u03c6 \u2264 fun a => f a\nh : \u00ac\u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u03c6 a \u2260 \u22a4\n\u22a2 SimpleFunc.lintegral \u03c6 \u03bc \u2264\n    \u2a06 \u03c6, \u2a06 (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc"}, {"tactic": "let \u03c8 := \u03c6.map ENNReal.toNNReal", "annotated_tactic": ["let \u03c8 := \u03c6.map <a>ENNReal.toNNReal</a>", [{"full_name": "ENNReal.toNNReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [164, 15], "def_end_pos": [164, 23]}]], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : Measure \u03b1\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\u221e\nh\u03c6 : \u2191\u03c6 \u2264 fun a => f a\nh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u03c6 a \u2260 \u22a4\n\u22a2 SimpleFunc.lintegral \u03c6 \u03bc \u2264\n    \u2a06 \u03c6, \u2a06 (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : Measure \u03b1\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\u221e\nh\u03c6 : \u2191\u03c6 \u2264 fun a => f a\nh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u03c6 a \u2260 \u22a4\n\u03c8 : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map ENNReal.toNNReal \u03c6\n\u22a2 SimpleFunc.lintegral \u03c6 \u03bc \u2264\n    \u2a06 \u03c6, \u2a06 (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc"}, {"tactic": "replace h : \u03c8.map ((\u2191) : \u211d\u22650 \u2192 \u211d\u22650\u221e) =\u1d50[\u03bc] \u03c6 := h.mono fun a => ENNReal.coe_toNNReal", "annotated_tactic": ["replace h : \u03c8.map ((\u2191) : \u211d\u22650 \u2192 \u211d\u22650\u221e) =\u1d50[\u03bc] \u03c6 := h.mono fun a => <a>ENNReal.coe_toNNReal</a>", [{"full_name": "ENNReal.coe_toNNReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [180, 9], "def_end_pos": [180, 21]}]], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : Measure \u03b1\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\u221e\nh\u03c6 : \u2191\u03c6 \u2264 fun a => f a\nh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u03c6 a \u2260 \u22a4\n\u03c8 : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map ENNReal.toNNReal \u03c6\n\u22a2 SimpleFunc.lintegral \u03c6 \u03bc \u2264\n    \u2a06 \u03c6, \u2a06 (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : Measure \u03b1\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\u221e\nh\u03c6 : \u2191\u03c6 \u2264 fun a => f a\n\u03c8 : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map ENNReal.toNNReal \u03c6\nh : \u2191(SimpleFunc.map ENNReal.some \u03c8) =\u1d50[\u03bc] \u2191\u03c6\n\u22a2 SimpleFunc.lintegral \u03c6 \u03bc \u2264\n    \u2a06 \u03c6, \u2a06 (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc"}, {"tactic": "have : \u2200 x, \u2191(\u03c8 x) \u2264 f x := fun x => le_trans ENNReal.coe_toNNReal_le_self (h\u03c6 x)", "annotated_tactic": ["have : \u2200 x, \u2191(\u03c8 x) \u2264 f x := fun x => <a>le_trans</a> <a>ENNReal.coe_toNNReal_le_self</a> (h\u03c6 x)", [{"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}, {"full_name": "ENNReal.coe_toNNReal_le_self", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [198, 9], "def_end_pos": [198, 29]}]], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : Measure \u03b1\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\u221e\nh\u03c6 : \u2191\u03c6 \u2264 fun a => f a\n\u03c8 : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map ENNReal.toNNReal \u03c6\nh : \u2191(SimpleFunc.map ENNReal.some \u03c8) =\u1d50[\u03bc] \u2191\u03c6\n\u22a2 SimpleFunc.lintegral \u03c6 \u03bc \u2264\n    \u2a06 \u03c6, \u2a06 (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : Measure \u03b1\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\u221e\nh\u03c6 : \u2191\u03c6 \u2264 fun a => f a\n\u03c8 : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map ENNReal.toNNReal \u03c6\nh : \u2191(SimpleFunc.map ENNReal.some \u03c8) =\u1d50[\u03bc] \u2191\u03c6\nthis : \u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x\n\u22a2 SimpleFunc.lintegral \u03c6 \u03bc \u2264\n    \u2a06 \u03c6, \u2a06 (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc"}, {"tactic": "exact\n  le_iSup_of_le (\u03c6.map ENNReal.toNNReal) (le_iSup_of_le this (ge_of_eq <| lintegral_congr h))", "annotated_tactic": ["exact\n      <a>le_iSup_of_le</a> (\u03c6.map <a>ENNReal.toNNReal</a>) (<a>le_iSup_of_le</a> this (<a>ge_of_eq</a> <| <a>lintegral_congr</a> h))", [{"full_name": "le_iSup_of_le", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [849, 9], "def_end_pos": [849, 22]}, {"full_name": "ENNReal.toNNReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [164, 15], "def_end_pos": [164, 23]}, {"full_name": "le_iSup_of_le", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [849, 9], "def_end_pos": [849, 22]}, {"full_name": "ge_of_eq", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [352, 9], "def_end_pos": [352, 17]}, {"full_name": "MeasureTheory.SimpleFunc.lintegral_congr", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [1126, 9], "def_end_pos": [1126, 24]}]], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : Measure \u03b1\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\u221e\nh\u03c6 : \u2191\u03c6 \u2264 fun a => f a\n\u03c8 : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map ENNReal.toNNReal \u03c6\nh : \u2191(SimpleFunc.map ENNReal.some \u03c8) =\u1d50[\u03bc] \u2191\u03c6\nthis : \u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x\n\u22a2 SimpleFunc.lintegral \u03c6 \u03bc \u2264\n    \u2a06 \u03c6, \u2a06 (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc", "state_after": "no goals"}, {"tactic": "have h_meas : \u03bc (\u03c6 \u207b\u00b9' {\u221e}) \u2260 0 := mt measure_zero_iff_ae_nmem.1 h", "annotated_tactic": ["have h_meas : \u03bc (\u03c6 \u207b\u00b9' {\u221e}) \u2260 0 := <a>mt</a> <a>measure_zero_iff_ae_nmem</a>.1 h", [{"full_name": "mt", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [516, 9], "def_end_pos": [516, 11]}, {"full_name": "MeasureTheory.measure_zero_iff_ae_nmem", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [403, 9], "def_end_pos": [403, 33]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : Measure \u03b1\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\u221e\nh\u03c6 : \u2191\u03c6 \u2264 fun a => f a\nh : \u00ac\u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u03c6 a \u2260 \u22a4\n\u22a2 SimpleFunc.lintegral \u03c6 \u03bc \u2264\n    \u2a06 \u03c6, \u2a06 (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc", "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : Measure \u03b1\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\u221e\nh\u03c6 : \u2191\u03c6 \u2264 fun a => f a\nh : \u00ac\u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u03c6 a \u2260 \u22a4\nh_meas : \u2191\u2191\u03bc (\u2191\u03c6 \u207b\u00b9' {\u22a4}) \u2260 0\n\u22a2 SimpleFunc.lintegral \u03c6 \u03bc \u2264\n    \u2a06 \u03c6, \u2a06 (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc"}, {"tactic": "refine' le_trans le_top (ge_of_eq <| (iSup_eq_top _).2 fun b hb => _)", "annotated_tactic": ["refine' <a>le_trans</a> <a>le_top</a> (<a>ge_of_eq</a> <| (<a>iSup_eq_top</a> _).2 fun b hb => _)", [{"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}, {"full_name": "le_top", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [98, 9], "def_end_pos": [98, 15]}, {"full_name": "ge_of_eq", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [352, 9], "def_end_pos": [352, 17]}, {"full_name": "iSup_eq_top", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [1766, 9], "def_end_pos": [1766, 20]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : Measure \u03b1\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\u221e\nh\u03c6 : \u2191\u03c6 \u2264 fun a => f a\nh : \u00ac\u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u03c6 a \u2260 \u22a4\nh_meas : \u2191\u2191\u03bc (\u2191\u03c6 \u207b\u00b9' {\u22a4}) \u2260 0\n\u22a2 SimpleFunc.lintegral \u03c6 \u03bc \u2264\n    \u2a06 \u03c6, \u2a06 (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc", "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : Measure \u03b1\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\u221e\nh\u03c6 : \u2191\u03c6 \u2264 fun a => f a\nh : \u00ac\u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u03c6 a \u2260 \u22a4\nh_meas : \u2191\u2191\u03bc (\u2191\u03c6 \u207b\u00b9' {\u22a4}) \u2260 0\nb : \u211d\u22650\u221e\nhb : b < \u22a4\n\u22a2 \u2203 i, b < \u2a06 (_ : \u2200 (x : \u03b1), \u2191(\u2191i x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some i) \u03bc"}, {"tactic": "obtain \u27e8n, hn\u27e9 : \u2203 n : \u2115, b < n * \u03bc (\u03c6 \u207b\u00b9' {\u221e})", "annotated_tactic": ["obtain \u27e8n, hn\u27e9 : \u2203 n : \u2115, b < n * \u03bc (\u03c6 \u207b\u00b9' {\u221e})", []], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : Measure \u03b1\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\u221e\nh\u03c6 : \u2191\u03c6 \u2264 fun a => f a\nh : \u00ac\u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u03c6 a \u2260 \u22a4\nh_meas : \u2191\u2191\u03bc (\u2191\u03c6 \u207b\u00b9' {\u22a4}) \u2260 0\nb : \u211d\u22650\u221e\nhb : b < \u22a4\n\u22a2 \u2203 i, b < \u2a06 (_ : \u2200 (x : \u03b1), \u2191(\u2191i x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some i) \u03bc", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : Measure \u03b1\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\u221e\nh\u03c6 : \u2191\u03c6 \u2264 fun a => f a\nh : \u00ac\u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u03c6 a \u2260 \u22a4\nh_meas : \u2191\u2191\u03bc (\u2191\u03c6 \u207b\u00b9' {\u22a4}) \u2260 0\nb : \u211d\u22650\u221e\nhb : b < \u22a4\n\u22a2 \u2203 n, b < \u2191n * \u2191\u2191\u03bc (\u2191\u03c6 \u207b\u00b9' {\u22a4})\n\ncase neg.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : Measure \u03b1\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\u221e\nh\u03c6 : \u2191\u03c6 \u2264 fun a => f a\nh : \u00ac\u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u03c6 a \u2260 \u22a4\nh_meas : \u2191\u2191\u03bc (\u2191\u03c6 \u207b\u00b9' {\u22a4}) \u2260 0\nb : \u211d\u22650\u221e\nhb : b < \u22a4\nn : \u2115\nhn : b < \u2191n * \u2191\u2191\u03bc (\u2191\u03c6 \u207b\u00b9' {\u22a4})\n\u22a2 \u2203 i, b < \u2a06 (_ : \u2200 (x : \u03b1), \u2191(\u2191i x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some i) \u03bc"}, {"tactic": "exact exists_nat_mul_gt h_meas (ne_of_lt hb)", "annotated_tactic": ["exact <a>exists_nat_mul_gt</a> h_meas (<a>ne_of_lt</a> hb)", [{"full_name": "ENNReal.exists_nat_mul_gt", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1880, 9], "def_end_pos": [1880, 26]}, {"full_name": "ne_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [101, 9], "def_end_pos": [101, 17]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : Measure \u03b1\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\u221e\nh\u03c6 : \u2191\u03c6 \u2264 fun a => f a\nh : \u00ac\u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u03c6 a \u2260 \u22a4\nh_meas : \u2191\u2191\u03bc (\u2191\u03c6 \u207b\u00b9' {\u22a4}) \u2260 0\nb : \u211d\u22650\u221e\nhb : b < \u22a4\n\u22a2 \u2203 n, b < \u2191n * \u2191\u2191\u03bc (\u2191\u03c6 \u207b\u00b9' {\u22a4})\n\ncase neg.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : Measure \u03b1\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\u221e\nh\u03c6 : \u2191\u03c6 \u2264 fun a => f a\nh : \u00ac\u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u03c6 a \u2260 \u22a4\nh_meas : \u2191\u2191\u03bc (\u2191\u03c6 \u207b\u00b9' {\u22a4}) \u2260 0\nb : \u211d\u22650\u221e\nhb : b < \u22a4\nn : \u2115\nhn : b < \u2191n * \u2191\u2191\u03bc (\u2191\u03c6 \u207b\u00b9' {\u22a4})\n\u22a2 \u2203 i, b < \u2a06 (_ : \u2200 (x : \u03b1), \u2191(\u2191i x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some i) \u03bc", "state_after": "case neg.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : Measure \u03b1\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\u221e\nh\u03c6 : \u2191\u03c6 \u2264 fun a => f a\nh : \u00ac\u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u03c6 a \u2260 \u22a4\nh_meas : \u2191\u2191\u03bc (\u2191\u03c6 \u207b\u00b9' {\u22a4}) \u2260 0\nb : \u211d\u22650\u221e\nhb : b < \u22a4\nn : \u2115\nhn : b < \u2191n * \u2191\u2191\u03bc (\u2191\u03c6 \u207b\u00b9' {\u22a4})\n\u22a2 \u2203 i, b < \u2a06 (_ : \u2200 (x : \u03b1), \u2191(\u2191i x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some i) \u03bc"}, {"tactic": "use (const \u03b1 (n : \u211d\u22650)).restrict (\u03c6 \u207b\u00b9' {\u221e})", "annotated_tactic": ["use (<a>const</a> \u03b1 (n : \u211d\u22650)).<a>restrict</a> (\u03c6 \u207b\u00b9' {\u221e})", [{"full_name": "MeasureTheory.SimpleFunc.const", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [145, 5], "def_end_pos": [145, 10]}, {"full_name": "MeasureTheory.SimpleFunc.restrict", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [728, 5], "def_end_pos": [728, 13]}]], "state_before": "case neg.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : Measure \u03b1\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\u221e\nh\u03c6 : \u2191\u03c6 \u2264 fun a => f a\nh : \u00ac\u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u03c6 a \u2260 \u22a4\nh_meas : \u2191\u2191\u03bc (\u2191\u03c6 \u207b\u00b9' {\u22a4}) \u2260 0\nb : \u211d\u22650\u221e\nhb : b < \u22a4\nn : \u2115\nhn : b < \u2191n * \u2191\u2191\u03bc (\u2191\u03c6 \u207b\u00b9' {\u22a4})\n\u22a2 \u2203 i, b < \u2a06 (_ : \u2200 (x : \u03b1), \u2191(\u2191i x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some i) \u03bc", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : Measure \u03b1\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\u221e\nh\u03c6 : \u2191\u03c6 \u2264 fun a => f a\nh : \u00ac\u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u03c6 a \u2260 \u22a4\nh_meas : \u2191\u2191\u03bc (\u2191\u03c6 \u207b\u00b9' {\u22a4}) \u2260 0\nb : \u211d\u22650\u221e\nhb : b < \u22a4\nn : \u2115\nhn : b < \u2191n * \u2191\u2191\u03bc (\u2191\u03c6 \u207b\u00b9' {\u22a4})\n\u22a2 b <\n    \u2a06 (_ : \u2200 (x : \u03b1), \u2191(\u2191(restrict (const \u03b1 \u2191n) (\u2191\u03c6 \u207b\u00b9' {\u22a4})) x) \u2264 f x),\n      SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (restrict (const \u03b1 \u2191n) (\u2191\u03c6 \u207b\u00b9' {\u22a4}))) \u03bc"}, {"tactic": "simp only [lt_iSup_iff, exists_prop, coe_restrict, \u03c6.measurableSet_preimage, coe_const,\n  ENNReal.coe_indicator, map_coe_ennreal_restrict, SimpleFunc.map_const, ENNReal.coe_nat,\n  restrict_const_lintegral]", "annotated_tactic": ["simp only [<a>lt_iSup_iff</a>, <a>exists_prop</a>, <a>coe_restrict</a>, \u03c6.measurableSet_preimage, <a>coe_const</a>,\n      <a>ENNReal.coe_indicator</a>, <a>map_coe_ennreal_restrict</a>, <a>SimpleFunc.map_const</a>, <a>ENNReal.coe_nat</a>,\n      <a>restrict_const_lintegral</a>]", [{"full_name": "lt_iSup_iff", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [668, 9], "def_end_pos": [668, 20]}, {"full_name": "exists_prop", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [485, 17], "def_end_pos": [485, 28]}, {"full_name": "MeasureTheory.SimpleFunc.coe_restrict", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [738, 9], "def_end_pos": [738, 21]}, {"full_name": "MeasureTheory.SimpleFunc.coe_const", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [158, 9], "def_end_pos": [158, 18]}, {"full_name": "ENNReal.coe_indicator", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [548, 9], "def_end_pos": [548, 22]}, {"full_name": "MeasureTheory.SimpleFunc.map_coe_ennreal_restrict", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [758, 9], "def_end_pos": [758, 33]}, {"full_name": "MeasureTheory.SimpleFunc.map_const", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [313, 9], "def_end_pos": [313, 18]}, {"full_name": "ENNReal.coe_nat", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [707, 9], "def_end_pos": [707, 16]}, {"full_name": "MeasureTheory.SimpleFunc.restrict_const_lintegral", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [1085, 9], "def_end_pos": [1085, 33]}]], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : Measure \u03b1\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\u221e\nh\u03c6 : \u2191\u03c6 \u2264 fun a => f a\nh : \u00ac\u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u03c6 a \u2260 \u22a4\nh_meas : \u2191\u2191\u03bc (\u2191\u03c6 \u207b\u00b9' {\u22a4}) \u2260 0\nb : \u211d\u22650\u221e\nhb : b < \u22a4\nn : \u2115\nhn : b < \u2191n * \u2191\u2191\u03bc (\u2191\u03c6 \u207b\u00b9' {\u22a4})\n\u22a2 b <\n    \u2a06 (_ : \u2200 (x : \u03b1), \u2191(\u2191(restrict (const \u03b1 \u2191n) (\u2191\u03c6 \u207b\u00b9' {\u22a4})) x) \u2264 f x),\n      SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (restrict (const \u03b1 \u2191n) (\u2191\u03c6 \u207b\u00b9' {\u22a4}))) \u03bc", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : Measure \u03b1\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\u221e\nh\u03c6 : \u2191\u03c6 \u2264 fun a => f a\nh : \u00ac\u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u03c6 a \u2260 \u22a4\nh_meas : \u2191\u2191\u03bc (\u2191\u03c6 \u207b\u00b9' {\u22a4}) \u2260 0\nb : \u211d\u22650\u221e\nhb : b < \u22a4\nn : \u2115\nhn : b < \u2191n * \u2191\u2191\u03bc (\u2191\u03c6 \u207b\u00b9' {\u22a4})\n\u22a2 (\u2200 (x : \u03b1), indicator (\u2191\u03c6 \u207b\u00b9' {\u22a4}) (fun x => \u2191(Function.const \u03b1 (\u2191n) x)) x \u2264 f x) \u2227 b < \u2191n * \u2191\u2191\u03bc (\u2191\u03c6 \u207b\u00b9' {\u22a4})"}, {"tactic": "refine' \u27e8indicator_le fun x hx => le_trans _ (h\u03c6 _), hn\u27e9", "annotated_tactic": ["refine' \u27e8<a>indicator_le</a> fun x hx => <a>le_trans</a> _ (h\u03c6 _), hn\u27e9", [{"full_name": "Set.indicator_le", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [913, 3], "def_end_pos": [913, 14]}, {"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}]], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : Measure \u03b1\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\u221e\nh\u03c6 : \u2191\u03c6 \u2264 fun a => f a\nh : \u00ac\u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u03c6 a \u2260 \u22a4\nh_meas : \u2191\u2191\u03bc (\u2191\u03c6 \u207b\u00b9' {\u22a4}) \u2260 0\nb : \u211d\u22650\u221e\nhb : b < \u22a4\nn : \u2115\nhn : b < \u2191n * \u2191\u2191\u03bc (\u2191\u03c6 \u207b\u00b9' {\u22a4})\n\u22a2 (\u2200 (x : \u03b1), indicator (\u2191\u03c6 \u207b\u00b9' {\u22a4}) (fun x => \u2191(Function.const \u03b1 (\u2191n) x)) x \u2264 f x) \u2227 b < \u2191n * \u2191\u2191\u03bc (\u2191\u03c6 \u207b\u00b9' {\u22a4})", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : Measure \u03b1\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\u221e\nh\u03c6 : \u2191\u03c6 \u2264 fun a => f a\nh : \u00ac\u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u03c6 a \u2260 \u22a4\nh_meas : \u2191\u2191\u03bc (\u2191\u03c6 \u207b\u00b9' {\u22a4}) \u2260 0\nb : \u211d\u22650\u221e\nhb : b < \u22a4\nn : \u2115\nhn : b < \u2191n * \u2191\u2191\u03bc (\u2191\u03c6 \u207b\u00b9' {\u22a4})\nx : \u03b1\nhx : x \u2208 \u2191\u03c6 \u207b\u00b9' {\u22a4}\n\u22a2 \u2191(Function.const \u03b1 (\u2191n) x) \u2264 \u2191\u03c6 x"}, {"tactic": "simp only [mem_preimage, mem_singleton_iff] at hx", "annotated_tactic": ["simp only [<a>mem_preimage</a>, <a>mem_singleton_iff</a>] at hx", [{"full_name": "Set.mem_preimage", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [64, 9], "def_end_pos": [64, 21]}, {"full_name": "Set.mem_singleton_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1273, 9], "def_end_pos": [1273, 26]}]], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : Measure \u03b1\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\u221e\nh\u03c6 : \u2191\u03c6 \u2264 fun a => f a\nh : \u00ac\u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u03c6 a \u2260 \u22a4\nh_meas : \u2191\u2191\u03bc (\u2191\u03c6 \u207b\u00b9' {\u22a4}) \u2260 0\nb : \u211d\u22650\u221e\nhb : b < \u22a4\nn : \u2115\nhn : b < \u2191n * \u2191\u2191\u03bc (\u2191\u03c6 \u207b\u00b9' {\u22a4})\nx : \u03b1\nhx : x \u2208 \u2191\u03c6 \u207b\u00b9' {\u22a4}\n\u22a2 \u2191(Function.const \u03b1 (\u2191n) x) \u2264 \u2191\u03c6 x", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : Measure \u03b1\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\u221e\nh\u03c6 : \u2191\u03c6 \u2264 fun a => f a\nh : \u00ac\u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u03c6 a \u2260 \u22a4\nh_meas : \u2191\u2191\u03bc (\u2191\u03c6 \u207b\u00b9' {\u22a4}) \u2260 0\nb : \u211d\u22650\u221e\nhb : b < \u22a4\nn : \u2115\nhn : b < \u2191n * \u2191\u2191\u03bc (\u2191\u03c6 \u207b\u00b9' {\u22a4})\nx : \u03b1\nhx : \u2191\u03c6 x = \u22a4\n\u22a2 \u2191(Function.const \u03b1 (\u2191n) x) \u2264 \u2191\u03c6 x"}, {"tactic": "simp only [hx, le_top]", "annotated_tactic": ["simp only [hx, <a>le_top</a>]", [{"full_name": "le_top", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [98, 9], "def_end_pos": [98, 15]}]], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : Measure \u03b1\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\u221e\nh\u03c6 : \u2191\u03c6 \u2264 fun a => f a\nh : \u00ac\u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u03c6 a \u2260 \u22a4\nh_meas : \u2191\u2191\u03bc (\u2191\u03c6 \u207b\u00b9' {\u22a4}) \u2260 0\nb : \u211d\u22650\u221e\nhb : b < \u22a4\nn : \u2115\nhn : b < \u2191n * \u2191\u2191\u03bc (\u2191\u03c6 \u207b\u00b9' {\u22a4})\nx : \u03b1\nhx : \u2191\u03c6 x = \u22a4\n\u22a2 \u2191(Function.const \u03b1 (\u2191n) x) \u2264 \u2191\u03c6 x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Function.lean", "full_name": "Set.pi_piecewise", "start": [1561, 1], "end": [1563, 14], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean", "full_name": "MeasureTheory.condexpInd_nonneg", "start": [359, 1], "end": [363, 72], "traced_tactics": [{"tactic": "rw [\u2190 coeFn_le]", "annotated_tactic": ["rw [\u2190 <a>coeFn_le</a>]", [{"full_name": "MeasureTheory.Lp.coeFn_le", "def_path": "Mathlib/MeasureTheory/Function/LpOrder.lean", "def_pos": [42, 9], "def_end_pos": [42, 17]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2074 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9\u2070 : NormedSpace \u211d F'\ninst\u271d\u2079 : CompleteSpace F'\ninst\u271d\u2078 : NormedAddCommGroup G\ninst\u271d\u2077 : NormedAddCommGroup G'\ninst\u271d\u2076 : NormedSpace \u211d G'\ninst\u271d\u2075 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2074 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d\u00b3 : SigmaFinite (Measure.trim \u03bc hm)\nE : Type u_8\ninst\u271d\u00b2 : NormedLatticeAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : OrderedSMul \u211d E\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx : E\nhx : 0 \u2264 x\n\u22a2 0 \u2264 \u2191(condexpInd E hm \u03bc s) x", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2074 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9\u2070 : NormedSpace \u211d F'\ninst\u271d\u2079 : CompleteSpace F'\ninst\u271d\u2078 : NormedAddCommGroup G\ninst\u271d\u2077 : NormedAddCommGroup G'\ninst\u271d\u2076 : NormedSpace \u211d G'\ninst\u271d\u2075 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2074 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d\u00b3 : SigmaFinite (Measure.trim \u03bc hm)\nE : Type u_8\ninst\u271d\u00b2 : NormedLatticeAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : OrderedSMul \u211d E\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx : E\nhx : 0 \u2264 x\n\u22a2 \u2191\u21910 \u2264\u1d50[\u03bc] \u2191\u2191(\u2191(condexpInd E hm \u03bc s) x)"}, {"tactic": "refine' EventuallyLE.trans_eq _ (condexpInd_ae_eq_condexpIndSMul hm hs h\u03bcs x).symm", "annotated_tactic": ["refine' <a>EventuallyLE.trans_eq</a> _ (<a>condexpInd_ae_eq_condexpIndSMul</a> hm hs h\u03bcs x).<a>symm</a>", [{"full_name": "Filter.EventuallyLE.trans_eq", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1693, 9], "def_end_pos": [1693, 30]}, {"full_name": "MeasureTheory.condexpInd_ae_eq_condexpIndSMul", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean", "def_pos": [276, 9], "def_end_pos": [276, 40]}, {"full_name": "Filter.EventuallyEq.symm", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1498, 9], "def_end_pos": [1498, 26]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2074 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9\u2070 : NormedSpace \u211d F'\ninst\u271d\u2079 : CompleteSpace F'\ninst\u271d\u2078 : NormedAddCommGroup G\ninst\u271d\u2077 : NormedAddCommGroup G'\ninst\u271d\u2076 : NormedSpace \u211d G'\ninst\u271d\u2075 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2074 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d\u00b3 : SigmaFinite (Measure.trim \u03bc hm)\nE : Type u_8\ninst\u271d\u00b2 : NormedLatticeAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : OrderedSMul \u211d E\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx : E\nhx : 0 \u2264 x\n\u22a2 \u2191\u21910 \u2264\u1d50[\u03bc] \u2191\u2191(\u2191(condexpInd E hm \u03bc s) x)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2074 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9\u2070 : NormedSpace \u211d F'\ninst\u271d\u2079 : CompleteSpace F'\ninst\u271d\u2078 : NormedAddCommGroup G\ninst\u271d\u2077 : NormedAddCommGroup G'\ninst\u271d\u2076 : NormedSpace \u211d G'\ninst\u271d\u2075 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2074 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d\u00b3 : SigmaFinite (Measure.trim \u03bc hm)\nE : Type u_8\ninst\u271d\u00b2 : NormedLatticeAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : OrderedSMul \u211d E\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx : E\nhx : 0 \u2264 x\n\u22a2 \u2191\u21910 \u2264\u1d50[\u03bc] \u2191\u2191(condexpIndSMul hm hs h\u03bcs x)"}, {"tactic": "exact (coeFn_zero E 1 \u03bc).trans_le (condexpIndSMul_nonneg hs h\u03bcs x hx)", "annotated_tactic": ["exact (<a>coeFn_zero</a> E 1 \u03bc).<a>trans_le</a> (<a>condexpIndSMul_nonneg</a> hs h\u03bcs x hx)", [{"full_name": "MeasureTheory.Lp.coeFn_zero", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [222, 9], "def_end_pos": [222, 19]}, {"full_name": "Filter.EventuallyEq.trans_le", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1684, 9], "def_end_pos": [1684, 30]}, {"full_name": "MeasureTheory.condexpIndSMul_nonneg", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL2.lean", "def_pos": [516, 9], "def_end_pos": [516, 30]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2074 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9\u2070 : NormedSpace \u211d F'\ninst\u271d\u2079 : CompleteSpace F'\ninst\u271d\u2078 : NormedAddCommGroup G\ninst\u271d\u2077 : NormedAddCommGroup G'\ninst\u271d\u2076 : NormedSpace \u211d G'\ninst\u271d\u2075 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2074 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d\u00b3 : SigmaFinite (Measure.trim \u03bc hm)\nE : Type u_8\ninst\u271d\u00b2 : NormedLatticeAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : OrderedSMul \u211d E\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx : E\nhx : 0 \u2264 x\n\u22a2 \u2191\u21910 \u2264\u1d50[\u03bc] \u2191\u2191(condexpIndSMul hm hs h\u03bcs x)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/ProbabilityMassFunction/Basic.lean", "full_name": "PMF.toMeasure_apply_eq_of_inter_support_eq", "start": [297, 1], "end": [300, 52], "traced_tactics": [{"tactic": "simpa only [p.toMeasure_apply_eq_toOuterMeasure_apply, hs, ht] using\n  toOuterMeasure_apply_eq_of_inter_support_eq p h", "annotated_tactic": ["simpa only [p.toMeasure_apply_eq_toOuterMeasure_apply, hs, ht] using\n    <a>toOuterMeasure_apply_eq_of_inter_support_eq</a> p h", [{"full_name": "PMF.toOuterMeasure_apply_eq_of_inter_support_eq", "def_path": "Mathlib/Probability/ProbabilityMassFunction/Basic.lean", "def_pos": [226, 9], "def_end_pos": [226, 52]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b1\np : PMF \u03b1\ns\u271d t\u271d s t : Set \u03b1\nhs : MeasurableSet s\nht : MeasurableSet t\nh : s \u2229 support p = t \u2229 support p\n\u22a2 \u2191\u2191(toMeasure p) s = \u2191\u2191(toMeasure p) t", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "full_name": "MeasureTheory.snormEssSup_lt_top_of_ae_bound", "start": [458, 1], "end": [460, 66], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Intervals/WithBotTop.lean", "full_name": "WithBot.image_coe_Iio", "start": [197, 1], "end": [198, 95], "traced_tactics": [{"tactic": "rw [\u2190 preimage_coe_Iio, image_preimage_eq_inter_range, range_coe, inter_comm, Ioi_inter_Iio]", "annotated_tactic": ["rw [\u2190 <a>preimage_coe_Iio</a>, <a>image_preimage_eq_inter_range</a>, <a>range_coe</a>, <a>inter_comm</a>, <a>Ioi_inter_Iio</a>]", [{"full_name": "WithBot.preimage_coe_Iio", "def_path": "Mathlib/Data/Set/Intervals/WithBotTop.lean", "def_pos": [157, 9], "def_end_pos": [157, 25]}, {"full_name": "Set.image_preimage_eq_inter_range", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [796, 9], "def_end_pos": [796, 38]}, {"full_name": "WithBot.range_coe", "def_path": "Mathlib/Data/Set/Intervals/WithBotTop.lean", "def_pos": [142, 9], "def_end_pos": [142, 18]}, {"full_name": "Set.inter_comm", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [940, 9], "def_end_pos": [940, 19]}, {"full_name": "Set.Ioi_inter_Iio", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [630, 9], "def_end_pos": [630, 22]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : PartialOrder \u03b1\na b : \u03b1\n\u22a2 some '' Iio a = Ioo \u22a5 \u2191a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/HashMap/WF.lean", "full_name": "Std.HashMap.Imp.insert_WF", "start": [221, 1], "end": [239, 32], "traced_tactics": [{"tactic": "dsimp [insert, cond]", "annotated_tactic": ["dsimp [<a>insert</a>, <a>cond</a>]", [{"full_name": "Std.HashMap.Imp.insert", "def_path": "lake-packages/std/Std/Data/HashMap/Basic.lean", "def_pos": [160, 15], "def_end_pos": [160, 21]}, {"full_name": "cond", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [971, 21], "def_end_pos": [971, 25]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nk : \u03b1\nv : \u03b2\nh : Buckets.WF m.buckets\n\u22a2 Buckets.WF (insert m k v).buckets", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nk : \u03b1\nv : \u03b2\nh : Buckets.WF m.buckets\n\u22a2 Buckets.WF\n    (match\n        AssocList.contains k\n          m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] with\n      | true =>\n        { size := m.size,\n          buckets :=\n            Buckets.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n              (AssocList.replace k v\n                m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n              (_ :\n                USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n                  Array.size m.buckets.val) }\n      | false =>\n        if numBucketsForCapacity (m.size + 1) \u2264 Array.size m.buckets.val then\n          { size := m.size + 1,\n            buckets :=\n              Buckets.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n                (AssocList.cons k v\n                  m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n                (_ :\n                  USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n                    Array.size m.buckets.val) }\n        else\n          expand (m.size + 1)\n            (Buckets.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n              (AssocList.cons k v\n                m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n              (_ :\n                USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n                  Array.size m.buckets.val))).buckets"}, {"tactic": "split", "annotated_tactic": ["split", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nk : \u03b1\nv : \u03b2\nh : Buckets.WF m.buckets\n\u22a2 Buckets.WF\n    (match\n        AssocList.contains k\n          m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] with\n      | true =>\n        { size := m.size,\n          buckets :=\n            Buckets.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n              (AssocList.replace k v\n                m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n              (_ :\n                USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n                  Array.size m.buckets.val) }\n      | false =>\n        if numBucketsForCapacity (m.size + 1) \u2264 Array.size m.buckets.val then\n          { size := m.size + 1,\n            buckets :=\n              Buckets.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n                (AssocList.cons k v\n                  m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n                (_ :\n                  USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n                    Array.size m.buckets.val) }\n        else\n          expand (m.size + 1)\n            (Buckets.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n              (AssocList.cons k v\n                m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n              (_ :\n                USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n                  Array.size m.buckets.val))).buckets", "state_after": "case h_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nk : \u03b1\nv : \u03b2\nh : Buckets.WF m.buckets\nc\u271d : Bool\nheq\u271d :\n  AssocList.contains k\n      m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] =\n    true\n\u22a2 Buckets.WF\n    { size := m.size,\n        buckets :=\n          Buckets.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n            (AssocList.replace k v\n              m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n            (_ :\n              USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n                Array.size m.buckets.val) }.buckets\n\ncase h_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nk : \u03b1\nv : \u03b2\nh : Buckets.WF m.buckets\nc\u271d : Bool\nheq\u271d :\n  AssocList.contains k\n      m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] =\n    false\n\u22a2 Buckets.WF\n    (if numBucketsForCapacity (m.size + 1) \u2264 Array.size m.buckets.val then\n        { size := m.size + 1,\n          buckets :=\n            Buckets.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n              (AssocList.cons k v\n                m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n              (_ :\n                USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n                  Array.size m.buckets.val) }\n      else\n        expand (m.size + 1)\n          (Buckets.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n            (AssocList.cons k v\n              m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n            (_ :\n              USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n                Array.size m.buckets.val))).buckets"}, {"tactic": "simp at h\u2081", "annotated_tactic": ["simp at h\u2081", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nk : \u03b1\nv : \u03b2\nh : Buckets.WF m.buckets\nc\u271d : Bool\nh\u2081 :\n  AssocList.contains k\n      m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] =\n    true\n\u22a2 Buckets.WF\n    { size := m.size,\n        buckets :=\n          Buckets.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n            (AssocList.replace k v\n              m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n            (_ :\n              USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n                Array.size m.buckets.val) }.buckets", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nk : \u03b1\nv : \u03b2\nh : Buckets.WF m.buckets\nc\u271d : Bool\nh\u2081 :\n  \u2203 x,\n    x \u2208\n        AssocList.toList\n          m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] \u2227\n      (x.fst == k) = true\n\u22a2 Buckets.WF\n    { size := m.size,\n        buckets :=\n          Buckets.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n            (AssocList.replace k v\n              m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n            (_ :\n              USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n                Array.size m.buckets.val) }.buckets"}, {"tactic": "have \u27e8x, hx\u2081, hx\u2082\u27e9 := h\u2081", "annotated_tactic": ["have \u27e8x, hx\u2081, hx\u2082\u27e9 := h\u2081", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nk : \u03b1\nv : \u03b2\nh : Buckets.WF m.buckets\nc\u271d : Bool\nh\u2081 :\n  \u2203 x,\n    x \u2208\n        AssocList.toList\n          m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] \u2227\n      (x.fst == k) = true\n\u22a2 Buckets.WF\n    { size := m.size,\n        buckets :=\n          Buckets.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n            (AssocList.replace k v\n              m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n            (_ :\n              USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n                Array.size m.buckets.val) }.buckets", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nk : \u03b1\nv : \u03b2\nh : Buckets.WF m.buckets\nc\u271d : Bool\nh\u2081 :\n  \u2203 x,\n    x \u2208\n        AssocList.toList\n          m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] \u2227\n      (x.fst == k) = true\nx : \u03b1 \u00d7 \u03b2\nhx\u2081 :\n  x \u2208\n    AssocList.toList m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val]\nhx\u2082 : (x.fst == k) = true\n\u22a2 Buckets.WF\n    { size := m.size,\n        buckets :=\n          Buckets.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n            (AssocList.replace k v\n              m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n            (_ :\n              USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n                Array.size m.buckets.val) }.buckets"}, {"tactic": "refine h.update (fun H => ?_) (fun H a h => ?_)", "annotated_tactic": ["refine h.update (fun H => ?_) (fun H a h => ?_)", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nk : \u03b1\nv : \u03b2\nh : Buckets.WF m.buckets\nc\u271d : Bool\nh\u2081 :\n  \u2203 x,\n    x \u2208\n        AssocList.toList\n          m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] \u2227\n      (x.fst == k) = true\nx : \u03b1 \u00d7 \u03b2\nhx\u2081 :\n  x \u2208\n    AssocList.toList m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val]\nhx\u2082 : (x.fst == k) = true\n\u22a2 Buckets.WF\n    { size := m.size,\n        buckets :=\n          Buckets.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n            (AssocList.replace k v\n              m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n            (_ :\n              USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n                Array.size m.buckets.val) }.buckets", "state_after": "case refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : BEq \u03b1\ninst\u271d\u00b2 : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nk : \u03b1\nv : \u03b2\nh : Buckets.WF m.buckets\nc\u271d : Bool\nh\u2081 :\n  \u2203 x,\n    x \u2208\n        AssocList.toList\n          m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] \u2227\n      (x.fst == k) = true\nx : \u03b1 \u00d7 \u03b2\nhx\u2081 :\n  x \u2208\n    AssocList.toList m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val]\nhx\u2082 : (x.fst == k) = true\ninst\u271d\u00b9 : PartialEquivBEq \u03b1\ninst\u271d : LawfulHashable \u03b1\nH :\n  List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true)\n    (AssocList.toList m.buckets.val[(mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n\u22a2 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true)\n    (AssocList.toList\n      (AssocList.replace k v\n        m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val]))\n\ncase refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nk : \u03b1\nv : \u03b2\nh\u271d : Buckets.WF m.buckets\nc\u271d : Bool\nh\u2081 :\n  \u2203 x,\n    x \u2208\n        AssocList.toList\n          m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] \u2227\n      (x.fst == k) = true\nx : \u03b1 \u00d7 \u03b2\nhx\u2081 :\n  x \u2208\n    AssocList.toList m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val]\nhx\u2082 : (x.fst == k) = true\nH :\n  AssocList.All\n    (fun k_1 x =>\n      USize.toNat (UInt64.toUSize (hash k_1) % Array.size m.buckets.val) =\n        USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val)\n    m.buckets.val[(mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val]\na : \u03b1 \u00d7 \u03b2\nh :\n  a \u2208\n    AssocList.toList\n      (AssocList.replace k v\n        m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n\u22a2 (fun k_1 x =>\n      USize.toNat (UInt64.toUSize (hash k_1) % Array.size m.buckets.val) =\n        USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val)\n    a.fst a.snd"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : BEq \u03b1\ninst\u271d\u00b2 : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nk : \u03b1\nv : \u03b2\nh : Buckets.WF m.buckets\nc\u271d : Bool\nh\u2081 :\n  \u2203 x,\n    x \u2208\n        AssocList.toList\n          m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] \u2227\n      (x.fst == k) = true\nx : \u03b1 \u00d7 \u03b2\nhx\u2081 :\n  x \u2208\n    AssocList.toList m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val]\nhx\u2082 : (x.fst == k) = true\ninst\u271d\u00b9 : PartialEquivBEq \u03b1\ninst\u271d : LawfulHashable \u03b1\nH :\n  List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true)\n    (AssocList.toList m.buckets.val[(mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n\u22a2 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true)\n    (AssocList.toList\n      (AssocList.replace k v\n        m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val]))", "state_after": "case refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : BEq \u03b1\ninst\u271d\u00b2 : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nk : \u03b1\nv : \u03b2\nh : Buckets.WF m.buckets\nc\u271d : Bool\nh\u2081 :\n  \u2203 x,\n    x \u2208\n        AssocList.toList\n          m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] \u2227\n      (x.fst == k) = true\nx : \u03b1 \u00d7 \u03b2\nhx\u2081 :\n  x \u2208\n    AssocList.toList m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val]\nhx\u2082 : (x.fst == k) = true\ninst\u271d\u00b9 : PartialEquivBEq \u03b1\ninst\u271d : LawfulHashable \u03b1\nH :\n  List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true)\n    (AssocList.toList m.buckets.val[(mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n\u22a2 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true)\n    (List.replaceF (fun x => bif x.fst == k then some (k, v) else none)\n      (AssocList.toList\n        m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val]))"}, {"tactic": "exact pairwise_replaceF H", "annotated_tactic": ["exact <a>pairwise_replaceF</a> H", [{"full_name": "_private.\u00ablake-packages\u00bb.std.Std.Data.HashMap.WF.0.Std.HashMap.Imp.pairwise_replaceF", "def_path": "lake-packages/std/Std/Data/HashMap/WF.lean", "def_pos": [203, 17], "def_end_pos": [203, 34]}]], "state_before": "case refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : BEq \u03b1\ninst\u271d\u00b2 : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nk : \u03b1\nv : \u03b2\nh : Buckets.WF m.buckets\nc\u271d : Bool\nh\u2081 :\n  \u2203 x,\n    x \u2208\n        AssocList.toList\n          m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] \u2227\n      (x.fst == k) = true\nx : \u03b1 \u00d7 \u03b2\nhx\u2081 :\n  x \u2208\n    AssocList.toList m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val]\nhx\u2082 : (x.fst == k) = true\ninst\u271d\u00b9 : PartialEquivBEq \u03b1\ninst\u271d : LawfulHashable \u03b1\nH :\n  List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true)\n    (AssocList.toList m.buckets.val[(mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n\u22a2 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true)\n    (List.replaceF (fun x => bif x.fst == k then some (k, v) else none)\n      (AssocList.toList\n        m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val]))", "state_after": "no goals"}, {"tactic": "simp [AssocList.All] at H h \u22a2", "annotated_tactic": ["simp [<a>AssocList.All</a>] at H h \u22a2", [{"full_name": "Std.AssocList.All", "def_path": "lake-packages/std/Std/Data/AssocList.lean", "def_pos": [145, 5], "def_end_pos": [145, 8]}]], "state_before": "case refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nk : \u03b1\nv : \u03b2\nh\u271d : Buckets.WF m.buckets\nc\u271d : Bool\nh\u2081 :\n  \u2203 x,\n    x \u2208\n        AssocList.toList\n          m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] \u2227\n      (x.fst == k) = true\nx : \u03b1 \u00d7 \u03b2\nhx\u2081 :\n  x \u2208\n    AssocList.toList m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val]\nhx\u2082 : (x.fst == k) = true\nH :\n  AssocList.All\n    (fun k_1 x =>\n      USize.toNat (UInt64.toUSize (hash k_1) % Array.size m.buckets.val) =\n        USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val)\n    m.buckets.val[(mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val]\na : \u03b1 \u00d7 \u03b2\nh :\n  a \u2208\n    AssocList.toList\n      (AssocList.replace k v\n        m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n\u22a2 (fun k_1 x =>\n      USize.toNat (UInt64.toUSize (hash k_1) % Array.size m.buckets.val) =\n        USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val)\n    a.fst a.snd", "state_after": "case refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nk : \u03b1\nv : \u03b2\nh\u271d : Buckets.WF m.buckets\nc\u271d : Bool\nh\u2081 :\n  \u2203 x,\n    x \u2208\n        AssocList.toList\n          m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] \u2227\n      (x.fst == k) = true\nx : \u03b1 \u00d7 \u03b2\nhx\u2081 :\n  x \u2208\n    AssocList.toList m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val]\nhx\u2082 : (x.fst == k) = true\nH :\n  \u2200 (a : \u03b1 \u00d7 \u03b2),\n    a \u2208\n        AssocList.toList\n          m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] \u2192\n      USize.toNat (UInt64.toUSize (hash a.fst) % Array.size m.buckets.val) =\n        USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\na : \u03b1 \u00d7 \u03b2\nh :\n  a \u2208\n    List.replaceF (fun x => bif x.fst == k then some (k, v) else none)\n      (AssocList.toList\n        m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n\u22a2 USize.toNat (UInt64.toUSize (hash a.fst) % Array.size m.buckets.val) =\n    USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val"}, {"tactic": "match mem_replaceF h with\n| .inl rfl => rfl\n| .inr h => exact H _ h", "annotated_tactic": ["match <a>mem_replaceF</a> h with\n      | .inl <a>rfl</a> => rfl\n      | .inr h => exact H _ h", [{"full_name": "_private.\u00ablake-packages\u00bb.std.Std.Data.HashMap.WF.0.Std.HashMap.Imp.mem_replaceF", "def_path": "lake-packages/std/Std/Data/HashMap/WF.lean", "def_pos": [189, 17], "def_end_pos": [189, 29]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "case refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nk : \u03b1\nv : \u03b2\nh\u271d : Buckets.WF m.buckets\nc\u271d : Bool\nh\u2081 :\n  \u2203 x,\n    x \u2208\n        AssocList.toList\n          m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] \u2227\n      (x.fst == k) = true\nx : \u03b1 \u00d7 \u03b2\nhx\u2081 :\n  x \u2208\n    AssocList.toList m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val]\nhx\u2082 : (x.fst == k) = true\nH :\n  \u2200 (a : \u03b1 \u00d7 \u03b2),\n    a \u2208\n        AssocList.toList\n          m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] \u2192\n      USize.toNat (UInt64.toUSize (hash a.fst) % Array.size m.buckets.val) =\n        USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\na : \u03b1 \u00d7 \u03b2\nh :\n  a \u2208\n    List.replaceF (fun x => bif x.fst == k then some (k, v) else none)\n      (AssocList.toList\n        m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n\u22a2 USize.toNat (UInt64.toUSize (hash a.fst) % Array.size m.buckets.val) =\n    USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val", "state_after": "no goals"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nv : \u03b2\nh\u271d : Buckets.WF m.buckets\nc\u271d : Bool\nx a : \u03b1 \u00d7 \u03b2\nh\u2081 :\n  \u2203 x,\n    x \u2208\n        AssocList.toList\n          m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash a.fst))).val] \u2227\n      (x.fst == a.fst) = true\nhx\u2081 :\n  x \u2208\n    AssocList.toList\n      m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash a.fst))).val]\nhx\u2082 : (x.fst == a.fst) = true\nH :\n  \u2200 (a_1 : \u03b1 \u00d7 \u03b2),\n    a_1 \u2208\n        AssocList.toList\n          m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash a.fst))).val] \u2192\n      USize.toNat (UInt64.toUSize (hash a_1.fst) % Array.size m.buckets.val) =\n        USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash a.fst))).val\nh :\n  a \u2208\n    List.replaceF (fun x => bif x.fst == a.fst then some (a.fst, v) else none)\n      (AssocList.toList\n        m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash a.fst))).val])\n\u22a2 USize.toNat (UInt64.toUSize (hash a.fst) % Array.size m.buckets.val) =\n    USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash a.fst))).val", "state_after": "no goals"}, {"tactic": "exact H _ h", "annotated_tactic": ["exact H _ h", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nv : \u03b2\nh\u271d\u00b9 : Buckets.WF m.buckets\nc\u271d : Bool\nx a : \u03b1 \u00d7 \u03b2\nk : \u03b1\nh :\n  a \u2208\n    AssocList.toList m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val]\nh\u2081 :\n  \u2203 x,\n    x \u2208\n        AssocList.toList\n          m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] \u2227\n      (x.fst == k) = true\nhx\u2081 :\n  x \u2208\n    AssocList.toList m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val]\nhx\u2082 : (x.fst == k) = true\nH :\n  \u2200 (a : \u03b1 \u00d7 \u03b2),\n    a \u2208\n        AssocList.toList\n          m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] \u2192\n      USize.toNat (UInt64.toUSize (hash a.fst) % Array.size m.buckets.val) =\n        USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\nh\u271d :\n  a \u2208\n    List.replaceF (fun x => bif x.fst == k then some (k, v) else none)\n      (AssocList.toList\n        m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n\u22a2 USize.toNat (UInt64.toUSize (hash a.fst) % Array.size m.buckets.val) =\n    USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val", "state_after": "no goals"}, {"tactic": "rw [Bool.eq_false_iff] at h\u2081", "annotated_tactic": ["rw [<a>Bool.eq_false_iff</a>] at h\u2081", [{"full_name": "Bool.eq_false_iff", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [809, 9], "def_end_pos": [809, 26]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nk : \u03b1\nv : \u03b2\nh : Buckets.WF m.buckets\nc\u271d : Bool\nh\u2081 :\n  AssocList.contains k\n      m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] =\n    false\n\u22a2 Buckets.WF\n    (if numBucketsForCapacity (m.size + 1) \u2264 Array.size m.buckets.val then\n        { size := m.size + 1,\n          buckets :=\n            Buckets.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n              (AssocList.cons k v\n                m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n              (_ :\n                USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n                  Array.size m.buckets.val) }\n      else\n        expand (m.size + 1)\n          (Buckets.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n            (AssocList.cons k v\n              m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n            (_ :\n              USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n                Array.size m.buckets.val))).buckets", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nk : \u03b1\nv : \u03b2\nh : Buckets.WF m.buckets\nc\u271d : Bool\nh\u2081 :\n  AssocList.contains k\n      m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] \u2260\n    true\n\u22a2 Buckets.WF\n    (if numBucketsForCapacity (m.size + 1) \u2264 Array.size m.buckets.val then\n        { size := m.size + 1,\n          buckets :=\n            Buckets.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n              (AssocList.cons k v\n                m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n              (_ :\n                USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n                  Array.size m.buckets.val) }\n      else\n        expand (m.size + 1)\n          (Buckets.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n            (AssocList.cons k v\n              m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n            (_ :\n              USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n                Array.size m.buckets.val))).buckets"}, {"tactic": "simp at h\u2081", "annotated_tactic": ["simp at h\u2081", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nk : \u03b1\nv : \u03b2\nh : Buckets.WF m.buckets\nc\u271d : Bool\nh\u2081 :\n  AssocList.contains k\n      m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] \u2260\n    true\n\u22a2 Buckets.WF\n    (if numBucketsForCapacity (m.size + 1) \u2264 Array.size m.buckets.val then\n        { size := m.size + 1,\n          buckets :=\n            Buckets.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n              (AssocList.cons k v\n                m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n              (_ :\n                USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n                  Array.size m.buckets.val) }\n      else\n        expand (m.size + 1)\n          (Buckets.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n            (AssocList.cons k v\n              m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n            (_ :\n              USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n                Array.size m.buckets.val))).buckets", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nk : \u03b1\nv : \u03b2\nh : Buckets.WF m.buckets\nc\u271d : Bool\nh\u2081 :\n  \u2200 (x : \u03b1 \u00d7 \u03b2),\n    x \u2208\n        AssocList.toList\n          m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] \u2192\n      \u00ac(x.fst == k) = true\n\u22a2 Buckets.WF\n    (if numBucketsForCapacity (m.size + 1) \u2264 Array.size m.buckets.val then\n        { size := m.size + 1,\n          buckets :=\n            Buckets.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n              (AssocList.cons k v\n                m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n              (_ :\n                USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n                  Array.size m.buckets.val) }\n      else\n        expand (m.size + 1)\n          (Buckets.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n            (AssocList.cons k v\n              m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n            (_ :\n              USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n                Array.size m.buckets.val))).buckets"}, {"tactic": "suffices _ by split <;> [exact this; refine expand_WF this]", "annotated_tactic": ["suffices _ by split <;> [exact this; refine <a>expand_WF</a> this]", [{"full_name": "Std.HashMap.Imp.expand_WF", "def_path": "lake-packages/std/Std/Data/HashMap/WF.lean", "def_pos": [143, 9], "def_end_pos": [143, 18]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nk : \u03b1\nv : \u03b2\nh : Buckets.WF m.buckets\nc\u271d : Bool\nh\u2081 :\n  \u2200 (x : \u03b1 \u00d7 \u03b2),\n    x \u2208\n        AssocList.toList\n          m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] \u2192\n      \u00ac(x.fst == k) = true\n\u22a2 Buckets.WF\n    (if numBucketsForCapacity (m.size + 1) \u2264 Array.size m.buckets.val then\n        { size := m.size + 1,\n          buckets :=\n            Buckets.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n              (AssocList.cons k v\n                m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n              (_ :\n                USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n                  Array.size m.buckets.val) }\n      else\n        expand (m.size + 1)\n          (Buckets.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n            (AssocList.cons k v\n              m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n            (_ :\n              USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n                Array.size m.buckets.val))).buckets", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nk : \u03b1\nv : \u03b2\nh : Buckets.WF m.buckets\nc\u271d : Bool\nh\u2081 :\n  \u2200 (x : \u03b1 \u00d7 \u03b2),\n    x \u2208\n        AssocList.toList\n          m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] \u2192\n      \u00ac(x.fst == k) = true\n\u22a2 Buckets.WF\n    { size := m.size + 1,\n        buckets :=\n          Buckets.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n            (AssocList.cons k v\n              m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n            (_ :\n              USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n                Array.size m.buckets.val) }.buckets"}, {"tactic": "refine h.update (.cons ?_) (fun H a h => ?_)", "annotated_tactic": ["refine h.update (.cons ?_) (fun H a h => ?_)", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nk : \u03b1\nv : \u03b2\nh : Buckets.WF m.buckets\nc\u271d : Bool\nh\u2081 :\n  \u2200 (x : \u03b1 \u00d7 \u03b2),\n    x \u2208\n        AssocList.toList\n          m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] \u2192\n      \u00ac(x.fst == k) = true\n\u22a2 Buckets.WF\n    { size := m.size + 1,\n        buckets :=\n          Buckets.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n            (AssocList.cons k v\n              m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n            (_ :\n              USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n                Array.size m.buckets.val) }.buckets", "state_after": "case refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : BEq \u03b1\ninst\u271d\u00b2 : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nk : \u03b1\nv : \u03b2\nh : Buckets.WF m.buckets\nc\u271d : Bool\nh\u2081 :\n  \u2200 (x : \u03b1 \u00d7 \u03b2),\n    x \u2208\n        AssocList.toList\n          m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] \u2192\n      \u00ac(x.fst == k) = true\ninst\u271d\u00b9 : PartialEquivBEq \u03b1\ninst\u271d : LawfulHashable \u03b1\n\u22a2 \u2200 (a' : \u03b1 \u00d7 \u03b2),\n    a' \u2208 AssocList.toList m.buckets.val[(mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] \u2192\n      \u00ac((k, v).fst == a'.fst) = true\n\ncase refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nk : \u03b1\nv : \u03b2\nh\u271d : Buckets.WF m.buckets\nc\u271d : Bool\nh\u2081 :\n  \u2200 (x : \u03b1 \u00d7 \u03b2),\n    x \u2208\n        AssocList.toList\n          m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] \u2192\n      \u00ac(x.fst == k) = true\nH :\n  AssocList.All\n    (fun k_1 x =>\n      USize.toNat (UInt64.toUSize (hash k_1) % Array.size m.buckets.val) =\n        USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val)\n    m.buckets.val[(mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val]\na : \u03b1 \u00d7 \u03b2\nh :\n  a \u2208\n    AssocList.toList\n      (AssocList.cons k v\n        m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n\u22a2 (fun k_1 x =>\n      USize.toNat (UInt64.toUSize (hash k_1) % Array.size m.buckets.val) =\n        USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val)\n    a.fst a.snd"}, {"tactic": "split <;> [exact this; refine expand_WF this]", "annotated_tactic": ["split <;> [exact this; refine <a>expand_WF</a> this]", [{"full_name": "Std.HashMap.Imp.expand_WF", "def_path": "lake-packages/std/Std/Data/HashMap/WF.lean", "def_pos": [143, 9], "def_end_pos": [143, 18]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nk : \u03b1\nv : \u03b2\nh : Buckets.WF m.buckets\nc\u271d : Bool\nh\u2081 :\n  \u2200 (x : \u03b1 \u00d7 \u03b2),\n    x \u2208\n        AssocList.toList\n          m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] \u2192\n      \u00ac(x.fst == k) = true\nthis : ?m.86533\n\u22a2 Buckets.WF\n    (if numBucketsForCapacity (m.size + 1) \u2264 Array.size m.buckets.val then\n        { size := m.size + 1,\n          buckets :=\n            Buckets.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n              (AssocList.cons k v\n                m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n              (_ :\n                USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n                  Array.size m.buckets.val) }\n      else\n        expand (m.size + 1)\n          (Buckets.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n            (AssocList.cons k v\n              m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n            (_ :\n              USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n                Array.size m.buckets.val))).buckets", "state_after": "no goals"}, {"tactic": "exact fun a h h' => h\u2081 a h (PartialEquivBEq.symm h')", "annotated_tactic": ["exact fun a h h' => h\u2081 a h (<a>PartialEquivBEq.symm</a> h')", [{"full_name": "PartialEquivBEq.symm", "def_path": "lake-packages/std/Std/Classes/BEq.lean", "def_pos": [16, 3], "def_end_pos": [16, 7]}]], "state_before": "case refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : BEq \u03b1\ninst\u271d\u00b2 : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nk : \u03b1\nv : \u03b2\nh : Buckets.WF m.buckets\nc\u271d : Bool\nh\u2081 :\n  \u2200 (x : \u03b1 \u00d7 \u03b2),\n    x \u2208\n        AssocList.toList\n          m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] \u2192\n      \u00ac(x.fst == k) = true\ninst\u271d\u00b9 : PartialEquivBEq \u03b1\ninst\u271d : LawfulHashable \u03b1\n\u22a2 \u2200 (a' : \u03b1 \u00d7 \u03b2),\n    a' \u2208 AssocList.toList m.buckets.val[(mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] \u2192\n      \u00ac((k, v).fst == a'.fst) = true", "state_after": "no goals"}, {"tactic": "cases h with\n| head => rfl\n| tail _ h => exact H _ h", "annotated_tactic": ["cases h with\n      | <a>head</a> => rfl\n      | <a>tail</a> _ h => exact H _ h", [{"full_name": "List.Mem.head", "def_path": "lake-packages/lean4/src/lean/Init/Data/List/Basic.lean", "def_pos": [352, 5], "def_end_pos": [352, 9]}, {"full_name": "List.Mem.tail", "def_path": "lake-packages/lean4/src/lean/Init/Data/List/Basic.lean", "def_pos": [354, 5], "def_end_pos": [354, 9]}]], "state_before": "case refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nk : \u03b1\nv : \u03b2\nh\u271d : Buckets.WF m.buckets\nc\u271d : Bool\nh\u2081 :\n  \u2200 (x : \u03b1 \u00d7 \u03b2),\n    x \u2208\n        AssocList.toList\n          m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] \u2192\n      \u00ac(x.fst == k) = true\nH :\n  AssocList.All\n    (fun k_1 x =>\n      USize.toNat (UInt64.toUSize (hash k_1) % Array.size m.buckets.val) =\n        USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val)\n    m.buckets.val[(mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val]\na : \u03b1 \u00d7 \u03b2\nh :\n  a \u2208\n    AssocList.toList\n      (AssocList.cons k v\n        m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n\u22a2 (fun k_1 x =>\n      USize.toNat (UInt64.toUSize (hash k_1) % Array.size m.buckets.val) =\n        USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val)\n    a.fst a.snd", "state_after": "no goals"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case refine_2.head\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nk : \u03b1\nv : \u03b2\nh : Buckets.WF m.buckets\nc\u271d : Bool\nh\u2081 :\n  \u2200 (x : \u03b1 \u00d7 \u03b2),\n    x \u2208\n        AssocList.toList\n          m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] \u2192\n      \u00ac(x.fst == k) = true\nH :\n  AssocList.All\n    (fun k_1 x =>\n      USize.toNat (UInt64.toUSize (hash k_1) % Array.size m.buckets.val) =\n        USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val)\n    m.buckets.val[(mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val]\n\u22a2 USize.toNat (UInt64.toUSize (hash (k, v).fst) % Array.size m.buckets.val) =\n    USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val", "state_after": "no goals"}, {"tactic": "exact H _ h", "annotated_tactic": ["exact H _ h", []], "state_before": "case refine_2.tail\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nk : \u03b1\nv : \u03b2\nh\u271d : Buckets.WF m.buckets\nc\u271d : Bool\nh\u2081 :\n  \u2200 (x : \u03b1 \u00d7 \u03b2),\n    x \u2208\n        AssocList.toList\n          m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] \u2192\n      \u00ac(x.fst == k) = true\nH :\n  AssocList.All\n    (fun k_1 x =>\n      USize.toNat (UInt64.toUSize (hash k_1) % Array.size m.buckets.val) =\n        USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val)\n    m.buckets.val[(mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val]\na : \u03b1 \u00d7 \u03b2\nh :\n  List.Mem a\n    (AssocList.toList\n      m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n\u22a2 USize.toNat (UInt64.toUSize (hash a.fst) % Array.size m.buckets.val) =\n    USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Haar/Basic.lean", "full_name": "MeasureTheory.Measure.div_mem_nhds_one_of_haar_pos", "start": [772, 1], "end": [813, 71], "traced_tactics": [{"tactic": "obtain \u27e8L, hL, hLE, hLpos, hLtop\u27e9 : \u2203 L : Set G, MeasurableSet L \u2227 L \u2286 E \u2227 0 < \u03bc L \u2227 \u03bc L < \u221e :=\n  exists_subset_measure_lt_top hE hEpos", "annotated_tactic": ["obtain \u27e8L, hL, hLE, hLpos, hLtop\u27e9 : \u2203 L : <a>Set</a> G, <a>MeasurableSet</a> L \u2227 L \u2286 E \u2227 0 < \u03bc L \u2227 \u03bc L < \u221e :=\n    <a>exists_subset_measure_lt_top</a> hE hEpos", [{"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}, {"full_name": "MeasurableSet", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [64, 5], "def_end_pos": [64, 18]}, {"full_name": "MeasureTheory.Measure.exists_subset_measure_lt_top", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3396, 9], "def_end_pos": [3396, 37]}]], "state_before": "G : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : LocallyCompactSpace G\nE : Set G\nhE : MeasurableSet E\nhEpos : 0 < \u2191\u2191\u03bc E\n\u22a2 E / E \u2208 \ud835\udcdd 1", "state_after": "case intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : LocallyCompactSpace G\nE : Set G\nhE : MeasurableSet E\nhEpos : 0 < \u2191\u2191\u03bc E\nL : Set G\nhL : MeasurableSet L\nhLE : L \u2286 E\nhLpos : 0 < \u2191\u2191\u03bc L\nhLtop : \u2191\u2191\u03bc L < \u22a4\n\u22a2 E / E \u2208 \ud835\udcdd 1"}, {"tactic": "obtain \u27e8K, hKL, hK, hKpos\u27e9 : \u2203 (K : Set G), K \u2286 L \u2227 IsCompact K \u2227 0 < \u03bc K :=\n  MeasurableSet.exists_lt_isCompact_of_ne_top hL (ne_of_lt hLtop) hLpos", "annotated_tactic": ["obtain \u27e8K, hKL, hK, hKpos\u27e9 : \u2203 (K : <a>Set</a> G), K \u2286 L \u2227 <a>IsCompact</a> K \u2227 0 < \u03bc K :=\n    <a>MeasurableSet.exists_lt_isCompact_of_ne_top</a> hL (<a>ne_of_lt</a> hLtop) hLpos", [{"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}, {"full_name": "IsCompact", "def_path": "Mathlib/Topology/Compactness/Compact.lean", "def_pos": [40, 5], "def_end_pos": [40, 14]}, {"full_name": "MeasurableSet.exists_lt_isCompact_of_ne_top", "def_path": "Mathlib/MeasureTheory/Measure/Regular.lean", "def_pos": [518, 9], "def_end_pos": [518, 59]}, {"full_name": "ne_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [101, 9], "def_end_pos": [101, 17]}]], "state_before": "case intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : LocallyCompactSpace G\nE : Set G\nhE : MeasurableSet E\nhEpos : 0 < \u2191\u2191\u03bc E\nL : Set G\nhL : MeasurableSet L\nhLE : L \u2286 E\nhLpos : 0 < \u2191\u2191\u03bc L\nhLtop : \u2191\u2191\u03bc L < \u22a4\n\u22a2 E / E \u2208 \ud835\udcdd 1", "state_after": "case intro.intro.intro.intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : LocallyCompactSpace G\nE : Set G\nhE : MeasurableSet E\nhEpos : 0 < \u2191\u2191\u03bc E\nL : Set G\nhL : MeasurableSet L\nhLE : L \u2286 E\nhLpos : 0 < \u2191\u2191\u03bc L\nhLtop : \u2191\u2191\u03bc L < \u22a4\nK : Set G\nhKL : K \u2286 L\nhK : IsCompact K\nhKpos : 0 < \u2191\u2191\u03bc K\n\u22a2 E / E \u2208 \ud835\udcdd 1"}, {"tactic": "have hKtop : \u03bc K \u2260 \u221e := by\n  apply ne_top_of_le_ne_top (ne_of_lt hLtop)\n  apply measure_mono hKL", "annotated_tactic": ["have hKtop : \u03bc K \u2260 \u221e := by\n    apply <a>ne_top_of_le_ne_top</a> (<a>ne_of_lt</a> hLtop)\n    apply <a>measure_mono</a> hKL", [{"full_name": "ne_top_of_le_ne_top", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [194, 9], "def_end_pos": [194, 28]}, {"full_name": "ne_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [101, 9], "def_end_pos": [101, 17]}, {"full_name": "MeasureTheory.measure_mono", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [193, 9], "def_end_pos": [193, 21]}]], "state_before": "case intro.intro.intro.intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : LocallyCompactSpace G\nE : Set G\nhE : MeasurableSet E\nhEpos : 0 < \u2191\u2191\u03bc E\nL : Set G\nhL : MeasurableSet L\nhLE : L \u2286 E\nhLpos : 0 < \u2191\u2191\u03bc L\nhLtop : \u2191\u2191\u03bc L < \u22a4\nK : Set G\nhKL : K \u2286 L\nhK : IsCompact K\nhKpos : 0 < \u2191\u2191\u03bc K\n\u22a2 E / E \u2208 \ud835\udcdd 1", "state_after": "case intro.intro.intro.intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : LocallyCompactSpace G\nE : Set G\nhE : MeasurableSet E\nhEpos : 0 < \u2191\u2191\u03bc E\nL : Set G\nhL : MeasurableSet L\nhLE : L \u2286 E\nhLpos : 0 < \u2191\u2191\u03bc L\nhLtop : \u2191\u2191\u03bc L < \u22a4\nK : Set G\nhKL : K \u2286 L\nhK : IsCompact K\nhKpos : 0 < \u2191\u2191\u03bc K\nhKtop : \u2191\u2191\u03bc K \u2260 \u22a4\n\u22a2 E / E \u2208 \ud835\udcdd 1"}, {"tactic": "obtain \u27e8U, hUK, hU, h\u03bcUK\u27e9 : \u2203 (U : Set G), U \u2287 K \u2227 IsOpen U \u2227 \u03bc U < \u03bc K + \u03bc K :=\n  Set.exists_isOpen_lt_add K hKtop hKpos.ne'", "annotated_tactic": ["obtain \u27e8U, hUK, hU, h\u03bcUK\u27e9 : \u2203 (U : <a>Set</a> G), U \u2287 K \u2227 <a>IsOpen</a> U \u2227 \u03bc U < \u03bc K + \u03bc K :=\n    <a>Set.exists_isOpen_lt_add</a> K hKtop hKpos.ne'", [{"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}, {"full_name": "IsOpen", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [101, 5], "def_end_pos": [101, 11]}, {"full_name": "Set.exists_isOpen_lt_add", "def_path": "Mathlib/MeasureTheory/Measure/Regular.lean", "def_pos": [259, 9], "def_end_pos": [259, 40]}]], "state_before": "case intro.intro.intro.intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : LocallyCompactSpace G\nE : Set G\nhE : MeasurableSet E\nhEpos : 0 < \u2191\u2191\u03bc E\nL : Set G\nhL : MeasurableSet L\nhLE : L \u2286 E\nhLpos : 0 < \u2191\u2191\u03bc L\nhLtop : \u2191\u2191\u03bc L < \u22a4\nK : Set G\nhKL : K \u2286 L\nhK : IsCompact K\nhKpos : 0 < \u2191\u2191\u03bc K\nhKtop : \u2191\u2191\u03bc K \u2260 \u22a4\n\u22a2 E / E \u2208 \ud835\udcdd 1", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : LocallyCompactSpace G\nE : Set G\nhE : MeasurableSet E\nhEpos : 0 < \u2191\u2191\u03bc E\nL : Set G\nhL : MeasurableSet L\nhLE : L \u2286 E\nhLpos : 0 < \u2191\u2191\u03bc L\nhLtop : \u2191\u2191\u03bc L < \u22a4\nK : Set G\nhKL : K \u2286 L\nhK : IsCompact K\nhKpos : 0 < \u2191\u2191\u03bc K\nhKtop : \u2191\u2191\u03bc K \u2260 \u22a4\nU : Set G\nhUK : U \u2287 K\nhU : IsOpen U\nh\u03bcUK : \u2191\u2191\u03bc U < \u2191\u2191\u03bc K + \u2191\u2191\u03bc K\n\u22a2 E / E \u2208 \ud835\udcdd 1"}, {"tactic": "obtain \u27e8V, hV1, hVKU\u27e9 : \u2203 V \u2208 \ud835\udcdd (1 : G), V * K \u2286 U :=\n  compact_open_separated_mul_left hK hU hUK", "annotated_tactic": ["obtain \u27e8V, hV1, hVKU\u27e9 : \u2203 V \u2208 \ud835\udcdd (1 : G), V * K \u2286 U :=\n    <a>compact_open_separated_mul_left</a> hK hU hUK", [{"full_name": "compact_open_separated_mul_left", "def_path": "Mathlib/Topology/Algebra/Group/Basic.lean", "def_pos": [1645, 9], "def_end_pos": [1645, 40]}]], "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : LocallyCompactSpace G\nE : Set G\nhE : MeasurableSet E\nhEpos : 0 < \u2191\u2191\u03bc E\nL : Set G\nhL : MeasurableSet L\nhLE : L \u2286 E\nhLpos : 0 < \u2191\u2191\u03bc L\nhLtop : \u2191\u2191\u03bc L < \u22a4\nK : Set G\nhKL : K \u2286 L\nhK : IsCompact K\nhKpos : 0 < \u2191\u2191\u03bc K\nhKtop : \u2191\u2191\u03bc K \u2260 \u22a4\nU : Set G\nhUK : U \u2287 K\nhU : IsOpen U\nh\u03bcUK : \u2191\u2191\u03bc U < \u2191\u2191\u03bc K + \u2191\u2191\u03bc K\n\u22a2 E / E \u2208 \ud835\udcdd 1", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : LocallyCompactSpace G\nE : Set G\nhE : MeasurableSet E\nhEpos : 0 < \u2191\u2191\u03bc E\nL : Set G\nhL : MeasurableSet L\nhLE : L \u2286 E\nhLpos : 0 < \u2191\u2191\u03bc L\nhLtop : \u2191\u2191\u03bc L < \u22a4\nK : Set G\nhKL : K \u2286 L\nhK : IsCompact K\nhKpos : 0 < \u2191\u2191\u03bc K\nhKtop : \u2191\u2191\u03bc K \u2260 \u22a4\nU : Set G\nhUK : U \u2287 K\nhU : IsOpen U\nh\u03bcUK : \u2191\u2191\u03bc U < \u2191\u2191\u03bc K + \u2191\u2191\u03bc K\nV : Set G\nhV1 : V \u2208 \ud835\udcdd 1\nhVKU : V * K \u2286 U\n\u22a2 E / E \u2208 \ud835\udcdd 1"}, {"tactic": "have hv : \u2200 v : G, v \u2208 V \u2192 \u00acDisjoint ({v} * K) K := by\n  intro v hv hKv\n  have hKvsub : {v} * K \u222a K \u2286 U := by\n    apply Set.union_subset _ hUK\n    apply _root_.subset_trans _ hVKU\n    apply Set.mul_subset_mul _ (Set.Subset.refl K)\n    simp only [Set.singleton_subset_iff, hv]\n  replace hKvsub := @measure_mono _ _ \u03bc _ _ hKvsub\n  have hcontr := lt_of_le_of_lt hKvsub h\u03bcUK\n  rw [measure_union hKv (IsCompact.measurableSet hK)] at hcontr\n  have hKtranslate : \u03bc ({v} * K) = \u03bc K := by\n    simp only [singleton_mul, image_mul_left, measure_preimage_mul]\n  rw [hKtranslate, lt_self_iff_false] at hcontr\n  assumption", "annotated_tactic": ["have hv : \u2200 v : G, v \u2208 V \u2192 \u00ac<a>Disjoint</a> ({v} * K) K := by\n    intro v hv hKv\n    have hKvsub : {v} * K \u222a K \u2286 U := by\n      apply <a>Set.union_subset</a> _ hUK\n      apply <a>_root_.subset_trans</a> _ hVKU\n      apply <a>Set.mul_subset_mul</a> _ (<a>Set.Subset.refl</a> K)\n      simp only [<a>Set.singleton_subset_iff</a>, hv]\n    replace hKvsub := @<a>measure_mono</a> _ _ \u03bc _ _ hKvsub\n    have hcontr := <a>lt_of_le_of_lt</a> hKvsub h\u03bcUK\n    rw [<a>measure_union</a> hKv (<a>IsCompact.measurableSet</a> hK)] at hcontr\n    have hKtranslate : \u03bc ({v} * K) = \u03bc K := by\n      simp only [<a>singleton_mul</a>, <a>image_mul_left</a>, <a>measure_preimage_mul</a>]\n    rw [hKtranslate, <a>lt_self_iff_false</a>] at hcontr\n    assumption", [{"full_name": "Disjoint", "def_path": "Mathlib/Order/Disjoint.lean", "def_pos": [41, 5], "def_end_pos": [41, 13]}, {"full_name": "Set.union_subset", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [836, 9], "def_end_pos": [836, 21]}, {"full_name": "subset_trans", "def_path": "Mathlib/Order/RelClasses.lean", "def_pos": [643, 7], "def_end_pos": [643, 19]}, {"full_name": "Set.mul_subset_mul", "def_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "def_pos": [416, 9], "def_end_pos": [416, 23]}, {"full_name": "Set.Subset.refl", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [354, 9], "def_end_pos": [354, 20]}, {"full_name": "Set.singleton_subset_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1330, 9], "def_end_pos": [1330, 29]}, {"full_name": "MeasureTheory.measure_mono", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [193, 9], "def_end_pos": [193, 21]}, {"full_name": "lt_of_le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [122, 9], "def_end_pos": [122, 23]}, {"full_name": "MeasureTheory.measure_union", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [124, 9], "def_end_pos": [124, 22]}, {"full_name": "IsCompact.measurableSet", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [344, 9], "def_end_pos": [344, 32]}, {"full_name": "Set.singleton_mul", "def_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "def_pos": [403, 9], "def_end_pos": [403, 22]}, {"full_name": "Set.image_mul_left", "def_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "def_pos": [1199, 9], "def_end_pos": [1199, 23]}, {"full_name": "MeasureTheory.measure_preimage_mul", "def_path": "Mathlib/MeasureTheory/Group/Measure.lean", "def_pos": [320, 9], "def_end_pos": [320, 29]}, {"full_name": "lt_self_iff_false", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [175, 9], "def_end_pos": [175, 26]}]], "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : LocallyCompactSpace G\nE : Set G\nhE : MeasurableSet E\nhEpos : 0 < \u2191\u2191\u03bc E\nL : Set G\nhL : MeasurableSet L\nhLE : L \u2286 E\nhLpos : 0 < \u2191\u2191\u03bc L\nhLtop : \u2191\u2191\u03bc L < \u22a4\nK : Set G\nhKL : K \u2286 L\nhK : IsCompact K\nhKpos : 0 < \u2191\u2191\u03bc K\nhKtop : \u2191\u2191\u03bc K \u2260 \u22a4\nU : Set G\nhUK : U \u2287 K\nhU : IsOpen U\nh\u03bcUK : \u2191\u2191\u03bc U < \u2191\u2191\u03bc K + \u2191\u2191\u03bc K\nV : Set G\nhV1 : V \u2208 \ud835\udcdd 1\nhVKU : V * K \u2286 U\n\u22a2 E / E \u2208 \ud835\udcdd 1", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : LocallyCompactSpace G\nE : Set G\nhE : MeasurableSet E\nhEpos : 0 < \u2191\u2191\u03bc E\nL : Set G\nhL : MeasurableSet L\nhLE : L \u2286 E\nhLpos : 0 < \u2191\u2191\u03bc L\nhLtop : \u2191\u2191\u03bc L < \u22a4\nK : Set G\nhKL : K \u2286 L\nhK : IsCompact K\nhKpos : 0 < \u2191\u2191\u03bc K\nhKtop : \u2191\u2191\u03bc K \u2260 \u22a4\nU : Set G\nhUK : U \u2287 K\nhU : IsOpen U\nh\u03bcUK : \u2191\u2191\u03bc U < \u2191\u2191\u03bc K + \u2191\u2191\u03bc K\nV : Set G\nhV1 : V \u2208 \ud835\udcdd 1\nhVKU : V * K \u2286 U\nhv : \u2200 (v : G), v \u2208 V \u2192 \u00acDisjoint ({v} * K) K\n\u22a2 E / E \u2208 \ud835\udcdd 1"}, {"tactic": "suffices V \u2286 E / E from Filter.mem_of_superset hV1 this", "annotated_tactic": ["suffices V \u2286 E / E from <a>Filter.mem_of_superset</a> hV1 this", [{"full_name": "Filter.mem_of_superset", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [152, 9], "def_end_pos": [152, 24]}]], "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : LocallyCompactSpace G\nE : Set G\nhE : MeasurableSet E\nhEpos : 0 < \u2191\u2191\u03bc E\nL : Set G\nhL : MeasurableSet L\nhLE : L \u2286 E\nhLpos : 0 < \u2191\u2191\u03bc L\nhLtop : \u2191\u2191\u03bc L < \u22a4\nK : Set G\nhKL : K \u2286 L\nhK : IsCompact K\nhKpos : 0 < \u2191\u2191\u03bc K\nhKtop : \u2191\u2191\u03bc K \u2260 \u22a4\nU : Set G\nhUK : U \u2287 K\nhU : IsOpen U\nh\u03bcUK : \u2191\u2191\u03bc U < \u2191\u2191\u03bc K + \u2191\u2191\u03bc K\nV : Set G\nhV1 : V \u2208 \ud835\udcdd 1\nhVKU : V * K \u2286 U\nhv : \u2200 (v : G), v \u2208 V \u2192 \u00acDisjoint ({v} * K) K\n\u22a2 E / E \u2208 \ud835\udcdd 1", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : LocallyCompactSpace G\nE : Set G\nhE : MeasurableSet E\nhEpos : 0 < \u2191\u2191\u03bc E\nL : Set G\nhL : MeasurableSet L\nhLE : L \u2286 E\nhLpos : 0 < \u2191\u2191\u03bc L\nhLtop : \u2191\u2191\u03bc L < \u22a4\nK : Set G\nhKL : K \u2286 L\nhK : IsCompact K\nhKpos : 0 < \u2191\u2191\u03bc K\nhKtop : \u2191\u2191\u03bc K \u2260 \u22a4\nU : Set G\nhUK : U \u2287 K\nhU : IsOpen U\nh\u03bcUK : \u2191\u2191\u03bc U < \u2191\u2191\u03bc K + \u2191\u2191\u03bc K\nV : Set G\nhV1 : V \u2208 \ud835\udcdd 1\nhVKU : V * K \u2286 U\nhv : \u2200 (v : G), v \u2208 V \u2192 \u00acDisjoint ({v} * K) K\n\u22a2 V \u2286 E / E"}, {"tactic": "intro v hvV", "annotated_tactic": ["intro v hvV", []], "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : LocallyCompactSpace G\nE : Set G\nhE : MeasurableSet E\nhEpos : 0 < \u2191\u2191\u03bc E\nL : Set G\nhL : MeasurableSet L\nhLE : L \u2286 E\nhLpos : 0 < \u2191\u2191\u03bc L\nhLtop : \u2191\u2191\u03bc L < \u22a4\nK : Set G\nhKL : K \u2286 L\nhK : IsCompact K\nhKpos : 0 < \u2191\u2191\u03bc K\nhKtop : \u2191\u2191\u03bc K \u2260 \u22a4\nU : Set G\nhUK : U \u2287 K\nhU : IsOpen U\nh\u03bcUK : \u2191\u2191\u03bc U < \u2191\u2191\u03bc K + \u2191\u2191\u03bc K\nV : Set G\nhV1 : V \u2208 \ud835\udcdd 1\nhVKU : V * K \u2286 U\nhv : \u2200 (v : G), v \u2208 V \u2192 \u00acDisjoint ({v} * K) K\n\u22a2 V \u2286 E / E", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : LocallyCompactSpace G\nE : Set G\nhE : MeasurableSet E\nhEpos : 0 < \u2191\u2191\u03bc E\nL : Set G\nhL : MeasurableSet L\nhLE : L \u2286 E\nhLpos : 0 < \u2191\u2191\u03bc L\nhLtop : \u2191\u2191\u03bc L < \u22a4\nK : Set G\nhKL : K \u2286 L\nhK : IsCompact K\nhKpos : 0 < \u2191\u2191\u03bc K\nhKtop : \u2191\u2191\u03bc K \u2260 \u22a4\nU : Set G\nhUK : U \u2287 K\nhU : IsOpen U\nh\u03bcUK : \u2191\u2191\u03bc U < \u2191\u2191\u03bc K + \u2191\u2191\u03bc K\nV : Set G\nhV1 : V \u2208 \ud835\udcdd 1\nhVKU : V * K \u2286 U\nhv : \u2200 (v : G), v \u2208 V \u2192 \u00acDisjoint ({v} * K) K\nv : G\nhvV : v \u2208 V\n\u22a2 v \u2208 E / E"}, {"tactic": "obtain \u27e8x, hxK, hxvK\u27e9 : \u2203 x : G, x \u2208 {v} * K \u2227 x \u2208 K := Set.not_disjoint_iff.1 (hv v hvV)", "annotated_tactic": ["obtain \u27e8x, hxK, hxvK\u27e9 : \u2203 x : G, x \u2208 {v} * K \u2227 x \u2208 K := <a>Set.not_disjoint_iff</a>.1 (hv v hvV)", [{"full_name": "Set.not_disjoint_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1553, 7], "def_end_pos": [1553, 23]}]], "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : LocallyCompactSpace G\nE : Set G\nhE : MeasurableSet E\nhEpos : 0 < \u2191\u2191\u03bc E\nL : Set G\nhL : MeasurableSet L\nhLE : L \u2286 E\nhLpos : 0 < \u2191\u2191\u03bc L\nhLtop : \u2191\u2191\u03bc L < \u22a4\nK : Set G\nhKL : K \u2286 L\nhK : IsCompact K\nhKpos : 0 < \u2191\u2191\u03bc K\nhKtop : \u2191\u2191\u03bc K \u2260 \u22a4\nU : Set G\nhUK : U \u2287 K\nhU : IsOpen U\nh\u03bcUK : \u2191\u2191\u03bc U < \u2191\u2191\u03bc K + \u2191\u2191\u03bc K\nV : Set G\nhV1 : V \u2208 \ud835\udcdd 1\nhVKU : V * K \u2286 U\nhv : \u2200 (v : G), v \u2208 V \u2192 \u00acDisjoint ({v} * K) K\nv : G\nhvV : v \u2208 V\n\u22a2 v \u2208 E / E", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : LocallyCompactSpace G\nE : Set G\nhE : MeasurableSet E\nhEpos : 0 < \u2191\u2191\u03bc E\nL : Set G\nhL : MeasurableSet L\nhLE : L \u2286 E\nhLpos : 0 < \u2191\u2191\u03bc L\nhLtop : \u2191\u2191\u03bc L < \u22a4\nK : Set G\nhKL : K \u2286 L\nhK : IsCompact K\nhKpos : 0 < \u2191\u2191\u03bc K\nhKtop : \u2191\u2191\u03bc K \u2260 \u22a4\nU : Set G\nhUK : U \u2287 K\nhU : IsOpen U\nh\u03bcUK : \u2191\u2191\u03bc U < \u2191\u2191\u03bc K + \u2191\u2191\u03bc K\nV : Set G\nhV1 : V \u2208 \ud835\udcdd 1\nhVKU : V * K \u2286 U\nhv : \u2200 (v : G), v \u2208 V \u2192 \u00acDisjoint ({v} * K) K\nv : G\nhvV : v \u2208 V\nx : G\nhxK : x \u2208 {v} * K\nhxvK : x \u2208 K\n\u22a2 v \u2208 E / E"}, {"tactic": "refine' \u27e8x, v\u207b\u00b9 * x, hLE (hKL hxvK), _, _\u27e9", "annotated_tactic": ["refine' \u27e8x, v\u207b\u00b9 * x, hLE (hKL hxvK), _, _\u27e9", []], "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : LocallyCompactSpace G\nE : Set G\nhE : MeasurableSet E\nhEpos : 0 < \u2191\u2191\u03bc E\nL : Set G\nhL : MeasurableSet L\nhLE : L \u2286 E\nhLpos : 0 < \u2191\u2191\u03bc L\nhLtop : \u2191\u2191\u03bc L < \u22a4\nK : Set G\nhKL : K \u2286 L\nhK : IsCompact K\nhKpos : 0 < \u2191\u2191\u03bc K\nhKtop : \u2191\u2191\u03bc K \u2260 \u22a4\nU : Set G\nhUK : U \u2287 K\nhU : IsOpen U\nh\u03bcUK : \u2191\u2191\u03bc U < \u2191\u2191\u03bc K + \u2191\u2191\u03bc K\nV : Set G\nhV1 : V \u2208 \ud835\udcdd 1\nhVKU : V * K \u2286 U\nhv : \u2200 (v : G), v \u2208 V \u2192 \u00acDisjoint ({v} * K) K\nv : G\nhvV : v \u2208 V\nx : G\nhxK : x \u2208 {v} * K\nhxvK : x \u2208 K\n\u22a2 v \u2208 E / E", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_1\nG : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : LocallyCompactSpace G\nE : Set G\nhE : MeasurableSet E\nhEpos : 0 < \u2191\u2191\u03bc E\nL : Set G\nhL : MeasurableSet L\nhLE : L \u2286 E\nhLpos : 0 < \u2191\u2191\u03bc L\nhLtop : \u2191\u2191\u03bc L < \u22a4\nK : Set G\nhKL : K \u2286 L\nhK : IsCompact K\nhKpos : 0 < \u2191\u2191\u03bc K\nhKtop : \u2191\u2191\u03bc K \u2260 \u22a4\nU : Set G\nhUK : U \u2287 K\nhU : IsOpen U\nh\u03bcUK : \u2191\u2191\u03bc U < \u2191\u2191\u03bc K + \u2191\u2191\u03bc K\nV : Set G\nhV1 : V \u2208 \ud835\udcdd 1\nhVKU : V * K \u2286 U\nhv : \u2200 (v : G), v \u2208 V \u2192 \u00acDisjoint ({v} * K) K\nv : G\nhvV : v \u2208 V\nx : G\nhxK : x \u2208 {v} * K\nhxvK : x \u2208 K\n\u22a2 v\u207b\u00b9 * x \u2208 E\n\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_2\nG : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : LocallyCompactSpace G\nE : Set G\nhE : MeasurableSet E\nhEpos : 0 < \u2191\u2191\u03bc E\nL : Set G\nhL : MeasurableSet L\nhLE : L \u2286 E\nhLpos : 0 < \u2191\u2191\u03bc L\nhLtop : \u2191\u2191\u03bc L < \u22a4\nK : Set G\nhKL : K \u2286 L\nhK : IsCompact K\nhKpos : 0 < \u2191\u2191\u03bc K\nhKtop : \u2191\u2191\u03bc K \u2260 \u22a4\nU : Set G\nhUK : U \u2287 K\nhU : IsOpen U\nh\u03bcUK : \u2191\u2191\u03bc U < \u2191\u2191\u03bc K + \u2191\u2191\u03bc K\nV : Set G\nhV1 : V \u2208 \ud835\udcdd 1\nhVKU : V * K \u2286 U\nhv : \u2200 (v : G), v \u2208 V \u2192 \u00acDisjoint ({v} * K) K\nv : G\nhvV : v \u2208 V\nx : G\nhxK : x \u2208 {v} * K\nhxvK : x \u2208 K\n\u22a2 (fun x x_1 => x / x_1) x (v\u207b\u00b9 * x) = v"}, {"tactic": "apply ne_top_of_le_ne_top (ne_of_lt hLtop)", "annotated_tactic": ["apply <a>ne_top_of_le_ne_top</a> (<a>ne_of_lt</a> hLtop)", [{"full_name": "ne_top_of_le_ne_top", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [194, 9], "def_end_pos": [194, 28]}, {"full_name": "ne_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [101, 9], "def_end_pos": [101, 17]}]], "state_before": "G : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : LocallyCompactSpace G\nE : Set G\nhE : MeasurableSet E\nhEpos : 0 < \u2191\u2191\u03bc E\nL : Set G\nhL : MeasurableSet L\nhLE : L \u2286 E\nhLpos : 0 < \u2191\u2191\u03bc L\nhLtop : \u2191\u2191\u03bc L < \u22a4\nK : Set G\nhKL : K \u2286 L\nhK : IsCompact K\nhKpos : 0 < \u2191\u2191\u03bc K\n\u22a2 \u2191\u2191\u03bc K \u2260 \u22a4", "state_after": "G : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : LocallyCompactSpace G\nE : Set G\nhE : MeasurableSet E\nhEpos : 0 < \u2191\u2191\u03bc E\nL : Set G\nhL : MeasurableSet L\nhLE : L \u2286 E\nhLpos : 0 < \u2191\u2191\u03bc L\nhLtop : \u2191\u2191\u03bc L < \u22a4\nK : Set G\nhKL : K \u2286 L\nhK : IsCompact K\nhKpos : 0 < \u2191\u2191\u03bc K\n\u22a2 \u2191\u2191\u03bc K \u2264 \u2191\u2191\u03bc L"}, {"tactic": "apply measure_mono hKL", "annotated_tactic": ["apply <a>measure_mono</a> hKL", [{"full_name": "MeasureTheory.measure_mono", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [193, 9], "def_end_pos": [193, 21]}]], "state_before": "G : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : LocallyCompactSpace G\nE : Set G\nhE : MeasurableSet E\nhEpos : 0 < \u2191\u2191\u03bc E\nL : Set G\nhL : MeasurableSet L\nhLE : L \u2286 E\nhLpos : 0 < \u2191\u2191\u03bc L\nhLtop : \u2191\u2191\u03bc L < \u22a4\nK : Set G\nhKL : K \u2286 L\nhK : IsCompact K\nhKpos : 0 < \u2191\u2191\u03bc K\n\u22a2 \u2191\u2191\u03bc K \u2264 \u2191\u2191\u03bc L", "state_after": "no goals"}, {"tactic": "intro v hv hKv", "annotated_tactic": ["intro v hv hKv", []], "state_before": "G : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : LocallyCompactSpace G\nE : Set G\nhE : MeasurableSet E\nhEpos : 0 < \u2191\u2191\u03bc E\nL : Set G\nhL : MeasurableSet L\nhLE : L \u2286 E\nhLpos : 0 < \u2191\u2191\u03bc L\nhLtop : \u2191\u2191\u03bc L < \u22a4\nK : Set G\nhKL : K \u2286 L\nhK : IsCompact K\nhKpos : 0 < \u2191\u2191\u03bc K\nhKtop : \u2191\u2191\u03bc K \u2260 \u22a4\nU : Set G\nhUK : U \u2287 K\nhU : IsOpen U\nh\u03bcUK : \u2191\u2191\u03bc U < \u2191\u2191\u03bc K + \u2191\u2191\u03bc K\nV : Set G\nhV1 : V \u2208 \ud835\udcdd 1\nhVKU : V * K \u2286 U\n\u22a2 \u2200 (v : G), v \u2208 V \u2192 \u00acDisjoint ({v} * K) K", "state_after": "G : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : LocallyCompactSpace G\nE : Set G\nhE : MeasurableSet E\nhEpos : 0 < \u2191\u2191\u03bc E\nL : Set G\nhL : MeasurableSet L\nhLE : L \u2286 E\nhLpos : 0 < \u2191\u2191\u03bc L\nhLtop : \u2191\u2191\u03bc L < \u22a4\nK : Set G\nhKL : K \u2286 L\nhK : IsCompact K\nhKpos : 0 < \u2191\u2191\u03bc K\nhKtop : \u2191\u2191\u03bc K \u2260 \u22a4\nU : Set G\nhUK : U \u2287 K\nhU : IsOpen U\nh\u03bcUK : \u2191\u2191\u03bc U < \u2191\u2191\u03bc K + \u2191\u2191\u03bc K\nV : Set G\nhV1 : V \u2208 \ud835\udcdd 1\nhVKU : V * K \u2286 U\nv : G\nhv : v \u2208 V\nhKv : Disjoint ({v} * K) K\n\u22a2 False"}, {"tactic": "have hKvsub : {v} * K \u222a K \u2286 U := by\n  apply Set.union_subset _ hUK\n  apply _root_.subset_trans _ hVKU\n  apply Set.mul_subset_mul _ (Set.Subset.refl K)\n  simp only [Set.singleton_subset_iff, hv]", "annotated_tactic": ["have hKvsub : {v} * K \u222a K \u2286 U := by\n      apply <a>Set.union_subset</a> _ hUK\n      apply <a>_root_.subset_trans</a> _ hVKU\n      apply <a>Set.mul_subset_mul</a> _ (<a>Set.Subset.refl</a> K)\n      simp only [<a>Set.singleton_subset_iff</a>, hv]", [{"full_name": "Set.union_subset", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [836, 9], "def_end_pos": [836, 21]}, {"full_name": "subset_trans", "def_path": "Mathlib/Order/RelClasses.lean", "def_pos": [643, 7], "def_end_pos": [643, 19]}, {"full_name": "Set.mul_subset_mul", "def_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "def_pos": [416, 9], "def_end_pos": [416, 23]}, {"full_name": "Set.Subset.refl", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [354, 9], "def_end_pos": [354, 20]}, {"full_name": "Set.singleton_subset_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1330, 9], "def_end_pos": [1330, 29]}]], "state_before": "G : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : LocallyCompactSpace G\nE : Set G\nhE : MeasurableSet E\nhEpos : 0 < \u2191\u2191\u03bc E\nL : Set G\nhL : MeasurableSet L\nhLE : L \u2286 E\nhLpos : 0 < \u2191\u2191\u03bc L\nhLtop : \u2191\u2191\u03bc L < \u22a4\nK : Set G\nhKL : K \u2286 L\nhK : IsCompact K\nhKpos : 0 < \u2191\u2191\u03bc K\nhKtop : \u2191\u2191\u03bc K \u2260 \u22a4\nU : Set G\nhUK : U \u2287 K\nhU : IsOpen U\nh\u03bcUK : \u2191\u2191\u03bc U < \u2191\u2191\u03bc K + \u2191\u2191\u03bc K\nV : Set G\nhV1 : V \u2208 \ud835\udcdd 1\nhVKU : V * K \u2286 U\nv : G\nhv : v \u2208 V\nhKv : Disjoint ({v} * K) K\n\u22a2 False", "state_after": "G : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : LocallyCompactSpace G\nE : Set G\nhE : MeasurableSet E\nhEpos : 0 < \u2191\u2191\u03bc E\nL : Set G\nhL : MeasurableSet L\nhLE : L \u2286 E\nhLpos : 0 < \u2191\u2191\u03bc L\nhLtop : \u2191\u2191\u03bc L < \u22a4\nK : Set G\nhKL : K \u2286 L\nhK : IsCompact K\nhKpos : 0 < \u2191\u2191\u03bc K\nhKtop : \u2191\u2191\u03bc K \u2260 \u22a4\nU : Set G\nhUK : U \u2287 K\nhU : IsOpen U\nh\u03bcUK : \u2191\u2191\u03bc U < \u2191\u2191\u03bc K + \u2191\u2191\u03bc K\nV : Set G\nhV1 : V \u2208 \ud835\udcdd 1\nhVKU : V * K \u2286 U\nv : G\nhv : v \u2208 V\nhKv : Disjoint ({v} * K) K\nhKvsub : {v} * K \u222a K \u2286 U\n\u22a2 False"}, {"tactic": "replace hKvsub := @measure_mono _ _ \u03bc _ _ hKvsub", "annotated_tactic": ["replace hKvsub := @<a>measure_mono</a> _ _ \u03bc _ _ hKvsub", [{"full_name": "MeasureTheory.measure_mono", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [193, 9], "def_end_pos": [193, 21]}]], "state_before": "G : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : LocallyCompactSpace G\nE : Set G\nhE : MeasurableSet E\nhEpos : 0 < \u2191\u2191\u03bc E\nL : Set G\nhL : MeasurableSet L\nhLE : L \u2286 E\nhLpos : 0 < \u2191\u2191\u03bc L\nhLtop : \u2191\u2191\u03bc L < \u22a4\nK : Set G\nhKL : K \u2286 L\nhK : IsCompact K\nhKpos : 0 < \u2191\u2191\u03bc K\nhKtop : \u2191\u2191\u03bc K \u2260 \u22a4\nU : Set G\nhUK : U \u2287 K\nhU : IsOpen U\nh\u03bcUK : \u2191\u2191\u03bc U < \u2191\u2191\u03bc K + \u2191\u2191\u03bc K\nV : Set G\nhV1 : V \u2208 \ud835\udcdd 1\nhVKU : V * K \u2286 U\nv : G\nhv : v \u2208 V\nhKv : Disjoint ({v} * K) K\nhKvsub : {v} * K \u222a K \u2286 U\n\u22a2 False", "state_after": "G : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : LocallyCompactSpace G\nE : Set G\nhE : MeasurableSet E\nhEpos : 0 < \u2191\u2191\u03bc E\nL : Set G\nhL : MeasurableSet L\nhLE : L \u2286 E\nhLpos : 0 < \u2191\u2191\u03bc L\nhLtop : \u2191\u2191\u03bc L < \u22a4\nK : Set G\nhKL : K \u2286 L\nhK : IsCompact K\nhKpos : 0 < \u2191\u2191\u03bc K\nhKtop : \u2191\u2191\u03bc K \u2260 \u22a4\nU : Set G\nhUK : U \u2287 K\nhU : IsOpen U\nh\u03bcUK : \u2191\u2191\u03bc U < \u2191\u2191\u03bc K + \u2191\u2191\u03bc K\nV : Set G\nhV1 : V \u2208 \ud835\udcdd 1\nhVKU : V * K \u2286 U\nv : G\nhv : v \u2208 V\nhKv : Disjoint ({v} * K) K\nhKvsub : \u2191\u2191\u03bc ({v} * K \u222a K) \u2264 \u2191\u2191\u03bc U\n\u22a2 False"}, {"tactic": "have hcontr := lt_of_le_of_lt hKvsub h\u03bcUK", "annotated_tactic": ["have hcontr := <a>lt_of_le_of_lt</a> hKvsub h\u03bcUK", [{"full_name": "lt_of_le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [122, 9], "def_end_pos": [122, 23]}]], "state_before": "G : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : LocallyCompactSpace G\nE : Set G\nhE : MeasurableSet E\nhEpos : 0 < \u2191\u2191\u03bc E\nL : Set G\nhL : MeasurableSet L\nhLE : L \u2286 E\nhLpos : 0 < \u2191\u2191\u03bc L\nhLtop : \u2191\u2191\u03bc L < \u22a4\nK : Set G\nhKL : K \u2286 L\nhK : IsCompact K\nhKpos : 0 < \u2191\u2191\u03bc K\nhKtop : \u2191\u2191\u03bc K \u2260 \u22a4\nU : Set G\nhUK : U \u2287 K\nhU : IsOpen U\nh\u03bcUK : \u2191\u2191\u03bc U < \u2191\u2191\u03bc K + \u2191\u2191\u03bc K\nV : Set G\nhV1 : V \u2208 \ud835\udcdd 1\nhVKU : V * K \u2286 U\nv : G\nhv : v \u2208 V\nhKv : Disjoint ({v} * K) K\nhKvsub : \u2191\u2191\u03bc ({v} * K \u222a K) \u2264 \u2191\u2191\u03bc U\n\u22a2 False", "state_after": "G : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : LocallyCompactSpace G\nE : Set G\nhE : MeasurableSet E\nhEpos : 0 < \u2191\u2191\u03bc E\nL : Set G\nhL : MeasurableSet L\nhLE : L \u2286 E\nhLpos : 0 < \u2191\u2191\u03bc L\nhLtop : \u2191\u2191\u03bc L < \u22a4\nK : Set G\nhKL : K \u2286 L\nhK : IsCompact K\nhKpos : 0 < \u2191\u2191\u03bc K\nhKtop : \u2191\u2191\u03bc K \u2260 \u22a4\nU : Set G\nhUK : U \u2287 K\nhU : IsOpen U\nh\u03bcUK : \u2191\u2191\u03bc U < \u2191\u2191\u03bc K + \u2191\u2191\u03bc K\nV : Set G\nhV1 : V \u2208 \ud835\udcdd 1\nhVKU : V * K \u2286 U\nv : G\nhv : v \u2208 V\nhKv : Disjoint ({v} * K) K\nhKvsub : \u2191\u2191\u03bc ({v} * K \u222a K) \u2264 \u2191\u2191\u03bc U\nhcontr : \u2191\u2191\u03bc ({v} * K \u222a K) < \u2191\u2191\u03bc K + \u2191\u2191\u03bc K\n\u22a2 False"}, {"tactic": "rw [measure_union hKv (IsCompact.measurableSet hK)] at hcontr", "annotated_tactic": ["rw [<a>measure_union</a> hKv (<a>IsCompact.measurableSet</a> hK)] at hcontr", [{"full_name": "MeasureTheory.measure_union", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [124, 9], "def_end_pos": [124, 22]}, {"full_name": "IsCompact.measurableSet", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [344, 9], "def_end_pos": [344, 32]}]], "state_before": "G : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : LocallyCompactSpace G\nE : Set G\nhE : MeasurableSet E\nhEpos : 0 < \u2191\u2191\u03bc E\nL : Set G\nhL : MeasurableSet L\nhLE : L \u2286 E\nhLpos : 0 < \u2191\u2191\u03bc L\nhLtop : \u2191\u2191\u03bc L < \u22a4\nK : Set G\nhKL : K \u2286 L\nhK : IsCompact K\nhKpos : 0 < \u2191\u2191\u03bc K\nhKtop : \u2191\u2191\u03bc K \u2260 \u22a4\nU : Set G\nhUK : U \u2287 K\nhU : IsOpen U\nh\u03bcUK : \u2191\u2191\u03bc U < \u2191\u2191\u03bc K + \u2191\u2191\u03bc K\nV : Set G\nhV1 : V \u2208 \ud835\udcdd 1\nhVKU : V * K \u2286 U\nv : G\nhv : v \u2208 V\nhKv : Disjoint ({v} * K) K\nhKvsub : \u2191\u2191\u03bc ({v} * K \u222a K) \u2264 \u2191\u2191\u03bc U\nhcontr : \u2191\u2191\u03bc ({v} * K \u222a K) < \u2191\u2191\u03bc K + \u2191\u2191\u03bc K\n\u22a2 False", "state_after": "G : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : LocallyCompactSpace G\nE : Set G\nhE : MeasurableSet E\nhEpos : 0 < \u2191\u2191\u03bc E\nL : Set G\nhL : MeasurableSet L\nhLE : L \u2286 E\nhLpos : 0 < \u2191\u2191\u03bc L\nhLtop : \u2191\u2191\u03bc L < \u22a4\nK : Set G\nhKL : K \u2286 L\nhK : IsCompact K\nhKpos : 0 < \u2191\u2191\u03bc K\nhKtop : \u2191\u2191\u03bc K \u2260 \u22a4\nU : Set G\nhUK : U \u2287 K\nhU : IsOpen U\nh\u03bcUK : \u2191\u2191\u03bc U < \u2191\u2191\u03bc K + \u2191\u2191\u03bc K\nV : Set G\nhV1 : V \u2208 \ud835\udcdd 1\nhVKU : V * K \u2286 U\nv : G\nhv : v \u2208 V\nhKv : Disjoint ({v} * K) K\nhKvsub : \u2191\u2191\u03bc ({v} * K \u222a K) \u2264 \u2191\u2191\u03bc U\nhcontr : \u2191\u2191\u03bc ({v} * K) + \u2191\u2191\u03bc K < \u2191\u2191\u03bc K + \u2191\u2191\u03bc K\n\u22a2 False"}, {"tactic": "have hKtranslate : \u03bc ({v} * K) = \u03bc K := by\n  simp only [singleton_mul, image_mul_left, measure_preimage_mul]", "annotated_tactic": ["have hKtranslate : \u03bc ({v} * K) = \u03bc K := by\n      simp only [<a>singleton_mul</a>, <a>image_mul_left</a>, <a>measure_preimage_mul</a>]", [{"full_name": "Set.singleton_mul", "def_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "def_pos": [403, 9], "def_end_pos": [403, 22]}, {"full_name": "Set.image_mul_left", "def_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "def_pos": [1199, 9], "def_end_pos": [1199, 23]}, {"full_name": "MeasureTheory.measure_preimage_mul", "def_path": "Mathlib/MeasureTheory/Group/Measure.lean", "def_pos": [320, 9], "def_end_pos": [320, 29]}]], "state_before": "G : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : LocallyCompactSpace G\nE : Set G\nhE : MeasurableSet E\nhEpos : 0 < \u2191\u2191\u03bc E\nL : Set G\nhL : MeasurableSet L\nhLE : L \u2286 E\nhLpos : 0 < \u2191\u2191\u03bc L\nhLtop : \u2191\u2191\u03bc L < \u22a4\nK : Set G\nhKL : K \u2286 L\nhK : IsCompact K\nhKpos : 0 < \u2191\u2191\u03bc K\nhKtop : \u2191\u2191\u03bc K \u2260 \u22a4\nU : Set G\nhUK : U \u2287 K\nhU : IsOpen U\nh\u03bcUK : \u2191\u2191\u03bc U < \u2191\u2191\u03bc K + \u2191\u2191\u03bc K\nV : Set G\nhV1 : V \u2208 \ud835\udcdd 1\nhVKU : V * K \u2286 U\nv : G\nhv : v \u2208 V\nhKv : Disjoint ({v} * K) K\nhKvsub : \u2191\u2191\u03bc ({v} * K \u222a K) \u2264 \u2191\u2191\u03bc U\nhcontr : \u2191\u2191\u03bc ({v} * K) + \u2191\u2191\u03bc K < \u2191\u2191\u03bc K + \u2191\u2191\u03bc K\n\u22a2 False", "state_after": "G : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : LocallyCompactSpace G\nE : Set G\nhE : MeasurableSet E\nhEpos : 0 < \u2191\u2191\u03bc E\nL : Set G\nhL : MeasurableSet L\nhLE : L \u2286 E\nhLpos : 0 < \u2191\u2191\u03bc L\nhLtop : \u2191\u2191\u03bc L < \u22a4\nK : Set G\nhKL : K \u2286 L\nhK : IsCompact K\nhKpos : 0 < \u2191\u2191\u03bc K\nhKtop : \u2191\u2191\u03bc K \u2260 \u22a4\nU : Set G\nhUK : U \u2287 K\nhU : IsOpen U\nh\u03bcUK : \u2191\u2191\u03bc U < \u2191\u2191\u03bc K + \u2191\u2191\u03bc K\nV : Set G\nhV1 : V \u2208 \ud835\udcdd 1\nhVKU : V * K \u2286 U\nv : G\nhv : v \u2208 V\nhKv : Disjoint ({v} * K) K\nhKvsub : \u2191\u2191\u03bc ({v} * K \u222a K) \u2264 \u2191\u2191\u03bc U\nhcontr : \u2191\u2191\u03bc ({v} * K) + \u2191\u2191\u03bc K < \u2191\u2191\u03bc K + \u2191\u2191\u03bc K\nhKtranslate : \u2191\u2191\u03bc ({v} * K) = \u2191\u2191\u03bc K\n\u22a2 False"}, {"tactic": "rw [hKtranslate, lt_self_iff_false] at hcontr", "annotated_tactic": ["rw [hKtranslate, <a>lt_self_iff_false</a>] at hcontr", [{"full_name": "lt_self_iff_false", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [175, 9], "def_end_pos": [175, 26]}]], "state_before": "G : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : LocallyCompactSpace G\nE : Set G\nhE : MeasurableSet E\nhEpos : 0 < \u2191\u2191\u03bc E\nL : Set G\nhL : MeasurableSet L\nhLE : L \u2286 E\nhLpos : 0 < \u2191\u2191\u03bc L\nhLtop : \u2191\u2191\u03bc L < \u22a4\nK : Set G\nhKL : K \u2286 L\nhK : IsCompact K\nhKpos : 0 < \u2191\u2191\u03bc K\nhKtop : \u2191\u2191\u03bc K \u2260 \u22a4\nU : Set G\nhUK : U \u2287 K\nhU : IsOpen U\nh\u03bcUK : \u2191\u2191\u03bc U < \u2191\u2191\u03bc K + \u2191\u2191\u03bc K\nV : Set G\nhV1 : V \u2208 \ud835\udcdd 1\nhVKU : V * K \u2286 U\nv : G\nhv : v \u2208 V\nhKv : Disjoint ({v} * K) K\nhKvsub : \u2191\u2191\u03bc ({v} * K \u222a K) \u2264 \u2191\u2191\u03bc U\nhcontr : \u2191\u2191\u03bc ({v} * K) + \u2191\u2191\u03bc K < \u2191\u2191\u03bc K + \u2191\u2191\u03bc K\nhKtranslate : \u2191\u2191\u03bc ({v} * K) = \u2191\u2191\u03bc K\n\u22a2 False", "state_after": "G : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : LocallyCompactSpace G\nE : Set G\nhE : MeasurableSet E\nhEpos : 0 < \u2191\u2191\u03bc E\nL : Set G\nhL : MeasurableSet L\nhLE : L \u2286 E\nhLpos : 0 < \u2191\u2191\u03bc L\nhLtop : \u2191\u2191\u03bc L < \u22a4\nK : Set G\nhKL : K \u2286 L\nhK : IsCompact K\nhKpos : 0 < \u2191\u2191\u03bc K\nhKtop : \u2191\u2191\u03bc K \u2260 \u22a4\nU : Set G\nhUK : U \u2287 K\nhU : IsOpen U\nh\u03bcUK : \u2191\u2191\u03bc U < \u2191\u2191\u03bc K + \u2191\u2191\u03bc K\nV : Set G\nhV1 : V \u2208 \ud835\udcdd 1\nhVKU : V * K \u2286 U\nv : G\nhv : v \u2208 V\nhKv : Disjoint ({v} * K) K\nhKvsub : \u2191\u2191\u03bc ({v} * K \u222a K) \u2264 \u2191\u2191\u03bc U\nhcontr : False\nhKtranslate : \u2191\u2191\u03bc ({v} * K) = \u2191\u2191\u03bc K\n\u22a2 False"}, {"tactic": "assumption", "annotated_tactic": ["assumption", []], "state_before": "G : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : LocallyCompactSpace G\nE : Set G\nhE : MeasurableSet E\nhEpos : 0 < \u2191\u2191\u03bc E\nL : Set G\nhL : MeasurableSet L\nhLE : L \u2286 E\nhLpos : 0 < \u2191\u2191\u03bc L\nhLtop : \u2191\u2191\u03bc L < \u22a4\nK : Set G\nhKL : K \u2286 L\nhK : IsCompact K\nhKpos : 0 < \u2191\u2191\u03bc K\nhKtop : \u2191\u2191\u03bc K \u2260 \u22a4\nU : Set G\nhUK : U \u2287 K\nhU : IsOpen U\nh\u03bcUK : \u2191\u2191\u03bc U < \u2191\u2191\u03bc K + \u2191\u2191\u03bc K\nV : Set G\nhV1 : V \u2208 \ud835\udcdd 1\nhVKU : V * K \u2286 U\nv : G\nhv : v \u2208 V\nhKv : Disjoint ({v} * K) K\nhKvsub : \u2191\u2191\u03bc ({v} * K \u222a K) \u2264 \u2191\u2191\u03bc U\nhcontr : False\nhKtranslate : \u2191\u2191\u03bc ({v} * K) = \u2191\u2191\u03bc K\n\u22a2 False", "state_after": "no goals"}, {"tactic": "apply Set.union_subset _ hUK", "annotated_tactic": ["apply <a>Set.union_subset</a> _ hUK", [{"full_name": "Set.union_subset", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [836, 9], "def_end_pos": [836, 21]}]], "state_before": "G : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : LocallyCompactSpace G\nE : Set G\nhE : MeasurableSet E\nhEpos : 0 < \u2191\u2191\u03bc E\nL : Set G\nhL : MeasurableSet L\nhLE : L \u2286 E\nhLpos : 0 < \u2191\u2191\u03bc L\nhLtop : \u2191\u2191\u03bc L < \u22a4\nK : Set G\nhKL : K \u2286 L\nhK : IsCompact K\nhKpos : 0 < \u2191\u2191\u03bc K\nhKtop : \u2191\u2191\u03bc K \u2260 \u22a4\nU : Set G\nhUK : U \u2287 K\nhU : IsOpen U\nh\u03bcUK : \u2191\u2191\u03bc U < \u2191\u2191\u03bc K + \u2191\u2191\u03bc K\nV : Set G\nhV1 : V \u2208 \ud835\udcdd 1\nhVKU : V * K \u2286 U\nv : G\nhv : v \u2208 V\nhKv : Disjoint ({v} * K) K\n\u22a2 {v} * K \u222a K \u2286 U", "state_after": "G : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : LocallyCompactSpace G\nE : Set G\nhE : MeasurableSet E\nhEpos : 0 < \u2191\u2191\u03bc E\nL : Set G\nhL : MeasurableSet L\nhLE : L \u2286 E\nhLpos : 0 < \u2191\u2191\u03bc L\nhLtop : \u2191\u2191\u03bc L < \u22a4\nK : Set G\nhKL : K \u2286 L\nhK : IsCompact K\nhKpos : 0 < \u2191\u2191\u03bc K\nhKtop : \u2191\u2191\u03bc K \u2260 \u22a4\nU : Set G\nhUK : U \u2287 K\nhU : IsOpen U\nh\u03bcUK : \u2191\u2191\u03bc U < \u2191\u2191\u03bc K + \u2191\u2191\u03bc K\nV : Set G\nhV1 : V \u2208 \ud835\udcdd 1\nhVKU : V * K \u2286 U\nv : G\nhv : v \u2208 V\nhKv : Disjoint ({v} * K) K\n\u22a2 {v} * K \u2286 U"}, {"tactic": "apply _root_.subset_trans _ hVKU", "annotated_tactic": ["apply <a>_root_.subset_trans</a> _ hVKU", [{"full_name": "subset_trans", "def_path": "Mathlib/Order/RelClasses.lean", "def_pos": [643, 7], "def_end_pos": [643, 19]}]], "state_before": "G : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : LocallyCompactSpace G\nE : Set G\nhE : MeasurableSet E\nhEpos : 0 < \u2191\u2191\u03bc E\nL : Set G\nhL : MeasurableSet L\nhLE : L \u2286 E\nhLpos : 0 < \u2191\u2191\u03bc L\nhLtop : \u2191\u2191\u03bc L < \u22a4\nK : Set G\nhKL : K \u2286 L\nhK : IsCompact K\nhKpos : 0 < \u2191\u2191\u03bc K\nhKtop : \u2191\u2191\u03bc K \u2260 \u22a4\nU : Set G\nhUK : U \u2287 K\nhU : IsOpen U\nh\u03bcUK : \u2191\u2191\u03bc U < \u2191\u2191\u03bc K + \u2191\u2191\u03bc K\nV : Set G\nhV1 : V \u2208 \ud835\udcdd 1\nhVKU : V * K \u2286 U\nv : G\nhv : v \u2208 V\nhKv : Disjoint ({v} * K) K\n\u22a2 {v} * K \u2286 U", "state_after": "G : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : LocallyCompactSpace G\nE : Set G\nhE : MeasurableSet E\nhEpos : 0 < \u2191\u2191\u03bc E\nL : Set G\nhL : MeasurableSet L\nhLE : L \u2286 E\nhLpos : 0 < \u2191\u2191\u03bc L\nhLtop : \u2191\u2191\u03bc L < \u22a4\nK : Set G\nhKL : K \u2286 L\nhK : IsCompact K\nhKpos : 0 < \u2191\u2191\u03bc K\nhKtop : \u2191\u2191\u03bc K \u2260 \u22a4\nU : Set G\nhUK : U \u2287 K\nhU : IsOpen U\nh\u03bcUK : \u2191\u2191\u03bc U < \u2191\u2191\u03bc K + \u2191\u2191\u03bc K\nV : Set G\nhV1 : V \u2208 \ud835\udcdd 1\nhVKU : V * K \u2286 U\nv : G\nhv : v \u2208 V\nhKv : Disjoint ({v} * K) K\n\u22a2 {v} * K \u2286 V * K"}, {"tactic": "apply Set.mul_subset_mul _ (Set.Subset.refl K)", "annotated_tactic": ["apply <a>Set.mul_subset_mul</a> _ (<a>Set.Subset.refl</a> K)", [{"full_name": "Set.mul_subset_mul", "def_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "def_pos": [416, 9], "def_end_pos": [416, 23]}, {"full_name": "Set.Subset.refl", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [354, 9], "def_end_pos": [354, 20]}]], "state_before": "G : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : LocallyCompactSpace G\nE : Set G\nhE : MeasurableSet E\nhEpos : 0 < \u2191\u2191\u03bc E\nL : Set G\nhL : MeasurableSet L\nhLE : L \u2286 E\nhLpos : 0 < \u2191\u2191\u03bc L\nhLtop : \u2191\u2191\u03bc L < \u22a4\nK : Set G\nhKL : K \u2286 L\nhK : IsCompact K\nhKpos : 0 < \u2191\u2191\u03bc K\nhKtop : \u2191\u2191\u03bc K \u2260 \u22a4\nU : Set G\nhUK : U \u2287 K\nhU : IsOpen U\nh\u03bcUK : \u2191\u2191\u03bc U < \u2191\u2191\u03bc K + \u2191\u2191\u03bc K\nV : Set G\nhV1 : V \u2208 \ud835\udcdd 1\nhVKU : V * K \u2286 U\nv : G\nhv : v \u2208 V\nhKv : Disjoint ({v} * K) K\n\u22a2 {v} * K \u2286 V * K", "state_after": "G : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : LocallyCompactSpace G\nE : Set G\nhE : MeasurableSet E\nhEpos : 0 < \u2191\u2191\u03bc E\nL : Set G\nhL : MeasurableSet L\nhLE : L \u2286 E\nhLpos : 0 < \u2191\u2191\u03bc L\nhLtop : \u2191\u2191\u03bc L < \u22a4\nK : Set G\nhKL : K \u2286 L\nhK : IsCompact K\nhKpos : 0 < \u2191\u2191\u03bc K\nhKtop : \u2191\u2191\u03bc K \u2260 \u22a4\nU : Set G\nhUK : U \u2287 K\nhU : IsOpen U\nh\u03bcUK : \u2191\u2191\u03bc U < \u2191\u2191\u03bc K + \u2191\u2191\u03bc K\nV : Set G\nhV1 : V \u2208 \ud835\udcdd 1\nhVKU : V * K \u2286 U\nv : G\nhv : v \u2208 V\nhKv : Disjoint ({v} * K) K\n\u22a2 {v} \u2286 V"}, {"tactic": "simp only [Set.singleton_subset_iff, hv]", "annotated_tactic": ["simp only [<a>Set.singleton_subset_iff</a>, hv]", [{"full_name": "Set.singleton_subset_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1330, 9], "def_end_pos": [1330, 29]}]], "state_before": "G : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : LocallyCompactSpace G\nE : Set G\nhE : MeasurableSet E\nhEpos : 0 < \u2191\u2191\u03bc E\nL : Set G\nhL : MeasurableSet L\nhLE : L \u2286 E\nhLpos : 0 < \u2191\u2191\u03bc L\nhLtop : \u2191\u2191\u03bc L < \u22a4\nK : Set G\nhKL : K \u2286 L\nhK : IsCompact K\nhKpos : 0 < \u2191\u2191\u03bc K\nhKtop : \u2191\u2191\u03bc K \u2260 \u22a4\nU : Set G\nhUK : U \u2287 K\nhU : IsOpen U\nh\u03bcUK : \u2191\u2191\u03bc U < \u2191\u2191\u03bc K + \u2191\u2191\u03bc K\nV : Set G\nhV1 : V \u2208 \ud835\udcdd 1\nhVKU : V * K \u2286 U\nv : G\nhv : v \u2208 V\nhKv : Disjoint ({v} * K) K\n\u22a2 {v} \u2286 V", "state_after": "no goals"}, {"tactic": "simp only [singleton_mul, image_mul_left, measure_preimage_mul]", "annotated_tactic": ["simp only [<a>singleton_mul</a>, <a>image_mul_left</a>, <a>measure_preimage_mul</a>]", [{"full_name": "Set.singleton_mul", "def_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "def_pos": [403, 9], "def_end_pos": [403, 22]}, {"full_name": "Set.image_mul_left", "def_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "def_pos": [1199, 9], "def_end_pos": [1199, 23]}, {"full_name": "MeasureTheory.measure_preimage_mul", "def_path": "Mathlib/MeasureTheory/Group/Measure.lean", "def_pos": [320, 9], "def_end_pos": [320, 29]}]], "state_before": "G : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : LocallyCompactSpace G\nE : Set G\nhE : MeasurableSet E\nhEpos : 0 < \u2191\u2191\u03bc E\nL : Set G\nhL : MeasurableSet L\nhLE : L \u2286 E\nhLpos : 0 < \u2191\u2191\u03bc L\nhLtop : \u2191\u2191\u03bc L < \u22a4\nK : Set G\nhKL : K \u2286 L\nhK : IsCompact K\nhKpos : 0 < \u2191\u2191\u03bc K\nhKtop : \u2191\u2191\u03bc K \u2260 \u22a4\nU : Set G\nhUK : U \u2287 K\nhU : IsOpen U\nh\u03bcUK : \u2191\u2191\u03bc U < \u2191\u2191\u03bc K + \u2191\u2191\u03bc K\nV : Set G\nhV1 : V \u2208 \ud835\udcdd 1\nhVKU : V * K \u2286 U\nv : G\nhv : v \u2208 V\nhKv : Disjoint ({v} * K) K\nhKvsub : \u2191\u2191\u03bc ({v} * K \u222a K) \u2264 \u2191\u2191\u03bc U\nhcontr : \u2191\u2191\u03bc ({v} * K) + \u2191\u2191\u03bc K < \u2191\u2191\u03bc K + \u2191\u2191\u03bc K\n\u22a2 \u2191\u2191\u03bc ({v} * K) = \u2191\u2191\u03bc K", "state_after": "no goals"}, {"tactic": "apply hKL.trans hLE", "annotated_tactic": ["apply hKL.trans hLE", []], "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_1\nG : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : LocallyCompactSpace G\nE : Set G\nhE : MeasurableSet E\nhEpos : 0 < \u2191\u2191\u03bc E\nL : Set G\nhL : MeasurableSet L\nhLE : L \u2286 E\nhLpos : 0 < \u2191\u2191\u03bc L\nhLtop : \u2191\u2191\u03bc L < \u22a4\nK : Set G\nhKL : K \u2286 L\nhK : IsCompact K\nhKpos : 0 < \u2191\u2191\u03bc K\nhKtop : \u2191\u2191\u03bc K \u2260 \u22a4\nU : Set G\nhUK : U \u2287 K\nhU : IsOpen U\nh\u03bcUK : \u2191\u2191\u03bc U < \u2191\u2191\u03bc K + \u2191\u2191\u03bc K\nV : Set G\nhV1 : V \u2208 \ud835\udcdd 1\nhVKU : V * K \u2286 U\nhv : \u2200 (v : G), v \u2208 V \u2192 \u00acDisjoint ({v} * K) K\nv : G\nhvV : v \u2208 V\nx : G\nhxK : x \u2208 {v} * K\nhxvK : x \u2208 K\n\u22a2 v\u207b\u00b9 * x \u2208 E", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_1.a\nG : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : LocallyCompactSpace G\nE : Set G\nhE : MeasurableSet E\nhEpos : 0 < \u2191\u2191\u03bc E\nL : Set G\nhL : MeasurableSet L\nhLE : L \u2286 E\nhLpos : 0 < \u2191\u2191\u03bc L\nhLtop : \u2191\u2191\u03bc L < \u22a4\nK : Set G\nhKL : K \u2286 L\nhK : IsCompact K\nhKpos : 0 < \u2191\u2191\u03bc K\nhKtop : \u2191\u2191\u03bc K \u2260 \u22a4\nU : Set G\nhUK : U \u2287 K\nhU : IsOpen U\nh\u03bcUK : \u2191\u2191\u03bc U < \u2191\u2191\u03bc K + \u2191\u2191\u03bc K\nV : Set G\nhV1 : V \u2208 \ud835\udcdd 1\nhVKU : V * K \u2286 U\nhv : \u2200 (v : G), v \u2208 V \u2192 \u00acDisjoint ({v} * K) K\nv : G\nhvV : v \u2208 V\nx : G\nhxK : x \u2208 {v} * K\nhxvK : x \u2208 K\n\u22a2 v\u207b\u00b9 * x \u2208 K"}, {"tactic": "simpa only [singleton_mul, image_mul_left, mem_preimage] using hxK", "annotated_tactic": ["simpa only [<a>singleton_mul</a>, <a>image_mul_left</a>, <a>mem_preimage</a>] using hxK", [{"full_name": "Set.singleton_mul", "def_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "def_pos": [403, 9], "def_end_pos": [403, 22]}, {"full_name": "Set.image_mul_left", "def_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "def_pos": [1199, 9], "def_end_pos": [1199, 23]}, {"full_name": "Set.mem_preimage", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [64, 9], "def_end_pos": [64, 21]}]], "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_1.a\nG : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : LocallyCompactSpace G\nE : Set G\nhE : MeasurableSet E\nhEpos : 0 < \u2191\u2191\u03bc E\nL : Set G\nhL : MeasurableSet L\nhLE : L \u2286 E\nhLpos : 0 < \u2191\u2191\u03bc L\nhLtop : \u2191\u2191\u03bc L < \u22a4\nK : Set G\nhKL : K \u2286 L\nhK : IsCompact K\nhKpos : 0 < \u2191\u2191\u03bc K\nhKtop : \u2191\u2191\u03bc K \u2260 \u22a4\nU : Set G\nhUK : U \u2287 K\nhU : IsOpen U\nh\u03bcUK : \u2191\u2191\u03bc U < \u2191\u2191\u03bc K + \u2191\u2191\u03bc K\nV : Set G\nhV1 : V \u2208 \ud835\udcdd 1\nhVKU : V * K \u2286 U\nhv : \u2200 (v : G), v \u2208 V \u2192 \u00acDisjoint ({v} * K) K\nv : G\nhvV : v \u2208 V\nx : G\nhxK : x \u2208 {v} * K\nhxvK : x \u2208 K\n\u22a2 v\u207b\u00b9 * x \u2208 K", "state_after": "no goals"}, {"tactic": "simp only [div_eq_iff_eq_mul, \u2190 mul_assoc, mul_right_inv, one_mul]", "annotated_tactic": ["simp only [<a>div_eq_iff_eq_mul</a>, \u2190 <a>mul_assoc</a>, <a>mul_right_inv</a>, <a>one_mul</a>]", [{"full_name": "div_eq_iff_eq_mul", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [830, 9], "def_end_pos": [830, 26]}, {"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [264, 9], "def_end_pos": [264, 18]}, {"full_name": "mul_right_inv", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [1135, 9], "def_end_pos": [1135, 22]}, {"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [464, 9], "def_end_pos": [464, 16]}]], "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_2\nG : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : LocallyCompactSpace G\nE : Set G\nhE : MeasurableSet E\nhEpos : 0 < \u2191\u2191\u03bc E\nL : Set G\nhL : MeasurableSet L\nhLE : L \u2286 E\nhLpos : 0 < \u2191\u2191\u03bc L\nhLtop : \u2191\u2191\u03bc L < \u22a4\nK : Set G\nhKL : K \u2286 L\nhK : IsCompact K\nhKpos : 0 < \u2191\u2191\u03bc K\nhKtop : \u2191\u2191\u03bc K \u2260 \u22a4\nU : Set G\nhUK : U \u2287 K\nhU : IsOpen U\nh\u03bcUK : \u2191\u2191\u03bc U < \u2191\u2191\u03bc K + \u2191\u2191\u03bc K\nV : Set G\nhV1 : V \u2208 \ud835\udcdd 1\nhVKU : V * K \u2286 U\nhv : \u2200 (v : G), v \u2208 V \u2192 \u00acDisjoint ({v} * K) K\nv : G\nhvV : v \u2208 V\nx : G\nhxK : x \u2208 {v} * K\nhxvK : x \u2208 K\n\u22a2 (fun x x_1 => x / x_1) x (v\u207b\u00b9 * x) = v", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Lattice.lean", "full_name": "Finset.toDual_min'", "start": [1538, 1], "end": [1543, 6], "traced_tactics": [{"tactic": "rw [\u2190 WithBot.coe_eq_coe]", "annotated_tactic": ["rw [\u2190 <a>WithBot.coe_eq_coe</a>]", [{"full_name": "WithBot.coe_eq_coe", "def_path": "Mathlib/Order/WithBot.lean", "def_pos": [130, 9], "def_end_pos": [130, 19]}]], "state_before": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Type u_5\n\u03ba : Type u_6\ninst\u271d : LinearOrder \u03b1\ns\u271d : Finset \u03b1\nH : Finset.Nonempty s\u271d\nx : \u03b1\ns : Finset \u03b1\nhs : Finset.Nonempty s\n\u22a2 \u2191toDual (min' s hs) = max' (image (\u2191toDual) s) (_ : Finset.Nonempty (image (\u2191toDual) s))", "state_after": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Type u_5\n\u03ba : Type u_6\ninst\u271d : LinearOrder \u03b1\ns\u271d : Finset \u03b1\nH : Finset.Nonempty s\u271d\nx : \u03b1\ns : Finset \u03b1\nhs : Finset.Nonempty s\n\u22a2 \u2191(\u2191toDual (min' s hs)) = \u2191(max' (image (\u2191toDual) s) (_ : Finset.Nonempty (image (\u2191toDual) s)))"}, {"tactic": "simp only [min'_eq_inf', id_eq, toDual_inf', Function.comp_apply, coe_sup', max'_eq_sup',\n  sup_image]", "annotated_tactic": ["simp only [<a>min'_eq_inf'</a>, <a>id_eq</a>, <a>toDual_inf'</a>, <a>Function.comp_apply</a>, <a>coe_sup'</a>, <a>max'_eq_sup'</a>,\n    <a>sup_image</a>]", [{"full_name": "Finset.min'_eq_inf'", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [1476, 9], "def_end_pos": [1476, 21]}, {"full_name": "id_eq", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [284, 17], "def_end_pos": [284, 22]}, {"full_name": "Finset.toDual_inf'", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [1118, 9], "def_end_pos": [1118, 20]}, {"full_name": "Function.comp_apply", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [33, 17], "def_end_pos": [33, 36]}, {"full_name": "Finset.coe_sup'", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [772, 9], "def_end_pos": [772, 17]}, {"full_name": "Finset.max'_eq_sup'", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [1472, 9], "def_end_pos": [1472, 21]}, {"full_name": "Finset.sup_image", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [63, 9], "def_end_pos": [63, 18]}]], "state_before": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Type u_5\n\u03ba : Type u_6\ninst\u271d : LinearOrder \u03b1\ns\u271d : Finset \u03b1\nH : Finset.Nonempty s\u271d\nx : \u03b1\ns : Finset \u03b1\nhs : Finset.Nonempty s\n\u22a2 \u2191(\u2191toDual (min' s hs)) = \u2191(max' (image (\u2191toDual) s) (_ : Finset.Nonempty (image (\u2191toDual) s)))", "state_after": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Type u_5\n\u03ba : Type u_6\ninst\u271d : LinearOrder \u03b1\ns\u271d : Finset \u03b1\nH : Finset.Nonempty s\u271d\nx : \u03b1\ns : Finset \u03b1\nhs : Finset.Nonempty s\n\u22a2 sup s (WithBot.some \u2218 fun x => \u2191toDual x) = sup s ((WithBot.some \u2218 fun x => x) \u2218 \u2191toDual)"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Type u_5\n\u03ba : Type u_6\ninst\u271d : LinearOrder \u03b1\ns\u271d : Finset \u03b1\nH : Finset.Nonempty s\u271d\nx : \u03b1\ns : Finset \u03b1\nhs : Finset.Nonempty s\n\u22a2 sup s (WithBot.some \u2218 fun x => \u2191toDual x) = sup s ((WithBot.some \u2218 fun x => x) \u2218 \u2191toDual)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Setoid/Basic.lean", "full_name": "Setoid.eqvGen_eq", "start": [201, 1], "end": [207, 44], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/GiryMonad.lean", "full_name": "MeasureTheory.Measure.dirac_bind", "start": [202, 1], "end": [205, 81], "traced_tactics": [{"tactic": "ext1 s hs", "annotated_tactic": ["ext1 s hs", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\nm : Measure \u03b1\n\u22a2 bind m dirac = m", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\nm : Measure \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(bind m dirac) s = \u2191\u2191m s"}, {"tactic": "simp only [bind_apply hs measurable_dirac, dirac_apply' _ hs, lintegral_indicator 1 hs,\n  Pi.one_apply, lintegral_one, restrict_apply, MeasurableSet.univ, univ_inter]", "annotated_tactic": ["simp only [<a>bind_apply</a> hs <a>measurable_dirac</a>, <a>dirac_apply'</a> _ hs, <a>lintegral_indicator</a> 1 hs,\n    <a>Pi.one_apply</a>, <a>lintegral_one</a>, <a>restrict_apply</a>, <a>MeasurableSet.univ</a>, <a>univ_inter</a>]", [{"full_name": "MeasureTheory.Measure.bind_apply", "def_path": "Mathlib/MeasureTheory/Measure/GiryMonad.lean", "def_pos": [174, 9], "def_end_pos": [174, 19]}, {"full_name": "MeasureTheory.Measure.measurable_dirac", "def_path": "Mathlib/MeasureTheory/Measure/GiryMonad.lean", "def_pos": [83, 9], "def_end_pos": [83, 25]}, {"full_name": "MeasureTheory.Measure.dirac_apply'", "def_path": "Mathlib/MeasureTheory/Measure/Dirac.lean", "def_pos": [39, 9], "def_end_pos": [39, 21]}, {"full_name": "MeasureTheory.lintegral_indicator", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [762, 9], "def_end_pos": [762, 28]}, {"full_name": "Pi.one_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [47, 9], "def_end_pos": [47, 18]}, {"full_name": "MeasureTheory.lintegral_one", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [149, 9], "def_end_pos": [149, 22]}, {"full_name": "MeasureTheory.Measure.restrict_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1533, 9], "def_end_pos": [1533, 23]}, {"full_name": "MeasurableSet.univ", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [101, 19], "def_end_pos": [101, 37]}, {"full_name": "Set.univ_inter", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1017, 9], "def_end_pos": [1017, 19]}]], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\nm : Measure \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(bind m dirac) s = \u2191\u2191m s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "full_name": "MeasureTheory.FinStronglyMeasurable.add", "start": [1070, 11], "end": [1076, 62], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "full_name": "MeasureTheory.integral_eq_lintegral_pos_part_sub_lintegral_neg_part", "start": [1128, 1], "end": [1155, 42], "traced_tactics": [{"tactic": "let f\u2081 := hf.toL1 f", "annotated_tactic": ["let f\u2081 := hf.toL1 f", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u22a2 \u222b (a : \u03b1), f a \u2202\u03bc =\n    ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc) - ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (-f a) \u2202\u03bc)", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\nf\u2081 : { x // x \u2208 Lp \u211d 1 } := Integrable.toL1 f hf\n\u22a2 \u222b (a : \u03b1), f a \u2202\u03bc =\n    ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc) - ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (-f a) \u2202\u03bc)"}, {"tactic": "have eq\u2081 : ENNReal.toReal (\u222b\u207b a, ENNReal.ofReal (f a) \u2202\u03bc) = \u2016Lp.posPart f\u2081\u2016 := by\n  rw [L1.norm_def]\n  congr 1\n  apply lintegral_congr_ae\n  filter_upwards [Lp.coeFn_posPart f\u2081, hf.coeFn_toL1] with _ h\u2081 h\u2082\n  rw [h\u2081, h\u2082, ENNReal.ofReal]\n  congr 1\n  apply NNReal.eq\n  rw [Real.nnnorm_of_nonneg (le_max_right _ _)]\n  rw [Real.coe_toNNReal', NNReal.coe_mk]", "annotated_tactic": ["have eq\u2081 : <a>ENNReal.toReal</a> (\u222b\u207b a, <a>ENNReal.ofReal</a> (f a) \u2202\u03bc) = \u2016<a>Lp.posPart</a> f\u2081\u2016 := by\n    rw [<a>L1.norm_def</a>]\n    congr 1\n    apply <a>lintegral_congr_ae</a>\n    filter_upwards [<a>Lp.coeFn_posPart</a> f\u2081, hf.coeFn_toL1] with _ h\u2081 h\u2082\n    rw [h\u2081, h\u2082, <a>ENNReal.ofReal</a>]\n    congr 1\n    apply <a>NNReal.eq</a>\n    rw [<a>Real.nnnorm_of_nonneg</a> (<a>le_max_right</a> _ _)]\n    rw [<a>Real.coe_toNNReal'</a>, <a>NNReal.coe_mk</a>]", [{"full_name": "ENNReal.toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [168, 15], "def_end_pos": [168, 21]}, {"full_name": "ENNReal.ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [172, 29], "def_end_pos": [172, 35]}, {"full_name": "MeasureTheory.Lp.posPart", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [1240, 5], "def_end_pos": [1240, 12]}, {"full_name": "MeasureTheory.L1.norm_def", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [1361, 9], "def_end_pos": [1361, 17]}, {"full_name": "MeasureTheory.lintegral_congr_ae", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [304, 9], "def_end_pos": [304, 27]}, {"full_name": "MeasureTheory.Lp.coeFn_posPart", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [1254, 9], "def_end_pos": [1254, 22]}, {"full_name": "ENNReal.ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [172, 29], "def_end_pos": [172, 35]}, {"full_name": "NNReal.eq", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [95, 26], "def_end_pos": [95, 28]}, {"full_name": "Real.nnnorm_of_nonneg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [1800, 9], "def_end_pos": [1800, 25]}, {"full_name": "le_max_right", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [61, 9], "def_end_pos": [61, 21]}, {"full_name": "Real.coe_toNNReal'", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [615, 9], "def_end_pos": [615, 22]}, {"full_name": "NNReal.coe_mk", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [137, 28], "def_end_pos": [137, 34]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\nf\u2081 : { x // x \u2208 Lp \u211d 1 } := Integrable.toL1 f hf\n\u22a2 \u222b (a : \u03b1), f a \u2202\u03bc =\n    ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc) - ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (-f a) \u2202\u03bc)", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\nf\u2081 : { x // x \u2208 Lp \u211d 1 } := Integrable.toL1 f hf\neq\u2081 : ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc) = \u2016Lp.posPart f\u2081\u2016\n\u22a2 \u222b (a : \u03b1), f a \u2202\u03bc =\n    ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc) - ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (-f a) \u2202\u03bc)"}, {"tactic": "have eq\u2082 : ENNReal.toReal (\u222b\u207b a, ENNReal.ofReal (-f a) \u2202\u03bc) = \u2016Lp.negPart f\u2081\u2016 := by\n  rw [L1.norm_def]\n  congr 1\n  apply lintegral_congr_ae\n  filter_upwards [Lp.coeFn_negPart f\u2081, hf.coeFn_toL1] with _ h\u2081 h\u2082\n  rw [h\u2081, h\u2082, ENNReal.ofReal]\n  congr 1\n  apply NNReal.eq\n  simp only [Real.coe_toNNReal', coe_nnnorm, nnnorm_neg]\n  rw [Real.norm_of_nonpos (min_le_right _ _), \u2190 max_neg_neg, neg_zero]", "annotated_tactic": ["have eq\u2082 : <a>ENNReal.toReal</a> (\u222b\u207b a, <a>ENNReal.ofReal</a> (-f a) \u2202\u03bc) = \u2016<a>Lp.negPart</a> f\u2081\u2016 := by\n    rw [<a>L1.norm_def</a>]\n    congr 1\n    apply <a>lintegral_congr_ae</a>\n    filter_upwards [<a>Lp.coeFn_negPart</a> f\u2081, hf.coeFn_toL1] with _ h\u2081 h\u2082\n    rw [h\u2081, h\u2082, <a>ENNReal.ofReal</a>]\n    congr 1\n    apply <a>NNReal.eq</a>\n    simp only [<a>Real.coe_toNNReal'</a>, <a>coe_nnnorm</a>, <a>nnnorm_neg</a>]\n    rw [<a>Real.norm_of_nonpos</a> (<a>min_le_right</a> _ _), \u2190 <a>max_neg_neg</a>, <a>neg_zero</a>]", [{"full_name": "ENNReal.toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [168, 15], "def_end_pos": [168, 21]}, {"full_name": "ENNReal.ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [172, 29], "def_end_pos": [172, 35]}, {"full_name": "MeasureTheory.Lp.negPart", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [1245, 5], "def_end_pos": [1245, 12]}, {"full_name": "MeasureTheory.L1.norm_def", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [1361, 9], "def_end_pos": [1361, 17]}, {"full_name": "MeasureTheory.lintegral_congr_ae", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [304, 9], "def_end_pos": [304, 27]}, {"full_name": "MeasureTheory.Lp.coeFn_negPart", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [1264, 9], "def_end_pos": [1264, 22]}, {"full_name": "ENNReal.ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [172, 29], "def_end_pos": [172, 35]}, {"full_name": "NNReal.eq", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [95, 26], "def_end_pos": [95, 28]}, {"full_name": "Real.coe_toNNReal'", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [615, 9], "def_end_pos": [615, 22]}, {"full_name": "coe_nnnorm", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [905, 41], "def_end_pos": [905, 51]}, {"full_name": "nnnorm_neg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [952, 30], "def_end_pos": [952, 40]}, {"full_name": "Real.norm_of_nonpos", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [1772, 9], "def_end_pos": [1772, 23]}, {"full_name": "min_le_right", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [40, 9], "def_end_pos": [40, 21]}, {"full_name": "max_neg_neg", "def_path": "Mathlib/Algebra/Order/Group/MinMax.lean", "def_pos": [43, 15], "def_end_pos": [43, 26]}, {"full_name": "neg_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [1014, 3], "def_end_pos": [1014, 14]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\nf\u2081 : { x // x \u2208 Lp \u211d 1 } := Integrable.toL1 f hf\neq\u2081 : ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc) = \u2016Lp.posPart f\u2081\u2016\n\u22a2 \u222b (a : \u03b1), f a \u2202\u03bc =\n    ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc) - ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (-f a) \u2202\u03bc)", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\nf\u2081 : { x // x \u2208 Lp \u211d 1 } := Integrable.toL1 f hf\neq\u2081 : ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc) = \u2016Lp.posPart f\u2081\u2016\neq\u2082 : ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (-f a) \u2202\u03bc) = \u2016Lp.negPart f\u2081\u2016\n\u22a2 \u222b (a : \u03b1), f a \u2202\u03bc =\n    ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc) - ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (-f a) \u2202\u03bc)"}, {"tactic": "rw [eq\u2081, eq\u2082, integral, dif_pos, dif_pos]", "annotated_tactic": ["rw [eq\u2081, eq\u2082, <a>integral</a>, <a>dif_pos</a>, <a>dif_pos</a>]", [{"full_name": "MeasureTheory.integral", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [791, 17], "def_end_pos": [791, 25]}, {"full_name": "dif_pos", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [807, 9], "def_end_pos": [807, 16]}, {"full_name": "dif_pos", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [807, 9], "def_end_pos": [807, 16]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\nf\u2081 : { x // x \u2208 Lp \u211d 1 } := Integrable.toL1 f hf\neq\u2081 : ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc) = \u2016Lp.posPart f\u2081\u2016\neq\u2082 : ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (-f a) \u2202\u03bc) = \u2016Lp.negPart f\u2081\u2016\n\u22a2 \u222b (a : \u03b1), f a \u2202\u03bc =\n    ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc) - ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (-f a) \u2202\u03bc)", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\nf\u2081 : { x // x \u2208 Lp \u211d 1 } := Integrable.toL1 f hf\neq\u2081 : ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc) = \u2016Lp.posPart f\u2081\u2016\neq\u2082 : ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (-f a) \u2202\u03bc) = \u2016Lp.negPart f\u2081\u2016\n\u22a2 L1.integral (Integrable.toL1 (fun a => f a) ?hc) = \u2016Lp.posPart f\u2081\u2016 - \u2016Lp.negPart f\u2081\u2016\n\ncase hc\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\nf\u2081 : { x // x \u2208 Lp \u211d 1 } := Integrable.toL1 f hf\neq\u2081 : ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc) = \u2016Lp.posPart f\u2081\u2016\neq\u2082 : ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (-f a) \u2202\u03bc) = \u2016Lp.negPart f\u2081\u2016\n\u22a2 Integrable fun a => f a\n\ncase hc\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\nf\u2081 : { x // x \u2208 Lp \u211d 1 } := Integrable.toL1 f hf\neq\u2081 : ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc) = \u2016Lp.posPart f\u2081\u2016\neq\u2082 : ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (-f a) \u2202\u03bc) = \u2016Lp.negPart f\u2081\u2016\n\u22a2 CompleteSpace \u211d\n\ncase hc\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\nf\u2081 : { x // x \u2208 Lp \u211d 1 } := Integrable.toL1 f hf\neq\u2081 : ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc) = \u2016Lp.posPart f\u2081\u2016\neq\u2082 : ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (-f a) \u2202\u03bc) = \u2016Lp.negPart f\u2081\u2016\n\u22a2 CompleteSpace \u211d"}, {"tactic": "exact L1.integral_eq_norm_posPart_sub _", "annotated_tactic": ["exact <a>L1.integral_eq_norm_posPart_sub</a> _", [{"full_name": "MeasureTheory.L1.integral_eq_norm_posPart_sub", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [753, 9], "def_end_pos": [753, 37]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\nf\u2081 : { x // x \u2208 Lp \u211d 1 } := Integrable.toL1 f hf\neq\u2081 : ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc) = \u2016Lp.posPart f\u2081\u2016\neq\u2082 : ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (-f a) \u2202\u03bc) = \u2016Lp.negPart f\u2081\u2016\n\u22a2 L1.integral (Integrable.toL1 (fun a => f a) ?hc) = \u2016Lp.posPart f\u2081\u2016 - \u2016Lp.negPart f\u2081\u2016\n\ncase hc\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\nf\u2081 : { x // x \u2208 Lp \u211d 1 } := Integrable.toL1 f hf\neq\u2081 : ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc) = \u2016Lp.posPart f\u2081\u2016\neq\u2082 : ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (-f a) \u2202\u03bc) = \u2016Lp.negPart f\u2081\u2016\n\u22a2 Integrable fun a => f a\n\ncase hc\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\nf\u2081 : { x // x \u2208 Lp \u211d 1 } := Integrable.toL1 f hf\neq\u2081 : ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc) = \u2016Lp.posPart f\u2081\u2016\neq\u2082 : ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (-f a) \u2202\u03bc) = \u2016Lp.negPart f\u2081\u2016\n\u22a2 CompleteSpace \u211d\n\ncase hc\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\nf\u2081 : { x // x \u2208 Lp \u211d 1 } := Integrable.toL1 f hf\neq\u2081 : ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc) = \u2016Lp.posPart f\u2081\u2016\neq\u2082 : ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (-f a) \u2202\u03bc) = \u2016Lp.negPart f\u2081\u2016\n\u22a2 CompleteSpace \u211d", "state_after": "no goals"}, {"tactic": "rw [L1.norm_def]", "annotated_tactic": ["rw [<a>L1.norm_def</a>]", [{"full_name": "MeasureTheory.L1.norm_def", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [1361, 9], "def_end_pos": [1361, 17]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\nf\u2081 : { x // x \u2208 Lp \u211d 1 } := Integrable.toL1 f hf\n\u22a2 ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc) = \u2016Lp.posPart f\u2081\u2016", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\nf\u2081 : { x // x \u2208 Lp \u211d 1 } := Integrable.toL1 f hf\n\u22a2 ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc) = ENNReal.toReal (\u222b\u207b (a : \u03b1), \u2191\u2016\u2191\u2191(Lp.posPart f\u2081) a\u2016\u208a \u2202\u03bc)"}, {"tactic": "congr 1", "annotated_tactic": ["congr 1", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\nf\u2081 : { x // x \u2208 Lp \u211d 1 } := Integrable.toL1 f hf\n\u22a2 ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc) = ENNReal.toReal (\u222b\u207b (a : \u03b1), \u2191\u2016\u2191\u2191(Lp.posPart f\u2081) a\u2016\u208a \u2202\u03bc)", "state_after": "case e_a\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\nf\u2081 : { x // x \u2208 Lp \u211d 1 } := Integrable.toL1 f hf\n\u22a2 \u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc = \u222b\u207b (a : \u03b1), \u2191\u2016\u2191\u2191(Lp.posPart f\u2081) a\u2016\u208a \u2202\u03bc"}, {"tactic": "apply lintegral_congr_ae", "annotated_tactic": ["apply <a>lintegral_congr_ae</a>", [{"full_name": "MeasureTheory.lintegral_congr_ae", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [304, 9], "def_end_pos": [304, 27]}]], "state_before": "case e_a\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\nf\u2081 : { x // x \u2208 Lp \u211d 1 } := Integrable.toL1 f hf\n\u22a2 \u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc = \u222b\u207b (a : \u03b1), \u2191\u2016\u2191\u2191(Lp.posPart f\u2081) a\u2016\u208a \u2202\u03bc", "state_after": "case e_a.h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\nf\u2081 : { x // x \u2208 Lp \u211d 1 } := Integrable.toL1 f hf\n\u22a2 (fun a => ENNReal.ofReal (f a)) =\u1d50[\u03bc] fun a => \u2191\u2016\u2191\u2191(Lp.posPart f\u2081) a\u2016\u208a"}, {"tactic": "filter_upwards [Lp.coeFn_posPart f\u2081, hf.coeFn_toL1] with _ h\u2081 h\u2082", "annotated_tactic": ["filter_upwards [<a>Lp.coeFn_posPart</a> f\u2081, hf.coeFn_toL1] with _ h\u2081 h\u2082", [{"full_name": "MeasureTheory.Lp.coeFn_posPart", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [1254, 9], "def_end_pos": [1254, 22]}]], "state_before": "case e_a.h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\nf\u2081 : { x // x \u2208 Lp \u211d 1 } := Integrable.toL1 f hf\n\u22a2 (fun a => ENNReal.ofReal (f a)) =\u1d50[\u03bc] fun a => \u2191\u2016\u2191\u2191(Lp.posPart f\u2081) a\u2016\u208a", "state_after": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\nf\u2081 : { x // x \u2208 Lp \u211d 1 } := Integrable.toL1 f hf\na\u271d : \u03b1\nh\u2081 : \u2191\u2191(Lp.posPart f\u2081) a\u271d = max (\u2191\u2191f\u2081 a\u271d) 0\nh\u2082 : \u2191\u2191(Integrable.toL1 f hf) a\u271d = f a\u271d\n\u22a2 ENNReal.ofReal (f a\u271d) = \u2191\u2016\u2191\u2191(Lp.posPart f\u2081) a\u271d\u2016\u208a"}, {"tactic": "rw [h\u2081, h\u2082, ENNReal.ofReal]", "annotated_tactic": ["rw [h\u2081, h\u2082, <a>ENNReal.ofReal</a>]", [{"full_name": "ENNReal.ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [172, 29], "def_end_pos": [172, 35]}]], "state_before": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\nf\u2081 : { x // x \u2208 Lp \u211d 1 } := Integrable.toL1 f hf\na\u271d : \u03b1\nh\u2081 : \u2191\u2191(Lp.posPart f\u2081) a\u271d = max (\u2191\u2191f\u2081 a\u271d) 0\nh\u2082 : \u2191\u2191(Integrable.toL1 f hf) a\u271d = f a\u271d\n\u22a2 ENNReal.ofReal (f a\u271d) = \u2191\u2016\u2191\u2191(Lp.posPart f\u2081) a\u271d\u2016\u208a", "state_after": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\nf\u2081 : { x // x \u2208 Lp \u211d 1 } := Integrable.toL1 f hf\na\u271d : \u03b1\nh\u2081 : \u2191\u2191(Lp.posPart f\u2081) a\u271d = max (\u2191\u2191f\u2081 a\u271d) 0\nh\u2082 : \u2191\u2191(Integrable.toL1 f hf) a\u271d = f a\u271d\n\u22a2 \u2191(Real.toNNReal (f a\u271d)) = \u2191\u2016max (f a\u271d) 0\u2016\u208a"}, {"tactic": "congr 1", "annotated_tactic": ["congr 1", []], "state_before": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\nf\u2081 : { x // x \u2208 Lp \u211d 1 } := Integrable.toL1 f hf\na\u271d : \u03b1\nh\u2081 : \u2191\u2191(Lp.posPart f\u2081) a\u271d = max (\u2191\u2191f\u2081 a\u271d) 0\nh\u2082 : \u2191\u2191(Integrable.toL1 f hf) a\u271d = f a\u271d\n\u22a2 \u2191(Real.toNNReal (f a\u271d)) = \u2191\u2016max (f a\u271d) 0\u2016\u208a", "state_after": "case h.e_a\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\nf\u2081 : { x // x \u2208 Lp \u211d 1 } := Integrable.toL1 f hf\na\u271d : \u03b1\nh\u2081 : \u2191\u2191(Lp.posPart f\u2081) a\u271d = max (\u2191\u2191f\u2081 a\u271d) 0\nh\u2082 : \u2191\u2191(Integrable.toL1 f hf) a\u271d = f a\u271d\n\u22a2 Real.toNNReal (f a\u271d) = \u2016max (f a\u271d) 0\u2016\u208a"}, {"tactic": "apply NNReal.eq", "annotated_tactic": ["apply <a>NNReal.eq</a>", [{"full_name": "NNReal.eq", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [95, 26], "def_end_pos": [95, 28]}]], "state_before": "case h.e_a\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\nf\u2081 : { x // x \u2208 Lp \u211d 1 } := Integrable.toL1 f hf\na\u271d : \u03b1\nh\u2081 : \u2191\u2191(Lp.posPart f\u2081) a\u271d = max (\u2191\u2191f\u2081 a\u271d) 0\nh\u2082 : \u2191\u2191(Integrable.toL1 f hf) a\u271d = f a\u271d\n\u22a2 Real.toNNReal (f a\u271d) = \u2016max (f a\u271d) 0\u2016\u208a", "state_after": "case h.e_a.a\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\nf\u2081 : { x // x \u2208 Lp \u211d 1 } := Integrable.toL1 f hf\na\u271d : \u03b1\nh\u2081 : \u2191\u2191(Lp.posPart f\u2081) a\u271d = max (\u2191\u2191f\u2081 a\u271d) 0\nh\u2082 : \u2191\u2191(Integrable.toL1 f hf) a\u271d = f a\u271d\n\u22a2 \u2191(Real.toNNReal (f a\u271d)) = \u2191\u2016max (f a\u271d) 0\u2016\u208a"}, {"tactic": "rw [Real.nnnorm_of_nonneg (le_max_right _ _)]", "annotated_tactic": ["rw [<a>Real.nnnorm_of_nonneg</a> (<a>le_max_right</a> _ _)]", [{"full_name": "Real.nnnorm_of_nonneg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [1800, 9], "def_end_pos": [1800, 25]}, {"full_name": "le_max_right", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [61, 9], "def_end_pos": [61, 21]}]], "state_before": "case h.e_a.a\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\nf\u2081 : { x // x \u2208 Lp \u211d 1 } := Integrable.toL1 f hf\na\u271d : \u03b1\nh\u2081 : \u2191\u2191(Lp.posPart f\u2081) a\u271d = max (\u2191\u2191f\u2081 a\u271d) 0\nh\u2082 : \u2191\u2191(Integrable.toL1 f hf) a\u271d = f a\u271d\n\u22a2 \u2191(Real.toNNReal (f a\u271d)) = \u2191\u2016max (f a\u271d) 0\u2016\u208a", "state_after": "case h.e_a.a\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\nf\u2081 : { x // x \u2208 Lp \u211d 1 } := Integrable.toL1 f hf\na\u271d : \u03b1\nh\u2081 : \u2191\u2191(Lp.posPart f\u2081) a\u271d = max (\u2191\u2191f\u2081 a\u271d) 0\nh\u2082 : \u2191\u2191(Integrable.toL1 f hf) a\u271d = f a\u271d\n\u22a2 \u2191(Real.toNNReal (f a\u271d)) = \u2191{ val := max (f a\u271d) 0, property := (_ : 0 \u2264 max (f a\u271d) 0) }"}, {"tactic": "rw [Real.coe_toNNReal', NNReal.coe_mk]", "annotated_tactic": ["rw [<a>Real.coe_toNNReal'</a>, <a>NNReal.coe_mk</a>]", [{"full_name": "Real.coe_toNNReal'", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [615, 9], "def_end_pos": [615, 22]}, {"full_name": "NNReal.coe_mk", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [137, 28], "def_end_pos": [137, 34]}]], "state_before": "case h.e_a.a\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\nf\u2081 : { x // x \u2208 Lp \u211d 1 } := Integrable.toL1 f hf\na\u271d : \u03b1\nh\u2081 : \u2191\u2191(Lp.posPart f\u2081) a\u271d = max (\u2191\u2191f\u2081 a\u271d) 0\nh\u2082 : \u2191\u2191(Integrable.toL1 f hf) a\u271d = f a\u271d\n\u22a2 \u2191(Real.toNNReal (f a\u271d)) = \u2191{ val := max (f a\u271d) 0, property := (_ : 0 \u2264 max (f a\u271d) 0) }", "state_after": "no goals"}, {"tactic": "rw [L1.norm_def]", "annotated_tactic": ["rw [<a>L1.norm_def</a>]", [{"full_name": "MeasureTheory.L1.norm_def", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [1361, 9], "def_end_pos": [1361, 17]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\nf\u2081 : { x // x \u2208 Lp \u211d 1 } := Integrable.toL1 f hf\neq\u2081 : ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc) = \u2016Lp.posPart f\u2081\u2016\n\u22a2 ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (-f a) \u2202\u03bc) = \u2016Lp.negPart f\u2081\u2016", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\nf\u2081 : { x // x \u2208 Lp \u211d 1 } := Integrable.toL1 f hf\neq\u2081 : ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc) = \u2016Lp.posPart f\u2081\u2016\n\u22a2 ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (-f a) \u2202\u03bc) = ENNReal.toReal (\u222b\u207b (a : \u03b1), \u2191\u2016\u2191\u2191(Lp.negPart f\u2081) a\u2016\u208a \u2202\u03bc)"}, {"tactic": "congr 1", "annotated_tactic": ["congr 1", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\nf\u2081 : { x // x \u2208 Lp \u211d 1 } := Integrable.toL1 f hf\neq\u2081 : ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc) = \u2016Lp.posPart f\u2081\u2016\n\u22a2 ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (-f a) \u2202\u03bc) = ENNReal.toReal (\u222b\u207b (a : \u03b1), \u2191\u2016\u2191\u2191(Lp.negPart f\u2081) a\u2016\u208a \u2202\u03bc)", "state_after": "case e_a\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\nf\u2081 : { x // x \u2208 Lp \u211d 1 } := Integrable.toL1 f hf\neq\u2081 : ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc) = \u2016Lp.posPart f\u2081\u2016\n\u22a2 \u222b\u207b (a : \u03b1), ENNReal.ofReal (-f a) \u2202\u03bc = \u222b\u207b (a : \u03b1), \u2191\u2016\u2191\u2191(Lp.negPart f\u2081) a\u2016\u208a \u2202\u03bc"}, {"tactic": "apply lintegral_congr_ae", "annotated_tactic": ["apply <a>lintegral_congr_ae</a>", [{"full_name": "MeasureTheory.lintegral_congr_ae", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [304, 9], "def_end_pos": [304, 27]}]], "state_before": "case e_a\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\nf\u2081 : { x // x \u2208 Lp \u211d 1 } := Integrable.toL1 f hf\neq\u2081 : ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc) = \u2016Lp.posPart f\u2081\u2016\n\u22a2 \u222b\u207b (a : \u03b1), ENNReal.ofReal (-f a) \u2202\u03bc = \u222b\u207b (a : \u03b1), \u2191\u2016\u2191\u2191(Lp.negPart f\u2081) a\u2016\u208a \u2202\u03bc", "state_after": "case e_a.h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\nf\u2081 : { x // x \u2208 Lp \u211d 1 } := Integrable.toL1 f hf\neq\u2081 : ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc) = \u2016Lp.posPart f\u2081\u2016\n\u22a2 (fun a => ENNReal.ofReal (-f a)) =\u1d50[\u03bc] fun a => \u2191\u2016\u2191\u2191(Lp.negPart f\u2081) a\u2016\u208a"}, {"tactic": "filter_upwards [Lp.coeFn_negPart f\u2081, hf.coeFn_toL1] with _ h\u2081 h\u2082", "annotated_tactic": ["filter_upwards [<a>Lp.coeFn_negPart</a> f\u2081, hf.coeFn_toL1] with _ h\u2081 h\u2082", [{"full_name": "MeasureTheory.Lp.coeFn_negPart", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [1264, 9], "def_end_pos": [1264, 22]}]], "state_before": "case e_a.h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\nf\u2081 : { x // x \u2208 Lp \u211d 1 } := Integrable.toL1 f hf\neq\u2081 : ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc) = \u2016Lp.posPart f\u2081\u2016\n\u22a2 (fun a => ENNReal.ofReal (-f a)) =\u1d50[\u03bc] fun a => \u2191\u2016\u2191\u2191(Lp.negPart f\u2081) a\u2016\u208a", "state_after": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\nf\u2081 : { x // x \u2208 Lp \u211d 1 } := Integrable.toL1 f hf\neq\u2081 : ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc) = \u2016Lp.posPart f\u2081\u2016\na\u271d : \u03b1\nh\u2081 : \u2191\u2191(Lp.negPart f\u2081) a\u271d = -min (\u2191\u2191f\u2081 a\u271d) 0\nh\u2082 : \u2191\u2191(Integrable.toL1 f hf) a\u271d = f a\u271d\n\u22a2 ENNReal.ofReal (-f a\u271d) = \u2191\u2016\u2191\u2191(Lp.negPart f\u2081) a\u271d\u2016\u208a"}, {"tactic": "rw [h\u2081, h\u2082, ENNReal.ofReal]", "annotated_tactic": ["rw [h\u2081, h\u2082, <a>ENNReal.ofReal</a>]", [{"full_name": "ENNReal.ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [172, 29], "def_end_pos": [172, 35]}]], "state_before": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\nf\u2081 : { x // x \u2208 Lp \u211d 1 } := Integrable.toL1 f hf\neq\u2081 : ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc) = \u2016Lp.posPart f\u2081\u2016\na\u271d : \u03b1\nh\u2081 : \u2191\u2191(Lp.negPart f\u2081) a\u271d = -min (\u2191\u2191f\u2081 a\u271d) 0\nh\u2082 : \u2191\u2191(Integrable.toL1 f hf) a\u271d = f a\u271d\n\u22a2 ENNReal.ofReal (-f a\u271d) = \u2191\u2016\u2191\u2191(Lp.negPart f\u2081) a\u271d\u2016\u208a", "state_after": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\nf\u2081 : { x // x \u2208 Lp \u211d 1 } := Integrable.toL1 f hf\neq\u2081 : ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc) = \u2016Lp.posPart f\u2081\u2016\na\u271d : \u03b1\nh\u2081 : \u2191\u2191(Lp.negPart f\u2081) a\u271d = -min (\u2191\u2191f\u2081 a\u271d) 0\nh\u2082 : \u2191\u2191(Integrable.toL1 f hf) a\u271d = f a\u271d\n\u22a2 \u2191(Real.toNNReal (-f a\u271d)) = \u2191\u2016-min (f a\u271d) 0\u2016\u208a"}, {"tactic": "congr 1", "annotated_tactic": ["congr 1", []], "state_before": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\nf\u2081 : { x // x \u2208 Lp \u211d 1 } := Integrable.toL1 f hf\neq\u2081 : ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc) = \u2016Lp.posPart f\u2081\u2016\na\u271d : \u03b1\nh\u2081 : \u2191\u2191(Lp.negPart f\u2081) a\u271d = -min (\u2191\u2191f\u2081 a\u271d) 0\nh\u2082 : \u2191\u2191(Integrable.toL1 f hf) a\u271d = f a\u271d\n\u22a2 \u2191(Real.toNNReal (-f a\u271d)) = \u2191\u2016-min (f a\u271d) 0\u2016\u208a", "state_after": "case h.e_a\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\nf\u2081 : { x // x \u2208 Lp \u211d 1 } := Integrable.toL1 f hf\neq\u2081 : ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc) = \u2016Lp.posPart f\u2081\u2016\na\u271d : \u03b1\nh\u2081 : \u2191\u2191(Lp.negPart f\u2081) a\u271d = -min (\u2191\u2191f\u2081 a\u271d) 0\nh\u2082 : \u2191\u2191(Integrable.toL1 f hf) a\u271d = f a\u271d\n\u22a2 Real.toNNReal (-f a\u271d) = \u2016-min (f a\u271d) 0\u2016\u208a"}, {"tactic": "apply NNReal.eq", "annotated_tactic": ["apply <a>NNReal.eq</a>", [{"full_name": "NNReal.eq", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [95, 26], "def_end_pos": [95, 28]}]], "state_before": "case h.e_a\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\nf\u2081 : { x // x \u2208 Lp \u211d 1 } := Integrable.toL1 f hf\neq\u2081 : ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc) = \u2016Lp.posPart f\u2081\u2016\na\u271d : \u03b1\nh\u2081 : \u2191\u2191(Lp.negPart f\u2081) a\u271d = -min (\u2191\u2191f\u2081 a\u271d) 0\nh\u2082 : \u2191\u2191(Integrable.toL1 f hf) a\u271d = f a\u271d\n\u22a2 Real.toNNReal (-f a\u271d) = \u2016-min (f a\u271d) 0\u2016\u208a", "state_after": "case h.e_a.a\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\nf\u2081 : { x // x \u2208 Lp \u211d 1 } := Integrable.toL1 f hf\neq\u2081 : ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc) = \u2016Lp.posPart f\u2081\u2016\na\u271d : \u03b1\nh\u2081 : \u2191\u2191(Lp.negPart f\u2081) a\u271d = -min (\u2191\u2191f\u2081 a\u271d) 0\nh\u2082 : \u2191\u2191(Integrable.toL1 f hf) a\u271d = f a\u271d\n\u22a2 \u2191(Real.toNNReal (-f a\u271d)) = \u2191\u2016-min (f a\u271d) 0\u2016\u208a"}, {"tactic": "simp only [Real.coe_toNNReal', coe_nnnorm, nnnorm_neg]", "annotated_tactic": ["simp only [<a>Real.coe_toNNReal'</a>, <a>coe_nnnorm</a>, <a>nnnorm_neg</a>]", [{"full_name": "Real.coe_toNNReal'", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [615, 9], "def_end_pos": [615, 22]}, {"full_name": "coe_nnnorm", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [905, 41], "def_end_pos": [905, 51]}, {"full_name": "nnnorm_neg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [952, 30], "def_end_pos": [952, 40]}]], "state_before": "case h.e_a.a\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\nf\u2081 : { x // x \u2208 Lp \u211d 1 } := Integrable.toL1 f hf\neq\u2081 : ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc) = \u2016Lp.posPart f\u2081\u2016\na\u271d : \u03b1\nh\u2081 : \u2191\u2191(Lp.negPart f\u2081) a\u271d = -min (\u2191\u2191f\u2081 a\u271d) 0\nh\u2082 : \u2191\u2191(Integrable.toL1 f hf) a\u271d = f a\u271d\n\u22a2 \u2191(Real.toNNReal (-f a\u271d)) = \u2191\u2016-min (f a\u271d) 0\u2016\u208a", "state_after": "case h.e_a.a\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\nf\u2081 : { x // x \u2208 Lp \u211d 1 } := Integrable.toL1 f hf\neq\u2081 : ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc) = \u2016Lp.posPart f\u2081\u2016\na\u271d : \u03b1\nh\u2081 : \u2191\u2191(Lp.negPart f\u2081) a\u271d = -min (\u2191\u2191f\u2081 a\u271d) 0\nh\u2082 : \u2191\u2191(Integrable.toL1 f hf) a\u271d = f a\u271d\n\u22a2 max (-f a\u271d) 0 = \u2016min (f a\u271d) 0\u2016"}, {"tactic": "rw [Real.norm_of_nonpos (min_le_right _ _), \u2190 max_neg_neg, neg_zero]", "annotated_tactic": ["rw [<a>Real.norm_of_nonpos</a> (<a>min_le_right</a> _ _), \u2190 <a>max_neg_neg</a>, <a>neg_zero</a>]", [{"full_name": "Real.norm_of_nonpos", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [1772, 9], "def_end_pos": [1772, 23]}, {"full_name": "min_le_right", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [40, 9], "def_end_pos": [40, 21]}, {"full_name": "max_neg_neg", "def_path": "Mathlib/Algebra/Order/Group/MinMax.lean", "def_pos": [43, 15], "def_end_pos": [43, 26]}, {"full_name": "neg_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [1014, 3], "def_end_pos": [1014, 14]}]], "state_before": "case h.e_a.a\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\nf\u2081 : { x // x \u2208 Lp \u211d 1 } := Integrable.toL1 f hf\neq\u2081 : ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc) = \u2016Lp.posPart f\u2081\u2016\na\u271d : \u03b1\nh\u2081 : \u2191\u2191(Lp.negPart f\u2081) a\u271d = -min (\u2191\u2191f\u2081 a\u271d) 0\nh\u2082 : \u2191\u2191(Integrable.toL1 f hf) a\u271d = f a\u271d\n\u22a2 max (-f a\u271d) 0 = \u2016min (f a\u271d) 0\u2016", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "full_name": "MeasureTheory.HasFiniteIntegral.tendsto_set_integral_nhds_zero", "start": [987, 1], "end": [995, 53], "traced_tactics": [{"tactic": "rw [tendsto_zero_iff_norm_tendsto_zero]", "annotated_tactic": ["rw [<a>tendsto_zero_iff_norm_tendsto_zero</a>]", [{"full_name": "tendsto_zero_iff_norm_tendsto_zero", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [1086, 3], "def_end_pos": [1086, 14]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b9 : Type u_6\nf : \u03b1 \u2192 G\nhf : HasFiniteIntegral f\nl : Filter \u03b9\ns : \u03b9 \u2192 Set \u03b1\nhs : Tendsto (\u2191\u2191\u03bc \u2218 s) l (\ud835\udcdd 0)\n\u22a2 Tendsto (fun i => \u222b (x : \u03b1) in s i, f x \u2202\u03bc) l (\ud835\udcdd 0)", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b9 : Type u_6\nf : \u03b1 \u2192 G\nhf : HasFiniteIntegral f\nl : Filter \u03b9\ns : \u03b9 \u2192 Set \u03b1\nhs : Tendsto (\u2191\u2191\u03bc \u2218 s) l (\ud835\udcdd 0)\n\u22a2 Tendsto (fun x => \u2016\u222b (x : \u03b1) in s x, f x \u2202\u03bc\u2016) l (\ud835\udcdd 0)"}, {"tactic": "simp_rw [\u2190 coe_nnnorm, \u2190 NNReal.coe_zero, NNReal.tendsto_coe, \u2190 ENNReal.tendsto_coe,\n  ENNReal.coe_zero]", "annotated_tactic": ["simp_rw [\u2190 <a>coe_nnnorm</a>, \u2190 <a>NNReal.coe_zero</a>, <a>NNReal.tendsto_coe</a>, \u2190 <a>ENNReal.tendsto_coe</a>,\n    <a>ENNReal.coe_zero</a>]", [{"full_name": "coe_nnnorm", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [905, 41], "def_end_pos": [905, 51]}, {"full_name": "NNReal.coe_zero", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [171, 19], "def_end_pos": [171, 27]}, {"full_name": "NNReal.tendsto_coe", "def_path": "Mathlib/Topology/Instances/NNReal.lean", "def_pos": [104, 9], "def_end_pos": [104, 20]}, {"full_name": "ENNReal.tendsto_coe", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [71, 9], "def_end_pos": [71, 20]}, {"full_name": "ENNReal.coe_zero", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [215, 28], "def_end_pos": [215, 36]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b9 : Type u_6\nf : \u03b1 \u2192 G\nhf : HasFiniteIntegral f\nl : Filter \u03b9\ns : \u03b9 \u2192 Set \u03b1\nhs : Tendsto (\u2191\u2191\u03bc \u2218 s) l (\ud835\udcdd 0)\n\u22a2 Tendsto (fun x => \u2016\u222b (x : \u03b1) in s x, f x \u2202\u03bc\u2016) l (\ud835\udcdd 0)", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b9 : Type u_6\nf : \u03b1 \u2192 G\nhf : HasFiniteIntegral f\nl : Filter \u03b9\ns : \u03b9 \u2192 Set \u03b1\nhs : Tendsto (\u2191\u2191\u03bc \u2218 s) l (\ud835\udcdd 0)\n\u22a2 Tendsto (fun a => \u2191\u2016\u222b (x : \u03b1) in s a, f x \u2202\u03bc\u2016\u208a) l (\ud835\udcdd 0)"}, {"tactic": "exact tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds\n  (tendsto_set_lintegral_zero (ne_of_lt hf) hs) (fun i => zero_le _)\n  fun i => ennnorm_integral_le_lintegral_ennnorm _", "annotated_tactic": ["exact <a>tendsto_of_tendsto_of_tendsto_of_le_of_le</a> <a>tendsto_const_nhds</a>\n    (<a>tendsto_set_lintegral_zero</a> (<a>ne_of_lt</a> hf) hs) (fun i => <a>zero_le</a> _)\n    fun i => <a>ennnorm_integral_le_lintegral_ennnorm</a> _", [{"full_name": "tendsto_of_tendsto_of_tendsto_of_le_of_le", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [955, 9], "def_end_pos": [955, 50]}, {"full_name": "tendsto_const_nhds", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [1049, 9], "def_end_pos": [1049, 27]}, {"full_name": "MeasureTheory.tendsto_set_lintegral_zero", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [499, 9], "def_end_pos": [499, 35]}, {"full_name": "ne_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [101, 9], "def_end_pos": [101, 17]}, {"full_name": "zero_le", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [217, 30], "def_end_pos": [217, 37]}, {"full_name": "MeasureTheory.ennnorm_integral_le_lintegral_ennnorm", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [974, 9], "def_end_pos": [974, 46]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b9 : Type u_6\nf : \u03b1 \u2192 G\nhf : HasFiniteIntegral f\nl : Filter \u03b9\ns : \u03b9 \u2192 Set \u03b1\nhs : Tendsto (\u2191\u2191\u03bc \u2218 s) l (\ud835\udcdd 0)\n\u22a2 Tendsto (fun a => \u2191\u2016\u222b (x : \u03b1) in s a, f x \u2202\u03bc\u2016\u208a) l (\ud835\udcdd 0)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Sigma.lean", "full_name": "Set.sigma_subset_iff", "start": [75, 1], "end": [76, 72], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Num/Lemmas.lean", "full_name": "Num.to_of_nat", "start": [488, 1], "end": [490, 70], "traced_tactics": [{"tactic": "rw [Nat.cast_zero, cast_zero]", "annotated_tactic": ["rw [<a>Nat.cast_zero</a>, <a>cast_zero</a>]", [{"full_name": "Nat.cast_zero", "def_path": "Mathlib/Data/Nat/Cast/Defs.lean", "def_pos": [114, 9], "def_end_pos": [114, 18]}, {"full_name": "Num.cast_zero", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [272, 9], "def_end_pos": [272, 18]}]], "state_before": "\u22a2 \u2191\u21910 = 0", "state_after": "no goals"}, {"tactic": "rw [Nat.cast_succ, add_one, succ_to_nat, to_of_nat n]", "annotated_tactic": ["rw [<a>Nat.cast_succ</a>, <a>add_one</a>, <a>succ_to_nat</a>, to_of_nat n]", [{"full_name": "Nat.cast_succ", "def_path": "Mathlib/Data/Nat/Cast/Defs.lean", "def_pos": [122, 9], "def_end_pos": [122, 18]}, {"full_name": "Num.add_one", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [214, 9], "def_end_pos": [214, 16]}, {"full_name": "Num.succ_to_nat", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [296, 9], "def_end_pos": [296, 20]}]], "state_before": "n : \u2115\n\u22a2 \u2191\u2191(n + 1) = n + 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Condensed/Equivalence.lean", "full_name": "Condensed.StoneanProfinite.coverDense.inducedTopology_Sieve_iff_EffectiveEpiFamily", "start": [170, 1], "end": [194, 8], "traced_tactics": [{"tactic": "refine \u27e8fun \u27e8\u03b1, _, Y, \u03c0, \u27e8H\u2081, H\u2082\u27e9\u27e9 \u21a6 ?_, fun hS \u21a6 ?_\u27e9", "annotated_tactic": ["refine \u27e8fun \u27e8\u03b1, _, Y, \u03c0, \u27e8H\u2081, H\u2082\u27e9\u27e9 \u21a6 ?_, fun hS \u21a6 ?_\u27e9", []], "state_before": "X : Stonean\nS : Sieve X\n\u22a2 (\u2203 \u03b1 x Y \u03c0, EffectiveEpiFamily Y \u03c0 \u2227 \u2200 (a : \u03b1), S.arrows (\u03c0 a)) \u2194\n    S \u2208 GrothendieckTopology.sieves (CoverDense.inducedTopology coverDense) X", "state_after": "case refine_1\nX : Stonean\nS : Sieve X\nx\u271d : \u2203 \u03b1 x Y \u03c0, EffectiveEpiFamily Y \u03c0 \u2227 \u2200 (a : \u03b1), S.arrows (\u03c0 a)\n\u03b1 : Type\nw\u271d : Fintype \u03b1\nY : \u03b1 \u2192 Stonean\n\u03c0 : (a : \u03b1) \u2192 Y a \u27f6 X\nH\u2081 : EffectiveEpiFamily Y \u03c0\nH\u2082 : \u2200 (a : \u03b1), S.arrows (\u03c0 a)\n\u22a2 S \u2208 GrothendieckTopology.sieves (CoverDense.inducedTopology coverDense) X\n\ncase refine_2\nX : Stonean\nS : Sieve X\nhS : S \u2208 GrothendieckTopology.sieves (CoverDense.inducedTopology coverDense) X\n\u22a2 \u2203 \u03b1 x Y \u03c0, EffectiveEpiFamily Y \u03c0 \u2227 \u2200 (a : \u03b1), S.arrows (\u03c0 a)"}, {"tactic": "apply (coherentTopology.mem_sieves_iff_hasEffectiveEpiFamily (Sieve.functorPushforward _ S)).mpr", "annotated_tactic": ["apply (<a>coherentTopology.mem_sieves_iff_hasEffectiveEpiFamily</a> (<a>Sieve.functorPushforward</a> _ S)).<a>mpr</a>", [{"full_name": "CategoryTheory.coherentTopology.mem_sieves_iff_hasEffectiveEpiFamily", "def_path": "Mathlib/CategoryTheory/Sites/Coherent.lean", "def_pos": [185, 9], "def_end_pos": [185, 62]}, {"full_name": "CategoryTheory.Sieve.functorPushforward", "def_path": "Mathlib/CategoryTheory/Sites/Sieves.lean", "def_pos": [632, 5], "def_end_pos": [632, 23]}, {"full_name": "Iff.mpr", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [92, 3], "def_end_pos": [92, 6]}]], "state_before": "case refine_1\nX : Stonean\nS : Sieve X\nx\u271d : \u2203 \u03b1 x Y \u03c0, EffectiveEpiFamily Y \u03c0 \u2227 \u2200 (a : \u03b1), S.arrows (\u03c0 a)\n\u03b1 : Type\nw\u271d : Fintype \u03b1\nY : \u03b1 \u2192 Stonean\n\u03c0 : (a : \u03b1) \u2192 Y a \u27f6 X\nH\u2081 : EffectiveEpiFamily Y \u03c0\nH\u2082 : \u2200 (a : \u03b1), S.arrows (\u03c0 a)\n\u22a2 S \u2208 GrothendieckTopology.sieves (CoverDense.inducedTopology coverDense) X", "state_after": "case refine_1\nX : Stonean\nS : Sieve X\nx\u271d : \u2203 \u03b1 x Y \u03c0, EffectiveEpiFamily Y \u03c0 \u2227 \u2200 (a : \u03b1), S.arrows (\u03c0 a)\n\u03b1 : Type\nw\u271d : Fintype \u03b1\nY : \u03b1 \u2192 Stonean\n\u03c0 : (a : \u03b1) \u2192 Y a \u27f6 X\nH\u2081 : EffectiveEpiFamily Y \u03c0\nH\u2082 : \u2200 (a : \u03b1), S.arrows (\u03c0 a)\n\u22a2 \u2203 \u03b1 x Y \u03c0, EffectiveEpiFamily Y \u03c0 \u2227 \u2200 (a : \u03b1), (Sieve.functorPushforward Stonean.toProfinite S).arrows (\u03c0 a)"}, {"tactic": "refine \u27e8\u03b1, inferInstance, fun i => Stonean.toProfinite.obj (Y i),\n  fun i => Stonean.toProfinite.map (\u03c0 i), \u27e8?_,\n  fun a => Sieve.image_mem_functorPushforward Stonean.toCompHaus S (H\u2082 a)\u27e9\u27e9", "annotated_tactic": ["refine \u27e8\u03b1, <a>inferInstance</a>, fun i => Stonean.toProfinite.obj (Y i),\n      fun i => Stonean.toProfinite.map (\u03c0 i), \u27e8?_,\n      fun a => <a>Sieve.image_mem_functorPushforward</a> <a>Stonean.toCompHaus</a> S (H\u2082 a)\u27e9\u27e9", [{"full_name": "inferInstance", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [86, 8], "def_end_pos": [86, 21]}, {"full_name": "CategoryTheory.Sieve.image_mem_functorPushforward", "def_path": "Mathlib/CategoryTheory/Sites/Sieves.lean", "def_pos": [729, 9], "def_end_pos": [729, 37]}, {"full_name": "Stonean.toCompHaus", "def_path": "Mathlib/Topology/Category/Stonean/Basic.lean", "def_pos": [86, 5], "def_end_pos": [86, 15]}]], "state_before": "case refine_1\nX : Stonean\nS : Sieve X\nx\u271d : \u2203 \u03b1 x Y \u03c0, EffectiveEpiFamily Y \u03c0 \u2227 \u2200 (a : \u03b1), S.arrows (\u03c0 a)\n\u03b1 : Type\nw\u271d : Fintype \u03b1\nY : \u03b1 \u2192 Stonean\n\u03c0 : (a : \u03b1) \u2192 Y a \u27f6 X\nH\u2081 : EffectiveEpiFamily Y \u03c0\nH\u2082 : \u2200 (a : \u03b1), S.arrows (\u03c0 a)\n\u22a2 \u2203 \u03b1 x Y \u03c0, EffectiveEpiFamily Y \u03c0 \u2227 \u2200 (a : \u03b1), (Sieve.functorPushforward Stonean.toProfinite S).arrows (\u03c0 a)", "state_after": "case refine_1\nX : Stonean\nS : Sieve X\nx\u271d : \u2203 \u03b1 x Y \u03c0, EffectiveEpiFamily Y \u03c0 \u2227 \u2200 (a : \u03b1), S.arrows (\u03c0 a)\n\u03b1 : Type\nw\u271d : Fintype \u03b1\nY : \u03b1 \u2192 Stonean\n\u03c0 : (a : \u03b1) \u2192 Y a \u27f6 X\nH\u2081 : EffectiveEpiFamily Y \u03c0\nH\u2082 : \u2200 (a : \u03b1), S.arrows (\u03c0 a)\n\u22a2 EffectiveEpiFamily (fun i => Stonean.toProfinite.obj (Y i)) fun i => Stonean.toProfinite.map (\u03c0 i)"}, {"tactic": "simp only [(Stonean.effectiveEpiFamily_tfae _ _).out 0 2] at H\u2081", "annotated_tactic": ["simp only [(<a>Stonean.effectiveEpiFamily_tfae</a> _ _).<a>out</a> 0 2] at H\u2081", [{"full_name": "Stonean.effectiveEpiFamily_tfae", "def_path": "Mathlib/Topology/Category/Stonean/EffectiveEpi.lean", "def_pos": [132, 9], "def_end_pos": [132, 32]}, {"full_name": "List.TFAE.out", "def_path": "Mathlib/Data/List/TFAE.lean", "def_pos": [72, 9], "def_end_pos": [72, 17]}]], "state_before": "case refine_1\nX : Stonean\nS : Sieve X\nx\u271d : \u2203 \u03b1 x Y \u03c0, EffectiveEpiFamily Y \u03c0 \u2227 \u2200 (a : \u03b1), S.arrows (\u03c0 a)\n\u03b1 : Type\nw\u271d : Fintype \u03b1\nY : \u03b1 \u2192 Stonean\n\u03c0 : (a : \u03b1) \u2192 Y a \u27f6 X\nH\u2081 : EffectiveEpiFamily Y \u03c0\nH\u2082 : \u2200 (a : \u03b1), S.arrows (\u03c0 a)\n\u22a2 EffectiveEpiFamily (fun i => Stonean.toProfinite.obj (Y i)) fun i => Stonean.toProfinite.map (\u03c0 i)", "state_after": "case refine_1\nX : Stonean\nS : Sieve X\nx\u271d : \u2203 \u03b1 x Y \u03c0, EffectiveEpiFamily Y \u03c0 \u2227 \u2200 (a : \u03b1), S.arrows (\u03c0 a)\n\u03b1 : Type\nw\u271d : Fintype \u03b1\nY : \u03b1 \u2192 Stonean\n\u03c0 : (a : \u03b1) \u2192 Y a \u27f6 X\nH\u2082 : \u2200 (a : \u03b1), S.arrows (\u03c0 a)\nH\u2081 : \u2200 (b : CoeSort.coe X), \u2203 a x, \u2191(\u03c0 a) x = b\n\u22a2 EffectiveEpiFamily (fun i => Stonean.toProfinite.obj (Y i)) fun i => Stonean.toProfinite.map (\u03c0 i)"}, {"tactic": "exact Profinite.effectiveEpiFamily_of_jointly_surjective\n    (fun i => Stonean.toProfinite.obj (Y i)) (fun i => Stonean.toProfinite.map (\u03c0 i)) H\u2081", "annotated_tactic": ["exact <a>Profinite.effectiveEpiFamily_of_jointly_surjective</a>\n        (fun i => Stonean.toProfinite.obj (Y i)) (fun i => Stonean.toProfinite.map (\u03c0 i)) H\u2081", [{"full_name": "Profinite.effectiveEpiFamily_of_jointly_surjective", "def_path": "Mathlib/Topology/Category/Profinite/EffectiveEpi.lean", "def_pos": [215, 9], "def_end_pos": [215, 49]}]], "state_before": "case refine_1\nX : Stonean\nS : Sieve X\nx\u271d : \u2203 \u03b1 x Y \u03c0, EffectiveEpiFamily Y \u03c0 \u2227 \u2200 (a : \u03b1), S.arrows (\u03c0 a)\n\u03b1 : Type\nw\u271d : Fintype \u03b1\nY : \u03b1 \u2192 Stonean\n\u03c0 : (a : \u03b1) \u2192 Y a \u27f6 X\nH\u2082 : \u2200 (a : \u03b1), S.arrows (\u03c0 a)\nH\u2081 : \u2200 (b : CoeSort.coe X), \u2203 a x, \u2191(\u03c0 a) x = b\n\u22a2 EffectiveEpiFamily (fun i => Stonean.toProfinite.obj (Y i)) fun i => Stonean.toProfinite.map (\u03c0 i)", "state_after": "no goals"}, {"tactic": "obtain \u27e8\u03b1, _, Y, \u03c0, \u27e8H\u2081, H\u2082\u27e9\u27e9 := (coherentTopology.mem_sieves_iff_hasEffectiveEpiFamily _).mp hS", "annotated_tactic": ["obtain \u27e8\u03b1, _, Y, \u03c0, \u27e8H\u2081, H\u2082\u27e9\u27e9 := (<a>coherentTopology.mem_sieves_iff_hasEffectiveEpiFamily</a> _).<a>mp</a> hS", [{"full_name": "CategoryTheory.coherentTopology.mem_sieves_iff_hasEffectiveEpiFamily", "def_path": "Mathlib/CategoryTheory/Sites/Coherent.lean", "def_pos": [185, 9], "def_end_pos": [185, 62]}, {"full_name": "Iff.mp", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [90, 3], "def_end_pos": [90, 5]}]], "state_before": "case refine_2\nX : Stonean\nS : Sieve X\nhS : S \u2208 GrothendieckTopology.sieves (CoverDense.inducedTopology coverDense) X\n\u22a2 \u2203 \u03b1 x Y \u03c0, EffectiveEpiFamily Y \u03c0 \u2227 \u2200 (a : \u03b1), S.arrows (\u03c0 a)", "state_after": "case refine_2.intro.intro.intro.intro.intro\nX : Stonean\nS : Sieve X\nhS : S \u2208 GrothendieckTopology.sieves (CoverDense.inducedTopology coverDense) X\n\u03b1 : Type\nw\u271d : Fintype \u03b1\nY : \u03b1 \u2192 Profinite\n\u03c0 : (a : \u03b1) \u2192 Y a \u27f6 Stonean.toProfinite.obj X\nH\u2081 : EffectiveEpiFamily Y \u03c0\nH\u2082 : \u2200 (a : \u03b1), (Sieve.functorPushforward Stonean.toProfinite S).arrows (\u03c0 a)\n\u22a2 \u2203 \u03b1 x Y \u03c0, EffectiveEpiFamily Y \u03c0 \u2227 \u2200 (a : \u03b1), S.arrows (\u03c0 a)"}, {"tactic": "refine \u27e8\u03b1, inferInstance, ?_\u27e9", "annotated_tactic": ["refine \u27e8\u03b1, <a>inferInstance</a>, ?_\u27e9", [{"full_name": "inferInstance", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [86, 8], "def_end_pos": [86, 21]}]], "state_before": "case refine_2.intro.intro.intro.intro.intro\nX : Stonean\nS : Sieve X\nhS : S \u2208 GrothendieckTopology.sieves (CoverDense.inducedTopology coverDense) X\n\u03b1 : Type\nw\u271d : Fintype \u03b1\nY : \u03b1 \u2192 Profinite\n\u03c0 : (a : \u03b1) \u2192 Y a \u27f6 Stonean.toProfinite.obj X\nH\u2081 : EffectiveEpiFamily Y \u03c0\nH\u2082 : \u2200 (a : \u03b1), (Sieve.functorPushforward Stonean.toProfinite S).arrows (\u03c0 a)\n\u22a2 \u2203 \u03b1 x Y \u03c0, EffectiveEpiFamily Y \u03c0 \u2227 \u2200 (a : \u03b1), S.arrows (\u03c0 a)", "state_after": "case refine_2.intro.intro.intro.intro.intro\nX : Stonean\nS : Sieve X\nhS : S \u2208 GrothendieckTopology.sieves (CoverDense.inducedTopology coverDense) X\n\u03b1 : Type\nw\u271d : Fintype \u03b1\nY : \u03b1 \u2192 Profinite\n\u03c0 : (a : \u03b1) \u2192 Y a \u27f6 Stonean.toProfinite.obj X\nH\u2081 : EffectiveEpiFamily Y \u03c0\nH\u2082 : \u2200 (a : \u03b1), (Sieve.functorPushforward Stonean.toProfinite S).arrows (\u03c0 a)\n\u22a2 \u2203 Y \u03c0, EffectiveEpiFamily Y \u03c0 \u2227 \u2200 (a : \u03b1), S.arrows (\u03c0 a)"}, {"tactic": "obtain \u27e8Y\u2080, H\u2082\u27e9 := Classical.skolem.mp H\u2082", "annotated_tactic": ["obtain \u27e8Y\u2080, H\u2082\u27e9 := Classical.skolem.mp H\u2082", []], "state_before": "case refine_2.intro.intro.intro.intro.intro\nX : Stonean\nS : Sieve X\nhS : S \u2208 GrothendieckTopology.sieves (CoverDense.inducedTopology coverDense) X\n\u03b1 : Type\nw\u271d : Fintype \u03b1\nY : \u03b1 \u2192 Profinite\n\u03c0 : (a : \u03b1) \u2192 Y a \u27f6 Stonean.toProfinite.obj X\nH\u2081 : EffectiveEpiFamily Y \u03c0\nH\u2082 : \u2200 (a : \u03b1), (Sieve.functorPushforward Stonean.toProfinite S).arrows (\u03c0 a)\n\u22a2 \u2203 Y \u03c0, EffectiveEpiFamily Y \u03c0 \u2227 \u2200 (a : \u03b1), S.arrows (\u03c0 a)", "state_after": "case refine_2.intro.intro.intro.intro.intro.intro\nX : Stonean\nS : Sieve X\nhS : S \u2208 GrothendieckTopology.sieves (CoverDense.inducedTopology coverDense) X\n\u03b1 : Type\nw\u271d : Fintype \u03b1\nY : \u03b1 \u2192 Profinite\n\u03c0 : (a : \u03b1) \u2192 Y a \u27f6 Stonean.toProfinite.obj X\nH\u2081 : EffectiveEpiFamily Y \u03c0\nH\u2082\u271d : \u2200 (a : \u03b1), (Sieve.functorPushforward Stonean.toProfinite S).arrows (\u03c0 a)\nY\u2080 : \u03b1 \u2192 Stonean\nH\u2082 : \u2200 (x : \u03b1), \u2203 g h, S.arrows g \u2227 \u03c0 x = h \u226b Stonean.toProfinite.map g\n\u22a2 \u2203 Y \u03c0, EffectiveEpiFamily Y \u03c0 \u2227 \u2200 (a : \u03b1), S.arrows (\u03c0 a)"}, {"tactic": "obtain \u27e8\u03c0\u2080, H\u2082\u27e9 := Classical.skolem.mp H\u2082", "annotated_tactic": ["obtain \u27e8\u03c0\u2080, H\u2082\u27e9 := Classical.skolem.mp H\u2082", []], "state_before": "case refine_2.intro.intro.intro.intro.intro.intro\nX : Stonean\nS : Sieve X\nhS : S \u2208 GrothendieckTopology.sieves (CoverDense.inducedTopology coverDense) X\n\u03b1 : Type\nw\u271d : Fintype \u03b1\nY : \u03b1 \u2192 Profinite\n\u03c0 : (a : \u03b1) \u2192 Y a \u27f6 Stonean.toProfinite.obj X\nH\u2081 : EffectiveEpiFamily Y \u03c0\nH\u2082\u271d : \u2200 (a : \u03b1), (Sieve.functorPushforward Stonean.toProfinite S).arrows (\u03c0 a)\nY\u2080 : \u03b1 \u2192 Stonean\nH\u2082 : \u2200 (x : \u03b1), \u2203 g h, S.arrows g \u2227 \u03c0 x = h \u226b Stonean.toProfinite.map g\n\u22a2 \u2203 Y \u03c0, EffectiveEpiFamily Y \u03c0 \u2227 \u2200 (a : \u03b1), S.arrows (\u03c0 a)", "state_after": "case refine_2.intro.intro.intro.intro.intro.intro.intro\nX : Stonean\nS : Sieve X\nhS : S \u2208 GrothendieckTopology.sieves (CoverDense.inducedTopology coverDense) X\n\u03b1 : Type\nw\u271d : Fintype \u03b1\nY : \u03b1 \u2192 Profinite\n\u03c0 : (a : \u03b1) \u2192 Y a \u27f6 Stonean.toProfinite.obj X\nH\u2081 : EffectiveEpiFamily Y \u03c0\nH\u2082\u271d\u00b9 : \u2200 (a : \u03b1), (Sieve.functorPushforward Stonean.toProfinite S).arrows (\u03c0 a)\nY\u2080 : \u03b1 \u2192 Stonean\nH\u2082\u271d : \u2200 (x : \u03b1), \u2203 g h, S.arrows g \u2227 \u03c0 x = h \u226b Stonean.toProfinite.map g\n\u03c0\u2080 : (x : \u03b1) \u2192 Y\u2080 x \u27f6 X\nH\u2082 : \u2200 (x : \u03b1), \u2203 h, S.arrows (\u03c0\u2080 x) \u2227 \u03c0 x = h \u226b Stonean.toProfinite.map (\u03c0\u2080 x)\n\u22a2 \u2203 Y \u03c0, EffectiveEpiFamily Y \u03c0 \u2227 \u2200 (a : \u03b1), S.arrows (\u03c0 a)"}, {"tactic": "obtain \u27e8f\u2080, H\u2082\u27e9 := Classical.skolem.mp H\u2082", "annotated_tactic": ["obtain \u27e8f\u2080, H\u2082\u27e9 := Classical.skolem.mp H\u2082", []], "state_before": "case refine_2.intro.intro.intro.intro.intro.intro.intro\nX : Stonean\nS : Sieve X\nhS : S \u2208 GrothendieckTopology.sieves (CoverDense.inducedTopology coverDense) X\n\u03b1 : Type\nw\u271d : Fintype \u03b1\nY : \u03b1 \u2192 Profinite\n\u03c0 : (a : \u03b1) \u2192 Y a \u27f6 Stonean.toProfinite.obj X\nH\u2081 : EffectiveEpiFamily Y \u03c0\nH\u2082\u271d\u00b9 : \u2200 (a : \u03b1), (Sieve.functorPushforward Stonean.toProfinite S).arrows (\u03c0 a)\nY\u2080 : \u03b1 \u2192 Stonean\nH\u2082\u271d : \u2200 (x : \u03b1), \u2203 g h, S.arrows g \u2227 \u03c0 x = h \u226b Stonean.toProfinite.map g\n\u03c0\u2080 : (x : \u03b1) \u2192 Y\u2080 x \u27f6 X\nH\u2082 : \u2200 (x : \u03b1), \u2203 h, S.arrows (\u03c0\u2080 x) \u2227 \u03c0 x = h \u226b Stonean.toProfinite.map (\u03c0\u2080 x)\n\u22a2 \u2203 Y \u03c0, EffectiveEpiFamily Y \u03c0 \u2227 \u2200 (a : \u03b1), S.arrows (\u03c0 a)", "state_after": "case refine_2.intro.intro.intro.intro.intro.intro.intro.intro\nX : Stonean\nS : Sieve X\nhS : S \u2208 GrothendieckTopology.sieves (CoverDense.inducedTopology coverDense) X\n\u03b1 : Type\nw\u271d : Fintype \u03b1\nY : \u03b1 \u2192 Profinite\n\u03c0 : (a : \u03b1) \u2192 Y a \u27f6 Stonean.toProfinite.obj X\nH\u2081 : EffectiveEpiFamily Y \u03c0\nH\u2082\u271d\u00b2 : \u2200 (a : \u03b1), (Sieve.functorPushforward Stonean.toProfinite S).arrows (\u03c0 a)\nY\u2080 : \u03b1 \u2192 Stonean\nH\u2082\u271d\u00b9 : \u2200 (x : \u03b1), \u2203 g h, S.arrows g \u2227 \u03c0 x = h \u226b Stonean.toProfinite.map g\n\u03c0\u2080 : (x : \u03b1) \u2192 Y\u2080 x \u27f6 X\nH\u2082\u271d : \u2200 (x : \u03b1), \u2203 h, S.arrows (\u03c0\u2080 x) \u2227 \u03c0 x = h \u226b Stonean.toProfinite.map (\u03c0\u2080 x)\nf\u2080 : (x : \u03b1) \u2192 Y x \u27f6 Stonean.toProfinite.obj (Y\u2080 x)\nH\u2082 : \u2200 (x : \u03b1), S.arrows (\u03c0\u2080 x) \u2227 \u03c0 x = f\u2080 x \u226b Stonean.toProfinite.map (\u03c0\u2080 x)\n\u22a2 \u2203 Y \u03c0, EffectiveEpiFamily Y \u03c0 \u2227 \u2200 (a : \u03b1), S.arrows (\u03c0 a)"}, {"tactic": "refine \u27e8Y\u2080 , \u03c0\u2080, \u27e8?_, fun i \u21a6 (H\u2082 i).1\u27e9\u27e9", "annotated_tactic": ["refine \u27e8Y\u2080 , \u03c0\u2080, \u27e8?_, fun i \u21a6 (H\u2082 i).1\u27e9\u27e9", []], "state_before": "case refine_2.intro.intro.intro.intro.intro.intro.intro.intro\nX : Stonean\nS : Sieve X\nhS : S \u2208 GrothendieckTopology.sieves (CoverDense.inducedTopology coverDense) X\n\u03b1 : Type\nw\u271d : Fintype \u03b1\nY : \u03b1 \u2192 Profinite\n\u03c0 : (a : \u03b1) \u2192 Y a \u27f6 Stonean.toProfinite.obj X\nH\u2081 : EffectiveEpiFamily Y \u03c0\nH\u2082\u271d\u00b2 : \u2200 (a : \u03b1), (Sieve.functorPushforward Stonean.toProfinite S).arrows (\u03c0 a)\nY\u2080 : \u03b1 \u2192 Stonean\nH\u2082\u271d\u00b9 : \u2200 (x : \u03b1), \u2203 g h, S.arrows g \u2227 \u03c0 x = h \u226b Stonean.toProfinite.map g\n\u03c0\u2080 : (x : \u03b1) \u2192 Y\u2080 x \u27f6 X\nH\u2082\u271d : \u2200 (x : \u03b1), \u2203 h, S.arrows (\u03c0\u2080 x) \u2227 \u03c0 x = h \u226b Stonean.toProfinite.map (\u03c0\u2080 x)\nf\u2080 : (x : \u03b1) \u2192 Y x \u27f6 Stonean.toProfinite.obj (Y\u2080 x)\nH\u2082 : \u2200 (x : \u03b1), S.arrows (\u03c0\u2080 x) \u2227 \u03c0 x = f\u2080 x \u226b Stonean.toProfinite.map (\u03c0\u2080 x)\n\u22a2 \u2203 Y \u03c0, EffectiveEpiFamily Y \u03c0 \u2227 \u2200 (a : \u03b1), S.arrows (\u03c0 a)", "state_after": "case refine_2.intro.intro.intro.intro.intro.intro.intro.intro\nX : Stonean\nS : Sieve X\nhS : S \u2208 GrothendieckTopology.sieves (CoverDense.inducedTopology coverDense) X\n\u03b1 : Type\nw\u271d : Fintype \u03b1\nY : \u03b1 \u2192 Profinite\n\u03c0 : (a : \u03b1) \u2192 Y a \u27f6 Stonean.toProfinite.obj X\nH\u2081 : EffectiveEpiFamily Y \u03c0\nH\u2082\u271d\u00b2 : \u2200 (a : \u03b1), (Sieve.functorPushforward Stonean.toProfinite S).arrows (\u03c0 a)\nY\u2080 : \u03b1 \u2192 Stonean\nH\u2082\u271d\u00b9 : \u2200 (x : \u03b1), \u2203 g h, S.arrows g \u2227 \u03c0 x = h \u226b Stonean.toProfinite.map g\n\u03c0\u2080 : (x : \u03b1) \u2192 Y\u2080 x \u27f6 X\nH\u2082\u271d : \u2200 (x : \u03b1), \u2203 h, S.arrows (\u03c0\u2080 x) \u2227 \u03c0 x = h \u226b Stonean.toProfinite.map (\u03c0\u2080 x)\nf\u2080 : (x : \u03b1) \u2192 Y x \u27f6 Stonean.toProfinite.obj (Y\u2080 x)\nH\u2082 : \u2200 (x : \u03b1), S.arrows (\u03c0\u2080 x) \u2227 \u03c0 x = f\u2080 x \u226b Stonean.toProfinite.map (\u03c0\u2080 x)\n\u22a2 EffectiveEpiFamily Y\u2080 \u03c0\u2080"}, {"tactic": "simp only [(Stonean.effectiveEpiFamily_tfae _ _).out 0 2]", "annotated_tactic": ["simp only [(<a>Stonean.effectiveEpiFamily_tfae</a> _ _).<a>out</a> 0 2]", [{"full_name": "Stonean.effectiveEpiFamily_tfae", "def_path": "Mathlib/Topology/Category/Stonean/EffectiveEpi.lean", "def_pos": [132, 9], "def_end_pos": [132, 32]}, {"full_name": "List.TFAE.out", "def_path": "Mathlib/Data/List/TFAE.lean", "def_pos": [72, 9], "def_end_pos": [72, 17]}]], "state_before": "case refine_2.intro.intro.intro.intro.intro.intro.intro.intro\nX : Stonean\nS : Sieve X\nhS : S \u2208 GrothendieckTopology.sieves (CoverDense.inducedTopology coverDense) X\n\u03b1 : Type\nw\u271d : Fintype \u03b1\nY : \u03b1 \u2192 Profinite\n\u03c0 : (a : \u03b1) \u2192 Y a \u27f6 Stonean.toProfinite.obj X\nH\u2081 : EffectiveEpiFamily Y \u03c0\nH\u2082\u271d\u00b2 : \u2200 (a : \u03b1), (Sieve.functorPushforward Stonean.toProfinite S).arrows (\u03c0 a)\nY\u2080 : \u03b1 \u2192 Stonean\nH\u2082\u271d\u00b9 : \u2200 (x : \u03b1), \u2203 g h, S.arrows g \u2227 \u03c0 x = h \u226b Stonean.toProfinite.map g\n\u03c0\u2080 : (x : \u03b1) \u2192 Y\u2080 x \u27f6 X\nH\u2082\u271d : \u2200 (x : \u03b1), \u2203 h, S.arrows (\u03c0\u2080 x) \u2227 \u03c0 x = h \u226b Stonean.toProfinite.map (\u03c0\u2080 x)\nf\u2080 : (x : \u03b1) \u2192 Y x \u27f6 Stonean.toProfinite.obj (Y\u2080 x)\nH\u2082 : \u2200 (x : \u03b1), S.arrows (\u03c0\u2080 x) \u2227 \u03c0 x = f\u2080 x \u226b Stonean.toProfinite.map (\u03c0\u2080 x)\n\u22a2 EffectiveEpiFamily Y\u2080 \u03c0\u2080", "state_after": "case refine_2.intro.intro.intro.intro.intro.intro.intro.intro\nX : Stonean\nS : Sieve X\nhS : S \u2208 GrothendieckTopology.sieves (CoverDense.inducedTopology coverDense) X\n\u03b1 : Type\nw\u271d : Fintype \u03b1\nY : \u03b1 \u2192 Profinite\n\u03c0 : (a : \u03b1) \u2192 Y a \u27f6 Stonean.toProfinite.obj X\nH\u2081 : EffectiveEpiFamily Y \u03c0\nH\u2082\u271d\u00b2 : \u2200 (a : \u03b1), (Sieve.functorPushforward Stonean.toProfinite S).arrows (\u03c0 a)\nY\u2080 : \u03b1 \u2192 Stonean\nH\u2082\u271d\u00b9 : \u2200 (x : \u03b1), \u2203 g h, S.arrows g \u2227 \u03c0 x = h \u226b Stonean.toProfinite.map g\n\u03c0\u2080 : (x : \u03b1) \u2192 Y\u2080 x \u27f6 X\nH\u2082\u271d : \u2200 (x : \u03b1), \u2203 h, S.arrows (\u03c0\u2080 x) \u2227 \u03c0 x = h \u226b Stonean.toProfinite.map (\u03c0\u2080 x)\nf\u2080 : (x : \u03b1) \u2192 Y x \u27f6 Stonean.toProfinite.obj (Y\u2080 x)\nH\u2082 : \u2200 (x : \u03b1), S.arrows (\u03c0\u2080 x) \u2227 \u03c0 x = f\u2080 x \u226b Stonean.toProfinite.map (\u03c0\u2080 x)\n\u22a2 \u2200 (b : CoeSort.coe X), \u2203 a x, \u2191(\u03c0\u2080 a) x = b"}, {"tactic": "simp only [(Profinite.effectiveEpiFamily_tfae _ _).out 0 2] at H\u2081", "annotated_tactic": ["simp only [(<a>Profinite.effectiveEpiFamily_tfae</a> _ _).<a>out</a> 0 2] at H\u2081", [{"full_name": "Profinite.effectiveEpiFamily_tfae", "def_path": "Mathlib/Topology/Category/Profinite/EffectiveEpi.lean", "def_pos": [229, 9], "def_end_pos": [229, 32]}, {"full_name": "List.TFAE.out", "def_path": "Mathlib/Data/List/TFAE.lean", "def_pos": [72, 9], "def_end_pos": [72, 17]}]], "state_before": "case refine_2.intro.intro.intro.intro.intro.intro.intro.intro\nX : Stonean\nS : Sieve X\nhS : S \u2208 GrothendieckTopology.sieves (CoverDense.inducedTopology coverDense) X\n\u03b1 : Type\nw\u271d : Fintype \u03b1\nY : \u03b1 \u2192 Profinite\n\u03c0 : (a : \u03b1) \u2192 Y a \u27f6 Stonean.toProfinite.obj X\nH\u2081 : EffectiveEpiFamily Y \u03c0\nH\u2082\u271d\u00b2 : \u2200 (a : \u03b1), (Sieve.functorPushforward Stonean.toProfinite S).arrows (\u03c0 a)\nY\u2080 : \u03b1 \u2192 Stonean\nH\u2082\u271d\u00b9 : \u2200 (x : \u03b1), \u2203 g h, S.arrows g \u2227 \u03c0 x = h \u226b Stonean.toProfinite.map g\n\u03c0\u2080 : (x : \u03b1) \u2192 Y\u2080 x \u27f6 X\nH\u2082\u271d : \u2200 (x : \u03b1), \u2203 h, S.arrows (\u03c0\u2080 x) \u2227 \u03c0 x = h \u226b Stonean.toProfinite.map (\u03c0\u2080 x)\nf\u2080 : (x : \u03b1) \u2192 Y x \u27f6 Stonean.toProfinite.obj (Y\u2080 x)\nH\u2082 : \u2200 (x : \u03b1), S.arrows (\u03c0\u2080 x) \u2227 \u03c0 x = f\u2080 x \u226b Stonean.toProfinite.map (\u03c0\u2080 x)\n\u22a2 \u2200 (b : CoeSort.coe X), \u2203 a x, \u2191(\u03c0\u2080 a) x = b", "state_after": "case refine_2.intro.intro.intro.intro.intro.intro.intro.intro\nX : Stonean\nS : Sieve X\nhS : S \u2208 GrothendieckTopology.sieves (CoverDense.inducedTopology coverDense) X\n\u03b1 : Type\nw\u271d : Fintype \u03b1\nY : \u03b1 \u2192 Profinite\n\u03c0 : (a : \u03b1) \u2192 Y a \u27f6 Stonean.toProfinite.obj X\nH\u2082\u271d\u00b2 : \u2200 (a : \u03b1), (Sieve.functorPushforward Stonean.toProfinite S).arrows (\u03c0 a)\nY\u2080 : \u03b1 \u2192 Stonean\nH\u2082\u271d\u00b9 : \u2200 (x : \u03b1), \u2203 g h, S.arrows g \u2227 \u03c0 x = h \u226b Stonean.toProfinite.map g\n\u03c0\u2080 : (x : \u03b1) \u2192 Y\u2080 x \u27f6 X\nH\u2082\u271d : \u2200 (x : \u03b1), \u2203 h, S.arrows (\u03c0\u2080 x) \u2227 \u03c0 x = h \u226b Stonean.toProfinite.map (\u03c0\u2080 x)\nf\u2080 : (x : \u03b1) \u2192 Y x \u27f6 Stonean.toProfinite.obj (Y\u2080 x)\nH\u2082 : \u2200 (x : \u03b1), S.arrows (\u03c0\u2080 x) \u2227 \u03c0 x = f\u2080 x \u226b Stonean.toProfinite.map (\u03c0\u2080 x)\nH\u2081 : \u2200 (b : \u2191(Stonean.toProfinite.obj X).toCompHaus.toTop), \u2203 a x, \u2191(\u03c0 a) x = b\n\u22a2 \u2200 (b : CoeSort.coe X), \u2203 a x, \u2191(\u03c0\u2080 a) x = b"}, {"tactic": "intro b", "annotated_tactic": ["intro b", []], "state_before": "case refine_2.intro.intro.intro.intro.intro.intro.intro.intro\nX : Stonean\nS : Sieve X\nhS : S \u2208 GrothendieckTopology.sieves (CoverDense.inducedTopology coverDense) X\n\u03b1 : Type\nw\u271d : Fintype \u03b1\nY : \u03b1 \u2192 Profinite\n\u03c0 : (a : \u03b1) \u2192 Y a \u27f6 Stonean.toProfinite.obj X\nH\u2082\u271d\u00b2 : \u2200 (a : \u03b1), (Sieve.functorPushforward Stonean.toProfinite S).arrows (\u03c0 a)\nY\u2080 : \u03b1 \u2192 Stonean\nH\u2082\u271d\u00b9 : \u2200 (x : \u03b1), \u2203 g h, S.arrows g \u2227 \u03c0 x = h \u226b Stonean.toProfinite.map g\n\u03c0\u2080 : (x : \u03b1) \u2192 Y\u2080 x \u27f6 X\nH\u2082\u271d : \u2200 (x : \u03b1), \u2203 h, S.arrows (\u03c0\u2080 x) \u2227 \u03c0 x = h \u226b Stonean.toProfinite.map (\u03c0\u2080 x)\nf\u2080 : (x : \u03b1) \u2192 Y x \u27f6 Stonean.toProfinite.obj (Y\u2080 x)\nH\u2082 : \u2200 (x : \u03b1), S.arrows (\u03c0\u2080 x) \u2227 \u03c0 x = f\u2080 x \u226b Stonean.toProfinite.map (\u03c0\u2080 x)\nH\u2081 : \u2200 (b : \u2191(Stonean.toProfinite.obj X).toCompHaus.toTop), \u2203 a x, \u2191(\u03c0 a) x = b\n\u22a2 \u2200 (b : CoeSort.coe X), \u2203 a x, \u2191(\u03c0\u2080 a) x = b", "state_after": "case refine_2.intro.intro.intro.intro.intro.intro.intro.intro\nX : Stonean\nS : Sieve X\nhS : S \u2208 GrothendieckTopology.sieves (CoverDense.inducedTopology coverDense) X\n\u03b1 : Type\nw\u271d : Fintype \u03b1\nY : \u03b1 \u2192 Profinite\n\u03c0 : (a : \u03b1) \u2192 Y a \u27f6 Stonean.toProfinite.obj X\nH\u2082\u271d\u00b2 : \u2200 (a : \u03b1), (Sieve.functorPushforward Stonean.toProfinite S).arrows (\u03c0 a)\nY\u2080 : \u03b1 \u2192 Stonean\nH\u2082\u271d\u00b9 : \u2200 (x : \u03b1), \u2203 g h, S.arrows g \u2227 \u03c0 x = h \u226b Stonean.toProfinite.map g\n\u03c0\u2080 : (x : \u03b1) \u2192 Y\u2080 x \u27f6 X\nH\u2082\u271d : \u2200 (x : \u03b1), \u2203 h, S.arrows (\u03c0\u2080 x) \u2227 \u03c0 x = h \u226b Stonean.toProfinite.map (\u03c0\u2080 x)\nf\u2080 : (x : \u03b1) \u2192 Y x \u27f6 Stonean.toProfinite.obj (Y\u2080 x)\nH\u2082 : \u2200 (x : \u03b1), S.arrows (\u03c0\u2080 x) \u2227 \u03c0 x = f\u2080 x \u226b Stonean.toProfinite.map (\u03c0\u2080 x)\nH\u2081 : \u2200 (b : \u2191(Stonean.toProfinite.obj X).toCompHaus.toTop), \u2203 a x, \u2191(\u03c0 a) x = b\nb : CoeSort.coe X\n\u22a2 \u2203 a x, \u2191(\u03c0\u2080 a) x = b"}, {"tactic": "obtain \u27e8i, x, H\u2081\u27e9 := H\u2081 b", "annotated_tactic": ["obtain \u27e8i, x, H\u2081\u27e9 := H\u2081 b", []], "state_before": "case refine_2.intro.intro.intro.intro.intro.intro.intro.intro\nX : Stonean\nS : Sieve X\nhS : S \u2208 GrothendieckTopology.sieves (CoverDense.inducedTopology coverDense) X\n\u03b1 : Type\nw\u271d : Fintype \u03b1\nY : \u03b1 \u2192 Profinite\n\u03c0 : (a : \u03b1) \u2192 Y a \u27f6 Stonean.toProfinite.obj X\nH\u2082\u271d\u00b2 : \u2200 (a : \u03b1), (Sieve.functorPushforward Stonean.toProfinite S).arrows (\u03c0 a)\nY\u2080 : \u03b1 \u2192 Stonean\nH\u2082\u271d\u00b9 : \u2200 (x : \u03b1), \u2203 g h, S.arrows g \u2227 \u03c0 x = h \u226b Stonean.toProfinite.map g\n\u03c0\u2080 : (x : \u03b1) \u2192 Y\u2080 x \u27f6 X\nH\u2082\u271d : \u2200 (x : \u03b1), \u2203 h, S.arrows (\u03c0\u2080 x) \u2227 \u03c0 x = h \u226b Stonean.toProfinite.map (\u03c0\u2080 x)\nf\u2080 : (x : \u03b1) \u2192 Y x \u27f6 Stonean.toProfinite.obj (Y\u2080 x)\nH\u2082 : \u2200 (x : \u03b1), S.arrows (\u03c0\u2080 x) \u2227 \u03c0 x = f\u2080 x \u226b Stonean.toProfinite.map (\u03c0\u2080 x)\nH\u2081 : \u2200 (b : \u2191(Stonean.toProfinite.obj X).toCompHaus.toTop), \u2203 a x, \u2191(\u03c0 a) x = b\nb : CoeSort.coe X\n\u22a2 \u2203 a x, \u2191(\u03c0\u2080 a) x = b", "state_after": "case refine_2.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nX : Stonean\nS : Sieve X\nhS : S \u2208 GrothendieckTopology.sieves (CoverDense.inducedTopology coverDense) X\n\u03b1 : Type\nw\u271d : Fintype \u03b1\nY : \u03b1 \u2192 Profinite\n\u03c0 : (a : \u03b1) \u2192 Y a \u27f6 Stonean.toProfinite.obj X\nH\u2082\u271d\u00b2 : \u2200 (a : \u03b1), (Sieve.functorPushforward Stonean.toProfinite S).arrows (\u03c0 a)\nY\u2080 : \u03b1 \u2192 Stonean\nH\u2082\u271d\u00b9 : \u2200 (x : \u03b1), \u2203 g h, S.arrows g \u2227 \u03c0 x = h \u226b Stonean.toProfinite.map g\n\u03c0\u2080 : (x : \u03b1) \u2192 Y\u2080 x \u27f6 X\nH\u2082\u271d : \u2200 (x : \u03b1), \u2203 h, S.arrows (\u03c0\u2080 x) \u2227 \u03c0 x = h \u226b Stonean.toProfinite.map (\u03c0\u2080 x)\nf\u2080 : (x : \u03b1) \u2192 Y x \u27f6 Stonean.toProfinite.obj (Y\u2080 x)\nH\u2082 : \u2200 (x : \u03b1), S.arrows (\u03c0\u2080 x) \u2227 \u03c0 x = f\u2080 x \u226b Stonean.toProfinite.map (\u03c0\u2080 x)\nH\u2081\u271d : \u2200 (b : \u2191(Stonean.toProfinite.obj X).toCompHaus.toTop), \u2203 a x, \u2191(\u03c0 a) x = b\nb : CoeSort.coe X\ni : \u03b1\nx : \u2191(Y i).toCompHaus.toTop\nH\u2081 : \u2191(\u03c0 i) x = b\n\u22a2 \u2203 a x, \u2191(\u03c0\u2080 a) x = b"}, {"tactic": "refine \u27e8i, f\u2080 i x, ?_\u27e9", "annotated_tactic": ["refine \u27e8i, f\u2080 i x, ?_\u27e9", []], "state_before": "case refine_2.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nX : Stonean\nS : Sieve X\nhS : S \u2208 GrothendieckTopology.sieves (CoverDense.inducedTopology coverDense) X\n\u03b1 : Type\nw\u271d : Fintype \u03b1\nY : \u03b1 \u2192 Profinite\n\u03c0 : (a : \u03b1) \u2192 Y a \u27f6 Stonean.toProfinite.obj X\nH\u2082\u271d\u00b2 : \u2200 (a : \u03b1), (Sieve.functorPushforward Stonean.toProfinite S).arrows (\u03c0 a)\nY\u2080 : \u03b1 \u2192 Stonean\nH\u2082\u271d\u00b9 : \u2200 (x : \u03b1), \u2203 g h, S.arrows g \u2227 \u03c0 x = h \u226b Stonean.toProfinite.map g\n\u03c0\u2080 : (x : \u03b1) \u2192 Y\u2080 x \u27f6 X\nH\u2082\u271d : \u2200 (x : \u03b1), \u2203 h, S.arrows (\u03c0\u2080 x) \u2227 \u03c0 x = h \u226b Stonean.toProfinite.map (\u03c0\u2080 x)\nf\u2080 : (x : \u03b1) \u2192 Y x \u27f6 Stonean.toProfinite.obj (Y\u2080 x)\nH\u2082 : \u2200 (x : \u03b1), S.arrows (\u03c0\u2080 x) \u2227 \u03c0 x = f\u2080 x \u226b Stonean.toProfinite.map (\u03c0\u2080 x)\nH\u2081\u271d : \u2200 (b : \u2191(Stonean.toProfinite.obj X).toCompHaus.toTop), \u2203 a x, \u2191(\u03c0 a) x = b\nb : CoeSort.coe X\ni : \u03b1\nx : \u2191(Y i).toCompHaus.toTop\nH\u2081 : \u2191(\u03c0 i) x = b\n\u22a2 \u2203 a x, \u2191(\u03c0\u2080 a) x = b", "state_after": "case refine_2.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nX : Stonean\nS : Sieve X\nhS : S \u2208 GrothendieckTopology.sieves (CoverDense.inducedTopology coverDense) X\n\u03b1 : Type\nw\u271d : Fintype \u03b1\nY : \u03b1 \u2192 Profinite\n\u03c0 : (a : \u03b1) \u2192 Y a \u27f6 Stonean.toProfinite.obj X\nH\u2082\u271d\u00b2 : \u2200 (a : \u03b1), (Sieve.functorPushforward Stonean.toProfinite S).arrows (\u03c0 a)\nY\u2080 : \u03b1 \u2192 Stonean\nH\u2082\u271d\u00b9 : \u2200 (x : \u03b1), \u2203 g h, S.arrows g \u2227 \u03c0 x = h \u226b Stonean.toProfinite.map g\n\u03c0\u2080 : (x : \u03b1) \u2192 Y\u2080 x \u27f6 X\nH\u2082\u271d : \u2200 (x : \u03b1), \u2203 h, S.arrows (\u03c0\u2080 x) \u2227 \u03c0 x = h \u226b Stonean.toProfinite.map (\u03c0\u2080 x)\nf\u2080 : (x : \u03b1) \u2192 Y x \u27f6 Stonean.toProfinite.obj (Y\u2080 x)\nH\u2082 : \u2200 (x : \u03b1), S.arrows (\u03c0\u2080 x) \u2227 \u03c0 x = f\u2080 x \u226b Stonean.toProfinite.map (\u03c0\u2080 x)\nH\u2081\u271d : \u2200 (b : \u2191(Stonean.toProfinite.obj X).toCompHaus.toTop), \u2203 a x, \u2191(\u03c0 a) x = b\nb : CoeSort.coe X\ni : \u03b1\nx : \u2191(Y i).toCompHaus.toTop\nH\u2081 : \u2191(\u03c0 i) x = b\n\u22a2 \u2191(\u03c0\u2080 i) (\u2191(f\u2080 i) x) = b"}, {"tactic": "rw [\u2190 H\u2081, (H\u2082 i).2]", "annotated_tactic": ["rw [\u2190 H\u2081, (H\u2082 i).2]", []], "state_before": "case refine_2.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nX : Stonean\nS : Sieve X\nhS : S \u2208 GrothendieckTopology.sieves (CoverDense.inducedTopology coverDense) X\n\u03b1 : Type\nw\u271d : Fintype \u03b1\nY : \u03b1 \u2192 Profinite\n\u03c0 : (a : \u03b1) \u2192 Y a \u27f6 Stonean.toProfinite.obj X\nH\u2082\u271d\u00b2 : \u2200 (a : \u03b1), (Sieve.functorPushforward Stonean.toProfinite S).arrows (\u03c0 a)\nY\u2080 : \u03b1 \u2192 Stonean\nH\u2082\u271d\u00b9 : \u2200 (x : \u03b1), \u2203 g h, S.arrows g \u2227 \u03c0 x = h \u226b Stonean.toProfinite.map g\n\u03c0\u2080 : (x : \u03b1) \u2192 Y\u2080 x \u27f6 X\nH\u2082\u271d : \u2200 (x : \u03b1), \u2203 h, S.arrows (\u03c0\u2080 x) \u2227 \u03c0 x = h \u226b Stonean.toProfinite.map (\u03c0\u2080 x)\nf\u2080 : (x : \u03b1) \u2192 Y x \u27f6 Stonean.toProfinite.obj (Y\u2080 x)\nH\u2082 : \u2200 (x : \u03b1), S.arrows (\u03c0\u2080 x) \u2227 \u03c0 x = f\u2080 x \u226b Stonean.toProfinite.map (\u03c0\u2080 x)\nH\u2081\u271d : \u2200 (b : \u2191(Stonean.toProfinite.obj X).toCompHaus.toTop), \u2203 a x, \u2191(\u03c0 a) x = b\nb : CoeSort.coe X\ni : \u03b1\nx : \u2191(Y i).toCompHaus.toTop\nH\u2081 : \u2191(\u03c0 i) x = b\n\u22a2 \u2191(\u03c0\u2080 i) (\u2191(f\u2080 i) x) = b", "state_after": "case refine_2.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nX : Stonean\nS : Sieve X\nhS : S \u2208 GrothendieckTopology.sieves (CoverDense.inducedTopology coverDense) X\n\u03b1 : Type\nw\u271d : Fintype \u03b1\nY : \u03b1 \u2192 Profinite\n\u03c0 : (a : \u03b1) \u2192 Y a \u27f6 Stonean.toProfinite.obj X\nH\u2082\u271d\u00b2 : \u2200 (a : \u03b1), (Sieve.functorPushforward Stonean.toProfinite S).arrows (\u03c0 a)\nY\u2080 : \u03b1 \u2192 Stonean\nH\u2082\u271d\u00b9 : \u2200 (x : \u03b1), \u2203 g h, S.arrows g \u2227 \u03c0 x = h \u226b Stonean.toProfinite.map g\n\u03c0\u2080 : (x : \u03b1) \u2192 Y\u2080 x \u27f6 X\nH\u2082\u271d : \u2200 (x : \u03b1), \u2203 h, S.arrows (\u03c0\u2080 x) \u2227 \u03c0 x = h \u226b Stonean.toProfinite.map (\u03c0\u2080 x)\nf\u2080 : (x : \u03b1) \u2192 Y x \u27f6 Stonean.toProfinite.obj (Y\u2080 x)\nH\u2082 : \u2200 (x : \u03b1), S.arrows (\u03c0\u2080 x) \u2227 \u03c0 x = f\u2080 x \u226b Stonean.toProfinite.map (\u03c0\u2080 x)\nH\u2081\u271d : \u2200 (b : \u2191(Stonean.toProfinite.obj X).toCompHaus.toTop), \u2203 a x, \u2191(\u03c0 a) x = b\nb : CoeSort.coe X\ni : \u03b1\nx : \u2191(Y i).toCompHaus.toTop\nH\u2081 : \u2191(\u03c0 i) x = b\n\u22a2 \u2191(\u03c0\u2080 i) (\u2191(f\u2080 i) x) = \u2191(f\u2080 i \u226b Stonean.toProfinite.map (\u03c0\u2080 i)) x"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case refine_2.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nX : Stonean\nS : Sieve X\nhS : S \u2208 GrothendieckTopology.sieves (CoverDense.inducedTopology coverDense) X\n\u03b1 : Type\nw\u271d : Fintype \u03b1\nY : \u03b1 \u2192 Profinite\n\u03c0 : (a : \u03b1) \u2192 Y a \u27f6 Stonean.toProfinite.obj X\nH\u2082\u271d\u00b2 : \u2200 (a : \u03b1), (Sieve.functorPushforward Stonean.toProfinite S).arrows (\u03c0 a)\nY\u2080 : \u03b1 \u2192 Stonean\nH\u2082\u271d\u00b9 : \u2200 (x : \u03b1), \u2203 g h, S.arrows g \u2227 \u03c0 x = h \u226b Stonean.toProfinite.map g\n\u03c0\u2080 : (x : \u03b1) \u2192 Y\u2080 x \u27f6 X\nH\u2082\u271d : \u2200 (x : \u03b1), \u2203 h, S.arrows (\u03c0\u2080 x) \u2227 \u03c0 x = h \u226b Stonean.toProfinite.map (\u03c0\u2080 x)\nf\u2080 : (x : \u03b1) \u2192 Y x \u27f6 Stonean.toProfinite.obj (Y\u2080 x)\nH\u2082 : \u2200 (x : \u03b1), S.arrows (\u03c0\u2080 x) \u2227 \u03c0 x = f\u2080 x \u226b Stonean.toProfinite.map (\u03c0\u2080 x)\nH\u2081\u271d : \u2200 (b : \u2191(Stonean.toProfinite.obj X).toCompHaus.toTop), \u2203 a x, \u2191(\u03c0 a) x = b\nb : CoeSort.coe X\ni : \u03b1\nx : \u2191(Y i).toCompHaus.toTop\nH\u2081 : \u2191(\u03c0 i) x = b\n\u22a2 \u2191(\u03c0\u2080 i) (\u2191(f\u2080 i) x) = \u2191(f\u2080 i \u226b Stonean.toProfinite.map (\u03c0\u2080 i)) x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "full_name": "Substring.ValidFor.foldr", "start": [941, 1], "end": [942, 78], "traced_tactics": [{"tactic": "simp [-List.append_assoc, Substring.foldr, foldrAux_of_valid]", "annotated_tactic": ["simp [-<a>List.append_assoc</a>, <a>Substring.foldr</a>, <a>foldrAux_of_valid</a>]", [{"full_name": "List.append_assoc", "def_path": "lake-packages/lean4/src/lean/Init/Data/List/Basic.lean", "def_pos": [103, 9], "def_end_pos": [103, 21]}, {"full_name": "Substring.foldr", "def_path": "lake-packages/lean4/src/lean/Init/Data/String/Basic.lean", "def_pos": [622, 15], "def_end_pos": [622, 20]}, {"full_name": "String.foldrAux_of_valid", "def_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "def_pos": [710, 9], "def_end_pos": [710, 26]}]], "state_before": "\u03b1 : Type u_1\nl m r : List Char\nf : Char \u2192 \u03b1 \u2192 \u03b1\ninit : \u03b1\n\u22a2 Substring.foldr f init\n      { str := { data := l ++ m ++ r }, startPos := { byteIdx := utf8Len l },\n        stopPos := { byteIdx := utf8Len l + utf8Len m } } =\n    List.foldr f init m", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/StrongLaw.lean", "full_name": "ProbabilityTheory.strong_law_aux3", "start": [545, 1], "end": [555, 70], "traced_tactics": [{"tactic": "have A : Tendsto (fun i => \ud835\udd3c[truncation (X i) i]) atTop (\ud835\udcdd \ud835\udd3c[X 0]) := by\n  convert (tendsto_integral_truncation hint).comp tendsto_nat_cast_atTop_atTop using 1\n  ext i\n  exact (hident i).truncation.integral_eq", "annotated_tactic": ["have A : <a>Tendsto</a> (fun i => \ud835\udd3c[<a>truncation</a> (X i) i]) <a>atTop</a> (\ud835\udcdd \ud835\udd3c[X 0]) := by\n    convert (<a>tendsto_integral_truncation</a> hint).<a>comp</a> <a>tendsto_nat_cast_atTop_atTop</a> using 1\n    ext i\n    exact (hident i).truncation.integral_eq", [{"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2939, 5], "def_end_pos": [2939, 12]}, {"full_name": "ProbabilityTheory.truncation", "def_path": "Mathlib/Probability/StrongLaw.lean", "def_pos": [78, 5], "def_end_pos": [78, 15]}, {"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}, {"full_name": "ProbabilityTheory.tendsto_integral_truncation", "def_path": "Mathlib/Probability/StrongLaw.lean", "def_pos": [203, 9], "def_end_pos": [203, 36]}, {"full_name": "Filter.Tendsto.comp", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [3032, 9], "def_end_pos": [3032, 21]}, {"full_name": "tendsto_nat_cast_atTop_atTop", "def_path": "Mathlib/Order/Filter/Archimedean.lean", "def_pos": [37, 9], "def_end_pos": [37, 37]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\n\u22a2 (fun n => (\u222b (a : \u03a9), Finset.sum (range n) (fun i => truncation (X i) \u2191i) a) - \u2191n * \u222b (a : \u03a9), X 0 a) =o[atTop]\n    Nat.cast", "state_after": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nA : Tendsto (fun i => \u222b (a : \u03a9), truncation (X i) (\u2191i) a) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a))\n\u22a2 (fun n => (\u222b (a : \u03a9), Finset.sum (range n) (fun i => truncation (X i) \u2191i) a) - \u2191n * \u222b (a : \u03a9), X 0 a) =o[atTop]\n    Nat.cast"}, {"tactic": "convert Asymptotics.isLittleO_sum_range_of_tendsto_zero (tendsto_sub_nhds_zero_iff.2 A) using 1", "annotated_tactic": ["convert <a>Asymptotics.isLittleO_sum_range_of_tendsto_zero</a> (<a>tendsto_sub_nhds_zero_iff</a>.2 A) using 1", [{"full_name": "Asymptotics.isLittleO_sum_range_of_tendsto_zero", "def_path": "Mathlib/Analysis/Asymptotics/SpecificAsymptotics.lean", "def_pos": [139, 9], "def_end_pos": [139, 56]}, {"full_name": "tendsto_sub_nhds_zero_iff", "def_path": "Mathlib/Topology/Algebra/Group/Basic.lean", "def_pos": [1192, 3], "def_end_pos": [1192, 14]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nA : Tendsto (fun i => \u222b (a : \u03a9), truncation (X i) (\u2191i) a) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a))\n\u22a2 (fun n => (\u222b (a : \u03a9), Finset.sum (range n) (fun i => truncation (X i) \u2191i) a) - \u2191n * \u222b (a : \u03a9), X 0 a) =o[atTop]\n    Nat.cast", "state_after": "case h.e'_7\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nA : Tendsto (fun i => \u222b (a : \u03a9), truncation (X i) (\u2191i) a) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a))\n\u22a2 (fun n => (\u222b (a : \u03a9), Finset.sum (range n) (fun i => truncation (X i) \u2191i) a) - \u2191n * \u222b (a : \u03a9), X 0 a) = fun n =>\n    \u2211 i in range n, ((\u222b (a : \u03a9), truncation (X i) (\u2191i) a) - \u222b (a : \u03a9), X 0 a)"}, {"tactic": "ext1 n", "annotated_tactic": ["ext1 n", []], "state_before": "case h.e'_7\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nA : Tendsto (fun i => \u222b (a : \u03a9), truncation (X i) (\u2191i) a) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a))\n\u22a2 (fun n => (\u222b (a : \u03a9), Finset.sum (range n) (fun i => truncation (X i) \u2191i) a) - \u2191n * \u222b (a : \u03a9), X 0 a) = fun n =>\n    \u2211 i in range n, ((\u222b (a : \u03a9), truncation (X i) (\u2191i) a) - \u222b (a : \u03a9), X 0 a)", "state_after": "case h.e'_7.h\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nA : Tendsto (fun i => \u222b (a : \u03a9), truncation (X i) (\u2191i) a) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a))\nn : \u2115\n\u22a2 (\u222b (a : \u03a9), Finset.sum (range n) (fun i => truncation (X i) \u2191i) a) - \u2191n * \u222b (a : \u03a9), X 0 a =\n    \u2211 i in range n, ((\u222b (a : \u03a9), truncation (X i) (\u2191i) a) - \u222b (a : \u03a9), X 0 a)"}, {"tactic": "simp only [sum_sub_distrib, sum_const, card_range, nsmul_eq_mul, sum_apply, sub_left_inj]", "annotated_tactic": ["simp only [<a>sum_sub_distrib</a>, <a>sum_const</a>, <a>card_range</a>, <a>nsmul_eq_mul</a>, <a>sum_apply</a>, <a>sub_left_inj</a>]", [{"full_name": "Finset.sum_sub_distrib", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [1820, 3], "def_end_pos": [1820, 14]}, {"full_name": "Finset.sum_const", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [1440, 3], "def_end_pos": [1440, 14]}, {"full_name": "Finset.card_range", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [177, 9], "def_end_pos": [177, 19]}, {"full_name": "nsmul_eq_mul", "def_path": "Mathlib/Algebra/GroupPower/Lemmas.lean", "def_pos": [509, 9], "def_end_pos": [509, 21]}, {"full_name": "Finset.sum_apply", "def_path": "Mathlib/Algebra/BigOperators/Pi.lean", "def_pos": [41, 3], "def_end_pos": [41, 14]}, {"full_name": "sub_left_inj", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [782, 3], "def_end_pos": [782, 14]}]], "state_before": "case h.e'_7.h\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nA : Tendsto (fun i => \u222b (a : \u03a9), truncation (X i) (\u2191i) a) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a))\nn : \u2115\n\u22a2 (\u222b (a : \u03a9), Finset.sum (range n) (fun i => truncation (X i) \u2191i) a) - \u2191n * \u222b (a : \u03a9), X 0 a =\n    \u2211 i in range n, ((\u222b (a : \u03a9), truncation (X i) (\u2191i) a) - \u222b (a : \u03a9), X 0 a)", "state_after": "case h.e'_7.h\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nA : Tendsto (fun i => \u222b (a : \u03a9), truncation (X i) (\u2191i) a) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a))\nn : \u2115\n\u22a2 \u222b (a : \u03a9), \u2211 c in range n, truncation (X c) (\u2191c) a = \u2211 x in range n, \u222b (a : \u03a9), truncation (X x) (\u2191x) a"}, {"tactic": "rw [integral_finset_sum _ fun i _ => ?_]", "annotated_tactic": ["rw [<a>integral_finset_sum</a> _ fun i _ => ?_]", [{"full_name": "MeasureTheory.integral_finset_sum", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [881, 9], "def_end_pos": [881, 28]}]], "state_before": "case h.e'_7.h\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nA : Tendsto (fun i => \u222b (a : \u03a9), truncation (X i) (\u2191i) a) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a))\nn : \u2115\n\u22a2 \u222b (a : \u03a9), \u2211 c in range n, truncation (X c) (\u2191c) a = \u2211 x in range n, \u222b (a : \u03a9), truncation (X x) (\u2191x) a", "state_after": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nA : Tendsto (fun i => \u222b (a : \u03a9), truncation (X i) (\u2191i) a) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a))\nn i : \u2115\nx\u271d : i \u2208 range n\n\u22a2 Integrable fun a => truncation (X i) (\u2191i) a"}, {"tactic": "exact ((hident i).symm.integrable_snd hint).1.integrable_truncation", "annotated_tactic": ["exact ((hident i).symm.integrable_snd hint).1.<a>integrable_truncation</a>", [{"full_name": "MeasureTheory.AEStronglyMeasurable.integrable_truncation", "def_path": "Mathlib/Probability/StrongLaw.lean", "def_pos": [137, 9], "def_end_pos": [137, 72]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nA : Tendsto (fun i => \u222b (a : \u03a9), truncation (X i) (\u2191i) a) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a))\nn i : \u2115\nx\u271d : i \u2208 range n\n\u22a2 Integrable fun a => truncation (X i) (\u2191i) a", "state_after": "no goals"}, {"tactic": "convert (tendsto_integral_truncation hint).comp tendsto_nat_cast_atTop_atTop using 1", "annotated_tactic": ["convert (<a>tendsto_integral_truncation</a> hint).<a>comp</a> <a>tendsto_nat_cast_atTop_atTop</a> using 1", [{"full_name": "ProbabilityTheory.tendsto_integral_truncation", "def_path": "Mathlib/Probability/StrongLaw.lean", "def_pos": [203, 9], "def_end_pos": [203, 36]}, {"full_name": "Filter.Tendsto.comp", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [3032, 9], "def_end_pos": [3032, 21]}, {"full_name": "tendsto_nat_cast_atTop_atTop", "def_path": "Mathlib/Order/Filter/Archimedean.lean", "def_pos": [37, 9], "def_end_pos": [37, 37]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\n\u22a2 Tendsto (fun i => \u222b (a : \u03a9), truncation (X i) (\u2191i) a) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a))", "state_after": "case h.e'_3\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\n\u22a2 (fun i => \u222b (a : \u03a9), truncation (X i) (\u2191i) a) = (fun A => \u222b (x : \u03a9), truncation (X 0) A x) \u2218 Nat.cast"}, {"tactic": "ext i", "annotated_tactic": ["ext i", []], "state_before": "case h.e'_3\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\n\u22a2 (fun i => \u222b (a : \u03a9), truncation (X i) (\u2191i) a) = (fun A => \u222b (x : \u03a9), truncation (X 0) A x) \u2218 Nat.cast", "state_after": "case h.e'_3.h\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\ni : \u2115\n\u22a2 \u222b (a : \u03a9), truncation (X i) (\u2191i) a = ((fun A => \u222b (x : \u03a9), truncation (X 0) A x) \u2218 Nat.cast) i"}, {"tactic": "exact (hident i).truncation.integral_eq", "annotated_tactic": ["exact (hident i).truncation.integral_eq", []], "state_before": "case h.e'_3.h\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\ni : \u2115\n\u22a2 \u222b (a : \u03a9), truncation (X i) (\u2191i) a = ((fun A => \u222b (x : \u03a9), truncation (X 0) A x) \u2218 Nat.cast) i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "full_name": "ContinuousMap.toLp_norm_le", "start": [1902, 1], "end": [1905, 49], "traced_tactics": [{"tactic": "rw [toLp_norm_eq_toLp_norm_coe]", "annotated_tactic": ["rw [<a>toLp_norm_eq_toLp_norm_coe</a>]", [{"full_name": "ContinuousMap.toLp_norm_eq_toLp_norm_coe", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [1895, 9], "def_end_pos": [1895, 35]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedAddCommGroup G\ninst\u271d\u2077 : TopologicalSpace \u03b1\ninst\u271d\u2076 : BorelSpace \u03b1\ninst\u271d\u2075 : SecondCountableTopologyEither \u03b1 E\ninst\u271d\u2074 : CompactSpace \u03b1\ninst\u271d\u00b3 : IsFiniteMeasure \u03bc\n\ud835\udd5c : Type u_5\ninst\u271d\u00b2 : Fact (1 \u2264 p)\ninst\u271d\u00b9 : NontriviallyNormedField \ud835\udd5c\ninst\u271d : NormedSpace \ud835\udd5c E\n\u22a2 \u2016toLp p \u03bc \ud835\udd5c\u2016 \u2264 \u2191(measureUnivNNReal \u03bc ^ (ENNReal.toReal p)\u207b\u00b9)", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedAddCommGroup G\ninst\u271d\u2077 : TopologicalSpace \u03b1\ninst\u271d\u2076 : BorelSpace \u03b1\ninst\u271d\u2075 : SecondCountableTopologyEither \u03b1 E\ninst\u271d\u2074 : CompactSpace \u03b1\ninst\u271d\u00b3 : IsFiniteMeasure \u03bc\n\ud835\udd5c : Type u_5\ninst\u271d\u00b2 : Fact (1 \u2264 p)\ninst\u271d\u00b9 : NontriviallyNormedField \ud835\udd5c\ninst\u271d : NormedSpace \ud835\udd5c E\n\u22a2 \u2016BoundedContinuousFunction.toLp p \u03bc \ud835\udd5c\u2016 \u2264 \u2191(measureUnivNNReal \u03bc ^ (ENNReal.toReal p)\u207b\u00b9)"}, {"tactic": "exact BoundedContinuousFunction.toLp_norm_le \u03bc", "annotated_tactic": ["exact <a>BoundedContinuousFunction.toLp_norm_le</a> \u03bc", [{"full_name": "BoundedContinuousFunction.toLp_norm_le", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [1800, 9], "def_end_pos": [1800, 21]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedAddCommGroup G\ninst\u271d\u2077 : TopologicalSpace \u03b1\ninst\u271d\u2076 : BorelSpace \u03b1\ninst\u271d\u2075 : SecondCountableTopologyEither \u03b1 E\ninst\u271d\u2074 : CompactSpace \u03b1\ninst\u271d\u00b3 : IsFiniteMeasure \u03bc\n\ud835\udd5c : Type u_5\ninst\u271d\u00b2 : Fact (1 \u2264 p)\ninst\u271d\u00b9 : NontriviallyNormedField \ud835\udd5c\ninst\u271d : NormedSpace \ud835\udd5c E\n\u22a2 \u2016BoundedContinuousFunction.toLp p \u03bc \ud835\udd5c\u2016 \u2264 \u2191(measureUnivNNReal \u03bc ^ (ENNReal.toReal p)\u207b\u00b9)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Constructions/Prod/Basic.lean", "full_name": "MeasureTheory.lintegral_prod_symm", "start": [862, 1], "end": [865, 38], "traced_tactics": [{"tactic": "simp_rw [\u2190 lintegral_prod_swap f]", "annotated_tactic": ["simp_rw [\u2190 <a>lintegral_prod_swap</a> f]", [{"full_name": "MeasureTheory.lintegral_prod_swap", "def_path": "Mathlib/MeasureTheory/Constructions/Prod/Basic.lean", "def_pos": [814, 9], "def_end_pos": [814, 28]}]], "state_before": "\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2'\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : SigmaFinite \u03bd\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u00d7 \u03b2 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\n\u22a2 \u222b\u207b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (y : \u03b2), \u222b\u207b (x : \u03b1), f (x, y) \u2202\u03bc \u2202\u03bd", "state_after": "\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2'\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : SigmaFinite \u03bd\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u00d7 \u03b2 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\n\u22a2 \u222b\u207b (z : \u03b2 \u00d7 \u03b1), f (Prod.swap z) \u2202Measure.prod \u03bd \u03bc = \u222b\u207b (y : \u03b2), \u222b\u207b (x : \u03b1), f (x, y) \u2202\u03bc \u2202\u03bd"}, {"tactic": "exact lintegral_prod _ hf.prod_swap", "annotated_tactic": ["exact <a>lintegral_prod</a> _ hf.prod_swap", [{"full_name": "MeasureTheory.lintegral_prod", "def_path": "Mathlib/MeasureTheory/Constructions/Prod/Basic.lean", "def_pos": [851, 9], "def_end_pos": [851, 23]}]], "state_before": "\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2'\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : SigmaFinite \u03bd\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u00d7 \u03b2 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\n\u22a2 \u222b\u207b (z : \u03b2 \u00d7 \u03b1), f (Prod.swap z) \u2202Measure.prod \u03bd \u03bc = \u222b\u207b (y : \u03b2), \u222b\u207b (x : \u03b1), f (x, y) \u2202\u03bc \u2202\u03bd", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Int/ModEq.lean", "full_name": "Int.ModEq.mul", "start": [204, 11], "end": [205, 41], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Intervals/Monoid.lean", "full_name": "Set.image_const_add_Ioi", "start": [119, 1], "end": [120, 46], "traced_tactics": [{"tactic": "simp only [add_comm a, image_add_const_Ioi]", "annotated_tactic": ["simp only [<a>add_comm</a> a, <a>image_add_const_Ioi</a>]", [{"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [301, 3], "def_end_pos": [301, 14]}, {"full_name": "Set.image_add_const_Ioi", "def_path": "Mathlib/Data/Set/Intervals/Monoid.lean", "def_pos": [84, 9], "def_end_pos": [84, 28]}]], "state_before": "M : Type u_1\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid M\ninst\u271d : ExistsAddOfLE M\na b c d : M\n\u22a2 (fun x => a + x) '' Ioi b = Ioi (a + b)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "Finset.disjoint_or_nonempty_inter", "start": [1859, 1], "end": [1861, 13], "traced_tactics": [{"tactic": "rw [\u2190 not_disjoint_iff_nonempty_inter]", "annotated_tactic": ["rw [\u2190 <a>not_disjoint_iff_nonempty_inter</a>]", [{"full_name": "Finset.not_disjoint_iff_nonempty_inter", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1852, 9], "def_end_pos": [1852, 40]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns\u271d s\u2081 s\u2082 t\u271d t\u2081 t\u2082 u v : Finset \u03b1\na b : \u03b1\ns t : Finset \u03b1\n\u22a2 _root_.Disjoint s t \u2228 Finset.Nonempty (s \u2229 t)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns\u271d s\u2081 s\u2082 t\u271d t\u2081 t\u2082 u v : Finset \u03b1\na b : \u03b1\ns t : Finset \u03b1\n\u22a2 _root_.Disjoint s t \u2228 \u00ac_root_.Disjoint s t"}, {"tactic": "exact em _", "annotated_tactic": ["exact <a>em</a> _", [{"full_name": "em", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [194, 7], "def_end_pos": [194, 9]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns\u271d s\u2081 s\u2082 t\u271d t\u2081 t\u2082 u v : Finset \u03b1\na b : \u03b1\ns t : Finset \u03b1\n\u22a2 _root_.Disjoint s t \u2228 \u00ac_root_.Disjoint s t", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean", "full_name": "Real.map_matrix_volume_pi_eq_smul_volume_pi", "start": [423, 1], "end": [441, 39], "traced_tactics": [{"tactic": "apply diagonal_transvection_induction_of_det_ne_zero _ M hM", "annotated_tactic": ["apply <a>diagonal_transvection_induction_of_det_ne_zero</a> _ M hM", [{"full_name": "Matrix.diagonal_transvection_induction_of_det_ne_zero", "def_path": "Mathlib/LinearAlgebra/Matrix/Transvection.lean", "def_pos": [733, 9], "def_end_pos": [733, 55]}]], "state_before": "\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nM : Matrix \u03b9 \u03b9 \u211d\nhM : det M \u2260 0\n\u22a2 Measure.map (\u2191(\u2191toLin' M)) volume = ofReal |(det M)\u207b\u00b9| \u2022 volume", "state_after": "case hdiag\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nM : Matrix \u03b9 \u03b9 \u211d\nhM : det M \u2260 0\n\u22a2 \u2200 (D : \u03b9 \u2192 \u211d),\n    det (Matrix.diagonal D) \u2260 0 \u2192\n      Measure.map (\u2191(\u2191toLin' (Matrix.diagonal D))) volume = ofReal |(det (Matrix.diagonal D))\u207b\u00b9| \u2022 volume\n\ncase htransvec\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nM : Matrix \u03b9 \u03b9 \u211d\nhM : det M \u2260 0\n\u22a2 \u2200 (t : TransvectionStruct \u03b9 \u211d),\n    Measure.map (\u2191(\u2191toLin' (TransvectionStruct.toMatrix t))) volume =\n      ofReal |(det (TransvectionStruct.toMatrix t))\u207b\u00b9| \u2022 volume\n\ncase hmul\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nM : Matrix \u03b9 \u03b9 \u211d\nhM : det M \u2260 0\n\u22a2 \u2200 (A B : Matrix \u03b9 \u03b9 \u211d),\n    det A \u2260 0 \u2192\n      det B \u2260 0 \u2192\n        Measure.map (\u2191(\u2191toLin' A)) volume = ofReal |(det A)\u207b\u00b9| \u2022 volume \u2192\n          Measure.map (\u2191(\u2191toLin' B)) volume = ofReal |(det B)\u207b\u00b9| \u2022 volume \u2192\n            Measure.map (\u2191(\u2191toLin' (A * B))) volume = ofReal |(det (A * B))\u207b\u00b9| \u2022 volume"}, {"tactic": "intro D hD", "annotated_tactic": ["intro D hD", []], "state_before": "case hdiag\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nM : Matrix \u03b9 \u03b9 \u211d\nhM : det M \u2260 0\n\u22a2 \u2200 (D : \u03b9 \u2192 \u211d),\n    det (Matrix.diagonal D) \u2260 0 \u2192\n      Measure.map (\u2191(\u2191toLin' (Matrix.diagonal D))) volume = ofReal |(det (Matrix.diagonal D))\u207b\u00b9| \u2022 volume", "state_after": "case hdiag\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nM : Matrix \u03b9 \u03b9 \u211d\nhM : det M \u2260 0\nD : \u03b9 \u2192 \u211d\nhD : det (Matrix.diagonal D) \u2260 0\n\u22a2 Measure.map (\u2191(\u2191toLin' (Matrix.diagonal D))) volume = ofReal |(det (Matrix.diagonal D))\u207b\u00b9| \u2022 volume"}, {"tactic": "conv_rhs => rw [\u2190 smul_map_diagonal_volume_pi hD]", "annotated_tactic": ["conv_rhs => rw [\u2190 <a>smul_map_diagonal_volume_pi</a> hD]", [{"full_name": "Real.smul_map_diagonal_volume_pi", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean", "def_pos": [353, 9], "def_end_pos": [353, 36]}]], "state_before": "case hdiag\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nM : Matrix \u03b9 \u03b9 \u211d\nhM : det M \u2260 0\nD : \u03b9 \u2192 \u211d\nhD : det (Matrix.diagonal D) \u2260 0\n\u22a2 Measure.map (\u2191(\u2191toLin' (Matrix.diagonal D))) volume = ofReal |(det (Matrix.diagonal D))\u207b\u00b9| \u2022 volume", "state_after": "case hdiag\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nM : Matrix \u03b9 \u03b9 \u211d\nhM : det M \u2260 0\nD : \u03b9 \u2192 \u211d\nhD : det (Matrix.diagonal D) \u2260 0\n\u22a2 Measure.map (\u2191(\u2191toLin' (Matrix.diagonal D))) volume =\n    ofReal |(det (Matrix.diagonal D))\u207b\u00b9| \u2022\n      ofReal |det (Matrix.diagonal D)| \u2022 Measure.map (\u2191(\u2191toLin' (Matrix.diagonal D))) volume"}, {"tactic": "rw [smul_smul, \u2190 ENNReal.ofReal_mul (abs_nonneg _), \u2190 abs_mul, inv_mul_cancel hD, abs_one,\n  ENNReal.ofReal_one, one_smul]", "annotated_tactic": ["rw [<a>smul_smul</a>, \u2190 <a>ENNReal.ofReal_mul</a> (<a>abs_nonneg</a> _), \u2190 <a>abs_mul</a>, <a>inv_mul_cancel</a> hD, <a>abs_one</a>,\n      <a>ENNReal.ofReal_one</a>, <a>one_smul</a>]", [{"full_name": "smul_smul", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [484, 9], "def_end_pos": [484, 18]}, {"full_name": "ENNReal.ofReal_mul", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2225, 9], "def_end_pos": [2225, 19]}, {"full_name": "abs_nonneg", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [169, 9], "def_end_pos": [169, 19]}, {"full_name": "abs_mul", "def_path": "Mathlib/Algebra/Order/Ring/Abs.lean", "def_pos": [33, 9], "def_end_pos": [33, 16]}, {"full_name": "inv_mul_cancel", "def_path": "Mathlib/Algebra/GroupWithZero/NeZero.lean", "def_pos": [55, 9], "def_end_pos": [55, 23]}, {"full_name": "abs_one", "def_path": "Mathlib/Algebra/Order/Ring/Abs.lean", "def_pos": [24, 9], "def_end_pos": [24, 16]}, {"full_name": "ENNReal.ofReal_one", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [248, 17], "def_end_pos": [248, 27]}, {"full_name": "one_smul", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [492, 9], "def_end_pos": [492, 17]}]], "state_before": "case hdiag\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nM : Matrix \u03b9 \u03b9 \u211d\nhM : det M \u2260 0\nD : \u03b9 \u2192 \u211d\nhD : det (Matrix.diagonal D) \u2260 0\n\u22a2 Measure.map (\u2191(\u2191toLin' (Matrix.diagonal D))) volume =\n    ofReal |(det (Matrix.diagonal D))\u207b\u00b9| \u2022\n      ofReal |det (Matrix.diagonal D)| \u2022 Measure.map (\u2191(\u2191toLin' (Matrix.diagonal D))) volume", "state_after": "no goals"}, {"tactic": "intro t", "annotated_tactic": ["intro t", []], "state_before": "case htransvec\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nM : Matrix \u03b9 \u03b9 \u211d\nhM : det M \u2260 0\n\u22a2 \u2200 (t : TransvectionStruct \u03b9 \u211d),\n    Measure.map (\u2191(\u2191toLin' (TransvectionStruct.toMatrix t))) volume =\n      ofReal |(det (TransvectionStruct.toMatrix t))\u207b\u00b9| \u2022 volume", "state_after": "case htransvec\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nM : Matrix \u03b9 \u03b9 \u211d\nhM : det M \u2260 0\nt : TransvectionStruct \u03b9 \u211d\n\u22a2 Measure.map (\u2191(\u2191toLin' (TransvectionStruct.toMatrix t))) volume =\n    ofReal |(det (TransvectionStruct.toMatrix t))\u207b\u00b9| \u2022 volume"}, {"tactic": "simp only [Matrix.TransvectionStruct.det, ENNReal.ofReal_one,\n  (volume_preserving_transvectionStruct _).map_eq, one_smul, _root_.inv_one, abs_one]", "annotated_tactic": ["simp only [<a>Matrix.TransvectionStruct.det</a>, <a>ENNReal.ofReal_one</a>,\n      (<a>volume_preserving_transvectionStruct</a> _).<a>map_eq</a>, <a>one_smul</a>, <a>_root_.inv_one</a>, <a>abs_one</a>]", [{"full_name": "Matrix.TransvectionStruct.det", "def_path": "Mathlib/LinearAlgebra/Matrix/Transvection.lean", "def_pos": [180, 19], "def_end_pos": [180, 22]}, {"full_name": "ENNReal.ofReal_one", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [248, 17], "def_end_pos": [248, 27]}, {"full_name": "Real.volume_preserving_transvectionStruct", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean", "def_pos": [379, 9], "def_end_pos": [379, 45]}, {"full_name": "MeasureTheory.MeasurePreserving.map_eq", "def_path": "Mathlib/Dynamics/Ergodic/MeasurePreserving.lean", "def_pos": [45, 13], "def_end_pos": [45, 19]}, {"full_name": "one_smul", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [492, 9], "def_end_pos": [492, 17]}, {"full_name": "inv_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [1015, 9], "def_end_pos": [1015, 16]}, {"full_name": "abs_one", "def_path": "Mathlib/Algebra/Order/Ring/Abs.lean", "def_pos": [24, 9], "def_end_pos": [24, 16]}]], "state_before": "case htransvec\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nM : Matrix \u03b9 \u03b9 \u211d\nhM : det M \u2260 0\nt : TransvectionStruct \u03b9 \u211d\n\u22a2 Measure.map (\u2191(\u2191toLin' (TransvectionStruct.toMatrix t))) volume =\n    ofReal |(det (TransvectionStruct.toMatrix t))\u207b\u00b9| \u2022 volume", "state_after": "no goals"}, {"tactic": "intro A B _ _ IHA IHB", "annotated_tactic": ["intro A B _ _ IHA IHB", []], "state_before": "case hmul\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nM : Matrix \u03b9 \u03b9 \u211d\nhM : det M \u2260 0\n\u22a2 \u2200 (A B : Matrix \u03b9 \u03b9 \u211d),\n    det A \u2260 0 \u2192\n      det B \u2260 0 \u2192\n        Measure.map (\u2191(\u2191toLin' A)) volume = ofReal |(det A)\u207b\u00b9| \u2022 volume \u2192\n          Measure.map (\u2191(\u2191toLin' B)) volume = ofReal |(det B)\u207b\u00b9| \u2022 volume \u2192\n            Measure.map (\u2191(\u2191toLin' (A * B))) volume = ofReal |(det (A * B))\u207b\u00b9| \u2022 volume", "state_after": "case hmul\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nM : Matrix \u03b9 \u03b9 \u211d\nhM : det M \u2260 0\nA B : Matrix \u03b9 \u03b9 \u211d\na\u271d\u00b9 : det A \u2260 0\na\u271d : det B \u2260 0\nIHA : Measure.map (\u2191(\u2191toLin' A)) volume = ofReal |(det A)\u207b\u00b9| \u2022 volume\nIHB : Measure.map (\u2191(\u2191toLin' B)) volume = ofReal |(det B)\u207b\u00b9| \u2022 volume\n\u22a2 Measure.map (\u2191(\u2191toLin' (A * B))) volume = ofReal |(det (A * B))\u207b\u00b9| \u2022 volume"}, {"tactic": "rw [toLin'_mul, det_mul, LinearMap.coe_comp, \u2190 Measure.map_map, IHB, Measure.map_smul, IHA,\n  smul_smul, \u2190 ENNReal.ofReal_mul (abs_nonneg _), \u2190 abs_mul, mul_comm, mul_inv]", "annotated_tactic": ["rw [<a>toLin'_mul</a>, <a>det_mul</a>, <a>LinearMap.coe_comp</a>, \u2190 <a>Measure.map_map</a>, IHB, <a>Measure.map_smul</a>, IHA,\n      <a>smul_smul</a>, \u2190 <a>ENNReal.ofReal_mul</a> (<a>abs_nonneg</a> _), \u2190 <a>abs_mul</a>, <a>mul_comm</a>, <a>mul_inv</a>]", [{"full_name": "Matrix.toLin'_mul", "def_path": "Mathlib/LinearAlgebra/Matrix/ToLin.lean", "def_pos": [369, 9], "def_end_pos": [369, 26]}, {"full_name": "Matrix.det_mul", "def_path": "Mathlib/LinearAlgebra/Matrix/Determinant.lean", "def_pos": [151, 9], "def_end_pos": [151, 16]}, {"full_name": "LinearMap.coe_comp", "def_path": "Mathlib/Algebra/Module/LinearMap.lean", "def_pos": [554, 9], "def_end_pos": [554, 17]}, {"full_name": "MeasureTheory.Measure.map_map", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1258, 9], "def_end_pos": [1258, 16]}, {"full_name": "MeasureTheory.Measure.map_smul", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1202, 19], "def_end_pos": [1202, 27]}, {"full_name": "smul_smul", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [484, 9], "def_end_pos": [484, 18]}, {"full_name": "ENNReal.ofReal_mul", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2225, 9], "def_end_pos": [2225, 19]}, {"full_name": "abs_nonneg", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [169, 9], "def_end_pos": [169, 19]}, {"full_name": "abs_mul", "def_path": "Mathlib/Algebra/Order/Ring/Abs.lean", "def_pos": [33, 9], "def_end_pos": [33, 16]}, {"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [302, 9], "def_end_pos": [302, 17]}, {"full_name": "mul_inv", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [482, 9], "def_end_pos": [482, 16]}]], "state_before": "case hmul\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nM : Matrix \u03b9 \u03b9 \u211d\nhM : det M \u2260 0\nA B : Matrix \u03b9 \u03b9 \u211d\na\u271d\u00b9 : det A \u2260 0\na\u271d : det B \u2260 0\nIHA : Measure.map (\u2191(\u2191toLin' A)) volume = ofReal |(det A)\u207b\u00b9| \u2022 volume\nIHB : Measure.map (\u2191(\u2191toLin' B)) volume = ofReal |(det B)\u207b\u00b9| \u2022 volume\n\u22a2 Measure.map (\u2191(\u2191toLin' (A * B))) volume = ofReal |(det (A * B))\u207b\u00b9| \u2022 volume", "state_after": "case hmul.hg\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nM : Matrix \u03b9 \u03b9 \u211d\nhM : det M \u2260 0\nA B : Matrix \u03b9 \u03b9 \u211d\na\u271d\u00b9 : det A \u2260 0\na\u271d : det B \u2260 0\nIHA : Measure.map (\u2191(\u2191toLin' A)) volume = ofReal |(det A)\u207b\u00b9| \u2022 volume\nIHB : Measure.map (\u2191(\u2191toLin' B)) volume = ofReal |(det B)\u207b\u00b9| \u2022 volume\n\u22a2 Measurable \u2191(\u2191toLin' A)\n\ncase hmul.hf\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nM : Matrix \u03b9 \u03b9 \u211d\nhM : det M \u2260 0\nA B : Matrix \u03b9 \u03b9 \u211d\na\u271d\u00b9 : det A \u2260 0\na\u271d : det B \u2260 0\nIHA : Measure.map (\u2191(\u2191toLin' A)) volume = ofReal |(det A)\u207b\u00b9| \u2022 volume\nIHB : Measure.map (\u2191(\u2191toLin' B)) volume = ofReal |(det B)\u207b\u00b9| \u2022 volume\n\u22a2 Measurable \u2191(\u2191toLin' B)"}, {"tactic": "apply Continuous.measurable", "annotated_tactic": ["apply <a>Continuous.measurable</a>", [{"full_name": "Continuous.measurable", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [891, 9], "def_end_pos": [891, 30]}]], "state_before": "case hmul.hg\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nM : Matrix \u03b9 \u03b9 \u211d\nhM : det M \u2260 0\nA B : Matrix \u03b9 \u03b9 \u211d\na\u271d\u00b9 : det A \u2260 0\na\u271d : det B \u2260 0\nIHA : Measure.map (\u2191(\u2191toLin' A)) volume = ofReal |(det A)\u207b\u00b9| \u2022 volume\nIHB : Measure.map (\u2191(\u2191toLin' B)) volume = ofReal |(det B)\u207b\u00b9| \u2022 volume\n\u22a2 Measurable \u2191(\u2191toLin' A)", "state_after": "case hmul.hg.hf\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nM : Matrix \u03b9 \u03b9 \u211d\nhM : det M \u2260 0\nA B : Matrix \u03b9 \u03b9 \u211d\na\u271d\u00b9 : det A \u2260 0\na\u271d : det B \u2260 0\nIHA : Measure.map (\u2191(\u2191toLin' A)) volume = ofReal |(det A)\u207b\u00b9| \u2022 volume\nIHB : Measure.map (\u2191(\u2191toLin' B)) volume = ofReal |(det B)\u207b\u00b9| \u2022 volume\n\u22a2 Continuous \u2191(\u2191toLin' A)"}, {"tactic": "apply LinearMap.continuous_on_pi", "annotated_tactic": ["apply <a>LinearMap.continuous_on_pi</a>", [{"full_name": "LinearMap.continuous_on_pi", "def_path": "Mathlib/Topology/Algebra/Module/Basic.lean", "def_pos": [229, 9], "def_end_pos": [229, 35]}]], "state_before": "case hmul.hg.hf\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nM : Matrix \u03b9 \u03b9 \u211d\nhM : det M \u2260 0\nA B : Matrix \u03b9 \u03b9 \u211d\na\u271d\u00b9 : det A \u2260 0\na\u271d : det B \u2260 0\nIHA : Measure.map (\u2191(\u2191toLin' A)) volume = ofReal |(det A)\u207b\u00b9| \u2022 volume\nIHB : Measure.map (\u2191(\u2191toLin' B)) volume = ofReal |(det B)\u207b\u00b9| \u2022 volume\n\u22a2 Continuous \u2191(\u2191toLin' A)", "state_after": "no goals"}, {"tactic": "apply Continuous.measurable", "annotated_tactic": ["apply <a>Continuous.measurable</a>", [{"full_name": "Continuous.measurable", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [891, 9], "def_end_pos": [891, 30]}]], "state_before": "case hmul.hf\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nM : Matrix \u03b9 \u03b9 \u211d\nhM : det M \u2260 0\nA B : Matrix \u03b9 \u03b9 \u211d\na\u271d\u00b9 : det A \u2260 0\na\u271d : det B \u2260 0\nIHA : Measure.map (\u2191(\u2191toLin' A)) volume = ofReal |(det A)\u207b\u00b9| \u2022 volume\nIHB : Measure.map (\u2191(\u2191toLin' B)) volume = ofReal |(det B)\u207b\u00b9| \u2022 volume\n\u22a2 Measurable \u2191(\u2191toLin' B)", "state_after": "case hmul.hf.hf\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nM : Matrix \u03b9 \u03b9 \u211d\nhM : det M \u2260 0\nA B : Matrix \u03b9 \u03b9 \u211d\na\u271d\u00b9 : det A \u2260 0\na\u271d : det B \u2260 0\nIHA : Measure.map (\u2191(\u2191toLin' A)) volume = ofReal |(det A)\u207b\u00b9| \u2022 volume\nIHB : Measure.map (\u2191(\u2191toLin' B)) volume = ofReal |(det B)\u207b\u00b9| \u2022 volume\n\u22a2 Continuous \u2191(\u2191toLin' B)"}, {"tactic": "apply LinearMap.continuous_on_pi", "annotated_tactic": ["apply <a>LinearMap.continuous_on_pi</a>", [{"full_name": "LinearMap.continuous_on_pi", "def_path": "Mathlib/Topology/Algebra/Module/Basic.lean", "def_pos": [229, 9], "def_end_pos": [229, 35]}]], "state_before": "case hmul.hf.hf\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nM : Matrix \u03b9 \u03b9 \u211d\nhM : det M \u2260 0\nA B : Matrix \u03b9 \u03b9 \u211d\na\u271d\u00b9 : det A \u2260 0\na\u271d : det B \u2260 0\nIHA : Measure.map (\u2191(\u2191toLin' A)) volume = ofReal |(det A)\u207b\u00b9| \u2022 volume\nIHB : Measure.map (\u2191(\u2191toLin' B)) volume = ofReal |(det B)\u207b\u00b9| \u2022 volume\n\u22a2 Continuous \u2191(\u2191toLin' B)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/NullMeasurable.lean", "full_name": "MeasureTheory.measurableSet_of_null", "start": [459, 1], "end": [460, 45], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/RBMap/Lemmas.lean", "full_name": "Std.RBNode.Path.insertNew_toList", "start": [528, 9], "end": [529, 19], "traced_tactics": [{"tactic": "simp [insertNew]", "annotated_tactic": ["simp [<a>insertNew</a>]", [{"full_name": "Std.RBNode.Path.insertNew", "def_path": "lake-packages/std/Std/Data/RBMap/Basic.lean", "def_pos": [474, 15], "def_end_pos": [474, 29]}]], "state_before": "\u03b1 : Type u_1\nv : \u03b1\np : Path \u03b1\n\u22a2 toList (insertNew p v) = withList p [v]", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Int/GCD.lean", "full_name": "Nat.exists_mul_emod_eq_one_of_coprime", "start": [170, 1], "end": [173, 22], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/Layercake.lean", "full_name": "MeasureTheory.countable_meas_le_ne_meas_lt", "start": [73, 1], "end": [82, 86], "traced_tactics": [{"tactic": "let F : R \u2192 \u211d\u22650\u221e := fun t \u21a6 \u03bc {a : \u03b1 | t \u2264 g a}", "annotated_tactic": ["let F : R \u2192 \u211d\u22650\u221e := fun t \u21a6 \u03bc {a : \u03b1 | t \u2264 g a}", []], "state_before": "\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : LinearOrder R\ng : \u03b1 \u2192 R\n\u22a2 Set.Countable {t | \u2191\u2191\u03bc {a | t \u2264 g a} \u2260 \u2191\u2191\u03bc {a | t < g a}}", "state_after": "\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : LinearOrder R\ng : \u03b1 \u2192 R\nF : R \u2192 \u211d\u22650\u221e := fun t => \u2191\u2191\u03bc {a | t \u2264 g a}\n\u22a2 Set.Countable {t | \u2191\u2191\u03bc {a | t \u2264 g a} \u2260 \u2191\u2191\u03bc {a | t < g a}}"}, {"tactic": "apply (countable_image_gt_image_Ioi F).mono", "annotated_tactic": ["apply (<a>countable_image_gt_image_Ioi</a> F).<a>mono</a>", [{"full_name": "countable_image_gt_image_Ioi", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [1471, 9], "def_end_pos": [1471, 37]}, {"full_name": "Set.Countable.mono", "def_path": "Mathlib/Data/Set/Countable.lean", "def_pos": [85, 9], "def_end_pos": [85, 23]}]], "state_before": "\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : LinearOrder R\ng : \u03b1 \u2192 R\nF : R \u2192 \u211d\u22650\u221e := fun t => \u2191\u2191\u03bc {a | t \u2264 g a}\n\u22a2 Set.Countable {t | \u2191\u2191\u03bc {a | t \u2264 g a} \u2260 \u2191\u2191\u03bc {a | t < g a}}", "state_after": "\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : LinearOrder R\ng : \u03b1 \u2192 R\nF : R \u2192 \u211d\u22650\u221e := fun t => \u2191\u2191\u03bc {a | t \u2264 g a}\n\u22a2 {t | \u2191\u2191\u03bc {a | t \u2264 g a} \u2260 \u2191\u2191\u03bc {a | t < g a}} \u2286 {x | \u2203 z, z < F x \u2227 \u2200 (y : R), x < y \u2192 F y \u2264 z}"}, {"tactic": "intro t ht", "annotated_tactic": ["intro t ht", []], "state_before": "\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : LinearOrder R\ng : \u03b1 \u2192 R\nF : R \u2192 \u211d\u22650\u221e := fun t => \u2191\u2191\u03bc {a | t \u2264 g a}\n\u22a2 {t | \u2191\u2191\u03bc {a | t \u2264 g a} \u2260 \u2191\u2191\u03bc {a | t < g a}} \u2286 {x | \u2203 z, z < F x \u2227 \u2200 (y : R), x < y \u2192 F y \u2264 z}", "state_after": "\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : LinearOrder R\ng : \u03b1 \u2192 R\nF : R \u2192 \u211d\u22650\u221e := fun t => \u2191\u2191\u03bc {a | t \u2264 g a}\nt : R\nht : t \u2208 {t | \u2191\u2191\u03bc {a | t \u2264 g a} \u2260 \u2191\u2191\u03bc {a | t < g a}}\n\u22a2 t \u2208 {x | \u2203 z, z < F x \u2227 \u2200 (y : R), x < y \u2192 F y \u2264 z}"}, {"tactic": "have : \u03bc {a | t < g a} < \u03bc {a | t \u2264 g a} :=\n  lt_of_le_of_ne (measure_mono (fun a ha \u21a6 le_of_lt ha)) (Ne.symm ht)", "annotated_tactic": ["have : \u03bc {a | t < g a} < \u03bc {a | t \u2264 g a} :=\n    <a>lt_of_le_of_ne</a> (<a>measure_mono</a> (fun a ha \u21a6 <a>le_of_lt</a> ha)) (<a>Ne.symm</a> ht)", [{"full_name": "lt_of_le_of_ne", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [196, 9], "def_end_pos": [196, 23]}, {"full_name": "MeasureTheory.measure_mono", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [193, 9], "def_end_pos": [193, 21]}, {"full_name": "le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [110, 9], "def_end_pos": [110, 17]}, {"full_name": "Ne.symm", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [575, 9], "def_end_pos": [575, 16]}]], "state_before": "\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : LinearOrder R\ng : \u03b1 \u2192 R\nF : R \u2192 \u211d\u22650\u221e := fun t => \u2191\u2191\u03bc {a | t \u2264 g a}\nt : R\nht : t \u2208 {t | \u2191\u2191\u03bc {a | t \u2264 g a} \u2260 \u2191\u2191\u03bc {a | t < g a}}\n\u22a2 t \u2208 {x | \u2203 z, z < F x \u2227 \u2200 (y : R), x < y \u2192 F y \u2264 z}", "state_after": "\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : LinearOrder R\ng : \u03b1 \u2192 R\nF : R \u2192 \u211d\u22650\u221e := fun t => \u2191\u2191\u03bc {a | t \u2264 g a}\nt : R\nht : t \u2208 {t | \u2191\u2191\u03bc {a | t \u2264 g a} \u2260 \u2191\u2191\u03bc {a | t < g a}}\nthis : \u2191\u2191\u03bc {a | t < g a} < \u2191\u2191\u03bc {a | t \u2264 g a}\n\u22a2 t \u2208 {x | \u2203 z, z < F x \u2227 \u2200 (y : R), x < y \u2192 F y \u2264 z}"}, {"tactic": "refine \u27e8\u03bc {a | t < g a}, this, fun s hs \u21a6 measure_mono (fun a ha \u21a6 hs.trans_le ha)\u27e9", "annotated_tactic": ["refine \u27e8\u03bc {a | t < g a}, this, fun s hs \u21a6 <a>measure_mono</a> (fun a ha \u21a6 hs.trans_le ha)\u27e9", [{"full_name": "MeasureTheory.measure_mono", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [193, 9], "def_end_pos": [193, 21]}]], "state_before": "\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : LinearOrder R\ng : \u03b1 \u2192 R\nF : R \u2192 \u211d\u22650\u221e := fun t => \u2191\u2191\u03bc {a | t \u2264 g a}\nt : R\nht : t \u2208 {t | \u2191\u2191\u03bc {a | t \u2264 g a} \u2260 \u2191\u2191\u03bc {a | t < g a}}\nthis : \u2191\u2191\u03bc {a | t < g a} < \u2191\u2191\u03bc {a | t \u2264 g a}\n\u22a2 t \u2208 {x | \u2203 z, z < F x \u2227 \u2200 (y : R), x < y \u2192 F y \u2264 z}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "full_name": "Int.le_add_of_neg_le_sub_right", "start": [1024, 11], "end": [1025, 59], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "full_name": "tendsto_measure_cthickening_of_isCompact", "start": [1840, 1], "end": [1845, 77], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Doubling.lean", "full_name": "IsUnifLocDoublingMeasure.exists_measure_closedBall_le_mul'", "start": [67, 1], "end": [69, 62], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/CircleIntegral.lean", "full_name": "deriv_circleMap_ne_zero", "start": [203, 1], "end": [205, 38], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Sym.lean", "full_name": "Finset.sym_fill_mem", "start": [225, 1], "end": [228, 79], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "full_name": "MeasureTheory.finStronglyMeasurable_iff_stronglyMeasurable_and_exists_set_sigmaFinite", "start": [1137, 1], "end": [1144, 28], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "full_name": "intervalIntegral.integral_congr_ae", "start": [990, 1], "end": [992, 63], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Martingale/Upcrossing.lean", "full_name": "MeasureTheory.upperCrossingTime_lt_succ", "start": [263, 1], "end": [266, 52], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Martingale/Centering.lean", "full_name": "MeasureTheory.martingalePart_add_ae_eq", "start": [135, 1], "end": [153, 13], "traced_tactics": [{"tactic": "set h := f - martingalePart (f + g) \u2131 \u03bc with hhdef", "annotated_tactic": ["set h := f - <a>martingalePart</a> (f + g) \u2131 \u03bc with hhdef", [{"full_name": "MeasureTheory.martingalePart", "def_path": "Mathlib/Probability/Martingale/Centering.lean", "def_pos": [66, 19], "def_end_pos": [66, 33]}]], "state_before": "\u03a9 : Type u_1\nE : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn\u271d : \u2115\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\nf g : \u2115 \u2192 \u03a9 \u2192 E\nhf : Martingale f \u2131 \u03bc\nhg : Adapted \u2131 fun n => g (n + 1)\nhg0 : g 0 = 0\nhgint : \u2200 (n : \u2115), Integrable (g n)\nn : \u2115\n\u22a2 martingalePart (f + g) \u2131 \u03bc n =\u1d50[\u03bc] f n", "state_after": "\u03a9 : Type u_1\nE : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn\u271d : \u2115\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\nf g : \u2115 \u2192 \u03a9 \u2192 E\nhf : Martingale f \u2131 \u03bc\nhg : Adapted \u2131 fun n => g (n + 1)\nhg0 : g 0 = 0\nhgint : \u2200 (n : \u2115), Integrable (g n)\nn : \u2115\nh : \u2115 \u2192 \u03a9 \u2192 E := f - martingalePart (f + g) \u2131 \u03bc\nhhdef : h = f - martingalePart (f + g) \u2131 \u03bc\n\u22a2 martingalePart (f + g) \u2131 \u03bc n =\u1d50[\u03bc] f n"}, {"tactic": "have hh : h = predictablePart (f + g) \u2131 \u03bc - g := by\n  rw [hhdef, sub_eq_sub_iff_add_eq_add, add_comm (predictablePart (f + g) \u2131 \u03bc),\n    martingalePart_add_predictablePart]", "annotated_tactic": ["have hh : h = <a>predictablePart</a> (f + g) \u2131 \u03bc - g := by\n    rw [hhdef, <a>sub_eq_sub_iff_add_eq_add</a>, <a>add_comm</a> (<a>predictablePart</a> (f + g) \u2131 \u03bc),\n      <a>martingalePart_add_predictablePart</a>]", [{"full_name": "MeasureTheory.predictablePart", "def_path": "Mathlib/Probability/Martingale/Centering.lean", "def_pos": [45, 19], "def_end_pos": [45, 34]}, {"full_name": "sub_eq_sub_iff_add_eq_add", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [1003, 3], "def_end_pos": [1003, 14]}, {"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [301, 3], "def_end_pos": [301, 14]}, {"full_name": "MeasureTheory.predictablePart", "def_path": "Mathlib/Probability/Martingale/Centering.lean", "def_pos": [45, 19], "def_end_pos": [45, 34]}, {"full_name": "MeasureTheory.martingalePart_add_predictablePart", "def_path": "Mathlib/Probability/Martingale/Centering.lean", "def_pos": [70, 9], "def_end_pos": [70, 43]}]], "state_before": "\u03a9 : Type u_1\nE : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn\u271d : \u2115\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\nf g : \u2115 \u2192 \u03a9 \u2192 E\nhf : Martingale f \u2131 \u03bc\nhg : Adapted \u2131 fun n => g (n + 1)\nhg0 : g 0 = 0\nhgint : \u2200 (n : \u2115), Integrable (g n)\nn : \u2115\nh : \u2115 \u2192 \u03a9 \u2192 E := f - martingalePart (f + g) \u2131 \u03bc\nhhdef : h = f - martingalePart (f + g) \u2131 \u03bc\n\u22a2 martingalePart (f + g) \u2131 \u03bc n =\u1d50[\u03bc] f n", "state_after": "\u03a9 : Type u_1\nE : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn\u271d : \u2115\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\nf g : \u2115 \u2192 \u03a9 \u2192 E\nhf : Martingale f \u2131 \u03bc\nhg : Adapted \u2131 fun n => g (n + 1)\nhg0 : g 0 = 0\nhgint : \u2200 (n : \u2115), Integrable (g n)\nn : \u2115\nh : \u2115 \u2192 \u03a9 \u2192 E := f - martingalePart (f + g) \u2131 \u03bc\nhhdef : h = f - martingalePart (f + g) \u2131 \u03bc\nhh : h = predictablePart (f + g) \u2131 \u03bc - g\n\u22a2 martingalePart (f + g) \u2131 \u03bc n =\u1d50[\u03bc] f n"}, {"tactic": "have hhpred : Adapted \u2131 fun n => h (n + 1) := by\n  rw [hh]\n  exact adapted_predictablePart.sub hg", "annotated_tactic": ["have hhpred : <a>Adapted</a> \u2131 fun n => h (n + 1) := by\n    rw [hh]\n    exact adapted_predictablePart.sub hg", [{"full_name": "MeasureTheory.Adapted", "def_path": "Mathlib/Probability/Process/Adapted.lean", "def_pos": [49, 5], "def_end_pos": [49, 12]}]], "state_before": "\u03a9 : Type u_1\nE : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn\u271d : \u2115\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\nf g : \u2115 \u2192 \u03a9 \u2192 E\nhf : Martingale f \u2131 \u03bc\nhg : Adapted \u2131 fun n => g (n + 1)\nhg0 : g 0 = 0\nhgint : \u2200 (n : \u2115), Integrable (g n)\nn : \u2115\nh : \u2115 \u2192 \u03a9 \u2192 E := f - martingalePart (f + g) \u2131 \u03bc\nhhdef : h = f - martingalePart (f + g) \u2131 \u03bc\nhh : h = predictablePart (f + g) \u2131 \u03bc - g\n\u22a2 martingalePart (f + g) \u2131 \u03bc n =\u1d50[\u03bc] f n", "state_after": "\u03a9 : Type u_1\nE : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn\u271d : \u2115\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\nf g : \u2115 \u2192 \u03a9 \u2192 E\nhf : Martingale f \u2131 \u03bc\nhg : Adapted \u2131 fun n => g (n + 1)\nhg0 : g 0 = 0\nhgint : \u2200 (n : \u2115), Integrable (g n)\nn : \u2115\nh : \u2115 \u2192 \u03a9 \u2192 E := f - martingalePart (f + g) \u2131 \u03bc\nhhdef : h = f - martingalePart (f + g) \u2131 \u03bc\nhh : h = predictablePart (f + g) \u2131 \u03bc - g\nhhpred : Adapted \u2131 fun n => h (n + 1)\n\u22a2 martingalePart (f + g) \u2131 \u03bc n =\u1d50[\u03bc] f n"}, {"tactic": "have hhmgle : Martingale h \u2131 \u03bc := hf.sub (martingale_martingalePart\n  (hf.adapted.add <| Predictable.adapted hg <| hg0.symm \u25b8 stronglyMeasurable_zero) fun n =>\n  (hf.integrable n).add <| hgint n)", "annotated_tactic": ["have hhmgle : <a>Martingale</a> h \u2131 \u03bc := hf.sub (<a>martingale_martingalePart</a>\n    (hf.adapted.add <| <a>Predictable.adapted</a> hg <| hg0.symm \u25b8 <a>stronglyMeasurable_zero</a>) fun n =>\n    (hf.integrable n).<a>add</a> <| hgint n)", [{"full_name": "MeasureTheory.Martingale", "def_path": "Mathlib/Probability/Martingale/Basic.lean", "def_pos": [52, 5], "def_end_pos": [52, 15]}, {"full_name": "MeasureTheory.martingale_martingalePart", "def_path": "Mathlib/Probability/Martingale/Centering.lean", "def_pos": [93, 9], "def_end_pos": [93, 34]}, {"full_name": "MeasureTheory.Predictable.adapted", "def_path": "Mathlib/Probability/Process/Adapted.lean", "def_pos": [225, 9], "def_end_pos": [225, 28]}, {"full_name": "MeasureTheory.stronglyMeasurable_zero", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [148, 3], "def_end_pos": [148, 14]}, {"full_name": "MeasureTheory.Integrable.add", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [677, 9], "def_end_pos": [677, 23]}]], "state_before": "\u03a9 : Type u_1\nE : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn\u271d : \u2115\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\nf g : \u2115 \u2192 \u03a9 \u2192 E\nhf : Martingale f \u2131 \u03bc\nhg : Adapted \u2131 fun n => g (n + 1)\nhg0 : g 0 = 0\nhgint : \u2200 (n : \u2115), Integrable (g n)\nn : \u2115\nh : \u2115 \u2192 \u03a9 \u2192 E := f - martingalePart (f + g) \u2131 \u03bc\nhhdef : h = f - martingalePart (f + g) \u2131 \u03bc\nhh : h = predictablePart (f + g) \u2131 \u03bc - g\nhhpred : Adapted \u2131 fun n => h (n + 1)\n\u22a2 martingalePart (f + g) \u2131 \u03bc n =\u1d50[\u03bc] f n", "state_after": "\u03a9 : Type u_1\nE : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn\u271d : \u2115\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\nf g : \u2115 \u2192 \u03a9 \u2192 E\nhf : Martingale f \u2131 \u03bc\nhg : Adapted \u2131 fun n => g (n + 1)\nhg0 : g 0 = 0\nhgint : \u2200 (n : \u2115), Integrable (g n)\nn : \u2115\nh : \u2115 \u2192 \u03a9 \u2192 E := f - martingalePart (f + g) \u2131 \u03bc\nhhdef : h = f - martingalePart (f + g) \u2131 \u03bc\nhh : h = predictablePart (f + g) \u2131 \u03bc - g\nhhpred : Adapted \u2131 fun n => h (n + 1)\nhhmgle : Martingale h \u2131 \u03bc\n\u22a2 martingalePart (f + g) \u2131 \u03bc n =\u1d50[\u03bc] f n"}, {"tactic": "refine' (eventuallyEq_iff_sub.2 _).symm", "annotated_tactic": ["refine' (<a>eventuallyEq_iff_sub</a>.2 _).<a>symm</a>", [{"full_name": "Filter.eventuallyEq_iff_sub", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1628, 9], "def_end_pos": [1628, 29]}, {"full_name": "Filter.EventuallyEq.symm", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1498, 9], "def_end_pos": [1498, 26]}]], "state_before": "\u03a9 : Type u_1\nE : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn\u271d : \u2115\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\nf g : \u2115 \u2192 \u03a9 \u2192 E\nhf : Martingale f \u2131 \u03bc\nhg : Adapted \u2131 fun n => g (n + 1)\nhg0 : g 0 = 0\nhgint : \u2200 (n : \u2115), Integrable (g n)\nn : \u2115\nh : \u2115 \u2192 \u03a9 \u2192 E := f - martingalePart (f + g) \u2131 \u03bc\nhhdef : h = f - martingalePart (f + g) \u2131 \u03bc\nhh : h = predictablePart (f + g) \u2131 \u03bc - g\nhhpred : Adapted \u2131 fun n => h (n + 1)\nhhmgle : Martingale h \u2131 \u03bc\n\u22a2 martingalePart (f + g) \u2131 \u03bc n =\u1d50[\u03bc] f n", "state_after": "\u03a9 : Type u_1\nE : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn\u271d : \u2115\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\nf g : \u2115 \u2192 \u03a9 \u2192 E\nhf : Martingale f \u2131 \u03bc\nhg : Adapted \u2131 fun n => g (n + 1)\nhg0 : g 0 = 0\nhgint : \u2200 (n : \u2115), Integrable (g n)\nn : \u2115\nh : \u2115 \u2192 \u03a9 \u2192 E := f - martingalePart (f + g) \u2131 \u03bc\nhhdef : h = f - martingalePart (f + g) \u2131 \u03bc\nhh : h = predictablePart (f + g) \u2131 \u03bc - g\nhhpred : Adapted \u2131 fun n => h (n + 1)\nhhmgle : Martingale h \u2131 \u03bc\n\u22a2 f n - martingalePart (f + g) \u2131 \u03bc n =\u1d50[\u03bc] 0"}, {"tactic": "filter_upwards [hhmgle.eq_zero_of_predictable hhpred n] with \u03c9 h\u03c9", "annotated_tactic": ["filter_upwards [hhmgle.eq_zero_of_predictable hhpred n] with \u03c9 h\u03c9", []], "state_before": "\u03a9 : Type u_1\nE : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn\u271d : \u2115\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\nf g : \u2115 \u2192 \u03a9 \u2192 E\nhf : Martingale f \u2131 \u03bc\nhg : Adapted \u2131 fun n => g (n + 1)\nhg0 : g 0 = 0\nhgint : \u2200 (n : \u2115), Integrable (g n)\nn : \u2115\nh : \u2115 \u2192 \u03a9 \u2192 E := f - martingalePart (f + g) \u2131 \u03bc\nhhdef : h = f - martingalePart (f + g) \u2131 \u03bc\nhh : h = predictablePart (f + g) \u2131 \u03bc - g\nhhpred : Adapted \u2131 fun n => h (n + 1)\nhhmgle : Martingale h \u2131 \u03bc\n\u22a2 f n - martingalePart (f + g) \u2131 \u03bc n =\u1d50[\u03bc] 0", "state_after": "case h\n\u03a9 : Type u_1\nE : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn\u271d : \u2115\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\nf g : \u2115 \u2192 \u03a9 \u2192 E\nhf : Martingale f \u2131 \u03bc\nhg : Adapted \u2131 fun n => g (n + 1)\nhg0 : g 0 = 0\nhgint : \u2200 (n : \u2115), Integrable (g n)\nn : \u2115\nh : \u2115 \u2192 \u03a9 \u2192 E := f - martingalePart (f + g) \u2131 \u03bc\nhhdef : h = f - martingalePart (f + g) \u2131 \u03bc\nhh : h = predictablePart (f + g) \u2131 \u03bc - g\nhhpred : Adapted \u2131 fun n => h (n + 1)\nhhmgle : Martingale h \u2131 \u03bc\n\u03c9 : \u03a9\nh\u03c9 : h n \u03c9 = h 0 \u03c9\n\u22a2 (f n - martingalePart (f + g) \u2131 \u03bc n) \u03c9 = OfNat.ofNat 0 \u03c9"}, {"tactic": "unfold_let h at h\u03c9", "annotated_tactic": ["unfold_let h at h\u03c9", []], "state_before": "case h\n\u03a9 : Type u_1\nE : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn\u271d : \u2115\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\nf g : \u2115 \u2192 \u03a9 \u2192 E\nhf : Martingale f \u2131 \u03bc\nhg : Adapted \u2131 fun n => g (n + 1)\nhg0 : g 0 = 0\nhgint : \u2200 (n : \u2115), Integrable (g n)\nn : \u2115\nh : \u2115 \u2192 \u03a9 \u2192 E := f - martingalePart (f + g) \u2131 \u03bc\nhhdef : h = f - martingalePart (f + g) \u2131 \u03bc\nhh : h = predictablePart (f + g) \u2131 \u03bc - g\nhhpred : Adapted \u2131 fun n => h (n + 1)\nhhmgle : Martingale h \u2131 \u03bc\n\u03c9 : \u03a9\nh\u03c9 : h n \u03c9 = h 0 \u03c9\n\u22a2 (f n - martingalePart (f + g) \u2131 \u03bc n) \u03c9 = OfNat.ofNat 0 \u03c9", "state_after": "case h\n\u03a9 : Type u_1\nE : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn\u271d : \u2115\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\nf g : \u2115 \u2192 \u03a9 \u2192 E\nhf : Martingale f \u2131 \u03bc\nhg : Adapted \u2131 fun n => g (n + 1)\nhg0 : g 0 = 0\nhgint : \u2200 (n : \u2115), Integrable (g n)\nn : \u2115\nh : \u2115 \u2192 \u03a9 \u2192 E := f - martingalePart (f + g) \u2131 \u03bc\nhhdef : h = f - martingalePart (f + g) \u2131 \u03bc\nhh : h = predictablePart (f + g) \u2131 \u03bc - g\nhhpred : Adapted \u2131 fun n => h (n + 1)\nhhmgle : Martingale h \u2131 \u03bc\n\u03c9 : \u03a9\nh\u03c9 : (f - martingalePart (f + g) \u2131 \u03bc) n \u03c9 = (f - martingalePart (f + g) \u2131 \u03bc) 0 \u03c9\n\u22a2 (f n - martingalePart (f + g) \u2131 \u03bc n) \u03c9 = OfNat.ofNat 0 \u03c9"}, {"tactic": "rw [Pi.sub_apply] at h\u03c9", "annotated_tactic": ["rw [<a>Pi.sub_apply</a>] at h\u03c9", [{"full_name": "Pi.sub_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [200, 3], "def_end_pos": [200, 14]}]], "state_before": "case h\n\u03a9 : Type u_1\nE : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn\u271d : \u2115\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\nf g : \u2115 \u2192 \u03a9 \u2192 E\nhf : Martingale f \u2131 \u03bc\nhg : Adapted \u2131 fun n => g (n + 1)\nhg0 : g 0 = 0\nhgint : \u2200 (n : \u2115), Integrable (g n)\nn : \u2115\nh : \u2115 \u2192 \u03a9 \u2192 E := f - martingalePart (f + g) \u2131 \u03bc\nhhdef : h = f - martingalePart (f + g) \u2131 \u03bc\nhh : h = predictablePart (f + g) \u2131 \u03bc - g\nhhpred : Adapted \u2131 fun n => h (n + 1)\nhhmgle : Martingale h \u2131 \u03bc\n\u03c9 : \u03a9\nh\u03c9 : (f - martingalePart (f + g) \u2131 \u03bc) n \u03c9 = (f - martingalePart (f + g) \u2131 \u03bc) 0 \u03c9\n\u22a2 (f n - martingalePart (f + g) \u2131 \u03bc n) \u03c9 = OfNat.ofNat 0 \u03c9", "state_after": "case h\n\u03a9 : Type u_1\nE : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn\u271d : \u2115\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\nf g : \u2115 \u2192 \u03a9 \u2192 E\nhf : Martingale f \u2131 \u03bc\nhg : Adapted \u2131 fun n => g (n + 1)\nhg0 : g 0 = 0\nhgint : \u2200 (n : \u2115), Integrable (g n)\nn : \u2115\nh : \u2115 \u2192 \u03a9 \u2192 E := f - martingalePart (f + g) \u2131 \u03bc\nhhdef : h = f - martingalePart (f + g) \u2131 \u03bc\nhh : h = predictablePart (f + g) \u2131 \u03bc - g\nhhpred : Adapted \u2131 fun n => h (n + 1)\nhhmgle : Martingale h \u2131 \u03bc\n\u03c9 : \u03a9\nh\u03c9 : (f n - martingalePart (f + g) \u2131 \u03bc n) \u03c9 = (f - martingalePart (f + g) \u2131 \u03bc) 0 \u03c9\n\u22a2 (f n - martingalePart (f + g) \u2131 \u03bc n) \u03c9 = OfNat.ofNat 0 \u03c9"}, {"tactic": "rw [h\u03c9, Pi.sub_apply, martingalePart]", "annotated_tactic": ["rw [h\u03c9, <a>Pi.sub_apply</a>, <a>martingalePart</a>]", [{"full_name": "Pi.sub_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [200, 3], "def_end_pos": [200, 14]}, {"full_name": "MeasureTheory.martingalePart", "def_path": "Mathlib/Probability/Martingale/Centering.lean", "def_pos": [66, 19], "def_end_pos": [66, 33]}]], "state_before": "case h\n\u03a9 : Type u_1\nE : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn\u271d : \u2115\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\nf g : \u2115 \u2192 \u03a9 \u2192 E\nhf : Martingale f \u2131 \u03bc\nhg : Adapted \u2131 fun n => g (n + 1)\nhg0 : g 0 = 0\nhgint : \u2200 (n : \u2115), Integrable (g n)\nn : \u2115\nh : \u2115 \u2192 \u03a9 \u2192 E := f - martingalePart (f + g) \u2131 \u03bc\nhhdef : h = f - martingalePart (f + g) \u2131 \u03bc\nhh : h = predictablePart (f + g) \u2131 \u03bc - g\nhhpred : Adapted \u2131 fun n => h (n + 1)\nhhmgle : Martingale h \u2131 \u03bc\n\u03c9 : \u03a9\nh\u03c9 : (f n - martingalePart (f + g) \u2131 \u03bc n) \u03c9 = (f - martingalePart (f + g) \u2131 \u03bc) 0 \u03c9\n\u22a2 (f n - martingalePart (f + g) \u2131 \u03bc n) \u03c9 = OfNat.ofNat 0 \u03c9", "state_after": "case h\n\u03a9 : Type u_1\nE : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn\u271d : \u2115\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\nf g : \u2115 \u2192 \u03a9 \u2192 E\nhf : Martingale f \u2131 \u03bc\nhg : Adapted \u2131 fun n => g (n + 1)\nhg0 : g 0 = 0\nhgint : \u2200 (n : \u2115), Integrable (g n)\nn : \u2115\nh : \u2115 \u2192 \u03a9 \u2192 E := f - martingalePart (f + g) \u2131 \u03bc\nhhdef : h = f - martingalePart (f + g) \u2131 \u03bc\nhh : h = predictablePart (f + g) \u2131 \u03bc - g\nhhpred : Adapted \u2131 fun n => h (n + 1)\nhhmgle : Martingale h \u2131 \u03bc\n\u03c9 : \u03a9\nh\u03c9 : (f n - martingalePart (f + g) \u2131 \u03bc n) \u03c9 = (f - martingalePart (f + g) \u2131 \u03bc) 0 \u03c9\n\u22a2 (f 0 - ((f + g) 0 - predictablePart (f + g) \u2131 \u03bc 0)) \u03c9 = OfNat.ofNat 0 \u03c9"}, {"tactic": "simp [hg0]", "annotated_tactic": ["simp [hg0]", []], "state_before": "case h\n\u03a9 : Type u_1\nE : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn\u271d : \u2115\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\nf g : \u2115 \u2192 \u03a9 \u2192 E\nhf : Martingale f \u2131 \u03bc\nhg : Adapted \u2131 fun n => g (n + 1)\nhg0 : g 0 = 0\nhgint : \u2200 (n : \u2115), Integrable (g n)\nn : \u2115\nh : \u2115 \u2192 \u03a9 \u2192 E := f - martingalePart (f + g) \u2131 \u03bc\nhhdef : h = f - martingalePart (f + g) \u2131 \u03bc\nhh : h = predictablePart (f + g) \u2131 \u03bc - g\nhhpred : Adapted \u2131 fun n => h (n + 1)\nhhmgle : Martingale h \u2131 \u03bc\n\u03c9 : \u03a9\nh\u03c9 : (f n - martingalePart (f + g) \u2131 \u03bc n) \u03c9 = (f - martingalePart (f + g) \u2131 \u03bc) 0 \u03c9\n\u22a2 (f 0 - ((f + g) 0 - predictablePart (f + g) \u2131 \u03bc 0)) \u03c9 = OfNat.ofNat 0 \u03c9", "state_after": "no goals"}, {"tactic": "rw [hhdef, sub_eq_sub_iff_add_eq_add, add_comm (predictablePart (f + g) \u2131 \u03bc),\n  martingalePart_add_predictablePart]", "annotated_tactic": ["rw [hhdef, <a>sub_eq_sub_iff_add_eq_add</a>, <a>add_comm</a> (<a>predictablePart</a> (f + g) \u2131 \u03bc),\n      <a>martingalePart_add_predictablePart</a>]", [{"full_name": "sub_eq_sub_iff_add_eq_add", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [1003, 3], "def_end_pos": [1003, 14]}, {"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [301, 3], "def_end_pos": [301, 14]}, {"full_name": "MeasureTheory.predictablePart", "def_path": "Mathlib/Probability/Martingale/Centering.lean", "def_pos": [45, 19], "def_end_pos": [45, 34]}, {"full_name": "MeasureTheory.martingalePart_add_predictablePart", "def_path": "Mathlib/Probability/Martingale/Centering.lean", "def_pos": [70, 9], "def_end_pos": [70, 43]}]], "state_before": "\u03a9 : Type u_1\nE : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn\u271d : \u2115\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\nf g : \u2115 \u2192 \u03a9 \u2192 E\nhf : Martingale f \u2131 \u03bc\nhg : Adapted \u2131 fun n => g (n + 1)\nhg0 : g 0 = 0\nhgint : \u2200 (n : \u2115), Integrable (g n)\nn : \u2115\nh : \u2115 \u2192 \u03a9 \u2192 E := f - martingalePart (f + g) \u2131 \u03bc\nhhdef : h = f - martingalePart (f + g) \u2131 \u03bc\n\u22a2 h = predictablePart (f + g) \u2131 \u03bc - g", "state_after": "no goals"}, {"tactic": "rw [hh]", "annotated_tactic": ["rw [hh]", []], "state_before": "\u03a9 : Type u_1\nE : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn\u271d : \u2115\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\nf g : \u2115 \u2192 \u03a9 \u2192 E\nhf : Martingale f \u2131 \u03bc\nhg : Adapted \u2131 fun n => g (n + 1)\nhg0 : g 0 = 0\nhgint : \u2200 (n : \u2115), Integrable (g n)\nn : \u2115\nh : \u2115 \u2192 \u03a9 \u2192 E := f - martingalePart (f + g) \u2131 \u03bc\nhhdef : h = f - martingalePart (f + g) \u2131 \u03bc\nhh : h = predictablePart (f + g) \u2131 \u03bc - g\n\u22a2 Adapted \u2131 fun n => h (n + 1)", "state_after": "\u03a9 : Type u_1\nE : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn\u271d : \u2115\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\nf g : \u2115 \u2192 \u03a9 \u2192 E\nhf : Martingale f \u2131 \u03bc\nhg : Adapted \u2131 fun n => g (n + 1)\nhg0 : g 0 = 0\nhgint : \u2200 (n : \u2115), Integrable (g n)\nn : \u2115\nh : \u2115 \u2192 \u03a9 \u2192 E := f - martingalePart (f + g) \u2131 \u03bc\nhhdef : h = f - martingalePart (f + g) \u2131 \u03bc\nhh : h = predictablePart (f + g) \u2131 \u03bc - g\n\u22a2 Adapted \u2131 fun n => (predictablePart (f + g) \u2131 \u03bc - g) (n + 1)"}, {"tactic": "exact adapted_predictablePart.sub hg", "annotated_tactic": ["exact adapted_predictablePart.sub hg", []], "state_before": "\u03a9 : Type u_1\nE : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn\u271d : \u2115\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\nf g : \u2115 \u2192 \u03a9 \u2192 E\nhf : Martingale f \u2131 \u03bc\nhg : Adapted \u2131 fun n => g (n + 1)\nhg0 : g 0 = 0\nhgint : \u2200 (n : \u2115), Integrable (g n)\nn : \u2115\nh : \u2115 \u2192 \u03a9 \u2192 E := f - martingalePart (f + g) \u2131 \u03bc\nhhdef : h = f - martingalePart (f + g) \u2131 \u03bc\nhh : h = predictablePart (f + g) \u2131 \u03bc - g\n\u22a2 Adapted \u2131 fun n => (predictablePart (f + g) \u2131 \u03bc - g) (n + 1)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "full_name": "intervalIntegral.integral_lt_integral_of_continuousOn_of_le_of_exists_lt", "start": [1337, 1], "end": [1351, 80], "traced_tactics": [{"tactic": "apply integral_lt_integral_of_ae_le_of_measure_setOf_lt_ne_zero hab.le\n  (hfc.intervalIntegrable_of_Icc hab.le) (hgc.intervalIntegrable_of_Icc hab.le)", "annotated_tactic": ["apply <a>integral_lt_integral_of_ae_le_of_measure_setOf_lt_ne_zero</a> hab.le\n    (hfc.intervalIntegrable_of_Icc hab.le) (hgc.intervalIntegrable_of_Icc hab.le)", [{"full_name": "intervalIntegral.integral_lt_integral_of_ae_le_of_measure_setOf_lt_ne_zero", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [1324, 9], "def_end_pos": [1324, 66]}]], "state_before": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf\u271d g\u271d : \u211d \u2192 \u211d\na\u271d b\u271d : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 \u211d\na b : \u211d\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhgc : ContinuousOn g (Icc a b)\nhle : \u2200 (x : \u211d), x \u2208 Ioc a b \u2192 f x \u2264 g x\nhlt : \u2203 c, c \u2208 Icc a b \u2227 f c < g c\n\u22a2 \u222b (x : \u211d) in a..b, f x < \u222b (x : \u211d) in a..b, g x", "state_after": "case hle\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf\u271d g\u271d : \u211d \u2192 \u211d\na\u271d b\u271d : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 \u211d\na b : \u211d\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhgc : ContinuousOn g (Icc a b)\nhle : \u2200 (x : \u211d), x \u2208 Ioc a b \u2192 f x \u2264 g x\nhlt : \u2203 c, c \u2208 Icc a b \u2227 f c < g c\n\u22a2 f \u2264\u1d50[Measure.restrict volume (Ioc a b)] g\n\ncase hlt\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf\u271d g\u271d : \u211d \u2192 \u211d\na\u271d b\u271d : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 \u211d\na b : \u211d\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhgc : ContinuousOn g (Icc a b)\nhle : \u2200 (x : \u211d), x \u2208 Ioc a b \u2192 f x \u2264 g x\nhlt : \u2203 c, c \u2208 Icc a b \u2227 f c < g c\n\u22a2 \u2191\u2191(Measure.restrict volume (Ioc a b)) {x | f x < g x} \u2260 0"}, {"tactic": "contrapose! hlt", "annotated_tactic": ["contrapose! hlt", []], "state_before": "case hlt\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf\u271d g\u271d : \u211d \u2192 \u211d\na\u271d b\u271d : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 \u211d\na b : \u211d\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhgc : ContinuousOn g (Icc a b)\nhle : \u2200 (x : \u211d), x \u2208 Ioc a b \u2192 f x \u2264 g x\nhlt : \u2203 c, c \u2208 Icc a b \u2227 f c < g c\n\u22a2 \u2191\u2191(Measure.restrict volume (Ioc a b)) {x | f x < g x} \u2260 0", "state_after": "case hlt\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf\u271d g\u271d : \u211d \u2192 \u211d\na\u271d b\u271d : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 \u211d\na b : \u211d\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhgc : ContinuousOn g (Icc a b)\nhle : \u2200 (x : \u211d), x \u2208 Ioc a b \u2192 f x \u2264 g x\nhlt : \u2191\u2191(Measure.restrict volume (Ioc a b)) {x | f x < g x} = 0\n\u22a2 \u2200 (c : \u211d), c \u2208 Icc a b \u2192 g c \u2264 f c"}, {"tactic": "have h_eq : f =\u1d50[volume.restrict (Ioc a b)] g := by\n  simp only [\u2190 not_le, \u2190 ae_iff] at hlt\n  exact EventuallyLE.antisymm ((ae_restrict_iff' measurableSet_Ioc).2 <|\n    eventually_of_forall hle) hlt", "annotated_tactic": ["have h_eq : f =\u1d50[volume.restrict (<a>Ioc</a> a b)] g := by\n    simp only [\u2190 <a>not_le</a>, \u2190 <a>ae_iff</a>] at hlt\n    exact <a>EventuallyLE.antisymm</a> ((<a>ae_restrict_iff'</a> <a>measurableSet_Ioc</a>).2 <|\n      <a>eventually_of_forall</a> hle) hlt", [{"full_name": "Set.Ioc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [69, 5], "def_end_pos": [69, 8]}, {"full_name": "not_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [373, 9], "def_end_pos": [373, 15]}, {"full_name": "MeasureTheory.ae_iff", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [388, 9], "def_end_pos": [388, 15]}, {"full_name": "Filter.EventuallyLE.antisymm", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1703, 9], "def_end_pos": [1703, 30]}, {"full_name": "MeasureTheory.ae_restrict_iff'", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2572, 9], "def_end_pos": [2572, 25]}, {"full_name": "measurableSet_Ioc", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [589, 9], "def_end_pos": [589, 26]}, {"full_name": "Filter.eventually_of_forall", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1111, 9], "def_end_pos": [1111, 29]}]], "state_before": "case hlt\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf\u271d g\u271d : \u211d \u2192 \u211d\na\u271d b\u271d : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 \u211d\na b : \u211d\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhgc : ContinuousOn g (Icc a b)\nhle : \u2200 (x : \u211d), x \u2208 Ioc a b \u2192 f x \u2264 g x\nhlt : \u2191\u2191(Measure.restrict volume (Ioc a b)) {x | f x < g x} = 0\n\u22a2 \u2200 (c : \u211d), c \u2208 Icc a b \u2192 g c \u2264 f c", "state_after": "case hlt\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf\u271d g\u271d : \u211d \u2192 \u211d\na\u271d b\u271d : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 \u211d\na b : \u211d\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhgc : ContinuousOn g (Icc a b)\nhle : \u2200 (x : \u211d), x \u2208 Ioc a b \u2192 f x \u2264 g x\nhlt : \u2191\u2191(Measure.restrict volume (Ioc a b)) {x | f x < g x} = 0\nh_eq : f =\u1d50[Measure.restrict volume (Ioc a b)] g\n\u22a2 \u2200 (c : \u211d), c \u2208 Icc a b \u2192 g c \u2264 f c"}, {"tactic": "rw [Measure.restrict_congr_set Ioc_ae_eq_Icc] at h_eq", "annotated_tactic": ["rw [<a>Measure.restrict_congr_set</a> <a>Ioc_ae_eq_Icc</a>] at h_eq", [{"full_name": "MeasureTheory.Measure.restrict_congr_set", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1559, 9], "def_end_pos": [1559, 27]}, {"full_name": "MeasureTheory.Ioc_ae_eq_Icc", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3172, 9], "def_end_pos": [3172, 22]}]], "state_before": "case hlt\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf\u271d g\u271d : \u211d \u2192 \u211d\na\u271d b\u271d : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 \u211d\na b : \u211d\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhgc : ContinuousOn g (Icc a b)\nhle : \u2200 (x : \u211d), x \u2208 Ioc a b \u2192 f x \u2264 g x\nhlt : \u2191\u2191(Measure.restrict volume (Ioc a b)) {x | f x < g x} = 0\nh_eq : f =\u1d50[Measure.restrict volume (Ioc a b)] g\n\u22a2 \u2200 (c : \u211d), c \u2208 Icc a b \u2192 g c \u2264 f c", "state_after": "case hlt\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf\u271d g\u271d : \u211d \u2192 \u211d\na\u271d b\u271d : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 \u211d\na b : \u211d\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhgc : ContinuousOn g (Icc a b)\nhle : \u2200 (x : \u211d), x \u2208 Ioc a b \u2192 f x \u2264 g x\nhlt : \u2191\u2191(Measure.restrict volume (Ioc a b)) {x | f x < g x} = 0\nh_eq : f =\u1d50[Measure.restrict volume (Icc a b)] g\n\u22a2 \u2200 (c : \u211d), c \u2208 Icc a b \u2192 g c \u2264 f c"}, {"tactic": "exact fun c hc \u21a6 (Measure.eqOn_Icc_of_ae_eq volume hab.ne h_eq hfc hgc hc).ge", "annotated_tactic": ["exact fun c hc \u21a6 (<a>Measure.eqOn_Icc_of_ae_eq</a> <a>volume</a> hab.ne h_eq hfc hgc hc).<a>ge</a>", [{"full_name": "MeasureTheory.Measure.eqOn_Icc_of_ae_eq", "def_path": "Mathlib/MeasureTheory/Measure/OpenPos.lean", "def_pos": [203, 9], "def_end_pos": [203, 26]}, {"full_name": "MeasureTheory.MeasureSpace.volume", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [663, 3], "def_end_pos": [663, 9]}, {"full_name": "Eq.ge", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [239, 19], "def_end_pos": [239, 21]}]], "state_before": "case hlt\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf\u271d g\u271d : \u211d \u2192 \u211d\na\u271d b\u271d : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 \u211d\na b : \u211d\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhgc : ContinuousOn g (Icc a b)\nhle : \u2200 (x : \u211d), x \u2208 Ioc a b \u2192 f x \u2264 g x\nhlt : \u2191\u2191(Measure.restrict volume (Ioc a b)) {x | f x < g x} = 0\nh_eq : f =\u1d50[Measure.restrict volume (Icc a b)] g\n\u22a2 \u2200 (c : \u211d), c \u2208 Icc a b \u2192 g c \u2264 f c", "state_after": "no goals"}, {"tactic": "simpa only [gt_iff_lt, not_lt, ge_iff_le, measurableSet_Ioc, ae_restrict_eq, le_principal_iff]\n  using (ae_restrict_mem measurableSet_Ioc).mono hle", "annotated_tactic": ["simpa only [<a>gt_iff_lt</a>, <a>not_lt</a>, <a>ge_iff_le</a>, <a>measurableSet_Ioc</a>, <a>ae_restrict_eq</a>, <a>le_principal_iff</a>]\n      using (<a>ae_restrict_mem</a> <a>measurableSet_Ioc</a>).<a>mono</a> hle", [{"full_name": "gt_iff_lt", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [366, 9], "def_end_pos": [366, 18]}, {"full_name": "not_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [368, 9], "def_end_pos": [368, 15]}, {"full_name": "ge_iff_le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [359, 9], "def_end_pos": [359, 18]}, {"full_name": "measurableSet_Ioc", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [589, 9], "def_end_pos": [589, 26]}, {"full_name": "MeasureTheory.ae_restrict_eq", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2665, 9], "def_end_pos": [2665, 23]}, {"full_name": "Filter.le_principal_iff", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [655, 9], "def_end_pos": [655, 25]}, {"full_name": "MeasureTheory.ae_restrict_mem", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2586, 9], "def_end_pos": [2586, 24]}, {"full_name": "measurableSet_Ioc", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [589, 9], "def_end_pos": [589, 26]}, {"full_name": "Filter.Eventually.mono", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1140, 9], "def_end_pos": [1140, 24]}]], "state_before": "case hle\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf\u271d g\u271d : \u211d \u2192 \u211d\na\u271d b\u271d : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 \u211d\na b : \u211d\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhgc : ContinuousOn g (Icc a b)\nhle : \u2200 (x : \u211d), x \u2208 Ioc a b \u2192 f x \u2264 g x\nhlt : \u2203 c, c \u2208 Icc a b \u2227 f c < g c\n\u22a2 f \u2264\u1d50[Measure.restrict volume (Ioc a b)] g", "state_after": "no goals"}, {"tactic": "simp only [\u2190 not_le, \u2190 ae_iff] at hlt", "annotated_tactic": ["simp only [\u2190 <a>not_le</a>, \u2190 <a>ae_iff</a>] at hlt", [{"full_name": "not_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [373, 9], "def_end_pos": [373, 15]}, {"full_name": "MeasureTheory.ae_iff", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [388, 9], "def_end_pos": [388, 15]}]], "state_before": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf\u271d g\u271d : \u211d \u2192 \u211d\na\u271d b\u271d : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 \u211d\na b : \u211d\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhgc : ContinuousOn g (Icc a b)\nhle : \u2200 (x : \u211d), x \u2208 Ioc a b \u2192 f x \u2264 g x\nhlt : \u2191\u2191(Measure.restrict volume (Ioc a b)) {x | f x < g x} = 0\n\u22a2 f =\u1d50[Measure.restrict volume (Ioc a b)] g", "state_after": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf\u271d g\u271d : \u211d \u2192 \u211d\na\u271d b\u271d : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 \u211d\na b : \u211d\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhgc : ContinuousOn g (Icc a b)\nhle : \u2200 (x : \u211d), x \u2208 Ioc a b \u2192 f x \u2264 g x\nhlt : \u2200\u1d50 (a : \u211d) \u2202Measure.restrict volume (Ioc a b), g a \u2264 f a\n\u22a2 f =\u1d50[Measure.restrict volume (Ioc a b)] g"}, {"tactic": "exact EventuallyLE.antisymm ((ae_restrict_iff' measurableSet_Ioc).2 <|\n  eventually_of_forall hle) hlt", "annotated_tactic": ["exact <a>EventuallyLE.antisymm</a> ((<a>ae_restrict_iff'</a> <a>measurableSet_Ioc</a>).2 <|\n      <a>eventually_of_forall</a> hle) hlt", [{"full_name": "Filter.EventuallyLE.antisymm", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1703, 9], "def_end_pos": [1703, 30]}, {"full_name": "MeasureTheory.ae_restrict_iff'", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2572, 9], "def_end_pos": [2572, 25]}, {"full_name": "measurableSet_Ioc", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [589, 9], "def_end_pos": [589, 26]}, {"full_name": "Filter.eventually_of_forall", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1111, 9], "def_end_pos": [1111, 29]}]], "state_before": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf\u271d g\u271d : \u211d \u2192 \u211d\na\u271d b\u271d : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 \u211d\na b : \u211d\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhgc : ContinuousOn g (Icc a b)\nhle : \u2200 (x : \u211d), x \u2208 Ioc a b \u2192 f x \u2264 g x\nhlt : \u2200\u1d50 (a : \u211d) \u2202Measure.restrict volume (Ioc a b), g a \u2264 f a\n\u22a2 f =\u1d50[Measure.restrict volume (Ioc a b)] g", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "full_name": "MeasureTheory.Measure.toSignedMeasure_toMeasureOfZeroLE", "start": [1415, 1], "end": [1422, 18], "traced_tactics": [{"tactic": "refine' Measure.ext fun i hi => _", "annotated_tactic": ["refine' <a>Measure.ext</a> fun i hi => _", [{"full_name": "MeasureTheory.Measure.ext", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [135, 9], "def_end_pos": [135, 12]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\n\u22a2 SignedMeasure.toMeasureOfZeroLE (toSignedMeasure \u03bc) univ (_ : MeasurableSet univ)\n      (_ : VectorMeasure.restrict 0 univ \u2264 VectorMeasure.restrict (toSignedMeasure \u03bc) univ) =\n    \u03bc", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 \u2191\u2191(SignedMeasure.toMeasureOfZeroLE (toSignedMeasure \u03bc) univ (_ : MeasurableSet univ)\n            (_ : VectorMeasure.restrict 0 univ \u2264 VectorMeasure.restrict (toSignedMeasure \u03bc) univ))\n      i =\n    \u2191\u2191\u03bc i"}, {"tactic": "lift \u03bc i to \u211d\u22650 using (measure_lt_top _ _).ne with m hm", "annotated_tactic": ["lift \u03bc i to \u211d\u22650 using (<a>measure_lt_top</a> _ _).<a>ne</a> with m hm", [{"full_name": "MeasureTheory.measure_lt_top", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2866, 9], "def_end_pos": [2866, 23]}, {"full_name": "LT.lt.ne", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [152, 7], "def_end_pos": [152, 15]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 \u2191\u2191(SignedMeasure.toMeasureOfZeroLE (toSignedMeasure \u03bc) univ (_ : MeasurableSet univ)\n            (_ : VectorMeasure.restrict 0 univ \u2264 VectorMeasure.restrict (toSignedMeasure \u03bc) univ))\n      i =\n    \u2191\u2191\u03bc i", "state_after": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\ni : Set \u03b1\nhi : MeasurableSet i\nm : \u211d\u22650\nhm : \u2191m = \u2191\u2191\u03bc i\n\u22a2 \u2191\u2191(SignedMeasure.toMeasureOfZeroLE (toSignedMeasure \u03bc) univ (_ : MeasurableSet univ)\n            (_ : VectorMeasure.restrict 0 univ \u2264 VectorMeasure.restrict (toSignedMeasure \u03bc) univ))\n      i =\n    \u2191m"}, {"tactic": "rw [SignedMeasure.toMeasureOfZeroLE_apply _ _ _ hi, coe_eq_coe]", "annotated_tactic": ["rw [<a>SignedMeasure.toMeasureOfZeroLE_apply</a> _ _ _ hi, <a>coe_eq_coe</a>]", [{"full_name": "MeasureTheory.SignedMeasure.toMeasureOfZeroLE_apply", "def_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "def_pos": [1349, 9], "def_end_pos": [1349, 32]}, {"full_name": "ENNReal.coe_eq_coe", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [346, 28], "def_end_pos": [346, 38]}]], "state_before": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\ni : Set \u03b1\nhi : MeasurableSet i\nm : \u211d\u22650\nhm : \u2191m = \u2191\u2191\u03bc i\n\u22a2 \u2191\u2191(SignedMeasure.toMeasureOfZeroLE (toSignedMeasure \u03bc) univ (_ : MeasurableSet univ)\n            (_ : VectorMeasure.restrict 0 univ \u2264 VectorMeasure.restrict (toSignedMeasure \u03bc) univ))\n      i =\n    \u2191m", "state_after": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\ni : Set \u03b1\nhi : MeasurableSet i\nm : \u211d\u22650\nhm : \u2191m = \u2191\u2191\u03bc i\n\u22a2 { val := \u2191(toSignedMeasure \u03bc) (univ \u2229 i), property := (_ : 0 \u2264 \u2191(toSignedMeasure \u03bc) (univ \u2229 i)) } = m"}, {"tactic": "congr", "annotated_tactic": ["congr", []], "state_before": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\ni : Set \u03b1\nhi : MeasurableSet i\nm : \u211d\u22650\nhm : \u2191m = \u2191\u2191\u03bc i\n\u22a2 { val := \u2191(toSignedMeasure \u03bc) (univ \u2229 i), property := (_ : 0 \u2264 \u2191(toSignedMeasure \u03bc) (univ \u2229 i)) } = m", "state_after": "case intro.e_val\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\ni : Set \u03b1\nhi : MeasurableSet i\nm : \u211d\u22650\nhm : \u2191m = \u2191\u2191\u03bc i\n\u22a2 \u2191(toSignedMeasure \u03bc) (univ \u2229 i) = \u2191m"}, {"tactic": "simp [hi, \u2190 hm]", "annotated_tactic": ["simp [hi, \u2190 hm]", []], "state_before": "case intro.e_val\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\ni : Set \u03b1\nhi : MeasurableSet i\nm : \u211d\u22650\nhm : \u2191m = \u2191\u2191\u03bc i\n\u22a2 \u2191(toSignedMeasure \u03bc) (univ \u2229 i) = \u2191m", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Semiquot.lean", "full_name": "Semiquot.get_mem", "start": [216, 1], "end": [218, 51], "traced_tactics": [{"tactic": "let \u27e8a, h\u27e9 := exists_mem q", "annotated_tactic": ["let \u27e8a, h\u27e9 := <a>exists_mem</a> q", [{"full_name": "Semiquot.exists_mem", "def_path": "Mathlib/Data/Semiquot.lean", "def_pos": [57, 9], "def_end_pos": [57, 19]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nq : Semiquot \u03b1\np : IsPure q\n\u22a2 get q p \u2208 q", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nq : Semiquot \u03b1\np : IsPure q\na : \u03b1\nh : a \u2208 q\n\u22a2 get q p \u2208 q"}, {"tactic": "unfold get", "annotated_tactic": ["unfold <a>get</a>", [{"full_name": "Semiquot.get", "def_path": "Mathlib/Data/Semiquot.lean", "def_pos": [212, 5], "def_end_pos": [212, 8]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nq : Semiquot \u03b1\np : IsPure q\na : \u03b1\nh : a \u2208 q\n\u22a2 get q p \u2208 q", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nq : Semiquot \u03b1\np : IsPure q\na : \u03b1\nh : a \u2208 q\n\u22a2 liftOn q id p \u2208 q"}, {"tactic": "rw [liftOn_ofMem q _ _ a h]", "annotated_tactic": ["rw [<a>liftOn_ofMem</a> q _ _ a h]", [{"full_name": "Semiquot.liftOn_ofMem", "def_path": "Mathlib/Data/Semiquot.lean", "def_pos": [119, 9], "def_end_pos": [119, 21]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nq : Semiquot \u03b1\np : IsPure q\na : \u03b1\nh : a \u2208 q\n\u22a2 liftOn q id p \u2208 q", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nq : Semiquot \u03b1\np : IsPure q\na : \u03b1\nh : a \u2208 q\n\u22a2 id a \u2208 q"}, {"tactic": "exact h", "annotated_tactic": ["exact h", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nq : Semiquot \u03b1\np : IsPure q\na : \u03b1\nh : a \u2208 q\n\u22a2 id a \u2208 q", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/ProbabilityMassFunction/Monad.lean", "full_name": "PMF.toPMF_dirac", "start": [96, 1], "end": [98, 49], "traced_tactics": [{"tactic": "rw [toPMF_eq_iff_toMeasure_eq, toMeasure_pure]", "annotated_tactic": ["rw [<a>toPMF_eq_iff_toMeasure_eq</a>, <a>toMeasure_pure</a>]", [{"full_name": "PMF.toPMF_eq_iff_toMeasure_eq", "def_path": "Mathlib/Probability/ProbabilityMassFunction/Basic.lean", "def_pos": [403, 9], "def_end_pos": [403, 34]}, {"full_name": "PMF.toMeasure_pure", "def_path": "Mathlib/Probability/ProbabilityMassFunction/Monad.lean", "def_pos": [91, 9], "def_end_pos": [91, 23]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na a' : \u03b1\ns : Set \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : Countable \u03b1\nh : MeasurableSingletonClass \u03b1\n\u22a2 Measure.toPMF (Measure.dirac a) = pure a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "full_name": "Measurable.iInf_Prop", "start": [1349, 1], "end": [1352, 64], "traced_tactics": [{"tactic": "convert hf", "annotated_tactic": ["convert hf", []], "state_before": "\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns t u : Set \u03b1\u271d\ninst\u271d\u00b9\u00b9 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b1\u271d\ninst\u271d\u2079 : BorelSpace \u03b1\u271d\ninst\u271d\u2078 : TopologicalSpace \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : BorelSpace \u03b2\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : MeasurableSpace \u03b3\ninst\u271d\u00b3 : BorelSpace \u03b3\ninst\u271d\u00b2 : MeasurableSpace \u03b4\n\u03b1 : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : ConditionallyCompleteLattice \u03b1\np : Prop\nf : \u03b4 \u2192 \u03b1\nhf : Measurable f\nh : p\n\u22a2 Measurable fun b => \u2a05 (_ : p), f b", "state_after": "case h.e'_5.h\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns t u : Set \u03b1\u271d\ninst\u271d\u00b9\u00b9 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b1\u271d\ninst\u271d\u2079 : BorelSpace \u03b1\u271d\ninst\u271d\u2078 : TopologicalSpace \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : BorelSpace \u03b2\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : MeasurableSpace \u03b3\ninst\u271d\u00b3 : BorelSpace \u03b3\ninst\u271d\u00b2 : MeasurableSpace \u03b4\n\u03b1 : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : ConditionallyCompleteLattice \u03b1\np : Prop\nf : \u03b4 \u2192 \u03b1\nhf : Measurable f\nh : p\nx\u271d : \u03b4\n\u22a2 \u2a05 (_ : p), f x\u271d = f x\u271d"}, {"tactic": "exact ciInf_pos h", "annotated_tactic": ["exact <a>ciInf_pos</a> h", [{"full_name": "ciInf_pos", "def_path": "Mathlib/Order/ConditionallyCompleteLattice/Basic.lean", "def_pos": [880, 9], "def_end_pos": [880, 18]}]], "state_before": "case h.e'_5.h\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns t u : Set \u03b1\u271d\ninst\u271d\u00b9\u00b9 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b1\u271d\ninst\u271d\u2079 : BorelSpace \u03b1\u271d\ninst\u271d\u2078 : TopologicalSpace \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : BorelSpace \u03b2\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : MeasurableSpace \u03b3\ninst\u271d\u00b3 : BorelSpace \u03b3\ninst\u271d\u00b2 : MeasurableSpace \u03b4\n\u03b1 : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : ConditionallyCompleteLattice \u03b1\np : Prop\nf : \u03b4 \u2192 \u03b1\nhf : Measurable f\nh : p\nx\u271d : \u03b4\n\u22a2 \u2a05 (_ : p), f x\u271d = f x\u271d", "state_after": "no goals"}, {"tactic": "convert measurable_const using 1", "annotated_tactic": ["convert <a>measurable_const</a> using 1", [{"full_name": "measurable_const", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [570, 9], "def_end_pos": [570, 25]}]], "state_before": "\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns t u : Set \u03b1\u271d\ninst\u271d\u00b9\u00b9 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b1\u271d\ninst\u271d\u2079 : BorelSpace \u03b1\u271d\ninst\u271d\u2078 : TopologicalSpace \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : BorelSpace \u03b2\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : MeasurableSpace \u03b3\ninst\u271d\u00b3 : BorelSpace \u03b3\ninst\u271d\u00b2 : MeasurableSpace \u03b4\n\u03b1 : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : ConditionallyCompleteLattice \u03b1\np : Prop\nf : \u03b4 \u2192 \u03b1\nhf : Measurable f\nh : \u00acp\n\u22a2 Measurable fun b => \u2a05 (_ : p), f b", "state_after": "case h.e'_5\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns t u : Set \u03b1\u271d\ninst\u271d\u00b9\u00b9 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b1\u271d\ninst\u271d\u2079 : BorelSpace \u03b1\u271d\ninst\u271d\u2078 : TopologicalSpace \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : BorelSpace \u03b2\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : MeasurableSpace \u03b3\ninst\u271d\u00b3 : BorelSpace \u03b3\ninst\u271d\u00b2 : MeasurableSpace \u03b4\n\u03b1 : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : ConditionallyCompleteLattice \u03b1\np : Prop\nf : \u03b4 \u2192 \u03b1\nhf : Measurable f\nh : \u00acp\n\u22a2 (fun b => \u2a05 (_ : p), f b) = fun x => ?convert_5\n\ncase convert_5\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns t u : Set \u03b1\u271d\ninst\u271d\u00b9\u00b9 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b1\u271d\ninst\u271d\u2079 : BorelSpace \u03b1\u271d\ninst\u271d\u2078 : TopologicalSpace \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : BorelSpace \u03b2\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : MeasurableSpace \u03b3\ninst\u271d\u00b3 : BorelSpace \u03b3\ninst\u271d\u00b2 : MeasurableSpace \u03b4\n\u03b1 : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : ConditionallyCompleteLattice \u03b1\np : Prop\nf : \u03b4 \u2192 \u03b1\nhf : Measurable f\nh : \u00acp\n\u22a2 \u03b1"}, {"tactic": "funext", "annotated_tactic": ["funext", []], "state_before": "case h.e'_5\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns t u : Set \u03b1\u271d\ninst\u271d\u00b9\u00b9 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b1\u271d\ninst\u271d\u2079 : BorelSpace \u03b1\u271d\ninst\u271d\u2078 : TopologicalSpace \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : BorelSpace \u03b2\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : MeasurableSpace \u03b3\ninst\u271d\u00b3 : BorelSpace \u03b3\ninst\u271d\u00b2 : MeasurableSpace \u03b4\n\u03b1 : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : ConditionallyCompleteLattice \u03b1\np : Prop\nf : \u03b4 \u2192 \u03b1\nhf : Measurable f\nh : \u00acp\n\u22a2 (fun b => \u2a05 (_ : p), f b) = fun x => ?convert_5\n\ncase convert_5\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns t u : Set \u03b1\u271d\ninst\u271d\u00b9\u00b9 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b1\u271d\ninst\u271d\u2079 : BorelSpace \u03b1\u271d\ninst\u271d\u2078 : TopologicalSpace \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : BorelSpace \u03b2\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : MeasurableSpace \u03b3\ninst\u271d\u00b3 : BorelSpace \u03b3\ninst\u271d\u00b2 : MeasurableSpace \u03b4\n\u03b1 : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : ConditionallyCompleteLattice \u03b1\np : Prop\nf : \u03b4 \u2192 \u03b1\nhf : Measurable f\nh : \u00acp\n\u22a2 \u03b1", "state_after": "case h.e'_5.h\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns t u : Set \u03b1\u271d\ninst\u271d\u00b9\u00b9 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b1\u271d\ninst\u271d\u2079 : BorelSpace \u03b1\u271d\ninst\u271d\u2078 : TopologicalSpace \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : BorelSpace \u03b2\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : MeasurableSpace \u03b3\ninst\u271d\u00b3 : BorelSpace \u03b3\ninst\u271d\u00b2 : MeasurableSpace \u03b4\n\u03b1 : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : ConditionallyCompleteLattice \u03b1\np : Prop\nf : \u03b4 \u2192 \u03b1\nhf : Measurable f\nh : \u00acp\nx\u271d : \u03b4\n\u22a2 \u2a05 (_ : p), f x\u271d = ?convert_5\n\ncase convert_5\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns t u : Set \u03b1\u271d\ninst\u271d\u00b9\u00b9 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b1\u271d\ninst\u271d\u2079 : BorelSpace \u03b1\u271d\ninst\u271d\u2078 : TopologicalSpace \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : BorelSpace \u03b2\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : MeasurableSpace \u03b3\ninst\u271d\u00b3 : BorelSpace \u03b3\ninst\u271d\u00b2 : MeasurableSpace \u03b4\n\u03b1 : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : ConditionallyCompleteLattice \u03b1\np : Prop\nf : \u03b4 \u2192 \u03b1\nhf : Measurable f\nh : \u00acp\n\u22a2 \u03b1"}, {"tactic": "exact ciInf_neg h", "annotated_tactic": ["exact <a>ciInf_neg</a> h", [{"full_name": "ciInf_neg", "def_path": "Mathlib/Order/ConditionallyCompleteLattice/Basic.lean", "def_pos": [890, 7], "def_end_pos": [890, 16]}]], "state_before": "case h.e'_5.h\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns t u : Set \u03b1\u271d\ninst\u271d\u00b9\u00b9 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b1\u271d\ninst\u271d\u2079 : BorelSpace \u03b1\u271d\ninst\u271d\u2078 : TopologicalSpace \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : BorelSpace \u03b2\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : MeasurableSpace \u03b3\ninst\u271d\u00b3 : BorelSpace \u03b3\ninst\u271d\u00b2 : MeasurableSpace \u03b4\n\u03b1 : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : ConditionallyCompleteLattice \u03b1\np : Prop\nf : \u03b4 \u2192 \u03b1\nhf : Measurable f\nh : \u00acp\nx\u271d : \u03b4\n\u22a2 \u2a05 (_ : p), f x\u271d = ?convert_5\n\ncase convert_5\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns t u : Set \u03b1\u271d\ninst\u271d\u00b9\u00b9 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b1\u271d\ninst\u271d\u2079 : BorelSpace \u03b1\u271d\ninst\u271d\u2078 : TopologicalSpace \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : BorelSpace \u03b2\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : MeasurableSpace \u03b3\ninst\u271d\u00b3 : BorelSpace \u03b3\ninst\u271d\u00b2 : MeasurableSpace \u03b4\n\u03b1 : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : ConditionallyCompleteLattice \u03b1\np : Prop\nf : \u03b4 \u2192 \u03b1\nhf : Measurable f\nh : \u00acp\n\u22a2 \u03b1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Finite.lean", "full_name": "Set.infinite_prod", "start": [943, 11], "end": [951, 31], "traced_tactics": [{"tactic": "refine' \u27e8fun h => _, _\u27e9", "annotated_tactic": ["refine' \u27e8fun h => _, _\u27e9", []], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\n\u03b3 : Type x\ns : Set \u03b1\nt : Set \u03b2\n\u22a2 Set.Infinite (s \u00d7\u02e2 t) \u2194 Set.Infinite s \u2227 Set.Nonempty t \u2228 Set.Infinite t \u2227 Set.Nonempty s", "state_after": "case refine'_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\n\u03b3 : Type x\ns : Set \u03b1\nt : Set \u03b2\nh : Set.Infinite (s \u00d7\u02e2 t)\n\u22a2 Set.Infinite s \u2227 Set.Nonempty t \u2228 Set.Infinite t \u2227 Set.Nonempty s\n\ncase refine'_2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\n\u03b3 : Type x\ns : Set \u03b1\nt : Set \u03b2\n\u22a2 Set.Infinite s \u2227 Set.Nonempty t \u2228 Set.Infinite t \u2227 Set.Nonempty s \u2192 Set.Infinite (s \u00d7\u02e2 t)"}, {"tactic": "simp_rw [Set.Infinite, @and_comm \u00ac_, \u2190 not_imp]", "annotated_tactic": ["simp_rw [<a>Set.Infinite</a>, @<a>and_comm</a> \u00ac_, \u2190 <a>not_imp</a>]", [{"full_name": "Set.Infinite", "def_path": "Mathlib/Data/Set/Finite.lean", "def_pos": [142, 15], "def_end_pos": [142, 23]}, {"full_name": "and_comm", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [159, 9], "def_end_pos": [159, 17]}, {"full_name": "not_imp", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [433, 9], "def_end_pos": [433, 16]}]], "state_before": "case refine'_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\n\u03b3 : Type x\ns : Set \u03b1\nt : Set \u03b2\nh : Set.Infinite (s \u00d7\u02e2 t)\n\u22a2 Set.Infinite s \u2227 Set.Nonempty t \u2228 Set.Infinite t \u2227 Set.Nonempty s", "state_after": "case refine'_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\n\u03b3 : Type x\ns : Set \u03b1\nt : Set \u03b2\nh : Set.Infinite (s \u00d7\u02e2 t)\n\u22a2 \u00ac(Set.Nonempty t \u2192 Set.Finite s) \u2228 \u00ac(Set.Nonempty s \u2192 Set.Finite t)"}, {"tactic": "by_contra'", "annotated_tactic": ["by_contra'", []], "state_before": "case refine'_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\n\u03b3 : Type x\ns : Set \u03b1\nt : Set \u03b2\nh : Set.Infinite (s \u00d7\u02e2 t)\n\u22a2 \u00ac(Set.Nonempty t \u2192 Set.Finite s) \u2228 \u00ac(Set.Nonempty s \u2192 Set.Finite t)", "state_after": "case refine'_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\n\u03b3 : Type x\ns : Set \u03b1\nt : Set \u03b2\nh : Set.Infinite (s \u00d7\u02e2 t)\nthis : (Set.Nonempty t \u2192 Set.Finite s) \u2227 (Set.Nonempty s \u2192 Set.Finite t)\n\u22a2 False"}, {"tactic": "exact h ((this.1 h.nonempty.snd).prod $ this.2 h.nonempty.fst)", "annotated_tactic": ["exact h ((this.1 h.nonempty.snd).<a>prod</a> $ this.2 h.nonempty.fst)", [{"full_name": "Set.Finite.prod", "def_path": "Mathlib/Data/Set/Finite.lean", "def_pos": [921, 19], "def_end_pos": [921, 30]}]], "state_before": "case refine'_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\n\u03b3 : Type x\ns : Set \u03b1\nt : Set \u03b2\nh : Set.Infinite (s \u00d7\u02e2 t)\nthis : (Set.Nonempty t \u2192 Set.Finite s) \u2227 (Set.Nonempty s \u2192 Set.Finite t)\n\u22a2 False", "state_after": "no goals"}, {"tactic": "rintro (h | h)", "annotated_tactic": ["rintro (h | h)", []], "state_before": "case refine'_2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\n\u03b3 : Type x\ns : Set \u03b1\nt : Set \u03b2\n\u22a2 Set.Infinite s \u2227 Set.Nonempty t \u2228 Set.Infinite t \u2227 Set.Nonempty s \u2192 Set.Infinite (s \u00d7\u02e2 t)", "state_after": "case refine'_2.inl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\n\u03b3 : Type x\ns : Set \u03b1\nt : Set \u03b2\nh : Set.Infinite s \u2227 Set.Nonempty t\n\u22a2 Set.Infinite (s \u00d7\u02e2 t)\n\ncase refine'_2.inr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\n\u03b3 : Type x\ns : Set \u03b1\nt : Set \u03b2\nh : Set.Infinite t \u2227 Set.Nonempty s\n\u22a2 Set.Infinite (s \u00d7\u02e2 t)"}, {"tactic": "exact h.1.prod_left h.2", "annotated_tactic": ["exact h.1.<a>prod_left</a> h.2", [{"full_name": "Set.Infinite.prod_left", "def_path": "Mathlib/Data/Set/Finite.lean", "def_pos": [935, 19], "def_end_pos": [935, 37]}]], "state_before": "case refine'_2.inl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\n\u03b3 : Type x\ns : Set \u03b1\nt : Set \u03b2\nh : Set.Infinite s \u2227 Set.Nonempty t\n\u22a2 Set.Infinite (s \u00d7\u02e2 t)", "state_after": "no goals"}, {"tactic": "exact h.1.prod_right h.2", "annotated_tactic": ["exact h.1.<a>prod_right</a> h.2", [{"full_name": "Set.Infinite.prod_right", "def_path": "Mathlib/Data/Set/Finite.lean", "def_pos": [939, 19], "def_end_pos": [939, 38]}]], "state_before": "case refine'_2.inr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\n\u03b3 : Type x\ns : Set \u03b1\nt : Set \u03b2\nh : Set.Infinite t \u2227 Set.Nonempty s\n\u22a2 Set.Infinite (s \u00d7\u02e2 t)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/BinaryHeap.lean", "full_name": "BinaryHeap.size_pos_of_max", "start": [130, 1], "end": [131, 95], "traced_tactics": [{"tactic": "simp [BinaryHeap.max, Array.get?, h] at e", "annotated_tactic": ["simp [<a>BinaryHeap.max</a>, <a>Array.get?</a>, h] at e", [{"full_name": "BinaryHeap.max", "def_path": "Mathlib/Data/BinaryHeap.lean", "def_pos": [103, 5], "def_end_pos": [103, 8]}, {"full_name": "Array.get?", "def_path": "lake-packages/lean4/src/lean/Init/Data/Array/Basic.lean", "def_pos": [50, 5], "def_end_pos": [50, 9]}]], "state_before": "\u03b1 : Type u_1\nx : \u03b1\nlt : \u03b1 \u2192 \u03b1 \u2192 Bool\nself : BinaryHeap \u03b1 lt\ne : max self = some x\nh : \u00ac0 < Array.size self.arr\n\u22a2 False", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/CircleTransform.lean", "full_name": "Complex.continuousOn_abs_circleTransformBoundingFunction", "start": [113, 1], "end": [125, 16], "traced_tactics": [{"tactic": "have : ContinuousOn (circleTransformBoundingFunction R z) (closedBall z r \u00d7\u02e2 (\u22a4 : Set \u211d)) := by\n  apply_rules [ContinuousOn.smul, continuousOn_const]\n  simp only [deriv_circleMap]\n  have c := (continuous_circleMap 0 R).continuousOn (s := \u22a4)\n  apply_rules [ContinuousOn.mul, c.comp continuousOn_snd fun _ => And.right, continuousOn_const]\n  simp_rw [\u2190 inv_pow]\n  apply continuousOn_prod_circle_transform_function hr", "annotated_tactic": ["have : <a>ContinuousOn</a> (<a>circleTransformBoundingFunction</a> R z) (<a>closedBall</a> z r \u00d7\u02e2 (\u22a4 : <a>Set</a> \u211d)) := by\n    apply_rules [<a>ContinuousOn.smul</a>, <a>continuousOn_const</a>]\n    simp only [<a>deriv_circleMap</a>]\n    have c := (<a>continuous_circleMap</a> 0 R).<a>continuousOn</a> (s := \u22a4)\n    apply_rules [<a>ContinuousOn.mul</a>, c.comp <a>continuousOn_snd</a> fun _ => <a>And.right</a>, <a>continuousOn_const</a>]\n    simp_rw [\u2190 <a>inv_pow</a>]\n    apply <a>continuousOn_prod_circle_transform_function</a> hr", [{"full_name": "ContinuousOn", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [532, 5], "def_end_pos": [532, 17]}, {"full_name": "Complex.circleTransformBoundingFunction", "def_path": "Mathlib/MeasureTheory/Integral/CircleTransform.lean", "def_pos": [96, 5], "def_end_pos": [96, 36]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}, {"full_name": "ContinuousOn.smul", "def_path": "Mathlib/Topology/Algebra/MulAction.lean", "def_pos": [114, 9], "def_end_pos": [114, 26]}, {"full_name": "continuousOn_const", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [1025, 9], "def_end_pos": [1025, 27]}, {"full_name": "deriv_circleMap", "def_path": "Mathlib/MeasureTheory/Integral/CircleIntegral.lean", "def_pos": [195, 9], "def_end_pos": [195, 24]}, {"full_name": "continuous_circleMap", "def_path": "Mathlib/MeasureTheory/Integral/CircleIntegral.lean", "def_pos": [185, 9], "def_end_pos": [185, 29]}, {"full_name": "Continuous.continuousOn", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [947, 9], "def_end_pos": [947, 32]}, {"full_name": "ContinuousOn.mul", "def_path": "Mathlib/Topology/Algebra/Monoid.lean", "def_pos": [106, 9], "def_end_pos": [106, 25]}, {"full_name": "continuousOn_snd", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [1330, 9], "def_end_pos": [1330, 25]}, {"full_name": "And.right", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [507, 3], "def_end_pos": [507, 8]}, {"full_name": "continuousOn_const", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [1025, 9], "def_end_pos": [1025, 27]}, {"full_name": "inv_pow", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [317, 9], "def_end_pos": [317, 16]}, {"full_name": "Complex.continuousOn_prod_circle_transform_function", "def_path": "Mathlib/MeasureTheory/Integral/CircleTransform.lean", "def_pos": [100, 9], "def_end_pos": [100, 52]}]], "state_before": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR r : \u211d\nhr : r < R\nz : \u2102\n\u22a2 ContinuousOn (\u2191abs \u2218 fun t => circleTransformBoundingFunction R z t) (closedBall z r \u00d7\u02e2 univ)", "state_after": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR r : \u211d\nhr : r < R\nz : \u2102\nthis : ContinuousOn (circleTransformBoundingFunction R z) (closedBall z r \u00d7\u02e2 \u22a4)\n\u22a2 ContinuousOn (\u2191abs \u2218 fun t => circleTransformBoundingFunction R z t) (closedBall z r \u00d7\u02e2 univ)"}, {"tactic": "refine' continuous_abs.continuousOn (s := \u22a4).comp this _", "annotated_tactic": ["refine' continuous_abs.continuousOn (s := \u22a4).<a>comp</a> this _", [{"full_name": "ContinuousOn.comp", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [929, 9], "def_end_pos": [929, 26]}]], "state_before": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR r : \u211d\nhr : r < R\nz : \u2102\nthis : ContinuousOn (circleTransformBoundingFunction R z) (closedBall z r \u00d7\u02e2 \u22a4)\n\u22a2 ContinuousOn (\u2191abs \u2218 fun t => circleTransformBoundingFunction R z t) (closedBall z r \u00d7\u02e2 univ)", "state_after": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR r : \u211d\nhr : r < R\nz : \u2102\nthis : ContinuousOn (circleTransformBoundingFunction R z) (closedBall z r \u00d7\u02e2 \u22a4)\n\u22a2 MapsTo (fun t => circleTransformBoundingFunction R z t) (closedBall z r \u00d7\u02e2 univ) \u22a4"}, {"tactic": "simp [MapsTo]", "annotated_tactic": ["simp [<a>MapsTo</a>]", [{"full_name": "Set.MapsTo", "def_path": "Mathlib/Data/Set/Function.lean", "def_pos": [348, 5], "def_end_pos": [348, 11]}]], "state_before": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR r : \u211d\nhr : r < R\nz : \u2102\nthis : ContinuousOn (circleTransformBoundingFunction R z) (closedBall z r \u00d7\u02e2 \u22a4)\n\u22a2 MapsTo (fun t => circleTransformBoundingFunction R z t) (closedBall z r \u00d7\u02e2 univ) \u22a4", "state_after": "no goals"}, {"tactic": "apply_rules [ContinuousOn.smul, continuousOn_const]", "annotated_tactic": ["apply_rules [<a>ContinuousOn.smul</a>, <a>continuousOn_const</a>]", [{"full_name": "ContinuousOn.smul", "def_path": "Mathlib/Topology/Algebra/MulAction.lean", "def_pos": [114, 9], "def_end_pos": [114, 26]}, {"full_name": "continuousOn_const", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [1025, 9], "def_end_pos": [1025, 27]}]], "state_before": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR r : \u211d\nhr : r < R\nz : \u2102\n\u22a2 ContinuousOn (circleTransformBoundingFunction R z) (closedBall z r \u00d7\u02e2 \u22a4)", "state_after": "case hg.hf\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR r : \u211d\nhr : r < R\nz : \u2102\n\u22a2 ContinuousOn (fun x => deriv (circleMap z R) x.2) (closedBall z r \u00d7\u02e2 \u22a4)\n\ncase hg.hg.hf\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR r : \u211d\nhr : r < R\nz : \u2102\n\u22a2 ContinuousOn (fun x => ((circleMap z R x.2 - x.1) ^ 2)\u207b\u00b9) (closedBall z r \u00d7\u02e2 \u22a4)"}, {"tactic": "simp only [deriv_circleMap]", "annotated_tactic": ["simp only [<a>deriv_circleMap</a>]", [{"full_name": "deriv_circleMap", "def_path": "Mathlib/MeasureTheory/Integral/CircleIntegral.lean", "def_pos": [195, 9], "def_end_pos": [195, 24]}]], "state_before": "case hg.hf\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR r : \u211d\nhr : r < R\nz : \u2102\n\u22a2 ContinuousOn (fun x => deriv (circleMap z R) x.2) (closedBall z r \u00d7\u02e2 \u22a4)\n\ncase hg.hg.hf\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR r : \u211d\nhr : r < R\nz : \u2102\n\u22a2 ContinuousOn (fun x => ((circleMap z R x.2 - x.1) ^ 2)\u207b\u00b9) (closedBall z r \u00d7\u02e2 \u22a4)", "state_after": "case hg.hf\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR r : \u211d\nhr : r < R\nz : \u2102\n\u22a2 ContinuousOn (fun x => circleMap 0 R x.2 * I) (closedBall z r \u00d7\u02e2 \u22a4)\n\ncase hg.hg.hf\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR r : \u211d\nhr : r < R\nz : \u2102\n\u22a2 ContinuousOn (fun x => ((circleMap z R x.2 - x.1) ^ 2)\u207b\u00b9) (closedBall z r \u00d7\u02e2 \u22a4)"}, {"tactic": "have c := (continuous_circleMap 0 R).continuousOn (s := \u22a4)", "annotated_tactic": ["have c := (<a>continuous_circleMap</a> 0 R).<a>continuousOn</a> (s := \u22a4)", [{"full_name": "continuous_circleMap", "def_path": "Mathlib/MeasureTheory/Integral/CircleIntegral.lean", "def_pos": [185, 9], "def_end_pos": [185, 29]}, {"full_name": "Continuous.continuousOn", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [947, 9], "def_end_pos": [947, 32]}]], "state_before": "case hg.hf\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR r : \u211d\nhr : r < R\nz : \u2102\n\u22a2 ContinuousOn (fun x => circleMap 0 R x.2 * I) (closedBall z r \u00d7\u02e2 \u22a4)\n\ncase hg.hg.hf\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR r : \u211d\nhr : r < R\nz : \u2102\n\u22a2 ContinuousOn (fun x => ((circleMap z R x.2 - x.1) ^ 2)\u207b\u00b9) (closedBall z r \u00d7\u02e2 \u22a4)", "state_after": "case hg.hf\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR r : \u211d\nhr : r < R\nz : \u2102\nc : ContinuousOn (circleMap 0 R) \u22a4\n\u22a2 ContinuousOn (fun x => circleMap 0 R x.2 * I) (closedBall z r \u00d7\u02e2 \u22a4)\n\ncase hg.hg.hf\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR r : \u211d\nhr : r < R\nz : \u2102\n\u22a2 ContinuousOn (fun x => ((circleMap z R x.2 - x.1) ^ 2)\u207b\u00b9) (closedBall z r \u00d7\u02e2 \u22a4)"}, {"tactic": "apply_rules [ContinuousOn.mul, c.comp continuousOn_snd fun _ => And.right, continuousOn_const]", "annotated_tactic": ["apply_rules [<a>ContinuousOn.mul</a>, c.comp <a>continuousOn_snd</a> fun _ => <a>And.right</a>, <a>continuousOn_const</a>]", [{"full_name": "ContinuousOn.mul", "def_path": "Mathlib/Topology/Algebra/Monoid.lean", "def_pos": [106, 9], "def_end_pos": [106, 25]}, {"full_name": "continuousOn_snd", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [1330, 9], "def_end_pos": [1330, 25]}, {"full_name": "And.right", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [507, 3], "def_end_pos": [507, 8]}, {"full_name": "continuousOn_const", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [1025, 9], "def_end_pos": [1025, 27]}]], "state_before": "case hg.hf\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR r : \u211d\nhr : r < R\nz : \u2102\nc : ContinuousOn (circleMap 0 R) \u22a4\n\u22a2 ContinuousOn (fun x => circleMap 0 R x.2 * I) (closedBall z r \u00d7\u02e2 \u22a4)\n\ncase hg.hg.hf\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR r : \u211d\nhr : r < R\nz : \u2102\n\u22a2 ContinuousOn (fun x => ((circleMap z R x.2 - x.1) ^ 2)\u207b\u00b9) (closedBall z r \u00d7\u02e2 \u22a4)", "state_after": "case hg.hg.hf\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR r : \u211d\nhr : r < R\nz : \u2102\n\u22a2 ContinuousOn (fun x => ((circleMap z R x.2 - x.1) ^ 2)\u207b\u00b9) (closedBall z r \u00d7\u02e2 \u22a4)"}, {"tactic": "simp_rw [\u2190 inv_pow]", "annotated_tactic": ["simp_rw [\u2190 <a>inv_pow</a>]", [{"full_name": "inv_pow", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [317, 9], "def_end_pos": [317, 16]}]], "state_before": "case hg.hg.hf\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR r : \u211d\nhr : r < R\nz : \u2102\n\u22a2 ContinuousOn (fun x => ((circleMap z R x.2 - x.1) ^ 2)\u207b\u00b9) (closedBall z r \u00d7\u02e2 \u22a4)", "state_after": "case hg.hg.hf\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR r : \u211d\nhr : r < R\nz : \u2102\n\u22a2 ContinuousOn (fun x => (circleMap z R x.2 - x.1)\u207b\u00b9 ^ 2) (closedBall z r \u00d7\u02e2 \u22a4)"}, {"tactic": "apply continuousOn_prod_circle_transform_function hr", "annotated_tactic": ["apply <a>continuousOn_prod_circle_transform_function</a> hr", [{"full_name": "Complex.continuousOn_prod_circle_transform_function", "def_path": "Mathlib/MeasureTheory/Integral/CircleTransform.lean", "def_pos": [100, 9], "def_end_pos": [100, 52]}]], "state_before": "case hg.hg.hf\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR r : \u211d\nhr : r < R\nz : \u2102\n\u22a2 ContinuousOn (fun x => (circleMap z R x.2 - x.1)\u207b\u00b9 ^ 2) (closedBall z r \u00d7\u02e2 \u22a4)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Martingale/Basic.lean", "full_name": "MeasureTheory.martingale_of_set_integral_eq_succ", "start": [415, 1], "end": [420, 82], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "full_name": "AEMeasurable.nnreal_tsum", "start": [2166, 1], "end": [2170, 87], "traced_tactics": [{"tactic": "simp_rw [NNReal.tsum_eq_toNNReal_tsum]", "annotated_tactic": ["simp_rw [<a>NNReal.tsum_eq_toNNReal_tsum</a>]", [{"full_name": "NNReal.tsum_eq_toNNReal_tsum", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [1095, 9], "def_end_pos": [1095, 30]}]], "state_before": "\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9\u271d : Sort y\ns t u : Set \u03b1\u271d\ninst\u271d\u00b2 : MeasurableSpace \u03b1\u271d\n\u03b1 : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03b9 : Type u_7\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 \u211d\u22650\n\u03bc : Measure \u03b1\nh : \u2200 (i : \u03b9), AEMeasurable (f i)\n\u22a2 AEMeasurable fun x => \u2211' (i : \u03b9), f i x", "state_after": "\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9\u271d : Sort y\ns t u : Set \u03b1\u271d\ninst\u271d\u00b2 : MeasurableSpace \u03b1\u271d\n\u03b1 : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03b9 : Type u_7\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 \u211d\u22650\n\u03bc : Measure \u03b1\nh : \u2200 (i : \u03b9), AEMeasurable (f i)\n\u22a2 AEMeasurable fun x => ENNReal.toNNReal (\u2211' (b : \u03b9), \u2191(f b x))"}, {"tactic": "exact (AEMeasurable.ennreal_tsum fun i => (h i).coe_nnreal_ennreal).ennreal_toNNReal", "annotated_tactic": ["exact (<a>AEMeasurable.ennreal_tsum</a> fun i => (h i).<a>coe_nnreal_ennreal</a>).<a>ennreal_toNNReal</a>", [{"full_name": "AEMeasurable.ennreal_tsum", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [2158, 9], "def_end_pos": [2158, 34]}, {"full_name": "AEMeasurable.coe_nnreal_ennreal", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [1999, 9], "def_end_pos": [1999, 40]}, {"full_name": "AEMeasurable.ennreal_toNNReal", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [2105, 9], "def_end_pos": [2105, 38]}]], "state_before": "\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9\u271d : Sort y\ns t u : Set \u03b1\u271d\ninst\u271d\u00b2 : MeasurableSpace \u03b1\u271d\n\u03b1 : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03b9 : Type u_7\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 \u211d\u22650\n\u03bc : Measure \u03b1\nh : \u2200 (i : \u03b9), AEMeasurable (f i)\n\u22a2 AEMeasurable fun x => ENNReal.toNNReal (\u2211' (b : \u03b9), \u2191(f b x))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/MeasurableSpace/Card.lean", "full_name": "MeasurableSpace.self_subset_generateMeasurableRec", "start": [55, 1], "end": [59, 19], "traced_tactics": [{"tactic": "unfold generateMeasurableRec", "annotated_tactic": ["unfold <a>generateMeasurableRec</a>", [{"full_name": "MeasurableSpace.generateMeasurableRec", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Card.lean", "def_pos": [47, 5], "def_end_pos": [47, 26]}]], "state_before": "\u03b1 : Type u\ns : Set (Set \u03b1)\ni : (Quotient.out (ord (aleph 1))).\u03b1\n\u22a2 s \u2286 generateMeasurableRec s i", "state_after": "\u03b1 : Type u\ns : Set (Set \u03b1)\ni : (Quotient.out (ord (aleph 1))).\u03b1\n\u22a2 s \u2286\n    let i := i;\n    let S := \u22c3 j, generateMeasurableRec s \u2191j;\n    s \u222a {\u2205} \u222a compl '' S \u222a range fun f => \u22c3 n, \u2191(f n)"}, {"tactic": "apply_rules [subset_union_of_subset_left]", "annotated_tactic": ["apply_rules [<a>subset_union_of_subset_left</a>]", [{"full_name": "Set.subset_union_of_subset_left", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [860, 9], "def_end_pos": [860, 36]}]], "state_before": "\u03b1 : Type u\ns : Set (Set \u03b1)\ni : (Quotient.out (ord (aleph 1))).\u03b1\n\u22a2 s \u2286\n    let i := i;\n    let S := \u22c3 j, generateMeasurableRec s \u2191j;\n    s \u222a {\u2205} \u222a compl '' S \u222a range fun f => \u22c3 n, \u2191(f n)", "state_after": "case h.h.h\n\u03b1 : Type u\ns : Set (Set \u03b1)\ni : (Quotient.out (ord (aleph 1))).\u03b1\n\u22a2 s \u2286 s"}, {"tactic": "exact subset_rfl", "annotated_tactic": ["exact <a>subset_rfl</a>", [{"full_name": "subset_rfl", "def_path": "Mathlib/Order/RelClasses.lean", "def_pos": [627, 7], "def_end_pos": [627, 17]}]], "state_before": "case h.h.h\n\u03b1 : Type u\ns : Set (Set \u03b1)\ni : (Quotient.out (ord (aleph 1))).\u03b1\n\u22a2 s \u2286 s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Real.lean", "full_name": "MeasureTheory.Integrable.uniformIntegrable_condexp", "start": [186, 1], "end": [224, 37], "traced_tactics": [{"tactic": "have hmeas : \u2200 n, \u2200 C, MeasurableSet {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a} := fun n C =>\n  measurableSet_le measurable_const (stronglyMeasurable_condexp.mono (h\u2131 n)).measurable.nnnorm", "annotated_tactic": ["have hmeas : \u2200 n, \u2200 C, <a>MeasurableSet</a> {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a} := fun n C =>\n    <a>measurableSet_le</a> <a>measurable_const</a> (stronglyMeasurable_condexp.mono (h\u2131 n)).measurable.nnnorm", [{"full_name": "MeasurableSet", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [64, 5], "def_end_pos": [64, 18]}, {"full_name": "measurableSet_le", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [559, 9], "def_end_pos": [559, 25]}, {"full_name": "measurable_const", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [570, 9], "def_end_pos": [570, 25]}]], "state_before": "\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b9 : Type u_2\ninst\u271d : IsFiniteMeasure \u03bc\ng : \u03b1 \u2192 \u211d\nhint : Integrable g\n\u2131 : \u03b9 \u2192 MeasurableSpace \u03b1\nh\u2131 : \u2200 (i : \u03b9), \u2131 i \u2264 m0\n\u22a2 UniformIntegrable (fun i => \u03bc[g|\u2131 i]) 1 \u03bc", "state_after": "\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b9 : Type u_2\ninst\u271d : IsFiniteMeasure \u03bc\ng : \u03b1 \u2192 \u211d\nhint : Integrable g\n\u2131 : \u03b9 \u2192 MeasurableSpace \u03b1\nh\u2131 : \u2200 (i : \u03b9), \u2131 i \u2264 m0\nhmeas : \u2200 (n : \u03b9) (C : \u211d\u22650), MeasurableSet {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a}\n\u22a2 UniformIntegrable (fun i => \u03bc[g|\u2131 i]) 1 \u03bc"}, {"tactic": "have hg : Mem\u2112p g 1 \u03bc := mem\u2112p_one_iff_integrable.2 hint", "annotated_tactic": ["have hg : <a>Mem\u2112p</a> g 1 \u03bc := <a>mem\u2112p_one_iff_integrable</a>.2 hint", [{"full_name": "MeasureTheory.Mem\u2112p", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [108, 5], "def_end_pos": [108, 10]}, {"full_name": "MeasureTheory.mem\u2112p_one_iff_integrable", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [453, 9], "def_end_pos": [453, 33]}]], "state_before": "\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b9 : Type u_2\ninst\u271d : IsFiniteMeasure \u03bc\ng : \u03b1 \u2192 \u211d\nhint : Integrable g\n\u2131 : \u03b9 \u2192 MeasurableSpace \u03b1\nh\u2131 : \u2200 (i : \u03b9), \u2131 i \u2264 m0\nhmeas : \u2200 (n : \u03b9) (C : \u211d\u22650), MeasurableSet {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a}\n\u22a2 UniformIntegrable (fun i => \u03bc[g|\u2131 i]) 1 \u03bc", "state_after": "\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b9 : Type u_2\ninst\u271d : IsFiniteMeasure \u03bc\ng : \u03b1 \u2192 \u211d\nhint : Integrable g\n\u2131 : \u03b9 \u2192 MeasurableSpace \u03b1\nh\u2131 : \u2200 (i : \u03b9), \u2131 i \u2264 m0\nhmeas : \u2200 (n : \u03b9) (C : \u211d\u22650), MeasurableSet {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a}\nhg : Mem\u2112p g 1\n\u22a2 UniformIntegrable (fun i => \u03bc[g|\u2131 i]) 1 \u03bc"}, {"tactic": "refine' uniformIntegrable_of le_rfl ENNReal.one_ne_top\n  (fun n => (stronglyMeasurable_condexp.mono (h\u2131 n)).aestronglyMeasurable) fun \u03b5 h\u03b5 => _", "annotated_tactic": ["refine' <a>uniformIntegrable_of</a> <a>le_rfl</a> <a>ENNReal.one_ne_top</a>\n    (fun n => (stronglyMeasurable_condexp.mono (h\u2131 n)).<a>aestronglyMeasurable</a>) fun \u03b5 h\u03b5 => _", [{"full_name": "MeasureTheory.uniformIntegrable_of", "def_path": "Mathlib/MeasureTheory/Function/UniformIntegrable.lean", "def_pos": [836, 9], "def_end_pos": [836, 29]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}, {"full_name": "ENNReal.one_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [340, 17], "def_end_pos": [340, 27]}, {"full_name": "MeasureTheory.StronglyMeasurable.aestronglyMeasurable", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [110, 19], "def_end_pos": [110, 58]}]], "state_before": "\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b9 : Type u_2\ninst\u271d : IsFiniteMeasure \u03bc\ng : \u03b1 \u2192 \u211d\nhint : Integrable g\n\u2131 : \u03b9 \u2192 MeasurableSpace \u03b1\nh\u2131 : \u2200 (i : \u03b9), \u2131 i \u2264 m0\nhmeas : \u2200 (n : \u03b9) (C : \u211d\u22650), MeasurableSet {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a}\nhg : Mem\u2112p g 1\n\u22a2 UniformIntegrable (fun i => \u03bc[g|\u2131 i]) 1 \u03bc", "state_after": "\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b9 : Type u_2\ninst\u271d : IsFiniteMeasure \u03bc\ng : \u03b1 \u2192 \u211d\nhint : Integrable g\n\u2131 : \u03b9 \u2192 MeasurableSpace \u03b1\nh\u2131 : \u2200 (i : \u03b9), \u2131 i \u2264 m0\nhmeas : \u2200 (n : \u03b9) (C : \u211d\u22650), MeasurableSet {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a}\nhg : Mem\u2112p g 1\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u22a2 \u2203 C, \u2200 (i : \u03b9), snorm (Set.indicator {x | C \u2264 \u2016(\u03bc[g|\u2131 i]) x\u2016\u208a} (\u03bc[g|\u2131 i])) 1 \u03bc \u2264 ENNReal.ofReal \u03b5"}, {"tactic": "by_cases hne : snorm g 1 \u03bc = 0", "annotated_tactic": ["by_cases hne : <a>snorm</a> g 1 \u03bc = 0", [{"full_name": "MeasureTheory.snorm", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [84, 5], "def_end_pos": [84, 10]}]], "state_before": "\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b9 : Type u_2\ninst\u271d : IsFiniteMeasure \u03bc\ng : \u03b1 \u2192 \u211d\nhint : Integrable g\n\u2131 : \u03b9 \u2192 MeasurableSpace \u03b1\nh\u2131 : \u2200 (i : \u03b9), \u2131 i \u2264 m0\nhmeas : \u2200 (n : \u03b9) (C : \u211d\u22650), MeasurableSet {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a}\nhg : Mem\u2112p g 1\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u22a2 \u2203 C, \u2200 (i : \u03b9), snorm (Set.indicator {x | C \u2264 \u2016(\u03bc[g|\u2131 i]) x\u2016\u208a} (\u03bc[g|\u2131 i])) 1 \u03bc \u2264 ENNReal.ofReal \u03b5", "state_after": "case pos\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b9 : Type u_2\ninst\u271d : IsFiniteMeasure \u03bc\ng : \u03b1 \u2192 \u211d\nhint : Integrable g\n\u2131 : \u03b9 \u2192 MeasurableSpace \u03b1\nh\u2131 : \u2200 (i : \u03b9), \u2131 i \u2264 m0\nhmeas : \u2200 (n : \u03b9) (C : \u211d\u22650), MeasurableSet {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a}\nhg : Mem\u2112p g 1\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhne : snorm g 1 \u03bc = 0\n\u22a2 \u2203 C, \u2200 (i : \u03b9), snorm (Set.indicator {x | C \u2264 \u2016(\u03bc[g|\u2131 i]) x\u2016\u208a} (\u03bc[g|\u2131 i])) 1 \u03bc \u2264 ENNReal.ofReal \u03b5\n\ncase neg\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b9 : Type u_2\ninst\u271d : IsFiniteMeasure \u03bc\ng : \u03b1 \u2192 \u211d\nhint : Integrable g\n\u2131 : \u03b9 \u2192 MeasurableSpace \u03b1\nh\u2131 : \u2200 (i : \u03b9), \u2131 i \u2264 m0\nhmeas : \u2200 (n : \u03b9) (C : \u211d\u22650), MeasurableSet {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a}\nhg : Mem\u2112p g 1\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhne : \u00acsnorm g 1 \u03bc = 0\n\u22a2 \u2203 C, \u2200 (i : \u03b9), snorm (Set.indicator {x | C \u2264 \u2016(\u03bc[g|\u2131 i]) x\u2016\u208a} (\u03bc[g|\u2131 i])) 1 \u03bc \u2264 ENNReal.ofReal \u03b5"}, {"tactic": "obtain \u27e8\u03b4, h\u03b4, h\u27e9 := hg.snorm_indicator_le \u03bc le_rfl ENNReal.one_ne_top h\u03b5", "annotated_tactic": ["obtain \u27e8\u03b4, h\u03b4, h\u27e9 := hg.snorm_indicator_le \u03bc <a>le_rfl</a> <a>ENNReal.one_ne_top</a> h\u03b5", [{"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}, {"full_name": "ENNReal.one_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [340, 17], "def_end_pos": [340, 27]}]], "state_before": "case neg\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b9 : Type u_2\ninst\u271d : IsFiniteMeasure \u03bc\ng : \u03b1 \u2192 \u211d\nhint : Integrable g\n\u2131 : \u03b9 \u2192 MeasurableSpace \u03b1\nh\u2131 : \u2200 (i : \u03b9), \u2131 i \u2264 m0\nhmeas : \u2200 (n : \u03b9) (C : \u211d\u22650), MeasurableSet {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a}\nhg : Mem\u2112p g 1\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhne : \u00acsnorm g 1 \u03bc = 0\n\u22a2 \u2203 C, \u2200 (i : \u03b9), snorm (Set.indicator {x | C \u2264 \u2016(\u03bc[g|\u2131 i]) x\u2016\u208a} (\u03bc[g|\u2131 i])) 1 \u03bc \u2264 ENNReal.ofReal \u03b5", "state_after": "case neg.intro.intro\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b9 : Type u_2\ninst\u271d : IsFiniteMeasure \u03bc\ng : \u03b1 \u2192 \u211d\nhint : Integrable g\n\u2131 : \u03b9 \u2192 MeasurableSpace \u03b1\nh\u2131 : \u2200 (i : \u03b9), \u2131 i \u2264 m0\nhmeas : \u2200 (n : \u03b9) (C : \u211d\u22650), MeasurableSet {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a}\nhg : Mem\u2112p g 1\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhne : \u00acsnorm g 1 \u03bc = 0\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (Set.indicator s g) 1 \u03bc \u2264 ENNReal.ofReal \u03b5\n\u22a2 \u2203 C, \u2200 (i : \u03b9), snorm (Set.indicator {x | C \u2264 \u2016(\u03bc[g|\u2131 i]) x\u2016\u208a} (\u03bc[g|\u2131 i])) 1 \u03bc \u2264 ENNReal.ofReal \u03b5"}, {"tactic": "set C : \u211d\u22650 := \u27e8\u03b4, h\u03b4.le\u27e9\u207b\u00b9 * (snorm g 1 \u03bc).toNNReal with hC", "annotated_tactic": ["set C : \u211d\u22650 := \u27e8\u03b4, h\u03b4.le\u27e9\u207b\u00b9 * (<a>snorm</a> g 1 \u03bc).<a>toNNReal</a> with hC", [{"full_name": "MeasureTheory.snorm", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [84, 5], "def_end_pos": [84, 10]}, {"full_name": "ENNReal.toNNReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [164, 15], "def_end_pos": [164, 23]}]], "state_before": "case neg.intro.intro\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b9 : Type u_2\ninst\u271d : IsFiniteMeasure \u03bc\ng : \u03b1 \u2192 \u211d\nhint : Integrable g\n\u2131 : \u03b9 \u2192 MeasurableSpace \u03b1\nh\u2131 : \u2200 (i : \u03b9), \u2131 i \u2264 m0\nhmeas : \u2200 (n : \u03b9) (C : \u211d\u22650), MeasurableSet {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a}\nhg : Mem\u2112p g 1\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhne : \u00acsnorm g 1 \u03bc = 0\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (Set.indicator s g) 1 \u03bc \u2264 ENNReal.ofReal \u03b5\n\u22a2 \u2203 C, \u2200 (i : \u03b9), snorm (Set.indicator {x | C \u2264 \u2016(\u03bc[g|\u2131 i]) x\u2016\u208a} (\u03bc[g|\u2131 i])) 1 \u03bc \u2264 ENNReal.ofReal \u03b5", "state_after": "case neg.intro.intro\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b9 : Type u_2\ninst\u271d : IsFiniteMeasure \u03bc\ng : \u03b1 \u2192 \u211d\nhint : Integrable g\n\u2131 : \u03b9 \u2192 MeasurableSpace \u03b1\nh\u2131 : \u2200 (i : \u03b9), \u2131 i \u2264 m0\nhmeas : \u2200 (n : \u03b9) (C : \u211d\u22650), MeasurableSet {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a}\nhg : Mem\u2112p g 1\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhne : \u00acsnorm g 1 \u03bc = 0\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (Set.indicator s g) 1 \u03bc \u2264 ENNReal.ofReal \u03b5\nC : \u211d\u22650 := { val := \u03b4, property := (_ : 0 \u2264 \u03b4) }\u207b\u00b9 * ENNReal.toNNReal (snorm g 1 \u03bc)\nhC : C = { val := \u03b4, property := (_ : 0 \u2264 \u03b4) }\u207b\u00b9 * ENNReal.toNNReal (snorm g 1 \u03bc)\n\u22a2 \u2203 C, \u2200 (i : \u03b9), snorm (Set.indicator {x | C \u2264 \u2016(\u03bc[g|\u2131 i]) x\u2016\u208a} (\u03bc[g|\u2131 i])) 1 \u03bc \u2264 ENNReal.ofReal \u03b5"}, {"tactic": "have hCpos : 0 < C := mul_pos (inv_pos.2 h\u03b4) (ENNReal.toNNReal_pos hne hg.snorm_lt_top.ne)", "annotated_tactic": ["have hCpos : 0 < C := <a>mul_pos</a> (<a>inv_pos</a>.2 h\u03b4) (<a>ENNReal.toNNReal_pos</a> hne hg.snorm_lt_top.ne)", [{"full_name": "mul_pos", "def_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "def_pos": [345, 7], "def_end_pos": [345, 14]}, {"full_name": "inv_pos", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [49, 9], "def_end_pos": [49, 16]}, {"full_name": "ENNReal.toNNReal_pos", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2123, 9], "def_end_pos": [2123, 21]}]], "state_before": "case neg.intro.intro\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b9 : Type u_2\ninst\u271d : IsFiniteMeasure \u03bc\ng : \u03b1 \u2192 \u211d\nhint : Integrable g\n\u2131 : \u03b9 \u2192 MeasurableSpace \u03b1\nh\u2131 : \u2200 (i : \u03b9), \u2131 i \u2264 m0\nhmeas : \u2200 (n : \u03b9) (C : \u211d\u22650), MeasurableSet {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a}\nhg : Mem\u2112p g 1\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhne : \u00acsnorm g 1 \u03bc = 0\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (Set.indicator s g) 1 \u03bc \u2264 ENNReal.ofReal \u03b5\nC : \u211d\u22650 := { val := \u03b4, property := (_ : 0 \u2264 \u03b4) }\u207b\u00b9 * ENNReal.toNNReal (snorm g 1 \u03bc)\nhC : C = { val := \u03b4, property := (_ : 0 \u2264 \u03b4) }\u207b\u00b9 * ENNReal.toNNReal (snorm g 1 \u03bc)\n\u22a2 \u2203 C, \u2200 (i : \u03b9), snorm (Set.indicator {x | C \u2264 \u2016(\u03bc[g|\u2131 i]) x\u2016\u208a} (\u03bc[g|\u2131 i])) 1 \u03bc \u2264 ENNReal.ofReal \u03b5", "state_after": "case neg.intro.intro\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b9 : Type u_2\ninst\u271d : IsFiniteMeasure \u03bc\ng : \u03b1 \u2192 \u211d\nhint : Integrable g\n\u2131 : \u03b9 \u2192 MeasurableSpace \u03b1\nh\u2131 : \u2200 (i : \u03b9), \u2131 i \u2264 m0\nhmeas : \u2200 (n : \u03b9) (C : \u211d\u22650), MeasurableSet {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a}\nhg : Mem\u2112p g 1\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhne : \u00acsnorm g 1 \u03bc = 0\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (Set.indicator s g) 1 \u03bc \u2264 ENNReal.ofReal \u03b5\nC : \u211d\u22650 := { val := \u03b4, property := (_ : 0 \u2264 \u03b4) }\u207b\u00b9 * ENNReal.toNNReal (snorm g 1 \u03bc)\nhC : C = { val := \u03b4, property := (_ : 0 \u2264 \u03b4) }\u207b\u00b9 * ENNReal.toNNReal (snorm g 1 \u03bc)\nhCpos : 0 < C\n\u22a2 \u2203 C, \u2200 (i : \u03b9), snorm (Set.indicator {x | C \u2264 \u2016(\u03bc[g|\u2131 i]) x\u2016\u208a} (\u03bc[g|\u2131 i])) 1 \u03bc \u2264 ENNReal.ofReal \u03b5"}, {"tactic": "have : \u2200 n, \u03bc {x : \u03b1 | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a} \u2264 ENNReal.ofReal \u03b4 := by\n  intro n\n  have := mul_meas_ge_le_pow_snorm' \u03bc one_ne_zero ENNReal.one_ne_top\n    ((@stronglyMeasurable_condexp _ _ _ _ _ (\u2131 n) _ \u03bc g).mono (h\u2131 n)).aestronglyMeasurable C\n  rw [ENNReal.one_toReal, ENNReal.rpow_one, ENNReal.rpow_one, mul_comm, \u2190\n    ENNReal.le_div_iff_mul_le (Or.inl (ENNReal.coe_ne_zero.2 hCpos.ne.symm))\n      (Or.inl ENNReal.coe_lt_top.ne)] at this\n  simp_rw [ENNReal.coe_le_coe] at this\n  refine' this.trans _\n  rw [ENNReal.div_le_iff_le_mul (Or.inl (ENNReal.coe_ne_zero.2 hCpos.ne.symm))\n      (Or.inl ENNReal.coe_lt_top.ne),\n    hC, Nonneg.inv_mk, ENNReal.coe_mul, ENNReal.coe_toNNReal hg.snorm_lt_top.ne, \u2190 mul_assoc, \u2190\n    ENNReal.ofReal_eq_coe_nnreal, \u2190 ENNReal.ofReal_mul h\u03b4.le, mul_inv_cancel h\u03b4.ne.symm,\n    ENNReal.ofReal_one, one_mul]\n  exact snorm_one_condexp_le_snorm _", "annotated_tactic": ["have : \u2200 n, \u03bc {x : \u03b1 | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a} \u2264 <a>ENNReal.ofReal</a> \u03b4 := by\n    intro n\n    have := <a>mul_meas_ge_le_pow_snorm'</a> \u03bc <a>one_ne_zero</a> <a>ENNReal.one_ne_top</a>\n      ((@<a>stronglyMeasurable_condexp</a> _ _ _ _ _ (\u2131 n) _ \u03bc g).<a>mono</a> (h\u2131 n)).<a>aestronglyMeasurable</a> C\n    rw [<a>ENNReal.one_toReal</a>, <a>ENNReal.rpow_one</a>, <a>ENNReal.rpow_one</a>, <a>mul_comm</a>, \u2190\n      <a>ENNReal.le_div_iff_mul_le</a> (<a>Or.inl</a> (<a>ENNReal.coe_ne_zero</a>.2 hCpos.ne.symm))\n        (<a>Or.inl</a> ENNReal.coe_lt_top.ne)] at this\n    simp_rw [<a>ENNReal.coe_le_coe</a>] at this\n    refine' this.trans _\n    rw [<a>ENNReal.div_le_iff_le_mul</a> (<a>Or.inl</a> (<a>ENNReal.coe_ne_zero</a>.2 hCpos.ne.symm))\n        (<a>Or.inl</a> ENNReal.coe_lt_top.ne),\n      hC, <a>Nonneg.inv_mk</a>, <a>ENNReal.coe_mul</a>, <a>ENNReal.coe_toNNReal</a> hg.snorm_lt_top.ne, \u2190 <a>mul_assoc</a>, \u2190\n      <a>ENNReal.ofReal_eq_coe_nnreal</a>, \u2190 <a>ENNReal.ofReal_mul</a> h\u03b4.le, <a>mul_inv_cancel</a> h\u03b4.ne.symm,\n      <a>ENNReal.ofReal_one</a>, <a>one_mul</a>]\n    exact <a>snorm_one_condexp_le_snorm</a> _", [{"full_name": "ENNReal.ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [172, 29], "def_end_pos": [172, 35]}, {"full_name": "MeasureTheory.mul_meas_ge_le_pow_snorm'", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [1164, 9], "def_end_pos": [1164, 34]}, {"full_name": "one_ne_zero", "def_path": "Mathlib/Algebra/NeZero.lean", "def_pos": [55, 15], "def_end_pos": [55, 26]}, {"full_name": "ENNReal.one_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [340, 17], "def_end_pos": [340, 27]}, {"full_name": "MeasureTheory.stronglyMeasurable_condexp", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean", "def_pos": [179, 9], "def_end_pos": [179, 35]}, {"full_name": "MeasureTheory.StronglyMeasurable.mono", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [369, 19], "def_end_pos": [369, 23]}, {"full_name": "MeasureTheory.StronglyMeasurable.aestronglyMeasurable", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [110, 19], "def_end_pos": [110, 58]}, {"full_name": "ENNReal.one_toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [230, 17], "def_end_pos": [230, 27]}, {"full_name": "ENNReal.rpow_one", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [450, 9], "def_end_pos": [450, 17]}, {"full_name": "ENNReal.rpow_one", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [450, 9], "def_end_pos": [450, 17]}, {"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [302, 9], "def_end_pos": [302, 17]}, {"full_name": "ENNReal.le_div_iff_mul_le", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1611, 19], "def_end_pos": [1611, 36]}, {"full_name": "Or.inl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [517, 5], "def_end_pos": [517, 8]}, {"full_name": "ENNReal.coe_ne_zero", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [383, 9], "def_end_pos": [383, 20]}, {"full_name": "Or.inl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [517, 5], "def_end_pos": [517, 8]}, {"full_name": "ENNReal.coe_le_coe", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [349, 28], "def_end_pos": [349, 38]}, {"full_name": "ENNReal.div_le_iff_le_mul", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1624, 19], "def_end_pos": [1624, 36]}, {"full_name": "Or.inl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [517, 5], "def_end_pos": [517, 8]}, {"full_name": "ENNReal.coe_ne_zero", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [383, 9], "def_end_pos": [383, 20]}, {"full_name": "Or.inl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [517, 5], "def_end_pos": [517, 8]}, {"full_name": "Nonneg.inv_mk", "def_path": "Mathlib/Algebra/Order/Nonneg/Field.lean", "def_pos": [47, 9], "def_end_pos": [47, 15]}, {"full_name": "ENNReal.coe_mul", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [390, 9], "def_end_pos": [390, 16]}, {"full_name": "ENNReal.coe_toNNReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [180, 9], "def_end_pos": [180, 21]}, {"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [264, 9], "def_end_pos": [264, 18]}, {"full_name": "ENNReal.ofReal_eq_coe_nnreal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [207, 9], "def_end_pos": [207, 29]}, {"full_name": "ENNReal.ofReal_mul", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2225, 9], "def_end_pos": [2225, 19]}, {"full_name": "mul_inv_cancel", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [194, 15], "def_end_pos": [194, 29]}, {"full_name": "ENNReal.ofReal_one", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [248, 17], "def_end_pos": [248, 27]}, {"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [464, 9], "def_end_pos": [464, 16]}, {"full_name": "MeasureTheory.snorm_one_condexp_le_snorm", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Real.lean", "def_pos": [60, 9], "def_end_pos": [60, 35]}]], "state_before": "case neg.intro.intro\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b9 : Type u_2\ninst\u271d : IsFiniteMeasure \u03bc\ng : \u03b1 \u2192 \u211d\nhint : Integrable g\n\u2131 : \u03b9 \u2192 MeasurableSpace \u03b1\nh\u2131 : \u2200 (i : \u03b9), \u2131 i \u2264 m0\nhmeas : \u2200 (n : \u03b9) (C : \u211d\u22650), MeasurableSet {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a}\nhg : Mem\u2112p g 1\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhne : \u00acsnorm g 1 \u03bc = 0\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (Set.indicator s g) 1 \u03bc \u2264 ENNReal.ofReal \u03b5\nC : \u211d\u22650 := { val := \u03b4, property := (_ : 0 \u2264 \u03b4) }\u207b\u00b9 * ENNReal.toNNReal (snorm g 1 \u03bc)\nhC : C = { val := \u03b4, property := (_ : 0 \u2264 \u03b4) }\u207b\u00b9 * ENNReal.toNNReal (snorm g 1 \u03bc)\nhCpos : 0 < C\n\u22a2 \u2203 C, \u2200 (i : \u03b9), snorm (Set.indicator {x | C \u2264 \u2016(\u03bc[g|\u2131 i]) x\u2016\u208a} (\u03bc[g|\u2131 i])) 1 \u03bc \u2264 ENNReal.ofReal \u03b5", "state_after": "case neg.intro.intro\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b9 : Type u_2\ninst\u271d : IsFiniteMeasure \u03bc\ng : \u03b1 \u2192 \u211d\nhint : Integrable g\n\u2131 : \u03b9 \u2192 MeasurableSpace \u03b1\nh\u2131 : \u2200 (i : \u03b9), \u2131 i \u2264 m0\nhmeas : \u2200 (n : \u03b9) (C : \u211d\u22650), MeasurableSet {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a}\nhg : Mem\u2112p g 1\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhne : \u00acsnorm g 1 \u03bc = 0\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (Set.indicator s g) 1 \u03bc \u2264 ENNReal.ofReal \u03b5\nC : \u211d\u22650 := { val := \u03b4, property := (_ : 0 \u2264 \u03b4) }\u207b\u00b9 * ENNReal.toNNReal (snorm g 1 \u03bc)\nhC : C = { val := \u03b4, property := (_ : 0 \u2264 \u03b4) }\u207b\u00b9 * ENNReal.toNNReal (snorm g 1 \u03bc)\nhCpos : 0 < C\nthis : \u2200 (n : \u03b9), \u2191\u2191\u03bc {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a} \u2264 ENNReal.ofReal \u03b4\n\u22a2 \u2203 C, \u2200 (i : \u03b9), snorm (Set.indicator {x | C \u2264 \u2016(\u03bc[g|\u2131 i]) x\u2016\u208a} (\u03bc[g|\u2131 i])) 1 \u03bc \u2264 ENNReal.ofReal \u03b5"}, {"tactic": "refine' \u27e8C, fun n => le_trans _ (h {x : \u03b1 | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a} (hmeas n C) (this n))\u27e9", "annotated_tactic": ["refine' \u27e8C, fun n => <a>le_trans</a> _ (h {x : \u03b1 | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a} (hmeas n C) (this n))\u27e9", [{"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}]], "state_before": "case neg.intro.intro\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b9 : Type u_2\ninst\u271d : IsFiniteMeasure \u03bc\ng : \u03b1 \u2192 \u211d\nhint : Integrable g\n\u2131 : \u03b9 \u2192 MeasurableSpace \u03b1\nh\u2131 : \u2200 (i : \u03b9), \u2131 i \u2264 m0\nhmeas : \u2200 (n : \u03b9) (C : \u211d\u22650), MeasurableSet {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a}\nhg : Mem\u2112p g 1\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhne : \u00acsnorm g 1 \u03bc = 0\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (Set.indicator s g) 1 \u03bc \u2264 ENNReal.ofReal \u03b5\nC : \u211d\u22650 := { val := \u03b4, property := (_ : 0 \u2264 \u03b4) }\u207b\u00b9 * ENNReal.toNNReal (snorm g 1 \u03bc)\nhC : C = { val := \u03b4, property := (_ : 0 \u2264 \u03b4) }\u207b\u00b9 * ENNReal.toNNReal (snorm g 1 \u03bc)\nhCpos : 0 < C\nthis : \u2200 (n : \u03b9), \u2191\u2191\u03bc {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a} \u2264 ENNReal.ofReal \u03b4\n\u22a2 \u2203 C, \u2200 (i : \u03b9), snorm (Set.indicator {x | C \u2264 \u2016(\u03bc[g|\u2131 i]) x\u2016\u208a} (\u03bc[g|\u2131 i])) 1 \u03bc \u2264 ENNReal.ofReal \u03b5", "state_after": "case neg.intro.intro\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b9 : Type u_2\ninst\u271d : IsFiniteMeasure \u03bc\ng : \u03b1 \u2192 \u211d\nhint : Integrable g\n\u2131 : \u03b9 \u2192 MeasurableSpace \u03b1\nh\u2131 : \u2200 (i : \u03b9), \u2131 i \u2264 m0\nhmeas : \u2200 (n : \u03b9) (C : \u211d\u22650), MeasurableSet {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a}\nhg : Mem\u2112p g 1\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhne : \u00acsnorm g 1 \u03bc = 0\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (Set.indicator s g) 1 \u03bc \u2264 ENNReal.ofReal \u03b5\nC : \u211d\u22650 := { val := \u03b4, property := (_ : 0 \u2264 \u03b4) }\u207b\u00b9 * ENNReal.toNNReal (snorm g 1 \u03bc)\nhC : C = { val := \u03b4, property := (_ : 0 \u2264 \u03b4) }\u207b\u00b9 * ENNReal.toNNReal (snorm g 1 \u03bc)\nhCpos : 0 < C\nthis : \u2200 (n : \u03b9), \u2191\u2191\u03bc {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a} \u2264 ENNReal.ofReal \u03b4\nn : \u03b9\n\u22a2 snorm (Set.indicator {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a} (\u03bc[g|\u2131 n])) 1 \u03bc \u2264 snorm (Set.indicator {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a} g) 1 \u03bc"}, {"tactic": "have hmeas\u2131 : MeasurableSet[\u2131 n] {x : \u03b1 | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a} :=\n  @measurableSet_le _ _ _ _ _ (\u2131 n) _ _ _ _ _ measurable_const\n    (@Measurable.nnnorm _ _ _ _ _ (\u2131 n) _ stronglyMeasurable_condexp.measurable)", "annotated_tactic": ["have hmeas\u2131 : MeasurableSet[\u2131 n] {x : \u03b1 | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a} :=\n    @<a>measurableSet_le</a> _ _ _ _ _ (\u2131 n) _ _ _ _ _ <a>measurable_const</a>\n      (@<a>Measurable.nnnorm</a> _ _ _ _ _ (\u2131 n) _ stronglyMeasurable_condexp.measurable)", [{"full_name": "measurableSet_le", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [559, 9], "def_end_pos": [559, 25]}, {"full_name": "measurable_const", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [570, 9], "def_end_pos": [570, 25]}, {"full_name": "Measurable.nnnorm", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [2261, 9], "def_end_pos": [2261, 26]}]], "state_before": "case neg.intro.intro\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b9 : Type u_2\ninst\u271d : IsFiniteMeasure \u03bc\ng : \u03b1 \u2192 \u211d\nhint : Integrable g\n\u2131 : \u03b9 \u2192 MeasurableSpace \u03b1\nh\u2131 : \u2200 (i : \u03b9), \u2131 i \u2264 m0\nhmeas : \u2200 (n : \u03b9) (C : \u211d\u22650), MeasurableSet {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a}\nhg : Mem\u2112p g 1\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhne : \u00acsnorm g 1 \u03bc = 0\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (Set.indicator s g) 1 \u03bc \u2264 ENNReal.ofReal \u03b5\nC : \u211d\u22650 := { val := \u03b4, property := (_ : 0 \u2264 \u03b4) }\u207b\u00b9 * ENNReal.toNNReal (snorm g 1 \u03bc)\nhC : C = { val := \u03b4, property := (_ : 0 \u2264 \u03b4) }\u207b\u00b9 * ENNReal.toNNReal (snorm g 1 \u03bc)\nhCpos : 0 < C\nthis : \u2200 (n : \u03b9), \u2191\u2191\u03bc {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a} \u2264 ENNReal.ofReal \u03b4\nn : \u03b9\n\u22a2 snorm (Set.indicator {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a} (\u03bc[g|\u2131 n])) 1 \u03bc \u2264 snorm (Set.indicator {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a} g) 1 \u03bc", "state_after": "case neg.intro.intro\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b9 : Type u_2\ninst\u271d : IsFiniteMeasure \u03bc\ng : \u03b1 \u2192 \u211d\nhint : Integrable g\n\u2131 : \u03b9 \u2192 MeasurableSpace \u03b1\nh\u2131 : \u2200 (i : \u03b9), \u2131 i \u2264 m0\nhmeas : \u2200 (n : \u03b9) (C : \u211d\u22650), MeasurableSet {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a}\nhg : Mem\u2112p g 1\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhne : \u00acsnorm g 1 \u03bc = 0\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (Set.indicator s g) 1 \u03bc \u2264 ENNReal.ofReal \u03b5\nC : \u211d\u22650 := { val := \u03b4, property := (_ : 0 \u2264 \u03b4) }\u207b\u00b9 * ENNReal.toNNReal (snorm g 1 \u03bc)\nhC : C = { val := \u03b4, property := (_ : 0 \u2264 \u03b4) }\u207b\u00b9 * ENNReal.toNNReal (snorm g 1 \u03bc)\nhCpos : 0 < C\nthis : \u2200 (n : \u03b9), \u2191\u2191\u03bc {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a} \u2264 ENNReal.ofReal \u03b4\nn : \u03b9\nhmeas\u2131 : MeasurableSet {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a}\n\u22a2 snorm (Set.indicator {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a} (\u03bc[g|\u2131 n])) 1 \u03bc \u2264 snorm (Set.indicator {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a} g) 1 \u03bc"}, {"tactic": "rw [\u2190 snorm_congr_ae (condexp_indicator hint hmeas\u2131)]", "annotated_tactic": ["rw [\u2190 <a>snorm_congr_ae</a> (<a>condexp_indicator</a> hint hmeas\u2131)]", [{"full_name": "MeasureTheory.snorm_congr_ae", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [549, 9], "def_end_pos": [549, 23]}, {"full_name": "MeasureTheory.condexp_indicator", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Indicator.lean", "def_pos": [75, 9], "def_end_pos": [75, 26]}]], "state_before": "case neg.intro.intro\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b9 : Type u_2\ninst\u271d : IsFiniteMeasure \u03bc\ng : \u03b1 \u2192 \u211d\nhint : Integrable g\n\u2131 : \u03b9 \u2192 MeasurableSpace \u03b1\nh\u2131 : \u2200 (i : \u03b9), \u2131 i \u2264 m0\nhmeas : \u2200 (n : \u03b9) (C : \u211d\u22650), MeasurableSet {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a}\nhg : Mem\u2112p g 1\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhne : \u00acsnorm g 1 \u03bc = 0\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (Set.indicator s g) 1 \u03bc \u2264 ENNReal.ofReal \u03b5\nC : \u211d\u22650 := { val := \u03b4, property := (_ : 0 \u2264 \u03b4) }\u207b\u00b9 * ENNReal.toNNReal (snorm g 1 \u03bc)\nhC : C = { val := \u03b4, property := (_ : 0 \u2264 \u03b4) }\u207b\u00b9 * ENNReal.toNNReal (snorm g 1 \u03bc)\nhCpos : 0 < C\nthis : \u2200 (n : \u03b9), \u2191\u2191\u03bc {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a} \u2264 ENNReal.ofReal \u03b4\nn : \u03b9\nhmeas\u2131 : MeasurableSet {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a}\n\u22a2 snorm (Set.indicator {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a} (\u03bc[g|\u2131 n])) 1 \u03bc \u2264 snorm (Set.indicator {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a} g) 1 \u03bc", "state_after": "case neg.intro.intro\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b9 : Type u_2\ninst\u271d : IsFiniteMeasure \u03bc\ng : \u03b1 \u2192 \u211d\nhint : Integrable g\n\u2131 : \u03b9 \u2192 MeasurableSpace \u03b1\nh\u2131 : \u2200 (i : \u03b9), \u2131 i \u2264 m0\nhmeas : \u2200 (n : \u03b9) (C : \u211d\u22650), MeasurableSet {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a}\nhg : Mem\u2112p g 1\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhne : \u00acsnorm g 1 \u03bc = 0\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (Set.indicator s g) 1 \u03bc \u2264 ENNReal.ofReal \u03b5\nC : \u211d\u22650 := { val := \u03b4, property := (_ : 0 \u2264 \u03b4) }\u207b\u00b9 * ENNReal.toNNReal (snorm g 1 \u03bc)\nhC : C = { val := \u03b4, property := (_ : 0 \u2264 \u03b4) }\u207b\u00b9 * ENNReal.toNNReal (snorm g 1 \u03bc)\nhCpos : 0 < C\nthis : \u2200 (n : \u03b9), \u2191\u2191\u03bc {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a} \u2264 ENNReal.ofReal \u03b4\nn : \u03b9\nhmeas\u2131 : MeasurableSet {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a}\n\u22a2 snorm (\u03bc[Set.indicator {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a} g|\u2131 n]) 1 \u03bc \u2264 snorm (Set.indicator {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a} g) 1 \u03bc"}, {"tactic": "exact snorm_one_condexp_le_snorm _", "annotated_tactic": ["exact <a>snorm_one_condexp_le_snorm</a> _", [{"full_name": "MeasureTheory.snorm_one_condexp_le_snorm", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Real.lean", "def_pos": [60, 9], "def_end_pos": [60, 35]}]], "state_before": "case neg.intro.intro\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b9 : Type u_2\ninst\u271d : IsFiniteMeasure \u03bc\ng : \u03b1 \u2192 \u211d\nhint : Integrable g\n\u2131 : \u03b9 \u2192 MeasurableSpace \u03b1\nh\u2131 : \u2200 (i : \u03b9), \u2131 i \u2264 m0\nhmeas : \u2200 (n : \u03b9) (C : \u211d\u22650), MeasurableSet {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a}\nhg : Mem\u2112p g 1\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhne : \u00acsnorm g 1 \u03bc = 0\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (Set.indicator s g) 1 \u03bc \u2264 ENNReal.ofReal \u03b5\nC : \u211d\u22650 := { val := \u03b4, property := (_ : 0 \u2264 \u03b4) }\u207b\u00b9 * ENNReal.toNNReal (snorm g 1 \u03bc)\nhC : C = { val := \u03b4, property := (_ : 0 \u2264 \u03b4) }\u207b\u00b9 * ENNReal.toNNReal (snorm g 1 \u03bc)\nhCpos : 0 < C\nthis : \u2200 (n : \u03b9), \u2191\u2191\u03bc {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a} \u2264 ENNReal.ofReal \u03b4\nn : \u03b9\nhmeas\u2131 : MeasurableSet {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a}\n\u22a2 snorm (\u03bc[Set.indicator {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a} g|\u2131 n]) 1 \u03bc \u2264 snorm (Set.indicator {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a} g) 1 \u03bc", "state_after": "no goals"}, {"tactic": "rw [snorm_eq_zero_iff hg.1 one_ne_zero] at hne", "annotated_tactic": ["rw [<a>snorm_eq_zero_iff</a> hg.1 <a>one_ne_zero</a>] at hne", [{"full_name": "MeasureTheory.snorm_eq_zero_iff", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [764, 9], "def_end_pos": [764, 26]}, {"full_name": "one_ne_zero", "def_path": "Mathlib/Algebra/NeZero.lean", "def_pos": [55, 15], "def_end_pos": [55, 26]}]], "state_before": "case pos\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b9 : Type u_2\ninst\u271d : IsFiniteMeasure \u03bc\ng : \u03b1 \u2192 \u211d\nhint : Integrable g\n\u2131 : \u03b9 \u2192 MeasurableSpace \u03b1\nh\u2131 : \u2200 (i : \u03b9), \u2131 i \u2264 m0\nhmeas : \u2200 (n : \u03b9) (C : \u211d\u22650), MeasurableSet {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a}\nhg : Mem\u2112p g 1\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhne : snorm g 1 \u03bc = 0\n\u22a2 \u2203 C, \u2200 (i : \u03b9), snorm (Set.indicator {x | C \u2264 \u2016(\u03bc[g|\u2131 i]) x\u2016\u208a} (\u03bc[g|\u2131 i])) 1 \u03bc \u2264 ENNReal.ofReal \u03b5", "state_after": "case pos\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b9 : Type u_2\ninst\u271d : IsFiniteMeasure \u03bc\ng : \u03b1 \u2192 \u211d\nhint : Integrable g\n\u2131 : \u03b9 \u2192 MeasurableSpace \u03b1\nh\u2131 : \u2200 (i : \u03b9), \u2131 i \u2264 m0\nhmeas : \u2200 (n : \u03b9) (C : \u211d\u22650), MeasurableSet {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a}\nhg : Mem\u2112p g 1\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhne : g =\u1d50[\u03bc] 0\n\u22a2 \u2203 C, \u2200 (i : \u03b9), snorm (Set.indicator {x | C \u2264 \u2016(\u03bc[g|\u2131 i]) x\u2016\u208a} (\u03bc[g|\u2131 i])) 1 \u03bc \u2264 ENNReal.ofReal \u03b5"}, {"tactic": "refine' \u27e80, fun n => (le_of_eq <|\n  (snorm_eq_zero_iff ((stronglyMeasurable_condexp.mono (h\u2131 n)).aestronglyMeasurable.indicator\n    (hmeas n 0)) one_ne_zero).2 _).trans (zero_le _)\u27e9", "annotated_tactic": ["refine' \u27e80, fun n => (<a>le_of_eq</a> <|\n      (<a>snorm_eq_zero_iff</a> ((stronglyMeasurable_condexp.mono (h\u2131 n)).aestronglyMeasurable.indicator\n        (hmeas n 0)) <a>one_ne_zero</a>).2 _).<a>trans</a> (<a>zero_le</a> _)\u27e9", [{"full_name": "le_of_eq", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [72, 9], "def_end_pos": [72, 17]}, {"full_name": "MeasureTheory.snorm_eq_zero_iff", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [764, 9], "def_end_pos": [764, 26]}, {"full_name": "one_ne_zero", "def_path": "Mathlib/Algebra/NeZero.lean", "def_pos": [55, 15], "def_end_pos": [55, 26]}, {"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [120, 7], "def_end_pos": [120, 18]}, {"full_name": "zero_le", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [217, 30], "def_end_pos": [217, 37]}]], "state_before": "case pos\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b9 : Type u_2\ninst\u271d : IsFiniteMeasure \u03bc\ng : \u03b1 \u2192 \u211d\nhint : Integrable g\n\u2131 : \u03b9 \u2192 MeasurableSpace \u03b1\nh\u2131 : \u2200 (i : \u03b9), \u2131 i \u2264 m0\nhmeas : \u2200 (n : \u03b9) (C : \u211d\u22650), MeasurableSet {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a}\nhg : Mem\u2112p g 1\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhne : g =\u1d50[\u03bc] 0\n\u22a2 \u2203 C, \u2200 (i : \u03b9), snorm (Set.indicator {x | C \u2264 \u2016(\u03bc[g|\u2131 i]) x\u2016\u208a} (\u03bc[g|\u2131 i])) 1 \u03bc \u2264 ENNReal.ofReal \u03b5", "state_after": "case pos\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b9 : Type u_2\ninst\u271d : IsFiniteMeasure \u03bc\ng : \u03b1 \u2192 \u211d\nhint : Integrable g\n\u2131 : \u03b9 \u2192 MeasurableSpace \u03b1\nh\u2131 : \u2200 (i : \u03b9), \u2131 i \u2264 m0\nhmeas : \u2200 (n : \u03b9) (C : \u211d\u22650), MeasurableSet {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a}\nhg : Mem\u2112p g 1\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhne : g =\u1d50[\u03bc] 0\nn : \u03b9\n\u22a2 Set.indicator {x | 0 \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a} (\u03bc[g|\u2131 n]) =\u1d50[\u03bc] 0"}, {"tactic": "filter_upwards [@condexp_congr_ae _ _ _ _ _ (\u2131 n) m0 \u03bc _ _ hne] with x hx", "annotated_tactic": ["filter_upwards [@<a>condexp_congr_ae</a> _ _ _ _ _ (\u2131 n) m0 \u03bc _ _ hne] with x hx", [{"full_name": "MeasureTheory.condexp_congr_ae", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean", "def_pos": [192, 9], "def_end_pos": [192, 25]}]], "state_before": "case pos\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b9 : Type u_2\ninst\u271d : IsFiniteMeasure \u03bc\ng : \u03b1 \u2192 \u211d\nhint : Integrable g\n\u2131 : \u03b9 \u2192 MeasurableSpace \u03b1\nh\u2131 : \u2200 (i : \u03b9), \u2131 i \u2264 m0\nhmeas : \u2200 (n : \u03b9) (C : \u211d\u22650), MeasurableSet {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a}\nhg : Mem\u2112p g 1\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhne : g =\u1d50[\u03bc] 0\nn : \u03b9\n\u22a2 Set.indicator {x | 0 \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a} (\u03bc[g|\u2131 n]) =\u1d50[\u03bc] 0", "state_after": "case h\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b9 : Type u_2\ninst\u271d : IsFiniteMeasure \u03bc\ng : \u03b1 \u2192 \u211d\nhint : Integrable g\n\u2131 : \u03b9 \u2192 MeasurableSpace \u03b1\nh\u2131 : \u2200 (i : \u03b9), \u2131 i \u2264 m0\nhmeas : \u2200 (n : \u03b9) (C : \u211d\u22650), MeasurableSet {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a}\nhg : Mem\u2112p g 1\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhne : g =\u1d50[\u03bc] 0\nn : \u03b9\nx : \u03b1\nhx : (\u03bc[g|\u2131 n]) x = (\u03bc[0|\u2131 n]) x\n\u22a2 Set.indicator {x | 0 \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a} (\u03bc[g|\u2131 n]) x = OfNat.ofNat 0 x"}, {"tactic": "simp only [zero_le', Set.setOf_true, Set.indicator_univ, Pi.zero_apply, hx, condexp_zero]", "annotated_tactic": ["simp only [<a>zero_le'</a>, <a>Set.setOf_true</a>, <a>Set.indicator_univ</a>, <a>Pi.zero_apply</a>, hx, <a>condexp_zero</a>]", [{"full_name": "zero_le'", "def_path": "Mathlib/Algebra/Order/WithZero.lean", "def_pos": [93, 9], "def_end_pos": [93, 17]}, {"full_name": "Set.setOf_true", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [669, 9], "def_end_pos": [669, 19]}, {"full_name": "Set.indicator_univ", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [187, 3], "def_end_pos": [187, 14]}, {"full_name": "Pi.zero_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [46, 3], "def_end_pos": [46, 14]}, {"full_name": "MeasureTheory.condexp_zero", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean", "def_pos": [169, 9], "def_end_pos": [169, 21]}]], "state_before": "case h\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b9 : Type u_2\ninst\u271d : IsFiniteMeasure \u03bc\ng : \u03b1 \u2192 \u211d\nhint : Integrable g\n\u2131 : \u03b9 \u2192 MeasurableSpace \u03b1\nh\u2131 : \u2200 (i : \u03b9), \u2131 i \u2264 m0\nhmeas : \u2200 (n : \u03b9) (C : \u211d\u22650), MeasurableSet {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a}\nhg : Mem\u2112p g 1\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhne : g =\u1d50[\u03bc] 0\nn : \u03b9\nx : \u03b1\nhx : (\u03bc[g|\u2131 n]) x = (\u03bc[0|\u2131 n]) x\n\u22a2 Set.indicator {x | 0 \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a} (\u03bc[g|\u2131 n]) x = OfNat.ofNat 0 x", "state_after": "no goals"}, {"tactic": "intro n", "annotated_tactic": ["intro n", []], "state_before": "\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b9 : Type u_2\ninst\u271d : IsFiniteMeasure \u03bc\ng : \u03b1 \u2192 \u211d\nhint : Integrable g\n\u2131 : \u03b9 \u2192 MeasurableSpace \u03b1\nh\u2131 : \u2200 (i : \u03b9), \u2131 i \u2264 m0\nhmeas : \u2200 (n : \u03b9) (C : \u211d\u22650), MeasurableSet {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a}\nhg : Mem\u2112p g 1\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhne : \u00acsnorm g 1 \u03bc = 0\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (Set.indicator s g) 1 \u03bc \u2264 ENNReal.ofReal \u03b5\nC : \u211d\u22650 := { val := \u03b4, property := (_ : 0 \u2264 \u03b4) }\u207b\u00b9 * ENNReal.toNNReal (snorm g 1 \u03bc)\nhC : C = { val := \u03b4, property := (_ : 0 \u2264 \u03b4) }\u207b\u00b9 * ENNReal.toNNReal (snorm g 1 \u03bc)\nhCpos : 0 < C\n\u22a2 \u2200 (n : \u03b9), \u2191\u2191\u03bc {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a} \u2264 ENNReal.ofReal \u03b4", "state_after": "\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b9 : Type u_2\ninst\u271d : IsFiniteMeasure \u03bc\ng : \u03b1 \u2192 \u211d\nhint : Integrable g\n\u2131 : \u03b9 \u2192 MeasurableSpace \u03b1\nh\u2131 : \u2200 (i : \u03b9), \u2131 i \u2264 m0\nhmeas : \u2200 (n : \u03b9) (C : \u211d\u22650), MeasurableSet {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a}\nhg : Mem\u2112p g 1\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhne : \u00acsnorm g 1 \u03bc = 0\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (Set.indicator s g) 1 \u03bc \u2264 ENNReal.ofReal \u03b5\nC : \u211d\u22650 := { val := \u03b4, property := (_ : 0 \u2264 \u03b4) }\u207b\u00b9 * ENNReal.toNNReal (snorm g 1 \u03bc)\nhC : C = { val := \u03b4, property := (_ : 0 \u2264 \u03b4) }\u207b\u00b9 * ENNReal.toNNReal (snorm g 1 \u03bc)\nhCpos : 0 < C\nn : \u03b9\n\u22a2 \u2191\u2191\u03bc {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a} \u2264 ENNReal.ofReal \u03b4"}, {"tactic": "have := mul_meas_ge_le_pow_snorm' \u03bc one_ne_zero ENNReal.one_ne_top\n  ((@stronglyMeasurable_condexp _ _ _ _ _ (\u2131 n) _ \u03bc g).mono (h\u2131 n)).aestronglyMeasurable C", "annotated_tactic": ["have := <a>mul_meas_ge_le_pow_snorm'</a> \u03bc <a>one_ne_zero</a> <a>ENNReal.one_ne_top</a>\n      ((@<a>stronglyMeasurable_condexp</a> _ _ _ _ _ (\u2131 n) _ \u03bc g).<a>mono</a> (h\u2131 n)).<a>aestronglyMeasurable</a> C", [{"full_name": "MeasureTheory.mul_meas_ge_le_pow_snorm'", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [1164, 9], "def_end_pos": [1164, 34]}, {"full_name": "one_ne_zero", "def_path": "Mathlib/Algebra/NeZero.lean", "def_pos": [55, 15], "def_end_pos": [55, 26]}, {"full_name": "ENNReal.one_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [340, 17], "def_end_pos": [340, 27]}, {"full_name": "MeasureTheory.stronglyMeasurable_condexp", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean", "def_pos": [179, 9], "def_end_pos": [179, 35]}, {"full_name": "MeasureTheory.StronglyMeasurable.mono", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [369, 19], "def_end_pos": [369, 23]}, {"full_name": "MeasureTheory.StronglyMeasurable.aestronglyMeasurable", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [110, 19], "def_end_pos": [110, 58]}]], "state_before": "\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b9 : Type u_2\ninst\u271d : IsFiniteMeasure \u03bc\ng : \u03b1 \u2192 \u211d\nhint : Integrable g\n\u2131 : \u03b9 \u2192 MeasurableSpace \u03b1\nh\u2131 : \u2200 (i : \u03b9), \u2131 i \u2264 m0\nhmeas : \u2200 (n : \u03b9) (C : \u211d\u22650), MeasurableSet {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a}\nhg : Mem\u2112p g 1\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhne : \u00acsnorm g 1 \u03bc = 0\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (Set.indicator s g) 1 \u03bc \u2264 ENNReal.ofReal \u03b5\nC : \u211d\u22650 := { val := \u03b4, property := (_ : 0 \u2264 \u03b4) }\u207b\u00b9 * ENNReal.toNNReal (snorm g 1 \u03bc)\nhC : C = { val := \u03b4, property := (_ : 0 \u2264 \u03b4) }\u207b\u00b9 * ENNReal.toNNReal (snorm g 1 \u03bc)\nhCpos : 0 < C\nn : \u03b9\n\u22a2 \u2191\u2191\u03bc {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a} \u2264 ENNReal.ofReal \u03b4", "state_after": "\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b9 : Type u_2\ninst\u271d : IsFiniteMeasure \u03bc\ng : \u03b1 \u2192 \u211d\nhint : Integrable g\n\u2131 : \u03b9 \u2192 MeasurableSpace \u03b1\nh\u2131 : \u2200 (i : \u03b9), \u2131 i \u2264 m0\nhmeas : \u2200 (n : \u03b9) (C : \u211d\u22650), MeasurableSet {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a}\nhg : Mem\u2112p g 1\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhne : \u00acsnorm g 1 \u03bc = 0\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (Set.indicator s g) 1 \u03bc \u2264 ENNReal.ofReal \u03b5\nC : \u211d\u22650 := { val := \u03b4, property := (_ : 0 \u2264 \u03b4) }\u207b\u00b9 * ENNReal.toNNReal (snorm g 1 \u03bc)\nhC : C = { val := \u03b4, property := (_ : 0 \u2264 \u03b4) }\u207b\u00b9 * ENNReal.toNNReal (snorm g 1 \u03bc)\nhCpos : 0 < C\nn : \u03b9\nthis : \u2191C ^ ENNReal.toReal 1 * \u2191\u2191\u03bc {x | \u2191C \u2264 \u2191\u2016(\u03bc[g|\u2131 n]) x\u2016\u208a} \u2264 snorm (\u03bc[g|\u2131 n]) 1 \u03bc ^ ENNReal.toReal 1\n\u22a2 \u2191\u2191\u03bc {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a} \u2264 ENNReal.ofReal \u03b4"}, {"tactic": "rw [ENNReal.one_toReal, ENNReal.rpow_one, ENNReal.rpow_one, mul_comm, \u2190\n  ENNReal.le_div_iff_mul_le (Or.inl (ENNReal.coe_ne_zero.2 hCpos.ne.symm))\n    (Or.inl ENNReal.coe_lt_top.ne)] at this", "annotated_tactic": ["rw [<a>ENNReal.one_toReal</a>, <a>ENNReal.rpow_one</a>, <a>ENNReal.rpow_one</a>, <a>mul_comm</a>, \u2190\n      <a>ENNReal.le_div_iff_mul_le</a> (<a>Or.inl</a> (<a>ENNReal.coe_ne_zero</a>.2 hCpos.ne.symm))\n        (<a>Or.inl</a> ENNReal.coe_lt_top.ne)] at this", [{"full_name": "ENNReal.one_toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [230, 17], "def_end_pos": [230, 27]}, {"full_name": "ENNReal.rpow_one", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [450, 9], "def_end_pos": [450, 17]}, {"full_name": "ENNReal.rpow_one", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [450, 9], "def_end_pos": [450, 17]}, {"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [302, 9], "def_end_pos": [302, 17]}, {"full_name": "ENNReal.le_div_iff_mul_le", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1611, 19], "def_end_pos": [1611, 36]}, {"full_name": "Or.inl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [517, 5], "def_end_pos": [517, 8]}, {"full_name": "ENNReal.coe_ne_zero", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [383, 9], "def_end_pos": [383, 20]}, {"full_name": "Or.inl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [517, 5], "def_end_pos": [517, 8]}]], "state_before": "\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b9 : Type u_2\ninst\u271d : IsFiniteMeasure \u03bc\ng : \u03b1 \u2192 \u211d\nhint : Integrable g\n\u2131 : \u03b9 \u2192 MeasurableSpace \u03b1\nh\u2131 : \u2200 (i : \u03b9), \u2131 i \u2264 m0\nhmeas : \u2200 (n : \u03b9) (C : \u211d\u22650), MeasurableSet {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a}\nhg : Mem\u2112p g 1\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhne : \u00acsnorm g 1 \u03bc = 0\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (Set.indicator s g) 1 \u03bc \u2264 ENNReal.ofReal \u03b5\nC : \u211d\u22650 := { val := \u03b4, property := (_ : 0 \u2264 \u03b4) }\u207b\u00b9 * ENNReal.toNNReal (snorm g 1 \u03bc)\nhC : C = { val := \u03b4, property := (_ : 0 \u2264 \u03b4) }\u207b\u00b9 * ENNReal.toNNReal (snorm g 1 \u03bc)\nhCpos : 0 < C\nn : \u03b9\nthis : \u2191C ^ ENNReal.toReal 1 * \u2191\u2191\u03bc {x | \u2191C \u2264 \u2191\u2016(\u03bc[g|\u2131 n]) x\u2016\u208a} \u2264 snorm (\u03bc[g|\u2131 n]) 1 \u03bc ^ ENNReal.toReal 1\n\u22a2 \u2191\u2191\u03bc {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a} \u2264 ENNReal.ofReal \u03b4", "state_after": "\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b9 : Type u_2\ninst\u271d : IsFiniteMeasure \u03bc\ng : \u03b1 \u2192 \u211d\nhint : Integrable g\n\u2131 : \u03b9 \u2192 MeasurableSpace \u03b1\nh\u2131 : \u2200 (i : \u03b9), \u2131 i \u2264 m0\nhmeas : \u2200 (n : \u03b9) (C : \u211d\u22650), MeasurableSet {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a}\nhg : Mem\u2112p g 1\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhne : \u00acsnorm g 1 \u03bc = 0\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (Set.indicator s g) 1 \u03bc \u2264 ENNReal.ofReal \u03b5\nC : \u211d\u22650 := { val := \u03b4, property := (_ : 0 \u2264 \u03b4) }\u207b\u00b9 * ENNReal.toNNReal (snorm g 1 \u03bc)\nhC : C = { val := \u03b4, property := (_ : 0 \u2264 \u03b4) }\u207b\u00b9 * ENNReal.toNNReal (snorm g 1 \u03bc)\nhCpos : 0 < C\nn : \u03b9\nthis : \u2191\u2191\u03bc {x | \u2191C \u2264 \u2191\u2016(\u03bc[g|\u2131 n]) x\u2016\u208a} \u2264 snorm (\u03bc[g|\u2131 n]) 1 \u03bc / \u2191C\n\u22a2 \u2191\u2191\u03bc {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a} \u2264 ENNReal.ofReal \u03b4"}, {"tactic": "simp_rw [ENNReal.coe_le_coe] at this", "annotated_tactic": ["simp_rw [<a>ENNReal.coe_le_coe</a>] at this", [{"full_name": "ENNReal.coe_le_coe", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [349, 28], "def_end_pos": [349, 38]}]], "state_before": "\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b9 : Type u_2\ninst\u271d : IsFiniteMeasure \u03bc\ng : \u03b1 \u2192 \u211d\nhint : Integrable g\n\u2131 : \u03b9 \u2192 MeasurableSpace \u03b1\nh\u2131 : \u2200 (i : \u03b9), \u2131 i \u2264 m0\nhmeas : \u2200 (n : \u03b9) (C : \u211d\u22650), MeasurableSet {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a}\nhg : Mem\u2112p g 1\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhne : \u00acsnorm g 1 \u03bc = 0\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (Set.indicator s g) 1 \u03bc \u2264 ENNReal.ofReal \u03b5\nC : \u211d\u22650 := { val := \u03b4, property := (_ : 0 \u2264 \u03b4) }\u207b\u00b9 * ENNReal.toNNReal (snorm g 1 \u03bc)\nhC : C = { val := \u03b4, property := (_ : 0 \u2264 \u03b4) }\u207b\u00b9 * ENNReal.toNNReal (snorm g 1 \u03bc)\nhCpos : 0 < C\nn : \u03b9\nthis : \u2191\u2191\u03bc {x | \u2191C \u2264 \u2191\u2016(\u03bc[g|\u2131 n]) x\u2016\u208a} \u2264 snorm (\u03bc[g|\u2131 n]) 1 \u03bc / \u2191C\n\u22a2 \u2191\u2191\u03bc {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a} \u2264 ENNReal.ofReal \u03b4", "state_after": "\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b9 : Type u_2\ninst\u271d : IsFiniteMeasure \u03bc\ng : \u03b1 \u2192 \u211d\nhint : Integrable g\n\u2131 : \u03b9 \u2192 MeasurableSpace \u03b1\nh\u2131 : \u2200 (i : \u03b9), \u2131 i \u2264 m0\nhmeas : \u2200 (n : \u03b9) (C : \u211d\u22650), MeasurableSet {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a}\nhg : Mem\u2112p g 1\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhne : \u00acsnorm g 1 \u03bc = 0\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (Set.indicator s g) 1 \u03bc \u2264 ENNReal.ofReal \u03b5\nC : \u211d\u22650 := { val := \u03b4, property := (_ : 0 \u2264 \u03b4) }\u207b\u00b9 * ENNReal.toNNReal (snorm g 1 \u03bc)\nhC : C = { val := \u03b4, property := (_ : 0 \u2264 \u03b4) }\u207b\u00b9 * ENNReal.toNNReal (snorm g 1 \u03bc)\nhCpos : 0 < C\nn : \u03b9\nthis :\n  \u2191\u2191\u03bc {x | { val := \u03b4, property := (_ : 0 \u2264 \u03b4) }\u207b\u00b9 * ENNReal.toNNReal (snorm g 1 \u03bc) \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a} \u2264\n    snorm (\u03bc[g|\u2131 n]) 1 \u03bc / \u2191({ val := \u03b4, property := (_ : 0 \u2264 \u03b4) }\u207b\u00b9 * ENNReal.toNNReal (snorm g 1 \u03bc))\n\u22a2 \u2191\u2191\u03bc {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a} \u2264 ENNReal.ofReal \u03b4"}, {"tactic": "refine' this.trans _", "annotated_tactic": ["refine' this.trans _", []], "state_before": "\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b9 : Type u_2\ninst\u271d : IsFiniteMeasure \u03bc\ng : \u03b1 \u2192 \u211d\nhint : Integrable g\n\u2131 : \u03b9 \u2192 MeasurableSpace \u03b1\nh\u2131 : \u2200 (i : \u03b9), \u2131 i \u2264 m0\nhmeas : \u2200 (n : \u03b9) (C : \u211d\u22650), MeasurableSet {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a}\nhg : Mem\u2112p g 1\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhne : \u00acsnorm g 1 \u03bc = 0\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (Set.indicator s g) 1 \u03bc \u2264 ENNReal.ofReal \u03b5\nC : \u211d\u22650 := { val := \u03b4, property := (_ : 0 \u2264 \u03b4) }\u207b\u00b9 * ENNReal.toNNReal (snorm g 1 \u03bc)\nhC : C = { val := \u03b4, property := (_ : 0 \u2264 \u03b4) }\u207b\u00b9 * ENNReal.toNNReal (snorm g 1 \u03bc)\nhCpos : 0 < C\nn : \u03b9\nthis :\n  \u2191\u2191\u03bc {x | { val := \u03b4, property := (_ : 0 \u2264 \u03b4) }\u207b\u00b9 * ENNReal.toNNReal (snorm g 1 \u03bc) \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a} \u2264\n    snorm (\u03bc[g|\u2131 n]) 1 \u03bc / \u2191({ val := \u03b4, property := (_ : 0 \u2264 \u03b4) }\u207b\u00b9 * ENNReal.toNNReal (snorm g 1 \u03bc))\n\u22a2 \u2191\u2191\u03bc {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a} \u2264 ENNReal.ofReal \u03b4", "state_after": "\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b9 : Type u_2\ninst\u271d : IsFiniteMeasure \u03bc\ng : \u03b1 \u2192 \u211d\nhint : Integrable g\n\u2131 : \u03b9 \u2192 MeasurableSpace \u03b1\nh\u2131 : \u2200 (i : \u03b9), \u2131 i \u2264 m0\nhmeas : \u2200 (n : \u03b9) (C : \u211d\u22650), MeasurableSet {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a}\nhg : Mem\u2112p g 1\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhne : \u00acsnorm g 1 \u03bc = 0\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (Set.indicator s g) 1 \u03bc \u2264 ENNReal.ofReal \u03b5\nC : \u211d\u22650 := { val := \u03b4, property := (_ : 0 \u2264 \u03b4) }\u207b\u00b9 * ENNReal.toNNReal (snorm g 1 \u03bc)\nhC : C = { val := \u03b4, property := (_ : 0 \u2264 \u03b4) }\u207b\u00b9 * ENNReal.toNNReal (snorm g 1 \u03bc)\nhCpos : 0 < C\nn : \u03b9\nthis :\n  \u2191\u2191\u03bc {x | { val := \u03b4, property := (_ : 0 \u2264 \u03b4) }\u207b\u00b9 * ENNReal.toNNReal (snorm g 1 \u03bc) \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a} \u2264\n    snorm (\u03bc[g|\u2131 n]) 1 \u03bc / \u2191({ val := \u03b4, property := (_ : 0 \u2264 \u03b4) }\u207b\u00b9 * ENNReal.toNNReal (snorm g 1 \u03bc))\n\u22a2 snorm (\u03bc[g|\u2131 n]) 1 \u03bc / \u2191({ val := \u03b4, property := (_ : 0 \u2264 \u03b4) }\u207b\u00b9 * ENNReal.toNNReal (snorm g 1 \u03bc)) \u2264 ENNReal.ofReal \u03b4"}, {"tactic": "rw [ENNReal.div_le_iff_le_mul (Or.inl (ENNReal.coe_ne_zero.2 hCpos.ne.symm))\n    (Or.inl ENNReal.coe_lt_top.ne),\n  hC, Nonneg.inv_mk, ENNReal.coe_mul, ENNReal.coe_toNNReal hg.snorm_lt_top.ne, \u2190 mul_assoc, \u2190\n  ENNReal.ofReal_eq_coe_nnreal, \u2190 ENNReal.ofReal_mul h\u03b4.le, mul_inv_cancel h\u03b4.ne.symm,\n  ENNReal.ofReal_one, one_mul]", "annotated_tactic": ["rw [<a>ENNReal.div_le_iff_le_mul</a> (<a>Or.inl</a> (<a>ENNReal.coe_ne_zero</a>.2 hCpos.ne.symm))\n        (<a>Or.inl</a> ENNReal.coe_lt_top.ne),\n      hC, <a>Nonneg.inv_mk</a>, <a>ENNReal.coe_mul</a>, <a>ENNReal.coe_toNNReal</a> hg.snorm_lt_top.ne, \u2190 <a>mul_assoc</a>, \u2190\n      <a>ENNReal.ofReal_eq_coe_nnreal</a>, \u2190 <a>ENNReal.ofReal_mul</a> h\u03b4.le, <a>mul_inv_cancel</a> h\u03b4.ne.symm,\n      <a>ENNReal.ofReal_one</a>, <a>one_mul</a>]", [{"full_name": "ENNReal.div_le_iff_le_mul", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1624, 19], "def_end_pos": [1624, 36]}, {"full_name": "Or.inl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [517, 5], "def_end_pos": [517, 8]}, {"full_name": "ENNReal.coe_ne_zero", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [383, 9], "def_end_pos": [383, 20]}, {"full_name": "Or.inl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [517, 5], "def_end_pos": [517, 8]}, {"full_name": "Nonneg.inv_mk", "def_path": "Mathlib/Algebra/Order/Nonneg/Field.lean", "def_pos": [47, 9], "def_end_pos": [47, 15]}, {"full_name": "ENNReal.coe_mul", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [390, 9], "def_end_pos": [390, 16]}, {"full_name": "ENNReal.coe_toNNReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [180, 9], "def_end_pos": [180, 21]}, {"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [264, 9], "def_end_pos": [264, 18]}, {"full_name": "ENNReal.ofReal_eq_coe_nnreal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [207, 9], "def_end_pos": [207, 29]}, {"full_name": "ENNReal.ofReal_mul", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2225, 9], "def_end_pos": [2225, 19]}, {"full_name": "mul_inv_cancel", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [194, 15], "def_end_pos": [194, 29]}, {"full_name": "ENNReal.ofReal_one", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [248, 17], "def_end_pos": [248, 27]}, {"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [464, 9], "def_end_pos": [464, 16]}]], "state_before": "\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b9 : Type u_2\ninst\u271d : IsFiniteMeasure \u03bc\ng : \u03b1 \u2192 \u211d\nhint : Integrable g\n\u2131 : \u03b9 \u2192 MeasurableSpace \u03b1\nh\u2131 : \u2200 (i : \u03b9), \u2131 i \u2264 m0\nhmeas : \u2200 (n : \u03b9) (C : \u211d\u22650), MeasurableSet {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a}\nhg : Mem\u2112p g 1\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhne : \u00acsnorm g 1 \u03bc = 0\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (Set.indicator s g) 1 \u03bc \u2264 ENNReal.ofReal \u03b5\nC : \u211d\u22650 := { val := \u03b4, property := (_ : 0 \u2264 \u03b4) }\u207b\u00b9 * ENNReal.toNNReal (snorm g 1 \u03bc)\nhC : C = { val := \u03b4, property := (_ : 0 \u2264 \u03b4) }\u207b\u00b9 * ENNReal.toNNReal (snorm g 1 \u03bc)\nhCpos : 0 < C\nn : \u03b9\nthis :\n  \u2191\u2191\u03bc {x | { val := \u03b4, property := (_ : 0 \u2264 \u03b4) }\u207b\u00b9 * ENNReal.toNNReal (snorm g 1 \u03bc) \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a} \u2264\n    snorm (\u03bc[g|\u2131 n]) 1 \u03bc / \u2191({ val := \u03b4, property := (_ : 0 \u2264 \u03b4) }\u207b\u00b9 * ENNReal.toNNReal (snorm g 1 \u03bc))\n\u22a2 snorm (\u03bc[g|\u2131 n]) 1 \u03bc / \u2191({ val := \u03b4, property := (_ : 0 \u2264 \u03b4) }\u207b\u00b9 * ENNReal.toNNReal (snorm g 1 \u03bc)) \u2264 ENNReal.ofReal \u03b4", "state_after": "\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b9 : Type u_2\ninst\u271d : IsFiniteMeasure \u03bc\ng : \u03b1 \u2192 \u211d\nhint : Integrable g\n\u2131 : \u03b9 \u2192 MeasurableSpace \u03b1\nh\u2131 : \u2200 (i : \u03b9), \u2131 i \u2264 m0\nhmeas : \u2200 (n : \u03b9) (C : \u211d\u22650), MeasurableSet {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a}\nhg : Mem\u2112p g 1\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhne : \u00acsnorm g 1 \u03bc = 0\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (Set.indicator s g) 1 \u03bc \u2264 ENNReal.ofReal \u03b5\nC : \u211d\u22650 := { val := \u03b4, property := (_ : 0 \u2264 \u03b4) }\u207b\u00b9 * ENNReal.toNNReal (snorm g 1 \u03bc)\nhC : C = { val := \u03b4, property := (_ : 0 \u2264 \u03b4) }\u207b\u00b9 * ENNReal.toNNReal (snorm g 1 \u03bc)\nhCpos : 0 < C\nn : \u03b9\nthis :\n  \u2191\u2191\u03bc {x | { val := \u03b4, property := (_ : 0 \u2264 \u03b4) }\u207b\u00b9 * ENNReal.toNNReal (snorm g 1 \u03bc) \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a} \u2264\n    snorm (\u03bc[g|\u2131 n]) 1 \u03bc / \u2191({ val := \u03b4, property := (_ : 0 \u2264 \u03b4) }\u207b\u00b9 * ENNReal.toNNReal (snorm g 1 \u03bc))\n\u22a2 snorm (\u03bc[g|\u2131 n]) 1 \u03bc \u2264 snorm g 1 \u03bc"}, {"tactic": "exact snorm_one_condexp_le_snorm _", "annotated_tactic": ["exact <a>snorm_one_condexp_le_snorm</a> _", [{"full_name": "MeasureTheory.snorm_one_condexp_le_snorm", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Real.lean", "def_pos": [60, 9], "def_end_pos": [60, 35]}]], "state_before": "\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b9 : Type u_2\ninst\u271d : IsFiniteMeasure \u03bc\ng : \u03b1 \u2192 \u211d\nhint : Integrable g\n\u2131 : \u03b9 \u2192 MeasurableSpace \u03b1\nh\u2131 : \u2200 (i : \u03b9), \u2131 i \u2264 m0\nhmeas : \u2200 (n : \u03b9) (C : \u211d\u22650), MeasurableSet {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a}\nhg : Mem\u2112p g 1\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhne : \u00acsnorm g 1 \u03bc = 0\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (Set.indicator s g) 1 \u03bc \u2264 ENNReal.ofReal \u03b5\nC : \u211d\u22650 := { val := \u03b4, property := (_ : 0 \u2264 \u03b4) }\u207b\u00b9 * ENNReal.toNNReal (snorm g 1 \u03bc)\nhC : C = { val := \u03b4, property := (_ : 0 \u2264 \u03b4) }\u207b\u00b9 * ENNReal.toNNReal (snorm g 1 \u03bc)\nhCpos : 0 < C\nn : \u03b9\nthis :\n  \u2191\u2191\u03bc {x | { val := \u03b4, property := (_ : 0 \u2264 \u03b4) }\u207b\u00b9 * ENNReal.toNNReal (snorm g 1 \u03bc) \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a} \u2264\n    snorm (\u03bc[g|\u2131 n]) 1 \u03bc / \u2191({ val := \u03b4, property := (_ : 0 \u2264 \u03b4) }\u207b\u00b9 * ENNReal.toNNReal (snorm g 1 \u03bc))\n\u22a2 snorm (\u03bc[g|\u2131 n]) 1 \u03bc \u2264 snorm g 1 \u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "full_name": "integrableOn_Iic_iff_integrableOn_Iio", "start": [728, 1], "end": [730, 96], "traced_tactics": [{"tactic": "rw [measure_singleton]", "annotated_tactic": ["rw [<a>measure_singleton</a>]", [{"full_name": "MeasureTheory.NoAtoms.measure_singleton", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3103, 3], "def_end_pos": [3103, 20]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : MeasurableSingletonClass \u03b1\nf : \u03b1 \u2192 E\n\u03bc : Measure \u03b1\na b : \u03b1\ninst\u271d : NoAtoms \u03bc\n\u22a2 \u2191\u2191\u03bc {b} \u2260 \u22a4", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : MeasurableSingletonClass \u03b1\nf : \u03b1 \u2192 E\n\u03bc : Measure \u03b1\na b : \u03b1\ninst\u271d : NoAtoms \u03bc\n\u22a2 0 \u2260 \u22a4"}, {"tactic": "exact ENNReal.zero_ne_top", "annotated_tactic": ["exact <a>ENNReal.zero_ne_top</a>", [{"full_name": "ENNReal.zero_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [334, 17], "def_end_pos": [334, 28]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : MeasurableSingletonClass \u03b1\nf : \u03b1 \u2192 E\n\u03bc : Measure \u03b1\na b : \u03b1\ninst\u271d : NoAtoms \u03bc\n\u22a2 0 \u2260 \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/ZMod/Basic.lean", "full_name": "ZMod.neg_val", "start": [919, 1], "end": [926, 38], "traced_tactics": [{"tactic": "rw [neg_val']", "annotated_tactic": ["rw [<a>neg_val'</a>]", [{"full_name": "ZMod.neg_val'", "def_path": "Mathlib/Data/ZMod/Basic.lean", "def_pos": [909, 9], "def_end_pos": [909, 17]}]], "state_before": "n : \u2115\ninst\u271d : NeZero n\na : ZMod n\n\u22a2 val (-a) = if a = 0 then 0 else n - val a", "state_after": "n : \u2115\ninst\u271d : NeZero n\na : ZMod n\n\u22a2 (n - val a) % n = if a = 0 then 0 else n - val a"}, {"tactic": "by_cases h : a = 0", "annotated_tactic": ["by_cases h : a = 0", []], "state_before": "n : \u2115\ninst\u271d : NeZero n\na : ZMod n\n\u22a2 (n - val a) % n = if a = 0 then 0 else n - val a", "state_after": "case pos\nn : \u2115\ninst\u271d : NeZero n\na : ZMod n\nh : a = 0\n\u22a2 (n - val a) % n = if a = 0 then 0 else n - val a\n\ncase neg\nn : \u2115\ninst\u271d : NeZero n\na : ZMod n\nh : \u00aca = 0\n\u22a2 (n - val a) % n = if a = 0 then 0 else n - val a"}, {"tactic": "rw [if_neg h]", "annotated_tactic": ["rw [<a>if_neg</a> h]", [{"full_name": "if_neg", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [795, 9], "def_end_pos": [795, 15]}]], "state_before": "case neg\nn : \u2115\ninst\u271d : NeZero n\na : ZMod n\nh : \u00aca = 0\n\u22a2 (n - val a) % n = if a = 0 then 0 else n - val a", "state_after": "case neg\nn : \u2115\ninst\u271d : NeZero n\na : ZMod n\nh : \u00aca = 0\n\u22a2 (n - val a) % n = n - val a"}, {"tactic": "apply Nat.mod_eq_of_lt", "annotated_tactic": ["apply <a>Nat.mod_eq_of_lt</a>", [{"full_name": "Nat.mod_eq_of_lt", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Div.lean", "def_pos": [111, 9], "def_end_pos": [111, 21]}]], "state_before": "case neg\nn : \u2115\ninst\u271d : NeZero n\na : ZMod n\nh : \u00aca = 0\n\u22a2 (n - val a) % n = n - val a", "state_after": "case neg.h\nn : \u2115\ninst\u271d : NeZero n\na : ZMod n\nh : \u00aca = 0\n\u22a2 n - val a < n"}, {"tactic": "apply Nat.sub_lt (NeZero.pos n)", "annotated_tactic": ["apply <a>Nat.sub_lt</a> (<a>NeZero.pos</a> n)", [{"full_name": "Nat.sub_lt", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [236, 9], "def_end_pos": [236, 15]}, {"full_name": "NeZero.pos", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [311, 9], "def_end_pos": [311, 12]}]], "state_before": "case neg.h\nn : \u2115\ninst\u271d : NeZero n\na : ZMod n\nh : \u00aca = 0\n\u22a2 n - val a < n", "state_after": "case neg.h\nn : \u2115\ninst\u271d : NeZero n\na : ZMod n\nh : \u00aca = 0\n\u22a2 0 < val a"}, {"tactic": "contrapose! h", "annotated_tactic": ["contrapose! h", []], "state_before": "case neg.h\nn : \u2115\ninst\u271d : NeZero n\na : ZMod n\nh : \u00aca = 0\n\u22a2 0 < val a", "state_after": "case neg.h\nn : \u2115\ninst\u271d : NeZero n\na : ZMod n\nh : val a \u2264 0\n\u22a2 a = 0"}, {"tactic": "rwa [le_zero_iff, val_eq_zero] at h", "annotated_tactic": ["rwa [<a>le_zero_iff</a>, <a>val_eq_zero</a>] at h", [{"full_name": "le_zero_iff", "def_path": "Mathlib/Algebra/Order/WithZero.lean", "def_pos": [102, 9], "def_end_pos": [102, 20]}, {"full_name": "ZMod.val_eq_zero", "def_path": "Mathlib/Data/ZMod/Basic.lean", "def_pos": [867, 9], "def_end_pos": [867, 20]}]], "state_before": "case neg.h\nn : \u2115\ninst\u271d : NeZero n\na : ZMod n\nh : val a \u2264 0\n\u22a2 a = 0", "state_after": "no goals"}, {"tactic": "rw [if_pos h, h, val_zero, tsub_zero, Nat.mod_self]", "annotated_tactic": ["rw [<a>if_pos</a> h, h, <a>val_zero</a>, <a>tsub_zero</a>, <a>Nat.mod_self</a>]", [{"full_name": "if_pos", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [790, 9], "def_end_pos": [790, 15]}, {"full_name": "ZMod.val_zero", "def_path": "Mathlib/Data/ZMod/Basic.lean", "def_pos": [63, 9], "def_end_pos": [63, 17]}, {"full_name": "tsub_zero", "def_path": "Mathlib/Algebra/Order/Sub/Defs.lean", "def_pos": [448, 9], "def_end_pos": [448, 18]}, {"full_name": "Nat.mod_self", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Div.lean", "def_pos": [154, 17], "def_end_pos": [154, 25]}]], "state_before": "case pos\nn : \u2115\ninst\u271d : NeZero n\na : ZMod n\nh : a = 0\n\u22a2 (n - val a) % n = if a = 0 then 0 else n - val a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Ackermann.lean", "full_name": "not_primrec_ack_self", "start": [386, 1], "end": [388, 33], "traced_tactics": [{"tactic": "rw [Primrec.nat_iff]", "annotated_tactic": ["rw [<a>Primrec.nat_iff</a>]", [{"full_name": "Primrec.nat_iff", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [233, 9], "def_end_pos": [233, 16]}]], "state_before": "\u22a2 \u00acPrimrec fun n => ack n n", "state_after": "\u22a2 \u00acNat.Primrec fun n => ack n n"}, {"tactic": "exact not_nat_primrec_ack_self", "annotated_tactic": ["exact <a>not_nat_primrec_ack_self</a>", [{"full_name": "not_nat_primrec_ack_self", "def_path": "Mathlib/Computability/Ackermann.lean", "def_pos": [381, 9], "def_end_pos": [381, 33]}]], "state_before": "\u22a2 \u00acNat.Primrec fun n => ack n n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Option/NAry.lean", "full_name": "Option.map\u2082_curry", "start": [106, 1], "end": [107, 80], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/RBMap/Lemmas.lean", "full_name": "Std.RBNode.Ordered.memP_iff_lowerBound?", "start": [333, 1], "end": [343, 75], "traced_tactics": [{"tactic": "refine memP_def.trans \u27e8fun \u27e8y, hy, ey\u27e9 => ?_, fun \u27e8x, hx, e\u27e9 => \u27e8_, lowerBound?_mem hx, e\u27e9\u27e9", "annotated_tactic": ["refine memP_def.trans \u27e8fun \u27e8y, hy, ey\u27e9 => ?_, fun \u27e8x, hx, e\u27e9 => \u27e8_, <a>lowerBound?_mem</a> hx, e\u27e9\u27e9", [{"full_name": "Std.RBNode.lowerBound?_mem", "def_path": "lake-packages/std/Std/Data/RBMap/Lemmas.lean", "def_pos": [280, 9], "def_end_pos": [280, 24]}]], "state_before": "\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\ncut : \u03b1 \u2192 Ordering\nt : RBNode \u03b1\ninst\u271d\u00b9 : TransCmp cmp\ninst\u271d : IsCut cmp cut\nht : Ordered cmp t\n\u22a2 MemP cut t \u2194 \u2203 x, lowerBound? cut t none = some x \u2227 cut x = Ordering.eq", "state_after": "\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\ncut : \u03b1 \u2192 Ordering\nt : RBNode \u03b1\ninst\u271d\u00b9 : TransCmp cmp\ninst\u271d : IsCut cmp cut\nht : Ordered cmp t\nx\u271d : \u2203 x, x \u2208 t \u2227 cut x = Ordering.eq\ny : \u03b1\nhy : y \u2208 t\ney : cut y = Ordering.eq\n\u22a2 \u2203 x, lowerBound? cut t none = some x \u2227 cut x = Ordering.eq"}, {"tactic": "have \u27e8x, hx\u27e9 := ht.lowerBound?_exists.2 \u27e8_, hy, fun h => nomatch ey.symm.trans h\u27e9", "annotated_tactic": ["have \u27e8x, hx\u27e9 := ht.lowerBound?_exists.2 \u27e8_, hy, fun h => nomatch ey.symm.trans h\u27e9", []], "state_before": "\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\ncut : \u03b1 \u2192 Ordering\nt : RBNode \u03b1\ninst\u271d\u00b9 : TransCmp cmp\ninst\u271d : IsCut cmp cut\nht : Ordered cmp t\nx\u271d : \u2203 x, x \u2208 t \u2227 cut x = Ordering.eq\ny : \u03b1\nhy : y \u2208 t\ney : cut y = Ordering.eq\n\u22a2 \u2203 x, lowerBound? cut t none = some x \u2227 cut x = Ordering.eq", "state_after": "\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\ncut : \u03b1 \u2192 Ordering\nt : RBNode \u03b1\ninst\u271d\u00b9 : TransCmp cmp\ninst\u271d : IsCut cmp cut\nht : Ordered cmp t\nx\u271d : \u2203 x, x \u2208 t \u2227 cut x = Ordering.eq\ny : \u03b1\nhy : y \u2208 t\ney : cut y = Ordering.eq\nx : \u03b1\nhx : lowerBound? cut t none = some x\n\u22a2 \u2203 x, lowerBound? cut t none = some x \u2227 cut x = Ordering.eq"}, {"tactic": "refine \u27e8x, hx, ?_\u27e9", "annotated_tactic": ["refine \u27e8x, hx, ?_\u27e9", []], "state_before": "\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\ncut : \u03b1 \u2192 Ordering\nt : RBNode \u03b1\ninst\u271d\u00b9 : TransCmp cmp\ninst\u271d : IsCut cmp cut\nht : Ordered cmp t\nx\u271d : \u2203 x, x \u2208 t \u2227 cut x = Ordering.eq\ny : \u03b1\nhy : y \u2208 t\ney : cut y = Ordering.eq\nx : \u03b1\nhx : lowerBound? cut t none = some x\n\u22a2 \u2203 x, lowerBound? cut t none = some x \u2227 cut x = Ordering.eq", "state_after": "\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\ncut : \u03b1 \u2192 Ordering\nt : RBNode \u03b1\ninst\u271d\u00b9 : TransCmp cmp\ninst\u271d : IsCut cmp cut\nht : Ordered cmp t\nx\u271d : \u2203 x, x \u2208 t \u2227 cut x = Ordering.eq\ny : \u03b1\nhy : y \u2208 t\ney : cut y = Ordering.eq\nx : \u03b1\nhx : lowerBound? cut t none = some x\n\u22a2 cut x = Ordering.eq"}, {"tactic": "cases ex : cut x", "annotated_tactic": ["cases ex : cut x", []], "state_before": "\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\ncut : \u03b1 \u2192 Ordering\nt : RBNode \u03b1\ninst\u271d\u00b9 : TransCmp cmp\ninst\u271d : IsCut cmp cut\nht : Ordered cmp t\nx\u271d : \u2203 x, x \u2208 t \u2227 cut x = Ordering.eq\ny : \u03b1\nhy : y \u2208 t\ney : cut y = Ordering.eq\nx : \u03b1\nhx : lowerBound? cut t none = some x\n\u22a2 cut x = Ordering.eq", "state_after": "case lt\n\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\ncut : \u03b1 \u2192 Ordering\nt : RBNode \u03b1\ninst\u271d\u00b9 : TransCmp cmp\ninst\u271d : IsCut cmp cut\nht : Ordered cmp t\nx\u271d : \u2203 x, x \u2208 t \u2227 cut x = Ordering.eq\ny : \u03b1\nhy : y \u2208 t\ney : cut y = Ordering.eq\nx : \u03b1\nhx : lowerBound? cut t none = some x\nex : cut x = Ordering.lt\n\u22a2 Ordering.lt = Ordering.eq\n\ncase eq\n\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\ncut : \u03b1 \u2192 Ordering\nt : RBNode \u03b1\ninst\u271d\u00b9 : TransCmp cmp\ninst\u271d : IsCut cmp cut\nht : Ordered cmp t\nx\u271d : \u2203 x, x \u2208 t \u2227 cut x = Ordering.eq\ny : \u03b1\nhy : y \u2208 t\ney : cut y = Ordering.eq\nx : \u03b1\nhx : lowerBound? cut t none = some x\nex : cut x = Ordering.eq\n\u22a2 Ordering.eq = Ordering.eq\n\ncase gt\n\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\ncut : \u03b1 \u2192 Ordering\nt : RBNode \u03b1\ninst\u271d\u00b9 : TransCmp cmp\ninst\u271d : IsCut cmp cut\nht : Ordered cmp t\nx\u271d : \u2203 x, x \u2208 t \u2227 cut x = Ordering.eq\ny : \u03b1\nhy : y \u2208 t\ney : cut y = Ordering.eq\nx : \u03b1\nhx : lowerBound? cut t none = some x\nex : cut x = Ordering.gt\n\u22a2 Ordering.gt = Ordering.eq"}, {"tactic": "cases lowerBound?_le hx ex", "annotated_tactic": ["cases <a>lowerBound?_le</a> hx ex", [{"full_name": "Std.RBNode.lowerBound?_le", "def_path": "lake-packages/std/Std/Data/RBMap/Lemmas.lean", "def_pos": [260, 9], "def_end_pos": [260, 23]}]], "state_before": "case lt\n\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\ncut : \u03b1 \u2192 Ordering\nt : RBNode \u03b1\ninst\u271d\u00b9 : TransCmp cmp\ninst\u271d : IsCut cmp cut\nht : Ordered cmp t\nx\u271d : \u2203 x, x \u2208 t \u2227 cut x = Ordering.eq\ny : \u03b1\nhy : y \u2208 t\ney : cut y = Ordering.eq\nx : \u03b1\nhx : lowerBound? cut t none = some x\nex : cut x = Ordering.lt\n\u22a2 Ordering.lt = Ordering.eq", "state_after": "no goals"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case eq\n\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\ncut : \u03b1 \u2192 Ordering\nt : RBNode \u03b1\ninst\u271d\u00b9 : TransCmp cmp\ninst\u271d : IsCut cmp cut\nht : Ordered cmp t\nx\u271d : \u2203 x, x \u2208 t \u2227 cut x = Ordering.eq\ny : \u03b1\nhy : y \u2208 t\ney : cut y = Ordering.eq\nx : \u03b1\nhx : lowerBound? cut t none = some x\nex : cut x = Ordering.eq\n\u22a2 Ordering.eq = Ordering.eq", "state_after": "no goals"}, {"tactic": "cases e : cmp x y", "annotated_tactic": ["cases e : cmp x y", []], "state_before": "case gt\n\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\ncut : \u03b1 \u2192 Ordering\nt : RBNode \u03b1\ninst\u271d\u00b9 : TransCmp cmp\ninst\u271d : IsCut cmp cut\nht : Ordered cmp t\nx\u271d : \u2203 x, x \u2208 t \u2227 cut x = Ordering.eq\ny : \u03b1\nhy : y \u2208 t\ney : cut y = Ordering.eq\nx : \u03b1\nhx : lowerBound? cut t none = some x\nex : cut x = Ordering.gt\n\u22a2 Ordering.gt = Ordering.eq", "state_after": "case gt.lt\n\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\ncut : \u03b1 \u2192 Ordering\nt : RBNode \u03b1\ninst\u271d\u00b9 : TransCmp cmp\ninst\u271d : IsCut cmp cut\nht : Ordered cmp t\nx\u271d : \u2203 x, x \u2208 t \u2227 cut x = Ordering.eq\ny : \u03b1\nhy : y \u2208 t\ney : cut y = Ordering.eq\nx : \u03b1\nhx : lowerBound? cut t none = some x\nex : cut x = Ordering.gt\ne : cmp x y = Ordering.lt\n\u22a2 Ordering.gt = Ordering.eq\n\ncase gt.eq\n\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\ncut : \u03b1 \u2192 Ordering\nt : RBNode \u03b1\ninst\u271d\u00b9 : TransCmp cmp\ninst\u271d : IsCut cmp cut\nht : Ordered cmp t\nx\u271d : \u2203 x, x \u2208 t \u2227 cut x = Ordering.eq\ny : \u03b1\nhy : y \u2208 t\ney : cut y = Ordering.eq\nx : \u03b1\nhx : lowerBound? cut t none = some x\nex : cut x = Ordering.gt\ne : cmp x y = Ordering.eq\n\u22a2 Ordering.gt = Ordering.eq\n\ncase gt.gt\n\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\ncut : \u03b1 \u2192 Ordering\nt : RBNode \u03b1\ninst\u271d\u00b9 : TransCmp cmp\ninst\u271d : IsCut cmp cut\nht : Ordered cmp t\nx\u271d : \u2203 x, x \u2208 t \u2227 cut x = Ordering.eq\ny : \u03b1\nhy : y \u2208 t\ney : cut y = Ordering.eq\nx : \u03b1\nhx : lowerBound? cut t none = some x\nex : cut x = Ordering.gt\ne : cmp x y = Ordering.gt\n\u22a2 Ordering.gt = Ordering.eq"}, {"tactic": "cases ey.symm.trans <| ht.lowerBound?_least hx hy e ex", "annotated_tactic": ["cases ey.symm.trans <| ht.lowerBound?_least hx hy e ex", []], "state_before": "case gt.lt\n\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\ncut : \u03b1 \u2192 Ordering\nt : RBNode \u03b1\ninst\u271d\u00b9 : TransCmp cmp\ninst\u271d : IsCut cmp cut\nht : Ordered cmp t\nx\u271d : \u2203 x, x \u2208 t \u2227 cut x = Ordering.eq\ny : \u03b1\nhy : y \u2208 t\ney : cut y = Ordering.eq\nx : \u03b1\nhx : lowerBound? cut t none = some x\nex : cut x = Ordering.gt\ne : cmp x y = Ordering.lt\n\u22a2 Ordering.gt = Ordering.eq", "state_after": "no goals"}, {"tactic": "cases ey.symm.trans <| IsCut.congr e |>.symm.trans ex", "annotated_tactic": ["cases ey.symm.trans <| <a>IsCut.congr</a> e |>.symm.trans ex", [{"full_name": "Std.RBNode.IsCut.congr", "def_path": "lake-packages/std/Std/Data/RBMap/Lemmas.lean", "def_pos": [160, 9], "def_end_pos": [160, 20]}]], "state_before": "case gt.eq\n\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\ncut : \u03b1 \u2192 Ordering\nt : RBNode \u03b1\ninst\u271d\u00b9 : TransCmp cmp\ninst\u271d : IsCut cmp cut\nht : Ordered cmp t\nx\u271d : \u2203 x, x \u2208 t \u2227 cut x = Ordering.eq\ny : \u03b1\nhy : y \u2208 t\ney : cut y = Ordering.eq\nx : \u03b1\nhx : lowerBound? cut t none = some x\nex : cut x = Ordering.gt\ne : cmp x y = Ordering.eq\n\u22a2 Ordering.gt = Ordering.eq", "state_after": "no goals"}, {"tactic": "cases ey.symm.trans <| IsCut.gt_trans (OrientedCmp.cmp_eq_gt.1 e) ex", "annotated_tactic": ["cases ey.symm.trans <| <a>IsCut.gt_trans</a> (<a>OrientedCmp.cmp_eq_gt</a>.1 e) ex", [{"full_name": "Std.RBNode.IsCut.gt_trans", "def_path": "lake-packages/std/Std/Data/RBMap/Lemmas.lean", "def_pos": [156, 9], "def_end_pos": [156, 23]}, {"full_name": "Std.OrientedCmp.cmp_eq_gt", "def_path": "lake-packages/std/Std/Classes/Order.lean", "def_pos": [31, 9], "def_end_pos": [31, 18]}]], "state_before": "case gt.gt\n\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\ncut : \u03b1 \u2192 Ordering\nt : RBNode \u03b1\ninst\u271d\u00b9 : TransCmp cmp\ninst\u271d : IsCut cmp cut\nht : Ordered cmp t\nx\u271d : \u2203 x, x \u2208 t \u2227 cut x = Ordering.eq\ny : \u03b1\nhy : y \u2208 t\ney : cut y = Ordering.eq\nx : \u03b1\nhx : lowerBound? cut t none = some x\nex : cut x = Ordering.gt\ne : cmp x y = Ordering.gt\n\u22a2 Ordering.gt = Ordering.eq", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Num/Lemmas.lean", "full_name": "Num.cast_le", "start": [869, 1], "end": [870, 41], "traced_tactics": [{"tactic": "rw [\u2190 not_lt]", "annotated_tactic": ["rw [\u2190 <a>not_lt</a>]", [{"full_name": "not_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [368, 9], "def_end_pos": [368, 15]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : LinearOrderedSemiring \u03b1\nm n : Num\n\u22a2 \u2191m \u2264 \u2191n \u2194 m \u2264 n", "state_after": "\u03b1 : Type u_1\ninst\u271d : LinearOrderedSemiring \u03b1\nm n : Num\n\u22a2 \u00ac\u2191n < \u2191m \u2194 m \u2264 n"}, {"tactic": "exact not_congr cast_lt", "annotated_tactic": ["exact <a>not_congr</a> <a>cast_lt</a>", [{"full_name": "not_congr", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [22, 9], "def_end_pos": [22, 18]}, {"full_name": "Num.cast_lt", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [864, 9], "def_end_pos": [864, 16]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : LinearOrderedSemiring \u03b1\nm n : Num\n\u22a2 \u00ac\u2191n < \u2191m \u2194 m \u2264 n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Martingale/Basic.lean", "full_name": "MeasureTheory.martingale_nat", "start": [442, 1], "end": [445, 52], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "full_name": "MeasureTheory.L1.setToL1_congr_left", "start": [1054, 1], "end": [1062, 47], "traced_tactics": [{"tactic": "suffices setToL1 hT = setToL1 hT' by rw [this]", "annotated_tactic": ["suffices <a>setToL1</a> hT = <a>setToL1</a> hT' by rw [this]", [{"full_name": "MeasureTheory.L1.setToL1", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [1019, 5], "def_end_pos": [1019, 12]}, {"full_name": "MeasureTheory.L1.setToL1", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [1019, 5], "def_end_pos": [1019, 12]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : NormedAddCommGroup F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nT\u271d T'\u271d T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC\u271d C'\u271d C'' : \u211d\nT T' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc T' C'\nh : T = T'\nf : { x // x \u2208 Lp E 1 }\n\u22a2 \u2191(setToL1 hT) f = \u2191(setToL1 hT') f", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : NormedAddCommGroup F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nT\u271d T'\u271d T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC\u271d C'\u271d C'' : \u211d\nT T' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc T' C'\nh : T = T'\nf : { x // x \u2208 Lp E 1 }\n\u22a2 setToL1 hT = setToL1 hT'"}, {"tactic": "refine' ContinuousLinearMap.extend_unique (setToL1SCLM \u03b1 E \u03bc hT) _ _ _ _ _", "annotated_tactic": ["refine' <a>ContinuousLinearMap.extend_unique</a> (<a>setToL1SCLM</a> \u03b1 E \u03bc hT) _ _ _ _ _", [{"full_name": "ContinuousLinearMap.extend_unique", "def_path": "Mathlib/Analysis/NormedSpace/OperatorNorm.lean", "def_pos": [1746, 9], "def_end_pos": [1746, 22]}, {"full_name": "MeasureTheory.L1.SimpleFunc.setToL1SCLM", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [877, 5], "def_end_pos": [877, 16]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : NormedAddCommGroup F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nT\u271d T'\u271d T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC\u271d C'\u271d C'' : \u211d\nT T' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc T' C'\nh : T = T'\nf : { x // x \u2208 Lp E 1 }\n\u22a2 setToL1 hT = setToL1 hT'", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : NormedAddCommGroup F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nT\u271d T'\u271d T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC\u271d C'\u271d C'' : \u211d\nT T' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc T' C'\nh : T = T'\nf : { x // x \u2208 Lp E 1 }\n\u22a2 ContinuousLinearMap.comp (setToL1 hT') (coeToLp \u03b1 E \u211d) = setToL1SCLM \u03b1 E \u03bc hT"}, {"tactic": "ext1 f", "annotated_tactic": ["ext1 f", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : NormedAddCommGroup F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nT\u271d T'\u271d T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC\u271d C'\u271d C'' : \u211d\nT T' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc T' C'\nh : T = T'\nf : { x // x \u2208 Lp E 1 }\n\u22a2 ContinuousLinearMap.comp (setToL1 hT') (coeToLp \u03b1 E \u211d) = setToL1SCLM \u03b1 E \u03bc hT", "state_after": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : NormedAddCommGroup F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nT\u271d T'\u271d T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC\u271d C'\u271d C'' : \u211d\nT T' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc T' C'\nh : T = T'\nf\u271d : { x // x \u2208 Lp E 1 }\nf : { x // x \u2208 simpleFunc E 1 \u03bc }\n\u22a2 \u2191(ContinuousLinearMap.comp (setToL1 hT') (coeToLp \u03b1 E \u211d)) f = \u2191(setToL1SCLM \u03b1 E \u03bc hT) f"}, {"tactic": "suffices setToL1 hT' f = setToL1SCLM \u03b1 E \u03bc hT f by rw [\u2190 this]; rfl", "annotated_tactic": ["suffices <a>setToL1</a> hT' f = <a>setToL1SCLM</a> \u03b1 E \u03bc hT f by rw [\u2190 this]; rfl", [{"full_name": "MeasureTheory.L1.setToL1", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [1019, 5], "def_end_pos": [1019, 12]}, {"full_name": "MeasureTheory.L1.SimpleFunc.setToL1SCLM", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [877, 5], "def_end_pos": [877, 16]}]], "state_before": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : NormedAddCommGroup F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nT\u271d T'\u271d T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC\u271d C'\u271d C'' : \u211d\nT T' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc T' C'\nh : T = T'\nf\u271d : { x // x \u2208 Lp E 1 }\nf : { x // x \u2208 simpleFunc E 1 \u03bc }\n\u22a2 \u2191(ContinuousLinearMap.comp (setToL1 hT') (coeToLp \u03b1 E \u211d)) f = \u2191(setToL1SCLM \u03b1 E \u03bc hT) f", "state_after": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : NormedAddCommGroup F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nT\u271d T'\u271d T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC\u271d C'\u271d C'' : \u211d\nT T' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc T' C'\nh : T = T'\nf\u271d : { x // x \u2208 Lp E 1 }\nf : { x // x \u2208 simpleFunc E 1 \u03bc }\n\u22a2 \u2191(setToL1 hT') \u2191f = \u2191(setToL1SCLM \u03b1 E \u03bc hT) f"}, {"tactic": "rw [setToL1_eq_setToL1SCLM]", "annotated_tactic": ["rw [<a>setToL1_eq_setToL1SCLM</a>]", [{"full_name": "MeasureTheory.L1.setToL1_eq_setToL1SCLM", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [1024, 9], "def_end_pos": [1024, 31]}]], "state_before": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : NormedAddCommGroup F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nT\u271d T'\u271d T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC\u271d C'\u271d C'' : \u211d\nT T' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc T' C'\nh : T = T'\nf\u271d : { x // x \u2208 Lp E 1 }\nf : { x // x \u2208 simpleFunc E 1 \u03bc }\n\u22a2 \u2191(setToL1 hT') \u2191f = \u2191(setToL1SCLM \u03b1 E \u03bc hT) f", "state_after": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : NormedAddCommGroup F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nT\u271d T'\u271d T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC\u271d C'\u271d C'' : \u211d\nT T' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc T' C'\nh : T = T'\nf\u271d : { x // x \u2208 Lp E 1 }\nf : { x // x \u2208 simpleFunc E 1 \u03bc }\n\u22a2 \u2191(setToL1SCLM \u03b1 E \u03bc hT') f = \u2191(setToL1SCLM \u03b1 E \u03bc hT) f"}, {"tactic": "exact setToL1SCLM_congr_left hT' hT h.symm f", "annotated_tactic": ["exact <a>setToL1SCLM_congr_left</a> hT' hT h.symm f", [{"full_name": "MeasureTheory.L1.SimpleFunc.setToL1SCLM_congr_left", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [901, 9], "def_end_pos": [901, 31]}]], "state_before": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : NormedAddCommGroup F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nT\u271d T'\u271d T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC\u271d C'\u271d C'' : \u211d\nT T' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc T' C'\nh : T = T'\nf\u271d : { x // x \u2208 Lp E 1 }\nf : { x // x \u2208 simpleFunc E 1 \u03bc }\n\u22a2 \u2191(setToL1SCLM \u03b1 E \u03bc hT') f = \u2191(setToL1SCLM \u03b1 E \u03bc hT) f", "state_after": "no goals"}, {"tactic": "rw [this]", "annotated_tactic": ["rw [this]", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : NormedAddCommGroup F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nT\u271d T'\u271d T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC\u271d C'\u271d C'' : \u211d\nT T' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc T' C'\nh : T = T'\nf : { x // x \u2208 Lp E 1 }\nthis : setToL1 hT = setToL1 hT'\n\u22a2 \u2191(setToL1 hT) f = \u2191(setToL1 hT') f", "state_after": "no goals"}, {"tactic": "rw [\u2190 this]", "annotated_tactic": ["rw [\u2190 this]", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : NormedAddCommGroup F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nT\u271d T'\u271d T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC\u271d C'\u271d C'' : \u211d\nT T' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc T' C'\nh : T = T'\nf\u271d : { x // x \u2208 Lp E 1 }\nf : { x // x \u2208 simpleFunc E 1 \u03bc }\nthis : \u2191(setToL1 hT') \u2191f = \u2191(setToL1SCLM \u03b1 E \u03bc hT) f\n\u22a2 \u2191(ContinuousLinearMap.comp (setToL1 hT') (coeToLp \u03b1 E \u211d)) f = \u2191(setToL1SCLM \u03b1 E \u03bc hT) f", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : NormedAddCommGroup F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nT\u271d T'\u271d T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC\u271d C'\u271d C'' : \u211d\nT T' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc T' C'\nh : T = T'\nf\u271d : { x // x \u2208 Lp E 1 }\nf : { x // x \u2208 simpleFunc E 1 \u03bc }\nthis : \u2191(setToL1 hT') \u2191f = \u2191(setToL1SCLM \u03b1 E \u03bc hT) f\n\u22a2 \u2191(ContinuousLinearMap.comp (setToL1 hT') (coeToLp \u03b1 E \u211d)) f = \u2191(setToL1 hT') \u2191f"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : NormedAddCommGroup F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nT\u271d T'\u271d T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC\u271d C'\u271d C'' : \u211d\nT T' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc T' C'\nh : T = T'\nf\u271d : { x // x \u2208 Lp E 1 }\nf : { x // x \u2208 simpleFunc E 1 \u03bc }\nthis : \u2191(setToL1 hT') \u2191f = \u2191(setToL1SCLM \u03b1 E \u03bc hT) f\n\u22a2 \u2191(ContinuousLinearMap.comp (setToL1 hT') (coeToLp \u03b1 E \u211d)) f = \u2191(setToL1 hT') \u2191f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "full_name": "List.leftpad_length", "start": [1633, 1], "end": [1635, 75], "traced_tactics": [{"tactic": "simp only [leftpad, length_append, length_replicate, Nat.sub_add_eq_max]", "annotated_tactic": ["simp only [<a>leftpad</a>, <a>length_append</a>, <a>length_replicate</a>, <a>Nat.sub_add_eq_max</a>]", [{"full_name": "List.leftpad", "def_path": "lake-packages/std/Std/Data/List/Basic.lean", "def_pos": [628, 5], "def_end_pos": [628, 12]}, {"full_name": "List.length_append", "def_path": "lake-packages/lean4/src/lean/Init/Data/List/Basic.lean", "def_pos": [790, 17], "def_end_pos": [790, 30]}, {"full_name": "List.length_replicate", "def_path": "lake-packages/lean4/src/lean/Init/Data/List/Basic.lean", "def_pos": [767, 17], "def_end_pos": [767, 33]}, {"full_name": "Nat.sub_add_eq_max", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [493, 19], "def_end_pos": [493, 33]}]], "state_before": "\u03b1 : Type u_1\nn : Nat\na : \u03b1\nl : List \u03b1\n\u22a2 length (leftpad n a l) = max n (length l)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean", "full_name": "MeasureTheory.integrable_of_intervalIntegral_norm_tendsto", "start": [531, 1], "end": [536, 63], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Pointwise.lean", "full_name": "Finset.mul_subset_iff_left", "start": [1846, 1], "end": [1847, 25], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "full_name": "MeasureTheory.Lp.coeFn_posPart", "start": [1254, 1], "end": [1255, 26], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Category/MeasCat.lean", "full_name": "MeasCat.coe_of", "start": [61, 1], "end": [62, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Finite.lean", "full_name": "Set.Finite.toFinset_eq_univ", "start": [298, 11], "end": [300, 36], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/UniformIntegrable.lean", "full_name": "MeasureTheory.tendsto_Lp_of_tendstoInMeasure", "start": [608, 1], "end": [617, 12], "traced_tactics": [{"tactic": "refine' tendsto_of_subseq_tendsto fun ns hns => _", "annotated_tactic": ["refine' <a>tendsto_of_subseq_tendsto</a> fun ns hns => _", [{"full_name": "Filter.tendsto_of_subseq_tendsto", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [1984, 9], "def_end_pos": [1984, 34]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhg : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : TendstoInMeasure \u03bc f atTop g\n\u22a2 Tendsto (fun n => snorm (f n - g) p \u03bc) atTop (\ud835\udcdd 0)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhg : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : TendstoInMeasure \u03bc f atTop g\nns : \u2115 \u2192 \u2115\nhns : Tendsto ns atTop atTop\n\u22a2 \u2203 ms, Tendsto (fun n => snorm (f (ns (ms n)) - g) p \u03bc) atTop (\ud835\udcdd 0)"}, {"tactic": "obtain \u27e8ms, _, hms'\u27e9 := TendstoInMeasure.exists_seq_tendsto_ae fun \u03b5 h\u03b5 => (hfg \u03b5 h\u03b5).comp hns", "annotated_tactic": ["obtain \u27e8ms, _, hms'\u27e9 := <a>TendstoInMeasure.exists_seq_tendsto_ae</a> fun \u03b5 h\u03b5 => (hfg \u03b5 h\u03b5).<a>comp</a> hns", [{"full_name": "MeasureTheory.TendstoInMeasure.exists_seq_tendsto_ae", "def_path": "Mathlib/MeasureTheory/Function/ConvergenceInMeasure.lean", "def_pos": [192, 9], "def_end_pos": [192, 47]}, {"full_name": "Filter.Tendsto.comp", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [3032, 9], "def_end_pos": [3032, 21]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhg : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : TendstoInMeasure \u03bc f atTop g\nns : \u2115 \u2192 \u2115\nhns : Tendsto ns atTop atTop\n\u22a2 \u2203 ms, Tendsto (fun n => snorm (f (ns (ms n)) - g) p \u03bc) atTop (\ud835\udcdd 0)", "state_after": "case intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhg : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : TendstoInMeasure \u03bc f atTop g\nns : \u2115 \u2192 \u2115\nhns : Tendsto ns atTop atTop\nms : \u2115 \u2192 \u2115\nleft\u271d : StrictMono ms\nhms' : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun i => f (ns (ms i)) x) atTop (\ud835\udcdd (g x))\n\u22a2 \u2203 ms, Tendsto (fun n => snorm (f (ns (ms n)) - g) p \u03bc) atTop (\ud835\udcdd 0)"}, {"tactic": "exact \u27e8ms,\n  tendsto_Lp_of_tendsto_ae \u03bc hp hp' (fun _ => hf _) hg (fun \u03b5 h\u03b5 =>\n    let \u27e8\u03b4, h\u03b4, h\u03b4'\u27e9 := hui h\u03b5\n    \u27e8\u03b4, h\u03b4, fun i s hs h\u03bcs => h\u03b4' _ s hs h\u03bcs\u27e9)\n    hms'\u27e9", "annotated_tactic": ["exact \u27e8ms,\n    <a>tendsto_Lp_of_tendsto_ae</a> \u03bc hp hp' (fun _ => hf _) hg (fun \u03b5 h\u03b5 =>\n      let \u27e8\u03b4, h\u03b4, h\u03b4'\u27e9 := hui h\u03b5\n      \u27e8\u03b4, h\u03b4, fun i s hs h\u03bcs => h\u03b4' _ s hs h\u03bcs\u27e9)\n      hms'\u27e9", [{"full_name": "MeasureTheory.tendsto_Lp_of_tendsto_ae", "def_path": "Mathlib/MeasureTheory/Function/UniformIntegrable.lean", "def_pos": [555, 9], "def_end_pos": [555, 33]}]], "state_before": "case intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhg : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : TendstoInMeasure \u03bc f atTop g\nns : \u2115 \u2192 \u2115\nhns : Tendsto ns atTop atTop\nms : \u2115 \u2192 \u2115\nleft\u271d : StrictMono ms\nhms' : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun i => f (ns (ms i)) x) atTop (\ud835\udcdd (g x))\n\u22a2 \u2203 ms, Tendsto (fun n => snorm (f (ns (ms n)) - g) p \u03bc) atTop (\ud835\udcdd 0)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Vector/Basic.lean", "full_name": "Vector.scanl_head", "start": [373, 1], "end": [379, 16], "traced_tactics": [{"tactic": "cases n", "annotated_tactic": ["cases n", []], "state_before": "n : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b2 \u2192 \u03b1 \u2192 \u03b2\nb : \u03b2\nv : Vector \u03b1 n\n\u22a2 head (scanl f b v) = b", "state_after": "case zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b2 \u2192 \u03b1 \u2192 \u03b2\nb : \u03b2\nv : Vector \u03b1 Nat.zero\n\u22a2 head (scanl f b v) = b\n\ncase succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b2 \u2192 \u03b1 \u2192 \u03b2\nb : \u03b2\nn\u271d : \u2115\nv : Vector \u03b1 (Nat.succ n\u271d)\n\u22a2 head (scanl f b v) = b"}, {"tactic": "have : v = nil := by simp only [Nat.zero_eq, eq_iff_true_of_subsingleton]", "annotated_tactic": ["have : v = <a>nil</a> := by simp only [<a>Nat.zero_eq</a>, <a>eq_iff_true_of_subsingleton</a>]", [{"full_name": "Vector.nil", "def_path": "Mathlib/Data/Vector.lean", "def_pos": [37, 5], "def_end_pos": [37, 8]}, {"full_name": "Nat.zero_eq", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [83, 17], "def_end_pos": [83, 24]}, {"full_name": "eq_iff_true_of_subsingleton", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [790, 9], "def_end_pos": [790, 36]}]], "state_before": "case zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b2 \u2192 \u03b1 \u2192 \u03b2\nb : \u03b2\nv : Vector \u03b1 Nat.zero\n\u22a2 head (scanl f b v) = b", "state_after": "case zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b2 \u2192 \u03b1 \u2192 \u03b2\nb : \u03b2\nv : Vector \u03b1 Nat.zero\nthis : v = nil\n\u22a2 head (scanl f b v) = b"}, {"tactic": "simp only [this, scanl_nil, head_cons]", "annotated_tactic": ["simp only [this, <a>scanl_nil</a>, <a>head_cons</a>]", [{"full_name": "Vector.scanl_nil", "def_path": "Mathlib/Data/Vector/Basic.lean", "def_pos": [327, 9], "def_end_pos": [327, 18]}, {"full_name": "Vector.head_cons", "def_path": "Mathlib/Data/Vector.lean", "def_pos": [64, 9], "def_end_pos": [64, 18]}]], "state_before": "case zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b2 \u2192 \u03b1 \u2192 \u03b2\nb : \u03b2\nv : Vector \u03b1 Nat.zero\nthis : v = nil\n\u22a2 head (scanl f b v) = b", "state_after": "no goals"}, {"tactic": "simp only [Nat.zero_eq, eq_iff_true_of_subsingleton]", "annotated_tactic": ["simp only [<a>Nat.zero_eq</a>, <a>eq_iff_true_of_subsingleton</a>]", [{"full_name": "Nat.zero_eq", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [83, 17], "def_end_pos": [83, 24]}, {"full_name": "eq_iff_true_of_subsingleton", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [790, 9], "def_end_pos": [790, 36]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b2 \u2192 \u03b1 \u2192 \u03b2\nb : \u03b2\nv : Vector \u03b1 Nat.zero\n\u22a2 v = nil", "state_after": "no goals"}, {"tactic": "rw [\u2190 cons_head_tail v]", "annotated_tactic": ["rw [\u2190 <a>cons_head_tail</a> v]", [{"full_name": "Vector.cons_head_tail", "def_path": "Mathlib/Data/Vector.lean", "def_pos": [81, 9], "def_end_pos": [81, 23]}]], "state_before": "case succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b2 \u2192 \u03b1 \u2192 \u03b2\nb : \u03b2\nn\u271d : \u2115\nv : Vector \u03b1 (Nat.succ n\u271d)\n\u22a2 head (scanl f b v) = b", "state_after": "case succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b2 \u2192 \u03b1 \u2192 \u03b2\nb : \u03b2\nn\u271d : \u2115\nv : Vector \u03b1 (Nat.succ n\u271d)\n\u22a2 head (scanl f b (head v ::\u1d65 tail v)) = b"}, {"tactic": "simp only [\u2190 get_zero, get_eq_get, toList_scanl, toList_cons, List.scanl, Fin.val_zero,\n  List.get]", "annotated_tactic": ["simp only [\u2190 <a>get_zero</a>, <a>get_eq_get</a>, <a>toList_scanl</a>, <a>toList_cons</a>, <a>List.scanl</a>, <a>Fin.val_zero</a>,\n      <a>List.get</a>]", [{"full_name": "Vector.get_zero", "def_path": "Mathlib/Data/Vector/Basic.lean", "def_pos": [264, 9], "def_end_pos": [264, 17]}, {"full_name": "Vector.get_eq_get", "def_path": "Mathlib/Data/Vector/Basic.lean", "def_pos": [116, 9], "def_end_pos": [116, 19]}, {"full_name": "Vector.toList_scanl", "def_path": "Mathlib/Data/Vector/Basic.lean", "def_pos": [355, 9], "def_end_pos": [355, 21]}, {"full_name": "Vector.toList_cons", "def_path": "Mathlib/Data/Vector.lean", "def_pos": [264, 9], "def_end_pos": [264, 20]}, {"full_name": "List.scanl", "def_path": "lake-packages/std/Std/Data/List/Basic.lean", "def_pos": [643, 13], "def_end_pos": [643, 18]}, {"full_name": "Fin.val_zero", "def_path": "lake-packages/std/Std/Data/Fin/Lemmas.lean", "def_pos": [105, 17], "def_end_pos": [105, 25]}, {"full_name": "List.get", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2261, 5], "def_end_pos": [2261, 13]}]], "state_before": "case succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b2 \u2192 \u03b1 \u2192 \u03b2\nb : \u03b2\nn\u271d : \u2115\nv : Vector \u03b1 (Nat.succ n\u271d)\n\u22a2 head (scanl f b (head v ::\u1d65 tail v)) = b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Sum.lean", "full_name": "Finset.disjSum_empty", "start": [45, 1], "end": [46, 39], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "full_name": "Int.lt_add_of_neg_lt_sub_left", "start": [1106, 11], "end": [1107, 63], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/AEEqOfIntegral.lean", "full_name": "MeasureTheory.ae_le_of_forall_set_lintegral_le_of_sigmaFinite", "start": [165, 1], "end": [222, 42], "traced_tactics": [{"tactic": "have A :\n  \u2200 (\u03b5 N : \u211d\u22650) (p : \u2115), 0 < \u03b5 \u2192 \u03bc ({x | g x + \u03b5 \u2264 f x \u2227 g x \u2264 N} \u2229 spanningSets \u03bc p) = 0 := by\n  intro \u03b5 N p \u03b5pos\n  let s := {x | g x + \u03b5 \u2264 f x \u2227 g x \u2264 N} \u2229 spanningSets \u03bc p\n  have s_meas : MeasurableSet s := by\n    have A : MeasurableSet {x | g x + \u03b5 \u2264 f x} := measurableSet_le (hg.add measurable_const) hf\n    have B : MeasurableSet {x | g x \u2264 N} := measurableSet_le hg measurable_const\n    exact (A.inter B).inter (measurable_spanningSets \u03bc p)\n  have s_lt_top : \u03bc s < \u221e :=\n    (measure_mono (Set.inter_subset_right _ _)).trans_lt (measure_spanningSets_lt_top \u03bc p)\n  have A : (\u222b\u207b x in s, g x \u2202\u03bc) + \u03b5 * \u03bc s \u2264 (\u222b\u207b x in s, g x \u2202\u03bc) + 0 :=\n    calc\n      (\u222b\u207b x in s, g x \u2202\u03bc) + \u03b5 * \u03bc s = (\u222b\u207b x in s, g x \u2202\u03bc) + \u222b\u207b _ in s, \u03b5 \u2202\u03bc := by\n        simp only [lintegral_const, Set.univ_inter, MeasurableSet.univ, Measure.restrict_apply]\n      _ = \u222b\u207b x in s, g x + \u03b5 \u2202\u03bc := (lintegral_add_right _ measurable_const).symm\n      _ \u2264 \u222b\u207b x in s, f x \u2202\u03bc :=\n        (set_lintegral_mono (hg.add measurable_const) hf fun x hx => hx.1.1)\n      _ \u2264 (\u222b\u207b x in s, g x \u2202\u03bc) + 0 := by rw [add_zero]; exact h s s_meas s_lt_top\n  have B : (\u222b\u207b x in s, g x \u2202\u03bc) \u2260 \u221e := by\n    apply ne_of_lt\n    calc\n      (\u222b\u207b x in s, g x \u2202\u03bc) \u2264 \u222b\u207b _ in s, N \u2202\u03bc :=\n        set_lintegral_mono hg measurable_const fun x hx => hx.1.2\n      _ = N * \u03bc s := by\n        simp only [lintegral_const, Set.univ_inter, MeasurableSet.univ, Measure.restrict_apply]\n      _ < \u221e := by\n        simp only [lt_top_iff_ne_top, s_lt_top.ne, and_false_iff, ENNReal.coe_ne_top,\n          ENNReal.mul_eq_top, Ne.def, not_false_iff, false_and_iff, or_self_iff]\n  have : (\u03b5 : \u211d\u22650\u221e) * \u03bc s \u2264 0 := ENNReal.le_of_add_le_add_left B A\n  simpa only [ENNReal.coe_eq_zero, nonpos_iff_eq_zero, mul_eq_zero, \u03b5pos.ne', false_or_iff]", "annotated_tactic": ["have A :\n    \u2200 (\u03b5 N : \u211d\u22650) (p : \u2115), 0 < \u03b5 \u2192 \u03bc ({x | g x + \u03b5 \u2264 f x \u2227 g x \u2264 N} \u2229 <a>spanningSets</a> \u03bc p) = 0 := by\n    intro \u03b5 N p \u03b5pos\n    let s := {x | g x + \u03b5 \u2264 f x \u2227 g x \u2264 N} \u2229 <a>spanningSets</a> \u03bc p\n    have s_meas : <a>MeasurableSet</a> s := by\n      have A : <a>MeasurableSet</a> {x | g x + \u03b5 \u2264 f x} := <a>measurableSet_le</a> (hg.add <a>measurable_const</a>) hf\n      have B : <a>MeasurableSet</a> {x | g x \u2264 N} := <a>measurableSet_le</a> hg <a>measurable_const</a>\n      exact (A.inter B).<a>inter</a> (<a>measurable_spanningSets</a> \u03bc p)\n    have s_lt_top : \u03bc s < \u221e :=\n      (<a>measure_mono</a> (<a>Set.inter_subset_right</a> _ _)).<a>trans_lt</a> (<a>measure_spanningSets_lt_top</a> \u03bc p)\n    have A : (\u222b\u207b x in s, g x \u2202\u03bc) + \u03b5 * \u03bc s \u2264 (\u222b\u207b x in s, g x \u2202\u03bc) + 0 :=\n      calc\n        (\u222b\u207b x in s, g x \u2202\u03bc) + \u03b5 * \u03bc s = (\u222b\u207b x in s, g x \u2202\u03bc) + \u222b\u207b _ in s, \u03b5 \u2202\u03bc := by\n          simp only [<a>lintegral_const</a>, <a>Set.univ_inter</a>, <a>MeasurableSet.univ</a>, <a>Measure.restrict_apply</a>]\n        _ = \u222b\u207b x in s, g x + \u03b5 \u2202\u03bc := (<a>lintegral_add_right</a> _ <a>measurable_const</a>).<a>symm</a>\n        _ \u2264 \u222b\u207b x in s, f x \u2202\u03bc :=\n          (<a>set_lintegral_mono</a> (hg.add <a>measurable_const</a>) hf fun x hx => hx.1.1)\n        _ \u2264 (\u222b\u207b x in s, g x \u2202\u03bc) + 0 := by rw [<a>add_zero</a>]; exact h s s_meas s_lt_top\n    have B : (\u222b\u207b x in s, g x \u2202\u03bc) \u2260 \u221e := by\n      apply <a>ne_of_lt</a>\n      calc\n        (\u222b\u207b x in s, g x \u2202\u03bc) \u2264 \u222b\u207b _ in s, N \u2202\u03bc :=\n          <a>set_lintegral_mono</a> hg <a>measurable_const</a> fun x hx => hx.1.2\n        _ = N * \u03bc s := by\n          simp only [<a>lintegral_const</a>, <a>Set.univ_inter</a>, <a>MeasurableSet.univ</a>, <a>Measure.restrict_apply</a>]\n        _ < \u221e := by\n          simp only [<a>lt_top_iff_ne_top</a>, s_lt_top.ne, <a>and_false_iff</a>, <a>ENNReal.coe_ne_top</a>,\n            <a>ENNReal.mul_eq_top</a>, <a>Ne.def</a>, <a>not_false_iff</a>, <a>false_and_iff</a>, <a>or_self_iff</a>]\n    have : (\u03b5 : \u211d\u22650\u221e) * \u03bc s \u2264 0 := <a>ENNReal.le_of_add_le_add_left</a> B A\n    simpa only [<a>ENNReal.coe_eq_zero</a>, <a>nonpos_iff_eq_zero</a>, <a>mul_eq_zero</a>, \u03b5pos.ne', <a>false_or_iff</a>]", [{"full_name": "MeasureTheory.spanningSets", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3316, 5], "def_end_pos": [3316, 17]}, {"full_name": "MeasureTheory.spanningSets", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3316, 5], "def_end_pos": [3316, 17]}, {"full_name": "MeasurableSet", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [64, 5], "def_end_pos": [64, 18]}, {"full_name": "MeasurableSet", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [64, 5], "def_end_pos": [64, 18]}, {"full_name": "measurableSet_le", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [559, 9], "def_end_pos": [559, 25]}, {"full_name": "measurable_const", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [570, 9], "def_end_pos": [570, 25]}, {"full_name": "MeasurableSet", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [64, 5], "def_end_pos": [64, 18]}, {"full_name": "measurableSet_le", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [559, 9], "def_end_pos": [559, 25]}, {"full_name": "measurable_const", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [570, 9], "def_end_pos": [570, 25]}, {"full_name": "MeasurableSet.inter", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [198, 19], "def_end_pos": [198, 38]}, {"full_name": "MeasureTheory.measurable_spanningSets", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3324, 9], "def_end_pos": [3324, 32]}, {"full_name": "MeasureTheory.measure_mono", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [193, 9], "def_end_pos": [193, 21]}, {"full_name": "Set.inter_subset_right", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [969, 9], "def_end_pos": [969, 27]}, {"full_name": "LE.le.trans_lt", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [124, 7], "def_end_pos": [124, 21]}, {"full_name": "MeasureTheory.measure_spanningSets_lt_top", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3329, 9], "def_end_pos": [3329, 36]}, {"full_name": "MeasureTheory.lintegral_const", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [136, 9], "def_end_pos": [136, 24]}, {"full_name": "Set.univ_inter", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1017, 9], "def_end_pos": [1017, 19]}, {"full_name": "MeasurableSet.univ", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [101, 19], "def_end_pos": [101, 37]}, {"full_name": "MeasureTheory.Measure.restrict_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1533, 9], "def_end_pos": [1533, 23]}, {"full_name": "MeasureTheory.lintegral_add_right", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [581, 9], "def_end_pos": [581, 28]}, {"full_name": "measurable_const", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [570, 9], "def_end_pos": [570, 25]}, {"full_name": "Eq.symm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [310, 9], "def_end_pos": [310, 16]}, {"full_name": "MeasureTheory.set_lintegral_mono", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [287, 9], "def_end_pos": [287, 27]}, {"full_name": "measurable_const", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [570, 9], "def_end_pos": [570, 25]}, {"full_name": "add_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [469, 3], "def_end_pos": [469, 14]}, {"full_name": "ne_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [101, 9], "def_end_pos": [101, 17]}, {"full_name": "MeasureTheory.set_lintegral_mono", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [287, 9], "def_end_pos": [287, 27]}, {"full_name": "measurable_const", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [570, 9], "def_end_pos": [570, 25]}, {"full_name": "MeasureTheory.lintegral_const", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [136, 9], "def_end_pos": [136, 24]}, {"full_name": "Set.univ_inter", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1017, 9], "def_end_pos": [1017, 19]}, {"full_name": "MeasurableSet.univ", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [101, 19], "def_end_pos": [101, 37]}, {"full_name": "MeasureTheory.Measure.restrict_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1533, 9], "def_end_pos": [1533, 23]}, {"full_name": "lt_top_iff_ne_top", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [173, 9], "def_end_pos": [173, 26]}, {"full_name": "and_false_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [149, 9], "def_end_pos": [149, 22]}, {"full_name": "ENNReal.coe_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [302, 17], "def_end_pos": [302, 27]}, {"full_name": "ENNReal.mul_eq_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [608, 9], "def_end_pos": [608, 19]}, {"full_name": "Ne.def", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [59, 9], "def_end_pos": [59, 15]}, {"full_name": "not_false_iff", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [82, 9], "def_end_pos": [82, 22]}, {"full_name": "false_and_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [151, 9], "def_end_pos": [151, 22]}, {"full_name": "or_self_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [190, 9], "def_end_pos": [190, 20]}, {"full_name": "ENNReal.le_of_add_le_add_left", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [785, 19], "def_end_pos": [785, 40]}, {"full_name": "ENNReal.coe_eq_zero", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [368, 28], "def_end_pos": [368, 39]}, {"full_name": "nonpos_iff_eq_zero", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [237, 3], "def_end_pos": [237, 14]}, {"full_name": "mul_eq_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [240, 9], "def_end_pos": [240, 20]}, {"full_name": "false_or_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [186, 9], "def_end_pos": [186, 21]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\n\u22a2 f \u2264\u1d50[\u03bc] g", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\nA : \u2200 (\u03b5 N : \u211d\u22650) (p : \u2115), 0 < \u03b5 \u2192 \u2191\u2191\u03bc ({x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p) = 0\n\u22a2 f \u2264\u1d50[\u03bc] g"}, {"tactic": "obtain \u27e8u, _, u_pos, u_lim\u27e9 :\n  \u2203 u : \u2115 \u2192 \u211d\u22650, StrictAnti u \u2227 (\u2200 n, 0 < u n) \u2227 Tendsto u atTop (nhds 0) :=\n  exists_seq_strictAnti_tendsto (0 : \u211d\u22650)", "annotated_tactic": ["obtain \u27e8u, _, u_pos, u_lim\u27e9 :\n    \u2203 u : \u2115 \u2192 \u211d\u22650, <a>StrictAnti</a> u \u2227 (\u2200 n, 0 < u n) \u2227 <a>Tendsto</a> u <a>atTop</a> (<a>nhds</a> 0) :=\n    <a>exists_seq_strictAnti_tendsto</a> (0 : \u211d\u22650)", [{"full_name": "StrictAnti", "def_path": "Mathlib/Order/Monotone/Basic.lean", "def_pos": [102, 5], "def_end_pos": [102, 15]}, {"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2939, 5], "def_end_pos": [2939, 12]}, {"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}, {"full_name": "nhds", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [831, 17], "def_end_pos": [831, 21]}, {"full_name": "exists_seq_strictAnti_tendsto", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [2258, 9], "def_end_pos": [2258, 38]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\nA : \u2200 (\u03b5 N : \u211d\u22650) (p : \u2115), 0 < \u03b5 \u2192 \u2191\u2191\u03bc ({x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p) = 0\n\u22a2 f \u2264\u1d50[\u03bc] g", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\nA : \u2200 (\u03b5 N : \u211d\u22650) (p : \u2115), 0 < \u03b5 \u2192 \u2191\u2191\u03bc ({x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p) = 0\nu : \u2115 \u2192 \u211d\u22650\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\n\u22a2 f \u2264\u1d50[\u03bc] g"}, {"tactic": "let s := fun n : \u2115 => {x | g x + u n \u2264 f x \u2227 g x \u2264 (n : \u211d\u22650)} \u2229 spanningSets \u03bc n", "annotated_tactic": ["let s := fun n : \u2115 => {x | g x + u n \u2264 f x \u2227 g x \u2264 (n : \u211d\u22650)} \u2229 <a>spanningSets</a> \u03bc n", [{"full_name": "MeasureTheory.spanningSets", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3316, 5], "def_end_pos": [3316, 17]}]], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\nA : \u2200 (\u03b5 N : \u211d\u22650) (p : \u2115), 0 < \u03b5 \u2192 \u2191\u2191\u03bc ({x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p) = 0\nu : \u2115 \u2192 \u211d\u22650\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\n\u22a2 f \u2264\u1d50[\u03bc] g", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\nA : \u2200 (\u03b5 N : \u211d\u22650) (p : \u2115), 0 < \u03b5 \u2192 \u2191\u2191\u03bc ({x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p) = 0\nu : \u2115 \u2192 \u211d\u22650\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\ns : \u2115 \u2192 Set \u03b1 := fun n => {x | g x + \u2191(u n) \u2264 f x \u2227 g x \u2264 \u2191\u2191n} \u2229 spanningSets \u03bc n\n\u22a2 f \u2264\u1d50[\u03bc] g"}, {"tactic": "have \u03bcs : \u2200 n, \u03bc (s n) = 0 := fun n => A _ _ _ (u_pos n)", "annotated_tactic": ["have \u03bcs : \u2200 n, \u03bc (s n) = 0 := fun n => A _ _ _ (u_pos n)", []], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\nA : \u2200 (\u03b5 N : \u211d\u22650) (p : \u2115), 0 < \u03b5 \u2192 \u2191\u2191\u03bc ({x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p) = 0\nu : \u2115 \u2192 \u211d\u22650\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\ns : \u2115 \u2192 Set \u03b1 := fun n => {x | g x + \u2191(u n) \u2264 f x \u2227 g x \u2264 \u2191\u2191n} \u2229 spanningSets \u03bc n\n\u22a2 f \u2264\u1d50[\u03bc] g", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\nA : \u2200 (\u03b5 N : \u211d\u22650) (p : \u2115), 0 < \u03b5 \u2192 \u2191\u2191\u03bc ({x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p) = 0\nu : \u2115 \u2192 \u211d\u22650\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\ns : \u2115 \u2192 Set \u03b1 := fun n => {x | g x + \u2191(u n) \u2264 f x \u2227 g x \u2264 \u2191\u2191n} \u2229 spanningSets \u03bc n\n\u03bcs : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) = 0\n\u22a2 f \u2264\u1d50[\u03bc] g"}, {"tactic": "have B : {x | f x \u2264 g x}\u1d9c \u2286 \u22c3 n, s n := by\n  intro x hx\n  simp only [Set.mem_compl_iff, Set.mem_setOf, not_le] at hx\n  have L1 : \u2200\u1da0 n in atTop, g x + u n \u2264 f x := by\n    have : Tendsto (fun n => g x + u n) atTop (\ud835\udcdd (g x + (0 : \u211d\u22650))) :=\n      tendsto_const_nhds.add (ENNReal.tendsto_coe.2 u_lim)\n    simp only [ENNReal.coe_zero, add_zero] at this\n    exact eventually_le_of_tendsto_lt hx this\n  have L2 : \u2200\u1da0 n : \u2115 in (atTop : Filter \u2115), g x \u2264 (n : \u211d\u22650) :=\n    haveI : Tendsto (fun n : \u2115 => ((n : \u211d\u22650) : \u211d\u22650\u221e)) atTop (\ud835\udcdd \u221e) := by\n      simp only [ENNReal.coe_nat]\n      exact ENNReal.tendsto_nat_nhds_top\n    eventually_ge_of_tendsto_gt (hx.trans_le le_top) this\n  apply Set.mem_iUnion.2\n  exact ((L1.and L2).and (eventually_mem_spanningSets \u03bc x)).exists", "annotated_tactic": ["have B : {x | f x \u2264 g x}\u1d9c \u2286 \u22c3 n, s n := by\n    intro x hx\n    simp only [<a>Set.mem_compl_iff</a>, <a>Set.mem_setOf</a>, <a>not_le</a>] at hx\n    have L1 : \u2200\u1da0 n in <a>atTop</a>, g x + u n \u2264 f x := by\n      have : <a>Tendsto</a> (fun n => g x + u n) <a>atTop</a> (\ud835\udcdd (g x + (0 : \u211d\u22650))) :=\n        tendsto_const_nhds.add (<a>ENNReal.tendsto_coe</a>.2 u_lim)\n      simp only [<a>ENNReal.coe_zero</a>, <a>add_zero</a>] at this\n      exact <a>eventually_le_of_tendsto_lt</a> hx this\n    have L2 : \u2200\u1da0 n : \u2115 in (<a>atTop</a> : <a>Filter</a> \u2115), g x \u2264 (n : \u211d\u22650) :=\n      haveI : <a>Tendsto</a> (fun n : \u2115 => ((n : \u211d\u22650) : \u211d\u22650\u221e)) <a>atTop</a> (\ud835\udcdd \u221e) := by\n        simp only [<a>ENNReal.coe_nat</a>]\n        exact <a>ENNReal.tendsto_nat_nhds_top</a>\n      <a>eventually_ge_of_tendsto_gt</a> (hx.trans_le <a>le_top</a>) this\n    apply <a>Set.mem_iUnion</a>.2\n    exact ((L1.and L2).<a>and</a> (<a>eventually_mem_spanningSets</a> \u03bc x)).<a>exists</a>", [{"full_name": "Set.mem_compl_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1658, 9], "def_end_pos": [1658, 22]}, {"full_name": "Set.mem_setOf", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [259, 9], "def_end_pos": [259, 18]}, {"full_name": "not_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [373, 9], "def_end_pos": [373, 15]}, {"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}, {"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2939, 5], "def_end_pos": [2939, 12]}, {"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}, {"full_name": "ENNReal.tendsto_coe", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [71, 9], "def_end_pos": [71, 20]}, {"full_name": "ENNReal.coe_zero", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [215, 28], "def_end_pos": [215, 36]}, {"full_name": "add_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [469, 3], "def_end_pos": [469, 14]}, {"full_name": "eventually_le_of_tendsto_lt", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [401, 9], "def_end_pos": [401, 36]}, {"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}, {"full_name": "Filter", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [91, 11], "def_end_pos": [91, 17]}, {"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2939, 5], "def_end_pos": [2939, 12]}, {"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}, {"full_name": "ENNReal.coe_nat", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [707, 9], "def_end_pos": [707, 16]}, {"full_name": "ENNReal.tendsto_nat_nhds_top", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [178, 9], "def_end_pos": [178, 29]}, {"full_name": "eventually_ge_of_tendsto_gt", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [406, 9], "def_end_pos": [406, 36]}, {"full_name": "le_top", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [98, 9], "def_end_pos": [98, 15]}, {"full_name": "Set.mem_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [201, 9], "def_end_pos": [201, 19]}, {"full_name": "Filter.Eventually.and", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1103, 19], "def_end_pos": [1103, 33]}, {"full_name": "MeasureTheory.eventually_mem_spanningSets", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3378, 9], "def_end_pos": [3378, 36]}, {"full_name": "Filter.Eventually.exists", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1308, 9], "def_end_pos": [1308, 26]}]], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\nA : \u2200 (\u03b5 N : \u211d\u22650) (p : \u2115), 0 < \u03b5 \u2192 \u2191\u2191\u03bc ({x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p) = 0\nu : \u2115 \u2192 \u211d\u22650\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\ns : \u2115 \u2192 Set \u03b1 := fun n => {x | g x + \u2191(u n) \u2264 f x \u2227 g x \u2264 \u2191\u2191n} \u2229 spanningSets \u03bc n\n\u03bcs : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) = 0\n\u22a2 f \u2264\u1d50[\u03bc] g", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\nA : \u2200 (\u03b5 N : \u211d\u22650) (p : \u2115), 0 < \u03b5 \u2192 \u2191\u2191\u03bc ({x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p) = 0\nu : \u2115 \u2192 \u211d\u22650\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\ns : \u2115 \u2192 Set \u03b1 := fun n => {x | g x + \u2191(u n) \u2264 f x \u2227 g x \u2264 \u2191\u2191n} \u2229 spanningSets \u03bc n\n\u03bcs : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) = 0\nB : {x | f x \u2264 g x}\u1d9c \u2286 \u22c3 n, s n\n\u22a2 f \u2264\u1d50[\u03bc] g"}, {"tactic": "refine' le_antisymm _ bot_le", "annotated_tactic": ["refine' <a>le_antisymm</a> _ <a>bot_le</a>", [{"full_name": "le_antisymm", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [188, 9], "def_end_pos": [188, 20]}, {"full_name": "bot_le", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [256, 9], "def_end_pos": [256, 15]}]], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\nA : \u2200 (\u03b5 N : \u211d\u22650) (p : \u2115), 0 < \u03b5 \u2192 \u2191\u2191\u03bc ({x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p) = 0\nu : \u2115 \u2192 \u211d\u22650\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\ns : \u2115 \u2192 Set \u03b1 := fun n => {x | g x + \u2191(u n) \u2264 f x \u2227 g x \u2264 \u2191\u2191n} \u2229 spanningSets \u03bc n\n\u03bcs : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) = 0\nB : {x | f x \u2264 g x}\u1d9c \u2286 \u22c3 n, s n\n\u22a2 f \u2264\u1d50[\u03bc] g", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\nA : \u2200 (\u03b5 N : \u211d\u22650) (p : \u2115), 0 < \u03b5 \u2192 \u2191\u2191\u03bc ({x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p) = 0\nu : \u2115 \u2192 \u211d\u22650\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\ns : \u2115 \u2192 Set \u03b1 := fun n => {x | g x + \u2191(u n) \u2264 f x \u2227 g x \u2264 \u2191\u2191n} \u2229 spanningSets \u03bc n\n\u03bcs : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) = 0\nB : {x | f x \u2264 g x}\u1d9c \u2286 \u22c3 n, s n\n\u22a2 \u2191\u2191\u03bc {x | (fun x => f x \u2264 g x) x}\u1d9c \u2264 0"}, {"tactic": "calc\n  \u03bc {x : \u03b1 | (fun x : \u03b1 => f x \u2264 g x) x}\u1d9c \u2264 \u03bc (\u22c3 n, s n) := measure_mono B\n  _ \u2264 \u2211' n, \u03bc (s n) := (measure_iUnion_le _)\n  _ = 0 := by simp only [\u03bcs, tsum_zero]", "annotated_tactic": ["calc\n    \u03bc {x : \u03b1 | (fun x : \u03b1 => f x \u2264 g x) x}\u1d9c \u2264 \u03bc (\u22c3 n, s n) := <a>measure_mono</a> B\n    _ \u2264 \u2211' n, \u03bc (s n) := (<a>measure_iUnion_le</a> _)\n    _ = 0 := by simp only [\u03bcs, <a>tsum_zero</a>]", [{"full_name": "MeasureTheory.measure_mono", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [193, 9], "def_end_pos": [193, 21]}, {"full_name": "MeasureTheory.measure_iUnion_le", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [240, 9], "def_end_pos": [240, 26]}, {"full_name": "tsum_zero", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/Basic.lean", "def_pos": [489, 9], "def_end_pos": [489, 18]}]], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\nA : \u2200 (\u03b5 N : \u211d\u22650) (p : \u2115), 0 < \u03b5 \u2192 \u2191\u2191\u03bc ({x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p) = 0\nu : \u2115 \u2192 \u211d\u22650\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\ns : \u2115 \u2192 Set \u03b1 := fun n => {x | g x + \u2191(u n) \u2264 f x \u2227 g x \u2264 \u2191\u2191n} \u2229 spanningSets \u03bc n\n\u03bcs : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) = 0\nB : {x | f x \u2264 g x}\u1d9c \u2286 \u22c3 n, s n\n\u22a2 \u2191\u2191\u03bc {x | (fun x => f x \u2264 g x) x}\u1d9c \u2264 0", "state_after": "no goals"}, {"tactic": "intro \u03b5 N p \u03b5pos", "annotated_tactic": ["intro \u03b5 N p \u03b5pos", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\n\u22a2 \u2200 (\u03b5 N : \u211d\u22650) (p : \u2115), 0 < \u03b5 \u2192 \u2191\u2191\u03bc ({x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p) = 0", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np\u271d : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\n\u03b5 N : \u211d\u22650\np : \u2115\n\u03b5pos : 0 < \u03b5\n\u22a2 \u2191\u2191\u03bc ({x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p) = 0"}, {"tactic": "let s := {x | g x + \u03b5 \u2264 f x \u2227 g x \u2264 N} \u2229 spanningSets \u03bc p", "annotated_tactic": ["let s := {x | g x + \u03b5 \u2264 f x \u2227 g x \u2264 N} \u2229 <a>spanningSets</a> \u03bc p", [{"full_name": "MeasureTheory.spanningSets", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3316, 5], "def_end_pos": [3316, 17]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np\u271d : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\n\u03b5 N : \u211d\u22650\np : \u2115\n\u03b5pos : 0 < \u03b5\n\u22a2 \u2191\u2191\u03bc ({x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p) = 0", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np\u271d : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\n\u03b5 N : \u211d\u22650\np : \u2115\n\u03b5pos : 0 < \u03b5\ns : Set \u03b1 := {x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p\n\u22a2 \u2191\u2191\u03bc ({x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p) = 0"}, {"tactic": "have s_meas : MeasurableSet s := by\n  have A : MeasurableSet {x | g x + \u03b5 \u2264 f x} := measurableSet_le (hg.add measurable_const) hf\n  have B : MeasurableSet {x | g x \u2264 N} := measurableSet_le hg measurable_const\n  exact (A.inter B).inter (measurable_spanningSets \u03bc p)", "annotated_tactic": ["have s_meas : <a>MeasurableSet</a> s := by\n      have A : <a>MeasurableSet</a> {x | g x + \u03b5 \u2264 f x} := <a>measurableSet_le</a> (hg.add <a>measurable_const</a>) hf\n      have B : <a>MeasurableSet</a> {x | g x \u2264 N} := <a>measurableSet_le</a> hg <a>measurable_const</a>\n      exact (A.inter B).<a>inter</a> (<a>measurable_spanningSets</a> \u03bc p)", [{"full_name": "MeasurableSet", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [64, 5], "def_end_pos": [64, 18]}, {"full_name": "MeasurableSet", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [64, 5], "def_end_pos": [64, 18]}, {"full_name": "measurableSet_le", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [559, 9], "def_end_pos": [559, 25]}, {"full_name": "measurable_const", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [570, 9], "def_end_pos": [570, 25]}, {"full_name": "MeasurableSet", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [64, 5], "def_end_pos": [64, 18]}, {"full_name": "measurableSet_le", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [559, 9], "def_end_pos": [559, 25]}, {"full_name": "measurable_const", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [570, 9], "def_end_pos": [570, 25]}, {"full_name": "MeasurableSet.inter", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [198, 19], "def_end_pos": [198, 38]}, {"full_name": "MeasureTheory.measurable_spanningSets", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3324, 9], "def_end_pos": [3324, 32]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np\u271d : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\n\u03b5 N : \u211d\u22650\np : \u2115\n\u03b5pos : 0 < \u03b5\ns : Set \u03b1 := {x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p\n\u22a2 \u2191\u2191\u03bc ({x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p) = 0", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np\u271d : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\n\u03b5 N : \u211d\u22650\np : \u2115\n\u03b5pos : 0 < \u03b5\ns : Set \u03b1 := {x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p\ns_meas : MeasurableSet s\n\u22a2 \u2191\u2191\u03bc ({x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p) = 0"}, {"tactic": "have s_lt_top : \u03bc s < \u221e :=\n  (measure_mono (Set.inter_subset_right _ _)).trans_lt (measure_spanningSets_lt_top \u03bc p)", "annotated_tactic": ["have s_lt_top : \u03bc s < \u221e :=\n      (<a>measure_mono</a> (<a>Set.inter_subset_right</a> _ _)).<a>trans_lt</a> (<a>measure_spanningSets_lt_top</a> \u03bc p)", [{"full_name": "MeasureTheory.measure_mono", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [193, 9], "def_end_pos": [193, 21]}, {"full_name": "Set.inter_subset_right", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [969, 9], "def_end_pos": [969, 27]}, {"full_name": "LE.le.trans_lt", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [124, 7], "def_end_pos": [124, 21]}, {"full_name": "MeasureTheory.measure_spanningSets_lt_top", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3329, 9], "def_end_pos": [3329, 36]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np\u271d : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\n\u03b5 N : \u211d\u22650\np : \u2115\n\u03b5pos : 0 < \u03b5\ns : Set \u03b1 := {x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p\ns_meas : MeasurableSet s\n\u22a2 \u2191\u2191\u03bc ({x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p) = 0", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np\u271d : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\n\u03b5 N : \u211d\u22650\np : \u2115\n\u03b5pos : 0 < \u03b5\ns : Set \u03b1 := {x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p\ns_meas : MeasurableSet s\ns_lt_top : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u2191\u2191\u03bc ({x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p) = 0"}, {"tactic": "have A : (\u222b\u207b x in s, g x \u2202\u03bc) + \u03b5 * \u03bc s \u2264 (\u222b\u207b x in s, g x \u2202\u03bc) + 0 :=\n  calc\n    (\u222b\u207b x in s, g x \u2202\u03bc) + \u03b5 * \u03bc s = (\u222b\u207b x in s, g x \u2202\u03bc) + \u222b\u207b _ in s, \u03b5 \u2202\u03bc := by\n      simp only [lintegral_const, Set.univ_inter, MeasurableSet.univ, Measure.restrict_apply]\n    _ = \u222b\u207b x in s, g x + \u03b5 \u2202\u03bc := (lintegral_add_right _ measurable_const).symm\n    _ \u2264 \u222b\u207b x in s, f x \u2202\u03bc :=\n      (set_lintegral_mono (hg.add measurable_const) hf fun x hx => hx.1.1)\n    _ \u2264 (\u222b\u207b x in s, g x \u2202\u03bc) + 0 := by rw [add_zero]; exact h s s_meas s_lt_top", "annotated_tactic": ["have A : (\u222b\u207b x in s, g x \u2202\u03bc) + \u03b5 * \u03bc s \u2264 (\u222b\u207b x in s, g x \u2202\u03bc) + 0 :=\n      calc\n        (\u222b\u207b x in s, g x \u2202\u03bc) + \u03b5 * \u03bc s = (\u222b\u207b x in s, g x \u2202\u03bc) + \u222b\u207b _ in s, \u03b5 \u2202\u03bc := by\n          simp only [<a>lintegral_const</a>, <a>Set.univ_inter</a>, <a>MeasurableSet.univ</a>, <a>Measure.restrict_apply</a>]\n        _ = \u222b\u207b x in s, g x + \u03b5 \u2202\u03bc := (<a>lintegral_add_right</a> _ <a>measurable_const</a>).<a>symm</a>\n        _ \u2264 \u222b\u207b x in s, f x \u2202\u03bc :=\n          (<a>set_lintegral_mono</a> (hg.add <a>measurable_const</a>) hf fun x hx => hx.1.1)\n        _ \u2264 (\u222b\u207b x in s, g x \u2202\u03bc) + 0 := by rw [<a>add_zero</a>]; exact h s s_meas s_lt_top", [{"full_name": "MeasureTheory.lintegral_const", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [136, 9], "def_end_pos": [136, 24]}, {"full_name": "Set.univ_inter", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1017, 9], "def_end_pos": [1017, 19]}, {"full_name": "MeasurableSet.univ", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [101, 19], "def_end_pos": [101, 37]}, {"full_name": "MeasureTheory.Measure.restrict_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1533, 9], "def_end_pos": [1533, 23]}, {"full_name": "MeasureTheory.lintegral_add_right", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [581, 9], "def_end_pos": [581, 28]}, {"full_name": "measurable_const", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [570, 9], "def_end_pos": [570, 25]}, {"full_name": "Eq.symm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [310, 9], "def_end_pos": [310, 16]}, {"full_name": "MeasureTheory.set_lintegral_mono", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [287, 9], "def_end_pos": [287, 27]}, {"full_name": "measurable_const", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [570, 9], "def_end_pos": [570, 25]}, {"full_name": "add_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [469, 3], "def_end_pos": [469, 14]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np\u271d : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\n\u03b5 N : \u211d\u22650\np : \u2115\n\u03b5pos : 0 < \u03b5\ns : Set \u03b1 := {x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p\ns_meas : MeasurableSet s\ns_lt_top : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u2191\u2191\u03bc ({x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p) = 0", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np\u271d : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\n\u03b5 N : \u211d\u22650\np : \u2115\n\u03b5pos : 0 < \u03b5\ns : Set \u03b1 := {x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p\ns_meas : MeasurableSet s\ns_lt_top : \u2191\u2191\u03bc s < \u22a4\nA : \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc + \u2191\u03b5 * \u2191\u2191\u03bc s \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc + 0\n\u22a2 \u2191\u2191\u03bc ({x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p) = 0"}, {"tactic": "have B : (\u222b\u207b x in s, g x \u2202\u03bc) \u2260 \u221e := by\n  apply ne_of_lt\n  calc\n    (\u222b\u207b x in s, g x \u2202\u03bc) \u2264 \u222b\u207b _ in s, N \u2202\u03bc :=\n      set_lintegral_mono hg measurable_const fun x hx => hx.1.2\n    _ = N * \u03bc s := by\n      simp only [lintegral_const, Set.univ_inter, MeasurableSet.univ, Measure.restrict_apply]\n    _ < \u221e := by\n      simp only [lt_top_iff_ne_top, s_lt_top.ne, and_false_iff, ENNReal.coe_ne_top,\n        ENNReal.mul_eq_top, Ne.def, not_false_iff, false_and_iff, or_self_iff]", "annotated_tactic": ["have B : (\u222b\u207b x in s, g x \u2202\u03bc) \u2260 \u221e := by\n      apply <a>ne_of_lt</a>\n      calc\n        (\u222b\u207b x in s, g x \u2202\u03bc) \u2264 \u222b\u207b _ in s, N \u2202\u03bc :=\n          <a>set_lintegral_mono</a> hg <a>measurable_const</a> fun x hx => hx.1.2\n        _ = N * \u03bc s := by\n          simp only [<a>lintegral_const</a>, <a>Set.univ_inter</a>, <a>MeasurableSet.univ</a>, <a>Measure.restrict_apply</a>]\n        _ < \u221e := by\n          simp only [<a>lt_top_iff_ne_top</a>, s_lt_top.ne, <a>and_false_iff</a>, <a>ENNReal.coe_ne_top</a>,\n            <a>ENNReal.mul_eq_top</a>, <a>Ne.def</a>, <a>not_false_iff</a>, <a>false_and_iff</a>, <a>or_self_iff</a>]", [{"full_name": "ne_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [101, 9], "def_end_pos": [101, 17]}, {"full_name": "MeasureTheory.set_lintegral_mono", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [287, 9], "def_end_pos": [287, 27]}, {"full_name": "measurable_const", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [570, 9], "def_end_pos": [570, 25]}, {"full_name": "MeasureTheory.lintegral_const", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [136, 9], "def_end_pos": [136, 24]}, {"full_name": "Set.univ_inter", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1017, 9], "def_end_pos": [1017, 19]}, {"full_name": "MeasurableSet.univ", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [101, 19], "def_end_pos": [101, 37]}, {"full_name": "MeasureTheory.Measure.restrict_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1533, 9], "def_end_pos": [1533, 23]}, {"full_name": "lt_top_iff_ne_top", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [173, 9], "def_end_pos": [173, 26]}, {"full_name": "and_false_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [149, 9], "def_end_pos": [149, 22]}, {"full_name": "ENNReal.coe_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [302, 17], "def_end_pos": [302, 27]}, {"full_name": "ENNReal.mul_eq_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [608, 9], "def_end_pos": [608, 19]}, {"full_name": "Ne.def", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [59, 9], "def_end_pos": [59, 15]}, {"full_name": "not_false_iff", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [82, 9], "def_end_pos": [82, 22]}, {"full_name": "false_and_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [151, 9], "def_end_pos": [151, 22]}, {"full_name": "or_self_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [190, 9], "def_end_pos": [190, 20]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np\u271d : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\n\u03b5 N : \u211d\u22650\np : \u2115\n\u03b5pos : 0 < \u03b5\ns : Set \u03b1 := {x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p\ns_meas : MeasurableSet s\ns_lt_top : \u2191\u2191\u03bc s < \u22a4\nA : \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc + \u2191\u03b5 * \u2191\u2191\u03bc s \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc + 0\n\u22a2 \u2191\u2191\u03bc ({x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p) = 0", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np\u271d : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\n\u03b5 N : \u211d\u22650\np : \u2115\n\u03b5pos : 0 < \u03b5\ns : Set \u03b1 := {x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p\ns_meas : MeasurableSet s\ns_lt_top : \u2191\u2191\u03bc s < \u22a4\nA : \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc + \u2191\u03b5 * \u2191\u2191\u03bc s \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc + 0\nB : \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc \u2260 \u22a4\n\u22a2 \u2191\u2191\u03bc ({x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p) = 0"}, {"tactic": "have : (\u03b5 : \u211d\u22650\u221e) * \u03bc s \u2264 0 := ENNReal.le_of_add_le_add_left B A", "annotated_tactic": ["have : (\u03b5 : \u211d\u22650\u221e) * \u03bc s \u2264 0 := <a>ENNReal.le_of_add_le_add_left</a> B A", [{"full_name": "ENNReal.le_of_add_le_add_left", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [785, 19], "def_end_pos": [785, 40]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np\u271d : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\n\u03b5 N : \u211d\u22650\np : \u2115\n\u03b5pos : 0 < \u03b5\ns : Set \u03b1 := {x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p\ns_meas : MeasurableSet s\ns_lt_top : \u2191\u2191\u03bc s < \u22a4\nA : \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc + \u2191\u03b5 * \u2191\u2191\u03bc s \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc + 0\nB : \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc \u2260 \u22a4\n\u22a2 \u2191\u2191\u03bc ({x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p) = 0", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np\u271d : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\n\u03b5 N : \u211d\u22650\np : \u2115\n\u03b5pos : 0 < \u03b5\ns : Set \u03b1 := {x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p\ns_meas : MeasurableSet s\ns_lt_top : \u2191\u2191\u03bc s < \u22a4\nA : \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc + \u2191\u03b5 * \u2191\u2191\u03bc s \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc + 0\nB : \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc \u2260 \u22a4\nthis : \u2191\u03b5 * \u2191\u2191\u03bc s \u2264 0\n\u22a2 \u2191\u2191\u03bc ({x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p) = 0"}, {"tactic": "simpa only [ENNReal.coe_eq_zero, nonpos_iff_eq_zero, mul_eq_zero, \u03b5pos.ne', false_or_iff]", "annotated_tactic": ["simpa only [<a>ENNReal.coe_eq_zero</a>, <a>nonpos_iff_eq_zero</a>, <a>mul_eq_zero</a>, \u03b5pos.ne', <a>false_or_iff</a>]", [{"full_name": "ENNReal.coe_eq_zero", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [368, 28], "def_end_pos": [368, 39]}, {"full_name": "nonpos_iff_eq_zero", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [237, 3], "def_end_pos": [237, 14]}, {"full_name": "mul_eq_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [240, 9], "def_end_pos": [240, 20]}, {"full_name": "false_or_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [186, 9], "def_end_pos": [186, 21]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np\u271d : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\n\u03b5 N : \u211d\u22650\np : \u2115\n\u03b5pos : 0 < \u03b5\ns : Set \u03b1 := {x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p\ns_meas : MeasurableSet s\ns_lt_top : \u2191\u2191\u03bc s < \u22a4\nA : \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc + \u2191\u03b5 * \u2191\u2191\u03bc s \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc + 0\nB : \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc \u2260 \u22a4\nthis : \u2191\u03b5 * \u2191\u2191\u03bc s \u2264 0\n\u22a2 \u2191\u2191\u03bc ({x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p) = 0", "state_after": "no goals"}, {"tactic": "have A : MeasurableSet {x | g x + \u03b5 \u2264 f x} := measurableSet_le (hg.add measurable_const) hf", "annotated_tactic": ["have A : <a>MeasurableSet</a> {x | g x + \u03b5 \u2264 f x} := <a>measurableSet_le</a> (hg.add <a>measurable_const</a>) hf", [{"full_name": "MeasurableSet", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [64, 5], "def_end_pos": [64, 18]}, {"full_name": "measurableSet_le", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [559, 9], "def_end_pos": [559, 25]}, {"full_name": "measurable_const", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [570, 9], "def_end_pos": [570, 25]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np\u271d : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\n\u03b5 N : \u211d\u22650\np : \u2115\n\u03b5pos : 0 < \u03b5\ns : Set \u03b1 := {x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p\n\u22a2 MeasurableSet s", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np\u271d : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\n\u03b5 N : \u211d\u22650\np : \u2115\n\u03b5pos : 0 < \u03b5\ns : Set \u03b1 := {x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p\nA : MeasurableSet {x | g x + \u2191\u03b5 \u2264 f x}\n\u22a2 MeasurableSet s"}, {"tactic": "have B : MeasurableSet {x | g x \u2264 N} := measurableSet_le hg measurable_const", "annotated_tactic": ["have B : <a>MeasurableSet</a> {x | g x \u2264 N} := <a>measurableSet_le</a> hg <a>measurable_const</a>", [{"full_name": "MeasurableSet", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [64, 5], "def_end_pos": [64, 18]}, {"full_name": "measurableSet_le", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [559, 9], "def_end_pos": [559, 25]}, {"full_name": "measurable_const", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [570, 9], "def_end_pos": [570, 25]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np\u271d : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\n\u03b5 N : \u211d\u22650\np : \u2115\n\u03b5pos : 0 < \u03b5\ns : Set \u03b1 := {x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p\nA : MeasurableSet {x | g x + \u2191\u03b5 \u2264 f x}\n\u22a2 MeasurableSet s", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np\u271d : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\n\u03b5 N : \u211d\u22650\np : \u2115\n\u03b5pos : 0 < \u03b5\ns : Set \u03b1 := {x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p\nA : MeasurableSet {x | g x + \u2191\u03b5 \u2264 f x}\nB : MeasurableSet {x | g x \u2264 \u2191N}\n\u22a2 MeasurableSet s"}, {"tactic": "exact (A.inter B).inter (measurable_spanningSets \u03bc p)", "annotated_tactic": ["exact (A.inter B).<a>inter</a> (<a>measurable_spanningSets</a> \u03bc p)", [{"full_name": "MeasurableSet.inter", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [198, 19], "def_end_pos": [198, 38]}, {"full_name": "MeasureTheory.measurable_spanningSets", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3324, 9], "def_end_pos": [3324, 32]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np\u271d : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\n\u03b5 N : \u211d\u22650\np : \u2115\n\u03b5pos : 0 < \u03b5\ns : Set \u03b1 := {x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p\nA : MeasurableSet {x | g x + \u2191\u03b5 \u2264 f x}\nB : MeasurableSet {x | g x \u2264 \u2191N}\n\u22a2 MeasurableSet s", "state_after": "no goals"}, {"tactic": "simp only [lintegral_const, Set.univ_inter, MeasurableSet.univ, Measure.restrict_apply]", "annotated_tactic": ["simp only [<a>lintegral_const</a>, <a>Set.univ_inter</a>, <a>MeasurableSet.univ</a>, <a>Measure.restrict_apply</a>]", [{"full_name": "MeasureTheory.lintegral_const", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [136, 9], "def_end_pos": [136, 24]}, {"full_name": "Set.univ_inter", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1017, 9], "def_end_pos": [1017, 19]}, {"full_name": "MeasurableSet.univ", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [101, 19], "def_end_pos": [101, 37]}, {"full_name": "MeasureTheory.Measure.restrict_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1533, 9], "def_end_pos": [1533, 23]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np\u271d : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\n\u03b5 N : \u211d\u22650\np : \u2115\n\u03b5pos : 0 < \u03b5\ns : Set \u03b1 := {x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p\ns_meas : MeasurableSet s\ns_lt_top : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc + \u2191\u03b5 * \u2191\u2191\u03bc s = \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc + \u222b\u207b (x : \u03b1) in s, \u2191\u03b5 \u2202\u03bc", "state_after": "no goals"}, {"tactic": "rw [add_zero]", "annotated_tactic": ["rw [<a>add_zero</a>]", [{"full_name": "add_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [469, 3], "def_end_pos": [469, 14]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np\u271d : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\n\u03b5 N : \u211d\u22650\np : \u2115\n\u03b5pos : 0 < \u03b5\ns : Set \u03b1 := {x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p\ns_meas : MeasurableSet s\ns_lt_top : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc + 0", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np\u271d : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\n\u03b5 N : \u211d\u22650\np : \u2115\n\u03b5pos : 0 < \u03b5\ns : Set \u03b1 := {x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p\ns_meas : MeasurableSet s\ns_lt_top : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc"}, {"tactic": "exact h s s_meas s_lt_top", "annotated_tactic": ["exact h s s_meas s_lt_top", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np\u271d : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\n\u03b5 N : \u211d\u22650\np : \u2115\n\u03b5pos : 0 < \u03b5\ns : Set \u03b1 := {x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p\ns_meas : MeasurableSet s\ns_lt_top : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc", "state_after": "no goals"}, {"tactic": "apply ne_of_lt", "annotated_tactic": ["apply <a>ne_of_lt</a>", [{"full_name": "ne_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [101, 9], "def_end_pos": [101, 17]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np\u271d : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\n\u03b5 N : \u211d\u22650\np : \u2115\n\u03b5pos : 0 < \u03b5\ns : Set \u03b1 := {x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p\ns_meas : MeasurableSet s\ns_lt_top : \u2191\u2191\u03bc s < \u22a4\nA : \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc + \u2191\u03b5 * \u2191\u2191\u03bc s \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc + 0\n\u22a2 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc \u2260 \u22a4", "state_after": "case h\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np\u271d : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\n\u03b5 N : \u211d\u22650\np : \u2115\n\u03b5pos : 0 < \u03b5\ns : Set \u03b1 := {x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p\ns_meas : MeasurableSet s\ns_lt_top : \u2191\u2191\u03bc s < \u22a4\nA : \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc + \u2191\u03b5 * \u2191\u2191\u03bc s \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc + 0\n\u22a2 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc < \u22a4"}, {"tactic": "calc\n  (\u222b\u207b x in s, g x \u2202\u03bc) \u2264 \u222b\u207b _ in s, N \u2202\u03bc :=\n    set_lintegral_mono hg measurable_const fun x hx => hx.1.2\n  _ = N * \u03bc s := by\n    simp only [lintegral_const, Set.univ_inter, MeasurableSet.univ, Measure.restrict_apply]\n  _ < \u221e := by\n    simp only [lt_top_iff_ne_top, s_lt_top.ne, and_false_iff, ENNReal.coe_ne_top,\n      ENNReal.mul_eq_top, Ne.def, not_false_iff, false_and_iff, or_self_iff]", "annotated_tactic": ["calc\n        (\u222b\u207b x in s, g x \u2202\u03bc) \u2264 \u222b\u207b _ in s, N \u2202\u03bc :=\n          <a>set_lintegral_mono</a> hg <a>measurable_const</a> fun x hx => hx.1.2\n        _ = N * \u03bc s := by\n          simp only [<a>lintegral_const</a>, <a>Set.univ_inter</a>, <a>MeasurableSet.univ</a>, <a>Measure.restrict_apply</a>]\n        _ < \u221e := by\n          simp only [<a>lt_top_iff_ne_top</a>, s_lt_top.ne, <a>and_false_iff</a>, <a>ENNReal.coe_ne_top</a>,\n            <a>ENNReal.mul_eq_top</a>, <a>Ne.def</a>, <a>not_false_iff</a>, <a>false_and_iff</a>, <a>or_self_iff</a>]", [{"full_name": "MeasureTheory.set_lintegral_mono", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [287, 9], "def_end_pos": [287, 27]}, {"full_name": "measurable_const", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [570, 9], "def_end_pos": [570, 25]}, {"full_name": "MeasureTheory.lintegral_const", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [136, 9], "def_end_pos": [136, 24]}, {"full_name": "Set.univ_inter", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1017, 9], "def_end_pos": [1017, 19]}, {"full_name": "MeasurableSet.univ", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [101, 19], "def_end_pos": [101, 37]}, {"full_name": "MeasureTheory.Measure.restrict_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1533, 9], "def_end_pos": [1533, 23]}, {"full_name": "lt_top_iff_ne_top", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [173, 9], "def_end_pos": [173, 26]}, {"full_name": "and_false_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [149, 9], "def_end_pos": [149, 22]}, {"full_name": "ENNReal.coe_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [302, 17], "def_end_pos": [302, 27]}, {"full_name": "ENNReal.mul_eq_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [608, 9], "def_end_pos": [608, 19]}, {"full_name": "Ne.def", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [59, 9], "def_end_pos": [59, 15]}, {"full_name": "not_false_iff", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [82, 9], "def_end_pos": [82, 22]}, {"full_name": "false_and_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [151, 9], "def_end_pos": [151, 22]}, {"full_name": "or_self_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [190, 9], "def_end_pos": [190, 20]}]], "state_before": "case h\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np\u271d : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\n\u03b5 N : \u211d\u22650\np : \u2115\n\u03b5pos : 0 < \u03b5\ns : Set \u03b1 := {x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p\ns_meas : MeasurableSet s\ns_lt_top : \u2191\u2191\u03bc s < \u22a4\nA : \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc + \u2191\u03b5 * \u2191\u2191\u03bc s \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc + 0\n\u22a2 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc < \u22a4", "state_after": "no goals"}, {"tactic": "simp only [lintegral_const, Set.univ_inter, MeasurableSet.univ, Measure.restrict_apply]", "annotated_tactic": ["simp only [<a>lintegral_const</a>, <a>Set.univ_inter</a>, <a>MeasurableSet.univ</a>, <a>Measure.restrict_apply</a>]", [{"full_name": "MeasureTheory.lintegral_const", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [136, 9], "def_end_pos": [136, 24]}, {"full_name": "Set.univ_inter", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1017, 9], "def_end_pos": [1017, 19]}, {"full_name": "MeasurableSet.univ", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [101, 19], "def_end_pos": [101, 37]}, {"full_name": "MeasureTheory.Measure.restrict_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1533, 9], "def_end_pos": [1533, 23]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np\u271d : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\n\u03b5 N : \u211d\u22650\np : \u2115\n\u03b5pos : 0 < \u03b5\ns : Set \u03b1 := {x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p\ns_meas : MeasurableSet s\ns_lt_top : \u2191\u2191\u03bc s < \u22a4\nA : \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc + \u2191\u03b5 * \u2191\u2191\u03bc s \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc + 0\n\u22a2 \u222b\u207b (x : \u03b1) in s, \u2191N \u2202\u03bc = \u2191N * \u2191\u2191\u03bc s", "state_after": "no goals"}, {"tactic": "simp only [lt_top_iff_ne_top, s_lt_top.ne, and_false_iff, ENNReal.coe_ne_top,\n  ENNReal.mul_eq_top, Ne.def, not_false_iff, false_and_iff, or_self_iff]", "annotated_tactic": ["simp only [<a>lt_top_iff_ne_top</a>, s_lt_top.ne, <a>and_false_iff</a>, <a>ENNReal.coe_ne_top</a>,\n            <a>ENNReal.mul_eq_top</a>, <a>Ne.def</a>, <a>not_false_iff</a>, <a>false_and_iff</a>, <a>or_self_iff</a>]", [{"full_name": "lt_top_iff_ne_top", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [173, 9], "def_end_pos": [173, 26]}, {"full_name": "and_false_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [149, 9], "def_end_pos": [149, 22]}, {"full_name": "ENNReal.coe_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [302, 17], "def_end_pos": [302, 27]}, {"full_name": "ENNReal.mul_eq_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [608, 9], "def_end_pos": [608, 19]}, {"full_name": "Ne.def", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [59, 9], "def_end_pos": [59, 15]}, {"full_name": "not_false_iff", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [82, 9], "def_end_pos": [82, 22]}, {"full_name": "false_and_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [151, 9], "def_end_pos": [151, 22]}, {"full_name": "or_self_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [190, 9], "def_end_pos": [190, 20]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np\u271d : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\n\u03b5 N : \u211d\u22650\np : \u2115\n\u03b5pos : 0 < \u03b5\ns : Set \u03b1 := {x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p\ns_meas : MeasurableSet s\ns_lt_top : \u2191\u2191\u03bc s < \u22a4\nA : \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc + \u2191\u03b5 * \u2191\u2191\u03bc s \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc + 0\n\u22a2 \u2191N * \u2191\u2191\u03bc s < \u22a4", "state_after": "no goals"}, {"tactic": "intro x hx", "annotated_tactic": ["intro x hx", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\nA : \u2200 (\u03b5 N : \u211d\u22650) (p : \u2115), 0 < \u03b5 \u2192 \u2191\u2191\u03bc ({x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p) = 0\nu : \u2115 \u2192 \u211d\u22650\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\ns : \u2115 \u2192 Set \u03b1 := fun n => {x | g x + \u2191(u n) \u2264 f x \u2227 g x \u2264 \u2191\u2191n} \u2229 spanningSets \u03bc n\n\u03bcs : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) = 0\n\u22a2 {x | f x \u2264 g x}\u1d9c \u2286 \u22c3 n, s n", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\nA : \u2200 (\u03b5 N : \u211d\u22650) (p : \u2115), 0 < \u03b5 \u2192 \u2191\u2191\u03bc ({x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p) = 0\nu : \u2115 \u2192 \u211d\u22650\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\ns : \u2115 \u2192 Set \u03b1 := fun n => {x | g x + \u2191(u n) \u2264 f x \u2227 g x \u2264 \u2191\u2191n} \u2229 spanningSets \u03bc n\n\u03bcs : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) = 0\nx : \u03b1\nhx : x \u2208 {x | f x \u2264 g x}\u1d9c\n\u22a2 x \u2208 \u22c3 n, s n"}, {"tactic": "simp only [Set.mem_compl_iff, Set.mem_setOf, not_le] at hx", "annotated_tactic": ["simp only [<a>Set.mem_compl_iff</a>, <a>Set.mem_setOf</a>, <a>not_le</a>] at hx", [{"full_name": "Set.mem_compl_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1658, 9], "def_end_pos": [1658, 22]}, {"full_name": "Set.mem_setOf", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [259, 9], "def_end_pos": [259, 18]}, {"full_name": "not_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [373, 9], "def_end_pos": [373, 15]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\nA : \u2200 (\u03b5 N : \u211d\u22650) (p : \u2115), 0 < \u03b5 \u2192 \u2191\u2191\u03bc ({x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p) = 0\nu : \u2115 \u2192 \u211d\u22650\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\ns : \u2115 \u2192 Set \u03b1 := fun n => {x | g x + \u2191(u n) \u2264 f x \u2227 g x \u2264 \u2191\u2191n} \u2229 spanningSets \u03bc n\n\u03bcs : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) = 0\nx : \u03b1\nhx : x \u2208 {x | f x \u2264 g x}\u1d9c\n\u22a2 x \u2208 \u22c3 n, s n", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\nA : \u2200 (\u03b5 N : \u211d\u22650) (p : \u2115), 0 < \u03b5 \u2192 \u2191\u2191\u03bc ({x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p) = 0\nu : \u2115 \u2192 \u211d\u22650\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\ns : \u2115 \u2192 Set \u03b1 := fun n => {x | g x + \u2191(u n) \u2264 f x \u2227 g x \u2264 \u2191\u2191n} \u2229 spanningSets \u03bc n\n\u03bcs : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) = 0\nx : \u03b1\nhx : g x < f x\n\u22a2 x \u2208 \u22c3 n, s n"}, {"tactic": "have L1 : \u2200\u1da0 n in atTop, g x + u n \u2264 f x := by\n  have : Tendsto (fun n => g x + u n) atTop (\ud835\udcdd (g x + (0 : \u211d\u22650))) :=\n    tendsto_const_nhds.add (ENNReal.tendsto_coe.2 u_lim)\n  simp only [ENNReal.coe_zero, add_zero] at this\n  exact eventually_le_of_tendsto_lt hx this", "annotated_tactic": ["have L1 : \u2200\u1da0 n in <a>atTop</a>, g x + u n \u2264 f x := by\n      have : <a>Tendsto</a> (fun n => g x + u n) <a>atTop</a> (\ud835\udcdd (g x + (0 : \u211d\u22650))) :=\n        tendsto_const_nhds.add (<a>ENNReal.tendsto_coe</a>.2 u_lim)\n      simp only [<a>ENNReal.coe_zero</a>, <a>add_zero</a>] at this\n      exact <a>eventually_le_of_tendsto_lt</a> hx this", [{"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}, {"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2939, 5], "def_end_pos": [2939, 12]}, {"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}, {"full_name": "ENNReal.tendsto_coe", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [71, 9], "def_end_pos": [71, 20]}, {"full_name": "ENNReal.coe_zero", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [215, 28], "def_end_pos": [215, 36]}, {"full_name": "add_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [469, 3], "def_end_pos": [469, 14]}, {"full_name": "eventually_le_of_tendsto_lt", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [401, 9], "def_end_pos": [401, 36]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\nA : \u2200 (\u03b5 N : \u211d\u22650) (p : \u2115), 0 < \u03b5 \u2192 \u2191\u2191\u03bc ({x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p) = 0\nu : \u2115 \u2192 \u211d\u22650\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\ns : \u2115 \u2192 Set \u03b1 := fun n => {x | g x + \u2191(u n) \u2264 f x \u2227 g x \u2264 \u2191\u2191n} \u2229 spanningSets \u03bc n\n\u03bcs : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) = 0\nx : \u03b1\nhx : g x < f x\n\u22a2 x \u2208 \u22c3 n, s n", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\nA : \u2200 (\u03b5 N : \u211d\u22650) (p : \u2115), 0 < \u03b5 \u2192 \u2191\u2191\u03bc ({x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p) = 0\nu : \u2115 \u2192 \u211d\u22650\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\ns : \u2115 \u2192 Set \u03b1 := fun n => {x | g x + \u2191(u n) \u2264 f x \u2227 g x \u2264 \u2191\u2191n} \u2229 spanningSets \u03bc n\n\u03bcs : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) = 0\nx : \u03b1\nhx : g x < f x\nL1 : \u2200\u1da0 (n : \u2115) in atTop, g x + \u2191(u n) \u2264 f x\n\u22a2 x \u2208 \u22c3 n, s n"}, {"tactic": "have L2 : \u2200\u1da0 n : \u2115 in (atTop : Filter \u2115), g x \u2264 (n : \u211d\u22650) :=\n  haveI : Tendsto (fun n : \u2115 => ((n : \u211d\u22650) : \u211d\u22650\u221e)) atTop (\ud835\udcdd \u221e) := by\n    simp only [ENNReal.coe_nat]\n    exact ENNReal.tendsto_nat_nhds_top\n  eventually_ge_of_tendsto_gt (hx.trans_le le_top) this", "annotated_tactic": ["have L2 : \u2200\u1da0 n : \u2115 in (<a>atTop</a> : <a>Filter</a> \u2115), g x \u2264 (n : \u211d\u22650) :=\n      haveI : <a>Tendsto</a> (fun n : \u2115 => ((n : \u211d\u22650) : \u211d\u22650\u221e)) <a>atTop</a> (\ud835\udcdd \u221e) := by\n        simp only [<a>ENNReal.coe_nat</a>]\n        exact <a>ENNReal.tendsto_nat_nhds_top</a>\n      <a>eventually_ge_of_tendsto_gt</a> (hx.trans_le <a>le_top</a>) this", [{"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}, {"full_name": "Filter", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [91, 11], "def_end_pos": [91, 17]}, {"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2939, 5], "def_end_pos": [2939, 12]}, {"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}, {"full_name": "ENNReal.coe_nat", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [707, 9], "def_end_pos": [707, 16]}, {"full_name": "ENNReal.tendsto_nat_nhds_top", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [178, 9], "def_end_pos": [178, 29]}, {"full_name": "eventually_ge_of_tendsto_gt", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [406, 9], "def_end_pos": [406, 36]}, {"full_name": "le_top", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [98, 9], "def_end_pos": [98, 15]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\nA : \u2200 (\u03b5 N : \u211d\u22650) (p : \u2115), 0 < \u03b5 \u2192 \u2191\u2191\u03bc ({x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p) = 0\nu : \u2115 \u2192 \u211d\u22650\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\ns : \u2115 \u2192 Set \u03b1 := fun n => {x | g x + \u2191(u n) \u2264 f x \u2227 g x \u2264 \u2191\u2191n} \u2229 spanningSets \u03bc n\n\u03bcs : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) = 0\nx : \u03b1\nhx : g x < f x\nL1 : \u2200\u1da0 (n : \u2115) in atTop, g x + \u2191(u n) \u2264 f x\n\u22a2 x \u2208 \u22c3 n, s n", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\nA : \u2200 (\u03b5 N : \u211d\u22650) (p : \u2115), 0 < \u03b5 \u2192 \u2191\u2191\u03bc ({x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p) = 0\nu : \u2115 \u2192 \u211d\u22650\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\ns : \u2115 \u2192 Set \u03b1 := fun n => {x | g x + \u2191(u n) \u2264 f x \u2227 g x \u2264 \u2191\u2191n} \u2229 spanningSets \u03bc n\n\u03bcs : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) = 0\nx : \u03b1\nhx : g x < f x\nL1 : \u2200\u1da0 (n : \u2115) in atTop, g x + \u2191(u n) \u2264 f x\nL2 : \u2200\u1da0 (n : \u2115) in atTop, g x \u2264 \u2191\u2191n\n\u22a2 x \u2208 \u22c3 n, s n"}, {"tactic": "apply Set.mem_iUnion.2", "annotated_tactic": ["apply <a>Set.mem_iUnion</a>.2", [{"full_name": "Set.mem_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [201, 9], "def_end_pos": [201, 19]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\nA : \u2200 (\u03b5 N : \u211d\u22650) (p : \u2115), 0 < \u03b5 \u2192 \u2191\u2191\u03bc ({x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p) = 0\nu : \u2115 \u2192 \u211d\u22650\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\ns : \u2115 \u2192 Set \u03b1 := fun n => {x | g x + \u2191(u n) \u2264 f x \u2227 g x \u2264 \u2191\u2191n} \u2229 spanningSets \u03bc n\n\u03bcs : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) = 0\nx : \u03b1\nhx : g x < f x\nL1 : \u2200\u1da0 (n : \u2115) in atTop, g x + \u2191(u n) \u2264 f x\nL2 : \u2200\u1da0 (n : \u2115) in atTop, g x \u2264 \u2191\u2191n\n\u22a2 x \u2208 \u22c3 n, s n", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\nA : \u2200 (\u03b5 N : \u211d\u22650) (p : \u2115), 0 < \u03b5 \u2192 \u2191\u2191\u03bc ({x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p) = 0\nu : \u2115 \u2192 \u211d\u22650\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\ns : \u2115 \u2192 Set \u03b1 := fun n => {x | g x + \u2191(u n) \u2264 f x \u2227 g x \u2264 \u2191\u2191n} \u2229 spanningSets \u03bc n\n\u03bcs : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) = 0\nx : \u03b1\nhx : g x < f x\nL1 : \u2200\u1da0 (n : \u2115) in atTop, g x + \u2191(u n) \u2264 f x\nL2 : \u2200\u1da0 (n : \u2115) in atTop, g x \u2264 \u2191\u2191n\n\u22a2 \u2203 i, x \u2208 s i"}, {"tactic": "exact ((L1.and L2).and (eventually_mem_spanningSets \u03bc x)).exists", "annotated_tactic": ["exact ((L1.and L2).<a>and</a> (<a>eventually_mem_spanningSets</a> \u03bc x)).<a>exists</a>", [{"full_name": "Filter.Eventually.and", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1103, 19], "def_end_pos": [1103, 33]}, {"full_name": "MeasureTheory.eventually_mem_spanningSets", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3378, 9], "def_end_pos": [3378, 36]}, {"full_name": "Filter.Eventually.exists", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1308, 9], "def_end_pos": [1308, 26]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\nA : \u2200 (\u03b5 N : \u211d\u22650) (p : \u2115), 0 < \u03b5 \u2192 \u2191\u2191\u03bc ({x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p) = 0\nu : \u2115 \u2192 \u211d\u22650\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\ns : \u2115 \u2192 Set \u03b1 := fun n => {x | g x + \u2191(u n) \u2264 f x \u2227 g x \u2264 \u2191\u2191n} \u2229 spanningSets \u03bc n\n\u03bcs : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) = 0\nx : \u03b1\nhx : g x < f x\nL1 : \u2200\u1da0 (n : \u2115) in atTop, g x + \u2191(u n) \u2264 f x\nL2 : \u2200\u1da0 (n : \u2115) in atTop, g x \u2264 \u2191\u2191n\n\u22a2 \u2203 i, x \u2208 s i", "state_after": "no goals"}, {"tactic": "have : Tendsto (fun n => g x + u n) atTop (\ud835\udcdd (g x + (0 : \u211d\u22650))) :=\n  tendsto_const_nhds.add (ENNReal.tendsto_coe.2 u_lim)", "annotated_tactic": ["have : <a>Tendsto</a> (fun n => g x + u n) <a>atTop</a> (\ud835\udcdd (g x + (0 : \u211d\u22650))) :=\n        tendsto_const_nhds.add (<a>ENNReal.tendsto_coe</a>.2 u_lim)", [{"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2939, 5], "def_end_pos": [2939, 12]}, {"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}, {"full_name": "ENNReal.tendsto_coe", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [71, 9], "def_end_pos": [71, 20]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\nA : \u2200 (\u03b5 N : \u211d\u22650) (p : \u2115), 0 < \u03b5 \u2192 \u2191\u2191\u03bc ({x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p) = 0\nu : \u2115 \u2192 \u211d\u22650\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\ns : \u2115 \u2192 Set \u03b1 := fun n => {x | g x + \u2191(u n) \u2264 f x \u2227 g x \u2264 \u2191\u2191n} \u2229 spanningSets \u03bc n\n\u03bcs : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) = 0\nx : \u03b1\nhx : g x < f x\n\u22a2 \u2200\u1da0 (n : \u2115) in atTop, g x + \u2191(u n) \u2264 f x", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\nA : \u2200 (\u03b5 N : \u211d\u22650) (p : \u2115), 0 < \u03b5 \u2192 \u2191\u2191\u03bc ({x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p) = 0\nu : \u2115 \u2192 \u211d\u22650\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\ns : \u2115 \u2192 Set \u03b1 := fun n => {x | g x + \u2191(u n) \u2264 f x \u2227 g x \u2264 \u2191\u2191n} \u2229 spanningSets \u03bc n\n\u03bcs : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) = 0\nx : \u03b1\nhx : g x < f x\nthis : Tendsto (fun n => g x + \u2191(u n)) atTop (\ud835\udcdd (g x + \u21910))\n\u22a2 \u2200\u1da0 (n : \u2115) in atTop, g x + \u2191(u n) \u2264 f x"}, {"tactic": "simp only [ENNReal.coe_zero, add_zero] at this", "annotated_tactic": ["simp only [<a>ENNReal.coe_zero</a>, <a>add_zero</a>] at this", [{"full_name": "ENNReal.coe_zero", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [215, 28], "def_end_pos": [215, 36]}, {"full_name": "add_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [469, 3], "def_end_pos": [469, 14]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\nA : \u2200 (\u03b5 N : \u211d\u22650) (p : \u2115), 0 < \u03b5 \u2192 \u2191\u2191\u03bc ({x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p) = 0\nu : \u2115 \u2192 \u211d\u22650\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\ns : \u2115 \u2192 Set \u03b1 := fun n => {x | g x + \u2191(u n) \u2264 f x \u2227 g x \u2264 \u2191\u2191n} \u2229 spanningSets \u03bc n\n\u03bcs : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) = 0\nx : \u03b1\nhx : g x < f x\nthis : Tendsto (fun n => g x + \u2191(u n)) atTop (\ud835\udcdd (g x + \u21910))\n\u22a2 \u2200\u1da0 (n : \u2115) in atTop, g x + \u2191(u n) \u2264 f x", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\nA : \u2200 (\u03b5 N : \u211d\u22650) (p : \u2115), 0 < \u03b5 \u2192 \u2191\u2191\u03bc ({x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p) = 0\nu : \u2115 \u2192 \u211d\u22650\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\ns : \u2115 \u2192 Set \u03b1 := fun n => {x | g x + \u2191(u n) \u2264 f x \u2227 g x \u2264 \u2191\u2191n} \u2229 spanningSets \u03bc n\n\u03bcs : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) = 0\nx : \u03b1\nhx : g x < f x\nthis : Tendsto (fun n => g x + \u2191(u n)) atTop (\ud835\udcdd (g x))\n\u22a2 \u2200\u1da0 (n : \u2115) in atTop, g x + \u2191(u n) \u2264 f x"}, {"tactic": "exact eventually_le_of_tendsto_lt hx this", "annotated_tactic": ["exact <a>eventually_le_of_tendsto_lt</a> hx this", [{"full_name": "eventually_le_of_tendsto_lt", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [401, 9], "def_end_pos": [401, 36]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\nA : \u2200 (\u03b5 N : \u211d\u22650) (p : \u2115), 0 < \u03b5 \u2192 \u2191\u2191\u03bc ({x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p) = 0\nu : \u2115 \u2192 \u211d\u22650\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\ns : \u2115 \u2192 Set \u03b1 := fun n => {x | g x + \u2191(u n) \u2264 f x \u2227 g x \u2264 \u2191\u2191n} \u2229 spanningSets \u03bc n\n\u03bcs : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) = 0\nx : \u03b1\nhx : g x < f x\nthis : Tendsto (fun n => g x + \u2191(u n)) atTop (\ud835\udcdd (g x))\n\u22a2 \u2200\u1da0 (n : \u2115) in atTop, g x + \u2191(u n) \u2264 f x", "state_after": "no goals"}, {"tactic": "simp only [ENNReal.coe_nat]", "annotated_tactic": ["simp only [<a>ENNReal.coe_nat</a>]", [{"full_name": "ENNReal.coe_nat", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [707, 9], "def_end_pos": [707, 16]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\nA : \u2200 (\u03b5 N : \u211d\u22650) (p : \u2115), 0 < \u03b5 \u2192 \u2191\u2191\u03bc ({x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p) = 0\nu : \u2115 \u2192 \u211d\u22650\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\ns : \u2115 \u2192 Set \u03b1 := fun n => {x | g x + \u2191(u n) \u2264 f x \u2227 g x \u2264 \u2191\u2191n} \u2229 spanningSets \u03bc n\n\u03bcs : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) = 0\nx : \u03b1\nhx : g x < f x\nL1 : \u2200\u1da0 (n : \u2115) in atTop, g x + \u2191(u n) \u2264 f x\n\u22a2 Tendsto (fun n => \u2191\u2191n) atTop (\ud835\udcdd \u22a4)", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\nA : \u2200 (\u03b5 N : \u211d\u22650) (p : \u2115), 0 < \u03b5 \u2192 \u2191\u2191\u03bc ({x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p) = 0\nu : \u2115 \u2192 \u211d\u22650\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\ns : \u2115 \u2192 Set \u03b1 := fun n => {x | g x + \u2191(u n) \u2264 f x \u2227 g x \u2264 \u2191\u2191n} \u2229 spanningSets \u03bc n\n\u03bcs : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) = 0\nx : \u03b1\nhx : g x < f x\nL1 : \u2200\u1da0 (n : \u2115) in atTop, g x + \u2191(u n) \u2264 f x\n\u22a2 Tendsto (fun n => \u2191n) atTop (\ud835\udcdd \u22a4)"}, {"tactic": "exact ENNReal.tendsto_nat_nhds_top", "annotated_tactic": ["exact <a>ENNReal.tendsto_nat_nhds_top</a>", [{"full_name": "ENNReal.tendsto_nat_nhds_top", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [178, 9], "def_end_pos": [178, 29]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\nA : \u2200 (\u03b5 N : \u211d\u22650) (p : \u2115), 0 < \u03b5 \u2192 \u2191\u2191\u03bc ({x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p) = 0\nu : \u2115 \u2192 \u211d\u22650\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\ns : \u2115 \u2192 Set \u03b1 := fun n => {x | g x + \u2191(u n) \u2264 f x \u2227 g x \u2264 \u2191\u2191n} \u2229 spanningSets \u03bc n\n\u03bcs : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) = 0\nx : \u03b1\nhx : g x < f x\nL1 : \u2200\u1da0 (n : \u2115) in atTop, g x + \u2191(u n) \u2264 f x\n\u22a2 Tendsto (fun n => \u2191n) atTop (\ud835\udcdd \u22a4)", "state_after": "no goals"}, {"tactic": "simp only [\u03bcs, tsum_zero]", "annotated_tactic": ["simp only [\u03bcs, <a>tsum_zero</a>]", [{"full_name": "tsum_zero", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/Basic.lean", "def_pos": [489, 9], "def_end_pos": [489, 18]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\nA : \u2200 (\u03b5 N : \u211d\u22650) (p : \u2115), 0 < \u03b5 \u2192 \u2191\u2191\u03bc ({x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p) = 0\nu : \u2115 \u2192 \u211d\u22650\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\ns : \u2115 \u2192 Set \u03b1 := fun n => {x | g x + \u2191(u n) \u2264 f x \u2227 g x \u2264 \u2191\u2191n} \u2229 spanningSets \u03bc n\n\u03bcs : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) = 0\nB : {x | f x \u2264 g x}\u1d9c \u2286 \u22c3 n, s n\n\u22a2 \u2211' (n : \u2115), \u2191\u2191\u03bc (s n) = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/Basic.lean", "full_name": "MvPolynomial.support_mul_X", "start": [739, 1], "end": [741, 54], "traced_tactics": [{"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "R : Type u\nS\u2081 : Type v\nS\u2082 : Type w\nS\u2083 : Type x\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\nn m : \u03c3\ns\u271d : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : CommSemiring S\u2081\np\u271d q : MvPolynomial \u03c3 R\ns : \u03c3\np : MvPolynomial \u03c3 R\n\u22a2 \u2200 (y : R), y * 1 = 0 \u2194 y = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean", "full_name": "MeasureTheory.norm_condexpIndL1_le", "start": [232, 1], "end": [241, 43], "traced_tactics": [{"tactic": "by_cases hs : MeasurableSet s", "annotated_tactic": ["by_cases hs : <a>MeasurableSet</a> s", [{"full_name": "MeasurableSet", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [64, 5], "def_end_pos": [64, 18]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : CompleteSpace F'\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \u211d G'\ninst\u271d\u00b2 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nx : G\n\u22a2 \u2016condexpIndL1 hm \u03bc s x\u2016 \u2264 ENNReal.toReal (\u2191\u2191\u03bc s) * \u2016x\u2016", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : CompleteSpace F'\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \u211d G'\ninst\u271d\u00b2 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nx : G\nhs : MeasurableSet s\n\u22a2 \u2016condexpIndL1 hm \u03bc s x\u2016 \u2264 ENNReal.toReal (\u2191\u2191\u03bc s) * \u2016x\u2016\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : CompleteSpace F'\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \u211d G'\ninst\u271d\u00b2 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nx : G\nhs : \u00acMeasurableSet s\n\u22a2 \u2016condexpIndL1 hm \u03bc s x\u2016 \u2264 ENNReal.toReal (\u2191\u2191\u03bc s) * \u2016x\u2016"}, {"tactic": "swap", "annotated_tactic": ["swap", []], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : CompleteSpace F'\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \u211d G'\ninst\u271d\u00b2 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nx : G\nhs : MeasurableSet s\n\u22a2 \u2016condexpIndL1 hm \u03bc s x\u2016 \u2264 ENNReal.toReal (\u2191\u2191\u03bc s) * \u2016x\u2016\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : CompleteSpace F'\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \u211d G'\ninst\u271d\u00b2 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nx : G\nhs : \u00acMeasurableSet s\n\u22a2 \u2016condexpIndL1 hm \u03bc s x\u2016 \u2264 ENNReal.toReal (\u2191\u2191\u03bc s) * \u2016x\u2016", "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : CompleteSpace F'\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \u211d G'\ninst\u271d\u00b2 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nx : G\nhs : \u00acMeasurableSet s\n\u22a2 \u2016condexpIndL1 hm \u03bc s x\u2016 \u2264 ENNReal.toReal (\u2191\u2191\u03bc s) * \u2016x\u2016\n\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : CompleteSpace F'\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \u211d G'\ninst\u271d\u00b2 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nx : G\nhs : MeasurableSet s\n\u22a2 \u2016condexpIndL1 hm \u03bc s x\u2016 \u2264 ENNReal.toReal (\u2191\u2191\u03bc s) * \u2016x\u2016"}, {"tactic": "by_cases h\u03bcs : \u03bc s = \u221e", "annotated_tactic": ["by_cases h\u03bcs : \u03bc s = \u221e", []], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : CompleteSpace F'\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \u211d G'\ninst\u271d\u00b2 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nx : G\nhs : MeasurableSet s\n\u22a2 \u2016condexpIndL1 hm \u03bc s x\u2016 \u2264 ENNReal.toReal (\u2191\u2191\u03bc s) * \u2016x\u2016", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : CompleteSpace F'\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \u211d G'\ninst\u271d\u00b2 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nx : G\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s = \u22a4\n\u22a2 \u2016condexpIndL1 hm \u03bc s x\u2016 \u2264 ENNReal.toReal (\u2191\u2191\u03bc s) * \u2016x\u2016\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : CompleteSpace F'\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \u211d G'\ninst\u271d\u00b2 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nx : G\nhs : MeasurableSet s\nh\u03bcs : \u00ac\u2191\u2191\u03bc s = \u22a4\n\u22a2 \u2016condexpIndL1 hm \u03bc s x\u2016 \u2264 ENNReal.toReal (\u2191\u2191\u03bc s) * \u2016x\u2016"}, {"tactic": "simp_rw [condexpIndL1_of_not_measurableSet hs]", "annotated_tactic": ["simp_rw [<a>condexpIndL1_of_not_measurableSet</a> hs]", [{"full_name": "MeasureTheory.condexpIndL1_of_not_measurableSet", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean", "def_pos": [197, 9], "def_end_pos": [197, 42]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : CompleteSpace F'\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \u211d G'\ninst\u271d\u00b2 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nx : G\nhs : \u00acMeasurableSet s\n\u22a2 \u2016condexpIndL1 hm \u03bc s x\u2016 \u2264 ENNReal.toReal (\u2191\u2191\u03bc s) * \u2016x\u2016", "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : CompleteSpace F'\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \u211d G'\ninst\u271d\u00b2 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nx : G\nhs : \u00acMeasurableSet s\n\u22a2 \u20160\u2016 \u2264 ENNReal.toReal (\u2191\u2191\u03bc s) * \u2016x\u2016"}, {"tactic": "rw [Lp.norm_zero]", "annotated_tactic": ["rw [<a>Lp.norm_zero</a>]", [{"full_name": "MeasureTheory.Lp.norm_zero", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [328, 9], "def_end_pos": [328, 18]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : CompleteSpace F'\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \u211d G'\ninst\u271d\u00b2 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nx : G\nhs : \u00acMeasurableSet s\n\u22a2 \u20160\u2016 \u2264 ENNReal.toReal (\u2191\u2191\u03bc s) * \u2016x\u2016", "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : CompleteSpace F'\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \u211d G'\ninst\u271d\u00b2 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nx : G\nhs : \u00acMeasurableSet s\n\u22a2 0 \u2264 ENNReal.toReal (\u2191\u2191\u03bc s) * \u2016x\u2016"}, {"tactic": "exact mul_nonneg ENNReal.toReal_nonneg (norm_nonneg _)", "annotated_tactic": ["exact <a>mul_nonneg</a> <a>ENNReal.toReal_nonneg</a> (<a>norm_nonneg</a> _)", [{"full_name": "mul_nonneg", "def_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "def_pos": [380, 7], "def_end_pos": [380, 17]}, {"full_name": "ENNReal.toReal_nonneg", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [221, 17], "def_end_pos": [221, 30]}, {"full_name": "norm_nonneg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [500, 30], "def_end_pos": [500, 41]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : CompleteSpace F'\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \u211d G'\ninst\u271d\u00b2 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nx : G\nhs : \u00acMeasurableSet s\n\u22a2 0 \u2264 ENNReal.toReal (\u2191\u2191\u03bc s) * \u2016x\u2016", "state_after": "no goals"}, {"tactic": "rw [condexpIndL1_of_measure_eq_top h\u03bcs x, Lp.norm_zero]", "annotated_tactic": ["rw [<a>condexpIndL1_of_measure_eq_top</a> h\u03bcs x, <a>Lp.norm_zero</a>]", [{"full_name": "MeasureTheory.condexpIndL1_of_measure_eq_top", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean", "def_pos": [192, 9], "def_end_pos": [192, 39]}, {"full_name": "MeasureTheory.Lp.norm_zero", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [328, 9], "def_end_pos": [328, 18]}]], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : CompleteSpace F'\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \u211d G'\ninst\u271d\u00b2 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nx : G\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s = \u22a4\n\u22a2 \u2016condexpIndL1 hm \u03bc s x\u2016 \u2264 ENNReal.toReal (\u2191\u2191\u03bc s) * \u2016x\u2016", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : CompleteSpace F'\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \u211d G'\ninst\u271d\u00b2 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nx : G\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s = \u22a4\n\u22a2 0 \u2264 ENNReal.toReal (\u2191\u2191\u03bc s) * \u2016x\u2016"}, {"tactic": "exact mul_nonneg ENNReal.toReal_nonneg (norm_nonneg _)", "annotated_tactic": ["exact <a>mul_nonneg</a> <a>ENNReal.toReal_nonneg</a> (<a>norm_nonneg</a> _)", [{"full_name": "mul_nonneg", "def_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "def_pos": [380, 7], "def_end_pos": [380, 17]}, {"full_name": "ENNReal.toReal_nonneg", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [221, 17], "def_end_pos": [221, 30]}, {"full_name": "norm_nonneg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [500, 30], "def_end_pos": [500, 41]}]], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : CompleteSpace F'\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \u211d G'\ninst\u271d\u00b2 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nx : G\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s = \u22a4\n\u22a2 0 \u2264 ENNReal.toReal (\u2191\u2191\u03bc s) * \u2016x\u2016", "state_after": "no goals"}, {"tactic": "rw [condexpIndL1_of_measurableSet_of_measure_ne_top hs h\u03bcs x]", "annotated_tactic": ["rw [<a>condexpIndL1_of_measurableSet_of_measure_ne_top</a> hs h\u03bcs x]", [{"full_name": "MeasureTheory.condexpIndL1_of_measurableSet_of_measure_ne_top", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean", "def_pos": [187, 9], "def_end_pos": [187, 56]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : CompleteSpace F'\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \u211d G'\ninst\u271d\u00b2 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nx : G\nhs : MeasurableSet s\nh\u03bcs : \u00ac\u2191\u2191\u03bc s = \u22a4\n\u22a2 \u2016condexpIndL1 hm \u03bc s x\u2016 \u2264 ENNReal.toReal (\u2191\u2191\u03bc s) * \u2016x\u2016", "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : CompleteSpace F'\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \u211d G'\ninst\u271d\u00b2 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nx : G\nhs : MeasurableSet s\nh\u03bcs : \u00ac\u2191\u2191\u03bc s = \u22a4\n\u22a2 \u2016condexpIndL1Fin hm hs h\u03bcs x\u2016 \u2264 ENNReal.toReal (\u2191\u2191\u03bc s) * \u2016x\u2016"}, {"tactic": "exact norm_condexpIndL1Fin_le hs h\u03bcs x", "annotated_tactic": ["exact <a>norm_condexpIndL1Fin_le</a> hs h\u03bcs x", [{"full_name": "MeasureTheory.norm_condexpIndL1Fin_le", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean", "def_pos": [128, 9], "def_end_pos": [128, 32]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : CompleteSpace F'\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \u211d G'\ninst\u271d\u00b2 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nx : G\nhs : MeasurableSet s\nh\u03bcs : \u00ac\u2191\u2191\u03bc s = \u22a4\n\u22a2 \u2016condexpIndL1Fin hm hs h\u03bcs x\u2016 \u2264 ENNReal.toReal (\u2191\u2191\u03bc s) * \u2016x\u2016", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "full_name": "MeasureTheory.OuterMeasure.restrict_sInf_eq_sInf_restrict", "start": [1281, 1], "end": [1283, 59], "traced_tactics": [{"tactic": "simp only [sInf_eq_iInf, restrict_biInf, hm, iInf_image]", "annotated_tactic": ["simp only [<a>sInf_eq_iInf</a>, <a>restrict_biInf</a>, hm, <a>iInf_image</a>]", [{"full_name": "sInf_eq_iInf", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [993, 9], "def_end_pos": [993, 21]}, {"full_name": "MeasureTheory.OuterMeasure.restrict_biInf", "def_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "def_pos": [1272, 9], "def_end_pos": [1272, 23]}, {"full_name": "iInf_image", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [1538, 9], "def_end_pos": [1538, 19]}]], "state_before": "\u03b1 : Type u_1\nm : Set (OuterMeasure \u03b1)\ns : Set \u03b1\nhm : Set.Nonempty m\n\u22a2 \u2191(restrict s) (sInf m) = sInf (\u2191(restrict s) '' m)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "full_name": "set_integral_withDensity_eq_set_integral_smul\u2080", "start": [1339, 1], "end": [1342, 74], "traced_tactics": [{"tactic": "rw [restrict_withDensity hs, integral_withDensity_eq_integral_smul\u2080 hf]", "annotated_tactic": ["rw [<a>restrict_withDensity</a> hs, <a>integral_withDensity_eq_integral_smul\u2080</a> hf]", [{"full_name": "MeasureTheory.restrict_withDensity", "def_path": "Mathlib/MeasureTheory/Measure/WithDensity.lean", "def_pos": [176, 9], "def_end_pos": [176, 29]}, {"full_name": "integral_withDensity_eq_integral_smul\u2080", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [1317, 9], "def_end_pos": [1317, 47]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\ns : Set \u03b1\nhf : AEMeasurable f\ng : \u03b1 \u2192 E\nhs : MeasurableSet s\n\u22a2 (\u222b (a : \u03b1) in s, g a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1) in s, f a \u2022 g a \u2202\u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "full_name": "List.disjoint_take_drop", "start": [1927, 1], "end": [1935, 61], "traced_tactics": [{"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03b1 : Type u_1\nm n : Nat\nx\u271d\u00b9 : Nodup []\nx\u271d : m \u2264 n\n\u22a2 Disjoint (take m []) (drop n [])", "state_after": "no goals"}, {"tactic": "cases m <;> cases n <;> simp only [disjoint_cons_left, mem_cons, disjoint_cons_right,\n  drop, true_or, eq_self_iff_true, not_true, false_and, not_mem_nil, disjoint_nil_left, take]", "annotated_tactic": ["cases m <;> cases n <;> simp only [<a>disjoint_cons_left</a>, <a>mem_cons</a>, <a>disjoint_cons_right</a>,\n      <a>drop</a>, <a>true_or</a>, <a>eq_self_iff_true</a>, <a>not_true</a>, <a>false_and</a>, <a>not_mem_nil</a>, <a>disjoint_nil_left</a>, <a>take</a>]", [{"full_name": "List.disjoint_cons_left", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [1559, 17], "def_end_pos": [1559, 35]}, {"full_name": "List.mem_cons", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [62, 17], "def_end_pos": [62, 25]}, {"full_name": "List.disjoint_cons_right", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [1562, 17], "def_end_pos": [1562, 36]}, {"full_name": "List.drop", "def_path": "lake-packages/lean4/src/lean/Init/Data/List/Basic.lean", "def_pos": [475, 5], "def_end_pos": [475, 9]}, {"full_name": "true_or", "def_path": "lake-packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [89, 17], "def_end_pos": [89, 24]}, {"full_name": "eq_self_iff_true", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [86, 9], "def_end_pos": [86, 25]}, {"full_name": "not_true", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [80, 17], "def_end_pos": [80, 25]}, {"full_name": "false_and", "def_path": "lake-packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [86, 17], "def_end_pos": [86, 26]}, {"full_name": "List.not_mem_nil", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [58, 17], "def_end_pos": [58, 28]}, {"full_name": "List.disjoint_nil_left", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [1543, 17], "def_end_pos": [1543, 34]}, {"full_name": "List.take", "def_path": "lake-packages/lean4/src/lean/Init/Data/List/Basic.lean", "def_pos": [494, 5], "def_end_pos": [494, 9]}]], "state_before": "\u03b1 : Type u_1\nm n : Nat\nx : \u03b1\nxs : List \u03b1\nhl : Nodup (x :: xs)\nh : m \u2264 n\n\u22a2 Disjoint (take m (x :: xs)) (drop n (x :: xs))", "state_after": "case succ.zero\n\u03b1 : Type u_1\nx : \u03b1\nxs : List \u03b1\nhl : Nodup (x :: xs)\nn\u271d : Nat\nh : succ n\u271d \u2264 zero\n\u22a2 False\n\ncase succ.succ\n\u03b1 : Type u_1\nx : \u03b1\nxs : List \u03b1\nhl : Nodup (x :: xs)\nn\u271d\u00b9 n\u271d : Nat\nh : succ n\u271d\u00b9 \u2264 succ n\u271d\n\u22a2 \u00acx \u2208 drop n\u271d xs \u2227 Disjoint (take n\u271d\u00b9 xs) (drop n\u271d xs)"}, {"tactic": "case succ.zero => cases h", "annotated_tactic": ["case succ.zero => cases h", []], "state_before": "case succ.zero\n\u03b1 : Type u_1\nx : \u03b1\nxs : List \u03b1\nhl : Nodup (x :: xs)\nn\u271d : Nat\nh : succ n\u271d \u2264 zero\n\u22a2 False", "state_after": "no goals"}, {"tactic": "cases h", "annotated_tactic": ["cases h", []], "state_before": "\u03b1 : Type u_1\nx : \u03b1\nxs : List \u03b1\nhl : Nodup (x :: xs)\nn\u271d : Nat\nh : succ n\u271d \u2264 zero\n\u22a2 False", "state_after": "no goals"}, {"tactic": "cases hl with | cons h\u2080 h\u2081 =>\nrefine \u27e8fun h => h\u2080 _ (mem_of_mem_drop h) rfl, ?_\u27e9\nexact disjoint_take_drop h\u2081 (Nat.le_of_succ_le_succ h)", "annotated_tactic": ["cases hl with | <a>cons</a> h\u2080 h\u2081 =>\n      refine \u27e8fun h => h\u2080 _ (<a>mem_of_mem_drop</a> h) <a>rfl</a>, ?_\u27e9\n      exact disjoint_take_drop h\u2081 (<a>Nat.le_of_succ_le_succ</a> h)", [{"full_name": "List.Pairwise.cons", "def_path": "lake-packages/std/Std/Data/List/Basic.lean", "def_pos": [1123, 5], "def_end_pos": [1123, 9]}, {"full_name": "List.mem_of_mem_drop", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [1925, 9], "def_end_pos": [1925, 24]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}, {"full_name": "Nat.le_of_succ_le_succ", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1628, 9], "def_end_pos": [1628, 31]}]], "state_before": "case succ.succ\n\u03b1 : Type u_1\nx : \u03b1\nxs : List \u03b1\nhl : Nodup (x :: xs)\nn\u271d\u00b9 n\u271d : Nat\nh : succ n\u271d\u00b9 \u2264 succ n\u271d\n\u22a2 \u00acx \u2208 drop n\u271d xs \u2227 Disjoint (take n\u271d\u00b9 xs) (drop n\u271d xs)", "state_after": "no goals"}, {"tactic": "refine \u27e8fun h => h\u2080 _ (mem_of_mem_drop h) rfl, ?_\u27e9", "annotated_tactic": ["refine \u27e8fun h => h\u2080 _ (<a>mem_of_mem_drop</a> h) <a>rfl</a>, ?_\u27e9", [{"full_name": "List.mem_of_mem_drop", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [1925, 9], "def_end_pos": [1925, 24]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "case succ.succ.cons\n\u03b1 : Type u_1\nx : \u03b1\nxs : List \u03b1\nn\u271d\u00b9 n\u271d : Nat\nh : succ n\u271d\u00b9 \u2264 succ n\u271d\nh\u2081 : Pairwise (fun x x_1 => x \u2260 x_1) xs\nh\u2080 : \u2200 (a' : \u03b1), a' \u2208 xs \u2192 x \u2260 a'\n\u22a2 \u00acx \u2208 drop n\u271d xs \u2227 Disjoint (take n\u271d\u00b9 xs) (drop n\u271d xs)", "state_after": "case succ.succ.cons\n\u03b1 : Type u_1\nx : \u03b1\nxs : List \u03b1\nn\u271d\u00b9 n\u271d : Nat\nh : succ n\u271d\u00b9 \u2264 succ n\u271d\nh\u2081 : Pairwise (fun x x_1 => x \u2260 x_1) xs\nh\u2080 : \u2200 (a' : \u03b1), a' \u2208 xs \u2192 x \u2260 a'\n\u22a2 Disjoint (take n\u271d\u00b9 xs) (drop n\u271d xs)"}, {"tactic": "exact disjoint_take_drop h\u2081 (Nat.le_of_succ_le_succ h)", "annotated_tactic": ["exact disjoint_take_drop h\u2081 (<a>Nat.le_of_succ_le_succ</a> h)", [{"full_name": "Nat.le_of_succ_le_succ", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1628, 9], "def_end_pos": [1628, 31]}]], "state_before": "case succ.succ.cons\n\u03b1 : Type u_1\nx : \u03b1\nxs : List \u03b1\nn\u271d\u00b9 n\u271d : Nat\nh : succ n\u271d\u00b9 \u2264 succ n\u271d\nh\u2081 : Pairwise (fun x x_1 => x \u2260 x_1) xs\nh\u2080 : \u2200 (a' : \u03b1), a' \u2208 xs \u2192 x \u2260 a'\n\u22a2 Disjoint (take n\u271d\u00b9 xs) (drop n\u271d xs)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/ProbabilityMassFunction/Basic.lean", "full_name": "PMF.support_countable", "start": [103, 1], "end": [104, 57], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Ackermann.lean", "full_name": "ack_strictMono_right", "start": [136, 1], "end": [145, 56], "traced_tactics": [{"tactic": "simpa using h", "annotated_tactic": ["simpa using h", []], "state_before": "n\u2081 n\u2082 : \u2115\nh : n\u2081 < n\u2082\n\u22a2 ack 0 n\u2081 < ack 0 n\u2082", "state_after": "no goals"}, {"tactic": "rw [ack_succ_zero, ack_succ_succ]", "annotated_tactic": ["rw [<a>ack_succ_zero</a>, <a>ack_succ_succ</a>]", [{"full_name": "ack_succ_zero", "def_path": "Mathlib/Computability/Ackermann.lean", "def_pos": [74, 9], "def_end_pos": [74, 22]}, {"full_name": "ack_succ_succ", "def_path": "Mathlib/Computability/Ackermann.lean", "def_pos": [78, 9], "def_end_pos": [78, 22]}]], "state_before": "m n : \u2115\n_h : 0 < n + 1\n\u22a2 ack (m + 1) 0 < ack (m + 1) (n + 1)", "state_after": "m n : \u2115\n_h : 0 < n + 1\n\u22a2 ack m 1 < ack m (ack (m + 1) n)"}, {"tactic": "exact ack_strictMono_right _ (one_lt_ack_succ_left m n)", "annotated_tactic": ["exact ack_strictMono_right _ (<a>one_lt_ack_succ_left</a> m n)", [{"full_name": "one_lt_ack_succ_left", "def_path": "Mathlib/Computability/Ackermann.lean", "def_pos": [117, 9], "def_end_pos": [117, 29]}]], "state_before": "m n : \u2115\n_h : 0 < n + 1\n\u22a2 ack m 1 < ack m (ack (m + 1) n)", "state_after": "no goals"}, {"tactic": "rw [ack_succ_succ, ack_succ_succ]", "annotated_tactic": ["rw [<a>ack_succ_succ</a>, <a>ack_succ_succ</a>]", [{"full_name": "ack_succ_succ", "def_path": "Mathlib/Computability/Ackermann.lean", "def_pos": [78, 9], "def_end_pos": [78, 22]}, {"full_name": "ack_succ_succ", "def_path": "Mathlib/Computability/Ackermann.lean", "def_pos": [78, 9], "def_end_pos": [78, 22]}]], "state_before": "m n\u2081 n\u2082 : \u2115\nh : n\u2081 + 1 < n\u2082 + 1\n\u22a2 ack (m + 1) (n\u2081 + 1) < ack (m + 1) (n\u2082 + 1)", "state_after": "m n\u2081 n\u2082 : \u2115\nh : n\u2081 + 1 < n\u2082 + 1\n\u22a2 ack m (ack (m + 1) n\u2081) < ack m (ack (m + 1) n\u2082)"}, {"tactic": "apply ack_strictMono_right _ (ack_strictMono_right _ _)", "annotated_tactic": ["apply ack_strictMono_right _ (ack_strictMono_right _ _)", []], "state_before": "m n\u2081 n\u2082 : \u2115\nh : n\u2081 + 1 < n\u2082 + 1\n\u22a2 ack m (ack (m + 1) n\u2081) < ack m (ack (m + 1) n\u2082)", "state_after": "m n\u2081 n\u2082 : \u2115\nh : n\u2081 + 1 < n\u2082 + 1\n\u22a2 n\u2081 < n\u2082"}, {"tactic": "rwa [add_lt_add_iff_right] at h", "annotated_tactic": ["rwa [<a>add_lt_add_iff_right</a>] at h", [{"full_name": "add_lt_add_iff_right", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [112, 3], "def_end_pos": [112, 14]}]], "state_before": "m n\u2081 n\u2082 : \u2115\nh : n\u2081 + 1 < n\u2082 + 1\n\u22a2 n\u2081 < n\u2082", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/ProbabilityMeasure.lean", "full_name": "MeasureTheory.FiniteMeasure.tendsto_testAgainstNN_of_tendsto_normalize_testAgainstNN_of_tendsto_mass", "start": [419, 1], "end": [435, 34], "traced_tactics": [{"tactic": "by_cases h_mass : \u03bc.mass = 0", "annotated_tactic": ["by_cases h_mass : \u03bc.mass = 0", []], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b2 : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_2\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\n\u03bcs_lim : Tendsto (fun i => normalize (\u03bcs i)) F (\ud835\udcdd (normalize \u03bc))\nmass_lim : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd (mass \u03bc))\nf : \u03a9 \u2192\u1d47 \u211d\u22650\n\u22a2 Tendsto (fun i => testAgainstNN (\u03bcs i) f) F (\ud835\udcdd (testAgainstNN \u03bc f))", "state_after": "case pos\n\u03a9 : Type u_1\ninst\u271d\u00b2 : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_2\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\n\u03bcs_lim : Tendsto (fun i => normalize (\u03bcs i)) F (\ud835\udcdd (normalize \u03bc))\nmass_lim : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd (mass \u03bc))\nf : \u03a9 \u2192\u1d47 \u211d\u22650\nh_mass : mass \u03bc = 0\n\u22a2 Tendsto (fun i => testAgainstNN (\u03bcs i) f) F (\ud835\udcdd (testAgainstNN \u03bc f))\n\ncase neg\n\u03a9 : Type u_1\ninst\u271d\u00b2 : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_2\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\n\u03bcs_lim : Tendsto (fun i => normalize (\u03bcs i)) F (\ud835\udcdd (normalize \u03bc))\nmass_lim : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd (mass \u03bc))\nf : \u03a9 \u2192\u1d47 \u211d\u22650\nh_mass : \u00acmass \u03bc = 0\n\u22a2 Tendsto (fun i => testAgainstNN (\u03bcs i) f) F (\ud835\udcdd (testAgainstNN \u03bc f))"}, {"tactic": "simp_rw [fun i => (\u03bcs i).testAgainstNN_eq_mass_mul f, \u03bc.testAgainstNN_eq_mass_mul f]", "annotated_tactic": ["simp_rw [fun i => (\u03bcs i).<a>testAgainstNN_eq_mass_mul</a> f, \u03bc.testAgainstNN_eq_mass_mul f]", [{"full_name": "MeasureTheory.FiniteMeasure.testAgainstNN_eq_mass_mul", "def_path": "Mathlib/MeasureTheory/Measure/ProbabilityMeasure.lean", "def_pos": [404, 9], "def_end_pos": [404, 34]}]], "state_before": "case neg\n\u03a9 : Type u_1\ninst\u271d\u00b2 : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_2\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\n\u03bcs_lim : Tendsto (fun i => normalize (\u03bcs i)) F (\ud835\udcdd (normalize \u03bc))\nmass_lim : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd (mass \u03bc))\nf : \u03a9 \u2192\u1d47 \u211d\u22650\nh_mass : \u00acmass \u03bc = 0\n\u22a2 Tendsto (fun i => testAgainstNN (\u03bcs i) f) F (\ud835\udcdd (testAgainstNN \u03bc f))", "state_after": "case neg\n\u03a9 : Type u_1\ninst\u271d\u00b2 : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_2\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\n\u03bcs_lim : Tendsto (fun i => normalize (\u03bcs i)) F (\ud835\udcdd (normalize \u03bc))\nmass_lim : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd (mass \u03bc))\nf : \u03a9 \u2192\u1d47 \u211d\u22650\nh_mass : \u00acmass \u03bc = 0\n\u22a2 Tendsto (fun i => mass (\u03bcs i) * testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize (\u03bcs i))) f) F\n    (\ud835\udcdd (mass \u03bc * testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize \u03bc)) f))"}, {"tactic": "rw [ProbabilityMeasure.tendsto_nhds_iff_toFiniteMeasure_tendsto_nhds] at \u03bcs_lim", "annotated_tactic": ["rw [<a>ProbabilityMeasure.tendsto_nhds_iff_toFiniteMeasure_tendsto_nhds</a>] at \u03bcs_lim", [{"full_name": "MeasureTheory.ProbabilityMeasure.tendsto_nhds_iff_toFiniteMeasure_tendsto_nhds", "def_path": "Mathlib/MeasureTheory/Measure/ProbabilityMeasure.lean", "def_pos": [272, 9], "def_end_pos": [272, 54]}]], "state_before": "case neg\n\u03a9 : Type u_1\ninst\u271d\u00b2 : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_2\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\n\u03bcs_lim : Tendsto (fun i => normalize (\u03bcs i)) F (\ud835\udcdd (normalize \u03bc))\nmass_lim : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd (mass \u03bc))\nf : \u03a9 \u2192\u1d47 \u211d\u22650\nh_mass : \u00acmass \u03bc = 0\n\u22a2 Tendsto (fun i => mass (\u03bcs i) * testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize (\u03bcs i))) f) F\n    (\ud835\udcdd (mass \u03bc * testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize \u03bc)) f))", "state_after": "case neg\n\u03a9 : Type u_1\ninst\u271d\u00b2 : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_2\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\n\u03bcs_lim :\n  Tendsto (ProbabilityMeasure.toFiniteMeasure \u2218 fun i => normalize (\u03bcs i)) F\n    (\ud835\udcdd (ProbabilityMeasure.toFiniteMeasure (normalize \u03bc)))\nmass_lim : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd (mass \u03bc))\nf : \u03a9 \u2192\u1d47 \u211d\u22650\nh_mass : \u00acmass \u03bc = 0\n\u22a2 Tendsto (fun i => mass (\u03bcs i) * testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize (\u03bcs i))) f) F\n    (\ud835\udcdd (mass \u03bc * testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize \u03bc)) f))"}, {"tactic": "rw [tendsto_iff_forall_testAgainstNN_tendsto] at \u03bcs_lim", "annotated_tactic": ["rw [<a>tendsto_iff_forall_testAgainstNN_tendsto</a>] at \u03bcs_lim", [{"full_name": "MeasureTheory.FiniteMeasure.tendsto_iff_forall_testAgainstNN_tendsto", "def_path": "Mathlib/MeasureTheory/Measure/FiniteMeasure.lean", "def_pos": [494, 9], "def_end_pos": [494, 49]}]], "state_before": "case neg\n\u03a9 : Type u_1\ninst\u271d\u00b2 : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_2\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\n\u03bcs_lim :\n  Tendsto (ProbabilityMeasure.toFiniteMeasure \u2218 fun i => normalize (\u03bcs i)) F\n    (\ud835\udcdd (ProbabilityMeasure.toFiniteMeasure (normalize \u03bc)))\nmass_lim : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd (mass \u03bc))\nf : \u03a9 \u2192\u1d47 \u211d\u22650\nh_mass : \u00acmass \u03bc = 0\n\u22a2 Tendsto (fun i => mass (\u03bcs i) * testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize (\u03bcs i))) f) F\n    (\ud835\udcdd (mass \u03bc * testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize \u03bc)) f))", "state_after": "case neg\n\u03a9 : Type u_1\ninst\u271d\u00b2 : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_2\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\n\u03bcs_lim :\n  \u2200 (f : \u03a9 \u2192\u1d47 \u211d\u22650),\n    Tendsto (fun i => testAgainstNN ((ProbabilityMeasure.toFiniteMeasure \u2218 fun i => normalize (\u03bcs i)) i) f) F\n      (\ud835\udcdd (testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize \u03bc)) f))\nmass_lim : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd (mass \u03bc))\nf : \u03a9 \u2192\u1d47 \u211d\u22650\nh_mass : \u00acmass \u03bc = 0\n\u22a2 Tendsto (fun i => mass (\u03bcs i) * testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize (\u03bcs i))) f) F\n    (\ud835\udcdd (mass \u03bc * testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize \u03bc)) f))"}, {"tactic": "have lim_pair :\n  Tendsto (fun i => (\u27e8(\u03bcs i).mass, (\u03bcs i).normalize.toFiniteMeasure.testAgainstNN f\u27e9 : \u211d\u22650 \u00d7 \u211d\u22650))\n    F (\ud835\udcdd \u27e8\u03bc.mass, \u03bc.normalize.toFiniteMeasure.testAgainstNN f\u27e9) :=\n  (Prod.tendsto_iff _ _).mpr \u27e8mass_lim, \u03bcs_lim f\u27e9", "annotated_tactic": ["have lim_pair :\n    <a>Tendsto</a> (fun i => (\u27e8(\u03bcs i).<a>mass</a>, (\u03bcs i).normalize.toFiniteMeasure.testAgainstNN f\u27e9 : \u211d\u22650 \u00d7 \u211d\u22650))\n      F (\ud835\udcdd \u27e8\u03bc.mass, \u03bc.normalize.toFiniteMeasure.testAgainstNN f\u27e9) :=\n    (<a>Prod.tendsto_iff</a> _ _).<a>mpr</a> \u27e8mass_lim, \u03bcs_lim f\u27e9", [{"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2939, 5], "def_end_pos": [2939, 12]}, {"full_name": "MeasureTheory.FiniteMeasure.mass", "def_path": "Mathlib/MeasureTheory/Measure/FiniteMeasure.lean", "def_pos": [163, 5], "def_end_pos": [163, 9]}, {"full_name": "Prod.tendsto_iff", "def_path": "Mathlib/Topology/Constructions.lean", "def_pos": [554, 9], "def_end_pos": [554, 25]}, {"full_name": "Iff.mpr", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [92, 3], "def_end_pos": [92, 6]}]], "state_before": "case neg\n\u03a9 : Type u_1\ninst\u271d\u00b2 : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_2\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\n\u03bcs_lim :\n  \u2200 (f : \u03a9 \u2192\u1d47 \u211d\u22650),\n    Tendsto (fun i => testAgainstNN ((ProbabilityMeasure.toFiniteMeasure \u2218 fun i => normalize (\u03bcs i)) i) f) F\n      (\ud835\udcdd (testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize \u03bc)) f))\nmass_lim : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd (mass \u03bc))\nf : \u03a9 \u2192\u1d47 \u211d\u22650\nh_mass : \u00acmass \u03bc = 0\n\u22a2 Tendsto (fun i => mass (\u03bcs i) * testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize (\u03bcs i))) f) F\n    (\ud835\udcdd (mass \u03bc * testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize \u03bc)) f))", "state_after": "case neg\n\u03a9 : Type u_1\ninst\u271d\u00b2 : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_2\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\n\u03bcs_lim :\n  \u2200 (f : \u03a9 \u2192\u1d47 \u211d\u22650),\n    Tendsto (fun i => testAgainstNN ((ProbabilityMeasure.toFiniteMeasure \u2218 fun i => normalize (\u03bcs i)) i) f) F\n      (\ud835\udcdd (testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize \u03bc)) f))\nmass_lim : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd (mass \u03bc))\nf : \u03a9 \u2192\u1d47 \u211d\u22650\nh_mass : \u00acmass \u03bc = 0\nlim_pair :\n  Tendsto (fun i => (mass (\u03bcs i), testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize (\u03bcs i))) f)) F\n    (\ud835\udcdd (mass \u03bc, testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize \u03bc)) f))\n\u22a2 Tendsto (fun i => mass (\u03bcs i) * testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize (\u03bcs i))) f) F\n    (\ud835\udcdd (mass \u03bc * testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize \u03bc)) f))"}, {"tactic": "exact tendsto_mul.comp lim_pair", "annotated_tactic": ["exact tendsto_mul.comp lim_pair", []], "state_before": "case neg\n\u03a9 : Type u_1\ninst\u271d\u00b2 : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_2\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\n\u03bcs_lim :\n  \u2200 (f : \u03a9 \u2192\u1d47 \u211d\u22650),\n    Tendsto (fun i => testAgainstNN ((ProbabilityMeasure.toFiniteMeasure \u2218 fun i => normalize (\u03bcs i)) i) f) F\n      (\ud835\udcdd (testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize \u03bc)) f))\nmass_lim : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd (mass \u03bc))\nf : \u03a9 \u2192\u1d47 \u211d\u22650\nh_mass : \u00acmass \u03bc = 0\nlim_pair :\n  Tendsto (fun i => (mass (\u03bcs i), testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize (\u03bcs i))) f)) F\n    (\ud835\udcdd (mass \u03bc, testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize \u03bc)) f))\n\u22a2 Tendsto (fun i => mass (\u03bcs i) * testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize (\u03bcs i))) f) F\n    (\ud835\udcdd (mass \u03bc * testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize \u03bc)) f))", "state_after": "no goals"}, {"tactic": "simp only [\u03bc.mass_zero_iff.mp h_mass, zero_testAgainstNN_apply, zero_mass,\n  eq_self_iff_true] at *", "annotated_tactic": ["simp only [\u03bc.mass_zero_iff.mp h_mass, <a>zero_testAgainstNN_apply</a>, <a>zero_mass</a>,\n      <a>eq_self_iff_true</a>] at *", [{"full_name": "MeasureTheory.FiniteMeasure.zero_testAgainstNN_apply", "def_path": "Mathlib/MeasureTheory/Measure/FiniteMeasure.lean", "def_pos": [354, 9], "def_end_pos": [354, 33]}, {"full_name": "MeasureTheory.FiniteMeasure.zero_mass", "def_path": "Mathlib/MeasureTheory/Measure/FiniteMeasure.lean", "def_pos": [179, 9], "def_end_pos": [179, 18]}, {"full_name": "eq_self_iff_true", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [86, 9], "def_end_pos": [86, 25]}]], "state_before": "case pos\n\u03a9 : Type u_1\ninst\u271d\u00b2 : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_2\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\n\u03bcs_lim : Tendsto (fun i => normalize (\u03bcs i)) F (\ud835\udcdd (normalize \u03bc))\nmass_lim : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd (mass \u03bc))\nf : \u03a9 \u2192\u1d47 \u211d\u22650\nh_mass : mass \u03bc = 0\n\u22a2 Tendsto (fun i => testAgainstNN (\u03bcs i) f) F (\ud835\udcdd (testAgainstNN \u03bc f))", "state_after": "case pos\n\u03a9 : Type u_1\ninst\u271d\u00b2 : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_2\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\nf : \u03a9 \u2192\u1d47 \u211d\u22650\n\u03bcs_lim : Tendsto (fun i => normalize (\u03bcs i)) F (\ud835\udcdd (normalize 0))\nmass_lim : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd 0)\nh_mass : True\n\u22a2 Tendsto (fun i => testAgainstNN (\u03bcs i) f) F (\ud835\udcdd 0)"}, {"tactic": "exact tendsto_zero_testAgainstNN_of_tendsto_zero_mass mass_lim f", "annotated_tactic": ["exact <a>tendsto_zero_testAgainstNN_of_tendsto_zero_mass</a> mass_lim f", [{"full_name": "MeasureTheory.FiniteMeasure.tendsto_zero_testAgainstNN_of_tendsto_zero_mass", "def_path": "Mathlib/MeasureTheory/Measure/FiniteMeasure.lean", "def_pos": [505, 9], "def_end_pos": [505, 56]}]], "state_before": "case pos\n\u03a9 : Type u_1\ninst\u271d\u00b2 : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_2\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\nf : \u03a9 \u2192\u1d47 \u211d\u22650\n\u03bcs_lim : Tendsto (fun i => normalize (\u03bcs i)) F (\ud835\udcdd (normalize 0))\nmass_lim : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd 0)\nh_mass : True\n\u22a2 Tendsto (fun i => testAgainstNN (\u03bcs i) f) F (\ud835\udcdd 0)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Finite.lean", "full_name": "Set.Infinite.exists_ne_map_eq_of_mapsTo", "start": [1372, 1], "end": [1375, 87], "traced_tactics": [{"tactic": "contrapose! ht", "annotated_tactic": ["contrapose! ht", []], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\n\u03b3 : Type x\ns : Set \u03b1\nt : Set \u03b2\nf : \u03b1 \u2192 \u03b2\nhs : Set.Infinite s\nhf : MapsTo f s t\nht : Set.Finite t\n\u22a2 \u2203 x, x \u2208 s \u2227 \u2203 y, y \u2208 s \u2227 x \u2260 y \u2227 f x = f y", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\n\u03b3 : Type x\ns : Set \u03b1\nt : Set \u03b2\nf : \u03b1 \u2192 \u03b2\nhs : Set.Infinite s\nhf : MapsTo f s t\nht : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 x \u2260 y \u2192 f x \u2260 f y\n\u22a2 \u00acSet.Finite t"}, {"tactic": "exact infinite_of_injOn_mapsTo (fun x hx y hy => not_imp_not.1 (ht x hx y hy)) hf hs", "annotated_tactic": ["exact <a>infinite_of_injOn_mapsTo</a> (fun x hx y hy => <a>not_imp_not</a>.1 (ht x hx y hy)) hf hs", [{"full_name": "Set.infinite_of_injOn_mapsTo", "def_path": "Mathlib/Data/Set/Finite.lean", "def_pos": [1367, 9], "def_end_pos": [1367, 33]}, {"full_name": "not_imp_not", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [383, 9], "def_end_pos": [383, 20]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\n\u03b3 : Type x\ns : Set \u03b1\nt : Set \u03b2\nf : \u03b1 \u2192 \u03b2\nhs : Set.Infinite s\nhf : MapsTo f s t\nht : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 x \u2260 y \u2192 f x \u2260 f y\n\u22a2 \u00acSet.Finite t", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "full_name": "MeasureTheory.snorm_one_smul_measure", "start": [667, 1], "end": [669, 7], "traced_tactics": [{"tactic": "rw [@snorm_smul_measure_of_ne_top _ _ _ \u03bc _ 1 (@ENNReal.coe_ne_top 1) f c]", "annotated_tactic": ["rw [@<a>snorm_smul_measure_of_ne_top</a> _ _ _ \u03bc _ 1 (@<a>ENNReal.coe_ne_top</a> 1) f c]", [{"full_name": "MeasureTheory.snorm_smul_measure_of_ne_top", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [660, 9], "def_end_pos": [660, 37]}, {"full_name": "ENNReal.coe_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [302, 17], "def_end_pos": [302, 27]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nc : \u211d\u22650\u221e\n\u22a2 snorm f 1 (c \u2022 \u03bc) = c * snorm f 1 \u03bc", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nc : \u211d\u22650\u221e\n\u22a2 c ^ ENNReal.toReal (1 / 1) \u2022 snorm f 1 \u03bc = c * snorm f 1 \u03bc"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nc : \u211d\u22650\u221e\n\u22a2 c ^ ENNReal.toReal (1 / 1) \u2022 snorm f 1 \u03bc = c * snorm f 1 \u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "full_name": "Int.dvd_iff_dvd_of_dvd_add", "start": [632, 11], "end": [633, 74], "traced_tactics": [{"tactic": "rw [\u2190 Int.sub_neg] at H", "annotated_tactic": ["rw [\u2190 <a>Int.sub_neg</a>] at H", [{"full_name": "Int.sub_neg", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [370, 19], "def_end_pos": [370, 26]}]], "state_before": "a b c : Int\nH : a \u2223 b + c\n\u22a2 a \u2223 b \u2194 a \u2223 c", "state_after": "a b c : Int\nH : a \u2223 b - -c\n\u22a2 a \u2223 b \u2194 a \u2223 c"}, {"tactic": "rw [Int.dvd_iff_dvd_of_dvd_sub H, Int.dvd_neg]", "annotated_tactic": ["rw [<a>Int.dvd_iff_dvd_of_dvd_sub</a> H, <a>Int.dvd_neg</a>]", [{"full_name": "Int.dvd_iff_dvd_of_dvd_sub", "def_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "def_pos": [628, 19], "def_end_pos": [628, 41]}, {"full_name": "Int.dvd_neg", "def_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "def_pos": [608, 19], "def_end_pos": [608, 26]}]], "state_before": "a b c : Int\nH : a \u2223 b - -c\n\u22a2 a \u2223 b \u2194 a \u2223 c", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "full_name": "MeasureTheory.ae_restrict_uIoc_eq", "start": [2548, 1], "end": [2550, 50], "traced_tactics": [{"tactic": "simp only [uIoc_eq_union, ae_restrict_union_eq]", "annotated_tactic": ["simp only [<a>uIoc_eq_union</a>, <a>ae_restrict_union_eq</a>]", [{"full_name": "Set.uIoc_eq_union", "def_path": "Mathlib/Data/Set/Intervals/UnorderedInterval.lean", "def_pos": [293, 7], "def_end_pos": [293, 20]}, {"full_name": "MeasureTheory.ae_restrict_union_eq", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2498, 9], "def_end_pos": [2498, 29]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\n\u22a2 ae (Measure.restrict \u03bc (\u0399 a b)) = ae (Measure.restrict \u03bc (Ioc a b)) \u2294 ae (Measure.restrict \u03bc (Ioc b a))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/W/Cardinal.lean", "full_name": "WType.cardinal_mk_le_of_le", "start": [47, 1], "end": [52, 65], "traced_tactics": [{"tactic": "induction' \u03ba using Cardinal.inductionOn with \u03b3", "annotated_tactic": ["induction' \u03ba using <a>Cardinal.inductionOn</a> with \u03b3", [{"full_name": "Cardinal.inductionOn", "def_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "def_pos": [127, 9], "def_end_pos": [127, 20]}]], "state_before": "\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type u\n\u03ba : Cardinal.{u}\nh\u03ba : (sum fun a => \u03ba ^ #(\u03b2 a)) \u2264 \u03ba\n\u22a2 #(WType \u03b2) \u2264 \u03ba", "state_after": "case h\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type u\n\u03ba : Cardinal.{u}\nh\u03ba\u271d : (sum fun a => \u03ba ^ #(\u03b2 a)) \u2264 \u03ba\n\u03b3 : Type u\nh\u03ba : (sum fun a => #\u03b3 ^ #(\u03b2 a)) \u2264 #\u03b3\n\u22a2 #(WType \u03b2) \u2264 #\u03b3"}, {"tactic": "simp only [Cardinal.power_def, \u2190 Cardinal.mk_sigma, Cardinal.le_def] at h\u03ba", "annotated_tactic": ["simp only [<a>Cardinal.power_def</a>, \u2190 <a>Cardinal.mk_sigma</a>, <a>Cardinal.le_def</a>] at h\u03ba", [{"full_name": "Cardinal.power_def", "def_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "def_pos": [496, 9], "def_end_pos": [496, 18]}, {"full_name": "Cardinal.mk_sigma", "def_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "def_pos": [873, 9], "def_end_pos": [873, 17]}, {"full_name": "Cardinal.le_def", "def_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "def_pos": [266, 9], "def_end_pos": [266, 15]}]], "state_before": "case h\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type u\n\u03ba : Cardinal.{u}\nh\u03ba\u271d : (sum fun a => \u03ba ^ #(\u03b2 a)) \u2264 \u03ba\n\u03b3 : Type u\nh\u03ba : (sum fun a => #\u03b3 ^ #(\u03b2 a)) \u2264 #\u03b3\n\u22a2 #(WType \u03b2) \u2264 #\u03b3", "state_after": "case h\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type u\n\u03ba : Cardinal.{u}\nh\u03ba\u271d : (sum fun a => \u03ba ^ #(\u03b2 a)) \u2264 \u03ba\n\u03b3 : Type u\nh\u03ba : Nonempty ((i : \u03b1) \u00d7 (\u03b2 i \u2192 \u03b3) \u21aa \u03b3)\n\u22a2 #(WType \u03b2) \u2264 #\u03b3"}, {"tactic": "cases' h\u03ba with h\u03ba", "annotated_tactic": ["cases' h\u03ba with h\u03ba", []], "state_before": "case h\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type u\n\u03ba : Cardinal.{u}\nh\u03ba\u271d : (sum fun a => \u03ba ^ #(\u03b2 a)) \u2264 \u03ba\n\u03b3 : Type u\nh\u03ba : Nonempty ((i : \u03b1) \u00d7 (\u03b2 i \u2192 \u03b3) \u21aa \u03b3)\n\u22a2 #(WType \u03b2) \u2264 #\u03b3", "state_after": "case h.intro\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type u\n\u03ba : Cardinal.{u}\nh\u03ba\u271d : (sum fun a => \u03ba ^ #(\u03b2 a)) \u2264 \u03ba\n\u03b3 : Type u\nh\u03ba : (i : \u03b1) \u00d7 (\u03b2 i \u2192 \u03b3) \u21aa \u03b3\n\u22a2 #(WType \u03b2) \u2264 #\u03b3"}, {"tactic": "exact Cardinal.mk_le_of_injective (elim_injective _ h\u03ba.1 h\u03ba.2)", "annotated_tactic": ["exact <a>Cardinal.mk_le_of_injective</a> (<a>elim_injective</a> _ h\u03ba.1 h\u03ba.2)", [{"full_name": "Cardinal.mk_le_of_injective", "def_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "def_pos": [270, 9], "def_end_pos": [270, 27]}, {"full_name": "WType.elim_injective", "def_path": "Mathlib/Data/W/Basic.lean", "def_pos": [93, 9], "def_end_pos": [93, 23]}]], "state_before": "case h.intro\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type u\n\u03ba : Cardinal.{u}\nh\u03ba\u271d : (sum fun a => \u03ba ^ #(\u03b2 a)) \u2264 \u03ba\n\u03b3 : Type u\nh\u03ba : (i : \u03b1) \u00d7 (\u03b2 i \u2192 \u03b3) \u21aa \u03b3\n\u22a2 #(WType \u03b2) \u2264 #\u03b3", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Kernel/IntegralCompProd.lean", "full_name": "ProbabilityTheory.kernel.continuous_integral_integral", "start": [217, 1], "end": [242, 19], "traced_tactics": [{"tactic": "rw [continuous_iff_continuousAt]", "annotated_tactic": ["rw [<a>continuous_iff_continuousAt</a>]", [{"full_name": "continuous_iff_continuousAt", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [1713, 9], "def_end_pos": [1713, 36]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nE : Type u_4\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d\u2077 : NormedAddCommGroup E\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u2076 : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d\u2075 : IsSFiniteKernel \u03b7\na : \u03b1\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\nE' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : CompleteSpace E'\ninst\u271d : NormedSpace \u211d E'\n\u22a2 Continuous fun f => \u222b (x : \u03b2), \u222b (y : \u03b3), \u2191\u2191f (x, y) \u2202\u2191\u03b7 (a, x) \u2202\u2191\u03ba a", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nE : Type u_4\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d\u2077 : NormedAddCommGroup E\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u2076 : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d\u2075 : IsSFiniteKernel \u03b7\na : \u03b1\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\nE' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : CompleteSpace E'\ninst\u271d : NormedSpace \u211d E'\n\u22a2 \u2200 (x : { x // x \u2208 Lp E 1 }), ContinuousAt (fun f => \u222b (x : \u03b2), \u222b (y : \u03b3), \u2191\u2191f (x, y) \u2202\u2191\u03b7 (a, x) \u2202\u2191\u03ba a) x"}, {"tactic": "intro g", "annotated_tactic": ["intro g", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nE : Type u_4\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d\u2077 : NormedAddCommGroup E\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u2076 : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d\u2075 : IsSFiniteKernel \u03b7\na : \u03b1\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\nE' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : CompleteSpace E'\ninst\u271d : NormedSpace \u211d E'\n\u22a2 \u2200 (x : { x // x \u2208 Lp E 1 }), ContinuousAt (fun f => \u222b (x : \u03b2), \u222b (y : \u03b3), \u2191\u2191f (x, y) \u2202\u2191\u03b7 (a, x) \u2202\u2191\u03ba a) x", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nE : Type u_4\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d\u2077 : NormedAddCommGroup E\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u2076 : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d\u2075 : IsSFiniteKernel \u03b7\na : \u03b1\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\nE' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : CompleteSpace E'\ninst\u271d : NormedSpace \u211d E'\ng : { x // x \u2208 Lp E 1 }\n\u22a2 ContinuousAt (fun f => \u222b (x : \u03b2), \u222b (y : \u03b3), \u2191\u2191f (x, y) \u2202\u2191\u03b7 (a, x) \u2202\u2191\u03ba a) g"}, {"tactic": "refine'\n  tendsto_integral_of_L1 _ (L1.integrable_coeFn g).integral_compProd\n    (eventually_of_forall fun h => (L1.integrable_coeFn h).integral_compProd) _", "annotated_tactic": ["refine'\n    <a>tendsto_integral_of_L1</a> _ (<a>L1.integrable_coeFn</a> g).<a>integral_compProd</a>\n      (<a>eventually_of_forall</a> fun h => (<a>L1.integrable_coeFn</a> h).<a>integral_compProd</a>) _", [{"full_name": "MeasureTheory.tendsto_integral_of_L1", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1007, 9], "def_end_pos": [1007, 31]}, {"full_name": "MeasureTheory.L1.integrable_coeFn", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [1324, 9], "def_end_pos": [1324, 25]}, {"full_name": "MeasureTheory.Integrable.integral_compProd", "def_path": "Mathlib/Probability/Kernel/IntegralCompProd.lean", "def_pos": [142, 9], "def_end_pos": [142, 58]}, {"full_name": "Filter.eventually_of_forall", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1111, 9], "def_end_pos": [1111, 29]}, {"full_name": "MeasureTheory.L1.integrable_coeFn", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [1324, 9], "def_end_pos": [1324, 25]}, {"full_name": "MeasureTheory.Integrable.integral_compProd", "def_path": "Mathlib/Probability/Kernel/IntegralCompProd.lean", "def_pos": [142, 9], "def_end_pos": [142, 58]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nE : Type u_4\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d\u2077 : NormedAddCommGroup E\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u2076 : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d\u2075 : IsSFiniteKernel \u03b7\na : \u03b1\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\nE' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : CompleteSpace E'\ninst\u271d : NormedSpace \u211d E'\ng : { x // x \u2208 Lp E 1 }\n\u22a2 ContinuousAt (fun f => \u222b (x : \u03b2), \u222b (y : \u03b3), \u2191\u2191f (x, y) \u2202\u2191\u03b7 (a, x) \u2202\u2191\u03ba a) g", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nE : Type u_4\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d\u2077 : NormedAddCommGroup E\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u2076 : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d\u2075 : IsSFiniteKernel \u03b7\na : \u03b1\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\nE' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : CompleteSpace E'\ninst\u271d : NormedSpace \u211d E'\ng : { x // x \u2208 Lp E 1 }\n\u22a2 Tendsto (fun i => \u222b\u207b (x : \u03b2), \u2191\u2016\u222b (y : \u03b3), \u2191\u2191i (x, y) \u2202\u2191\u03b7 (a, x) - \u222b (y : \u03b3), \u2191\u2191g (x, y) \u2202\u2191\u03b7 (a, x)\u2016\u208a \u2202\u2191\u03ba a) (\ud835\udcdd g)\n    (\ud835\udcdd 0)"}, {"tactic": "simp_rw [\u2190\n  kernel.lintegral_fn_integral_sub (fun x => (\u2016x\u2016\u208a : \u211d\u22650\u221e)) (L1.integrable_coeFn _)\n    (L1.integrable_coeFn g)]", "annotated_tactic": ["simp_rw [\u2190\n    <a>kernel.lintegral_fn_integral_sub</a> (fun x => (\u2016x\u2016\u208a : \u211d\u22650\u221e)) (<a>L1.integrable_coeFn</a> _)\n      (<a>L1.integrable_coeFn</a> g)]", [{"full_name": "ProbabilityTheory.kernel.lintegral_fn_integral_sub", "def_path": "Mathlib/Probability/Kernel/IntegralCompProd.lean", "def_pos": [177, 9], "def_end_pos": [177, 41]}, {"full_name": "MeasureTheory.L1.integrable_coeFn", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [1324, 9], "def_end_pos": [1324, 25]}, {"full_name": "MeasureTheory.L1.integrable_coeFn", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [1324, 9], "def_end_pos": [1324, 25]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nE : Type u_4\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d\u2077 : NormedAddCommGroup E\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u2076 : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d\u2075 : IsSFiniteKernel \u03b7\na : \u03b1\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\nE' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : CompleteSpace E'\ninst\u271d : NormedSpace \u211d E'\ng : { x // x \u2208 Lp E 1 }\n\u22a2 Tendsto (fun i => \u222b\u207b (x : \u03b2), \u2191\u2016\u222b (y : \u03b3), \u2191\u2191i (x, y) \u2202\u2191\u03b7 (a, x) - \u222b (y : \u03b3), \u2191\u2191g (x, y) \u2202\u2191\u03b7 (a, x)\u2016\u208a \u2202\u2191\u03ba a) (\ud835\udcdd g)\n    (\ud835\udcdd 0)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nE : Type u_4\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d\u2077 : NormedAddCommGroup E\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u2076 : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d\u2075 : IsSFiniteKernel \u03b7\na : \u03b1\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\nE' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : CompleteSpace E'\ninst\u271d : NormedSpace \u211d E'\ng : { x // x \u2208 Lp E 1 }\n\u22a2 Tendsto (fun i => \u222b\u207b (x : \u03b2), \u2191\u2016\u222b (y : \u03b3), \u2191\u2191i (x, y) - \u2191\u2191g (x, y) \u2202\u2191\u03b7 (a, x)\u2016\u208a \u2202\u2191\u03ba a) (\ud835\udcdd g) (\ud835\udcdd 0)"}, {"tactic": "refine' tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds _ (fun i => zero_le _) _", "annotated_tactic": ["refine' <a>tendsto_of_tendsto_of_tendsto_of_le_of_le</a> <a>tendsto_const_nhds</a> _ (fun i => <a>zero_le</a> _) _", [{"full_name": "tendsto_of_tendsto_of_tendsto_of_le_of_le", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [955, 9], "def_end_pos": [955, 50]}, {"full_name": "tendsto_const_nhds", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [1049, 9], "def_end_pos": [1049, 27]}, {"full_name": "zero_le", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [217, 30], "def_end_pos": [217, 37]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nE : Type u_4\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d\u2077 : NormedAddCommGroup E\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u2076 : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d\u2075 : IsSFiniteKernel \u03b7\na : \u03b1\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\nE' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : CompleteSpace E'\ninst\u271d : NormedSpace \u211d E'\ng : { x // x \u2208 Lp E 1 }\n\u22a2 Tendsto (fun i => \u222b\u207b (x : \u03b2), \u2191\u2016\u222b (y : \u03b3), \u2191\u2191i (x, y) - \u2191\u2191g (x, y) \u2202\u2191\u03b7 (a, x)\u2016\u208a \u2202\u2191\u03ba a) (\ud835\udcdd g) (\ud835\udcdd 0)", "state_after": "case refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nE : Type u_4\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d\u2077 : NormedAddCommGroup E\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u2076 : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d\u2075 : IsSFiniteKernel \u03b7\na : \u03b1\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\nE' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : CompleteSpace E'\ninst\u271d : NormedSpace \u211d E'\ng : { x // x \u2208 Lp E 1 }\n\u22a2 { x // x \u2208 Lp E 1 } \u2192 \u211d\u22650\u221e\n\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nE : Type u_4\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d\u2077 : NormedAddCommGroup E\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u2076 : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d\u2075 : IsSFiniteKernel \u03b7\na : \u03b1\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\nE' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : CompleteSpace E'\ninst\u271d : NormedSpace \u211d E'\ng : { x // x \u2208 Lp E 1 }\n\u22a2 Tendsto ?refine'_1 (\ud835\udcdd g) (\ud835\udcdd 0)\n\ncase refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nE : Type u_4\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d\u2077 : NormedAddCommGroup E\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u2076 : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d\u2075 : IsSFiniteKernel \u03b7\na : \u03b1\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\nE' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : CompleteSpace E'\ninst\u271d : NormedSpace \u211d E'\ng : { x // x \u2208 Lp E 1 }\n\u22a2 (fun i => \u222b\u207b (x : \u03b2), \u2191\u2016\u222b (y : \u03b3), \u2191\u2191i (x, y) - \u2191\u2191g (x, y) \u2202\u2191\u03b7 (a, x)\u2016\u208a \u2202\u2191\u03ba a) \u2264 ?refine'_1"}, {"tactic": "swap", "annotated_tactic": ["swap", []], "state_before": "case refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nE : Type u_4\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d\u2077 : NormedAddCommGroup E\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u2076 : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d\u2075 : IsSFiniteKernel \u03b7\na : \u03b1\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\nE' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : CompleteSpace E'\ninst\u271d : NormedSpace \u211d E'\ng : { x // x \u2208 Lp E 1 }\n\u22a2 Tendsto (fun i => \u222b\u207b (x : \u03b2), \u222b\u207b (y : \u03b3), \u2191\u2016\u2191\u2191i (x, y) - \u2191\u2191g (x, y)\u2016\u208a \u2202\u2191\u03b7 (a, x) \u2202\u2191\u03ba a) (\ud835\udcdd g) (\ud835\udcdd 0)\n\ncase refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nE : Type u_4\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d\u2077 : NormedAddCommGroup E\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u2076 : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d\u2075 : IsSFiniteKernel \u03b7\na : \u03b1\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\nE' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : CompleteSpace E'\ninst\u271d : NormedSpace \u211d E'\ng : { x // x \u2208 Lp E 1 }\n\u22a2 (fun i => \u222b\u207b (x : \u03b2), \u2191\u2016\u222b (y : \u03b3), \u2191\u2191i (x, y) - \u2191\u2191g (x, y) \u2202\u2191\u03b7 (a, x)\u2016\u208a \u2202\u2191\u03ba a) \u2264 fun i =>\n    \u222b\u207b (x : \u03b2), \u222b\u207b (y : \u03b3), \u2191\u2016\u2191\u2191i (x, y) - \u2191\u2191g (x, y)\u2016\u208a \u2202\u2191\u03b7 (a, x) \u2202\u2191\u03ba a", "state_after": "case refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nE : Type u_4\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d\u2077 : NormedAddCommGroup E\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u2076 : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d\u2075 : IsSFiniteKernel \u03b7\na : \u03b1\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\nE' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : CompleteSpace E'\ninst\u271d : NormedSpace \u211d E'\ng : { x // x \u2208 Lp E 1 }\n\u22a2 (fun i => \u222b\u207b (x : \u03b2), \u2191\u2016\u222b (y : \u03b3), \u2191\u2191i (x, y) - \u2191\u2191g (x, y) \u2202\u2191\u03b7 (a, x)\u2016\u208a \u2202\u2191\u03ba a) \u2264 fun i =>\n    \u222b\u207b (x : \u03b2), \u222b\u207b (y : \u03b3), \u2191\u2016\u2191\u2191i (x, y) - \u2191\u2191g (x, y)\u2016\u208a \u2202\u2191\u03b7 (a, x) \u2202\u2191\u03ba a\n\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nE : Type u_4\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d\u2077 : NormedAddCommGroup E\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u2076 : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d\u2075 : IsSFiniteKernel \u03b7\na : \u03b1\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\nE' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : CompleteSpace E'\ninst\u271d : NormedSpace \u211d E'\ng : { x // x \u2208 Lp E 1 }\n\u22a2 Tendsto (fun i => \u222b\u207b (x : \u03b2), \u222b\u207b (y : \u03b3), \u2191\u2016\u2191\u2191i (x, y) - \u2191\u2191g (x, y)\u2016\u208a \u2202\u2191\u03b7 (a, x) \u2202\u2191\u03ba a) (\ud835\udcdd g) (\ud835\udcdd 0)"}, {"tactic": "have : \u2200 i : (MeasureTheory.Lp (\u03b1 := \u03b2 \u00d7 \u03b3) E 1 (((\u03ba \u2297\u2096 \u03b7) a) : Measure (\u03b2 \u00d7 \u03b3))),\n    Measurable fun z => (\u2016i z - g z\u2016\u208a : \u211d\u22650\u221e) := fun i =>\n  ((Lp.stronglyMeasurable i).sub (Lp.stronglyMeasurable g)).ennnorm", "annotated_tactic": ["have : \u2200 i : (<a>MeasureTheory.Lp</a> (\u03b1 := \u03b2 \u00d7 \u03b3) E 1 (((\u03ba \u2297\u2096 \u03b7) a) : <a>Measure</a> (\u03b2 \u00d7 \u03b3))),\n      <a>Measurable</a> fun z => (\u2016i z - g z\u2016\u208a : \u211d\u22650\u221e) := fun i =>\n    ((<a>Lp.stronglyMeasurable</a> i).<a>sub</a> (<a>Lp.stronglyMeasurable</a> g)).<a>ennnorm</a>", [{"full_name": "MeasureTheory.Lp", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [98, 5], "def_end_pos": [98, 7]}, {"full_name": "MeasureTheory.Measure", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [74, 11], "def_end_pos": [74, 18]}, {"full_name": "Measurable", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [535, 5], "def_end_pos": [535, 15]}, {"full_name": "MeasureTheory.Lp.stronglyMeasurable", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [207, 19], "def_end_pos": [207, 37]}, {"full_name": "MeasureTheory.StronglyMeasurable.sub", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [436, 3], "def_end_pos": [436, 14]}, {"full_name": "MeasureTheory.Lp.stronglyMeasurable", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [207, 19], "def_end_pos": [207, 37]}, {"full_name": "MeasureTheory.StronglyMeasurable.ennnorm", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [851, 19], "def_end_pos": [851, 26]}]], "state_before": "case refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nE : Type u_4\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d\u2077 : NormedAddCommGroup E\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u2076 : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d\u2075 : IsSFiniteKernel \u03b7\na : \u03b1\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\nE' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : CompleteSpace E'\ninst\u271d : NormedSpace \u211d E'\ng : { x // x \u2208 Lp E 1 }\n\u22a2 Tendsto (fun i => \u222b\u207b (x : \u03b2), \u222b\u207b (y : \u03b3), \u2191\u2016\u2191\u2191i (x, y) - \u2191\u2191g (x, y)\u2016\u208a \u2202\u2191\u03b7 (a, x) \u2202\u2191\u03ba a) (\ud835\udcdd g) (\ud835\udcdd 0)", "state_after": "case refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nE : Type u_4\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d\u2077 : NormedAddCommGroup E\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u2076 : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d\u2075 : IsSFiniteKernel \u03b7\na : \u03b1\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\nE' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : CompleteSpace E'\ninst\u271d : NormedSpace \u211d E'\ng : { x // x \u2208 Lp E 1 }\nthis : \u2200 (i : { x // x \u2208 Lp E 1 }), Measurable fun z => \u2191\u2016\u2191\u2191i z - \u2191\u2191g z\u2016\u208a\n\u22a2 Tendsto (fun i => \u222b\u207b (x : \u03b2), \u222b\u207b (y : \u03b3), \u2191\u2016\u2191\u2191i (x, y) - \u2191\u2191g (x, y)\u2016\u208a \u2202\u2191\u03b7 (a, x) \u2202\u2191\u03ba a) (\ud835\udcdd g) (\ud835\udcdd 0)"}, {"tactic": "simp_rw [\u2190 kernel.lintegral_compProd _ _ _ (this _), \u2190 L1.ofReal_norm_sub_eq_lintegral, \u2190\n  ofReal_zero]", "annotated_tactic": ["simp_rw [\u2190 <a>kernel.lintegral_compProd</a> _ _ _ (this _), \u2190 <a>L1.ofReal_norm_sub_eq_lintegral</a>, \u2190\n    <a>ofReal_zero</a>]", [{"full_name": "ProbabilityTheory.kernel.lintegral_compProd", "def_path": "Mathlib/Probability/Kernel/Composition.lean", "def_pos": [432, 9], "def_end_pos": [432, 27]}, {"full_name": "MeasureTheory.L1.ofReal_norm_sub_eq_lintegral", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [1386, 9], "def_end_pos": [1386, 37]}, {"full_name": "ENNReal.ofReal_zero", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [245, 17], "def_end_pos": [245, 28]}]], "state_before": "case refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nE : Type u_4\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d\u2077 : NormedAddCommGroup E\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u2076 : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d\u2075 : IsSFiniteKernel \u03b7\na : \u03b1\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\nE' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : CompleteSpace E'\ninst\u271d : NormedSpace \u211d E'\ng : { x // x \u2208 Lp E 1 }\nthis : \u2200 (i : { x // x \u2208 Lp E 1 }), Measurable fun z => \u2191\u2016\u2191\u2191i z - \u2191\u2191g z\u2016\u208a\n\u22a2 Tendsto (fun i => \u222b\u207b (x : \u03b2), \u222b\u207b (y : \u03b3), \u2191\u2016\u2191\u2191i (x, y) - \u2191\u2191g (x, y)\u2016\u208a \u2202\u2191\u03b7 (a, x) \u2202\u2191\u03ba a) (\ud835\udcdd g) (\ud835\udcdd 0)", "state_after": "case refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nE : Type u_4\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d\u2077 : NormedAddCommGroup E\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u2076 : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d\u2075 : IsSFiniteKernel \u03b7\na : \u03b1\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\nE' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : CompleteSpace E'\ninst\u271d : NormedSpace \u211d E'\ng : { x // x \u2208 Lp E 1 }\nthis : \u2200 (i : { x // x \u2208 Lp E 1 }), Measurable fun z => \u2191\u2016\u2191\u2191i z - \u2191\u2191g z\u2016\u208a\n\u22a2 Tendsto (fun i => ENNReal.ofReal \u2016i - g\u2016) (\ud835\udcdd g) (\ud835\udcdd (ENNReal.ofReal 0))"}, {"tactic": "refine' (continuous_ofReal.tendsto 0).comp _", "annotated_tactic": ["refine' (continuous_ofReal.tendsto 0).<a>comp</a> _", [{"full_name": "Filter.Tendsto.comp", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [3032, 9], "def_end_pos": [3032, 21]}]], "state_before": "case refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nE : Type u_4\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d\u2077 : NormedAddCommGroup E\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u2076 : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d\u2075 : IsSFiniteKernel \u03b7\na : \u03b1\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\nE' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : CompleteSpace E'\ninst\u271d : NormedSpace \u211d E'\ng : { x // x \u2208 Lp E 1 }\nthis : \u2200 (i : { x // x \u2208 Lp E 1 }), Measurable fun z => \u2191\u2016\u2191\u2191i z - \u2191\u2191g z\u2016\u208a\n\u22a2 Tendsto (fun i => ENNReal.ofReal \u2016i - g\u2016) (\ud835\udcdd g) (\ud835\udcdd (ENNReal.ofReal 0))", "state_after": "case refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nE : Type u_4\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d\u2077 : NormedAddCommGroup E\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u2076 : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d\u2075 : IsSFiniteKernel \u03b7\na : \u03b1\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\nE' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : CompleteSpace E'\ninst\u271d : NormedSpace \u211d E'\ng : { x // x \u2208 Lp E 1 }\nthis : \u2200 (i : { x // x \u2208 Lp E 1 }), Measurable fun z => \u2191\u2016\u2191\u2191i z - \u2191\u2191g z\u2016\u208a\n\u22a2 Tendsto (fun i => \u2016i - g\u2016) (\ud835\udcdd g) (\ud835\udcdd 0)"}, {"tactic": "rw [\u2190 tendsto_iff_norm_sub_tendsto_zero]", "annotated_tactic": ["rw [\u2190 <a>tendsto_iff_norm_sub_tendsto_zero</a>]", [{"full_name": "tendsto_iff_norm_sub_tendsto_zero", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [1079, 3], "def_end_pos": [1079, 14]}]], "state_before": "case refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nE : Type u_4\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d\u2077 : NormedAddCommGroup E\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u2076 : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d\u2075 : IsSFiniteKernel \u03b7\na : \u03b1\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\nE' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : CompleteSpace E'\ninst\u271d : NormedSpace \u211d E'\ng : { x // x \u2208 Lp E 1 }\nthis : \u2200 (i : { x // x \u2208 Lp E 1 }), Measurable fun z => \u2191\u2016\u2191\u2191i z - \u2191\u2191g z\u2016\u208a\n\u22a2 Tendsto (fun i => \u2016i - g\u2016) (\ud835\udcdd g) (\ud835\udcdd 0)", "state_after": "case refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nE : Type u_4\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d\u2077 : NormedAddCommGroup E\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u2076 : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d\u2075 : IsSFiniteKernel \u03b7\na : \u03b1\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\nE' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : CompleteSpace E'\ninst\u271d : NormedSpace \u211d E'\ng : { x // x \u2208 Lp E 1 }\nthis : \u2200 (i : { x // x \u2208 Lp E 1 }), Measurable fun z => \u2191\u2016\u2191\u2191i z - \u2191\u2191g z\u2016\u208a\n\u22a2 Tendsto (fun i => i) (\ud835\udcdd g) (\ud835\udcdd g)"}, {"tactic": "exact tendsto_id", "annotated_tactic": ["exact <a>tendsto_id</a>", [{"full_name": "Filter.tendsto_id", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [3028, 9], "def_end_pos": [3028, 19]}]], "state_before": "case refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nE : Type u_4\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d\u2077 : NormedAddCommGroup E\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u2076 : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d\u2075 : IsSFiniteKernel \u03b7\na : \u03b1\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\nE' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : CompleteSpace E'\ninst\u271d : NormedSpace \u211d E'\ng : { x // x \u2208 Lp E 1 }\nthis : \u2200 (i : { x // x \u2208 Lp E 1 }), Measurable fun z => \u2191\u2016\u2191\u2191i z - \u2191\u2191g z\u2016\u208a\n\u22a2 Tendsto (fun i => i) (\ud835\udcdd g) (\ud835\udcdd g)", "state_after": "no goals"}, {"tactic": "exact fun i => \u222b\u207b x, \u222b\u207b y, \u2016i (x, y) - g (x, y)\u2016\u208a \u2202\u03b7 (a, x) \u2202\u03ba a", "annotated_tactic": ["exact fun i => \u222b\u207b x, \u222b\u207b y, \u2016i (x, y) - g (x, y)\u2016\u208a \u2202\u03b7 (a, x) \u2202\u03ba a", []], "state_before": "case refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nE : Type u_4\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d\u2077 : NormedAddCommGroup E\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u2076 : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d\u2075 : IsSFiniteKernel \u03b7\na : \u03b1\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\nE' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : CompleteSpace E'\ninst\u271d : NormedSpace \u211d E'\ng : { x // x \u2208 Lp E 1 }\n\u22a2 { x // x \u2208 Lp E 1 } \u2192 \u211d\u22650\u221e", "state_after": "no goals"}, {"tactic": "exact fun i => lintegral_mono fun x => ennnorm_integral_le_lintegral_ennnorm _", "annotated_tactic": ["exact fun i => <a>lintegral_mono</a> fun x => <a>ennnorm_integral_le_lintegral_ennnorm</a> _", [{"full_name": "MeasureTheory.lintegral_mono", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [99, 9], "def_end_pos": [99, 23]}, {"full_name": "MeasureTheory.ennnorm_integral_le_lintegral_ennnorm", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [974, 9], "def_end_pos": [974, 46]}]], "state_before": "case refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nE : Type u_4\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d\u2077 : NormedAddCommGroup E\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u2076 : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d\u2075 : IsSFiniteKernel \u03b7\na : \u03b1\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\nE' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : CompleteSpace E'\ninst\u271d : NormedSpace \u211d E'\ng : { x // x \u2208 Lp E 1 }\n\u22a2 (fun i => \u222b\u207b (x : \u03b2), \u2191\u2016\u222b (y : \u03b3), \u2191\u2191i (x, y) - \u2191\u2191g (x, y) \u2202\u2191\u03b7 (a, x)\u2016\u208a \u2202\u2191\u03ba a) \u2264 fun i =>\n    \u222b\u207b (x : \u03b2), \u222b\u207b (y : \u03b3), \u2191\u2016\u2191\u2191i (x, y) - \u2191\u2191g (x, y)\u2016\u208a \u2202\u2191\u03b7 (a, x) \u2202\u2191\u03ba a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "Finset.empty_subset", "start": [571, 1], "end": [572, 16], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/RBMap/Lemmas.lean", "full_name": "Std.RBNode.Path.ordered_iff", "start": [496, 1], "end": [515, 78], "traced_tactics": [{"tactic": "induction p with\n| root => simp\n| left _ _ x _ ih | right _ _ x _ ih => ?_", "annotated_tactic": ["induction p with\n  | <a>root</a> => simp\n  | <a>left</a> _ _ x _ ih | <a>right</a> _ _ x _ ih => ?_", [{"full_name": "Std.RBNode.Path.root", "def_path": "lake-packages/std/Std/Data/RBMap/Basic.lean", "def_pos": [435, 5], "def_end_pos": [435, 9]}, {"full_name": "Std.RBNode.Path.left", "def_path": "lake-packages/std/Std/Data/RBMap/Basic.lean", "def_pos": [437, 5], "def_end_pos": [437, 9]}, {"full_name": "Std.RBNode.Path.right", "def_path": "lake-packages/std/Std/Data/RBMap/Basic.lean", "def_pos": [439, 5], "def_end_pos": [439, 10]}]], "state_before": "\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\np : Path \u03b1\n\u22a2 Ordered cmp p \u2194\n    List.Pairwise (cmpLT cmp) (listL p) \u2227\n      List.Pairwise (cmpLT cmp) (listR p) \u2227 \u2200 (x : \u03b1), x \u2208 listL p \u2192 \u2200 (y : \u03b1), y \u2208 listR p \u2192 cmpLT cmp x y", "state_after": "case left\n\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nc\u271d : RBColor\nparent\u271d : Path \u03b1\nx : \u03b1\nr\u271d : RBNode \u03b1\nih :\n  Ordered cmp parent\u271d \u2194\n    List.Pairwise (cmpLT cmp) (listL parent\u271d) \u2227\n      List.Pairwise (cmpLT cmp) (listR parent\u271d) \u2227\n        \u2200 (x : \u03b1), x \u2208 listL parent\u271d \u2192 \u2200 (y : \u03b1), y \u2208 listR parent\u271d \u2192 cmpLT cmp x y\n\u22a2 Ordered cmp (left c\u271d parent\u271d x r\u271d) \u2194\n    List.Pairwise (cmpLT cmp) (listL (left c\u271d parent\u271d x r\u271d)) \u2227\n      List.Pairwise (cmpLT cmp) (listR (left c\u271d parent\u271d x r\u271d)) \u2227\n        \u2200 (x_1 : \u03b1), x_1 \u2208 listL (left c\u271d parent\u271d x r\u271d) \u2192 \u2200 (y : \u03b1), y \u2208 listR (left c\u271d parent\u271d x r\u271d) \u2192 cmpLT cmp x_1 y\n\ncase right\n\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nc\u271d : RBColor\nl\u271d : RBNode \u03b1\nx : \u03b1\nparent\u271d : Path \u03b1\nih :\n  Ordered cmp parent\u271d \u2194\n    List.Pairwise (cmpLT cmp) (listL parent\u271d) \u2227\n      List.Pairwise (cmpLT cmp) (listR parent\u271d) \u2227\n        \u2200 (x : \u03b1), x \u2208 listL parent\u271d \u2192 \u2200 (y : \u03b1), y \u2208 listR parent\u271d \u2192 cmpLT cmp x y\n\u22a2 Ordered cmp (right c\u271d l\u271d x parent\u271d) \u2194\n    List.Pairwise (cmpLT cmp) (listL (right c\u271d l\u271d x parent\u271d)) \u2227\n      List.Pairwise (cmpLT cmp) (listR (right c\u271d l\u271d x parent\u271d)) \u2227\n        \u2200 (x_1 : \u03b1),\n          x_1 \u2208 listL (right c\u271d l\u271d x parent\u271d) \u2192 \u2200 (y : \u03b1), y \u2208 listR (right c\u271d l\u271d x parent\u271d) \u2192 cmpLT cmp x_1 y"}, {"tactic": "all_goals\n  rw [Ordered, and_congr_right_eq fun h => by simp [All_def, rootOrdered_iff h]; rfl]\n  simp [List.pairwise_append, or_imp, forall_and, ih, RBNode.ordered_iff]", "annotated_tactic": ["all_goals\n    rw [<a>Ordered</a>, <a>and_congr_right_eq</a> fun h => by simp [<a>All_def</a>, <a>rootOrdered_iff</a> h]; rfl]\n    simp [<a>List.pairwise_append</a>, <a>or_imp</a>, <a>forall_and</a>, ih, <a>RBNode.ordered_iff</a>]", [{"full_name": "Std.RBNode.Path.Ordered", "def_path": "lake-packages/std/Std/Data/RBMap/Alter.lean", "def_pos": [259, 5], "def_end_pos": [259, 12]}, {"full_name": "and_congr_right_eq", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [171, 9], "def_end_pos": [171, 27]}, {"full_name": "Std.RBNode.All_def", "def_path": "lake-packages/std/Std/Data/RBMap/Lemmas.lean", "def_pos": [100, 9], "def_end_pos": [100, 16]}, {"full_name": "Std.RBNode.Path.rootOrdered_iff", "def_path": "lake-packages/std/Std/Data/RBMap/Lemmas.lean", "def_pos": [489, 9], "def_end_pos": [489, 24]}, {"full_name": "List.pairwise_append", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [1457, 9], "def_end_pos": [1457, 24]}, {"full_name": "or_imp", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [337, 9], "def_end_pos": [337, 15]}, {"full_name": "forall_and", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [426, 9], "def_end_pos": [426, 19]}, {"full_name": "Std.RBNode.ordered_iff", "def_path": "lake-packages/std/Std/Data/RBMap/Lemmas.lean", "def_pos": [431, 9], "def_end_pos": [431, 20]}]], "state_before": "case left\n\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nc\u271d : RBColor\nparent\u271d : Path \u03b1\nx : \u03b1\nr\u271d : RBNode \u03b1\nih :\n  Ordered cmp parent\u271d \u2194\n    List.Pairwise (cmpLT cmp) (listL parent\u271d) \u2227\n      List.Pairwise (cmpLT cmp) (listR parent\u271d) \u2227\n        \u2200 (x : \u03b1), x \u2208 listL parent\u271d \u2192 \u2200 (y : \u03b1), y \u2208 listR parent\u271d \u2192 cmpLT cmp x y\n\u22a2 Ordered cmp (left c\u271d parent\u271d x r\u271d) \u2194\n    List.Pairwise (cmpLT cmp) (listL (left c\u271d parent\u271d x r\u271d)) \u2227\n      List.Pairwise (cmpLT cmp) (listR (left c\u271d parent\u271d x r\u271d)) \u2227\n        \u2200 (x_1 : \u03b1), x_1 \u2208 listL (left c\u271d parent\u271d x r\u271d) \u2192 \u2200 (y : \u03b1), y \u2208 listR (left c\u271d parent\u271d x r\u271d) \u2192 cmpLT cmp x_1 y\n\ncase right\n\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nc\u271d : RBColor\nl\u271d : RBNode \u03b1\nx : \u03b1\nparent\u271d : Path \u03b1\nih :\n  Ordered cmp parent\u271d \u2194\n    List.Pairwise (cmpLT cmp) (listL parent\u271d) \u2227\n      List.Pairwise (cmpLT cmp) (listR parent\u271d) \u2227\n        \u2200 (x : \u03b1), x \u2208 listL parent\u271d \u2192 \u2200 (y : \u03b1), y \u2208 listR parent\u271d \u2192 cmpLT cmp x y\n\u22a2 Ordered cmp (right c\u271d l\u271d x parent\u271d) \u2194\n    List.Pairwise (cmpLT cmp) (listL (right c\u271d l\u271d x parent\u271d)) \u2227\n      List.Pairwise (cmpLT cmp) (listR (right c\u271d l\u271d x parent\u271d)) \u2227\n        \u2200 (x_1 : \u03b1),\n          x_1 \u2208 listL (right c\u271d l\u271d x parent\u271d) \u2192 \u2200 (y : \u03b1), y \u2208 listR (right c\u271d l\u271d x parent\u271d) \u2192 cmpLT cmp x_1 y", "state_after": "case left\n\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nc\u271d : RBColor\nparent\u271d : Path \u03b1\nx : \u03b1\nr\u271d : RBNode \u03b1\nih :\n  Ordered cmp parent\u271d \u2194\n    List.Pairwise (cmpLT cmp) (listL parent\u271d) \u2227\n      List.Pairwise (cmpLT cmp) (listR parent\u271d) \u2227\n        \u2200 (x : \u03b1), x \u2208 listL parent\u271d \u2192 \u2200 (y : \u03b1), y \u2208 listR parent\u271d \u2192 cmpLT cmp x y\n\u22a2 (List.Pairwise (cmpLT cmp) (listL parent\u271d) \u2227\n        List.Pairwise (cmpLT cmp) (listR parent\u271d) \u2227\n          \u2200 (x : \u03b1), x \u2208 listL parent\u271d \u2192 \u2200 (y : \u03b1), y \u2208 listR parent\u271d \u2192 cmpLT cmp x y) \u2227\n      (\u2200 (x_1 : \u03b1), x_1 \u2208 r\u271d \u2192 cmpLT cmp x x_1) \u2227\n        ((\u2200 (a : \u03b1), a \u2208 listL parent\u271d \u2192 cmpLT cmp a x) \u2227 \u2200 (a : \u03b1), a \u2208 listR parent\u271d \u2192 cmpLT cmp x a) \u2227\n          ((\u2200 (x : \u03b1), x \u2208 r\u271d \u2192 \u2200 (a : \u03b1), a \u2208 listL parent\u271d \u2192 cmpLT cmp a x) \u2227\n              \u2200 (x : \u03b1), x \u2208 r\u271d \u2192 \u2200 (a : \u03b1), a \u2208 listR parent\u271d \u2192 cmpLT cmp x a) \u2227\n            List.Pairwise (cmpLT cmp) (toList r\u271d) \u2194\n    List.Pairwise (cmpLT cmp) (listL parent\u271d) \u2227\n      (((\u2200 (x_1 : \u03b1), x_1 \u2208 r\u271d \u2192 cmpLT cmp x x_1) \u2227 \u2200 (a : \u03b1), a \u2208 listR parent\u271d \u2192 cmpLT cmp x a) \u2227\n          List.Pairwise (cmpLT cmp) (toList r\u271d) \u2227\n            List.Pairwise (cmpLT cmp) (listR parent\u271d) \u2227\n              \u2200 (x : \u03b1), x \u2208 r\u271d \u2192 \u2200 (a : \u03b1), a \u2208 listR parent\u271d \u2192 cmpLT cmp x a) \u2227\n        (\u2200 (a : \u03b1), a \u2208 listL parent\u271d \u2192 cmpLT cmp a x) \u2227\n          (\u2200 (x : \u03b1), x \u2208 listL parent\u271d \u2192 \u2200 (x_2 : \u03b1), x_2 \u2208 r\u271d \u2192 cmpLT cmp x x_2) \u2227\n            \u2200 (x : \u03b1), x \u2208 listL parent\u271d \u2192 \u2200 (y : \u03b1), y \u2208 listR parent\u271d \u2192 cmpLT cmp x y\n\ncase right\n\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nc\u271d : RBColor\nl\u271d : RBNode \u03b1\nx : \u03b1\nparent\u271d : Path \u03b1\nih :\n  Ordered cmp parent\u271d \u2194\n    List.Pairwise (cmpLT cmp) (listL parent\u271d) \u2227\n      List.Pairwise (cmpLT cmp) (listR parent\u271d) \u2227\n        \u2200 (x : \u03b1), x \u2208 listL parent\u271d \u2192 \u2200 (y : \u03b1), y \u2208 listR parent\u271d \u2192 cmpLT cmp x y\n\u22a2 (List.Pairwise (cmpLT cmp) (listL parent\u271d) \u2227\n        List.Pairwise (cmpLT cmp) (listR parent\u271d) \u2227\n          \u2200 (x : \u03b1), x \u2208 listL parent\u271d \u2192 \u2200 (y : \u03b1), y \u2208 listR parent\u271d \u2192 cmpLT cmp x y) \u2227\n      (\u2200 (x_1 : \u03b1), x_1 \u2208 l\u271d \u2192 cmpLT cmp x_1 x) \u2227\n        ((\u2200 (a : \u03b1), a \u2208 listL parent\u271d \u2192 cmpLT cmp a x) \u2227 \u2200 (a : \u03b1), a \u2208 listR parent\u271d \u2192 cmpLT cmp x a) \u2227\n          ((\u2200 (x : \u03b1), x \u2208 l\u271d \u2192 \u2200 (a : \u03b1), a \u2208 listL parent\u271d \u2192 cmpLT cmp a x) \u2227\n              \u2200 (x : \u03b1), x \u2208 l\u271d \u2192 \u2200 (a : \u03b1), a \u2208 listR parent\u271d \u2192 cmpLT cmp x a) \u2227\n            List.Pairwise (cmpLT cmp) (toList l\u271d) \u2194\n    (List.Pairwise (cmpLT cmp) (listL parent\u271d) \u2227\n        (List.Pairwise (cmpLT cmp) (toList l\u271d) \u2227 \u2200 (x_1 : \u03b1), x_1 \u2208 l\u271d \u2192 cmpLT cmp x_1 x) \u2227\n          (\u2200 (x : \u03b1), x \u2208 listL parent\u271d \u2192 \u2200 (x_2 : \u03b1), x_2 \u2208 l\u271d \u2192 cmpLT cmp x x_2) \u2227\n            \u2200 (a : \u03b1), a \u2208 listL parent\u271d \u2192 cmpLT cmp a x) \u2227\n      List.Pairwise (cmpLT cmp) (listR parent\u271d) \u2227\n        (\u2200 (x : \u03b1), x \u2208 listL parent\u271d \u2192 \u2200 (y : \u03b1), y \u2208 listR parent\u271d \u2192 cmpLT cmp x y) \u2227\n          (\u2200 (x : \u03b1), x \u2208 l\u271d \u2192 \u2200 (a : \u03b1), a \u2208 listR parent\u271d \u2192 cmpLT cmp x a) \u2227\n            \u2200 (a : \u03b1), a \u2208 listR parent\u271d \u2192 cmpLT cmp x a"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case root\n\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\n\u22a2 Ordered cmp root \u2194\n    List.Pairwise (cmpLT cmp) (listL root) \u2227\n      List.Pairwise (cmpLT cmp) (listR root) \u2227 \u2200 (x : \u03b1), x \u2208 listL root \u2192 \u2200 (y : \u03b1), y \u2208 listR root \u2192 cmpLT cmp x y", "state_after": "no goals"}, {"tactic": "rw [Ordered, and_congr_right_eq fun h => by simp [All_def, rootOrdered_iff h]; rfl]", "annotated_tactic": ["rw [<a>Ordered</a>, <a>and_congr_right_eq</a> fun h => by simp [<a>All_def</a>, <a>rootOrdered_iff</a> h]; rfl]", [{"full_name": "Std.RBNode.Path.Ordered", "def_path": "lake-packages/std/Std/Data/RBMap/Alter.lean", "def_pos": [259, 5], "def_end_pos": [259, 12]}, {"full_name": "and_congr_right_eq", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [171, 9], "def_end_pos": [171, 27]}, {"full_name": "Std.RBNode.All_def", "def_path": "lake-packages/std/Std/Data/RBMap/Lemmas.lean", "def_pos": [100, 9], "def_end_pos": [100, 16]}, {"full_name": "Std.RBNode.Path.rootOrdered_iff", "def_path": "lake-packages/std/Std/Data/RBMap/Lemmas.lean", "def_pos": [489, 9], "def_end_pos": [489, 24]}]], "state_before": "case right\n\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nc\u271d : RBColor\nl\u271d : RBNode \u03b1\nx : \u03b1\nparent\u271d : Path \u03b1\nih :\n  Ordered cmp parent\u271d \u2194\n    List.Pairwise (cmpLT cmp) (listL parent\u271d) \u2227\n      List.Pairwise (cmpLT cmp) (listR parent\u271d) \u2227\n        \u2200 (x : \u03b1), x \u2208 listL parent\u271d \u2192 \u2200 (y : \u03b1), y \u2208 listR parent\u271d \u2192 cmpLT cmp x y\n\u22a2 Ordered cmp (right c\u271d l\u271d x parent\u271d) \u2194\n    List.Pairwise (cmpLT cmp) (listL (right c\u271d l\u271d x parent\u271d)) \u2227\n      List.Pairwise (cmpLT cmp) (listR (right c\u271d l\u271d x parent\u271d)) \u2227\n        \u2200 (x_1 : \u03b1),\n          x_1 \u2208 listL (right c\u271d l\u271d x parent\u271d) \u2192 \u2200 (y : \u03b1), y \u2208 listR (right c\u271d l\u271d x parent\u271d) \u2192 cmpLT cmp x_1 y", "state_after": "case right\n\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nc\u271d : RBColor\nl\u271d : RBNode \u03b1\nx : \u03b1\nparent\u271d : Path \u03b1\nih :\n  Ordered cmp parent\u271d \u2194\n    List.Pairwise (cmpLT cmp) (listL parent\u271d) \u2227\n      List.Pairwise (cmpLT cmp) (listR parent\u271d) \u2227\n        \u2200 (x : \u03b1), x \u2208 listL parent\u271d \u2192 \u2200 (y : \u03b1), y \u2208 listR parent\u271d \u2192 cmpLT cmp x y\n\u22a2 Ordered cmp parent\u271d \u2227\n      (\u2200 (x_1 : \u03b1), x_1 \u2208 l\u271d \u2192 cmpLT cmp x_1 x) \u2227\n        ((\u2200 (a : \u03b1), a \u2208 listL parent\u271d \u2192 cmpLT cmp a x) \u2227 \u2200 (a : \u03b1), a \u2208 listR parent\u271d \u2192 cmpLT cmp x a) \u2227\n          (\u2200 (x : \u03b1),\n              x \u2208 l\u271d \u2192 (\u2200 (a : \u03b1), a \u2208 listL parent\u271d \u2192 cmpLT cmp a x) \u2227 \u2200 (a : \u03b1), a \u2208 listR parent\u271d \u2192 cmpLT cmp x a) \u2227\n            RBNode.Ordered cmp l\u271d \u2194\n    List.Pairwise (cmpLT cmp) (listL (right c\u271d l\u271d x parent\u271d)) \u2227\n      List.Pairwise (cmpLT cmp) (listR (right c\u271d l\u271d x parent\u271d)) \u2227\n        \u2200 (x_1 : \u03b1),\n          x_1 \u2208 listL (right c\u271d l\u271d x parent\u271d) \u2192 \u2200 (y : \u03b1), y \u2208 listR (right c\u271d l\u271d x parent\u271d) \u2192 cmpLT cmp x_1 y"}, {"tactic": "simp [List.pairwise_append, or_imp, forall_and, ih, RBNode.ordered_iff]", "annotated_tactic": ["simp [<a>List.pairwise_append</a>, <a>or_imp</a>, <a>forall_and</a>, ih, <a>RBNode.ordered_iff</a>]", [{"full_name": "List.pairwise_append", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [1457, 9], "def_end_pos": [1457, 24]}, {"full_name": "or_imp", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [337, 9], "def_end_pos": [337, 15]}, {"full_name": "forall_and", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [426, 9], "def_end_pos": [426, 19]}, {"full_name": "Std.RBNode.ordered_iff", "def_path": "lake-packages/std/Std/Data/RBMap/Lemmas.lean", "def_pos": [431, 9], "def_end_pos": [431, 20]}]], "state_before": "case right\n\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nc\u271d : RBColor\nl\u271d : RBNode \u03b1\nx : \u03b1\nparent\u271d : Path \u03b1\nih :\n  Ordered cmp parent\u271d \u2194\n    List.Pairwise (cmpLT cmp) (listL parent\u271d) \u2227\n      List.Pairwise (cmpLT cmp) (listR parent\u271d) \u2227\n        \u2200 (x : \u03b1), x \u2208 listL parent\u271d \u2192 \u2200 (y : \u03b1), y \u2208 listR parent\u271d \u2192 cmpLT cmp x y\n\u22a2 Ordered cmp parent\u271d \u2227\n      (\u2200 (x_1 : \u03b1), x_1 \u2208 l\u271d \u2192 cmpLT cmp x_1 x) \u2227\n        ((\u2200 (a : \u03b1), a \u2208 listL parent\u271d \u2192 cmpLT cmp a x) \u2227 \u2200 (a : \u03b1), a \u2208 listR parent\u271d \u2192 cmpLT cmp x a) \u2227\n          (\u2200 (x : \u03b1),\n              x \u2208 l\u271d \u2192 (\u2200 (a : \u03b1), a \u2208 listL parent\u271d \u2192 cmpLT cmp a x) \u2227 \u2200 (a : \u03b1), a \u2208 listR parent\u271d \u2192 cmpLT cmp x a) \u2227\n            RBNode.Ordered cmp l\u271d \u2194\n    List.Pairwise (cmpLT cmp) (listL (right c\u271d l\u271d x parent\u271d)) \u2227\n      List.Pairwise (cmpLT cmp) (listR (right c\u271d l\u271d x parent\u271d)) \u2227\n        \u2200 (x_1 : \u03b1),\n          x_1 \u2208 listL (right c\u271d l\u271d x parent\u271d) \u2192 \u2200 (y : \u03b1), y \u2208 listR (right c\u271d l\u271d x parent\u271d) \u2192 cmpLT cmp x_1 y", "state_after": "case right\n\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nc\u271d : RBColor\nl\u271d : RBNode \u03b1\nx : \u03b1\nparent\u271d : Path \u03b1\nih :\n  Ordered cmp parent\u271d \u2194\n    List.Pairwise (cmpLT cmp) (listL parent\u271d) \u2227\n      List.Pairwise (cmpLT cmp) (listR parent\u271d) \u2227\n        \u2200 (x : \u03b1), x \u2208 listL parent\u271d \u2192 \u2200 (y : \u03b1), y \u2208 listR parent\u271d \u2192 cmpLT cmp x y\n\u22a2 (List.Pairwise (cmpLT cmp) (listL parent\u271d) \u2227\n        List.Pairwise (cmpLT cmp) (listR parent\u271d) \u2227\n          \u2200 (x : \u03b1), x \u2208 listL parent\u271d \u2192 \u2200 (y : \u03b1), y \u2208 listR parent\u271d \u2192 cmpLT cmp x y) \u2227\n      (\u2200 (x_1 : \u03b1), x_1 \u2208 l\u271d \u2192 cmpLT cmp x_1 x) \u2227\n        ((\u2200 (a : \u03b1), a \u2208 listL parent\u271d \u2192 cmpLT cmp a x) \u2227 \u2200 (a : \u03b1), a \u2208 listR parent\u271d \u2192 cmpLT cmp x a) \u2227\n          ((\u2200 (x : \u03b1), x \u2208 l\u271d \u2192 \u2200 (a : \u03b1), a \u2208 listL parent\u271d \u2192 cmpLT cmp a x) \u2227\n              \u2200 (x : \u03b1), x \u2208 l\u271d \u2192 \u2200 (a : \u03b1), a \u2208 listR parent\u271d \u2192 cmpLT cmp x a) \u2227\n            List.Pairwise (cmpLT cmp) (toList l\u271d) \u2194\n    (List.Pairwise (cmpLT cmp) (listL parent\u271d) \u2227\n        (List.Pairwise (cmpLT cmp) (toList l\u271d) \u2227 \u2200 (x_1 : \u03b1), x_1 \u2208 l\u271d \u2192 cmpLT cmp x_1 x) \u2227\n          (\u2200 (x : \u03b1), x \u2208 listL parent\u271d \u2192 \u2200 (x_2 : \u03b1), x_2 \u2208 l\u271d \u2192 cmpLT cmp x x_2) \u2227\n            \u2200 (a : \u03b1), a \u2208 listL parent\u271d \u2192 cmpLT cmp a x) \u2227\n      List.Pairwise (cmpLT cmp) (listR parent\u271d) \u2227\n        (\u2200 (x : \u03b1), x \u2208 listL parent\u271d \u2192 \u2200 (y : \u03b1), y \u2208 listR parent\u271d \u2192 cmpLT cmp x y) \u2227\n          (\u2200 (x : \u03b1), x \u2208 l\u271d \u2192 \u2200 (a : \u03b1), a \u2208 listR parent\u271d \u2192 cmpLT cmp x a) \u2227\n            \u2200 (a : \u03b1), a \u2208 listR parent\u271d \u2192 cmpLT cmp x a"}, {"tactic": "simp [All_def, rootOrdered_iff h]", "annotated_tactic": ["simp [<a>All_def</a>, <a>rootOrdered_iff</a> h]", [{"full_name": "Std.RBNode.All_def", "def_path": "lake-packages/std/Std/Data/RBMap/Lemmas.lean", "def_pos": [100, 9], "def_end_pos": [100, 16]}, {"full_name": "Std.RBNode.Path.rootOrdered_iff", "def_path": "lake-packages/std/Std/Data/RBMap/Lemmas.lean", "def_pos": [489, 9], "def_end_pos": [489, 24]}]], "state_before": "\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nc\u271d : RBColor\nl\u271d : RBNode \u03b1\nx : \u03b1\nparent\u271d : Path \u03b1\nih :\n  Ordered cmp parent\u271d \u2194\n    List.Pairwise (cmpLT cmp) (listL parent\u271d) \u2227\n      List.Pairwise (cmpLT cmp) (listR parent\u271d) \u2227\n        \u2200 (x : \u03b1), x \u2208 listL parent\u271d \u2192 \u2200 (y : \u03b1), y \u2208 listR parent\u271d \u2192 cmpLT cmp x y\nh : Ordered cmp parent\u271d\n\u22a2 (All (fun x_1 => cmpLT cmp x_1 x) l\u271d \u2227\n      RootOrdered cmp parent\u271d x \u2227 All (RootOrdered cmp parent\u271d) l\u271d \u2227 RBNode.Ordered cmp l\u271d) =\n    ?m.118001", "state_after": "\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nc\u271d : RBColor\nl\u271d : RBNode \u03b1\nx : \u03b1\nparent\u271d : Path \u03b1\nih :\n  Ordered cmp parent\u271d \u2194\n    List.Pairwise (cmpLT cmp) (listL parent\u271d) \u2227\n      List.Pairwise (cmpLT cmp) (listR parent\u271d) \u2227\n        \u2200 (x : \u03b1), x \u2208 listL parent\u271d \u2192 \u2200 (y : \u03b1), y \u2208 listR parent\u271d \u2192 cmpLT cmp x y\nh : Ordered cmp parent\u271d\n\u22a2 (\u2200 (x_1 : \u03b1), x_1 \u2208 l\u271d \u2192 cmpLT cmp x_1 x) \u2227\n      ((\u2200 (a : \u03b1), a \u2208 listL parent\u271d \u2192 cmpLT cmp a x) \u2227 \u2200 (a : \u03b1), a \u2208 listR parent\u271d \u2192 cmpLT cmp x a) \u2227\n        (\u2200 (x : \u03b1),\n            x \u2208 l\u271d \u2192 (\u2200 (a : \u03b1), a \u2208 listL parent\u271d \u2192 cmpLT cmp a x) \u2227 \u2200 (a : \u03b1), a \u2208 listR parent\u271d \u2192 cmpLT cmp x a) \u2227\n          RBNode.Ordered cmp l\u271d \u2194\n    ?m.118001"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nc\u271d : RBColor\nl\u271d : RBNode \u03b1\nx : \u03b1\nparent\u271d : Path \u03b1\nih :\n  Ordered cmp parent\u271d \u2194\n    List.Pairwise (cmpLT cmp) (listL parent\u271d) \u2227\n      List.Pairwise (cmpLT cmp) (listR parent\u271d) \u2227\n        \u2200 (x : \u03b1), x \u2208 listL parent\u271d \u2192 \u2200 (y : \u03b1), y \u2208 listR parent\u271d \u2192 cmpLT cmp x y\nh : Ordered cmp parent\u271d\n\u22a2 (\u2200 (x_1 : \u03b1), x_1 \u2208 l\u271d \u2192 cmpLT cmp x_1 x) \u2227\n      ((\u2200 (a : \u03b1), a \u2208 listL parent\u271d \u2192 cmpLT cmp a x) \u2227 \u2200 (a : \u03b1), a \u2208 listR parent\u271d \u2192 cmpLT cmp x a) \u2227\n        (\u2200 (x : \u03b1),\n            x \u2208 l\u271d \u2192 (\u2200 (a : \u03b1), a \u2208 listL parent\u271d \u2192 cmpLT cmp a x) \u2227 \u2200 (a : \u03b1), a \u2208 listR parent\u271d \u2192 cmpLT cmp x a) \u2227\n          RBNode.Ordered cmp l\u271d \u2194\n    ?m.118001", "state_after": "no goals"}, {"tactic": "exact \u27e8\n  fun \u27e8\u27e8hL, hR, LR\u27e9, xr, \u27e8Lx, xR\u27e9, \u27e8rL, rR\u27e9, hr\u27e9 =>\n    \u27e8hL, \u27e8\u27e8xr, xR\u27e9, hr, hR, rR\u27e9, Lx, fun _ ha _ hb => rL _ hb _ ha, LR\u27e9,\n  fun \u27e8hL, \u27e8\u27e8xr, xR\u27e9, hr, hR, rR\u27e9, Lx, Lr, LR\u27e9 =>\n    \u27e8\u27e8hL, hR, LR\u27e9, xr, \u27e8Lx, xR\u27e9, \u27e8fun _ ha _ hb => Lr _ hb _ ha, rR\u27e9, hr\u27e9\u27e9", "annotated_tactic": ["exact \u27e8\n      fun \u27e8\u27e8hL, hR, LR\u27e9, xr, \u27e8Lx, xR\u27e9, \u27e8rL, rR\u27e9, hr\u27e9 =>\n        \u27e8hL, \u27e8\u27e8xr, xR\u27e9, hr, hR, rR\u27e9, Lx, fun _ ha _ hb => rL _ hb _ ha, LR\u27e9,\n      fun \u27e8hL, \u27e8\u27e8xr, xR\u27e9, hr, hR, rR\u27e9, Lx, Lr, LR\u27e9 =>\n        \u27e8\u27e8hL, hR, LR\u27e9, xr, \u27e8Lx, xR\u27e9, \u27e8fun _ ha _ hb => Lr _ hb _ ha, rR\u27e9, hr\u27e9\u27e9", []], "state_before": "case left\n\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nc\u271d : RBColor\nparent\u271d : Path \u03b1\nx : \u03b1\nr\u271d : RBNode \u03b1\nih :\n  Ordered cmp parent\u271d \u2194\n    List.Pairwise (cmpLT cmp) (listL parent\u271d) \u2227\n      List.Pairwise (cmpLT cmp) (listR parent\u271d) \u2227\n        \u2200 (x : \u03b1), x \u2208 listL parent\u271d \u2192 \u2200 (y : \u03b1), y \u2208 listR parent\u271d \u2192 cmpLT cmp x y\n\u22a2 (List.Pairwise (cmpLT cmp) (listL parent\u271d) \u2227\n        List.Pairwise (cmpLT cmp) (listR parent\u271d) \u2227\n          \u2200 (x : \u03b1), x \u2208 listL parent\u271d \u2192 \u2200 (y : \u03b1), y \u2208 listR parent\u271d \u2192 cmpLT cmp x y) \u2227\n      (\u2200 (x_1 : \u03b1), x_1 \u2208 r\u271d \u2192 cmpLT cmp x x_1) \u2227\n        ((\u2200 (a : \u03b1), a \u2208 listL parent\u271d \u2192 cmpLT cmp a x) \u2227 \u2200 (a : \u03b1), a \u2208 listR parent\u271d \u2192 cmpLT cmp x a) \u2227\n          ((\u2200 (x : \u03b1), x \u2208 r\u271d \u2192 \u2200 (a : \u03b1), a \u2208 listL parent\u271d \u2192 cmpLT cmp a x) \u2227\n              \u2200 (x : \u03b1), x \u2208 r\u271d \u2192 \u2200 (a : \u03b1), a \u2208 listR parent\u271d \u2192 cmpLT cmp x a) \u2227\n            List.Pairwise (cmpLT cmp) (toList r\u271d) \u2194\n    List.Pairwise (cmpLT cmp) (listL parent\u271d) \u2227\n      (((\u2200 (x_1 : \u03b1), x_1 \u2208 r\u271d \u2192 cmpLT cmp x x_1) \u2227 \u2200 (a : \u03b1), a \u2208 listR parent\u271d \u2192 cmpLT cmp x a) \u2227\n          List.Pairwise (cmpLT cmp) (toList r\u271d) \u2227\n            List.Pairwise (cmpLT cmp) (listR parent\u271d) \u2227\n              \u2200 (x : \u03b1), x \u2208 r\u271d \u2192 \u2200 (a : \u03b1), a \u2208 listR parent\u271d \u2192 cmpLT cmp x a) \u2227\n        (\u2200 (a : \u03b1), a \u2208 listL parent\u271d \u2192 cmpLT cmp a x) \u2227\n          (\u2200 (x : \u03b1), x \u2208 listL parent\u271d \u2192 \u2200 (x_2 : \u03b1), x_2 \u2208 r\u271d \u2192 cmpLT cmp x x_2) \u2227\n            \u2200 (x : \u03b1), x \u2208 listL parent\u271d \u2192 \u2200 (y : \u03b1), y \u2208 listR parent\u271d \u2192 cmpLT cmp x y", "state_after": "no goals"}, {"tactic": "exact \u27e8\n  fun \u27e8\u27e8hL, hR, LR\u27e9, lx, \u27e8Lx, xR\u27e9, \u27e8lL, lR\u27e9, hl\u27e9 =>\n    \u27e8\u27e8hL, \u27e8hl, lx\u27e9, fun _ ha _ hb => lL _ hb _ ha, Lx\u27e9, hR, LR, lR, xR\u27e9,\n  fun \u27e8\u27e8hL, \u27e8hl, lx\u27e9, Ll, Lx\u27e9, hR, LR, lR, xR\u27e9 =>\n   \u27e8\u27e8hL, hR, LR\u27e9, lx, \u27e8Lx, xR\u27e9, \u27e8fun _ ha _ hb => Ll _ hb _ ha, lR\u27e9, hl\u27e9\u27e9", "annotated_tactic": ["exact \u27e8\n      fun \u27e8\u27e8hL, hR, LR\u27e9, lx, \u27e8Lx, xR\u27e9, \u27e8lL, lR\u27e9, hl\u27e9 =>\n        \u27e8\u27e8hL, \u27e8hl, lx\u27e9, fun _ ha _ hb => lL _ hb _ ha, Lx\u27e9, hR, LR, lR, xR\u27e9,\n      fun \u27e8\u27e8hL, \u27e8hl, lx\u27e9, Ll, Lx\u27e9, hR, LR, lR, xR\u27e9 =>\n       \u27e8\u27e8hL, hR, LR\u27e9, lx, \u27e8Lx, xR\u27e9, \u27e8fun _ ha _ hb => Ll _ hb _ ha, lR\u27e9, hl\u27e9\u27e9", []], "state_before": "case right\n\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nc\u271d : RBColor\nl\u271d : RBNode \u03b1\nx : \u03b1\nparent\u271d : Path \u03b1\nih :\n  Ordered cmp parent\u271d \u2194\n    List.Pairwise (cmpLT cmp) (listL parent\u271d) \u2227\n      List.Pairwise (cmpLT cmp) (listR parent\u271d) \u2227\n        \u2200 (x : \u03b1), x \u2208 listL parent\u271d \u2192 \u2200 (y : \u03b1), y \u2208 listR parent\u271d \u2192 cmpLT cmp x y\n\u22a2 (List.Pairwise (cmpLT cmp) (listL parent\u271d) \u2227\n        List.Pairwise (cmpLT cmp) (listR parent\u271d) \u2227\n          \u2200 (x : \u03b1), x \u2208 listL parent\u271d \u2192 \u2200 (y : \u03b1), y \u2208 listR parent\u271d \u2192 cmpLT cmp x y) \u2227\n      (\u2200 (x_1 : \u03b1), x_1 \u2208 l\u271d \u2192 cmpLT cmp x_1 x) \u2227\n        ((\u2200 (a : \u03b1), a \u2208 listL parent\u271d \u2192 cmpLT cmp a x) \u2227 \u2200 (a : \u03b1), a \u2208 listR parent\u271d \u2192 cmpLT cmp x a) \u2227\n          ((\u2200 (x : \u03b1), x \u2208 l\u271d \u2192 \u2200 (a : \u03b1), a \u2208 listL parent\u271d \u2192 cmpLT cmp a x) \u2227\n              \u2200 (x : \u03b1), x \u2208 l\u271d \u2192 \u2200 (a : \u03b1), a \u2208 listR parent\u271d \u2192 cmpLT cmp x a) \u2227\n            List.Pairwise (cmpLT cmp) (toList l\u271d) \u2194\n    (List.Pairwise (cmpLT cmp) (listL parent\u271d) \u2227\n        (List.Pairwise (cmpLT cmp) (toList l\u271d) \u2227 \u2200 (x_1 : \u03b1), x_1 \u2208 l\u271d \u2192 cmpLT cmp x_1 x) \u2227\n          (\u2200 (x : \u03b1), x \u2208 listL parent\u271d \u2192 \u2200 (x_2 : \u03b1), x_2 \u2208 l\u271d \u2192 cmpLT cmp x x_2) \u2227\n            \u2200 (a : \u03b1), a \u2208 listL parent\u271d \u2192 cmpLT cmp a x) \u2227\n      List.Pairwise (cmpLT cmp) (listR parent\u271d) \u2227\n        (\u2200 (x : \u03b1), x \u2208 listL parent\u271d \u2192 \u2200 (y : \u03b1), y \u2208 listR parent\u271d \u2192 cmpLT cmp x y) \u2227\n          (\u2200 (x : \u03b1), x \u2208 l\u271d \u2192 \u2200 (a : \u03b1), a \u2208 listR parent\u271d \u2192 cmpLT cmp x a) \u2227\n            \u2200 (a : \u03b1), a \u2208 listR parent\u271d \u2192 cmpLT cmp x a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Quot.lean", "full_name": "Quotient.out_inj", "start": [408, 1], "end": [409, 54], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "full_name": "MeasureTheory.stronglyMeasurable_of_isEmpty", "start": [133, 1], "end": [135, 47], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Sort.lean", "full_name": "Finset.mem_sort", "start": [58, 1], "end": [59, 22], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/PDeriv.lean", "full_name": "MvPolynomial.pderiv_C", "start": [82, 1], "end": [83, 19], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Dirac.lean", "full_name": "MeasureTheory.ae_dirac_iff", "start": [111, 1], "end": [113, 22], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Finite.lean", "full_name": "Set.empty_card'", "start": [1205, 1], "end": [1206, 62], "traced_tactics": [{"tactic": "congr", "annotated_tactic": ["congr", []], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\n\u03b3 : Type x\nh : Fintype \u2191\u2205\n\u22a2 Fintype.card \u2191\u2205 = Fintype.card \u2191\u2205", "state_after": "case h.e_2.h\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\n\u03b3 : Type x\nh : Fintype \u2191\u2205\n\u22a2 h = fintypeEmpty"}, {"tactic": "exact Subsingleton.elim _ _", "annotated_tactic": ["exact <a>Subsingleton.elim</a> _ _", [{"full_name": "Subsingleton.elim", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [873, 19], "def_end_pos": [873, 36]}]], "state_before": "case h.e_2.h\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\n\u03b3 : Type x\nh : Fintype \u2191\u2205\n\u22a2 h = fintypeEmpty", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Num/Lemmas.lean", "full_name": "Num.div_to_nat", "start": [1639, 1], "end": [1643, 43], "traced_tactics": [{"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u22a2 \u2191(0 / 0) = \u21910 / \u21910", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Independence/ZeroOne.lean", "full_name": "ProbabilityTheory.measure_eq_zero_or_one_of_indepSetCat_self", "start": [45, 1], "end": [48, 43], "traced_tactics": [{"tactic": "have h_0_1_top := measure_eq_zero_or_one_or_top_of_indepSet_self h_indep", "annotated_tactic": ["have h_0_1_top := <a>measure_eq_zero_or_one_or_top_of_indepSet_self</a> h_indep", [{"full_name": "ProbabilityTheory.measure_eq_zero_or_one_or_top_of_indepSet_self", "def_path": "Mathlib/Probability/Independence/ZeroOne.lean", "def_pos": [32, 9], "def_end_pos": [32, 55]}]], "state_before": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm m0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d : IsFiniteMeasure \u03bc\nt : Set \u03a9\nh_indep : IndepSet t t\n\u22a2 \u2191\u2191\u03bc t = 0 \u2228 \u2191\u2191\u03bc t = 1", "state_after": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm m0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d : IsFiniteMeasure \u03bc\nt : Set \u03a9\nh_indep : IndepSet t t\nh_0_1_top : \u2191\u2191\u03bc t = 0 \u2228 \u2191\u2191\u03bc t = 1 \u2228 \u2191\u2191\u03bc t = \u22a4\n\u22a2 \u2191\u2191\u03bc t = 0 \u2228 \u2191\u2191\u03bc t = 1"}, {"tactic": "simpa [measure_ne_top \u03bc] using h_0_1_top", "annotated_tactic": ["simpa [<a>measure_ne_top</a> \u03bc] using h_0_1_top", [{"full_name": "MeasureTheory.measure_ne_top", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2875, 9], "def_end_pos": [2875, 23]}]], "state_before": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm m0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d : IsFiniteMeasure \u03bc\nt : Set \u03a9\nh_indep : IndepSet t t\nh_0_1_top : \u2191\u2191\u03bc t = 0 \u2228 \u2191\u2191\u03bc t = 1 \u2228 \u2191\u2191\u03bc t = \u22a4\n\u22a2 \u2191\u2191\u03bc t = 0 \u2228 \u2191\u2191\u03bc t = 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean", "full_name": "MeasureTheory.Measure.map_linearMap_addHaar_eq_smul_addHaar", "start": [242, 1], "end": [267, 68], "traced_tactics": [{"tactic": "let \u03b9 := Fin (finrank \u211d E)", "annotated_tactic": ["let \u03b9 := <a>Fin</a> (<a>finrank</a> \u211d E)", [{"full_name": "Fin", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1745, 11], "def_end_pos": [1745, 14]}, {"full_name": "FiniteDimensional.finrank", "def_path": "Mathlib/LinearAlgebra/Finrank.lean", "def_pos": [58, 19], "def_end_pos": [58, 26]}]], "state_before": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf : E \u2192\u2097[\u211d] E\nhf : \u2191LinearMap.det f \u2260 0\n\u22a2 map (\u2191f) \u03bc = ENNReal.ofReal |(\u2191LinearMap.det f)\u207b\u00b9| \u2022 \u03bc", "state_after": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf : E \u2192\u2097[\u211d] E\nhf : \u2191LinearMap.det f \u2260 0\n\u03b9 : Type := Fin (finrank \u211d E)\n\u22a2 map (\u2191f) \u03bc = ENNReal.ofReal |(\u2191LinearMap.det f)\u207b\u00b9| \u2022 \u03bc"}, {"tactic": "haveI : FiniteDimensional \u211d (\u03b9 \u2192 \u211d) := by infer_instance", "annotated_tactic": ["haveI : <a>FiniteDimensional</a> \u211d (\u03b9 \u2192 \u211d) := by infer_instance", [{"full_name": "FiniteDimensional", "def_path": "Mathlib/LinearAlgebra/FiniteDimensional.lean", "def_pos": [87, 5], "def_end_pos": [87, 22]}]], "state_before": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf : E \u2192\u2097[\u211d] E\nhf : \u2191LinearMap.det f \u2260 0\n\u03b9 : Type := Fin (finrank \u211d E)\n\u22a2 map (\u2191f) \u03bc = ENNReal.ofReal |(\u2191LinearMap.det f)\u207b\u00b9| \u2022 \u03bc", "state_after": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf : E \u2192\u2097[\u211d] E\nhf : \u2191LinearMap.det f \u2260 0\n\u03b9 : Type := Fin (finrank \u211d E)\nthis : FiniteDimensional \u211d (\u03b9 \u2192 \u211d)\n\u22a2 map (\u2191f) \u03bc = ENNReal.ofReal |(\u2191LinearMap.det f)\u207b\u00b9| \u2022 \u03bc"}, {"tactic": "have : finrank \u211d E = finrank \u211d (\u03b9 \u2192 \u211d) := by simp", "annotated_tactic": ["have : <a>finrank</a> \u211d E = <a>finrank</a> \u211d (\u03b9 \u2192 \u211d) := by simp", [{"full_name": "FiniteDimensional.finrank", "def_path": "Mathlib/LinearAlgebra/Finrank.lean", "def_pos": [58, 19], "def_end_pos": [58, 26]}, {"full_name": "FiniteDimensional.finrank", "def_path": "Mathlib/LinearAlgebra/Finrank.lean", "def_pos": [58, 19], "def_end_pos": [58, 26]}]], "state_before": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf : E \u2192\u2097[\u211d] E\nhf : \u2191LinearMap.det f \u2260 0\n\u03b9 : Type := Fin (finrank \u211d E)\nthis : FiniteDimensional \u211d (\u03b9 \u2192 \u211d)\n\u22a2 map (\u2191f) \u03bc = ENNReal.ofReal |(\u2191LinearMap.det f)\u207b\u00b9| \u2022 \u03bc", "state_after": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf : E \u2192\u2097[\u211d] E\nhf : \u2191LinearMap.det f \u2260 0\n\u03b9 : Type := Fin (finrank \u211d E)\nthis\u271d : FiniteDimensional \u211d (\u03b9 \u2192 \u211d)\nthis : finrank \u211d E = finrank \u211d (\u03b9 \u2192 \u211d)\n\u22a2 map (\u2191f) \u03bc = ENNReal.ofReal |(\u2191LinearMap.det f)\u207b\u00b9| \u2022 \u03bc"}, {"tactic": "have e : E \u2243\u2097[\u211d] \u03b9 \u2192 \u211d := LinearEquiv.ofFinrankEq E (\u03b9 \u2192 \u211d) this", "annotated_tactic": ["have e : E \u2243\u2097[\u211d] \u03b9 \u2192 \u211d := <a>LinearEquiv.ofFinrankEq</a> E (\u03b9 \u2192 \u211d) this", [{"full_name": "LinearEquiv.ofFinrankEq", "def_path": "Mathlib/LinearAlgebra/FreeModule/Finite/Rank.lean", "def_pos": [153, 19], "def_end_pos": [153, 49]}]], "state_before": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf : E \u2192\u2097[\u211d] E\nhf : \u2191LinearMap.det f \u2260 0\n\u03b9 : Type := Fin (finrank \u211d E)\nthis\u271d : FiniteDimensional \u211d (\u03b9 \u2192 \u211d)\nthis : finrank \u211d E = finrank \u211d (\u03b9 \u2192 \u211d)\n\u22a2 map (\u2191f) \u03bc = ENNReal.ofReal |(\u2191LinearMap.det f)\u207b\u00b9| \u2022 \u03bc", "state_after": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf : E \u2192\u2097[\u211d] E\nhf : \u2191LinearMap.det f \u2260 0\n\u03b9 : Type := Fin (finrank \u211d E)\nthis\u271d : FiniteDimensional \u211d (\u03b9 \u2192 \u211d)\nthis : finrank \u211d E = finrank \u211d (\u03b9 \u2192 \u211d)\ne : E \u2243\u2097[\u211d] \u03b9 \u2192 \u211d\n\u22a2 map (\u2191f) \u03bc = ENNReal.ofReal |(\u2191LinearMap.det f)\u207b\u00b9| \u2022 \u03bc"}, {"tactic": "obtain \u27e8g, hg\u27e9 : \u2203 g, g = (e : E \u2192\u2097[\u211d] \u03b9 \u2192 \u211d).comp (f.comp (e.symm : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] E)) := \u27e8_, rfl\u27e9", "annotated_tactic": ["obtain \u27e8g, hg\u27e9 : \u2203 g, g = (e : E \u2192\u2097[\u211d] \u03b9 \u2192 \u211d).<a>comp</a> (f.comp (e.symm : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] E)) := \u27e8_, <a>rfl</a>\u27e9", [{"full_name": "LinearMap.comp", "def_path": "Mathlib/Algebra/Module/LinearMap.lean", "def_pos": [536, 5], "def_end_pos": [536, 9]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf : E \u2192\u2097[\u211d] E\nhf : \u2191LinearMap.det f \u2260 0\n\u03b9 : Type := Fin (finrank \u211d E)\nthis\u271d : FiniteDimensional \u211d (\u03b9 \u2192 \u211d)\nthis : finrank \u211d E = finrank \u211d (\u03b9 \u2192 \u211d)\ne : E \u2243\u2097[\u211d] \u03b9 \u2192 \u211d\n\u22a2 map (\u2191f) \u03bc = ENNReal.ofReal |(\u2191LinearMap.det f)\u207b\u00b9| \u2022 \u03bc", "state_after": "case intro\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf : E \u2192\u2097[\u211d] E\nhf : \u2191LinearMap.det f \u2260 0\n\u03b9 : Type := Fin (finrank \u211d E)\nthis\u271d : FiniteDimensional \u211d (\u03b9 \u2192 \u211d)\nthis : finrank \u211d E = finrank \u211d (\u03b9 \u2192 \u211d)\ne : E \u2243\u2097[\u211d] \u03b9 \u2192 \u211d\ng : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d\nhg : g = LinearMap.comp (\u2191e) (LinearMap.comp f \u2191(LinearEquiv.symm e))\n\u22a2 map (\u2191f) \u03bc = ENNReal.ofReal |(\u2191LinearMap.det f)\u207b\u00b9| \u2022 \u03bc"}, {"tactic": "have gdet : LinearMap.det g = LinearMap.det f := by rw [hg]; exact LinearMap.det_conj f e", "annotated_tactic": ["have gdet : <a>LinearMap.det</a> g = <a>LinearMap.det</a> f := by rw [hg]; exact <a>LinearMap.det_conj</a> f e", [{"full_name": "LinearMap.det", "def_path": "Mathlib/LinearAlgebra/Determinant.lean", "def_pos": [178, 27], "def_end_pos": [178, 30]}, {"full_name": "LinearMap.det", "def_path": "Mathlib/LinearAlgebra/Determinant.lean", "def_pos": [178, 27], "def_end_pos": [178, 30]}, {"full_name": "LinearMap.det_conj", "def_path": "Mathlib/LinearAlgebra/Determinant.lean", "def_pos": [310, 9], "def_end_pos": [310, 17]}]], "state_before": "case intro\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf : E \u2192\u2097[\u211d] E\nhf : \u2191LinearMap.det f \u2260 0\n\u03b9 : Type := Fin (finrank \u211d E)\nthis\u271d : FiniteDimensional \u211d (\u03b9 \u2192 \u211d)\nthis : finrank \u211d E = finrank \u211d (\u03b9 \u2192 \u211d)\ne : E \u2243\u2097[\u211d] \u03b9 \u2192 \u211d\ng : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d\nhg : g = LinearMap.comp (\u2191e) (LinearMap.comp f \u2191(LinearEquiv.symm e))\n\u22a2 map (\u2191f) \u03bc = ENNReal.ofReal |(\u2191LinearMap.det f)\u207b\u00b9| \u2022 \u03bc", "state_after": "case intro\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf : E \u2192\u2097[\u211d] E\nhf : \u2191LinearMap.det f \u2260 0\n\u03b9 : Type := Fin (finrank \u211d E)\nthis\u271d : FiniteDimensional \u211d (\u03b9 \u2192 \u211d)\nthis : finrank \u211d E = finrank \u211d (\u03b9 \u2192 \u211d)\ne : E \u2243\u2097[\u211d] \u03b9 \u2192 \u211d\ng : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d\nhg : g = LinearMap.comp (\u2191e) (LinearMap.comp f \u2191(LinearEquiv.symm e))\ngdet : \u2191LinearMap.det g = \u2191LinearMap.det f\n\u22a2 map (\u2191f) \u03bc = ENNReal.ofReal |(\u2191LinearMap.det f)\u207b\u00b9| \u2022 \u03bc"}, {"tactic": "rw [\u2190 gdet] at hf \u22a2", "annotated_tactic": ["rw [\u2190 gdet] at hf \u22a2", []], "state_before": "case intro\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf : E \u2192\u2097[\u211d] E\nhf : \u2191LinearMap.det f \u2260 0\n\u03b9 : Type := Fin (finrank \u211d E)\nthis\u271d : FiniteDimensional \u211d (\u03b9 \u2192 \u211d)\nthis : finrank \u211d E = finrank \u211d (\u03b9 \u2192 \u211d)\ne : E \u2243\u2097[\u211d] \u03b9 \u2192 \u211d\ng : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d\nhg : g = LinearMap.comp (\u2191e) (LinearMap.comp f \u2191(LinearEquiv.symm e))\ngdet : \u2191LinearMap.det g = \u2191LinearMap.det f\n\u22a2 map (\u2191f) \u03bc = ENNReal.ofReal |(\u2191LinearMap.det f)\u207b\u00b9| \u2022 \u03bc", "state_after": "case intro\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf : E \u2192\u2097[\u211d] E\n\u03b9 : Type := Fin (finrank \u211d E)\nthis\u271d : FiniteDimensional \u211d (\u03b9 \u2192 \u211d)\nthis : finrank \u211d E = finrank \u211d (\u03b9 \u2192 \u211d)\ne : E \u2243\u2097[\u211d] \u03b9 \u2192 \u211d\ng : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d\nhf : \u2191LinearMap.det g \u2260 0\nhg : g = LinearMap.comp (\u2191e) (LinearMap.comp f \u2191(LinearEquiv.symm e))\ngdet : \u2191LinearMap.det g = \u2191LinearMap.det f\n\u22a2 map (\u2191f) \u03bc = ENNReal.ofReal |(\u2191LinearMap.det g)\u207b\u00b9| \u2022 \u03bc"}, {"tactic": "have fg : f = (e.symm : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] E).comp (g.comp (e : E \u2192\u2097[\u211d] \u03b9 \u2192 \u211d)) := by\n  ext x\n  simp only [LinearEquiv.coe_coe, Function.comp_apply, LinearMap.coe_comp,\n    LinearEquiv.symm_apply_apply, hg]", "annotated_tactic": ["have fg : f = (e.symm : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] E).<a>comp</a> (g.comp (e : E \u2192\u2097[\u211d] \u03b9 \u2192 \u211d)) := by\n    ext x\n    simp only [<a>LinearEquiv.coe_coe</a>, <a>Function.comp_apply</a>, <a>LinearMap.coe_comp</a>,\n      <a>LinearEquiv.symm_apply_apply</a>, hg]", [{"full_name": "LinearMap.comp", "def_path": "Mathlib/Algebra/Module/LinearMap.lean", "def_pos": [536, 5], "def_end_pos": [536, 9]}, {"full_name": "LinearEquiv.coe_coe", "def_path": "Mathlib/Algebra/Module/Equiv.lean", "def_pos": [211, 9], "def_end_pos": [211, 16]}, {"full_name": "Function.comp_apply", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [33, 17], "def_end_pos": [33, 36]}, {"full_name": "LinearMap.coe_comp", "def_path": "Mathlib/Algebra/Module/LinearMap.lean", "def_pos": [554, 9], "def_end_pos": [554, 17]}, {"full_name": "LinearEquiv.symm_apply_apply", "def_path": "Mathlib/Algebra/Module/Equiv.lean", "def_pos": [379, 9], "def_end_pos": [379, 25]}]], "state_before": "case intro\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf : E \u2192\u2097[\u211d] E\n\u03b9 : Type := Fin (finrank \u211d E)\nthis\u271d : FiniteDimensional \u211d (\u03b9 \u2192 \u211d)\nthis : finrank \u211d E = finrank \u211d (\u03b9 \u2192 \u211d)\ne : E \u2243\u2097[\u211d] \u03b9 \u2192 \u211d\ng : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d\nhf : \u2191LinearMap.det g \u2260 0\nhg : g = LinearMap.comp (\u2191e) (LinearMap.comp f \u2191(LinearEquiv.symm e))\ngdet : \u2191LinearMap.det g = \u2191LinearMap.det f\n\u22a2 map (\u2191f) \u03bc = ENNReal.ofReal |(\u2191LinearMap.det g)\u207b\u00b9| \u2022 \u03bc", "state_after": "case intro\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf : E \u2192\u2097[\u211d] E\n\u03b9 : Type := Fin (finrank \u211d E)\nthis\u271d : FiniteDimensional \u211d (\u03b9 \u2192 \u211d)\nthis : finrank \u211d E = finrank \u211d (\u03b9 \u2192 \u211d)\ne : E \u2243\u2097[\u211d] \u03b9 \u2192 \u211d\ng : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d\nhf : \u2191LinearMap.det g \u2260 0\nhg : g = LinearMap.comp (\u2191e) (LinearMap.comp f \u2191(LinearEquiv.symm e))\ngdet : \u2191LinearMap.det g = \u2191LinearMap.det f\nfg : f = LinearMap.comp (\u2191(LinearEquiv.symm e)) (LinearMap.comp g \u2191e)\n\u22a2 map (\u2191f) \u03bc = ENNReal.ofReal |(\u2191LinearMap.det g)\u207b\u00b9| \u2022 \u03bc"}, {"tactic": "simp only [fg, LinearEquiv.coe_coe, LinearMap.coe_comp]", "annotated_tactic": ["simp only [fg, <a>LinearEquiv.coe_coe</a>, <a>LinearMap.coe_comp</a>]", [{"full_name": "LinearEquiv.coe_coe", "def_path": "Mathlib/Algebra/Module/Equiv.lean", "def_pos": [211, 9], "def_end_pos": [211, 16]}, {"full_name": "LinearMap.coe_comp", "def_path": "Mathlib/Algebra/Module/LinearMap.lean", "def_pos": [554, 9], "def_end_pos": [554, 17]}]], "state_before": "case intro\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf : E \u2192\u2097[\u211d] E\n\u03b9 : Type := Fin (finrank \u211d E)\nthis\u271d : FiniteDimensional \u211d (\u03b9 \u2192 \u211d)\nthis : finrank \u211d E = finrank \u211d (\u03b9 \u2192 \u211d)\ne : E \u2243\u2097[\u211d] \u03b9 \u2192 \u211d\ng : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d\nhf : \u2191LinearMap.det g \u2260 0\nhg : g = LinearMap.comp (\u2191e) (LinearMap.comp f \u2191(LinearEquiv.symm e))\ngdet : \u2191LinearMap.det g = \u2191LinearMap.det f\nfg : f = LinearMap.comp (\u2191(LinearEquiv.symm e)) (LinearMap.comp g \u2191e)\n\u22a2 map (\u2191f) \u03bc = ENNReal.ofReal |(\u2191LinearMap.det g)\u207b\u00b9| \u2022 \u03bc", "state_after": "case intro\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf : E \u2192\u2097[\u211d] E\n\u03b9 : Type := Fin (finrank \u211d E)\nthis\u271d : FiniteDimensional \u211d (\u03b9 \u2192 \u211d)\nthis : finrank \u211d E = finrank \u211d (\u03b9 \u2192 \u211d)\ne : E \u2243\u2097[\u211d] \u03b9 \u2192 \u211d\ng : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d\nhf : \u2191LinearMap.det g \u2260 0\nhg : g = LinearMap.comp (\u2191e) (LinearMap.comp f \u2191(LinearEquiv.symm e))\ngdet : \u2191LinearMap.det g = \u2191LinearMap.det f\nfg : f = LinearMap.comp (\u2191(LinearEquiv.symm e)) (LinearMap.comp g \u2191e)\n\u22a2 map (\u2191(LinearEquiv.symm e) \u2218 \u2191g \u2218 \u2191e) \u03bc = ENNReal.ofReal |(\u2191LinearMap.det g)\u207b\u00b9| \u2022 \u03bc"}, {"tactic": "have Ce : Continuous e := (e : E \u2192\u2097[\u211d] \u03b9 \u2192 \u211d).continuous_of_finiteDimensional", "annotated_tactic": ["have Ce : <a>Continuous</a> e := (e : E \u2192\u2097[\u211d] \u03b9 \u2192 \u211d).<a>continuous_of_finiteDimensional</a>", [{"full_name": "Continuous", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [1591, 11], "def_end_pos": [1591, 21]}, {"full_name": "LinearMap.continuous_of_finiteDimensional", "def_path": "Mathlib/Topology/Algebra/Module/FiniteDimension.lean", "def_pos": [246, 9], "def_end_pos": [246, 50]}]], "state_before": "case intro\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf : E \u2192\u2097[\u211d] E\n\u03b9 : Type := Fin (finrank \u211d E)\nthis\u271d : FiniteDimensional \u211d (\u03b9 \u2192 \u211d)\nthis : finrank \u211d E = finrank \u211d (\u03b9 \u2192 \u211d)\ne : E \u2243\u2097[\u211d] \u03b9 \u2192 \u211d\ng : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d\nhf : \u2191LinearMap.det g \u2260 0\nhg : g = LinearMap.comp (\u2191e) (LinearMap.comp f \u2191(LinearEquiv.symm e))\ngdet : \u2191LinearMap.det g = \u2191LinearMap.det f\nfg : f = LinearMap.comp (\u2191(LinearEquiv.symm e)) (LinearMap.comp g \u2191e)\n\u22a2 map (\u2191(LinearEquiv.symm e) \u2218 \u2191g \u2218 \u2191e) \u03bc = ENNReal.ofReal |(\u2191LinearMap.det g)\u207b\u00b9| \u2022 \u03bc", "state_after": "case intro\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf : E \u2192\u2097[\u211d] E\n\u03b9 : Type := Fin (finrank \u211d E)\nthis\u271d : FiniteDimensional \u211d (\u03b9 \u2192 \u211d)\nthis : finrank \u211d E = finrank \u211d (\u03b9 \u2192 \u211d)\ne : E \u2243\u2097[\u211d] \u03b9 \u2192 \u211d\ng : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d\nhf : \u2191LinearMap.det g \u2260 0\nhg : g = LinearMap.comp (\u2191e) (LinearMap.comp f \u2191(LinearEquiv.symm e))\ngdet : \u2191LinearMap.det g = \u2191LinearMap.det f\nfg : f = LinearMap.comp (\u2191(LinearEquiv.symm e)) (LinearMap.comp g \u2191e)\nCe : Continuous \u2191e\n\u22a2 map (\u2191(LinearEquiv.symm e) \u2218 \u2191g \u2218 \u2191e) \u03bc = ENNReal.ofReal |(\u2191LinearMap.det g)\u207b\u00b9| \u2022 \u03bc"}, {"tactic": "have Cg : Continuous g := LinearMap.continuous_of_finiteDimensional g", "annotated_tactic": ["have Cg : <a>Continuous</a> g := <a>LinearMap.continuous_of_finiteDimensional</a> g", [{"full_name": "Continuous", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [1591, 11], "def_end_pos": [1591, 21]}, {"full_name": "LinearMap.continuous_of_finiteDimensional", "def_path": "Mathlib/Topology/Algebra/Module/FiniteDimension.lean", "def_pos": [246, 9], "def_end_pos": [246, 50]}]], "state_before": "case intro\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf : E \u2192\u2097[\u211d] E\n\u03b9 : Type := Fin (finrank \u211d E)\nthis\u271d : FiniteDimensional \u211d (\u03b9 \u2192 \u211d)\nthis : finrank \u211d E = finrank \u211d (\u03b9 \u2192 \u211d)\ne : E \u2243\u2097[\u211d] \u03b9 \u2192 \u211d\ng : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d\nhf : \u2191LinearMap.det g \u2260 0\nhg : g = LinearMap.comp (\u2191e) (LinearMap.comp f \u2191(LinearEquiv.symm e))\ngdet : \u2191LinearMap.det g = \u2191LinearMap.det f\nfg : f = LinearMap.comp (\u2191(LinearEquiv.symm e)) (LinearMap.comp g \u2191e)\nCe : Continuous \u2191e\n\u22a2 map (\u2191(LinearEquiv.symm e) \u2218 \u2191g \u2218 \u2191e) \u03bc = ENNReal.ofReal |(\u2191LinearMap.det g)\u207b\u00b9| \u2022 \u03bc", "state_after": "case intro\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf : E \u2192\u2097[\u211d] E\n\u03b9 : Type := Fin (finrank \u211d E)\nthis\u271d : FiniteDimensional \u211d (\u03b9 \u2192 \u211d)\nthis : finrank \u211d E = finrank \u211d (\u03b9 \u2192 \u211d)\ne : E \u2243\u2097[\u211d] \u03b9 \u2192 \u211d\ng : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d\nhf : \u2191LinearMap.det g \u2260 0\nhg : g = LinearMap.comp (\u2191e) (LinearMap.comp f \u2191(LinearEquiv.symm e))\ngdet : \u2191LinearMap.det g = \u2191LinearMap.det f\nfg : f = LinearMap.comp (\u2191(LinearEquiv.symm e)) (LinearMap.comp g \u2191e)\nCe : Continuous \u2191e\nCg : Continuous \u2191g\n\u22a2 map (\u2191(LinearEquiv.symm e) \u2218 \u2191g \u2218 \u2191e) \u03bc = ENNReal.ofReal |(\u2191LinearMap.det g)\u207b\u00b9| \u2022 \u03bc"}, {"tactic": "have Cesymm : Continuous e.symm := (e.symm : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] E).continuous_of_finiteDimensional", "annotated_tactic": ["have Cesymm : <a>Continuous</a> e.symm := (e.symm : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] E).<a>continuous_of_finiteDimensional</a>", [{"full_name": "Continuous", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [1591, 11], "def_end_pos": [1591, 21]}, {"full_name": "LinearMap.continuous_of_finiteDimensional", "def_path": "Mathlib/Topology/Algebra/Module/FiniteDimension.lean", "def_pos": [246, 9], "def_end_pos": [246, 50]}]], "state_before": "case intro\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf : E \u2192\u2097[\u211d] E\n\u03b9 : Type := Fin (finrank \u211d E)\nthis\u271d : FiniteDimensional \u211d (\u03b9 \u2192 \u211d)\nthis : finrank \u211d E = finrank \u211d (\u03b9 \u2192 \u211d)\ne : E \u2243\u2097[\u211d] \u03b9 \u2192 \u211d\ng : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d\nhf : \u2191LinearMap.det g \u2260 0\nhg : g = LinearMap.comp (\u2191e) (LinearMap.comp f \u2191(LinearEquiv.symm e))\ngdet : \u2191LinearMap.det g = \u2191LinearMap.det f\nfg : f = LinearMap.comp (\u2191(LinearEquiv.symm e)) (LinearMap.comp g \u2191e)\nCe : Continuous \u2191e\nCg : Continuous \u2191g\n\u22a2 map (\u2191(LinearEquiv.symm e) \u2218 \u2191g \u2218 \u2191e) \u03bc = ENNReal.ofReal |(\u2191LinearMap.det g)\u207b\u00b9| \u2022 \u03bc", "state_after": "case intro\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf : E \u2192\u2097[\u211d] E\n\u03b9 : Type := Fin (finrank \u211d E)\nthis\u271d : FiniteDimensional \u211d (\u03b9 \u2192 \u211d)\nthis : finrank \u211d E = finrank \u211d (\u03b9 \u2192 \u211d)\ne : E \u2243\u2097[\u211d] \u03b9 \u2192 \u211d\ng : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d\nhf : \u2191LinearMap.det g \u2260 0\nhg : g = LinearMap.comp (\u2191e) (LinearMap.comp f \u2191(LinearEquiv.symm e))\ngdet : \u2191LinearMap.det g = \u2191LinearMap.det f\nfg : f = LinearMap.comp (\u2191(LinearEquiv.symm e)) (LinearMap.comp g \u2191e)\nCe : Continuous \u2191e\nCg : Continuous \u2191g\nCesymm : Continuous \u2191(LinearEquiv.symm e)\n\u22a2 map (\u2191(LinearEquiv.symm e) \u2218 \u2191g \u2218 \u2191e) \u03bc = ENNReal.ofReal |(\u2191LinearMap.det g)\u207b\u00b9| \u2022 \u03bc"}, {"tactic": "rw [\u2190 map_map Cesymm.measurable (Cg.comp Ce).measurable, \u2190 map_map Cg.measurable Ce.measurable]", "annotated_tactic": ["rw [\u2190 <a>map_map</a> Cesymm.measurable (Cg.comp Ce).<a>measurable</a>, \u2190 <a>map_map</a> Cg.measurable Ce.measurable]", [{"full_name": "MeasureTheory.Measure.map_map", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1258, 9], "def_end_pos": [1258, 16]}, {"full_name": "Continuous.measurable", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [891, 9], "def_end_pos": [891, 30]}, {"full_name": "MeasureTheory.Measure.map_map", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1258, 9], "def_end_pos": [1258, 16]}]], "state_before": "case intro\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf : E \u2192\u2097[\u211d] E\n\u03b9 : Type := Fin (finrank \u211d E)\nthis\u271d : FiniteDimensional \u211d (\u03b9 \u2192 \u211d)\nthis : finrank \u211d E = finrank \u211d (\u03b9 \u2192 \u211d)\ne : E \u2243\u2097[\u211d] \u03b9 \u2192 \u211d\ng : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d\nhf : \u2191LinearMap.det g \u2260 0\nhg : g = LinearMap.comp (\u2191e) (LinearMap.comp f \u2191(LinearEquiv.symm e))\ngdet : \u2191LinearMap.det g = \u2191LinearMap.det f\nfg : f = LinearMap.comp (\u2191(LinearEquiv.symm e)) (LinearMap.comp g \u2191e)\nCe : Continuous \u2191e\nCg : Continuous \u2191g\nCesymm : Continuous \u2191(LinearEquiv.symm e)\n\u22a2 map (\u2191(LinearEquiv.symm e) \u2218 \u2191g \u2218 \u2191e) \u03bc = ENNReal.ofReal |(\u2191LinearMap.det g)\u207b\u00b9| \u2022 \u03bc", "state_after": "case intro\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf : E \u2192\u2097[\u211d] E\n\u03b9 : Type := Fin (finrank \u211d E)\nthis\u271d : FiniteDimensional \u211d (\u03b9 \u2192 \u211d)\nthis : finrank \u211d E = finrank \u211d (\u03b9 \u2192 \u211d)\ne : E \u2243\u2097[\u211d] \u03b9 \u2192 \u211d\ng : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d\nhf : \u2191LinearMap.det g \u2260 0\nhg : g = LinearMap.comp (\u2191e) (LinearMap.comp f \u2191(LinearEquiv.symm e))\ngdet : \u2191LinearMap.det g = \u2191LinearMap.det f\nfg : f = LinearMap.comp (\u2191(LinearEquiv.symm e)) (LinearMap.comp g \u2191e)\nCe : Continuous \u2191e\nCg : Continuous \u2191g\nCesymm : Continuous \u2191(LinearEquiv.symm e)\n\u22a2 map (\u2191(LinearEquiv.symm e)) (map (\u2191g) (map (\u2191e) \u03bc)) = ENNReal.ofReal |(\u2191LinearMap.det g)\u207b\u00b9| \u2022 \u03bc"}, {"tactic": "haveI : IsAddHaarMeasure (map e \u03bc) := (e : E \u2243+ (\u03b9 \u2192 \u211d)).isAddHaarMeasure_map \u03bc Ce Cesymm", "annotated_tactic": ["haveI : <a>IsAddHaarMeasure</a> (<a>map</a> e \u03bc) := (e : E \u2243+ (\u03b9 \u2192 \u211d)).<a>isAddHaarMeasure_map</a> \u03bc Ce Cesymm", [{"full_name": "MeasureTheory.Measure.IsAddHaarMeasure", "def_path": "Mathlib/MeasureTheory/Group/Measure.lean", "def_pos": [725, 7], "def_end_pos": [725, 23]}, {"full_name": "MeasureTheory.Measure.map", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1163, 17], "def_end_pos": [1163, 20]}, {"full_name": "AddEquiv.isAddHaarMeasure_map", "def_path": "Mathlib/MeasureTheory/Group/Measure.lean", "def_pos": [827, 3], "def_end_pos": [827, 14]}]], "state_before": "case intro\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf : E \u2192\u2097[\u211d] E\n\u03b9 : Type := Fin (finrank \u211d E)\nthis\u271d : FiniteDimensional \u211d (\u03b9 \u2192 \u211d)\nthis : finrank \u211d E = finrank \u211d (\u03b9 \u2192 \u211d)\ne : E \u2243\u2097[\u211d] \u03b9 \u2192 \u211d\ng : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d\nhf : \u2191LinearMap.det g \u2260 0\nhg : g = LinearMap.comp (\u2191e) (LinearMap.comp f \u2191(LinearEquiv.symm e))\ngdet : \u2191LinearMap.det g = \u2191LinearMap.det f\nfg : f = LinearMap.comp (\u2191(LinearEquiv.symm e)) (LinearMap.comp g \u2191e)\nCe : Continuous \u2191e\nCg : Continuous \u2191g\nCesymm : Continuous \u2191(LinearEquiv.symm e)\n\u22a2 map (\u2191(LinearEquiv.symm e)) (map (\u2191g) (map (\u2191e) \u03bc)) = ENNReal.ofReal |(\u2191LinearMap.det g)\u207b\u00b9| \u2022 \u03bc", "state_after": "case intro\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf : E \u2192\u2097[\u211d] E\n\u03b9 : Type := Fin (finrank \u211d E)\nthis\u271d\u00b9 : FiniteDimensional \u211d (\u03b9 \u2192 \u211d)\nthis\u271d : finrank \u211d E = finrank \u211d (\u03b9 \u2192 \u211d)\ne : E \u2243\u2097[\u211d] \u03b9 \u2192 \u211d\ng : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d\nhf : \u2191LinearMap.det g \u2260 0\nhg : g = LinearMap.comp (\u2191e) (LinearMap.comp f \u2191(LinearEquiv.symm e))\ngdet : \u2191LinearMap.det g = \u2191LinearMap.det f\nfg : f = LinearMap.comp (\u2191(LinearEquiv.symm e)) (LinearMap.comp g \u2191e)\nCe : Continuous \u2191e\nCg : Continuous \u2191g\nCesymm : Continuous \u2191(LinearEquiv.symm e)\nthis : IsAddHaarMeasure (map (\u2191e) \u03bc)\n\u22a2 map (\u2191(LinearEquiv.symm e)) (map (\u2191g) (map (\u2191e) \u03bc)) = ENNReal.ofReal |(\u2191LinearMap.det g)\u207b\u00b9| \u2022 \u03bc"}, {"tactic": "have ecomp : e.symm \u2218 e = id := by\n  ext x; simp only [id.def, Function.comp_apply, LinearEquiv.symm_apply_apply]", "annotated_tactic": ["have ecomp : e.symm \u2218 e = <a>id</a> := by\n    ext x; simp only [<a>id.def</a>, <a>Function.comp_apply</a>, <a>LinearEquiv.symm_apply_apply</a>]", [{"full_name": "id", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [33, 15], "def_end_pos": [33, 17]}, {"full_name": "id.def", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [527, 9], "def_end_pos": [527, 15]}, {"full_name": "Function.comp_apply", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [33, 17], "def_end_pos": [33, 36]}, {"full_name": "LinearEquiv.symm_apply_apply", "def_path": "Mathlib/Algebra/Module/Equiv.lean", "def_pos": [379, 9], "def_end_pos": [379, 25]}]], "state_before": "case intro\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf : E \u2192\u2097[\u211d] E\n\u03b9 : Type := Fin (finrank \u211d E)\nthis\u271d\u00b9 : FiniteDimensional \u211d (\u03b9 \u2192 \u211d)\nthis\u271d : finrank \u211d E = finrank \u211d (\u03b9 \u2192 \u211d)\ne : E \u2243\u2097[\u211d] \u03b9 \u2192 \u211d\ng : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d\nhf : \u2191LinearMap.det g \u2260 0\nhg : g = LinearMap.comp (\u2191e) (LinearMap.comp f \u2191(LinearEquiv.symm e))\ngdet : \u2191LinearMap.det g = \u2191LinearMap.det f\nfg : f = LinearMap.comp (\u2191(LinearEquiv.symm e)) (LinearMap.comp g \u2191e)\nCe : Continuous \u2191e\nCg : Continuous \u2191g\nCesymm : Continuous \u2191(LinearEquiv.symm e)\nthis : IsAddHaarMeasure (map (\u2191e) \u03bc)\n\u22a2 map (\u2191(LinearEquiv.symm e)) (map (\u2191g) (map (\u2191e) \u03bc)) = ENNReal.ofReal |(\u2191LinearMap.det g)\u207b\u00b9| \u2022 \u03bc", "state_after": "case intro\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf : E \u2192\u2097[\u211d] E\n\u03b9 : Type := Fin (finrank \u211d E)\nthis\u271d\u00b9 : FiniteDimensional \u211d (\u03b9 \u2192 \u211d)\nthis\u271d : finrank \u211d E = finrank \u211d (\u03b9 \u2192 \u211d)\ne : E \u2243\u2097[\u211d] \u03b9 \u2192 \u211d\ng : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d\nhf : \u2191LinearMap.det g \u2260 0\nhg : g = LinearMap.comp (\u2191e) (LinearMap.comp f \u2191(LinearEquiv.symm e))\ngdet : \u2191LinearMap.det g = \u2191LinearMap.det f\nfg : f = LinearMap.comp (\u2191(LinearEquiv.symm e)) (LinearMap.comp g \u2191e)\nCe : Continuous \u2191e\nCg : Continuous \u2191g\nCesymm : Continuous \u2191(LinearEquiv.symm e)\nthis : IsAddHaarMeasure (map (\u2191e) \u03bc)\necomp : \u2191(LinearEquiv.symm e) \u2218 \u2191e = id\n\u22a2 map (\u2191(LinearEquiv.symm e)) (map (\u2191g) (map (\u2191e) \u03bc)) = ENNReal.ofReal |(\u2191LinearMap.det g)\u207b\u00b9| \u2022 \u03bc"}, {"tactic": "rw [map_linearMap_addHaar_pi_eq_smul_addHaar hf (map e \u03bc), Measure.map_smul,\n  map_map Cesymm.measurable Ce.measurable, ecomp, Measure.map_id]", "annotated_tactic": ["rw [<a>map_linearMap_addHaar_pi_eq_smul_addHaar</a> hf (<a>map</a> e \u03bc), <a>Measure.map_smul</a>,\n    <a>map_map</a> Cesymm.measurable Ce.measurable, ecomp, <a>Measure.map_id</a>]", [{"full_name": "MeasureTheory.Measure.map_linearMap_addHaar_pi_eq_smul_addHaar", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean", "def_pos": [227, 9], "def_end_pos": [227, 49]}, {"full_name": "MeasureTheory.Measure.map", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1163, 17], "def_end_pos": [1163, 20]}, {"full_name": "MeasureTheory.Measure.map_smul", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1202, 19], "def_end_pos": [1202, 27]}, {"full_name": "MeasureTheory.Measure.map_map", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1258, 9], "def_end_pos": [1258, 16]}, {"full_name": "MeasureTheory.Measure.map_id", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1249, 9], "def_end_pos": [1249, 15]}]], "state_before": "case intro\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf : E \u2192\u2097[\u211d] E\n\u03b9 : Type := Fin (finrank \u211d E)\nthis\u271d\u00b9 : FiniteDimensional \u211d (\u03b9 \u2192 \u211d)\nthis\u271d : finrank \u211d E = finrank \u211d (\u03b9 \u2192 \u211d)\ne : E \u2243\u2097[\u211d] \u03b9 \u2192 \u211d\ng : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d\nhf : \u2191LinearMap.det g \u2260 0\nhg : g = LinearMap.comp (\u2191e) (LinearMap.comp f \u2191(LinearEquiv.symm e))\ngdet : \u2191LinearMap.det g = \u2191LinearMap.det f\nfg : f = LinearMap.comp (\u2191(LinearEquiv.symm e)) (LinearMap.comp g \u2191e)\nCe : Continuous \u2191e\nCg : Continuous \u2191g\nCesymm : Continuous \u2191(LinearEquiv.symm e)\nthis : IsAddHaarMeasure (map (\u2191e) \u03bc)\necomp : \u2191(LinearEquiv.symm e) \u2218 \u2191e = id\n\u22a2 map (\u2191(LinearEquiv.symm e)) (map (\u2191g) (map (\u2191e) \u03bc)) = ENNReal.ofReal |(\u2191LinearMap.det g)\u207b\u00b9| \u2022 \u03bc", "state_after": "no goals"}, {"tactic": "infer_instance", "annotated_tactic": ["infer_instance", []], "state_before": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf : E \u2192\u2097[\u211d] E\nhf : \u2191LinearMap.det f \u2260 0\n\u03b9 : Type := Fin (finrank \u211d E)\n\u22a2 FiniteDimensional \u211d (\u03b9 \u2192 \u211d)", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf : E \u2192\u2097[\u211d] E\nhf : \u2191LinearMap.det f \u2260 0\n\u03b9 : Type := Fin (finrank \u211d E)\nthis : FiniteDimensional \u211d (\u03b9 \u2192 \u211d)\n\u22a2 finrank \u211d E = finrank \u211d (\u03b9 \u2192 \u211d)", "state_after": "no goals"}, {"tactic": "rw [hg]", "annotated_tactic": ["rw [hg]", []], "state_before": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf : E \u2192\u2097[\u211d] E\nhf : \u2191LinearMap.det f \u2260 0\n\u03b9 : Type := Fin (finrank \u211d E)\nthis\u271d : FiniteDimensional \u211d (\u03b9 \u2192 \u211d)\nthis : finrank \u211d E = finrank \u211d (\u03b9 \u2192 \u211d)\ne : E \u2243\u2097[\u211d] \u03b9 \u2192 \u211d\ng : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d\nhg : g = LinearMap.comp (\u2191e) (LinearMap.comp f \u2191(LinearEquiv.symm e))\n\u22a2 \u2191LinearMap.det g = \u2191LinearMap.det f", "state_after": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf : E \u2192\u2097[\u211d] E\nhf : \u2191LinearMap.det f \u2260 0\n\u03b9 : Type := Fin (finrank \u211d E)\nthis\u271d : FiniteDimensional \u211d (\u03b9 \u2192 \u211d)\nthis : finrank \u211d E = finrank \u211d (\u03b9 \u2192 \u211d)\ne : E \u2243\u2097[\u211d] \u03b9 \u2192 \u211d\ng : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d\nhg : g = LinearMap.comp (\u2191e) (LinearMap.comp f \u2191(LinearEquiv.symm e))\n\u22a2 \u2191LinearMap.det (LinearMap.comp (\u2191e) (LinearMap.comp f \u2191(LinearEquiv.symm e))) = \u2191LinearMap.det f"}, {"tactic": "exact LinearMap.det_conj f e", "annotated_tactic": ["exact <a>LinearMap.det_conj</a> f e", [{"full_name": "LinearMap.det_conj", "def_path": "Mathlib/LinearAlgebra/Determinant.lean", "def_pos": [310, 9], "def_end_pos": [310, 17]}]], "state_before": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf : E \u2192\u2097[\u211d] E\nhf : \u2191LinearMap.det f \u2260 0\n\u03b9 : Type := Fin (finrank \u211d E)\nthis\u271d : FiniteDimensional \u211d (\u03b9 \u2192 \u211d)\nthis : finrank \u211d E = finrank \u211d (\u03b9 \u2192 \u211d)\ne : E \u2243\u2097[\u211d] \u03b9 \u2192 \u211d\ng : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d\nhg : g = LinearMap.comp (\u2191e) (LinearMap.comp f \u2191(LinearEquiv.symm e))\n\u22a2 \u2191LinearMap.det (LinearMap.comp (\u2191e) (LinearMap.comp f \u2191(LinearEquiv.symm e))) = \u2191LinearMap.det f", "state_after": "no goals"}, {"tactic": "ext x", "annotated_tactic": ["ext x", []], "state_before": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf : E \u2192\u2097[\u211d] E\n\u03b9 : Type := Fin (finrank \u211d E)\nthis\u271d : FiniteDimensional \u211d (\u03b9 \u2192 \u211d)\nthis : finrank \u211d E = finrank \u211d (\u03b9 \u2192 \u211d)\ne : E \u2243\u2097[\u211d] \u03b9 \u2192 \u211d\ng : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d\nhf : \u2191LinearMap.det g \u2260 0\nhg : g = LinearMap.comp (\u2191e) (LinearMap.comp f \u2191(LinearEquiv.symm e))\ngdet : \u2191LinearMap.det g = \u2191LinearMap.det f\n\u22a2 f = LinearMap.comp (\u2191(LinearEquiv.symm e)) (LinearMap.comp g \u2191e)", "state_after": "case h\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf : E \u2192\u2097[\u211d] E\n\u03b9 : Type := Fin (finrank \u211d E)\nthis\u271d : FiniteDimensional \u211d (\u03b9 \u2192 \u211d)\nthis : finrank \u211d E = finrank \u211d (\u03b9 \u2192 \u211d)\ne : E \u2243\u2097[\u211d] \u03b9 \u2192 \u211d\ng : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d\nhf : \u2191LinearMap.det g \u2260 0\nhg : g = LinearMap.comp (\u2191e) (LinearMap.comp f \u2191(LinearEquiv.symm e))\ngdet : \u2191LinearMap.det g = \u2191LinearMap.det f\nx : E\n\u22a2 \u2191f x = \u2191(LinearMap.comp (\u2191(LinearEquiv.symm e)) (LinearMap.comp g \u2191e)) x"}, {"tactic": "simp only [LinearEquiv.coe_coe, Function.comp_apply, LinearMap.coe_comp,\n  LinearEquiv.symm_apply_apply, hg]", "annotated_tactic": ["simp only [<a>LinearEquiv.coe_coe</a>, <a>Function.comp_apply</a>, <a>LinearMap.coe_comp</a>,\n      <a>LinearEquiv.symm_apply_apply</a>, hg]", [{"full_name": "LinearEquiv.coe_coe", "def_path": "Mathlib/Algebra/Module/Equiv.lean", "def_pos": [211, 9], "def_end_pos": [211, 16]}, {"full_name": "Function.comp_apply", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [33, 17], "def_end_pos": [33, 36]}, {"full_name": "LinearMap.coe_comp", "def_path": "Mathlib/Algebra/Module/LinearMap.lean", "def_pos": [554, 9], "def_end_pos": [554, 17]}, {"full_name": "LinearEquiv.symm_apply_apply", "def_path": "Mathlib/Algebra/Module/Equiv.lean", "def_pos": [379, 9], "def_end_pos": [379, 25]}]], "state_before": "case h\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf : E \u2192\u2097[\u211d] E\n\u03b9 : Type := Fin (finrank \u211d E)\nthis\u271d : FiniteDimensional \u211d (\u03b9 \u2192 \u211d)\nthis : finrank \u211d E = finrank \u211d (\u03b9 \u2192 \u211d)\ne : E \u2243\u2097[\u211d] \u03b9 \u2192 \u211d\ng : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d\nhf : \u2191LinearMap.det g \u2260 0\nhg : g = LinearMap.comp (\u2191e) (LinearMap.comp f \u2191(LinearEquiv.symm e))\ngdet : \u2191LinearMap.det g = \u2191LinearMap.det f\nx : E\n\u22a2 \u2191f x = \u2191(LinearMap.comp (\u2191(LinearEquiv.symm e)) (LinearMap.comp g \u2191e)) x", "state_after": "no goals"}, {"tactic": "ext x", "annotated_tactic": ["ext x", []], "state_before": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf : E \u2192\u2097[\u211d] E\n\u03b9 : Type := Fin (finrank \u211d E)\nthis\u271d\u00b9 : FiniteDimensional \u211d (\u03b9 \u2192 \u211d)\nthis\u271d : finrank \u211d E = finrank \u211d (\u03b9 \u2192 \u211d)\ne : E \u2243\u2097[\u211d] \u03b9 \u2192 \u211d\ng : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d\nhf : \u2191LinearMap.det g \u2260 0\nhg : g = LinearMap.comp (\u2191e) (LinearMap.comp f \u2191(LinearEquiv.symm e))\ngdet : \u2191LinearMap.det g = \u2191LinearMap.det f\nfg : f = LinearMap.comp (\u2191(LinearEquiv.symm e)) (LinearMap.comp g \u2191e)\nCe : Continuous \u2191e\nCg : Continuous \u2191g\nCesymm : Continuous \u2191(LinearEquiv.symm e)\nthis : IsAddHaarMeasure (map (\u2191e) \u03bc)\n\u22a2 \u2191(LinearEquiv.symm e) \u2218 \u2191e = id", "state_after": "case h\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf : E \u2192\u2097[\u211d] E\n\u03b9 : Type := Fin (finrank \u211d E)\nthis\u271d\u00b9 : FiniteDimensional \u211d (\u03b9 \u2192 \u211d)\nthis\u271d : finrank \u211d E = finrank \u211d (\u03b9 \u2192 \u211d)\ne : E \u2243\u2097[\u211d] \u03b9 \u2192 \u211d\ng : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d\nhf : \u2191LinearMap.det g \u2260 0\nhg : g = LinearMap.comp (\u2191e) (LinearMap.comp f \u2191(LinearEquiv.symm e))\ngdet : \u2191LinearMap.det g = \u2191LinearMap.det f\nfg : f = LinearMap.comp (\u2191(LinearEquiv.symm e)) (LinearMap.comp g \u2191e)\nCe : Continuous \u2191e\nCg : Continuous \u2191g\nCesymm : Continuous \u2191(LinearEquiv.symm e)\nthis : IsAddHaarMeasure (map (\u2191e) \u03bc)\nx : E\n\u22a2 (\u2191(LinearEquiv.symm e) \u2218 \u2191e) x = id x"}, {"tactic": "simp only [id.def, Function.comp_apply, LinearEquiv.symm_apply_apply]", "annotated_tactic": ["simp only [<a>id.def</a>, <a>Function.comp_apply</a>, <a>LinearEquiv.symm_apply_apply</a>]", [{"full_name": "id.def", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [527, 9], "def_end_pos": [527, 15]}, {"full_name": "Function.comp_apply", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [33, 17], "def_end_pos": [33, 36]}, {"full_name": "LinearEquiv.symm_apply_apply", "def_path": "Mathlib/Algebra/Module/Equiv.lean", "def_pos": [379, 9], "def_end_pos": [379, 25]}]], "state_before": "case h\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf : E \u2192\u2097[\u211d] E\n\u03b9 : Type := Fin (finrank \u211d E)\nthis\u271d\u00b9 : FiniteDimensional \u211d (\u03b9 \u2192 \u211d)\nthis\u271d : finrank \u211d E = finrank \u211d (\u03b9 \u2192 \u211d)\ne : E \u2243\u2097[\u211d] \u03b9 \u2192 \u211d\ng : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d\nhf : \u2191LinearMap.det g \u2260 0\nhg : g = LinearMap.comp (\u2191e) (LinearMap.comp f \u2191(LinearEquiv.symm e))\ngdet : \u2191LinearMap.det g = \u2191LinearMap.det f\nfg : f = LinearMap.comp (\u2191(LinearEquiv.symm e)) (LinearMap.comp g \u2191e)\nCe : Continuous \u2191e\nCg : Continuous \u2191g\nCesymm : Continuous \u2191(LinearEquiv.symm e)\nthis : IsAddHaarMeasure (map (\u2191e) \u03bc)\nx : E\n\u22a2 (\u2191(LinearEquiv.symm e) \u2218 \u2191e) x = id x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "full_name": "NNReal.count_const_le_le_of_tsum_le", "start": [1448, 1], "end": [1459, 45], "traced_tactics": [{"tactic": "rw [show (fun i => \u03b5 \u2264 a i) = fun i => (\u03b5 : \u211d\u22650\u221e) \u2264 ((\u2191) \u2218 a) i by\n    funext i\n    simp only [ENNReal.coe_le_coe, Function.comp]]", "annotated_tactic": ["rw [show (fun i => \u03b5 \u2264 a i) = fun i => (\u03b5 : \u211d\u22650\u221e) \u2264 ((\u2191) \u2218 a) i by\n      funext i\n      simp only [<a>ENNReal.coe_le_coe</a>, <a>Function.comp</a>]]", [{"full_name": "ENNReal.coe_le_coe", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [349, 28], "def_end_pos": [349, 38]}, {"full_name": "Function.comp", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [52, 15], "def_end_pos": [52, 28]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSingletonClass \u03b1\na : \u03b1 \u2192 \u211d\u22650\na_mble : Measurable a\na_summable : Summable a\nc : \u211d\u22650\ntsum_le_c : \u2211' (i : \u03b1), a i \u2264 c\n\u03b5 : \u211d\u22650\n\u03b5_ne_zero : \u03b5 \u2260 0\n\u22a2 \u2191\u2191count {i | \u03b5 \u2264 a i} \u2264 \u2191c / \u2191\u03b5", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSingletonClass \u03b1\na : \u03b1 \u2192 \u211d\u22650\na_mble : Measurable a\na_summable : Summable a\nc : \u211d\u22650\ntsum_le_c : \u2211' (i : \u03b1), a i \u2264 c\n\u03b5 : \u211d\u22650\n\u03b5_ne_zero : \u03b5 \u2260 0\n\u22a2 \u2191\u2191count {i | \u2191\u03b5 \u2264 (ENNReal.some \u2218 a) i} \u2264 \u2191c / \u2191\u03b5"}, {"tactic": "apply\n  ENNReal.count_const_le_le_of_tsum_le (measurable_coe_nnreal_ennreal.comp a_mble) _\n    (by exact_mod_cast \u03b5_ne_zero) (@ENNReal.coe_ne_top \u03b5)", "annotated_tactic": ["apply\n    <a>ENNReal.count_const_le_le_of_tsum_le</a> (measurable_coe_nnreal_ennreal.comp a_mble) _\n      (by exact_mod_cast \u03b5_ne_zero) (@<a>ENNReal.coe_ne_top</a> \u03b5)", [{"full_name": "ENNReal.count_const_le_le_of_tsum_le", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [1439, 9], "def_end_pos": [1439, 52]}, {"full_name": "ENNReal.coe_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [302, 17], "def_end_pos": [302, 27]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSingletonClass \u03b1\na : \u03b1 \u2192 \u211d\u22650\na_mble : Measurable a\na_summable : Summable a\nc : \u211d\u22650\ntsum_le_c : \u2211' (i : \u03b1), a i \u2264 c\n\u03b5 : \u211d\u22650\n\u03b5_ne_zero : \u03b5 \u2260 0\n\u22a2 \u2191\u2191count {i | \u2191\u03b5 \u2264 (ENNReal.some \u2218 a) i} \u2264 \u2191c / \u2191\u03b5", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSingletonClass \u03b1\na : \u03b1 \u2192 \u211d\u22650\na_mble : Measurable a\na_summable : Summable a\nc : \u211d\u22650\ntsum_le_c : \u2211' (i : \u03b1), a i \u2264 c\n\u03b5 : \u211d\u22650\n\u03b5_ne_zero : \u03b5 \u2260 0\n\u22a2 \u2211' (i : \u03b1), (ENNReal.some \u2218 a) i \u2264 \u2191c"}, {"tactic": "convert ENNReal.coe_le_coe.mpr tsum_le_c", "annotated_tactic": ["convert ENNReal.coe_le_coe.mpr tsum_le_c", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSingletonClass \u03b1\na : \u03b1 \u2192 \u211d\u22650\na_mble : Measurable a\na_summable : Summable a\nc : \u211d\u22650\ntsum_le_c : \u2211' (i : \u03b1), a i \u2264 c\n\u03b5 : \u211d\u22650\n\u03b5_ne_zero : \u03b5 \u2260 0\n\u22a2 \u2211' (i : \u03b1), (ENNReal.some \u2218 a) i \u2264 \u2191c", "state_after": "case h.e'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSingletonClass \u03b1\na : \u03b1 \u2192 \u211d\u22650\na_mble : Measurable a\na_summable : Summable a\nc : \u211d\u22650\ntsum_le_c : \u2211' (i : \u03b1), a i \u2264 c\n\u03b5 : \u211d\u22650\n\u03b5_ne_zero : \u03b5 \u2260 0\n\u22a2 \u2211' (i : \u03b1), (ENNReal.some \u2218 a) i = \u2191(\u2211' (i : \u03b1), a i)"}, {"tactic": "simp_rw [Function.comp_apply]", "annotated_tactic": ["simp_rw [<a>Function.comp_apply</a>]", [{"full_name": "Function.comp_apply", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [33, 17], "def_end_pos": [33, 36]}]], 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: \u03b1\n\u22a2 (\u03b5 \u2264 a i) = (\u2191\u03b5 \u2264 (ENNReal.some \u2218 a) i)"}, {"tactic": "simp only [ENNReal.coe_le_coe, Function.comp]", "annotated_tactic": ["simp only [<a>ENNReal.coe_le_coe</a>, <a>Function.comp</a>]", [{"full_name": "ENNReal.coe_le_coe", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [349, 28], "def_end_pos": [349, 38]}, {"full_name": "Function.comp", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [52, 15], "def_end_pos": [52, 28]}]], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSingletonClass \u03b1\na : \u03b1 \u2192 \u211d\u22650\na_mble : Measurable a\na_summable : Summable a\nc : \u211d\u22650\ntsum_le_c : \u2211' (i : \u03b1), a i \u2264 c\n\u03b5 : \u211d\u22650\n\u03b5_ne_zero : \u03b5 \u2260 0\ni : \u03b1\n\u22a2 (\u03b5 \u2264 a i) = (\u2191\u03b5 \u2264 (ENNReal.some \u2218 a) i)", "state_after": "no goals"}, {"tactic": "exact_mod_cast \u03b5_ne_zero", "annotated_tactic": ["exact_mod_cast \u03b5_ne_zero", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSingletonClass \u03b1\na : \u03b1 \u2192 \u211d\u22650\na_mble : Measurable a\na_summable : Summable a\nc : \u211d\u22650\ntsum_le_c : \u2211' (i : \u03b1), a i \u2264 c\n\u03b5 : \u211d\u22650\n\u03b5_ne_zero : \u03b5 \u2260 0\n\u22a2 \u2191\u03b5 \u2260 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "Finset.sdiff_union_erase_cancel", "start": [2342, 1], "end": [2343, 79], "traced_tactics": [{"tactic": "simp_rw [erase_eq, sdiff_union_sdiff_cancel hts (singleton_subset_iff.2 ha)]", "annotated_tactic": ["simp_rw [<a>erase_eq</a>, <a>sdiff_union_sdiff_cancel</a> hts (<a>singleton_subset_iff</a>.2 ha)]", [{"full_name": "Finset.erase_eq", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2264, 9], "def_end_pos": [2264, 17]}, {"full_name": "Finset.sdiff_union_sdiff_cancel", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2338, 9], "def_end_pos": [2338, 33]}, {"full_name": "Finset.singleton_subset_iff", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [772, 9], "def_end_pos": [772, 29]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns t u v : Finset \u03b1\na b : \u03b1\nhts : t \u2286 s\nha : a \u2208 t\n\u22a2 s \\ t \u222a erase t a = erase s a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Haar/Basic.lean", "full_name": "MeasureTheory.Measure.haar.prehaar_mono", "start": [316, 1], "end": [320, 33], "traced_tactics": [{"tactic": "simp only [prehaar]", "annotated_tactic": ["simp only [<a>prehaar</a>]", [{"full_name": "MeasureTheory.Measure.haar.prehaar", "def_path": "Mathlib/MeasureTheory/Measure/Haar/Basic.lean", "def_pos": [116, 19], "def_end_pos": [116, 26]}]], "state_before": "G : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nU : Set G\nhU : Set.Nonempty (interior U)\nK\u2081 K\u2082 : Compacts G\nh : \u2191K\u2081 \u2286 K\u2082.carrier\n\u22a2 prehaar (\u2191K\u2080) U K\u2081 \u2264 prehaar (\u2191K\u2080) U K\u2082", "state_after": "G : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nU : Set G\nhU : Set.Nonempty (interior U)\nK\u2081 K\u2082 : Compacts G\nh : \u2191K\u2081 \u2286 K\u2082.carrier\n\u22a2 \u2191(index (\u2191K\u2081) U) / \u2191(index (\u2191K\u2080) U) \u2264 \u2191(index (\u2191K\u2082) U) / \u2191(index (\u2191K\u2080) U)"}, {"tactic": "rw [div_le_div_right]", "annotated_tactic": ["rw [<a>div_le_div_right</a>]", [{"full_name": "div_le_div_right", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [363, 9], "def_end_pos": [363, 25]}]], "state_before": "G : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nU : Set G\nhU : Set.Nonempty (interior U)\nK\u2081 K\u2082 : Compacts G\nh : \u2191K\u2081 \u2286 K\u2082.carrier\n\u22a2 \u2191(index (\u2191K\u2081) U) / \u2191(index (\u2191K\u2080) U) \u2264 \u2191(index (\u2191K\u2082) U) / \u2191(index (\u2191K\u2080) U)", "state_after": "G : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nU : Set G\nhU : Set.Nonempty (interior U)\nK\u2081 K\u2082 : Compacts G\nh : \u2191K\u2081 \u2286 K\u2082.carrier\n\u22a2 \u2191(index (\u2191K\u2081) U) \u2264 \u2191(index (\u2191K\u2082) U)\n\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nU : Set G\nhU : Set.Nonempty (interior U)\nK\u2081 K\u2082 : Compacts G\nh : \u2191K\u2081 \u2286 K\u2082.carrier\n\u22a2 0 < \u2191(index (\u2191K\u2080) U)"}, {"tactic": "exact_mod_cast index_mono K\u2082.2 h hU", "annotated_tactic": ["exact_mod_cast <a>index_mono</a> K\u2082.2 h hU", [{"full_name": "MeasureTheory.Measure.haar.index_mono", "def_path": "Mathlib/MeasureTheory/Measure/Haar/Basic.lean", "def_pos": [214, 9], "def_end_pos": [214, 19]}]], "state_before": "G : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nU : Set G\nhU : Set.Nonempty (interior U)\nK\u2081 K\u2082 : Compacts G\nh : \u2191K\u2081 \u2286 K\u2082.carrier\n\u22a2 \u2191(index (\u2191K\u2081) U) \u2264 \u2191(index (\u2191K\u2082) U)\n\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nU : Set G\nhU : Set.Nonempty (interior U)\nK\u2081 K\u2082 : Compacts G\nh : \u2191K\u2081 \u2286 K\u2082.carrier\n\u22a2 0 < \u2191(index (\u2191K\u2080) U)", "state_after": "G : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nU : Set G\nhU : Set.Nonempty (interior U)\nK\u2081 K\u2082 : Compacts G\nh : \u2191K\u2081 \u2286 K\u2082.carrier\n\u22a2 0 < \u2191(index (\u2191K\u2080) U)"}, {"tactic": "exact_mod_cast index_pos K\u2080 hU", "annotated_tactic": ["exact_mod_cast <a>index_pos</a> K\u2080 hU", [{"full_name": "MeasureTheory.Measure.haar.index_pos", "def_path": 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88], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Process/Stopping.lean", "full_name": "MeasureTheory.IsStoppingTime.measurableSet_lt_of_pred", "start": [72, 1], "end": [82, 61], "traced_tactics": [{"tactic": "by_cases hi_min : IsMin i", "annotated_tactic": ["by_cases hi_min : <a>IsMin</a> i", [{"full_name": "IsMin", "def_path": "Mathlib/Order/Max.lean", "def_pos": [202, 5], "def_end_pos": [202, 10]}]], "state_before": "\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u00b9 : Preorder \u03b9\nf : Filtration \u03b9 m\n\u03c4 : \u03a9 \u2192 \u03b9\ninst\u271d : PredOrder \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\ni : \u03b9\n\u22a2 MeasurableSet {\u03c9 | \u03c4 \u03c9 < i}", "state_after": "case pos\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u00b9 : Preorder \u03b9\nf : Filtration \u03b9 m\n\u03c4 : \u03a9 \u2192 \u03b9\ninst\u271d : PredOrder \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\ni : \u03b9\nhi_min : IsMin i\n\u22a2 MeasurableSet {\u03c9 | \u03c4 \u03c9 < i}\n\ncase neg\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u00b9 : Preorder \u03b9\nf : Filtration \u03b9 m\n\u03c4 : \u03a9 \u2192 \u03b9\ninst\u271d : PredOrder \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\ni : \u03b9\nhi_min : \u00acIsMin i\n\u22a2 MeasurableSet {\u03c9 | \u03c4 \u03c9 < i}"}, {"tactic": "have : {\u03c9 : \u03a9 | \u03c4 \u03c9 < i} = \u03c4 \u207b\u00b9' Set.Iio i := rfl", "annotated_tactic": ["have : {\u03c9 : \u03a9 | \u03c4 \u03c9 < i} = \u03c4 \u207b\u00b9' <a>Set.Iio</a> i := <a>rfl</a>", [{"full_name": "Set.Iio", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [54, 5], "def_end_pos": [54, 8]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "case neg\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u00b9 : Preorder \u03b9\nf : Filtration \u03b9 m\n\u03c4 : \u03a9 \u2192 \u03b9\ninst\u271d : PredOrder \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\ni : \u03b9\nhi_min : \u00acIsMin i\n\u22a2 MeasurableSet {\u03c9 | \u03c4 \u03c9 < i}", "state_after": "case neg\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u00b9 : Preorder \u03b9\nf : Filtration \u03b9 m\n\u03c4 : \u03a9 \u2192 \u03b9\ninst\u271d : PredOrder \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\ni : \u03b9\nhi_min : \u00acIsMin i\nthis : {\u03c9 | \u03c4 \u03c9 < i} = \u03c4 \u207b\u00b9' Set.Iio i\n\u22a2 MeasurableSet {\u03c9 | \u03c4 \u03c9 < i}"}, {"tactic": "rw [this, \u2190 Iic_pred_of_not_isMin hi_min]", "annotated_tactic": ["rw [this, \u2190 <a>Iic_pred_of_not_isMin</a> hi_min]", [{"full_name": "Order.Iic_pred_of_not_isMin", "def_path": "Mathlib/Order/SuccPred/Basic.lean", "def_pos": [676, 9], "def_end_pos": [676, 30]}]], "state_before": "case neg\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u00b9 : Preorder \u03b9\nf : Filtration \u03b9 m\n\u03c4 : \u03a9 \u2192 \u03b9\ninst\u271d : PredOrder \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\ni : \u03b9\nhi_min : \u00acIsMin i\nthis : {\u03c9 | \u03c4 \u03c9 < i} = \u03c4 \u207b\u00b9' Set.Iio i\n\u22a2 MeasurableSet {\u03c9 | \u03c4 \u03c9 < i}", "state_after": "case neg\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u00b9 : Preorder \u03b9\nf : Filtration \u03b9 m\n\u03c4 : \u03a9 \u2192 \u03b9\ninst\u271d : PredOrder \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\ni : \u03b9\nhi_min : \u00acIsMin i\nthis : {\u03c9 | \u03c4 \u03c9 < i} = \u03c4 \u207b\u00b9' Set.Iio i\n\u22a2 MeasurableSet (\u03c4 \u207b\u00b9' Set.Iic (pred i))"}, {"tactic": "exact f.mono (pred_le i) _ (h\u03c4.measurableSet_le <| pred i)", "annotated_tactic": ["exact f.mono (<a>pred_le</a> i) _ (h\u03c4.measurableSet_le <| <a>pred</a> i)", [{"full_name": "Order.pred_le", "def_path": "Mathlib/Order/SuccPred/Basic.lean", "def_pos": [601, 9], "def_end_pos": [601, 16]}, {"full_name": "Order.pred", "def_path": "Mathlib/Order/SuccPred/Basic.lean", "def_pos": [597, 5], "def_end_pos": [597, 9]}]], "state_before": "case neg\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u00b9 : Preorder \u03b9\nf : Filtration \u03b9 m\n\u03c4 : \u03a9 \u2192 \u03b9\ninst\u271d : PredOrder \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\ni : \u03b9\nhi_min : \u00acIsMin i\nthis : {\u03c9 | \u03c4 \u03c9 < i} = \u03c4 \u207b\u00b9' Set.Iio i\n\u22a2 MeasurableSet (\u03c4 \u207b\u00b9' Set.Iic (pred i))", "state_after": "no goals"}, {"tactic": "suffices {\u03c9 : \u03a9 | \u03c4 \u03c9 < i} = \u2205 by rw [this]; exact @MeasurableSet.empty _ (f i)", "annotated_tactic": ["suffices {\u03c9 : \u03a9 | \u03c4 \u03c9 < i} = \u2205 by rw [this]; exact @<a>MeasurableSet.empty</a> _ (f i)", [{"full_name": "MeasurableSet.empty", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [80, 9], "def_end_pos": [80, 28]}]], "state_before": "case pos\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u00b9 : Preorder \u03b9\nf : Filtration \u03b9 m\n\u03c4 : \u03a9 \u2192 \u03b9\ninst\u271d : PredOrder \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\ni : \u03b9\nhi_min : IsMin i\n\u22a2 MeasurableSet {\u03c9 | \u03c4 \u03c9 < i}", "state_after": "case pos\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u00b9 : Preorder \u03b9\nf : Filtration \u03b9 m\n\u03c4 : \u03a9 \u2192 \u03b9\ninst\u271d : PredOrder \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\ni : \u03b9\nhi_min : IsMin i\n\u22a2 {\u03c9 | \u03c4 \u03c9 < i} = \u2205"}, {"tactic": "ext1 \u03c9", "annotated_tactic": ["ext1 \u03c9", []], "state_before": "case pos\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u00b9 : Preorder \u03b9\nf : Filtration \u03b9 m\n\u03c4 : \u03a9 \u2192 \u03b9\ninst\u271d : PredOrder \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\ni : \u03b9\nhi_min : IsMin i\n\u22a2 {\u03c9 | \u03c4 \u03c9 < i} = \u2205", "state_after": "case pos.h\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u00b9 : Preorder \u03b9\nf : Filtration \u03b9 m\n\u03c4 : \u03a9 \u2192 \u03b9\ninst\u271d : PredOrder \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\ni : \u03b9\nhi_min : IsMin i\n\u03c9 : \u03a9\n\u22a2 \u03c9 \u2208 {\u03c9 | \u03c4 \u03c9 < i} \u2194 \u03c9 \u2208 \u2205"}, {"tactic": "simp only [Set.mem_setOf_eq, Set.mem_empty_iff_false, iff_false_iff]", "annotated_tactic": ["simp only [<a>Set.mem_setOf_eq</a>, <a>Set.mem_empty_iff_false</a>, <a>iff_false_iff</a>]", [{"full_name": "Set.mem_setOf_eq", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [256, 29], "def_end_pos": [256, 41]}, {"full_name": "Set.mem_empty_iff_false", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [562, 9], "def_end_pos": [562, 28]}, {"full_name": "iff_false_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [201, 9], "def_end_pos": [201, 22]}]], "state_before": "case pos.h\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u00b9 : Preorder \u03b9\nf : Filtration \u03b9 m\n\u03c4 : \u03a9 \u2192 \u03b9\ninst\u271d : PredOrder \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\ni : \u03b9\nhi_min : IsMin i\n\u03c9 : \u03a9\n\u22a2 \u03c9 \u2208 {\u03c9 | \u03c4 \u03c9 < i} \u2194 \u03c9 \u2208 \u2205", "state_after": "case pos.h\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u00b9 : Preorder \u03b9\nf : Filtration \u03b9 m\n\u03c4 : \u03a9 \u2192 \u03b9\ninst\u271d : PredOrder \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\ni : \u03b9\nhi_min : IsMin i\n\u03c9 : \u03a9\n\u22a2 \u00ac\u03c4 \u03c9 < i"}, {"tactic": "rw [isMin_iff_forall_not_lt] at hi_min", "annotated_tactic": ["rw [<a>isMin_iff_forall_not_lt</a>] at hi_min", [{"full_name": "isMin_iff_forall_not_lt", "def_path": "Mathlib/Order/Max.lean", "def_pos": [331, 9], "def_end_pos": [331, 32]}]], "state_before": "case pos.h\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u00b9 : Preorder \u03b9\nf : Filtration \u03b9 m\n\u03c4 : \u03a9 \u2192 \u03b9\ninst\u271d : PredOrder \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\ni : \u03b9\nhi_min : IsMin i\n\u03c9 : \u03a9\n\u22a2 \u00ac\u03c4 \u03c9 < i", "state_after": "case pos.h\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u00b9 : Preorder \u03b9\nf : Filtration \u03b9 m\n\u03c4 : \u03a9 \u2192 \u03b9\ninst\u271d : PredOrder \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\ni : \u03b9\nhi_min : \u2200 (b : \u03b9), \u00acb < i\n\u03c9 : \u03a9\n\u22a2 \u00ac\u03c4 \u03c9 < i"}, {"tactic": "exact hi_min (\u03c4 \u03c9)", "annotated_tactic": ["exact hi_min (\u03c4 \u03c9)", []], "state_before": "case pos.h\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u00b9 : Preorder \u03b9\nf : Filtration \u03b9 m\n\u03c4 : \u03a9 \u2192 \u03b9\ninst\u271d : PredOrder \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\ni : \u03b9\nhi_min : \u2200 (b : \u03b9), \u00acb < i\n\u03c9 : \u03a9\n\u22a2 \u00ac\u03c4 \u03c9 < i", "state_after": "no goals"}, {"tactic": "rw [this]", "annotated_tactic": ["rw [this]", []], "state_before": "\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u00b9 : Preorder \u03b9\nf : Filtration \u03b9 m\n\u03c4 : \u03a9 \u2192 \u03b9\ninst\u271d : PredOrder \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\ni : \u03b9\nhi_min : IsMin i\nthis : {\u03c9 | \u03c4 \u03c9 < i} = \u2205\n\u22a2 MeasurableSet {\u03c9 | \u03c4 \u03c9 < i}", "state_after": "\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u00b9 : Preorder \u03b9\nf : Filtration \u03b9 m\n\u03c4 : \u03a9 \u2192 \u03b9\ninst\u271d : PredOrder \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\ni : \u03b9\nhi_min : IsMin i\nthis : {\u03c9 | \u03c4 \u03c9 < i} = \u2205\n\u22a2 MeasurableSet \u2205"}, {"tactic": "exact @MeasurableSet.empty _ (f i)", "annotated_tactic": ["exact @<a>MeasurableSet.empty</a> _ (f i)", [{"full_name": "MeasurableSet.empty", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [80, 9], "def_end_pos": [80, 28]}]], "state_before": "\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u00b9 : Preorder \u03b9\nf : Filtration \u03b9 m\n\u03c4 : \u03a9 \u2192 \u03b9\ninst\u271d : PredOrder \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\ni : \u03b9\nhi_min : IsMin i\nthis : {\u03c9 | \u03c4 \u03c9 < i} = \u2205\n\u22a2 MeasurableSet \u2205", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/LocallyFinite.lean", "full_name": "Finset.image_add_right_Ioc", "start": [1165, 1], "end": [1166, 80], "traced_tactics": [{"tactic": "rw [\u2190 map_add_right_Ioc, map_eq_image, addRightEmbedding, Embedding.coeFn_mk]", "annotated_tactic": ["rw [\u2190 <a>map_add_right_Ioc</a>, <a>map_eq_image</a>, <a>addRightEmbedding</a>, <a>Embedding.coeFn_mk</a>]", [{"full_name": "Finset.map_add_right_Ioc", "def_path": "Mathlib/Data/Finset/LocallyFinite.lean", "def_pos": [1114, 9], "def_end_pos": [1114, 26]}, {"full_name": "Finset.map_eq_image", "def_path": "Mathlib/Data/Finset/Image.lean", "def_pos": [342, 9], "def_end_pos": [342, 21]}, {"full_name": "addRightEmbedding", "def_path": "Mathlib/Algebra/Hom/Embedding.lean", "def_pos": [37, 3], "def_end_pos": [37, 14]}, {"full_name": "Function.Embedding.coeFn_mk", "def_path": "Mathlib/Logic/Embedding/Basic.lean", "def_pos": [115, 9], "def_end_pos": [115, 17]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : OrderedCancelAddCommMonoid \u03b1\ninst\u271d\u00b2 : ExistsAddOfLE \u03b1\ninst\u271d\u00b9 : LocallyFiniteOrder \u03b1\ninst\u271d : DecidableEq \u03b1\na b c : \u03b1\n\u22a2 image (fun x => x + c) (Ioc a b) = Ioc (a + c) (b + c)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Constructions/Prod/Integral.lean", "full_name": "MeasureTheory.hasFiniteIntegral_prod_iff'", "start": [254, 1], "end": [267, 53], "traced_tactics": [{"tactic": "rw [hasFiniteIntegral_congr h1f.ae_eq_mk,\n  hasFiniteIntegral_prod_iff h1f.stronglyMeasurable_mk]", "annotated_tactic": ["rw [<a>hasFiniteIntegral_congr</a> h1f.ae_eq_mk,\n    <a>hasFiniteIntegral_prod_iff</a> h1f.stronglyMeasurable_mk]", [{"full_name": "MeasureTheory.hasFiniteIntegral_congr", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [166, 9], "def_end_pos": [166, 32]}, {"full_name": "MeasureTheory.hasFiniteIntegral_prod_iff", "def_path": "Mathlib/MeasureTheory/Constructions/Prod/Integral.lean", "def_pos": [233, 9], "def_end_pos": [233, 35]}]], "state_before": "\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u00d7 \u03b2 \u2192 E\nh1f : AEStronglyMeasurable f (Measure.prod \u03bc \u03bd)\n\u22a2 HasFiniteIntegral f \u2194\n    (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, HasFiniteIntegral fun y => f (x, y)) \u2227 HasFiniteIntegral fun x => \u222b (y : \u03b2), \u2016f (x, y)\u2016 \u2202\u03bd", "state_after": "\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u00d7 \u03b2 \u2192 E\nh1f : AEStronglyMeasurable f (Measure.prod \u03bc \u03bd)\n\u22a2 ((\u2200\u1d50 (x : \u03b1) \u2202\u03bc, HasFiniteIntegral fun y => AEStronglyMeasurable.mk f h1f (x, y)) \u2227\n      HasFiniteIntegral fun x => \u222b (y : \u03b2), \u2016AEStronglyMeasurable.mk f h1f (x, y)\u2016 \u2202\u03bd) \u2194\n    (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, HasFiniteIntegral fun y => f (x, y)) \u2227 HasFiniteIntegral fun x => \u222b (y : \u03b2), \u2016f (x, y)\u2016 \u2202\u03bd"}, {"tactic": "apply and_congr", "annotated_tactic": ["apply <a>and_congr</a>", [{"full_name": "and_congr", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [156, 9], "def_end_pos": [156, 18]}]], "state_before": "\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u00d7 \u03b2 \u2192 E\nh1f : AEStronglyMeasurable f (Measure.prod \u03bc \u03bd)\n\u22a2 ((\u2200\u1d50 (x : \u03b1) \u2202\u03bc, HasFiniteIntegral fun y => AEStronglyMeasurable.mk f h1f (x, y)) \u2227\n      HasFiniteIntegral fun x => \u222b (y : \u03b2), \u2016AEStronglyMeasurable.mk f h1f (x, y)\u2016 \u2202\u03bd) \u2194\n    (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, HasFiniteIntegral fun y => f (x, y)) \u2227 HasFiniteIntegral fun x => \u222b (y : \u03b2), \u2016f (x, y)\u2016 \u2202\u03bd", "state_after": "case h\u2081\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u00d7 \u03b2 \u2192 E\nh1f : AEStronglyMeasurable f (Measure.prod \u03bc \u03bd)\n\u22a2 (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, HasFiniteIntegral fun y => AEStronglyMeasurable.mk f h1f (x, y)) \u2194\n    \u2200\u1d50 (x : \u03b1) \u2202\u03bc, HasFiniteIntegral fun y => f (x, y)\n\ncase h\u2082\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u00d7 \u03b2 \u2192 E\nh1f : AEStronglyMeasurable f (Measure.prod \u03bc \u03bd)\n\u22a2 (HasFiniteIntegral fun x => \u222b (y : \u03b2), \u2016AEStronglyMeasurable.mk f h1f (x, y)\u2016 \u2202\u03bd) \u2194\n    HasFiniteIntegral fun x => \u222b (y : \u03b2), \u2016f (x, y)\u2016 \u2202\u03bd"}, {"tactic": "apply eventually_congr", "annotated_tactic": ["apply <a>eventually_congr</a>", [{"full_name": "Filter.eventually_congr", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1161, 9], "def_end_pos": [1161, 25]}]], "state_before": "case h\u2081\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u00d7 \u03b2 \u2192 E\nh1f : AEStronglyMeasurable f (Measure.prod \u03bc \u03bd)\n\u22a2 (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, HasFiniteIntegral fun y => AEStronglyMeasurable.mk f h1f (x, y)) \u2194\n    \u2200\u1d50 (x : \u03b1) \u2202\u03bc, HasFiniteIntegral fun y => f (x, y)", "state_after": "case h\u2081.h\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u00d7 \u03b2 \u2192 E\nh1f : AEStronglyMeasurable f (Measure.prod \u03bc \u03bd)\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, (HasFiniteIntegral fun y => AEStronglyMeasurable.mk f h1f (x, y)) \u2194 HasFiniteIntegral fun y => f (x, y)"}, {"tactic": "filter_upwards [ae_ae_of_ae_prod h1f.ae_eq_mk.symm]", "annotated_tactic": ["filter_upwards [<a>ae_ae_of_ae_prod</a> h1f.ae_eq_mk.symm]", [{"full_name": "MeasureTheory.Measure.ae_ae_of_ae_prod", "def_path": "Mathlib/MeasureTheory/Constructions/Prod/Basic.lean", "def_pos": [456, 9], "def_end_pos": [456, 25]}]], "state_before": "case h\u2081.h\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u00d7 \u03b2 \u2192 E\nh1f : AEStronglyMeasurable f (Measure.prod \u03bc \u03bd)\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, (HasFiniteIntegral fun y => AEStronglyMeasurable.mk f h1f (x, y)) \u2194 HasFiniteIntegral fun y => f (x, y)", "state_after": "case h\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u00d7 \u03b2 \u2192 E\nh1f : AEStronglyMeasurable f (Measure.prod \u03bc \u03bd)\n\u22a2 \u2200 (a : \u03b1),\n    (\u2200\u1d50 (y : \u03b2) \u2202\u03bd, AEStronglyMeasurable.mk f h1f (a, y) = f (a, y)) \u2192\n      ((HasFiniteIntegral fun y => AEStronglyMeasurable.mk f h1f (a, y)) \u2194 HasFiniteIntegral fun y => f (a, y))"}, {"tactic": "intro x hx", "annotated_tactic": ["intro x hx", []], "state_before": "case h\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u00d7 \u03b2 \u2192 E\nh1f : AEStronglyMeasurable f (Measure.prod \u03bc \u03bd)\n\u22a2 \u2200 (a : \u03b1),\n    (\u2200\u1d50 (y : \u03b2) \u2202\u03bd, AEStronglyMeasurable.mk f h1f (a, y) = f (a, y)) \u2192\n      ((HasFiniteIntegral fun y => AEStronglyMeasurable.mk f h1f (a, y)) \u2194 HasFiniteIntegral fun y => f (a, y))", "state_after": "case h\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u00d7 \u03b2 \u2192 E\nh1f : AEStronglyMeasurable f (Measure.prod \u03bc \u03bd)\nx : \u03b1\nhx : \u2200\u1d50 (y : \u03b2) \u2202\u03bd, AEStronglyMeasurable.mk f h1f (x, y) = f (x, y)\n\u22a2 (HasFiniteIntegral fun y => AEStronglyMeasurable.mk f h1f (x, y)) \u2194 HasFiniteIntegral fun y => f (x, y)"}, {"tactic": "exact hasFiniteIntegral_congr hx", "annotated_tactic": ["exact <a>hasFiniteIntegral_congr</a> hx", [{"full_name": "MeasureTheory.hasFiniteIntegral_congr", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [166, 9], "def_end_pos": [166, 32]}]], "state_before": "case h\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u00d7 \u03b2 \u2192 E\nh1f : AEStronglyMeasurable f (Measure.prod \u03bc \u03bd)\nx : \u03b1\nhx : \u2200\u1d50 (y : \u03b2) \u2202\u03bd, AEStronglyMeasurable.mk f h1f (x, y) = f (x, y)\n\u22a2 (HasFiniteIntegral fun y => AEStronglyMeasurable.mk f h1f (x, y)) \u2194 HasFiniteIntegral fun y => f (x, y)", "state_after": "no goals"}, {"tactic": "apply hasFiniteIntegral_congr", "annotated_tactic": ["apply <a>hasFiniteIntegral_congr</a>", [{"full_name": "MeasureTheory.hasFiniteIntegral_congr", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [166, 9], "def_end_pos": [166, 32]}]], "state_before": "case h\u2082\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u00d7 \u03b2 \u2192 E\nh1f : AEStronglyMeasurable f (Measure.prod \u03bc \u03bd)\n\u22a2 (HasFiniteIntegral fun x => \u222b (y : \u03b2), \u2016AEStronglyMeasurable.mk f h1f (x, y)\u2016 \u2202\u03bd) \u2194\n    HasFiniteIntegral fun x => \u222b (y : \u03b2), \u2016f (x, y)\u2016 \u2202\u03bd", "state_after": "case h\u2082.h\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u00d7 \u03b2 \u2192 E\nh1f : AEStronglyMeasurable f (Measure.prod \u03bc \u03bd)\n\u22a2 (fun x => \u222b (y : \u03b2), \u2016AEStronglyMeasurable.mk f h1f (x, y)\u2016 \u2202\u03bd) =\u1da0[ae \u03bc] fun x => \u222b (y : \u03b2), \u2016f (x, y)\u2016 \u2202\u03bd"}, {"tactic": "filter_upwards [ae_ae_of_ae_prod h1f.ae_eq_mk.symm] with _ hx using\n  integral_congr_ae (EventuallyEq.fun_comp hx _)", "annotated_tactic": ["filter_upwards [<a>ae_ae_of_ae_prod</a> h1f.ae_eq_mk.symm] with _ hx using\n      <a>integral_congr_ae</a> (<a>EventuallyEq.fun_comp</a> hx _)", [{"full_name": "MeasureTheory.Measure.ae_ae_of_ae_prod", "def_path": "Mathlib/MeasureTheory/Constructions/Prod/Basic.lean", "def_pos": [456, 9], "def_end_pos": [456, 25]}, {"full_name": "MeasureTheory.integral_congr_ae", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [938, 9], "def_end_pos": [938, 26]}, {"full_name": "Filter.EventuallyEq.fun_comp", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1520, 9], "def_end_pos": [1520, 30]}]], "state_before": "case h\u2082.h\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u00d7 \u03b2 \u2192 E\nh1f : AEStronglyMeasurable f (Measure.prod \u03bc \u03bd)\n\u22a2 (fun x => \u222b (y : \u03b2), \u2016AEStronglyMeasurable.mk f h1f (x, y)\u2016 \u2202\u03bd) =\u1da0[ae \u03bc] fun x => \u222b (y : \u03b2), \u2016f (x, y)\u2016 \u2202\u03bd", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Covering/Differentiation.lean", "full_name": "VitaliFamily.ae_tendsto_measure_inter_div_of_measurableSet", "start": [739, 1], "end": [745, 60], "traced_tactics": [{"tactic": "haveI : IsLocallyFiniteMeasure (\u03bc.restrict s) :=\n  isLocallyFiniteMeasure_of_le restrict_le_self", "annotated_tactic": ["haveI : <a>IsLocallyFiniteMeasure</a> (\u03bc.restrict s) :=\n    <a>isLocallyFiniteMeasure_of_le</a> <a>restrict_le_self</a>", [{"full_name": "MeasureTheory.IsLocallyFiniteMeasure", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3804, 7], "def_end_pos": [3804, 29]}, {"full_name": "MeasureTheory.Measure.isLocallyFiniteMeasure_of_le", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [4084, 9], "def_end_pos": [4084, 37]}, {"full_name": "MeasureTheory.Measure.restrict_le_self", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1578, 9], "def_end_pos": [1578, 25]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun a => \u2191\u2191\u03bc (s \u2229 a) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd (indicator s 1 x))", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nthis : IsLocallyFiniteMeasure (Measure.restrict \u03bc s)\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun a => \u2191\u2191\u03bc (s \u2229 a) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd (indicator s 1 x))"}, {"tactic": "filter_upwards [ae_tendsto_rnDeriv v (\u03bc.restrict s), rnDeriv_restrict \u03bc hs]", "annotated_tactic": ["filter_upwards [<a>ae_tendsto_rnDeriv</a> v (\u03bc.restrict s), <a>rnDeriv_restrict</a> \u03bc hs]", [{"full_name": "VitaliFamily.ae_tendsto_rnDeriv", "def_path": "Mathlib/MeasureTheory/Covering/Differentiation.lean", "def_pos": [714, 9], "def_end_pos": [714, 27]}, {"full_name": "MeasureTheory.Measure.rnDeriv_restrict", "def_path": "Mathlib/MeasureTheory/Decomposition/Lebesgue.lean", "def_pos": [377, 9], "def_end_pos": [377, 25]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nthis : IsLocallyFiniteMeasure (Measure.restrict \u03bc s)\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun a => \u2191\u2191\u03bc (s \u2229 a) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd (indicator s 1 x))", "state_after": "case h\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nthis : IsLocallyFiniteMeasure (Measure.restrict \u03bc s)\n\u22a2 \u2200 (a : \u03b1),\n    Tendsto (fun a => \u2191\u2191(Measure.restrict \u03bc s) a / \u2191\u2191\u03bc a) (filterAt v a) (\ud835\udcdd (rnDeriv (Measure.restrict \u03bc s) \u03bc a)) \u2192\n      rnDeriv (Measure.restrict \u03bc s) \u03bc a = indicator s 1 a \u2192\n        Tendsto (fun a => \u2191\u2191\u03bc (s \u2229 a) / \u2191\u2191\u03bc a) (filterAt v a) (\ud835\udcdd (indicator s 1 a))"}, {"tactic": "intro x hx h'x", "annotated_tactic": ["intro x hx h'x", []], "state_before": "case h\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nthis : IsLocallyFiniteMeasure (Measure.restrict \u03bc s)\n\u22a2 \u2200 (a : \u03b1),\n    Tendsto (fun a => \u2191\u2191(Measure.restrict \u03bc s) a / \u2191\u2191\u03bc a) (filterAt v a) (\ud835\udcdd (rnDeriv (Measure.restrict \u03bc s) \u03bc a)) \u2192\n      rnDeriv (Measure.restrict \u03bc s) \u03bc a = indicator s 1 a \u2192\n        Tendsto (fun a => \u2191\u2191\u03bc (s \u2229 a) / \u2191\u2191\u03bc a) (filterAt v a) (\ud835\udcdd (indicator s 1 a))", "state_after": "case h\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nthis : IsLocallyFiniteMeasure (Measure.restrict \u03bc s)\nx : \u03b1\nhx : Tendsto (fun a => \u2191\u2191(Measure.restrict \u03bc s) a / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd (rnDeriv (Measure.restrict \u03bc s) \u03bc x))\nh'x : rnDeriv (Measure.restrict \u03bc s) \u03bc x = indicator s 1 x\n\u22a2 Tendsto (fun a => \u2191\u2191\u03bc (s \u2229 a) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd (indicator s 1 x))"}, {"tactic": "simpa only [h'x, restrict_apply' hs, inter_comm] using hx", "annotated_tactic": ["simpa only [h'x, <a>restrict_apply'</a> hs, <a>inter_comm</a>] using hx", [{"full_name": "MeasureTheory.Measure.restrict_apply'", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1567, 9], "def_end_pos": [1567, 24]}, {"full_name": "Set.inter_comm", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [940, 9], "def_end_pos": [940, 19]}]], "state_before": "case h\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nthis : IsLocallyFiniteMeasure (Measure.restrict \u03bc s)\nx : \u03b1\nhx : Tendsto (fun a => \u2191\u2191(Measure.restrict \u03bc s) a / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd (rnDeriv (Measure.restrict \u03bc s) \u03bc x))\nh'x : rnDeriv (Measure.restrict \u03bc s) \u03bc x = indicator s 1 x\n\u22a2 Tendsto (fun a => \u2191\u2191\u03bc (s \u2229 a) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd (indicator s 1 x))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "full_name": "MeasureTheory.prob_compl_eq_zero_iff", "start": [3083, 1], "end": [3085, 70], "traced_tactics": [{"tactic": "rw [prob_compl_eq_one_sub hs, tsub_eq_zero_iff_le, one_le_prob_iff]", "annotated_tactic": ["rw [<a>prob_compl_eq_one_sub</a> hs, <a>tsub_eq_zero_iff_le</a>, <a>one_le_prob_iff</a>]", [{"full_name": "MeasureTheory.prob_compl_eq_one_sub", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3078, 9], "def_end_pos": [3078, 30]}, {"full_name": "tsub_eq_zero_iff_le", "def_path": "Mathlib/Algebra/Order/Sub/Canonical.lean", "def_pos": [324, 9], "def_end_pos": [324, 28]}, {"full_name": "MeasureTheory.one_le_prob_iff", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3071, 9], "def_end_pos": [3071, 24]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d : IsProbabilityMeasure \u03bc\nhs : MeasurableSet s\n\u22a2 \u2191\u2191\u03bc s\u1d9c = 0 \u2194 \u2191\u2191\u03bc s = 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Pointwise/SMul.lean", "full_name": "Set.zero_smul_set", "start": [812, 1], "end": [813, 80], "traced_tactics": [{"tactic": "simp only [\u2190 image_smul, image_eta, zero_smul, h.image_const, singleton_zero]", "annotated_tactic": ["simp only [\u2190 <a>image_smul</a>, <a>image_eta</a>, <a>zero_smul</a>, h.image_const, <a>singleton_zero</a>]", [{"full_name": "Set.image_smul", "def_path": "Mathlib/Data/Set/Pointwise/SMul.lean", "def_pos": [310, 9], "def_end_pos": [310, 19]}, {"full_name": "Set.image_eta", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [235, 9], "def_end_pos": [235, 18]}, {"full_name": "zero_smul", "def_path": "Mathlib/Algebra/SMulWithZero.lean", "def_pos": [70, 9], "def_end_pos": [70, 18]}, {"full_name": "Set.singleton_zero", "def_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "def_pos": [93, 3], "def_end_pos": [93, 14]}]], "state_before": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : Zero \u03b1\ninst\u271d\u00b9 : Zero \u03b2\ninst\u271d : SMulWithZero \u03b1 \u03b2\ns\u271d : Set \u03b1\nt s : Set \u03b2\nh : Set.Nonempty s\n\u22a2 0 \u2022 s = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Intervals/OrdConnected.lean", "full_name": "Set.ordConnected_image", "start": [202, 1], "end": [205, 48], "traced_tactics": [{"tactic": "erw [(e : \u03b1 \u2243o \u03b2).image_eq_preimage]", "annotated_tactic": ["erw [(e : \u03b1 \u2243o \u03b2).<a>image_eq_preimage</a>]", [{"full_name": "OrderIso.image_eq_preimage", "def_path": "Mathlib/Order/Hom/Set.lean", "def_pos": [41, 9], "def_end_pos": [41, 26]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ns\u271d t : Set \u03b1\nE : Type u_3\ninst\u271d : OrderIsoClass E \u03b1 \u03b2\ne : E\ns : Set \u03b1\nhs : OrdConnected s\n\u22a2 OrdConnected (\u2191e '' s)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ns\u271d t : Set \u03b1\nE : Type u_3\ninst\u271d : OrderIsoClass E \u03b1 \u03b2\ne : E\ns : Set \u03b1\nhs : OrdConnected s\n\u22a2 OrdConnected (\u2191(OrderIso.symm \u2191e) \u207b\u00b9' s)"}, {"tactic": "apply ordConnected_preimage (e : \u03b1 \u2243o \u03b2).symm", "annotated_tactic": ["apply <a>ordConnected_preimage</a> (e : \u03b1 \u2243o \u03b2).<a>symm</a>", [{"full_name": "Set.ordConnected_preimage", "def_path": "Mathlib/Data/Set/Intervals/OrdConnected.lean", "def_pos": [196, 9], "def_end_pos": [196, 30]}, {"full_name": "OrderIso.symm", "def_path": "Mathlib/Order/Hom/Basic.lean", "def_pos": [849, 5], "def_end_pos": [849, 9]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ns\u271d t : Set \u03b1\nE : Type u_3\ninst\u271d : OrderIsoClass E \u03b1 \u03b2\ne : E\ns : Set \u03b1\nhs : OrdConnected s\n\u22a2 OrdConnected (\u2191(OrderIso.symm \u2191e) \u207b\u00b9' s)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "full_name": "MeasureTheory.integral_Ico_eq_integral_Ioo", "start": [687, 1], "end": [688, 55], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Hausdorff.lean", "full_name": "MeasureTheory.hausdorffMeasure_measurePreserving_piFinTwo", "start": [1050, 1], "end": [1054, 66], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "full_name": "MeasureTheory.Lp.coe_nnnorm_toLp", "start": [287, 1], "end": [288, 54], "traced_tactics": [{"tactic": "rw [nnnorm_toLp f hf, ENNReal.coe_toNNReal hf.2.ne]", "annotated_tactic": ["rw [<a>nnnorm_toLp</a> f hf, <a>ENNReal.coe_toNNReal</a> hf.2.<a>ne</a>]", [{"full_name": "MeasureTheory.Lp.nnnorm_toLp", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [282, 9], "def_end_pos": [282, 20]}, {"full_name": "ENNReal.coe_toNNReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [180, 9], "def_end_pos": [180, 21]}, {"full_name": "LT.lt.ne", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [152, 7], "def_end_pos": [152, 15]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u22a2 \u2191\u2016Mem\u2112p.toLp f hf\u2016\u208a = snorm f p \u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Sign.lean", "full_name": "exists_signed_sum", "start": [506, 1], "end": [513, 81], "traced_tactics": [{"tactic": "simp [ht]", "annotated_tactic": ["simp [ht]", []], "state_before": "\u03b1\u271d \u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\nf : \u03b1 \u2192 \u2124\n\u03b2 : Type u_1\nt : Finset \u03b2\nsgn : \u03b2 \u2192 SignType\ng : \u03b2 \u2192 \u03b1\nhg : \u2200 (b : \u03b2), g b \u2208 s\nht : card t = \u2211 a in s, Int.natAbs (f a)\nhf : \u2200 (a : \u03b1), a \u2208 s \u2192 (\u2211 b in t, if g b = a then \u2191(sgn b) else 0) = f a\n\u22a2 Fintype.card { x // x \u2208 t } = \u2211 a in s, Int.natAbs (f a)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Haar/Basic.lean", "full_name": "MeasureTheory.Measure.haar.chaar_self", "start": [442, 1], "end": [450, 68], "traced_tactics": [{"tactic": "let eval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2080.toCompacts", "annotated_tactic": ["let eval : (<a>Compacts</a> G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2080.toCompacts", [{"full_name": "TopologicalSpace.Compacts", "def_path": "Mathlib/Topology/Sets/Compacts.lean", "def_pos": [36, 11], "def_end_pos": [36, 19]}]], "state_before": "G : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\n\u22a2 chaar K\u2080 K\u2080.toCompacts = 1", "state_after": "G : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2080.toCompacts\n\u22a2 chaar K\u2080 K\u2080.toCompacts = 1"}, {"tactic": "have : Continuous eval := continuous_apply _", "annotated_tactic": ["have : <a>Continuous</a> eval := <a>continuous_apply</a> _", [{"full_name": "Continuous", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [1591, 11], "def_end_pos": [1591, 21]}, {"full_name": "continuous_apply", "def_path": "Mathlib/Topology/Constructions.lean", "def_pos": [1208, 9], "def_end_pos": [1208, 25]}]], "state_before": "G : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2080.toCompacts\n\u22a2 chaar K\u2080 K\u2080.toCompacts = 1", "state_after": "G : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2080.toCompacts\nthis : Continuous eval\n\u22a2 chaar K\u2080 K\u2080.toCompacts = 1"}, {"tactic": "show chaar K\u2080 \u2208 eval \u207b\u00b9' {(1 : \u211d)}", "annotated_tactic": ["show <a>chaar</a> K\u2080 \u2208 eval \u207b\u00b9' {(1 : \u211d)}", [{"full_name": "MeasureTheory.Measure.haar.chaar", "def_path": "Mathlib/MeasureTheory/Measure/Haar/Basic.lean", "def_pos": [404, 19], "def_end_pos": [404, 24]}]], "state_before": "G : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2080.toCompacts\nthis : Continuous eval\n\u22a2 chaar K\u2080 K\u2080.toCompacts = 1", "state_after": "G : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2080.toCompacts\nthis : Continuous eval\n\u22a2 chaar K\u2080 \u2208 eval \u207b\u00b9' {1}"}, {"tactic": "apply mem_of_subset_of_mem _ (chaar_mem_clPrehaar K\u2080 \u22a4)", "annotated_tactic": ["apply <a>mem_of_subset_of_mem</a> _ (<a>chaar_mem_clPrehaar</a> K\u2080 \u22a4)", [{"full_name": "Set.mem_of_subset_of_mem", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [383, 9], "def_end_pos": [383, 29]}, {"full_name": "MeasureTheory.Measure.haar.chaar_mem_clPrehaar", "def_path": "Mathlib/MeasureTheory/Measure/Haar/Basic.lean", "def_pos": [416, 9], "def_end_pos": [416, 28]}]], "state_before": "G : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2080.toCompacts\nthis : Continuous eval\n\u22a2 chaar K\u2080 \u2208 eval \u207b\u00b9' {1}", "state_after": "G : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2080.toCompacts\nthis : Continuous eval\n\u22a2 clPrehaar \u2191K\u2080 \u22a4 \u2286 eval \u207b\u00b9' {1}"}, {"tactic": "unfold clPrehaar", "annotated_tactic": ["unfold <a>clPrehaar</a>", [{"full_name": "MeasureTheory.Measure.haar.clPrehaar", "def_path": "Mathlib/MeasureTheory/Measure/Haar/Basic.lean", "def_pos": [154, 5], "def_end_pos": [154, 14]}]], "state_before": "G : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2080.toCompacts\nthis : Continuous eval\n\u22a2 clPrehaar \u2191K\u2080 \u22a4 \u2286 eval \u207b\u00b9' {1}", "state_after": "G : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2080.toCompacts\nthis : Continuous eval\n\u22a2 closure (prehaar \u2191K\u2080 '' {U | U \u2286 \u2191\u22a4.toOpens \u2227 IsOpen U \u2227 1 \u2208 U}) \u2286 eval \u207b\u00b9' {1}"}, {"tactic": "rw [IsClosed.closure_subset_iff]", "annotated_tactic": ["rw [<a>IsClosed.closure_subset_iff</a>]", [{"full_name": "IsClosed.closure_subset_iff", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [465, 9], "def_end_pos": [465, 36]}]], "state_before": "G : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2080.toCompacts\nthis : Continuous eval\n\u22a2 closure (prehaar \u2191K\u2080 '' {U | U \u2286 \u2191\u22a4.toOpens \u2227 IsOpen U \u2227 1 \u2208 U}) \u2286 eval \u207b\u00b9' {1}", "state_after": "G : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2080.toCompacts\nthis : Continuous eval\n\u22a2 prehaar \u2191K\u2080 '' {U | U \u2286 \u2191\u22a4.toOpens \u2227 IsOpen U \u2227 1 \u2208 U} \u2286 eval \u207b\u00b9' {1}\n\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2080.toCompacts\nthis : Continuous eval\n\u22a2 IsClosed (eval \u207b\u00b9' {1})"}, {"tactic": "rintro _ \u27e8U, \u27e8_, h2U, h3U\u27e9, rfl\u27e9", "annotated_tactic": ["rintro _ \u27e8U, \u27e8_, h2U, h3U\u27e9, rfl\u27e9", []], "state_before": "G : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2080.toCompacts\nthis : Continuous eval\n\u22a2 prehaar \u2191K\u2080 '' {U | U \u2286 \u2191\u22a4.toOpens \u2227 IsOpen U \u2227 1 \u2208 U} \u2286 eval \u207b\u00b9' {1}", "state_after": "case intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2080.toCompacts\nthis : Continuous eval\nU : Set G\nleft\u271d : U \u2286 \u2191\u22a4.toOpens\nh2U : IsOpen U\nh3U : 1 \u2208 U\n\u22a2 prehaar (\u2191K\u2080) U \u2208 eval \u207b\u00b9' {1}"}, {"tactic": "apply prehaar_self", "annotated_tactic": ["apply <a>prehaar_self</a>", [{"full_name": "MeasureTheory.Measure.haar.prehaar_self", "def_path": "Mathlib/MeasureTheory/Measure/Haar/Basic.lean", "def_pos": [325, 9], "def_end_pos": [325, 21]}]], "state_before": "case intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2080.toCompacts\nthis : Continuous eval\nU : Set G\nleft\u271d : U \u2286 \u2191\u22a4.toOpens\nh2U : IsOpen U\nh3U : 1 \u2208 U\n\u22a2 prehaar (\u2191K\u2080) U \u2208 eval \u207b\u00b9' {1}", "state_after": "case intro.intro.intro.intro.hU\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2080.toCompacts\nthis : Continuous eval\nU : Set G\nleft\u271d : U \u2286 \u2191\u22a4.toOpens\nh2U : IsOpen U\nh3U : 1 \u2208 U\n\u22a2 Set.Nonempty (interior U)"}, {"tactic": "rw [h2U.interior_eq]", "annotated_tactic": ["rw [h2U.interior_eq]", []], "state_before": "case intro.intro.intro.intro.hU\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2080.toCompacts\nthis : Continuous eval\nU : Set G\nleft\u271d : U \u2286 \u2191\u22a4.toOpens\nh2U : IsOpen U\nh3U : 1 \u2208 U\n\u22a2 Set.Nonempty (interior U)", "state_after": "case intro.intro.intro.intro.hU\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2080.toCompacts\nthis : Continuous eval\nU : Set G\nleft\u271d : U \u2286 \u2191\u22a4.toOpens\nh2U : IsOpen U\nh3U : 1 \u2208 U\n\u22a2 Set.Nonempty U"}, {"tactic": "exact \u27e81, h3U\u27e9", "annotated_tactic": ["exact \u27e81, h3U\u27e9", []], "state_before": "case intro.intro.intro.intro.hU\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2080.toCompacts\nthis : Continuous eval\nU : Set G\nleft\u271d : U \u2286 \u2191\u22a4.toOpens\nh2U : IsOpen U\nh3U : 1 \u2208 U\n\u22a2 Set.Nonempty U", "state_after": "no goals"}, {"tactic": "apply continuous_iff_isClosed.mp this", "annotated_tactic": ["apply continuous_iff_isClosed.mp this", []], "state_before": "G : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2080.toCompacts\nthis : Continuous eval\n\u22a2 IsClosed (eval \u207b\u00b9' {1})", "state_after": "case a\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2080.toCompacts\nthis : Continuous eval\n\u22a2 IsClosed {1}"}, {"tactic": "exact isClosed_singleton", "annotated_tactic": ["exact <a>isClosed_singleton</a>", [{"full_name": "isClosed_singleton", "def_path": "Mathlib/Topology/Separation.lean", "def_pos": [384, 9], "def_end_pos": [384, 27]}]], "state_before": "case a\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2080.toCompacts\nthis : Continuous eval\n\u22a2 IsClosed {1}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "full_name": "MeasurableSet.induction_on_open", "start": [299, 1], "end": [307, 51], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Halting.lean", "full_name": "ComputablePred.computable_iff_re_compl_re'", "start": [261, 1], "end": [263, 45], "traced_tactics": [{"tactic": "classical exact computable_iff_re_compl_re", "annotated_tactic": ["classical exact <a>computable_iff_re_compl_re</a>", [{"full_name": "ComputablePred.computable_iff_re_compl_re", "def_path": "Mathlib/Computability/Halting.lean", "def_pos": [244, 9], "def_end_pos": [244, 35]}]], "state_before": "\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\np : \u03b1 \u2192 Prop\n\u22a2 ComputablePred p \u2194 RePred p \u2227 RePred fun a => \u00acp a", "state_after": "no goals"}, {"tactic": "exact computable_iff_re_compl_re", "annotated_tactic": ["exact <a>computable_iff_re_compl_re</a>", [{"full_name": "ComputablePred.computable_iff_re_compl_re", "def_path": "Mathlib/Computability/Halting.lean", "def_pos": [244, 9], "def_end_pos": [244, 35]}]], "state_before": "\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\np : \u03b1 \u2192 Prop\n\u22a2 ComputablePred p \u2194 RePred p \u2227 RePred fun a => \u00acp a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/QPF/Multivariate/Constructions/Cofix.lean", "full_name": "MvQPF.Cofix.bisim_aux", "start": [214, 9], "end": [255, 26], "traced_tactics": [{"tactic": "intro x", "annotated_tactic": ["intro x", []], "state_before": "n : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh' : \u2200 (x : Cofix F \u03b1), r x x\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\n\u22a2 \u2200 (x y : Cofix F \u03b1), r x y \u2192 x = y", "state_after": "n : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh' : \u2200 (x : Cofix F \u03b1), r x x\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nx : Cofix F \u03b1\n\u22a2 \u2200 (y : Cofix F \u03b1), r x y \u2192 x = y"}, {"tactic": "rcases x", "annotated_tactic": ["rcases x", []], "state_before": "n : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh' : \u2200 (x : Cofix F \u03b1), r x x\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nx : Cofix F \u03b1\n\u22a2 \u2200 (y : Cofix F \u03b1), r x y \u2192 x = y", "state_after": "case mk\nn : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh' : \u2200 (x : Cofix F \u03b1), r x x\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nx : Cofix F \u03b1\na\u271d : M (P F) \u03b1\n\u22a2 \u2200 (y : Cofix F \u03b1), r (Quot.mk Mcongr a\u271d) y \u2192 Quot.mk Mcongr a\u271d = y"}, {"tactic": "clear x", "annotated_tactic": ["clear x", []], "state_before": "case mk\nn : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh' : \u2200 (x : Cofix F \u03b1), r x x\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nx : Cofix F \u03b1\na\u271d : M (P F) \u03b1\n\u22a2 \u2200 (y : Cofix F \u03b1), r (Quot.mk Mcongr a\u271d) y \u2192 Quot.mk Mcongr a\u271d = y", "state_after": "case mk\nn : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh' : \u2200 (x : Cofix F \u03b1), r x x\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\na\u271d : M (P F) \u03b1\n\u22a2 \u2200 (y : Cofix F \u03b1), r (Quot.mk Mcongr a\u271d) y \u2192 Quot.mk Mcongr a\u271d = y"}, {"tactic": "rename M (P F) \u03b1 => x", "annotated_tactic": ["rename <a>M</a> (<a>P</a> F) \u03b1 => x", [{"full_name": "MvPFunctor.M", "def_path": "Mathlib/Data/PFunctor/Multivariate/M.lean", "def_pos": [101, 5], "def_end_pos": [101, 6]}, {"full_name": "MvQPF.P", "def_path": "Mathlib/Data/QPF/Multivariate/Basic.lean", "def_pos": [86, 3], "def_end_pos": [86, 4]}]], "state_before": "case mk\nn : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh' : \u2200 (x : Cofix F \u03b1), r x x\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\na\u271d : M (P F) \u03b1\n\u22a2 \u2200 (y : Cofix F \u03b1), r (Quot.mk Mcongr a\u271d) y \u2192 Quot.mk Mcongr a\u271d = y", "state_after": "case mk\nn : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh' : \u2200 (x : Cofix F \u03b1), r x x\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nx : M (P F) \u03b1\n\u22a2 \u2200 (y : Cofix F \u03b1), r (Quot.mk Mcongr x) y \u2192 Quot.mk Mcongr x = y"}, {"tactic": "intro y", "annotated_tactic": ["intro y", []], "state_before": "case mk\nn : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh' : \u2200 (x : Cofix F \u03b1), r x x\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nx : M (P F) \u03b1\n\u22a2 \u2200 (y : Cofix F \u03b1), r (Quot.mk Mcongr x) y \u2192 Quot.mk Mcongr x = y", "state_after": "case mk\nn : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh' : \u2200 (x : Cofix F \u03b1), r x x\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nx : M (P F) \u03b1\ny : Cofix F \u03b1\n\u22a2 r (Quot.mk Mcongr x) y \u2192 Quot.mk Mcongr x = y"}, {"tactic": "rcases y", "annotated_tactic": ["rcases y", []], "state_before": "case mk\nn : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh' : \u2200 (x : Cofix F \u03b1), r x x\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nx : M (P F) \u03b1\ny : Cofix F \u03b1\n\u22a2 r (Quot.mk Mcongr x) y \u2192 Quot.mk Mcongr x = y", "state_after": "case mk.mk\nn : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh' : \u2200 (x : Cofix F \u03b1), r x x\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nx : M (P F) \u03b1\ny : Cofix F \u03b1\na\u271d : M (P F) \u03b1\n\u22a2 r (Quot.mk Mcongr x) (Quot.mk Mcongr a\u271d) \u2192 Quot.mk Mcongr x = Quot.mk Mcongr a\u271d"}, {"tactic": "clear y", "annotated_tactic": ["clear y", []], "state_before": "case mk.mk\nn : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh' : \u2200 (x : Cofix F \u03b1), r x x\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nx : M (P F) \u03b1\ny : Cofix F \u03b1\na\u271d : M (P F) \u03b1\n\u22a2 r (Quot.mk Mcongr x) (Quot.mk Mcongr a\u271d) \u2192 Quot.mk Mcongr x = Quot.mk Mcongr a\u271d", "state_after": "case mk.mk\nn : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh' : \u2200 (x : Cofix F \u03b1), r x x\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nx a\u271d : M (P F) \u03b1\n\u22a2 r (Quot.mk Mcongr x) (Quot.mk Mcongr a\u271d) \u2192 Quot.mk Mcongr x = Quot.mk Mcongr a\u271d"}, {"tactic": "rename M (P F) \u03b1 => y", "annotated_tactic": ["rename <a>M</a> (<a>P</a> F) \u03b1 => y", [{"full_name": "MvPFunctor.M", "def_path": "Mathlib/Data/PFunctor/Multivariate/M.lean", "def_pos": [101, 5], "def_end_pos": [101, 6]}, {"full_name": "MvQPF.P", "def_path": "Mathlib/Data/QPF/Multivariate/Basic.lean", "def_pos": [86, 3], "def_end_pos": [86, 4]}]], "state_before": "case mk.mk\nn : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh' : \u2200 (x : Cofix F \u03b1), r x x\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nx a\u271d : M (P F) \u03b1\n\u22a2 r (Quot.mk Mcongr x) (Quot.mk Mcongr a\u271d) \u2192 Quot.mk Mcongr x = Quot.mk Mcongr a\u271d", "state_after": "case mk.mk\nn : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh' : \u2200 (x : Cofix F \u03b1), r x x\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nx y : M (P F) \u03b1\n\u22a2 r (Quot.mk Mcongr x) (Quot.mk Mcongr y) \u2192 Quot.mk Mcongr x = Quot.mk Mcongr y"}, {"tactic": "intro rxy", "annotated_tactic": ["intro rxy", []], "state_before": "case mk.mk\nn : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh' : \u2200 (x : Cofix F \u03b1), r x x\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nx y : M (P F) \u03b1\n\u22a2 r (Quot.mk Mcongr x) (Quot.mk Mcongr y) \u2192 Quot.mk Mcongr x = Quot.mk Mcongr y", "state_after": "case mk.mk\nn : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh' : \u2200 (x : Cofix F \u03b1), r x x\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nx y : M (P F) \u03b1\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\n\u22a2 Quot.mk Mcongr x = Quot.mk Mcongr y"}, {"tactic": "apply Quot.sound", "annotated_tactic": ["apply <a>Quot.sound</a>", [{"full_name": "Quot.sound", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [1209, 7], "def_end_pos": [1209, 12]}]], "state_before": "case mk.mk\nn : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh' : \u2200 (x : Cofix F \u03b1), r x x\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nx y : M (P F) \u03b1\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\n\u22a2 Quot.mk Mcongr x = Quot.mk Mcongr y", "state_after": "case mk.mk.a\nn : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh' : \u2200 (x : Cofix F \u03b1), r x x\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nx y : M (P F) \u03b1\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\n\u22a2 Mcongr x y"}, {"tactic": "let r' := fun x y => r (Quot.mk _ x) (Quot.mk _ y)", "annotated_tactic": ["let r' := fun x y => r (<a>Quot.mk</a> _ x) (<a>Quot.mk</a> _ y)", [{"full_name": "Quot.mk", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [406, 14], "def_end_pos": [406, 21]}, {"full_name": "Quot.mk", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [406, 14], "def_end_pos": [406, 21]}]], "state_before": "case mk.mk.a\nn : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh' : \u2200 (x : Cofix F \u03b1), r x x\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nx y : M (P F) \u03b1\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\n\u22a2 Mcongr x y", "state_after": "case mk.mk.a\nn : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh' : \u2200 (x : Cofix F \u03b1), r x x\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nx y : M (P F) \u03b1\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\n\u22a2 Mcongr x y"}, {"tactic": "have hr' : r' = fun x y => r (Quot.mk _ x) (Quot.mk _ y) := rfl", "annotated_tactic": ["have hr' : r' = fun x y => r (<a>Quot.mk</a> _ x) (<a>Quot.mk</a> _ y) := <a>rfl</a>", [{"full_name": "Quot.mk", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [406, 14], "def_end_pos": [406, 21]}, {"full_name": "Quot.mk", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [406, 14], "def_end_pos": [406, 21]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "case mk.mk.a\nn : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh' : \u2200 (x : Cofix F \u03b1), r x x\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nx y : M (P F) \u03b1\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\n\u22a2 Mcongr x y", "state_after": "case mk.mk.a\nn : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh' : \u2200 (x : Cofix F \u03b1), r x x\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nx y : M (P F) \u03b1\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nhr' : r' = fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\n\u22a2 Mcongr x y"}, {"tactic": "have : IsPrecongr r' := by\n  intro a b r'ab\n  have h\u2080 :\n    appendFun id (Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest q.P a) =\n      appendFun id (Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest q.P b) :=\n    by rw [appendFun_comp_id, comp_map, comp_map]; exact h _ _ r'ab\n  have h\u2081 : \u2200 u v : q.P.M \u03b1, Mcongr u v \u2192 Quot.mk r' u = Quot.mk r' v := by\n    intro u v cuv\n    apply Quot.sound\n    dsimp [hr']\n    rw [Quot.sound cuv]\n    apply h'\n  let f : Quot r \u2192 Quot r' :=\n    Quot.lift (Quot.lift (Quot.mk r') h\u2081)\n      (by\n        intro c\n        apply Quot.inductionOn\n          (motive := fun c =>\n            \u2200b, r c b \u2192 Quot.lift (Quot.mk r') h\u2081 c = Quot.lift (Quot.mk r') h\u2081 b) c\n        clear c\n        intro c d\n        apply Quot.inductionOn\n          (motive := fun d => r (Quot.mk Mcongr c) d \u2192\n            Quot.lift (Quot.mk r') h\u2081 (Quot.mk Mcongr c) = Quot.lift (Quot.mk r') h\u2081 d) d\n        clear d\n        intro d rcd; apply Quot.sound; apply rcd)\n  have : f \u2218 Quot.mk r \u2218 Quot.mk Mcongr = Quot.mk r' := rfl\n  rw [\u2190 this, appendFun_comp_id, q.P.comp_map, q.P.comp_map, abs_map, abs_map, abs_map, abs_map,\n    h\u2080]", "annotated_tactic": ["have : <a>IsPrecongr</a> r' := by\n    intro a b r'ab\n    have h\u2080 :\n      <a>appendFun</a> <a>id</a> (<a>Quot.mk</a> r \u2218 <a>Quot.mk</a> <a>Mcongr</a>) <$$> <a>MvQPF.abs</a> (<a>M.dest</a> q.P a) =\n        <a>appendFun</a> <a>id</a> (<a>Quot.mk</a> r \u2218 <a>Quot.mk</a> <a>Mcongr</a>) <$$> <a>MvQPF.abs</a> (<a>M.dest</a> q.P b) :=\n      by rw [<a>appendFun_comp_id</a>, <a>comp_map</a>, <a>comp_map</a>]; exact h _ _ r'ab\n    have h\u2081 : \u2200 u v : q.P.M \u03b1, <a>Mcongr</a> u v \u2192 <a>Quot.mk</a> r' u = <a>Quot.mk</a> r' v := by\n      intro u v cuv\n      apply <a>Quot.sound</a>\n      dsimp [hr']\n      rw [<a>Quot.sound</a> cuv]\n      apply h'\n    let f : <a>Quot</a> r \u2192 <a>Quot</a> r' :=\n      <a>Quot.lift</a> (<a>Quot.lift</a> (<a>Quot.mk</a> r') h\u2081)\n        (by\n          intro c\n          apply <a>Quot.inductionOn</a>\n            (motive := fun c =>\n              \u2200b, r c b \u2192 <a>Quot.lift</a> (<a>Quot.mk</a> r') h\u2081 c = <a>Quot.lift</a> (<a>Quot.mk</a> r') h\u2081 b) c\n          clear c\n          intro c d\n          apply <a>Quot.inductionOn</a>\n            (motive := fun d => r (<a>Quot.mk</a> <a>Mcongr</a> c) d \u2192\n              <a>Quot.lift</a> (<a>Quot.mk</a> r') h\u2081 (<a>Quot.mk</a> <a>Mcongr</a> c) = <a>Quot.lift</a> (<a>Quot.mk</a> r') h\u2081 d) d\n          clear d\n          intro d rcd; apply <a>Quot.sound</a>; apply rcd)\n    have : f \u2218 <a>Quot.mk</a> r \u2218 <a>Quot.mk</a> <a>Mcongr</a> = <a>Quot.mk</a> r' := <a>rfl</a>\n    rw [\u2190 this, <a>appendFun_comp_id</a>, q.P.comp_map, q.P.comp_map, <a>abs_map</a>, <a>abs_map</a>, <a>abs_map</a>, <a>abs_map</a>,\n      h\u2080]", [{"full_name": "MvQPF.IsPrecongr", "def_path": "Mathlib/Data/QPF/Multivariate/Constructions/Cofix.lean", "def_pos": [71, 5], "def_end_pos": [71, 15]}, {"full_name": "TypeVec.appendFun", "def_path": "Mathlib/Data/TypeVec.lean", "def_pos": [150, 5], "def_end_pos": [150, 14]}, {"full_name": "TypeVec.id", "def_path": "Mathlib/Data/TypeVec.lean", "def_pos": [69, 5], "def_end_pos": [69, 7]}, {"full_name": "Quot.mk", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [406, 14], "def_end_pos": [406, 21]}, {"full_name": "Quot.mk", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [406, 14], "def_end_pos": [406, 21]}, {"full_name": "MvQPF.Mcongr", "def_path": "Mathlib/Data/QPF/Multivariate/Constructions/Cofix.lean", "def_pos": [79, 5], "def_end_pos": [79, 11]}, {"full_name": "MvQPF.abs", "def_path": "Mathlib/Data/QPF/Multivariate/Basic.lean", "def_pos": [87, 3], "def_end_pos": [87, 6]}, {"full_name": "MvPFunctor.M.dest", "def_path": "Mathlib/Data/PFunctor/Multivariate/M.lean", "def_pos": [186, 5], "def_end_pos": [186, 11]}, {"full_name": "TypeVec.appendFun", "def_path": "Mathlib/Data/TypeVec.lean", "def_pos": [150, 5], "def_end_pos": [150, 14]}, {"full_name": "TypeVec.id", "def_path": "Mathlib/Data/TypeVec.lean", "def_pos": [69, 5], "def_end_pos": [69, 7]}, {"full_name": "Quot.mk", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [406, 14], "def_end_pos": [406, 21]}, {"full_name": "Quot.mk", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [406, 14], "def_end_pos": [406, 21]}, {"full_name": "MvQPF.Mcongr", "def_path": "Mathlib/Data/QPF/Multivariate/Constructions/Cofix.lean", "def_pos": [79, 5], "def_end_pos": [79, 11]}, {"full_name": "MvQPF.abs", "def_path": "Mathlib/Data/QPF/Multivariate/Basic.lean", "def_pos": [87, 3], "def_end_pos": [87, 6]}, {"full_name": "MvPFunctor.M.dest", "def_path": "Mathlib/Data/PFunctor/Multivariate/M.lean", "def_pos": [186, 5], "def_end_pos": [186, 11]}, {"full_name": "TypeVec.appendFun_comp_id", "def_path": "Mathlib/Data/TypeVec.lean", "def_pos": [270, 9], "def_end_pos": [270, 26]}, {"full_name": "MvQPF.comp_map", "def_path": "Mathlib/Data/QPF/Multivariate/Basic.lean", "def_pos": [112, 9], "def_end_pos": [112, 17]}, {"full_name": "MvQPF.comp_map", "def_path": "Mathlib/Data/QPF/Multivariate/Basic.lean", "def_pos": [112, 9], "def_end_pos": [112, 17]}, {"full_name": "MvQPF.Mcongr", "def_path": "Mathlib/Data/QPF/Multivariate/Constructions/Cofix.lean", "def_pos": [79, 5], "def_end_pos": [79, 11]}, {"full_name": "Quot.mk", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [406, 14], "def_end_pos": [406, 21]}, {"full_name": "Quot.mk", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [406, 14], "def_end_pos": [406, 21]}, {"full_name": "Quot.sound", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [1209, 7], "def_end_pos": [1209, 12]}, {"full_name": "Quot.sound", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [1209, 7], "def_end_pos": [1209, 12]}, {"full_name": "Quot", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [398, 14], "def_end_pos": [398, 18]}, {"full_name": "Quot", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [398, 14], "def_end_pos": [398, 18]}, {"full_name": "Quot.lift", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [424, 14], "def_end_pos": [424, 23]}, {"full_name": "Quot.lift", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [424, 14], "def_end_pos": [424, 23]}, {"full_name": "Quot.mk", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [406, 14], "def_end_pos": [406, 21]}, {"full_name": "Quot.inductionOn", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [1233, 19], "def_end_pos": [1233, 30]}, {"full_name": "Quot.lift", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [424, 14], "def_end_pos": [424, 23]}, {"full_name": "Quot.mk", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [406, 14], "def_end_pos": [406, 21]}, {"full_name": "Quot.lift", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [424, 14], "def_end_pos": [424, 23]}, {"full_name": "Quot.mk", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [406, 14], "def_end_pos": [406, 21]}, {"full_name": "Quot.inductionOn", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [1233, 19], "def_end_pos": [1233, 30]}, {"full_name": "Quot.mk", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [406, 14], "def_end_pos": [406, 21]}, {"full_name": "MvQPF.Mcongr", "def_path": "Mathlib/Data/QPF/Multivariate/Constructions/Cofix.lean", "def_pos": [79, 5], "def_end_pos": [79, 11]}, {"full_name": "Quot.lift", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [424, 14], "def_end_pos": [424, 23]}, {"full_name": "Quot.mk", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [406, 14], "def_end_pos": [406, 21]}, {"full_name": "Quot.mk", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [406, 14], "def_end_pos": [406, 21]}, {"full_name": "MvQPF.Mcongr", "def_path": "Mathlib/Data/QPF/Multivariate/Constructions/Cofix.lean", "def_pos": [79, 5], "def_end_pos": [79, 11]}, {"full_name": "Quot.lift", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [424, 14], "def_end_pos": [424, 23]}, {"full_name": "Quot.mk", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [406, 14], "def_end_pos": [406, 21]}, {"full_name": "Quot.sound", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [1209, 7], "def_end_pos": [1209, 12]}, {"full_name": "Quot.mk", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [406, 14], "def_end_pos": [406, 21]}, {"full_name": "Quot.mk", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [406, 14], "def_end_pos": [406, 21]}, {"full_name": "MvQPF.Mcongr", "def_path": "Mathlib/Data/QPF/Multivariate/Constructions/Cofix.lean", "def_pos": [79, 5], "def_end_pos": [79, 11]}, {"full_name": "Quot.mk", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [406, 14], "def_end_pos": [406, 21]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}, {"full_name": "TypeVec.appendFun_comp_id", "def_path": "Mathlib/Data/TypeVec.lean", "def_pos": [270, 9], "def_end_pos": [270, 26]}, {"full_name": "MvQPF.abs_map", "def_path": "Mathlib/Data/QPF/Multivariate/Basic.lean", "def_pos": [90, 3], "def_end_pos": [90, 10]}, {"full_name": "MvQPF.abs_map", "def_path": "Mathlib/Data/QPF/Multivariate/Basic.lean", "def_pos": [90, 3], "def_end_pos": [90, 10]}, {"full_name": "MvQPF.abs_map", "def_path": "Mathlib/Data/QPF/Multivariate/Basic.lean", "def_pos": [90, 3], "def_end_pos": [90, 10]}, {"full_name": "MvQPF.abs_map", "def_path": "Mathlib/Data/QPF/Multivariate/Basic.lean", "def_pos": [90, 3], "def_end_pos": [90, 10]}]], "state_before": "case mk.mk.a\nn : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh' : \u2200 (x : Cofix F \u03b1), r x x\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nx y : M (P F) \u03b1\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nhr' : r' = fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\n\u22a2 Mcongr x y", "state_after": "case mk.mk.a\nn : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh' : \u2200 (x : Cofix F \u03b1), r x x\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nx y : M (P F) \u03b1\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nhr' : r' = fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nthis : IsPrecongr r'\n\u22a2 Mcongr x y"}, {"tactic": "refine' \u27e8r', this, rxy\u27e9", "annotated_tactic": ["refine' \u27e8r', this, rxy\u27e9", []], "state_before": "case mk.mk.a\nn : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh' : \u2200 (x : Cofix F \u03b1), r x x\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nx y : M (P F) \u03b1\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nhr' : r' = fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nthis : IsPrecongr r'\n\u22a2 Mcongr x y", "state_after": "no goals"}, {"tactic": "intro a b r'ab", "annotated_tactic": ["intro a b r'ab", []], "state_before": "n : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh' : \u2200 (x : Cofix F \u03b1), r x x\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nx y : M (P F) \u03b1\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nhr' : r' = fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\n\u22a2 IsPrecongr r'", "state_after": "n : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh' : \u2200 (x : Cofix F \u03b1), r x x\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nx y : M (P F) \u03b1\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nhr' : r' = fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : M (P F) \u03b1\nr'ab : r' a b\n\u22a2 MvQPF.abs ((TypeVec.id ::: Quot.mk r') <$$> M.dest (P F) a) =\n    MvQPF.abs ((TypeVec.id ::: Quot.mk r') <$$> M.dest (P F) b)"}, {"tactic": "have h\u2080 :\n  appendFun id (Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest q.P a) =\n    appendFun id (Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest q.P b) :=\n  by rw [appendFun_comp_id, comp_map, comp_map]; exact h _ _ r'ab", "annotated_tactic": ["have h\u2080 :\n      <a>appendFun</a> <a>id</a> (<a>Quot.mk</a> r \u2218 <a>Quot.mk</a> <a>Mcongr</a>) <$$> <a>MvQPF.abs</a> (<a>M.dest</a> q.P a) =\n        <a>appendFun</a> <a>id</a> (<a>Quot.mk</a> r \u2218 <a>Quot.mk</a> <a>Mcongr</a>) <$$> <a>MvQPF.abs</a> (<a>M.dest</a> q.P b) :=\n      by rw [<a>appendFun_comp_id</a>, <a>comp_map</a>, <a>comp_map</a>]; exact h _ _ r'ab", [{"full_name": "TypeVec.appendFun", "def_path": "Mathlib/Data/TypeVec.lean", "def_pos": [150, 5], "def_end_pos": [150, 14]}, {"full_name": "TypeVec.id", "def_path": "Mathlib/Data/TypeVec.lean", "def_pos": [69, 5], "def_end_pos": [69, 7]}, {"full_name": "Quot.mk", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [406, 14], "def_end_pos": [406, 21]}, {"full_name": "Quot.mk", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [406, 14], "def_end_pos": [406, 21]}, {"full_name": "MvQPF.Mcongr", "def_path": "Mathlib/Data/QPF/Multivariate/Constructions/Cofix.lean", "def_pos": [79, 5], "def_end_pos": [79, 11]}, {"full_name": "MvQPF.abs", "def_path": "Mathlib/Data/QPF/Multivariate/Basic.lean", "def_pos": [87, 3], "def_end_pos": [87, 6]}, {"full_name": "MvPFunctor.M.dest", "def_path": "Mathlib/Data/PFunctor/Multivariate/M.lean", "def_pos": [186, 5], "def_end_pos": [186, 11]}, {"full_name": "TypeVec.appendFun", "def_path": "Mathlib/Data/TypeVec.lean", "def_pos": [150, 5], "def_end_pos": [150, 14]}, {"full_name": "TypeVec.id", "def_path": "Mathlib/Data/TypeVec.lean", "def_pos": [69, 5], "def_end_pos": [69, 7]}, {"full_name": "Quot.mk", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [406, 14], "def_end_pos": [406, 21]}, {"full_name": "Quot.mk", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [406, 14], "def_end_pos": [406, 21]}, {"full_name": "MvQPF.Mcongr", "def_path": "Mathlib/Data/QPF/Multivariate/Constructions/Cofix.lean", "def_pos": [79, 5], "def_end_pos": [79, 11]}, {"full_name": "MvQPF.abs", "def_path": "Mathlib/Data/QPF/Multivariate/Basic.lean", "def_pos": [87, 3], "def_end_pos": [87, 6]}, {"full_name": "MvPFunctor.M.dest", "def_path": "Mathlib/Data/PFunctor/Multivariate/M.lean", "def_pos": [186, 5], "def_end_pos": [186, 11]}, {"full_name": "TypeVec.appendFun_comp_id", "def_path": "Mathlib/Data/TypeVec.lean", "def_pos": [270, 9], "def_end_pos": [270, 26]}, {"full_name": "MvQPF.comp_map", "def_path": "Mathlib/Data/QPF/Multivariate/Basic.lean", "def_pos": [112, 9], "def_end_pos": [112, 17]}, {"full_name": "MvQPF.comp_map", "def_path": "Mathlib/Data/QPF/Multivariate/Basic.lean", "def_pos": [112, 9], "def_end_pos": [112, 17]}]], "state_before": "n : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh' : \u2200 (x : Cofix F \u03b1), r x x\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nx y : M (P F) \u03b1\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nhr' : r' = fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : M (P F) \u03b1\nr'ab : r' a b\n\u22a2 MvQPF.abs ((TypeVec.id ::: Quot.mk r') <$$> M.dest (P F) a) =\n    MvQPF.abs ((TypeVec.id ::: Quot.mk r') <$$> M.dest (P F) b)", "state_after": "n : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh' : \u2200 (x : Cofix F \u03b1), r x x\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nx y : M (P F) \u03b1\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nhr' : r' = fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : M (P F) \u03b1\nr'ab : r' a b\nh\u2080 :\n  (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) a) =\n    (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) b)\n\u22a2 MvQPF.abs ((TypeVec.id ::: Quot.mk r') <$$> M.dest (P F) a) =\n    MvQPF.abs ((TypeVec.id ::: Quot.mk r') <$$> M.dest (P F) b)"}, {"tactic": "have h\u2081 : \u2200 u v : q.P.M \u03b1, Mcongr u v \u2192 Quot.mk r' u = Quot.mk r' v := by\n  intro u v cuv\n  apply Quot.sound\n  dsimp [hr']\n  rw [Quot.sound cuv]\n  apply h'", "annotated_tactic": ["have h\u2081 : \u2200 u v : q.P.M \u03b1, <a>Mcongr</a> u v \u2192 <a>Quot.mk</a> r' u = <a>Quot.mk</a> r' v := by\n      intro u v cuv\n      apply <a>Quot.sound</a>\n      dsimp [hr']\n      rw [<a>Quot.sound</a> cuv]\n      apply h'", [{"full_name": "MvQPF.Mcongr", "def_path": "Mathlib/Data/QPF/Multivariate/Constructions/Cofix.lean", "def_pos": [79, 5], "def_end_pos": [79, 11]}, {"full_name": "Quot.mk", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [406, 14], "def_end_pos": [406, 21]}, {"full_name": "Quot.mk", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [406, 14], "def_end_pos": [406, 21]}, {"full_name": "Quot.sound", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [1209, 7], "def_end_pos": [1209, 12]}, {"full_name": "Quot.sound", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [1209, 7], "def_end_pos": [1209, 12]}]], "state_before": "n : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh' : \u2200 (x : Cofix F \u03b1), r x x\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nx y : M (P F) \u03b1\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nhr' : r' = fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : M (P F) \u03b1\nr'ab : r' a b\nh\u2080 :\n  (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) a) =\n    (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) b)\n\u22a2 MvQPF.abs ((TypeVec.id ::: Quot.mk r') <$$> M.dest (P F) a) =\n    MvQPF.abs ((TypeVec.id ::: Quot.mk r') <$$> M.dest (P F) b)", "state_after": "n : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh' : \u2200 (x : Cofix F \u03b1), r x x\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nx y : M (P F) \u03b1\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nhr' : r' = fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : M (P F) \u03b1\nr'ab : r' a b\nh\u2080 :\n  (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) a) =\n    (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) b)\nh\u2081 : \u2200 (u v : M (P F) \u03b1), Mcongr u v \u2192 Quot.mk r' u = Quot.mk r' v\n\u22a2 MvQPF.abs ((TypeVec.id ::: Quot.mk r') <$$> M.dest (P F) a) =\n    MvQPF.abs ((TypeVec.id ::: Quot.mk r') <$$> M.dest (P F) b)"}, {"tactic": "let f : Quot r \u2192 Quot r' :=\n  Quot.lift (Quot.lift (Quot.mk r') h\u2081)\n    (by\n      intro c\n      apply Quot.inductionOn\n        (motive := fun c =>\n          \u2200b, r c b \u2192 Quot.lift (Quot.mk r') h\u2081 c = Quot.lift (Quot.mk r') h\u2081 b) c\n      clear c\n      intro c d\n      apply Quot.inductionOn\n        (motive := fun d => r (Quot.mk Mcongr c) d \u2192\n          Quot.lift (Quot.mk r') h\u2081 (Quot.mk Mcongr c) = Quot.lift (Quot.mk r') h\u2081 d) d\n      clear d\n      intro d rcd; apply Quot.sound; apply rcd)", "annotated_tactic": ["let f : <a>Quot</a> r \u2192 <a>Quot</a> r' :=\n      <a>Quot.lift</a> (<a>Quot.lift</a> (<a>Quot.mk</a> r') h\u2081)\n        (by\n          intro c\n          apply <a>Quot.inductionOn</a>\n            (motive := fun c =>\n              \u2200b, r c b \u2192 <a>Quot.lift</a> (<a>Quot.mk</a> r') h\u2081 c = <a>Quot.lift</a> (<a>Quot.mk</a> r') h\u2081 b) c\n          clear c\n          intro c d\n          apply <a>Quot.inductionOn</a>\n            (motive := fun d => r (<a>Quot.mk</a> <a>Mcongr</a> c) d \u2192\n              <a>Quot.lift</a> (<a>Quot.mk</a> r') h\u2081 (<a>Quot.mk</a> <a>Mcongr</a> c) = <a>Quot.lift</a> (<a>Quot.mk</a> r') h\u2081 d) d\n          clear d\n          intro d rcd; apply <a>Quot.sound</a>; apply rcd)", [{"full_name": "Quot", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [398, 14], "def_end_pos": [398, 18]}, {"full_name": "Quot", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [398, 14], "def_end_pos": [398, 18]}, {"full_name": "Quot.lift", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [424, 14], "def_end_pos": [424, 23]}, {"full_name": "Quot.lift", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [424, 14], "def_end_pos": [424, 23]}, {"full_name": "Quot.mk", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [406, 14], "def_end_pos": [406, 21]}, {"full_name": "Quot.inductionOn", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [1233, 19], "def_end_pos": [1233, 30]}, {"full_name": "Quot.lift", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [424, 14], "def_end_pos": [424, 23]}, {"full_name": "Quot.mk", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [406, 14], "def_end_pos": [406, 21]}, {"full_name": "Quot.lift", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [424, 14], "def_end_pos": [424, 23]}, {"full_name": "Quot.mk", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [406, 14], "def_end_pos": [406, 21]}, {"full_name": "Quot.inductionOn", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [1233, 19], "def_end_pos": [1233, 30]}, {"full_name": "Quot.mk", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [406, 14], "def_end_pos": [406, 21]}, {"full_name": "MvQPF.Mcongr", "def_path": "Mathlib/Data/QPF/Multivariate/Constructions/Cofix.lean", "def_pos": [79, 5], "def_end_pos": [79, 11]}, {"full_name": "Quot.lift", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [424, 14], "def_end_pos": [424, 23]}, {"full_name": "Quot.mk", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [406, 14], "def_end_pos": [406, 21]}, {"full_name": "Quot.mk", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [406, 14], "def_end_pos": [406, 21]}, {"full_name": "MvQPF.Mcongr", "def_path": "Mathlib/Data/QPF/Multivariate/Constructions/Cofix.lean", "def_pos": [79, 5], "def_end_pos": [79, 11]}, {"full_name": "Quot.lift", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [424, 14], "def_end_pos": [424, 23]}, {"full_name": "Quot.mk", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [406, 14], "def_end_pos": [406, 21]}, {"full_name": "Quot.sound", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [1209, 7], "def_end_pos": [1209, 12]}]], "state_before": "n : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh' : \u2200 (x : Cofix F \u03b1), r x x\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nx y : M (P F) \u03b1\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nhr' : r' = fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : M (P F) \u03b1\nr'ab : r' a b\nh\u2080 :\n  (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) a) =\n    (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) b)\nh\u2081 : \u2200 (u v : M (P F) \u03b1), Mcongr u v \u2192 Quot.mk r' u = Quot.mk r' v\n\u22a2 MvQPF.abs ((TypeVec.id ::: Quot.mk r') <$$> M.dest (P F) a) =\n    MvQPF.abs ((TypeVec.id ::: Quot.mk r') <$$> M.dest (P F) b)", "state_after": "n : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh' : \u2200 (x : Cofix F \u03b1), r x x\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nx y : M (P F) \u03b1\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nhr' : r' = fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : M (P F) \u03b1\nr'ab : r' a b\nh\u2080 :\n  (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) a) =\n    (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) b)\nh\u2081 : \u2200 (u v : M (P F) \u03b1), Mcongr u v \u2192 Quot.mk r' u = Quot.mk r' v\nf : Quot r \u2192 Quot r' :=\n  Quot.lift (Quot.lift (Quot.mk r') h\u2081)\n    (_ : \u2200 (c b : Cofix F \u03b1), r c b \u2192 Quot.lift (Quot.mk r') h\u2081 c = Quot.lift (Quot.mk r') h\u2081 b)\n\u22a2 MvQPF.abs ((TypeVec.id ::: Quot.mk r') <$$> M.dest (P F) a) =\n    MvQPF.abs ((TypeVec.id ::: Quot.mk r') <$$> M.dest (P F) b)"}, {"tactic": "have : f \u2218 Quot.mk r \u2218 Quot.mk Mcongr = Quot.mk r' := rfl", "annotated_tactic": ["have : f \u2218 <a>Quot.mk</a> r \u2218 <a>Quot.mk</a> <a>Mcongr</a> = <a>Quot.mk</a> r' := <a>rfl</a>", [{"full_name": "Quot.mk", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [406, 14], "def_end_pos": [406, 21]}, {"full_name": "Quot.mk", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [406, 14], "def_end_pos": [406, 21]}, {"full_name": "MvQPF.Mcongr", "def_path": "Mathlib/Data/QPF/Multivariate/Constructions/Cofix.lean", "def_pos": [79, 5], "def_end_pos": [79, 11]}, {"full_name": "Quot.mk", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [406, 14], "def_end_pos": [406, 21]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "n : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh' : \u2200 (x : Cofix F \u03b1), r x x\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nx y : M (P F) \u03b1\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nhr' : r' = fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : M (P F) \u03b1\nr'ab : r' a b\nh\u2080 :\n  (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) a) =\n    (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) b)\nh\u2081 : \u2200 (u v : M (P F) \u03b1), Mcongr u v \u2192 Quot.mk r' u = Quot.mk r' v\nf : Quot r \u2192 Quot r' :=\n  Quot.lift (Quot.lift (Quot.mk r') h\u2081)\n    (_ : \u2200 (c b : Cofix F \u03b1), r c b \u2192 Quot.lift (Quot.mk r') h\u2081 c = Quot.lift (Quot.mk r') h\u2081 b)\n\u22a2 MvQPF.abs ((TypeVec.id ::: Quot.mk r') <$$> M.dest (P F) a) =\n    MvQPF.abs ((TypeVec.id ::: Quot.mk r') <$$> M.dest (P F) b)", "state_after": "n : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh' : \u2200 (x : Cofix F \u03b1), r x x\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nx y : M (P F) \u03b1\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nhr' : r' = fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : M (P F) \u03b1\nr'ab : r' a b\nh\u2080 :\n  (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) a) =\n    (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) b)\nh\u2081 : \u2200 (u v : M (P F) \u03b1), Mcongr u v \u2192 Quot.mk r' u = Quot.mk r' v\nf : Quot r \u2192 Quot r' :=\n  Quot.lift (Quot.lift (Quot.mk r') h\u2081)\n    (_ : \u2200 (c b : Cofix F \u03b1), r c b \u2192 Quot.lift (Quot.mk r') h\u2081 c = Quot.lift (Quot.mk r') h\u2081 b)\nthis : f \u2218 Quot.mk r \u2218 Quot.mk Mcongr = Quot.mk r'\n\u22a2 MvQPF.abs ((TypeVec.id ::: Quot.mk r') <$$> M.dest (P F) a) =\n    MvQPF.abs ((TypeVec.id ::: Quot.mk r') <$$> M.dest (P F) b)"}, {"tactic": "rw [\u2190 this, appendFun_comp_id, q.P.comp_map, q.P.comp_map, abs_map, abs_map, abs_map, abs_map,\n  h\u2080]", "annotated_tactic": ["rw [\u2190 this, <a>appendFun_comp_id</a>, q.P.comp_map, q.P.comp_map, <a>abs_map</a>, <a>abs_map</a>, <a>abs_map</a>, <a>abs_map</a>,\n      h\u2080]", [{"full_name": "TypeVec.appendFun_comp_id", "def_path": "Mathlib/Data/TypeVec.lean", "def_pos": [270, 9], "def_end_pos": [270, 26]}, {"full_name": "MvQPF.abs_map", "def_path": "Mathlib/Data/QPF/Multivariate/Basic.lean", "def_pos": [90, 3], "def_end_pos": [90, 10]}, {"full_name": "MvQPF.abs_map", "def_path": "Mathlib/Data/QPF/Multivariate/Basic.lean", "def_pos": [90, 3], "def_end_pos": [90, 10]}, {"full_name": "MvQPF.abs_map", "def_path": "Mathlib/Data/QPF/Multivariate/Basic.lean", "def_pos": [90, 3], "def_end_pos": [90, 10]}, {"full_name": "MvQPF.abs_map", "def_path": "Mathlib/Data/QPF/Multivariate/Basic.lean", "def_pos": [90, 3], "def_end_pos": [90, 10]}]], "state_before": "n : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh' : \u2200 (x : Cofix F \u03b1), r x x\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nx y : M (P F) \u03b1\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nhr' : r' = fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : M (P F) \u03b1\nr'ab : r' a b\nh\u2080 :\n  (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) a) =\n    (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) b)\nh\u2081 : \u2200 (u v : M (P F) \u03b1), Mcongr u v \u2192 Quot.mk r' u = Quot.mk r' v\nf : Quot r \u2192 Quot r' :=\n  Quot.lift (Quot.lift (Quot.mk r') h\u2081)\n    (_ : \u2200 (c b : Cofix F \u03b1), r c b \u2192 Quot.lift (Quot.mk r') h\u2081 c = Quot.lift (Quot.mk r') h\u2081 b)\nthis : f \u2218 Quot.mk r \u2218 Quot.mk Mcongr = Quot.mk r'\n\u22a2 MvQPF.abs ((TypeVec.id ::: Quot.mk r') <$$> M.dest (P F) a) =\n    MvQPF.abs ((TypeVec.id ::: Quot.mk r') <$$> M.dest (P F) b)", "state_after": "no goals"}, {"tactic": "rw [appendFun_comp_id, comp_map, comp_map]", "annotated_tactic": ["rw [<a>appendFun_comp_id</a>, <a>comp_map</a>, <a>comp_map</a>]", [{"full_name": "TypeVec.appendFun_comp_id", "def_path": "Mathlib/Data/TypeVec.lean", "def_pos": [270, 9], "def_end_pos": [270, 26]}, {"full_name": "MvQPF.comp_map", "def_path": "Mathlib/Data/QPF/Multivariate/Basic.lean", "def_pos": [112, 9], "def_end_pos": [112, 17]}, {"full_name": "MvQPF.comp_map", "def_path": "Mathlib/Data/QPF/Multivariate/Basic.lean", "def_pos": [112, 9], "def_end_pos": [112, 17]}]], "state_before": "n : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh' : \u2200 (x : Cofix F \u03b1), r x x\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nx y : M (P F) \u03b1\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nhr' : r' = fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : M (P F) \u03b1\nr'ab : r' a b\n\u22a2 (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) a) =\n    (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) b)", "state_after": "n : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh' : \u2200 (x : Cofix F \u03b1), r x x\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nx y : M (P F) \u03b1\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nhr' : r' = fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : M (P F) \u03b1\nr'ab : r' a b\n\u22a2 (TypeVec.id ::: Quot.mk r) <$$> (TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) a) =\n    (TypeVec.id ::: Quot.mk r) <$$> (TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) b)"}, {"tactic": "exact h _ _ r'ab", "annotated_tactic": ["exact h _ _ r'ab", []], "state_before": "n : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh' : \u2200 (x : Cofix F \u03b1), r x x\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nx y : M (P F) \u03b1\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nhr' : r' = fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : M (P F) \u03b1\nr'ab : r' a b\n\u22a2 (TypeVec.id ::: Quot.mk r) <$$> (TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) a) =\n    (TypeVec.id ::: Quot.mk r) <$$> (TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) b)", "state_after": "no goals"}, {"tactic": "intro u v cuv", "annotated_tactic": ["intro u v cuv", []], "state_before": "n : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh' : \u2200 (x : Cofix F \u03b1), r x x\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nx y : M (P F) \u03b1\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nhr' : r' = fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : M (P F) \u03b1\nr'ab : r' a b\nh\u2080 :\n  (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) a) =\n    (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) b)\n\u22a2 \u2200 (u v : M (P F) \u03b1), Mcongr u v \u2192 Quot.mk r' u = Quot.mk r' v", "state_after": "n : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh' : \u2200 (x : Cofix F \u03b1), r x x\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nx y : M (P F) \u03b1\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nhr' : r' = fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : M (P F) \u03b1\nr'ab : r' a b\nh\u2080 :\n  (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) a) =\n    (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) b)\nu v : M (P F) \u03b1\ncuv : Mcongr u v\n\u22a2 Quot.mk r' u = Quot.mk r' v"}, {"tactic": "apply Quot.sound", "annotated_tactic": ["apply <a>Quot.sound</a>", [{"full_name": "Quot.sound", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [1209, 7], "def_end_pos": [1209, 12]}]], "state_before": "n : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh' : \u2200 (x : Cofix F \u03b1), r x x\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nx y : M (P F) \u03b1\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nhr' : r' = fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : M (P F) \u03b1\nr'ab : r' a b\nh\u2080 :\n  (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) a) =\n    (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) b)\nu v : M (P F) \u03b1\ncuv : Mcongr u v\n\u22a2 Quot.mk r' u = Quot.mk r' v", "state_after": "case a\nn : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh' : \u2200 (x : Cofix F \u03b1), r x x\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nx y : M (P F) \u03b1\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nhr' : r' = fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : M (P F) \u03b1\nr'ab : r' a b\nh\u2080 :\n  (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) a) =\n    (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) b)\nu v : M (P F) \u03b1\ncuv : Mcongr u v\n\u22a2 r' u v"}, {"tactic": "dsimp [hr']", "annotated_tactic": ["dsimp [hr']", []], "state_before": "case a\nn : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh' : \u2200 (x : Cofix F \u03b1), r x x\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nx y : M (P F) \u03b1\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nhr' : r' = fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : M (P F) \u03b1\nr'ab : r' a b\nh\u2080 :\n  (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) a) =\n    (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) b)\nu v : M (P F) \u03b1\ncuv : Mcongr u v\n\u22a2 r' u v", "state_after": "case a\nn : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh' : \u2200 (x : Cofix F \u03b1), r x x\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nx y : M (P F) \u03b1\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nhr' : r' = fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : M (P F) \u03b1\nr'ab : r' a b\nh\u2080 :\n  (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) a) =\n    (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) b)\nu v : M (P F) \u03b1\ncuv : Mcongr u v\n\u22a2 r (Quot.mk Mcongr u) (Quot.mk Mcongr v)"}, {"tactic": "rw [Quot.sound cuv]", "annotated_tactic": ["rw [<a>Quot.sound</a> cuv]", [{"full_name": "Quot.sound", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [1209, 7], "def_end_pos": [1209, 12]}]], "state_before": "case a\nn : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh' : \u2200 (x : Cofix F \u03b1), r x x\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nx y : M (P F) \u03b1\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nhr' : r' = fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : M (P F) \u03b1\nr'ab : r' a b\nh\u2080 :\n  (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) a) =\n    (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) b)\nu v : M (P F) \u03b1\ncuv : Mcongr u v\n\u22a2 r (Quot.mk Mcongr u) (Quot.mk Mcongr v)", "state_after": "case a\nn : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh' : \u2200 (x : Cofix F \u03b1), r x x\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nx y : M (P F) \u03b1\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nhr' : r' = fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : M (P F) \u03b1\nr'ab : r' a b\nh\u2080 :\n  (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) a) =\n    (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) b)\nu v : M (P F) \u03b1\ncuv : Mcongr u v\n\u22a2 r (Quot.mk Mcongr v) (Quot.mk Mcongr v)"}, {"tactic": "apply h'", "annotated_tactic": ["apply h'", []], "state_before": "case a\nn : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh' : \u2200 (x : Cofix F \u03b1), r x x\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nx y : M (P F) \u03b1\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nhr' : r' = fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : M (P F) \u03b1\nr'ab : r' a b\nh\u2080 :\n  (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) a) =\n    (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) b)\nu v : M (P F) \u03b1\ncuv : Mcongr u v\n\u22a2 r (Quot.mk Mcongr v) (Quot.mk Mcongr v)", "state_after": "no goals"}, {"tactic": "intro c", "annotated_tactic": ["intro c", []], "state_before": "n : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh' : \u2200 (x : Cofix F \u03b1), r x x\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nx y : M (P F) \u03b1\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nhr' : r' = fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : M (P F) \u03b1\nr'ab : r' a b\nh\u2080 :\n  (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) a) =\n    (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) b)\nh\u2081 : \u2200 (u v : M (P F) \u03b1), Mcongr u v \u2192 Quot.mk r' u = Quot.mk r' v\n\u22a2 \u2200 (a b : Cofix F \u03b1), r a b \u2192 Quot.lift (Quot.mk r') h\u2081 a = Quot.lift (Quot.mk r') h\u2081 b", "state_after": "n : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh' : \u2200 (x : Cofix F \u03b1), r x x\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nx y : M (P F) \u03b1\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nhr' : r' = fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : M (P F) \u03b1\nr'ab : r' a b\nh\u2080 :\n  (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) a) =\n    (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) b)\nh\u2081 : \u2200 (u v : M (P F) \u03b1), Mcongr u v \u2192 Quot.mk r' u = Quot.mk r' v\nc : Cofix F \u03b1\n\u22a2 \u2200 (b : Cofix F \u03b1), r c b \u2192 Quot.lift (Quot.mk r') h\u2081 c = Quot.lift (Quot.mk r') h\u2081 b"}, {"tactic": "apply Quot.inductionOn\n  (motive := fun c =>\n    \u2200b, r c b \u2192 Quot.lift (Quot.mk r') h\u2081 c = Quot.lift (Quot.mk r') h\u2081 b) c", "annotated_tactic": ["apply <a>Quot.inductionOn</a>\n            (motive := fun c =>\n              \u2200b, r c b \u2192 <a>Quot.lift</a> (<a>Quot.mk</a> r') h\u2081 c = <a>Quot.lift</a> (<a>Quot.mk</a> r') h\u2081 b) c", [{"full_name": "Quot.inductionOn", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [1233, 19], "def_end_pos": [1233, 30]}, {"full_name": "Quot.lift", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [424, 14], "def_end_pos": [424, 23]}, {"full_name": "Quot.mk", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [406, 14], "def_end_pos": [406, 21]}, {"full_name": "Quot.lift", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [424, 14], "def_end_pos": [424, 23]}, {"full_name": "Quot.mk", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [406, 14], "def_end_pos": [406, 21]}]], "state_before": "n : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh' : \u2200 (x : Cofix F \u03b1), r x x\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nx y : M (P F) \u03b1\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nhr' : r' = fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : M (P F) \u03b1\nr'ab : r' a b\nh\u2080 :\n  (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) a) =\n    (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) b)\nh\u2081 : \u2200 (u v : M (P F) \u03b1), Mcongr u v \u2192 Quot.mk r' u = Quot.mk r' v\nc : Cofix F \u03b1\n\u22a2 \u2200 (b : Cofix F \u03b1), r c b \u2192 Quot.lift (Quot.mk r') h\u2081 c = Quot.lift (Quot.mk r') h\u2081 b", "state_after": "n : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh' : \u2200 (x : Cofix F \u03b1), r x x\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nx y : M (P F) \u03b1\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nhr' : r' = fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : M (P F) \u03b1\nr'ab : r' a b\nh\u2080 :\n  (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) a) =\n    (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) b)\nh\u2081 : \u2200 (u v : M (P F) \u03b1), Mcongr u v \u2192 Quot.mk r' u = Quot.mk r' v\nc : Cofix F \u03b1\n\u22a2 \u2200 (a : M (P F) \u03b1) (b : Cofix F \u03b1),\n    r (Quot.mk Mcongr a) b \u2192 Quot.lift (Quot.mk r') h\u2081 (Quot.mk Mcongr a) = Quot.lift (Quot.mk r') h\u2081 b"}, {"tactic": "clear c", "annotated_tactic": ["clear c", []], "state_before": "n : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh' : \u2200 (x : Cofix F \u03b1), r x x\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nx y : M (P F) \u03b1\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nhr' : r' = fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : M (P F) \u03b1\nr'ab : r' a b\nh\u2080 :\n  (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) a) =\n    (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) b)\nh\u2081 : \u2200 (u v : M (P F) \u03b1), Mcongr u v \u2192 Quot.mk r' u = Quot.mk r' v\nc : Cofix F \u03b1\n\u22a2 \u2200 (a : M (P F) \u03b1) (b : Cofix F \u03b1),\n    r (Quot.mk Mcongr a) b \u2192 Quot.lift (Quot.mk r') h\u2081 (Quot.mk Mcongr a) = Quot.lift (Quot.mk r') h\u2081 b", "state_after": "n : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh' : \u2200 (x : Cofix F \u03b1), r x x\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nx y : M (P F) \u03b1\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nhr' : r' = fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : M (P F) \u03b1\nr'ab : r' a b\nh\u2080 :\n  (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) a) =\n    (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) b)\nh\u2081 : \u2200 (u v : M (P F) \u03b1), Mcongr u v \u2192 Quot.mk r' u = Quot.mk r' v\n\u22a2 \u2200 (a : M (P F) \u03b1) (b : Cofix F \u03b1),\n    r (Quot.mk Mcongr a) b \u2192 Quot.lift (Quot.mk r') h\u2081 (Quot.mk Mcongr a) = Quot.lift (Quot.mk r') h\u2081 b"}, {"tactic": "intro c d", "annotated_tactic": ["intro c d", []], "state_before": "n : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh' : \u2200 (x : Cofix F \u03b1), r x x\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nx y : M (P F) \u03b1\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nhr' : r' = fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : M (P F) \u03b1\nr'ab : r' a b\nh\u2080 :\n  (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) a) =\n    (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) b)\nh\u2081 : \u2200 (u v : M (P F) \u03b1), Mcongr u v \u2192 Quot.mk r' u = Quot.mk r' v\n\u22a2 \u2200 (a : M (P F) \u03b1) (b : Cofix F \u03b1),\n    r (Quot.mk Mcongr a) b \u2192 Quot.lift (Quot.mk r') h\u2081 (Quot.mk Mcongr a) = Quot.lift (Quot.mk r') h\u2081 b", "state_after": "n : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh' : \u2200 (x : Cofix F \u03b1), r x x\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nx y : M (P F) \u03b1\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nhr' : r' = fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : M (P F) \u03b1\nr'ab : r' a b\nh\u2080 :\n  (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) a) =\n    (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) b)\nh\u2081 : \u2200 (u v : M (P F) \u03b1), Mcongr u v \u2192 Quot.mk r' u = Quot.mk r' v\nc : M (P F) \u03b1\nd : Cofix F \u03b1\n\u22a2 r (Quot.mk Mcongr c) d \u2192 Quot.lift (Quot.mk r') h\u2081 (Quot.mk Mcongr c) = Quot.lift (Quot.mk r') h\u2081 d"}, {"tactic": "apply Quot.inductionOn\n  (motive := fun d => r (Quot.mk Mcongr c) d \u2192\n    Quot.lift (Quot.mk r') h\u2081 (Quot.mk Mcongr c) = Quot.lift (Quot.mk r') h\u2081 d) d", "annotated_tactic": ["apply <a>Quot.inductionOn</a>\n            (motive := fun d => r (<a>Quot.mk</a> <a>Mcongr</a> c) d \u2192\n              <a>Quot.lift</a> (<a>Quot.mk</a> r') h\u2081 (<a>Quot.mk</a> <a>Mcongr</a> c) = <a>Quot.lift</a> (<a>Quot.mk</a> r') h\u2081 d) d", [{"full_name": "Quot.inductionOn", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [1233, 19], "def_end_pos": [1233, 30]}, {"full_name": "Quot.mk", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [406, 14], "def_end_pos": [406, 21]}, {"full_name": "MvQPF.Mcongr", "def_path": "Mathlib/Data/QPF/Multivariate/Constructions/Cofix.lean", "def_pos": [79, 5], "def_end_pos": [79, 11]}, {"full_name": "Quot.lift", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [424, 14], "def_end_pos": [424, 23]}, {"full_name": "Quot.mk", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [406, 14], "def_end_pos": [406, 21]}, {"full_name": "Quot.mk", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [406, 14], "def_end_pos": [406, 21]}, {"full_name": "MvQPF.Mcongr", "def_path": "Mathlib/Data/QPF/Multivariate/Constructions/Cofix.lean", "def_pos": [79, 5], "def_end_pos": [79, 11]}, {"full_name": "Quot.lift", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [424, 14], "def_end_pos": [424, 23]}, {"full_name": "Quot.mk", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [406, 14], "def_end_pos": [406, 21]}]], "state_before": "n : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh' : \u2200 (x : Cofix F \u03b1), r x x\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nx y : M (P F) \u03b1\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nhr' : r' = fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : M (P F) \u03b1\nr'ab : r' a b\nh\u2080 :\n  (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) a) =\n    (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) b)\nh\u2081 : \u2200 (u v : M (P F) \u03b1), Mcongr u v \u2192 Quot.mk r' u = Quot.mk r' v\nc : M (P F) \u03b1\nd : Cofix F \u03b1\n\u22a2 r (Quot.mk Mcongr c) d \u2192 Quot.lift (Quot.mk r') h\u2081 (Quot.mk Mcongr c) = Quot.lift (Quot.mk r') h\u2081 d", "state_after": "n : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh' : \u2200 (x : Cofix F \u03b1), r x x\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nx y : M (P F) \u03b1\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nhr' : r' = fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : M (P F) \u03b1\nr'ab : r' a b\nh\u2080 :\n  (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) a) =\n    (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) b)\nh\u2081 : \u2200 (u v : M (P F) \u03b1), Mcongr u v \u2192 Quot.mk r' u = Quot.mk r' v\nc : M (P F) \u03b1\nd : Cofix F \u03b1\n\u22a2 \u2200 (a : M (P F) \u03b1),\n    r (Quot.mk Mcongr c) (Quot.mk Mcongr a) \u2192\n      Quot.lift (Quot.mk r') h\u2081 (Quot.mk Mcongr c) = Quot.lift (Quot.mk r') h\u2081 (Quot.mk Mcongr a)"}, {"tactic": "clear d", "annotated_tactic": ["clear d", []], "state_before": "n : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh' : \u2200 (x : Cofix F \u03b1), r x x\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nx y : M (P F) \u03b1\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nhr' : r' = fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : M (P F) \u03b1\nr'ab : r' a b\nh\u2080 :\n  (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) a) =\n    (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) b)\nh\u2081 : \u2200 (u v : M (P F) \u03b1), Mcongr u v \u2192 Quot.mk r' u = Quot.mk r' v\nc : M (P F) \u03b1\nd : Cofix F \u03b1\n\u22a2 \u2200 (a : M (P F) \u03b1),\n    r (Quot.mk Mcongr c) (Quot.mk Mcongr a) \u2192\n      Quot.lift (Quot.mk r') h\u2081 (Quot.mk Mcongr c) = Quot.lift (Quot.mk r') h\u2081 (Quot.mk Mcongr a)", "state_after": "n : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh' : \u2200 (x : Cofix F \u03b1), r x x\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nx y : M (P F) \u03b1\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nhr' : r' = fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : M (P F) \u03b1\nr'ab : r' a b\nh\u2080 :\n  (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) a) =\n    (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) b)\nh\u2081 : \u2200 (u v : M (P F) \u03b1), Mcongr u v \u2192 Quot.mk r' u = Quot.mk r' v\nc : M (P F) \u03b1\n\u22a2 \u2200 (a : M (P F) \u03b1),\n    r (Quot.mk Mcongr c) (Quot.mk Mcongr a) \u2192\n      Quot.lift (Quot.mk r') h\u2081 (Quot.mk Mcongr c) = Quot.lift (Quot.mk r') h\u2081 (Quot.mk Mcongr a)"}, {"tactic": "intro d rcd", "annotated_tactic": ["intro d rcd", []], "state_before": "n : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh' : \u2200 (x : Cofix F \u03b1), r x x\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nx y : M (P F) \u03b1\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nhr' : r' = fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : M (P F) \u03b1\nr'ab : r' a b\nh\u2080 :\n  (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) a) =\n    (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) b)\nh\u2081 : \u2200 (u v : M (P F) \u03b1), Mcongr u v \u2192 Quot.mk r' u = Quot.mk r' v\nc : M (P F) \u03b1\n\u22a2 \u2200 (a : M (P F) \u03b1),\n    r (Quot.mk Mcongr c) (Quot.mk Mcongr a) \u2192\n      Quot.lift (Quot.mk r') h\u2081 (Quot.mk Mcongr c) = Quot.lift (Quot.mk r') h\u2081 (Quot.mk Mcongr a)", "state_after": "n : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh' : \u2200 (x : Cofix F \u03b1), r x x\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nx y : M (P F) \u03b1\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nhr' : r' = fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : M (P F) \u03b1\nr'ab : r' a b\nh\u2080 :\n  (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) a) =\n    (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) b)\nh\u2081 : \u2200 (u v : M (P F) \u03b1), Mcongr u v \u2192 Quot.mk r' u = Quot.mk r' v\nc d : M (P F) \u03b1\nrcd : r (Quot.mk Mcongr c) (Quot.mk Mcongr d)\n\u22a2 Quot.lift (Quot.mk r') h\u2081 (Quot.mk Mcongr c) = Quot.lift (Quot.mk r') h\u2081 (Quot.mk Mcongr d)"}, {"tactic": "apply Quot.sound", "annotated_tactic": ["apply <a>Quot.sound</a>", [{"full_name": "Quot.sound", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [1209, 7], "def_end_pos": [1209, 12]}]], "state_before": "n : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh' : \u2200 (x : Cofix F \u03b1), r x x\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nx y : M (P F) \u03b1\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nhr' : r' = fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : M (P F) \u03b1\nr'ab : r' a b\nh\u2080 :\n  (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) a) =\n    (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) b)\nh\u2081 : \u2200 (u v : M (P F) \u03b1), Mcongr u v \u2192 Quot.mk r' u = Quot.mk r' v\nc d : M (P F) \u03b1\nrcd : r (Quot.mk Mcongr c) (Quot.mk Mcongr d)\n\u22a2 Quot.lift (Quot.mk r') h\u2081 (Quot.mk Mcongr c) = Quot.lift (Quot.mk r') h\u2081 (Quot.mk Mcongr d)", "state_after": "case a\nn : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh' : \u2200 (x : Cofix F \u03b1), r x x\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nx y : M (P F) \u03b1\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nhr' : r' = fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : M (P F) \u03b1\nr'ab : r' a b\nh\u2080 :\n  (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) a) =\n    (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) b)\nh\u2081 : \u2200 (u v : M (P F) \u03b1), Mcongr u v \u2192 Quot.mk r' u = Quot.mk r' v\nc d : M (P F) \u03b1\nrcd : r (Quot.mk Mcongr c) (Quot.mk Mcongr d)\n\u22a2 r' c d"}, {"tactic": "apply rcd", "annotated_tactic": ["apply rcd", []], "state_before": "case a\nn : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh' : \u2200 (x : Cofix F \u03b1), r x x\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nx y : M (P F) \u03b1\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nhr' : r' = fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : M (P F) \u03b1\nr'ab : r' a b\nh\u2080 :\n  (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) a) =\n    (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) b)\nh\u2081 : \u2200 (u v : M (P F) \u03b1), Mcongr u v \u2192 Quot.mk r' u = Quot.mk r' v\nc d : M (P F) \u03b1\nrcd : r (Quot.mk Mcongr c) (Quot.mk Mcongr d)\n\u22a2 r' c d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Int/Bitwise.lean", "full_name": "Int.bodd_add", "start": [75, 1], "end": [81, 32], "traced_tactics": [{"tactic": "cases' m with m m <;>\ncases' n with n n <;>\nsimp only [ofNat_eq_coe, ofNat_add_negSucc, negSucc_add_ofNat,\n           negSucc_add_negSucc, bodd_subNatNat] <;>\nsimp only [negSucc_coe, bodd_neg, bodd_coe, \u2190Nat.bodd_add, Bool.xor_comm, \u2190Nat.cast_add]", "annotated_tactic": ["cases' m with m m <;>\n  cases' n with n n <;>\n  simp only [<a>ofNat_eq_coe</a>, <a>ofNat_add_negSucc</a>, <a>negSucc_add_ofNat</a>,\n             <a>negSucc_add_negSucc</a>, <a>bodd_subNatNat</a>] <;>\n  simp only [<a>negSucc_coe</a>, <a>bodd_neg</a>, <a>bodd_coe</a>, \u2190<a>Nat.bodd_add</a>, <a>Bool.xor_comm</a>, \u2190<a>Nat.cast_add</a>]", [{"full_name": "Int.ofNat_eq_coe", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [17, 17], "def_end_pos": [17, 29]}, {"full_name": "Int.ofNat_add_negSucc", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [52, 23], "def_end_pos": [52, 40]}, {"full_name": "Int.negSucc_add_ofNat", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [53, 23], "def_end_pos": [53, 40]}, {"full_name": "Int.negSucc_add_negSucc", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [54, 23], "def_end_pos": [54, 42]}, {"full_name": "Int.bodd_subNatNat", "def_path": "Mathlib/Data/Int/Bitwise.lean", "def_pos": [47, 9], "def_end_pos": [47, 23]}, {"full_name": "Int.negSucc_coe", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [43, 9], "def_end_pos": [43, 20]}, {"full_name": "Int.bodd_neg", "def_path": "Mathlib/Data/Int/Bitwise.lean", "def_pos": [61, 9], "def_end_pos": [61, 17]}, {"full_name": "Int.bodd_coe", "def_path": "Mathlib/Data/Int/Bitwise.lean", "def_pos": [42, 9], "def_end_pos": [42, 17]}, {"full_name": "Nat.bodd_add", "def_path": "Mathlib/Init/Data/Nat/Bitwise.lean", "def_pos": [75, 9], "def_end_pos": [75, 17]}, {"full_name": "Bool.xor_comm", "def_path": "Mathlib/Data/Bool/Basic.lean", "def_pos": [264, 9], "def_end_pos": [264, 17]}, {"full_name": "Nat.cast_add", "def_path": "Mathlib/Data/Nat/Cast/Defs.lean", "def_pos": [146, 9], "def_end_pos": [146, 17]}]], "state_before": "m n : \u2124\n\u22a2 bodd (m + n) = xor (bodd m) (bodd n)", "state_after": "case negSucc.negSucc\nm n : \u2115\n\u22a2 Nat.bodd (Nat.succ (m + n) + 1) = Nat.bodd (m + 1 + (n + 1))"}, {"tactic": "rw [\u2190Nat.succ_add, add_assoc]", "annotated_tactic": ["rw [\u2190<a>Nat.succ_add</a>, <a>add_assoc</a>]", [{"full_name": "Nat.succ_add", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [118, 9], "def_end_pos": [118, 17]}, {"full_name": "add_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [263, 3], "def_end_pos": [263, 14]}]], "state_before": "case negSucc.negSucc\nm n : \u2115\n\u22a2 Nat.bodd (Nat.succ (m + n) + 1) = Nat.bodd (m + 1 + (n + 1))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "full_name": "MeasureTheory.tendsto_setToFun_of_dominated_convergence", "start": [1714, 1], "end": [1753, 51], "traced_tactics": [{"tactic": "have f_measurable : AEStronglyMeasurable f \u03bc :=\n  aestronglyMeasurable_of_tendsto_ae _ fs_measurable h_lim", "annotated_tactic": ["have f_measurable : <a>AEStronglyMeasurable</a> f \u03bc :=\n    <a>aestronglyMeasurable_of_tendsto_ae</a> _ fs_measurable h_lim", [{"full_name": "MeasureTheory.AEStronglyMeasurable", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [93, 5], "def_end_pos": [93, 25]}, {"full_name": "aestronglyMeasurable_of_tendsto_ae", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1664, 9], "def_end_pos": [1664, 50]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\nfs : \u2115 \u2192 \u03b1 \u2192 E\nf : \u03b1 \u2192 E\nbound : \u03b1 \u2192 \u211d\nfs_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (fs n) \u03bc\nbound_integrable : Integrable bound\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n a) atTop (\ud835\udcdd (f a))\n\u22a2 Tendsto (fun n => setToFun \u03bc T hT (fs n)) atTop (\ud835\udcdd (setToFun \u03bc T hT f))", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\nfs : \u2115 \u2192 \u03b1 \u2192 E\nf : \u03b1 \u2192 E\nbound : \u03b1 \u2192 \u211d\nfs_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (fs n) \u03bc\nbound_integrable : Integrable bound\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n a) atTop (\ud835\udcdd (f a))\nf_measurable : AEStronglyMeasurable f \u03bc\n\u22a2 Tendsto (fun n => setToFun \u03bc T hT (fs n)) atTop (\ud835\udcdd (setToFun \u03bc T hT f))"}, {"tactic": "have fs_int : \u2200 n, Integrable (fs n) \u03bc := fun n =>\n  bound_integrable.mono' (fs_measurable n) (h_bound _)", "annotated_tactic": ["have fs_int : \u2200 n, <a>Integrable</a> (fs n) \u03bc := fun n =>\n    bound_integrable.mono' (fs_measurable n) (h_bound _)", [{"full_name": "MeasureTheory.Integrable", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [442, 5], "def_end_pos": [442, 15]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\nfs : \u2115 \u2192 \u03b1 \u2192 E\nf : \u03b1 \u2192 E\nbound : \u03b1 \u2192 \u211d\nfs_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (fs n) \u03bc\nbound_integrable : Integrable bound\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n a) atTop (\ud835\udcdd (f a))\nf_measurable : AEStronglyMeasurable f \u03bc\n\u22a2 Tendsto (fun n => setToFun \u03bc T hT (fs n)) atTop (\ud835\udcdd (setToFun \u03bc T hT f))", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\nfs : \u2115 \u2192 \u03b1 \u2192 E\nf : \u03b1 \u2192 E\nbound : \u03b1 \u2192 \u211d\nfs_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (fs n) \u03bc\nbound_integrable : Integrable bound\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n a) atTop (\ud835\udcdd (f a))\nf_measurable : AEStronglyMeasurable f \u03bc\nfs_int : \u2200 (n : \u2115), Integrable (fs n)\n\u22a2 Tendsto (fun n => setToFun \u03bc T hT (fs n)) atTop (\ud835\udcdd (setToFun \u03bc T hT f))"}, {"tactic": "have f_int : Integrable f \u03bc :=\n  \u27e8f_measurable,\n    hasFiniteIntegral_of_dominated_convergence bound_integrable.hasFiniteIntegral h_bound\n      h_lim\u27e9", "annotated_tactic": ["have f_int : <a>Integrable</a> f \u03bc :=\n    \u27e8f_measurable,\n      <a>hasFiniteIntegral_of_dominated_convergence</a> bound_integrable.hasFiniteIntegral h_bound\n        h_lim\u27e9", [{"full_name": "MeasureTheory.Integrable", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [442, 5], "def_end_pos": [442, 15]}, {"full_name": "MeasureTheory.hasFiniteIntegral_of_dominated_convergence", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [311, 9], "def_end_pos": [311, 51]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\nfs : \u2115 \u2192 \u03b1 \u2192 E\nf : \u03b1 \u2192 E\nbound : \u03b1 \u2192 \u211d\nfs_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (fs n) \u03bc\nbound_integrable : Integrable bound\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n a) atTop (\ud835\udcdd (f a))\nf_measurable : AEStronglyMeasurable f \u03bc\nfs_int : \u2200 (n : \u2115), Integrable (fs n)\n\u22a2 Tendsto (fun n => setToFun \u03bc T hT (fs n)) atTop (\ud835\udcdd (setToFun \u03bc T hT f))", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\nfs : \u2115 \u2192 \u03b1 \u2192 E\nf : \u03b1 \u2192 E\nbound : \u03b1 \u2192 \u211d\nfs_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (fs n) \u03bc\nbound_integrable : Integrable bound\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n a) atTop (\ud835\udcdd (f a))\nf_measurable : AEStronglyMeasurable f \u03bc\nfs_int : \u2200 (n : \u2115), Integrable (fs n)\nf_int : Integrable f\n\u22a2 Tendsto (fun n => setToFun \u03bc T hT (fs n)) atTop (\ud835\udcdd (setToFun \u03bc T hT f))"}, {"tactic": "refine' L1.tendsto_setToL1 hT _ _ _", "annotated_tactic": ["refine' <a>L1.tendsto_setToL1</a> hT _ _ _", [{"full_name": "MeasureTheory.L1.tendsto_setToL1", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [1250, 9], "def_end_pos": [1250, 24]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\nfs : \u2115 \u2192 \u03b1 \u2192 E\nf : \u03b1 \u2192 E\nbound : \u03b1 \u2192 \u211d\nfs_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (fs n) \u03bc\nbound_integrable : Integrable bound\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n a) atTop (\ud835\udcdd (f a))\nf_measurable : AEStronglyMeasurable f \u03bc\nfs_int : \u2200 (n : \u2115), Integrable (fs n)\nf_int : Integrable f\n\u22a2 Tendsto (fun n => \u2191(L1.setToL1 hT) (Integrable.toL1 (fs n) (_ : Integrable (fs n)))) atTop\n    (\ud835\udcdd (\u2191(L1.setToL1 hT) (Integrable.toL1 f f_int)))", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\nfs : \u2115 \u2192 \u03b1 \u2192 E\nf : \u03b1 \u2192 E\nbound : \u03b1 \u2192 \u211d\nfs_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (fs n) \u03bc\nbound_integrable : Integrable bound\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n a) atTop (\ud835\udcdd (f a))\nf_measurable : AEStronglyMeasurable f \u03bc\nfs_int : \u2200 (n : \u2115), Integrable (fs n)\nf_int : Integrable f\n\u22a2 Tendsto (fun n => Integrable.toL1 (fs n) (_ : Integrable (fs n))) atTop (\ud835\udcdd (Integrable.toL1 f f_int))"}, {"tactic": "rw [tendsto_iff_norm_sub_tendsto_zero]", "annotated_tactic": ["rw [<a>tendsto_iff_norm_sub_tendsto_zero</a>]", [{"full_name": "tendsto_iff_norm_sub_tendsto_zero", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [1079, 3], "def_end_pos": [1079, 14]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\nfs : \u2115 \u2192 \u03b1 \u2192 E\nf : \u03b1 \u2192 E\nbound : \u03b1 \u2192 \u211d\nfs_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (fs n) \u03bc\nbound_integrable : Integrable bound\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n a) atTop (\ud835\udcdd (f a))\nf_measurable : AEStronglyMeasurable f \u03bc\nfs_int : \u2200 (n : \u2115), Integrable (fs n)\nf_int : Integrable f\n\u22a2 Tendsto (fun n => Integrable.toL1 (fs n) (_ : Integrable (fs n))) atTop (\ud835\udcdd (Integrable.toL1 f f_int))", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\nfs : \u2115 \u2192 \u03b1 \u2192 E\nf : \u03b1 \u2192 E\nbound : \u03b1 \u2192 \u211d\nfs_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (fs n) \u03bc\nbound_integrable : Integrable bound\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n a) atTop (\ud835\udcdd (f a))\nf_measurable : AEStronglyMeasurable f \u03bc\nfs_int : \u2200 (n : \u2115), Integrable (fs n)\nf_int : Integrable f\n\u22a2 Tendsto (fun e => \u2016Integrable.toL1 (fs e) (_ : Integrable (fs e)) - Integrable.toL1 f f_int\u2016) atTop (\ud835\udcdd 0)"}, {"tactic": "have lintegral_norm_tendsto_zero :\n  Tendsto (fun n => ENNReal.toReal <| \u222b\u207b a, ENNReal.ofReal \u2016fs n a - f a\u2016 \u2202\u03bc) atTop (\ud835\udcdd 0) :=\n  (tendsto_toReal zero_ne_top).comp\n    (tendsto_lintegral_norm_of_dominated_convergence fs_measurable\n      bound_integrable.hasFiniteIntegral h_bound h_lim)", "annotated_tactic": ["have lintegral_norm_tendsto_zero :\n    <a>Tendsto</a> (fun n => <a>ENNReal.toReal</a> <| \u222b\u207b a, <a>ENNReal.ofReal</a> \u2016fs n a - f a\u2016 \u2202\u03bc) <a>atTop</a> (\ud835\udcdd 0) :=\n    (<a>tendsto_toReal</a> <a>zero_ne_top</a>).<a>comp</a>\n      (<a>tendsto_lintegral_norm_of_dominated_convergence</a> fs_measurable\n        bound_integrable.hasFiniteIntegral h_bound h_lim)", [{"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2939, 5], "def_end_pos": [2939, 12]}, {"full_name": "ENNReal.toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [168, 15], "def_end_pos": [168, 21]}, {"full_name": "ENNReal.ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [172, 29], "def_end_pos": [172, 35]}, {"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}, {"full_name": "ENNReal.tendsto_toReal", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [131, 9], "def_end_pos": [131, 23]}, {"full_name": "ENNReal.zero_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [334, 17], "def_end_pos": [334, 28]}, {"full_name": "Filter.Tendsto.comp", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [3032, 9], "def_end_pos": [3032, 21]}, {"full_name": "MeasureTheory.tendsto_lintegral_norm_of_dominated_convergence", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [327, 9], "def_end_pos": [327, 56]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\nfs : \u2115 \u2192 \u03b1 \u2192 E\nf : \u03b1 \u2192 E\nbound : \u03b1 \u2192 \u211d\nfs_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (fs n) \u03bc\nbound_integrable : Integrable bound\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n a) atTop (\ud835\udcdd (f a))\nf_measurable : AEStronglyMeasurable f \u03bc\nfs_int : \u2200 (n : \u2115), Integrable (fs n)\nf_int : Integrable f\n\u22a2 Tendsto (fun e => \u2016Integrable.toL1 (fs e) (_ : Integrable (fs e)) - Integrable.toL1 f f_int\u2016) atTop (\ud835\udcdd 0)", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\nfs : \u2115 \u2192 \u03b1 \u2192 E\nf : \u03b1 \u2192 E\nbound : \u03b1 \u2192 \u211d\nfs_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (fs n) \u03bc\nbound_integrable : Integrable bound\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n a) atTop (\ud835\udcdd (f a))\nf_measurable : AEStronglyMeasurable f \u03bc\nfs_int : \u2200 (n : \u2115), Integrable (fs n)\nf_int : Integrable f\nlintegral_norm_tendsto_zero :\n  Tendsto (fun n => ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016fs n a - f a\u2016 \u2202\u03bc)) atTop (\ud835\udcdd 0)\n\u22a2 Tendsto (fun e => \u2016Integrable.toL1 (fs e) (_ : Integrable (fs e)) - Integrable.toL1 f f_int\u2016) atTop (\ud835\udcdd 0)"}, {"tactic": "convert lintegral_norm_tendsto_zero with n", "annotated_tactic": ["convert lintegral_norm_tendsto_zero with n", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\nfs : \u2115 \u2192 \u03b1 \u2192 E\nf : \u03b1 \u2192 E\nbound : \u03b1 \u2192 \u211d\nfs_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (fs n) \u03bc\nbound_integrable : Integrable bound\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n a) atTop (\ud835\udcdd (f a))\nf_measurable : AEStronglyMeasurable f \u03bc\nfs_int : \u2200 (n : \u2115), Integrable (fs n)\nf_int : Integrable f\nlintegral_norm_tendsto_zero :\n  Tendsto (fun n => ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016fs n a - f a\u2016 \u2202\u03bc)) atTop (\ud835\udcdd 0)\n\u22a2 Tendsto (fun e => \u2016Integrable.toL1 (fs e) (_ : Integrable (fs e)) - Integrable.toL1 f f_int\u2016) atTop (\ud835\udcdd 0)", "state_after": "case h.e'_3.h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\nfs : \u2115 \u2192 \u03b1 \u2192 E\nf : \u03b1 \u2192 E\nbound : \u03b1 \u2192 \u211d\nfs_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (fs n) \u03bc\nbound_integrable : Integrable bound\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n a) atTop (\ud835\udcdd (f a))\nf_measurable : AEStronglyMeasurable f \u03bc\nfs_int : \u2200 (n : \u2115), Integrable (fs n)\nf_int : Integrable f\nlintegral_norm_tendsto_zero :\n  Tendsto (fun n => ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016fs n a - f a\u2016 \u2202\u03bc)) atTop (\ud835\udcdd 0)\nn : \u2115\n\u22a2 \u2016Integrable.toL1 (fs n) (_ : Integrable (fs n)) - Integrable.toL1 f f_int\u2016 =\n    ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016fs n a - f a\u2016 \u2202\u03bc)"}, {"tactic": "rw [L1.norm_def]", "annotated_tactic": ["rw [<a>L1.norm_def</a>]", [{"full_name": "MeasureTheory.L1.norm_def", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [1361, 9], "def_end_pos": [1361, 17]}]], "state_before": "case h.e'_3.h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\nfs : \u2115 \u2192 \u03b1 \u2192 E\nf : \u03b1 \u2192 E\nbound : \u03b1 \u2192 \u211d\nfs_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (fs n) \u03bc\nbound_integrable : Integrable bound\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n a) atTop (\ud835\udcdd (f a))\nf_measurable : AEStronglyMeasurable f \u03bc\nfs_int : \u2200 (n : \u2115), Integrable (fs n)\nf_int : Integrable f\nlintegral_norm_tendsto_zero :\n  Tendsto (fun n => ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016fs n a - f a\u2016 \u2202\u03bc)) atTop (\ud835\udcdd 0)\nn : \u2115\n\u22a2 \u2016Integrable.toL1 (fs n) (_ : Integrable (fs n)) - Integrable.toL1 f f_int\u2016 =\n    ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016fs n a - f a\u2016 \u2202\u03bc)", "state_after": "case h.e'_3.h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\nfs : \u2115 \u2192 \u03b1 \u2192 E\nf : \u03b1 \u2192 E\nbound : \u03b1 \u2192 \u211d\nfs_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (fs n) \u03bc\nbound_integrable : Integrable bound\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n a) atTop (\ud835\udcdd (f a))\nf_measurable : AEStronglyMeasurable f \u03bc\nfs_int : \u2200 (n : \u2115), Integrable (fs n)\nf_int : Integrable f\nlintegral_norm_tendsto_zero :\n  Tendsto (fun n => ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016fs n a - f a\u2016 \u2202\u03bc)) atTop (\ud835\udcdd 0)\nn : \u2115\n\u22a2 ENNReal.toReal (\u222b\u207b (a : \u03b1), \u2191\u2016\u2191\u2191(Integrable.toL1 (fs n) (_ : Integrable (fs n)) - Integrable.toL1 f f_int) a\u2016\u208a \u2202\u03bc) =\n    ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016fs n a - f a\u2016 \u2202\u03bc)"}, {"tactic": "congr 1", "annotated_tactic": ["congr 1", []], "state_before": "case h.e'_3.h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\nfs : \u2115 \u2192 \u03b1 \u2192 E\nf : \u03b1 \u2192 E\nbound : \u03b1 \u2192 \u211d\nfs_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (fs n) \u03bc\nbound_integrable : Integrable bound\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n a) atTop (\ud835\udcdd (f a))\nf_measurable : AEStronglyMeasurable f \u03bc\nfs_int : \u2200 (n : \u2115), Integrable (fs n)\nf_int : Integrable f\nlintegral_norm_tendsto_zero :\n  Tendsto (fun n => ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016fs n a - f a\u2016 \u2202\u03bc)) atTop (\ud835\udcdd 0)\nn : \u2115\n\u22a2 ENNReal.toReal (\u222b\u207b (a : \u03b1), \u2191\u2016\u2191\u2191(Integrable.toL1 (fs n) (_ : Integrable (fs n)) - Integrable.toL1 f f_int) a\u2016\u208a \u2202\u03bc) =\n    ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016fs n a - f a\u2016 \u2202\u03bc)", "state_after": "case h.e'_3.h.e_a\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\nfs : \u2115 \u2192 \u03b1 \u2192 E\nf : \u03b1 \u2192 E\nbound : \u03b1 \u2192 \u211d\nfs_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (fs n) \u03bc\nbound_integrable : Integrable bound\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n a) atTop (\ud835\udcdd (f a))\nf_measurable : AEStronglyMeasurable f \u03bc\nfs_int : \u2200 (n : \u2115), Integrable (fs n)\nf_int : Integrable f\nlintegral_norm_tendsto_zero :\n  Tendsto (fun n => ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016fs n a - f a\u2016 \u2202\u03bc)) atTop (\ud835\udcdd 0)\nn : \u2115\n\u22a2 \u222b\u207b (a : \u03b1), \u2191\u2016\u2191\u2191(Integrable.toL1 (fs n) (_ : Integrable (fs n)) - Integrable.toL1 f f_int) a\u2016\u208a \u2202\u03bc =\n    \u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016fs n a - f a\u2016 \u2202\u03bc"}, {"tactic": "refine' lintegral_congr_ae _", "annotated_tactic": ["refine' <a>lintegral_congr_ae</a> _", [{"full_name": "MeasureTheory.lintegral_congr_ae", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [304, 9], "def_end_pos": [304, 27]}]], "state_before": "case h.e'_3.h.e_a\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\nfs : \u2115 \u2192 \u03b1 \u2192 E\nf : \u03b1 \u2192 E\nbound : \u03b1 \u2192 \u211d\nfs_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (fs n) \u03bc\nbound_integrable : Integrable bound\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n a) atTop (\ud835\udcdd (f a))\nf_measurable : AEStronglyMeasurable f \u03bc\nfs_int : \u2200 (n : \u2115), Integrable (fs n)\nf_int : Integrable f\nlintegral_norm_tendsto_zero :\n  Tendsto (fun n => ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016fs n a - f a\u2016 \u2202\u03bc)) atTop (\ud835\udcdd 0)\nn : \u2115\n\u22a2 \u222b\u207b (a : \u03b1), \u2191\u2016\u2191\u2191(Integrable.toL1 (fs n) (_ : Integrable (fs n)) - Integrable.toL1 f f_int) a\u2016\u208a \u2202\u03bc =\n    \u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016fs n a - f a\u2016 \u2202\u03bc", "state_after": "case h.e'_3.h.e_a\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\nfs : \u2115 \u2192 \u03b1 \u2192 E\nf : \u03b1 \u2192 E\nbound : \u03b1 \u2192 \u211d\nfs_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (fs n) \u03bc\nbound_integrable : Integrable bound\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n a) atTop (\ud835\udcdd (f a))\nf_measurable : AEStronglyMeasurable f \u03bc\nfs_int : \u2200 (n : \u2115), Integrable (fs n)\nf_int : Integrable f\nlintegral_norm_tendsto_zero :\n  Tendsto (fun n => ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016fs n a - f a\u2016 \u2202\u03bc)) atTop (\ud835\udcdd 0)\nn : \u2115\n\u22a2 (fun a => \u2191\u2016\u2191\u2191(Integrable.toL1 (fs n) (_ : Integrable (fs n)) - Integrable.toL1 f f_int) a\u2016\u208a) =\u1d50[\u03bc] fun a =>\n    ENNReal.ofReal \u2016fs n a - f a\u2016"}, {"tactic": "rw [\u2190 Integrable.toL1_sub]", "annotated_tactic": ["rw [\u2190 <a>Integrable.toL1_sub</a>]", [{"full_name": "MeasureTheory.Integrable.toL1_sub", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [1441, 9], "def_end_pos": [1441, 17]}]], "state_before": "case h.e'_3.h.e_a\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\nfs : \u2115 \u2192 \u03b1 \u2192 E\nf : \u03b1 \u2192 E\nbound : \u03b1 \u2192 \u211d\nfs_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (fs n) \u03bc\nbound_integrable : Integrable bound\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n a) atTop (\ud835\udcdd (f a))\nf_measurable : AEStronglyMeasurable f \u03bc\nfs_int : \u2200 (n : \u2115), Integrable (fs n)\nf_int : Integrable f\nlintegral_norm_tendsto_zero :\n  Tendsto (fun n => ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016fs n a - f a\u2016 \u2202\u03bc)) atTop (\ud835\udcdd 0)\nn : \u2115\n\u22a2 (fun a => \u2191\u2016\u2191\u2191(Integrable.toL1 (fs n) (_ : Integrable (fs n)) - Integrable.toL1 f f_int) a\u2016\u208a) =\u1d50[\u03bc] fun a =>\n    ENNReal.ofReal \u2016fs n a - f a\u2016", "state_after": "case h.e'_3.h.e_a\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\nfs : \u2115 \u2192 \u03b1 \u2192 E\nf : \u03b1 \u2192 E\nbound : \u03b1 \u2192 \u211d\nfs_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (fs n) \u03bc\nbound_integrable : Integrable bound\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n a) atTop (\ud835\udcdd (f a))\nf_measurable : AEStronglyMeasurable f \u03bc\nfs_int : \u2200 (n : \u2115), Integrable (fs n)\nf_int : Integrable f\nlintegral_norm_tendsto_zero :\n  Tendsto (fun n => ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016fs n a - f a\u2016 \u2202\u03bc)) atTop (\ud835\udcdd 0)\nn : \u2115\n\u22a2 (fun a => \u2191\u2016\u2191\u2191(Integrable.toL1 (fs n - f) (_ : Integrable (fs n - f))) a\u2016\u208a) =\u1d50[\u03bc] fun a =>\n    ENNReal.ofReal \u2016fs n a - f a\u2016"}, {"tactic": "refine' ((fs_int n).sub f_int).coeFn_toL1.mono fun x hx => _", "annotated_tactic": ["refine' ((fs_int n).<a>sub</a> f_int).coeFn_toL1.mono fun x hx => _", [{"full_name": "MeasureTheory.Integrable.sub", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [702, 9], "def_end_pos": [702, 23]}]], "state_before": "case h.e'_3.h.e_a\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\nfs : \u2115 \u2192 \u03b1 \u2192 E\nf : \u03b1 \u2192 E\nbound : \u03b1 \u2192 \u211d\nfs_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (fs n) \u03bc\nbound_integrable : Integrable bound\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n a) atTop (\ud835\udcdd (f a))\nf_measurable : AEStronglyMeasurable f \u03bc\nfs_int : \u2200 (n : \u2115), Integrable (fs n)\nf_int : Integrable f\nlintegral_norm_tendsto_zero :\n  Tendsto (fun n => ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016fs n a - f a\u2016 \u2202\u03bc)) atTop (\ud835\udcdd 0)\nn : \u2115\n\u22a2 (fun a => \u2191\u2016\u2191\u2191(Integrable.toL1 (fs n - f) (_ : Integrable (fs n - f))) a\u2016\u208a) =\u1d50[\u03bc] fun a =>\n    ENNReal.ofReal \u2016fs n a - f a\u2016", "state_after": "case h.e'_3.h.e_a\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\nfs : \u2115 \u2192 \u03b1 \u2192 E\nf : \u03b1 \u2192 E\nbound : \u03b1 \u2192 \u211d\nfs_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (fs n) \u03bc\nbound_integrable : Integrable bound\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n a) atTop (\ud835\udcdd (f a))\nf_measurable : AEStronglyMeasurable f \u03bc\nfs_int : \u2200 (n : \u2115), Integrable (fs n)\nf_int : Integrable f\nlintegral_norm_tendsto_zero :\n  Tendsto (fun n => ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016fs n a - f a\u2016 \u2202\u03bc)) atTop (\ud835\udcdd 0)\nn : \u2115\nx : \u03b1\nhx : \u2191\u2191(Integrable.toL1 (fs n - f) (_ : Integrable (fs n - f))) x = (fs n - f) x\n\u22a2 (fun a => \u2191\u2016\u2191\u2191(Integrable.toL1 (fs n - f) (_ : Integrable (fs n - f))) a\u2016\u208a) x =\n    (fun a => ENNReal.ofReal \u2016fs n a - f a\u2016) x"}, {"tactic": "dsimp only", "annotated_tactic": ["dsimp only", []], "state_before": "case h.e'_3.h.e_a\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\nfs : \u2115 \u2192 \u03b1 \u2192 E\nf : \u03b1 \u2192 E\nbound : \u03b1 \u2192 \u211d\nfs_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (fs n) \u03bc\nbound_integrable : Integrable bound\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n a) atTop (\ud835\udcdd (f a))\nf_measurable : AEStronglyMeasurable f \u03bc\nfs_int : \u2200 (n : \u2115), Integrable (fs n)\nf_int : Integrable f\nlintegral_norm_tendsto_zero :\n  Tendsto (fun n => ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016fs n a - f a\u2016 \u2202\u03bc)) atTop (\ud835\udcdd 0)\nn : \u2115\nx : \u03b1\nhx : \u2191\u2191(Integrable.toL1 (fs n - f) (_ : Integrable (fs n - f))) x = (fs n - f) x\n\u22a2 (fun a => \u2191\u2016\u2191\u2191(Integrable.toL1 (fs n - f) (_ : Integrable (fs n - f))) a\u2016\u208a) x =\n    (fun a => ENNReal.ofReal \u2016fs n a - f a\u2016) x", "state_after": "case h.e'_3.h.e_a\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\nfs : \u2115 \u2192 \u03b1 \u2192 E\nf : \u03b1 \u2192 E\nbound : \u03b1 \u2192 \u211d\nfs_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (fs n) \u03bc\nbound_integrable : Integrable bound\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n a) atTop (\ud835\udcdd (f a))\nf_measurable : AEStronglyMeasurable f \u03bc\nfs_int : \u2200 (n : \u2115), Integrable (fs n)\nf_int : Integrable f\nlintegral_norm_tendsto_zero :\n  Tendsto (fun n => ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016fs n a - f a\u2016 \u2202\u03bc)) atTop (\ud835\udcdd 0)\nn : \u2115\nx : \u03b1\nhx : \u2191\u2191(Integrable.toL1 (fs n - f) (_ : Integrable (fs n - f))) x = (fs n - f) x\n\u22a2 \u2191\u2016\u2191\u2191(Integrable.toL1 (fs n - f) (_ : Integrable (fs n - f))) x\u2016\u208a = ENNReal.ofReal \u2016fs n x - f x\u2016"}, {"tactic": "rw [hx, ofReal_norm_eq_coe_nnnorm, Pi.sub_apply]", "annotated_tactic": ["rw [hx, <a>ofReal_norm_eq_coe_nnnorm</a>, <a>Pi.sub_apply</a>]", [{"full_name": "ofReal_norm_eq_coe_nnnorm", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [999, 15], "def_end_pos": [999, 40]}, {"full_name": "Pi.sub_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [200, 3], "def_end_pos": [200, 14]}]], "state_before": "case h.e'_3.h.e_a\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\nfs : \u2115 \u2192 \u03b1 \u2192 E\nf : \u03b1 \u2192 E\nbound : \u03b1 \u2192 \u211d\nfs_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (fs n) \u03bc\nbound_integrable : Integrable bound\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n a) atTop (\ud835\udcdd (f a))\nf_measurable : AEStronglyMeasurable f \u03bc\nfs_int : \u2200 (n : \u2115), Integrable (fs n)\nf_int : Integrable f\nlintegral_norm_tendsto_zero :\n  Tendsto (fun n => ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016fs n a - f a\u2016 \u2202\u03bc)) atTop (\ud835\udcdd 0)\nn : \u2115\nx : \u03b1\nhx : \u2191\u2191(Integrable.toL1 (fs n - f) (_ : Integrable (fs n - f))) x = (fs n - f) x\n\u22a2 \u2191\u2016\u2191\u2191(Integrable.toL1 (fs n - f) (_ : Integrable (fs n - f))) x\u2016\u208a = ENNReal.ofReal \u2016fs n x - f x\u2016", "state_after": "no goals"}, {"tactic": "convert this with n", "annotated_tactic": ["convert this with n", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\nfs : \u2115 \u2192 \u03b1 \u2192 E\nf : \u03b1 \u2192 E\nbound : \u03b1 \u2192 \u211d\nfs_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (fs n) \u03bc\nbound_integrable : Integrable bound\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n a) atTop (\ud835\udcdd (f a))\nf_measurable : AEStronglyMeasurable f \u03bc\nfs_int : \u2200 (n : \u2115), Integrable (fs n)\nf_int : Integrable f\nthis :\n  Tendsto (fun n => \u2191(L1.setToL1 hT) (Integrable.toL1 (fs n) (_ : Integrable (fs n)))) atTop\n    (\ud835\udcdd (\u2191(L1.setToL1 hT) (Integrable.toL1 f f_int)))\n\u22a2 Tendsto (fun n => setToFun \u03bc T hT (fs n)) atTop (\ud835\udcdd (setToFun \u03bc T hT f))", "state_after": "case h.e'_3.h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\nfs : \u2115 \u2192 \u03b1 \u2192 E\nf : \u03b1 \u2192 E\nbound : \u03b1 \u2192 \u211d\nfs_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (fs n) \u03bc\nbound_integrable : Integrable bound\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n a) atTop (\ud835\udcdd (f a))\nf_measurable : AEStronglyMeasurable f \u03bc\nfs_int : \u2200 (n : \u2115), Integrable (fs n)\nf_int : Integrable f\nthis :\n  Tendsto (fun n => \u2191(L1.setToL1 hT) (Integrable.toL1 (fs n) (_ : Integrable (fs n)))) atTop\n    (\ud835\udcdd (\u2191(L1.setToL1 hT) (Integrable.toL1 f f_int)))\nn : \u2115\n\u22a2 setToFun \u03bc T hT (fs n) = \u2191(L1.setToL1 hT) (Integrable.toL1 (fs n) (_ : Integrable (fs n)))\n\ncase h.e'_5.h.e'_3\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\nfs : \u2115 \u2192 \u03b1 \u2192 E\nf : \u03b1 \u2192 E\nbound : \u03b1 \u2192 \u211d\nfs_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (fs n) \u03bc\nbound_integrable : Integrable bound\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n a) atTop (\ud835\udcdd (f a))\nf_measurable : AEStronglyMeasurable f \u03bc\nfs_int : \u2200 (n : \u2115), Integrable (fs n)\nf_int : Integrable f\nthis :\n  Tendsto (fun n => \u2191(L1.setToL1 hT) (Integrable.toL1 (fs n) (_ : Integrable (fs n)))) atTop\n    (\ud835\udcdd (\u2191(L1.setToL1 hT) (Integrable.toL1 f f_int)))\n\u22a2 setToFun \u03bc T hT f = \u2191(L1.setToL1 hT) (Integrable.toL1 f f_int)"}, {"tactic": "exact setToFun_eq hT (fs_int n)", "annotated_tactic": ["exact <a>setToFun_eq</a> hT (fs_int n)", [{"full_name": "MeasureTheory.setToFun_eq", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [1276, 9], "def_end_pos": [1276, 20]}]], "state_before": "case h.e'_3.h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\nfs : \u2115 \u2192 \u03b1 \u2192 E\nf : \u03b1 \u2192 E\nbound : \u03b1 \u2192 \u211d\nfs_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (fs n) \u03bc\nbound_integrable : Integrable bound\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n a) atTop (\ud835\udcdd (f a))\nf_measurable : AEStronglyMeasurable f \u03bc\nfs_int : \u2200 (n : \u2115), Integrable (fs n)\nf_int : Integrable f\nthis :\n  Tendsto (fun n => \u2191(L1.setToL1 hT) (Integrable.toL1 (fs n) (_ : Integrable (fs n)))) atTop\n    (\ud835\udcdd (\u2191(L1.setToL1 hT) (Integrable.toL1 f f_int)))\nn : \u2115\n\u22a2 setToFun \u03bc T hT (fs n) = \u2191(L1.setToL1 hT) (Integrable.toL1 (fs n) (_ : Integrable (fs n)))", "state_after": "no goals"}, {"tactic": "exact setToFun_eq hT f_int", "annotated_tactic": ["exact <a>setToFun_eq</a> hT f_int", [{"full_name": "MeasureTheory.setToFun_eq", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [1276, 9], "def_end_pos": [1276, 20]}]], "state_before": "case h.e'_5.h.e'_3\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\nfs : \u2115 \u2192 \u03b1 \u2192 E\nf : \u03b1 \u2192 E\nbound : \u03b1 \u2192 \u211d\nfs_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (fs n) \u03bc\nbound_integrable : Integrable bound\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n a) atTop (\ud835\udcdd (f a))\nf_measurable : AEStronglyMeasurable f \u03bc\nfs_int : \u2200 (n : \u2115), Integrable (fs n)\nf_int : Integrable f\nthis :\n  Tendsto (fun n => \u2191(L1.setToL1 hT) (Integrable.toL1 (fs n) (_ : Integrable (fs n)))) atTop\n    (\ud835\udcdd (\u2191(L1.setToL1 hT) (Integrable.toL1 f f_int)))\n\u22a2 setToFun \u03bc T hT f = \u2191(L1.setToL1 hT) (Integrable.toL1 f f_int)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/TMToPartrec.lean", "full_name": "Turing.PartrecToTM2.tr_read", "start": [1110, 1], "end": [1110, 53], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Fin/Lemmas.lean", "full_name": "Fin.lastCases_last", "start": [680, 9], "end": [682, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "full_name": "List.length_erase_of_mem", "start": [1185, 9], "end": [1187, 74], "traced_tactics": [{"tactic": "rw [erase_eq_eraseP]", "annotated_tactic": ["rw [<a>erase_eq_eraseP</a>]", [{"full_name": "List.erase_eq_eraseP", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [1166, 9], "def_end_pos": [1166, 24]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl : List \u03b1\nh : a \u2208 l\n\u22a2 length (List.erase l a) = pred (length l)", "state_after": "\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl : List \u03b1\nh : a \u2208 l\n\u22a2 length (eraseP (fun b => decide (a = b)) l) = pred (length l)"}, {"tactic": "exact length_eraseP_of_mem h (decide_eq_true rfl)", "annotated_tactic": ["exact <a>length_eraseP_of_mem</a> h (<a>decide_eq_true</a> <a>rfl</a>)", [{"full_name": "List.length_eraseP_of_mem", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [1088, 17], "def_end_pos": [1088, 37]}, {"full_name": "decide_eq_true", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [847, 9], "def_end_pos": [847, 23]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl : List \u03b1\nh : a \u2208 l\n\u22a2 length (eraseP (fun b => decide (a = b)) l) = pred (length l)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "full_name": "MeasureTheory.OuterMeasure.restrict_le_self", "start": [595, 1], "end": [596, 19], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Setoid/Partition.lean", "full_name": "Setoid.mkClasses_classes", "start": [189, 1], "end": [192, 73], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Sort.lean", "full_name": "Finset.sort_sorted_lt", "start": [88, 1], "end": [89, 46], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Powerset.lean", "full_name": "Finset.powersetCard_nonempty", "start": [254, 1], "end": [266, 37], "traced_tactics": [{"tactic": "induction' s using Finset.induction_on with x s hx IH generalizing n", "annotated_tactic": ["induction' s using <a>Finset.induction_on</a> with x s hx IH generalizing n", [{"full_name": "Finset.induction_on", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1251, 19], "def_end_pos": [1251, 31]}]], "state_before": "\u03b1 : Type u_1\ns\u271d t : Finset \u03b1\nn : \u2115\ns : Finset \u03b1\nh : n \u2264 card s\n\u22a2 Finset.Nonempty (powersetCard n s)", "state_after": "case empty\n\u03b1 : Type u_1\ns\u271d t : Finset \u03b1\nn\u271d : \u2115\ns : Finset \u03b1\nh\u271d : n\u271d \u2264 card s\nn : \u2115\nh : n \u2264 card \u2205\n\u22a2 Finset.Nonempty (powersetCard n \u2205)\n\ncase insert\n\u03b1 : Type u_1\ns\u271d\u00b9 t : Finset \u03b1\nn\u271d : \u2115\ns\u271d : Finset \u03b1\nh\u271d : n\u271d \u2264 card s\u271d\nx : \u03b1\ns : Finset \u03b1\nhx : \u00acx \u2208 s\nIH : \u2200 {n : \u2115}, n \u2264 card s \u2192 Finset.Nonempty (powersetCard n s)\nn : \u2115\nh : n \u2264 card (insert x s)\n\u22a2 Finset.Nonempty (powersetCard n (insert x s))"}, {"tactic": "rw [card_empty, le_zero_iff] at h", "annotated_tactic": ["rw [<a>card_empty</a>, <a>le_zero_iff</a>] at h", [{"full_name": "Finset.card_empty", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [57, 9], "def_end_pos": [57, 19]}, {"full_name": "le_zero_iff", "def_path": "Mathlib/Algebra/Order/WithZero.lean", "def_pos": [102, 9], "def_end_pos": [102, 20]}]], "state_before": "case empty\n\u03b1 : Type u_1\ns\u271d t : Finset \u03b1\nn\u271d : \u2115\ns : Finset \u03b1\nh\u271d : n\u271d \u2264 card s\nn : \u2115\nh : n \u2264 card \u2205\n\u22a2 Finset.Nonempty (powersetCard n \u2205)", "state_after": "case empty\n\u03b1 : Type u_1\ns\u271d t : Finset \u03b1\nn\u271d : \u2115\ns : Finset \u03b1\nh\u271d : n\u271d \u2264 card s\nn : \u2115\nh : n = 0\n\u22a2 Finset.Nonempty (powersetCard n \u2205)"}, {"tactic": "rw [h, powersetCard_zero]", "annotated_tactic": ["rw [h, <a>powersetCard_zero</a>]", [{"full_name": "Finset.powersetCard_zero", "def_path": "Mathlib/Data/Finset/Powerset.lean", "def_pos": [221, 9], "def_end_pos": [221, 26]}]], "state_before": "case empty\n\u03b1 : Type u_1\ns\u271d t : Finset \u03b1\nn\u271d : \u2115\ns : Finset \u03b1\nh\u271d : n\u271d \u2264 card s\nn : \u2115\nh : n = 0\n\u22a2 Finset.Nonempty (powersetCard n \u2205)", "state_after": "case empty\n\u03b1 : Type u_1\ns\u271d t : Finset \u03b1\nn\u271d : \u2115\ns : Finset \u03b1\nh\u271d : n\u271d \u2264 card s\nn : \u2115\nh : n = 0\n\u22a2 Finset.Nonempty {\u2205}"}, {"tactic": "exact Finset.singleton_nonempty _", "annotated_tactic": ["exact <a>Finset.singleton_nonempty</a> _", [{"full_name": "Finset.singleton_nonempty", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [711, 9], "def_end_pos": [711, 27]}]], "state_before": "case empty\n\u03b1 : Type u_1\ns\u271d t : Finset \u03b1\nn\u271d : \u2115\ns : Finset \u03b1\nh\u271d : n\u271d \u2264 card s\nn : \u2115\nh : n = 0\n\u22a2 Finset.Nonempty {\u2205}", "state_after": "no goals"}, {"tactic": "cases n", "annotated_tactic": ["cases n", []], "state_before": "case insert\n\u03b1 : Type u_1\ns\u271d\u00b9 t : Finset \u03b1\nn\u271d : \u2115\ns\u271d : Finset \u03b1\nh\u271d : n\u271d \u2264 card s\u271d\nx : \u03b1\ns : Finset \u03b1\nhx : \u00acx \u2208 s\nIH : \u2200 {n : \u2115}, n \u2264 card s \u2192 Finset.Nonempty (powersetCard n s)\nn : \u2115\nh : n \u2264 card (insert x s)\n\u22a2 Finset.Nonempty (powersetCard n (insert x s))", "state_after": "case insert.zero\n\u03b1 : Type u_1\ns\u271d\u00b9 t : Finset \u03b1\nn : \u2115\ns\u271d : Finset \u03b1\nh\u271d : n \u2264 card s\u271d\nx : \u03b1\ns : Finset \u03b1\nhx : \u00acx \u2208 s\nIH : \u2200 {n : \u2115}, n \u2264 card s \u2192 Finset.Nonempty (powersetCard n s)\nh : Nat.zero \u2264 card (insert x s)\n\u22a2 Finset.Nonempty (powersetCard Nat.zero (insert x s))\n\ncase insert.succ\n\u03b1 : Type u_1\ns\u271d\u00b9 t : Finset \u03b1\nn : \u2115\ns\u271d : Finset \u03b1\nh\u271d : n \u2264 card s\u271d\nx : \u03b1\ns : Finset \u03b1\nhx : \u00acx \u2208 s\nIH : \u2200 {n : \u2115}, n \u2264 card s \u2192 Finset.Nonempty (powersetCard n s)\nn\u271d : \u2115\nh : Nat.succ n\u271d \u2264 card (insert x s)\n\u22a2 Finset.Nonempty (powersetCard (Nat.succ n\u271d) (insert x s))"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case insert.zero\n\u03b1 : Type u_1\ns\u271d\u00b9 t : Finset \u03b1\nn : \u2115\ns\u271d : Finset \u03b1\nh\u271d : n \u2264 card s\u271d\nx : \u03b1\ns : Finset \u03b1\nhx : \u00acx \u2208 s\nIH : \u2200 {n : \u2115}, n \u2264 card s \u2192 Finset.Nonempty (powersetCard n s)\nh : Nat.zero \u2264 card (insert x s)\n\u22a2 Finset.Nonempty (powersetCard Nat.zero (insert x s))", "state_after": "no goals"}, {"tactic": "rw [card_insert_of_not_mem hx, Nat.succ_le_succ_iff] at h", "annotated_tactic": ["rw [<a>card_insert_of_not_mem</a> hx, <a>Nat.succ_le_succ_iff</a>] at h", [{"full_name": "Finset.card_insert_of_not_mem", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [106, 9], "def_end_pos": [106, 31]}, {"full_name": "Nat.succ_le_succ_iff", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [257, 9], "def_end_pos": [257, 25]}]], "state_before": "case insert.succ\n\u03b1 : Type u_1\ns\u271d\u00b9 t : Finset \u03b1\nn : \u2115\ns\u271d : Finset \u03b1\nh\u271d : n \u2264 card s\u271d\nx : \u03b1\ns : Finset \u03b1\nhx : \u00acx \u2208 s\nIH : \u2200 {n : \u2115}, n \u2264 card s \u2192 Finset.Nonempty (powersetCard n s)\nn\u271d : \u2115\nh : Nat.succ n\u271d \u2264 card (insert x s)\n\u22a2 Finset.Nonempty (powersetCard (Nat.succ n\u271d) (insert x s))", "state_after": "case insert.succ\n\u03b1 : Type u_1\ns\u271d\u00b9 t : Finset \u03b1\nn : \u2115\ns\u271d : Finset \u03b1\nh\u271d : n \u2264 card s\u271d\nx : \u03b1\ns : Finset \u03b1\nhx : \u00acx \u2208 s\nIH : \u2200 {n : \u2115}, n \u2264 card s \u2192 Finset.Nonempty (powersetCard n s)\nn\u271d : \u2115\nh : n\u271d \u2264 card s\n\u22a2 Finset.Nonempty (powersetCard (Nat.succ n\u271d) (insert x s))"}, {"tactic": "rw [powersetCard_succ_insert hx]", "annotated_tactic": ["rw [<a>powersetCard_succ_insert</a> hx]", [{"full_name": "Finset.powersetCard_succ_insert", "def_path": "Mathlib/Data/Finset/Powerset.lean", "def_pos": [240, 9], "def_end_pos": [240, 33]}]], "state_before": "case insert.succ\n\u03b1 : Type u_1\ns\u271d\u00b9 t : Finset \u03b1\nn : \u2115\ns\u271d : Finset \u03b1\nh\u271d : n \u2264 card s\u271d\nx : \u03b1\ns : Finset \u03b1\nhx : \u00acx \u2208 s\nIH : \u2200 {n : \u2115}, n \u2264 card s \u2192 Finset.Nonempty (powersetCard n s)\nn\u271d : \u2115\nh : n\u271d \u2264 card s\n\u22a2 Finset.Nonempty (powersetCard (Nat.succ n\u271d) (insert x s))", "state_after": "case insert.succ\n\u03b1 : Type u_1\ns\u271d\u00b9 t : Finset \u03b1\nn : \u2115\ns\u271d : Finset \u03b1\nh\u271d : n \u2264 card s\u271d\nx : \u03b1\ns : Finset \u03b1\nhx : \u00acx \u2208 s\nIH : \u2200 {n : \u2115}, n \u2264 card s \u2192 Finset.Nonempty (powersetCard n s)\nn\u271d : \u2115\nh : n\u271d \u2264 card s\n\u22a2 Finset.Nonempty (powersetCard (Nat.succ n\u271d) s \u222a image (insert x) (powersetCard n\u271d s))"}, {"tactic": "refine' Nonempty.mono _ ((IH h).image (insert x))", "annotated_tactic": ["refine' <a>Nonempty.mono</a> _ ((IH h).<a>image</a> (<a>insert</a> x))", [{"full_name": "Finset.Nonempty.mono", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [507, 9], "def_end_pos": [507, 22]}, {"full_name": "Finset.Nonempty.image", "def_path": "Mathlib/Data/Finset/Image.lean", "def_pos": [396, 19], "def_end_pos": [396, 33]}, {"full_name": "Insert.insert", "def_path": "lake-packages/std/Std/Classes/SetNotation.lean", "def_pos": [69, 3], "def_end_pos": [69, 9]}]], "state_before": "case insert.succ\n\u03b1 : Type u_1\ns\u271d\u00b9 t : Finset \u03b1\nn : \u2115\ns\u271d : Finset \u03b1\nh\u271d : n \u2264 card s\u271d\nx : \u03b1\ns : Finset \u03b1\nhx : \u00acx \u2208 s\nIH : \u2200 {n : \u2115}, n \u2264 card s \u2192 Finset.Nonempty (powersetCard n s)\nn\u271d : \u2115\nh : n\u271d \u2264 card s\n\u22a2 Finset.Nonempty (powersetCard (Nat.succ n\u271d) s \u222a image (insert x) (powersetCard n\u271d s))", "state_after": "case insert.succ\n\u03b1 : Type u_1\ns\u271d\u00b9 t : Finset \u03b1\nn : \u2115\ns\u271d : Finset \u03b1\nh\u271d : n \u2264 card s\u271d\nx : \u03b1\ns : Finset \u03b1\nhx : \u00acx \u2208 s\nIH : \u2200 {n : \u2115}, n \u2264 card s \u2192 Finset.Nonempty (powersetCard n s)\nn\u271d : \u2115\nh : n\u271d \u2264 card s\n\u22a2 image (insert x) (powersetCard n\u271d s) \u2286 powersetCard (Nat.succ n\u271d) s \u222a image (insert x) (powersetCard n\u271d s)"}, {"tactic": "exact subset_union_right _ _", "annotated_tactic": ["exact <a>subset_union_right</a> _ _", [{"full_name": "Finset.subset_union_right", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1410, 9], "def_end_pos": [1410, 27]}]], "state_before": "case insert.succ\n\u03b1 : Type u_1\ns\u271d\u00b9 t : Finset \u03b1\nn : \u2115\ns\u271d : Finset \u03b1\nh\u271d : n \u2264 card s\u271d\nx : \u03b1\ns : Finset \u03b1\nhx : \u00acx \u2208 s\nIH : \u2200 {n : \u2115}, n \u2264 card s \u2192 Finset.Nonempty (powersetCard n s)\nn\u271d : \u2115\nh : n\u271d \u2264 card s\n\u22a2 image (insert x) (powersetCard n\u271d s) \u2286 powersetCard (Nat.succ n\u271d) s \u222a image (insert x) (powersetCard n\u271d s)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Haar/Basic.lean", "full_name": "MeasureTheory.Measure.haar.index_pos", "start": [201, 1], "end": [209, 39], "traced_tactics": [{"tactic": "unfold index", "annotated_tactic": ["unfold <a>index</a>", [{"full_name": "MeasureTheory.Measure.haar.index", "def_path": "Mathlib/MeasureTheory/Measure/Haar/Basic.lean", "def_pos": [96, 19], "def_end_pos": [96, 24]}]], "state_before": "G : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK : PositiveCompacts G\nV : Set G\nhV : Set.Nonempty (interior V)\n\u22a2 0 < index (\u2191K) V", "state_after": "G : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK : PositiveCompacts G\nV : Set G\nhV : Set.Nonempty (interior V)\n\u22a2 0 < sInf (Finset.card '' {t | \u2191K \u2286 \u22c3 g \u2208 t, (fun h => g * h) \u207b\u00b9' V})"}, {"tactic": "rw [Nat.sInf_def, Nat.find_pos, mem_image]", "annotated_tactic": ["rw [<a>Nat.sInf_def</a>, <a>Nat.find_pos</a>, <a>mem_image</a>]", [{"full_name": "Nat.sInf_def", "def_path": "Mathlib/Data/Nat/Lattice.lean", "def_pos": [33, 9], "def_end_pos": [33, 17]}, {"full_name": "Nat.find_pos", "def_path": "Mathlib/Data/Nat/Order/Basic.lean", "def_pos": [569, 9], "def_end_pos": [569, 17]}, {"full_name": "Set.mem_image", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [231, 9], "def_end_pos": [231, 18]}]], "state_before": "G : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK : PositiveCompacts G\nV : Set G\nhV : Set.Nonempty (interior V)\n\u22a2 0 < sInf (Finset.card '' {t | \u2191K \u2286 \u22c3 g \u2208 t, (fun h => g * h) \u207b\u00b9' V})", "state_after": "G : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK : PositiveCompacts G\nV : Set G\nhV : Set.Nonempty (interior V)\n\u22a2 \u00ac\u2203 x, x \u2208 {t | \u2191K \u2286 \u22c3 g \u2208 t, (fun h => g * h) \u207b\u00b9' V} \u2227 Finset.card x = 0\n\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK : PositiveCompacts G\nV : Set G\nhV : Set.Nonempty (interior V)\n\u22a2 \u2203 n, n \u2208 Finset.card '' {t | \u2191K \u2286 \u22c3 g \u2208 t, (fun h => g * h) \u207b\u00b9' V}"}, {"tactic": "rintro \u27e8t, h1t, h2t\u27e9", "annotated_tactic": ["rintro \u27e8t, h1t, h2t\u27e9", []], "state_before": "G : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK : PositiveCompacts G\nV : Set G\nhV : Set.Nonempty (interior V)\n\u22a2 \u00ac\u2203 x, x \u2208 {t | \u2191K \u2286 \u22c3 g \u2208 t, (fun h => g * h) \u207b\u00b9' V} \u2227 Finset.card x = 0", "state_after": "case intro.intro\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK : PositiveCompacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nt : Finset G\nh1t : t \u2208 {t | \u2191K \u2286 \u22c3 g \u2208 t, (fun h => g * h) \u207b\u00b9' V}\nh2t : Finset.card t = 0\n\u22a2 False"}, {"tactic": "rw [Finset.card_eq_zero] at h2t", "annotated_tactic": ["rw [<a>Finset.card_eq_zero</a>] at h2t", [{"full_name": "Finset.card_eq_zero", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [70, 9], "def_end_pos": [70, 21]}]], "state_before": "case intro.intro\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK : PositiveCompacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nt : Finset G\nh1t : t \u2208 {t | \u2191K \u2286 \u22c3 g \u2208 t, (fun h => g * h) \u207b\u00b9' V}\nh2t : Finset.card t = 0\n\u22a2 False", "state_after": "case intro.intro\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK : PositiveCompacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nt : Finset G\nh1t : t \u2208 {t | \u2191K \u2286 \u22c3 g \u2208 t, (fun h => g * h) \u207b\u00b9' V}\nh2t : t = \u2205\n\u22a2 False"}, {"tactic": "subst h2t", "annotated_tactic": ["subst h2t", []], "state_before": "case intro.intro\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK : PositiveCompacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nt : Finset G\nh1t : t \u2208 {t | \u2191K \u2286 \u22c3 g \u2208 t, (fun h => g * h) \u207b\u00b9' V}\nh2t : t = \u2205\n\u22a2 False", "state_after": "case intro.intro\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK : PositiveCompacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh1t : \u2205 \u2208 {t | \u2191K \u2286 \u22c3 g \u2208 t, (fun h => g * h) \u207b\u00b9' V}\n\u22a2 False"}, {"tactic": "obtain \u27e8g, hg\u27e9 := K.interior_nonempty", "annotated_tactic": ["obtain \u27e8g, hg\u27e9 := K.interior_nonempty", []], "state_before": "case intro.intro\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK : PositiveCompacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh1t : \u2205 \u2208 {t | \u2191K \u2286 \u22c3 g \u2208 t, (fun h => g * h) \u207b\u00b9' V}\n\u22a2 False", "state_after": "case intro.intro.intro\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK : PositiveCompacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh1t : \u2205 \u2208 {t | \u2191K \u2286 \u22c3 g \u2208 t, (fun h => g * h) \u207b\u00b9' V}\ng : G\nhg : g \u2208 interior \u2191K\n\u22a2 False"}, {"tactic": "show g \u2208 (\u2205 : Set G)", "annotated_tactic": ["show g \u2208 (\u2205 : <a>Set</a> G)", [{"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}]], "state_before": "case intro.intro.intro\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK : PositiveCompacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh1t : \u2205 \u2208 {t | \u2191K \u2286 \u22c3 g \u2208 t, (fun h => g * h) \u207b\u00b9' V}\ng : G\nhg : g \u2208 interior \u2191K\n\u22a2 False", "state_after": "case intro.intro.intro\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK : PositiveCompacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh1t : \u2205 \u2208 {t | \u2191K \u2286 \u22c3 g \u2208 t, (fun h => g * h) \u207b\u00b9' V}\ng : G\nhg : g \u2208 interior \u2191K\n\u22a2 g \u2208 \u2205"}, {"tactic": "convert h1t (interior_subset hg)", "annotated_tactic": ["convert h1t (<a>interior_subset</a> hg)", [{"full_name": "interior_subset", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [302, 9], "def_end_pos": [302, 24]}]], "state_before": "case intro.intro.intro\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK : PositiveCompacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh1t : \u2205 \u2208 {t | \u2191K \u2286 \u22c3 g \u2208 t, (fun h => g * h) \u207b\u00b9' V}\ng : G\nhg : g \u2208 interior \u2191K\n\u22a2 g \u2208 \u2205", "state_after": "case h.e'_5\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK : PositiveCompacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh1t : \u2205 \u2208 {t | \u2191K \u2286 \u22c3 g \u2208 t, (fun h => g * h) \u207b\u00b9' V}\ng : G\nhg : g \u2208 interior \u2191K\n\u22a2 \u2205 = \u22c3 g \u2208 \u2205, (fun h => g * h) \u207b\u00b9' V"}, {"tactic": "symm", "annotated_tactic": ["symm", []], "state_before": "case h.e'_5\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK : PositiveCompacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh1t : \u2205 \u2208 {t | \u2191K \u2286 \u22c3 g \u2208 t, (fun h => g * h) \u207b\u00b9' V}\ng : G\nhg : g \u2208 interior \u2191K\n\u22a2 \u2205 = \u22c3 g \u2208 \u2205, (fun h => g * h) \u207b\u00b9' V", "state_after": "case h.e'_5\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK : PositiveCompacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh1t : \u2205 \u2208 {t | \u2191K \u2286 \u22c3 g \u2208 t, (fun h => g * h) \u207b\u00b9' V}\ng : G\nhg : g \u2208 interior \u2191K\n\u22a2 \u22c3 g \u2208 \u2205, (fun h => g * h) \u207b\u00b9' V = \u2205"}, {"tactic": "simp only [Finset.not_mem_empty, iUnion_of_empty, iUnion_empty]", "annotated_tactic": ["simp only [<a>Finset.not_mem_empty</a>, <a>iUnion_of_empty</a>, <a>iUnion_empty</a>]", [{"full_name": "Finset.not_mem_empty", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [548, 9], "def_end_pos": [548, 22]}, {"full_name": "Set.iUnion_of_empty", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [1469, 9], "def_end_pos": [1469, 24]}, {"full_name": "Set.iUnion_empty", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [810, 9], "def_end_pos": [810, 21]}]], "state_before": "case h.e'_5\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK : PositiveCompacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh1t : \u2205 \u2208 {t | \u2191K \u2286 \u22c3 g \u2208 t, (fun h => g * h) \u207b\u00b9' V}\ng : G\nhg : g \u2208 interior \u2191K\n\u22a2 \u22c3 g \u2208 \u2205, (fun h => g * h) \u207b\u00b9' V = \u2205", "state_after": "no goals"}, {"tactic": "exact index_defined K.isCompact hV", "annotated_tactic": ["exact <a>index_defined</a> K.isCompact hV", [{"full_name": "MeasureTheory.Measure.haar.index_defined", "def_path": "Mathlib/MeasureTheory/Measure/Haar/Basic.lean", "def_pos": [171, 9], "def_end_pos": [171, 22]}]], "state_before": "G : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK : PositiveCompacts G\nV : Set G\nhV : Set.Nonempty (interior V)\n\u22a2 \u2203 n, n \u2208 Finset.card '' {t | \u2191K \u2286 \u22c3 g \u2208 t, (fun h => g * h) \u207b\u00b9' V}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/LocallyFinite.lean", "full_name": "Finset.card_Iio_eq_card_Iic_sub_one", "start": [750, 1], "end": [751, 57], "traced_tactics": [{"tactic": "rw [Iic_eq_cons_Iio, card_cons, add_tsub_cancel_right]", "annotated_tactic": ["rw [<a>Iic_eq_cons_Iio</a>, <a>card_cons</a>, <a>add_tsub_cancel_right</a>]", [{"full_name": "Finset.Iic_eq_cons_Iio", "def_path": "Mathlib/Data/Finset/LocallyFinite.lean", "def_pos": [746, 9], "def_end_pos": [746, 24]}, {"full_name": "Finset.card_cons", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [97, 9], "def_end_pos": [97, 18]}, {"full_name": "add_tsub_cancel_right", "def_path": "Mathlib/Algebra/Order/Sub/Defs.lean", "def_pos": [356, 9], "def_end_pos": [356, 30]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : LocallyFiniteOrderBot \u03b1\na : \u03b1\n\u22a2 card (Iio a) = card (Iic a) - 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/PartrecCode.lean", "full_name": "Nat.Partrec.Code.exists_code", "start": [699, 1], "end": [728, 42], "traced_tactics": [{"tactic": "induction h", "annotated_tactic": ["induction h", []], "state_before": "f : \u2115 \u2192. \u2115\nh : Partrec f\n\u22a2 \u2203 c, eval c = f", "state_after": "case zero\nf : \u2115 \u2192. \u2115\n\u22a2 \u2203 c, eval c = pure 0\n\ncase succ\nf : \u2115 \u2192. \u2115\n\u22a2 \u2203 c, eval c = \u2191Nat.succ\n\ncase left\nf : \u2115 \u2192. \u2115\n\u22a2 \u2203 c, eval c = \u2191fun n => (unpair n).1\n\ncase right\nf : \u2115 \u2192. \u2115\n\u22a2 \u2203 c, eval c = \u2191fun n => (unpair n).2\n\ncase pair\nf f\u271d g\u271d : \u2115 \u2192. \u2115\na\u271d\u00b9 : Partrec f\u271d\na\u271d : Partrec g\u271d\na_ih\u271d\u00b9 : \u2203 c, eval c = f\u271d\na_ih\u271d : \u2203 c, eval c = g\u271d\n\u22a2 \u2203 c, eval c = fun n => Seq.seq (Nat.pair <$> f\u271d n) fun x => g\u271d n\n\ncase comp\nf f\u271d g\u271d : \u2115 \u2192. \u2115\na\u271d\u00b9 : Partrec f\u271d\na\u271d : Partrec g\u271d\na_ih\u271d\u00b9 : \u2203 c, eval c = f\u271d\na_ih\u271d : \u2203 c, eval c = g\u271d\n\u22a2 \u2203 c, eval c = fun n => g\u271d n >>= f\u271d\n\ncase prec\nf f\u271d g\u271d : \u2115 \u2192. \u2115\na\u271d\u00b9 : Partrec f\u271d\na\u271d : Partrec g\u271d\na_ih\u271d\u00b9 : \u2203 c, eval c = f\u271d\na_ih\u271d : \u2203 c, eval c = g\u271d\n\u22a2 \u2203 c,\n    eval c =\n      unpaired fun a n =>\n        Nat.rec (f\u271d a)\n          (fun y IH => do\n            let i \u2190 IH\n            g\u271d (Nat.pair a (Nat.pair y i)))\n          n\n\ncase rfind\nf f\u271d : \u2115 \u2192. \u2115\na\u271d : Partrec f\u271d\na_ih\u271d : \u2203 c, eval c = f\u271d\n\u22a2 \u2203 c, eval c = fun a => Nat.rfind fun n => (fun m => decide (m = 0)) <$> f\u271d (Nat.pair a n)"}, {"tactic": "case zero => exact \u27e8zero, rfl\u27e9", "annotated_tactic": ["case zero => exact \u27e8<a>zero</a>, <a>rfl</a>\u27e9", [{"full_name": "Nat.Partrec.Code.zero", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [77, 5], "def_end_pos": [77, 9]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "case zero\nf : \u2115 \u2192. \u2115\n\u22a2 \u2203 c, eval c = pure 0\n\ncase succ\nf : \u2115 \u2192. \u2115\n\u22a2 \u2203 c, eval c = \u2191Nat.succ\n\ncase left\nf : \u2115 \u2192. \u2115\n\u22a2 \u2203 c, eval c = \u2191fun n => (unpair n).1\n\ncase right\nf : \u2115 \u2192. \u2115\n\u22a2 \u2203 c, eval c = \u2191fun n => (unpair n).2\n\ncase pair\nf f\u271d g\u271d : \u2115 \u2192. \u2115\na\u271d\u00b9 : Partrec f\u271d\na\u271d : Partrec g\u271d\na_ih\u271d\u00b9 : \u2203 c, eval c = f\u271d\na_ih\u271d : \u2203 c, eval c = g\u271d\n\u22a2 \u2203 c, eval c = fun n => Seq.seq (Nat.pair <$> f\u271d n) fun x => g\u271d n\n\ncase comp\nf f\u271d g\u271d : \u2115 \u2192. \u2115\na\u271d\u00b9 : Partrec f\u271d\na\u271d : Partrec g\u271d\na_ih\u271d\u00b9 : \u2203 c, eval c = f\u271d\na_ih\u271d : \u2203 c, eval c = g\u271d\n\u22a2 \u2203 c, eval c = fun n => g\u271d n >>= f\u271d\n\ncase prec\nf f\u271d g\u271d : \u2115 \u2192. \u2115\na\u271d\u00b9 : Partrec f\u271d\na\u271d : Partrec g\u271d\na_ih\u271d\u00b9 : \u2203 c, eval c = f\u271d\na_ih\u271d : \u2203 c, eval c = g\u271d\n\u22a2 \u2203 c,\n    eval c =\n      unpaired fun a n =>\n        Nat.rec (f\u271d a)\n          (fun y IH => do\n            let i \u2190 IH\n            g\u271d (Nat.pair a (Nat.pair y i)))\n          n\n\ncase rfind\nf f\u271d : \u2115 \u2192. \u2115\na\u271d : Partrec f\u271d\na_ih\u271d : \u2203 c, eval c = f\u271d\n\u22a2 \u2203 c, eval c = fun a => Nat.rfind fun n => (fun m => decide (m = 0)) <$> f\u271d (Nat.pair a n)", "state_after": "case succ\nf : \u2115 \u2192. \u2115\n\u22a2 \u2203 c, eval c = \u2191Nat.succ\n\ncase left\nf : \u2115 \u2192. \u2115\n\u22a2 \u2203 c, eval c = \u2191fun n => (unpair n).1\n\ncase right\nf : \u2115 \u2192. \u2115\n\u22a2 \u2203 c, eval c = \u2191fun n => (unpair n).2\n\ncase pair\nf f\u271d g\u271d : \u2115 \u2192. \u2115\na\u271d\u00b9 : Partrec f\u271d\na\u271d : Partrec g\u271d\na_ih\u271d\u00b9 : \u2203 c, eval c = f\u271d\na_ih\u271d : \u2203 c, eval c = g\u271d\n\u22a2 \u2203 c, eval c = fun n => Seq.seq (Nat.pair <$> f\u271d n) fun x => g\u271d n\n\ncase comp\nf f\u271d g\u271d : \u2115 \u2192. \u2115\na\u271d\u00b9 : Partrec f\u271d\na\u271d : Partrec g\u271d\na_ih\u271d\u00b9 : \u2203 c, eval c = f\u271d\na_ih\u271d : \u2203 c, eval c = g\u271d\n\u22a2 \u2203 c, eval c = fun n => g\u271d n >>= f\u271d\n\ncase prec\nf f\u271d g\u271d : \u2115 \u2192. \u2115\na\u271d\u00b9 : Partrec f\u271d\na\u271d : Partrec g\u271d\na_ih\u271d\u00b9 : \u2203 c, eval c = f\u271d\na_ih\u271d : \u2203 c, eval c = g\u271d\n\u22a2 \u2203 c,\n    eval c =\n      unpaired fun a n =>\n        Nat.rec (f\u271d a)\n          (fun y IH => do\n            let i \u2190 IH\n            g\u271d (Nat.pair a (Nat.pair y i)))\n          n\n\ncase rfind\nf f\u271d : \u2115 \u2192. \u2115\na\u271d : Partrec f\u271d\na_ih\u271d : \u2203 c, eval c = f\u271d\n\u22a2 \u2203 c, eval c = fun a => Nat.rfind fun n => (fun m => decide (m = 0)) <$> f\u271d (Nat.pair a n)"}, {"tactic": "case succ => exact \u27e8succ, rfl\u27e9", "annotated_tactic": ["case succ => exact \u27e8<a>succ</a>, <a>rfl</a>\u27e9", [{"full_name": "Nat.Partrec.Code.succ", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [78, 5], "def_end_pos": [78, 9]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "case succ\nf : \u2115 \u2192. \u2115\n\u22a2 \u2203 c, eval c = \u2191Nat.succ\n\ncase left\nf : \u2115 \u2192. \u2115\n\u22a2 \u2203 c, eval c = \u2191fun n => (unpair n).1\n\ncase right\nf : \u2115 \u2192. \u2115\n\u22a2 \u2203 c, eval c = \u2191fun n => (unpair n).2\n\ncase pair\nf f\u271d g\u271d : \u2115 \u2192. \u2115\na\u271d\u00b9 : Partrec f\u271d\na\u271d : Partrec g\u271d\na_ih\u271d\u00b9 : \u2203 c, eval c = f\u271d\na_ih\u271d : \u2203 c, eval c = g\u271d\n\u22a2 \u2203 c, eval c = fun n => Seq.seq (Nat.pair <$> f\u271d n) fun x => g\u271d n\n\ncase comp\nf f\u271d g\u271d : \u2115 \u2192. \u2115\na\u271d\u00b9 : Partrec f\u271d\na\u271d : Partrec g\u271d\na_ih\u271d\u00b9 : \u2203 c, eval c = f\u271d\na_ih\u271d : \u2203 c, eval c = g\u271d\n\u22a2 \u2203 c, eval c = fun n => g\u271d n >>= f\u271d\n\ncase prec\nf f\u271d g\u271d : \u2115 \u2192. \u2115\na\u271d\u00b9 : Partrec f\u271d\na\u271d : Partrec g\u271d\na_ih\u271d\u00b9 : \u2203 c, eval c = f\u271d\na_ih\u271d : \u2203 c, eval c = g\u271d\n\u22a2 \u2203 c,\n    eval c =\n      unpaired fun a n =>\n        Nat.rec (f\u271d a)\n          (fun y IH => do\n            let i \u2190 IH\n            g\u271d (Nat.pair a (Nat.pair y i)))\n          n\n\ncase rfind\nf f\u271d : \u2115 \u2192. \u2115\na\u271d : Partrec f\u271d\na_ih\u271d : \u2203 c, eval c = f\u271d\n\u22a2 \u2203 c, eval c = fun a => Nat.rfind fun n => (fun m => decide (m = 0)) <$> f\u271d (Nat.pair a n)", "state_after": "case left\nf : \u2115 \u2192. \u2115\n\u22a2 \u2203 c, eval c = \u2191fun n => (unpair n).1\n\ncase right\nf : \u2115 \u2192. \u2115\n\u22a2 \u2203 c, eval c = \u2191fun n => (unpair n).2\n\ncase pair\nf f\u271d g\u271d : \u2115 \u2192. \u2115\na\u271d\u00b9 : Partrec f\u271d\na\u271d : Partrec g\u271d\na_ih\u271d\u00b9 : \u2203 c, eval c = f\u271d\na_ih\u271d : \u2203 c, eval c = g\u271d\n\u22a2 \u2203 c, eval c = fun n => Seq.seq (Nat.pair <$> f\u271d n) fun x => g\u271d n\n\ncase comp\nf f\u271d g\u271d : \u2115 \u2192. \u2115\na\u271d\u00b9 : Partrec f\u271d\na\u271d : Partrec g\u271d\na_ih\u271d\u00b9 : \u2203 c, eval c = f\u271d\na_ih\u271d : \u2203 c, eval c = g\u271d\n\u22a2 \u2203 c, eval c = fun n => g\u271d n >>= f\u271d\n\ncase prec\nf f\u271d g\u271d : \u2115 \u2192. \u2115\na\u271d\u00b9 : Partrec f\u271d\na\u271d : Partrec g\u271d\na_ih\u271d\u00b9 : \u2203 c, eval c = f\u271d\na_ih\u271d : \u2203 c, eval c = g\u271d\n\u22a2 \u2203 c,\n    eval c =\n      unpaired fun a n =>\n        Nat.rec (f\u271d a)\n          (fun y IH => do\n            let i \u2190 IH\n            g\u271d (Nat.pair a (Nat.pair y i)))\n          n\n\ncase rfind\nf f\u271d : \u2115 \u2192. \u2115\na\u271d : Partrec f\u271d\na_ih\u271d : \u2203 c, eval c = f\u271d\n\u22a2 \u2203 c, eval c = fun a => Nat.rfind fun n => (fun m => decide (m = 0)) <$> f\u271d (Nat.pair a n)"}, {"tactic": "case left => exact \u27e8left, rfl\u27e9", "annotated_tactic": ["case left => exact \u27e8<a>left</a>, <a>rfl</a>\u27e9", [{"full_name": "Nat.Partrec.Code.left", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [79, 5], "def_end_pos": [79, 9]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "case left\nf : \u2115 \u2192. \u2115\n\u22a2 \u2203 c, eval c = \u2191fun n => (unpair n).1\n\ncase right\nf : \u2115 \u2192. \u2115\n\u22a2 \u2203 c, eval c = \u2191fun n => (unpair n).2\n\ncase pair\nf f\u271d g\u271d : \u2115 \u2192. \u2115\na\u271d\u00b9 : Partrec f\u271d\na\u271d : Partrec g\u271d\na_ih\u271d\u00b9 : \u2203 c, eval c = f\u271d\na_ih\u271d : \u2203 c, eval c = g\u271d\n\u22a2 \u2203 c, eval c = fun n => Seq.seq (Nat.pair <$> f\u271d n) fun x => g\u271d n\n\ncase comp\nf f\u271d g\u271d : \u2115 \u2192. \u2115\na\u271d\u00b9 : Partrec f\u271d\na\u271d : Partrec g\u271d\na_ih\u271d\u00b9 : \u2203 c, eval c = f\u271d\na_ih\u271d : \u2203 c, eval c = g\u271d\n\u22a2 \u2203 c, eval c = fun n => g\u271d n >>= f\u271d\n\ncase prec\nf f\u271d g\u271d : \u2115 \u2192. \u2115\na\u271d\u00b9 : Partrec f\u271d\na\u271d : Partrec g\u271d\na_ih\u271d\u00b9 : \u2203 c, eval c = f\u271d\na_ih\u271d : \u2203 c, eval c = g\u271d\n\u22a2 \u2203 c,\n    eval c =\n      unpaired fun a n =>\n        Nat.rec (f\u271d a)\n          (fun y IH => do\n            let i \u2190 IH\n            g\u271d (Nat.pair a (Nat.pair y i)))\n          n\n\ncase rfind\nf f\u271d : \u2115 \u2192. \u2115\na\u271d : Partrec f\u271d\na_ih\u271d : \u2203 c, eval c = f\u271d\n\u22a2 \u2203 c, eval c = fun a => Nat.rfind fun n => (fun m => decide (m = 0)) <$> f\u271d (Nat.pair a n)", "state_after": "case right\nf : \u2115 \u2192. \u2115\n\u22a2 \u2203 c, eval c = \u2191fun n => (unpair n).2\n\ncase pair\nf f\u271d g\u271d : \u2115 \u2192. \u2115\na\u271d\u00b9 : Partrec f\u271d\na\u271d : Partrec g\u271d\na_ih\u271d\u00b9 : \u2203 c, eval c = f\u271d\na_ih\u271d : \u2203 c, eval c = g\u271d\n\u22a2 \u2203 c, eval c = fun n => Seq.seq (Nat.pair <$> f\u271d n) fun x => g\u271d n\n\ncase comp\nf f\u271d g\u271d : \u2115 \u2192. \u2115\na\u271d\u00b9 : Partrec f\u271d\na\u271d : Partrec g\u271d\na_ih\u271d\u00b9 : \u2203 c, eval c = f\u271d\na_ih\u271d : \u2203 c, eval c = g\u271d\n\u22a2 \u2203 c, eval c = fun n => g\u271d n >>= f\u271d\n\ncase prec\nf f\u271d g\u271d : \u2115 \u2192. \u2115\na\u271d\u00b9 : Partrec f\u271d\na\u271d : Partrec g\u271d\na_ih\u271d\u00b9 : \u2203 c, eval c = f\u271d\na_ih\u271d : \u2203 c, eval c = g\u271d\n\u22a2 \u2203 c,\n    eval c =\n      unpaired fun a n =>\n        Nat.rec (f\u271d a)\n          (fun y IH => do\n            let i \u2190 IH\n            g\u271d (Nat.pair a (Nat.pair y i)))\n          n\n\ncase rfind\nf f\u271d : \u2115 \u2192. \u2115\na\u271d : Partrec f\u271d\na_ih\u271d : \u2203 c, eval c = f\u271d\n\u22a2 \u2203 c, eval c = fun a => Nat.rfind fun n => (fun m => decide (m = 0)) <$> f\u271d (Nat.pair a n)"}, {"tactic": "case right => exact \u27e8right, rfl\u27e9", "annotated_tactic": ["case right => exact \u27e8<a>right</a>, <a>rfl</a>\u27e9", [{"full_name": "Nat.Partrec.Code.right", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [80, 5], "def_end_pos": [80, 10]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "case right\nf : \u2115 \u2192. \u2115\n\u22a2 \u2203 c, eval c = \u2191fun n => (unpair n).2\n\ncase pair\nf f\u271d g\u271d : \u2115 \u2192. \u2115\na\u271d\u00b9 : Partrec f\u271d\na\u271d : Partrec g\u271d\na_ih\u271d\u00b9 : \u2203 c, eval c = f\u271d\na_ih\u271d : \u2203 c, eval c = g\u271d\n\u22a2 \u2203 c, eval c = fun n => Seq.seq (Nat.pair <$> f\u271d n) fun x => g\u271d n\n\ncase comp\nf f\u271d g\u271d : \u2115 \u2192. \u2115\na\u271d\u00b9 : Partrec f\u271d\na\u271d : Partrec g\u271d\na_ih\u271d\u00b9 : \u2203 c, eval c = f\u271d\na_ih\u271d : \u2203 c, eval c = g\u271d\n\u22a2 \u2203 c, eval c = fun n => g\u271d n >>= f\u271d\n\ncase prec\nf f\u271d g\u271d : \u2115 \u2192. \u2115\na\u271d\u00b9 : Partrec f\u271d\na\u271d : Partrec g\u271d\na_ih\u271d\u00b9 : \u2203 c, eval c = f\u271d\na_ih\u271d : \u2203 c, eval c = g\u271d\n\u22a2 \u2203 c,\n    eval c =\n      unpaired fun a n =>\n        Nat.rec (f\u271d a)\n          (fun y IH => do\n            let i \u2190 IH\n            g\u271d (Nat.pair a (Nat.pair y i)))\n          n\n\ncase rfind\nf f\u271d : \u2115 \u2192. \u2115\na\u271d : Partrec f\u271d\na_ih\u271d : \u2203 c, eval c = f\u271d\n\u22a2 \u2203 c, eval c = fun a => Nat.rfind fun n => (fun m => decide (m = 0)) <$> f\u271d (Nat.pair a n)", "state_after": "case pair\nf f\u271d g\u271d : \u2115 \u2192. \u2115\na\u271d\u00b9 : Partrec f\u271d\na\u271d : Partrec g\u271d\na_ih\u271d\u00b9 : \u2203 c, eval c = f\u271d\na_ih\u271d : \u2203 c, eval c = g\u271d\n\u22a2 \u2203 c, eval c = fun n => Seq.seq (Nat.pair <$> f\u271d n) fun x => g\u271d n\n\ncase comp\nf f\u271d g\u271d : \u2115 \u2192. \u2115\na\u271d\u00b9 : Partrec f\u271d\na\u271d : Partrec g\u271d\na_ih\u271d\u00b9 : \u2203 c, eval c = f\u271d\na_ih\u271d : \u2203 c, eval c = g\u271d\n\u22a2 \u2203 c, eval c = fun n => g\u271d n >>= f\u271d\n\ncase prec\nf f\u271d g\u271d : \u2115 \u2192. \u2115\na\u271d\u00b9 : Partrec f\u271d\na\u271d : Partrec g\u271d\na_ih\u271d\u00b9 : \u2203 c, eval c = f\u271d\na_ih\u271d : \u2203 c, eval c = g\u271d\n\u22a2 \u2203 c,\n    eval c =\n      unpaired fun a n =>\n        Nat.rec (f\u271d a)\n          (fun y IH => do\n            let i \u2190 IH\n            g\u271d (Nat.pair a (Nat.pair y i)))\n          n\n\ncase rfind\nf f\u271d : \u2115 \u2192. \u2115\na\u271d : Partrec f\u271d\na_ih\u271d : \u2203 c, eval c = f\u271d\n\u22a2 \u2203 c, eval c = fun a => Nat.rfind fun n => (fun m => decide (m = 0)) <$> f\u271d (Nat.pair a n)"}, {"tactic": "case pair f g pf pg hf hg =>\n  rcases hf with \u27e8cf, rfl\u27e9; rcases hg with \u27e8cg, rfl\u27e9\n  exact \u27e8pair cf cg, rfl\u27e9", "annotated_tactic": ["case pair f g pf pg hf hg =>\n      rcases hf with \u27e8cf, rfl\u27e9; rcases hg with \u27e8cg, rfl\u27e9\n      exact \u27e8<a>pair</a> cf cg, <a>rfl</a>\u27e9", [{"full_name": "Nat.Partrec.Code.pair", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [81, 5], "def_end_pos": [81, 9]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "case pair\nf f\u271d g\u271d : \u2115 \u2192. \u2115\na\u271d\u00b9 : Partrec f\u271d\na\u271d : Partrec g\u271d\na_ih\u271d\u00b9 : \u2203 c, eval c = f\u271d\na_ih\u271d : \u2203 c, eval c = g\u271d\n\u22a2 \u2203 c, eval c = fun n => Seq.seq (Nat.pair <$> f\u271d n) fun x => g\u271d n\n\ncase comp\nf f\u271d g\u271d : \u2115 \u2192. \u2115\na\u271d\u00b9 : Partrec f\u271d\na\u271d : Partrec g\u271d\na_ih\u271d\u00b9 : \u2203 c, eval c = f\u271d\na_ih\u271d : \u2203 c, eval c = g\u271d\n\u22a2 \u2203 c, eval c = fun n => g\u271d n >>= f\u271d\n\ncase prec\nf f\u271d g\u271d : \u2115 \u2192. \u2115\na\u271d\u00b9 : Partrec f\u271d\na\u271d : Partrec g\u271d\na_ih\u271d\u00b9 : \u2203 c, eval c = f\u271d\na_ih\u271d : \u2203 c, eval c = g\u271d\n\u22a2 \u2203 c,\n    eval c =\n      unpaired fun a n =>\n        Nat.rec (f\u271d a)\n          (fun y IH => do\n            let i \u2190 IH\n            g\u271d (Nat.pair a (Nat.pair y i)))\n          n\n\ncase rfind\nf f\u271d : \u2115 \u2192. \u2115\na\u271d : Partrec f\u271d\na_ih\u271d : \u2203 c, eval c = f\u271d\n\u22a2 \u2203 c, eval c = fun a => Nat.rfind fun n => (fun m => decide (m = 0)) <$> f\u271d (Nat.pair a n)", "state_after": "case comp\nf f\u271d g\u271d : \u2115 \u2192. \u2115\na\u271d\u00b9 : Partrec f\u271d\na\u271d : Partrec g\u271d\na_ih\u271d\u00b9 : \u2203 c, eval c = f\u271d\na_ih\u271d : \u2203 c, eval c = g\u271d\n\u22a2 \u2203 c, eval c = fun n => g\u271d n >>= f\u271d\n\ncase prec\nf f\u271d g\u271d : \u2115 \u2192. \u2115\na\u271d\u00b9 : Partrec f\u271d\na\u271d : Partrec g\u271d\na_ih\u271d\u00b9 : \u2203 c, eval c = f\u271d\na_ih\u271d : \u2203 c, eval c = g\u271d\n\u22a2 \u2203 c,\n    eval c =\n      unpaired fun a n =>\n        Nat.rec (f\u271d a)\n          (fun y IH => do\n            let i \u2190 IH\n            g\u271d (Nat.pair a (Nat.pair y i)))\n          n\n\ncase rfind\nf f\u271d : \u2115 \u2192. \u2115\na\u271d : Partrec f\u271d\na_ih\u271d : \u2203 c, eval c = f\u271d\n\u22a2 \u2203 c, eval c = fun a => Nat.rfind fun n => (fun m => decide (m = 0)) <$> f\u271d (Nat.pair a n)"}, {"tactic": "case comp f g pf pg hf hg =>\n  rcases hf with \u27e8cf, rfl\u27e9; rcases hg with \u27e8cg, rfl\u27e9\n  exact \u27e8comp cf cg, rfl\u27e9", "annotated_tactic": ["case comp f g pf pg hf hg =>\n      rcases hf with \u27e8cf, rfl\u27e9; rcases hg with \u27e8cg, rfl\u27e9\n      exact \u27e8<a>comp</a> cf cg, <a>rfl</a>\u27e9", [{"full_name": "Nat.Partrec.Code.comp", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [82, 5], "def_end_pos": [82, 9]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "case comp\nf f\u271d g\u271d : \u2115 \u2192. \u2115\na\u271d\u00b9 : Partrec f\u271d\na\u271d : Partrec g\u271d\na_ih\u271d\u00b9 : \u2203 c, eval c = f\u271d\na_ih\u271d : \u2203 c, eval c = g\u271d\n\u22a2 \u2203 c, eval c = fun n => g\u271d n >>= f\u271d\n\ncase prec\nf f\u271d g\u271d : \u2115 \u2192. \u2115\na\u271d\u00b9 : Partrec f\u271d\na\u271d : Partrec g\u271d\na_ih\u271d\u00b9 : \u2203 c, eval c = f\u271d\na_ih\u271d : \u2203 c, eval c = g\u271d\n\u22a2 \u2203 c,\n    eval c =\n      unpaired fun a n =>\n        Nat.rec (f\u271d a)\n          (fun y IH => do\n            let i \u2190 IH\n            g\u271d (Nat.pair a (Nat.pair y i)))\n          n\n\ncase rfind\nf f\u271d : \u2115 \u2192. \u2115\na\u271d : Partrec f\u271d\na_ih\u271d : \u2203 c, eval c = f\u271d\n\u22a2 \u2203 c, eval c = fun a => Nat.rfind fun n => (fun m => decide (m = 0)) <$> f\u271d (Nat.pair a n)", "state_after": "case prec\nf f\u271d g\u271d : \u2115 \u2192. \u2115\na\u271d\u00b9 : Partrec f\u271d\na\u271d : Partrec g\u271d\na_ih\u271d\u00b9 : \u2203 c, eval c = f\u271d\na_ih\u271d : \u2203 c, eval c = g\u271d\n\u22a2 \u2203 c,\n    eval c =\n      unpaired fun a n =>\n        Nat.rec (f\u271d a)\n          (fun y IH => do\n            let i \u2190 IH\n            g\u271d (Nat.pair a (Nat.pair y i)))\n          n\n\ncase rfind\nf f\u271d : \u2115 \u2192. \u2115\na\u271d : Partrec f\u271d\na_ih\u271d : \u2203 c, eval c = f\u271d\n\u22a2 \u2203 c, eval c = fun a => Nat.rfind fun n => (fun m => decide (m = 0)) <$> f\u271d (Nat.pair a n)"}, {"tactic": "case prec f g pf pg hf hg =>\n  rcases hf with \u27e8cf, rfl\u27e9; rcases hg with \u27e8cg, rfl\u27e9\n  exact \u27e8prec cf cg, rfl\u27e9", "annotated_tactic": ["case prec f g pf pg hf hg =>\n      rcases hf with \u27e8cf, rfl\u27e9; rcases hg with \u27e8cg, rfl\u27e9\n      exact \u27e8<a>prec</a> cf cg, <a>rfl</a>\u27e9", [{"full_name": "Nat.Partrec.Code.prec", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [83, 5], "def_end_pos": [83, 9]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "case prec\nf f\u271d g\u271d : \u2115 \u2192. \u2115\na\u271d\u00b9 : Partrec f\u271d\na\u271d : Partrec g\u271d\na_ih\u271d\u00b9 : \u2203 c, eval c = f\u271d\na_ih\u271d : \u2203 c, eval c = g\u271d\n\u22a2 \u2203 c,\n    eval c =\n      unpaired fun a n =>\n        Nat.rec (f\u271d a)\n          (fun y IH => do\n            let i \u2190 IH\n            g\u271d (Nat.pair a (Nat.pair y i)))\n          n\n\ncase rfind\nf f\u271d : \u2115 \u2192. \u2115\na\u271d : Partrec f\u271d\na_ih\u271d : \u2203 c, eval c = f\u271d\n\u22a2 \u2203 c, eval c = fun a => Nat.rfind fun n => (fun m => decide (m = 0)) <$> f\u271d (Nat.pair a n)", "state_after": "case rfind\nf f\u271d : \u2115 \u2192. \u2115\na\u271d : Partrec f\u271d\na_ih\u271d : \u2203 c, eval c = f\u271d\n\u22a2 \u2203 c, eval c = fun a => Nat.rfind fun n => (fun m => decide (m = 0)) <$> f\u271d (Nat.pair a n)"}, {"tactic": "case rfind f pf hf =>\n  rcases hf with \u27e8cf, rfl\u27e9\n  refine' \u27e8comp (rfind' cf) (pair Code.id zero), _\u27e9\n  simp [eval, Seq.seq, pure, PFun.pure, Part.map_id']", "annotated_tactic": ["case rfind f pf hf =>\n      rcases hf with \u27e8cf, rfl\u27e9\n      refine' \u27e8<a>comp</a> (<a>rfind'</a> cf) (<a>pair</a> <a>Code.id</a> <a>zero</a>), _\u27e9\n      simp [<a>eval</a>, <a>Seq.seq</a>, <a>pure</a>, <a>PFun.pure</a>, <a>Part.map_id'</a>]", [{"full_name": "Nat.Partrec.Code.comp", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [82, 5], "def_end_pos": [82, 9]}, {"full_name": "Nat.Partrec.Code.rfind'", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [84, 5], "def_end_pos": [84, 11]}, {"full_name": "Nat.Partrec.Code.pair", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [81, 5], "def_end_pos": [81, 9]}, {"full_name": "Nat.Partrec.Code.id", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [117, 15], "def_end_pos": [117, 17]}, {"full_name": "Nat.Partrec.Code.zero", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [77, 5], "def_end_pos": [77, 9]}, {"full_name": "Nat.Partrec.Code.eval", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [620, 5], "def_end_pos": [620, 9]}, {"full_name": "Seq.seq", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2698, 3], "def_end_pos": [2698, 6]}, {"full_name": "Pure.pure", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2670, 3], "def_end_pos": [2670, 7]}, {"full_name": "PFun.pure", "def_path": "Mathlib/Data/PFun.lean", "def_pos": [206, 15], "def_end_pos": [206, 19]}, {"full_name": "Part.map_id'", "def_path": "Mathlib/Data/Part.lean", "def_pos": [589, 9], "def_end_pos": [589, 16]}]], "state_before": "case rfind\nf f\u271d : \u2115 \u2192. \u2115\na\u271d : Partrec f\u271d\na_ih\u271d : \u2203 c, eval c = f\u271d\n\u22a2 \u2203 c, eval c = fun a => Nat.rfind fun n => (fun m => decide (m = 0)) <$> f\u271d (Nat.pair a n)", "state_after": "no goals"}, {"tactic": "exact \u27e8zero, rfl\u27e9", "annotated_tactic": ["exact \u27e8<a>zero</a>, <a>rfl</a>\u27e9", [{"full_name": "Nat.Partrec.Code.zero", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [77, 5], "def_end_pos": [77, 9]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "f : \u2115 \u2192. \u2115\n\u22a2 \u2203 c, eval c = pure 0", "state_after": "no goals"}, {"tactic": "exact \u27e8succ, rfl\u27e9", "annotated_tactic": ["exact \u27e8<a>succ</a>, <a>rfl</a>\u27e9", [{"full_name": "Nat.Partrec.Code.succ", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [78, 5], "def_end_pos": [78, 9]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "f : \u2115 \u2192. \u2115\n\u22a2 \u2203 c, eval c = \u2191Nat.succ", "state_after": "no goals"}, {"tactic": "exact \u27e8left, rfl\u27e9", "annotated_tactic": ["exact \u27e8<a>left</a>, <a>rfl</a>\u27e9", [{"full_name": "Nat.Partrec.Code.left", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [79, 5], "def_end_pos": [79, 9]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "f : \u2115 \u2192. \u2115\n\u22a2 \u2203 c, eval c = \u2191fun n => (unpair n).1", "state_after": "no goals"}, {"tactic": "exact \u27e8right, rfl\u27e9", "annotated_tactic": ["exact \u27e8<a>right</a>, <a>rfl</a>\u27e9", [{"full_name": "Nat.Partrec.Code.right", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [80, 5], "def_end_pos": [80, 10]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "f : \u2115 \u2192. \u2115\n\u22a2 \u2203 c, eval c = \u2191fun n => (unpair n).2", "state_after": "no goals"}, {"tactic": "rcases hf with \u27e8cf, rfl\u27e9", "annotated_tactic": ["rcases hf with \u27e8cf, rfl\u27e9", []], "state_before": "f\u271d f g : \u2115 \u2192. \u2115\npf : Partrec f\npg : Partrec g\nhf : \u2203 c, eval c = f\nhg : \u2203 c, eval c = g\n\u22a2 \u2203 c, eval c = fun n => Seq.seq (Nat.pair <$> f n) fun x => g n", "state_after": "case intro\nf g : \u2115 \u2192. \u2115\npg : Partrec g\nhg : \u2203 c, eval c = g\ncf : Code\npf : Partrec (eval cf)\n\u22a2 \u2203 c, eval c = fun n => Seq.seq (Nat.pair <$> eval cf n) fun x => g n"}, {"tactic": "rcases hg with \u27e8cg, rfl\u27e9", "annotated_tactic": ["rcases hg with \u27e8cg, rfl\u27e9", []], "state_before": "case intro\nf g : \u2115 \u2192. \u2115\npg : Partrec g\nhg : \u2203 c, eval c = g\ncf : Code\npf : Partrec (eval cf)\n\u22a2 \u2203 c, eval c = fun n => Seq.seq (Nat.pair <$> eval cf n) fun x => g n", "state_after": "case intro.intro\nf : \u2115 \u2192. \u2115\ncf : Code\npf : Partrec (eval cf)\ncg : Code\npg : Partrec (eval cg)\n\u22a2 \u2203 c, eval c = fun n => Seq.seq (Nat.pair <$> eval cf n) fun x => eval cg n"}, {"tactic": "exact \u27e8pair cf cg, rfl\u27e9", "annotated_tactic": ["exact \u27e8<a>pair</a> cf cg, <a>rfl</a>\u27e9", [{"full_name": "Nat.Partrec.Code.pair", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [81, 5], "def_end_pos": [81, 9]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "case intro.intro\nf : \u2115 \u2192. \u2115\ncf : Code\npf : Partrec (eval cf)\ncg : Code\npg : Partrec (eval cg)\n\u22a2 \u2203 c, eval c = fun n => Seq.seq (Nat.pair <$> eval cf n) fun x => eval cg n", "state_after": "no goals"}, {"tactic": "rcases hf with \u27e8cf, rfl\u27e9", "annotated_tactic": ["rcases hf with \u27e8cf, rfl\u27e9", []], "state_before": "f\u271d f g : \u2115 \u2192. \u2115\npf : Partrec f\npg : Partrec g\nhf : \u2203 c, eval c = f\nhg : \u2203 c, eval c = g\n\u22a2 \u2203 c, eval c = fun n => g n >>= f", "state_after": "case intro\nf g : \u2115 \u2192. \u2115\npg : Partrec g\nhg : \u2203 c, eval c = g\ncf : Code\npf : Partrec (eval cf)\n\u22a2 \u2203 c, eval c = fun n => g n >>= eval cf"}, {"tactic": "rcases hg with \u27e8cg, rfl\u27e9", "annotated_tactic": ["rcases hg with \u27e8cg, rfl\u27e9", []], "state_before": "case intro\nf g : \u2115 \u2192. \u2115\npg : Partrec g\nhg : \u2203 c, eval c = g\ncf : Code\npf : Partrec (eval cf)\n\u22a2 \u2203 c, eval c = fun n => g n >>= eval cf", "state_after": "case intro.intro\nf : \u2115 \u2192. \u2115\ncf : Code\npf : Partrec (eval cf)\ncg : Code\npg : Partrec (eval cg)\n\u22a2 \u2203 c, eval c = fun n => eval cg n >>= eval cf"}, {"tactic": "exact \u27e8comp cf cg, rfl\u27e9", "annotated_tactic": ["exact \u27e8<a>comp</a> cf cg, <a>rfl</a>\u27e9", [{"full_name": "Nat.Partrec.Code.comp", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [82, 5], "def_end_pos": [82, 9]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "case intro.intro\nf : \u2115 \u2192. \u2115\ncf : Code\npf : Partrec (eval cf)\ncg : Code\npg : Partrec (eval cg)\n\u22a2 \u2203 c, eval c = fun n => eval cg n >>= eval cf", "state_after": "no goals"}, {"tactic": "rcases hf with \u27e8cf, rfl\u27e9", "annotated_tactic": ["rcases hf with \u27e8cf, rfl\u27e9", []], "state_before": "f\u271d f g : \u2115 \u2192. \u2115\npf : Partrec f\npg : Partrec g\nhf : \u2203 c, eval c = f\nhg : \u2203 c, eval c = g\n\u22a2 \u2203 c,\n    eval c =\n      unpaired fun a n =>\n        Nat.rec (f a)\n          (fun y IH => do\n            let i \u2190 IH\n            g (Nat.pair a (Nat.pair y i)))\n          n", "state_after": "case intro\nf g : \u2115 \u2192. \u2115\npg : Partrec g\nhg : \u2203 c, eval c = g\ncf : Code\npf : Partrec (eval cf)\n\u22a2 \u2203 c,\n    eval c =\n      unpaired fun a n =>\n        Nat.rec (eval cf a)\n          (fun y IH => do\n            let i \u2190 IH\n            g (Nat.pair a (Nat.pair y i)))\n          n"}, {"tactic": "rcases hg with \u27e8cg, rfl\u27e9", "annotated_tactic": ["rcases hg with \u27e8cg, rfl\u27e9", []], "state_before": "case intro\nf g : \u2115 \u2192. \u2115\npg : Partrec g\nhg : \u2203 c, eval c = g\ncf : Code\npf : Partrec (eval cf)\n\u22a2 \u2203 c,\n    eval c =\n      unpaired fun a n =>\n        Nat.rec (eval cf a)\n          (fun y IH => do\n            let i \u2190 IH\n            g (Nat.pair a (Nat.pair y i)))\n          n", "state_after": "case intro.intro\nf : \u2115 \u2192. \u2115\ncf : Code\npf : Partrec (eval cf)\ncg : Code\npg : Partrec (eval cg)\n\u22a2 \u2203 c,\n    eval c =\n      unpaired fun a n =>\n        Nat.rec (eval cf a)\n          (fun y IH => do\n            let i \u2190 IH\n            eval cg (Nat.pair a (Nat.pair y i)))\n          n"}, {"tactic": "exact \u27e8prec cf cg, rfl\u27e9", "annotated_tactic": ["exact \u27e8<a>prec</a> cf cg, <a>rfl</a>\u27e9", [{"full_name": "Nat.Partrec.Code.prec", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [83, 5], "def_end_pos": [83, 9]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "case intro.intro\nf : \u2115 \u2192. \u2115\ncf : Code\npf : Partrec (eval cf)\ncg : Code\npg : Partrec (eval cg)\n\u22a2 \u2203 c,\n    eval c =\n      unpaired fun a n =>\n        Nat.rec (eval cf a)\n          (fun y IH => do\n            let i \u2190 IH\n            eval cg (Nat.pair a (Nat.pair y i)))\n          n", "state_after": "no goals"}, {"tactic": "rcases hf with \u27e8cf, rfl\u27e9", "annotated_tactic": ["rcases hf with \u27e8cf, rfl\u27e9", []], "state_before": "f\u271d f : \u2115 \u2192. \u2115\npf : Partrec f\nhf : \u2203 c, eval c = f\n\u22a2 \u2203 c, eval c = fun a => Nat.rfind fun n => (fun m => decide (m = 0)) <$> f (Nat.pair a n)", "state_after": "case intro\nf : \u2115 \u2192. \u2115\ncf : Code\npf : Partrec (eval cf)\n\u22a2 \u2203 c, eval c = fun a => Nat.rfind fun n => (fun m => decide (m = 0)) <$> eval cf (Nat.pair a n)"}, {"tactic": "refine' \u27e8comp (rfind' cf) (pair Code.id zero), _\u27e9", "annotated_tactic": ["refine' \u27e8<a>comp</a> (<a>rfind'</a> cf) (<a>pair</a> <a>Code.id</a> <a>zero</a>), _\u27e9", [{"full_name": "Nat.Partrec.Code.comp", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [82, 5], "def_end_pos": [82, 9]}, {"full_name": "Nat.Partrec.Code.rfind'", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [84, 5], "def_end_pos": [84, 11]}, {"full_name": "Nat.Partrec.Code.pair", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [81, 5], "def_end_pos": [81, 9]}, {"full_name": "Nat.Partrec.Code.id", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [117, 15], "def_end_pos": [117, 17]}, {"full_name": "Nat.Partrec.Code.zero", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [77, 5], "def_end_pos": [77, 9]}]], "state_before": "case intro\nf : \u2115 \u2192. \u2115\ncf : Code\npf : Partrec (eval cf)\n\u22a2 \u2203 c, eval c = fun a => Nat.rfind fun n => (fun m => decide (m = 0)) <$> eval cf (Nat.pair a n)", "state_after": "case intro\nf : \u2115 \u2192. \u2115\ncf : Code\npf : Partrec (eval cf)\n\u22a2 eval (comp (rfind' cf) (pair Code.id zero)) = fun a =>\n    Nat.rfind fun n => (fun m => decide (m = 0)) <$> eval cf (Nat.pair a n)"}, {"tactic": "simp [eval, Seq.seq, pure, PFun.pure, Part.map_id']", "annotated_tactic": ["simp [<a>eval</a>, <a>Seq.seq</a>, <a>pure</a>, <a>PFun.pure</a>, <a>Part.map_id'</a>]", [{"full_name": "Nat.Partrec.Code.eval", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [620, 5], "def_end_pos": [620, 9]}, {"full_name": "Seq.seq", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2698, 3], "def_end_pos": [2698, 6]}, {"full_name": "Pure.pure", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2670, 3], "def_end_pos": [2670, 7]}, {"full_name": "PFun.pure", "def_path": "Mathlib/Data/PFun.lean", "def_pos": [206, 15], "def_end_pos": [206, 19]}, {"full_name": "Part.map_id'", "def_path": "Mathlib/Data/Part.lean", "def_pos": [589, 9], "def_end_pos": [589, 16]}]], "state_before": "case intro\nf : \u2115 \u2192. \u2115\ncf : Code\npf : Partrec (eval cf)\n\u22a2 eval (comp (rfind' cf) (pair Code.id zero)) = fun a =>\n    Nat.rfind fun n => (fun m => decide (m = 0)) <$> eval cf (Nat.pair a n)", "state_after": "no goals"}, {"tactic": "rcases h with \u27e8c, rfl\u27e9", "annotated_tactic": ["rcases h with \u27e8c, rfl\u27e9", []], "state_before": "f : \u2115 \u2192. \u2115\nh : \u2203 c, eval c = f\n\u22a2 Partrec f", "state_after": "case intro\nc : Code\n\u22a2 Partrec (eval c)"}, {"tactic": "induction c", "annotated_tactic": ["induction c", []], "state_before": "case intro\nc : Code\n\u22a2 Partrec (eval c)", "state_after": "case intro.zero\n\n\u22a2 Partrec (eval zero)\n\ncase intro.succ\n\n\u22a2 Partrec (eval succ)\n\ncase intro.left\n\n\u22a2 Partrec (eval left)\n\ncase intro.right\n\n\u22a2 Partrec (eval right)\n\ncase intro.pair\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 : Partrec (eval a\u271d\u00b9)\na_ih\u271d : Partrec (eval a\u271d)\n\u22a2 Partrec (eval (pair a\u271d\u00b9 a\u271d))\n\ncase intro.comp\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 : Partrec (eval a\u271d\u00b9)\na_ih\u271d : Partrec (eval a\u271d)\n\u22a2 Partrec (eval (comp a\u271d\u00b9 a\u271d))\n\ncase intro.prec\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 : Partrec (eval a\u271d\u00b9)\na_ih\u271d : Partrec (eval a\u271d)\n\u22a2 Partrec (eval (prec a\u271d\u00b9 a\u271d))\n\ncase intro.rfind'\na\u271d : Code\na_ih\u271d : Partrec (eval a\u271d)\n\u22a2 Partrec (eval (rfind' a\u271d))"}, {"tactic": "case zero => exact Nat.Partrec.zero", "annotated_tactic": ["case zero => exact <a>Nat.Partrec.zero</a>", [{"full_name": "Nat.Partrec.zero", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [156, 5], "def_end_pos": [156, 9]}]], "state_before": "case intro.zero\n\n\u22a2 Partrec (eval zero)\n\ncase intro.succ\n\n\u22a2 Partrec (eval succ)\n\ncase intro.left\n\n\u22a2 Partrec (eval left)\n\ncase intro.right\n\n\u22a2 Partrec (eval right)\n\ncase intro.pair\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 : Partrec (eval a\u271d\u00b9)\na_ih\u271d : Partrec (eval a\u271d)\n\u22a2 Partrec (eval (pair a\u271d\u00b9 a\u271d))\n\ncase intro.comp\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 : Partrec (eval a\u271d\u00b9)\na_ih\u271d : Partrec (eval a\u271d)\n\u22a2 Partrec (eval (comp a\u271d\u00b9 a\u271d))\n\ncase intro.prec\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 : Partrec (eval a\u271d\u00b9)\na_ih\u271d : Partrec (eval a\u271d)\n\u22a2 Partrec (eval (prec a\u271d\u00b9 a\u271d))\n\ncase intro.rfind'\na\u271d : Code\na_ih\u271d : Partrec (eval a\u271d)\n\u22a2 Partrec (eval (rfind' a\u271d))", "state_after": "case intro.succ\n\n\u22a2 Partrec (eval succ)\n\ncase intro.left\n\n\u22a2 Partrec (eval left)\n\ncase intro.right\n\n\u22a2 Partrec (eval right)\n\ncase intro.pair\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 : Partrec (eval a\u271d\u00b9)\na_ih\u271d : Partrec (eval a\u271d)\n\u22a2 Partrec (eval (pair a\u271d\u00b9 a\u271d))\n\ncase intro.comp\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 : Partrec (eval a\u271d\u00b9)\na_ih\u271d : Partrec (eval a\u271d)\n\u22a2 Partrec (eval (comp a\u271d\u00b9 a\u271d))\n\ncase intro.prec\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 : Partrec (eval a\u271d\u00b9)\na_ih\u271d : Partrec (eval a\u271d)\n\u22a2 Partrec (eval (prec a\u271d\u00b9 a\u271d))\n\ncase intro.rfind'\na\u271d : Code\na_ih\u271d : Partrec (eval a\u271d)\n\u22a2 Partrec (eval (rfind' a\u271d))"}, {"tactic": "case succ => exact Nat.Partrec.succ", "annotated_tactic": ["case succ => exact <a>Nat.Partrec.succ</a>", [{"full_name": "Nat.Partrec.succ", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [157, 5], "def_end_pos": [157, 9]}]], "state_before": "case intro.succ\n\n\u22a2 Partrec (eval succ)\n\ncase intro.left\n\n\u22a2 Partrec (eval left)\n\ncase intro.right\n\n\u22a2 Partrec (eval right)\n\ncase intro.pair\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 : Partrec (eval a\u271d\u00b9)\na_ih\u271d : Partrec (eval a\u271d)\n\u22a2 Partrec (eval (pair a\u271d\u00b9 a\u271d))\n\ncase intro.comp\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 : Partrec (eval a\u271d\u00b9)\na_ih\u271d : Partrec (eval a\u271d)\n\u22a2 Partrec (eval (comp a\u271d\u00b9 a\u271d))\n\ncase intro.prec\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 : Partrec (eval a\u271d\u00b9)\na_ih\u271d : Partrec (eval a\u271d)\n\u22a2 Partrec (eval (prec a\u271d\u00b9 a\u271d))\n\ncase intro.rfind'\na\u271d : Code\na_ih\u271d : Partrec (eval a\u271d)\n\u22a2 Partrec (eval (rfind' a\u271d))", "state_after": "case intro.left\n\n\u22a2 Partrec (eval left)\n\ncase intro.right\n\n\u22a2 Partrec (eval right)\n\ncase intro.pair\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 : Partrec (eval a\u271d\u00b9)\na_ih\u271d : Partrec (eval a\u271d)\n\u22a2 Partrec (eval (pair a\u271d\u00b9 a\u271d))\n\ncase intro.comp\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 : Partrec (eval a\u271d\u00b9)\na_ih\u271d : Partrec (eval a\u271d)\n\u22a2 Partrec (eval (comp a\u271d\u00b9 a\u271d))\n\ncase intro.prec\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 : Partrec (eval a\u271d\u00b9)\na_ih\u271d : Partrec (eval a\u271d)\n\u22a2 Partrec (eval (prec a\u271d\u00b9 a\u271d))\n\ncase intro.rfind'\na\u271d : Code\na_ih\u271d : Partrec (eval a\u271d)\n\u22a2 Partrec (eval (rfind' a\u271d))"}, {"tactic": "case left => exact Nat.Partrec.left", "annotated_tactic": ["case left => exact <a>Nat.Partrec.left</a>", [{"full_name": "Nat.Partrec.left", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [158, 5], "def_end_pos": [158, 9]}]], "state_before": "case intro.left\n\n\u22a2 Partrec (eval left)\n\ncase intro.right\n\n\u22a2 Partrec (eval right)\n\ncase intro.pair\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 : Partrec (eval a\u271d\u00b9)\na_ih\u271d : Partrec (eval a\u271d)\n\u22a2 Partrec (eval (pair a\u271d\u00b9 a\u271d))\n\ncase intro.comp\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 : Partrec (eval a\u271d\u00b9)\na_ih\u271d : Partrec (eval a\u271d)\n\u22a2 Partrec (eval (comp a\u271d\u00b9 a\u271d))\n\ncase intro.prec\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 : Partrec (eval a\u271d\u00b9)\na_ih\u271d : Partrec (eval a\u271d)\n\u22a2 Partrec (eval (prec a\u271d\u00b9 a\u271d))\n\ncase intro.rfind'\na\u271d : Code\na_ih\u271d : Partrec (eval a\u271d)\n\u22a2 Partrec (eval (rfind' a\u271d))", "state_after": "case intro.right\n\n\u22a2 Partrec (eval right)\n\ncase intro.pair\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 : Partrec (eval a\u271d\u00b9)\na_ih\u271d : Partrec (eval a\u271d)\n\u22a2 Partrec (eval (pair a\u271d\u00b9 a\u271d))\n\ncase intro.comp\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 : Partrec (eval a\u271d\u00b9)\na_ih\u271d : Partrec (eval a\u271d)\n\u22a2 Partrec (eval (comp a\u271d\u00b9 a\u271d))\n\ncase intro.prec\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 : Partrec (eval a\u271d\u00b9)\na_ih\u271d : Partrec (eval a\u271d)\n\u22a2 Partrec (eval (prec a\u271d\u00b9 a\u271d))\n\ncase intro.rfind'\na\u271d : Code\na_ih\u271d : Partrec (eval a\u271d)\n\u22a2 Partrec (eval (rfind' a\u271d))"}, {"tactic": "case right => exact Nat.Partrec.right", "annotated_tactic": ["case right => exact <a>Nat.Partrec.right</a>", [{"full_name": "Nat.Partrec.right", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [159, 5], "def_end_pos": [159, 10]}]], "state_before": "case intro.right\n\n\u22a2 Partrec (eval right)\n\ncase intro.pair\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 : Partrec (eval a\u271d\u00b9)\na_ih\u271d : Partrec (eval a\u271d)\n\u22a2 Partrec (eval (pair a\u271d\u00b9 a\u271d))\n\ncase intro.comp\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 : Partrec (eval a\u271d\u00b9)\na_ih\u271d : Partrec (eval a\u271d)\n\u22a2 Partrec (eval (comp a\u271d\u00b9 a\u271d))\n\ncase intro.prec\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 : Partrec (eval a\u271d\u00b9)\na_ih\u271d : Partrec (eval a\u271d)\n\u22a2 Partrec (eval (prec a\u271d\u00b9 a\u271d))\n\ncase intro.rfind'\na\u271d : Code\na_ih\u271d : Partrec (eval a\u271d)\n\u22a2 Partrec (eval (rfind' a\u271d))", "state_after": "case intro.pair\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 : Partrec (eval a\u271d\u00b9)\na_ih\u271d : Partrec (eval a\u271d)\n\u22a2 Partrec (eval (pair a\u271d\u00b9 a\u271d))\n\ncase intro.comp\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 : Partrec (eval a\u271d\u00b9)\na_ih\u271d : Partrec (eval a\u271d)\n\u22a2 Partrec (eval (comp a\u271d\u00b9 a\u271d))\n\ncase intro.prec\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 : Partrec (eval a\u271d\u00b9)\na_ih\u271d : Partrec (eval a\u271d)\n\u22a2 Partrec (eval (prec a\u271d\u00b9 a\u271d))\n\ncase intro.rfind'\na\u271d : Code\na_ih\u271d : Partrec (eval a\u271d)\n\u22a2 Partrec (eval (rfind' a\u271d))"}, {"tactic": "case pair cf cg pf pg => exact pf.pair pg", "annotated_tactic": ["case pair cf cg pf pg => exact pf.pair pg", []], "state_before": "case intro.pair\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 : Partrec (eval a\u271d\u00b9)\na_ih\u271d : Partrec (eval a\u271d)\n\u22a2 Partrec (eval (pair a\u271d\u00b9 a\u271d))\n\ncase intro.comp\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 : Partrec (eval a\u271d\u00b9)\na_ih\u271d : Partrec (eval a\u271d)\n\u22a2 Partrec (eval (comp a\u271d\u00b9 a\u271d))\n\ncase intro.prec\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 : Partrec (eval a\u271d\u00b9)\na_ih\u271d : Partrec (eval a\u271d)\n\u22a2 Partrec (eval (prec a\u271d\u00b9 a\u271d))\n\ncase intro.rfind'\na\u271d : Code\na_ih\u271d : Partrec (eval a\u271d)\n\u22a2 Partrec (eval (rfind' a\u271d))", "state_after": "case intro.comp\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 : Partrec (eval a\u271d\u00b9)\na_ih\u271d : Partrec (eval a\u271d)\n\u22a2 Partrec (eval (comp a\u271d\u00b9 a\u271d))\n\ncase intro.prec\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 : Partrec (eval a\u271d\u00b9)\na_ih\u271d : Partrec (eval a\u271d)\n\u22a2 Partrec (eval (prec a\u271d\u00b9 a\u271d))\n\ncase intro.rfind'\na\u271d : Code\na_ih\u271d : Partrec (eval a\u271d)\n\u22a2 Partrec (eval (rfind' a\u271d))"}, {"tactic": "case comp cf cg pf pg => exact pf.comp pg", "annotated_tactic": ["case comp cf cg pf pg => exact pf.comp pg", []], "state_before": "case intro.comp\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 : Partrec (eval a\u271d\u00b9)\na_ih\u271d : Partrec (eval a\u271d)\n\u22a2 Partrec (eval (comp a\u271d\u00b9 a\u271d))\n\ncase intro.prec\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 : Partrec (eval a\u271d\u00b9)\na_ih\u271d : Partrec (eval a\u271d)\n\u22a2 Partrec (eval (prec a\u271d\u00b9 a\u271d))\n\ncase intro.rfind'\na\u271d : Code\na_ih\u271d : Partrec (eval a\u271d)\n\u22a2 Partrec (eval (rfind' a\u271d))", "state_after": "case intro.prec\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 : Partrec (eval a\u271d\u00b9)\na_ih\u271d : Partrec (eval a\u271d)\n\u22a2 Partrec (eval (prec a\u271d\u00b9 a\u271d))\n\ncase intro.rfind'\na\u271d : Code\na_ih\u271d : Partrec (eval a\u271d)\n\u22a2 Partrec (eval (rfind' a\u271d))"}, {"tactic": "case prec cf cg pf pg => exact pf.prec pg", "annotated_tactic": ["case prec cf cg pf pg => exact pf.prec pg", []], "state_before": "case intro.prec\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 : Partrec (eval a\u271d\u00b9)\na_ih\u271d : Partrec (eval a\u271d)\n\u22a2 Partrec (eval (prec a\u271d\u00b9 a\u271d))\n\ncase intro.rfind'\na\u271d : Code\na_ih\u271d : Partrec (eval a\u271d)\n\u22a2 Partrec (eval (rfind' a\u271d))", "state_after": "case intro.rfind'\na\u271d : Code\na_ih\u271d : Partrec (eval a\u271d)\n\u22a2 Partrec (eval (rfind' a\u271d))"}, {"tactic": "case rfind' cf pf => exact pf.rfind'", "annotated_tactic": ["case rfind' cf pf => exact pf.rfind'", []], "state_before": "case intro.rfind'\na\u271d : Code\na_ih\u271d : Partrec (eval a\u271d)\n\u22a2 Partrec (eval (rfind' a\u271d))", "state_after": "no goals"}, {"tactic": "exact Nat.Partrec.zero", "annotated_tactic": ["exact <a>Nat.Partrec.zero</a>", [{"full_name": "Nat.Partrec.zero", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [156, 5], "def_end_pos": [156, 9]}]], "state_before": "\u22a2 Partrec (eval zero)", "state_after": "no goals"}, {"tactic": "exact Nat.Partrec.succ", "annotated_tactic": ["exact <a>Nat.Partrec.succ</a>", [{"full_name": "Nat.Partrec.succ", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [157, 5], "def_end_pos": [157, 9]}]], "state_before": "\u22a2 Partrec (eval succ)", "state_after": "no goals"}, {"tactic": "exact Nat.Partrec.left", "annotated_tactic": ["exact <a>Nat.Partrec.left</a>", [{"full_name": "Nat.Partrec.left", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [158, 5], "def_end_pos": [158, 9]}]], "state_before": "\u22a2 Partrec (eval left)", "state_after": "no goals"}, {"tactic": "exact Nat.Partrec.right", "annotated_tactic": ["exact <a>Nat.Partrec.right</a>", [{"full_name": "Nat.Partrec.right", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [159, 5], "def_end_pos": [159, 10]}]], "state_before": "\u22a2 Partrec (eval right)", "state_after": "no goals"}, {"tactic": "exact pf.pair pg", "annotated_tactic": ["exact pf.pair pg", []], "state_before": "cf cg : Code\npf : Partrec (eval cf)\npg : Partrec (eval cg)\n\u22a2 Partrec (eval (pair cf cg))", "state_after": "no goals"}, {"tactic": "exact pf.comp pg", "annotated_tactic": ["exact pf.comp pg", []], "state_before": "cf cg : Code\npf : Partrec (eval cf)\npg : Partrec (eval cg)\n\u22a2 Partrec (eval (comp cf cg))", "state_after": "no goals"}, {"tactic": "exact pf.prec pg", "annotated_tactic": ["exact pf.prec pg", []], "state_before": "cf cg : Code\npf : Partrec (eval cf)\npg : Partrec (eval cg)\n\u22a2 Partrec (eval (prec cf cg))", "state_after": "no goals"}, {"tactic": "exact pf.rfind'", "annotated_tactic": ["exact pf.rfind'", []], "state_before": "cf : Code\npf : Partrec (eval cf)\n\u22a2 Partrec (eval (rfind' cf))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/LocallyFinite.lean", "full_name": "Finset.Icc_eq_cons_Ioc", "start": [634, 1], "end": [635, 51], "traced_tactics": [{"tactic": "classical rw [cons_eq_insert, Ioc_insert_left h]", "annotated_tactic": ["classical rw [<a>cons_eq_insert</a>, <a>Ioc_insert_left</a> h]", [{"full_name": "Finset.cons_eq_insert", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1108, 9], "def_end_pos": [1108, 23]}, {"full_name": "Finset.Ioc_insert_left", "def_path": "Mathlib/Data/Finset/LocallyFinite.lean", "def_pos": [585, 9], "def_end_pos": [585, 24]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na b c : \u03b1\nh : a \u2264 b\n\u22a2 Icc a b = cons a (Ioc a b) (_ : \u00aca \u2208 Ioc a b)", "state_after": "no goals"}, {"tactic": "rw [cons_eq_insert, Ioc_insert_left h]", "annotated_tactic": ["rw [<a>cons_eq_insert</a>, <a>Ioc_insert_left</a> h]", [{"full_name": "Finset.cons_eq_insert", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1108, 9], "def_end_pos": [1108, 23]}, {"full_name": "Finset.Ioc_insert_left", "def_path": "Mathlib/Data/Finset/LocallyFinite.lean", "def_pos": [585, 9], "def_end_pos": [585, 24]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na b c : \u03b1\nh : a \u2264 b\n\u22a2 Icc a b = cons a (Ioc a b) (_ : \u00aca \u2208 Ioc a b)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/TuringMachine.lean", "full_name": "Turing.TM0.map_init", "start": [1169, 1], "end": [1170, 56], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "full_name": "MeasurableSet.measurableSet_liminf", "start": [2138, 1], "end": [2140, 74], "traced_tactics": [{"tactic": "simpa only [\u2190 bliminf_true] using measurableSet_bliminf fun n _ => hs n", "annotated_tactic": ["simpa only [\u2190 <a>bliminf_true</a>] using <a>measurableSet_bliminf</a> fun n _ => hs n", [{"full_name": "Filter.bliminf_true", "def_path": "Mathlib/Order/LiminfLimsup.lean", "def_pos": [470, 9], "def_end_pos": [470, 21]}, {"full_name": "MeasurableSet.measurableSet_bliminf", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [2125, 9], "def_end_pos": [2125, 30]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u\u03b9\ns\u271d t u : Set \u03b1\ninst\u271d : MeasurableSpace \u03b1\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (n : \u2115), MeasurableSet (s n)\n\u22a2 MeasurableSet (liminf s atTop)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/List/Init/Lemmas.lean", "full_name": "List.isPrefixOf_cons\u2082_self", "start": [381, 9], "end": [382, 80], "traced_tactics": [{"tactic": "simp [isPrefixOf_cons\u2082]", "annotated_tactic": ["simp [<a>isPrefixOf_cons\u2082</a>]", [{"full_name": "List.isPrefixOf_cons\u2082", "def_path": "lake-packages/std/Std/Data/List/Init/Lemmas.lean", "def_pos": [379, 9], "def_end_pos": [379, 25]}]], "state_before": "\u03b1 : Type u_1\nas bs : List \u03b1\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : LawfulBEq \u03b1\na : \u03b1\n\u22a2 isPrefixOf (a :: as) (a :: bs) = isPrefixOf as bs", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Sigma.lean", "full_name": "Set.sigma_insert", "start": [151, 1], "end": [153, 52], "traced_tactics": [{"tactic": "simp_rw [insert_eq, sigma_union, sigma_singleton]", "annotated_tactic": ["simp_rw [<a>insert_eq</a>, <a>sigma_union</a>, <a>sigma_singleton</a>]", [{"full_name": "Set.insert_eq", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1310, 9], "def_end_pos": [1310, 18]}, {"full_name": "Set.sigma_union", "def_path": "Mathlib/Data/Set/Sigma.lean", "def_pos": [137, 9], "def_end_pos": [137, 20]}, {"full_name": "Set.sigma_singleton", "def_path": "Mathlib/Data/Set/Sigma.lean", "def_pos": [120, 9], "def_end_pos": [120, 24]}]], "state_before": "\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : \u03b9 \u2192 Type u_3\n\u03b2 : \u03b9 \u2192 Type u_4\ns s\u2081 s\u2082 : Set \u03b9\nt t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\nu : Set ((i : \u03b9) \u00d7 \u03b1 i)\nx : (i : \u03b9) \u00d7 \u03b1 i\ni j : \u03b9\na\u271d : \u03b1 i\na : (i : \u03b9) \u2192 \u03b1 i\n\u22a2 (Set.Sigma s fun i => insert (a i) (t i)) = (fun i => { fst := i, snd := a i }) '' s \u222a Set.Sigma s t", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/LocallyIntegrable.lean", "full_name": "Monotone.locallyIntegrable", "start": [484, 1], "end": [494, 61], "traced_tactics": [{"tactic": "intro x", "annotated_tactic": ["intro x", []], "state_before": "X : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u00b9\u00b9 : MeasurableSpace X\ninst\u271d\u00b9\u2070 : TopologicalSpace X\ninst\u271d\u2079 : MeasurableSpace Y\ninst\u271d\u2078 : TopologicalSpace Y\ninst\u271d\u2077 : NormedAddCommGroup E\nf g : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u2076 : BorelSpace X\ninst\u271d\u2075 : ConditionallyCompleteLinearOrder X\ninst\u271d\u2074 : ConditionallyCompleteLinearOrder E\ninst\u271d\u00b3 : OrderTopology X\ninst\u271d\u00b2 : OrderTopology E\ninst\u271d\u00b9 : SecondCountableTopology E\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhmono : Monotone f\n\u22a2 LocallyIntegrable f", "state_after": "X : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u00b9\u00b9 : MeasurableSpace X\ninst\u271d\u00b9\u2070 : TopologicalSpace X\ninst\u271d\u2079 : MeasurableSpace Y\ninst\u271d\u2078 : TopologicalSpace Y\ninst\u271d\u2077 : NormedAddCommGroup E\nf g : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u2076 : BorelSpace X\ninst\u271d\u2075 : ConditionallyCompleteLinearOrder X\ninst\u271d\u2074 : ConditionallyCompleteLinearOrder E\ninst\u271d\u00b3 : OrderTopology X\ninst\u271d\u00b2 : OrderTopology E\ninst\u271d\u00b9 : SecondCountableTopology E\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhmono : Monotone f\nx : X\n\u22a2 IntegrableAtFilter f (\ud835\udcdd x)"}, {"tactic": "rcases \u03bc.finiteAt_nhds x with \u27e8U, hU, h'U\u27e9", "annotated_tactic": ["rcases \u03bc.finiteAt_nhds x with \u27e8U, hU, h'U\u27e9", []], "state_before": "X : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u00b9\u00b9 : MeasurableSpace X\ninst\u271d\u00b9\u2070 : TopologicalSpace X\ninst\u271d\u2079 : MeasurableSpace Y\ninst\u271d\u2078 : TopologicalSpace Y\ninst\u271d\u2077 : NormedAddCommGroup E\nf g : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u2076 : BorelSpace X\ninst\u271d\u2075 : ConditionallyCompleteLinearOrder X\ninst\u271d\u2074 : ConditionallyCompleteLinearOrder E\ninst\u271d\u00b3 : OrderTopology X\ninst\u271d\u00b2 : OrderTopology E\ninst\u271d\u00b9 : SecondCountableTopology E\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhmono : Monotone f\nx : X\n\u22a2 IntegrableAtFilter f (\ud835\udcdd x)", "state_after": "case intro.intro\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u00b9\u00b9 : MeasurableSpace X\ninst\u271d\u00b9\u2070 : TopologicalSpace X\ninst\u271d\u2079 : MeasurableSpace Y\ninst\u271d\u2078 : TopologicalSpace Y\ninst\u271d\u2077 : NormedAddCommGroup E\nf g : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u2076 : BorelSpace X\ninst\u271d\u2075 : ConditionallyCompleteLinearOrder X\ninst\u271d\u2074 : ConditionallyCompleteLinearOrder E\ninst\u271d\u00b3 : OrderTopology X\ninst\u271d\u00b2 : OrderTopology E\ninst\u271d\u00b9 : SecondCountableTopology E\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhmono : Monotone f\nx : X\nU : Set X\nhU : U \u2208 \ud835\udcdd x\nh'U : \u2191\u2191\u03bc U < \u22a4\n\u22a2 IntegrableAtFilter f (\ud835\udcdd x)"}, {"tactic": "obtain \u27e8a, b, xab, hab, abU\u27e9 : \u2203 a b : X, x \u2208 Icc a b \u2227 Icc a b \u2208 \ud835\udcdd x \u2227 Icc a b \u2286 U", "annotated_tactic": ["obtain \u27e8a, b, xab, hab, abU\u27e9 : \u2203 a b : X, x \u2208 <a>Icc</a> a b \u2227 <a>Icc</a> a b \u2208 \ud835\udcdd x \u2227 <a>Icc</a> a b \u2286 U", [{"full_name": "Set.Icc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [59, 5], "def_end_pos": [59, 8]}, {"full_name": "Set.Icc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [59, 5], "def_end_pos": [59, 8]}, {"full_name": "Set.Icc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [59, 5], "def_end_pos": [59, 8]}]], "state_before": "case intro.intro\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u00b9\u00b9 : MeasurableSpace X\ninst\u271d\u00b9\u2070 : TopologicalSpace X\ninst\u271d\u2079 : MeasurableSpace Y\ninst\u271d\u2078 : TopologicalSpace Y\ninst\u271d\u2077 : NormedAddCommGroup E\nf g : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u2076 : BorelSpace X\ninst\u271d\u2075 : ConditionallyCompleteLinearOrder X\ninst\u271d\u2074 : ConditionallyCompleteLinearOrder E\ninst\u271d\u00b3 : OrderTopology X\ninst\u271d\u00b2 : OrderTopology E\ninst\u271d\u00b9 : SecondCountableTopology E\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhmono : Monotone f\nx : X\nU : Set X\nhU : U \u2208 \ud835\udcdd x\nh'U : \u2191\u2191\u03bc U < \u22a4\n\u22a2 IntegrableAtFilter f (\ud835\udcdd x)", "state_after": "X : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u00b9\u00b9 : MeasurableSpace X\ninst\u271d\u00b9\u2070 : TopologicalSpace X\ninst\u271d\u2079 : MeasurableSpace Y\ninst\u271d\u2078 : TopologicalSpace Y\ninst\u271d\u2077 : NormedAddCommGroup E\nf g : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u2076 : BorelSpace X\ninst\u271d\u2075 : ConditionallyCompleteLinearOrder X\ninst\u271d\u2074 : ConditionallyCompleteLinearOrder E\ninst\u271d\u00b3 : OrderTopology X\ninst\u271d\u00b2 : OrderTopology E\ninst\u271d\u00b9 : SecondCountableTopology E\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhmono : Monotone f\nx : X\nU : Set X\nhU : U \u2208 \ud835\udcdd x\nh'U : \u2191\u2191\u03bc U < \u22a4\n\u22a2 \u2203 a b, x \u2208 Icc a b \u2227 Icc a b \u2208 \ud835\udcdd x \u2227 Icc a b \u2286 U\n\ncase intro.intro.intro.intro.intro.intro\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u00b9\u00b9 : MeasurableSpace X\ninst\u271d\u00b9\u2070 : TopologicalSpace X\ninst\u271d\u2079 : MeasurableSpace Y\ninst\u271d\u2078 : TopologicalSpace Y\ninst\u271d\u2077 : NormedAddCommGroup E\nf g : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u2076 : BorelSpace X\ninst\u271d\u2075 : ConditionallyCompleteLinearOrder X\ninst\u271d\u2074 : ConditionallyCompleteLinearOrder E\ninst\u271d\u00b3 : OrderTopology X\ninst\u271d\u00b2 : OrderTopology E\ninst\u271d\u00b9 : SecondCountableTopology E\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhmono : Monotone f\nx : X\nU : Set X\nhU : U \u2208 \ud835\udcdd x\nh'U : \u2191\u2191\u03bc U < \u22a4\na b : X\nxab : x \u2208 Icc a b\nhab : Icc a b \u2208 \ud835\udcdd x\nabU : Icc a b \u2286 U\n\u22a2 IntegrableAtFilter f (\ud835\udcdd x)"}, {"tactic": "exact exists_Icc_mem_subset_of_mem_nhds hU", "annotated_tactic": ["exact <a>exists_Icc_mem_subset_of_mem_nhds</a> hU", [{"full_name": "exists_Icc_mem_subset_of_mem_nhds", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [1253, 9], "def_end_pos": [1253, 42]}]], "state_before": "X : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u00b9\u00b9 : MeasurableSpace X\ninst\u271d\u00b9\u2070 : TopologicalSpace X\ninst\u271d\u2079 : MeasurableSpace Y\ninst\u271d\u2078 : TopologicalSpace Y\ninst\u271d\u2077 : NormedAddCommGroup E\nf g : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u2076 : BorelSpace X\ninst\u271d\u2075 : ConditionallyCompleteLinearOrder X\ninst\u271d\u2074 : ConditionallyCompleteLinearOrder E\ninst\u271d\u00b3 : OrderTopology X\ninst\u271d\u00b2 : OrderTopology E\ninst\u271d\u00b9 : SecondCountableTopology E\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhmono : Monotone f\nx : X\nU : Set X\nhU : U \u2208 \ud835\udcdd x\nh'U : \u2191\u2191\u03bc U < \u22a4\n\u22a2 \u2203 a b, x \u2208 Icc a b \u2227 Icc a b \u2208 \ud835\udcdd x \u2227 Icc a b \u2286 U\n\ncase intro.intro.intro.intro.intro.intro\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u00b9\u00b9 : MeasurableSpace X\ninst\u271d\u00b9\u2070 : TopologicalSpace X\ninst\u271d\u2079 : MeasurableSpace Y\ninst\u271d\u2078 : TopologicalSpace Y\ninst\u271d\u2077 : NormedAddCommGroup E\nf g : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u2076 : BorelSpace X\ninst\u271d\u2075 : ConditionallyCompleteLinearOrder X\ninst\u271d\u2074 : ConditionallyCompleteLinearOrder E\ninst\u271d\u00b3 : OrderTopology X\ninst\u271d\u00b2 : OrderTopology E\ninst\u271d\u00b9 : SecondCountableTopology E\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhmono : Monotone f\nx : X\nU : Set X\nhU : U \u2208 \ud835\udcdd x\nh'U : \u2191\u2191\u03bc U < \u22a4\na b : X\nxab : x \u2208 Icc a b\nhab : Icc a b \u2208 \ud835\udcdd x\nabU : Icc a b \u2286 U\n\u22a2 IntegrableAtFilter f (\ud835\udcdd x)", "state_after": "case intro.intro.intro.intro.intro.intro\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u00b9\u00b9 : MeasurableSpace X\ninst\u271d\u00b9\u2070 : TopologicalSpace X\ninst\u271d\u2079 : MeasurableSpace Y\ninst\u271d\u2078 : TopologicalSpace Y\ninst\u271d\u2077 : NormedAddCommGroup E\nf g : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u2076 : BorelSpace X\ninst\u271d\u2075 : ConditionallyCompleteLinearOrder X\ninst\u271d\u2074 : ConditionallyCompleteLinearOrder E\ninst\u271d\u00b3 : OrderTopology X\ninst\u271d\u00b2 : OrderTopology E\ninst\u271d\u00b9 : SecondCountableTopology E\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhmono : Monotone f\nx : X\nU : Set X\nhU : U \u2208 \ud835\udcdd x\nh'U : \u2191\u2191\u03bc U < \u22a4\na b : X\nxab : x \u2208 Icc a b\nhab : Icc a b \u2208 \ud835\udcdd x\nabU : Icc a b \u2286 U\n\u22a2 IntegrableAtFilter f (\ud835\udcdd x)"}, {"tactic": "have ab : a \u2264 b := xab.1.trans xab.2", "annotated_tactic": ["have ab : a \u2264 b := xab.1.<a>trans</a> xab.2", [{"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [120, 7], "def_end_pos": [120, 18]}]], "state_before": "case intro.intro.intro.intro.intro.intro\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u00b9\u00b9 : MeasurableSpace X\ninst\u271d\u00b9\u2070 : TopologicalSpace X\ninst\u271d\u2079 : MeasurableSpace Y\ninst\u271d\u2078 : TopologicalSpace Y\ninst\u271d\u2077 : NormedAddCommGroup E\nf g : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u2076 : BorelSpace X\ninst\u271d\u2075 : ConditionallyCompleteLinearOrder X\ninst\u271d\u2074 : ConditionallyCompleteLinearOrder E\ninst\u271d\u00b3 : OrderTopology X\ninst\u271d\u00b2 : OrderTopology E\ninst\u271d\u00b9 : SecondCountableTopology E\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhmono : Monotone f\nx : X\nU : Set X\nhU : U \u2208 \ud835\udcdd x\nh'U : \u2191\u2191\u03bc U < \u22a4\na b : X\nxab : x \u2208 Icc a b\nhab : Icc a b \u2208 \ud835\udcdd x\nabU : Icc a b \u2286 U\n\u22a2 IntegrableAtFilter f (\ud835\udcdd x)", "state_after": "case intro.intro.intro.intro.intro.intro\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u00b9\u00b9 : MeasurableSpace X\ninst\u271d\u00b9\u2070 : TopologicalSpace X\ninst\u271d\u2079 : MeasurableSpace Y\ninst\u271d\u2078 : TopologicalSpace Y\ninst\u271d\u2077 : NormedAddCommGroup E\nf g : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u2076 : BorelSpace X\ninst\u271d\u2075 : ConditionallyCompleteLinearOrder X\ninst\u271d\u2074 : ConditionallyCompleteLinearOrder E\ninst\u271d\u00b3 : OrderTopology X\ninst\u271d\u00b2 : OrderTopology E\ninst\u271d\u00b9 : SecondCountableTopology E\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhmono : Monotone f\nx : X\nU : Set X\nhU : U \u2208 \ud835\udcdd x\nh'U : \u2191\u2191\u03bc U < \u22a4\na b : X\nxab : x \u2208 Icc a b\nhab : Icc a b \u2208 \ud835\udcdd x\nabU : Icc a b \u2286 U\nab : a \u2264 b\n\u22a2 IntegrableAtFilter f (\ud835\udcdd x)"}, {"tactic": "refine' \u27e8Icc a b, hab, _\u27e9", "annotated_tactic": ["refine' \u27e8<a>Icc</a> a b, hab, _\u27e9", [{"full_name": "Set.Icc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [59, 5], "def_end_pos": [59, 8]}]], "state_before": "case intro.intro.intro.intro.intro.intro\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u00b9\u00b9 : MeasurableSpace X\ninst\u271d\u00b9\u2070 : TopologicalSpace X\ninst\u271d\u2079 : MeasurableSpace Y\ninst\u271d\u2078 : TopologicalSpace Y\ninst\u271d\u2077 : NormedAddCommGroup E\nf g : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u2076 : BorelSpace X\ninst\u271d\u2075 : ConditionallyCompleteLinearOrder X\ninst\u271d\u2074 : ConditionallyCompleteLinearOrder E\ninst\u271d\u00b3 : OrderTopology X\ninst\u271d\u00b2 : OrderTopology E\ninst\u271d\u00b9 : SecondCountableTopology E\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhmono : Monotone f\nx : X\nU : Set X\nhU : U \u2208 \ud835\udcdd x\nh'U : \u2191\u2191\u03bc U < \u22a4\na b : X\nxab : x \u2208 Icc a b\nhab : Icc a b \u2208 \ud835\udcdd x\nabU : Icc a b \u2286 U\nab : a \u2264 b\n\u22a2 IntegrableAtFilter f (\ud835\udcdd x)", "state_after": "case intro.intro.intro.intro.intro.intro\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u00b9\u00b9 : MeasurableSpace X\ninst\u271d\u00b9\u2070 : TopologicalSpace X\ninst\u271d\u2079 : MeasurableSpace Y\ninst\u271d\u2078 : TopologicalSpace Y\ninst\u271d\u2077 : NormedAddCommGroup E\nf g : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u2076 : BorelSpace X\ninst\u271d\u2075 : ConditionallyCompleteLinearOrder X\ninst\u271d\u2074 : ConditionallyCompleteLinearOrder E\ninst\u271d\u00b3 : OrderTopology X\ninst\u271d\u00b2 : OrderTopology E\ninst\u271d\u00b9 : SecondCountableTopology E\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhmono : Monotone f\nx : X\nU : Set X\nhU : U \u2208 \ud835\udcdd x\nh'U : \u2191\u2191\u03bc U < \u22a4\na b : X\nxab : x \u2208 Icc a b\nhab : Icc a b \u2208 \ud835\udcdd x\nabU : Icc a b \u2286 U\nab : a \u2264 b\n\u22a2 IntegrableOn f (Icc a b)"}, {"tactic": "exact\n  (hmono.monotoneOn _).integrableOn_of_measure_ne_top (isLeast_Icc ab) (isGreatest_Icc ab)\n    ((measure_mono abU).trans_lt h'U).ne measurableSet_Icc", "annotated_tactic": ["exact\n    (hmono.monotoneOn _).<a>integrableOn_of_measure_ne_top</a> (<a>isLeast_Icc</a> ab) (<a>isGreatest_Icc</a> ab)\n      ((<a>measure_mono</a> abU).<a>trans_lt</a> h'U).<a>ne</a> <a>measurableSet_Icc</a>", [{"full_name": "MonotoneOn.integrableOn_of_measure_ne_top", "def_path": "Mathlib/MeasureTheory/Function/LocallyIntegrable.lean", "def_pos": [447, 9], "def_end_pos": [447, 50]}, {"full_name": "isLeast_Icc", "def_path": "Mathlib/Order/Bounds/Basic.lean", "def_pos": [699, 9], "def_end_pos": [699, 20]}, {"full_name": "isGreatest_Icc", "def_path": "Mathlib/Order/Bounds/Basic.lean", "def_pos": [687, 9], "def_end_pos": [687, 23]}, {"full_name": "MeasureTheory.measure_mono", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [193, 9], "def_end_pos": [193, 21]}, {"full_name": "LE.le.trans_lt", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [124, 7], "def_end_pos": [124, 21]}, {"full_name": "LT.lt.ne", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [152, 7], "def_end_pos": [152, 15]}, {"full_name": "measurableSet_Icc", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [520, 9], "def_end_pos": [520, 26]}]], "state_before": "case intro.intro.intro.intro.intro.intro\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u00b9\u00b9 : MeasurableSpace X\ninst\u271d\u00b9\u2070 : TopologicalSpace X\ninst\u271d\u2079 : MeasurableSpace Y\ninst\u271d\u2078 : TopologicalSpace Y\ninst\u271d\u2077 : NormedAddCommGroup E\nf g : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u2076 : BorelSpace X\ninst\u271d\u2075 : ConditionallyCompleteLinearOrder X\ninst\u271d\u2074 : ConditionallyCompleteLinearOrder E\ninst\u271d\u00b3 : OrderTopology X\ninst\u271d\u00b2 : OrderTopology E\ninst\u271d\u00b9 : SecondCountableTopology E\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhmono : Monotone f\nx : X\nU : Set X\nhU : U \u2208 \ud835\udcdd x\nh'U : \u2191\u2191\u03bc U < \u22a4\na b : X\nxab : x \u2208 Icc a b\nhab : Icc a b \u2208 \ud835\udcdd x\nabU : Icc a b \u2286 U\nab : a \u2264 b\n\u22a2 IntegrableOn f (Icc a b)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Finite.lean", "full_name": "Set.Finite.finite_subsets", "start": [1007, 1], "end": [1010, 50], "traced_tactics": [{"tactic": "simpa [\u2190 @exists_finite_iff_finset \u03b1 fun t => t \u2286 a \u2227 t = s, Finite.subset_toFinset, \u2190\n  and_assoc, Finset.coeEmb] using h.subset", "annotated_tactic": ["simpa [\u2190 @<a>exists_finite_iff_finset</a> \u03b1 fun t => t \u2286 a \u2227 t = s, <a>Finite.subset_toFinset</a>, \u2190\n        <a>and_assoc</a>, <a>Finset.coeEmb</a>] using h.subset", [{"full_name": "Set.exists_finite_iff_finset", "def_path": "Mathlib/Data/Set/Finite.lean", "def_pos": [1000, 9], "def_end_pos": [1000, 33]}, {"full_name": "Set.Finite.subset_toFinset", "def_path": "Mathlib/Data/Set/Finite.lean", "def_pos": [208, 9], "def_end_pos": [208, 24]}, {"full_name": "and_assoc", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [177, 9], "def_end_pos": [177, 18]}, {"full_name": "Finset.coeEmb", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [459, 5], "def_end_pos": [459, 11]}]], "state_before": "\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\n\u03b3 : Type x\n\u03b1 : Type u\na : Set \u03b1\nh : Set.Finite a\ns : Set \u03b1\n\u22a2 s \u2208 Finset.map Finset.coeEmb.toEmbedding (Finset.powerset (Finite.toFinset h)) \u2194 s \u2208 {b | b \u2286 a}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Card.lean", "full_name": "Set.Finite.injOn_of_encard_image_eq", "start": [401, 1], "end": [407, 81], "traced_tactics": [{"tactic": "obtain (h' | hne) := isEmpty_or_nonempty \u03b1", "annotated_tactic": ["obtain (h' | hne) := <a>isEmpty_or_nonempty</a> \u03b1", [{"full_name": "isEmpty_or_nonempty", "def_path": "Mathlib/Logic/IsEmpty.lean", "def_pos": [207, 9], "def_end_pos": [207, 28]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ns\u271d t\u271d s : Set \u03b1\nt : Set \u03b2\nf : \u03b1 \u2192 \u03b2\nhs : Set.Finite s\nh : encard (f '' s) = encard s\n\u22a2 InjOn f s", "state_after": "case inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns\u271d t\u271d s : Set \u03b1\nt : Set \u03b2\nf : \u03b1 \u2192 \u03b2\nhs : Set.Finite s\nh : encard (f '' s) = encard s\nh' : IsEmpty \u03b1\n\u22a2 InjOn f s\n\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns\u271d t\u271d s : Set \u03b1\nt : Set \u03b2\nf : \u03b1 \u2192 \u03b2\nhs : Set.Finite s\nh : encard (f '' s) = encard s\nhne : Nonempty \u03b1\n\u22a2 InjOn f s"}, {"tactic": "rw [\u2190(f.invFunOn_injOn_image s).encard_image] at h", "annotated_tactic": ["rw [\u2190(f.invFunOn_injOn_image s).<a>encard_image</a>] at h", [{"full_name": "Set.InjOn.encard_image", "def_path": "Mathlib/Data/Set/Card.lean", "def_pos": [380, 9], "def_end_pos": [380, 27]}]], "state_before": "case inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns\u271d t\u271d s : Set \u03b1\nt : Set \u03b2\nf : \u03b1 \u2192 \u03b2\nhs : Set.Finite s\nh : encard (f '' s) = encard s\nhne : Nonempty \u03b1\n\u22a2 InjOn f s", "state_after": "case inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns\u271d t\u271d s : Set \u03b1\nt : Set \u03b2\nf : \u03b1 \u2192 \u03b2\nhs : Set.Finite s\nhne : Nonempty \u03b1\nh : encard (Function.invFunOn f s '' (f '' s)) = encard s\n\u22a2 InjOn f s"}, {"tactic": "rw [injOn_iff_invFunOn_image_image_eq_self]", "annotated_tactic": ["rw [<a>injOn_iff_invFunOn_image_image_eq_self</a>]", [{"full_name": "Set.injOn_iff_invFunOn_image_image_eq_self", "def_path": "Mathlib/Data/Set/Function.lean", "def_pos": [1278, 9], "def_end_pos": [1278, 47]}]], "state_before": "case inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns\u271d t\u271d s : Set \u03b1\nt : Set \u03b2\nf : \u03b1 \u2192 \u03b2\nhs : Set.Finite s\nhne : Nonempty \u03b1\nh : encard (Function.invFunOn f s '' (f '' s)) = encard s\n\u22a2 InjOn f s", "state_after": "case inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns\u271d t\u271d s : Set \u03b1\nt : Set \u03b2\nf : \u03b1 \u2192 \u03b2\nhs : Set.Finite s\nhne : Nonempty \u03b1\nh : encard (Function.invFunOn f s '' (f '' s)) = encard s\n\u22a2 Function.invFunOn f s '' (f '' s) = s"}, {"tactic": "exact hs.eq_of_subset_of_encard_le (f.invFunOn_image_image_subset s) h.symm.le", "annotated_tactic": ["exact hs.eq_of_subset_of_encard_le (f.invFunOn_image_image_subset s) h.symm.le", []], "state_before": "case inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns\u271d t\u271d s : Set \u03b1\nt : Set \u03b2\nf : \u03b1 \u2192 \u03b2\nhs : Set.Finite s\nhne : Nonempty \u03b1\nh : encard (Function.invFunOn f s '' (f '' s)) = encard s\n\u22a2 Function.invFunOn f s '' (f '' s) = s", "state_after": "no goals"}, {"tactic": "rw [s.eq_empty_of_isEmpty]", "annotated_tactic": ["rw [s.eq_empty_of_isEmpty]", []], "state_before": "case inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns\u271d t\u271d s : Set \u03b1\nt : Set \u03b2\nf : \u03b1 \u2192 \u03b2\nhs : Set.Finite s\nh : encard (f '' s) = encard s\nh' : IsEmpty \u03b1\n\u22a2 InjOn f s", "state_after": "case inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns\u271d t\u271d s : Set \u03b1\nt : Set \u03b2\nf : \u03b1 \u2192 \u03b2\nhs : Set.Finite s\nh : encard (f '' s) = encard s\nh' : IsEmpty \u03b1\n\u22a2 InjOn f \u2205"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns\u271d t\u271d s : Set \u03b1\nt : Set \u03b2\nf : \u03b1 \u2192 \u03b2\nhs : Set.Finite s\nh : encard (f '' s) = encard s\nh' : IsEmpty \u03b1\n\u22a2 InjOn f \u2205", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/NAry.lean", "full_name": "Finset.image\u2082_insert_right", "start": [188, 1], "end": [192, 30], "traced_tactics": [{"tactic": "push_cast", "annotated_tactic": ["push_cast", []], "state_before": "\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\ninst\u271d\u2078 : DecidableEq \u03b1'\ninst\u271d\u2077 : DecidableEq \u03b2'\ninst\u271d\u2076 : DecidableEq \u03b3\ninst\u271d\u2075 : DecidableEq \u03b3'\ninst\u271d\u2074 : DecidableEq \u03b4\ninst\u271d\u00b3 : DecidableEq \u03b4'\ninst\u271d\u00b2 : DecidableEq \u03b5\ninst\u271d\u00b9 : DecidableEq \u03b5'\nf f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Finset \u03b1\nt t' : Finset \u03b2\nu u' : Finset \u03b3\na a' : \u03b1\nb b' : \u03b2\nc : \u03b3\ninst\u271d : DecidableEq \u03b2\n\u22a2 \u2191(image\u2082 f s (insert b t)) = \u2191(image (fun a => f a b) s \u222a image\u2082 f s t)", "state_after": "\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\ninst\u271d\u2078 : DecidableEq \u03b1'\ninst\u271d\u2077 : DecidableEq \u03b2'\ninst\u271d\u2076 : DecidableEq \u03b3\ninst\u271d\u2075 : DecidableEq \u03b3'\ninst\u271d\u2074 : DecidableEq \u03b4\ninst\u271d\u00b3 : DecidableEq \u03b4'\ninst\u271d\u00b2 : DecidableEq \u03b5\ninst\u271d\u00b9 : DecidableEq \u03b5'\nf f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Finset \u03b1\nt t' : Finset \u03b2\nu u' : Finset \u03b3\na a' : \u03b1\nb b' : \u03b2\nc : \u03b3\ninst\u271d : DecidableEq \u03b2\n\u22a2 image2 f (\u2191s) (insert b \u2191t) = (fun a => f a b) '' \u2191s \u222a image2 f \u2191s \u2191t"}, {"tactic": "exact image2_insert_right", "annotated_tactic": ["exact <a>image2_insert_right</a>", [{"full_name": "Set.image2_insert_right", "def_path": "Mathlib/Data/Set/NAry.lean", "def_pos": [213, 9], "def_end_pos": [213, 28]}]], "state_before": "\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\ninst\u271d\u2078 : DecidableEq \u03b1'\ninst\u271d\u2077 : DecidableEq \u03b2'\ninst\u271d\u2076 : DecidableEq \u03b3\ninst\u271d\u2075 : DecidableEq \u03b3'\ninst\u271d\u2074 : DecidableEq \u03b4\ninst\u271d\u00b3 : DecidableEq \u03b4'\ninst\u271d\u00b2 : DecidableEq \u03b5\ninst\u271d\u00b9 : DecidableEq \u03b5'\nf f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Finset \u03b1\nt t' : Finset \u03b2\nu u' : Finset \u03b3\na a' : \u03b1\nb b' : \u03b2\nc : \u03b3\ninst\u271d : DecidableEq \u03b2\n\u22a2 image2 f (\u2191s) (insert b \u2191t) = (fun a => f a b) '' \u2191s \u222a image2 f \u2191s \u2191t", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Countable.lean", "full_name": "Set.exists_seq_cover_iff_countable", "start": [174, 1], "end": [177, 41], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "full_name": "MeasureTheory.snorm_trim", "start": [1020, 1], "end": [1026, 64], "traced_tactics": [{"tactic": "by_cases h0 : p = 0", "annotated_tactic": ["by_cases h0 : p = 0", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nhm : m \u2264 m0\nf : \u03b1 \u2192 E\nhf : StronglyMeasurable f\n\u22a2 snorm f p (Measure.trim \u03bd hm) = snorm f p \u03bd", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nhm : m \u2264 m0\nf : \u03b1 \u2192 E\nhf : StronglyMeasurable f\nh0 : p = 0\n\u22a2 snorm f p (Measure.trim \u03bd hm) = snorm f p \u03bd\n\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nhm : m \u2264 m0\nf : \u03b1 \u2192 E\nhf : StronglyMeasurable f\nh0 : \u00acp = 0\n\u22a2 snorm f p (Measure.trim \u03bd hm) = snorm f p \u03bd"}, {"tactic": "by_cases h_top : p = \u221e", "annotated_tactic": ["by_cases h_top : p = \u221e", []], "state_before": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nhm : m \u2264 m0\nf : \u03b1 \u2192 E\nhf : StronglyMeasurable f\nh0 : \u00acp = 0\n\u22a2 snorm f p (Measure.trim \u03bd hm) = snorm f p \u03bd", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nhm : m \u2264 m0\nf : \u03b1 \u2192 E\nhf : StronglyMeasurable f\nh0 : \u00acp = 0\nh_top : p = \u22a4\n\u22a2 snorm f p (Measure.trim \u03bd hm) = snorm f p \u03bd\n\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nhm : m \u2264 m0\nf : \u03b1 \u2192 E\nhf : StronglyMeasurable f\nh0 : \u00acp = 0\nh_top : \u00acp = \u22a4\n\u22a2 snorm f p (Measure.trim \u03bd hm) = snorm f p \u03bd"}, {"tactic": "simpa only [snorm_eq_snorm' h0 h_top] using snorm'_trim hm hf", "annotated_tactic": ["simpa only [<a>snorm_eq_snorm'</a> h0 h_top] using <a>snorm'_trim</a> hm hf", [{"full_name": "MeasureTheory.snorm_eq_snorm'", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [88, 9], "def_end_pos": [88, 24]}, {"full_name": "MeasureTheory.snorm'_trim", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [986, 9], "def_end_pos": [986, 20]}]], "state_before": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nhm : m \u2264 m0\nf : \u03b1 \u2192 E\nhf : StronglyMeasurable f\nh0 : \u00acp = 0\nh_top : \u00acp = \u22a4\n\u22a2 snorm f p (Measure.trim \u03bd hm) = snorm f p \u03bd", "state_after": "no goals"}, {"tactic": "simp [h0]", "annotated_tactic": ["simp [h0]", []], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nhm : m \u2264 m0\nf : \u03b1 \u2192 E\nhf : StronglyMeasurable f\nh0 : p = 0\n\u22a2 snorm f p (Measure.trim \u03bd hm) = snorm f p \u03bd", "state_after": "no goals"}, {"tactic": "simpa only [h_top, snorm_exponent_top] using snormEssSup_trim hm hf", "annotated_tactic": ["simpa only [h_top, <a>snorm_exponent_top</a>] using <a>snormEssSup_trim</a> hm hf", [{"full_name": "MeasureTheory.snorm_exponent_top", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [103, 9], "def_end_pos": [103, 27]}, {"full_name": "MeasureTheory.snormEssSup_trim", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [1015, 9], "def_end_pos": [1015, 25]}]], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nhm : m \u2264 m0\nf : \u03b1 \u2192 E\nhf : StronglyMeasurable f\nh0 : \u00acp = 0\nh_top : p = \u22a4\n\u22a2 snorm f p (Measure.trim \u03bd hm) = snorm f p \u03bd", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "full_name": "MeasureTheory.Measure.FiniteAtFilter.integrableAtFilter", "start": [461, 1], "end": [470, 32], "traced_tactics": [{"tactic": "obtain \u27e8C, hC\u27e9 : \u2203 C, \u2200\u1da0 s in l.smallSets, \u2200 x \u2208 s, \u2016f x\u2016 \u2264 C :=\n  hf.imp fun C hC => eventually_smallSets.2 \u27e8_, hC, fun t => id\u27e9", "annotated_tactic": ["obtain \u27e8C, hC\u27e9 : \u2203 C, \u2200\u1da0 s in l.smallSets, \u2200 x \u2208 s, \u2016f x\u2016 \u2264 C :=\n    hf.imp fun C hC => <a>eventually_smallSets</a>.2 \u27e8_, hC, fun t => <a>id</a>\u27e9", [{"full_name": "Filter.eventually_smallSets", "def_path": "Mathlib/Order/Filter/SmallSets.lean", "def_pos": [63, 9], "def_end_pos": [63, 29]}, {"full_name": "id", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [33, 15], "def_end_pos": [33, 17]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\nf g : \u03b1 \u2192 E\ns t : Set \u03b1\n\u03bc \u03bd : Measure \u03b1\nl\u271d l' l : Filter \u03b1\ninst\u271d : IsMeasurablyGenerated l\nhfm : StronglyMeasurableAtFilter f l\nh\u03bc : FiniteAtFilter \u03bc l\nhf : IsBoundedUnder (fun x x_1 => x \u2264 x_1) l (norm \u2218 f)\n\u22a2 IntegrableAtFilter f l", "state_after": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\nf g : \u03b1 \u2192 E\ns t : Set \u03b1\n\u03bc \u03bd : Measure \u03b1\nl\u271d l' l : Filter \u03b1\ninst\u271d : IsMeasurablyGenerated l\nhfm : StronglyMeasurableAtFilter f l\nh\u03bc : FiniteAtFilter \u03bc l\nhf : IsBoundedUnder (fun x x_1 => x \u2264 x_1) l (norm \u2218 f)\nC : \u211d\nhC : \u2200\u1da0 (s : Set \u03b1) in smallSets l, \u2200 (x : \u03b1), x \u2208 s \u2192 \u2016f x\u2016 \u2264 C\n\u22a2 IntegrableAtFilter f l"}, {"tactic": "rcases (hfm.eventually.and (h\u03bc.eventually.and hC)).exists_measurable_mem_of_smallSets with\n  \u27e8s, hsl, hsm, hfm, h\u03bc, hC\u27e9", "annotated_tactic": ["rcases (hfm.eventually.and (h\u03bc.eventually.and hC)).<a>exists_measurable_mem_of_smallSets</a> with\n    \u27e8s, hsl, hsm, hfm, h\u03bc, hC\u27e9", [{"full_name": "Filter.Eventually.exists_measurable_mem_of_smallSets", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [1934, 9], "def_end_pos": [1934, 54]}]], "state_before": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\nf g : \u03b1 \u2192 E\ns t : Set \u03b1\n\u03bc \u03bd : Measure \u03b1\nl\u271d l' l : Filter \u03b1\ninst\u271d : IsMeasurablyGenerated l\nhfm : StronglyMeasurableAtFilter f l\nh\u03bc : FiniteAtFilter \u03bc l\nhf : IsBoundedUnder (fun x x_1 => x \u2264 x_1) l (norm \u2218 f)\nC : \u211d\nhC : \u2200\u1da0 (s : Set \u03b1) in smallSets l, \u2200 (x : \u03b1), x \u2208 s \u2192 \u2016f x\u2016 \u2264 C\n\u22a2 IntegrableAtFilter f l", "state_after": "case intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\nf g : \u03b1 \u2192 E\ns\u271d t : Set \u03b1\n\u03bc \u03bd : Measure \u03b1\nl\u271d l' l : Filter \u03b1\ninst\u271d : IsMeasurablyGenerated l\nhfm\u271d : StronglyMeasurableAtFilter f l\nh\u03bc\u271d : FiniteAtFilter \u03bc l\nhf : IsBoundedUnder (fun x x_1 => x \u2264 x_1) l (norm \u2218 f)\nC : \u211d\nhC\u271d : \u2200\u1da0 (s : Set \u03b1) in smallSets l, \u2200 (x : \u03b1), x \u2208 s \u2192 \u2016f x\u2016 \u2264 C\ns : Set \u03b1\nhsl : s \u2208 l\nhsm : MeasurableSet s\nhfm : AEStronglyMeasurable f (restrict \u03bc s)\nh\u03bc : \u2191\u2191\u03bc s < \u22a4\nhC : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2016f x\u2016 \u2264 C\n\u22a2 IntegrableAtFilter f l"}, {"tactic": "refine' \u27e8s, hsl, \u27e8hfm, hasFiniteIntegral_restrict_of_bounded h\u03bc (C := C) _\u27e9\u27e9", "annotated_tactic": ["refine' \u27e8s, hsl, \u27e8hfm, <a>hasFiniteIntegral_restrict_of_bounded</a> h\u03bc (C := C) _\u27e9\u27e9", [{"full_name": "MeasureTheory.hasFiniteIntegral_restrict_of_bounded", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [79, 9], "def_end_pos": [79, 46]}]], "state_before": "case intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\nf g : \u03b1 \u2192 E\ns\u271d t : Set \u03b1\n\u03bc \u03bd : Measure \u03b1\nl\u271d l' l : Filter \u03b1\ninst\u271d : IsMeasurablyGenerated l\nhfm\u271d : StronglyMeasurableAtFilter f l\nh\u03bc\u271d : FiniteAtFilter \u03bc l\nhf : IsBoundedUnder (fun x x_1 => x \u2264 x_1) l (norm \u2218 f)\nC : \u211d\nhC\u271d : \u2200\u1da0 (s : Set \u03b1) in smallSets l, \u2200 (x : \u03b1), x \u2208 s \u2192 \u2016f x\u2016 \u2264 C\ns : Set \u03b1\nhsl : s \u2208 l\nhsm : MeasurableSet s\nhfm : AEStronglyMeasurable f (restrict \u03bc s)\nh\u03bc : \u2191\u2191\u03bc s < \u22a4\nhC : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2016f x\u2016 \u2264 C\n\u22a2 IntegrableAtFilter f l", "state_after": "case intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\nf g : \u03b1 \u2192 E\ns\u271d t : Set \u03b1\n\u03bc \u03bd : Measure \u03b1\nl\u271d l' l : Filter \u03b1\ninst\u271d : IsMeasurablyGenerated l\nhfm\u271d : StronglyMeasurableAtFilter f l\nh\u03bc\u271d : FiniteAtFilter \u03bc l\nhf : IsBoundedUnder (fun x x_1 => x \u2264 x_1) l (norm \u2218 f)\nC : \u211d\nhC\u271d : \u2200\u1da0 (s : Set \u03b1) in smallSets l, \u2200 (x : \u03b1), x \u2208 s \u2192 \u2016f x\u2016 \u2264 C\ns : Set \u03b1\nhsl : s \u2208 l\nhsm : MeasurableSet s\nhfm : AEStronglyMeasurable f (restrict \u03bc s)\nh\u03bc : \u2191\u2191\u03bc s < \u22a4\nhC : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2016f x\u2016 \u2264 C\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202restrict \u03bc s, \u2016f x\u2016 \u2264 C"}, {"tactic": "rw [ae_restrict_eq hsm, eventually_inf_principal]", "annotated_tactic": ["rw [<a>ae_restrict_eq</a> hsm, <a>eventually_inf_principal</a>]", [{"full_name": "MeasureTheory.ae_restrict_eq", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2665, 9], "def_end_pos": [2665, 23]}, {"full_name": "Filter.eventually_inf_principal", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1250, 9], "def_end_pos": [1250, 33]}]], "state_before": "case intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\nf g : \u03b1 \u2192 E\ns\u271d t : Set \u03b1\n\u03bc \u03bd : Measure \u03b1\nl\u271d l' l : Filter \u03b1\ninst\u271d : IsMeasurablyGenerated l\nhfm\u271d : StronglyMeasurableAtFilter f l\nh\u03bc\u271d : FiniteAtFilter \u03bc l\nhf : IsBoundedUnder (fun x x_1 => x \u2264 x_1) l (norm \u2218 f)\nC : \u211d\nhC\u271d : \u2200\u1da0 (s : Set \u03b1) in smallSets l, \u2200 (x : \u03b1), x \u2208 s \u2192 \u2016f x\u2016 \u2264 C\ns : Set \u03b1\nhsl : s \u2208 l\nhsm : MeasurableSet s\nhfm : AEStronglyMeasurable f (restrict \u03bc s)\nh\u03bc : \u2191\u2191\u03bc s < \u22a4\nhC : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2016f x\u2016 \u2264 C\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202restrict \u03bc s, \u2016f x\u2016 \u2264 C", "state_after": "case intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\nf g : \u03b1 \u2192 E\ns\u271d t : Set \u03b1\n\u03bc \u03bd : Measure \u03b1\nl\u271d l' l : Filter \u03b1\ninst\u271d : IsMeasurablyGenerated l\nhfm\u271d : StronglyMeasurableAtFilter f l\nh\u03bc\u271d : FiniteAtFilter \u03bc l\nhf : IsBoundedUnder (fun x x_1 => x \u2264 x_1) l (norm \u2218 f)\nC : \u211d\nhC\u271d : \u2200\u1da0 (s : Set \u03b1) in smallSets l, \u2200 (x : \u03b1), x \u2208 s \u2192 \u2016f x\u2016 \u2264 C\ns : Set \u03b1\nhsl : s \u2208 l\nhsm : MeasurableSet s\nhfm : AEStronglyMeasurable f (restrict \u03bc s)\nh\u03bc : \u2191\u2191\u03bc s < \u22a4\nhC : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2016f x\u2016 \u2264 C\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 \u2016f x\u2016 \u2264 C"}, {"tactic": "exact eventually_of_forall hC", "annotated_tactic": ["exact <a>eventually_of_forall</a> hC", [{"full_name": "Filter.eventually_of_forall", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1111, 9], "def_end_pos": [1111, 29]}]], "state_before": "case intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\nf g : \u03b1 \u2192 E\ns\u271d t : Set \u03b1\n\u03bc \u03bd : Measure \u03b1\nl\u271d l' l : Filter \u03b1\ninst\u271d : IsMeasurablyGenerated l\nhfm\u271d : StronglyMeasurableAtFilter f l\nh\u03bc\u271d : FiniteAtFilter \u03bc l\nhf : IsBoundedUnder (fun x x_1 => x \u2264 x_1) l (norm \u2218 f)\nC : \u211d\nhC\u271d : \u2200\u1da0 (s : Set \u03b1) in smallSets l, \u2200 (x : \u03b1), x \u2208 s \u2192 \u2016f x\u2016 \u2264 C\ns : Set \u03b1\nhsl : s \u2208 l\nhsm : MeasurableSet s\nhfm : AEStronglyMeasurable f (restrict \u03bc s)\nh\u03bc : \u2191\u2191\u03bc s < \u22a4\nhC : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2016f x\u2016 \u2264 C\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 \u2016f x\u2016 \u2264 C", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Kernel/CondDistrib.lean", "full_name": "ProbabilityTheory.condexp_ae_eq_integral_condDistrib'", "start": [282, 1], "end": [286, 91], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finmap.lean", "full_name": "Multiset.coe_keys", "start": [32, 1], "end": [33, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/UniformIntegrable.lean", "full_name": "MeasureTheory.UniformIntegrable.spec'", "start": [856, 1], "end": [889, 97], "traced_tactics": [{"tactic": "obtain \u27e8-, hfu, M, hM\u27e9 := hfu", "annotated_tactic": ["obtain \u27e8-, hfu, M, hM\u27e9 := hfu", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhp : p \u2260 0\nhp' : p \u2260 \u22a4\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nhfu : UniformIntegrable f p \u03bc\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u22a2 \u2203 C, \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhp : p \u2260 0\nhp' : p \u2260 \u22a4\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhfu : UnifIntegrable f p \u03bc\nM : \u211d\u22650\nhM : \u2200 (i : \u03b9), snorm (f i) p \u03bc \u2264 \u2191M\n\u22a2 \u2203 C, \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5"}, {"tactic": "obtain \u27e8\u03b4, h\u03b4pos, h\u03b4\u27e9 := hfu h\u03b5", "annotated_tactic": ["obtain \u27e8\u03b4, h\u03b4pos, h\u03b4\u27e9 := hfu h\u03b5", []], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhp : p \u2260 0\nhp' : p \u2260 \u22a4\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhfu : UnifIntegrable f p \u03bc\nM : \u211d\u22650\nhM : \u2200 (i : \u03b9), snorm (f i) p \u03bc \u2264 \u2191M\n\u22a2 \u2203 C, \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5", "state_after": "case intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhp : p \u2260 0\nhp' : p \u2260 \u22a4\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhfu : UnifIntegrable f p \u03bc\nM : \u211d\u22650\nhM : \u2200 (i : \u03b9), snorm (f i) p \u03bc \u2264 \u2191M\n\u03b4 : \u211d\nh\u03b4pos : 0 < \u03b4\nh\u03b4 :\n  \u2200 (i : \u03b9) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\n\u22a2 \u2203 C, \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5"}, {"tactic": "obtain \u27e8C, hC\u27e9 : \u2203 C : \u211d\u22650, \u2200 i, \u03bc { x | C \u2264 \u2016f i x\u2016\u208a } \u2264 ENNReal.ofReal \u03b4 := by\n  by_contra hcon; push_neg at hcon\n  choose \u2110 h\u2110 using hcon\n  lift \u03b4 to \u211d\u22650 using h\u03b4pos.le\n  have : \u2200 C : \u211d\u22650, C \u2022 (\u03b4 : \u211d\u22650\u221e) ^ (1 / p.toReal) \u2264 snorm (f (\u2110 C)) p \u03bc := by\n    intro C\n    calc\n      C \u2022 (\u03b4 : \u211d\u22650\u221e) ^ (1 / p.toReal) \u2264 C \u2022 \u03bc { x | C \u2264 \u2016f (\u2110 C) x\u2016\u208a } ^ (1 / p.toReal) := by\n        rw [ENNReal.smul_def, ENNReal.smul_def, smul_eq_mul, smul_eq_mul]\n        simp_rw [ENNReal.ofReal_coe_nnreal] at h\u2110\n        refine' mul_le_mul' le_rfl\n          (ENNReal.rpow_le_rpow (h\u2110 C).le (one_div_nonneg.2 ENNReal.toReal_nonneg))\n      _ \u2264 snorm ({ x | C \u2264 \u2016f (\u2110 C) x\u2016\u208a }.indicator (f (\u2110 C))) p \u03bc := by\n        refine' snorm_indicator_ge_of_bdd_below hp hp' _\n          (measurableSet_le measurable_const (hf _).nnnorm.measurable)\n          (eventually_of_forall fun x hx => _)\n        rwa [nnnorm_indicator_eq_indicator_nnnorm, Set.indicator_of_mem hx]\n      _ \u2264 snorm (f (\u2110 C)) p \u03bc := snorm_indicator_le _\n  specialize this (2 * max M 1 * HPow.hPow \u03b4\u207b\u00b9 (1 / p.toReal))\n  rw [ENNReal.coe_rpow_of_nonneg _ (one_div_nonneg.2 ENNReal.toReal_nonneg), \u2190 ENNReal.coe_smul,\n    smul_eq_mul, mul_assoc, NNReal.inv_rpow,\n    inv_mul_cancel (NNReal.rpow_pos (NNReal.coe_pos.1 h\u03b4pos)).ne.symm, mul_one, ENNReal.coe_mul,\n    \u2190 NNReal.inv_rpow] at this\n  refine' (lt_of_le_of_lt (le_trans\n    (hM <| \u2110 <| 2 * max M 1 * HPow.hPow \u03b4\u207b\u00b9 (1 / p.toReal)) (le_max_left (M : \u211d\u22650\u221e) 1))\n      (lt_of_lt_of_le _ this)).ne rfl\n  rw [\u2190 ENNReal.coe_one, \u2190 ENNReal.coe_max, \u2190 ENNReal.coe_mul, ENNReal.coe_lt_coe]\n  exact lt_two_mul_self (lt_max_of_lt_right one_pos)", "annotated_tactic": ["obtain \u27e8C, hC\u27e9 : \u2203 C : \u211d\u22650, \u2200 i, \u03bc { x | C \u2264 \u2016f i x\u2016\u208a } \u2264 <a>ENNReal.ofReal</a> \u03b4 := by\n    by_contra hcon; push_neg at hcon\n    choose \u2110 h\u2110 using hcon\n    lift \u03b4 to \u211d\u22650 using h\u03b4pos.le\n    have : \u2200 C : \u211d\u22650, C \u2022 (\u03b4 : \u211d\u22650\u221e) ^ (1 / p.toReal) \u2264 <a>snorm</a> (f (\u2110 C)) p \u03bc := by\n      intro C\n      calc\n        C \u2022 (\u03b4 : \u211d\u22650\u221e) ^ (1 / p.toReal) \u2264 C \u2022 \u03bc { x | C \u2264 \u2016f (\u2110 C) x\u2016\u208a } ^ (1 / p.toReal) := by\n          rw [<a>ENNReal.smul_def</a>, <a>ENNReal.smul_def</a>, <a>smul_eq_mul</a>, <a>smul_eq_mul</a>]\n          simp_rw [<a>ENNReal.ofReal_coe_nnreal</a>] at h\u2110\n          refine' <a>mul_le_mul'</a> <a>le_rfl</a>\n            (<a>ENNReal.rpow_le_rpow</a> (h\u2110 C).<a>le</a> (<a>one_div_nonneg</a>.2 <a>ENNReal.toReal_nonneg</a>))\n        _ \u2264 <a>snorm</a> ({ x | C \u2264 \u2016f (\u2110 C) x\u2016\u208a }.<a>indicator</a> (f (\u2110 C))) p \u03bc := by\n          refine' <a>snorm_indicator_ge_of_bdd_below</a> hp hp' _\n            (<a>measurableSet_le</a> <a>measurable_const</a> (hf _).nnnorm.measurable)\n            (<a>eventually_of_forall</a> fun x hx => _)\n          rwa [<a>nnnorm_indicator_eq_indicator_nnnorm</a>, <a>Set.indicator_of_mem</a> hx]\n        _ \u2264 <a>snorm</a> (f (\u2110 C)) p \u03bc := <a>snorm_indicator_le</a> _\n    specialize this (2 * <a>max</a> M 1 * <a>HPow.hPow</a> \u03b4\u207b\u00b9 (1 / p.toReal))\n    rw [<a>ENNReal.coe_rpow_of_nonneg</a> _ (<a>one_div_nonneg</a>.2 <a>ENNReal.toReal_nonneg</a>), \u2190 <a>ENNReal.coe_smul</a>,\n      <a>smul_eq_mul</a>, <a>mul_assoc</a>, <a>NNReal.inv_rpow</a>,\n      <a>inv_mul_cancel</a> (<a>NNReal.rpow_pos</a> (<a>NNReal.coe_pos</a>.1 h\u03b4pos)).ne.symm, <a>mul_one</a>, <a>ENNReal.coe_mul</a>,\n      \u2190 <a>NNReal.inv_rpow</a>] at this\n    refine' (<a>lt_of_le_of_lt</a> (<a>le_trans</a>\n      (hM <| \u2110 <| 2 * <a>max</a> M 1 * <a>HPow.hPow</a> \u03b4\u207b\u00b9 (1 / p.toReal)) (<a>le_max_left</a> (M : \u211d\u22650\u221e) 1))\n        (<a>lt_of_lt_of_le</a> _ this)).<a>ne</a> <a>rfl</a>\n    rw [\u2190 <a>ENNReal.coe_one</a>, \u2190 <a>ENNReal.coe_max</a>, \u2190 <a>ENNReal.coe_mul</a>, <a>ENNReal.coe_lt_coe</a>]\n    exact <a>lt_two_mul_self</a> (<a>lt_max_of_lt_right</a> <a>one_pos</a>)", [{"full_name": "ENNReal.ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [172, 29], "def_end_pos": [172, 35]}, {"full_name": "MeasureTheory.snorm", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [84, 5], "def_end_pos": [84, 10]}, {"full_name": "ENNReal.smul_def", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [503, 9], "def_end_pos": [503, 17]}, {"full_name": "ENNReal.smul_def", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [503, 9], "def_end_pos": [503, 17]}, {"full_name": "smul_eq_mul", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [93, 9], "def_end_pos": [93, 20]}, {"full_name": "smul_eq_mul", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [93, 9], "def_end_pos": [93, 20]}, {"full_name": "ENNReal.ofReal_coe_nnreal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [212, 17], "def_end_pos": [212, 34]}, {"full_name": "mul_le_mul'", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [206, 9], "def_end_pos": [206, 20]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}, {"full_name": "ENNReal.rpow_le_rpow", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [642, 9], "def_end_pos": [642, 21]}, {"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [142, 7], "def_end_pos": [142, 15]}, {"full_name": "one_div_nonneg", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [81, 9], "def_end_pos": [81, 23]}, {"full_name": "ENNReal.toReal_nonneg", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [221, 17], "def_end_pos": [221, 30]}, {"full_name": "MeasureTheory.snorm", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [84, 5], "def_end_pos": [84, 10]}, {"full_name": "Set.indicator", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [46, 3], "def_end_pos": [46, 14]}, {"full_name": "MeasureTheory.snorm_indicator_ge_of_bdd_below", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [1589, 9], "def_end_pos": [1589, 40]}, {"full_name": "measurableSet_le", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [559, 9], "def_end_pos": [559, 25]}, {"full_name": "measurable_const", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [570, 9], "def_end_pos": [570, 25]}, {"full_name": "Filter.eventually_of_forall", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1111, 9], "def_end_pos": [1111, 29]}, {"full_name": "nnnorm_indicator_eq_indicator_nnnorm", "def_path": "Mathlib/Analysis/NormedSpace/IndicatorFunction.lean", "def_pos": [29, 9], "def_end_pos": [29, 45]}, {"full_name": "Set.indicator_of_mem", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [67, 3], "def_end_pos": [67, 14]}, {"full_name": "MeasureTheory.snorm", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [84, 5], "def_end_pos": [84, 10]}, {"full_name": "MeasureTheory.snorm_indicator_le", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [586, 9], "def_end_pos": [586, 27]}, {"full_name": "Max.max", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1090, 3], "def_end_pos": [1090, 6]}, {"full_name": "HPow.hPow", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1192, 3], "def_end_pos": [1192, 7]}, {"full_name": "ENNReal.coe_rpow_of_nonneg", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [436, 9], "def_end_pos": [436, 27]}, {"full_name": "one_div_nonneg", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [81, 9], "def_end_pos": [81, 23]}, {"full_name": "ENNReal.toReal_nonneg", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [221, 17], "def_end_pos": [221, 30]}, {"full_name": "ENNReal.coe_smul", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [539, 9], "def_end_pos": [539, 17]}, {"full_name": "smul_eq_mul", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [93, 9], "def_end_pos": [93, 20]}, {"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [264, 9], "def_end_pos": [264, 18]}, {"full_name": "NNReal.inv_rpow", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [112, 9], "def_end_pos": [112, 17]}, {"full_name": "inv_mul_cancel", "def_path": "Mathlib/Algebra/GroupWithZero/NeZero.lean", "def_pos": [55, 9], "def_end_pos": [55, 23]}, {"full_name": "NNReal.rpow_pos", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [245, 9], "def_end_pos": [245, 17]}, {"full_name": "NNReal.coe_pos", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [376, 19], "def_end_pos": [376, 26]}, {"full_name": "mul_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [470, 9], "def_end_pos": [470, 16]}, {"full_name": "ENNReal.coe_mul", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [390, 9], "def_end_pos": [390, 16]}, {"full_name": "NNReal.inv_rpow", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [112, 9], "def_end_pos": [112, 17]}, {"full_name": "lt_of_le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [122, 9], "def_end_pos": [122, 23]}, {"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}, {"full_name": "Max.max", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1090, 3], "def_end_pos": [1090, 6]}, {"full_name": "HPow.hPow", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1192, 3], "def_end_pos": [1192, 7]}, {"full_name": "le_max_left", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [54, 9], "def_end_pos": [54, 20]}, {"full_name": "lt_of_lt_of_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [115, 9], "def_end_pos": [115, 23]}, {"full_name": "LT.lt.ne", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [152, 7], "def_end_pos": [152, 15]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}, {"full_name": "ENNReal.coe_one", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [218, 28], "def_end_pos": [218, 35]}, {"full_name": "ENNReal.coe_max", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [945, 9], "def_end_pos": [945, 16]}, {"full_name": "ENNReal.coe_mul", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [390, 9], "def_end_pos": [390, 16]}, {"full_name": "ENNReal.coe_lt_coe", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [352, 28], "def_end_pos": [352, 38]}, {"full_name": "lt_two_mul_self", "def_path": "Mathlib/Algebra/Order/Ring/Defs.lean", "def_pos": [622, 9], "def_end_pos": [622, 24]}, {"full_name": "lt_max_of_lt_right", "def_path": "Mathlib/Order/MinMax.lean", "def_pos": [102, 9], "def_end_pos": [102, 27]}, {"full_name": "one_pos", "def_path": "Mathlib/Algebra/Order/ZeroLEOne.lean", "def_pos": [50, 7], "def_end_pos": [50, 14]}]], "state_before": "case intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhp : p \u2260 0\nhp' : p \u2260 \u22a4\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhfu : UnifIntegrable f p \u03bc\nM : \u211d\u22650\nhM : \u2200 (i : \u03b9), snorm (f i) p \u03bc \u2264 \u2191M\n\u03b4 : \u211d\nh\u03b4pos : 0 < \u03b4\nh\u03b4 :\n  \u2200 (i : \u03b9) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\n\u22a2 \u2203 C, \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5", "state_after": "case intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhp : p \u2260 0\nhp' : p \u2260 \u22a4\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhfu : UnifIntegrable f p \u03bc\nM : \u211d\u22650\nhM : \u2200 (i : \u03b9), snorm (f i) p \u03bc \u2264 \u2191M\n\u03b4 : \u211d\nh\u03b4pos : 0 < \u03b4\nh\u03b4 :\n  \u2200 (i : \u03b9) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nC : \u211d\u22650\nhC : \u2200 (i : \u03b9), \u2191\u2191\u03bc {x | C \u2264 \u2016f i x\u2016\u208a} \u2264 ENNReal.ofReal \u03b4\n\u22a2 \u2203 C, \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5"}, {"tactic": "exact \u27e8C, fun i => h\u03b4 i _ (measurableSet_le measurable_const (hf i).nnnorm.measurable) (hC i)\u27e9", "annotated_tactic": ["exact \u27e8C, fun i => h\u03b4 i _ (<a>measurableSet_le</a> <a>measurable_const</a> (hf i).nnnorm.measurable) (hC i)\u27e9", [{"full_name": "measurableSet_le", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [559, 9], "def_end_pos": [559, 25]}, {"full_name": "measurable_const", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [570, 9], "def_end_pos": [570, 25]}]], "state_before": "case intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhp : p \u2260 0\nhp' : p \u2260 \u22a4\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhfu : UnifIntegrable f p \u03bc\nM : \u211d\u22650\nhM : \u2200 (i : \u03b9), snorm (f i) p \u03bc \u2264 \u2191M\n\u03b4 : \u211d\nh\u03b4pos : 0 < \u03b4\nh\u03b4 :\n  \u2200 (i : \u03b9) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nC : \u211d\u22650\nhC : \u2200 (i : \u03b9), \u2191\u2191\u03bc {x | C \u2264 \u2016f i x\u2016\u208a} \u2264 ENNReal.ofReal \u03b4\n\u22a2 \u2203 C, \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5", "state_after": "no goals"}, {"tactic": "by_contra hcon", "annotated_tactic": ["by_contra hcon", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhp : p \u2260 0\nhp' : p \u2260 \u22a4\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhfu : UnifIntegrable f p \u03bc\nM : \u211d\u22650\nhM : \u2200 (i : \u03b9), snorm (f i) p \u03bc \u2264 \u2191M\n\u03b4 : \u211d\nh\u03b4pos : 0 < \u03b4\nh\u03b4 :\n  \u2200 (i : \u03b9) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\n\u22a2 \u2203 C, \u2200 (i : \u03b9), \u2191\u2191\u03bc {x | C \u2264 \u2016f i x\u2016\u208a} \u2264 ENNReal.ofReal \u03b4", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhp : p \u2260 0\nhp' : p \u2260 \u22a4\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhfu : UnifIntegrable f p \u03bc\nM : \u211d\u22650\nhM : \u2200 (i : \u03b9), snorm (f i) p \u03bc \u2264 \u2191M\n\u03b4 : \u211d\nh\u03b4pos : 0 < \u03b4\nh\u03b4 :\n  \u2200 (i : \u03b9) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nhcon : \u00ac\u2203 C, \u2200 (i : \u03b9), \u2191\u2191\u03bc {x | C \u2264 \u2016f i x\u2016\u208a} \u2264 ENNReal.ofReal \u03b4\n\u22a2 False"}, {"tactic": "push_neg at hcon", "annotated_tactic": ["push_neg at hcon", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhp : p \u2260 0\nhp' : p \u2260 \u22a4\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhfu : UnifIntegrable f p \u03bc\nM : \u211d\u22650\nhM : \u2200 (i : \u03b9), snorm (f i) p \u03bc \u2264 \u2191M\n\u03b4 : \u211d\nh\u03b4pos : 0 < \u03b4\nh\u03b4 :\n  \u2200 (i : \u03b9) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nhcon : \u00ac\u2203 C, \u2200 (i : \u03b9), \u2191\u2191\u03bc {x | C \u2264 \u2016f i x\u2016\u208a} \u2264 ENNReal.ofReal \u03b4\n\u22a2 False", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhp : p \u2260 0\nhp' : p \u2260 \u22a4\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhfu : UnifIntegrable f p \u03bc\nM : \u211d\u22650\nhM : \u2200 (i : \u03b9), snorm (f i) p \u03bc \u2264 \u2191M\n\u03b4 : \u211d\nh\u03b4pos : 0 < \u03b4\nh\u03b4 :\n  \u2200 (i : \u03b9) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nhcon : \u2200 (C : \u211d\u22650), \u2203 i, ENNReal.ofReal \u03b4 < \u2191\u2191\u03bc {x | C \u2264 \u2016f i x\u2016\u208a}\n\u22a2 False"}, {"tactic": "choose \u2110 h\u2110 using hcon", "annotated_tactic": ["choose \u2110 h\u2110 using hcon", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhp : p \u2260 0\nhp' : p \u2260 \u22a4\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhfu : UnifIntegrable f p \u03bc\nM : \u211d\u22650\nhM : \u2200 (i : \u03b9), snorm (f i) p \u03bc \u2264 \u2191M\n\u03b4 : \u211d\nh\u03b4pos : 0 < \u03b4\nh\u03b4 :\n  \u2200 (i : \u03b9) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nhcon : \u2200 (C : \u211d\u22650), \u2203 i, ENNReal.ofReal \u03b4 < \u2191\u2191\u03bc {x | C \u2264 \u2016f i x\u2016\u208a}\n\u22a2 False", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhp : p \u2260 0\nhp' : p \u2260 \u22a4\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhfu : UnifIntegrable f p \u03bc\nM : \u211d\u22650\nhM : \u2200 (i : \u03b9), snorm (f i) p \u03bc \u2264 \u2191M\n\u03b4 : \u211d\nh\u03b4pos : 0 < \u03b4\nh\u03b4 :\n  \u2200 (i : \u03b9) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\n\u2110 : \u211d\u22650 \u2192 \u03b9\nh\u2110 : \u2200 (C : \u211d\u22650), ENNReal.ofReal \u03b4 < \u2191\u2191\u03bc {x | C \u2264 \u2016f (\u2110 C) x\u2016\u208a}\n\u22a2 False"}, {"tactic": "lift \u03b4 to \u211d\u22650 using h\u03b4pos.le", "annotated_tactic": ["lift \u03b4 to \u211d\u22650 using h\u03b4pos.le", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhp : p \u2260 0\nhp' : p \u2260 \u22a4\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhfu : UnifIntegrable f p \u03bc\nM : \u211d\u22650\nhM : \u2200 (i : \u03b9), snorm (f i) p \u03bc \u2264 \u2191M\n\u03b4 : \u211d\nh\u03b4pos : 0 < \u03b4\nh\u03b4 :\n  \u2200 (i : \u03b9) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\n\u2110 : \u211d\u22650 \u2192 \u03b9\nh\u2110 : \u2200 (C : \u211d\u22650), ENNReal.ofReal \u03b4 < \u2191\u2191\u03bc {x | C \u2264 \u2016f (\u2110 C) x\u2016\u208a}\n\u22a2 False", "state_after": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhp : p \u2260 0\nhp' : p \u2260 \u22a4\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhfu : UnifIntegrable f p \u03bc\nM : \u211d\u22650\nhM : \u2200 (i : \u03b9), snorm (f i) p \u03bc \u2264 \u2191M\n\u2110 : \u211d\u22650 \u2192 \u03b9\n\u03b4 : \u211d\u22650\nh\u03b4pos : 0 < \u2191\u03b4\nh\u03b4 :\n  \u2200 (i : \u03b9) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u2191\u03b4 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nh\u2110 : \u2200 (C : \u211d\u22650), ENNReal.ofReal \u2191\u03b4 < \u2191\u2191\u03bc {x | C \u2264 \u2016f (\u2110 C) x\u2016\u208a}\n\u22a2 False"}, {"tactic": "have : \u2200 C : \u211d\u22650, C \u2022 (\u03b4 : \u211d\u22650\u221e) ^ (1 / p.toReal) \u2264 snorm (f (\u2110 C)) p \u03bc := by\n  intro C\n  calc\n    C \u2022 (\u03b4 : \u211d\u22650\u221e) ^ (1 / p.toReal) \u2264 C \u2022 \u03bc { x | C \u2264 \u2016f (\u2110 C) x\u2016\u208a } ^ (1 / p.toReal) := by\n      rw [ENNReal.smul_def, ENNReal.smul_def, smul_eq_mul, smul_eq_mul]\n      simp_rw [ENNReal.ofReal_coe_nnreal] at h\u2110\n      refine' mul_le_mul' le_rfl\n        (ENNReal.rpow_le_rpow (h\u2110 C).le (one_div_nonneg.2 ENNReal.toReal_nonneg))\n    _ \u2264 snorm ({ x | C \u2264 \u2016f (\u2110 C) x\u2016\u208a }.indicator (f (\u2110 C))) p \u03bc := by\n      refine' snorm_indicator_ge_of_bdd_below hp hp' _\n        (measurableSet_le measurable_const (hf _).nnnorm.measurable)\n        (eventually_of_forall fun x hx => _)\n      rwa [nnnorm_indicator_eq_indicator_nnnorm, Set.indicator_of_mem hx]\n    _ \u2264 snorm (f (\u2110 C)) p \u03bc := snorm_indicator_le _", "annotated_tactic": ["have : \u2200 C : \u211d\u22650, C \u2022 (\u03b4 : \u211d\u22650\u221e) ^ (1 / p.toReal) \u2264 <a>snorm</a> (f (\u2110 C)) p \u03bc := by\n      intro C\n      calc\n        C \u2022 (\u03b4 : \u211d\u22650\u221e) ^ (1 / p.toReal) \u2264 C \u2022 \u03bc { x | C \u2264 \u2016f (\u2110 C) x\u2016\u208a } ^ (1 / p.toReal) := by\n          rw [<a>ENNReal.smul_def</a>, <a>ENNReal.smul_def</a>, <a>smul_eq_mul</a>, <a>smul_eq_mul</a>]\n          simp_rw [<a>ENNReal.ofReal_coe_nnreal</a>] at h\u2110\n          refine' <a>mul_le_mul'</a> <a>le_rfl</a>\n            (<a>ENNReal.rpow_le_rpow</a> (h\u2110 C).<a>le</a> (<a>one_div_nonneg</a>.2 <a>ENNReal.toReal_nonneg</a>))\n        _ \u2264 <a>snorm</a> ({ x | C \u2264 \u2016f (\u2110 C) x\u2016\u208a }.<a>indicator</a> (f (\u2110 C))) p \u03bc := by\n          refine' <a>snorm_indicator_ge_of_bdd_below</a> hp hp' _\n            (<a>measurableSet_le</a> <a>measurable_const</a> (hf _).nnnorm.measurable)\n            (<a>eventually_of_forall</a> fun x hx => _)\n          rwa [<a>nnnorm_indicator_eq_indicator_nnnorm</a>, <a>Set.indicator_of_mem</a> hx]\n        _ \u2264 <a>snorm</a> (f (\u2110 C)) p \u03bc := <a>snorm_indicator_le</a> _", [{"full_name": "MeasureTheory.snorm", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [84, 5], "def_end_pos": [84, 10]}, {"full_name": "ENNReal.smul_def", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [503, 9], "def_end_pos": [503, 17]}, {"full_name": "ENNReal.smul_def", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [503, 9], "def_end_pos": [503, 17]}, {"full_name": "smul_eq_mul", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [93, 9], "def_end_pos": [93, 20]}, {"full_name": "smul_eq_mul", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [93, 9], "def_end_pos": [93, 20]}, {"full_name": "ENNReal.ofReal_coe_nnreal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [212, 17], "def_end_pos": [212, 34]}, {"full_name": "mul_le_mul'", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [206, 9], "def_end_pos": [206, 20]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}, {"full_name": "ENNReal.rpow_le_rpow", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [642, 9], "def_end_pos": [642, 21]}, {"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [142, 7], "def_end_pos": [142, 15]}, {"full_name": "one_div_nonneg", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [81, 9], "def_end_pos": [81, 23]}, {"full_name": "ENNReal.toReal_nonneg", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [221, 17], "def_end_pos": [221, 30]}, {"full_name": "MeasureTheory.snorm", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [84, 5], "def_end_pos": [84, 10]}, {"full_name": "Set.indicator", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [46, 3], "def_end_pos": [46, 14]}, {"full_name": "MeasureTheory.snorm_indicator_ge_of_bdd_below", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [1589, 9], "def_end_pos": [1589, 40]}, {"full_name": "measurableSet_le", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [559, 9], "def_end_pos": [559, 25]}, {"full_name": "measurable_const", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [570, 9], "def_end_pos": [570, 25]}, {"full_name": "Filter.eventually_of_forall", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1111, 9], "def_end_pos": [1111, 29]}, {"full_name": "nnnorm_indicator_eq_indicator_nnnorm", "def_path": "Mathlib/Analysis/NormedSpace/IndicatorFunction.lean", "def_pos": [29, 9], "def_end_pos": [29, 45]}, {"full_name": "Set.indicator_of_mem", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [67, 3], "def_end_pos": [67, 14]}, {"full_name": "MeasureTheory.snorm", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [84, 5], "def_end_pos": [84, 10]}, {"full_name": "MeasureTheory.snorm_indicator_le", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [586, 9], "def_end_pos": [586, 27]}]], "state_before": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhp : p \u2260 0\nhp' : p \u2260 \u22a4\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhfu : UnifIntegrable f p \u03bc\nM : \u211d\u22650\nhM : \u2200 (i : \u03b9), snorm (f i) p \u03bc \u2264 \u2191M\n\u2110 : \u211d\u22650 \u2192 \u03b9\n\u03b4 : \u211d\u22650\nh\u03b4pos : 0 < \u2191\u03b4\nh\u03b4 :\n  \u2200 (i : \u03b9) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u2191\u03b4 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nh\u2110 : \u2200 (C : \u211d\u22650), ENNReal.ofReal \u2191\u03b4 < \u2191\u2191\u03bc {x | C \u2264 \u2016f (\u2110 C) x\u2016\u208a}\n\u22a2 False", "state_after": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhp : p \u2260 0\nhp' : p \u2260 \u22a4\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhfu : UnifIntegrable f p \u03bc\nM : \u211d\u22650\nhM : \u2200 (i : \u03b9), snorm (f i) p \u03bc \u2264 \u2191M\n\u2110 : \u211d\u22650 \u2192 \u03b9\n\u03b4 : \u211d\u22650\nh\u03b4pos : 0 < \u2191\u03b4\nh\u03b4 :\n  \u2200 (i : \u03b9) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u2191\u03b4 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nh\u2110 : \u2200 (C : \u211d\u22650), ENNReal.ofReal \u2191\u03b4 < \u2191\u2191\u03bc {x | C \u2264 \u2016f (\u2110 C) x\u2016\u208a}\nthis : \u2200 (C : \u211d\u22650), C \u2022 \u2191\u03b4 ^ (1 / ENNReal.toReal p) \u2264 snorm (f (\u2110 C)) p \u03bc\n\u22a2 False"}, {"tactic": "specialize this (2 * max M 1 * HPow.hPow \u03b4\u207b\u00b9 (1 / p.toReal))", "annotated_tactic": ["specialize this (2 * <a>max</a> M 1 * <a>HPow.hPow</a> \u03b4\u207b\u00b9 (1 / p.toReal))", [{"full_name": "Max.max", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1090, 3], "def_end_pos": [1090, 6]}, {"full_name": "HPow.hPow", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1192, 3], "def_end_pos": [1192, 7]}]], "state_before": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhp : p \u2260 0\nhp' : p \u2260 \u22a4\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhfu : UnifIntegrable f p \u03bc\nM : \u211d\u22650\nhM : \u2200 (i : \u03b9), snorm (f i) p \u03bc \u2264 \u2191M\n\u2110 : \u211d\u22650 \u2192 \u03b9\n\u03b4 : \u211d\u22650\nh\u03b4pos : 0 < \u2191\u03b4\nh\u03b4 :\n  \u2200 (i : \u03b9) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u2191\u03b4 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nh\u2110 : \u2200 (C : \u211d\u22650), ENNReal.ofReal \u2191\u03b4 < \u2191\u2191\u03bc {x | C \u2264 \u2016f (\u2110 C) x\u2016\u208a}\nthis : \u2200 (C : \u211d\u22650), C \u2022 \u2191\u03b4 ^ (1 / ENNReal.toReal p) \u2264 snorm (f (\u2110 C)) p \u03bc\n\u22a2 False", "state_after": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhp : p \u2260 0\nhp' : p \u2260 \u22a4\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhfu : UnifIntegrable f p \u03bc\nM : \u211d\u22650\nhM : \u2200 (i : \u03b9), snorm (f i) p \u03bc \u2264 \u2191M\n\u2110 : \u211d\u22650 \u2192 \u03b9\n\u03b4 : \u211d\u22650\nh\u03b4pos : 0 < \u2191\u03b4\nh\u03b4 :\n  \u2200 (i : \u03b9) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u2191\u03b4 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nh\u2110 : \u2200 (C : \u211d\u22650), ENNReal.ofReal \u2191\u03b4 < \u2191\u2191\u03bc {x | C \u2264 \u2016f (\u2110 C) x\u2016\u208a}\nthis :\n  (2 * max M 1 * \u03b4\u207b\u00b9 ^ (1 / ENNReal.toReal p)) \u2022 \u2191\u03b4 ^ (1 / ENNReal.toReal p) \u2264\n    snorm (f (\u2110 (2 * max M 1 * \u03b4\u207b\u00b9 ^ (1 / ENNReal.toReal p)))) p \u03bc\n\u22a2 False"}, {"tactic": "rw [ENNReal.coe_rpow_of_nonneg _ (one_div_nonneg.2 ENNReal.toReal_nonneg), \u2190 ENNReal.coe_smul,\n  smul_eq_mul, mul_assoc, NNReal.inv_rpow,\n  inv_mul_cancel (NNReal.rpow_pos (NNReal.coe_pos.1 h\u03b4pos)).ne.symm, mul_one, ENNReal.coe_mul,\n  \u2190 NNReal.inv_rpow] at this", "annotated_tactic": ["rw [<a>ENNReal.coe_rpow_of_nonneg</a> _ (<a>one_div_nonneg</a>.2 <a>ENNReal.toReal_nonneg</a>), \u2190 <a>ENNReal.coe_smul</a>,\n      <a>smul_eq_mul</a>, <a>mul_assoc</a>, <a>NNReal.inv_rpow</a>,\n      <a>inv_mul_cancel</a> (<a>NNReal.rpow_pos</a> (<a>NNReal.coe_pos</a>.1 h\u03b4pos)).ne.symm, <a>mul_one</a>, <a>ENNReal.coe_mul</a>,\n      \u2190 <a>NNReal.inv_rpow</a>] at this", [{"full_name": "ENNReal.coe_rpow_of_nonneg", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [436, 9], "def_end_pos": [436, 27]}, {"full_name": "one_div_nonneg", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [81, 9], "def_end_pos": [81, 23]}, {"full_name": "ENNReal.toReal_nonneg", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [221, 17], "def_end_pos": [221, 30]}, {"full_name": "ENNReal.coe_smul", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [539, 9], "def_end_pos": [539, 17]}, {"full_name": "smul_eq_mul", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [93, 9], "def_end_pos": [93, 20]}, {"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [264, 9], "def_end_pos": [264, 18]}, {"full_name": "NNReal.inv_rpow", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [112, 9], "def_end_pos": [112, 17]}, {"full_name": "inv_mul_cancel", "def_path": "Mathlib/Algebra/GroupWithZero/NeZero.lean", "def_pos": [55, 9], "def_end_pos": [55, 23]}, {"full_name": "NNReal.rpow_pos", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [245, 9], "def_end_pos": [245, 17]}, {"full_name": "NNReal.coe_pos", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [376, 19], "def_end_pos": [376, 26]}, {"full_name": "mul_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [470, 9], "def_end_pos": [470, 16]}, {"full_name": "ENNReal.coe_mul", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [390, 9], "def_end_pos": [390, 16]}, {"full_name": "NNReal.inv_rpow", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [112, 9], "def_end_pos": [112, 17]}]], "state_before": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhp : p \u2260 0\nhp' : p \u2260 \u22a4\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhfu : UnifIntegrable f p \u03bc\nM : \u211d\u22650\nhM : \u2200 (i : \u03b9), snorm (f i) p \u03bc \u2264 \u2191M\n\u2110 : \u211d\u22650 \u2192 \u03b9\n\u03b4 : \u211d\u22650\nh\u03b4pos : 0 < \u2191\u03b4\nh\u03b4 :\n  \u2200 (i : \u03b9) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u2191\u03b4 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nh\u2110 : \u2200 (C : \u211d\u22650), ENNReal.ofReal \u2191\u03b4 < \u2191\u2191\u03bc {x | C \u2264 \u2016f (\u2110 C) x\u2016\u208a}\nthis :\n  (2 * max M 1 * \u03b4\u207b\u00b9 ^ (1 / ENNReal.toReal p)) \u2022 \u2191\u03b4 ^ (1 / ENNReal.toReal p) \u2264\n    snorm (f (\u2110 (2 * max M 1 * \u03b4\u207b\u00b9 ^ (1 / ENNReal.toReal p)))) p \u03bc\n\u22a2 False", "state_after": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhp : p \u2260 0\nhp' : p \u2260 \u22a4\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhfu : UnifIntegrable f p \u03bc\nM : \u211d\u22650\nhM : \u2200 (i : \u03b9), snorm (f i) p \u03bc \u2264 \u2191M\n\u2110 : \u211d\u22650 \u2192 \u03b9\n\u03b4 : \u211d\u22650\nh\u03b4pos : 0 < \u2191\u03b4\nh\u03b4 :\n  \u2200 (i : \u03b9) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u2191\u03b4 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nh\u2110 : \u2200 (C : \u211d\u22650), ENNReal.ofReal \u2191\u03b4 < \u2191\u2191\u03bc {x | C \u2264 \u2016f (\u2110 C) x\u2016\u208a}\nthis : \u21912 * \u2191(max M 1) \u2264 snorm (f (\u2110 (2 * max M 1 * \u03b4\u207b\u00b9 ^ (1 / ENNReal.toReal p)))) p \u03bc\n\u22a2 False"}, {"tactic": "refine' (lt_of_le_of_lt (le_trans\n  (hM <| \u2110 <| 2 * max M 1 * HPow.hPow \u03b4\u207b\u00b9 (1 / p.toReal)) (le_max_left (M : \u211d\u22650\u221e) 1))\n    (lt_of_lt_of_le _ this)).ne rfl", "annotated_tactic": ["refine' (<a>lt_of_le_of_lt</a> (<a>le_trans</a>\n      (hM <| \u2110 <| 2 * <a>max</a> M 1 * <a>HPow.hPow</a> \u03b4\u207b\u00b9 (1 / p.toReal)) (<a>le_max_left</a> (M : \u211d\u22650\u221e) 1))\n        (<a>lt_of_lt_of_le</a> _ this)).<a>ne</a> <a>rfl</a>", [{"full_name": "lt_of_le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [122, 9], "def_end_pos": [122, 23]}, {"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}, {"full_name": "Max.max", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1090, 3], "def_end_pos": [1090, 6]}, {"full_name": "HPow.hPow", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1192, 3], "def_end_pos": [1192, 7]}, {"full_name": "le_max_left", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [54, 9], "def_end_pos": [54, 20]}, {"full_name": "lt_of_lt_of_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [115, 9], "def_end_pos": [115, 23]}, {"full_name": "LT.lt.ne", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [152, 7], "def_end_pos": [152, 15]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhp : p \u2260 0\nhp' : p \u2260 \u22a4\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhfu : UnifIntegrable f p \u03bc\nM : \u211d\u22650\nhM : \u2200 (i : \u03b9), snorm (f i) p \u03bc \u2264 \u2191M\n\u2110 : \u211d\u22650 \u2192 \u03b9\n\u03b4 : \u211d\u22650\nh\u03b4pos : 0 < \u2191\u03b4\nh\u03b4 :\n  \u2200 (i : \u03b9) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u2191\u03b4 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nh\u2110 : \u2200 (C : \u211d\u22650), ENNReal.ofReal \u2191\u03b4 < \u2191\u2191\u03bc {x | C \u2264 \u2016f (\u2110 C) x\u2016\u208a}\nthis : \u21912 * \u2191(max M 1) \u2264 snorm (f (\u2110 (2 * max M 1 * \u03b4\u207b\u00b9 ^ (1 / ENNReal.toReal p)))) p \u03bc\n\u22a2 False", "state_after": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhp : p \u2260 0\nhp' : p \u2260 \u22a4\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhfu : UnifIntegrable f p \u03bc\nM : \u211d\u22650\nhM : \u2200 (i : \u03b9), snorm (f i) p \u03bc \u2264 \u2191M\n\u2110 : \u211d\u22650 \u2192 \u03b9\n\u03b4 : \u211d\u22650\nh\u03b4pos : 0 < \u2191\u03b4\nh\u03b4 :\n  \u2200 (i : \u03b9) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u2191\u03b4 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nh\u2110 : \u2200 (C : \u211d\u22650), ENNReal.ofReal \u2191\u03b4 < \u2191\u2191\u03bc {x | C \u2264 \u2016f (\u2110 C) x\u2016\u208a}\nthis : \u21912 * \u2191(max M 1) \u2264 snorm (f (\u2110 (2 * max M 1 * \u03b4\u207b\u00b9 ^ (1 / ENNReal.toReal p)))) p \u03bc\n\u22a2 max (\u2191M) 1 < \u21912 * \u2191(max M 1)"}, {"tactic": "rw [\u2190 ENNReal.coe_one, \u2190 ENNReal.coe_max, \u2190 ENNReal.coe_mul, ENNReal.coe_lt_coe]", "annotated_tactic": ["rw [\u2190 <a>ENNReal.coe_one</a>, \u2190 <a>ENNReal.coe_max</a>, \u2190 <a>ENNReal.coe_mul</a>, <a>ENNReal.coe_lt_coe</a>]", [{"full_name": "ENNReal.coe_one", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [218, 28], "def_end_pos": [218, 35]}, {"full_name": "ENNReal.coe_max", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [945, 9], "def_end_pos": [945, 16]}, {"full_name": "ENNReal.coe_mul", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [390, 9], "def_end_pos": [390, 16]}, {"full_name": "ENNReal.coe_lt_coe", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [352, 28], "def_end_pos": [352, 38]}]], "state_before": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhp : p \u2260 0\nhp' : p \u2260 \u22a4\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhfu : UnifIntegrable f p \u03bc\nM : \u211d\u22650\nhM : \u2200 (i : \u03b9), snorm (f i) p \u03bc \u2264 \u2191M\n\u2110 : \u211d\u22650 \u2192 \u03b9\n\u03b4 : \u211d\u22650\nh\u03b4pos : 0 < \u2191\u03b4\nh\u03b4 :\n  \u2200 (i : \u03b9) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u2191\u03b4 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nh\u2110 : \u2200 (C : \u211d\u22650), ENNReal.ofReal \u2191\u03b4 < \u2191\u2191\u03bc {x | C \u2264 \u2016f (\u2110 C) x\u2016\u208a}\nthis : \u21912 * \u2191(max M 1) \u2264 snorm (f (\u2110 (2 * max M 1 * \u03b4\u207b\u00b9 ^ (1 / ENNReal.toReal p)))) p \u03bc\n\u22a2 max (\u2191M) 1 < \u21912 * \u2191(max M 1)", "state_after": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhp : p \u2260 0\nhp' : p \u2260 \u22a4\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhfu : UnifIntegrable f p \u03bc\nM : \u211d\u22650\nhM : \u2200 (i : \u03b9), snorm (f i) p \u03bc \u2264 \u2191M\n\u2110 : \u211d\u22650 \u2192 \u03b9\n\u03b4 : \u211d\u22650\nh\u03b4pos : 0 < \u2191\u03b4\nh\u03b4 :\n  \u2200 (i : \u03b9) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u2191\u03b4 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nh\u2110 : \u2200 (C : \u211d\u22650), ENNReal.ofReal \u2191\u03b4 < \u2191\u2191\u03bc {x | C \u2264 \u2016f (\u2110 C) x\u2016\u208a}\nthis : \u21912 * \u2191(max M 1) \u2264 snorm (f (\u2110 (2 * max M 1 * \u03b4\u207b\u00b9 ^ (1 / ENNReal.toReal p)))) p \u03bc\n\u22a2 max M 1 < 2 * max M 1"}, {"tactic": "exact lt_two_mul_self (lt_max_of_lt_right one_pos)", "annotated_tactic": ["exact <a>lt_two_mul_self</a> (<a>lt_max_of_lt_right</a> <a>one_pos</a>)", [{"full_name": "lt_two_mul_self", "def_path": "Mathlib/Algebra/Order/Ring/Defs.lean", "def_pos": [622, 9], "def_end_pos": [622, 24]}, {"full_name": "lt_max_of_lt_right", "def_path": "Mathlib/Order/MinMax.lean", "def_pos": [102, 9], "def_end_pos": [102, 27]}, {"full_name": "one_pos", "def_path": "Mathlib/Algebra/Order/ZeroLEOne.lean", "def_pos": [50, 7], "def_end_pos": [50, 14]}]], "state_before": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhp : p \u2260 0\nhp' : p \u2260 \u22a4\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhfu : UnifIntegrable f p \u03bc\nM : \u211d\u22650\nhM : \u2200 (i : \u03b9), snorm (f i) p \u03bc \u2264 \u2191M\n\u2110 : \u211d\u22650 \u2192 \u03b9\n\u03b4 : \u211d\u22650\nh\u03b4pos : 0 < \u2191\u03b4\nh\u03b4 :\n  \u2200 (i : \u03b9) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u2191\u03b4 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nh\u2110 : \u2200 (C : \u211d\u22650), ENNReal.ofReal \u2191\u03b4 < \u2191\u2191\u03bc {x | C \u2264 \u2016f (\u2110 C) x\u2016\u208a}\nthis : \u21912 * \u2191(max M 1) \u2264 snorm (f (\u2110 (2 * max M 1 * \u03b4\u207b\u00b9 ^ (1 / ENNReal.toReal p)))) p \u03bc\n\u22a2 max M 1 < 2 * max M 1", "state_after": "no goals"}, {"tactic": "intro C", "annotated_tactic": ["intro C", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhp : p \u2260 0\nhp' : p \u2260 \u22a4\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhfu : UnifIntegrable f p \u03bc\nM : \u211d\u22650\nhM : \u2200 (i : \u03b9), snorm (f i) p \u03bc \u2264 \u2191M\n\u2110 : \u211d\u22650 \u2192 \u03b9\n\u03b4 : \u211d\u22650\nh\u03b4pos : 0 < \u2191\u03b4\nh\u03b4 :\n  \u2200 (i : \u03b9) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u2191\u03b4 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nh\u2110 : \u2200 (C : \u211d\u22650), ENNReal.ofReal \u2191\u03b4 < \u2191\u2191\u03bc {x | C \u2264 \u2016f (\u2110 C) x\u2016\u208a}\n\u22a2 \u2200 (C : \u211d\u22650), C \u2022 \u2191\u03b4 ^ (1 / ENNReal.toReal p) \u2264 snorm (f (\u2110 C)) p \u03bc", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhp : p \u2260 0\nhp' : p \u2260 \u22a4\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhfu : UnifIntegrable f p \u03bc\nM : \u211d\u22650\nhM : \u2200 (i : \u03b9), snorm (f i) p \u03bc \u2264 \u2191M\n\u2110 : \u211d\u22650 \u2192 \u03b9\n\u03b4 : \u211d\u22650\nh\u03b4pos : 0 < \u2191\u03b4\nh\u03b4 :\n  \u2200 (i : \u03b9) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u2191\u03b4 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nh\u2110 : \u2200 (C : \u211d\u22650), ENNReal.ofReal \u2191\u03b4 < \u2191\u2191\u03bc {x | C \u2264 \u2016f (\u2110 C) x\u2016\u208a}\nC : \u211d\u22650\n\u22a2 C \u2022 \u2191\u03b4 ^ (1 / ENNReal.toReal p) \u2264 snorm (f (\u2110 C)) p \u03bc"}, {"tactic": "calc\n  C \u2022 (\u03b4 : \u211d\u22650\u221e) ^ (1 / p.toReal) \u2264 C \u2022 \u03bc { x | C \u2264 \u2016f (\u2110 C) x\u2016\u208a } ^ (1 / p.toReal) := by\n    rw [ENNReal.smul_def, ENNReal.smul_def, smul_eq_mul, smul_eq_mul]\n    simp_rw [ENNReal.ofReal_coe_nnreal] at h\u2110\n    refine' mul_le_mul' le_rfl\n      (ENNReal.rpow_le_rpow (h\u2110 C).le (one_div_nonneg.2 ENNReal.toReal_nonneg))\n  _ \u2264 snorm ({ x | C \u2264 \u2016f (\u2110 C) x\u2016\u208a }.indicator (f (\u2110 C))) p \u03bc := by\n    refine' snorm_indicator_ge_of_bdd_below hp hp' _\n      (measurableSet_le measurable_const (hf _).nnnorm.measurable)\n      (eventually_of_forall fun x hx => _)\n    rwa [nnnorm_indicator_eq_indicator_nnnorm, Set.indicator_of_mem hx]\n  _ \u2264 snorm (f (\u2110 C)) p \u03bc := snorm_indicator_le _", "annotated_tactic": ["calc\n        C \u2022 (\u03b4 : \u211d\u22650\u221e) ^ (1 / p.toReal) \u2264 C \u2022 \u03bc { x | C \u2264 \u2016f (\u2110 C) x\u2016\u208a } ^ (1 / p.toReal) := by\n          rw [<a>ENNReal.smul_def</a>, <a>ENNReal.smul_def</a>, <a>smul_eq_mul</a>, <a>smul_eq_mul</a>]\n          simp_rw [<a>ENNReal.ofReal_coe_nnreal</a>] at h\u2110\n          refine' <a>mul_le_mul'</a> <a>le_rfl</a>\n            (<a>ENNReal.rpow_le_rpow</a> (h\u2110 C).<a>le</a> (<a>one_div_nonneg</a>.2 <a>ENNReal.toReal_nonneg</a>))\n        _ \u2264 <a>snorm</a> ({ x | C \u2264 \u2016f (\u2110 C) x\u2016\u208a }.<a>indicator</a> (f (\u2110 C))) p \u03bc := by\n          refine' <a>snorm_indicator_ge_of_bdd_below</a> hp hp' _\n            (<a>measurableSet_le</a> <a>measurable_const</a> (hf _).nnnorm.measurable)\n            (<a>eventually_of_forall</a> fun x hx => _)\n          rwa [<a>nnnorm_indicator_eq_indicator_nnnorm</a>, <a>Set.indicator_of_mem</a> hx]\n        _ \u2264 <a>snorm</a> (f (\u2110 C)) p \u03bc := <a>snorm_indicator_le</a> _", [{"full_name": "ENNReal.smul_def", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [503, 9], "def_end_pos": [503, 17]}, {"full_name": "ENNReal.smul_def", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [503, 9], "def_end_pos": [503, 17]}, {"full_name": "smul_eq_mul", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [93, 9], "def_end_pos": [93, 20]}, {"full_name": "smul_eq_mul", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [93, 9], "def_end_pos": [93, 20]}, {"full_name": "ENNReal.ofReal_coe_nnreal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [212, 17], "def_end_pos": [212, 34]}, {"full_name": "mul_le_mul'", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [206, 9], "def_end_pos": [206, 20]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}, {"full_name": "ENNReal.rpow_le_rpow", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [642, 9], "def_end_pos": [642, 21]}, {"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [142, 7], "def_end_pos": [142, 15]}, {"full_name": "one_div_nonneg", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [81, 9], "def_end_pos": [81, 23]}, {"full_name": "ENNReal.toReal_nonneg", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [221, 17], "def_end_pos": [221, 30]}, {"full_name": "MeasureTheory.snorm", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [84, 5], "def_end_pos": [84, 10]}, {"full_name": "Set.indicator", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [46, 3], "def_end_pos": [46, 14]}, {"full_name": "MeasureTheory.snorm_indicator_ge_of_bdd_below", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [1589, 9], "def_end_pos": [1589, 40]}, {"full_name": "measurableSet_le", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [559, 9], "def_end_pos": [559, 25]}, {"full_name": "measurable_const", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [570, 9], "def_end_pos": [570, 25]}, {"full_name": "Filter.eventually_of_forall", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1111, 9], "def_end_pos": [1111, 29]}, {"full_name": "nnnorm_indicator_eq_indicator_nnnorm", "def_path": "Mathlib/Analysis/NormedSpace/IndicatorFunction.lean", "def_pos": [29, 9], "def_end_pos": [29, 45]}, {"full_name": "Set.indicator_of_mem", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [67, 3], "def_end_pos": [67, 14]}, {"full_name": "MeasureTheory.snorm", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [84, 5], "def_end_pos": [84, 10]}, {"full_name": "MeasureTheory.snorm_indicator_le", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [586, 9], "def_end_pos": [586, 27]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhp : p \u2260 0\nhp' : p \u2260 \u22a4\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhfu : UnifIntegrable f p \u03bc\nM : \u211d\u22650\nhM : \u2200 (i : \u03b9), snorm (f i) p \u03bc \u2264 \u2191M\n\u2110 : \u211d\u22650 \u2192 \u03b9\n\u03b4 : \u211d\u22650\nh\u03b4pos : 0 < \u2191\u03b4\nh\u03b4 :\n  \u2200 (i : \u03b9) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u2191\u03b4 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nh\u2110 : \u2200 (C : \u211d\u22650), ENNReal.ofReal \u2191\u03b4 < \u2191\u2191\u03bc {x | C \u2264 \u2016f (\u2110 C) x\u2016\u208a}\nC : \u211d\u22650\n\u22a2 C \u2022 \u2191\u03b4 ^ (1 / ENNReal.toReal p) \u2264 snorm (f (\u2110 C)) p \u03bc", "state_after": "no goals"}, {"tactic": "rw [ENNReal.smul_def, ENNReal.smul_def, smul_eq_mul, smul_eq_mul]", "annotated_tactic": ["rw [<a>ENNReal.smul_def</a>, <a>ENNReal.smul_def</a>, <a>smul_eq_mul</a>, <a>smul_eq_mul</a>]", [{"full_name": "ENNReal.smul_def", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [503, 9], "def_end_pos": [503, 17]}, {"full_name": "ENNReal.smul_def", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [503, 9], "def_end_pos": [503, 17]}, {"full_name": "smul_eq_mul", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [93, 9], "def_end_pos": [93, 20]}, {"full_name": "smul_eq_mul", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [93, 9], "def_end_pos": [93, 20]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhp : p \u2260 0\nhp' : p \u2260 \u22a4\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhfu : UnifIntegrable f p \u03bc\nM : \u211d\u22650\nhM : \u2200 (i : \u03b9), snorm (f i) p \u03bc \u2264 \u2191M\n\u2110 : \u211d\u22650 \u2192 \u03b9\n\u03b4 : \u211d\u22650\nh\u03b4pos : 0 < \u2191\u03b4\nh\u03b4 :\n  \u2200 (i : \u03b9) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u2191\u03b4 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nh\u2110 : \u2200 (C : \u211d\u22650), ENNReal.ofReal \u2191\u03b4 < \u2191\u2191\u03bc {x | C \u2264 \u2016f (\u2110 C) x\u2016\u208a}\nC : \u211d\u22650\n\u22a2 C \u2022 \u2191\u03b4 ^ (1 / ENNReal.toReal p) \u2264 C \u2022 \u2191\u2191\u03bc {x | C \u2264 \u2016f (\u2110 C) x\u2016\u208a} ^ (1 / ENNReal.toReal p)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhp : p \u2260 0\nhp' : p \u2260 \u22a4\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhfu : UnifIntegrable f p \u03bc\nM : \u211d\u22650\nhM : \u2200 (i : \u03b9), snorm (f i) p \u03bc \u2264 \u2191M\n\u2110 : \u211d\u22650 \u2192 \u03b9\n\u03b4 : \u211d\u22650\nh\u03b4pos : 0 < \u2191\u03b4\nh\u03b4 :\n  \u2200 (i : \u03b9) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u2191\u03b4 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nh\u2110 : \u2200 (C : \u211d\u22650), ENNReal.ofReal \u2191\u03b4 < \u2191\u2191\u03bc {x | C \u2264 \u2016f (\u2110 C) x\u2016\u208a}\nC : \u211d\u22650\n\u22a2 \u2191C * \u2191\u03b4 ^ (1 / ENNReal.toReal p) \u2264 \u2191C * \u2191\u2191\u03bc {x | C \u2264 \u2016f (\u2110 C) x\u2016\u208a} ^ (1 / ENNReal.toReal p)"}, {"tactic": "simp_rw [ENNReal.ofReal_coe_nnreal] at h\u2110", "annotated_tactic": ["simp_rw [<a>ENNReal.ofReal_coe_nnreal</a>] at h\u2110", [{"full_name": "ENNReal.ofReal_coe_nnreal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [212, 17], "def_end_pos": [212, 34]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhp : p \u2260 0\nhp' : p \u2260 \u22a4\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhfu : UnifIntegrable f p \u03bc\nM : \u211d\u22650\nhM : \u2200 (i : \u03b9), snorm (f i) p \u03bc \u2264 \u2191M\n\u2110 : \u211d\u22650 \u2192 \u03b9\n\u03b4 : \u211d\u22650\nh\u03b4pos : 0 < \u2191\u03b4\nh\u03b4 :\n  \u2200 (i : \u03b9) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u2191\u03b4 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nh\u2110 : \u2200 (C : \u211d\u22650), ENNReal.ofReal \u2191\u03b4 < \u2191\u2191\u03bc {x | C \u2264 \u2016f (\u2110 C) x\u2016\u208a}\nC : \u211d\u22650\n\u22a2 \u2191C * \u2191\u03b4 ^ (1 / ENNReal.toReal p) \u2264 \u2191C * \u2191\u2191\u03bc {x | C \u2264 \u2016f (\u2110 C) x\u2016\u208a} ^ (1 / ENNReal.toReal p)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhp : p \u2260 0\nhp' : p \u2260 \u22a4\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhfu : UnifIntegrable f p \u03bc\nM : \u211d\u22650\nhM : \u2200 (i : \u03b9), snorm (f i) p \u03bc \u2264 \u2191M\n\u2110 : \u211d\u22650 \u2192 \u03b9\n\u03b4 : \u211d\u22650\nh\u03b4pos : 0 < \u2191\u03b4\nh\u03b4 :\n  \u2200 (i : \u03b9) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u2191\u03b4 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nC : \u211d\u22650\nh\u2110 : \u2200 (C : \u211d\u22650), \u2191\u03b4 < \u2191\u2191\u03bc {x | C \u2264 \u2016f (\u2110 C) x\u2016\u208a}\n\u22a2 \u2191C * \u2191\u03b4 ^ (1 / ENNReal.toReal p) \u2264 \u2191C * \u2191\u2191\u03bc {x | C \u2264 \u2016f (\u2110 C) x\u2016\u208a} ^ (1 / ENNReal.toReal p)"}, {"tactic": "refine' mul_le_mul' le_rfl\n  (ENNReal.rpow_le_rpow (h\u2110 C).le (one_div_nonneg.2 ENNReal.toReal_nonneg))", "annotated_tactic": ["refine' <a>mul_le_mul'</a> <a>le_rfl</a>\n            (<a>ENNReal.rpow_le_rpow</a> (h\u2110 C).<a>le</a> (<a>one_div_nonneg</a>.2 <a>ENNReal.toReal_nonneg</a>))", [{"full_name": "mul_le_mul'", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [206, 9], "def_end_pos": [206, 20]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}, {"full_name": "ENNReal.rpow_le_rpow", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [642, 9], "def_end_pos": [642, 21]}, {"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [142, 7], "def_end_pos": [142, 15]}, {"full_name": "one_div_nonneg", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [81, 9], "def_end_pos": [81, 23]}, {"full_name": "ENNReal.toReal_nonneg", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [221, 17], "def_end_pos": [221, 30]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhp : p \u2260 0\nhp' : p \u2260 \u22a4\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhfu : UnifIntegrable f p \u03bc\nM : \u211d\u22650\nhM : \u2200 (i : \u03b9), snorm (f i) p \u03bc \u2264 \u2191M\n\u2110 : \u211d\u22650 \u2192 \u03b9\n\u03b4 : \u211d\u22650\nh\u03b4pos : 0 < \u2191\u03b4\nh\u03b4 :\n  \u2200 (i : \u03b9) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u2191\u03b4 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nC : \u211d\u22650\nh\u2110 : \u2200 (C : \u211d\u22650), \u2191\u03b4 < \u2191\u2191\u03bc {x | C \u2264 \u2016f (\u2110 C) x\u2016\u208a}\n\u22a2 \u2191C * \u2191\u03b4 ^ (1 / ENNReal.toReal p) \u2264 \u2191C * \u2191\u2191\u03bc {x | C \u2264 \u2016f (\u2110 C) x\u2016\u208a} ^ (1 / ENNReal.toReal p)", "state_after": "no goals"}, {"tactic": "refine' snorm_indicator_ge_of_bdd_below hp hp' _\n  (measurableSet_le measurable_const (hf _).nnnorm.measurable)\n  (eventually_of_forall fun x hx => _)", "annotated_tactic": ["refine' <a>snorm_indicator_ge_of_bdd_below</a> hp hp' _\n            (<a>measurableSet_le</a> <a>measurable_const</a> (hf _).nnnorm.measurable)\n            (<a>eventually_of_forall</a> fun x hx => _)", [{"full_name": "MeasureTheory.snorm_indicator_ge_of_bdd_below", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [1589, 9], "def_end_pos": [1589, 40]}, {"full_name": "measurableSet_le", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [559, 9], "def_end_pos": [559, 25]}, {"full_name": "measurable_const", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [570, 9], "def_end_pos": [570, 25]}, {"full_name": "Filter.eventually_of_forall", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1111, 9], "def_end_pos": [1111, 29]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhp : p \u2260 0\nhp' : p \u2260 \u22a4\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhfu : UnifIntegrable f p \u03bc\nM : \u211d\u22650\nhM : \u2200 (i : \u03b9), snorm (f i) p \u03bc \u2264 \u2191M\n\u2110 : \u211d\u22650 \u2192 \u03b9\n\u03b4 : \u211d\u22650\nh\u03b4pos : 0 < \u2191\u03b4\nh\u03b4 :\n  \u2200 (i : \u03b9) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u2191\u03b4 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nh\u2110 : \u2200 (C : \u211d\u22650), ENNReal.ofReal \u2191\u03b4 < \u2191\u2191\u03bc {x | C \u2264 \u2016f (\u2110 C) x\u2016\u208a}\nC : \u211d\u22650\n\u22a2 C \u2022 \u2191\u2191\u03bc {x | C \u2264 \u2016f (\u2110 C) x\u2016\u208a} ^ (1 / ENNReal.toReal p) \u2264 snorm (indicator {x | C \u2264 \u2016f (\u2110 C) x\u2016\u208a} (f (\u2110 C))) p \u03bc", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhp : p \u2260 0\nhp' : p \u2260 \u22a4\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhfu : UnifIntegrable f p \u03bc\nM : \u211d\u22650\nhM : \u2200 (i : \u03b9), snorm (f i) p \u03bc \u2264 \u2191M\n\u2110 : \u211d\u22650 \u2192 \u03b9\n\u03b4 : \u211d\u22650\nh\u03b4pos : 0 < \u2191\u03b4\nh\u03b4 :\n  \u2200 (i : \u03b9) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u2191\u03b4 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nh\u2110 : \u2200 (C : \u211d\u22650), ENNReal.ofReal \u2191\u03b4 < \u2191\u2191\u03bc {x | C \u2264 \u2016f (\u2110 C) x\u2016\u208a}\nC : \u211d\u22650\nx : \u03b1\nhx : x \u2208 {x | C \u2264 \u2016f (\u2110 C) x\u2016\u208a}\n\u22a2 C \u2264 \u2016indicator {x | C \u2264 \u2016f (\u2110 C) x\u2016\u208a} (f (\u2110 C)) x\u2016\u208a"}, {"tactic": "rwa [nnnorm_indicator_eq_indicator_nnnorm, Set.indicator_of_mem hx]", "annotated_tactic": ["rwa [<a>nnnorm_indicator_eq_indicator_nnnorm</a>, <a>Set.indicator_of_mem</a> hx]", [{"full_name": "nnnorm_indicator_eq_indicator_nnnorm", "def_path": "Mathlib/Analysis/NormedSpace/IndicatorFunction.lean", "def_pos": [29, 9], "def_end_pos": [29, 45]}, {"full_name": "Set.indicator_of_mem", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [67, 3], "def_end_pos": [67, 14]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhp : p \u2260 0\nhp' : p \u2260 \u22a4\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhfu : UnifIntegrable f p \u03bc\nM : \u211d\u22650\nhM : \u2200 (i : \u03b9), snorm (f i) p \u03bc \u2264 \u2191M\n\u2110 : \u211d\u22650 \u2192 \u03b9\n\u03b4 : \u211d\u22650\nh\u03b4pos : 0 < \u2191\u03b4\nh\u03b4 :\n  \u2200 (i : \u03b9) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u2191\u03b4 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nh\u2110 : \u2200 (C : \u211d\u22650), ENNReal.ofReal \u2191\u03b4 < \u2191\u2191\u03bc {x | C \u2264 \u2016f (\u2110 C) x\u2016\u208a}\nC : \u211d\u22650\nx : \u03b1\nhx : x \u2208 {x | C \u2264 \u2016f (\u2110 C) x\u2016\u208a}\n\u22a2 C \u2264 \u2016indicator {x | C \u2264 \u2016f (\u2110 C) x\u2016\u208a} (f (\u2110 C)) x\u2016\u208a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Primrec.lean", "full_name": "Primrec.fin_val_iff", "start": [1268, 1], "end": [1270, 68], "traced_tactics": [{"tactic": "letI : Primcodable { a // id a < n } := Primcodable.subtype (nat_lt.comp .id (const _))", "annotated_tactic": ["letI : <a>Primcodable</a> { a // <a>id</a> a < n } := <a>Primcodable.subtype</a> (nat_lt.comp .id (<a>const</a> _))", [{"full_name": "Primcodable", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [156, 7], "def_end_pos": [156, 18]}, {"full_name": "id", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [33, 15], "def_end_pos": [33, 17]}, {"full_name": "Primcodable.subtype", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [1169, 5], "def_end_pos": [1169, 12]}, {"full_name": "Primrec.const", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [250, 9], "def_end_pos": [250, 14]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nn : \u2115\nf : \u03b1 \u2192 Fin n\n\u22a2 (Primrec fun a => \u2191(f a)) \u2194 Primrec f", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nn : \u2115\nf : \u03b1 \u2192 Fin n\nthis : Primcodable { a // id a < n } := Primcodable.subtype (_ : PrimrecPred fun a => id a < n)\n\u22a2 (Primrec fun a => \u2191(f a)) \u2194 Primrec f"}, {"tactic": "exact (Iff.trans (by rfl) subtype_val_iff).trans (of_equiv_iff _)", "annotated_tactic": ["exact (<a>Iff.trans</a> (by rfl) <a>subtype_val_iff</a>).<a>trans</a> (<a>of_equiv_iff</a> _)", [{"full_name": "Iff.trans", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [666, 9], "def_end_pos": [666, 18]}, {"full_name": "Primrec.subtype_val_iff", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [1232, 9], "def_end_pos": [1232, 24]}, {"full_name": "Iff.trans", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [666, 9], "def_end_pos": [666, 18]}, {"full_name": "Primrec.of_equiv_iff", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [303, 9], "def_end_pos": [303, 21]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nn : \u2115\nf : \u03b1 \u2192 Fin n\nthis : Primcodable { a // id a < n } := Primcodable.subtype (_ : PrimrecPred fun a => id a < n)\n\u22a2 (Primrec fun a => \u2191(f a)) \u2194 Primrec f", "state_after": "no goals"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nn : \u2115\nf : \u03b1 \u2192 Fin n\nthis : Primcodable { a // id a < n } := Primcodable.subtype (_ : PrimrecPred fun a => id a < n)\n\u22a2 (Primrec fun a => \u2191(f a)) \u2194 Primrec fun a => \u2191(\u2191Fin.equivSubtype (f a))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/TypeVec.lean", "full_name": "TypeVec.append1_cases_append1", "start": [138, 1], "end": [140, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Intervals/Group.lean", "full_name": "Set.pairwise_disjoint_Ioo_zpow", "start": [221, 1], "end": [223, 64], "traced_tactics": [{"tactic": "simpa only [one_mul] using pairwise_disjoint_Ioo_mul_zpow 1 b", "annotated_tactic": ["simpa only [<a>one_mul</a>] using <a>pairwise_disjoint_Ioo_mul_zpow</a> 1 b", [{"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [464, 9], "def_end_pos": [464, 16]}, {"full_name": "Set.pairwise_disjoint_Ioo_mul_zpow", "def_path": "Mathlib/Data/Set/Intervals/Group.lean", "def_pos": [200, 9], "def_end_pos": [200, 39]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : OrderedCommGroup \u03b1\na b : \u03b1\n\u22a2 Pairwise (Disjoint on fun n => Ioo (b ^ n) (b ^ (n + 1)))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Countable.lean", "full_name": "Set.countable_isBot", "start": [252, 1], "end": [253, 29], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Intervals/OrdConnected.lean", "full_name": "Set.ordConnected_iff_uIcc_subset_left", "start": [278, 1], "end": [280, 66], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "full_name": "Int.dvd_add_right", "start": [625, 11], "end": [626, 40], "traced_tactics": [{"tactic": "rw [Int.add_comm, Int.dvd_add_left H]", "annotated_tactic": ["rw [<a>Int.add_comm</a>, <a>Int.dvd_add_left</a> H]", [{"full_name": "Int.add_comm", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [218, 19], "def_end_pos": [218, 27]}, {"full_name": "Int.dvd_add_left", "def_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "def_pos": [622, 19], "def_end_pos": [622, 31]}]], "state_before": "a b c : Int\nH : a \u2223 b\n\u22a2 a \u2223 b + c \u2194 a \u2223 c", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean", "full_name": "MeasureTheory.set_integral_condexpL1Clm", "start": [430, 1], "end": [462, 43], "traced_tactics": [{"tactic": "let S := spanningSets (\u03bc.trim hm)", "annotated_tactic": ["let S := <a>spanningSets</a> (\u03bc.trim hm)", [{"full_name": "MeasureTheory.spanningSets", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3316, 5], "def_end_pos": [3316, 17]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup F'\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2076 : NormedSpace \u211d F'\ninst\u271d\u2075 : CompleteSpace F'\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedAddCommGroup G'\ninst\u271d\u00b2 : NormedSpace \u211d G'\ninst\u271d\u00b9 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nf\u271d g : \u03b1 \u2192 F'\ns : Set \u03b1\nf : { x // x \u2208 Lp F' 1 }\nhs : MeasurableSet s\n\u22a2 \u222b (x : \u03b1) in s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) f) x \u2202\u03bc = \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup F'\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2076 : NormedSpace \u211d F'\ninst\u271d\u2075 : CompleteSpace F'\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedAddCommGroup G'\ninst\u271d\u00b2 : NormedSpace \u211d G'\ninst\u271d\u00b9 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nf\u271d g : \u03b1 \u2192 F'\ns : Set \u03b1\nf : { x // x \u2208 Lp F' 1 }\nhs : MeasurableSet s\nS : \u2115 \u2192 Set \u03b1 := spanningSets (Measure.trim \u03bc hm)\n\u22a2 \u222b (x : \u03b1) in s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) f) x \u2202\u03bc = \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc"}, {"tactic": "have hS_meas : \u2200 i, MeasurableSet[m] (S i) := measurable_spanningSets (\u03bc.trim hm)", "annotated_tactic": ["have hS_meas : \u2200 i, MeasurableSet[m] (S i) := <a>measurable_spanningSets</a> (\u03bc.trim hm)", [{"full_name": "MeasureTheory.measurable_spanningSets", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3324, 9], "def_end_pos": [3324, 32]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup F'\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2076 : NormedSpace \u211d F'\ninst\u271d\u2075 : CompleteSpace F'\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedAddCommGroup G'\ninst\u271d\u00b2 : NormedSpace \u211d G'\ninst\u271d\u00b9 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nf\u271d g : \u03b1 \u2192 F'\ns : Set \u03b1\nf : { x // x \u2208 Lp F' 1 }\nhs : MeasurableSet s\nS : \u2115 \u2192 Set \u03b1 := spanningSets (Measure.trim \u03bc hm)\n\u22a2 \u222b (x : \u03b1) in s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) f) x \u2202\u03bc = \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup F'\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2076 : NormedSpace \u211d F'\ninst\u271d\u2075 : CompleteSpace F'\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedAddCommGroup G'\ninst\u271d\u00b2 : NormedSpace \u211d G'\ninst\u271d\u00b9 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nf\u271d g : \u03b1 \u2192 F'\ns : Set \u03b1\nf : { x // x \u2208 Lp F' 1 }\nhs : MeasurableSet s\nS : \u2115 \u2192 Set \u03b1 := spanningSets (Measure.trim \u03bc hm)\nhS_meas : \u2200 (i : \u2115), MeasurableSet (S i)\n\u22a2 \u222b (x : \u03b1) in s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) f) x \u2202\u03bc = \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc"}, {"tactic": "have hS_meas0 : \u2200 i, MeasurableSet (S i) := fun i => hm _ (hS_meas i)", "annotated_tactic": ["have hS_meas0 : \u2200 i, <a>MeasurableSet</a> (S i) := fun i => hm _ (hS_meas i)", [{"full_name": "MeasurableSet", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [64, 5], "def_end_pos": [64, 18]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup F'\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2076 : NormedSpace \u211d F'\ninst\u271d\u2075 : CompleteSpace F'\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedAddCommGroup G'\ninst\u271d\u00b2 : NormedSpace \u211d G'\ninst\u271d\u00b9 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nf\u271d g : \u03b1 \u2192 F'\ns : Set \u03b1\nf : { x // x \u2208 Lp F' 1 }\nhs : MeasurableSet s\nS : \u2115 \u2192 Set \u03b1 := spanningSets (Measure.trim \u03bc hm)\nhS_meas : \u2200 (i : \u2115), MeasurableSet (S i)\n\u22a2 \u222b (x : \u03b1) in s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) f) x \u2202\u03bc = \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup F'\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2076 : NormedSpace \u211d F'\ninst\u271d\u2075 : CompleteSpace F'\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedAddCommGroup G'\ninst\u271d\u00b2 : NormedSpace \u211d G'\ninst\u271d\u00b9 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nf\u271d g : \u03b1 \u2192 F'\ns : Set \u03b1\nf : { x // x \u2208 Lp F' 1 }\nhs : MeasurableSet s\nS : \u2115 \u2192 Set \u03b1 := spanningSets (Measure.trim \u03bc hm)\nhS_meas : \u2200 (i : \u2115), MeasurableSet (S i)\nhS_meas0 : \u2200 (i : \u2115), MeasurableSet (S i)\n\u22a2 \u222b (x : \u03b1) in s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) f) x \u2202\u03bc = \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc"}, {"tactic": "have hs_eq : s = \u22c3 i, S i \u2229 s := by\n  simp_rw [Set.inter_comm]\n  rw [\u2190 Set.inter_iUnion, iUnion_spanningSets (\u03bc.trim hm), Set.inter_univ]", "annotated_tactic": ["have hs_eq : s = \u22c3 i, S i \u2229 s := by\n    simp_rw [<a>Set.inter_comm</a>]\n    rw [\u2190 <a>Set.inter_iUnion</a>, <a>iUnion_spanningSets</a> (\u03bc.trim hm), <a>Set.inter_univ</a>]", [{"full_name": "Set.inter_comm", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [940, 9], "def_end_pos": [940, 19]}, {"full_name": "Set.inter_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [635, 9], "def_end_pos": [635, 21]}, {"full_name": "MeasureTheory.iUnion_spanningSets", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3334, 9], "def_end_pos": [3334, 28]}, {"full_name": "Set.inter_univ", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1012, 9], "def_end_pos": [1012, 19]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup F'\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2076 : NormedSpace \u211d F'\ninst\u271d\u2075 : CompleteSpace F'\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedAddCommGroup G'\ninst\u271d\u00b2 : NormedSpace \u211d G'\ninst\u271d\u00b9 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nf\u271d g : \u03b1 \u2192 F'\ns : Set \u03b1\nf : { x // x \u2208 Lp F' 1 }\nhs : MeasurableSet s\nS : \u2115 \u2192 Set \u03b1 := spanningSets (Measure.trim \u03bc hm)\nhS_meas : \u2200 (i : \u2115), MeasurableSet (S i)\nhS_meas0 : \u2200 (i : \u2115), MeasurableSet (S i)\n\u22a2 \u222b (x : \u03b1) in s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) f) x \u2202\u03bc = \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup F'\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2076 : NormedSpace \u211d F'\ninst\u271d\u2075 : CompleteSpace F'\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedAddCommGroup G'\ninst\u271d\u00b2 : NormedSpace \u211d G'\ninst\u271d\u00b9 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nf\u271d g : \u03b1 \u2192 F'\ns : Set \u03b1\nf : { x // x \u2208 Lp F' 1 }\nhs : MeasurableSet s\nS : \u2115 \u2192 Set \u03b1 := spanningSets (Measure.trim \u03bc hm)\nhS_meas : \u2200 (i : \u2115), MeasurableSet (S i)\nhS_meas0 : \u2200 (i : \u2115), MeasurableSet (S i)\nhs_eq : s = \u22c3 i, S i \u2229 s\n\u22a2 \u222b (x : \u03b1) in s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) f) x \u2202\u03bc = \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc"}, {"tactic": "have hS_finite : \u2200 i, \u03bc (S i \u2229 s) < \u221e := by\n  refine' fun i => (measure_mono (Set.inter_subset_left _ _)).trans_lt _\n  have hS_finite_trim := measure_spanningSets_lt_top (\u03bc.trim hm) i\n  rwa [trim_measurableSet_eq hm (hS_meas i)] at hS_finite_trim", "annotated_tactic": ["have hS_finite : \u2200 i, \u03bc (S i \u2229 s) < \u221e := by\n    refine' fun i => (<a>measure_mono</a> (<a>Set.inter_subset_left</a> _ _)).<a>trans_lt</a> _\n    have hS_finite_trim := <a>measure_spanningSets_lt_top</a> (\u03bc.trim hm) i\n    rwa [<a>trim_measurableSet_eq</a> hm (hS_meas i)] at hS_finite_trim", [{"full_name": "MeasureTheory.measure_mono", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [193, 9], "def_end_pos": [193, 21]}, {"full_name": "Set.inter_subset_left", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [965, 9], "def_end_pos": [965, 26]}, {"full_name": "LE.le.trans_lt", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [124, 7], "def_end_pos": [124, 21]}, {"full_name": "MeasureTheory.measure_spanningSets_lt_top", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3329, 9], "def_end_pos": [3329, 36]}, {"full_name": "MeasureTheory.trim_measurableSet_eq", "def_path": "Mathlib/MeasureTheory/Measure/Trim.lean", "def_pos": [53, 9], "def_end_pos": [53, 30]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup F'\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2076 : NormedSpace \u211d F'\ninst\u271d\u2075 : CompleteSpace F'\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedAddCommGroup G'\ninst\u271d\u00b2 : NormedSpace \u211d G'\ninst\u271d\u00b9 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nf\u271d g : \u03b1 \u2192 F'\ns : Set \u03b1\nf : { x // x \u2208 Lp F' 1 }\nhs : MeasurableSet s\nS : \u2115 \u2192 Set \u03b1 := spanningSets (Measure.trim \u03bc hm)\nhS_meas : \u2200 (i : \u2115), MeasurableSet (S i)\nhS_meas0 : \u2200 (i : \u2115), MeasurableSet (S i)\nhs_eq : s = \u22c3 i, S i \u2229 s\n\u22a2 \u222b (x : \u03b1) in s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) f) x \u2202\u03bc = \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup F'\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2076 : NormedSpace \u211d F'\ninst\u271d\u2075 : CompleteSpace F'\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedAddCommGroup G'\ninst\u271d\u00b2 : NormedSpace \u211d G'\ninst\u271d\u00b9 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nf\u271d g : \u03b1 \u2192 F'\ns : Set \u03b1\nf : { x // x \u2208 Lp F' 1 }\nhs : MeasurableSet s\nS : \u2115 \u2192 Set \u03b1 := spanningSets (Measure.trim \u03bc hm)\nhS_meas : \u2200 (i : \u2115), MeasurableSet (S i)\nhS_meas0 : \u2200 (i : \u2115), MeasurableSet (S i)\nhs_eq : s = \u22c3 i, S i \u2229 s\nhS_finite : \u2200 (i : \u2115), \u2191\u2191\u03bc (S i \u2229 s) < \u22a4\n\u22a2 \u222b (x : \u03b1) in s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) f) x \u2202\u03bc = \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc"}, {"tactic": "have h_mono : Monotone fun i => S i \u2229 s := by\n  intro i j hij x\n  simp_rw [Set.mem_inter_iff]\n  exact fun h => \u27e8monotone_spanningSets (\u03bc.trim hm) hij h.1, h.2\u27e9", "annotated_tactic": ["have h_mono : <a>Monotone</a> fun i => S i \u2229 s := by\n    intro i j hij x\n    simp_rw [<a>Set.mem_inter_iff</a>]\n    exact fun h => \u27e8<a>monotone_spanningSets</a> (\u03bc.trim hm) hij h.1, h.2\u27e9", [{"full_name": "Monotone", "def_path": "Mathlib/Order/Monotone/Basic.lean", "def_pos": [77, 5], "def_end_pos": [77, 13]}, {"full_name": "Set.mem_inter_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [909, 9], "def_end_pos": [909, 22]}, {"full_name": "MeasureTheory.monotone_spanningSets", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3320, 9], "def_end_pos": [3320, 30]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup F'\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2076 : NormedSpace \u211d F'\ninst\u271d\u2075 : CompleteSpace F'\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedAddCommGroup G'\ninst\u271d\u00b2 : NormedSpace \u211d G'\ninst\u271d\u00b9 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nf\u271d g : \u03b1 \u2192 F'\ns : Set \u03b1\nf : { x // x \u2208 Lp F' 1 }\nhs : MeasurableSet s\nS : \u2115 \u2192 Set \u03b1 := spanningSets (Measure.trim \u03bc hm)\nhS_meas : \u2200 (i : \u2115), MeasurableSet (S i)\nhS_meas0 : \u2200 (i : \u2115), MeasurableSet (S i)\nhs_eq : s = \u22c3 i, S i \u2229 s\nhS_finite : \u2200 (i : \u2115), \u2191\u2191\u03bc (S i \u2229 s) < \u22a4\n\u22a2 \u222b (x : \u03b1) in s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) f) x \u2202\u03bc = \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup F'\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2076 : NormedSpace \u211d F'\ninst\u271d\u2075 : CompleteSpace F'\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedAddCommGroup G'\ninst\u271d\u00b2 : NormedSpace \u211d G'\ninst\u271d\u00b9 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nf\u271d g : \u03b1 \u2192 F'\ns : Set \u03b1\nf : { x // x \u2208 Lp F' 1 }\nhs : MeasurableSet s\nS : \u2115 \u2192 Set \u03b1 := spanningSets (Measure.trim \u03bc hm)\nhS_meas : \u2200 (i : \u2115), MeasurableSet (S i)\nhS_meas0 : \u2200 (i : \u2115), MeasurableSet (S i)\nhs_eq : s = \u22c3 i, S i \u2229 s\nhS_finite : \u2200 (i : \u2115), \u2191\u2191\u03bc (S i \u2229 s) < \u22a4\nh_mono : Monotone fun i => S i \u2229 s\n\u22a2 \u222b (x : \u03b1) in s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) f) x \u2202\u03bc = \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc"}, {"tactic": "have h_eq_forall :\n  (fun i => \u222b x in S i \u2229 s, condexpL1Clm F' hm \u03bc f x \u2202\u03bc) = fun i => \u222b x in S i \u2229 s, f x \u2202\u03bc :=\n  funext fun i =>\n    set_integral_condexpL1Clm_of_measure_ne_top f (@MeasurableSet.inter \u03b1 m _ _ (hS_meas i) hs)\n      (hS_finite i).ne", "annotated_tactic": ["have h_eq_forall :\n    (fun i => \u222b x in S i \u2229 s, <a>condexpL1Clm</a> F' hm \u03bc f x \u2202\u03bc) = fun i => \u222b x in S i \u2229 s, f x \u2202\u03bc :=\n    <a>funext</a> fun i =>\n      <a>set_integral_condexpL1Clm_of_measure_ne_top</a> f (@<a>MeasurableSet.inter</a> \u03b1 m _ _ (hS_meas i) hs)\n        (hS_finite i).<a>ne</a>", [{"full_name": "MeasureTheory.condexpL1Clm", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean", "def_pos": [380, 5], "def_end_pos": [380, 17]}, {"full_name": "funext", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [1555, 9], "def_end_pos": [1555, 15]}, {"full_name": "MeasureTheory.set_integral_condexpL1Clm_of_measure_ne_top", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean", "def_pos": [404, 9], "def_end_pos": [404, 52]}, {"full_name": "MeasurableSet.inter", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [198, 19], "def_end_pos": [198, 38]}, {"full_name": "LT.lt.ne", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [152, 7], "def_end_pos": [152, 15]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup F'\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2076 : NormedSpace \u211d F'\ninst\u271d\u2075 : CompleteSpace F'\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedAddCommGroup G'\ninst\u271d\u00b2 : NormedSpace \u211d G'\ninst\u271d\u00b9 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nf\u271d g : \u03b1 \u2192 F'\ns : Set \u03b1\nf : { x // x \u2208 Lp F' 1 }\nhs : MeasurableSet s\nS : \u2115 \u2192 Set \u03b1 := spanningSets (Measure.trim \u03bc hm)\nhS_meas : \u2200 (i : \u2115), MeasurableSet (S i)\nhS_meas0 : \u2200 (i : \u2115), MeasurableSet (S i)\nhs_eq : s = \u22c3 i, S i \u2229 s\nhS_finite : \u2200 (i : \u2115), \u2191\u2191\u03bc (S i \u2229 s) < \u22a4\nh_mono : Monotone fun i => S i \u2229 s\n\u22a2 \u222b (x : \u03b1) in s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) f) x \u2202\u03bc = \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup F'\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2076 : NormedSpace \u211d F'\ninst\u271d\u2075 : CompleteSpace F'\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedAddCommGroup G'\ninst\u271d\u00b2 : NormedSpace \u211d G'\ninst\u271d\u00b9 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nf\u271d g : \u03b1 \u2192 F'\ns : Set \u03b1\nf : { x // x \u2208 Lp F' 1 }\nhs : MeasurableSet s\nS : \u2115 \u2192 Set \u03b1 := spanningSets (Measure.trim \u03bc hm)\nhS_meas : \u2200 (i : \u2115), MeasurableSet (S i)\nhS_meas0 : \u2200 (i : \u2115), MeasurableSet (S i)\nhs_eq : s = \u22c3 i, S i \u2229 s\nhS_finite : \u2200 (i : \u2115), \u2191\u2191\u03bc (S i \u2229 s) < \u22a4\nh_mono : Monotone fun i => S i \u2229 s\nh_eq_forall :\n  (fun i => \u222b (x : \u03b1) in S i \u2229 s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) f) x \u2202\u03bc) = fun i => \u222b (x : \u03b1) in S i \u2229 s, \u2191\u2191f x \u2202\u03bc\n\u22a2 \u222b (x : \u03b1) in s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) f) x \u2202\u03bc = \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc"}, {"tactic": "have h_right : Tendsto (fun i => \u222b x in S i \u2229 s, f x \u2202\u03bc) atTop (\ud835\udcdd (\u222b x in s, f x \u2202\u03bc)) := by\n  have h :=\n    tendsto_set_integral_of_monotone (fun i => (hS_meas0 i).inter (hm s hs)) h_mono\n      (L1.integrable_coeFn f).integrableOn\n  rwa [\u2190 hs_eq] at h", "annotated_tactic": ["have h_right : <a>Tendsto</a> (fun i => \u222b x in S i \u2229 s, f x \u2202\u03bc) <a>atTop</a> (\ud835\udcdd (\u222b x in s, f x \u2202\u03bc)) := by\n    have h :=\n      <a>tendsto_set_integral_of_monotone</a> (fun i => (hS_meas0 i).<a>inter</a> (hm s hs)) h_mono\n        (<a>L1.integrable_coeFn</a> f).<a>integrableOn</a>\n    rwa [\u2190 hs_eq] at h", [{"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2939, 5], "def_end_pos": [2939, 12]}, {"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}, {"full_name": "MeasureTheory.tendsto_set_integral_of_monotone", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [212, 9], "def_end_pos": [212, 41]}, {"full_name": "MeasurableSet.inter", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [198, 19], "def_end_pos": [198, 38]}, {"full_name": "MeasureTheory.L1.integrable_coeFn", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [1324, 9], "def_end_pos": [1324, 25]}, {"full_name": "MeasureTheory.Integrable.integrableOn", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [163, 9], "def_end_pos": [163, 32]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup F'\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2076 : NormedSpace \u211d F'\ninst\u271d\u2075 : CompleteSpace F'\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedAddCommGroup G'\ninst\u271d\u00b2 : NormedSpace \u211d G'\ninst\u271d\u00b9 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nf\u271d g : \u03b1 \u2192 F'\ns : Set \u03b1\nf : { x // x \u2208 Lp F' 1 }\nhs : MeasurableSet s\nS : \u2115 \u2192 Set \u03b1 := spanningSets (Measure.trim \u03bc hm)\nhS_meas : \u2200 (i : \u2115), MeasurableSet (S i)\nhS_meas0 : \u2200 (i : \u2115), MeasurableSet (S i)\nhs_eq : s = \u22c3 i, S i \u2229 s\nhS_finite : \u2200 (i : \u2115), \u2191\u2191\u03bc (S i \u2229 s) < \u22a4\nh_mono : Monotone fun i => S i \u2229 s\nh_eq_forall :\n  (fun i => \u222b (x : \u03b1) in S i \u2229 s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) f) x \u2202\u03bc) = fun i => \u222b (x : \u03b1) in S i \u2229 s, \u2191\u2191f x \u2202\u03bc\n\u22a2 \u222b (x : \u03b1) in s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) f) x \u2202\u03bc = \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup F'\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2076 : NormedSpace \u211d F'\ninst\u271d\u2075 : CompleteSpace F'\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedAddCommGroup G'\ninst\u271d\u00b2 : NormedSpace \u211d G'\ninst\u271d\u00b9 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nf\u271d g : \u03b1 \u2192 F'\ns : Set \u03b1\nf : { x // x \u2208 Lp F' 1 }\nhs : MeasurableSet s\nS : \u2115 \u2192 Set \u03b1 := spanningSets (Measure.trim \u03bc hm)\nhS_meas : \u2200 (i : \u2115), MeasurableSet (S i)\nhS_meas0 : \u2200 (i : \u2115), MeasurableSet (S i)\nhs_eq : s = \u22c3 i, S i \u2229 s\nhS_finite : \u2200 (i : \u2115), \u2191\u2191\u03bc (S i \u2229 s) < \u22a4\nh_mono : Monotone fun i => S i \u2229 s\nh_eq_forall :\n  (fun i => \u222b (x : \u03b1) in S i \u2229 s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) f) x \u2202\u03bc) = fun i => \u222b (x : \u03b1) in S i \u2229 s, \u2191\u2191f x \u2202\u03bc\nh_right : Tendsto (fun i => \u222b (x : \u03b1) in S i \u2229 s, \u2191\u2191f x \u2202\u03bc) atTop (\ud835\udcdd (\u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc))\n\u22a2 \u222b (x : \u03b1) in s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) f) x \u2202\u03bc = \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc"}, {"tactic": "have h_left : Tendsto (fun i => \u222b x in S i \u2229 s, condexpL1Clm F' hm \u03bc f x \u2202\u03bc) atTop\n    (\ud835\udcdd (\u222b x in s, condexpL1Clm F' hm \u03bc f x \u2202\u03bc)) := by\n  have h := tendsto_set_integral_of_monotone (fun i => (hS_meas0 i).inter (hm s hs)) h_mono\n    (L1.integrable_coeFn (condexpL1Clm F' hm \u03bc f)).integrableOn\n  rwa [\u2190 hs_eq] at h", "annotated_tactic": ["have h_left : <a>Tendsto</a> (fun i => \u222b x in S i \u2229 s, <a>condexpL1Clm</a> F' hm \u03bc f x \u2202\u03bc) <a>atTop</a>\n      (\ud835\udcdd (\u222b x in s, <a>condexpL1Clm</a> F' hm \u03bc f x \u2202\u03bc)) := by\n    have h := <a>tendsto_set_integral_of_monotone</a> (fun i => (hS_meas0 i).<a>inter</a> (hm s hs)) h_mono\n      (<a>L1.integrable_coeFn</a> (<a>condexpL1Clm</a> F' hm \u03bc f)).<a>integrableOn</a>\n    rwa [\u2190 hs_eq] at h", [{"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2939, 5], "def_end_pos": [2939, 12]}, {"full_name": "MeasureTheory.condexpL1Clm", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean", "def_pos": [380, 5], "def_end_pos": [380, 17]}, {"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}, {"full_name": "MeasureTheory.condexpL1Clm", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean", "def_pos": [380, 5], "def_end_pos": [380, 17]}, {"full_name": "MeasureTheory.tendsto_set_integral_of_monotone", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [212, 9], "def_end_pos": [212, 41]}, {"full_name": "MeasurableSet.inter", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [198, 19], "def_end_pos": [198, 38]}, {"full_name": "MeasureTheory.L1.integrable_coeFn", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [1324, 9], "def_end_pos": [1324, 25]}, {"full_name": "MeasureTheory.condexpL1Clm", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean", "def_pos": [380, 5], "def_end_pos": [380, 17]}, {"full_name": "MeasureTheory.Integrable.integrableOn", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [163, 9], "def_end_pos": [163, 32]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup F'\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2076 : NormedSpace \u211d F'\ninst\u271d\u2075 : CompleteSpace F'\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedAddCommGroup G'\ninst\u271d\u00b2 : NormedSpace \u211d G'\ninst\u271d\u00b9 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nf\u271d g : \u03b1 \u2192 F'\ns : Set \u03b1\nf : { x // x \u2208 Lp F' 1 }\nhs : MeasurableSet s\nS : \u2115 \u2192 Set \u03b1 := spanningSets (Measure.trim \u03bc hm)\nhS_meas : \u2200 (i : \u2115), MeasurableSet (S i)\nhS_meas0 : \u2200 (i : \u2115), MeasurableSet (S i)\nhs_eq : s = \u22c3 i, S i \u2229 s\nhS_finite : \u2200 (i : \u2115), \u2191\u2191\u03bc (S i \u2229 s) < \u22a4\nh_mono : Monotone fun i => S i \u2229 s\nh_eq_forall :\n  (fun i => \u222b (x : \u03b1) in S i \u2229 s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) f) x \u2202\u03bc) = fun i => \u222b (x : \u03b1) in S i \u2229 s, \u2191\u2191f x \u2202\u03bc\nh_right : Tendsto (fun i => \u222b (x : \u03b1) in S i \u2229 s, \u2191\u2191f x \u2202\u03bc) atTop (\ud835\udcdd (\u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc))\n\u22a2 \u222b (x : \u03b1) in s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) f) x \u2202\u03bc = \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup F'\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2076 : NormedSpace \u211d F'\ninst\u271d\u2075 : CompleteSpace F'\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedAddCommGroup G'\ninst\u271d\u00b2 : NormedSpace \u211d G'\ninst\u271d\u00b9 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nf\u271d g : \u03b1 \u2192 F'\ns : Set \u03b1\nf : { x // x \u2208 Lp F' 1 }\nhs : MeasurableSet s\nS : \u2115 \u2192 Set \u03b1 := spanningSets (Measure.trim \u03bc hm)\nhS_meas : \u2200 (i : \u2115), MeasurableSet (S i)\nhS_meas0 : \u2200 (i : \u2115), MeasurableSet (S i)\nhs_eq : s = \u22c3 i, S i \u2229 s\nhS_finite : \u2200 (i : \u2115), \u2191\u2191\u03bc (S i \u2229 s) < \u22a4\nh_mono : Monotone fun i => S i \u2229 s\nh_eq_forall :\n  (fun i => \u222b (x : \u03b1) in S i \u2229 s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) f) x \u2202\u03bc) = fun i => \u222b (x : \u03b1) in S i \u2229 s, \u2191\u2191f x \u2202\u03bc\nh_right : Tendsto (fun i => \u222b (x : \u03b1) in S i \u2229 s, \u2191\u2191f x \u2202\u03bc) atTop (\ud835\udcdd (\u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc))\nh_left :\n  Tendsto (fun i => \u222b (x : \u03b1) in S i \u2229 s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) f) x \u2202\u03bc) atTop\n    (\ud835\udcdd (\u222b (x : \u03b1) in s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) f) x \u2202\u03bc))\n\u22a2 \u222b (x : \u03b1) in s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) f) x \u2202\u03bc = \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc"}, {"tactic": "rw [h_eq_forall] at h_left", "annotated_tactic": ["rw [h_eq_forall] at h_left", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup F'\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2076 : NormedSpace \u211d F'\ninst\u271d\u2075 : CompleteSpace F'\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedAddCommGroup G'\ninst\u271d\u00b2 : NormedSpace \u211d G'\ninst\u271d\u00b9 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nf\u271d g : \u03b1 \u2192 F'\ns : Set \u03b1\nf : { x // x \u2208 Lp F' 1 }\nhs : MeasurableSet s\nS : \u2115 \u2192 Set \u03b1 := spanningSets (Measure.trim \u03bc hm)\nhS_meas : \u2200 (i : \u2115), MeasurableSet (S i)\nhS_meas0 : \u2200 (i : \u2115), MeasurableSet (S i)\nhs_eq : s = \u22c3 i, S i \u2229 s\nhS_finite : \u2200 (i : \u2115), \u2191\u2191\u03bc (S i \u2229 s) < \u22a4\nh_mono : Monotone fun i => S i \u2229 s\nh_eq_forall :\n  (fun i => \u222b (x : \u03b1) in S i \u2229 s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) f) x \u2202\u03bc) = fun i => \u222b (x : \u03b1) in S i \u2229 s, \u2191\u2191f x \u2202\u03bc\nh_right : Tendsto (fun i => \u222b (x : \u03b1) in S i \u2229 s, \u2191\u2191f x \u2202\u03bc) atTop (\ud835\udcdd (\u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc))\nh_left :\n  Tendsto (fun i => \u222b (x : \u03b1) in S i \u2229 s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) f) x \u2202\u03bc) atTop\n    (\ud835\udcdd (\u222b (x : \u03b1) in s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) f) x \u2202\u03bc))\n\u22a2 \u222b (x : \u03b1) in s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) f) x \u2202\u03bc = \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup F'\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2076 : NormedSpace \u211d F'\ninst\u271d\u2075 : CompleteSpace F'\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedAddCommGroup G'\ninst\u271d\u00b2 : NormedSpace \u211d G'\ninst\u271d\u00b9 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nf\u271d g : \u03b1 \u2192 F'\ns : Set \u03b1\nf : { x // x \u2208 Lp F' 1 }\nhs : MeasurableSet s\nS : \u2115 \u2192 Set \u03b1 := spanningSets (Measure.trim \u03bc hm)\nhS_meas : \u2200 (i : \u2115), MeasurableSet (S i)\nhS_meas0 : \u2200 (i : \u2115), MeasurableSet (S i)\nhs_eq : s = \u22c3 i, S i \u2229 s\nhS_finite : \u2200 (i : \u2115), \u2191\u2191\u03bc (S i \u2229 s) < \u22a4\nh_mono : Monotone fun i => S i \u2229 s\nh_eq_forall :\n  (fun i => \u222b (x : \u03b1) in S i \u2229 s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) f) x \u2202\u03bc) = fun i => \u222b (x : \u03b1) in S i \u2229 s, \u2191\u2191f x \u2202\u03bc\nh_right : Tendsto (fun i => \u222b (x : \u03b1) in S i \u2229 s, \u2191\u2191f x \u2202\u03bc) atTop (\ud835\udcdd (\u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc))\nh_left :\n  Tendsto (fun i => \u222b (x : \u03b1) in S i \u2229 s, \u2191\u2191f x \u2202\u03bc) atTop (\ud835\udcdd (\u222b (x : \u03b1) in s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) f) x \u2202\u03bc))\n\u22a2 \u222b (x : \u03b1) in s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) f) x \u2202\u03bc = \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc"}, {"tactic": "exact tendsto_nhds_unique h_left h_right", "annotated_tactic": ["exact <a>tendsto_nhds_unique</a> h_left h_right", [{"full_name": "tendsto_nhds_unique", "def_path": "Mathlib/Topology/Separation.lean", "def_pos": [994, 9], "def_end_pos": [994, 28]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup F'\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2076 : NormedSpace \u211d F'\ninst\u271d\u2075 : CompleteSpace F'\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedAddCommGroup G'\ninst\u271d\u00b2 : NormedSpace \u211d G'\ninst\u271d\u00b9 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nf\u271d g : \u03b1 \u2192 F'\ns : Set \u03b1\nf : { x // x \u2208 Lp F' 1 }\nhs : MeasurableSet s\nS : \u2115 \u2192 Set \u03b1 := spanningSets (Measure.trim \u03bc hm)\nhS_meas : \u2200 (i : \u2115), MeasurableSet (S i)\nhS_meas0 : \u2200 (i : \u2115), MeasurableSet (S i)\nhs_eq : s = \u22c3 i, S i \u2229 s\nhS_finite : \u2200 (i : \u2115), \u2191\u2191\u03bc (S i \u2229 s) < \u22a4\nh_mono : Monotone fun i => S i \u2229 s\nh_eq_forall :\n  (fun i => \u222b (x : \u03b1) in S i \u2229 s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) f) x \u2202\u03bc) = fun i => \u222b (x : \u03b1) in S i \u2229 s, \u2191\u2191f x \u2202\u03bc\nh_right : Tendsto (fun i => \u222b (x : \u03b1) in S i \u2229 s, \u2191\u2191f x \u2202\u03bc) atTop (\ud835\udcdd (\u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc))\nh_left :\n  Tendsto (fun i => \u222b (x : \u03b1) in S i \u2229 s, \u2191\u2191f x \u2202\u03bc) atTop (\ud835\udcdd (\u222b (x : \u03b1) in s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) f) x \u2202\u03bc))\n\u22a2 \u222b (x : \u03b1) in s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) f) x \u2202\u03bc = \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc", "state_after": "no goals"}, {"tactic": "simp_rw [Set.inter_comm]", "annotated_tactic": ["simp_rw [<a>Set.inter_comm</a>]", [{"full_name": "Set.inter_comm", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [940, 9], "def_end_pos": [940, 19]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup F'\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2076 : NormedSpace \u211d F'\ninst\u271d\u2075 : CompleteSpace F'\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedAddCommGroup G'\ninst\u271d\u00b2 : NormedSpace \u211d G'\ninst\u271d\u00b9 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nf\u271d g : \u03b1 \u2192 F'\ns : Set \u03b1\nf : { x // x \u2208 Lp F' 1 }\nhs : MeasurableSet s\nS : \u2115 \u2192 Set \u03b1 := spanningSets (Measure.trim \u03bc hm)\nhS_meas : \u2200 (i : \u2115), MeasurableSet (S i)\nhS_meas0 : \u2200 (i : \u2115), MeasurableSet (S i)\n\u22a2 s = \u22c3 i, S i \u2229 s", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup F'\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2076 : NormedSpace \u211d F'\ninst\u271d\u2075 : CompleteSpace F'\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedAddCommGroup G'\ninst\u271d\u00b2 : NormedSpace \u211d G'\ninst\u271d\u00b9 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nf\u271d g : \u03b1 \u2192 F'\ns : Set \u03b1\nf : { x // x \u2208 Lp F' 1 }\nhs : MeasurableSet s\nS : \u2115 \u2192 Set \u03b1 := spanningSets (Measure.trim \u03bc hm)\nhS_meas : \u2200 (i : \u2115), MeasurableSet (S i)\nhS_meas0 : \u2200 (i : \u2115), MeasurableSet (S i)\n\u22a2 s = \u22c3 i, s \u2229 spanningSets (Measure.trim \u03bc hm) i"}, {"tactic": "rw [\u2190 Set.inter_iUnion, iUnion_spanningSets (\u03bc.trim hm), Set.inter_univ]", "annotated_tactic": ["rw [\u2190 <a>Set.inter_iUnion</a>, <a>iUnion_spanningSets</a> (\u03bc.trim hm), <a>Set.inter_univ</a>]", [{"full_name": "Set.inter_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [635, 9], "def_end_pos": [635, 21]}, {"full_name": "MeasureTheory.iUnion_spanningSets", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3334, 9], "def_end_pos": [3334, 28]}, {"full_name": "Set.inter_univ", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1012, 9], "def_end_pos": [1012, 19]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup F'\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2076 : NormedSpace \u211d F'\ninst\u271d\u2075 : CompleteSpace F'\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedAddCommGroup G'\ninst\u271d\u00b2 : NormedSpace \u211d G'\ninst\u271d\u00b9 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nf\u271d g : \u03b1 \u2192 F'\ns : Set \u03b1\nf : { x // x \u2208 Lp F' 1 }\nhs : MeasurableSet s\nS : \u2115 \u2192 Set \u03b1 := spanningSets (Measure.trim \u03bc hm)\nhS_meas : \u2200 (i : \u2115), MeasurableSet (S i)\nhS_meas0 : \u2200 (i : \u2115), MeasurableSet (S i)\n\u22a2 s = \u22c3 i, s \u2229 spanningSets (Measure.trim \u03bc hm) i", "state_after": "no goals"}, {"tactic": "refine' fun i => (measure_mono (Set.inter_subset_left _ _)).trans_lt _", "annotated_tactic": ["refine' fun i => (<a>measure_mono</a> (<a>Set.inter_subset_left</a> _ _)).<a>trans_lt</a> _", [{"full_name": "MeasureTheory.measure_mono", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [193, 9], "def_end_pos": [193, 21]}, {"full_name": "Set.inter_subset_left", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [965, 9], "def_end_pos": [965, 26]}, {"full_name": "LE.le.trans_lt", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [124, 7], "def_end_pos": [124, 21]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup F'\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2076 : NormedSpace \u211d F'\ninst\u271d\u2075 : CompleteSpace F'\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedAddCommGroup G'\ninst\u271d\u00b2 : NormedSpace \u211d G'\ninst\u271d\u00b9 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nf\u271d g : \u03b1 \u2192 F'\ns : Set \u03b1\nf : { x // x \u2208 Lp F' 1 }\nhs : MeasurableSet s\nS : \u2115 \u2192 Set \u03b1 := spanningSets (Measure.trim \u03bc hm)\nhS_meas : \u2200 (i : \u2115), MeasurableSet (S i)\nhS_meas0 : \u2200 (i : \u2115), MeasurableSet (S i)\nhs_eq : s = \u22c3 i, S i \u2229 s\n\u22a2 \u2200 (i : \u2115), \u2191\u2191\u03bc (S i \u2229 s) < \u22a4", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup F'\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2076 : NormedSpace \u211d F'\ninst\u271d\u2075 : CompleteSpace F'\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedAddCommGroup G'\ninst\u271d\u00b2 : NormedSpace \u211d G'\ninst\u271d\u00b9 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nf\u271d g : \u03b1 \u2192 F'\ns : Set \u03b1\nf : { x // x \u2208 Lp F' 1 }\nhs : MeasurableSet s\nS : \u2115 \u2192 Set \u03b1 := spanningSets (Measure.trim \u03bc hm)\nhS_meas : \u2200 (i : \u2115), MeasurableSet (S i)\nhS_meas0 : \u2200 (i : \u2115), MeasurableSet (S i)\nhs_eq : s = \u22c3 i, S i \u2229 s\ni : \u2115\n\u22a2 \u2191\u2191\u03bc (S i) < \u22a4"}, {"tactic": "have hS_finite_trim := measure_spanningSets_lt_top (\u03bc.trim hm) i", "annotated_tactic": ["have hS_finite_trim := <a>measure_spanningSets_lt_top</a> (\u03bc.trim hm) i", [{"full_name": "MeasureTheory.measure_spanningSets_lt_top", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3329, 9], "def_end_pos": [3329, 36]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup F'\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2076 : NormedSpace \u211d F'\ninst\u271d\u2075 : CompleteSpace F'\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedAddCommGroup G'\ninst\u271d\u00b2 : NormedSpace \u211d G'\ninst\u271d\u00b9 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nf\u271d g : \u03b1 \u2192 F'\ns : Set \u03b1\nf : { x // x \u2208 Lp F' 1 }\nhs : MeasurableSet s\nS : \u2115 \u2192 Set \u03b1 := spanningSets (Measure.trim \u03bc hm)\nhS_meas : \u2200 (i : \u2115), MeasurableSet (S i)\nhS_meas0 : \u2200 (i : \u2115), MeasurableSet (S i)\nhs_eq : s = \u22c3 i, S i \u2229 s\ni : \u2115\n\u22a2 \u2191\u2191\u03bc (S i) < \u22a4", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup F'\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2076 : NormedSpace \u211d F'\ninst\u271d\u2075 : CompleteSpace F'\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedAddCommGroup G'\ninst\u271d\u00b2 : NormedSpace \u211d G'\ninst\u271d\u00b9 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nf\u271d g : \u03b1 \u2192 F'\ns : Set \u03b1\nf : { x // x \u2208 Lp F' 1 }\nhs : MeasurableSet s\nS : \u2115 \u2192 Set \u03b1 := spanningSets (Measure.trim \u03bc hm)\nhS_meas : \u2200 (i : \u2115), MeasurableSet (S i)\nhS_meas0 : \u2200 (i : \u2115), MeasurableSet (S i)\nhs_eq : s = \u22c3 i, S i \u2229 s\ni : \u2115\nhS_finite_trim : \u2191\u2191(Measure.trim \u03bc hm) (spanningSets (Measure.trim \u03bc hm) i) < \u22a4\n\u22a2 \u2191\u2191\u03bc (S i) < \u22a4"}, {"tactic": "rwa [trim_measurableSet_eq hm (hS_meas i)] at hS_finite_trim", "annotated_tactic": ["rwa [<a>trim_measurableSet_eq</a> hm (hS_meas i)] at hS_finite_trim", [{"full_name": "MeasureTheory.trim_measurableSet_eq", "def_path": "Mathlib/MeasureTheory/Measure/Trim.lean", "def_pos": [53, 9], "def_end_pos": [53, 30]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup F'\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2076 : NormedSpace \u211d F'\ninst\u271d\u2075 : CompleteSpace F'\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedAddCommGroup G'\ninst\u271d\u00b2 : NormedSpace \u211d G'\ninst\u271d\u00b9 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nf\u271d g : \u03b1 \u2192 F'\ns : Set \u03b1\nf : { x // x \u2208 Lp F' 1 }\nhs : MeasurableSet s\nS : \u2115 \u2192 Set \u03b1 := spanningSets (Measure.trim \u03bc hm)\nhS_meas : \u2200 (i : \u2115), MeasurableSet (S i)\nhS_meas0 : \u2200 (i : \u2115), MeasurableSet (S i)\nhs_eq : s = \u22c3 i, S i \u2229 s\ni : \u2115\nhS_finite_trim : \u2191\u2191(Measure.trim \u03bc hm) (spanningSets (Measure.trim \u03bc hm) i) < \u22a4\n\u22a2 \u2191\u2191\u03bc (S i) < \u22a4", "state_after": "no goals"}, {"tactic": "intro i j hij x", "annotated_tactic": ["intro i j hij x", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup F'\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2076 : NormedSpace \u211d F'\ninst\u271d\u2075 : CompleteSpace F'\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedAddCommGroup G'\ninst\u271d\u00b2 : NormedSpace \u211d G'\ninst\u271d\u00b9 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nf\u271d g : \u03b1 \u2192 F'\ns : Set \u03b1\nf : { x // x \u2208 Lp F' 1 }\nhs : MeasurableSet s\nS : \u2115 \u2192 Set \u03b1 := spanningSets (Measure.trim \u03bc hm)\nhS_meas : \u2200 (i : \u2115), MeasurableSet (S i)\nhS_meas0 : \u2200 (i : \u2115), MeasurableSet (S i)\nhs_eq : s = \u22c3 i, S i \u2229 s\nhS_finite : \u2200 (i : \u2115), \u2191\u2191\u03bc (S i \u2229 s) < \u22a4\n\u22a2 Monotone fun i => S i \u2229 s", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup F'\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2076 : NormedSpace \u211d F'\ninst\u271d\u2075 : CompleteSpace F'\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedAddCommGroup G'\ninst\u271d\u00b2 : NormedSpace \u211d G'\ninst\u271d\u00b9 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nf\u271d g : \u03b1 \u2192 F'\ns : Set \u03b1\nf : { x // x \u2208 Lp F' 1 }\nhs : MeasurableSet s\nS : \u2115 \u2192 Set \u03b1 := spanningSets (Measure.trim \u03bc hm)\nhS_meas : \u2200 (i : \u2115), MeasurableSet (S i)\nhS_meas0 : \u2200 (i : \u2115), MeasurableSet (S i)\nhs_eq : s = \u22c3 i, S i \u2229 s\nhS_finite : \u2200 (i : \u2115), \u2191\u2191\u03bc (S i \u2229 s) < \u22a4\ni j : \u2115\nhij : i \u2264 j\nx : \u03b1\n\u22a2 x \u2208 (fun i => S i \u2229 s) i \u2192 x \u2208 (fun i => S i \u2229 s) j"}, {"tactic": "simp_rw [Set.mem_inter_iff]", "annotated_tactic": ["simp_rw [<a>Set.mem_inter_iff</a>]", [{"full_name": "Set.mem_inter_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [909, 9], "def_end_pos": [909, 22]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup F'\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2076 : NormedSpace \u211d F'\ninst\u271d\u2075 : CompleteSpace F'\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedAddCommGroup G'\ninst\u271d\u00b2 : NormedSpace \u211d G'\ninst\u271d\u00b9 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nf\u271d g : \u03b1 \u2192 F'\ns : Set \u03b1\nf : { x // x \u2208 Lp F' 1 }\nhs : MeasurableSet s\nS : \u2115 \u2192 Set \u03b1 := spanningSets (Measure.trim \u03bc hm)\nhS_meas : \u2200 (i : \u2115), MeasurableSet (S i)\nhS_meas0 : \u2200 (i : \u2115), MeasurableSet (S i)\nhs_eq : s = \u22c3 i, S i \u2229 s\nhS_finite : \u2200 (i : \u2115), \u2191\u2191\u03bc (S i \u2229 s) < \u22a4\ni j : \u2115\nhij : i \u2264 j\nx : \u03b1\n\u22a2 x \u2208 (fun i => S i \u2229 s) i \u2192 x \u2208 (fun i => S i \u2229 s) j", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup F'\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2076 : NormedSpace \u211d F'\ninst\u271d\u2075 : CompleteSpace F'\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedAddCommGroup G'\ninst\u271d\u00b2 : NormedSpace \u211d G'\ninst\u271d\u00b9 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nf\u271d g : \u03b1 \u2192 F'\ns : Set \u03b1\nf : { x // x \u2208 Lp F' 1 }\nhs : MeasurableSet s\nS : \u2115 \u2192 Set \u03b1 := spanningSets (Measure.trim \u03bc hm)\nhS_meas : \u2200 (i : \u2115), MeasurableSet (S i)\nhS_meas0 : \u2200 (i : \u2115), MeasurableSet (S i)\nhs_eq : s = \u22c3 i, S i \u2229 s\nhS_finite : \u2200 (i : \u2115), \u2191\u2191\u03bc (S i \u2229 s) < \u22a4\ni j : \u2115\nhij : i \u2264 j\nx : \u03b1\n\u22a2 x \u2208 spanningSets (Measure.trim \u03bc hm) i \u2227 x \u2208 s \u2192 x \u2208 spanningSets (Measure.trim \u03bc hm) j \u2227 x \u2208 s"}, {"tactic": "exact fun h => \u27e8monotone_spanningSets (\u03bc.trim hm) hij h.1, h.2\u27e9", "annotated_tactic": ["exact fun h => \u27e8<a>monotone_spanningSets</a> (\u03bc.trim hm) hij h.1, h.2\u27e9", [{"full_name": "MeasureTheory.monotone_spanningSets", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3320, 9], "def_end_pos": [3320, 30]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup F'\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2076 : NormedSpace \u211d F'\ninst\u271d\u2075 : CompleteSpace F'\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedAddCommGroup G'\ninst\u271d\u00b2 : NormedSpace \u211d G'\ninst\u271d\u00b9 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nf\u271d g : \u03b1 \u2192 F'\ns : Set \u03b1\nf : { x // x \u2208 Lp F' 1 }\nhs : MeasurableSet s\nS : \u2115 \u2192 Set \u03b1 := spanningSets (Measure.trim \u03bc hm)\nhS_meas : \u2200 (i : \u2115), MeasurableSet (S i)\nhS_meas0 : \u2200 (i : \u2115), MeasurableSet (S i)\nhs_eq : s = \u22c3 i, S i \u2229 s\nhS_finite : \u2200 (i : \u2115), \u2191\u2191\u03bc (S i \u2229 s) < \u22a4\ni j : \u2115\nhij : i \u2264 j\nx : \u03b1\n\u22a2 x \u2208 spanningSets (Measure.trim \u03bc hm) i \u2227 x \u2208 s \u2192 x \u2208 spanningSets (Measure.trim \u03bc hm) j \u2227 x \u2208 s", "state_after": "no goals"}, {"tactic": "have h :=\n  tendsto_set_integral_of_monotone (fun i => (hS_meas0 i).inter (hm s hs)) h_mono\n    (L1.integrable_coeFn f).integrableOn", "annotated_tactic": ["have h :=\n      <a>tendsto_set_integral_of_monotone</a> (fun i => (hS_meas0 i).<a>inter</a> (hm s hs)) h_mono\n        (<a>L1.integrable_coeFn</a> f).<a>integrableOn</a>", [{"full_name": "MeasureTheory.tendsto_set_integral_of_monotone", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [212, 9], "def_end_pos": [212, 41]}, {"full_name": "MeasurableSet.inter", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [198, 19], "def_end_pos": [198, 38]}, {"full_name": "MeasureTheory.L1.integrable_coeFn", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [1324, 9], "def_end_pos": [1324, 25]}, {"full_name": "MeasureTheory.Integrable.integrableOn", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [163, 9], "def_end_pos": [163, 32]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup F'\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2076 : NormedSpace \u211d F'\ninst\u271d\u2075 : CompleteSpace F'\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedAddCommGroup G'\ninst\u271d\u00b2 : NormedSpace \u211d G'\ninst\u271d\u00b9 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nf\u271d g : \u03b1 \u2192 F'\ns : Set \u03b1\nf : { x // x \u2208 Lp F' 1 }\nhs : MeasurableSet s\nS : \u2115 \u2192 Set \u03b1 := spanningSets (Measure.trim \u03bc hm)\nhS_meas : \u2200 (i : \u2115), MeasurableSet (S i)\nhS_meas0 : \u2200 (i : \u2115), MeasurableSet (S i)\nhs_eq : s = \u22c3 i, S i \u2229 s\nhS_finite : \u2200 (i : \u2115), \u2191\u2191\u03bc (S i \u2229 s) < \u22a4\nh_mono : Monotone fun i => S i \u2229 s\nh_eq_forall :\n  (fun i => \u222b (x : \u03b1) in S i \u2229 s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) f) x \u2202\u03bc) = fun i => \u222b (x : \u03b1) in S i \u2229 s, \u2191\u2191f x \u2202\u03bc\n\u22a2 Tendsto (fun i => \u222b (x : \u03b1) in S i \u2229 s, \u2191\u2191f x \u2202\u03bc) atTop (\ud835\udcdd (\u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc))", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup F'\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2076 : NormedSpace \u211d F'\ninst\u271d\u2075 : CompleteSpace F'\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedAddCommGroup G'\ninst\u271d\u00b2 : NormedSpace \u211d G'\ninst\u271d\u00b9 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nf\u271d g : \u03b1 \u2192 F'\ns : Set \u03b1\nf : { x // x \u2208 Lp F' 1 }\nhs : MeasurableSet s\nS : \u2115 \u2192 Set \u03b1 := spanningSets (Measure.trim \u03bc hm)\nhS_meas : \u2200 (i : \u2115), MeasurableSet (S i)\nhS_meas0 : \u2200 (i : \u2115), MeasurableSet (S i)\nhs_eq : s = \u22c3 i, S i \u2229 s\nhS_finite : \u2200 (i : \u2115), \u2191\u2191\u03bc (S i \u2229 s) < \u22a4\nh_mono : Monotone fun i => S i \u2229 s\nh_eq_forall :\n  (fun i => \u222b (x : \u03b1) in S i \u2229 s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) f) x \u2202\u03bc) = fun i => \u222b (x : \u03b1) in S i \u2229 s, \u2191\u2191f x \u2202\u03bc\nh : Tendsto (fun i => \u222b (a : \u03b1) in S i \u2229 s, \u2191\u2191f a \u2202\u03bc) atTop (\ud835\udcdd (\u222b (a : \u03b1) in \u22c3 n, S n \u2229 s, \u2191\u2191f a \u2202\u03bc))\n\u22a2 Tendsto (fun i => \u222b (x : \u03b1) in S i \u2229 s, \u2191\u2191f x \u2202\u03bc) atTop (\ud835\udcdd (\u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc))"}, {"tactic": "rwa [\u2190 hs_eq] at h", "annotated_tactic": ["rwa [\u2190 hs_eq] at h", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup F'\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2076 : NormedSpace \u211d F'\ninst\u271d\u2075 : CompleteSpace F'\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedAddCommGroup G'\ninst\u271d\u00b2 : NormedSpace \u211d G'\ninst\u271d\u00b9 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nf\u271d g : \u03b1 \u2192 F'\ns : Set \u03b1\nf : { x // x \u2208 Lp F' 1 }\nhs : MeasurableSet s\nS : \u2115 \u2192 Set \u03b1 := spanningSets (Measure.trim \u03bc hm)\nhS_meas : \u2200 (i : \u2115), MeasurableSet (S i)\nhS_meas0 : \u2200 (i : \u2115), MeasurableSet (S i)\nhs_eq : s = \u22c3 i, S i \u2229 s\nhS_finite : \u2200 (i : \u2115), \u2191\u2191\u03bc (S i \u2229 s) < \u22a4\nh_mono : Monotone fun i => S i \u2229 s\nh_eq_forall :\n  (fun i => \u222b (x : \u03b1) in S i \u2229 s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) f) x \u2202\u03bc) = fun i => \u222b (x : \u03b1) in S i \u2229 s, \u2191\u2191f x \u2202\u03bc\nh : Tendsto (fun i => \u222b (a : \u03b1) in S i \u2229 s, \u2191\u2191f a \u2202\u03bc) atTop (\ud835\udcdd (\u222b (a : \u03b1) in \u22c3 n, S n \u2229 s, \u2191\u2191f a \u2202\u03bc))\n\u22a2 Tendsto (fun i => \u222b (x : \u03b1) in S i \u2229 s, \u2191\u2191f x \u2202\u03bc) atTop (\ud835\udcdd (\u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc))", "state_after": "no goals"}, {"tactic": "have h := tendsto_set_integral_of_monotone (fun i => (hS_meas0 i).inter (hm s hs)) h_mono\n  (L1.integrable_coeFn (condexpL1Clm F' hm \u03bc f)).integrableOn", "annotated_tactic": ["have h := <a>tendsto_set_integral_of_monotone</a> (fun i => (hS_meas0 i).<a>inter</a> (hm s hs)) h_mono\n      (<a>L1.integrable_coeFn</a> (<a>condexpL1Clm</a> F' hm \u03bc f)).<a>integrableOn</a>", [{"full_name": "MeasureTheory.tendsto_set_integral_of_monotone", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [212, 9], "def_end_pos": [212, 41]}, {"full_name": "MeasurableSet.inter", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [198, 19], "def_end_pos": [198, 38]}, {"full_name": "MeasureTheory.L1.integrable_coeFn", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [1324, 9], "def_end_pos": [1324, 25]}, {"full_name": "MeasureTheory.condexpL1Clm", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean", "def_pos": [380, 5], "def_end_pos": [380, 17]}, {"full_name": "MeasureTheory.Integrable.integrableOn", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [163, 9], "def_end_pos": [163, 32]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup F'\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2076 : NormedSpace \u211d F'\ninst\u271d\u2075 : CompleteSpace F'\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedAddCommGroup G'\ninst\u271d\u00b2 : NormedSpace \u211d G'\ninst\u271d\u00b9 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nf\u271d g : \u03b1 \u2192 F'\ns : Set \u03b1\nf : { x // x \u2208 Lp F' 1 }\nhs : MeasurableSet s\nS : \u2115 \u2192 Set \u03b1 := spanningSets (Measure.trim \u03bc hm)\nhS_meas : \u2200 (i : \u2115), MeasurableSet (S i)\nhS_meas0 : \u2200 (i : \u2115), MeasurableSet (S i)\nhs_eq : s = \u22c3 i, S i \u2229 s\nhS_finite : \u2200 (i : \u2115), \u2191\u2191\u03bc (S i \u2229 s) < \u22a4\nh_mono : Monotone fun i => S i \u2229 s\nh_eq_forall :\n  (fun i => \u222b (x : \u03b1) in S i \u2229 s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) f) x \u2202\u03bc) = fun i => \u222b (x : \u03b1) in S i \u2229 s, \u2191\u2191f x \u2202\u03bc\nh_right : Tendsto (fun i => \u222b (x : \u03b1) in S i \u2229 s, \u2191\u2191f x \u2202\u03bc) atTop (\ud835\udcdd (\u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc))\n\u22a2 Tendsto (fun i => \u222b (x : \u03b1) in S i \u2229 s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) f) x \u2202\u03bc) atTop\n    (\ud835\udcdd (\u222b (x : \u03b1) in s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) f) x \u2202\u03bc))", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup F'\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2076 : NormedSpace \u211d F'\ninst\u271d\u2075 : CompleteSpace F'\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedAddCommGroup G'\ninst\u271d\u00b2 : NormedSpace \u211d G'\ninst\u271d\u00b9 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nf\u271d g : \u03b1 \u2192 F'\ns : Set \u03b1\nf : { x // x \u2208 Lp F' 1 }\nhs : MeasurableSet s\nS : \u2115 \u2192 Set \u03b1 := spanningSets (Measure.trim \u03bc hm)\nhS_meas : \u2200 (i : \u2115), MeasurableSet (S i)\nhS_meas0 : \u2200 (i : \u2115), MeasurableSet (S i)\nhs_eq : s = \u22c3 i, S i \u2229 s\nhS_finite : \u2200 (i : \u2115), \u2191\u2191\u03bc (S i \u2229 s) < \u22a4\nh_mono : Monotone fun i => S i \u2229 s\nh_eq_forall :\n  (fun i => \u222b (x : \u03b1) in S i \u2229 s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) f) x \u2202\u03bc) = fun i => \u222b (x : \u03b1) in S i \u2229 s, \u2191\u2191f x \u2202\u03bc\nh_right : Tendsto (fun i => \u222b (x : \u03b1) in S i \u2229 s, \u2191\u2191f x \u2202\u03bc) atTop (\ud835\udcdd (\u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc))\nh :\n  Tendsto (fun i => \u222b (a : \u03b1) in S i \u2229 s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) f) a \u2202\u03bc) atTop\n    (\ud835\udcdd (\u222b (a : \u03b1) in \u22c3 n, S n \u2229 s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) f) a \u2202\u03bc))\n\u22a2 Tendsto (fun i => \u222b (x : \u03b1) in S i \u2229 s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) f) x \u2202\u03bc) atTop\n    (\ud835\udcdd (\u222b (x : \u03b1) in s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) f) x \u2202\u03bc))"}, {"tactic": "rwa [\u2190 hs_eq] at h", "annotated_tactic": ["rwa [\u2190 hs_eq] at h", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup F'\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2076 : NormedSpace \u211d F'\ninst\u271d\u2075 : CompleteSpace F'\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedAddCommGroup G'\ninst\u271d\u00b2 : NormedSpace \u211d G'\ninst\u271d\u00b9 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nf\u271d g : \u03b1 \u2192 F'\ns : Set \u03b1\nf : { x // x \u2208 Lp F' 1 }\nhs : MeasurableSet s\nS : \u2115 \u2192 Set \u03b1 := spanningSets (Measure.trim \u03bc hm)\nhS_meas : \u2200 (i : \u2115), MeasurableSet (S i)\nhS_meas0 : \u2200 (i : \u2115), MeasurableSet (S i)\nhs_eq : s = \u22c3 i, S i \u2229 s\nhS_finite : \u2200 (i : \u2115), \u2191\u2191\u03bc (S i \u2229 s) < \u22a4\nh_mono : Monotone fun i => S i \u2229 s\nh_eq_forall :\n  (fun i => \u222b (x : \u03b1) in S i \u2229 s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) f) x \u2202\u03bc) = fun i => \u222b (x : \u03b1) in S i \u2229 s, \u2191\u2191f x \u2202\u03bc\nh_right : Tendsto (fun i => \u222b (x : \u03b1) in S i \u2229 s, \u2191\u2191f x \u2202\u03bc) atTop (\ud835\udcdd (\u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc))\nh :\n  Tendsto (fun i => \u222b (a : \u03b1) in S i \u2229 s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) f) a \u2202\u03bc) atTop\n    (\ud835\udcdd (\u222b (a : \u03b1) in \u22c3 n, S n \u2229 s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) f) a \u2202\u03bc))\n\u22a2 Tendsto (fun i => \u222b (x : \u03b1) in S i \u2229 s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) f) x \u2202\u03bc) atTop\n    (\ud835\udcdd (\u222b (x : \u03b1) in s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) f) x \u2202\u03bc))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "full_name": "Int.le_total", "start": [570, 11], "end": [572, 80], "traced_tactics": [{"tactic": "rwa [show -(b - a) = a - b by simp [Int.add_comm, Int.sub_eq_add_neg]] at H", "annotated_tactic": ["rwa [show -(b - a) = a - b by simp [<a>Int.add_comm</a>, <a>Int.sub_eq_add_neg</a>]] at H", [{"full_name": "Int.add_comm", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [218, 19], "def_end_pos": [218, 27]}, {"full_name": "Int.sub_eq_add_neg", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [94, 19], "def_end_pos": [94, 33]}]], "state_before": "a b : Int\nH : NonNeg (-(b - a))\n\u22a2 b \u2264 a", "state_after": "no goals"}, {"tactic": "simp [Int.add_comm, Int.sub_eq_add_neg]", "annotated_tactic": ["simp [<a>Int.add_comm</a>, <a>Int.sub_eq_add_neg</a>]", [{"full_name": "Int.add_comm", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [218, 19], "def_end_pos": [218, 27]}, {"full_name": "Int.sub_eq_add_neg", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [94, 19], "def_end_pos": [94, 33]}]], "state_before": "a b : Int\nH : NonNeg (-(b - a))\n\u22a2 -(b - a) = a - b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Setoid/Partition.lean", "full_name": "Setoid.IsPartition.pairwiseDisjoint", "start": [220, 1], "end": [222, 28], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Analysis/Topology.lean", "full_name": "Ctop.Realizer.ofEquiv_F", "start": [204, 1], "end": [205, 22], "traced_tactics": [{"tactic": "delta ofEquiv", "annotated_tactic": ["delta <a>ofEquiv</a>", [{"full_name": "Ctop.Realizer.ofEquiv", "def_path": "Mathlib/Data/Analysis/Topology.lean", "def_pos": [192, 5], "def_end_pos": [192, 12]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03c3 : Type u_3\n\u03c4 : Type u_4\ninst\u271d : TopologicalSpace \u03b1\nF : Realizer \u03b1\nE : F.\u03c3 \u2243 \u03c4\ns : \u03c4\n\u22a2 f (ofEquiv F E).F s = f F.F (\u2191E.symm s)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03c3 : Type u_3\n\u03c4 : Type u_4\ninst\u271d : TopologicalSpace \u03b1\nF : Realizer \u03b1\nE : F.\u03c3 \u2243 \u03c4\ns : \u03c4\n\u22a2 f { \u03c3 := \u03c4, F := Ctop.ofEquiv E F.F, eq := (_ : toTopsp (Ctop.ofEquiv E F.F) = inst\u271d) }.F s = f F.F (\u2191E.symm s)"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03c3 : Type u_3\n\u03c4 : Type u_4\ninst\u271d : TopologicalSpace \u03b1\nF : Realizer \u03b1\nE : F.\u03c3 \u2243 \u03c4\ns : \u03c4\n\u22a2 f { \u03c3 := \u03c4, F := Ctop.ofEquiv E F.F, eq := (_ : toTopsp (Ctop.ofEquiv E F.F) = inst\u271d) }.F s = f F.F (\u2191E.symm s)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "full_name": "MeasureTheory.integrable_withDensity_iff_integrable_smul'", "start": [923, 1], "end": [929, 30], "traced_tactics": [{"tactic": "rw [\u2190 withDensity_congr_ae (coe_toNNReal_ae_eq hflt),\n  integrable_withDensity_iff_integrable_smul]", "annotated_tactic": ["rw [\u2190 <a>withDensity_congr_ae</a> (<a>coe_toNNReal_ae_eq</a> hflt),\n    <a>integrable_withDensity_iff_integrable_smul</a>]", [{"full_name": "MeasureTheory.withDensity_congr_ae", "def_path": "Mathlib/MeasureTheory/Measure/WithDensity.lean", "def_pos": [50, 9], "def_end_pos": [50, 29]}, {"full_name": "MeasureTheory.coe_toNNReal_ae_eq", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [889, 9], "def_end_pos": [889, 27]}, {"full_name": "MeasureTheory.integrable_withDensity_iff_integrable_smul", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [918, 9], "def_end_pos": [918, 51]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b4\ninst\u271d\u00b3 : NormedAddCommGroup \u03b2\ninst\u271d\u00b2 : NormedAddCommGroup \u03b3\nE : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhflt : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4\ng : \u03b1 \u2192 E\n\u22a2 Integrable g \u2194 Integrable fun x => ENNReal.toReal (f x) \u2022 g x", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b4\ninst\u271d\u00b3 : NormedAddCommGroup \u03b2\ninst\u271d\u00b2 : NormedAddCommGroup \u03b3\nE : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhflt : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4\ng : \u03b1 \u2192 E\n\u22a2 (Integrable fun x => ENNReal.toNNReal (f x) \u2022 g x) \u2194 Integrable fun x => ENNReal.toReal (f x) \u2022 g x\n\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b4\ninst\u271d\u00b3 : NormedAddCommGroup \u03b2\ninst\u271d\u00b2 : NormedAddCommGroup \u03b3\nE : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhflt : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4\ng : \u03b1 \u2192 E\n\u22a2 Measurable fun x => ENNReal.toNNReal (f x)"}, {"tactic": "simp_rw [NNReal.smul_def, ENNReal.toReal]", "annotated_tactic": ["simp_rw [<a>NNReal.smul_def</a>, <a>ENNReal.toReal</a>]", [{"full_name": "NNReal.smul_def", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [247, 9], "def_end_pos": [247, 17]}, {"full_name": "ENNReal.toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [168, 15], "def_end_pos": [168, 21]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b4\ninst\u271d\u00b3 : NormedAddCommGroup \u03b2\ninst\u271d\u00b2 : NormedAddCommGroup \u03b3\nE : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhflt : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4\ng : \u03b1 \u2192 E\n\u22a2 (Integrable fun x => ENNReal.toNNReal (f x) \u2022 g x) \u2194 Integrable fun x => ENNReal.toReal (f x) \u2022 g x", "state_after": "no goals"}, {"tactic": "exact hf.ennreal_toNNReal", "annotated_tactic": ["exact hf.ennreal_toNNReal", []], "state_before": "case hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b4\ninst\u271d\u00b3 : NormedAddCommGroup \u03b2\ninst\u271d\u00b2 : NormedAddCommGroup \u03b3\nE : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhflt : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4\ng : \u03b1 \u2192 E\n\u22a2 Measurable fun x => ENNReal.toNNReal (f x)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "full_name": "Set.isUnit_iff", "start": [1072, 1], "end": [1080, 17], "traced_tactics": [{"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : DivisionMonoid \u03b1\ns t : Set \u03b1\n\u22a2 IsUnit s \u2194 \u2203 a, s = {a} \u2227 IsUnit a", "state_after": "case mp\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : DivisionMonoid \u03b1\ns t : Set \u03b1\n\u22a2 IsUnit s \u2192 \u2203 a, s = {a} \u2227 IsUnit a\n\ncase mpr\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : DivisionMonoid \u03b1\ns t : Set \u03b1\n\u22a2 (\u2203 a, s = {a} \u2227 IsUnit a) \u2192 IsUnit s"}, {"tactic": "rintro \u27e8u, rfl\u27e9", "annotated_tactic": ["rintro \u27e8u, rfl\u27e9", []], "state_before": "case mp\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : DivisionMonoid \u03b1\ns t : Set \u03b1\n\u22a2 IsUnit s \u2192 \u2203 a, s = {a} \u2227 IsUnit a", "state_after": "case mp.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : DivisionMonoid \u03b1\nt : Set \u03b1\nu : (Set \u03b1)\u02e3\n\u22a2 \u2203 a, \u2191u = {a} \u2227 IsUnit a"}, {"tactic": "obtain \u27e8a, b, ha, hb, h\u27e9 := Set.mul_eq_one_iff.1 u.mul_inv", "annotated_tactic": ["obtain \u27e8a, b, ha, hb, h\u27e9 := <a>Set.mul_eq_one_iff</a>.1 u.mul_inv", [{"full_name": "Set.mul_eq_one_iff", "def_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "def_pos": [1039, 19], "def_end_pos": [1039, 33]}]], "state_before": "case mp.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : DivisionMonoid \u03b1\nt : Set \u03b1\nu : (Set \u03b1)\u02e3\n\u22a2 \u2203 a, \u2191u = {a} \u2227 IsUnit a", "state_after": "case mp.intro.intro.intro.intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : DivisionMonoid \u03b1\nt : Set \u03b1\nu : (Set \u03b1)\u02e3\na b : \u03b1\nha : \u2191u = {a}\nhb : \u2191u\u207b\u00b9 = {b}\nh : a * b = 1\n\u22a2 \u2203 a, \u2191u = {a} \u2227 IsUnit a"}, {"tactic": "refine' \u27e8a, ha, \u27e8a, b, h, singleton_injective _\u27e9, rfl\u27e9", "annotated_tactic": ["refine' \u27e8a, ha, \u27e8a, b, h, <a>singleton_injective</a> _\u27e9, <a>rfl</a>\u27e9", [{"full_name": "Set.singleton_injective", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1302, 9], "def_end_pos": [1302, 28]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "case mp.intro.intro.intro.intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : DivisionMonoid \u03b1\nt : Set \u03b1\nu : (Set \u03b1)\u02e3\na b : \u03b1\nha : \u2191u = {a}\nhb : \u2191u\u207b\u00b9 = {b}\nh : a * b = 1\n\u22a2 \u2203 a, \u2191u = {a} \u2227 IsUnit a", "state_after": "case mp.intro.intro.intro.intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : DivisionMonoid \u03b1\nt : Set \u03b1\nu : (Set \u03b1)\u02e3\na b : \u03b1\nha : \u2191u = {a}\nhb : \u2191u\u207b\u00b9 = {b}\nh : a * b = 1\n\u22a2 {b * a} = {1}"}, {"tactic": "rw [\u2190 singleton_mul_singleton, \u2190 ha, \u2190 hb]", "annotated_tactic": ["rw [\u2190 <a>singleton_mul_singleton</a>, \u2190 ha, \u2190 hb]", [{"full_name": "Set.singleton_mul_singleton", "def_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "def_pos": [410, 9], "def_end_pos": [410, 32]}]], "state_before": "case mp.intro.intro.intro.intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : DivisionMonoid \u03b1\nt : Set \u03b1\nu : (Set \u03b1)\u02e3\na b : \u03b1\nha : \u2191u = {a}\nhb : \u2191u\u207b\u00b9 = {b}\nh : a * b = 1\n\u22a2 {b * a} = {1}", "state_after": "case mp.intro.intro.intro.intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : DivisionMonoid \u03b1\nt : Set \u03b1\nu : (Set \u03b1)\u02e3\na b : \u03b1\nha : \u2191u = {a}\nhb : \u2191u\u207b\u00b9 = {b}\nh : a * b = 1\n\u22a2 \u2191u\u207b\u00b9 * \u2191u = {1}"}, {"tactic": "exact u.inv_mul", "annotated_tactic": ["exact u.inv_mul", []], "state_before": "case mp.intro.intro.intro.intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : DivisionMonoid \u03b1\nt : Set \u03b1\nu : (Set \u03b1)\u02e3\na b : \u03b1\nha : \u2191u = {a}\nhb : \u2191u\u207b\u00b9 = {b}\nh : a * b = 1\n\u22a2 \u2191u\u207b\u00b9 * \u2191u = {1}", "state_after": "no goals"}, {"tactic": "rintro \u27e8a, rfl, ha\u27e9", "annotated_tactic": ["rintro \u27e8a, rfl, ha\u27e9", []], "state_before": "case mpr\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : DivisionMonoid \u03b1\ns t : Set \u03b1\n\u22a2 (\u2203 a, s = {a} \u2227 IsUnit a) \u2192 IsUnit s", "state_after": "case mpr.intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : DivisionMonoid \u03b1\nt : Set \u03b1\na : \u03b1\nha : IsUnit a\n\u22a2 IsUnit {a}"}, {"tactic": "exact ha.set", "annotated_tactic": ["exact ha.set", []], "state_before": "case mpr.intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : DivisionMonoid \u03b1\nt : Set \u03b1\na : \u03b1\nha : IsUnit a\n\u22a2 IsUnit {a}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Martingale/OptionalStopping.lean", "full_name": "MeasureTheory.smul_le_stoppedValue_hitting", "start": [112, 1], "end": [133, 69], "traced_tactics": [{"tactic": "have hn : Set.Icc 0 n = {k | k \u2264 n} := by ext x; simp", "annotated_tactic": ["have hn : <a>Set.Icc</a> 0 n = {k | k \u2264 n} := by ext x; simp", [{"full_name": "Set.Icc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [59, 5], "def_end_pos": [59, 8]}]], "state_before": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\n\u03b5 : \u211d\u22650\nn : \u2115\n\u22a2 \u03b5 \u2022 \u2191\u2191\u03bc {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9} \u2264\n    ENNReal.ofReal\n      (\u222b (\u03c9 : \u03a9) in {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9},\n        stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) \u03c9 \u2202\u03bc)", "state_after": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\n\u03b5 : \u211d\u22650\nn : \u2115\nhn : Set.Icc 0 n = {k | k \u2264 n}\n\u22a2 \u03b5 \u2022 \u2191\u2191\u03bc {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9} \u2264\n    ENNReal.ofReal\n      (\u222b (\u03c9 : \u03a9) in {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9},\n        stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) \u03c9 \u2202\u03bc)"}, {"tactic": "have : \u2200 \u03c9, ((\u03b5 : \u211d) \u2264 (range (n + 1)).sup' nonempty_range_succ fun k => f k \u03c9) \u2192\n    (\u03b5 : \u211d) \u2264 stoppedValue f (hitting f {y : \u211d | \u2191\u03b5 \u2264 y} 0 n) \u03c9 := by\n  intro x hx\n  simp_rw [le_sup'_iff, mem_range, Nat.lt_succ_iff] at hx\n  refine' stoppedValue_hitting_mem _\n  simp only [Set.mem_setOf_eq, exists_prop, hn]\n  exact\n    let \u27e8j, hj\u2081, hj\u2082\u27e9 := hx\n    \u27e8j, hj\u2081, hj\u2082\u27e9", "annotated_tactic": ["have : \u2200 \u03c9, ((\u03b5 : \u211d) \u2264 (<a>range</a> (n + 1)).<a>sup'</a> <a>nonempty_range_succ</a> fun k => f k \u03c9) \u2192\n      (\u03b5 : \u211d) \u2264 <a>stoppedValue</a> f (<a>hitting</a> f {y : \u211d | \u2191\u03b5 \u2264 y} 0 n) \u03c9 := by\n    intro x hx\n    simp_rw [<a>le_sup'_iff</a>, <a>mem_range</a>, <a>Nat.lt_succ_iff</a>] at hx\n    refine' <a>stoppedValue_hitting_mem</a> _\n    simp only [<a>Set.mem_setOf_eq</a>, <a>exists_prop</a>, hn]\n    exact\n      let \u27e8j, hj\u2081, hj\u2082\u27e9 := hx\n      \u27e8j, hj\u2081, hj\u2082\u27e9", [{"full_name": "Finset.range", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3027, 5], "def_end_pos": [3027, 10]}, {"full_name": "Finset.sup'", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [765, 5], "def_end_pos": [765, 9]}, {"full_name": "Finset.nonempty_range_succ", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3105, 9], "def_end_pos": [3105, 28]}, {"full_name": "MeasureTheory.stoppedValue", "def_path": "Mathlib/Probability/Process/Stopping.lean", "def_pos": [768, 5], "def_end_pos": [768, 17]}, {"full_name": "MeasureTheory.hitting", "def_path": "Mathlib/Probability/Process/HittingTime.lean", "def_pos": [51, 19], "def_end_pos": [51, 26]}, {"full_name": "Finset.le_sup'_iff", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [1175, 9], "def_end_pos": [1175, 20]}, {"full_name": "Finset.mem_range", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3037, 9], "def_end_pos": [3037, 18]}, {"full_name": "Nat.lt_succ_iff", "def_path": "Mathlib/Data/Nat/Basic.lean", "def_pos": [207, 9], "def_end_pos": [207, 20]}, {"full_name": "MeasureTheory.stoppedValue_hitting_mem", "def_path": "Mathlib/Probability/Process/HittingTime.lean", "def_pos": [244, 9], "def_end_pos": [244, 33]}, {"full_name": "Set.mem_setOf_eq", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [256, 29], "def_end_pos": [256, 41]}, {"full_name": "exists_prop", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [485, 17], "def_end_pos": [485, 28]}]], "state_before": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\n\u03b5 : \u211d\u22650\nn : \u2115\nhn : Set.Icc 0 n = {k | k \u2264 n}\n\u22a2 \u03b5 \u2022 \u2191\u2191\u03bc {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9} \u2264\n    ENNReal.ofReal\n      (\u222b (\u03c9 : \u03a9) in {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9},\n        stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) \u03c9 \u2202\u03bc)", "state_after": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\n\u03b5 : \u211d\u22650\nn : \u2115\nhn : Set.Icc 0 n = {k | k \u2264 n}\nthis :\n  \u2200 (\u03c9 : \u03a9),\n    (\u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) \u2192\n      \u2191\u03b5 \u2264 stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) \u03c9\n\u22a2 \u03b5 \u2022 \u2191\u2191\u03bc {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9} \u2264\n    ENNReal.ofReal\n      (\u222b (\u03c9 : \u03a9) in {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9},\n        stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) \u03c9 \u2202\u03bc)"}, {"tactic": "have h := set_integral_ge_of_const_le (measurableSet_le measurable_const\n  (Finset.measurable_range_sup'' fun n _ => (hsub.stronglyMeasurable n).measurable.le (\ud835\udca2.le n)))\n    (measure_ne_top _ _) this (Integrable.integrableOn (hsub.integrable_stoppedValue\n      (hitting_isStoppingTime hsub.adapted measurableSet_Ici) hitting_le))", "annotated_tactic": ["have h := <a>set_integral_ge_of_const_le</a> (<a>measurableSet_le</a> <a>measurable_const</a>\n    (<a>Finset.measurable_range_sup''</a> fun n _ => (hsub.stronglyMeasurable n).measurable.le (\ud835\udca2.le n)))\n      (<a>measure_ne_top</a> _ _) this (<a>Integrable.integrableOn</a> (hsub.integrable_stoppedValue\n        (<a>hitting_isStoppingTime</a> hsub.adapted <a>measurableSet_Ici</a>) <a>hitting_le</a>))", [{"full_name": "MeasureTheory.set_integral_ge_of_const_le", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [745, 9], "def_end_pos": [745, 36]}, {"full_name": "measurableSet_le", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [559, 9], "def_end_pos": [559, 25]}, {"full_name": "measurable_const", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [570, 9], "def_end_pos": [570, 25]}, {"full_name": "Finset.measurable_range_sup''", "def_path": "Mathlib/MeasureTheory/Lattice.lean", "def_pos": [256, 9], "def_end_pos": [256, 38]}, {"full_name": "MeasureTheory.measure_ne_top", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2875, 9], "def_end_pos": [2875, 23]}, {"full_name": "MeasureTheory.Integrable.integrableOn", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [163, 9], "def_end_pos": [163, 32]}, {"full_name": "MeasureTheory.hitting_isStoppingTime", "def_path": "Mathlib/Probability/Process/HittingTime.lean", "def_pos": [227, 9], "def_end_pos": [227, 31]}, {"full_name": "measurableSet_Ici", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [510, 9], "def_end_pos": [510, 26]}, {"full_name": "MeasureTheory.hitting_le", "def_path": "Mathlib/Probability/Process/HittingTime.lean", "def_pos": [71, 9], "def_end_pos": [71, 19]}]], "state_before": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\n\u03b5 : \u211d\u22650\nn : \u2115\nhn : Set.Icc 0 n = {k | k \u2264 n}\nthis :\n  \u2200 (\u03c9 : \u03a9),\n    (\u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) \u2192\n      \u2191\u03b5 \u2264 stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) \u03c9\n\u22a2 \u03b5 \u2022 \u2191\u2191\u03bc {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9} \u2264\n    ENNReal.ofReal\n      (\u222b (\u03c9 : \u03a9) in {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9},\n        stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) \u03c9 \u2202\u03bc)", "state_after": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\n\u03b5 : \u211d\u22650\nn : \u2115\nhn : Set.Icc 0 n = {k | k \u2264 n}\nthis :\n  \u2200 (\u03c9 : \u03a9),\n    (\u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) \u2192\n      \u2191\u03b5 \u2264 stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) \u03c9\nh :\n  \u2191\u03b5 * ENNReal.toReal (\u2191\u2191\u03bc {a | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k a}) \u2264\n    \u222b (x : \u03a9) in {a | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k a},\n      stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) x \u2202\u03bc\n\u22a2 \u03b5 \u2022 \u2191\u2191\u03bc {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9} \u2264\n    ENNReal.ofReal\n      (\u222b (\u03c9 : \u03a9) in {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9},\n        stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) \u03c9 \u2202\u03bc)"}, {"tactic": "rw [ENNReal.le_ofReal_iff_toReal_le, ENNReal.toReal_smul]", "annotated_tactic": ["rw [<a>ENNReal.le_ofReal_iff_toReal_le</a>, <a>ENNReal.toReal_smul</a>]", [{"full_name": "ENNReal.le_ofReal_iff_toReal_le", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2204, 9], "def_end_pos": [2204, 32]}, {"full_name": "ENNReal.toReal_smul", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2333, 9], "def_end_pos": [2333, 20]}]], "state_before": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\n\u03b5 : \u211d\u22650\nn : \u2115\nhn : Set.Icc 0 n = {k | k \u2264 n}\nthis :\n  \u2200 (\u03c9 : \u03a9),\n    (\u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) \u2192\n      \u2191\u03b5 \u2264 stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) \u03c9\nh :\n  \u2191\u03b5 * ENNReal.toReal (\u2191\u2191\u03bc {a | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k a}) \u2264\n    \u222b (x : \u03a9) in {a | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k a},\n      stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) x \u2202\u03bc\n\u22a2 \u03b5 \u2022 \u2191\u2191\u03bc {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9} \u2264\n    ENNReal.ofReal\n      (\u222b (\u03c9 : \u03a9) in {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9},\n        stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) \u03c9 \u2202\u03bc)", "state_after": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\n\u03b5 : \u211d\u22650\nn : \u2115\nhn : Set.Icc 0 n = {k | k \u2264 n}\nthis :\n  \u2200 (\u03c9 : \u03a9),\n    (\u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) \u2192\n      \u2191\u03b5 \u2264 stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) \u03c9\nh :\n  \u2191\u03b5 * ENNReal.toReal (\u2191\u2191\u03bc {a | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k a}) \u2264\n    \u222b (x : \u03a9) in {a | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k a},\n      stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) x \u2202\u03bc\n\u22a2 \u03b5 \u2022 ENNReal.toReal (\u2191\u2191\u03bc {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9}) \u2264\n    \u222b (\u03c9 : \u03a9) in {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9},\n      stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) \u03c9 \u2202\u03bc\n\ncase ha\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\n\u03b5 : \u211d\u22650\nn : \u2115\nhn : Set.Icc 0 n = {k | k \u2264 n}\nthis :\n  \u2200 (\u03c9 : \u03a9),\n    (\u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) \u2192\n      \u2191\u03b5 \u2264 stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) \u03c9\nh :\n  \u2191\u03b5 * ENNReal.toReal (\u2191\u2191\u03bc {a | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k a}) \u2264\n    \u222b (x : \u03a9) in {a | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k a},\n      stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) x \u2202\u03bc\n\u22a2 \u03b5 \u2022 \u2191\u2191\u03bc {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9} \u2260 \u22a4\n\ncase hb\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\n\u03b5 : \u211d\u22650\nn : \u2115\nhn : Set.Icc 0 n = {k | k \u2264 n}\nthis :\n  \u2200 (\u03c9 : \u03a9),\n    (\u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) \u2192\n      \u2191\u03b5 \u2264 stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) \u03c9\nh :\n  \u2191\u03b5 * ENNReal.toReal (\u2191\u2191\u03bc {a | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k a}) \u2264\n    \u222b (x : \u03a9) in {a | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k a},\n      stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) x \u2202\u03bc\n\u22a2 0 \u2264\n    \u222b (\u03c9 : \u03a9) in {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9},\n      stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) \u03c9 \u2202\u03bc"}, {"tactic": "ext x", "annotated_tactic": ["ext x", []], "state_before": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\n\u03b5 : \u211d\u22650\nn : \u2115\n\u22a2 Set.Icc 0 n = {k | k \u2264 n}", "state_after": "case h\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\n\u03b5 : \u211d\u22650\nn x : \u2115\n\u22a2 x \u2208 Set.Icc 0 n \u2194 x \u2208 {k | k \u2264 n}"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case h\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\n\u03b5 : \u211d\u22650\nn x : \u2115\n\u22a2 x \u2208 Set.Icc 0 n \u2194 x \u2208 {k | k \u2264 n}", "state_after": "no goals"}, {"tactic": "intro x hx", "annotated_tactic": ["intro x hx", []], "state_before": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\n\u03b5 : \u211d\u22650\nn : \u2115\nhn : Set.Icc 0 n = {k | k \u2264 n}\n\u22a2 \u2200 (\u03c9 : \u03a9),\n    (\u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) \u2192\n      \u2191\u03b5 \u2264 stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) \u03c9", "state_after": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\n\u03b5 : \u211d\u22650\nn : \u2115\nhn : Set.Icc 0 n = {k | k \u2264 n}\nx : \u03a9\nhx : \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k x\n\u22a2 \u2191\u03b5 \u2264 stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) x"}, {"tactic": "simp_rw [le_sup'_iff, mem_range, Nat.lt_succ_iff] at hx", "annotated_tactic": ["simp_rw [<a>le_sup'_iff</a>, <a>mem_range</a>, <a>Nat.lt_succ_iff</a>] at hx", [{"full_name": "Finset.le_sup'_iff", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [1175, 9], "def_end_pos": [1175, 20]}, {"full_name": "Finset.mem_range", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3037, 9], "def_end_pos": [3037, 18]}, {"full_name": "Nat.lt_succ_iff", "def_path": "Mathlib/Data/Nat/Basic.lean", "def_pos": [207, 9], "def_end_pos": [207, 20]}]], "state_before": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\n\u03b5 : \u211d\u22650\nn : \u2115\nhn : Set.Icc 0 n = {k | k \u2264 n}\nx : \u03a9\nhx : \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k x\n\u22a2 \u2191\u03b5 \u2264 stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) x", "state_after": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\n\u03b5 : \u211d\u22650\nn : \u2115\nhn : Set.Icc 0 n = {k | k \u2264 n}\nx : \u03a9\nhx : \u2203 b, b \u2264 n \u2227 \u2191\u03b5 \u2264 f b x\n\u22a2 \u2191\u03b5 \u2264 stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) x"}, {"tactic": "refine' stoppedValue_hitting_mem _", "annotated_tactic": ["refine' <a>stoppedValue_hitting_mem</a> _", [{"full_name": "MeasureTheory.stoppedValue_hitting_mem", "def_path": "Mathlib/Probability/Process/HittingTime.lean", "def_pos": [244, 9], "def_end_pos": [244, 33]}]], "state_before": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\n\u03b5 : \u211d\u22650\nn : \u2115\nhn : Set.Icc 0 n = {k | k \u2264 n}\nx : \u03a9\nhx : \u2203 b, b \u2264 n \u2227 \u2191\u03b5 \u2264 f b x\n\u22a2 \u2191\u03b5 \u2264 stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) x", "state_after": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\n\u03b5 : \u211d\u22650\nn : \u2115\nhn : Set.Icc 0 n = {k | k \u2264 n}\nx : \u03a9\nhx : \u2203 b, b \u2264 n \u2227 \u2191\u03b5 \u2264 f b x\n\u22a2 \u2203 j, j \u2208 Set.Icc 0 n \u2227 f j x \u2208 {y | \u2191\u03b5 \u2264 y}"}, {"tactic": "simp only [Set.mem_setOf_eq, exists_prop, hn]", "annotated_tactic": ["simp only [<a>Set.mem_setOf_eq</a>, <a>exists_prop</a>, hn]", [{"full_name": "Set.mem_setOf_eq", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [256, 29], "def_end_pos": [256, 41]}, {"full_name": "exists_prop", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [485, 17], "def_end_pos": [485, 28]}]], "state_before": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\n\u03b5 : \u211d\u22650\nn : \u2115\nhn : Set.Icc 0 n = {k | k \u2264 n}\nx : \u03a9\nhx : \u2203 b, b \u2264 n \u2227 \u2191\u03b5 \u2264 f b x\n\u22a2 \u2203 j, j \u2208 Set.Icc 0 n \u2227 f j x \u2208 {y | \u2191\u03b5 \u2264 y}", "state_after": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\n\u03b5 : \u211d\u22650\nn : \u2115\nhn : Set.Icc 0 n = {k | k \u2264 n}\nx : \u03a9\nhx : \u2203 b, b \u2264 n \u2227 \u2191\u03b5 \u2264 f b x\n\u22a2 \u2203 j, j \u2264 n \u2227 \u2191\u03b5 \u2264 f j x"}, {"tactic": "exact\n  let \u27e8j, hj\u2081, hj\u2082\u27e9 := hx\n  \u27e8j, hj\u2081, hj\u2082\u27e9", "annotated_tactic": ["exact\n      let \u27e8j, hj\u2081, hj\u2082\u27e9 := hx\n      \u27e8j, hj\u2081, hj\u2082\u27e9", []], "state_before": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\n\u03b5 : \u211d\u22650\nn : \u2115\nhn : Set.Icc 0 n = {k | k \u2264 n}\nx : \u03a9\nhx : \u2203 b, b \u2264 n \u2227 \u2191\u03b5 \u2264 f b x\n\u22a2 \u2203 j, j \u2264 n \u2227 \u2191\u03b5 \u2264 f j x", "state_after": "no goals"}, {"tactic": "exact h", "annotated_tactic": ["exact h", []], "state_before": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\n\u03b5 : \u211d\u22650\nn : \u2115\nhn : Set.Icc 0 n = {k | k \u2264 n}\nthis :\n  \u2200 (\u03c9 : \u03a9),\n    (\u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) \u2192\n      \u2191\u03b5 \u2264 stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) \u03c9\nh :\n  \u2191\u03b5 * ENNReal.toReal (\u2191\u2191\u03bc {a | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k a}) \u2264\n    \u222b (x : \u03a9) in {a | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k a},\n      stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) x \u2202\u03bc\n\u22a2 \u03b5 \u2022 ENNReal.toReal (\u2191\u2191\u03bc {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9}) \u2264\n    \u222b (\u03c9 : \u03a9) in {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9},\n      stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) \u03c9 \u2202\u03bc", "state_after": "no goals"}, {"tactic": "exact ENNReal.mul_ne_top (by simp) (measure_ne_top _ _)", "annotated_tactic": ["exact <a>ENNReal.mul_ne_top</a> (by simp) (<a>measure_ne_top</a> _ _)", [{"full_name": "ENNReal.mul_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [615, 9], "def_end_pos": [615, 19]}, {"full_name": "MeasureTheory.measure_ne_top", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2875, 9], "def_end_pos": [2875, 23]}]], "state_before": "case ha\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\n\u03b5 : \u211d\u22650\nn : \u2115\nhn : Set.Icc 0 n = {k | k \u2264 n}\nthis :\n  \u2200 (\u03c9 : \u03a9),\n    (\u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) \u2192\n      \u2191\u03b5 \u2264 stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) \u03c9\nh :\n  \u2191\u03b5 * ENNReal.toReal (\u2191\u2191\u03bc {a | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k a}) \u2264\n    \u222b (x : \u03a9) in {a | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k a},\n      stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) x \u2202\u03bc\n\u22a2 \u03b5 \u2022 \u2191\u2191\u03bc {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9} \u2260 \u22a4", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\n\u03b5 : \u211d\u22650\nn : \u2115\nhn : Set.Icc 0 n = {k | k \u2264 n}\nthis :\n  \u2200 (\u03c9 : \u03a9),\n    (\u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) \u2192\n      \u2191\u03b5 \u2264 stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) \u03c9\nh :\n  \u2191\u03b5 * ENNReal.toReal (\u2191\u2191\u03bc {a | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k a}) \u2264\n    \u222b (x : \u03a9) in {a | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k a},\n      stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) x \u2202\u03bc\n\u22a2 \u2191(RingHom.id \u211d\u22650\u221e) (\u2191\u2191(RingHom.toMonoidWithZeroHom ENNReal.ofNNRealHom) \u03b5) \u2260 \u22a4", "state_after": "no goals"}, {"tactic": "exact le_trans (mul_nonneg \u03b5.coe_nonneg ENNReal.toReal_nonneg) h", "annotated_tactic": ["exact <a>le_trans</a> (<a>mul_nonneg</a> \u03b5.coe_nonneg <a>ENNReal.toReal_nonneg</a>) h", [{"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}, {"full_name": "mul_nonneg", "def_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "def_pos": [380, 7], "def_end_pos": [380, 17]}, {"full_name": "ENNReal.toReal_nonneg", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [221, 17], "def_end_pos": [221, 30]}]], "state_before": "case hb\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\n\u03b5 : \u211d\u22650\nn : \u2115\nhn : Set.Icc 0 n = {k | k \u2264 n}\nthis :\n  \u2200 (\u03c9 : \u03a9),\n    (\u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) \u2192\n      \u2191\u03b5 \u2264 stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) \u03c9\nh :\n  \u2191\u03b5 * ENNReal.toReal (\u2191\u2191\u03bc {a | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k a}) \u2264\n    \u222b (x : \u03a9) in {a | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k a},\n      stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) x \u2202\u03bc\n\u22a2 0 \u2264\n    \u222b (\u03c9 : \u03a9) in {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9},\n      stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) \u03c9 \u2202\u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "full_name": "Int.le_add_of_neg_le_sub_left", "start": [1017, 11], "end": [1018, 63], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "full_name": "MeasurableSet.measurableSet_bliminf", "start": [2125, 1], "end": [2128, 69], "traced_tactics": [{"tactic": "simp only [Filter.bliminf_eq_iSup_biInf_of_nat, iInf_eq_iInter, iSup_eq_iUnion]", "annotated_tactic": ["simp only [<a>Filter.bliminf_eq_iSup_biInf_of_nat</a>, <a>iInf_eq_iInter</a>, <a>iSup_eq_iUnion</a>]", [{"full_name": "Filter.bliminf_eq_iSup_biInf_of_nat", "def_path": "Mathlib/Order/LiminfLimsup.lean", "def_pos": [873, 9], "def_end_pos": [873, 37]}, {"full_name": "Set.iInf_eq_iInter", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [196, 9], "def_end_pos": [196, 23]}, {"full_name": "Set.iSup_eq_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [191, 9], "def_end_pos": [191, 23]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u\u03b9\ns\u271d t u : Set \u03b1\ninst\u271d : MeasurableSpace \u03b1\ns : \u2115 \u2192 Set \u03b1\np : \u2115 \u2192 Prop\nh : \u2200 (n : \u2115), p n \u2192 MeasurableSet (s n)\n\u22a2 MeasurableSet (bliminf s atTop p)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u\u03b9\ns\u271d t u : Set \u03b1\ninst\u271d : MeasurableSpace \u03b1\ns : \u2115 \u2192 Set \u03b1\np : \u2115 \u2192 Prop\nh : \u2200 (n : \u2115), p n \u2192 MeasurableSet (s n)\n\u22a2 MeasurableSet (\u22c3 i, \u22c2 j, \u22c2 (_ : p j \u2227 i \u2264 j), s j)"}, {"tactic": "exact .iUnion fun n => .iInter fun m => .iInter fun hm => h m hm.1", "annotated_tactic": ["exact .iUnion fun n => .iInter fun m => .iInter fun hm => h m hm.1", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u\u03b9\ns\u271d t u : Set \u03b1\ninst\u271d : MeasurableSpace \u03b1\ns : \u2115 \u2192 Set \u03b1\np : \u2115 \u2192 Prop\nh : \u2200 (n : \u2115), p n \u2192 MeasurableSet (s n)\n\u22a2 MeasurableSet (\u22c3 i, \u22c2 j, \u22c2 (_ : p j \u2227 i \u2264 j), s j)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/LpSpace/DomAct/Basic.lean", "full_name": "DomMulAct.edist_smul_Lp", "start": [103, 1], "end": [104, 42], "traced_tactics": [{"tactic": "simp only [Lp.edist_dist, dist_smul_Lp]", "annotated_tactic": ["simp only [<a>Lp.edist_dist</a>, <a>dist_smul_Lp</a>]", [{"full_name": "MeasureTheory.Lp.edist_dist", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [300, 19], "def_end_pos": [300, 29]}, {"full_name": "DomMulAct.dist_smul_Lp", "def_path": "Mathlib/MeasureTheory/Function/LpSpace/DomAct/Basic.lean", "def_pos": [99, 9], "def_end_pos": [99, 21]}]], "state_before": "M : Type u_1\nN : Type u_2\n\u03b1 : Type u_3\nE : Type u_4\ninst\u271d\u2076 : MeasurableSpace M\ninst\u271d\u2075 : MeasurableSpace N\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b2 : SMul M \u03b1\ninst\u271d\u00b9 : SMulInvariantMeasure M \u03b1 \u03bc\ninst\u271d : MeasurableSMul M \u03b1\nc : M\u1d48\u1d50\u1d43\nf g : { x // x \u2208 Lp E p }\n\u22a2 edist (c \u2022 f) (c \u2022 g) = edist f g", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "full_name": "MeasureTheory.integral_Iic_eq_integral_Iio", "start": [695, 1], "end": [696, 55], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "full_name": "String.Iterator.ValidFor.remainingBytes", "start": [546, 1], "end": [547, 72], "traced_tactics": [{"tactic": "simp [Iterator.remainingBytes, Nat.add_sub_cancel_left]", "annotated_tactic": ["simp [<a>Iterator.remainingBytes</a>, <a>Nat.add_sub_cancel_left</a>]", [{"full_name": "String.Iterator.remainingBytes", "def_path": "lake-packages/lean4/src/lean/Init/Data/String/Basic.lean", "def_pos": [314, 5], "def_end_pos": [314, 19]}, {"full_name": "Nat.add_sub_cancel_left", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [598, 19], "def_end_pos": [598, 38]}]], "state_before": "l r : List Char\n\u22a2 Iterator.remainingBytes { s := { data := List.reverseAux l r }, i := { byteIdx := utf8Len l } } = utf8Len r", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Card.lean", "full_name": "Set.encard_exchange'", "start": [260, 1], "end": [261, 87], "traced_tactics": [{"tactic": "rw [\u2190insert_diff_singleton_comm (by rintro rfl; exact ha hb), encard_exchange ha hb]", "annotated_tactic": ["rw [\u2190<a>insert_diff_singleton_comm</a> (by rintro rfl; exact ha hb), <a>encard_exchange</a> ha hb]", [{"full_name": "Set.insert_diff_singleton_comm", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [2082, 9], "def_end_pos": [2082, 35]}, {"full_name": "Set.encard_exchange", "def_path": "Mathlib/Data/Set/Card.lean", "def_pos": [256, 9], "def_end_pos": [256, 24]}]], "state_before": "\u03b1 : Type u_1\ns t : Set \u03b1\na b : \u03b1\nha : \u00aca \u2208 s\nhb : b \u2208 s\n\u22a2 encard (insert a s \\ {b}) = encard s", "state_after": "no goals"}, {"tactic": "rintro rfl", "annotated_tactic": ["rintro rfl", []], "state_before": "\u03b1 : Type u_1\ns t : Set \u03b1\na b : \u03b1\nha : \u00aca \u2208 s\nhb : b \u2208 s\n\u22a2 a \u2260 b", "state_after": "\u03b1 : Type u_1\ns t : Set \u03b1\na : \u03b1\nha : \u00aca \u2208 s\nhb : a \u2208 s\n\u22a2 False"}, {"tactic": "exact ha hb", "annotated_tactic": ["exact ha hb", []], "state_before": "\u03b1 : Type u_1\ns t : Set \u03b1\na : \u03b1\nha : \u00aca \u2208 s\nhb : a \u2208 s\n\u22a2 False", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Logic.lean", "full_name": "Decidable.not_imp_self", "start": [555, 1], "end": [556, 52], "traced_tactics": [{"tactic": "have := @imp_not_self (\u00aca)", "annotated_tactic": ["have := @<a>imp_not_self</a> (\u00aca)", [{"full_name": "imp_not_self", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [30, 17], "def_end_pos": [30, 29]}]], "state_before": "a : Prop\ninst\u271d : Decidable a\n\u22a2 \u00aca \u2192 a \u2194 a", "state_after": "a : Prop\ninst\u271d : Decidable a\nthis : \u00aca \u2192 \u00ac\u00aca \u2194 \u00ac\u00aca\n\u22a2 \u00aca \u2192 a \u2194 a"}, {"tactic": "rwa [not_not] at this", "annotated_tactic": ["rwa [<a>not_not</a>] at this", [{"full_name": "Decidable.not_not", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [513, 9], "def_end_pos": [513, 26]}]], "state_before": "a : Prop\ninst\u271d : Decidable a\nthis : \u00aca \u2192 \u00ac\u00aca \u2194 \u00ac\u00aca\n\u22a2 \u00aca \u2192 a \u2194 a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Image.lean", "full_name": "Set.preimage_surjective", "start": [1558, 1], "end": [1561, 87], "traced_tactics": [{"tactic": "refine' \u27e8fun h x x' hx => _, Injective.preimage_surjective\u27e9", "annotated_tactic": ["refine' \u27e8fun h x x' hx => _, <a>Injective.preimage_surjective</a>\u27e9", [{"full_name": "Function.Injective.preimage_surjective", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [1304, 9], "def_end_pos": [1304, 38]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\n\u22a2 Surjective (preimage f) \u2194 Injective f", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\nh : Surjective (preimage f)\nx x' : \u03b1\nhx : f x = f x'\n\u22a2 x = x'"}, {"tactic": "cases' h {x} with s hs", "annotated_tactic": ["cases' h {x} with s hs", []], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\nh : Surjective (preimage f)\nx x' : \u03b1\nhx : f x = f x'\n\u22a2 x = x'", "state_after": "case intro\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\nh : Surjective (preimage f)\nx x' : \u03b1\nhx : f x = f x'\ns : Set \u03b2\nhs : f \u207b\u00b9' s = {x}\n\u22a2 x = x'"}, {"tactic": "have := mem_singleton x", "annotated_tactic": ["have := <a>mem_singleton</a> x", [{"full_name": "Set.mem_singleton", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1289, 9], "def_end_pos": [1289, 22]}]], "state_before": "case intro\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\nh : Surjective (preimage f)\nx x' : \u03b1\nhx : f x = f x'\ns : Set \u03b2\nhs : f \u207b\u00b9' s = {x}\n\u22a2 x = x'", "state_after": "case intro\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\nh : Surjective (preimage f)\nx x' : \u03b1\nhx : f x = f x'\ns : Set \u03b2\nhs : f \u207b\u00b9' s = {x}\nthis : x \u2208 {x}\n\u22a2 x = x'"}, {"tactic": "rwa [\u2190 hs, mem_preimage, hx, \u2190 mem_preimage, hs, mem_singleton_iff, eq_comm] at this", "annotated_tactic": ["rwa [\u2190 hs, <a>mem_preimage</a>, hx, \u2190 <a>mem_preimage</a>, hs, <a>mem_singleton_iff</a>, <a>eq_comm</a>] at this", [{"full_name": "Set.mem_preimage", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [64, 9], "def_end_pos": [64, 21]}, {"full_name": "Set.mem_preimage", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [64, 9], "def_end_pos": [64, 21]}, {"full_name": "Set.mem_singleton_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1273, 9], "def_end_pos": [1273, 26]}, {"full_name": "eq_comm", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [104, 9], "def_end_pos": [104, 16]}]], "state_before": "case intro\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\nh : Surjective (preimage f)\nx x' : \u03b1\nhx : f x = f x'\ns : Set \u03b2\nhs : f \u207b\u00b9' s = {x}\nthis : x \u2208 {x}\n\u22a2 x = x'", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "full_name": "MeasureTheory.Measure.sigmaFinite_of_le", "start": [3680, 1], "end": [3681, 46], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Prod.lean", "full_name": "Set.pi_eq_empty_iff", "start": [727, 1], "end": [731, 88], "traced_tactics": [{"tactic": "rw [\u2190 not_nonempty_iff_eq_empty, pi_nonempty_iff]", "annotated_tactic": ["rw [\u2190 <a>not_nonempty_iff_eq_empty</a>, <a>pi_nonempty_iff</a>]", [{"full_name": "Set.not_nonempty_iff_eq_empty", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [605, 9], "def_end_pos": [605, 34]}, {"full_name": "Set.pi_nonempty_iff", "def_path": "Mathlib/Data/Set/Prod.lean", "def_pos": [719, 9], "def_end_pos": [719, 24]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\ns s\u2081 s\u2082 : Set \u03b9\nt t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\ni : \u03b9\n\u22a2 pi s t = \u2205 \u2194 \u2203 i, IsEmpty (\u03b1 i) \u2228 i \u2208 s \u2227 t i = \u2205", "state_after": "\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\ns s\u2081 s\u2082 : Set \u03b9\nt t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\ni : \u03b9\n\u22a2 (\u00ac\u2200 (i : \u03b9), \u2203 x, i \u2208 s \u2192 x \u2208 t i) \u2194 \u2203 i, IsEmpty (\u03b1 i) \u2228 i \u2208 s \u2227 t i = \u2205"}, {"tactic": "push_neg", "annotated_tactic": ["push_neg", []], "state_before": "\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\ns s\u2081 s\u2082 : Set \u03b9\nt t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\ni : \u03b9\n\u22a2 (\u00ac\u2200 (i : \u03b9), \u2203 x, i \u2208 s \u2192 x \u2208 t i) \u2194 \u2203 i, IsEmpty (\u03b1 i) \u2228 i \u2208 s \u2227 t i = \u2205", "state_after": "\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\ns s\u2081 s\u2082 : Set \u03b9\nt t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\ni : \u03b9\n\u22a2 (\u2203 i, \u2200 (x : \u03b1 i), i \u2208 s \u2227 \u00acx \u2208 t i) \u2194 \u2203 i, IsEmpty (\u03b1 i) \u2228 i \u2208 s \u2227 t i = \u2205"}, {"tactic": "refine' exists_congr fun i => _", "annotated_tactic": ["refine' <a>exists_congr</a> fun i => _", [{"full_name": "exists_congr", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [379, 9], "def_end_pos": [379, 21]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\ns s\u2081 s\u2082 : Set \u03b9\nt t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\ni : \u03b9\n\u22a2 (\u2203 i, \u2200 (x : \u03b1 i), i \u2208 s \u2227 \u00acx \u2208 t i) \u2194 \u2203 i, IsEmpty (\u03b1 i) \u2228 i \u2208 s \u2227 t i = \u2205", "state_after": "\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\ns s\u2081 s\u2082 : Set \u03b9\nt t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\ni\u271d i : \u03b9\n\u22a2 (\u2200 (x : \u03b1 i), i \u2208 s \u2227 \u00acx \u2208 t i) \u2194 IsEmpty (\u03b1 i) \u2228 i \u2208 s \u2227 t i = \u2205"}, {"tactic": "cases isEmpty_or_nonempty (\u03b1 i) <;> simp [*, forall_and, eq_empty_iff_forall_not_mem]", "annotated_tactic": ["cases <a>isEmpty_or_nonempty</a> (\u03b1 i) <;> simp [*, <a>forall_and</a>, <a>eq_empty_iff_forall_not_mem</a>]", [{"full_name": "isEmpty_or_nonempty", "def_path": "Mathlib/Logic/IsEmpty.lean", "def_pos": [207, 9], "def_end_pos": [207, 28]}, {"full_name": "forall_and", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [426, 9], "def_end_pos": [426, 19]}, {"full_name": "Set.eq_empty_iff_forall_not_mem", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [582, 9], "def_end_pos": [582, 36]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\ns s\u2081 s\u2082 : Set \u03b9\nt t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\ni\u271d i : \u03b9\n\u22a2 (\u2200 (x : \u03b1 i), i \u2208 s \u2227 \u00acx \u2208 t i) \u2194 IsEmpty (\u03b1 i) \u2228 i \u2208 s \u2227 t i = \u2205", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "full_name": "MeasurableEquiv.measurableSet_preimage", "start": [1435, 1], "end": [1438, 20], "traced_tactics": [{"tactic": "simpa only [symm_preimage_preimage] using e.symm.measurable h", "annotated_tactic": ["simpa only [<a>symm_preimage_preimage</a>] using e.symm.measurable h", [{"full_name": "MeasurableEquiv.symm_preimage_preimage", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [1409, 9], "def_end_pos": [1409, 31]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u\u03b9\ns\u271d t u : Set \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\ninst\u271d : MeasurableSpace \u03b4\ne : \u03b1 \u2243\u1d50 \u03b2\ns : Set \u03b2\nh : MeasurableSet (\u2191e \u207b\u00b9' s)\n\u22a2 MeasurableSet s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "full_name": "MeasureTheory.Measure.ae_sum_iff", "start": [2021, 1], "end": [2023, 20], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Pointwise.lean", "full_name": "Set.toFinset_smul_set", "start": [2272, 1], "end": [2274, 21], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Kernel/CondCdf.lean", "full_name": "ProbabilityTheory.monotone_preCdf", "start": [318, 1], "end": [326, 22], "traced_tactics": [{"tactic": "simp_rw [Monotone, ae_all_iff]", "annotated_tactic": ["simp_rw [<a>Monotone</a>, <a>ae_all_iff</a>]", [{"full_name": "Monotone", "def_path": "Mathlib/Order/Monotone/Basic.lean", "def_pos": [77, 5], "def_end_pos": [77, 13]}, {"full_name": "MeasureTheory.ae_all_iff", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [422, 9], "def_end_pos": [422, 19]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\n\u22a2 \u2200 (i i_1 : \u211a), i \u2264 i_1 \u2192 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, preCdf \u03c1 i a \u2264 preCdf \u03c1 i_1 a"}, {"tactic": "refine' fun r r' hrr' =>\n  ae_le_of_forall_set_lintegral_le_of_sigmaFinite measurable_preCdf measurable_preCdf\n    fun s hs _ => _", "annotated_tactic": ["refine' fun r r' hrr' =>\n    <a>ae_le_of_forall_set_lintegral_le_of_sigmaFinite</a> <a>measurable_preCdf</a> <a>measurable_preCdf</a>\n      fun s hs _ => _", [{"full_name": "MeasureTheory.ae_le_of_forall_set_lintegral_le_of_sigmaFinite", "def_path": "Mathlib/MeasureTheory/Function/AEEqOfIntegral.lean", "def_pos": [165, 9], "def_end_pos": [165, 56]}, {"full_name": "ProbabilityTheory.measurable_preCdf", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [298, 9], "def_end_pos": [298, 26]}, {"full_name": "ProbabilityTheory.measurable_preCdf", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [298, 9], "def_end_pos": [298, 26]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\n\u22a2 \u2200 (i i_1 : \u211a), i \u2264 i_1 \u2192 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, preCdf \u03c1 i a \u2264 preCdf \u03c1 i_1 a", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nr r' : \u211a\nhrr' : r \u2264 r'\ns : Set \u03b1\nhs : MeasurableSet s\nx\u271d : \u2191\u2191(Measure.fst \u03c1) s < \u22a4\n\u22a2 \u222b\u207b (x : \u03b1) in s, preCdf \u03c1 r x \u2202Measure.fst \u03c1 \u2264 \u222b\u207b (x : \u03b1) in s, preCdf \u03c1 r' x \u2202Measure.fst \u03c1"}, {"tactic": "rw [set_lintegral_preCdf_fst \u03c1 r hs, set_lintegral_preCdf_fst \u03c1 r' hs]", "annotated_tactic": ["rw [<a>set_lintegral_preCdf_fst</a> \u03c1 r hs, <a>set_lintegral_preCdf_fst</a> \u03c1 r' hs]", [{"full_name": "ProbabilityTheory.set_lintegral_preCdf_fst", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [307, 9], "def_end_pos": [307, 33]}, {"full_name": "ProbabilityTheory.set_lintegral_preCdf_fst", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [307, 9], "def_end_pos": [307, 33]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nr r' : \u211a\nhrr' : r \u2264 r'\ns : Set \u03b1\nhs : MeasurableSet s\nx\u271d : \u2191\u2191(Measure.fst \u03c1) s < \u22a4\n\u22a2 \u222b\u207b (x : \u03b1) in s, preCdf \u03c1 r x \u2202Measure.fst \u03c1 \u2264 \u222b\u207b (x : \u03b1) in s, preCdf \u03c1 r' x \u2202Measure.fst \u03c1", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nr r' : \u211a\nhrr' : r \u2264 r'\ns : Set \u03b1\nhs : MeasurableSet s\nx\u271d : \u2191\u2191(Measure.fst \u03c1) s < \u22a4\n\u22a2 \u2191\u2191(Measure.IicSnd \u03c1 \u2191r) s \u2264 \u2191\u2191(Measure.IicSnd \u03c1 \u2191r') s"}, {"tactic": "refine' Measure.IicSnd_mono \u03c1 _ s hs", "annotated_tactic": ["refine' <a>Measure.IicSnd_mono</a> \u03c1 _ s hs", [{"full_name": "MeasureTheory.Measure.IicSnd_mono", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [194, 9], "def_end_pos": [194, 20]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nr r' : \u211a\nhrr' : r \u2264 r'\ns : Set \u03b1\nhs : MeasurableSet s\nx\u271d : \u2191\u2191(Measure.fst \u03c1) s < \u22a4\n\u22a2 \u2191\u2191(Measure.IicSnd \u03c1 \u2191r) s \u2264 \u2191\u2191(Measure.IicSnd \u03c1 \u2191r') s", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nr r' : \u211a\nhrr' : r \u2264 r'\ns : Set \u03b1\nhs : MeasurableSet s\nx\u271d : \u2191\u2191(Measure.fst \u03c1) s < \u22a4\n\u22a2 \u2191r \u2264 \u2191r'"}, {"tactic": "exact_mod_cast hrr'", "annotated_tactic": ["exact_mod_cast hrr'", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nr r' : \u211a\nhrr' : r \u2264 r'\ns : Set \u03b1\nhs : MeasurableSet s\nx\u271d : \u2191\u2191(Measure.fst \u03c1) s < \u22a4\n\u22a2 \u2191r \u2264 \u2191r'", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Basic.lean", "full_name": "Set.nontrivial_coe_sort", "start": [2603, 1], "end": [2611, 83], "traced_tactics": [{"tactic": "rw [\u2190 nontrivial_univ_iff, Set.Nontrivial, Set.Nontrivial]", "annotated_tactic": ["rw [\u2190 <a>nontrivial_univ_iff</a>, <a>Set.Nontrivial</a>, <a>Set.Nontrivial</a>]", [{"full_name": "Set.nontrivial_univ_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [2590, 9], "def_end_pos": [2590, 28]}, {"full_name": "Set.Nontrivial", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [2455, 15], "def_end_pos": [2455, 25]}, {"full_name": "Set.Nontrivial", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [2455, 15], "def_end_pos": [2455, 25]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\na b : \u03b1\ns\u271d s\u2081 s\u2082 t t\u2081 t\u2082 u s : Set \u03b1\n\u22a2 Nontrivial \u2191s \u2194 Set.Nontrivial s", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\na b : \u03b1\ns\u271d s\u2081 s\u2082 t t\u2081 t\u2082 u s : Set \u03b1\n\u22a2 (\u2203 x, x \u2208 univ \u2227 \u2203 y, y \u2208 univ \u2227 x \u2260 y) \u2194 \u2203 x, x \u2208 s \u2227 \u2203 y, y \u2208 s \u2227 x \u2260 y"}, {"tactic": "apply Iff.intro", "annotated_tactic": ["apply <a>Iff.intro</a>", [{"full_name": "Iff.intro", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [88, 3], "def_end_pos": [88, 8]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\na b : \u03b1\ns\u271d s\u2081 s\u2082 t t\u2081 t\u2082 u s : Set \u03b1\n\u22a2 (\u2203 x, x \u2208 univ \u2227 \u2203 y, y \u2208 univ \u2227 x \u2260 y) \u2194 \u2203 x, x \u2208 s \u2227 \u2203 y, y \u2208 s \u2227 x \u2260 y", "state_after": "case mp\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\na b : \u03b1\ns\u271d s\u2081 s\u2082 t t\u2081 t\u2082 u s : Set \u03b1\n\u22a2 (\u2203 x, x \u2208 univ \u2227 \u2203 y, y \u2208 univ \u2227 x \u2260 y) \u2192 \u2203 x, x \u2208 s \u2227 \u2203 y, y \u2208 s \u2227 x \u2260 y\n\ncase mpr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\na b : \u03b1\ns\u271d s\u2081 s\u2082 t t\u2081 t\u2082 u s : Set \u03b1\n\u22a2 (\u2203 x, x \u2208 s \u2227 \u2203 y, y \u2208 s \u2227 x \u2260 y) \u2192 \u2203 x, x \u2208 univ \u2227 \u2203 y, y \u2208 univ \u2227 x \u2260 y"}, {"tactic": "rintro \u27e8x, _, y, _, hxy\u27e9", "annotated_tactic": ["rintro \u27e8x, _, y, _, hxy\u27e9", []], "state_before": "case mp\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\na b : \u03b1\ns\u271d s\u2081 s\u2082 t t\u2081 t\u2082 u s : Set \u03b1\n\u22a2 (\u2203 x, x \u2208 univ \u2227 \u2203 y, y \u2208 univ \u2227 x \u2260 y) \u2192 \u2203 x, x \u2208 s \u2227 \u2203 y, y \u2208 s \u2227 x \u2260 y", "state_after": "case mp.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\na b : \u03b1\ns\u271d s\u2081 s\u2082 t t\u2081 t\u2082 u s : Set \u03b1\nx : \u2191s\nleft\u271d\u00b9 : x \u2208 univ\ny : \u2191s\nleft\u271d : y \u2208 univ\nhxy : x \u2260 y\n\u22a2 \u2203 x, x \u2208 s \u2227 \u2203 y, y \u2208 s \u2227 x \u2260 y"}, {"tactic": "exact \u27e8x, Subtype.prop x, y, Subtype.prop y, fun h => hxy (Subtype.coe_injective h)\u27e9", "annotated_tactic": ["exact \u27e8x, <a>Subtype.prop</a> x, y, <a>Subtype.prop</a> y, fun h => hxy (<a>Subtype.coe_injective</a> h)\u27e9", [{"full_name": "Subtype.prop", "def_path": "Mathlib/Data/Subtype.lean", "def_pos": [38, 9], "def_end_pos": [38, 13]}, {"full_name": "Subtype.prop", "def_path": "Mathlib/Data/Subtype.lean", "def_pos": [38, 9], "def_end_pos": [38, 13]}, {"full_name": "Subtype.coe_injective", "def_path": "Mathlib/Data/Subtype.lean", "def_pos": [119, 9], "def_end_pos": [119, 22]}]], "state_before": "case mp.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\na b : \u03b1\ns\u271d s\u2081 s\u2082 t t\u2081 t\u2082 u s : Set \u03b1\nx : \u2191s\nleft\u271d\u00b9 : x \u2208 univ\ny : \u2191s\nleft\u271d : y \u2208 univ\nhxy : x \u2260 y\n\u22a2 \u2203 x, x \u2208 s \u2227 \u2203 y, y \u2208 s \u2227 x \u2260 y", "state_after": "no goals"}, {"tactic": "rintro \u27e8x, hx, y, hy, hxy\u27e9", "annotated_tactic": ["rintro \u27e8x, hx, y, hy, hxy\u27e9", []], "state_before": "case mpr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\na b : \u03b1\ns\u271d s\u2081 s\u2082 t t\u2081 t\u2082 u s : Set \u03b1\n\u22a2 (\u2203 x, x \u2208 s \u2227 \u2203 y, y \u2208 s \u2227 x \u2260 y) \u2192 \u2203 x, x \u2208 univ \u2227 \u2203 y, y \u2208 univ \u2227 x \u2260 y", "state_after": "case mpr.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\na b : \u03b1\ns\u271d s\u2081 s\u2082 t t\u2081 t\u2082 u s : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\ny : \u03b1\nhy : y \u2208 s\nhxy : x \u2260 y\n\u22a2 \u2203 x, x \u2208 univ \u2227 \u2203 y, y \u2208 univ \u2227 x \u2260 y"}, {"tactic": "exact \u27e8\u27e8x, hx\u27e9, mem_univ _, \u27e8y, hy\u27e9, mem_univ _, Subtype.mk_eq_mk.not.mpr hxy\u27e9", "annotated_tactic": ["exact \u27e8\u27e8x, hx\u27e9, <a>mem_univ</a> _, \u27e8y, hy\u27e9, <a>mem_univ</a> _, Subtype.mk_eq_mk.not.mpr hxy\u27e9", [{"full_name": "Set.mem_univ", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [676, 9], "def_end_pos": [676, 17]}, {"full_name": "Set.mem_univ", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [676, 9], "def_end_pos": [676, 17]}]], "state_before": "case mpr.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\na b : \u03b1\ns\u271d s\u2081 s\u2082 t t\u2081 t\u2082 u s : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\ny : \u03b1\nhy : y \u2208 s\nhxy : x \u2260 y\n\u22a2 \u2203 x, x \u2208 univ \u2227 \u2203 y, y \u2208 univ \u2227 x \u2260 y", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/PartrecCode.lean", "full_name": "Nat.Partrec.Code.encode_ofNatCode", "start": [165, 9], "end": [185, 62], "traced_tactics": [{"tactic": "simp [ofNatCode, encodeCode]", "annotated_tactic": ["simp [<a>ofNatCode</a>, <a>encodeCode</a>]", [{"full_name": "Nat.Partrec.Code.ofNatCode", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [143, 5], "def_end_pos": [143, 14]}, {"full_name": "Nat.Partrec.Code.encodeCode", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [129, 5], "def_end_pos": [129, 15]}]], "state_before": "\u22a2 encodeCode (ofNatCode 0) = 0", "state_after": "no goals"}, {"tactic": "simp [ofNatCode, encodeCode]", "annotated_tactic": ["simp [<a>ofNatCode</a>, <a>encodeCode</a>]", [{"full_name": "Nat.Partrec.Code.ofNatCode", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [143, 5], "def_end_pos": [143, 14]}, {"full_name": "Nat.Partrec.Code.encodeCode", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [129, 5], "def_end_pos": [129, 15]}]], "state_before": "\u22a2 encodeCode (ofNatCode 1) = 1", "state_after": "no goals"}, {"tactic": "simp [ofNatCode, encodeCode]", "annotated_tactic": ["simp [<a>ofNatCode</a>, <a>encodeCode</a>]", [{"full_name": "Nat.Partrec.Code.ofNatCode", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [143, 5], "def_end_pos": [143, 14]}, {"full_name": "Nat.Partrec.Code.encodeCode", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [129, 5], "def_end_pos": [129, 15]}]], "state_before": "\u22a2 encodeCode (ofNatCode 2) = 2", "state_after": "no goals"}, {"tactic": "simp [ofNatCode, encodeCode]", "annotated_tactic": ["simp [<a>ofNatCode</a>, <a>encodeCode</a>]", [{"full_name": "Nat.Partrec.Code.ofNatCode", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [143, 5], "def_end_pos": [143, 14]}, {"full_name": "Nat.Partrec.Code.encodeCode", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [129, 5], "def_end_pos": [129, 15]}]], "state_before": "\u22a2 encodeCode (ofNatCode 3) = 3", "state_after": "no goals"}, {"tactic": "let m := n.div2.div2", "annotated_tactic": ["let m := n.div2.div2", []], "state_before": "n : \u2115\n\u22a2 encodeCode (ofNatCode (n + 4)) = n + 4", "state_after": "n : \u2115\nm : \u2115 := div2 (div2 n)\n\u22a2 encodeCode (ofNatCode (n + 4)) = n + 4"}, {"tactic": "have hm : m < n + 4 := by\n  simp only [div2_val]\n  exact\n    lt_of_le_of_lt (le_trans (Nat.div_le_self _ _) (Nat.div_le_self _ _))\n      (Nat.succ_le_succ (Nat.le_add_right _ _))", "annotated_tactic": ["have hm : m < n + 4 := by\n      simp only [<a>div2_val</a>]\n      exact\n        <a>lt_of_le_of_lt</a> (<a>le_trans</a> (<a>Nat.div_le_self</a> _ _) (<a>Nat.div_le_self</a> _ _))\n          (<a>Nat.succ_le_succ</a> (<a>Nat.le_add_right</a> _ _))", [{"full_name": "Nat.div2_val", "def_path": "Mathlib/Init/Data/Nat/Bitwise.lean", "def_pos": [139, 9], "def_end_pos": [139, 17]}, {"full_name": "lt_of_le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [122, 9], "def_end_pos": [122, 23]}, {"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}, {"full_name": "Nat.div_le_self", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Div.lean", "def_pos": [42, 9], "def_end_pos": [42, 20]}, {"full_name": "Nat.div_le_self", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Div.lean", "def_pos": [42, 9], "def_end_pos": [42, 20]}, {"full_name": "Nat.succ_le_succ", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1582, 9], "def_end_pos": [1582, 25]}, {"full_name": "Nat.le_add_right", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [340, 9], "def_end_pos": [340, 21]}]], "state_before": "n : \u2115\nm : \u2115 := div2 (div2 n)\n\u22a2 encodeCode (ofNatCode (n + 4)) = n + 4", "state_after": "n : \u2115\nm : \u2115 := div2 (div2 n)\nhm : m < n + 4\n\u22a2 encodeCode (ofNatCode (n + 4)) = n + 4"}, {"tactic": "have _m1 : m.unpair.1 < n + 4 := lt_of_le_of_lt m.unpair_left_le hm", "annotated_tactic": ["have _m1 : m.unpair.1 < n + 4 := <a>lt_of_le_of_lt</a> m.unpair_left_le hm", [{"full_name": "lt_of_le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [122, 9], "def_end_pos": [122, 23]}]], "state_before": "n : \u2115\nm : \u2115 := div2 (div2 n)\nhm : m < n + 4\n\u22a2 encodeCode (ofNatCode (n + 4)) = n + 4", "state_after": "n : \u2115\nm : \u2115 := div2 (div2 n)\nhm : m < n + 4\n_m1 : (unpair m).1 < n + 4\n\u22a2 encodeCode (ofNatCode (n + 4)) = n + 4"}, {"tactic": "have _m2 : m.unpair.2 < n + 4 := lt_of_le_of_lt m.unpair_right_le hm", "annotated_tactic": ["have _m2 : m.unpair.2 < n + 4 := <a>lt_of_le_of_lt</a> m.unpair_right_le hm", [{"full_name": "lt_of_le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [122, 9], "def_end_pos": [122, 23]}]], "state_before": "n : \u2115\nm : \u2115 := div2 (div2 n)\nhm : m < n + 4\n_m1 : (unpair m).1 < n + 4\n\u22a2 encodeCode (ofNatCode (n + 4)) = n + 4", "state_after": "n : \u2115\nm : \u2115 := div2 (div2 n)\nhm : m < n + 4\n_m1 : (unpair m).1 < n + 4\n_m2 : (unpair m).2 < n + 4\n\u22a2 encodeCode (ofNatCode (n + 4)) = n + 4"}, {"tactic": "have IH := encode_ofNatCode m", "annotated_tactic": ["have IH := encode_ofNatCode m", []], "state_before": "n : \u2115\nm : \u2115 := div2 (div2 n)\nhm : m < n + 4\n_m1 : (unpair m).1 < n + 4\n_m2 : (unpair m).2 < n + 4\n\u22a2 encodeCode (ofNatCode (n + 4)) = n + 4", "state_after": "n : \u2115\nm : \u2115 := div2 (div2 n)\nhm : m < n + 4\n_m1 : (unpair m).1 < n + 4\n_m2 : (unpair m).2 < n + 4\nIH : encodeCode (ofNatCode m) = m\n\u22a2 encodeCode (ofNatCode (n + 4)) = n + 4"}, {"tactic": "have IH1 := encode_ofNatCode m.unpair.1", "annotated_tactic": ["have IH1 := encode_ofNatCode m.unpair.1", []], "state_before": "n : \u2115\nm : \u2115 := div2 (div2 n)\nhm : m < n + 4\n_m1 : (unpair m).1 < n + 4\n_m2 : (unpair m).2 < n + 4\nIH : encodeCode (ofNatCode m) = m\n\u22a2 encodeCode (ofNatCode (n + 4)) = n + 4", "state_after": "n : \u2115\nm : \u2115 := div2 (div2 n)\nhm : m < n + 4\n_m1 : (unpair m).1 < n + 4\n_m2 : (unpair m).2 < n + 4\nIH : encodeCode (ofNatCode m) = m\nIH1 : encodeCode (ofNatCode (unpair m).1) = (unpair m).1\n\u22a2 encodeCode (ofNatCode (n + 4)) = n + 4"}, {"tactic": "have IH2 := encode_ofNatCode m.unpair.2", "annotated_tactic": ["have IH2 := encode_ofNatCode m.unpair.2", []], "state_before": "n : \u2115\nm : \u2115 := div2 (div2 n)\nhm : m < n + 4\n_m1 : (unpair m).1 < n + 4\n_m2 : (unpair m).2 < n + 4\nIH : encodeCode (ofNatCode m) = m\nIH1 : encodeCode (ofNatCode (unpair m).1) = (unpair m).1\n\u22a2 encodeCode (ofNatCode (n + 4)) = n + 4", "state_after": "n : \u2115\nm : \u2115 := div2 (div2 n)\nhm : m < n + 4\n_m1 : (unpair m).1 < n + 4\n_m2 : (unpair m).2 < n + 4\nIH : encodeCode (ofNatCode m) = m\nIH1 : encodeCode (ofNatCode (unpair m).1) = (unpair m).1\nIH2 : encodeCode (ofNatCode (unpair m).2) = (unpair m).2\n\u22a2 encodeCode (ofNatCode (n + 4)) = n + 4"}, {"tactic": "conv_rhs => rw [\u2190 Nat.bit_decomp n, \u2190 Nat.bit_decomp n.div2]", "annotated_tactic": ["conv_rhs => rw [\u2190 <a>Nat.bit_decomp</a> n, \u2190 <a>Nat.bit_decomp</a> n.div2]", [{"full_name": "Nat.bit_decomp", "def_path": "Mathlib/Init/Data/Nat/Bitwise.lean", "def_pos": [169, 9], "def_end_pos": [169, 19]}, {"full_name": "Nat.bit_decomp", "def_path": "Mathlib/Init/Data/Nat/Bitwise.lean", "def_pos": [169, 9], "def_end_pos": [169, 19]}]], "state_before": "n : \u2115\nm : \u2115 := div2 (div2 n)\nhm : m < n + 4\n_m1 : (unpair m).1 < n + 4\n_m2 : (unpair m).2 < n + 4\nIH : encodeCode (ofNatCode m) = m\nIH1 : encodeCode (ofNatCode (unpair m).1) = (unpair m).1\nIH2 : encodeCode (ofNatCode (unpair m).2) = (unpair m).2\n\u22a2 encodeCode (ofNatCode (n + 4)) = n + 4", "state_after": "n : \u2115\nm : \u2115 := div2 (div2 n)\nhm : m < n + 4\n_m1 : (unpair m).1 < n + 4\n_m2 : (unpair m).2 < n + 4\nIH : encodeCode (ofNatCode m) = m\nIH1 : encodeCode (ofNatCode (unpair m).1) = (unpair m).1\nIH2 : encodeCode (ofNatCode (unpair m).2) = (unpair m).2\n\u22a2 encodeCode (ofNatCode (n + 4)) = bit (bodd n) (bit (bodd (div2 n)) (div2 (div2 n))) + 4"}, {"tactic": "simp only [ofNatCode._eq_5]", "annotated_tactic": ["simp only [ofNatCode._eq_5]", []], "state_before": "n : \u2115\nm : \u2115 := div2 (div2 n)\nhm : m < n + 4\n_m1 : (unpair m).1 < n + 4\n_m2 : (unpair m).2 < n + 4\nIH : encodeCode (ofNatCode m) = m\nIH1 : encodeCode (ofNatCode (unpair m).1) = (unpair m).1\nIH2 : encodeCode (ofNatCode (unpair m).2) = (unpair m).2\n\u22a2 encodeCode (ofNatCode (n + 4)) = bit (bodd n) (bit (bodd (div2 n)) (div2 (div2 n))) + 4", "state_after": "n : \u2115\nm : \u2115 := div2 (div2 n)\nhm : m < n + 4\n_m1 : (unpair m).1 < n + 4\n_m2 : (unpair m).2 < n + 4\nIH : encodeCode (ofNatCode m) = m\nIH1 : encodeCode (ofNatCode (unpair m).1) = (unpair m).1\nIH2 : encodeCode (ofNatCode (unpair m).2) = (unpair m).2\n\u22a2 encodeCode\n      (match bodd n, bodd (div2 n) with\n      | false, false => pair (ofNatCode (unpair (div2 (div2 n))).1) (ofNatCode (unpair (div2 (div2 n))).2)\n      | false, true => comp (ofNatCode (unpair (div2 (div2 n))).1) (ofNatCode (unpair (div2 (div2 n))).2)\n      | true, false => prec (ofNatCode (unpair (div2 (div2 n))).1) (ofNatCode (unpair (div2 (div2 n))).2)\n      | true, true => rfind' (ofNatCode (div2 (div2 n)))) =\n    bit (bodd n) (bit (bodd (div2 n)) (div2 (div2 n))) + 4"}, {"tactic": "cases n.bodd <;> cases n.div2.bodd <;>\n  simp [encodeCode, ofNatCode, IH, IH1, IH2, Nat.bit_val]", "annotated_tactic": ["cases n.bodd <;> cases n.div2.bodd <;>\n      simp [<a>encodeCode</a>, <a>ofNatCode</a>, IH, IH1, IH2, <a>Nat.bit_val</a>]", [{"full_name": "Nat.Partrec.Code.encodeCode", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [129, 5], "def_end_pos": [129, 15]}, {"full_name": "Nat.Partrec.Code.ofNatCode", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [143, 5], "def_end_pos": [143, 14]}, {"full_name": "Nat.bit_val", "def_path": "Mathlib/Init/Data/Nat/Bitwise.lean", "def_pos": [163, 9], "def_end_pos": [163, 16]}]], "state_before": "n : \u2115\nm : \u2115 := div2 (div2 n)\nhm : m < n + 4\n_m1 : (unpair m).1 < n + 4\n_m2 : (unpair m).2 < n + 4\nIH : encodeCode (ofNatCode m) = m\nIH1 : encodeCode (ofNatCode (unpair m).1) = (unpair m).1\nIH2 : encodeCode (ofNatCode (unpair m).2) = (unpair m).2\n\u22a2 encodeCode\n      (match bodd n, bodd (div2 n) with\n      | false, false => pair (ofNatCode (unpair (div2 (div2 n))).1) (ofNatCode (unpair (div2 (div2 n))).2)\n      | false, true => comp (ofNatCode (unpair (div2 (div2 n))).1) (ofNatCode (unpair (div2 (div2 n))).2)\n      | true, false => prec (ofNatCode (unpair (div2 (div2 n))).1) (ofNatCode (unpair (div2 (div2 n))).2)\n      | true, true => rfind' (ofNatCode (div2 (div2 n)))) =\n    bit (bodd n) (bit (bodd (div2 n)) (div2 (div2 n))) + 4", "state_after": "no goals"}, {"tactic": "simp only [div2_val]", "annotated_tactic": ["simp only [<a>div2_val</a>]", [{"full_name": "Nat.div2_val", "def_path": "Mathlib/Init/Data/Nat/Bitwise.lean", "def_pos": [139, 9], "def_end_pos": [139, 17]}]], "state_before": "n : \u2115\nm : \u2115 := div2 (div2 n)\n\u22a2 m < n + 4", "state_after": "n : \u2115\nm : \u2115 := div2 (div2 n)\n\u22a2 n / 2 / 2 < n + 4"}, {"tactic": "exact\n  lt_of_le_of_lt (le_trans (Nat.div_le_self _ _) (Nat.div_le_self _ _))\n    (Nat.succ_le_succ (Nat.le_add_right _ _))", "annotated_tactic": ["exact\n        <a>lt_of_le_of_lt</a> (<a>le_trans</a> (<a>Nat.div_le_self</a> _ _) (<a>Nat.div_le_self</a> _ _))\n          (<a>Nat.succ_le_succ</a> (<a>Nat.le_add_right</a> _ _))", [{"full_name": "lt_of_le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [122, 9], "def_end_pos": [122, 23]}, {"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}, {"full_name": "Nat.div_le_self", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Div.lean", "def_pos": [42, 9], "def_end_pos": [42, 20]}, {"full_name": "Nat.div_le_self", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Div.lean", "def_pos": [42, 9], "def_end_pos": [42, 20]}, {"full_name": "Nat.succ_le_succ", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1582, 9], "def_end_pos": [1582, 25]}, {"full_name": "Nat.le_add_right", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [340, 9], "def_end_pos": [340, 21]}]], "state_before": "n : \u2115\nm : \u2115 := div2 (div2 n)\n\u22a2 n / 2 / 2 < n + 4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Group/FundamentalDomain.lean", "full_name": "MeasureTheory.IsFundamentalDomain.essSup_measure_restrict", "start": [517, 1], "end": [528, 37], "traced_tactics": [{"tactic": "refine' le_antisymm (essSup_mono_measure' Measure.restrict_le_self) _", "annotated_tactic": ["refine' <a>le_antisymm</a> (<a>essSup_mono_measure'</a> <a>Measure.restrict_le_self</a>) _", [{"full_name": "le_antisymm", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [188, 9], "def_end_pos": [188, 20]}, {"full_name": "essSup_mono_measure'", "def_path": "Mathlib/MeasureTheory/Function/EssSup.lean", "def_pos": [197, 9], "def_end_pos": [197, 29]}, {"full_name": "MeasureTheory.Measure.restrict_le_self", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1578, 9], "def_end_pos": [1578, 25]}]], "state_before": "G : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : Group H\ninst\u271d\u00b9\u2070 : MulAction G \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MulAction H \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : MeasurableSMul G \u03b1\ninst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b2 : Countable G\n\u03bd : Measure \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhs : IsFundamentalDomain G s\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (\u03b3 : G) (x : \u03b1), f (\u03b3 \u2022 x) = f x\n\u22a2 essSup f (Measure.restrict \u03bc s) = essSup f \u03bc", "state_after": "G : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : Group H\ninst\u271d\u00b9\u2070 : MulAction G \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MulAction H \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : MeasurableSMul G \u03b1\ninst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b2 : Countable G\n\u03bd : Measure \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhs : IsFundamentalDomain G s\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (\u03b3 : G) (x : \u03b1), f (\u03b3 \u2022 x) = f x\n\u22a2 essSup f \u03bc \u2264 essSup f (Measure.restrict \u03bc s)"}, {"tactic": "rw [essSup_eq_sInf (\u03bc.restrict s) f, essSup_eq_sInf \u03bc f]", "annotated_tactic": ["rw [<a>essSup_eq_sInf</a> (\u03bc.restrict s) f, <a>essSup_eq_sInf</a> \u03bc f]", [{"full_name": "essSup_eq_sInf", "def_path": "Mathlib/MeasureTheory/Function/EssSup.lean", "def_pos": [86, 9], "def_end_pos": [86, 23]}, {"full_name": "essSup_eq_sInf", "def_path": "Mathlib/MeasureTheory/Function/EssSup.lean", "def_pos": [86, 9], "def_end_pos": [86, 23]}]], "state_before": "G : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : Group H\ninst\u271d\u00b9\u2070 : MulAction G \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MulAction H \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : MeasurableSMul G \u03b1\ninst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b2 : Countable G\n\u03bd : Measure \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhs : IsFundamentalDomain G s\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (\u03b3 : G) (x : \u03b1), f (\u03b3 \u2022 x) = f x\n\u22a2 essSup f \u03bc \u2264 essSup f (Measure.restrict \u03bc s)", "state_after": "G : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : Group H\ninst\u271d\u00b9\u2070 : MulAction G \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MulAction H \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : MeasurableSMul G \u03b1\ninst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b2 : Countable G\n\u03bd : Measure \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhs : IsFundamentalDomain G s\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (\u03b3 : G) (x : \u03b1), f (\u03b3 \u2022 x) = f x\n\u22a2 sInf {a | \u2191\u2191\u03bc {x | a < f x} = 0} \u2264 sInf {a | \u2191\u2191(Measure.restrict \u03bc s) {x | a < f x} = 0}"}, {"tactic": "refine' sInf_le_sInf _", "annotated_tactic": ["refine' <a>sInf_le_sInf</a> _", [{"full_name": "sInf_le_sInf", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [286, 9], "def_end_pos": [286, 21]}]], "state_before": "G : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : Group H\ninst\u271d\u00b9\u2070 : MulAction G \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MulAction H \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : MeasurableSMul G \u03b1\ninst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b2 : Countable G\n\u03bd : Measure \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhs : IsFundamentalDomain G s\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (\u03b3 : G) (x : \u03b1), f (\u03b3 \u2022 x) = f x\n\u22a2 sInf {a | \u2191\u2191\u03bc {x | a < f x} = 0} \u2264 sInf {a | \u2191\u2191(Measure.restrict \u03bc s) {x | a < f x} = 0}", "state_after": "G : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : Group H\ninst\u271d\u00b9\u2070 : MulAction G \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MulAction H \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : MeasurableSMul G \u03b1\ninst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b2 : Countable G\n\u03bd : Measure \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhs : IsFundamentalDomain G s\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (\u03b3 : G) (x : \u03b1), f (\u03b3 \u2022 x) = f x\n\u22a2 {a | \u2191\u2191(Measure.restrict \u03bc s) {x | a < f x} = 0} \u2286 {a | \u2191\u2191\u03bc {x | a < f x} = 0}"}, {"tactic": "rintro a (ha : (\u03bc.restrict s) {x : \u03b1 | a < f x} = 0)", "annotated_tactic": ["rintro a (ha : (\u03bc.restrict s) {x : \u03b1 | a < f x} = 0)", []], "state_before": "G : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : Group H\ninst\u271d\u00b9\u2070 : MulAction G \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MulAction H \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : MeasurableSMul G \u03b1\ninst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b2 : Countable G\n\u03bd : Measure \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhs : IsFundamentalDomain G s\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (\u03b3 : G) (x : \u03b1), f (\u03b3 \u2022 x) = f x\n\u22a2 {a | \u2191\u2191(Measure.restrict \u03bc s) {x | a < f x} = 0} \u2286 {a | \u2191\u2191\u03bc {x | a < f x} = 0}", "state_after": "G : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : Group H\ninst\u271d\u00b9\u2070 : MulAction G \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MulAction H \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : MeasurableSMul G \u03b1\ninst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b2 : Countable G\n\u03bd : Measure \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhs : IsFundamentalDomain G s\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (\u03b3 : G) (x : \u03b1), f (\u03b3 \u2022 x) = f x\na : \u211d\u22650\u221e\nha : \u2191\u2191(Measure.restrict \u03bc s) {x | a < f x} = 0\n\u22a2 a \u2208 {a | \u2191\u2191\u03bc {x | a < f x} = 0}"}, {"tactic": "rw [Measure.restrict_apply\u2080' hs.nullMeasurableSet] at ha", "annotated_tactic": ["rw [<a>Measure.restrict_apply\u2080'</a> hs.nullMeasurableSet] at ha", [{"full_name": "MeasureTheory.Measure.restrict_apply\u2080'", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1572, 9], "def_end_pos": [1572, 25]}]], "state_before": "G : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : Group H\ninst\u271d\u00b9\u2070 : MulAction G \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MulAction H \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : MeasurableSMul G \u03b1\ninst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b2 : Countable G\n\u03bd : Measure \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhs : IsFundamentalDomain G s\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (\u03b3 : G) (x : \u03b1), f (\u03b3 \u2022 x) = f x\na : \u211d\u22650\u221e\nha : \u2191\u2191(Measure.restrict \u03bc s) {x | a < f x} = 0\n\u22a2 a \u2208 {a | \u2191\u2191\u03bc {x | a < f x} = 0}", "state_after": "G : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : Group H\ninst\u271d\u00b9\u2070 : MulAction G \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MulAction H \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : MeasurableSMul G \u03b1\ninst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b2 : Countable G\n\u03bd : Measure \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhs : IsFundamentalDomain G s\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (\u03b3 : G) (x : \u03b1), f (\u03b3 \u2022 x) = f x\na : \u211d\u22650\u221e\nha : \u2191\u2191\u03bc ({x | a < f x} \u2229 s) = 0\n\u22a2 a \u2208 {a | \u2191\u2191\u03bc {x | a < f x} = 0}"}, {"tactic": "refine' measure_zero_of_invariant hs _ _ ha", "annotated_tactic": ["refine' <a>measure_zero_of_invariant</a> hs _ _ ha", [{"full_name": "MeasureTheory.IsFundamentalDomain.measure_zero_of_invariant", "def_path": "Mathlib/MeasureTheory/Group/FundamentalDomain.lean", "def_pos": [307, 9], "def_end_pos": [307, 34]}]], "state_before": "G : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : Group H\ninst\u271d\u00b9\u2070 : MulAction G \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MulAction H \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : MeasurableSMul G \u03b1\ninst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b2 : Countable G\n\u03bd : Measure \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhs : IsFundamentalDomain G s\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (\u03b3 : G) (x : \u03b1), f (\u03b3 \u2022 x) = f x\na : \u211d\u22650\u221e\nha : \u2191\u2191\u03bc ({x | a < f x} \u2229 s) = 0\n\u22a2 a \u2208 {a | \u2191\u2191\u03bc {x | a < f x} = 0}", "state_after": "G : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : Group H\ninst\u271d\u00b9\u2070 : MulAction G \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MulAction H \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : MeasurableSMul G \u03b1\ninst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b2 : Countable G\n\u03bd : Measure \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhs : IsFundamentalDomain G s\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (\u03b3 : G) (x : \u03b1), f (\u03b3 \u2022 x) = f x\na : \u211d\u22650\u221e\nha : \u2191\u2191\u03bc ({x | a < f x} \u2229 s) = 0\n\u22a2 \u2200 (g : G), g \u2022 {x | a < f x} = {x | a < f x}"}, {"tactic": "intro \u03b3", "annotated_tactic": ["intro \u03b3", []], "state_before": "G : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : Group H\ninst\u271d\u00b9\u2070 : MulAction G \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MulAction H \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : MeasurableSMul G \u03b1\ninst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b2 : Countable G\n\u03bd : Measure \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhs : IsFundamentalDomain G s\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (\u03b3 : G) (x : \u03b1), f (\u03b3 \u2022 x) = f x\na : \u211d\u22650\u221e\nha : \u2191\u2191\u03bc ({x | a < f x} \u2229 s) = 0\n\u22a2 \u2200 (g : G), g \u2022 {x | a < f x} = {x | a < f x}", "state_after": "G : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : Group H\ninst\u271d\u00b9\u2070 : MulAction G \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MulAction H \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : MeasurableSMul G \u03b1\ninst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b2 : Countable G\n\u03bd : Measure \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhs : IsFundamentalDomain G s\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (\u03b3 : G) (x : \u03b1), f (\u03b3 \u2022 x) = f x\na : \u211d\u22650\u221e\nha : \u2191\u2191\u03bc ({x | a < f x} \u2229 s) = 0\n\u03b3 : G\n\u22a2 \u03b3 \u2022 {x | a < f x} = {x | a < f x}"}, {"tactic": "ext x", "annotated_tactic": ["ext x", []], "state_before": "G : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : Group H\ninst\u271d\u00b9\u2070 : MulAction G \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MulAction H \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : MeasurableSMul G \u03b1\ninst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b2 : Countable G\n\u03bd : Measure \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhs : IsFundamentalDomain G s\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (\u03b3 : G) (x : \u03b1), f (\u03b3 \u2022 x) = f x\na : \u211d\u22650\u221e\nha : \u2191\u2191\u03bc ({x | a < f x} \u2229 s) = 0\n\u03b3 : G\n\u22a2 \u03b3 \u2022 {x | a < f x} = {x | a < f x}", "state_after": "case h\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : Group H\ninst\u271d\u00b9\u2070 : MulAction G \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MulAction H \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : MeasurableSMul G \u03b1\ninst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b2 : Countable G\n\u03bd : Measure \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhs : IsFundamentalDomain G s\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (\u03b3 : G) (x : \u03b1), f (\u03b3 \u2022 x) = f x\na : \u211d\u22650\u221e\nha : \u2191\u2191\u03bc ({x | a < f x} \u2229 s) = 0\n\u03b3 : G\nx : \u03b1\n\u22a2 x \u2208 \u03b3 \u2022 {x | a < f x} \u2194 x \u2208 {x | a < f x}"}, {"tactic": "rw [mem_smul_set_iff_inv_smul_mem]", "annotated_tactic": ["rw [<a>mem_smul_set_iff_inv_smul_mem</a>]", [{"full_name": "Set.mem_smul_set_iff_inv_smul_mem", "def_path": "Mathlib/Data/Set/Pointwise/SMul.lean", "def_pos": [887, 9], "def_end_pos": [887, 38]}]], "state_before": "case h\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : Group H\ninst\u271d\u00b9\u2070 : MulAction G \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MulAction H \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : MeasurableSMul G \u03b1\ninst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b2 : Countable G\n\u03bd : Measure \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhs : IsFundamentalDomain G s\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (\u03b3 : G) (x : \u03b1), f (\u03b3 \u2022 x) = f x\na : \u211d\u22650\u221e\nha : \u2191\u2191\u03bc ({x | a < f x} \u2229 s) = 0\n\u03b3 : G\nx : \u03b1\n\u22a2 x \u2208 \u03b3 \u2022 {x | a < f x} \u2194 x \u2208 {x | a < f x}", "state_after": "case h\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : Group H\ninst\u271d\u00b9\u2070 : MulAction G \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MulAction H \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : MeasurableSMul G \u03b1\ninst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b2 : Countable G\n\u03bd : Measure \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhs : IsFundamentalDomain G s\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (\u03b3 : G) (x : \u03b1), f (\u03b3 \u2022 x) = f x\na : \u211d\u22650\u221e\nha : \u2191\u2191\u03bc ({x | a < f x} \u2229 s) = 0\n\u03b3 : G\nx : \u03b1\n\u22a2 \u03b3\u207b\u00b9 \u2022 x \u2208 {x | a < f x} \u2194 x \u2208 {x | a < f x}"}, {"tactic": "simp only [mem_setOf_eq, hf \u03b3\u207b\u00b9 x]", "annotated_tactic": ["simp only [<a>mem_setOf_eq</a>, hf \u03b3\u207b\u00b9 x]", [{"full_name": "Set.mem_setOf_eq", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [256, 29], "def_end_pos": [256, 41]}]], "state_before": "case h\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : Group H\ninst\u271d\u00b9\u2070 : MulAction G \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MulAction H \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : MeasurableSMul G \u03b1\ninst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b2 : Countable G\n\u03bd : Measure \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhs : IsFundamentalDomain G s\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (\u03b3 : G) (x : \u03b1), f (\u03b3 \u2022 x) = f x\na : \u211d\u22650\u221e\nha : \u2191\u2191\u03bc ({x | a < f x} \u2229 s) = 0\n\u03b3 : G\nx : \u03b1\n\u22a2 \u03b3\u207b\u00b9 \u2022 x \u2208 {x | a < f x} \u2194 x \u2208 {x | a < f x}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Prod.lean", "full_name": "Finset.offDiag_inter", "start": [389, 1], "end": [392, 32], "traced_tactics": [{"tactic": "push_cast", "annotated_tactic": ["push_cast", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\nx : \u03b1 \u00d7 \u03b1\n\u22a2 \u2191(offDiag (s \u2229 t)) = \u2191(offDiag s \u2229 offDiag t)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\nx : \u03b1 \u00d7 \u03b1\n\u22a2 Set.offDiag (\u2191s \u2229 \u2191t) = Set.offDiag \u2191s \u2229 Set.offDiag \u2191t"}, {"tactic": "exact Set.offDiag_inter _ _", "annotated_tactic": ["exact <a>Set.offDiag_inter</a> _ _", [{"full_name": "Set.offDiag_inter", "def_path": "Mathlib/Data/Set/Prod.lean", "def_pos": [623, 9], "def_end_pos": [623, 22]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\nx : \u03b1 \u00d7 \u03b1\n\u22a2 Set.offDiag (\u2191s \u2229 \u2191t) = Set.offDiag \u2191s \u2229 Set.offDiag \u2191t", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/Division.lean", "full_name": "MvPolynomial.modMonomial_add_divMonomial_single", "start": [205, 1], "end": [207, 34], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Martingale/BorelCantelli.lean", "full_name": "MeasureTheory.Submartingale.bddAbove_iff_exists_tendsto", "start": [181, 1], "end": [203, 34], "traced_tactics": [{"tactic": "set g : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n \u03c9 => f n \u03c9 - f 0 \u03c9", "annotated_tactic": ["set g : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n \u03c9 => f n \u03c9 - f 0 \u03c9", []], "state_before": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, BddAbove (Set.range fun n => f n \u03c9) \u2194 \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c)", "state_after": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\ng : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n \u03c9 => f n \u03c9 - f 0 \u03c9\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, BddAbove (Set.range fun n => f n \u03c9) \u2194 \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c)"}, {"tactic": "have hg : Submartingale g \u2131 \u03bc :=\n  hf.sub_martingale (martingale_const_fun _ _ (hf.adapted 0) (hf.integrable 0))", "annotated_tactic": ["have hg : <a>Submartingale</a> g \u2131 \u03bc :=\n    hf.sub_martingale (<a>martingale_const_fun</a> _ _ (hf.adapted 0) (hf.integrable 0))", [{"full_name": "MeasureTheory.Submartingale", "def_path": "Mathlib/Probability/Martingale/Basic.lean", "def_pos": [66, 5], "def_end_pos": [66, 18]}, {"full_name": "MeasureTheory.martingale_const_fun", "def_path": "Mathlib/Probability/Martingale/Basic.lean", "def_pos": [75, 9], "def_end_pos": [75, 29]}]], "state_before": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\ng : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n \u03c9 => f n \u03c9 - f 0 \u03c9\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, BddAbove (Set.range fun n => f n \u03c9) \u2194 \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c)", "state_after": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\ng : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n \u03c9 => f n \u03c9 - f 0 \u03c9\nhg : Submartingale g \u2131 \u03bc\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, BddAbove (Set.range fun n => f n \u03c9) \u2194 \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c)"}, {"tactic": "have hg0 : g 0 = 0 := by\n  ext \u03c9\n  simp only [sub_self, Pi.zero_apply]", "annotated_tactic": ["have hg0 : g 0 = 0 := by\n    ext \u03c9\n    simp only [<a>sub_self</a>, <a>Pi.zero_apply</a>]", [{"full_name": "sub_self", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [734, 30], "def_end_pos": [734, 38]}, {"full_name": "Pi.zero_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [46, 3], "def_end_pos": [46, 14]}]], "state_before": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\ng : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n \u03c9 => f n \u03c9 - f 0 \u03c9\nhg : Submartingale g \u2131 \u03bc\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, BddAbove (Set.range fun n => f n \u03c9) \u2194 \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c)", "state_after": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\ng : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n \u03c9 => f n \u03c9 - f 0 \u03c9\nhg : Submartingale g \u2131 \u03bc\nhg0 : g 0 = 0\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, BddAbove (Set.range fun n => f n \u03c9) \u2194 \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c)"}, {"tactic": "have hgbdd : \u2200\u1d50 \u03c9 \u2202\u03bc, \u2200 i : \u2115, |g (i + 1) \u03c9 - g i \u03c9| \u2264 \u2191R := by\n  simpa only [sub_sub_sub_cancel_right]", "annotated_tactic": ["have hgbdd : \u2200\u1d50 \u03c9 \u2202\u03bc, \u2200 i : \u2115, |g (i + 1) \u03c9 - g i \u03c9| \u2264 \u2191R := by\n    simpa only [<a>sub_sub_sub_cancel_right</a>]", [{"full_name": "sub_sub_sub_cancel_right", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [795, 30], "def_end_pos": [795, 54]}]], "state_before": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\ng : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n \u03c9 => f n \u03c9 - f 0 \u03c9\nhg : Submartingale g \u2131 \u03bc\nhg0 : g 0 = 0\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, BddAbove (Set.range fun n => f n \u03c9) \u2194 \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c)", "state_after": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\ng : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n \u03c9 => f n \u03c9 - f 0 \u03c9\nhg : Submartingale g \u2131 \u03bc\nhg0 : g 0 = 0\nhgbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |g (i + 1) \u03c9 - g i \u03c9| \u2264 \u2191R\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, BddAbove (Set.range fun n => f n \u03c9) \u2194 \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c)"}, {"tactic": "filter_upwards [hg.bddAbove_iff_exists_tendsto_aux hg0 hgbdd] with \u03c9 h\u03c9", "annotated_tactic": ["filter_upwards [hg.bddAbove_iff_exists_tendsto_aux hg0 hgbdd] with \u03c9 h\u03c9", []], "state_before": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\ng : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n \u03c9 => f n \u03c9 - f 0 \u03c9\nhg : Submartingale g \u2131 \u03bc\nhg0 : g 0 = 0\nhgbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |g (i + 1) \u03c9 - g i \u03c9| \u2264 \u2191R\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, BddAbove (Set.range fun n => f n \u03c9) \u2194 \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c)", "state_after": "case h\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\ng : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n \u03c9 => f n \u03c9 - f 0 \u03c9\nhg : Submartingale g \u2131 \u03bc\nhg0 : g 0 = 0\nhgbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |g (i + 1) \u03c9 - g i \u03c9| \u2264 \u2191R\n\u03c9 : \u03a9\nh\u03c9 : BddAbove (Set.range fun n => g n \u03c9) \u2194 \u2203 c, Tendsto (fun n => g n \u03c9) atTop (\ud835\udcdd c)\n\u22a2 BddAbove (Set.range fun n => f n \u03c9) \u2194 \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c)"}, {"tactic": "convert h\u03c9 using 1", "annotated_tactic": ["convert h\u03c9 using 1", []], "state_before": "case h\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\ng : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n \u03c9 => f n \u03c9 - f 0 \u03c9\nhg : Submartingale g \u2131 \u03bc\nhg0 : g 0 = 0\nhgbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |g (i + 1) \u03c9 - g i \u03c9| \u2264 \u2191R\n\u03c9 : \u03a9\nh\u03c9 : BddAbove (Set.range fun n => g n \u03c9) \u2194 \u2203 c, Tendsto (fun n => g n \u03c9) atTop (\ud835\udcdd c)\n\u22a2 BddAbove (Set.range fun n => f n \u03c9) \u2194 \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c)", "state_after": "case h.e'_1.a\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\ng : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n \u03c9 => f n \u03c9 - f 0 \u03c9\nhg : Submartingale g \u2131 \u03bc\nhg0 : g 0 = 0\nhgbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |g (i + 1) \u03c9 - g i \u03c9| \u2264 \u2191R\n\u03c9 : \u03a9\nh\u03c9 : BddAbove (Set.range fun n => g n \u03c9) \u2194 \u2203 c, Tendsto (fun n => g n \u03c9) atTop (\ud835\udcdd c)\n\u22a2 BddAbove (Set.range fun n => f n \u03c9) \u2194 BddAbove (Set.range fun n => g n \u03c9)\n\ncase h.e'_2.a\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\ng : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n \u03c9 => f n \u03c9 - f 0 \u03c9\nhg : Submartingale g \u2131 \u03bc\nhg0 : g 0 = 0\nhgbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |g (i + 1) \u03c9 - g i \u03c9| \u2264 \u2191R\n\u03c9 : \u03a9\nh\u03c9 : BddAbove (Set.range fun n => g n \u03c9) \u2194 \u2203 c, Tendsto (fun n => g n \u03c9) atTop (\ud835\udcdd c)\n\u22a2 (\u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c)) \u2194 \u2203 c, Tendsto (fun n => g n \u03c9) atTop (\ud835\udcdd c)"}, {"tactic": "ext \u03c9", "annotated_tactic": ["ext \u03c9", []], "state_before": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\ng : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n \u03c9 => f n \u03c9 - f 0 \u03c9\nhg : Submartingale g \u2131 \u03bc\n\u22a2 g 0 = 0", "state_after": "case h\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\ng : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n \u03c9 => f n \u03c9 - f 0 \u03c9\nhg : Submartingale g \u2131 \u03bc\n\u03c9 : \u03a9\n\u22a2 g 0 \u03c9 = OfNat.ofNat 0 \u03c9"}, {"tactic": "simp only [sub_self, Pi.zero_apply]", "annotated_tactic": ["simp only [<a>sub_self</a>, <a>Pi.zero_apply</a>]", [{"full_name": "sub_self", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [734, 30], "def_end_pos": [734, 38]}, {"full_name": "Pi.zero_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [46, 3], "def_end_pos": [46, 14]}]], "state_before": "case h\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\ng : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n \u03c9 => f n \u03c9 - f 0 \u03c9\nhg : Submartingale g \u2131 \u03bc\n\u03c9 : \u03a9\n\u22a2 g 0 \u03c9 = OfNat.ofNat 0 \u03c9", "state_after": "no goals"}, {"tactic": "simpa only [sub_sub_sub_cancel_right]", "annotated_tactic": ["simpa only [<a>sub_sub_sub_cancel_right</a>]", [{"full_name": "sub_sub_sub_cancel_right", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [795, 30], "def_end_pos": [795, 54]}]], "state_before": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\ng : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n \u03c9 => f n \u03c9 - f 0 \u03c9\nhg : Submartingale g \u2131 \u03bc\nhg0 : g 0 = 0\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |g (i + 1) \u03c9 - g i \u03c9| \u2264 \u2191R", "state_after": "no goals"}, {"tactic": "refine' \u27e8fun h => _, fun h => _\u27e9 <;> obtain \u27e8b, hb\u27e9 := h <;>\nrefine' \u27e8b + |f 0 \u03c9|, fun y hy => _\u27e9 <;> obtain \u27e8n, rfl\u27e9 := hy", "annotated_tactic": ["refine' \u27e8fun h => _, fun h => _\u27e9 <;> obtain \u27e8b, hb\u27e9 := h <;>\n    refine' \u27e8b + |f 0 \u03c9|, fun y hy => _\u27e9 <;> obtain \u27e8n, rfl\u27e9 := hy", []], "state_before": "case h.e'_1.a\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\ng : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n \u03c9 => f n \u03c9 - f 0 \u03c9\nhg : Submartingale g \u2131 \u03bc\nhg0 : g 0 = 0\nhgbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |g (i + 1) \u03c9 - g i \u03c9| \u2264 \u2191R\n\u03c9 : \u03a9\nh\u03c9 : BddAbove (Set.range fun n => g n \u03c9) \u2194 \u2203 c, Tendsto (fun n => g n \u03c9) atTop (\ud835\udcdd c)\n\u22a2 BddAbove (Set.range fun n => f n \u03c9) \u2194 BddAbove (Set.range fun n => g n \u03c9)", "state_after": "case h.e'_1.a.refine'_1.intro.intro\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\ng : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n \u03c9 => f n \u03c9 - f 0 \u03c9\nhg : Submartingale g \u2131 \u03bc\nhg0 : g 0 = 0\nhgbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |g (i + 1) \u03c9 - g i \u03c9| \u2264 \u2191R\n\u03c9 : \u03a9\nh\u03c9 : BddAbove (Set.range fun n => g n \u03c9) \u2194 \u2203 c, Tendsto (fun n => g n \u03c9) atTop (\ud835\udcdd c)\nb : \u211d\nhb : b \u2208 upperBounds (Set.range fun n => f n \u03c9)\nn : \u2115\n\u22a2 (fun n => g n \u03c9) n \u2264 b + |f 0 \u03c9|\n\ncase h.e'_1.a.refine'_2.intro.intro\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\ng : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n \u03c9 => f n \u03c9 - f 0 \u03c9\nhg : Submartingale g \u2131 \u03bc\nhg0 : g 0 = 0\nhgbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |g (i + 1) \u03c9 - g i \u03c9| \u2264 \u2191R\n\u03c9 : \u03a9\nh\u03c9 : BddAbove (Set.range fun n => g n \u03c9) \u2194 \u2203 c, Tendsto (fun n => g n \u03c9) atTop (\ud835\udcdd c)\nb : \u211d\nhb : b \u2208 upperBounds (Set.range fun n => g n \u03c9)\nn : \u2115\n\u22a2 (fun n => f n \u03c9) n \u2264 b + |f 0 \u03c9|"}, {"tactic": "simp_rw [sub_eq_add_neg]", "annotated_tactic": ["simp_rw [<a>sub_eq_add_neg</a>]", [{"full_name": "sub_eq_add_neg", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [975, 3], "def_end_pos": [975, 14]}]], "state_before": "case h.e'_1.a.refine'_1.intro.intro\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\ng : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n \u03c9 => f n \u03c9 - f 0 \u03c9\nhg : Submartingale g \u2131 \u03bc\nhg0 : g 0 = 0\nhgbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |g (i + 1) \u03c9 - g i \u03c9| \u2264 \u2191R\n\u03c9 : \u03a9\nh\u03c9 : BddAbove (Set.range fun n => g n \u03c9) \u2194 \u2203 c, Tendsto (fun n => g n \u03c9) atTop (\ud835\udcdd c)\nb : \u211d\nhb : b \u2208 upperBounds (Set.range fun n => f n \u03c9)\nn : \u2115\n\u22a2 (fun n => g n \u03c9) n \u2264 b + |f 0 \u03c9|", "state_after": "case h.e'_1.a.refine'_1.intro.intro\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\ng : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n \u03c9 => f n \u03c9 - f 0 \u03c9\nhg : Submartingale g \u2131 \u03bc\nhg0 : g 0 = 0\nhgbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |g (i + 1) \u03c9 - g i \u03c9| \u2264 \u2191R\n\u03c9 : \u03a9\nh\u03c9 : BddAbove (Set.range fun n => g n \u03c9) \u2194 \u2203 c, Tendsto (fun n => g n \u03c9) atTop (\ud835\udcdd c)\nb : \u211d\nhb : b \u2208 upperBounds (Set.range fun n => f n \u03c9)\nn : \u2115\n\u22a2 f n \u03c9 + -f 0 \u03c9 \u2264 b + |f 0 \u03c9|"}, {"tactic": "exact add_le_add (hb \u27e8n, rfl\u27e9) (neg_le_abs_self _)", "annotated_tactic": ["exact <a>add_le_add</a> (hb \u27e8n, <a>rfl</a>\u27e9) (<a>neg_le_abs_self</a> _)", [{"full_name": "add_le_add", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [205, 15], "def_end_pos": [205, 25]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}, {"full_name": "neg_le_abs_self", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [61, 9], "def_end_pos": [61, 24]}]], "state_before": "case h.e'_1.a.refine'_1.intro.intro\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\ng : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n \u03c9 => f n \u03c9 - f 0 \u03c9\nhg : Submartingale g \u2131 \u03bc\nhg0 : g 0 = 0\nhgbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |g (i + 1) \u03c9 - g i \u03c9| \u2264 \u2191R\n\u03c9 : \u03a9\nh\u03c9 : BddAbove (Set.range fun n => g n \u03c9) \u2194 \u2203 c, Tendsto (fun n => g n \u03c9) atTop (\ud835\udcdd c)\nb : \u211d\nhb : b \u2208 upperBounds (Set.range fun n => f n \u03c9)\nn : \u2115\n\u22a2 f n \u03c9 + -f 0 \u03c9 \u2264 b + |f 0 \u03c9|", "state_after": "no goals"}, {"tactic": "exact sub_le_iff_le_add.1 (le_trans (sub_le_sub_left (le_abs_self _) _) (hb \u27e8n, rfl\u27e9))", "annotated_tactic": ["exact <a>sub_le_iff_le_add</a>.1 (<a>le_trans</a> (<a>sub_le_sub_left</a> (<a>le_abs_self</a> _) _) (hb \u27e8n, <a>rfl</a>\u27e9))", [{"full_name": "sub_le_iff_le_add", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [750, 3], "def_end_pos": [750, 14]}, {"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}, {"full_name": "sub_le_sub_left", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [779, 15], "def_end_pos": [779, 30]}, {"full_name": "le_abs_self", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [57, 9], "def_end_pos": [57, 20]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "case h.e'_1.a.refine'_2.intro.intro\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\ng : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n \u03c9 => f n \u03c9 - f 0 \u03c9\nhg : Submartingale g \u2131 \u03bc\nhg0 : g 0 = 0\nhgbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |g (i + 1) \u03c9 - g i \u03c9| \u2264 \u2191R\n\u03c9 : \u03a9\nh\u03c9 : BddAbove (Set.range fun n => g n \u03c9) \u2194 \u2203 c, Tendsto (fun n => g n \u03c9) atTop (\ud835\udcdd c)\nb : \u211d\nhb : b \u2208 upperBounds (Set.range fun n => g n \u03c9)\nn : \u2115\n\u22a2 (fun n => f n \u03c9) n \u2264 b + |f 0 \u03c9|", "state_after": "no goals"}, {"tactic": "refine' \u27e8fun h => _, fun h => _\u27e9 <;> obtain \u27e8c, hc\u27e9 := h", "annotated_tactic": ["refine' \u27e8fun h => _, fun h => _\u27e9 <;> obtain \u27e8c, hc\u27e9 := h", []], "state_before": "case h.e'_2.a\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\ng : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n \u03c9 => f n \u03c9 - f 0 \u03c9\nhg : Submartingale g \u2131 \u03bc\nhg0 : g 0 = 0\nhgbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |g (i + 1) \u03c9 - g i \u03c9| \u2264 \u2191R\n\u03c9 : \u03a9\nh\u03c9 : BddAbove (Set.range fun n => g n \u03c9) \u2194 \u2203 c, Tendsto (fun n => g n \u03c9) atTop (\ud835\udcdd c)\n\u22a2 (\u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c)) \u2194 \u2203 c, Tendsto (fun n => g n \u03c9) atTop (\ud835\udcdd c)", "state_after": "case h.e'_2.a.refine'_1.intro\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\ng : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n \u03c9 => f n \u03c9 - f 0 \u03c9\nhg : Submartingale g \u2131 \u03bc\nhg0 : g 0 = 0\nhgbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |g (i + 1) \u03c9 - g i \u03c9| \u2264 \u2191R\n\u03c9 : \u03a9\nh\u03c9 : BddAbove (Set.range fun n => g n \u03c9) \u2194 \u2203 c, Tendsto (fun n => g n \u03c9) atTop (\ud835\udcdd c)\nc : \u211d\nhc : Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c)\n\u22a2 \u2203 c, Tendsto (fun n => g n \u03c9) atTop (\ud835\udcdd c)\n\ncase h.e'_2.a.refine'_2.intro\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\ng : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n \u03c9 => f n \u03c9 - f 0 \u03c9\nhg : Submartingale g \u2131 \u03bc\nhg0 : g 0 = 0\nhgbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |g (i + 1) \u03c9 - g i \u03c9| \u2264 \u2191R\n\u03c9 : \u03a9\nh\u03c9 : BddAbove (Set.range fun n => g n \u03c9) \u2194 \u2203 c, Tendsto (fun n => g n \u03c9) atTop (\ud835\udcdd c)\nc : \u211d\nhc : Tendsto (fun n => g n \u03c9) atTop (\ud835\udcdd c)\n\u22a2 \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c)"}, {"tactic": "exact \u27e8c - f 0 \u03c9, hc.sub_const _\u27e9", "annotated_tactic": ["exact \u27e8c - f 0 \u03c9, hc.sub_const _\u27e9", []], "state_before": "case h.e'_2.a.refine'_1.intro\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\ng : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n \u03c9 => f n \u03c9 - f 0 \u03c9\nhg : Submartingale g \u2131 \u03bc\nhg0 : g 0 = 0\nhgbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |g (i + 1) \u03c9 - g i \u03c9| \u2264 \u2191R\n\u03c9 : \u03a9\nh\u03c9 : BddAbove (Set.range fun n => g n \u03c9) \u2194 \u2203 c, Tendsto (fun n => g n \u03c9) atTop (\ud835\udcdd c)\nc : \u211d\nhc : Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c)\n\u22a2 \u2203 c, Tendsto (fun n => g n \u03c9) atTop (\ud835\udcdd c)", "state_after": "no goals"}, {"tactic": "refine' \u27e8c + f 0 \u03c9, _\u27e9", "annotated_tactic": ["refine' \u27e8c + f 0 \u03c9, _\u27e9", []], "state_before": "case h.e'_2.a.refine'_2.intro\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\ng : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n \u03c9 => f n \u03c9 - f 0 \u03c9\nhg : Submartingale g \u2131 \u03bc\nhg0 : g 0 = 0\nhgbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |g (i + 1) \u03c9 - g i \u03c9| \u2264 \u2191R\n\u03c9 : \u03a9\nh\u03c9 : BddAbove (Set.range fun n => g n \u03c9) \u2194 \u2203 c, Tendsto (fun n => g n \u03c9) atTop (\ud835\udcdd c)\nc : \u211d\nhc : Tendsto (fun n => g n \u03c9) atTop (\ud835\udcdd c)\n\u22a2 \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c)", "state_after": "case h.e'_2.a.refine'_2.intro\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\ng : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n \u03c9 => f n \u03c9 - f 0 \u03c9\nhg : Submartingale g \u2131 \u03bc\nhg0 : g 0 = 0\nhgbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |g (i + 1) \u03c9 - g i \u03c9| \u2264 \u2191R\n\u03c9 : \u03a9\nh\u03c9 : BddAbove (Set.range fun n => g n \u03c9) \u2194 \u2203 c, Tendsto (fun n => g n \u03c9) atTop (\ud835\udcdd c)\nc : \u211d\nhc : Tendsto (fun n => g n \u03c9) atTop (\ud835\udcdd c)\n\u22a2 Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd (c + f 0 \u03c9))"}, {"tactic": "have := hc.add_const (f 0 \u03c9)", "annotated_tactic": ["have := hc.add_const (f 0 \u03c9)", []], "state_before": "case h.e'_2.a.refine'_2.intro\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\ng : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n \u03c9 => f n \u03c9 - f 0 \u03c9\nhg : Submartingale g \u2131 \u03bc\nhg0 : g 0 = 0\nhgbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |g (i + 1) \u03c9 - g i \u03c9| \u2264 \u2191R\n\u03c9 : \u03a9\nh\u03c9 : BddAbove (Set.range fun n => g n \u03c9) \u2194 \u2203 c, Tendsto (fun n => g n \u03c9) atTop (\ud835\udcdd c)\nc : \u211d\nhc : Tendsto (fun n => g n \u03c9) atTop (\ud835\udcdd c)\n\u22a2 Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd (c + f 0 \u03c9))", "state_after": "case h.e'_2.a.refine'_2.intro\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\ng : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n \u03c9 => f n \u03c9 - f 0 \u03c9\nhg : Submartingale g \u2131 \u03bc\nhg0 : g 0 = 0\nhgbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |g (i + 1) \u03c9 - g i \u03c9| \u2264 \u2191R\n\u03c9 : \u03a9\nh\u03c9 : BddAbove (Set.range fun n => g n \u03c9) \u2194 \u2203 c, Tendsto (fun n => g n \u03c9) atTop (\ud835\udcdd c)\nc : \u211d\nhc : Tendsto (fun n => g n \u03c9) atTop (\ud835\udcdd c)\nthis : Tendsto (fun k => g k \u03c9 + f 0 \u03c9) atTop (\ud835\udcdd (c + f 0 \u03c9))\n\u22a2 Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd (c + f 0 \u03c9))"}, {"tactic": "simpa only [sub_add_cancel]", "annotated_tactic": ["simpa only [<a>sub_add_cancel</a>]", [{"full_name": "sub_add_cancel", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [728, 30], "def_end_pos": [728, 44]}]], "state_before": "case h.e'_2.a.refine'_2.intro\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\ng : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n \u03c9 => f n \u03c9 - f 0 \u03c9\nhg : Submartingale g \u2131 \u03bc\nhg0 : g 0 = 0\nhgbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |g (i + 1) \u03c9 - g i \u03c9| \u2264 \u2191R\n\u03c9 : \u03a9\nh\u03c9 : BddAbove (Set.range fun n => g n \u03c9) \u2194 \u2203 c, Tendsto (fun n => g n \u03c9) atTop (\ud835\udcdd c)\nc : \u211d\nhc : Tendsto (fun n => g n \u03c9) atTop (\ud835\udcdd c)\nthis : Tendsto (fun k => g k \u03c9 + f 0 \u03c9) atTop (\ud835\udcdd (c + f 0 \u03c9))\n\u22a2 Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd (c + f 0 \u03c9))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Constructions/Polish.lean", "full_name": "MeasureTheory.exists_subset_real_measurableEquiv", "start": [1048, 1], "end": [1068, 38], "traced_tactics": [{"tactic": "by_cases h\u03b1 : Countable \u03b1", "annotated_tactic": ["by_cases h\u03b1 : <a>Countable</a> \u03b1", [{"full_name": "Countable", "def_path": "Mathlib/Data/Countable/Defs.lean", "def_pos": [34, 7], "def_end_pos": [34, 16]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : StandardBorelSpace \u03b1\n\u22a2 \u2203 s, MeasurableSet s \u2227 Nonempty (\u03b1 \u2243\u1d50 \u2191s)", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : StandardBorelSpace \u03b1\nh\u03b1 : Countable \u03b1\n\u22a2 \u2203 s, MeasurableSet s \u2227 Nonempty (\u03b1 \u2243\u1d50 \u2191s)\n\ncase neg\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : StandardBorelSpace \u03b1\nh\u03b1 : \u00acCountable \u03b1\n\u22a2 \u2203 s, MeasurableSet s \u2227 Nonempty (\u03b1 \u2243\u1d50 \u2191s)"}, {"tactic": "cases finite_or_infinite \u03b1", "annotated_tactic": ["cases <a>finite_or_infinite</a> \u03b1", [{"full_name": "finite_or_infinite", "def_path": "Mathlib/Data/Finite/Defs.lean", "def_pos": [120, 9], "def_end_pos": [120, 27]}]], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : StandardBorelSpace \u03b1\nh\u03b1 : Countable \u03b1\n\u22a2 \u2203 s, MeasurableSet s \u2227 Nonempty (\u03b1 \u2243\u1d50 \u2191s)", "state_after": "case pos.inl\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : StandardBorelSpace \u03b1\nh\u03b1 : Countable \u03b1\nh\u271d : Finite \u03b1\n\u22a2 \u2203 s, MeasurableSet s \u2227 Nonempty (\u03b1 \u2243\u1d50 \u2191s)\n\ncase pos.inr\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : StandardBorelSpace \u03b1\nh\u03b1 : Countable \u03b1\nh\u271d : Infinite \u03b1\n\u22a2 \u2203 s, MeasurableSet s \u2227 Nonempty (\u03b1 \u2243\u1d50 \u2191s)"}, {"tactic": "obtain \u27e8n, h_nonempty_equiv\u27e9 := exists_nat_measurableEquiv_range_coe_fin_of_finite \u03b1", "annotated_tactic": ["obtain \u27e8n, h_nonempty_equiv\u27e9 := <a>exists_nat_measurableEquiv_range_coe_fin_of_finite</a> \u03b1", [{"full_name": "MeasureTheory.exists_nat_measurableEquiv_range_coe_fin_of_finite", "def_path": "Mathlib/MeasureTheory/Constructions/Polish.lean", "def_pos": [1031, 9], "def_end_pos": [1031, 59]}]], "state_before": "case pos.inl\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : StandardBorelSpace \u03b1\nh\u03b1 : Countable \u03b1\nh\u271d : Finite \u03b1\n\u22a2 \u2203 s, MeasurableSet s \u2227 Nonempty (\u03b1 \u2243\u1d50 \u2191s)", "state_after": "case pos.inl.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : StandardBorelSpace \u03b1\nh\u03b1 : Countable \u03b1\nh\u271d : Finite \u03b1\nn : \u2115\nh_nonempty_equiv : Nonempty (\u03b1 \u2243\u1d50 \u2191(range fun x => \u2191\u2191x))\n\u22a2 \u2203 s, MeasurableSet s \u2227 Nonempty (\u03b1 \u2243\u1d50 \u2191s)"}, {"tactic": "refine' \u27e8_, _, h_nonempty_equiv\u27e9", "annotated_tactic": ["refine' \u27e8_, _, h_nonempty_equiv\u27e9", []], "state_before": "case pos.inl.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : StandardBorelSpace \u03b1\nh\u03b1 : Countable \u03b1\nh\u271d : Finite \u03b1\nn : \u2115\nh_nonempty_equiv : Nonempty (\u03b1 \u2243\u1d50 \u2191(range fun x => \u2191\u2191x))\n\u22a2 \u2203 s, MeasurableSet s \u2227 Nonempty (\u03b1 \u2243\u1d50 \u2191s)", "state_after": "case pos.inl.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : StandardBorelSpace \u03b1\nh\u03b1 : Countable \u03b1\nh\u271d : Finite \u03b1\nn : \u2115\nh_nonempty_equiv : Nonempty (\u03b1 \u2243\u1d50 \u2191(range fun x => \u2191\u2191x))\n\u22a2 MeasurableSet (range fun x => \u2191\u2191x)"}, {"tactic": "letI : MeasurableSpace (Fin n) := borel (Fin n)", "annotated_tactic": ["letI : <a>MeasurableSpace</a> (<a>Fin</a> n) := <a>borel</a> (<a>Fin</a> n)", [{"full_name": "MeasurableSpace", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [48, 20], "def_end_pos": [48, 35]}, {"full_name": "Fin", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1745, 11], "def_end_pos": [1745, 14]}, {"full_name": "borel", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [59, 5], "def_end_pos": [59, 10]}, {"full_name": "Fin", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1745, 11], "def_end_pos": [1745, 14]}]], "state_before": "case pos.inl.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : StandardBorelSpace \u03b1\nh\u03b1 : Countable \u03b1\nh\u271d : Finite \u03b1\nn : \u2115\nh_nonempty_equiv : Nonempty (\u03b1 \u2243\u1d50 \u2191(range fun x => \u2191\u2191x))\n\u22a2 MeasurableSet (range fun x => \u2191\u2191x)", "state_after": "case pos.inl.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : StandardBorelSpace \u03b1\nh\u03b1 : Countable \u03b1\nh\u271d : Finite \u03b1\nn : \u2115\nh_nonempty_equiv : Nonempty (\u03b1 \u2243\u1d50 \u2191(range fun x => \u2191\u2191x))\nthis : MeasurableSpace (Fin n) := borel (Fin n)\n\u22a2 MeasurableSet (range fun x => \u2191\u2191x)"}, {"tactic": "haveI : BorelSpace (Fin n) := \u27e8rfl\u27e9", "annotated_tactic": ["haveI : <a>BorelSpace</a> (<a>Fin</a> n) := \u27e8<a>rfl</a>\u27e9", [{"full_name": "BorelSpace", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [202, 7], "def_end_pos": [202, 17]}, {"full_name": "Fin", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1745, 11], "def_end_pos": [1745, 14]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "case pos.inl.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : StandardBorelSpace \u03b1\nh\u03b1 : Countable \u03b1\nh\u271d : Finite \u03b1\nn : \u2115\nh_nonempty_equiv : Nonempty (\u03b1 \u2243\u1d50 \u2191(range fun x => \u2191\u2191x))\nthis : MeasurableSpace (Fin n) := borel (Fin n)\n\u22a2 MeasurableSet (range fun x => \u2191\u2191x)", "state_after": "case pos.inl.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : StandardBorelSpace \u03b1\nh\u03b1 : Countable \u03b1\nh\u271d : Finite \u03b1\nn : \u2115\nh_nonempty_equiv : Nonempty (\u03b1 \u2243\u1d50 \u2191(range fun x => \u2191\u2191x))\nthis\u271d : MeasurableSpace (Fin n) := borel (Fin n)\nthis : BorelSpace (Fin n)\n\u22a2 MeasurableSet (range fun x => \u2191\u2191x)"}, {"tactic": "refine' MeasurableEmbedding.measurableSet_range _", "annotated_tactic": ["refine' <a>MeasurableEmbedding.measurableSet_range</a> _", [{"full_name": "MeasurableEmbedding.measurableSet_range", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [1214, 9], "def_end_pos": [1214, 28]}]], "state_before": "case pos.inl.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : StandardBorelSpace \u03b1\nh\u03b1 : Countable \u03b1\nh\u271d : Finite \u03b1\nn : \u2115\nh_nonempty_equiv : Nonempty (\u03b1 \u2243\u1d50 \u2191(range fun x => \u2191\u2191x))\nthis\u271d : MeasurableSpace (Fin n) := borel (Fin n)\nthis : BorelSpace (Fin n)\n\u22a2 MeasurableSet (range fun x => \u2191\u2191x)", "state_after": "case pos.inl.intro.refine'_1\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : StandardBorelSpace \u03b1\nh\u03b1 : Countable \u03b1\nh\u271d : Finite \u03b1\nn : \u2115\nh_nonempty_equiv : Nonempty (\u03b1 \u2243\u1d50 \u2191(range fun x => \u2191\u2191x))\nthis\u271d : MeasurableSpace (Fin n) := borel (Fin n)\nthis : BorelSpace (Fin n)\n\u22a2 MeasurableSpace (Fin n)\n\ncase pos.inl.intro.refine'_2\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : StandardBorelSpace \u03b1\nh\u03b1 : Countable \u03b1\nh\u271d : Finite \u03b1\nn : \u2115\nh_nonempty_equiv : Nonempty (\u03b1 \u2243\u1d50 \u2191(range fun x => \u2191\u2191x))\nthis\u271d : MeasurableSpace (Fin n) := borel (Fin n)\nthis : BorelSpace (Fin n)\n\u22a2 MeasurableEmbedding fun x => \u2191\u2191x"}, {"tactic": "infer_instance", "annotated_tactic": ["infer_instance", []], "state_before": "case pos.inl.intro.refine'_1\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : StandardBorelSpace \u03b1\nh\u03b1 : Countable \u03b1\nh\u271d : Finite \u03b1\nn : \u2115\nh_nonempty_equiv : Nonempty (\u03b1 \u2243\u1d50 \u2191(range fun x => \u2191\u2191x))\nthis\u271d : MeasurableSpace (Fin n) := borel (Fin n)\nthis : BorelSpace (Fin n)\n\u22a2 MeasurableSpace (Fin n)", "state_after": "no goals"}, {"tactic": "exact\n  continuous_of_discreteTopology.measurableEmbedding\n    (Nat.cast_injective.comp Fin.val_injective)", "annotated_tactic": ["exact\n          continuous_of_discreteTopology.measurableEmbedding\n            (Nat.cast_injective.comp <a>Fin.val_injective</a>)", [{"full_name": "Fin.val_injective", "def_path": "Mathlib/Data/Fin/Basic.lean", "def_pos": [103, 9], "def_end_pos": [103, 22]}]], "state_before": "case pos.inl.intro.refine'_2\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : StandardBorelSpace \u03b1\nh\u03b1 : Countable \u03b1\nh\u271d : Finite \u03b1\nn : \u2115\nh_nonempty_equiv : Nonempty (\u03b1 \u2243\u1d50 \u2191(range fun x => \u2191\u2191x))\nthis\u271d : MeasurableSpace (Fin n) := borel (Fin n)\nthis : BorelSpace (Fin n)\n\u22a2 MeasurableEmbedding fun x => \u2191\u2191x", "state_after": "no goals"}, {"tactic": "refine' \u27e8_, _, measurableEquiv_range_coe_nat_of_infinite_of_countable \u03b1\u27e9", "annotated_tactic": ["refine' \u27e8_, _, <a>measurableEquiv_range_coe_nat_of_infinite_of_countable</a> \u03b1\u27e9", [{"full_name": "MeasureTheory.measurableEquiv_range_coe_nat_of_infinite_of_countable", "def_path": "Mathlib/MeasureTheory/Constructions/Polish.lean", "def_pos": [1038, 9], "def_end_pos": [1038, 63]}]], "state_before": "case pos.inr\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : StandardBorelSpace \u03b1\nh\u03b1 : Countable \u03b1\nh\u271d : Infinite \u03b1\n\u22a2 \u2203 s, MeasurableSet s \u2227 Nonempty (\u03b1 \u2243\u1d50 \u2191s)", "state_after": "case pos.inr\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : StandardBorelSpace \u03b1\nh\u03b1 : Countable \u03b1\nh\u271d : Infinite \u03b1\n\u22a2 MeasurableSet (range Nat.cast)"}, {"tactic": "refine' MeasurableEmbedding.measurableSet_range _", "annotated_tactic": ["refine' <a>MeasurableEmbedding.measurableSet_range</a> _", [{"full_name": "MeasurableEmbedding.measurableSet_range", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [1214, 9], "def_end_pos": [1214, 28]}]], "state_before": "case pos.inr\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : StandardBorelSpace \u03b1\nh\u03b1 : Countable \u03b1\nh\u271d : Infinite \u03b1\n\u22a2 MeasurableSet (range Nat.cast)", "state_after": "case pos.inr.refine'_1\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : StandardBorelSpace \u03b1\nh\u03b1 : Countable \u03b1\nh\u271d : Infinite \u03b1\n\u22a2 MeasurableSpace \u2115\n\ncase pos.inr.refine'_2\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : StandardBorelSpace \u03b1\nh\u03b1 : Countable \u03b1\nh\u271d : Infinite \u03b1\n\u22a2 MeasurableEmbedding Nat.cast"}, {"tactic": "infer_instance", "annotated_tactic": ["infer_instance", []], "state_before": "case pos.inr.refine'_1\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : StandardBorelSpace \u03b1\nh\u03b1 : Countable \u03b1\nh\u271d : Infinite \u03b1\n\u22a2 MeasurableSpace \u2115", "state_after": "no goals"}, {"tactic": "exact continuous_of_discreteTopology.measurableEmbedding Nat.cast_injective", "annotated_tactic": ["exact continuous_of_discreteTopology.measurableEmbedding <a>Nat.cast_injective</a>", [{"full_name": "Nat.cast_injective", "def_path": "Mathlib/Algebra/CharZero/Defs.lean", "def_pos": [70, 9], "def_end_pos": [70, 23]}]], "state_before": "case pos.inr.refine'_2\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : StandardBorelSpace \u03b1\nh\u03b1 : Countable \u03b1\nh\u271d : Infinite \u03b1\n\u22a2 MeasurableEmbedding Nat.cast", "state_after": "no goals"}, {"tactic": "refine'\n  \u27e8univ, MeasurableSet.univ,\n    \u27e8(PolishSpace.measurableEquivOfNotCountable h\u03b1 _ : \u03b1 \u2243\u1d50 (univ : Set \u211d))\u27e9\u27e9", "annotated_tactic": ["refine'\n      \u27e8<a>univ</a>, <a>MeasurableSet.univ</a>,\n        \u27e8(<a>PolishSpace.measurableEquivOfNotCountable</a> h\u03b1 _ : \u03b1 \u2243\u1d50 (<a>univ</a> : <a>Set</a> \u211d))\u27e9\u27e9", [{"full_name": "Set.univ", "def_path": "Mathlib/Init/Set.lean", "def_pos": [90, 5], "def_end_pos": [90, 9]}, {"full_name": "MeasurableSet.univ", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [101, 19], "def_end_pos": [101, 37]}, {"full_name": "PolishSpace.measurableEquivOfNotCountable", "def_path": "Mathlib/MeasureTheory/Constructions/Polish.lean", "def_pos": [1010, 19], "def_end_pos": [1010, 48]}, {"full_name": "Set.univ", "def_path": "Mathlib/Init/Set.lean", "def_pos": [90, 5], "def_end_pos": [90, 9]}, {"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : StandardBorelSpace \u03b1\nh\u03b1 : \u00acCountable \u03b1\n\u22a2 \u2203 s, MeasurableSet s \u2227 Nonempty (\u03b1 \u2243\u1d50 \u2191s)", "state_after": "case neg\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : StandardBorelSpace \u03b1\nh\u03b1 : \u00acCountable \u03b1\n\u22a2 \u00acCountable \u2191univ"}, {"tactic": "rw [countable_coe_iff]", "annotated_tactic": ["rw [<a>countable_coe_iff</a>]", [{"full_name": "Set.countable_coe_iff", "def_path": "Mathlib/Data/Set/Countable.lean", "def_pos": [36, 9], "def_end_pos": [36, 26]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : StandardBorelSpace \u03b1\nh\u03b1 : \u00acCountable \u03b1\n\u22a2 \u00acCountable \u2191univ", "state_after": "case neg\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : StandardBorelSpace \u03b1\nh\u03b1 : \u00acCountable \u03b1\n\u22a2 \u00acSet.Countable univ"}, {"tactic": "exact Cardinal.not_countable_real", "annotated_tactic": ["exact <a>Cardinal.not_countable_real</a>", [{"full_name": "Cardinal.not_countable_real", "def_path": "Mathlib/Data/Real/Cardinality.lean", "def_pos": [218, 9], "def_end_pos": [218, 27]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : StandardBorelSpace \u03b1\nh\u03b1 : \u00acCountable \u03b1\n\u22a2 \u00acSet.Countable univ", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Regular.lean", "full_name": "MeasurableSet.exists_lt_isCompact_of_ne_top", "start": [518, 1], "end": [521, 49], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finmap.lean", "full_name": "Finmap.not_mem_erase_self", "start": [439, 1], "end": [442, 6], "traced_tactics": [{"tactic": "rw [mem_erase, not_and_or, not_not]", "annotated_tactic": ["rw [<a>mem_erase</a>, <a>not_and_or</a>, <a>not_not</a>]", [{"full_name": "Finmap.mem_erase", "def_path": "Mathlib/Data/Finmap.lean", "def_pos": [435, 9], "def_end_pos": [435, 18]}, {"full_name": "not_and_or", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [473, 9], "def_end_pos": [473, 19]}, {"full_name": "Classical.not_not", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [683, 24], "def_end_pos": [683, 31]}]], "state_before": "\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\na : \u03b1\ns : Finmap \u03b2\n\u22a2 \u00aca \u2208 erase a s", "state_after": "\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\na : \u03b1\ns : Finmap \u03b2\n\u22a2 a = a \u2228 \u00aca \u2208 s"}, {"tactic": "left", "annotated_tactic": ["left", []], "state_before": "\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\na : \u03b1\ns : Finmap \u03b2\n\u22a2 a = a \u2228 \u00aca \u2208 s", "state_after": "case h\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\na : \u03b1\ns : Finmap \u03b2\n\u22a2 a = a"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case h\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\na : \u03b1\ns : Finmap \u03b2\n\u22a2 a = a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "full_name": "MeasureTheory.continuous_of_dominated", "start": [1115, 1], "end": [1123, 42], "traced_tactics": [{"tactic": "by_cases hG : CompleteSpace G", "annotated_tactic": ["by_cases hG : <a>CompleteSpace</a> G", [{"full_name": "CompleteSpace", "def_path": "Mathlib/Topology/UniformSpace/Cauchy.lean", "def_pos": [397, 7], "def_end_pos": [397, 20]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF\u271d : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\u271d\ninst\u271d\u2075 : NormedSpace \u211d F\u271d\ninst\u271d\u2074 : CompleteSpace F\u271d\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nF : X \u2192 \u03b1 \u2192 G\nbound : \u03b1 \u2192 \u211d\nhF_meas : \u2200 (x : X), AEStronglyMeasurable (F x) \u03bc\nh_bound : \u2200 (x : X), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016F x a\u2016 \u2264 bound a\nbound_integrable : Integrable bound\nh_cont : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Continuous fun x => F x a\n\u22a2 Continuous fun x => \u222b (a : \u03b1), F x a \u2202\u03bc", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF\u271d : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\u271d\ninst\u271d\u2075 : NormedSpace \u211d F\u271d\ninst\u271d\u2074 : CompleteSpace F\u271d\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nF : X \u2192 \u03b1 \u2192 G\nbound : \u03b1 \u2192 \u211d\nhF_meas : \u2200 (x : X), AEStronglyMeasurable (F x) \u03bc\nh_bound : \u2200 (x : X), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016F x a\u2016 \u2264 bound a\nbound_integrable : Integrable bound\nh_cont : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Continuous fun x => F x a\nhG : CompleteSpace G\n\u22a2 Continuous fun x => \u222b (a : \u03b1), F x a \u2202\u03bc\n\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF\u271d : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\u271d\ninst\u271d\u2075 : NormedSpace \u211d F\u271d\ninst\u271d\u2074 : CompleteSpace F\u271d\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nF : X \u2192 \u03b1 \u2192 G\nbound : \u03b1 \u2192 \u211d\nhF_meas : \u2200 (x : X), AEStronglyMeasurable (F x) \u03bc\nh_bound : \u2200 (x : X), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016F x a\u2016 \u2264 bound a\nbound_integrable : Integrable bound\nh_cont : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Continuous fun x => F x a\nhG : \u00acCompleteSpace G\n\u22a2 Continuous fun x => \u222b (a : \u03b1), F x a \u2202\u03bc"}, {"tactic": "simp only [integral, hG, L1.integral]", "annotated_tactic": ["simp only [<a>integral</a>, hG, <a>L1.integral</a>]", [{"full_name": "MeasureTheory.integral", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [791, 17], "def_end_pos": [791, 25]}, {"full_name": "MeasureTheory.L1.integral", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [666, 17], "def_end_pos": [666, 25]}]], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF\u271d : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\u271d\ninst\u271d\u2075 : NormedSpace \u211d F\u271d\ninst\u271d\u2074 : CompleteSpace F\u271d\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nF : X \u2192 \u03b1 \u2192 G\nbound : \u03b1 \u2192 \u211d\nhF_meas : \u2200 (x : X), AEStronglyMeasurable (F x) \u03bc\nh_bound : \u2200 (x : X), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016F x a\u2016 \u2264 bound a\nbound_integrable : Integrable bound\nh_cont : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Continuous fun x => F x a\nhG : CompleteSpace G\n\u22a2 Continuous fun x => \u222b (a : \u03b1), F x a \u2202\u03bc", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF\u271d : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\u271d\ninst\u271d\u2075 : NormedSpace \u211d F\u271d\ninst\u271d\u2074 : CompleteSpace F\u271d\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nF : X \u2192 \u03b1 \u2192 G\nbound : \u03b1 \u2192 \u211d\nhF_meas : \u2200 (x : X), AEStronglyMeasurable (F x) \u03bc\nh_bound : \u2200 (x : X), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016F x a\u2016 \u2264 bound a\nbound_integrable : Integrable bound\nh_cont : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Continuous fun x => F x a\nhG : CompleteSpace G\n\u22a2 Continuous fun x =>\n    if h : True then if hf : Integrable fun a => F x a then \u2191L1.integralCLM (Integrable.toL1 (fun a => F x a) hf) else 0\n    else 0"}, {"tactic": "exact continuous_setToFun_of_dominated (dominatedFinMeasAdditive_weightedSMul \u03bc)\n  hF_meas h_bound bound_integrable h_cont", "annotated_tactic": ["exact <a>continuous_setToFun_of_dominated</a> (<a>dominatedFinMeasAdditive_weightedSMul</a> \u03bc)\n      hF_meas h_bound bound_integrable h_cont", [{"full_name": "MeasureTheory.continuous_setToFun_of_dominated", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [1812, 9], "def_end_pos": [1812, 41]}, {"full_name": "MeasureTheory.dominatedFinMeasAdditive_weightedSMul", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [243, 9], "def_end_pos": [243, 46]}]], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF\u271d : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\u271d\ninst\u271d\u2075 : NormedSpace \u211d F\u271d\ninst\u271d\u2074 : CompleteSpace F\u271d\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nF : X \u2192 \u03b1 \u2192 G\nbound : \u03b1 \u2192 \u211d\nhF_meas : \u2200 (x : X), AEStronglyMeasurable (F x) \u03bc\nh_bound : \u2200 (x : X), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016F x a\u2016 \u2264 bound a\nbound_integrable : Integrable bound\nh_cont : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Continuous fun x => F x a\nhG : CompleteSpace G\n\u22a2 Continuous fun x =>\n    if h : True then if hf : Integrable fun a => F x a then \u2191L1.integralCLM (Integrable.toL1 (fun a => F x a) hf) else 0\n    else 0", "state_after": "no goals"}, {"tactic": "simp [integral, hG, continuous_const]", "annotated_tactic": ["simp [<a>integral</a>, hG, <a>continuous_const</a>]", [{"full_name": "MeasureTheory.integral", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [791, 17], "def_end_pos": [791, 25]}, {"full_name": "continuous_const", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [1723, 9], "def_end_pos": [1723, 25]}]], "state_before": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF\u271d : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\u271d\ninst\u271d\u2075 : NormedSpace \u211d F\u271d\ninst\u271d\u2074 : CompleteSpace F\u271d\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nF : X \u2192 \u03b1 \u2192 G\nbound : \u03b1 \u2192 \u211d\nhF_meas : \u2200 (x : X), AEStronglyMeasurable (F x) \u03bc\nh_bound : \u2200 (x : X), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016F x a\u2016 \u2264 bound a\nbound_integrable : Integrable bound\nh_cont : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Continuous fun x => F x a\nhG : \u00acCompleteSpace G\n\u22a2 Continuous fun x => \u222b (a : \u03b1), F x a \u2202\u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/Supported.lean", "full_name": "MvPolynomial.mem_supported", "start": [75, 1], "end": [83, 97], "traced_tactics": [{"tactic": "rw [supported_eq_range_rename, AlgHom.mem_range]", "annotated_tactic": ["rw [<a>supported_eq_range_rename</a>, <a>AlgHom.mem_range</a>]", [{"full_name": "MvPolynomial.supported_eq_range_rename", "def_path": "Mathlib/Data/MvPolynomial/Supported.lean", "def_pos": [47, 9], "def_end_pos": [47, 34]}, {"full_name": "AlgHom.mem_range", "def_path": "Mathlib/Algebra/Algebra/Subalgebra/Basic.lean", "def_pos": [615, 9], "def_end_pos": [615, 18]}]], "state_before": "\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u\nS : Type v\nr : R\ne : \u2115\nn m : \u03c3\ninst\u271d : CommSemiring R\np q : MvPolynomial \u03c3 R\ns t : Set \u03c3\n\u22a2 p \u2208 supported R s \u2194 \u2191(vars p) \u2286 s", "state_after": "\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u\nS : Type v\nr : R\ne : \u2115\nn m : \u03c3\ninst\u271d : CommSemiring R\np q : MvPolynomial \u03c3 R\ns t : Set \u03c3\n\u22a2 (\u2203 x, \u2191(rename Subtype.val) x = p) \u2194 \u2191(vars p) \u2286 s"}, {"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u\nS : Type v\nr : R\ne : \u2115\nn m : \u03c3\ninst\u271d : CommSemiring R\np q : MvPolynomial \u03c3 R\ns t : Set \u03c3\n\u22a2 (\u2203 x, \u2191(rename Subtype.val) x = p) \u2194 \u2191(vars p) \u2286 s", "state_after": "case mp\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u\nS : Type v\nr : R\ne : \u2115\nn m : \u03c3\ninst\u271d : CommSemiring R\np q : MvPolynomial \u03c3 R\ns t : Set \u03c3\n\u22a2 (\u2203 x, \u2191(rename Subtype.val) x = p) \u2192 \u2191(vars p) \u2286 s\n\ncase mpr\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u\nS : Type v\nr : R\ne : \u2115\nn m : \u03c3\ninst\u271d : CommSemiring R\np q : MvPolynomial \u03c3 R\ns t : Set \u03c3\n\u22a2 \u2191(vars p) \u2286 s \u2192 \u2203 x, \u2191(rename Subtype.val) x = p"}, {"tactic": "rintro \u27e8p, rfl\u27e9", "annotated_tactic": ["rintro \u27e8p, rfl\u27e9", []], "state_before": "case mp\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u\nS : Type v\nr : R\ne : \u2115\nn m : \u03c3\ninst\u271d : CommSemiring R\np q : MvPolynomial \u03c3 R\ns t : Set \u03c3\n\u22a2 (\u2203 x, \u2191(rename Subtype.val) x = p) \u2192 \u2191(vars p) \u2286 s", "state_after": "case mp.intro\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u\nS : Type v\nr : R\ne : \u2115\nn m : \u03c3\ninst\u271d : CommSemiring R\nq : MvPolynomial \u03c3 R\ns t : Set \u03c3\np : MvPolynomial { x // x \u2208 s } R\n\u22a2 \u2191(vars (\u2191(rename Subtype.val) p)) \u2286 s"}, {"tactic": "refine' _root_.trans (Finset.coe_subset.2 (vars_rename _ _)) _", "annotated_tactic": ["refine' <a>_root_.trans</a> (<a>Finset.coe_subset</a>.2 (<a>vars_rename</a> _ _)) _", [{"full_name": "trans", "def_path": "Mathlib/Init/Algebra/Classes.lean", "def_pos": [308, 9], "def_end_pos": [308, 14]}, {"full_name": "Finset.coe_subset", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [376, 9], "def_end_pos": [376, 19]}, {"full_name": "MvPolynomial.vars_rename", "def_path": "Mathlib/Data/MvPolynomial/Variables.lean", "def_pos": [868, 9], "def_end_pos": [868, 20]}]], "state_before": "case mp.intro\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u\nS : Type v\nr : R\ne : \u2115\nn m : \u03c3\ninst\u271d : CommSemiring R\nq : MvPolynomial \u03c3 R\ns t : Set \u03c3\np : MvPolynomial { x // x \u2208 s } R\n\u22a2 \u2191(vars (\u2191(rename Subtype.val) p)) \u2286 s", "state_after": "case mp.intro\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u\nS : Type v\nr : R\ne : \u2115\nn m : \u03c3\ninst\u271d : CommSemiring R\nq : MvPolynomial \u03c3 R\ns t : Set \u03c3\np : MvPolynomial { x // x \u2208 s } R\n\u22a2 \u2191(Finset.image Subtype.val (vars p)) \u2286 s"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case mp.intro\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u\nS : Type v\nr : R\ne : \u2115\nn m : \u03c3\ninst\u271d : CommSemiring R\nq : MvPolynomial \u03c3 R\ns t : Set \u03c3\np : MvPolynomial { x // x \u2208 s } R\n\u22a2 \u2191(Finset.image Subtype.val (vars p)) \u2286 s", "state_after": "no goals"}, {"tactic": "intro hs", "annotated_tactic": ["intro hs", []], "state_before": "case mpr\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u\nS : Type v\nr : R\ne : \u2115\nn m : \u03c3\ninst\u271d : CommSemiring R\np q : MvPolynomial \u03c3 R\ns t : Set \u03c3\n\u22a2 \u2191(vars p) \u2286 s \u2192 \u2203 x, \u2191(rename Subtype.val) x = p", "state_after": "case mpr\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u\nS : Type v\nr : R\ne : \u2115\nn m : \u03c3\ninst\u271d : CommSemiring R\np q : MvPolynomial \u03c3 R\ns t : Set \u03c3\nhs : \u2191(vars p) \u2286 s\n\u22a2 \u2203 x, \u2191(rename Subtype.val) x = p"}, {"tactic": "exact exists_rename_eq_of_vars_subset_range p ((\u2191) : s \u2192 \u03c3) Subtype.val_injective (by simpa)", "annotated_tactic": ["exact <a>exists_rename_eq_of_vars_subset_range</a> p ((\u2191) : s \u2192 \u03c3) <a>Subtype.val_injective</a> (by simpa)", [{"full_name": "MvPolynomial.exists_rename_eq_of_vars_subset_range", "def_path": "Mathlib/Data/MvPolynomial/Variables.lean", "def_pos": [854, 9], "def_end_pos": [854, 46]}, {"full_name": "Subtype.val_injective", "def_path": "Mathlib/Data/Subtype.lean", "def_pos": [122, 9], "def_end_pos": [122, 22]}]], "state_before": "case mpr\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u\nS : Type v\nr : R\ne : \u2115\nn m : \u03c3\ninst\u271d : CommSemiring R\np q : MvPolynomial \u03c3 R\ns t : Set \u03c3\nhs : \u2191(vars p) \u2286 s\n\u22a2 \u2203 x, \u2191(rename Subtype.val) x = p", "state_after": "no goals"}, {"tactic": "simpa", "annotated_tactic": ["simpa", []], "state_before": "\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u\nS : Type v\nr : R\ne : \u2115\nn m : \u03c3\ninst\u271d : CommSemiring R\np q : MvPolynomial \u03c3 R\ns t : Set \u03c3\nhs : \u2191(vars p) \u2286 s\n\u22a2 \u2191(vars p) \u2286 Set.range Subtype.val", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "full_name": "MeasureTheory.SimpleFunc.restrict_lintegral_eq_lintegral_restrict", "start": [1068, 1], "end": [1070, 51], "traced_tactics": [{"tactic": "rw [f.restrict_lintegral hs, lintegral_restrict]", "annotated_tactic": ["rw [f.restrict_lintegral hs, <a>lintegral_restrict</a>]", [{"full_name": "MeasureTheory.SimpleFunc.lintegral_restrict", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [1063, 9], "def_end_pos": [1063, 27]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192\u209b \u211d\u22650\u221e\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 lintegral (restrict f s) \u03bc = lintegral f (Measure.restrict \u03bc s)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/ProbabilityMassFunction/Constructions.lean", "full_name": "PMF.filter_apply", "start": [273, 1], "end": [275, 31], "traced_tactics": [{"tactic": "rw [filter, normalize_apply]", "annotated_tactic": ["rw [<a>filter</a>, <a>normalize_apply</a>]", [{"full_name": "PMF.filter", "def_path": "Mathlib/Probability/ProbabilityMassFunction/Constructions.lean", "def_pos": [266, 5], "def_end_pos": [266, 11]}, {"full_name": "PMF.normalize_apply", "def_path": "Mathlib/Probability/ProbabilityMassFunction/Constructions.lean", "def_pos": [250, 9], "def_end_pos": [250, 24]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\ns : Set \u03b1\nh : \u2203 a, a \u2208 s \u2227 a \u2208 support p\na : \u03b1\n\u22a2 \u2191(filter p s h) a = Set.indicator s (\u2191p) a * (\u2211' (a' : \u03b1), Set.indicator s (\u2191p) a')\u207b\u00b9", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/Basic.lean", "full_name": "MvPolynomial.coeff_X'", "start": [691, 1], "end": [693, 30], "traced_tactics": [{"tactic": "rw [\u2190 coeff_X_pow, pow_one]", "annotated_tactic": ["rw [\u2190 <a>coeff_X_pow</a>, <a>pow_one</a>]", [{"full_name": "MvPolynomial.coeff_X_pow", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [683, 9], "def_end_pos": [683, 20]}, {"full_name": "pow_one", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [97, 9], "def_end_pos": [97, 16]}]], "state_before": "R : Type u\nS\u2081 : Type v\nS\u2082 : Type w\nS\u2083 : Type x\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\nn m\u271d : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\u2081\np q : MvPolynomial \u03c3 R\ninst\u271d : DecidableEq \u03c3\ni : \u03c3\nm : \u03c3 \u2192\u2080 \u2115\n\u22a2 coeff m (X i) = if (fun\u2080 | i => 1) = m then 1 else 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finmap.lean", "full_name": "Finmap.mem_insert", "start": [493, 1], "end": [494, 34], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Content.lean", "full_name": "MeasureTheory.Content.measure_eq_content_of_regular", "start": [437, 1], "end": [453, 39], "traced_tactics": [{"tactic": "refine' le_antisymm _ _", "annotated_tactic": ["refine' <a>le_antisymm</a> _ _", [{"full_name": "le_antisymm", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [188, 9], "def_end_pos": [188, 20]}]], "state_before": "G : Type w\ninst\u271d\u00b3 : TopologicalSpace G\n\u03bc : Content G\ninst\u271d\u00b2 : MeasurableSpace G\ninst\u271d\u00b9 : T2Space G\ninst\u271d : BorelSpace G\nH : ContentRegular \u03bc\nK : Compacts G\n\u22a2 \u2191\u2191(Content.measure \u03bc) \u2191K = (fun s => \u2191(toFun \u03bc s)) K", "state_after": "case refine'_1\nG : Type w\ninst\u271d\u00b3 : TopologicalSpace G\n\u03bc : Content G\ninst\u271d\u00b2 : MeasurableSpace G\ninst\u271d\u00b9 : T2Space G\ninst\u271d : BorelSpace G\nH : ContentRegular \u03bc\nK : Compacts G\n\u22a2 \u2191\u2191(Content.measure \u03bc) \u2191K \u2264 (fun s => \u2191(toFun \u03bc s)) K\n\ncase refine'_2\nG : Type w\ninst\u271d\u00b3 : TopologicalSpace G\n\u03bc : Content G\ninst\u271d\u00b2 : MeasurableSpace G\ninst\u271d\u00b9 : T2Space G\ninst\u271d : BorelSpace G\nH : ContentRegular \u03bc\nK : Compacts G\n\u22a2 (fun s => \u2191(toFun \u03bc s)) K \u2264 \u2191\u2191(Content.measure \u03bc) \u2191K"}, {"tactic": "apply ENNReal.le_of_forall_pos_le_add", "annotated_tactic": ["apply <a>ENNReal.le_of_forall_pos_le_add</a>", [{"full_name": "ENNReal.le_of_forall_pos_le_add", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [867, 9], "def_end_pos": [867, 32]}]], "state_before": "case refine'_1\nG : Type w\ninst\u271d\u00b3 : TopologicalSpace G\n\u03bc : Content G\ninst\u271d\u00b2 : MeasurableSpace G\ninst\u271d\u00b9 : T2Space G\ninst\u271d : BorelSpace G\nH : ContentRegular \u03bc\nK : Compacts G\n\u22a2 \u2191\u2191(Content.measure \u03bc) \u2191K \u2264 (fun s => \u2191(toFun \u03bc s)) K", "state_after": "case refine'_1.h\nG : Type w\ninst\u271d\u00b3 : TopologicalSpace G\n\u03bc : Content G\ninst\u271d\u00b2 : MeasurableSpace G\ninst\u271d\u00b9 : T2Space G\ninst\u271d : BorelSpace G\nH : ContentRegular \u03bc\nK : Compacts G\n\u22a2 \u2200 (\u03b5 : \u211d\u22650), 0 < \u03b5 \u2192 (fun s => \u2191(toFun \u03bc s)) K < \u22a4 \u2192 \u2191\u2191(Content.measure \u03bc) \u2191K \u2264 (fun s => \u2191(toFun \u03bc s)) K + \u2191\u03b5"}, {"tactic": "intro \u03b5 \u03b5pos _", "annotated_tactic": ["intro \u03b5 \u03b5pos _", []], "state_before": "case refine'_1.h\nG : Type w\ninst\u271d\u00b3 : TopologicalSpace G\n\u03bc : Content G\ninst\u271d\u00b2 : MeasurableSpace G\ninst\u271d\u00b9 : T2Space G\ninst\u271d : BorelSpace G\nH : ContentRegular \u03bc\nK : Compacts G\n\u22a2 \u2200 (\u03b5 : \u211d\u22650), 0 < \u03b5 \u2192 (fun s => \u2191(toFun \u03bc s)) K < \u22a4 \u2192 \u2191\u2191(Content.measure \u03bc) \u2191K \u2264 (fun s => \u2191(toFun \u03bc s)) K + \u2191\u03b5", "state_after": "case refine'_1.h\nG : Type w\ninst\u271d\u00b3 : TopologicalSpace G\n\u03bc : Content G\ninst\u271d\u00b2 : MeasurableSpace G\ninst\u271d\u00b9 : T2Space G\ninst\u271d : BorelSpace G\nH : ContentRegular \u03bc\nK : Compacts G\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\na\u271d : (fun s => \u2191(toFun \u03bc s)) K < \u22a4\n\u22a2 \u2191\u2191(Content.measure \u03bc) \u2191K \u2264 (fun s => \u2191(toFun \u03bc s)) K + \u2191\u03b5"}, {"tactic": "obtain \u27e8K', K'_hyp\u27e9 := contentRegular_exists_compact \u03bc H K (ne_bot_of_gt \u03b5pos)", "annotated_tactic": ["obtain \u27e8K', K'_hyp\u27e9 := <a>contentRegular_exists_compact</a> \u03bc H K (<a>ne_bot_of_gt</a> \u03b5pos)", [{"full_name": "MeasureTheory.Content.contentRegular_exists_compact", "def_path": "Mathlib/MeasureTheory/Measure/Content.lean", "def_pos": [420, 9], "def_end_pos": [420, 38]}, {"full_name": "ne_bot_of_gt", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [320, 9], "def_end_pos": [320, 21]}]], "state_before": "case refine'_1.h\nG : Type w\ninst\u271d\u00b3 : TopologicalSpace G\n\u03bc : Content G\ninst\u271d\u00b2 : MeasurableSpace G\ninst\u271d\u00b9 : T2Space G\ninst\u271d : BorelSpace G\nH : ContentRegular \u03bc\nK : Compacts G\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\na\u271d : (fun s => \u2191(toFun \u03bc s)) K < \u22a4\n\u22a2 \u2191\u2191(Content.measure \u03bc) \u2191K \u2264 (fun s => \u2191(toFun \u03bc s)) K + \u2191\u03b5", "state_after": "case refine'_1.h.intro\nG : Type w\ninst\u271d\u00b3 : TopologicalSpace G\n\u03bc : Content G\ninst\u271d\u00b2 : MeasurableSpace G\ninst\u271d\u00b9 : T2Space G\ninst\u271d : BorelSpace G\nH : ContentRegular \u03bc\nK : Compacts G\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\na\u271d : (fun s => \u2191(toFun \u03bc s)) K < \u22a4\nK' : Compacts G\nK'_hyp : K.carrier \u2286 interior K'.carrier \u2227 (fun s => \u2191(toFun \u03bc s)) K' \u2264 (fun s => \u2191(toFun \u03bc s)) K + \u2191\u03b5\n\u22a2 \u2191\u2191(Content.measure \u03bc) \u2191K \u2264 (fun s => \u2191(toFun \u03bc s)) K + \u2191\u03b5"}, {"tactic": "calc\n  \u03bc.measure \u2191K \u2264 \u03bc.measure (interior \u2191K') := by\n    rw [\u03bc.measure_apply isOpen_interior.measurableSet,\n      \u03bc.measure_apply K.isCompact.measurableSet]\n    exact \u03bc.outerMeasure.mono K'_hyp.left\n  _ \u2264 \u03bc K' := by\n    rw [\u03bc.measure_apply (IsOpen.measurableSet isOpen_interior)]\n    exact \u03bc.outerMeasure_interior_compacts K'\n  _ \u2264 \u03bc K + \u03b5 := K'_hyp.right", "annotated_tactic": ["calc\n      \u03bc.measure \u2191K \u2264 \u03bc.measure (<a>interior</a> \u2191K') := by\n        rw [\u03bc.measure_apply isOpen_interior.measurableSet,\n          \u03bc.measure_apply K.isCompact.measurableSet]\n        exact \u03bc.outerMeasure.mono K'_hyp.left\n      _ \u2264 \u03bc K' := by\n        rw [\u03bc.measure_apply (<a>IsOpen.measurableSet</a> <a>isOpen_interior</a>)]\n        exact \u03bc.outerMeasure_interior_compacts K'\n      _ \u2264 \u03bc K + \u03b5 := K'_hyp.right", [{"full_name": "interior", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [288, 5], "def_end_pos": [288, 13]}, {"full_name": "IsOpen.measurableSet", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [316, 9], "def_end_pos": [316, 29]}, {"full_name": "isOpen_interior", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [298, 9], "def_end_pos": [298, 24]}]], "state_before": "case refine'_1.h.intro\nG : Type w\ninst\u271d\u00b3 : TopologicalSpace G\n\u03bc : Content G\ninst\u271d\u00b2 : MeasurableSpace G\ninst\u271d\u00b9 : T2Space G\ninst\u271d : BorelSpace G\nH : ContentRegular \u03bc\nK : Compacts G\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\na\u271d : (fun s => \u2191(toFun \u03bc s)) K < \u22a4\nK' : Compacts G\nK'_hyp : K.carrier \u2286 interior K'.carrier \u2227 (fun s => \u2191(toFun \u03bc s)) K' \u2264 (fun s => \u2191(toFun \u03bc s)) K + \u2191\u03b5\n\u22a2 \u2191\u2191(Content.measure \u03bc) \u2191K \u2264 (fun s => \u2191(toFun \u03bc s)) K + \u2191\u03b5", "state_after": "no goals"}, {"tactic": "rw [\u03bc.measure_apply isOpen_interior.measurableSet,\n  \u03bc.measure_apply K.isCompact.measurableSet]", "annotated_tactic": ["rw [\u03bc.measure_apply isOpen_interior.measurableSet,\n          \u03bc.measure_apply K.isCompact.measurableSet]", []], "state_before": "G : Type w\ninst\u271d\u00b3 : TopologicalSpace G\n\u03bc : Content G\ninst\u271d\u00b2 : MeasurableSpace G\ninst\u271d\u00b9 : T2Space G\ninst\u271d : BorelSpace G\nH : ContentRegular \u03bc\nK : Compacts G\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\na\u271d : (fun s => \u2191(toFun \u03bc s)) K < \u22a4\nK' : Compacts G\nK'_hyp : K.carrier \u2286 interior K'.carrier \u2227 (fun s => \u2191(toFun \u03bc s)) K' \u2264 (fun s => \u2191(toFun \u03bc s)) K + \u2191\u03b5\n\u22a2 \u2191\u2191(Content.measure \u03bc) \u2191K \u2264 \u2191\u2191(Content.measure \u03bc) (interior \u2191K')", "state_after": "G : Type w\ninst\u271d\u00b3 : TopologicalSpace G\n\u03bc : Content G\ninst\u271d\u00b2 : MeasurableSpace G\ninst\u271d\u00b9 : T2Space G\ninst\u271d : BorelSpace G\nH : ContentRegular \u03bc\nK : Compacts G\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\na\u271d : (fun s => \u2191(toFun \u03bc s)) K < \u22a4\nK' : Compacts G\nK'_hyp : K.carrier \u2286 interior K'.carrier \u2227 (fun s => \u2191(toFun \u03bc s)) K' \u2264 (fun s => \u2191(toFun \u03bc s)) K + \u2191\u03b5\n\u22a2 \u2191(Content.outerMeasure \u03bc) \u2191K \u2264 \u2191(Content.outerMeasure \u03bc) (interior \u2191K')"}, {"tactic": "exact \u03bc.outerMeasure.mono K'_hyp.left", "annotated_tactic": ["exact \u03bc.outerMeasure.mono K'_hyp.left", []], "state_before": "G : Type w\ninst\u271d\u00b3 : TopologicalSpace G\n\u03bc : Content G\ninst\u271d\u00b2 : MeasurableSpace G\ninst\u271d\u00b9 : T2Space G\ninst\u271d : BorelSpace G\nH : ContentRegular \u03bc\nK : Compacts G\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\na\u271d : (fun s => \u2191(toFun \u03bc s)) K < \u22a4\nK' : Compacts G\nK'_hyp : K.carrier \u2286 interior K'.carrier \u2227 (fun s => \u2191(toFun \u03bc s)) K' \u2264 (fun s => \u2191(toFun \u03bc s)) K + \u2191\u03b5\n\u22a2 \u2191(Content.outerMeasure \u03bc) \u2191K \u2264 \u2191(Content.outerMeasure \u03bc) (interior \u2191K')", "state_after": "no goals"}, {"tactic": "rw [\u03bc.measure_apply (IsOpen.measurableSet isOpen_interior)]", "annotated_tactic": ["rw [\u03bc.measure_apply (<a>IsOpen.measurableSet</a> <a>isOpen_interior</a>)]", [{"full_name": "IsOpen.measurableSet", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [316, 9], "def_end_pos": [316, 29]}, {"full_name": "isOpen_interior", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [298, 9], "def_end_pos": [298, 24]}]], "state_before": "G : Type w\ninst\u271d\u00b3 : TopologicalSpace G\n\u03bc : Content G\ninst\u271d\u00b2 : MeasurableSpace G\ninst\u271d\u00b9 : T2Space G\ninst\u271d : BorelSpace G\nH : ContentRegular \u03bc\nK : Compacts G\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\na\u271d : (fun s => \u2191(toFun \u03bc s)) K < \u22a4\nK' : Compacts G\nK'_hyp : K.carrier \u2286 interior K'.carrier \u2227 (fun s => \u2191(toFun \u03bc s)) K' \u2264 (fun s => \u2191(toFun \u03bc s)) K + \u2191\u03b5\n\u22a2 \u2191\u2191(Content.measure \u03bc) (interior \u2191K') \u2264 (fun s => \u2191(toFun \u03bc s)) K'", "state_after": "G : Type w\ninst\u271d\u00b3 : TopologicalSpace G\n\u03bc : Content G\ninst\u271d\u00b2 : MeasurableSpace G\ninst\u271d\u00b9 : T2Space G\ninst\u271d : BorelSpace G\nH : ContentRegular \u03bc\nK : Compacts G\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\na\u271d : (fun s => \u2191(toFun \u03bc s)) K < \u22a4\nK' : Compacts G\nK'_hyp : K.carrier \u2286 interior K'.carrier \u2227 (fun s => \u2191(toFun \u03bc s)) K' \u2264 (fun s => \u2191(toFun \u03bc s)) K + \u2191\u03b5\n\u22a2 \u2191(Content.outerMeasure \u03bc) (interior \u2191K') \u2264 (fun s => \u2191(toFun \u03bc s)) K'"}, {"tactic": "exact \u03bc.outerMeasure_interior_compacts K'", "annotated_tactic": ["exact \u03bc.outerMeasure_interior_compacts K'", []], "state_before": "G : Type w\ninst\u271d\u00b3 : TopologicalSpace G\n\u03bc : Content G\ninst\u271d\u00b2 : MeasurableSpace G\ninst\u271d\u00b9 : T2Space G\ninst\u271d : BorelSpace G\nH : ContentRegular \u03bc\nK : Compacts G\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\na\u271d : (fun s => \u2191(toFun \u03bc s)) K < \u22a4\nK' : Compacts G\nK'_hyp : K.carrier \u2286 interior K'.carrier \u2227 (fun s => \u2191(toFun \u03bc s)) K' \u2264 (fun s => \u2191(toFun \u03bc s)) K + \u2191\u03b5\n\u22a2 \u2191(Content.outerMeasure \u03bc) (interior \u2191K') \u2264 (fun s => \u2191(toFun \u03bc s)) K'", "state_after": "no goals"}, {"tactic": "rw [\u03bc.measure_apply (IsCompact.measurableSet K.isCompact)]", "annotated_tactic": ["rw [\u03bc.measure_apply (<a>IsCompact.measurableSet</a> K.isCompact)]", [{"full_name": "IsCompact.measurableSet", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [344, 9], "def_end_pos": [344, 32]}]], "state_before": "case refine'_2\nG : Type w\ninst\u271d\u00b3 : TopologicalSpace G\n\u03bc : Content G\ninst\u271d\u00b2 : MeasurableSpace G\ninst\u271d\u00b9 : T2Space G\ninst\u271d : BorelSpace G\nH : ContentRegular \u03bc\nK : Compacts G\n\u22a2 (fun s => \u2191(toFun \u03bc s)) K \u2264 \u2191\u2191(Content.measure \u03bc) \u2191K", "state_after": "case refine'_2\nG : Type w\ninst\u271d\u00b3 : TopologicalSpace G\n\u03bc : Content G\ninst\u271d\u00b2 : MeasurableSpace G\ninst\u271d\u00b9 : T2Space G\ninst\u271d : BorelSpace G\nH : ContentRegular \u03bc\nK : Compacts G\n\u22a2 (fun s => \u2191(toFun \u03bc s)) K \u2264 \u2191(Content.outerMeasure \u03bc) \u2191K"}, {"tactic": "exact \u03bc.le_outerMeasure_compacts K", "annotated_tactic": ["exact \u03bc.le_outerMeasure_compacts K", []], "state_before": "case refine'_2\nG : Type w\ninst\u271d\u00b3 : TopologicalSpace G\n\u03bc : Content G\ninst\u271d\u00b2 : MeasurableSpace G\ninst\u271d\u00b9 : T2Space G\ninst\u271d : BorelSpace G\nH : ContentRegular \u03bc\nK : Compacts G\n\u22a2 (fun s => \u2191(toFun \u03bc s)) K \u2264 \u2191(Content.outerMeasure \u03bc) \u2191K", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/NullMeasurable.lean", "full_name": "MeasureTheory.nullMeasurableSet_insert", "start": [352, 1], "end": [354, 23], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "full_name": "MeasureTheory.L1.SimpleFunc.setToL1SCLM_add_left", "start": [919, 1], "end": [922, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/AEEqFun.lean", "full_name": "MeasureTheory.AEEqFun.lintegral_coeFn", "start": [916, 1], "end": [917, 32], "traced_tactics": [{"tactic": "rw [\u2190 lintegral_mk, mk_coeFn]", "annotated_tactic": ["rw [\u2190 <a>lintegral_mk</a>, <a>mk_coeFn</a>]", [{"full_name": "MeasureTheory.AEEqFun.lintegral_mk", "def_path": "Mathlib/MeasureTheory/Function/AEEqFun.lean", "def_pos": [912, 9], "def_end_pos": [912, 21]}, {"full_name": "MeasureTheory.AEEqFun.mk_coeFn", "def_path": "Mathlib/MeasureTheory/Function/AEEqFun.lean", "def_pos": [165, 9], "def_end_pos": [165, 17]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : TopologicalSpace \u03b4\nf : \u03b1 \u2192\u2098[\u03bc] \u211d\u22650\u221e\n\u22a2 \u222b\u207b (a : \u03b1), \u2191f a \u2202\u03bc = lintegral f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "full_name": "String.Iterator.ValidFor.hasPrev", "start": [589, 1], "end": [590, 68], "traced_tactics": [{"tactic": "simp [Iterator.hasPrev, h.pos, Nat.pos_iff_ne_zero]", "annotated_tactic": ["simp [<a>Iterator.hasPrev</a>, h.pos, <a>Nat.pos_iff_ne_zero</a>]", [{"full_name": "String.Iterator.hasPrev", "def_path": "lake-packages/lean4/src/lean/Init/Data/String/Basic.lean", "def_pos": [335, 5], "def_end_pos": [335, 12]}, {"full_name": "Nat.pos_iff_ne_zero", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [204, 19], "def_end_pos": [204, 34]}]], "state_before": "l r : List Char\nit : Iterator\nh : ValidFor l r it\n\u22a2 Iterator.hasPrev it = true \u2194 l \u2260 []", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Nat/Gcd.lean", "full_name": "Nat.gcd_mul_gcd_of_coprime_of_mul_eq_mul", "start": [426, 1], "end": [434, 56], "traced_tactics": [{"tactic": "apply Nat.dvd_antisymm", "annotated_tactic": ["apply <a>Nat.dvd_antisymm</a>", [{"full_name": "Nat.dvd_antisymm", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [918, 19], "def_end_pos": [918, 31]}]], "state_before": "c d a b : Nat\ncop : Coprime c d\nh : a * b = c * d\n\u22a2 gcd a c * gcd b c = c", "state_after": "case a\nc d a b : Nat\ncop : Coprime c d\nh : a * b = c * d\n\u22a2 gcd a c * gcd b c \u2223 c\n\ncase a\nc d a b : Nat\ncop : Coprime c d\nh : a * b = c * d\n\u22a2 c \u2223 gcd a c * gcd b c"}, {"tactic": "apply ((cop.gcd_left _).mul (cop.gcd_left _)).dvd_of_dvd_mul_right", "annotated_tactic": ["apply ((cop.gcd_left _).<a>mul</a> (cop.gcd_left _)).<a>dvd_of_dvd_mul_right</a>", [{"full_name": "Nat.Coprime.mul", "def_path": "lake-packages/std/Std/Data/Nat/Gcd.lean", "def_pos": [306, 9], "def_end_pos": [306, 20]}, {"full_name": "Nat.Coprime.dvd_of_dvd_mul_right", "def_path": "lake-packages/std/Std/Data/Nat/Gcd.lean", "def_pos": [263, 9], "def_end_pos": [263, 37]}]], "state_before": "case a\nc d a b : Nat\ncop : Coprime c d\nh : a * b = c * d\n\u22a2 gcd a c * gcd b c \u2223 c", "state_after": "case a\nc d a b : Nat\ncop : Coprime c d\nh : a * b = c * d\n\u22a2 gcd a c * gcd b c \u2223 c * d"}, {"tactic": "rw [\u2190 h]", "annotated_tactic": ["rw [\u2190 h]", []], "state_before": "case a\nc d a b : Nat\ncop : Coprime c d\nh : a * b = c * d\n\u22a2 gcd a c * gcd b c \u2223 c * d", "state_after": "case a\nc d a b : Nat\ncop : Coprime c d\nh : a * b = c * d\n\u22a2 gcd a c * gcd b c \u2223 a * b"}, {"tactic": "apply Nat.mul_dvd_mul (gcd_dvd ..).1 (gcd_dvd ..).1", "annotated_tactic": ["apply <a>Nat.mul_dvd_mul</a> (<a>gcd_dvd</a> ..).1 (<a>gcd_dvd</a> ..).1", [{"full_name": "Nat.mul_dvd_mul", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [893, 19], "def_end_pos": [893, 30]}, {"full_name": "Nat.gcd_dvd", "def_path": "lake-packages/std/Std/Data/Nat/Gcd.lean", "def_pos": [37, 9], "def_end_pos": [37, 16]}, {"full_name": "Nat.gcd_dvd", "def_path": "lake-packages/std/Std/Data/Nat/Gcd.lean", "def_pos": [37, 9], "def_end_pos": [37, 16]}]], "state_before": "case a\nc d a b : Nat\ncop : Coprime c d\nh : a * b = c * d\n\u22a2 gcd a c * gcd b c \u2223 a * b", "state_after": "no goals"}, {"tactic": "rw [gcd_comm a, gcd_comm b]", "annotated_tactic": ["rw [<a>gcd_comm</a> a, <a>gcd_comm</a> b]", [{"full_name": "Nat.gcd_comm", "def_path": "lake-packages/std/Std/Data/Nat/Gcd.lean", "def_pos": [59, 9], "def_end_pos": [59, 17]}, {"full_name": "Nat.gcd_comm", "def_path": "lake-packages/std/Std/Data/Nat/Gcd.lean", "def_pos": [59, 9], "def_end_pos": [59, 17]}]], "state_before": "case a\nc d a b : Nat\ncop : Coprime c d\nh : a * b = c * d\n\u22a2 c \u2223 gcd a c * gcd b c", "state_after": "case a\nc d a b : Nat\ncop : Coprime c d\nh : a * b = c * d\n\u22a2 c \u2223 gcd c a * gcd c b"}, {"tactic": "refine Nat.dvd_trans ?_ (gcd_mul_dvd_mul_gcd ..)", "annotated_tactic": ["refine <a>Nat.dvd_trans</a> ?_ (<a>gcd_mul_dvd_mul_gcd</a> ..)", [{"full_name": "Nat.dvd_trans", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [870, 19], "def_end_pos": [870, 28]}, {"full_name": "Nat.gcd_mul_dvd_mul_gcd", "def_path": "lake-packages/std/Std/Data/Nat/Gcd.lean", "def_pos": [410, 9], "def_end_pos": [410, 28]}]], "state_before": "case a\nc d a b : Nat\ncop : Coprime c d\nh : a * b = c * d\n\u22a2 c \u2223 gcd c a * gcd c b", "state_after": "case a\nc d a b : Nat\ncop : Coprime c d\nh : a * b = c * d\n\u22a2 c \u2223 gcd c (a * b)"}, {"tactic": "rw [h, gcd_mul_right_right d c]", "annotated_tactic": ["rw [h, <a>gcd_mul_right_right</a> d c]", [{"full_name": "Nat.gcd_mul_right_right", "def_path": "lake-packages/std/Std/Data/Nat/Gcd.lean", "def_pos": [159, 17], "def_end_pos": [159, 36]}]], "state_before": "case a\nc d a b : Nat\ncop : Coprime c d\nh : a * b = c * d\n\u22a2 c \u2223 gcd c (a * b)", "state_after": "case a\nc d a b : Nat\ncop : Coprime c d\nh : a * b = c * d\n\u22a2 c \u2223 c"}, {"tactic": "apply Nat.dvd_refl", "annotated_tactic": ["apply <a>Nat.dvd_refl</a>", [{"full_name": "Nat.dvd_refl", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [862, 19], "def_end_pos": [862, 27]}]], "state_before": "case a\nc d a b : Nat\ncop : Coprime c d\nh : a * b = c * d\n\u22a2 c \u2223 c", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/TuringMachine.lean", "full_name": "Turing.TM2to1.trCfg_init", "start": [2723, 1], "end": [2743, 8], "traced_tactics": [{"tactic": "rw [(_ : TM1.init _ = _)]", "annotated_tactic": ["rw [(_ : <a>TM1.init</a> _ = _)]", [{"full_name": "Turing.TM1.init", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1400, 5], "def_end_pos": [1400, 9]}]], "state_before": "K : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nk : K\nL : List (\u0393 k)\n\u22a2 TrCfg (TM2.init k L) (TM1.init (trInit k L))", "state_after": "K : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nk : K\nL : List (\u0393 k)\n\u22a2 TrCfg (TM2.init k L) ?m.742315\n\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nk : K\nL : List (\u0393 k)\n\u22a2 TM1.Cfg \u0393' \u039b' \u03c3\n\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nk : K\nL : List (\u0393 k)\n\u22a2 TM1.init (trInit k L) = ?m.742315"}, {"tactic": "refine' \u27e8ListBlank.mk (L.reverse.map fun a \u21a6 update default k (some a)), fun k' \u21a6 _\u27e9", "annotated_tactic": ["refine' \u27e8<a>ListBlank.mk</a> (L.reverse.map fun a \u21a6 <a>update</a> <a>default</a> k (<a>some</a> a)), fun k' \u21a6 _\u27e9", [{"full_name": "Turing.ListBlank.mk", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [198, 5], "def_end_pos": [198, 17]}, {"full_name": "Function.update", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [550, 5], "def_end_pos": [550, 11]}, {"full_name": "Inhabited.default", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [674, 3], "def_end_pos": [674, 10]}, {"full_name": "Option.some", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2143, 5], "def_end_pos": [2143, 9]}]], "state_before": "K : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nk : K\nL : List (\u0393 k)\n\u22a2 TrCfg (TM2.init k L) ?m.742315", "state_after": "K : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nk : K\nL : List (\u0393 k)\nk' : K\n\u22a2 ListBlank.map (proj k') (ListBlank.mk (List.map (fun a => update default k (some a)) (List.reverse L))) =\n    ListBlank.mk (List.reverse (List.map some (update (fun x => []) k L k')))"}, {"tactic": "refine' ListBlank.ext fun i \u21a6 _", "annotated_tactic": ["refine' <a>ListBlank.ext</a> fun i \u21a6 _", [{"full_name": "Turing.ListBlank.ext", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [312, 9], "def_end_pos": [312, 22]}]], "state_before": "K : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nk : K\nL : List (\u0393 k)\nk' : K\n\u22a2 ListBlank.map (proj k') (ListBlank.mk (List.map (fun a => update default k (some a)) (List.reverse L))) =\n    ListBlank.mk (List.reverse (List.map some (update (fun x => []) k L k')))", "state_after": "K : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nk : K\nL : List (\u0393 k)\nk' : K\ni : \u2115\n\u22a2 ListBlank.nth\n      (ListBlank.map (proj k') (ListBlank.mk (List.map (fun a => update default k (some a)) (List.reverse L)))) i =\n    ListBlank.nth (ListBlank.mk (List.reverse (List.map some (update (fun x => []) k L k')))) i"}, {"tactic": "rw [ListBlank.map_mk, ListBlank.nth_mk, List.getI_eq_iget_get?, List.map_map]", "annotated_tactic": ["rw [<a>ListBlank.map_mk</a>, <a>ListBlank.nth_mk</a>, <a>List.getI_eq_iget_get?</a>, <a>List.map_map</a>]", [{"full_name": "Turing.ListBlank.map_mk", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [389, 9], "def_end_pos": [389, 25]}, {"full_name": "Turing.ListBlank.nth_mk", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [293, 9], "def_end_pos": [293, 25]}, {"full_name": "List.getI_eq_iget_get?", "def_path": "Mathlib/Data/List/Basic.lean", "def_pos": [4416, 9], "def_end_pos": [4416, 26]}, {"full_name": "List.map_map", "def_path": "lake-packages/std/Std/Data/List/Init/Lemmas.lean", "def_pos": [96, 17], "def_end_pos": [96, 24]}]], "state_before": "K : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nk : K\nL : List (\u0393 k)\nk' : K\ni : \u2115\n\u22a2 ListBlank.nth\n      (ListBlank.map (proj k') (ListBlank.mk (List.map (fun a => update default k (some a)) (List.reverse L)))) i =\n    ListBlank.nth (ListBlank.mk (List.reverse (List.map some (update (fun x => []) k L k')))) i", "state_after": "K : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nk : K\nL : List (\u0393 k)\nk' : K\ni : \u2115\n\u22a2 Option.iget (List.get? (List.map ((proj k').f \u2218 fun a => update default k (some a)) (List.reverse L)) i) =\n    ListBlank.nth (ListBlank.mk (List.reverse (List.map some (update (fun x => []) k L k')))) i"}, {"tactic": "have : ((proj k').f \u2218 fun a => update (\u03b2 := fun k => Option (\u0393 k)) default k (some a))\n  = fun a => (proj k').f (update (\u03b2 := fun k => Option (\u0393 k)) default k (some a)) := rfl", "annotated_tactic": ["have : ((<a>proj</a> k').<a>f</a> \u2218 fun a => <a>update</a> (\u03b2 := fun k => <a>Option</a> (\u0393 k)) <a>default</a> k (<a>some</a> a))\n      = fun a => (<a>proj</a> k').<a>f</a> (<a>update</a> (\u03b2 := fun k => <a>Option</a> (\u0393 k)) <a>default</a> k (<a>some</a> a)) := <a>rfl</a>", [{"full_name": "Turing.proj", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [428, 5], "def_end_pos": [428, 9]}, {"full_name": "Turing.PointedMap.f", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [351, 3], "def_end_pos": [351, 4]}, {"full_name": "Function.update", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [550, 5], "def_end_pos": [550, 11]}, {"full_name": "Option", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2139, 11], "def_end_pos": [2139, 17]}, {"full_name": "Inhabited.default", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [674, 3], "def_end_pos": [674, 10]}, {"full_name": "Option.some", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2143, 5], "def_end_pos": [2143, 9]}, {"full_name": "Turing.proj", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [428, 5], "def_end_pos": [428, 9]}, {"full_name": "Turing.PointedMap.f", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [351, 3], "def_end_pos": [351, 4]}, {"full_name": "Function.update", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [550, 5], "def_end_pos": [550, 11]}, {"full_name": "Option", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2139, 11], "def_end_pos": [2139, 17]}, {"full_name": "Inhabited.default", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [674, 3], "def_end_pos": [674, 10]}, {"full_name": "Option.some", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2143, 5], "def_end_pos": [2143, 9]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "K : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nk : K\nL : List (\u0393 k)\nk' : K\ni : \u2115\n\u22a2 Option.iget (List.get? (List.map ((proj k').f \u2218 fun a => update default k (some a)) (List.reverse L)) i) =\n    ListBlank.nth (ListBlank.mk (List.reverse (List.map some (update (fun x => []) k L k')))) i", "state_after": "K : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nk : K\nL : List (\u0393 k)\nk' : K\ni : \u2115\nthis : ((proj k').f \u2218 fun a => update default k (some a)) = fun a => PointedMap.f (proj k') (update default k (some a))\n\u22a2 Option.iget (List.get? (List.map ((proj k').f \u2218 fun a => update default k (some a)) (List.reverse L)) i) =\n    ListBlank.nth (ListBlank.mk (List.reverse (List.map some (update (fun x => []) k L k')))) i"}, {"tactic": "rw [this, List.get?_map, proj, PointedMap.mk_val]", "annotated_tactic": ["rw [this, <a>List.get?_map</a>, <a>proj</a>, <a>PointedMap.mk_val</a>]", [{"full_name": "List.get?_map", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [655, 17], "def_end_pos": [655, 25]}, {"full_name": "Turing.proj", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [428, 5], "def_end_pos": [428, 9]}, {"full_name": "Turing.PointedMap.mk_val", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [362, 9], "def_end_pos": [362, 26]}]], "state_before": "K : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nk : K\nL : List (\u0393 k)\nk' : K\ni : \u2115\nthis : ((proj k').f \u2218 fun a => update default k (some a)) = fun a => PointedMap.f (proj k') (update default k (some a))\n\u22a2 Option.iget (List.get? (List.map ((proj k').f \u2218 fun a => update default k (some a)) (List.reverse L)) i) =\n    ListBlank.nth (ListBlank.mk (List.reverse (List.map some (update (fun x => []) k L k')))) i", "state_after": "K : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nk : K\nL : List (\u0393 k)\nk' : K\ni : \u2115\nthis : ((proj k').f \u2218 fun a => update default k (some a)) = fun a => PointedMap.f (proj k') (update default k (some a))\n\u22a2 Option.iget (Option.map (fun a => update default k (some a) k') (List.get? (List.reverse L) i)) =\n    ListBlank.nth (ListBlank.mk (List.reverse (List.map some (update (fun x => []) k L k')))) i"}, {"tactic": "by_cases h : k' = k", "annotated_tactic": ["by_cases h : k' = k", []], "state_before": "K : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nk : K\nL : List (\u0393 k)\nk' : K\ni : \u2115\nthis : ((proj k').f \u2218 fun a => update default k (some a)) = fun a => PointedMap.f (proj k') (update default k (some a))\n\u22a2 Option.iget (Option.map (fun a => update default k (some a) k') (List.get? (List.reverse L) i)) =\n    ListBlank.nth (ListBlank.mk (List.reverse (List.map some (update (fun x => []) k L k')))) i", "state_after": "case pos\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nk : K\nL : List (\u0393 k)\nk' : K\ni : \u2115\nthis : ((proj k').f \u2218 fun a => update default k (some a)) = fun a => PointedMap.f (proj k') (update default k (some a))\nh : k' = k\n\u22a2 Option.iget (Option.map (fun a => update default k (some a) k') (List.get? (List.reverse L) i)) =\n    ListBlank.nth (ListBlank.mk (List.reverse (List.map some (update (fun x => []) k L k')))) i\n\ncase neg\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nk : K\nL : List (\u0393 k)\nk' : K\ni : \u2115\nthis : ((proj k').f \u2218 fun a => update default k (some a)) = fun a => PointedMap.f (proj k') (update default k (some a))\nh : \u00ack' = k\n\u22a2 Option.iget (Option.map (fun a => update default k (some a) k') (List.get? (List.reverse L) i)) =\n    ListBlank.nth (ListBlank.mk (List.reverse (List.map some (update (fun x => []) k L k')))) i"}, {"tactic": "subst k'", "annotated_tactic": ["subst k'", []], "state_before": "case pos\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nk : K\nL : List (\u0393 k)\nk' : K\ni : \u2115\nthis : ((proj k').f \u2218 fun a => update default k (some a)) = fun a => PointedMap.f (proj k') (update default k (some a))\nh : k' = k\n\u22a2 Option.iget (Option.map (fun a => update default k (some a) k') (List.get? (List.reverse L) i)) =\n    ListBlank.nth (ListBlank.mk (List.reverse (List.map some (update (fun x => []) k L k')))) i", "state_after": "case pos\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nk : K\nL : List (\u0393 k)\ni : \u2115\nthis : ((proj k).f \u2218 fun a => update default k (some a)) = fun a => PointedMap.f (proj k) (update default k (some a))\n\u22a2 Option.iget (Option.map (fun a => update default k (some a) k) (List.get? (List.reverse L) i)) =\n    ListBlank.nth (ListBlank.mk (List.reverse (List.map some (update (fun x => []) k L k)))) i"}, {"tactic": "simp only [Function.update_same]", "annotated_tactic": ["simp only [<a>Function.update_same</a>]", [{"full_name": "Function.update_same", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [555, 9], "def_end_pos": [555, 20]}]], "state_before": "case pos\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nk : K\nL : List (\u0393 k)\ni : \u2115\nthis : ((proj k).f \u2218 fun a => update default k (some a)) = fun a => PointedMap.f (proj k) (update default k (some a))\n\u22a2 Option.iget (Option.map (fun a => update default k (some a) k) (List.get? (List.reverse L) i)) =\n    ListBlank.nth (ListBlank.mk (List.reverse (List.map some (update (fun x => []) k L k)))) i", "state_after": "case pos\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nk : K\nL : List (\u0393 k)\ni : \u2115\nthis : ((proj k).f \u2218 fun a => update default k (some a)) = fun a => PointedMap.f (proj k) (update default k (some a))\n\u22a2 Option.iget (Option.map (fun a => some a) (List.get? (List.reverse L) i)) =\n    ListBlank.nth (ListBlank.mk (List.reverse (List.map some L))) i"}, {"tactic": "rw [ListBlank.nth_mk, List.getI_eq_iget_get?, \u2190 List.map_reverse, List.get?_map]", "annotated_tactic": ["rw [<a>ListBlank.nth_mk</a>, <a>List.getI_eq_iget_get?</a>, \u2190 <a>List.map_reverse</a>, <a>List.get?_map</a>]", [{"full_name": "Turing.ListBlank.nth_mk", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [293, 9], "def_end_pos": [293, 25]}, {"full_name": "List.getI_eq_iget_get?", "def_path": "Mathlib/Data/List/Basic.lean", "def_pos": [4416, 9], "def_end_pos": [4416, 26]}, {"full_name": "List.map_reverse", "def_path": "Mathlib/Data/List/Basic.lean", "def_pos": [618, 9], "def_end_pos": [618, 20]}, {"full_name": "List.get?_map", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [655, 17], "def_end_pos": [655, 25]}]], "state_before": "case pos\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nk : K\nL : List (\u0393 k)\ni : \u2115\nthis : ((proj k).f \u2218 fun a => update default k (some a)) = fun a => PointedMap.f (proj k) (update default k (some a))\n\u22a2 Option.iget (Option.map (fun a => some a) (List.get? (List.reverse L) i)) =\n    ListBlank.nth (ListBlank.mk (List.reverse (List.map some L))) i", "state_after": "no goals"}, {"tactic": "simp only [Function.update_noteq h]", "annotated_tactic": ["simp only [<a>Function.update_noteq</a> h]", [{"full_name": "Function.update_noteq", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [560, 9], "def_end_pos": [560, 21]}]], "state_before": "case neg\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nk : K\nL : List (\u0393 k)\nk' : K\ni : \u2115\nthis : ((proj k').f \u2218 fun a => update default k (some a)) = fun a => PointedMap.f (proj k') (update default k (some a))\nh : \u00ack' = k\n\u22a2 Option.iget (Option.map (fun a => update default k (some a) k') (List.get? (List.reverse L) i)) =\n    ListBlank.nth (ListBlank.mk (List.reverse (List.map some (update (fun x => []) k L k')))) i", "state_after": "case neg\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nk : K\nL : List (\u0393 k)\nk' : K\ni : \u2115\nthis : ((proj k').f \u2218 fun a => update default k (some a)) = fun a => PointedMap.f (proj k') (update default k (some a))\nh : \u00ack' = k\n\u22a2 Option.iget (Option.map (fun a => default k') (List.get? (List.reverse L) i)) =\n    ListBlank.nth (ListBlank.mk (List.reverse (List.map some []))) i"}, {"tactic": "rw [ListBlank.nth_mk, List.getI_eq_iget_get?, List.map, List.reverse_nil]", "annotated_tactic": ["rw [<a>ListBlank.nth_mk</a>, <a>List.getI_eq_iget_get?</a>, <a>List.map</a>, <a>List.reverse_nil</a>]", [{"full_name": "Turing.ListBlank.nth_mk", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [293, 9], "def_end_pos": [293, 25]}, {"full_name": "List.getI_eq_iget_get?", "def_path": "Mathlib/Data/List/Basic.lean", "def_pos": [4416, 9], "def_end_pos": [4416, 26]}, {"full_name": "List.map", "def_path": "lake-packages/lean4/src/lean/Init/Data/List/Basic.lean", "def_pos": [151, 19], "def_end_pos": [151, 22]}, {"full_name": "List.reverse_nil", "def_path": "lake-packages/lean4/src/lean/Init/Data/List/Basic.lean", "def_pos": [171, 17], "def_end_pos": [171, 28]}]], "state_before": "case neg\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nk : K\nL : List (\u0393 k)\nk' : K\ni : \u2115\nthis : ((proj k').f \u2218 fun a => update default k (some a)) = fun a => PointedMap.f (proj k') (update default k (some a))\nh : \u00ack' = k\n\u22a2 Option.iget (Option.map (fun a => default k') (List.get? (List.reverse L) i)) =\n    ListBlank.nth (ListBlank.mk (List.reverse (List.map some []))) i", "state_after": "case neg\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nk : K\nL : List (\u0393 k)\nk' : K\ni : \u2115\nthis : ((proj k').f \u2218 fun a => update default k (some a)) = fun a => PointedMap.f (proj k') (update default k (some a))\nh : \u00ack' = k\n\u22a2 Option.iget (Option.map (fun a => default k') (List.get? (List.reverse L) i)) = Option.iget (List.get? [] i)"}, {"tactic": "cases L.reverse.get? i <;> rfl", "annotated_tactic": ["cases L.reverse.get? i <;> rfl", []], "state_before": "case neg\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nk : K\nL : List (\u0393 k)\nk' : K\ni : \u2115\nthis : ((proj k').f \u2218 fun a => update default k (some a)) = fun a => PointedMap.f (proj k') (update default k (some a))\nh : \u00ack' = k\n\u22a2 Option.iget (Option.map (fun a => default k') (List.get? (List.reverse L) i)) = Option.iget (List.get? [] i)", "state_after": "no goals"}, {"tactic": "rw [trInit, TM1.init]", "annotated_tactic": ["rw [<a>trInit</a>, <a>TM1.init</a>]", [{"full_name": "Turing.TM2to1.trInit", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [2501, 5], "def_end_pos": [2501, 11]}, {"full_name": "Turing.TM1.init", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1400, 5], "def_end_pos": [1400, 9]}]], "state_before": "K : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nk : K\nL : List (\u0393 k)\n\u22a2 TM1.init (trInit k L) =\n    { l := Option.map normal (some default), var := default,\n      Tape := Tape.mk' \u2205 (addBottom (ListBlank.mk (List.map (fun a => update default k (some a)) (List.reverse L)))) }", "state_after": "K : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nk : K\nL : List (\u0393 k)\n\u22a2 { l := some default, var := default,\n      Tape :=\n        Tape.mk\u2081\n          (let L' := List.map (fun a => (false, update (fun x => none) k (some a))) (List.reverse L);\n          (true, (List.headI L').2) :: List.tail L') } =\n    { l := Option.map normal (some default), var := default,\n      Tape := Tape.mk' \u2205 (addBottom (ListBlank.mk (List.map (fun a => update default k (some a)) (List.reverse L)))) }"}, {"tactic": "dsimp only", "annotated_tactic": ["dsimp only", []], "state_before": "K : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nk : K\nL : List (\u0393 k)\n\u22a2 { l := some default, var := default,\n      Tape :=\n        Tape.mk\u2081\n          (let L' := List.map (fun a => (false, update (fun x => none) k (some a))) (List.reverse L);\n          (true, (List.headI L').2) :: List.tail L') } =\n    { l := Option.map normal (some default), var := default,\n      Tape := Tape.mk' \u2205 (addBottom (ListBlank.mk (List.map (fun a => update default k (some a)) (List.reverse L)))) }", "state_after": "K : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nk : K\nL : List (\u0393 k)\n\u22a2 { l := some default, var := default,\n      Tape :=\n        Tape.mk\u2081\n          ((true, (List.headI (List.map (fun a => (false, update (fun x => none) k (some a))) (List.reverse L))).2) ::\n            List.tail (List.map (fun a => (false, update (fun x => none) k (some a))) (List.reverse L))) } =\n    { l := Option.map normal (some default), var := default,\n      Tape := Tape.mk' \u2205 (addBottom (ListBlank.mk (List.map (fun a => update default k (some a)) (List.reverse L)))) }"}, {"tactic": "congr <;> cases L.reverse <;> try rfl", "annotated_tactic": ["congr <;> cases L.reverse <;> try rfl", []], "state_before": "K : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nk : K\nL : List (\u0393 k)\n\u22a2 { l := some default, var := default,\n      Tape :=\n        Tape.mk\u2081\n          ((true, (List.headI (List.map (fun a => (false, update (fun x => none) k (some a))) (List.reverse L))).2) ::\n            List.tail (List.map (fun a => (false, update (fun x => none) k (some a))) (List.reverse L))) } =\n    { l := Option.map normal (some default), var := default,\n      Tape := Tape.mk' \u2205 (addBottom (ListBlank.mk (List.map (fun a => update default k (some a)) (List.reverse L)))) }", "state_after": "case e_Tape.e_l.e_tail.cons\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nk : K\nL : List (\u0393 k)\nhead\u271d : \u0393 k\ntail\u271d : List (\u0393 k)\n\u22a2 List.tail (List.map (fun a => (false, update (fun x => none) k (some a))) (head\u271d :: tail\u271d)) =\n    List.map { f := Prod.mk false, map_pt' := (_ : (false, default) = (false, default)) }.f\n      (List.tail (List.map (fun a => update default k (some a)) (head\u271d :: tail\u271d)))"}, {"tactic": "simp only [List.map_map, List.tail_cons, List.map]", "annotated_tactic": ["simp only [<a>List.map_map</a>, <a>List.tail_cons</a>, <a>List.map</a>]", [{"full_name": "List.map_map", "def_path": "lake-packages/std/Std/Data/List/Init/Lemmas.lean", "def_pos": [96, 17], "def_end_pos": [96, 24]}, {"full_name": "List.tail_cons", "def_path": "lake-packages/std/Std/Data/List/Basic.lean", "def_pos": [316, 17], "def_end_pos": [316, 26]}, {"full_name": "List.map", "def_path": "lake-packages/lean4/src/lean/Init/Data/List/Basic.lean", "def_pos": [151, 19], "def_end_pos": [151, 22]}]], "state_before": "case e_Tape.e_l.e_tail.cons\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nk : K\nL : List (\u0393 k)\nhead\u271d : \u0393 k\ntail\u271d : List (\u0393 k)\n\u22a2 List.tail (List.map (fun a => (false, update (fun x => none) k (some a))) (head\u271d :: tail\u271d)) =\n    List.map { f := Prod.mk false, map_pt' := (_ : (false, default) = (false, default)) }.f\n      (List.tail (List.map (fun a => update default k (some a)) (head\u271d :: tail\u271d)))", "state_after": "case e_Tape.e_l.e_tail.cons\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nk : K\nL : List (\u0393 k)\nhead\u271d : \u0393 k\ntail\u271d : List (\u0393 k)\n\u22a2 List.map (fun a => (false, update (fun x => none) k (some a))) tail\u271d =\n    List.map (Prod.mk false \u2218 fun a => update default k (some a)) tail\u271d"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case e_Tape.e_l.e_tail.cons\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nk : K\nL : List (\u0393 k)\nhead\u271d : \u0393 k\ntail\u271d : List (\u0393 k)\n\u22a2 List.map (fun a => (false, update (fun x => none) k (some a))) tail\u271d =\n    List.map (Prod.mk false \u2218 fun a => update default k (some a)) tail\u271d", "state_after": "no goals"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case e_Tape.e_l.e_tail.nil\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nk : K\nL : List (\u0393 k)\n\u22a2 List.tail (List.map (fun a => (false, update (fun x => none) k (some a))) []) =\n    List.map { f := Prod.mk false, map_pt' := (_ : (false, default) = (false, default)) }.f\n      (List.tail (List.map (fun a => update default k (some a)) []))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Reduce.lean", "full_name": "OneOneReducible.of_equiv", "start": [108, 1], "end": [110, 37], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Vector/Mem.lean", "full_name": "Vector.not_mem_zero", "start": [44, 1], "end": [45, 41], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean", "full_name": "MeasureTheory.Measure.addHaar_affineSubspace", "start": [207, 1], "end": [215, 99], "traced_tactics": [{"tactic": "rcases s.eq_bot_or_nonempty with (rfl | hne)", "annotated_tactic": ["rcases s.eq_bot_or_nonempty with (rfl | hne)", []], "state_before": "E : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\ns : AffineSubspace \u211d E\nhs : s \u2260 \u22a4\n\u22a2 \u2191\u2191\u03bc \u2191s = 0", "state_after": "case inl\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : \u22a5 \u2260 \u22a4\n\u22a2 \u2191\u2191\u03bc \u2191\u22a5 = 0\n\ncase inr\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\ns : AffineSubspace \u211d E\nhs : s \u2260 \u22a4\nhne : Set.Nonempty \u2191s\n\u22a2 \u2191\u2191\u03bc \u2191s = 0"}, {"tactic": "rw [Ne.def, \u2190 AffineSubspace.direction_eq_top_iff_of_nonempty hne] at hs", "annotated_tactic": ["rw [<a>Ne.def</a>, \u2190 <a>AffineSubspace.direction_eq_top_iff_of_nonempty</a> hne] at hs", [{"full_name": "Ne.def", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [59, 9], "def_end_pos": [59, 15]}, {"full_name": "AffineSubspace.direction_eq_top_iff_of_nonempty", "def_path": "Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean", "def_pos": [887, 9], "def_end_pos": [887, 41]}]], "state_before": "case inr\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\ns : AffineSubspace \u211d E\nhs : s \u2260 \u22a4\nhne : Set.Nonempty \u2191s\n\u22a2 \u2191\u2191\u03bc \u2191s = 0", "state_after": "case inr\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\ns : AffineSubspace \u211d E\nhs : \u00acAffineSubspace.direction s = \u22a4\nhne : Set.Nonempty \u2191s\n\u22a2 \u2191\u2191\u03bc \u2191s = 0"}, {"tactic": "rcases hne with \u27e8x, hx : x \u2208 s\u27e9", "annotated_tactic": ["rcases hne with \u27e8x, hx : x \u2208 s\u27e9", []], "state_before": "case inr\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\ns : AffineSubspace \u211d E\nhs : \u00acAffineSubspace.direction s = \u22a4\nhne : Set.Nonempty \u2191s\n\u22a2 \u2191\u2191\u03bc \u2191s = 0", "state_after": "case inr.intro\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\ns : AffineSubspace \u211d E\nhs : \u00acAffineSubspace.direction s = \u22a4\nx : E\nhx : x \u2208 s\n\u22a2 \u2191\u2191\u03bc \u2191s = 0"}, {"tactic": "simpa only [AffineSubspace.coe_direction_eq_vsub_set_right hx, vsub_eq_sub, sub_eq_add_neg,\n  image_add_right, neg_neg, measure_preimage_add_right] using addHaar_submodule \u03bc s.direction hs", "annotated_tactic": ["simpa only [<a>AffineSubspace.coe_direction_eq_vsub_set_right</a> hx, <a>vsub_eq_sub</a>, <a>sub_eq_add_neg</a>,\n    <a>image_add_right</a>, <a>neg_neg</a>, <a>measure_preimage_add_right</a>] using <a>addHaar_submodule</a> \u03bc s.direction hs", [{"full_name": "AffineSubspace.coe_direction_eq_vsub_set_right", "def_path": "Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean", "def_pos": [292, 9], "def_end_pos": [292, 40]}, {"full_name": "vsub_eq_sub", "def_path": "Mathlib/Algebra/AddTorsor.lean", "def_pos": [76, 9], "def_end_pos": [76, 20]}, {"full_name": "sub_eq_add_neg", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [975, 3], "def_end_pos": [975, 14]}, {"full_name": "Set.image_add_right", "def_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "def_pos": [1204, 3], "def_end_pos": [1204, 14]}, {"full_name": "neg_neg", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [799, 3], "def_end_pos": [799, 14]}, {"full_name": "MeasureTheory.measure_preimage_add_right", "def_path": "Mathlib/MeasureTheory/Group/Measure.lean", "def_pos": [329, 3], "def_end_pos": [329, 14]}, {"full_name": "MeasureTheory.Measure.addHaar_submodule", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean", "def_pos": [179, 9], "def_end_pos": [179, 26]}]], "state_before": "case inr.intro\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\ns : AffineSubspace \u211d E\nhs : \u00acAffineSubspace.direction s = \u22a4\nx : E\nhx : x \u2208 s\n\u22a2 \u2191\u2191\u03bc \u2191s = 0", "state_after": "no goals"}, {"tactic": "rw [AffineSubspace.bot_coe, measure_empty]", "annotated_tactic": ["rw [<a>AffineSubspace.bot_coe</a>, <a>measure_empty</a>]", [{"full_name": "AffineSubspace.bot_coe", "def_path": "Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean", "def_pos": [780, 9], "def_end_pos": [780, 16]}, {"full_name": "MeasureTheory.measure_empty", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [185, 9], "def_end_pos": [185, 22]}]], "state_before": "case inl\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : \u22a5 \u2260 \u22a4\n\u22a2 \u2191\u2191\u03bc \u2191\u22a5 = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Constructions/Polish.lean", "full_name": "IsClosed.measurableSet_image_of_continuousOn_injOn", "start": [807, 1], "end": [815, 39], "traced_tactics": [{"tactic": "rw [image_eq_range]", "annotated_tactic": ["rw [<a>image_eq_range</a>]", [{"full_name": "Set.image_eq_range", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [1080, 9], "def_end_pos": [1080, 23]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\n\u03b2 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\ns : Set \u03b3\nhs : IsClosed s\nf : \u03b3 \u2192 \u03b2\nf_cont : ContinuousOn f s\nf_inj : InjOn f s\n\u22a2 MeasurableSet (f '' s)", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\n\u03b2 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\ns : Set \u03b3\nhs : IsClosed s\nf : \u03b3 \u2192 \u03b2\nf_cont : ContinuousOn f s\nf_inj : InjOn f s\n\u22a2 MeasurableSet (range fun x => f \u2191x)"}, {"tactic": "haveI : PolishSpace s := IsClosed.polishSpace hs", "annotated_tactic": ["haveI : <a>PolishSpace</a> s := <a>IsClosed.polishSpace</a> hs", [{"full_name": "PolishSpace", "def_path": "Mathlib/Topology/MetricSpace/Polish.lean", "def_pos": [65, 7], "def_end_pos": [65, 18]}, {"full_name": "IsClosed.polishSpace", "def_path": "Mathlib/Topology/MetricSpace/Polish.lean", "def_pos": [181, 9], "def_end_pos": [181, 36]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\n\u03b2 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\ns : Set \u03b3\nhs : IsClosed s\nf : \u03b3 \u2192 \u03b2\nf_cont : ContinuousOn f s\nf_inj : InjOn f s\n\u22a2 MeasurableSet (range fun x => f \u2191x)", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\n\u03b2 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\ns : Set \u03b3\nhs : IsClosed s\nf : \u03b3 \u2192 \u03b2\nf_cont : ContinuousOn f s\nf_inj : InjOn f s\nthis : PolishSpace \u2191s\n\u22a2 MeasurableSet (range fun x => f \u2191x)"}, {"tactic": "apply measurableSet_range_of_continuous_injective", "annotated_tactic": ["apply <a>measurableSet_range_of_continuous_injective</a>", [{"full_name": "MeasureTheory.measurableSet_range_of_continuous_injective", "def_path": "Mathlib/MeasureTheory/Constructions/Polish.lean", "def_pos": [656, 9], "def_end_pos": [656, 52]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\n\u03b2 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\ns : Set \u03b3\nhs : IsClosed s\nf : \u03b3 \u2192 \u03b2\nf_cont : ContinuousOn f s\nf_inj : InjOn f s\nthis : PolishSpace \u2191s\n\u22a2 MeasurableSet (range fun x => f \u2191x)", "state_after": "case f_cont\n\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\n\u03b2 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\ns : Set \u03b3\nhs : IsClosed s\nf : \u03b3 \u2192 \u03b2\nf_cont : ContinuousOn f s\nf_inj : InjOn f s\nthis : PolishSpace \u2191s\n\u22a2 Continuous fun x => f \u2191x\n\ncase f_inj\n\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\n\u03b2 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\ns : Set \u03b3\nhs : IsClosed s\nf : \u03b3 \u2192 \u03b2\nf_cont : ContinuousOn f s\nf_inj : InjOn f s\nthis : PolishSpace \u2191s\n\u22a2 Injective fun x => f \u2191x"}, {"tactic": "rwa [continuousOn_iff_continuous_restrict] at f_cont", "annotated_tactic": ["rwa [<a>continuousOn_iff_continuous_restrict</a>] at f_cont", [{"full_name": "continuousOn_iff_continuous_restrict", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [643, 9], "def_end_pos": [643, 45]}]], "state_before": "case f_cont\n\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\n\u03b2 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\ns : Set \u03b3\nhs : IsClosed s\nf : \u03b3 \u2192 \u03b2\nf_cont : ContinuousOn f s\nf_inj : InjOn f s\nthis : PolishSpace \u2191s\n\u22a2 Continuous fun x => f \u2191x", "state_after": "no goals"}, {"tactic": "rwa [injOn_iff_injective] at f_inj", "annotated_tactic": ["rwa [<a>injOn_iff_injective</a>] at f_inj", [{"full_name": "Set.injOn_iff_injective", "def_path": "Mathlib/Data/Set/Function.lean", "def_pos": [697, 9], "def_end_pos": [697, 28]}]], "state_before": "case f_inj\n\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\n\u03b2 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\ns : Set \u03b3\nhs : IsClosed s\nf : \u03b3 \u2192 \u03b2\nf_cont : ContinuousOn f s\nf_inj : InjOn f s\nthis : PolishSpace \u2191s\n\u22a2 Injective fun x => f \u2191x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Int/Interval.lean", "full_name": "Int.card_fintype_Ico_of_le", "start": [179, 1], "end": [180, 43], "traced_tactics": [{"tactic": "rw [card_fintype_Ico, toNat_sub_of_le h]", "annotated_tactic": ["rw [<a>card_fintype_Ico</a>, <a>toNat_sub_of_le</a> h]", [{"full_name": "Int.card_fintype_Ico", "def_path": "Mathlib/Data/Int/Interval.lean", "def_pos": [157, 9], "def_end_pos": [157, 25]}, {"full_name": "Int.toNat_sub_of_le", "def_path": "Mathlib/Data/Int/Order/Basic.lean", "def_pos": [547, 9], "def_end_pos": [547, 24]}]], "state_before": "a b : \u2124\nh : a \u2264 b\n\u22a2 \u2191(Fintype.card \u2191(Set.Ico a b)) = b - a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "full_name": "String.revPosOfAux_eq", "start": [330, 1], "end": [330, 78], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Card.lean", "full_name": "Set.two_lt_ncard", "start": [1070, 1], "end": [1072, 64], "traced_tactics": [{"tactic": "simp only [two_lt_ncard_iff hs, exists_and_left, exists_prop]", "annotated_tactic": ["simp only [<a>two_lt_ncard_iff</a> hs, <a>exists_and_left</a>, <a>exists_prop</a>]", [{"full_name": "Set.two_lt_ncard_iff", "def_path": "Mathlib/Data/Set/Card.lean", "def_pos": [1065, 9], "def_end_pos": [1065, 25]}, {"full_name": "exists_and_left", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [465, 17], "def_end_pos": [465, 32]}, {"full_name": "exists_prop", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [485, 17], "def_end_pos": [485, 28]}]], "state_before": "\u03b1 : Type u_1\ns t : Set \u03b1\nhs : autoParam (Set.Finite s) _auto\u271d\n\u22a2 2 < ncard s \u2194 \u2203 a, a \u2208 s \u2227 \u2203 b, b \u2208 s \u2227 \u2203 c, c \u2208 s \u2227 a \u2260 b \u2227 a \u2260 c \u2227 b \u2260 c", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Int/Bitwise.lean", "full_name": "Int.shiftLeft_sub", "start": [431, 1], "end": [432, 22], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Function.lean", "full_name": "Function.insert_injOn", "start": [1726, 1], "end": [1727, 20], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "full_name": "List.tail_eq_tail?", "start": [529, 1], "end": [529, 85], "traced_tactics": [{"tactic": "simp [tail_eq_tailD]", "annotated_tactic": ["simp [<a>tail_eq_tailD</a>]", [{"full_name": "List.tail_eq_tailD", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [527, 9], "def_end_pos": [527, 22]}]], "state_before": "\u03b1 : Type u_1\nl : List \u03b1\n\u22a2 tail l = Option.getD (tail? l) []", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/CondCount.lean", "full_name": "ProbabilityTheory.condCount_inter'", "start": [151, 1], "end": [154, 27], "traced_tactics": [{"tactic": "rw [\u2190 Set.inter_comm]", "annotated_tactic": ["rw [\u2190 <a>Set.inter_comm</a>]", [{"full_name": "Set.inter_comm", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [940, 9], "def_end_pos": [940, 19]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nhs : Set.Finite s\n\u22a2 \u2191\u2191(condCount s) (t \u2229 u) = \u2191\u2191(condCount (s \u2229 u)) t * \u2191\u2191(condCount s) u", "state_after": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nhs : Set.Finite s\n\u22a2 \u2191\u2191(condCount s) (u \u2229 t) = \u2191\u2191(condCount (s \u2229 u)) t * \u2191\u2191(condCount s) u"}, {"tactic": "exact condCount_inter hs", "annotated_tactic": ["exact <a>condCount_inter</a> hs", [{"full_name": "ProbabilityTheory.condCount_inter", "def_path": "Mathlib/Probability/CondCount.lean", "def_pos": [138, 9], "def_end_pos": [138, 24]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nhs : Set.Finite s\n\u22a2 \u2191\u2191(condCount s) (u \u2229 t) = \u2191\u2191(condCount (s \u2229 u)) t * \u2191\u2191(condCount s) u", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Kernel/Basic.lean", "full_name": "ProbabilityTheory.kernel.comapRight_apply'", "start": [577, 1], "end": [580, 83], "traced_tactics": [{"tactic": "rw [comapRight_apply,\n  Measure.comap_apply _ hf.injective (fun s => hf.measurableSet_image.mpr) _ ht]", "annotated_tactic": ["rw [<a>comapRight_apply</a>,\n    <a>Measure.comap_apply</a> _ hf.injective (fun s => hf.measurableSet_image.mpr) _ ht]", [{"full_name": "ProbabilityTheory.kernel.comapRight_apply", "def_path": "Mathlib/Probability/Kernel/Basic.lean", "def_pos": [572, 9], "def_end_pos": [572, 25]}, {"full_name": "MeasureTheory.Measure.comap_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1335, 9], "def_end_pos": [1335, 20]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b3 : Type u_4\nm\u03b3 : MeasurableSpace \u03b3\nf : \u03b3 \u2192 \u03b2\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\nhf : MeasurableEmbedding f\na : \u03b1\nt : Set \u03b3\nht : MeasurableSet t\n\u22a2 \u2191\u2191(\u2191(comapRight \u03ba hf) a) t = \u2191\u2191(\u2191\u03ba a) (f '' t)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "full_name": "MeasureTheory.measure_union_eq_top_iff", "start": [332, 1], "end": [333, 98], "traced_tactics": [{"tactic": "simp only [\u2190 lt_top_iff_ne_top, \u2190 Ne.def, not_or, measure_union_lt_top_iff]", "annotated_tactic": ["simp only [\u2190 <a>lt_top_iff_ne_top</a>, \u2190 <a>Ne.def</a>, <a>not_or</a>, <a>measure_union_lt_top_iff</a>]", [{"full_name": "lt_top_iff_ne_top", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [173, 9], "def_end_pos": [173, 26]}, {"full_name": "Ne.def", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [59, 9], "def_end_pos": [59, 15]}, {"full_name": "not_or", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [340, 9], "def_end_pos": [340, 15]}, {"full_name": "MeasureTheory.measure_union_lt_top_iff", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [318, 9], "def_end_pos": [318, 33]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\ninst\u271d : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns s\u2081 s\u2082 t : Set \u03b1\n\u22a2 \u00ac\u2191\u2191\u03bc (s \u222a t) = \u22a4 \u2194 \u00ac(\u2191\u2191\u03bc s = \u22a4 \u2228 \u2191\u2191\u03bc t = \u22a4)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/Basic.lean", "full_name": "MvPolynomial.induction_on''", "start": [438, 1], "end": [446, 67], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "full_name": "MeasureTheory.lintegral_inter_add_diff", "start": [1253, 1], "end": [1255, 61], "traced_tactics": [{"tactic": "rw [\u2190 lintegral_add_measure, restrict_inter_add_diff _ hB]", "annotated_tactic": ["rw [\u2190 <a>lintegral_add_measure</a>, <a>restrict_inter_add_diff</a> _ hB]", [{"full_name": "MeasureTheory.lintegral_add_measure", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [619, 9], "def_end_pos": [619, 30]}, {"full_name": "MeasureTheory.Measure.restrict_inter_add_diff", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1712, 9], "def_end_pos": [1712, 32]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nB : Set \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nA : Set \u03b1\nhB : MeasurableSet B\n\u22a2 \u222b\u207b (x : \u03b1) in A \u2229 B, f x \u2202\u03bc + \u222b\u207b (x : \u03b1) in A \\ B, f x \u2202\u03bc = \u222b\u207b (x : \u03b1) in A, f x \u2202\u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Card.lean", "full_name": "Finset.card_image_iff", "start": [246, 1], "end": [247, 48], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Finite.lean", "full_name": "Set.Finite.iSup_biInf_of_antitone", "start": [1484, 1], "end": [1487, 89], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Indicator.lean", "full_name": "MeasureTheory.condexp_ae_eq_restrict_zero", "start": [38, 1], "end": [59, 56], "traced_tactics": [{"tactic": "by_cases hm : m \u2264 m0", "annotated_tactic": ["by_cases hm : m \u2264 m0", []], "state_before": "\u03b1 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nm m0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhs : MeasurableSet s\nhf : f =\u1d50[Measure.restrict \u03bc s] 0\n\u22a2 \u03bc[f|m] =\u1d50[Measure.restrict \u03bc s] 0", "state_after": "case pos\n\u03b1 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nm m0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhs : MeasurableSet s\nhf : f =\u1d50[Measure.restrict \u03bc s] 0\nhm : m \u2264 m0\n\u22a2 \u03bc[f|m] =\u1d50[Measure.restrict \u03bc s] 0\n\ncase neg\n\u03b1 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nm m0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhs : MeasurableSet s\nhf : f =\u1d50[Measure.restrict \u03bc s] 0\nhm : \u00acm \u2264 m0\n\u22a2 \u03bc[f|m] =\u1d50[Measure.restrict \u03bc s] 0"}, {"tactic": "swap", "annotated_tactic": ["swap", []], "state_before": "case pos\n\u03b1 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nm m0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhs : MeasurableSet s\nhf : f =\u1d50[Measure.restrict \u03bc s] 0\nhm : m \u2264 m0\n\u22a2 \u03bc[f|m] =\u1d50[Measure.restrict \u03bc s] 0\n\ncase neg\n\u03b1 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nm m0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhs : MeasurableSet s\nhf : f =\u1d50[Measure.restrict \u03bc s] 0\nhm : \u00acm \u2264 m0\n\u22a2 \u03bc[f|m] =\u1d50[Measure.restrict \u03bc s] 0", "state_after": "case neg\n\u03b1 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nm m0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhs : MeasurableSet s\nhf : f =\u1d50[Measure.restrict \u03bc s] 0\nhm : \u00acm \u2264 m0\n\u22a2 \u03bc[f|m] =\u1d50[Measure.restrict \u03bc s] 0\n\ncase pos\n\u03b1 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nm m0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhs : MeasurableSet s\nhf : f =\u1d50[Measure.restrict \u03bc s] 0\nhm : m \u2264 m0\n\u22a2 \u03bc[f|m] =\u1d50[Measure.restrict \u03bc s] 0"}, {"tactic": "by_cases h\u03bcm : SigmaFinite (\u03bc.trim hm)", "annotated_tactic": ["by_cases h\u03bcm : <a>SigmaFinite</a> (\u03bc.trim hm)", [{"full_name": "MeasureTheory.SigmaFinite", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3289, 7], "def_end_pos": [3289, 18]}]], "state_before": "case pos\n\u03b1 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nm m0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhs : MeasurableSet s\nhf : f =\u1d50[Measure.restrict \u03bc s] 0\nhm : m \u2264 m0\n\u22a2 \u03bc[f|m] =\u1d50[Measure.restrict \u03bc s] 0", "state_after": "case pos\n\u03b1 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nm m0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhs : MeasurableSet s\nhf : f =\u1d50[Measure.restrict \u03bc s] 0\nhm : m \u2264 m0\nh\u03bcm : SigmaFinite (Measure.trim \u03bc hm)\n\u22a2 \u03bc[f|m] =\u1d50[Measure.restrict \u03bc s] 0\n\ncase neg\n\u03b1 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nm m0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhs : MeasurableSet s\nhf : f =\u1d50[Measure.restrict \u03bc s] 0\nhm : m \u2264 m0\nh\u03bcm : \u00acSigmaFinite (Measure.trim \u03bc hm)\n\u22a2 \u03bc[f|m] =\u1d50[Measure.restrict \u03bc s] 0"}, {"tactic": "swap", "annotated_tactic": ["swap", []], "state_before": "case pos\n\u03b1 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nm m0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhs : MeasurableSet s\nhf : f =\u1d50[Measure.restrict \u03bc s] 0\nhm : m \u2264 m0\nh\u03bcm : SigmaFinite (Measure.trim \u03bc hm)\n\u22a2 \u03bc[f|m] =\u1d50[Measure.restrict \u03bc s] 0\n\ncase neg\n\u03b1 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nm m0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhs : MeasurableSet s\nhf : f =\u1d50[Measure.restrict \u03bc s] 0\nhm : m \u2264 m0\nh\u03bcm : \u00acSigmaFinite (Measure.trim \u03bc hm)\n\u22a2 \u03bc[f|m] =\u1d50[Measure.restrict \u03bc s] 0", "state_after": "case neg\n\u03b1 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nm m0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhs : MeasurableSet s\nhf : f =\u1d50[Measure.restrict \u03bc s] 0\nhm : m \u2264 m0\nh\u03bcm : \u00acSigmaFinite (Measure.trim \u03bc hm)\n\u22a2 \u03bc[f|m] =\u1d50[Measure.restrict \u03bc s] 0\n\ncase pos\n\u03b1 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nm m0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhs : MeasurableSet s\nhf : f =\u1d50[Measure.restrict \u03bc s] 0\nhm : m \u2264 m0\nh\u03bcm : SigmaFinite (Measure.trim \u03bc hm)\n\u22a2 \u03bc[f|m] =\u1d50[Measure.restrict \u03bc s] 0"}, {"tactic": "haveI : SigmaFinite (\u03bc.trim hm) := h\u03bcm", "annotated_tactic": ["haveI : <a>SigmaFinite</a> (\u03bc.trim hm) := h\u03bcm", [{"full_name": "MeasureTheory.SigmaFinite", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3289, 7], "def_end_pos": [3289, 18]}]], "state_before": "case pos\n\u03b1 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nm m0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhs : MeasurableSet s\nhf : f =\u1d50[Measure.restrict \u03bc s] 0\nhm : m \u2264 m0\nh\u03bcm : SigmaFinite (Measure.trim \u03bc hm)\n\u22a2 \u03bc[f|m] =\u1d50[Measure.restrict \u03bc s] 0", "state_after": "case pos\n\u03b1 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nm m0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhs : MeasurableSet s\nhf : f =\u1d50[Measure.restrict \u03bc s] 0\nhm : m \u2264 m0\nh\u03bcm this : SigmaFinite (Measure.trim \u03bc hm)\n\u22a2 \u03bc[f|m] =\u1d50[Measure.restrict \u03bc s] 0"}, {"tactic": "have : SigmaFinite ((\u03bc.restrict s).trim hm) := by\n  rw [\u2190 restrict_trim hm _ hs]\n  exact Restrict.sigmaFinite _ s", "annotated_tactic": ["have : <a>SigmaFinite</a> ((\u03bc.restrict s).<a>trim</a> hm) := by\n    rw [\u2190 <a>restrict_trim</a> hm _ hs]\n    exact <a>Restrict.sigmaFinite</a> _ s", [{"full_name": "MeasureTheory.SigmaFinite", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3289, 7], "def_end_pos": [3289, 18]}, {"full_name": "MeasureTheory.Measure.trim", "def_path": "Mathlib/MeasureTheory/Measure/Trim.lean", "def_pos": [32, 5], "def_end_pos": [32, 17]}, {"full_name": "MeasureTheory.restrict_trim", "def_path": "Mathlib/MeasureTheory/Measure/Trim.lean", "def_pos": [93, 9], "def_end_pos": [93, 22]}, {"full_name": "MeasureTheory.Restrict.sigmaFinite", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3728, 10], "def_end_pos": [3728, 30]}]], "state_before": "case pos\n\u03b1 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nm m0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhs : MeasurableSet s\nhf : f =\u1d50[Measure.restrict \u03bc s] 0\nhm : m \u2264 m0\nh\u03bcm this : SigmaFinite (Measure.trim \u03bc hm)\n\u22a2 \u03bc[f|m] =\u1d50[Measure.restrict \u03bc s] 0", "state_after": "case pos\n\u03b1 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nm m0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhs : MeasurableSet s\nhf : f =\u1d50[Measure.restrict \u03bc s] 0\nhm : m \u2264 m0\nh\u03bcm this\u271d : SigmaFinite (Measure.trim \u03bc hm)\nthis : SigmaFinite (Measure.trim (Measure.restrict \u03bc s) hm)\n\u22a2 \u03bc[f|m] =\u1d50[Measure.restrict \u03bc s] 0"}, {"tactic": "by_cases hf_int : Integrable f \u03bc", "annotated_tactic": ["by_cases hf_int : <a>Integrable</a> f \u03bc", [{"full_name": "MeasureTheory.Integrable", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [442, 5], "def_end_pos": [442, 15]}]], "state_before": "case pos\n\u03b1 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nm m0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhs : MeasurableSet s\nhf : f =\u1d50[Measure.restrict \u03bc s] 0\nhm : m \u2264 m0\nh\u03bcm this\u271d : SigmaFinite (Measure.trim \u03bc hm)\nthis : SigmaFinite (Measure.trim (Measure.restrict \u03bc s) hm)\n\u22a2 \u03bc[f|m] =\u1d50[Measure.restrict \u03bc s] 0", "state_after": "case pos\n\u03b1 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nm m0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhs : MeasurableSet s\nhf : f =\u1d50[Measure.restrict \u03bc s] 0\nhm : m \u2264 m0\nh\u03bcm this\u271d : SigmaFinite (Measure.trim \u03bc hm)\nthis : SigmaFinite (Measure.trim (Measure.restrict \u03bc s) hm)\nhf_int : Integrable f\n\u22a2 \u03bc[f|m] =\u1d50[Measure.restrict \u03bc s] 0\n\ncase neg\n\u03b1 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nm m0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhs : MeasurableSet s\nhf : f =\u1d50[Measure.restrict \u03bc s] 0\nhm : m \u2264 m0\nh\u03bcm this\u271d : SigmaFinite (Measure.trim \u03bc hm)\nthis : SigmaFinite (Measure.trim (Measure.restrict \u03bc s) hm)\nhf_int : \u00acIntegrable f\n\u22a2 \u03bc[f|m] =\u1d50[Measure.restrict \u03bc s] 0"}, {"tactic": "swap", "annotated_tactic": ["swap", []], "state_before": "case pos\n\u03b1 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nm m0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhs : MeasurableSet s\nhf : f =\u1d50[Measure.restrict \u03bc s] 0\nhm : m \u2264 m0\nh\u03bcm this\u271d : SigmaFinite (Measure.trim \u03bc hm)\nthis : SigmaFinite (Measure.trim (Measure.restrict \u03bc s) hm)\nhf_int : Integrable f\n\u22a2 \u03bc[f|m] =\u1d50[Measure.restrict \u03bc s] 0\n\ncase neg\n\u03b1 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nm m0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhs : MeasurableSet s\nhf : f =\u1d50[Measure.restrict \u03bc s] 0\nhm : m \u2264 m0\nh\u03bcm this\u271d : SigmaFinite (Measure.trim \u03bc hm)\nthis : SigmaFinite (Measure.trim (Measure.restrict \u03bc s) hm)\nhf_int : \u00acIntegrable f\n\u22a2 \u03bc[f|m] =\u1d50[Measure.restrict \u03bc s] 0", "state_after": "case neg\n\u03b1 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nm m0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhs : MeasurableSet s\nhf : f =\u1d50[Measure.restrict \u03bc s] 0\nhm : m \u2264 m0\nh\u03bcm this\u271d : SigmaFinite (Measure.trim \u03bc hm)\nthis : SigmaFinite (Measure.trim (Measure.restrict \u03bc s) hm)\nhf_int : \u00acIntegrable f\n\u22a2 \u03bc[f|m] =\u1d50[Measure.restrict \u03bc s] 0\n\ncase pos\n\u03b1 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nm m0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhs : MeasurableSet s\nhf : f =\u1d50[Measure.restrict \u03bc s] 0\nhm : m \u2264 m0\nh\u03bcm this\u271d : SigmaFinite (Measure.trim \u03bc hm)\nthis : SigmaFinite (Measure.trim (Measure.restrict \u03bc s) hm)\nhf_int : Integrable f\n\u22a2 \u03bc[f|m] =\u1d50[Measure.restrict \u03bc s] 0"}, {"tactic": "refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' hm _ _ _ _ _", "annotated_tactic": ["refine' <a>ae_eq_of_forall_set_integral_eq_of_sigmaFinite'</a> hm _ _ _ _ _", [{"full_name": "MeasureTheory.ae_eq_of_forall_set_integral_eq_of_sigmaFinite'", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Unique.lean", "def_pos": [119, 9], "def_end_pos": [119, 56]}]], "state_before": "case pos\n\u03b1 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nm m0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhs : MeasurableSet s\nhf : f =\u1d50[Measure.restrict \u03bc s] 0\nhm : m \u2264 m0\nh\u03bcm this\u271d : SigmaFinite (Measure.trim \u03bc hm)\nthis : SigmaFinite (Measure.trim (Measure.restrict \u03bc s) hm)\nhf_int : Integrable f\n\u22a2 \u03bc[f|m] =\u1d50[Measure.restrict \u03bc s] 0", "state_after": "case pos.refine'_1\n\u03b1 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nm m0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhs : MeasurableSet s\nhf : f =\u1d50[Measure.restrict \u03bc s] 0\nhm : m \u2264 m0\nh\u03bcm this\u271d : SigmaFinite (Measure.trim \u03bc hm)\nthis : SigmaFinite (Measure.trim (Measure.restrict \u03bc s) hm)\nhf_int : Integrable f\n\u22a2 \u2200 (s_1 : Set \u03b1), MeasurableSet s_1 \u2192 \u2191\u2191(Measure.restrict \u03bc s) s_1 < \u22a4 \u2192 IntegrableOn (\u03bc[f|m]) s_1\n\ncase pos.refine'_2\n\u03b1 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nm m0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhs : MeasurableSet s\nhf : f =\u1d50[Measure.restrict \u03bc s] 0\nhm : m \u2264 m0\nh\u03bcm this\u271d : SigmaFinite (Measure.trim \u03bc hm)\nthis : SigmaFinite (Measure.trim (Measure.restrict \u03bc s) hm)\nhf_int : Integrable f\n\u22a2 \u2200 (s_1 : Set \u03b1), MeasurableSet s_1 \u2192 \u2191\u2191(Measure.restrict \u03bc s) s_1 < \u22a4 \u2192 IntegrableOn 0 s_1\n\ncase pos.refine'_3\n\u03b1 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nm m0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhs : MeasurableSet s\nhf : f =\u1d50[Measure.restrict \u03bc s] 0\nhm : m \u2264 m0\nh\u03bcm this\u271d : SigmaFinite (Measure.trim \u03bc hm)\nthis : SigmaFinite (Measure.trim (Measure.restrict \u03bc s) hm)\nhf_int : Integrable f\n\u22a2 \u2200 (s_1 : Set \u03b1),\n    MeasurableSet s_1 \u2192\n      \u2191\u2191(Measure.restrict \u03bc s) s_1 < \u22a4 \u2192\n        \u222b (x : \u03b1) in s_1, (\u03bc[f|m]) x \u2202Measure.restrict \u03bc s = \u222b (x : \u03b1) in s_1, OfNat.ofNat 0 x \u2202Measure.restrict \u03bc s\n\ncase pos.refine'_4\n\u03b1 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nm m0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhs : MeasurableSet s\nhf : f =\u1d50[Measure.restrict \u03bc s] 0\nhm : m \u2264 m0\nh\u03bcm this\u271d : SigmaFinite (Measure.trim \u03bc hm)\nthis : SigmaFinite (Measure.trim (Measure.restrict \u03bc s) hm)\nhf_int : Integrable f\n\u22a2 AEStronglyMeasurable' m (\u03bc[f|m]) (Measure.restrict \u03bc s)\n\ncase pos.refine'_5\n\u03b1 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nm m0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhs : MeasurableSet s\nhf : f =\u1d50[Measure.restrict \u03bc s] 0\nhm : m \u2264 m0\nh\u03bcm this\u271d : SigmaFinite (Measure.trim \u03bc hm)\nthis : SigmaFinite (Measure.trim (Measure.restrict \u03bc s) hm)\nhf_int : Integrable f\n\u22a2 AEStronglyMeasurable' m 0 (Measure.restrict \u03bc s)"}, {"tactic": "simp_rw [condexp_of_not_le hm]", "annotated_tactic": ["simp_rw [<a>condexp_of_not_le</a> hm]", [{"full_name": "MeasureTheory.condexp_of_not_le", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean", "def_pos": [106, 9], "def_end_pos": [106, 26]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nm m0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhs : MeasurableSet s\nhf : f =\u1d50[Measure.restrict \u03bc s] 0\nhm : \u00acm \u2264 m0\n\u22a2 \u03bc[f|m] =\u1d50[Measure.restrict \u03bc s] 0", "state_after": "case neg\n\u03b1 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nm m0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhs : MeasurableSet s\nhf : f =\u1d50[Measure.restrict \u03bc s] 0\nhm : \u00acm \u2264 m0\n\u22a2 0 =\u1d50[Measure.restrict \u03bc s] 0"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case neg\n\u03b1 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nm m0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhs : MeasurableSet s\nhf : f =\u1d50[Measure.restrict \u03bc s] 0\nhm : \u00acm \u2264 m0\n\u22a2 0 =\u1d50[Measure.restrict \u03bc s] 0", "state_after": "no goals"}, {"tactic": "simp_rw [condexp_of_not_sigmaFinite hm h\u03bcm]", "annotated_tactic": ["simp_rw [<a>condexp_of_not_sigmaFinite</a> hm h\u03bcm]", [{"full_name": "MeasureTheory.condexp_of_not_sigmaFinite", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean", "def_pos": [109, 9], "def_end_pos": [109, 35]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nm m0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhs : MeasurableSet s\nhf : f =\u1d50[Measure.restrict \u03bc s] 0\nhm : m \u2264 m0\nh\u03bcm : \u00acSigmaFinite (Measure.trim \u03bc hm)\n\u22a2 \u03bc[f|m] =\u1d50[Measure.restrict \u03bc s] 0", "state_after": "case neg\n\u03b1 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nm m0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhs : MeasurableSet s\nhf : f =\u1d50[Measure.restrict \u03bc s] 0\nhm : m \u2264 m0\nh\u03bcm : \u00acSigmaFinite (Measure.trim \u03bc hm)\n\u22a2 0 =\u1d50[Measure.restrict \u03bc s] 0"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case neg\n\u03b1 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nm m0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhs : MeasurableSet s\nhf : f =\u1d50[Measure.restrict \u03bc s] 0\nhm : m \u2264 m0\nh\u03bcm : \u00acSigmaFinite (Measure.trim \u03bc hm)\n\u22a2 0 =\u1d50[Measure.restrict \u03bc s] 0", "state_after": "no goals"}, {"tactic": "rw [\u2190 restrict_trim hm _ hs]", "annotated_tactic": ["rw [\u2190 <a>restrict_trim</a> hm _ hs]", [{"full_name": "MeasureTheory.restrict_trim", "def_path": "Mathlib/MeasureTheory/Measure/Trim.lean", "def_pos": [93, 9], "def_end_pos": [93, 22]}]], "state_before": "\u03b1 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nm m0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhs : MeasurableSet s\nhf : f =\u1d50[Measure.restrict \u03bc s] 0\nhm : m \u2264 m0\nh\u03bcm this : SigmaFinite (Measure.trim \u03bc hm)\n\u22a2 SigmaFinite (Measure.trim (Measure.restrict \u03bc s) hm)", "state_after": "\u03b1 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nm m0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhs : MeasurableSet s\nhf : f =\u1d50[Measure.restrict \u03bc s] 0\nhm : m \u2264 m0\nh\u03bcm this : SigmaFinite (Measure.trim \u03bc hm)\n\u22a2 SigmaFinite (Measure.restrict (Measure.trim \u03bc hm) s)"}, {"tactic": "exact Restrict.sigmaFinite _ s", "annotated_tactic": ["exact <a>Restrict.sigmaFinite</a> _ s", [{"full_name": "MeasureTheory.Restrict.sigmaFinite", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3728, 10], "def_end_pos": [3728, 30]}]], "state_before": "\u03b1 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nm m0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhs : MeasurableSet s\nhf : f =\u1d50[Measure.restrict \u03bc s] 0\nhm : m \u2264 m0\nh\u03bcm this : SigmaFinite (Measure.trim \u03bc hm)\n\u22a2 SigmaFinite (Measure.restrict (Measure.trim \u03bc hm) s)", "state_after": "no goals"}, {"tactic": "rw [condexp_undef hf_int]", "annotated_tactic": ["rw [<a>condexp_undef</a> hf_int]", [{"full_name": "MeasureTheory.condexp_undef", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean", "def_pos": [159, 9], "def_end_pos": [159, 22]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nm m0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhs : MeasurableSet s\nhf : f =\u1d50[Measure.restrict \u03bc s] 0\nhm : m \u2264 m0\nh\u03bcm this\u271d : SigmaFinite (Measure.trim \u03bc hm)\nthis : SigmaFinite (Measure.trim (Measure.restrict \u03bc s) hm)\nhf_int : \u00acIntegrable f\n\u22a2 \u03bc[f|m] =\u1d50[Measure.restrict \u03bc s] 0", "state_after": "no goals"}, {"tactic": "exact fun t _ _ => integrable_condexp.integrableOn.integrableOn", "annotated_tactic": ["exact fun t _ _ => integrable_condexp.integrableOn.integrableOn", []], "state_before": "case pos.refine'_1\n\u03b1 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nm m0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhs : MeasurableSet s\nhf : f =\u1d50[Measure.restrict \u03bc s] 0\nhm : m \u2264 m0\nh\u03bcm this\u271d : SigmaFinite (Measure.trim \u03bc hm)\nthis : SigmaFinite (Measure.trim (Measure.restrict \u03bc s) hm)\nhf_int : Integrable f\n\u22a2 \u2200 (s_1 : Set \u03b1), MeasurableSet s_1 \u2192 \u2191\u2191(Measure.restrict \u03bc s) s_1 < \u22a4 \u2192 IntegrableOn (\u03bc[f|m]) s_1", "state_after": "no goals"}, {"tactic": "exact fun t _ _ => (integrable_zero _ _ _).integrableOn", "annotated_tactic": ["exact fun t _ _ => (<a>integrable_zero</a> _ _ _).<a>integrableOn</a>", [{"full_name": "MeasureTheory.integrable_zero", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [662, 9], "def_end_pos": [662, 24]}, {"full_name": "MeasureTheory.Integrable.integrableOn", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [163, 9], "def_end_pos": [163, 32]}]], "state_before": "case pos.refine'_2\n\u03b1 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nm m0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhs : MeasurableSet s\nhf : f =\u1d50[Measure.restrict \u03bc s] 0\nhm : m \u2264 m0\nh\u03bcm this\u271d : SigmaFinite (Measure.trim \u03bc hm)\nthis : SigmaFinite (Measure.trim (Measure.restrict \u03bc s) hm)\nhf_int : Integrable f\n\u22a2 \u2200 (s_1 : Set \u03b1), MeasurableSet s_1 \u2192 \u2191\u2191(Measure.restrict \u03bc s) s_1 < \u22a4 \u2192 IntegrableOn 0 s_1", "state_after": "no goals"}, {"tactic": "intro t ht _", "annotated_tactic": ["intro t ht _", []], "state_before": "case pos.refine'_3\n\u03b1 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nm m0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhs : MeasurableSet s\nhf : f =\u1d50[Measure.restrict \u03bc s] 0\nhm : m \u2264 m0\nh\u03bcm this\u271d : SigmaFinite (Measure.trim \u03bc hm)\nthis : SigmaFinite (Measure.trim (Measure.restrict \u03bc s) hm)\nhf_int : Integrable f\n\u22a2 \u2200 (s_1 : Set \u03b1),\n    MeasurableSet s_1 \u2192\n      \u2191\u2191(Measure.restrict \u03bc s) s_1 < \u22a4 \u2192\n        \u222b (x : \u03b1) in s_1, (\u03bc[f|m]) x \u2202Measure.restrict \u03bc s = \u222b (x : \u03b1) in s_1, OfNat.ofNat 0 x \u2202Measure.restrict \u03bc s", "state_after": "case pos.refine'_3\n\u03b1 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nm m0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhs : MeasurableSet s\nhf : f =\u1d50[Measure.restrict \u03bc s] 0\nhm : m \u2264 m0\nh\u03bcm this\u271d : SigmaFinite (Measure.trim \u03bc hm)\nthis : SigmaFinite (Measure.trim (Measure.restrict \u03bc s) hm)\nhf_int : Integrable f\nt : Set \u03b1\nht : MeasurableSet t\na\u271d : \u2191\u2191(Measure.restrict \u03bc s) t < \u22a4\n\u22a2 \u222b (x : \u03b1) in t, (\u03bc[f|m]) x \u2202Measure.restrict \u03bc s = \u222b (x : \u03b1) in t, OfNat.ofNat 0 x \u2202Measure.restrict \u03bc s"}, {"tactic": "rw [Measure.restrict_restrict (hm _ ht), set_integral_condexp hm hf_int (ht.inter hs), \u2190\n  Measure.restrict_restrict (hm _ ht)]", "annotated_tactic": ["rw [<a>Measure.restrict_restrict</a> (hm _ ht), <a>set_integral_condexp</a> hm hf_int (ht.inter hs), \u2190\n      <a>Measure.restrict_restrict</a> (hm _ ht)]", [{"full_name": "MeasureTheory.Measure.restrict_restrict", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1645, 9], "def_end_pos": [1645, 26]}, {"full_name": "MeasureTheory.set_integral_condexp", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean", "def_pos": [221, 9], "def_end_pos": [221, 29]}, {"full_name": "MeasureTheory.Measure.restrict_restrict", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1645, 9], "def_end_pos": [1645, 26]}]], "state_before": "case pos.refine'_3\n\u03b1 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nm m0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhs : MeasurableSet s\nhf : f =\u1d50[Measure.restrict \u03bc s] 0\nhm : m \u2264 m0\nh\u03bcm this\u271d : SigmaFinite (Measure.trim \u03bc hm)\nthis : SigmaFinite (Measure.trim (Measure.restrict \u03bc s) hm)\nhf_int : Integrable f\nt : Set \u03b1\nht : MeasurableSet t\na\u271d : \u2191\u2191(Measure.restrict \u03bc s) t < \u22a4\n\u22a2 \u222b (x : \u03b1) in t, (\u03bc[f|m]) x \u2202Measure.restrict \u03bc s = \u222b (x : \u03b1) in t, OfNat.ofNat 0 x \u2202Measure.restrict \u03bc s", "state_after": "case pos.refine'_3\n\u03b1 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nm m0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhs : MeasurableSet s\nhf : f =\u1d50[Measure.restrict \u03bc s] 0\nhm : m \u2264 m0\nh\u03bcm this\u271d : SigmaFinite (Measure.trim \u03bc hm)\nthis : SigmaFinite (Measure.trim (Measure.restrict \u03bc s) hm)\nhf_int : Integrable f\nt : Set \u03b1\nht : MeasurableSet t\na\u271d : \u2191\u2191(Measure.restrict \u03bc s) t < \u22a4\n\u22a2 \u222b (x : \u03b1) in t, f x \u2202Measure.restrict \u03bc s = \u222b (x : \u03b1) in t, OfNat.ofNat 0 x \u2202Measure.restrict \u03bc s"}, {"tactic": "refine' set_integral_congr_ae (hm _ ht) _", "annotated_tactic": ["refine' <a>set_integral_congr_ae</a> (hm _ ht) _", [{"full_name": "MeasureTheory.set_integral_congr_ae", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [77, 9], "def_end_pos": [77, 30]}]], "state_before": "case pos.refine'_3\n\u03b1 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nm m0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhs : MeasurableSet s\nhf : f =\u1d50[Measure.restrict \u03bc s] 0\nhm : m \u2264 m0\nh\u03bcm this\u271d : SigmaFinite (Measure.trim \u03bc hm)\nthis : SigmaFinite (Measure.trim (Measure.restrict \u03bc s) hm)\nhf_int : Integrable f\nt : Set \u03b1\nht : MeasurableSet t\na\u271d : \u2191\u2191(Measure.restrict \u03bc s) t < \u22a4\n\u22a2 \u222b (x : \u03b1) in t, f x \u2202Measure.restrict \u03bc s = \u222b (x : \u03b1) in t, OfNat.ofNat 0 x \u2202Measure.restrict \u03bc s", "state_after": "case pos.refine'_3\n\u03b1 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nm m0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhs : MeasurableSet s\nhf : f =\u1d50[Measure.restrict \u03bc s] 0\nhm : m \u2264 m0\nh\u03bcm this\u271d : SigmaFinite (Measure.trim \u03bc hm)\nthis : SigmaFinite (Measure.trim (Measure.restrict \u03bc s) hm)\nhf_int : Integrable f\nt : Set \u03b1\nht : MeasurableSet t\na\u271d : \u2191\u2191(Measure.restrict \u03bc s) t < \u22a4\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc s, x \u2208 t \u2192 f x = OfNat.ofNat 0 x"}, {"tactic": "filter_upwards [hf] with x hx _ using hx", "annotated_tactic": ["filter_upwards [hf] with x hx _ using hx", []], "state_before": "case pos.refine'_3\n\u03b1 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nm m0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhs : MeasurableSet s\nhf : f =\u1d50[Measure.restrict \u03bc s] 0\nhm : m \u2264 m0\nh\u03bcm this\u271d : SigmaFinite (Measure.trim \u03bc hm)\nthis : SigmaFinite (Measure.trim (Measure.restrict \u03bc s) hm)\nhf_int : Integrable f\nt : Set \u03b1\nht : MeasurableSet t\na\u271d : \u2191\u2191(Measure.restrict \u03bc s) t < \u22a4\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc s, x \u2208 t \u2192 f x = OfNat.ofNat 0 x", "state_after": "no goals"}, {"tactic": "exact stronglyMeasurable_condexp.aeStronglyMeasurable'", "annotated_tactic": ["exact stronglyMeasurable_condexp.aeStronglyMeasurable'", []], "state_before": "case pos.refine'_4\n\u03b1 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nm m0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhs : MeasurableSet s\nhf : f =\u1d50[Measure.restrict \u03bc s] 0\nhm : m \u2264 m0\nh\u03bcm this\u271d : SigmaFinite (Measure.trim \u03bc hm)\nthis : SigmaFinite (Measure.trim (Measure.restrict \u03bc s) hm)\nhf_int : Integrable f\n\u22a2 AEStronglyMeasurable' m (\u03bc[f|m]) (Measure.restrict \u03bc s)", "state_after": "no goals"}, {"tactic": "exact stronglyMeasurable_zero.aeStronglyMeasurable'", "annotated_tactic": ["exact stronglyMeasurable_zero.aeStronglyMeasurable'", []], "state_before": "case pos.refine'_5\n\u03b1 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nm m0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhs : MeasurableSet s\nhf : f =\u1d50[Measure.restrict \u03bc s] 0\nhm : m \u2264 m0\nh\u03bcm this\u271d : SigmaFinite (Measure.trim \u03bc hm)\nthis : SigmaFinite (Measure.trim (Measure.restrict \u03bc s) hm)\nhf_int : Integrable f\n\u22a2 AEStronglyMeasurable' m 0 (Measure.restrict \u03bc s)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "full_name": "MeasureTheory.meas_ge_le_mul_pow_snorm", "start": [1172, 1], "end": [1181, 62], "traced_tactics": [{"tactic": "by_cases \u03b5 = \u221e", "annotated_tactic": ["by_cases \u03b5 = \u221e", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 E\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhf : AEStronglyMeasurable f \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u22a2 \u2191\u2191\u03bc {x | \u03b5 \u2264 \u2191\u2016f x\u2016\u208a} \u2264 \u03b5\u207b\u00b9 ^ ENNReal.toReal p * snorm f p \u03bc ^ ENNReal.toReal p", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 E\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhf : AEStronglyMeasurable f \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nh : \u03b5 = \u22a4\n\u22a2 \u2191\u2191\u03bc {x | \u03b5 \u2264 \u2191\u2016f x\u2016\u208a} \u2264 \u03b5\u207b\u00b9 ^ ENNReal.toReal p * snorm f p \u03bc ^ ENNReal.toReal p\n\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 E\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhf : AEStronglyMeasurable f \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nh : \u00ac\u03b5 = \u22a4\n\u22a2 \u2191\u2191\u03bc {x | \u03b5 \u2264 \u2191\u2016f x\u2016\u208a} \u2264 \u03b5\u207b\u00b9 ^ ENNReal.toReal p * snorm f p \u03bc ^ ENNReal.toReal p"}, {"tactic": "have h\u03b5pow : \u03b5 ^ p.toReal \u2260 0 := (ENNReal.rpow_pos (pos_iff_ne_zero.2 h\u03b5) h).ne.symm", "annotated_tactic": ["have h\u03b5pow : \u03b5 ^ p.toReal \u2260 0 := (<a>ENNReal.rpow_pos</a> (<a>pos_iff_ne_zero</a>.2 h\u03b5) h).ne.symm", [{"full_name": "ENNReal.rpow_pos", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [735, 9], "def_end_pos": [735, 17]}, {"full_name": "pos_iff_ne_zero", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [243, 3], "def_end_pos": [243, 14]}]], "state_before": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 E\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhf : AEStronglyMeasurable f \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nh : \u00ac\u03b5 = \u22a4\n\u22a2 \u2191\u2191\u03bc {x | \u03b5 \u2264 \u2191\u2016f x\u2016\u208a} \u2264 \u03b5\u207b\u00b9 ^ ENNReal.toReal p * snorm f p \u03bc ^ ENNReal.toReal p", "state_after": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 E\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhf : AEStronglyMeasurable f \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nh : \u00ac\u03b5 = \u22a4\nh\u03b5pow : \u03b5 ^ ENNReal.toReal p \u2260 0\n\u22a2 \u2191\u2191\u03bc {x | \u03b5 \u2264 \u2191\u2016f x\u2016\u208a} \u2264 \u03b5\u207b\u00b9 ^ ENNReal.toReal p * snorm f p \u03bc ^ ENNReal.toReal p"}, {"tactic": "have h\u03b5pow' : \u03b5 ^ p.toReal \u2260 \u221e := ENNReal.rpow_ne_top_of_nonneg ENNReal.toReal_nonneg h", "annotated_tactic": ["have h\u03b5pow' : \u03b5 ^ p.toReal \u2260 \u221e := <a>ENNReal.rpow_ne_top_of_nonneg</a> <a>ENNReal.toReal_nonneg</a> h", [{"full_name": "ENNReal.rpow_ne_top_of_nonneg", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [499, 9], "def_end_pos": [499, 30]}, {"full_name": "ENNReal.toReal_nonneg", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [221, 17], "def_end_pos": [221, 30]}]], "state_before": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 E\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhf : AEStronglyMeasurable f \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nh : \u00ac\u03b5 = \u22a4\nh\u03b5pow : \u03b5 ^ ENNReal.toReal p \u2260 0\n\u22a2 \u2191\u2191\u03bc {x | \u03b5 \u2264 \u2191\u2016f x\u2016\u208a} \u2264 \u03b5\u207b\u00b9 ^ ENNReal.toReal p * snorm f p \u03bc ^ ENNReal.toReal p", "state_after": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 E\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhf : AEStronglyMeasurable f \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nh : \u00ac\u03b5 = \u22a4\nh\u03b5pow : \u03b5 ^ ENNReal.toReal p \u2260 0\nh\u03b5pow' : \u03b5 ^ ENNReal.toReal p \u2260 \u22a4\n\u22a2 \u2191\u2191\u03bc {x | \u03b5 \u2264 \u2191\u2016f x\u2016\u208a} \u2264 \u03b5\u207b\u00b9 ^ ENNReal.toReal p * snorm f p \u03bc ^ ENNReal.toReal p"}, {"tactic": "rw [ENNReal.inv_rpow, \u2190 ENNReal.mul_le_mul_left h\u03b5pow h\u03b5pow', \u2190 mul_assoc,\n  ENNReal.mul_inv_cancel h\u03b5pow h\u03b5pow', one_mul]", "annotated_tactic": ["rw [<a>ENNReal.inv_rpow</a>, \u2190 <a>ENNReal.mul_le_mul_left</a> h\u03b5pow h\u03b5pow', \u2190 <a>mul_assoc</a>,\n    <a>ENNReal.mul_inv_cancel</a> h\u03b5pow h\u03b5pow', <a>one_mul</a>]", [{"full_name": "ENNReal.inv_rpow", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [598, 9], "def_end_pos": [598, 17]}, {"full_name": "ENNReal.mul_le_mul_left", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1065, 9], "def_end_pos": [1065, 24]}, {"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [264, 9], "def_end_pos": [264, 18]}, {"full_name": "ENNReal.mul_inv_cancel", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1418, 19], "def_end_pos": [1418, 33]}, {"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [464, 9], "def_end_pos": [464, 16]}]], "state_before": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 E\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhf : AEStronglyMeasurable f \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nh : \u00ac\u03b5 = \u22a4\nh\u03b5pow : \u03b5 ^ ENNReal.toReal p \u2260 0\nh\u03b5pow' : \u03b5 ^ ENNReal.toReal p \u2260 \u22a4\n\u22a2 \u2191\u2191\u03bc {x | \u03b5 \u2264 \u2191\u2016f x\u2016\u208a} \u2264 \u03b5\u207b\u00b9 ^ ENNReal.toReal p * snorm f p \u03bc ^ ENNReal.toReal p", "state_after": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 E\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhf : AEStronglyMeasurable f \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nh : \u00ac\u03b5 = \u22a4\nh\u03b5pow : \u03b5 ^ ENNReal.toReal p \u2260 0\nh\u03b5pow' : \u03b5 ^ ENNReal.toReal p \u2260 \u22a4\n\u22a2 \u03b5 ^ ENNReal.toReal p * \u2191\u2191\u03bc {x | \u03b5 \u2264 \u2191\u2016f x\u2016\u208a} \u2264 snorm f p \u03bc ^ ENNReal.toReal p"}, {"tactic": "exact mul_meas_ge_le_pow_snorm' \u03bc hp_ne_zero hp_ne_top hf \u03b5", "annotated_tactic": ["exact <a>mul_meas_ge_le_pow_snorm'</a> \u03bc hp_ne_zero hp_ne_top hf \u03b5", [{"full_name": "MeasureTheory.mul_meas_ge_le_pow_snorm'", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [1164, 9], "def_end_pos": [1164, 34]}]], "state_before": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 E\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhf : AEStronglyMeasurable f \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nh : \u00ac\u03b5 = \u22a4\nh\u03b5pow : \u03b5 ^ ENNReal.toReal p \u2260 0\nh\u03b5pow' : \u03b5 ^ ENNReal.toReal p \u2260 \u22a4\n\u22a2 \u03b5 ^ ENNReal.toReal p * \u2191\u2191\u03bc {x | \u03b5 \u2264 \u2191\u2016f x\u2016\u208a} \u2264 snorm f p \u03bc ^ ENNReal.toReal p", "state_after": "no goals"}, {"tactic": "simp [h]", "annotated_tactic": ["simp [h]", []], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 E\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhf : AEStronglyMeasurable f \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nh : \u03b5 = \u22a4\n\u22a2 \u2191\u2191\u03bc {x | \u03b5 \u2264 \u2191\u2016f x\u2016\u208a} \u2264 \u03b5\u207b\u00b9 ^ ENNReal.toReal p * snorm f p \u03bc ^ ENNReal.toReal p", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Setoid/Partition.lean", "full_name": "Setoid.eqv_classes_of_disjoint_union", "start": [175, 1], "end": [178, 75], "traced_tactics": [{"tactic": "rw [hu]", "annotated_tactic": ["rw [hu]", []], "state_before": "\u03b1 : Type u_1\nc : Set (Set \u03b1)\nhu : \u22c3\u2080 c = Set.univ\nH : Set.PairwiseDisjoint c id\na : \u03b1\n\u22a2 a \u2208 \u22c3\u2080 c", "state_after": "\u03b1 : Type u_1\nc : Set (Set \u03b1)\nhu : \u22c3\u2080 c = Set.univ\nH : Set.PairwiseDisjoint c id\na : \u03b1\n\u22a2 a \u2208 Set.univ"}, {"tactic": "exact Set.mem_univ a", "annotated_tactic": ["exact <a>Set.mem_univ</a> a", [{"full_name": "Set.mem_univ", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [676, 9], "def_end_pos": [676, 17]}]], "state_before": "\u03b1 : Type u_1\nc : Set (Set \u03b1)\nhu : \u22c3\u2080 c = Set.univ\nH : Set.PairwiseDisjoint c id\na : \u03b1\n\u22a2 a \u2208 Set.univ", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Setoid/Partition.lean", "full_name": "Finpartition.isPartition_parts", "start": [309, 1], "end": [313, 35], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "full_name": "String.offsetOfPos_of_valid", "start": [693, 1], "end": [694, 47], "traced_tactics": [{"tactic": "simpa using offsetOfPosAux_of_valid [] l r 0", "annotated_tactic": ["simpa using <a>offsetOfPosAux_of_valid</a> [] l r 0", [{"full_name": "String.offsetOfPosAux_of_valid", "def_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "def_pos": [682, 9], "def_end_pos": [682, 32]}]], "state_before": "l r : List Char\n\u22a2 offsetOfPos { data := l ++ r } { byteIdx := utf8Len l } = List.length l", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "Finset.erase_cons_of_ne", "start": [1952, 1], "end": [1954, 52], "traced_tactics": [{"tactic": "simp only [cons_eq_insert, erase_insert_of_ne hb]", "annotated_tactic": ["simp only [<a>cons_eq_insert</a>, <a>erase_insert_of_ne</a> hb]", [{"full_name": "Finset.cons_eq_insert", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1108, 9], "def_end_pos": [1108, 23]}, {"full_name": "Finset.erase_insert_of_ne", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1945, 9], "def_end_pos": [1945, 27]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns\u271d t u v : Finset \u03b1\na\u271d b\u271d a b : \u03b1\ns : Finset \u03b1\nha : \u00aca \u2208 s\nhb : a \u2260 b\n\u22a2 erase (cons a s ha) b = cons a (erase s b) (_ : a \u2208 erase s b \u2192 False)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/Average.lean", "full_name": "MeasureTheory.laverage_one", "start": [210, 1], "end": [211, 21], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Process/Stopping.lean", "full_name": "MeasureTheory.integrable_stoppedProcess", "start": [1011, 1], "end": [1013, 82], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "full_name": "intervalIntegral.integral_finset_sum", "start": [583, 8], "end": [587, 21], "traced_tactics": [{"tactic": "simp only [intervalIntegral_eq_integral_uIoc, integral_finset_sum s fun i hi => (h i hi).def,\n  Finset.smul_sum]", "annotated_tactic": ["simp only [<a>intervalIntegral_eq_integral_uIoc</a>, <a>integral_finset_sum</a> s fun i hi => (h i hi).<a>def</a>,\n    <a>Finset.smul_sum</a>]", [{"full_name": "intervalIntegral.intervalIntegral_eq_integral_uIoc", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [482, 9], "def_end_pos": [482, 42]}, {"full_name": "MeasureTheory.integral_finset_sum", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [881, 9], "def_end_pos": [881, 28]}, {"full_name": "IntervalIntegrable.def", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [87, 9], "def_end_pos": [87, 31]}, {"full_name": "Finset.smul_sum", "def_path": "Mathlib/GroupTheory/GroupAction/BigOperators.lean", "def_pos": [52, 9], "def_end_pos": [52, 24]}]], "state_before": "\u03b9\u271d : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\na b : \u211d\nf\u271d g : \u211d \u2192 E\n\u03bc : Measure \u211d\n\u03b9 : Type u_6\ns : Finset \u03b9\nf : \u03b9 \u2192 \u211d \u2192 E\nh : \u2200 (i : \u03b9), i \u2208 s \u2192 IntervalIntegrable (f i) \u03bc a b\n\u22a2 \u222b (x : \u211d) in a..b, \u2211 i in s, f i x \u2202\u03bc = \u2211 i in s, \u222b (x : \u211d) in a..b, f i x \u2202\u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/LocallyFinite.lean", "full_name": "Finset.Ici_eq_cons_Ioi", "start": [713, 1], "end": [714, 44], "traced_tactics": [{"tactic": "classical rw [cons_eq_insert, Ioi_insert]", "annotated_tactic": ["classical rw [<a>cons_eq_insert</a>, <a>Ioi_insert</a>]", [{"full_name": "Finset.cons_eq_insert", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1108, 9], "def_end_pos": [1108, 23]}, {"full_name": "Finset.Ioi_insert", "def_path": "Mathlib/Data/Finset/LocallyFinite.lean", "def_pos": [701, 9], "def_end_pos": [701, 19]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : LocallyFiniteOrderTop \u03b1\na : \u03b1\n\u22a2 Ici a = cons a (Ioi a) (_ : \u00aca \u2208 Ioi a)", "state_after": "no goals"}, {"tactic": "rw [cons_eq_insert, Ioi_insert]", "annotated_tactic": ["rw [<a>cons_eq_insert</a>, <a>Ioi_insert</a>]", [{"full_name": "Finset.cons_eq_insert", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1108, 9], "def_end_pos": [1108, 23]}, {"full_name": "Finset.Ioi_insert", "def_path": "Mathlib/Data/Finset/LocallyFinite.lean", "def_pos": [701, 9], "def_end_pos": [701, 19]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : LocallyFiniteOrderTop \u03b1\na : \u03b1\n\u22a2 Ici a = cons a (Ioi a) (_ : \u00aca \u2208 Ioi a)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Accumulate.lean", "full_name": "Set.iUnion_accumulate", "start": [51, 1], "end": [56, 49], "traced_tactics": [{"tactic": "apply Subset.antisymm", "annotated_tactic": ["apply <a>Subset.antisymm</a>", [{"full_name": "Set.Subset.antisymm", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [370, 9], "def_end_pos": [370, 24]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns : \u03b1 \u2192 Set \u03b2\nt : \u03b1 \u2192 Set \u03b3\ninst\u271d : Preorder \u03b1\n\u22a2 \u22c3 x, Accumulate s x = \u22c3 x, s x", "state_after": "case h\u2081\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns : \u03b1 \u2192 Set \u03b2\nt : \u03b1 \u2192 Set \u03b3\ninst\u271d : Preorder \u03b1\n\u22a2 \u22c3 x, Accumulate s x \u2286 \u22c3 x, s x\n\ncase h\u2082\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns : \u03b1 \u2192 Set \u03b2\nt : \u03b1 \u2192 Set \u03b3\ninst\u271d : Preorder \u03b1\n\u22a2 \u22c3 x, s x \u2286 \u22c3 x, Accumulate s x"}, {"tactic": "simp only [subset_def, mem_iUnion, exists_imp, mem_accumulate]", "annotated_tactic": ["simp only [<a>subset_def</a>, <a>mem_iUnion</a>, <a>exists_imp</a>, <a>mem_accumulate</a>]", [{"full_name": "Set.subset_def", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [345, 9], "def_end_pos": [345, 19]}, {"full_name": "Set.mem_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [201, 9], "def_end_pos": [201, 19]}, {"full_name": "exists_imp", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [367, 9], "def_end_pos": [367, 19]}, {"full_name": "Set.mem_accumulate", "def_path": "Mathlib/Data/Set/Accumulate.lean", "def_pos": [31, 9], "def_end_pos": [31, 23]}]], "state_before": "case h\u2081\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns : \u03b1 \u2192 Set \u03b2\nt : \u03b1 \u2192 Set \u03b3\ninst\u271d : Preorder \u03b1\n\u22a2 \u22c3 x, Accumulate s x \u2286 \u22c3 x, s x", "state_after": "case h\u2081\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns : \u03b1 \u2192 Set \u03b2\nt : \u03b1 \u2192 Set \u03b3\ninst\u271d : Preorder \u03b1\n\u22a2 \u2200 (x : \u03b2) (x_1 x_2 : \u03b1), x_2 \u2264 x_1 \u2227 x \u2208 s x_2 \u2192 \u2203 i, x \u2208 s i"}, {"tactic": "intro z x x' \u27e8_, hz\u27e9", "annotated_tactic": ["intro z x x' \u27e8_, hz\u27e9", []], "state_before": "case h\u2081\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns : \u03b1 \u2192 Set \u03b2\nt : \u03b1 \u2192 Set \u03b3\ninst\u271d : Preorder \u03b1\n\u22a2 \u2200 (x : \u03b2) (x_1 x_2 : \u03b1), x_2 \u2264 x_1 \u2227 x \u2208 s x_2 \u2192 \u2203 i, x \u2208 s i", "state_after": "case h\u2081\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns : \u03b1 \u2192 Set \u03b2\nt : \u03b1 \u2192 Set \u03b3\ninst\u271d : Preorder \u03b1\nz : \u03b2\nx x' : \u03b1\nleft\u271d : x' \u2264 x\nhz : z \u2208 s x'\n\u22a2 \u2203 i, z \u2208 s i"}, {"tactic": "exact \u27e8x', hz\u27e9", "annotated_tactic": ["exact \u27e8x', hz\u27e9", []], "state_before": "case h\u2081\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns : \u03b1 \u2192 Set \u03b2\nt : \u03b1 \u2192 Set \u03b3\ninst\u271d : Preorder \u03b1\nz : \u03b2\nx x' : \u03b1\nleft\u271d : x' \u2264 x\nhz : z \u2208 s x'\n\u22a2 \u2203 i, z \u2208 s i", "state_after": "no goals"}, {"tactic": "exact iUnion_mono fun i => subset_accumulate", "annotated_tactic": ["exact <a>iUnion_mono</a> fun i => <a>subset_accumulate</a>", [{"full_name": "Set.iUnion_mono", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [478, 9], "def_end_pos": [478, 20]}, {"full_name": "Set.subset_accumulate", "def_path": "Mathlib/Data/Set/Accumulate.lean", "def_pos": [35, 9], "def_end_pos": [35, 26]}]], "state_before": "case h\u2082\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns : \u03b1 \u2192 Set \u03b2\nt : \u03b1 \u2192 Set \u03b3\ninst\u271d : Preorder \u03b1\n\u22a2 \u22c3 x, s x \u2286 \u22c3 x, Accumulate s x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Encoding.lean", "full_name": "Computability.decode_encodeNat", "start": [151, 1], "end": [154, 55], "traced_tactics": [{"tactic": "intro n", "annotated_tactic": ["intro n", []], "state_before": "\u22a2 \u2200 (n : \u2115), decodeNat (encodeNat n) = n", "state_after": "n : \u2115\n\u22a2 decodeNat (encodeNat n) = n"}, {"tactic": "conv_rhs => rw [\u2190 Num.to_of_nat n]", "annotated_tactic": ["conv_rhs => rw [\u2190 <a>Num.to_of_nat</a> n]", [{"full_name": "Num.to_of_nat", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [488, 9], "def_end_pos": [488, 18]}]], "state_before": "n : \u2115\n\u22a2 decodeNat (encodeNat n) = n", "state_after": "n : \u2115\n\u22a2 decodeNat (encodeNat n) = \u2191\u2191n"}, {"tactic": "exact congr_arg ((\u2191) : Num \u2192 \u2115) (decode_encodeNum n)", "annotated_tactic": ["exact <a>congr_arg</a> ((\u2191) : <a>Num</a> \u2192 \u2115) (<a>decode_encodeNum</a> n)", [{"full_name": "congr_arg", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [43, 7], "def_end_pos": [43, 16]}, {"full_name": "Num", "def_path": "Mathlib/Data/Num/Basic.lean", "def_pos": [43, 11], "def_end_pos": [43, 14]}, {"full_name": "Computability.decode_encodeNum", "def_path": "Mathlib/Computability/Encoding.lean", "def_pos": [142, 9], "def_end_pos": [142, 25]}]], "state_before": "n : \u2115\n\u22a2 decodeNat (encodeNat n) = \u2191\u2191n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "full_name": "Int.lt.intro", "start": [594, 1], "end": [595, 22], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "full_name": "aestronglyMeasurable_const_smul_iff\u2080", "start": [1805, 1], "end": [1807, 56], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/NoncommProd.lean", "full_name": "Finset.noncommProd_insert_of_not_mem'", "start": [292, 1], "end": [303, 60], "traced_tactics": [{"tactic": "convert noncommProd_lemma _ f comm using 3", "annotated_tactic": ["convert <a>noncommProd_lemma</a> _ f comm using 3", [{"full_name": "Finset.noncommProd_lemma", "def_path": "Mathlib/Data/Finset/NoncommProd.lean", "def_pos": [230, 9], "def_end_pos": [230, 26]}]], "state_before": "F : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b2 : Monoid \u03b2\ninst\u271d\u00b9 : Monoid \u03b3\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\na : \u03b1\nf : \u03b1 \u2192 \u03b2\ncomm : Set.Pairwise \u2191(insert a s) fun a b => Commute (f a) (f b)\nha : \u00aca \u2208 s\n\u22a2 Set.Pairwise {x | x \u2208 f a ::\u2098 Multiset.map f s.val} Commute", "state_after": "case h.e'_2.h.e'_2.h.a\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b2 : Monoid \u03b2\ninst\u271d\u00b9 : Monoid \u03b3\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\na : \u03b1\nf : \u03b1 \u2192 \u03b2\ncomm : Set.Pairwise \u2191(insert a s) fun a b => Commute (f a) (f b)\nha : \u00aca \u2208 s\nx\u271d : \u03b2\n\u22a2 x\u271d \u2208 f a ::\u2098 Multiset.map f s.val \u2194 x\u271d \u2208 Multiset.map f (insert a s).val"}, {"tactic": "simp [@eq_comm _ (f a)]", "annotated_tactic": ["simp [@<a>eq_comm</a> _ (f a)]", [{"full_name": "eq_comm", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [104, 9], "def_end_pos": [104, 16]}]], "state_before": "case h.e'_2.h.e'_2.h.a\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b2 : Monoid \u03b2\ninst\u271d\u00b9 : Monoid \u03b3\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\na : \u03b1\nf : \u03b1 \u2192 \u03b2\ncomm : Set.Pairwise \u2191(insert a s) fun a b => Commute (f a) (f b)\nha : \u00aca \u2208 s\nx\u271d : \u03b2\n\u22a2 x\u271d \u2208 f a ::\u2098 Multiset.map f s.val \u2194 x\u271d \u2208 Multiset.map f (insert a s).val", "state_after": "no goals"}, {"tactic": "rw [Multiset.noncommProd_cons', noncommProd]", "annotated_tactic": ["rw [<a>Multiset.noncommProd_cons'</a>, <a>noncommProd</a>]", [{"full_name": "Multiset.noncommProd_cons'", "def_path": "Mathlib/Data/Finset/NoncommProd.lean", "def_pos": [150, 9], "def_end_pos": [150, 26]}, {"full_name": "Finset.noncommProd", "def_path": "Mathlib/Data/Finset/NoncommProd.lean", "def_pos": [242, 5], "def_end_pos": [242, 16]}]], "state_before": "F : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b2 : Monoid \u03b2\ninst\u271d\u00b9 : Monoid \u03b3\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\na : \u03b1\nf : \u03b1 \u2192 \u03b2\ncomm : Set.Pairwise \u2191(insert a s) fun a b => Commute (f a) (f b)\nha : \u00aca \u2208 s\n\u22a2 Multiset.noncommProd (f a ::\u2098 Multiset.map f s.val)\n      (_ : Set.Pairwise {x | x \u2208 f a ::\u2098 Multiset.map f s.val} Commute) =\n    noncommProd s f (_ : Set.Pairwise \u2191s fun a b => Commute (f a) (f b)) * f a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Basic.lean", "full_name": "Set.subset_union_compl_iff_inter_subset", "start": [1790, 1], "end": [1791, 54], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "full_name": "Int.mul_pos", "start": [650, 11], "end": [653, 49], "traced_tactics": [{"tactic": "let \u27e8n, hn\u27e9 := eq_succ_of_zero_lt ha", "annotated_tactic": ["let \u27e8n, hn\u27e9 := <a>eq_succ_of_zero_lt</a> ha", [{"full_name": "Int.eq_succ_of_zero_lt", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [587, 9], "def_end_pos": [587, 27]}]], "state_before": "a b : Int\nha : 0 < a\nhb : 0 < b\n\u22a2 0 < a * b", "state_after": "a b : Int\nha : 0 < a\nhb : 0 < b\nn : Nat\nhn : a = \u2191(succ n)\n\u22a2 0 < a * b"}, {"tactic": "let \u27e8m, hm\u27e9 := eq_succ_of_zero_lt hb", "annotated_tactic": ["let \u27e8m, hm\u27e9 := <a>eq_succ_of_zero_lt</a> hb", [{"full_name": "Int.eq_succ_of_zero_lt", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [587, 9], "def_end_pos": [587, 27]}]], "state_before": "a b : Int\nha : 0 < a\nhb : 0 < b\nn : Nat\nhn : a = \u2191(succ n)\n\u22a2 0 < a * b", "state_after": "a b : Int\nha : 0 < a\nhb : 0 < b\nn : Nat\nhn : a = \u2191(succ n)\nm : Nat\nhm : b = \u2191(succ m)\n\u22a2 0 < a * b"}, {"tactic": "rw [hn, hm, \u2190 ofNat_mul]", "annotated_tactic": ["rw [hn, hm, \u2190 <a>ofNat_mul</a>]", [{"full_name": "Int.ofNat_mul", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [36, 9], "def_end_pos": [36, 18]}]], "state_before": "a b : Int\nha : 0 < a\nhb : 0 < b\nn : Nat\nhn : a = \u2191(succ n)\nm : Nat\nhm : b = \u2191(succ m)\n\u22a2 0 < a * b", "state_after": "a b : Int\nha : 0 < a\nhb : 0 < b\nn : Nat\nhn : a = \u2191(succ n)\nm : Nat\nhm : b = \u2191(succ m)\n\u22a2 0 < \u2191(succ n * succ m)"}, {"tactic": "apply ofNat_succ_pos", "annotated_tactic": ["apply <a>ofNat_succ_pos</a>", [{"full_name": "Int.ofNat_succ_pos", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [607, 9], "def_end_pos": [607, 23]}]], "state_before": "a b : Int\nha : 0 < a\nhb : 0 < b\nn : Nat\nhn : a = \u2191(succ n)\nm : Nat\nhm : b = \u2191(succ m)\n\u22a2 0 < \u2191(succ n * succ m)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/QPF/Multivariate/Constructions/Fix.lean", "full_name": "MvQPF.Fix.dest_mk", "start": [297, 1], "end": [307, 29], "traced_tactics": [{"tactic": "unfold Fix.dest", "annotated_tactic": ["unfold <a>Fix.dest</a>", [{"full_name": "MvQPF.Fix.dest", "def_path": "Mathlib/Data/QPF/Multivariate/Constructions/Fix.lean", "def_pos": [221, 5], "def_end_pos": [221, 13]}]], "state_before": "n : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nx : F (\u03b1 ::: Fix F \u03b1)\n\u22a2 dest (mk x) = x", "state_after": "n : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nx : F (\u03b1 ::: Fix F \u03b1)\n\u22a2 rec (MvFunctor.map (TypeVec.id ::: mk)) (mk x) = x"}, {"tactic": "rw [Fix.rec_eq, \u2190 Fix.dest, \u2190 comp_map]", "annotated_tactic": ["rw [<a>Fix.rec_eq</a>, \u2190 <a>Fix.dest</a>, \u2190 <a>comp_map</a>]", [{"full_name": "MvQPF.Fix.rec_eq", "def_path": "Mathlib/Data/QPF/Multivariate/Constructions/Fix.lean", "def_pos": [225, 9], "def_end_pos": [225, 19]}, {"full_name": "MvQPF.Fix.dest", "def_path": "Mathlib/Data/QPF/Multivariate/Constructions/Fix.lean", "def_pos": [221, 5], "def_end_pos": [221, 13]}, {"full_name": "MvQPF.comp_map", "def_path": "Mathlib/Data/QPF/Multivariate/Basic.lean", "def_pos": [112, 9], "def_end_pos": [112, 17]}]], "state_before": "n : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nx : F (\u03b1 ::: Fix F \u03b1)\n\u22a2 rec (MvFunctor.map (TypeVec.id ::: mk)) (mk x) = x", "state_after": "n : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nx : F (\u03b1 ::: Fix F \u03b1)\n\u22a2 ((TypeVec.id ::: mk) \u229a (TypeVec.id ::: dest)) <$$> x = x"}, {"tactic": "conv =>\n  rhs\n  rw [\u2190 MvFunctor.id_map x]", "annotated_tactic": ["conv =>\n    rhs\n    rw [\u2190 <a>MvFunctor.id_map</a> x]", [{"full_name": "MvFunctor.id_map", "def_path": "Mathlib/Control/Functor/Multivariate.lean", "def_pos": [106, 9], "def_end_pos": [106, 15]}]], "state_before": "n : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nx : F (\u03b1 ::: Fix F \u03b1)\n\u22a2 ((TypeVec.id ::: mk) \u229a (TypeVec.id ::: dest)) <$$> x = x", "state_after": "n : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nx : F (\u03b1 ::: Fix F \u03b1)\n\u22a2 ((TypeVec.id ::: mk) \u229a (TypeVec.id ::: dest)) <$$> x = TypeVec.id <$$> x"}, {"tactic": "rw [\u2190 appendFun_comp, id_comp]", "annotated_tactic": ["rw [\u2190 <a>appendFun_comp</a>, <a>id_comp</a>]", [{"full_name": "TypeVec.appendFun_comp", "def_path": "Mathlib/Data/TypeVec.lean", "def_pos": [252, 9], "def_end_pos": [252, 23]}, {"full_name": "TypeVec.id_comp", "def_path": "Mathlib/Data/TypeVec.lean", "def_pos": [79, 9], "def_end_pos": [79, 16]}]], "state_before": "n : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nx : F (\u03b1 ::: Fix F \u03b1)\n\u22a2 ((TypeVec.id ::: mk) \u229a (TypeVec.id ::: dest)) <$$> x = TypeVec.id <$$> x", "state_after": "n : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nx : F (\u03b1 ::: Fix F \u03b1)\n\u22a2 (TypeVec.id ::: mk \u2218 dest) <$$> x = TypeVec.id <$$> x"}, {"tactic": "have : Fix.mk \u2218 Fix.dest = _root_.id := by\n  ext (x : Fix F \u03b1)\n  apply Fix.mk_dest", "annotated_tactic": ["have : <a>Fix.mk</a> \u2218 <a>Fix.dest</a> = <a>_root_.id</a> := by\n    ext (x : <a>Fix</a> F \u03b1)\n    apply <a>Fix.mk_dest</a>", [{"full_name": "MvQPF.Fix.mk", "def_path": "Mathlib/Data/QPF/Multivariate/Constructions/Fix.lean", "def_pos": [216, 5], "def_end_pos": [216, 11]}, {"full_name": "MvQPF.Fix.dest", "def_path": "Mathlib/Data/QPF/Multivariate/Constructions/Fix.lean", "def_pos": [221, 5], "def_end_pos": [221, 13]}, {"full_name": "id", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [33, 15], "def_end_pos": [33, 17]}, {"full_name": "MvQPF.Fix", "def_path": "Mathlib/Data/QPF/Multivariate/Constructions/Fix.lean", "def_pos": [188, 5], "def_end_pos": [188, 8]}, {"full_name": "MvQPF.Fix.mk_dest", "def_path": "Mathlib/Data/QPF/Multivariate/Constructions/Fix.lean", "def_pos": [287, 9], "def_end_pos": [287, 20]}]], "state_before": "n : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nx : F (\u03b1 ::: Fix F \u03b1)\n\u22a2 (TypeVec.id ::: mk \u2218 dest) <$$> x = TypeVec.id <$$> x", "state_after": "n : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nx : F (\u03b1 ::: Fix F \u03b1)\nthis : mk \u2218 dest = _root_.id\n\u22a2 (TypeVec.id ::: mk \u2218 dest) <$$> x = TypeVec.id <$$> x"}, {"tactic": "rw [this, appendFun_id_id]", "annotated_tactic": ["rw [this, <a>appendFun_id_id</a>]", [{"full_name": "TypeVec.appendFun_id_id", "def_path": "Mathlib/Data/TypeVec.lean", "def_pos": [292, 9], "def_end_pos": [292, 24]}]], "state_before": "n : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nx : F (\u03b1 ::: Fix F \u03b1)\nthis : mk \u2218 dest = _root_.id\n\u22a2 (TypeVec.id ::: mk \u2218 dest) <$$> x = TypeVec.id <$$> x", "state_after": "no goals"}, {"tactic": "ext (x : Fix F \u03b1)", "annotated_tactic": ["ext (x : <a>Fix</a> F \u03b1)", [{"full_name": "MvQPF.Fix", "def_path": "Mathlib/Data/QPF/Multivariate/Constructions/Fix.lean", "def_pos": [188, 5], "def_end_pos": [188, 8]}]], "state_before": "n : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nx : F (\u03b1 ::: Fix F \u03b1)\n\u22a2 mk \u2218 dest = _root_.id", "state_after": "case h\nn : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nx\u271d : F (\u03b1 ::: Fix F \u03b1)\nx : Fix F \u03b1\n\u22a2 (mk \u2218 dest) x = _root_.id x"}, {"tactic": "apply Fix.mk_dest", "annotated_tactic": ["apply <a>Fix.mk_dest</a>", [{"full_name": "MvQPF.Fix.mk_dest", "def_path": "Mathlib/Data/QPF/Multivariate/Constructions/Fix.lean", "def_pos": [287, 9], "def_end_pos": [287, 20]}]], "state_before": "case h\nn : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nx\u271d : F (\u03b1 ::: Fix F \u03b1)\nx : Fix F \u03b1\n\u22a2 (mk \u2218 dest) x = _root_.id x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Analysis/Filter.lean", "full_name": "CFilter.coe_mk", "start": [59, 1], "end": [60, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Constructions/Pi.lean", "full_name": "MeasureTheory.measurePreserving_piFinTwo", "start": [830, 1], "end": [836, 6], "traced_tactics": [{"tactic": "refine' \u27e8MeasurableEquiv.measurable _, (Measure.prod_eq fun s t _ _ => _).symm\u27e9", "annotated_tactic": ["refine' \u27e8<a>MeasurableEquiv.measurable</a> _, (<a>Measure.prod_eq</a> fun s t _ _ => _).<a>symm</a>\u27e9", [{"full_name": "MeasurableEquiv.measurable", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [1298, 19], "def_end_pos": [1298, 29]}, {"full_name": "MeasureTheory.Measure.prod_eq", "def_path": "Mathlib/MeasureTheory/Constructions/Prod/Basic.lean", "def_pos": [565, 9], "def_end_pos": [565, 16]}, {"full_name": "Eq.symm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [310, 9], "def_end_pos": [310, 16]}]], "state_before": "\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1\u271d : \u03b9 \u2192 Type u_3\ninst\u271d\u00b3 : Fintype \u03b9\nm\u271d\u00b9 : (i : \u03b9) \u2192 OuterMeasure (\u03b1\u271d i)\nm\u271d : (i : \u03b9) \u2192 MeasurableSpace (\u03b1\u271d i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (\u03b1\u271d i)\ninst\u271d\u00b2 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\ninst\u271d\u00b9 : Fintype \u03b9'\n\u03b1 : Fin 2 \u2192 Type u\nm : (i : Fin 2) \u2192 MeasurableSpace (\u03b1 i)\n\u03bc : (i : Fin 2) \u2192 Measure (\u03b1 i)\ninst\u271d : \u2200 (i : Fin 2), SigmaFinite (\u03bc i)\n\u22a2 MeasurePreserving \u2191(MeasurableEquiv.piFinTwo \u03b1)", "state_after": "\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1\u271d : \u03b9 \u2192 Type u_3\ninst\u271d\u00b3 : Fintype \u03b9\nm\u271d\u00b9 : (i : \u03b9) \u2192 OuterMeasure (\u03b1\u271d i)\nm\u271d : (i : \u03b9) \u2192 MeasurableSpace (\u03b1\u271d i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (\u03b1\u271d i)\ninst\u271d\u00b2 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\ninst\u271d\u00b9 : Fintype \u03b9'\n\u03b1 : Fin 2 \u2192 Type u\nm : (i : Fin 2) \u2192 MeasurableSpace (\u03b1 i)\n\u03bc : (i : Fin 2) \u2192 Measure (\u03b1 i)\ninst\u271d : \u2200 (i : Fin 2), SigmaFinite (\u03bc i)\ns : Set (\u03b1 0)\nt : Set (\u03b1 1)\nx\u271d\u00b9 : MeasurableSet s\nx\u271d : MeasurableSet t\n\u22a2 \u2191\u2191(Measure.map (\u2191(MeasurableEquiv.piFinTwo \u03b1)) (Measure.pi \u03bc)) (s \u00d7\u02e2 t) = \u2191\u2191(\u03bc 0) s * \u2191\u2191(\u03bc 1) t"}, {"tactic": "rw [MeasurableEquiv.map_apply, MeasurableEquiv.piFinTwo_apply, Fin.preimage_apply_01_prod,\n  Measure.pi_pi, Fin.prod_univ_two]", "annotated_tactic": ["rw [<a>MeasurableEquiv.map_apply</a>, <a>MeasurableEquiv.piFinTwo_apply</a>, <a>Fin.preimage_apply_01_prod</a>,\n    <a>Measure.pi_pi</a>, <a>Fin.prod_univ_two</a>]", [{"full_name": "MeasurableEquiv.map_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [4218, 19], "def_end_pos": [4218, 28]}, {"full_name": "MeasurableEquiv.piFinTwo_apply", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [1638, 3], "def_end_pos": [1638, 47]}, {"full_name": "Fin.preimage_apply_01_prod", "def_path": "Mathlib/Logic/Equiv/Fin.lean", "def_pos": [57, 9], "def_end_pos": [57, 35]}, {"full_name": "MeasureTheory.Measure.pi_pi", "def_path": "Mathlib/MeasureTheory/Constructions/Pi.lean", "def_pos": [394, 9], "def_end_pos": [394, 14]}, {"full_name": "Fin.prod_univ_two", "def_path": "Mathlib/Algebra/BigOperators/Fin.lean", "def_pos": [110, 9], "def_end_pos": [110, 22]}]], "state_before": "\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1\u271d : \u03b9 \u2192 Type u_3\ninst\u271d\u00b3 : Fintype \u03b9\nm\u271d\u00b9 : (i : \u03b9) \u2192 OuterMeasure (\u03b1\u271d i)\nm\u271d : (i : \u03b9) \u2192 MeasurableSpace (\u03b1\u271d i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (\u03b1\u271d i)\ninst\u271d\u00b2 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\ninst\u271d\u00b9 : Fintype \u03b9'\n\u03b1 : Fin 2 \u2192 Type u\nm : (i : Fin 2) \u2192 MeasurableSpace (\u03b1 i)\n\u03bc : (i : Fin 2) \u2192 Measure (\u03b1 i)\ninst\u271d : \u2200 (i : Fin 2), SigmaFinite (\u03bc i)\ns : Set (\u03b1 0)\nt : Set (\u03b1 1)\nx\u271d\u00b9 : MeasurableSet s\nx\u271d : MeasurableSet t\n\u22a2 \u2191\u2191(Measure.map (\u2191(MeasurableEquiv.piFinTwo \u03b1)) (Measure.pi \u03bc)) (s \u00d7\u02e2 t) = \u2191\u2191(\u03bc 0) s * \u2191\u2191(\u03bc 1) t", "state_after": "\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1\u271d : \u03b9 \u2192 Type u_3\ninst\u271d\u00b3 : Fintype \u03b9\nm\u271d\u00b9 : (i : \u03b9) \u2192 OuterMeasure (\u03b1\u271d i)\nm\u271d : (i : \u03b9) \u2192 MeasurableSpace (\u03b1\u271d i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (\u03b1\u271d i)\ninst\u271d\u00b2 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\ninst\u271d\u00b9 : Fintype \u03b9'\n\u03b1 : Fin 2 \u2192 Type u\nm : (i : Fin 2) \u2192 MeasurableSpace (\u03b1 i)\n\u03bc : (i : Fin 2) \u2192 Measure (\u03b1 i)\ninst\u271d : \u2200 (i : Fin 2), SigmaFinite (\u03bc i)\ns : Set (\u03b1 0)\nt : Set (\u03b1 1)\nx\u271d\u00b9 : MeasurableSet s\nx\u271d : MeasurableSet t\n\u22a2 \u2191\u2191(\u03bc 0) (Fin.cons s (Fin.cons t finZeroElim) 0) * \u2191\u2191(\u03bc 1) (Fin.cons s (Fin.cons t finZeroElim) 1) =\n    \u2191\u2191(\u03bc 0) s * \u2191\u2191(\u03bc 1) t"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1\u271d : \u03b9 \u2192 Type u_3\ninst\u271d\u00b3 : Fintype \u03b9\nm\u271d\u00b9 : (i : \u03b9) \u2192 OuterMeasure (\u03b1\u271d i)\nm\u271d : (i : \u03b9) \u2192 MeasurableSpace (\u03b1\u271d i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (\u03b1\u271d i)\ninst\u271d\u00b2 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\ninst\u271d\u00b9 : Fintype \u03b9'\n\u03b1 : Fin 2 \u2192 Type u\nm : (i : Fin 2) \u2192 MeasurableSpace (\u03b1 i)\n\u03bc : (i : Fin 2) \u2192 Measure (\u03b1 i)\ninst\u271d : \u2200 (i : Fin 2), SigmaFinite (\u03bc i)\ns : Set (\u03b1 0)\nt : Set (\u03b1 1)\nx\u271d\u00b9 : MeasurableSet s\nx\u271d : MeasurableSet t\n\u22a2 \u2191\u2191(\u03bc 0) (Fin.cons s (Fin.cons t finZeroElim) 0) * \u2191\u2191(\u03bc 1) (Fin.cons s (Fin.cons t finZeroElim) 1) =\n    \u2191\u2191(\u03bc 0) s * \u2191\u2191(\u03bc 1) t", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean", "full_name": "MeasureTheory.Measure.addHaar_image_linearMap", "start": [313, 1], "end": [323, 93], "traced_tactics": [{"tactic": "rcases ne_or_eq (LinearMap.det f) 0 with (hf | hf)", "annotated_tactic": ["rcases <a>ne_or_eq</a> (<a>LinearMap.det</a> f) 0 with (hf | hf)", [{"full_name": "ne_or_eq", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [212, 9], "def_end_pos": [212, 17]}, {"full_name": "LinearMap.det", "def_path": "Mathlib/LinearAlgebra/Determinant.lean", "def_pos": [178, 27], "def_end_pos": [178, 30]}]], "state_before": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf : E \u2192\u2097[\u211d] E\ns : Set E\n\u22a2 \u2191\u2191\u03bc (\u2191f '' s) = ENNReal.ofReal |\u2191LinearMap.det f| * \u2191\u2191\u03bc s", "state_after": "case inl\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf : E \u2192\u2097[\u211d] E\ns : Set E\nhf : \u2191LinearMap.det f \u2260 0\n\u22a2 \u2191\u2191\u03bc (\u2191f '' s) = ENNReal.ofReal |\u2191LinearMap.det f| * \u2191\u2191\u03bc s\n\ncase inr\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf : E \u2192\u2097[\u211d] E\ns : Set E\nhf : \u2191LinearMap.det f = 0\n\u22a2 \u2191\u2191\u03bc (\u2191f '' s) = ENNReal.ofReal |\u2191LinearMap.det f| * \u2191\u2191\u03bc s"}, {"tactic": "let g := (f.equivOfDetNeZero hf).toContinuousLinearEquiv", "annotated_tactic": ["let g := (f.equivOfDetNeZero hf).<a>toContinuousLinearEquiv</a>", [{"full_name": "LinearEquiv.toContinuousLinearEquiv", "def_path": "Mathlib/Topology/Algebra/Module/FiniteDimension.lean", "def_pos": [357, 5], "def_end_pos": [357, 28]}]], "state_before": "case inl\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf : E \u2192\u2097[\u211d] E\ns : Set E\nhf : \u2191LinearMap.det f \u2260 0\n\u22a2 \u2191\u2191\u03bc (\u2191f '' s) = ENNReal.ofReal |\u2191LinearMap.det f| * \u2191\u2191\u03bc s", "state_after": "case inl\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf : E \u2192\u2097[\u211d] E\ns : Set E\nhf : \u2191LinearMap.det f \u2260 0\ng : E \u2243L[\u211d] E := LinearEquiv.toContinuousLinearEquiv (LinearMap.equivOfDetNeZero f hf)\n\u22a2 \u2191\u2191\u03bc (\u2191f '' s) = ENNReal.ofReal |\u2191LinearMap.det f| * \u2191\u2191\u03bc s"}, {"tactic": "change \u03bc (g '' s) = _", "annotated_tactic": ["change \u03bc (g '' s) = _", []], "state_before": "case inl\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf : E \u2192\u2097[\u211d] E\ns : Set E\nhf : \u2191LinearMap.det f \u2260 0\ng : E \u2243L[\u211d] E := LinearEquiv.toContinuousLinearEquiv (LinearMap.equivOfDetNeZero f hf)\n\u22a2 \u2191\u2191\u03bc (\u2191f '' s) = ENNReal.ofReal |\u2191LinearMap.det f| * \u2191\u2191\u03bc s", "state_after": "case inl\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf : E \u2192\u2097[\u211d] E\ns : Set E\nhf : \u2191LinearMap.det f \u2260 0\ng : E \u2243L[\u211d] E := LinearEquiv.toContinuousLinearEquiv (LinearMap.equivOfDetNeZero f hf)\n\u22a2 \u2191\u2191\u03bc (\u2191g '' s) = ENNReal.ofReal |\u2191LinearMap.det f| * \u2191\u2191\u03bc s"}, {"tactic": "rw [ContinuousLinearEquiv.image_eq_preimage g s, addHaar_preimage_continuousLinearEquiv]", "annotated_tactic": ["rw [<a>ContinuousLinearEquiv.image_eq_preimage</a> g s, <a>addHaar_preimage_continuousLinearEquiv</a>]", [{"full_name": "ContinuousLinearEquiv.image_eq_preimage", "def_path": "Mathlib/Topology/Algebra/Module/Basic.lean", "def_pos": [2202, 19], "def_end_pos": [2202, 36]}, {"full_name": "MeasureTheory.Measure.addHaar_preimage_continuousLinearEquiv", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean", "def_pos": [305, 9], "def_end_pos": [305, 47]}]], "state_before": "case inl\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf : E \u2192\u2097[\u211d] E\ns : Set E\nhf : \u2191LinearMap.det f \u2260 0\ng : E \u2243L[\u211d] E := LinearEquiv.toContinuousLinearEquiv (LinearMap.equivOfDetNeZero f hf)\n\u22a2 \u2191\u2191\u03bc (\u2191g '' s) = ENNReal.ofReal |\u2191LinearMap.det f| * \u2191\u2191\u03bc s", "state_after": "case inl\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf : E \u2192\u2097[\u211d] E\ns : Set E\nhf : \u2191LinearMap.det f \u2260 0\ng : E \u2243L[\u211d] E := LinearEquiv.toContinuousLinearEquiv (LinearMap.equivOfDetNeZero f hf)\n\u22a2 ENNReal.ofReal |\u2191LinearMap.det \u2191\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.symm g))| * \u2191\u2191\u03bc s =\n    ENNReal.ofReal |\u2191LinearMap.det f| * \u2191\u2191\u03bc s"}, {"tactic": "congr", "annotated_tactic": ["congr", []], "state_before": "case inl\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf : E \u2192\u2097[\u211d] E\ns : Set E\nhf : \u2191LinearMap.det f \u2260 0\ng : E \u2243L[\u211d] E := LinearEquiv.toContinuousLinearEquiv (LinearMap.equivOfDetNeZero f hf)\n\u22a2 ENNReal.ofReal |\u2191LinearMap.det \u2191\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.symm g))| * \u2191\u2191\u03bc s =\n    ENNReal.ofReal |\u2191LinearMap.det f| * \u2191\u2191\u03bc s", "state_after": "no goals"}, {"tactic": "simp only [hf, zero_mul, ENNReal.ofReal_zero, abs_zero]", "annotated_tactic": ["simp only [hf, <a>zero_mul</a>, <a>ENNReal.ofReal_zero</a>, <a>abs_zero</a>]", [{"full_name": "MulZeroClass.zero_mul", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [36, 3], "def_end_pos": [36, 11]}, {"full_name": "ENNReal.ofReal_zero", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [245, 17], "def_end_pos": [245, 28]}, {"full_name": "abs_zero", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [128, 9], "def_end_pos": [128, 17]}]], "state_before": "case inr\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf : E \u2192\u2097[\u211d] E\ns : Set E\nhf : \u2191LinearMap.det f = 0\n\u22a2 \u2191\u2191\u03bc (\u2191f '' s) = ENNReal.ofReal |\u2191LinearMap.det f| * \u2191\u2191\u03bc s", "state_after": "case inr\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf : E \u2192\u2097[\u211d] E\ns : Set E\nhf : \u2191LinearMap.det f = 0\n\u22a2 \u2191\u2191\u03bc ((fun a => \u2191f a) '' s) = 0"}, {"tactic": "have : \u03bc (LinearMap.range f) = 0 :=\n  addHaar_submodule \u03bc _ (LinearMap.range_lt_top_of_det_eq_zero hf).ne", "annotated_tactic": ["have : \u03bc (<a>LinearMap.range</a> f) = 0 :=\n      <a>addHaar_submodule</a> \u03bc _ (<a>LinearMap.range_lt_top_of_det_eq_zero</a> hf).<a>ne</a>", [{"full_name": "LinearMap.range", "def_path": "Mathlib/LinearAlgebra/Basic.lean", "def_pos": [1080, 5], "def_end_pos": [1080, 10]}, {"full_name": "MeasureTheory.Measure.addHaar_submodule", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean", "def_pos": [179, 9], "def_end_pos": [179, 26]}, {"full_name": "LinearMap.range_lt_top_of_det_eq_zero", "def_path": "Mathlib/LinearAlgebra/Determinant.lean", "def_pos": [347, 9], "def_end_pos": [347, 36]}, {"full_name": "LT.lt.ne", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [152, 7], "def_end_pos": [152, 15]}]], "state_before": "case inr\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf : E \u2192\u2097[\u211d] E\ns : Set E\nhf : \u2191LinearMap.det f = 0\n\u22a2 \u2191\u2191\u03bc ((fun a => \u2191f a) '' s) = 0", "state_after": "case inr\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf : E \u2192\u2097[\u211d] E\ns : Set E\nhf : \u2191LinearMap.det f = 0\nthis : \u2191\u2191\u03bc \u2191(LinearMap.range f) = 0\n\u22a2 \u2191\u2191\u03bc ((fun a => \u2191f a) '' s) = 0"}, {"tactic": "exact le_antisymm (le_trans (measure_mono (image_subset_range _ _)) this.le) (zero_le _)", "annotated_tactic": ["exact <a>le_antisymm</a> (<a>le_trans</a> (<a>measure_mono</a> (<a>image_subset_range</a> _ _)) this.le) (<a>zero_le</a> _)", [{"full_name": "le_antisymm", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [188, 9], "def_end_pos": [188, 20]}, {"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}, {"full_name": "MeasureTheory.measure_mono", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [193, 9], "def_end_pos": [193, 21]}, {"full_name": "Set.image_subset_range", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [723, 9], "def_end_pos": [723, 27]}, {"full_name": "zero_le", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [217, 30], "def_end_pos": [217, 37]}]], "state_before": "case inr\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf : E \u2192\u2097[\u211d] E\ns : Set E\nhf : \u2191LinearMap.det f = 0\nthis : \u2191\u2191\u03bc \u2191(LinearMap.range f) = 0\n\u22a2 \u2191\u2191\u03bc ((fun a => \u2191f a) '' s) = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/L2Space.lean", "full_name": "MeasureTheory.L2.mem_L1_inner", "start": [184, 1], "end": [188, 84], "traced_tactics": [{"tactic": "simp_rw [mem_Lp_iff_snorm_lt_top, snorm_aeeqFun]", "annotated_tactic": ["simp_rw [<a>mem_Lp_iff_snorm_lt_top</a>, <a>snorm_aeeqFun</a>]", [{"full_name": "MeasureTheory.Lp.mem_Lp_iff_snorm_lt_top", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [171, 9], "def_end_pos": [171, 32]}, {"full_name": "MeasureTheory.snorm_aeeqFun", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [86, 9], "def_end_pos": [86, 22]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : NormedAddCommGroup F\nf g : { x // x \u2208 Lp E 2 }\n\u22a2 AEEqFun.mk (fun x => inner (\u2191\u2191f x) (\u2191\u2191g x)) (_ : AEStronglyMeasurable (fun x => inner (\u2191\u2191f x) (\u2191\u2191g x)) \u03bc) \u2208 Lp \ud835\udd5c 1", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : NormedAddCommGroup F\nf g : { x // x \u2208 Lp E 2 }\n\u22a2 snorm (fun x => inner (\u2191\u2191f x) (\u2191\u2191g x)) 1 \u03bc < \u22a4"}, {"tactic": "exact snorm_inner_lt_top f g", "annotated_tactic": ["exact <a>snorm_inner_lt_top</a> f g", [{"full_name": "MeasureTheory.L2.snorm_inner_lt_top", "def_path": "Mathlib/MeasureTheory/Function/L2Space.lean", "def_pos": [128, 9], "def_end_pos": [128, 27]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : NormedAddCommGroup F\nf g : { x // x \u2208 Lp E 2 }\n\u22a2 snorm (fun x => inner (\u2191\u2191f x) (\u2191\u2191g x)) 1 \u03bc < \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Independence/ZeroOne.lean", "full_name": "ProbabilityTheory.measure_zero_or_one_of_measurableSet_limsup_atTop", "start": [149, 1], "end": [153, 87], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "full_name": "MeasureTheory.Lp.norm_zero", "start": [328, 1], "end": [329, 40], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Finite.lean", "full_name": "Set.infinite_of_not_bddBelow", "start": [1648, 1], "end": [1648, 82], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Int/Bitwise.lean", "full_name": "Int.bit1_ne_bit0", "start": [186, 1], "end": [187, 26], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "full_name": "Int.lt_iff_le_and_ne", "start": [637, 11], "end": [641, 82], "traced_tactics": [{"tactic": "refine \u27e8fun h => \u27e8Int.le_of_lt h, Int.ne_of_lt h\u27e9, fun \u27e8aleb, aneb\u27e9 => ?_\u27e9", "annotated_tactic": ["refine \u27e8fun h => \u27e8<a>Int.le_of_lt</a> h, <a>Int.ne_of_lt</a> h\u27e9, fun \u27e8aleb, aneb\u27e9 => ?_\u27e9", [{"full_name": "Int.le_of_lt", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [634, 19], "def_end_pos": [634, 27]}, {"full_name": "Int.ne_of_lt", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [629, 19], "def_end_pos": [629, 27]}]], "state_before": "a b : Int\n\u22a2 a < b \u2194 a \u2264 b \u2227 a \u2260 b", "state_after": "a b : Int\nx\u271d : a \u2264 b \u2227 a \u2260 b\naleb : a \u2264 b\naneb : a \u2260 b\n\u22a2 a < b"}, {"tactic": "let \u27e8n, hn\u27e9 := le.dest aleb", "annotated_tactic": ["let \u27e8n, hn\u27e9 := <a>le.dest</a> aleb", [{"full_name": "Int.le.dest", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [566, 9], "def_end_pos": [566, 16]}]], "state_before": "a b : Int\nx\u271d : a \u2264 b \u2227 a \u2260 b\naleb : a \u2264 b\naneb : a \u2260 b\n\u22a2 a < b", "state_after": "a b : Int\nx\u271d : a \u2264 b \u2227 a \u2260 b\naleb : a \u2264 b\naneb : a \u2260 b\nn : Nat\nhn : a + \u2191n = b\n\u22a2 a < b"}, {"tactic": "have : n \u2260 0 := aneb.imp fun eq => by rw [\u2190 hn, eq, ofNat_zero, Int.add_zero]", "annotated_tactic": ["have : n \u2260 0 := aneb.imp fun eq => by rw [\u2190 hn, eq, <a>ofNat_zero</a>, <a>Int.add_zero</a>]", [{"full_name": "Int.ofNat_zero", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [19, 17], "def_end_pos": [19, 27]}, {"full_name": "Int.add_zero", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [224, 33], "def_end_pos": [224, 41]}]], "state_before": "a b : Int\nx\u271d : a \u2264 b \u2227 a \u2260 b\naleb : a \u2264 b\naneb : a \u2260 b\nn : Nat\nhn : a + \u2191n = b\n\u22a2 a < b", "state_after": "a b : Int\nx\u271d : a \u2264 b \u2227 a \u2260 b\naleb : a \u2264 b\naneb : a \u2260 b\nn : Nat\nhn : a + \u2191n = b\nthis : n \u2260 0\n\u22a2 a < b"}, {"tactic": "apply lt.intro", "annotated_tactic": ["apply <a>lt.intro</a>", [{"full_name": "Int.lt.intro", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [594, 9], "def_end_pos": [594, 17]}]], "state_before": "a b : Int\nx\u271d : a \u2264 b \u2227 a \u2260 b\naleb : a \u2264 b\naneb : a \u2260 b\nn : Nat\nhn : a + \u2191n = b\nthis : n \u2260 0\n\u22a2 a < b", "state_after": "case h\na b : Int\nx\u271d : a \u2264 b \u2227 a \u2260 b\naleb : a \u2264 b\naneb : a \u2260 b\nn : Nat\nhn : a + \u2191n = b\nthis : n \u2260 0\n\u22a2 a + \u2191(succ ?n) = b\n\ncase n\na b : Int\nx\u271d : a \u2264 b \u2227 a \u2260 b\naleb : a \u2264 b\naneb : a \u2260 b\nn : Nat\nhn : a + \u2191n = b\nthis : n \u2260 0\n\u22a2 Nat"}, {"tactic": "rwa [\u2190 Nat.succ_pred_eq_of_pos (Nat.pos_of_ne_zero this)] at hn", "annotated_tactic": ["rwa [\u2190 <a>Nat.succ_pred_eq_of_pos</a> (<a>Nat.pos_of_ne_zero</a> this)] at hn", [{"full_name": "Nat.succ_pred_eq_of_pos", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [232, 9], "def_end_pos": [232, 28]}, {"full_name": "Nat.pos_of_ne_zero", "def_path": "lake-packages/std/Std/Data/Nat/Init/Lemmas.lean", "def_pos": [25, 19], "def_end_pos": [25, 33]}]], "state_before": "case h\na b : Int\nx\u271d : a \u2264 b \u2227 a \u2260 b\naleb : a \u2264 b\naneb : a \u2260 b\nn : Nat\nhn : a + \u2191n = b\nthis : n \u2260 0\n\u22a2 a + \u2191(succ ?n) = b\n\ncase n\na b : Int\nx\u271d : a \u2264 b \u2227 a \u2260 b\naleb : a \u2264 b\naneb : a \u2260 b\nn : Nat\nhn : a + \u2191n = b\nthis : n \u2260 0\n\u22a2 Nat", "state_after": "no goals"}, {"tactic": "rw [\u2190 hn, eq, ofNat_zero, Int.add_zero]", "annotated_tactic": ["rw [\u2190 hn, eq, <a>ofNat_zero</a>, <a>Int.add_zero</a>]", [{"full_name": "Int.ofNat_zero", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [19, 17], "def_end_pos": [19, 27]}, {"full_name": "Int.add_zero", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [224, 33], "def_end_pos": [224, 41]}]], "state_before": "a b : Int\nx\u271d : a \u2264 b \u2227 a \u2260 b\naleb : a \u2264 b\naneb : a \u2260 b\nn : Nat\nhn : a + \u2191n = b\neq : n = 0\n\u22a2 a = b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "Finset.monotone_filter_right", "start": [2791, 1], "end": [2793, 65], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "full_name": "MeasureTheory.measure_biUnion_toMeasurable", "start": [379, 1], "end": [382, 61], "traced_tactics": [{"tactic": "haveI := hc.toEncodable", "annotated_tactic": ["haveI := hc.toEncodable", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\nI : Set \u03b2\nhc : Set.Countable I\ns : \u03b2 \u2192 Set \u03b1\n\u22a2 \u2191\u2191\u03bc (\u22c3 b \u2208 I, toMeasurable \u03bc (s b)) = \u2191\u2191\u03bc (\u22c3 b \u2208 I, s b)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\nI : Set \u03b2\nhc : Set.Countable I\ns : \u03b2 \u2192 Set \u03b1\nthis : Encodable \u2191I\n\u22a2 \u2191\u2191\u03bc (\u22c3 b \u2208 I, toMeasurable \u03bc (s b)) = \u2191\u2191\u03bc (\u22c3 b \u2208 I, s b)"}, {"tactic": "simp only [biUnion_eq_iUnion, measure_iUnion_toMeasurable]", "annotated_tactic": ["simp only [<a>biUnion_eq_iUnion</a>, <a>measure_iUnion_toMeasurable</a>]", [{"full_name": "Set.biUnion_eq_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [1010, 9], "def_end_pos": [1010, 26]}, {"full_name": "MeasureTheory.measure_iUnion_toMeasurable", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [372, 9], "def_end_pos": [372, 36]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\nI : Set \u03b2\nhc : Set.Countable I\ns : \u03b2 \u2192 Set \u03b1\nthis : Encodable \u2191I\n\u22a2 \u2191\u2191\u03bc (\u22c3 b \u2208 I, toMeasurable \u03bc (s b)) = \u2191\u2191\u03bc (\u22c3 b \u2208 I, s b)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Option/Basic.lean", "full_name": "Option.pbind_eq_bind", "start": [156, 1], "end": [157, 56], "traced_tactics": [{"tactic": "cases x <;> simp only [pbind, none_bind', some_bind']", "annotated_tactic": ["cases x <;> simp only [<a>pbind</a>, <a>none_bind'</a>, <a>some_bind'</a>]", [{"full_name": "Option.pbind", "def_path": "lake-packages/std/Std/Data/Option/Basic.lean", "def_pos": [94, 5], "def_end_pos": [94, 10]}, {"full_name": "Option.none_bind'", "def_path": "Mathlib/Data/Option/Basic.lean", "def_pos": [92, 9], "def_end_pos": [92, 19]}, {"full_name": "Option.some_bind'", "def_path": "Mathlib/Data/Option/Basic.lean", "def_pos": [97, 9], "def_end_pos": [97, 19]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\np : \u03b1 \u2192 Prop\nf\u271d : (a : \u03b1) \u2192 p a \u2192 \u03b2\nx\u271d : Option \u03b1\nf : \u03b1 \u2192 Option \u03b2\nx : Option \u03b1\n\u22a2 (pbind x fun a x => f a) = Option.bind x f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Intervals/OrdConnectedComponent.lean", "full_name": "Set.self_mem_ordConnectedComponent", "start": [52, 1], "end": [53, 66], "traced_tactics": [{"tactic": "rw [mem_ordConnectedComponent, uIcc_self, singleton_subset_iff]", "annotated_tactic": ["rw [<a>mem_ordConnectedComponent</a>, <a>uIcc_self</a>, <a>singleton_subset_iff</a>]", [{"full_name": "Set.mem_ordConnectedComponent", "def_path": "Mathlib/Data/Set/Intervals/OrdConnectedComponent.lean", "def_pos": [32, 9], "def_end_pos": [32, 34]}, {"full_name": "Set.uIcc_self", "def_path": "Mathlib/Data/Set/Intervals/UnorderedInterval.lean", "def_pos": [86, 7], "def_end_pos": [86, 16]}, {"full_name": "Set.singleton_subset_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1330, 9], "def_end_pos": [1330, 29]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\ns t : Set \u03b1\nx y z : \u03b1\n\u22a2 x \u2208 ordConnectedComponent s x \u2194 x \u2208 s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "full_name": "MeasureTheory.Measure.restrict_restrict'", "start": [1660, 1], "end": [1662, 43], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/PeakFunction.lean", "full_name": "tendsto_set_integral_pow_smul_of_unique_maximum_of_isCompact_of_measure_nhdsWithin_pos", "start": [195, 1], "end": [282, 39], "traced_tactics": [{"tactic": "let \u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b x in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n", "annotated_tactic": ["let \u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b x in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u22a2 Tendsto (fun n => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 \u2022 \u222b (x : \u03b1) in s, c x ^ n \u2022 g x \u2202\u03bc) atTop (\ud835\udcdd (g x\u2080))", "state_after": "\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\n\u22a2 Tendsto (fun n => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 \u2022 \u222b (x : \u03b1) in s, c x ^ n \u2022 g x \u2202\u03bc) atTop (\ud835\udcdd (g x\u2080))"}, {"tactic": "have hn\u03c6 : \u2200 n, \u2200 x \u2208 s, 0 \u2264 \u03c6 n x := by\n  intro n x hx\n  apply mul_nonneg (inv_nonneg.2 _) (pow_nonneg (hnc x hx) _)\n  exact set_integral_nonneg hs.measurableSet fun x hx => pow_nonneg (hnc x hx) _", "annotated_tactic": ["have hn\u03c6 : \u2200 n, \u2200 x \u2208 s, 0 \u2264 \u03c6 n x := by\n    intro n x hx\n    apply <a>mul_nonneg</a> (<a>inv_nonneg</a>.2 _) (<a>pow_nonneg</a> (hnc x hx) _)\n    exact <a>set_integral_nonneg</a> hs.measurableSet fun x hx => <a>pow_nonneg</a> (hnc x hx) _", [{"full_name": "mul_nonneg", "def_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "def_pos": [380, 7], "def_end_pos": [380, 17]}, {"full_name": "inv_nonneg", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [58, 9], "def_end_pos": [58, 19]}, {"full_name": "pow_nonneg", "def_path": "Mathlib/Algebra/Order/Ring/Defs.lean", "def_pos": [244, 9], "def_end_pos": [244, 19]}, {"full_name": "MeasureTheory.set_integral_nonneg", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [766, 9], "def_end_pos": [766, 28]}, {"full_name": "pow_nonneg", "def_path": "Mathlib/Algebra/Order/Ring/Defs.lean", "def_pos": [244, 9], "def_end_pos": [244, 19]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\n\u22a2 Tendsto (fun n => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 \u2022 \u222b (x : \u03b1) in s, c x ^ n \u2022 g x \u2202\u03bc) atTop (\ud835\udcdd (g x\u2080))", "state_after": "\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\n\u22a2 Tendsto (fun n => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 \u2022 \u222b (x : \u03b1) in s, c x ^ n \u2022 g x \u2202\u03bc) atTop (\ud835\udcdd (g x\u2080))"}, {"tactic": "have I : \u2200 n, IntegrableOn (fun x => c x ^ n) s \u03bc := fun n =>\n  ContinuousOn.integrableOn_compact hs (hc.pow n)", "annotated_tactic": ["have I : \u2200 n, <a>IntegrableOn</a> (fun x => c x ^ n) s \u03bc := fun n =>\n    <a>ContinuousOn.integrableOn_compact</a> hs (hc.pow n)", [{"full_name": "MeasureTheory.IntegrableOn", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [90, 5], "def_end_pos": [90, 17]}, {"full_name": "ContinuousOn.integrableOn_compact", "def_path": "Mathlib/MeasureTheory/Function/LocallyIntegrable.lean", "def_pos": [394, 9], "def_end_pos": [394, 42]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\n\u22a2 Tendsto (fun n => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 \u2022 \u222b (x : \u03b1) in s, c x ^ n \u2022 g x \u2202\u03bc) atTop (\ud835\udcdd (g x\u2080))", "state_after": "\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\n\u22a2 Tendsto (fun n => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 \u2022 \u222b (x : \u03b1) in s, c x ^ n \u2022 g x \u2202\u03bc) atTop (\ud835\udcdd (g x\u2080))"}, {"tactic": "have J : \u2200 n, 0 \u2264\u1d50[\u03bc.restrict s] fun x : \u03b1 => c x ^ n := by\n  intro n\n  filter_upwards [ae_restrict_mem hs.measurableSet] with x hx\n  exact pow_nonneg (hnc x hx) n", "annotated_tactic": ["have J : \u2200 n, 0 \u2264\u1d50[\u03bc.restrict s] fun x : \u03b1 => c x ^ n := by\n    intro n\n    filter_upwards [<a>ae_restrict_mem</a> hs.measurableSet] with x hx\n    exact <a>pow_nonneg</a> (hnc x hx) n", [{"full_name": "MeasureTheory.ae_restrict_mem", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2586, 9], "def_end_pos": [2586, 24]}, {"full_name": "pow_nonneg", "def_path": "Mathlib/Algebra/Order/Ring/Defs.lean", "def_pos": [244, 9], "def_end_pos": [244, 19]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\n\u22a2 Tendsto (fun n => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 \u2022 \u222b (x : \u03b1) in s, c x ^ n \u2022 g x \u2202\u03bc) atTop (\ud835\udcdd (g x\u2080))", "state_after": "\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\n\u22a2 Tendsto (fun n => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 \u2022 \u222b (x : \u03b1) in s, c x ^ n \u2022 g x \u2202\u03bc) atTop (\ud835\udcdd (g x\u2080))"}, {"tactic": "have P : \u2200 n, (0 : \u211d) < \u222b x in s, c x ^ n \u2202\u03bc := by\n  intro n\n  refine' (set_integral_pos_iff_support_of_nonneg_ae (J n) (I n)).2 _\n  obtain \u27e8u, u_open, x\u2080_u, hu\u27e9 : \u2203 u : Set \u03b1, IsOpen u \u2227 x\u2080 \u2208 u \u2227 u \u2229 s \u2286 c \u207b\u00b9' Ioi 0 :=\n    _root_.continuousOn_iff.1 hc x\u2080 h\u2080 (Ioi (0 : \u211d)) isOpen_Ioi hnc\u2080\n  apply (h\u03bc u u_open x\u2080_u).trans_le\n  exact measure_mono fun x hx => \u27e8ne_of_gt (pow_pos (a := c x) (hu hx) _), hx.2\u27e9", "annotated_tactic": ["have P : \u2200 n, (0 : \u211d) < \u222b x in s, c x ^ n \u2202\u03bc := by\n    intro n\n    refine' (<a>set_integral_pos_iff_support_of_nonneg_ae</a> (J n) (I n)).2 _\n    obtain \u27e8u, u_open, x\u2080_u, hu\u27e9 : \u2203 u : <a>Set</a> \u03b1, <a>IsOpen</a> u \u2227 x\u2080 \u2208 u \u2227 u \u2229 s \u2286 c \u207b\u00b9' <a>Ioi</a> 0 :=\n      <a>_root_.continuousOn_iff</a>.1 hc x\u2080 h\u2080 (<a>Ioi</a> (0 : \u211d)) <a>isOpen_Ioi</a> hnc\u2080\n    apply (h\u03bc u u_open x\u2080_u).<a>trans_le</a>\n    exact <a>measure_mono</a> fun x hx => \u27e8<a>ne_of_gt</a> (<a>pow_pos</a> (a := c x) (hu hx) _), hx.2\u27e9", [{"full_name": "MeasureTheory.set_integral_pos_iff_support_of_nonneg_ae", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [590, 9], "def_end_pos": [590, 50]}, {"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}, {"full_name": "IsOpen", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [101, 5], "def_end_pos": [101, 11]}, {"full_name": "Set.Ioi", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [79, 5], "def_end_pos": [79, 8]}, {"full_name": "continuousOn_iff", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [637, 9], "def_end_pos": [637, 25]}, {"full_name": "Set.Ioi", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [79, 5], "def_end_pos": [79, 8]}, {"full_name": "isOpen_Ioi", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [332, 9], "def_end_pos": [332, 19]}, {"full_name": "LT.lt.trans_le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [148, 7], "def_end_pos": [148, 21]}, {"full_name": "MeasureTheory.measure_mono", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [193, 9], "def_end_pos": [193, 21]}, {"full_name": "ne_of_gt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [104, 9], "def_end_pos": [104, 17]}, {"full_name": "pow_pos", "def_path": "Mathlib/Algebra/Order/Ring/Defs.lean", "def_pos": [530, 9], "def_end_pos": [530, 16]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\n\u22a2 Tendsto (fun n => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 \u2022 \u222b (x : \u03b1) in s, c x ^ n \u2022 g x \u2202\u03bc) atTop (\ud835\udcdd (g x\u2080))", "state_after": "\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nP : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\n\u22a2 Tendsto (fun n => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 \u2022 \u222b (x : \u03b1) in s, c x ^ n \u2022 g x \u2202\u03bc) atTop (\ud835\udcdd (g x\u2080))"}, {"tactic": "have hi\u03c6 : \u2200 n, \u222b x in s, \u03c6 n x \u2202\u03bc = 1 := fun n => by\n  rw [integral_mul_left, inv_mul_cancel (P n).ne']", "annotated_tactic": ["have hi\u03c6 : \u2200 n, \u222b x in s, \u03c6 n x \u2202\u03bc = 1 := fun n => by\n    rw [<a>integral_mul_left</a>, <a>inv_mul_cancel</a> (P n).<a>ne'</a>]", [{"full_name": "MeasureTheory.integral_mul_left", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [923, 9], "def_end_pos": [923, 26]}, {"full_name": "inv_mul_cancel", "def_path": "Mathlib/Algebra/GroupWithZero/NeZero.lean", "def_pos": [55, 9], "def_end_pos": [55, 23]}, {"full_name": "LT.lt.ne'", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [328, 9], "def_end_pos": [328, 12]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nP : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\n\u22a2 Tendsto (fun n => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 \u2022 \u222b (x : \u03b1) in s, c x ^ n \u2022 g x \u2202\u03bc) atTop (\ud835\udcdd (g x\u2080))", "state_after": "\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nP : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\nhi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1\n\u22a2 Tendsto (fun n => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 \u2022 \u222b (x : \u03b1) in s, c x ^ n \u2022 g x \u2202\u03bc) atTop (\ud835\udcdd (g x\u2080))"}, {"tactic": "have : Tendsto (fun i : \u2115 => \u222b x : \u03b1 in s, \u03c6 i x \u2022 g x \u2202\u03bc) atTop (\ud835\udcdd (g x\u2080)) :=\n  tendsto_set_integral_peak_smul_of_integrableOn_of_continuousWithinAt hs.measurableSet\n    hs.measure_lt_top.ne (eventually_of_forall hn\u03c6) A (eventually_of_forall hi\u03c6) hmg hcg", "annotated_tactic": ["have : <a>Tendsto</a> (fun i : \u2115 => \u222b x : \u03b1 in s, \u03c6 i x \u2022 g x \u2202\u03bc) <a>atTop</a> (\ud835\udcdd (g x\u2080)) :=\n    <a>tendsto_set_integral_peak_smul_of_integrableOn_of_continuousWithinAt</a> hs.measurableSet\n      hs.measure_lt_top.ne (<a>eventually_of_forall</a> hn\u03c6) A (<a>eventually_of_forall</a> hi\u03c6) hmg hcg", [{"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2939, 5], "def_end_pos": [2939, 12]}, {"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}, {"full_name": "tendsto_set_integral_peak_smul_of_integrableOn_of_continuousWithinAt", "def_path": "Mathlib/MeasureTheory/Integral/PeakFunction.lean", "def_pos": [160, 9], "def_end_pos": [160, 77]}, {"full_name": "Filter.eventually_of_forall", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1111, 9], "def_end_pos": [1111, 29]}, {"full_name": "Filter.eventually_of_forall", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1111, 9], "def_end_pos": [1111, 29]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nP : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\nhi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1\nA : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 atTop (s \\ u)\n\u22a2 Tendsto (fun n => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 \u2022 \u222b (x : \u03b1) in s, c x ^ n \u2022 g x \u2202\u03bc) atTop (\ud835\udcdd (g x\u2080))", "state_after": "\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nP : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\nhi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1\nA : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 atTop (s \\ u)\nthis : Tendsto (fun i => \u222b (x : \u03b1) in s, \u03c6 i x \u2022 g x \u2202\u03bc) atTop (\ud835\udcdd (g x\u2080))\n\u22a2 Tendsto (fun n => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 \u2022 \u222b (x : \u03b1) in s, c x ^ n \u2022 g x \u2202\u03bc) atTop (\ud835\udcdd (g x\u2080))"}, {"tactic": "convert this", "annotated_tactic": ["convert this", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nP : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\nhi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1\nA : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 atTop (s \\ u)\nthis : Tendsto (fun i => \u222b (x : \u03b1) in s, \u03c6 i x \u2022 g x \u2202\u03bc) atTop (\ud835\udcdd (g x\u2080))\n\u22a2 Tendsto (fun n => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 \u2022 \u222b (x : \u03b1) in s, c x ^ n \u2022 g x \u2202\u03bc) atTop (\ud835\udcdd (g x\u2080))", "state_after": "case h.e'_3.h\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nP : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\nhi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1\nA : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 atTop (s \\ u)\nthis : Tendsto (fun i => \u222b (x : \u03b1) in s, \u03c6 i x \u2022 g x \u2202\u03bc) atTop (\ud835\udcdd (g x\u2080))\nx\u271d : \u2115\n\u22a2 (\u222b (x : \u03b1) in s, c x ^ x\u271d \u2202\u03bc)\u207b\u00b9 \u2022 \u222b (x : \u03b1) in s, c x ^ x\u271d \u2022 g x \u2202\u03bc = \u222b (x : \u03b1) in s, \u03c6 x\u271d x \u2022 g x \u2202\u03bc"}, {"tactic": "simp_rw [\u2190 smul_smul, integral_smul]", "annotated_tactic": ["simp_rw [\u2190 <a>smul_smul</a>, <a>integral_smul</a>]", [{"full_name": "smul_smul", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [484, 9], "def_end_pos": [484, 18]}, {"full_name": "MeasureTheory.integral_smul", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [915, 9], "def_end_pos": [915, 22]}]], "state_before": "case h.e'_3.h\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nP : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\nhi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1\nA : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 atTop (s \\ u)\nthis : Tendsto (fun i => \u222b (x : \u03b1) in s, \u03c6 i x \u2022 g x \u2202\u03bc) atTop (\ud835\udcdd (g x\u2080))\nx\u271d : \u2115\n\u22a2 (\u222b (x : \u03b1) in s, c x ^ x\u271d \u2202\u03bc)\u207b\u00b9 \u2022 \u222b (x : \u03b1) in s, c x ^ x\u271d \u2022 g x \u2202\u03bc = \u222b (x : \u03b1) in s, \u03c6 x\u271d x \u2022 g x \u2202\u03bc", "state_after": "no goals"}, {"tactic": "intro n x hx", "annotated_tactic": ["intro n x hx", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\n\u22a2 \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x", "state_after": "\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nn : \u2115\nx : \u03b1\nhx : x \u2208 s\n\u22a2 0 \u2264 \u03c6 n x"}, {"tactic": "apply mul_nonneg (inv_nonneg.2 _) (pow_nonneg (hnc x hx) _)", "annotated_tactic": ["apply <a>mul_nonneg</a> (<a>inv_nonneg</a>.2 _) (<a>pow_nonneg</a> (hnc x hx) _)", [{"full_name": "mul_nonneg", "def_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "def_pos": [380, 7], "def_end_pos": [380, 17]}, {"full_name": "inv_nonneg", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [58, 9], "def_end_pos": [58, 19]}, {"full_name": "pow_nonneg", "def_path": "Mathlib/Algebra/Order/Ring/Defs.lean", "def_pos": [244, 9], "def_end_pos": [244, 19]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nn : \u2115\nx : \u03b1\nhx : x \u2208 s\n\u22a2 0 \u2264 \u03c6 n x", "state_after": "\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nn : \u2115\nx : \u03b1\nhx : x \u2208 s\n\u22a2 0 \u2264 \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc"}, {"tactic": "exact set_integral_nonneg hs.measurableSet fun x hx => pow_nonneg (hnc x hx) _", "annotated_tactic": ["exact <a>set_integral_nonneg</a> hs.measurableSet fun x hx => <a>pow_nonneg</a> (hnc x hx) _", [{"full_name": "MeasureTheory.set_integral_nonneg", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [766, 9], "def_end_pos": [766, 28]}, {"full_name": "pow_nonneg", "def_path": "Mathlib/Algebra/Order/Ring/Defs.lean", "def_pos": [244, 9], "def_end_pos": [244, 19]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nn : \u2115\nx : \u03b1\nhx : x \u2208 s\n\u22a2 0 \u2264 \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc", "state_after": "no goals"}, {"tactic": "intro n", "annotated_tactic": ["intro n", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\n\u22a2 \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n", "state_after": "\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nn : \u2115\n\u22a2 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n"}, {"tactic": "filter_upwards [ae_restrict_mem hs.measurableSet] with x hx", "annotated_tactic": ["filter_upwards [<a>ae_restrict_mem</a> hs.measurableSet] with x hx", [{"full_name": "MeasureTheory.ae_restrict_mem", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2586, 9], "def_end_pos": [2586, 24]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nn : \u2115\n\u22a2 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n", "state_after": "case h\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nn : \u2115\nx : \u03b1\nhx : x \u2208 s\n\u22a2 OfNat.ofNat 0 x \u2264 c x ^ n"}, {"tactic": "exact pow_nonneg (hnc x hx) n", "annotated_tactic": ["exact <a>pow_nonneg</a> (hnc x hx) n", [{"full_name": "pow_nonneg", "def_path": "Mathlib/Algebra/Order/Ring/Defs.lean", "def_pos": [244, 9], "def_end_pos": [244, 19]}]], "state_before": "case h\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nn : \u2115\nx : \u03b1\nhx : x \u2208 s\n\u22a2 OfNat.ofNat 0 x \u2264 c x ^ n", "state_after": "no goals"}, {"tactic": "intro n", "annotated_tactic": ["intro n", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\n\u22a2 \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc", "state_after": "\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nn : \u2115\n\u22a2 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc"}, {"tactic": "refine' (set_integral_pos_iff_support_of_nonneg_ae (J n) (I n)).2 _", "annotated_tactic": ["refine' (<a>set_integral_pos_iff_support_of_nonneg_ae</a> (J n) (I n)).2 _", [{"full_name": "MeasureTheory.set_integral_pos_iff_support_of_nonneg_ae", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [590, 9], "def_end_pos": [590, 50]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nn : \u2115\n\u22a2 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc", "state_after": "\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nn : \u2115\n\u22a2 0 < \u2191\u2191\u03bc ((Function.support fun x => c x ^ n) \u2229 s)"}, {"tactic": "obtain \u27e8u, u_open, x\u2080_u, hu\u27e9 : \u2203 u : Set \u03b1, IsOpen u \u2227 x\u2080 \u2208 u \u2227 u \u2229 s \u2286 c \u207b\u00b9' Ioi 0 :=\n  _root_.continuousOn_iff.1 hc x\u2080 h\u2080 (Ioi (0 : \u211d)) isOpen_Ioi hnc\u2080", "annotated_tactic": ["obtain \u27e8u, u_open, x\u2080_u, hu\u27e9 : \u2203 u : <a>Set</a> \u03b1, <a>IsOpen</a> u \u2227 x\u2080 \u2208 u \u2227 u \u2229 s \u2286 c \u207b\u00b9' <a>Ioi</a> 0 :=\n      <a>_root_.continuousOn_iff</a>.1 hc x\u2080 h\u2080 (<a>Ioi</a> (0 : \u211d)) <a>isOpen_Ioi</a> hnc\u2080", [{"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}, {"full_name": "IsOpen", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [101, 5], "def_end_pos": [101, 11]}, {"full_name": "Set.Ioi", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [79, 5], "def_end_pos": [79, 8]}, {"full_name": "continuousOn_iff", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [637, 9], "def_end_pos": [637, 25]}, {"full_name": "Set.Ioi", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [79, 5], "def_end_pos": [79, 8]}, {"full_name": "isOpen_Ioi", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [332, 9], "def_end_pos": [332, 19]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nn : \u2115\n\u22a2 0 < \u2191\u2191\u03bc ((Function.support fun x => c x ^ n) \u2229 s)", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nn : \u2115\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080_u : x\u2080 \u2208 u\nhu : u \u2229 s \u2286 c \u207b\u00b9' Ioi 0\n\u22a2 0 < \u2191\u2191\u03bc ((Function.support fun x => c x ^ n) \u2229 s)"}, {"tactic": "apply (h\u03bc u u_open x\u2080_u).trans_le", "annotated_tactic": ["apply (h\u03bc u u_open x\u2080_u).<a>trans_le</a>", [{"full_name": "LT.lt.trans_le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [148, 7], "def_end_pos": [148, 21]}]], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nn : \u2115\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080_u : x\u2080 \u2208 u\nhu : u \u2229 s \u2286 c \u207b\u00b9' Ioi 0\n\u22a2 0 < \u2191\u2191\u03bc ((Function.support fun x => c x ^ n) \u2229 s)", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nn : \u2115\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080_u : x\u2080 \u2208 u\nhu : u \u2229 s \u2286 c \u207b\u00b9' Ioi 0\n\u22a2 \u2191\u2191\u03bc (u \u2229 s) \u2264 \u2191\u2191\u03bc ((Function.support fun x => c x ^ n) \u2229 s)"}, {"tactic": "exact measure_mono fun x hx => \u27e8ne_of_gt (pow_pos (a := c x) (hu hx) _), hx.2\u27e9", "annotated_tactic": ["exact <a>measure_mono</a> fun x hx => \u27e8<a>ne_of_gt</a> (<a>pow_pos</a> (a := c x) (hu hx) _), hx.2\u27e9", [{"full_name": "MeasureTheory.measure_mono", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [193, 9], "def_end_pos": [193, 21]}, {"full_name": "ne_of_gt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [104, 9], "def_end_pos": [104, 17]}, {"full_name": "pow_pos", "def_path": "Mathlib/Algebra/Order/Ring/Defs.lean", "def_pos": [530, 9], "def_end_pos": [530, 16]}]], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nn : \u2115\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080_u : x\u2080 \u2208 u\nhu : u \u2229 s \u2286 c \u207b\u00b9' Ioi 0\n\u22a2 \u2191\u2191\u03bc (u \u2229 s) \u2264 \u2191\u2191\u03bc ((Function.support fun x => c x ^ n) \u2229 s)", "state_after": "no goals"}, {"tactic": "rw [integral_mul_left, inv_mul_cancel (P n).ne']", "annotated_tactic": ["rw [<a>integral_mul_left</a>, <a>inv_mul_cancel</a> (P n).<a>ne'</a>]", [{"full_name": "MeasureTheory.integral_mul_left", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [923, 9], "def_end_pos": [923, 26]}, {"full_name": "inv_mul_cancel", "def_path": "Mathlib/Algebra/GroupWithZero/NeZero.lean", "def_pos": [55, 9], "def_end_pos": [55, 23]}, {"full_name": "LT.lt.ne'", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [328, 9], "def_end_pos": [328, 12]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nP : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\nn : \u2115\n\u22a2 \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1", "state_after": "no goals"}, {"tactic": "intro u u_open x\u2080u", "annotated_tactic": ["intro u u_open x\u2080u", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nP : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\nhi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1\n\u22a2 \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 atTop (s \\ u)", "state_after": "\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nP : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\nhi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\n\u22a2 TendstoUniformlyOn \u03c6 0 atTop (s \\ u)"}, {"tactic": "obtain \u27e8t', tt', t'x\u2080\u27e9 : \u2203 t', t < t' \u2227 t' < c x\u2080 := exists_between tx\u2080", "annotated_tactic": ["obtain \u27e8t', tt', t'x\u2080\u27e9 : \u2203 t', t < t' \u2227 t' < c x\u2080 := <a>exists_between</a> tx\u2080", [{"full_name": "exists_between", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [1294, 9], "def_end_pos": [1294, 23]}]], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nP : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\nhi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nt : \u211d\nt_pos : 0 \u2264 t\ntx\u2080 : t < c x\u2080\nht : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t\n\u22a2 TendstoUniformlyOn \u03c6 0 atTop (s \\ u)", "state_after": "case intro.intro.intro.intro.intro\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nP : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\nhi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nt : \u211d\nt_pos : 0 \u2264 t\ntx\u2080 : t < c x\u2080\nht : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t\nt' : \u211d\ntt' : t < t'\nt'x\u2080 : t' < c x\u2080\n\u22a2 TendstoUniformlyOn \u03c6 0 atTop (s \\ u)"}, {"tactic": "have t'_pos : 0 < t' := t_pos.trans_lt tt'", "annotated_tactic": ["have t'_pos : 0 < t' := t_pos.trans_lt tt'", []], "state_before": "case intro.intro.intro.intro.intro\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nP : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\nhi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nt : \u211d\nt_pos : 0 \u2264 t\ntx\u2080 : t < c x\u2080\nht : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t\nt' : \u211d\ntt' : t < t'\nt'x\u2080 : t' < c x\u2080\n\u22a2 TendstoUniformlyOn \u03c6 0 atTop (s \\ u)", "state_after": "case intro.intro.intro.intro.intro\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nP : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\nhi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nt : \u211d\nt_pos : 0 \u2264 t\ntx\u2080 : t < c x\u2080\nht : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t\nt' : \u211d\ntt' : t < t'\nt'x\u2080 : t' < c x\u2080\nt'_pos : 0 < t'\n\u22a2 TendstoUniformlyOn \u03c6 0 atTop (s \\ u)"}, {"tactic": "obtain \u27e8v, v_open, x\u2080_v, hv\u27e9 : \u2203 v : Set \u03b1, IsOpen v \u2227 x\u2080 \u2208 v \u2227 v \u2229 s \u2286 c \u207b\u00b9' Ioi t' :=\n  _root_.continuousOn_iff.1 hc x\u2080 h\u2080 (Ioi t') isOpen_Ioi t'x\u2080", "annotated_tactic": ["obtain \u27e8v, v_open, x\u2080_v, hv\u27e9 : \u2203 v : <a>Set</a> \u03b1, <a>IsOpen</a> v \u2227 x\u2080 \u2208 v \u2227 v \u2229 s \u2286 c \u207b\u00b9' <a>Ioi</a> t' :=\n      <a>_root_.continuousOn_iff</a>.1 hc x\u2080 h\u2080 (<a>Ioi</a> t') <a>isOpen_Ioi</a> t'x\u2080", [{"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}, {"full_name": "IsOpen", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [101, 5], "def_end_pos": [101, 11]}, {"full_name": "Set.Ioi", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [79, 5], "def_end_pos": [79, 8]}, {"full_name": "continuousOn_iff", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [637, 9], "def_end_pos": [637, 25]}, {"full_name": "Set.Ioi", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [79, 5], "def_end_pos": [79, 8]}, {"full_name": "isOpen_Ioi", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [332, 9], "def_end_pos": [332, 19]}]], "state_before": "case intro.intro.intro.intro.intro\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nP : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\nhi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nt : \u211d\nt_pos : 0 \u2264 t\ntx\u2080 : t < c x\u2080\nht : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t\nt' : \u211d\ntt' : t < t'\nt'x\u2080 : t' < c x\u2080\nt'_pos : 0 < t'\n\u22a2 TendstoUniformlyOn \u03c6 0 atTop (s \\ u)", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nP : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\nhi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nt : \u211d\nt_pos : 0 \u2264 t\ntx\u2080 : t < c x\u2080\nht : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t\nt' : \u211d\ntt' : t < t'\nt'x\u2080 : t' < c x\u2080\nt'_pos : 0 < t'\nv : Set \u03b1\nv_open : IsOpen v\nx\u2080_v : x\u2080 \u2208 v\nhv : v \u2229 s \u2286 c \u207b\u00b9' Ioi t'\n\u22a2 TendstoUniformlyOn \u03c6 0 atTop (s \\ u)"}, {"tactic": "have N :\n  Tendsto (fun n => (\u03bc (v \u2229 s)).toReal\u207b\u00b9 * (t / t') ^ n) atTop\n    (\ud835\udcdd ((\u03bc (v \u2229 s)).toReal\u207b\u00b9 * 0)) := by\n  apply Tendsto.mul tendsto_const_nhds _\n  apply tendsto_pow_atTop_nhds_0_of_lt_1 (div_nonneg t_pos t'_pos.le)\n  exact (div_lt_one t'_pos).2 tt'", "annotated_tactic": ["have N :\n      <a>Tendsto</a> (fun n => (\u03bc (v \u2229 s)).<a>toReal</a>\u207b\u00b9 * (t / t') ^ n) <a>atTop</a>\n        (\ud835\udcdd ((\u03bc (v \u2229 s)).<a>toReal</a>\u207b\u00b9 * 0)) := by\n      apply <a>Tendsto.mul</a> <a>tendsto_const_nhds</a> _\n      apply <a>tendsto_pow_atTop_nhds_0_of_lt_1</a> (<a>div_nonneg</a> t_pos t'_pos.le)\n      exact (<a>div_lt_one</a> t'_pos).2 tt'", [{"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2939, 5], "def_end_pos": [2939, 12]}, {"full_name": "ENNReal.toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [168, 15], "def_end_pos": [168, 21]}, {"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}, {"full_name": "ENNReal.toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [168, 15], "def_end_pos": [168, 21]}, {"full_name": "Filter.Tendsto.mul", "def_path": "Mathlib/Topology/Algebra/Monoid.lean", "def_pos": [119, 9], "def_end_pos": [119, 27]}, {"full_name": "tendsto_const_nhds", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [1049, 9], "def_end_pos": [1049, 27]}, {"full_name": "tendsto_pow_atTop_nhds_0_of_lt_1", "def_path": "Mathlib/Analysis/SpecificLimits/Basic.lean", "def_pos": [110, 9], "def_end_pos": [110, 41]}, {"full_name": "div_nonneg", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [94, 9], "def_end_pos": [94, 19]}, {"full_name": "div_lt_one", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [431, 9], "def_end_pos": [431, 19]}]], "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nP : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\nhi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nt : \u211d\nt_pos : 0 \u2264 t\ntx\u2080 : t < c x\u2080\nht : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t\nt' : \u211d\ntt' : t < t'\nt'x\u2080 : t' < c x\u2080\nt'_pos : 0 < t'\nv : Set \u03b1\nv_open : IsOpen v\nx\u2080_v : x\u2080 \u2208 v\nhv : v \u2229 s \u2286 c \u207b\u00b9' Ioi t'\nM : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \\ u \u2192 \u03c6 n x \u2264 (ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s)))\u207b\u00b9 * (t / t') ^ n\n\u22a2 TendstoUniformlyOn \u03c6 0 atTop (s \\ u)", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nP : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\nhi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nt : \u211d\nt_pos : 0 \u2264 t\ntx\u2080 : t < c x\u2080\nht : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t\nt' : \u211d\ntt' : t < t'\nt'x\u2080 : t' < c x\u2080\nt'_pos : 0 < t'\nv : Set \u03b1\nv_open : IsOpen v\nx\u2080_v : x\u2080 \u2208 v\nhv : v \u2229 s \u2286 c \u207b\u00b9' Ioi t'\nM : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \\ u \u2192 \u03c6 n x \u2264 (ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s)))\u207b\u00b9 * (t / t') ^ n\nN : Tendsto (fun n => (ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s)))\u207b\u00b9 * (t / t') ^ n) atTop (\ud835\udcdd ((ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s)))\u207b\u00b9 * 0))\n\u22a2 TendstoUniformlyOn \u03c6 0 atTop (s \\ u)"}, {"tactic": "rw [mul_zero] at N", "annotated_tactic": ["rw [<a>mul_zero</a>] at N", [{"full_name": "MulZeroClass.mul_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [38, 3], "def_end_pos": [38, 11]}]], "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nP : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\nhi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nt : \u211d\nt_pos : 0 \u2264 t\ntx\u2080 : t < c x\u2080\nht : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t\nt' : \u211d\ntt' : t < t'\nt'x\u2080 : t' < c x\u2080\nt'_pos : 0 < t'\nv : Set \u03b1\nv_open : IsOpen v\nx\u2080_v : x\u2080 \u2208 v\nhv : v \u2229 s \u2286 c \u207b\u00b9' Ioi t'\nM : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \\ u \u2192 \u03c6 n x \u2264 (ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s)))\u207b\u00b9 * (t / t') ^ n\nN : Tendsto (fun n => (ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s)))\u207b\u00b9 * (t / t') ^ n) atTop (\ud835\udcdd ((ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s)))\u207b\u00b9 * 0))\n\u22a2 TendstoUniformlyOn \u03c6 0 atTop (s \\ u)", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nP : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\nhi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nt : \u211d\nt_pos : 0 \u2264 t\ntx\u2080 : t < c x\u2080\nht : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t\nt' : \u211d\ntt' : t < t'\nt'x\u2080 : t' < c x\u2080\nt'_pos : 0 < t'\nv : Set \u03b1\nv_open : IsOpen v\nx\u2080_v : x\u2080 \u2208 v\nhv : v \u2229 s \u2286 c \u207b\u00b9' Ioi t'\nM : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \\ u \u2192 \u03c6 n x \u2264 (ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s)))\u207b\u00b9 * (t / t') ^ n\nN : Tendsto (fun n => (ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s)))\u207b\u00b9 * (t / t') ^ n) atTop (\ud835\udcdd 0)\n\u22a2 TendstoUniformlyOn \u03c6 0 atTop (s \\ u)"}, {"tactic": "refine' tendstoUniformlyOn_iff.2 fun \u03b5 \u03b5pos => _", "annotated_tactic": ["refine' <a>tendstoUniformlyOn_iff</a>.2 fun \u03b5 \u03b5pos => _", [{"full_name": "Metric.tendstoUniformlyOn_iff", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [922, 9], "def_end_pos": [922, 31]}]], "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nP : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\nhi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nt : \u211d\nt_pos : 0 \u2264 t\ntx\u2080 : t < c x\u2080\nht : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t\nt' : \u211d\ntt' : t < t'\nt'x\u2080 : t' < c x\u2080\nt'_pos : 0 < t'\nv : Set \u03b1\nv_open : IsOpen v\nx\u2080_v : x\u2080 \u2208 v\nhv : v \u2229 s \u2286 c \u207b\u00b9' Ioi t'\nM : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \\ u \u2192 \u03c6 n x \u2264 (ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s)))\u207b\u00b9 * (t / t') ^ n\nN : Tendsto (fun n => (ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s)))\u207b\u00b9 * (t / t') ^ n) atTop (\ud835\udcdd 0)\n\u22a2 TendstoUniformlyOn \u03c6 0 atTop (s \\ u)", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nP : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\nhi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nt : \u211d\nt_pos : 0 \u2264 t\ntx\u2080 : t < c x\u2080\nht : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t\nt' : \u211d\ntt' : t < t'\nt'x\u2080 : t' < c x\u2080\nt'_pos : 0 < t'\nv : Set \u03b1\nv_open : IsOpen v\nx\u2080_v : x\u2080 \u2208 v\nhv : v \u2229 s \u2286 c \u207b\u00b9' Ioi t'\nM : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \\ u \u2192 \u03c6 n x \u2264 (ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s)))\u207b\u00b9 * (t / t') ^ n\nN : Tendsto (fun n => (ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s)))\u207b\u00b9 * (t / t') ^ n) atTop (\ud835\udcdd 0)\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u22a2 \u2200\u1da0 (n : \u2115) in atTop, \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 n x) < \u03b5"}, {"tactic": "filter_upwards [(tendsto_order.1 N).2 \u03b5 \u03b5pos] with n hn x hx", "annotated_tactic": ["filter_upwards [(<a>tendsto_order</a>.1 N).2 \u03b5 \u03b5pos] with n hn x hx", [{"full_name": "tendsto_order", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [919, 9], "def_end_pos": [919, 22]}]], "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nP : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\nhi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nt : \u211d\nt_pos : 0 \u2264 t\ntx\u2080 : t < c x\u2080\nht : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t\nt' : \u211d\ntt' : t < t'\nt'x\u2080 : t' < c x\u2080\nt'_pos : 0 < t'\nv : Set \u03b1\nv_open : IsOpen v\nx\u2080_v : x\u2080 \u2208 v\nhv : v \u2229 s \u2286 c \u207b\u00b9' Ioi t'\nM : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \\ u \u2192 \u03c6 n x \u2264 (ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s)))\u207b\u00b9 * (t / t') ^ n\nN : Tendsto (fun n => (ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s)))\u207b\u00b9 * (t / t') ^ n) atTop (\ud835\udcdd 0)\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u22a2 \u2200\u1da0 (n : \u2115) in atTop, \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 n x) < \u03b5", "state_after": "case h\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nP : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\nhi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nt : \u211d\nt_pos : 0 \u2264 t\ntx\u2080 : t < c x\u2080\nht : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t\nt' : \u211d\ntt' : t < t'\nt'x\u2080 : t' < c x\u2080\nt'_pos : 0 < t'\nv : Set \u03b1\nv_open : IsOpen v\nx\u2080_v : x\u2080 \u2208 v\nhv : v \u2229 s \u2286 c \u207b\u00b9' Ioi t'\nM : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \\ u \u2192 \u03c6 n x \u2264 (ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s)))\u207b\u00b9 * (t / t') ^ n\nN : Tendsto (fun n => (ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s)))\u207b\u00b9 * (t / t') ^ n) atTop (\ud835\udcdd 0)\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nn : \u2115\nhn : (ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s)))\u207b\u00b9 * (t / t') ^ n < \u03b5\nx : \u03b1\nhx : x \u2208 s \\ u\n\u22a2 dist (OfNat.ofNat 0 x) (\u03c6 n x) < \u03b5"}, {"tactic": "simp only [Pi.zero_apply, dist_zero_left, Real.norm_of_nonneg (hn\u03c6 n x hx.1)]", "annotated_tactic": ["simp only [<a>Pi.zero_apply</a>, <a>dist_zero_left</a>, <a>Real.norm_of_nonneg</a> (hn\u03c6 n x hx.1)]", [{"full_name": "Pi.zero_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [46, 3], "def_end_pos": [46, 14]}, {"full_name": "dist_zero_left", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [400, 3], "def_end_pos": [400, 14]}, {"full_name": "Real.norm_of_nonneg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [1768, 9], "def_end_pos": [1768, 23]}]], "state_before": "case h\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nP : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\nhi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nt : \u211d\nt_pos : 0 \u2264 t\ntx\u2080 : t < c x\u2080\nht : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t\nt' : \u211d\ntt' : t < t'\nt'x\u2080 : t' < c x\u2080\nt'_pos : 0 < t'\nv : Set \u03b1\nv_open : IsOpen v\nx\u2080_v : x\u2080 \u2208 v\nhv : v \u2229 s \u2286 c \u207b\u00b9' Ioi t'\nM : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \\ u \u2192 \u03c6 n x \u2264 (ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s)))\u207b\u00b9 * (t / t') ^ n\nN : Tendsto (fun n => (ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s)))\u207b\u00b9 * (t / t') ^ n) atTop (\ud835\udcdd 0)\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nn : \u2115\nhn : (ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s)))\u207b\u00b9 * (t / t') ^ n < \u03b5\nx : \u03b1\nhx : x \u2208 s \\ u\n\u22a2 dist (OfNat.ofNat 0 x) (\u03c6 n x) < \u03b5", "state_after": "case h\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nP : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\nhi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nt : \u211d\nt_pos : 0 \u2264 t\ntx\u2080 : t < c x\u2080\nht : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t\nt' : \u211d\ntt' : t < t'\nt'x\u2080 : t' < c x\u2080\nt'_pos : 0 < t'\nv : Set \u03b1\nv_open : IsOpen v\nx\u2080_v : x\u2080 \u2208 v\nhv : v \u2229 s \u2286 c \u207b\u00b9' Ioi t'\nM : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \\ u \u2192 \u03c6 n x \u2264 (ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s)))\u207b\u00b9 * (t / t') ^ n\nN : Tendsto (fun n => (ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s)))\u207b\u00b9 * (t / t') ^ n) atTop (\ud835\udcdd 0)\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nn : \u2115\nhn : (ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s)))\u207b\u00b9 * (t / t') ^ n < \u03b5\nx : \u03b1\nhx : x \u2208 s \\ u\n\u22a2 (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n < \u03b5"}, {"tactic": "exact (M n x hx).trans_lt hn", "annotated_tactic": ["exact (M n x hx).<a>trans_lt</a> hn", [{"full_name": "LE.le.trans_lt", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [124, 7], "def_end_pos": [124, 21]}]], "state_before": "case h\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nP : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\nhi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nt : \u211d\nt_pos : 0 \u2264 t\ntx\u2080 : t < c x\u2080\nht : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t\nt' : \u211d\ntt' : t < t'\nt'x\u2080 : t' < c x\u2080\nt'_pos : 0 < t'\nv : Set \u03b1\nv_open : IsOpen v\nx\u2080_v : x\u2080 \u2208 v\nhv : v \u2229 s \u2286 c \u207b\u00b9' Ioi t'\nM : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \\ u \u2192 \u03c6 n x \u2264 (ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s)))\u207b\u00b9 * (t / t') ^ n\nN : Tendsto (fun n => (ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s)))\u207b\u00b9 * (t / t') ^ n) atTop (\ud835\udcdd 0)\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nn : \u2115\nhn : (ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s)))\u207b\u00b9 * (t / t') ^ n < \u03b5\nx : \u03b1\nhx : x \u2208 s \\ u\n\u22a2 (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n < \u03b5", "state_after": "no goals"}, {"tactic": "rcases eq_empty_or_nonempty (s \\ u) with (h | h)", "annotated_tactic": ["rcases <a>eq_empty_or_nonempty</a> (s \\ u) with (h | h)", [{"full_name": "Set.eq_empty_or_nonempty", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [635, 9], "def_end_pos": [635, 29]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nP : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\nhi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\n\u22a2 \u2203 t, 0 \u2264 t \u2227 t < c x\u2080 \u2227 \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t", "state_after": "case inl\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nP : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\nhi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nh : s \\ u = \u2205\n\u22a2 \u2203 t, 0 \u2264 t \u2227 t < c x\u2080 \u2227 \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t\n\ncase inr\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nP : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\nhi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nh : Set.Nonempty (s \\ u)\n\u22a2 \u2203 t, 0 \u2264 t \u2227 t < c x\u2080 \u2227 \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t"}, {"tactic": "obtain \u27e8x, hx, h'x\u27e9 : \u2203 x \u2208 s \\ u, \u2200 y \u2208 s \\ u, c y \u2264 c x :=\n  IsCompact.exists_isMaxOn (hs.diff u_open) h (hc.mono (diff_subset _ _))", "annotated_tactic": ["obtain \u27e8x, hx, h'x\u27e9 : \u2203 x \u2208 s \\ u, \u2200 y \u2208 s \\ u, c y \u2264 c x :=\n        <a>IsCompact.exists_isMaxOn</a> (hs.diff u_open) h (hc.mono (<a>diff_subset</a> _ _))", [{"full_name": "IsCompact.exists_isMaxOn", "def_path": "Mathlib/Topology/Algebra/Order/Compact.lean", "def_pos": [212, 9], "def_end_pos": [212, 33]}, {"full_name": "Set.diff_subset", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1845, 9], "def_end_pos": [1845, 20]}]], "state_before": "case inr\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nP : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\nhi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nh : Set.Nonempty (s \\ u)\n\u22a2 \u2203 t, 0 \u2264 t \u2227 t < c x\u2080 \u2227 \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t", "state_after": "case inr.intro.intro\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nP : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\nhi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nh : Set.Nonempty (s \\ u)\nx : \u03b1\nhx : x \u2208 s \\ u\nh'x : \u2200 (y : \u03b1), y \u2208 s \\ u \u2192 c y \u2264 c x\n\u22a2 \u2203 t, 0 \u2264 t \u2227 t < c x\u2080 \u2227 \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t"}, {"tactic": "refine' \u27e8c x, hnc x hx.1, h'c x hx.1 _, h'x\u27e9", "annotated_tactic": ["refine' \u27e8c x, hnc x hx.1, h'c x hx.1 _, h'x\u27e9", []], "state_before": "case inr.intro.intro\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nP : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\nhi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nh : Set.Nonempty (s \\ u)\nx : \u03b1\nhx : x \u2208 s \\ u\nh'x : \u2200 (y : \u03b1), y \u2208 s \\ u \u2192 c y \u2264 c x\n\u22a2 \u2203 t, 0 \u2264 t \u2227 t < c x\u2080 \u2227 \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t", "state_after": "case inr.intro.intro\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nP : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\nhi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nh : Set.Nonempty (s \\ u)\nx : \u03b1\nhx : x \u2208 s \\ u\nh'x : \u2200 (y : \u03b1), y \u2208 s \\ u \u2192 c y \u2264 c x\n\u22a2 x \u2260 x\u2080"}, {"tactic": "rintro rfl", "annotated_tactic": ["rintro rfl", []], "state_before": "case inr.intro.intro\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nP : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\nhi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nh : Set.Nonempty (s \\ u)\nx : \u03b1\nhx : x \u2208 s \\ u\nh'x : \u2200 (y : \u03b1), y \u2208 s \\ u \u2192 c y \u2264 c x\n\u22a2 x \u2260 x\u2080", "state_after": "case inr.intro.intro\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhmg : IntegrableOn g s\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nP : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\nhi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1\nu : Set \u03b1\nu_open : IsOpen u\nh : Set.Nonempty (s \\ u)\nx : \u03b1\nhx : x \u2208 s \\ u\nh'x : \u2200 (y : \u03b1), y \u2208 s \\ u \u2192 c y \u2264 c x\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x \u2192 c y < c x\nhnc\u2080 : 0 < c x\nh\u2080 : x \u2208 s\nhcg : ContinuousWithinAt g s x\nx\u2080u : x \u2208 u\n\u22a2 False"}, {"tactic": "exact hx.2 x\u2080u", "annotated_tactic": ["exact hx.2 x\u2080u", []], "state_before": "case inr.intro.intro\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhmg : IntegrableOn g s\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nP : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\nhi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1\nu : Set \u03b1\nu_open : IsOpen u\nh : Set.Nonempty (s \\ u)\nx : \u03b1\nhx : x \u2208 s \\ u\nh'x : \u2200 (y : \u03b1), y \u2208 s \\ u \u2192 c y \u2264 c x\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x \u2192 c y < c x\nhnc\u2080 : 0 < c x\nh\u2080 : x \u2208 s\nhcg : ContinuousWithinAt g s x\nx\u2080u : x \u2208 u\n\u22a2 False", "state_after": "no goals"}, {"tactic": "exact\n  \u27e80, le_rfl, hnc\u2080, by simp only [h, mem_empty_iff_false, IsEmpty.forall_iff, imp_true_iff]\u27e9", "annotated_tactic": ["exact\n          \u27e80, <a>le_rfl</a>, hnc\u2080, by simp only [h, <a>mem_empty_iff_false</a>, <a>IsEmpty.forall_iff</a>, <a>imp_true_iff</a>]\u27e9", [{"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}, {"full_name": "Set.mem_empty_iff_false", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [562, 9], "def_end_pos": [562, 28]}, {"full_name": "IsEmpty.forall_iff", "def_path": "Mathlib/Logic/IsEmpty.lean", "def_pos": [121, 9], "def_end_pos": [121, 19]}, {"full_name": "imp_true_iff", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [116, 9], "def_end_pos": [116, 21]}]], "state_before": "case inl\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nP : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\nhi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nh : s \\ u = \u2205\n\u22a2 \u2203 t, 0 \u2264 t \u2227 t < c x\u2080 \u2227 \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t", "state_after": "no goals"}, {"tactic": "simp only [h, mem_empty_iff_false, IsEmpty.forall_iff, imp_true_iff]", "annotated_tactic": ["simp only [h, <a>mem_empty_iff_false</a>, <a>IsEmpty.forall_iff</a>, <a>imp_true_iff</a>]", [{"full_name": "Set.mem_empty_iff_false", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [562, 9], "def_end_pos": [562, 28]}, {"full_name": "IsEmpty.forall_iff", "def_path": "Mathlib/Logic/IsEmpty.lean", "def_pos": [121, 9], "def_end_pos": [121, 19]}, {"full_name": "imp_true_iff", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [116, 9], "def_end_pos": [116, 21]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nP : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\nhi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nh : s \\ u = \u2205\n\u22a2 \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 0", "state_after": "no goals"}, {"tactic": "intro n x hx", "annotated_tactic": ["intro n x hx", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nP : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\nhi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nt : \u211d\nt_pos : 0 \u2264 t\ntx\u2080 : t < c x\u2080\nht : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t\nt' : \u211d\ntt' : t < t'\nt'x\u2080 : t' < c x\u2080\nt'_pos : 0 < t'\nv : Set \u03b1\nv_open : IsOpen v\nx\u2080_v : x\u2080 \u2208 v\nhv : v \u2229 s \u2286 c \u207b\u00b9' Ioi t'\n\u22a2 \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \\ u \u2192 \u03c6 n x \u2264 (ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s)))\u207b\u00b9 * (t / t') ^ n", "state_after": "\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nP : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\nhi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nt : \u211d\nt_pos : 0 \u2264 t\ntx\u2080 : t < c x\u2080\nht : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t\nt' : \u211d\ntt' : t < t'\nt'x\u2080 : t' < c x\u2080\nt'_pos : 0 < t'\nv : Set \u03b1\nv_open : IsOpen v\nx\u2080_v : x\u2080 \u2208 v\nhv : v \u2229 s \u2286 c \u207b\u00b9' Ioi t'\nn : \u2115\nx : \u03b1\nhx : x \u2208 s \\ u\n\u22a2 \u03c6 n x \u2264 (ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s)))\u207b\u00b9 * (t / t') ^ n"}, {"tactic": "simp_rw [\u2190 div_eq_inv_mul, div_pow, div_div]", "annotated_tactic": ["simp_rw [\u2190 <a>div_eq_inv_mul</a>, <a>div_pow</a>, <a>div_div</a>]", [{"full_name": "div_eq_inv_mul", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [492, 9], "def_end_pos": [492, 23]}, {"full_name": "div_pow", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [391, 9], "def_end_pos": [391, 16]}, {"full_name": "div_div", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [527, 9], "def_end_pos": [527, 16]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nP : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\nhi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nt : \u211d\nt_pos : 0 \u2264 t\ntx\u2080 : t < c x\u2080\nht : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t\nt' : \u211d\ntt' : t < t'\nt'x\u2080 : t' < c x\u2080\nt'_pos : 0 < t'\nv : Set \u03b1\nv_open : IsOpen v\nx\u2080_v : x\u2080 \u2208 v\nhv : v \u2229 s \u2286 c \u207b\u00b9' Ioi t'\nn : \u2115\nx : \u03b1\nhx : x \u2208 s \\ u\nB : t' ^ n * ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s)) \u2264 \u222b (y : \u03b1) in s, c y ^ n \u2202\u03bc\n\u22a2 \u03c6 n x \u2264 (ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s)))\u207b\u00b9 * (t / t') ^ n", "state_after": "\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nP : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\nhi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nt : \u211d\nt_pos : 0 \u2264 t\ntx\u2080 : t < c x\u2080\nht : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t\nt' : \u211d\ntt' : t < t'\nt'x\u2080 : t' < c x\u2080\nt'_pos : 0 < t'\nv : Set \u03b1\nv_open : IsOpen v\nx\u2080_v : x\u2080 \u2208 v\nhv : v \u2229 s \u2286 c \u207b\u00b9' Ioi t'\nn : \u2115\nx : \u03b1\nhx : x \u2208 s \\ u\nB : t' ^ n * ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s)) \u2264 \u222b (y : \u03b1) in s, c y ^ n \u2202\u03bc\n\u22a2 c x ^ n / \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc \u2264 t ^ n / (t' ^ n * ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s)))"}, {"tactic": "apply div_le_div (pow_nonneg t_pos n) _ _ B", "annotated_tactic": ["apply <a>div_le_div</a> (<a>pow_nonneg</a> t_pos n) _ _ B", [{"full_name": "div_le_div", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [388, 9], "def_end_pos": [388, 19]}, {"full_name": "pow_nonneg", "def_path": "Mathlib/Algebra/Order/Ring/Defs.lean", "def_pos": [244, 9], "def_end_pos": [244, 19]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nP : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\nhi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nt : \u211d\nt_pos : 0 \u2264 t\ntx\u2080 : t < c x\u2080\nht : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t\nt' : \u211d\ntt' : t < t'\nt'x\u2080 : t' < c x\u2080\nt'_pos : 0 < t'\nv : Set \u03b1\nv_open : IsOpen v\nx\u2080_v : x\u2080 \u2208 v\nhv : v \u2229 s \u2286 c \u207b\u00b9' Ioi t'\nn : \u2115\nx : \u03b1\nhx : x \u2208 s \\ u\nB : t' ^ n * ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s)) \u2264 \u222b (y : \u03b1) in s, c y ^ n \u2202\u03bc\n\u22a2 c x ^ n / \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc \u2264 t ^ n / (t' ^ n * ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s)))", "state_after": "\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nP : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\nhi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nt : \u211d\nt_pos : 0 \u2264 t\ntx\u2080 : t < c x\u2080\nht : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t\nt' : \u211d\ntt' : t < t'\nt'x\u2080 : t' < c x\u2080\nt'_pos : 0 < t'\nv : Set \u03b1\nv_open : IsOpen v\nx\u2080_v : x\u2080 \u2208 v\nhv : v \u2229 s \u2286 c \u207b\u00b9' Ioi t'\nn : \u2115\nx : \u03b1\nhx : x \u2208 s \\ u\nB : t' ^ n * ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s)) \u2264 \u222b (y : \u03b1) in s, c y ^ n \u2202\u03bc\n\u22a2 c x ^ n \u2264 t ^ n\n\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nP : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\nhi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nt : \u211d\nt_pos : 0 \u2264 t\ntx\u2080 : t < c x\u2080\nht : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t\nt' : \u211d\ntt' : t < t'\nt'x\u2080 : t' < c x\u2080\nt'_pos : 0 < t'\nv : Set \u03b1\nv_open : IsOpen v\nx\u2080_v : x\u2080 \u2208 v\nhv : v \u2229 s \u2286 c \u207b\u00b9' Ioi t'\nn : \u2115\nx : \u03b1\nhx : x \u2208 s \\ u\nB : t' ^ n * ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s)) \u2264 \u222b (y : \u03b1) in s, c y ^ n \u2202\u03bc\n\u22a2 0 < t' ^ n * ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s))"}, {"tactic": "simp only [integral_const, Measure.restrict_apply, MeasurableSet.univ, univ_inter,\n  Algebra.id.smul_eq_mul, mul_comm]", "annotated_tactic": ["simp only [<a>integral_const</a>, <a>Measure.restrict_apply</a>, <a>MeasurableSet.univ</a>, <a>univ_inter</a>,\n              <a>Algebra.id.smul_eq_mul</a>, <a>mul_comm</a>]", [{"full_name": "MeasureTheory.integral_const", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1409, 9], "def_end_pos": [1409, 23]}, {"full_name": "MeasureTheory.Measure.restrict_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1533, 9], "def_end_pos": [1533, 23]}, {"full_name": "MeasurableSet.univ", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [101, 19], "def_end_pos": [101, 37]}, {"full_name": "Set.univ_inter", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1017, 9], "def_end_pos": [1017, 19]}, {"full_name": "Algebra.id.smul_eq_mul", "def_path": "Mathlib/Algebra/Algebra/Basic.lean", "def_pos": [453, 9], "def_end_pos": [453, 20]}, {"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [302, 9], "def_end_pos": [302, 17]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nP : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\nhi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nt : \u211d\nt_pos : 0 \u2264 t\ntx\u2080 : t < c x\u2080\nht : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t\nt' : \u211d\ntt' : t < t'\nt'x\u2080 : t' < c x\u2080\nt'_pos : 0 < t'\nv : Set \u03b1\nv_open : IsOpen v\nx\u2080_v : x\u2080 \u2208 v\nhv : v \u2229 s \u2286 c \u207b\u00b9' Ioi t'\nn : \u2115\nx : \u03b1\nhx : x \u2208 s \\ u\n\u22a2 t' ^ n * ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s)) = \u222b (x : \u03b1) in v \u2229 s, t' ^ n \u2202\u03bc", "state_after": "no goals"}, {"tactic": "apply set_integral_mono_on _ _ (v_open.measurableSet.inter hs.measurableSet) _", "annotated_tactic": ["apply <a>set_integral_mono_on</a> _ _ (v_open.measurableSet.inter hs.measurableSet) _", [{"full_name": "MeasureTheory.set_integral_mono_on", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [721, 9], "def_end_pos": [721, 29]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nP : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\nhi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nt : \u211d\nt_pos : 0 \u2264 t\ntx\u2080 : t < c x\u2080\nht : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t\nt' : \u211d\ntt' : t < t'\nt'x\u2080 : t' < c x\u2080\nt'_pos : 0 < t'\nv : Set \u03b1\nv_open : IsOpen v\nx\u2080_v : x\u2080 \u2208 v\nhv : v \u2229 s \u2286 c \u207b\u00b9' Ioi t'\nn : \u2115\nx : \u03b1\nhx : x \u2208 s \\ u\n\u22a2 \u222b (x : \u03b1) in v \u2229 s, t' ^ n \u2202\u03bc \u2264 \u222b (y : \u03b1) in v \u2229 s, c y ^ n \u2202\u03bc", "state_after": "\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nP : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\nhi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nt : \u211d\nt_pos : 0 \u2264 t\ntx\u2080 : t < c x\u2080\nht : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t\nt' : \u211d\ntt' : t < t'\nt'x\u2080 : t' < c x\u2080\nt'_pos : 0 < t'\nv : Set \u03b1\nv_open : IsOpen v\nx\u2080_v : x\u2080 \u2208 v\nhv : v \u2229 s \u2286 c \u207b\u00b9' Ioi t'\nn : \u2115\nx : \u03b1\nhx : x \u2208 s \\ u\n\u22a2 IntegrableOn (fun a => t' ^ n) (v \u2229 s)\n\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nP : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\nhi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nt : \u211d\nt_pos : 0 \u2264 t\ntx\u2080 : t < c x\u2080\nht : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t\nt' : \u211d\ntt' : t < t'\nt'x\u2080 : t' < c x\u2080\nt'_pos : 0 < t'\nv : Set \u03b1\nv_open : IsOpen v\nx\u2080_v : x\u2080 \u2208 v\nhv : v \u2229 s \u2286 c \u207b\u00b9' Ioi t'\nn : \u2115\nx : \u03b1\nhx : x \u2208 s \\ u\n\u22a2 IntegrableOn (fun a => c a ^ n) (v \u2229 s)\n\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nP : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\nhi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nt : \u211d\nt_pos : 0 \u2264 t\ntx\u2080 : t < c x\u2080\nht : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t\nt' : \u211d\ntt' : t < t'\nt'x\u2080 : t' < c x\u2080\nt'_pos : 0 < t'\nv : Set \u03b1\nv_open : IsOpen v\nx\u2080_v : x\u2080 \u2208 v\nhv : v \u2229 s \u2286 c \u207b\u00b9' Ioi t'\nn : \u2115\nx : \u03b1\nhx : x \u2208 s \\ u\n\u22a2 \u2200 (x : \u03b1), x \u2208 v \u2229 s \u2192 t' ^ n \u2264 c x ^ n"}, {"tactic": "apply integrableOn_const.2 (Or.inr _)", "annotated_tactic": ["apply <a>integrableOn_const</a>.2 (<a>Or.inr</a> _)", [{"full_name": "MeasureTheory.integrableOn_const", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [119, 9], "def_end_pos": [119, 27]}, {"full_name": "Or.inr", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [519, 5], "def_end_pos": [519, 8]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nP : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\nhi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nt : \u211d\nt_pos : 0 \u2264 t\ntx\u2080 : t < c x\u2080\nht : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t\nt' : \u211d\ntt' : t < t'\nt'x\u2080 : t' < c x\u2080\nt'_pos : 0 < t'\nv : Set \u03b1\nv_open : IsOpen v\nx\u2080_v : x\u2080 \u2208 v\nhv : v \u2229 s \u2286 c \u207b\u00b9' Ioi t'\nn : \u2115\nx : \u03b1\nhx : x \u2208 s \\ u\n\u22a2 IntegrableOn (fun a => t' ^ n) (v \u2229 s)", "state_after": "\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nP : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\nhi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nt : \u211d\nt_pos : 0 \u2264 t\ntx\u2080 : t < c x\u2080\nht : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t\nt' : \u211d\ntt' : t < t'\nt'x\u2080 : t' < c x\u2080\nt'_pos : 0 < t'\nv : Set \u03b1\nv_open : IsOpen v\nx\u2080_v : x\u2080 \u2208 v\nhv : v \u2229 s \u2286 c \u207b\u00b9' Ioi t'\nn : \u2115\nx : \u03b1\nhx : x \u2208 s \\ u\n\u22a2 \u2191\u2191\u03bc (v \u2229 s) < \u22a4"}, {"tactic": "exact lt_of_le_of_lt (measure_mono (inter_subset_right _ _)) hs.measure_lt_top", "annotated_tactic": ["exact <a>lt_of_le_of_lt</a> (<a>measure_mono</a> (<a>inter_subset_right</a> _ _)) hs.measure_lt_top", [{"full_name": "lt_of_le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [122, 9], "def_end_pos": [122, 23]}, {"full_name": "MeasureTheory.measure_mono", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [193, 9], "def_end_pos": [193, 21]}, {"full_name": "Set.inter_subset_right", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [969, 9], "def_end_pos": [969, 27]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nP : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\nhi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nt : \u211d\nt_pos : 0 \u2264 t\ntx\u2080 : t < c x\u2080\nht : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t\nt' : \u211d\ntt' : t < t'\nt'x\u2080 : t' < c x\u2080\nt'_pos : 0 < t'\nv : Set \u03b1\nv_open : IsOpen v\nx\u2080_v : x\u2080 \u2208 v\nhv : v \u2229 s \u2286 c \u207b\u00b9' Ioi t'\nn : \u2115\nx : \u03b1\nhx : x \u2208 s \\ u\n\u22a2 \u2191\u2191\u03bc (v \u2229 s) < \u22a4", "state_after": "no goals"}, {"tactic": "exact (I n).mono (inter_subset_right _ _) le_rfl", "annotated_tactic": ["exact (I n).<a>mono</a> (<a>inter_subset_right</a> _ _) <a>le_rfl</a>", [{"full_name": "MeasureTheory.IntegrableOn.mono", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [123, 9], "def_end_pos": [123, 26]}, {"full_name": "Set.inter_subset_right", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [969, 9], "def_end_pos": [969, 27]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nP : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\nhi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nt : \u211d\nt_pos : 0 \u2264 t\ntx\u2080 : t < c x\u2080\nht : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t\nt' : \u211d\ntt' : t < t'\nt'x\u2080 : t' < c x\u2080\nt'_pos : 0 < t'\nv : Set \u03b1\nv_open : IsOpen v\nx\u2080_v : x\u2080 \u2208 v\nhv : v \u2229 s \u2286 c \u207b\u00b9' Ioi t'\nn : \u2115\nx : \u03b1\nhx : x \u2208 s \\ u\n\u22a2 IntegrableOn (fun a => c a ^ n) (v \u2229 s)", "state_after": "no goals"}, {"tactic": "intro x hx", "annotated_tactic": ["intro x hx", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nP : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\nhi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nt : \u211d\nt_pos : 0 \u2264 t\ntx\u2080 : t < c x\u2080\nht : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t\nt' : \u211d\ntt' : t < t'\nt'x\u2080 : t' < c x\u2080\nt'_pos : 0 < t'\nv : Set \u03b1\nv_open : IsOpen v\nx\u2080_v : x\u2080 \u2208 v\nhv : v \u2229 s \u2286 c \u207b\u00b9' Ioi t'\nn : \u2115\nx : \u03b1\nhx : x \u2208 s \\ u\n\u22a2 \u2200 (x : \u03b1), x \u2208 v \u2229 s \u2192 t' ^ n \u2264 c x ^ n", "state_after": "\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nP : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\nhi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nt : \u211d\nt_pos : 0 \u2264 t\ntx\u2080 : t < c x\u2080\nht : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t\nt' : \u211d\ntt' : t < t'\nt'x\u2080 : t' < c x\u2080\nt'_pos : 0 < t'\nv : Set \u03b1\nv_open : IsOpen v\nx\u2080_v : x\u2080 \u2208 v\nhv : v \u2229 s \u2286 c \u207b\u00b9' Ioi t'\nn : \u2115\nx\u271d : \u03b1\nhx\u271d : x\u271d \u2208 s \\ u\nx : \u03b1\nhx : x \u2208 v \u2229 s\n\u22a2 t' ^ n \u2264 c x ^ n"}, {"tactic": "exact pow_le_pow_of_le_left t'_pos.le (le_of_lt (hv hx)) _", "annotated_tactic": ["exact <a>pow_le_pow_of_le_left</a> t'_pos.le (<a>le_of_lt</a> (hv hx)) _", [{"full_name": "pow_le_pow_of_le_left", "def_path": "Mathlib/Algebra/GroupPower/Order.lean", "def_pos": [446, 9], "def_end_pos": [446, 30]}, {"full_name": "le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [110, 9], "def_end_pos": [110, 17]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nP : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\nhi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nt : \u211d\nt_pos : 0 \u2264 t\ntx\u2080 : t < c x\u2080\nht : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t\nt' : \u211d\ntt' : t < t'\nt'x\u2080 : t' < c x\u2080\nt'_pos : 0 < t'\nv : Set \u03b1\nv_open : IsOpen v\nx\u2080_v : x\u2080 \u2208 v\nhv : v \u2229 s \u2286 c \u207b\u00b9' Ioi t'\nn : \u2115\nx\u271d : \u03b1\nhx\u271d : x\u271d \u2208 s \\ u\nx : \u03b1\nhx : x \u2208 v \u2229 s\n\u22a2 t' ^ n \u2264 c x ^ n", "state_after": "no goals"}, {"tactic": "exact pow_le_pow_of_le_left (hnc _ hx.1) (ht x hx) _", "annotated_tactic": ["exact <a>pow_le_pow_of_le_left</a> (hnc _ hx.1) (ht x hx) _", [{"full_name": "pow_le_pow_of_le_left", "def_path": "Mathlib/Algebra/GroupPower/Order.lean", "def_pos": [446, 9], "def_end_pos": [446, 30]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nP : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\nhi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nt : \u211d\nt_pos : 0 \u2264 t\ntx\u2080 : t < c x\u2080\nht : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t\nt' : \u211d\ntt' : t < t'\nt'x\u2080 : t' < c x\u2080\nt'_pos : 0 < t'\nv : Set \u03b1\nv_open : IsOpen v\nx\u2080_v : x\u2080 \u2208 v\nhv : v \u2229 s \u2286 c \u207b\u00b9' Ioi t'\nn : \u2115\nx : \u03b1\nhx : x \u2208 s \\ u\nB : t' ^ n * ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s)) \u2264 \u222b (y : \u03b1) in s, c y ^ n \u2202\u03bc\n\u22a2 c x ^ n \u2264 t ^ n", "state_after": "no goals"}, {"tactic": "apply mul_pos (pow_pos (t_pos.trans_lt tt') _) (ENNReal.toReal_pos (h\u03bc v v_open x\u2080_v).ne' _)", "annotated_tactic": ["apply <a>mul_pos</a> (<a>pow_pos</a> (t_pos.trans_lt tt') _) (<a>ENNReal.toReal_pos</a> (h\u03bc v v_open x\u2080_v).<a>ne'</a> _)", [{"full_name": "mul_pos", "def_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "def_pos": [345, 7], "def_end_pos": [345, 14]}, {"full_name": "pow_pos", "def_path": "Mathlib/Algebra/Order/Ring/Defs.lean", "def_pos": [530, 9], "def_end_pos": [530, 16]}, {"full_name": "ENNReal.toReal_pos", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2131, 9], "def_end_pos": [2131, 19]}, {"full_name": "LT.lt.ne'", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [328, 9], "def_end_pos": [328, 12]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nP : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\nhi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nt : \u211d\nt_pos : 0 \u2264 t\ntx\u2080 : t < c x\u2080\nht : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t\nt' : \u211d\ntt' : t < t'\nt'x\u2080 : t' < c x\u2080\nt'_pos : 0 < t'\nv : Set \u03b1\nv_open : IsOpen v\nx\u2080_v : x\u2080 \u2208 v\nhv : v \u2229 s \u2286 c \u207b\u00b9' Ioi t'\nn : \u2115\nx : \u03b1\nhx : x \u2208 s \\ u\nB : t' ^ n * ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s)) \u2264 \u222b (y : \u03b1) in s, c y ^ n \u2202\u03bc\n\u22a2 0 < t' ^ n * ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s))", "state_after": "\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nP : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\nhi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nt : \u211d\nt_pos : 0 \u2264 t\ntx\u2080 : t < c x\u2080\nht : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t\nt' : \u211d\ntt' : t < t'\nt'x\u2080 : t' < c x\u2080\nt'_pos : 0 < t'\nv : Set \u03b1\nv_open : IsOpen v\nx\u2080_v : x\u2080 \u2208 v\nhv : v \u2229 s \u2286 c \u207b\u00b9' Ioi t'\nn : \u2115\nx : \u03b1\nhx : x \u2208 s \\ u\nB : t' ^ n * ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s)) \u2264 \u222b (y : \u03b1) in s, c y ^ n \u2202\u03bc\n\u22a2 \u2191\u2191\u03bc (v \u2229 s) \u2260 \u22a4"}, {"tactic": "have : \u03bc (v \u2229 s) \u2264 \u03bc s := measure_mono (inter_subset_right _ _)", "annotated_tactic": ["have : \u03bc (v \u2229 s) \u2264 \u03bc s := <a>measure_mono</a> (<a>inter_subset_right</a> _ _)", [{"full_name": "MeasureTheory.measure_mono", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [193, 9], "def_end_pos": [193, 21]}, {"full_name": "Set.inter_subset_right", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [969, 9], "def_end_pos": [969, 27]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nP : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\nhi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nt : \u211d\nt_pos : 0 \u2264 t\ntx\u2080 : t < c x\u2080\nht : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t\nt' : \u211d\ntt' : t < t'\nt'x\u2080 : t' < c x\u2080\nt'_pos : 0 < t'\nv : Set \u03b1\nv_open : IsOpen v\nx\u2080_v : x\u2080 \u2208 v\nhv : v \u2229 s \u2286 c \u207b\u00b9' Ioi t'\nn : \u2115\nx : \u03b1\nhx : x \u2208 s \\ u\nB : t' ^ n * ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s)) \u2264 \u222b (y : \u03b1) in s, c y ^ n \u2202\u03bc\n\u22a2 \u2191\u2191\u03bc (v \u2229 s) \u2260 \u22a4", "state_after": "\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nP : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\nhi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nt : \u211d\nt_pos : 0 \u2264 t\ntx\u2080 : t < c x\u2080\nht : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t\nt' : \u211d\ntt' : t < t'\nt'x\u2080 : t' < c x\u2080\nt'_pos : 0 < t'\nv : Set \u03b1\nv_open : IsOpen v\nx\u2080_v : x\u2080 \u2208 v\nhv : v \u2229 s \u2286 c \u207b\u00b9' Ioi t'\nn : \u2115\nx : \u03b1\nhx : x \u2208 s \\ u\nB : t' ^ n * ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s)) \u2264 \u222b (y : \u03b1) in s, c y ^ n \u2202\u03bc\nthis : \u2191\u2191\u03bc (v \u2229 s) \u2264 \u2191\u2191\u03bc s\n\u22a2 \u2191\u2191\u03bc (v \u2229 s) \u2260 \u22a4"}, {"tactic": "exact ne_of_lt (lt_of_le_of_lt this hs.measure_lt_top)", "annotated_tactic": ["exact <a>ne_of_lt</a> (<a>lt_of_le_of_lt</a> this hs.measure_lt_top)", [{"full_name": "ne_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [101, 9], "def_end_pos": [101, 17]}, {"full_name": "lt_of_le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [122, 9], "def_end_pos": [122, 23]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nP : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\nhi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nt : \u211d\nt_pos : 0 \u2264 t\ntx\u2080 : t < c x\u2080\nht : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t\nt' : \u211d\ntt' : t < t'\nt'x\u2080 : t' < c x\u2080\nt'_pos : 0 < t'\nv : Set \u03b1\nv_open : IsOpen v\nx\u2080_v : x\u2080 \u2208 v\nhv : v \u2229 s \u2286 c \u207b\u00b9' Ioi t'\nn : \u2115\nx : \u03b1\nhx : x \u2208 s \\ u\nB : t' ^ n * ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s)) \u2264 \u222b (y : \u03b1) in s, c y ^ n \u2202\u03bc\nthis : \u2191\u2191\u03bc (v \u2229 s) \u2264 \u2191\u2191\u03bc s\n\u22a2 \u2191\u2191\u03bc (v \u2229 s) \u2260 \u22a4", "state_after": "no goals"}, {"tactic": "apply Tendsto.mul tendsto_const_nhds _", "annotated_tactic": ["apply <a>Tendsto.mul</a> <a>tendsto_const_nhds</a> _", [{"full_name": "Filter.Tendsto.mul", "def_path": "Mathlib/Topology/Algebra/Monoid.lean", "def_pos": [119, 9], "def_end_pos": [119, 27]}, {"full_name": "tendsto_const_nhds", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [1049, 9], "def_end_pos": [1049, 27]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nP : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\nhi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nt : \u211d\nt_pos : 0 \u2264 t\ntx\u2080 : t < c x\u2080\nht : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t\nt' : \u211d\ntt' : t < t'\nt'x\u2080 : t' < c x\u2080\nt'_pos : 0 < t'\nv : Set \u03b1\nv_open : IsOpen v\nx\u2080_v : x\u2080 \u2208 v\nhv : v \u2229 s \u2286 c \u207b\u00b9' Ioi t'\nM : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \\ u \u2192 \u03c6 n x \u2264 (ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s)))\u207b\u00b9 * (t / t') ^ n\n\u22a2 Tendsto (fun n => (ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s)))\u207b\u00b9 * (t / t') ^ n) atTop (\ud835\udcdd ((ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s)))\u207b\u00b9 * 0))", "state_after": "\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nP : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\nhi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nt : \u211d\nt_pos : 0 \u2264 t\ntx\u2080 : t < c x\u2080\nht : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t\nt' : \u211d\ntt' : t < t'\nt'x\u2080 : t' < c x\u2080\nt'_pos : 0 < t'\nv : Set \u03b1\nv_open : IsOpen v\nx\u2080_v : x\u2080 \u2208 v\nhv : v \u2229 s \u2286 c \u207b\u00b9' Ioi t'\nM : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \\ u \u2192 \u03c6 n x \u2264 (ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s)))\u207b\u00b9 * (t / t') ^ n\n\u22a2 Tendsto (fun x => (t / t') ^ x) atTop (\ud835\udcdd 0)"}, {"tactic": "apply tendsto_pow_atTop_nhds_0_of_lt_1 (div_nonneg t_pos t'_pos.le)", "annotated_tactic": ["apply <a>tendsto_pow_atTop_nhds_0_of_lt_1</a> (<a>div_nonneg</a> t_pos t'_pos.le)", [{"full_name": "tendsto_pow_atTop_nhds_0_of_lt_1", "def_path": "Mathlib/Analysis/SpecificLimits/Basic.lean", "def_pos": [110, 9], "def_end_pos": [110, 41]}, {"full_name": "div_nonneg", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [94, 9], "def_end_pos": [94, 19]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nP : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\nhi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nt : \u211d\nt_pos : 0 \u2264 t\ntx\u2080 : t < c x\u2080\nht : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t\nt' : \u211d\ntt' : t < t'\nt'x\u2080 : t' < c x\u2080\nt'_pos : 0 < t'\nv : Set \u03b1\nv_open : IsOpen v\nx\u2080_v : x\u2080 \u2208 v\nhv : v \u2229 s \u2286 c \u207b\u00b9' Ioi t'\nM : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \\ u \u2192 \u03c6 n x \u2264 (ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s)))\u207b\u00b9 * (t / t') ^ n\n\u22a2 Tendsto (fun x => (t / t') ^ x) atTop (\ud835\udcdd 0)", "state_after": "\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nP : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\nhi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nt : \u211d\nt_pos : 0 \u2264 t\ntx\u2080 : t < c x\u2080\nht : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t\nt' : \u211d\ntt' : t < t'\nt'x\u2080 : t' < c x\u2080\nt'_pos : 0 < t'\nv : Set \u03b1\nv_open : IsOpen v\nx\u2080_v : x\u2080 \u2208 v\nhv : v \u2229 s \u2286 c \u207b\u00b9' Ioi t'\nM : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \\ u \u2192 \u03c6 n x \u2264 (ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s)))\u207b\u00b9 * (t / t') ^ n\n\u22a2 t / t' < 1"}, {"tactic": "exact (div_lt_one t'_pos).2 tt'", "annotated_tactic": ["exact (<a>div_lt_one</a> t'_pos).2 tt'", [{"full_name": "div_lt_one", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [431, 9], "def_end_pos": [431, 19]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nP : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\nhi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nt : \u211d\nt_pos : 0 \u2264 t\ntx\u2080 : t < c x\u2080\nht : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t\nt' : \u211d\ntt' : t < t'\nt'x\u2080 : t' < c x\u2080\nt'_pos : 0 < t'\nv : Set \u03b1\nv_open : IsOpen v\nx\u2080_v : x\u2080 \u2208 v\nhv : v \u2229 s \u2286 c \u207b\u00b9' Ioi t'\nM : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \\ u \u2192 \u03c6 n x \u2264 (ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s)))\u207b\u00b9 * (t / t') ^ n\n\u22a2 t / t' < 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "full_name": "Nat.lt_min", "start": [460, 11], "end": [460, 83], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "full_name": "String.find_of_valid", "start": [349, 1], "end": [350, 43], "traced_tactics": [{"tactic": "simpa using findAux_of_valid p [] s.1 []", "annotated_tactic": ["simpa using <a>findAux_of_valid</a> p [] s.1 []", [{"full_name": "String.findAux_of_valid", "def_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "def_pos": [335, 9], "def_end_pos": [335, 25]}]], "state_before": "p : Char \u2192 Bool\ns : String\n\u22a2 find s p = { byteIdx := utf8Len (List.takeWhile (fun x => !p x) s.data) }", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "full_name": "IntervalIntegrable.div_const", "start": [286, 1], "end": [288, 52], "traced_tactics": [{"tactic": "simpa only [div_eq_mul_inv] using mul_const h c\u207b\u00b9", "annotated_tactic": ["simpa only [<a>div_eq_mul_inv</a>] using <a>mul_const</a> h c\u207b\u00b9", [{"full_name": "div_eq_mul_inv", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [977, 9], "def_end_pos": [977, 23]}, {"full_name": "IntervalIntegrable.mul_const", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [280, 9], "def_end_pos": [280, 18]}]], "state_before": "\u03b9 : Type u_1\n\ud835\udd5c\u271d : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedRing A\nf\u271d g : \u211d \u2192 E\na b : \u211d\n\u03bc : Measure \u211d\n\ud835\udd5c : Type u_6\nf : \u211d \u2192 \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c\nh : IntervalIntegrable f \u03bc a b\nc : \ud835\udd5c\n\u22a2 IntervalIntegrable (fun x => f x / c) \u03bc a b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Intervals/WithBotTop.lean", "full_name": "WithBot.image_coe_Ici", "start": [210, 1], "end": [212, 69], "traced_tactics": [{"tactic": "rw [\u2190 preimage_coe_Ici, image_preimage_eq_inter_range, range_coe,\n  inter_eq_self_of_subset_left (Ici_subset_Ioi.2 <| bot_lt_coe a)]", "annotated_tactic": ["rw [\u2190 <a>preimage_coe_Ici</a>, <a>image_preimage_eq_inter_range</a>, <a>range_coe</a>,\n    <a>inter_eq_self_of_subset_left</a> (<a>Ici_subset_Ioi</a>.2 <| <a>bot_lt_coe</a> a)]", [{"full_name": "WithBot.preimage_coe_Ici", "def_path": "Mathlib/Data/Set/Intervals/WithBotTop.lean", "def_pos": [152, 9], "def_end_pos": [152, 25]}, {"full_name": "Set.image_preimage_eq_inter_range", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [796, 9], "def_end_pos": [796, 38]}, {"full_name": "WithBot.range_coe", "def_path": "Mathlib/Data/Set/Intervals/WithBotTop.lean", "def_pos": [142, 9], "def_end_pos": [142, 18]}, {"full_name": "Set.inter_eq_self_of_subset_left", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [987, 9], "def_end_pos": [987, 37]}, {"full_name": "Set.Ici_subset_Ioi", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [423, 9], "def_end_pos": [423, 23]}, {"full_name": "WithBot.bot_lt_coe", "def_path": "Mathlib/Order/WithBot.lean", "def_pos": [289, 9], "def_end_pos": [289, 19]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : PartialOrder \u03b1\na b : \u03b1\n\u22a2 some '' Ici a = Ici \u2191a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/ProbabilityMeasure.lean", "full_name": "MeasureTheory.ProbabilityMeasure.toFiniteMeasure_embedding", "start": [265, 1], "end": [269, 78], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Decomposition/Lebesgue.lean", "full_name": "MeasureTheory.SignedMeasure.toJordanDecomposition_eq_of_eq_add_withDensity", "start": [971, 1], "end": [994, 34], "traced_tactics": [{"tactic": "haveI := isFiniteMeasure_withDensity_ofReal hfi.2", "annotated_tactic": ["haveI := <a>isFiniteMeasure_withDensity_ofReal</a> hfi.2", [{"full_name": "MeasureTheory.isFiniteMeasure_withDensity_ofReal", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [280, 9], "def_end_pos": [280, 43]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : Measurable f\nhfi : Integrable f\nht\u03bc : t \u27c2\u1d65 toENNRealVectorMeasure \u03bc\nhadd : s = t + withDensity\u1d65 \u03bc f\n\u22a2 IsFiniteMeasure ((toJordanDecomposition t).posPart + withDensity \u03bc fun x => ENNReal.ofReal (f x))", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : Measurable f\nhfi : Integrable f\nht\u03bc : t \u27c2\u1d65 toENNRealVectorMeasure \u03bc\nhadd : s = t + withDensity\u1d65 \u03bc f\nthis : IsFiniteMeasure (withDensity \u03bc fun x => ENNReal.ofReal (f x))\n\u22a2 IsFiniteMeasure ((toJordanDecomposition t).posPart + withDensity \u03bc fun x => ENNReal.ofReal (f x))"}, {"tactic": "infer_instance", "annotated_tactic": ["infer_instance", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : Measurable f\nhfi : Integrable f\nht\u03bc : t \u27c2\u1d65 toENNRealVectorMeasure \u03bc\nhadd : s = t + withDensity\u1d65 \u03bc f\nthis : IsFiniteMeasure (withDensity \u03bc fun x => ENNReal.ofReal (f x))\n\u22a2 IsFiniteMeasure ((toJordanDecomposition t).posPart + withDensity \u03bc fun x => ENNReal.ofReal (f x))", "state_after": "no goals"}, {"tactic": "haveI := isFiniteMeasure_withDensity_ofReal hfi.neg.2", "annotated_tactic": ["haveI := <a>isFiniteMeasure_withDensity_ofReal</a> hfi.neg.2", [{"full_name": "MeasureTheory.isFiniteMeasure_withDensity_ofReal", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [280, 9], "def_end_pos": [280, 43]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : Measurable f\nhfi : Integrable f\nht\u03bc : t \u27c2\u1d65 toENNRealVectorMeasure \u03bc\nhadd : s = t + withDensity\u1d65 \u03bc f\n\u22a2 IsFiniteMeasure ((toJordanDecomposition t).negPart + withDensity \u03bc fun x => ENNReal.ofReal (-f x))", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : Measurable f\nhfi : Integrable f\nht\u03bc : t \u27c2\u1d65 toENNRealVectorMeasure \u03bc\nhadd : s = t + withDensity\u1d65 \u03bc f\nthis : IsFiniteMeasure (withDensity \u03bc fun x => ENNReal.ofReal ((-f) x))\n\u22a2 IsFiniteMeasure ((toJordanDecomposition t).negPart + withDensity \u03bc fun x => ENNReal.ofReal (-f x))"}, {"tactic": "infer_instance", "annotated_tactic": ["infer_instance", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : Measurable f\nhfi : Integrable f\nht\u03bc : t \u27c2\u1d65 toENNRealVectorMeasure \u03bc\nhadd : s = t + withDensity\u1d65 \u03bc f\nthis : IsFiniteMeasure (withDensity \u03bc fun x => ENNReal.ofReal ((-f) x))\n\u22a2 IsFiniteMeasure ((toJordanDecomposition t).negPart + withDensity \u03bc fun x => ENNReal.ofReal (-f x))", "state_after": "no goals"}, {"tactic": "haveI := isFiniteMeasure_withDensity_ofReal hfi.2", "annotated_tactic": ["haveI := <a>isFiniteMeasure_withDensity_ofReal</a> hfi.2", [{"full_name": "MeasureTheory.isFiniteMeasure_withDensity_ofReal", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [280, 9], "def_end_pos": [280, 43]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : Measurable f\nhfi : Integrable f\nht\u03bc : t \u27c2\u1d65 toENNRealVectorMeasure \u03bc\nhadd : s = t + withDensity\u1d65 \u03bc f\n\u22a2 toJordanDecomposition s =\n    JordanDecomposition.mk ((toJordanDecomposition t).posPart + withDensity \u03bc fun x => ENNReal.ofReal (f x))\n      ((toJordanDecomposition t).negPart + withDensity \u03bc fun x => ENNReal.ofReal (-f x))\n      (_ :\n        ((toJordanDecomposition t).posPart + withDensity \u03bc fun x => ENNReal.ofReal (f x)) \u27c2\u2098\n          (toJordanDecomposition t).negPart + withDensity \u03bc fun x => ENNReal.ofReal (-f x))", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : Measurable f\nhfi : Integrable f\nht\u03bc : t \u27c2\u1d65 toENNRealVectorMeasure \u03bc\nhadd : s = t + withDensity\u1d65 \u03bc f\nthis : IsFiniteMeasure (withDensity \u03bc fun x => ENNReal.ofReal (f x))\n\u22a2 toJordanDecomposition s =\n    JordanDecomposition.mk ((toJordanDecomposition t).posPart + withDensity \u03bc fun x => ENNReal.ofReal (f x))\n      ((toJordanDecomposition t).negPart + withDensity \u03bc fun x => ENNReal.ofReal (-f x))\n      (_ :\n        ((toJordanDecomposition t).posPart + withDensity \u03bc fun x => ENNReal.ofReal (f x)) \u27c2\u2098\n          (toJordanDecomposition t).negPart + withDensity \u03bc fun x => ENNReal.ofReal (-f x))"}, {"tactic": "haveI := isFiniteMeasure_withDensity_ofReal hfi.neg.2", "annotated_tactic": ["haveI := <a>isFiniteMeasure_withDensity_ofReal</a> hfi.neg.2", [{"full_name": "MeasureTheory.isFiniteMeasure_withDensity_ofReal", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [280, 9], "def_end_pos": [280, 43]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : Measurable f\nhfi : Integrable f\nht\u03bc : t \u27c2\u1d65 toENNRealVectorMeasure \u03bc\nhadd : s = t + withDensity\u1d65 \u03bc f\nthis : IsFiniteMeasure (withDensity \u03bc fun x => ENNReal.ofReal (f x))\n\u22a2 toJordanDecomposition s =\n    JordanDecomposition.mk ((toJordanDecomposition t).posPart + withDensity \u03bc fun x => ENNReal.ofReal (f x))\n      ((toJordanDecomposition t).negPart + withDensity \u03bc fun x => ENNReal.ofReal (-f x))\n      (_ :\n        ((toJordanDecomposition t).posPart + withDensity \u03bc fun x => ENNReal.ofReal (f x)) \u27c2\u2098\n          (toJordanDecomposition t).negPart + withDensity \u03bc fun x => ENNReal.ofReal (-f x))", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : Measurable f\nhfi : Integrable f\nht\u03bc : t \u27c2\u1d65 toENNRealVectorMeasure \u03bc\nhadd : s = t + withDensity\u1d65 \u03bc f\nthis\u271d : IsFiniteMeasure (withDensity \u03bc fun x => ENNReal.ofReal (f x))\nthis : IsFiniteMeasure (withDensity \u03bc fun x => ENNReal.ofReal ((-f) x))\n\u22a2 toJordanDecomposition s =\n    JordanDecomposition.mk ((toJordanDecomposition t).posPart + withDensity \u03bc fun x => ENNReal.ofReal (f x))\n      ((toJordanDecomposition t).negPart + withDensity \u03bc fun x => ENNReal.ofReal (-f x))\n      (_ :\n        ((toJordanDecomposition t).posPart + withDensity \u03bc fun x => ENNReal.ofReal (f x)) \u27c2\u2098\n          (toJordanDecomposition t).negPart + withDensity \u03bc fun x => ENNReal.ofReal (-f x))"}, {"tactic": "refine' toJordanDecomposition_eq _", "annotated_tactic": ["refine' <a>toJordanDecomposition_eq</a> _", [{"full_name": "MeasureTheory.SignedMeasure.toJordanDecomposition_eq", "def_path": "Mathlib/MeasureTheory/Decomposition/Jordan.lean", "def_pos": [488, 9], "def_end_pos": [488, 33]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : Measurable f\nhfi : Integrable f\nht\u03bc : t \u27c2\u1d65 toENNRealVectorMeasure \u03bc\nhadd : s = t + withDensity\u1d65 \u03bc f\nthis\u271d : IsFiniteMeasure (withDensity \u03bc fun x => ENNReal.ofReal (f x))\nthis : IsFiniteMeasure (withDensity \u03bc fun x => ENNReal.ofReal ((-f) x))\n\u22a2 toJordanDecomposition s =\n    JordanDecomposition.mk ((toJordanDecomposition t).posPart + withDensity \u03bc fun x => ENNReal.ofReal (f x))\n      ((toJordanDecomposition t).negPart + withDensity \u03bc fun x => ENNReal.ofReal (-f x))\n      (_ :\n        ((toJordanDecomposition t).posPart + withDensity \u03bc fun x => ENNReal.ofReal (f x)) \u27c2\u2098\n          (toJordanDecomposition t).negPart + withDensity \u03bc fun x => ENNReal.ofReal (-f x))", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : Measurable f\nhfi : Integrable f\nht\u03bc : t \u27c2\u1d65 toENNRealVectorMeasure \u03bc\nhadd : s = t + withDensity\u1d65 \u03bc f\nthis\u271d : IsFiniteMeasure (withDensity \u03bc fun x => ENNReal.ofReal (f x))\nthis : IsFiniteMeasure (withDensity \u03bc fun x => ENNReal.ofReal ((-f) x))\n\u22a2 s =\n    JordanDecomposition.toSignedMeasure\n      (JordanDecomposition.mk ((toJordanDecomposition t).posPart + withDensity \u03bc fun x => ENNReal.ofReal (f x))\n        ((toJordanDecomposition t).negPart + withDensity \u03bc fun x => ENNReal.ofReal (-f x))\n        (_ :\n          ((toJordanDecomposition t).posPart + withDensity \u03bc fun x => ENNReal.ofReal (f x)) \u27c2\u2098\n            (toJordanDecomposition t).negPart + withDensity \u03bc fun x => ENNReal.ofReal (-f x)))"}, {"tactic": "simp_rw [JordanDecomposition.toSignedMeasure, hadd]", "annotated_tactic": ["simp_rw [<a>JordanDecomposition.toSignedMeasure</a>, hadd]", [{"full_name": "MeasureTheory.JordanDecomposition.toSignedMeasure", "def_path": "Mathlib/MeasureTheory/Decomposition/Jordan.lean", "def_pos": [169, 5], "def_end_pos": [169, 20]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : Measurable f\nhfi : Integrable f\nht\u03bc : t \u27c2\u1d65 toENNRealVectorMeasure \u03bc\nhadd : s = t + withDensity\u1d65 \u03bc f\nthis\u271d : IsFiniteMeasure (withDensity \u03bc fun x => ENNReal.ofReal (f x))\nthis : IsFiniteMeasure (withDensity \u03bc fun x => ENNReal.ofReal ((-f) x))\n\u22a2 s =\n    JordanDecomposition.toSignedMeasure\n      (JordanDecomposition.mk ((toJordanDecomposition t).posPart + withDensity \u03bc fun x => ENNReal.ofReal (f x))\n        ((toJordanDecomposition t).negPart + withDensity \u03bc fun x => ENNReal.ofReal (-f x))\n        (_ :\n          ((toJordanDecomposition t).posPart + withDensity \u03bc fun x => ENNReal.ofReal (f x)) \u27c2\u2098\n            (toJordanDecomposition t).negPart + withDensity \u03bc fun x => ENNReal.ofReal (-f x)))", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : Measurable f\nhfi : Integrable f\nht\u03bc : t \u27c2\u1d65 toENNRealVectorMeasure \u03bc\nhadd : s = t + withDensity\u1d65 \u03bc f\nthis\u271d : IsFiniteMeasure (withDensity \u03bc fun x => ENNReal.ofReal (f x))\nthis : IsFiniteMeasure (withDensity \u03bc fun x => ENNReal.ofReal ((-f) x))\n\u22a2 t + withDensity\u1d65 \u03bc f =\n    toSignedMeasure ((toJordanDecomposition t).posPart + withDensity \u03bc fun x => ENNReal.ofReal (f x)) -\n      toSignedMeasure ((toJordanDecomposition t).negPart + withDensity \u03bc fun x => ENNReal.ofReal (-f x))"}, {"tactic": "ext i hi", "annotated_tactic": ["ext i hi", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : Measurable f\nhfi : Integrable f\nht\u03bc : t \u27c2\u1d65 toENNRealVectorMeasure \u03bc\nhadd : s = t + withDensity\u1d65 \u03bc f\nthis\u271d : IsFiniteMeasure (withDensity \u03bc fun x => ENNReal.ofReal (f x))\nthis : IsFiniteMeasure (withDensity \u03bc fun x => ENNReal.ofReal ((-f) x))\n\u22a2 t + withDensity\u1d65 \u03bc f =\n    toSignedMeasure ((toJordanDecomposition t).posPart + withDensity \u03bc fun x => ENNReal.ofReal (f x)) -\n      toSignedMeasure ((toJordanDecomposition t).negPart + withDensity \u03bc fun x => ENNReal.ofReal (-f x))", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : Measurable f\nhfi : Integrable f\nht\u03bc : t \u27c2\u1d65 toENNRealVectorMeasure \u03bc\nhadd : s = t + withDensity\u1d65 \u03bc f\nthis\u271d : IsFiniteMeasure (withDensity \u03bc fun x => ENNReal.ofReal (f x))\nthis : IsFiniteMeasure (withDensity \u03bc fun x => ENNReal.ofReal ((-f) x))\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 \u2191(t + withDensity\u1d65 \u03bc f) i =\n    \u2191(toSignedMeasure ((toJordanDecomposition t).posPart + withDensity \u03bc fun x => ENNReal.ofReal (f x)) -\n          toSignedMeasure ((toJordanDecomposition t).negPart + withDensity \u03bc fun x => ENNReal.ofReal (-f x)))\n      i"}, {"tactic": "rw [VectorMeasure.sub_apply, toSignedMeasure_apply_measurable hi,\n    toSignedMeasure_apply_measurable hi, add_apply, add_apply, ENNReal.toReal_add,\n    ENNReal.toReal_add, add_sub_add_comm, \u2190 toSignedMeasure_apply_measurable hi,\n    \u2190 toSignedMeasure_apply_measurable hi, \u2190 VectorMeasure.sub_apply,\n    \u2190 JordanDecomposition.toSignedMeasure, toSignedMeasure_toJordanDecomposition,\n    VectorMeasure.add_apply, \u2190 toSignedMeasure_apply_measurable hi,\n    \u2190 toSignedMeasure_apply_measurable hi,\n    withDensity\u1d65_eq_withDensity_pos_part_sub_withDensity_neg_part hfi,\n    VectorMeasure.sub_apply] <;>\n  exact (measure_lt_top _ _).ne", "annotated_tactic": ["rw [<a>VectorMeasure.sub_apply</a>, <a>toSignedMeasure_apply_measurable</a> hi,\n      <a>toSignedMeasure_apply_measurable</a> hi, <a>add_apply</a>, <a>add_apply</a>, <a>ENNReal.toReal_add</a>,\n      <a>ENNReal.toReal_add</a>, <a>add_sub_add_comm</a>, \u2190 <a>toSignedMeasure_apply_measurable</a> hi,\n      \u2190 <a>toSignedMeasure_apply_measurable</a> hi, \u2190 <a>VectorMeasure.sub_apply</a>,\n      \u2190 <a>JordanDecomposition.toSignedMeasure</a>, <a>toSignedMeasure_toJordanDecomposition</a>,\n      <a>VectorMeasure.add_apply</a>, \u2190 <a>toSignedMeasure_apply_measurable</a> hi,\n      \u2190 <a>toSignedMeasure_apply_measurable</a> hi,\n      <a>withDensity\u1d65_eq_withDensity_pos_part_sub_withDensity_neg_part</a> hfi,\n      <a>VectorMeasure.sub_apply</a>] <;>\n    exact (<a>measure_lt_top</a> _ _).<a>ne</a>", [{"full_name": "MeasureTheory.VectorMeasure.sub_apply", "def_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "def_pos": [370, 9], "def_end_pos": [370, 18]}, {"full_name": "MeasureTheory.Measure.toSignedMeasure_apply_measurable", "def_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "def_pos": [435, 9], "def_end_pos": [435, 41]}, {"full_name": "MeasureTheory.Measure.toSignedMeasure_apply_measurable", "def_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "def_pos": [435, 9], "def_end_pos": [435, 41]}, {"full_name": "MeasureTheory.Measure.add_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [798, 9], "def_end_pos": [798, 18]}, {"full_name": "MeasureTheory.Measure.add_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [798, 9], "def_end_pos": [798, 18]}, {"full_name": "ENNReal.toReal_add", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1997, 9], "def_end_pos": [1997, 19]}, {"full_name": "ENNReal.toReal_add", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1997, 9], "def_end_pos": [1997, 19]}, {"full_name": "add_sub_add_comm", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [586, 3], "def_end_pos": [586, 14]}, {"full_name": "MeasureTheory.Measure.toSignedMeasure_apply_measurable", "def_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "def_pos": [435, 9], "def_end_pos": [435, 41]}, {"full_name": "MeasureTheory.Measure.toSignedMeasure_apply_measurable", "def_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "def_pos": [435, 9], "def_end_pos": [435, 41]}, {"full_name": "MeasureTheory.VectorMeasure.sub_apply", "def_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "def_pos": [370, 9], "def_end_pos": [370, 18]}, {"full_name": "MeasureTheory.JordanDecomposition.toSignedMeasure", "def_path": "Mathlib/MeasureTheory/Decomposition/Jordan.lean", "def_pos": [169, 5], "def_end_pos": [169, 20]}, {"full_name": "MeasureTheory.SignedMeasure.toSignedMeasure_toJordanDecomposition", "def_path": "Mathlib/MeasureTheory/Decomposition/Jordan.lean", "def_pos": [260, 9], "def_end_pos": [260, 46]}, {"full_name": "MeasureTheory.VectorMeasure.add_apply", "def_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "def_pos": [314, 9], "def_end_pos": [314, 18]}, {"full_name": "MeasureTheory.Measure.toSignedMeasure_apply_measurable", "def_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "def_pos": [435, 9], "def_end_pos": [435, 41]}, {"full_name": "MeasureTheory.Measure.toSignedMeasure_apply_measurable", "def_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "def_pos": [435, 9], "def_end_pos": [435, 41]}, {"full_name": "MeasureTheory.withDensity\u1d65_eq_withDensity_pos_part_sub_withDensity_neg_part", "def_path": "Mathlib/MeasureTheory/Measure/WithDensityVectorMeasure.lean", "def_pos": [179, 9], "def_end_pos": [179, 70]}, {"full_name": "MeasureTheory.VectorMeasure.sub_apply", "def_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "def_pos": [370, 9], "def_end_pos": [370, 18]}, {"full_name": "MeasureTheory.measure_lt_top", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2866, 9], "def_end_pos": [2866, 23]}, {"full_name": "LT.lt.ne", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [152, 7], "def_end_pos": [152, 15]}]], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : Measurable f\nhfi : Integrable f\nht\u03bc : t \u27c2\u1d65 toENNRealVectorMeasure \u03bc\nhadd : s = t + withDensity\u1d65 \u03bc f\nthis\u271d : IsFiniteMeasure (withDensity \u03bc fun x => ENNReal.ofReal (f x))\nthis : IsFiniteMeasure (withDensity \u03bc fun x => ENNReal.ofReal ((-f) x))\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 \u2191(t + withDensity\u1d65 \u03bc f) i =\n    \u2191(toSignedMeasure ((toJordanDecomposition t).posPart + withDensity \u03bc fun x => ENNReal.ofReal (f x)) -\n          toSignedMeasure ((toJordanDecomposition t).negPart + withDensity \u03bc fun x => ENNReal.ofReal (-f x)))\n      i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Part.lean", "full_name": "Part.bind_defined", "start": [659, 1], "end": [661, 17], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "full_name": "Int.ediv_add_emod'", "start": [402, 1], "end": [403, 41], "traced_tactics": [{"tactic": "rw [Int.mul_comm]", "annotated_tactic": ["rw [<a>Int.mul_comm</a>]", [{"full_name": "Int.mul_comm", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [380, 19], "def_end_pos": [380, 27]}]], "state_before": "m k : Int\n\u22a2 m / k * k + m % k = m", "state_after": "m k : Int\n\u22a2 k * (m / k) + m % k = m"}, {"tactic": "apply ediv_add_emod", "annotated_tactic": ["apply <a>ediv_add_emod</a>", [{"full_name": "Int.ediv_add_emod", "def_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "def_pos": [319, 9], "def_end_pos": [319, 22]}]], "state_before": "m k : Int\n\u22a2 k * (m / k) + m % k = m", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Ackermann.lean", "full_name": "ack_add_one_sq_lt_ack_add_four", "start": [297, 1], "end": [304, 61], "traced_tactics": [{"tactic": "linarith", "annotated_tactic": ["linarith", []], "state_before": "m n : \u2115\n\u22a2 m \u2264 m + 2", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/PartrecCode.lean", "full_name": "Nat.Partrec.Code.curry_inj", "start": [682, 1], "end": [686, 24], "traced_tactics": [{"tactic": "injection h", "annotated_tactic": ["injection h", []], "state_before": "c\u2081 c\u2082 : Code\nn\u2081 n\u2082 : \u2115\nh : curry c\u2081 n\u2081 = curry c\u2082 n\u2082\n\u22a2 c\u2081 = c\u2082", "state_after": "no goals"}, {"tactic": "injection h with h\u2081 h\u2082", "annotated_tactic": ["injection h with h\u2081 h\u2082", []], "state_before": "c\u2081 c\u2082 : Code\nn\u2081 n\u2082 : \u2115\nh : curry c\u2081 n\u2081 = curry c\u2082 n\u2082\n\u22a2 n\u2081 = n\u2082", "state_after": "c\u2081 c\u2082 : Code\nn\u2081 n\u2082 : \u2115\nh\u2081 : c\u2081 = c\u2082\nh\u2082 : pair (Code.const n\u2081) Code.id = pair (Code.const n\u2082) Code.id\n\u22a2 n\u2081 = n\u2082"}, {"tactic": "injection h\u2082 with h\u2083 h\u2084", "annotated_tactic": ["injection h\u2082 with h\u2083 h\u2084", []], "state_before": "c\u2081 c\u2082 : Code\nn\u2081 n\u2082 : \u2115\nh\u2081 : c\u2081 = c\u2082\nh\u2082 : pair (Code.const n\u2081) Code.id = pair (Code.const n\u2082) Code.id\n\u22a2 n\u2081 = n\u2082", "state_after": "c\u2081 c\u2082 : Code\nn\u2081 n\u2082 : \u2115\nh\u2081 : c\u2081 = c\u2082\nh\u2083 : Code.const n\u2081 = Code.const n\u2082\nh\u2084 : Code.id = Code.id\n\u22a2 n\u2081 = n\u2082"}, {"tactic": "exact const_inj h\u2083", "annotated_tactic": ["exact <a>const_inj</a> h\u2083", [{"full_name": "Nat.Partrec.Code.const_inj", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [108, 9], "def_end_pos": [108, 18]}]], "state_before": "c\u2081 c\u2082 : Code\nn\u2081 n\u2082 : \u2115\nh\u2081 : c\u2081 = c\u2082\nh\u2083 : Code.const n\u2081 = Code.const n\u2082\nh\u2084 : Code.id = Code.id\n\u22a2 n\u2081 = n\u2082", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "full_name": "MeasureTheory.OuterMeasure.restrict_iInf_restrict", "start": [1257, 1], "end": [1263, 85], "traced_tactics": [{"tactic": "rw [Subtype.range_coe]", "annotated_tactic": ["rw [<a>Subtype.range_coe</a>]", [{"full_name": "Subtype.range_coe", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [1409, 9], "def_end_pos": [1409, 18]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Sort u_2\ns : Set \u03b1\nm : \u03b9 \u2192 OuterMeasure \u03b1\n\u22a2 \u2191(restrict s) (\u2a05 i, \u2191(restrict s) (m i)) = \u2191(restrict (range Subtype.val)) (\u2a05 i, \u2191(restrict s) (m i))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Finsupp.lean", "full_name": "Finsupp.mem_pi", "start": [95, 1], "end": [96, 53], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Decomposition/Lebesgue.lean", "full_name": "MeasureTheory.SignedMeasure.eq_rnDeriv", "start": [1130, 1], "end": [1140, 45], "traced_tactics": [{"tactic": "set f' := hfi.1.mk f", "annotated_tactic": ["set f' := hfi.1.<a>mk</a> f", [{"full_name": "MeasureTheory.AEStronglyMeasurable.mk", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1202, 29], "def_end_pos": [1202, 31]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t\u271d t : SignedMeasure \u03b1\nf : \u03b1 \u2192 \u211d\nhfi : Integrable f\nht\u03bc : t \u27c2\u1d65 toENNRealVectorMeasure \u03bc\nhadd : s = t + withDensity\u1d65 \u03bc f\n\u22a2 f =\u1da0[ae \u03bc] rnDeriv s \u03bc", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t\u271d t : SignedMeasure \u03b1\nf : \u03b1 \u2192 \u211d\nhfi : Integrable f\nht\u03bc : t \u27c2\u1d65 toENNRealVectorMeasure \u03bc\nhadd : s = t + withDensity\u1d65 \u03bc f\nf' : \u03b1 \u2192 \u211d := AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f \u03bc)\n\u22a2 f =\u1da0[ae \u03bc] rnDeriv s \u03bc"}, {"tactic": "have hadd' : s = t + \u03bc.withDensity\u1d65 f' := by\n  convert hadd using 2\n  exact WithDensity\u1d65Eq.congr_ae hfi.1.ae_eq_mk.symm", "annotated_tactic": ["have hadd' : s = t + \u03bc.withDensity\u1d65 f' := by\n    convert hadd using 2\n    exact <a>WithDensity\u1d65Eq.congr_ae</a> hfi.1.ae_eq_mk.symm", [{"full_name": "MeasureTheory.WithDensity\u1d65Eq.congr_ae", "def_path": "Mathlib/MeasureTheory/Measure/WithDensityVectorMeasure.lean", "def_pos": [149, 9], "def_end_pos": [149, 32]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t\u271d t : SignedMeasure \u03b1\nf : \u03b1 \u2192 \u211d\nhfi : Integrable f\nht\u03bc : t \u27c2\u1d65 toENNRealVectorMeasure \u03bc\nhadd : s = t + withDensity\u1d65 \u03bc f\nf' : \u03b1 \u2192 \u211d := AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f \u03bc)\n\u22a2 f =\u1da0[ae \u03bc] rnDeriv s \u03bc", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t\u271d t : SignedMeasure \u03b1\nf : \u03b1 \u2192 \u211d\nhfi : Integrable f\nht\u03bc : t \u27c2\u1d65 toENNRealVectorMeasure \u03bc\nhadd : s = t + withDensity\u1d65 \u03bc f\nf' : \u03b1 \u2192 \u211d := AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f \u03bc)\nhadd' : s = t + withDensity\u1d65 \u03bc f'\n\u22a2 f =\u1da0[ae \u03bc] rnDeriv s \u03bc"}, {"tactic": "haveI := haveLebesgueDecomposition_mk \u03bc hfi.1.measurable_mk ht\u03bc hadd'", "annotated_tactic": ["haveI := <a>haveLebesgueDecomposition_mk</a> \u03bc hfi.1.<a>measurable_mk</a> ht\u03bc hadd'", [{"full_name": "MeasureTheory.SignedMeasure.haveLebesgueDecomposition_mk", "def_path": "Mathlib/MeasureTheory/Decomposition/Lebesgue.lean", "def_pos": [1014, 9], "def_end_pos": [1014, 37]}, {"full_name": "MeasureTheory.AEStronglyMeasurable.measurable_mk", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1210, 9], "def_end_pos": [1210, 22]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t\u271d t : SignedMeasure \u03b1\nf : \u03b1 \u2192 \u211d\nhfi : Integrable f\nht\u03bc : t \u27c2\u1d65 toENNRealVectorMeasure \u03bc\nhadd : s = t + withDensity\u1d65 \u03bc f\nf' : \u03b1 \u2192 \u211d := AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f \u03bc)\nhadd' : s = t + withDensity\u1d65 \u03bc f'\n\u22a2 f =\u1da0[ae \u03bc] rnDeriv s \u03bc", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t\u271d t : SignedMeasure \u03b1\nf : \u03b1 \u2192 \u211d\nhfi : Integrable f\nht\u03bc : t \u27c2\u1d65 toENNRealVectorMeasure \u03bc\nhadd : s = t + withDensity\u1d65 \u03bc f\nf' : \u03b1 \u2192 \u211d := AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f \u03bc)\nhadd' : s = t + withDensity\u1d65 \u03bc f'\nthis : HaveLebesgueDecomposition s \u03bc\n\u22a2 f =\u1da0[ae \u03bc] rnDeriv s \u03bc"}, {"tactic": "refine' (Integrable.ae_eq_of_withDensity\u1d65_eq (integrable_rnDeriv _ _) hfi _).symm", "annotated_tactic": ["refine' (<a>Integrable.ae_eq_of_withDensity\u1d65_eq</a> (<a>integrable_rnDeriv</a> _ _) hfi _).<a>symm</a>", [{"full_name": "MeasureTheory.Integrable.ae_eq_of_withDensity\u1d65_eq", "def_path": "Mathlib/MeasureTheory/Measure/WithDensityVectorMeasure.lean", "def_pos": [143, 9], "def_end_pos": [143, 44]}, {"full_name": "MeasureTheory.SignedMeasure.integrable_rnDeriv", "def_path": "Mathlib/MeasureTheory/Decomposition/Lebesgue.lean", "def_pos": [916, 9], "def_end_pos": [916, 27]}, {"full_name": "Filter.EventuallyEq.symm", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1498, 9], "def_end_pos": [1498, 26]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t\u271d t : SignedMeasure \u03b1\nf : \u03b1 \u2192 \u211d\nhfi : Integrable f\nht\u03bc : t \u27c2\u1d65 toENNRealVectorMeasure \u03bc\nhadd : s = t + withDensity\u1d65 \u03bc f\nf' : \u03b1 \u2192 \u211d := AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f \u03bc)\nhadd' : s = t + withDensity\u1d65 \u03bc f'\nthis : HaveLebesgueDecomposition s \u03bc\n\u22a2 f =\u1da0[ae \u03bc] rnDeriv s \u03bc", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t\u271d t : SignedMeasure \u03b1\nf : \u03b1 \u2192 \u211d\nhfi : Integrable f\nht\u03bc : t \u27c2\u1d65 toENNRealVectorMeasure \u03bc\nhadd : s = t + withDensity\u1d65 \u03bc f\nf' : \u03b1 \u2192 \u211d := AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f \u03bc)\nhadd' : s = t + withDensity\u1d65 \u03bc f'\nthis : HaveLebesgueDecomposition s \u03bc\n\u22a2 withDensity\u1d65 \u03bc (rnDeriv s \u03bc) = withDensity\u1d65 \u03bc f"}, {"tactic": "rw [\u2190 add_right_inj t, \u2190 hadd, eq_singularPart _ f ht\u03bc hadd,\n  singularPart_add_withDensity_rnDeriv_eq]", "annotated_tactic": ["rw [\u2190 <a>add_right_inj</a> t, \u2190 hadd, <a>eq_singularPart</a> _ f ht\u03bc hadd,\n    <a>singularPart_add_withDensity_rnDeriv_eq</a>]", [{"full_name": "add_right_inj", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [188, 3], "def_end_pos": [188, 14]}, {"full_name": "MeasureTheory.SignedMeasure.eq_singularPart", "def_path": "Mathlib/MeasureTheory/Decomposition/Lebesgue.lean", "def_pos": [1047, 9], "def_end_pos": [1047, 24]}, {"full_name": "MeasureTheory.SignedMeasure.singularPart_add_withDensity_rnDeriv_eq", "def_path": "Mathlib/MeasureTheory/Decomposition/Lebesgue.lean", "def_pos": [930, 9], "def_end_pos": [930, 48]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t\u271d t : SignedMeasure \u03b1\nf : \u03b1 \u2192 \u211d\nhfi : Integrable f\nht\u03bc : t \u27c2\u1d65 toENNRealVectorMeasure \u03bc\nhadd : s = t + withDensity\u1d65 \u03bc f\nf' : \u03b1 \u2192 \u211d := AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f \u03bc)\nhadd' : s = t + withDensity\u1d65 \u03bc f'\nthis : HaveLebesgueDecomposition s \u03bc\n\u22a2 withDensity\u1d65 \u03bc (rnDeriv s \u03bc) = withDensity\u1d65 \u03bc f", "state_after": "no goals"}, {"tactic": "convert hadd using 2", "annotated_tactic": ["convert hadd using 2", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t\u271d t : SignedMeasure \u03b1\nf : \u03b1 \u2192 \u211d\nhfi : Integrable f\nht\u03bc : t \u27c2\u1d65 toENNRealVectorMeasure \u03bc\nhadd : s = t + withDensity\u1d65 \u03bc f\nf' : \u03b1 \u2192 \u211d := AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f \u03bc)\n\u22a2 s = t + withDensity\u1d65 \u03bc f'", "state_after": "case h.e'_3.h.e'_6\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t\u271d t : SignedMeasure \u03b1\nf : \u03b1 \u2192 \u211d\nhfi : Integrable f\nht\u03bc : t \u27c2\u1d65 toENNRealVectorMeasure \u03bc\nhadd : s = t + withDensity\u1d65 \u03bc f\nf' : \u03b1 \u2192 \u211d := AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f \u03bc)\n\u22a2 withDensity\u1d65 \u03bc f' = withDensity\u1d65 \u03bc f"}, {"tactic": "exact WithDensity\u1d65Eq.congr_ae hfi.1.ae_eq_mk.symm", "annotated_tactic": ["exact <a>WithDensity\u1d65Eq.congr_ae</a> hfi.1.ae_eq_mk.symm", [{"full_name": "MeasureTheory.WithDensity\u1d65Eq.congr_ae", "def_path": "Mathlib/MeasureTheory/Measure/WithDensityVectorMeasure.lean", "def_pos": [149, 9], "def_end_pos": [149, 32]}]], "state_before": "case h.e'_3.h.e'_6\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t\u271d t : SignedMeasure \u03b1\nf : \u03b1 \u2192 \u211d\nhfi : Integrable f\nht\u03bc : t \u27c2\u1d65 toENNRealVectorMeasure \u03bc\nhadd : s = t + withDensity\u1d65 \u03bc f\nf' : \u03b1 \u2192 \u211d := AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f \u03bc)\n\u22a2 withDensity\u1d65 \u03bc f' = withDensity\u1d65 \u03bc f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "full_name": "MeasureTheory.set_integral_pos_iff_support_of_nonneg_ae", "start": [590, 1], "end": [594, 95], "traced_tactics": [{"tactic": "rw [integral_pos_iff_support_of_nonneg_ae hf hfi, Measure.restrict_apply\u2080]", "annotated_tactic": ["rw [<a>integral_pos_iff_support_of_nonneg_ae</a> hf hfi, <a>Measure.restrict_apply\u2080</a>]", [{"full_name": "MeasureTheory.integral_pos_iff_support_of_nonneg_ae", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1282, 9], "def_end_pos": [1282, 46]}, {"full_name": "MeasureTheory.Measure.restrict_apply\u2080", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1521, 9], "def_end_pos": [1521, 24]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\nf\u271d g : \u03b1 \u2192 E\ns t : Set \u03b1\n\u03bc \u03bd : Measure \u03b1\nl l' : Filter \u03b1\ninst\u271d : NormedSpace \u211d E\nf : \u03b1 \u2192 \u211d\nhf : 0 \u2264\u1d50[Measure.restrict \u03bc s] f\nhfi : IntegrableOn f s\n\u22a2 0 < \u222b (x : \u03b1) in s, f x \u2202\u03bc \u2194 0 < \u2191\u2191\u03bc (support f \u2229 s)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\nf\u271d g : \u03b1 \u2192 E\ns t : Set \u03b1\n\u03bc \u03bd : Measure \u03b1\nl l' : Filter \u03b1\ninst\u271d : NormedSpace \u211d E\nf : \u03b1 \u2192 \u211d\nhf : 0 \u2264\u1d50[Measure.restrict \u03bc s] f\nhfi : IntegrableOn f s\n\u22a2 NullMeasurableSet (support f)"}, {"tactic": "rw [support_eq_preimage]", "annotated_tactic": ["rw [<a>support_eq_preimage</a>]", [{"full_name": "Function.support_eq_preimage", "def_path": "Mathlib/Algebra/Support.lean", "def_pos": [47, 3], "def_end_pos": [47, 14]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\nf\u271d g : \u03b1 \u2192 E\ns t : Set \u03b1\n\u03bc \u03bd : Measure \u03b1\nl l' : Filter \u03b1\ninst\u271d : NormedSpace \u211d E\nf : \u03b1 \u2192 \u211d\nhf : 0 \u2264\u1d50[Measure.restrict \u03bc s] f\nhfi : IntegrableOn f s\n\u22a2 NullMeasurableSet (support f)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\nf\u271d g : \u03b1 \u2192 E\ns t : Set \u03b1\n\u03bc \u03bd : Measure \u03b1\nl l' : Filter \u03b1\ninst\u271d : NormedSpace \u211d E\nf : \u03b1 \u2192 \u211d\nhf : 0 \u2264\u1d50[Measure.restrict \u03bc s] f\nhfi : IntegrableOn f s\n\u22a2 NullMeasurableSet (f \u207b\u00b9' {0}\u1d9c)"}, {"tactic": "exact hfi.aestronglyMeasurable.aemeasurable.nullMeasurable (measurableSet_singleton 0).compl", "annotated_tactic": ["exact hfi.aestronglyMeasurable.aemeasurable.nullMeasurable (<a>measurableSet_singleton</a> 0).<a>compl</a>", [{"full_name": "MeasurableSingletonClass.measurableSet_singleton", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [269, 3], "def_end_pos": [269, 26]}, {"full_name": "MeasurableSet.compl", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [87, 19], "def_end_pos": [87, 38]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\nf\u271d g : \u03b1 \u2192 E\ns t : Set \u03b1\n\u03bc \u03bd : Measure \u03b1\nl l' : Filter \u03b1\ninst\u271d : NormedSpace \u211d E\nf : \u03b1 \u2192 \u211d\nhf : 0 \u2264\u1d50[Measure.restrict \u03bc s] f\nhfi : IntegrableOn f s\n\u22a2 NullMeasurableSet (f \u207b\u00b9' {0}\u1d9c)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "full_name": "String.contains_iff", "start": [744, 1], "end": [745, 27], "traced_tactics": [{"tactic": "simp [contains, any_iff]", "annotated_tactic": ["simp [<a>contains</a>, <a>any_iff</a>]", [{"full_name": "String.contains", "def_path": "lake-packages/lean4/src/lean/Init/Data/String/Basic.lean", "def_pos": [415, 5], "def_end_pos": [415, 13]}, {"full_name": "String.any_iff", "def_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "def_pos": [737, 9], "def_end_pos": [737, 16]}]], "state_before": "s : String\nc : Char\n\u22a2 contains s c = true \u2194 c \u2208 s.data", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Kernel/CondCdf.lean", "full_name": "MeasureTheory.Measure.IsFiniteMeasure.IicSnd", "start": [211, 1], "end": [213, 46], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "full_name": "MeasurableEmbedding.map_apply", "start": [4102, 8], "end": [4116, 86], "traced_tactics": [{"tactic": "refine' le_antisymm _ (le_map_apply hf.measurable.aemeasurable s)", "annotated_tactic": ["refine' <a>le_antisymm</a> _ (<a>le_map_apply</a> hf.measurable.aemeasurable s)", [{"full_name": "le_antisymm", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [188, 9], "def_end_pos": [188, 20]}, {"full_name": "MeasureTheory.Measure.le_map_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1270, 9], "def_end_pos": [1270, 21]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\nm1 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : MeasurableEmbedding f\n\u03bc : Measure \u03b1\ns : Set \u03b2\n\u22a2 \u2191\u2191(Measure.map f \u03bc) s = \u2191\u2191\u03bc (f \u207b\u00b9' s)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\nm1 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : MeasurableEmbedding f\n\u03bc : Measure \u03b1\ns : Set \u03b2\n\u22a2 \u2191\u2191(Measure.map f \u03bc) s \u2264 \u2191\u2191\u03bc (f \u207b\u00b9' s)"}, {"tactic": "set t := f '' toMeasurable \u03bc (f \u207b\u00b9' s) \u222a (range f)\u1d9c", "annotated_tactic": ["set t := f '' <a>toMeasurable</a> \u03bc (f \u207b\u00b9' s) \u222a (<a>range</a> f)\u1d9c", [{"full_name": "MeasureTheory.toMeasurable", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [626, 17], "def_end_pos": [626, 29]}, {"full_name": "Set.range", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [668, 5], "def_end_pos": [668, 10]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\nm1 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : MeasurableEmbedding f\n\u03bc : Measure \u03b1\ns : Set \u03b2\n\u22a2 \u2191\u2191(Measure.map f \u03bc) s \u2264 \u2191\u2191\u03bc (f \u207b\u00b9' s)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\nm1 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : MeasurableEmbedding f\n\u03bc : Measure \u03b1\ns : Set \u03b2\nt : Set \u03b2 := f '' toMeasurable \u03bc (f \u207b\u00b9' s) \u222a (range f)\u1d9c\n\u22a2 \u2191\u2191(Measure.map f \u03bc) s \u2264 \u2191\u2191\u03bc (f \u207b\u00b9' s)"}, {"tactic": "have htm : MeasurableSet t :=\n  (hf.measurableSet_image.2 <| measurableSet_toMeasurable _ _).union\n    hf.measurableSet_range.compl", "annotated_tactic": ["have htm : <a>MeasurableSet</a> t :=\n    (hf.measurableSet_image.2 <| <a>measurableSet_toMeasurable</a> _ _).<a>union</a>\n      hf.measurableSet_range.compl", [{"full_name": "MeasurableSet", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [64, 5], "def_end_pos": [64, 18]}, {"full_name": "MeasureTheory.measurableSet_toMeasurable", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [645, 9], "def_end_pos": [645, 35]}, {"full_name": "MeasurableSet.union", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [191, 19], "def_end_pos": [191, 38]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\nm1 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : MeasurableEmbedding f\n\u03bc : Measure \u03b1\ns : Set \u03b2\nt : Set \u03b2 := f '' toMeasurable \u03bc (f \u207b\u00b9' s) \u222a (range f)\u1d9c\n\u22a2 \u2191\u2191(Measure.map f \u03bc) s \u2264 \u2191\u2191\u03bc (f \u207b\u00b9' s)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\nm1 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : MeasurableEmbedding f\n\u03bc : Measure \u03b1\ns : Set \u03b2\nt : Set \u03b2 := f '' toMeasurable \u03bc (f \u207b\u00b9' s) \u222a (range f)\u1d9c\nhtm : MeasurableSet t\n\u22a2 \u2191\u2191(Measure.map f \u03bc) s \u2264 \u2191\u2191\u03bc (f \u207b\u00b9' s)"}, {"tactic": "have hst : s \u2286 t := by\n  rw [subset_union_compl_iff_inter_subset, \u2190 image_preimage_eq_inter_range]\n  exact image_subset _ (subset_toMeasurable _ _)", "annotated_tactic": ["have hst : s \u2286 t := by\n    rw [<a>subset_union_compl_iff_inter_subset</a>, \u2190 <a>image_preimage_eq_inter_range</a>]\n    exact <a>image_subset</a> _ (<a>subset_toMeasurable</a> _ _)", [{"full_name": "Set.subset_union_compl_iff_inter_subset", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1790, 9], "def_end_pos": [1790, 44]}, {"full_name": "Set.image_preimage_eq_inter_range", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [796, 9], "def_end_pos": [796, 38]}, {"full_name": "Set.image_subset", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [321, 9], "def_end_pos": [321, 21]}, {"full_name": "MeasureTheory.subset_toMeasurable", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [633, 9], "def_end_pos": [633, 28]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\nm1 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : MeasurableEmbedding f\n\u03bc : Measure \u03b1\ns : Set \u03b2\nt : Set \u03b2 := f '' toMeasurable \u03bc (f \u207b\u00b9' s) \u222a (range f)\u1d9c\nhtm : MeasurableSet t\n\u22a2 \u2191\u2191(Measure.map f \u03bc) s \u2264 \u2191\u2191\u03bc (f \u207b\u00b9' s)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\nm1 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : MeasurableEmbedding f\n\u03bc : Measure \u03b1\ns : Set \u03b2\nt : Set \u03b2 := f '' toMeasurable \u03bc (f \u207b\u00b9' s) \u222a (range f)\u1d9c\nhtm : MeasurableSet t\nhst : s \u2286 t\n\u22a2 \u2191\u2191(Measure.map f \u03bc) s \u2264 \u2191\u2191\u03bc (f \u207b\u00b9' s)"}, {"tactic": "have hft : f \u207b\u00b9' t = toMeasurable \u03bc (f \u207b\u00b9' s) := by\n  rw [preimage_union, preimage_compl, preimage_range, compl_univ, union_empty,\n    hf.injective.preimage_image]", "annotated_tactic": ["have hft : f \u207b\u00b9' t = <a>toMeasurable</a> \u03bc (f \u207b\u00b9' s) := by\n    rw [<a>preimage_union</a>, <a>preimage_compl</a>, <a>preimage_range</a>, <a>compl_univ</a>, <a>union_empty</a>,\n      hf.injective.preimage_image]", [{"full_name": "MeasureTheory.toMeasurable", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [626, 17], "def_end_pos": [626, 29]}, {"full_name": "Set.preimage_union", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [92, 9], "def_end_pos": [92, 23]}, {"full_name": "Set.preimage_compl", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [97, 9], "def_end_pos": [97, 23]}, {"full_name": "Set.preimage_range", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [1116, 9], "def_end_pos": [1116, 23]}, {"full_name": "Set.compl_univ", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1691, 9], "def_end_pos": [1691, 19]}, {"full_name": "Set.union_empty", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [777, 9], "def_end_pos": [777, 20]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\nm1 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : MeasurableEmbedding f\n\u03bc : Measure \u03b1\ns : Set \u03b2\nt : Set \u03b2 := f '' toMeasurable \u03bc (f \u207b\u00b9' s) \u222a (range f)\u1d9c\nhtm : MeasurableSet t\nhst : s \u2286 t\n\u22a2 \u2191\u2191(Measure.map f \u03bc) s \u2264 \u2191\u2191\u03bc (f \u207b\u00b9' s)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\nm1 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : MeasurableEmbedding f\n\u03bc : Measure \u03b1\ns : Set \u03b2\nt : Set \u03b2 := f '' toMeasurable \u03bc (f \u207b\u00b9' s) \u222a (range f)\u1d9c\nhtm : MeasurableSet t\nhst : s \u2286 t\nhft : f \u207b\u00b9' t = toMeasurable \u03bc (f \u207b\u00b9' s)\n\u22a2 \u2191\u2191(Measure.map f \u03bc) s \u2264 \u2191\u2191\u03bc (f \u207b\u00b9' s)"}, {"tactic": "calc\n  \u03bc.map f s \u2264 \u03bc.map f t := measure_mono hst\n  _ = \u03bc (f \u207b\u00b9' s) := by rw [map_apply hf.measurable htm, hft, measure_toMeasurable]", "annotated_tactic": ["calc\n    \u03bc.map f s \u2264 \u03bc.map f t := <a>measure_mono</a> hst\n    _ = \u03bc (f \u207b\u00b9' s) := by rw [<a>map_apply</a> hf.measurable htm, hft, <a>measure_toMeasurable</a>]", [{"full_name": "MeasureTheory.measure_mono", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [193, 9], "def_end_pos": [193, 21]}, {"full_name": "MeasureTheory.Measure.map_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1236, 9], "def_end_pos": [1236, 18]}, {"full_name": "MeasureTheory.measure_toMeasurable", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [653, 9], "def_end_pos": [653, 29]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\nm1 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : MeasurableEmbedding f\n\u03bc : Measure \u03b1\ns : Set \u03b2\nt : Set \u03b2 := f '' toMeasurable \u03bc (f \u207b\u00b9' s) \u222a (range f)\u1d9c\nhtm : MeasurableSet t\nhst : s \u2286 t\nhft : f \u207b\u00b9' t = toMeasurable \u03bc (f \u207b\u00b9' s)\n\u22a2 \u2191\u2191(Measure.map f \u03bc) s \u2264 \u2191\u2191\u03bc (f \u207b\u00b9' s)", "state_after": "no goals"}, {"tactic": "rw [subset_union_compl_iff_inter_subset, \u2190 image_preimage_eq_inter_range]", "annotated_tactic": ["rw [<a>subset_union_compl_iff_inter_subset</a>, \u2190 <a>image_preimage_eq_inter_range</a>]", [{"full_name": "Set.subset_union_compl_iff_inter_subset", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1790, 9], "def_end_pos": [1790, 44]}, {"full_name": "Set.image_preimage_eq_inter_range", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [796, 9], "def_end_pos": [796, 38]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\nm1 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : MeasurableEmbedding f\n\u03bc : Measure \u03b1\ns : Set \u03b2\nt : Set \u03b2 := f '' toMeasurable \u03bc (f \u207b\u00b9' s) \u222a (range f)\u1d9c\nhtm : MeasurableSet t\n\u22a2 s \u2286 t", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\nm1 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : MeasurableEmbedding f\n\u03bc : Measure \u03b1\ns : Set \u03b2\nt : Set \u03b2 := f '' toMeasurable \u03bc (f \u207b\u00b9' s) \u222a (range f)\u1d9c\nhtm : MeasurableSet t\n\u22a2 f '' (f \u207b\u00b9' s) \u2286 f '' toMeasurable \u03bc (f \u207b\u00b9' s)"}, {"tactic": "exact image_subset _ (subset_toMeasurable _ _)", "annotated_tactic": ["exact <a>image_subset</a> _ (<a>subset_toMeasurable</a> _ _)", [{"full_name": "Set.image_subset", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [321, 9], "def_end_pos": [321, 21]}, {"full_name": "MeasureTheory.subset_toMeasurable", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [633, 9], "def_end_pos": [633, 28]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\nm1 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : MeasurableEmbedding f\n\u03bc : Measure \u03b1\ns : Set \u03b2\nt : Set \u03b2 := f '' toMeasurable \u03bc (f \u207b\u00b9' s) \u222a (range f)\u1d9c\nhtm : MeasurableSet t\n\u22a2 f '' (f \u207b\u00b9' s) \u2286 f '' toMeasurable \u03bc (f \u207b\u00b9' s)", "state_after": "no goals"}, {"tactic": "rw [preimage_union, preimage_compl, preimage_range, compl_univ, union_empty,\n  hf.injective.preimage_image]", "annotated_tactic": ["rw [<a>preimage_union</a>, <a>preimage_compl</a>, <a>preimage_range</a>, <a>compl_univ</a>, <a>union_empty</a>,\n      hf.injective.preimage_image]", [{"full_name": "Set.preimage_union", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [92, 9], "def_end_pos": [92, 23]}, {"full_name": "Set.preimage_compl", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [97, 9], "def_end_pos": [97, 23]}, {"full_name": "Set.preimage_range", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [1116, 9], "def_end_pos": [1116, 23]}, {"full_name": "Set.compl_univ", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1691, 9], "def_end_pos": [1691, 19]}, {"full_name": "Set.union_empty", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [777, 9], "def_end_pos": [777, 20]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\nm1 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : MeasurableEmbedding f\n\u03bc : Measure \u03b1\ns : Set \u03b2\nt : Set \u03b2 := f '' toMeasurable \u03bc (f \u207b\u00b9' s) \u222a (range f)\u1d9c\nhtm : MeasurableSet t\nhst : s \u2286 t\n\u22a2 f \u207b\u00b9' t = toMeasurable \u03bc (f \u207b\u00b9' s)", "state_after": "no goals"}, {"tactic": "rw [map_apply hf.measurable htm, hft, measure_toMeasurable]", "annotated_tactic": ["rw [<a>map_apply</a> hf.measurable htm, hft, <a>measure_toMeasurable</a>]", [{"full_name": "MeasureTheory.Measure.map_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1236, 9], "def_end_pos": [1236, 18]}, {"full_name": "MeasureTheory.measure_toMeasurable", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [653, 9], "def_end_pos": [653, 29]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\nm1 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : MeasurableEmbedding f\n\u03bc : Measure \u03b1\ns : Set \u03b2\nt : Set \u03b2 := f '' toMeasurable \u03bc (f \u207b\u00b9' s) \u222a (range f)\u1d9c\nhtm : MeasurableSet t\nhst : s \u2286 t\nhft : f \u207b\u00b9' t = toMeasurable \u03bc (f \u207b\u00b9' s)\n\u22a2 \u2191\u2191(Measure.map f \u03bc) t = \u2191\u2191\u03bc (f \u207b\u00b9' s)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/PiInduction.lean", "full_name": "Finset.induction_on_pi_min", "start": [108, 1], "end": [114, 63], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "full_name": "MeasureTheory.OuterMeasure.comap_iInf", "start": [1220, 1], "end": [1227, 66], "traced_tactics": [{"tactic": "refine' ext_nonempty fun s hs => _", "annotated_tactic": ["refine' <a>ext_nonempty</a> fun s hs => _", [{"full_name": "MeasureTheory.OuterMeasure.ext_nonempty", "def_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "def_pos": [244, 9], "def_end_pos": [244, 21]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Sort u_2\n\u03b2 : Type u_3\nf : \u03b1 \u2192 \u03b2\nm : \u03b9 \u2192 OuterMeasure \u03b2\n\u22a2 \u2191(comap f) (\u2a05 i, m i) = \u2a05 i, \u2191(comap f) (m i)", "state_after": "\u03b1 : Type u_1\n\u03b9 : Sort u_2\n\u03b2 : Type u_3\nf : \u03b1 \u2192 \u03b2\nm : \u03b9 \u2192 OuterMeasure \u03b2\ns : Set \u03b1\nhs : Set.Nonempty s\n\u22a2 \u2191(\u2191(comap f) (\u2a05 i, m i)) s = \u2191(\u2a05 i, \u2191(comap f) (m i)) s"}, {"tactic": "refine' ((comap_mono f).map_iInf_le s).antisymm _", "annotated_tactic": ["refine' ((<a>comap_mono</a> f).<a>map_iInf_le</a> s).<a>antisymm</a> _", [{"full_name": "MeasureTheory.OuterMeasure.comap_mono", "def_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "def_pos": [546, 9], "def_end_pos": [546, 19]}, {"full_name": "Monotone.map_iInf_le", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [1070, 9], "def_end_pos": [1070, 29]}, {"full_name": "LE.le.antisymm", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [128, 7], "def_end_pos": [128, 21]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Sort u_2\n\u03b2 : Type u_3\nf : \u03b1 \u2192 \u03b2\nm : \u03b9 \u2192 OuterMeasure \u03b2\ns : Set \u03b1\nhs : Set.Nonempty s\n\u22a2 \u2191(\u2191(comap f) (\u2a05 i, m i)) s = \u2191(\u2a05 i, \u2191(comap f) (m i)) s", "state_after": "\u03b1 : Type u_1\n\u03b9 : Sort u_2\n\u03b2 : Type u_3\nf : \u03b1 \u2192 \u03b2\nm : \u03b9 \u2192 OuterMeasure \u03b2\ns : Set \u03b1\nhs : Set.Nonempty s\n\u22a2 \u2191(\u2a05 i, \u2191(comap f) (m i)) s \u2264 \u2191(\u2191(comap f) (\u2a05 i, m i)) s"}, {"tactic": "simp only [comap_apply, iInf_apply' _ hs, iInf_apply' _ (hs.image _), le_iInf_iff,\n  Set.image_subset_iff, preimage_iUnion]", "annotated_tactic": ["simp only [<a>comap_apply</a>, <a>iInf_apply'</a> _ hs, <a>iInf_apply'</a> _ (hs.image _), <a>le_iInf_iff</a>,\n    <a>Set.image_subset_iff</a>, <a>preimage_iUnion</a>]", [{"full_name": "MeasureTheory.OuterMeasure.comap_apply", "def_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "def_pos": [541, 9], "def_end_pos": [541, 20]}, {"full_name": "MeasureTheory.OuterMeasure.iInf_apply'", "def_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "def_pos": [1192, 9], "def_end_pos": [1192, 20]}, {"full_name": "MeasureTheory.OuterMeasure.iInf_apply'", "def_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "def_pos": [1192, 9], "def_end_pos": [1192, 20]}, {"full_name": "le_iInf_iff", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [969, 9], "def_end_pos": [969, 20]}, {"full_name": "Set.image_subset_iff", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [497, 9], "def_end_pos": [497, 25]}, {"full_name": "Set.preimage_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [1854, 9], "def_end_pos": [1854, 24]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Sort u_2\n\u03b2 : Type u_3\nf : \u03b1 \u2192 \u03b2\nm : \u03b9 \u2192 OuterMeasure \u03b2\ns : Set \u03b1\nhs : Set.Nonempty s\n\u22a2 \u2191(\u2a05 i, \u2191(comap f) (m i)) s \u2264 \u2191(\u2191(comap f) (\u2a05 i, m i)) s", "state_after": "\u03b1 : Type u_1\n\u03b9 : Sort u_2\n\u03b2 : Type u_3\nf : \u03b1 \u2192 \u03b2\nm : \u03b9 \u2192 OuterMeasure \u03b2\ns : Set \u03b1\nhs : Set.Nonempty s\n\u22a2 \u2200 (i : \u2115 \u2192 Set \u03b2),\n    s \u2286 \u22c3 i_1, f \u207b\u00b9' i i_1 \u2192\n      \u2a05 t, \u2a05 (_ : s \u2286 iUnion t), \u2211' (n : \u2115), \u2a05 i, \u2191(m i) (f '' t n) \u2264 \u2211' (n : \u2115), 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Fintype.card \u03b2", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Group/FundamentalDomain.lean", "full_name": "MeasureTheory.IsFundamentalDomain.aEStronglyMeasurable_on_iff", "start": [361, 11], "end": [383, 69], "traced_tactics": [{"tactic": "simp only [\u2190 ht.restrict_restrict,\n  ht.sum_restrict_of_ac restrict_le_self.absolutelyContinuous]", "annotated_tactic": ["simp only [\u2190 ht.restrict_restrict,\n        ht.sum_restrict_of_ac restrict_le_self.absolutelyContinuous]", []], "state_before": "G : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2\u271d : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : Group H\ninst\u271d\u00b9\u2070 : MulAction G \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MulAction H \u03b2\u271d\ninst\u271d\u2077 : MeasurableSpace \u03b2\u271d\ninst\u271d\u2076 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : MeasurableSMul G \u03b1\ninst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b2 : Countable G\n\u03bd : Measure \u03b1\n\u03b2 : Type u_6\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : PseudoMetrizableSpace \u03b2\nhs : IsFundamentalDomain G s\nht : IsFundamentalDomain G t\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 (g : G) (x : \u03b1), f (g \u2022 x) = f x\n\u22a2 AEStronglyMeasurable f (Measure.restrict \u03bc s) \u2194 AEStronglyMeasurable f (sum fun g => Measure.restrict \u03bc (g \u2022 t \u2229 s))", "state_after": "no goals"}, {"tactic": "simp only [smul_set_inter, inter_comm, smul_inv_smul, aestronglyMeasurable_sum_measure_iff]", "annotated_tactic": ["simp only [<a>smul_set_inter</a>, <a>inter_comm</a>, <a>smul_inv_smul</a>, <a>aestronglyMeasurable_sum_measure_iff</a>]", [{"full_name": "Set.smul_set_inter", "def_path": 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G\ninst\u271d\u2074 : MeasurableSMul G \u03b1\ninst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b2 : Countable G\n\u03bd : Measure \u03b1\n\u03b2 : Type u_6\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : PseudoMetrizableSpace \u03b2\nhs : IsFundamentalDomain G s\nht : IsFundamentalDomain G t\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 (g : G) (x : \u03b1), f (g \u2022 x) = f x\n\u22a2 AEStronglyMeasurable f (sum fun g => Measure.restrict \u03bc (g \u2022 t \u2229 s)) \u2194\n    \u2200 (g : G), AEStronglyMeasurable f (Measure.restrict \u03bc (g \u2022 (g\u207b\u00b9 \u2022 s \u2229 t)))", "state_after": "no goals"}, {"tactic": "simp only [inv_inv]", "annotated_tactic": ["simp only [<a>inv_inv</a>]", [{"full_name": "inv_inv", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [800, 9], "def_end_pos": [800, 16]}]], "state_before": "G : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2\u271d : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : Group H\ninst\u271d\u00b9\u2070 : MulAction G \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MulAction H \u03b2\u271d\ninst\u271d\u2077 : MeasurableSpace \u03b2\u271d\ninst\u271d\u2076 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : MeasurableSMul G \u03b1\ninst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b2 : Countable G\n\u03bd : Measure \u03b1\n\u03b2 : Type u_6\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : PseudoMetrizableSpace \u03b2\nhs : IsFundamentalDomain G s\nht : IsFundamentalDomain G t\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 (g : G) (x : \u03b1), f (g \u2022 x) = f x\n\u22a2 (\u2200 (g : G), AEStronglyMeasurable f (Measure.restrict \u03bc (g\u207b\u00b9 \u2022 (g\u207b\u00b9\u207b\u00b9 \u2022 s \u2229 t)))) \u2194\n    \u2200 (g : G), AEStronglyMeasurable f (Measure.restrict \u03bc (g\u207b\u00b9 \u2022 (g \u2022 s \u2229 t)))", "state_after": "no goals"}, {"tactic": "refine' forall_congr' fun g => _", "annotated_tactic": ["refine' <a>forall_congr'</a> fun g => _", [{"full_name": "forall_congr'", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [376, 9], "def_end_pos": [376, 22]}]], "state_before": "G : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2\u271d : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : Group H\ninst\u271d\u00b9\u2070 : MulAction G \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MulAction H \u03b2\u271d\ninst\u271d\u2077 : MeasurableSpace \u03b2\u271d\ninst\u271d\u2076 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : MeasurableSMul G \u03b1\ninst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b2 : Countable G\n\u03bd : Measure \u03b1\n\u03b2 : Type u_6\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : PseudoMetrizableSpace \u03b2\nhs : IsFundamentalDomain G s\nht : IsFundamentalDomain G t\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 (g : G) (x : \u03b1), f (g \u2022 x) = f x\n\u22a2 (\u2200 (g : G), AEStronglyMeasurable f (Measure.restrict \u03bc (g\u207b\u00b9 \u2022 (g \u2022 s \u2229 t)))) \u2194\n    \u2200 (g : G), AEStronglyMeasurable f (Measure.restrict \u03bc (g \u2022 s \u2229 t))", "state_after": "G : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2\u271d : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : Group H\ninst\u271d\u00b9\u2070 : MulAction G \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MulAction H \u03b2\u271d\ninst\u271d\u2077 : MeasurableSpace \u03b2\u271d\ninst\u271d\u2076 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : MeasurableSMul G \u03b1\ninst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b2 : Countable G\n\u03bd : Measure \u03b1\n\u03b2 : Type u_6\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : PseudoMetrizableSpace \u03b2\nhs : IsFundamentalDomain G s\nht : IsFundamentalDomain G t\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 (g : G) (x : \u03b1), f (g \u2022 x) = f x\ng : G\n\u22a2 AEStronglyMeasurable f (Measure.restrict \u03bc (g\u207b\u00b9 \u2022 (g \u2022 s \u2229 t))) \u2194\n    AEStronglyMeasurable f (Measure.restrict \u03bc (g \u2022 s \u2229 t))"}, {"tactic": "rw [\u2190 image_smul, \u2190 ((measurePreserving_smul g\u207b\u00b9 \u03bc).restrict_image_emb he\n  _).aestronglyMeasurable_comp_iff he]", "annotated_tactic": ["rw [\u2190 <a>image_smul</a>, \u2190 ((<a>measurePreserving_smul</a> g\u207b\u00b9 \u03bc).<a>restrict_image_emb</a> he\n        _).<a>aestronglyMeasurable_comp_iff</a> he]", [{"full_name": "Set.image_smul", "def_path": "Mathlib/Data/Set/Pointwise/SMul.lean", "def_pos": [310, 9], "def_end_pos": [310, 19]}, {"full_name": 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G\ninst\u271d\u2074 : MeasurableSMul G \u03b1\ninst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b2 : Countable G\n\u03bd : Measure \u03b1\n\u03b2 : Type u_6\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : PseudoMetrizableSpace \u03b2\nhs : IsFundamentalDomain G s\nht : IsFundamentalDomain G t\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 (g : G) (x : \u03b1), f (g \u2022 x) = f x\ng : G\nhe : MeasurableEmbedding ((fun x x_1 => x \u2022 x_1) g\u207b\u00b9)\n\u22a2 AEStronglyMeasurable f (Measure.restrict \u03bc (g\u207b\u00b9 \u2022 (g \u2022 s \u2229 t))) \u2194\n    AEStronglyMeasurable f (Measure.restrict \u03bc (g \u2022 s \u2229 t))", "state_after": "G : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2\u271d : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : Group H\ninst\u271d\u00b9\u2070 : MulAction G \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MulAction H \u03b2\u271d\ninst\u271d\u2077 : MeasurableSpace \u03b2\u271d\ninst\u271d\u2076 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : MeasurableSMul G \u03b1\ninst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b2 : Countable G\n\u03bd : Measure \u03b1\n\u03b2 : Type u_6\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : PseudoMetrizableSpace \u03b2\nhs : IsFundamentalDomain G s\nht : IsFundamentalDomain G t\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 (g : G) (x : \u03b1), f (g \u2022 x) = f x\ng : G\nhe : MeasurableEmbedding ((fun x x_1 => x \u2022 x_1) g\u207b\u00b9)\n\u22a2 AEStronglyMeasurable (f \u2218 fun x => g\u207b\u00b9 \u2022 x) (Measure.restrict \u03bc (g \u2022 s \u2229 t)) \u2194\n    AEStronglyMeasurable f (Measure.restrict \u03bc (g \u2022 s \u2229 t))"}, {"tactic": "simp only [\u2190 aestronglyMeasurable_sum_measure_iff, \u2190 hs.restrict_restrict,\n  hs.sum_restrict_of_ac restrict_le_self.absolutelyContinuous]", "annotated_tactic": ["simp only [\u2190 <a>aestronglyMeasurable_sum_measure_iff</a>, \u2190 hs.restrict_restrict,\n        hs.sum_restrict_of_ac restrict_le_self.absolutelyContinuous]", [{"full_name": "aestronglyMeasurable_sum_measure_iff", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1717, 9], "def_end_pos": [1717, 52]}]], "state_before": "G : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2\u271d : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : Group H\ninst\u271d\u00b9\u2070 : MulAction G \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MulAction H \u03b2\u271d\ninst\u271d\u2077 : MeasurableSpace \u03b2\u271d\ninst\u271d\u2076 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : MeasurableSMul G \u03b1\ninst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b2 : Countable G\n\u03bd : Measure \u03b1\n\u03b2 : Type u_6\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : PseudoMetrizableSpace \u03b2\nhs : IsFundamentalDomain G s\nht : IsFundamentalDomain G t\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 (g : G) (x : \u03b1), f (g \u2022 x) = f x\n\u22a2 (\u2200 (g : G), AEStronglyMeasurable f (Measure.restrict \u03bc (g \u2022 s \u2229 t))) \u2194 AEStronglyMeasurable f (Measure.restrict \u03bc t)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Lattice.lean", "full_name": "Finset.inf_attach", "start": [416, 1], "end": [417, 28], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean", "full_name": "MeasureTheory.Measure.addHaar_ball", "start": [459, 1], "end": [461, 42], "traced_tactics": [{"tactic": "rw [\u2190 addHaar_ball_mul \u03bc x hr, mul_one]", "annotated_tactic": ["rw [\u2190 <a>addHaar_ball_mul</a> \u03bc x hr, <a>mul_one</a>]", [{"full_name": "MeasureTheory.Measure.addHaar_ball_mul", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean", "def_pos": [451, 9], "def_end_pos": [451, 25]}, {"full_name": "mul_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [470, 9], "def_end_pos": [470, 16]}]], "state_before": "E : Type u_1\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\ninst\u271d\u2077 : MeasurableSpace E\ninst\u271d\u2076 : BorelSpace E\ninst\u271d\u2075 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u2074 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\ns : Set E\ninst\u271d : Nontrivial E\nx : E\nr : \u211d\nhr : 0 \u2264 r\n\u22a2 \u2191\u2191\u03bc (ball x r) = ENNReal.ofReal (r ^ finrank \u211d E) * \u2191\u2191\u03bc (ball 0 1)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Pairwise/Basic.lean", "full_name": "Set.PairwiseDisjoint.insert_of_not_mem", "start": [284, 1], "end": [286, 52], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/IdentDistrib.lean", "full_name": "ProbabilityTheory.IdentDistrib.snorm_eq", "start": [209, 1], "end": [221, 14], "traced_tactics": [{"tactic": "by_cases h0 : p = 0", "annotated_tactic": ["by_cases h0 : p = 0", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2075 : MeasurableSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b3\ninst\u271d\u00b2 : MeasurableSpace \u03b4\n\u03bc : Measure \u03b1\n\u03bd : Measure \u03b2\nf : \u03b1 \u2192 \u03b3\ng : \u03b2 \u2192 \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup \u03b3\ninst\u271d : OpensMeasurableSpace \u03b3\nh : IdentDistrib f g\np : \u211d\u22650\u221e\n\u22a2 snorm f p \u03bc = snorm g p \u03bd", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2075 : MeasurableSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b3\ninst\u271d\u00b2 : MeasurableSpace \u03b4\n\u03bc : Measure \u03b1\n\u03bd : Measure \u03b2\nf : \u03b1 \u2192 \u03b3\ng : \u03b2 \u2192 \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup \u03b3\ninst\u271d : OpensMeasurableSpace \u03b3\nh : IdentDistrib f g\np : \u211d\u22650\u221e\nh0 : p = 0\n\u22a2 snorm f p \u03bc = snorm g p \u03bd\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2075 : MeasurableSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b3\ninst\u271d\u00b2 : MeasurableSpace \u03b4\n\u03bc : Measure \u03b1\n\u03bd : Measure \u03b2\nf : \u03b1 \u2192 \u03b3\ng : \u03b2 \u2192 \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup \u03b3\ninst\u271d : OpensMeasurableSpace \u03b3\nh : IdentDistrib f g\np : \u211d\u22650\u221e\nh0 : \u00acp = 0\n\u22a2 snorm f p \u03bc = snorm g p \u03bd"}, {"tactic": "by_cases h_top : p = \u221e", "annotated_tactic": ["by_cases h_top : p = \u221e", []], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2075 : MeasurableSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b3\ninst\u271d\u00b2 : MeasurableSpace \u03b4\n\u03bc : Measure \u03b1\n\u03bd : Measure \u03b2\nf : \u03b1 \u2192 \u03b3\ng : \u03b2 \u2192 \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup \u03b3\ninst\u271d : OpensMeasurableSpace \u03b3\nh : IdentDistrib f g\np : \u211d\u22650\u221e\nh0 : \u00acp = 0\n\u22a2 snorm f p \u03bc = snorm g p \u03bd", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2075 : MeasurableSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b3\ninst\u271d\u00b2 : MeasurableSpace \u03b4\n\u03bc : Measure \u03b1\n\u03bd : Measure \u03b2\nf : \u03b1 \u2192 \u03b3\ng : \u03b2 \u2192 \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup \u03b3\ninst\u271d : OpensMeasurableSpace \u03b3\nh : IdentDistrib f g\np : \u211d\u22650\u221e\nh0 : \u00acp = 0\nh_top : p = \u22a4\n\u22a2 snorm f p \u03bc = snorm g p \u03bd\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2075 : MeasurableSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b3\ninst\u271d\u00b2 : MeasurableSpace \u03b4\n\u03bc : Measure \u03b1\n\u03bd : Measure \u03b2\nf : \u03b1 \u2192 \u03b3\ng : \u03b2 \u2192 \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup \u03b3\ninst\u271d : OpensMeasurableSpace \u03b3\nh : IdentDistrib f g\np : \u211d\u22650\u221e\nh0 : \u00acp = 0\nh_top : \u00acp = \u22a4\n\u22a2 snorm f p \u03bc = snorm g p \u03bd"}, {"tactic": "simp only [snorm_eq_snorm' h0 h_top, snorm', one_div]", "annotated_tactic": ["simp only [<a>snorm_eq_snorm'</a> h0 h_top, <a>snorm'</a>, <a>one_div</a>]", [{"full_name": "MeasureTheory.snorm_eq_snorm'", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [88, 9], "def_end_pos": [88, 24]}, {"full_name": "MeasureTheory.snorm'", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [73, 5], "def_end_pos": [73, 11]}, {"full_name": "one_div", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [318, 9], "def_end_pos": [318, 16]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2075 : MeasurableSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b3\ninst\u271d\u00b2 : MeasurableSpace \u03b4\n\u03bc : Measure \u03b1\n\u03bd : Measure \u03b2\nf : \u03b1 \u2192 \u03b3\ng : \u03b2 \u2192 \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup \u03b3\ninst\u271d : OpensMeasurableSpace \u03b3\nh : IdentDistrib f g\np : \u211d\u22650\u221e\nh0 : \u00acp = 0\nh_top : \u00acp = \u22a4\n\u22a2 snorm f p \u03bc = snorm g p \u03bd", "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2075 : MeasurableSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b3\ninst\u271d\u00b2 : MeasurableSpace \u03b4\n\u03bc : Measure \u03b1\n\u03bd : Measure \u03b2\nf : \u03b1 \u2192 \u03b3\ng : \u03b2 \u2192 \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup \u03b3\ninst\u271d : OpensMeasurableSpace \u03b3\nh : IdentDistrib f g\np : \u211d\u22650\u221e\nh0 : \u00acp = 0\nh_top : \u00acp = \u22a4\n\u22a2 (\u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) ^ (ENNReal.toReal p)\u207b\u00b9 =\n    (\u222b\u207b (a : \u03b2), \u2191\u2016g a\u2016\u208a ^ ENNReal.toReal p \u2202\u03bd) ^ (ENNReal.toReal p)\u207b\u00b9"}, {"tactic": "congr 1", "annotated_tactic": ["congr 1", []], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2075 : MeasurableSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b3\ninst\u271d\u00b2 : MeasurableSpace \u03b4\n\u03bc : Measure \u03b1\n\u03bd : Measure \u03b2\nf : \u03b1 \u2192 \u03b3\ng : \u03b2 \u2192 \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup \u03b3\ninst\u271d : OpensMeasurableSpace \u03b3\nh : IdentDistrib f g\np : \u211d\u22650\u221e\nh0 : \u00acp = 0\nh_top : \u00acp = \u22a4\n\u22a2 (\u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) ^ (ENNReal.toReal p)\u207b\u00b9 =\n    (\u222b\u207b (a : \u03b2), \u2191\u2016g a\u2016\u208a ^ ENNReal.toReal p \u2202\u03bd) ^ (ENNReal.toReal p)\u207b\u00b9", "state_after": "case neg.e_a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2075 : MeasurableSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b3\ninst\u271d\u00b2 : MeasurableSpace \u03b4\n\u03bc : Measure \u03b1\n\u03bd : Measure \u03b2\nf : \u03b1 \u2192 \u03b3\ng : \u03b2 \u2192 \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup \u03b3\ninst\u271d : OpensMeasurableSpace \u03b3\nh : IdentDistrib f g\np : \u211d\u22650\u221e\nh0 : \u00acp = 0\nh_top : \u00acp = \u22a4\n\u22a2 \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc = \u222b\u207b (a : \u03b2), \u2191\u2016g a\u2016\u208a ^ ENNReal.toReal p \u2202\u03bd"}, {"tactic": "apply lintegral_eq", "annotated_tactic": ["apply <a>lintegral_eq</a>", [{"full_name": "ProbabilityTheory.IdentDistrib.lintegral_eq", "def_path": "Mathlib/Probability/IdentDistrib.lean", "def_pos": [183, 9], "def_end_pos": [183, 21]}]], "state_before": "case neg.e_a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2075 : MeasurableSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b3\ninst\u271d\u00b2 : MeasurableSpace \u03b4\n\u03bc : Measure \u03b1\n\u03bd : Measure \u03b2\nf : \u03b1 \u2192 \u03b3\ng : \u03b2 \u2192 \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup \u03b3\ninst\u271d : OpensMeasurableSpace \u03b3\nh : IdentDistrib f g\np : \u211d\u22650\u221e\nh0 : \u00acp = 0\nh_top : \u00acp = \u22a4\n\u22a2 \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc = \u222b\u207b (a : \u03b2), \u2191\u2016g a\u2016\u208a ^ ENNReal.toReal p \u2202\u03bd", "state_after": "case neg.e_a.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2075 : MeasurableSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b3\ninst\u271d\u00b2 : MeasurableSpace \u03b4\n\u03bc : Measure \u03b1\n\u03bd : Measure \u03b2\nf : \u03b1 \u2192 \u03b3\ng : \u03b2 \u2192 \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup \u03b3\ninst\u271d : OpensMeasurableSpace \u03b3\nh : IdentDistrib f g\np : \u211d\u22650\u221e\nh0 : \u00acp = 0\nh_top : \u00acp = \u22a4\n\u22a2 IdentDistrib (fun x => \u2191\u2016f x\u2016\u208a ^ ENNReal.toReal p) fun x => \u2191\u2016g x\u2016\u208a ^ ENNReal.toReal p"}, {"tactic": "exact h.comp (Measurable.pow_const (measurable_coe_nnreal_ennreal.comp measurable_nnnorm)\n  p.toReal)", "annotated_tactic": ["exact h.comp (<a>Measurable.pow_const</a> (measurable_coe_nnreal_ennreal.comp <a>measurable_nnnorm</a>)\n    p.toReal)", [{"full_name": "Measurable.pow_const", "def_path": "Mathlib/MeasureTheory/Group/Arithmetic.lean", "def_pos": [228, 9], "def_end_pos": [228, 29]}, {"full_name": "measurable_nnnorm", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [2256, 9], "def_end_pos": [2256, 26]}]], "state_before": "case neg.e_a.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2075 : MeasurableSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b3\ninst\u271d\u00b2 : MeasurableSpace \u03b4\n\u03bc : Measure \u03b1\n\u03bd : Measure \u03b2\nf : \u03b1 \u2192 \u03b3\ng : \u03b2 \u2192 \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup \u03b3\ninst\u271d : OpensMeasurableSpace \u03b3\nh : IdentDistrib f g\np : \u211d\u22650\u221e\nh0 : \u00acp = 0\nh_top : \u00acp = \u22a4\n\u22a2 IdentDistrib (fun x => \u2191\u2016f x\u2016\u208a ^ ENNReal.toReal p) fun x => \u2191\u2016g x\u2016\u208a ^ ENNReal.toReal p", "state_after": "no goals"}, {"tactic": "simp [h0]", "annotated_tactic": ["simp [h0]", []], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2075 : MeasurableSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b3\ninst\u271d\u00b2 : MeasurableSpace \u03b4\n\u03bc : Measure \u03b1\n\u03bd : Measure \u03b2\nf : \u03b1 \u2192 \u03b3\ng : \u03b2 \u2192 \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup \u03b3\ninst\u271d : OpensMeasurableSpace \u03b3\nh : IdentDistrib f g\np : \u211d\u22650\u221e\nh0 : p = 0\n\u22a2 snorm f p \u03bc = snorm g p \u03bd", "state_after": "no goals"}, {"tactic": "simp only [h_top, snorm, snormEssSup, ENNReal.top_ne_zero, eq_self_iff_true, if_true, if_false]", "annotated_tactic": ["simp only [h_top, <a>snorm</a>, <a>snormEssSup</a>, <a>ENNReal.top_ne_zero</a>, <a>eq_self_iff_true</a>, <a>if_true</a>, <a>if_false</a>]", [{"full_name": "MeasureTheory.snorm", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [84, 5], "def_end_pos": [84, 10]}, {"full_name": "MeasureTheory.snormEssSup", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [78, 5], "def_end_pos": [78, 16]}, {"full_name": "ENNReal.top_ne_zero", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [337, 17], "def_end_pos": [337, 28]}, {"full_name": "eq_self_iff_true", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [86, 9], "def_end_pos": [86, 25]}, {"full_name": "if_true", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [727, 17], "def_end_pos": [727, 24]}, {"full_name": "if_false", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [729, 17], "def_end_pos": [729, 25]}]], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2075 : MeasurableSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b3\ninst\u271d\u00b2 : MeasurableSpace \u03b4\n\u03bc : Measure \u03b1\n\u03bd : Measure \u03b2\nf : \u03b1 \u2192 \u03b3\ng : \u03b2 \u2192 \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup \u03b3\ninst\u271d : OpensMeasurableSpace \u03b3\nh : IdentDistrib f g\np : \u211d\u22650\u221e\nh0 : \u00acp = 0\nh_top : p = \u22a4\n\u22a2 snorm f p \u03bc = snorm g p \u03bd", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2075 : MeasurableSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b3\ninst\u271d\u00b2 : MeasurableSpace \u03b4\n\u03bc : Measure \u03b1\n\u03bd : Measure \u03b2\nf : \u03b1 \u2192 \u03b3\ng : \u03b2 \u2192 \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup \u03b3\ninst\u271d : OpensMeasurableSpace \u03b3\nh : IdentDistrib f g\np : \u211d\u22650\u221e\nh0 : \u00acp = 0\nh_top : p = \u22a4\n\u22a2 essSup (fun x => \u2191\u2016f x\u2016\u208a) \u03bc = essSup (fun x => \u2191\u2016g x\u2016\u208a) \u03bd"}, {"tactic": "apply essSup_eq", "annotated_tactic": ["apply <a>essSup_eq</a>", [{"full_name": "ProbabilityTheory.IdentDistrib.essSup_eq", "def_path": "Mathlib/Probability/IdentDistrib.lean", "def_pos": [176, 9], "def_end_pos": [176, 18]}]], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2075 : MeasurableSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b3\ninst\u271d\u00b2 : MeasurableSpace \u03b4\n\u03bc : Measure \u03b1\n\u03bd : Measure \u03b2\nf : \u03b1 \u2192 \u03b3\ng : \u03b2 \u2192 \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup \u03b3\ninst\u271d : OpensMeasurableSpace \u03b3\nh : IdentDistrib f g\np : \u211d\u22650\u221e\nh0 : \u00acp = 0\nh_top : p = \u22a4\n\u22a2 essSup (fun x => \u2191\u2016f x\u2016\u208a) \u03bc = essSup (fun x => \u2191\u2016g x\u2016\u208a) \u03bd", "state_after": "case pos.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2075 : MeasurableSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b3\ninst\u271d\u00b2 : MeasurableSpace \u03b4\n\u03bc : Measure \u03b1\n\u03bd : Measure \u03b2\nf : \u03b1 \u2192 \u03b3\ng : \u03b2 \u2192 \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup \u03b3\ninst\u271d : OpensMeasurableSpace \u03b3\nh : IdentDistrib f g\np : \u211d\u22650\u221e\nh0 : \u00acp = 0\nh_top : p = \u22a4\n\u22a2 IdentDistrib (fun x => \u2191\u2016f x\u2016\u208a) fun x => \u2191\u2016g x\u2016\u208a"}, {"tactic": "exact h.comp (measurable_coe_nnreal_ennreal.comp measurable_nnnorm)", "annotated_tactic": ["exact h.comp (measurable_coe_nnreal_ennreal.comp <a>measurable_nnnorm</a>)", [{"full_name": "measurable_nnnorm", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [2256, 9], "def_end_pos": [2256, 26]}]], "state_before": "case pos.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2075 : MeasurableSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b3\ninst\u271d\u00b2 : MeasurableSpace \u03b4\n\u03bc : Measure \u03b1\n\u03bd : Measure \u03b2\nf : \u03b1 \u2192 \u03b3\ng : \u03b2 \u2192 \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup \u03b3\ninst\u271d : OpensMeasurableSpace \u03b3\nh : IdentDistrib f g\np : \u211d\u22650\u221e\nh0 : \u00acp = 0\nh_top : p = \u22a4\n\u22a2 IdentDistrib (fun x => \u2191\u2016f x\u2016\u208a) fun x => \u2191\u2016g x\u2016\u208a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Independence/Basic.lean", "full_name": "ProbabilityTheory.indep_of_indep_of_le_left", "start": [264, 1], "end": [267, 47], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/AEEqFun.lean", "full_name": "MeasureTheory.AEEqFun.coeFn_compMeasurable", "start": [324, 1], "end": [327, 17], "traced_tactics": [{"tactic": "rw [compMeasurable_eq_mk]", "annotated_tactic": ["rw [<a>compMeasurable_eq_mk</a>]", [{"full_name": "MeasureTheory.AEEqFun.compMeasurable_eq_mk", "def_path": "Mathlib/MeasureTheory/Function/AEEqFun.lean", "def_pos": [319, 9], "def_end_pos": [319, 29]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2079 : TopologicalSpace \u03b2\ninst\u271d\u2078 : TopologicalSpace \u03b3\ninst\u271d\u2077 : TopologicalSpace \u03b4\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : PseudoMetrizableSpace \u03b2\ninst\u271d\u2074 : BorelSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b3\ninst\u271d\u00b2 : PseudoMetrizableSpace \u03b3\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b3\ninst\u271d : SecondCountableTopology \u03b3\ng : \u03b2 \u2192 \u03b3\nhg : Measurable g\nf : \u03b1 \u2192\u2098[\u03bc] \u03b2\n\u22a2 \u2191(compMeasurable g hg f) =\u1d50[\u03bc] g \u2218 \u2191f", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2079 : TopologicalSpace \u03b2\ninst\u271d\u2078 : TopologicalSpace \u03b3\ninst\u271d\u2077 : TopologicalSpace \u03b4\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : PseudoMetrizableSpace \u03b2\ninst\u271d\u2074 : BorelSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b3\ninst\u271d\u00b2 : PseudoMetrizableSpace \u03b3\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b3\ninst\u271d : SecondCountableTopology \u03b3\ng : \u03b2 \u2192 \u03b3\nhg : Measurable g\nf : \u03b1 \u2192\u2098[\u03bc] \u03b2\n\u22a2 \u2191(mk (g \u2218 \u2191f) (_ : AEStronglyMeasurable (g \u2218 \u2191f) \u03bc)) =\u1d50[\u03bc] g \u2218 \u2191f"}, {"tactic": "apply coeFn_mk", "annotated_tactic": ["apply <a>coeFn_mk</a>", [{"full_name": "MeasureTheory.AEEqFun.coeFn_mk", "def_path": "Mathlib/MeasureTheory/Function/AEEqFun.lean", "def_pos": [182, 9], "def_end_pos": [182, 17]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2079 : TopologicalSpace \u03b2\ninst\u271d\u2078 : TopologicalSpace \u03b3\ninst\u271d\u2077 : TopologicalSpace \u03b4\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : PseudoMetrizableSpace \u03b2\ninst\u271d\u2074 : BorelSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b3\ninst\u271d\u00b2 : PseudoMetrizableSpace \u03b3\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b3\ninst\u271d : SecondCountableTopology \u03b3\ng : \u03b2 \u2192 \u03b3\nhg : Measurable g\nf : \u03b1 \u2192\u2098[\u03bc] \u03b2\n\u22a2 \u2191(mk (g \u2218 \u2191f) (_ : AEStronglyMeasurable (g \u2218 \u2191f) \u03bc)) =\u1d50[\u03bc] g \u2218 \u2191f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Rel.lean", "full_name": "Rel.preimage_def", "start": [182, 1], "end": [183, 38], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Covering/LiminfLimsup.lean", "full_name": "blimsup_cthickening_ae_le_of_eventually_mul_le_aux", "start": [42, 1], "end": [151, 46], "traced_tactics": [{"tactic": "set Y\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)", "annotated_tactic": ["set Y\u2081 : \u2115 \u2192 <a>Set</a> \u03b1 := fun i => <a>cthickening</a> (r\u2081 i) (s i)", [{"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}, {"full_name": "Metric.cthickening", "def_path": "Mathlib/Topology/MetricSpace/HausdorffDistance.lean", "def_pos": [1027, 5], "def_end_pos": [1027, 16]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhr : Tendsto r\u2081 atTop (\ud835\udcdd[Ioi 0] 0)\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nhMr : \u2200\u1da0 (i : \u2115) in atTop, M * r\u2081 i \u2264 r\u2082 i\n\u22a2 blimsup (fun i => cthickening (r\u2081 i) (s i)) atTop p \u2264\u1d50[\u03bc] blimsup (fun i => cthickening (r\u2082 i) (s i)) atTop p", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhr : Tendsto r\u2081 atTop (\ud835\udcdd[Ioi 0] 0)\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nhMr : \u2200\u1da0 (i : \u2115) in atTop, M * r\u2081 i \u2264 r\u2082 i\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\n\u22a2 blimsup Y\u2081 atTop p \u2264\u1d50[\u03bc] blimsup (fun i => cthickening (r\u2082 i) (s i)) atTop p"}, {"tactic": "set Y\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)", "annotated_tactic": ["set Y\u2082 : \u2115 \u2192 <a>Set</a> \u03b1 := fun i => <a>cthickening</a> (r\u2082 i) (s i)", [{"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}, {"full_name": "Metric.cthickening", "def_path": "Mathlib/Topology/MetricSpace/HausdorffDistance.lean", "def_pos": [1027, 5], "def_end_pos": [1027, 16]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhr : Tendsto r\u2081 atTop (\ud835\udcdd[Ioi 0] 0)\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nhMr : \u2200\u1da0 (i : \u2115) in atTop, M * r\u2081 i \u2264 r\u2082 i\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\n\u22a2 blimsup Y\u2081 atTop p \u2264\u1d50[\u03bc] blimsup (fun i => cthickening (r\u2082 i) (s i)) atTop p", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhr : Tendsto r\u2081 atTop (\ud835\udcdd[Ioi 0] 0)\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nhMr : \u2200\u1da0 (i : \u2115) in atTop, M * r\u2081 i \u2264 r\u2082 i\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\n\u22a2 blimsup Y\u2081 atTop p \u2264\u1d50[\u03bc] blimsup Y\u2082 atTop p"}, {"tactic": "let Z : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 (j) (_ : p j \u2227 i \u2264 j), Y\u2082 j", "annotated_tactic": ["let Z : \u2115 \u2192 <a>Set</a> \u03b1 := fun i => \u22c3 (j) (_ : p j \u2227 i \u2264 j), Y\u2082 j", [{"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhr : Tendsto r\u2081 atTop (\ud835\udcdd[Ioi 0] 0)\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nhMr : \u2200\u1da0 (i : \u2115) in atTop, M * r\u2081 i \u2264 r\u2082 i\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\n\u22a2 blimsup Y\u2081 atTop p \u2264\u1d50[\u03bc] blimsup Y\u2082 atTop p", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhr : Tendsto r\u2081 atTop (\ud835\udcdd[Ioi 0] 0)\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nhMr : \u2200\u1da0 (i : \u2115) in atTop, M * r\u2081 i \u2264 r\u2082 i\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\n\u22a2 blimsup Y\u2081 atTop p \u2264\u1d50[\u03bc] blimsup Y\u2082 atTop p"}, {"tactic": "suffices \u2200 i, \u03bc (atTop.blimsup Y\u2081 p \\ Z i) = 0 by\n  rwa [ae_le_set, @blimsup_eq_iInf_biSup_of_nat _ _ _ Y\u2082, iInf_eq_iInter, diff_iInter,\n    measure_iUnion_null_iff]", "annotated_tactic": ["suffices \u2200 i, \u03bc (atTop.blimsup Y\u2081 p \\ Z i) = 0 by\n    rwa [<a>ae_le_set</a>, @<a>blimsup_eq_iInf_biSup_of_nat</a> _ _ _ Y\u2082, <a>iInf_eq_iInter</a>, <a>diff_iInter</a>,\n      <a>measure_iUnion_null_iff</a>]", [{"full_name": "MeasureTheory.ae_le_set", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [466, 9], "def_end_pos": [466, 18]}, {"full_name": "Filter.blimsup_eq_iInf_biSup_of_nat", "def_path": "Mathlib/Order/LiminfLimsup.lean", "def_pos": [841, 9], "def_end_pos": [841, 37]}, {"full_name": "Set.iInf_eq_iInter", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [196, 9], "def_end_pos": [196, 23]}, {"full_name": "Set.diff_iInter", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [686, 9], "def_end_pos": [686, 20]}, {"full_name": "MeasureTheory.measure_iUnion_null_iff", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [275, 9], "def_end_pos": [275, 32]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhr : Tendsto r\u2081 atTop (\ud835\udcdd[Ioi 0] 0)\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nhMr : \u2200\u1da0 (i : \u2115) in atTop, M * r\u2081 i \u2264 r\u2082 i\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\n\u22a2 blimsup Y\u2081 atTop p \u2264\u1d50[\u03bc] blimsup Y\u2082 atTop p", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhr : Tendsto r\u2081 atTop (\ud835\udcdd[Ioi 0] 0)\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nhMr : \u2200\u1da0 (i : \u2115) in atTop, M * r\u2081 i \u2264 r\u2082 i\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\n\u22a2 \u2200 (i : \u2115), \u2191\u2191\u03bc (blimsup Y\u2081 atTop p \\ Z i) = 0"}, {"tactic": "intros i", "annotated_tactic": ["intros i", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhr : Tendsto r\u2081 atTop (\ud835\udcdd[Ioi 0] 0)\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nhMr : \u2200\u1da0 (i : \u2115) in atTop, M * r\u2081 i \u2264 r\u2082 i\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\n\u22a2 \u2200 (i : \u2115), \u2191\u2191\u03bc (blimsup Y\u2081 atTop p \\ Z i) = 0", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhr : Tendsto r\u2081 atTop (\ud835\udcdd[Ioi 0] 0)\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nhMr : \u2200\u1da0 (i : \u2115) in atTop, M * r\u2081 i \u2264 r\u2082 i\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\n\u22a2 \u2191\u2191\u03bc (blimsup Y\u2081 atTop p \\ Z i) = 0"}, {"tactic": "set W := atTop.blimsup Y\u2081 p \\ Z i", "annotated_tactic": ["set W := atTop.blimsup Y\u2081 p \\ Z i", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhr : Tendsto r\u2081 atTop (\ud835\udcdd[Ioi 0] 0)\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nhMr : \u2200\u1da0 (i : \u2115) in atTop, M * r\u2081 i \u2264 r\u2082 i\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\n\u22a2 \u2191\u2191\u03bc (blimsup Y\u2081 atTop p \\ Z i) = 0", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhr : Tendsto r\u2081 atTop (\ud835\udcdd[Ioi 0] 0)\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nhMr : \u2200\u1da0 (i : \u2115) in atTop, M * r\u2081 i \u2264 r\u2082 i\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\n\u22a2 \u2191\u2191\u03bc W = 0"}, {"tactic": "by_contra contra", "annotated_tactic": ["by_contra contra", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhr : Tendsto r\u2081 atTop (\ud835\udcdd[Ioi 0] 0)\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nhMr : \u2200\u1da0 (i : \u2115) in atTop, M * r\u2081 i \u2264 r\u2082 i\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\n\u22a2 \u2191\u2191\u03bc W = 0", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhr : Tendsto r\u2081 atTop (\ud835\udcdd[Ioi 0] 0)\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nhMr : \u2200\u1da0 (i : \u2115) in atTop, M * r\u2081 i \u2264 r\u2082 i\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\n\u22a2 False"}, {"tactic": "obtain \u27e8d, hd, hd'\u27e9 : \u2203 d, d \u2208 W \u2227 \u2200 {\u03b9 : Type _} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[>] 0) \u2192 (\u2200\u1da0 j in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n      Tendsto (fun j => \u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1) :=\n  Measure.exists_mem_of_measure_ne_zero_of_ae contra\n    (IsUnifLocDoublingMeasure.ae_tendsto_measure_inter_div \u03bc W 2)", "annotated_tactic": ["obtain \u27e8d, hd, hd'\u27e9 : \u2203 d, d \u2208 W \u2227 \u2200 {\u03b9 : Type _} {l : <a>Filter</a> \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n      <a>Tendsto</a> \u03b4 l (\ud835\udcdd[>] 0) \u2192 (\u2200\u1da0 j in l, d \u2208 <a>closedBall</a> (w j) (2 * \u03b4 j)) \u2192\n        <a>Tendsto</a> (fun j => \u03bc (W \u2229 <a>closedBall</a> (w j) (\u03b4 j)) / \u03bc (<a>closedBall</a> (w j) (\u03b4 j))) l (\ud835\udcdd 1) :=\n    <a>Measure.exists_mem_of_measure_ne_zero_of_ae</a> contra\n      (<a>IsUnifLocDoublingMeasure.ae_tendsto_measure_inter_div</a> \u03bc W 2)", [{"full_name": "Filter", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [91, 11], "def_end_pos": [91, 17]}, {"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2939, 5], "def_end_pos": [2939, 12]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2939, 5], "def_end_pos": [2939, 12]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "MeasureTheory.Measure.exists_mem_of_measure_ne_zero_of_ae", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1891, 9], "def_end_pos": [1891, 44]}, {"full_name": "IsUnifLocDoublingMeasure.ae_tendsto_measure_inter_div", "def_path": "Mathlib/MeasureTheory/Covering/DensityTheorem.lean", "def_pos": [146, 9], "def_end_pos": [146, 37]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhr : Tendsto r\u2081 atTop (\ud835\udcdd[Ioi 0] 0)\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nhMr : \u2200\u1da0 (i : \u2115) in atTop, M * r\u2081 i \u2264 r\u2082 i\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\n\u22a2 False", "state_after": "case intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhr : Tendsto r\u2081 atTop (\ud835\udcdd[Ioi 0] 0)\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nhMr : \u2200\u1da0 (i : \u2115) in atTop, M * r\u2081 i \u2264 r\u2082 i\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd : d \u2208 W\nhd' :\n  \u2200 {\u03b9 : Type ?u.5949} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\n\u22a2 False"}, {"tactic": "replace hd : d \u2208 blimsup Y\u2081 atTop p := ((mem_diff _).mp hd).1", "annotated_tactic": ["replace hd : d \u2208 <a>blimsup</a> Y\u2081 <a>atTop</a> p := ((<a>mem_diff</a> _).<a>mp</a> hd).1", [{"full_name": "Filter.blimsup", "def_path": "Mathlib/Order/LiminfLimsup.lean", "def_pos": [432, 5], "def_end_pos": [432, 12]}, {"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}, {"full_name": "Set.mem_diff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1819, 9], "def_end_pos": [1819, 17]}, {"full_name": "Iff.mp", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [90, 3], "def_end_pos": [90, 5]}]], "state_before": "case intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhr : Tendsto r\u2081 atTop (\ud835\udcdd[Ioi 0] 0)\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nhMr : \u2200\u1da0 (i : \u2115) in atTop, M * r\u2081 i \u2264 r\u2082 i\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd : d \u2208 W\nhd' :\n  \u2200 {\u03b9 : Type ?u.5949} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\n\u22a2 False", "state_after": "case intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhr : Tendsto r\u2081 atTop (\ud835\udcdd[Ioi 0] 0)\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nhMr : \u2200\u1da0 (i : \u2115) in atTop, M * r\u2081 i \u2264 r\u2082 i\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type ?u.5949} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\n\u22a2 False"}, {"tactic": "obtain \u27e8f : \u2115 \u2192 \u2115, hf\u27e9 := exists_forall_mem_of_hasBasis_mem_blimsup' atTop_basis hd", "annotated_tactic": ["obtain \u27e8f : \u2115 \u2192 \u2115, hf\u27e9 := <a>exists_forall_mem_of_hasBasis_mem_blimsup'</a> <a>atTop_basis</a> hd", [{"full_name": "Filter.exists_forall_mem_of_hasBasis_mem_blimsup'", "def_path": "Mathlib/Order/LiminfLimsup.lean", "def_pos": [1190, 9], "def_end_pos": [1190, 51]}, {"full_name": "Filter.atTop_basis", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [131, 9], "def_end_pos": [131, 20]}]], "state_before": "case intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhr : Tendsto r\u2081 atTop (\ud835\udcdd[Ioi 0] 0)\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nhMr : \u2200\u1da0 (i : \u2115) in atTop, M * r\u2081 i \u2264 r\u2082 i\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type ?u.5949} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\n\u22a2 False", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhr : Tendsto r\u2081 atTop (\ud835\udcdd[Ioi 0] 0)\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nhMr : \u2200\u1da0 (i : \u2115) in atTop, M * r\u2081 i \u2264 r\u2082 i\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type ?u.5949} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\nf : \u2115 \u2192 \u2115\nhf : \u2200 (i : \u2115), d \u2208 Y\u2081 (f i) \u2227 p (f i) \u2227 f i \u2208 Ici i\n\u22a2 False"}, {"tactic": "simp only [forall_and] at hf", "annotated_tactic": ["simp only [<a>forall_and</a>] at hf", [{"full_name": "forall_and", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [426, 9], "def_end_pos": [426, 19]}]], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhr : Tendsto r\u2081 atTop (\ud835\udcdd[Ioi 0] 0)\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nhMr : \u2200\u1da0 (i : \u2115) in atTop, M * r\u2081 i \u2264 r\u2082 i\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type ?u.5949} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\nf : \u2115 \u2192 \u2115\nhf : \u2200 (i : \u2115), d \u2208 Y\u2081 (f i) \u2227 p (f i) \u2227 f i \u2208 Ici i\n\u22a2 False", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhr : Tendsto r\u2081 atTop (\ud835\udcdd[Ioi 0] 0)\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nhMr : \u2200\u1da0 (i : \u2115) in atTop, M * r\u2081 i \u2264 r\u2082 i\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type ?u.5949} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\nf : \u2115 \u2192 \u2115\nhf : (\u2200 (x : \u2115), d \u2208 cthickening (r\u2081 (f x)) (s (f x))) \u2227 (\u2200 (x : \u2115), p (f x)) \u2227 \u2200 (x : \u2115), f x \u2208 Ici x\n\u22a2 False"}, {"tactic": "obtain \u27e8hf\u2080 : \u2200 j, d \u2208 cthickening (r\u2081 (f j)) (s (f j)), hf\u2081, hf\u2082 : \u2200 j, j \u2264 f j\u27e9 := hf", "annotated_tactic": ["obtain \u27e8hf\u2080 : \u2200 j, d \u2208 <a>cthickening</a> (r\u2081 (f j)) (s (f j)), hf\u2081, hf\u2082 : \u2200 j, j \u2264 f j\u27e9 := hf", [{"full_name": "Metric.cthickening", "def_path": "Mathlib/Topology/MetricSpace/HausdorffDistance.lean", "def_pos": [1027, 5], "def_end_pos": [1027, 16]}]], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhr : Tendsto r\u2081 atTop (\ud835\udcdd[Ioi 0] 0)\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nhMr : \u2200\u1da0 (i : \u2115) in atTop, M * r\u2081 i \u2264 r\u2082 i\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type ?u.5949} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\nf : \u2115 \u2192 \u2115\nhf : (\u2200 (x : \u2115), d \u2208 cthickening (r\u2081 (f x)) (s (f x))) \u2227 (\u2200 (x : \u2115), p (f x)) \u2227 \u2200 (x : \u2115), f x \u2208 Ici x\n\u22a2 False", "state_after": "case intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhr : Tendsto r\u2081 atTop (\ud835\udcdd[Ioi 0] 0)\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nhMr : \u2200\u1da0 (i : \u2115) in atTop, M * r\u2081 i \u2264 r\u2082 i\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type ?u.5949} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\nf : \u2115 \u2192 \u2115\nhf\u2080 : \u2200 (j : \u2115), d \u2208 cthickening (r\u2081 (f j)) (s (f j))\nhf\u2081 : \u2200 (x : \u2115), p (f x)\nhf\u2082 : \u2200 (j : \u2115), j \u2264 f j\n\u22a2 False"}, {"tactic": "have hf\u2083 : Tendsto f atTop atTop :=\n  tendsto_atTop_atTop.mpr fun j => \u27e8f j, fun i hi => (hf\u2082 j).trans (hi.trans <| hf\u2082 i)\u27e9", "annotated_tactic": ["have hf\u2083 : <a>Tendsto</a> f <a>atTop</a> <a>atTop</a> :=\n    tendsto_atTop_atTop.mpr fun j => \u27e8f j, fun i hi => (hf\u2082 j).<a>trans</a> (hi.trans <| hf\u2082 i)\u27e9", [{"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2939, 5], "def_end_pos": [2939, 12]}, {"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}, {"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}, {"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [120, 7], "def_end_pos": [120, 18]}]], "state_before": "case intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhr : Tendsto r\u2081 atTop (\ud835\udcdd[Ioi 0] 0)\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nhMr : \u2200\u1da0 (i : \u2115) in atTop, M * r\u2081 i \u2264 r\u2082 i\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type ?u.5949} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\nf : \u2115 \u2192 \u2115\nhf\u2080 : \u2200 (j : \u2115), d \u2208 cthickening (r\u2081 (f j)) (s (f j))\nhf\u2081 : \u2200 (x : \u2115), p (f x)\nhf\u2082 : \u2200 (j : \u2115), j \u2264 f j\n\u22a2 False", "state_after": "case intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhr : Tendsto r\u2081 atTop (\ud835\udcdd[Ioi 0] 0)\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nhMr : \u2200\u1da0 (i : \u2115) in atTop, M * r\u2081 i \u2264 r\u2082 i\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type ?u.5949} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\nf : \u2115 \u2192 \u2115\nhf\u2080 : \u2200 (j : \u2115), d \u2208 cthickening (r\u2081 (f j)) (s (f j))\nhf\u2081 : \u2200 (x : \u2115), p (f x)\nhf\u2082 : \u2200 (j : \u2115), j \u2264 f j\nhf\u2083 : Tendsto f atTop atTop\n\u22a2 False"}, {"tactic": "replace hr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[>] 0) := hr.comp hf\u2083", "annotated_tactic": ["replace hr : <a>Tendsto</a> (r\u2081 \u2218 f) <a>atTop</a> (\ud835\udcdd[>] 0) := hr.comp hf\u2083", [{"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2939, 5], "def_end_pos": [2939, 12]}, {"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}]], "state_before": "case intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhr : Tendsto r\u2081 atTop (\ud835\udcdd[Ioi 0] 0)\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nhMr : \u2200\u1da0 (i : \u2115) in atTop, M * r\u2081 i \u2264 r\u2082 i\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type ?u.5949} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\nf : \u2115 \u2192 \u2115\nhf\u2080 : \u2200 (j : \u2115), d \u2208 cthickening (r\u2081 (f j)) (s (f j))\nhf\u2081 : \u2200 (x : \u2115), p (f x)\nhf\u2082 : \u2200 (j : \u2115), j \u2264 f j\nhf\u2083 : Tendsto f atTop atTop\n\u22a2 False", "state_after": "case intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nhMr : \u2200\u1da0 (i : \u2115) in atTop, M * r\u2081 i \u2264 r\u2082 i\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type ?u.5949} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\nf : \u2115 \u2192 \u2115\nhf\u2080 : \u2200 (j : \u2115), d \u2208 cthickening (r\u2081 (f j)) (s (f j))\nhf\u2081 : \u2200 (x : \u2115), p (f x)\nhf\u2082 : \u2200 (j : \u2115), j \u2264 f j\nhf\u2083 : Tendsto f atTop atTop\nhr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0)\n\u22a2 False"}, {"tactic": "replace hMr : \u2200\u1da0 j in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j) := hf\u2083.eventually hMr", "annotated_tactic": ["replace hMr : \u2200\u1da0 j in <a>atTop</a>, M * r\u2081 (f j) \u2264 r\u2082 (f j) := hf\u2083.eventually hMr", [{"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}]], "state_before": "case intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nhMr : \u2200\u1da0 (i : \u2115) in atTop, M * r\u2081 i \u2264 r\u2082 i\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type ?u.5949} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\nf : \u2115 \u2192 \u2115\nhf\u2080 : \u2200 (j : \u2115), d \u2208 cthickening (r\u2081 (f j)) (s (f j))\nhf\u2081 : \u2200 (x : \u2115), p (f x)\nhf\u2082 : \u2200 (j : \u2115), j \u2264 f j\nhf\u2083 : Tendsto f atTop atTop\nhr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0)\n\u22a2 False", "state_after": "case intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type ?u.5949} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\nf : \u2115 \u2192 \u2115\nhf\u2080 : \u2200 (j : \u2115), d \u2208 cthickening (r\u2081 (f j)) (s (f j))\nhf\u2081 : \u2200 (x : \u2115), p (f x)\nhf\u2082 : \u2200 (j : \u2115), j \u2264 f j\nhf\u2083 : Tendsto f atTop atTop\nhr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j)\n\u22a2 False"}, {"tactic": "replace hf\u2080 : \u2200 j, \u2203 w \u2208 s (f j), d \u2208 closedBall w (2 * r\u2081 (f j))", "annotated_tactic": ["replace hf\u2080 : \u2200 j, \u2203 w \u2208 s (f j), d \u2208 <a>closedBall</a> w (2 * r\u2081 (f j))", [{"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}]], "state_before": "case intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type ?u.5949} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\nf : \u2115 \u2192 \u2115\nhf\u2080 : \u2200 (j : \u2115), d \u2208 cthickening (r\u2081 (f j)) (s (f j))\nhf\u2081 : \u2200 (x : \u2115), p (f x)\nhf\u2082 : \u2200 (j : \u2115), j \u2264 f j\nhf\u2083 : Tendsto f atTop atTop\nhr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j)\n\u22a2 False", "state_after": "case hf\u2080\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type ?u.5949} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\nf : \u2115 \u2192 \u2115\nhf\u2080 : \u2200 (j : \u2115), d \u2208 cthickening (r\u2081 (f j)) (s (f j))\nhf\u2081 : \u2200 (x : \u2115), p (f x)\nhf\u2082 : \u2200 (j : \u2115), j \u2264 f j\nhf\u2083 : Tendsto f atTop atTop\nhr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j)\n\u22a2 \u2200 (j : \u2115), \u2203 w, w \u2208 s (f j) \u2227 d \u2208 closedBall w (2 * r\u2081 (f j))\n\ncase intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type ?u.5949} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\nf : \u2115 \u2192 \u2115\nhf\u2081 : \u2200 (x : \u2115), p (f x)\nhf\u2082 : \u2200 (j : \u2115), j \u2264 f j\nhf\u2083 : Tendsto f atTop atTop\nhr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j)\nhf\u2080 : \u2200 (j : \u2115), \u2203 w, w \u2208 s (f j) \u2227 d \u2208 closedBall w (2 * r\u2081 (f j))\n\u22a2 False"}, {"tactic": "choose w hw hw' using hf\u2080", "annotated_tactic": ["choose w hw hw' using hf\u2080", []], "state_before": "case intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type ?u.5949} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\nf : \u2115 \u2192 \u2115\nhf\u2081 : \u2200 (x : \u2115), p (f x)\nhf\u2082 : \u2200 (j : \u2115), j \u2264 f j\nhf\u2083 : Tendsto f atTop atTop\nhr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j)\nhf\u2080 : \u2200 (j : \u2115), \u2203 w, w \u2208 s (f j) \u2227 d \u2208 closedBall w (2 * r\u2081 (f j))\n\u22a2 False", "state_after": "case intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type ?u.5949} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\nf : \u2115 \u2192 \u2115\nhf\u2081 : \u2200 (x : \u2115), p (f x)\nhf\u2082 : \u2200 (j : \u2115), j \u2264 f j\nhf\u2083 : Tendsto f atTop atTop\nhr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j)\nw : \u2115 \u2192 \u03b1\nhw : \u2200 (j : \u2115), w j \u2208 s (f j)\nhw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j))\n\u22a2 False"}, {"tactic": "let C := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9", "annotated_tactic": ["let C := <a>IsUnifLocDoublingMeasure.scalingConstantOf</a> \u03bc M\u207b\u00b9", [{"full_name": "IsUnifLocDoublingMeasure.scalingConstantOf", "def_path": "Mathlib/MeasureTheory/Measure/Doubling.lean", "def_pos": [108, 5], "def_end_pos": [108, 22]}]], "state_before": "case intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type ?u.5949} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\nf : \u2115 \u2192 \u2115\nhf\u2081 : \u2200 (x : \u2115), p (f x)\nhf\u2082 : \u2200 (j : \u2115), j \u2264 f j\nhf\u2083 : Tendsto f atTop atTop\nhr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j)\nw : \u2115 \u2192 \u03b1\nhw : \u2200 (j : \u2115), w j \u2208 s (f j)\nhw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j))\n\u22a2 False", "state_after": "case intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type ?u.5949} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\nf : \u2115 \u2192 \u2115\nhf\u2081 : \u2200 (x : \u2115), p (f x)\nhf\u2082 : \u2200 (j : \u2115), j \u2264 f j\nhf\u2083 : Tendsto f atTop atTop\nhr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j)\nw : \u2115 \u2192 \u03b1\nhw : \u2200 (j : \u2115), w j \u2208 s (f j)\nhw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j))\nC : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9\n\u22a2 False"}, {"tactic": "have hC : 0 < C :=\n  lt_of_lt_of_le zero_lt_one (IsUnifLocDoublingMeasure.one_le_scalingConstantOf \u03bc M\u207b\u00b9)", "annotated_tactic": ["have hC : 0 < C :=\n    <a>lt_of_lt_of_le</a> <a>zero_lt_one</a> (<a>IsUnifLocDoublingMeasure.one_le_scalingConstantOf</a> \u03bc M\u207b\u00b9)", [{"full_name": "lt_of_lt_of_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [115, 9], "def_end_pos": [115, 23]}, {"full_name": "zero_lt_one", "def_path": "Mathlib/Algebra/Order/ZeroLEOne.lean", "def_pos": [39, 15], "def_end_pos": [39, 26]}, {"full_name": "IsUnifLocDoublingMeasure.one_le_scalingConstantOf", "def_path": "Mathlib/MeasureTheory/Measure/Doubling.lean", "def_pos": [113, 9], "def_end_pos": [113, 33]}]], "state_before": "case intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type ?u.5949} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\nf : \u2115 \u2192 \u2115\nhf\u2081 : \u2200 (x : \u2115), p (f x)\nhf\u2082 : \u2200 (j : \u2115), j \u2264 f j\nhf\u2083 : Tendsto f atTop atTop\nhr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j)\nw : \u2115 \u2192 \u03b1\nhw : \u2200 (j : \u2115), w j \u2208 s (f j)\nhw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j))\nC : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9\n\u22a2 False", "state_after": "case intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type ?u.5949} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\nf : \u2115 \u2192 \u2115\nhf\u2081 : \u2200 (x : \u2115), p (f x)\nhf\u2082 : \u2200 (j : \u2115), j \u2264 f j\nhf\u2083 : Tendsto f atTop atTop\nhr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j)\nw : \u2115 \u2192 \u03b1\nhw : \u2200 (j : \u2115), w j \u2208 s (f j)\nhw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j))\nC : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9\nhC : 0 < C\n\u22a2 False"}, {"tactic": "suffices \u2203 \u03b7 < (1 : \u211d\u22650),\n    \u2200\u1da0 j in atTop, \u03bc (W \u2229 closedBall (w j) (r\u2081 (f j))) / \u03bc (closedBall (w j) (r\u2081 (f j))) \u2264 \u03b7 by\n  obtain \u27e8\u03b7, h\u03b7, h\u03b7'\u27e9 := this\n  replace h\u03b7' : 1 \u2264 \u03b7 := by\n    simpa only [ENNReal.one_le_coe_iff] using\n      le_of_tendsto (hd' w (fun j => r\u2081 (f j)) hr <| eventually_of_forall hw') h\u03b7'\n  exact (lt_self_iff_false _).mp (lt_of_lt_of_le h\u03b7 h\u03b7')", "annotated_tactic": ["suffices \u2203 \u03b7 < (1 : \u211d\u22650),\n      \u2200\u1da0 j in <a>atTop</a>, \u03bc (W \u2229 <a>closedBall</a> (w j) (r\u2081 (f j))) / \u03bc (<a>closedBall</a> (w j) (r\u2081 (f j))) \u2264 \u03b7 by\n    obtain \u27e8\u03b7, h\u03b7, h\u03b7'\u27e9 := this\n    replace h\u03b7' : 1 \u2264 \u03b7 := by\n      simpa only [<a>ENNReal.one_le_coe_iff</a>] using\n        <a>le_of_tendsto</a> (hd' w (fun j => r\u2081 (f j)) hr <| <a>eventually_of_forall</a> hw') h\u03b7'\n    exact (<a>lt_self_iff_false</a> _).<a>mp</a> (<a>lt_of_lt_of_le</a> h\u03b7 h\u03b7')", [{"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "ENNReal.one_le_coe_iff", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [691, 9], "def_end_pos": [691, 23]}, {"full_name": "le_of_tendsto", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [140, 9], "def_end_pos": [140, 22]}, {"full_name": "Filter.eventually_of_forall", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1111, 9], "def_end_pos": [1111, 29]}, {"full_name": "lt_self_iff_false", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [175, 9], "def_end_pos": [175, 26]}, {"full_name": "Iff.mp", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [90, 3], "def_end_pos": [90, 5]}, {"full_name": "lt_of_lt_of_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [115, 9], "def_end_pos": [115, 23]}]], "state_before": "case intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type ?u.5949} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\nf : \u2115 \u2192 \u2115\nhf\u2081 : \u2200 (x : \u2115), p (f x)\nhf\u2082 : \u2200 (j : \u2115), j \u2264 f j\nhf\u2083 : Tendsto f atTop atTop\nhr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j)\nw : \u2115 \u2192 \u03b1\nhw : \u2200 (j : \u2115), w j \u2208 s (f j)\nhw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j))\nC : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9\nhC : 0 < C\n\u22a2 False", "state_after": "case intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\nf : \u2115 \u2192 \u2115\nhf\u2081 : \u2200 (x : \u2115), p (f x)\nhf\u2082 : \u2200 (j : \u2115), j \u2264 f j\nhf\u2083 : Tendsto f atTop atTop\nhr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j)\nw : \u2115 \u2192 \u03b1\nhw : \u2200 (j : \u2115), w j \u2208 s (f j)\nhw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j))\nC : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9\nhC : 0 < C\n\u22a2 \u2203 \u03b7, \u03b7 < 1 \u2227 \u2200\u1da0 (j : \u2115) in atTop, \u2191\u2191\u03bc (W \u2229 closedBall (w j) (r\u2081 (f j))) / \u2191\u2191\u03bc (closedBall (w j) (r\u2081 (f j))) \u2264 \u2191\u03b7"}, {"tactic": "refine' \u27e81 - C\u207b\u00b9, tsub_lt_self zero_lt_one (inv_pos.mpr hC), _\u27e9", "annotated_tactic": ["refine' \u27e81 - C\u207b\u00b9, <a>tsub_lt_self</a> <a>zero_lt_one</a> (inv_pos.mpr hC), _\u27e9", [{"full_name": "tsub_lt_self", "def_path": "Mathlib/Algebra/Order/Sub/Canonical.lean", "def_pos": [474, 9], "def_end_pos": [474, 21]}, {"full_name": "zero_lt_one", "def_path": "Mathlib/Algebra/Order/ZeroLEOne.lean", "def_pos": [39, 15], "def_end_pos": [39, 26]}]], "state_before": "case intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\nf : \u2115 \u2192 \u2115\nhf\u2081 : \u2200 (x : \u2115), p (f x)\nhf\u2082 : \u2200 (j : \u2115), j \u2264 f j\nhf\u2083 : Tendsto f atTop atTop\nhr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j)\nw : \u2115 \u2192 \u03b1\nhw : \u2200 (j : \u2115), w j \u2208 s (f j)\nhw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j))\nC : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9\nhC : 0 < C\n\u22a2 \u2203 \u03b7, \u03b7 < 1 \u2227 \u2200\u1da0 (j : \u2115) in atTop, \u2191\u2191\u03bc (W \u2229 closedBall (w j) (r\u2081 (f j))) / \u2191\u2191\u03bc (closedBall (w j) (r\u2081 (f j))) \u2264 \u2191\u03b7", "state_after": "case intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\nf : \u2115 \u2192 \u2115\nhf\u2081 : \u2200 (x : \u2115), p (f x)\nhf\u2082 : \u2200 (j : \u2115), j \u2264 f j\nhf\u2083 : Tendsto f atTop atTop\nhr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j)\nw : \u2115 \u2192 \u03b1\nhw : \u2200 (j : \u2115), w j \u2208 s (f j)\nhw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j))\nC : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9\nhC : 0 < C\n\u22a2 \u2200\u1da0 (j : \u2115) in atTop, \u2191\u2191\u03bc (W \u2229 closedBall (w j) (r\u2081 (f j))) / \u2191\u2191\u03bc (closedBall (w j) (r\u2081 (f j))) \u2264 \u2191(1 - C\u207b\u00b9)"}, {"tactic": "replace hC : C \u2260 0 := ne_of_gt hC", "annotated_tactic": ["replace hC : C \u2260 0 := <a>ne_of_gt</a> hC", [{"full_name": "ne_of_gt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [104, 9], "def_end_pos": [104, 17]}]], "state_before": "case intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\nf : \u2115 \u2192 \u2115\nhf\u2081 : \u2200 (x : \u2115), p (f x)\nhf\u2082 : \u2200 (j : \u2115), j \u2264 f j\nhf\u2083 : Tendsto f atTop atTop\nhr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j)\nw : \u2115 \u2192 \u03b1\nhw : \u2200 (j : \u2115), w j \u2208 s (f j)\nhw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j))\nC : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9\nhC : 0 < C\n\u22a2 \u2200\u1da0 (j : \u2115) in atTop, \u2191\u2191\u03bc (W \u2229 closedBall (w j) (r\u2081 (f j))) / \u2191\u2191\u03bc (closedBall (w j) (r\u2081 (f j))) \u2264 \u2191(1 - C\u207b\u00b9)", "state_after": "case intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\nf : \u2115 \u2192 \u2115\nhf\u2081 : \u2200 (x : \u2115), p (f x)\nhf\u2082 : \u2200 (j : \u2115), j \u2264 f j\nhf\u2083 : Tendsto f atTop atTop\nhr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j)\nw : \u2115 \u2192 \u03b1\nhw : \u2200 (j : \u2115), w j \u2208 s (f j)\nhw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j))\nC : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9\nhC : C \u2260 0\n\u22a2 \u2200\u1da0 (j : \u2115) in atTop, \u2191\u2191\u03bc (W \u2229 closedBall (w j) (r\u2081 (f j))) / \u2191\u2191\u03bc (closedBall (w j) (r\u2081 (f j))) \u2264 \u2191(1 - C\u207b\u00b9)"}, {"tactic": "let b : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (M * r\u2081 (f j))", "annotated_tactic": ["let b : \u2115 \u2192 <a>Set</a> \u03b1 := fun j => <a>closedBall</a> (w j) (M * r\u2081 (f j))", [{"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}]], "state_before": "case intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\nf : \u2115 \u2192 \u2115\nhf\u2081 : \u2200 (x : \u2115), p (f x)\nhf\u2082 : \u2200 (j : \u2115), j \u2264 f j\nhf\u2083 : Tendsto f atTop atTop\nhr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j)\nw : \u2115 \u2192 \u03b1\nhw : \u2200 (j : \u2115), w j \u2208 s (f j)\nhw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j))\nC : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9\nhC : C \u2260 0\n\u22a2 \u2200\u1da0 (j : \u2115) in atTop, \u2191\u2191\u03bc (W \u2229 closedBall (w j) (r\u2081 (f j))) / \u2191\u2191\u03bc (closedBall (w j) (r\u2081 (f j))) \u2264 \u2191(1 - C\u207b\u00b9)", "state_after": "case intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\nf : \u2115 \u2192 \u2115\nhf\u2081 : \u2200 (x : \u2115), p (f x)\nhf\u2082 : \u2200 (j : \u2115), j \u2264 f j\nhf\u2083 : Tendsto f atTop atTop\nhr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j)\nw : \u2115 \u2192 \u03b1\nhw : \u2200 (j : \u2115), w j \u2208 s (f j)\nhw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j))\nC : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9\nhC : C \u2260 0\nb : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (M * r\u2081 (f j))\n\u22a2 \u2200\u1da0 (j : \u2115) in atTop, \u2191\u2191\u03bc (W \u2229 closedBall (w j) (r\u2081 (f j))) / \u2191\u2191\u03bc (closedBall (w j) (r\u2081 (f j))) \u2264 \u2191(1 - C\u207b\u00b9)"}, {"tactic": "let B : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (r\u2081 (f j))", "annotated_tactic": ["let B : \u2115 \u2192 <a>Set</a> \u03b1 := fun j => <a>closedBall</a> (w j) (r\u2081 (f j))", [{"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}]], "state_before": "case intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\nf : \u2115 \u2192 \u2115\nhf\u2081 : \u2200 (x : \u2115), p (f x)\nhf\u2082 : \u2200 (j : \u2115), j \u2264 f j\nhf\u2083 : Tendsto f atTop atTop\nhr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j)\nw : \u2115 \u2192 \u03b1\nhw : \u2200 (j : \u2115), w j \u2208 s (f j)\nhw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j))\nC : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9\nhC : C \u2260 0\nb : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (M * r\u2081 (f j))\n\u22a2 \u2200\u1da0 (j : \u2115) in atTop, \u2191\u2191\u03bc (W \u2229 closedBall (w j) (r\u2081 (f j))) / \u2191\u2191\u03bc (closedBall (w j) (r\u2081 (f j))) \u2264 \u2191(1 - C\u207b\u00b9)", "state_after": "case intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\nf : \u2115 \u2192 \u2115\nhf\u2081 : \u2200 (x : \u2115), p (f x)\nhf\u2082 : \u2200 (j : \u2115), j \u2264 f j\nhf\u2083 : Tendsto f atTop atTop\nhr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j)\nw : \u2115 \u2192 \u03b1\nhw : \u2200 (j : \u2115), w j \u2208 s (f j)\nhw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j))\nC : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9\nhC : C \u2260 0\nb : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (M * r\u2081 (f j))\nB : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (r\u2081 (f j))\n\u22a2 \u2200\u1da0 (j : \u2115) in atTop, \u2191\u2191\u03bc (W \u2229 closedBall (w j) (r\u2081 (f j))) / \u2191\u2191\u03bc (closedBall (w j) (r\u2081 (f j))) \u2264 \u2191(1 - C\u207b\u00b9)"}, {"tactic": "have h\u2081 : \u2200 j, b j \u2286 B j := fun j =>\n  closedBall_subset_closedBall (mul_le_of_le_one_left (hrp (f j)) hM'.le)", "annotated_tactic": ["have h\u2081 : \u2200 j, b j \u2286 B j := fun j =>\n    <a>closedBall_subset_closedBall</a> (<a>mul_le_of_le_one_left</a> (hrp (f j)) hM'.le)", [{"full_name": "Metric.closedBall_subset_closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [609, 9], "def_end_pos": [609, 37]}, {"full_name": "mul_le_of_le_one_left", "def_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "def_pos": [666, 9], "def_end_pos": [666, 30]}]], "state_before": "case intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\nf : \u2115 \u2192 \u2115\nhf\u2081 : \u2200 (x : \u2115), p (f x)\nhf\u2082 : \u2200 (j : \u2115), j \u2264 f j\nhf\u2083 : Tendsto f atTop atTop\nhr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j)\nw : \u2115 \u2192 \u03b1\nhw : \u2200 (j : \u2115), w j \u2208 s (f j)\nhw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j))\nC : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9\nhC : C \u2260 0\nb : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (M * r\u2081 (f j))\nB : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (r\u2081 (f j))\n\u22a2 \u2200\u1da0 (j : \u2115) in atTop, \u2191\u2191\u03bc (W \u2229 closedBall (w j) (r\u2081 (f j))) / \u2191\u2191\u03bc (closedBall (w j) (r\u2081 (f j))) \u2264 \u2191(1 - C\u207b\u00b9)", "state_after": "case intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\nf : \u2115 \u2192 \u2115\nhf\u2081 : \u2200 (x : \u2115), p (f x)\nhf\u2082 : \u2200 (j : \u2115), j \u2264 f j\nhf\u2083 : Tendsto f atTop atTop\nhr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j)\nw : \u2115 \u2192 \u03b1\nhw : \u2200 (j : \u2115), w j \u2208 s (f j)\nhw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j))\nC : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9\nhC : C \u2260 0\nb : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (M * r\u2081 (f j))\nB : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (r\u2081 (f j))\nh\u2081 : \u2200 (j : \u2115), b j \u2286 B j\n\u22a2 \u2200\u1da0 (j : \u2115) in atTop, \u2191\u2191\u03bc (W \u2229 closedBall (w j) (r\u2081 (f j))) / \u2191\u2191\u03bc (closedBall (w j) (r\u2081 (f j))) \u2264 \u2191(1 - C\u207b\u00b9)"}, {"tactic": "have h\u2082 : \u2200 j, W \u2229 B j \u2286 B j := fun j => inter_subset_right W (B j)", "annotated_tactic": ["have h\u2082 : \u2200 j, W \u2229 B j \u2286 B j := fun j => <a>inter_subset_right</a> W (B j)", [{"full_name": "Set.inter_subset_right", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [969, 9], "def_end_pos": [969, 27]}]], "state_before": "case intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\nf : \u2115 \u2192 \u2115\nhf\u2081 : \u2200 (x : \u2115), p (f x)\nhf\u2082 : \u2200 (j : \u2115), j \u2264 f j\nhf\u2083 : Tendsto f atTop atTop\nhr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j)\nw : \u2115 \u2192 \u03b1\nhw : \u2200 (j : \u2115), w j \u2208 s (f j)\nhw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j))\nC : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9\nhC : C \u2260 0\nb : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (M * r\u2081 (f j))\nB : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (r\u2081 (f j))\nh\u2081 : \u2200 (j : \u2115), b j \u2286 B j\n\u22a2 \u2200\u1da0 (j : \u2115) in atTop, \u2191\u2191\u03bc (W \u2229 closedBall (w j) (r\u2081 (f j))) / \u2191\u2191\u03bc (closedBall (w j) (r\u2081 (f j))) \u2264 \u2191(1 - C\u207b\u00b9)", "state_after": "case intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\nf : \u2115 \u2192 \u2115\nhf\u2081 : \u2200 (x : \u2115), p (f x)\nhf\u2082 : \u2200 (j : \u2115), j \u2264 f j\nhf\u2083 : Tendsto f atTop atTop\nhr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j)\nw : \u2115 \u2192 \u03b1\nhw : \u2200 (j : \u2115), w j \u2208 s (f j)\nhw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j))\nC : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9\nhC : C \u2260 0\nb : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (M * r\u2081 (f j))\nB : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (r\u2081 (f j))\nh\u2081 : \u2200 (j : \u2115), b j \u2286 B j\nh\u2082 : \u2200 (j : \u2115), W \u2229 B j \u2286 B j\n\u22a2 \u2200\u1da0 (j : \u2115) in atTop, \u2191\u2191\u03bc (W \u2229 closedBall (w j) (r\u2081 (f j))) / \u2191\u2191\u03bc (closedBall (w j) (r\u2081 (f j))) \u2264 \u2191(1 - C\u207b\u00b9)"}, {"tactic": "have h\u2083 : \u2200\u1da0 j in atTop, Disjoint (b j) (W \u2229 B j) := by\n  apply hMr.mp\n  rw [eventually_atTop]\n  refine'\n    \u27e8i, fun j hj hj' => Disjoint.inf_right (B j) <| Disjoint.inf_right' (blimsup Y\u2081 atTop p) _\u27e9\n  change Disjoint (b j) (Z i)\u1d9c\n  rw [disjoint_compl_right_iff_subset]\n  refine' (closedBall_subset_cthickening (hw j) (M * r\u2081 (f j))).trans\n    ((cthickening_mono hj' _).trans fun a ha => _)\n  simp only [mem_iUnion, exists_prop]\n  exact \u27e8f j, \u27e8hf\u2081 j, hj.le.trans (hf\u2082 j)\u27e9, ha\u27e9", "annotated_tactic": ["have h\u2083 : \u2200\u1da0 j in <a>atTop</a>, <a>Disjoint</a> (b j) (W \u2229 B j) := by\n    apply hMr.mp\n    rw [<a>eventually_atTop</a>]\n    refine'\n      \u27e8i, fun j hj hj' => <a>Disjoint.inf_right</a> (B j) <| <a>Disjoint.inf_right'</a> (<a>blimsup</a> Y\u2081 <a>atTop</a> p) _\u27e9\n    change <a>Disjoint</a> (b j) (Z i)\u1d9c\n    rw [<a>disjoint_compl_right_iff_subset</a>]\n    refine' (<a>closedBall_subset_cthickening</a> (hw j) (M * r\u2081 (f j))).<a>trans</a>\n      ((<a>cthickening_mono</a> hj' _).<a>trans</a> fun a ha => _)\n    simp only [<a>mem_iUnion</a>, <a>exists_prop</a>]\n    exact \u27e8f j, \u27e8hf\u2081 j, hj.le.trans (hf\u2082 j)\u27e9, ha\u27e9", [{"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}, {"full_name": "Disjoint", "def_path": "Mathlib/Order/Disjoint.lean", "def_pos": [41, 5], "def_end_pos": [41, 13]}, {"full_name": "Filter.eventually_atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [178, 9], "def_end_pos": [178, 25]}, {"full_name": "Disjoint.inf_right", "def_path": "Mathlib/Order/Disjoint.lean", "def_pos": [160, 9], "def_end_pos": [160, 27]}, {"full_name": "Disjoint.inf_right'", "def_path": "Mathlib/Order/Disjoint.lean", "def_pos": [164, 9], "def_end_pos": [164, 28]}, {"full_name": "Filter.blimsup", "def_path": "Mathlib/Order/LiminfLimsup.lean", "def_pos": [432, 5], "def_end_pos": [432, 12]}, {"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}, {"full_name": "Disjoint", "def_path": "Mathlib/Order/Disjoint.lean", "def_pos": [41, 5], "def_end_pos": [41, 13]}, {"full_name": "Set.disjoint_compl_right_iff_subset", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1774, 9], "def_end_pos": [1774, 40]}, {"full_name": "Metric.closedBall_subset_cthickening", "def_path": "Mathlib/Topology/MetricSpace/HausdorffDistance.lean", "def_pos": [1415, 9], "def_end_pos": [1415, 38]}, {"full_name": "HasSubset.Subset.trans", "def_path": "Mathlib/Order/RelClasses.lean", "def_pos": [664, 7], "def_end_pos": [664, 29]}, {"full_name": "Metric.cthickening_mono", "def_path": "Mathlib/Topology/MetricSpace/HausdorffDistance.lean", "def_pos": [1090, 9], "def_end_pos": [1090, 25]}, {"full_name": "HasSubset.Subset.trans", "def_path": "Mathlib/Order/RelClasses.lean", "def_pos": [664, 7], "def_end_pos": [664, 29]}, {"full_name": "Set.mem_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [201, 9], "def_end_pos": [201, 19]}, {"full_name": "exists_prop", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [485, 17], "def_end_pos": [485, 28]}]], "state_before": "case intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\nf : \u2115 \u2192 \u2115\nhf\u2081 : \u2200 (x : \u2115), p (f x)\nhf\u2082 : \u2200 (j : \u2115), j \u2264 f j\nhf\u2083 : Tendsto f atTop atTop\nhr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j)\nw : \u2115 \u2192 \u03b1\nhw : \u2200 (j : \u2115), w j \u2208 s (f j)\nhw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j))\nC : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9\nhC : C \u2260 0\nb : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (M * r\u2081 (f j))\nB : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (r\u2081 (f j))\nh\u2081 : \u2200 (j : \u2115), b j \u2286 B j\nh\u2082 : \u2200 (j : \u2115), W \u2229 B j \u2286 B j\n\u22a2 \u2200\u1da0 (j : \u2115) in atTop, \u2191\u2191\u03bc (W \u2229 closedBall (w j) (r\u2081 (f j))) / \u2191\u2191\u03bc (closedBall (w j) (r\u2081 (f j))) \u2264 \u2191(1 - C\u207b\u00b9)", "state_after": "case intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\nf : \u2115 \u2192 \u2115\nhf\u2081 : \u2200 (x : \u2115), p (f x)\nhf\u2082 : \u2200 (j : \u2115), j \u2264 f j\nhf\u2083 : Tendsto f atTop atTop\nhr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j)\nw : \u2115 \u2192 \u03b1\nhw : \u2200 (j : \u2115), w j \u2208 s (f j)\nhw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j))\nC : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9\nhC : C \u2260 0\nb : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (M * r\u2081 (f j))\nB : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (r\u2081 (f j))\nh\u2081 : \u2200 (j : \u2115), b j \u2286 B j\nh\u2082 : \u2200 (j : \u2115), W \u2229 B j \u2286 B j\nh\u2083 : \u2200\u1da0 (j : \u2115) in atTop, Disjoint (b j) (W \u2229 B j)\n\u22a2 \u2200\u1da0 (j : \u2115) in atTop, \u2191\u2191\u03bc (W \u2229 closedBall (w j) (r\u2081 (f j))) / \u2191\u2191\u03bc (closedBall (w j) (r\u2081 (f j))) \u2264 \u2191(1 - C\u207b\u00b9)"}, {"tactic": "have h\u2084 : \u2200\u1da0 j in atTop, \u03bc (B j) \u2264 C * \u03bc (b j) :=\n  (hr.eventually (IsUnifLocDoublingMeasure.eventually_measure_le_scaling_constant_mul'\n    \u03bc M hM)).mono fun j hj => hj (w j)", "annotated_tactic": ["have h\u2084 : \u2200\u1da0 j in <a>atTop</a>, \u03bc (B j) \u2264 C * \u03bc (b j) :=\n    (hr.eventually (<a>IsUnifLocDoublingMeasure.eventually_measure_le_scaling_constant_mul'</a>\n      \u03bc M hM)).<a>mono</a> fun j hj => hj (w j)", [{"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}, {"full_name": "IsUnifLocDoublingMeasure.eventually_measure_le_scaling_constant_mul'", "def_path": "Mathlib/MeasureTheory/Measure/Doubling.lean", "def_pos": [142, 9], "def_end_pos": [142, 52]}, {"full_name": "Filter.Eventually.mono", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1140, 9], "def_end_pos": [1140, 24]}]], "state_before": "case intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\nf : \u2115 \u2192 \u2115\nhf\u2081 : \u2200 (x : \u2115), p (f x)\nhf\u2082 : \u2200 (j : \u2115), j \u2264 f j\nhf\u2083 : Tendsto f atTop atTop\nhr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j)\nw : \u2115 \u2192 \u03b1\nhw : \u2200 (j : \u2115), w j \u2208 s (f j)\nhw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j))\nC : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9\nhC : C \u2260 0\nb : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (M * r\u2081 (f j))\nB : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (r\u2081 (f j))\nh\u2081 : \u2200 (j : \u2115), b j \u2286 B j\nh\u2082 : \u2200 (j : \u2115), W \u2229 B j \u2286 B j\nh\u2083 : \u2200\u1da0 (j : \u2115) in atTop, Disjoint (b j) (W \u2229 B j)\n\u22a2 \u2200\u1da0 (j : \u2115) in atTop, \u2191\u2191\u03bc (W \u2229 closedBall (w j) (r\u2081 (f j))) / \u2191\u2191\u03bc (closedBall (w j) (r\u2081 (f j))) \u2264 \u2191(1 - C\u207b\u00b9)", "state_after": "case intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\nf : \u2115 \u2192 \u2115\nhf\u2081 : \u2200 (x : \u2115), p (f x)\nhf\u2082 : \u2200 (j : \u2115), j \u2264 f j\nhf\u2083 : Tendsto f atTop atTop\nhr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j)\nw : \u2115 \u2192 \u03b1\nhw : \u2200 (j : \u2115), w j \u2208 s (f j)\nhw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j))\nC : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9\nhC : C \u2260 0\nb : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (M * r\u2081 (f j))\nB : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (r\u2081 (f j))\nh\u2081 : \u2200 (j : \u2115), b j \u2286 B j\nh\u2082 : \u2200 (j : \u2115), W \u2229 B j \u2286 B j\nh\u2083 : \u2200\u1da0 (j : \u2115) in atTop, Disjoint (b j) (W \u2229 B j)\nh\u2084 : \u2200\u1da0 (j : \u2115) in atTop, \u2191\u2191\u03bc (B j) \u2264 \u2191C * \u2191\u2191\u03bc (b j)\n\u22a2 \u2200\u1da0 (j : \u2115) in atTop, \u2191\u2191\u03bc (W \u2229 closedBall (w j) (r\u2081 (f j))) / \u2191\u2191\u03bc (closedBall (w j) (r\u2081 (f j))) \u2264 \u2191(1 - C\u207b\u00b9)"}, {"tactic": "refine' (h\u2083.and h\u2084).mono fun j hj\u2080 => _", "annotated_tactic": ["refine' (h\u2083.and h\u2084).<a>mono</a> fun j hj\u2080 => _", [{"full_name": "Filter.Eventually.mono", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1140, 9], "def_end_pos": [1140, 24]}]], "state_before": "case intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\nf : \u2115 \u2192 \u2115\nhf\u2081 : \u2200 (x : \u2115), p (f x)\nhf\u2082 : \u2200 (j : \u2115), j \u2264 f j\nhf\u2083 : Tendsto f atTop atTop\nhr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j)\nw : \u2115 \u2192 \u03b1\nhw : \u2200 (j : \u2115), w j \u2208 s (f j)\nhw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j))\nC : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9\nhC : C \u2260 0\nb : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (M * r\u2081 (f j))\nB : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (r\u2081 (f j))\nh\u2081 : \u2200 (j : \u2115), b j \u2286 B j\nh\u2082 : \u2200 (j : \u2115), W \u2229 B j \u2286 B j\nh\u2083 : \u2200\u1da0 (j : \u2115) in atTop, Disjoint (b j) (W \u2229 B j)\nh\u2084 : \u2200\u1da0 (j : \u2115) in atTop, \u2191\u2191\u03bc (B j) \u2264 \u2191C * \u2191\u2191\u03bc (b j)\n\u22a2 \u2200\u1da0 (j : \u2115) in atTop, \u2191\u2191\u03bc (W \u2229 closedBall (w j) (r\u2081 (f j))) / \u2191\u2191\u03bc (closedBall (w j) (r\u2081 (f j))) \u2264 \u2191(1 - C\u207b\u00b9)", "state_after": "case intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\nf : \u2115 \u2192 \u2115\nhf\u2081 : \u2200 (x : \u2115), p (f x)\nhf\u2082 : \u2200 (j : \u2115), j \u2264 f j\nhf\u2083 : Tendsto f atTop atTop\nhr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j)\nw : \u2115 \u2192 \u03b1\nhw : \u2200 (j : \u2115), w j \u2208 s (f j)\nhw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j))\nC : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9\nhC : C \u2260 0\nb : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (M * r\u2081 (f j))\nB : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (r\u2081 (f j))\nh\u2081 : \u2200 (j : \u2115), b j \u2286 B j\nh\u2082 : \u2200 (j : \u2115), W \u2229 B j \u2286 B j\nh\u2083 : \u2200\u1da0 (j : \u2115) in atTop, Disjoint (b j) (W \u2229 B j)\nh\u2084 : \u2200\u1da0 (j : \u2115) in atTop, \u2191\u2191\u03bc (B j) \u2264 \u2191C * \u2191\u2191\u03bc (b j)\nj : \u2115\nhj\u2080 : Disjoint (b j) (W \u2229 B j) \u2227 \u2191\u2191\u03bc (B j) \u2264 \u2191C * \u2191\u2191\u03bc (b j)\n\u22a2 \u2191\u2191\u03bc (W \u2229 closedBall (w j) (r\u2081 (f j))) / \u2191\u2191\u03bc (closedBall (w j) (r\u2081 (f j))) \u2264 \u2191(1 - C\u207b\u00b9)"}, {"tactic": "change \u03bc (W \u2229 B j) / \u03bc (B j) \u2264 \u2191(1 - C\u207b\u00b9)", "annotated_tactic": ["change \u03bc (W \u2229 B j) / \u03bc (B j) \u2264 \u2191(1 - C\u207b\u00b9)", []], "state_before": "case intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\nf : \u2115 \u2192 \u2115\nhf\u2081 : \u2200 (x : \u2115), p (f x)\nhf\u2082 : \u2200 (j : \u2115), j \u2264 f j\nhf\u2083 : Tendsto f atTop atTop\nhr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j)\nw : \u2115 \u2192 \u03b1\nhw : \u2200 (j : \u2115), w j \u2208 s (f j)\nhw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j))\nC : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9\nhC : C \u2260 0\nb : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (M * r\u2081 (f j))\nB : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (r\u2081 (f j))\nh\u2081 : \u2200 (j : \u2115), b j \u2286 B j\nh\u2082 : \u2200 (j : \u2115), W \u2229 B j \u2286 B j\nh\u2083 : \u2200\u1da0 (j : \u2115) in atTop, Disjoint (b j) (W \u2229 B j)\nh\u2084 : \u2200\u1da0 (j : \u2115) in atTop, \u2191\u2191\u03bc (B j) \u2264 \u2191C * \u2191\u2191\u03bc (b j)\nj : \u2115\nhj\u2080 : Disjoint (b j) (W \u2229 B j) \u2227 \u2191\u2191\u03bc (B j) \u2264 \u2191C * \u2191\u2191\u03bc (b j)\n\u22a2 \u2191\u2191\u03bc (W \u2229 closedBall (w j) (r\u2081 (f j))) / \u2191\u2191\u03bc (closedBall (w j) (r\u2081 (f j))) \u2264 \u2191(1 - C\u207b\u00b9)", "state_after": "case intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\nf : \u2115 \u2192 \u2115\nhf\u2081 : \u2200 (x : \u2115), p (f x)\nhf\u2082 : \u2200 (j : \u2115), j \u2264 f j\nhf\u2083 : Tendsto f atTop atTop\nhr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j)\nw : \u2115 \u2192 \u03b1\nhw : \u2200 (j : \u2115), w j \u2208 s (f j)\nhw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j))\nC : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9\nhC : C \u2260 0\nb : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (M * r\u2081 (f j))\nB : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (r\u2081 (f j))\nh\u2081 : \u2200 (j : \u2115), b j \u2286 B j\nh\u2082 : \u2200 (j : \u2115), W \u2229 B j \u2286 B j\nh\u2083 : \u2200\u1da0 (j : \u2115) in atTop, Disjoint (b j) (W \u2229 B j)\nh\u2084 : \u2200\u1da0 (j : \u2115) in atTop, \u2191\u2191\u03bc (B j) \u2264 \u2191C * \u2191\u2191\u03bc (b j)\nj : \u2115\nhj\u2080 : Disjoint (b j) (W \u2229 B j) \u2227 \u2191\u2191\u03bc (B j) \u2264 \u2191C * \u2191\u2191\u03bc (b j)\n\u22a2 \u2191\u2191\u03bc (W \u2229 B j) / \u2191\u2191\u03bc (B j) \u2264 \u2191(1 - C\u207b\u00b9)"}, {"tactic": "rcases eq_or_ne (\u03bc (B j)) \u221e with (hB | hB)", "annotated_tactic": ["rcases <a>eq_or_ne</a> (\u03bc (B j)) \u221e with (hB | hB)", [{"full_name": "eq_or_ne", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [209, 9], "def_end_pos": [209, 17]}]], "state_before": "case intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\nf : \u2115 \u2192 \u2115\nhf\u2081 : \u2200 (x : \u2115), p (f x)\nhf\u2082 : \u2200 (j : \u2115), j \u2264 f j\nhf\u2083 : Tendsto f atTop atTop\nhr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j)\nw : \u2115 \u2192 \u03b1\nhw : \u2200 (j : \u2115), w j \u2208 s (f j)\nhw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j))\nC : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9\nhC : C \u2260 0\nb : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (M * r\u2081 (f j))\nB : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (r\u2081 (f j))\nh\u2081 : \u2200 (j : \u2115), b j \u2286 B j\nh\u2082 : \u2200 (j : \u2115), W \u2229 B j \u2286 B j\nh\u2083 : \u2200\u1da0 (j : \u2115) in atTop, Disjoint (b j) (W \u2229 B j)\nh\u2084 : \u2200\u1da0 (j : \u2115) in atTop, \u2191\u2191\u03bc (B j) \u2264 \u2191C * \u2191\u2191\u03bc (b j)\nj : \u2115\nhj\u2080 : Disjoint (b j) (W \u2229 B j) \u2227 \u2191\u2191\u03bc (B j) \u2264 \u2191C * \u2191\u2191\u03bc (b j)\n\u22a2 \u2191\u2191\u03bc (W \u2229 B j) / \u2191\u2191\u03bc (B j) \u2264 \u2191(1 - C\u207b\u00b9)", "state_after": "case intro.intro.intro.intro.intro.inl\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\nf : \u2115 \u2192 \u2115\nhf\u2081 : \u2200 (x : \u2115), p (f x)\nhf\u2082 : \u2200 (j : \u2115), j \u2264 f j\nhf\u2083 : Tendsto f atTop atTop\nhr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j)\nw : \u2115 \u2192 \u03b1\nhw : \u2200 (j : \u2115), w j \u2208 s (f j)\nhw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j))\nC : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9\nhC : C \u2260 0\nb : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (M * r\u2081 (f j))\nB : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (r\u2081 (f j))\nh\u2081 : \u2200 (j : \u2115), b j \u2286 B j\nh\u2082 : \u2200 (j : \u2115), W \u2229 B j \u2286 B j\nh\u2083 : \u2200\u1da0 (j : \u2115) in atTop, Disjoint (b j) (W \u2229 B j)\nh\u2084 : \u2200\u1da0 (j : \u2115) in atTop, \u2191\u2191\u03bc (B j) \u2264 \u2191C * \u2191\u2191\u03bc (b j)\nj : \u2115\nhj\u2080 : Disjoint (b j) (W \u2229 B j) \u2227 \u2191\u2191\u03bc (B j) \u2264 \u2191C * \u2191\u2191\u03bc (b j)\nhB : \u2191\u2191\u03bc (B j) = \u22a4\n\u22a2 \u2191\u2191\u03bc (W \u2229 B j) / \u2191\u2191\u03bc (B j) \u2264 \u2191(1 - C\u207b\u00b9)\n\ncase intro.intro.intro.intro.intro.inr\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\nf : \u2115 \u2192 \u2115\nhf\u2081 : \u2200 (x : \u2115), p (f x)\nhf\u2082 : \u2200 (j : \u2115), j \u2264 f j\nhf\u2083 : Tendsto f atTop atTop\nhr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j)\nw : \u2115 \u2192 \u03b1\nhw : \u2200 (j : \u2115), w j \u2208 s (f j)\nhw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j))\nC : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9\nhC : C \u2260 0\nb : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (M * r\u2081 (f j))\nB : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (r\u2081 (f j))\nh\u2081 : \u2200 (j : \u2115), b j \u2286 B j\nh\u2082 : \u2200 (j : \u2115), W \u2229 B j \u2286 B j\nh\u2083 : \u2200\u1da0 (j : \u2115) in atTop, Disjoint (b j) (W \u2229 B j)\nh\u2084 : \u2200\u1da0 (j : \u2115) in atTop, \u2191\u2191\u03bc (B j) \u2264 \u2191C * \u2191\u2191\u03bc (b j)\nj : \u2115\nhj\u2080 : Disjoint (b j) (W \u2229 B j) \u2227 \u2191\u2191\u03bc (B j) \u2264 \u2191C * \u2191\u2191\u03bc (b j)\nhB : \u2191\u2191\u03bc (B j) \u2260 \u22a4\n\u22a2 \u2191\u2191\u03bc (W \u2229 B j) / \u2191\u2191\u03bc (B j) \u2264 \u2191(1 - C\u207b\u00b9)"}, {"tactic": "apply ENNReal.div_le_of_le_mul", "annotated_tactic": ["apply <a>ENNReal.div_le_of_le_mul</a>", [{"full_name": "ENNReal.div_le_of_le_mul", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1635, 9], "def_end_pos": [1635, 25]}]], "state_before": "case intro.intro.intro.intro.intro.inr\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\nf : \u2115 \u2192 \u2115\nhf\u2081 : \u2200 (x : \u2115), p (f x)\nhf\u2082 : \u2200 (j : \u2115), j \u2264 f j\nhf\u2083 : Tendsto f atTop atTop\nhr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j)\nw : \u2115 \u2192 \u03b1\nhw : \u2200 (j : \u2115), w j \u2208 s (f j)\nhw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j))\nC : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9\nhC : C \u2260 0\nb : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (M * r\u2081 (f j))\nB : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (r\u2081 (f j))\nh\u2081 : \u2200 (j : \u2115), b j \u2286 B j\nh\u2082 : \u2200 (j : \u2115), W \u2229 B j \u2286 B j\nh\u2083 : \u2200\u1da0 (j : \u2115) in atTop, Disjoint (b j) (W \u2229 B j)\nh\u2084 : \u2200\u1da0 (j : \u2115) in atTop, \u2191\u2191\u03bc (B j) \u2264 \u2191C * \u2191\u2191\u03bc (b j)\nj : \u2115\nhj\u2080 : Disjoint (b j) (W \u2229 B j) \u2227 \u2191\u2191\u03bc (B j) \u2264 \u2191C * \u2191\u2191\u03bc (b j)\nhB : \u2191\u2191\u03bc (B j) \u2260 \u22a4\n\u22a2 \u2191\u2191\u03bc (W \u2229 B j) / \u2191\u2191\u03bc (B j) \u2264 \u2191(1 - C\u207b\u00b9)", "state_after": "case intro.intro.intro.intro.intro.inr.h\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\nf : \u2115 \u2192 \u2115\nhf\u2081 : \u2200 (x : \u2115), p (f x)\nhf\u2082 : \u2200 (j : \u2115), j \u2264 f j\nhf\u2083 : Tendsto f atTop atTop\nhr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j)\nw : \u2115 \u2192 \u03b1\nhw : \u2200 (j : \u2115), w j \u2208 s (f j)\nhw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j))\nC : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9\nhC : C \u2260 0\nb : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (M * r\u2081 (f j))\nB : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (r\u2081 (f j))\nh\u2081 : \u2200 (j : \u2115), b j \u2286 B j\nh\u2082 : \u2200 (j : \u2115), W \u2229 B j \u2286 B j\nh\u2083 : \u2200\u1da0 (j : \u2115) in atTop, Disjoint (b j) (W \u2229 B j)\nh\u2084 : \u2200\u1da0 (j : \u2115) in atTop, \u2191\u2191\u03bc (B j) \u2264 \u2191C * \u2191\u2191\u03bc (b j)\nj : \u2115\nhj\u2080 : Disjoint (b j) (W \u2229 B j) \u2227 \u2191\u2191\u03bc (B j) \u2264 \u2191C * \u2191\u2191\u03bc (b j)\nhB : \u2191\u2191\u03bc (B j) \u2260 \u22a4\n\u22a2 \u2191\u2191\u03bc (W \u2229 B j) \u2264 \u2191(1 - C\u207b\u00b9) * \u2191\u2191\u03bc (B j)"}, {"tactic": "rw [ENNReal.coe_sub, ENNReal.coe_one, ENNReal.sub_mul fun _ _ => hB, one_mul]", "annotated_tactic": ["rw [<a>ENNReal.coe_sub</a>, <a>ENNReal.coe_one</a>, <a>ENNReal.sub_mul</a> fun _ _ => hB, <a>one_mul</a>]", [{"full_name": "ENNReal.coe_sub", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1137, 17], "def_end_pos": [1137, 24]}, {"full_name": "ENNReal.coe_one", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [218, 28], "def_end_pos": [218, 35]}, {"full_name": "ENNReal.sub_mul", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1235, 9], "def_end_pos": [1235, 16]}, {"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [464, 9], "def_end_pos": [464, 16]}]], "state_before": "case intro.intro.intro.intro.intro.inr.h\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\nf : \u2115 \u2192 \u2115\nhf\u2081 : \u2200 (x : \u2115), p (f x)\nhf\u2082 : \u2200 (j : \u2115), j \u2264 f j\nhf\u2083 : Tendsto f atTop atTop\nhr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j)\nw : \u2115 \u2192 \u03b1\nhw : \u2200 (j : \u2115), w j \u2208 s (f j)\nhw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j))\nC : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9\nhC : C \u2260 0\nb : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (M * r\u2081 (f j))\nB : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (r\u2081 (f j))\nh\u2081 : \u2200 (j : \u2115), b j \u2286 B j\nh\u2082 : \u2200 (j : \u2115), W \u2229 B j \u2286 B j\nh\u2083 : \u2200\u1da0 (j : \u2115) in atTop, Disjoint (b j) (W \u2229 B j)\nh\u2084 : \u2200\u1da0 (j : \u2115) in atTop, \u2191\u2191\u03bc (B j) \u2264 \u2191C * \u2191\u2191\u03bc (b j)\nj : \u2115\nhj\u2080 : Disjoint (b j) (W \u2229 B j) \u2227 \u2191\u2191\u03bc (B j) \u2264 \u2191C * \u2191\u2191\u03bc (b j)\nhB : \u2191\u2191\u03bc (B j) \u2260 \u22a4\n\u22a2 \u2191\u2191\u03bc (W \u2229 B j) \u2264 \u2191(1 - C\u207b\u00b9) * \u2191\u2191\u03bc (B j)", "state_after": "case intro.intro.intro.intro.intro.inr.h\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\nf : \u2115 \u2192 \u2115\nhf\u2081 : \u2200 (x : \u2115), p (f x)\nhf\u2082 : \u2200 (j : \u2115), j \u2264 f j\nhf\u2083 : Tendsto f atTop atTop\nhr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j)\nw : \u2115 \u2192 \u03b1\nhw : \u2200 (j : \u2115), w j \u2208 s (f j)\nhw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j))\nC : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9\nhC : C \u2260 0\nb : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (M * r\u2081 (f j))\nB : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (r\u2081 (f j))\nh\u2081 : \u2200 (j : \u2115), b j \u2286 B j\nh\u2082 : \u2200 (j : \u2115), W \u2229 B j \u2286 B j\nh\u2083 : \u2200\u1da0 (j : \u2115) in atTop, Disjoint (b j) (W \u2229 B j)\nh\u2084 : \u2200\u1da0 (j : \u2115) in atTop, \u2191\u2191\u03bc (B j) \u2264 \u2191C * \u2191\u2191\u03bc (b j)\nj : \u2115\nhj\u2080 : Disjoint (b j) (W \u2229 B j) \u2227 \u2191\u2191\u03bc (B j) \u2264 \u2191C * \u2191\u2191\u03bc (b j)\nhB : \u2191\u2191\u03bc (B j) \u2260 \u22a4\n\u22a2 \u2191\u2191\u03bc (W \u2229 B j) \u2264 \u2191\u2191\u03bc (B j) - \u2191C\u207b\u00b9 * \u2191\u2191\u03bc (B j)"}, {"tactic": "replace hB : \u2191C\u207b\u00b9 * \u03bc (B j) \u2260 \u221e", "annotated_tactic": ["replace hB : \u2191C\u207b\u00b9 * \u03bc (B j) \u2260 \u221e", []], "state_before": "case intro.intro.intro.intro.intro.inr.h\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\nf : \u2115 \u2192 \u2115\nhf\u2081 : \u2200 (x : \u2115), p (f x)\nhf\u2082 : \u2200 (j : \u2115), j \u2264 f j\nhf\u2083 : Tendsto f atTop atTop\nhr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j)\nw : \u2115 \u2192 \u03b1\nhw : \u2200 (j : \u2115), w j \u2208 s (f j)\nhw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j))\nC : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9\nhC : C \u2260 0\nb : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (M * r\u2081 (f j))\nB : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (r\u2081 (f j))\nh\u2081 : \u2200 (j : \u2115), b j \u2286 B j\nh\u2082 : \u2200 (j : \u2115), W \u2229 B j \u2286 B j\nh\u2083 : \u2200\u1da0 (j : \u2115) in atTop, Disjoint (b j) (W \u2229 B j)\nh\u2084 : \u2200\u1da0 (j : \u2115) in atTop, \u2191\u2191\u03bc (B j) \u2264 \u2191C * \u2191\u2191\u03bc (b j)\nj : \u2115\nhj\u2080 : Disjoint (b j) (W \u2229 B j) \u2227 \u2191\u2191\u03bc (B j) \u2264 \u2191C * \u2191\u2191\u03bc (b j)\nhB : \u2191\u2191\u03bc (B j) \u2260 \u22a4\n\u22a2 \u2191\u2191\u03bc (W \u2229 B j) \u2264 \u2191\u2191\u03bc (B j) - \u2191C\u207b\u00b9 * \u2191\u2191\u03bc (B j)", "state_after": "case hB\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\nf : \u2115 \u2192 \u2115\nhf\u2081 : \u2200 (x : \u2115), p (f x)\nhf\u2082 : \u2200 (j : \u2115), j \u2264 f j\nhf\u2083 : Tendsto f atTop atTop\nhr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j)\nw : \u2115 \u2192 \u03b1\nhw : \u2200 (j : \u2115), w j \u2208 s (f j)\nhw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j))\nC : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9\nhC : C \u2260 0\nb : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (M * r\u2081 (f j))\nB : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (r\u2081 (f j))\nh\u2081 : \u2200 (j : \u2115), b j \u2286 B j\nh\u2082 : \u2200 (j : \u2115), W \u2229 B j \u2286 B j\nh\u2083 : \u2200\u1da0 (j : \u2115) in atTop, Disjoint (b j) (W \u2229 B j)\nh\u2084 : \u2200\u1da0 (j : \u2115) in atTop, \u2191\u2191\u03bc (B j) \u2264 \u2191C * \u2191\u2191\u03bc (b j)\nj : \u2115\nhj\u2080 : Disjoint (b j) (W \u2229 B j) \u2227 \u2191\u2191\u03bc (B j) \u2264 \u2191C * \u2191\u2191\u03bc (b j)\nhB : \u2191\u2191\u03bc (B j) \u2260 \u22a4\n\u22a2 \u2191C\u207b\u00b9 * \u2191\u2191\u03bc (B j) \u2260 \u22a4\n\ncase intro.intro.intro.intro.intro.inr.h\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\nf : \u2115 \u2192 \u2115\nhf\u2081 : \u2200 (x : \u2115), p (f x)\nhf\u2082 : \u2200 (j : \u2115), j \u2264 f j\nhf\u2083 : Tendsto f atTop atTop\nhr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j)\nw : \u2115 \u2192 \u03b1\nhw : \u2200 (j : \u2115), w j \u2208 s (f j)\nhw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j))\nC : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9\nhC : C \u2260 0\nb : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (M * r\u2081 (f j))\nB : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (r\u2081 (f j))\nh\u2081 : \u2200 (j : \u2115), b j \u2286 B j\nh\u2082 : \u2200 (j : \u2115), W \u2229 B j \u2286 B j\nh\u2083 : \u2200\u1da0 (j : \u2115) in atTop, Disjoint (b j) (W \u2229 B j)\nh\u2084 : \u2200\u1da0 (j : \u2115) in atTop, \u2191\u2191\u03bc (B j) \u2264 \u2191C * \u2191\u2191\u03bc (b j)\nj : \u2115\nhj\u2080 : Disjoint (b j) (W \u2229 B j) \u2227 \u2191\u2191\u03bc (B j) \u2264 \u2191C * \u2191\u2191\u03bc (b j)\nhB : \u2191C\u207b\u00b9 * \u2191\u2191\u03bc (B j) \u2260 \u22a4\n\u22a2 \u2191\u2191\u03bc (W \u2229 B j) \u2264 \u2191\u2191\u03bc (B j) - \u2191C\u207b\u00b9 * \u2191\u2191\u03bc (B j)"}, {"tactic": "obtain \u27e8hj\u2081 : Disjoint (b j) (W \u2229 B j), hj\u2082 : \u03bc (B j) \u2264 C * \u03bc (b j)\u27e9 := hj\u2080", "annotated_tactic": ["obtain \u27e8hj\u2081 : <a>Disjoint</a> (b j) (W \u2229 B j), hj\u2082 : \u03bc (B j) \u2264 C * \u03bc (b j)\u27e9 := hj\u2080", [{"full_name": "Disjoint", "def_path": "Mathlib/Order/Disjoint.lean", "def_pos": [41, 5], "def_end_pos": [41, 13]}]], "state_before": "case intro.intro.intro.intro.intro.inr.h\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\nf : \u2115 \u2192 \u2115\nhf\u2081 : \u2200 (x : \u2115), p (f x)\nhf\u2082 : \u2200 (j : \u2115), j \u2264 f j\nhf\u2083 : Tendsto f atTop atTop\nhr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j)\nw : \u2115 \u2192 \u03b1\nhw : \u2200 (j : \u2115), w j \u2208 s (f j)\nhw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j))\nC : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9\nhC : C \u2260 0\nb : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (M * r\u2081 (f j))\nB : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (r\u2081 (f j))\nh\u2081 : \u2200 (j : \u2115), b j \u2286 B j\nh\u2082 : \u2200 (j : \u2115), W \u2229 B j \u2286 B j\nh\u2083 : \u2200\u1da0 (j : \u2115) in atTop, Disjoint (b j) (W \u2229 B j)\nh\u2084 : \u2200\u1da0 (j : \u2115) in atTop, \u2191\u2191\u03bc (B j) \u2264 \u2191C * \u2191\u2191\u03bc (b j)\nj : \u2115\nhj\u2080 : Disjoint (b j) (W \u2229 B j) \u2227 \u2191\u2191\u03bc (B j) \u2264 \u2191C * \u2191\u2191\u03bc (b j)\nhB : \u2191C\u207b\u00b9 * \u2191\u2191\u03bc (B j) \u2260 \u22a4\n\u22a2 \u2191\u2191\u03bc (W \u2229 B j) \u2264 \u2191\u2191\u03bc (B j) - \u2191C\u207b\u00b9 * \u2191\u2191\u03bc (B j)", "state_after": "case intro.intro.intro.intro.intro.inr.h.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\nf : \u2115 \u2192 \u2115\nhf\u2081 : \u2200 (x : \u2115), p (f x)\nhf\u2082 : \u2200 (j : \u2115), j \u2264 f j\nhf\u2083 : Tendsto f atTop atTop\nhr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j)\nw : \u2115 \u2192 \u03b1\nhw : \u2200 (j : \u2115), w j \u2208 s (f j)\nhw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j))\nC : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9\nhC : C \u2260 0\nb : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (M * r\u2081 (f j))\nB : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (r\u2081 (f j))\nh\u2081 : \u2200 (j : \u2115), b j \u2286 B j\nh\u2082 : \u2200 (j : \u2115), W \u2229 B j \u2286 B j\nh\u2083 : \u2200\u1da0 (j : \u2115) in atTop, Disjoint (b j) (W \u2229 B j)\nh\u2084 : \u2200\u1da0 (j : \u2115) in atTop, \u2191\u2191\u03bc (B j) \u2264 \u2191C * \u2191\u2191\u03bc (b j)\nj : \u2115\nhB : \u2191C\u207b\u00b9 * \u2191\u2191\u03bc (B j) \u2260 \u22a4\nhj\u2081 : Disjoint (b j) (W \u2229 B j)\nhj\u2082 : \u2191\u2191\u03bc (B j) \u2264 \u2191C * \u2191\u2191\u03bc (b j)\n\u22a2 \u2191\u2191\u03bc (W \u2229 B j) \u2264 \u2191\u2191\u03bc (B j) - \u2191C\u207b\u00b9 * \u2191\u2191\u03bc (B j)"}, {"tactic": "replace hj\u2082 : \u2191C\u207b\u00b9 * \u03bc (B j) \u2264 \u03bc (b j)", "annotated_tactic": ["replace hj\u2082 : \u2191C\u207b\u00b9 * \u03bc (B j) \u2264 \u03bc (b j)", []], "state_before": "case intro.intro.intro.intro.intro.inr.h.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\nf : \u2115 \u2192 \u2115\nhf\u2081 : \u2200 (x : \u2115), p (f x)\nhf\u2082 : \u2200 (j : \u2115), j \u2264 f j\nhf\u2083 : Tendsto f atTop atTop\nhr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j)\nw : \u2115 \u2192 \u03b1\nhw : \u2200 (j : \u2115), w j \u2208 s (f j)\nhw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j))\nC : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9\nhC : C \u2260 0\nb : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (M * r\u2081 (f j))\nB : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (r\u2081 (f j))\nh\u2081 : \u2200 (j : \u2115), b j \u2286 B j\nh\u2082 : \u2200 (j : \u2115), W \u2229 B j \u2286 B j\nh\u2083 : \u2200\u1da0 (j : \u2115) in atTop, Disjoint (b j) (W \u2229 B j)\nh\u2084 : \u2200\u1da0 (j : \u2115) in atTop, \u2191\u2191\u03bc (B j) \u2264 \u2191C * \u2191\u2191\u03bc (b j)\nj : \u2115\nhB : \u2191C\u207b\u00b9 * \u2191\u2191\u03bc (B j) \u2260 \u22a4\nhj\u2081 : Disjoint (b j) (W \u2229 B j)\nhj\u2082 : \u2191\u2191\u03bc (B j) \u2264 \u2191C * \u2191\u2191\u03bc (b j)\n\u22a2 \u2191\u2191\u03bc (W \u2229 B j) \u2264 \u2191\u2191\u03bc (B j) - \u2191C\u207b\u00b9 * \u2191\u2191\u03bc (B j)", "state_after": "case hj\u2082\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\nf : \u2115 \u2192 \u2115\nhf\u2081 : \u2200 (x : \u2115), p (f x)\nhf\u2082 : \u2200 (j : \u2115), j \u2264 f j\nhf\u2083 : Tendsto f atTop atTop\nhr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j)\nw : \u2115 \u2192 \u03b1\nhw : \u2200 (j : \u2115), w j \u2208 s (f j)\nhw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j))\nC : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9\nhC : C \u2260 0\nb : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (M * r\u2081 (f j))\nB : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (r\u2081 (f j))\nh\u2081 : \u2200 (j : \u2115), b j \u2286 B j\nh\u2082 : \u2200 (j : \u2115), W \u2229 B j \u2286 B j\nh\u2083 : \u2200\u1da0 (j : \u2115) in atTop, Disjoint (b j) (W \u2229 B j)\nh\u2084 : \u2200\u1da0 (j : \u2115) in atTop, \u2191\u2191\u03bc (B j) \u2264 \u2191C * \u2191\u2191\u03bc (b j)\nj : \u2115\nhB : \u2191C\u207b\u00b9 * \u2191\u2191\u03bc (B j) \u2260 \u22a4\nhj\u2081 : Disjoint (b j) (W \u2229 B j)\nhj\u2082 : \u2191\u2191\u03bc (B j) \u2264 \u2191C * \u2191\u2191\u03bc (b j)\n\u22a2 \u2191C\u207b\u00b9 * \u2191\u2191\u03bc (B j) \u2264 \u2191\u2191\u03bc (b j)\n\ncase intro.intro.intro.intro.intro.inr.h.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\nf : \u2115 \u2192 \u2115\nhf\u2081 : \u2200 (x : \u2115), p (f x)\nhf\u2082 : \u2200 (j : \u2115), j \u2264 f j\nhf\u2083 : Tendsto f atTop atTop\nhr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j)\nw : \u2115 \u2192 \u03b1\nhw : \u2200 (j : \u2115), w j \u2208 s (f j)\nhw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j))\nC : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9\nhC : C \u2260 0\nb : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (M * r\u2081 (f j))\nB : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (r\u2081 (f j))\nh\u2081 : \u2200 (j : \u2115), b j \u2286 B j\nh\u2082 : \u2200 (j : \u2115), W \u2229 B j \u2286 B j\nh\u2083 : \u2200\u1da0 (j : \u2115) in atTop, Disjoint (b j) (W \u2229 B j)\nh\u2084 : \u2200\u1da0 (j : \u2115) in atTop, \u2191\u2191\u03bc (B j) \u2264 \u2191C * \u2191\u2191\u03bc (b j)\nj : \u2115\nhB : \u2191C\u207b\u00b9 * \u2191\u2191\u03bc (B j) \u2260 \u22a4\nhj\u2081 : Disjoint (b j) (W \u2229 B j)\nhj\u2082 : \u2191C\u207b\u00b9 * \u2191\u2191\u03bc (B j) \u2264 \u2191\u2191\u03bc (b j)\n\u22a2 \u2191\u2191\u03bc (W \u2229 B j) \u2264 \u2191\u2191\u03bc (B j) - \u2191C\u207b\u00b9 * \u2191\u2191\u03bc (B j)"}, {"tactic": "have hj\u2083 : \u2191C\u207b\u00b9 * \u03bc (B j) + \u03bc (W \u2229 B j) \u2264 \u03bc (B j) := by\n  refine' le_trans (add_le_add_right hj\u2082 _) _\n  rw [\u2190 measure_union' hj\u2081 measurableSet_closedBall]\n  exact measure_mono (union_subset (h\u2081 j) (h\u2082 j))", "annotated_tactic": ["have hj\u2083 : \u2191C\u207b\u00b9 * \u03bc (B j) + \u03bc (W \u2229 B j) \u2264 \u03bc (B j) := by\n    refine' <a>le_trans</a> (<a>add_le_add_right</a> hj\u2082 _) _\n    rw [\u2190 <a>measure_union'</a> hj\u2081 <a>measurableSet_closedBall</a>]\n    exact <a>measure_mono</a> (<a>union_subset</a> (h\u2081 j) (h\u2082 j))", [{"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}, {"full_name": "add_le_add_right", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [66, 15], "def_end_pos": [66, 31]}, {"full_name": "MeasureTheory.measure_union'", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [128, 9], "def_end_pos": [128, 23]}, {"full_name": "measurableSet_closedBall", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [1681, 9], "def_end_pos": [1681, 33]}, {"full_name": "MeasureTheory.measure_mono", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [193, 9], "def_end_pos": [193, 21]}, {"full_name": "Set.union_subset", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [836, 9], "def_end_pos": [836, 21]}]], "state_before": "case intro.intro.intro.intro.intro.inr.h.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\nf : \u2115 \u2192 \u2115\nhf\u2081 : \u2200 (x : \u2115), p (f x)\nhf\u2082 : \u2200 (j : \u2115), j \u2264 f j\nhf\u2083 : Tendsto f atTop atTop\nhr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j)\nw : \u2115 \u2192 \u03b1\nhw : \u2200 (j : \u2115), w j \u2208 s (f j)\nhw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j))\nC : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9\nhC : C \u2260 0\nb : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (M * r\u2081 (f j))\nB : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (r\u2081 (f j))\nh\u2081 : \u2200 (j : \u2115), b j \u2286 B j\nh\u2082 : \u2200 (j : \u2115), W \u2229 B j \u2286 B j\nh\u2083 : \u2200\u1da0 (j : \u2115) in atTop, Disjoint (b j) (W \u2229 B j)\nh\u2084 : \u2200\u1da0 (j : \u2115) in atTop, \u2191\u2191\u03bc (B j) \u2264 \u2191C * \u2191\u2191\u03bc (b j)\nj : \u2115\nhB : \u2191C\u207b\u00b9 * \u2191\u2191\u03bc (B j) \u2260 \u22a4\nhj\u2081 : Disjoint (b j) (W \u2229 B j)\nhj\u2082 : \u2191C\u207b\u00b9 * \u2191\u2191\u03bc (B j) \u2264 \u2191\u2191\u03bc (b j)\n\u22a2 \u2191\u2191\u03bc (W \u2229 B j) \u2264 \u2191\u2191\u03bc (B j) - \u2191C\u207b\u00b9 * \u2191\u2191\u03bc (B j)", "state_after": "case intro.intro.intro.intro.intro.inr.h.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\nf : \u2115 \u2192 \u2115\nhf\u2081 : \u2200 (x : \u2115), p (f x)\nhf\u2082 : \u2200 (j : \u2115), j \u2264 f j\nhf\u2083 : Tendsto f atTop atTop\nhr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j)\nw : \u2115 \u2192 \u03b1\nhw : \u2200 (j : \u2115), w j \u2208 s (f j)\nhw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j))\nC : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9\nhC : C \u2260 0\nb : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (M * r\u2081 (f j))\nB : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (r\u2081 (f j))\nh\u2081 : \u2200 (j : \u2115), b j \u2286 B j\nh\u2082 : \u2200 (j : \u2115), W \u2229 B j \u2286 B j\nh\u2083 : \u2200\u1da0 (j : \u2115) in atTop, Disjoint (b j) (W \u2229 B j)\nh\u2084 : \u2200\u1da0 (j : \u2115) in atTop, \u2191\u2191\u03bc (B j) \u2264 \u2191C * \u2191\u2191\u03bc (b j)\nj : \u2115\nhB : \u2191C\u207b\u00b9 * \u2191\u2191\u03bc (B j) \u2260 \u22a4\nhj\u2081 : Disjoint (b j) (W \u2229 B j)\nhj\u2082 : \u2191C\u207b\u00b9 * \u2191\u2191\u03bc (B j) \u2264 \u2191\u2191\u03bc (b j)\nhj\u2083 : \u2191C\u207b\u00b9 * \u2191\u2191\u03bc (B j) + \u2191\u2191\u03bc (W \u2229 B j) \u2264 \u2191\u2191\u03bc (B j)\n\u22a2 \u2191\u2191\u03bc (W \u2229 B j) \u2264 \u2191\u2191\u03bc (B j) - \u2191C\u207b\u00b9 * \u2191\u2191\u03bc (B j)"}, {"tactic": "replace hj\u2083 := tsub_le_tsub_right hj\u2083 (\u2191C\u207b\u00b9 * \u03bc (B j))", "annotated_tactic": ["replace hj\u2083 := <a>tsub_le_tsub_right</a> hj\u2083 (\u2191C\u207b\u00b9 * \u03bc (B j))", [{"full_name": "tsub_le_tsub_right", "def_path": "Mathlib/Algebra/Order/Sub/Defs.lean", "def_pos": [106, 9], "def_end_pos": [106, 27]}]], "state_before": "case intro.intro.intro.intro.intro.inr.h.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\nf : \u2115 \u2192 \u2115\nhf\u2081 : \u2200 (x : \u2115), p (f x)\nhf\u2082 : \u2200 (j : \u2115), j \u2264 f j\nhf\u2083 : Tendsto f atTop atTop\nhr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j)\nw : \u2115 \u2192 \u03b1\nhw : \u2200 (j : \u2115), w j \u2208 s (f j)\nhw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j))\nC : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9\nhC : C \u2260 0\nb : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (M * r\u2081 (f j))\nB : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (r\u2081 (f j))\nh\u2081 : \u2200 (j : \u2115), b j \u2286 B j\nh\u2082 : \u2200 (j : \u2115), W \u2229 B j \u2286 B j\nh\u2083 : \u2200\u1da0 (j : \u2115) in atTop, Disjoint (b j) (W \u2229 B j)\nh\u2084 : \u2200\u1da0 (j : \u2115) in atTop, \u2191\u2191\u03bc (B j) \u2264 \u2191C * \u2191\u2191\u03bc (b j)\nj : \u2115\nhB : \u2191C\u207b\u00b9 * \u2191\u2191\u03bc (B j) \u2260 \u22a4\nhj\u2081 : Disjoint (b j) (W \u2229 B j)\nhj\u2082 : \u2191C\u207b\u00b9 * \u2191\u2191\u03bc (B j) \u2264 \u2191\u2191\u03bc (b j)\nhj\u2083 : \u2191C\u207b\u00b9 * \u2191\u2191\u03bc (B j) + \u2191\u2191\u03bc (W \u2229 B j) \u2264 \u2191\u2191\u03bc (B j)\n\u22a2 \u2191\u2191\u03bc (W \u2229 B j) \u2264 \u2191\u2191\u03bc (B j) - \u2191C\u207b\u00b9 * \u2191\u2191\u03bc (B j)", "state_after": "case intro.intro.intro.intro.intro.inr.h.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\nf : \u2115 \u2192 \u2115\nhf\u2081 : \u2200 (x : \u2115), p (f x)\nhf\u2082 : \u2200 (j : \u2115), j \u2264 f j\nhf\u2083 : Tendsto f atTop atTop\nhr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j)\nw : \u2115 \u2192 \u03b1\nhw : \u2200 (j : \u2115), w j \u2208 s (f j)\nhw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j))\nC : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9\nhC : C \u2260 0\nb : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (M * r\u2081 (f j))\nB : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (r\u2081 (f j))\nh\u2081 : \u2200 (j : \u2115), b j \u2286 B j\nh\u2082 : \u2200 (j : \u2115), W \u2229 B j \u2286 B j\nh\u2083 : \u2200\u1da0 (j : \u2115) in atTop, Disjoint (b j) (W \u2229 B j)\nh\u2084 : \u2200\u1da0 (j : \u2115) in atTop, \u2191\u2191\u03bc (B j) \u2264 \u2191C * \u2191\u2191\u03bc (b j)\nj : \u2115\nhB : \u2191C\u207b\u00b9 * \u2191\u2191\u03bc (B j) \u2260 \u22a4\nhj\u2081 : Disjoint (b j) (W \u2229 B j)\nhj\u2082 : \u2191C\u207b\u00b9 * \u2191\u2191\u03bc (B j) \u2264 \u2191\u2191\u03bc (b j)\nhj\u2083 : \u2191C\u207b\u00b9 * \u2191\u2191\u03bc (B j) + \u2191\u2191\u03bc (W \u2229 B j) - \u2191C\u207b\u00b9 * \u2191\u2191\u03bc (B j) \u2264 \u2191\u2191\u03bc (B j) - \u2191C\u207b\u00b9 * \u2191\u2191\u03bc (B j)\n\u22a2 \u2191\u2191\u03bc (W \u2229 B j) \u2264 \u2191\u2191\u03bc (B j) - \u2191C\u207b\u00b9 * \u2191\u2191\u03bc (B j)"}, {"tactic": "rwa [ENNReal.add_sub_cancel_left hB] at hj\u2083", "annotated_tactic": ["rwa [<a>ENNReal.add_sub_cancel_left</a> hB] at hj\u2083", [{"full_name": "ENNReal.add_sub_cancel_left", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1177, 19], "def_end_pos": [1177, 38]}]], "state_before": "case intro.intro.intro.intro.intro.inr.h.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\nf : \u2115 \u2192 \u2115\nhf\u2081 : \u2200 (x : \u2115), p (f x)\nhf\u2082 : \u2200 (j : \u2115), j \u2264 f j\nhf\u2083 : Tendsto f atTop atTop\nhr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j)\nw : \u2115 \u2192 \u03b1\nhw : \u2200 (j : \u2115), w j \u2208 s (f j)\nhw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j))\nC : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9\nhC : C \u2260 0\nb : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (M * r\u2081 (f j))\nB : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (r\u2081 (f j))\nh\u2081 : \u2200 (j : \u2115), b j \u2286 B j\nh\u2082 : \u2200 (j : \u2115), W \u2229 B j \u2286 B j\nh\u2083 : \u2200\u1da0 (j : \u2115) in atTop, Disjoint (b j) (W \u2229 B j)\nh\u2084 : \u2200\u1da0 (j : \u2115) in atTop, \u2191\u2191\u03bc (B j) \u2264 \u2191C * \u2191\u2191\u03bc (b j)\nj : \u2115\nhB : \u2191C\u207b\u00b9 * \u2191\u2191\u03bc (B j) \u2260 \u22a4\nhj\u2081 : Disjoint (b j) (W \u2229 B j)\nhj\u2082 : \u2191C\u207b\u00b9 * \u2191\u2191\u03bc (B j) \u2264 \u2191\u2191\u03bc (b j)\nhj\u2083 : \u2191C\u207b\u00b9 * \u2191\u2191\u03bc (B j) + \u2191\u2191\u03bc (W \u2229 B j) - \u2191C\u207b\u00b9 * \u2191\u2191\u03bc (B j) \u2264 \u2191\u2191\u03bc (B j) - \u2191C\u207b\u00b9 * \u2191\u2191\u03bc (B j)\n\u22a2 \u2191\u2191\u03bc (W \u2229 B j) \u2264 \u2191\u2191\u03bc (B j) - \u2191C\u207b\u00b9 * \u2191\u2191\u03bc (B j)", "state_after": "no goals"}, {"tactic": "rwa [ae_le_set, @blimsup_eq_iInf_biSup_of_nat _ _ _ Y\u2082, iInf_eq_iInter, diff_iInter,\n  measure_iUnion_null_iff]", "annotated_tactic": ["rwa [<a>ae_le_set</a>, @<a>blimsup_eq_iInf_biSup_of_nat</a> _ _ _ Y\u2082, <a>iInf_eq_iInter</a>, <a>diff_iInter</a>,\n      <a>measure_iUnion_null_iff</a>]", [{"full_name": "MeasureTheory.ae_le_set", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [466, 9], "def_end_pos": [466, 18]}, {"full_name": "Filter.blimsup_eq_iInf_biSup_of_nat", "def_path": "Mathlib/Order/LiminfLimsup.lean", "def_pos": [841, 9], "def_end_pos": [841, 37]}, {"full_name": "Set.iInf_eq_iInter", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [196, 9], "def_end_pos": [196, 23]}, {"full_name": "Set.diff_iInter", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [686, 9], "def_end_pos": [686, 20]}, {"full_name": "MeasureTheory.measure_iUnion_null_iff", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [275, 9], "def_end_pos": [275, 32]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhr : Tendsto r\u2081 atTop (\ud835\udcdd[Ioi 0] 0)\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nhMr : \u2200\u1da0 (i : \u2115) in atTop, M * r\u2081 i \u2264 r\u2082 i\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\nthis : \u2200 (i : \u2115), \u2191\u2191\u03bc (blimsup Y\u2081 atTop p \\ Z i) = 0\n\u22a2 blimsup Y\u2081 atTop p \u2264\u1d50[\u03bc] blimsup Y\u2082 atTop p", "state_after": "no goals"}, {"tactic": "intro j", "annotated_tactic": ["intro j", []], "state_before": "case hf\u2080\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type ?u.5949} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\nf : \u2115 \u2192 \u2115\nhf\u2080 : \u2200 (j : \u2115), d \u2208 cthickening (r\u2081 (f j)) (s (f j))\nhf\u2081 : \u2200 (x : \u2115), p (f x)\nhf\u2082 : \u2200 (j : \u2115), j \u2264 f j\nhf\u2083 : Tendsto f atTop atTop\nhr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j)\n\u22a2 \u2200 (j : \u2115), \u2203 w, w \u2208 s (f j) \u2227 d \u2208 closedBall w (2 * r\u2081 (f j))", "state_after": "case hf\u2080\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type ?u.5949} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\nf : \u2115 \u2192 \u2115\nhf\u2080 : \u2200 (j : \u2115), d \u2208 cthickening (r\u2081 (f j)) (s (f j))\nhf\u2081 : \u2200 (x : \u2115), p (f x)\nhf\u2082 : \u2200 (j : \u2115), j \u2264 f j\nhf\u2083 : Tendsto f atTop atTop\nhr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j)\nj : \u2115\n\u22a2 \u2203 w, w \u2208 s (f j) \u2227 d \u2208 closedBall w (2 * r\u2081 (f j))"}, {"tactic": "specialize hrp (f j)", "annotated_tactic": ["specialize hrp (f j)", []], "state_before": "case hf\u2080\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type ?u.5949} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\nf : \u2115 \u2192 \u2115\nhf\u2080 : \u2200 (j : \u2115), d \u2208 cthickening (r\u2081 (f j)) (s (f j))\nhf\u2081 : \u2200 (x : \u2115), p (f x)\nhf\u2082 : \u2200 (j : \u2115), j \u2264 f j\nhf\u2083 : Tendsto f atTop atTop\nhr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j)\nj : \u2115\n\u22a2 \u2203 w, w \u2208 s (f j) \u2227 d \u2208 closedBall w (2 * r\u2081 (f j))", "state_after": "case hf\u2080\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type ?u.5949} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\nf : \u2115 \u2192 \u2115\nhf\u2080 : \u2200 (j : \u2115), d \u2208 cthickening (r\u2081 (f j)) (s (f j))\nhf\u2081 : \u2200 (x : \u2115), p (f x)\nhf\u2082 : \u2200 (j : \u2115), j \u2264 f j\nhf\u2083 : Tendsto f atTop atTop\nhr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j)\nj : \u2115\nhrp : OfNat.ofNat 0 (f j) \u2264 r\u2081 (f j)\n\u22a2 \u2203 w, w \u2208 s (f j) \u2227 d \u2208 closedBall w (2 * r\u2081 (f j))"}, {"tactic": "rw [Pi.zero_apply] at hrp", "annotated_tactic": ["rw [<a>Pi.zero_apply</a>] at hrp", [{"full_name": "Pi.zero_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [46, 3], "def_end_pos": [46, 14]}]], "state_before": "case hf\u2080\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type ?u.5949} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\nf : \u2115 \u2192 \u2115\nhf\u2080 : \u2200 (j : \u2115), d \u2208 cthickening (r\u2081 (f j)) (s (f j))\nhf\u2081 : \u2200 (x : \u2115), p (f x)\nhf\u2082 : \u2200 (j : \u2115), j \u2264 f j\nhf\u2083 : Tendsto f atTop atTop\nhr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j)\nj : \u2115\nhrp : OfNat.ofNat 0 (f j) \u2264 r\u2081 (f j)\n\u22a2 \u2203 w, w \u2208 s (f j) \u2227 d \u2208 closedBall w (2 * r\u2081 (f j))", "state_after": "case hf\u2080\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type ?u.5949} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\nf : \u2115 \u2192 \u2115\nhf\u2080 : \u2200 (j : \u2115), d \u2208 cthickening (r\u2081 (f j)) (s (f j))\nhf\u2081 : \u2200 (x : \u2115), p (f x)\nhf\u2082 : \u2200 (j : \u2115), j \u2264 f j\nhf\u2083 : Tendsto f atTop atTop\nhr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j)\nj : \u2115\nhrp : 0 \u2264 r\u2081 (f j)\n\u22a2 \u2203 w, w \u2208 s (f j) \u2227 d \u2208 closedBall w (2 * r\u2081 (f j))"}, {"tactic": "rcases eq_or_lt_of_le hrp with (hr0 | hrp')", "annotated_tactic": ["rcases <a>eq_or_lt_of_le</a> hrp with (hr0 | hrp')", [{"full_name": "eq_or_lt_of_le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [414, 9], "def_end_pos": [414, 23]}]], "state_before": "case hf\u2080\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type ?u.5949} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\nf : \u2115 \u2192 \u2115\nhf\u2080 : \u2200 (j : \u2115), d \u2208 cthickening (r\u2081 (f j)) (s (f j))\nhf\u2081 : \u2200 (x : \u2115), p (f x)\nhf\u2082 : \u2200 (j : \u2115), j \u2264 f j\nhf\u2083 : Tendsto f atTop atTop\nhr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j)\nj : \u2115\nhrp : 0 \u2264 r\u2081 (f j)\n\u22a2 \u2203 w, w \u2208 s (f j) \u2227 d \u2208 closedBall w (2 * r\u2081 (f j))", "state_after": "case hf\u2080.inl\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type ?u.5949} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\nf : \u2115 \u2192 \u2115\nhf\u2080 : \u2200 (j : \u2115), d \u2208 cthickening (r\u2081 (f j)) (s (f j))\nhf\u2081 : \u2200 (x : \u2115), p (f x)\nhf\u2082 : \u2200 (j : \u2115), j \u2264 f j\nhf\u2083 : Tendsto f atTop atTop\nhr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j)\nj : \u2115\nhrp : 0 \u2264 r\u2081 (f j)\nhr0 : 0 = r\u2081 (f j)\n\u22a2 \u2203 w, w \u2208 s (f j) \u2227 d \u2208 closedBall w (2 * r\u2081 (f j))\n\ncase hf\u2080.inr\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type ?u.5949} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\nf : \u2115 \u2192 \u2115\nhf\u2080 : \u2200 (j : \u2115), d \u2208 cthickening (r\u2081 (f j)) (s (f j))\nhf\u2081 : \u2200 (x : \u2115), p (f x)\nhf\u2082 : \u2200 (j : \u2115), j \u2264 f j\nhf\u2083 : Tendsto f atTop atTop\nhr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j)\nj : \u2115\nhrp : 0 \u2264 r\u2081 (f j)\nhrp' : 0 < r\u2081 (f j)\n\u22a2 \u2203 w, w \u2208 s (f j) \u2227 d \u2208 closedBall w (2 * r\u2081 (f j))"}, {"tactic": "specialize hf\u2080 j", "annotated_tactic": ["specialize hf\u2080 j", []], "state_before": "case hf\u2080.inl\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type ?u.5949} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\nf : \u2115 \u2192 \u2115\nhf\u2080 : \u2200 (j : \u2115), d \u2208 cthickening (r\u2081 (f j)) (s (f j))\nhf\u2081 : \u2200 (x : \u2115), p (f x)\nhf\u2082 : \u2200 (j : \u2115), j \u2264 f j\nhf\u2083 : Tendsto f atTop atTop\nhr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j)\nj : \u2115\nhrp : 0 \u2264 r\u2081 (f j)\nhr0 : 0 = r\u2081 (f j)\n\u22a2 \u2203 w, w \u2208 s (f j) \u2227 d \u2208 closedBall w (2 * r\u2081 (f j))", "state_after": "case hf\u2080.inl\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type ?u.5949} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\nf : \u2115 \u2192 \u2115\nhf\u2081 : \u2200 (x : \u2115), p (f x)\nhf\u2082 : \u2200 (j : \u2115), j \u2264 f j\nhf\u2083 : Tendsto f atTop atTop\nhr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j)\nj : \u2115\nhrp : 0 \u2264 r\u2081 (f j)\nhr0 : 0 = r\u2081 (f j)\nhf\u2080 : d \u2208 cthickening (r\u2081 (f j)) (s (f j))\n\u22a2 \u2203 w, w \u2208 s (f j) \u2227 d \u2208 closedBall w (2 * r\u2081 (f j))"}, {"tactic": "rw [\u2190 hr0, cthickening_zero, (hs (f j)).closure_eq] at hf\u2080", "annotated_tactic": ["rw [\u2190 hr0, <a>cthickening_zero</a>, (hs (f j)).<a>closure_eq</a>] at hf\u2080", [{"full_name": "Metric.cthickening_zero", "def_path": "Mathlib/Topology/MetricSpace/HausdorffDistance.lean", "def_pos": [1080, 9], "def_end_pos": [1080, 25]}, {"full_name": "IsClosed.closure_eq", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [457, 9], "def_end_pos": [457, 28]}]], "state_before": "case hf\u2080.inl\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type ?u.5949} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\nf : \u2115 \u2192 \u2115\nhf\u2081 : \u2200 (x : \u2115), p (f x)\nhf\u2082 : \u2200 (j : \u2115), j \u2264 f j\nhf\u2083 : Tendsto f atTop atTop\nhr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j)\nj : \u2115\nhrp : 0 \u2264 r\u2081 (f j)\nhr0 : 0 = r\u2081 (f j)\nhf\u2080 : d \u2208 cthickening (r\u2081 (f j)) (s (f j))\n\u22a2 \u2203 w, w \u2208 s (f j) \u2227 d \u2208 closedBall w (2 * r\u2081 (f j))", "state_after": "case hf\u2080.inl\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type ?u.5949} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\nf : \u2115 \u2192 \u2115\nhf\u2081 : \u2200 (x : \u2115), p (f x)\nhf\u2082 : \u2200 (j : \u2115), j \u2264 f j\nhf\u2083 : Tendsto f atTop atTop\nhr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j)\nj : \u2115\nhrp : 0 \u2264 r\u2081 (f j)\nhr0 : 0 = r\u2081 (f j)\nhf\u2080 : d \u2208 s (f j)\n\u22a2 \u2203 w, w \u2208 s (f j) \u2227 d \u2208 closedBall w (2 * r\u2081 (f j))"}, {"tactic": "exact \u27e8d, hf\u2080, by simp [\u2190 hr0]\u27e9", "annotated_tactic": ["exact \u27e8d, hf\u2080, by simp [\u2190 hr0]\u27e9", []], "state_before": "case hf\u2080.inl\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type ?u.5949} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\nf : \u2115 \u2192 \u2115\nhf\u2081 : \u2200 (x : \u2115), p (f x)\nhf\u2082 : \u2200 (j : \u2115), j \u2264 f j\nhf\u2083 : Tendsto f atTop atTop\nhr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j)\nj : \u2115\nhrp : 0 \u2264 r\u2081 (f j)\nhr0 : 0 = r\u2081 (f j)\nhf\u2080 : d \u2208 s (f j)\n\u22a2 \u2203 w, w \u2208 s (f j) \u2227 d \u2208 closedBall w (2 * r\u2081 (f j))", "state_after": "no goals"}, {"tactic": "simp [\u2190 hr0]", "annotated_tactic": ["simp [\u2190 hr0]", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type ?u.5949} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\nf : \u2115 \u2192 \u2115\nhf\u2081 : \u2200 (x : \u2115), p (f x)\nhf\u2082 : \u2200 (j : \u2115), j \u2264 f j\nhf\u2083 : Tendsto f atTop atTop\nhr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j)\nj : \u2115\nhrp : 0 \u2264 r\u2081 (f j)\nhr0 : 0 = r\u2081 (f j)\nhf\u2080 : d \u2208 s (f j)\n\u22a2 d \u2208 closedBall d (2 * r\u2081 (f j))", "state_after": "no goals"}, {"tactic": "simpa using mem_iUnion\u2082.mp (cthickening_subset_iUnion_closedBall_of_lt (s (f j))\n  (by positivity) (lt_two_mul_self hrp') (hf\u2080 j))", "annotated_tactic": ["simpa using mem_iUnion\u2082.mp (<a>cthickening_subset_iUnion_closedBall_of_lt</a> (s (f j))\n        (by positivity) (<a>lt_two_mul_self</a> hrp') (hf\u2080 j))", [{"full_name": "Metric.cthickening_subset_iUnion_closedBall_of_lt", "def_path": "Mathlib/Topology/MetricSpace/HausdorffDistance.lean", "def_pos": [1421, 9], "def_end_pos": [1421, 51]}, {"full_name": "lt_two_mul_self", "def_path": "Mathlib/Algebra/Order/Ring/Defs.lean", "def_pos": [622, 9], "def_end_pos": [622, 24]}]], "state_before": "case hf\u2080.inr\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type ?u.5949} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\nf : \u2115 \u2192 \u2115\nhf\u2080 : \u2200 (j : \u2115), d \u2208 cthickening (r\u2081 (f j)) (s (f j))\nhf\u2081 : \u2200 (x : \u2115), p (f x)\nhf\u2082 : \u2200 (j : \u2115), j \u2264 f j\nhf\u2083 : Tendsto f atTop atTop\nhr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j)\nj : \u2115\nhrp : 0 \u2264 r\u2081 (f j)\nhrp' : 0 < r\u2081 (f j)\n\u22a2 \u2203 w, w \u2208 s (f j) \u2227 d \u2208 closedBall w (2 * r\u2081 (f j))", "state_after": "no goals"}, {"tactic": "positivity", "annotated_tactic": ["positivity", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type ?u.5949} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\nf : \u2115 \u2192 \u2115\nhf\u2080 : \u2200 (j : \u2115), d \u2208 cthickening (r\u2081 (f j)) (s (f j))\nhf\u2081 : \u2200 (x : \u2115), p (f x)\nhf\u2082 : \u2200 (j : \u2115), j \u2264 f j\nhf\u2083 : Tendsto f atTop atTop\nhr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j)\nj : \u2115\nhrp : 0 \u2264 r\u2081 (f j)\nhrp' : 0 < r\u2081 (f j)\n\u22a2 0 < 2 * r\u2081 (f j)", "state_after": "no goals"}, {"tactic": "obtain \u27e8\u03b7, h\u03b7, h\u03b7'\u27e9 := this", "annotated_tactic": ["obtain \u27e8\u03b7, h\u03b7, h\u03b7'\u27e9 := this", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type ?u.5949} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\nf : \u2115 \u2192 \u2115\nhf\u2081 : \u2200 (x : \u2115), p (f x)\nhf\u2082 : \u2200 (j : \u2115), j \u2264 f j\nhf\u2083 : Tendsto f atTop atTop\nhr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j)\nw : \u2115 \u2192 \u03b1\nhw : \u2200 (j : \u2115), w j \u2208 s (f j)\nhw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j))\nC : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9\nhC : 0 < C\nthis : \u2203 \u03b7, \u03b7 < 1 \u2227 \u2200\u1da0 (j : \u2115) in atTop, \u2191\u2191\u03bc (W \u2229 closedBall (w j) (r\u2081 (f j))) / \u2191\u2191\u03bc (closedBall (w j) (r\u2081 (f j))) \u2264 \u2191\u03b7\n\u22a2 False", "state_after": "case intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type ?u.5949} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\nf : \u2115 \u2192 \u2115\nhf\u2081 : \u2200 (x : \u2115), p (f x)\nhf\u2082 : \u2200 (j : \u2115), j \u2264 f j\nhf\u2083 : Tendsto f atTop atTop\nhr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j)\nw : \u2115 \u2192 \u03b1\nhw : \u2200 (j : \u2115), w j \u2208 s (f j)\nhw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j))\nC : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9\nhC : 0 < C\n\u03b7 : \u211d\u22650\nh\u03b7 : \u03b7 < 1\nh\u03b7' : \u2200\u1da0 (j : \u2115) in atTop, \u2191\u2191\u03bc (W \u2229 closedBall (w j) (r\u2081 (f j))) / \u2191\u2191\u03bc (closedBall (w j) (r\u2081 (f j))) \u2264 \u2191\u03b7\n\u22a2 False"}, {"tactic": "replace h\u03b7' : 1 \u2264 \u03b7 := by\n  simpa only [ENNReal.one_le_coe_iff] using\n    le_of_tendsto (hd' w (fun j => r\u2081 (f j)) hr <| eventually_of_forall hw') h\u03b7'", "annotated_tactic": ["replace h\u03b7' : 1 \u2264 \u03b7 := by\n      simpa only [<a>ENNReal.one_le_coe_iff</a>] using\n        <a>le_of_tendsto</a> (hd' w (fun j => r\u2081 (f j)) hr <| <a>eventually_of_forall</a> hw') h\u03b7'", [{"full_name": "ENNReal.one_le_coe_iff", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [691, 9], "def_end_pos": [691, 23]}, {"full_name": "le_of_tendsto", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [140, 9], "def_end_pos": [140, 22]}, {"full_name": "Filter.eventually_of_forall", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1111, 9], "def_end_pos": [1111, 29]}]], "state_before": "case intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type ?u.5949} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\nf : \u2115 \u2192 \u2115\nhf\u2081 : \u2200 (x : \u2115), p (f x)\nhf\u2082 : \u2200 (j : \u2115), j \u2264 f j\nhf\u2083 : Tendsto f atTop atTop\nhr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j)\nw : \u2115 \u2192 \u03b1\nhw : \u2200 (j : \u2115), w j \u2208 s (f j)\nhw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j))\nC : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9\nhC : 0 < C\n\u03b7 : \u211d\u22650\nh\u03b7 : \u03b7 < 1\nh\u03b7' : \u2200\u1da0 (j : \u2115) in atTop, \u2191\u2191\u03bc (W \u2229 closedBall (w j) (r\u2081 (f j))) / \u2191\u2191\u03bc (closedBall (w j) (r\u2081 (f j))) \u2264 \u2191\u03b7\n\u22a2 False", "state_after": "case intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\nf : \u2115 \u2192 \u2115\nhf\u2081 : \u2200 (x : \u2115), p (f x)\nhf\u2082 : \u2200 (j : \u2115), j \u2264 f j\nhf\u2083 : Tendsto f atTop atTop\nhr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j)\nw : \u2115 \u2192 \u03b1\nhw : \u2200 (j : \u2115), w j \u2208 s (f j)\nhw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j))\nC : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9\nhC : 0 < C\n\u03b7 : \u211d\u22650\nh\u03b7 : \u03b7 < 1\nh\u03b7' : 1 \u2264 \u03b7\n\u22a2 False"}, {"tactic": "exact (lt_self_iff_false _).mp (lt_of_lt_of_le h\u03b7 h\u03b7')", "annotated_tactic": ["exact (<a>lt_self_iff_false</a> _).<a>mp</a> (<a>lt_of_lt_of_le</a> h\u03b7 h\u03b7')", [{"full_name": "lt_self_iff_false", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [175, 9], "def_end_pos": [175, 26]}, {"full_name": "Iff.mp", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [90, 3], "def_end_pos": [90, 5]}, {"full_name": "lt_of_lt_of_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [115, 9], "def_end_pos": [115, 23]}]], "state_before": "case intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\nf : \u2115 \u2192 \u2115\nhf\u2081 : \u2200 (x : \u2115), p (f x)\nhf\u2082 : \u2200 (j : \u2115), j \u2264 f j\nhf\u2083 : Tendsto f atTop atTop\nhr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j)\nw : \u2115 \u2192 \u03b1\nhw : \u2200 (j : \u2115), w j \u2208 s (f j)\nhw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j))\nC : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9\nhC : 0 < C\n\u03b7 : \u211d\u22650\nh\u03b7 : \u03b7 < 1\nh\u03b7' : 1 \u2264 \u03b7\n\u22a2 False", "state_after": "no goals"}, {"tactic": "simpa only [ENNReal.one_le_coe_iff] using\n  le_of_tendsto (hd' w (fun j => r\u2081 (f j)) hr <| eventually_of_forall hw') h\u03b7'", "annotated_tactic": ["simpa only [<a>ENNReal.one_le_coe_iff</a>] using\n        <a>le_of_tendsto</a> (hd' w (fun j => r\u2081 (f j)) hr <| <a>eventually_of_forall</a> hw') h\u03b7'", [{"full_name": "ENNReal.one_le_coe_iff", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [691, 9], "def_end_pos": [691, 23]}, {"full_name": "le_of_tendsto", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [140, 9], "def_end_pos": [140, 22]}, {"full_name": "Filter.eventually_of_forall", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1111, 9], "def_end_pos": [1111, 29]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type ?u.5949} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\nf : \u2115 \u2192 \u2115\nhf\u2081 : \u2200 (x : \u2115), p (f x)\nhf\u2082 : \u2200 (j : \u2115), j \u2264 f j\nhf\u2083 : Tendsto f atTop atTop\nhr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j)\nw : \u2115 \u2192 \u03b1\nhw : \u2200 (j : \u2115), w j \u2208 s (f j)\nhw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j))\nC : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9\nhC : 0 < C\n\u03b7 : \u211d\u22650\nh\u03b7 : \u03b7 < 1\nh\u03b7' : \u2200\u1da0 (j : \u2115) in atTop, \u2191\u2191\u03bc (W \u2229 closedBall (w j) (r\u2081 (f j))) / \u2191\u2191\u03bc (closedBall (w j) (r\u2081 (f j))) \u2264 \u2191\u03b7\n\u22a2 1 \u2264 \u03b7", "state_after": "no goals"}, {"tactic": "apply hMr.mp", "annotated_tactic": ["apply hMr.mp", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\nf : \u2115 \u2192 \u2115\nhf\u2081 : \u2200 (x : \u2115), p (f x)\nhf\u2082 : \u2200 (j : \u2115), j \u2264 f j\nhf\u2083 : Tendsto f atTop atTop\nhr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j)\nw : \u2115 \u2192 \u03b1\nhw : \u2200 (j : \u2115), w j \u2208 s (f j)\nhw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j))\nC : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9\nhC : C \u2260 0\nb : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (M * r\u2081 (f j))\nB : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (r\u2081 (f j))\nh\u2081 : \u2200 (j : \u2115), b j \u2286 B j\nh\u2082 : \u2200 (j : \u2115), W \u2229 B j \u2286 B j\n\u22a2 \u2200\u1da0 (j : \u2115) in atTop, Disjoint (b j) (W \u2229 B j)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\nf : \u2115 \u2192 \u2115\nhf\u2081 : \u2200 (x : \u2115), p (f x)\nhf\u2082 : \u2200 (j : \u2115), j \u2264 f j\nhf\u2083 : Tendsto f atTop atTop\nhr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j)\nw : \u2115 \u2192 \u03b1\nhw : \u2200 (j : \u2115), w j \u2208 s (f j)\nhw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j))\nC : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9\nhC : C \u2260 0\nb : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (M * r\u2081 (f j))\nB : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (r\u2081 (f j))\nh\u2081 : \u2200 (j : \u2115), b j \u2286 B j\nh\u2082 : \u2200 (j : \u2115), W \u2229 B j \u2286 B j\n\u22a2 \u2200\u1da0 (x : \u2115) in atTop, M * r\u2081 (f x) \u2264 r\u2082 (f x) \u2192 Disjoint (b x) (W \u2229 B x)"}, {"tactic": "rw [eventually_atTop]", "annotated_tactic": ["rw [<a>eventually_atTop</a>]", [{"full_name": "Filter.eventually_atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [178, 9], "def_end_pos": [178, 25]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\nf : \u2115 \u2192 \u2115\nhf\u2081 : \u2200 (x : \u2115), p (f x)\nhf\u2082 : \u2200 (j : \u2115), j \u2264 f j\nhf\u2083 : Tendsto f atTop atTop\nhr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j)\nw : \u2115 \u2192 \u03b1\nhw : \u2200 (j : \u2115), w j \u2208 s (f j)\nhw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j))\nC : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9\nhC : C \u2260 0\nb : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (M * r\u2081 (f j))\nB : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (r\u2081 (f j))\nh\u2081 : \u2200 (j : \u2115), b j \u2286 B j\nh\u2082 : \u2200 (j : \u2115), W \u2229 B j \u2286 B j\n\u22a2 \u2200\u1da0 (x : \u2115) in atTop, M * r\u2081 (f x) \u2264 r\u2082 (f x) \u2192 Disjoint (b x) (W \u2229 B x)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\nf : \u2115 \u2192 \u2115\nhf\u2081 : \u2200 (x : \u2115), p (f x)\nhf\u2082 : \u2200 (j : \u2115), j \u2264 f j\nhf\u2083 : Tendsto f atTop atTop\nhr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j)\nw : \u2115 \u2192 \u03b1\nhw : \u2200 (j : \u2115), w j \u2208 s (f j)\nhw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j))\nC : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9\nhC : C \u2260 0\nb : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (M * r\u2081 (f j))\nB : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (r\u2081 (f j))\nh\u2081 : \u2200 (j : \u2115), b j \u2286 B j\nh\u2082 : \u2200 (j : \u2115), W \u2229 B j \u2286 B j\n\u22a2 \u2203 a, \u2200 (b_1 : \u2115), b_1 \u2265 a \u2192 M * r\u2081 (f b_1) \u2264 r\u2082 (f b_1) \u2192 Disjoint (b b_1) (W \u2229 B b_1)"}, {"tactic": "refine'\n  \u27e8i, fun j hj hj' => Disjoint.inf_right (B j) <| Disjoint.inf_right' (blimsup Y\u2081 atTop p) _\u27e9", "annotated_tactic": ["refine'\n      \u27e8i, fun j hj hj' => <a>Disjoint.inf_right</a> (B j) <| <a>Disjoint.inf_right'</a> (<a>blimsup</a> Y\u2081 <a>atTop</a> p) _\u27e9", [{"full_name": "Disjoint.inf_right", "def_path": "Mathlib/Order/Disjoint.lean", "def_pos": [160, 9], "def_end_pos": [160, 27]}, {"full_name": "Disjoint.inf_right'", "def_path": "Mathlib/Order/Disjoint.lean", "def_pos": [164, 9], "def_end_pos": [164, 28]}, {"full_name": "Filter.blimsup", "def_path": "Mathlib/Order/LiminfLimsup.lean", "def_pos": [432, 5], "def_end_pos": [432, 12]}, {"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\nf : \u2115 \u2192 \u2115\nhf\u2081 : \u2200 (x : \u2115), p (f x)\nhf\u2082 : \u2200 (j : \u2115), j \u2264 f j\nhf\u2083 : Tendsto f atTop atTop\nhr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j)\nw : \u2115 \u2192 \u03b1\nhw : \u2200 (j : \u2115), w j \u2208 s (f j)\nhw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j))\nC : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9\nhC : C \u2260 0\nb : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (M * r\u2081 (f j))\nB : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (r\u2081 (f j))\nh\u2081 : \u2200 (j : \u2115), b j \u2286 B j\nh\u2082 : \u2200 (j : \u2115), W \u2229 B j \u2286 B j\n\u22a2 \u2203 a, \u2200 (b_1 : \u2115), b_1 \u2265 a \u2192 M * r\u2081 (f b_1) \u2264 r\u2082 (f b_1) \u2192 Disjoint (b b_1) (W \u2229 B b_1)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\nf : \u2115 \u2192 \u2115\nhf\u2081 : \u2200 (x : \u2115), p (f x)\nhf\u2082 : \u2200 (j : \u2115), j \u2264 f j\nhf\u2083 : Tendsto f atTop atTop\nhr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j)\nw : \u2115 \u2192 \u03b1\nhw : \u2200 (j : \u2115), w j \u2208 s (f j)\nhw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j))\nC : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9\nhC : C \u2260 0\nb : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (M * r\u2081 (f j))\nB : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (r\u2081 (f j))\nh\u2081 : \u2200 (j : \u2115), b j \u2286 B j\nh\u2082 : \u2200 (j : \u2115), W \u2229 B j \u2286 B j\nj : \u2115\nhj : j \u2265 i\nhj' : M * r\u2081 (f j) \u2264 r\u2082 (f j)\n\u22a2 Disjoint (b j) fun a => a \u2208 Z i \u2192 False"}, {"tactic": "change Disjoint (b j) (Z i)\u1d9c", "annotated_tactic": ["change <a>Disjoint</a> (b j) (Z i)\u1d9c", [{"full_name": "Disjoint", "def_path": "Mathlib/Order/Disjoint.lean", "def_pos": [41, 5], "def_end_pos": [41, 13]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\nf : \u2115 \u2192 \u2115\nhf\u2081 : \u2200 (x : \u2115), p (f x)\nhf\u2082 : \u2200 (j : \u2115), j \u2264 f j\nhf\u2083 : Tendsto f atTop atTop\nhr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j)\nw : \u2115 \u2192 \u03b1\nhw : \u2200 (j : \u2115), w j \u2208 s (f j)\nhw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j))\nC : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9\nhC : C \u2260 0\nb : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (M * r\u2081 (f j))\nB : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (r\u2081 (f j))\nh\u2081 : \u2200 (j : \u2115), b j \u2286 B j\nh\u2082 : \u2200 (j : \u2115), W \u2229 B j \u2286 B j\nj : \u2115\nhj : j \u2265 i\nhj' : M * r\u2081 (f j) \u2264 r\u2082 (f j)\n\u22a2 Disjoint (b j) fun a => a \u2208 Z i \u2192 False", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\nf : \u2115 \u2192 \u2115\nhf\u2081 : \u2200 (x : \u2115), p (f x)\nhf\u2082 : \u2200 (j : \u2115), j \u2264 f j\nhf\u2083 : Tendsto f atTop atTop\nhr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j)\nw : \u2115 \u2192 \u03b1\nhw : \u2200 (j : \u2115), w j \u2208 s (f j)\nhw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j))\nC : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9\nhC : C \u2260 0\nb : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (M * r\u2081 (f j))\nB : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (r\u2081 (f j))\nh\u2081 : \u2200 (j : \u2115), b j \u2286 B j\nh\u2082 : \u2200 (j : \u2115), W \u2229 B j \u2286 B j\nj : \u2115\nhj : j \u2265 i\nhj' : M * r\u2081 (f j) \u2264 r\u2082 (f j)\n\u22a2 Disjoint (b j) (Z i)\u1d9c"}, {"tactic": "rw [disjoint_compl_right_iff_subset]", "annotated_tactic": ["rw [<a>disjoint_compl_right_iff_subset</a>]", [{"full_name": "Set.disjoint_compl_right_iff_subset", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1774, 9], "def_end_pos": [1774, 40]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\nf : \u2115 \u2192 \u2115\nhf\u2081 : \u2200 (x : \u2115), p (f x)\nhf\u2082 : \u2200 (j : \u2115), j \u2264 f j\nhf\u2083 : Tendsto f atTop atTop\nhr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j)\nw : \u2115 \u2192 \u03b1\nhw : \u2200 (j : \u2115), w j \u2208 s (f j)\nhw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j))\nC : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9\nhC : C \u2260 0\nb : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (M * r\u2081 (f j))\nB : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (r\u2081 (f j))\nh\u2081 : \u2200 (j : \u2115), b j \u2286 B j\nh\u2082 : \u2200 (j : \u2115), W \u2229 B j \u2286 B j\nj : \u2115\nhj : j \u2265 i\nhj' : M * r\u2081 (f j) \u2264 r\u2082 (f j)\n\u22a2 Disjoint (b j) (Z i)\u1d9c", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\nf : \u2115 \u2192 \u2115\nhf\u2081 : \u2200 (x : \u2115), p (f x)\nhf\u2082 : \u2200 (j : \u2115), j \u2264 f j\nhf\u2083 : Tendsto f atTop atTop\nhr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j)\nw : \u2115 \u2192 \u03b1\nhw : \u2200 (j : \u2115), w j \u2208 s (f j)\nhw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j))\nC : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9\nhC : C \u2260 0\nb : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (M * r\u2081 (f j))\nB : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (r\u2081 (f j))\nh\u2081 : \u2200 (j : \u2115), b j \u2286 B j\nh\u2082 : \u2200 (j : \u2115), W \u2229 B j \u2286 B j\nj : \u2115\nhj : j \u2265 i\nhj' : M * r\u2081 (f j) \u2264 r\u2082 (f j)\n\u22a2 b j \u2286 Z i"}, {"tactic": "refine' (closedBall_subset_cthickening (hw j) (M * r\u2081 (f j))).trans\n  ((cthickening_mono hj' _).trans fun a ha => _)", "annotated_tactic": ["refine' (<a>closedBall_subset_cthickening</a> (hw j) (M * r\u2081 (f j))).<a>trans</a>\n      ((<a>cthickening_mono</a> hj' _).<a>trans</a> fun a ha => _)", [{"full_name": "Metric.closedBall_subset_cthickening", "def_path": "Mathlib/Topology/MetricSpace/HausdorffDistance.lean", "def_pos": [1415, 9], "def_end_pos": [1415, 38]}, {"full_name": "HasSubset.Subset.trans", "def_path": "Mathlib/Order/RelClasses.lean", "def_pos": [664, 7], "def_end_pos": [664, 29]}, {"full_name": "Metric.cthickening_mono", "def_path": "Mathlib/Topology/MetricSpace/HausdorffDistance.lean", "def_pos": [1090, 9], "def_end_pos": [1090, 25]}, {"full_name": "HasSubset.Subset.trans", "def_path": "Mathlib/Order/RelClasses.lean", "def_pos": [664, 7], "def_end_pos": [664, 29]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\nf : \u2115 \u2192 \u2115\nhf\u2081 : \u2200 (x : \u2115), p (f x)\nhf\u2082 : \u2200 (j : \u2115), j \u2264 f j\nhf\u2083 : Tendsto f atTop atTop\nhr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j)\nw : \u2115 \u2192 \u03b1\nhw : \u2200 (j : \u2115), w j \u2208 s (f j)\nhw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j))\nC : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9\nhC : C \u2260 0\nb : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (M * r\u2081 (f j))\nB : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (r\u2081 (f j))\nh\u2081 : \u2200 (j : \u2115), b j \u2286 B j\nh\u2082 : \u2200 (j : \u2115), W \u2229 B j \u2286 B j\nj : \u2115\nhj : j \u2265 i\nhj' : M * r\u2081 (f j) \u2264 r\u2082 (f j)\n\u22a2 b j \u2286 Z i", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\nf : \u2115 \u2192 \u2115\nhf\u2081 : \u2200 (x : \u2115), p (f x)\nhf\u2082 : \u2200 (j : \u2115), j \u2264 f j\nhf\u2083 : Tendsto f atTop atTop\nhr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j)\nw : \u2115 \u2192 \u03b1\nhw : \u2200 (j : \u2115), w j \u2208 s (f j)\nhw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j))\nC : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9\nhC : C \u2260 0\nb : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (M * r\u2081 (f j))\nB : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (r\u2081 (f j))\nh\u2081 : \u2200 (j : \u2115), b j \u2286 B j\nh\u2082 : \u2200 (j : \u2115), W \u2229 B j \u2286 B j\nj : \u2115\nhj : j \u2265 i\nhj' : M * r\u2081 (f j) \u2264 r\u2082 (f j)\na : \u03b1\nha : a \u2208 cthickening (r\u2082 (f j)) (s (f j))\n\u22a2 a \u2208 Z i"}, {"tactic": "simp only [mem_iUnion, exists_prop]", "annotated_tactic": ["simp only [<a>mem_iUnion</a>, <a>exists_prop</a>]", [{"full_name": "Set.mem_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [201, 9], "def_end_pos": [201, 19]}, {"full_name": "exists_prop", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [485, 17], "def_end_pos": [485, 28]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\nf : \u2115 \u2192 \u2115\nhf\u2081 : \u2200 (x : \u2115), p (f x)\nhf\u2082 : \u2200 (j : \u2115), j \u2264 f j\nhf\u2083 : Tendsto f atTop atTop\nhr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j)\nw : \u2115 \u2192 \u03b1\nhw : \u2200 (j : \u2115), w j \u2208 s (f j)\nhw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j))\nC : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9\nhC : C \u2260 0\nb : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (M * r\u2081 (f j))\nB : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (r\u2081 (f j))\nh\u2081 : \u2200 (j : \u2115), b j \u2286 B j\nh\u2082 : \u2200 (j : \u2115), W \u2229 B j \u2286 B j\nj : \u2115\nhj : j \u2265 i\nhj' : M * r\u2081 (f j) \u2264 r\u2082 (f j)\na : \u03b1\nha : a \u2208 cthickening (r\u2082 (f j)) (s (f j))\n\u22a2 a \u2208 Z i", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\nf : \u2115 \u2192 \u2115\nhf\u2081 : \u2200 (x : \u2115), p (f x)\nhf\u2082 : \u2200 (j : \u2115), j \u2264 f j\nhf\u2083 : Tendsto f atTop atTop\nhr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j)\nw : \u2115 \u2192 \u03b1\nhw : \u2200 (j : \u2115), w j \u2208 s (f j)\nhw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j))\nC : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9\nhC : C \u2260 0\nb : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (M * r\u2081 (f j))\nB : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (r\u2081 (f j))\nh\u2081 : \u2200 (j : \u2115), b j \u2286 B j\nh\u2082 : \u2200 (j : \u2115), W \u2229 B j \u2286 B j\nj : \u2115\nhj : j \u2265 i\nhj' : M * r\u2081 (f j) \u2264 r\u2082 (f j)\na : \u03b1\nha : a \u2208 cthickening (r\u2082 (f j)) (s (f j))\n\u22a2 \u2203 i_1, (p i_1 \u2227 i \u2264 i_1) \u2227 a \u2208 cthickening (r\u2082 i_1) (s i_1)"}, {"tactic": "exact \u27e8f j, \u27e8hf\u2081 j, hj.le.trans (hf\u2082 j)\u27e9, ha\u27e9", "annotated_tactic": ["exact \u27e8f j, \u27e8hf\u2081 j, hj.le.trans (hf\u2082 j)\u27e9, ha\u27e9", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\nf : \u2115 \u2192 \u2115\nhf\u2081 : \u2200 (x : \u2115), p (f x)\nhf\u2082 : \u2200 (j : \u2115), j \u2264 f j\nhf\u2083 : Tendsto f atTop atTop\nhr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j)\nw : \u2115 \u2192 \u03b1\nhw : \u2200 (j : \u2115), w j \u2208 s (f j)\nhw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j))\nC : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9\nhC : C \u2260 0\nb : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (M * r\u2081 (f j))\nB : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (r\u2081 (f j))\nh\u2081 : \u2200 (j : \u2115), b j \u2286 B j\nh\u2082 : \u2200 (j : \u2115), W \u2229 B j \u2286 B j\nj : \u2115\nhj : j \u2265 i\nhj' : M * r\u2081 (f j) \u2264 r\u2082 (f j)\na : \u03b1\nha : a \u2208 cthickening (r\u2082 (f j)) (s (f j))\n\u22a2 \u2203 i_1, (p i_1 \u2227 i \u2264 i_1) \u2227 a \u2208 cthickening (r\u2082 i_1) (s i_1)", "state_after": "no goals"}, {"tactic": "simp [hB]", "annotated_tactic": ["simp [hB]", []], "state_before": "case intro.intro.intro.intro.intro.inl\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\nf : \u2115 \u2192 \u2115\nhf\u2081 : \u2200 (x : \u2115), p (f x)\nhf\u2082 : \u2200 (j : \u2115), j \u2264 f j\nhf\u2083 : Tendsto f atTop atTop\nhr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j)\nw : \u2115 \u2192 \u03b1\nhw : \u2200 (j : \u2115), w j \u2208 s (f j)\nhw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j))\nC : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9\nhC : C \u2260 0\nb : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (M * r\u2081 (f j))\nB : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (r\u2081 (f j))\nh\u2081 : \u2200 (j : \u2115), b j \u2286 B j\nh\u2082 : \u2200 (j : \u2115), W \u2229 B j \u2286 B j\nh\u2083 : \u2200\u1da0 (j : \u2115) in atTop, Disjoint (b j) (W \u2229 B j)\nh\u2084 : \u2200\u1da0 (j : \u2115) in atTop, \u2191\u2191\u03bc (B j) \u2264 \u2191C * \u2191\u2191\u03bc (b j)\nj : \u2115\nhj\u2080 : Disjoint (b j) (W \u2229 B j) \u2227 \u2191\u2191\u03bc (B j) \u2264 \u2191C * \u2191\u2191\u03bc (b j)\nhB : \u2191\u2191\u03bc (B j) = \u22a4\n\u22a2 \u2191\u2191\u03bc (W \u2229 B j) / \u2191\u2191\u03bc (B j) \u2264 \u2191(1 - C\u207b\u00b9)", "state_after": "no goals"}, {"tactic": "refine' ENNReal.mul_ne_top _ hB", "annotated_tactic": ["refine' <a>ENNReal.mul_ne_top</a> _ hB", [{"full_name": "ENNReal.mul_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [615, 9], "def_end_pos": [615, 19]}]], "state_before": "case hB\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\nf : \u2115 \u2192 \u2115\nhf\u2081 : \u2200 (x : \u2115), p (f x)\nhf\u2082 : \u2200 (j : \u2115), j \u2264 f j\nhf\u2083 : Tendsto f atTop atTop\nhr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j)\nw : \u2115 \u2192 \u03b1\nhw : \u2200 (j : \u2115), w j \u2208 s (f j)\nhw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j))\nC : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9\nhC : C \u2260 0\nb : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (M * r\u2081 (f j))\nB : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (r\u2081 (f j))\nh\u2081 : \u2200 (j : \u2115), b j \u2286 B j\nh\u2082 : \u2200 (j : \u2115), W \u2229 B j \u2286 B j\nh\u2083 : \u2200\u1da0 (j : \u2115) in atTop, Disjoint (b j) (W \u2229 B j)\nh\u2084 : \u2200\u1da0 (j : \u2115) in atTop, \u2191\u2191\u03bc (B j) \u2264 \u2191C * \u2191\u2191\u03bc (b j)\nj : \u2115\nhj\u2080 : Disjoint (b j) (W \u2229 B j) \u2227 \u2191\u2191\u03bc (B j) \u2264 \u2191C * \u2191\u2191\u03bc (b j)\nhB : \u2191\u2191\u03bc (B j) \u2260 \u22a4\n\u22a2 \u2191C\u207b\u00b9 * \u2191\u2191\u03bc (B j) \u2260 \u22a4", "state_after": "case hB\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\nf : \u2115 \u2192 \u2115\nhf\u2081 : \u2200 (x : \u2115), p (f x)\nhf\u2082 : \u2200 (j : \u2115), j \u2264 f j\nhf\u2083 : Tendsto f atTop atTop\nhr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j)\nw : \u2115 \u2192 \u03b1\nhw : \u2200 (j : \u2115), w j \u2208 s (f j)\nhw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j))\nC : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9\nhC : C \u2260 0\nb : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (M * r\u2081 (f j))\nB : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (r\u2081 (f j))\nh\u2081 : \u2200 (j : \u2115), b j \u2286 B j\nh\u2082 : \u2200 (j : \u2115), W \u2229 B j \u2286 B j\nh\u2083 : \u2200\u1da0 (j : \u2115) in atTop, Disjoint (b j) (W \u2229 B j)\nh\u2084 : \u2200\u1da0 (j : \u2115) in atTop, \u2191\u2191\u03bc (B j) \u2264 \u2191C * \u2191\u2191\u03bc (b j)\nj : \u2115\nhj\u2080 : Disjoint (b j) (W \u2229 B j) \u2227 \u2191\u2191\u03bc (B j) \u2264 \u2191C * \u2191\u2191\u03bc (b j)\nhB : \u2191\u2191\u03bc (B j) \u2260 \u22a4\n\u22a2 \u2191C\u207b\u00b9 \u2260 \u22a4"}, {"tactic": "rwa [ENNReal.coe_inv hC, Ne.def, ENNReal.inv_eq_top, ENNReal.coe_eq_zero]", "annotated_tactic": ["rwa [<a>ENNReal.coe_inv</a> hC, <a>Ne.def</a>, <a>ENNReal.inv_eq_top</a>, <a>ENNReal.coe_eq_zero</a>]", [{"full_name": "ENNReal.coe_inv", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1387, 9], "def_end_pos": [1387, 16]}, {"full_name": "Ne.def", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [59, 9], "def_end_pos": [59, 15]}, {"full_name": "ENNReal.inv_eq_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1449, 17], "def_end_pos": [1449, 27]}, {"full_name": "ENNReal.coe_eq_zero", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [368, 28], "def_end_pos": [368, 39]}]], "state_before": "case hB\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\nf : \u2115 \u2192 \u2115\nhf\u2081 : \u2200 (x : \u2115), p (f x)\nhf\u2082 : \u2200 (j : \u2115), j \u2264 f j\nhf\u2083 : Tendsto f atTop atTop\nhr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j)\nw : \u2115 \u2192 \u03b1\nhw : \u2200 (j : \u2115), w j \u2208 s (f j)\nhw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j))\nC : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9\nhC : C \u2260 0\nb : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (M * r\u2081 (f j))\nB : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (r\u2081 (f j))\nh\u2081 : \u2200 (j : \u2115), b j \u2286 B j\nh\u2082 : \u2200 (j : \u2115), W \u2229 B j \u2286 B j\nh\u2083 : \u2200\u1da0 (j : \u2115) in atTop, Disjoint (b j) (W \u2229 B j)\nh\u2084 : \u2200\u1da0 (j : \u2115) in atTop, \u2191\u2191\u03bc (B j) \u2264 \u2191C * \u2191\u2191\u03bc (b j)\nj : \u2115\nhj\u2080 : Disjoint (b j) (W \u2229 B j) \u2227 \u2191\u2191\u03bc (B j) \u2264 \u2191C * \u2191\u2191\u03bc (b j)\nhB : \u2191\u2191\u03bc (B j) \u2260 \u22a4\n\u22a2 \u2191C\u207b\u00b9 \u2260 \u22a4", "state_after": "no goals"}, {"tactic": "rw [ENNReal.coe_inv hC, \u2190 ENNReal.div_eq_inv_mul]", "annotated_tactic": ["rw [<a>ENNReal.coe_inv</a> hC, \u2190 <a>ENNReal.div_eq_inv_mul</a>]", [{"full_name": "ENNReal.coe_inv", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1387, 9], "def_end_pos": [1387, 16]}, {"full_name": "ENNReal.div_eq_inv_mul", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1367, 19], "def_end_pos": [1367, 33]}]], "state_before": "case hj\u2082\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\nf : \u2115 \u2192 \u2115\nhf\u2081 : \u2200 (x : \u2115), p (f x)\nhf\u2082 : \u2200 (j : \u2115), j \u2264 f j\nhf\u2083 : Tendsto f atTop atTop\nhr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j)\nw : \u2115 \u2192 \u03b1\nhw : \u2200 (j : \u2115), w j \u2208 s (f j)\nhw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j))\nC : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9\nhC : C \u2260 0\nb : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (M * r\u2081 (f j))\nB : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (r\u2081 (f j))\nh\u2081 : \u2200 (j : \u2115), b j \u2286 B j\nh\u2082 : \u2200 (j : \u2115), W \u2229 B j \u2286 B j\nh\u2083 : \u2200\u1da0 (j : \u2115) in atTop, Disjoint (b j) (W \u2229 B j)\nh\u2084 : \u2200\u1da0 (j : \u2115) in atTop, \u2191\u2191\u03bc (B j) \u2264 \u2191C * \u2191\u2191\u03bc (b j)\nj : \u2115\nhB : \u2191C\u207b\u00b9 * \u2191\u2191\u03bc (B j) \u2260 \u22a4\nhj\u2081 : Disjoint (b j) (W \u2229 B j)\nhj\u2082 : \u2191\u2191\u03bc (B j) \u2264 \u2191C * \u2191\u2191\u03bc (b j)\n\u22a2 \u2191C\u207b\u00b9 * \u2191\u2191\u03bc (B j) \u2264 \u2191\u2191\u03bc (b j)", "state_after": "case hj\u2082\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\nf : \u2115 \u2192 \u2115\nhf\u2081 : \u2200 (x : \u2115), p (f x)\nhf\u2082 : \u2200 (j : \u2115), j \u2264 f j\nhf\u2083 : Tendsto f atTop atTop\nhr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j)\nw : \u2115 \u2192 \u03b1\nhw : \u2200 (j : \u2115), w j \u2208 s (f j)\nhw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j))\nC : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9\nhC : C \u2260 0\nb : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (M * r\u2081 (f j))\nB : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (r\u2081 (f j))\nh\u2081 : \u2200 (j : \u2115), b j \u2286 B j\nh\u2082 : \u2200 (j : \u2115), W \u2229 B j \u2286 B j\nh\u2083 : \u2200\u1da0 (j : \u2115) in atTop, Disjoint (b j) (W \u2229 B j)\nh\u2084 : \u2200\u1da0 (j : \u2115) in atTop, \u2191\u2191\u03bc (B j) \u2264 \u2191C * \u2191\u2191\u03bc (b j)\nj : \u2115\nhB : \u2191C\u207b\u00b9 * \u2191\u2191\u03bc (B j) \u2260 \u22a4\nhj\u2081 : Disjoint (b j) (W \u2229 B j)\nhj\u2082 : \u2191\u2191\u03bc (B j) \u2264 \u2191C * \u2191\u2191\u03bc (b j)\n\u22a2 \u2191\u2191\u03bc (B j) / \u2191C \u2264 \u2191\u2191\u03bc (b j)"}, {"tactic": "exact ENNReal.div_le_of_le_mul' hj\u2082", "annotated_tactic": ["exact <a>ENNReal.div_le_of_le_mul'</a> hj\u2082", [{"full_name": "ENNReal.div_le_of_le_mul'", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1643, 9], "def_end_pos": [1643, 26]}]], "state_before": "case hj\u2082\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\nf : \u2115 \u2192 \u2115\nhf\u2081 : \u2200 (x : \u2115), p (f x)\nhf\u2082 : \u2200 (j : \u2115), j \u2264 f j\nhf\u2083 : Tendsto f atTop atTop\nhr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j)\nw : \u2115 \u2192 \u03b1\nhw : \u2200 (j : \u2115), w j \u2208 s (f j)\nhw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j))\nC : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9\nhC : C \u2260 0\nb : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (M * r\u2081 (f j))\nB : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (r\u2081 (f j))\nh\u2081 : \u2200 (j : \u2115), b j \u2286 B j\nh\u2082 : \u2200 (j : \u2115), W \u2229 B j \u2286 B j\nh\u2083 : \u2200\u1da0 (j : \u2115) in atTop, Disjoint (b j) (W \u2229 B j)\nh\u2084 : \u2200\u1da0 (j : \u2115) in atTop, \u2191\u2191\u03bc (B j) \u2264 \u2191C * \u2191\u2191\u03bc (b j)\nj : \u2115\nhB : \u2191C\u207b\u00b9 * \u2191\u2191\u03bc (B j) \u2260 \u22a4\nhj\u2081 : Disjoint (b j) (W \u2229 B j)\nhj\u2082 : \u2191\u2191\u03bc (B j) \u2264 \u2191C * \u2191\u2191\u03bc (b j)\n\u22a2 \u2191\u2191\u03bc (B j) / \u2191C \u2264 \u2191\u2191\u03bc (b j)", "state_after": "no goals"}, {"tactic": "refine' le_trans (add_le_add_right hj\u2082 _) _", "annotated_tactic": ["refine' <a>le_trans</a> (<a>add_le_add_right</a> hj\u2082 _) _", [{"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}, {"full_name": "add_le_add_right", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [66, 15], "def_end_pos": [66, 31]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\nf : \u2115 \u2192 \u2115\nhf\u2081 : \u2200 (x : \u2115), p (f x)\nhf\u2082 : \u2200 (j : \u2115), j \u2264 f j\nhf\u2083 : Tendsto f atTop atTop\nhr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j)\nw : \u2115 \u2192 \u03b1\nhw : \u2200 (j : \u2115), w j \u2208 s (f j)\nhw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j))\nC : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9\nhC : C \u2260 0\nb : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (M * r\u2081 (f j))\nB : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (r\u2081 (f j))\nh\u2081 : \u2200 (j : \u2115), b j \u2286 B j\nh\u2082 : \u2200 (j : \u2115), W \u2229 B j \u2286 B j\nh\u2083 : \u2200\u1da0 (j : \u2115) in atTop, Disjoint (b j) (W \u2229 B j)\nh\u2084 : \u2200\u1da0 (j : \u2115) in atTop, \u2191\u2191\u03bc (B j) \u2264 \u2191C * \u2191\u2191\u03bc (b j)\nj : \u2115\nhB : \u2191C\u207b\u00b9 * \u2191\u2191\u03bc (B j) \u2260 \u22a4\nhj\u2081 : Disjoint (b j) (W \u2229 B j)\nhj\u2082 : \u2191C\u207b\u00b9 * \u2191\u2191\u03bc (B j) \u2264 \u2191\u2191\u03bc (b j)\n\u22a2 \u2191C\u207b\u00b9 * \u2191\u2191\u03bc (B j) + \u2191\u2191\u03bc (W \u2229 B j) \u2264 \u2191\u2191\u03bc (B j)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\nf : \u2115 \u2192 \u2115\nhf\u2081 : \u2200 (x : \u2115), p (f x)\nhf\u2082 : \u2200 (j : \u2115), j \u2264 f j\nhf\u2083 : Tendsto f atTop atTop\nhr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j)\nw : \u2115 \u2192 \u03b1\nhw : \u2200 (j : \u2115), w j \u2208 s (f j)\nhw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j))\nC : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9\nhC : C \u2260 0\nb : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (M * r\u2081 (f j))\nB : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (r\u2081 (f j))\nh\u2081 : \u2200 (j : \u2115), b j \u2286 B j\nh\u2082 : \u2200 (j : \u2115), W \u2229 B j \u2286 B j\nh\u2083 : \u2200\u1da0 (j : \u2115) in atTop, Disjoint (b j) (W \u2229 B j)\nh\u2084 : \u2200\u1da0 (j : \u2115) in atTop, \u2191\u2191\u03bc (B j) \u2264 \u2191C * \u2191\u2191\u03bc (b j)\nj : \u2115\nhB : \u2191C\u207b\u00b9 * \u2191\u2191\u03bc (B j) \u2260 \u22a4\nhj\u2081 : Disjoint (b j) (W \u2229 B j)\nhj\u2082 : \u2191C\u207b\u00b9 * \u2191\u2191\u03bc (B j) \u2264 \u2191\u2191\u03bc (b j)\n\u22a2 \u2191\u2191\u03bc (b j) + \u2191\u2191\u03bc (W \u2229 B j) \u2264 \u2191\u2191\u03bc (B j)"}, {"tactic": "rw [\u2190 measure_union' hj\u2081 measurableSet_closedBall]", "annotated_tactic": ["rw [\u2190 <a>measure_union'</a> hj\u2081 <a>measurableSet_closedBall</a>]", [{"full_name": "MeasureTheory.measure_union'", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [128, 9], "def_end_pos": [128, 23]}, {"full_name": "measurableSet_closedBall", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [1681, 9], "def_end_pos": [1681, 33]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\nf : \u2115 \u2192 \u2115\nhf\u2081 : \u2200 (x : \u2115), p (f x)\nhf\u2082 : \u2200 (j : \u2115), j \u2264 f j\nhf\u2083 : Tendsto f atTop atTop\nhr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j)\nw : \u2115 \u2192 \u03b1\nhw : \u2200 (j : \u2115), w j \u2208 s (f j)\nhw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j))\nC : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9\nhC : C \u2260 0\nb : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (M * r\u2081 (f j))\nB : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (r\u2081 (f j))\nh\u2081 : \u2200 (j : \u2115), b j \u2286 B j\nh\u2082 : \u2200 (j : \u2115), W \u2229 B j \u2286 B j\nh\u2083 : \u2200\u1da0 (j : \u2115) in atTop, Disjoint (b j) (W \u2229 B j)\nh\u2084 : \u2200\u1da0 (j : \u2115) in atTop, \u2191\u2191\u03bc (B j) \u2264 \u2191C * \u2191\u2191\u03bc (b j)\nj : \u2115\nhB : \u2191C\u207b\u00b9 * \u2191\u2191\u03bc (B j) \u2260 \u22a4\nhj\u2081 : Disjoint (b j) (W \u2229 B j)\nhj\u2082 : \u2191C\u207b\u00b9 * \u2191\u2191\u03bc (B j) \u2264 \u2191\u2191\u03bc (b j)\n\u22a2 \u2191\u2191\u03bc (b j) + \u2191\u2191\u03bc (W \u2229 B j) \u2264 \u2191\u2191\u03bc (B j)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\nf : \u2115 \u2192 \u2115\nhf\u2081 : \u2200 (x : \u2115), p (f x)\nhf\u2082 : \u2200 (j : \u2115), j \u2264 f j\nhf\u2083 : Tendsto f atTop atTop\nhr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j)\nw : \u2115 \u2192 \u03b1\nhw : \u2200 (j : \u2115), w j \u2208 s (f j)\nhw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j))\nC : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9\nhC : C \u2260 0\nb : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (M * r\u2081 (f j))\nB : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (r\u2081 (f j))\nh\u2081 : \u2200 (j : \u2115), b j \u2286 B j\nh\u2082 : \u2200 (j : \u2115), W \u2229 B j \u2286 B j\nh\u2083 : \u2200\u1da0 (j : \u2115) in atTop, Disjoint (b j) (W \u2229 B j)\nh\u2084 : \u2200\u1da0 (j : \u2115) in atTop, \u2191\u2191\u03bc (B j) \u2264 \u2191C * \u2191\u2191\u03bc (b j)\nj : \u2115\nhB : \u2191C\u207b\u00b9 * \u2191\u2191\u03bc (B j) \u2260 \u22a4\nhj\u2081 : Disjoint (b j) (W \u2229 B j)\nhj\u2082 : \u2191C\u207b\u00b9 * \u2191\u2191\u03bc (B j) \u2264 \u2191\u2191\u03bc (b j)\n\u22a2 \u2191\u2191\u03bc (b j \u222a W \u2229 B j) \u2264 \u2191\u2191\u03bc (B j)"}, {"tactic": "exact measure_mono (union_subset (h\u2081 j) (h\u2082 j))", "annotated_tactic": ["exact <a>measure_mono</a> (<a>union_subset</a> (h\u2081 j) (h\u2082 j))", [{"full_name": "MeasureTheory.measure_mono", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [193, 9], "def_end_pos": [193, 21]}, {"full_name": "Set.union_subset", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [836, 9], "def_end_pos": [836, 21]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), IsClosed (s i)\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhrp : 0 \u2264 r\u2081\nM : \u211d\nhM : 0 < M\nhM' : M < 1\nY\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i)\nY\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i)\nZ : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j\ni : \u2115\nW : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i\ncontra : \u00ac\u2191\u2191\u03bc W = 0\nd : \u03b1\nhd' :\n  \u2200 {\u03b9 : Type} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192\n      (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n        Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1)\nhd : d \u2208 blimsup Y\u2081 atTop p\nf : \u2115 \u2192 \u2115\nhf\u2081 : \u2200 (x : \u2115), p (f x)\nhf\u2082 : \u2200 (j : \u2115), j \u2264 f j\nhf\u2083 : Tendsto f atTop atTop\nhr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j)\nw : \u2115 \u2192 \u03b1\nhw : \u2200 (j : \u2115), w j \u2208 s (f j)\nhw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j))\nC : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9\nhC : C \u2260 0\nb : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (M * r\u2081 (f j))\nB : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (r\u2081 (f j))\nh\u2081 : \u2200 (j : \u2115), b j \u2286 B j\nh\u2082 : \u2200 (j : \u2115), W \u2229 B j \u2286 B j\nh\u2083 : \u2200\u1da0 (j : \u2115) in atTop, Disjoint (b j) (W \u2229 B j)\nh\u2084 : \u2200\u1da0 (j : \u2115) in atTop, \u2191\u2191\u03bc (B j) \u2264 \u2191C * \u2191\u2191\u03bc (b j)\nj : \u2115\nhB : \u2191C\u207b\u00b9 * \u2191\u2191\u03bc (B j) \u2260 \u22a4\nhj\u2081 : Disjoint (b j) (W \u2229 B j)\nhj\u2082 : \u2191C\u207b\u00b9 * \u2191\u2191\u03bc (B j) \u2264 \u2191\u2191\u03bc (b j)\n\u22a2 \u2191\u2191\u03bc (b j \u222a W \u2229 B j) \u2264 \u2191\u2191\u03bc (B j)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Group/Action.lean", "full_name": "MeasureTheory.smulInvariantMeasure_tfae", "start": [156, 1], "end": [184, 14], "traced_tactics": [{"tactic": "tfae_have 1 \u2194 2", "annotated_tactic": ["tfae_have 1 \u2194 2", []], "state_before": "G : Type u\nM : Type v\n\u03b1 : Type w\ns : Set \u03b1\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : MulAction G \u03b1\ninst\u271d\u00b9 : MeasurableSpace G\ninst\u271d : MeasurableSMul G \u03b1\nc : G\n\u03bc : Measure \u03b1\n\u22a2 List.TFAE\n    [SMulInvariantMeasure G \u03b1 \u03bc, \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s,\n      \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s,\n      \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s, \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s,\n      \u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc, \u2200 (c : G), MeasurePreserving fun x => c \u2022 x]", "state_after": "case tfae_1_iff_2\nG : Type u\nM : Type v\n\u03b1 : Type w\ns : Set \u03b1\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : MulAction G \u03b1\ninst\u271d\u00b9 : MeasurableSpace G\ninst\u271d : MeasurableSMul G \u03b1\nc : G\n\u03bc : Measure \u03b1\n\u22a2 SMulInvariantMeasure G \u03b1 \u03bc \u2194 \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s\n\nG : Type u\nM : Type v\n\u03b1 : Type w\ns : Set \u03b1\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : MulAction G \u03b1\ninst\u271d\u00b9 : MeasurableSpace G\ninst\u271d : MeasurableSMul G \u03b1\nc : G\n\u03bc : Measure \u03b1\ntfae_1_iff_2 :\n  SMulInvariantMeasure G \u03b1 \u03bc \u2194 \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s\n\u22a2 List.TFAE\n    [SMulInvariantMeasure G \u03b1 \u03bc, \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s,\n      \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s,\n      \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s, \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s,\n      \u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc, \u2200 (c : G), MeasurePreserving fun x => c \u2022 x]"}, {"tactic": "tfae_have 1 \u2192 6", "annotated_tactic": ["tfae_have 1 \u2192 6", []], "state_before": "G : Type u\nM : Type v\n\u03b1 : Type w\ns : Set \u03b1\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : MulAction G \u03b1\ninst\u271d\u00b9 : MeasurableSpace G\ninst\u271d : MeasurableSMul G \u03b1\nc : G\n\u03bc : Measure \u03b1\ntfae_1_iff_2 :\n  SMulInvariantMeasure G \u03b1 \u03bc \u2194 \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s\n\u22a2 List.TFAE\n    [SMulInvariantMeasure G \u03b1 \u03bc, \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s,\n      \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s,\n      \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s, \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s,\n      \u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc, \u2200 (c : G), MeasurePreserving fun x => c \u2022 x]", "state_after": "case tfae_1_to_6\nG : Type u\nM : Type v\n\u03b1 : Type w\ns : Set \u03b1\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : MulAction G \u03b1\ninst\u271d\u00b9 : MeasurableSpace G\ninst\u271d : MeasurableSMul G \u03b1\nc : G\n\u03bc : Measure \u03b1\ntfae_1_iff_2 :\n  SMulInvariantMeasure G \u03b1 \u03bc \u2194 \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s\n\u22a2 SMulInvariantMeasure G \u03b1 \u03bc \u2192 \u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc\n\nG : Type u\nM : Type v\n\u03b1 : Type w\ns : Set \u03b1\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : MulAction G \u03b1\ninst\u271d\u00b9 : MeasurableSpace G\ninst\u271d : MeasurableSMul G \u03b1\nc : G\n\u03bc : Measure \u03b1\ntfae_1_iff_2 :\n  SMulInvariantMeasure G \u03b1 \u03bc \u2194 \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s\ntfae_1_to_6 : SMulInvariantMeasure G \u03b1 \u03bc \u2192 \u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc\n\u22a2 List.TFAE\n    [SMulInvariantMeasure G \u03b1 \u03bc, \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s,\n      \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s,\n      \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s, \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s,\n      \u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc, \u2200 (c : G), MeasurePreserving fun x => c \u2022 x]"}, {"tactic": "tfae_have 6 \u2192 7", "annotated_tactic": ["tfae_have 6 \u2192 7", []], "state_before": "G : Type u\nM : Type v\n\u03b1 : Type w\ns : Set \u03b1\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : MulAction G \u03b1\ninst\u271d\u00b9 : MeasurableSpace G\ninst\u271d : MeasurableSMul G \u03b1\nc : G\n\u03bc : Measure \u03b1\ntfae_1_iff_2 :\n  SMulInvariantMeasure G \u03b1 \u03bc \u2194 \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s\ntfae_1_to_6 : SMulInvariantMeasure G \u03b1 \u03bc \u2192 \u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc\n\u22a2 List.TFAE\n    [SMulInvariantMeasure G \u03b1 \u03bc, \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s,\n      \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s,\n      \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s, \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s,\n      \u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc, \u2200 (c : G), MeasurePreserving fun x => c \u2022 x]", "state_after": "case tfae_6_to_7\nG : Type u\nM : Type v\n\u03b1 : Type w\ns : Set \u03b1\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : MulAction G \u03b1\ninst\u271d\u00b9 : MeasurableSpace G\ninst\u271d : MeasurableSMul G \u03b1\nc : G\n\u03bc : Measure \u03b1\ntfae_1_iff_2 :\n  SMulInvariantMeasure G \u03b1 \u03bc \u2194 \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s\ntfae_1_to_6 : SMulInvariantMeasure G \u03b1 \u03bc \u2192 \u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc\n\u22a2 (\u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc) \u2192 \u2200 (c : G), MeasurePreserving fun x => c \u2022 x\n\nG : Type u\nM : Type v\n\u03b1 : Type w\ns : Set \u03b1\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : MulAction G \u03b1\ninst\u271d\u00b9 : MeasurableSpace G\ninst\u271d : MeasurableSMul G \u03b1\nc : G\n\u03bc : Measure \u03b1\ntfae_1_iff_2 :\n  SMulInvariantMeasure G \u03b1 \u03bc \u2194 \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s\ntfae_1_to_6 : SMulInvariantMeasure G \u03b1 \u03bc \u2192 \u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc\ntfae_6_to_7 : (\u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc) \u2192 \u2200 (c : G), MeasurePreserving fun x => c \u2022 x\n\u22a2 List.TFAE\n    [SMulInvariantMeasure G \u03b1 \u03bc, \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s,\n      \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s,\n      \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s, \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s,\n      \u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc, \u2200 (c : G), MeasurePreserving fun x => c \u2022 x]"}, {"tactic": "tfae_have 7 \u2192 4", "annotated_tactic": ["tfae_have 7 \u2192 4", []], "state_before": "G : Type u\nM : Type v\n\u03b1 : Type w\ns : Set \u03b1\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : MulAction G \u03b1\ninst\u271d\u00b9 : MeasurableSpace G\ninst\u271d : MeasurableSMul G \u03b1\nc : G\n\u03bc : Measure \u03b1\ntfae_1_iff_2 :\n  SMulInvariantMeasure G \u03b1 \u03bc \u2194 \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s\ntfae_1_to_6 : SMulInvariantMeasure G \u03b1 \u03bc \u2192 \u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc\ntfae_6_to_7 : (\u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc) \u2192 \u2200 (c : G), MeasurePreserving fun x => c \u2022 x\n\u22a2 List.TFAE\n    [SMulInvariantMeasure G \u03b1 \u03bc, \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s,\n      \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s,\n      \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s, \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s,\n      \u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc, \u2200 (c : G), MeasurePreserving fun x => c \u2022 x]", "state_after": "case tfae_7_to_4\nG : Type u\nM : Type v\n\u03b1 : Type w\ns : Set \u03b1\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : MulAction G \u03b1\ninst\u271d\u00b9 : MeasurableSpace G\ninst\u271d : MeasurableSMul G \u03b1\nc : G\n\u03bc : Measure \u03b1\ntfae_1_iff_2 :\n  SMulInvariantMeasure G \u03b1 \u03bc \u2194 \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s\ntfae_1_to_6 : SMulInvariantMeasure G \u03b1 \u03bc \u2192 \u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc\ntfae_6_to_7 : (\u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc) \u2192 \u2200 (c : G), MeasurePreserving fun x => c \u2022 x\n\u22a2 (\u2200 (c : G), MeasurePreserving fun x => c \u2022 x) \u2192 \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s\n\nG : Type u\nM : Type v\n\u03b1 : Type w\ns : Set \u03b1\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : MulAction G \u03b1\ninst\u271d\u00b9 : MeasurableSpace G\ninst\u271d : MeasurableSMul G \u03b1\nc : G\n\u03bc : Measure \u03b1\ntfae_1_iff_2 :\n  SMulInvariantMeasure G \u03b1 \u03bc \u2194 \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s\ntfae_1_to_6 : SMulInvariantMeasure G \u03b1 \u03bc \u2192 \u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc\ntfae_6_to_7 : (\u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc) \u2192 \u2200 (c : G), MeasurePreserving fun x => c \u2022 x\ntfae_7_to_4 :\n  (\u2200 (c : G), MeasurePreserving fun x => c \u2022 x) \u2192 \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s\n\u22a2 List.TFAE\n    [SMulInvariantMeasure G \u03b1 \u03bc, \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s,\n      \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s,\n      \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s, \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s,\n      \u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc, \u2200 (c : G), MeasurePreserving fun x => c \u2022 x]"}, {"tactic": "tfae_have 4 \u2192 5", "annotated_tactic": ["tfae_have 4 \u2192 5", []], "state_before": "G : Type u\nM : Type v\n\u03b1 : Type w\ns : Set \u03b1\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : MulAction G \u03b1\ninst\u271d\u00b9 : MeasurableSpace G\ninst\u271d : MeasurableSMul G \u03b1\nc : G\n\u03bc : Measure \u03b1\ntfae_1_iff_2 :\n  SMulInvariantMeasure G \u03b1 \u03bc \u2194 \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s\ntfae_1_to_6 : SMulInvariantMeasure G \u03b1 \u03bc \u2192 \u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc\ntfae_6_to_7 : (\u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc) \u2192 \u2200 (c : G), MeasurePreserving fun x => c \u2022 x\ntfae_7_to_4 :\n  (\u2200 (c : G), MeasurePreserving fun x => c \u2022 x) \u2192 \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s\n\u22a2 List.TFAE\n    [SMulInvariantMeasure G \u03b1 \u03bc, \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s,\n      \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s,\n      \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s, \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s,\n      \u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc, \u2200 (c : G), MeasurePreserving fun x => c \u2022 x]", "state_after": "case tfae_4_to_5\nG : Type u\nM : Type v\n\u03b1 : Type w\ns : Set \u03b1\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : MulAction G \u03b1\ninst\u271d\u00b9 : MeasurableSpace G\ninst\u271d : MeasurableSMul G \u03b1\nc : G\n\u03bc : Measure \u03b1\ntfae_1_iff_2 :\n  SMulInvariantMeasure G \u03b1 \u03bc \u2194 \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s\ntfae_1_to_6 : SMulInvariantMeasure G \u03b1 \u03bc \u2192 \u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc\ntfae_6_to_7 : (\u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc) \u2192 \u2200 (c : G), MeasurePreserving fun x => c \u2022 x\ntfae_7_to_4 :\n  (\u2200 (c : G), MeasurePreserving fun x => c \u2022 x) \u2192 \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s\n\u22a2 (\u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s) \u2192 \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s\n\nG : Type u\nM : Type v\n\u03b1 : Type w\ns : Set \u03b1\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : MulAction G \u03b1\ninst\u271d\u00b9 : MeasurableSpace G\ninst\u271d : MeasurableSMul G \u03b1\nc : G\n\u03bc : Measure \u03b1\ntfae_1_iff_2 :\n  SMulInvariantMeasure G \u03b1 \u03bc \u2194 \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s\ntfae_1_to_6 : SMulInvariantMeasure G \u03b1 \u03bc \u2192 \u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc\ntfae_6_to_7 : (\u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc) \u2192 \u2200 (c : G), MeasurePreserving fun x => c \u2022 x\ntfae_7_to_4 :\n  (\u2200 (c : G), MeasurePreserving fun x => c \u2022 x) \u2192 \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s\ntfae_4_to_5 : (\u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s) \u2192 \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s\n\u22a2 List.TFAE\n    [SMulInvariantMeasure G \u03b1 \u03bc, \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s,\n      \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s,\n      \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s, \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s,\n      \u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc, \u2200 (c : G), MeasurePreserving fun x => c \u2022 x]"}, {"tactic": "tfae_have 5 \u2192 3", "annotated_tactic": ["tfae_have 5 \u2192 3", []], "state_before": "G : Type u\nM : Type v\n\u03b1 : Type w\ns : Set \u03b1\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : MulAction G \u03b1\ninst\u271d\u00b9 : MeasurableSpace G\ninst\u271d : MeasurableSMul G \u03b1\nc : G\n\u03bc : Measure \u03b1\ntfae_1_iff_2 :\n  SMulInvariantMeasure G \u03b1 \u03bc \u2194 \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s\ntfae_1_to_6 : SMulInvariantMeasure G \u03b1 \u03bc \u2192 \u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc\ntfae_6_to_7 : (\u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc) \u2192 \u2200 (c : G), MeasurePreserving fun x => c \u2022 x\ntfae_7_to_4 :\n  (\u2200 (c : G), MeasurePreserving fun x => c \u2022 x) \u2192 \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s\ntfae_4_to_5 : (\u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s) \u2192 \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s\n\u22a2 List.TFAE\n    [SMulInvariantMeasure G \u03b1 \u03bc, \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s,\n      \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s,\n      \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s, \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s,\n      \u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc, \u2200 (c : G), MeasurePreserving fun x => c \u2022 x]", "state_after": "case tfae_5_to_3\nG : Type u\nM : Type v\n\u03b1 : Type w\ns : Set \u03b1\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : MulAction G \u03b1\ninst\u271d\u00b9 : MeasurableSpace G\ninst\u271d : MeasurableSMul G \u03b1\nc : G\n\u03bc : Measure \u03b1\ntfae_1_iff_2 :\n  SMulInvariantMeasure G \u03b1 \u03bc \u2194 \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s\ntfae_1_to_6 : SMulInvariantMeasure G \u03b1 \u03bc \u2192 \u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc\ntfae_6_to_7 : (\u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc) \u2192 \u2200 (c : G), MeasurePreserving fun x => c \u2022 x\ntfae_7_to_4 :\n  (\u2200 (c : G), MeasurePreserving fun x => c \u2022 x) \u2192 \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s\ntfae_4_to_5 : (\u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s) \u2192 \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s\n\u22a2 (\u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s) \u2192 \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s\n\nG : Type u\nM : Type v\n\u03b1 : Type w\ns : Set \u03b1\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : MulAction G \u03b1\ninst\u271d\u00b9 : MeasurableSpace G\ninst\u271d : MeasurableSMul G \u03b1\nc : G\n\u03bc : Measure \u03b1\ntfae_1_iff_2 :\n  SMulInvariantMeasure G \u03b1 \u03bc \u2194 \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s\ntfae_1_to_6 : SMulInvariantMeasure G \u03b1 \u03bc \u2192 \u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc\ntfae_6_to_7 : (\u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc) \u2192 \u2200 (c : G), MeasurePreserving fun x => c \u2022 x\ntfae_7_to_4 :\n  (\u2200 (c : G), MeasurePreserving fun x => c \u2022 x) \u2192 \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s\ntfae_4_to_5 : (\u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s) \u2192 \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s\ntfae_5_to_3 :\n  (\u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s) \u2192 \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s\n\u22a2 List.TFAE\n    [SMulInvariantMeasure G \u03b1 \u03bc, \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s,\n      \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s,\n      \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s, \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s,\n      \u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc, \u2200 (c : G), MeasurePreserving fun x => c \u2022 x]"}, {"tactic": "tfae_have 3 \u2192 2", "annotated_tactic": ["tfae_have 3 \u2192 2", []], "state_before": "G : Type u\nM : Type v\n\u03b1 : Type w\ns : Set \u03b1\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : MulAction G \u03b1\ninst\u271d\u00b9 : MeasurableSpace G\ninst\u271d : MeasurableSMul G \u03b1\nc : G\n\u03bc : Measure \u03b1\ntfae_1_iff_2 :\n  SMulInvariantMeasure G \u03b1 \u03bc \u2194 \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s\ntfae_1_to_6 : SMulInvariantMeasure G \u03b1 \u03bc \u2192 \u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc\ntfae_6_to_7 : (\u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc) \u2192 \u2200 (c : G), MeasurePreserving fun x => c \u2022 x\ntfae_7_to_4 :\n  (\u2200 (c : G), MeasurePreserving fun x => c \u2022 x) \u2192 \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s\ntfae_4_to_5 : (\u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s) \u2192 \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s\ntfae_5_to_3 :\n  (\u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s) \u2192 \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s\n\u22a2 List.TFAE\n    [SMulInvariantMeasure G \u03b1 \u03bc, \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s,\n      \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s,\n      \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s, \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s,\n      \u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc, \u2200 (c : G), MeasurePreserving fun x => c \u2022 x]", "state_after": "case tfae_3_to_2\nG : Type u\nM : Type v\n\u03b1 : Type w\ns : Set \u03b1\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : MulAction G \u03b1\ninst\u271d\u00b9 : MeasurableSpace G\ninst\u271d : MeasurableSMul G \u03b1\nc : G\n\u03bc : Measure \u03b1\ntfae_1_iff_2 :\n  SMulInvariantMeasure G \u03b1 \u03bc \u2194 \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s\ntfae_1_to_6 : SMulInvariantMeasure G \u03b1 \u03bc \u2192 \u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc\ntfae_6_to_7 : (\u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc) \u2192 \u2200 (c : G), MeasurePreserving fun x => c \u2022 x\ntfae_7_to_4 :\n  (\u2200 (c : G), MeasurePreserving fun x => c \u2022 x) \u2192 \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s\ntfae_4_to_5 : (\u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s) \u2192 \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s\ntfae_5_to_3 :\n  (\u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s) \u2192 \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s\n\u22a2 (\u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s) \u2192\n    \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s\n\nG : Type u\nM : Type v\n\u03b1 : Type w\ns : Set \u03b1\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : MulAction G \u03b1\ninst\u271d\u00b9 : MeasurableSpace G\ninst\u271d : MeasurableSMul G \u03b1\nc : G\n\u03bc : Measure \u03b1\ntfae_1_iff_2 :\n  SMulInvariantMeasure G \u03b1 \u03bc \u2194 \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s\ntfae_1_to_6 : SMulInvariantMeasure G \u03b1 \u03bc \u2192 \u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc\ntfae_6_to_7 : (\u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc) \u2192 \u2200 (c : G), MeasurePreserving fun x => c \u2022 x\ntfae_7_to_4 :\n  (\u2200 (c : G), MeasurePreserving fun x => c \u2022 x) \u2192 \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s\ntfae_4_to_5 : (\u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s) \u2192 \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s\ntfae_5_to_3 :\n  (\u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s) \u2192 \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s\ntfae_3_to_2 :\n  (\u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s) \u2192\n    \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s\n\u22a2 List.TFAE\n    [SMulInvariantMeasure G \u03b1 \u03bc, \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s,\n      \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s,\n      \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s, \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s,\n      \u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc, \u2200 (c : G), MeasurePreserving fun x => c \u2022 x]"}, {"tactic": "tfae_finish", "annotated_tactic": ["tfae_finish", []], "state_before": "G : Type u\nM : Type v\n\u03b1 : Type w\ns : Set \u03b1\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : MulAction G \u03b1\ninst\u271d\u00b9 : MeasurableSpace G\ninst\u271d : MeasurableSMul G \u03b1\nc : G\n\u03bc : Measure \u03b1\ntfae_1_iff_2 :\n  SMulInvariantMeasure G \u03b1 \u03bc \u2194 \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s\ntfae_1_to_6 : SMulInvariantMeasure G \u03b1 \u03bc \u2192 \u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc\ntfae_6_to_7 : (\u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc) \u2192 \u2200 (c : G), MeasurePreserving fun x => c \u2022 x\ntfae_7_to_4 :\n  (\u2200 (c : G), MeasurePreserving fun x => c \u2022 x) \u2192 \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s\ntfae_4_to_5 : (\u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s) \u2192 \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s\ntfae_5_to_3 :\n  (\u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s) \u2192 \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s\ntfae_3_to_2 :\n  (\u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s) \u2192\n    \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s\n\u22a2 List.TFAE\n    [SMulInvariantMeasure G \u03b1 \u03bc, \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s,\n      \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s,\n      \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s, \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s,\n      \u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc, \u2200 (c : G), MeasurePreserving fun x => c \u2022 x]", "state_after": "no goals"}, {"tactic": "exact \u27e8fun h => h.1, fun h => \u27e8h\u27e9\u27e9", "annotated_tactic": ["exact \u27e8fun h => h.1, fun h => \u27e8h\u27e9\u27e9", []], "state_before": "case tfae_1_iff_2\nG : Type u\nM : Type v\n\u03b1 : Type w\ns : Set \u03b1\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : MulAction G \u03b1\ninst\u271d\u00b9 : MeasurableSpace G\ninst\u271d : MeasurableSMul G \u03b1\nc : G\n\u03bc : Measure \u03b1\n\u22a2 SMulInvariantMeasure G \u03b1 \u03bc \u2194 \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s", "state_after": "no goals"}, {"tactic": "intro h c", "annotated_tactic": ["intro h c", []], "state_before": "case tfae_1_to_6\nG : Type u\nM : Type v\n\u03b1 : Type w\ns : Set \u03b1\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : MulAction G \u03b1\ninst\u271d\u00b9 : MeasurableSpace G\ninst\u271d : MeasurableSMul G \u03b1\nc : G\n\u03bc : Measure \u03b1\ntfae_1_iff_2 :\n  SMulInvariantMeasure G \u03b1 \u03bc \u2194 \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s\n\u22a2 SMulInvariantMeasure G \u03b1 \u03bc \u2192 \u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc", "state_after": "case tfae_1_to_6\nG : Type u\nM : Type v\n\u03b1 : Type w\ns : Set \u03b1\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : MulAction G \u03b1\ninst\u271d\u00b9 : MeasurableSpace G\ninst\u271d : MeasurableSMul G \u03b1\nc\u271d : G\n\u03bc : Measure \u03b1\ntfae_1_iff_2 :\n  SMulInvariantMeasure G \u03b1 \u03bc \u2194 \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s\nh : SMulInvariantMeasure G \u03b1 \u03bc\nc : G\n\u22a2 map (fun x => c \u2022 x) \u03bc = \u03bc"}, {"tactic": "exact (measurePreserving_smul c \u03bc).map_eq", "annotated_tactic": ["exact (<a>measurePreserving_smul</a> c \u03bc).<a>map_eq</a>", [{"full_name": "MeasureTheory.measurePreserving_smul", "def_path": "Mathlib/MeasureTheory/Group/Action.lean", "def_pos": [90, 9], "def_end_pos": [90, 31]}, {"full_name": "MeasureTheory.MeasurePreserving.map_eq", "def_path": "Mathlib/Dynamics/Ergodic/MeasurePreserving.lean", "def_pos": [45, 13], "def_end_pos": [45, 19]}]], "state_before": "case tfae_1_to_6\nG : Type u\nM : Type v\n\u03b1 : Type w\ns : Set \u03b1\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : MulAction G \u03b1\ninst\u271d\u00b9 : MeasurableSpace G\ninst\u271d : MeasurableSMul G \u03b1\nc\u271d : G\n\u03bc : Measure \u03b1\ntfae_1_iff_2 :\n  SMulInvariantMeasure G \u03b1 \u03bc \u2194 \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s\nh : SMulInvariantMeasure G \u03b1 \u03bc\nc : G\n\u22a2 map (fun x => c \u2022 x) \u03bc = \u03bc", "state_after": "no goals"}, {"tactic": "exact fun H c => \u27e8measurable_const_smul c, H c\u27e9", "annotated_tactic": ["exact fun H c => \u27e8<a>measurable_const_smul</a> c, H c\u27e9", [{"full_name": "MeasurableSMul.measurable_const_smul", "def_path": "Mathlib/MeasureTheory/Group/Arithmetic.lean", "def_pos": [542, 3], "def_end_pos": [542, 24]}]], "state_before": "case tfae_6_to_7\nG : Type u\nM : Type v\n\u03b1 : Type w\ns : Set \u03b1\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : MulAction G \u03b1\ninst\u271d\u00b9 : MeasurableSpace G\ninst\u271d : MeasurableSMul G \u03b1\nc : G\n\u03bc : Measure \u03b1\ntfae_1_iff_2 :\n  SMulInvariantMeasure G \u03b1 \u03bc \u2194 \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s\ntfae_1_to_6 : SMulInvariantMeasure G \u03b1 \u03bc \u2192 \u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc\n\u22a2 (\u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc) \u2192 \u2200 (c : G), MeasurePreserving fun x => c \u2022 x", "state_after": "no goals"}, {"tactic": "exact fun H c => (H c).measure_preimage_emb (measurableEmbedding_const_smul c)", "annotated_tactic": ["exact fun H c => (H c).<a>measure_preimage_emb</a> (<a>measurableEmbedding_const_smul</a> c)", [{"full_name": "MeasureTheory.MeasurePreserving.measure_preimage_emb", "def_path": "Mathlib/Dynamics/Ergodic/MeasurePreserving.lean", "def_pos": [130, 9], "def_end_pos": [130, 29]}, {"full_name": "measurableEmbedding_const_smul", "def_path": "Mathlib/MeasureTheory/Group/MeasurableEquiv.lean", "def_pos": [57, 9], "def_end_pos": [57, 46]}]], "state_before": "case tfae_7_to_4\nG : Type u\nM : Type v\n\u03b1 : Type w\ns : Set \u03b1\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : MulAction G \u03b1\ninst\u271d\u00b9 : MeasurableSpace G\ninst\u271d : MeasurableSMul G \u03b1\nc : G\n\u03bc : Measure \u03b1\ntfae_1_iff_2 :\n  SMulInvariantMeasure G \u03b1 \u03bc \u2194 \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s\ntfae_1_to_6 : SMulInvariantMeasure G \u03b1 \u03bc \u2192 \u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc\ntfae_6_to_7 : (\u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc) \u2192 \u2200 (c : G), MeasurePreserving fun x => c \u2022 x\n\u22a2 (\u2200 (c : G), MeasurePreserving fun x => c \u2022 x) \u2192 \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s", "state_after": "no goals"}, {"tactic": "exact fun H c s => by\n  rw [\u2190 preimage_smul_inv]\n  apply H", "annotated_tactic": ["exact fun H c s => by\n      rw [\u2190 <a>preimage_smul_inv</a>]\n      apply H", [{"full_name": "Set.preimage_smul_inv", "def_path": "Mathlib/Data/Set/Pointwise/SMul.lean", "def_pos": [905, 9], "def_end_pos": [905, 26]}]], "state_before": "case tfae_4_to_5\nG : Type u\nM : Type v\n\u03b1 : Type w\ns : Set \u03b1\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : MulAction G \u03b1\ninst\u271d\u00b9 : MeasurableSpace G\ninst\u271d : MeasurableSMul G \u03b1\nc : G\n\u03bc : Measure \u03b1\ntfae_1_iff_2 :\n  SMulInvariantMeasure G \u03b1 \u03bc \u2194 \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s\ntfae_1_to_6 : SMulInvariantMeasure G \u03b1 \u03bc \u2192 \u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc\ntfae_6_to_7 : (\u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc) \u2192 \u2200 (c : G), MeasurePreserving fun x => c \u2022 x\ntfae_7_to_4 :\n  (\u2200 (c : G), MeasurePreserving fun x => c \u2022 x) \u2192 \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s\n\u22a2 (\u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s) \u2192 \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s", "state_after": "no goals"}, {"tactic": "rw [\u2190 preimage_smul_inv]", "annotated_tactic": ["rw [\u2190 <a>preimage_smul_inv</a>]", [{"full_name": "Set.preimage_smul_inv", "def_path": "Mathlib/Data/Set/Pointwise/SMul.lean", "def_pos": [905, 9], "def_end_pos": [905, 26]}]], "state_before": "G : Type u\nM : Type v\n\u03b1 : Type w\ns\u271d : Set \u03b1\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : MulAction G \u03b1\ninst\u271d\u00b9 : MeasurableSpace G\ninst\u271d : MeasurableSMul G \u03b1\nc\u271d : G\n\u03bc : Measure \u03b1\ntfae_1_iff_2 :\n  SMulInvariantMeasure G \u03b1 \u03bc \u2194 \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s\ntfae_1_to_6 : SMulInvariantMeasure G \u03b1 \u03bc \u2192 \u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc\ntfae_6_to_7 : (\u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc) \u2192 \u2200 (c : G), MeasurePreserving fun x => c \u2022 x\ntfae_7_to_4 :\n  (\u2200 (c : G), MeasurePreserving fun x => c \u2022 x) \u2192 \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s\nH : \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s\nc : G\ns : Set \u03b1\n\u22a2 \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s", "state_after": "G : Type u\nM : Type v\n\u03b1 : Type w\ns\u271d : Set \u03b1\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : MulAction G \u03b1\ninst\u271d\u00b9 : MeasurableSpace G\ninst\u271d : MeasurableSMul G \u03b1\nc\u271d : G\n\u03bc : Measure \u03b1\ntfae_1_iff_2 :\n  SMulInvariantMeasure G \u03b1 \u03bc \u2194 \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s\ntfae_1_to_6 : SMulInvariantMeasure G \u03b1 \u03bc \u2192 \u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc\ntfae_6_to_7 : (\u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc) \u2192 \u2200 (c : G), MeasurePreserving fun x => c \u2022 x\ntfae_7_to_4 :\n  (\u2200 (c : G), MeasurePreserving fun x => c \u2022 x) \u2192 \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s\nH : \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s\nc : G\ns : Set \u03b1\n\u22a2 \u2191\u2191\u03bc ((fun x => c\u207b\u00b9 \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s"}, {"tactic": "apply H", "annotated_tactic": ["apply H", []], "state_before": "G : Type u\nM : Type v\n\u03b1 : Type w\ns\u271d : Set \u03b1\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : MulAction G \u03b1\ninst\u271d\u00b9 : MeasurableSpace G\ninst\u271d : MeasurableSMul G \u03b1\nc\u271d : G\n\u03bc : Measure \u03b1\ntfae_1_iff_2 :\n  SMulInvariantMeasure G \u03b1 \u03bc \u2194 \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s\ntfae_1_to_6 : SMulInvariantMeasure G \u03b1 \u03bc \u2192 \u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc\ntfae_6_to_7 : (\u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc) \u2192 \u2200 (c : G), MeasurePreserving fun x => c \u2022 x\ntfae_7_to_4 :\n  (\u2200 (c : G), MeasurePreserving fun x => c \u2022 x) \u2192 \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s\nH : \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s\nc : G\ns : Set \u03b1\n\u22a2 \u2191\u2191\u03bc ((fun x => c\u207b\u00b9 \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s", "state_after": "no goals"}, {"tactic": "exact fun H c s _ => H c s", "annotated_tactic": ["exact fun H c s _ => H c s", []], "state_before": "case tfae_5_to_3\nG : Type u\nM : Type v\n\u03b1 : Type w\ns : Set \u03b1\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : MulAction G \u03b1\ninst\u271d\u00b9 : MeasurableSpace G\ninst\u271d : MeasurableSMul G \u03b1\nc : G\n\u03bc : Measure \u03b1\ntfae_1_iff_2 :\n  SMulInvariantMeasure G \u03b1 \u03bc \u2194 \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s\ntfae_1_to_6 : SMulInvariantMeasure G \u03b1 \u03bc \u2192 \u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc\ntfae_6_to_7 : (\u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc) \u2192 \u2200 (c : G), MeasurePreserving fun x => c \u2022 x\ntfae_7_to_4 :\n  (\u2200 (c : G), MeasurePreserving fun x => c \u2022 x) \u2192 \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s\ntfae_4_to_5 : (\u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s) \u2192 \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s\n\u22a2 (\u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s) \u2192 \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s", "state_after": "no goals"}, {"tactic": "intro H c s hs", "annotated_tactic": ["intro H c s hs", []], "state_before": "case tfae_3_to_2\nG : Type u\nM : Type v\n\u03b1 : Type w\ns : Set \u03b1\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : MulAction G \u03b1\ninst\u271d\u00b9 : MeasurableSpace G\ninst\u271d : MeasurableSMul G \u03b1\nc : G\n\u03bc : Measure \u03b1\ntfae_1_iff_2 :\n  SMulInvariantMeasure G \u03b1 \u03bc \u2194 \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s\ntfae_1_to_6 : SMulInvariantMeasure G \u03b1 \u03bc \u2192 \u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc\ntfae_6_to_7 : (\u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc) \u2192 \u2200 (c : G), MeasurePreserving fun x => c \u2022 x\ntfae_7_to_4 :\n  (\u2200 (c : G), MeasurePreserving fun x => c \u2022 x) \u2192 \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s\ntfae_4_to_5 : (\u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s) \u2192 \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s\ntfae_5_to_3 :\n  (\u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s) \u2192 \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s\n\u22a2 (\u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s) \u2192\n    \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s", "state_after": "case tfae_3_to_2\nG : Type u\nM : Type v\n\u03b1 : Type w\ns\u271d : Set \u03b1\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : MulAction G \u03b1\ninst\u271d\u00b9 : MeasurableSpace G\ninst\u271d : MeasurableSMul G \u03b1\nc\u271d : G\n\u03bc : Measure \u03b1\ntfae_1_iff_2 :\n  SMulInvariantMeasure G \u03b1 \u03bc \u2194 \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s\ntfae_1_to_6 : SMulInvariantMeasure G \u03b1 \u03bc \u2192 \u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc\ntfae_6_to_7 : (\u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc) \u2192 \u2200 (c : G), MeasurePreserving fun x => c \u2022 x\ntfae_7_to_4 :\n  (\u2200 (c : G), MeasurePreserving fun x => c \u2022 x) \u2192 \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s\ntfae_4_to_5 : (\u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s) \u2192 \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s\ntfae_5_to_3 :\n  (\u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s) \u2192 \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s\nH : \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s\nc : G\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s"}, {"tactic": "rw [preimage_smul]", "annotated_tactic": ["rw [<a>preimage_smul</a>]", [{"full_name": "Set.preimage_smul", "def_path": "Mathlib/Data/Set/Pointwise/SMul.lean", "def_pos": [899, 9], "def_end_pos": [899, 22]}]], "state_before": "case tfae_3_to_2\nG : Type u\nM : Type v\n\u03b1 : Type w\ns\u271d : Set \u03b1\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : MulAction G \u03b1\ninst\u271d\u00b9 : MeasurableSpace G\ninst\u271d : MeasurableSMul G \u03b1\nc\u271d : G\n\u03bc : Measure \u03b1\ntfae_1_iff_2 :\n  SMulInvariantMeasure G \u03b1 \u03bc \u2194 \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s\ntfae_1_to_6 : SMulInvariantMeasure G \u03b1 \u03bc \u2192 \u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc\ntfae_6_to_7 : (\u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc) \u2192 \u2200 (c : G), MeasurePreserving fun x => c \u2022 x\ntfae_7_to_4 :\n  (\u2200 (c : G), MeasurePreserving fun x => c \u2022 x) \u2192 \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s\ntfae_4_to_5 : (\u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s) \u2192 \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s\ntfae_5_to_3 :\n  (\u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s) \u2192 \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s\nH : \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s\nc : G\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s", "state_after": "case tfae_3_to_2\nG : Type u\nM : Type v\n\u03b1 : Type w\ns\u271d : Set \u03b1\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : MulAction G \u03b1\ninst\u271d\u00b9 : MeasurableSpace G\ninst\u271d : MeasurableSMul G \u03b1\nc\u271d : G\n\u03bc : Measure \u03b1\ntfae_1_iff_2 :\n  SMulInvariantMeasure G \u03b1 \u03bc \u2194 \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s\ntfae_1_to_6 : SMulInvariantMeasure G \u03b1 \u03bc \u2192 \u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc\ntfae_6_to_7 : (\u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc) \u2192 \u2200 (c : G), MeasurePreserving fun x => c \u2022 x\ntfae_7_to_4 :\n  (\u2200 (c : G), MeasurePreserving fun x => c \u2022 x) \u2192 \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s\ntfae_4_to_5 : (\u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s) \u2192 \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s\ntfae_5_to_3 :\n  (\u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s) \u2192 \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s\nH : \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s\nc : G\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191\u2191\u03bc (c\u207b\u00b9 \u2022 s) = \u2191\u2191\u03bc s"}, {"tactic": "exact H c\u207b\u00b9 s hs", "annotated_tactic": ["exact H c\u207b\u00b9 s hs", []], "state_before": "case tfae_3_to_2\nG : Type u\nM : Type v\n\u03b1 : Type w\ns\u271d : Set \u03b1\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : MulAction G \u03b1\ninst\u271d\u00b9 : MeasurableSpace G\ninst\u271d : MeasurableSMul G \u03b1\nc\u271d : G\n\u03bc : Measure \u03b1\ntfae_1_iff_2 :\n  SMulInvariantMeasure G \u03b1 \u03bc \u2194 \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s\ntfae_1_to_6 : SMulInvariantMeasure G \u03b1 \u03bc \u2192 \u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc\ntfae_6_to_7 : (\u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc) \u2192 \u2200 (c : G), MeasurePreserving fun x => c \u2022 x\ntfae_7_to_4 :\n  (\u2200 (c : G), MeasurePreserving fun x => c \u2022 x) \u2192 \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s\ntfae_4_to_5 : (\u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s) \u2192 \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s\ntfae_5_to_3 :\n  (\u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s) \u2192 \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s\nH : \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s\nc : G\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191\u2191\u03bc (c\u207b\u00b9 \u2022 s) = \u2191\u2191\u03bc s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Nat/Gcd.lean", "full_name": "Nat.gcd_mul_right_right", "start": [159, 9], "end": [160, 36], "traced_tactics": [{"tactic": "rw [gcd_comm, gcd_mul_right_left]", "annotated_tactic": ["rw [<a>gcd_comm</a>, <a>gcd_mul_right_left</a>]", [{"full_name": "Nat.gcd_comm", "def_path": "lake-packages/std/Std/Data/Nat/Gcd.lean", "def_pos": [59, 9], "def_end_pos": [59, 17]}, {"full_name": "Nat.gcd_mul_right_left", "def_path": "lake-packages/std/Std/Data/Nat/Gcd.lean", "def_pos": [156, 17], "def_end_pos": [156, 35]}]], "state_before": "m n : Nat\n\u22a2 gcd n (n * m) = n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/FundThmCalculus.lean", "full_name": "intervalIntegral.measure_integral_sub_integral_sub_linear_isLittleO_of_tendsto_ae", "start": [408, 1], "end": [435, 64], "traced_tactics": [{"tactic": "haveI := FTCFilter.meas_gen la", "annotated_tactic": ["haveI := <a>FTCFilter.meas_gen</a> la", [{"full_name": "intervalIntegral.FTCFilter.meas_gen", "def_path": "Mathlib/MeasureTheory/Integral/FundThmCalculus.lean", "def_pos": [201, 4], "def_end_pos": [201, 12]}]], "state_before": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : NormedSpace \u211d E\nf : \u211d \u2192 E\na b : \u211d\nc ca cb : E\nl l' la la' lb lb' : Filter \u211d\nlt : Filter \u03b9\n\u03bc : Measure \u211d\nu v ua va ub vb : \u03b9 \u2192 \u211d\ninst\u271d\u00b2 : FTCFilter a la la'\ninst\u271d\u00b9 : FTCFilter b lb lb'\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhab : IntervalIntegrable f \u03bc a b\nhmeas_a : StronglyMeasurableAtFilter f la'\nhmeas_b : StronglyMeasurableAtFilter f lb'\nha_lim : Tendsto f (la' \u2293 Measure.ae \u03bc) (\ud835\udcdd ca)\nhb_lim : Tendsto f (lb' \u2293 Measure.ae \u03bc) (\ud835\udcdd cb)\nhua : Tendsto ua lt la\nhva : Tendsto va lt la\nhub : Tendsto ub lt lb\nhvb : Tendsto vb lt lb\n\u22a2 (fun t =>\n      \u222b (x : \u211d) in va t..vb t, f x \u2202\u03bc - \u222b (x : \u211d) in ua t..ub t, f x \u2202\u03bc -\n        (\u222b (x : \u211d) in ub t..vb t, cb \u2202\u03bc - \u222b (x : \u211d) in ua t..va t, ca \u2202\u03bc)) =o[lt]\n    fun t => \u2016\u222b (x : \u211d) in ua t..va t, 1 \u2202\u03bc\u2016 + \u2016\u222b (x : \u211d) in ub t..vb t, 1 \u2202\u03bc\u2016", "state_after": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : NormedSpace \u211d E\nf : \u211d \u2192 E\na b : \u211d\nc ca cb : E\nl l' la la' lb lb' : Filter \u211d\nlt : Filter \u03b9\n\u03bc : Measure \u211d\nu v ua va ub vb : \u03b9 \u2192 \u211d\ninst\u271d\u00b2 : FTCFilter a la la'\ninst\u271d\u00b9 : FTCFilter b lb lb'\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhab : IntervalIntegrable f \u03bc a b\nhmeas_a : StronglyMeasurableAtFilter f la'\nhmeas_b : StronglyMeasurableAtFilter f lb'\nha_lim : Tendsto f (la' \u2293 Measure.ae \u03bc) (\ud835\udcdd ca)\nhb_lim : Tendsto f (lb' \u2293 Measure.ae \u03bc) (\ud835\udcdd cb)\nhua : Tendsto ua lt la\nhva : Tendsto va lt la\nhub : Tendsto ub lt lb\nhvb : Tendsto vb lt lb\nthis : IsMeasurablyGenerated la'\n\u22a2 (fun t =>\n      \u222b (x : \u211d) in va t..vb t, f x \u2202\u03bc - \u222b (x : \u211d) in ua t..ub t, f x \u2202\u03bc -\n        (\u222b (x : \u211d) in ub t..vb t, cb \u2202\u03bc - \u222b (x : \u211d) in ua t..va t, ca \u2202\u03bc)) =o[lt]\n    fun t => \u2016\u222b (x : \u211d) in ua t..va t, 1 \u2202\u03bc\u2016 + \u2016\u222b (x : \u211d) in ub t..vb t, 1 \u2202\u03bc\u2016"}, {"tactic": "haveI := FTCFilter.meas_gen lb", "annotated_tactic": ["haveI := <a>FTCFilter.meas_gen</a> lb", [{"full_name": "intervalIntegral.FTCFilter.meas_gen", "def_path": "Mathlib/MeasureTheory/Integral/FundThmCalculus.lean", "def_pos": [201, 4], "def_end_pos": [201, 12]}]], "state_before": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : NormedSpace \u211d E\nf : \u211d \u2192 E\na b : \u211d\nc ca cb : E\nl l' la la' lb lb' : Filter \u211d\nlt : Filter \u03b9\n\u03bc : Measure \u211d\nu v ua va ub vb : \u03b9 \u2192 \u211d\ninst\u271d\u00b2 : FTCFilter a la la'\ninst\u271d\u00b9 : FTCFilter b lb lb'\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhab : IntervalIntegrable f \u03bc a b\nhmeas_a : StronglyMeasurableAtFilter f la'\nhmeas_b : StronglyMeasurableAtFilter f lb'\nha_lim : Tendsto f (la' \u2293 Measure.ae \u03bc) (\ud835\udcdd ca)\nhb_lim : Tendsto f (lb' \u2293 Measure.ae \u03bc) (\ud835\udcdd cb)\nhua : Tendsto ua lt la\nhva : Tendsto va lt la\nhub : Tendsto ub lt lb\nhvb : Tendsto vb lt lb\nthis : IsMeasurablyGenerated la'\n\u22a2 (fun t =>\n      \u222b (x : \u211d) in va t..vb t, f x \u2202\u03bc - \u222b (x : \u211d) in ua t..ub t, f x \u2202\u03bc -\n        (\u222b (x : \u211d) in ub t..vb t, cb \u2202\u03bc - \u222b (x : \u211d) in ua t..va t, ca \u2202\u03bc)) =o[lt]\n    fun t => \u2016\u222b (x : \u211d) in ua t..va t, 1 \u2202\u03bc\u2016 + \u2016\u222b (x : \u211d) in ub t..vb t, 1 \u2202\u03bc\u2016", "state_after": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : NormedSpace \u211d E\nf : \u211d \u2192 E\na b : \u211d\nc ca cb : E\nl l' la la' lb lb' : Filter \u211d\nlt : Filter \u03b9\n\u03bc : Measure \u211d\nu v ua va ub vb : \u03b9 \u2192 \u211d\ninst\u271d\u00b2 : FTCFilter a la la'\ninst\u271d\u00b9 : FTCFilter b lb lb'\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhab : IntervalIntegrable f \u03bc a b\nhmeas_a : StronglyMeasurableAtFilter f la'\nhmeas_b : StronglyMeasurableAtFilter f lb'\nha_lim : Tendsto f (la' \u2293 Measure.ae \u03bc) (\ud835\udcdd ca)\nhb_lim : Tendsto f (lb' \u2293 Measure.ae \u03bc) (\ud835\udcdd cb)\nhua : Tendsto ua lt la\nhva : Tendsto va lt la\nhub : Tendsto ub lt lb\nhvb : Tendsto vb lt lb\nthis\u271d : IsMeasurablyGenerated la'\nthis : IsMeasurablyGenerated lb'\n\u22a2 (fun t =>\n      \u222b (x : \u211d) in va t..vb t, f x \u2202\u03bc - \u222b (x : \u211d) in ua t..ub t, f x \u2202\u03bc -\n        (\u222b (x : \u211d) in ub t..vb t, cb \u2202\u03bc - \u222b (x : \u211d) in ua t..va t, ca \u2202\u03bc)) =o[lt]\n    fun t => \u2016\u222b (x : \u211d) in ua t..va t, 1 \u2202\u03bc\u2016 + \u2016\u222b (x : \u211d) in ub t..vb t, 1 \u2202\u03bc\u2016"}, {"tactic": "refine'\n  ((measure_integral_sub_linear_isLittleO_of_tendsto_ae hmeas_a ha_lim hua hva).neg_left.add_add\n        (measure_integral_sub_linear_isLittleO_of_tendsto_ae hmeas_b hb_lim hub hvb)).congr'\n    _ EventuallyEq.rfl", "annotated_tactic": ["refine'\n    ((<a>measure_integral_sub_linear_isLittleO_of_tendsto_ae</a> hmeas_a ha_lim hua hva).neg_left.add_add\n          (<a>measure_integral_sub_linear_isLittleO_of_tendsto_ae</a> hmeas_b hb_lim hub hvb)).<a>congr'</a>\n      _ <a>EventuallyEq.rfl</a>", [{"full_name": "intervalIntegral.measure_integral_sub_linear_isLittleO_of_tendsto_ae", "def_path": "Mathlib/MeasureTheory/Integral/FundThmCalculus.lean", "def_pos": [347, 9], "def_end_pos": [347, 60]}, {"full_name": "intervalIntegral.measure_integral_sub_linear_isLittleO_of_tendsto_ae", "def_path": "Mathlib/MeasureTheory/Integral/FundThmCalculus.lean", "def_pos": [347, 9], "def_end_pos": [347, 60]}, {"full_name": "Asymptotics.IsLittleO.congr'", "def_path": "Mathlib/Analysis/Asymptotics/Asymptotics.lean", "def_pos": [388, 9], "def_end_pos": [388, 25]}, {"full_name": "Filter.EventuallyEq.rfl", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1493, 9], "def_end_pos": [1493, 25]}]], "state_before": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : NormedSpace \u211d E\nf : \u211d \u2192 E\na b : \u211d\nc ca cb : E\nl l' la la' lb lb' : Filter \u211d\nlt : Filter \u03b9\n\u03bc : Measure \u211d\nu v ua va ub vb : \u03b9 \u2192 \u211d\ninst\u271d\u00b2 : FTCFilter a la la'\ninst\u271d\u00b9 : FTCFilter b lb lb'\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhab : IntervalIntegrable f \u03bc a b\nhmeas_a : StronglyMeasurableAtFilter f la'\nhmeas_b : StronglyMeasurableAtFilter f lb'\nha_lim : Tendsto f (la' \u2293 Measure.ae \u03bc) (\ud835\udcdd ca)\nhb_lim : Tendsto f (lb' \u2293 Measure.ae \u03bc) (\ud835\udcdd cb)\nhua : Tendsto ua lt la\nhva : Tendsto va lt la\nhub : Tendsto ub lt lb\nhvb : Tendsto vb lt lb\nthis\u271d : IsMeasurablyGenerated la'\nthis : IsMeasurablyGenerated lb'\n\u22a2 (fun t =>\n      \u222b (x : \u211d) in va t..vb t, f x \u2202\u03bc - \u222b (x : \u211d) in ua t..ub t, f x \u2202\u03bc -\n        (\u222b (x : \u211d) in ub t..vb t, cb \u2202\u03bc - \u222b (x : \u211d) in ua t..va t, ca \u2202\u03bc)) =o[lt]\n    fun t => \u2016\u222b (x : \u211d) in ua t..va t, 1 \u2202\u03bc\u2016 + \u2016\u222b (x : \u211d) in ub t..vb t, 1 \u2202\u03bc\u2016", "state_after": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : NormedSpace \u211d E\nf : \u211d \u2192 E\na b : \u211d\nc ca cb : E\nl l' la la' lb lb' : Filter \u211d\nlt : Filter \u03b9\n\u03bc : Measure \u211d\nu v ua va ub vb : \u03b9 \u2192 \u211d\ninst\u271d\u00b2 : FTCFilter a la la'\ninst\u271d\u00b9 : FTCFilter b lb lb'\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhab : IntervalIntegrable f \u03bc a b\nhmeas_a : StronglyMeasurableAtFilter f la'\nhmeas_b : StronglyMeasurableAtFilter f lb'\nha_lim : Tendsto f (la' \u2293 Measure.ae \u03bc) (\ud835\udcdd ca)\nhb_lim : Tendsto f (lb' \u2293 Measure.ae \u03bc) (\ud835\udcdd cb)\nhua : Tendsto ua lt la\nhva : Tendsto va lt la\nhub : Tendsto ub lt lb\nhvb : Tendsto vb lt lb\nthis\u271d : IsMeasurablyGenerated la'\nthis : IsMeasurablyGenerated lb'\n\u22a2 (fun x =>\n      -(\u222b (x : \u211d) in ua x..va x, f x \u2202\u03bc - \u222b (x : \u211d) in ua x..va x, ca \u2202\u03bc) +\n        (\u222b (x : \u211d) in ub x..vb x, f x \u2202\u03bc - \u222b (x : \u211d) in ub x..vb x, cb \u2202\u03bc)) =\u1da0[lt]\n    fun t =>\n    \u222b (x : \u211d) in va t..vb t, f x \u2202\u03bc - \u222b (x : \u211d) in ua t..ub t, f x \u2202\u03bc -\n      (\u222b (x : \u211d) in ub t..vb t, cb \u2202\u03bc - \u222b (x : \u211d) in ua t..va t, ca \u2202\u03bc)"}, {"tactic": "have A : \u2200\u1da0 t in lt, IntervalIntegrable f \u03bc (ua t) (va t) :=\n  ha_lim.eventually_intervalIntegrable_ae hmeas_a (FTCFilter.finiteAt_inner la) hua hva", "annotated_tactic": ["have A : \u2200\u1da0 t in lt, <a>IntervalIntegrable</a> f \u03bc (ua t) (va t) :=\n    ha_lim.eventually_intervalIntegrable_ae hmeas_a (<a>FTCFilter.finiteAt_inner</a> la) hua hva", [{"full_name": "IntervalIntegrable", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [70, 5], "def_end_pos": [70, 23]}, {"full_name": "intervalIntegral.FTCFilter.finiteAt_inner", "def_path": "Mathlib/MeasureTheory/Integral/FundThmCalculus.lean", "def_pos": [218, 9], "def_end_pos": [218, 23]}]], "state_before": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : NormedSpace \u211d E\nf : \u211d \u2192 E\na b : \u211d\nc ca cb : E\nl l' la la' lb lb' : Filter \u211d\nlt : Filter \u03b9\n\u03bc : Measure \u211d\nu v ua va ub vb : \u03b9 \u2192 \u211d\ninst\u271d\u00b2 : FTCFilter a la la'\ninst\u271d\u00b9 : FTCFilter b lb lb'\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhab : IntervalIntegrable f \u03bc a b\nhmeas_a : StronglyMeasurableAtFilter f la'\nhmeas_b : StronglyMeasurableAtFilter f lb'\nha_lim : Tendsto f (la' \u2293 Measure.ae \u03bc) (\ud835\udcdd ca)\nhb_lim : Tendsto f (lb' \u2293 Measure.ae \u03bc) (\ud835\udcdd cb)\nhua : Tendsto ua lt la\nhva : Tendsto va lt la\nhub : Tendsto ub lt lb\nhvb : Tendsto vb lt lb\nthis\u271d : IsMeasurablyGenerated la'\nthis : IsMeasurablyGenerated lb'\n\u22a2 (fun x =>\n      -(\u222b (x : \u211d) in ua x..va x, f x \u2202\u03bc - \u222b (x : \u211d) in ua x..va x, ca \u2202\u03bc) +\n        (\u222b (x : \u211d) in ub x..vb x, f x \u2202\u03bc - \u222b (x : \u211d) in ub x..vb x, cb \u2202\u03bc)) =\u1da0[lt]\n    fun t =>\n    \u222b (x : \u211d) in va t..vb t, f x \u2202\u03bc - \u222b (x : \u211d) in ua t..ub t, f x \u2202\u03bc -\n      (\u222b (x : \u211d) in ub t..vb t, cb \u2202\u03bc - \u222b (x : \u211d) in ua t..va t, ca \u2202\u03bc)", "state_after": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA\u271d : Type u_5\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : NormedSpace \u211d E\nf : \u211d \u2192 E\na b : \u211d\nc ca cb : E\nl l' la la' lb lb' : Filter \u211d\nlt : Filter \u03b9\n\u03bc : Measure \u211d\nu v ua va ub vb : \u03b9 \u2192 \u211d\ninst\u271d\u00b2 : FTCFilter a la la'\ninst\u271d\u00b9 : FTCFilter b lb lb'\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhab : IntervalIntegrable f \u03bc a b\nhmeas_a : StronglyMeasurableAtFilter f la'\nhmeas_b : StronglyMeasurableAtFilter f lb'\nha_lim : Tendsto f (la' \u2293 Measure.ae \u03bc) (\ud835\udcdd ca)\nhb_lim : Tendsto f (lb' \u2293 Measure.ae \u03bc) (\ud835\udcdd cb)\nhua : Tendsto ua lt la\nhva : Tendsto va lt la\nhub : Tendsto ub lt lb\nhvb : Tendsto vb lt lb\nthis\u271d : IsMeasurablyGenerated la'\nthis : IsMeasurablyGenerated lb'\nA : \u2200\u1da0 (t : \u03b9) in lt, IntervalIntegrable f \u03bc (ua t) (va t)\n\u22a2 (fun x =>\n      -(\u222b (x : \u211d) in ua x..va x, f x \u2202\u03bc - \u222b (x : \u211d) in ua x..va x, ca \u2202\u03bc) +\n        (\u222b (x : \u211d) in ub x..vb x, f x \u2202\u03bc - \u222b (x : \u211d) in ub x..vb x, cb \u2202\u03bc)) =\u1da0[lt]\n    fun t =>\n    \u222b (x : \u211d) in va t..vb t, f x \u2202\u03bc - \u222b (x : \u211d) in ua t..ub t, f x \u2202\u03bc -\n      (\u222b (x : \u211d) in ub t..vb t, cb \u2202\u03bc - \u222b (x : \u211d) in ua t..va t, ca \u2202\u03bc)"}, {"tactic": "have A' : \u2200\u1da0 t in lt, IntervalIntegrable f \u03bc a (ua t) :=\n  ha_lim.eventually_intervalIntegrable_ae hmeas_a (FTCFilter.finiteAt_inner la)\n    (tendsto_const_pure.mono_right FTCFilter.pure_le) hua", "annotated_tactic": ["have A' : \u2200\u1da0 t in lt, <a>IntervalIntegrable</a> f \u03bc a (ua t) :=\n    ha_lim.eventually_intervalIntegrable_ae hmeas_a (<a>FTCFilter.finiteAt_inner</a> la)\n      (tendsto_const_pure.mono_right <a>FTCFilter.pure_le</a>) hua", [{"full_name": "IntervalIntegrable", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [70, 5], "def_end_pos": [70, 23]}, {"full_name": "intervalIntegral.FTCFilter.finiteAt_inner", "def_path": "Mathlib/MeasureTheory/Integral/FundThmCalculus.lean", "def_pos": [218, 9], "def_end_pos": [218, 23]}, {"full_name": "intervalIntegral.FTCFilter.pure_le", "def_path": "Mathlib/MeasureTheory/Integral/FundThmCalculus.lean", "def_pos": [199, 3], "def_end_pos": [199, 10]}]], "state_before": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA\u271d : Type u_5\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : NormedSpace \u211d E\nf : \u211d \u2192 E\na b : \u211d\nc ca cb : E\nl l' la la' lb lb' : Filter \u211d\nlt : Filter \u03b9\n\u03bc : Measure \u211d\nu v ua va ub vb : \u03b9 \u2192 \u211d\ninst\u271d\u00b2 : FTCFilter a la la'\ninst\u271d\u00b9 : FTCFilter b lb lb'\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhab : IntervalIntegrable f \u03bc a b\nhmeas_a : StronglyMeasurableAtFilter f la'\nhmeas_b : StronglyMeasurableAtFilter f lb'\nha_lim : Tendsto f (la' \u2293 Measure.ae \u03bc) (\ud835\udcdd ca)\nhb_lim : Tendsto f (lb' \u2293 Measure.ae \u03bc) (\ud835\udcdd cb)\nhua : Tendsto ua lt la\nhva : Tendsto va lt la\nhub : Tendsto ub lt lb\nhvb : Tendsto vb lt lb\nthis\u271d : IsMeasurablyGenerated la'\nthis : IsMeasurablyGenerated lb'\nA : \u2200\u1da0 (t : \u03b9) in lt, IntervalIntegrable f \u03bc (ua t) (va t)\n\u22a2 (fun x =>\n      -(\u222b (x : \u211d) in ua x..va x, f x \u2202\u03bc - \u222b (x : \u211d) in ua x..va x, ca \u2202\u03bc) +\n        (\u222b (x : \u211d) in ub x..vb x, f x \u2202\u03bc - \u222b (x : \u211d) in ub x..vb x, cb \u2202\u03bc)) =\u1da0[lt]\n    fun t =>\n    \u222b (x : \u211d) in va t..vb t, f x \u2202\u03bc - \u222b (x : \u211d) in ua t..ub t, f x \u2202\u03bc -\n      (\u222b (x : \u211d) in ub t..vb t, cb \u2202\u03bc - \u222b (x : \u211d) in ua t..va t, ca \u2202\u03bc)", "state_after": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA\u271d : Type u_5\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : NormedSpace \u211d E\nf : \u211d \u2192 E\na b : \u211d\nc ca cb : E\nl l' la la' lb lb' : Filter \u211d\nlt : Filter \u03b9\n\u03bc : Measure \u211d\nu v ua va ub vb : \u03b9 \u2192 \u211d\ninst\u271d\u00b2 : FTCFilter a la la'\ninst\u271d\u00b9 : FTCFilter b lb lb'\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhab : IntervalIntegrable f \u03bc a b\nhmeas_a : StronglyMeasurableAtFilter f la'\nhmeas_b : StronglyMeasurableAtFilter f lb'\nha_lim : Tendsto f (la' \u2293 Measure.ae \u03bc) (\ud835\udcdd ca)\nhb_lim : Tendsto f (lb' \u2293 Measure.ae \u03bc) (\ud835\udcdd cb)\nhua : Tendsto ua lt la\nhva : Tendsto va lt la\nhub : Tendsto ub lt lb\nhvb : Tendsto vb lt lb\nthis\u271d : IsMeasurablyGenerated la'\nthis : IsMeasurablyGenerated lb'\nA : \u2200\u1da0 (t : \u03b9) in lt, IntervalIntegrable f \u03bc (ua t) (va t)\nA' : \u2200\u1da0 (t : \u03b9) in lt, IntervalIntegrable f \u03bc a (ua t)\n\u22a2 (fun x =>\n      -(\u222b (x : \u211d) in ua x..va x, f x \u2202\u03bc - \u222b (x : \u211d) in ua x..va x, ca \u2202\u03bc) +\n        (\u222b (x : \u211d) in ub x..vb x, f x \u2202\u03bc - \u222b (x : \u211d) in ub x..vb x, cb \u2202\u03bc)) =\u1da0[lt]\n    fun t =>\n    \u222b (x : \u211d) in va t..vb t, f x \u2202\u03bc - \u222b (x : \u211d) in ua t..ub t, f x \u2202\u03bc -\n      (\u222b (x : \u211d) in ub t..vb t, cb \u2202\u03bc - \u222b (x : \u211d) in ua t..va t, ca \u2202\u03bc)"}, {"tactic": "have B : \u2200\u1da0 t in lt, IntervalIntegrable f \u03bc (ub t) (vb t) :=\n  hb_lim.eventually_intervalIntegrable_ae hmeas_b (FTCFilter.finiteAt_inner lb) hub hvb", "annotated_tactic": ["have B : \u2200\u1da0 t in lt, <a>IntervalIntegrable</a> f \u03bc (ub t) (vb t) :=\n    hb_lim.eventually_intervalIntegrable_ae hmeas_b (<a>FTCFilter.finiteAt_inner</a> lb) hub hvb", [{"full_name": "IntervalIntegrable", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [70, 5], "def_end_pos": [70, 23]}, {"full_name": "intervalIntegral.FTCFilter.finiteAt_inner", "def_path": "Mathlib/MeasureTheory/Integral/FundThmCalculus.lean", "def_pos": [218, 9], "def_end_pos": [218, 23]}]], "state_before": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA\u271d : Type u_5\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : NormedSpace \u211d E\nf : \u211d \u2192 E\na b : \u211d\nc ca cb : E\nl l' la la' lb lb' : Filter \u211d\nlt : Filter \u03b9\n\u03bc : Measure \u211d\nu v ua va ub vb : \u03b9 \u2192 \u211d\ninst\u271d\u00b2 : FTCFilter a la la'\ninst\u271d\u00b9 : FTCFilter b lb lb'\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhab : IntervalIntegrable f \u03bc a b\nhmeas_a : StronglyMeasurableAtFilter f la'\nhmeas_b : StronglyMeasurableAtFilter f lb'\nha_lim : Tendsto f (la' \u2293 Measure.ae \u03bc) (\ud835\udcdd ca)\nhb_lim : Tendsto f (lb' \u2293 Measure.ae \u03bc) (\ud835\udcdd cb)\nhua : Tendsto ua lt la\nhva : Tendsto va lt la\nhub : Tendsto ub lt lb\nhvb : Tendsto vb lt lb\nthis\u271d : IsMeasurablyGenerated la'\nthis : IsMeasurablyGenerated lb'\nA : \u2200\u1da0 (t : \u03b9) in lt, IntervalIntegrable f \u03bc (ua t) (va t)\nA' : \u2200\u1da0 (t : \u03b9) in lt, IntervalIntegrable f \u03bc a (ua t)\n\u22a2 (fun x =>\n      -(\u222b (x : \u211d) in ua x..va x, f x \u2202\u03bc - \u222b (x : \u211d) in ua x..va x, ca \u2202\u03bc) +\n        (\u222b (x : \u211d) in ub x..vb x, f x \u2202\u03bc - \u222b (x : \u211d) in ub x..vb x, cb \u2202\u03bc)) =\u1da0[lt]\n    fun t =>\n    \u222b (x : \u211d) in va t..vb t, f x \u2202\u03bc - \u222b (x : \u211d) in ua t..ub t, f x \u2202\u03bc -\n      (\u222b (x : \u211d) in ub t..vb t, cb \u2202\u03bc - \u222b (x : \u211d) in ua t..va t, ca \u2202\u03bc)", "state_after": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA\u271d : Type u_5\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : NormedSpace \u211d E\nf : \u211d \u2192 E\na b : \u211d\nc ca cb : E\nl l' la la' lb lb' : Filter \u211d\nlt : Filter \u03b9\n\u03bc : Measure \u211d\nu v ua va ub vb : \u03b9 \u2192 \u211d\ninst\u271d\u00b2 : FTCFilter a la la'\ninst\u271d\u00b9 : FTCFilter b lb lb'\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhab : IntervalIntegrable f \u03bc a b\nhmeas_a : StronglyMeasurableAtFilter f la'\nhmeas_b : StronglyMeasurableAtFilter f lb'\nha_lim : Tendsto f (la' \u2293 Measure.ae \u03bc) (\ud835\udcdd ca)\nhb_lim : Tendsto f (lb' \u2293 Measure.ae \u03bc) (\ud835\udcdd cb)\nhua : Tendsto ua lt la\nhva : Tendsto va lt la\nhub : Tendsto ub lt lb\nhvb : Tendsto vb lt lb\nthis\u271d : IsMeasurablyGenerated la'\nthis : IsMeasurablyGenerated lb'\nA : \u2200\u1da0 (t : \u03b9) in lt, IntervalIntegrable f \u03bc (ua t) (va t)\nA' : \u2200\u1da0 (t : \u03b9) in lt, IntervalIntegrable f \u03bc a (ua t)\nB : \u2200\u1da0 (t : \u03b9) in lt, IntervalIntegrable f \u03bc (ub t) (vb t)\n\u22a2 (fun x =>\n      -(\u222b (x : \u211d) in ua x..va x, f x \u2202\u03bc - \u222b (x : \u211d) in ua x..va x, ca \u2202\u03bc) +\n        (\u222b (x : \u211d) in ub x..vb x, f x \u2202\u03bc - \u222b (x : \u211d) in ub x..vb x, cb \u2202\u03bc)) =\u1da0[lt]\n    fun t =>\n    \u222b (x : \u211d) in va t..vb t, f x \u2202\u03bc - \u222b (x : \u211d) in ua t..ub t, f x \u2202\u03bc -\n      (\u222b (x : \u211d) in ub t..vb t, cb \u2202\u03bc - \u222b (x : \u211d) in ua t..va t, ca \u2202\u03bc)"}, {"tactic": "have B' : \u2200\u1da0 t in lt, IntervalIntegrable f \u03bc b (ub t) :=\n  hb_lim.eventually_intervalIntegrable_ae hmeas_b (FTCFilter.finiteAt_inner lb)\n    (tendsto_const_pure.mono_right FTCFilter.pure_le) hub", "annotated_tactic": ["have B' : \u2200\u1da0 t in lt, <a>IntervalIntegrable</a> f \u03bc b (ub t) :=\n    hb_lim.eventually_intervalIntegrable_ae hmeas_b (<a>FTCFilter.finiteAt_inner</a> lb)\n      (tendsto_const_pure.mono_right <a>FTCFilter.pure_le</a>) hub", [{"full_name": "IntervalIntegrable", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [70, 5], "def_end_pos": [70, 23]}, {"full_name": "intervalIntegral.FTCFilter.finiteAt_inner", "def_path": "Mathlib/MeasureTheory/Integral/FundThmCalculus.lean", "def_pos": [218, 9], "def_end_pos": [218, 23]}, {"full_name": "intervalIntegral.FTCFilter.pure_le", "def_path": "Mathlib/MeasureTheory/Integral/FundThmCalculus.lean", "def_pos": [199, 3], "def_end_pos": [199, 10]}]], "state_before": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA\u271d : Type u_5\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : NormedSpace \u211d E\nf : \u211d \u2192 E\na b : \u211d\nc ca cb : E\nl l' la la' lb lb' : Filter \u211d\nlt : Filter \u03b9\n\u03bc : Measure \u211d\nu v ua va ub vb : \u03b9 \u2192 \u211d\ninst\u271d\u00b2 : FTCFilter a la la'\ninst\u271d\u00b9 : FTCFilter b lb lb'\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhab : IntervalIntegrable f \u03bc a b\nhmeas_a : StronglyMeasurableAtFilter f la'\nhmeas_b : StronglyMeasurableAtFilter f lb'\nha_lim : Tendsto f (la' \u2293 Measure.ae \u03bc) (\ud835\udcdd ca)\nhb_lim : Tendsto f (lb' \u2293 Measure.ae \u03bc) (\ud835\udcdd cb)\nhua : Tendsto ua lt la\nhva : Tendsto va lt la\nhub : Tendsto ub lt lb\nhvb : Tendsto vb lt lb\nthis\u271d : IsMeasurablyGenerated la'\nthis : IsMeasurablyGenerated lb'\nA : \u2200\u1da0 (t : \u03b9) in lt, IntervalIntegrable f \u03bc (ua t) (va t)\nA' : \u2200\u1da0 (t : \u03b9) in lt, IntervalIntegrable f \u03bc a (ua t)\nB : \u2200\u1da0 (t : \u03b9) in lt, IntervalIntegrable f \u03bc (ub t) (vb t)\n\u22a2 (fun x =>\n      -(\u222b (x : \u211d) in ua x..va x, f x \u2202\u03bc - \u222b (x : \u211d) in ua x..va x, ca \u2202\u03bc) +\n        (\u222b (x : \u211d) in ub x..vb x, f x \u2202\u03bc - \u222b (x : \u211d) in ub x..vb x, cb \u2202\u03bc)) =\u1da0[lt]\n    fun t =>\n    \u222b (x : \u211d) in va t..vb t, f x \u2202\u03bc - \u222b (x : \u211d) in ua t..ub t, f x \u2202\u03bc -\n      (\u222b (x : \u211d) in ub t..vb t, cb \u2202\u03bc - \u222b (x : \u211d) in ua t..va t, ca \u2202\u03bc)", "state_after": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA\u271d : Type u_5\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : NormedSpace \u211d E\nf : \u211d \u2192 E\na b : \u211d\nc ca cb : E\nl l' la la' lb lb' : Filter \u211d\nlt : Filter \u03b9\n\u03bc : Measure \u211d\nu v ua va ub vb : \u03b9 \u2192 \u211d\ninst\u271d\u00b2 : FTCFilter a la la'\ninst\u271d\u00b9 : FTCFilter b lb lb'\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhab : IntervalIntegrable f \u03bc a b\nhmeas_a : StronglyMeasurableAtFilter f la'\nhmeas_b : StronglyMeasurableAtFilter f lb'\nha_lim : Tendsto f (la' \u2293 Measure.ae \u03bc) (\ud835\udcdd ca)\nhb_lim : Tendsto f (lb' \u2293 Measure.ae \u03bc) (\ud835\udcdd cb)\nhua : Tendsto ua lt la\nhva : Tendsto va lt la\nhub : Tendsto ub lt lb\nhvb : Tendsto vb lt lb\nthis\u271d : IsMeasurablyGenerated la'\nthis : IsMeasurablyGenerated lb'\nA : \u2200\u1da0 (t : \u03b9) in lt, IntervalIntegrable f \u03bc (ua t) (va t)\nA' : \u2200\u1da0 (t : \u03b9) in lt, IntervalIntegrable f \u03bc a (ua t)\nB : \u2200\u1da0 (t : \u03b9) in lt, IntervalIntegrable f \u03bc (ub t) (vb t)\nB' : \u2200\u1da0 (t : \u03b9) in lt, IntervalIntegrable f \u03bc b (ub t)\n\u22a2 (fun x =>\n      -(\u222b (x : \u211d) in ua x..va x, f x \u2202\u03bc - \u222b (x : \u211d) in ua x..va x, ca \u2202\u03bc) +\n        (\u222b (x : \u211d) in ub x..vb x, f x \u2202\u03bc - \u222b (x : \u211d) in ub x..vb x, cb \u2202\u03bc)) =\u1da0[lt]\n    fun t =>\n    \u222b (x : \u211d) in va t..vb t, f x \u2202\u03bc - \u222b (x : \u211d) in ua t..ub t, f x \u2202\u03bc -\n      (\u222b (x : \u211d) in ub t..vb t, cb \u2202\u03bc - \u222b (x : \u211d) in ua t..va t, ca \u2202\u03bc)"}, {"tactic": "filter_upwards [A, A', B, B'] with _ ua_va a_ua ub_vb b_ub", "annotated_tactic": ["filter_upwards [A, A', B, B'] with _ ua_va a_ua ub_vb b_ub", []], "state_before": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA\u271d : Type u_5\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : NormedSpace \u211d E\nf : \u211d \u2192 E\na b : \u211d\nc ca cb : E\nl l' la la' lb lb' : Filter \u211d\nlt : Filter \u03b9\n\u03bc : Measure \u211d\nu v ua va ub vb : \u03b9 \u2192 \u211d\ninst\u271d\u00b2 : FTCFilter a la la'\ninst\u271d\u00b9 : FTCFilter b lb lb'\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhab : IntervalIntegrable f \u03bc a b\nhmeas_a : StronglyMeasurableAtFilter f la'\nhmeas_b : StronglyMeasurableAtFilter f lb'\nha_lim : Tendsto f (la' \u2293 Measure.ae \u03bc) (\ud835\udcdd ca)\nhb_lim : Tendsto f (lb' \u2293 Measure.ae \u03bc) (\ud835\udcdd cb)\nhua : Tendsto ua lt la\nhva : Tendsto va lt la\nhub : Tendsto ub lt lb\nhvb : Tendsto vb lt lb\nthis\u271d : IsMeasurablyGenerated la'\nthis : IsMeasurablyGenerated lb'\nA : \u2200\u1da0 (t : \u03b9) in lt, IntervalIntegrable f \u03bc (ua t) (va t)\nA' : \u2200\u1da0 (t : \u03b9) in lt, IntervalIntegrable f \u03bc a (ua t)\nB : \u2200\u1da0 (t : \u03b9) in lt, IntervalIntegrable f \u03bc (ub t) (vb t)\nB' : \u2200\u1da0 (t : \u03b9) in lt, IntervalIntegrable f \u03bc b (ub t)\n\u22a2 (fun x =>\n      -(\u222b (x : \u211d) in ua x..va x, f x \u2202\u03bc - \u222b (x : \u211d) in ua x..va x, ca \u2202\u03bc) +\n        (\u222b (x : \u211d) in ub x..vb x, f x \u2202\u03bc - \u222b (x : \u211d) in ub x..vb x, cb \u2202\u03bc)) =\u1da0[lt]\n    fun t =>\n    \u222b (x : \u211d) in va t..vb t, f x \u2202\u03bc - \u222b (x : \u211d) in ua t..ub t, f x \u2202\u03bc -\n      (\u222b (x : \u211d) in ub t..vb t, cb \u2202\u03bc - \u222b (x : \u211d) in ua t..va t, ca \u2202\u03bc)", "state_after": "case h\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA\u271d : Type u_5\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : NormedSpace \u211d E\nf : \u211d \u2192 E\na b : \u211d\nc ca cb : E\nl l' la la' lb lb' : Filter \u211d\nlt : Filter \u03b9\n\u03bc : Measure \u211d\nu v ua va ub vb : \u03b9 \u2192 \u211d\ninst\u271d\u00b2 : FTCFilter a la la'\ninst\u271d\u00b9 : FTCFilter b lb lb'\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhab : IntervalIntegrable f \u03bc a b\nhmeas_a : StronglyMeasurableAtFilter f la'\nhmeas_b : StronglyMeasurableAtFilter f lb'\nha_lim : Tendsto f (la' \u2293 Measure.ae \u03bc) (\ud835\udcdd ca)\nhb_lim : Tendsto f (lb' \u2293 Measure.ae \u03bc) (\ud835\udcdd cb)\nhua : Tendsto ua lt la\nhva : Tendsto va lt la\nhub : Tendsto ub lt lb\nhvb : Tendsto vb lt lb\nthis\u271d : IsMeasurablyGenerated la'\nthis : IsMeasurablyGenerated lb'\nA : \u2200\u1da0 (t : \u03b9) in lt, IntervalIntegrable f \u03bc (ua t) (va t)\nA' : \u2200\u1da0 (t : \u03b9) in lt, IntervalIntegrable f \u03bc a (ua t)\nB : \u2200\u1da0 (t : \u03b9) in lt, IntervalIntegrable f \u03bc (ub t) (vb t)\nB' : \u2200\u1da0 (t : \u03b9) in lt, IntervalIntegrable f \u03bc b (ub t)\na\u271d : \u03b9\nua_va : IntervalIntegrable f \u03bc (ua a\u271d) (va a\u271d)\na_ua : IntervalIntegrable f \u03bc a (ua a\u271d)\nub_vb : IntervalIntegrable f \u03bc (ub a\u271d) (vb a\u271d)\nb_ub : IntervalIntegrable f \u03bc b (ub a\u271d)\n\u22a2 -(\u222b (x : \u211d) in ua a\u271d..va a\u271d, f x \u2202\u03bc - \u222b (x : \u211d) in ua a\u271d..va a\u271d, ca \u2202\u03bc) +\n      (\u222b (x : \u211d) in ub a\u271d..vb a\u271d, f x \u2202\u03bc - \u222b (x : \u211d) in ub a\u271d..vb a\u271d, cb \u2202\u03bc) =\n    \u222b (x : \u211d) in va a\u271d..vb a\u271d, f x \u2202\u03bc - \u222b (x : \u211d) in ua a\u271d..ub a\u271d, f x \u2202\u03bc -\n      (\u222b (x : \u211d) in ub a\u271d..vb a\u271d, cb \u2202\u03bc - \u222b (x : \u211d) in ua a\u271d..va a\u271d, ca \u2202\u03bc)"}, {"tactic": "rw [\u2190 integral_interval_sub_interval_comm']", "annotated_tactic": ["rw [\u2190 <a>integral_interval_sub_interval_comm'</a>]", [{"full_name": "intervalIntegral.integral_interval_sub_interval_comm'", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [951, 9], "def_end_pos": [951, 45]}]], "state_before": "case h\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA\u271d : Type u_5\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : NormedSpace \u211d E\nf : \u211d \u2192 E\na b : \u211d\nc ca cb : E\nl l' la la' lb lb' : Filter \u211d\nlt : Filter \u03b9\n\u03bc : Measure \u211d\nu v ua va ub vb : \u03b9 \u2192 \u211d\ninst\u271d\u00b2 : FTCFilter a la la'\ninst\u271d\u00b9 : FTCFilter b lb lb'\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhab : IntervalIntegrable f \u03bc a b\nhmeas_a : StronglyMeasurableAtFilter f la'\nhmeas_b : StronglyMeasurableAtFilter f lb'\nha_lim : Tendsto f (la' \u2293 Measure.ae \u03bc) (\ud835\udcdd ca)\nhb_lim : Tendsto f (lb' \u2293 Measure.ae \u03bc) (\ud835\udcdd cb)\nhua : Tendsto ua lt la\nhva : Tendsto va lt la\nhub : Tendsto ub lt lb\nhvb : Tendsto vb lt lb\nthis\u271d : IsMeasurablyGenerated la'\nthis : IsMeasurablyGenerated lb'\nA : \u2200\u1da0 (t : \u03b9) in lt, IntervalIntegrable f \u03bc (ua t) (va t)\nA' : \u2200\u1da0 (t : \u03b9) in lt, IntervalIntegrable f \u03bc a (ua t)\nB : \u2200\u1da0 (t : \u03b9) in lt, IntervalIntegrable f \u03bc (ub t) (vb t)\nB' : \u2200\u1da0 (t : \u03b9) in lt, IntervalIntegrable f \u03bc b (ub t)\na\u271d : \u03b9\nua_va : IntervalIntegrable f \u03bc (ua a\u271d) (va a\u271d)\na_ua : IntervalIntegrable f \u03bc a (ua a\u271d)\nub_vb : IntervalIntegrable f \u03bc (ub a\u271d) (vb a\u271d)\nb_ub : IntervalIntegrable f \u03bc b (ub a\u271d)\n\u22a2 -(\u222b (x : \u211d) in ua a\u271d..va a\u271d, f x \u2202\u03bc - \u222b (x : \u211d) in ua a\u271d..va a\u271d, ca \u2202\u03bc) +\n      (\u222b (x : \u211d) in ub a\u271d..vb a\u271d, f x \u2202\u03bc - \u222b (x : \u211d) in ub a\u271d..vb a\u271d, cb \u2202\u03bc) =\n    \u222b (x : \u211d) in va a\u271d..vb a\u271d, f x \u2202\u03bc - \u222b (x : \u211d) in ua a\u271d..ub a\u271d, f x \u2202\u03bc -\n      (\u222b (x : \u211d) in ub a\u271d..vb a\u271d, cb \u2202\u03bc - \u222b (x : \u211d) in ua a\u271d..va a\u271d, ca \u2202\u03bc)", "state_after": "case h\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA\u271d : Type u_5\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : NormedSpace \u211d E\nf : \u211d \u2192 E\na b : \u211d\nc ca cb : E\nl l' la la' lb lb' : Filter \u211d\nlt : Filter \u03b9\n\u03bc : Measure \u211d\nu v ua va ub vb : \u03b9 \u2192 \u211d\ninst\u271d\u00b2 : FTCFilter a la la'\ninst\u271d\u00b9 : FTCFilter b lb lb'\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhab : IntervalIntegrable f \u03bc a b\nhmeas_a : StronglyMeasurableAtFilter f la'\nhmeas_b : StronglyMeasurableAtFilter f lb'\nha_lim : Tendsto f (la' \u2293 Measure.ae \u03bc) (\ud835\udcdd ca)\nhb_lim : Tendsto f (lb' \u2293 Measure.ae \u03bc) (\ud835\udcdd cb)\nhua : Tendsto ua lt la\nhva : Tendsto va lt la\nhub : Tendsto ub lt lb\nhvb : Tendsto vb lt lb\nthis\u271d : IsMeasurablyGenerated la'\nthis : IsMeasurablyGenerated lb'\nA : \u2200\u1da0 (t : \u03b9) in lt, IntervalIntegrable f \u03bc (ua t) (va t)\nA' : \u2200\u1da0 (t : \u03b9) in lt, IntervalIntegrable f \u03bc a (ua t)\nB : \u2200\u1da0 (t : \u03b9) in lt, IntervalIntegrable f \u03bc (ub t) (vb t)\nB' : \u2200\u1da0 (t : \u03b9) in lt, IntervalIntegrable f \u03bc b (ub t)\na\u271d : \u03b9\nua_va : IntervalIntegrable f \u03bc (ua a\u271d) (va a\u271d)\na_ua : IntervalIntegrable f \u03bc a (ua a\u271d)\nub_vb : IntervalIntegrable f \u03bc (ub a\u271d) (vb a\u271d)\nb_ub : IntervalIntegrable f \u03bc b (ub a\u271d)\n\u22a2 -(\u222b (x : \u211d) in ua a\u271d..va a\u271d, f x \u2202\u03bc - \u222b (x : \u211d) in ua a\u271d..va a\u271d, ca \u2202\u03bc) +\n      (\u222b (x : \u211d) in ub a\u271d..vb a\u271d, f x \u2202\u03bc - \u222b (x : \u211d) in ub a\u271d..vb a\u271d, cb \u2202\u03bc) =\n    \u222b (x : \u211d) in ub a\u271d..vb a\u271d, f x \u2202\u03bc - \u222b (x : \u211d) in ua a\u271d..va a\u271d, f x \u2202\u03bc -\n      (\u222b (x : \u211d) in ub a\u271d..vb a\u271d, cb \u2202\u03bc - \u222b (x : \u211d) in ua a\u271d..va a\u271d, ca \u2202\u03bc)\n\ncase h.hab\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA\u271d : Type u_5\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : NormedSpace \u211d E\nf : \u211d \u2192 E\na b : \u211d\nc ca cb : E\nl l' la la' lb lb' : Filter \u211d\nlt : Filter \u03b9\n\u03bc : Measure \u211d\nu v ua va ub vb : \u03b9 \u2192 \u211d\ninst\u271d\u00b2 : FTCFilter a la la'\ninst\u271d\u00b9 : FTCFilter b lb lb'\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhab : IntervalIntegrable f \u03bc a b\nhmeas_a : StronglyMeasurableAtFilter f la'\nhmeas_b : StronglyMeasurableAtFilter f lb'\nha_lim : Tendsto f (la' \u2293 Measure.ae \u03bc) (\ud835\udcdd ca)\nhb_lim : Tendsto f (lb' \u2293 Measure.ae \u03bc) (\ud835\udcdd cb)\nhua : Tendsto ua lt la\nhva : Tendsto va lt la\nhub : Tendsto ub lt lb\nhvb : Tendsto vb lt lb\nthis\u271d : IsMeasurablyGenerated la'\nthis : IsMeasurablyGenerated lb'\nA : \u2200\u1da0 (t : \u03b9) in lt, IntervalIntegrable f \u03bc (ua t) (va t)\nA' : \u2200\u1da0 (t : \u03b9) in lt, IntervalIntegrable f \u03bc a (ua t)\nB : \u2200\u1da0 (t : \u03b9) in lt, IntervalIntegrable f \u03bc (ub t) (vb t)\nB' : \u2200\u1da0 (t : \u03b9) in lt, IntervalIntegrable f \u03bc b (ub t)\na\u271d : \u03b9\nua_va : IntervalIntegrable f \u03bc (ua a\u271d) (va a\u271d)\na_ua : IntervalIntegrable f \u03bc a (ua a\u271d)\nub_vb : IntervalIntegrable f \u03bc (ub a\u271d) (vb a\u271d)\nb_ub : IntervalIntegrable f \u03bc b (ub a\u271d)\n\u22a2 IntervalIntegrable (fun x => f x) \u03bc (ub a\u271d) (vb a\u271d)\n\ncase h.hcd\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA\u271d : Type u_5\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : NormedSpace \u211d E\nf : \u211d \u2192 E\na b : \u211d\nc ca cb : E\nl l' la la' lb lb' : Filter \u211d\nlt : Filter \u03b9\n\u03bc : Measure \u211d\nu v ua va ub vb : \u03b9 \u2192 \u211d\ninst\u271d\u00b2 : FTCFilter a la la'\ninst\u271d\u00b9 : FTCFilter b lb lb'\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhab : IntervalIntegrable f \u03bc a b\nhmeas_a : StronglyMeasurableAtFilter f la'\nhmeas_b : StronglyMeasurableAtFilter f lb'\nha_lim : Tendsto f (la' \u2293 Measure.ae \u03bc) (\ud835\udcdd ca)\nhb_lim : Tendsto f (lb' \u2293 Measure.ae \u03bc) (\ud835\udcdd cb)\nhua : Tendsto ua lt la\nhva : Tendsto va lt la\nhub : Tendsto ub lt lb\nhvb : Tendsto vb lt lb\nthis\u271d : IsMeasurablyGenerated la'\nthis : IsMeasurablyGenerated lb'\nA : \u2200\u1da0 (t : \u03b9) in lt, IntervalIntegrable f \u03bc (ua t) (va t)\nA' : \u2200\u1da0 (t : \u03b9) in lt, IntervalIntegrable f \u03bc a (ua t)\nB : \u2200\u1da0 (t : \u03b9) in lt, IntervalIntegrable f \u03bc (ub t) (vb t)\nB' : \u2200\u1da0 (t : \u03b9) in lt, IntervalIntegrable f \u03bc b (ub t)\na\u271d : \u03b9\nua_va : IntervalIntegrable f \u03bc (ua a\u271d) (va a\u271d)\na_ua : IntervalIntegrable f \u03bc a (ua a\u271d)\nub_vb : IntervalIntegrable f \u03bc (ub a\u271d) (vb a\u271d)\nb_ub : IntervalIntegrable f \u03bc b (ub a\u271d)\n\u22a2 IntervalIntegrable (fun x => f x) \u03bc (ua a\u271d) (va a\u271d)\n\ncase h.hac\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA\u271d : Type u_5\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : NormedSpace \u211d E\nf : \u211d \u2192 E\na b : \u211d\nc ca cb : E\nl l' la la' lb lb' : Filter \u211d\nlt : Filter \u03b9\n\u03bc : Measure \u211d\nu v ua va ub vb : \u03b9 \u2192 \u211d\ninst\u271d\u00b2 : FTCFilter a la la'\ninst\u271d\u00b9 : FTCFilter b lb lb'\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhab : IntervalIntegrable f \u03bc a b\nhmeas_a : StronglyMeasurableAtFilter f la'\nhmeas_b : StronglyMeasurableAtFilter f lb'\nha_lim : Tendsto f (la' \u2293 Measure.ae \u03bc) (\ud835\udcdd ca)\nhb_lim : Tendsto f (lb' \u2293 Measure.ae \u03bc) (\ud835\udcdd cb)\nhua : Tendsto ua lt la\nhva : Tendsto va lt la\nhub : Tendsto ub lt lb\nhvb : Tendsto vb lt lb\nthis\u271d : IsMeasurablyGenerated la'\nthis : IsMeasurablyGenerated lb'\nA : \u2200\u1da0 (t : \u03b9) in lt, IntervalIntegrable f \u03bc (ua t) (va t)\nA' : \u2200\u1da0 (t : \u03b9) in lt, IntervalIntegrable f \u03bc a (ua t)\nB : \u2200\u1da0 (t : \u03b9) in lt, IntervalIntegrable f \u03bc (ub t) (vb t)\nB' : \u2200\u1da0 (t : \u03b9) in lt, IntervalIntegrable f \u03bc b (ub t)\na\u271d : \u03b9\nua_va : IntervalIntegrable f \u03bc (ua a\u271d) (va a\u271d)\na_ua : IntervalIntegrable f \u03bc a (ua a\u271d)\nub_vb : IntervalIntegrable f \u03bc (ub a\u271d) (vb a\u271d)\nb_ub : IntervalIntegrable f \u03bc b (ub a\u271d)\n\u22a2 IntervalIntegrable (fun x => f x) \u03bc (ub a\u271d) (ua a\u271d)"}, {"tactic": "exacts [ub_vb, ua_va, b_ub.symm.trans <| hab.symm.trans a_ua]", "annotated_tactic": ["exacts [ub_vb, ua_va, b_ub.symm.trans <| hab.symm.trans a_ua]", []], "state_before": "case h.hab\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA\u271d : Type u_5\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : NormedSpace \u211d E\nf : \u211d \u2192 E\na b : \u211d\nc ca cb : E\nl l' la la' lb lb' : Filter \u211d\nlt : Filter \u03b9\n\u03bc : Measure \u211d\nu v ua va ub vb : \u03b9 \u2192 \u211d\ninst\u271d\u00b2 : FTCFilter a la la'\ninst\u271d\u00b9 : FTCFilter b lb lb'\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhab : IntervalIntegrable f \u03bc a b\nhmeas_a : StronglyMeasurableAtFilter f la'\nhmeas_b : StronglyMeasurableAtFilter f lb'\nha_lim : Tendsto f (la' \u2293 Measure.ae \u03bc) (\ud835\udcdd ca)\nhb_lim : Tendsto f (lb' \u2293 Measure.ae \u03bc) (\ud835\udcdd cb)\nhua : Tendsto ua lt la\nhva : Tendsto va lt la\nhub : Tendsto ub lt lb\nhvb : Tendsto vb lt lb\nthis\u271d : IsMeasurablyGenerated la'\nthis : IsMeasurablyGenerated lb'\nA : \u2200\u1da0 (t : \u03b9) in lt, IntervalIntegrable f \u03bc (ua t) (va t)\nA' : \u2200\u1da0 (t : \u03b9) in lt, IntervalIntegrable f \u03bc a (ua t)\nB : \u2200\u1da0 (t : \u03b9) in lt, IntervalIntegrable f \u03bc (ub t) (vb t)\nB' : \u2200\u1da0 (t : \u03b9) in lt, IntervalIntegrable f \u03bc b (ub t)\na\u271d : \u03b9\nua_va : IntervalIntegrable f \u03bc (ua a\u271d) (va a\u271d)\na_ua : IntervalIntegrable f \u03bc a (ua a\u271d)\nub_vb : IntervalIntegrable f \u03bc (ub a\u271d) (vb a\u271d)\nb_ub : IntervalIntegrable f \u03bc b (ub a\u271d)\n\u22a2 IntervalIntegrable (fun x => f x) \u03bc (ub a\u271d) (vb a\u271d)\n\ncase h.hcd\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA\u271d : Type u_5\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : NormedSpace \u211d E\nf : \u211d \u2192 E\na b : \u211d\nc ca cb : E\nl l' la la' lb lb' : Filter \u211d\nlt : Filter \u03b9\n\u03bc : Measure \u211d\nu v ua va ub vb : \u03b9 \u2192 \u211d\ninst\u271d\u00b2 : FTCFilter a la la'\ninst\u271d\u00b9 : FTCFilter b lb lb'\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhab : IntervalIntegrable f \u03bc a b\nhmeas_a : StronglyMeasurableAtFilter f la'\nhmeas_b : StronglyMeasurableAtFilter f lb'\nha_lim : Tendsto f (la' \u2293 Measure.ae \u03bc) (\ud835\udcdd ca)\nhb_lim : Tendsto f (lb' \u2293 Measure.ae \u03bc) (\ud835\udcdd cb)\nhua : Tendsto ua lt la\nhva : Tendsto va lt la\nhub : Tendsto ub lt lb\nhvb : Tendsto vb lt lb\nthis\u271d : IsMeasurablyGenerated la'\nthis : IsMeasurablyGenerated lb'\nA : \u2200\u1da0 (t : \u03b9) in lt, IntervalIntegrable f \u03bc (ua t) (va t)\nA' : \u2200\u1da0 (t : \u03b9) in lt, IntervalIntegrable f \u03bc a (ua t)\nB : \u2200\u1da0 (t : \u03b9) in lt, IntervalIntegrable f \u03bc (ub t) (vb t)\nB' : \u2200\u1da0 (t : \u03b9) in lt, IntervalIntegrable f \u03bc b (ub t)\na\u271d : \u03b9\nua_va : IntervalIntegrable f \u03bc (ua a\u271d) (va a\u271d)\na_ua : IntervalIntegrable f \u03bc a (ua a\u271d)\nub_vb : IntervalIntegrable f \u03bc (ub a\u271d) (vb a\u271d)\nb_ub : IntervalIntegrable f \u03bc b (ub a\u271d)\n\u22a2 IntervalIntegrable (fun x => f x) \u03bc (ua a\u271d) (va a\u271d)\n\ncase h.hac\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA\u271d : Type u_5\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : NormedSpace \u211d E\nf : \u211d \u2192 E\na b : \u211d\nc ca cb : E\nl l' la la' lb lb' : Filter \u211d\nlt : Filter \u03b9\n\u03bc : Measure \u211d\nu v ua va ub vb : \u03b9 \u2192 \u211d\ninst\u271d\u00b2 : FTCFilter a la la'\ninst\u271d\u00b9 : FTCFilter b lb lb'\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhab : IntervalIntegrable f \u03bc a b\nhmeas_a : StronglyMeasurableAtFilter f la'\nhmeas_b : StronglyMeasurableAtFilter f lb'\nha_lim : Tendsto f (la' \u2293 Measure.ae \u03bc) (\ud835\udcdd ca)\nhb_lim : Tendsto f (lb' \u2293 Measure.ae \u03bc) (\ud835\udcdd cb)\nhua : Tendsto ua lt la\nhva : Tendsto va lt la\nhub : Tendsto ub lt lb\nhvb : Tendsto vb lt lb\nthis\u271d : IsMeasurablyGenerated la'\nthis : IsMeasurablyGenerated lb'\nA : \u2200\u1da0 (t : \u03b9) in lt, IntervalIntegrable f \u03bc (ua t) (va t)\nA' : \u2200\u1da0 (t : \u03b9) in lt, IntervalIntegrable f \u03bc a (ua t)\nB : \u2200\u1da0 (t : \u03b9) in lt, IntervalIntegrable f \u03bc (ub t) (vb t)\nB' : \u2200\u1da0 (t : \u03b9) in lt, IntervalIntegrable f \u03bc b (ub t)\na\u271d : \u03b9\nua_va : IntervalIntegrable f \u03bc (ua a\u271d) (va a\u271d)\na_ua : IntervalIntegrable f \u03bc a (ua a\u271d)\nub_vb : IntervalIntegrable f \u03bc (ub a\u271d) (vb a\u271d)\nb_ub : IntervalIntegrable f \u03bc b (ub a\u271d)\n\u22a2 IntervalIntegrable (fun x => f x) \u03bc (ub a\u271d) (ua a\u271d)", "state_after": "no goals"}, {"tactic": "abel", "annotated_tactic": ["abel", []], "state_before": "case h\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA\u271d : Type u_5\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : NormedSpace \u211d E\nf : \u211d \u2192 E\na b : \u211d\nc ca cb : E\nl l' la la' lb lb' : Filter \u211d\nlt : Filter \u03b9\n\u03bc : Measure \u211d\nu v ua va ub vb : \u03b9 \u2192 \u211d\ninst\u271d\u00b2 : FTCFilter a la la'\ninst\u271d\u00b9 : FTCFilter b lb lb'\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhab : IntervalIntegrable f \u03bc a b\nhmeas_a : StronglyMeasurableAtFilter f la'\nhmeas_b : StronglyMeasurableAtFilter f lb'\nha_lim : Tendsto f (la' \u2293 Measure.ae \u03bc) (\ud835\udcdd ca)\nhb_lim : Tendsto f (lb' \u2293 Measure.ae \u03bc) (\ud835\udcdd cb)\nhua : Tendsto ua lt la\nhva : Tendsto va lt la\nhub : Tendsto ub lt lb\nhvb : Tendsto vb lt lb\nthis\u271d : IsMeasurablyGenerated la'\nthis : IsMeasurablyGenerated lb'\nA : \u2200\u1da0 (t : \u03b9) in lt, IntervalIntegrable f \u03bc (ua t) (va t)\nA' : \u2200\u1da0 (t : \u03b9) in lt, IntervalIntegrable f \u03bc a (ua t)\nB : \u2200\u1da0 (t : \u03b9) in lt, IntervalIntegrable f \u03bc (ub t) (vb t)\nB' : \u2200\u1da0 (t : \u03b9) in lt, IntervalIntegrable f \u03bc b (ub t)\na\u271d : \u03b9\nua_va : IntervalIntegrable f \u03bc (ua a\u271d) (va a\u271d)\na_ua : IntervalIntegrable f \u03bc a (ua a\u271d)\nub_vb : IntervalIntegrable f \u03bc (ub a\u271d) (vb a\u271d)\nb_ub : IntervalIntegrable f \u03bc b (ub a\u271d)\n\u22a2 -(\u222b (x : \u211d) in ua a\u271d..va a\u271d, f x \u2202\u03bc - \u222b (x : \u211d) in ua a\u271d..va a\u271d, ca \u2202\u03bc) +\n      (\u222b (x : \u211d) in ub a\u271d..vb a\u271d, f x \u2202\u03bc - \u222b (x : \u211d) in ub a\u271d..vb a\u271d, cb \u2202\u03bc) =\n    \u222b (x : \u211d) in ub a\u271d..vb a\u271d, f x \u2202\u03bc - \u222b (x : \u211d) in ua a\u271d..va a\u271d, f x \u2202\u03bc -\n      (\u222b (x : \u211d) in ub a\u271d..vb a\u271d, cb \u2202\u03bc - \u222b (x : \u211d) in ua a\u271d..va a\u271d, ca \u2202\u03bc)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "full_name": "List.exists_of_modifyNthTail", "start": [864, 1], "end": [868, 58], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Intervals/Pi.lean", "full_name": "Set.image_mulSingle_Ioo_right", "start": [272, 1], "end": [274, 31], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "full_name": "MeasureTheory.lintegral_sum_measure", "start": [596, 1], "end": [610, 86], "traced_tactics": [{"tactic": "simp only [lintegral, iSup_subtype', SimpleFunc.lintegral_sum, ENNReal.tsum_eq_iSup_sum]", "annotated_tactic": ["simp only [<a>lintegral</a>, <a>iSup_subtype'</a>, <a>SimpleFunc.lintegral_sum</a>, <a>ENNReal.tsum_eq_iSup_sum</a>]", [{"full_name": "MeasureTheory.lintegral", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [60, 17], "def_end_pos": [60, 26]}, {"full_name": "iSup_subtype'", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [1266, 9], "def_end_pos": [1266, 22]}, {"full_name": "MeasureTheory.SimpleFunc.lintegral_sum", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [1041, 9], "def_end_pos": [1041, 22]}, {"full_name": "ENNReal.tsum_eq_iSup_sum", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [789, 19], "def_end_pos": [789, 35]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nm : MeasurableSpace \u03b1\n\u03b9 : Type u_5\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : \u03b9 \u2192 Measure \u03b1\n\u22a2 \u222b\u207b (a : \u03b1), f a \u2202Measure.sum \u03bc = \u2211' (i : \u03b9), \u222b\u207b (a : \u03b1), f a \u2202\u03bc i", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nm : MeasurableSpace \u03b1\n\u03b9 : Type u_5\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : \u03b9 \u2192 Measure \u03b1\n\u22a2 \u2a06 x, \u2a06 s, \u2211 i in s, SimpleFunc.lintegral (\u2191x) (\u03bc i) = \u2a06 s, \u2211 a in s, \u2a06 x, SimpleFunc.lintegral (\u2191x) (\u03bc a)"}, {"tactic": "rw [iSup_comm]", "annotated_tactic": ["rw [<a>iSup_comm</a>]", [{"full_name": "iSup_comm", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [1196, 9], "def_end_pos": [1196, 18]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nm : MeasurableSpace \u03b1\n\u03b9 : Type u_5\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : \u03b9 \u2192 Measure \u03b1\n\u22a2 \u2a06 x, \u2a06 s, \u2211 i in s, SimpleFunc.lintegral (\u2191x) (\u03bc i) = \u2a06 s, \u2211 a in s, \u2a06 x, SimpleFunc.lintegral (\u2191x) (\u03bc a)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nm : MeasurableSpace \u03b1\n\u03b9 : Type u_5\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : \u03b9 \u2192 Measure \u03b1\n\u22a2 \u2a06 j, \u2a06 i, \u2211 i_1 in j, SimpleFunc.lintegral (\u2191i) (\u03bc i_1) = \u2a06 s, \u2211 a in s, \u2a06 x, SimpleFunc.lintegral (\u2191x) (\u03bc a)"}, {"tactic": "congr", "annotated_tactic": ["congr", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nm : MeasurableSpace \u03b1\n\u03b9 : Type u_5\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : \u03b9 \u2192 Measure \u03b1\n\u22a2 \u2a06 j, \u2a06 i, \u2211 i_1 in j, SimpleFunc.lintegral (\u2191i) (\u03bc i_1) = \u2a06 s, \u2211 a in s, \u2a06 x, SimpleFunc.lintegral (\u2191x) (\u03bc a)", "state_after": "case e_s\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nm : MeasurableSpace \u03b1\n\u03b9 : Type u_5\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : \u03b9 \u2192 Measure \u03b1\n\u22a2 (fun j => \u2a06 i, \u2211 i_1 in j, SimpleFunc.lintegral (\u2191i) (\u03bc i_1)) = fun s =>\n    \u2211 a in s, \u2a06 x, SimpleFunc.lintegral (\u2191x) (\u03bc a)"}, {"tactic": "funext s", "annotated_tactic": ["funext s", []], "state_before": "case e_s\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nm : MeasurableSpace \u03b1\n\u03b9 : Type u_5\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : \u03b9 \u2192 Measure \u03b1\n\u22a2 (fun j => \u2a06 i, \u2211 i_1 in j, SimpleFunc.lintegral (\u2191i) (\u03bc i_1)) = fun s =>\n    \u2211 a in s, \u2a06 x, SimpleFunc.lintegral (\u2191x) (\u03bc a)", "state_after": "case e_s.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nm : MeasurableSpace \u03b1\n\u03b9 : Type u_5\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : \u03b9 \u2192 Measure \u03b1\ns : Finset \u03b9\n\u22a2 \u2a06 i, \u2211 i_1 in s, SimpleFunc.lintegral (\u2191i) (\u03bc i_1) = \u2211 a in s, \u2a06 x, SimpleFunc.lintegral (\u2191x) (\u03bc a)"}, {"tactic": "induction' s using Finset.induction_on with i s hi hs", "annotated_tactic": ["induction' s using <a>Finset.induction_on</a> with i s hi hs", [{"full_name": "Finset.induction_on", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1251, 19], "def_end_pos": [1251, 31]}]], "state_before": "case e_s.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nm : MeasurableSpace \u03b1\n\u03b9 : Type u_5\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : \u03b9 \u2192 Measure \u03b1\ns : Finset \u03b9\n\u22a2 \u2a06 i, \u2211 i_1 in s, SimpleFunc.lintegral (\u2191i) (\u03bc i_1) = \u2211 a in s, \u2a06 x, SimpleFunc.lintegral (\u2191x) (\u03bc a)", "state_after": "case e_s.h.empty\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nm : MeasurableSpace \u03b1\n\u03b9 : Type u_5\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : \u03b9 \u2192 Measure \u03b1\n\u22a2 \u2a06 i, \u2211 i_1 in \u2205, SimpleFunc.lintegral (\u2191i) (\u03bc i_1) = \u2211 a in \u2205, \u2a06 x, SimpleFunc.lintegral (\u2191x) (\u03bc a)\n\ncase e_s.h.insert\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nm : MeasurableSpace \u03b1\n\u03b9 : Type u_5\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : \u03b9 \u2192 Measure \u03b1\ni : \u03b9\ns : Finset \u03b9\nhi : \u00aci \u2208 s\nhs : \u2a06 i, \u2211 i_1 in s, SimpleFunc.lintegral (\u2191i) (\u03bc i_1) = \u2211 a in s, \u2a06 x, SimpleFunc.lintegral (\u2191x) (\u03bc a)\n\u22a2 \u2a06 i_1, \u2211 i in insert i s, SimpleFunc.lintegral (\u2191i_1) (\u03bc i) = \u2211 a in insert i s, \u2a06 x, SimpleFunc.lintegral (\u2191x) (\u03bc a)"}, {"tactic": "simp only [Finset.sum_insert hi, \u2190 hs]", "annotated_tactic": ["simp only [<a>Finset.sum_insert</a> hi, \u2190 hs]", [{"full_name": "Finset.sum_insert", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [316, 3], "def_end_pos": [316, 14]}]], "state_before": "case e_s.h.insert\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nm : MeasurableSpace \u03b1\n\u03b9 : Type u_5\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : \u03b9 \u2192 Measure \u03b1\ni : \u03b9\ns : Finset \u03b9\nhi : \u00aci \u2208 s\nhs : \u2a06 i, \u2211 i_1 in s, SimpleFunc.lintegral (\u2191i) (\u03bc i_1) = \u2211 a in s, \u2a06 x, SimpleFunc.lintegral (\u2191x) (\u03bc a)\n\u22a2 \u2a06 i_1, \u2211 i in insert i s, SimpleFunc.lintegral (\u2191i_1) (\u03bc i) = \u2211 a in insert i s, \u2a06 x, SimpleFunc.lintegral (\u2191x) (\u03bc a)", "state_after": "case e_s.h.insert\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nm : MeasurableSpace \u03b1\n\u03b9 : Type u_5\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : \u03b9 \u2192 Measure \u03b1\ni : \u03b9\ns : Finset \u03b9\nhi : \u00aci \u2208 s\nhs : \u2a06 i, \u2211 i_1 in s, SimpleFunc.lintegral (\u2191i) (\u03bc i_1) = \u2211 a in s, \u2a06 x, SimpleFunc.lintegral (\u2191x) (\u03bc a)\n\u22a2 \u2a06 i_1, SimpleFunc.lintegral (\u2191i_1) (\u03bc i) + \u2211 i in s, SimpleFunc.lintegral (\u2191i_1) (\u03bc i) =\n    (\u2a06 x, SimpleFunc.lintegral (\u2191x) (\u03bc i)) + \u2a06 i, \u2211 i_1 in s, SimpleFunc.lintegral (\u2191i) (\u03bc i_1)"}, {"tactic": "refine' (ENNReal.iSup_add_iSup _).symm", "annotated_tactic": ["refine' (<a>ENNReal.iSup_add_iSup</a> _).<a>symm</a>", [{"full_name": "ENNReal.iSup_add_iSup", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [612, 9], "def_end_pos": [612, 22]}, {"full_name": "Eq.symm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [310, 9], "def_end_pos": [310, 16]}]], "state_before": "case e_s.h.insert\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nm : MeasurableSpace \u03b1\n\u03b9 : Type u_5\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : \u03b9 \u2192 Measure \u03b1\ni : \u03b9\ns : Finset \u03b9\nhi : \u00aci \u2208 s\nhs : \u2a06 i, \u2211 i_1 in s, SimpleFunc.lintegral (\u2191i) (\u03bc i_1) = \u2211 a in s, \u2a06 x, SimpleFunc.lintegral (\u2191x) (\u03bc a)\n\u22a2 \u2a06 i_1, SimpleFunc.lintegral (\u2191i_1) (\u03bc i) + \u2211 i in s, SimpleFunc.lintegral (\u2191i_1) (\u03bc i) =\n    (\u2a06 x, SimpleFunc.lintegral (\u2191x) (\u03bc i)) + \u2a06 i, \u2211 i_1 in s, SimpleFunc.lintegral (\u2191i) (\u03bc i_1)", "state_after": "case e_s.h.insert\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nm : MeasurableSpace \u03b1\n\u03b9 : Type u_5\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : \u03b9 \u2192 Measure \u03b1\ni : \u03b9\ns : Finset \u03b9\nhi : \u00aci \u2208 s\nhs : \u2a06 i, \u2211 i_1 in s, SimpleFunc.lintegral (\u2191i) (\u03bc i_1) = \u2211 a in s, \u2a06 x, SimpleFunc.lintegral (\u2191x) (\u03bc a)\n\u22a2 \u2200 (i_1 j : { i // \u2191i \u2264 fun a => f a }),\n    \u2203 k,\n      SimpleFunc.lintegral (\u2191i_1) (\u03bc i) + \u2211 i in s, SimpleFunc.lintegral (\u2191j) (\u03bc i) \u2264\n        SimpleFunc.lintegral (\u2191k) (\u03bc i) + \u2211 i in s, SimpleFunc.lintegral (\u2191k) (\u03bc i)"}, {"tactic": "intro \u03c6 \u03c8", "annotated_tactic": ["intro \u03c6 \u03c8", []], "state_before": "case e_s.h.insert\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nm : MeasurableSpace \u03b1\n\u03b9 : Type u_5\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : \u03b9 \u2192 Measure \u03b1\ni : \u03b9\ns : Finset \u03b9\nhi : \u00aci \u2208 s\nhs : \u2a06 i, \u2211 i_1 in s, SimpleFunc.lintegral (\u2191i) (\u03bc i_1) = \u2211 a in s, \u2a06 x, SimpleFunc.lintegral (\u2191x) (\u03bc a)\n\u22a2 \u2200 (i_1 j : { i // \u2191i \u2264 fun a => f a }),\n    \u2203 k,\n      SimpleFunc.lintegral (\u2191i_1) (\u03bc i) + \u2211 i in s, SimpleFunc.lintegral (\u2191j) (\u03bc i) \u2264\n        SimpleFunc.lintegral (\u2191k) (\u03bc i) + \u2211 i in s, SimpleFunc.lintegral (\u2191k) (\u03bc i)", "state_after": "case e_s.h.insert\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nm : MeasurableSpace \u03b1\n\u03b9 : Type u_5\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : \u03b9 \u2192 Measure \u03b1\ni : \u03b9\ns : Finset \u03b9\nhi : \u00aci \u2208 s\nhs : \u2a06 i, \u2211 i_1 in s, SimpleFunc.lintegral (\u2191i) (\u03bc i_1) = \u2211 a in s, \u2a06 x, SimpleFunc.lintegral (\u2191x) (\u03bc a)\n\u03c6 \u03c8 : { i // \u2191i \u2264 fun a => f a }\n\u22a2 \u2203 k,\n    SimpleFunc.lintegral (\u2191\u03c6) (\u03bc i) + \u2211 i in s, SimpleFunc.lintegral (\u2191\u03c8) (\u03bc i) \u2264\n      SimpleFunc.lintegral (\u2191k) (\u03bc i) + \u2211 i in s, SimpleFunc.lintegral (\u2191k) (\u03bc i)"}, {"tactic": "exact\n  \u27e8\u27e8\u03c6 \u2294 \u03c8, fun x => sup_le (\u03c6.2 x) (\u03c8.2 x)\u27e9,\n    add_le_add (SimpleFunc.lintegral_mono le_sup_left le_rfl)\n      (Finset.sum_le_sum fun j _ => SimpleFunc.lintegral_mono le_sup_right le_rfl)\u27e9", "annotated_tactic": ["exact\n    \u27e8\u27e8\u03c6 \u2294 \u03c8, fun x => <a>sup_le</a> (\u03c6.2 x) (\u03c8.2 x)\u27e9,\n      <a>add_le_add</a> (<a>SimpleFunc.lintegral_mono</a> <a>le_sup_left</a> <a>le_rfl</a>)\n        (<a>Finset.sum_le_sum</a> fun j _ => <a>SimpleFunc.lintegral_mono</a> <a>le_sup_right</a> <a>le_rfl</a>)\u27e9", [{"full_name": "sup_le", "def_path": "Mathlib/Order/Lattice.lean", "def_pos": [167, 9], "def_end_pos": [167, 15]}, {"full_name": "add_le_add", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [205, 15], "def_end_pos": [205, 25]}, {"full_name": "MeasureTheory.SimpleFunc.lintegral_mono", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [1105, 9], "def_end_pos": [1105, 23]}, {"full_name": "le_sup_left", "def_path": "Mathlib/Order/Lattice.lean", "def_pos": [130, 9], "def_end_pos": [130, 20]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}, {"full_name": "Finset.sum_le_sum", "def_path": "Mathlib/Algebra/BigOperators/Order.lean", "def_pos": [111, 15], "def_end_pos": [111, 25]}, {"full_name": "MeasureTheory.SimpleFunc.lintegral_mono", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [1105, 9], "def_end_pos": [1105, 23]}, {"full_name": "le_sup_right", "def_path": "Mathlib/Order/Lattice.lean", "def_pos": [141, 9], "def_end_pos": [141, 21]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}]], "state_before": "case e_s.h.insert\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nm : MeasurableSpace \u03b1\n\u03b9 : Type u_5\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : \u03b9 \u2192 Measure \u03b1\ni : \u03b9\ns : Finset \u03b9\nhi : \u00aci \u2208 s\nhs : \u2a06 i, \u2211 i_1 in s, SimpleFunc.lintegral (\u2191i) (\u03bc i_1) = \u2211 a in s, \u2a06 x, SimpleFunc.lintegral (\u2191x) (\u03bc a)\n\u03c6 \u03c8 : { i // \u2191i \u2264 fun a => f a }\n\u22a2 \u2203 k,\n    SimpleFunc.lintegral (\u2191\u03c6) (\u03bc i) + \u2211 i in s, SimpleFunc.lintegral (\u2191\u03c8) (\u03bc i) \u2264\n      SimpleFunc.lintegral (\u2191k) (\u03bc i) + \u2211 i in s, SimpleFunc.lintegral (\u2191k) (\u03bc i)", "state_after": "no goals"}, {"tactic": "apply bot_unique", "annotated_tactic": ["apply <a>bot_unique</a>", [{"full_name": "bot_unique", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [359, 9], "def_end_pos": [359, 19]}]], "state_before": "case e_s.h.empty\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nm : MeasurableSpace \u03b1\n\u03b9 : Type u_5\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : \u03b9 \u2192 Measure \u03b1\n\u22a2 \u2a06 i, \u2211 i_1 in \u2205, SimpleFunc.lintegral (\u2191i) (\u03bc i_1) = \u2211 a in \u2205, \u2a06 x, SimpleFunc.lintegral (\u2191x) (\u03bc a)", "state_after": "case e_s.h.empty.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nm : MeasurableSpace \u03b1\n\u03b9 : Type u_5\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : \u03b9 \u2192 Measure \u03b1\n\u22a2 \u2a06 i, \u2211 i_1 in \u2205, SimpleFunc.lintegral (\u2191i) (\u03bc i_1) \u2264 \u22a5"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case e_s.h.empty.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nm : MeasurableSpace \u03b1\n\u03b9 : Type u_5\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : \u03b9 \u2192 Measure \u03b1\n\u22a2 \u2a06 i, \u2211 i_1 in \u2205, SimpleFunc.lintegral (\u2191i) (\u03bc i_1) \u2264 \u22a5", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/Basic.lean", "full_name": "MvPolynomial.mapAlgHom_coe_ringHom", "start": [1439, 1], "end": [1442, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Rat/Lemmas.lean", "full_name": "Rat.normalize_eq_zero", "start": [73, 9], "end": [75, 56], "traced_tactics": [{"tactic": "have' := normalize_eq_iff d0 Nat.one_ne_zero", "annotated_tactic": ["have' := <a>normalize_eq_iff</a> d0 <a>Nat.one_ne_zero</a>", [{"full_name": "Rat.normalize_eq_iff", "def_path": "lake-packages/std/Std/Data/Rat/Lemmas.lean", "def_pos": [51, 9], "def_end_pos": [51, 25]}, {"full_name": "Nat.one_ne_zero", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [426, 19], "def_end_pos": [426, 30]}]], "state_before": "d : Nat\nn : Int\nd0 : d \u2260 0\n\u22a2 normalize n d = 0 \u2194 n = 0", "state_after": "case refine'_3\nd : Nat\nn : Int\nd0 : d \u2260 0\nthis : normalize ?refine'_1 d = normalize ?refine'_2 1 \u2194 ?refine'_1 * \u21911 = ?refine'_2 * \u2191d\n\u22a2 normalize n d = 0 \u2194 n = 0\n\ncase refine'_1\nd : Nat\nn : Int\nd0 : d \u2260 0\n\u22a2 Int\n\ncase refine'_2\nd : Nat\nn : Int\nd0 : d \u2260 0\n\u22a2 Int"}, {"tactic": "rw [normalize_zero (d := 1)] at this", "annotated_tactic": ["rw [<a>normalize_zero</a> (d := 1)] at this", [{"full_name": "Rat.normalize_zero", "def_path": "lake-packages/std/Std/Data/Rat/Lemmas.lean", "def_pos": [33, 17], "def_end_pos": [33, 31]}]], "state_before": "case refine'_3\nd : Nat\nn : Int\nd0 : d \u2260 0\nthis : normalize ?refine'_1 d = normalize ?refine'_2 1 \u2194 ?refine'_1 * \u21911 = ?refine'_2 * \u2191d\n\u22a2 normalize n d = 0 \u2194 n = 0\n\ncase refine'_1\nd : Nat\nn : Int\nd0 : d \u2260 0\n\u22a2 Int\n\ncase refine'_2\nd : Nat\nn : Int\nd0 : d \u2260 0\n\u22a2 Int", "state_after": "case refine'_3\nd : Nat\nn : Int\nd0 : d \u2260 0\nthis : normalize ?refine'_1 d = 0 \u2194 ?refine'_1 * \u21911 = 0 * \u2191d\n\u22a2 normalize n d = 0 \u2194 n = 0\n\ncase refine'_1\nd : Nat\nn : Int\nd0 : d \u2260 0\n\u22a2 Int\n\ncase refine'_1\nd : Nat\nn : Int\nd0 : d \u2260 0\n\u22a2 Int"}, {"tactic": "rw [this]", "annotated_tactic": ["rw [this]", []], "state_before": "case refine'_3\nd : Nat\nn : Int\nd0 : d \u2260 0\nthis : normalize ?refine'_1 d = 0 \u2194 ?refine'_1 * \u21911 = 0 * \u2191d\n\u22a2 normalize n d = 0 \u2194 n = 0\n\ncase refine'_1\nd : Nat\nn : Int\nd0 : d \u2260 0\n\u22a2 Int\n\ncase refine'_1\nd : Nat\nn : Int\nd0 : d \u2260 0\n\u22a2 Int", "state_after": "case refine'_3\nd : Nat\nn : Int\nd0 : d \u2260 0\nthis : normalize n d = 0 \u2194 n * \u21911 = 0 * \u2191d\n\u22a2 n * \u21911 = 0 * \u2191d \u2194 n = 0"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case refine'_3\nd : Nat\nn : Int\nd0 : d \u2260 0\nthis : normalize n d = 0 \u2194 n * \u21911 = 0 * \u2191d\n\u22a2 n * \u21911 = 0 * \u2191d \u2194 n = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/RBMap/Lemmas.lean", "full_name": "Std.RBSet.toStream_toList", "start": [649, 9], "end": [650, 21], "traced_tactics": [{"tactic": "simp [toStream_eq]", "annotated_tactic": ["simp [<a>toStream_eq</a>]", [{"full_name": "Std.RBSet.toStream_eq", "def_path": "lake-packages/std/Std/Data/RBMap/Lemmas.lean", "def_pos": [647, 9], "def_end_pos": [647, 20]}]], "state_before": "\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nt : RBSet \u03b1 cmp\n\u22a2 RBNode.Stream.toList (toStream t) = toList t", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Group/LIntegral.lean", "full_name": "MeasureTheory.lintegral_div_right_eq_self", "start": [54, 1], "end": [56, 62], "traced_tactics": [{"tactic": "simp_rw [div_eq_mul_inv, lintegral_mul_right_eq_self f g\u207b\u00b9]", "annotated_tactic": ["simp_rw [<a>div_eq_mul_inv</a>, <a>lintegral_mul_right_eq_self</a> f g\u207b\u00b9]", [{"full_name": "div_eq_mul_inv", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [977, 9], "def_end_pos": [977, 23]}, {"full_name": "MeasureTheory.lintegral_mul_right_eq_self", "def_path": "Mathlib/MeasureTheory/Group/LIntegral.lean", "def_pos": [46, 9], "def_end_pos": [46, 36]}]], "state_before": "G : Type u_1\ninst\u271d\u00b3 : MeasurableSpace G\n\u03bc : Measure G\ng\u271d : G\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : MeasurableMul G\ninst\u271d : IsMulRightInvariant \u03bc\nf : G \u2192 \u211d\u22650\u221e\ng : G\n\u22a2 \u222b\u207b (x : G), f (x / g) \u2202\u03bc = \u222b\u207b (x : G), f x \u2202\u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/ProbabilityMassFunction/Monad.lean", "full_name": "PMF.toMeasure_pure", "start": [91, 1], "end": [92, 94], "traced_tactics": [{"tactic": "rw [toMeasure_pure_apply a s hs, Measure.dirac_apply' a hs]", "annotated_tactic": ["rw [<a>toMeasure_pure_apply</a> a s hs, <a>Measure.dirac_apply'</a> a hs]", [{"full_name": "PMF.toMeasure_pure_apply", "def_path": "Mathlib/Probability/ProbabilityMassFunction/Monad.lean", "def_pos": [86, 9], "def_end_pos": [86, 29]}, {"full_name": "MeasureTheory.Measure.dirac_apply'", "def_path": "Mathlib/MeasureTheory/Measure/Dirac.lean", "def_pos": [39, 9], "def_end_pos": [39, 21]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na a' : \u03b1\ns\u271d : Set \u03b1\ninst\u271d : MeasurableSpace \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(toMeasure (pure a)) s = \u2191\u2191(Measure.dirac a) s", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na a' : \u03b1\ns\u271d : Set \u03b1\ninst\u271d : MeasurableSpace \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 (if a \u2208 s then 1 else 0) = Set.indicator s 1 a"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na a' : \u03b1\ns\u271d : Set \u03b1\ninst\u271d : MeasurableSpace \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 (if a \u2208 s then 1 else 0) = Set.indicator s 1 a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Holor.lean", "full_name": "Holor.mul_assoc", "start": [204, 1], "end": [205, 89], "traced_tactics": [{"tactic": "simp [cast_heq, mul_assoc0, assocLeft]", "annotated_tactic": ["simp [<a>cast_heq</a>, <a>mul_assoc0</a>, <a>assocLeft</a>]", 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"869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "full_name": "String.foldl_eq", "start": [706, 1], "end": [707, 46], "traced_tactics": [{"tactic": "simpa using foldlAux_of_valid f [] s.1 [] a", "annotated_tactic": ["simpa using <a>foldlAux_of_valid</a> f [] s.1 [] a", [{"full_name": "String.foldlAux_of_valid", "def_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "def_pos": [697, 9], "def_end_pos": [697, 26]}]], "state_before": "\u03b1 : Type u_1\nf : \u03b1 \u2192 Char \u2192 \u03b1\ns : String\na : \u03b1\n\u22a2 foldl f a s = List.foldl f a s.data", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/TorusIntegral.lean", "full_name": "torusIntegral_const_mul", "start": [187, 1], "end": [189, 29], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/FundThmCalculus.lean", "full_name": "intervalIntegral.integral_hasDerivAt_left", "start": [789, 1], "end": [792, 58], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Kernel/Composition.lean", "full_name": "ProbabilityTheory.kernel.compProd_eq_sum_compProd_left", "start": [489, 1], "end": [498, 37], "traced_tactics": [{"tactic": "by_cases h : IsSFiniteKernel \u03b7", "annotated_tactic": ["by_cases h : <a>IsSFiniteKernel</a> \u03b7", [{"full_name": "ProbabilityTheory.IsSFiniteKernel", "def_path": "Mathlib/Probability/Kernel/Basic.lean", "def_pos": [278, 7], "def_end_pos": [278, 47]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03b3 : Type u_4\nm\u03b3 : MeasurableSpace \u03b3\ns : Set (\u03b2 \u00d7 \u03b3)\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\n\u22a2 \u03ba \u2297\u2096 \u03b7 = kernel.sum fun n => seq \u03ba n \u2297\u2096 \u03b7", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03b3 : Type u_4\nm\u03b3 : MeasurableSpace \u03b3\ns : Set (\u03b2 \u00d7 \u03b3)\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\nh : IsSFiniteKernel \u03b7\n\u22a2 \u03ba \u2297\u2096 \u03b7 = kernel.sum fun n => seq \u03ba n \u2297\u2096 \u03b7\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03b3 : Type u_4\nm\u03b3 : MeasurableSpace \u03b3\ns : Set (\u03b2 \u00d7 \u03b3)\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\nh : \u00acIsSFiniteKernel \u03b7\n\u22a2 \u03ba \u2297\u2096 \u03b7 = kernel.sum fun n => seq \u03ba n \u2297\u2096 \u03b7"}, {"tactic": "swap", "annotated_tactic": ["swap", []], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03b3 : Type u_4\nm\u03b3 : MeasurableSpace \u03b3\ns : Set (\u03b2 \u00d7 \u03b3)\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\nh : IsSFiniteKernel \u03b7\n\u22a2 \u03ba \u2297\u2096 \u03b7 = kernel.sum fun n => seq \u03ba n \u2297\u2096 \u03b7\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03b3 : Type u_4\nm\u03b3 : MeasurableSpace \u03b3\ns : Set (\u03b2 \u00d7 \u03b3)\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\nh : \u00acIsSFiniteKernel \u03b7\n\u22a2 \u03ba \u2297\u2096 \u03b7 = kernel.sum fun n => seq \u03ba n \u2297\u2096 \u03b7", "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03b3 : Type u_4\nm\u03b3 : MeasurableSpace \u03b3\ns : Set (\u03b2 \u00d7 \u03b3)\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\nh : \u00acIsSFiniteKernel \u03b7\n\u22a2 \u03ba \u2297\u2096 \u03b7 = kernel.sum fun n => seq \u03ba n \u2297\u2096 \u03b7\n\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03b3 : Type u_4\nm\u03b3 : MeasurableSpace \u03b3\ns : Set (\u03b2 \u00d7 \u03b3)\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\nh : IsSFiniteKernel \u03b7\n\u22a2 \u03ba \u2297\u2096 \u03b7 = kernel.sum fun n => seq \u03ba n \u2297\u2096 \u03b7"}, {"tactic": "rw [compProd_eq_sum_compProd]", "annotated_tactic": ["rw [<a>compProd_eq_sum_compProd</a>]", [{"full_name": "ProbabilityTheory.kernel.compProd_eq_sum_compProd", "def_path": "Mathlib/Probability/Kernel/Composition.lean", "def_pos": [484, 9], "def_end_pos": [484, 33]}]], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03b3 : Type u_4\nm\u03b3 : MeasurableSpace \u03b3\ns : Set (\u03b2 \u00d7 \u03b3)\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\nh : IsSFiniteKernel \u03b7\n\u22a2 \u03ba \u2297\u2096 \u03b7 = kernel.sum fun n => seq \u03ba n \u2297\u2096 \u03b7", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03b3 : Type u_4\nm\u03b3 : MeasurableSpace \u03b3\ns : Set (\u03b2 \u00d7 \u03b3)\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\nh : IsSFiniteKernel \u03b7\n\u22a2 (kernel.sum fun n => kernel.sum fun m => seq \u03ba n \u2297\u2096 seq \u03b7 m) = kernel.sum fun n => seq \u03ba n \u2297\u2096 \u03b7"}, {"tactic": "congr with n a s hs", "annotated_tactic": ["congr with n a s hs", []], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03b3 : Type u_4\nm\u03b3 : MeasurableSpace \u03b3\ns : Set (\u03b2 \u00d7 \u03b3)\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\nh : IsSFiniteKernel \u03b7\n\u22a2 (kernel.sum fun n => kernel.sum fun m => seq \u03ba n \u2297\u2096 seq \u03b7 m) = kernel.sum fun n => seq \u03ba n \u2297\u2096 \u03b7", "state_after": "case pos.e_\u03ba.h.h.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03b3 : Type u_4\nm\u03b3 : MeasurableSpace \u03b3\ns\u271d : Set (\u03b2 \u00d7 \u03b3)\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\nh : IsSFiniteKernel \u03b7\nn : \u2115\na : \u03b1\ns : Set (\u03b2 \u00d7 \u03b3)\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(\u2191(kernel.sum fun m => seq \u03ba n \u2297\u2096 seq \u03b7 m) a) s = \u2191\u2191(\u2191(seq \u03ba n \u2297\u2096 \u03b7) a) s"}, {"tactic": "simp_rw [kernel.sum_apply' _ _ hs, compProd_apply_eq_compProdFun _ _ _ hs,\n  compProdFun_tsum_right _ \u03b7 a hs]", "annotated_tactic": ["simp_rw [<a>kernel.sum_apply'</a> _ _ hs, <a>compProd_apply_eq_compProdFun</a> _ _ _ hs,\n    <a>compProdFun_tsum_right</a> _ \u03b7 a hs]", [{"full_name": "ProbabilityTheory.kernel.sum_apply'", "def_path": "Mathlib/Probability/Kernel/Basic.lean", "def_pos": [244, 9], "def_end_pos": [244, 19]}, {"full_name": "ProbabilityTheory.kernel.compProd_apply_eq_compProdFun", "def_path": "Mathlib/Probability/Kernel/Composition.lean", "def_pos": [216, 9], "def_end_pos": [216, 38]}, {"full_name": "ProbabilityTheory.kernel.compProdFun_tsum_right", "def_path": "Mathlib/Probability/Kernel/Composition.lean", "def_pos": [131, 9], "def_end_pos": [131, 31]}]], "state_before": "case pos.e_\u03ba.h.h.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03b3 : Type u_4\nm\u03b3 : MeasurableSpace \u03b3\ns\u271d : Set (\u03b2 \u00d7 \u03b3)\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\nh : IsSFiniteKernel \u03b7\nn : \u2115\na : \u03b1\ns : Set (\u03b2 \u00d7 \u03b3)\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(\u2191(kernel.sum fun m => seq \u03ba n \u2297\u2096 seq \u03b7 m) a) s = \u2191\u2191(\u2191(seq \u03ba n \u2297\u2096 \u03b7) a) s", "state_after": "no goals"}, {"tactic": "simp_rw [compProd_of_not_isSFiniteKernel_right _ _ h]", "annotated_tactic": ["simp_rw [<a>compProd_of_not_isSFiniteKernel_right</a> _ _ h]", [{"full_name": "ProbabilityTheory.kernel.compProd_of_not_isSFiniteKernel_right", "def_path": "Mathlib/Probability/Kernel/Composition.lean", "def_pos": [236, 9], "def_end_pos": [236, 46]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03b3 : Type u_4\nm\u03b3 : MeasurableSpace \u03b3\ns : Set (\u03b2 \u00d7 \u03b3)\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\nh : \u00acIsSFiniteKernel \u03b7\n\u22a2 \u03ba \u2297\u2096 \u03b7 = kernel.sum fun n => seq \u03ba n \u2297\u2096 \u03b7", "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03b3 : Type u_4\nm\u03b3 : MeasurableSpace \u03b3\ns : Set (\u03b2 \u00d7 \u03b3)\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\nh : \u00acIsSFiniteKernel \u03b7\n\u22a2 0 = kernel.sum fun n => 0"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03b3 : Type u_4\nm\u03b3 : MeasurableSpace \u03b3\ns : Set (\u03b2 \u00d7 \u03b3)\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\nh : \u00acIsSFiniteKernel \u03b7\n\u22a2 0 = kernel.sum fun n => 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "full_name": "MeasureTheory.lintegral_add_left", "start": [554, 1], "end": [563, 80], "traced_tactics": [{"tactic": "refine' le_antisymm _ (le_lintegral_add _ _)", "annotated_tactic": ["refine' <a>le_antisymm</a> _ (<a>le_lintegral_add</a> _ _)", [{"full_name": "le_antisymm", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [188, 9], "def_end_pos": [188, 20]}, {"full_name": "MeasureTheory.le_lintegral_add", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [511, 9], "def_end_pos": [511, 25]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\ng : \u03b1 \u2192 \u211d\u22650\u221e\n\u22a2 \u222b\u207b (a : \u03b1), f a + g a \u2202\u03bc = \u222b\u207b (a : \u03b1), f a \u2202\u03bc + \u222b\u207b (a : \u03b1), g a \u2202\u03bc", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\ng : \u03b1 \u2192 \u211d\u22650\u221e\n\u22a2 \u222b\u207b (a : \u03b1), f a + g a \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), f a \u2202\u03bc + \u222b\u207b (a : \u03b1), g a \u2202\u03bc"}, {"tactic": "rcases exists_measurable_le_lintegral_eq \u03bc fun a => f a + g a with \u27e8\u03c6, h\u03c6m, h\u03c6_le, h\u03c6_eq\u27e9", "annotated_tactic": ["rcases <a>exists_measurable_le_lintegral_eq</a> \u03bc fun a => f a + g a with \u27e8\u03c6, h\u03c6m, h\u03c6_le, h\u03c6_eq\u27e9", [{"full_name": "MeasureTheory.exists_measurable_le_lintegral_eq", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [175, 9], "def_end_pos": [175, 42]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\ng : \u03b1 \u2192 \u211d\u22650\u221e\n\u22a2 \u222b\u207b (a : \u03b1), f a + g a \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), f a \u2202\u03bc + \u222b\u207b (a : \u03b1), g a \u2202\u03bc", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\ng \u03c6 : \u03b1 \u2192 \u211d\u22650\u221e\nh\u03c6m : Measurable \u03c6\nh\u03c6_le : \u03c6 \u2264 fun a => f a + g a\nh\u03c6_eq : \u222b\u207b (a : \u03b1), f a + g a \u2202\u03bc = \u222b\u207b (a : \u03b1), \u03c6 a \u2202\u03bc\n\u22a2 \u222b\u207b (a : \u03b1), f a + g a \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), f a \u2202\u03bc + \u222b\u207b (a : \u03b1), g a \u2202\u03bc"}, {"tactic": "calc\n  \u222b\u207b a, f a + g a \u2202\u03bc = \u222b\u207b a, \u03c6 a \u2202\u03bc := h\u03c6_eq\n  _ \u2264 \u222b\u207b a, f a + (\u03c6 a - f a) \u2202\u03bc := (lintegral_mono fun a => le_add_tsub)\n  _ = \u222b\u207b a, f a \u2202\u03bc + \u222b\u207b a, \u03c6 a - f a \u2202\u03bc := (lintegral_add_aux hf (h\u03c6m.sub hf))\n  _ \u2264 \u222b\u207b a, f a \u2202\u03bc + \u222b\u207b a, g a \u2202\u03bc :=\n    add_le_add_left (lintegral_mono fun a => tsub_le_iff_left.2 <| h\u03c6_le a) _", "annotated_tactic": ["calc\n    \u222b\u207b a, f a + g a \u2202\u03bc = \u222b\u207b a, \u03c6 a \u2202\u03bc := h\u03c6_eq\n    _ \u2264 \u222b\u207b a, f a + (\u03c6 a - f a) \u2202\u03bc := (<a>lintegral_mono</a> fun a => <a>le_add_tsub</a>)\n    _ = \u222b\u207b a, f a \u2202\u03bc + \u222b\u207b a, \u03c6 a - f a \u2202\u03bc := (<a>lintegral_add_aux</a> hf (h\u03c6m.sub hf))\n    _ \u2264 \u222b\u207b a, f a \u2202\u03bc + \u222b\u207b a, g a \u2202\u03bc :=\n      <a>add_le_add_left</a> (<a>lintegral_mono</a> fun a => <a>tsub_le_iff_left</a>.2 <| h\u03c6_le a) _", [{"full_name": "MeasureTheory.lintegral_mono", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [99, 9], "def_end_pos": [99, 23]}, {"full_name": "le_add_tsub", "def_path": "Mathlib/Algebra/Order/Sub/Defs.lean", "def_pos": [97, 9], "def_end_pos": [97, 20]}, {"full_name": "MeasureTheory.lintegral_add_aux", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [520, 9], "def_end_pos": [520, 26]}, {"full_name": "add_le_add_left", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [49, 15], "def_end_pos": [49, 30]}, {"full_name": "MeasureTheory.lintegral_mono", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [99, 9], "def_end_pos": [99, 23]}, {"full_name": "tsub_le_iff_left", "def_path": "Mathlib/Algebra/Order/Sub/Defs.lean", "def_pos": [94, 9], "def_end_pos": [94, 25]}]], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\ng \u03c6 : \u03b1 \u2192 \u211d\u22650\u221e\nh\u03c6m : Measurable \u03c6\nh\u03c6_le : \u03c6 \u2264 fun a => f a + g a\nh\u03c6_eq : \u222b\u207b (a : \u03b1), f a + g a \u2202\u03bc = \u222b\u207b (a : \u03b1), \u03c6 a \u2202\u03bc\n\u22a2 \u222b\u207b (a : \u03b1), f a + g a \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), f a \u2202\u03bc + \u222b\u207b (a : \u03b1), g a \u2202\u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Card.lean", "full_name": "Finset.card_erase_lt_of_mem", "start": [154, 1], "end": [155, 32], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Num/Lemmas.lean", "full_name": "Num.cast_bit1", "start": [738, 1], "end": [739, 75], "traced_tactics": [{"tactic": "rw [\u2190 bit1_of_bit1, _root_.bit1, bit0_of_bit0, cast_add, cast_bit0]", "annotated_tactic": ["rw [\u2190 <a>bit1_of_bit1</a>, <a>_root_.bit1</a>, <a>bit0_of_bit0</a>, <a>cast_add</a>, <a>cast_bit0</a>]", [{"full_name": "Num.bit1_of_bit1", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [230, 9], "def_end_pos": [230, 21]}, {"full_name": "bit1", "def_path": "Mathlib/Init/ZeroOne.lean", "def_pos": [39, 34], "def_end_pos": [39, 38]}, {"full_name": "Num.bit0_of_bit0", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [225, 9], "def_end_pos": [225, 21]}, {"full_name": "Num.cast_add", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [728, 9], "def_end_pos": [728, 17]}, {"full_name": "Num.cast_bit0", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [733, 9], "def_end_pos": [733, 18]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : Semiring \u03b1\nn : Num\n\u22a2 \u2191(Num.bit1 n) = _root_.bit1 \u2191n", "state_after": "\u03b1 : Type u_1\ninst\u271d : Semiring \u03b1\nn : Num\n\u22a2 _root_.bit0 \u2191n + \u21911 = _root_.bit1 \u2191n"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\u03b1 : Type u_1\ninst\u271d : Semiring \u03b1\nn : Num\n\u22a2 _root_.bit0 \u2191n + \u21911 = _root_.bit1 \u2191n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "full_name": "Nat.le_of_mod_lt", "start": [707, 1], "end": [708, 65], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Pairwise/Lattice.lean", "full_name": "Pairwise.subset_of_biUnion_subset_biUnion", "start": [156, 1], "end": [158, 81], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Card.lean", "full_name": "Set.ncard_pos", "start": [538, 1], "end": [539, 72], "traced_tactics": [{"tactic": "rw [pos_iff_ne_zero, Ne.def, ncard_eq_zero hs, nonempty_iff_ne_empty]", "annotated_tactic": ["rw [<a>pos_iff_ne_zero</a>, <a>Ne.def</a>, <a>ncard_eq_zero</a> hs, <a>nonempty_iff_ne_empty</a>]", [{"full_name": "pos_iff_ne_zero", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [243, 3], "def_end_pos": [243, 14]}, {"full_name": "Ne.def", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [59, 9], "def_end_pos": [59, 15]}, {"full_name": "Set.ncard_eq_zero", "def_path": "Mathlib/Data/Set/Card.lean", "def_pos": [517, 17], "def_end_pos": [517, 30]}, {"full_name": "Set.nonempty_iff_ne_empty", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [610, 9], "def_end_pos": [610, 30]}]], "state_before": "\u03b1 : Type u_1\ns t : Set \u03b1\nhs : autoParam (Set.Finite s) _auto\u271d\n\u22a2 0 < ncard s \u2194 Set.Nonempty s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/AEMeasurable.lean", "full_name": "AEMeasurable.add_measure", "start": [128, 1], "end": [130, 42], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean", "full_name": "Real.volume_preimage_mul_right", "start": [335, 1], "end": [340, 83], "traced_tactics": [{"tactic": "rw [map_volume_mul_right h]", "annotated_tactic": ["rw [<a>map_volume_mul_right</a> h]", [{"full_name": "Real.map_volume_mul_right", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean", "def_pos": [329, 9], "def_end_pos": [329, 29]}]], "state_before": "\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\na : \u211d\nh : a \u2260 0\ns : Set \u211d\n\u22a2 \u2191\u2191(Measure.map (fun x => x * a) volume) s = ofReal |a\u207b\u00b9| * \u2191\u2191volume s", "state_after": "\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\na : \u211d\nh : a \u2260 0\ns : Set \u211d\n\u22a2 \u2191\u2191(ofReal |a\u207b\u00b9| \u2022 volume) s = ofReal |a\u207b\u00b9| * \u2191\u2191volume s"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\na : \u211d\nh : a \u2260 0\ns : Set \u211d\n\u22a2 \u2191\u2191(ofReal |a\u207b\u00b9| \u2022 volume) s = ofReal |a\u207b\u00b9| * \u2191\u2191volume s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/PartrecCode.lean", "full_name": "Nat.Partrec.Code.hG", "start": [973, 9], "end": [1073, 18], "traced_tactics": [{"tactic": "have a := (Primrec.ofNat (\u2115 \u00d7 Code)).comp (Primrec.list_length (\u03b1 := List (Option \u2115)))", "annotated_tactic": ["have a := (<a>Primrec.ofNat</a> (\u2115 \u00d7 <a>Code</a>)).<a>comp</a> (<a>Primrec.list_length</a> (\u03b1 := <a>List</a> (<a>Option</a> \u2115)))", [{"full_name": "Primrec.ofNat", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [281, 19], "def_end_pos": [281, 24]}, {"full_name": "Nat.Partrec.Code", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [76, 11], "def_end_pos": [76, 15]}, {"full_name": "Primrec.comp", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [259, 9], "def_end_pos": [259, 13]}, {"full_name": "Primrec.list_length", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [1124, 9], "def_end_pos": [1124, 20]}, {"full_name": "List", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2182, 11], "def_end_pos": [2182, 15]}, {"full_name": "Option", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2139, 11], "def_end_pos": [2139, 17]}]], "state_before": "\u22a2 Primrec Nat.Partrec.Code.G", "state_after": "a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\n\u22a2 Primrec Nat.Partrec.Code.G"}, {"tactic": "have k := Primrec.fst.comp a", "annotated_tactic": ["have k := Primrec.fst.comp a", []], "state_before": "a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\n\u22a2 Primrec Nat.Partrec.Code.G", "state_after": "a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a)).1\n\u22a2 Primrec Nat.Partrec.Code.G"}, {"tactic": "refine' Primrec.option_some.comp (Primrec.list_map (Primrec.list_range.comp k) (_ : Primrec _))", "annotated_tactic": ["refine' Primrec.option_some.comp (<a>Primrec.list_map</a> (Primrec.list_range.comp k) (_ : <a>Primrec</a> _))", [{"full_name": "Primrec.list_map", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [1107, 9], "def_end_pos": [1107, 17]}, {"full_name": "Primrec", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [207, 5], "def_end_pos": [207, 12]}]], "state_before": "a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a)).1\n\u22a2 Primrec Nat.Partrec.Code.G", "state_after": "a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a)).1\n\u22a2 Primrec fun p =>\n    (fun a n =>\n        Nat.casesOn (ofNat (\u2115 \u00d7 Code) (List.length a)).1 Option.none fun k' =>\n          Code.recOn (ofNat (\u2115 \u00d7 Code) (List.length a)).2 (some 0) (some (Nat.succ n)) (some (unpair n).1)\n            (some (unpair n).2)\n            (fun cf cg x x => do\n              let x \u2190 Nat.Partrec.Code.lup a ((ofNat (\u2115 \u00d7 Code) (List.length a)).1, cf) n\n              let y \u2190 Nat.Partrec.Code.lup a ((ofNat (\u2115 \u00d7 Code) (List.length a)).1, cg) n\n              some (Nat.pair x y))\n            (fun cf cg x x => do\n              let x \u2190 Nat.Partrec.Code.lup a ((ofNat (\u2115 \u00d7 Code) (List.length a)).1, cg) n\n              Nat.Partrec.Code.lup a ((ofNat (\u2115 \u00d7 Code) (List.length a)).1, cf) x)\n            (fun cf cg x x =>\n              let z := (unpair n).1;\n              Nat.casesOn (unpair n).2 (Nat.Partrec.Code.lup a ((ofNat (\u2115 \u00d7 Code) (List.length a)).1, cf) z) fun y => do\n                let i \u2190 Nat.Partrec.Code.lup a (k', (ofNat (\u2115 \u00d7 Code) (List.length a)).2) (Nat.pair z y)\n                Nat.Partrec.Code.lup a ((ofNat (\u2115 \u00d7 Code) (List.length a)).1, cg) (Nat.pair z (Nat.pair y i)))\n            fun cf x =>\n            let z := (unpair n).1;\n            let m := (unpair n).2;\n            do\n            let x \u2190 Nat.Partrec.Code.lup a ((ofNat (\u2115 \u00d7 Code) (List.length a)).1, cf) (Nat.pair z m)\n            Nat.casesOn x (some m) fun x =>\n                Nat.Partrec.Code.lup a (k', (ofNat (\u2115 \u00d7 Code) (List.length a)).2) (Nat.pair z (m + 1)))\n      p.1 p.2"}, {"tactic": "replace k := k.comp (Primrec.fst (\u03b2 := \u2115))", "annotated_tactic": ["replace k := k.comp (<a>Primrec.fst</a> (\u03b2 := \u2115))", [{"full_name": "Primrec.fst", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [340, 9], "def_end_pos": [340, 12]}]], "state_before": "a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a)).1\n\u22a2 Primrec fun p =>\n    (fun a n =>\n        Nat.casesOn (ofNat (\u2115 \u00d7 Code) (List.length a)).1 Option.none fun k' =>\n          Code.recOn (ofNat (\u2115 \u00d7 Code) (List.length a)).2 (some 0) (some (Nat.succ n)) (some (unpair n).1)\n            (some (unpair n).2)\n            (fun cf cg x x => do\n              let x \u2190 Nat.Partrec.Code.lup a ((ofNat (\u2115 \u00d7 Code) (List.length a)).1, cf) n\n              let y \u2190 Nat.Partrec.Code.lup a ((ofNat (\u2115 \u00d7 Code) (List.length a)).1, cg) n\n              some (Nat.pair x y))\n            (fun cf cg x x => do\n              let x \u2190 Nat.Partrec.Code.lup a ((ofNat (\u2115 \u00d7 Code) (List.length a)).1, cg) n\n              Nat.Partrec.Code.lup a ((ofNat (\u2115 \u00d7 Code) (List.length a)).1, cf) x)\n            (fun cf cg x x =>\n              let z := (unpair n).1;\n              Nat.casesOn (unpair n).2 (Nat.Partrec.Code.lup a ((ofNat (\u2115 \u00d7 Code) (List.length a)).1, cf) z) fun y => do\n                let i \u2190 Nat.Partrec.Code.lup a (k', (ofNat (\u2115 \u00d7 Code) (List.length a)).2) (Nat.pair z y)\n                Nat.Partrec.Code.lup a ((ofNat (\u2115 \u00d7 Code) (List.length a)).1, cg) (Nat.pair z (Nat.pair y i)))\n            fun cf x =>\n            let z := (unpair n).1;\n            let m := (unpair n).2;\n            do\n            let x \u2190 Nat.Partrec.Code.lup a ((ofNat (\u2115 \u00d7 Code) (List.length a)).1, cf) (Nat.pair z m)\n            Nat.casesOn x (some m) fun x =>\n                Nat.Partrec.Code.lup a (k', (ofNat (\u2115 \u00d7 Code) (List.length a)).2) (Nat.pair z (m + 1)))\n      p.1 p.2", "state_after": "a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1\n\u22a2 Primrec fun p =>\n    (fun a n =>\n        Nat.casesOn (ofNat (\u2115 \u00d7 Code) (List.length a)).1 Option.none fun k' =>\n          Code.recOn (ofNat (\u2115 \u00d7 Code) (List.length a)).2 (some 0) (some (Nat.succ n)) (some (unpair n).1)\n            (some (unpair n).2)\n            (fun cf cg x x => do\n              let x \u2190 Nat.Partrec.Code.lup a ((ofNat (\u2115 \u00d7 Code) (List.length a)).1, cf) n\n              let y \u2190 Nat.Partrec.Code.lup a ((ofNat (\u2115 \u00d7 Code) (List.length a)).1, cg) n\n              some (Nat.pair x y))\n            (fun cf cg x x => do\n              let x \u2190 Nat.Partrec.Code.lup a ((ofNat (\u2115 \u00d7 Code) (List.length a)).1, cg) n\n              Nat.Partrec.Code.lup a ((ofNat (\u2115 \u00d7 Code) (List.length a)).1, cf) x)\n            (fun cf cg x x =>\n              let z := (unpair n).1;\n              Nat.casesOn (unpair n).2 (Nat.Partrec.Code.lup a ((ofNat (\u2115 \u00d7 Code) (List.length a)).1, cf) z) fun y => do\n                let i \u2190 Nat.Partrec.Code.lup a (k', (ofNat (\u2115 \u00d7 Code) (List.length a)).2) (Nat.pair z y)\n                Nat.Partrec.Code.lup a ((ofNat (\u2115 \u00d7 Code) (List.length a)).1, cg) (Nat.pair z (Nat.pair y i)))\n            fun cf x =>\n            let z := (unpair n).1;\n            let m := (unpair n).2;\n            do\n            let x \u2190 Nat.Partrec.Code.lup a ((ofNat (\u2115 \u00d7 Code) (List.length a)).1, cf) (Nat.pair z m)\n            Nat.casesOn x (some m) fun x =>\n                Nat.Partrec.Code.lup a (k', (ofNat (\u2115 \u00d7 Code) (List.length a)).2) (Nat.pair z (m + 1)))\n      p.1 p.2"}, {"tactic": "have n := Primrec.snd (\u03b1 := List (List (Option \u2115))) (\u03b2 := \u2115)", "annotated_tactic": ["have n := <a>Primrec.snd</a> (\u03b1 := <a>List</a> (<a>List</a> (<a>Option</a> \u2115))) (\u03b2 := \u2115)", [{"full_name": "Primrec.snd", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [351, 9], "def_end_pos": [351, 12]}, {"full_name": "List", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2182, 11], "def_end_pos": [2182, 15]}, {"full_name": "List", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2182, 11], "def_end_pos": [2182, 15]}, {"full_name": "Option", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2139, 11], "def_end_pos": [2139, 17]}]], "state_before": "a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1\n\u22a2 Primrec fun p =>\n    (fun a n =>\n        Nat.casesOn (ofNat (\u2115 \u00d7 Code) (List.length a)).1 Option.none fun k' =>\n          Code.recOn (ofNat (\u2115 \u00d7 Code) (List.length a)).2 (some 0) (some (Nat.succ n)) (some (unpair n).1)\n            (some (unpair n).2)\n            (fun cf cg x x => do\n              let x \u2190 Nat.Partrec.Code.lup a ((ofNat (\u2115 \u00d7 Code) (List.length a)).1, cf) n\n              let y \u2190 Nat.Partrec.Code.lup a ((ofNat (\u2115 \u00d7 Code) (List.length a)).1, cg) n\n              some (Nat.pair x y))\n            (fun cf cg x x => do\n              let x \u2190 Nat.Partrec.Code.lup a ((ofNat (\u2115 \u00d7 Code) (List.length a)).1, cg) n\n              Nat.Partrec.Code.lup a ((ofNat (\u2115 \u00d7 Code) (List.length a)).1, cf) x)\n            (fun cf cg x x =>\n              let z := (unpair n).1;\n              Nat.casesOn (unpair n).2 (Nat.Partrec.Code.lup a ((ofNat (\u2115 \u00d7 Code) (List.length a)).1, cf) z) fun y => do\n                let i \u2190 Nat.Partrec.Code.lup a (k', (ofNat (\u2115 \u00d7 Code) (List.length a)).2) (Nat.pair z y)\n                Nat.Partrec.Code.lup a ((ofNat (\u2115 \u00d7 Code) (List.length a)).1, cg) (Nat.pair z (Nat.pair y i)))\n            fun cf x =>\n            let z := (unpair n).1;\n            let m := (unpair n).2;\n            do\n            let x \u2190 Nat.Partrec.Code.lup a ((ofNat (\u2115 \u00d7 Code) (List.length a)).1, cf) (Nat.pair z m)\n            Nat.casesOn x (some m) fun x =>\n                Nat.Partrec.Code.lup a (k', (ofNat (\u2115 \u00d7 Code) (List.length a)).2) (Nat.pair z (m + 1)))\n      p.1 p.2", "state_after": "a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1\nn : Primrec Prod.snd\n\u22a2 Primrec fun p =>\n    (fun a n =>\n        Nat.casesOn (ofNat (\u2115 \u00d7 Code) (List.length a)).1 Option.none fun k' =>\n          Code.recOn (ofNat (\u2115 \u00d7 Code) (List.length a)).2 (some 0) (some (Nat.succ n)) (some (unpair n).1)\n            (some (unpair n).2)\n            (fun cf cg x x => do\n              let x \u2190 Nat.Partrec.Code.lup a ((ofNat (\u2115 \u00d7 Code) (List.length a)).1, cf) n\n              let y \u2190 Nat.Partrec.Code.lup a ((ofNat (\u2115 \u00d7 Code) (List.length a)).1, cg) n\n              some (Nat.pair x y))\n            (fun cf cg x x => do\n              let x \u2190 Nat.Partrec.Code.lup a ((ofNat (\u2115 \u00d7 Code) (List.length a)).1, cg) n\n              Nat.Partrec.Code.lup a ((ofNat (\u2115 \u00d7 Code) (List.length a)).1, cf) x)\n            (fun cf cg x x =>\n              let z := (unpair n).1;\n              Nat.casesOn (unpair n).2 (Nat.Partrec.Code.lup a ((ofNat (\u2115 \u00d7 Code) (List.length a)).1, cf) z) fun y => do\n                let i \u2190 Nat.Partrec.Code.lup a (k', (ofNat (\u2115 \u00d7 Code) (List.length a)).2) (Nat.pair z y)\n                Nat.Partrec.Code.lup a ((ofNat (\u2115 \u00d7 Code) (List.length a)).1, cg) (Nat.pair z (Nat.pair y i)))\n            fun cf x =>\n            let z := (unpair n).1;\n            let m := (unpair n).2;\n            do\n            let x \u2190 Nat.Partrec.Code.lup a ((ofNat (\u2115 \u00d7 Code) (List.length a)).1, cf) (Nat.pair z m)\n            Nat.casesOn x (some m) fun x =>\n                Nat.Partrec.Code.lup a (k', (ofNat (\u2115 \u00d7 Code) (List.length a)).2) (Nat.pair z (m + 1)))\n      p.1 p.2"}, {"tactic": "refine' Primrec.nat_casesOn k (_root_.Primrec.const Option.none) (_ : Primrec _)", "annotated_tactic": ["refine' <a>Primrec.nat_casesOn</a> k (<a>_root_.Primrec.const</a> <a>Option.none</a>) (_ : <a>Primrec</a> _)", [{"full_name": "Primrec.nat_casesOn", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [594, 9], "def_end_pos": [594, 20]}, {"full_name": "Primrec.const", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [250, 9], "def_end_pos": [250, 14]}, {"full_name": "Option.none", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2141, 5], "def_end_pos": [2141, 9]}, {"full_name": "Primrec", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [207, 5], "def_end_pos": [207, 12]}]], "state_before": "a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1\nn : Primrec Prod.snd\n\u22a2 Primrec fun p =>\n    (fun a n =>\n        Nat.casesOn (ofNat (\u2115 \u00d7 Code) (List.length a)).1 Option.none fun k' =>\n          Code.recOn (ofNat (\u2115 \u00d7 Code) (List.length a)).2 (some 0) (some (Nat.succ n)) (some (unpair n).1)\n            (some (unpair n).2)\n            (fun cf cg x x => do\n              let x \u2190 Nat.Partrec.Code.lup a ((ofNat (\u2115 \u00d7 Code) (List.length a)).1, cf) n\n              let y \u2190 Nat.Partrec.Code.lup a ((ofNat (\u2115 \u00d7 Code) (List.length a)).1, cg) n\n              some (Nat.pair x y))\n            (fun cf cg x x => do\n              let x \u2190 Nat.Partrec.Code.lup a ((ofNat (\u2115 \u00d7 Code) (List.length a)).1, cg) n\n              Nat.Partrec.Code.lup a ((ofNat (\u2115 \u00d7 Code) (List.length a)).1, cf) x)\n            (fun cf cg x x =>\n              let z := (unpair n).1;\n              Nat.casesOn (unpair n).2 (Nat.Partrec.Code.lup a ((ofNat (\u2115 \u00d7 Code) (List.length a)).1, cf) z) fun y => do\n                let i \u2190 Nat.Partrec.Code.lup a (k', (ofNat (\u2115 \u00d7 Code) (List.length a)).2) (Nat.pair z y)\n                Nat.Partrec.Code.lup a ((ofNat (\u2115 \u00d7 Code) (List.length a)).1, cg) (Nat.pair z (Nat.pair y i)))\n            fun cf x =>\n            let z := (unpair n).1;\n            let m := (unpair n).2;\n            do\n            let x \u2190 Nat.Partrec.Code.lup a ((ofNat (\u2115 \u00d7 Code) (List.length a)).1, cf) (Nat.pair z m)\n            Nat.casesOn x (some m) fun x =>\n                Nat.Partrec.Code.lup a (k', (ofNat (\u2115 \u00d7 Code) (List.length a)).2) (Nat.pair z (m + 1)))\n      p.1 p.2", "state_after": "a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1\nn : Primrec Prod.snd\n\u22a2 Primrec fun p =>\n    (fun p n =>\n        (fun k' =>\n            Code.recOn (ofNat (\u2115 \u00d7 Code) (List.length p.1)).2 (some 0) (some (Nat.succ p.2)) (some (unpair p.2).1)\n              (some (unpair p.2).2)\n              (fun cf cg x x => do\n                let x \u2190 Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cf) p.2\n                let y \u2190 Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cg) p.2\n                some (Nat.pair x y))\n              (fun cf cg x x => do\n                let x \u2190 Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cg) p.2\n                Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cf) x)\n              (fun cf cg x x =>\n                let z := (unpair p.2).1;\n                Nat.casesOn (unpair p.2).2 (Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cf) z)\n                  fun y => do\n                  let i \u2190 Nat.Partrec.Code.lup p.1 (k', (ofNat (\u2115 \u00d7 Code) (List.length p.1)).2) (Nat.pair z y)\n                  Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cg) (Nat.pair z (Nat.pair y i)))\n              fun cf x =>\n              let z := (unpair p.2).1;\n              let m := (unpair p.2).2;\n              do\n              let x \u2190 Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cf) (Nat.pair z m)\n              Nat.casesOn x (some m) fun x =>\n                  Nat.Partrec.Code.lup p.1 (k', (ofNat (\u2115 \u00d7 Code) (List.length p.1)).2) (Nat.pair z (m + 1)))\n          n)\n      p.1 p.2"}, {"tactic": "have k := k.comp (Primrec.fst (\u03b2 := \u2115))", "annotated_tactic": ["have k := k.comp (<a>Primrec.fst</a> (\u03b2 := \u2115))", [{"full_name": "Primrec.fst", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [340, 9], "def_end_pos": [340, 12]}]], "state_before": "a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1\nn : Primrec Prod.snd\n\u22a2 Primrec fun p =>\n    (fun p n =>\n        (fun k' =>\n            Code.recOn (ofNat (\u2115 \u00d7 Code) (List.length p.1)).2 (some 0) (some (Nat.succ p.2)) (some (unpair p.2).1)\n              (some (unpair p.2).2)\n              (fun cf cg x x => do\n                let x \u2190 Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cf) p.2\n                let y \u2190 Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cg) p.2\n                some (Nat.pair x y))\n              (fun cf cg x x => do\n                let x \u2190 Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cg) p.2\n                Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cf) x)\n              (fun cf cg x x =>\n                let z := (unpair p.2).1;\n                Nat.casesOn (unpair p.2).2 (Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cf) z)\n                  fun y => do\n                  let i \u2190 Nat.Partrec.Code.lup p.1 (k', (ofNat (\u2115 \u00d7 Code) (List.length p.1)).2) (Nat.pair z y)\n                  Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cg) (Nat.pair z (Nat.pair y i)))\n              fun cf x =>\n              let z := (unpair p.2).1;\n              let m := (unpair p.2).2;\n              do\n              let x \u2190 Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cf) (Nat.pair z m)\n              Nat.casesOn x (some m) fun x =>\n                  Nat.Partrec.Code.lup p.1 (k', (ofNat (\u2115 \u00d7 Code) (List.length p.1)).2) (Nat.pair z (m + 1)))\n          n)\n      p.1 p.2", "state_after": "a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1\nn : Primrec Prod.snd\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1\n\u22a2 Primrec fun p =>\n    (fun p n =>\n        (fun k' =>\n            Code.recOn (ofNat (\u2115 \u00d7 Code) (List.length p.1)).2 (some 0) (some (Nat.succ p.2)) (some (unpair p.2).1)\n              (some (unpair p.2).2)\n              (fun cf cg x x => do\n                let x \u2190 Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cf) p.2\n                let y \u2190 Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cg) p.2\n                some (Nat.pair x y))\n              (fun cf cg x x => do\n                let x \u2190 Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cg) p.2\n                Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cf) x)\n              (fun cf cg x x =>\n                let z := (unpair p.2).1;\n                Nat.casesOn (unpair p.2).2 (Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cf) z)\n                  fun y => do\n                  let i \u2190 Nat.Partrec.Code.lup p.1 (k', (ofNat (\u2115 \u00d7 Code) (List.length p.1)).2) (Nat.pair z y)\n                  Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cg) (Nat.pair z (Nat.pair y i)))\n              fun cf x =>\n              let z := (unpair p.2).1;\n              let m := (unpair p.2).2;\n              do\n              let x \u2190 Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cf) (Nat.pair z m)\n              Nat.casesOn x (some m) fun x =>\n                  Nat.Partrec.Code.lup p.1 (k', (ofNat (\u2115 \u00d7 Code) (List.length p.1)).2) (Nat.pair z (m + 1)))\n          n)\n      p.1 p.2"}, {"tactic": "have n := n.comp (Primrec.fst (\u03b2 := \u2115))", "annotated_tactic": ["have n := n.comp (<a>Primrec.fst</a> (\u03b2 := \u2115))", [{"full_name": "Primrec.fst", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [340, 9], "def_end_pos": [340, 12]}]], "state_before": "a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1\nn : Primrec Prod.snd\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1\n\u22a2 Primrec fun p =>\n    (fun p n =>\n        (fun k' =>\n            Code.recOn (ofNat (\u2115 \u00d7 Code) (List.length p.1)).2 (some 0) (some (Nat.succ p.2)) (some (unpair p.2).1)\n              (some (unpair p.2).2)\n              (fun cf cg x x => do\n                let x \u2190 Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cf) p.2\n                let y \u2190 Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cg) p.2\n                some (Nat.pair x y))\n              (fun cf cg x x => do\n                let x \u2190 Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cg) p.2\n                Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cf) x)\n              (fun cf cg x x =>\n                let z := (unpair p.2).1;\n                Nat.casesOn (unpair p.2).2 (Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cf) z)\n                  fun y => do\n                  let i \u2190 Nat.Partrec.Code.lup p.1 (k', (ofNat (\u2115 \u00d7 Code) (List.length p.1)).2) (Nat.pair z y)\n                  Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cg) (Nat.pair z (Nat.pair y i)))\n              fun cf x =>\n              let z := (unpair p.2).1;\n              let m := (unpair p.2).2;\n              do\n              let x \u2190 Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cf) (Nat.pair z m)\n              Nat.casesOn x (some m) fun x =>\n                  Nat.Partrec.Code.lup p.1 (k', (ofNat (\u2115 \u00d7 Code) (List.length p.1)).2) (Nat.pair z (m + 1)))\n          n)\n      p.1 p.2", "state_after": "a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1\nn\u271d : Primrec Prod.snd\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1\nn : Primrec fun a => a.1.2\n\u22a2 Primrec fun p =>\n    (fun p n =>\n        (fun k' =>\n            Code.recOn (ofNat (\u2115 \u00d7 Code) (List.length p.1)).2 (some 0) (some (Nat.succ p.2)) (some (unpair p.2).1)\n              (some (unpair p.2).2)\n              (fun cf cg x x => do\n                let x \u2190 Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cf) p.2\n                let y \u2190 Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cg) p.2\n                some (Nat.pair x y))\n              (fun cf cg x x => do\n                let x \u2190 Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cg) p.2\n                Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cf) x)\n              (fun cf cg x x =>\n                let z := (unpair p.2).1;\n                Nat.casesOn (unpair p.2).2 (Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cf) z)\n                  fun y => do\n                  let i \u2190 Nat.Partrec.Code.lup p.1 (k', (ofNat (\u2115 \u00d7 Code) (List.length p.1)).2) (Nat.pair z y)\n                  Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cg) (Nat.pair z (Nat.pair y i)))\n              fun cf x =>\n              let z := (unpair p.2).1;\n              let m := (unpair p.2).2;\n              do\n              let x \u2190 Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cf) (Nat.pair z m)\n              Nat.casesOn x (some m) fun x =>\n                  Nat.Partrec.Code.lup p.1 (k', (ofNat (\u2115 \u00d7 Code) (List.length p.1)).2) (Nat.pair z (m + 1)))\n          n)\n      p.1 p.2"}, {"tactic": "have k' := Primrec.snd (\u03b1 := List (List (Option \u2115)) \u00d7 \u2115) (\u03b2 := \u2115)", "annotated_tactic": ["have k' := <a>Primrec.snd</a> (\u03b1 := <a>List</a> (<a>List</a> (<a>Option</a> \u2115)) \u00d7 \u2115) (\u03b2 := \u2115)", [{"full_name": "Primrec.snd", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [351, 9], "def_end_pos": [351, 12]}, {"full_name": "List", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2182, 11], "def_end_pos": [2182, 15]}, {"full_name": "List", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2182, 11], "def_end_pos": [2182, 15]}, {"full_name": "Option", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2139, 11], "def_end_pos": [2139, 17]}]], "state_before": "a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1\nn\u271d : Primrec Prod.snd\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1\nn : Primrec fun a => a.1.2\n\u22a2 Primrec fun p =>\n    (fun p n =>\n        (fun k' =>\n            Code.recOn (ofNat (\u2115 \u00d7 Code) (List.length p.1)).2 (some 0) (some (Nat.succ p.2)) (some (unpair p.2).1)\n              (some (unpair p.2).2)\n              (fun cf cg x x => do\n                let x \u2190 Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cf) p.2\n                let y \u2190 Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cg) p.2\n                some (Nat.pair x y))\n              (fun cf cg x x => do\n                let x \u2190 Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cg) p.2\n                Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cf) x)\n              (fun cf cg x x =>\n                let z := (unpair p.2).1;\n                Nat.casesOn (unpair p.2).2 (Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cf) z)\n                  fun y => do\n                  let i \u2190 Nat.Partrec.Code.lup p.1 (k', (ofNat (\u2115 \u00d7 Code) (List.length p.1)).2) (Nat.pair z y)\n                  Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cg) (Nat.pair z (Nat.pair y i)))\n              fun cf x =>\n              let z := (unpair p.2).1;\n              let m := (unpair p.2).2;\n              do\n              let x \u2190 Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cf) (Nat.pair z m)\n              Nat.casesOn x (some m) fun x =>\n                  Nat.Partrec.Code.lup p.1 (k', (ofNat (\u2115 \u00d7 Code) (List.length p.1)).2) (Nat.pair z (m + 1)))\n          n)\n      p.1 p.2", "state_after": "a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1\nn\u271d : Primrec Prod.snd\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1\nn : Primrec fun a => a.1.2\nk' : Primrec Prod.snd\n\u22a2 Primrec fun p =>\n    (fun p n =>\n        (fun k' =>\n            Code.recOn (ofNat (\u2115 \u00d7 Code) (List.length p.1)).2 (some 0) (some (Nat.succ p.2)) (some (unpair p.2).1)\n              (some (unpair p.2).2)\n              (fun cf cg x x => do\n                let x \u2190 Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cf) p.2\n                let y \u2190 Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cg) p.2\n                some (Nat.pair x y))\n              (fun cf cg x x => do\n                let x \u2190 Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cg) p.2\n                Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cf) x)\n              (fun cf cg x x =>\n                let z := (unpair p.2).1;\n                Nat.casesOn (unpair p.2).2 (Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cf) z)\n                  fun y => do\n                  let i \u2190 Nat.Partrec.Code.lup p.1 (k', (ofNat (\u2115 \u00d7 Code) (List.length p.1)).2) (Nat.pair z y)\n                  Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cg) (Nat.pair z (Nat.pair y i)))\n              fun cf x =>\n              let z := (unpair p.2).1;\n              let m := (unpair p.2).2;\n              do\n              let x \u2190 Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cf) (Nat.pair z m)\n              Nat.casesOn x (some m) fun x =>\n                  Nat.Partrec.Code.lup p.1 (k', (ofNat (\u2115 \u00d7 Code) (List.length p.1)).2) (Nat.pair z (m + 1)))\n          n)\n      p.1 p.2"}, {"tactic": "have c := Primrec.snd.comp (a.comp <| (Primrec.fst (\u03b2 := \u2115)).comp (Primrec.fst (\u03b2 := \u2115)))", "annotated_tactic": ["have c := Primrec.snd.comp (a.comp <| (<a>Primrec.fst</a> (\u03b2 := \u2115)).<a>comp</a> (<a>Primrec.fst</a> (\u03b2 := \u2115)))", [{"full_name": "Primrec.fst", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [340, 9], "def_end_pos": [340, 12]}, {"full_name": "Primrec.comp", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [259, 9], "def_end_pos": [259, 13]}, {"full_name": "Primrec.fst", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [340, 9], "def_end_pos": [340, 12]}]], "state_before": "a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1\nn\u271d : Primrec Prod.snd\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1\nn : Primrec fun a => a.1.2\nk' : Primrec Prod.snd\n\u22a2 Primrec fun p =>\n    (fun p n =>\n        (fun k' =>\n            Code.recOn (ofNat (\u2115 \u00d7 Code) (List.length p.1)).2 (some 0) (some (Nat.succ p.2)) (some (unpair p.2).1)\n              (some (unpair p.2).2)\n              (fun cf cg x x => do\n                let x \u2190 Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cf) p.2\n                let y \u2190 Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cg) p.2\n                some (Nat.pair x y))\n              (fun cf cg x x => do\n                let x \u2190 Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cg) p.2\n                Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cf) x)\n              (fun cf cg x x =>\n                let z := (unpair p.2).1;\n                Nat.casesOn (unpair p.2).2 (Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cf) z)\n                  fun y => do\n                  let i \u2190 Nat.Partrec.Code.lup p.1 (k', (ofNat (\u2115 \u00d7 Code) (List.length p.1)).2) (Nat.pair z y)\n                  Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cg) (Nat.pair z (Nat.pair y i)))\n              fun cf x =>\n              let z := (unpair p.2).1;\n              let m := (unpair p.2).2;\n              do\n              let x \u2190 Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cf) (Nat.pair z m)\n              Nat.casesOn x (some m) fun x =>\n                  Nat.Partrec.Code.lup p.1 (k', (ofNat (\u2115 \u00d7 Code) (List.length p.1)).2) (Nat.pair z (m + 1)))\n          n)\n      p.1 p.2", "state_after": "a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1\nn\u271d : Primrec Prod.snd\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1\nn : Primrec fun a => a.1.2\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2\n\u22a2 Primrec fun p =>\n    (fun p n =>\n        (fun k' =>\n            Code.recOn (ofNat (\u2115 \u00d7 Code) (List.length p.1)).2 (some 0) (some (Nat.succ p.2)) (some (unpair p.2).1)\n              (some (unpair p.2).2)\n              (fun cf cg x x => do\n                let x \u2190 Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cf) p.2\n                let y \u2190 Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cg) p.2\n                some (Nat.pair x y))\n              (fun cf cg x x => do\n                let x \u2190 Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cg) p.2\n                Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cf) x)\n              (fun cf cg x x =>\n                let z := (unpair p.2).1;\n                Nat.casesOn (unpair p.2).2 (Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cf) z)\n                  fun y => do\n                  let i \u2190 Nat.Partrec.Code.lup p.1 (k', (ofNat (\u2115 \u00d7 Code) (List.length p.1)).2) (Nat.pair z y)\n                  Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cg) (Nat.pair z (Nat.pair y i)))\n              fun cf x =>\n              let z := (unpair p.2).1;\n              let m := (unpair p.2).2;\n              do\n              let x \u2190 Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cf) (Nat.pair z m)\n              Nat.casesOn x (some m) fun x =>\n                  Nat.Partrec.Code.lup p.1 (k', (ofNat (\u2115 \u00d7 Code) (List.length p.1)).2) (Nat.pair z (m + 1)))\n          n)\n      p.1 p.2"}, {"tactic": "apply\n  Nat.Partrec.Code.rec_prim c\n    (_root_.Primrec.const (some 0))\n    (Primrec.option_some.comp (_root_.Primrec.succ.comp n))\n    (Primrec.option_some.comp (Primrec.fst.comp <| Primrec.unpair.comp n))\n    (Primrec.option_some.comp (Primrec.snd.comp <| Primrec.unpair.comp n))", "annotated_tactic": ["apply\n    <a>Nat.Partrec.Code.rec_prim</a> c\n      (<a>_root_.Primrec.const</a> (<a>some</a> 0))\n      (Primrec.option_some.comp (_root_.Primrec.succ.comp n))\n      (Primrec.option_some.comp (Primrec.fst.comp <| Primrec.unpair.comp n))\n      (Primrec.option_some.comp (Primrec.snd.comp <| Primrec.unpair.comp n))", [{"full_name": "Nat.Partrec.Code.rec_prim", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [386, 9], "def_end_pos": [386, 17]}, {"full_name": "Primrec.const", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [250, 9], "def_end_pos": [250, 14]}, {"full_name": "Option.some", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2143, 5], "def_end_pos": [2143, 9]}]], "state_before": "a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1\nn\u271d : Primrec Prod.snd\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1\nn : Primrec fun a => a.1.2\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2\n\u22a2 Primrec fun p =>\n    (fun p n =>\n        (fun k' =>\n            Code.recOn (ofNat (\u2115 \u00d7 Code) (List.length p.1)).2 (some 0) (some (Nat.succ p.2)) (some (unpair p.2).1)\n              (some (unpair p.2).2)\n              (fun cf cg x x => do\n                let x \u2190 Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cf) p.2\n                let y \u2190 Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cg) p.2\n                some (Nat.pair x y))\n              (fun cf cg x x => do\n                let x \u2190 Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cg) p.2\n                Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cf) x)\n              (fun cf cg x x =>\n                let z := (unpair p.2).1;\n                Nat.casesOn (unpair p.2).2 (Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cf) z)\n                  fun y => do\n                  let i \u2190 Nat.Partrec.Code.lup p.1 (k', (ofNat (\u2115 \u00d7 Code) (List.length p.1)).2) (Nat.pair z y)\n                  Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cg) (Nat.pair z (Nat.pair y i)))\n              fun cf x =>\n              let z := (unpair p.2).1;\n              let m := (unpair p.2).2;\n              do\n              let x \u2190 Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cf) (Nat.pair z m)\n              Nat.casesOn x (some m) fun x =>\n                  Nat.Partrec.Code.lup p.1 (k', (ofNat (\u2115 \u00d7 Code) (List.length p.1)).2) (Nat.pair z (m + 1)))\n          n)\n      p.1 p.2", "state_after": "case hpr\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1\nn\u271d : Primrec Prod.snd\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1\nn : Primrec fun a => a.1.2\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2\n\u22a2 Primrec fun a => do\n    let x \u2190 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) a.1.1.2\n    let y \u2190 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.2.1) a.1.1.2\n    some (Nat.pair x y)\n\ncase hco\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1\nn\u271d : Primrec Prod.snd\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1\nn : Primrec fun a => a.1.2\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2\n\u22a2 Primrec fun a => do\n    let x \u2190 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.2.1) a.1.1.2\n    Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) x\n\ncase hpc\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1\nn\u271d : Primrec Prod.snd\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1\nn : Primrec fun a => a.1.2\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2\n\u22a2 Primrec fun a =>\n    let z := (unpair a.1.1.2).1;\n    Nat.casesOn (unpair a.1.1.2).2 (Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) z)\n      fun y => do\n      let i \u2190 Nat.Partrec.Code.lup a.1.1.1 (a.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).2) (Nat.pair z y)\n      Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.2.1) (Nat.pair z (Nat.pair y i))\n\ncase hrf\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1\nn\u271d : Primrec Prod.snd\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1\nn : Primrec fun a => a.1.2\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2\n\u22a2 Primrec fun a =>\n    let z := (unpair a.1.1.2).1;\n    let m := (unpair a.1.1.2).2;\n    do\n    let x \u2190 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) (Nat.pair z m)\n    Nat.casesOn x (some m) fun x =>\n        Nat.Partrec.Code.lup a.1.1.1 (a.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).2) (Nat.pair z (m + 1))"}, {"tactic": "have L := (Primrec.fst.comp Primrec.fst).comp\n  (Primrec.fst (\u03b1 := (List (List (Option \u2115)) \u00d7 \u2115) \u00d7 \u2115)\n    (\u03b2 := Code \u00d7 Code \u00d7 Option \u2115 \u00d7 Option \u2115))", "annotated_tactic": ["have L := (Primrec.fst.comp <a>Primrec.fst</a>).<a>comp</a>\n      (<a>Primrec.fst</a> (\u03b1 := (<a>List</a> (<a>List</a> (<a>Option</a> \u2115)) \u00d7 \u2115) \u00d7 \u2115)\n        (\u03b2 := <a>Code</a> \u00d7 <a>Code</a> \u00d7 <a>Option</a> \u2115 \u00d7 <a>Option</a> \u2115))", [{"full_name": "Primrec.fst", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [340, 9], "def_end_pos": [340, 12]}, {"full_name": "Primrec.comp", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [259, 9], "def_end_pos": [259, 13]}, {"full_name": "Primrec.fst", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [340, 9], "def_end_pos": [340, 12]}, {"full_name": "List", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2182, 11], "def_end_pos": [2182, 15]}, {"full_name": "List", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2182, 11], "def_end_pos": [2182, 15]}, {"full_name": "Option", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2139, 11], "def_end_pos": [2139, 17]}, {"full_name": "Nat.Partrec.Code", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [76, 11], "def_end_pos": [76, 15]}, {"full_name": "Nat.Partrec.Code", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [76, 11], "def_end_pos": [76, 15]}, {"full_name": "Option", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2139, 11], "def_end_pos": [2139, 17]}, {"full_name": "Option", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2139, 11], "def_end_pos": [2139, 17]}]], "state_before": "case hpr\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1\nn\u271d : Primrec Prod.snd\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1\nn : Primrec fun a => a.1.2\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2\n\u22a2 Primrec fun a => do\n    let x \u2190 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) a.1.1.2\n    let y \u2190 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.2.1) a.1.1.2\n    some (Nat.pair x y)", "state_after": "case hpr\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1\nn\u271d : Primrec Prod.snd\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1\nn : Primrec fun a => a.1.2\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2\nL : Primrec fun a => a.1.1.1\n\u22a2 Primrec fun a => do\n    let x \u2190 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) a.1.1.2\n    let y \u2190 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.2.1) a.1.1.2\n    some (Nat.pair x y)"}, {"tactic": "have k := k.comp (Primrec.fst (\u03b2 := Code \u00d7 Code \u00d7 Option \u2115 \u00d7 Option \u2115))", "annotated_tactic": ["have k := k.comp (<a>Primrec.fst</a> (\u03b2 := <a>Code</a> \u00d7 <a>Code</a> \u00d7 <a>Option</a> \u2115 \u00d7 <a>Option</a> \u2115))", [{"full_name": "Primrec.fst", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [340, 9], "def_end_pos": [340, 12]}, {"full_name": "Nat.Partrec.Code", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [76, 11], "def_end_pos": [76, 15]}, {"full_name": "Nat.Partrec.Code", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [76, 11], "def_end_pos": [76, 15]}, {"full_name": "Option", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2139, 11], "def_end_pos": [2139, 17]}, {"full_name": "Option", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2139, 11], "def_end_pos": [2139, 17]}]], "state_before": "case hpr\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1\nn\u271d : Primrec Prod.snd\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1\nn : Primrec fun a => a.1.2\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2\nL : Primrec fun a => a.1.1.1\n\u22a2 Primrec fun a => do\n    let x \u2190 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) a.1.1.2\n    let y \u2190 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.2.1) a.1.1.2\n    some (Nat.pair x y)", "state_after": "case hpr\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1\nn\u271d : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1\nn : Primrec fun a => a.1.2\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2\nL : Primrec fun a => a.1.1.1\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1\n\u22a2 Primrec fun a => do\n    let x \u2190 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) a.1.1.2\n    let y \u2190 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.2.1) a.1.1.2\n    some (Nat.pair x y)"}, {"tactic": "have n := n.comp (Primrec.fst (\u03b2 := Code \u00d7 Code \u00d7 Option \u2115 \u00d7 Option \u2115))", "annotated_tactic": ["have n := n.comp (<a>Primrec.fst</a> (\u03b2 := <a>Code</a> \u00d7 <a>Code</a> \u00d7 <a>Option</a> \u2115 \u00d7 <a>Option</a> \u2115))", [{"full_name": "Primrec.fst", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [340, 9], "def_end_pos": [340, 12]}, {"full_name": "Nat.Partrec.Code", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [76, 11], "def_end_pos": [76, 15]}, {"full_name": "Nat.Partrec.Code", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [76, 11], "def_end_pos": [76, 15]}, {"full_name": "Option", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2139, 11], "def_end_pos": [2139, 17]}, {"full_name": "Option", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2139, 11], "def_end_pos": [2139, 17]}]], "state_before": "case hpr\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1\nn\u271d : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1\nn : Primrec fun a => a.1.2\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2\nL : Primrec fun a => a.1.1.1\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1\n\u22a2 Primrec fun a => do\n    let x \u2190 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) a.1.1.2\n    let y \u2190 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.2.1) a.1.1.2\n    some (Nat.pair x y)", "state_after": "case hpr\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1\nn\u271d : Primrec fun a => a.1.2\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2\nL : Primrec fun a => a.1.1.1\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1\nn : Primrec fun a => a.1.1.2\n\u22a2 Primrec fun a => do\n    let x \u2190 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) a.1.1.2\n    let y \u2190 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.2.1) a.1.1.2\n    some (Nat.pair x y)"}, {"tactic": "have cf := Primrec.fst.comp (Primrec.snd (\u03b1 := (List (List (Option \u2115)) \u00d7 \u2115) \u00d7 \u2115)\n    (\u03b2 := Code \u00d7 Code \u00d7 Option \u2115 \u00d7 Option \u2115))", "annotated_tactic": ["have cf := Primrec.fst.comp (<a>Primrec.snd</a> (\u03b1 := (<a>List</a> (<a>List</a> (<a>Option</a> \u2115)) \u00d7 \u2115) \u00d7 \u2115)\n        (\u03b2 := <a>Code</a> \u00d7 <a>Code</a> \u00d7 <a>Option</a> \u2115 \u00d7 <a>Option</a> \u2115))", [{"full_name": "Primrec.snd", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [351, 9], "def_end_pos": [351, 12]}, {"full_name": "List", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2182, 11], "def_end_pos": [2182, 15]}, {"full_name": "List", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2182, 11], "def_end_pos": [2182, 15]}, {"full_name": "Option", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2139, 11], "def_end_pos": [2139, 17]}, {"full_name": "Nat.Partrec.Code", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [76, 11], "def_end_pos": [76, 15]}, {"full_name": "Nat.Partrec.Code", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [76, 11], "def_end_pos": [76, 15]}, {"full_name": "Option", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2139, 11], "def_end_pos": [2139, 17]}, {"full_name": "Option", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2139, 11], "def_end_pos": [2139, 17]}]], "state_before": "case hpr\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1\nn\u271d : Primrec fun a => a.1.2\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2\nL : Primrec fun a => a.1.1.1\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1\nn : Primrec fun a => a.1.1.2\n\u22a2 Primrec fun a => do\n    let x \u2190 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) a.1.1.2\n    let y \u2190 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.2.1) a.1.1.2\n    some (Nat.pair x y)", "state_after": "case hpr\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1\nn\u271d : Primrec fun a => a.1.2\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2\nL : Primrec fun a => a.1.1.1\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1\nn : Primrec fun a => a.1.1.2\ncf : Primrec fun a => a.2.1\n\u22a2 Primrec fun a => do\n    let x \u2190 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) a.1.1.2\n    let y \u2190 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.2.1) a.1.1.2\n    some (Nat.pair x y)"}, {"tactic": "have cg := (Primrec.fst.comp Primrec.snd).comp\n  (Primrec.snd (\u03b1 := (List (List (Option \u2115)) \u00d7 \u2115) \u00d7 \u2115)\n    (\u03b2 := Code \u00d7 Code \u00d7 Option \u2115 \u00d7 Option \u2115))", "annotated_tactic": ["have cg := (Primrec.fst.comp <a>Primrec.snd</a>).<a>comp</a>\n      (<a>Primrec.snd</a> (\u03b1 := (<a>List</a> (<a>List</a> (<a>Option</a> \u2115)) \u00d7 \u2115) \u00d7 \u2115)\n        (\u03b2 := <a>Code</a> \u00d7 <a>Code</a> \u00d7 <a>Option</a> \u2115 \u00d7 <a>Option</a> \u2115))", [{"full_name": "Primrec.snd", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [351, 9], "def_end_pos": [351, 12]}, {"full_name": "Primrec.comp", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [259, 9], "def_end_pos": [259, 13]}, {"full_name": "Primrec.snd", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [351, 9], "def_end_pos": [351, 12]}, {"full_name": "List", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2182, 11], "def_end_pos": [2182, 15]}, {"full_name": "List", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2182, 11], "def_end_pos": [2182, 15]}, {"full_name": "Option", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2139, 11], "def_end_pos": [2139, 17]}, {"full_name": "Nat.Partrec.Code", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [76, 11], "def_end_pos": [76, 15]}, {"full_name": "Nat.Partrec.Code", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [76, 11], "def_end_pos": [76, 15]}, {"full_name": "Option", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2139, 11], "def_end_pos": [2139, 17]}, {"full_name": "Option", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2139, 11], "def_end_pos": [2139, 17]}]], "state_before": "case hpr\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1\nn\u271d : Primrec fun a => a.1.2\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2\nL : Primrec fun a => a.1.1.1\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1\nn : Primrec fun a => a.1.1.2\ncf : Primrec fun a => a.2.1\n\u22a2 Primrec fun a => do\n    let x \u2190 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) a.1.1.2\n    let y \u2190 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.2.1) a.1.1.2\n    some (Nat.pair x y)", "state_after": "case hpr\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1\nn\u271d : Primrec fun a => a.1.2\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2\nL : Primrec fun a => a.1.1.1\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1\nn : Primrec fun a => a.1.1.2\ncf : Primrec fun a => a.2.1\ncg : Primrec fun a => a.2.2.1\n\u22a2 Primrec fun a => do\n    let x \u2190 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) a.1.1.2\n    let y \u2190 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.2.1) a.1.1.2\n    some (Nat.pair x y)"}, {"tactic": "refine Primrec.option_bind (hlup.comp <| L.pair <| (k.pair cf).pair n) ?_", "annotated_tactic": ["refine <a>Primrec.option_bind</a> (hlup.comp <| L.pair <| (k.pair cf).<a>pair</a> n) ?_", [{"full_name": "Primrec.option_bind", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [622, 9], "def_end_pos": [622, 20]}, {"full_name": "Primrec.pair", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [362, 9], "def_end_pos": [362, 13]}]], "state_before": "case hpr\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1\nn\u271d : Primrec fun a => a.1.2\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2\nL : Primrec fun a => a.1.1.1\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1\nn : Primrec fun a => a.1.1.2\ncf : Primrec fun a => a.2.1\ncg : Primrec fun a => a.2.2.1\n\u22a2 Primrec fun a => do\n    let x \u2190 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) a.1.1.2\n    let y \u2190 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.2.1) a.1.1.2\n    some (Nat.pair x y)", "state_after": "case hpr\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1\nn\u271d : Primrec fun a => a.1.2\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2\nL : Primrec fun a => a.1.1.1\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1\nn : Primrec fun a => a.1.1.2\ncf : Primrec fun a => a.2.1\ncg : Primrec fun a => a.2.2.1\n\u22a2 Primrec\u2082 fun a x => do\n    let y \u2190 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.2.1) a.1.1.2\n    some (Nat.pair x y)"}, {"tactic": "unfold Primrec\u2082", "annotated_tactic": ["unfold <a>Primrec\u2082</a>", [{"full_name": "Primrec\u2082", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [389, 5], "def_end_pos": [389, 13]}]], "state_before": "case hpr\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1\nn\u271d : Primrec fun a => a.1.2\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2\nL : Primrec fun a => a.1.1.1\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1\nn : Primrec fun a => a.1.1.2\ncf : Primrec fun a => a.2.1\ncg : Primrec fun a => a.2.2.1\n\u22a2 Primrec\u2082 fun a x => do\n    let y \u2190 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.2.1) a.1.1.2\n    some (Nat.pair x y)", "state_after": "case hpr\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1\nn\u271d : Primrec fun a => a.1.2\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2\nL : Primrec fun a => a.1.1.1\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1\nn : Primrec fun a => a.1.1.2\ncf : Primrec fun a => a.2.1\ncg : Primrec fun a => a.2.2.1\n\u22a2 Primrec fun p =>\n    (fun a x => do\n        let y \u2190 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.2.1) a.1.1.2\n        some (Nat.pair x y))\n      p.1 p.2"}, {"tactic": "refine Primrec.option_map ((hlup.comp <| L.pair <| (k.pair cg).pair n).comp Primrec.fst) ?_", "annotated_tactic": ["refine <a>Primrec.option_map</a> ((hlup.comp <| L.pair <| (k.pair cg).<a>pair</a> n).<a>comp</a> <a>Primrec.fst</a>) ?_", [{"full_name": "Primrec.option_map", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [631, 9], "def_end_pos": [631, 19]}, {"full_name": "Primrec.pair", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [362, 9], "def_end_pos": [362, 13]}, {"full_name": "Primrec.comp", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [259, 9], "def_end_pos": [259, 13]}, {"full_name": "Primrec.fst", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [340, 9], "def_end_pos": [340, 12]}]], "state_before": "case hpr\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1\nn\u271d : Primrec fun a => a.1.2\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2\nL : Primrec fun a => a.1.1.1\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1\nn : Primrec fun a => a.1.1.2\ncf : Primrec fun a => a.2.1\ncg : Primrec fun a => a.2.2.1\n\u22a2 Primrec fun p =>\n    Option.map (fun y => Nat.pair p.2 y)\n      (Nat.Partrec.Code.lup p.1.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1.1.1.1)).1, p.1.2.2.1) p.1.1.1.2)", "state_after": "case hpr\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1\nn\u271d : Primrec fun a => a.1.2\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2\nL : Primrec fun a => a.1.1.1\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1\nn : Primrec fun a => a.1.1.2\ncf : Primrec fun a => a.2.1\ncg : Primrec fun a => a.2.2.1\n\u22a2 Primrec\u2082 fun p y => Nat.pair p.2 y"}, {"tactic": "unfold Primrec\u2082", "annotated_tactic": ["unfold <a>Primrec\u2082</a>", [{"full_name": "Primrec\u2082", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [389, 5], "def_end_pos": [389, 13]}]], "state_before": "case hpr\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1\nn\u271d : Primrec fun a => a.1.2\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2\nL : Primrec fun a => a.1.1.1\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1\nn : Primrec fun a => a.1.1.2\ncf : Primrec fun a => a.2.1\ncg : Primrec fun a => a.2.2.1\n\u22a2 Primrec\u2082 fun p y => Nat.pair p.2 y", "state_after": "case hpr\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1\nn\u271d : Primrec fun a => a.1.2\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2\nL : Primrec fun a => a.1.1.1\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1\nn : Primrec fun a => a.1.1.2\ncf : Primrec fun a => a.2.1\ncg : Primrec fun a => a.2.2.1\n\u22a2 Primrec fun p => (fun p y => Nat.pair p.2 y) p.1 p.2"}, {"tactic": "exact Primrec\u2082.natPair.comp (Primrec.snd.comp Primrec.fst) Primrec.snd", "annotated_tactic": ["exact Primrec\u2082.natPair.comp (Primrec.snd.comp <a>Primrec.fst</a>) <a>Primrec.snd</a>", [{"full_name": "Primrec.fst", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [340, 9], "def_end_pos": [340, 12]}, {"full_name": "Primrec.snd", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [351, 9], "def_end_pos": [351, 12]}]], "state_before": "case hpr\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1\nn\u271d : Primrec fun a => a.1.2\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2\nL : Primrec fun a => a.1.1.1\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1\nn : Primrec fun a => a.1.1.2\ncf : Primrec fun a => a.2.1\ncg : Primrec fun a => a.2.2.1\n\u22a2 Primrec fun p => (fun p y => Nat.pair p.2 y) p.1 p.2", "state_after": "no goals"}, {"tactic": "have L := (Primrec.fst.comp Primrec.fst).comp\n  (Primrec.fst (\u03b1 := (List (List (Option \u2115)) \u00d7 \u2115) \u00d7 \u2115)\n    (\u03b2 := Code \u00d7 Code \u00d7 Option \u2115 \u00d7 Option \u2115))", "annotated_tactic": ["have L := (Primrec.fst.comp <a>Primrec.fst</a>).<a>comp</a>\n      (<a>Primrec.fst</a> (\u03b1 := (<a>List</a> (<a>List</a> (<a>Option</a> \u2115)) \u00d7 \u2115) \u00d7 \u2115)\n        (\u03b2 := <a>Code</a> \u00d7 <a>Code</a> \u00d7 <a>Option</a> \u2115 \u00d7 <a>Option</a> \u2115))", [{"full_name": "Primrec.fst", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [340, 9], "def_end_pos": [340, 12]}, {"full_name": "Primrec.comp", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [259, 9], "def_end_pos": [259, 13]}, {"full_name": "Primrec.fst", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [340, 9], "def_end_pos": [340, 12]}, {"full_name": "List", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2182, 11], "def_end_pos": [2182, 15]}, {"full_name": "List", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2182, 11], "def_end_pos": [2182, 15]}, {"full_name": "Option", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2139, 11], "def_end_pos": [2139, 17]}, {"full_name": "Nat.Partrec.Code", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [76, 11], "def_end_pos": [76, 15]}, {"full_name": "Nat.Partrec.Code", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [76, 11], "def_end_pos": [76, 15]}, {"full_name": "Option", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2139, 11], "def_end_pos": [2139, 17]}, {"full_name": "Option", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2139, 11], "def_end_pos": [2139, 17]}]], "state_before": "case hco\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1\nn\u271d : Primrec Prod.snd\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1\nn : Primrec fun a => a.1.2\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2\n\u22a2 Primrec fun a => do\n    let x \u2190 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.2.1) a.1.1.2\n    Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) x", "state_after": "case hco\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1\nn\u271d : Primrec Prod.snd\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1\nn : Primrec fun a => a.1.2\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2\nL : Primrec fun a => a.1.1.1\n\u22a2 Primrec fun a => do\n    let x \u2190 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.2.1) a.1.1.2\n    Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) x"}, {"tactic": "have k := k.comp (Primrec.fst (\u03b2 := Code \u00d7 Code \u00d7 Option \u2115 \u00d7 Option \u2115))", "annotated_tactic": ["have k := k.comp (<a>Primrec.fst</a> (\u03b2 := <a>Code</a> \u00d7 <a>Code</a> \u00d7 <a>Option</a> \u2115 \u00d7 <a>Option</a> \u2115))", [{"full_name": "Primrec.fst", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [340, 9], "def_end_pos": [340, 12]}, {"full_name": "Nat.Partrec.Code", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [76, 11], "def_end_pos": [76, 15]}, {"full_name": "Nat.Partrec.Code", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [76, 11], "def_end_pos": [76, 15]}, {"full_name": "Option", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2139, 11], "def_end_pos": [2139, 17]}, {"full_name": "Option", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2139, 11], "def_end_pos": [2139, 17]}]], "state_before": "case hco\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1\nn\u271d : Primrec Prod.snd\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1\nn : Primrec fun a => a.1.2\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2\nL : Primrec fun a => a.1.1.1\n\u22a2 Primrec fun a => do\n    let x \u2190 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.2.1) a.1.1.2\n    Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) x", "state_after": "case hco\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1\nn\u271d : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1\nn : Primrec fun a => a.1.2\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2\nL : Primrec fun a => a.1.1.1\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1\n\u22a2 Primrec fun a => do\n    let x \u2190 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.2.1) a.1.1.2\n    Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) x"}, {"tactic": "have n := n.comp (Primrec.fst (\u03b2 := Code \u00d7 Code \u00d7 Option \u2115 \u00d7 Option \u2115))", "annotated_tactic": ["have n := n.comp (<a>Primrec.fst</a> (\u03b2 := <a>Code</a> \u00d7 <a>Code</a> \u00d7 <a>Option</a> \u2115 \u00d7 <a>Option</a> \u2115))", [{"full_name": "Primrec.fst", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [340, 9], "def_end_pos": [340, 12]}, {"full_name": "Nat.Partrec.Code", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [76, 11], "def_end_pos": [76, 15]}, {"full_name": "Nat.Partrec.Code", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [76, 11], "def_end_pos": [76, 15]}, {"full_name": "Option", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2139, 11], "def_end_pos": [2139, 17]}, {"full_name": "Option", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2139, 11], "def_end_pos": [2139, 17]}]], "state_before": "case hco\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1\nn\u271d : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1\nn : Primrec fun a => a.1.2\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2\nL : Primrec fun a => a.1.1.1\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1\n\u22a2 Primrec fun a => do\n    let x \u2190 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.2.1) a.1.1.2\n    Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) x", "state_after": "case hco\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1\nn\u271d : Primrec fun a => a.1.2\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2\nL : Primrec fun a => a.1.1.1\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1\nn : Primrec fun a => a.1.1.2\n\u22a2 Primrec fun a => do\n    let x \u2190 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.2.1) a.1.1.2\n    Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) x"}, {"tactic": "have cf := Primrec.fst.comp (Primrec.snd (\u03b1 := (List (List (Option \u2115)) \u00d7 \u2115) \u00d7 \u2115)\n    (\u03b2 := Code \u00d7 Code \u00d7 Option \u2115 \u00d7 Option \u2115))", "annotated_tactic": ["have cf := Primrec.fst.comp (<a>Primrec.snd</a> (\u03b1 := (<a>List</a> (<a>List</a> (<a>Option</a> \u2115)) \u00d7 \u2115) \u00d7 \u2115)\n        (\u03b2 := <a>Code</a> \u00d7 <a>Code</a> \u00d7 <a>Option</a> \u2115 \u00d7 <a>Option</a> \u2115))", [{"full_name": "Primrec.snd", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [351, 9], "def_end_pos": [351, 12]}, {"full_name": "List", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2182, 11], "def_end_pos": [2182, 15]}, {"full_name": "List", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2182, 11], "def_end_pos": [2182, 15]}, {"full_name": "Option", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2139, 11], "def_end_pos": [2139, 17]}, {"full_name": "Nat.Partrec.Code", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [76, 11], "def_end_pos": [76, 15]}, {"full_name": "Nat.Partrec.Code", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [76, 11], "def_end_pos": [76, 15]}, {"full_name": "Option", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2139, 11], "def_end_pos": [2139, 17]}, {"full_name": "Option", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2139, 11], "def_end_pos": [2139, 17]}]], "state_before": "case hco\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1\nn\u271d : Primrec fun a => a.1.2\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2\nL : Primrec fun a => a.1.1.1\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1\nn : Primrec fun a => a.1.1.2\n\u22a2 Primrec fun a => do\n    let x \u2190 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.2.1) a.1.1.2\n    Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) x", "state_after": "case hco\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1\nn\u271d : Primrec fun a => a.1.2\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2\nL : Primrec fun a => a.1.1.1\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1\nn : Primrec fun a => a.1.1.2\ncf : Primrec fun a => a.2.1\n\u22a2 Primrec fun a => do\n    let x \u2190 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.2.1) a.1.1.2\n    Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) x"}, {"tactic": "have cg := (Primrec.fst.comp Primrec.snd).comp\n  (Primrec.snd (\u03b1 := (List (List (Option \u2115)) \u00d7 \u2115) \u00d7 \u2115)\n    (\u03b2 := Code \u00d7 Code \u00d7 Option \u2115 \u00d7 Option \u2115))", "annotated_tactic": ["have cg := (Primrec.fst.comp <a>Primrec.snd</a>).<a>comp</a>\n      (<a>Primrec.snd</a> (\u03b1 := (<a>List</a> (<a>List</a> (<a>Option</a> \u2115)) \u00d7 \u2115) \u00d7 \u2115)\n        (\u03b2 := <a>Code</a> \u00d7 <a>Code</a> \u00d7 <a>Option</a> \u2115 \u00d7 <a>Option</a> \u2115))", [{"full_name": "Primrec.snd", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [351, 9], "def_end_pos": [351, 12]}, {"full_name": "Primrec.comp", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [259, 9], "def_end_pos": [259, 13]}, {"full_name": "Primrec.snd", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [351, 9], "def_end_pos": [351, 12]}, {"full_name": "List", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2182, 11], "def_end_pos": [2182, 15]}, {"full_name": "List", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2182, 11], "def_end_pos": [2182, 15]}, {"full_name": "Option", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2139, 11], "def_end_pos": [2139, 17]}, {"full_name": "Nat.Partrec.Code", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [76, 11], "def_end_pos": [76, 15]}, {"full_name": "Nat.Partrec.Code", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [76, 11], "def_end_pos": [76, 15]}, {"full_name": "Option", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2139, 11], "def_end_pos": [2139, 17]}, {"full_name": "Option", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2139, 11], "def_end_pos": [2139, 17]}]], "state_before": "case hco\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1\nn\u271d : Primrec fun a => a.1.2\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2\nL : Primrec fun a => a.1.1.1\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1\nn : Primrec fun a => a.1.1.2\ncf : Primrec fun a => a.2.1\n\u22a2 Primrec fun a => do\n    let x \u2190 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.2.1) a.1.1.2\n    Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) x", "state_after": "case hco\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1\nn\u271d : Primrec fun a => a.1.2\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2\nL : Primrec fun a => a.1.1.1\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1\nn : Primrec fun a => a.1.1.2\ncf : Primrec fun a => a.2.1\ncg : Primrec fun a => a.2.2.1\n\u22a2 Primrec fun a => do\n    let x \u2190 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.2.1) a.1.1.2\n    Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) x"}, {"tactic": "refine Primrec.option_bind (hlup.comp <| L.pair <| (k.pair cg).pair n) ?_", "annotated_tactic": ["refine <a>Primrec.option_bind</a> (hlup.comp <| L.pair <| (k.pair cg).<a>pair</a> n) ?_", [{"full_name": "Primrec.option_bind", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [622, 9], "def_end_pos": [622, 20]}, {"full_name": "Primrec.pair", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [362, 9], "def_end_pos": [362, 13]}]], "state_before": "case hco\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1\nn\u271d : Primrec fun a => a.1.2\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2\nL : Primrec fun a => a.1.1.1\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1\nn : Primrec fun a => a.1.1.2\ncf : Primrec fun a => a.2.1\ncg : Primrec fun a => a.2.2.1\n\u22a2 Primrec fun a => do\n    let x \u2190 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.2.1) a.1.1.2\n    Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) x", "state_after": "case hco\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1\nn\u271d : Primrec fun a => a.1.2\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2\nL : Primrec fun a => a.1.1.1\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1\nn : Primrec fun a => a.1.1.2\ncf : Primrec fun a => a.2.1\ncg : Primrec fun a => a.2.2.1\n\u22a2 Primrec\u2082 fun a x => Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) x"}, {"tactic": "unfold Primrec\u2082", "annotated_tactic": ["unfold <a>Primrec\u2082</a>", [{"full_name": "Primrec\u2082", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [389, 5], "def_end_pos": [389, 13]}]], "state_before": "case hco\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1\nn\u271d : Primrec fun a => a.1.2\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2\nL : Primrec fun a => a.1.1.1\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1\nn : Primrec fun a => a.1.1.2\ncf : Primrec fun a => a.2.1\ncg : Primrec fun a => a.2.2.1\n\u22a2 Primrec\u2082 fun a x => Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) x", "state_after": "case hco\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1\nn\u271d : Primrec fun a => a.1.2\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2\nL : Primrec fun a => a.1.1.1\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1\nn : Primrec fun a => a.1.1.2\ncf : Primrec fun a => a.2.1\ncg : Primrec fun a => a.2.2.1\n\u22a2 Primrec fun p =>\n    (fun a x => Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) x) p.1 p.2"}, {"tactic": "have h :=\n  hlup.comp ((L.comp Primrec.fst).pair <| ((k.pair cf).comp Primrec.fst).pair Primrec.snd)", "annotated_tactic": ["have h :=\n      hlup.comp ((L.comp <a>Primrec.fst</a>).<a>pair</a> <| ((k.pair cf).<a>comp</a> <a>Primrec.fst</a>).<a>pair</a> <a>Primrec.snd</a>)", [{"full_name": "Primrec.fst", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [340, 9], "def_end_pos": [340, 12]}, {"full_name": "Primrec.pair", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [362, 9], "def_end_pos": [362, 13]}, {"full_name": "Primrec.comp", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [259, 9], "def_end_pos": [259, 13]}, {"full_name": "Primrec.fst", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [340, 9], "def_end_pos": [340, 12]}, {"full_name": "Primrec.pair", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [362, 9], "def_end_pos": [362, 13]}, {"full_name": "Primrec.snd", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [351, 9], "def_end_pos": [351, 12]}]], "state_before": "case hco\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1\nn\u271d : Primrec fun a => a.1.2\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2\nL : Primrec fun a => a.1.1.1\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1\nn : Primrec fun a => a.1.1.2\ncf : Primrec fun a => a.2.1\ncg : Primrec fun a => a.2.2.1\n\u22a2 Primrec fun p =>\n    (fun a x => Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) x) p.1 p.2", "state_after": "case hco\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1\nn\u271d : Primrec fun a => a.1.2\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2\nL : Primrec fun a => a.1.1.1\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1\nn : Primrec fun a => a.1.1.2\ncf : Primrec fun a => a.2.1\ncg : Primrec fun a => a.2.2.1\nh :\n  Primrec fun a =>\n    Nat.Partrec.Code.lup (a.1.1.1.1, ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1.1)).1, a.1.2.1), a.2).1\n      (a.1.1.1.1, ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1.1)).1, a.1.2.1), a.2).2.1\n      (a.1.1.1.1, ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1.1)).1, a.1.2.1), a.2).2.2\n\u22a2 Primrec fun p =>\n    (fun a x => Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) x) p.1 p.2"}, {"tactic": "exact h", "annotated_tactic": ["exact h", []], "state_before": "case hco\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1\nn\u271d : Primrec fun a => a.1.2\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2\nL : Primrec fun a => a.1.1.1\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1\nn : Primrec fun a => a.1.1.2\ncf : Primrec fun a => a.2.1\ncg : Primrec fun a => a.2.2.1\nh :\n  Primrec fun a =>\n    Nat.Partrec.Code.lup (a.1.1.1.1, ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1.1)).1, a.1.2.1), a.2).1\n      (a.1.1.1.1, ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1.1)).1, a.1.2.1), a.2).2.1\n      (a.1.1.1.1, ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1.1)).1, a.1.2.1), a.2).2.2\n\u22a2 Primrec fun p =>\n    (fun a x => Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) x) p.1 p.2", "state_after": "no goals"}, {"tactic": "have L := (Primrec.fst.comp Primrec.fst).comp\n  (Primrec.fst (\u03b1 := (List (List (Option \u2115)) \u00d7 \u2115) \u00d7 \u2115)\n    (\u03b2 := Code \u00d7 Code \u00d7 Option \u2115 \u00d7 Option \u2115))", "annotated_tactic": ["have L := (Primrec.fst.comp <a>Primrec.fst</a>).<a>comp</a>\n      (<a>Primrec.fst</a> (\u03b1 := (<a>List</a> (<a>List</a> (<a>Option</a> \u2115)) \u00d7 \u2115) \u00d7 \u2115)\n        (\u03b2 := <a>Code</a> \u00d7 <a>Code</a> \u00d7 <a>Option</a> \u2115 \u00d7 <a>Option</a> \u2115))", [{"full_name": "Primrec.fst", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [340, 9], "def_end_pos": [340, 12]}, {"full_name": "Primrec.comp", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [259, 9], "def_end_pos": [259, 13]}, {"full_name": "Primrec.fst", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [340, 9], "def_end_pos": [340, 12]}, {"full_name": "List", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2182, 11], "def_end_pos": [2182, 15]}, {"full_name": "List", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2182, 11], "def_end_pos": [2182, 15]}, {"full_name": "Option", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2139, 11], "def_end_pos": [2139, 17]}, {"full_name": "Nat.Partrec.Code", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [76, 11], "def_end_pos": [76, 15]}, {"full_name": "Nat.Partrec.Code", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [76, 11], "def_end_pos": [76, 15]}, {"full_name": "Option", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2139, 11], "def_end_pos": [2139, 17]}, {"full_name": "Option", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2139, 11], "def_end_pos": [2139, 17]}]], "state_before": "case hpc\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1\nn\u271d : Primrec Prod.snd\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1\nn : Primrec fun a => a.1.2\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2\n\u22a2 Primrec fun a =>\n    let z := (unpair a.1.1.2).1;\n    Nat.casesOn (unpair a.1.1.2).2 (Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) z)\n      fun y => do\n      let i \u2190 Nat.Partrec.Code.lup a.1.1.1 (a.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).2) (Nat.pair z y)\n      Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.2.1) (Nat.pair z (Nat.pair y i))", "state_after": "case hpc\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1\nn\u271d : Primrec Prod.snd\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1\nn : Primrec fun a => a.1.2\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2\nL : Primrec fun a => a.1.1.1\n\u22a2 Primrec fun a =>\n    let z := (unpair a.1.1.2).1;\n    Nat.casesOn (unpair a.1.1.2).2 (Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) z)\n      fun y => do\n      let i \u2190 Nat.Partrec.Code.lup a.1.1.1 (a.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).2) (Nat.pair z y)\n      Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.2.1) (Nat.pair z (Nat.pair y i))"}, {"tactic": "have k := k.comp (Primrec.fst (\u03b2 := Code \u00d7 Code \u00d7 Option \u2115 \u00d7 Option \u2115))", "annotated_tactic": ["have k := k.comp (<a>Primrec.fst</a> (\u03b2 := <a>Code</a> \u00d7 <a>Code</a> \u00d7 <a>Option</a> \u2115 \u00d7 <a>Option</a> \u2115))", [{"full_name": "Primrec.fst", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [340, 9], "def_end_pos": [340, 12]}, {"full_name": "Nat.Partrec.Code", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [76, 11], "def_end_pos": [76, 15]}, {"full_name": "Nat.Partrec.Code", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [76, 11], "def_end_pos": [76, 15]}, {"full_name": "Option", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2139, 11], "def_end_pos": [2139, 17]}, {"full_name": "Option", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2139, 11], "def_end_pos": [2139, 17]}]], "state_before": "case hpc\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1\nn\u271d : Primrec Prod.snd\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1\nn : Primrec fun a => a.1.2\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2\nL : Primrec fun a => a.1.1.1\n\u22a2 Primrec fun a =>\n    let z := (unpair a.1.1.2).1;\n    Nat.casesOn (unpair a.1.1.2).2 (Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) z)\n      fun y => do\n      let i \u2190 Nat.Partrec.Code.lup a.1.1.1 (a.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).2) (Nat.pair z y)\n      Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.2.1) (Nat.pair z (Nat.pair y i))", "state_after": "case hpc\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1\nn\u271d : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1\nn : Primrec fun a => a.1.2\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2\nL : Primrec fun a => a.1.1.1\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1\n\u22a2 Primrec fun a =>\n    let z := (unpair a.1.1.2).1;\n    Nat.casesOn (unpair a.1.1.2).2 (Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) z)\n      fun y => do\n      let i \u2190 Nat.Partrec.Code.lup a.1.1.1 (a.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).2) (Nat.pair z y)\n      Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.2.1) (Nat.pair z (Nat.pair y i))"}, {"tactic": "have n := n.comp (Primrec.fst (\u03b2 := Code \u00d7 Code \u00d7 Option \u2115 \u00d7 Option \u2115))", "annotated_tactic": ["have n := n.comp (<a>Primrec.fst</a> (\u03b2 := <a>Code</a> \u00d7 <a>Code</a> \u00d7 <a>Option</a> \u2115 \u00d7 <a>Option</a> \u2115))", [{"full_name": "Primrec.fst", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [340, 9], "def_end_pos": [340, 12]}, {"full_name": "Nat.Partrec.Code", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [76, 11], "def_end_pos": [76, 15]}, {"full_name": "Nat.Partrec.Code", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [76, 11], "def_end_pos": [76, 15]}, {"full_name": "Option", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2139, 11], "def_end_pos": [2139, 17]}, {"full_name": "Option", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2139, 11], "def_end_pos": [2139, 17]}]], "state_before": "case hpc\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1\nn\u271d : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1\nn : Primrec fun a => a.1.2\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2\nL : Primrec fun a => a.1.1.1\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1\n\u22a2 Primrec fun a =>\n    let z := (unpair a.1.1.2).1;\n    Nat.casesOn (unpair a.1.1.2).2 (Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) z)\n      fun y => do\n      let i \u2190 Nat.Partrec.Code.lup a.1.1.1 (a.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).2) (Nat.pair z y)\n      Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.2.1) (Nat.pair z (Nat.pair y i))", "state_after": "case hpc\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1\nn\u271d : Primrec fun a => a.1.2\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2\nL : Primrec fun a => a.1.1.1\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1\nn : Primrec fun a => a.1.1.2\n\u22a2 Primrec fun a =>\n    let z := (unpair a.1.1.2).1;\n    Nat.casesOn (unpair a.1.1.2).2 (Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) z)\n      fun y => do\n      let i \u2190 Nat.Partrec.Code.lup a.1.1.1 (a.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).2) (Nat.pair z y)\n      Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.2.1) (Nat.pair z (Nat.pair y i))"}, {"tactic": "have cf := Primrec.fst.comp (Primrec.snd (\u03b1 := (List (List (Option \u2115)) \u00d7 \u2115) \u00d7 \u2115)\n    (\u03b2 := Code \u00d7 Code \u00d7 Option \u2115 \u00d7 Option \u2115))", "annotated_tactic": ["have cf := Primrec.fst.comp (<a>Primrec.snd</a> (\u03b1 := (<a>List</a> (<a>List</a> (<a>Option</a> \u2115)) \u00d7 \u2115) \u00d7 \u2115)\n        (\u03b2 := <a>Code</a> \u00d7 <a>Code</a> \u00d7 <a>Option</a> \u2115 \u00d7 <a>Option</a> \u2115))", [{"full_name": "Primrec.snd", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [351, 9], "def_end_pos": [351, 12]}, {"full_name": "List", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2182, 11], "def_end_pos": [2182, 15]}, {"full_name": "List", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2182, 11], "def_end_pos": [2182, 15]}, {"full_name": "Option", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2139, 11], "def_end_pos": [2139, 17]}, {"full_name": "Nat.Partrec.Code", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [76, 11], "def_end_pos": [76, 15]}, {"full_name": "Nat.Partrec.Code", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [76, 11], "def_end_pos": [76, 15]}, {"full_name": "Option", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2139, 11], "def_end_pos": [2139, 17]}, {"full_name": "Option", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2139, 11], "def_end_pos": [2139, 17]}]], "state_before": "case hpc\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1\nn\u271d : Primrec fun a => a.1.2\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2\nL : Primrec fun a => a.1.1.1\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1\nn : Primrec fun a => a.1.1.2\n\u22a2 Primrec fun a =>\n    let z := (unpair a.1.1.2).1;\n    Nat.casesOn (unpair a.1.1.2).2 (Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) z)\n      fun y => do\n      let i \u2190 Nat.Partrec.Code.lup a.1.1.1 (a.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).2) (Nat.pair z y)\n      Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.2.1) (Nat.pair z (Nat.pair y i))", "state_after": "case hpc\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1\nn\u271d : Primrec fun a => a.1.2\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2\nL : Primrec fun a => a.1.1.1\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1\nn : Primrec fun a => a.1.1.2\ncf : Primrec fun a => a.2.1\n\u22a2 Primrec fun a =>\n    let z := (unpair a.1.1.2).1;\n    Nat.casesOn (unpair a.1.1.2).2 (Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) z)\n      fun y => do\n      let i \u2190 Nat.Partrec.Code.lup a.1.1.1 (a.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).2) (Nat.pair z y)\n      Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.2.1) (Nat.pair z (Nat.pair y i))"}, {"tactic": "have cg := (Primrec.fst.comp Primrec.snd).comp\n  (Primrec.snd (\u03b1 := (List (List (Option \u2115)) \u00d7 \u2115) \u00d7 \u2115)\n    (\u03b2 := Code \u00d7 Code \u00d7 Option \u2115 \u00d7 Option \u2115))", "annotated_tactic": ["have cg := (Primrec.fst.comp <a>Primrec.snd</a>).<a>comp</a>\n      (<a>Primrec.snd</a> (\u03b1 := (<a>List</a> (<a>List</a> (<a>Option</a> \u2115)) \u00d7 \u2115) \u00d7 \u2115)\n        (\u03b2 := <a>Code</a> \u00d7 <a>Code</a> \u00d7 <a>Option</a> \u2115 \u00d7 <a>Option</a> \u2115))", [{"full_name": "Primrec.snd", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [351, 9], "def_end_pos": [351, 12]}, {"full_name": "Primrec.comp", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [259, 9], "def_end_pos": [259, 13]}, {"full_name": "Primrec.snd", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [351, 9], "def_end_pos": [351, 12]}, {"full_name": "List", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2182, 11], "def_end_pos": [2182, 15]}, {"full_name": "List", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2182, 11], "def_end_pos": [2182, 15]}, {"full_name": "Option", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2139, 11], "def_end_pos": [2139, 17]}, {"full_name": "Nat.Partrec.Code", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [76, 11], "def_end_pos": [76, 15]}, {"full_name": "Nat.Partrec.Code", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [76, 11], "def_end_pos": [76, 15]}, {"full_name": "Option", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2139, 11], "def_end_pos": [2139, 17]}, {"full_name": "Option", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2139, 11], "def_end_pos": [2139, 17]}]], "state_before": "case hpc\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1\nn\u271d : Primrec fun a => a.1.2\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2\nL : Primrec fun a => a.1.1.1\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1\nn : Primrec fun a => a.1.1.2\ncf : Primrec fun a => a.2.1\n\u22a2 Primrec fun a =>\n    let z := (unpair a.1.1.2).1;\n    Nat.casesOn (unpair a.1.1.2).2 (Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) z)\n      fun y => do\n      let i \u2190 Nat.Partrec.Code.lup a.1.1.1 (a.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).2) (Nat.pair z y)\n      Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.2.1) (Nat.pair z (Nat.pair y i))", "state_after": "case hpc\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1\nn\u271d : Primrec fun a => a.1.2\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2\nL : Primrec fun a => a.1.1.1\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1\nn : Primrec fun a => a.1.1.2\ncf : Primrec fun a => a.2.1\ncg : Primrec fun a => a.2.2.1\n\u22a2 Primrec fun a =>\n    let z := (unpair a.1.1.2).1;\n    Nat.casesOn (unpair a.1.1.2).2 (Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) z)\n      fun y => do\n      let i \u2190 Nat.Partrec.Code.lup a.1.1.1 (a.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).2) (Nat.pair z y)\n      Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.2.1) (Nat.pair z (Nat.pair y i))"}, {"tactic": "have z := Primrec.fst.comp (Primrec.unpair.comp n)", "annotated_tactic": ["have z := Primrec.fst.comp (Primrec.unpair.comp n)", []], "state_before": "case hpc\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1\nn\u271d : Primrec fun a => a.1.2\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2\nL : Primrec fun a => a.1.1.1\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1\nn : Primrec fun a => a.1.1.2\ncf : Primrec fun a => a.2.1\ncg : Primrec fun a => a.2.2.1\n\u22a2 Primrec fun a =>\n    let z := (unpair a.1.1.2).1;\n    Nat.casesOn (unpair a.1.1.2).2 (Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) z)\n      fun y => do\n      let i \u2190 Nat.Partrec.Code.lup a.1.1.1 (a.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).2) (Nat.pair z y)\n      Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.2.1) (Nat.pair z (Nat.pair y i))", "state_after": "case hpc\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1\nn\u271d : Primrec fun a => a.1.2\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2\nL : Primrec fun a => a.1.1.1\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1\nn : Primrec fun a => a.1.1.2\ncf : Primrec fun a => a.2.1\ncg : Primrec fun a => a.2.2.1\nz : Primrec fun a => (unpair a.1.1.2).1\n\u22a2 Primrec fun a =>\n    let z := (unpair a.1.1.2).1;\n    Nat.casesOn (unpair a.1.1.2).2 (Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) z)\n      fun y => do\n      let i \u2190 Nat.Partrec.Code.lup a.1.1.1 (a.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).2) (Nat.pair z y)\n      Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.2.1) (Nat.pair z (Nat.pair y i))"}, {"tactic": "refine'\n  Primrec.nat_casesOn (Primrec.snd.comp (Primrec.unpair.comp n))\n    (hlup.comp <| L.pair <| (k.pair cf).pair z)\n    (_ : Primrec _)", "annotated_tactic": ["refine'\n      <a>Primrec.nat_casesOn</a> (Primrec.snd.comp (Primrec.unpair.comp n))\n        (hlup.comp <| L.pair <| (k.pair cf).<a>pair</a> z)\n        (_ : <a>Primrec</a> _)", [{"full_name": "Primrec.nat_casesOn", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [594, 9], "def_end_pos": [594, 20]}, {"full_name": "Primrec.pair", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [362, 9], "def_end_pos": [362, 13]}, {"full_name": "Primrec", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [207, 5], "def_end_pos": [207, 12]}]], "state_before": "case hpc\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1\nn\u271d : Primrec fun a => a.1.2\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2\nL : Primrec fun a => a.1.1.1\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1\nn : Primrec fun a => a.1.1.2\ncf : Primrec fun a => a.2.1\ncg : Primrec fun a => a.2.2.1\nz : Primrec fun a => (unpair a.1.1.2).1\n\u22a2 Primrec fun a =>\n    let z := (unpair a.1.1.2).1;\n    Nat.casesOn (unpair a.1.1.2).2 (Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) z)\n      fun y => do\n      let i \u2190 Nat.Partrec.Code.lup a.1.1.1 (a.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).2) (Nat.pair z y)\n      Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.2.1) (Nat.pair z (Nat.pair y i))", "state_after": "case hpc\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1\nn\u271d : Primrec fun a => a.1.2\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2\nL : Primrec fun a => a.1.1.1\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1\nn : Primrec fun a => a.1.1.2\ncf : Primrec fun a => a.2.1\ncg : Primrec fun a => a.2.2.1\nz : Primrec fun a => (unpair a.1.1.2).1\n\u22a2 Primrec fun p =>\n    (fun a n =>\n        (fun y => do\n            let i \u2190\n              Nat.Partrec.Code.lup a.1.1.1 (a.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).2)\n                  (Nat.pair (unpair a.1.1.2).1 y)\n            Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.2.1)\n                (Nat.pair (unpair a.1.1.2).1 (Nat.pair y i)))\n          n)\n      p.1 p.2"}, {"tactic": "have L := L.comp (Primrec.fst (\u03b2 := \u2115))", "annotated_tactic": ["have L := L.comp (<a>Primrec.fst</a> (\u03b2 := \u2115))", [{"full_name": "Primrec.fst", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [340, 9], "def_end_pos": [340, 12]}]], "state_before": "case hpc\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1\nn\u271d : Primrec fun a => a.1.2\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2\nL : Primrec fun a => a.1.1.1\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1\nn : Primrec fun a => a.1.1.2\ncf : Primrec fun a => a.2.1\ncg : Primrec fun a => a.2.2.1\nz : Primrec fun a => (unpair a.1.1.2).1\n\u22a2 Primrec fun p =>\n    (fun a n =>\n        (fun y => do\n            let i \u2190\n              Nat.Partrec.Code.lup a.1.1.1 (a.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).2)\n                  (Nat.pair (unpair a.1.1.2).1 y)\n            Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.2.1)\n                (Nat.pair (unpair a.1.1.2).1 (Nat.pair y i)))\n          n)\n      p.1 p.2", "state_after": "case hpc\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1\nn\u271d : Primrec fun a => a.1.2\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2\nL\u271d : Primrec fun a => a.1.1.1\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1\nn : Primrec fun a => a.1.1.2\ncf : Primrec fun a => a.2.1\ncg : Primrec fun a => a.2.2.1\nz : Primrec fun a => (unpair a.1.1.2).1\nL : Primrec fun a => a.1.1.1.1\n\u22a2 Primrec fun p =>\n    (fun a n =>\n        (fun y => do\n            let i \u2190\n              Nat.Partrec.Code.lup a.1.1.1 (a.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).2)\n                  (Nat.pair (unpair a.1.1.2).1 y)\n            Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.2.1)\n                (Nat.pair (unpair a.1.1.2).1 (Nat.pair y i)))\n          n)\n      p.1 p.2"}, {"tactic": "have z := z.comp (Primrec.fst (\u03b2 := \u2115))", "annotated_tactic": ["have z := z.comp (<a>Primrec.fst</a> (\u03b2 := \u2115))", [{"full_name": "Primrec.fst", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [340, 9], "def_end_pos": [340, 12]}]], "state_before": "case hpc\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1\nn\u271d : Primrec fun a => a.1.2\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2\nL\u271d : Primrec fun a => a.1.1.1\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1\nn : Primrec fun a => a.1.1.2\ncf : Primrec fun a => a.2.1\ncg : Primrec fun a => a.2.2.1\nz : Primrec fun a => (unpair a.1.1.2).1\nL : Primrec fun a => a.1.1.1.1\n\u22a2 Primrec fun p =>\n    (fun a n =>\n        (fun y => do\n            let i \u2190\n              Nat.Partrec.Code.lup a.1.1.1 (a.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).2)\n                  (Nat.pair (unpair a.1.1.2).1 y)\n            Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.2.1)\n                (Nat.pair (unpair a.1.1.2).1 (Nat.pair y i)))\n          n)\n      p.1 p.2", "state_after": "case hpc\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1\nn\u271d : Primrec fun a => a.1.2\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2\nL\u271d : Primrec fun a => a.1.1.1\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1\nn : Primrec fun a => a.1.1.2\ncf : Primrec fun a => a.2.1\ncg : Primrec fun a => a.2.2.1\nz\u271d : Primrec fun a => (unpair a.1.1.2).1\nL : Primrec fun a => a.1.1.1.1\nz : Primrec fun a => (unpair a.1.1.1.2).1\n\u22a2 Primrec fun p =>\n    (fun a n =>\n        (fun y => do\n            let i \u2190\n              Nat.Partrec.Code.lup a.1.1.1 (a.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).2)\n                  (Nat.pair (unpair a.1.1.2).1 y)\n            Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.2.1)\n                (Nat.pair (unpair a.1.1.2).1 (Nat.pair y i)))\n          n)\n      p.1 p.2"}, {"tactic": "have y := Primrec.snd\n  (\u03b1 := ((List (List (Option \u2115)) \u00d7 \u2115) \u00d7 \u2115) \u00d7 Code \u00d7 Code \u00d7 Option \u2115 \u00d7 Option \u2115) (\u03b2 := \u2115)", "annotated_tactic": ["have y := <a>Primrec.snd</a>\n      (\u03b1 := ((<a>List</a> (<a>List</a> (<a>Option</a> \u2115)) \u00d7 \u2115) \u00d7 \u2115) \u00d7 <a>Code</a> \u00d7 <a>Code</a> \u00d7 <a>Option</a> \u2115 \u00d7 <a>Option</a> \u2115) (\u03b2 := \u2115)", [{"full_name": "Primrec.snd", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [351, 9], "def_end_pos": [351, 12]}, {"full_name": "List", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2182, 11], "def_end_pos": [2182, 15]}, {"full_name": "List", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2182, 11], "def_end_pos": [2182, 15]}, {"full_name": "Option", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2139, 11], "def_end_pos": [2139, 17]}, {"full_name": "Nat.Partrec.Code", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [76, 11], "def_end_pos": [76, 15]}, {"full_name": "Nat.Partrec.Code", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [76, 11], "def_end_pos": [76, 15]}, {"full_name": "Option", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2139, 11], "def_end_pos": [2139, 17]}, {"full_name": "Option", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2139, 11], "def_end_pos": [2139, 17]}]], "state_before": "case hpc\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1\nn\u271d : Primrec fun a => a.1.2\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2\nL\u271d : Primrec fun a => a.1.1.1\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1\nn : Primrec fun a => a.1.1.2\ncf : Primrec fun a => a.2.1\ncg : Primrec fun a => a.2.2.1\nz\u271d : Primrec fun a => (unpair a.1.1.2).1\nL : Primrec fun a => a.1.1.1.1\nz : Primrec fun a => (unpair a.1.1.1.2).1\n\u22a2 Primrec fun p =>\n    (fun a n =>\n        (fun y => do\n            let i \u2190\n              Nat.Partrec.Code.lup a.1.1.1 (a.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).2)\n                  (Nat.pair (unpair a.1.1.2).1 y)\n            Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.2.1)\n                (Nat.pair (unpair a.1.1.2).1 (Nat.pair y i)))\n          n)\n      p.1 p.2", "state_after": "case hpc\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1\nn\u271d : Primrec fun a => a.1.2\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2\nL\u271d : Primrec fun a => a.1.1.1\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1\nn : Primrec fun a => a.1.1.2\ncf : Primrec fun a => a.2.1\ncg : Primrec fun a => a.2.2.1\nz\u271d : Primrec fun a => (unpair a.1.1.2).1\nL : Primrec fun a => a.1.1.1.1\nz : Primrec fun a => (unpair a.1.1.1.2).1\ny : Primrec Prod.snd\n\u22a2 Primrec fun p =>\n    (fun a n =>\n        (fun y => do\n            let i \u2190\n              Nat.Partrec.Code.lup a.1.1.1 (a.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).2)\n                  (Nat.pair (unpair a.1.1.2).1 y)\n            Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.2.1)\n                (Nat.pair (unpair a.1.1.2).1 (Nat.pair y i)))\n          n)\n      p.1 p.2"}, {"tactic": "have h\u2081 := hlup.comp <| L.pair <| (((k'.pair c).comp Primrec.fst).comp Primrec.fst).pair\n  (Primrec\u2082.natPair.comp z y)", "annotated_tactic": ["have h\u2081 := hlup.comp <| L.pair <| (((k'.pair c).<a>comp</a> <a>Primrec.fst</a>).<a>comp</a> <a>Primrec.fst</a>).<a>pair</a>\n      (Primrec\u2082.natPair.comp z y)", [{"full_name": "Primrec.comp", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [259, 9], "def_end_pos": [259, 13]}, {"full_name": "Primrec.fst", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [340, 9], "def_end_pos": [340, 12]}, {"full_name": "Primrec.comp", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [259, 9], "def_end_pos": [259, 13]}, {"full_name": "Primrec.fst", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [340, 9], "def_end_pos": [340, 12]}, {"full_name": "Primrec.pair", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [362, 9], "def_end_pos": [362, 13]}]], "state_before": "case hpc\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1\nn\u271d : Primrec fun a => a.1.2\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2\nL\u271d : Primrec fun a => a.1.1.1\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1\nn : Primrec fun a => a.1.1.2\ncf : Primrec fun a => a.2.1\ncg : Primrec fun a => a.2.2.1\nz\u271d : Primrec fun a => (unpair a.1.1.2).1\nL : Primrec fun a => a.1.1.1.1\nz : Primrec fun a => (unpair a.1.1.1.2).1\ny : Primrec Prod.snd\n\u22a2 Primrec fun p =>\n    (fun a n =>\n        (fun y => do\n            let i \u2190\n              Nat.Partrec.Code.lup a.1.1.1 (a.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).2)\n                  (Nat.pair (unpair a.1.1.2).1 y)\n            Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.2.1)\n                (Nat.pair (unpair a.1.1.2).1 (Nat.pair y i)))\n          n)\n      p.1 p.2", "state_after": "case hpc\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1\nn\u271d : Primrec fun a => a.1.2\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2\nL\u271d : Primrec fun a => a.1.1.1\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1\nn : Primrec fun a => a.1.1.2\ncf : Primrec fun a => a.2.1\ncg : Primrec fun a => a.2.2.1\nz\u271d : Primrec fun a => (unpair a.1.1.2).1\nL : Primrec fun a => a.1.1.1.1\nz : Primrec fun a => (unpair a.1.1.1.2).1\ny : Primrec Prod.snd\nh\u2081 :\n  Primrec fun a =>\n    Nat.Partrec.Code.lup\n      (a.1.1.1.1, (a.1.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1.1)).2), Nat.pair (unpair a.1.1.1.2).1 a.2).1\n      (a.1.1.1.1, (a.1.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1.1)).2), Nat.pair (unpair a.1.1.1.2).1 a.2).2.1\n      (a.1.1.1.1, (a.1.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1.1)).2), Nat.pair (unpair a.1.1.1.2).1 a.2).2.2\n\u22a2 Primrec fun p =>\n    (fun a n =>\n        (fun y => do\n            let i \u2190\n              Nat.Partrec.Code.lup a.1.1.1 (a.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).2)\n                  (Nat.pair (unpair a.1.1.2).1 y)\n            Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.2.1)\n                (Nat.pair (unpair a.1.1.2).1 (Nat.pair y i)))\n          n)\n      p.1 p.2"}, {"tactic": "refine' Primrec.option_bind h\u2081 (_ : Primrec _)", "annotated_tactic": ["refine' <a>Primrec.option_bind</a> h\u2081 (_ : <a>Primrec</a> _)", [{"full_name": "Primrec.option_bind", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [622, 9], "def_end_pos": [622, 20]}, {"full_name": "Primrec", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [207, 5], "def_end_pos": [207, 12]}]], "state_before": "case hpc\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1\nn\u271d : Primrec fun a => a.1.2\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2\nL\u271d : Primrec fun a => a.1.1.1\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1\nn : Primrec fun a => a.1.1.2\ncf : Primrec fun a => a.2.1\ncg : Primrec fun a => a.2.2.1\nz\u271d : Primrec fun a => (unpair a.1.1.2).1\nL : Primrec fun a => a.1.1.1.1\nz : Primrec fun a => (unpair a.1.1.1.2).1\ny : Primrec Prod.snd\nh\u2081 :\n  Primrec fun a =>\n    Nat.Partrec.Code.lup\n      (a.1.1.1.1, (a.1.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1.1)).2), Nat.pair (unpair a.1.1.1.2).1 a.2).1\n      (a.1.1.1.1, (a.1.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1.1)).2), Nat.pair (unpair a.1.1.1.2).1 a.2).2.1\n      (a.1.1.1.1, (a.1.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1.1)).2), Nat.pair (unpair a.1.1.1.2).1 a.2).2.2\n\u22a2 Primrec fun p =>\n    (fun a n =>\n        (fun y => do\n            let i \u2190\n              Nat.Partrec.Code.lup a.1.1.1 (a.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).2)\n                  (Nat.pair (unpair a.1.1.2).1 y)\n            Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.2.1)\n                (Nat.pair (unpair a.1.1.2).1 (Nat.pair y i)))\n          n)\n      p.1 p.2", "state_after": "case hpc\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1\nn\u271d : Primrec fun a => a.1.2\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2\nL\u271d : Primrec fun a => a.1.1.1\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1\nn : Primrec fun a => a.1.1.2\ncf : Primrec fun a => a.2.1\ncg : Primrec fun a => a.2.2.1\nz\u271d : Primrec fun a => (unpair a.1.1.2).1\nL : Primrec fun a => a.1.1.1.1\nz : Primrec fun a => (unpair a.1.1.1.2).1\ny : Primrec Prod.snd\nh\u2081 :\n  Primrec fun a =>\n    Nat.Partrec.Code.lup\n      (a.1.1.1.1, (a.1.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1.1)).2), Nat.pair (unpair a.1.1.1.2).1 a.2).1\n      (a.1.1.1.1, (a.1.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1.1)).2), Nat.pair (unpair a.1.1.1.2).1 a.2).2.1\n      (a.1.1.1.1, (a.1.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1.1)).2), Nat.pair (unpair a.1.1.1.2).1 a.2).2.2\n\u22a2 Primrec fun p =>\n    (fun p i =>\n        Nat.Partrec.Code.lup p.1.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1.1.1.1)).1, p.1.2.2.1)\n          (Nat.pair (unpair p.1.1.1.2).1 (Nat.pair p.2 i)))\n      p.1 p.2"}, {"tactic": "have z := z.comp (Primrec.fst (\u03b2 := \u2115))", "annotated_tactic": ["have z := z.comp (<a>Primrec.fst</a> (\u03b2 := \u2115))", [{"full_name": "Primrec.fst", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [340, 9], "def_end_pos": [340, 12]}]], "state_before": "case hpc\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1\nn\u271d : Primrec fun a => a.1.2\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2\nL\u271d : Primrec fun a => a.1.1.1\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1\nn : Primrec fun a => a.1.1.2\ncf : Primrec fun a => a.2.1\ncg : Primrec fun a => a.2.2.1\nz\u271d : Primrec fun a => (unpair a.1.1.2).1\nL : Primrec fun a => a.1.1.1.1\nz : Primrec fun a => (unpair a.1.1.1.2).1\ny : Primrec Prod.snd\nh\u2081 :\n  Primrec fun a =>\n    Nat.Partrec.Code.lup\n      (a.1.1.1.1, (a.1.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1.1)).2), Nat.pair (unpair a.1.1.1.2).1 a.2).1\n      (a.1.1.1.1, (a.1.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1.1)).2), Nat.pair (unpair a.1.1.1.2).1 a.2).2.1\n      (a.1.1.1.1, (a.1.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1.1)).2), Nat.pair (unpair a.1.1.1.2).1 a.2).2.2\n\u22a2 Primrec fun p =>\n    (fun p i =>\n        Nat.Partrec.Code.lup p.1.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1.1.1.1)).1, p.1.2.2.1)\n          (Nat.pair (unpair p.1.1.1.2).1 (Nat.pair p.2 i)))\n      p.1 p.2", "state_after": "case hpc\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1\nn\u271d : Primrec fun a => a.1.2\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2\nL\u271d : Primrec fun a => a.1.1.1\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1\nn : Primrec fun a => a.1.1.2\ncf : Primrec fun a => a.2.1\ncg : Primrec fun a => a.2.2.1\nz\u271d\u00b9 : Primrec fun a => (unpair a.1.1.2).1\nL : Primrec fun a => a.1.1.1.1\nz\u271d : Primrec fun a => (unpair a.1.1.1.2).1\ny : Primrec Prod.snd\nh\u2081 :\n  Primrec fun a =>\n    Nat.Partrec.Code.lup\n      (a.1.1.1.1, (a.1.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1.1)).2), Nat.pair (unpair a.1.1.1.2).1 a.2).1\n      (a.1.1.1.1, (a.1.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1.1)).2), Nat.pair (unpair a.1.1.1.2).1 a.2).2.1\n      (a.1.1.1.1, (a.1.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1.1)).2), Nat.pair (unpair a.1.1.1.2).1 a.2).2.2\nz : Primrec fun a => (unpair a.1.1.1.1.2).1\n\u22a2 Primrec fun p =>\n    (fun p i =>\n        Nat.Partrec.Code.lup p.1.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1.1.1.1)).1, p.1.2.2.1)\n          (Nat.pair (unpair p.1.1.1.2).1 (Nat.pair p.2 i)))\n      p.1 p.2"}, {"tactic": "have y := y.comp (Primrec.fst (\u03b2 := \u2115))", "annotated_tactic": ["have y := y.comp (<a>Primrec.fst</a> (\u03b2 := \u2115))", [{"full_name": "Primrec.fst", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [340, 9], "def_end_pos": [340, 12]}]], "state_before": "case hpc\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1\nn\u271d : Primrec fun a => a.1.2\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2\nL\u271d : Primrec fun a => a.1.1.1\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1\nn : Primrec fun a => a.1.1.2\ncf : Primrec fun a => a.2.1\ncg : Primrec fun a => a.2.2.1\nz\u271d\u00b9 : Primrec fun a => (unpair a.1.1.2).1\nL : Primrec fun a => a.1.1.1.1\nz\u271d : Primrec fun a => (unpair a.1.1.1.2).1\ny : Primrec Prod.snd\nh\u2081 :\n  Primrec fun a =>\n    Nat.Partrec.Code.lup\n      (a.1.1.1.1, (a.1.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1.1)).2), Nat.pair (unpair a.1.1.1.2).1 a.2).1\n      (a.1.1.1.1, (a.1.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1.1)).2), Nat.pair (unpair a.1.1.1.2).1 a.2).2.1\n      (a.1.1.1.1, (a.1.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1.1)).2), Nat.pair (unpair a.1.1.1.2).1 a.2).2.2\nz : Primrec fun a => (unpair a.1.1.1.1.2).1\n\u22a2 Primrec fun p =>\n    (fun p i =>\n        Nat.Partrec.Code.lup p.1.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1.1.1.1)).1, p.1.2.2.1)\n          (Nat.pair (unpair p.1.1.1.2).1 (Nat.pair p.2 i)))\n      p.1 p.2", "state_after": "case hpc\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1\nn\u271d : Primrec fun a => a.1.2\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2\nL\u271d : Primrec fun a => a.1.1.1\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1\nn : Primrec fun a => a.1.1.2\ncf : Primrec fun a => a.2.1\ncg : Primrec fun a => a.2.2.1\nz\u271d\u00b9 : Primrec fun a => (unpair a.1.1.2).1\nL : Primrec fun a => a.1.1.1.1\nz\u271d : Primrec fun a => (unpair a.1.1.1.2).1\ny\u271d : Primrec Prod.snd\nh\u2081 :\n  Primrec fun a =>\n    Nat.Partrec.Code.lup\n      (a.1.1.1.1, (a.1.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1.1)).2), Nat.pair (unpair a.1.1.1.2).1 a.2).1\n      (a.1.1.1.1, (a.1.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1.1)).2), Nat.pair (unpair a.1.1.1.2).1 a.2).2.1\n      (a.1.1.1.1, (a.1.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1.1)).2), Nat.pair (unpair a.1.1.1.2).1 a.2).2.2\nz : Primrec fun a => (unpair a.1.1.1.1.2).1\ny : Primrec fun a => a.1.2\n\u22a2 Primrec fun p =>\n    (fun p i =>\n        Nat.Partrec.Code.lup p.1.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1.1.1.1)).1, p.1.2.2.1)\n          (Nat.pair (unpair p.1.1.1.2).1 (Nat.pair p.2 i)))\n      p.1 p.2"}, {"tactic": "have i := Primrec.snd\n  (\u03b1 := (((List (List (Option \u2115)) \u00d7 \u2115) \u00d7 \u2115) \u00d7 Code \u00d7 Code \u00d7 Option \u2115 \u00d7 Option \u2115) \u00d7 \u2115)\n  (\u03b2 := \u2115)", "annotated_tactic": ["have i := <a>Primrec.snd</a>\n      (\u03b1 := (((<a>List</a> (<a>List</a> (<a>Option</a> \u2115)) \u00d7 \u2115) \u00d7 \u2115) \u00d7 <a>Code</a> \u00d7 <a>Code</a> \u00d7 <a>Option</a> \u2115 \u00d7 <a>Option</a> \u2115) \u00d7 \u2115)\n      (\u03b2 := \u2115)", [{"full_name": "Primrec.snd", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [351, 9], "def_end_pos": [351, 12]}, {"full_name": "List", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2182, 11], "def_end_pos": [2182, 15]}, {"full_name": "List", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2182, 11], "def_end_pos": [2182, 15]}, {"full_name": "Option", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2139, 11], "def_end_pos": [2139, 17]}, {"full_name": "Nat.Partrec.Code", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [76, 11], "def_end_pos": [76, 15]}, {"full_name": "Nat.Partrec.Code", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [76, 11], "def_end_pos": [76, 15]}, {"full_name": "Option", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2139, 11], "def_end_pos": [2139, 17]}, {"full_name": "Option", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2139, 11], "def_end_pos": [2139, 17]}]], "state_before": "case hpc\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1\nn\u271d : Primrec fun a => a.1.2\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2\nL\u271d : Primrec fun a => a.1.1.1\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1\nn : Primrec fun a => a.1.1.2\ncf : Primrec fun a => a.2.1\ncg : Primrec fun a => a.2.2.1\nz\u271d\u00b9 : Primrec fun a => (unpair a.1.1.2).1\nL : Primrec fun a => a.1.1.1.1\nz\u271d : Primrec fun a => (unpair a.1.1.1.2).1\ny\u271d : Primrec Prod.snd\nh\u2081 :\n  Primrec fun a =>\n    Nat.Partrec.Code.lup\n      (a.1.1.1.1, (a.1.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1.1)).2), Nat.pair (unpair a.1.1.1.2).1 a.2).1\n      (a.1.1.1.1, (a.1.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1.1)).2), Nat.pair (unpair a.1.1.1.2).1 a.2).2.1\n      (a.1.1.1.1, (a.1.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1.1)).2), Nat.pair (unpair a.1.1.1.2).1 a.2).2.2\nz : Primrec fun a => (unpair a.1.1.1.1.2).1\ny : Primrec fun a => a.1.2\n\u22a2 Primrec fun p =>\n    (fun p i =>\n        Nat.Partrec.Code.lup p.1.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1.1.1.1)).1, p.1.2.2.1)\n          (Nat.pair (unpair p.1.1.1.2).1 (Nat.pair p.2 i)))\n      p.1 p.2", "state_after": "case hpc\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1\nn\u271d : Primrec fun a => a.1.2\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2\nL\u271d : Primrec fun a => a.1.1.1\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1\nn : Primrec fun a => a.1.1.2\ncf : Primrec fun a => a.2.1\ncg : Primrec fun a => a.2.2.1\nz\u271d\u00b9 : Primrec fun a => (unpair a.1.1.2).1\nL : Primrec fun a => a.1.1.1.1\nz\u271d : Primrec fun a => (unpair a.1.1.1.2).1\ny\u271d : Primrec Prod.snd\nh\u2081 :\n  Primrec fun a =>\n    Nat.Partrec.Code.lup\n      (a.1.1.1.1, (a.1.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1.1)).2), Nat.pair (unpair a.1.1.1.2).1 a.2).1\n      (a.1.1.1.1, (a.1.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1.1)).2), Nat.pair (unpair a.1.1.1.2).1 a.2).2.1\n      (a.1.1.1.1, (a.1.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1.1)).2), Nat.pair (unpair a.1.1.1.2).1 a.2).2.2\nz : Primrec fun a => (unpair a.1.1.1.1.2).1\ny : Primrec fun a => a.1.2\ni : Primrec Prod.snd\n\u22a2 Primrec fun p =>\n    (fun p i =>\n        Nat.Partrec.Code.lup p.1.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1.1.1.1)).1, p.1.2.2.1)\n          (Nat.pair (unpair p.1.1.1.2).1 (Nat.pair p.2 i)))\n      p.1 p.2"}, {"tactic": "have h\u2082 := hlup.comp ((L.comp Primrec.fst).pair <|\n  ((k.pair cg).comp <| Primrec.fst.comp Primrec.fst).pair <|\n    Primrec\u2082.natPair.comp z <| Primrec\u2082.natPair.comp y i)", "annotated_tactic": ["have h\u2082 := hlup.comp ((L.comp <a>Primrec.fst</a>).<a>pair</a> <|\n      ((k.pair cg).<a>comp</a> <| Primrec.fst.comp <a>Primrec.fst</a>).<a>pair</a> <|\n        Primrec\u2082.natPair.comp z <| Primrec\u2082.natPair.comp y i)", [{"full_name": "Primrec.fst", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [340, 9], "def_end_pos": [340, 12]}, {"full_name": "Primrec.pair", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [362, 9], "def_end_pos": [362, 13]}, {"full_name": "Primrec.comp", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [259, 9], "def_end_pos": [259, 13]}, {"full_name": "Primrec.fst", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [340, 9], "def_end_pos": [340, 12]}, {"full_name": "Primrec.pair", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [362, 9], "def_end_pos": [362, 13]}]], "state_before": "case hpc\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1\nn\u271d : Primrec fun a => a.1.2\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2\nL\u271d : Primrec fun a => a.1.1.1\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1\nn : Primrec fun a => a.1.1.2\ncf : Primrec fun a => a.2.1\ncg : Primrec fun a => a.2.2.1\nz\u271d\u00b9 : Primrec fun a => (unpair a.1.1.2).1\nL : Primrec fun a => a.1.1.1.1\nz\u271d : Primrec fun a => (unpair a.1.1.1.2).1\ny\u271d : Primrec Prod.snd\nh\u2081 :\n  Primrec fun a =>\n    Nat.Partrec.Code.lup\n      (a.1.1.1.1, (a.1.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1.1)).2), Nat.pair (unpair a.1.1.1.2).1 a.2).1\n      (a.1.1.1.1, (a.1.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1.1)).2), Nat.pair (unpair a.1.1.1.2).1 a.2).2.1\n      (a.1.1.1.1, (a.1.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1.1)).2), Nat.pair (unpair a.1.1.1.2).1 a.2).2.2\nz : Primrec fun a => (unpair a.1.1.1.1.2).1\ny : Primrec fun a => a.1.2\ni : Primrec Prod.snd\n\u22a2 Primrec fun p =>\n    (fun p i =>\n        Nat.Partrec.Code.lup p.1.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1.1.1.1)).1, p.1.2.2.1)\n          (Nat.pair (unpair p.1.1.1.2).1 (Nat.pair p.2 i)))\n      p.1 p.2", "state_after": "case hpc\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1\nn\u271d : Primrec fun a => a.1.2\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2\nL\u271d : Primrec fun a => a.1.1.1\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1\nn : Primrec fun a => a.1.1.2\ncf : Primrec fun a => a.2.1\ncg : Primrec fun a => a.2.2.1\nz\u271d\u00b9 : Primrec fun a => (unpair a.1.1.2).1\nL : Primrec fun a => a.1.1.1.1\nz\u271d : Primrec fun a => (unpair a.1.1.1.2).1\ny\u271d : Primrec Prod.snd\nh\u2081 :\n  Primrec fun a =>\n    Nat.Partrec.Code.lup\n      (a.1.1.1.1, (a.1.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1.1)).2), Nat.pair (unpair a.1.1.1.2).1 a.2).1\n      (a.1.1.1.1, (a.1.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1.1)).2), Nat.pair (unpair a.1.1.1.2).1 a.2).2.1\n      (a.1.1.1.1, (a.1.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1.1)).2), Nat.pair (unpair a.1.1.1.2).1 a.2).2.2\nz : Primrec fun a => (unpair a.1.1.1.1.2).1\ny : Primrec fun a => a.1.2\ni : Primrec Prod.snd\nh\u2082 :\n  Primrec fun a =>\n    Nat.Partrec.Code.lup\n      (a.1.1.1.1.1, ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1.1.1)).1, a.1.1.2.2.1),\n          Nat.pair (unpair a.1.1.1.1.2).1 (Nat.pair a.1.2 a.2)).1\n      (a.1.1.1.1.1, ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1.1.1)).1, a.1.1.2.2.1),\n            Nat.pair (unpair a.1.1.1.1.2).1 (Nat.pair a.1.2 a.2)).2.1\n      (a.1.1.1.1.1, ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1.1.1)).1, a.1.1.2.2.1),\n            Nat.pair (unpair a.1.1.1.1.2).1 (Nat.pair a.1.2 a.2)).2.2\n\u22a2 Primrec fun p =>\n    (fun p i =>\n        Nat.Partrec.Code.lup p.1.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1.1.1.1)).1, p.1.2.2.1)\n          (Nat.pair (unpair p.1.1.1.2).1 (Nat.pair p.2 i)))\n      p.1 p.2"}, {"tactic": "exact h\u2082", "annotated_tactic": ["exact h\u2082", []], "state_before": "case hpc\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1\nn\u271d : Primrec fun a => a.1.2\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2\nL\u271d : Primrec fun a => a.1.1.1\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1\nn : Primrec fun a => a.1.1.2\ncf : Primrec fun a => a.2.1\ncg : Primrec fun a => a.2.2.1\nz\u271d\u00b9 : Primrec fun a => (unpair a.1.1.2).1\nL : Primrec fun a => a.1.1.1.1\nz\u271d : Primrec fun a => (unpair a.1.1.1.2).1\ny\u271d : Primrec Prod.snd\nh\u2081 :\n  Primrec fun a =>\n    Nat.Partrec.Code.lup\n      (a.1.1.1.1, (a.1.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1.1)).2), Nat.pair (unpair a.1.1.1.2).1 a.2).1\n      (a.1.1.1.1, (a.1.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1.1)).2), Nat.pair (unpair a.1.1.1.2).1 a.2).2.1\n      (a.1.1.1.1, (a.1.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1.1)).2), Nat.pair (unpair a.1.1.1.2).1 a.2).2.2\nz : Primrec fun a => (unpair a.1.1.1.1.2).1\ny : Primrec fun a => a.1.2\ni : Primrec Prod.snd\nh\u2082 :\n  Primrec fun a =>\n    Nat.Partrec.Code.lup\n      (a.1.1.1.1.1, ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1.1.1)).1, a.1.1.2.2.1),\n          Nat.pair (unpair a.1.1.1.1.2).1 (Nat.pair a.1.2 a.2)).1\n      (a.1.1.1.1.1, ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1.1.1)).1, a.1.1.2.2.1),\n            Nat.pair (unpair a.1.1.1.1.2).1 (Nat.pair a.1.2 a.2)).2.1\n      (a.1.1.1.1.1, ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1.1.1)).1, a.1.1.2.2.1),\n            Nat.pair (unpair a.1.1.1.1.2).1 (Nat.pair a.1.2 a.2)).2.2\n\u22a2 Primrec fun p =>\n    (fun p i =>\n        Nat.Partrec.Code.lup p.1.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1.1.1.1)).1, p.1.2.2.1)\n          (Nat.pair (unpair p.1.1.1.2).1 (Nat.pair p.2 i)))\n      p.1 p.2", "state_after": "no goals"}, {"tactic": "have L := (Primrec.fst.comp Primrec.fst).comp\n  (Primrec.fst (\u03b1 := (List (List (Option \u2115)) \u00d7 \u2115) \u00d7 \u2115)\n    (\u03b2 := Code \u00d7 Option \u2115))", "annotated_tactic": ["have L := (Primrec.fst.comp <a>Primrec.fst</a>).<a>comp</a>\n      (<a>Primrec.fst</a> (\u03b1 := (<a>List</a> (<a>List</a> (<a>Option</a> \u2115)) \u00d7 \u2115) \u00d7 \u2115)\n        (\u03b2 := <a>Code</a> \u00d7 <a>Option</a> \u2115))", [{"full_name": "Primrec.fst", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [340, 9], "def_end_pos": [340, 12]}, {"full_name": "Primrec.comp", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [259, 9], "def_end_pos": [259, 13]}, {"full_name": "Primrec.fst", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [340, 9], "def_end_pos": [340, 12]}, {"full_name": "List", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2182, 11], "def_end_pos": [2182, 15]}, {"full_name": "List", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2182, 11], "def_end_pos": [2182, 15]}, {"full_name": "Option", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2139, 11], "def_end_pos": [2139, 17]}, {"full_name": "Nat.Partrec.Code", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [76, 11], "def_end_pos": [76, 15]}, {"full_name": "Option", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2139, 11], "def_end_pos": [2139, 17]}]], "state_before": "case hrf\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1\nn\u271d : Primrec Prod.snd\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1\nn : Primrec fun a => a.1.2\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2\n\u22a2 Primrec fun a =>\n    let z := (unpair a.1.1.2).1;\n    let m := (unpair a.1.1.2).2;\n    do\n    let x \u2190 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) (Nat.pair z m)\n    Nat.casesOn x (some m) fun x =>\n        Nat.Partrec.Code.lup a.1.1.1 (a.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).2) (Nat.pair z (m + 1))", "state_after": "case hrf\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1\nn\u271d : Primrec Prod.snd\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1\nn : Primrec fun a => a.1.2\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2\nL : Primrec fun a => a.1.1.1\n\u22a2 Primrec fun a =>\n    let z := (unpair a.1.1.2).1;\n    let m := (unpair a.1.1.2).2;\n    do\n    let x \u2190 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) (Nat.pair z m)\n    Nat.casesOn x (some m) fun x =>\n        Nat.Partrec.Code.lup a.1.1.1 (a.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).2) (Nat.pair z (m + 1))"}, {"tactic": "have k := k.comp (Primrec.fst (\u03b2 := Code \u00d7 Option \u2115))", "annotated_tactic": ["have k := k.comp (<a>Primrec.fst</a> (\u03b2 := <a>Code</a> \u00d7 <a>Option</a> \u2115))", [{"full_name": "Primrec.fst", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [340, 9], "def_end_pos": [340, 12]}, {"full_name": "Nat.Partrec.Code", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [76, 11], "def_end_pos": [76, 15]}, {"full_name": "Option", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2139, 11], "def_end_pos": [2139, 17]}]], "state_before": "case hrf\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1\nn\u271d : Primrec Prod.snd\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1\nn : Primrec fun a => a.1.2\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2\nL : Primrec fun a => a.1.1.1\n\u22a2 Primrec fun a =>\n    let z := (unpair a.1.1.2).1;\n    let m := (unpair a.1.1.2).2;\n    do\n    let x \u2190 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) (Nat.pair z m)\n    Nat.casesOn x (some m) fun x =>\n        Nat.Partrec.Code.lup a.1.1.1 (a.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).2) (Nat.pair z (m + 1))", "state_after": "case hrf\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1\nn\u271d : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1\nn : Primrec fun a => a.1.2\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2\nL : Primrec fun a => a.1.1.1\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1\n\u22a2 Primrec fun a =>\n    let z := (unpair a.1.1.2).1;\n    let m := (unpair a.1.1.2).2;\n    do\n    let x \u2190 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) (Nat.pair z m)\n    Nat.casesOn x (some m) fun x =>\n        Nat.Partrec.Code.lup a.1.1.1 (a.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).2) (Nat.pair z (m + 1))"}, {"tactic": "have n := n.comp (Primrec.fst (\u03b2 := Code \u00d7 Option \u2115))", "annotated_tactic": ["have n := n.comp (<a>Primrec.fst</a> (\u03b2 := <a>Code</a> \u00d7 <a>Option</a> \u2115))", [{"full_name": "Primrec.fst", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [340, 9], "def_end_pos": [340, 12]}, {"full_name": "Nat.Partrec.Code", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [76, 11], "def_end_pos": [76, 15]}, {"full_name": "Option", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2139, 11], "def_end_pos": [2139, 17]}]], "state_before": "case hrf\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1\nn\u271d : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1\nn : Primrec fun a => a.1.2\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2\nL : Primrec fun a => a.1.1.1\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1\n\u22a2 Primrec fun a =>\n    let z := (unpair a.1.1.2).1;\n    let m := (unpair a.1.1.2).2;\n    do\n    let x \u2190 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) (Nat.pair z m)\n    Nat.casesOn x (some m) fun x =>\n        Nat.Partrec.Code.lup a.1.1.1 (a.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).2) (Nat.pair z (m + 1))", "state_after": "case hrf\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1\nn\u271d : Primrec fun a => a.1.2\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2\nL : Primrec fun a => a.1.1.1\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1\nn : Primrec fun a => a.1.1.2\n\u22a2 Primrec fun a =>\n    let z := (unpair a.1.1.2).1;\n    let m := (unpair a.1.1.2).2;\n    do\n    let x \u2190 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) (Nat.pair z m)\n    Nat.casesOn x (some m) fun x =>\n        Nat.Partrec.Code.lup a.1.1.1 (a.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).2) (Nat.pair z (m + 1))"}, {"tactic": "have cf := Primrec.fst.comp (Primrec.snd (\u03b1 := (List (List (Option \u2115)) \u00d7 \u2115) \u00d7 \u2115)\n    (\u03b2 := Code \u00d7 Option \u2115))", "annotated_tactic": ["have cf := Primrec.fst.comp (<a>Primrec.snd</a> (\u03b1 := (<a>List</a> (<a>List</a> (<a>Option</a> \u2115)) \u00d7 \u2115) \u00d7 \u2115)\n        (\u03b2 := <a>Code</a> \u00d7 <a>Option</a> \u2115))", [{"full_name": "Primrec.snd", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [351, 9], "def_end_pos": [351, 12]}, {"full_name": "List", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2182, 11], "def_end_pos": [2182, 15]}, {"full_name": "List", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2182, 11], "def_end_pos": [2182, 15]}, {"full_name": "Option", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2139, 11], "def_end_pos": [2139, 17]}, {"full_name": "Nat.Partrec.Code", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [76, 11], "def_end_pos": [76, 15]}, {"full_name": "Option", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2139, 11], "def_end_pos": [2139, 17]}]], "state_before": "case hrf\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1\nn\u271d : Primrec fun a => a.1.2\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2\nL : Primrec fun a => a.1.1.1\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1\nn : Primrec fun a => a.1.1.2\n\u22a2 Primrec fun a =>\n    let z := (unpair a.1.1.2).1;\n    let m := (unpair a.1.1.2).2;\n    do\n    let x \u2190 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) (Nat.pair z m)\n    Nat.casesOn x (some m) fun x =>\n        Nat.Partrec.Code.lup a.1.1.1 (a.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).2) (Nat.pair z (m + 1))", "state_after": "case hrf\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1\nn\u271d : Primrec fun a => a.1.2\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2\nL : Primrec fun a => a.1.1.1\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1\nn : Primrec fun a => a.1.1.2\ncf : Primrec fun a => a.2.1\n\u22a2 Primrec fun a =>\n    let z := (unpair a.1.1.2).1;\n    let m := (unpair a.1.1.2).2;\n    do\n    let x \u2190 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) (Nat.pair z m)\n    Nat.casesOn x (some m) fun x =>\n        Nat.Partrec.Code.lup a.1.1.1 (a.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).2) (Nat.pair z (m + 1))"}, {"tactic": "have z := Primrec.fst.comp (Primrec.unpair.comp n)", "annotated_tactic": ["have z := Primrec.fst.comp (Primrec.unpair.comp n)", []], "state_before": "case hrf\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1\nn\u271d : Primrec fun a => a.1.2\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2\nL : Primrec fun a => a.1.1.1\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1\nn : Primrec fun a => a.1.1.2\ncf : Primrec fun a => a.2.1\n\u22a2 Primrec fun a =>\n    let z := (unpair a.1.1.2).1;\n    let m := (unpair a.1.1.2).2;\n    do\n    let x \u2190 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) (Nat.pair z m)\n    Nat.casesOn x (some m) fun x =>\n        Nat.Partrec.Code.lup a.1.1.1 (a.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).2) (Nat.pair z (m + 1))", "state_after": "case hrf\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1\nn\u271d : Primrec fun a => a.1.2\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2\nL : Primrec fun a => a.1.1.1\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1\nn : Primrec fun a => a.1.1.2\ncf : Primrec fun a => a.2.1\nz : Primrec fun a => (unpair a.1.1.2).1\n\u22a2 Primrec fun a =>\n    let z := (unpair a.1.1.2).1;\n    let m := (unpair a.1.1.2).2;\n    do\n    let x \u2190 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) (Nat.pair z m)\n    Nat.casesOn x (some m) fun x =>\n        Nat.Partrec.Code.lup a.1.1.1 (a.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).2) (Nat.pair z (m + 1))"}, {"tactic": "have m := Primrec.snd.comp (Primrec.unpair.comp n)", "annotated_tactic": ["have m := Primrec.snd.comp (Primrec.unpair.comp n)", []], "state_before": "case hrf\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1\nn\u271d : Primrec fun a => a.1.2\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2\nL : Primrec fun a => a.1.1.1\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1\nn : Primrec fun a => a.1.1.2\ncf : Primrec fun a => a.2.1\nz : Primrec fun a => (unpair a.1.1.2).1\n\u22a2 Primrec fun a =>\n    let z := (unpair a.1.1.2).1;\n    let m := (unpair a.1.1.2).2;\n    do\n    let x \u2190 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) (Nat.pair z m)\n    Nat.casesOn x (some m) fun x =>\n        Nat.Partrec.Code.lup a.1.1.1 (a.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).2) (Nat.pair z (m + 1))", "state_after": "case hrf\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1\nn\u271d : Primrec fun a => a.1.2\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2\nL : Primrec fun a => a.1.1.1\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1\nn : Primrec fun a => a.1.1.2\ncf : Primrec fun a => a.2.1\nz : Primrec fun a => (unpair a.1.1.2).1\nm : Primrec fun a => (unpair a.1.1.2).2\n\u22a2 Primrec fun a =>\n    let z := (unpair a.1.1.2).1;\n    let m := (unpair a.1.1.2).2;\n    do\n    let x \u2190 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) (Nat.pair z m)\n    Nat.casesOn x (some m) fun x =>\n        Nat.Partrec.Code.lup a.1.1.1 (a.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).2) (Nat.pair z (m + 1))"}, {"tactic": "have h\u2081 := hlup.comp <| L.pair <| (k.pair cf).pair (Primrec\u2082.natPair.comp z m)", "annotated_tactic": ["have h\u2081 := hlup.comp <| L.pair <| (k.pair cf).<a>pair</a> (Primrec\u2082.natPair.comp z m)", [{"full_name": "Primrec.pair", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [362, 9], "def_end_pos": [362, 13]}]], "state_before": "case hrf\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1\nn\u271d : Primrec fun a => a.1.2\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2\nL : Primrec fun a => a.1.1.1\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1\nn : Primrec fun a => a.1.1.2\ncf : Primrec fun a => a.2.1\nz : Primrec fun a => (unpair a.1.1.2).1\nm : Primrec fun a => (unpair a.1.1.2).2\n\u22a2 Primrec fun a =>\n    let z := (unpair a.1.1.2).1;\n    let m := (unpair a.1.1.2).2;\n    do\n    let x \u2190 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) (Nat.pair z m)\n    Nat.casesOn x (some m) fun x =>\n        Nat.Partrec.Code.lup a.1.1.1 (a.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).2) (Nat.pair z (m + 1))", "state_after": "case hrf\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1\nn\u271d : Primrec fun a => a.1.2\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2\nL : Primrec fun a => a.1.1.1\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1\nn : Primrec fun a => a.1.1.2\ncf : Primrec fun a => a.2.1\nz : Primrec fun a => (unpair a.1.1.2).1\nm : Primrec fun a => (unpair a.1.1.2).2\nh\u2081 :\n  Primrec fun a =>\n    Nat.Partrec.Code.lup\n      (a.1.1.1, ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1), Nat.pair (unpair a.1.1.2).1 (unpair a.1.1.2).2).1\n      (a.1.1.1, ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1), Nat.pair (unpair a.1.1.2).1 (unpair a.1.1.2).2).2.1\n      (a.1.1.1, ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1), Nat.pair (unpair a.1.1.2).1 (unpair a.1.1.2).2).2.2\n\u22a2 Primrec fun a =>\n    let z := (unpair a.1.1.2).1;\n    let m := (unpair a.1.1.2).2;\n    do\n    let x \u2190 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) (Nat.pair z m)\n    Nat.casesOn x (some m) fun x =>\n        Nat.Partrec.Code.lup a.1.1.1 (a.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).2) (Nat.pair z (m + 1))"}, {"tactic": "refine' Primrec.option_bind h\u2081 (_ : Primrec _)", "annotated_tactic": ["refine' <a>Primrec.option_bind</a> h\u2081 (_ : <a>Primrec</a> _)", [{"full_name": "Primrec.option_bind", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [622, 9], "def_end_pos": [622, 20]}, {"full_name": "Primrec", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [207, 5], "def_end_pos": [207, 12]}]], "state_before": "case hrf\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1\nn\u271d : Primrec fun a => a.1.2\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2\nL : Primrec fun a => a.1.1.1\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1\nn : Primrec fun a => a.1.1.2\ncf : Primrec fun a => a.2.1\nz : Primrec fun a => (unpair a.1.1.2).1\nm : Primrec fun a => (unpair a.1.1.2).2\nh\u2081 :\n  Primrec fun a =>\n    Nat.Partrec.Code.lup\n      (a.1.1.1, ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1), Nat.pair (unpair a.1.1.2).1 (unpair a.1.1.2).2).1\n      (a.1.1.1, ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1), Nat.pair (unpair a.1.1.2).1 (unpair a.1.1.2).2).2.1\n      (a.1.1.1, ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1), Nat.pair (unpair a.1.1.2).1 (unpair a.1.1.2).2).2.2\n\u22a2 Primrec fun a =>\n    let z := (unpair a.1.1.2).1;\n    let m := (unpair a.1.1.2).2;\n    do\n    let x \u2190 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) (Nat.pair z m)\n    Nat.casesOn x (some m) fun x =>\n        Nat.Partrec.Code.lup a.1.1.1 (a.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).2) (Nat.pair z (m + 1))", "state_after": "case hrf\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1\nn\u271d : Primrec fun a => a.1.2\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2\nL : Primrec fun a => a.1.1.1\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1\nn : Primrec fun a => a.1.1.2\ncf : Primrec fun a => a.2.1\nz : Primrec fun a => (unpair a.1.1.2).1\nm : Primrec fun a => (unpair a.1.1.2).2\nh\u2081 :\n  Primrec fun a =>\n    Nat.Partrec.Code.lup\n      (a.1.1.1, ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1), Nat.pair (unpair a.1.1.2).1 (unpair a.1.1.2).2).1\n      (a.1.1.1, ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1), Nat.pair (unpair a.1.1.2).1 (unpair a.1.1.2).2).2.1\n      (a.1.1.1, ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1), Nat.pair (unpair a.1.1.2).1 (unpair a.1.1.2).2).2.2\n\u22a2 Primrec fun p =>\n    (fun a x =>\n        Nat.casesOn x (some (unpair a.1.1.2).2) fun x =>\n          Nat.Partrec.Code.lup a.1.1.1 (a.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).2)\n            (Nat.pair (unpair a.1.1.2).1 ((unpair a.1.1.2).2 + 1)))\n      p.1 p.2"}, {"tactic": "have m := m.comp (Primrec.fst (\u03b2 := \u2115))", "annotated_tactic": ["have m := m.comp (<a>Primrec.fst</a> (\u03b2 := \u2115))", [{"full_name": "Primrec.fst", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [340, 9], "def_end_pos": [340, 12]}]], "state_before": "case hrf\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1\nn\u271d : Primrec fun a => a.1.2\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2\nL : Primrec fun a => a.1.1.1\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1\nn : Primrec fun a => a.1.1.2\ncf : Primrec fun a => a.2.1\nz : Primrec fun a => (unpair a.1.1.2).1\nm : Primrec fun a => (unpair a.1.1.2).2\nh\u2081 :\n  Primrec fun a =>\n    Nat.Partrec.Code.lup\n      (a.1.1.1, ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1), Nat.pair (unpair a.1.1.2).1 (unpair a.1.1.2).2).1\n      (a.1.1.1, ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1), Nat.pair (unpair a.1.1.2).1 (unpair a.1.1.2).2).2.1\n      (a.1.1.1, ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1), Nat.pair (unpair a.1.1.2).1 (unpair a.1.1.2).2).2.2\n\u22a2 Primrec fun p =>\n    (fun a x =>\n        Nat.casesOn x (some (unpair a.1.1.2).2) fun x =>\n          Nat.Partrec.Code.lup a.1.1.1 (a.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).2)\n            (Nat.pair (unpair a.1.1.2).1 ((unpair a.1.1.2).2 + 1)))\n      p.1 p.2", "state_after": "case hrf\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1\nn\u271d : Primrec fun a => a.1.2\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2\nL : Primrec fun a => a.1.1.1\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1\nn : Primrec fun a => a.1.1.2\ncf : Primrec fun a => a.2.1\nz : Primrec fun a => (unpair a.1.1.2).1\nm\u271d : Primrec fun a => (unpair a.1.1.2).2\nh\u2081 :\n  Primrec fun a =>\n    Nat.Partrec.Code.lup\n      (a.1.1.1, ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1), Nat.pair (unpair a.1.1.2).1 (unpair a.1.1.2).2).1\n      (a.1.1.1, ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1), Nat.pair (unpair a.1.1.2).1 (unpair a.1.1.2).2).2.1\n      (a.1.1.1, ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1), Nat.pair (unpair a.1.1.2).1 (unpair a.1.1.2).2).2.2\nm : Primrec fun a => (unpair a.1.1.1.2).2\n\u22a2 Primrec fun p =>\n    (fun a x =>\n        Nat.casesOn x (some (unpair a.1.1.2).2) fun x =>\n          Nat.Partrec.Code.lup a.1.1.1 (a.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).2)\n            (Nat.pair (unpair a.1.1.2).1 ((unpair a.1.1.2).2 + 1)))\n      p.1 p.2"}, {"tactic": "refine Primrec.nat_casesOn Primrec.snd (Primrec.option_some.comp m) ?_", "annotated_tactic": ["refine <a>Primrec.nat_casesOn</a> <a>Primrec.snd</a> (Primrec.option_some.comp m) ?_", [{"full_name": "Primrec.nat_casesOn", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [594, 9], "def_end_pos": [594, 20]}, {"full_name": "Primrec.snd", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [351, 9], "def_end_pos": [351, 12]}]], "state_before": "case hrf\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1\nn\u271d : Primrec fun a => a.1.2\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2\nL : Primrec fun a => a.1.1.1\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1\nn : Primrec fun a => a.1.1.2\ncf : Primrec fun a => a.2.1\nz : Primrec fun a => (unpair a.1.1.2).1\nm\u271d : Primrec fun a => (unpair a.1.1.2).2\nh\u2081 :\n  Primrec fun a =>\n    Nat.Partrec.Code.lup\n      (a.1.1.1, ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1), Nat.pair (unpair a.1.1.2).1 (unpair a.1.1.2).2).1\n      (a.1.1.1, ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1), Nat.pair (unpair a.1.1.2).1 (unpair a.1.1.2).2).2.1\n      (a.1.1.1, ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1), Nat.pair (unpair a.1.1.2).1 (unpair a.1.1.2).2).2.2\nm : Primrec fun a => (unpair a.1.1.1.2).2\n\u22a2 Primrec fun p =>\n    (fun a x =>\n        Nat.casesOn x (some (unpair a.1.1.2).2) fun x =>\n          Nat.Partrec.Code.lup a.1.1.1 (a.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).2)\n            (Nat.pair (unpair a.1.1.2).1 ((unpair a.1.1.2).2 + 1)))\n      p.1 p.2", "state_after": "case hrf\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1\nn\u271d : Primrec fun a => a.1.2\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2\nL : Primrec fun a => a.1.1.1\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1\nn : Primrec fun a => a.1.1.2\ncf : Primrec fun a => a.2.1\nz : Primrec fun a => (unpair a.1.1.2).1\nm\u271d : Primrec fun a => (unpair a.1.1.2).2\nh\u2081 :\n  Primrec fun a =>\n    Nat.Partrec.Code.lup\n      (a.1.1.1, ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1), Nat.pair (unpair a.1.1.2).1 (unpair a.1.1.2).2).1\n      (a.1.1.1, ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1), Nat.pair (unpair a.1.1.2).1 (unpair a.1.1.2).2).2.1\n      (a.1.1.1, ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1), Nat.pair (unpair a.1.1.2).1 (unpair a.1.1.2).2).2.2\nm : Primrec fun a => (unpair a.1.1.1.2).2\n\u22a2 Primrec\u2082 fun p n =>\n    (fun x =>\n        Nat.Partrec.Code.lup p.1.1.1.1 (p.1.1.2, (ofNat (\u2115 \u00d7 Code) (List.length p.1.1.1.1)).2)\n          (Nat.pair (unpair p.1.1.1.2).1 ((unpair p.1.1.1.2).2 + 1)))\n      n"}, {"tactic": "unfold Primrec\u2082", "annotated_tactic": ["unfold <a>Primrec\u2082</a>", [{"full_name": "Primrec\u2082", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [389, 5], "def_end_pos": [389, 13]}]], "state_before": "case hrf\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1\nn\u271d : Primrec fun a => a.1.2\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2\nL : Primrec fun a => a.1.1.1\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1\nn : Primrec fun a => a.1.1.2\ncf : Primrec fun a => a.2.1\nz : Primrec fun a => (unpair a.1.1.2).1\nm\u271d : Primrec fun a => (unpair a.1.1.2).2\nh\u2081 :\n  Primrec fun a =>\n    Nat.Partrec.Code.lup\n      (a.1.1.1, ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1), Nat.pair (unpair a.1.1.2).1 (unpair a.1.1.2).2).1\n      (a.1.1.1, ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1), Nat.pair (unpair a.1.1.2).1 (unpair a.1.1.2).2).2.1\n      (a.1.1.1, ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1), Nat.pair (unpair a.1.1.2).1 (unpair a.1.1.2).2).2.2\nm : Primrec fun a => (unpair a.1.1.1.2).2\n\u22a2 Primrec\u2082 fun p n =>\n    (fun x =>\n        Nat.Partrec.Code.lup p.1.1.1.1 (p.1.1.2, (ofNat (\u2115 \u00d7 Code) (List.length p.1.1.1.1)).2)\n          (Nat.pair (unpair p.1.1.1.2).1 ((unpair p.1.1.1.2).2 + 1)))\n      n", "state_after": "case hrf\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1\nn\u271d : Primrec fun a => a.1.2\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2\nL : Primrec fun a => a.1.1.1\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1\nn : Primrec fun a => a.1.1.2\ncf : Primrec fun a => a.2.1\nz : Primrec fun a => (unpair a.1.1.2).1\nm\u271d : Primrec fun a => (unpair a.1.1.2).2\nh\u2081 :\n  Primrec fun a =>\n    Nat.Partrec.Code.lup\n      (a.1.1.1, ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1), Nat.pair (unpair a.1.1.2).1 (unpair a.1.1.2).2).1\n      (a.1.1.1, ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1), Nat.pair (unpair a.1.1.2).1 (unpair a.1.1.2).2).2.1\n      (a.1.1.1, ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1), Nat.pair (unpair a.1.1.2).1 (unpair a.1.1.2).2).2.2\nm : Primrec fun a => (unpair a.1.1.1.2).2\n\u22a2 Primrec fun p =>\n    (fun p n =>\n        (fun x =>\n            Nat.Partrec.Code.lup p.1.1.1.1 (p.1.1.2, (ofNat (\u2115 \u00d7 Code) (List.length p.1.1.1.1)).2)\n              (Nat.pair (unpair p.1.1.1.2).1 ((unpair p.1.1.1.2).2 + 1)))\n          n)\n      p.1 p.2"}, {"tactic": "exact (hlup.comp ((L.comp Primrec.fst).pair <|\n  ((k'.pair c).comp <| Primrec.fst.comp Primrec.fst).pair\n    (Primrec\u2082.natPair.comp (z.comp Primrec.fst) (_root_.Primrec.succ.comp m)))).comp\n  Primrec.fst", "annotated_tactic": ["exact (hlup.comp ((L.comp <a>Primrec.fst</a>).<a>pair</a> <|\n      ((k'.pair c).<a>comp</a> <| Primrec.fst.comp <a>Primrec.fst</a>).<a>pair</a>\n        (Primrec\u2082.natPair.comp (z.comp <a>Primrec.fst</a>) (_root_.Primrec.succ.comp m)))).<a>comp</a>\n      <a>Primrec.fst</a>", [{"full_name": "Primrec.fst", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [340, 9], "def_end_pos": [340, 12]}, {"full_name": "Primrec.pair", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [362, 9], "def_end_pos": [362, 13]}, {"full_name": "Primrec.comp", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [259, 9], "def_end_pos": [259, 13]}, {"full_name": "Primrec.fst", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [340, 9], "def_end_pos": [340, 12]}, {"full_name": "Primrec.pair", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [362, 9], "def_end_pos": [362, 13]}, {"full_name": "Primrec.fst", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [340, 9], "def_end_pos": [340, 12]}, {"full_name": "Primrec.comp", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [259, 9], "def_end_pos": [259, 13]}, {"full_name": "Primrec.fst", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [340, 9], "def_end_pos": [340, 12]}]], "state_before": "case hrf\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1\nn\u271d : Primrec fun a => a.1.2\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2\nL : Primrec fun a => a.1.1.1\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1\nn : Primrec fun a => a.1.1.2\ncf : Primrec fun a => a.2.1\nz : Primrec fun a => (unpair a.1.1.2).1\nm\u271d : Primrec fun a => (unpair a.1.1.2).2\nh\u2081 :\n  Primrec fun a =>\n    Nat.Partrec.Code.lup\n      (a.1.1.1, ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1), Nat.pair (unpair a.1.1.2).1 (unpair a.1.1.2).2).1\n      (a.1.1.1, ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1), Nat.pair (unpair a.1.1.2).1 (unpair a.1.1.2).2).2.1\n      (a.1.1.1, ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1), Nat.pair (unpair a.1.1.2).1 (unpair a.1.1.2).2).2.2\nm : Primrec fun a => (unpair a.1.1.1.2).2\n\u22a2 Primrec fun p =>\n    (fun p n =>\n        (fun x =>\n            Nat.Partrec.Code.lup p.1.1.1.1 (p.1.1.2, (ofNat (\u2115 \u00d7 Code) (List.length p.1.1.1.1)).2)\n              (Nat.pair (unpair p.1.1.1.2).1 ((unpair p.1.1.1.2).2 + 1)))\n          n)\n      p.1 p.2", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "full_name": "ContinuousLinearMap.integral_comp_comm'", "start": [1146, 1], "end": [1153, 33], "traced_tactics": [{"tactic": "by_cases h : Integrable \u03c6 \u03bc", "annotated_tactic": ["by_cases h : <a>Integrable</a> \u03c6 \u03bc", [{"full_name": "MeasureTheory.Integrable", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [442, 5], "def_end_pos": [442, 15]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2079 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2078 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : NormedSpace \u211d E\nL : E \u2192L[\ud835\udd5c] F\nK : \u211d\u22650\nhL : AntilipschitzWith K \u2191L\n\u03c6 : \u03b1 \u2192 E\n\u22a2 \u222b (a : \u03b1), \u2191L (\u03c6 a) \u2202\u03bc = \u2191L (\u222b (a : \u03b1), \u03c6 a \u2202\u03bc)", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2079 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2078 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : NormedSpace \u211d E\nL : E \u2192L[\ud835\udd5c] F\nK : \u211d\u22650\nhL : AntilipschitzWith K \u2191L\n\u03c6 : \u03b1 \u2192 E\nh : Integrable \u03c6\n\u22a2 \u222b (a : \u03b1), \u2191L (\u03c6 a) \u2202\u03bc = \u2191L (\u222b (a : \u03b1), \u03c6 a \u2202\u03bc)\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2079 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2078 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : NormedSpace \u211d E\nL : E \u2192L[\ud835\udd5c] F\nK : \u211d\u22650\nhL : AntilipschitzWith K \u2191L\n\u03c6 : \u03b1 \u2192 E\nh : \u00acIntegrable \u03c6\n\u22a2 \u222b (a : \u03b1), \u2191L (\u03c6 a) \u2202\u03bc = \u2191L (\u222b (a : \u03b1), \u03c6 a \u2202\u03bc)"}, {"tactic": "have : \u00acIntegrable (fun a => L (\u03c6 a)) \u03bc := by\n  erw [LipschitzWith.integrable_comp_iff_of_antilipschitz L.lipschitz hL L.map_zero]\n  assumption", "annotated_tactic": ["have : \u00ac<a>Integrable</a> (fun a => L (\u03c6 a)) \u03bc := by\n    erw [<a>LipschitzWith.integrable_comp_iff_of_antilipschitz</a> L.lipschitz hL L.map_zero]\n    assumption", [{"full_name": "MeasureTheory.Integrable", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [442, 5], "def_end_pos": [442, 15]}, {"full_name": "MeasureTheory.LipschitzWith.integrable_comp_iff_of_antilipschitz", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [865, 9], "def_end_pos": [865, 59]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2079 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2078 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : NormedSpace \u211d E\nL : E \u2192L[\ud835\udd5c] F\nK : \u211d\u22650\nhL : AntilipschitzWith K \u2191L\n\u03c6 : \u03b1 \u2192 E\nh : \u00acIntegrable \u03c6\n\u22a2 \u222b (a : \u03b1), \u2191L (\u03c6 a) \u2202\u03bc = \u2191L (\u222b (a : \u03b1), \u03c6 a \u2202\u03bc)", "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2079 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2078 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : NormedSpace \u211d E\nL : E \u2192L[\ud835\udd5c] F\nK : \u211d\u22650\nhL : AntilipschitzWith K \u2191L\n\u03c6 : \u03b1 \u2192 E\nh : \u00acIntegrable \u03c6\nthis : \u00acIntegrable fun a => \u2191L (\u03c6 a)\n\u22a2 \u222b (a : \u03b1), \u2191L (\u03c6 a) \u2202\u03bc = \u2191L (\u222b (a : \u03b1), \u03c6 a \u2202\u03bc)"}, {"tactic": "simp [integral_undef, h, this]", "annotated_tactic": ["simp [<a>integral_undef</a>, h, this]", [{"full_name": "MeasureTheory.integral_undef", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [836, 9], "def_end_pos": [836, 23]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2079 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2078 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : NormedSpace \u211d E\nL : E \u2192L[\ud835\udd5c] F\nK : \u211d\u22650\nhL : AntilipschitzWith K \u2191L\n\u03c6 : \u03b1 \u2192 E\nh : \u00acIntegrable \u03c6\nthis : \u00acIntegrable fun a => \u2191L (\u03c6 a)\n\u22a2 \u222b (a : \u03b1), \u2191L (\u03c6 a) \u2202\u03bc = \u2191L (\u222b (a : \u03b1), \u03c6 a \u2202\u03bc)", "state_after": "no goals"}, {"tactic": "exact integral_comp_comm L h", "annotated_tactic": ["exact <a>integral_comp_comm</a> L h", [{"full_name": "ContinuousLinearMap.integral_comp_comm", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [1120, 9], "def_end_pos": [1120, 27]}]], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2079 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2078 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : NormedSpace \u211d E\nL : E \u2192L[\ud835\udd5c] F\nK : \u211d\u22650\nhL : AntilipschitzWith K \u2191L\n\u03c6 : \u03b1 \u2192 E\nh : Integrable \u03c6\n\u22a2 \u222b (a : \u03b1), \u2191L (\u03c6 a) \u2202\u03bc = \u2191L (\u222b (a : \u03b1), \u03c6 a \u2202\u03bc)", "state_after": "no goals"}, {"tactic": "erw [LipschitzWith.integrable_comp_iff_of_antilipschitz L.lipschitz hL L.map_zero]", "annotated_tactic": ["erw [<a>LipschitzWith.integrable_comp_iff_of_antilipschitz</a> L.lipschitz hL L.map_zero]", [{"full_name": "MeasureTheory.LipschitzWith.integrable_comp_iff_of_antilipschitz", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [865, 9], "def_end_pos": [865, 59]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2079 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2078 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : NormedSpace \u211d E\nL : E \u2192L[\ud835\udd5c] F\nK : \u211d\u22650\nhL : AntilipschitzWith K \u2191L\n\u03c6 : \u03b1 \u2192 E\nh : \u00acIntegrable \u03c6\n\u22a2 \u00acIntegrable fun a => \u2191L (\u03c6 a)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2079 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2078 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : NormedSpace \u211d E\nL : E \u2192L[\ud835\udd5c] F\nK : \u211d\u22650\nhL : AntilipschitzWith K \u2191L\n\u03c6 : \u03b1 \u2192 E\nh : \u00acIntegrable \u03c6\n\u22a2 \u00acIntegrable fun a => \u03c6 a"}, {"tactic": "assumption", "annotated_tactic": ["assumption", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2079 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2078 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : NormedSpace \u211d E\nL : E \u2192L[\ud835\udd5c] F\nK : \u211d\u22650\nhL : AntilipschitzWith K \u2191L\n\u03c6 : \u03b1 \u2192 E\nh : \u00acIntegrable \u03c6\n\u22a2 \u00acIntegrable fun a => \u03c6 a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Lattice.lean", "full_name": "map_finset_inf", "start": [358, 1], "end": [361, 61], "traced_tactics": [{"tactic": "rw [inf_cons, inf_cons, map_inf, h, Function.comp_apply]", "annotated_tactic": ["rw [<a>inf_cons</a>, <a>inf_cons</a>, <a>map_inf</a>, h, <a>Function.comp_apply</a>]", [{"full_name": "Finset.inf_cons", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [323, 9], "def_end_pos": [323, 17]}, {"full_name": "Finset.inf_cons", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [323, 9], "def_end_pos": [323, 17]}, {"full_name": "InfHomClass.map_inf", "def_path": "Mathlib/Order/Hom/Lattice.lean", "def_pos": [114, 3], "def_end_pos": [114, 10]}, {"full_name": "Function.comp_apply", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [33, 17], "def_end_pos": [33, 36]}]], "state_before": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Type u_5\n\u03ba : Type u_6\ninst\u271d\u2074 : SemilatticeInf \u03b1\ninst\u271d\u00b3 : OrderTop \u03b1\ns\u271d\u00b9 s\u2081 s\u2082 : Finset \u03b2\nf\u271d g\u271d : \u03b2 \u2192 \u03b1\na : \u03b1\ninst\u271d\u00b2 : SemilatticeInf \u03b2\ninst\u271d\u00b9 : OrderTop \u03b2\ninst\u271d : InfTopHomClass F \u03b1 \u03b2\nf : F\ns\u271d : Finset \u03b9\ng : \u03b9 \u2192 \u03b1\ni : \u03b9\ns : Finset \u03b9\nx\u271d : \u00aci \u2208 s\nh : \u2191f (inf s g) = inf s (\u2191f \u2218 g)\n\u22a2 \u2191f (inf (cons i s x\u271d) g) = inf (cons i s x\u271d) (\u2191f \u2218 g)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Process/Stopping.lean", "full_name": "MeasureTheory.progMeasurable_min_stopping_time", "start": [807, 1], "end": [842, 60], "traced_tactics": [{"tactic": "intro i", "annotated_tactic": ["intro i", []], "state_before": "\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2077 : LinearOrder \u03b9\ninst\u271d\u2076 : MeasurableSpace \u03b9\ninst\u271d\u2075 : TopologicalSpace \u03b9\ninst\u271d\u2074 : OrderTopology \u03b9\ninst\u271d\u00b3 : SecondCountableTopology \u03b9\ninst\u271d\u00b2 : BorelSpace \u03b9\ninst\u271d\u00b9 : TopologicalSpace \u03b2\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\n\u03c4 : \u03a9 \u2192 \u03b9\nf : Filtration \u03b9 m\ninst\u271d : MetrizableSpace \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\n\u22a2 ProgMeasurable f fun i \u03c9 => min i (\u03c4 \u03c9)", "state_after": "\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2077 : LinearOrder \u03b9\ninst\u271d\u2076 : MeasurableSpace \u03b9\ninst\u271d\u2075 : TopologicalSpace \u03b9\ninst\u271d\u2074 : OrderTopology \u03b9\ninst\u271d\u00b3 : SecondCountableTopology \u03b9\ninst\u271d\u00b2 : BorelSpace \u03b9\ninst\u271d\u00b9 : TopologicalSpace \u03b2\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\n\u03c4 : \u03a9 \u2192 \u03b9\nf : Filtration \u03b9 m\ninst\u271d : MetrizableSpace \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\ni : \u03b9\n\u22a2 StronglyMeasurable fun p => (fun i \u03c9 => min i (\u03c4 \u03c9)) (\u2191p.1) p.2"}, {"tactic": "let m_prod : MeasurableSpace (Set.Iic i \u00d7 \u03a9) := Subtype.instMeasurableSpace.prod (f i)", "annotated_tactic": ["let m_prod : <a>MeasurableSpace</a> (<a>Set.Iic</a> i \u00d7 \u03a9) := Subtype.instMeasurableSpace.prod (f i)", [{"full_name": "MeasurableSpace", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [48, 20], "def_end_pos": [48, 35]}, {"full_name": "Set.Iic", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [64, 5], "def_end_pos": [64, 8]}]], "state_before": "\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2077 : LinearOrder \u03b9\ninst\u271d\u2076 : MeasurableSpace \u03b9\ninst\u271d\u2075 : TopologicalSpace \u03b9\ninst\u271d\u2074 : OrderTopology \u03b9\ninst\u271d\u00b3 : SecondCountableTopology \u03b9\ninst\u271d\u00b2 : BorelSpace \u03b9\ninst\u271d\u00b9 : TopologicalSpace \u03b2\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\n\u03c4 : \u03a9 \u2192 \u03b9\nf : Filtration \u03b9 m\ninst\u271d : MetrizableSpace \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\ni : \u03b9\n\u22a2 StronglyMeasurable fun p => (fun i \u03c9 => min i (\u03c4 \u03c9)) (\u2191p.1) p.2", "state_after": "\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2077 : LinearOrder \u03b9\ninst\u271d\u2076 : MeasurableSpace \u03b9\ninst\u271d\u2075 : TopologicalSpace \u03b9\ninst\u271d\u2074 : OrderTopology \u03b9\ninst\u271d\u00b3 : SecondCountableTopology \u03b9\ninst\u271d\u00b2 : BorelSpace \u03b9\ninst\u271d\u00b9 : TopologicalSpace \u03b2\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\n\u03c4 : \u03a9 \u2192 \u03b9\nf : Filtration \u03b9 m\ninst\u271d : MetrizableSpace \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\ni : \u03b9\nm_prod : MeasurableSpace (\u2191(Set.Iic i) \u00d7 \u03a9) := MeasurableSpace.prod Subtype.instMeasurableSpace (\u2191f i)\n\u22a2 StronglyMeasurable fun p => (fun i \u03c9 => min i (\u03c4 \u03c9)) (\u2191p.1) p.2"}, {"tactic": "let m_set : \u2200 t : Set (Set.Iic i \u00d7 \u03a9), MeasurableSpace t := fun _ =>\n  @Subtype.instMeasurableSpace (Set.Iic i \u00d7 \u03a9) _ m_prod", "annotated_tactic": ["let m_set : \u2200 t : <a>Set</a> (<a>Set.Iic</a> i \u00d7 \u03a9), <a>MeasurableSpace</a> t := fun _ =>\n    @<a>Subtype.instMeasurableSpace</a> (<a>Set.Iic</a> i \u00d7 \u03a9) _ m_prod", [{"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}, {"full_name": "Set.Iic", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [64, 5], "def_end_pos": [64, 8]}, {"full_name": "MeasurableSpace", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [48, 20], "def_end_pos": [48, 35]}, {"full_name": "Subtype.instMeasurableSpace", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [567, 10], "def_end_pos": [567, 37]}, {"full_name": "Set.Iic", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [64, 5], "def_end_pos": [64, 8]}]], "state_before": "\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2077 : LinearOrder \u03b9\ninst\u271d\u2076 : MeasurableSpace \u03b9\ninst\u271d\u2075 : TopologicalSpace \u03b9\ninst\u271d\u2074 : OrderTopology \u03b9\ninst\u271d\u00b3 : SecondCountableTopology \u03b9\ninst\u271d\u00b2 : BorelSpace \u03b9\ninst\u271d\u00b9 : TopologicalSpace \u03b2\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\n\u03c4 : \u03a9 \u2192 \u03b9\nf : Filtration \u03b9 m\ninst\u271d : MetrizableSpace \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\ni : \u03b9\nm_prod : MeasurableSpace (\u2191(Set.Iic i) \u00d7 \u03a9) := MeasurableSpace.prod Subtype.instMeasurableSpace (\u2191f i)\n\u22a2 StronglyMeasurable fun p => (fun i \u03c9 => min i (\u03c4 \u03c9)) (\u2191p.1) p.2", "state_after": "\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2077 : LinearOrder \u03b9\ninst\u271d\u2076 : MeasurableSpace \u03b9\ninst\u271d\u2075 : TopologicalSpace \u03b9\ninst\u271d\u2074 : OrderTopology \u03b9\ninst\u271d\u00b3 : SecondCountableTopology \u03b9\ninst\u271d\u00b2 : BorelSpace \u03b9\ninst\u271d\u00b9 : TopologicalSpace \u03b2\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\n\u03c4 : \u03a9 \u2192 \u03b9\nf : Filtration \u03b9 m\ninst\u271d : MetrizableSpace \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\ni : \u03b9\nm_prod : MeasurableSpace (\u2191(Set.Iic i) \u00d7 \u03a9) := MeasurableSpace.prod Subtype.instMeasurableSpace (\u2191f i)\nm_set : (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)) \u2192 MeasurableSpace \u2191t := fun x => Subtype.instMeasurableSpace\n\u22a2 StronglyMeasurable fun p => (fun i \u03c9 => min i (\u03c4 \u03c9)) (\u2191p.1) p.2"}, {"tactic": "let s := {p : Set.Iic i \u00d7 \u03a9 | \u03c4 p.2 \u2264 i}", "annotated_tactic": ["let s := {p : <a>Set.Iic</a> i \u00d7 \u03a9 | \u03c4 p.2 \u2264 i}", [{"full_name": "Set.Iic", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [64, 5], "def_end_pos": [64, 8]}]], "state_before": "\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2077 : LinearOrder \u03b9\ninst\u271d\u2076 : MeasurableSpace \u03b9\ninst\u271d\u2075 : TopologicalSpace \u03b9\ninst\u271d\u2074 : OrderTopology \u03b9\ninst\u271d\u00b3 : SecondCountableTopology \u03b9\ninst\u271d\u00b2 : BorelSpace \u03b9\ninst\u271d\u00b9 : TopologicalSpace \u03b2\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\n\u03c4 : \u03a9 \u2192 \u03b9\nf : Filtration \u03b9 m\ninst\u271d : MetrizableSpace \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\ni : \u03b9\nm_prod : MeasurableSpace (\u2191(Set.Iic i) \u00d7 \u03a9) := MeasurableSpace.prod Subtype.instMeasurableSpace (\u2191f i)\nm_set : (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)) \u2192 MeasurableSpace \u2191t := fun x => Subtype.instMeasurableSpace\n\u22a2 StronglyMeasurable fun p => (fun i \u03c9 => min i (\u03c4 \u03c9)) (\u2191p.1) p.2", "state_after": "\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2077 : LinearOrder \u03b9\ninst\u271d\u2076 : MeasurableSpace \u03b9\ninst\u271d\u2075 : TopologicalSpace \u03b9\ninst\u271d\u2074 : OrderTopology \u03b9\ninst\u271d\u00b3 : SecondCountableTopology \u03b9\ninst\u271d\u00b2 : BorelSpace \u03b9\ninst\u271d\u00b9 : TopologicalSpace \u03b2\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\n\u03c4 : \u03a9 \u2192 \u03b9\nf : Filtration \u03b9 m\ninst\u271d : MetrizableSpace \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\ni : \u03b9\nm_prod : MeasurableSpace (\u2191(Set.Iic i) \u00d7 \u03a9) := MeasurableSpace.prod Subtype.instMeasurableSpace (\u2191f i)\nm_set : (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)) \u2192 MeasurableSpace \u2191t := fun x => Subtype.instMeasurableSpace\ns : Set (\u2191(Set.Iic i) \u00d7 \u03a9) := {p | \u03c4 p.2 \u2264 i}\n\u22a2 StronglyMeasurable fun p => (fun i \u03c9 => min i (\u03c4 \u03c9)) (\u2191p.1) p.2"}, {"tactic": "have hs : MeasurableSet[m_prod] s := @measurable_snd (Set.Iic i) \u03a9 _ (f i) _ (h\u03c4 i)", "annotated_tactic": ["have hs : MeasurableSet[m_prod] s := @<a>measurable_snd</a> (<a>Set.Iic</a> i) \u03a9 _ (f i) _ (h\u03c4 i)", [{"full_name": "measurable_snd", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [698, 9], "def_end_pos": [698, 23]}, {"full_name": "Set.Iic", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [64, 5], "def_end_pos": [64, 8]}]], "state_before": "\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2077 : LinearOrder \u03b9\ninst\u271d\u2076 : MeasurableSpace \u03b9\ninst\u271d\u2075 : TopologicalSpace \u03b9\ninst\u271d\u2074 : OrderTopology \u03b9\ninst\u271d\u00b3 : SecondCountableTopology \u03b9\ninst\u271d\u00b2 : BorelSpace \u03b9\ninst\u271d\u00b9 : TopologicalSpace \u03b2\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\n\u03c4 : \u03a9 \u2192 \u03b9\nf : Filtration \u03b9 m\ninst\u271d : MetrizableSpace \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\ni : \u03b9\nm_prod : MeasurableSpace (\u2191(Set.Iic i) \u00d7 \u03a9) := MeasurableSpace.prod Subtype.instMeasurableSpace (\u2191f i)\nm_set : (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)) \u2192 MeasurableSpace \u2191t := fun x => Subtype.instMeasurableSpace\ns : Set (\u2191(Set.Iic i) \u00d7 \u03a9) := {p | \u03c4 p.2 \u2264 i}\n\u22a2 StronglyMeasurable fun p => (fun i \u03c9 => min i (\u03c4 \u03c9)) (\u2191p.1) p.2", "state_after": "\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2077 : LinearOrder \u03b9\ninst\u271d\u2076 : MeasurableSpace \u03b9\ninst\u271d\u2075 : TopologicalSpace \u03b9\ninst\u271d\u2074 : OrderTopology \u03b9\ninst\u271d\u00b3 : SecondCountableTopology \u03b9\ninst\u271d\u00b2 : BorelSpace \u03b9\ninst\u271d\u00b9 : TopologicalSpace \u03b2\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\n\u03c4 : \u03a9 \u2192 \u03b9\nf : Filtration \u03b9 m\ninst\u271d : MetrizableSpace \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\ni : \u03b9\nm_prod : MeasurableSpace (\u2191(Set.Iic i) \u00d7 \u03a9) := MeasurableSpace.prod Subtype.instMeasurableSpace (\u2191f i)\nm_set : (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)) \u2192 MeasurableSpace \u2191t := fun x => Subtype.instMeasurableSpace\ns : Set (\u2191(Set.Iic i) \u00d7 \u03a9) := {p | \u03c4 p.2 \u2264 i}\nhs : MeasurableSet s\n\u22a2 StronglyMeasurable fun p => (fun i \u03c9 => min i (\u03c4 \u03c9)) (\u2191p.1) p.2"}, {"tactic": "have h_meas_fst : \u2200 t : Set (Set.Iic i \u00d7 \u03a9),\n    Measurable[m_set t] fun x : t => ((x : Set.Iic i \u00d7 \u03a9).fst : \u03b9) :=\n  fun t => (@measurable_subtype_coe (Set.Iic i \u00d7 \u03a9) m_prod _).fst.subtype_val", "annotated_tactic": ["have h_meas_fst : \u2200 t : <a>Set</a> (<a>Set.Iic</a> i \u00d7 \u03a9),\n      Measurable[m_set t] fun x : t => ((x : <a>Set.Iic</a> i \u00d7 \u03a9).<a>fst</a> : \u03b9) :=\n    fun t => (@<a>measurable_subtype_coe</a> (<a>Set.Iic</a> i \u00d7 \u03a9) m_prod _).fst.subtype_val", [{"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}, {"full_name": "Set.Iic", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [64, 5], "def_end_pos": [64, 8]}, {"full_name": "Set.Iic", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [64, 5], "def_end_pos": [64, 8]}, {"full_name": "Prod.fst", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [468, 3], "def_end_pos": [468, 6]}, {"full_name": "measurable_subtype_coe", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [577, 9], "def_end_pos": [577, 31]}, {"full_name": "Set.Iic", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [64, 5], "def_end_pos": [64, 8]}]], "state_before": "\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2077 : LinearOrder \u03b9\ninst\u271d\u2076 : MeasurableSpace \u03b9\ninst\u271d\u2075 : TopologicalSpace \u03b9\ninst\u271d\u2074 : OrderTopology \u03b9\ninst\u271d\u00b3 : SecondCountableTopology \u03b9\ninst\u271d\u00b2 : BorelSpace \u03b9\ninst\u271d\u00b9 : TopologicalSpace \u03b2\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\n\u03c4 : \u03a9 \u2192 \u03b9\nf : Filtration \u03b9 m\ninst\u271d : MetrizableSpace \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\ni : \u03b9\nm_prod : MeasurableSpace (\u2191(Set.Iic i) \u00d7 \u03a9) := MeasurableSpace.prod Subtype.instMeasurableSpace (\u2191f i)\nm_set : (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)) \u2192 MeasurableSpace \u2191t := fun x => Subtype.instMeasurableSpace\ns : Set (\u2191(Set.Iic i) \u00d7 \u03a9) := {p | \u03c4 p.2 \u2264 i}\nhs : MeasurableSet s\n\u22a2 StronglyMeasurable fun p => (fun i \u03c9 => min i (\u03c4 \u03c9)) (\u2191p.1) p.2", "state_after": "\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2077 : LinearOrder \u03b9\ninst\u271d\u2076 : MeasurableSpace \u03b9\ninst\u271d\u2075 : TopologicalSpace \u03b9\ninst\u271d\u2074 : OrderTopology \u03b9\ninst\u271d\u00b3 : SecondCountableTopology \u03b9\ninst\u271d\u00b2 : BorelSpace \u03b9\ninst\u271d\u00b9 : TopologicalSpace \u03b2\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\n\u03c4 : \u03a9 \u2192 \u03b9\nf : Filtration \u03b9 m\ninst\u271d : MetrizableSpace \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\ni : \u03b9\nm_prod : MeasurableSpace (\u2191(Set.Iic i) \u00d7 \u03a9) := MeasurableSpace.prod Subtype.instMeasurableSpace (\u2191f i)\nm_set : (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)) \u2192 MeasurableSpace \u2191t := fun x => Subtype.instMeasurableSpace\ns : Set (\u2191(Set.Iic i) \u00d7 \u03a9) := {p | \u03c4 p.2 \u2264 i}\nhs : MeasurableSet s\nh_meas_fst : \u2200 (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)), Measurable fun x => \u2191(\u2191x).1\n\u22a2 StronglyMeasurable fun p => (fun i \u03c9 => min i (\u03c4 \u03c9)) (\u2191p.1) p.2"}, {"tactic": "apply Measurable.stronglyMeasurable", "annotated_tactic": ["apply <a>Measurable.stronglyMeasurable</a>", [{"full_name": "Measurable.stronglyMeasurable", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [653, 9], "def_end_pos": [653, 45]}]], "state_before": "\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2077 : LinearOrder \u03b9\ninst\u271d\u2076 : MeasurableSpace \u03b9\ninst\u271d\u2075 : TopologicalSpace \u03b9\ninst\u271d\u2074 : OrderTopology \u03b9\ninst\u271d\u00b3 : SecondCountableTopology \u03b9\ninst\u271d\u00b2 : BorelSpace \u03b9\ninst\u271d\u00b9 : TopologicalSpace \u03b2\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\n\u03c4 : \u03a9 \u2192 \u03b9\nf : Filtration \u03b9 m\ninst\u271d : MetrizableSpace \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\ni : \u03b9\nm_prod : MeasurableSpace (\u2191(Set.Iic i) \u00d7 \u03a9) := MeasurableSpace.prod Subtype.instMeasurableSpace (\u2191f i)\nm_set : (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)) \u2192 MeasurableSpace \u2191t := fun x => Subtype.instMeasurableSpace\ns : Set (\u2191(Set.Iic i) \u00d7 \u03a9) := {p | \u03c4 p.2 \u2264 i}\nhs : MeasurableSet s\nh_meas_fst : \u2200 (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)), Measurable fun x => \u2191(\u2191x).1\n\u22a2 StronglyMeasurable fun p => (fun i \u03c9 => min i (\u03c4 \u03c9)) (\u2191p.1) p.2", "state_after": "case hf\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2077 : LinearOrder \u03b9\ninst\u271d\u2076 : MeasurableSpace \u03b9\ninst\u271d\u2075 : TopologicalSpace \u03b9\ninst\u271d\u2074 : OrderTopology \u03b9\ninst\u271d\u00b3 : SecondCountableTopology \u03b9\ninst\u271d\u00b2 : BorelSpace \u03b9\ninst\u271d\u00b9 : TopologicalSpace \u03b2\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\n\u03c4 : \u03a9 \u2192 \u03b9\nf : Filtration \u03b9 m\ninst\u271d : MetrizableSpace \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\ni : \u03b9\nm_prod : MeasurableSpace (\u2191(Set.Iic i) \u00d7 \u03a9) := MeasurableSpace.prod Subtype.instMeasurableSpace (\u2191f i)\nm_set : (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)) \u2192 MeasurableSpace \u2191t := fun x => Subtype.instMeasurableSpace\ns : Set (\u2191(Set.Iic i) \u00d7 \u03a9) := {p | \u03c4 p.2 \u2264 i}\nhs : MeasurableSet s\nh_meas_fst : \u2200 (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)), Measurable fun x => \u2191(\u2191x).1\n\u22a2 Measurable fun p => (fun i \u03c9 => min i (\u03c4 \u03c9)) (\u2191p.1) p.2"}, {"tactic": "refine' measurable_of_restrict_of_restrict_compl hs _ _", "annotated_tactic": ["refine' <a>measurable_of_restrict_of_restrict_compl</a> hs _ _", [{"full_name": "measurable_of_restrict_of_restrict_compl", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [654, 9], "def_end_pos": [654, 49]}]], "state_before": "case hf\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2077 : LinearOrder \u03b9\ninst\u271d\u2076 : MeasurableSpace \u03b9\ninst\u271d\u2075 : TopologicalSpace \u03b9\ninst\u271d\u2074 : OrderTopology \u03b9\ninst\u271d\u00b3 : SecondCountableTopology \u03b9\ninst\u271d\u00b2 : BorelSpace \u03b9\ninst\u271d\u00b9 : TopologicalSpace \u03b2\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\n\u03c4 : \u03a9 \u2192 \u03b9\nf : Filtration \u03b9 m\ninst\u271d : MetrizableSpace \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\ni : \u03b9\nm_prod : MeasurableSpace (\u2191(Set.Iic i) \u00d7 \u03a9) := MeasurableSpace.prod Subtype.instMeasurableSpace (\u2191f i)\nm_set : (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)) \u2192 MeasurableSpace \u2191t := fun x => Subtype.instMeasurableSpace\ns : Set (\u2191(Set.Iic i) \u00d7 \u03a9) := {p | \u03c4 p.2 \u2264 i}\nhs : MeasurableSet s\nh_meas_fst : \u2200 (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)), Measurable fun x => \u2191(\u2191x).1\n\u22a2 Measurable fun p => (fun i \u03c9 => min i (\u03c4 \u03c9)) (\u2191p.1) p.2", "state_after": "case hf.refine'_1\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2077 : LinearOrder \u03b9\ninst\u271d\u2076 : MeasurableSpace \u03b9\ninst\u271d\u2075 : TopologicalSpace \u03b9\ninst\u271d\u2074 : OrderTopology \u03b9\ninst\u271d\u00b3 : SecondCountableTopology \u03b9\ninst\u271d\u00b2 : BorelSpace \u03b9\ninst\u271d\u00b9 : TopologicalSpace \u03b2\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\n\u03c4 : \u03a9 \u2192 \u03b9\nf : Filtration \u03b9 m\ninst\u271d : MetrizableSpace \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\ni : \u03b9\nm_prod : MeasurableSpace (\u2191(Set.Iic i) \u00d7 \u03a9) := MeasurableSpace.prod Subtype.instMeasurableSpace (\u2191f i)\nm_set : (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)) \u2192 MeasurableSpace \u2191t := fun x => Subtype.instMeasurableSpace\ns : Set (\u2191(Set.Iic i) \u00d7 \u03a9) := {p | \u03c4 p.2 \u2264 i}\nhs : MeasurableSet s\nh_meas_fst : \u2200 (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)), Measurable fun x => \u2191(\u2191x).1\n\u22a2 Measurable (Set.restrict s fun p => (fun i \u03c9 => min i (\u03c4 \u03c9)) (\u2191p.1) p.2)\n\ncase hf.refine'_2\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2077 : LinearOrder \u03b9\ninst\u271d\u2076 : MeasurableSpace \u03b9\ninst\u271d\u2075 : TopologicalSpace \u03b9\ninst\u271d\u2074 : OrderTopology \u03b9\ninst\u271d\u00b3 : SecondCountableTopology \u03b9\ninst\u271d\u00b2 : BorelSpace \u03b9\ninst\u271d\u00b9 : TopologicalSpace \u03b2\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\n\u03c4 : \u03a9 \u2192 \u03b9\nf : Filtration \u03b9 m\ninst\u271d : MetrizableSpace \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\ni : \u03b9\nm_prod : MeasurableSpace (\u2191(Set.Iic i) \u00d7 \u03a9) := MeasurableSpace.prod Subtype.instMeasurableSpace (\u2191f i)\nm_set : (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)) \u2192 MeasurableSpace \u2191t := fun x => Subtype.instMeasurableSpace\ns : Set (\u2191(Set.Iic i) \u00d7 \u03a9) := {p | \u03c4 p.2 \u2264 i}\nhs : MeasurableSet s\nh_meas_fst : \u2200 (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)), Measurable fun x => \u2191(\u2191x).1\n\u22a2 Measurable (Set.restrict s\u1d9c fun p => (fun i \u03c9 => min i (\u03c4 \u03c9)) (\u2191p.1) p.2)"}, {"tactic": "refine @Measurable.min _ _ _ _ _ (m_set s) _ _ _ _ _ (h_meas_fst s) ?_", "annotated_tactic": ["refine @<a>Measurable.min</a> _ _ _ _ _ (m_set s) _ _ _ _ _ (h_meas_fst s) ?_", [{"full_name": "Measurable.min", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [875, 9], "def_end_pos": [875, 23]}]], "state_before": "case hf.refine'_1\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2077 : LinearOrder \u03b9\ninst\u271d\u2076 : MeasurableSpace \u03b9\ninst\u271d\u2075 : TopologicalSpace \u03b9\ninst\u271d\u2074 : OrderTopology \u03b9\ninst\u271d\u00b3 : SecondCountableTopology \u03b9\ninst\u271d\u00b2 : BorelSpace \u03b9\ninst\u271d\u00b9 : TopologicalSpace \u03b2\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\n\u03c4 : \u03a9 \u2192 \u03b9\nf : Filtration \u03b9 m\ninst\u271d : MetrizableSpace \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\ni : \u03b9\nm_prod : MeasurableSpace (\u2191(Set.Iic i) \u00d7 \u03a9) := MeasurableSpace.prod Subtype.instMeasurableSpace (\u2191f i)\nm_set : (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)) \u2192 MeasurableSpace \u2191t := fun x => Subtype.instMeasurableSpace\ns : Set (\u2191(Set.Iic i) \u00d7 \u03a9) := {p | \u03c4 p.2 \u2264 i}\nhs : MeasurableSet s\nh_meas_fst : \u2200 (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)), Measurable fun x => \u2191(\u2191x).1\n\u22a2 Measurable (Set.restrict s fun p => (fun i \u03c9 => min i (\u03c4 \u03c9)) (\u2191p.1) p.2)", "state_after": "case hf.refine'_1\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2077 : LinearOrder \u03b9\ninst\u271d\u2076 : MeasurableSpace \u03b9\ninst\u271d\u2075 : TopologicalSpace \u03b9\ninst\u271d\u2074 : OrderTopology \u03b9\ninst\u271d\u00b3 : SecondCountableTopology \u03b9\ninst\u271d\u00b2 : BorelSpace \u03b9\ninst\u271d\u00b9 : TopologicalSpace \u03b2\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\n\u03c4 : \u03a9 \u2192 \u03b9\nf : Filtration \u03b9 m\ninst\u271d : MetrizableSpace \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\ni : \u03b9\nm_prod : MeasurableSpace (\u2191(Set.Iic i) \u00d7 \u03a9) := MeasurableSpace.prod Subtype.instMeasurableSpace (\u2191f i)\nm_set : (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)) \u2192 MeasurableSpace \u2191t := fun x => Subtype.instMeasurableSpace\ns : Set (\u2191(Set.Iic i) \u00d7 \u03a9) := {p | \u03c4 p.2 \u2264 i}\nhs : MeasurableSet s\nh_meas_fst : \u2200 (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)), Measurable fun x => \u2191(\u2191x).1\n\u22a2 Measurable fun a => \u03c4 (\u2191a).2"}, {"tactic": "refine' @measurable_of_Iic \u03b9 s _ _ _ (m_set s) _ _ _ _ fun j => _", "annotated_tactic": ["refine' @<a>measurable_of_Iic</a> \u03b9 s _ _ _ (m_set s) _ _ _ _ fun j => _", [{"full_name": "measurable_of_Iic", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [1131, 9], "def_end_pos": [1131, 26]}]], "state_before": "case hf.refine'_1\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2077 : LinearOrder \u03b9\ninst\u271d\u2076 : MeasurableSpace \u03b9\ninst\u271d\u2075 : TopologicalSpace \u03b9\ninst\u271d\u2074 : OrderTopology \u03b9\ninst\u271d\u00b3 : SecondCountableTopology \u03b9\ninst\u271d\u00b2 : BorelSpace \u03b9\ninst\u271d\u00b9 : TopologicalSpace \u03b2\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\n\u03c4 : \u03a9 \u2192 \u03b9\nf : Filtration \u03b9 m\ninst\u271d : MetrizableSpace \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\ni : \u03b9\nm_prod : MeasurableSpace (\u2191(Set.Iic i) \u00d7 \u03a9) := MeasurableSpace.prod Subtype.instMeasurableSpace (\u2191f i)\nm_set : (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)) \u2192 MeasurableSpace \u2191t := fun x => Subtype.instMeasurableSpace\ns : Set (\u2191(Set.Iic i) \u00d7 \u03a9) := {p | \u03c4 p.2 \u2264 i}\nhs : MeasurableSet s\nh_meas_fst : \u2200 (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)), Measurable fun x => \u2191(\u2191x).1\n\u22a2 Measurable fun a => \u03c4 (\u2191a).2", "state_after": "case hf.refine'_1\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2077 : LinearOrder \u03b9\ninst\u271d\u2076 : MeasurableSpace \u03b9\ninst\u271d\u2075 : TopologicalSpace \u03b9\ninst\u271d\u2074 : OrderTopology \u03b9\ninst\u271d\u00b3 : SecondCountableTopology \u03b9\ninst\u271d\u00b2 : BorelSpace \u03b9\ninst\u271d\u00b9 : TopologicalSpace \u03b2\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\n\u03c4 : \u03a9 \u2192 \u03b9\nf : Filtration \u03b9 m\ninst\u271d : MetrizableSpace \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\ni : \u03b9\nm_prod : MeasurableSpace (\u2191(Set.Iic i) \u00d7 \u03a9) := MeasurableSpace.prod Subtype.instMeasurableSpace (\u2191f i)\nm_set : (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)) \u2192 MeasurableSpace \u2191t := fun x => Subtype.instMeasurableSpace\ns : Set (\u2191(Set.Iic i) \u00d7 \u03a9) := {p | \u03c4 p.2 \u2264 i}\nhs : MeasurableSet s\nh_meas_fst : \u2200 (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)), Measurable fun x => \u2191(\u2191x).1\nj : \u03b9\n\u22a2 MeasurableSet ((fun a => \u03c4 (\u2191a).2) \u207b\u00b9' Set.Iic j)"}, {"tactic": "have h_set_eq : (fun x : s => \u03c4 (x : Set.Iic i \u00d7 \u03a9).snd) \u207b\u00b9' Set.Iic j =\n    (fun x : s => (x : Set.Iic i \u00d7 \u03a9).snd) \u207b\u00b9' {\u03c9 | \u03c4 \u03c9 \u2264 min i j} := by\n  ext1 \u03c9\n  simp only [Set.mem_preimage, Set.mem_Iic, iff_and_self, le_min_iff, Set.mem_setOf_eq]\n  exact fun _ => \u03c9.prop", "annotated_tactic": ["have h_set_eq : (fun x : s => \u03c4 (x : <a>Set.Iic</a> i \u00d7 \u03a9).<a>snd</a>) \u207b\u00b9' <a>Set.Iic</a> j =\n        (fun x : s => (x : <a>Set.Iic</a> i \u00d7 \u03a9).<a>snd</a>) \u207b\u00b9' {\u03c9 | \u03c4 \u03c9 \u2264 <a>min</a> i j} := by\n      ext1 \u03c9\n      simp only [<a>Set.mem_preimage</a>, <a>Set.mem_Iic</a>, <a>iff_and_self</a>, <a>le_min_iff</a>, <a>Set.mem_setOf_eq</a>]\n      exact fun _ => \u03c9.prop", [{"full_name": "Set.Iic", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [64, 5], "def_end_pos": [64, 8]}, {"full_name": "Prod.snd", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [470, 3], "def_end_pos": [470, 6]}, {"full_name": "Set.Iic", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [64, 5], "def_end_pos": [64, 8]}, {"full_name": "Set.Iic", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [64, 5], "def_end_pos": [64, 8]}, {"full_name": "Prod.snd", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [470, 3], "def_end_pos": [470, 6]}, {"full_name": "Min.min", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1103, 3], "def_end_pos": [1103, 6]}, {"full_name": "Set.mem_preimage", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [64, 9], "def_end_pos": [64, 21]}, {"full_name": "Set.mem_Iic", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [136, 9], "def_end_pos": [136, 16]}, {"full_name": "iff_and_self", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [217, 17], "def_end_pos": [217, 29]}, {"full_name": "le_min_iff", "def_path": "Mathlib/Order/MinMax.lean", "def_pos": [33, 9], "def_end_pos": [33, 19]}, {"full_name": "Set.mem_setOf_eq", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [256, 29], "def_end_pos": [256, 41]}]], "state_before": "case hf.refine'_1\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2077 : LinearOrder \u03b9\ninst\u271d\u2076 : MeasurableSpace \u03b9\ninst\u271d\u2075 : TopologicalSpace \u03b9\ninst\u271d\u2074 : OrderTopology \u03b9\ninst\u271d\u00b3 : SecondCountableTopology \u03b9\ninst\u271d\u00b2 : BorelSpace \u03b9\ninst\u271d\u00b9 : TopologicalSpace \u03b2\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\n\u03c4 : \u03a9 \u2192 \u03b9\nf : Filtration \u03b9 m\ninst\u271d : MetrizableSpace \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\ni : \u03b9\nm_prod : MeasurableSpace (\u2191(Set.Iic i) \u00d7 \u03a9) := MeasurableSpace.prod Subtype.instMeasurableSpace (\u2191f i)\nm_set : (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)) \u2192 MeasurableSpace \u2191t := fun x => Subtype.instMeasurableSpace\ns : Set (\u2191(Set.Iic i) \u00d7 \u03a9) := {p | \u03c4 p.2 \u2264 i}\nhs : MeasurableSet s\nh_meas_fst : \u2200 (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)), Measurable fun x => \u2191(\u2191x).1\nj : \u03b9\n\u22a2 MeasurableSet ((fun a => \u03c4 (\u2191a).2) \u207b\u00b9' Set.Iic j)", "state_after": "case hf.refine'_1\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2077 : LinearOrder \u03b9\ninst\u271d\u2076 : MeasurableSpace \u03b9\ninst\u271d\u2075 : TopologicalSpace \u03b9\ninst\u271d\u2074 : OrderTopology \u03b9\ninst\u271d\u00b3 : SecondCountableTopology \u03b9\ninst\u271d\u00b2 : BorelSpace \u03b9\ninst\u271d\u00b9 : TopologicalSpace \u03b2\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\n\u03c4 : \u03a9 \u2192 \u03b9\nf : Filtration \u03b9 m\ninst\u271d : MetrizableSpace \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\ni : \u03b9\nm_prod : MeasurableSpace (\u2191(Set.Iic i) \u00d7 \u03a9) := MeasurableSpace.prod Subtype.instMeasurableSpace (\u2191f i)\nm_set : (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)) \u2192 MeasurableSpace \u2191t := fun x => Subtype.instMeasurableSpace\ns : Set (\u2191(Set.Iic i) \u00d7 \u03a9) := {p | \u03c4 p.2 \u2264 i}\nhs : MeasurableSet s\nh_meas_fst : \u2200 (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)), Measurable fun x => \u2191(\u2191x).1\nj : \u03b9\nh_set_eq : (fun x => \u03c4 (\u2191x).2) \u207b\u00b9' Set.Iic j = (fun x => (\u2191x).2) \u207b\u00b9' {\u03c9 | \u03c4 \u03c9 \u2264 min i j}\n\u22a2 MeasurableSet ((fun a => \u03c4 (\u2191a).2) \u207b\u00b9' Set.Iic j)"}, {"tactic": "rw [h_set_eq]", "annotated_tactic": ["rw [h_set_eq]", []], "state_before": "case hf.refine'_1\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2077 : LinearOrder \u03b9\ninst\u271d\u2076 : MeasurableSpace \u03b9\ninst\u271d\u2075 : TopologicalSpace \u03b9\ninst\u271d\u2074 : OrderTopology \u03b9\ninst\u271d\u00b3 : SecondCountableTopology \u03b9\ninst\u271d\u00b2 : BorelSpace \u03b9\ninst\u271d\u00b9 : TopologicalSpace \u03b2\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\n\u03c4 : \u03a9 \u2192 \u03b9\nf : Filtration \u03b9 m\ninst\u271d : MetrizableSpace \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\ni : \u03b9\nm_prod : MeasurableSpace (\u2191(Set.Iic i) \u00d7 \u03a9) := MeasurableSpace.prod Subtype.instMeasurableSpace (\u2191f i)\nm_set : (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)) \u2192 MeasurableSpace \u2191t := fun x => Subtype.instMeasurableSpace\ns : Set (\u2191(Set.Iic i) \u00d7 \u03a9) := {p | \u03c4 p.2 \u2264 i}\nhs : MeasurableSet s\nh_meas_fst : \u2200 (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)), Measurable fun x => \u2191(\u2191x).1\nj : \u03b9\nh_set_eq : (fun x => \u03c4 (\u2191x).2) \u207b\u00b9' Set.Iic j = (fun x => (\u2191x).2) \u207b\u00b9' {\u03c9 | \u03c4 \u03c9 \u2264 min i j}\n\u22a2 MeasurableSet ((fun a => \u03c4 (\u2191a).2) \u207b\u00b9' Set.Iic j)", "state_after": "case hf.refine'_1\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2077 : LinearOrder \u03b9\ninst\u271d\u2076 : MeasurableSpace \u03b9\ninst\u271d\u2075 : TopologicalSpace \u03b9\ninst\u271d\u2074 : OrderTopology \u03b9\ninst\u271d\u00b3 : SecondCountableTopology \u03b9\ninst\u271d\u00b2 : BorelSpace \u03b9\ninst\u271d\u00b9 : TopologicalSpace \u03b2\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\n\u03c4 : \u03a9 \u2192 \u03b9\nf : Filtration \u03b9 m\ninst\u271d : MetrizableSpace \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\ni : \u03b9\nm_prod : MeasurableSpace (\u2191(Set.Iic i) \u00d7 \u03a9) := MeasurableSpace.prod Subtype.instMeasurableSpace (\u2191f i)\nm_set : (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)) \u2192 MeasurableSpace \u2191t := fun x => Subtype.instMeasurableSpace\ns : Set (\u2191(Set.Iic i) \u00d7 \u03a9) := {p | \u03c4 p.2 \u2264 i}\nhs : MeasurableSet s\nh_meas_fst : \u2200 (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)), Measurable fun x => \u2191(\u2191x).1\nj : \u03b9\nh_set_eq : (fun x => \u03c4 (\u2191x).2) \u207b\u00b9' Set.Iic j = (fun x => (\u2191x).2) \u207b\u00b9' {\u03c9 | \u03c4 \u03c9 \u2264 min i j}\n\u22a2 MeasurableSet ((fun x => (\u2191x).2) \u207b\u00b9' {\u03c9 | \u03c4 \u03c9 \u2264 min i j})"}, {"tactic": "suffices h_meas : @Measurable _ _ (m_set s) (f i) fun x : s => (x : Set.Iic i \u00d7 \u03a9).snd", "annotated_tactic": ["suffices h_meas : @<a>Measurable</a> _ _ (m_set s) (f i) fun x : s => (x : <a>Set.Iic</a> i \u00d7 \u03a9).<a>snd</a>", [{"full_name": "Measurable", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [535, 5], "def_end_pos": [535, 15]}, {"full_name": "Set.Iic", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [64, 5], "def_end_pos": [64, 8]}, {"full_name": "Prod.snd", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [470, 3], "def_end_pos": [470, 6]}]], "state_before": "case hf.refine'_1\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2077 : LinearOrder \u03b9\ninst\u271d\u2076 : MeasurableSpace \u03b9\ninst\u271d\u2075 : TopologicalSpace \u03b9\ninst\u271d\u2074 : OrderTopology \u03b9\ninst\u271d\u00b3 : SecondCountableTopology \u03b9\ninst\u271d\u00b2 : BorelSpace \u03b9\ninst\u271d\u00b9 : TopologicalSpace \u03b2\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\n\u03c4 : \u03a9 \u2192 \u03b9\nf : Filtration \u03b9 m\ninst\u271d : MetrizableSpace \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\ni : \u03b9\nm_prod : MeasurableSpace (\u2191(Set.Iic i) \u00d7 \u03a9) := MeasurableSpace.prod Subtype.instMeasurableSpace (\u2191f i)\nm_set : (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)) \u2192 MeasurableSpace \u2191t := fun x => Subtype.instMeasurableSpace\ns : Set (\u2191(Set.Iic i) \u00d7 \u03a9) := {p | \u03c4 p.2 \u2264 i}\nhs : MeasurableSet s\nh_meas_fst : \u2200 (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)), Measurable fun x => \u2191(\u2191x).1\nj : \u03b9\nh_set_eq : (fun x => \u03c4 (\u2191x).2) \u207b\u00b9' Set.Iic j = (fun x => (\u2191x).2) \u207b\u00b9' {\u03c9 | \u03c4 \u03c9 \u2264 min i j}\n\u22a2 MeasurableSet ((fun x => (\u2191x).2) \u207b\u00b9' {\u03c9 | \u03c4 \u03c9 \u2264 min i j})", "state_after": "case hf.refine'_1\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2077 : LinearOrder \u03b9\ninst\u271d\u2076 : MeasurableSpace \u03b9\ninst\u271d\u2075 : TopologicalSpace \u03b9\ninst\u271d\u2074 : OrderTopology \u03b9\ninst\u271d\u00b3 : SecondCountableTopology \u03b9\ninst\u271d\u00b2 : BorelSpace \u03b9\ninst\u271d\u00b9 : TopologicalSpace \u03b2\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\n\u03c4 : \u03a9 \u2192 \u03b9\nf : Filtration \u03b9 m\ninst\u271d : MetrizableSpace \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\ni : \u03b9\nm_prod : MeasurableSpace (\u2191(Set.Iic i) \u00d7 \u03a9) := MeasurableSpace.prod Subtype.instMeasurableSpace (\u2191f i)\nm_set : (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)) \u2192 MeasurableSpace \u2191t := fun x => Subtype.instMeasurableSpace\ns : Set (\u2191(Set.Iic i) \u00d7 \u03a9) := {p | \u03c4 p.2 \u2264 i}\nhs : MeasurableSet s\nh_meas_fst : \u2200 (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)), Measurable fun x => \u2191(\u2191x).1\nj : \u03b9\nh_set_eq : (fun x => \u03c4 (\u2191x).2) \u207b\u00b9' Set.Iic j = (fun x => (\u2191x).2) \u207b\u00b9' {\u03c9 | \u03c4 \u03c9 \u2264 min i j}\nh_meas : Measurable fun x => (\u2191x).2\n\u22a2 MeasurableSet ((fun x => (\u2191x).2) \u207b\u00b9' {\u03c9 | \u03c4 \u03c9 \u2264 min i j})\n\ncase h_meas\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2077 : LinearOrder \u03b9\ninst\u271d\u2076 : MeasurableSpace \u03b9\ninst\u271d\u2075 : TopologicalSpace \u03b9\ninst\u271d\u2074 : OrderTopology \u03b9\ninst\u271d\u00b3 : SecondCountableTopology \u03b9\ninst\u271d\u00b2 : BorelSpace \u03b9\ninst\u271d\u00b9 : TopologicalSpace \u03b2\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\n\u03c4 : \u03a9 \u2192 \u03b9\nf : Filtration \u03b9 m\ninst\u271d : MetrizableSpace \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\ni : \u03b9\nm_prod : MeasurableSpace (\u2191(Set.Iic i) \u00d7 \u03a9) := MeasurableSpace.prod Subtype.instMeasurableSpace (\u2191f i)\nm_set : (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)) \u2192 MeasurableSpace \u2191t := fun x => Subtype.instMeasurableSpace\ns : Set (\u2191(Set.Iic i) \u00d7 \u03a9) := {p | \u03c4 p.2 \u2264 i}\nhs : MeasurableSet s\nh_meas_fst : \u2200 (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)), Measurable fun x => \u2191(\u2191x).1\nj : \u03b9\nh_set_eq : (fun x => \u03c4 (\u2191x).2) \u207b\u00b9' Set.Iic j = (fun x => (\u2191x).2) \u207b\u00b9' {\u03c9 | \u03c4 \u03c9 \u2264 min i j}\n\u22a2 Measurable fun x => (\u2191x).2"}, {"tactic": "exact h_meas (f.mono (min_le_left _ _) _ (h\u03c4.measurableSet_le (min i j)))", "annotated_tactic": ["exact h_meas (f.mono (<a>min_le_left</a> _ _) _ (h\u03c4.measurableSet_le (<a>min</a> i j)))", [{"full_name": "min_le_left", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [33, 9], "def_end_pos": [33, 20]}, {"full_name": "Min.min", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1103, 3], "def_end_pos": [1103, 6]}]], "state_before": "case hf.refine'_1\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2077 : LinearOrder \u03b9\ninst\u271d\u2076 : MeasurableSpace \u03b9\ninst\u271d\u2075 : TopologicalSpace \u03b9\ninst\u271d\u2074 : OrderTopology \u03b9\ninst\u271d\u00b3 : SecondCountableTopology \u03b9\ninst\u271d\u00b2 : BorelSpace \u03b9\ninst\u271d\u00b9 : TopologicalSpace \u03b2\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\n\u03c4 : \u03a9 \u2192 \u03b9\nf : Filtration \u03b9 m\ninst\u271d : MetrizableSpace \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\ni : \u03b9\nm_prod : MeasurableSpace (\u2191(Set.Iic i) \u00d7 \u03a9) := MeasurableSpace.prod Subtype.instMeasurableSpace (\u2191f i)\nm_set : (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)) \u2192 MeasurableSpace \u2191t := fun x => Subtype.instMeasurableSpace\ns : Set (\u2191(Set.Iic i) \u00d7 \u03a9) := {p | \u03c4 p.2 \u2264 i}\nhs : MeasurableSet s\nh_meas_fst : \u2200 (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)), Measurable fun x => \u2191(\u2191x).1\nj : \u03b9\nh_set_eq : (fun x => \u03c4 (\u2191x).2) \u207b\u00b9' Set.Iic j = (fun x => (\u2191x).2) \u207b\u00b9' {\u03c9 | \u03c4 \u03c9 \u2264 min i j}\nh_meas : Measurable fun x => (\u2191x).2\n\u22a2 MeasurableSet ((fun x => (\u2191x).2) \u207b\u00b9' {\u03c9 | \u03c4 \u03c9 \u2264 min i j})\n\ncase h_meas\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2077 : LinearOrder \u03b9\ninst\u271d\u2076 : MeasurableSpace \u03b9\ninst\u271d\u2075 : TopologicalSpace \u03b9\ninst\u271d\u2074 : OrderTopology \u03b9\ninst\u271d\u00b3 : SecondCountableTopology \u03b9\ninst\u271d\u00b2 : BorelSpace \u03b9\ninst\u271d\u00b9 : TopologicalSpace \u03b2\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\n\u03c4 : \u03a9 \u2192 \u03b9\nf : Filtration \u03b9 m\ninst\u271d : MetrizableSpace \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\ni : \u03b9\nm_prod : MeasurableSpace (\u2191(Set.Iic i) \u00d7 \u03a9) := MeasurableSpace.prod Subtype.instMeasurableSpace (\u2191f i)\nm_set : (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)) \u2192 MeasurableSpace \u2191t := fun x => Subtype.instMeasurableSpace\ns : Set (\u2191(Set.Iic i) \u00d7 \u03a9) := {p | \u03c4 p.2 \u2264 i}\nhs : MeasurableSet s\nh_meas_fst : \u2200 (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)), Measurable fun x => \u2191(\u2191x).1\nj : \u03b9\nh_set_eq : (fun x => \u03c4 (\u2191x).2) \u207b\u00b9' Set.Iic j = (fun x => (\u2191x).2) \u207b\u00b9' {\u03c9 | \u03c4 \u03c9 \u2264 min i j}\n\u22a2 Measurable fun x => (\u2191x).2", "state_after": "case h_meas\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2077 : LinearOrder \u03b9\ninst\u271d\u2076 : MeasurableSpace \u03b9\ninst\u271d\u2075 : TopologicalSpace \u03b9\ninst\u271d\u2074 : OrderTopology \u03b9\ninst\u271d\u00b3 : SecondCountableTopology \u03b9\ninst\u271d\u00b2 : BorelSpace \u03b9\ninst\u271d\u00b9 : TopologicalSpace \u03b2\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\n\u03c4 : \u03a9 \u2192 \u03b9\nf : Filtration \u03b9 m\ninst\u271d : MetrizableSpace \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\ni : \u03b9\nm_prod : MeasurableSpace (\u2191(Set.Iic i) \u00d7 \u03a9) := MeasurableSpace.prod Subtype.instMeasurableSpace (\u2191f i)\nm_set : (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)) \u2192 MeasurableSpace \u2191t := fun x => Subtype.instMeasurableSpace\ns : Set (\u2191(Set.Iic i) \u00d7 \u03a9) := {p | \u03c4 p.2 \u2264 i}\nhs : MeasurableSet s\nh_meas_fst : \u2200 (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)), Measurable fun x => \u2191(\u2191x).1\nj : \u03b9\nh_set_eq : (fun x => \u03c4 (\u2191x).2) \u207b\u00b9' Set.Iic j = (fun x => (\u2191x).2) \u207b\u00b9' {\u03c9 | \u03c4 \u03c9 \u2264 min i j}\n\u22a2 Measurable fun x => (\u2191x).2"}, {"tactic": "exact measurable_snd.comp (@measurable_subtype_coe _ m_prod _)", "annotated_tactic": ["exact measurable_snd.comp (@<a>measurable_subtype_coe</a> _ m_prod _)", [{"full_name": "measurable_subtype_coe", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [577, 9], "def_end_pos": [577, 31]}]], "state_before": "case h_meas\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2077 : LinearOrder \u03b9\ninst\u271d\u2076 : MeasurableSpace \u03b9\ninst\u271d\u2075 : TopologicalSpace \u03b9\ninst\u271d\u2074 : OrderTopology \u03b9\ninst\u271d\u00b3 : SecondCountableTopology \u03b9\ninst\u271d\u00b2 : BorelSpace \u03b9\ninst\u271d\u00b9 : TopologicalSpace \u03b2\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\n\u03c4 : \u03a9 \u2192 \u03b9\nf : Filtration \u03b9 m\ninst\u271d : MetrizableSpace \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\ni : \u03b9\nm_prod : MeasurableSpace (\u2191(Set.Iic i) \u00d7 \u03a9) := MeasurableSpace.prod Subtype.instMeasurableSpace (\u2191f i)\nm_set : (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)) \u2192 MeasurableSpace \u2191t := fun x => Subtype.instMeasurableSpace\ns : Set (\u2191(Set.Iic i) \u00d7 \u03a9) := {p | \u03c4 p.2 \u2264 i}\nhs : MeasurableSet s\nh_meas_fst : \u2200 (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)), Measurable fun x => \u2191(\u2191x).1\nj : \u03b9\nh_set_eq : (fun x => \u03c4 (\u2191x).2) \u207b\u00b9' Set.Iic j = (fun x => (\u2191x).2) \u207b\u00b9' {\u03c9 | \u03c4 \u03c9 \u2264 min i j}\n\u22a2 Measurable fun x => (\u2191x).2", "state_after": "no goals"}, {"tactic": "ext1 \u03c9", "annotated_tactic": ["ext1 \u03c9", []], "state_before": "\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2077 : LinearOrder \u03b9\ninst\u271d\u2076 : MeasurableSpace \u03b9\ninst\u271d\u2075 : TopologicalSpace \u03b9\ninst\u271d\u2074 : OrderTopology \u03b9\ninst\u271d\u00b3 : SecondCountableTopology \u03b9\ninst\u271d\u00b2 : BorelSpace \u03b9\ninst\u271d\u00b9 : TopologicalSpace \u03b2\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\n\u03c4 : \u03a9 \u2192 \u03b9\nf : Filtration \u03b9 m\ninst\u271d : MetrizableSpace \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\ni : \u03b9\nm_prod : MeasurableSpace (\u2191(Set.Iic i) \u00d7 \u03a9) := MeasurableSpace.prod Subtype.instMeasurableSpace (\u2191f i)\nm_set : (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)) \u2192 MeasurableSpace \u2191t := fun x => Subtype.instMeasurableSpace\ns : Set (\u2191(Set.Iic i) \u00d7 \u03a9) := {p | \u03c4 p.2 \u2264 i}\nhs : MeasurableSet s\nh_meas_fst : \u2200 (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)), Measurable fun x => \u2191(\u2191x).1\nj : \u03b9\n\u22a2 (fun x => \u03c4 (\u2191x).2) \u207b\u00b9' Set.Iic j = (fun x => (\u2191x).2) \u207b\u00b9' {\u03c9 | \u03c4 \u03c9 \u2264 min i j}", "state_after": "case h\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2077 : LinearOrder \u03b9\ninst\u271d\u2076 : MeasurableSpace \u03b9\ninst\u271d\u2075 : TopologicalSpace \u03b9\ninst\u271d\u2074 : OrderTopology \u03b9\ninst\u271d\u00b3 : SecondCountableTopology \u03b9\ninst\u271d\u00b2 : BorelSpace \u03b9\ninst\u271d\u00b9 : TopologicalSpace \u03b2\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\n\u03c4 : \u03a9 \u2192 \u03b9\nf : Filtration \u03b9 m\ninst\u271d : MetrizableSpace \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\ni : \u03b9\nm_prod : MeasurableSpace (\u2191(Set.Iic i) \u00d7 \u03a9) := MeasurableSpace.prod Subtype.instMeasurableSpace (\u2191f i)\nm_set : (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)) \u2192 MeasurableSpace \u2191t := fun x => Subtype.instMeasurableSpace\ns : Set (\u2191(Set.Iic i) \u00d7 \u03a9) := {p | \u03c4 p.2 \u2264 i}\nhs : MeasurableSet s\nh_meas_fst : \u2200 (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)), Measurable fun x => \u2191(\u2191x).1\nj : \u03b9\n\u03c9 : \u2191s\n\u22a2 \u03c9 \u2208 (fun x => \u03c4 (\u2191x).2) \u207b\u00b9' Set.Iic j \u2194 \u03c9 \u2208 (fun x => (\u2191x).2) \u207b\u00b9' {\u03c9 | \u03c4 \u03c9 \u2264 min i j}"}, {"tactic": "simp only [Set.mem_preimage, Set.mem_Iic, iff_and_self, le_min_iff, Set.mem_setOf_eq]", "annotated_tactic": ["simp only [<a>Set.mem_preimage</a>, <a>Set.mem_Iic</a>, <a>iff_and_self</a>, <a>le_min_iff</a>, <a>Set.mem_setOf_eq</a>]", [{"full_name": "Set.mem_preimage", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [64, 9], "def_end_pos": [64, 21]}, {"full_name": "Set.mem_Iic", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [136, 9], "def_end_pos": [136, 16]}, {"full_name": "iff_and_self", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [217, 17], "def_end_pos": [217, 29]}, {"full_name": "le_min_iff", "def_path": "Mathlib/Order/MinMax.lean", "def_pos": [33, 9], "def_end_pos": [33, 19]}, {"full_name": "Set.mem_setOf_eq", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [256, 29], "def_end_pos": [256, 41]}]], "state_before": "case h\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2077 : LinearOrder \u03b9\ninst\u271d\u2076 : MeasurableSpace \u03b9\ninst\u271d\u2075 : TopologicalSpace \u03b9\ninst\u271d\u2074 : OrderTopology \u03b9\ninst\u271d\u00b3 : SecondCountableTopology \u03b9\ninst\u271d\u00b2 : BorelSpace \u03b9\ninst\u271d\u00b9 : TopologicalSpace \u03b2\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\n\u03c4 : \u03a9 \u2192 \u03b9\nf : Filtration \u03b9 m\ninst\u271d : MetrizableSpace \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\ni : \u03b9\nm_prod : MeasurableSpace (\u2191(Set.Iic i) \u00d7 \u03a9) := MeasurableSpace.prod Subtype.instMeasurableSpace (\u2191f i)\nm_set : (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)) \u2192 MeasurableSpace \u2191t := fun x => Subtype.instMeasurableSpace\ns : Set (\u2191(Set.Iic i) \u00d7 \u03a9) := {p | \u03c4 p.2 \u2264 i}\nhs : MeasurableSet s\nh_meas_fst : \u2200 (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)), Measurable fun x => \u2191(\u2191x).1\nj : \u03b9\n\u03c9 : \u2191s\n\u22a2 \u03c9 \u2208 (fun x => \u03c4 (\u2191x).2) \u207b\u00b9' Set.Iic j \u2194 \u03c9 \u2208 (fun x => (\u2191x).2) \u207b\u00b9' {\u03c9 | \u03c4 \u03c9 \u2264 min i j}", "state_after": "case h\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2077 : LinearOrder \u03b9\ninst\u271d\u2076 : MeasurableSpace \u03b9\ninst\u271d\u2075 : TopologicalSpace \u03b9\ninst\u271d\u2074 : OrderTopology \u03b9\ninst\u271d\u00b3 : SecondCountableTopology \u03b9\ninst\u271d\u00b2 : BorelSpace \u03b9\ninst\u271d\u00b9 : TopologicalSpace \u03b2\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\n\u03c4 : \u03a9 \u2192 \u03b9\nf : Filtration \u03b9 m\ninst\u271d : MetrizableSpace \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\ni : \u03b9\nm_prod : MeasurableSpace (\u2191(Set.Iic i) \u00d7 \u03a9) := MeasurableSpace.prod Subtype.instMeasurableSpace (\u2191f i)\nm_set : (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)) \u2192 MeasurableSpace \u2191t := fun x => Subtype.instMeasurableSpace\ns : Set (\u2191(Set.Iic i) \u00d7 \u03a9) := {p | \u03c4 p.2 \u2264 i}\nhs : MeasurableSet s\nh_meas_fst : \u2200 (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)), Measurable fun x => \u2191(\u2191x).1\nj : \u03b9\n\u03c9 : \u2191s\n\u22a2 \u03c4 (\u2191\u03c9).2 \u2264 j \u2192 \u03c4 (\u2191\u03c9).2 \u2264 i"}, {"tactic": "exact fun _ => \u03c9.prop", "annotated_tactic": ["exact fun _ => \u03c9.prop", []], "state_before": "case h\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2077 : LinearOrder \u03b9\ninst\u271d\u2076 : MeasurableSpace \u03b9\ninst\u271d\u2075 : TopologicalSpace \u03b9\ninst\u271d\u2074 : OrderTopology \u03b9\ninst\u271d\u00b3 : SecondCountableTopology \u03b9\ninst\u271d\u00b2 : BorelSpace \u03b9\ninst\u271d\u00b9 : TopologicalSpace \u03b2\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\n\u03c4 : \u03a9 \u2192 \u03b9\nf : Filtration \u03b9 m\ninst\u271d : MetrizableSpace \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\ni : \u03b9\nm_prod : MeasurableSpace (\u2191(Set.Iic i) \u00d7 \u03a9) := MeasurableSpace.prod Subtype.instMeasurableSpace (\u2191f i)\nm_set : (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)) \u2192 MeasurableSpace \u2191t := fun x => Subtype.instMeasurableSpace\ns : Set (\u2191(Set.Iic i) \u00d7 \u03a9) := {p | \u03c4 p.2 \u2264 i}\nhs : MeasurableSet s\nh_meas_fst : \u2200 (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)), Measurable fun x => \u2191(\u2191x).1\nj : \u03b9\n\u03c9 : \u2191s\n\u22a2 \u03c4 (\u2191\u03c9).2 \u2264 j \u2192 \u03c4 (\u2191\u03c9).2 \u2264 i", "state_after": "no goals"}, {"tactic": "letI sc := s\u1d9c", "annotated_tactic": ["letI sc := s\u1d9c", []], "state_before": "case hf.refine'_2\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2077 : LinearOrder \u03b9\ninst\u271d\u2076 : MeasurableSpace \u03b9\ninst\u271d\u2075 : TopologicalSpace \u03b9\ninst\u271d\u2074 : OrderTopology \u03b9\ninst\u271d\u00b3 : SecondCountableTopology \u03b9\ninst\u271d\u00b2 : BorelSpace \u03b9\ninst\u271d\u00b9 : TopologicalSpace \u03b2\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\n\u03c4 : \u03a9 \u2192 \u03b9\nf : Filtration \u03b9 m\ninst\u271d : MetrizableSpace \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\ni : \u03b9\nm_prod : MeasurableSpace (\u2191(Set.Iic i) \u00d7 \u03a9) := MeasurableSpace.prod Subtype.instMeasurableSpace (\u2191f i)\nm_set : (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)) \u2192 MeasurableSpace \u2191t := fun x => Subtype.instMeasurableSpace\ns : Set (\u2191(Set.Iic i) \u00d7 \u03a9) := {p | \u03c4 p.2 \u2264 i}\nhs : MeasurableSet s\nh_meas_fst : \u2200 (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)), Measurable fun x => \u2191(\u2191x).1\n\u22a2 Measurable (Set.restrict s\u1d9c fun p => (fun i \u03c9 => min i (\u03c4 \u03c9)) (\u2191p.1) p.2)", "state_after": "case hf.refine'_2\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2077 : LinearOrder \u03b9\ninst\u271d\u2076 : MeasurableSpace \u03b9\ninst\u271d\u2075 : TopologicalSpace \u03b9\ninst\u271d\u2074 : OrderTopology \u03b9\ninst\u271d\u00b3 : SecondCountableTopology \u03b9\ninst\u271d\u00b2 : BorelSpace \u03b9\ninst\u271d\u00b9 : TopologicalSpace \u03b2\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\n\u03c4 : \u03a9 \u2192 \u03b9\nf : Filtration \u03b9 m\ninst\u271d : MetrizableSpace \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\ni : \u03b9\nm_prod : MeasurableSpace (\u2191(Set.Iic i) \u00d7 \u03a9) := MeasurableSpace.prod Subtype.instMeasurableSpace (\u2191f i)\nm_set : (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)) \u2192 MeasurableSpace \u2191t := fun x => Subtype.instMeasurableSpace\ns : Set (\u2191(Set.Iic i) \u00d7 \u03a9) := {p | \u03c4 p.2 \u2264 i}\nhs : MeasurableSet s\nh_meas_fst : \u2200 (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)), Measurable fun x => \u2191(\u2191x).1\nsc : Set (\u2191(Set.Iic i) \u00d7 \u03a9) := s\u1d9c\n\u22a2 Measurable (Set.restrict s\u1d9c fun p => (fun i \u03c9 => min i (\u03c4 \u03c9)) (\u2191p.1) p.2)"}, {"tactic": "suffices h_min_eq_left :\n  (fun x : sc => min (\u2191(x : Set.Iic i \u00d7 \u03a9).fst) (\u03c4 (x : Set.Iic i \u00d7 \u03a9).snd)) = fun x : sc =>\n    \u2191(x : Set.Iic i \u00d7 \u03a9).fst", "annotated_tactic": ["suffices h_min_eq_left :\n      (fun x : sc => <a>min</a> (\u2191(x : <a>Set.Iic</a> i \u00d7 \u03a9).<a>fst</a>) (\u03c4 (x : <a>Set.Iic</a> i \u00d7 \u03a9).<a>snd</a>)) = fun x : sc =>\n        \u2191(x : <a>Set.Iic</a> i \u00d7 \u03a9).<a>fst</a>", [{"full_name": "Min.min", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1103, 3], "def_end_pos": [1103, 6]}, {"full_name": "Set.Iic", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [64, 5], "def_end_pos": [64, 8]}, {"full_name": "Prod.fst", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [468, 3], "def_end_pos": [468, 6]}, {"full_name": "Set.Iic", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [64, 5], "def_end_pos": [64, 8]}, {"full_name": "Prod.snd", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [470, 3], "def_end_pos": [470, 6]}, {"full_name": "Set.Iic", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [64, 5], "def_end_pos": [64, 8]}, {"full_name": "Prod.fst", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [468, 3], "def_end_pos": [468, 6]}]], "state_before": "case hf.refine'_2\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2077 : LinearOrder \u03b9\ninst\u271d\u2076 : MeasurableSpace \u03b9\ninst\u271d\u2075 : TopologicalSpace \u03b9\ninst\u271d\u2074 : OrderTopology \u03b9\ninst\u271d\u00b3 : SecondCountableTopology \u03b9\ninst\u271d\u00b2 : BorelSpace \u03b9\ninst\u271d\u00b9 : TopologicalSpace \u03b2\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\n\u03c4 : \u03a9 \u2192 \u03b9\nf : Filtration \u03b9 m\ninst\u271d : MetrizableSpace \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\ni : \u03b9\nm_prod : MeasurableSpace (\u2191(Set.Iic i) \u00d7 \u03a9) := MeasurableSpace.prod Subtype.instMeasurableSpace (\u2191f i)\nm_set : (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)) \u2192 MeasurableSpace \u2191t := fun x => Subtype.instMeasurableSpace\ns : Set (\u2191(Set.Iic i) \u00d7 \u03a9) := {p | \u03c4 p.2 \u2264 i}\nhs : MeasurableSet s\nh_meas_fst : \u2200 (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)), Measurable fun x => \u2191(\u2191x).1\nsc : Set (\u2191(Set.Iic i) \u00d7 \u03a9) := s\u1d9c\n\u22a2 Measurable (Set.restrict s\u1d9c fun p => (fun i \u03c9 => min i (\u03c4 \u03c9)) (\u2191p.1) p.2)", "state_after": "case hf.refine'_2\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2077 : LinearOrder \u03b9\ninst\u271d\u2076 : MeasurableSpace \u03b9\ninst\u271d\u2075 : TopologicalSpace \u03b9\ninst\u271d\u2074 : OrderTopology \u03b9\ninst\u271d\u00b3 : SecondCountableTopology \u03b9\ninst\u271d\u00b2 : BorelSpace \u03b9\ninst\u271d\u00b9 : TopologicalSpace \u03b2\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\n\u03c4 : \u03a9 \u2192 \u03b9\nf : Filtration \u03b9 m\ninst\u271d : MetrizableSpace \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\ni : \u03b9\nm_prod : MeasurableSpace (\u2191(Set.Iic i) \u00d7 \u03a9) := MeasurableSpace.prod Subtype.instMeasurableSpace (\u2191f i)\nm_set : (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)) \u2192 MeasurableSpace \u2191t := fun x => Subtype.instMeasurableSpace\ns : Set (\u2191(Set.Iic i) \u00d7 \u03a9) := {p | \u03c4 p.2 \u2264 i}\nhs : MeasurableSet s\nh_meas_fst : \u2200 (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)), Measurable fun x => \u2191(\u2191x).1\nsc : Set (\u2191(Set.Iic i) \u00d7 \u03a9) := s\u1d9c\nh_min_eq_left : (fun x => min (\u2191(\u2191x).1) (\u03c4 (\u2191x).2)) = fun x => \u2191(\u2191x).1\n\u22a2 Measurable (Set.restrict s\u1d9c fun p => (fun i \u03c9 => min i (\u03c4 \u03c9)) (\u2191p.1) p.2)\n\ncase h_min_eq_left\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2077 : LinearOrder \u03b9\ninst\u271d\u2076 : MeasurableSpace \u03b9\ninst\u271d\u2075 : TopologicalSpace \u03b9\ninst\u271d\u2074 : OrderTopology \u03b9\ninst\u271d\u00b3 : SecondCountableTopology \u03b9\ninst\u271d\u00b2 : BorelSpace \u03b9\ninst\u271d\u00b9 : TopologicalSpace \u03b2\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\n\u03c4 : \u03a9 \u2192 \u03b9\nf : Filtration \u03b9 m\ninst\u271d : MetrizableSpace \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\ni : \u03b9\nm_prod : MeasurableSpace (\u2191(Set.Iic i) \u00d7 \u03a9) := MeasurableSpace.prod Subtype.instMeasurableSpace (\u2191f i)\nm_set : (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)) \u2192 MeasurableSpace \u2191t := fun x => Subtype.instMeasurableSpace\ns : Set (\u2191(Set.Iic i) \u00d7 \u03a9) := {p | \u03c4 p.2 \u2264 i}\nhs : MeasurableSet s\nh_meas_fst : \u2200 (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)), Measurable fun x => \u2191(\u2191x).1\nsc : Set (\u2191(Set.Iic i) \u00d7 \u03a9) := s\u1d9c\n\u22a2 (fun x => min (\u2191(\u2191x).1) (\u03c4 (\u2191x).2)) = fun x => \u2191(\u2191x).1"}, {"tactic": "ext1 \u03c9", "annotated_tactic": ["ext1 \u03c9", []], "state_before": "case h_min_eq_left\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2077 : LinearOrder \u03b9\ninst\u271d\u2076 : MeasurableSpace \u03b9\ninst\u271d\u2075 : TopologicalSpace \u03b9\ninst\u271d\u2074 : OrderTopology \u03b9\ninst\u271d\u00b3 : SecondCountableTopology \u03b9\ninst\u271d\u00b2 : BorelSpace \u03b9\ninst\u271d\u00b9 : TopologicalSpace \u03b2\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\n\u03c4 : \u03a9 \u2192 \u03b9\nf : Filtration \u03b9 m\ninst\u271d : MetrizableSpace \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\ni : \u03b9\nm_prod : MeasurableSpace (\u2191(Set.Iic i) \u00d7 \u03a9) := MeasurableSpace.prod Subtype.instMeasurableSpace (\u2191f i)\nm_set : (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)) \u2192 MeasurableSpace \u2191t := fun x => Subtype.instMeasurableSpace\ns : Set (\u2191(Set.Iic i) \u00d7 \u03a9) := {p | \u03c4 p.2 \u2264 i}\nhs : MeasurableSet s\nh_meas_fst : \u2200 (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)), Measurable fun x => \u2191(\u2191x).1\nsc : Set (\u2191(Set.Iic i) \u00d7 \u03a9) := s\u1d9c\n\u22a2 (fun x => min (\u2191(\u2191x).1) (\u03c4 (\u2191x).2)) = fun x => \u2191(\u2191x).1", "state_after": "case h_min_eq_left.h\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2077 : LinearOrder \u03b9\ninst\u271d\u2076 : MeasurableSpace \u03b9\ninst\u271d\u2075 : TopologicalSpace \u03b9\ninst\u271d\u2074 : OrderTopology \u03b9\ninst\u271d\u00b3 : SecondCountableTopology \u03b9\ninst\u271d\u00b2 : BorelSpace \u03b9\ninst\u271d\u00b9 : TopologicalSpace \u03b2\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\n\u03c4 : \u03a9 \u2192 \u03b9\nf : Filtration \u03b9 m\ninst\u271d : MetrizableSpace \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\ni : \u03b9\nm_prod : MeasurableSpace (\u2191(Set.Iic i) \u00d7 \u03a9) := MeasurableSpace.prod Subtype.instMeasurableSpace (\u2191f i)\nm_set : (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)) \u2192 MeasurableSpace \u2191t := fun x => Subtype.instMeasurableSpace\ns : Set (\u2191(Set.Iic i) \u00d7 \u03a9) := {p | \u03c4 p.2 \u2264 i}\nhs : MeasurableSet s\nh_meas_fst : \u2200 (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)), Measurable fun x => \u2191(\u2191x).1\nsc : Set (\u2191(Set.Iic i) \u00d7 \u03a9) := s\u1d9c\n\u03c9 : \u2191sc\n\u22a2 min (\u2191(\u2191\u03c9).1) (\u03c4 (\u2191\u03c9).2) = \u2191(\u2191\u03c9).1"}, {"tactic": "rw [min_eq_left]", "annotated_tactic": ["rw [<a>min_eq_left</a>]", [{"full_name": "min_eq_left", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [100, 9], "def_end_pos": [100, 20]}]], "state_before": "case h_min_eq_left.h\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2077 : LinearOrder \u03b9\ninst\u271d\u2076 : MeasurableSpace \u03b9\ninst\u271d\u2075 : TopologicalSpace \u03b9\ninst\u271d\u2074 : OrderTopology \u03b9\ninst\u271d\u00b3 : SecondCountableTopology \u03b9\ninst\u271d\u00b2 : BorelSpace \u03b9\ninst\u271d\u00b9 : TopologicalSpace \u03b2\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\n\u03c4 : \u03a9 \u2192 \u03b9\nf : Filtration \u03b9 m\ninst\u271d : MetrizableSpace \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\ni : \u03b9\nm_prod : MeasurableSpace (\u2191(Set.Iic i) \u00d7 \u03a9) := MeasurableSpace.prod Subtype.instMeasurableSpace (\u2191f i)\nm_set : (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)) \u2192 MeasurableSpace \u2191t := fun x => Subtype.instMeasurableSpace\ns : Set (\u2191(Set.Iic i) \u00d7 \u03a9) := {p | \u03c4 p.2 \u2264 i}\nhs : MeasurableSet s\nh_meas_fst : \u2200 (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)), Measurable fun x => \u2191(\u2191x).1\nsc : Set (\u2191(Set.Iic i) \u00d7 \u03a9) := s\u1d9c\n\u03c9 : \u2191sc\n\u22a2 min (\u2191(\u2191\u03c9).1) (\u03c4 (\u2191\u03c9).2) = \u2191(\u2191\u03c9).1", "state_after": "case h_min_eq_left.h\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2077 : LinearOrder \u03b9\ninst\u271d\u2076 : MeasurableSpace \u03b9\ninst\u271d\u2075 : TopologicalSpace \u03b9\ninst\u271d\u2074 : OrderTopology \u03b9\ninst\u271d\u00b3 : SecondCountableTopology \u03b9\ninst\u271d\u00b2 : BorelSpace \u03b9\ninst\u271d\u00b9 : TopologicalSpace \u03b2\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\n\u03c4 : \u03a9 \u2192 \u03b9\nf : Filtration \u03b9 m\ninst\u271d : MetrizableSpace \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\ni : \u03b9\nm_prod : MeasurableSpace (\u2191(Set.Iic i) \u00d7 \u03a9) := MeasurableSpace.prod Subtype.instMeasurableSpace (\u2191f i)\nm_set : (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)) \u2192 MeasurableSpace \u2191t := fun x => Subtype.instMeasurableSpace\ns : Set (\u2191(Set.Iic i) \u00d7 \u03a9) := {p | \u03c4 p.2 \u2264 i}\nhs : MeasurableSet s\nh_meas_fst : \u2200 (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)), Measurable fun x => \u2191(\u2191x).1\nsc : Set (\u2191(Set.Iic i) \u00d7 \u03a9) := s\u1d9c\n\u03c9 : \u2191sc\n\u22a2 \u2191(\u2191\u03c9).1 \u2264 \u03c4 (\u2191\u03c9).2"}, {"tactic": "have hx_fst_le : \u2191(\u03c9 : Set.Iic i \u00d7 \u03a9).fst \u2264 i := (\u03c9 : Set.Iic i \u00d7 \u03a9).fst.prop", "annotated_tactic": ["have hx_fst_le : \u2191(\u03c9 : <a>Set.Iic</a> i \u00d7 \u03a9).<a>fst</a> \u2264 i := (\u03c9 : <a>Set.Iic</a> i \u00d7 \u03a9).fst.prop", [{"full_name": "Set.Iic", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [64, 5], "def_end_pos": [64, 8]}, {"full_name": "Prod.fst", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [468, 3], "def_end_pos": [468, 6]}, {"full_name": "Set.Iic", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [64, 5], "def_end_pos": [64, 8]}]], "state_before": "case h_min_eq_left.h\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2077 : LinearOrder \u03b9\ninst\u271d\u2076 : MeasurableSpace \u03b9\ninst\u271d\u2075 : TopologicalSpace \u03b9\ninst\u271d\u2074 : OrderTopology \u03b9\ninst\u271d\u00b3 : SecondCountableTopology \u03b9\ninst\u271d\u00b2 : BorelSpace \u03b9\ninst\u271d\u00b9 : TopologicalSpace \u03b2\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\n\u03c4 : \u03a9 \u2192 \u03b9\nf : Filtration \u03b9 m\ninst\u271d : MetrizableSpace \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\ni : \u03b9\nm_prod : MeasurableSpace (\u2191(Set.Iic i) \u00d7 \u03a9) := MeasurableSpace.prod Subtype.instMeasurableSpace (\u2191f i)\nm_set : (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)) \u2192 MeasurableSpace \u2191t := fun x => Subtype.instMeasurableSpace\ns : Set (\u2191(Set.Iic i) \u00d7 \u03a9) := {p | \u03c4 p.2 \u2264 i}\nhs : MeasurableSet s\nh_meas_fst : \u2200 (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)), Measurable fun x => \u2191(\u2191x).1\nsc : Set (\u2191(Set.Iic i) \u00d7 \u03a9) := s\u1d9c\n\u03c9 : \u2191sc\n\u22a2 \u2191(\u2191\u03c9).1 \u2264 \u03c4 (\u2191\u03c9).2", "state_after": "case h_min_eq_left.h\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2077 : LinearOrder \u03b9\ninst\u271d\u2076 : MeasurableSpace \u03b9\ninst\u271d\u2075 : TopologicalSpace \u03b9\ninst\u271d\u2074 : OrderTopology \u03b9\ninst\u271d\u00b3 : SecondCountableTopology \u03b9\ninst\u271d\u00b2 : BorelSpace \u03b9\ninst\u271d\u00b9 : TopologicalSpace \u03b2\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\n\u03c4 : \u03a9 \u2192 \u03b9\nf : Filtration \u03b9 m\ninst\u271d : MetrizableSpace \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\ni : \u03b9\nm_prod : MeasurableSpace (\u2191(Set.Iic i) \u00d7 \u03a9) := MeasurableSpace.prod Subtype.instMeasurableSpace (\u2191f i)\nm_set : (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)) \u2192 MeasurableSpace \u2191t := fun x => Subtype.instMeasurableSpace\ns : Set (\u2191(Set.Iic i) \u00d7 \u03a9) := {p | \u03c4 p.2 \u2264 i}\nhs : MeasurableSet s\nh_meas_fst : \u2200 (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)), Measurable fun x => \u2191(\u2191x).1\nsc : Set (\u2191(Set.Iic i) \u00d7 \u03a9) := s\u1d9c\n\u03c9 : \u2191sc\nhx_fst_le : \u2191(\u2191\u03c9).1 \u2264 i\n\u22a2 \u2191(\u2191\u03c9).1 \u2264 \u03c4 (\u2191\u03c9).2"}, {"tactic": "refine' hx_fst_le.trans (le_of_lt _)", "annotated_tactic": ["refine' hx_fst_le.trans (<a>le_of_lt</a> _)", [{"full_name": "le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [110, 9], "def_end_pos": [110, 17]}]], "state_before": "case h_min_eq_left.h\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2077 : LinearOrder \u03b9\ninst\u271d\u2076 : MeasurableSpace \u03b9\ninst\u271d\u2075 : TopologicalSpace \u03b9\ninst\u271d\u2074 : OrderTopology \u03b9\ninst\u271d\u00b3 : SecondCountableTopology \u03b9\ninst\u271d\u00b2 : BorelSpace \u03b9\ninst\u271d\u00b9 : TopologicalSpace \u03b2\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\n\u03c4 : \u03a9 \u2192 \u03b9\nf : Filtration \u03b9 m\ninst\u271d : MetrizableSpace \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\ni : \u03b9\nm_prod : MeasurableSpace (\u2191(Set.Iic i) \u00d7 \u03a9) := MeasurableSpace.prod Subtype.instMeasurableSpace (\u2191f i)\nm_set : (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)) \u2192 MeasurableSpace \u2191t := fun x => Subtype.instMeasurableSpace\ns : Set (\u2191(Set.Iic i) \u00d7 \u03a9) := {p | \u03c4 p.2 \u2264 i}\nhs : MeasurableSet s\nh_meas_fst : \u2200 (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)), Measurable fun x => \u2191(\u2191x).1\nsc : Set (\u2191(Set.Iic i) \u00d7 \u03a9) := s\u1d9c\n\u03c9 : \u2191sc\nhx_fst_le : \u2191(\u2191\u03c9).1 \u2264 i\n\u22a2 \u2191(\u2191\u03c9).1 \u2264 \u03c4 (\u2191\u03c9).2", "state_after": "case h_min_eq_left.h\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2077 : LinearOrder \u03b9\ninst\u271d\u2076 : MeasurableSpace \u03b9\ninst\u271d\u2075 : TopologicalSpace \u03b9\ninst\u271d\u2074 : OrderTopology \u03b9\ninst\u271d\u00b3 : SecondCountableTopology \u03b9\ninst\u271d\u00b2 : BorelSpace \u03b9\ninst\u271d\u00b9 : TopologicalSpace \u03b2\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\n\u03c4 : \u03a9 \u2192 \u03b9\nf : Filtration \u03b9 m\ninst\u271d : MetrizableSpace \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\ni : \u03b9\nm_prod : MeasurableSpace (\u2191(Set.Iic i) \u00d7 \u03a9) := MeasurableSpace.prod Subtype.instMeasurableSpace (\u2191f i)\nm_set : (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)) \u2192 MeasurableSpace \u2191t := fun x => Subtype.instMeasurableSpace\ns : Set (\u2191(Set.Iic i) \u00d7 \u03a9) := {p | \u03c4 p.2 \u2264 i}\nhs : MeasurableSet s\nh_meas_fst : \u2200 (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)), Measurable fun x => \u2191(\u2191x).1\nsc : Set (\u2191(Set.Iic i) \u00d7 \u03a9) := s\u1d9c\n\u03c9 : \u2191sc\nhx_fst_le : \u2191(\u2191\u03c9).1 \u2264 i\n\u22a2 i < \u03c4 (\u2191\u03c9).2"}, {"tactic": "convert \u03c9.prop", "annotated_tactic": ["convert \u03c9.prop", []], "state_before": "case h_min_eq_left.h\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2077 : LinearOrder \u03b9\ninst\u271d\u2076 : MeasurableSpace \u03b9\ninst\u271d\u2075 : TopologicalSpace \u03b9\ninst\u271d\u2074 : OrderTopology \u03b9\ninst\u271d\u00b3 : SecondCountableTopology \u03b9\ninst\u271d\u00b2 : BorelSpace \u03b9\ninst\u271d\u00b9 : TopologicalSpace \u03b2\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\n\u03c4 : \u03a9 \u2192 \u03b9\nf : Filtration \u03b9 m\ninst\u271d : MetrizableSpace \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\ni : \u03b9\nm_prod : MeasurableSpace (\u2191(Set.Iic i) \u00d7 \u03a9) := MeasurableSpace.prod Subtype.instMeasurableSpace (\u2191f i)\nm_set : (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)) \u2192 MeasurableSpace \u2191t := fun x => Subtype.instMeasurableSpace\ns : Set (\u2191(Set.Iic i) \u00d7 \u03a9) := {p | \u03c4 p.2 \u2264 i}\nhs : MeasurableSet s\nh_meas_fst : \u2200 (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)), Measurable fun x => \u2191(\u2191x).1\nsc : Set (\u2191(Set.Iic i) \u00d7 \u03a9) := s\u1d9c\n\u03c9 : \u2191sc\nhx_fst_le : \u2191(\u2191\u03c9).1 \u2264 i\n\u22a2 i < \u03c4 (\u2191\u03c9).2", "state_after": "case a\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2077 : LinearOrder \u03b9\ninst\u271d\u2076 : MeasurableSpace \u03b9\ninst\u271d\u2075 : TopologicalSpace \u03b9\ninst\u271d\u2074 : OrderTopology \u03b9\ninst\u271d\u00b3 : SecondCountableTopology \u03b9\ninst\u271d\u00b2 : BorelSpace \u03b9\ninst\u271d\u00b9 : TopologicalSpace \u03b2\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\n\u03c4 : \u03a9 \u2192 \u03b9\nf : Filtration \u03b9 m\ninst\u271d : MetrizableSpace \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\ni : \u03b9\nm_prod : MeasurableSpace (\u2191(Set.Iic i) \u00d7 \u03a9) := MeasurableSpace.prod Subtype.instMeasurableSpace (\u2191f i)\nm_set : (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)) \u2192 MeasurableSpace \u2191t := fun x => Subtype.instMeasurableSpace\ns : Set (\u2191(Set.Iic i) \u00d7 \u03a9) := {p | \u03c4 p.2 \u2264 i}\nhs : MeasurableSet s\nh_meas_fst : \u2200 (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)), Measurable fun x => \u2191(\u2191x).1\nsc : Set (\u2191(Set.Iic i) \u00d7 \u03a9) := s\u1d9c\n\u03c9 : \u2191sc\nhx_fst_le : \u2191(\u2191\u03c9).1 \u2264 i\n\u22a2 i < \u03c4 (\u2191\u03c9).2 \u2194 \u2191\u03c9 \u2208 sc"}, {"tactic": "simp only [not_le, Set.mem_compl_iff, Set.mem_setOf_eq]", "annotated_tactic": ["simp only [<a>not_le</a>, <a>Set.mem_compl_iff</a>, <a>Set.mem_setOf_eq</a>]", [{"full_name": "not_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [373, 9], "def_end_pos": [373, 15]}, {"full_name": "Set.mem_compl_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1658, 9], "def_end_pos": [1658, 22]}, {"full_name": "Set.mem_setOf_eq", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [256, 29], "def_end_pos": [256, 41]}]], "state_before": "case a\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2077 : LinearOrder \u03b9\ninst\u271d\u2076 : MeasurableSpace \u03b9\ninst\u271d\u2075 : TopologicalSpace \u03b9\ninst\u271d\u2074 : OrderTopology \u03b9\ninst\u271d\u00b3 : SecondCountableTopology \u03b9\ninst\u271d\u00b2 : BorelSpace \u03b9\ninst\u271d\u00b9 : TopologicalSpace \u03b2\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\n\u03c4 : \u03a9 \u2192 \u03b9\nf : Filtration \u03b9 m\ninst\u271d : MetrizableSpace \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\ni : \u03b9\nm_prod : MeasurableSpace (\u2191(Set.Iic i) \u00d7 \u03a9) := MeasurableSpace.prod Subtype.instMeasurableSpace (\u2191f i)\nm_set : (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)) \u2192 MeasurableSpace \u2191t := fun x => Subtype.instMeasurableSpace\ns : Set (\u2191(Set.Iic i) \u00d7 \u03a9) := {p | \u03c4 p.2 \u2264 i}\nhs : MeasurableSet s\nh_meas_fst : \u2200 (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)), Measurable fun x => \u2191(\u2191x).1\nsc : Set (\u2191(Set.Iic i) \u00d7 \u03a9) := s\u1d9c\n\u03c9 : \u2191sc\nhx_fst_le : \u2191(\u2191\u03c9).1 \u2264 i\n\u22a2 i < \u03c4 (\u2191\u03c9).2 \u2194 \u2191\u03c9 \u2208 sc", "state_after": "no goals"}, {"tactic": "simp_rw [Set.restrict, h_min_eq_left]", "annotated_tactic": ["simp_rw [<a>Set.restrict</a>, h_min_eq_left]", [{"full_name": "Set.restrict", "def_path": "Mathlib/Data/Set/Function.lean", "def_pos": [47, 5], "def_end_pos": [47, 13]}]], "state_before": "case hf.refine'_2\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2077 : LinearOrder \u03b9\ninst\u271d\u2076 : MeasurableSpace \u03b9\ninst\u271d\u2075 : TopologicalSpace \u03b9\ninst\u271d\u2074 : OrderTopology \u03b9\ninst\u271d\u00b3 : SecondCountableTopology \u03b9\ninst\u271d\u00b2 : BorelSpace \u03b9\ninst\u271d\u00b9 : TopologicalSpace \u03b2\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\n\u03c4 : \u03a9 \u2192 \u03b9\nf : Filtration \u03b9 m\ninst\u271d : MetrizableSpace \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\ni : \u03b9\nm_prod : MeasurableSpace (\u2191(Set.Iic i) \u00d7 \u03a9) := MeasurableSpace.prod Subtype.instMeasurableSpace (\u2191f i)\nm_set : (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)) \u2192 MeasurableSpace \u2191t := fun x => Subtype.instMeasurableSpace\ns : Set (\u2191(Set.Iic i) \u00d7 \u03a9) := {p | \u03c4 p.2 \u2264 i}\nhs : MeasurableSet s\nh_meas_fst : \u2200 (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)), Measurable fun x => \u2191(\u2191x).1\nsc : Set (\u2191(Set.Iic i) \u00d7 \u03a9) := s\u1d9c\nh_min_eq_left : (fun x => min (\u2191(\u2191x).1) (\u03c4 (\u2191x).2)) = fun x => \u2191(\u2191x).1\n\u22a2 Measurable (Set.restrict s\u1d9c fun p => (fun i \u03c9 => min i (\u03c4 \u03c9)) (\u2191p.1) p.2)", "state_after": "case hf.refine'_2\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2077 : LinearOrder \u03b9\ninst\u271d\u2076 : MeasurableSpace \u03b9\ninst\u271d\u2075 : TopologicalSpace \u03b9\ninst\u271d\u2074 : OrderTopology \u03b9\ninst\u271d\u00b3 : SecondCountableTopology \u03b9\ninst\u271d\u00b2 : BorelSpace \u03b9\ninst\u271d\u00b9 : TopologicalSpace \u03b2\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\n\u03c4 : \u03a9 \u2192 \u03b9\nf : Filtration \u03b9 m\ninst\u271d : MetrizableSpace \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\ni : \u03b9\nm_prod : MeasurableSpace (\u2191(Set.Iic i) \u00d7 \u03a9) := MeasurableSpace.prod Subtype.instMeasurableSpace (\u2191f i)\nm_set : (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)) \u2192 MeasurableSpace \u2191t := fun x => Subtype.instMeasurableSpace\ns : Set (\u2191(Set.Iic i) \u00d7 \u03a9) := {p | \u03c4 p.2 \u2264 i}\nhs : MeasurableSet s\nh_meas_fst : \u2200 (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)), Measurable fun x => \u2191(\u2191x).1\nsc : Set (\u2191(Set.Iic i) \u00d7 \u03a9) := s\u1d9c\nh_min_eq_left : (fun x => min (\u2191(\u2191x).1) (\u03c4 (\u2191x).2)) = fun x => \u2191(\u2191x).1\n\u22a2 Measurable fun x => \u2191(\u2191x).1"}, {"tactic": "exact h_meas_fst _", "annotated_tactic": ["exact h_meas_fst _", []], "state_before": "case hf.refine'_2\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2077 : LinearOrder \u03b9\ninst\u271d\u2076 : MeasurableSpace \u03b9\ninst\u271d\u2075 : TopologicalSpace \u03b9\ninst\u271d\u2074 : OrderTopology \u03b9\ninst\u271d\u00b3 : SecondCountableTopology \u03b9\ninst\u271d\u00b2 : BorelSpace \u03b9\ninst\u271d\u00b9 : TopologicalSpace \u03b2\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\n\u03c4 : \u03a9 \u2192 \u03b9\nf : Filtration \u03b9 m\ninst\u271d : MetrizableSpace \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\ni : \u03b9\nm_prod : MeasurableSpace (\u2191(Set.Iic i) \u00d7 \u03a9) := MeasurableSpace.prod Subtype.instMeasurableSpace (\u2191f i)\nm_set : (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)) \u2192 MeasurableSpace \u2191t := fun x => Subtype.instMeasurableSpace\ns : Set (\u2191(Set.Iic i) \u00d7 \u03a9) := {p | \u03c4 p.2 \u2264 i}\nhs : MeasurableSet s\nh_meas_fst : \u2200 (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)), Measurable fun x => \u2191(\u2191x).1\nsc : Set (\u2191(Set.Iic i) \u00d7 \u03a9) := s\u1d9c\nh_min_eq_left : (fun x => min (\u2191(\u2191x).1) (\u03c4 (\u2191x).2)) = fun x => \u2191(\u2191x).1\n\u22a2 Measurable fun x => \u2191(\u2191x).1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/BinomialHeap/Basic.lean", "full_name": "Std.BinomialHeap.Imp.HeapNode.rankTR_eq", "start": [61, 18], "end": [66, 60], "traced_tactics": [{"tactic": "funext \u03b1 s", "annotated_tactic": ["funext \u03b1 s", []], "state_before": "\u22a2 @Std.BinomialHeap.Imp.HeapNode.rankTR = @rank", "state_after": "case h.h\n\u03b1 : Type u_1\ns : HeapNode \u03b1\n\u22a2 Std.BinomialHeap.Imp.HeapNode.rankTR s = rank s"}, {"tactic": "exact go s 0", "annotated_tactic": ["exact go s 0", []], "state_before": "case h.h\n\u03b1 : Type u_1\ns : HeapNode \u03b1\n\u22a2 Std.BinomialHeap.Imp.HeapNode.rankTR s = rank s", "state_after": "no goals"}, {"tactic": "simp_arith only [rankTR.go, go, rank]", "annotated_tactic": ["simp_arith only [<a>rankTR.go</a>, go, <a>rank</a>]", [{"full_name": "_private.\u00ablake-packages\u00bb.std.Std.Data.BinomialHeap.Basic.0.Std.BinomialHeap.Imp.HeapNode.rankTR.go", "def_path": "lake-packages/std/Std/Data/BinomialHeap/Basic.lean", "def_pos": [57, 3], "def_end_pos": [57, 5]}, {"full_name": "Std.BinomialHeap.Imp.HeapNode.rank", "def_path": "lake-packages/std/Std/Data/BinomialHeap/Basic.lean", "def_pos": [50, 5], "def_end_pos": [50, 18]}]], "state_before": "\u03b1 : Type ?u.3227\na\u271d : \u03b1\nchild\u271d sibling\u271d : HeapNode \u03b1\nx\u271d : Nat\n\u22a2 Std.BinomialHeap.Imp.HeapNode.rankTR.go (node a\u271d child\u271d sibling\u271d) x\u271d = rank (node a\u271d child\u271d sibling\u271d) + x\u271d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Countable.lean", "full_name": "Set.countable_univ_iff", "start": [116, 1], "end": [117, 64], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/PeakFunction.lean", "full_name": "tendsto_set_integral_pow_smul_of_unique_maximum_of_isCompact_of_continuousOn", "start": [316, 1], "end": [324, 48], "traced_tactics": [{"tactic": "rw [\u2190 hs.isClosed.closure_eq]", "annotated_tactic": ["rw [\u2190 hs.isClosed.closure_eq]", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2077 : TopologicalSpace \u03b1\ninst\u271d\u2076 : BorelSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : MetrizableSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsOpenPosMeasure \u03bc\nhs : IsCompact s\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 closure (interior s)\nhmg : ContinuousOn g s\n\u22a2 x\u2080 \u2208 s", "state_after": "\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2077 : TopologicalSpace \u03b1\ninst\u271d\u2076 : BorelSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : MetrizableSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsOpenPosMeasure \u03bc\nhs : IsCompact s\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 closure (interior s)\nhmg : ContinuousOn g s\n\u22a2 x\u2080 \u2208 closure s"}, {"tactic": "exact closure_mono interior_subset h\u2080", "annotated_tactic": ["exact <a>closure_mono</a> <a>interior_subset</a> h\u2080", [{"full_name": "closure_mono", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [475, 9], "def_end_pos": [475, 21]}, {"full_name": "interior_subset", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [302, 9], "def_end_pos": [302, 24]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2077 : TopologicalSpace \u03b1\ninst\u271d\u2076 : BorelSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : MetrizableSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsOpenPosMeasure \u03bc\nhs : IsCompact s\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 closure (interior s)\nhmg : ContinuousOn g s\n\u22a2 x\u2080 \u2208 closure s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "full_name": "Real.isPiSystem_Ioo_rat", "start": [1897, 1], "end": [1901, 17], "traced_tactics": [{"tactic": "convert isPiSystem_Ioo ((\u2191) : \u211a \u2192 \u211d) ((\u2191) : \u211a \u2192 \u211d)", "annotated_tactic": ["convert <a>isPiSystem_Ioo</a> ((\u2191) : \u211a \u2192 \u211d) ((\u2191) : \u211a \u2192 \u211d)", [{"full_name": "isPiSystem_Ioo", "def_path": "Mathlib/MeasureTheory/PiSystem.lean", "def_pos": [192, 9], "def_end_pos": [192, 23]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns t u : Set \u03b1\n\u22a2 IsPiSystem (\u22c3 a, \u22c3 b, \u22c3 (_ : a < b), {Ioo \u2191a \u2191b})", "state_after": "case h.e'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns t u : Set \u03b1\n\u22a2 \u22c3 a, \u22c3 b, \u22c3 (_ : a < b), {Ioo \u2191a \u2191b} = {S | \u2203 l u, \u2191l < \u2191u \u2227 Ioo \u2191l \u2191u = S}"}, {"tactic": "ext x", "annotated_tactic": ["ext x", []], "state_before": "case h.e'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns t u : Set \u03b1\n\u22a2 \u22c3 a, \u22c3 b, \u22c3 (_ : a < b), {Ioo \u2191a \u2191b} = {S | \u2203 l u, \u2191l < \u2191u \u2227 Ioo \u2191l \u2191u = S}", "state_after": "case h.e'_2.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns t u : Set \u03b1\nx : Set \u211d\n\u22a2 x \u2208 \u22c3 a, \u22c3 b, \u22c3 (_ : a < b), {Ioo \u2191a \u2191b} \u2194 x \u2208 {S | \u2203 l u, \u2191l < \u2191u \u2227 Ioo \u2191l \u2191u = S}"}, {"tactic": "simp [eq_comm]", "annotated_tactic": ["simp [<a>eq_comm</a>]", [{"full_name": "eq_comm", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [104, 9], "def_end_pos": [104, 16]}]], "state_before": "case h.e'_2.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns t u : Set \u03b1\nx : Set \u211d\n\u22a2 x \u2208 \u22c3 a, \u22c3 b, \u22c3 (_ : a < b), {Ioo \u2191a \u2191b} \u2194 x \u2208 {S | \u2203 l u, \u2191l < \u2191u \u2227 Ioo \u2191l \u2191u = S}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "full_name": "IsUnit.aestronglyMeasurable_const_smul_iff", "start": [1799, 8], "end": [1802, 45], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/ContinuousMapDense.lean", "full_name": "MeasureTheory.Lp.boundedContinuousFunction_dense", "start": [331, 1], "end": [352, 65], "traced_tactics": [{"tactic": "rw [AddSubgroup.eq_top_iff']", "annotated_tactic": ["rw [<a>AddSubgroup.eq_top_iff'</a>]", [{"full_name": "AddSubgroup.eq_top_iff'", "def_path": "Mathlib/GroupTheory/Subgroup/Basic.lean", "def_pos": [1092, 3], "def_end_pos": [1092, 14]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : T4Space \u03b1\ninst\u271d\u2074 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : SecondCountableTopologyEither \u03b1 E\n_i : Fact (1 \u2264 p)\nhp : p \u2260 \u22a4\ninst\u271d : Measure.WeaklyRegular \u03bc\n\u22a2 AddSubgroup.topologicalClosure (boundedContinuousFunction E p \u03bc) = \u22a4", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : T4Space \u03b1\ninst\u271d\u2074 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : SecondCountableTopologyEither \u03b1 E\n_i : Fact (1 \u2264 p)\nhp : p \u2260 \u22a4\ninst\u271d : Measure.WeaklyRegular \u03bc\n\u22a2 \u2200 (x : { x // x \u2208 Lp E p }), x \u2208 AddSubgroup.topologicalClosure (boundedContinuousFunction E p \u03bc)"}, {"tactic": "intro f", "annotated_tactic": ["intro f", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : T4Space \u03b1\ninst\u271d\u2074 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : SecondCountableTopologyEither \u03b1 E\n_i : Fact (1 \u2264 p)\nhp : p \u2260 \u22a4\ninst\u271d : Measure.WeaklyRegular \u03bc\n\u22a2 \u2200 (x : { x // x \u2208 Lp E p }), x \u2208 AddSubgroup.topologicalClosure (boundedContinuousFunction E p \u03bc)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : T4Space \u03b1\ninst\u271d\u2074 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : SecondCountableTopologyEither \u03b1 E\n_i : Fact (1 \u2264 p)\nhp : p \u2260 \u22a4\ninst\u271d : Measure.WeaklyRegular \u03bc\nf : { x // x \u2208 Lp E p }\n\u22a2 f \u2208 AddSubgroup.topologicalClosure (boundedContinuousFunction E p \u03bc)"}, {"tactic": "refine' mem_closure_iff_frequently.mpr _", "annotated_tactic": ["refine' mem_closure_iff_frequently.mpr _", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : T4Space \u03b1\ninst\u271d\u2074 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : SecondCountableTopologyEither \u03b1 E\n_i : Fact (1 \u2264 p)\nhp : p \u2260 \u22a4\ninst\u271d : Measure.WeaklyRegular \u03bc\nf : { x // x \u2208 Lp E p }\n\u22a2 f \u2208 AddSubgroup.topologicalClosure (boundedContinuousFunction E p \u03bc)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : T4Space \u03b1\ninst\u271d\u2074 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : SecondCountableTopologyEither \u03b1 E\n_i : Fact (1 \u2264 p)\nhp : p \u2260 \u22a4\ninst\u271d : Measure.WeaklyRegular \u03bc\nf : { x // x \u2208 Lp E p }\n\u22a2 \u2203\u1da0 (x : { x // x \u2208 Lp E p }) in \ud835\udcdd f, x \u2208 \u2191(boundedContinuousFunction E p \u03bc)"}, {"tactic": "rw [Metric.nhds_basis_closedBall.frequently_iff]", "annotated_tactic": ["rw [Metric.nhds_basis_closedBall.frequently_iff]", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : T4Space \u03b1\ninst\u271d\u2074 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : SecondCountableTopologyEither \u03b1 E\n_i : Fact (1 \u2264 p)\nhp : p \u2260 \u22a4\ninst\u271d : Measure.WeaklyRegular \u03bc\nf : { x // x \u2208 Lp E p }\n\u22a2 \u2203\u1da0 (x : { x // x \u2208 Lp E p }) in \ud835\udcdd f, x \u2208 \u2191(boundedContinuousFunction E p \u03bc)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : T4Space \u03b1\ninst\u271d\u2074 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : SecondCountableTopologyEither \u03b1 E\n_i : Fact (1 \u2264 p)\nhp : p \u2260 \u22a4\ninst\u271d : Measure.WeaklyRegular \u03bc\nf : { x // x \u2208 Lp E p }\n\u22a2 \u2200 (i : \u211d), 0 < i \u2192 \u2203 x, x \u2208 Metric.closedBall f i \u2227 x \u2208 \u2191(boundedContinuousFunction E p \u03bc)"}, {"tactic": "intro \u03b5 h\u03b5", "annotated_tactic": ["intro \u03b5 h\u03b5", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : T4Space \u03b1\ninst\u271d\u2074 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : SecondCountableTopologyEither \u03b1 E\n_i : Fact (1 \u2264 p)\nhp : p \u2260 \u22a4\ninst\u271d : Measure.WeaklyRegular \u03bc\nf : { x // x \u2208 Lp E p }\n\u22a2 \u2200 (i : \u211d), 0 < i \u2192 \u2203 x, x \u2208 Metric.closedBall f i \u2227 x \u2208 \u2191(boundedContinuousFunction E p \u03bc)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : T4Space \u03b1\ninst\u271d\u2074 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : SecondCountableTopologyEither \u03b1 E\n_i : Fact (1 \u2264 p)\nhp : p \u2260 \u22a4\ninst\u271d : Measure.WeaklyRegular \u03bc\nf : { x // x \u2208 Lp E p }\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u22a2 \u2203 x, x \u2208 Metric.closedBall f \u03b5 \u2227 x \u2208 \u2191(boundedContinuousFunction E p \u03bc)"}, {"tactic": "have A : ENNReal.ofReal \u03b5 \u2260 0 := by simp only [Ne.def, ENNReal.ofReal_eq_zero, not_le, h\u03b5]", "annotated_tactic": ["have A : <a>ENNReal.ofReal</a> \u03b5 \u2260 0 := by simp only [<a>Ne.def</a>, <a>ENNReal.ofReal_eq_zero</a>, <a>not_le</a>, h\u03b5]", [{"full_name": "ENNReal.ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [172, 29], "def_end_pos": [172, 35]}, {"full_name": "Ne.def", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [59, 9], "def_end_pos": [59, 15]}, {"full_name": "ENNReal.ofReal_eq_zero", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2170, 9], "def_end_pos": [2170, 23]}, {"full_name": "not_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [373, 9], "def_end_pos": [373, 15]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : T4Space \u03b1\ninst\u271d\u2074 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : SecondCountableTopologyEither \u03b1 E\n_i : Fact (1 \u2264 p)\nhp : p \u2260 \u22a4\ninst\u271d : Measure.WeaklyRegular \u03bc\nf : { x // x \u2208 Lp E p }\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u22a2 \u2203 x, x \u2208 Metric.closedBall f \u03b5 \u2227 x \u2208 \u2191(boundedContinuousFunction E p \u03bc)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : T4Space \u03b1\ninst\u271d\u2074 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : SecondCountableTopologyEither \u03b1 E\n_i : Fact (1 \u2264 p)\nhp : p \u2260 \u22a4\ninst\u271d : Measure.WeaklyRegular \u03bc\nf : { x // x \u2208 Lp E p }\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nA : ENNReal.ofReal \u03b5 \u2260 0\n\u22a2 \u2203 x, x \u2208 Metric.closedBall f \u03b5 \u2227 x \u2208 \u2191(boundedContinuousFunction E p \u03bc)"}, {"tactic": "obtain \u27e8g, hg, g_mem\u27e9 :\n    \u2203 g : \u03b1 \u2192\u1d47 E, snorm ((f : \u03b1 \u2192 E) - (g : \u03b1 \u2192 E)) p \u03bc \u2264 ENNReal.ofReal \u03b5 \u2227 Mem\u2112p g p \u03bc :=\n  (Lp.mem\u2112p f).exists_boundedContinuous_snorm_sub_le hp A", "annotated_tactic": ["obtain \u27e8g, hg, g_mem\u27e9 :\n      \u2203 g : \u03b1 \u2192\u1d47 E, <a>snorm</a> ((f : \u03b1 \u2192 E) - (g : \u03b1 \u2192 E)) p \u03bc \u2264 <a>ENNReal.ofReal</a> \u03b5 \u2227 <a>Mem\u2112p</a> g p \u03bc :=\n    (<a>Lp.mem\u2112p</a> f).<a>exists_boundedContinuous_snorm_sub_le</a> hp A", [{"full_name": "MeasureTheory.snorm", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [84, 5], "def_end_pos": [84, 10]}, {"full_name": "ENNReal.ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [172, 29], "def_end_pos": [172, 35]}, {"full_name": "MeasureTheory.Mem\u2112p", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [108, 5], "def_end_pos": [108, 10]}, {"full_name": "MeasureTheory.Lp.mem\u2112p", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [216, 19], "def_end_pos": [216, 24]}, {"full_name": "MeasureTheory.Mem\u2112p.exists_boundedContinuous_snorm_sub_le", "def_path": "Mathlib/MeasureTheory/Function/ContinuousMapDense.lean", "def_pos": [239, 9], "def_end_pos": [239, 52]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : T4Space \u03b1\ninst\u271d\u2074 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : SecondCountableTopologyEither \u03b1 E\n_i : Fact (1 \u2264 p)\nhp : p \u2260 \u22a4\ninst\u271d : Measure.WeaklyRegular \u03bc\nf : { x // x \u2208 Lp E p }\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nA : ENNReal.ofReal \u03b5 \u2260 0\n\u22a2 \u2203 x, x \u2208 Metric.closedBall f \u03b5 \u2227 x \u2208 \u2191(boundedContinuousFunction E p \u03bc)", "state_after": "case intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : T4Space \u03b1\ninst\u271d\u2074 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : SecondCountableTopologyEither \u03b1 E\n_i : Fact (1 \u2264 p)\nhp : p \u2260 \u22a4\ninst\u271d : Measure.WeaklyRegular \u03bc\nf : { x // x \u2208 Lp E p }\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nA : ENNReal.ofReal \u03b5 \u2260 0\ng : \u03b1 \u2192\u1d47 E\nhg : snorm (\u2191\u2191f - \u2191g) p \u03bc \u2264 ENNReal.ofReal \u03b5\ng_mem : Mem\u2112p (\u2191g) p\n\u22a2 \u2203 x, x \u2208 Metric.closedBall f \u03b5 \u2227 x \u2208 \u2191(boundedContinuousFunction E p \u03bc)"}, {"tactic": "refine' \u27e8g_mem.toLp _, _, \u27e8g, rfl\u27e9\u27e9", "annotated_tactic": ["refine' \u27e8g_mem.toLp _, _, \u27e8g, <a>rfl</a>\u27e9\u27e9", [{"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "case intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : T4Space \u03b1\ninst\u271d\u2074 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : SecondCountableTopologyEither \u03b1 E\n_i : Fact (1 \u2264 p)\nhp : p \u2260 \u22a4\ninst\u271d : Measure.WeaklyRegular \u03bc\nf : { x // x \u2208 Lp E p }\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nA : ENNReal.ofReal \u03b5 \u2260 0\ng : \u03b1 \u2192\u1d47 E\nhg : snorm (\u2191\u2191f - \u2191g) p \u03bc \u2264 ENNReal.ofReal \u03b5\ng_mem : Mem\u2112p (\u2191g) p\n\u22a2 \u2203 x, x \u2208 Metric.closedBall f \u03b5 \u2227 x \u2208 \u2191(boundedContinuousFunction E p \u03bc)", "state_after": "case intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : T4Space \u03b1\ninst\u271d\u2074 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : SecondCountableTopologyEither \u03b1 E\n_i : Fact (1 \u2264 p)\nhp : p \u2260 \u22a4\ninst\u271d : Measure.WeaklyRegular \u03bc\nf : { x // x \u2208 Lp E p }\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nA : ENNReal.ofReal \u03b5 \u2260 0\ng : \u03b1 \u2192\u1d47 E\nhg : snorm (\u2191\u2191f - \u2191g) p \u03bc \u2264 ENNReal.ofReal \u03b5\ng_mem : Mem\u2112p (\u2191g) p\n\u22a2 Mem\u2112p.toLp (\u2191g) g_mem \u2208 Metric.closedBall f \u03b5"}, {"tactic": "simp only [dist_eq_norm, Metric.mem_closedBall']", "annotated_tactic": ["simp only [<a>dist_eq_norm</a>, <a>Metric.mem_closedBall'</a>]", [{"full_name": "dist_eq_norm", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [383, 7], "def_end_pos": [383, 19]}, {"full_name": "Metric.mem_closedBall'", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [481, 9], "def_end_pos": [481, 24]}]], "state_before": "case intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : T4Space \u03b1\ninst\u271d\u2074 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : SecondCountableTopologyEither \u03b1 E\n_i : Fact (1 \u2264 p)\nhp : p \u2260 \u22a4\ninst\u271d : Measure.WeaklyRegular \u03bc\nf : { x // x \u2208 Lp E p }\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nA : ENNReal.ofReal \u03b5 \u2260 0\ng : \u03b1 \u2192\u1d47 E\nhg : snorm (\u2191\u2191f - \u2191g) p \u03bc \u2264 ENNReal.ofReal \u03b5\ng_mem : Mem\u2112p (\u2191g) p\n\u22a2 Mem\u2112p.toLp (\u2191g) g_mem \u2208 Metric.closedBall f \u03b5", "state_after": "case intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : T4Space \u03b1\ninst\u271d\u2074 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : SecondCountableTopologyEither \u03b1 E\n_i : Fact (1 \u2264 p)\nhp : p \u2260 \u22a4\ninst\u271d : Measure.WeaklyRegular \u03bc\nf : { x // x \u2208 Lp E p }\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nA : ENNReal.ofReal \u03b5 \u2260 0\ng : \u03b1 \u2192\u1d47 E\nhg : snorm (\u2191\u2191f - \u2191g) p \u03bc \u2264 ENNReal.ofReal \u03b5\ng_mem : Mem\u2112p (\u2191g) p\n\u22a2 \u2016f - Mem\u2112p.toLp (\u2191g) g_mem\u2016 \u2264 \u03b5"}, {"tactic": "rw [Lp.norm_def]", "annotated_tactic": ["rw [<a>Lp.norm_def</a>]", [{"full_name": "MeasureTheory.Lp.norm_def", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [259, 9], "def_end_pos": [259, 17]}]], "state_before": "case intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : T4Space \u03b1\ninst\u271d\u2074 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : SecondCountableTopologyEither \u03b1 E\n_i : Fact (1 \u2264 p)\nhp : p \u2260 \u22a4\ninst\u271d : Measure.WeaklyRegular \u03bc\nf : { x // x \u2208 Lp E p }\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nA : ENNReal.ofReal \u03b5 \u2260 0\ng : \u03b1 \u2192\u1d47 E\nhg : snorm (\u2191\u2191f - \u2191g) p \u03bc \u2264 ENNReal.ofReal \u03b5\ng_mem : Mem\u2112p (\u2191g) p\n\u22a2 \u2016f - Mem\u2112p.toLp (\u2191g) g_mem\u2016 \u2264 \u03b5", "state_after": "case intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : T4Space \u03b1\ninst\u271d\u2074 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : SecondCountableTopologyEither \u03b1 E\n_i : Fact (1 \u2264 p)\nhp : p \u2260 \u22a4\ninst\u271d : Measure.WeaklyRegular \u03bc\nf : { x // x \u2208 Lp E p }\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nA : ENNReal.ofReal \u03b5 \u2260 0\ng : \u03b1 \u2192\u1d47 E\nhg : snorm (\u2191\u2191f - \u2191g) p \u03bc \u2264 ENNReal.ofReal \u03b5\ng_mem : Mem\u2112p (\u2191g) p\n\u22a2 ENNReal.toReal (snorm (\u2191\u2191(f - Mem\u2112p.toLp (\u2191g) g_mem)) p \u03bc) \u2264 \u03b5"}, {"tactic": "have key : snorm ((f : \u03b1 \u2192 E) - (g : \u03b1 \u2192 E)) p \u03bc = snorm (f - Mem\u2112p.toLp (\u2191g) g_mem) p \u03bc := by\n  apply snorm_congr_ae\n  filter_upwards [coeFn_sub f (g_mem.toLp g), g_mem.coeFn_toLp] with x hx h'x\n  simp only [hx, Pi.sub_apply, sub_right_inj, h'x]", "annotated_tactic": ["have key : <a>snorm</a> ((f : \u03b1 \u2192 E) - (g : \u03b1 \u2192 E)) p \u03bc = <a>snorm</a> (f - <a>Mem\u2112p.toLp</a> (\u2191g) g_mem) p \u03bc := by\n    apply <a>snorm_congr_ae</a>\n    filter_upwards [<a>coeFn_sub</a> f (g_mem.toLp g), g_mem.coeFn_toLp] with x hx h'x\n    simp only [hx, <a>Pi.sub_apply</a>, <a>sub_right_inj</a>, h'x]", [{"full_name": "MeasureTheory.snorm", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [84, 5], "def_end_pos": [84, 10]}, {"full_name": "MeasureTheory.snorm", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [84, 5], "def_end_pos": [84, 10]}, {"full_name": "MeasureTheory.Mem\u2112p.toLp", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [115, 5], "def_end_pos": [115, 9]}, {"full_name": "MeasureTheory.snorm_congr_ae", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [549, 9], "def_end_pos": [549, 23]}, {"full_name": "MeasureTheory.Lp.coeFn_sub", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [236, 9], "def_end_pos": [236, 18]}, {"full_name": "Pi.sub_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [200, 3], "def_end_pos": [200, 14]}, {"full_name": "sub_right_inj", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [776, 3], "def_end_pos": [776, 14]}]], "state_before": "case intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : T4Space \u03b1\ninst\u271d\u2074 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : SecondCountableTopologyEither \u03b1 E\n_i : Fact (1 \u2264 p)\nhp : p \u2260 \u22a4\ninst\u271d : Measure.WeaklyRegular \u03bc\nf : { x // x \u2208 Lp E p }\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nA : ENNReal.ofReal \u03b5 \u2260 0\ng : \u03b1 \u2192\u1d47 E\nhg : snorm (\u2191\u2191f - \u2191g) p \u03bc \u2264 ENNReal.ofReal \u03b5\ng_mem : Mem\u2112p (\u2191g) p\n\u22a2 ENNReal.toReal (snorm (\u2191\u2191(f - Mem\u2112p.toLp (\u2191g) g_mem)) p \u03bc) \u2264 \u03b5", "state_after": "case intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : T4Space \u03b1\ninst\u271d\u2074 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : SecondCountableTopologyEither \u03b1 E\n_i : Fact (1 \u2264 p)\nhp : p \u2260 \u22a4\ninst\u271d : Measure.WeaklyRegular \u03bc\nf : { x // x \u2208 Lp E p }\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nA : ENNReal.ofReal \u03b5 \u2260 0\ng : \u03b1 \u2192\u1d47 E\nhg : snorm (\u2191\u2191f - \u2191g) p \u03bc \u2264 ENNReal.ofReal \u03b5\ng_mem : Mem\u2112p (\u2191g) p\nkey : snorm (\u2191\u2191f - \u2191g) p \u03bc = snorm (\u2191\u2191(f - Mem\u2112p.toLp (\u2191g) g_mem)) p \u03bc\n\u22a2 ENNReal.toReal (snorm (\u2191\u2191(f - Mem\u2112p.toLp (\u2191g) g_mem)) p \u03bc) \u2264 \u03b5"}, {"tactic": "simpa only [key] using ENNReal.toReal_le_of_le_ofReal h\u03b5.le hg", "annotated_tactic": ["simpa only [key] using <a>ENNReal.toReal_le_of_le_ofReal</a> h\u03b5.le hg", [{"full_name": "ENNReal.toReal_le_of_le_ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2210, 9], "def_end_pos": [2210, 31]}]], "state_before": "case intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : T4Space \u03b1\ninst\u271d\u2074 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : SecondCountableTopologyEither \u03b1 E\n_i : Fact (1 \u2264 p)\nhp : p \u2260 \u22a4\ninst\u271d : Measure.WeaklyRegular \u03bc\nf : { x // x \u2208 Lp E p }\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nA : ENNReal.ofReal \u03b5 \u2260 0\ng : \u03b1 \u2192\u1d47 E\nhg : snorm (\u2191\u2191f - \u2191g) p \u03bc \u2264 ENNReal.ofReal \u03b5\ng_mem : Mem\u2112p (\u2191g) p\nkey : snorm (\u2191\u2191f - \u2191g) p \u03bc = snorm (\u2191\u2191(f - Mem\u2112p.toLp (\u2191g) g_mem)) p \u03bc\n\u22a2 ENNReal.toReal (snorm (\u2191\u2191(f - Mem\u2112p.toLp (\u2191g) g_mem)) p \u03bc) \u2264 \u03b5", "state_after": "no goals"}, {"tactic": "simp only [Ne.def, ENNReal.ofReal_eq_zero, not_le, h\u03b5]", "annotated_tactic": ["simp only [<a>Ne.def</a>, <a>ENNReal.ofReal_eq_zero</a>, <a>not_le</a>, h\u03b5]", [{"full_name": "Ne.def", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [59, 9], "def_end_pos": [59, 15]}, {"full_name": "ENNReal.ofReal_eq_zero", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2170, 9], "def_end_pos": [2170, 23]}, {"full_name": "not_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [373, 9], "def_end_pos": [373, 15]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : T4Space \u03b1\ninst\u271d\u2074 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : SecondCountableTopologyEither \u03b1 E\n_i : Fact (1 \u2264 p)\nhp : p \u2260 \u22a4\ninst\u271d : Measure.WeaklyRegular \u03bc\nf : { x // x \u2208 Lp E p }\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u22a2 ENNReal.ofReal \u03b5 \u2260 0", "state_after": "no goals"}, {"tactic": "apply snorm_congr_ae", "annotated_tactic": ["apply <a>snorm_congr_ae</a>", [{"full_name": "MeasureTheory.snorm_congr_ae", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [549, 9], "def_end_pos": [549, 23]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : T4Space \u03b1\ninst\u271d\u2074 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : SecondCountableTopologyEither \u03b1 E\n_i : Fact (1 \u2264 p)\nhp : p \u2260 \u22a4\ninst\u271d : Measure.WeaklyRegular \u03bc\nf : { x // x \u2208 Lp E p }\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nA : ENNReal.ofReal \u03b5 \u2260 0\ng : \u03b1 \u2192\u1d47 E\nhg : snorm (\u2191\u2191f - \u2191g) p \u03bc \u2264 ENNReal.ofReal \u03b5\ng_mem : Mem\u2112p (\u2191g) p\n\u22a2 snorm (\u2191\u2191f - \u2191g) p \u03bc = snorm (\u2191\u2191(f - Mem\u2112p.toLp (\u2191g) g_mem)) p \u03bc", "state_after": "case hfg\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : T4Space \u03b1\ninst\u271d\u2074 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : SecondCountableTopologyEither \u03b1 E\n_i : Fact (1 \u2264 p)\nhp : p \u2260 \u22a4\ninst\u271d : Measure.WeaklyRegular \u03bc\nf : { x // x \u2208 Lp E p }\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nA : ENNReal.ofReal \u03b5 \u2260 0\ng : \u03b1 \u2192\u1d47 E\nhg : snorm (\u2191\u2191f - \u2191g) p \u03bc \u2264 ENNReal.ofReal \u03b5\ng_mem : Mem\u2112p (\u2191g) p\n\u22a2 \u2191\u2191f - \u2191g =\u1d50[\u03bc] \u2191\u2191(f - Mem\u2112p.toLp (\u2191g) g_mem)"}, {"tactic": "filter_upwards [coeFn_sub f (g_mem.toLp g), g_mem.coeFn_toLp] with x hx h'x", "annotated_tactic": ["filter_upwards [<a>coeFn_sub</a> f (g_mem.toLp g), g_mem.coeFn_toLp] with x hx h'x", [{"full_name": "MeasureTheory.Lp.coeFn_sub", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [236, 9], "def_end_pos": [236, 18]}]], "state_before": "case hfg\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : T4Space \u03b1\ninst\u271d\u2074 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : SecondCountableTopologyEither \u03b1 E\n_i : Fact (1 \u2264 p)\nhp : p \u2260 \u22a4\ninst\u271d : Measure.WeaklyRegular \u03bc\nf : { x // x \u2208 Lp E p }\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nA : ENNReal.ofReal \u03b5 \u2260 0\ng : \u03b1 \u2192\u1d47 E\nhg : snorm (\u2191\u2191f - \u2191g) p \u03bc \u2264 ENNReal.ofReal \u03b5\ng_mem : Mem\u2112p (\u2191g) p\n\u22a2 \u2191\u2191f - \u2191g =\u1d50[\u03bc] \u2191\u2191(f - Mem\u2112p.toLp (\u2191g) g_mem)", "state_after": "case h\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : T4Space \u03b1\ninst\u271d\u2074 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : SecondCountableTopologyEither \u03b1 E\n_i : Fact (1 \u2264 p)\nhp : p \u2260 \u22a4\ninst\u271d : Measure.WeaklyRegular \u03bc\nf : { x // x \u2208 Lp E p }\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nA : ENNReal.ofReal \u03b5 \u2260 0\ng : \u03b1 \u2192\u1d47 E\nhg : snorm (\u2191\u2191f - \u2191g) p \u03bc \u2264 ENNReal.ofReal \u03b5\ng_mem : Mem\u2112p (\u2191g) p\nx : \u03b1\nhx : \u2191\u2191(f - Mem\u2112p.toLp (\u2191g) g_mem) x = (\u2191\u2191f - \u2191\u2191(Mem\u2112p.toLp (\u2191g) g_mem)) x\nh'x : \u2191\u2191(Mem\u2112p.toLp (\u2191g) g_mem) x = \u2191g x\n\u22a2 (\u2191\u2191f - \u2191g) x = \u2191\u2191(f - Mem\u2112p.toLp (\u2191g) g_mem) x"}, {"tactic": "simp only [hx, Pi.sub_apply, sub_right_inj, h'x]", "annotated_tactic": ["simp only [hx, <a>Pi.sub_apply</a>, <a>sub_right_inj</a>, h'x]", [{"full_name": "Pi.sub_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [200, 3], "def_end_pos": [200, 14]}, {"full_name": "sub_right_inj", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [776, 3], "def_end_pos": [776, 14]}]], "state_before": "case h\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : T4Space \u03b1\ninst\u271d\u2074 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : SecondCountableTopologyEither \u03b1 E\n_i : Fact (1 \u2264 p)\nhp : p \u2260 \u22a4\ninst\u271d : Measure.WeaklyRegular \u03bc\nf : { x // x \u2208 Lp E p }\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nA : ENNReal.ofReal \u03b5 \u2260 0\ng : \u03b1 \u2192\u1d47 E\nhg : snorm (\u2191\u2191f - \u2191g) p \u03bc \u2264 ENNReal.ofReal \u03b5\ng_mem : Mem\u2112p (\u2191g) p\nx : \u03b1\nhx : \u2191\u2191(f - Mem\u2112p.toLp (\u2191g) g_mem) x = (\u2191\u2191f - \u2191\u2191(Mem\u2112p.toLp (\u2191g) g_mem)) x\nh'x : \u2191\u2191(Mem\u2112p.toLp (\u2191g) g_mem) x = \u2191g x\n\u22a2 (\u2191\u2191f - \u2191g) x = \u2191\u2191(f - Mem\u2112p.toLp (\u2191g) g_mem) x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finmap.lean", "full_name": "Finmap.erase_erase", "start": [455, 1], "end": [456, 81], "traced_tactics": [{"tactic": "simp only [AList.erase_erase, erase_toFinmap]", "annotated_tactic": ["simp only [<a>AList.erase_erase</a>, <a>erase_toFinmap</a>]", [{"full_name": "AList.erase_erase", "def_path": "Mathlib/Data/List/AList.lean", "def_pos": [260, 9], "def_end_pos": [260, 20]}, {"full_name": "Finmap.erase_toFinmap", "def_path": "Mathlib/Data/Finmap.lean", "def_pos": [420, 9], "def_end_pos": [420, 23]}]], "state_before": "\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\na a' : \u03b1\ns\u271d : Finmap \u03b2\ns : AList \u03b2\n\u22a2 (erase a (erase a' \u27e6s\u27e7)).entries = (erase a' (erase a \u27e6s\u27e7)).entries", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Sum/Lemmas.lean", "full_name": "Sum.liftRel_subrelation_lex", "start": [200, 1], "end": [200, 85], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/Jacobian.lean", "full_name": "MeasureTheory.addHaar_image_eq_zero_of_det_fderivWithin_eq_zero", "start": [659, 1], "end": [686, 10], "traced_tactics": [{"tactic": "suffices H : \u2200 R, \u03bc (f '' (s \u2229 closedBall 0 R)) = 0", "annotated_tactic": ["suffices H : \u2200 R, \u03bc (f '' (s \u2229 <a>closedBall</a> 0 R)) = 0", [{"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\n\u22a2 \u2191\u2191\u03bc (f '' s) = 0", "state_after": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nH : \u2200 (R : \u211d), \u2191\u2191\u03bc (f '' (s \u2229 closedBall 0 R)) = 0\n\u22a2 \u2191\u2191\u03bc (f '' s) = 0\n\ncase H\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\n\u22a2 \u2200 (R : \u211d), \u2191\u2191\u03bc (f '' (s \u2229 closedBall 0 R)) = 0"}, {"tactic": "intro R", "annotated_tactic": ["intro R", []], "state_before": "case H\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\n\u22a2 \u2200 (R : \u211d), \u2191\u2191\u03bc (f '' (s \u2229 closedBall 0 R)) = 0", "state_after": "case H\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nR : \u211d\n\u22a2 \u2191\u2191\u03bc (f '' (s \u2229 closedBall 0 R)) = 0"}, {"tactic": "have A : \u2200 (\u03b5 : \u211d\u22650), 0 < \u03b5 \u2192 \u03bc (f '' (s \u2229 closedBall 0 R)) \u2264 \u03b5 * \u03bc (closedBall 0 R) :=\n  fun \u03b5 \u03b5pos =>\n  addHaar_image_eq_zero_of_det_fderivWithin_eq_zero_aux \u03bc\n    (fun x hx => (hf' x hx.1).mono (inter_subset_left _ _)) R (inter_subset_right _ _) \u03b5 \u03b5pos\n    fun x hx => h'f' x hx.1", "annotated_tactic": ["have A : \u2200 (\u03b5 : \u211d\u22650), 0 < \u03b5 \u2192 \u03bc (f '' (s \u2229 <a>closedBall</a> 0 R)) \u2264 \u03b5 * \u03bc (<a>closedBall</a> 0 R) :=\n    fun \u03b5 \u03b5pos =>\n    <a>addHaar_image_eq_zero_of_det_fderivWithin_eq_zero_aux</a> \u03bc\n      (fun x hx => (hf' x hx.1).<a>mono</a> (<a>inter_subset_left</a> _ _)) R (<a>inter_subset_right</a> _ _) \u03b5 \u03b5pos\n      fun x hx => h'f' x hx.1", [{"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "MeasureTheory.addHaar_image_eq_zero_of_det_fderivWithin_eq_zero_aux", "def_path": "Mathlib/MeasureTheory/Function/Jacobian.lean", "def_pos": [606, 9], "def_end_pos": [606, 62]}, {"full_name": "HasFDerivWithinAt.mono", "def_path": "Mathlib/Analysis/Calculus/FDeriv/Basic.lean", "def_pos": [380, 16], "def_end_pos": [380, 38]}, {"full_name": "Set.inter_subset_left", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [965, 9], "def_end_pos": [965, 26]}, {"full_name": "Set.inter_subset_right", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [969, 9], "def_end_pos": [969, 27]}]], "state_before": "case H\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nR : \u211d\n\u22a2 \u2191\u2191\u03bc (f '' (s \u2229 closedBall 0 R)) = 0", "state_after": "case H\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nR : \u211d\nA : \u2200 (\u03b5 : \u211d\u22650), 0 < \u03b5 \u2192 \u2191\u2191\u03bc (f '' (s \u2229 closedBall 0 R)) \u2264 \u2191\u03b5 * \u2191\u2191\u03bc (closedBall 0 R)\n\u22a2 \u2191\u2191\u03bc (f '' (s \u2229 closedBall 0 R)) = 0"}, {"tactic": "have B : Tendsto (fun \u03b5 : \u211d\u22650 => (\u03b5 : \u211d\u22650\u221e) * \u03bc (closedBall 0 R)) (\ud835\udcdd[>] 0) (\ud835\udcdd 0) := by\n  have :\n    Tendsto (fun \u03b5 : \u211d\u22650 => (\u03b5 : \u211d\u22650\u221e) * \u03bc (closedBall 0 R)) (\ud835\udcdd 0)\n      (\ud835\udcdd (((0 : \u211d\u22650) : \u211d\u22650\u221e) * \u03bc (closedBall 0 R))) :=\n    ENNReal.Tendsto.mul_const (ENNReal.tendsto_coe.2 tendsto_id)\n      (Or.inr measure_closedBall_lt_top.ne)\n  simp only [zero_mul, ENNReal.coe_zero] at this\n  exact Tendsto.mono_left this nhdsWithin_le_nhds", "annotated_tactic": ["have B : <a>Tendsto</a> (fun \u03b5 : \u211d\u22650 => (\u03b5 : \u211d\u22650\u221e) * \u03bc (<a>closedBall</a> 0 R)) (\ud835\udcdd[>] 0) (\ud835\udcdd 0) := by\n    have :\n      <a>Tendsto</a> (fun \u03b5 : \u211d\u22650 => (\u03b5 : \u211d\u22650\u221e) * \u03bc (<a>closedBall</a> 0 R)) (\ud835\udcdd 0)\n        (\ud835\udcdd (((0 : \u211d\u22650) : \u211d\u22650\u221e) * \u03bc (<a>closedBall</a> 0 R))) :=\n      <a>ENNReal.Tendsto.mul_const</a> (<a>ENNReal.tendsto_coe</a>.2 <a>tendsto_id</a>)\n        (<a>Or.inr</a> measure_closedBall_lt_top.ne)\n    simp only [<a>zero_mul</a>, <a>ENNReal.coe_zero</a>] at this\n    exact <a>Tendsto.mono_left</a> this <a>nhdsWithin_le_nhds</a>", [{"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2939, 5], "def_end_pos": [2939, 12]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2939, 5], "def_end_pos": [2939, 12]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "ENNReal.Tendsto.mul_const", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [379, 19], "def_end_pos": [379, 36]}, {"full_name": "ENNReal.tendsto_coe", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [71, 9], "def_end_pos": [71, 20]}, {"full_name": "Filter.tendsto_id", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [3028, 9], "def_end_pos": [3028, 19]}, {"full_name": "Or.inr", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [519, 5], "def_end_pos": [519, 8]}, {"full_name": "MulZeroClass.zero_mul", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [36, 3], "def_end_pos": [36, 11]}, {"full_name": "ENNReal.coe_zero", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [215, 28], "def_end_pos": [215, 36]}, {"full_name": "Filter.Tendsto.mono_left", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [3036, 9], "def_end_pos": [3036, 26]}, {"full_name": "nhdsWithin_le_nhds", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [204, 9], "def_end_pos": [204, 27]}]], "state_before": "case H\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nR : \u211d\nA : \u2200 (\u03b5 : \u211d\u22650), 0 < \u03b5 \u2192 \u2191\u2191\u03bc (f '' (s \u2229 closedBall 0 R)) \u2264 \u2191\u03b5 * \u2191\u2191\u03bc (closedBall 0 R)\n\u22a2 \u2191\u2191\u03bc (f '' (s \u2229 closedBall 0 R)) = 0", "state_after": "case H\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nR : \u211d\nA : \u2200 (\u03b5 : \u211d\u22650), 0 < \u03b5 \u2192 \u2191\u2191\u03bc (f '' (s \u2229 closedBall 0 R)) \u2264 \u2191\u03b5 * \u2191\u2191\u03bc (closedBall 0 R)\nB : Tendsto (fun \u03b5 => \u2191\u03b5 * \u2191\u2191\u03bc (closedBall 0 R)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\n\u22a2 \u2191\u2191\u03bc (f '' (s \u2229 closedBall 0 R)) = 0"}, {"tactic": "apply le_antisymm _ (zero_le _)", "annotated_tactic": ["apply <a>le_antisymm</a> _ (<a>zero_le</a> _)", [{"full_name": "le_antisymm", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [188, 9], "def_end_pos": [188, 20]}, {"full_name": "zero_le", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [217, 30], "def_end_pos": [217, 37]}]], "state_before": "case H\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nR : \u211d\nA : \u2200 (\u03b5 : \u211d\u22650), 0 < \u03b5 \u2192 \u2191\u2191\u03bc (f '' (s \u2229 closedBall 0 R)) \u2264 \u2191\u03b5 * \u2191\u2191\u03bc (closedBall 0 R)\nB : Tendsto (fun \u03b5 => \u2191\u03b5 * \u2191\u2191\u03bc (closedBall 0 R)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\n\u22a2 \u2191\u2191\u03bc (f '' (s \u2229 closedBall 0 R)) = 0", "state_after": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nR : \u211d\nA : \u2200 (\u03b5 : \u211d\u22650), 0 < \u03b5 \u2192 \u2191\u2191\u03bc (f '' (s \u2229 closedBall 0 R)) \u2264 \u2191\u03b5 * \u2191\u2191\u03bc (closedBall 0 R)\nB : Tendsto (fun \u03b5 => \u2191\u03b5 * \u2191\u2191\u03bc (closedBall 0 R)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\n\u22a2 \u2191\u2191\u03bc (f '' (s \u2229 closedBall 0 R)) \u2264 0"}, {"tactic": "apply ge_of_tendsto B", "annotated_tactic": ["apply <a>ge_of_tendsto</a> B", [{"full_name": "ge_of_tendsto", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [168, 9], "def_end_pos": [168, 22]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nR : \u211d\nA : \u2200 (\u03b5 : \u211d\u22650), 0 < \u03b5 \u2192 \u2191\u2191\u03bc (f '' (s \u2229 closedBall 0 R)) \u2264 \u2191\u03b5 * \u2191\u2191\u03bc (closedBall 0 R)\nB : Tendsto (fun \u03b5 => \u2191\u03b5 * \u2191\u2191\u03bc (closedBall 0 R)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\n\u22a2 \u2191\u2191\u03bc (f '' (s \u2229 closedBall 0 R)) \u2264 0", "state_after": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nR : \u211d\nA : \u2200 (\u03b5 : \u211d\u22650), 0 < \u03b5 \u2192 \u2191\u2191\u03bc (f '' (s \u2229 closedBall 0 R)) \u2264 \u2191\u03b5 * \u2191\u2191\u03bc (closedBall 0 R)\nB : Tendsto (fun \u03b5 => \u2191\u03b5 * \u2191\u2191\u03bc (closedBall 0 R)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\n\u22a2 \u2200\u1da0 (c : \u211d\u22650) in \ud835\udcdd[Ioi 0] 0, \u2191\u2191\u03bc (f '' (s \u2229 closedBall 0 R)) \u2264 \u2191c * \u2191\u2191\u03bc (closedBall 0 R)"}, {"tactic": "filter_upwards [self_mem_nhdsWithin]", "annotated_tactic": ["filter_upwards [<a>self_mem_nhdsWithin</a>]", [{"full_name": "self_mem_nhdsWithin", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [151, 9], "def_end_pos": [151, 28]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nR : \u211d\nA : \u2200 (\u03b5 : \u211d\u22650), 0 < \u03b5 \u2192 \u2191\u2191\u03bc (f '' (s \u2229 closedBall 0 R)) \u2264 \u2191\u03b5 * \u2191\u2191\u03bc (closedBall 0 R)\nB : Tendsto (fun \u03b5 => \u2191\u03b5 * \u2191\u2191\u03bc (closedBall 0 R)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\n\u22a2 \u2200\u1da0 (c : \u211d\u22650) in \ud835\udcdd[Ioi 0] 0, \u2191\u2191\u03bc (f '' (s \u2229 closedBall 0 R)) \u2264 \u2191c * \u2191\u2191\u03bc (closedBall 0 R)", "state_after": "case h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nR : \u211d\nA : \u2200 (\u03b5 : \u211d\u22650), 0 < \u03b5 \u2192 \u2191\u2191\u03bc (f '' (s \u2229 closedBall 0 R)) \u2264 \u2191\u03b5 * \u2191\u2191\u03bc (closedBall 0 R)\nB : Tendsto (fun \u03b5 => \u2191\u03b5 * \u2191\u2191\u03bc (closedBall 0 R)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\n\u22a2 \u2200 (a : \u211d\u22650), a \u2208 Ioi 0 \u2192 \u2191\u2191\u03bc (f '' (s \u2229 closedBall 0 R)) \u2264 \u2191a * \u2191\u2191\u03bc (closedBall 0 R)"}, {"tactic": "exact A", "annotated_tactic": ["exact A", []], "state_before": "case h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nR : \u211d\nA : \u2200 (\u03b5 : \u211d\u22650), 0 < \u03b5 \u2192 \u2191\u2191\u03bc (f '' (s \u2229 closedBall 0 R)) \u2264 \u2191\u03b5 * \u2191\u2191\u03bc (closedBall 0 R)\nB : Tendsto (fun \u03b5 => \u2191\u03b5 * \u2191\u2191\u03bc (closedBall 0 R)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\n\u22a2 \u2200 (a : \u211d\u22650), a \u2208 Ioi 0 \u2192 \u2191\u2191\u03bc (f '' (s \u2229 closedBall 0 R)) \u2264 \u2191a * \u2191\u2191\u03bc (closedBall 0 R)", "state_after": "no goals"}, {"tactic": "apply le_antisymm _ (zero_le _)", "annotated_tactic": ["apply <a>le_antisymm</a> _ (<a>zero_le</a> _)", [{"full_name": "le_antisymm", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [188, 9], "def_end_pos": [188, 20]}, {"full_name": "zero_le", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [217, 30], "def_end_pos": [217, 37]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nH : \u2200 (R : \u211d), \u2191\u2191\u03bc (f '' (s \u2229 closedBall 0 R)) = 0\n\u22a2 \u2191\u2191\u03bc (f '' s) = 0", "state_after": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nH : \u2200 (R : \u211d), \u2191\u2191\u03bc (f '' (s \u2229 closedBall 0 R)) = 0\n\u22a2 \u2191\u2191\u03bc (f '' s) \u2264 0"}, {"tactic": "rw [\u2190 iUnion_inter_closedBall_nat s 0]", "annotated_tactic": ["rw [\u2190 <a>iUnion_inter_closedBall_nat</a> s 0]", [{"full_name": "Metric.iUnion_inter_closedBall_nat", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [667, 9], "def_end_pos": [667, 36]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nH : \u2200 (R : \u211d), \u2191\u2191\u03bc (f '' (s \u2229 closedBall 0 R)) = 0\n\u22a2 \u2191\u2191\u03bc (f '' s) \u2264 0", "state_after": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nH : \u2200 (R : \u211d), \u2191\u2191\u03bc (f '' (s \u2229 closedBall 0 R)) = 0\n\u22a2 \u2191\u2191\u03bc (f '' \u22c3 n, s \u2229 closedBall 0 \u2191n) \u2264 0"}, {"tactic": "calc\n  \u03bc (f '' \u22c3 n : \u2115, s \u2229 closedBall 0 n) \u2264 \u2211' n : \u2115, \u03bc (f '' (s \u2229 closedBall 0 n)) := by\n    rw [image_iUnion]; exact measure_iUnion_le _\n  _ \u2264 0 := by simp only [H, tsum_zero, nonpos_iff_eq_zero]", "annotated_tactic": ["calc\n      \u03bc (f '' \u22c3 n : \u2115, s \u2229 <a>closedBall</a> 0 n) \u2264 \u2211' n : \u2115, \u03bc (f '' (s \u2229 <a>closedBall</a> 0 n)) := by\n        rw [<a>image_iUnion</a>]; exact <a>measure_iUnion_le</a> _\n      _ \u2264 0 := by simp only [H, <a>tsum_zero</a>, <a>nonpos_iff_eq_zero</a>]", [{"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "Set.image_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [1791, 9], "def_end_pos": [1791, 21]}, {"full_name": "MeasureTheory.measure_iUnion_le", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [240, 9], "def_end_pos": [240, 26]}, {"full_name": "tsum_zero", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/Basic.lean", "def_pos": [489, 9], "def_end_pos": [489, 18]}, {"full_name": "nonpos_iff_eq_zero", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [237, 3], "def_end_pos": [237, 14]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nH : \u2200 (R : \u211d), \u2191\u2191\u03bc (f '' (s \u2229 closedBall 0 R)) = 0\n\u22a2 \u2191\u2191\u03bc (f '' \u22c3 n, s \u2229 closedBall 0 \u2191n) \u2264 0", "state_after": "no goals"}, {"tactic": "rw [image_iUnion]", "annotated_tactic": ["rw [<a>image_iUnion</a>]", [{"full_name": "Set.image_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [1791, 9], "def_end_pos": [1791, 21]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nH : \u2200 (R : \u211d), \u2191\u2191\u03bc (f '' (s \u2229 closedBall 0 R)) = 0\n\u22a2 \u2191\u2191\u03bc (f '' \u22c3 n, s \u2229 closedBall 0 \u2191n) \u2264 \u2211' (n : \u2115), \u2191\u2191\u03bc (f '' (s \u2229 closedBall 0 \u2191n))", "state_after": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nH : \u2200 (R : \u211d), \u2191\u2191\u03bc (f '' (s \u2229 closedBall 0 R)) = 0\n\u22a2 \u2191\u2191\u03bc (\u22c3 i, f '' (s \u2229 closedBall 0 \u2191i)) \u2264 \u2211' (n : \u2115), \u2191\u2191\u03bc (f '' (s \u2229 closedBall 0 \u2191n))"}, {"tactic": "exact measure_iUnion_le _", "annotated_tactic": ["exact <a>measure_iUnion_le</a> _", [{"full_name": "MeasureTheory.measure_iUnion_le", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [240, 9], "def_end_pos": [240, 26]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nH : \u2200 (R : \u211d), \u2191\u2191\u03bc (f '' (s \u2229 closedBall 0 R)) = 0\n\u22a2 \u2191\u2191\u03bc (\u22c3 i, f '' (s \u2229 closedBall 0 \u2191i)) \u2264 \u2211' (n : \u2115), \u2191\u2191\u03bc (f '' (s \u2229 closedBall 0 \u2191n))", "state_after": "no goals"}, {"tactic": "simp only [H, tsum_zero, nonpos_iff_eq_zero]", "annotated_tactic": ["simp only [H, <a>tsum_zero</a>, <a>nonpos_iff_eq_zero</a>]", [{"full_name": "tsum_zero", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/Basic.lean", "def_pos": [489, 9], "def_end_pos": [489, 18]}, {"full_name": "nonpos_iff_eq_zero", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [237, 3], "def_end_pos": [237, 14]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nH : \u2200 (R : \u211d), \u2191\u2191\u03bc (f '' (s \u2229 closedBall 0 R)) = 0\n\u22a2 \u2211' (n : \u2115), \u2191\u2191\u03bc (f '' (s \u2229 closedBall 0 \u2191n)) \u2264 0", "state_after": "no goals"}, {"tactic": "have :\n  Tendsto (fun \u03b5 : \u211d\u22650 => (\u03b5 : \u211d\u22650\u221e) * \u03bc (closedBall 0 R)) (\ud835\udcdd 0)\n    (\ud835\udcdd (((0 : \u211d\u22650) : \u211d\u22650\u221e) * \u03bc (closedBall 0 R))) :=\n  ENNReal.Tendsto.mul_const (ENNReal.tendsto_coe.2 tendsto_id)\n    (Or.inr measure_closedBall_lt_top.ne)", "annotated_tactic": ["have :\n      <a>Tendsto</a> (fun \u03b5 : \u211d\u22650 => (\u03b5 : \u211d\u22650\u221e) * \u03bc (<a>closedBall</a> 0 R)) (\ud835\udcdd 0)\n        (\ud835\udcdd (((0 : \u211d\u22650) : \u211d\u22650\u221e) * \u03bc (<a>closedBall</a> 0 R))) :=\n      <a>ENNReal.Tendsto.mul_const</a> (<a>ENNReal.tendsto_coe</a>.2 <a>tendsto_id</a>)\n        (<a>Or.inr</a> measure_closedBall_lt_top.ne)", [{"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2939, 5], "def_end_pos": [2939, 12]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "ENNReal.Tendsto.mul_const", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [379, 19], "def_end_pos": [379, 36]}, {"full_name": "ENNReal.tendsto_coe", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [71, 9], "def_end_pos": [71, 20]}, {"full_name": "Filter.tendsto_id", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [3028, 9], "def_end_pos": [3028, 19]}, {"full_name": "Or.inr", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [519, 5], "def_end_pos": [519, 8]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nR : \u211d\nA : \u2200 (\u03b5 : \u211d\u22650), 0 < \u03b5 \u2192 \u2191\u2191\u03bc (f '' (s \u2229 closedBall 0 R)) \u2264 \u2191\u03b5 * \u2191\u2191\u03bc (closedBall 0 R)\n\u22a2 Tendsto (fun \u03b5 => \u2191\u03b5 * \u2191\u2191\u03bc (closedBall 0 R)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)", "state_after": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nR : \u211d\nA : \u2200 (\u03b5 : \u211d\u22650), 0 < \u03b5 \u2192 \u2191\u2191\u03bc (f '' (s \u2229 closedBall 0 R)) \u2264 \u2191\u03b5 * \u2191\u2191\u03bc (closedBall 0 R)\nthis : Tendsto (fun \u03b5 => \u2191\u03b5 * \u2191\u2191\u03bc (closedBall 0 R)) (\ud835\udcdd 0) (\ud835\udcdd (\u21910 * \u2191\u2191\u03bc (closedBall 0 R)))\n\u22a2 Tendsto (fun \u03b5 => \u2191\u03b5 * \u2191\u2191\u03bc (closedBall 0 R)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)"}, {"tactic": "simp only [zero_mul, ENNReal.coe_zero] at this", "annotated_tactic": ["simp only [<a>zero_mul</a>, <a>ENNReal.coe_zero</a>] at this", [{"full_name": "MulZeroClass.zero_mul", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [36, 3], "def_end_pos": [36, 11]}, {"full_name": "ENNReal.coe_zero", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [215, 28], "def_end_pos": [215, 36]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nR : \u211d\nA : \u2200 (\u03b5 : \u211d\u22650), 0 < \u03b5 \u2192 \u2191\u2191\u03bc (f '' (s \u2229 closedBall 0 R)) \u2264 \u2191\u03b5 * \u2191\u2191\u03bc (closedBall 0 R)\nthis : Tendsto (fun \u03b5 => \u2191\u03b5 * \u2191\u2191\u03bc (closedBall 0 R)) (\ud835\udcdd 0) (\ud835\udcdd (\u21910 * \u2191\u2191\u03bc (closedBall 0 R)))\n\u22a2 Tendsto (fun \u03b5 => \u2191\u03b5 * \u2191\u2191\u03bc (closedBall 0 R)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)", "state_after": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nR : \u211d\nA : \u2200 (\u03b5 : \u211d\u22650), 0 < \u03b5 \u2192 \u2191\u2191\u03bc (f '' (s \u2229 closedBall 0 R)) \u2264 \u2191\u03b5 * \u2191\u2191\u03bc (closedBall 0 R)\nthis : Tendsto (fun \u03b5 => \u2191\u03b5 * \u2191\u2191\u03bc (closedBall 0 R)) (\ud835\udcdd 0) (\ud835\udcdd 0)\n\u22a2 Tendsto (fun \u03b5 => \u2191\u03b5 * \u2191\u2191\u03bc (closedBall 0 R)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)"}, {"tactic": "exact Tendsto.mono_left this nhdsWithin_le_nhds", "annotated_tactic": ["exact <a>Tendsto.mono_left</a> this <a>nhdsWithin_le_nhds</a>", [{"full_name": "Filter.Tendsto.mono_left", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [3036, 9], "def_end_pos": [3036, 26]}, {"full_name": "nhdsWithin_le_nhds", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [204, 9], "def_end_pos": [204, 27]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nR : \u211d\nA : \u2200 (\u03b5 : \u211d\u22650), 0 < \u03b5 \u2192 \u2191\u2191\u03bc (f '' (s \u2229 closedBall 0 R)) \u2264 \u2191\u03b5 * \u2191\u2191\u03bc (closedBall 0 R)\nthis : Tendsto (fun \u03b5 => \u2191\u03b5 * \u2191\u2191\u03bc (closedBall 0 R)) (\ud835\udcdd 0) (\ud835\udcdd 0)\n\u22a2 Tendsto (fun \u03b5 => \u2191\u03b5 * \u2191\u2191\u03bc (closedBall 0 R)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Card.lean", "full_name": "Set.encard_le_coe_iff_finite_ncard_le", "start": [499, 1], "end": [502, 54], "traced_tactics": [{"tactic": "rw [encard_le_coe_iff, and_congr_right_iff]", "annotated_tactic": ["rw [<a>encard_le_coe_iff</a>, <a>and_congr_right_iff</a>]", [{"full_name": "Set.encard_le_coe_iff", "def_path": "Mathlib/Data/Set/Card.lean", "def_pos": [148, 9], "def_end_pos": [148, 26]}, {"full_name": "and_congr_right_iff", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [219, 17], "def_end_pos": [219, 36]}]], "state_before": "\u03b1 : Type u_1\ns t : Set \u03b1\nk : \u2115\n\u22a2 encard s \u2264 \u2191k \u2194 Set.Finite s \u2227 ncard s \u2264 k", "state_after": "\u03b1 : Type u_1\ns t : Set \u03b1\nk : \u2115\n\u22a2 Set.Finite s \u2192 ((\u2203 n\u2080, encard s = \u2191n\u2080 \u2227 n\u2080 \u2264 k) \u2194 ncard s \u2264 k)"}, {"tactic": "exact fun hfin \u21a6 \u27e8fun \u27e8n\u2080, hn\u2080, hle\u27e9 \u21a6 by rwa [ncard_def, hn\u2080, ENat.toNat_coe],\n  fun h \u21a6 \u27e8s.ncard, by rw [hfin.cast_ncard_eq], h\u27e9\u27e9", "annotated_tactic": ["exact fun hfin \u21a6 \u27e8fun \u27e8n\u2080, hn\u2080, hle\u27e9 \u21a6 by rwa [<a>ncard_def</a>, hn\u2080, <a>ENat.toNat_coe</a>],\n    fun h \u21a6 \u27e8s.ncard, by rw [hfin.cast_ncard_eq], h\u27e9\u27e9", [{"full_name": "Set.ncard_def", "def_path": "Mathlib/Data/Set/Card.lean", "def_pos": [475, 9], "def_end_pos": [475, 18]}, {"full_name": "ENat.toNat_coe", "def_path": "Mathlib/Data/ENat/Basic.lean", "def_pos": [109, 9], "def_end_pos": [109, 18]}]], "state_before": "\u03b1 : Type u_1\ns t : Set \u03b1\nk : \u2115\n\u22a2 Set.Finite s \u2192 ((\u2203 n\u2080, encard s = \u2191n\u2080 \u2227 n\u2080 \u2264 k) \u2194 ncard s \u2264 k)", "state_after": "no goals"}, {"tactic": "rwa [ncard_def, hn\u2080, ENat.toNat_coe]", "annotated_tactic": ["rwa [<a>ncard_def</a>, hn\u2080, <a>ENat.toNat_coe</a>]", [{"full_name": "Set.ncard_def", "def_path": "Mathlib/Data/Set/Card.lean", "def_pos": [475, 9], "def_end_pos": [475, 18]}, {"full_name": "ENat.toNat_coe", "def_path": "Mathlib/Data/ENat/Basic.lean", "def_pos": [109, 9], "def_end_pos": [109, 18]}]], "state_before": "\u03b1 : Type u_1\ns t : Set \u03b1\nk : \u2115\nhfin : Set.Finite s\nx\u271d : \u2203 n\u2080, encard s = \u2191n\u2080 \u2227 n\u2080 \u2264 k\nn\u2080 : \u2115\nhn\u2080 : encard s = \u2191n\u2080\nhle : n\u2080 \u2264 k\n\u22a2 ncard s \u2264 k", "state_after": "no goals"}, {"tactic": "rw [hfin.cast_ncard_eq]", "annotated_tactic": ["rw [hfin.cast_ncard_eq]", []], "state_before": "\u03b1 : Type u_1\ns t : Set \u03b1\nk : \u2115\nhfin : Set.Finite s\nh : ncard s \u2264 k\n\u22a2 encard s = \u2191(ncard s)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Sign.lean", "full_name": "sign_sum", "start": [465, 1], "end": [473, 30], "traced_tactics": [{"tactic": "cases t", "annotated_tactic": ["cases t", []], "state_before": "\u03b1 : Type u_1\ninst\u271d : LinearOrderedAddCommGroup \u03b1\n\u03b9 : Type u_2\ns : Finset \u03b9\nf : \u03b9 \u2192 \u03b1\nhs : Finset.Nonempty s\nt : SignType\nh : \u2200 (i : \u03b9), i \u2208 s \u2192 \u2191sign (f i) = t\n\u22a2 \u2191sign (\u2211 i in s, f i) = t", "state_after": "case zero\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedAddCommGroup \u03b1\n\u03b9 : Type u_2\ns : Finset \u03b9\nf : \u03b9 \u2192 \u03b1\nhs : Finset.Nonempty s\nh : \u2200 (i : \u03b9), i \u2208 s \u2192 \u2191sign (f i) = zero\n\u22a2 \u2191sign (\u2211 i in s, f i) = zero\n\ncase neg\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedAddCommGroup \u03b1\n\u03b9 : Type u_2\ns : Finset \u03b9\nf : \u03b9 \u2192 \u03b1\nhs : Finset.Nonempty s\nh : \u2200 (i : \u03b9), i \u2208 s \u2192 \u2191sign (f i) = neg\n\u22a2 \u2191sign (\u2211 i in s, f i) = neg\n\ncase pos\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedAddCommGroup \u03b1\n\u03b9 : Type u_2\ns : Finset \u03b9\nf : \u03b9 \u2192 \u03b1\nhs : Finset.Nonempty s\nh : \u2200 (i : \u03b9), i \u2208 s \u2192 \u2191sign (f i) = pos\n\u22a2 \u2191sign (\u2211 i in s, f i) = pos"}, {"tactic": "simp_rw [zero_eq_zero, sign_eq_zero_iff] at h \u22a2", "annotated_tactic": ["simp_rw [<a>zero_eq_zero</a>, <a>sign_eq_zero_iff</a>] at h \u22a2", [{"full_name": "SignType.zero_eq_zero", "def_path": "Mathlib/Data/Sign.lean", "def_pos": [53, 9], "def_end_pos": [53, 21]}, {"full_name": "sign_eq_zero_iff", "def_path": "Mathlib/Data/Sign.lean", "def_pos": [347, 9], "def_end_pos": [347, 25]}]], "state_before": "case zero\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedAddCommGroup \u03b1\n\u03b9 : Type u_2\ns : Finset \u03b9\nf : \u03b9 \u2192 \u03b1\nhs : Finset.Nonempty s\nh : \u2200 (i : \u03b9), i \u2208 s \u2192 \u2191sign (f i) = zero\n\u22a2 \u2191sign (\u2211 i in s, f i) = zero", "state_after": "case zero\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedAddCommGroup \u03b1\n\u03b9 : Type u_2\ns : Finset \u03b9\nf : \u03b9 \u2192 \u03b1\nhs : Finset.Nonempty s\nh : \u2200 (i : \u03b9), i \u2208 s \u2192 f i = 0\n\u22a2 \u2211 i in s, f i = 0"}, {"tactic": "exact Finset.sum_eq_zero h", "annotated_tactic": ["exact <a>Finset.sum_eq_zero</a> h", [{"full_name": "Finset.sum_eq_zero", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [728, 3], "def_end_pos": [728, 14]}]], "state_before": "case zero\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedAddCommGroup \u03b1\n\u03b9 : Type u_2\ns : Finset \u03b9\nf : \u03b9 \u2192 \u03b1\nhs : Finset.Nonempty s\nh : \u2200 (i : \u03b9), i \u2208 s \u2192 f i = 0\n\u22a2 \u2211 i in s, f i = 0", "state_after": "no goals"}, {"tactic": "simp_rw [neg_eq_neg_one, sign_eq_neg_one_iff] at h \u22a2", "annotated_tactic": ["simp_rw [<a>neg_eq_neg_one</a>, <a>sign_eq_neg_one_iff</a>] at h \u22a2", [{"full_name": "SignType.neg_eq_neg_one", "def_path": "Mathlib/Data/Sign.lean", "def_pos": [58, 9], "def_end_pos": [58, 23]}, {"full_name": "sign_eq_neg_one_iff", "def_path": "Mathlib/Data/Sign.lean", "def_pos": [333, 9], "def_end_pos": [333, 28]}]], "state_before": "case neg\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedAddCommGroup \u03b1\n\u03b9 : Type u_2\ns : Finset \u03b9\nf : \u03b9 \u2192 \u03b1\nhs : Finset.Nonempty s\nh : \u2200 (i : \u03b9), i \u2208 s \u2192 \u2191sign (f i) = neg\n\u22a2 \u2191sign (\u2211 i in s, f i) = neg", "state_after": "case neg\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedAddCommGroup \u03b1\n\u03b9 : Type u_2\ns : Finset \u03b9\nf : \u03b9 \u2192 \u03b1\nhs : Finset.Nonempty s\nh : \u2200 (i : \u03b9), i \u2208 s \u2192 f i < 0\n\u22a2 \u2211 i in s, f i < 0"}, {"tactic": "exact Finset.sum_neg h hs", "annotated_tactic": ["exact <a>Finset.sum_neg</a> h hs", [{"full_name": "Finset.sum_neg", "def_path": "Mathlib/Algebra/BigOperators/Order.lean", "def_pos": [510, 3], "def_end_pos": [510, 14]}]], "state_before": "case neg\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedAddCommGroup \u03b1\n\u03b9 : Type u_2\ns : Finset \u03b9\nf : \u03b9 \u2192 \u03b1\nhs : Finset.Nonempty s\nh : \u2200 (i : \u03b9), i \u2208 s \u2192 f i < 0\n\u22a2 \u2211 i in s, f i < 0", "state_after": "no goals"}, {"tactic": "simp_rw [pos_eq_one, sign_eq_one_iff] at h \u22a2", "annotated_tactic": ["simp_rw [<a>pos_eq_one</a>, <a>sign_eq_one_iff</a>] at h \u22a2", [{"full_name": "SignType.pos_eq_one", "def_path": "Mathlib/Data/Sign.lean", "def_pos": [63, 9], "def_end_pos": [63, 19]}, {"full_name": "sign_eq_one_iff", "def_path": "Mathlib/Data/Sign.lean", "def_pos": [326, 9], "def_end_pos": [326, 24]}]], "state_before": "case pos\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedAddCommGroup \u03b1\n\u03b9 : Type u_2\ns : Finset \u03b9\nf : \u03b9 \u2192 \u03b1\nhs : Finset.Nonempty s\nh : \u2200 (i : \u03b9), i \u2208 s \u2192 \u2191sign (f i) = pos\n\u22a2 \u2191sign (\u2211 i in s, f i) = pos", "state_after": "case pos\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedAddCommGroup \u03b1\n\u03b9 : Type u_2\ns : Finset \u03b9\nf : \u03b9 \u2192 \u03b1\nhs : Finset.Nonempty s\nh : \u2200 (i : \u03b9), i \u2208 s \u2192 0 < f i\n\u22a2 0 < \u2211 i in s, f i"}, {"tactic": "exact Finset.sum_pos h hs", "annotated_tactic": ["exact <a>Finset.sum_pos</a> h hs", [{"full_name": "Finset.sum_pos", "def_path": "Mathlib/Algebra/BigOperators/Order.lean", "def_pos": [504, 15], "def_end_pos": [504, 22]}]], "state_before": "case pos\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedAddCommGroup \u03b1\n\u03b9 : Type u_2\ns : Finset \u03b9\nf : \u03b9 \u2192 \u03b1\nhs : Finset.Nonempty s\nh : \u2200 (i : \u03b9), i \u2208 s \u2192 0 < f i\n\u22a2 0 < \u2211 i in s, f i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "full_name": "List.range_eq_range'", "start": [2064, 1], "end": [2065, 56], "traced_tactics": [{"tactic": "rw [Nat.zero_add]", "annotated_tactic": ["rw [<a>Nat.zero_add</a>]", [{"full_name": "Nat.zero_add", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [114, 27], "def_end_pos": [114, 35]}]], "state_before": "n : Nat\n\u22a2 range' 0 (0 + n) = range' 0 n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Vector/Basic.lean", "full_name": "Vector.not_empty_toList", "start": [226, 1], "end": [227, 69], "traced_tactics": [{"tactic": "simp only [empty_toList_eq_ff, Bool.coe_sort_false, not_false_iff]", "annotated_tactic": ["simp only [<a>empty_toList_eq_ff</a>, <a>Bool.coe_sort_false</a>, <a>not_false_iff</a>]", [{"full_name": "Vector.empty_toList_eq_ff", "def_path": "Mathlib/Data/Vector/Basic.lean", "def_pos": [221, 9], "def_end_pos": [221, 27]}, {"full_name": "Bool.coe_sort_false", "def_path": "Mathlib/Init/Data/Bool/Lemmas.lean", "def_pos": [120, 9], "def_end_pos": [120, 23]}, {"full_name": "not_false_iff", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [82, 9], "def_end_pos": [82, 22]}]], "state_before": "n : \u2115\n\u03b1 : Type u_1\nv : Vector \u03b1 (n + 1)\n\u22a2 \u00acList.isEmpty (toList v) = true", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Sups.lean", "full_name": "lowerClosure_infs", "start": [433, 1], "end": [442, 51], "traced_tactics": [{"tactic": "ext a", "annotated_tactic": ["ext a", []], "state_before": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : SemilatticeInf \u03b1\ns t : Set \u03b1\n\u22a2 lowerClosure (s \u22bc t) = lowerClosure s \u2293 lowerClosure t", "state_after": "case a.h\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : SemilatticeInf \u03b1\ns t : Set \u03b1\na : \u03b1\n\u22a2 a \u2208 \u2191(lowerClosure (s \u22bc t)) \u2194 a \u2208 \u2191(lowerClosure s \u2293 lowerClosure t)"}, {"tactic": "simp only [SetLike.mem_coe, mem_lowerClosure, Set.mem_infs, exists_and_left, exists_prop,\n  LowerSet.coe_sup, Set.mem_inter_iff]", "annotated_tactic": ["simp only [<a>SetLike.mem_coe</a>, <a>mem_lowerClosure</a>, <a>Set.mem_infs</a>, <a>exists_and_left</a>, <a>exists_prop</a>,\n    <a>LowerSet.coe_sup</a>, <a>Set.mem_inter_iff</a>]", [{"full_name": "SetLike.mem_coe", "def_path": "Mathlib/Data/SetLike/Basic.lean", "def_pos": [163, 9], "def_end_pos": [163, 16]}, {"full_name": "mem_lowerClosure", "def_path": "Mathlib/Order/UpperLower/Basic.lean", "def_pos": [1379, 9], "def_end_pos": [1379, 25]}, {"full_name": "Set.mem_infs", "def_path": "Mathlib/Data/Set/Sups.lean", "def_pos": [241, 9], "def_end_pos": [241, 17]}, {"full_name": "exists_and_left", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [465, 17], "def_end_pos": [465, 32]}, {"full_name": "exists_prop", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [485, 17], "def_end_pos": [485, 28]}, {"full_name": "LowerSet.coe_sup", "def_path": "Mathlib/Order/UpperLower/Basic.lean", "def_pos": [728, 9], "def_end_pos": [728, 16]}, {"full_name": "Set.mem_inter_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [909, 9], "def_end_pos": [909, 22]}]], "state_before": "case a.h\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : SemilatticeInf \u03b1\ns t : Set \u03b1\na : \u03b1\n\u22a2 a \u2208 \u2191(lowerClosure (s \u22bc t)) \u2194 a \u2208 \u2191(lowerClosure s \u2293 lowerClosure t)", "state_after": "case a.h\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : SemilatticeInf \u03b1\ns t : Set \u03b1\na : \u03b1\n\u22a2 (\u2203 a_1, (\u2203 a, a \u2208 s \u2227 \u2203 b, b \u2208 t \u2227 a \u2293 b = a_1) \u2227 a \u2264 a_1) \u2194 a \u2208 lowerClosure s \u2293 lowerClosure t"}, {"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "case a.h\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : SemilatticeInf \u03b1\ns t : Set \u03b1\na : \u03b1\n\u22a2 (\u2203 a_1, (\u2203 a, a \u2208 s \u2227 \u2203 b, b \u2208 t \u2227 a \u2293 b = a_1) \u2227 a \u2264 a_1) \u2194 a \u2208 lowerClosure s \u2293 lowerClosure t", "state_after": "case a.h.mp\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : SemilatticeInf \u03b1\ns t : Set \u03b1\na : \u03b1\n\u22a2 (\u2203 a_1, (\u2203 a, a \u2208 s \u2227 \u2203 b, b \u2208 t \u2227 a \u2293 b = a_1) \u2227 a \u2264 a_1) \u2192 a \u2208 lowerClosure s \u2293 lowerClosure t\n\ncase a.h.mpr\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : SemilatticeInf \u03b1\ns t : Set \u03b1\na : \u03b1\n\u22a2 a \u2208 lowerClosure s \u2293 lowerClosure t \u2192 \u2203 a_2, (\u2203 a, a \u2208 s \u2227 \u2203 b, b \u2208 t \u2227 a \u2293 b = a_2) \u2227 a \u2264 a_2"}, {"tactic": "rintro \u27e8_, \u27e8b, hb, c, hc, rfl\u27e9, ha\u27e9", "annotated_tactic": ["rintro \u27e8_, \u27e8b, hb, c, hc, rfl\u27e9, ha\u27e9", []], "state_before": "case a.h.mp\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : SemilatticeInf \u03b1\ns t : Set \u03b1\na : \u03b1\n\u22a2 (\u2203 a_1, (\u2203 a, a \u2208 s \u2227 \u2203 b, b \u2208 t \u2227 a \u2293 b = a_1) \u2227 a \u2264 a_1) \u2192 a \u2208 lowerClosure s \u2293 lowerClosure t", "state_after": "case a.h.mp.intro.intro.intro.intro.intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : SemilatticeInf \u03b1\ns t : Set \u03b1\na b : \u03b1\nhb : b \u2208 s\nc : \u03b1\nhc : c \u2208 t\nha : a \u2264 b \u2293 c\n\u22a2 a \u2208 lowerClosure s \u2293 lowerClosure t"}, {"tactic": "exact \u27e8\u27e8b, hb, ha.trans inf_le_left\u27e9, c, hc, ha.trans inf_le_right\u27e9", "annotated_tactic": ["exact \u27e8\u27e8b, hb, ha.trans <a>inf_le_left</a>\u27e9, c, hc, ha.trans <a>inf_le_right</a>\u27e9", [{"full_name": "inf_le_left", "def_path": "Mathlib/Order/Lattice.lean", "def_pos": [388, 9], "def_end_pos": [388, 20]}, {"full_name": "inf_le_right", "def_path": "Mathlib/Order/Lattice.lean", "def_pos": [399, 9], "def_end_pos": [399, 21]}]], "state_before": "case a.h.mp.intro.intro.intro.intro.intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : SemilatticeInf \u03b1\ns t : Set \u03b1\na b : \u03b1\nhb : b \u2208 s\nc : \u03b1\nhc : c \u2208 t\nha : a \u2264 b \u2293 c\n\u22a2 a \u2208 lowerClosure s \u2293 lowerClosure t", "state_after": "no goals"}, {"tactic": "rintro \u27e8\u27e8b, hb, hab\u27e9, c, hc, hac\u27e9", "annotated_tactic": ["rintro \u27e8\u27e8b, hb, hab\u27e9, c, hc, hac\u27e9", []], "state_before": "case a.h.mpr\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : SemilatticeInf \u03b1\ns t : Set \u03b1\na : \u03b1\n\u22a2 a \u2208 lowerClosure s \u2293 lowerClosure t \u2192 \u2203 a_2, (\u2203 a, a \u2208 s \u2227 \u2203 b, b \u2208 t \u2227 a \u2293 b = a_2) \u2227 a \u2264 a_2", "state_after": "case a.h.mpr.intro.intro.intro.intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : SemilatticeInf \u03b1\ns t : Set \u03b1\na b : \u03b1\nhb : b \u2208 s\nhab : a \u2264 b\nc : \u03b1\nhc : c \u2208 t\nhac : a \u2264 c\n\u22a2 \u2203 a_1, (\u2203 a, a \u2208 s \u2227 \u2203 b, b \u2208 t \u2227 a \u2293 b = a_1) \u2227 a \u2264 a_1"}, {"tactic": "exact \u27e8_, \u27e8b, hb, c, hc, rfl\u27e9, le_inf hab hac\u27e9", "annotated_tactic": ["exact \u27e8_, \u27e8b, hb, c, hc, <a>rfl</a>\u27e9, <a>le_inf</a> hab hac\u27e9", [{"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}, {"full_name": "le_inf", "def_path": "Mathlib/Order/Lattice.lean", "def_pos": [409, 9], "def_end_pos": [409, 15]}]], "state_before": "case a.h.mpr.intro.intro.intro.intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : SemilatticeInf \u03b1\ns t : Set \u03b1\na b : \u03b1\nhb : b \u2208 s\nhab : a \u2264 b\nc : \u03b1\nhc : c \u2208 t\nhac : a \u2264 c\n\u22a2 \u2203 a_1, (\u2203 a, a \u2208 s \u2227 \u2203 b, b \u2208 t \u2227 a \u2293 b = a_1) \u2227 a \u2264 a_1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/Jacobian.lean", "full_name": "MeasureTheory.addHaar_image_eq_zero_of_det_fderivWithin_eq_zero_aux", "start": [606, 1], "end": [654, 70], "traced_tactics": [{"tactic": "rcases eq_empty_or_nonempty s with (rfl | h's)", "annotated_tactic": ["rcases <a>eq_empty_or_nonempty</a> s with (rfl | h's)", [{"full_name": "Set.eq_empty_or_nonempty", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [635, 9], "def_end_pos": [635, 29]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nR : \u211d\nhs : s \u2286 closedBall 0 R\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\n\u22a2 \u2191\u2191\u03bc (f '' s) \u2264 \u2191\u03b5 * \u2191\u2191\u03bc (closedBall 0 R)", "state_after": "case inl\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nR : \u211d\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nhf' : \u2200 (x : E), x \u2208 \u2205 \u2192 HasFDerivWithinAt f (f' x) \u2205 x\nhs : \u2205 \u2286 closedBall 0 R\nh'f' : \u2200 (x : E), x \u2208 \u2205 \u2192 ContinuousLinearMap.det (f' x) = 0\n\u22a2 \u2191\u2191\u03bc (f '' \u2205) \u2264 \u2191\u03b5 * \u2191\u2191\u03bc (closedBall 0 R)\n\ncase inr\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nR : \u211d\nhs : s \u2286 closedBall 0 R\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nh's : Set.Nonempty s\n\u22a2 \u2191\u2191\u03bc (f '' s) \u2264 \u2191\u03b5 * \u2191\u2191\u03bc (closedBall 0 R)"}, {"tactic": "have :\n    \u2200 A : E \u2192L[\u211d] E, \u2203 \u03b4 : \u211d\u22650, 0 < \u03b4 \u2227\n      \u2200 (t : Set E), ApproximatesLinearOn f A t \u03b4 \u2192\n        \u03bc (f '' t) \u2264 (Real.toNNReal |A.det| + \u03b5 : \u211d\u22650) * \u03bc t := by\n  intro A\n  let m : \u211d\u22650 := Real.toNNReal |A.det| + \u03b5\n  have I : ENNReal.ofReal |A.det| < m := by\n    simp only [ENNReal.ofReal, lt_add_iff_pos_right, \u03b5pos, ENNReal.coe_lt_coe]\n  rcases ((addHaar_image_le_mul_of_det_lt \u03bc A I).and self_mem_nhdsWithin).exists with \u27e8\u03b4, h, h'\u27e9\n  exact \u27e8\u03b4, h', fun t ht => h t f ht\u27e9", "annotated_tactic": ["have :\n      \u2200 A : E \u2192L[\u211d] E, \u2203 \u03b4 : \u211d\u22650, 0 < \u03b4 \u2227\n        \u2200 (t : <a>Set</a> E), <a>ApproximatesLinearOn</a> f A t \u03b4 \u2192\n          \u03bc (f '' t) \u2264 (<a>Real.toNNReal</a> |A.det| + \u03b5 : \u211d\u22650) * \u03bc t := by\n    intro A\n    let m : \u211d\u22650 := <a>Real.toNNReal</a> |A.det| + \u03b5\n    have I : <a>ENNReal.ofReal</a> |A.det| < m := by\n      simp only [<a>ENNReal.ofReal</a>, <a>lt_add_iff_pos_right</a>, \u03b5pos, <a>ENNReal.coe_lt_coe</a>]\n    rcases ((<a>addHaar_image_le_mul_of_det_lt</a> \u03bc A I).<a>and</a> <a>self_mem_nhdsWithin</a>).<a>exists</a> with \u27e8\u03b4, h, h'\u27e9\n    exact \u27e8\u03b4, h', fun t ht => h t f ht\u27e9", [{"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}, {"full_name": "ApproximatesLinearOn", "def_path": "Mathlib/Analysis/Calculus/Inverse.lean", "def_pos": [116, 5], "def_end_pos": [116, 25]}, {"full_name": "Real.toNNReal", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [118, 19], "def_end_pos": [118, 39]}, {"full_name": "Real.toNNReal", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [118, 19], "def_end_pos": [118, 39]}, {"full_name": "ENNReal.ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [172, 29], "def_end_pos": [172, 35]}, {"full_name": "ENNReal.ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [172, 29], "def_end_pos": [172, 35]}, {"full_name": "lt_add_iff_pos_right", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [563, 30], "def_end_pos": [563, 50]}, {"full_name": "ENNReal.coe_lt_coe", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [352, 28], "def_end_pos": [352, 38]}, {"full_name": "MeasureTheory.addHaar_image_le_mul_of_det_lt", "def_path": "Mathlib/MeasureTheory/Function/Jacobian.lean", "def_pos": [285, 9], "def_end_pos": [285, 39]}, {"full_name": "Filter.Eventually.and", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1103, 19], "def_end_pos": [1103, 33]}, {"full_name": "self_mem_nhdsWithin", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [151, 9], "def_end_pos": [151, 28]}, {"full_name": "Filter.Eventually.exists", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1308, 9], "def_end_pos": [1308, 26]}]], "state_before": "case inr\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nR : \u211d\nhs : s \u2286 closedBall 0 R\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nh's : Set.Nonempty s\n\u22a2 \u2191\u2191\u03bc (f '' s) \u2264 \u2191\u03b5 * \u2191\u2191\u03bc (closedBall 0 R)", "state_after": "case inr\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nR : \u211d\nhs : s \u2286 closedBall 0 R\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nh's : Set.Nonempty s\nthis :\n  \u2200 (A : E \u2192L[\u211d] E),\n    \u2203 \u03b4,\n      0 < \u03b4 \u2227\n        \u2200 (t : Set E),\n          ApproximatesLinearOn f A t \u03b4 \u2192 \u2191\u2191\u03bc (f '' t) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det A| + \u03b5) * \u2191\u2191\u03bc t\n\u22a2 \u2191\u2191\u03bc (f '' s) \u2264 \u2191\u03b5 * \u2191\u2191\u03bc (closedBall 0 R)"}, {"tactic": "choose \u03b4 h\u03b4 using this", "annotated_tactic": ["choose \u03b4 h\u03b4 using this", []], "state_before": "case inr\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nR : \u211d\nhs : s \u2286 closedBall 0 R\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nh's : Set.Nonempty s\nthis :\n  \u2200 (A : E \u2192L[\u211d] E),\n    \u2203 \u03b4,\n      0 < \u03b4 \u2227\n        \u2200 (t : Set E),\n          ApproximatesLinearOn f A t \u03b4 \u2192 \u2191\u2191\u03bc (f '' t) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det A| + \u03b5) * \u2191\u2191\u03bc t\n\u22a2 \u2191\u2191\u03bc (f '' s) \u2264 \u2191\u03b5 * \u2191\u2191\u03bc (closedBall 0 R)", "state_after": "case inr\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nR : \u211d\nhs : s \u2286 closedBall 0 R\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nh's : Set.Nonempty s\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n  \u2200 (A : E \u2192L[\u211d] E),\n    0 < \u03b4 A \u2227\n      \u2200 (t : Set E),\n        ApproximatesLinearOn f A t (\u03b4 A) \u2192 \u2191\u2191\u03bc (f '' t) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det A| + \u03b5) * \u2191\u2191\u03bc t\n\u22a2 \u2191\u2191\u03bc (f '' s) \u2264 \u2191\u03b5 * \u2191\u2191\u03bc (closedBall 0 R)"}, {"tactic": "obtain \u27e8t, A, t_disj, t_meas, t_cover, ht, Af'\u27e9 :\n  \u2203 (t : \u2115 \u2192 Set E) (A : \u2115 \u2192 E \u2192L[\u211d] E),\n    Pairwise (Disjoint on t) \u2227\n      (\u2200 n : \u2115, MeasurableSet (t n)) \u2227\n        (s \u2286 \u22c3 n : \u2115, t n) \u2227\n          (\u2200 n : \u2115, ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))) \u2227\n            (s.Nonempty \u2192 \u2200 n, \u2203 y \u2208 s, A n = f' y) :=\n  exists_partition_approximatesLinearOn_of_hasFDerivWithinAt f s f' hf' \u03b4 fun A => (h\u03b4 A).1.ne'", "annotated_tactic": ["obtain \u27e8t, A, t_disj, t_meas, t_cover, ht, Af'\u27e9 :\n    \u2203 (t : \u2115 \u2192 <a>Set</a> E) (A : \u2115 \u2192 E \u2192L[\u211d] E),\n      <a>Pairwise</a> (<a>Disjoint</a> on t) \u2227\n        (\u2200 n : \u2115, <a>MeasurableSet</a> (t n)) \u2227\n          (s \u2286 \u22c3 n : \u2115, t n) \u2227\n            (\u2200 n : \u2115, <a>ApproximatesLinearOn</a> f (A n) (s \u2229 t n) (\u03b4 (A n))) \u2227\n              (s.Nonempty \u2192 \u2200 n, \u2203 y \u2208 s, A n = f' y) :=\n    <a>exists_partition_approximatesLinearOn_of_hasFDerivWithinAt</a> f s f' hf' \u03b4 fun A => (h\u03b4 A).1.<a>ne'</a>", [{"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}, {"full_name": "Pairwise", "def_path": "Mathlib/Logic/Pairwise.lean", "def_pos": [34, 5], "def_end_pos": [34, 13]}, {"full_name": "Disjoint", "def_path": "Mathlib/Order/Disjoint.lean", "def_pos": [41, 5], "def_end_pos": [41, 13]}, {"full_name": "MeasurableSet", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [64, 5], "def_end_pos": [64, 18]}, {"full_name": "ApproximatesLinearOn", "def_path": "Mathlib/Analysis/Calculus/Inverse.lean", "def_pos": [116, 5], "def_end_pos": [116, 25]}, {"full_name": "exists_partition_approximatesLinearOn_of_hasFDerivWithinAt", "def_path": "Mathlib/MeasureTheory/Function/Jacobian.lean", "def_pos": [254, 9], "def_end_pos": [254, 67]}, {"full_name": "LT.lt.ne'", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [328, 9], "def_end_pos": [328, 12]}]], "state_before": "case inr\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nR : \u211d\nhs : s \u2286 closedBall 0 R\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nh's : Set.Nonempty s\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n  \u2200 (A : E \u2192L[\u211d] E),\n    0 < \u03b4 A \u2227\n      \u2200 (t : Set E),\n        ApproximatesLinearOn f A t (\u03b4 A) \u2192 \u2191\u2191\u03bc (f '' t) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det A| + \u03b5) * \u2191\u2191\u03bc t\n\u22a2 \u2191\u2191\u03bc (f '' s) \u2264 \u2191\u03b5 * \u2191\u2191\u03bc (closedBall 0 R)", "state_after": "case inr.intro.intro.intro.intro.intro.intro\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nR : \u211d\nhs : s \u2286 closedBall 0 R\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nh's : Set.Nonempty s\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n  \u2200 (A : E \u2192L[\u211d] E),\n    0 < \u03b4 A \u2227\n      \u2200 (t : Set E),\n        ApproximatesLinearOn f A t (\u03b4 A) \u2192 \u2191\u2191\u03bc (f '' t) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det A| + \u03b5) * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 n, t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\nAf' : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\n\u22a2 \u2191\u2191\u03bc (f '' s) \u2264 \u2191\u03b5 * \u2191\u2191\u03bc (closedBall 0 R)"}, {"tactic": "simp only [measure_empty, zero_le, image_empty]", "annotated_tactic": ["simp only [<a>measure_empty</a>, <a>zero_le</a>, <a>image_empty</a>]", [{"full_name": "MeasureTheory.measure_empty", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [185, 9], "def_end_pos": [185, 22]}, {"full_name": "zero_le", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [217, 30], "def_end_pos": [217, 37]}, {"full_name": "Set.image_empty", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [337, 9], "def_end_pos": [337, 20]}]], "state_before": "case inl\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nR : \u211d\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nhf' : \u2200 (x : E), x \u2208 \u2205 \u2192 HasFDerivWithinAt f (f' x) \u2205 x\nhs : \u2205 \u2286 closedBall 0 R\nh'f' : \u2200 (x : E), x \u2208 \u2205 \u2192 ContinuousLinearMap.det (f' x) = 0\n\u22a2 \u2191\u2191\u03bc (f '' \u2205) \u2264 \u2191\u03b5 * \u2191\u2191\u03bc (closedBall 0 R)", "state_after": "no goals"}, {"tactic": "intro A", "annotated_tactic": ["intro A", []], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nR : \u211d\nhs : s \u2286 closedBall 0 R\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nh's : Set.Nonempty s\n\u22a2 \u2200 (A : E \u2192L[\u211d] E),\n    \u2203 \u03b4,\n      0 < \u03b4 \u2227\n        \u2200 (t : Set E),\n          ApproximatesLinearOn f A t \u03b4 \u2192 \u2191\u2191\u03bc (f '' t) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det A| + \u03b5) * \u2191\u2191\u03bc t", "state_after": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nR : \u211d\nhs : s \u2286 closedBall 0 R\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nh's : Set.Nonempty s\nA : E \u2192L[\u211d] E\n\u22a2 \u2203 \u03b4,\n    0 < \u03b4 \u2227\n      \u2200 (t : Set E),\n        ApproximatesLinearOn f A t \u03b4 \u2192 \u2191\u2191\u03bc (f '' t) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det A| + \u03b5) * \u2191\u2191\u03bc t"}, {"tactic": "let m : \u211d\u22650 := Real.toNNReal |A.det| + \u03b5", "annotated_tactic": ["let m : \u211d\u22650 := <a>Real.toNNReal</a> |A.det| + \u03b5", [{"full_name": "Real.toNNReal", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [118, 19], "def_end_pos": [118, 39]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nR : \u211d\nhs : s \u2286 closedBall 0 R\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nh's : Set.Nonempty s\nA : E \u2192L[\u211d] E\n\u22a2 \u2203 \u03b4,\n    0 < \u03b4 \u2227\n      \u2200 (t : Set E),\n        ApproximatesLinearOn f A t \u03b4 \u2192 \u2191\u2191\u03bc (f '' t) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det A| + \u03b5) * \u2191\u2191\u03bc t", "state_after": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nR : \u211d\nhs : s \u2286 closedBall 0 R\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nh's : Set.Nonempty s\nA : E \u2192L[\u211d] E\nm : \u211d\u22650 := Real.toNNReal |ContinuousLinearMap.det A| + \u03b5\n\u22a2 \u2203 \u03b4,\n    0 < \u03b4 \u2227\n      \u2200 (t : Set E),\n        ApproximatesLinearOn f A t \u03b4 \u2192 \u2191\u2191\u03bc (f '' t) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det A| + \u03b5) * \u2191\u2191\u03bc t"}, {"tactic": "have I : ENNReal.ofReal |A.det| < m := by\n  simp only [ENNReal.ofReal, lt_add_iff_pos_right, \u03b5pos, ENNReal.coe_lt_coe]", "annotated_tactic": ["have I : <a>ENNReal.ofReal</a> |A.det| < m := by\n      simp only [<a>ENNReal.ofReal</a>, <a>lt_add_iff_pos_right</a>, \u03b5pos, <a>ENNReal.coe_lt_coe</a>]", [{"full_name": "ENNReal.ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [172, 29], "def_end_pos": [172, 35]}, {"full_name": "ENNReal.ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [172, 29], "def_end_pos": [172, 35]}, {"full_name": "lt_add_iff_pos_right", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [563, 30], "def_end_pos": [563, 50]}, {"full_name": "ENNReal.coe_lt_coe", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [352, 28], "def_end_pos": [352, 38]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nR : \u211d\nhs : s \u2286 closedBall 0 R\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nh's : Set.Nonempty s\nA : E \u2192L[\u211d] E\nm : \u211d\u22650 := Real.toNNReal |ContinuousLinearMap.det A| + \u03b5\n\u22a2 \u2203 \u03b4,\n    0 < \u03b4 \u2227\n      \u2200 (t : Set E),\n        ApproximatesLinearOn f A t \u03b4 \u2192 \u2191\u2191\u03bc (f '' t) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det A| + \u03b5) * \u2191\u2191\u03bc t", "state_after": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nR : \u211d\nhs : s \u2286 closedBall 0 R\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nh's : Set.Nonempty s\nA : E \u2192L[\u211d] E\nm : \u211d\u22650 := Real.toNNReal |ContinuousLinearMap.det A| + \u03b5\nI : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\n\u22a2 \u2203 \u03b4,\n    0 < \u03b4 \u2227\n      \u2200 (t : Set E),\n        ApproximatesLinearOn f A t \u03b4 \u2192 \u2191\u2191\u03bc (f '' t) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det A| + \u03b5) * \u2191\u2191\u03bc t"}, {"tactic": "rcases ((addHaar_image_le_mul_of_det_lt \u03bc A I).and self_mem_nhdsWithin).exists with \u27e8\u03b4, h, h'\u27e9", "annotated_tactic": ["rcases ((<a>addHaar_image_le_mul_of_det_lt</a> \u03bc A I).<a>and</a> <a>self_mem_nhdsWithin</a>).<a>exists</a> with \u27e8\u03b4, h, h'\u27e9", [{"full_name": "MeasureTheory.addHaar_image_le_mul_of_det_lt", "def_path": "Mathlib/MeasureTheory/Function/Jacobian.lean", "def_pos": [285, 9], "def_end_pos": [285, 39]}, {"full_name": "Filter.Eventually.and", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1103, 19], "def_end_pos": [1103, 33]}, {"full_name": "self_mem_nhdsWithin", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [151, 9], "def_end_pos": [151, 28]}, {"full_name": "Filter.Eventually.exists", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1308, 9], "def_end_pos": [1308, 26]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nR : \u211d\nhs : s \u2286 closedBall 0 R\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nh's : Set.Nonempty s\nA : E \u2192L[\u211d] E\nm : \u211d\u22650 := Real.toNNReal |ContinuousLinearMap.det A| + \u03b5\nI : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\n\u22a2 \u2203 \u03b4,\n    0 < \u03b4 \u2227\n      \u2200 (t : Set E),\n        ApproximatesLinearOn f A t \u03b4 \u2192 \u2191\u2191\u03bc (f '' t) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det A| + \u03b5) * \u2191\u2191\u03bc t", "state_after": "case intro.intro\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nR : \u211d\nhs : s \u2286 closedBall 0 R\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nh's : Set.Nonempty s\nA : E \u2192L[\u211d] E\nm : \u211d\u22650 := Real.toNNReal |ContinuousLinearMap.det A| + \u03b5\nI : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\n\u03b4 : \u211d\u22650\nh : \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u2191\u2191\u03bc (f '' s) \u2264 \u2191m * \u2191\u2191\u03bc s\nh' : 0 < \u03b4\n\u22a2 \u2203 \u03b4,\n    0 < \u03b4 \u2227\n      \u2200 (t : Set E),\n        ApproximatesLinearOn f A t \u03b4 \u2192 \u2191\u2191\u03bc (f '' t) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det A| + \u03b5) * \u2191\u2191\u03bc t"}, {"tactic": "exact \u27e8\u03b4, h', fun t ht => h t f ht\u27e9", "annotated_tactic": ["exact \u27e8\u03b4, h', fun t ht => h t f ht\u27e9", []], "state_before": "case intro.intro\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nR : \u211d\nhs : s \u2286 closedBall 0 R\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nh's : Set.Nonempty s\nA : E \u2192L[\u211d] E\nm : \u211d\u22650 := Real.toNNReal |ContinuousLinearMap.det A| + \u03b5\nI : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\n\u03b4 : \u211d\u22650\nh : \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u2191\u2191\u03bc (f '' s) \u2264 \u2191m * \u2191\u2191\u03bc s\nh' : 0 < \u03b4\n\u22a2 \u2203 \u03b4,\n    0 < \u03b4 \u2227\n      \u2200 (t : Set E),\n        ApproximatesLinearOn f A t \u03b4 \u2192 \u2191\u2191\u03bc (f '' t) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det A| + \u03b5) * \u2191\u2191\u03bc t", "state_after": "no goals"}, {"tactic": "simp only [ENNReal.ofReal, lt_add_iff_pos_right, \u03b5pos, ENNReal.coe_lt_coe]", "annotated_tactic": ["simp only [<a>ENNReal.ofReal</a>, <a>lt_add_iff_pos_right</a>, \u03b5pos, <a>ENNReal.coe_lt_coe</a>]", [{"full_name": "ENNReal.ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [172, 29], "def_end_pos": [172, 35]}, {"full_name": "lt_add_iff_pos_right", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [563, 30], "def_end_pos": [563, 50]}, {"full_name": "ENNReal.coe_lt_coe", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [352, 28], "def_end_pos": [352, 38]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nR : \u211d\nhs : s \u2286 closedBall 0 R\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nh's : Set.Nonempty s\nA : E \u2192L[\u211d] E\nm : \u211d\u22650 := Real.toNNReal |ContinuousLinearMap.det A| + \u03b5\n\u22a2 ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m", "state_after": "no goals"}, {"tactic": "apply measure_mono", "annotated_tactic": ["apply <a>measure_mono</a>", [{"full_name": "MeasureTheory.measure_mono", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [193, 9], "def_end_pos": [193, 21]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nR : \u211d\nhs : s \u2286 closedBall 0 R\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nh's : Set.Nonempty s\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n  \u2200 (A : E \u2192L[\u211d] E),\n    0 < \u03b4 A \u2227\n      \u2200 (t : Set E),\n        ApproximatesLinearOn f A t (\u03b4 A) \u2192 \u2191\u2191\u03bc (f '' t) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det A| + \u03b5) * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 n, t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\nAf' : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\n\u22a2 \u2191\u2191\u03bc (f '' s) \u2264 \u2191\u2191\u03bc (\u22c3 n, f '' (s \u2229 t n))", "state_after": "case h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nR : \u211d\nhs : s \u2286 closedBall 0 R\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nh's : Set.Nonempty s\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n  \u2200 (A : E \u2192L[\u211d] E),\n    0 < \u03b4 A \u2227\n      \u2200 (t : Set E),\n        ApproximatesLinearOn f A t (\u03b4 A) \u2192 \u2191\u2191\u03bc (f '' t) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det A| + \u03b5) * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 n, t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\nAf' : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\n\u22a2 f '' s \u2286 \u22c3 n, f '' (s \u2229 t n)"}, {"tactic": "rw [\u2190 image_iUnion, \u2190 inter_iUnion]", "annotated_tactic": ["rw [\u2190 <a>image_iUnion</a>, \u2190 <a>inter_iUnion</a>]", [{"full_name": "Set.image_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [1791, 9], "def_end_pos": [1791, 21]}, {"full_name": "Set.inter_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [635, 9], "def_end_pos": [635, 21]}]], "state_before": "case h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nR : \u211d\nhs : s \u2286 closedBall 0 R\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nh's : Set.Nonempty s\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n  \u2200 (A : E \u2192L[\u211d] E),\n    0 < \u03b4 A \u2227\n      \u2200 (t : Set E),\n        ApproximatesLinearOn f A t (\u03b4 A) \u2192 \u2191\u2191\u03bc (f '' t) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det A| + \u03b5) * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 n, t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\nAf' : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\n\u22a2 f '' s \u2286 \u22c3 n, f '' (s \u2229 t n)", "state_after": "case h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nR : \u211d\nhs : s \u2286 closedBall 0 R\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nh's : Set.Nonempty s\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n  \u2200 (A : E \u2192L[\u211d] E),\n    0 < \u03b4 A \u2227\n      \u2200 (t : Set E),\n        ApproximatesLinearOn f A t (\u03b4 A) \u2192 \u2191\u2191\u03bc (f '' t) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det A| + \u03b5) * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 n, t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\nAf' : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\n\u22a2 f '' s \u2286 f '' (s \u2229 \u22c3 i, t i)"}, {"tactic": "exact image_subset f (subset_inter Subset.rfl t_cover)", "annotated_tactic": ["exact <a>image_subset</a> f (<a>subset_inter</a> <a>Subset.rfl</a> t_cover)", [{"full_name": "Set.image_subset", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [321, 9], "def_end_pos": [321, 21]}, {"full_name": "Set.subset_inter", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [972, 9], "def_end_pos": [972, 21]}, {"full_name": "Set.Subset.rfl", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [357, 9], "def_end_pos": [357, 19]}]], "state_before": "case h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nR : \u211d\nhs : s \u2286 closedBall 0 R\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nh's : Set.Nonempty s\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n  \u2200 (A : E \u2192L[\u211d] E),\n    0 < \u03b4 A \u2227\n      \u2200 (t : Set E),\n        ApproximatesLinearOn f A t (\u03b4 A) \u2192 \u2191\u2191\u03bc (f '' t) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det A| + \u03b5) * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 n, t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\nAf' : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\n\u22a2 f '' s \u2286 f '' (s \u2229 \u22c3 i, t i)", "state_after": "no goals"}, {"tactic": "apply ENNReal.tsum_le_tsum fun n => ?_", "annotated_tactic": ["apply <a>ENNReal.tsum_le_tsum</a> fun n => ?_", [{"full_name": "ENNReal.tsum_le_tsum", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [827, 19], "def_end_pos": [827, 31]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nR : \u211d\nhs : s \u2286 closedBall 0 R\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nh's : Set.Nonempty s\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n  \u2200 (A : E \u2192L[\u211d] E),\n    0 < \u03b4 A \u2227\n      \u2200 (t : Set E),\n        ApproximatesLinearOn f A t (\u03b4 A) \u2192 \u2191\u2191\u03bc (f '' t) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det A| + \u03b5) * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 n, t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\nAf' : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\n\u22a2 \u2211' (n : \u2115), \u2191\u2191\u03bc (f '' (s \u2229 t n)) \u2264 \u2211' (n : \u2115), \u2191(Real.toNNReal |ContinuousLinearMap.det (A n)| + \u03b5) * \u2191\u2191\u03bc (s \u2229 t n)", "state_after": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nR : \u211d\nhs : s \u2286 closedBall 0 R\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nh's : Set.Nonempty s\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n  \u2200 (A : E \u2192L[\u211d] E),\n    0 < \u03b4 A \u2227\n      \u2200 (t : Set E),\n        ApproximatesLinearOn f A t (\u03b4 A) \u2192 \u2191\u2191\u03bc (f '' t) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det A| + \u03b5) * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 n, t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\nAf' : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\nn : \u2115\n\u22a2 \u2191\u2191\u03bc (f '' (s \u2229 t n)) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det (A n)| + \u03b5) * \u2191\u2191\u03bc (s \u2229 t n)"}, {"tactic": "apply (h\u03b4 (A n)).2", "annotated_tactic": ["apply (h\u03b4 (A n)).2", []], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nR : \u211d\nhs : s \u2286 closedBall 0 R\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nh's : Set.Nonempty s\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n  \u2200 (A : E \u2192L[\u211d] E),\n    0 < \u03b4 A \u2227\n      \u2200 (t : Set E),\n        ApproximatesLinearOn f A t (\u03b4 A) \u2192 \u2191\u2191\u03bc (f '' t) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det A| + \u03b5) * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 n, t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\nAf' : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\nn : \u2115\n\u22a2 \u2191\u2191\u03bc (f '' (s \u2229 t n)) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det (A n)| + \u03b5) * \u2191\u2191\u03bc (s \u2229 t n)", "state_after": "case a\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nR : \u211d\nhs : s \u2286 closedBall 0 R\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nh's : Set.Nonempty s\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n  \u2200 (A : E \u2192L[\u211d] E),\n    0 < \u03b4 A \u2227\n      \u2200 (t : Set E),\n        ApproximatesLinearOn f A t (\u03b4 A) \u2192 \u2191\u2191\u03bc (f '' t) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det A| + \u03b5) * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 n, t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\nAf' : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\nn : \u2115\n\u22a2 ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))"}, {"tactic": "exact ht n", "annotated_tactic": ["exact ht n", []], "state_before": "case a\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nR : \u211d\nhs : s \u2286 closedBall 0 R\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nh's : Set.Nonempty s\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n  \u2200 (A : E \u2192L[\u211d] E),\n    0 < \u03b4 A \u2227\n      \u2200 (t : Set E),\n        ApproximatesLinearOn f A t (\u03b4 A) \u2192 \u2191\u2191\u03bc (f '' t) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det A| + \u03b5) * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 n, t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\nAf' : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\nn : \u2115\n\u22a2 ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))", "state_after": "no goals"}, {"tactic": "congr with n", "annotated_tactic": ["congr with n", []], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nR : \u211d\nhs : s \u2286 closedBall 0 R\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nh's : Set.Nonempty s\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n  \u2200 (A : E \u2192L[\u211d] E),\n    0 < \u03b4 A \u2227\n      \u2200 (t : Set E),\n        ApproximatesLinearOn f A t (\u03b4 A) \u2192 \u2191\u2191\u03bc (f '' t) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det A| + \u03b5) * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 n, t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\nAf' : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\n\u22a2 \u2211' (n : \u2115), \u2191(Real.toNNReal |ContinuousLinearMap.det (A n)| + \u03b5) * \u2191\u2191\u03bc (s \u2229 t n) = \u2211' (n : \u2115), \u2191\u03b5 * \u2191\u2191\u03bc (s \u2229 t n)", "state_after": "case e_f.h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nR : \u211d\nhs : s \u2286 closedBall 0 R\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nh's : Set.Nonempty s\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n  \u2200 (A : E \u2192L[\u211d] E),\n    0 < \u03b4 A \u2227\n      \u2200 (t : Set E),\n        ApproximatesLinearOn f A t (\u03b4 A) \u2192 \u2191\u2191\u03bc (f '' t) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det A| + \u03b5) * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 n, t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\nAf' : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\nn : \u2115\n\u22a2 \u2191(Real.toNNReal |ContinuousLinearMap.det (A n)| + \u03b5) * \u2191\u2191\u03bc (s \u2229 t n) = \u2191\u03b5 * \u2191\u2191\u03bc (s \u2229 t n)"}, {"tactic": "rcases Af' h's n with \u27e8y, ys, hy\u27e9", "annotated_tactic": ["rcases Af' h's n with \u27e8y, ys, hy\u27e9", []], "state_before": "case e_f.h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nR : \u211d\nhs : s \u2286 closedBall 0 R\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nh's : Set.Nonempty s\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n  \u2200 (A : E \u2192L[\u211d] E),\n    0 < \u03b4 A \u2227\n      \u2200 (t : Set E),\n        ApproximatesLinearOn f A t (\u03b4 A) \u2192 \u2191\u2191\u03bc (f '' t) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det A| + \u03b5) * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 n, t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\nAf' : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\nn : \u2115\n\u22a2 \u2191(Real.toNNReal |ContinuousLinearMap.det (A n)| + \u03b5) * \u2191\u2191\u03bc (s \u2229 t n) = \u2191\u03b5 * \u2191\u2191\u03bc (s \u2229 t n)", "state_after": "case e_f.h.intro.intro\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nR : \u211d\nhs : s \u2286 closedBall 0 R\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nh's : Set.Nonempty s\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n  \u2200 (A : E \u2192L[\u211d] E),\n    0 < \u03b4 A \u2227\n      \u2200 (t : Set E),\n        ApproximatesLinearOn f A t (\u03b4 A) \u2192 \u2191\u2191\u03bc (f '' t) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det A| + \u03b5) * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 n, t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\nAf' : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\nn : \u2115\ny : E\nys : y \u2208 s\nhy : A n = f' y\n\u22a2 \u2191(Real.toNNReal |ContinuousLinearMap.det (A n)| + \u03b5) * \u2191\u2191\u03bc (s \u2229 t n) = \u2191\u03b5 * \u2191\u2191\u03bc (s \u2229 t n)"}, {"tactic": "simp only [hy, h'f' y ys, Real.toNNReal_zero, abs_zero, zero_add]", "annotated_tactic": ["simp only [hy, h'f' y ys, <a>Real.toNNReal_zero</a>, <a>abs_zero</a>, <a>zero_add</a>]", [{"full_name": "Real.toNNReal_zero", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [620, 9], "def_end_pos": [620, 22]}, {"full_name": "abs_zero", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [128, 9], "def_end_pos": [128, 17]}, {"full_name": "zero_add", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [463, 3], "def_end_pos": [463, 14]}]], "state_before": "case e_f.h.intro.intro\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nR : \u211d\nhs : s \u2286 closedBall 0 R\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nh's : Set.Nonempty s\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n  \u2200 (A : E \u2192L[\u211d] E),\n    0 < \u03b4 A \u2227\n      \u2200 (t : Set E),\n        ApproximatesLinearOn f A t (\u03b4 A) \u2192 \u2191\u2191\u03bc (f '' t) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det A| + \u03b5) * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 n, t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\nAf' : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\nn : \u2115\ny : E\nys : y \u2208 s\nhy : A n = f' y\n\u22a2 \u2191(Real.toNNReal |ContinuousLinearMap.det (A n)| + \u03b5) * \u2191\u2191\u03bc (s \u2229 t n) = \u2191\u03b5 * \u2191\u2191\u03bc (s \u2229 t n)", "state_after": "no goals"}, {"tactic": "rw [ENNReal.tsum_mul_left]", "annotated_tactic": ["rw [<a>ENNReal.tsum_mul_left</a>]", [{"full_name": "ENNReal.tsum_mul_left", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [897, 19], "def_end_pos": [897, 32]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nR : \u211d\nhs : s \u2286 closedBall 0 R\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nh's : Set.Nonempty s\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n  \u2200 (A : E \u2192L[\u211d] E),\n    0 < \u03b4 A \u2227\n      \u2200 (t : Set E),\n        ApproximatesLinearOn f A t (\u03b4 A) \u2192 \u2191\u2191\u03bc (f '' t) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det A| + \u03b5) * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 n, t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\nAf' : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\n\u22a2 \u2211' (n : \u2115), \u2191\u03b5 * \u2191\u2191\u03bc (s \u2229 t n) \u2264 \u2191\u03b5 * \u2211' (n : \u2115), \u2191\u2191\u03bc (closedBall 0 R \u2229 t n)", "state_after": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nR : \u211d\nhs : s \u2286 closedBall 0 R\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nh's : Set.Nonempty s\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n  \u2200 (A : E \u2192L[\u211d] E),\n    0 < \u03b4 A \u2227\n      \u2200 (t : Set E),\n        ApproximatesLinearOn f A t (\u03b4 A) \u2192 \u2191\u2191\u03bc (f '' t) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det A| + \u03b5) * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 n, t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\nAf' : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\n\u22a2 \u2191\u03b5 * \u2211' (i : \u2115), \u2191\u2191\u03bc (s \u2229 t i) \u2264 \u2191\u03b5 * \u2211' (n : \u2115), \u2191\u2191\u03bc (closedBall 0 R \u2229 t n)"}, {"tactic": "refine' mul_le_mul_left' (ENNReal.tsum_le_tsum fun n => measure_mono _) _", "annotated_tactic": ["refine' <a>mul_le_mul_left'</a> (<a>ENNReal.tsum_le_tsum</a> fun n => <a>measure_mono</a> _) _", [{"full_name": "mul_le_mul_left'", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [50, 9], "def_end_pos": [50, 25]}, {"full_name": "ENNReal.tsum_le_tsum", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [827, 19], "def_end_pos": [827, 31]}, {"full_name": "MeasureTheory.measure_mono", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [193, 9], "def_end_pos": [193, 21]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nR : \u211d\nhs : s \u2286 closedBall 0 R\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nh's : Set.Nonempty s\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n  \u2200 (A : E \u2192L[\u211d] E),\n    0 < \u03b4 A \u2227\n      \u2200 (t : Set E),\n        ApproximatesLinearOn f A t (\u03b4 A) \u2192 \u2191\u2191\u03bc (f '' t) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det A| + \u03b5) * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 n, t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\nAf' : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\n\u22a2 \u2191\u03b5 * \u2211' (i : \u2115), \u2191\u2191\u03bc (s \u2229 t i) \u2264 \u2191\u03b5 * \u2211' (n : \u2115), \u2191\u2191\u03bc (closedBall 0 R \u2229 t n)", "state_after": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nR : \u211d\nhs : s \u2286 closedBall 0 R\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nh's : Set.Nonempty s\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n  \u2200 (A : E \u2192L[\u211d] E),\n    0 < \u03b4 A \u2227\n      \u2200 (t : Set E),\n        ApproximatesLinearOn f A t (\u03b4 A) \u2192 \u2191\u2191\u03bc (f '' t) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det A| + \u03b5) * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 n, t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\nAf' : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\nn : \u2115\n\u22a2 s \u2229 t n \u2286 closedBall 0 R \u2229 t n"}, {"tactic": "exact inter_subset_inter_left _ hs", "annotated_tactic": ["exact <a>inter_subset_inter_left</a> _ hs", [{"full_name": "Set.inter_subset_inter_left", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1027, 9], "def_end_pos": [1027, 32]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nR : \u211d\nhs : s \u2286 closedBall 0 R\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nh's : Set.Nonempty s\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n  \u2200 (A : E \u2192L[\u211d] E),\n    0 < \u03b4 A \u2227\n      \u2200 (t : Set E),\n        ApproximatesLinearOn f A t (\u03b4 A) \u2192 \u2191\u2191\u03bc (f '' t) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det A| + \u03b5) * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 n, t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\nAf' : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\nn : \u2115\n\u22a2 s \u2229 t n \u2286 closedBall 0 R \u2229 t n", "state_after": "no goals"}, {"tactic": "rw [measure_iUnion]", "annotated_tactic": ["rw [<a>measure_iUnion</a>]", [{"full_name": "MeasureTheory.measure_iUnion", "def_path": "Mathlib/MeasureTheory/Measure/NullMeasurable.lean", "def_pos": [272, 9], "def_end_pos": [272, 23]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nR : \u211d\nhs : s \u2286 closedBall 0 R\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nh's : Set.Nonempty s\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n  \u2200 (A : E \u2192L[\u211d] E),\n    0 < \u03b4 A \u2227\n      \u2200 (t : Set E),\n        ApproximatesLinearOn f A t (\u03b4 A) \u2192 \u2191\u2191\u03bc (f '' t) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det A| + \u03b5) * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 n, t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\nAf' : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\n\u22a2 \u2191\u03b5 * \u2211' (n : \u2115), \u2191\u2191\u03bc (closedBall 0 R \u2229 t n) = \u2191\u03b5 * \u2191\u2191\u03bc (\u22c3 n, closedBall 0 R \u2229 t n)", "state_after": "case hn\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nR : \u211d\nhs : s \u2286 closedBall 0 R\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nh's : Set.Nonempty s\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n  \u2200 (A : E \u2192L[\u211d] E),\n    0 < \u03b4 A \u2227\n      \u2200 (t : Set E),\n        ApproximatesLinearOn f A t (\u03b4 A) \u2192 \u2191\u2191\u03bc (f '' t) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det A| + \u03b5) * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 n, t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\nAf' : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\n\u22a2 Pairwise (Disjoint on fun n => closedBall 0 R \u2229 t n)\n\ncase h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nR : \u211d\nhs : s \u2286 closedBall 0 R\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nh's : Set.Nonempty s\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n  \u2200 (A : E \u2192L[\u211d] E),\n    0 < \u03b4 A \u2227\n      \u2200 (t : Set E),\n        ApproximatesLinearOn f A t (\u03b4 A) \u2192 \u2191\u2191\u03bc (f '' t) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det A| + \u03b5) * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 n, t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\nAf' : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\n\u22a2 \u2200 (i : \u2115), MeasurableSet (closedBall 0 R \u2229 t i)"}, {"tactic": "exact pairwise_disjoint_mono t_disj fun n => inter_subset_right _ _", "annotated_tactic": ["exact <a>pairwise_disjoint_mono</a> t_disj fun n => <a>inter_subset_right</a> _ _", [{"full_name": "pairwise_disjoint_mono", "def_path": "Mathlib/Data/Set/Pairwise/Basic.lean", "def_pos": [60, 9], "def_end_pos": [60, 31]}, {"full_name": "Set.inter_subset_right", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [969, 9], "def_end_pos": [969, 27]}]], "state_before": "case hn\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nR : \u211d\nhs : s \u2286 closedBall 0 R\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nh's : Set.Nonempty s\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n  \u2200 (A : E \u2192L[\u211d] E),\n    0 < \u03b4 A \u2227\n      \u2200 (t : Set E),\n        ApproximatesLinearOn f A t (\u03b4 A) \u2192 \u2191\u2191\u03bc (f '' t) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det A| + \u03b5) * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 n, t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\nAf' : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\n\u22a2 Pairwise (Disjoint on fun n => closedBall 0 R \u2229 t n)", "state_after": "no goals"}, {"tactic": "intro n", "annotated_tactic": ["intro n", []], "state_before": "case h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nR : \u211d\nhs : s \u2286 closedBall 0 R\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nh's : Set.Nonempty s\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n  \u2200 (A : E \u2192L[\u211d] E),\n    0 < \u03b4 A \u2227\n      \u2200 (t : Set E),\n        ApproximatesLinearOn f A t (\u03b4 A) \u2192 \u2191\u2191\u03bc (f '' t) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det A| + \u03b5) * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 n, t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\nAf' : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\n\u22a2 \u2200 (i : \u2115), MeasurableSet (closedBall 0 R \u2229 t i)", "state_after": "case h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nR : \u211d\nhs : s \u2286 closedBall 0 R\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nh's : Set.Nonempty s\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n  \u2200 (A : E \u2192L[\u211d] E),\n    0 < \u03b4 A \u2227\n      \u2200 (t : Set E),\n        ApproximatesLinearOn f A t (\u03b4 A) \u2192 \u2191\u2191\u03bc (f '' t) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det A| + \u03b5) * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 n, t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\nAf' : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\nn : \u2115\n\u22a2 MeasurableSet (closedBall 0 R \u2229 t n)"}, {"tactic": "exact measurableSet_closedBall.inter (t_meas n)", "annotated_tactic": ["exact measurableSet_closedBall.inter (t_meas n)", []], "state_before": "case h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nR : \u211d\nhs : s \u2286 closedBall 0 R\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nh's : Set.Nonempty s\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n  \u2200 (A : E \u2192L[\u211d] E),\n    0 < \u03b4 A \u2227\n      \u2200 (t : Set E),\n        ApproximatesLinearOn f A t (\u03b4 A) \u2192 \u2191\u2191\u03bc (f '' t) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det A| + \u03b5) * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 n, t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\nAf' : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\nn : \u2115\n\u22a2 MeasurableSet (closedBall 0 R \u2229 t n)", "state_after": "no goals"}, {"tactic": "rw [\u2190 inter_iUnion]", "annotated_tactic": ["rw [\u2190 <a>inter_iUnion</a>]", [{"full_name": "Set.inter_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [635, 9], "def_end_pos": [635, 21]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nR : \u211d\nhs : s \u2286 closedBall 0 R\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nh's : Set.Nonempty s\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n  \u2200 (A : E \u2192L[\u211d] E),\n    0 < \u03b4 A \u2227\n      \u2200 (t : Set E),\n        ApproximatesLinearOn f A t (\u03b4 A) \u2192 \u2191\u2191\u03bc (f '' t) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det A| + \u03b5) * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 n, t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\nAf' : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\n\u22a2 \u2191\u03b5 * \u2191\u2191\u03bc (\u22c3 n, closedBall 0 R \u2229 t n) \u2264 \u2191\u03b5 * \u2191\u2191\u03bc (closedBall 0 R)", "state_after": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nR : \u211d\nhs : s \u2286 closedBall 0 R\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nh's : Set.Nonempty s\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n  \u2200 (A : E \u2192L[\u211d] E),\n    0 < \u03b4 A \u2227\n      \u2200 (t : Set E),\n        ApproximatesLinearOn f A t (\u03b4 A) \u2192 \u2191\u2191\u03bc (f '' t) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det A| + \u03b5) * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 n, t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\nAf' : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\n\u22a2 \u2191\u03b5 * \u2191\u2191\u03bc (closedBall 0 R \u2229 \u22c3 i, t i) \u2264 \u2191\u03b5 * \u2191\u2191\u03bc (closedBall 0 R)"}, {"tactic": "exact mul_le_mul_left' (measure_mono (inter_subset_left _ _)) _", "annotated_tactic": ["exact <a>mul_le_mul_left'</a> (<a>measure_mono</a> (<a>inter_subset_left</a> _ _)) _", [{"full_name": "mul_le_mul_left'", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [50, 9], "def_end_pos": [50, 25]}, {"full_name": "MeasureTheory.measure_mono", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [193, 9], "def_end_pos": [193, 21]}, {"full_name": "Set.inter_subset_left", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [965, 9], "def_end_pos": [965, 26]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nR : \u211d\nhs : s \u2286 closedBall 0 R\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nh's : Set.Nonempty s\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n  \u2200 (A : E \u2192L[\u211d] E),\n    0 < \u03b4 A \u2227\n      \u2200 (t : Set E),\n        ApproximatesLinearOn f A t (\u03b4 A) \u2192 \u2191\u2191\u03bc (f '' t) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det A| + \u03b5) * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 n, t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\nAf' : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\n\u22a2 \u2191\u03b5 * \u2191\u2191\u03bc (closedBall 0 R \u2229 \u22c3 i, t i) \u2264 \u2191\u03b5 * \u2191\u2191\u03bc (closedBall 0 R)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Sort.lean", "full_name": "Finset.orderEmbOfFin_zero", "start": [192, 1], "end": [194, 67], "traced_tactics": [{"tactic": "simp only [orderEmbOfFin_apply, Fin.val_mk, sorted_zero_eq_min']", "annotated_tactic": ["simp only [<a>orderEmbOfFin_apply</a>, <a>Fin.val_mk</a>, <a>sorted_zero_eq_min'</a>]", [{"full_name": "Finset.orderEmbOfFin_apply", "def_path": "Mathlib/Data/Finset/Sort.lean", "def_pos": [170, 9], "def_end_pos": [170, 28]}, {"full_name": "Fin.val_mk", "def_path": "lake-packages/std/Std/Data/Fin/Lemmas.lean", "def_pos": [51, 9], "def_end_pos": [51, 15]}, {"full_name": "Finset.sorted_zero_eq_min'", "def_path": "Mathlib/Data/Finset/Sort.lean", "def_pos": [104, 9], "def_end_pos": [104, 28]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\ns : Finset \u03b1\nk : \u2115\nh : card s = k\nhz : 0 < k\n\u22a2 \u2191(orderEmbOfFin s h) { val := 0, isLt := hz } = min' s (_ : Finset.Nonempty s)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/PiSystem.lean", "full_name": "generatePiSystem_eq", "start": [258, 1], "end": [259, 91], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Int/Lemmas.lean", "full_name": "Int.natAbs_eq_iff_sq_eq", "start": [46, 1], "end": [48, 34], "traced_tactics": [{"tactic": "rw [sq, sq]", "annotated_tactic": ["rw [<a>sq</a>, <a>sq</a>]", [{"full_name": "sq", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [106, 7], "def_end_pos": [106, 9]}, {"full_name": "sq", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [106, 7], "def_end_pos": [106, 9]}]], "state_before": "a\u271d b\u271d : \u2124\nn : \u2115\na b : \u2124\n\u22a2 natAbs a = natAbs b \u2194 a ^ 2 = b ^ 2", "state_after": "a\u271d b\u271d : \u2124\nn : \u2115\na b : \u2124\n\u22a2 natAbs a = natAbs b \u2194 a * a = b * b"}, {"tactic": "exact natAbs_eq_iff_mul_self_eq", "annotated_tactic": ["exact <a>natAbs_eq_iff_mul_self_eq</a>", [{"full_name": "Int.natAbs_eq_iff_mul_self_eq", "def_path": "Mathlib/Data/Int/Order/Lemmas.lean", "def_pos": [29, 9], "def_end_pos": [29, 34]}]], "state_before": "a\u271d b\u271d : \u2124\nn : \u2115\na b : \u2124\n\u22a2 natAbs a = natAbs b \u2194 a * a = b * b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/ProbabilityMassFunction/Monad.lean", "full_name": "PMF.bind_const", "start": [146, 1], "end": [147, 81], "traced_tactics": [{"tactic": "rw [bind_apply, ENNReal.tsum_mul_right, tsum_coe, one_mul]", "annotated_tactic": ["rw [<a>bind_apply</a>, <a>ENNReal.tsum_mul_right</a>, <a>tsum_coe</a>, <a>one_mul</a>]", [{"full_name": "PMF.bind_apply", "def_path": "Mathlib/Probability/ProbabilityMassFunction/Monad.lean", "def_pos": [117, 9], "def_end_pos": [117, 19]}, {"full_name": "ENNReal.tsum_mul_right", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [907, 19], "def_end_pos": [907, 33]}, {"full_name": "PMF.tsum_coe", "def_path": "Mathlib/Probability/ProbabilityMassFunction/Basic.lean", "def_pos": [68, 9], "def_end_pos": [68, 17]}, {"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [464, 9], "def_end_pos": [464, 16]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np\u271d : PMF \u03b1\nf : \u03b1 \u2192 PMF \u03b2\ng : \u03b2 \u2192 PMF \u03b3\np : PMF \u03b1\nq : PMF \u03b2\nx : \u03b2\n\u22a2 \u2191(bind p fun x => q) x = \u2191q x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "full_name": "Real.borel_eq_generateFrom_Iio_rat", "start": [1857, 1], "end": [1867, 64], "traced_tactics": [{"tactic": "rw [borel_eq_generateFrom_Iio]", "annotated_tactic": ["rw [<a>borel_eq_generateFrom_Iio</a>]", [{"full_name": "borel_eq_generateFrom_Iio", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [124, 9], "def_end_pos": [124, 34]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns t u : Set \u03b1\n\u22a2 borel \u211d = MeasurableSpace.generateFrom (\u22c3 a, {Iio \u2191a})", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns t u : Set \u03b1\n\u22a2 MeasurableSpace.generateFrom (range Iio) = MeasurableSpace.generateFrom (\u22c3 a, {Iio \u2191a})"}, {"tactic": "refine le_antisymm\n  (generateFrom_le ?_)\n  (generateFrom_mono <| iUnion_subset fun q \u21a6 singleton_subset_iff.mpr <| mem_range_self _)", "annotated_tactic": ["refine <a>le_antisymm</a>\n    (<a>generateFrom_le</a> ?_)\n    (<a>generateFrom_mono</a> <| <a>iUnion_subset</a> fun q \u21a6 singleton_subset_iff.mpr <| <a>mem_range_self</a> _)", [{"full_name": "le_antisymm", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [188, 9], "def_end_pos": [188, 20]}, {"full_name": "MeasurableSpace.generateFrom_le", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [384, 9], "def_end_pos": [384, 24]}, {"full_name": "MeasurableSpace.generateFrom_mono", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [438, 9], "def_end_pos": [438, 26]}, {"full_name": "Set.iUnion_subset", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [390, 9], "def_end_pos": [390, 22]}, {"full_name": "Set.mem_range_self", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [680, 9], "def_end_pos": [680, 23]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns t u : Set \u03b1\n\u22a2 MeasurableSpace.generateFrom (range Iio) = MeasurableSpace.generateFrom (\u22c3 a, {Iio \u2191a})", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns t u : Set \u03b1\n\u22a2 \u2200 (t : Set \u211d), t \u2208 range Iio \u2192 MeasurableSet t"}, {"tactic": "rintro _ \u27e8a, rfl\u27e9", "annotated_tactic": ["rintro _ \u27e8a, rfl\u27e9", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns t u : Set \u03b1\n\u22a2 \u2200 (t : Set \u211d), t \u2208 range Iio \u2192 MeasurableSet t", "state_after": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns t u : Set \u03b1\na : \u211d\n\u22a2 MeasurableSet (Iio a)"}, {"tactic": "have : IsLUB (range ((\u2191) : \u211a \u2192 \u211d) \u2229 Iio a) a := by\n  simp [isLUB_iff_le_iff, mem_upperBounds, \u2190 le_iff_forall_rat_lt_imp_le]", "annotated_tactic": ["have : <a>IsLUB</a> (<a>range</a> ((\u2191) : \u211a \u2192 \u211d) \u2229 <a>Iio</a> a) a := by\n    simp [<a>isLUB_iff_le_iff</a>, <a>mem_upperBounds</a>, \u2190 <a>le_iff_forall_rat_lt_imp_le</a>]", [{"full_name": "IsLUB", "def_path": "Mathlib/Order/Bounds/Basic.lean", "def_pos": [76, 5], "def_end_pos": [76, 10]}, {"full_name": "Set.range", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [668, 5], "def_end_pos": [668, 10]}, {"full_name": "Set.Iio", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [54, 5], "def_end_pos": [54, 8]}, {"full_name": "isLUB_iff_le_iff", "def_path": "Mathlib/Order/Bounds/Basic.lean", "def_pos": [319, 9], "def_end_pos": [319, 25]}, {"full_name": "mem_upperBounds", "def_path": "Mathlib/Order/Bounds/Basic.lean", "def_pos": [85, 9], "def_end_pos": [85, 24]}, {"full_name": "le_iff_forall_rat_lt_imp_le", "def_path": "Mathlib/Algebra/Order/Archimedean.lean", "def_pos": [296, 9], "def_end_pos": [296, 36]}]], "state_before": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns t u : Set \u03b1\na : \u211d\n\u22a2 MeasurableSet (Iio a)", "state_after": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns t u : Set \u03b1\na : \u211d\nthis : IsLUB (range Rat.cast \u2229 Iio a) a\n\u22a2 MeasurableSet (Iio a)"}, {"tactic": "rw [\u2190 this.biUnion_Iio_eq, \u2190 image_univ, \u2190 image_inter_preimage, univ_inter, biUnion_image]", "annotated_tactic": ["rw [\u2190 this.biUnion_Iio_eq, \u2190 <a>image_univ</a>, \u2190 <a>image_inter_preimage</a>, <a>univ_inter</a>, <a>biUnion_image</a>]", [{"full_name": "Set.image_univ", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [718, 9], "def_end_pos": [718, 19]}, {"full_name": "Set.image_inter_preimage", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [528, 9], "def_end_pos": [528, 29]}, {"full_name": "Set.univ_inter", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1017, 9], "def_end_pos": [1017, 19]}, {"full_name": "Set.biUnion_image", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [1838, 9], "def_end_pos": [1838, 22]}]], "state_before": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns t u : Set \u03b1\na : \u211d\nthis : IsLUB (range Rat.cast \u2229 Iio a) a\n\u22a2 MeasurableSet (Iio a)", "state_after": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns t u : Set \u03b1\na : \u211d\nthis : IsLUB (range Rat.cast \u2229 Iio a) a\n\u22a2 MeasurableSet (\u22c3 y \u2208 Rat.cast \u207b\u00b9' Iio a, Iio \u2191y)"}, {"tactic": "exact MeasurableSet.biUnion (to_countable _)\n  fun b _ => GenerateMeasurable.basic (Iio (b : \u211d)) (by simp)", "annotated_tactic": ["exact <a>MeasurableSet.biUnion</a> (<a>to_countable</a> _)\n    fun b _ => <a>GenerateMeasurable.basic</a> (<a>Iio</a> (b : \u211d)) (by simp)", [{"full_name": "MeasurableSet.biUnion", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [130, 19], "def_end_pos": [130, 40]}, {"full_name": "Set.to_countable", "def_path": "Mathlib/Data/Set/Countable.lean", "def_pos": [41, 9], "def_end_pos": [41, 21]}, {"full_name": "MeasurableSpace.GenerateMeasurable.basic", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [355, 15], "def_end_pos": [355, 20]}, {"full_name": "Set.Iio", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [54, 5], "def_end_pos": [54, 8]}]], "state_before": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns t u : Set \u03b1\na : \u211d\nthis : IsLUB (range Rat.cast \u2229 Iio a) a\n\u22a2 MeasurableSet (\u22c3 y \u2208 Rat.cast \u207b\u00b9' Iio a, Iio \u2191y)", "state_after": "no goals"}, {"tactic": "simp [isLUB_iff_le_iff, mem_upperBounds, \u2190 le_iff_forall_rat_lt_imp_le]", "annotated_tactic": ["simp [<a>isLUB_iff_le_iff</a>, <a>mem_upperBounds</a>, \u2190 <a>le_iff_forall_rat_lt_imp_le</a>]", [{"full_name": "isLUB_iff_le_iff", "def_path": "Mathlib/Order/Bounds/Basic.lean", "def_pos": [319, 9], "def_end_pos": [319, 25]}, {"full_name": "mem_upperBounds", "def_path": "Mathlib/Order/Bounds/Basic.lean", "def_pos": [85, 9], "def_end_pos": [85, 24]}, {"full_name": "le_iff_forall_rat_lt_imp_le", "def_path": "Mathlib/Algebra/Order/Archimedean.lean", "def_pos": [296, 9], "def_end_pos": [296, 36]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns t u : Set \u03b1\na : \u211d\n\u22a2 IsLUB (range Rat.cast \u2229 Iio a) a", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns t u : Set \u03b1\na : \u211d\nthis : IsLUB (range Rat.cast \u2229 Iio a) a\nb : \u211a\nx\u271d : b \u2208 Rat.cast \u207b\u00b9' Iio a\n\u22a2 Iio \u2191b \u2208 \u22c3 a, {Iio \u2191a}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Sups.lean", "full_name": "Finset.disjSups_subset_iff", "start": [515, 1], "end": [516, 22], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Int/Basic.lean", "full_name": "Int.sign_coe_nat_of_nonzero", "start": [290, 1], "end": [290, 98], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Decomposition/Jordan.lean", "full_name": "MeasureTheory.JordanDecomposition.eq_of_posPart_eq_posPart", "start": [361, 9], "end": [369, 14], "traced_tactics": [{"tactic": "ext1", "annotated_tactic": ["ext1", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj\u2081 j\u2082 : JordanDecomposition \u03b1\nhj : j\u2081.posPart = j\u2082.posPart\nhj' : toSignedMeasure j\u2081 = toSignedMeasure j\u2082\n\u22a2 j\u2081 = j\u2082", "state_after": "case posPart\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj\u2081 j\u2082 : JordanDecomposition \u03b1\nhj : j\u2081.posPart = j\u2082.posPart\nhj' : toSignedMeasure j\u2081 = toSignedMeasure j\u2082\n\u22a2 j\u2081.posPart = j\u2082.posPart\n\ncase negPart\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj\u2081 j\u2082 : JordanDecomposition \u03b1\nhj : j\u2081.posPart = j\u2082.posPart\nhj' : toSignedMeasure j\u2081 = toSignedMeasure j\u2082\n\u22a2 j\u2081.negPart = j\u2082.negPart"}, {"tactic": "exact hj", "annotated_tactic": ["exact hj", []], "state_before": "case posPart\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj\u2081 j\u2082 : JordanDecomposition \u03b1\nhj : j\u2081.posPart = j\u2082.posPart\nhj' : toSignedMeasure j\u2081 = toSignedMeasure j\u2082\n\u22a2 j\u2081.posPart = j\u2082.posPart", "state_after": "no goals"}, {"tactic": "rw [\u2190 toSignedMeasure_eq_toSignedMeasure_iff]", "annotated_tactic": ["rw [\u2190 <a>toSignedMeasure_eq_toSignedMeasure_iff</a>]", [{"full_name": "MeasureTheory.Measure.toSignedMeasure_eq_toSignedMeasure_iff", "def_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "def_pos": [447, 9], "def_end_pos": [447, 47]}]], "state_before": "case negPart\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj\u2081 j\u2082 : JordanDecomposition \u03b1\nhj : j\u2081.posPart = j\u2082.posPart\nhj' : toSignedMeasure j\u2081 = toSignedMeasure j\u2082\n\u22a2 j\u2081.negPart = j\u2082.negPart", "state_after": "case negPart\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj\u2081 j\u2082 : JordanDecomposition \u03b1\nhj : j\u2081.posPart = j\u2082.posPart\nhj' : toSignedMeasure j\u2081 = toSignedMeasure j\u2082\n\u22a2 Measure.toSignedMeasure j\u2081.negPart = Measure.toSignedMeasure j\u2082.negPart"}, {"tactic": "unfold toSignedMeasure at hj'", "annotated_tactic": ["unfold <a>toSignedMeasure</a> at hj'", [{"full_name": "MeasureTheory.JordanDecomposition.toSignedMeasure", "def_path": "Mathlib/MeasureTheory/Decomposition/Jordan.lean", "def_pos": [169, 5], "def_end_pos": [169, 20]}]], "state_before": "case negPart\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj\u2081 j\u2082 : JordanDecomposition \u03b1\nhj : j\u2081.posPart = j\u2082.posPart\nhj' : toSignedMeasure j\u2081 = toSignedMeasure j\u2082\n\u22a2 Measure.toSignedMeasure j\u2081.negPart = Measure.toSignedMeasure j\u2082.negPart", "state_after": "case negPart\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj\u2081 j\u2082 : JordanDecomposition \u03b1\nhj : j\u2081.posPart = j\u2082.posPart\nhj' :\n  Measure.toSignedMeasure j\u2081.posPart - Measure.toSignedMeasure j\u2081.negPart =\n    Measure.toSignedMeasure j\u2082.posPart - Measure.toSignedMeasure j\u2082.negPart\n\u22a2 Measure.toSignedMeasure j\u2081.negPart = Measure.toSignedMeasure j\u2082.negPart"}, {"tactic": "simp_rw [hj, sub_right_inj] at hj'", "annotated_tactic": ["simp_rw [hj, <a>sub_right_inj</a>] at hj'", [{"full_name": "sub_right_inj", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [776, 3], "def_end_pos": [776, 14]}]], "state_before": "case negPart\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj\u2081 j\u2082 : JordanDecomposition \u03b1\nhj : j\u2081.posPart = j\u2082.posPart\nhj' :\n  Measure.toSignedMeasure j\u2081.posPart - Measure.toSignedMeasure j\u2081.negPart =\n    Measure.toSignedMeasure j\u2082.posPart - Measure.toSignedMeasure j\u2082.negPart\n\u22a2 Measure.toSignedMeasure j\u2081.negPart = Measure.toSignedMeasure j\u2082.negPart", "state_after": "case negPart\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj\u2081 j\u2082 : JordanDecomposition \u03b1\nhj : j\u2081.posPart = j\u2082.posPart\nhj' : Measure.toSignedMeasure j\u2081.negPart = Measure.toSignedMeasure j\u2082.negPart\n\u22a2 Measure.toSignedMeasure j\u2081.negPart = Measure.toSignedMeasure j\u2082.negPart"}, {"tactic": "exact hj'", "annotated_tactic": ["exact hj'", []], "state_before": "case negPart\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj\u2081 j\u2082 : JordanDecomposition \u03b1\nhj : j\u2081.posPart = j\u2082.posPart\nhj' : Measure.toSignedMeasure j\u2081.negPart = Measure.toSignedMeasure j\u2082.negPart\n\u22a2 Measure.toSignedMeasure j\u2081.negPart = Measure.toSignedMeasure j\u2082.negPart", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean", "full_name": "MeasureTheory.Measure.tendsto_addHaar_inter_smul_zero_of_density_zero_aux1", "start": [628, 1], "end": [677, 17], "traced_tactics": [{"tactic": "have A : Tendsto (fun r : \u211d => \u03bc (s \u2229 ({x} + r \u2022 t)) / \u03bc (closedBall x r)) (\ud835\udcdd[>] 0) (\ud835\udcdd 0) := by\n  apply\n    tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_const_nhds h\n      (eventually_of_forall fun b => zero_le _)\n  filter_upwards [self_mem_nhdsWithin]\n  rintro r (rpos : 0 < r)\n  apply mul_le_mul_right' (measure_mono (inter_subset_inter_right _ _)) _\n  intro y hy\n  have : y - x \u2208 r \u2022 closedBall (0 : E) 1 := by\n    apply smul_set_mono t_bound\n    simpa [neg_add_eq_sub] using hy\n  simpa only [smul_closedBall _ _ zero_le_one, Real.norm_of_nonneg rpos.le,\n    mem_closedBall_iff_norm, mul_one, sub_zero, smul_zero]", "annotated_tactic": ["have A : <a>Tendsto</a> (fun r : \u211d => \u03bc (s \u2229 ({x} + r \u2022 t)) / \u03bc (<a>closedBall</a> x r)) (\ud835\udcdd[>] 0) (\ud835\udcdd 0) := by\n    apply\n      <a>tendsto_of_tendsto_of_tendsto_of_le_of_le'</a> <a>tendsto_const_nhds</a> h\n        (<a>eventually_of_forall</a> fun b => <a>zero_le</a> _)\n    filter_upwards [<a>self_mem_nhdsWithin</a>]\n    rintro r (rpos : 0 < r)\n    apply <a>mul_le_mul_right'</a> (<a>measure_mono</a> (<a>inter_subset_inter_right</a> _ _)) _\n    intro y hy\n    have : y - x \u2208 r \u2022 <a>closedBall</a> (0 : E) 1 := by\n      apply <a>smul_set_mono</a> t_bound\n      simpa [<a>neg_add_eq_sub</a>] using hy\n    simpa only [<a>smul_closedBall</a> _ _ <a>zero_le_one</a>, <a>Real.norm_of_nonneg</a> rpos.le,\n      <a>mem_closedBall_iff_norm</a>, <a>mul_one</a>, <a>sub_zero</a>, <a>smul_zero</a>]", [{"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2939, 5], "def_end_pos": [2939, 12]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "tendsto_of_tendsto_of_tendsto_of_le_of_le'", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [947, 9], "def_end_pos": [947, 51]}, {"full_name": "tendsto_const_nhds", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [1049, 9], "def_end_pos": [1049, 27]}, {"full_name": "Filter.eventually_of_forall", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1111, 9], "def_end_pos": [1111, 29]}, {"full_name": "zero_le", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [217, 30], "def_end_pos": [217, 37]}, {"full_name": "self_mem_nhdsWithin", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [151, 9], "def_end_pos": [151, 28]}, {"full_name": "mul_le_mul_right'", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [67, 9], "def_end_pos": [67, 26]}, {"full_name": "MeasureTheory.measure_mono", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [193, 9], "def_end_pos": [193, 21]}, {"full_name": "Set.inter_subset_inter_right", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1032, 9], "def_end_pos": [1032, 33]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "Set.smul_set_mono", "def_path": "Mathlib/Data/Set/Pointwise/SMul.lean", "def_pos": [354, 9], "def_end_pos": [354, 22]}, {"full_name": "neg_add_eq_sub", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [496, 3], "def_end_pos": [496, 14]}, {"full_name": "smul_closedBall", "def_path": "Mathlib/Analysis/NormedSpace/Pointwise.lean", "def_pos": [388, 9], "def_end_pos": [388, 24]}, {"full_name": "zero_le_one", "def_path": "Mathlib/Algebra/Order/ZeroLEOne.lean", "def_pos": [26, 15], "def_end_pos": [26, 26]}, {"full_name": "Real.norm_of_nonneg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [1768, 9], "def_end_pos": [1768, 23]}, {"full_name": "mem_closedBall_iff_norm", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [646, 15], "def_end_pos": [646, 38]}, {"full_name": "mul_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [470, 9], "def_end_pos": [470, 16]}, {"full_name": "sub_zero", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [339, 3], "def_end_pos": [339, 14]}, {"full_name": "smul_zero", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [732, 9], "def_end_pos": [732, 18]}]], "state_before": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt u : Set E\nh'u : \u2191\u2191\u03bc u \u2260 0\nt_bound : t \u2286 closedBall 0 1\n\u22a2 Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 u)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)", "state_after": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt u : Set E\nh'u : \u2191\u2191\u03bc u \u2260 0\nt_bound : t \u2286 closedBall 0 1\nA : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\n\u22a2 Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 u)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)"}, {"tactic": "have B :\n  Tendsto (fun r : \u211d => \u03bc (closedBall x r) / \u03bc ({x} + r \u2022 u)) (\ud835\udcdd[>] 0)\n    (\ud835\udcdd (\u03bc (closedBall x 1) / \u03bc ({x} + u))) := by\n  apply tendsto_const_nhds.congr' _\n  filter_upwards [self_mem_nhdsWithin]\n  rintro r (rpos : 0 < r)\n  have : closedBall x r = {x} + r \u2022 closedBall (0 : E) 1 := by\n    simp only [_root_.smul_closedBall, Real.norm_of_nonneg rpos.le, zero_le_one, add_zero,\n      mul_one, singleton_add_closedBall, smul_zero]\n  simp only [this, addHaar_singleton_add_smul_div_singleton_add_smul \u03bc rpos.ne']\n  simp only [addHaar_closedBall_center, image_add_left, measure_preimage_add, singleton_add]", "annotated_tactic": ["have B :\n    <a>Tendsto</a> (fun r : \u211d => \u03bc (<a>closedBall</a> x r) / \u03bc ({x} + r \u2022 u)) (\ud835\udcdd[>] 0)\n      (\ud835\udcdd (\u03bc (<a>closedBall</a> x 1) / \u03bc ({x} + u))) := by\n    apply tendsto_const_nhds.congr' _\n    filter_upwards [<a>self_mem_nhdsWithin</a>]\n    rintro r (rpos : 0 < r)\n    have : <a>closedBall</a> x r = {x} + r \u2022 <a>closedBall</a> (0 : E) 1 := by\n      simp only [<a>_root_.smul_closedBall</a>, <a>Real.norm_of_nonneg</a> rpos.le, <a>zero_le_one</a>, <a>add_zero</a>,\n        <a>mul_one</a>, <a>singleton_add_closedBall</a>, <a>smul_zero</a>]\n    simp only [this, <a>addHaar_singleton_add_smul_div_singleton_add_smul</a> \u03bc rpos.ne']\n    simp only [<a>addHaar_closedBall_center</a>, <a>image_add_left</a>, <a>measure_preimage_add</a>, <a>singleton_add</a>]", [{"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2939, 5], "def_end_pos": [2939, 12]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "self_mem_nhdsWithin", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [151, 9], "def_end_pos": [151, 28]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "smul_closedBall", "def_path": "Mathlib/Analysis/NormedSpace/Pointwise.lean", "def_pos": [388, 9], "def_end_pos": [388, 24]}, {"full_name": "Real.norm_of_nonneg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [1768, 9], "def_end_pos": [1768, 23]}, {"full_name": "zero_le_one", "def_path": "Mathlib/Algebra/Order/ZeroLEOne.lean", "def_pos": [26, 15], "def_end_pos": [26, 26]}, {"full_name": "add_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [469, 3], "def_end_pos": [469, 14]}, {"full_name": "mul_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [470, 9], "def_end_pos": [470, 16]}, {"full_name": "singleton_add_closedBall", "def_path": "Mathlib/Analysis/Normed/Group/Pointwise.lean", "def_pos": [153, 3], "def_end_pos": [153, 14]}, {"full_name": "smul_zero", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [732, 9], "def_end_pos": [732, 18]}, {"full_name": "MeasureTheory.Measure.addHaar_singleton_add_smul_div_singleton_add_smul", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean", "def_pos": [529, 9], "def_end_pos": [529, 58]}, {"full_name": "MeasureTheory.Measure.addHaar_closedBall_center", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean", "def_pos": [432, 9], "def_end_pos": [432, 34]}, {"full_name": "Set.image_add_left", "def_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "def_pos": [1198, 3], "def_end_pos": [1198, 14]}, {"full_name": "MeasureTheory.measure_preimage_add", "def_path": "Mathlib/MeasureTheory/Group/Measure.lean", "def_pos": [317, 3], "def_end_pos": [317, 14]}, {"full_name": "Set.singleton_add", "def_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "def_pos": [402, 3], "def_end_pos": [402, 14]}]], "state_before": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt u : Set E\nh'u : \u2191\u2191\u03bc u \u2260 0\nt_bound : t \u2286 closedBall 0 1\nA : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\n\u22a2 Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 u)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)", "state_after": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt u : Set E\nh'u : \u2191\u2191\u03bc u \u2260 0\nt_bound : t \u2286 closedBall 0 1\nA : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nB : Tendsto (fun r => \u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc ({x} + r \u2022 u)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (closedBall x 1) / \u2191\u2191\u03bc ({x} + u)))\n\u22a2 Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 u)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)"}, {"tactic": "have C : Tendsto (fun r : \u211d =>\n      \u03bc (s \u2229 ({x} + r \u2022 t)) / \u03bc (closedBall x r) * (\u03bc (closedBall x r) / \u03bc ({x} + r \u2022 u)))\n    (\ud835\udcdd[>] 0) (\ud835\udcdd (0 * (\u03bc (closedBall x 1) / \u03bc ({x} + u)))) := by\n  apply ENNReal.Tendsto.mul A _ B (Or.inr ENNReal.zero_ne_top)\n  simp only [ne_eq, not_true, singleton_add, image_add_left, measure_preimage_add, false_or,\n    ENNReal.div_eq_top, h'u, false_or_iff, not_and, and_false_iff]\n  intro aux\n  exact (measure_closedBall_lt_top.ne aux).elim", "annotated_tactic": ["have C : <a>Tendsto</a> (fun r : \u211d =>\n        \u03bc (s \u2229 ({x} + r \u2022 t)) / \u03bc (<a>closedBall</a> x r) * (\u03bc (<a>closedBall</a> x r) / \u03bc ({x} + r \u2022 u)))\n      (\ud835\udcdd[>] 0) (\ud835\udcdd (0 * (\u03bc (<a>closedBall</a> x 1) / \u03bc ({x} + u)))) := by\n    apply <a>ENNReal.Tendsto.mul</a> A _ B (<a>Or.inr</a> <a>ENNReal.zero_ne_top</a>)\n    simp only [<a>ne_eq</a>, <a>not_true</a>, <a>singleton_add</a>, <a>image_add_left</a>, <a>measure_preimage_add</a>, <a>false_or</a>,\n      <a>ENNReal.div_eq_top</a>, h'u, <a>false_or_iff</a>, <a>not_and</a>, <a>and_false_iff</a>]\n    intro aux\n    exact (measure_closedBall_lt_top.ne aux).<a>elim</a>", [{"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2939, 5], "def_end_pos": [2939, 12]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "ENNReal.Tendsto.mul", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [353, 19], "def_end_pos": [353, 30]}, {"full_name": "Or.inr", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [519, 5], "def_end_pos": [519, 8]}, {"full_name": "ENNReal.zero_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [334, 17], "def_end_pos": [334, 28]}, {"full_name": "ne_eq", "def_path": "lake-packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [76, 17], "def_end_pos": [76, 22]}, {"full_name": "not_true", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [80, 17], "def_end_pos": [80, 25]}, {"full_name": "Set.singleton_add", "def_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "def_pos": [402, 3], "def_end_pos": [402, 14]}, {"full_name": "Set.image_add_left", "def_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "def_pos": [1198, 3], "def_end_pos": [1198, 14]}, {"full_name": "MeasureTheory.measure_preimage_add", "def_path": "Mathlib/MeasureTheory/Group/Measure.lean", "def_pos": [317, 3], "def_end_pos": [317, 14]}, {"full_name": "false_or", "def_path": "lake-packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [91, 17], "def_end_pos": [91, 25]}, {"full_name": "ENNReal.div_eq_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1607, 9], "def_end_pos": [1607, 19]}, {"full_name": "false_or_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [186, 9], "def_end_pos": [186, 21]}, {"full_name": "not_and", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [316, 17], "def_end_pos": [316, 24]}, {"full_name": "and_false_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [149, 9], "def_end_pos": [149, 22]}, {"full_name": "False.elim", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [223, 21], "def_end_pos": [223, 31]}]], "state_before": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt u : Set E\nh'u : \u2191\u2191\u03bc u \u2260 0\nt_bound : t \u2286 closedBall 0 1\nA : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nB : Tendsto (fun r => \u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc ({x} + r \u2022 u)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (closedBall x 1) / \u2191\u2191\u03bc ({x} + u)))\n\u22a2 Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 u)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)", "state_after": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt u : Set E\nh'u : \u2191\u2191\u03bc u \u2260 0\nt_bound : t \u2286 closedBall 0 1\nA : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nB : Tendsto (fun r => \u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc ({x} + r \u2022 u)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (closedBall x 1) / \u2191\u2191\u03bc ({x} + u)))\nC :\n  Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r) * (\u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc ({x} + r \u2022 u)))\n    (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (0 * (\u2191\u2191\u03bc (closedBall x 1) / \u2191\u2191\u03bc ({x} + u))))\n\u22a2 Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 u)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)"}, {"tactic": "simp only [zero_mul] at C", "annotated_tactic": ["simp only [<a>zero_mul</a>] at C", [{"full_name": "MulZeroClass.zero_mul", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [36, 3], "def_end_pos": [36, 11]}]], "state_before": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt u : Set E\nh'u : \u2191\u2191\u03bc u \u2260 0\nt_bound : t \u2286 closedBall 0 1\nA : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nB : Tendsto (fun r => \u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc ({x} + r \u2022 u)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (closedBall x 1) / \u2191\u2191\u03bc ({x} + u)))\nC :\n  Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r) * (\u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc ({x} + r \u2022 u)))\n    (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (0 * (\u2191\u2191\u03bc (closedBall x 1) / \u2191\u2191\u03bc ({x} + u))))\n\u22a2 Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 u)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)", "state_after": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt u : Set E\nh'u : \u2191\u2191\u03bc u \u2260 0\nt_bound : t \u2286 closedBall 0 1\nA : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nB : Tendsto (fun r => \u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc ({x} + r \u2022 u)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (closedBall x 1) / \u2191\u2191\u03bc ({x} + u)))\nC :\n  Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r) * (\u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc ({x} + r \u2022 u)))\n    (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\n\u22a2 Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 u)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)"}, {"tactic": "apply C.congr' _", "annotated_tactic": ["apply C.congr' _", []], "state_before": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt u : Set E\nh'u : \u2191\u2191\u03bc u \u2260 0\nt_bound : t \u2286 closedBall 0 1\nA : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nB : Tendsto (fun r => \u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc ({x} + r \u2022 u)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (closedBall x 1) / \u2191\u2191\u03bc ({x} + u)))\nC :\n  Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r) * (\u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc ({x} + r \u2022 u)))\n    (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\n\u22a2 Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 u)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)", "state_after": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt u : Set E\nh'u : \u2191\u2191\u03bc u \u2260 0\nt_bound : t \u2286 closedBall 0 1\nA : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nB : Tendsto (fun r => \u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc ({x} + r \u2022 u)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (closedBall x 1) / \u2191\u2191\u03bc ({x} + u)))\nC :\n  Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r) * (\u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc ({x} + r \u2022 u)))\n    (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\n\u22a2 (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r) * (\u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc ({x} + r \u2022 u))) =\u1da0[\ud835\udcdd[Ioi 0] 0]\n    fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 u)"}, {"tactic": "filter_upwards [self_mem_nhdsWithin]", "annotated_tactic": ["filter_upwards [<a>self_mem_nhdsWithin</a>]", [{"full_name": "self_mem_nhdsWithin", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [151, 9], "def_end_pos": [151, 28]}]], "state_before": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt u : Set E\nh'u : \u2191\u2191\u03bc u \u2260 0\nt_bound : t \u2286 closedBall 0 1\nA : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nB : Tendsto (fun r => \u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc ({x} + r \u2022 u)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (closedBall x 1) / \u2191\u2191\u03bc ({x} + u)))\nC :\n  Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r) * (\u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc ({x} + r \u2022 u)))\n    (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\n\u22a2 (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r) * (\u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc ({x} + r \u2022 u))) =\u1da0[\ud835\udcdd[Ioi 0] 0]\n    fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 u)", "state_after": "case h\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt u : Set E\nh'u : \u2191\u2191\u03bc u \u2260 0\nt_bound : t \u2286 closedBall 0 1\nA : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nB : Tendsto (fun r => \u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc ({x} + r \u2022 u)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (closedBall x 1) / \u2191\u2191\u03bc ({x} + u)))\nC :\n  Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r) * (\u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc ({x} + r \u2022 u)))\n    (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\n\u22a2 \u2200 (a : \u211d),\n    a \u2208 Ioi 0 \u2192\n      \u2191\u2191\u03bc (s \u2229 ({x} + a \u2022 t)) / \u2191\u2191\u03bc (closedBall x a) * (\u2191\u2191\u03bc (closedBall x a) / \u2191\u2191\u03bc ({x} + a \u2022 u)) =\n        \u2191\u2191\u03bc (s \u2229 ({x} + a \u2022 t)) / \u2191\u2191\u03bc ({x} + a \u2022 u)"}, {"tactic": "rintro r (rpos : 0 < r)", "annotated_tactic": ["rintro r (rpos : 0 < r)", []], "state_before": "case h\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt u : Set E\nh'u : \u2191\u2191\u03bc u \u2260 0\nt_bound : t \u2286 closedBall 0 1\nA : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nB : Tendsto (fun r => \u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc ({x} + r \u2022 u)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (closedBall x 1) / \u2191\u2191\u03bc ({x} + u)))\nC :\n  Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r) * (\u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc ({x} + r \u2022 u)))\n    (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\n\u22a2 \u2200 (a : \u211d),\n    a \u2208 Ioi 0 \u2192\n      \u2191\u2191\u03bc (s \u2229 ({x} + a \u2022 t)) / \u2191\u2191\u03bc (closedBall x a) * (\u2191\u2191\u03bc (closedBall x a) / \u2191\u2191\u03bc ({x} + a \u2022 u)) =\n        \u2191\u2191\u03bc (s \u2229 ({x} + a \u2022 t)) / \u2191\u2191\u03bc ({x} + a \u2022 u)", "state_after": "case h\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt u : Set E\nh'u : \u2191\u2191\u03bc u \u2260 0\nt_bound : t \u2286 closedBall 0 1\nA : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nB : Tendsto (fun r => \u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc ({x} + r \u2022 u)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (closedBall x 1) / \u2191\u2191\u03bc ({x} + u)))\nC :\n  Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r) * (\u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc ({x} + r \u2022 u)))\n    (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nr : \u211d\nrpos : 0 < r\n\u22a2 \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r) * (\u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc ({x} + r \u2022 u)) =\n    \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 u)"}, {"tactic": "calc\n  \u03bc (s \u2229 ({x} + r \u2022 t)) / \u03bc (closedBall x r) * (\u03bc (closedBall x r) / \u03bc ({x} + r \u2022 u)) =\n      \u03bc (closedBall x r) * (\u03bc (closedBall x r))\u207b\u00b9 * (\u03bc (s \u2229 ({x} + r \u2022 t)) / \u03bc ({x} + r \u2022 u)) :=\n    by simp only [div_eq_mul_inv]; ring\n  _ = \u03bc (s \u2229 ({x} + r \u2022 t)) / \u03bc ({x} + r \u2022 u) := by\n    rw [ENNReal.mul_inv_cancel (measure_closedBall_pos \u03bc x rpos).ne'\n        measure_closedBall_lt_top.ne,\n      one_mul]", "annotated_tactic": ["calc\n    \u03bc (s \u2229 ({x} + r \u2022 t)) / \u03bc (<a>closedBall</a> x r) * (\u03bc (<a>closedBall</a> x r) / \u03bc ({x} + r \u2022 u)) =\n        \u03bc (<a>closedBall</a> x r) * (\u03bc (<a>closedBall</a> x r))\u207b\u00b9 * (\u03bc (s \u2229 ({x} + r \u2022 t)) / \u03bc ({x} + r \u2022 u)) :=\n      by simp only [<a>div_eq_mul_inv</a>]; ring\n    _ = \u03bc (s \u2229 ({x} + r \u2022 t)) / \u03bc ({x} + r \u2022 u) := by\n      rw [<a>ENNReal.mul_inv_cancel</a> (<a>measure_closedBall_pos</a> \u03bc x rpos).<a>ne'</a>\n          measure_closedBall_lt_top.ne,\n        <a>one_mul</a>]", [{"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "div_eq_mul_inv", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [977, 9], "def_end_pos": [977, 23]}, {"full_name": "ENNReal.mul_inv_cancel", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1418, 19], "def_end_pos": [1418, 33]}, {"full_name": "Metric.measure_closedBall_pos", "def_path": "Mathlib/MeasureTheory/Measure/OpenPos.lean", "def_pos": [227, 9], "def_end_pos": [227, 31]}, {"full_name": "LT.lt.ne'", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [328, 9], "def_end_pos": [328, 12]}, {"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [464, 9], "def_end_pos": [464, 16]}]], "state_before": "case h\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt u : Set E\nh'u : \u2191\u2191\u03bc u \u2260 0\nt_bound : t \u2286 closedBall 0 1\nA : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nB : Tendsto (fun r => \u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc ({x} + r \u2022 u)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (closedBall x 1) / \u2191\u2191\u03bc ({x} + u)))\nC :\n  Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r) * (\u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc ({x} + r \u2022 u)))\n    (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nr : \u211d\nrpos : 0 < r\n\u22a2 \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r) * (\u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc ({x} + r \u2022 u)) =\n    \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 u)", "state_after": "no goals"}, {"tactic": "apply\n  tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_const_nhds h\n    (eventually_of_forall fun b => zero_le _)", "annotated_tactic": ["apply\n      <a>tendsto_of_tendsto_of_tendsto_of_le_of_le'</a> <a>tendsto_const_nhds</a> h\n        (<a>eventually_of_forall</a> fun b => <a>zero_le</a> _)", [{"full_name": "tendsto_of_tendsto_of_tendsto_of_le_of_le'", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [947, 9], "def_end_pos": [947, 51]}, {"full_name": "tendsto_const_nhds", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [1049, 9], "def_end_pos": [1049, 27]}, {"full_name": "Filter.eventually_of_forall", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1111, 9], "def_end_pos": [1111, 29]}, {"full_name": "zero_le", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [217, 30], "def_end_pos": [217, 37]}]], "state_before": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt u : Set E\nh'u : \u2191\u2191\u03bc u \u2260 0\nt_bound : t \u2286 closedBall 0 1\n\u22a2 Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)", "state_after": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt u : Set E\nh'u : \u2191\u2191\u03bc u \u2260 0\nt_bound : t \u2286 closedBall 0 1\n\u22a2 \u2200\u1da0 (b : \u211d) in \ud835\udcdd[Ioi 0] 0,\n    \u2191\u2191\u03bc (s \u2229 ({x} + b \u2022 t)) / \u2191\u2191\u03bc (closedBall x b) \u2264 \u2191\u2191\u03bc (s \u2229 closedBall x b) / \u2191\u2191\u03bc (closedBall x b)"}, {"tactic": "filter_upwards [self_mem_nhdsWithin]", "annotated_tactic": ["filter_upwards [<a>self_mem_nhdsWithin</a>]", [{"full_name": "self_mem_nhdsWithin", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [151, 9], "def_end_pos": [151, 28]}]], "state_before": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt u : Set E\nh'u : \u2191\u2191\u03bc u \u2260 0\nt_bound : t \u2286 closedBall 0 1\n\u22a2 \u2200\u1da0 (b : \u211d) in \ud835\udcdd[Ioi 0] 0,\n    \u2191\u2191\u03bc (s \u2229 ({x} + b \u2022 t)) / \u2191\u2191\u03bc (closedBall x b) \u2264 \u2191\u2191\u03bc (s \u2229 closedBall x b) / \u2191\u2191\u03bc (closedBall x b)", "state_after": "case h\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt u : Set E\nh'u : \u2191\u2191\u03bc u \u2260 0\nt_bound : t \u2286 closedBall 0 1\n\u22a2 \u2200 (a : \u211d),\n    a \u2208 Ioi 0 \u2192 \u2191\u2191\u03bc (s \u2229 ({x} + a \u2022 t)) / \u2191\u2191\u03bc (closedBall x a) \u2264 \u2191\u2191\u03bc (s \u2229 closedBall x a) / \u2191\u2191\u03bc (closedBall x a)"}, {"tactic": "rintro r (rpos : 0 < r)", "annotated_tactic": ["rintro r (rpos : 0 < r)", []], "state_before": "case h\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt u : Set E\nh'u : \u2191\u2191\u03bc u \u2260 0\nt_bound : t \u2286 closedBall 0 1\n\u22a2 \u2200 (a : \u211d),\n    a \u2208 Ioi 0 \u2192 \u2191\u2191\u03bc (s \u2229 ({x} + a \u2022 t)) / \u2191\u2191\u03bc (closedBall x a) \u2264 \u2191\u2191\u03bc (s \u2229 closedBall x a) / \u2191\u2191\u03bc (closedBall x a)", "state_after": "case h\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt u : Set E\nh'u : \u2191\u2191\u03bc u \u2260 0\nt_bound : t \u2286 closedBall 0 1\nr : \u211d\nrpos : 0 < r\n\u22a2 \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r) \u2264 \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)"}, {"tactic": "apply mul_le_mul_right' (measure_mono (inter_subset_inter_right _ _)) _", "annotated_tactic": ["apply <a>mul_le_mul_right'</a> (<a>measure_mono</a> (<a>inter_subset_inter_right</a> _ _)) _", [{"full_name": "mul_le_mul_right'", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [67, 9], "def_end_pos": [67, 26]}, {"full_name": "MeasureTheory.measure_mono", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [193, 9], "def_end_pos": [193, 21]}, {"full_name": "Set.inter_subset_inter_right", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1032, 9], "def_end_pos": [1032, 33]}]], "state_before": "case h\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt u : Set E\nh'u : \u2191\u2191\u03bc u \u2260 0\nt_bound : t \u2286 closedBall 0 1\nr : \u211d\nrpos : 0 < r\n\u22a2 \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r) \u2264 \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)", "state_after": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt u : Set E\nh'u : \u2191\u2191\u03bc u \u2260 0\nt_bound : t \u2286 closedBall 0 1\nr : \u211d\nrpos : 0 < r\n\u22a2 {x} + r \u2022 t \u2286 closedBall x r"}, {"tactic": "intro y hy", "annotated_tactic": ["intro y hy", []], "state_before": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt u : Set E\nh'u : \u2191\u2191\u03bc u \u2260 0\nt_bound : t \u2286 closedBall 0 1\nr : \u211d\nrpos : 0 < r\n\u22a2 {x} + r \u2022 t \u2286 closedBall x r", "state_after": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt u : Set E\nh'u : \u2191\u2191\u03bc u \u2260 0\nt_bound : t \u2286 closedBall 0 1\nr : \u211d\nrpos : 0 < r\ny : E\nhy : y \u2208 {x} + r \u2022 t\n\u22a2 y \u2208 closedBall x r"}, {"tactic": "have : y - x \u2208 r \u2022 closedBall (0 : E) 1 := by\n  apply smul_set_mono t_bound\n  simpa [neg_add_eq_sub] using hy", "annotated_tactic": ["have : y - x \u2208 r \u2022 <a>closedBall</a> (0 : E) 1 := by\n      apply <a>smul_set_mono</a> t_bound\n      simpa [<a>neg_add_eq_sub</a>] using hy", [{"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "Set.smul_set_mono", "def_path": "Mathlib/Data/Set/Pointwise/SMul.lean", "def_pos": [354, 9], "def_end_pos": [354, 22]}, {"full_name": "neg_add_eq_sub", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [496, 3], "def_end_pos": [496, 14]}]], "state_before": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt u : Set E\nh'u : \u2191\u2191\u03bc u \u2260 0\nt_bound : t \u2286 closedBall 0 1\nr : \u211d\nrpos : 0 < r\ny : E\nhy : y \u2208 {x} + r \u2022 t\n\u22a2 y \u2208 closedBall x r", "state_after": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt u : Set E\nh'u : \u2191\u2191\u03bc u \u2260 0\nt_bound : t \u2286 closedBall 0 1\nr : \u211d\nrpos : 0 < r\ny : E\nhy : y \u2208 {x} + r \u2022 t\nthis : y - x \u2208 r \u2022 closedBall 0 1\n\u22a2 y \u2208 closedBall x r"}, {"tactic": "simpa only [smul_closedBall _ _ zero_le_one, Real.norm_of_nonneg rpos.le,\n  mem_closedBall_iff_norm, mul_one, sub_zero, smul_zero]", "annotated_tactic": ["simpa only [<a>smul_closedBall</a> _ _ <a>zero_le_one</a>, <a>Real.norm_of_nonneg</a> rpos.le,\n      <a>mem_closedBall_iff_norm</a>, <a>mul_one</a>, <a>sub_zero</a>, <a>smul_zero</a>]", [{"full_name": "smul_closedBall", "def_path": "Mathlib/Analysis/NormedSpace/Pointwise.lean", "def_pos": [388, 9], "def_end_pos": [388, 24]}, {"full_name": "zero_le_one", "def_path": "Mathlib/Algebra/Order/ZeroLEOne.lean", "def_pos": [26, 15], "def_end_pos": [26, 26]}, {"full_name": "Real.norm_of_nonneg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [1768, 9], "def_end_pos": [1768, 23]}, {"full_name": "mem_closedBall_iff_norm", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [646, 15], "def_end_pos": [646, 38]}, {"full_name": "mul_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [470, 9], "def_end_pos": [470, 16]}, {"full_name": "sub_zero", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [339, 3], "def_end_pos": [339, 14]}, {"full_name": "smul_zero", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [732, 9], "def_end_pos": [732, 18]}]], "state_before": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt u : Set E\nh'u : \u2191\u2191\u03bc u \u2260 0\nt_bound : t \u2286 closedBall 0 1\nr : \u211d\nrpos : 0 < r\ny : E\nhy : y \u2208 {x} + r \u2022 t\nthis : y - x \u2208 r \u2022 closedBall 0 1\n\u22a2 y \u2208 closedBall x r", "state_after": "no goals"}, {"tactic": "apply smul_set_mono t_bound", "annotated_tactic": ["apply <a>smul_set_mono</a> t_bound", [{"full_name": "Set.smul_set_mono", "def_path": "Mathlib/Data/Set/Pointwise/SMul.lean", "def_pos": [354, 9], "def_end_pos": [354, 22]}]], "state_before": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt u : Set E\nh'u : \u2191\u2191\u03bc u \u2260 0\nt_bound : t \u2286 closedBall 0 1\nr : \u211d\nrpos : 0 < r\ny : E\nhy : y \u2208 {x} + r \u2022 t\n\u22a2 y - x \u2208 r \u2022 closedBall 0 1", "state_after": "case a\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt u : Set E\nh'u : \u2191\u2191\u03bc u \u2260 0\nt_bound : t \u2286 closedBall 0 1\nr : \u211d\nrpos : 0 < r\ny : E\nhy : y \u2208 {x} + r \u2022 t\n\u22a2 y - x \u2208 r \u2022 t"}, {"tactic": "simpa [neg_add_eq_sub] using hy", "annotated_tactic": ["simpa [<a>neg_add_eq_sub</a>] using hy", [{"full_name": "neg_add_eq_sub", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [496, 3], "def_end_pos": [496, 14]}]], "state_before": "case a\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt u : Set E\nh'u : \u2191\u2191\u03bc u \u2260 0\nt_bound : t \u2286 closedBall 0 1\nr : \u211d\nrpos : 0 < r\ny : E\nhy : y \u2208 {x} + r \u2022 t\n\u22a2 y - x \u2208 r \u2022 t", "state_after": "no goals"}, {"tactic": "apply tendsto_const_nhds.congr' _", "annotated_tactic": ["apply tendsto_const_nhds.congr' _", []], "state_before": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt u : Set E\nh'u : \u2191\u2191\u03bc u \u2260 0\nt_bound : t \u2286 closedBall 0 1\nA : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\n\u22a2 Tendsto (fun r => \u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc ({x} + r \u2022 u)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (closedBall x 1) / \u2191\u2191\u03bc ({x} + u)))", "state_after": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt u : Set E\nh'u : \u2191\u2191\u03bc u \u2260 0\nt_bound : t \u2286 closedBall 0 1\nA : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\n\u22a2 (fun x_1 => \u2191\u2191\u03bc (closedBall x 1) / \u2191\u2191\u03bc ({x} + u)) =\u1da0[\ud835\udcdd[Ioi 0] 0] fun r => \u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc ({x} + r \u2022 u)"}, {"tactic": "filter_upwards [self_mem_nhdsWithin]", "annotated_tactic": ["filter_upwards [<a>self_mem_nhdsWithin</a>]", [{"full_name": "self_mem_nhdsWithin", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [151, 9], "def_end_pos": [151, 28]}]], "state_before": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt u : Set E\nh'u : \u2191\u2191\u03bc u \u2260 0\nt_bound : t \u2286 closedBall 0 1\nA : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\n\u22a2 (fun x_1 => \u2191\u2191\u03bc (closedBall x 1) / \u2191\u2191\u03bc ({x} + u)) =\u1da0[\ud835\udcdd[Ioi 0] 0] fun r => \u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc ({x} + r \u2022 u)", "state_after": "case h\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt u : Set E\nh'u : \u2191\u2191\u03bc u \u2260 0\nt_bound : t \u2286 closedBall 0 1\nA : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\n\u22a2 \u2200 (a : \u211d), a \u2208 Ioi 0 \u2192 \u2191\u2191\u03bc (closedBall x 1) / \u2191\u2191\u03bc ({x} + u) = \u2191\u2191\u03bc (closedBall x a) / \u2191\u2191\u03bc ({x} + a \u2022 u)"}, {"tactic": "rintro r (rpos : 0 < r)", "annotated_tactic": ["rintro r (rpos : 0 < r)", []], "state_before": "case h\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt u : Set E\nh'u : \u2191\u2191\u03bc u \u2260 0\nt_bound : t \u2286 closedBall 0 1\nA : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\n\u22a2 \u2200 (a : \u211d), a \u2208 Ioi 0 \u2192 \u2191\u2191\u03bc (closedBall x 1) / \u2191\u2191\u03bc ({x} + u) = \u2191\u2191\u03bc (closedBall x a) / \u2191\u2191\u03bc ({x} + a \u2022 u)", "state_after": "case h\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt u : Set E\nh'u : \u2191\u2191\u03bc u \u2260 0\nt_bound : t \u2286 closedBall 0 1\nA : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nr : \u211d\nrpos : 0 < r\n\u22a2 \u2191\u2191\u03bc (closedBall x 1) / \u2191\u2191\u03bc ({x} + u) = \u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc ({x} + r \u2022 u)"}, {"tactic": "have : closedBall x r = {x} + r \u2022 closedBall (0 : E) 1 := by\n  simp only [_root_.smul_closedBall, Real.norm_of_nonneg rpos.le, zero_le_one, add_zero,\n    mul_one, singleton_add_closedBall, smul_zero]", "annotated_tactic": ["have : <a>closedBall</a> x r = {x} + r \u2022 <a>closedBall</a> (0 : E) 1 := by\n      simp only [<a>_root_.smul_closedBall</a>, <a>Real.norm_of_nonneg</a> rpos.le, <a>zero_le_one</a>, <a>add_zero</a>,\n        <a>mul_one</a>, <a>singleton_add_closedBall</a>, <a>smul_zero</a>]", [{"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "smul_closedBall", "def_path": "Mathlib/Analysis/NormedSpace/Pointwise.lean", "def_pos": [388, 9], "def_end_pos": [388, 24]}, {"full_name": "Real.norm_of_nonneg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [1768, 9], "def_end_pos": [1768, 23]}, {"full_name": "zero_le_one", "def_path": "Mathlib/Algebra/Order/ZeroLEOne.lean", "def_pos": [26, 15], "def_end_pos": [26, 26]}, {"full_name": "add_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [469, 3], "def_end_pos": [469, 14]}, {"full_name": "mul_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [470, 9], "def_end_pos": [470, 16]}, {"full_name": "singleton_add_closedBall", "def_path": "Mathlib/Analysis/Normed/Group/Pointwise.lean", "def_pos": [153, 3], "def_end_pos": [153, 14]}, {"full_name": "smul_zero", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [732, 9], "def_end_pos": [732, 18]}]], "state_before": "case h\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt u : Set E\nh'u : \u2191\u2191\u03bc u \u2260 0\nt_bound : t \u2286 closedBall 0 1\nA : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nr : \u211d\nrpos : 0 < r\n\u22a2 \u2191\u2191\u03bc (closedBall x 1) / \u2191\u2191\u03bc ({x} + u) = \u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc ({x} + r \u2022 u)", "state_after": "case h\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt u : Set E\nh'u : \u2191\u2191\u03bc u \u2260 0\nt_bound : t \u2286 closedBall 0 1\nA : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nr : \u211d\nrpos : 0 < r\nthis : closedBall x r = {x} + r \u2022 closedBall 0 1\n\u22a2 \u2191\u2191\u03bc (closedBall x 1) / \u2191\u2191\u03bc ({x} + u) = \u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc ({x} + r \u2022 u)"}, {"tactic": "simp only [this, addHaar_singleton_add_smul_div_singleton_add_smul \u03bc rpos.ne']", "annotated_tactic": ["simp only [this, <a>addHaar_singleton_add_smul_div_singleton_add_smul</a> \u03bc rpos.ne']", [{"full_name": "MeasureTheory.Measure.addHaar_singleton_add_smul_div_singleton_add_smul", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean", "def_pos": [529, 9], "def_end_pos": [529, 58]}]], "state_before": "case h\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt u : Set E\nh'u : \u2191\u2191\u03bc u \u2260 0\nt_bound : t \u2286 closedBall 0 1\nA : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nr : \u211d\nrpos : 0 < r\nthis : closedBall x r = {x} + r \u2022 closedBall 0 1\n\u22a2 \u2191\u2191\u03bc (closedBall x 1) / \u2191\u2191\u03bc ({x} + u) = \u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc ({x} + r \u2022 u)", "state_after": "case h\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt u : Set E\nh'u : \u2191\u2191\u03bc u \u2260 0\nt_bound : t \u2286 closedBall 0 1\nA : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nr : \u211d\nrpos : 0 < r\nthis : closedBall x r = {x} + r \u2022 closedBall 0 1\n\u22a2 \u2191\u2191\u03bc (closedBall x 1) / \u2191\u2191\u03bc ({x} + u) = \u2191\u2191\u03bc (closedBall 0 1) / \u2191\u2191\u03bc u"}, {"tactic": "simp only [addHaar_closedBall_center, image_add_left, measure_preimage_add, singleton_add]", "annotated_tactic": ["simp only [<a>addHaar_closedBall_center</a>, <a>image_add_left</a>, <a>measure_preimage_add</a>, <a>singleton_add</a>]", [{"full_name": "MeasureTheory.Measure.addHaar_closedBall_center", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean", "def_pos": [432, 9], "def_end_pos": [432, 34]}, {"full_name": "Set.image_add_left", "def_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "def_pos": [1198, 3], "def_end_pos": [1198, 14]}, {"full_name": "MeasureTheory.measure_preimage_add", "def_path": "Mathlib/MeasureTheory/Group/Measure.lean", "def_pos": [317, 3], "def_end_pos": [317, 14]}, {"full_name": "Set.singleton_add", "def_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "def_pos": [402, 3], "def_end_pos": [402, 14]}]], "state_before": "case h\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt u : Set E\nh'u : \u2191\u2191\u03bc u \u2260 0\nt_bound : t \u2286 closedBall 0 1\nA : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nr : \u211d\nrpos : 0 < r\nthis : closedBall x r = {x} + r \u2022 closedBall 0 1\n\u22a2 \u2191\u2191\u03bc (closedBall x 1) / \u2191\u2191\u03bc ({x} + u) = \u2191\u2191\u03bc (closedBall 0 1) / \u2191\u2191\u03bc u", "state_after": "no goals"}, {"tactic": "simp only [_root_.smul_closedBall, Real.norm_of_nonneg rpos.le, zero_le_one, add_zero,\n  mul_one, singleton_add_closedBall, smul_zero]", "annotated_tactic": ["simp only [<a>_root_.smul_closedBall</a>, <a>Real.norm_of_nonneg</a> rpos.le, <a>zero_le_one</a>, <a>add_zero</a>,\n        <a>mul_one</a>, <a>singleton_add_closedBall</a>, <a>smul_zero</a>]", [{"full_name": "smul_closedBall", "def_path": "Mathlib/Analysis/NormedSpace/Pointwise.lean", "def_pos": [388, 9], "def_end_pos": [388, 24]}, {"full_name": "Real.norm_of_nonneg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [1768, 9], "def_end_pos": [1768, 23]}, {"full_name": "zero_le_one", "def_path": "Mathlib/Algebra/Order/ZeroLEOne.lean", "def_pos": [26, 15], "def_end_pos": [26, 26]}, {"full_name": "add_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [469, 3], "def_end_pos": [469, 14]}, {"full_name": "mul_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [470, 9], "def_end_pos": [470, 16]}, {"full_name": "singleton_add_closedBall", "def_path": "Mathlib/Analysis/Normed/Group/Pointwise.lean", "def_pos": [153, 3], "def_end_pos": [153, 14]}, {"full_name": "smul_zero", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [732, 9], "def_end_pos": [732, 18]}]], "state_before": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt u : Set E\nh'u : \u2191\u2191\u03bc u \u2260 0\nt_bound : t \u2286 closedBall 0 1\nA : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nr : \u211d\nrpos : 0 < r\n\u22a2 closedBall x r = {x} + r \u2022 closedBall 0 1", "state_after": "no goals"}, {"tactic": "apply ENNReal.Tendsto.mul A _ B (Or.inr ENNReal.zero_ne_top)", "annotated_tactic": ["apply <a>ENNReal.Tendsto.mul</a> A _ B (<a>Or.inr</a> <a>ENNReal.zero_ne_top</a>)", [{"full_name": "ENNReal.Tendsto.mul", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [353, 19], "def_end_pos": [353, 30]}, {"full_name": "Or.inr", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [519, 5], "def_end_pos": [519, 8]}, {"full_name": "ENNReal.zero_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [334, 17], "def_end_pos": [334, 28]}]], "state_before": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt u : Set E\nh'u : \u2191\u2191\u03bc u \u2260 0\nt_bound : t \u2286 closedBall 0 1\nA : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nB : Tendsto (fun r => \u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc ({x} + r \u2022 u)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (closedBall x 1) / \u2191\u2191\u03bc ({x} + u)))\n\u22a2 Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r) * (\u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc ({x} + r \u2022 u)))\n    (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (0 * (\u2191\u2191\u03bc (closedBall x 1) / \u2191\u2191\u03bc ({x} + u))))", "state_after": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt u : Set E\nh'u : \u2191\u2191\u03bc u \u2260 0\nt_bound : t \u2286 closedBall 0 1\nA : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nB : Tendsto (fun r => \u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc ({x} + r \u2022 u)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (closedBall x 1) / \u2191\u2191\u03bc ({x} + u)))\n\u22a2 0 \u2260 0 \u2228 \u2191\u2191\u03bc (closedBall x 1) / \u2191\u2191\u03bc ({x} + u) \u2260 \u22a4"}, {"tactic": "simp only [ne_eq, not_true, singleton_add, image_add_left, measure_preimage_add, false_or,\n  ENNReal.div_eq_top, h'u, false_or_iff, not_and, and_false_iff]", "annotated_tactic": ["simp only [<a>ne_eq</a>, <a>not_true</a>, <a>singleton_add</a>, <a>image_add_left</a>, <a>measure_preimage_add</a>, <a>false_or</a>,\n      <a>ENNReal.div_eq_top</a>, h'u, <a>false_or_iff</a>, <a>not_and</a>, <a>and_false_iff</a>]", [{"full_name": "ne_eq", "def_path": "lake-packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [76, 17], "def_end_pos": [76, 22]}, {"full_name": "not_true", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [80, 17], "def_end_pos": [80, 25]}, {"full_name": "Set.singleton_add", "def_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "def_pos": [402, 3], "def_end_pos": [402, 14]}, {"full_name": "Set.image_add_left", "def_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "def_pos": [1198, 3], "def_end_pos": [1198, 14]}, {"full_name": "MeasureTheory.measure_preimage_add", "def_path": "Mathlib/MeasureTheory/Group/Measure.lean", "def_pos": [317, 3], "def_end_pos": [317, 14]}, {"full_name": "false_or", "def_path": "lake-packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [91, 17], "def_end_pos": [91, 25]}, {"full_name": "ENNReal.div_eq_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1607, 9], "def_end_pos": [1607, 19]}, {"full_name": "false_or_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [186, 9], "def_end_pos": [186, 21]}, {"full_name": "not_and", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [316, 17], "def_end_pos": [316, 24]}, {"full_name": "and_false_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [149, 9], "def_end_pos": [149, 22]}]], "state_before": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt u : Set E\nh'u : \u2191\u2191\u03bc u \u2260 0\nt_bound : t \u2286 closedBall 0 1\nA : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nB : Tendsto (fun r => \u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc ({x} + r \u2022 u)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (closedBall x 1) / \u2191\u2191\u03bc ({x} + u)))\n\u22a2 0 \u2260 0 \u2228 \u2191\u2191\u03bc (closedBall x 1) / \u2191\u2191\u03bc ({x} + u) \u2260 \u22a4", "state_after": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt u : Set E\nh'u : \u2191\u2191\u03bc u \u2260 0\nt_bound : t \u2286 closedBall 0 1\nA : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nB : Tendsto (fun r => \u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc ({x} + r \u2022 u)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (closedBall x 1) / \u2191\u2191\u03bc ({x} + u)))\n\u22a2 \u2191\u2191\u03bc (closedBall x 1) = \u22a4 \u2192 \u00ac\u00ac\u2191\u2191\u03bc u = \u22a4"}, {"tactic": "intro aux", "annotated_tactic": ["intro aux", []], "state_before": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt u : Set E\nh'u : \u2191\u2191\u03bc u \u2260 0\nt_bound : t \u2286 closedBall 0 1\nA : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nB : Tendsto (fun r => \u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc ({x} + r \u2022 u)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (closedBall x 1) / \u2191\u2191\u03bc ({x} + u)))\n\u22a2 \u2191\u2191\u03bc (closedBall x 1) = \u22a4 \u2192 \u00ac\u00ac\u2191\u2191\u03bc u = \u22a4", "state_after": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt u : Set E\nh'u : \u2191\u2191\u03bc u \u2260 0\nt_bound : t \u2286 closedBall 0 1\nA : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nB : Tendsto (fun r => \u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc ({x} + r \u2022 u)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (closedBall x 1) / \u2191\u2191\u03bc ({x} + u)))\naux : \u2191\u2191\u03bc (closedBall x 1) = \u22a4\n\u22a2 \u00ac\u00ac\u2191\u2191\u03bc u = \u22a4"}, {"tactic": "exact (measure_closedBall_lt_top.ne aux).elim", "annotated_tactic": ["exact (measure_closedBall_lt_top.ne aux).<a>elim</a>", [{"full_name": "False.elim", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [223, 21], "def_end_pos": [223, 31]}]], "state_before": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt u : Set E\nh'u : \u2191\u2191\u03bc u \u2260 0\nt_bound : t \u2286 closedBall 0 1\nA : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nB : Tendsto (fun r => \u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc ({x} + r \u2022 u)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (closedBall x 1) / \u2191\u2191\u03bc ({x} + u)))\naux : \u2191\u2191\u03bc (closedBall x 1) = \u22a4\n\u22a2 \u00ac\u00ac\u2191\u2191\u03bc u = \u22a4", "state_after": "no goals"}, {"tactic": "simp only [div_eq_mul_inv]", "annotated_tactic": ["simp only [<a>div_eq_mul_inv</a>]", [{"full_name": "div_eq_mul_inv", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [977, 9], "def_end_pos": [977, 23]}]], "state_before": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt u : Set E\nh'u : \u2191\u2191\u03bc u \u2260 0\nt_bound : t \u2286 closedBall 0 1\nA : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nB : Tendsto (fun r => \u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc ({x} + r \u2022 u)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (closedBall x 1) / \u2191\u2191\u03bc ({x} + u)))\nC :\n  Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r) * (\u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc ({x} + r \u2022 u)))\n    (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nr : \u211d\nrpos : 0 < r\n\u22a2 \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r) * (\u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc ({x} + r \u2022 u)) =\n    \u2191\u2191\u03bc (closedBall x r) * (\u2191\u2191\u03bc (closedBall x r))\u207b\u00b9 * (\u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 u))", "state_after": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt u : Set E\nh'u : \u2191\u2191\u03bc u \u2260 0\nt_bound : t \u2286 closedBall 0 1\nA : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nB : Tendsto (fun r => \u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc ({x} + r \u2022 u)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (closedBall x 1) / \u2191\u2191\u03bc ({x} + u)))\nC :\n  Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r) * (\u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc ({x} + r \u2022 u)))\n    (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nr : \u211d\nrpos : 0 < r\n\u22a2 \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) * (\u2191\u2191\u03bc (closedBall x r))\u207b\u00b9 * (\u2191\u2191\u03bc (closedBall x r) * (\u2191\u2191\u03bc ({x} + r \u2022 u))\u207b\u00b9) =\n    \u2191\u2191\u03bc (closedBall x r) * (\u2191\u2191\u03bc (closedBall x r))\u207b\u00b9 * (\u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) * (\u2191\u2191\u03bc ({x} + r \u2022 u))\u207b\u00b9)"}, {"tactic": "ring", "annotated_tactic": ["ring", []], "state_before": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt u : Set E\nh'u : \u2191\u2191\u03bc u \u2260 0\nt_bound : t \u2286 closedBall 0 1\nA : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nB : Tendsto (fun r => \u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc ({x} + r \u2022 u)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (closedBall x 1) / \u2191\u2191\u03bc ({x} + u)))\nC :\n  Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r) * (\u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc ({x} + r \u2022 u)))\n    (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nr : \u211d\nrpos : 0 < r\n\u22a2 \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) * (\u2191\u2191\u03bc (closedBall x r))\u207b\u00b9 * (\u2191\u2191\u03bc (closedBall x r) * (\u2191\u2191\u03bc ({x} + r \u2022 u))\u207b\u00b9) =\n    \u2191\u2191\u03bc (closedBall x r) * (\u2191\u2191\u03bc (closedBall x r))\u207b\u00b9 * (\u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) * (\u2191\u2191\u03bc ({x} + r \u2022 u))\u207b\u00b9)", "state_after": "no goals"}, {"tactic": "rw [ENNReal.mul_inv_cancel (measure_closedBall_pos \u03bc x rpos).ne'\n    measure_closedBall_lt_top.ne,\n  one_mul]", "annotated_tactic": ["rw [<a>ENNReal.mul_inv_cancel</a> (<a>measure_closedBall_pos</a> \u03bc x rpos).<a>ne'</a>\n          measure_closedBall_lt_top.ne,\n        <a>one_mul</a>]", [{"full_name": "ENNReal.mul_inv_cancel", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1418, 19], "def_end_pos": [1418, 33]}, {"full_name": "Metric.measure_closedBall_pos", "def_path": "Mathlib/MeasureTheory/Measure/OpenPos.lean", "def_pos": [227, 9], "def_end_pos": [227, 31]}, {"full_name": "LT.lt.ne'", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [328, 9], "def_end_pos": [328, 12]}, {"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [464, 9], "def_end_pos": [464, 16]}]], "state_before": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt u : Set E\nh'u : \u2191\u2191\u03bc u \u2260 0\nt_bound : t \u2286 closedBall 0 1\nA : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nB : Tendsto (fun r => \u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc ({x} + r \u2022 u)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (closedBall x 1) / \u2191\u2191\u03bc ({x} + u)))\nC :\n  Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r) * (\u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc ({x} + r \u2022 u)))\n    (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nr : \u211d\nrpos : 0 < r\n\u22a2 \u2191\u2191\u03bc (closedBall x r) * (\u2191\u2191\u03bc (closedBall x r))\u207b\u00b9 * (\u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 u)) =\n    \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 u)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Group/Prod.lean", "full_name": "MeasureTheory.quasiMeasurePreserving_div_of_right_invariant", "start": [466, 1], "end": [469, 65], "traced_tactics": [{"tactic": "refine' QuasiMeasurePreserving.prod_of_left measurable_div (eventually_of_forall fun y => _)", "annotated_tactic": ["refine' <a>QuasiMeasurePreserving.prod_of_left</a> <a>measurable_div</a> (<a>eventually_of_forall</a> fun y => _)", [{"full_name": "MeasureTheory.QuasiMeasurePreserving.prod_of_left", "def_path": "Mathlib/MeasureTheory/Constructions/Prod/Basic.lean", "def_pos": [770, 9], "def_end_pos": [770, 21]}, {"full_name": "MeasurableDiv\u2082.measurable_div", "def_path": "Mathlib/MeasureTheory/Group/Arithmetic.lean", "def_pos": [287, 3], "def_end_pos": [287, 17]}, {"full_name": "Filter.eventually_of_forall", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1111, 9], "def_end_pos": [1111, 29]}]], "state_before": "G : Type u_1\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b3 : SigmaFinite \u03bd\ninst\u271d\u00b2 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b9 : MeasurableInv G\ninst\u271d : IsMulRightInvariant \u03bc\n\u22a2 QuasiMeasurePreserving fun p => p.1 / p.2", "state_after": "G : Type u_1\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b3 : SigmaFinite \u03bd\ninst\u271d\u00b2 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b9 : MeasurableInv G\ninst\u271d : IsMulRightInvariant \u03bc\ny : G\n\u22a2 QuasiMeasurePreserving fun x => (x, y).1 / (x, y).2"}, {"tactic": "exact (measurePreserving_div_right \u03bc y).quasiMeasurePreserving", "annotated_tactic": ["exact (<a>measurePreserving_div_right</a> \u03bc y).<a>quasiMeasurePreserving</a>", [{"full_name": "MeasureTheory.measurePreserving_div_right", "def_path": "Mathlib/MeasureTheory/Group/Measure.lean", "def_pos": [310, 9], "def_end_pos": [310, 36]}, {"full_name": "MeasureTheory.MeasurePreserving.quasiMeasurePreserving", "def_path": "Mathlib/Dynamics/Ergodic/MeasurePreserving.lean", "def_pos": [97, 19], "def_end_pos": [97, 41]}]], "state_before": "G : Type u_1\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b3 : SigmaFinite \u03bd\ninst\u271d\u00b2 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b9 : MeasurableInv G\ninst\u271d : IsMulRightInvariant \u03bc\ny : G\n\u22a2 QuasiMeasurePreserving fun x => (x, y).1 / (x, y).2", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/Variables.lean", "full_name": "MvPolynomial.aeval_eq_constantCoeff_of_vars", "start": [818, 1], "end": [820, 41], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "full_name": "MeasureTheory.Lp.ae_tendsto_of_cauchy_snorm'", "start": [1547, 1], "end": [1587, 44], "traced_tactics": [{"tactic": "have h_summable : \u2200\u1d50 x \u2202\u03bc, Summable fun i : \u2115 => f (i + 1) x - f i x := by\n  have h1 :\n    \u2200 n, snorm' (fun x => \u2211 i in Finset.range (n + 1), \u2016f (i + 1) x - f i x\u2016) p \u03bc \u2264 \u2211' i, B i :=\n    snorm'_sum_norm_sub_le_tsum_of_cauchy_snorm' hf hp1 h_cau\n  have h2 :\n    \u2200 n,\n      (\u222b\u207b a, (\u2211 i in Finset.range (n + 1), \u2016f (i + 1) a - f i a\u2016\u208a : \u211d\u22650\u221e) ^ p \u2202\u03bc) \u2264\n        (\u2211' i, B i) ^ p :=\n    fun n => lintegral_rpow_sum_coe_nnnorm_sub_le_rpow_tsum hp1 n (h1 n)\n  have h3 : (\u222b\u207b a, (\u2211' i, \u2016f (i + 1) a - f i a\u2016\u208a : \u211d\u22650\u221e) ^ p \u2202\u03bc) ^ (1 / p) \u2264 \u2211' i, B i :=\n    lintegral_rpow_tsum_coe_nnnorm_sub_le_tsum hf hp1 h2\n  have h4 : \u2200\u1d50 x \u2202\u03bc, (\u2211' i, \u2016f (i + 1) x - f i x\u2016\u208a : \u211d\u22650\u221e) < \u221e :=\n    tsum_nnnorm_sub_ae_lt_top hf hp1 hB h3\n  exact\n    h4.mono fun x hx =>\n      summable_of_summable_nnnorm\n        (ENNReal.tsum_coe_ne_top_iff_summable.mp (lt_top_iff_ne_top.mp hx))", "annotated_tactic": ["have h_summable : \u2200\u1d50 x \u2202\u03bc, <a>Summable</a> fun i : \u2115 => f (i + 1) x - f i x := by\n    have h1 :\n      \u2200 n, <a>snorm'</a> (fun x => \u2211 i in <a>Finset.range</a> (n + 1), \u2016f (i + 1) x - f i x\u2016) p \u03bc \u2264 \u2211' i, B i :=\n      <a>snorm'_sum_norm_sub_le_tsum_of_cauchy_snorm'</a> hf hp1 h_cau\n    have h2 :\n      \u2200 n,\n        (\u222b\u207b a, (\u2211 i in <a>Finset.range</a> (n + 1), \u2016f (i + 1) a - f i a\u2016\u208a : \u211d\u22650\u221e) ^ p \u2202\u03bc) \u2264\n          (\u2211' i, B i) ^ p :=\n      fun n => <a>lintegral_rpow_sum_coe_nnnorm_sub_le_rpow_tsum</a> hp1 n (h1 n)\n    have h3 : (\u222b\u207b a, (\u2211' i, \u2016f (i + 1) a - f i a\u2016\u208a : \u211d\u22650\u221e) ^ p \u2202\u03bc) ^ (1 / p) \u2264 \u2211' i, B i :=\n      <a>lintegral_rpow_tsum_coe_nnnorm_sub_le_tsum</a> hf hp1 h2\n    have h4 : \u2200\u1d50 x \u2202\u03bc, (\u2211' i, \u2016f (i + 1) x - f i x\u2016\u208a : \u211d\u22650\u221e) < \u221e :=\n      <a>tsum_nnnorm_sub_ae_lt_top</a> hf hp1 hB h3\n    exact\n      h4.mono fun x hx =>\n        <a>summable_of_summable_nnnorm</a>\n          (ENNReal.tsum_coe_ne_top_iff_summable.mp (lt_top_iff_ne_top.mp hx))", [{"full_name": "Summable", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/Basic.lean", "def_pos": [62, 5], "def_end_pos": [62, 13]}, {"full_name": "MeasureTheory.snorm'", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [73, 5], "def_end_pos": [73, 11]}, {"full_name": "Finset.range", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3027, 5], "def_end_pos": [3027, 10]}, {"full_name": "_private.Mathlib.MeasureTheory.Function.LpSpace.0.MeasureTheory.Lp.snorm'_sum_norm_sub_le_tsum_of_cauchy_snorm'", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [1457, 17], "def_end_pos": [1457, 61]}, {"full_name": "Finset.range", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3027, 5], "def_end_pos": [3027, 10]}, {"full_name": "_private.Mathlib.MeasureTheory.Function.LpSpace.0.MeasureTheory.Lp.lintegral_rpow_sum_coe_nnnorm_sub_le_rpow_tsum", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [1473, 17], "def_end_pos": [1473, 63]}, {"full_name": "_private.Mathlib.MeasureTheory.Function.LpSpace.0.MeasureTheory.Lp.lintegral_rpow_tsum_coe_nnnorm_sub_le_tsum", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [1497, 17], "def_end_pos": [1497, 59]}, {"full_name": "_private.Mathlib.MeasureTheory.Function.LpSpace.0.MeasureTheory.Lp.tsum_nnnorm_sub_ae_lt_top", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [1531, 17], "def_end_pos": [1531, 42]}, {"full_name": "summable_of_summable_nnnorm", "def_path": "Mathlib/Analysis/Normed/Group/InfiniteSum.lean", "def_pos": [177, 9], "def_end_pos": [177, 36]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) atTop (\ud835\udcdd l)", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nh_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) atTop (\ud835\udcdd l)"}, {"tactic": "have h :\n  \u2200\u1d50 x \u2202\u03bc, \u2203 l : E,\n    atTop.Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) (\ud835\udcdd l) := by\n  refine' h_summable.mono fun x hx => _\n  let hx_sum := hx.hasSum.tendsto_sum_nat\n  exact \u27e8\u2211' i, (f (i + 1) x - f i x), hx_sum\u27e9", "annotated_tactic": ["have h :\n    \u2200\u1d50 x \u2202\u03bc, \u2203 l : E,\n      atTop.Tendsto (fun n => \u2211 i in <a>Finset.range</a> n, (f (i + 1) x - f i x)) (\ud835\udcdd l) := by\n    refine' h_summable.mono fun x hx => _\n    let hx_sum := hx.hasSum.tendsto_sum_nat\n    exact \u27e8\u2211' i, (f (i + 1) x - f i x), hx_sum\u27e9", [{"full_name": "Finset.range", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3027, 5], "def_end_pos": [3027, 10]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nh_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) atTop (\ud835\udcdd l)", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nh_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) atTop (\ud835\udcdd l)"}, {"tactic": "refine' h.mono fun x hx => _", "annotated_tactic": ["refine' h.mono fun x hx => _", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nh_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) atTop (\ud835\udcdd l)", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nh_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)\nx : \u03b1\nhx : \u2203 l, Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)\n\u22a2 \u2203 l, Tendsto (fun n => f n x) atTop (\ud835\udcdd l)"}, {"tactic": "cases' hx with l hx", "annotated_tactic": ["cases' hx with l hx", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nh_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)\nx : \u03b1\nhx : \u2203 l, Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)\n\u22a2 \u2203 l, Tendsto (fun n => f n x) atTop (\ud835\udcdd l)", "state_after": "case intro\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nh_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)\nx : \u03b1\nl : E\nhx : Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)\n\u22a2 \u2203 l, Tendsto (fun n => f n x) atTop (\ud835\udcdd l)"}, {"tactic": "have h_rw_sum :\n    (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) = fun n => f n x - f 0 x := by\n  ext1 n\n  change\n    (\u2211 i : \u2115 in Finset.range n, ((fun m => f m x) (i + 1) - (fun m => f m x) i)) = f n x - f 0 x\n  rw [Finset.sum_range_sub (fun m => f m x)]", "annotated_tactic": ["have h_rw_sum :\n      (fun n => \u2211 i in <a>Finset.range</a> n, (f (i + 1) x - f i x)) = fun n => f n x - f 0 x := by\n    ext1 n\n    change\n      (\u2211 i : \u2115 in <a>Finset.range</a> n, ((fun m => f m x) (i + 1) - (fun m => f m x) i)) = f n x - f 0 x\n    rw [<a>Finset.sum_range_sub</a> (fun m => f m x)]", [{"full_name": "Finset.range", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3027, 5], "def_end_pos": [3027, 10]}, {"full_name": "Finset.range", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3027, 5], "def_end_pos": [3027, 10]}, {"full_name": "Finset.sum_range_sub", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [1397, 3], "def_end_pos": [1397, 14]}]], "state_before": "case intro\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nh_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)\nx : \u03b1\nl : E\nhx : Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)\n\u22a2 \u2203 l, Tendsto (fun n => f n x) atTop (\ud835\udcdd l)", "state_after": "case intro\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nh_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)\nx : \u03b1\nl : E\nhx : Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)\nh_rw_sum : (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) = fun n => f n x - f 0 x\n\u22a2 \u2203 l, Tendsto (fun n => f n x) atTop (\ud835\udcdd l)"}, {"tactic": "rw [h_rw_sum] at hx", "annotated_tactic": ["rw [h_rw_sum] at hx", []], "state_before": "case intro\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nh_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)\nx : \u03b1\nl : E\nhx : Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)\nh_rw_sum : (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) = fun n => f n x - f 0 x\n\u22a2 \u2203 l, Tendsto (fun n => f n x) atTop (\ud835\udcdd l)", "state_after": "case intro\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nh_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)\nx : \u03b1\nl : E\nhx : Tendsto (fun n => f n x - f 0 x) atTop (\ud835\udcdd l)\nh_rw_sum : (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) = fun n => f n x - f 0 x\n\u22a2 \u2203 l, Tendsto (fun n => f n x) atTop (\ud835\udcdd l)"}, {"tactic": "have hf_rw : (fun n => f n x) = fun n => f n x - f 0 x + f 0 x := by\n  ext1 n\n  abel", "annotated_tactic": ["have hf_rw : (fun n => f n x) = fun n => f n x - f 0 x + f 0 x := by\n    ext1 n\n    abel", []], "state_before": "case intro\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nh_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)\nx : \u03b1\nl : E\nhx : Tendsto (fun n => f n x - f 0 x) atTop (\ud835\udcdd l)\nh_rw_sum : (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) = fun n => f n x - f 0 x\n\u22a2 \u2203 l, Tendsto (fun n => f n x) atTop (\ud835\udcdd l)", "state_after": "case intro\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nh_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)\nx : \u03b1\nl : E\nhx : Tendsto (fun n => f n x - f 0 x) atTop (\ud835\udcdd l)\nh_rw_sum : (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) = fun n => f n x - f 0 x\nhf_rw : (fun n => f n x) = fun n => f n x - f 0 x + f 0 x\n\u22a2 \u2203 l, Tendsto (fun n => f n x) atTop (\ud835\udcdd l)"}, {"tactic": "rw [hf_rw]", "annotated_tactic": ["rw [hf_rw]", []], "state_before": "case intro\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nh_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)\nx : \u03b1\nl : E\nhx : Tendsto (fun n => f n x - f 0 x) atTop (\ud835\udcdd l)\nh_rw_sum : (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) = fun n => f n x - f 0 x\nhf_rw : (fun n => f n x) = fun n => f n x - f 0 x + f 0 x\n\u22a2 \u2203 l, Tendsto (fun n => f n x) atTop (\ud835\udcdd l)", "state_after": "case intro\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nh_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)\nx : \u03b1\nl : E\nhx : Tendsto (fun n => f n x - f 0 x) atTop (\ud835\udcdd l)\nh_rw_sum : (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) = fun n => f n x - f 0 x\nhf_rw : (fun n => f n x) = fun n => f n x - f 0 x + f 0 x\n\u22a2 \u2203 l, Tendsto (fun n => f n x - f 0 x + f 0 x) atTop (\ud835\udcdd l)"}, {"tactic": "exact \u27e8l + f 0 x, Tendsto.add_const _ hx\u27e9", "annotated_tactic": ["exact \u27e8l + f 0 x, <a>Tendsto.add_const</a> _ hx\u27e9", [{"full_name": "Filter.Tendsto.add_const", "def_path": "Mathlib/Topology/Algebra/Monoid.lean", "def_pos": [132, 3], "def_end_pos": [132, 14]}]], "state_before": "case intro\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nh_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)\nx : \u03b1\nl : E\nhx : Tendsto (fun n => f n x - f 0 x) atTop (\ud835\udcdd l)\nh_rw_sum : (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) = fun n => f n x - f 0 x\nhf_rw : (fun n => f n x) = fun n => f n x - f 0 x + f 0 x\n\u22a2 \u2203 l, Tendsto (fun n => f n x - f 0 x + f 0 x) atTop (\ud835\udcdd l)", "state_after": "no goals"}, {"tactic": "have h1 :\n  \u2200 n, snorm' (fun x => \u2211 i in Finset.range (n + 1), \u2016f (i + 1) x - f i x\u2016) p \u03bc \u2264 \u2211' i, B i :=\n  snorm'_sum_norm_sub_le_tsum_of_cauchy_snorm' hf hp1 h_cau", "annotated_tactic": ["have h1 :\n      \u2200 n, <a>snorm'</a> (fun x => \u2211 i in <a>Finset.range</a> (n + 1), \u2016f (i + 1) x - f i x\u2016) p \u03bc \u2264 \u2211' i, B i :=\n      <a>snorm'_sum_norm_sub_le_tsum_of_cauchy_snorm'</a> hf hp1 h_cau", [{"full_name": "MeasureTheory.snorm'", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [73, 5], "def_end_pos": [73, 11]}, {"full_name": "Finset.range", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3027, 5], "def_end_pos": [3027, 10]}, {"full_name": "_private.Mathlib.MeasureTheory.Function.LpSpace.0.MeasureTheory.Lp.snorm'_sum_norm_sub_le_tsum_of_cauchy_snorm'", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [1457, 17], "def_end_pos": [1457, 61]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nh1 : \u2200 (n : \u2115), snorm' (fun x => \u2211 i in Finset.range (n + 1), \u2016f (i + 1) x - f i x\u2016) p \u03bc \u2264 \u2211' (i : \u2115), B i\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x"}, {"tactic": "have h2 :\n  \u2200 n,\n    (\u222b\u207b a, (\u2211 i in Finset.range (n + 1), \u2016f (i + 1) a - f i a\u2016\u208a : \u211d\u22650\u221e) ^ p \u2202\u03bc) \u2264\n      (\u2211' i, B i) ^ p :=\n  fun n => lintegral_rpow_sum_coe_nnnorm_sub_le_rpow_tsum hp1 n (h1 n)", "annotated_tactic": ["have h2 :\n      \u2200 n,\n        (\u222b\u207b a, (\u2211 i in <a>Finset.range</a> (n + 1), \u2016f (i + 1) a - f i a\u2016\u208a : \u211d\u22650\u221e) ^ p \u2202\u03bc) \u2264\n          (\u2211' i, B i) ^ p :=\n      fun n => <a>lintegral_rpow_sum_coe_nnnorm_sub_le_rpow_tsum</a> hp1 n (h1 n)", [{"full_name": "Finset.range", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3027, 5], "def_end_pos": [3027, 10]}, {"full_name": "_private.Mathlib.MeasureTheory.Function.LpSpace.0.MeasureTheory.Lp.lintegral_rpow_sum_coe_nnnorm_sub_le_rpow_tsum", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [1473, 17], "def_end_pos": [1473, 63]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nh1 : \u2200 (n : \u2115), snorm' (fun x => \u2211 i in Finset.range (n + 1), \u2016f (i + 1) x - f i x\u2016) p \u03bc \u2264 \u2211' (i : \u2115), B i\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nh1 : \u2200 (n : \u2115), snorm' (fun x => \u2211 i in Finset.range (n + 1), \u2016f (i + 1) x - f i x\u2016) p \u03bc \u2264 \u2211' (i : \u2115), B i\nh2 : \u2200 (n : \u2115), \u222b\u207b (a : \u03b1), (\u2211 i in Finset.range (n + 1), \u2191\u2016f (i + 1) a - f i a\u2016\u208a) ^ p \u2202\u03bc \u2264 (\u2211' (i : \u2115), B i) ^ p\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x"}, {"tactic": "have h3 : (\u222b\u207b a, (\u2211' i, \u2016f (i + 1) a - f i a\u2016\u208a : \u211d\u22650\u221e) ^ p \u2202\u03bc) ^ (1 / p) \u2264 \u2211' i, B i :=\n  lintegral_rpow_tsum_coe_nnnorm_sub_le_tsum hf hp1 h2", "annotated_tactic": ["have h3 : (\u222b\u207b a, (\u2211' i, \u2016f (i + 1) a - f i a\u2016\u208a : \u211d\u22650\u221e) ^ p \u2202\u03bc) ^ (1 / p) \u2264 \u2211' i, B i :=\n      <a>lintegral_rpow_tsum_coe_nnnorm_sub_le_tsum</a> hf hp1 h2", [{"full_name": "_private.Mathlib.MeasureTheory.Function.LpSpace.0.MeasureTheory.Lp.lintegral_rpow_tsum_coe_nnnorm_sub_le_tsum", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [1497, 17], "def_end_pos": [1497, 59]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nh1 : \u2200 (n : \u2115), snorm' (fun x => \u2211 i in Finset.range (n + 1), \u2016f (i + 1) x - f i x\u2016) p \u03bc \u2264 \u2211' (i : \u2115), B i\nh2 : \u2200 (n : \u2115), \u222b\u207b (a : \u03b1), (\u2211 i in Finset.range (n + 1), \u2191\u2016f (i + 1) a - f i a\u2016\u208a) ^ p \u2202\u03bc \u2264 (\u2211' (i : \u2115), B i) ^ p\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nh1 : \u2200 (n : \u2115), snorm' (fun x => \u2211 i in Finset.range (n + 1), \u2016f (i + 1) x - f i x\u2016) p \u03bc \u2264 \u2211' (i : \u2115), B i\nh2 : \u2200 (n : \u2115), \u222b\u207b (a : \u03b1), (\u2211 i in Finset.range (n + 1), \u2191\u2016f (i + 1) a - f i a\u2016\u208a) ^ p \u2202\u03bc \u2264 (\u2211' (i : \u2115), B i) ^ p\nh3 : (\u222b\u207b (a : \u03b1), (\u2211' (i : \u2115), \u2191\u2016f (i + 1) a - f i a\u2016\u208a) ^ p \u2202\u03bc) ^ (1 / p) \u2264 \u2211' (i : \u2115), B i\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x"}, {"tactic": "have h4 : \u2200\u1d50 x \u2202\u03bc, (\u2211' i, \u2016f (i + 1) x - f i x\u2016\u208a : \u211d\u22650\u221e) < \u221e :=\n  tsum_nnnorm_sub_ae_lt_top hf hp1 hB h3", "annotated_tactic": ["have h4 : \u2200\u1d50 x \u2202\u03bc, (\u2211' i, \u2016f (i + 1) x - f i x\u2016\u208a : \u211d\u22650\u221e) < \u221e :=\n      <a>tsum_nnnorm_sub_ae_lt_top</a> hf hp1 hB h3", [{"full_name": "_private.Mathlib.MeasureTheory.Function.LpSpace.0.MeasureTheory.Lp.tsum_nnnorm_sub_ae_lt_top", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [1531, 17], "def_end_pos": [1531, 42]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nh1 : \u2200 (n : \u2115), snorm' (fun x => \u2211 i in Finset.range (n + 1), \u2016f (i + 1) x - f i x\u2016) p \u03bc \u2264 \u2211' (i : \u2115), B i\nh2 : \u2200 (n : \u2115), \u222b\u207b (a : \u03b1), (\u2211 i in Finset.range (n + 1), \u2191\u2016f (i + 1) a - f i a\u2016\u208a) ^ p \u2202\u03bc \u2264 (\u2211' (i : \u2115), B i) ^ p\nh3 : (\u222b\u207b (a : \u03b1), (\u2211' (i : \u2115), \u2191\u2016f (i + 1) a - f i a\u2016\u208a) ^ p \u2202\u03bc) ^ (1 / p) \u2264 \u2211' (i : \u2115), B i\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nh1 : \u2200 (n : \u2115), snorm' (fun x => \u2211 i in Finset.range (n + 1), \u2016f (i + 1) x - f i x\u2016) p \u03bc \u2264 \u2211' (i : \u2115), B i\nh2 : \u2200 (n : \u2115), \u222b\u207b (a : \u03b1), (\u2211 i in Finset.range (n + 1), \u2191\u2016f (i + 1) a - f i a\u2016\u208a) ^ p \u2202\u03bc \u2264 (\u2211' (i : \u2115), B i) ^ p\nh3 : (\u222b\u207b (a : \u03b1), (\u2211' (i : \u2115), \u2191\u2016f (i + 1) a - f i a\u2016\u208a) ^ p \u2202\u03bc) ^ (1 / p) \u2264 \u2211' (i : \u2115), B i\nh4 : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2211' (i : \u2115), \u2191\u2016f (i + 1) x - f i x\u2016\u208a < \u22a4\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x"}, {"tactic": "exact\n  h4.mono fun x hx =>\n    summable_of_summable_nnnorm\n      (ENNReal.tsum_coe_ne_top_iff_summable.mp (lt_top_iff_ne_top.mp hx))", "annotated_tactic": ["exact\n      h4.mono fun x hx =>\n        <a>summable_of_summable_nnnorm</a>\n          (ENNReal.tsum_coe_ne_top_iff_summable.mp (lt_top_iff_ne_top.mp hx))", [{"full_name": "summable_of_summable_nnnorm", "def_path": "Mathlib/Analysis/Normed/Group/InfiniteSum.lean", "def_pos": [177, 9], "def_end_pos": [177, 36]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nh1 : \u2200 (n : \u2115), snorm' (fun x => \u2211 i in Finset.range (n + 1), \u2016f (i + 1) x - f i x\u2016) p \u03bc \u2264 \u2211' (i : \u2115), B i\nh2 : \u2200 (n : \u2115), \u222b\u207b (a : \u03b1), (\u2211 i in Finset.range (n + 1), \u2191\u2016f (i + 1) a - f i a\u2016\u208a) ^ p \u2202\u03bc \u2264 (\u2211' (i : \u2115), B i) ^ p\nh3 : (\u222b\u207b (a : \u03b1), (\u2211' (i : \u2115), \u2191\u2016f (i + 1) a - f i a\u2016\u208a) ^ p \u2202\u03bc) ^ (1 / p) \u2264 \u2211' (i : \u2115), B i\nh4 : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2211' (i : \u2115), \u2191\u2016f (i + 1) x - f i x\u2016\u208a < \u22a4\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x", "state_after": "no goals"}, {"tactic": "refine' h_summable.mono fun x hx => _", "annotated_tactic": ["refine' h_summable.mono fun x hx => _", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nh_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nh_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x\nx : \u03b1\nhx : Summable fun i => f (i + 1) x - f i x\n\u22a2 \u2203 l, Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)"}, {"tactic": "let hx_sum := hx.hasSum.tendsto_sum_nat", "annotated_tactic": ["let hx_sum := hx.hasSum.tendsto_sum_nat", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nh_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x\nx : \u03b1\nhx : Summable fun i => f (i + 1) x - f i x\n\u22a2 \u2203 l, Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nh_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x\nx : \u03b1\nhx : Summable fun i => f (i + 1) x - f i x\nhx_sum : Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) atTop\n  (\ud835\udcdd (\u2211' (b : \u2115), (f (b + 1) x - f b x))) :=\n  HasSum.tendsto_sum_nat (Summable.hasSum hx)\n\u22a2 \u2203 l, Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)"}, {"tactic": "exact \u27e8\u2211' i, (f (i + 1) x - f i x), hx_sum\u27e9", "annotated_tactic": ["exact \u27e8\u2211' i, (f (i + 1) x - f i x), hx_sum\u27e9", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nh_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x\nx : \u03b1\nhx : Summable fun i => f (i + 1) x - f i x\nhx_sum : Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) atTop\n  (\ud835\udcdd (\u2211' (b : \u2115), (f (b + 1) x - f b x))) :=\n  HasSum.tendsto_sum_nat (Summable.hasSum hx)\n\u22a2 \u2203 l, Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)", "state_after": "no goals"}, {"tactic": "ext1 n", "annotated_tactic": ["ext1 n", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nh_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)\nx : \u03b1\nl : E\nhx : Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)\n\u22a2 (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) = fun n => f n x - f 0 x", "state_after": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nh_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)\nx : \u03b1\nl : E\nhx : Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)\nn : \u2115\n\u22a2 \u2211 i in Finset.range n, (f (i + 1) x - f i x) = f n x - f 0 x"}, {"tactic": "change\n  (\u2211 i : \u2115 in Finset.range n, ((fun m => f m x) (i + 1) - (fun m => f m x) i)) = f n x - f 0 x", "annotated_tactic": ["change\n      (\u2211 i : \u2115 in <a>Finset.range</a> n, ((fun m => f m x) (i + 1) - (fun m => f m x) i)) = f n x - f 0 x", [{"full_name": "Finset.range", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3027, 5], "def_end_pos": [3027, 10]}]], "state_before": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nh_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)\nx : \u03b1\nl : E\nhx : Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)\nn : \u2115\n\u22a2 \u2211 i in Finset.range n, (f (i + 1) x - f i x) = f n x - f 0 x", "state_after": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nh_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)\nx : \u03b1\nl : E\nhx : Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)\nn : \u2115\n\u22a2 \u2211 i in Finset.range n, ((fun m => f m x) (i + 1) - (fun m => f m x) i) = f n x - f 0 x"}, {"tactic": "rw [Finset.sum_range_sub (fun m => f m x)]", "annotated_tactic": ["rw [<a>Finset.sum_range_sub</a> (fun m => f m x)]", [{"full_name": "Finset.sum_range_sub", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [1397, 3], "def_end_pos": [1397, 14]}]], "state_before": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nh_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)\nx : \u03b1\nl : E\nhx : Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)\nn : \u2115\n\u22a2 \u2211 i in Finset.range n, ((fun m => f m x) (i + 1) - (fun m => f m x) i) = f n x - f 0 x", "state_after": "no goals"}, {"tactic": "ext1 n", "annotated_tactic": ["ext1 n", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nh_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)\nx : \u03b1\nl : E\nhx : Tendsto (fun n => f n x - f 0 x) atTop (\ud835\udcdd l)\nh_rw_sum : (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) = fun n => f n x - f 0 x\n\u22a2 (fun n => f n x) = fun n => f n x - f 0 x + f 0 x", "state_after": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nh_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)\nx : \u03b1\nl : E\nhx : Tendsto (fun n => f n x - f 0 x) atTop (\ud835\udcdd l)\nh_rw_sum : (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) = fun n => f n x - f 0 x\nn : \u2115\n\u22a2 f n x = f n x - f 0 x + f 0 x"}, {"tactic": "abel", "annotated_tactic": ["abel", []], "state_before": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nh_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)\nx : \u03b1\nl : E\nhx : Tendsto (fun n => f n x - f 0 x) atTop (\ud835\udcdd l)\nh_rw_sum : (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) = fun n => f n x - f 0 x\nn : \u2115\n\u22a2 f n x = f n x - f 0 x + f 0 x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Pointwise/Interval.lean", "full_name": "Set.preimage_const_sub_uIcc", "start": [447, 1], "end": [449, 91], "traced_tactics": [{"tactic": "simp_rw [\u2190 Icc_min_max, preimage_const_sub_Icc]", "annotated_tactic": ["simp_rw [\u2190 <a>Icc_min_max</a>, <a>preimage_const_sub_Icc</a>]", [{"full_name": "Set.Icc_min_max", "def_path": "Mathlib/Data/Set/Intervals/UnorderedInterval.lean", "def_pos": [220, 9], "def_end_pos": [220, 20]}, {"full_name": "Set.preimage_const_sub_Icc", "def_path": "Mathlib/Data/Set/Pointwise/Interval.lean", "def_pos": [238, 9], "def_end_pos": [238, 31]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : LinearOrderedAddCommGroup \u03b1\na b c d : \u03b1\n\u22a2 (fun x => a - x) \u207b\u00b9' [[b, c]] = [[a - b, a - c]]", "state_after": "\u03b1 : Type u_1\ninst\u271d : LinearOrderedAddCommGroup \u03b1\na b c d : \u03b1\n\u22a2 Icc (a - max b c) (a - min b c) = Icc (min (a - b) (a - c)) (max (a - b) (a - c))"}, {"tactic": "simp only [sub_eq_add_neg, min_add_add_left, max_add_add_left, min_neg_neg, max_neg_neg]", "annotated_tactic": ["simp only [<a>sub_eq_add_neg</a>, <a>min_add_add_left</a>, <a>max_add_add_left</a>, <a>min_neg_neg</a>, <a>max_neg_neg</a>]", [{"full_name": "sub_eq_add_neg", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [975, 3], "def_end_pos": [975, 14]}, {"full_name": "min_add_add_left", "def_path": "Mathlib/Algebra/Order/Monoid/MinMax.lean", "def_pos": [48, 3], "def_end_pos": [48, 14]}, {"full_name": "max_add_add_left", "def_path": "Mathlib/Algebra/Order/Monoid/MinMax.lean", "def_pos": [54, 3], "def_end_pos": [54, 14]}, {"full_name": "min_neg_neg", "def_path": "Mathlib/Algebra/Order/Group/MinMax.lean", "def_pos": [35, 15], "def_end_pos": [35, 26]}, {"full_name": "max_neg_neg", "def_path": "Mathlib/Algebra/Order/Group/MinMax.lean", "def_pos": [43, 15], "def_end_pos": [43, 26]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : LinearOrderedAddCommGroup \u03b1\na b c d : \u03b1\n\u22a2 Icc (a - max b c) (a - min b c) = Icc (min (a - b) (a - c)) (max (a - b) (a - c))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/List/Basic.lean", "full_name": "List.takeD_eq_takeDTR", "start": [621, 10], "end": [622, 48], "traced_tactics": [{"tactic": "funext \u03b1 f n l", "annotated_tactic": ["funext \u03b1 f n l", []], "state_before": "\u22a2 @takeD = @takeDTR", "state_after": "case h.h.h.h\n\u03b1 : Type u_1\nf : Nat\nn : List \u03b1\nl : \u03b1\n\u22a2 takeD f n l = takeDTR f n l"}, {"tactic": "simp [takeDTR, takeDTR_go_eq]", "annotated_tactic": ["simp [<a>takeDTR</a>, <a>takeDTR_go_eq</a>]", [{"full_name": "List.takeDTR", "def_path": "lake-packages/std/Std/Data/List/Basic.lean", "def_pos": [609, 5], "def_end_pos": [609, 12]}, {"full_name": "List.takeDTR_go_eq", "def_path": "lake-packages/std/Std/Data/List/Basic.lean", "def_pos": [616, 9], "def_end_pos": [616, 22]}]], "state_before": "case h.h.h.h\n\u03b1 : Type u_1\nf : Nat\nn : List \u03b1\nl : \u03b1\n\u22a2 takeD f n l = takeDTR f n l", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL2.lean", "full_name": "MeasureTheory.condexpL2_const_inner", "start": [228, 1], "end": [252, 57], "traced_tactics": [{"tactic": "have h_mem_Lp : Mem\u2112p (fun a => \u27eac, (condexpL2 E \ud835\udd5c hm f : \u03b1 \u2192 E) a\u27eb) 2 \u03bc := by\n  refine' Mem\u2112p.const_inner _ _; rw [lpMeas_coe]; exact Lp.mem\u2112p _", "annotated_tactic": ["have h_mem_Lp : <a>Mem\u2112p</a> (fun a => \u27eac, (<a>condexpL2</a> E \ud835\udd5c hm f : \u03b1 \u2192 E) a\u27eb) 2 \u03bc := by\n    refine' <a>Mem\u2112p.const_inner</a> _ _; rw [<a>lpMeas_coe</a>]; exact <a>Lp.mem\u2112p</a> _", [{"full_name": "MeasureTheory.Mem\u2112p", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [108, 5], "def_end_pos": [108, 10]}, {"full_name": "MeasureTheory.condexpL2", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL2.lean", "def_pos": [71, 19], "def_end_pos": [71, 28]}, {"full_name": "MeasureTheory.Mem\u2112p.const_inner", "def_path": "Mathlib/MeasureTheory/Function/L2Space.lean", "def_pos": [73, 9], "def_end_pos": [73, 26]}, {"full_name": "MeasureTheory.lpMeas_coe", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean", "def_pos": [257, 9], "def_end_pos": [257, 19]}, {"full_name": "MeasureTheory.Lp.mem\u2112p", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [216, 19], "def_end_pos": [216, 24]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 Lp E 2 }\nc : E\n\u22a2 \u2191\u2191\u2191(\u2191(condexpL2 \ud835\udd5c \ud835\udd5c hm) (Mem\u2112p.toLp (fun a => inner c (\u2191\u2191f a)) (_ : Mem\u2112p (fun a => inner c (\u2191\u2191f a)) 2))) =\u1d50[\u03bc]\n    fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 Lp E 2 }\nc : E\nh_mem_Lp : Mem\u2112p (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) 2\n\u22a2 \u2191\u2191\u2191(\u2191(condexpL2 \ud835\udd5c \ud835\udd5c hm) (Mem\u2112p.toLp (fun a => inner c (\u2191\u2191f a)) (_ : Mem\u2112p (fun a => inner c (\u2191\u2191f a)) 2))) =\u1d50[\u03bc]\n    fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)"}, {"tactic": "have h_eq : h_mem_Lp.toLp _ =\u1d50[\u03bc] fun a => \u27eac, (condexpL2 E \ud835\udd5c hm f : \u03b1 \u2192 E) a\u27eb :=\n  h_mem_Lp.coeFn_toLp", "annotated_tactic": ["have h_eq : h_mem_Lp.toLp _ =\u1d50[\u03bc] fun a => \u27eac, (<a>condexpL2</a> E \ud835\udd5c hm f : \u03b1 \u2192 E) a\u27eb :=\n    h_mem_Lp.coeFn_toLp", [{"full_name": "MeasureTheory.condexpL2", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL2.lean", "def_pos": [71, 19], "def_end_pos": [71, 28]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 Lp E 2 }\nc : E\nh_mem_Lp : Mem\u2112p (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) 2\n\u22a2 \u2191\u2191\u2191(\u2191(condexpL2 \ud835\udd5c \ud835\udd5c hm) (Mem\u2112p.toLp (fun a => inner c (\u2191\u2191f a)) (_ : Mem\u2112p (fun a => inner c (\u2191\u2191f a)) 2))) =\u1d50[\u03bc]\n    fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 Lp E 2 }\nc : E\nh_mem_Lp : Mem\u2112p (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) 2\nh_eq :\n  \u2191\u2191(Mem\u2112p.toLp (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) h_mem_Lp) =\u1d50[\u03bc] fun a =>\n    inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)\n\u22a2 \u2191\u2191\u2191(\u2191(condexpL2 \ud835\udd5c \ud835\udd5c hm) (Mem\u2112p.toLp (fun a => inner c (\u2191\u2191f a)) (_ : Mem\u2112p (fun a => inner c (\u2191\u2191f a)) 2))) =\u1d50[\u03bc]\n    fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)"}, {"tactic": "refine' EventuallyEq.trans _ h_eq", "annotated_tactic": ["refine' <a>EventuallyEq.trans</a> _ h_eq", [{"full_name": "Filter.EventuallyEq.trans", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1503, 9], "def_end_pos": [1503, 27]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 Lp E 2 }\nc : E\nh_mem_Lp : Mem\u2112p (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) 2\nh_eq :\n  \u2191\u2191(Mem\u2112p.toLp (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) h_mem_Lp) =\u1d50[\u03bc] fun a =>\n    inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)\n\u22a2 \u2191\u2191\u2191(\u2191(condexpL2 \ud835\udd5c \ud835\udd5c hm) (Mem\u2112p.toLp (fun a => inner c (\u2191\u2191f a)) (_ : Mem\u2112p (fun a => inner c (\u2191\u2191f a)) 2))) =\u1d50[\u03bc]\n    fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 Lp E 2 }\nc : E\nh_mem_Lp : Mem\u2112p (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) 2\nh_eq :\n  \u2191\u2191(Mem\u2112p.toLp (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) h_mem_Lp) =\u1d50[\u03bc] fun a =>\n    inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)\n\u22a2 \u2191\u2191\u2191(\u2191(condexpL2 \ud835\udd5c \ud835\udd5c hm) (Mem\u2112p.toLp (fun a => inner c (\u2191\u2191f a)) (_ : Mem\u2112p (fun a => inner c (\u2191\u2191f a)) 2))) =\u1d50[\u03bc]\n    \u2191\u2191(Mem\u2112p.toLp (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) h_mem_Lp)"}, {"tactic": "refine' Lp.ae_eq_of_forall_set_integral_eq' \ud835\udd5c hm _ _ two_ne_zero ENNReal.coe_ne_top\n  (fun s _ h\u03bcs => integrableOn_condexpL2_of_measure_ne_top hm h\u03bcs.ne _) _ _ _ _", "annotated_tactic": ["refine' <a>Lp.ae_eq_of_forall_set_integral_eq'</a> \ud835\udd5c hm _ _ <a>two_ne_zero</a> <a>ENNReal.coe_ne_top</a>\n    (fun s _ h\u03bcs => <a>integrableOn_condexpL2_of_measure_ne_top</a> hm h\u03bcs.ne _) _ _ _ _", [{"full_name": "MeasureTheory.Lp.ae_eq_of_forall_set_integral_eq'", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Unique.lean", "def_pos": [93, 9], "def_end_pos": [93, 44]}, {"full_name": "two_ne_zero", "def_path": "Mathlib/Algebra/NeZero.lean", "def_pos": [62, 7], "def_end_pos": [62, 18]}, {"full_name": "ENNReal.coe_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [302, 17], "def_end_pos": [302, 27]}, {"full_name": "MeasureTheory.integrableOn_condexpL2_of_measure_ne_top", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL2.lean", "def_pos": [84, 9], "def_end_pos": [84, 49]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 Lp E 2 }\nc : E\nh_mem_Lp : Mem\u2112p (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) 2\nh_eq :\n  \u2191\u2191(Mem\u2112p.toLp (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) h_mem_Lp) =\u1d50[\u03bc] fun a =>\n    inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)\n\u22a2 \u2191\u2191\u2191(\u2191(condexpL2 \ud835\udd5c \ud835\udd5c hm) (Mem\u2112p.toLp (fun a => inner c (\u2191\u2191f a)) (_ : Mem\u2112p (fun a => inner c (\u2191\u2191f a)) 2))) =\u1d50[\u03bc]\n    \u2191\u2191(Mem\u2112p.toLp (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) h_mem_Lp)", "state_after": "case refine'_1\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 Lp E 2 }\nc : E\nh_mem_Lp : Mem\u2112p (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) 2\nh_eq :\n  \u2191\u2191(Mem\u2112p.toLp (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) h_mem_Lp) =\u1d50[\u03bc] fun a =>\n    inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)\n\u22a2 \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn (\u2191\u2191(Mem\u2112p.toLp (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) h_mem_Lp)) s\n\ncase refine'_2\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 Lp E 2 }\nc : E\nh_mem_Lp : Mem\u2112p (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) 2\nh_eq :\n  \u2191\u2191(Mem\u2112p.toLp (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) h_mem_Lp) =\u1d50[\u03bc] fun a =>\n    inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)\n\u22a2 \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s < \u22a4 \u2192\n        \u222b (x : \u03b1) in s,\n            \u2191\u2191\u2191(\u2191(condexpL2 \ud835\udd5c \ud835\udd5c hm) (Mem\u2112p.toLp (fun a => inner c (\u2191\u2191f a)) (_ : Mem\u2112p (fun a => inner c (\u2191\u2191f a)) 2)))\n              x \u2202\u03bc =\n          \u222b (x : \u03b1) in s, \u2191\u2191(Mem\u2112p.toLp (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) h_mem_Lp) x \u2202\u03bc\n\ncase refine'_3\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 Lp E 2 }\nc : E\nh_mem_Lp : Mem\u2112p (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) 2\nh_eq :\n  \u2191\u2191(Mem\u2112p.toLp (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) h_mem_Lp) =\u1d50[\u03bc] fun a =>\n    inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)\n\u22a2 AEStronglyMeasurable' m\n    (\u2191\u2191\u2191(\u2191(condexpL2 \ud835\udd5c \ud835\udd5c hm) (Mem\u2112p.toLp (fun a => inner c (\u2191\u2191f a)) (_ : Mem\u2112p (fun a => inner c (\u2191\u2191f a)) 2)))) \u03bc\n\ncase refine'_4\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 Lp E 2 }\nc : E\nh_mem_Lp : Mem\u2112p (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) 2\nh_eq :\n  \u2191\u2191(Mem\u2112p.toLp (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) h_mem_Lp) =\u1d50[\u03bc] fun a =>\n    inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)\n\u22a2 AEStronglyMeasurable' m (\u2191\u2191(Mem\u2112p.toLp (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) h_mem_Lp)) \u03bc"}, {"tactic": "refine' Mem\u2112p.const_inner _ _", "annotated_tactic": ["refine' <a>Mem\u2112p.const_inner</a> _ _", [{"full_name": "MeasureTheory.Mem\u2112p.const_inner", "def_path": "Mathlib/MeasureTheory/Function/L2Space.lean", "def_pos": [73, 9], "def_end_pos": [73, 26]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 Lp E 2 }\nc : E\n\u22a2 Mem\u2112p (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) 2", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 Lp E 2 }\nc : E\n\u22a2 Mem\u2112p (fun a => \u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a) 2"}, {"tactic": "exact Lp.mem\u2112p _", "annotated_tactic": ["exact <a>Lp.mem\u2112p</a> _", [{"full_name": "MeasureTheory.Lp.mem\u2112p", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [216, 19], "def_end_pos": [216, 24]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 Lp E 2 }\nc : E\n\u22a2 Mem\u2112p (fun a => \u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a) 2", "state_after": "no goals"}, {"tactic": "intro s _ h\u03bcs", "annotated_tactic": ["intro s _ h\u03bcs", []], "state_before": "case refine'_1\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 Lp E 2 }\nc : E\nh_mem_Lp : Mem\u2112p (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) 2\nh_eq :\n  \u2191\u2191(Mem\u2112p.toLp (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) h_mem_Lp) =\u1d50[\u03bc] fun a =>\n    inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)\n\u22a2 \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn (\u2191\u2191(Mem\u2112p.toLp (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) h_mem_Lp)) s", "state_after": "case refine'_1\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 Lp E 2 }\nc : E\nh_mem_Lp : Mem\u2112p (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) 2\nh_eq :\n  \u2191\u2191(Mem\u2112p.toLp (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) h_mem_Lp) =\u1d50[\u03bc] fun a =>\n    inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)\ns : Set \u03b1\na\u271d : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 IntegrableOn (\u2191\u2191(Mem\u2112p.toLp (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) h_mem_Lp)) s"}, {"tactic": "rw [IntegrableOn, integrable_congr (ae_restrict_of_ae h_eq)]", "annotated_tactic": ["rw [<a>IntegrableOn</a>, <a>integrable_congr</a> (<a>ae_restrict_of_ae</a> h_eq)]", [{"full_name": "MeasureTheory.IntegrableOn", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [90, 5], "def_end_pos": [90, 17]}, {"full_name": "MeasureTheory.integrable_congr", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [496, 9], "def_end_pos": [496, 25]}, {"full_name": "MeasureTheory.ae_restrict_of_ae", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2596, 9], "def_end_pos": [2596, 26]}]], "state_before": "case refine'_1\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 Lp E 2 }\nc : E\nh_mem_Lp : Mem\u2112p (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) 2\nh_eq :\n  \u2191\u2191(Mem\u2112p.toLp (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) h_mem_Lp) =\u1d50[\u03bc] fun a =>\n    inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)\ns : Set \u03b1\na\u271d : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 IntegrableOn (\u2191\u2191(Mem\u2112p.toLp (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) h_mem_Lp)) s", "state_after": "case refine'_1\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 Lp E 2 }\nc : E\nh_mem_Lp : Mem\u2112p (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) 2\nh_eq :\n  \u2191\u2191(Mem\u2112p.toLp (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) h_mem_Lp) =\u1d50[\u03bc] fun a =>\n    inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)\ns : Set \u03b1\na\u271d : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 Integrable fun x => (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) x"}, {"tactic": "exact (integrableOn_condexpL2_of_measure_ne_top hm h\u03bcs.ne _).const_inner _", "annotated_tactic": ["exact (<a>integrableOn_condexpL2_of_measure_ne_top</a> hm h\u03bcs.ne _).<a>const_inner</a> _", [{"full_name": "MeasureTheory.integrableOn_condexpL2_of_measure_ne_top", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL2.lean", "def_pos": [84, 9], "def_end_pos": [84, 49]}, {"full_name": "MeasureTheory.Integrable.const_inner", "def_path": "Mathlib/MeasureTheory/Function/L2Space.lean", "def_pos": [85, 9], "def_end_pos": [85, 31]}]], "state_before": "case refine'_1\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 Lp E 2 }\nc : E\nh_mem_Lp : Mem\u2112p (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) 2\nh_eq :\n  \u2191\u2191(Mem\u2112p.toLp (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) h_mem_Lp) =\u1d50[\u03bc] fun a =>\n    inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)\ns : Set \u03b1\na\u271d : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 Integrable fun x => (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) x", "state_after": "no goals"}, {"tactic": "intro s hs h\u03bcs", "annotated_tactic": ["intro s hs h\u03bcs", []], "state_before": "case refine'_2\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 Lp E 2 }\nc : E\nh_mem_Lp : Mem\u2112p (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) 2\nh_eq :\n  \u2191\u2191(Mem\u2112p.toLp (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) h_mem_Lp) =\u1d50[\u03bc] fun a =>\n    inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)\n\u22a2 \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s < \u22a4 \u2192\n        \u222b (x : \u03b1) in s,\n            \u2191\u2191\u2191(\u2191(condexpL2 \ud835\udd5c \ud835\udd5c hm) (Mem\u2112p.toLp (fun a => inner c (\u2191\u2191f a)) (_ : Mem\u2112p (fun a => inner c (\u2191\u2191f a)) 2)))\n              x \u2202\u03bc =\n          \u222b (x : \u03b1) in s, \u2191\u2191(Mem\u2112p.toLp (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) h_mem_Lp) x \u2202\u03bc", "state_after": "case refine'_2\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 Lp E 2 }\nc : E\nh_mem_Lp : Mem\u2112p (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) 2\nh_eq :\n  \u2191\u2191(Mem\u2112p.toLp (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) h_mem_Lp) =\u1d50[\u03bc] fun a =>\n    inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u222b (x : \u03b1) in s,\n      \u2191\u2191\u2191(\u2191(condexpL2 \ud835\udd5c \ud835\udd5c hm) (Mem\u2112p.toLp (fun a => inner c (\u2191\u2191f a)) (_ : Mem\u2112p (fun a => inner c (\u2191\u2191f a)) 2))) x \u2202\u03bc =\n    \u222b (x : \u03b1) in s, \u2191\u2191(Mem\u2112p.toLp (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) h_mem_Lp) x \u2202\u03bc"}, {"tactic": "rw [\u2190 lpMeas_coe, integral_condexpL2_eq_of_fin_meas_real _ hs h\u03bcs.ne,\n  integral_congr_ae (ae_restrict_of_ae h_eq), lpMeas_coe, \u2190\n  L2.inner_indicatorConstLp_eq_set_integral_inner \ud835\udd5c (\u2191(condexpL2 E \ud835\udd5c hm f)) (hm s hs) c h\u03bcs.ne,\n  \u2190 inner_condexpL2_left_eq_right, condexpL2_indicator_of_measurable _ hs,\n  L2.inner_indicatorConstLp_eq_set_integral_inner \ud835\udd5c f (hm s hs) c h\u03bcs.ne,\n  set_integral_congr_ae (hm s hs)\n    ((Mem\u2112p.coeFn_toLp ((Lp.mem\u2112p f).const_inner c)).mono fun x hx _ => hx)]", "annotated_tactic": ["rw [\u2190 <a>lpMeas_coe</a>, <a>integral_condexpL2_eq_of_fin_meas_real</a> _ hs h\u03bcs.ne,\n      <a>integral_congr_ae</a> (<a>ae_restrict_of_ae</a> h_eq), <a>lpMeas_coe</a>, \u2190\n      <a>L2.inner_indicatorConstLp_eq_set_integral_inner</a> \ud835\udd5c (\u2191(<a>condexpL2</a> E \ud835\udd5c hm f)) (hm s hs) c h\u03bcs.ne,\n      \u2190 <a>inner_condexpL2_left_eq_right</a>, <a>condexpL2_indicator_of_measurable</a> _ hs,\n      <a>L2.inner_indicatorConstLp_eq_set_integral_inner</a> \ud835\udd5c f (hm s hs) c h\u03bcs.ne,\n      <a>set_integral_congr_ae</a> (hm s hs)\n        ((<a>Mem\u2112p.coeFn_toLp</a> ((<a>Lp.mem\u2112p</a> f).<a>const_inner</a> c)).<a>mono</a> fun x hx _ => hx)]", [{"full_name": "MeasureTheory.lpMeas_coe", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean", "def_pos": [257, 9], "def_end_pos": [257, 19]}, {"full_name": "MeasureTheory.integral_condexpL2_eq_of_fin_meas_real", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL2.lean", "def_pos": [151, 9], "def_end_pos": [151, 47]}, {"full_name": "MeasureTheory.integral_congr_ae", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [938, 9], "def_end_pos": [938, 26]}, {"full_name": "MeasureTheory.ae_restrict_of_ae", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2596, 9], "def_end_pos": [2596, 26]}, {"full_name": "MeasureTheory.lpMeas_coe", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean", "def_pos": [257, 9], "def_end_pos": [257, 19]}, {"full_name": "MeasureTheory.L2.inner_indicatorConstLp_eq_set_integral_inner", "def_path": "Mathlib/MeasureTheory/Function/L2Space.lean", "def_pos": [234, 9], "def_end_pos": [234, 53]}, {"full_name": "MeasureTheory.condexpL2", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL2.lean", "def_pos": [71, 19], "def_end_pos": [71, 28]}, {"full_name": "MeasureTheory.inner_condexpL2_left_eq_right", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL2.lean", "def_pos": [119, 9], "def_end_pos": [119, 38]}, {"full_name": "MeasureTheory.condexpL2_indicator_of_measurable", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL2.lean", "def_pos": [125, 9], "def_end_pos": [125, 42]}, {"full_name": "MeasureTheory.L2.inner_indicatorConstLp_eq_set_integral_inner", "def_path": "Mathlib/MeasureTheory/Function/L2Space.lean", "def_pos": [234, 9], "def_end_pos": [234, 53]}, {"full_name": "MeasureTheory.set_integral_congr_ae", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [77, 9], "def_end_pos": [77, 30]}, {"full_name": "MeasureTheory.Mem\u2112p.coeFn_toLp", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [119, 9], "def_end_pos": [119, 19]}, {"full_name": "MeasureTheory.Lp.mem\u2112p", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [216, 19], "def_end_pos": [216, 24]}, {"full_name": "MeasureTheory.Mem\u2112p.const_inner", "def_path": "Mathlib/MeasureTheory/Function/L2Space.lean", "def_pos": [73, 9], "def_end_pos": [73, 26]}, {"full_name": "Filter.Eventually.mono", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1140, 9], "def_end_pos": [1140, 24]}]], "state_before": "case refine'_2\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 Lp E 2 }\nc : E\nh_mem_Lp : Mem\u2112p (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) 2\nh_eq :\n  \u2191\u2191(Mem\u2112p.toLp (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) h_mem_Lp) =\u1d50[\u03bc] fun a =>\n    inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u222b (x : \u03b1) in s,\n      \u2191\u2191\u2191(\u2191(condexpL2 \ud835\udd5c \ud835\udd5c hm) (Mem\u2112p.toLp (fun a => inner c (\u2191\u2191f a)) (_ : Mem\u2112p (fun a => inner c (\u2191\u2191f a)) 2))) x \u2202\u03bc =\n    \u222b (x : \u03b1) in s, \u2191\u2191(Mem\u2112p.toLp (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) h_mem_Lp) x \u2202\u03bc", "state_after": "no goals"}, {"tactic": "exact lpMeas.aeStronglyMeasurable' _", "annotated_tactic": ["exact <a>lpMeas.aeStronglyMeasurable'</a> _", [{"full_name": "MeasureTheory.lpMeas.aeStronglyMeasurable'", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean", "def_pos": [242, 9], "def_end_pos": [242, 37]}]], "state_before": "case refine'_3\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 Lp E 2 }\nc : E\nh_mem_Lp : Mem\u2112p (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) 2\nh_eq :\n  \u2191\u2191(Mem\u2112p.toLp (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) h_mem_Lp) =\u1d50[\u03bc] fun a =>\n    inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)\n\u22a2 AEStronglyMeasurable' m\n    (\u2191\u2191\u2191(\u2191(condexpL2 \ud835\udd5c \ud835\udd5c hm) (Mem\u2112p.toLp (fun a => inner c (\u2191\u2191f a)) (_ : Mem\u2112p (fun a => inner c (\u2191\u2191f a)) 2)))) \u03bc", "state_after": "no goals"}, {"tactic": "refine' AEStronglyMeasurable'.congr _ h_eq.symm", "annotated_tactic": ["refine' <a>AEStronglyMeasurable'.congr</a> _ h_eq.symm", [{"full_name": "MeasureTheory.AEStronglyMeasurable'.congr", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean", "def_pos": [59, 9], "def_end_pos": [59, 14]}]], "state_before": "case refine'_4\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 Lp E 2 }\nc : E\nh_mem_Lp : Mem\u2112p (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) 2\nh_eq :\n  \u2191\u2191(Mem\u2112p.toLp (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) h_mem_Lp) =\u1d50[\u03bc] fun a =>\n    inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)\n\u22a2 AEStronglyMeasurable' m (\u2191\u2191(Mem\u2112p.toLp (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) h_mem_Lp)) \u03bc", "state_after": "case refine'_4\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 Lp E 2 }\nc : E\nh_mem_Lp : Mem\u2112p (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) 2\nh_eq :\n  \u2191\u2191(Mem\u2112p.toLp (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) h_mem_Lp) =\u1d50[\u03bc] fun a =>\n    inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)\n\u22a2 AEStronglyMeasurable' m (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) \u03bc"}, {"tactic": "exact (lpMeas.aeStronglyMeasurable' _).const_inner _", "annotated_tactic": ["exact (<a>lpMeas.aeStronglyMeasurable'</a> _).<a>const_inner</a> _", [{"full_name": "MeasureTheory.lpMeas.aeStronglyMeasurable'", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean", "def_pos": [242, 9], "def_end_pos": [242, 37]}, {"full_name": "MeasureTheory.AEStronglyMeasurable'.const_inner", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean", "def_pos": [98, 9], "def_end_pos": [98, 20]}]], "state_before": "case refine'_4\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 Lp E 2 }\nc : E\nh_mem_Lp : Mem\u2112p (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) 2\nh_eq :\n  \u2191\u2191(Mem\u2112p.toLp (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) h_mem_Lp) =\u1d50[\u03bc] fun a =>\n    inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)\n\u22a2 AEStronglyMeasurable' m (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) \u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/List/Init/Lemmas.lean", "full_name": "List.tail!_cons", "start": [32, 9], "end": [32, 56], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/EssSup.lean", "full_name": "essSup_indicator_eq_essSup_restrict", "start": [268, 1], "end": [295, 25], "traced_tactics": [{"tactic": "refine'\n  le_antisymm _\n    (limsSup_le_limsSup_of_le (map_restrict_ae_le_map_indicator_ae hs)\n      (by isBoundedDefault) (by isBoundedDefault) )", "annotated_tactic": ["refine'\n    <a>le_antisymm</a> _\n      (<a>limsSup_le_limsSup_of_le</a> (<a>map_restrict_ae_le_map_indicator_ae</a> hs)\n        (by isBoundedDefault) (by isBoundedDefault) )", [{"full_name": "le_antisymm", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [188, 9], "def_end_pos": [188, 20]}, {"full_name": "Filter.limsSup_le_limsSup_of_le", "def_path": "Mathlib/Order/LiminfLimsup.lean", "def_pos": [594, 9], "def_end_pos": [594, 33]}, {"full_name": "map_restrict_ae_le_map_indicator_ae", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [4473, 9], "def_end_pos": [4473, 44]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : CompleteLinearOrder \u03b2\ninst\u271d : Zero \u03b2\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : 0 \u2264\u1d50[Measure.restrict \u03bc s] f\nhs : MeasurableSet s\nhs_not_null : \u2191\u2191\u03bc s \u2260 0\n\u22a2 essSup (indicator s f) \u03bc = essSup f (Measure.restrict \u03bc s)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : CompleteLinearOrder \u03b2\ninst\u271d : Zero \u03b2\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : 0 \u2264\u1d50[Measure.restrict \u03bc s] f\nhs : MeasurableSet s\nhs_not_null : \u2191\u2191\u03bc s \u2260 0\n\u22a2 essSup (indicator s f) \u03bc \u2264 essSup f (Measure.restrict \u03bc s)"}, {"tactic": "refine' limsSup_le_limsSup (by isBoundedDefault) (by isBoundedDefault) (fun c h_restrict_le => _)", "annotated_tactic": ["refine' <a>limsSup_le_limsSup</a> (by isBoundedDefault) (by isBoundedDefault) (fun c h_restrict_le => _)", [{"full_name": "Filter.limsSup_le_limsSup", "def_path": "Mathlib/Order/LiminfLimsup.lean", "def_pos": [562, 9], "def_end_pos": [562, 27]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : CompleteLinearOrder \u03b2\ninst\u271d : Zero \u03b2\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : 0 \u2264\u1d50[Measure.restrict \u03bc s] f\nhs : MeasurableSet s\nhs_not_null : \u2191\u2191\u03bc s \u2260 0\n\u22a2 essSup (indicator s f) \u03bc \u2264 essSup f (Measure.restrict \u03bc s)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : CompleteLinearOrder \u03b2\ninst\u271d : Zero \u03b2\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : 0 \u2264\u1d50[Measure.restrict \u03bc s] f\nhs : MeasurableSet s\nhs_not_null : \u2191\u2191\u03bc s \u2260 0\nc : \u03b2\nh_restrict_le : \u2200\u1da0 (n : \u03b2) in map f (Measure.ae (Measure.restrict \u03bc s)), n \u2264 c\n\u22a2 \u2200\u1da0 (n : \u03b2) in map (indicator s f) (Measure.ae \u03bc), n \u2264 c"}, {"tactic": "rw [eventually_map] at h_restrict_le \u22a2", "annotated_tactic": ["rw [<a>eventually_map</a>] at h_restrict_le \u22a2", [{"full_name": "Filter.eventually_map", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1845, 9], "def_end_pos": [1845, 23]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : CompleteLinearOrder \u03b2\ninst\u271d : Zero \u03b2\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : 0 \u2264\u1d50[Measure.restrict \u03bc s] f\nhs : MeasurableSet s\nhs_not_null : \u2191\u2191\u03bc s \u2260 0\nc : \u03b2\nh_restrict_le : \u2200\u1da0 (n : \u03b2) in map f (Measure.ae (Measure.restrict \u03bc s)), n \u2264 c\n\u22a2 \u2200\u1da0 (n : \u03b2) in map (indicator s f) (Measure.ae \u03bc), n \u2264 c", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : CompleteLinearOrder \u03b2\ninst\u271d : Zero \u03b2\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : 0 \u2264\u1d50[Measure.restrict \u03bc s] f\nhs : MeasurableSet s\nhs_not_null : \u2191\u2191\u03bc s \u2260 0\nc : \u03b2\nh_restrict_le : \u2200\u1d50 (a : \u03b1) \u2202Measure.restrict \u03bc s, f a \u2264 c\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, indicator s f a \u2264 c"}, {"tactic": "rw [ae_restrict_iff' hs] at h_restrict_le", "annotated_tactic": ["rw [<a>ae_restrict_iff'</a> hs] at h_restrict_le", [{"full_name": "MeasureTheory.ae_restrict_iff'", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2572, 9], "def_end_pos": [2572, 25]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : CompleteLinearOrder \u03b2\ninst\u271d : Zero \u03b2\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : 0 \u2264\u1d50[Measure.restrict \u03bc s] f\nhs : MeasurableSet s\nhs_not_null : \u2191\u2191\u03bc s \u2260 0\nc : \u03b2\nh_restrict_le : \u2200\u1d50 (a : \u03b1) \u2202Measure.restrict \u03bc s, f a \u2264 c\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, indicator s f a \u2264 c", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : CompleteLinearOrder \u03b2\ninst\u271d : Zero \u03b2\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : 0 \u2264\u1d50[Measure.restrict \u03bc s] f\nhs : MeasurableSet s\nhs_not_null : \u2191\u2191\u03bc s \u2260 0\nc : \u03b2\nh_restrict_le : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 f x \u2264 c\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, indicator s f a \u2264 c"}, {"tactic": "refine' h_restrict_le.mono fun x hxc => _", "annotated_tactic": ["refine' h_restrict_le.mono fun x hxc => _", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : CompleteLinearOrder \u03b2\ninst\u271d : Zero \u03b2\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : 0 \u2264\u1d50[Measure.restrict \u03bc s] f\nhs : MeasurableSet s\nhs_not_null : \u2191\u2191\u03bc s \u2260 0\nc : \u03b2\nh_restrict_le : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 f x \u2264 c\nhc : 0 \u2264 c\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, indicator s f a \u2264 c", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : CompleteLinearOrder \u03b2\ninst\u271d : Zero \u03b2\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : 0 \u2264\u1d50[Measure.restrict \u03bc s] f\nhs : MeasurableSet s\nhs_not_null : \u2191\u2191\u03bc s \u2260 0\nc : \u03b2\nh_restrict_le : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 f x \u2264 c\nhc : 0 \u2264 c\nx : \u03b1\nhxc : x \u2208 s \u2192 f x \u2264 c\n\u22a2 indicator s f x \u2264 c"}, {"tactic": "by_cases hxs : x \u2208 s", "annotated_tactic": ["by_cases hxs : x \u2208 s", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : CompleteLinearOrder \u03b2\ninst\u271d : Zero \u03b2\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : 0 \u2264\u1d50[Measure.restrict \u03bc s] f\nhs : MeasurableSet s\nhs_not_null : \u2191\u2191\u03bc s \u2260 0\nc : \u03b2\nh_restrict_le : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 f x \u2264 c\nhc : 0 \u2264 c\nx : \u03b1\nhxc : x \u2208 s \u2192 f x \u2264 c\n\u22a2 indicator s f x \u2264 c", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : CompleteLinearOrder \u03b2\ninst\u271d : Zero \u03b2\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : 0 \u2264\u1d50[Measure.restrict \u03bc s] f\nhs : MeasurableSet s\nhs_not_null : \u2191\u2191\u03bc s \u2260 0\nc : \u03b2\nh_restrict_le : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 f x \u2264 c\nhc : 0 \u2264 c\nx : \u03b1\nhxc : x \u2208 s \u2192 f x \u2264 c\nhxs : x \u2208 s\n\u22a2 indicator s f x \u2264 c\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : CompleteLinearOrder \u03b2\ninst\u271d : Zero \u03b2\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : 0 \u2264\u1d50[Measure.restrict \u03bc s] f\nhs : MeasurableSet s\nhs_not_null : \u2191\u2191\u03bc s \u2260 0\nc : \u03b2\nh_restrict_le : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 f x \u2264 c\nhc : 0 \u2264 c\nx : \u03b1\nhxc : x \u2208 s \u2192 f x \u2264 c\nhxs : \u00acx \u2208 s\n\u22a2 indicator s f x \u2264 c"}, {"tactic": "isBoundedDefault", "annotated_tactic": ["isBoundedDefault", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : CompleteLinearOrder \u03b2\ninst\u271d : Zero \u03b2\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : 0 \u2264\u1d50[Measure.restrict \u03bc s] f\nhs : MeasurableSet s\nhs_not_null : \u2191\u2191\u03bc s \u2260 0\n\u22a2 IsCobounded (fun x x_1 => x \u2264 x_1) (map f (Measure.ae (Measure.restrict \u03bc s)))", "state_after": "no goals"}, {"tactic": "isBoundedDefault", "annotated_tactic": ["isBoundedDefault", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : CompleteLinearOrder \u03b2\ninst\u271d : Zero \u03b2\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : 0 \u2264\u1d50[Measure.restrict \u03bc s] f\nhs : MeasurableSet s\nhs_not_null : \u2191\u2191\u03bc s \u2260 0\n\u22a2 IsBounded (fun x x_1 => x \u2264 x_1) (map (indicator s f) (Measure.ae \u03bc))", "state_after": "no goals"}, {"tactic": "isBoundedDefault", "annotated_tactic": ["isBoundedDefault", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : CompleteLinearOrder \u03b2\ninst\u271d : Zero \u03b2\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : 0 \u2264\u1d50[Measure.restrict \u03bc s] f\nhs : MeasurableSet s\nhs_not_null : \u2191\u2191\u03bc s \u2260 0\n\u22a2 IsCobounded (fun x x_1 => x \u2264 x_1) (map (indicator s f) (Measure.ae \u03bc))", "state_after": "no goals"}, {"tactic": "isBoundedDefault", "annotated_tactic": ["isBoundedDefault", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : CompleteLinearOrder \u03b2\ninst\u271d : Zero \u03b2\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : 0 \u2264\u1d50[Measure.restrict \u03bc s] f\nhs : MeasurableSet s\nhs_not_null : \u2191\u2191\u03bc s \u2260 0\n\u22a2 IsBounded (fun x x_1 => x \u2264 x_1) (map f (Measure.ae (Measure.restrict \u03bc s)))", "state_after": "no goals"}, {"tactic": "rsuffices \u27e8x, hx\u27e9 : \u2203 x, 0 \u2264 f x \u2227 f x \u2264 c", "annotated_tactic": ["rsuffices \u27e8x, hx\u27e9 : \u2203 x, 0 \u2264 f x \u2227 f x \u2264 c", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : CompleteLinearOrder \u03b2\ninst\u271d : Zero \u03b2\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : 0 \u2264\u1d50[Measure.restrict \u03bc s] f\nhs : MeasurableSet s\nhs_not_null : \u2191\u2191\u03bc s \u2260 0\nc : \u03b2\nh_restrict_le : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 f x \u2264 c\n\u22a2 0 \u2264 c", "state_after": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : CompleteLinearOrder \u03b2\ninst\u271d : Zero \u03b2\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : 0 \u2264\u1d50[Measure.restrict \u03bc s] f\nhs : MeasurableSet s\nhs_not_null : \u2191\u2191\u03bc s \u2260 0\nc : \u03b2\nh_restrict_le : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 f x \u2264 c\nx : \u03b1\nhx : 0 \u2264 f x \u2227 f x \u2264 c\n\u22a2 0 \u2264 c\n\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : CompleteLinearOrder \u03b2\ninst\u271d : Zero \u03b2\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : 0 \u2264\u1d50[Measure.restrict \u03bc s] f\nhs : MeasurableSet s\nhs_not_null : \u2191\u2191\u03bc s \u2260 0\nc : \u03b2\nh_restrict_le : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 f x \u2264 c\n\u22a2 \u2203 x, 0 \u2264 f x \u2227 f x \u2264 c"}, {"tactic": "exact hx.1.trans hx.2", "annotated_tactic": ["exact hx.1.<a>trans</a> hx.2", [{"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [120, 7], "def_end_pos": [120, 18]}]], "state_before": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : CompleteLinearOrder \u03b2\ninst\u271d : Zero \u03b2\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : 0 \u2264\u1d50[Measure.restrict \u03bc s] f\nhs : MeasurableSet s\nhs_not_null : \u2191\u2191\u03bc s \u2260 0\nc : \u03b2\nh_restrict_le : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 f x \u2264 c\nx : \u03b1\nhx : 0 \u2264 f x \u2227 f x \u2264 c\n\u22a2 0 \u2264 c\n\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : CompleteLinearOrder \u03b2\ninst\u271d : Zero \u03b2\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : 0 \u2264\u1d50[Measure.restrict \u03bc s] f\nhs : MeasurableSet s\nhs_not_null : \u2191\u2191\u03bc s \u2260 0\nc : \u03b2\nh_restrict_le : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 f x \u2264 c\n\u22a2 \u2203 x, 0 \u2264 f x \u2227 f x \u2264 c", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : CompleteLinearOrder \u03b2\ninst\u271d : Zero \u03b2\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : 0 \u2264\u1d50[Measure.restrict \u03bc s] f\nhs : MeasurableSet s\nhs_not_null : \u2191\u2191\u03bc s \u2260 0\nc : \u03b2\nh_restrict_le : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 f x \u2264 c\n\u22a2 \u2203 x, 0 \u2264 f x \u2227 f x \u2264 c"}, {"tactic": "refine' Frequently.exists _", "annotated_tactic": ["refine' <a>Frequently.exists</a> _", [{"full_name": "Filter.Frequently.exists", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1302, 9], "def_end_pos": [1302, 26]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : CompleteLinearOrder \u03b2\ninst\u271d : Zero \u03b2\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : 0 \u2264\u1d50[Measure.restrict \u03bc s] f\nhs : MeasurableSet s\nhs_not_null : \u2191\u2191\u03bc s \u2260 0\nc : \u03b2\nh_restrict_le : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 f x \u2264 c\n\u22a2 \u2203 x, 0 \u2264 f x \u2227 f x \u2264 c", "state_after": "case refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : CompleteLinearOrder \u03b2\ninst\u271d : Zero \u03b2\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : 0 \u2264\u1d50[Measure.restrict \u03bc s] f\nhs : MeasurableSet s\nhs_not_null : \u2191\u2191\u03bc s \u2260 0\nc : \u03b2\nh_restrict_le : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 f x \u2264 c\n\u22a2 Filter \u03b1\n\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : CompleteLinearOrder \u03b2\ninst\u271d : Zero \u03b2\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : 0 \u2264\u1d50[Measure.restrict \u03bc s] f\nhs : MeasurableSet s\nhs_not_null : \u2191\u2191\u03bc s \u2260 0\nc : \u03b2\nh_restrict_le : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 f x \u2264 c\n\u22a2 \u2203\u1da0 (x : \u03b1) in ?refine'_1, 0 \u2264 f x \u2227 f x \u2264 c"}, {"tactic": "rw [EventuallyLE, ae_restrict_iff' hs] at hf", "annotated_tactic": ["rw [<a>EventuallyLE</a>, <a>ae_restrict_iff'</a> hs] at hf", [{"full_name": "Filter.EventuallyLE", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1638, 5], "def_end_pos": [1638, 17]}, {"full_name": "MeasureTheory.ae_restrict_iff'", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2572, 9], "def_end_pos": [2572, 25]}]], "state_before": "case refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : CompleteLinearOrder \u03b2\ninst\u271d : Zero \u03b2\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : 0 \u2264\u1d50[Measure.restrict \u03bc s] f\nhs : MeasurableSet s\nhs_not_null : \u2191\u2191\u03bc s \u2260 0\nc : \u03b2\nh_restrict_le : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 f x \u2264 c\n\u22a2 \u2203\u1d50 (x : \u03b1) \u2202\u03bc, 0 \u2264 f x \u2227 f x \u2264 c", "state_after": "case refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : CompleteLinearOrder \u03b2\ninst\u271d : Zero \u03b2\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 OfNat.ofNat 0 x \u2264 f x\nhs : MeasurableSet s\nhs_not_null : \u2191\u2191\u03bc s \u2260 0\nc : \u03b2\nh_restrict_le : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 f x \u2264 c\n\u22a2 \u2203\u1d50 (x : \u03b1) \u2202\u03bc, 0 \u2264 f x \u2227 f x \u2264 c"}, {"tactic": "have hs' : \u2203\u1d50 x \u2202\u03bc, x \u2208 s := by\n  contrapose! hs_not_null\n  rw [not_frequently, ae_iff] at hs_not_null\n  suffices { a : \u03b1 | \u00aca \u2209 s } = s by rwa [\u2190 this]\n  simp", "annotated_tactic": ["have hs' : \u2203\u1d50 x \u2202\u03bc, x \u2208 s := by\n      contrapose! hs_not_null\n      rw [<a>not_frequently</a>, <a>ae_iff</a>] at hs_not_null\n      suffices { a : \u03b1 | \u00aca \u2209 s } = s by rwa [\u2190 this]\n      simp", [{"full_name": "Filter.not_frequently", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1331, 9], "def_end_pos": [1331, 23]}, {"full_name": "MeasureTheory.ae_iff", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [388, 9], "def_end_pos": [388, 15]}]], "state_before": "case refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : CompleteLinearOrder \u03b2\ninst\u271d : Zero \u03b2\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 OfNat.ofNat 0 x \u2264 f x\nhs : MeasurableSet s\nhs_not_null : \u2191\u2191\u03bc s \u2260 0\nc : \u03b2\nh_restrict_le : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 f x \u2264 c\n\u22a2 \u2203\u1d50 (x : \u03b1) \u2202\u03bc, 0 \u2264 f x \u2227 f x \u2264 c", "state_after": "case refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : CompleteLinearOrder \u03b2\ninst\u271d : Zero \u03b2\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 OfNat.ofNat 0 x \u2264 f x\nhs : MeasurableSet s\nhs_not_null : \u2191\u2191\u03bc s \u2260 0\nc : \u03b2\nh_restrict_le : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 f x \u2264 c\nhs' : \u2203\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s\n\u22a2 \u2203\u1d50 (x : \u03b1) \u2202\u03bc, 0 \u2264 f x \u2227 f x \u2264 c"}, {"tactic": "refine' hs'.mp (hf.mp (h_restrict_le.mono fun x hxs_imp_c hxf_nonneg hxs => _))", "annotated_tactic": ["refine' hs'.mp (hf.mp (h_restrict_le.mono fun x hxs_imp_c hxf_nonneg hxs => _))", []], "state_before": "case refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : CompleteLinearOrder \u03b2\ninst\u271d : Zero \u03b2\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 OfNat.ofNat 0 x \u2264 f x\nhs : MeasurableSet s\nhs_not_null : \u2191\u2191\u03bc s \u2260 0\nc : \u03b2\nh_restrict_le : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 f x \u2264 c\nhs' : \u2203\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s\n\u22a2 \u2203\u1d50 (x : \u03b1) \u2202\u03bc, 0 \u2264 f x \u2227 f x \u2264 c", "state_after": "case refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : CompleteLinearOrder \u03b2\ninst\u271d : Zero \u03b2\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 OfNat.ofNat 0 x \u2264 f x\nhs : MeasurableSet s\nhs_not_null : \u2191\u2191\u03bc s \u2260 0\nc : \u03b2\nh_restrict_le : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 f x \u2264 c\nhs' : \u2203\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s\nx : \u03b1\nhxs_imp_c : x \u2208 s \u2192 f x \u2264 c\nhxf_nonneg : x \u2208 s \u2192 OfNat.ofNat 0 x \u2264 f x\nhxs : x \u2208 s\n\u22a2 0 \u2264 f x \u2227 f x \u2264 c"}, {"tactic": "rw [Pi.zero_apply] at hxf_nonneg", "annotated_tactic": ["rw [<a>Pi.zero_apply</a>] at hxf_nonneg", [{"full_name": "Pi.zero_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [46, 3], "def_end_pos": [46, 14]}]], "state_before": "case refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : CompleteLinearOrder \u03b2\ninst\u271d : Zero \u03b2\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 OfNat.ofNat 0 x \u2264 f x\nhs : MeasurableSet s\nhs_not_null : \u2191\u2191\u03bc s \u2260 0\nc : \u03b2\nh_restrict_le : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 f x \u2264 c\nhs' : \u2203\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s\nx : \u03b1\nhxs_imp_c : x \u2208 s \u2192 f x \u2264 c\nhxf_nonneg : x \u2208 s \u2192 OfNat.ofNat 0 x \u2264 f x\nhxs : x \u2208 s\n\u22a2 0 \u2264 f x \u2227 f x \u2264 c", "state_after": "case refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : CompleteLinearOrder \u03b2\ninst\u271d : Zero \u03b2\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 OfNat.ofNat 0 x \u2264 f x\nhs : MeasurableSet s\nhs_not_null : \u2191\u2191\u03bc s \u2260 0\nc : \u03b2\nh_restrict_le : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 f x \u2264 c\nhs' : \u2203\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s\nx : \u03b1\nhxs_imp_c : x \u2208 s \u2192 f x \u2264 c\nhxf_nonneg : x \u2208 s \u2192 0 \u2264 f x\nhxs : x \u2208 s\n\u22a2 0 \u2264 f x \u2227 f x \u2264 c"}, {"tactic": "exact \u27e8hxf_nonneg hxs, hxs_imp_c hxs\u27e9", "annotated_tactic": ["exact \u27e8hxf_nonneg hxs, hxs_imp_c hxs\u27e9", []], "state_before": "case refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : CompleteLinearOrder \u03b2\ninst\u271d : Zero \u03b2\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 OfNat.ofNat 0 x \u2264 f x\nhs : MeasurableSet s\nhs_not_null : \u2191\u2191\u03bc s \u2260 0\nc : \u03b2\nh_restrict_le : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 f x \u2264 c\nhs' : \u2203\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s\nx : \u03b1\nhxs_imp_c : x \u2208 s \u2192 f x \u2264 c\nhxf_nonneg : x \u2208 s \u2192 0 \u2264 f x\nhxs : x \u2208 s\n\u22a2 0 \u2264 f x \u2227 f x \u2264 c", "state_after": "no goals"}, {"tactic": "exact \u03bc.ae", "annotated_tactic": ["exact \u03bc.ae", []], "state_before": "case refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : CompleteLinearOrder \u03b2\ninst\u271d : Zero \u03b2\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : 0 \u2264\u1d50[Measure.restrict \u03bc s] f\nhs : MeasurableSet s\nhs_not_null : \u2191\u2191\u03bc s \u2260 0\nc : \u03b2\nh_restrict_le : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 f x \u2264 c\n\u22a2 Filter \u03b1", "state_after": "no goals"}, {"tactic": "contrapose! hs_not_null", "annotated_tactic": ["contrapose! hs_not_null", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : CompleteLinearOrder \u03b2\ninst\u271d : Zero \u03b2\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 OfNat.ofNat 0 x \u2264 f x\nhs : MeasurableSet s\nhs_not_null : \u2191\u2191\u03bc s \u2260 0\nc : \u03b2\nh_restrict_le : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 f x \u2264 c\n\u22a2 \u2203\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : CompleteLinearOrder \u03b2\ninst\u271d : Zero \u03b2\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 OfNat.ofNat 0 x \u2264 f x\nhs : MeasurableSet s\nc : \u03b2\nh_restrict_le : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 f x \u2264 c\nhs_not_null : \u00ac\u2203\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s\n\u22a2 \u2191\u2191\u03bc s = 0"}, {"tactic": "rw [not_frequently, ae_iff] at hs_not_null", "annotated_tactic": ["rw [<a>not_frequently</a>, <a>ae_iff</a>] at hs_not_null", [{"full_name": "Filter.not_frequently", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1331, 9], "def_end_pos": [1331, 23]}, {"full_name": "MeasureTheory.ae_iff", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [388, 9], "def_end_pos": [388, 15]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : CompleteLinearOrder \u03b2\ninst\u271d : Zero \u03b2\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 OfNat.ofNat 0 x \u2264 f x\nhs : MeasurableSet s\nc : \u03b2\nh_restrict_le : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 f x \u2264 c\nhs_not_null : \u00ac\u2203\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s\n\u22a2 \u2191\u2191\u03bc s = 0", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : CompleteLinearOrder \u03b2\ninst\u271d : Zero \u03b2\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 OfNat.ofNat 0 x \u2264 f x\nhs : MeasurableSet s\nc : \u03b2\nh_restrict_le : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 f x \u2264 c\nhs_not_null : \u2191\u2191\u03bc {a | \u00ac\u00aca \u2208 s} = 0\n\u22a2 \u2191\u2191\u03bc s = 0"}, {"tactic": "suffices { a : \u03b1 | \u00aca \u2209 s } = s by rwa [\u2190 this]", "annotated_tactic": ["suffices { a : \u03b1 | \u00aca \u2209 s } = s by rwa [\u2190 this]", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : CompleteLinearOrder \u03b2\ninst\u271d : Zero \u03b2\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 OfNat.ofNat 0 x \u2264 f x\nhs : MeasurableSet s\nc : \u03b2\nh_restrict_le : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 f x \u2264 c\nhs_not_null : \u2191\u2191\u03bc {a | \u00ac\u00aca \u2208 s} = 0\n\u22a2 \u2191\u2191\u03bc s = 0", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : CompleteLinearOrder \u03b2\ninst\u271d : Zero \u03b2\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 OfNat.ofNat 0 x \u2264 f x\nhs : MeasurableSet s\nc : \u03b2\nh_restrict_le : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 f x \u2264 c\nhs_not_null : \u2191\u2191\u03bc {a | \u00ac\u00aca \u2208 s} = 0\n\u22a2 {a | \u00ac\u00aca \u2208 s} = s"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : CompleteLinearOrder \u03b2\ninst\u271d : Zero \u03b2\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 OfNat.ofNat 0 x \u2264 f x\nhs : MeasurableSet s\nc : \u03b2\nh_restrict_le : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 f x \u2264 c\nhs_not_null : \u2191\u2191\u03bc {a | \u00ac\u00aca \u2208 s} = 0\n\u22a2 {a | \u00ac\u00aca \u2208 s} = s", "state_after": "no goals"}, {"tactic": "rwa [\u2190 this]", "annotated_tactic": ["rwa [\u2190 this]", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : CompleteLinearOrder \u03b2\ninst\u271d : Zero \u03b2\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 OfNat.ofNat 0 x \u2264 f x\nhs : MeasurableSet s\nc : \u03b2\nh_restrict_le : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 f x \u2264 c\nhs_not_null : \u2191\u2191\u03bc {a | \u00ac\u00aca \u2208 s} = 0\nthis : {a | \u00ac\u00aca \u2208 s} = s\n\u22a2 \u2191\u2191\u03bc s = 0", "state_after": "no goals"}, {"tactic": "simpa [hxs] using hxc hxs", "annotated_tactic": ["simpa [hxs] using hxc hxs", []], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : CompleteLinearOrder \u03b2\ninst\u271d : Zero \u03b2\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : 0 \u2264\u1d50[Measure.restrict \u03bc s] f\nhs : MeasurableSet s\nhs_not_null : \u2191\u2191\u03bc s \u2260 0\nc : \u03b2\nh_restrict_le : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 f x \u2264 c\nhc : 0 \u2264 c\nx : \u03b1\nhxc : x \u2208 s \u2192 f x \u2264 c\nhxs : x \u2208 s\n\u22a2 indicator s f x \u2264 c", "state_after": "no goals"}, {"tactic": "simpa [hxs] using hc", "annotated_tactic": ["simpa [hxs] using hc", []], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : CompleteLinearOrder \u03b2\ninst\u271d : Zero \u03b2\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : 0 \u2264\u1d50[Measure.restrict \u03bc s] f\nhs : MeasurableSet s\nhs_not_null : \u2191\u2191\u03bc s \u2260 0\nc : \u03b2\nh_restrict_le : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 f x \u2264 c\nhc : 0 \u2264 c\nx : \u03b1\nhxc : x \u2208 s \u2192 f x \u2264 c\nhxs : \u00acx \u2208 s\n\u22a2 indicator s f x \u2264 c", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/CircleIntegral.lean", "full_name": "circleMap_ne_mem_ball", "start": [135, 1], "end": [137, 64], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Array/Lemmas.lean", "full_name": "Array.size_swap!", "start": [217, 9], "end": [218, 59], "traced_tactics": [{"tactic": "simp [swap!, hi, hj]", "annotated_tactic": ["simp [<a>swap!</a>, hi, hj]", [{"full_name": "Array.swap!", "def_path": "lake-packages/lean4/src/lean/Init/Data/Array/Basic.lean", "def_pos": [82, 5], "def_end_pos": [82, 10]}]], "state_before": "\u03b1 : Type u_1\na : Array \u03b1\ni j : Nat\nhi : i < size a\nhj : j < size a\n\u22a2 size (swap! a i j) = size a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/ProbabilityMassFunction/Monad.lean", "full_name": "PMF.toMeasure_pure_apply", "start": [86, 1], "end": [88, 96], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "full_name": "MeasureTheory.OuterMeasure.isCaratheodory_inter", "start": [982, 1], "end": [985, 87], "traced_tactics": [{"tactic": "rw [\u2190 isCaratheodory_compl_iff, Set.compl_inter]", "annotated_tactic": ["rw [\u2190 <a>isCaratheodory_compl_iff</a>, <a>Set.compl_inter</a>]", [{"full_name": "MeasureTheory.OuterMeasure.isCaratheodory_compl_iff", "def_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "def_pos": [955, 9], "def_end_pos": [955, 33]}, {"full_name": "Set.compl_inter", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1686, 9], "def_end_pos": [1686, 20]}]], "state_before": "\u03b1 : Type u\nm : OuterMeasure \u03b1\ns s\u2081 s\u2082 : Set \u03b1\nh\u2081 : IsCaratheodory m s\u2081\nh\u2082 : IsCaratheodory m s\u2082\n\u22a2 IsCaratheodory m (s\u2081 \u2229 s\u2082)", "state_after": "\u03b1 : Type u\nm : OuterMeasure \u03b1\ns s\u2081 s\u2082 : Set \u03b1\nh\u2081 : IsCaratheodory m s\u2081\nh\u2082 : IsCaratheodory m s\u2082\n\u22a2 IsCaratheodory m (s\u2081\u1d9c \u222a s\u2082\u1d9c)"}, {"tactic": "exact isCaratheodory_union _ (isCaratheodory_compl _ h\u2081) (isCaratheodory_compl _ h\u2082)", "annotated_tactic": ["exact <a>isCaratheodory_union</a> _ (<a>isCaratheodory_compl</a> _ h\u2081) (<a>isCaratheodory_compl</a> _ h\u2082)", [{"full_name": "MeasureTheory.OuterMeasure.isCaratheodory_union", "def_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "def_pos": [959, 9], "def_end_pos": [959, 29]}, {"full_name": "MeasureTheory.OuterMeasure.isCaratheodory_compl", "def_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "def_pos": [950, 9], "def_end_pos": [950, 29]}, {"full_name": "MeasureTheory.OuterMeasure.isCaratheodory_compl", "def_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "def_pos": [950, 9], "def_end_pos": [950, 29]}]], "state_before": "\u03b1 : Type u\nm : OuterMeasure \u03b1\ns s\u2081 s\u2082 : Set \u03b1\nh\u2081 : IsCaratheodory m s\u2081\nh\u2082 : IsCaratheodory m s\u2082\n\u22a2 IsCaratheodory m (s\u2081\u1d9c \u222a s\u2082\u1d9c)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Card.lean", "full_name": "Finset.card_erase_add_one", "start": [150, 1], "end": [151, 30], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/RBMap/Lemmas.lean", "full_name": "Std.RBNode.insert_toList_zoom", "start": [537, 1], "end": [540, 87], "traced_tactics": [{"tactic": "rw [\u2190 setBlack_toList, \u2190 Path.zoom_insert ht e, setBlack_toList, Path.insert_toList]", "annotated_tactic": ["rw [\u2190 <a>setBlack_toList</a>, \u2190 <a>Path.zoom_insert</a> ht e, <a>setBlack_toList</a>, <a>Path.insert_toList</a>]", [{"full_name": "Std.RBNode.setBlack_toList", "def_path": "lake-packages/std/Std/Data/RBMap/Lemmas.lean", "def_pos": [443, 17], "def_end_pos": [443, 32]}, {"full_name": "Std.RBNode.Path.zoom_insert", "def_path": "lake-packages/std/Std/Data/RBMap/Alter.lean", "def_pos": [169, 9], "def_end_pos": [169, 20]}, {"full_name": "Std.RBNode.setBlack_toList", "def_path": "lake-packages/std/Std/Data/RBMap/Lemmas.lean", "def_pos": [443, 17], "def_end_pos": [443, 32]}, {"full_name": "Std.RBNode.Path.insert_toList", "def_path": "lake-packages/std/Std/Data/RBMap/Lemmas.lean", "def_pos": [531, 9], "def_end_pos": [531, 22]}]], "state_before": "\u03b1 : Type u_1\nc : RBColor\nn : Nat\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nt' : RBNode \u03b1\np : Path \u03b1\nv : \u03b1\nt : RBNode \u03b1\nht : Balanced t c n\ne : zoom (cmp v) t Path.root = (t', p)\n\u22a2 toList (insert cmp t v) = Path.withList p (toList (setRoot v t'))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Constructions/Pi.lean", "full_name": "MeasureTheory.measurePreserving_funUnique", "start": [813, 1], "end": [822, 60], "traced_tactics": [{"tactic": "set e := MeasurableEquiv.funUnique \u03b1 \u03b2", "annotated_tactic": ["set e := <a>MeasurableEquiv.funUnique</a> \u03b1 \u03b2", [{"full_name": "MeasurableEquiv.funUnique", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [1631, 5], "def_end_pos": [1631, 14]}]], "state_before": "\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1\u271d : \u03b9 \u2192 Type u_3\ninst\u271d\u00b3 : Fintype \u03b9\nm\u271d\u00b9 : (i : \u03b9) \u2192 OuterMeasure (\u03b1\u271d i)\nm\u271d : (i : \u03b9) \u2192 MeasurableSpace (\u03b1\u271d i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (\u03b1\u271d i)\ninst\u271d\u00b2 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\ninst\u271d\u00b9 : Fintype \u03b9'\n\u03b2 : Type u\nm : MeasurableSpace \u03b2\n\u03bc : Measure \u03b2\n\u03b1 : Type v\ninst\u271d : Unique \u03b1\n\u22a2 MeasurePreserving \u2191(MeasurableEquiv.funUnique \u03b1 \u03b2)", "state_after": "\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1\u271d : \u03b9 \u2192 Type u_3\ninst\u271d\u00b3 : Fintype \u03b9\nm\u271d\u00b9 : (i : \u03b9) \u2192 OuterMeasure (\u03b1\u271d i)\nm\u271d : (i : \u03b9) \u2192 MeasurableSpace (\u03b1\u271d i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (\u03b1\u271d i)\ninst\u271d\u00b2 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\ninst\u271d\u00b9 : Fintype \u03b9'\n\u03b2 : Type u\nm : MeasurableSpace \u03b2\n\u03bc : Measure \u03b2\n\u03b1 : Type v\ninst\u271d : Unique \u03b1\ne : (\u03b1 \u2192 \u03b2) \u2243\u1d50 \u03b2 := MeasurableEquiv.funUnique \u03b1 \u03b2\n\u22a2 MeasurePreserving \u2191e"}, {"tactic": "have : (piPremeasure fun _ : \u03b1 => \u03bc.toOuterMeasure) = Measure.map e.symm \u03bc := by\n  ext1 s\n  rw [piPremeasure, Fintype.prod_unique, e.symm.map_apply]\n  congr 1; exact e.toEquiv.image_eq_preimage s", "annotated_tactic": ["have : (<a>piPremeasure</a> fun _ : \u03b1 => \u03bc.toOuterMeasure) = <a>Measure.map</a> e.symm \u03bc := by\n    ext1 s\n    rw [<a>piPremeasure</a>, <a>Fintype.prod_unique</a>, e.symm.map_apply]\n    congr 1; exact e.toEquiv.image_eq_preimage s", [{"full_name": "MeasureTheory.piPremeasure", "def_path": "Mathlib/MeasureTheory/Constructions/Pi.lean", "def_pos": [159, 5], "def_end_pos": [159, 17]}, {"full_name": "MeasureTheory.Measure.map", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1163, 17], "def_end_pos": [1163, 20]}, {"full_name": "MeasureTheory.piPremeasure", "def_path": "Mathlib/MeasureTheory/Constructions/Pi.lean", "def_pos": [159, 5], "def_end_pos": [159, 17]}, {"full_name": "Fintype.prod_unique", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [2012, 9], "def_end_pos": [2012, 20]}]], "state_before": "\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1\u271d : \u03b9 \u2192 Type u_3\ninst\u271d\u00b3 : Fintype \u03b9\nm\u271d\u00b9 : (i : \u03b9) \u2192 OuterMeasure (\u03b1\u271d i)\nm\u271d : (i : \u03b9) \u2192 MeasurableSpace (\u03b1\u271d i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (\u03b1\u271d i)\ninst\u271d\u00b2 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\ninst\u271d\u00b9 : Fintype \u03b9'\n\u03b2 : Type u\nm : MeasurableSpace \u03b2\n\u03bc : Measure \u03b2\n\u03b1 : Type v\ninst\u271d : Unique \u03b1\ne : (\u03b1 \u2192 \u03b2) \u2243\u1d50 \u03b2 := MeasurableEquiv.funUnique \u03b1 \u03b2\n\u22a2 MeasurePreserving \u2191e", "state_after": "\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1\u271d : \u03b9 \u2192 Type u_3\ninst\u271d\u00b3 : Fintype \u03b9\nm\u271d\u00b9 : (i : \u03b9) \u2192 OuterMeasure (\u03b1\u271d i)\nm\u271d : (i : \u03b9) \u2192 MeasurableSpace (\u03b1\u271d i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (\u03b1\u271d i)\ninst\u271d\u00b2 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\ninst\u271d\u00b9 : Fintype \u03b9'\n\u03b2 : Type u\nm : MeasurableSpace \u03b2\n\u03bc : Measure \u03b2\n\u03b1 : Type v\ninst\u271d : Unique \u03b1\ne : (\u03b1 \u2192 \u03b2) \u2243\u1d50 \u03b2 := MeasurableEquiv.funUnique \u03b1 \u03b2\nthis : (piPremeasure fun x => \u2191\u03bc) = \u2191\u2191(Measure.map (\u2191(MeasurableEquiv.symm e)) \u03bc)\n\u22a2 MeasurePreserving \u2191e"}, {"tactic": "simp only [Measure.pi, OuterMeasure.pi, this, boundedBy_measure, toOuterMeasure_toMeasure]", "annotated_tactic": ["simp only [<a>Measure.pi</a>, <a>OuterMeasure.pi</a>, this, <a>boundedBy_measure</a>, <a>toOuterMeasure_toMeasure</a>]", [{"full_name": "MeasureTheory.Measure.pi", "def_path": "Mathlib/MeasureTheory/Constructions/Pi.lean", "def_pos": [303, 27], "def_end_pos": [303, 29]}, {"full_name": "MeasureTheory.OuterMeasure.pi", "def_path": "Mathlib/MeasureTheory/Constructions/Pi.lean", "def_pos": [194, 15], "def_end_pos": [194, 17]}, {"full_name": "MeasureTheory.boundedBy_measure", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [702, 9], "def_end_pos": [702, 26]}, {"full_name": "MeasureTheory.toOuterMeasure_toMeasure", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [696, 9], "def_end_pos": [696, 33]}]], "state_before": "\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1\u271d : \u03b9 \u2192 Type u_3\ninst\u271d\u00b3 : Fintype \u03b9\nm\u271d\u00b9 : (i : \u03b9) \u2192 OuterMeasure (\u03b1\u271d i)\nm\u271d : (i : \u03b9) \u2192 MeasurableSpace (\u03b1\u271d i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (\u03b1\u271d i)\ninst\u271d\u00b2 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\ninst\u271d\u00b9 : Fintype \u03b9'\n\u03b2 : Type u\nm : MeasurableSpace \u03b2\n\u03bc : Measure \u03b2\n\u03b1 : Type v\ninst\u271d : Unique \u03b1\ne : (\u03b1 \u2192 \u03b2) \u2243\u1d50 \u03b2 := MeasurableEquiv.funUnique \u03b1 \u03b2\nthis : (piPremeasure fun x => \u2191\u03bc) = \u2191\u2191(Measure.map (\u2191(MeasurableEquiv.symm e)) \u03bc)\n\u22a2 MeasurePreserving \u2191e", "state_after": "\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1\u271d : \u03b9 \u2192 Type u_3\ninst\u271d\u00b3 : Fintype \u03b9\nm\u271d\u00b9 : (i : \u03b9) \u2192 OuterMeasure (\u03b1\u271d i)\nm\u271d : (i : \u03b9) \u2192 MeasurableSpace (\u03b1\u271d i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (\u03b1\u271d i)\ninst\u271d\u00b2 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\ninst\u271d\u00b9 : Fintype \u03b9'\n\u03b2 : Type u\nm : MeasurableSpace \u03b2\n\u03bc : Measure \u03b2\n\u03b1 : Type v\ninst\u271d : Unique \u03b1\ne : (\u03b1 \u2192 \u03b2) \u2243\u1d50 \u03b2 := MeasurableEquiv.funUnique \u03b1 \u03b2\nthis : (piPremeasure fun x => \u2191\u03bc) = \u2191\u2191(Measure.map (\u2191(MeasurableEquiv.symm e)) \u03bc)\n\u22a2 MeasurePreserving \u2191(MeasurableEquiv.funUnique \u03b1 \u03b2)"}, {"tactic": "exact (e.symm.measurable.measurePreserving _).symm e.symm", "annotated_tactic": ["exact (e.symm.measurable.measurePreserving _).<a>symm</a> e.symm", [{"full_name": "MeasureTheory.MeasurePreserving.symm", "def_path": "Mathlib/Dynamics/Ergodic/MeasurePreserving.lean", "def_pos": [70, 9], "def_end_pos": [70, 13]}]], "state_before": "\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1\u271d : \u03b9 \u2192 Type u_3\ninst\u271d\u00b3 : Fintype \u03b9\nm\u271d\u00b9 : (i : \u03b9) \u2192 OuterMeasure (\u03b1\u271d i)\nm\u271d : (i : \u03b9) \u2192 MeasurableSpace (\u03b1\u271d i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (\u03b1\u271d i)\ninst\u271d\u00b2 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\ninst\u271d\u00b9 : Fintype \u03b9'\n\u03b2 : Type u\nm : MeasurableSpace \u03b2\n\u03bc : Measure \u03b2\n\u03b1 : Type v\ninst\u271d : Unique \u03b1\ne : (\u03b1 \u2192 \u03b2) \u2243\u1d50 \u03b2 := MeasurableEquiv.funUnique \u03b1 \u03b2\nthis : (piPremeasure fun x => \u2191\u03bc) = \u2191\u2191(Measure.map (\u2191(MeasurableEquiv.symm e)) \u03bc)\n\u22a2 MeasurePreserving \u2191(MeasurableEquiv.funUnique \u03b1 \u03b2)", "state_after": "no goals"}, {"tactic": "ext1 s", "annotated_tactic": ["ext1 s", []], "state_before": "\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1\u271d : \u03b9 \u2192 Type u_3\ninst\u271d\u00b3 : Fintype \u03b9\nm\u271d\u00b9 : (i : \u03b9) \u2192 OuterMeasure (\u03b1\u271d i)\nm\u271d : (i : \u03b9) \u2192 MeasurableSpace (\u03b1\u271d i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (\u03b1\u271d i)\ninst\u271d\u00b2 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\ninst\u271d\u00b9 : Fintype \u03b9'\n\u03b2 : Type u\nm : MeasurableSpace \u03b2\n\u03bc : Measure \u03b2\n\u03b1 : Type v\ninst\u271d : Unique \u03b1\ne : (\u03b1 \u2192 \u03b2) \u2243\u1d50 \u03b2 := MeasurableEquiv.funUnique \u03b1 \u03b2\n\u22a2 (piPremeasure fun x => \u2191\u03bc) = \u2191\u2191(Measure.map (\u2191(MeasurableEquiv.symm e)) \u03bc)", "state_after": "case h\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1\u271d : \u03b9 \u2192 Type u_3\ninst\u271d\u00b3 : Fintype \u03b9\nm\u271d\u00b9 : (i : \u03b9) \u2192 OuterMeasure (\u03b1\u271d i)\nm\u271d : (i : \u03b9) \u2192 MeasurableSpace (\u03b1\u271d i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (\u03b1\u271d i)\ninst\u271d\u00b2 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\ninst\u271d\u00b9 : Fintype \u03b9'\n\u03b2 : Type u\nm : MeasurableSpace \u03b2\n\u03bc : Measure \u03b2\n\u03b1 : Type v\ninst\u271d : Unique \u03b1\ne : (\u03b1 \u2192 \u03b2) \u2243\u1d50 \u03b2 := MeasurableEquiv.funUnique \u03b1 \u03b2\ns : Set (\u03b1 \u2192 \u03b2)\n\u22a2 piPremeasure (fun x => \u2191\u03bc) s = \u2191\u2191(Measure.map (\u2191(MeasurableEquiv.symm e)) \u03bc) s"}, {"tactic": "rw [piPremeasure, Fintype.prod_unique, e.symm.map_apply]", "annotated_tactic": ["rw [<a>piPremeasure</a>, <a>Fintype.prod_unique</a>, e.symm.map_apply]", [{"full_name": "MeasureTheory.piPremeasure", "def_path": "Mathlib/MeasureTheory/Constructions/Pi.lean", "def_pos": [159, 5], "def_end_pos": [159, 17]}, {"full_name": "Fintype.prod_unique", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [2012, 9], "def_end_pos": [2012, 20]}]], "state_before": "case h\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1\u271d : \u03b9 \u2192 Type u_3\ninst\u271d\u00b3 : Fintype \u03b9\nm\u271d\u00b9 : (i : \u03b9) \u2192 OuterMeasure (\u03b1\u271d i)\nm\u271d : (i : \u03b9) \u2192 MeasurableSpace (\u03b1\u271d i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (\u03b1\u271d i)\ninst\u271d\u00b2 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\ninst\u271d\u00b9 : Fintype \u03b9'\n\u03b2 : Type u\nm : MeasurableSpace \u03b2\n\u03bc : Measure \u03b2\n\u03b1 : Type v\ninst\u271d : Unique \u03b1\ne : (\u03b1 \u2192 \u03b2) \u2243\u1d50 \u03b2 := MeasurableEquiv.funUnique \u03b1 \u03b2\ns : Set (\u03b1 \u2192 \u03b2)\n\u22a2 piPremeasure (fun x => \u2191\u03bc) s = \u2191\u2191(Measure.map (\u2191(MeasurableEquiv.symm e)) \u03bc) s", "state_after": "case h\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1\u271d : \u03b9 \u2192 Type u_3\ninst\u271d\u00b3 : Fintype \u03b9\nm\u271d\u00b9 : (i : \u03b9) \u2192 OuterMeasure (\u03b1\u271d i)\nm\u271d : (i : \u03b9) \u2192 MeasurableSpace (\u03b1\u271d i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (\u03b1\u271d i)\ninst\u271d\u00b2 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\ninst\u271d\u00b9 : Fintype \u03b9'\n\u03b2 : Type u\nm : MeasurableSpace \u03b2\n\u03bc : Measure \u03b2\n\u03b1 : Type v\ninst\u271d : Unique \u03b1\ne : (\u03b1 \u2192 \u03b2) \u2243\u1d50 \u03b2 := MeasurableEquiv.funUnique \u03b1 \u03b2\ns : Set (\u03b1 \u2192 \u03b2)\n\u22a2 \u2191\u2191\u03bc (eval default '' s) = \u2191\u2191\u03bc (\u2191(MeasurableEquiv.symm e) \u207b\u00b9' s)"}, {"tactic": "congr 1", "annotated_tactic": ["congr 1", []], "state_before": "case h\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1\u271d : \u03b9 \u2192 Type u_3\ninst\u271d\u00b3 : Fintype \u03b9\nm\u271d\u00b9 : (i : \u03b9) \u2192 OuterMeasure (\u03b1\u271d i)\nm\u271d : (i : \u03b9) \u2192 MeasurableSpace (\u03b1\u271d i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (\u03b1\u271d i)\ninst\u271d\u00b2 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\ninst\u271d\u00b9 : Fintype \u03b9'\n\u03b2 : Type u\nm : MeasurableSpace \u03b2\n\u03bc : Measure \u03b2\n\u03b1 : Type v\ninst\u271d : Unique \u03b1\ne : (\u03b1 \u2192 \u03b2) \u2243\u1d50 \u03b2 := MeasurableEquiv.funUnique \u03b1 \u03b2\ns : Set (\u03b1 \u2192 \u03b2)\n\u22a2 \u2191\u2191\u03bc (eval default '' s) = \u2191\u2191\u03bc (\u2191(MeasurableEquiv.symm e) \u207b\u00b9' s)", "state_after": "case h.e_a\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1\u271d : \u03b9 \u2192 Type u_3\ninst\u271d\u00b3 : Fintype \u03b9\nm\u271d\u00b9 : (i : \u03b9) \u2192 OuterMeasure (\u03b1\u271d i)\nm\u271d : (i : \u03b9) \u2192 MeasurableSpace (\u03b1\u271d i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (\u03b1\u271d i)\ninst\u271d\u00b2 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\ninst\u271d\u00b9 : Fintype \u03b9'\n\u03b2 : Type u\nm : MeasurableSpace \u03b2\n\u03bc : Measure \u03b2\n\u03b1 : Type v\ninst\u271d : Unique \u03b1\ne : (\u03b1 \u2192 \u03b2) \u2243\u1d50 \u03b2 := MeasurableEquiv.funUnique \u03b1 \u03b2\ns : Set (\u03b1 \u2192 \u03b2)\n\u22a2 eval default '' s = \u2191(MeasurableEquiv.symm e) \u207b\u00b9' s"}, {"tactic": "exact e.toEquiv.image_eq_preimage s", "annotated_tactic": ["exact e.toEquiv.image_eq_preimage s", []], "state_before": "case h.e_a\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1\u271d : \u03b9 \u2192 Type u_3\ninst\u271d\u00b3 : Fintype \u03b9\nm\u271d\u00b9 : (i : \u03b9) \u2192 OuterMeasure (\u03b1\u271d i)\nm\u271d : (i : \u03b9) \u2192 MeasurableSpace (\u03b1\u271d i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (\u03b1\u271d i)\ninst\u271d\u00b2 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\ninst\u271d\u00b9 : Fintype \u03b9'\n\u03b2 : Type u\nm : MeasurableSpace \u03b2\n\u03bc : Measure \u03b2\n\u03b1 : Type v\ninst\u271d : Unique \u03b1\ne : (\u03b1 \u2192 \u03b2) \u2243\u1d50 \u03b2 := MeasurableEquiv.funUnique \u03b1 \u03b2\ns : Set (\u03b1 \u2192 \u03b2)\n\u22a2 eval default '' s = \u2191(MeasurableEquiv.symm e) \u207b\u00b9' s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "full_name": "MeasureTheory.L1.SimpleFunc.setToL1S_smul_real", "start": [778, 1], "end": [784, 30], "traced_tactics": [{"tactic": "simp_rw [setToL1S]", "annotated_tactic": ["simp_rw [<a>setToL1S</a>]", [{"full_name": "MeasureTheory.L1.SimpleFunc.setToL1S", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [685, 5], "def_end_pos": [685, 13]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedSpace \ud835\udd5c E\nT : Set \u03b1 \u2192 E \u2192L[\u211d] F\nh_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s = 0 \u2192 T s = 0\nh_add : FinMeasAdditive \u03bc T\nc : \u211d\nf : { x // x \u2208 simpleFunc E 1 \u03bc }\n\u22a2 setToL1S T (c \u2022 f) = c \u2022 setToL1S T f", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedSpace \ud835\udd5c E\nT : Set \u03b1 \u2192 E \u2192L[\u211d] F\nh_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s = 0 \u2192 T s = 0\nh_add : FinMeasAdditive \u03bc T\nc : \u211d\nf : { x // x \u2208 simpleFunc E 1 \u03bc }\n\u22a2 SimpleFunc.setToSimpleFunc T (toSimpleFunc (c \u2022 f)) = c \u2022 SimpleFunc.setToSimpleFunc T (toSimpleFunc f)"}, {"tactic": "rw [\u2190 SimpleFunc.setToSimpleFunc_smul_real T h_add c (SimpleFunc.integrable f)]", "annotated_tactic": ["rw [\u2190 <a>SimpleFunc.setToSimpleFunc_smul_real</a> T h_add c (<a>SimpleFunc.integrable</a> f)]", [{"full_name": "MeasureTheory.SimpleFunc.setToSimpleFunc_smul_real", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [489, 9], "def_end_pos": [489, 34]}, {"full_name": "MeasureTheory.L1.SimpleFunc.integrable", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "def_pos": [1040, 19], "def_end_pos": [1040, 43]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedSpace \ud835\udd5c E\nT : Set \u03b1 \u2192 E \u2192L[\u211d] F\nh_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s = 0 \u2192 T s = 0\nh_add : FinMeasAdditive \u03bc T\nc : \u211d\nf : { x // x \u2208 simpleFunc E 1 \u03bc }\n\u22a2 SimpleFunc.setToSimpleFunc T (toSimpleFunc (c \u2022 f)) = c \u2022 SimpleFunc.setToSimpleFunc T (toSimpleFunc f)", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedSpace \ud835\udd5c E\nT : Set \u03b1 \u2192 E \u2192L[\u211d] F\nh_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s = 0 \u2192 T s = 0\nh_add : FinMeasAdditive \u03bc T\nc : \u211d\nf : { x // x \u2208 simpleFunc E 1 \u03bc }\n\u22a2 SimpleFunc.setToSimpleFunc T (toSimpleFunc (c \u2022 f)) = SimpleFunc.setToSimpleFunc T (c \u2022 toSimpleFunc f)"}, {"tactic": "refine' SimpleFunc.setToSimpleFunc_congr T h_zero h_add (SimpleFunc.integrable _) _", "annotated_tactic": ["refine' <a>SimpleFunc.setToSimpleFunc_congr</a> T h_zero h_add (<a>SimpleFunc.integrable</a> _) _", [{"full_name": "MeasureTheory.SimpleFunc.setToSimpleFunc_congr", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [380, 9], "def_end_pos": [380, 30]}, {"full_name": "MeasureTheory.L1.SimpleFunc.integrable", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "def_pos": [1040, 19], "def_end_pos": [1040, 43]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedSpace \ud835\udd5c E\nT : Set \u03b1 \u2192 E \u2192L[\u211d] F\nh_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s = 0 \u2192 T s = 0\nh_add : FinMeasAdditive \u03bc T\nc : \u211d\nf : { x // x \u2208 simpleFunc E 1 \u03bc }\n\u22a2 SimpleFunc.setToSimpleFunc T (toSimpleFunc (c \u2022 f)) = SimpleFunc.setToSimpleFunc T (c \u2022 toSimpleFunc f)", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedSpace \ud835\udd5c E\nT : Set \u03b1 \u2192 E \u2192L[\u211d] F\nh_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s = 0 \u2192 T s = 0\nh_add : FinMeasAdditive \u03bc T\nc : \u211d\nf : { x // x \u2208 simpleFunc E 1 \u03bc }\n\u22a2 \u2191(toSimpleFunc (c \u2022 f)) =\u1d50[\u03bc] \u2191(c \u2022 toSimpleFunc f)"}, {"tactic": "exact smul_toSimpleFunc c f", "annotated_tactic": ["exact <a>smul_toSimpleFunc</a> c f", [{"full_name": "MeasureTheory.Lp.simpleFunc.smul_toSimpleFunc", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "def_pos": [670, 9], "def_end_pos": [670, 26]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedSpace \ud835\udd5c E\nT : Set \u03b1 \u2192 E \u2192L[\u211d] F\nh_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s = 0 \u2192 T s = 0\nh_add : FinMeasAdditive \u03bc T\nc : \u211d\nf : { x // x \u2208 simpleFunc E 1 \u03bc }\n\u22a2 \u2191(toSimpleFunc (c \u2022 f)) =\u1d50[\u03bc] \u2191(c \u2022 toSimpleFunc f)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Vector/MapLemmas.lean", "full_name": "Vector.mapAccumr_bisim_tail", "start": [185, 1], "end": [190, 36], "traced_tactics": [{"tactic": "rcases h with \u27e8R, h\u2080, hR\u27e9", "annotated_tactic": ["rcases h with \u27e8R, h\u2080, hR\u27e9", []], "state_before": "\u03b1 : Type u_2\nn : \u2115\nxs : Vector \u03b1 n\n\u03c3\u2081 : Type\n\u03b2 : Type u_1\n\u03c3\u2082 : Type\nf\u2081 : \u03b1 \u2192 \u03c3\u2081 \u2192 \u03c3\u2081 \u00d7 \u03b2\nf\u2082 : \u03b1 \u2192 \u03c3\u2082 \u2192 \u03c3\u2082 \u00d7 \u03b2\ns\u2081 : \u03c3\u2081\ns\u2082 : \u03c3\u2082\nh : \u2203 R, R s\u2081 s\u2082 \u2227 \u2200 {s : \u03c3\u2081} {q : \u03c3\u2082} (a : \u03b1), R s q \u2192 R (f\u2081 a s).1 (f\u2082 a q).1 \u2227 (f\u2081 a s).2 = (f\u2082 a q).2\n\u22a2 (mapAccumr f\u2081 xs s\u2081).2 = (mapAccumr f\u2082 xs s\u2082).2", "state_after": "case intro.intro\n\u03b1 : Type u_2\nn : \u2115\nxs : Vector \u03b1 n\n\u03c3\u2081 : Type\n\u03b2 : Type u_1\n\u03c3\u2082 : Type\nf\u2081 : \u03b1 \u2192 \u03c3\u2081 \u2192 \u03c3\u2081 \u00d7 \u03b2\nf\u2082 : \u03b1 \u2192 \u03c3\u2082 \u2192 \u03c3\u2082 \u00d7 \u03b2\ns\u2081 : \u03c3\u2081\ns\u2082 : \u03c3\u2082\nR : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop\nh\u2080 : R s\u2081 s\u2082\nhR : \u2200 {s : \u03c3\u2081} {q : \u03c3\u2082} (a : \u03b1), R s q \u2192 R (f\u2081 a s).1 (f\u2082 a q).1 \u2227 (f\u2081 a s).2 = (f\u2082 a q).2\n\u22a2 (mapAccumr f\u2081 xs s\u2081).2 = (mapAccumr f\u2082 xs s\u2082).2"}, {"tactic": "exact (mapAccumr_bisim R h\u2080 hR).2", "annotated_tactic": ["exact (<a>mapAccumr_bisim</a> R h\u2080 hR).2", [{"full_name": "Vector.mapAccumr_bisim", "def_path": "Mathlib/Data/Vector/MapLemmas.lean", "def_pos": [173, 9], "def_end_pos": [173, 24]}]], "state_before": "case intro.intro\n\u03b1 : Type u_2\nn : \u2115\nxs : Vector \u03b1 n\n\u03c3\u2081 : Type\n\u03b2 : Type u_1\n\u03c3\u2082 : Type\nf\u2081 : \u03b1 \u2192 \u03c3\u2081 \u2192 \u03c3\u2081 \u00d7 \u03b2\nf\u2082 : \u03b1 \u2192 \u03c3\u2082 \u2192 \u03c3\u2082 \u00d7 \u03b2\ns\u2081 : \u03c3\u2081\ns\u2082 : \u03c3\u2082\nR : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop\nh\u2080 : R s\u2081 s\u2082\nhR : \u2200 {s : \u03c3\u2081} {q : \u03c3\u2082} (a : \u03b1), R s q \u2192 R (f\u2081 a s).1 (f\u2082 a q).1 \u2227 (f\u2081 a s).2 = (f\u2082 a q).2\n\u22a2 (mapAccumr f\u2081 xs s\u2081).2 = (mapAccumr f\u2082 xs s\u2082).2", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "full_name": "BoundedContinuousFunction.toLp_inj", "start": [1805, 1], "end": [1810, 9], "traced_tactics": [{"tactic": "refine' \u27e8fun h => _, by tauto\u27e9", "annotated_tactic": ["refine' \u27e8fun h => _, by tauto\u27e9", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedAddCommGroup G\ninst\u271d\u2077 : TopologicalSpace \u03b1\ninst\u271d\u2076 : BorelSpace \u03b1\ninst\u271d\u2075 : SecondCountableTopologyEither \u03b1 E\ninst\u271d\u2074 : IsFiniteMeasure \u03bc\n\ud835\udd5c : Type u_5\ninst\u271d\u00b3 : Fact (1 \u2264 p)\nf g : \u03b1 \u2192\u1d47 E\ninst\u271d\u00b2 : Measure.IsOpenPosMeasure \u03bc\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedSpace \ud835\udd5c E\n\u22a2 \u2191(toLp p \u03bc \ud835\udd5c) f = \u2191(toLp p \u03bc \ud835\udd5c) g \u2194 f = g", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedAddCommGroup G\ninst\u271d\u2077 : TopologicalSpace \u03b1\ninst\u271d\u2076 : BorelSpace \u03b1\ninst\u271d\u2075 : SecondCountableTopologyEither \u03b1 E\ninst\u271d\u2074 : IsFiniteMeasure \u03bc\n\ud835\udd5c : Type u_5\ninst\u271d\u00b3 : Fact (1 \u2264 p)\nf g : \u03b1 \u2192\u1d47 E\ninst\u271d\u00b2 : Measure.IsOpenPosMeasure \u03bc\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedSpace \ud835\udd5c E\nh : \u2191(toLp p \u03bc \ud835\udd5c) f = \u2191(toLp p \u03bc \ud835\udd5c) g\n\u22a2 f = g"}, {"tactic": "rw [\u2190 FunLike.coe_fn_eq, \u2190 (map_continuous f).ae_eq_iff_eq \u03bc (map_continuous g)]", "annotated_tactic": ["rw [\u2190 <a>FunLike.coe_fn_eq</a>, \u2190 (<a>map_continuous</a> f).<a>ae_eq_iff_eq</a> \u03bc (<a>map_continuous</a> g)]", [{"full_name": "FunLike.coe_fn_eq", "def_path": "Mathlib/Data/FunLike/Basic.lean", "def_pos": [165, 9], "def_end_pos": [165, 18]}, {"full_name": "ContinuousMapClass.map_continuous", "def_path": "Mathlib/Topology/ContinuousFunction/Basic.lean", "def_pos": [47, 3], "def_end_pos": [47, 17]}, {"full_name": "Continuous.ae_eq_iff_eq", "def_path": "Mathlib/MeasureTheory/Measure/OpenPos.lean", "def_pos": [147, 9], "def_end_pos": [147, 39]}, {"full_name": "ContinuousMapClass.map_continuous", "def_path": "Mathlib/Topology/ContinuousFunction/Basic.lean", "def_pos": [47, 3], "def_end_pos": [47, 17]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedAddCommGroup G\ninst\u271d\u2077 : TopologicalSpace \u03b1\ninst\u271d\u2076 : BorelSpace \u03b1\ninst\u271d\u2075 : SecondCountableTopologyEither \u03b1 E\ninst\u271d\u2074 : IsFiniteMeasure \u03bc\n\ud835\udd5c : Type u_5\ninst\u271d\u00b3 : Fact (1 \u2264 p)\nf g : \u03b1 \u2192\u1d47 E\ninst\u271d\u00b2 : Measure.IsOpenPosMeasure \u03bc\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedSpace \ud835\udd5c E\nh : \u2191(toLp p \u03bc \ud835\udd5c) f = \u2191(toLp p \u03bc \ud835\udd5c) g\n\u22a2 f = g", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedAddCommGroup G\ninst\u271d\u2077 : TopologicalSpace \u03b1\ninst\u271d\u2076 : BorelSpace \u03b1\ninst\u271d\u2075 : SecondCountableTopologyEither \u03b1 E\ninst\u271d\u2074 : IsFiniteMeasure \u03bc\n\ud835\udd5c : Type u_5\ninst\u271d\u00b3 : Fact (1 \u2264 p)\nf g : \u03b1 \u2192\u1d47 E\ninst\u271d\u00b2 : Measure.IsOpenPosMeasure \u03bc\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedSpace \ud835\udd5c E\nh : \u2191(toLp p \u03bc \ud835\udd5c) f = \u2191(toLp p \u03bc \ud835\udd5c) g\n\u22a2 \u2191f =\u1d50[\u03bc] \u2191g"}, {"tactic": "refine' (coeFn_toLp p \u03bc \ud835\udd5c f).symm.trans (EventuallyEq.trans _ <| coeFn_toLp p \u03bc \ud835\udd5c g)", "annotated_tactic": ["refine' (<a>coeFn_toLp</a> p \u03bc \ud835\udd5c f).symm.trans (<a>EventuallyEq.trans</a> _ <| <a>coeFn_toLp</a> p \u03bc \ud835\udd5c g)", [{"full_name": "BoundedContinuousFunction.coeFn_toLp", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [1785, 9], "def_end_pos": [1785, 19]}, {"full_name": "Filter.EventuallyEq.trans", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1503, 9], "def_end_pos": [1503, 27]}, {"full_name": "BoundedContinuousFunction.coeFn_toLp", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [1785, 9], "def_end_pos": [1785, 19]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedAddCommGroup G\ninst\u271d\u2077 : TopologicalSpace \u03b1\ninst\u271d\u2076 : BorelSpace \u03b1\ninst\u271d\u2075 : SecondCountableTopologyEither \u03b1 E\ninst\u271d\u2074 : IsFiniteMeasure \u03bc\n\ud835\udd5c : Type u_5\ninst\u271d\u00b3 : Fact (1 \u2264 p)\nf g : \u03b1 \u2192\u1d47 E\ninst\u271d\u00b2 : Measure.IsOpenPosMeasure \u03bc\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedSpace \ud835\udd5c E\nh : \u2191(toLp p \u03bc \ud835\udd5c) f = \u2191(toLp p \u03bc \ud835\udd5c) g\n\u22a2 \u2191f =\u1d50[\u03bc] \u2191g", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedAddCommGroup G\ninst\u271d\u2077 : TopologicalSpace \u03b1\ninst\u271d\u2076 : BorelSpace \u03b1\ninst\u271d\u2075 : SecondCountableTopologyEither \u03b1 E\ninst\u271d\u2074 : IsFiniteMeasure \u03bc\n\ud835\udd5c : Type u_5\ninst\u271d\u00b3 : Fact (1 \u2264 p)\nf g : \u03b1 \u2192\u1d47 E\ninst\u271d\u00b2 : Measure.IsOpenPosMeasure \u03bc\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedSpace \ud835\udd5c E\nh : \u2191(toLp p \u03bc \ud835\udd5c) f = \u2191(toLp p \u03bc \ud835\udd5c) g\n\u22a2 \u2191\u2191(\u2191(toLp p \u03bc \ud835\udd5c) f) =\u1d50[\u03bc] \u2191\u2191(\u2191(toLp p \u03bc \ud835\udd5c) g)"}, {"tactic": "rw [h]", "annotated_tactic": ["rw [h]", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedAddCommGroup G\ninst\u271d\u2077 : TopologicalSpace \u03b1\ninst\u271d\u2076 : BorelSpace \u03b1\ninst\u271d\u2075 : SecondCountableTopologyEither \u03b1 E\ninst\u271d\u2074 : IsFiniteMeasure \u03bc\n\ud835\udd5c : Type u_5\ninst\u271d\u00b3 : Fact (1 \u2264 p)\nf g : \u03b1 \u2192\u1d47 E\ninst\u271d\u00b2 : Measure.IsOpenPosMeasure \u03bc\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedSpace \ud835\udd5c E\nh : \u2191(toLp p \u03bc \ud835\udd5c) f = \u2191(toLp p \u03bc \ud835\udd5c) g\n\u22a2 \u2191\u2191(\u2191(toLp p \u03bc \ud835\udd5c) f) =\u1d50[\u03bc] \u2191\u2191(\u2191(toLp p \u03bc \ud835\udd5c) g)", "state_after": "no goals"}, {"tactic": "tauto", "annotated_tactic": ["tauto", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedAddCommGroup G\ninst\u271d\u2077 : TopologicalSpace \u03b1\ninst\u271d\u2076 : BorelSpace \u03b1\ninst\u271d\u2075 : SecondCountableTopologyEither \u03b1 E\ninst\u271d\u2074 : IsFiniteMeasure \u03bc\n\ud835\udd5c : Type u_5\ninst\u271d\u00b3 : Fact (1 \u2264 p)\nf g : \u03b1 \u2192\u1d47 E\ninst\u271d\u00b2 : Measure.IsOpenPosMeasure \u03bc\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedSpace \ud835\udd5c E\n\u22a2 f = g \u2192 \u2191(toLp p \u03bc \ud835\udd5c) f = \u2191(toLp p \u03bc \ud835\udd5c) g", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/MulAntidiagonal.lean", "full_name": "Finset.mulAntidiagonal_mono_left", "start": [79, 1], "end": [80, 62], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/PeakFunction.lean", "full_name": "tendsto_set_integral_peak_smul_of_integrableOn_of_continuousWithinAt", "start": [160, 1], "end": [183, 68], "traced_tactics": [{"tactic": "let h := g - fun _ => g x\u2080", "annotated_tactic": ["let h := g - fun _ => g x\u2080", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d : CompleteSpace E\nhs : MeasurableSet s\nh's : \u2191\u2191\u03bc s \u2260 \u22a4\nhn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : (fun i => \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) =\u1da0[l] 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u22a2 Tendsto (fun i => \u222b (x : \u03b1) in s, \u03c6 i x \u2022 g x \u2202\u03bc) l (\ud835\udcdd (g x\u2080))", "state_after": "\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d : CompleteSpace E\nhs : MeasurableSet s\nh's : \u2191\u2191\u03bc s \u2260 \u22a4\nhn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : (fun i => \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) =\u1da0[l] 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nh : \u03b1 \u2192 E := g - fun x => g x\u2080\n\u22a2 Tendsto (fun i => \u222b (x : \u03b1) in s, \u03c6 i x \u2022 g x \u2202\u03bc) l (\ud835\udcdd (g x\u2080))"}, {"tactic": "simp only [one_smul, zero_add] at A", "annotated_tactic": ["simp only [<a>one_smul</a>, <a>zero_add</a>] at A", [{"full_name": "one_smul", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [492, 9], "def_end_pos": [492, 17]}, {"full_name": "zero_add", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [463, 3], "def_end_pos": [463, 14]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d : CompleteSpace E\nhs : MeasurableSet s\nh's : \u2191\u2191\u03bc s \u2260 \u22a4\nhn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : (fun i => \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) =\u1da0[l] 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nh : \u03b1 \u2192 E := g - fun x => g x\u2080\nA : Tendsto (fun i => \u222b (x : \u03b1) in s, \u03c6 i x \u2022 h x \u2202\u03bc + (\u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) \u2022 g x\u2080) l (\ud835\udcdd (0 + 1 \u2022 g x\u2080))\n\u22a2 Tendsto (fun i => \u222b (x : \u03b1) in s, \u03c6 i x \u2022 g x \u2202\u03bc) l (\ud835\udcdd (g x\u2080))", "state_after": "\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d : CompleteSpace E\nhs : MeasurableSet s\nh's : \u2191\u2191\u03bc s \u2260 \u22a4\nhn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : (fun i => \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) =\u1da0[l] 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nh : \u03b1 \u2192 E := g - fun x => g x\u2080\nA : Tendsto (fun i => \u222b (x : \u03b1) in s, \u03c6 i x \u2022 (g - fun x => g x\u2080) x \u2202\u03bc + (\u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) \u2022 g x\u2080) l (\ud835\udcdd (g x\u2080))\n\u22a2 Tendsto (fun i => \u222b (x : \u03b1) in s, \u03c6 i x \u2022 g x \u2202\u03bc) l (\ud835\udcdd (g x\u2080))"}, {"tactic": "refine' Tendsto.congr' _ A", "annotated_tactic": ["refine' <a>Tendsto.congr'</a> _ A", [{"full_name": "Filter.Tendsto.congr'", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [3009, 9], "def_end_pos": [3009, 23]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d : CompleteSpace E\nhs : MeasurableSet s\nh's : \u2191\u2191\u03bc s \u2260 \u22a4\nhn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : (fun i => \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) =\u1da0[l] 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nh : \u03b1 \u2192 E := g - fun x => g x\u2080\nA : Tendsto (fun i => \u222b (x : \u03b1) in s, \u03c6 i x \u2022 (g - fun x => g x\u2080) x \u2202\u03bc + (\u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) \u2022 g x\u2080) l (\ud835\udcdd (g x\u2080))\n\u22a2 Tendsto (fun i => \u222b (x : \u03b1) in s, \u03c6 i x \u2022 g x \u2202\u03bc) l (\ud835\udcdd (g x\u2080))", "state_after": "\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d : CompleteSpace E\nhs : MeasurableSet s\nh's : \u2191\u2191\u03bc s \u2260 \u22a4\nhn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : (fun i => \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) =\u1da0[l] 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nh : \u03b1 \u2192 E := g - fun x => g x\u2080\nA : Tendsto (fun i => \u222b (x : \u03b1) in s, \u03c6 i x \u2022 (g - fun x => g x\u2080) x \u2202\u03bc + (\u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) \u2022 g x\u2080) l (\ud835\udcdd (g x\u2080))\n\u22a2 (fun i => \u222b (x : \u03b1) in s, \u03c6 i x \u2022 (g - fun x => g x\u2080) x \u2202\u03bc + (\u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) \u2022 g x\u2080) =\u1da0[l] fun i =>\n    \u222b (x : \u03b1) in s, \u03c6 i x \u2022 g x \u2202\u03bc"}, {"tactic": "filter_upwards [integrableOn_peak_smul_of_integrableOn_of_continuousWithinAt hs hl\u03c6 hi\u03c6 hmg hcg,\n  hi\u03c6] with i hi h'i", "annotated_tactic": ["filter_upwards [<a>integrableOn_peak_smul_of_integrableOn_of_continuousWithinAt</a> hs hl\u03c6 hi\u03c6 hmg hcg,\n    hi\u03c6] with i hi h'i", [{"full_name": "integrableOn_peak_smul_of_integrableOn_of_continuousWithinAt", "def_path": "Mathlib/MeasureTheory/Integral/PeakFunction.lean", "def_pos": [54, 9], "def_end_pos": [54, 69]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d : CompleteSpace E\nhs : MeasurableSet s\nh's : \u2191\u2191\u03bc s \u2260 \u22a4\nhn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : (fun i => \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) =\u1da0[l] 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nh : \u03b1 \u2192 E := g - fun x => g x\u2080\nA : Tendsto (fun i => \u222b (x : \u03b1) in s, \u03c6 i x \u2022 (g - fun x => g x\u2080) x \u2202\u03bc + (\u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) \u2022 g x\u2080) l (\ud835\udcdd (g x\u2080))\n\u22a2 (fun i => \u222b (x : \u03b1) in s, \u03c6 i x \u2022 (g - fun x => g x\u2080) x \u2202\u03bc + (\u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) \u2022 g x\u2080) =\u1da0[l] fun i =>\n    \u222b (x : \u03b1) in s, \u03c6 i x \u2022 g x \u2202\u03bc", "state_after": "case h\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d : CompleteSpace E\nhs : MeasurableSet s\nh's : \u2191\u2191\u03bc s \u2260 \u22a4\nhn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : (fun i => \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) =\u1da0[l] 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nh : \u03b1 \u2192 E := g - fun x => g x\u2080\nA : Tendsto (fun i => \u222b (x : \u03b1) in s, \u03c6 i x \u2022 (g - fun x => g x\u2080) x \u2202\u03bc + (\u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) \u2022 g x\u2080) l (\ud835\udcdd (g x\u2080))\ni : \u03b9\nhi : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = OfNat.ofNat 1 i\n\u22a2 \u222b (x : \u03b1) in s, \u03c6 i x \u2022 (g - fun x => g x\u2080) x \u2202\u03bc + (\u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) \u2022 g x\u2080 = \u222b (x : \u03b1) in s, \u03c6 i x \u2022 g x \u2202\u03bc"}, {"tactic": "simp only [Pi.sub_apply, smul_sub]", "annotated_tactic": ["simp only [<a>Pi.sub_apply</a>, <a>smul_sub</a>]", [{"full_name": "Pi.sub_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [200, 3], "def_end_pos": [200, 14]}, {"full_name": "smul_sub", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [988, 9], "def_end_pos": [988, 17]}]], "state_before": "case h\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d : CompleteSpace E\nhs : MeasurableSet s\nh's : \u2191\u2191\u03bc s \u2260 \u22a4\nhn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : (fun i => \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) =\u1da0[l] 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nh : \u03b1 \u2192 E := g - fun x => g x\u2080\nA : Tendsto (fun i => \u222b (x : \u03b1) in s, \u03c6 i x \u2022 (g - fun x => g x\u2080) x \u2202\u03bc + (\u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) \u2022 g x\u2080) l (\ud835\udcdd (g x\u2080))\ni : \u03b9\nhi : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = OfNat.ofNat 1 i\n\u22a2 \u222b (x : \u03b1) in s, \u03c6 i x \u2022 (g - fun x => g x\u2080) x \u2202\u03bc + (\u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) \u2022 g x\u2080 = \u222b (x : \u03b1) in s, \u03c6 i x \u2022 g x \u2202\u03bc", "state_after": "case h\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d : CompleteSpace E\nhs : MeasurableSet s\nh's : \u2191\u2191\u03bc s \u2260 \u22a4\nhn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : (fun i => \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) =\u1da0[l] 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nh : \u03b1 \u2192 E := g - fun x => g x\u2080\nA : Tendsto (fun i => \u222b (x : \u03b1) in s, \u03c6 i x \u2022 (g - fun x => g x\u2080) x \u2202\u03bc + (\u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) \u2022 g x\u2080) l (\ud835\udcdd (g x\u2080))\ni : \u03b9\nhi : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = OfNat.ofNat 1 i\n\u22a2 \u222b (x : \u03b1) in s, \u03c6 i x \u2022 g x - \u03c6 i x \u2022 g x\u2080 \u2202\u03bc + (\u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) \u2022 g x\u2080 = \u222b (x : \u03b1) in s, \u03c6 i x \u2022 g x \u2202\u03bc"}, {"tactic": "rw [integral_sub hi, integral_smul_const, sub_add_cancel]", "annotated_tactic": ["rw [<a>integral_sub</a> hi, <a>integral_smul_const</a>, <a>sub_add_cancel</a>]", [{"full_name": "MeasureTheory.integral_sub", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [901, 9], "def_end_pos": [901, 21]}, {"full_name": "integral_smul_const", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [1257, 9], "def_end_pos": [1257, 28]}, {"full_name": "sub_add_cancel", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [728, 30], "def_end_pos": [728, 44]}]], "state_before": "case h\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d : CompleteSpace E\nhs : MeasurableSet s\nh's : \u2191\u2191\u03bc s \u2260 \u22a4\nhn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : (fun i => \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) =\u1da0[l] 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nh : \u03b1 \u2192 E := g - fun x => g x\u2080\nA : Tendsto (fun i => \u222b (x : \u03b1) in s, \u03c6 i x \u2022 (g - fun x => g x\u2080) x \u2202\u03bc + (\u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) \u2022 g x\u2080) l (\ud835\udcdd (g x\u2080))\ni : \u03b9\nhi : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = OfNat.ofNat 1 i\n\u22a2 \u222b (x : \u03b1) in s, \u03c6 i x \u2022 g x - \u03c6 i x \u2022 g x\u2080 \u2202\u03bc + (\u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) \u2022 g x\u2080 = \u222b (x : \u03b1) in s, \u03c6 i x \u2022 g x \u2202\u03bc", "state_after": "case h\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d : CompleteSpace E\nhs : MeasurableSet s\nh's : \u2191\u2191\u03bc s \u2260 \u22a4\nhn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : (fun i => \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) =\u1da0[l] 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nh : \u03b1 \u2192 E := g - fun x => g x\u2080\nA : Tendsto (fun i => \u222b (x : \u03b1) in s, \u03c6 i x \u2022 (g - fun x => g x\u2080) x \u2202\u03bc + (\u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) \u2022 g x\u2080) l (\ud835\udcdd (g x\u2080))\ni : \u03b9\nhi : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = OfNat.ofNat 1 i\n\u22a2 Integrable fun x => \u03c6 i x \u2022 g x\u2080"}, {"tactic": "exact Integrable.smul_const (integrable_of_integral_eq_one h'i) _", "annotated_tactic": ["exact <a>Integrable.smul_const</a> (<a>integrable_of_integral_eq_one</a> h'i) _", [{"full_name": "MeasureTheory.Integrable.smul_const", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [1100, 9], "def_end_pos": [1100, 30]}, {"full_name": "MeasureTheory.integrable_of_integral_eq_one", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [864, 9], "def_end_pos": [864, 38]}]], "state_before": "case h\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d : CompleteSpace E\nhs : MeasurableSet s\nh's : \u2191\u2191\u03bc s \u2260 \u22a4\nhn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : (fun i => \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) =\u1da0[l] 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nh : \u03b1 \u2192 E := g - fun x => g x\u2080\nA : Tendsto (fun i => \u222b (x : \u03b1) in s, \u03c6 i x \u2022 (g - fun x => g x\u2080) x \u2202\u03bc + (\u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) \u2022 g x\u2080) l (\ud835\udcdd (g x\u2080))\ni : \u03b9\nhi : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = OfNat.ofNat 1 i\n\u22a2 Integrable fun x => \u03c6 i x \u2022 g x\u2080", "state_after": "no goals"}, {"tactic": "refine' Tendsto.add _ (Tendsto.smul (tendsto_const_nhds.congr' hi\u03c6.symm) tendsto_const_nhds)", "annotated_tactic": ["refine' <a>Tendsto.add</a> _ (<a>Tendsto.smul</a> (tendsto_const_nhds.congr' hi\u03c6.symm) <a>tendsto_const_nhds</a>)", [{"full_name": "Filter.Tendsto.add", "def_path": "Mathlib/Topology/Algebra/Monoid.lean", "def_pos": [118, 3], "def_end_pos": [118, 14]}, {"full_name": "Filter.Tendsto.smul", "def_path": "Mathlib/Topology/Algebra/MulAction.lean", "def_pos": [83, 9], "def_end_pos": [83, 28]}, {"full_name": "tendsto_const_nhds", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [1049, 9], "def_end_pos": [1049, 27]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d : CompleteSpace E\nhs : MeasurableSet s\nh's : \u2191\u2191\u03bc s \u2260 \u22a4\nhn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : (fun i => \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) =\u1da0[l] 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nh : \u03b1 \u2192 E := g - fun x => g x\u2080\n\u22a2 Tendsto (fun i => \u222b (x : \u03b1) in s, \u03c6 i x \u2022 h x \u2202\u03bc + (\u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) \u2022 g x\u2080) l (\ud835\udcdd (0 + 1 \u2022 g x\u2080))", "state_after": "\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d : CompleteSpace E\nhs : MeasurableSet s\nh's : \u2191\u2191\u03bc s \u2260 \u22a4\nhn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : (fun i => \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) =\u1da0[l] 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nh : \u03b1 \u2192 E := g - fun x => g x\u2080\n\u22a2 Tendsto (fun i => \u222b (x : \u03b1) in s, \u03c6 i x \u2022 h x \u2202\u03bc) l (\ud835\udcdd 0)"}, {"tactic": "apply tendsto_set_integral_peak_smul_of_integrableOn_of_continuousWithinAt_aux hs hn\u03c6 hl\u03c6 hi\u03c6", "annotated_tactic": ["apply <a>tendsto_set_integral_peak_smul_of_integrableOn_of_continuousWithinAt_aux</a> hs hn\u03c6 hl\u03c6 hi\u03c6", [{"full_name": "tendsto_set_integral_peak_smul_of_integrableOn_of_continuousWithinAt_aux", "def_path": "Mathlib/MeasureTheory/Integral/PeakFunction.lean", "def_pos": [85, 9], "def_end_pos": [85, 81]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d : CompleteSpace E\nhs : MeasurableSet s\nh's : \u2191\u2191\u03bc s \u2260 \u22a4\nhn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : (fun i => \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) =\u1da0[l] 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nh : \u03b1 \u2192 E := g - fun x => g x\u2080\n\u22a2 Tendsto (fun i => \u222b (x : \u03b1) in s, \u03c6 i x \u2022 h x \u2202\u03bc) l (\ud835\udcdd 0)", "state_after": "case hmg\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d : CompleteSpace E\nhs : MeasurableSet s\nh's : \u2191\u2191\u03bc s \u2260 \u22a4\nhn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : (fun i => \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) =\u1da0[l] 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nh : \u03b1 \u2192 E := g - fun x => g x\u2080\n\u22a2 IntegrableOn (fun x => h x) s\n\ncase h'g\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d : CompleteSpace E\nhs : MeasurableSet s\nh's : \u2191\u2191\u03bc s \u2260 \u22a4\nhn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : (fun i => \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) =\u1da0[l] 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nh : \u03b1 \u2192 E := g - fun x => g x\u2080\n\u22a2 h x\u2080 = 0\n\ncase hcg\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d : CompleteSpace E\nhs : MeasurableSet s\nh's : \u2191\u2191\u03bc s \u2260 \u22a4\nhn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : (fun i => \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) =\u1da0[l] 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nh : \u03b1 \u2192 E := g - fun x => g x\u2080\n\u22a2 ContinuousWithinAt (fun x => h x) s x\u2080"}, {"tactic": "apply Integrable.sub hmg", "annotated_tactic": ["apply <a>Integrable.sub</a> hmg", [{"full_name": "MeasureTheory.Integrable.sub", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [702, 9], "def_end_pos": [702, 23]}]], "state_before": "case hmg\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d : CompleteSpace E\nhs : MeasurableSet s\nh's : \u2191\u2191\u03bc s \u2260 \u22a4\nhn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : (fun i => \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) =\u1da0[l] 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nh : \u03b1 \u2192 E := g - fun x => g x\u2080\n\u22a2 IntegrableOn (fun x => h x) s", "state_after": "case hmg\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d : CompleteSpace E\nhs : MeasurableSet s\nh's : \u2191\u2191\u03bc s \u2260 \u22a4\nhn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : (fun i => \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) =\u1da0[l] 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nh : \u03b1 \u2192 E := g - fun x => g x\u2080\n\u22a2 Integrable fun x => g x\u2080"}, {"tactic": "apply integrableOn_const.2", "annotated_tactic": ["apply <a>integrableOn_const</a>.2", [{"full_name": "MeasureTheory.integrableOn_const", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [119, 9], "def_end_pos": [119, 27]}]], "state_before": "case hmg\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d : CompleteSpace E\nhs : MeasurableSet s\nh's : \u2191\u2191\u03bc s \u2260 \u22a4\nhn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : (fun i => \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) =\u1da0[l] 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nh : \u03b1 \u2192 E := g - fun x => g x\u2080\n\u22a2 Integrable fun x => g x\u2080", "state_after": "case hmg\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d : CompleteSpace E\nhs : MeasurableSet s\nh's : \u2191\u2191\u03bc s \u2260 \u22a4\nhn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : (fun i => \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) =\u1da0[l] 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nh : \u03b1 \u2192 E := g - fun x => g x\u2080\n\u22a2 g x\u2080 = 0 \u2228 \u2191\u2191\u03bc s < \u22a4"}, {"tactic": "simp only [h's.lt_top, or_true_iff]", "annotated_tactic": ["simp only [h's.lt_top, <a>or_true_iff</a>]", [{"full_name": "or_true_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [184, 9], "def_end_pos": [184, 20]}]], "state_before": "case hmg\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d : CompleteSpace E\nhs : MeasurableSet s\nh's : \u2191\u2191\u03bc s \u2260 \u22a4\nhn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : (fun i => \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) =\u1da0[l] 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nh : \u03b1 \u2192 E := g - fun x => g x\u2080\n\u22a2 g x\u2080 = 0 \u2228 \u2191\u2191\u03bc s < \u22a4", "state_after": "no goals"}, {"tactic": "simp only [Pi.sub_apply, sub_self]", "annotated_tactic": ["simp only [<a>Pi.sub_apply</a>, <a>sub_self</a>]", [{"full_name": "Pi.sub_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [200, 3], "def_end_pos": [200, 14]}, {"full_name": "sub_self", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [734, 30], "def_end_pos": [734, 38]}]], "state_before": "case h'g\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d : CompleteSpace E\nhs : MeasurableSet s\nh's : \u2191\u2191\u03bc s \u2260 \u22a4\nhn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : (fun i => \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) =\u1da0[l] 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nh : \u03b1 \u2192 E := g - fun x => g x\u2080\n\u22a2 h x\u2080 = 0", "state_after": "no goals"}, {"tactic": "exact hcg.sub continuousWithinAt_const", "annotated_tactic": ["exact hcg.sub <a>continuousWithinAt_const</a>", [{"full_name": "continuousWithinAt_const", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [1029, 9], "def_end_pos": [1029, 33]}]], "state_before": "case hcg\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d : CompleteSpace E\nhs : MeasurableSet s\nh's : \u2191\u2191\u03bc s \u2260 \u22a4\nhn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : (fun i => \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) =\u1da0[l] 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nh : \u03b1 \u2192 E := g - fun x => g x\u2080\n\u22a2 ContinuousWithinAt (fun x => h x) s x\u2080", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "Finset.piecewise_idem_left", "start": [2563, 1], "end": [2565, 59], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "full_name": "MeasureTheory.L1.SimpleFunc.setToL1SCLM_zero_left'", "start": [895, 1], "end": [898, 31], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Process/Adapted.lean", "full_name": "MeasureTheory.Adapted.progMeasurable_of_continuous", "start": [207, 1], "end": [212, 99], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/Variables.lean", "full_name": "MvPolynomial.degreeOf_mul_X_ne", "start": [552, 1], "end": [561, 87], "traced_tactics": [{"tactic": "classical\nrepeat' rw [degreeOf_eq_sup (R := R) i]\nrw [support_mul_X]\nsimp only [Finset.sup_map]\ncongr\next\nsimp only [Finsupp.single, Nat.one_ne_zero, add_right_eq_self, addRightEmbedding_apply, coe_mk,\n  Pi.add_apply, comp_apply, ite_eq_right_iff, Finsupp.coe_add, Pi.single_eq_of_ne h]", "annotated_tactic": ["classical\n  repeat' rw [<a>degreeOf_eq_sup</a> (R := R) i]\n  rw [<a>support_mul_X</a>]\n  simp only [<a>Finset.sup_map</a>]\n  congr\n  ext\n  simp only [<a>Finsupp.single</a>, <a>Nat.one_ne_zero</a>, <a>add_right_eq_self</a>, <a>addRightEmbedding_apply</a>, <a>coe_mk</a>,\n    <a>Pi.add_apply</a>, <a>comp_apply</a>, <a>ite_eq_right_iff</a>, <a>Finsupp.coe_add</a>, <a>Pi.single_eq_of_ne</a> h]", [{"full_name": "MvPolynomial.degreeOf_eq_sup", "def_path": "Mathlib/Data/MvPolynomial/Variables.lean", "def_pos": [494, 9], "def_end_pos": [494, 24]}, {"full_name": "MvPolynomial.support_mul_X", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [739, 9], "def_end_pos": [739, 22]}, {"full_name": "Finset.sup_map", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [69, 9], "def_end_pos": [69, 16]}, {"full_name": "Finsupp.single", "def_path": "Mathlib/Data/Finsupp/Defs.lean", "def_pos": [289, 5], "def_end_pos": [289, 11]}, {"full_name": "Nat.one_ne_zero", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [426, 19], "def_end_pos": [426, 30]}, {"full_name": "add_right_eq_self", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [180, 3], "def_end_pos": [180, 14]}, {"full_name": "addRightEmbedding_apply", "def_path": "Mathlib/Algebra/Hom/Embedding.lean", "def_pos": [37, 24], "def_end_pos": [37, 29]}, {"full_name": "Finsupp.coe_mk", "def_path": "Mathlib/Data/Finsupp/Defs.lean", "def_pos": [159, 9], "def_end_pos": [159, 15]}, {"full_name": "Pi.add_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [82, 3], "def_end_pos": [82, 14]}, {"full_name": "Function.comp_apply", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [33, 17], "def_end_pos": [33, 36]}, {"full_name": "ite_eq_right_iff", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [1162, 17], "def_end_pos": [1162, 33]}, {"full_name": "Finsupp.coe_add", "def_path": "Mathlib/Data/Finsupp/Defs.lean", "def_pos": [972, 9], "def_end_pos": [972, 16]}, {"full_name": "Pi.single_eq_of_ne", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [243, 3], "def_end_pos": [243, 14]}]], "state_before": "R : Type u\nS : Type v\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nr : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\np q : MvPolynomial \u03c3 R\ni j : \u03c3\nf : MvPolynomial \u03c3 R\nh : i \u2260 j\n\u22a2 degreeOf i (f * X j) = degreeOf i f", "state_after": "no goals"}, {"tactic": "repeat' rw [degreeOf_eq_sup (R := R) i]", "annotated_tactic": ["repeat' rw [<a>degreeOf_eq_sup</a> (R := R) i]", [{"full_name": "MvPolynomial.degreeOf_eq_sup", "def_path": "Mathlib/Data/MvPolynomial/Variables.lean", "def_pos": [494, 9], "def_end_pos": [494, 24]}]], "state_before": "R : Type u\nS : Type v\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nr : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\np q : MvPolynomial \u03c3 R\ni j : \u03c3\nf : MvPolynomial \u03c3 R\nh : i \u2260 j\n\u22a2 degreeOf i (f * X j) = degreeOf i f", "state_after": "R : Type u\nS : Type v\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nr : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\np q : MvPolynomial \u03c3 R\ni j : \u03c3\nf : MvPolynomial \u03c3 R\nh : i \u2260 j\n\u22a2 (Finset.sup (support (f * X j)) fun m => \u2191m i) = Finset.sup (support f) fun m => \u2191m i"}, {"tactic": "rw [support_mul_X]", "annotated_tactic": ["rw [<a>support_mul_X</a>]", [{"full_name": "MvPolynomial.support_mul_X", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [739, 9], "def_end_pos": [739, 22]}]], "state_before": "R : Type u\nS : Type v\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nr : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\np q : MvPolynomial \u03c3 R\ni j : \u03c3\nf : MvPolynomial \u03c3 R\nh : i \u2260 j\n\u22a2 (Finset.sup (support (f * X j)) fun m => \u2191m i) = Finset.sup (support f) fun m => \u2191m i", "state_after": "R : Type u\nS : Type v\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nr : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\np q : MvPolynomial \u03c3 R\ni j : \u03c3\nf : MvPolynomial \u03c3 R\nh : i \u2260 j\n\u22a2 (Finset.sup (Finset.map (addRightEmbedding fun\u2080 | j => 1) (support f)) fun m => \u2191m i) =\n    Finset.sup (support f) fun m => \u2191m i"}, {"tactic": "simp only [Finset.sup_map]", "annotated_tactic": ["simp only [<a>Finset.sup_map</a>]", [{"full_name": "Finset.sup_map", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [69, 9], "def_end_pos": [69, 16]}]], "state_before": "R : Type u\nS : Type v\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nr : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\np q : MvPolynomial \u03c3 R\ni j : \u03c3\nf : MvPolynomial \u03c3 R\nh : i \u2260 j\n\u22a2 (Finset.sup (Finset.map (addRightEmbedding fun\u2080 | j => 1) (support f)) fun m => \u2191m i) =\n    Finset.sup (support f) fun m => \u2191m i", "state_after": "R : Type u\nS : Type v\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nr : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\np q : MvPolynomial \u03c3 R\ni j : \u03c3\nf : MvPolynomial \u03c3 R\nh : i \u2260 j\n\u22a2 Finset.sup (support f) ((fun m => \u2191m i) \u2218 \u2191(addRightEmbedding fun\u2080 | j => 1)) = Finset.sup (support f) fun m => \u2191m i"}, {"tactic": "congr", "annotated_tactic": ["congr", []], "state_before": "R : Type u\nS : Type v\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nr : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\np q : MvPolynomial \u03c3 R\ni j : \u03c3\nf : MvPolynomial \u03c3 R\nh : i \u2260 j\n\u22a2 Finset.sup (support f) ((fun m => \u2191m i) \u2218 \u2191(addRightEmbedding fun\u2080 | j => 1)) = Finset.sup (support f) fun m => \u2191m i", "state_after": "case e_f\nR : Type u\nS : Type v\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nr : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\np q : MvPolynomial \u03c3 R\ni j : \u03c3\nf : MvPolynomial \u03c3 R\nh : i \u2260 j\n\u22a2 (fun m => \u2191m i) \u2218 \u2191(addRightEmbedding fun\u2080 | j => 1) = fun m => \u2191m i"}, {"tactic": "ext", "annotated_tactic": ["ext", []], "state_before": "case e_f\nR : Type u\nS : Type v\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nr : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\np q : MvPolynomial \u03c3 R\ni j : \u03c3\nf : MvPolynomial \u03c3 R\nh : i \u2260 j\n\u22a2 (fun m => \u2191m i) \u2218 \u2191(addRightEmbedding fun\u2080 | j => 1) = fun m => \u2191m i", "state_after": "case e_f.h\nR : Type u\nS : Type v\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nr : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\np q : MvPolynomial \u03c3 R\ni j : \u03c3\nf : MvPolynomial \u03c3 R\nh : i \u2260 j\nx\u271d : \u03c3 \u2192\u2080 \u2115\n\u22a2 ((fun m => \u2191m i) \u2218 \u2191(addRightEmbedding fun\u2080 | j => 1)) x\u271d = \u2191x\u271d i"}, {"tactic": "simp only [Finsupp.single, Nat.one_ne_zero, add_right_eq_self, addRightEmbedding_apply, coe_mk,\n  Pi.add_apply, comp_apply, ite_eq_right_iff, Finsupp.coe_add, Pi.single_eq_of_ne h]", "annotated_tactic": ["simp only [<a>Finsupp.single</a>, <a>Nat.one_ne_zero</a>, <a>add_right_eq_self</a>, <a>addRightEmbedding_apply</a>, <a>coe_mk</a>,\n    <a>Pi.add_apply</a>, <a>comp_apply</a>, <a>ite_eq_right_iff</a>, <a>Finsupp.coe_add</a>, <a>Pi.single_eq_of_ne</a> h]", [{"full_name": "Finsupp.single", "def_path": "Mathlib/Data/Finsupp/Defs.lean", "def_pos": [289, 5], "def_end_pos": [289, 11]}, {"full_name": "Nat.one_ne_zero", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [426, 19], "def_end_pos": [426, 30]}, {"full_name": "add_right_eq_self", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [180, 3], "def_end_pos": [180, 14]}, {"full_name": "addRightEmbedding_apply", "def_path": "Mathlib/Algebra/Hom/Embedding.lean", "def_pos": [37, 24], "def_end_pos": [37, 29]}, {"full_name": "Finsupp.coe_mk", "def_path": "Mathlib/Data/Finsupp/Defs.lean", "def_pos": [159, 9], "def_end_pos": [159, 15]}, {"full_name": "Pi.add_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [82, 3], "def_end_pos": [82, 14]}, {"full_name": "Function.comp_apply", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [33, 17], "def_end_pos": [33, 36]}, {"full_name": "ite_eq_right_iff", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [1162, 17], "def_end_pos": [1162, 33]}, {"full_name": "Finsupp.coe_add", "def_path": "Mathlib/Data/Finsupp/Defs.lean", "def_pos": [972, 9], "def_end_pos": [972, 16]}, {"full_name": "Pi.single_eq_of_ne", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [243, 3], "def_end_pos": [243, 14]}]], "state_before": "case e_f.h\nR : Type u\nS : Type v\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nr : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\np q : MvPolynomial \u03c3 R\ni j : \u03c3\nf : MvPolynomial \u03c3 R\nh : i \u2260 j\nx\u271d : \u03c3 \u2192\u2080 \u2115\n\u22a2 ((fun m => \u2191m i) \u2218 \u2191(addRightEmbedding fun\u2080 | j => 1)) x\u271d = \u2191x\u271d i", "state_after": "no goals"}, {"tactic": "rw [degreeOf_eq_sup (R := R) i]", "annotated_tactic": ["rw [<a>degreeOf_eq_sup</a> (R := R) i]", [{"full_name": "MvPolynomial.degreeOf_eq_sup", "def_path": "Mathlib/Data/MvPolynomial/Variables.lean", "def_pos": [494, 9], "def_end_pos": [494, 24]}]], "state_before": "R : Type u\nS : Type v\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nr : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\np q : MvPolynomial \u03c3 R\ni j : \u03c3\nf : MvPolynomial \u03c3 R\nh : i \u2260 j\n\u22a2 (Finset.sup (support (f * X j)) fun m => \u2191m i) = degreeOf i f", "state_after": "R : Type u\nS : Type v\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nr : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\np q : MvPolynomial \u03c3 R\ni j : \u03c3\nf : MvPolynomial \u03c3 R\nh : i \u2260 j\n\u22a2 (Finset.sup (support (f * X j)) fun m => \u2191m i) = Finset.sup (support f) fun m => \u2191m i"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "full_name": "MeasureTheory.snorm_zero", "start": [201, 1], "end": [207, 63], "traced_tactics": [{"tactic": "by_cases h0 : p = 0", "annotated_tactic": ["by_cases h0 : p = 0", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\n\u22a2 snorm 0 p \u03bc = 0", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nh0 : p = 0\n\u22a2 snorm 0 p \u03bc = 0\n\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nh0 : \u00acp = 0\n\u22a2 snorm 0 p \u03bc = 0"}, {"tactic": "by_cases h_top : p = \u221e", "annotated_tactic": ["by_cases h_top : p = \u221e", []], "state_before": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nh0 : \u00acp = 0\n\u22a2 snorm 0 p \u03bc = 0", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nh0 : \u00acp = 0\nh_top : p = \u22a4\n\u22a2 snorm 0 p \u03bc = 0\n\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nh0 : \u00acp = 0\nh_top : \u00acp = \u22a4\n\u22a2 snorm 0 p \u03bc = 0"}, {"tactic": "rw [\u2190 Ne.def] at h0", "annotated_tactic": ["rw [\u2190 <a>Ne.def</a>] at h0", [{"full_name": "Ne.def", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [59, 9], "def_end_pos": [59, 15]}]], "state_before": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nh0 : \u00acp = 0\nh_top : \u00acp = \u22a4\n\u22a2 snorm 0 p \u03bc = 0", "state_after": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nh0 : p \u2260 0\nh_top : \u00acp = \u22a4\n\u22a2 snorm 0 p \u03bc = 0"}, {"tactic": "simp [snorm_eq_snorm' h0 h_top, ENNReal.toReal_pos h0 h_top]", "annotated_tactic": ["simp [<a>snorm_eq_snorm'</a> h0 h_top, <a>ENNReal.toReal_pos</a> h0 h_top]", [{"full_name": "MeasureTheory.snorm_eq_snorm'", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [88, 9], "def_end_pos": [88, 24]}, {"full_name": "ENNReal.toReal_pos", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2131, 9], "def_end_pos": [2131, 19]}]], "state_before": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nh0 : p \u2260 0\nh_top : \u00acp = \u22a4\n\u22a2 snorm 0 p \u03bc = 0", "state_after": "no goals"}, {"tactic": "simp [h0]", "annotated_tactic": ["simp [h0]", []], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nh0 : p = 0\n\u22a2 snorm 0 p \u03bc = 0", "state_after": "no goals"}, {"tactic": "simp only [h_top, snorm_exponent_top, snormEssSup_zero]", "annotated_tactic": ["simp only [h_top, <a>snorm_exponent_top</a>, <a>snormEssSup_zero</a>]", [{"full_name": "MeasureTheory.snorm_exponent_top", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [103, 9], "def_end_pos": [103, 27]}, {"full_name": "MeasureTheory.snormEssSup_zero", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [195, 9], "def_end_pos": [195, 25]}]], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nh0 : \u00acp = 0\nh_top : p = \u22a4\n\u22a2 snorm 0 p \u03bc = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "full_name": 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\u03b2 \u2192 E\ns : Set \u03b2\nhs : MeasurableSet s\nhf : AEStronglyMeasurable f (Measure.map g \u03bc)\nhg : AEMeasurable g\n\u22a2 Measure.map g (Measure.restrict \u03bc (g \u207b\u00b9' s)) \u2264 Measure.map g \u03bc"}, {"tactic": "exact Measure.map_mono_of_aemeasurable Measure.restrict_le_self hg", "annotated_tactic": ["exact <a>Measure.map_mono_of_aemeasurable</a> <a>Measure.restrict_le_self</a> hg", [{"full_name": "MeasureTheory.Measure.map_mono_of_aemeasurable", "def_path": "Mathlib/MeasureTheory/Measure/AEMeasurable.lean", "def_pos": [397, 9], "def_end_pos": [397, 55]}, {"full_name": "MeasureTheory.Measure.restrict_le_self", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1578, 9], "def_end_pos": [1578, 25]}]], "state_before": "\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\nf\u271d g\u271d : \u03b1 \u2192 E\ns\u271d t : Set \u03b1\n\u03bc 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List.dropWhile p s.data", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/ContextFreeGrammar.lean", "full_name": "ContextFreeRule.rewrites_iff", "start": [74, 1], "end": [77, 86], "traced_tactics": [{"tactic": "rintro \u27e8p, q, rfl, rfl\u27e9", "annotated_tactic": ["rintro \u27e8p, q, rfl, rfl\u27e9", []], "state_before": "T : Type uT\nN : Type uN\nr : ContextFreeRule T N\nu v : List (Symbol T N)\n\u22a2 (\u2203 p q, u = p ++ [Symbol.nonterminal r.input] ++ q \u2227 v = p ++ r.output ++ q) \u2192 Rewrites r u v", "state_after": "case intro.intro.intro\nT : Type uT\nN : Type uN\nr : ContextFreeRule T N\np q : List (Symbol T N)\n\u22a2 Rewrites r (p ++ [Symbol.nonterminal r.input] ++ q) (p ++ r.output ++ q)"}, {"tactic": "apply rewrites_of_exists_parts", "annotated_tactic": ["apply <a>rewrites_of_exists_parts</a>", [{"full_name": "ContextFreeRule.rewrites_of_exists_parts", "def_path": "Mathlib/Computability/ContextFreeGrammar.lean", "def_pos": [65, 7], "def_end_pos": [65, 31]}]], "state_before": "case intro.intro.intro\nT : Type uT\nN : Type uN\nr : ContextFreeRule T N\np q : List (Symbol T N)\n\u22a2 Rewrites r (p ++ [Symbol.nonterminal r.input] ++ q) (p ++ r.output ++ q)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/PEquiv.lean", "full_name": "PEquiv.single_trans_of_eq_none", "start": [400, 1], "end": [402, 48], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/FundThmCalculus.lean", "full_name": "intervalIntegral.integral_sub_integral_sub_linear_isLittleO_of_tendsto_ae_right", "start": [544, 1], "end": [549, 94], "traced_tactics": [{"tactic": "simpa only 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6], "traced_tactics": [{"tactic": "cases v", "annotated_tactic": ["cases v", []], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03c6 : Type w\nn\u271d n m : \u2115\nv : Vector \u03b1 m\n\u22a2 toList (drop n v) = List.drop n (toList v)", "state_after": "case mk\n\u03b1 : Type u\n\u03b2 : Type v\n\u03c6 : Type w\nn\u271d n m : \u2115\nval\u271d : List \u03b1\nproperty\u271d : List.length val\u271d = m\n\u22a2 toList (drop n { val := val\u271d, property := property\u271d }) = List.drop n (toList { val := val\u271d, property := property\u271d })"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case mk\n\u03b1 : Type u\n\u03b2 : Type v\n\u03c6 : Type w\nn\u271d n m : \u2115\nval\u271d : List \u03b1\nproperty\u271d : List.length val\u271d = m\n\u22a2 toList (drop n { val := val\u271d, property := property\u271d }) = List.drop n (toList { val := val\u271d, property := property\u271d })", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Kernel/Basic.lean", "full_name": "ProbabilityTheory.kernel.isSFiniteKernel_sum", "start": [343, 1], "end": [349, 47], "traced_tactics": [{"tactic": "cases fintypeOrInfinite \u03b9", "annotated_tactic": ["cases <a>fintypeOrInfinite</a> \u03b9", [{"full_name": "fintypeOrInfinite", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [975, 19], "def_end_pos": [975, 36]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : Countable \u03b9\n\u03bas : \u03b9 \u2192 { x // x \u2208 kernel \u03b1 \u03b2 }\nh\u03bas : \u2200 (n : \u03b9), IsSFiniteKernel (\u03bas n)\n\u22a2 IsSFiniteKernel (kernel.sum \u03bas)", "state_after": "case inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : Countable \u03b9\n\u03bas : \u03b9 \u2192 { x // x \u2208 kernel \u03b1 \u03b2 }\nh\u03bas : \u2200 (n : \u03b9), IsSFiniteKernel (\u03bas n)\nval\u271d : Fintype \u03b9\n\u22a2 IsSFiniteKernel (kernel.sum \u03bas)\n\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : Countable \u03b9\n\u03bas : \u03b9 \u2192 { x // x \u2208 kernel \u03b1 \u03b2 }\nh\u03bas : \u2200 (n : \u03b9), IsSFiniteKernel (\u03bas n)\nval\u271d : Infinite \u03b9\n\u22a2 IsSFiniteKernel (kernel.sum \u03bas)"}, {"tactic": "cases nonempty_denumerable \u03b9", "annotated_tactic": ["cases <a>nonempty_denumerable</a> \u03b9", [{"full_name": "nonempty_denumerable", "def_path": "Mathlib/Logic/Denumerable.lean", "def_pos": [378, 9], "def_end_pos": [378, 29]}]], "state_before": "case inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : Countable \u03b9\n\u03bas : \u03b9 \u2192 { x // x \u2208 kernel \u03b1 \u03b2 }\nh\u03bas : \u2200 (n : \u03b9), IsSFiniteKernel (\u03bas n)\nval\u271d : Infinite \u03b9\n\u22a2 IsSFiniteKernel (kernel.sum \u03bas)", "state_after": "case inr.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : Countable \u03b9\n\u03bas : \u03b9 \u2192 { x // x \u2208 kernel \u03b1 \u03b2 }\nh\u03bas : \u2200 (n : \u03b9), IsSFiniteKernel (\u03bas n)\nval\u271d\u00b9 : Infinite \u03b9\nval\u271d : Denumerable \u03b9\n\u22a2 IsSFiniteKernel (kernel.sum \u03bas)"}, {"tactic": "exact isSFiniteKernel_sum_of_denumerable h\u03bas", "annotated_tactic": ["exact <a>isSFiniteKernel_sum_of_denumerable</a> h\u03bas", [{"full_name": "ProbabilityTheory.kernel.isSFiniteKernel_sum_of_denumerable", "def_path": "Mathlib/Probability/Kernel/Basic.lean", "def_pos": [329, 9], "def_end_pos": [329, 43]}]], "state_before": "case inr.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : Countable \u03b9\n\u03bas : \u03b9 \u2192 { x // x \u2208 kernel \u03b1 \u03b2 }\nh\u03bas : \u2200 (n : \u03b9), IsSFiniteKernel (\u03bas n)\nval\u271d\u00b9 : Infinite \u03b9\nval\u271d : Denumerable \u03b9\n\u22a2 IsSFiniteKernel (kernel.sum \u03bas)", "state_after": "no goals"}, {"tactic": "rw [sum_fintype]", "annotated_tactic": ["rw [<a>sum_fintype</a>]", [{"full_name": "ProbabilityTheory.kernel.sum_fintype", "def_path": "Mathlib/Probability/Kernel/Basic.lean", "def_pos": [261, 9], "def_end_pos": [261, 20]}]], "state_before": "case inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : Countable \u03b9\n\u03bas : \u03b9 \u2192 { x // x \u2208 kernel \u03b1 \u03b2 }\nh\u03bas : \u2200 (n : \u03b9), IsSFiniteKernel (\u03bas n)\nval\u271d : Fintype \u03b9\n\u22a2 IsSFiniteKernel (kernel.sum \u03bas)", "state_after": "case inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : Countable \u03b9\n\u03bas : \u03b9 \u2192 { x // x \u2208 kernel \u03b1 \u03b2 }\nh\u03bas : \u2200 (n : \u03b9), IsSFiniteKernel (\u03bas n)\nval\u271d : Fintype \u03b9\n\u22a2 IsSFiniteKernel (\u2211 i : \u03b9, \u03bas i)"}, {"tactic": "exact IsSFiniteKernel.finset_sum Finset.univ fun i _ => h\u03bas i", "annotated_tactic": ["exact <a>IsSFiniteKernel.finset_sum</a> <a>Finset.univ</a> fun i _ => h\u03bas i", [{"full_name": "ProbabilityTheory.kernel.IsSFiniteKernel.finset_sum", "def_path": "Mathlib/Probability/Kernel/Basic.lean", "def_pos": [317, 9], "def_end_pos": [317, 35]}, {"full_name": "Finset.univ", "def_path": "Mathlib/Data/Fintype/Basic.lean", "def_pos": [67, 5], "def_end_pos": [67, 9]}]], "state_before": "case inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : Countable \u03b9\n\u03bas : \u03b9 \u2192 { x // x \u2208 kernel \u03b1 \u03b2 }\nh\u03bas : \u2200 (n : \u03b9), IsSFiniteKernel (\u03bas n)\nval\u271d : Fintype \u03b9\n\u22a2 IsSFiniteKernel (\u2211 i : \u03b9, \u03bas i)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Decomposition/Lebesgue.lean", "full_name": "MeasureTheory.ComplexMeasure.singularPart_add_withDensity_rnDeriv_eq", "start": [1223, 1], "end": [1240, 49], "traced_tactics": [{"tactic": "conv_rhs => rw [\u2190 c.toComplexMeasure_to_signedMeasure]", "annotated_tactic": ["conv_rhs => rw [\u2190 c.toComplexMeasure_to_signedMeasure]", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nc : ComplexMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition c \u03bc\n\u22a2 singularPart c \u03bc + Measure.withDensity\u1d65 \u03bc (rnDeriv c \u03bc) = c", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nc : ComplexMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition c \u03bc\n\u22a2 singularPart c \u03bc + Measure.withDensity\u1d65 \u03bc (rnDeriv c \u03bc) = SignedMeasure.toComplexMeasure (\u2191re c) (\u2191im c)"}, {"tactic": "ext i hi : 1", "annotated_tactic": ["ext i hi : 1", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nc : ComplexMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition c \u03bc\n\u22a2 singularPart c \u03bc + Measure.withDensity\u1d65 \u03bc (rnDeriv c \u03bc) = SignedMeasure.toComplexMeasure (\u2191re c) (\u2191im c)", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nc : ComplexMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition c \u03bc\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 \u2191(singularPart c \u03bc + Measure.withDensity\u1d65 \u03bc (rnDeriv c \u03bc)) i = \u2191(SignedMeasure.toComplexMeasure (\u2191re c) (\u2191im c)) i"}, {"tactic": "rw [VectorMeasure.add_apply, SignedMeasure.toComplexMeasure_apply]", "annotated_tactic": ["rw [<a>VectorMeasure.add_apply</a>, <a>SignedMeasure.toComplexMeasure_apply</a>]", [{"full_name": "MeasureTheory.VectorMeasure.add_apply", "def_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "def_pos": [314, 9], "def_end_pos": [314, 18]}, {"full_name": "MeasureTheory.SignedMeasure.toComplexMeasure_apply", "def_path": "Mathlib/MeasureTheory/Measure/Complex.lean", "def_pos": [71, 9], "def_end_pos": [71, 66]}]], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nc : ComplexMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition c \u03bc\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 \u2191(singularPart c \u03bc + Measure.withDensity\u1d65 \u03bc (rnDeriv c \u03bc)) i = \u2191(SignedMeasure.toComplexMeasure (\u2191re c) (\u2191im c)) i", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nc : ComplexMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition c \u03bc\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 \u2191(singularPart c \u03bc) i + \u2191(Measure.withDensity\u1d65 \u03bc (rnDeriv c \u03bc)) i = { re := \u2191(\u2191re c) i, im := \u2191(\u2191im c) i }"}, {"tactic": "ext", "annotated_tactic": ["ext", []], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nc : ComplexMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition c \u03bc\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 \u2191(singularPart c \u03bc) i + \u2191(Measure.withDensity\u1d65 \u03bc (rnDeriv c \u03bc)) i = { re := \u2191(\u2191re c) i, im := \u2191(\u2191im c) i }", "state_after": "case h.a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nc : ComplexMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition c \u03bc\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 (\u2191(singularPart c \u03bc) i + \u2191(Measure.withDensity\u1d65 \u03bc (rnDeriv c \u03bc)) i).re = { re := \u2191(\u2191re c) i, im := \u2191(\u2191im c) i }.re\n\ncase h.a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nc : ComplexMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition c \u03bc\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 (\u2191(singularPart c \u03bc) i + \u2191(Measure.withDensity\u1d65 \u03bc (rnDeriv c \u03bc)) i).im = { re := \u2191(\u2191re c) i, im := \u2191(\u2191im c) i }.im"}, {"tactic": "rw [Complex.add_re, withDensity\u1d65_apply (c.integrable_rnDeriv \u03bc) hi, \u2190 IsROrC.re_eq_complex_re,\n  \u2190 integral_re (c.integrable_rnDeriv \u03bc).integrableOn, IsROrC.re_eq_complex_re,\n  \u2190 withDensity\u1d65_apply _ hi]", "annotated_tactic": ["rw [<a>Complex.add_re</a>, <a>withDensity\u1d65_apply</a> (c.integrable_rnDeriv \u03bc) hi, \u2190 <a>IsROrC.re_eq_complex_re</a>,\n      \u2190 <a>integral_re</a> (c.integrable_rnDeriv \u03bc).<a>integrableOn</a>, <a>IsROrC.re_eq_complex_re</a>,\n      \u2190 <a>withDensity\u1d65_apply</a> _ hi]", [{"full_name": "Complex.add_re", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [192, 9], "def_end_pos": [192, 15]}, {"full_name": "MeasureTheory.withDensity\u1d65_apply", "def_path": "Mathlib/MeasureTheory/Measure/WithDensityVectorMeasure.lean", "def_pos": [60, 9], "def_end_pos": [60, 27]}, {"full_name": "IsROrC.re_eq_complex_re", "def_path": "Mathlib/Analysis/Complex/Basic.lean", "def_pos": [429, 9], "def_end_pos": [429, 39]}, {"full_name": "integral_re", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [1195, 9], "def_end_pos": [1195, 20]}, {"full_name": "MeasureTheory.Integrable.integrableOn", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [163, 9], "def_end_pos": [163, 32]}, {"full_name": "IsROrC.re_eq_complex_re", "def_path": "Mathlib/Analysis/Complex/Basic.lean", "def_pos": [429, 9], "def_end_pos": [429, 39]}, {"full_name": "MeasureTheory.withDensity\u1d65_apply", "def_path": "Mathlib/MeasureTheory/Measure/WithDensityVectorMeasure.lean", "def_pos": [60, 9], "def_end_pos": [60, 27]}]], "state_before": "case h.a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nc : ComplexMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition c \u03bc\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 (\u2191(singularPart c \u03bc) i + \u2191(Measure.withDensity\u1d65 \u03bc (rnDeriv c \u03bc)) i).re = { re := \u2191(\u2191re c) i, im := \u2191(\u2191im c) i }.re", "state_after": "case h.a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nc : ComplexMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition c \u03bc\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 (\u2191(singularPart c \u03bc) i).re + \u2191(Measure.withDensity\u1d65 \u03bc fun a => (rnDeriv c \u03bc a).re) i =\n    { re := \u2191(\u2191re c) i, im := \u2191(\u2191im c) i }.re\n\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nc : ComplexMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition c \u03bc\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 Integrable fun a => (rnDeriv c \u03bc a).re"}, {"tactic": "change (c.re.singularPart \u03bc + \u03bc.withDensity\u1d65 (c.re.rnDeriv \u03bc)) i = _", "annotated_tactic": ["change (c.re.singularPart \u03bc + \u03bc.withDensity\u1d65 (c.re.rnDeriv \u03bc)) i = _", []], "state_before": "case h.a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nc : ComplexMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition c \u03bc\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 (\u2191(singularPart c \u03bc) i).re + \u2191(Measure.withDensity\u1d65 \u03bc fun a => (rnDeriv c \u03bc a).re) i =\n    { re := \u2191(\u2191re c) i, im := \u2191(\u2191im c) i }.re", "state_after": "case h.a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nc : ComplexMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition c \u03bc\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 \u2191(SignedMeasure.singularPart (\u2191re c) \u03bc + Measure.withDensity\u1d65 \u03bc (SignedMeasure.rnDeriv (\u2191re c) \u03bc)) i =\n    { re := \u2191(\u2191re c) i, im := \u2191(\u2191im c) i }.re"}, {"tactic": "rw [c.re.singularPart_add_withDensity_rnDeriv_eq \u03bc]", "annotated_tactic": ["rw [c.re.singularPart_add_withDensity_rnDeriv_eq \u03bc]", []], "state_before": "case h.a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nc : ComplexMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition c \u03bc\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 \u2191(SignedMeasure.singularPart (\u2191re c) \u03bc + Measure.withDensity\u1d65 \u03bc (SignedMeasure.rnDeriv (\u2191re c) \u03bc)) i =\n    { re := \u2191(\u2191re c) i, im := \u2191(\u2191im c) i }.re", "state_after": "no goals"}, {"tactic": "exact SignedMeasure.integrable_rnDeriv _ _", "annotated_tactic": ["exact <a>SignedMeasure.integrable_rnDeriv</a> _ _", [{"full_name": "MeasureTheory.SignedMeasure.integrable_rnDeriv", "def_path": "Mathlib/MeasureTheory/Decomposition/Lebesgue.lean", "def_pos": [916, 9], "def_end_pos": [916, 27]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nc : ComplexMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition c \u03bc\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 Integrable fun a => (rnDeriv c \u03bc a).re", "state_after": "no goals"}, {"tactic": "rw [Complex.add_im, withDensity\u1d65_apply (c.integrable_rnDeriv \u03bc) hi, \u2190 IsROrC.im_eq_complex_im,\n  \u2190 integral_im (c.integrable_rnDeriv \u03bc).integrableOn, IsROrC.im_eq_complex_im,\n  \u2190 withDensity\u1d65_apply _ hi]", "annotated_tactic": ["rw [<a>Complex.add_im</a>, <a>withDensity\u1d65_apply</a> (c.integrable_rnDeriv \u03bc) hi, \u2190 <a>IsROrC.im_eq_complex_im</a>,\n      \u2190 <a>integral_im</a> (c.integrable_rnDeriv \u03bc).<a>integrableOn</a>, <a>IsROrC.im_eq_complex_im</a>,\n      \u2190 <a>withDensity\u1d65_apply</a> _ hi]", [{"full_name": "Complex.add_im", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [197, 9], "def_end_pos": [197, 15]}, {"full_name": "MeasureTheory.withDensity\u1d65_apply", "def_path": "Mathlib/MeasureTheory/Measure/WithDensityVectorMeasure.lean", "def_pos": [60, 9], "def_end_pos": [60, 27]}, {"full_name": "IsROrC.im_eq_complex_im", "def_path": "Mathlib/Analysis/Complex/Basic.lean", "def_pos": [433, 9], "def_end_pos": [433, 39]}, {"full_name": "integral_im", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [1200, 9], "def_end_pos": [1200, 20]}, {"full_name": "MeasureTheory.Integrable.integrableOn", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [163, 9], "def_end_pos": [163, 32]}, {"full_name": "IsROrC.im_eq_complex_im", "def_path": "Mathlib/Analysis/Complex/Basic.lean", "def_pos": [433, 9], "def_end_pos": [433, 39]}, {"full_name": "MeasureTheory.withDensity\u1d65_apply", "def_path": "Mathlib/MeasureTheory/Measure/WithDensityVectorMeasure.lean", "def_pos": [60, 9], "def_end_pos": [60, 27]}]], "state_before": "case h.a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nc : ComplexMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition c \u03bc\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 (\u2191(singularPart c \u03bc) i + \u2191(Measure.withDensity\u1d65 \u03bc (rnDeriv c \u03bc)) i).im = { re := \u2191(\u2191re c) i, im := \u2191(\u2191im c) i }.im", "state_after": "case h.a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nc : ComplexMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition c \u03bc\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 (\u2191(singularPart c \u03bc) i).im + \u2191(Measure.withDensity\u1d65 \u03bc fun a => (rnDeriv c \u03bc a).im) i =\n    { re := \u2191(\u2191re c) i, im := \u2191(\u2191im c) i }.im\n\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nc : ComplexMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition c \u03bc\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 Integrable fun a => (rnDeriv c \u03bc a).im"}, {"tactic": "change (c.im.singularPart \u03bc + \u03bc.withDensity\u1d65 (c.im.rnDeriv \u03bc)) i = _", "annotated_tactic": ["change (c.im.singularPart \u03bc + \u03bc.withDensity\u1d65 (c.im.rnDeriv \u03bc)) i = _", []], "state_before": "case h.a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nc : ComplexMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition c \u03bc\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 (\u2191(singularPart c \u03bc) i).im + \u2191(Measure.withDensity\u1d65 \u03bc fun a => (rnDeriv c \u03bc a).im) i =\n    { re := \u2191(\u2191re c) i, im := \u2191(\u2191im c) i }.im", "state_after": "case h.a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nc : ComplexMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition c \u03bc\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 \u2191(SignedMeasure.singularPart (\u2191im c) \u03bc + Measure.withDensity\u1d65 \u03bc (SignedMeasure.rnDeriv (\u2191im c) \u03bc)) i =\n    { re := \u2191(\u2191re c) i, im := \u2191(\u2191im c) i }.im"}, {"tactic": "rw [c.im.singularPart_add_withDensity_rnDeriv_eq \u03bc]", "annotated_tactic": ["rw [c.im.singularPart_add_withDensity_rnDeriv_eq \u03bc]", []], "state_before": "case h.a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nc : ComplexMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition c \u03bc\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 \u2191(SignedMeasure.singularPart (\u2191im c) \u03bc + Measure.withDensity\u1d65 \u03bc (SignedMeasure.rnDeriv (\u2191im c) \u03bc)) i =\n    { re := \u2191(\u2191re c) i, im := \u2191(\u2191im c) i }.im", "state_after": "no goals"}, {"tactic": "exact SignedMeasure.integrable_rnDeriv _ _", "annotated_tactic": ["exact <a>SignedMeasure.integrable_rnDeriv</a> _ _", [{"full_name": "MeasureTheory.SignedMeasure.integrable_rnDeriv", "def_path": "Mathlib/MeasureTheory/Decomposition/Lebesgue.lean", "def_pos": [916, 9], "def_end_pos": [916, 27]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nc : ComplexMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition c \u03bc\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 Integrable fun a => (rnDeriv c \u03bc a).im", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Lattice.lean", "full_name": "Finset.sup_himp_left", "start": [638, 1], "end": [640, 36], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean", "full_name": "MeasureTheory.Measure.addHaar_preimage_continuousLinearEquiv", "start": [305, 1], "end": [307, 37], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Kernel/Composition.lean", "full_name": "ProbabilityTheory.kernel.comp_eq_snd_compProd", "start": [859, 1], "end": [865, 74], "traced_tactics": [{"tactic": "ext a s hs", "annotated_tactic": ["ext a s hs", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03b3 : Type u_4\nm\u03b3 : MeasurableSpace \u03b3\nf : \u03b2 \u2192 \u03b3\ng : \u03b3 \u2192 \u03b1\n\u03b7 : { x // x \u2208 kernel \u03b2 \u03b3 }\ninst\u271d\u00b9 : IsSFiniteKernel \u03b7\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsSFiniteKernel \u03ba\n\u22a2 \u03b7 \u2218\u2096 \u03ba = snd (\u03ba \u2297\u2096 prodMkLeft \u03b1 \u03b7)", "state_after": "case h.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03b3 : Type u_4\nm\u03b3 : MeasurableSpace \u03b3\nf : \u03b2 \u2192 \u03b3\ng : \u03b3 \u2192 \u03b1\n\u03b7 : { x // x \u2208 kernel \u03b2 \u03b3 }\ninst\u271d\u00b9 : IsSFiniteKernel \u03b7\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsSFiniteKernel \u03ba\na : \u03b1\ns : Set \u03b3\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(\u2191(\u03b7 \u2218\u2096 \u03ba) a) s = \u2191\u2191(\u2191(snd (\u03ba \u2297\u2096 prodMkLeft \u03b1 \u03b7)) a) s"}, {"tactic": "rw [comp_apply' _ _ _ hs, snd_apply' _ _ hs, compProd_apply]", "annotated_tactic": ["rw [<a>comp_apply'</a> _ _ _ hs, <a>snd_apply'</a> _ _ hs, <a>compProd_apply</a>]", [{"full_name": "ProbabilityTheory.kernel.comp_apply'", "def_path": "Mathlib/Probability/Kernel/Composition.lean", "def_pos": [854, 9], "def_end_pos": [854, 20]}, {"full_name": "ProbabilityTheory.kernel.snd_apply'", "def_path": "Mathlib/Probability/Kernel/Composition.lean", "def_pos": [812, 9], "def_end_pos": [812, 19]}, {"full_name": "ProbabilityTheory.kernel.compProd_apply", "def_path": "Mathlib/Probability/Kernel/Composition.lean", "def_pos": [242, 9], "def_end_pos": [242, 23]}]], "state_before": "case h.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03b3 : Type u_4\nm\u03b3 : MeasurableSpace \u03b3\nf : \u03b2 \u2192 \u03b3\ng : \u03b3 \u2192 \u03b1\n\u03b7 : { x // x \u2208 kernel \u03b2 \u03b3 }\ninst\u271d\u00b9 : IsSFiniteKernel \u03b7\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsSFiniteKernel \u03ba\na : \u03b1\ns : Set \u03b3\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(\u2191(\u03b7 \u2218\u2096 \u03ba) a) s = \u2191\u2191(\u2191(snd (\u03ba \u2297\u2096 prodMkLeft \u03b1 \u03b7)) a) s", "state_after": "case h.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03b3 : Type u_4\nm\u03b3 : MeasurableSpace \u03b3\nf : \u03b2 \u2192 \u03b3\ng : \u03b3 \u2192 \u03b1\n\u03b7 : { x // x \u2208 kernel \u03b2 \u03b3 }\ninst\u271d\u00b9 : IsSFiniteKernel \u03b7\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsSFiniteKernel \u03ba\na : \u03b1\ns : Set \u03b3\nhs : MeasurableSet s\n\u22a2 \u222b\u207b (b : \u03b2), \u2191\u2191(\u2191\u03b7 b) s \u2202\u2191\u03ba a = \u222b\u207b (b : \u03b2), \u2191\u2191(\u2191(prodMkLeft \u03b1 \u03b7) (a, b)) {c | (b, c) \u2208 {p | p.2 \u2208 s}} \u2202\u2191\u03ba a\n\ncase h.h.hs\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03b3 : Type u_4\nm\u03b3 : MeasurableSpace \u03b3\nf : \u03b2 \u2192 \u03b3\ng : \u03b3 \u2192 \u03b1\n\u03b7 : { x // x \u2208 kernel \u03b2 \u03b3 }\ninst\u271d\u00b9 : IsSFiniteKernel \u03b7\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsSFiniteKernel \u03ba\na : \u03b1\ns : Set \u03b3\nhs : MeasurableSet s\n\u22a2 MeasurableSet {p | p.2 \u2208 s}"}, {"tactic": "swap", "annotated_tactic": ["swap", []], "state_before": "case h.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03b3 : Type u_4\nm\u03b3 : MeasurableSpace \u03b3\nf : \u03b2 \u2192 \u03b3\ng : \u03b3 \u2192 \u03b1\n\u03b7 : { x // x \u2208 kernel \u03b2 \u03b3 }\ninst\u271d\u00b9 : IsSFiniteKernel \u03b7\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsSFiniteKernel \u03ba\na : \u03b1\ns : Set \u03b3\nhs : MeasurableSet s\n\u22a2 \u222b\u207b (b : \u03b2), \u2191\u2191(\u2191\u03b7 b) s \u2202\u2191\u03ba a = \u222b\u207b (b : \u03b2), \u2191\u2191(\u2191(prodMkLeft \u03b1 \u03b7) (a, b)) {c | (b, c) \u2208 {p | p.2 \u2208 s}} \u2202\u2191\u03ba a\n\ncase h.h.hs\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03b3 : Type u_4\nm\u03b3 : MeasurableSpace \u03b3\nf : \u03b2 \u2192 \u03b3\ng : \u03b3 \u2192 \u03b1\n\u03b7 : { x // x \u2208 kernel \u03b2 \u03b3 }\ninst\u271d\u00b9 : IsSFiniteKernel \u03b7\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsSFiniteKernel \u03ba\na : \u03b1\ns : Set \u03b3\nhs : MeasurableSet s\n\u22a2 MeasurableSet {p | p.2 \u2208 s}", "state_after": "case h.h.hs\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03b3 : Type u_4\nm\u03b3 : MeasurableSpace \u03b3\nf : \u03b2 \u2192 \u03b3\ng : \u03b3 \u2192 \u03b1\n\u03b7 : { x // x \u2208 kernel \u03b2 \u03b3 }\ninst\u271d\u00b9 : IsSFiniteKernel \u03b7\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsSFiniteKernel \u03ba\na : \u03b1\ns : Set \u03b3\nhs : MeasurableSet s\n\u22a2 MeasurableSet {p | p.2 \u2208 s}\n\ncase h.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03b3 : Type u_4\nm\u03b3 : MeasurableSpace \u03b3\nf : \u03b2 \u2192 \u03b3\ng : \u03b3 \u2192 \u03b1\n\u03b7 : { x // x \u2208 kernel \u03b2 \u03b3 }\ninst\u271d\u00b9 : IsSFiniteKernel \u03b7\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsSFiniteKernel \u03ba\na : \u03b1\ns : Set \u03b3\nhs : MeasurableSet s\n\u22a2 \u222b\u207b (b : \u03b2), \u2191\u2191(\u2191\u03b7 b) s \u2202\u2191\u03ba a = \u222b\u207b (b : \u03b2), \u2191\u2191(\u2191(prodMkLeft \u03b1 \u03b7) (a, b)) {c | (b, c) \u2208 {p | p.2 \u2208 s}} \u2202\u2191\u03ba a"}, {"tactic": "simp only [Set.mem_setOf_eq, Set.setOf_mem_eq, prodMkLeft_apply' _ _ s]", "annotated_tactic": ["simp only [<a>Set.mem_setOf_eq</a>, <a>Set.setOf_mem_eq</a>, <a>prodMkLeft_apply'</a> _ _ s]", [{"full_name": "Set.mem_setOf_eq", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [256, 29], "def_end_pos": [256, 41]}, {"full_name": "Set.setOf_mem_eq", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [275, 9], "def_end_pos": [275, 21]}, {"full_name": "ProbabilityTheory.kernel.prodMkLeft_apply'", "def_path": "Mathlib/Probability/Kernel/Composition.lean", "def_pos": [689, 9], "def_end_pos": [689, 26]}]], "state_before": "case h.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03b3 : Type u_4\nm\u03b3 : MeasurableSpace \u03b3\nf : \u03b2 \u2192 \u03b3\ng : \u03b3 \u2192 \u03b1\n\u03b7 : { x // x \u2208 kernel \u03b2 \u03b3 }\ninst\u271d\u00b9 : IsSFiniteKernel \u03b7\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsSFiniteKernel \u03ba\na : \u03b1\ns : Set \u03b3\nhs : MeasurableSet s\n\u22a2 \u222b\u207b (b : \u03b2), \u2191\u2191(\u2191\u03b7 b) s \u2202\u2191\u03ba a = \u222b\u207b (b : \u03b2), \u2191\u2191(\u2191(prodMkLeft \u03b1 \u03b7) (a, b)) {c | (b, c) \u2208 {p | p.2 \u2208 s}} \u2202\u2191\u03ba a", "state_after": "no goals"}, {"tactic": "exact measurable_snd hs", "annotated_tactic": ["exact <a>measurable_snd</a> hs", [{"full_name": "measurable_snd", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [698, 9], "def_end_pos": [698, 23]}]], "state_before": "case h.h.hs\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03b3 : Type u_4\nm\u03b3 : MeasurableSpace \u03b3\nf : \u03b2 \u2192 \u03b3\ng : \u03b3 \u2192 \u03b1\n\u03b7 : { x // x \u2208 kernel \u03b2 \u03b3 }\ninst\u271d\u00b9 : IsSFiniteKernel \u03b7\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsSFiniteKernel \u03ba\na : \u03b1\ns : Set \u03b3\nhs : MeasurableSet s\n\u22a2 MeasurableSet {p | p.2 \u2208 s}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Sym.lean", "full_name": "Finset.mem_sym_iff", "start": [144, 1], "end": [160, 77], "traced_tactics": [{"tactic": "induction' n with n ih", "annotated_tactic": ["induction' n with n ih", []], "state_before": "\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na b : \u03b1\nn : \u2115\nm : Sym \u03b1 n\n\u22a2 m \u2208 Finset.sym s n \u2194 \u2200 (a : \u03b1), a \u2208 m \u2192 a \u2208 s", "state_after": "case zero\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na b : \u03b1\nn : \u2115\nm\u271d : Sym \u03b1 n\nm : Sym \u03b1 Nat.zero\n\u22a2 m \u2208 Finset.sym s Nat.zero \u2194 \u2200 (a : \u03b1), a \u2208 m \u2192 a \u2208 s\n\ncase succ\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na b : \u03b1\nn\u271d : \u2115\nm\u271d : Sym \u03b1 n\u271d\nn : \u2115\nih : \u2200 {m : Sym \u03b1 n}, m \u2208 Finset.sym s n \u2194 \u2200 (a : \u03b1), a \u2208 m \u2192 a \u2208 s\nm : Sym \u03b1 (Nat.succ n)\n\u22a2 m \u2208 Finset.sym s (Nat.succ n) \u2194 \u2200 (a : \u03b1), a \u2208 m \u2192 a \u2208 s"}, {"tactic": "refine' mem_sup.trans \u27e8_, fun h \u21a6 _\u27e9", "annotated_tactic": ["refine' mem_sup.trans \u27e8_, fun h \u21a6 _\u27e9", []], "state_before": "case succ\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na b : \u03b1\nn\u271d : \u2115\nm\u271d : Sym \u03b1 n\u271d\nn : \u2115\nih : \u2200 {m : Sym \u03b1 n}, m \u2208 Finset.sym s n \u2194 \u2200 (a : \u03b1), a \u2208 m \u2192 a \u2208 s\nm : Sym \u03b1 (Nat.succ n)\n\u22a2 m \u2208 Finset.sym s (Nat.succ n) \u2194 \u2200 (a : \u03b1), a \u2208 m \u2192 a \u2208 s", "state_after": "case succ.refine'_1\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na b : \u03b1\nn\u271d : \u2115\nm\u271d : Sym \u03b1 n\u271d\nn : \u2115\nih : \u2200 {m : Sym \u03b1 n}, m \u2208 Finset.sym s n \u2194 \u2200 (a : \u03b1), a \u2208 m \u2192 a \u2208 s\nm : Sym \u03b1 (Nat.succ n)\n\u22a2 (\u2203 v, v \u2208 s \u2227 m \u2208 image (Sym.cons v) (Finset.sym s n)) \u2192 \u2200 (a : \u03b1), a \u2208 m \u2192 a \u2208 s\n\ncase succ.refine'_2\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na b : \u03b1\nn\u271d : \u2115\nm\u271d : Sym \u03b1 n\u271d\nn : \u2115\nih : \u2200 {m : Sym \u03b1 n}, m \u2208 Finset.sym s n \u2194 \u2200 (a : \u03b1), a \u2208 m \u2192 a \u2208 s\nm : Sym \u03b1 (Nat.succ n)\nh : \u2200 (a : \u03b1), a \u2208 m \u2192 a \u2208 s\n\u22a2 \u2203 v, v \u2208 s \u2227 m \u2208 image (Sym.cons v) (Finset.sym s n)"}, {"tactic": "refine' mem_singleton.trans \u27e8_, fun _ \u21a6 Sym.eq_nil_of_card_zero _\u27e9", "annotated_tactic": ["refine' mem_singleton.trans \u27e8_, fun _ \u21a6 <a>Sym.eq_nil_of_card_zero</a> _\u27e9", [{"full_name": "Sym.eq_nil_of_card_zero", "def_path": "Mathlib/Data/Sym/Basic.lean", "def_pos": [266, 9], "def_end_pos": [266, 28]}]], "state_before": "case zero\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na b : \u03b1\nn : \u2115\nm\u271d : Sym \u03b1 n\nm : Sym \u03b1 Nat.zero\n\u22a2 m \u2208 Finset.sym s Nat.zero \u2194 \u2200 (a : \u03b1), a \u2208 m \u2192 a \u2208 s", "state_after": "case zero\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na b : \u03b1\nn : \u2115\nm\u271d : Sym \u03b1 n\nm : Sym \u03b1 Nat.zero\n\u22a2 m = \u2205 \u2192 \u2200 (a : \u03b1), a \u2208 m \u2192 a \u2208 s"}, {"tactic": "rintro rfl", "annotated_tactic": ["rintro rfl", []], "state_before": "case zero\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na b : \u03b1\nn : \u2115\nm\u271d : Sym \u03b1 n\nm : Sym \u03b1 Nat.zero\n\u22a2 m = \u2205 \u2192 \u2200 (a : \u03b1), a \u2208 m \u2192 a \u2208 s", "state_after": "case zero\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na b : \u03b1\nn : \u2115\nm : Sym \u03b1 n\n\u22a2 \u2200 (a : \u03b1), a \u2208 \u2205 \u2192 a \u2208 s"}, {"tactic": "exact fun a ha \u21a6 (Finset.not_mem_empty _ ha).elim", "annotated_tactic": ["exact fun a ha \u21a6 (<a>Finset.not_mem_empty</a> _ ha).<a>elim</a>", [{"full_name": "Finset.not_mem_empty", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [548, 9], "def_end_pos": [548, 22]}, {"full_name": "False.elim", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [223, 21], "def_end_pos": [223, 31]}]], "state_before": "case zero\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na b : \u03b1\nn : \u2115\nm : Sym \u03b1 n\n\u22a2 \u2200 (a : \u03b1), a \u2208 \u2205 \u2192 a \u2208 s", "state_after": "no goals"}, {"tactic": "rintro \u27e8a, ha, he\u27e9 b hb", "annotated_tactic": ["rintro \u27e8a, ha, he\u27e9 b hb", []], "state_before": "case succ.refine'_1\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na b : \u03b1\nn\u271d : \u2115\nm\u271d : Sym \u03b1 n\u271d\nn : \u2115\nih : \u2200 {m : Sym \u03b1 n}, m \u2208 Finset.sym s n \u2194 \u2200 (a : \u03b1), a \u2208 m \u2192 a \u2208 s\nm : Sym \u03b1 (Nat.succ n)\n\u22a2 (\u2203 v, v \u2208 s \u2227 m \u2208 image (Sym.cons v) (Finset.sym s n)) \u2192 \u2200 (a : \u03b1), a \u2208 m \u2192 a \u2208 s", "state_after": "case succ.refine'_1.intro.intro\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na\u271d b\u271d : \u03b1\nn\u271d : \u2115\nm\u271d : Sym \u03b1 n\u271d\nn : \u2115\nih : \u2200 {m : Sym \u03b1 n}, m \u2208 Finset.sym s n \u2194 \u2200 (a : \u03b1), a \u2208 m \u2192 a \u2208 s\nm : Sym \u03b1 (Nat.succ n)\na : \u03b1\nha : a \u2208 s\nhe : m \u2208 image (Sym.cons a) (Finset.sym s n)\nb : \u03b1\nhb : b \u2208 m\n\u22a2 b \u2208 s"}, {"tactic": "rw [mem_image] at he", "annotated_tactic": ["rw [<a>mem_image</a>] at he", [{"full_name": "Finset.mem_image", "def_path": "Mathlib/Data/Finset/Image.lean", "def_pos": [330, 9], "def_end_pos": [330, 18]}]], "state_before": "case succ.refine'_1.intro.intro\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na\u271d b\u271d : \u03b1\nn\u271d : \u2115\nm\u271d : Sym \u03b1 n\u271d\nn : \u2115\nih : \u2200 {m : Sym \u03b1 n}, m \u2208 Finset.sym s n \u2194 \u2200 (a : \u03b1), a \u2208 m \u2192 a \u2208 s\nm : Sym \u03b1 (Nat.succ n)\na : \u03b1\nha : a \u2208 s\nhe : m \u2208 image (Sym.cons a) (Finset.sym s n)\nb : \u03b1\nhb : b \u2208 m\n\u22a2 b \u2208 s", "state_after": "case succ.refine'_1.intro.intro\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na\u271d b\u271d : \u03b1\nn\u271d : \u2115\nm\u271d : Sym \u03b1 n\u271d\nn : \u2115\nih : \u2200 {m : Sym \u03b1 n}, m \u2208 Finset.sym s n \u2194 \u2200 (a : \u03b1), a \u2208 m \u2192 a \u2208 s\nm : Sym \u03b1 (Nat.succ n)\na : \u03b1\nha : a \u2208 s\nhe : \u2203 a_1, a_1 \u2208 Finset.sym s n \u2227 a ::\u209b a_1 = m\nb : \u03b1\nhb : b \u2208 m\n\u22a2 b \u2208 s"}, {"tactic": "obtain \u27e8m, he, rfl\u27e9 := he", "annotated_tactic": ["obtain \u27e8m, he, rfl\u27e9 := he", []], "state_before": "case succ.refine'_1.intro.intro\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na\u271d b\u271d : \u03b1\nn\u271d : \u2115\nm\u271d : Sym \u03b1 n\u271d\nn : \u2115\nih : \u2200 {m : Sym \u03b1 n}, m \u2208 Finset.sym s n \u2194 \u2200 (a : \u03b1), a \u2208 m \u2192 a \u2208 s\nm : Sym \u03b1 (Nat.succ n)\na : \u03b1\nha : a \u2208 s\nhe : \u2203 a_1, a_1 \u2208 Finset.sym s n \u2227 a ::\u209b a_1 = m\nb : \u03b1\nhb : b \u2208 m\n\u22a2 b \u2208 s", "state_after": "case succ.refine'_1.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na\u271d b\u271d : \u03b1\nn\u271d : \u2115\nm\u271d : Sym \u03b1 n\u271d\nn : \u2115\nih : \u2200 {m : Sym \u03b1 n}, m \u2208 Finset.sym s n \u2194 \u2200 (a : \u03b1), a \u2208 m \u2192 a \u2208 s\na : \u03b1\nha : a \u2208 s\nb : \u03b1\nm : Sym \u03b1 n\nhe : m \u2208 Finset.sym s n\nhb : b \u2208 a ::\u209b m\n\u22a2 b \u2208 s"}, {"tactic": "rw [Sym.mem_cons] at hb", "annotated_tactic": ["rw [<a>Sym.mem_cons</a>] at hb", [{"full_name": "Sym.mem_cons", "def_path": "Mathlib/Data/Sym/Basic.lean", "def_pos": [174, 9], "def_end_pos": [174, 17]}]], "state_before": "case succ.refine'_1.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na\u271d b\u271d : \u03b1\nn\u271d : \u2115\nm\u271d : Sym \u03b1 n\u271d\nn : \u2115\nih : \u2200 {m : Sym \u03b1 n}, m \u2208 Finset.sym s n \u2194 \u2200 (a : \u03b1), a \u2208 m \u2192 a \u2208 s\na : \u03b1\nha : a \u2208 s\nb : \u03b1\nm : Sym \u03b1 n\nhe : m \u2208 Finset.sym s n\nhb : b \u2208 a ::\u209b m\n\u22a2 b \u2208 s", "state_after": "case succ.refine'_1.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na\u271d b\u271d : \u03b1\nn\u271d : \u2115\nm\u271d : Sym \u03b1 n\u271d\nn : \u2115\nih : \u2200 {m : Sym \u03b1 n}, m \u2208 Finset.sym s n \u2194 \u2200 (a : \u03b1), a \u2208 m \u2192 a \u2208 s\na : \u03b1\nha : a \u2208 s\nb : \u03b1\nm : Sym \u03b1 n\nhe : m \u2208 Finset.sym s n\nhb : b = a \u2228 b \u2208 m\n\u22a2 b \u2208 s"}, {"tactic": "obtain rfl | hb := hb", "annotated_tactic": ["obtain rfl | hb := hb", []], "state_before": "case succ.refine'_1.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na\u271d b\u271d : \u03b1\nn\u271d : \u2115\nm\u271d : Sym \u03b1 n\u271d\nn : \u2115\nih : \u2200 {m : Sym \u03b1 n}, m \u2208 Finset.sym s n \u2194 \u2200 (a : \u03b1), a \u2208 m \u2192 a \u2208 s\na : \u03b1\nha : a \u2208 s\nb : \u03b1\nm : Sym \u03b1 n\nhe : m \u2208 Finset.sym s n\nhb : b = a \u2228 b \u2208 m\n\u22a2 b \u2208 s", "state_after": "case succ.refine'_1.intro.intro.intro.intro.inl\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na b\u271d : \u03b1\nn\u271d : \u2115\nm\u271d : Sym \u03b1 n\u271d\nn : \u2115\nih : \u2200 {m : Sym \u03b1 n}, m \u2208 Finset.sym s n \u2194 \u2200 (a : \u03b1), a \u2208 m \u2192 a \u2208 s\nb : \u03b1\nm : Sym \u03b1 n\nhe : m \u2208 Finset.sym s n\nha : b \u2208 s\n\u22a2 b \u2208 s\n\ncase succ.refine'_1.intro.intro.intro.intro.inr\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na\u271d b\u271d : \u03b1\nn\u271d : \u2115\nm\u271d : Sym \u03b1 n\u271d\nn : \u2115\nih : \u2200 {m : Sym \u03b1 n}, m \u2208 Finset.sym s n \u2194 \u2200 (a : \u03b1), a \u2208 m \u2192 a \u2208 s\na : \u03b1\nha : a \u2208 s\nb : \u03b1\nm : Sym \u03b1 n\nhe : m \u2208 Finset.sym s n\nhb : b \u2208 m\n\u22a2 b \u2208 s"}, {"tactic": "exact ha", "annotated_tactic": ["exact ha", []], "state_before": "case succ.refine'_1.intro.intro.intro.intro.inl\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na b\u271d : \u03b1\nn\u271d : \u2115\nm\u271d : Sym \u03b1 n\u271d\nn : \u2115\nih : \u2200 {m : Sym \u03b1 n}, m \u2208 Finset.sym s n \u2194 \u2200 (a : \u03b1), a \u2208 m \u2192 a \u2208 s\nb : \u03b1\nm : Sym \u03b1 n\nhe : m \u2208 Finset.sym s n\nha : b \u2208 s\n\u22a2 b \u2208 s", "state_after": "no goals"}, {"tactic": "exact ih.1 he _ hb", "annotated_tactic": ["exact ih.1 he _ hb", []], "state_before": "case succ.refine'_1.intro.intro.intro.intro.inr\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na\u271d b\u271d : \u03b1\nn\u271d : \u2115\nm\u271d : Sym \u03b1 n\u271d\nn : \u2115\nih : \u2200 {m : Sym \u03b1 n}, m \u2208 Finset.sym s n \u2194 \u2200 (a : \u03b1), a \u2208 m \u2192 a \u2208 s\na : \u03b1\nha : a \u2208 s\nb : \u03b1\nm : Sym \u03b1 n\nhe : m \u2208 Finset.sym s n\nhb : b \u2208 m\n\u22a2 b \u2208 s", "state_after": "no goals"}, {"tactic": "obtain \u27e8a, m, rfl\u27e9 := m.exists_eq_cons_of_succ", "annotated_tactic": ["obtain \u27e8a, m, rfl\u27e9 := m.exists_eq_cons_of_succ", []], "state_before": "case succ.refine'_2\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na b : \u03b1\nn\u271d : \u2115\nm\u271d : Sym \u03b1 n\u271d\nn : \u2115\nih : \u2200 {m : Sym \u03b1 n}, m \u2208 Finset.sym s n \u2194 \u2200 (a : \u03b1), a \u2208 m \u2192 a \u2208 s\nm : Sym \u03b1 (Nat.succ n)\nh : \u2200 (a : \u03b1), a \u2208 m \u2192 a \u2208 s\n\u22a2 \u2203 v, v \u2208 s \u2227 m \u2208 image (Sym.cons v) (Finset.sym s n)", "state_after": "case succ.refine'_2.intro.intro\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na\u271d b : \u03b1\nn\u271d : \u2115\nm\u271d : Sym \u03b1 n\u271d\nn : \u2115\nih : \u2200 {m : Sym \u03b1 n}, m \u2208 Finset.sym s n \u2194 \u2200 (a : \u03b1), a \u2208 m \u2192 a \u2208 s\na : \u03b1\nm : Sym \u03b1 n\nh : \u2200 (a_1 : \u03b1), a_1 \u2208 a ::\u209b m \u2192 a_1 \u2208 s\n\u22a2 \u2203 v, v \u2208 s \u2227 a ::\u209b m \u2208 image (Sym.cons v) (Finset.sym s n)"}, {"tactic": "exact\n  \u27e8a, h _ <| Sym.mem_cons_self _ _,\n    mem_image_of_mem _ <| ih.2 fun b hb \u21a6 h _ <| Sym.mem_cons_of_mem hb\u27e9", "annotated_tactic": ["exact\n      \u27e8a, h _ <| <a>Sym.mem_cons_self</a> _ _,\n        <a>mem_image_of_mem</a> _ <| ih.2 fun b hb \u21a6 h _ <| <a>Sym.mem_cons_of_mem</a> hb\u27e9", [{"full_name": "Sym.mem_cons_self", "def_path": "Mathlib/Data/Sym/Basic.lean", "def_pos": [188, 9], "def_end_pos": [188, 22]}, {"full_name": "Finset.mem_image_of_mem", "def_path": "Mathlib/Data/Finset/Image.lean", "def_pos": [334, 9], "def_end_pos": [334, 25]}, {"full_name": "Sym.mem_cons_of_mem", "def_path": "Mathlib/Data/Sym/Basic.lean", "def_pos": [183, 9], "def_end_pos": [183, 24]}]], "state_before": "case succ.refine'_2.intro.intro\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na\u271d b : \u03b1\nn\u271d : \u2115\nm\u271d : Sym \u03b1 n\u271d\nn : \u2115\nih : \u2200 {m : Sym \u03b1 n}, m \u2208 Finset.sym s n \u2194 \u2200 (a : \u03b1), a \u2208 m \u2192 a \u2208 s\na : \u03b1\nm : Sym \u03b1 n\nh : \u2200 (a_1 : \u03b1), a_1 \u2208 a ::\u209b m \u2192 a_1 \u2208 s\n\u22a2 \u2203 v, v \u2208 s \u2227 a ::\u209b m \u2208 image (Sym.cons v) (Finset.sym s n)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Density.lean", "full_name": "MeasureTheory.pdf.quasiMeasurePreserving_hasPDF'", "start": [260, 1], "end": [262, 49], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/LazyList/Basic.lean", "full_name": "LazyList.append_bind", "start": [159, 1], "end": [167, 28], "traced_tactics": [{"tactic": "match xs with\n| LazyList.nil => rfl\n| LazyList.cons x xs =>\n  simp only [append, Thunk.get, LazyList.bind]\n  have := append_bind xs.get ys f\n  simp only [Thunk.get] at this\n  rw [this, append_assoc]", "annotated_tactic": ["match xs with\n  | <a>LazyList.nil</a> => rfl\n  | <a>LazyList.cons</a> x xs =>\n    simp only [<a>append</a>, <a>Thunk.get</a>, <a>LazyList.bind</a>]\n    have := append_bind xs.get ys f\n    simp only [<a>Thunk.get</a>] at this\n    rw [this, <a>append_assoc</a>]", [{"full_name": "LazyList.nil", "def_path": "Mathlib/Data/LazyList.lean", "def_pos": [27, 5], "def_end_pos": [27, 8]}, {"full_name": "LazyList.cons", "def_path": "Mathlib/Data/LazyList.lean", "def_pos": [28, 5], "def_end_pos": [28, 9]}, {"full_name": "LazyList.append", "def_path": "Mathlib/Data/LazyList.lean", "def_pos": [74, 5], "def_end_pos": [74, 11]}, {"full_name": "Thunk.get", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [60, 46], "def_end_pos": [60, 55]}, {"full_name": "LazyList.bind", "def_path": "Mathlib/Data/LazyList/Basic.lean", "def_pos": [125, 15], "def_end_pos": [125, 19]}, {"full_name": "Thunk.get", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [60, 46], "def_end_pos": [60, 55]}, {"full_name": "LazyList.append_assoc", "def_path": "Mathlib/Data/LazyList/Basic.lean", "def_pos": [150, 9], "def_end_pos": [150, 21]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nxs : LazyList \u03b1\nys : Thunk (LazyList \u03b1)\nf : \u03b1 \u2192 LazyList \u03b2\n\u22a2 LazyList.bind (append xs ys) f = append (LazyList.bind xs f) { fn := fun x => LazyList.bind (Thunk.get ys) f }", "state_after": "no goals"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nxs : LazyList \u03b1\nys : Thunk (LazyList \u03b1)\nf : \u03b1 \u2192 LazyList \u03b2\n\u22a2 LazyList.bind (append nil ys) f = append (LazyList.bind nil f) { fn := fun x => LazyList.bind (Thunk.get ys) f }", "state_after": "no goals"}, {"tactic": "simp only [append, Thunk.get, LazyList.bind]", "annotated_tactic": ["simp only [<a>append</a>, <a>Thunk.get</a>, <a>LazyList.bind</a>]", [{"full_name": "LazyList.append", "def_path": "Mathlib/Data/LazyList.lean", "def_pos": [74, 5], "def_end_pos": [74, 11]}, {"full_name": "Thunk.get", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [60, 46], "def_end_pos": [60, 55]}, {"full_name": "LazyList.bind", "def_path": "Mathlib/Data/LazyList/Basic.lean", "def_pos": [125, 15], "def_end_pos": [125, 19]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nxs\u271d : LazyList \u03b1\nys : Thunk (LazyList \u03b1)\nf : \u03b1 \u2192 LazyList \u03b2\nx : \u03b1\nxs : Thunk (LazyList \u03b1)\n\u22a2 LazyList.bind (append (cons x xs) ys) f =\n    append (LazyList.bind (cons x xs) f) { fn := fun x => LazyList.bind (Thunk.get ys) f }", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nxs\u271d : LazyList \u03b1\nys : Thunk (LazyList \u03b1)\nf : \u03b1 \u2192 LazyList \u03b2\nx : \u03b1\nxs : Thunk (LazyList \u03b1)\n\u22a2 append (f x) { fn := fun x => LazyList.bind (append (Thunk.fn xs ()) ys) f } =\n    append (append (f x) { fn := fun x => LazyList.bind (Thunk.fn xs ()) f })\n      { fn := fun x => LazyList.bind (Thunk.fn ys ()) f }"}, {"tactic": "have := append_bind xs.get ys f", "annotated_tactic": ["have := append_bind xs.get ys f", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nxs\u271d : LazyList \u03b1\nys : Thunk (LazyList \u03b1)\nf : \u03b1 \u2192 LazyList \u03b2\nx : \u03b1\nxs : Thunk (LazyList \u03b1)\n\u22a2 append (f x) { fn := fun x => LazyList.bind (append (Thunk.fn xs ()) ys) f } =\n    append (append (f x) { fn := fun x => LazyList.bind (Thunk.fn xs ()) f })\n      { fn := fun x => LazyList.bind (Thunk.fn ys ()) f }", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nxs\u271d : LazyList \u03b1\nys : Thunk (LazyList \u03b1)\nf : \u03b1 \u2192 LazyList \u03b2\nx : \u03b1\nxs : Thunk (LazyList \u03b1)\nthis :\n  LazyList.bind (append (Thunk.get xs) ys) f =\n    append (LazyList.bind (Thunk.get xs) f) { fn := fun x => LazyList.bind (Thunk.get ys) f }\n\u22a2 append (f x) { fn := fun x => LazyList.bind (append (Thunk.fn xs ()) ys) f } =\n    append (append (f x) { fn := fun x => LazyList.bind (Thunk.fn xs ()) f })\n      { fn := fun x => LazyList.bind (Thunk.fn ys ()) f }"}, {"tactic": "simp only [Thunk.get] at this", "annotated_tactic": ["simp only [<a>Thunk.get</a>] at this", [{"full_name": "Thunk.get", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [60, 46], "def_end_pos": [60, 55]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nxs\u271d : LazyList \u03b1\nys : Thunk (LazyList \u03b1)\nf : \u03b1 \u2192 LazyList \u03b2\nx : \u03b1\nxs : Thunk (LazyList \u03b1)\nthis :\n  LazyList.bind (append (Thunk.get xs) ys) f =\n    append (LazyList.bind (Thunk.get xs) f) { fn := fun x => LazyList.bind (Thunk.get ys) f }\n\u22a2 append (f x) { fn := fun x => LazyList.bind (append (Thunk.fn xs ()) ys) f } =\n    append (append (f x) { fn := fun x => LazyList.bind (Thunk.fn xs ()) f })\n      { fn := fun x => LazyList.bind (Thunk.fn ys ()) f }", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nxs\u271d : LazyList \u03b1\nys : Thunk (LazyList \u03b1)\nf : \u03b1 \u2192 LazyList \u03b2\nx : \u03b1\nxs : Thunk (LazyList \u03b1)\nthis :\n  LazyList.bind (append (Thunk.fn xs ()) ys) f =\n    append (LazyList.bind (Thunk.fn xs ()) f) { fn := fun x => LazyList.bind (Thunk.fn ys ()) f }\n\u22a2 append (f x) { fn := fun x => LazyList.bind (append (Thunk.fn xs ()) ys) f } =\n    append (append (f x) { fn := fun x => LazyList.bind (Thunk.fn xs ()) f })\n      { fn := fun x => LazyList.bind (Thunk.fn ys ()) f }"}, {"tactic": "rw [this, append_assoc]", "annotated_tactic": ["rw [this, <a>append_assoc</a>]", [{"full_name": "LazyList.append_assoc", "def_path": "Mathlib/Data/LazyList/Basic.lean", "def_pos": [150, 9], "def_end_pos": [150, 21]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nxs\u271d : LazyList \u03b1\nys : Thunk (LazyList \u03b1)\nf : \u03b1 \u2192 LazyList \u03b2\nx : \u03b1\nxs : Thunk (LazyList \u03b1)\nthis :\n  LazyList.bind (append (Thunk.fn xs ()) ys) f =\n    append (LazyList.bind (Thunk.fn xs ()) f) { fn := fun x => LazyList.bind (Thunk.fn ys ()) f }\n\u22a2 append (f x) { fn := fun x => LazyList.bind (append (Thunk.fn xs ()) ys) f } =\n    append (append (f x) { fn := fun x => LazyList.bind (Thunk.fn xs ()) f })\n      { fn := fun x => LazyList.bind (Thunk.fn ys ()) f }", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL2.lean", "full_name": "MeasureTheory.norm_condexpL2_coe_le", "start": [112, 1], "end": [116, 26], "traced_tactics": [{"tactic": "rw [Lp.norm_def, Lp.norm_def, \u2190 lpMeas_coe]", "annotated_tactic": ["rw [<a>Lp.norm_def</a>, <a>Lp.norm_def</a>, \u2190 <a>lpMeas_coe</a>]", [{"full_name": "MeasureTheory.Lp.norm_def", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [259, 9], "def_end_pos": [259, 17]}, {"full_name": "MeasureTheory.Lp.norm_def", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [259, 9], "def_end_pos": [259, 17]}, {"full_name": "MeasureTheory.lpMeas_coe", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean", "def_pos": [257, 9], "def_end_pos": [257, 19]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 Lp E 2 }\n\u22a2 \u2016\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f)\u2016 \u2264 \u2016f\u2016", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 Lp E 2 }\n\u22a2 ENNReal.toReal (snorm (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f)) 2 \u03bc) \u2264 ENNReal.toReal (snorm (\u2191\u2191f) 2 \u03bc)"}, {"tactic": "refine' (ENNReal.toReal_le_toReal _ (Lp.snorm_ne_top _)).mpr (snorm_condexpL2_le hm f)", "annotated_tactic": ["refine' (<a>ENNReal.toReal_le_toReal</a> _ (<a>Lp.snorm_ne_top</a> _)).<a>mpr</a> (<a>snorm_condexpL2_le</a> hm f)", [{"full_name": "ENNReal.toReal_le_toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2036, 9], "def_end_pos": [2036, 25]}, {"full_name": "MeasureTheory.Lp.snorm_ne_top", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [202, 9], "def_end_pos": [202, 21]}, {"full_name": "Iff.mpr", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [92, 3], "def_end_pos": [92, 6]}, {"full_name": "MeasureTheory.snorm_condexpL2_le", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL2.lean", "def_pos": [105, 9], "def_end_pos": [105, 27]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 Lp E 2 }\n\u22a2 ENNReal.toReal (snorm (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f)) 2 \u03bc) \u2264 ENNReal.toReal (snorm (\u2191\u2191f) 2 \u03bc)", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 Lp E 2 }\n\u22a2 snorm (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f)) 2 \u03bc \u2260 \u22a4"}, {"tactic": "exact Lp.snorm_ne_top _", "annotated_tactic": ["exact <a>Lp.snorm_ne_top</a> _", [{"full_name": "MeasureTheory.Lp.snorm_ne_top", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [202, 9], "def_end_pos": [202, 21]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 Lp E 2 }\n\u22a2 snorm (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f)) 2 \u03bc \u2260 \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Int/NatPrime.lean", "full_name": "Int.succ_dvd_or_succ_dvd_of_succ_sum_dvd_mul", "start": [24, 1], "end": [33, 81], "traced_tactics": [{"tactic": "rw [\u2190 Int.natAbs_mul]", "annotated_tactic": ["rw [\u2190 <a>Int.natAbs_mul</a>]", [{"full_name": "Int.natAbs_mul", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [171, 9], "def_end_pos": [171, 19]}]], "state_before": "p : \u2115\np_prime : Nat.Prime p\nm n : \u2124\nk l : \u2115\nhpm : \u2191(p ^ k) \u2223 m\nhpn : \u2191(p ^ l) \u2223 n\nhpmn : \u2191(p ^ (k + l + 1)) \u2223 m * n\nhpm' : p ^ k \u2223 natAbs m\nhpn' : p ^ l \u2223 natAbs n\n\u22a2 p ^ (k + l + 1) \u2223 natAbs m * natAbs n", "state_after": "p : \u2115\np_prime : Nat.Prime p\nm n : \u2124\nk l : \u2115\nhpm : \u2191(p ^ k) \u2223 m\nhpn : \u2191(p ^ l) \u2223 n\nhpmn : \u2191(p ^ (k + l + 1)) \u2223 m * n\nhpm' : p ^ k \u2223 natAbs m\nhpn' : p ^ l \u2223 natAbs n\n\u22a2 p ^ (k + l + 1) \u2223 natAbs (m * n)"}, {"tactic": "apply Int.coe_nat_dvd.1 <| Int.dvd_natAbs.2 hpmn", "annotated_tactic": ["apply <a>Int.coe_nat_dvd</a>.1 <| <a>Int.dvd_natAbs</a>.2 hpmn", [{"full_name": "Int.coe_nat_dvd", "def_path": "Mathlib/Data/Int/Dvd/Basic.lean", "def_pos": [21, 9], "def_end_pos": [21, 20]}, {"full_name": "Int.dvd_natAbs", "def_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "def_pos": [657, 9], "def_end_pos": [657, 19]}]], "state_before": "p : \u2115\np_prime : Nat.Prime p\nm n : \u2124\nk l : \u2115\nhpm : \u2191(p ^ k) \u2223 m\nhpn : \u2191(p ^ l) \u2223 n\nhpmn : \u2191(p ^ (k + l + 1)) \u2223 m * n\nhpm' : p ^ k \u2223 natAbs m\nhpn' : p ^ l \u2223 natAbs n\n\u22a2 p ^ (k + l + 1) \u2223 natAbs (m * n)", "state_after": "no goals"}, {"tactic": "apply Int.dvd_natAbs.1", "annotated_tactic": ["apply <a>Int.dvd_natAbs</a>.1", [{"full_name": "Int.dvd_natAbs", "def_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "def_pos": [657, 9], "def_end_pos": [657, 19]}]], "state_before": "p : \u2115\np_prime : Nat.Prime p\nm n : \u2124\nk l : \u2115\nhpm : \u2191(p ^ k) \u2223 m\nhpn : \u2191(p ^ l) \u2223 n\nhpmn : \u2191(p ^ (k + l + 1)) \u2223 m * n\nhpm' : p ^ k \u2223 natAbs m\nhpn' : p ^ l \u2223 natAbs n\nhpmn' : p ^ (k + l + 1) \u2223 natAbs m * natAbs n\nhsd : p ^ (k + 1) \u2223 natAbs m \u2228 p ^ (l + 1) \u2223 natAbs n :=\n  Nat.succ_dvd_or_succ_dvd_of_succ_sum_dvd_mul p_prime hpm' hpn' hpmn'\nhsd1 : p ^ (k + 1) \u2223 natAbs m\n\u22a2 \u2191(p ^ (k + 1)) \u2223 m", "state_after": "p : \u2115\np_prime : Nat.Prime p\nm n : \u2124\nk l : \u2115\nhpm : \u2191(p ^ k) \u2223 m\nhpn : \u2191(p ^ l) \u2223 n\nhpmn : \u2191(p ^ (k + l + 1)) \u2223 m * n\nhpm' : p ^ k \u2223 natAbs m\nhpn' : p ^ l \u2223 natAbs n\nhpmn' : p ^ (k + l + 1) \u2223 natAbs m * natAbs n\nhsd : p ^ (k + 1) \u2223 natAbs m \u2228 p ^ (l + 1) \u2223 natAbs n :=\n  Nat.succ_dvd_or_succ_dvd_of_succ_sum_dvd_mul p_prime hpm' hpn' hpmn'\nhsd1 : p ^ (k + 1) \u2223 natAbs m\n\u22a2 \u2191(p ^ (k + 1)) \u2223 \u2191(natAbs m)"}, {"tactic": "apply Int.coe_nat_dvd.2 hsd1", "annotated_tactic": ["apply <a>Int.coe_nat_dvd</a>.2 hsd1", [{"full_name": "Int.coe_nat_dvd", "def_path": "Mathlib/Data/Int/Dvd/Basic.lean", "def_pos": [21, 9], "def_end_pos": [21, 20]}]], "state_before": "p : \u2115\np_prime : Nat.Prime p\nm n : \u2124\nk l : \u2115\nhpm : \u2191(p ^ k) \u2223 m\nhpn : \u2191(p ^ l) \u2223 n\nhpmn : \u2191(p ^ (k + l + 1)) \u2223 m * n\nhpm' : p ^ k \u2223 natAbs m\nhpn' : p ^ l \u2223 natAbs n\nhpmn' : p ^ (k + l + 1) \u2223 natAbs m * natAbs n\nhsd : p ^ (k + 1) \u2223 natAbs m \u2228 p ^ (l + 1) \u2223 natAbs n :=\n  Nat.succ_dvd_or_succ_dvd_of_succ_sum_dvd_mul p_prime hpm' hpn' hpmn'\nhsd1 : p ^ (k + 1) \u2223 natAbs m\n\u22a2 \u2191(p ^ (k + 1)) \u2223 \u2191(natAbs m)", "state_after": "no goals"}, {"tactic": "apply Int.dvd_natAbs.1", "annotated_tactic": ["apply <a>Int.dvd_natAbs</a>.1", [{"full_name": "Int.dvd_natAbs", "def_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "def_pos": [657, 9], "def_end_pos": [657, 19]}]], "state_before": "p : \u2115\np_prime : Nat.Prime p\nm n : \u2124\nk l : \u2115\nhpm : \u2191(p ^ k) \u2223 m\nhpn : \u2191(p ^ l) \u2223 n\nhpmn : \u2191(p ^ (k + l + 1)) \u2223 m * n\nhpm' : p ^ k \u2223 natAbs m\nhpn' : p ^ l \u2223 natAbs n\nhpmn' : p ^ (k + l + 1) \u2223 natAbs m * natAbs n\nhsd : p ^ (k + 1) \u2223 natAbs m \u2228 p ^ (l + 1) \u2223 natAbs n :=\n  Nat.succ_dvd_or_succ_dvd_of_succ_sum_dvd_mul p_prime hpm' hpn' hpmn'\nhsd2 : p ^ (l + 1) \u2223 natAbs n\n\u22a2 \u2191(p ^ (l + 1)) \u2223 n", "state_after": "p : \u2115\np_prime : Nat.Prime p\nm n : \u2124\nk l : \u2115\nhpm : \u2191(p ^ k) \u2223 m\nhpn : \u2191(p ^ l) \u2223 n\nhpmn : \u2191(p ^ (k + l + 1)) \u2223 m * n\nhpm' : p ^ k \u2223 natAbs m\nhpn' : p ^ l \u2223 natAbs n\nhpmn' : p ^ (k + l + 1) \u2223 natAbs m * natAbs n\nhsd : p ^ (k + 1) \u2223 natAbs m \u2228 p ^ (l + 1) \u2223 natAbs n :=\n  Nat.succ_dvd_or_succ_dvd_of_succ_sum_dvd_mul p_prime hpm' hpn' hpmn'\nhsd2 : p ^ (l + 1) \u2223 natAbs n\n\u22a2 \u2191(p ^ (l + 1)) \u2223 \u2191(natAbs n)"}, {"tactic": "apply Int.coe_nat_dvd.2 hsd2", "annotated_tactic": ["apply <a>Int.coe_nat_dvd</a>.2 hsd2", [{"full_name": "Int.coe_nat_dvd", "def_path": "Mathlib/Data/Int/Dvd/Basic.lean", "def_pos": [21, 9], "def_end_pos": [21, 20]}]], "state_before": "p : \u2115\np_prime : Nat.Prime p\nm n : \u2124\nk l : \u2115\nhpm : \u2191(p ^ k) \u2223 m\nhpn : \u2191(p ^ l) \u2223 n\nhpmn : \u2191(p ^ (k + l + 1)) \u2223 m * n\nhpm' : p ^ k \u2223 natAbs m\nhpn' : p ^ l \u2223 natAbs n\nhpmn' : p ^ (k + l + 1) \u2223 natAbs m * natAbs n\nhsd : p ^ (k + 1) \u2223 natAbs m \u2228 p ^ (l + 1) \u2223 natAbs n :=\n  Nat.succ_dvd_or_succ_dvd_of_succ_sum_dvd_mul p_prime hpm' hpn' hpmn'\nhsd2 : p ^ (l + 1) \u2223 natAbs n\n\u22a2 \u2191(p ^ (l + 1)) \u2223 \u2191(natAbs n)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Kernel/Basic.lean", "full_name": "ProbabilityTheory.kernel.set_lintegral_deterministic", "start": [400, 1], "end": [403, 59], "traced_tactics": [{"tactic": "rw [kernel.deterministic_apply, set_lintegral_dirac f s]", "annotated_tactic": ["rw [<a>kernel.deterministic_apply</a>, <a>set_lintegral_dirac</a> f s]", [{"full_name": "ProbabilityTheory.kernel.deterministic_apply", "def_path": "Mathlib/Probability/Kernel/Basic.lean", "def_pos": [365, 9], "def_end_pos": [365, 28]}, {"full_name": "MeasureTheory.set_lintegral_dirac", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [1413, 9], "def_end_pos": [1413, 28]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\nf : \u03b2 \u2192 \u211d\u22650\u221e\ng : \u03b1 \u2192 \u03b2\na : \u03b1\nhg : Measurable g\ninst\u271d\u00b9 : MeasurableSingletonClass \u03b2\ns : Set \u03b2\ninst\u271d : Decidable (g a \u2208 s)\n\u22a2 \u222b\u207b (x : \u03b2) in s, f x \u2202\u2191(deterministic g hg) a = if g a \u2208 s then f (g a) else 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/Division.lean", "full_name": "MvPolynomial.modMonomial_add_divMonomial", "start": [142, 1], "end": [144, 39], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/RBMap/Lemmas.lean", "full_name": "Std.RBNode.exists_insert_toList_zoom_node", "start": [555, 1], "end": [559, 55], "traced_tactics": [{"tactic": "refine \u27e8p.listL ++ l.toList, r.toList ++ p.listR, ?_\u27e9", "annotated_tactic": ["refine \u27e8p.listL ++ l.toList, r.toList ++ p.listR, ?_\u27e9", []], "state_before": "\u03b1 : Type u_1\nc : RBColor\nn : Nat\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nc' : RBColor\nl : RBNode \u03b1\nv' : \u03b1\nr : RBNode \u03b1\np : Path \u03b1\nv : \u03b1\nt : RBNode \u03b1\nht : Balanced t c n\ne : zoom (cmp v) t Path.root = (node c' l v' r, p)\n\u22a2 \u2203 L R, toList t = L ++ v' :: R \u2227 toList (insert cmp t v) = L ++ v :: R", "state_after": "\u03b1 : Type u_1\nc : RBColor\nn : Nat\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nc' : RBColor\nl : RBNode \u03b1\nv' : \u03b1\nr : RBNode \u03b1\np : Path \u03b1\nv : \u03b1\nt : RBNode \u03b1\nht : Balanced t c n\ne : zoom (cmp v) t Path.root = (node c' l v' r, p)\n\u22a2 toList t = Path.listL p ++ toList l ++ v' :: (toList r ++ Path.listR p) \u2227\n    toList (insert cmp t v) = Path.listL p ++ toList l ++ v :: (toList r ++ Path.listR p)"}, {"tactic": "simp [\u2190 zoom_toList e, insert_toList_zoom_node ht e]", "annotated_tactic": ["simp [\u2190 <a>zoom_toList</a> e, <a>insert_toList_zoom_node</a> ht e]", [{"full_name": "Std.RBNode.zoom_toList", "def_path": "lake-packages/std/Std/Data/RBMap/Lemmas.lean", "def_pos": [520, 9], "def_end_pos": [520, 38]}, {"full_name": "Std.RBNode.insert_toList_zoom_node", "def_path": "lake-packages/std/Std/Data/RBMap/Lemmas.lean", "def_pos": [551, 9], "def_end_pos": [551, 32]}]], "state_before": "\u03b1 : Type u_1\nc : RBColor\nn : Nat\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nc' : RBColor\nl : RBNode \u03b1\nv' : \u03b1\nr : RBNode \u03b1\np : Path \u03b1\nv : \u03b1\nt : RBNode \u03b1\nht : Balanced t c n\ne : zoom (cmp v) t Path.root = (node c' l v' r, p)\n\u22a2 toList t = Path.listL p ++ toList l ++ v' :: (toList r ++ Path.listR p) \u2227\n    toList (insert cmp t v) = Path.listL p ++ toList l ++ v :: (toList r ++ Path.listR p)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Kernel/Composition.lean", "full_name": "ProbabilityTheory.kernel.ae_null_of_compProd_null", "start": [306, 1], "end": [314, 54], "traced_tactics": [{"tactic": "obtain \u27e8t, hst, mt, ht\u27e9 := exists_measurable_superset_of_null h", "annotated_tactic": ["obtain \u27e8t, hst, mt, ht\u27e9 := <a>exists_measurable_superset_of_null</a> h", [{"full_name": "MeasureTheory.exists_measurable_superset_of_null", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [231, 9], "def_end_pos": [231, 43]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03b3 : Type u_4\nm\u03b3 : MeasurableSpace \u03b3\ns : Set (\u03b2 \u00d7 \u03b3)\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d : IsSFiniteKernel \u03b7\na : \u03b1\nh : \u2191\u2191(\u2191(\u03ba \u2297\u2096 \u03b7) a) s = 0\n\u22a2 (fun b => \u2191\u2191(\u2191\u03b7 (a, b)) (Prod.mk b \u207b\u00b9' s)) =\u1d50[\u2191\u03ba a] 0", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03b3 : Type u_4\nm\u03b3 : MeasurableSpace \u03b3\ns : Set (\u03b2 \u00d7 \u03b3)\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d : IsSFiniteKernel \u03b7\na : \u03b1\nh : \u2191\u2191(\u2191(\u03ba \u2297\u2096 \u03b7) a) s = 0\nt : Set (\u03b2 \u00d7 \u03b3)\nhst : s \u2286 t\nmt : MeasurableSet t\nht : \u2191\u2191(\u2191(\u03ba \u2297\u2096 \u03b7) a) t = 0\n\u22a2 (fun b => \u2191\u2191(\u2191\u03b7 (a, b)) (Prod.mk b \u207b\u00b9' s)) =\u1d50[\u2191\u03ba a] 0"}, {"tactic": "simp_rw [compProd_null a mt] at ht", "annotated_tactic": ["simp_rw [<a>compProd_null</a> a mt] at ht", [{"full_name": "ProbabilityTheory.kernel.compProd_null", "def_path": "Mathlib/Probability/Kernel/Composition.lean", "def_pos": [299, 9], "def_end_pos": [299, 22]}]], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03b3 : Type u_4\nm\u03b3 : MeasurableSpace \u03b3\ns : Set (\u03b2 \u00d7 \u03b3)\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d : IsSFiniteKernel \u03b7\na : \u03b1\nh : \u2191\u2191(\u2191(\u03ba \u2297\u2096 \u03b7) a) s = 0\nt : Set (\u03b2 \u00d7 \u03b3)\nhst : s \u2286 t\nmt : MeasurableSet t\nht : \u2191\u2191(\u2191(\u03ba \u2297\u2096 \u03b7) a) t = 0\n\u22a2 (fun b => \u2191\u2191(\u2191\u03b7 (a, b)) (Prod.mk b \u207b\u00b9' s)) =\u1d50[\u2191\u03ba a] 0", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03b3 : Type u_4\nm\u03b3 : MeasurableSpace \u03b3\ns : Set (\u03b2 \u00d7 \u03b3)\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d : IsSFiniteKernel \u03b7\na : \u03b1\nh : \u2191\u2191(\u2191(\u03ba \u2297\u2096 \u03b7) a) s = 0\nt : Set (\u03b2 \u00d7 \u03b3)\nhst : s \u2286 t\nmt : MeasurableSet t\nht : (fun b => \u2191\u2191(\u2191\u03b7 (a, b)) (Prod.mk b \u207b\u00b9' t)) =\u1d50[\u2191\u03ba a] 0\n\u22a2 (fun b => \u2191\u2191(\u2191\u03b7 (a, b)) (Prod.mk b \u207b\u00b9' s)) =\u1d50[\u2191\u03ba a] 0"}, {"tactic": "rw [Filter.eventuallyLE_antisymm_iff]", "annotated_tactic": ["rw [<a>Filter.eventuallyLE_antisymm_iff</a>]", [{"full_name": "Filter.eventuallyLE_antisymm_iff", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1708, 9], "def_end_pos": [1708, 34]}]], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03b3 : Type u_4\nm\u03b3 : MeasurableSpace \u03b3\ns : Set (\u03b2 \u00d7 \u03b3)\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d : IsSFiniteKernel \u03b7\na : \u03b1\nh : \u2191\u2191(\u2191(\u03ba \u2297\u2096 \u03b7) a) s = 0\nt : Set (\u03b2 \u00d7 \u03b3)\nhst : s \u2286 t\nmt : MeasurableSet t\nht : (fun b => \u2191\u2191(\u2191\u03b7 (a, b)) (Prod.mk b \u207b\u00b9' t)) =\u1d50[\u2191\u03ba a] 0\n\u22a2 (fun b => \u2191\u2191(\u2191\u03b7 (a, b)) (Prod.mk b \u207b\u00b9' s)) =\u1d50[\u2191\u03ba a] 0", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03b3 : Type u_4\nm\u03b3 : MeasurableSpace \u03b3\ns : Set (\u03b2 \u00d7 \u03b3)\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d : IsSFiniteKernel \u03b7\na : \u03b1\nh : \u2191\u2191(\u2191(\u03ba \u2297\u2096 \u03b7) a) s = 0\nt : Set (\u03b2 \u00d7 \u03b3)\nhst : s \u2286 t\nmt : MeasurableSet t\nht : (fun b => \u2191\u2191(\u2191\u03b7 (a, b)) (Prod.mk b \u207b\u00b9' t)) =\u1d50[\u2191\u03ba a] 0\n\u22a2 (fun b => \u2191\u2191(\u2191\u03b7 (a, b)) (Prod.mk b \u207b\u00b9' s)) \u2264\u1d50[\u2191\u03ba a] 0 \u2227 0 \u2264\u1d50[\u2191\u03ba a] fun b => \u2191\u2191(\u2191\u03b7 (a, b)) (Prod.mk b \u207b\u00b9' s)"}, {"tactic": "exact\n  \u27e8Filter.EventuallyLE.trans_eq\n      (Filter.eventually_of_forall fun x => (measure_mono (Set.preimage_mono hst) : _)) ht,\n    Filter.eventually_of_forall fun x => zero_le _\u27e9", "annotated_tactic": ["exact\n    \u27e8<a>Filter.EventuallyLE.trans_eq</a>\n        (<a>Filter.eventually_of_forall</a> fun x => (<a>measure_mono</a> (<a>Set.preimage_mono</a> hst) : _)) ht,\n      <a>Filter.eventually_of_forall</a> fun x => <a>zero_le</a> _\u27e9", [{"full_name": "Filter.EventuallyLE.trans_eq", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1693, 9], "def_end_pos": [1693, 30]}, {"full_name": "Filter.eventually_of_forall", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1111, 9], "def_end_pos": [1111, 29]}, {"full_name": "MeasureTheory.measure_mono", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [193, 9], "def_end_pos": [193, 21]}, {"full_name": "Set.preimage_mono", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [74, 9], "def_end_pos": [74, 22]}, {"full_name": "Filter.eventually_of_forall", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1111, 9], "def_end_pos": [1111, 29]}, {"full_name": "zero_le", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [217, 30], "def_end_pos": [217, 37]}]], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03b3 : Type u_4\nm\u03b3 : MeasurableSpace \u03b3\ns : Set (\u03b2 \u00d7 \u03b3)\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d : IsSFiniteKernel \u03b7\na : \u03b1\nh : \u2191\u2191(\u2191(\u03ba \u2297\u2096 \u03b7) a) s = 0\nt : Set (\u03b2 \u00d7 \u03b3)\nhst : s \u2286 t\nmt : MeasurableSet t\nht : (fun b => \u2191\u2191(\u2191\u03b7 (a, b)) (Prod.mk b \u207b\u00b9' t)) =\u1d50[\u2191\u03ba a] 0\n\u22a2 (fun b => \u2191\u2191(\u2191\u03b7 (a, b)) (Prod.mk b \u207b\u00b9' s)) \u2264\u1d50[\u2191\u03ba a] 0 \u2227 0 \u2264\u1d50[\u2191\u03ba a] fun b => \u2191\u2191(\u2191\u03b7 (a, b)) (Prod.mk b \u207b\u00b9' s)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/ProbabilityMassFunction/Monad.lean", "full_name": "PMF.mem_support_bind_iff", "start": [125, 1], "end": [127, 74], "traced_tactics": [{"tactic": "simp only [support_bind, Set.mem_iUnion, Set.mem_setOf_eq, exists_prop]", "annotated_tactic": ["simp only [<a>support_bind</a>, <a>Set.mem_iUnion</a>, <a>Set.mem_setOf_eq</a>, <a>exists_prop</a>]", [{"full_name": "PMF.support_bind", "def_path": "Mathlib/Probability/ProbabilityMassFunction/Monad.lean", "def_pos": [121, 9], "def_end_pos": [121, 21]}, {"full_name": "Set.mem_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [201, 9], "def_end_pos": [201, 19]}, {"full_name": "Set.mem_setOf_eq", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [256, 29], "def_end_pos": [256, 41]}, {"full_name": "exists_prop", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [485, 17], "def_end_pos": [485, 28]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\nf : \u03b1 \u2192 PMF \u03b2\ng : \u03b2 \u2192 PMF \u03b3\nb : \u03b2\n\u22a2 b \u2208 support (bind p f) \u2194 \u2203 a, a \u2208 support p \u2227 b \u2208 support (f a)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Group/Prod.lean", "full_name": "MeasureTheory.measurePreserving_prod_div", "start": [394, 1], "end": [396, 85], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Haar/Basic.lean", "full_name": "MeasureTheory.Measure.haar.prehaar_le_index", "start": [300, 1], "end": [304, 26], "traced_tactics": [{"tactic": "unfold prehaar", "annotated_tactic": ["unfold <a>prehaar</a>", [{"full_name": "MeasureTheory.Measure.haar.prehaar", "def_path": "Mathlib/MeasureTheory/Measure/Haar/Basic.lean", "def_pos": [116, 19], "def_end_pos": [116, 26]}]], "state_before": "G : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nU : Set G\nK : Compacts G\nhU : Set.Nonempty (interior U)\n\u22a2 prehaar (\u2191K\u2080) U K \u2264 \u2191(index \u2191K \u2191K\u2080)", "state_after": "G : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nU : Set G\nK : Compacts G\nhU : Set.Nonempty (interior U)\n\u22a2 \u2191(index (\u2191K) U) / \u2191(index (\u2191K\u2080) U) \u2264 \u2191(index \u2191K \u2191K\u2080)"}, {"tactic": "rw [div_le_iff] <;> norm_cast", "annotated_tactic": ["rw [<a>div_le_iff</a>] <;> norm_cast", [{"full_name": "div_le_iff", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [144, 9], "def_end_pos": [144, 19]}]], "state_before": "G : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nU : Set G\nK : Compacts G\nhU : Set.Nonempty (interior U)\n\u22a2 \u2191(index (\u2191K) U) / \u2191(index (\u2191K\u2080) U) \u2264 \u2191(index \u2191K \u2191K\u2080)", "state_after": "G : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nU : Set G\nK : Compacts G\nhU : Set.Nonempty (interior U)\n\u22a2 index (\u2191K) U \u2264 index \u2191K \u2191K\u2080 * index (\u2191K\u2080) U\n\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nU : Set G\nK : Compacts G\nhU : Set.Nonempty (interior U)\n\u22a2 0 < index (\u2191K\u2080) U"}, {"tactic": "apply le_index_mul K\u2080 K hU", "annotated_tactic": ["apply <a>le_index_mul</a> K\u2080 K hU", [{"full_name": "MeasureTheory.Measure.haar.le_index_mul", "def_path": "Mathlib/MeasureTheory/Measure/Haar/Basic.lean", "def_pos": [185, 9], "def_end_pos": [185, 21]}]], "state_before": "G : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nU : Set G\nK : Compacts G\nhU : Set.Nonempty (interior U)\n\u22a2 index (\u2191K) U \u2264 index \u2191K \u2191K\u2080 * index (\u2191K\u2080) U", "state_after": "no goals"}, {"tactic": "exact index_pos K\u2080 hU", "annotated_tactic": ["exact <a>index_pos</a> K\u2080 hU", [{"full_name": "MeasureTheory.Measure.haar.index_pos", "def_path": "Mathlib/MeasureTheory/Measure/Haar/Basic.lean", "def_pos": [201, 9], "def_end_pos": [201, 18]}]], "state_before": "G : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nU : Set G\nK : Compacts G\nhU : Set.Nonempty (interior U)\n\u22a2 0 < index (\u2191K\u2080) U", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Encoding.lean", "full_name": "Computability.FinEncoding.card_le_aleph0", "start": [256, 1], "end": [258, 30], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Group/Prod.lean", "full_name": "MeasureTheory.measurePreserving_prod_inv_mul", "start": [124, 1], "end": [126, 75], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/List/Count.lean", "full_name": "List.count_replicate_self", "start": [185, 9], "end": [186, 93], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Lattice.lean", "full_name": "map_finset_sup", "start": [92, 1], "end": [95, 61], "traced_tactics": [{"tactic": "rw [sup_cons, sup_cons, map_sup, h, Function.comp_apply]", "annotated_tactic": ["rw [<a>sup_cons</a>, <a>sup_cons</a>, <a>map_sup</a>, h, <a>Function.comp_apply</a>]", [{"full_name": "Finset.sup_cons", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [54, 9], "def_end_pos": [54, 17]}, {"full_name": "Finset.sup_cons", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [54, 9], "def_end_pos": [54, 17]}, {"full_name": "SupHomClass.map_sup", "def_path": "Mathlib/Order/Hom/Lattice.lean", "def_pos": [105, 3], "def_end_pos": [105, 10]}, {"full_name": "Function.comp_apply", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [33, 17], "def_end_pos": [33, 36]}]], "state_before": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Type u_5\n\u03ba : Type u_6\ninst\u271d\u2074 : SemilatticeSup \u03b1\ninst\u271d\u00b3 : OrderBot \u03b1\ns\u271d\u00b9 s\u2081 s\u2082 : Finset \u03b2\nf\u271d g\u271d : \u03b2 \u2192 \u03b1\na : \u03b1\ninst\u271d\u00b2 : SemilatticeSup \u03b2\ninst\u271d\u00b9 : OrderBot \u03b2\ninst\u271d : SupBotHomClass F \u03b1 \u03b2\nf : F\ns\u271d : Finset \u03b9\ng : \u03b9 \u2192 \u03b1\ni : \u03b9\ns : Finset \u03b9\nx\u271d : \u00aci \u2208 s\nh : \u2191f (sup s g) = sup s (\u2191f \u2218 g)\n\u22a2 \u2191f (sup (cons i s x\u271d) g) = sup (cons i s x\u271d) (\u2191f \u2218 g)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/DList/Defs.lean", "full_name": "Std.DList.toList_empty", "start": [69, 1], "end": [69, 57], "traced_tactics": [{"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03b1 : Type u\n\u22a2 toList empty = []", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Intervals/Pi.lean", "full_name": "Set.image_mulSingle_Ico_right", "start": [258, 1], "end": [260, 31], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Hausdorff.lean", "full_name": "MeasureTheory.OuterMeasure.mkMetric_top", "start": [357, 1], "end": [361, 19], "traced_tactics": [{"tactic": "simp_rw [mkMetric, mkMetric', mkMetric'.pre, extend_top, boundedBy_top, eq_top_iff]", "annotated_tactic": ["simp_rw [<a>mkMetric</a>, <a>mkMetric'</a>, <a>mkMetric'.pre</a>, <a>extend_top</a>, <a>boundedBy_top</a>, <a>eq_top_iff</a>]", [{"full_name": "MeasureTheory.OuterMeasure.mkMetric", "def_path": "Mathlib/MeasureTheory/Measure/Hausdorff.lean", "def_pos": [264, 5], "def_end_pos": [264, 13]}, {"full_name": "MeasureTheory.OuterMeasure.mkMetric'", "def_path": "Mathlib/MeasureTheory/Measure/Hausdorff.lean", "def_pos": [258, 5], "def_end_pos": [258, 14]}, {"full_name": "MeasureTheory.OuterMeasure.mkMetric'.pre", "def_path": "Mathlib/MeasureTheory/Measure/Hausdorff.lean", "def_pos": [251, 5], "def_end_pos": [251, 18]}, {"full_name": "MeasureTheory.extend_top", "def_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "def_pos": [1346, 9], "def_end_pos": [1346, 19]}, {"full_name": "MeasureTheory.OuterMeasure.boundedBy_top", "def_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "def_pos": [879, 9], "def_end_pos": [879, 22]}, {"full_name": "eq_top_iff", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [165, 9], "def_end_pos": [165, 19]}]], "state_before": "\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u22a2 (mkMetric fun x => \u22a4) = \u22a4", "state_after": "\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u22a2 \u22a4 \u2264 \u2a06 r, \u2a06 (_ : r > 0), \u22a4"}, {"tactic": "rw [le_iSup_iff]", "annotated_tactic": ["rw [<a>le_iSup_iff</a>]", [{"full_name": "le_iSup_iff", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [230, 9], "def_end_pos": [230, 20]}]], "state_before": "\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u22a2 \u22a4 \u2264 \u2a06 r, \u2a06 (_ : r > 0), \u22a4", "state_after": "\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u22a2 \u2200 (b : OuterMeasure X), (\u2200 (i : \u211d\u22650\u221e), \u2a06 (_ : i > 0), \u22a4 \u2264 b) \u2192 \u22a4 \u2264 b"}, {"tactic": "intro b hb", "annotated_tactic": ["intro b hb", []], "state_before": "\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u22a2 \u2200 (b : OuterMeasure X), (\u2200 (i : \u211d\u22650\u221e), \u2a06 (_ : i > 0), \u22a4 \u2264 b) \u2192 \u22a4 \u2264 b", "state_after": "\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nb : OuterMeasure X\nhb : \u2200 (i : \u211d\u22650\u221e), \u2a06 (_ : i > 0), \u22a4 \u2264 b\n\u22a2 \u22a4 \u2264 b"}, {"tactic": "simpa using hb \u22a4", "annotated_tactic": ["simpa using hb \u22a4", []], "state_before": "\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nb : OuterMeasure X\nhb : \u2200 (i : \u211d\u22650\u221e), \u2a06 (_ : i > 0), \u22a4 \u2264 b\n\u22a2 \u22a4 \u2264 b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "full_name": "MeasureTheory.setToFun_zero_left'", "start": [1358, 1], "end": [1362, 31], "traced_tactics": [{"tactic": "by_cases hf : Integrable f \u03bc", "annotated_tactic": ["by_cases hf : <a>Integrable</a> f \u03bc", [{"full_name": "MeasureTheory.Integrable", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [442, 5], "def_end_pos": [442, 15]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\nh_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T s = 0\n\u22a2 setToFun \u03bc T hT f = 0", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\nh_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T s = 0\nhf : Integrable f\n\u22a2 setToFun \u03bc T hT f = 0\n\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\nh_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T s = 0\nhf : \u00acIntegrable f\n\u22a2 setToFun \u03bc T hT f = 0"}, {"tactic": "rw [setToFun_eq hT hf]", "annotated_tactic": ["rw [<a>setToFun_eq</a> hT hf]", [{"full_name": "MeasureTheory.setToFun_eq", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [1276, 9], "def_end_pos": [1276, 20]}]], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\nh_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T s = 0\nhf : Integrable f\n\u22a2 setToFun \u03bc T hT f = 0", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\nh_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T s = 0\nhf : Integrable f\n\u22a2 \u2191(L1.setToL1 hT) (Integrable.toL1 f hf) = 0"}, {"tactic": "exact L1.setToL1_zero_left' hT h_zero _", "annotated_tactic": ["exact <a>L1.setToL1_zero_left'</a> hT h_zero _", [{"full_name": "MeasureTheory.L1.setToL1_zero_left'", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [1045, 9], "def_end_pos": [1045, 27]}]], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\nh_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T s = 0\nhf : Integrable f\n\u22a2 \u2191(L1.setToL1 hT) (Integrable.toL1 f hf) = 0", "state_after": "no goals"}, {"tactic": "exact setToFun_undef hT hf", "annotated_tactic": ["exact <a>setToFun_undef</a> hT hf", [{"full_name": "MeasureTheory.setToFun_undef", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [1286, 9], "def_end_pos": [1286, 23]}]], "state_before": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\nh_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T s = 0\nhf : \u00acIntegrable f\n\u22a2 setToFun \u03bc T hT f = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/Monad.lean", "full_name": "MvPolynomial.aeval_bind\u2081", "start": [308, 1], "end": [310, 25], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Constructions/Polish.lean", "full_name": "MeasureTheory.analyticSet_iff_exists_polishSpace_range", "start": [194, 1], "end": [209, 50], "traced_tactics": [{"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns : Set \u03b1\n\u22a2 AnalyticSet s \u2194 \u2203 \u03b2 h x f, Continuous f \u2227 range f = s", "state_after": "case mp\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns : Set \u03b1\n\u22a2 AnalyticSet s \u2192 \u2203 \u03b2 h x f, Continuous f \u2227 range f = s\n\ncase mpr\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns : Set \u03b1\n\u22a2 (\u2203 \u03b2 h x f, Continuous f \u2227 range f = s) \u2192 AnalyticSet s"}, {"tactic": "intro h", "annotated_tactic": ["intro h", []], "state_before": "case mp\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns : Set \u03b1\n\u22a2 AnalyticSet s \u2192 \u2203 \u03b2 h x f, Continuous f \u2227 range f = s", "state_after": "case mp\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns : Set \u03b1\nh : AnalyticSet s\n\u22a2 \u2203 \u03b2 h x f, Continuous f \u2227 range f = s"}, {"tactic": "rw [AnalyticSet] at h", "annotated_tactic": ["rw [<a>AnalyticSet</a>] at h", [{"full_name": "MeasureTheory.AnalyticSet", "def_path": "Mathlib/MeasureTheory/Constructions/Polish.lean", "def_pos": [164, 17], "def_end_pos": [164, 28]}]], "state_before": "case mp\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns : Set \u03b1\nh : AnalyticSet s\n\u22a2 \u2203 \u03b2 h x f, Continuous f \u2227 range f = s", "state_after": "case mp\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns : Set \u03b1\nh : s = \u2205 \u2228 \u2203 f, Continuous f \u2227 range f = s\n\u22a2 \u2203 \u03b2 h x f, Continuous f \u2227 range f = s"}, {"tactic": "cases' h with h h", "annotated_tactic": ["cases' h with h h", []], "state_before": "case mp\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns : Set \u03b1\nh : s = \u2205 \u2228 \u2203 f, Continuous f \u2227 range f = s\n\u22a2 \u2203 \u03b2 h x f, Continuous f \u2227 range f = s", "state_after": "case mp.inl\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns : Set \u03b1\nh : s = \u2205\n\u22a2 \u2203 \u03b2 h x f, Continuous f \u2227 range f = s\n\ncase mp.inr\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns : Set \u03b1\nh : \u2203 f, Continuous f \u2227 range f = s\n\u22a2 \u2203 \u03b2 h x f, Continuous f \u2227 range f = s"}, {"tactic": "refine' \u27e8Empty, inferInstance, inferInstance, Empty.elim, continuous_bot, _\u27e9", "annotated_tactic": ["refine' \u27e8<a>Empty</a>, <a>inferInstance</a>, <a>inferInstance</a>, <a>Empty.elim</a>, <a>continuous_bot</a>, _\u27e9", [{"full_name": "Empty", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [195, 11], "def_end_pos": [195, 16]}, {"full_name": "inferInstance", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [86, 8], "def_end_pos": [86, 21]}, {"full_name": "inferInstance", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [86, 8], "def_end_pos": [86, 21]}, {"full_name": "Empty.elim", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [757, 5], "def_end_pos": [757, 15]}, {"full_name": "continuous_bot", "def_path": "Mathlib/Topology/Order.lean", "def_pos": [834, 9], "def_end_pos": [834, 23]}]], "state_before": "case mp.inl\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns : Set \u03b1\nh : s = \u2205\n\u22a2 \u2203 \u03b2 h x f, Continuous f \u2227 range f = s", "state_after": "case mp.inl\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns : Set \u03b1\nh : s = \u2205\n\u22a2 range Empty.elim = s"}, {"tactic": "rw [h]", "annotated_tactic": ["rw [h]", []], "state_before": "case mp.inl\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns : Set \u03b1\nh : s = \u2205\n\u22a2 range Empty.elim = s", "state_after": "case mp.inl\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns : Set \u03b1\nh : s = \u2205\n\u22a2 range Empty.elim = \u2205"}, {"tactic": "exact range_eq_empty _", "annotated_tactic": ["exact <a>range_eq_empty</a> _", [{"full_name": "Set.range_eq_empty", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [780, 9], "def_end_pos": [780, 23]}]], "state_before": "case mp.inl\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns : Set \u03b1\nh : s = \u2205\n\u22a2 range Empty.elim = \u2205", "state_after": "no goals"}, {"tactic": "exact \u27e8\u2115 \u2192 \u2115, inferInstance, inferInstance, h\u27e9", "annotated_tactic": ["exact \u27e8\u2115 \u2192 \u2115, <a>inferInstance</a>, <a>inferInstance</a>, h\u27e9", [{"full_name": "inferInstance", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [86, 8], "def_end_pos": [86, 21]}, {"full_name": "inferInstance", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [86, 8], "def_end_pos": [86, 21]}]], "state_before": "case mp.inr\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns : Set \u03b1\nh : \u2203 f, Continuous f \u2227 range f = s\n\u22a2 \u2203 \u03b2 h x f, Continuous f \u2227 range f = s", "state_after": "no goals"}, {"tactic": "rintro \u27e8\u03b2, h, h', f, f_cont, f_range\u27e9", "annotated_tactic": ["rintro \u27e8\u03b2, h, h', f, f_cont, f_range\u27e9", []], "state_before": "case mpr\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns : Set \u03b1\n\u22a2 (\u2203 \u03b2 h x f, Continuous f \u2227 range f = s) \u2192 AnalyticSet s", "state_after": "case mpr.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns : Set \u03b1\n\u03b2 : Type\nh : TopologicalSpace \u03b2\nh' : PolishSpace \u03b2\nf : \u03b2 \u2192 \u03b1\nf_cont : Continuous f\nf_range : range f = s\n\u22a2 AnalyticSet s"}, {"tactic": "rw [\u2190 f_range]", "annotated_tactic": ["rw [\u2190 f_range]", []], "state_before": "case mpr.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns : Set \u03b1\n\u03b2 : Type\nh : TopologicalSpace \u03b2\nh' : PolishSpace \u03b2\nf : \u03b2 \u2192 \u03b1\nf_cont : Continuous f\nf_range : range f = s\n\u22a2 AnalyticSet s", "state_after": "case mpr.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns : Set \u03b1\n\u03b2 : Type\nh : TopologicalSpace \u03b2\nh' : PolishSpace \u03b2\nf : \u03b2 \u2192 \u03b1\nf_cont : Continuous f\nf_range : range f = s\n\u22a2 AnalyticSet (range f)"}, {"tactic": "exact analyticSet_range_of_polishSpace f_cont", "annotated_tactic": ["exact <a>analyticSet_range_of_polishSpace</a> f_cont", [{"full_name": "MeasureTheory.analyticSet_range_of_polishSpace", "def_path": "Mathlib/MeasureTheory/Constructions/Polish.lean", "def_pos": [173, 9], "def_end_pos": [173, 41]}]], "state_before": "case mpr.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns : Set \u03b1\n\u03b2 : Type\nh : TopologicalSpace \u03b2\nh' : PolishSpace \u03b2\nf : \u03b2 \u2192 \u03b1\nf_cont : Continuous f\nf_range : range f = s\n\u22a2 AnalyticSet (range f)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/ConvergenceInMeasure.lean", "full_name": "MeasureTheory.tendstoInMeasure_of_tendsto_snorm_of_ne_top", "start": [304, 1], "end": [313, 12], "traced_tactics": [{"tactic": "refine' TendstoInMeasure.congr (fun i => (hf i).ae_eq_mk.symm) hg.ae_eq_mk.symm _", "annotated_tactic": ["refine' <a>TendstoInMeasure.congr</a> (fun i => (hf i).ae_eq_mk.symm) hg.ae_eq_mk.symm _", [{"full_name": "MeasureTheory.TendstoInMeasure.congr", "def_path": "Mathlib/MeasureTheory/Function/ConvergenceInMeasure.lean", "def_pos": [84, 19], "def_end_pos": [84, 24]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\nE : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup E\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhf : \u2200 (n : \u03b9), AEStronglyMeasurable (f n) \u03bc\nhg : AEStronglyMeasurable g \u03bc\nl : Filter \u03b9\nhfg : Tendsto (fun n => snorm (f n - g) p \u03bc) l (\ud835\udcdd 0)\n\u22a2 TendstoInMeasure \u03bc f l g", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\nE : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup E\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhf : \u2200 (n : \u03b9), AEStronglyMeasurable (f n) \u03bc\nhg : AEStronglyMeasurable g \u03bc\nl : Filter \u03b9\nhfg : Tendsto (fun n => snorm (f n - g) p \u03bc) l (\ud835\udcdd 0)\n\u22a2 TendstoInMeasure \u03bc (fun i => AEStronglyMeasurable.mk (f i) (_ : AEStronglyMeasurable (f i) \u03bc)) l\n    (AEStronglyMeasurable.mk g hg)"}, {"tactic": "refine' tendstoInMeasure_of_tendsto_snorm_of_stronglyMeasurable\n  hp_ne_zero hp_ne_top (fun i => (hf i).stronglyMeasurable_mk) hg.stronglyMeasurable_mk _", "annotated_tactic": ["refine' <a>tendstoInMeasure_of_tendsto_snorm_of_stronglyMeasurable</a>\n    hp_ne_zero hp_ne_top (fun i => (hf i).<a>stronglyMeasurable_mk</a>) hg.stronglyMeasurable_mk _", [{"full_name": "MeasureTheory.tendstoInMeasure_of_tendsto_snorm_of_stronglyMeasurable", "def_path": "Mathlib/MeasureTheory/Function/ConvergenceInMeasure.lean", "def_pos": [278, 9], "def_end_pos": [278, 64]}, {"full_name": "MeasureTheory.AEStronglyMeasurable.stronglyMeasurable_mk", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1206, 9], "def_end_pos": [1206, 30]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\nE : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup E\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhf : \u2200 (n : \u03b9), AEStronglyMeasurable (f n) \u03bc\nhg : AEStronglyMeasurable g \u03bc\nl : Filter \u03b9\nhfg : Tendsto (fun n => snorm (f n - g) p \u03bc) l (\ud835\udcdd 0)\n\u22a2 TendstoInMeasure \u03bc (fun i => AEStronglyMeasurable.mk (f i) (_ : AEStronglyMeasurable (f i) \u03bc)) l\n    (AEStronglyMeasurable.mk g hg)", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\nE : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup E\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhf : \u2200 (n : \u03b9), AEStronglyMeasurable (f n) \u03bc\nhg : AEStronglyMeasurable g \u03bc\nl : Filter \u03b9\nhfg : Tendsto (fun n => snorm (f n - g) p \u03bc) l (\ud835\udcdd 0)\n\u22a2 Tendsto\n    (fun n =>\n      snorm (AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) - AEStronglyMeasurable.mk g hg) p \u03bc)\n    l (\ud835\udcdd 0)"}, {"tactic": "have : (fun n => snorm ((hf n).mk (f n) - hg.mk g) p \u03bc) = fun n => snorm (f n - g) p \u03bc := by\n  ext1 n; refine' snorm_congr_ae (EventuallyEq.sub (hf n).ae_eq_mk.symm hg.ae_eq_mk.symm)", "annotated_tactic": ["have : (fun n => <a>snorm</a> ((hf n).<a>mk</a> (f n) - hg.mk g) p \u03bc) = fun n => <a>snorm</a> (f n - g) p \u03bc := by\n    ext1 n; refine' <a>snorm_congr_ae</a> (<a>EventuallyEq.sub</a> (hf n).ae_eq_mk.symm hg.ae_eq_mk.symm)", [{"full_name": "MeasureTheory.snorm", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [84, 5], "def_end_pos": [84, 10]}, {"full_name": "MeasureTheory.AEStronglyMeasurable.mk", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1202, 29], "def_end_pos": [1202, 31]}, {"full_name": "MeasureTheory.snorm", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [84, 5], "def_end_pos": [84, 10]}, {"full_name": "MeasureTheory.snorm_congr_ae", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [549, 9], "def_end_pos": [549, 23]}, {"full_name": "Filter.EventuallyEq.sub", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1544, 3], "def_end_pos": [1544, 14]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\nE : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup E\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhf : \u2200 (n : \u03b9), AEStronglyMeasurable (f n) \u03bc\nhg : AEStronglyMeasurable g \u03bc\nl : Filter \u03b9\nhfg : Tendsto (fun n => snorm (f n - g) p \u03bc) l (\ud835\udcdd 0)\n\u22a2 Tendsto\n    (fun n =>\n      snorm (AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) - AEStronglyMeasurable.mk g hg) p \u03bc)\n    l (\ud835\udcdd 0)", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\nE : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup E\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhf : \u2200 (n : \u03b9), AEStronglyMeasurable (f n) \u03bc\nhg : AEStronglyMeasurable g \u03bc\nl : Filter \u03b9\nhfg : Tendsto (fun n => snorm (f n - g) p \u03bc) l (\ud835\udcdd 0)\nthis :\n  (fun n =>\n      snorm (AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) - AEStronglyMeasurable.mk g hg) p \u03bc) =\n    fun n => snorm (f n - g) p \u03bc\n\u22a2 Tendsto\n    (fun n =>\n      snorm (AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) - AEStronglyMeasurable.mk g hg) p \u03bc)\n    l (\ud835\udcdd 0)"}, {"tactic": "rw [this]", "annotated_tactic": ["rw [this]", []], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\nE : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup E\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhf : \u2200 (n : \u03b9), AEStronglyMeasurable (f n) \u03bc\nhg : AEStronglyMeasurable g \u03bc\nl : Filter \u03b9\nhfg : Tendsto (fun n => snorm (f n - g) p \u03bc) l (\ud835\udcdd 0)\nthis :\n  (fun n =>\n      snorm (AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) - AEStronglyMeasurable.mk g hg) p \u03bc) =\n    fun n => snorm (f n - g) p \u03bc\n\u22a2 Tendsto\n    (fun n =>\n      snorm (AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) - AEStronglyMeasurable.mk g hg) p \u03bc)\n    l (\ud835\udcdd 0)", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\nE : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup E\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhf : \u2200 (n : \u03b9), AEStronglyMeasurable (f n) \u03bc\nhg : AEStronglyMeasurable g \u03bc\nl : Filter \u03b9\nhfg : Tendsto (fun n => snorm (f n - g) p \u03bc) l (\ud835\udcdd 0)\nthis :\n  (fun n =>\n      snorm (AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) - AEStronglyMeasurable.mk g hg) p \u03bc) =\n    fun n => snorm (f n - g) p \u03bc\n\u22a2 Tendsto (fun n => snorm (f n - g) p \u03bc) l (\ud835\udcdd 0)"}, {"tactic": "exact hfg", "annotated_tactic": ["exact hfg", []], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\nE : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup E\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhf : \u2200 (n : \u03b9), AEStronglyMeasurable (f n) \u03bc\nhg : AEStronglyMeasurable g \u03bc\nl : Filter \u03b9\nhfg : Tendsto (fun n => snorm (f n - g) p \u03bc) l (\ud835\udcdd 0)\nthis :\n  (fun n =>\n      snorm (AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) - AEStronglyMeasurable.mk g hg) p \u03bc) =\n    fun n => snorm (f n - g) p \u03bc\n\u22a2 Tendsto (fun n => snorm (f n - g) p \u03bc) l (\ud835\udcdd 0)", "state_after": "no goals"}, {"tactic": "ext1 n", "annotated_tactic": ["ext1 n", []], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\nE : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup E\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhf : \u2200 (n : \u03b9), AEStronglyMeasurable (f n) \u03bc\nhg : AEStronglyMeasurable g \u03bc\nl : Filter \u03b9\nhfg : Tendsto (fun n => snorm (f n - g) p \u03bc) l (\ud835\udcdd 0)\n\u22a2 (fun n =>\n      snorm (AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) - AEStronglyMeasurable.mk g hg) p \u03bc) =\n    fun n => snorm (f n - g) p \u03bc", "state_after": "case h\n\u03b1 : Type u_1\n\u03b9 : Type u_2\nE : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup E\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhf : \u2200 (n : \u03b9), AEStronglyMeasurable (f n) \u03bc\nhg : AEStronglyMeasurable g \u03bc\nl : Filter \u03b9\nhfg : Tendsto (fun n => snorm (f n - g) p \u03bc) l (\ud835\udcdd 0)\nn : \u03b9\n\u22a2 snorm (AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) - AEStronglyMeasurable.mk g hg) p \u03bc =\n    snorm (f n - g) p \u03bc"}, {"tactic": "refine' snorm_congr_ae (EventuallyEq.sub (hf n).ae_eq_mk.symm hg.ae_eq_mk.symm)", "annotated_tactic": ["refine' <a>snorm_congr_ae</a> (<a>EventuallyEq.sub</a> (hf n).ae_eq_mk.symm hg.ae_eq_mk.symm)", [{"full_name": "MeasureTheory.snorm_congr_ae", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [549, 9], "def_end_pos": [549, 23]}, {"full_name": "Filter.EventuallyEq.sub", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1544, 3], "def_end_pos": [1544, 14]}]], "state_before": "case h\n\u03b1 : Type u_1\n\u03b9 : Type u_2\nE : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup E\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhf : \u2200 (n : \u03b9), AEStronglyMeasurable (f n) \u03bc\nhg : AEStronglyMeasurable g \u03bc\nl : Filter \u03b9\nhfg : Tendsto (fun n => snorm (f n - g) p \u03bc) l (\ud835\udcdd 0)\nn : \u03b9\n\u22a2 snorm (AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) - AEStronglyMeasurable.mk g hg) p \u03bc =\n    snorm (f n - g) p \u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Portmanteau.lean", "full_name": "MeasureTheory.le_measure_liminf_of_limsup_measure_compl_le", "start": [131, 1], "end": [135, 94], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "full_name": "MeasureTheory.Integrable.norm_toL1_eq_lintegral_norm", "start": [1457, 1], "end": [1459, 52], "traced_tactics": [{"tactic": "rw [norm_toL1, lintegral_norm_eq_lintegral_edist]", "annotated_tactic": ["rw [<a>norm_toL1</a>, <a>lintegral_norm_eq_lintegral_edist</a>]", [{"full_name": "MeasureTheory.Integrable.norm_toL1", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [1446, 9], "def_end_pos": [1446, 18]}, {"full_name": "MeasureTheory.lintegral_norm_eq_lintegral_edist", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [71, 9], "def_end_pos": [71, 42]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : NormedAddCommGroup \u03b3\nf : \u03b1 \u2192 \u03b2\nhf : Integrable f\n\u22a2 \u2016toL1 f hf\u2016 = ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016f a\u2016 \u2202\u03bc)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "full_name": "MeasurableSet.image_inclusion", "start": [637, 1], "end": [640, 52], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "full_name": "MeasureTheory.SimpleFunc.FinMeasSupp.mul", "start": [1224, 11], "end": [1227, 32], "traced_tactics": [{"tactic": "rw [mul_eq_map\u2082]", "annotated_tactic": ["rw [<a>mul_eq_map\u2082</a>]", [{"full_name": "MeasureTheory.SimpleFunc.mul_eq_map\u2082", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [550, 9], "def_end_pos": [550, 20]}]], "state_before": "\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\ninst\u271d\u00b2 : Zero \u03b2\u271d\ninst\u271d\u00b9 : Zero \u03b3\n\u03bc : Measure \u03b1\nf\u271d : \u03b1 \u2192\u209b \u03b2\u271d\n\u03b2 : Type u_5\ninst\u271d : MonoidWithZero \u03b2\nf g : \u03b1 \u2192\u209b \u03b2\nhf : SimpleFunc.FinMeasSupp f \u03bc\nhg : SimpleFunc.FinMeasSupp g \u03bc\n\u22a2 SimpleFunc.FinMeasSupp (f * g) \u03bc", "state_after": "\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\ninst\u271d\u00b2 : Zero \u03b2\u271d\ninst\u271d\u00b9 : Zero \u03b3\n\u03bc : Measure \u03b1\nf\u271d : \u03b1 \u2192\u209b \u03b2\u271d\n\u03b2 : Type u_5\ninst\u271d : MonoidWithZero \u03b2\nf g : \u03b1 \u2192\u209b \u03b2\nhf : SimpleFunc.FinMeasSupp f \u03bc\nhg : SimpleFunc.FinMeasSupp g \u03bc\n\u22a2 SimpleFunc.FinMeasSupp (map (fun p => p.1 * p.2) (pair f g)) \u03bc"}, {"tactic": "exact hf.map\u2082 hg (zero_mul 0)", "annotated_tactic": ["exact hf.map\u2082 hg (<a>zero_mul</a> 0)", [{"full_name": "MulZeroClass.zero_mul", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [36, 3], "def_end_pos": [36, 11]}]], "state_before": "\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\ninst\u271d\u00b2 : Zero \u03b2\u271d\ninst\u271d\u00b9 : Zero \u03b3\n\u03bc : Measure \u03b1\nf\u271d : \u03b1 \u2192\u209b \u03b2\u271d\n\u03b2 : Type u_5\ninst\u271d : MonoidWithZero \u03b2\nf g : \u03b1 \u2192\u209b \u03b2\nhf : SimpleFunc.FinMeasSupp f \u03bc\nhg : SimpleFunc.FinMeasSupp g \u03bc\n\u22a2 SimpleFunc.FinMeasSupp (map (fun p => p.1 * p.2) (pair f g)) \u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/Layercake.lean", "full_name": "MeasureTheory.lintegral_comp_eq_lintegral_meas_le_mul", "start": [395, 1], "end": [437, 46], "traced_tactics": [{"tactic": "obtain \u27e8G, G_mble, G_nn, g_eq_G\u27e9 : \u2203 G : \u211d \u2192 \u211d, Measurable G \u2227 0 \u2264 G\n    \u2227 g =\u1d50[volume.restrict (Ioi 0)] G := by\n  refine' AEMeasurable.exists_measurable_nonneg _ g_nn\n  exact aemeasurable_Ioi_of_forall_Ioc fun t ht => (g_intble t ht).1.1.aemeasurable", "annotated_tactic": ["obtain \u27e8G, G_mble, G_nn, g_eq_G\u27e9 : \u2203 G : \u211d \u2192 \u211d, <a>Measurable</a> G \u2227 0 \u2264 G\n      \u2227 g =\u1d50[volume.restrict (<a>Ioi</a> 0)] G := by\n    refine' <a>AEMeasurable.exists_measurable_nonneg</a> _ g_nn\n    exact <a>aemeasurable_Ioi_of_forall_Ioc</a> fun t ht => (g_intble t ht).1.1.<a>aemeasurable</a>", [{"full_name": "Measurable", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [535, 5], "def_end_pos": [535, 15]}, {"full_name": "Set.Ioi", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [79, 5], "def_end_pos": [79, 8]}, {"full_name": "AEMeasurable.exists_measurable_nonneg", "def_path": "Mathlib/MeasureTheory/Measure/AEMeasurable.lean", "def_pos": [221, 9], "def_end_pos": [221, 33]}, {"full_name": "aemeasurable_Ioi_of_forall_Ioc", "def_path": "Mathlib/MeasureTheory/Measure/AEMeasurable.lean", "def_pos": [317, 9], "def_end_pos": [317, 39]}, {"full_name": "MeasureTheory.AEStronglyMeasurable.aemeasurable", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1220, 19], "def_end_pos": [1220, 31]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nf_mble : AEMeasurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t\n\u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =\n    \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nf_mble : AEMeasurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t\nG : \u211d \u2192 \u211d\nG_mble : Measurable G\nG_nn : 0 \u2264 G\ng_eq_G : g =\u1da0[ae (Measure.restrict volume (Ioi 0))] G\n\u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =\n    \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)"}, {"tactic": "have g_eq_G_on : \u2200 t, g =\u1d50[volume.restrict (Ioc 0 t)] G := fun t =>\n  ae_mono (Measure.restrict_mono Ioc_subset_Ioi_self le_rfl) g_eq_G", "annotated_tactic": ["have g_eq_G_on : \u2200 t, g =\u1d50[volume.restrict (<a>Ioc</a> 0 t)] G := fun t =>\n    <a>ae_mono</a> (<a>Measure.restrict_mono</a> <a>Ioc_subset_Ioi_self</a> <a>le_rfl</a>) g_eq_G", [{"full_name": "Set.Ioc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [69, 5], "def_end_pos": [69, 8]}, {"full_name": "MeasureTheory.ae_mono", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2456, 9], "def_end_pos": [2456, 16]}, {"full_name": "MeasureTheory.Measure.restrict_mono", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1550, 9], "def_end_pos": [1550, 22]}, {"full_name": "Set.Ioc_subset_Ioi_self", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [526, 9], "def_end_pos": [526, 28]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}]], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nf_mble : AEMeasurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t\nG : \u211d \u2192 \u211d\nG_mble : Measurable G\nG_nn : 0 \u2264 G\ng_eq_G : g =\u1da0[ae (Measure.restrict volume (Ioi 0))] G\n\u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =\n    \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nf_mble : AEMeasurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t\nG : \u211d \u2192 \u211d\nG_mble : Measurable G\nG_nn : 0 \u2264 G\ng_eq_G : g =\u1da0[ae (Measure.restrict volume (Ioi 0))] G\ng_eq_G_on : \u2200 (t : \u211d), g =\u1da0[ae (Measure.restrict volume (Ioc 0 t))] G\n\u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =\n    \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)"}, {"tactic": "have G_intble : \u2200 t > 0, IntervalIntegrable G volume 0 t := by\n  refine' fun t t_pos => \u27e8(g_intble t t_pos).1.congr_fun_ae (g_eq_G_on t), _\u27e9\n  rw [Ioc_eq_empty_of_le t_pos.lt.le]\n  exact integrableOn_empty", "annotated_tactic": ["have G_intble : \u2200 t > 0, <a>IntervalIntegrable</a> G <a>volume</a> 0 t := by\n    refine' fun t t_pos => \u27e8(g_intble t t_pos).1.<a>congr_fun_ae</a> (g_eq_G_on t), _\u27e9\n    rw [<a>Ioc_eq_empty_of_le</a> t_pos.lt.le]\n    exact <a>integrableOn_empty</a>", [{"full_name": "IntervalIntegrable", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [70, 5], "def_end_pos": [70, 23]}, {"full_name": "MeasureTheory.MeasureSpace.volume", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [663, 3], "def_end_pos": [663, 9]}, {"full_name": "MeasureTheory.IntegrableOn.congr_fun_ae", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [143, 9], "def_end_pos": [143, 34]}, {"full_name": "Set.Ioc_eq_empty_of_le", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [388, 9], "def_end_pos": [388, 27]}, {"full_name": "MeasureTheory.integrableOn_empty", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [106, 9], "def_end_pos": [106, 27]}]], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nf_mble : AEMeasurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t\nG : \u211d \u2192 \u211d\nG_mble : Measurable G\nG_nn : 0 \u2264 G\ng_eq_G : g =\u1da0[ae (Measure.restrict volume (Ioi 0))] G\ng_eq_G_on : \u2200 (t : \u211d), g =\u1da0[ae (Measure.restrict volume (Ioc 0 t))] G\n\u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =\n    \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nf_mble : AEMeasurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t\nG : \u211d \u2192 \u211d\nG_mble : Measurable G\nG_nn : 0 \u2264 G\ng_eq_G : g =\u1da0[ae (Measure.restrict volume (Ioi 0))] G\ng_eq_G_on : \u2200 (t : \u211d), g =\u1da0[ae (Measure.restrict volume (Ioc 0 t))] G\nG_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable G volume 0 t\n\u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =\n    \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)"}, {"tactic": "obtain \u27e8F, F_mble, F_nn, f_eq_F\u27e9 : \u2203 F : \u03b1 \u2192 \u211d, Measurable F \u2227 0 \u2264 F \u2227 f =\u1d50[\u03bc] F := by\n  refine \u27e8fun \u03c9 \u21a6 max (f_mble.mk f \u03c9) 0, f_mble.measurable_mk.max measurable_const,\n      fun \u03c9 \u21a6 le_max_right _ _, ?_\u27e9\n  filter_upwards [f_mble.ae_eq_mk, f_nn] with \u03c9 h\u03c9 h'\u03c9\n  rw [\u2190 h\u03c9]\n  exact (max_eq_left h'\u03c9).symm", "annotated_tactic": ["obtain \u27e8F, F_mble, F_nn, f_eq_F\u27e9 : \u2203 F : \u03b1 \u2192 \u211d, <a>Measurable</a> F \u2227 0 \u2264 F \u2227 f =\u1d50[\u03bc] F := by\n    refine \u27e8fun \u03c9 \u21a6 <a>max</a> (f_mble.mk f \u03c9) 0, f_mble.measurable_mk.max <a>measurable_const</a>,\n        fun \u03c9 \u21a6 <a>le_max_right</a> _ _, ?_\u27e9\n    filter_upwards [f_mble.ae_eq_mk, f_nn] with \u03c9 h\u03c9 h'\u03c9\n    rw [\u2190 h\u03c9]\n    exact (<a>max_eq_left</a> h'\u03c9).<a>symm</a>", [{"full_name": "Measurable", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [535, 5], "def_end_pos": [535, 15]}, {"full_name": "Max.max", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1090, 3], "def_end_pos": [1090, 6]}, {"full_name": "measurable_const", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [570, 9], "def_end_pos": [570, 25]}, {"full_name": "le_max_right", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [61, 9], "def_end_pos": [61, 21]}, {"full_name": "max_eq_left", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [133, 9], "def_end_pos": [133, 20]}, {"full_name": "Eq.symm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [310, 9], "def_end_pos": [310, 16]}]], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nf_mble : AEMeasurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t\nG : \u211d \u2192 \u211d\nG_mble : Measurable G\nG_nn : 0 \u2264 G\ng_eq_G : g =\u1da0[ae (Measure.restrict volume (Ioi 0))] G\ng_eq_G_on : \u2200 (t : \u211d), g =\u1da0[ae (Measure.restrict volume (Ioc 0 t))] G\nG_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable G volume 0 t\n\u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =\n    \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)", "state_after": "case intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nf_mble : AEMeasurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t\nG : \u211d \u2192 \u211d\nG_mble : Measurable G\nG_nn : 0 \u2264 G\ng_eq_G : g =\u1da0[ae (Measure.restrict volume (Ioi 0))] G\ng_eq_G_on : \u2200 (t : \u211d), g =\u1da0[ae (Measure.restrict volume (Ioc 0 t))] G\nG_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable G volume 0 t\nF : \u03b1 \u2192 \u211d\nF_mble : Measurable F\nF_nn : 0 \u2264 F\nf_eq_F : f =\u1da0[ae \u03bc] F\n\u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =\n    \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)"}, {"tactic": "have eq\u2081 :\n  (\u222b\u207b t in Ioi 0, \u03bc {a : \u03b1 | t \u2264 f a} * ENNReal.ofReal (g t)) =\n    \u222b\u207b t in Ioi 0, \u03bc {a : \u03b1 | t \u2264 F a} * ENNReal.ofReal (G t) := by\n  apply lintegral_congr_ae\n  filter_upwards [g_eq_G] with t ht\n  rw [ht]\n  congr 1\n  apply measure_congr\n  filter_upwards [f_eq_F] with a ha using by simp [setOf, ha]", "annotated_tactic": ["have eq\u2081 :\n    (\u222b\u207b t in <a>Ioi</a> 0, \u03bc {a : \u03b1 | t \u2264 f a} * <a>ENNReal.ofReal</a> (g t)) =\n      \u222b\u207b t in <a>Ioi</a> 0, \u03bc {a : \u03b1 | t \u2264 F a} * <a>ENNReal.ofReal</a> (G t) := by\n    apply <a>lintegral_congr_ae</a>\n    filter_upwards [g_eq_G] with t ht\n    rw [ht]\n    congr 1\n    apply <a>measure_congr</a>\n    filter_upwards [f_eq_F] with a ha using by simp [<a>setOf</a>, ha]", [{"full_name": "Set.Ioi", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [79, 5], "def_end_pos": [79, 8]}, {"full_name": "ENNReal.ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [172, 29], "def_end_pos": [172, 35]}, {"full_name": "Set.Ioi", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [79, 5], "def_end_pos": [79, 8]}, {"full_name": "ENNReal.ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [172, 29], "def_end_pos": [172, 35]}, {"full_name": "MeasureTheory.lintegral_congr_ae", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [304, 9], "def_end_pos": [304, 27]}, {"full_name": "MeasureTheory.measure_congr", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [608, 9], "def_end_pos": [608, 22]}, {"full_name": "setOf", "def_path": "Mathlib/Init/Set.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}]], "state_before": "case intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nf_mble : AEMeasurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t\nG : \u211d \u2192 \u211d\nG_mble : Measurable G\nG_nn : 0 \u2264 G\ng_eq_G : g =\u1da0[ae (Measure.restrict volume (Ioi 0))] G\ng_eq_G_on : \u2200 (t : \u211d), g =\u1da0[ae (Measure.restrict volume (Ioc 0 t))] G\nG_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable G volume 0 t\nF : \u03b1 \u2192 \u211d\nF_mble : Measurable F\nF_nn : 0 \u2264 F\nf_eq_F : f =\u1da0[ae \u03bc] F\n\u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =\n    \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)", "state_after": "case intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nf_mble : AEMeasurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t\nG : \u211d \u2192 \u211d\nG_mble : Measurable G\nG_nn : 0 \u2264 G\ng_eq_G : g =\u1da0[ae (Measure.restrict volume (Ioi 0))] G\ng_eq_G_on : \u2200 (t : \u211d), g =\u1da0[ae (Measure.restrict volume (Ioc 0 t))] G\nG_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable G volume 0 t\nF : \u03b1 \u2192 \u211d\nF_mble : Measurable F\nF_nn : 0 \u2264 F\nf_eq_F : f =\u1da0[ae \u03bc] F\neq\u2081 :\n  \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) =\n    \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 F a} * ENNReal.ofReal (G t)\n\u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =\n    \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)"}, {"tactic": "have eq\u2082 : \u2200\u1d50 \u03c9 \u2202\u03bc,\n    ENNReal.ofReal (\u222b t in (0)..f \u03c9, g t) = ENNReal.ofReal (\u222b t in (0)..F \u03c9, G t) := by\n  filter_upwards [f_eq_F] with \u03c9 f\u03c9_nn\n  rw [f\u03c9_nn]\n  congr 1\n  refine' intervalIntegral.integral_congr_ae _\n  have f\u03c9_nn : 0 \u2264 F \u03c9 := F_nn \u03c9\n  rw [uIoc_of_le f\u03c9_nn, \u2190\n    ae_restrict_iff' (measurableSet_Ioc : MeasurableSet (Ioc (0 : \u211d) (F \u03c9)))]\n  exact g_eq_G_on (F \u03c9)", "annotated_tactic": ["have eq\u2082 : \u2200\u1d50 \u03c9 \u2202\u03bc,\n      <a>ENNReal.ofReal</a> (\u222b t in (0)..f \u03c9, g t) = <a>ENNReal.ofReal</a> (\u222b t in (0)..F \u03c9, G t) := by\n    filter_upwards [f_eq_F] with \u03c9 f\u03c9_nn\n    rw [f\u03c9_nn]\n    congr 1\n    refine' <a>intervalIntegral.integral_congr_ae</a> _\n    have f\u03c9_nn : 0 \u2264 F \u03c9 := F_nn \u03c9\n    rw [<a>uIoc_of_le</a> f\u03c9_nn, \u2190\n      <a>ae_restrict_iff'</a> (<a>measurableSet_Ioc</a> : <a>MeasurableSet</a> (<a>Ioc</a> (0 : \u211d) (F \u03c9)))]\n    exact g_eq_G_on (F \u03c9)", [{"full_name": "ENNReal.ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [172, 29], "def_end_pos": [172, 35]}, {"full_name": "ENNReal.ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [172, 29], "def_end_pos": [172, 35]}, {"full_name": "intervalIntegral.integral_congr_ae", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [990, 9], "def_end_pos": [990, 26]}, {"full_name": "Set.uIoc_of_le", "def_path": "Mathlib/Data/Set/Intervals/UnorderedInterval.lean", "def_pos": [288, 15], "def_end_pos": [288, 25]}, {"full_name": "MeasureTheory.ae_restrict_iff'", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2572, 9], "def_end_pos": [2572, 25]}, {"full_name": "measurableSet_Ioc", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [589, 9], "def_end_pos": [589, 26]}, {"full_name": "MeasurableSet", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [64, 5], "def_end_pos": [64, 18]}, {"full_name": "Set.Ioc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [69, 5], "def_end_pos": [69, 8]}]], "state_before": "case intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nf_mble : AEMeasurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t\nG : \u211d \u2192 \u211d\nG_mble : Measurable G\nG_nn : 0 \u2264 G\ng_eq_G : g =\u1da0[ae (Measure.restrict volume (Ioi 0))] G\ng_eq_G_on : \u2200 (t : \u211d), g =\u1da0[ae (Measure.restrict volume (Ioc 0 t))] G\nG_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable G volume 0 t\nF : \u03b1 \u2192 \u211d\nF_mble : Measurable F\nF_nn : 0 \u2264 F\nf_eq_F : f =\u1da0[ae \u03bc] F\neq\u2081 :\n  \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) =\n    \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 F a} * ENNReal.ofReal (G t)\n\u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =\n    \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)", "state_after": "case intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nf_mble : AEMeasurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t\nG : \u211d \u2192 \u211d\nG_mble : Measurable G\nG_nn : 0 \u2264 G\ng_eq_G : g =\u1da0[ae (Measure.restrict volume (Ioi 0))] G\ng_eq_G_on : \u2200 (t : \u211d), g =\u1da0[ae (Measure.restrict volume (Ioc 0 t))] G\nG_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable G volume 0 t\nF : \u03b1 \u2192 \u211d\nF_mble : Measurable F\nF_nn : 0 \u2264 F\nf_eq_F : f =\u1da0[ae \u03bc] F\neq\u2081 :\n  \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) =\n    \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 F a} * ENNReal.ofReal (G t)\neq\u2082 : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = ENNReal.ofReal (\u222b (t : \u211d) in 0 ..F \u03c9, G t)\n\u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =\n    \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)"}, {"tactic": "simp_rw [lintegral_congr_ae eq\u2082, eq\u2081]", "annotated_tactic": ["simp_rw [<a>lintegral_congr_ae</a> eq\u2082, eq\u2081]", [{"full_name": "MeasureTheory.lintegral_congr_ae", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [304, 9], "def_end_pos": [304, 27]}]], "state_before": "case intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nf_mble : AEMeasurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t\nG : \u211d \u2192 \u211d\nG_mble : Measurable G\nG_nn : 0 \u2264 G\ng_eq_G : g =\u1da0[ae (Measure.restrict volume (Ioi 0))] G\ng_eq_G_on : \u2200 (t : \u211d), g =\u1da0[ae (Measure.restrict volume (Ioc 0 t))] G\nG_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable G volume 0 t\nF : \u03b1 \u2192 \u211d\nF_mble : Measurable F\nF_nn : 0 \u2264 F\nf_eq_F : f =\u1da0[ae \u03bc] F\neq\u2081 :\n  \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) =\n    \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 F a} * ENNReal.ofReal (G t)\neq\u2082 : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = ENNReal.ofReal (\u222b (t : \u211d) in 0 ..F \u03c9, G t)\n\u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =\n    \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)", "state_after": "case intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nf_mble : AEMeasurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t\nG : \u211d \u2192 \u211d\nG_mble : Measurable G\nG_nn : 0 \u2264 G\ng_eq_G : g =\u1da0[ae (Measure.restrict volume (Ioi 0))] G\ng_eq_G_on : \u2200 (t : \u211d), g =\u1da0[ae (Measure.restrict volume (Ioc 0 t))] G\nG_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable G volume 0 t\nF : \u03b1 \u2192 \u211d\nF_mble : Measurable F\nF_nn : 0 \u2264 F\nf_eq_F : f =\u1da0[ae \u03bc] F\neq\u2081 :\n  \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) =\n    \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 F a} * ENNReal.ofReal (G t)\neq\u2082 : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = ENNReal.ofReal (\u222b (t : \u211d) in 0 ..F \u03c9, G t)\n\u22a2 \u222b\u207b (a : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..F a, G t) \u2202\u03bc =\n    \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 F a} * ENNReal.ofReal (G t)"}, {"tactic": "exact lintegral_comp_eq_lintegral_meas_le_mul_of_measurable \u03bc F_nn F_mble\n        G_intble G_mble (fun t _ => G_nn t)", "annotated_tactic": ["exact <a>lintegral_comp_eq_lintegral_meas_le_mul_of_measurable</a> \u03bc F_nn F_mble\n          G_intble G_mble (fun t _ => G_nn t)", [{"full_name": "MeasureTheory.lintegral_comp_eq_lintegral_meas_le_mul_of_measurable", "def_path": "Mathlib/MeasureTheory/Integral/Layercake.lean", "def_pos": [195, 9], "def_end_pos": [195, 62]}]], "state_before": "case intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nf_mble : AEMeasurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t\nG : \u211d \u2192 \u211d\nG_mble : Measurable G\nG_nn : 0 \u2264 G\ng_eq_G : g =\u1da0[ae (Measure.restrict volume (Ioi 0))] G\ng_eq_G_on : \u2200 (t : \u211d), g =\u1da0[ae (Measure.restrict volume (Ioc 0 t))] G\nG_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable G volume 0 t\nF : \u03b1 \u2192 \u211d\nF_mble : Measurable F\nF_nn : 0 \u2264 F\nf_eq_F : f =\u1da0[ae \u03bc] F\neq\u2081 :\n  \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) =\n    \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 F a} * ENNReal.ofReal (G t)\neq\u2082 : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = ENNReal.ofReal (\u222b (t : \u211d) in 0 ..F \u03c9, G t)\n\u22a2 \u222b\u207b (a : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..F a, G t) \u2202\u03bc =\n    \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 F a} * ENNReal.ofReal (G t)", "state_after": "no goals"}, {"tactic": "refine' AEMeasurable.exists_measurable_nonneg _ g_nn", "annotated_tactic": ["refine' <a>AEMeasurable.exists_measurable_nonneg</a> _ g_nn", [{"full_name": "AEMeasurable.exists_measurable_nonneg", "def_path": "Mathlib/MeasureTheory/Measure/AEMeasurable.lean", "def_pos": [221, 9], "def_end_pos": [221, 33]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nf_mble : AEMeasurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t\n\u22a2 \u2203 G, Measurable G \u2227 0 \u2264 G \u2227 g =\u1da0[ae (Measure.restrict volume (Ioi 0))] G", "state_after": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nf_mble : AEMeasurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t\n\u22a2 AEMeasurable g"}, {"tactic": "exact aemeasurable_Ioi_of_forall_Ioc fun t ht => (g_intble t ht).1.1.aemeasurable", "annotated_tactic": ["exact <a>aemeasurable_Ioi_of_forall_Ioc</a> fun t ht => (g_intble t ht).1.1.<a>aemeasurable</a>", [{"full_name": "aemeasurable_Ioi_of_forall_Ioc", "def_path": "Mathlib/MeasureTheory/Measure/AEMeasurable.lean", "def_pos": [317, 9], "def_end_pos": [317, 39]}, {"full_name": "MeasureTheory.AEStronglyMeasurable.aemeasurable", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1220, 19], "def_end_pos": [1220, 31]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nf_mble : AEMeasurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t\n\u22a2 AEMeasurable g", "state_after": "no goals"}, {"tactic": "refine' fun t t_pos => \u27e8(g_intble t t_pos).1.congr_fun_ae (g_eq_G_on t), _\u27e9", "annotated_tactic": ["refine' fun t t_pos => \u27e8(g_intble t t_pos).1.<a>congr_fun_ae</a> (g_eq_G_on t), _\u27e9", [{"full_name": "MeasureTheory.IntegrableOn.congr_fun_ae", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [143, 9], "def_end_pos": [143, 34]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nf_mble : AEMeasurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t\nG : \u211d \u2192 \u211d\nG_mble : Measurable G\nG_nn : 0 \u2264 G\ng_eq_G : g =\u1da0[ae (Measure.restrict volume (Ioi 0))] G\ng_eq_G_on : \u2200 (t : \u211d), g =\u1da0[ae (Measure.restrict volume (Ioc 0 t))] G\n\u22a2 \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable G volume 0 t", "state_after": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nf_mble : AEMeasurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t\nG : \u211d \u2192 \u211d\nG_mble : Measurable G\nG_nn : 0 \u2264 G\ng_eq_G : g =\u1da0[ae (Measure.restrict volume (Ioi 0))] G\ng_eq_G_on : \u2200 (t : \u211d), g =\u1da0[ae (Measure.restrict volume (Ioc 0 t))] G\nt : \u211d\nt_pos : t > 0\n\u22a2 IntegrableOn G (Ioc t 0)"}, {"tactic": "rw [Ioc_eq_empty_of_le t_pos.lt.le]", "annotated_tactic": ["rw [<a>Ioc_eq_empty_of_le</a> t_pos.lt.le]", [{"full_name": "Set.Ioc_eq_empty_of_le", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [388, 9], "def_end_pos": [388, 27]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nf_mble : AEMeasurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t\nG : \u211d \u2192 \u211d\nG_mble : Measurable G\nG_nn : 0 \u2264 G\ng_eq_G : g =\u1da0[ae (Measure.restrict volume (Ioi 0))] G\ng_eq_G_on : \u2200 (t : \u211d), g =\u1da0[ae (Measure.restrict volume (Ioc 0 t))] G\nt : \u211d\nt_pos : t > 0\n\u22a2 IntegrableOn G (Ioc t 0)", "state_after": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nf_mble : AEMeasurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t\nG : \u211d \u2192 \u211d\nG_mble : Measurable G\nG_nn : 0 \u2264 G\ng_eq_G : g =\u1da0[ae (Measure.restrict volume (Ioi 0))] G\ng_eq_G_on : \u2200 (t : \u211d), g =\u1da0[ae (Measure.restrict volume (Ioc 0 t))] G\nt : \u211d\nt_pos : t > 0\n\u22a2 IntegrableOn G \u2205"}, {"tactic": "exact integrableOn_empty", "annotated_tactic": ["exact <a>integrableOn_empty</a>", [{"full_name": "MeasureTheory.integrableOn_empty", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [106, 9], "def_end_pos": [106, 27]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nf_mble : AEMeasurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t\nG : \u211d \u2192 \u211d\nG_mble : Measurable G\nG_nn : 0 \u2264 G\ng_eq_G : g =\u1da0[ae (Measure.restrict volume (Ioi 0))] G\ng_eq_G_on : \u2200 (t : \u211d), g =\u1da0[ae (Measure.restrict volume (Ioc 0 t))] G\nt : \u211d\nt_pos : t > 0\n\u22a2 IntegrableOn G \u2205", "state_after": "no goals"}, {"tactic": "refine \u27e8fun \u03c9 \u21a6 max (f_mble.mk f \u03c9) 0, f_mble.measurable_mk.max measurable_const,\n    fun \u03c9 \u21a6 le_max_right _ _, ?_\u27e9", "annotated_tactic": ["refine \u27e8fun \u03c9 \u21a6 <a>max</a> (f_mble.mk f \u03c9) 0, f_mble.measurable_mk.max <a>measurable_const</a>,\n        fun \u03c9 \u21a6 <a>le_max_right</a> _ _, ?_\u27e9", [{"full_name": "Max.max", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1090, 3], "def_end_pos": [1090, 6]}, {"full_name": "measurable_const", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [570, 9], "def_end_pos": [570, 25]}, {"full_name": "le_max_right", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [61, 9], "def_end_pos": [61, 21]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nf_mble : AEMeasurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t\nG : \u211d \u2192 \u211d\nG_mble : Measurable G\nG_nn : 0 \u2264 G\ng_eq_G : g =\u1da0[ae (Measure.restrict volume (Ioi 0))] G\ng_eq_G_on : \u2200 (t : \u211d), g =\u1da0[ae (Measure.restrict volume (Ioc 0 t))] G\nG_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable G volume 0 t\n\u22a2 \u2203 F, Measurable F \u2227 0 \u2264 F \u2227 f =\u1da0[ae \u03bc] F", "state_after": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nf_mble : AEMeasurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t\nG : \u211d \u2192 \u211d\nG_mble : Measurable G\nG_nn : 0 \u2264 G\ng_eq_G : g =\u1da0[ae (Measure.restrict volume (Ioi 0))] G\ng_eq_G_on : \u2200 (t : \u211d), g =\u1da0[ae (Measure.restrict volume (Ioc 0 t))] G\nG_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable G volume 0 t\n\u22a2 f =\u1da0[ae \u03bc] fun \u03c9 => max (AEMeasurable.mk f f_mble \u03c9) 0"}, {"tactic": "filter_upwards [f_mble.ae_eq_mk, f_nn] with \u03c9 h\u03c9 h'\u03c9", "annotated_tactic": ["filter_upwards [f_mble.ae_eq_mk, f_nn] with \u03c9 h\u03c9 h'\u03c9", []], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nf_mble : AEMeasurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t\nG : \u211d \u2192 \u211d\nG_mble : Measurable G\nG_nn : 0 \u2264 G\ng_eq_G : g =\u1da0[ae (Measure.restrict volume (Ioi 0))] G\ng_eq_G_on : \u2200 (t : \u211d), g =\u1da0[ae (Measure.restrict volume (Ioc 0 t))] G\nG_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable G volume 0 t\n\u22a2 f =\u1da0[ae \u03bc] fun \u03c9 => max (AEMeasurable.mk f f_mble \u03c9) 0", "state_after": "case h\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nf_mble : AEMeasurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t\nG : \u211d \u2192 \u211d\nG_mble : Measurable G\nG_nn : 0 \u2264 G\ng_eq_G : g =\u1da0[ae (Measure.restrict volume (Ioi 0))] G\ng_eq_G_on : \u2200 (t : \u211d), g =\u1da0[ae (Measure.restrict volume (Ioc 0 t))] G\nG_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable G volume 0 t\n\u03c9 : \u03b1\nh\u03c9 : f \u03c9 = AEMeasurable.mk f f_mble \u03c9\nh'\u03c9 : OfNat.ofNat 0 \u03c9 \u2264 f \u03c9\n\u22a2 f \u03c9 = max (AEMeasurable.mk f f_mble \u03c9) 0"}, {"tactic": "rw [\u2190 h\u03c9]", "annotated_tactic": ["rw [\u2190 h\u03c9]", []], "state_before": "case h\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nf_mble : AEMeasurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t\nG : \u211d \u2192 \u211d\nG_mble : Measurable G\nG_nn : 0 \u2264 G\ng_eq_G : g =\u1da0[ae (Measure.restrict volume (Ioi 0))] G\ng_eq_G_on : \u2200 (t : \u211d), g =\u1da0[ae (Measure.restrict volume (Ioc 0 t))] G\nG_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable G volume 0 t\n\u03c9 : \u03b1\nh\u03c9 : f \u03c9 = AEMeasurable.mk f f_mble \u03c9\nh'\u03c9 : OfNat.ofNat 0 \u03c9 \u2264 f \u03c9\n\u22a2 f \u03c9 = max (AEMeasurable.mk f f_mble \u03c9) 0", "state_after": "case h\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nf_mble : AEMeasurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t\nG : \u211d \u2192 \u211d\nG_mble : Measurable G\nG_nn : 0 \u2264 G\ng_eq_G : g =\u1da0[ae (Measure.restrict volume (Ioi 0))] G\ng_eq_G_on : \u2200 (t : \u211d), g =\u1da0[ae (Measure.restrict volume (Ioc 0 t))] G\nG_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable G volume 0 t\n\u03c9 : \u03b1\nh\u03c9 : f \u03c9 = AEMeasurable.mk f f_mble \u03c9\nh'\u03c9 : OfNat.ofNat 0 \u03c9 \u2264 f \u03c9\n\u22a2 f \u03c9 = max (f \u03c9) 0"}, {"tactic": "exact (max_eq_left h'\u03c9).symm", "annotated_tactic": ["exact (<a>max_eq_left</a> h'\u03c9).<a>symm</a>", [{"full_name": "max_eq_left", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [133, 9], "def_end_pos": [133, 20]}, {"full_name": "Eq.symm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [310, 9], "def_end_pos": [310, 16]}]], "state_before": "case h\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nf_mble : AEMeasurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t\nG : \u211d \u2192 \u211d\nG_mble : Measurable G\nG_nn : 0 \u2264 G\ng_eq_G : g =\u1da0[ae (Measure.restrict volume (Ioi 0))] G\ng_eq_G_on : \u2200 (t : \u211d), g =\u1da0[ae (Measure.restrict volume (Ioc 0 t))] G\nG_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable G volume 0 t\n\u03c9 : \u03b1\nh\u03c9 : f \u03c9 = AEMeasurable.mk f f_mble \u03c9\nh'\u03c9 : OfNat.ofNat 0 \u03c9 \u2264 f \u03c9\n\u22a2 f \u03c9 = max (f \u03c9) 0", "state_after": "no goals"}, {"tactic": "apply lintegral_congr_ae", "annotated_tactic": ["apply <a>lintegral_congr_ae</a>", [{"full_name": "MeasureTheory.lintegral_congr_ae", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [304, 9], "def_end_pos": [304, 27]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nf_mble : AEMeasurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t\nG : \u211d \u2192 \u211d\nG_mble : Measurable G\nG_nn : 0 \u2264 G\ng_eq_G : g =\u1da0[ae (Measure.restrict volume (Ioi 0))] G\ng_eq_G_on : \u2200 (t : \u211d), g =\u1da0[ae (Measure.restrict volume (Ioc 0 t))] G\nG_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable G volume 0 t\nF : \u03b1 \u2192 \u211d\nF_mble : Measurable F\nF_nn : 0 \u2264 F\nf_eq_F : f =\u1da0[ae \u03bc] F\n\u22a2 \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) =\n    \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 F a} * ENNReal.ofReal (G t)", "state_after": "case h\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nf_mble : AEMeasurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t\nG : \u211d \u2192 \u211d\nG_mble : Measurable G\nG_nn : 0 \u2264 G\ng_eq_G : g =\u1da0[ae (Measure.restrict volume (Ioi 0))] G\ng_eq_G_on : \u2200 (t : \u211d), g =\u1da0[ae (Measure.restrict volume (Ioc 0 t))] G\nG_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable G volume 0 t\nF : \u03b1 \u2192 \u211d\nF_mble : Measurable F\nF_nn : 0 \u2264 F\nf_eq_F : f =\u1da0[ae \u03bc] F\n\u22a2 (fun a => \u2191\u2191\u03bc {a_1 | a \u2264 f a_1} * ENNReal.ofReal (g a)) =\u1da0[ae (Measure.restrict volume (Ioi 0))] fun a =>\n    \u2191\u2191\u03bc {a_1 | a \u2264 F a_1} * ENNReal.ofReal (G a)"}, {"tactic": "filter_upwards [g_eq_G] with t ht", "annotated_tactic": ["filter_upwards [g_eq_G] with t ht", []], "state_before": "case h\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nf_mble : AEMeasurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t\nG : \u211d \u2192 \u211d\nG_mble : Measurable G\nG_nn : 0 \u2264 G\ng_eq_G : g =\u1da0[ae (Measure.restrict volume (Ioi 0))] G\ng_eq_G_on : \u2200 (t : \u211d), g =\u1da0[ae (Measure.restrict volume (Ioc 0 t))] G\nG_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable G volume 0 t\nF : \u03b1 \u2192 \u211d\nF_mble : Measurable F\nF_nn : 0 \u2264 F\nf_eq_F : f =\u1da0[ae \u03bc] F\n\u22a2 (fun a => \u2191\u2191\u03bc {a_1 | a \u2264 f a_1} * ENNReal.ofReal (g a)) =\u1da0[ae (Measure.restrict volume (Ioi 0))] fun a =>\n    \u2191\u2191\u03bc {a_1 | a \u2264 F a_1} * ENNReal.ofReal (G a)", "state_after": "case h\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nf_mble : AEMeasurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t\nG : \u211d \u2192 \u211d\nG_mble : Measurable G\nG_nn : 0 \u2264 G\ng_eq_G : g =\u1da0[ae (Measure.restrict volume (Ioi 0))] G\ng_eq_G_on : \u2200 (t : \u211d), g =\u1da0[ae (Measure.restrict volume (Ioc 0 t))] G\nG_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable G volume 0 t\nF : \u03b1 \u2192 \u211d\nF_mble : Measurable F\nF_nn : 0 \u2264 F\nf_eq_F : f =\u1da0[ae \u03bc] F\nt : \u211d\nht : g t = G t\n\u22a2 \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) = \u2191\u2191\u03bc {a | t \u2264 F a} * ENNReal.ofReal (G t)"}, {"tactic": "rw [ht]", "annotated_tactic": ["rw [ht]", []], "state_before": "case h\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nf_mble : AEMeasurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t\nG : \u211d \u2192 \u211d\nG_mble : Measurable G\nG_nn : 0 \u2264 G\ng_eq_G : g =\u1da0[ae (Measure.restrict volume (Ioi 0))] G\ng_eq_G_on : \u2200 (t : \u211d), g =\u1da0[ae (Measure.restrict volume (Ioc 0 t))] G\nG_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable G volume 0 t\nF : \u03b1 \u2192 \u211d\nF_mble : Measurable F\nF_nn : 0 \u2264 F\nf_eq_F : f =\u1da0[ae \u03bc] F\nt : \u211d\nht : g t = G t\n\u22a2 \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) = \u2191\u2191\u03bc {a | t \u2264 F a} * ENNReal.ofReal (G t)", "state_after": "case h\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nf_mble : AEMeasurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t\nG : \u211d \u2192 \u211d\nG_mble : Measurable G\nG_nn : 0 \u2264 G\ng_eq_G : g =\u1da0[ae (Measure.restrict volume (Ioi 0))] G\ng_eq_G_on : \u2200 (t : \u211d), g =\u1da0[ae (Measure.restrict volume (Ioc 0 t))] G\nG_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable G volume 0 t\nF : \u03b1 \u2192 \u211d\nF_mble : Measurable F\nF_nn : 0 \u2264 F\nf_eq_F : f =\u1da0[ae \u03bc] F\nt : \u211d\nht : g t = G t\n\u22a2 \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (G t) = \u2191\u2191\u03bc {a | t \u2264 F a} * ENNReal.ofReal (G t)"}, {"tactic": "congr 1", "annotated_tactic": ["congr 1", []], "state_before": "case h\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nf_mble : AEMeasurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t\nG : \u211d \u2192 \u211d\nG_mble : Measurable G\nG_nn : 0 \u2264 G\ng_eq_G : g =\u1da0[ae (Measure.restrict volume (Ioi 0))] G\ng_eq_G_on : \u2200 (t : \u211d), g =\u1da0[ae (Measure.restrict volume (Ioc 0 t))] G\nG_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable G volume 0 t\nF : \u03b1 \u2192 \u211d\nF_mble : Measurable F\nF_nn : 0 \u2264 F\nf_eq_F : f =\u1da0[ae \u03bc] F\nt : \u211d\nht : g t = G t\n\u22a2 \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (G t) = \u2191\u2191\u03bc {a | t \u2264 F a} * ENNReal.ofReal (G t)", "state_after": "case h.e_a\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nf_mble : AEMeasurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t\nG : \u211d \u2192 \u211d\nG_mble : Measurable G\nG_nn : 0 \u2264 G\ng_eq_G : g =\u1da0[ae (Measure.restrict volume (Ioi 0))] G\ng_eq_G_on : \u2200 (t : \u211d), g =\u1da0[ae (Measure.restrict volume (Ioc 0 t))] G\nG_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable G volume 0 t\nF : \u03b1 \u2192 \u211d\nF_mble : Measurable F\nF_nn : 0 \u2264 F\nf_eq_F : f =\u1da0[ae \u03bc] F\nt : \u211d\nht : g t = G t\n\u22a2 \u2191\u2191\u03bc {a | t \u2264 f a} = \u2191\u2191\u03bc {a | t \u2264 F a}"}, {"tactic": "apply measure_congr", "annotated_tactic": ["apply <a>measure_congr</a>", [{"full_name": "MeasureTheory.measure_congr", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [608, 9], "def_end_pos": [608, 22]}]], "state_before": "case h.e_a\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nf_mble : AEMeasurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t\nG : \u211d \u2192 \u211d\nG_mble : Measurable G\nG_nn : 0 \u2264 G\ng_eq_G : g =\u1da0[ae (Measure.restrict volume (Ioi 0))] G\ng_eq_G_on : \u2200 (t : \u211d), g =\u1da0[ae (Measure.restrict volume (Ioc 0 t))] G\nG_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable G volume 0 t\nF : \u03b1 \u2192 \u211d\nF_mble : Measurable F\nF_nn : 0 \u2264 F\nf_eq_F : f =\u1da0[ae \u03bc] F\nt : \u211d\nht : g t = G t\n\u22a2 \u2191\u2191\u03bc {a | t \u2264 f a} = \u2191\u2191\u03bc {a | t \u2264 F a}", "state_after": "case h.e_a.H\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nf_mble : AEMeasurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t\nG : \u211d \u2192 \u211d\nG_mble : Measurable G\nG_nn : 0 \u2264 G\ng_eq_G : g =\u1da0[ae (Measure.restrict volume (Ioi 0))] G\ng_eq_G_on : \u2200 (t : \u211d), g =\u1da0[ae (Measure.restrict volume (Ioc 0 t))] G\nG_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable G volume 0 t\nF : \u03b1 \u2192 \u211d\nF_mble : Measurable F\nF_nn : 0 \u2264 F\nf_eq_F : f =\u1da0[ae \u03bc] F\nt : \u211d\nht : g t = G t\n\u22a2 {a | t \u2264 f a} =\u1da0[ae \u03bc] {a | t \u2264 F a}"}, {"tactic": "filter_upwards [f_eq_F] with a ha using by simp [setOf, ha]", "annotated_tactic": ["filter_upwards [f_eq_F] with a ha using by simp [<a>setOf</a>, ha]", [{"full_name": "setOf", "def_path": "Mathlib/Init/Set.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}]], "state_before": "case h.e_a.H\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nf_mble : AEMeasurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t\nG : \u211d \u2192 \u211d\nG_mble : Measurable G\nG_nn : 0 \u2264 G\ng_eq_G : g =\u1da0[ae (Measure.restrict volume (Ioi 0))] G\ng_eq_G_on : \u2200 (t : \u211d), g =\u1da0[ae (Measure.restrict volume (Ioc 0 t))] G\nG_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable G volume 0 t\nF : \u03b1 \u2192 \u211d\nF_mble : Measurable F\nF_nn : 0 \u2264 F\nf_eq_F : f =\u1da0[ae \u03bc] F\nt : \u211d\nht : g t = G t\n\u22a2 {a | t \u2264 f a} =\u1da0[ae \u03bc] {a | t \u2264 F a}", "state_after": "no goals"}, {"tactic": "simp [setOf, ha]", "annotated_tactic": ["simp [<a>setOf</a>, ha]", [{"full_name": "setOf", "def_path": "Mathlib/Init/Set.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nf_mble : AEMeasurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t\nG : \u211d \u2192 \u211d\nG_mble : Measurable G\nG_nn : 0 \u2264 G\ng_eq_G : g =\u1da0[ae (Measure.restrict volume (Ioi 0))] G\ng_eq_G_on : \u2200 (t : \u211d), g =\u1da0[ae (Measure.restrict volume (Ioc 0 t))] G\nG_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable G volume 0 t\nF : \u03b1 \u2192 \u211d\nF_mble : Measurable F\nF_nn : 0 \u2264 F\nf_eq_F : f =\u1da0[ae \u03bc] F\nt : \u211d\nht : g t = G t\na : \u03b1\nha : f a = F a\n\u22a2 setOf (fun a => t \u2264 f a) a = setOf (fun a => t \u2264 F a) a", "state_after": "no goals"}, {"tactic": "filter_upwards [f_eq_F] with \u03c9 f\u03c9_nn", "annotated_tactic": ["filter_upwards [f_eq_F] with \u03c9 f\u03c9_nn", []], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nf_mble : AEMeasurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t\nG : \u211d \u2192 \u211d\nG_mble : Measurable G\nG_nn : 0 \u2264 G\ng_eq_G : g =\u1da0[ae (Measure.restrict volume (Ioi 0))] G\ng_eq_G_on : \u2200 (t : \u211d), g =\u1da0[ae (Measure.restrict volume (Ioc 0 t))] G\nG_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable G volume 0 t\nF : \u03b1 \u2192 \u211d\nF_mble : Measurable F\nF_nn : 0 \u2264 F\nf_eq_F : f =\u1da0[ae \u03bc] F\neq\u2081 :\n  \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) =\n    \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 F a} * ENNReal.ofReal (G t)\n\u22a2 \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = ENNReal.ofReal (\u222b (t : \u211d) in 0 ..F \u03c9, G t)", "state_after": "case h\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nf_mble : AEMeasurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t\nG : \u211d \u2192 \u211d\nG_mble : Measurable G\nG_nn : 0 \u2264 G\ng_eq_G : g =\u1da0[ae (Measure.restrict volume (Ioi 0))] G\ng_eq_G_on : \u2200 (t : \u211d), g =\u1da0[ae (Measure.restrict volume (Ioc 0 t))] G\nG_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable G volume 0 t\nF : \u03b1 \u2192 \u211d\nF_mble : Measurable F\nF_nn : 0 \u2264 F\nf_eq_F : f =\u1da0[ae \u03bc] F\neq\u2081 :\n  \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) =\n    \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 F a} * ENNReal.ofReal (G t)\n\u03c9 : \u03b1\nf\u03c9_nn : f \u03c9 = F \u03c9\n\u22a2 ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = ENNReal.ofReal (\u222b (t : \u211d) in 0 ..F \u03c9, G t)"}, {"tactic": "rw [f\u03c9_nn]", "annotated_tactic": ["rw [f\u03c9_nn]", []], "state_before": "case h\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nf_mble : AEMeasurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t\nG : \u211d \u2192 \u211d\nG_mble : Measurable G\nG_nn : 0 \u2264 G\ng_eq_G : g =\u1da0[ae (Measure.restrict volume (Ioi 0))] G\ng_eq_G_on : \u2200 (t : \u211d), g =\u1da0[ae (Measure.restrict volume (Ioc 0 t))] G\nG_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable G volume 0 t\nF : \u03b1 \u2192 \u211d\nF_mble : Measurable F\nF_nn : 0 \u2264 F\nf_eq_F : f =\u1da0[ae \u03bc] F\neq\u2081 :\n  \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) =\n    \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 F a} * ENNReal.ofReal (G t)\n\u03c9 : \u03b1\nf\u03c9_nn : f \u03c9 = F \u03c9\n\u22a2 ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = ENNReal.ofReal (\u222b (t : \u211d) in 0 ..F \u03c9, G t)", "state_after": "case h\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nf_mble : AEMeasurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t\nG : \u211d \u2192 \u211d\nG_mble : Measurable G\nG_nn : 0 \u2264 G\ng_eq_G : g =\u1da0[ae (Measure.restrict volume (Ioi 0))] G\ng_eq_G_on : \u2200 (t : \u211d), g =\u1da0[ae (Measure.restrict volume (Ioc 0 t))] G\nG_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable G volume 0 t\nF : \u03b1 \u2192 \u211d\nF_mble : Measurable F\nF_nn : 0 \u2264 F\nf_eq_F : f =\u1da0[ae \u03bc] F\neq\u2081 :\n  \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) =\n    \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 F a} * ENNReal.ofReal (G t)\n\u03c9 : \u03b1\nf\u03c9_nn : f \u03c9 = F \u03c9\n\u22a2 ENNReal.ofReal (\u222b (t : \u211d) in 0 ..F \u03c9, g t) = ENNReal.ofReal (\u222b (t : \u211d) in 0 ..F \u03c9, G t)"}, {"tactic": "congr 1", "annotated_tactic": ["congr 1", []], "state_before": "case h\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nf_mble : AEMeasurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t\nG : \u211d \u2192 \u211d\nG_mble : Measurable G\nG_nn : 0 \u2264 G\ng_eq_G : g =\u1da0[ae (Measure.restrict volume (Ioi 0))] G\ng_eq_G_on : \u2200 (t : \u211d), g =\u1da0[ae (Measure.restrict volume (Ioc 0 t))] G\nG_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable G volume 0 t\nF : \u03b1 \u2192 \u211d\nF_mble : Measurable F\nF_nn : 0 \u2264 F\nf_eq_F : f =\u1da0[ae \u03bc] F\neq\u2081 :\n  \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) =\n    \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 F a} * ENNReal.ofReal (G t)\n\u03c9 : \u03b1\nf\u03c9_nn : f \u03c9 = F \u03c9\n\u22a2 ENNReal.ofReal (\u222b (t : \u211d) in 0 ..F \u03c9, g t) = ENNReal.ofReal (\u222b (t : \u211d) in 0 ..F \u03c9, G t)", "state_after": "case h.e_r\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nf_mble : AEMeasurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t\nG : \u211d \u2192 \u211d\nG_mble : Measurable G\nG_nn : 0 \u2264 G\ng_eq_G : g =\u1da0[ae (Measure.restrict volume (Ioi 0))] G\ng_eq_G_on : \u2200 (t : \u211d), g =\u1da0[ae (Measure.restrict volume (Ioc 0 t))] G\nG_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable G volume 0 t\nF : \u03b1 \u2192 \u211d\nF_mble : Measurable F\nF_nn : 0 \u2264 F\nf_eq_F : f =\u1da0[ae \u03bc] F\neq\u2081 :\n  \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) =\n    \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 F a} * ENNReal.ofReal (G t)\n\u03c9 : \u03b1\nf\u03c9_nn : f \u03c9 = F \u03c9\n\u22a2 \u222b (t : \u211d) in 0 ..F \u03c9, g t = \u222b (t : \u211d) in 0 ..F \u03c9, G t"}, {"tactic": "refine' intervalIntegral.integral_congr_ae _", "annotated_tactic": ["refine' <a>intervalIntegral.integral_congr_ae</a> _", [{"full_name": "intervalIntegral.integral_congr_ae", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [990, 9], "def_end_pos": [990, 26]}]], "state_before": "case h.e_r\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nf_mble : AEMeasurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t\nG : \u211d \u2192 \u211d\nG_mble : Measurable G\nG_nn : 0 \u2264 G\ng_eq_G : g =\u1da0[ae (Measure.restrict volume (Ioi 0))] G\ng_eq_G_on : \u2200 (t : \u211d), g =\u1da0[ae (Measure.restrict volume (Ioc 0 t))] G\nG_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable G volume 0 t\nF : \u03b1 \u2192 \u211d\nF_mble : Measurable F\nF_nn : 0 \u2264 F\nf_eq_F : f =\u1da0[ae \u03bc] F\neq\u2081 :\n  \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) =\n    \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 F a} * ENNReal.ofReal (G t)\n\u03c9 : \u03b1\nf\u03c9_nn : f \u03c9 = F \u03c9\n\u22a2 \u222b (t : \u211d) in 0 ..F \u03c9, g t = \u222b (t : \u211d) in 0 ..F \u03c9, G t", "state_after": "case h.e_r\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nf_mble : AEMeasurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t\nG : \u211d \u2192 \u211d\nG_mble : Measurable G\nG_nn : 0 \u2264 G\ng_eq_G : g =\u1da0[ae (Measure.restrict volume (Ioi 0))] G\ng_eq_G_on : \u2200 (t : \u211d), g =\u1da0[ae (Measure.restrict volume (Ioc 0 t))] G\nG_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable G volume 0 t\nF : \u03b1 \u2192 \u211d\nF_mble : Measurable F\nF_nn : 0 \u2264 F\nf_eq_F : f =\u1da0[ae \u03bc] F\neq\u2081 :\n  \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) =\n    \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 F a} * ENNReal.ofReal (G t)\n\u03c9 : \u03b1\nf\u03c9_nn : f \u03c9 = F \u03c9\n\u22a2 \u2200\u1d50 (x : \u211d), x \u2208 \u0399 0 (F \u03c9) \u2192 g x = G x"}, {"tactic": "have f\u03c9_nn : 0 \u2264 F \u03c9 := F_nn \u03c9", "annotated_tactic": ["have f\u03c9_nn : 0 \u2264 F \u03c9 := F_nn \u03c9", []], "state_before": "case h.e_r\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nf_mble : AEMeasurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t\nG : \u211d \u2192 \u211d\nG_mble : Measurable G\nG_nn : 0 \u2264 G\ng_eq_G : g =\u1da0[ae (Measure.restrict volume (Ioi 0))] G\ng_eq_G_on : \u2200 (t : \u211d), g =\u1da0[ae (Measure.restrict volume (Ioc 0 t))] G\nG_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable G volume 0 t\nF : \u03b1 \u2192 \u211d\nF_mble : Measurable F\nF_nn : 0 \u2264 F\nf_eq_F : f =\u1da0[ae \u03bc] F\neq\u2081 :\n  \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) =\n    \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 F a} * ENNReal.ofReal (G t)\n\u03c9 : \u03b1\nf\u03c9_nn : f \u03c9 = F \u03c9\n\u22a2 \u2200\u1d50 (x : \u211d), x \u2208 \u0399 0 (F \u03c9) \u2192 g x = G x", "state_after": "case h.e_r\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nf_mble : AEMeasurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t\nG : \u211d \u2192 \u211d\nG_mble : Measurable G\nG_nn : 0 \u2264 G\ng_eq_G : g =\u1da0[ae (Measure.restrict volume (Ioi 0))] G\ng_eq_G_on : \u2200 (t : \u211d), g =\u1da0[ae (Measure.restrict volume (Ioc 0 t))] G\nG_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable G volume 0 t\nF : \u03b1 \u2192 \u211d\nF_mble : Measurable F\nF_nn : 0 \u2264 F\nf_eq_F : f =\u1da0[ae \u03bc] F\neq\u2081 :\n  \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) =\n    \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 F a} * ENNReal.ofReal (G t)\n\u03c9 : \u03b1\nf\u03c9_nn\u271d : f \u03c9 = F \u03c9\nf\u03c9_nn : 0 \u2264 F \u03c9\n\u22a2 \u2200\u1d50 (x : \u211d), x \u2208 \u0399 0 (F \u03c9) \u2192 g x = G x"}, {"tactic": "rw [uIoc_of_le f\u03c9_nn, \u2190\n  ae_restrict_iff' (measurableSet_Ioc : MeasurableSet (Ioc (0 : \u211d) (F \u03c9)))]", "annotated_tactic": ["rw [<a>uIoc_of_le</a> f\u03c9_nn, \u2190\n      <a>ae_restrict_iff'</a> (<a>measurableSet_Ioc</a> : <a>MeasurableSet</a> (<a>Ioc</a> (0 : \u211d) (F \u03c9)))]", [{"full_name": "Set.uIoc_of_le", "def_path": "Mathlib/Data/Set/Intervals/UnorderedInterval.lean", "def_pos": [288, 15], "def_end_pos": [288, 25]}, {"full_name": "MeasureTheory.ae_restrict_iff'", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2572, 9], "def_end_pos": [2572, 25]}, {"full_name": "measurableSet_Ioc", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [589, 9], "def_end_pos": [589, 26]}, {"full_name": "MeasurableSet", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [64, 5], "def_end_pos": [64, 18]}, {"full_name": "Set.Ioc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [69, 5], "def_end_pos": [69, 8]}]], "state_before": "case h.e_r\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nf_mble : AEMeasurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t\nG : \u211d \u2192 \u211d\nG_mble : Measurable G\nG_nn : 0 \u2264 G\ng_eq_G : g =\u1da0[ae (Measure.restrict volume (Ioi 0))] G\ng_eq_G_on : \u2200 (t : \u211d), g =\u1da0[ae (Measure.restrict volume (Ioc 0 t))] G\nG_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable G volume 0 t\nF : \u03b1 \u2192 \u211d\nF_mble : Measurable F\nF_nn : 0 \u2264 F\nf_eq_F : f =\u1da0[ae \u03bc] F\neq\u2081 :\n  \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) =\n    \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 F a} * ENNReal.ofReal (G t)\n\u03c9 : \u03b1\nf\u03c9_nn\u271d : f \u03c9 = F \u03c9\nf\u03c9_nn : 0 \u2264 F \u03c9\n\u22a2 \u2200\u1d50 (x : \u211d), x \u2208 \u0399 0 (F \u03c9) \u2192 g x = G x", "state_after": "case h.e_r\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nf_mble : AEMeasurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t\nG : \u211d \u2192 \u211d\nG_mble : Measurable G\nG_nn : 0 \u2264 G\ng_eq_G : g =\u1da0[ae (Measure.restrict volume (Ioi 0))] G\ng_eq_G_on : \u2200 (t : \u211d), g =\u1da0[ae (Measure.restrict volume (Ioc 0 t))] G\nG_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable G volume 0 t\nF : \u03b1 \u2192 \u211d\nF_mble : Measurable F\nF_nn : 0 \u2264 F\nf_eq_F : f =\u1da0[ae \u03bc] F\neq\u2081 :\n  \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) =\n    \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 F a} * ENNReal.ofReal (G t)\n\u03c9 : \u03b1\nf\u03c9_nn\u271d : f \u03c9 = F \u03c9\nf\u03c9_nn : 0 \u2264 F \u03c9\n\u22a2 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Ioc 0 (F \u03c9)), g x = G x"}, {"tactic": "exact g_eq_G_on (F \u03c9)", "annotated_tactic": ["exact g_eq_G_on (F \u03c9)", []], "state_before": "case h.e_r\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nf_mble : AEMeasurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t\nG : \u211d \u2192 \u211d\nG_mble : Measurable G\nG_nn : 0 \u2264 G\ng_eq_G : g =\u1da0[ae (Measure.restrict volume (Ioi 0))] G\ng_eq_G_on : \u2200 (t : \u211d), g =\u1da0[ae (Measure.restrict volume (Ioc 0 t))] G\nG_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable G volume 0 t\nF : \u03b1 \u2192 \u211d\nF_mble : Measurable F\nF_nn : 0 \u2264 F\nf_eq_F : f =\u1da0[ae \u03bc] F\neq\u2081 :\n  \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) =\n    \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 F a} * ENNReal.ofReal (G t)\n\u03c9 : \u03b1\nf\u03c9_nn\u271d : f \u03c9 = F \u03c9\nf\u03c9_nn : 0 \u2264 F \u03c9\n\u22a2 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Ioc 0 (F \u03c9)), g x = G x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/WithDensity.lean", "full_name": "MeasureTheory.lintegral_withDensity_eq_lintegral_mul\u2080'", "start": [298, 1], "end": [327, 35], "traced_tactics": [{"tactic": "let f' := hf.mk f", "annotated_tactic": ["let f' := hf.mk f", []], "state_before": "\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\ng : \u03b1 \u2192 \u211d\u22650\u221e\nhg : AEMeasurable g\n\u22a2 \u222b\u207b (a : \u03b1), g a \u2202withDensity \u03bc f = \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc", "state_after": "\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\ng : \u03b1 \u2192 \u211d\u22650\u221e\nhg : AEMeasurable g\nf' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk f hf\n\u22a2 \u222b\u207b (a : \u03b1), g a \u2202withDensity \u03bc f = \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc"}, {"tactic": "have : \u03bc.withDensity f = \u03bc.withDensity f' := withDensity_congr_ae hf.ae_eq_mk", "annotated_tactic": ["have : \u03bc.withDensity f = \u03bc.withDensity f' := <a>withDensity_congr_ae</a> hf.ae_eq_mk", [{"full_name": "MeasureTheory.withDensity_congr_ae", "def_path": "Mathlib/MeasureTheory/Measure/WithDensity.lean", "def_pos": [50, 9], "def_end_pos": [50, 29]}]], "state_before": "\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\ng : \u03b1 \u2192 \u211d\u22650\u221e\nhg : AEMeasurable g\nf' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk f hf\n\u22a2 \u222b\u207b (a : \u03b1), g a \u2202withDensity \u03bc f = \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc", "state_after": "\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\ng : \u03b1 \u2192 \u211d\u22650\u221e\nhg : AEMeasurable g\nf' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk f hf\nthis : withDensity \u03bc f = withDensity \u03bc f'\n\u22a2 \u222b\u207b (a : \u03b1), g a \u2202withDensity \u03bc f = \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc"}, {"tactic": "rw [this] at hg \u22a2", "annotated_tactic": ["rw [this] at hg \u22a2", []], "state_before": "\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\ng : \u03b1 \u2192 \u211d\u22650\u221e\nhg : AEMeasurable g\nf' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk f hf\nthis : withDensity \u03bc f = withDensity \u03bc f'\n\u22a2 \u222b\u207b (a : \u03b1), g a \u2202withDensity \u03bc f = \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc", "state_after": "\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk f hf\nhg : AEMeasurable g\nthis : withDensity \u03bc f = withDensity \u03bc f'\n\u22a2 \u222b\u207b (a : \u03b1), g a \u2202withDensity \u03bc f' = \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc"}, {"tactic": "let g' := hg.mk g", "annotated_tactic": ["let g' := hg.mk g", []], "state_before": "\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk f hf\nhg : AEMeasurable g\nthis : withDensity \u03bc f = withDensity \u03bc f'\n\u22a2 \u222b\u207b (a : \u03b1), g a \u2202withDensity \u03bc f' = \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc", "state_after": "\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk f hf\nhg : AEMeasurable g\nthis : withDensity \u03bc f = withDensity \u03bc f'\ng' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk g hg\n\u22a2 \u222b\u207b (a : \u03b1), g a \u2202withDensity \u03bc f' = \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc"}, {"tactic": "apply lintegral_congr_ae", "annotated_tactic": ["apply <a>lintegral_congr_ae</a>", [{"full_name": "MeasureTheory.lintegral_congr_ae", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [304, 9], "def_end_pos": [304, 27]}]], "state_before": "\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk f hf\nhg : AEMeasurable g\nthis : withDensity \u03bc f = withDensity \u03bc f'\ng' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk g hg\n\u22a2 \u222b\u207b (a : \u03b1), (f' * g') a \u2202\u03bc = \u222b\u207b (a : \u03b1), (f' * g) a \u2202\u03bc", "state_after": "case h\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk f hf\nhg : AEMeasurable g\nthis : withDensity \u03bc f = withDensity \u03bc f'\ng' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk g hg\n\u22a2 (fun a => (f' * g') a) =\u1da0[ae \u03bc] fun a => (f' * g) a"}, {"tactic": "apply ae_of_ae_restrict_of_ae_restrict_compl { x | f' x \u2260 0 }", "annotated_tactic": ["apply <a>ae_of_ae_restrict_of_ae_restrict_compl</a> { x | f' x \u2260 0 }", [{"full_name": "MeasureTheory.ae_of_ae_restrict_of_ae_restrict_compl", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2611, 9], "def_end_pos": [2611, 47]}]], "state_before": "case h\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk f hf\nhg : AEMeasurable g\nthis : withDensity \u03bc f = withDensity \u03bc f'\ng' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk g hg\n\u22a2 (fun a => (f' * g') a) =\u1da0[ae \u03bc] fun a => (f' * g) a", "state_after": "case h.ht\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk f hf\nhg : AEMeasurable g\nthis : withDensity \u03bc f = withDensity \u03bc f'\ng' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk g hg\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc {x | f' x \u2260 0}, (fun a => (f' * g') a) x = (fun a => (f' * g) a) x\n\ncase h.htc\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk f hf\nhg : AEMeasurable g\nthis : withDensity \u03bc f = withDensity \u03bc f'\ng' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk g hg\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc {x | f' x \u2260 0}\u1d9c, (fun a => (f' * g') a) x = (fun a => (f' * g) a) x"}, {"tactic": "have Z := hg.ae_eq_mk", "annotated_tactic": ["have Z := hg.ae_eq_mk", []], "state_before": "case h.ht\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk f hf\nhg : AEMeasurable g\nthis : withDensity \u03bc f = withDensity \u03bc f'\ng' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk g hg\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc {x | f' x \u2260 0}, (fun a => (f' * g') a) x = (fun a => (f' * g) a) x", "state_after": "case h.ht\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk f hf\nhg : AEMeasurable g\nthis : withDensity \u03bc f = withDensity \u03bc f'\ng' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk g hg\nZ : g =\u1da0[ae (withDensity \u03bc f')] AEMeasurable.mk g hg\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc {x | f' x \u2260 0}, (fun a => (f' * g') a) x = (fun a => (f' * g) a) x"}, {"tactic": "rw [EventuallyEq, ae_withDensity_iff_ae_restrict hf.measurable_mk] at Z", "annotated_tactic": ["rw [<a>EventuallyEq</a>, <a>ae_withDensity_iff_ae_restrict</a> hf.measurable_mk] at Z", [{"full_name": "Filter.EventuallyEq", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1438, 5], "def_end_pos": [1438, 17]}, {"full_name": "MeasureTheory.ae_withDensity_iff_ae_restrict", "def_path": "Mathlib/MeasureTheory/Measure/WithDensity.lean", "def_pos": [231, 9], "def_end_pos": [231, 39]}]], "state_before": "case h.ht\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk f hf\nhg : AEMeasurable g\nthis : withDensity \u03bc f = withDensity \u03bc f'\ng' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk g hg\nZ : g =\u1da0[ae (withDensity \u03bc f')] AEMeasurable.mk g hg\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc {x | f' x \u2260 0}, (fun a => (f' * g') a) x = (fun a => (f' * g) a) x", "state_after": "case h.ht\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk f hf\nhg : AEMeasurable g\nthis : withDensity \u03bc f = withDensity \u03bc f'\ng' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk g hg\nZ : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc {x | AEMeasurable.mk f hf x \u2260 0}, g x = AEMeasurable.mk g hg x\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc {x | f' x \u2260 0}, (fun a => (f' * g') a) x = (fun a => (f' * g) a) x"}, {"tactic": "filter_upwards [Z]", "annotated_tactic": ["filter_upwards [Z]", []], "state_before": "case h.ht\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk f hf\nhg : AEMeasurable g\nthis : withDensity \u03bc f = withDensity \u03bc f'\ng' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk g hg\nZ : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc {x | AEMeasurable.mk f hf x \u2260 0}, g x = AEMeasurable.mk g hg x\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc {x | f' x \u2260 0}, (fun a => (f' * g') a) x = (fun a => (f' * g) a) x", "state_after": "case h\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk f hf\nhg : AEMeasurable g\nthis : withDensity \u03bc f = withDensity \u03bc f'\ng' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk g hg\nZ : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc {x | AEMeasurable.mk f hf x \u2260 0}, g x = AEMeasurable.mk g hg x\n\u22a2 \u2200 (a : \u03b1), g a = AEMeasurable.mk g hg a \u2192 (f' * g') a = (f' * g) a"}, {"tactic": "intro x hx", "annotated_tactic": ["intro x hx", []], "state_before": "case h\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk f hf\nhg : AEMeasurable g\nthis : withDensity \u03bc f = withDensity \u03bc f'\ng' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk g hg\nZ : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc {x | AEMeasurable.mk f hf x \u2260 0}, g x = AEMeasurable.mk g hg x\n\u22a2 \u2200 (a : \u03b1), g a = AEMeasurable.mk g hg a \u2192 (f' * g') a = (f' * g) a", "state_after": "case h\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk f hf\nhg : AEMeasurable g\nthis : withDensity \u03bc f = withDensity \u03bc f'\ng' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk g hg\nZ : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc {x | AEMeasurable.mk f hf x \u2260 0}, g x = AEMeasurable.mk g hg x\nx : \u03b1\nhx : g x = AEMeasurable.mk g hg x\n\u22a2 (f' * g') x = (f' * g) x"}, {"tactic": "simp only [hx, Pi.mul_apply]", "annotated_tactic": ["simp only [hx, <a>Pi.mul_apply</a>]", [{"full_name": "Pi.mul_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [83, 9], "def_end_pos": [83, 18]}]], "state_before": "case h\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk f hf\nhg : AEMeasurable g\nthis : withDensity \u03bc f = withDensity \u03bc f'\ng' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk g hg\nZ : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc {x | AEMeasurable.mk f hf x \u2260 0}, g x = AEMeasurable.mk g hg x\nx : \u03b1\nhx : g x = AEMeasurable.mk g hg x\n\u22a2 (f' * g') x = (f' * g) x", "state_after": "no goals"}, {"tactic": "have M : MeasurableSet { x : \u03b1 | f' x \u2260 0 }\u1d9c :=\n  (hf.measurable_mk (measurableSet_singleton 0).compl).compl", "annotated_tactic": ["have M : <a>MeasurableSet</a> { x : \u03b1 | f' x \u2260 0 }\u1d9c :=\n          (hf.measurable_mk (<a>measurableSet_singleton</a> 0).<a>compl</a>).<a>compl</a>", [{"full_name": "MeasurableSet", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [64, 5], "def_end_pos": [64, 18]}, {"full_name": "MeasurableSingletonClass.measurableSet_singleton", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [269, 3], "def_end_pos": [269, 26]}, {"full_name": "MeasurableSet.compl", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [87, 19], "def_end_pos": [87, 38]}, {"full_name": "MeasurableSet.compl", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [87, 19], "def_end_pos": [87, 38]}]], "state_before": "case h.htc\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk f hf\nhg : AEMeasurable g\nthis : withDensity \u03bc f = withDensity \u03bc f'\ng' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk g hg\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc {x | f' x \u2260 0}\u1d9c, (fun a => (f' * g') a) x = (fun a => (f' * g) a) x", "state_after": "case h.htc\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk f hf\nhg : AEMeasurable g\nthis : withDensity \u03bc f = withDensity \u03bc f'\ng' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk g hg\nM : MeasurableSet {x | f' x \u2260 0}\u1d9c\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc {x | f' x \u2260 0}\u1d9c, (fun a => (f' * g') a) x = (fun a => (f' * g) a) x"}, {"tactic": "filter_upwards [ae_restrict_mem M]", "annotated_tactic": ["filter_upwards [<a>ae_restrict_mem</a> M]", [{"full_name": "MeasureTheory.ae_restrict_mem", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2586, 9], "def_end_pos": [2586, 24]}]], "state_before": "case h.htc\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk f hf\nhg : AEMeasurable g\nthis : withDensity \u03bc f = withDensity \u03bc f'\ng' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk g hg\nM : MeasurableSet {x | f' x \u2260 0}\u1d9c\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc {x | f' x \u2260 0}\u1d9c, (fun a => (f' * g') a) x = (fun a => (f' * g) a) x", "state_after": "case h\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk f hf\nhg : AEMeasurable g\nthis : withDensity \u03bc f = withDensity \u03bc f'\ng' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk g hg\nM : MeasurableSet {x | f' x \u2260 0}\u1d9c\n\u22a2 \u2200 (a : \u03b1), a \u2208 {x | f' x \u2260 0}\u1d9c \u2192 (f' * g') a = (f' * g) a"}, {"tactic": "intro x hx", "annotated_tactic": ["intro x hx", []], "state_before": "case h\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk f hf\nhg : AEMeasurable g\nthis : withDensity \u03bc f = withDensity \u03bc f'\ng' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk g hg\nM : MeasurableSet {x | f' x \u2260 0}\u1d9c\n\u22a2 \u2200 (a : \u03b1), a \u2208 {x | f' x \u2260 0}\u1d9c \u2192 (f' * g') a = (f' * g) a", "state_after": "case h\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk f hf\nhg : AEMeasurable g\nthis : withDensity \u03bc f = withDensity \u03bc f'\ng' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk g hg\nM : MeasurableSet {x | f' x \u2260 0}\u1d9c\nx : \u03b1\nhx : x \u2208 {x | f' x \u2260 0}\u1d9c\n\u22a2 (f' * g') x = (f' * g) x"}, {"tactic": "simp only [Classical.not_not, mem_setOf_eq, mem_compl_iff] at hx", "annotated_tactic": ["simp only [<a>Classical.not_not</a>, <a>mem_setOf_eq</a>, <a>mem_compl_iff</a>] at hx", [{"full_name": "Classical.not_not", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [683, 24], "def_end_pos": [683, 31]}, {"full_name": "Set.mem_setOf_eq", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [256, 29], "def_end_pos": [256, 41]}, {"full_name": "Set.mem_compl_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1658, 9], "def_end_pos": [1658, 22]}]], "state_before": "case h\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk f hf\nhg : AEMeasurable g\nthis : withDensity \u03bc f = withDensity \u03bc f'\ng' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk g hg\nM : MeasurableSet {x | f' x \u2260 0}\u1d9c\nx : \u03b1\nhx : x \u2208 {x | f' x \u2260 0}\u1d9c\n\u22a2 (f' * g') x = (f' * g) x", "state_after": "case h\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk f hf\nhg : AEMeasurable g\nthis : withDensity \u03bc f = withDensity \u03bc f'\ng' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk g hg\nM : MeasurableSet {x | f' x \u2260 0}\u1d9c\nx : \u03b1\nhx : AEMeasurable.mk f hf x = 0\n\u22a2 (f' * g') x = (f' * g) x"}, {"tactic": "simp only [hx, zero_mul, Pi.mul_apply]", "annotated_tactic": ["simp only [hx, <a>zero_mul</a>, <a>Pi.mul_apply</a>]", [{"full_name": "MulZeroClass.zero_mul", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [36, 3], "def_end_pos": [36, 11]}, {"full_name": "Pi.mul_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [83, 9], "def_end_pos": [83, 18]}]], "state_before": "case h\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk f hf\nhg : AEMeasurable g\nthis : withDensity \u03bc f = withDensity \u03bc f'\ng' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk g hg\nM : MeasurableSet {x | f' x \u2260 0}\u1d9c\nx : \u03b1\nhx : AEMeasurable.mk f hf x = 0\n\u22a2 (f' * g') x = (f' * g) x", "state_after": "no goals"}, {"tactic": "apply lintegral_congr_ae", "annotated_tactic": ["apply <a>lintegral_congr_ae</a>", [{"full_name": "MeasureTheory.lintegral_congr_ae", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [304, 9], "def_end_pos": [304, 27]}]], "state_before": "\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk f hf\nhg : AEMeasurable g\nthis : withDensity \u03bc f = withDensity \u03bc f'\ng' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk g hg\n\u22a2 \u222b\u207b (a : \u03b1), (f' * g) a \u2202\u03bc = \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc", "state_after": "case h\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk f hf\nhg : AEMeasurable g\nthis : withDensity \u03bc f = withDensity \u03bc f'\ng' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk g hg\n\u22a2 (fun a => (f' * g) a) =\u1da0[ae \u03bc] fun a => (f * g) a"}, {"tactic": "filter_upwards [hf.ae_eq_mk]", "annotated_tactic": ["filter_upwards [hf.ae_eq_mk]", []], "state_before": "case h\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk f hf\nhg : AEMeasurable g\nthis : withDensity \u03bc f = withDensity \u03bc f'\ng' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk g hg\n\u22a2 (fun a => (f' * g) a) =\u1da0[ae \u03bc] fun a => (f * g) a", "state_after": "case h\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk f hf\nhg : AEMeasurable g\nthis : withDensity \u03bc f = withDensity \u03bc f'\ng' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk g hg\n\u22a2 \u2200 (a : \u03b1), f a = AEMeasurable.mk f hf a \u2192 (f' * g) a = (f * g) a"}, {"tactic": "intro x hx", "annotated_tactic": ["intro x hx", []], "state_before": "case h\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk f hf\nhg : AEMeasurable g\nthis : withDensity \u03bc f = withDensity \u03bc f'\ng' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk g hg\n\u22a2 \u2200 (a : \u03b1), f a = AEMeasurable.mk f hf a \u2192 (f' * g) a = (f * g) a", "state_after": "case h\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk f hf\nhg : AEMeasurable g\nthis : withDensity \u03bc f = withDensity \u03bc f'\ng' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk g hg\nx : \u03b1\nhx : f x = AEMeasurable.mk f hf x\n\u22a2 (f' * g) x = (f * g) x"}, {"tactic": "simp only [hx, Pi.mul_apply]", "annotated_tactic": ["simp only [hx, <a>Pi.mul_apply</a>]", [{"full_name": "Pi.mul_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [83, 9], "def_end_pos": [83, 18]}]], "state_before": "case h\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk f hf\nhg : AEMeasurable g\nthis : withDensity \u03bc f = withDensity \u03bc f'\ng' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk g hg\nx : \u03b1\nhx : f x = AEMeasurable.mk f hf x\n\u22a2 (f' * g) x = (f * g) x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Haar/Basic.lean", "full_name": "MeasureTheory.Measure.haar.is_left_invariant_haarContent", "start": [569, 1], "end": [572, 32], "traced_tactics": [{"tactic": "simpa only [ENNReal.coe_eq_coe, \u2190 NNReal.coe_eq, haarContent_apply] using\n  is_left_invariant_chaar g K", "annotated_tactic": ["simpa only [<a>ENNReal.coe_eq_coe</a>, \u2190 <a>NNReal.coe_eq</a>, <a>haarContent_apply</a>] using\n    <a>is_left_invariant_chaar</a> g K", [{"full_name": "ENNReal.coe_eq_coe", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [346, 28], "def_end_pos": [346, 38]}, {"full_name": "NNReal.coe_eq", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [166, 19], "def_end_pos": [166, 25]}, {"full_name": "MeasureTheory.Measure.haar.haarContent_apply", "def_path": "Mathlib/MeasureTheory/Measure/Haar/Basic.lean", "def_pos": [554, 9], "def_end_pos": [554, 26]}, {"full_name": "MeasureTheory.Measure.haar.is_left_invariant_chaar", "def_path": "Mathlib/MeasureTheory/Measure/Haar/Basic.lean", "def_pos": [518, 9], "def_end_pos": [518, 32]}]], "state_before": "G : Type u_1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : TopologicalSpace G\ninst\u271d\u00b9 : TopologicalGroup G\ninst\u271d : T2Space G\nK\u2080 : PositiveCompacts G\ng : G\nK : Compacts G\n\u22a2 (fun s => \u2191(Content.toFun (haarContent K\u2080) s)) (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) =\n    (fun s => \u2191(Content.toFun (haarContent K\u2080) s)) K", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/TMToPartrec.lean", "full_name": "Turing.PartrecToTM2.trNat_natEnd", "start": [1465, 1], "end": [1466, 17], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Setoid/Basic.lean", "full_name": "Setoid.mapOfSurjective_eq_map", "start": [380, 1], "end": [382, 58], "traced_tactics": [{"tactic": "rw [\u2190 eqvGen_of_setoid (mapOfSurjective r f h hf)]", "annotated_tactic": ["rw [\u2190 <a>eqvGen_of_setoid</a> (<a>mapOfSurjective</a> r f h hf)]", [{"full_name": "Setoid.eqvGen_of_setoid", "def_path": "Mathlib/Data/Setoid/Basic.lean", "def_pos": [246, 9], "def_end_pos": [246, 25]}, {"full_name": "Setoid.mapOfSurjective", "def_path": "Mathlib/Data/Setoid/Basic.lean", "def_pos": [369, 5], "def_end_pos": [369, 20]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : Setoid \u03b1\nf : \u03b1 \u2192 \u03b2\nh : ker f \u2264 r\nhf : Surjective f\n\u22a2 map r f = mapOfSurjective r f h hf", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : Setoid \u03b1\nf : \u03b1 \u2192 \u03b2\nh : ker f \u2264 r\nhf : Surjective f\n\u22a2 map r f = EqvGen.Setoid Setoid.r"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : Setoid \u03b1\nf : \u03b1 \u2192 \u03b2\nh : ker f \u2264 r\nhf : Surjective f\n\u22a2 map r f = EqvGen.Setoid Setoid.r", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "full_name": "MeasureTheory.OuterMeasure.map_iInf_comap", "start": [1236, 1], "end": [1247, 24], "traced_tactics": [{"tactic": "refine' (map_iInf_le _ _).antisymm fun s => _", "annotated_tactic": ["refine' (<a>map_iInf_le</a> _ _).<a>antisymm</a> fun s => _", [{"full_name": "MeasureTheory.OuterMeasure.map_iInf_le", "def_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "def_pos": [1215, 9], "def_end_pos": [1215, 20]}, {"full_name": "LE.le.antisymm", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [128, 7], "def_end_pos": [128, 21]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Sort u_2\n\u03b2 : Type u_3\ninst\u271d : Nonempty \u03b9\nf : \u03b1 \u2192 \u03b2\nm : \u03b9 \u2192 OuterMeasure \u03b2\n\u22a2 \u2191(map f) (\u2a05 i, \u2191(comap f) (m i)) = \u2a05 i, \u2191(map f) (\u2191(comap f) (m i))", "state_after": "\u03b1 : Type u_1\n\u03b9 : Sort u_2\n\u03b2 : Type u_3\ninst\u271d : Nonempty \u03b9\nf : \u03b1 \u2192 \u03b2\nm : \u03b9 \u2192 OuterMeasure \u03b2\ns : Set \u03b2\n\u22a2 \u2191(\u2a05 i, \u2191(map f) (\u2191(comap f) (m i))) s \u2264 \u2191(\u2191(map f) (\u2a05 i, \u2191(comap f) (m i))) s"}, {"tactic": "simp only [map_apply, comap_apply, iInf_apply, le_iInf_iff]", "annotated_tactic": ["simp only [<a>map_apply</a>, <a>comap_apply</a>, <a>iInf_apply</a>, <a>le_iInf_iff</a>]", [{"full_name": "MeasureTheory.OuterMeasure.map_apply", "def_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "def_pos": [451, 9], "def_end_pos": [451, 18]}, {"full_name": "MeasureTheory.OuterMeasure.comap_apply", "def_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "def_pos": [541, 9], "def_end_pos": [541, 20]}, {"full_name": "MeasureTheory.OuterMeasure.iInf_apply", "def_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "def_pos": [1183, 9], "def_end_pos": [1183, 19]}, {"full_name": "le_iInf_iff", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [969, 9], "def_end_pos": [969, 20]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Sort u_2\n\u03b2 : Type u_3\ninst\u271d : Nonempty \u03b9\nf : \u03b1 \u2192 \u03b2\nm : \u03b9 \u2192 OuterMeasure \u03b2\ns : Set \u03b2\n\u22a2 \u2191(\u2a05 i, \u2191(map f) (\u2191(comap f) (m i))) s \u2264 \u2191(\u2191(map f) (\u2a05 i, \u2191(comap f) (m i))) s", "state_after": "\u03b1 : Type u_1\n\u03b9 : Sort u_2\n\u03b2 : Type u_3\ninst\u271d : Nonempty \u03b9\nf : \u03b1 \u2192 \u03b2\nm : \u03b9 \u2192 OuterMeasure \u03b2\ns : Set \u03b2\n\u22a2 \u2200 (i : \u2115 \u2192 Set \u03b1),\n    f \u207b\u00b9' s \u2286 iUnion i \u2192\n      \u2a05 t, \u2a05 (_ : s \u2286 iUnion t), \u2211' (n : \u2115), \u2a05 i, \u2191(m i) (f '' (f \u207b\u00b9' t n)) \u2264 \u2211' (n : \u2115), \u2a05 i_2, \u2191(m i_2) (f '' i n)"}, {"tactic": "refine' fun t ht => iInf_le_of_le (fun n => f '' t n \u222a (range f)\u1d9c) (iInf_le_of_le _ _)", "annotated_tactic": ["refine' fun t ht => <a>iInf_le_of_le</a> (fun n => f '' t n \u222a (<a>range</a> f)\u1d9c) (<a>iInf_le_of_le</a> _ _)", [{"full_name": "iInf_le_of_le", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [853, 9], "def_end_pos": [853, 22]}, {"full_name": "Set.range", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [668, 5], "def_end_pos": [668, 10]}, {"full_name": "iInf_le_of_le", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [853, 9], "def_end_pos": [853, 22]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Sort u_2\n\u03b2 : Type u_3\ninst\u271d : Nonempty \u03b9\nf : \u03b1 \u2192 \u03b2\nm : \u03b9 \u2192 OuterMeasure \u03b2\ns : Set \u03b2\n\u22a2 \u2200 (i : \u2115 \u2192 Set \u03b1),\n    f \u207b\u00b9' s \u2286 iUnion i \u2192\n      \u2a05 t, \u2a05 (_ : s \u2286 iUnion t), \u2211' (n : \u2115), \u2a05 i, \u2191(m i) (f '' (f \u207b\u00b9' t n)) \u2264 \u2211' (n : \u2115), \u2a05 i_2, \u2191(m i_2) (f '' i n)", "state_after": "case refine'_1\n\u03b1 : Type u_1\n\u03b9 : Sort u_2\n\u03b2 : Type u_3\ninst\u271d : Nonempty \u03b9\nf : \u03b1 \u2192 \u03b2\nm : \u03b9 \u2192 OuterMeasure \u03b2\ns : Set \u03b2\nt : \u2115 \u2192 Set \u03b1\nht : f \u207b\u00b9' s \u2286 iUnion t\n\u22a2 s \u2286 \u22c3 n, f '' t n \u222a (range f)\u1d9c\n\ncase refine'_2\n\u03b1 : Type u_1\n\u03b9 : Sort u_2\n\u03b2 : Type u_3\ninst\u271d : Nonempty \u03b9\nf : \u03b1 \u2192 \u03b2\nm : \u03b9 \u2192 OuterMeasure \u03b2\ns : Set \u03b2\nt : \u2115 \u2192 Set \u03b1\nht : f \u207b\u00b9' s \u2286 iUnion t\n\u22a2 \u2211' (n : \u2115), \u2a05 i, \u2191(m i) (f '' (f \u207b\u00b9' (fun n => f '' t n \u222a (range f)\u1d9c) n)) \u2264 \u2211' (n : \u2115), \u2a05 i, \u2191(m i) (f '' t n)"}, {"tactic": "rw [\u2190 iUnion_union, Set.union_comm, \u2190 inter_subset, \u2190 image_iUnion, \u2190\n  image_preimage_eq_inter_range]", "annotated_tactic": ["rw [\u2190 <a>iUnion_union</a>, <a>Set.union_comm</a>, \u2190 <a>inter_subset</a>, \u2190 <a>image_iUnion</a>, \u2190\n      <a>image_preimage_eq_inter_range</a>]", [{"full_name": "Set.iUnion_union", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [657, 9], "def_end_pos": [657, 21]}, {"full_name": "Set.union_comm", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [786, 9], "def_end_pos": [786, 19]}, {"full_name": "Set.inter_subset", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1803, 9], "def_end_pos": [1803, 21]}, {"full_name": "Set.image_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [1791, 9], "def_end_pos": [1791, 21]}, {"full_name": "Set.image_preimage_eq_inter_range", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [796, 9], "def_end_pos": [796, 38]}]], "state_before": "case refine'_1\n\u03b1 : Type u_1\n\u03b9 : Sort u_2\n\u03b2 : Type u_3\ninst\u271d : Nonempty \u03b9\nf : \u03b1 \u2192 \u03b2\nm : \u03b9 \u2192 OuterMeasure \u03b2\ns : Set \u03b2\nt : \u2115 \u2192 Set \u03b1\nht : f \u207b\u00b9' s \u2286 iUnion t\n\u22a2 s \u2286 \u22c3 n, f '' t n \u222a (range f)\u1d9c", "state_after": "case refine'_1\n\u03b1 : Type u_1\n\u03b9 : Sort u_2\n\u03b2 : Type u_3\ninst\u271d : Nonempty \u03b9\nf : \u03b1 \u2192 \u03b2\nm : \u03b9 \u2192 OuterMeasure \u03b2\ns : Set \u03b2\nt : \u2115 \u2192 Set \u03b1\nht : f \u207b\u00b9' s \u2286 iUnion t\n\u22a2 f '' (f \u207b\u00b9' s) \u2286 f '' \u22c3 i, t i"}, {"tactic": "exact image_subset _ ht", "annotated_tactic": ["exact <a>image_subset</a> _ ht", [{"full_name": "Set.image_subset", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [321, 9], "def_end_pos": [321, 21]}]], "state_before": "case refine'_1\n\u03b1 : Type u_1\n\u03b9 : Sort u_2\n\u03b2 : Type u_3\ninst\u271d : Nonempty \u03b9\nf : \u03b1 \u2192 \u03b2\nm : \u03b9 \u2192 OuterMeasure \u03b2\ns : Set \u03b2\nt : \u2115 \u2192 Set \u03b1\nht : f \u207b\u00b9' s \u2286 iUnion t\n\u22a2 f '' (f \u207b\u00b9' s) \u2286 f '' \u22c3 i, t i", "state_after": "no goals"}, {"tactic": "refine' ENNReal.tsum_le_tsum fun n => iInf_mono fun i => (m i).mono _", "annotated_tactic": ["refine' <a>ENNReal.tsum_le_tsum</a> fun n => <a>iInf_mono</a> fun i => (m i).<a>mono</a> _", [{"full_name": "ENNReal.tsum_le_tsum", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [827, 19], "def_end_pos": [827, 31]}, {"full_name": "iInf_mono", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [905, 9], "def_end_pos": [905, 18]}, {"full_name": "MeasureTheory.OuterMeasure.mono", "def_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "def_pos": [70, 3], "def_end_pos": [70, 7]}]], "state_before": "case refine'_2\n\u03b1 : Type u_1\n\u03b9 : Sort u_2\n\u03b2 : Type u_3\ninst\u271d : Nonempty \u03b9\nf : \u03b1 \u2192 \u03b2\nm : \u03b9 \u2192 OuterMeasure \u03b2\ns : Set \u03b2\nt : \u2115 \u2192 Set \u03b1\nht : f \u207b\u00b9' s \u2286 iUnion t\n\u22a2 \u2211' (n : \u2115), \u2a05 i, \u2191(m i) (f '' (f \u207b\u00b9' (fun n => f '' t n \u222a (range f)\u1d9c) n)) \u2264 \u2211' (n : \u2115), \u2a05 i, \u2191(m i) (f '' t n)", "state_after": "case refine'_2\n\u03b1 : Type u_1\n\u03b9 : Sort u_2\n\u03b2 : Type u_3\ninst\u271d : Nonempty \u03b9\nf : \u03b1 \u2192 \u03b2\nm : \u03b9 \u2192 OuterMeasure \u03b2\ns : Set \u03b2\nt : \u2115 \u2192 Set \u03b1\nht : f \u207b\u00b9' s \u2286 iUnion t\nn : \u2115\ni : \u03b9\n\u22a2 f '' (f \u207b\u00b9' (fun n => f '' t n \u222a (range f)\u1d9c) n) \u2286 f '' t n"}, {"tactic": "simp only [preimage_union, preimage_compl, preimage_range, compl_univ, union_empty,\n  image_subset_iff]", "annotated_tactic": ["simp only [<a>preimage_union</a>, <a>preimage_compl</a>, <a>preimage_range</a>, <a>compl_univ</a>, <a>union_empty</a>,\n      <a>image_subset_iff</a>]", [{"full_name": "Set.preimage_union", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [92, 9], "def_end_pos": [92, 23]}, {"full_name": "Set.preimage_compl", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [97, 9], "def_end_pos": [97, 23]}, {"full_name": "Set.preimage_range", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [1116, 9], "def_end_pos": [1116, 23]}, {"full_name": "Set.compl_univ", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1691, 9], "def_end_pos": [1691, 19]}, {"full_name": "Set.union_empty", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [777, 9], "def_end_pos": [777, 20]}, {"full_name": "Set.image_subset_iff", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [497, 9], "def_end_pos": [497, 25]}]], 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refine'_2\n\u03b1 : Type u_1\n\u03b9 : Sort u_2\n\u03b2 : Type u_3\ninst\u271d : Nonempty \u03b9\nf : \u03b1 \u2192 \u03b2\nm : \u03b9 \u2192 OuterMeasure \u03b2\ns : Set \u03b2\nt : \u2115 \u2192 Set \u03b1\nht : f \u207b\u00b9' s \u2286 iUnion t\nn : \u2115\ni : \u03b9\n\u22a2 f \u207b\u00b9' (f '' t n) \u2286 f \u207b\u00b9' (f '' t n)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Kernel/CondDistrib.lean", "full_name": "ProbabilityTheory.aestronglyMeasurable'_integral_condDistrib", "start": [101, 1], "end": [104, 58], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "full_name": "MeasureTheory.lintegral_finset", "start": [1504, 1], "end": [1506, 78], "traced_tactics": [{"tactic": "simp only [lintegral_countable _ s.countable_toSet, \u2190 Finset.tsum_subtype']", "annotated_tactic": ["simp only [<a>lintegral_countable</a> _ s.countable_toSet, \u2190 <a>Finset.tsum_subtype'</a>]", [{"full_name": "MeasureTheory.lintegral_countable", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [1488, 9], "def_end_pos": [1488, 28]}, {"full_name": "Finset.tsum_subtype'", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/Basic.lean", "def_pos": [555, 9], "def_end_pos": [555, 29]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : MeasurableSingletonClass \u03b1\ns : Finset \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u22a2 \u222b\u207b (x : \u03b1) in \u2191s, f x \u2202\u03bc = \u2211 x in s, f x * \u2191\u2191\u03bc {x}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Vector/Mem.lean", "full_name": "Vector.not_mem_map_zero", "start": [81, 1], "end": [82, 91], "traced_tactics": [{"tactic": "simpa only [Vector.eq_nil v, Vector.map_nil, Vector.toList_nil] using List.not_mem_nil b", "annotated_tactic": ["simpa only [<a>Vector.eq_nil</a> v, <a>Vector.map_nil</a>, <a>Vector.toList_nil</a>] using <a>List.not_mem_nil</a> b", [{"full_name": "Vector.eq_nil", "def_path": "Mathlib/Data/Vector.lean", "def_pos": [238, 19], "def_end_pos": [238, 25]}, {"full_name": "Vector.map_nil", "def_path": "Mathlib/Data/Vector.lean", "def_pos": [134, 9], "def_end_pos": [134, 16]}, {"full_name": "Vector.toList_nil", "def_path": "Mathlib/Data/Vector.lean", "def_pos": [251, 9], "def_end_pos": [251, 19]}, {"full_name": "List.not_mem_nil", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [58, 17], "def_end_pos": [58, 28]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nn : \u2115\na a' : \u03b1\nb : \u03b2\nv : Vector \u03b1 0\nf : \u03b1 \u2192 \u03b2\n\u22a2 \u00acb \u2208 toList (map f v)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Ackermann.lean", "full_name": "ack_strictMono_left", "start": [218, 1], "end": [219, 26], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Constructions/Pi.lean", "full_name": "MeasureTheory.Measure.univ_pi_Iio_ae_eq_Iic", "start": [495, 1], "end": [497, 48], "traced_tactics": [{"tactic": "rw [\u2190 pi_univ_Iic]", "annotated_tactic": ["rw [\u2190 <a>pi_univ_Iic</a>]", [{"full_name": "Set.pi_univ_Iic", "def_path": "Mathlib/Data/Set/Intervals/Pi.lean", "def_pos": [38, 9], "def_end_pos": [38, 20]}]], "state_before": "\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : \u03b9 \u2192 Type u_3\ninst\u271d\u2074 : Fintype \u03b9\nm : (i : \u03b9) \u2192 OuterMeasure (\u03b1 i)\ninst\u271d\u00b3 : (i : \u03b9) \u2192 MeasurableSpace (\u03b1 i)\n\u03bc : (i : \u03b9) \u2192 Measure (\u03b1 i)\ninst\u271d\u00b2 : \u2200 (i : \u03b9), SigmaFinite (\u03bc i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 PartialOrder (\u03b1 i)\ninst\u271d : \u2200 (i : \u03b9), NoAtoms (\u03bc i)\nf : (i : \u03b9) \u2192 \u03b1 i\n\u22a2 (Set.pi univ fun i => Iio (f i)) =\u1da0[ae (Measure.pi \u03bc)] Iic f", "state_after": "\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : \u03b9 \u2192 Type u_3\ninst\u271d\u2074 : Fintype \u03b9\nm : (i : \u03b9) \u2192 OuterMeasure (\u03b1 i)\ninst\u271d\u00b3 : (i : \u03b9) \u2192 MeasurableSpace (\u03b1 i)\n\u03bc : (i : \u03b9) \u2192 Measure (\u03b1 i)\ninst\u271d\u00b2 : \u2200 (i : \u03b9), SigmaFinite (\u03bc i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 PartialOrder (\u03b1 i)\ninst\u271d : \u2200 (i : \u03b9), NoAtoms (\u03bc i)\nf : (i : \u03b9) \u2192 \u03b1 i\n\u22a2 (Set.pi univ fun i => Iio (f i)) =\u1da0[ae (Measure.pi \u03bc)] Set.pi univ fun i => Iic (f i)"}, {"tactic": "exact pi_Iio_ae_eq_pi_Iic", "annotated_tactic": ["exact <a>pi_Iio_ae_eq_pi_Iic</a>", [{"full_name": "MeasureTheory.Measure.pi_Iio_ae_eq_pi_Iic", "def_path": "Mathlib/MeasureTheory/Constructions/Pi.lean", "def_pos": [485, 9], "def_end_pos": [485, 28]}]], "state_before": "\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : \u03b9 \u2192 Type u_3\ninst\u271d\u2074 : Fintype \u03b9\nm : (i : \u03b9) \u2192 OuterMeasure (\u03b1 i)\ninst\u271d\u00b3 : (i : \u03b9) \u2192 MeasurableSpace (\u03b1 i)\n\u03bc : (i : \u03b9) \u2192 Measure (\u03b1 i)\ninst\u271d\u00b2 : \u2200 (i : \u03b9), SigmaFinite (\u03bc i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 PartialOrder (\u03b1 i)\ninst\u271d : \u2200 (i : \u03b9), NoAtoms (\u03bc i)\nf : (i : \u03b9) \u2192 \u03b1 i\n\u22a2 (Set.pi univ fun i => Iio (f i)) =\u1da0[ae (Measure.pi \u03bc)] Set.pi univ fun i => Iic 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u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf\u271d : \u03b1 \u2192. \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nt : Set \u03b2\nx : \u03b1\n\u22a2 x \u2208 core (res f s) t \u2194 x \u2208 s \u2192 f x \u2208 t", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Fin/Lemmas.lean", "full_name": "Fin.ofNat'_zero_val", "start": [58, 9], "end": [58, 77], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Group/Integral.lean", "full_name": "MeasureTheory.integral_div_left_eq_self", "start": [152, 1], "end": [156, 84], "traced_tactics": [{"tactic": "simp_rw [div_eq_mul_inv]", "annotated_tactic": ["simp_rw [<a>div_eq_mul_inv</a>]", [{"full_name": "div_eq_mul_inv", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [977, 9], "def_end_pos": [977, 23]}]], "state_before": "\ud835\udd5c : Type u_1\nM : Type u_2\n\u03b1 : Type u_3\nG : Type u_4\nE : Type u_5\nF : Type u_6\ninst\u271d\u2079 : MeasurableSpace G\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : CompleteSpace E\ninst\u271d\u2075 : NormedAddCommGroup F\n\u03bc\u271d : Measure G\nf\u271d : G \u2192 E\ng : G\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : MeasurableMul G\ninst\u271d\u00b2 : MeasurableInv G\nf : G \u2192 E\n\u03bc : Measure G\ninst\u271d\u00b9 : IsInvInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bc\nx' : G\n\u22a2 \u222b (x : G), f (x' / x) \u2202\u03bc = \u222b (x : G), f x \u2202\u03bc", "state_after": "\ud835\udd5c : Type u_1\nM : Type u_2\n\u03b1 : Type u_3\nG : Type u_4\nE : Type u_5\nF : Type u_6\ninst\u271d\u2079 : MeasurableSpace G\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : CompleteSpace E\ninst\u271d\u2075 : 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u_2\n\u03b1 : Type u_3\nG : Type u_4\nE : Type u_5\nF : Type u_6\ninst\u271d\u2079 : MeasurableSpace G\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : CompleteSpace E\ninst\u271d\u2075 : NormedAddCommGroup F\n\u03bc\u271d : Measure G\nf\u271d : G \u2192 E\ng : G\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : MeasurableMul G\ninst\u271d\u00b2 : MeasurableInv G\nf : G \u2192 E\n\u03bc : Measure G\ninst\u271d\u00b9 : IsInvInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bc\nx' : G\n\u22a2 \u222b (x : G), f (x' * x\u207b\u00b9) \u2202\u03bc = \u222b (x : G), f x \u2202\u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/RBMap/Lemmas.lean", "full_name": "Std.RBSet.mem_toList_unique", "start": [664, 1], "end": [666, 57], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/Derivation.lean", "full_name": "MvPolynomial.derivation_eq_of_forall_mem_vars", "start": [80, 1], "end": [82, 51], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Card.lean", "full_name": "Set.encard_le_one_iff", "start": [299, 1], "end": [304, 36], "traced_tactics": [{"tactic": "rw [encard_le_one_iff_eq, or_iff_not_imp_left, \u2190Ne.def, \u2190nonempty_iff_ne_empty]", "annotated_tactic": ["rw [<a>encard_le_one_iff_eq</a>, <a>or_iff_not_imp_left</a>, \u2190<a>Ne.def</a>, \u2190<a>nonempty_iff_ne_empty</a>]", [{"full_name": "Set.encard_le_one_iff_eq", "def_path": "Mathlib/Data/Set/Card.lean", "def_pos": [295, 9], "def_end_pos": [295, 29]}, {"full_name": "or_iff_not_imp_left", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [360, 9], "def_end_pos": [360, 28]}, {"full_name": "Ne.def", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [59, 9], "def_end_pos": [59, 15]}, {"full_name": "Set.nonempty_iff_ne_empty", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [610, 9], "def_end_pos": [610, 30]}]], "state_before": "\u03b1 : Type u_1\ns t : Set \u03b1\n\u22a2 encard s \u2264 1 \u2194 \u2200 (a b : \u03b1), a \u2208 s \u2192 b \u2208 s \u2192 a = b", "state_after": "\u03b1 : Type u_1\ns t : Set \u03b1\n\u22a2 (Set.Nonempty s \u2192 \u2203 x, s = {x}) \u2194 \u2200 (a b : \u03b1), a \u2208 s \u2192 b \u2208 s \u2192 a = b"}, {"tactic": "refine' \u27e8fun h a b has hbs \u21a6 _,\n  fun h \u27e8x, hx\u27e9 \u21a6 \u27e8x, ((singleton_subset_iff.2 hx).antisymm' (fun y hy \u21a6 h _ _ hy hx))\u27e9\u27e9", "annotated_tactic": ["refine' \u27e8fun h a b has hbs \u21a6 _,\n    fun h \u27e8x, hx\u27e9 \u21a6 \u27e8x, ((<a>singleton_subset_iff</a>.2 hx).<a>antisymm'</a> (fun y hy \u21a6 h _ _ hy hx))\u27e9\u27e9", [{"full_name": "Set.singleton_subset_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1330, 9], "def_end_pos": [1330, 29]}, {"full_name": "HasSubset.Subset.antisymm'", "def_path": "Mathlib/Order/RelClasses.lean", "def_pos": [670, 7], "def_end_pos": [670, 33]}]], "state_before": "\u03b1 : Type u_1\ns t : Set \u03b1\n\u22a2 (Set.Nonempty s \u2192 \u2203 x, s = {x}) \u2194 \u2200 (a b : \u03b1), a \u2208 s \u2192 b \u2208 s \u2192 a = b", "state_after": "\u03b1 : Type u_1\ns t : Set \u03b1\nh : Set.Nonempty s \u2192 \u2203 x, s = {x}\na b : \u03b1\nhas : a \u2208 s\nhbs : b \u2208 s\n\u22a2 a = b"}, {"tactic": "obtain \u27e8x, rfl\u27e9 := h \u27e8_, has\u27e9", "annotated_tactic": ["obtain \u27e8x, rfl\u27e9 := h \u27e8_, has\u27e9", []], "state_before": "\u03b1 : Type u_1\ns t : Set \u03b1\nh : Set.Nonempty s \u2192 \u2203 x, s = {x}\na b : \u03b1\nhas : a \u2208 s\nhbs : b \u2208 s\n\u22a2 a = b", "state_after": "case intro\n\u03b1 : Type u_1\nt : Set \u03b1\na b x : \u03b1\nh : Set.Nonempty {x} \u2192 \u2203 x_1, {x} = {x_1}\nhas : a \u2208 {x}\nhbs : b \u2208 {x}\n\u22a2 a = b"}, {"tactic": "rw [(has : a = x), (hbs : b = x)]", "annotated_tactic": ["rw [(has : a = x), (hbs : b = x)]", []], "state_before": "case intro\n\u03b1 : Type u_1\nt : Set \u03b1\na b x : \u03b1\nh : Set.Nonempty {x} \u2192 \u2203 x_1, {x} = {x_1}\nhas : a \u2208 {x}\nhbs : b \u2208 {x}\n\u22a2 a = b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Decomposition/Lebesgue.lean", "full_name": "MeasureTheory.Measure.haveLebesgueDecomposition_of_finiteMeasure", "start": [553, 1], "end": [652, 43], "traced_tactics": [{"tactic": "have h :=\n  @exists_seq_tendsto_sSup _ _ _ _ _ (measurableLEEval \u03bd \u03bc)\n    \u27e80, 0, zero_mem_measurableLE, by simp\u27e9 (OrderTop.bddAbove _)", "annotated_tactic": ["have h :=\n      @<a>exists_seq_tendsto_sSup</a> _ _ _ _ _ (<a>measurableLEEval</a> \u03bd \u03bc)\n        \u27e80, 0, <a>zero_mem_measurableLE</a>, by simp\u27e9 (<a>OrderTop.bddAbove</a> _)", [{"full_name": "exists_seq_tendsto_sSup", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [2233, 9], "def_end_pos": [2233, 32]}, {"full_name": "MeasureTheory.Measure.LebesgueDecomposition.measurableLEEval", "def_path": "Mathlib/MeasureTheory/Decomposition/Lebesgue.lean", "def_pos": [539, 5], "def_end_pos": [539, 21]}, {"full_name": "MeasureTheory.Measure.LebesgueDecomposition.zero_mem_measurableLE", "def_path": "Mathlib/MeasureTheory/Decomposition/Lebesgue.lean", "def_pos": [462, 9], "def_end_pos": [462, 30]}, {"full_name": "OrderTop.bddAbove", "def_path": "Mathlib/Order/Bounds/Basic.lean", "def_pos": [984, 19], "def_end_pos": [984, 36]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\n\u22a2 \u2203 p, Measurable p.2 \u2227 p.1 \u27c2\u2098 \u03bd \u2227 \u03bc = p.1 + withDensity \u03bd p.2", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nh :\n  \u2203 u,\n    Monotone u \u2227\n      Filter.Tendsto u Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc))) \u2227 \u2200 (n : \u2115), u n \u2208 measurableLEEval \u03bd \u03bc\n\u22a2 \u2203 p, Measurable p.2 \u2227 p.1 \u27c2\u2098 \u03bd \u2227 \u03bc = p.1 + withDensity \u03bd p.2"}, {"tactic": "choose g _ hg\u2082 f hf\u2081 hf\u2082 using h", "annotated_tactic": ["choose g _ hg\u2082 f hf\u2081 hf\u2082 using h", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nh :\n  \u2203 u,\n    Monotone u \u2227\n      Filter.Tendsto u Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc))) \u2227 \u2200 (n : \u2115), u n \u2208 measurableLEEval \u03bd \u03bc\n\u22a2 \u2203 p, Measurable p.2 \u2227 p.1 \u27c2\u2098 \u03bd \u2227 \u03bc = p.1 + withDensity \u03bd p.2", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u22a2 \u2203 p, Measurable p.2 \u2227 p.1 \u27c2\u2098 \u03bd \u2227 \u03bc = p.1 + withDensity \u03bd p.2"}, {"tactic": "set \u03be := \u2a06 (n) (k) (_ : k \u2264 n), f k with h\u03be", "annotated_tactic": ["set \u03be := \u2a06 (n) (k) (_ : k \u2264 n), f k with h\u03be", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u22a2 \u2203 p, Measurable p.2 \u2227 p.1 \u27c2\u2098 \u03bd \u2227 \u03bc = p.1 + withDensity \u03bd p.2", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\n\u22a2 \u2203 p, Measurable p.2 \u2227 p.1 \u27c2\u2098 \u03bd \u2227 \u03bc = p.1 + withDensity \u03bd p.2"}, {"tactic": "have h\u03bem : Measurable \u03be := by\n  convert measurable_iSup fun n => (iSup_mem_measurableLE _ hf\u2081 n).1\n  refine Option.ext fun x => ?_; simp [h\u03be]", "annotated_tactic": ["have h\u03bem : <a>Measurable</a> \u03be := by\n      convert <a>measurable_iSup</a> fun n => (<a>iSup_mem_measurableLE</a> _ hf\u2081 n).1\n      refine <a>Option.ext</a> fun x => ?_; simp [h\u03be]", [{"full_name": "Measurable", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [535, 5], "def_end_pos": [535, 15]}, {"full_name": "measurable_iSup", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [1360, 9], "def_end_pos": [1360, 24]}, {"full_name": "MeasureTheory.Measure.LebesgueDecomposition.iSup_mem_measurableLE", "def_path": "Mathlib/MeasureTheory/Decomposition/Lebesgue.lean", "def_pos": [498, 9], "def_end_pos": [498, 30]}, {"full_name": "Option.ext", "def_path": "lake-packages/std/Std/Data/Option/Lemmas.lean", "def_pos": [43, 16], "def_end_pos": [43, 19]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd\n\u22a2 \u2203 p, Measurable p.2 \u2227 p.1 \u27c2\u2098 \u03bd \u2227 \u03bc = p.1 + withDensity \u03bd p.2", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd\nh\u03bem : Measurable \u03be\n\u22a2 \u2203 p, Measurable p.2 \u2227 p.1 \u27c2\u2098 \u03bd \u2227 \u03bc = p.1 + withDensity \u03bd p.2"}, {"tactic": "set \u03bc\u2081 := \u03bc - \u03bd.withDensity \u03be with h\u03bc\u2081", "annotated_tactic": ["set \u03bc\u2081 := \u03bc - \u03bd.withDensity \u03be with h\u03bc\u2081", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd\nh\u03bem : Measurable \u03be\n\u22a2 \u2203 p, Measurable p.2 \u2227 p.1 \u27c2\u2098 \u03bd \u2227 \u03bc = p.1 + withDensity \u03bd p.2", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd\nh\u03bem : Measurable \u03be\n\u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be\nh\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be\n\u22a2 \u2203 p, Measurable p.2 \u2227 p.1 \u27c2\u2098 \u03bd \u2227 \u03bc = p.1 + withDensity \u03bd p.2"}, {"tactic": "have hle : \u03bd.withDensity \u03be \u2264 \u03bc := by\n  intro B hB\n  rw [h\u03be, withDensity_apply _ hB]\n  simp_rw [iSup_apply]\n  rw [lintegral_iSup (fun i => (iSup_mem_measurableLE _ hf\u2081 i).1) (iSup_monotone _)]\n  exact iSup_le fun i => (iSup_mem_measurableLE _ hf\u2081 i).2 B hB", "annotated_tactic": ["have hle : \u03bd.withDensity \u03be \u2264 \u03bc := by\n      intro B hB\n      rw [h\u03be, <a>withDensity_apply</a> _ hB]\n      simp_rw [<a>iSup_apply</a>]\n      rw [<a>lintegral_iSup</a> (fun i => (<a>iSup_mem_measurableLE</a> _ hf\u2081 i).1) (<a>iSup_monotone</a> _)]\n      exact <a>iSup_le</a> fun i => (<a>iSup_mem_measurableLE</a> _ hf\u2081 i).2 B hB", [{"full_name": "MeasureTheory.withDensity_apply", "def_path": "Mathlib/MeasureTheory/Measure/WithDensity.lean", "def_pos": [39, 9], "def_end_pos": [39, 26]}, {"full_name": "iSup_apply", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [1844, 9], "def_end_pos": [1844, 19]}, {"full_name": "MeasureTheory.lintegral_iSup", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [345, 9], "def_end_pos": [345, 23]}, {"full_name": "MeasureTheory.Measure.LebesgueDecomposition.iSup_mem_measurableLE", "def_path": "Mathlib/MeasureTheory/Decomposition/Lebesgue.lean", "def_pos": [498, 9], "def_end_pos": [498, 30]}, {"full_name": "MeasureTheory.Measure.LebesgueDecomposition.iSup_monotone", "def_path": "Mathlib/MeasureTheory/Decomposition/Lebesgue.lean", "def_pos": [522, 9], "def_end_pos": [522, 22]}, {"full_name": "iSup_le", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [875, 9], "def_end_pos": [875, 16]}, {"full_name": "MeasureTheory.Measure.LebesgueDecomposition.iSup_mem_measurableLE", "def_path": "Mathlib/MeasureTheory/Decomposition/Lebesgue.lean", "def_pos": [498, 9], "def_end_pos": [498, 30]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd\nh\u03bem : Measurable \u03be\n\u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be\nh\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be\n\u22a2 \u2203 p, Measurable p.2 \u2227 p.1 \u27c2\u2098 \u03bd \u2227 \u03bc = p.1 + withDensity \u03bd p.2", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd\nh\u03bem : Measurable \u03be\n\u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be\nh\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be\nhle : withDensity \u03bd \u03be \u2264 \u03bc\n\u22a2 \u2203 p, Measurable p.2 \u2227 p.1 \u27c2\u2098 \u03bd \u2227 \u03bc = p.1 + withDensity \u03bd p.2"}, {"tactic": "have : IsFiniteMeasure (\u03bd.withDensity \u03be) := by\n  refine' isFiniteMeasure_withDensity _\n  have hle' := hle univ MeasurableSet.univ\n  rw [withDensity_apply _ MeasurableSet.univ, Measure.restrict_univ] at hle'\n  exact ne_top_of_le_ne_top (measure_ne_top _ _) hle'", "annotated_tactic": ["have : <a>IsFiniteMeasure</a> (\u03bd.withDensity \u03be) := by\n      refine' <a>isFiniteMeasure_withDensity</a> _\n      have hle' := hle <a>univ</a> <a>MeasurableSet.univ</a>\n      rw [<a>withDensity_apply</a> _ <a>MeasurableSet.univ</a>, <a>Measure.restrict_univ</a>] at hle'\n      exact <a>ne_top_of_le_ne_top</a> (<a>measure_ne_top</a> _ _) hle'", [{"full_name": "MeasureTheory.IsFiniteMeasure", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2850, 7], "def_end_pos": [2850, 22]}, {"full_name": "MeasureTheory.isFiniteMeasure_withDensity", "def_path": "Mathlib/MeasureTheory/Measure/WithDensity.lean", "def_pos": [111, 9], "def_end_pos": [111, 36]}, {"full_name": "Set.univ", "def_path": "Mathlib/Init/Set.lean", "def_pos": [90, 5], "def_end_pos": [90, 9]}, {"full_name": "MeasurableSet.univ", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [101, 19], "def_end_pos": [101, 37]}, {"full_name": "MeasureTheory.withDensity_apply", "def_path": "Mathlib/MeasureTheory/Measure/WithDensity.lean", "def_pos": [39, 9], "def_end_pos": [39, 26]}, {"full_name": "MeasurableSet.univ", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [101, 19], "def_end_pos": [101, 37]}, {"full_name": "MeasureTheory.Measure.restrict_univ", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1701, 9], "def_end_pos": [1701, 22]}, {"full_name": "ne_top_of_le_ne_top", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [194, 9], "def_end_pos": [194, 28]}, {"full_name": "MeasureTheory.measure_ne_top", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2875, 9], "def_end_pos": [2875, 23]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd\nh\u03bem : Measurable \u03be\n\u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be\nh\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be\nhle : withDensity \u03bd \u03be \u2264 \u03bc\n\u22a2 \u2203 p, Measurable p.2 \u2227 p.1 \u27c2\u2098 \u03bd \u2227 \u03bc = p.1 + withDensity \u03bd p.2", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd\nh\u03bem : Measurable \u03be\n\u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be\nh\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be\nhle : withDensity \u03bd \u03be \u2264 \u03bc\nthis : IsFiniteMeasure (withDensity \u03bd \u03be)\n\u22a2 \u2203 p, Measurable p.2 \u2227 p.1 \u27c2\u2098 \u03bd \u2227 \u03bc = p.1 + withDensity \u03bd p.2"}, {"tactic": "refine' \u27e8\u27e8\u03bc\u2081, \u03be\u27e9, h\u03bem, _, _\u27e9", "annotated_tactic": ["refine' \u27e8\u27e8\u03bc\u2081, \u03be\u27e9, h\u03bem, _, _\u27e9", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd\nh\u03bem : Measurable \u03be\n\u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be\nh\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be\nhle : withDensity \u03bd \u03be \u2264 \u03bc\nthis : IsFiniteMeasure (withDensity \u03bd \u03be)\n\u22a2 \u2203 p, Measurable p.2 \u2227 p.1 \u27c2\u2098 \u03bd \u2227 \u03bc = p.1 + withDensity \u03bd p.2", "state_after": "case refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd\nh\u03bem : Measurable \u03be\n\u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be\nh\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be\nhle : withDensity \u03bd \u03be \u2264 \u03bc\nthis : IsFiniteMeasure (withDensity \u03bd \u03be)\n\u22a2 (\u03bc\u2081, \u03be).1 \u27c2\u2098 \u03bd\n\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd\nh\u03bem : Measurable \u03be\n\u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be\nh\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be\nhle : withDensity \u03bd \u03be \u2264 \u03bc\nthis : IsFiniteMeasure (withDensity \u03bd \u03be)\n\u22a2 \u03bc = (\u03bc\u2081, \u03be).1 + withDensity \u03bd (\u03bc\u2081, \u03be).2"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\n\u22a2 (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) 0 = 0", "state_after": "no goals"}, {"tactic": "have :=\n  @lintegral_tendsto_of_tendsto_of_monotone _ _ \u03bd (fun n => \u2a06 (k) (_ : k \u2264 n), f k)\n    (\u2a06 (n) (k) (_ : k \u2264 n), f k) ?_ ?_ ?_", "annotated_tactic": ["have :=\n        @<a>lintegral_tendsto_of_tendsto_of_monotone</a> _ _ \u03bd (fun n => \u2a06 (k) (_ : k \u2264 n), f k)\n          (\u2a06 (n) (k) (_ : k \u2264 n), f k) ?_ ?_ ?_", [{"full_name": "MeasureTheory.lintegral_tendsto_of_tendsto_of_monotone", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [435, 9], "def_end_pos": [435, 49]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\n\u22a2 sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd", "state_after": "case refine_4\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nthis :\n  Filter.Tendsto (fun n => \u222b\u207b (x : \u03b1), iSup (fun k => \u2a06 (_ : k \u2264 n), f k) x \u2202\u03bd) Filter.atTop\n    (nhds (\u222b\u207b (x : \u03b1), iSup (fun n => \u2a06 k, \u2a06 (_ : k \u2264 n), f k) x \u2202\u03bd))\n\u22a2 sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd\n\ncase refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\n\u22a2 \u2200 (n : \u2115), AEMeasurable ((fun n => \u2a06 k, \u2a06 (_ : k \u2264 n), f k) n)\n\ncase refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bd, Monotone fun n => (fun n => \u2a06 k, \u2a06 (_ : k \u2264 n), f k) n x\n\ncase refine_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bd,\n    Filter.Tendsto (fun n => (fun n => \u2a06 k, \u2a06 (_ : k \u2264 n), f k) n x) Filter.atTop\n      (nhds (iSup (fun n => \u2a06 k, \u2a06 (_ : k \u2264 n), f k) x))"}, {"tactic": "refine' tendsto_nhds_unique _ this", "annotated_tactic": ["refine' <a>tendsto_nhds_unique</a> _ this", [{"full_name": "tendsto_nhds_unique", "def_path": "Mathlib/Topology/Separation.lean", "def_pos": [994, 9], "def_end_pos": [994, 28]}]], "state_before": "case refine_4\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nthis :\n  Filter.Tendsto (fun n => \u222b\u207b (x : \u03b1), iSup (fun k => \u2a06 (_ : k \u2264 n), f k) x \u2202\u03bd) Filter.atTop\n    (nhds (\u222b\u207b (x : \u03b1), iSup (fun n => \u2a06 k, \u2a06 (_ : k \u2264 n), f k) x \u2202\u03bd))\n\u22a2 sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd", "state_after": "case refine_4\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nthis :\n  Filter.Tendsto (fun n => \u222b\u207b (x : \u03b1), iSup (fun k => \u2a06 (_ : k \u2264 n), f k) x \u2202\u03bd) Filter.atTop\n    (nhds (\u222b\u207b (x : \u03b1), iSup (fun n => \u2a06 k, \u2a06 (_ : k \u2264 n), f k) x \u2202\u03bd))\n\u22a2 Filter.Tendsto (fun n => \u222b\u207b (x : \u03b1), iSup (fun k => \u2a06 (_ : k \u2264 n), f k) x \u2202\u03bd) Filter.atTop\n    (nhds (sSup (measurableLEEval \u03bd \u03bc)))"}, {"tactic": "refine' tendsto_of_tendsto_of_tendsto_of_le_of_le hg\u2082 tendsto_const_nhds _ _", "annotated_tactic": ["refine' <a>tendsto_of_tendsto_of_tendsto_of_le_of_le</a> hg\u2082 <a>tendsto_const_nhds</a> _ _", [{"full_name": "tendsto_of_tendsto_of_tendsto_of_le_of_le", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [955, 9], "def_end_pos": [955, 50]}, {"full_name": "tendsto_const_nhds", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [1049, 9], "def_end_pos": [1049, 27]}]], "state_before": "case refine_4\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nthis :\n  Filter.Tendsto (fun n => \u222b\u207b (x : \u03b1), iSup (fun k => \u2a06 (_ : k \u2264 n), f k) x \u2202\u03bd) Filter.atTop\n    (nhds (\u222b\u207b (x : \u03b1), iSup (fun n => \u2a06 k, \u2a06 (_ : k \u2264 n), f k) x \u2202\u03bd))\n\u22a2 Filter.Tendsto (fun n => \u222b\u207b (x : \u03b1), iSup (fun k => \u2a06 (_ : k \u2264 n), f k) x \u2202\u03bd) Filter.atTop\n    (nhds (sSup (measurableLEEval \u03bd \u03bc)))", "state_after": "case refine_4.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nthis :\n  Filter.Tendsto (fun n => \u222b\u207b (x : \u03b1), iSup (fun k => \u2a06 (_ : k \u2264 n), f k) x \u2202\u03bd) Filter.atTop\n    (nhds (\u222b\u207b (x : \u03b1), iSup (fun n => \u2a06 k, \u2a06 (_ : k \u2264 n), f k) x \u2202\u03bd))\n\u22a2 g \u2264 fun n => \u222b\u207b (x : \u03b1), iSup (fun k => \u2a06 (_ : k \u2264 n), f k) x \u2202\u03bd\n\ncase refine_4.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nthis :\n  Filter.Tendsto (fun n => \u222b\u207b (x : \u03b1), iSup (fun k => \u2a06 (_ : k \u2264 n), f k) x \u2202\u03bd) Filter.atTop\n    (nhds (\u222b\u207b (x : \u03b1), iSup (fun n => \u2a06 k, \u2a06 (_ : k \u2264 n), f k) x \u2202\u03bd))\n\u22a2 (fun n => \u222b\u207b (x : \u03b1), iSup (fun k => \u2a06 (_ : k \u2264 n), f k) x \u2202\u03bd) \u2264 fun x => sSup (measurableLEEval \u03bd \u03bc)"}, {"tactic": "intro n", "annotated_tactic": ["intro n", []], "state_before": "case refine_4.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nthis :\n  Filter.Tendsto (fun n => \u222b\u207b (x : \u03b1), iSup (fun k => \u2a06 (_ : k \u2264 n), f k) x \u2202\u03bd) Filter.atTop\n    (nhds (\u222b\u207b (x : \u03b1), iSup (fun n => \u2a06 k, \u2a06 (_ : k \u2264 n), f k) x \u2202\u03bd))\n\u22a2 g \u2264 fun n => \u222b\u207b (x : \u03b1), iSup (fun k => \u2a06 (_ : k \u2264 n), f k) x \u2202\u03bd", "state_after": "case refine_4.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nthis :\n  Filter.Tendsto (fun n => \u222b\u207b (x : \u03b1), iSup (fun k => \u2a06 (_ : k \u2264 n), f k) x \u2202\u03bd) Filter.atTop\n    (nhds (\u222b\u207b (x : \u03b1), iSup (fun n => \u2a06 k, \u2a06 (_ : k \u2264 n), f k) x \u2202\u03bd))\nn : \u2115\n\u22a2 g n \u2264 (fun n => \u222b\u207b (x : \u03b1), iSup (fun k => \u2a06 (_ : k \u2264 n), f k) x \u2202\u03bd) n"}, {"tactic": "rw [\u2190 hf\u2082 n]", "annotated_tactic": ["rw [\u2190 hf\u2082 n]", []], "state_before": "case refine_4.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nthis :\n  Filter.Tendsto (fun n => \u222b\u207b (x : \u03b1), iSup (fun k => \u2a06 (_ : k \u2264 n), f k) x \u2202\u03bd) Filter.atTop\n    (nhds (\u222b\u207b (x : \u03b1), iSup (fun n => \u2a06 k, \u2a06 (_ : k \u2264 n), f k) x \u2202\u03bd))\nn : \u2115\n\u22a2 g n \u2264 (fun n => \u222b\u207b (x : \u03b1), iSup (fun k => \u2a06 (_ : k \u2264 n), f k) x \u2202\u03bd) n", "state_after": "case refine_4.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nthis :\n  Filter.Tendsto (fun n => \u222b\u207b (x : \u03b1), iSup (fun k => \u2a06 (_ : k \u2264 n), f k) x \u2202\u03bd) Filter.atTop\n    (nhds (\u222b\u207b (x : \u03b1), iSup (fun n => \u2a06 k, \u2a06 (_ : k \u2264 n), f k) x \u2202\u03bd))\nn : \u2115\n\u22a2 (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) \u2264 (fun n => \u222b\u207b (x : \u03b1), iSup (fun k => \u2a06 (_ : k \u2264 n), f k) x \u2202\u03bd) n"}, {"tactic": "apply lintegral_mono", "annotated_tactic": ["apply <a>lintegral_mono</a>", [{"full_name": "MeasureTheory.lintegral_mono", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [99, 9], "def_end_pos": [99, 23]}]], "state_before": "case refine_4.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nthis :\n  Filter.Tendsto (fun n => \u222b\u207b (x : \u03b1), iSup (fun k => \u2a06 (_ : k \u2264 n), f k) x \u2202\u03bd) Filter.atTop\n    (nhds (\u222b\u207b (x : \u03b1), iSup (fun n => \u2a06 k, \u2a06 (_ : k \u2264 n), f k) x \u2202\u03bd))\nn : \u2115\n\u22a2 (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) \u2264 (fun n => \u222b\u207b (x : \u03b1), iSup (fun k => \u2a06 (_ : k \u2264 n), f k) x \u2202\u03bd) n", "state_after": "case refine_4.refine'_1.hfg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nthis :\n  Filter.Tendsto (fun n => \u222b\u207b (x : \u03b1), iSup (fun k => \u2a06 (_ : k \u2264 n), f k) x \u2202\u03bd) Filter.atTop\n    (nhds (\u222b\u207b (x : \u03b1), iSup (fun n => \u2a06 k, \u2a06 (_ : k \u2264 n), f k) x \u2202\u03bd))\nn : \u2115\n\u22a2 (fun a => f n a) \u2264 fun a => iSup (fun k => \u2a06 (_ : k \u2264 n), f k) a"}, {"tactic": "simp only [iSup_apply, iSup_le_le f n n le_rfl]", "annotated_tactic": ["simp only [<a>iSup_apply</a>, <a>iSup_le_le</a> f n n <a>le_rfl</a>]", [{"full_name": "iSup_apply", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [1844, 9], "def_end_pos": [1844, 19]}, {"full_name": "MeasureTheory.Measure.LebesgueDecomposition.iSup_le_le", "def_path": "Mathlib/MeasureTheory/Decomposition/Lebesgue.lean", "def_pos": [531, 9], "def_end_pos": [531, 19]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}]], "state_before": "case refine_4.refine'_1.hfg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nthis :\n  Filter.Tendsto (fun n => \u222b\u207b (x : \u03b1), iSup (fun k => \u2a06 (_ : k \u2264 n), f k) x \u2202\u03bd) Filter.atTop\n    (nhds (\u222b\u207b (x : \u03b1), iSup (fun n => \u2a06 k, \u2a06 (_ : k \u2264 n), f k) x \u2202\u03bd))\nn : \u2115\n\u22a2 (fun a => f n a) \u2264 fun a => iSup (fun k => \u2a06 (_ : k \u2264 n), f k) a", "state_after": "no goals"}, {"tactic": "intro n", "annotated_tactic": ["intro n", []], "state_before": "case refine_4.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nthis :\n  Filter.Tendsto (fun n => \u222b\u207b (x : \u03b1), iSup (fun k => \u2a06 (_ : k \u2264 n), f k) x \u2202\u03bd) Filter.atTop\n    (nhds (\u222b\u207b (x : \u03b1), iSup (fun n => \u2a06 k, \u2a06 (_ : k \u2264 n), f k) x \u2202\u03bd))\n\u22a2 (fun n => \u222b\u207b (x : \u03b1), iSup (fun k => \u2a06 (_ : k \u2264 n), f k) x \u2202\u03bd) \u2264 fun x => sSup (measurableLEEval \u03bd \u03bc)", "state_after": "case refine_4.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nthis :\n  Filter.Tendsto (fun n => \u222b\u207b (x : \u03b1), iSup (fun k => \u2a06 (_ : k \u2264 n), f k) x \u2202\u03bd) Filter.atTop\n    (nhds (\u222b\u207b (x : \u03b1), iSup (fun n => \u2a06 k, \u2a06 (_ : k \u2264 n), f k) x \u2202\u03bd))\nn : \u2115\n\u22a2 (fun n => \u222b\u207b (x : \u03b1), iSup (fun k => \u2a06 (_ : k \u2264 n), f k) x \u2202\u03bd) n \u2264 (fun x => sSup (measurableLEEval \u03bd \u03bc)) n"}, {"tactic": "exact le_sSup \u27e8\u2a06 (k : \u2115) (_ : k \u2264 n), f k, iSup_mem_measurableLE' _ hf\u2081 _, rfl\u27e9", "annotated_tactic": ["exact <a>le_sSup</a> \u27e8\u2a06 (k : \u2115) (_ : k \u2264 n), f k, <a>iSup_mem_measurableLE'</a> _ hf\u2081 _, <a>rfl</a>\u27e9", [{"full_name": "le_sSup", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [196, 9], "def_end_pos": [196, 16]}, {"full_name": "MeasureTheory.Measure.LebesgueDecomposition.iSup_mem_measurableLE'", "def_path": "Mathlib/MeasureTheory/Decomposition/Lebesgue.lean", "def_pos": [512, 9], "def_end_pos": [512, 31]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "case refine_4.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nthis :\n  Filter.Tendsto (fun n => \u222b\u207b (x : \u03b1), iSup (fun k => \u2a06 (_ : k \u2264 n), f k) x \u2202\u03bd) Filter.atTop\n    (nhds (\u222b\u207b (x : \u03b1), iSup (fun n => \u2a06 k, \u2a06 (_ : k \u2264 n), f k) x \u2202\u03bd))\nn : \u2115\n\u22a2 (fun n => \u222b\u207b (x : \u03b1), iSup (fun k => \u2a06 (_ : k \u2264 n), f k) x \u2202\u03bd) n \u2264 (fun x => sSup (measurableLEEval \u03bd \u03bc)) n", "state_after": "no goals"}, {"tactic": "intro n", "annotated_tactic": ["intro n", []], "state_before": "case refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\n\u22a2 \u2200 (n : \u2115), AEMeasurable ((fun n => \u2a06 k, \u2a06 (_ : k \u2264 n), f k) n)", "state_after": "case refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nn : \u2115\n\u22a2 AEMeasurable ((fun n => \u2a06 k, \u2a06 (_ : k \u2264 n), f k) n)"}, {"tactic": "refine' Measurable.aemeasurable _", "annotated_tactic": ["refine' <a>Measurable.aemeasurable</a> _", [{"full_name": "Measurable.aemeasurable", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [713, 9], "def_end_pos": [713, 32]}]], "state_before": "case refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nn : \u2115\n\u22a2 AEMeasurable ((fun n => \u2a06 k, \u2a06 (_ : k \u2264 n), f k) n)", "state_after": "case refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nn : \u2115\n\u22a2 Measurable ((fun n => \u2a06 k, \u2a06 (_ : k \u2264 n), f k) n)"}, {"tactic": "convert (iSup_mem_measurableLE _ hf\u2081 n).1", "annotated_tactic": ["convert (<a>iSup_mem_measurableLE</a> _ hf\u2081 n).1", [{"full_name": "MeasureTheory.Measure.LebesgueDecomposition.iSup_mem_measurableLE", "def_path": "Mathlib/MeasureTheory/Decomposition/Lebesgue.lean", "def_pos": [498, 9], "def_end_pos": [498, 30]}]], "state_before": "case refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nn : \u2115\n\u22a2 Measurable ((fun n => \u2a06 k, \u2a06 (_ : k \u2264 n), f k) n)", "state_after": "case h.e'_5.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nn : \u2115\nx\u271d : \u03b1\n\u22a2 (fun n => \u2a06 k, \u2a06 (_ : k \u2264 n), f k) n x\u271d = \u2a06 k, \u2a06 (_ : k \u2264 n), f k x\u271d"}, {"tactic": "refine Option.ext fun x => ?_", "annotated_tactic": ["refine <a>Option.ext</a> fun x => ?_", [{"full_name": "Option.ext", "def_path": "lake-packages/std/Std/Data/Option/Lemmas.lean", "def_pos": [43, 16], "def_end_pos": [43, 19]}]], "state_before": "case h.e'_5.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nn : \u2115\nx\u271d : \u03b1\n\u22a2 (fun n => \u2a06 k, \u2a06 (_ : k \u2264 n), f k) n x\u271d = \u2a06 k, \u2a06 (_ : k \u2264 n), f k x\u271d", "state_after": "case h.e'_5.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nn : \u2115\nx\u271d : \u03b1\nx : \u211d\u22650\n\u22a2 x \u2208 (fun n => \u2a06 k, \u2a06 (_ : k \u2264 n), f k) n x\u271d \u2194 x \u2208 \u2a06 k, \u2a06 (_ : k \u2264 n), f k x\u271d"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case h.e'_5.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nn : \u2115\nx\u271d : \u03b1\nx : \u211d\u22650\n\u22a2 x \u2208 (fun n => \u2a06 k, \u2a06 (_ : k \u2264 n), f k) n x\u271d \u2194 x \u2208 \u2a06 k, \u2a06 (_ : k \u2264 n), f k x\u271d", "state_after": "no goals"}, {"tactic": "refine' Filter.eventually_of_forall fun a => _", "annotated_tactic": ["refine' <a>Filter.eventually_of_forall</a> fun a => _", [{"full_name": "Filter.eventually_of_forall", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1111, 9], "def_end_pos": [1111, 29]}]], "state_before": "case refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bd, Monotone fun n => (fun n => \u2a06 k, \u2a06 (_ : k \u2264 n), f k) n x", "state_after": "case refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\na : \u03b1\n\u22a2 Monotone fun n => (fun n => \u2a06 k, \u2a06 (_ : k \u2264 n), f k) n a"}, {"tactic": "simp [iSup_monotone' f _]", "annotated_tactic": ["simp [<a>iSup_monotone'</a> f _]", [{"full_name": "MeasureTheory.Measure.LebesgueDecomposition.iSup_monotone'", "def_path": "Mathlib/MeasureTheory/Decomposition/Lebesgue.lean", "def_pos": [527, 9], "def_end_pos": [527, 23]}]], "state_before": "case refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\na : \u03b1\n\u22a2 Monotone fun n => (fun n => \u2a06 k, \u2a06 (_ : k \u2264 n), f k) n a", "state_after": "no goals"}, {"tactic": "refine' Filter.eventually_of_forall fun a => _", "annotated_tactic": ["refine' <a>Filter.eventually_of_forall</a> fun a => _", [{"full_name": "Filter.eventually_of_forall", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1111, 9], "def_end_pos": [1111, 29]}]], "state_before": "case refine_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bd,\n    Filter.Tendsto (fun n => (fun n => \u2a06 k, \u2a06 (_ : k \u2264 n), f k) n x) Filter.atTop\n      (nhds (iSup (fun n => \u2a06 k, \u2a06 (_ : k \u2264 n), f k) x))", "state_after": "case refine_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\na : \u03b1\n\u22a2 Filter.Tendsto (fun n => (fun n => \u2a06 k, \u2a06 (_ : k \u2264 n), f k) n a) Filter.atTop\n    (nhds (iSup (fun n => \u2a06 k, \u2a06 (_ : k \u2264 n), f k) a))"}, {"tactic": "simp [tendsto_atTop_iSup (iSup_monotone' f a)]", "annotated_tactic": ["simp [<a>tendsto_atTop_iSup</a> (<a>iSup_monotone'</a> f a)]", [{"full_name": "tendsto_atTop_iSup", "def_path": "Mathlib/Topology/Algebra/Order/MonotoneConvergence.lean", "def_pos": [156, 9], "def_end_pos": [156, 27]}, {"full_name": "MeasureTheory.Measure.LebesgueDecomposition.iSup_monotone'", "def_path": "Mathlib/MeasureTheory/Decomposition/Lebesgue.lean", "def_pos": [527, 9], "def_end_pos": [527, 23]}]], "state_before": "case refine_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\na : \u03b1\n\u22a2 Filter.Tendsto (fun n => (fun n => \u2a06 k, \u2a06 (_ : k \u2264 n), f k) n a) Filter.atTop\n    (nhds (iSup (fun n => \u2a06 k, \u2a06 (_ : k \u2264 n), f k) a))", "state_after": "no goals"}, {"tactic": "convert measurable_iSup fun n => (iSup_mem_measurableLE _ hf\u2081 n).1", "annotated_tactic": ["convert <a>measurable_iSup</a> fun n => (<a>iSup_mem_measurableLE</a> _ hf\u2081 n).1", [{"full_name": "measurable_iSup", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [1360, 9], "def_end_pos": [1360, 24]}, {"full_name": "MeasureTheory.Measure.LebesgueDecomposition.iSup_mem_measurableLE", "def_path": "Mathlib/MeasureTheory/Decomposition/Lebesgue.lean", "def_pos": [498, 9], "def_end_pos": [498, 30]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd\n\u22a2 Measurable \u03be", "state_after": "case h.e'_5.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd\nx\u271d : \u03b1\n\u22a2 \u03be x\u271d = \u2a06 i, \u2a06 k, \u2a06 (_ : k \u2264 i), f k x\u271d"}, {"tactic": "refine Option.ext fun x => ?_", "annotated_tactic": ["refine <a>Option.ext</a> fun x => ?_", [{"full_name": "Option.ext", "def_path": "lake-packages/std/Std/Data/Option/Lemmas.lean", "def_pos": [43, 16], "def_end_pos": [43, 19]}]], "state_before": "case h.e'_5.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd\nx\u271d : \u03b1\n\u22a2 \u03be x\u271d = \u2a06 i, \u2a06 k, \u2a06 (_ : k \u2264 i), f k x\u271d", "state_after": "case h.e'_5.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd\nx\u271d : \u03b1\nx : \u211d\u22650\n\u22a2 x \u2208 \u03be x\u271d \u2194 x \u2208 \u2a06 i, \u2a06 k, \u2a06 (_ : k \u2264 i), f k x\u271d"}, {"tactic": "simp [h\u03be]", "annotated_tactic": ["simp [h\u03be]", []], "state_before": "case h.e'_5.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd\nx\u271d : \u03b1\nx : \u211d\u22650\n\u22a2 x \u2208 \u03be x\u271d \u2194 x \u2208 \u2a06 i, \u2a06 k, \u2a06 (_ : k \u2264 i), f k x\u271d", "state_after": "no goals"}, {"tactic": "intro B hB", "annotated_tactic": ["intro B hB", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd\nh\u03bem : Measurable \u03be\n\u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be\nh\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be\n\u22a2 withDensity \u03bd \u03be \u2264 \u03bc", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd\nh\u03bem : Measurable \u03be\n\u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be\nh\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be\nB : Set \u03b1\nhB : MeasurableSet B\n\u22a2 \u2191\u2191(withDensity \u03bd \u03be) B \u2264 \u2191\u2191\u03bc B"}, {"tactic": "rw [h\u03be, withDensity_apply _ hB]", "annotated_tactic": ["rw [h\u03be, <a>withDensity_apply</a> _ hB]", [{"full_name": "MeasureTheory.withDensity_apply", "def_path": "Mathlib/MeasureTheory/Measure/WithDensity.lean", "def_pos": [39, 9], "def_end_pos": [39, 26]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd\nh\u03bem : Measurable \u03be\n\u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be\nh\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be\nB : Set \u03b1\nhB : MeasurableSet B\n\u22a2 \u2191\u2191(withDensity \u03bd \u03be) B \u2264 \u2191\u2191\u03bc B", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd\nh\u03bem : Measurable \u03be\n\u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be\nh\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be\nB : Set \u03b1\nhB : MeasurableSet B\n\u22a2 \u222b\u207b (a : \u03b1) in B, iSup (fun n => \u2a06 k, \u2a06 (_ : k \u2264 n), f k) a \u2202\u03bd \u2264 \u2191\u2191\u03bc B"}, {"tactic": "simp_rw [iSup_apply]", "annotated_tactic": ["simp_rw [<a>iSup_apply</a>]", [{"full_name": "iSup_apply", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [1844, 9], "def_end_pos": [1844, 19]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd\nh\u03bem : Measurable \u03be\n\u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be\nh\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be\nB : Set \u03b1\nhB : MeasurableSet B\n\u22a2 \u222b\u207b (a : \u03b1) in B, iSup (fun n => \u2a06 k, \u2a06 (_ : k \u2264 n), f k) a \u2202\u03bd \u2264 \u2191\u2191\u03bc B", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd\nh\u03bem : Measurable \u03be\n\u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be\nh\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be\nB : Set \u03b1\nhB : MeasurableSet B\n\u22a2 \u222b\u207b (a : \u03b1) in B, \u2a06 i, \u2a06 i_1, \u2a06 (_ : i_1 \u2264 i), f i_1 a \u2202\u03bd \u2264 \u2191\u2191\u03bc B"}, {"tactic": "rw [lintegral_iSup (fun i => (iSup_mem_measurableLE _ hf\u2081 i).1) (iSup_monotone _)]", "annotated_tactic": ["rw [<a>lintegral_iSup</a> (fun i => (<a>iSup_mem_measurableLE</a> _ hf\u2081 i).1) (<a>iSup_monotone</a> _)]", [{"full_name": "MeasureTheory.lintegral_iSup", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [345, 9], "def_end_pos": [345, 23]}, {"full_name": "MeasureTheory.Measure.LebesgueDecomposition.iSup_mem_measurableLE", "def_path": "Mathlib/MeasureTheory/Decomposition/Lebesgue.lean", "def_pos": [498, 9], "def_end_pos": [498, 30]}, {"full_name": "MeasureTheory.Measure.LebesgueDecomposition.iSup_monotone", "def_path": "Mathlib/MeasureTheory/Decomposition/Lebesgue.lean", "def_pos": [522, 9], "def_end_pos": [522, 22]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd\nh\u03bem : Measurable \u03be\n\u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be\nh\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be\nB : Set \u03b1\nhB : MeasurableSet B\n\u22a2 \u222b\u207b (a : \u03b1) in B, \u2a06 i, \u2a06 i_1, \u2a06 (_ : i_1 \u2264 i), f i_1 a \u2202\u03bd \u2264 \u2191\u2191\u03bc B", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd\nh\u03bem : Measurable \u03be\n\u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be\nh\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be\nB : Set \u03b1\nhB : MeasurableSet B\n\u22a2 \u2a06 n, \u222b\u207b (a : \u03b1) in B, \u2a06 k, \u2a06 (_ : k \u2264 n), f k a \u2202\u03bd \u2264 \u2191\u2191\u03bc B"}, {"tactic": "exact iSup_le fun i => (iSup_mem_measurableLE _ hf\u2081 i).2 B hB", "annotated_tactic": ["exact <a>iSup_le</a> fun i => (<a>iSup_mem_measurableLE</a> _ hf\u2081 i).2 B hB", [{"full_name": "iSup_le", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [875, 9], "def_end_pos": [875, 16]}, {"full_name": "MeasureTheory.Measure.LebesgueDecomposition.iSup_mem_measurableLE", "def_path": "Mathlib/MeasureTheory/Decomposition/Lebesgue.lean", "def_pos": [498, 9], "def_end_pos": [498, 30]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd\nh\u03bem : Measurable \u03be\n\u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be\nh\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be\nB : Set \u03b1\nhB : MeasurableSet B\n\u22a2 \u2a06 n, \u222b\u207b (a : \u03b1) in B, \u2a06 k, \u2a06 (_ : k \u2264 n), f k a \u2202\u03bd \u2264 \u2191\u2191\u03bc B", "state_after": "no goals"}, {"tactic": "refine' isFiniteMeasure_withDensity _", "annotated_tactic": ["refine' <a>isFiniteMeasure_withDensity</a> _", [{"full_name": "MeasureTheory.isFiniteMeasure_withDensity", "def_path": "Mathlib/MeasureTheory/Measure/WithDensity.lean", "def_pos": [111, 9], "def_end_pos": [111, 36]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd\nh\u03bem : Measurable \u03be\n\u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be\nh\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be\nhle : withDensity \u03bd \u03be \u2264 \u03bc\n\u22a2 IsFiniteMeasure (withDensity \u03bd \u03be)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd\nh\u03bem : Measurable \u03be\n\u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be\nh\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be\nhle : withDensity \u03bd \u03be \u2264 \u03bc\n\u22a2 \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd \u2260 \u22a4"}, {"tactic": "have hle' := hle univ MeasurableSet.univ", "annotated_tactic": ["have hle' := hle <a>univ</a> <a>MeasurableSet.univ</a>", [{"full_name": "Set.univ", "def_path": "Mathlib/Init/Set.lean", "def_pos": [90, 5], "def_end_pos": [90, 9]}, {"full_name": "MeasurableSet.univ", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [101, 19], "def_end_pos": [101, 37]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd\nh\u03bem : Measurable \u03be\n\u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be\nh\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be\nhle : withDensity \u03bd \u03be \u2264 \u03bc\n\u22a2 \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd \u2260 \u22a4", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd\nh\u03bem : Measurable \u03be\n\u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be\nh\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be\nhle : withDensity \u03bd \u03be \u2264 \u03bc\nhle' : \u2191\u2191(withDensity \u03bd \u03be) univ \u2264 \u2191\u2191\u03bc univ\n\u22a2 \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd \u2260 \u22a4"}, {"tactic": "rw [withDensity_apply _ MeasurableSet.univ, Measure.restrict_univ] at hle'", "annotated_tactic": ["rw [<a>withDensity_apply</a> _ <a>MeasurableSet.univ</a>, <a>Measure.restrict_univ</a>] at hle'", [{"full_name": "MeasureTheory.withDensity_apply", "def_path": "Mathlib/MeasureTheory/Measure/WithDensity.lean", "def_pos": [39, 9], "def_end_pos": [39, 26]}, {"full_name": "MeasurableSet.univ", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [101, 19], "def_end_pos": [101, 37]}, {"full_name": "MeasureTheory.Measure.restrict_univ", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1701, 9], "def_end_pos": [1701, 22]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd\nh\u03bem : Measurable \u03be\n\u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be\nh\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be\nhle : withDensity \u03bd \u03be \u2264 \u03bc\nhle' : \u2191\u2191(withDensity \u03bd \u03be) univ \u2264 \u2191\u2191\u03bc univ\n\u22a2 \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd \u2260 \u22a4", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd\nh\u03bem : Measurable \u03be\n\u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be\nh\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be\nhle : withDensity \u03bd \u03be \u2264 \u03bc\nhle' : \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc univ\n\u22a2 \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd \u2260 \u22a4"}, {"tactic": "exact ne_top_of_le_ne_top (measure_ne_top _ _) hle'", "annotated_tactic": ["exact <a>ne_top_of_le_ne_top</a> (<a>measure_ne_top</a> _ _) hle'", [{"full_name": "ne_top_of_le_ne_top", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [194, 9], "def_end_pos": [194, 28]}, {"full_name": "MeasureTheory.measure_ne_top", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2875, 9], "def_end_pos": [2875, 23]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd\nh\u03bem : Measurable \u03be\n\u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be\nh\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be\nhle : withDensity \u03bd \u03be \u2264 \u03bc\nhle' : \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc univ\n\u22a2 \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd \u2260 \u22a4", "state_after": "no goals"}, {"tactic": "by_contra h", "annotated_tactic": ["by_contra h", []], "state_before": "case refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd\nh\u03bem : Measurable \u03be\n\u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be\nh\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be\nhle : withDensity \u03bd \u03be \u2264 \u03bc\nthis : IsFiniteMeasure (withDensity \u03bd \u03be)\n\u22a2 (\u03bc\u2081, \u03be).1 \u27c2\u2098 \u03bd", "state_after": "case refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd\nh\u03bem : Measurable \u03be\n\u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be\nh\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be\nhle : withDensity \u03bd \u03be \u2264 \u03bc\nthis : IsFiniteMeasure (withDensity \u03bd \u03be)\nh : \u00ac(\u03bc\u2081, \u03be).1 \u27c2\u2098 \u03bd\n\u22a2 False"}, {"tactic": "obtain \u27e8\u03b5, h\u03b5\u2081, E, hE\u2081, hE\u2082, hE\u2083\u27e9 := exists_positive_of_not_mutuallySingular \u03bc\u2081 \u03bd h", "annotated_tactic": ["obtain \u27e8\u03b5, h\u03b5\u2081, E, hE\u2081, hE\u2082, hE\u2083\u27e9 := <a>exists_positive_of_not_mutuallySingular</a> \u03bc\u2081 \u03bd h", [{"full_name": "MeasureTheory.Measure.exists_positive_of_not_mutuallySingular", "def_path": "Mathlib/MeasureTheory/Decomposition/Lebesgue.lean", "def_pos": [389, 9], "def_end_pos": [389, 48]}]], "state_before": "case refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd\nh\u03bem : Measurable \u03be\n\u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be\nh\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be\nhle : withDensity \u03bd \u03be \u2264 \u03bc\nthis : IsFiniteMeasure (withDensity \u03bd \u03be)\nh : \u00ac(\u03bc\u2081, \u03be).1 \u27c2\u2098 \u03bd\n\u22a2 False", "state_after": "case refine'_1.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd\nh\u03bem : Measurable \u03be\n\u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be\nh\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be\nhle : withDensity \u03bd \u03be \u2264 \u03bc\nthis : IsFiniteMeasure (withDensity \u03bd \u03be)\nh : \u00ac(\u03bc\u2081, \u03be).1 \u27c2\u2098 \u03bd\n\u03b5 : \u211d\u22650\nh\u03b5\u2081 : 0 < \u03b5\nE : Set \u03b1\nhE\u2081 : MeasurableSet E\nhE\u2082 : 0 < \u2191\u2191\u03bd E\nhE\u2083 : VectorMeasure.restrict 0 E \u2264 VectorMeasure.restrict (toSignedMeasure \u03bc\u2081 - toSignedMeasure (\u03b5 \u2022 \u03bd)) E\n\u22a2 False"}, {"tactic": "simp_rw [h\u03bc\u2081] at hE\u2083", "annotated_tactic": ["simp_rw [h\u03bc\u2081] at hE\u2083", []], "state_before": "case refine'_1.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd\nh\u03bem : Measurable \u03be\n\u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be\nh\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be\nhle : withDensity \u03bd \u03be \u2264 \u03bc\nthis : IsFiniteMeasure (withDensity \u03bd \u03be)\nh : \u00ac(\u03bc\u2081, \u03be).1 \u27c2\u2098 \u03bd\n\u03b5 : \u211d\u22650\nh\u03b5\u2081 : 0 < \u03b5\nE : Set \u03b1\nhE\u2081 : MeasurableSet E\nhE\u2082 : 0 < \u2191\u2191\u03bd E\nhE\u2083 : VectorMeasure.restrict 0 E \u2264 VectorMeasure.restrict (toSignedMeasure \u03bc\u2081 - toSignedMeasure (\u03b5 \u2022 \u03bd)) E\n\u22a2 False", "state_after": "case refine'_1.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd\nh\u03bem : Measurable \u03be\n\u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be\nh\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be\nhle : withDensity \u03bd \u03be \u2264 \u03bc\nthis : IsFiniteMeasure (withDensity \u03bd \u03be)\nh : \u00ac(\u03bc\u2081, \u03be).1 \u27c2\u2098 \u03bd\n\u03b5 : \u211d\u22650\nh\u03b5\u2081 : 0 < \u03b5\nE : Set \u03b1\nhE\u2081 : MeasurableSet E\nhE\u2082 : 0 < \u2191\u2191\u03bd E\nhE\u2083 :\n  VectorMeasure.restrict 0 E \u2264\n    VectorMeasure.restrict (toSignedMeasure (\u03bc - withDensity \u03bd \u03be) - toSignedMeasure (\u03b5 \u2022 \u03bd)) E\n\u22a2 False"}, {"tactic": "have h\u03bele : \u2200 A, MeasurableSet A \u2192 (\u222b\u207b a in A, \u03be a \u2202\u03bd) \u2264 \u03bc A := by\n  intro A hA; rw [h\u03be]\n  simp_rw [iSup_apply]\n  rw [lintegral_iSup (fun n => (iSup_mem_measurableLE _ hf\u2081 n).1) (iSup_monotone _)]\n  exact iSup_le fun n => (iSup_mem_measurableLE _ hf\u2081 n).2 A hA", "annotated_tactic": ["have h\u03bele : \u2200 A, <a>MeasurableSet</a> A \u2192 (\u222b\u207b a in A, \u03be a \u2202\u03bd) \u2264 \u03bc A := by\n        intro A hA; rw [h\u03be]\n        simp_rw [<a>iSup_apply</a>]\n        rw [<a>lintegral_iSup</a> (fun n => (<a>iSup_mem_measurableLE</a> _ hf\u2081 n).1) (<a>iSup_monotone</a> _)]\n        exact <a>iSup_le</a> fun n => (<a>iSup_mem_measurableLE</a> _ hf\u2081 n).2 A hA", [{"full_name": "MeasurableSet", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [64, 5], "def_end_pos": [64, 18]}, {"full_name": "iSup_apply", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [1844, 9], "def_end_pos": [1844, 19]}, {"full_name": "MeasureTheory.lintegral_iSup", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [345, 9], "def_end_pos": [345, 23]}, {"full_name": "MeasureTheory.Measure.LebesgueDecomposition.iSup_mem_measurableLE", "def_path": "Mathlib/MeasureTheory/Decomposition/Lebesgue.lean", "def_pos": [498, 9], "def_end_pos": [498, 30]}, {"full_name": "MeasureTheory.Measure.LebesgueDecomposition.iSup_monotone", "def_path": "Mathlib/MeasureTheory/Decomposition/Lebesgue.lean", "def_pos": [522, 9], "def_end_pos": [522, 22]}, {"full_name": "iSup_le", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [875, 9], "def_end_pos": [875, 16]}, {"full_name": "MeasureTheory.Measure.LebesgueDecomposition.iSup_mem_measurableLE", "def_path": "Mathlib/MeasureTheory/Decomposition/Lebesgue.lean", "def_pos": [498, 9], "def_end_pos": [498, 30]}]], "state_before": "case refine'_1.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd\nh\u03bem : Measurable \u03be\n\u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be\nh\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be\nhle : withDensity \u03bd \u03be \u2264 \u03bc\nthis : IsFiniteMeasure (withDensity \u03bd \u03be)\nh : \u00ac(\u03bc\u2081, \u03be).1 \u27c2\u2098 \u03bd\n\u03b5 : \u211d\u22650\nh\u03b5\u2081 : 0 < \u03b5\nE : Set \u03b1\nhE\u2081 : MeasurableSet E\nhE\u2082 : 0 < \u2191\u2191\u03bd E\nhE\u2083 :\n  VectorMeasure.restrict 0 E \u2264\n    VectorMeasure.restrict (toSignedMeasure (\u03bc - withDensity \u03bd \u03be) - toSignedMeasure (\u03b5 \u2022 \u03bd)) E\n\u22a2 False", "state_after": "case refine'_1.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd\nh\u03bem : Measurable \u03be\n\u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be\nh\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be\nhle : withDensity \u03bd \u03be \u2264 \u03bc\nthis : IsFiniteMeasure (withDensity \u03bd \u03be)\nh : \u00ac(\u03bc\u2081, \u03be).1 \u27c2\u2098 \u03bd\n\u03b5 : \u211d\u22650\nh\u03b5\u2081 : 0 < \u03b5\nE : Set \u03b1\nhE\u2081 : MeasurableSet E\nhE\u2082 : 0 < \u2191\u2191\u03bd E\nhE\u2083 :\n  VectorMeasure.restrict 0 E \u2264\n    VectorMeasure.restrict (toSignedMeasure (\u03bc - withDensity \u03bd \u03be) - toSignedMeasure (\u03b5 \u2022 \u03bd)) E\nh\u03bele : \u2200 (A : Set \u03b1), MeasurableSet A \u2192 \u222b\u207b (a : \u03b1) in A, \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc A\n\u22a2 False"}, {"tactic": "have h\u03be\u03b5 : (\u03be + E.indicator fun _ => (\u03b5 : \u211d\u22650\u221e)) \u2208 measurableLE \u03bd \u03bc := by\n  refine' \u27e8Measurable.add h\u03bem (Measurable.indicator measurable_const hE\u2081), fun A hA => _\u27e9\n  have :\n    (\u222b\u207b a in A, (\u03be + E.indicator fun _ => (\u03b5 : \u211d\u22650\u221e)) a \u2202\u03bd) =\n      (\u222b\u207b a in A \u2229 E, \u03b5 + \u03be a \u2202\u03bd) + \u222b\u207b a in A \\ E, \u03be a \u2202\u03bd := by\n    simp only [lintegral_add_left measurable_const, lintegral_add_left h\u03bem,\n      set_lintegral_const, add_assoc, lintegral_inter_add_diff _ _ hE\u2081, Pi.add_apply,\n      lintegral_indicator _ hE\u2081, restrict_apply hE\u2081]\n    rw [inter_comm, add_comm]\n  rw [this, \u2190 measure_inter_add_diff A hE\u2081]\n  exact add_le_add (h\u03b5\u2082 A hA) (h\u03bele (A \\ E) (hA.diff hE\u2081))", "annotated_tactic": ["have h\u03be\u03b5 : (\u03be + E.indicator fun _ => (\u03b5 : \u211d\u22650\u221e)) \u2208 <a>measurableLE</a> \u03bd \u03bc := by\n        refine' \u27e8<a>Measurable.add</a> h\u03bem (<a>Measurable.indicator</a> <a>measurable_const</a> hE\u2081), fun A hA => _\u27e9\n        have :\n          (\u222b\u207b a in A, (\u03be + E.indicator fun _ => (\u03b5 : \u211d\u22650\u221e)) a \u2202\u03bd) =\n            (\u222b\u207b a in A \u2229 E, \u03b5 + \u03be a \u2202\u03bd) + \u222b\u207b a in A \\ E, \u03be a \u2202\u03bd := by\n          simp only [<a>lintegral_add_left</a> <a>measurable_const</a>, <a>lintegral_add_left</a> h\u03bem,\n            <a>set_lintegral_const</a>, <a>add_assoc</a>, <a>lintegral_inter_add_diff</a> _ _ hE\u2081, <a>Pi.add_apply</a>,\n            <a>lintegral_indicator</a> _ hE\u2081, <a>restrict_apply</a> hE\u2081]\n          rw [<a>inter_comm</a>, <a>add_comm</a>]\n        rw [this, \u2190 <a>measure_inter_add_diff</a> A hE\u2081]\n        exact <a>add_le_add</a> (h\u03b5\u2082 A hA) (h\u03bele (A \\ E) (hA.diff hE\u2081))", [{"full_name": "MeasureTheory.Measure.LebesgueDecomposition.measurableLE", "def_path": "Mathlib/MeasureTheory/Decomposition/Lebesgue.lean", "def_pos": [458, 5], "def_end_pos": [458, 17]}, {"full_name": "Measurable.add", "def_path": "Mathlib/MeasureTheory/Group/Arithmetic.lean", "def_pos": [140, 3], "def_end_pos": [140, 14]}, {"full_name": "Measurable.indicator", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [353, 9], "def_end_pos": [353, 29]}, {"full_name": "measurable_const", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [570, 9], "def_end_pos": [570, 25]}, {"full_name": "MeasureTheory.lintegral_add_left", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [554, 9], "def_end_pos": [554, 27]}, {"full_name": "measurable_const", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [570, 9], "def_end_pos": [570, 25]}, {"full_name": "MeasureTheory.lintegral_add_left", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [554, 9], "def_end_pos": [554, 27]}, {"full_name": "MeasureTheory.set_lintegral_const", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [152, 9], "def_end_pos": [152, 28]}, {"full_name": "add_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [263, 3], "def_end_pos": [263, 14]}, {"full_name": "MeasureTheory.lintegral_inter_add_diff", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [1253, 9], "def_end_pos": [1253, 33]}, {"full_name": "Pi.add_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [82, 3], "def_end_pos": [82, 14]}, {"full_name": "MeasureTheory.lintegral_indicator", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [762, 9], "def_end_pos": [762, 28]}, {"full_name": "MeasureTheory.Measure.restrict_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1533, 9], "def_end_pos": [1533, 23]}, {"full_name": "Set.inter_comm", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [940, 9], "def_end_pos": [940, 19]}, {"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [301, 3], "def_end_pos": [301, 14]}, {"full_name": "MeasureTheory.measure_inter_add_diff", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [132, 9], "def_end_pos": [132, 31]}, {"full_name": "add_le_add", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [205, 15], "def_end_pos": [205, 25]}]], "state_before": "case refine'_1.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd\nh\u03bem : Measurable \u03be\n\u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be\nh\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be\nhle : withDensity \u03bd \u03be \u2264 \u03bc\nthis : IsFiniteMeasure (withDensity \u03bd \u03be)\nh : \u00ac(\u03bc\u2081, \u03be).1 \u27c2\u2098 \u03bd\n\u03b5 : \u211d\u22650\nh\u03b5\u2081 : 0 < \u03b5\nE : Set \u03b1\nhE\u2081 : MeasurableSet E\nhE\u2082 : 0 < \u2191\u2191\u03bd E\nhE\u2083 :\n  VectorMeasure.restrict 0 E \u2264\n    VectorMeasure.restrict (toSignedMeasure (\u03bc - withDensity \u03bd \u03be) - toSignedMeasure (\u03b5 \u2022 \u03bd)) E\nh\u03bele : \u2200 (A : Set \u03b1), MeasurableSet A \u2192 \u222b\u207b (a : \u03b1) in A, \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc A\nh\u03b5\u2082 : \u2200 (A : Set \u03b1), MeasurableSet A \u2192 \u222b\u207b (a : \u03b1) in A \u2229 E, \u2191\u03b5 + \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc (A \u2229 E)\n\u22a2 False", "state_after": "case refine'_1.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd\nh\u03bem : Measurable \u03be\n\u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be\nh\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be\nhle : withDensity \u03bd \u03be \u2264 \u03bc\nthis : IsFiniteMeasure (withDensity \u03bd \u03be)\nh : \u00ac(\u03bc\u2081, \u03be).1 \u27c2\u2098 \u03bd\n\u03b5 : \u211d\u22650\nh\u03b5\u2081 : 0 < \u03b5\nE : Set \u03b1\nhE\u2081 : MeasurableSet E\nhE\u2082 : 0 < \u2191\u2191\u03bd E\nhE\u2083 :\n  VectorMeasure.restrict 0 E \u2264\n    VectorMeasure.restrict (toSignedMeasure (\u03bc - withDensity \u03bd \u03be) - toSignedMeasure (\u03b5 \u2022 \u03bd)) E\nh\u03bele : \u2200 (A : Set \u03b1), MeasurableSet A \u2192 \u222b\u207b (a : \u03b1) in A, \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc A\nh\u03b5\u2082 : \u2200 (A : Set \u03b1), MeasurableSet A \u2192 \u222b\u207b (a : \u03b1) in A \u2229 E, \u2191\u03b5 + \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc (A \u2229 E)\nh\u03be\u03b5 : (\u03be + indicator E fun x => \u2191\u03b5) \u2208 measurableLE \u03bd \u03bc\n\u22a2 False"}, {"tactic": "have : (\u222b\u207b a, \u03be a + E.indicator (fun _ => (\u03b5 : \u211d\u22650\u221e)) a \u2202\u03bd) \u2264 sSup (measurableLEEval \u03bd \u03bc) :=\n  le_sSup \u27e8\u03be + E.indicator fun _ => (\u03b5 : \u211d\u22650\u221e), h\u03be\u03b5, rfl\u27e9", "annotated_tactic": ["have : (\u222b\u207b a, \u03be a + E.indicator (fun _ => (\u03b5 : \u211d\u22650\u221e)) a \u2202\u03bd) \u2264 <a>sSup</a> (<a>measurableLEEval</a> \u03bd \u03bc) :=\n        <a>le_sSup</a> \u27e8\u03be + E.indicator fun _ => (\u03b5 : \u211d\u22650\u221e), h\u03be\u03b5, <a>rfl</a>\u27e9", [{"full_name": "SupSet.sSup", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [55, 3], "def_end_pos": [55, 7]}, {"full_name": "MeasureTheory.Measure.LebesgueDecomposition.measurableLEEval", "def_path": "Mathlib/MeasureTheory/Decomposition/Lebesgue.lean", "def_pos": [539, 5], "def_end_pos": [539, 21]}, {"full_name": "le_sSup", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [196, 9], "def_end_pos": [196, 16]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "case refine'_1.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd\nh\u03bem : Measurable \u03be\n\u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be\nh\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be\nhle : withDensity \u03bd \u03be \u2264 \u03bc\nthis : IsFiniteMeasure (withDensity \u03bd \u03be)\nh : \u00ac(\u03bc\u2081, \u03be).1 \u27c2\u2098 \u03bd\n\u03b5 : \u211d\u22650\nh\u03b5\u2081 : 0 < \u03b5\nE : Set \u03b1\nhE\u2081 : MeasurableSet E\nhE\u2082 : 0 < \u2191\u2191\u03bd E\nhE\u2083 :\n  VectorMeasure.restrict 0 E \u2264\n    VectorMeasure.restrict (toSignedMeasure (\u03bc - withDensity \u03bd \u03be) - toSignedMeasure (\u03b5 \u2022 \u03bd)) E\nh\u03bele : \u2200 (A : Set \u03b1), MeasurableSet A \u2192 \u222b\u207b (a : \u03b1) in A, \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc A\nh\u03b5\u2082 : \u2200 (A : Set \u03b1), MeasurableSet A \u2192 \u222b\u207b (a : \u03b1) in A \u2229 E, \u2191\u03b5 + \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc (A \u2229 E)\nh\u03be\u03b5 : (\u03be + indicator E fun x => \u2191\u03b5) \u2208 measurableLE \u03bd \u03bc\n\u22a2 False", "state_after": "case refine'_1.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd\nh\u03bem : Measurable \u03be\n\u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be\nh\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be\nhle : withDensity \u03bd \u03be \u2264 \u03bc\nthis\u271d : IsFiniteMeasure (withDensity \u03bd \u03be)\nh : \u00ac(\u03bc\u2081, \u03be).1 \u27c2\u2098 \u03bd\n\u03b5 : \u211d\u22650\nh\u03b5\u2081 : 0 < \u03b5\nE : Set \u03b1\nhE\u2081 : MeasurableSet E\nhE\u2082 : 0 < \u2191\u2191\u03bd E\nhE\u2083 :\n  VectorMeasure.restrict 0 E \u2264\n    VectorMeasure.restrict (toSignedMeasure (\u03bc - withDensity \u03bd \u03be) - toSignedMeasure (\u03b5 \u2022 \u03bd)) E\nh\u03bele : \u2200 (A : Set \u03b1), MeasurableSet A \u2192 \u222b\u207b (a : \u03b1) in A, \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc A\nh\u03b5\u2082 : \u2200 (A : Set \u03b1), MeasurableSet A \u2192 \u222b\u207b (a : \u03b1) in A \u2229 E, \u2191\u03b5 + \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc (A \u2229 E)\nh\u03be\u03b5 : (\u03be + indicator E fun x => \u2191\u03b5) \u2208 measurableLE \u03bd \u03bc\nthis : \u222b\u207b (a : \u03b1), \u03be a + indicator E (fun x => \u2191\u03b5) a \u2202\u03bd \u2264 sSup (measurableLEEval \u03bd \u03bc)\n\u22a2 False"}, {"tactic": "refine' not_lt.2 this _", "annotated_tactic": ["refine' <a>not_lt</a>.2 this _", [{"full_name": "not_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [368, 9], "def_end_pos": [368, 15]}]], "state_before": "case refine'_1.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd\nh\u03bem : Measurable \u03be\n\u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be\nh\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be\nhle : withDensity \u03bd \u03be \u2264 \u03bc\nthis\u271d : IsFiniteMeasure (withDensity \u03bd \u03be)\nh : \u00ac(\u03bc\u2081, \u03be).1 \u27c2\u2098 \u03bd\n\u03b5 : \u211d\u22650\nh\u03b5\u2081 : 0 < \u03b5\nE : Set \u03b1\nhE\u2081 : MeasurableSet E\nhE\u2082 : 0 < \u2191\u2191\u03bd E\nhE\u2083 :\n  VectorMeasure.restrict 0 E \u2264\n    VectorMeasure.restrict (toSignedMeasure (\u03bc - withDensity \u03bd \u03be) - toSignedMeasure (\u03b5 \u2022 \u03bd)) E\nh\u03bele : \u2200 (A : Set \u03b1), MeasurableSet A \u2192 \u222b\u207b (a : \u03b1) in A, \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc A\nh\u03b5\u2082 : \u2200 (A : Set \u03b1), MeasurableSet A \u2192 \u222b\u207b (a : \u03b1) in A \u2229 E, \u2191\u03b5 + \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc (A \u2229 E)\nh\u03be\u03b5 : (\u03be + indicator E fun x => \u2191\u03b5) \u2208 measurableLE \u03bd \u03bc\nthis : \u222b\u207b (a : \u03b1), \u03be a + indicator E (fun x => \u2191\u03b5) a \u2202\u03bd \u2264 sSup (measurableLEEval \u03bd \u03bc)\n\u22a2 False", "state_after": "case refine'_1.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd\nh\u03bem : Measurable \u03be\n\u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be\nh\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be\nhle : withDensity \u03bd \u03be \u2264 \u03bc\nthis\u271d : IsFiniteMeasure (withDensity \u03bd \u03be)\nh : \u00ac(\u03bc\u2081, \u03be).1 \u27c2\u2098 \u03bd\n\u03b5 : \u211d\u22650\nh\u03b5\u2081 : 0 < \u03b5\nE : Set \u03b1\nhE\u2081 : MeasurableSet E\nhE\u2082 : 0 < \u2191\u2191\u03bd E\nhE\u2083 :\n  VectorMeasure.restrict 0 E \u2264\n    VectorMeasure.restrict (toSignedMeasure (\u03bc - withDensity \u03bd \u03be) - toSignedMeasure (\u03b5 \u2022 \u03bd)) E\nh\u03bele : \u2200 (A : Set \u03b1), MeasurableSet A \u2192 \u222b\u207b (a : \u03b1) in A, \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc A\nh\u03b5\u2082 : \u2200 (A : Set \u03b1), MeasurableSet A \u2192 \u222b\u207b (a : \u03b1) in A \u2229 E, \u2191\u03b5 + \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc (A \u2229 E)\nh\u03be\u03b5 : (\u03be + indicator E fun x => \u2191\u03b5) \u2208 measurableLE \u03bd \u03bc\nthis : \u222b\u207b (a : \u03b1), \u03be a + indicator E (fun x => \u2191\u03b5) a \u2202\u03bd \u2264 sSup (measurableLEEval \u03bd \u03bc)\n\u22a2 sSup (measurableLEEval \u03bd \u03bc) < \u222b\u207b (a : \u03b1), \u03be a + indicator E (fun x => \u2191\u03b5) a \u2202\u03bd"}, {"tactic": "rw [h\u03be\u2081, lintegral_add_left h\u03bem, lintegral_indicator _ hE\u2081, set_lintegral_const]", "annotated_tactic": ["rw [h\u03be\u2081, <a>lintegral_add_left</a> h\u03bem, <a>lintegral_indicator</a> _ hE\u2081, <a>set_lintegral_const</a>]", [{"full_name": "MeasureTheory.lintegral_add_left", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [554, 9], "def_end_pos": [554, 27]}, {"full_name": "MeasureTheory.lintegral_indicator", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [762, 9], "def_end_pos": [762, 28]}, {"full_name": "MeasureTheory.set_lintegral_const", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [152, 9], "def_end_pos": [152, 28]}]], "state_before": "case refine'_1.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd\nh\u03bem : Measurable \u03be\n\u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be\nh\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be\nhle : withDensity \u03bd \u03be \u2264 \u03bc\nthis\u271d : IsFiniteMeasure (withDensity \u03bd \u03be)\nh : \u00ac(\u03bc\u2081, \u03be).1 \u27c2\u2098 \u03bd\n\u03b5 : \u211d\u22650\nh\u03b5\u2081 : 0 < \u03b5\nE : Set \u03b1\nhE\u2081 : MeasurableSet E\nhE\u2082 : 0 < \u2191\u2191\u03bd E\nhE\u2083 :\n  VectorMeasure.restrict 0 E \u2264\n    VectorMeasure.restrict (toSignedMeasure (\u03bc - withDensity \u03bd \u03be) - toSignedMeasure (\u03b5 \u2022 \u03bd)) E\nh\u03bele : \u2200 (A : Set \u03b1), MeasurableSet A \u2192 \u222b\u207b (a : \u03b1) in A, \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc A\nh\u03b5\u2082 : \u2200 (A : Set \u03b1), MeasurableSet A \u2192 \u222b\u207b (a : \u03b1) in A \u2229 E, \u2191\u03b5 + \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc (A \u2229 E)\nh\u03be\u03b5 : (\u03be + indicator E fun x => \u2191\u03b5) \u2208 measurableLE \u03bd \u03bc\nthis : \u222b\u207b (a : \u03b1), \u03be a + indicator E (fun x => \u2191\u03b5) a \u2202\u03bd \u2264 sSup (measurableLEEval \u03bd \u03bc)\n\u22a2 sSup (measurableLEEval \u03bd \u03bc) < \u222b\u207b (a : \u03b1), \u03be a + indicator E (fun x => \u2191\u03b5) a \u2202\u03bd", "state_after": "case refine'_1.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd\nh\u03bem : Measurable \u03be\n\u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be\nh\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be\nhle : withDensity \u03bd \u03be \u2264 \u03bc\nthis\u271d : IsFiniteMeasure (withDensity \u03bd \u03be)\nh : \u00ac(\u03bc\u2081, \u03be).1 \u27c2\u2098 \u03bd\n\u03b5 : \u211d\u22650\nh\u03b5\u2081 : 0 < \u03b5\nE : Set \u03b1\nhE\u2081 : MeasurableSet E\nhE\u2082 : 0 < \u2191\u2191\u03bd E\nhE\u2083 :\n  VectorMeasure.restrict 0 E \u2264\n    VectorMeasure.restrict (toSignedMeasure (\u03bc - withDensity \u03bd \u03be) - toSignedMeasure (\u03b5 \u2022 \u03bd)) E\nh\u03bele : \u2200 (A : Set \u03b1), MeasurableSet A \u2192 \u222b\u207b (a : \u03b1) in A, \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc A\nh\u03b5\u2082 : \u2200 (A : Set \u03b1), MeasurableSet A \u2192 \u222b\u207b (a : \u03b1) in A \u2229 E, \u2191\u03b5 + \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc (A \u2229 E)\nh\u03be\u03b5 : (\u03be + indicator E fun x => \u2191\u03b5) \u2208 measurableLE \u03bd \u03bc\nthis : \u222b\u207b (a : \u03b1), \u03be a + indicator E (fun x => \u2191\u03b5) a \u2202\u03bd \u2264 sSup (measurableLEEval \u03bd \u03bc)\n\u22a2 \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd < \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd + \u2191\u03b5 * \u2191\u2191\u03bd E"}, {"tactic": "refine' ENNReal.lt_add_right _ (ENNReal.mul_pos_iff.2 \u27e8ENNReal.coe_pos.2 h\u03b5\u2081, hE\u2082\u27e9).ne'", "annotated_tactic": ["refine' <a>ENNReal.lt_add_right</a> _ (<a>ENNReal.mul_pos_iff</a>.2 \u27e8<a>ENNReal.coe_pos</a>.2 h\u03b5\u2081, hE\u2082\u27e9).<a>ne'</a>", [{"full_name": "ENNReal.lt_add_right", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [829, 9], "def_end_pos": [829, 21]}, {"full_name": "ENNReal.mul_pos_iff", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [641, 9], "def_end_pos": [641, 20]}, {"full_name": "ENNReal.coe_pos", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [380, 28], "def_end_pos": [380, 35]}, {"full_name": "LT.lt.ne'", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [328, 9], "def_end_pos": [328, 12]}]], "state_before": "case refine'_1.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd\nh\u03bem : Measurable \u03be\n\u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be\nh\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be\nhle : withDensity \u03bd \u03be \u2264 \u03bc\nthis\u271d : IsFiniteMeasure (withDensity \u03bd \u03be)\nh : \u00ac(\u03bc\u2081, \u03be).1 \u27c2\u2098 \u03bd\n\u03b5 : \u211d\u22650\nh\u03b5\u2081 : 0 < \u03b5\nE : Set \u03b1\nhE\u2081 : MeasurableSet E\nhE\u2082 : 0 < \u2191\u2191\u03bd E\nhE\u2083 :\n  VectorMeasure.restrict 0 E \u2264\n    VectorMeasure.restrict (toSignedMeasure (\u03bc - withDensity \u03bd \u03be) - toSignedMeasure (\u03b5 \u2022 \u03bd)) E\nh\u03bele : \u2200 (A : Set \u03b1), MeasurableSet A \u2192 \u222b\u207b (a : \u03b1) in A, \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc A\nh\u03b5\u2082 : \u2200 (A : Set \u03b1), MeasurableSet A \u2192 \u222b\u207b (a : \u03b1) in A \u2229 E, \u2191\u03b5 + \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc (A \u2229 E)\nh\u03be\u03b5 : (\u03be + indicator E fun x => \u2191\u03b5) \u2208 measurableLE \u03bd \u03bc\nthis : \u222b\u207b (a : \u03b1), \u03be a + indicator E (fun x => \u2191\u03b5) a \u2202\u03bd \u2264 sSup (measurableLEEval \u03bd \u03bc)\n\u22a2 \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd < \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd + \u2191\u03b5 * \u2191\u2191\u03bd E", "state_after": "case refine'_1.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd\nh\u03bem : Measurable \u03be\n\u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be\nh\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be\nhle : withDensity \u03bd \u03be \u2264 \u03bc\nthis\u271d : IsFiniteMeasure (withDensity \u03bd \u03be)\nh : \u00ac(\u03bc\u2081, \u03be).1 \u27c2\u2098 \u03bd\n\u03b5 : \u211d\u22650\nh\u03b5\u2081 : 0 < \u03b5\nE : Set \u03b1\nhE\u2081 : MeasurableSet E\nhE\u2082 : 0 < \u2191\u2191\u03bd E\nhE\u2083 :\n  VectorMeasure.restrict 0 E \u2264\n    VectorMeasure.restrict (toSignedMeasure (\u03bc - withDensity \u03bd \u03be) - toSignedMeasure (\u03b5 \u2022 \u03bd)) E\nh\u03bele : \u2200 (A : Set \u03b1), MeasurableSet A \u2192 \u222b\u207b (a : \u03b1) in A, \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc A\nh\u03b5\u2082 : \u2200 (A : Set \u03b1), MeasurableSet A \u2192 \u222b\u207b (a : \u03b1) in A \u2229 E, \u2191\u03b5 + \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc (A \u2229 E)\nh\u03be\u03b5 : (\u03be + indicator E fun x => \u2191\u03b5) \u2208 measurableLE \u03bd \u03bc\nthis : \u222b\u207b (a : \u03b1), \u03be a + indicator E (fun x => \u2191\u03b5) a \u2202\u03bd \u2264 sSup (measurableLEEval \u03bd \u03bc)\n\u22a2 \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd \u2260 \u22a4"}, {"tactic": "have := measure_ne_top (\u03bd.withDensity \u03be) univ", "annotated_tactic": ["have := <a>measure_ne_top</a> (\u03bd.withDensity \u03be) <a>univ</a>", [{"full_name": "MeasureTheory.measure_ne_top", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2875, 9], "def_end_pos": [2875, 23]}, {"full_name": "Set.univ", "def_path": "Mathlib/Init/Set.lean", "def_pos": [90, 5], "def_end_pos": [90, 9]}]], "state_before": "case refine'_1.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd\nh\u03bem : Measurable \u03be\n\u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be\nh\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be\nhle : withDensity \u03bd \u03be \u2264 \u03bc\nthis\u271d : IsFiniteMeasure (withDensity \u03bd \u03be)\nh : \u00ac(\u03bc\u2081, \u03be).1 \u27c2\u2098 \u03bd\n\u03b5 : \u211d\u22650\nh\u03b5\u2081 : 0 < \u03b5\nE : Set \u03b1\nhE\u2081 : MeasurableSet E\nhE\u2082 : 0 < \u2191\u2191\u03bd E\nhE\u2083 :\n  VectorMeasure.restrict 0 E \u2264\n    VectorMeasure.restrict (toSignedMeasure (\u03bc - withDensity \u03bd \u03be) - toSignedMeasure (\u03b5 \u2022 \u03bd)) E\nh\u03bele : \u2200 (A : Set \u03b1), MeasurableSet A \u2192 \u222b\u207b (a : \u03b1) in A, \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc A\nh\u03b5\u2082 : \u2200 (A : Set \u03b1), MeasurableSet A \u2192 \u222b\u207b (a : \u03b1) in A \u2229 E, \u2191\u03b5 + \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc (A \u2229 E)\nh\u03be\u03b5 : (\u03be + indicator E fun x => \u2191\u03b5) \u2208 measurableLE \u03bd \u03bc\nthis : \u222b\u207b (a : \u03b1), \u03be a + indicator E (fun x => \u2191\u03b5) a \u2202\u03bd \u2264 sSup (measurableLEEval \u03bd \u03bc)\n\u22a2 \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd \u2260 \u22a4", "state_after": "case refine'_1.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd\nh\u03bem : Measurable \u03be\n\u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be\nh\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be\nhle : withDensity \u03bd \u03be \u2264 \u03bc\nthis\u271d\u00b9 : IsFiniteMeasure (withDensity \u03bd \u03be)\nh : \u00ac(\u03bc\u2081, \u03be).1 \u27c2\u2098 \u03bd\n\u03b5 : \u211d\u22650\nh\u03b5\u2081 : 0 < \u03b5\nE : Set \u03b1\nhE\u2081 : MeasurableSet E\nhE\u2082 : 0 < \u2191\u2191\u03bd E\nhE\u2083 :\n  VectorMeasure.restrict 0 E \u2264\n    VectorMeasure.restrict (toSignedMeasure (\u03bc - withDensity \u03bd \u03be) - toSignedMeasure (\u03b5 \u2022 \u03bd)) E\nh\u03bele : \u2200 (A : Set \u03b1), MeasurableSet A \u2192 \u222b\u207b (a : \u03b1) in A, \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc A\nh\u03b5\u2082 : \u2200 (A : Set \u03b1), MeasurableSet A \u2192 \u222b\u207b (a : \u03b1) in A \u2229 E, \u2191\u03b5 + \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc (A \u2229 E)\nh\u03be\u03b5 : (\u03be + indicator E fun x => \u2191\u03b5) \u2208 measurableLE \u03bd \u03bc\nthis\u271d : \u222b\u207b (a : \u03b1), \u03be a + indicator E (fun x => \u2191\u03b5) a \u2202\u03bd \u2264 sSup (measurableLEEval \u03bd \u03bc)\nthis : \u2191\u2191(withDensity \u03bd \u03be) univ \u2260 \u22a4\n\u22a2 \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd \u2260 \u22a4"}, {"tactic": "rwa [withDensity_apply _ MeasurableSet.univ, Measure.restrict_univ] at this", "annotated_tactic": ["rwa [<a>withDensity_apply</a> _ <a>MeasurableSet.univ</a>, <a>Measure.restrict_univ</a>] at this", [{"full_name": "MeasureTheory.withDensity_apply", "def_path": "Mathlib/MeasureTheory/Measure/WithDensity.lean", "def_pos": [39, 9], "def_end_pos": [39, 26]}, {"full_name": "MeasurableSet.univ", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [101, 19], "def_end_pos": [101, 37]}, {"full_name": "MeasureTheory.Measure.restrict_univ", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1701, 9], "def_end_pos": [1701, 22]}]], "state_before": "case refine'_1.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd\nh\u03bem : Measurable \u03be\n\u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be\nh\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be\nhle : withDensity \u03bd \u03be \u2264 \u03bc\nthis\u271d\u00b9 : IsFiniteMeasure (withDensity \u03bd \u03be)\nh : \u00ac(\u03bc\u2081, \u03be).1 \u27c2\u2098 \u03bd\n\u03b5 : \u211d\u22650\nh\u03b5\u2081 : 0 < \u03b5\nE : Set \u03b1\nhE\u2081 : MeasurableSet E\nhE\u2082 : 0 < \u2191\u2191\u03bd E\nhE\u2083 :\n  VectorMeasure.restrict 0 E \u2264\n    VectorMeasure.restrict (toSignedMeasure (\u03bc - withDensity \u03bd \u03be) - toSignedMeasure (\u03b5 \u2022 \u03bd)) E\nh\u03bele : \u2200 (A : Set \u03b1), MeasurableSet A \u2192 \u222b\u207b (a : \u03b1) in A, \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc A\nh\u03b5\u2082 : \u2200 (A : Set \u03b1), MeasurableSet A \u2192 \u222b\u207b (a : \u03b1) in A \u2229 E, \u2191\u03b5 + \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc (A \u2229 E)\nh\u03be\u03b5 : (\u03be + indicator E fun x => \u2191\u03b5) \u2208 measurableLE \u03bd \u03bc\nthis\u271d : \u222b\u207b (a : \u03b1), \u03be a + indicator E (fun x => \u2191\u03b5) a \u2202\u03bd \u2264 sSup (measurableLEEval \u03bd \u03bc)\nthis : \u2191\u2191(withDensity \u03bd \u03be) univ \u2260 \u22a4\n\u22a2 \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd \u2260 \u22a4", "state_after": "no goals"}, {"tactic": "intro A hA", "annotated_tactic": ["intro A hA", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd\nh\u03bem : Measurable \u03be\n\u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be\nh\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be\nhle : withDensity \u03bd \u03be \u2264 \u03bc\nthis : IsFiniteMeasure (withDensity \u03bd \u03be)\nh : \u00ac(\u03bc\u2081, \u03be).1 \u27c2\u2098 \u03bd\n\u03b5 : \u211d\u22650\nh\u03b5\u2081 : 0 < \u03b5\nE : Set \u03b1\nhE\u2081 : MeasurableSet E\nhE\u2082 : 0 < \u2191\u2191\u03bd E\nhE\u2083 :\n  VectorMeasure.restrict 0 E \u2264\n    VectorMeasure.restrict (toSignedMeasure (\u03bc - withDensity \u03bd \u03be) - toSignedMeasure (\u03b5 \u2022 \u03bd)) E\n\u22a2 \u2200 (A : Set \u03b1), MeasurableSet A \u2192 \u222b\u207b (a : \u03b1) in A, \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc A", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd\nh\u03bem : Measurable \u03be\n\u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be\nh\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be\nhle : withDensity \u03bd \u03be \u2264 \u03bc\nthis : IsFiniteMeasure (withDensity \u03bd \u03be)\nh : \u00ac(\u03bc\u2081, \u03be).1 \u27c2\u2098 \u03bd\n\u03b5 : \u211d\u22650\nh\u03b5\u2081 : 0 < \u03b5\nE : Set \u03b1\nhE\u2081 : MeasurableSet E\nhE\u2082 : 0 < \u2191\u2191\u03bd E\nhE\u2083 :\n  VectorMeasure.restrict 0 E \u2264\n    VectorMeasure.restrict (toSignedMeasure (\u03bc - withDensity \u03bd \u03be) - toSignedMeasure (\u03b5 \u2022 \u03bd)) E\nA : Set \u03b1\nhA : MeasurableSet A\n\u22a2 \u222b\u207b (a : \u03b1) in A, \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc A"}, {"tactic": "rw [h\u03be]", "annotated_tactic": ["rw [h\u03be]", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd\nh\u03bem : Measurable \u03be\n\u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be\nh\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be\nhle : withDensity \u03bd \u03be \u2264 \u03bc\nthis : IsFiniteMeasure (withDensity \u03bd \u03be)\nh : \u00ac(\u03bc\u2081, \u03be).1 \u27c2\u2098 \u03bd\n\u03b5 : \u211d\u22650\nh\u03b5\u2081 : 0 < \u03b5\nE : Set \u03b1\nhE\u2081 : MeasurableSet E\nhE\u2082 : 0 < \u2191\u2191\u03bd E\nhE\u2083 :\n  VectorMeasure.restrict 0 E \u2264\n    VectorMeasure.restrict (toSignedMeasure (\u03bc - withDensity \u03bd \u03be) - toSignedMeasure (\u03b5 \u2022 \u03bd)) E\nA : Set \u03b1\nhA : MeasurableSet A\n\u22a2 \u222b\u207b (a : \u03b1) in A, \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc A", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd\nh\u03bem : Measurable \u03be\n\u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be\nh\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be\nhle : withDensity \u03bd \u03be \u2264 \u03bc\nthis : IsFiniteMeasure (withDensity \u03bd \u03be)\nh : \u00ac(\u03bc\u2081, \u03be).1 \u27c2\u2098 \u03bd\n\u03b5 : \u211d\u22650\nh\u03b5\u2081 : 0 < \u03b5\nE : Set \u03b1\nhE\u2081 : MeasurableSet E\nhE\u2082 : 0 < \u2191\u2191\u03bd E\nhE\u2083 :\n  VectorMeasure.restrict 0 E \u2264\n    VectorMeasure.restrict (toSignedMeasure (\u03bc - withDensity \u03bd \u03be) - toSignedMeasure (\u03b5 \u2022 \u03bd)) E\nA : Set \u03b1\nhA : MeasurableSet A\n\u22a2 \u222b\u207b (a : \u03b1) in A, iSup (fun n => \u2a06 k, \u2a06 (_ : k \u2264 n), f k) a \u2202\u03bd \u2264 \u2191\u2191\u03bc A"}, {"tactic": "simp_rw [iSup_apply]", "annotated_tactic": ["simp_rw [<a>iSup_apply</a>]", [{"full_name": "iSup_apply", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [1844, 9], "def_end_pos": [1844, 19]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd\nh\u03bem : Measurable \u03be\n\u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be\nh\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be\nhle : withDensity \u03bd \u03be \u2264 \u03bc\nthis : IsFiniteMeasure (withDensity \u03bd \u03be)\nh : \u00ac(\u03bc\u2081, \u03be).1 \u27c2\u2098 \u03bd\n\u03b5 : \u211d\u22650\nh\u03b5\u2081 : 0 < \u03b5\nE : Set \u03b1\nhE\u2081 : MeasurableSet E\nhE\u2082 : 0 < \u2191\u2191\u03bd E\nhE\u2083 :\n  VectorMeasure.restrict 0 E \u2264\n    VectorMeasure.restrict (toSignedMeasure (\u03bc - withDensity \u03bd \u03be) - toSignedMeasure (\u03b5 \u2022 \u03bd)) E\nA : Set \u03b1\nhA : MeasurableSet A\n\u22a2 \u222b\u207b (a : \u03b1) in A, iSup (fun n => \u2a06 k, \u2a06 (_ : k \u2264 n), f k) a \u2202\u03bd \u2264 \u2191\u2191\u03bc A", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd\nh\u03bem : Measurable \u03be\n\u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be\nh\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be\nhle : withDensity \u03bd \u03be \u2264 \u03bc\nthis : IsFiniteMeasure (withDensity \u03bd \u03be)\nh : \u00ac(\u03bc\u2081, \u03be).1 \u27c2\u2098 \u03bd\n\u03b5 : \u211d\u22650\nh\u03b5\u2081 : 0 < \u03b5\nE : Set \u03b1\nhE\u2081 : MeasurableSet E\nhE\u2082 : 0 < \u2191\u2191\u03bd E\nhE\u2083 :\n  VectorMeasure.restrict 0 E \u2264\n    VectorMeasure.restrict (toSignedMeasure (\u03bc - withDensity \u03bd \u03be) - toSignedMeasure (\u03b5 \u2022 \u03bd)) E\nA : Set \u03b1\nhA : MeasurableSet A\n\u22a2 \u222b\u207b (a : \u03b1) in A, \u2a06 i, \u2a06 i_1, \u2a06 (_ : i_1 \u2264 i), f i_1 a \u2202\u03bd \u2264 \u2191\u2191\u03bc A"}, {"tactic": "rw [lintegral_iSup (fun n => (iSup_mem_measurableLE _ hf\u2081 n).1) (iSup_monotone _)]", "annotated_tactic": ["rw [<a>lintegral_iSup</a> (fun n => (<a>iSup_mem_measurableLE</a> _ hf\u2081 n).1) (<a>iSup_monotone</a> _)]", [{"full_name": "MeasureTheory.lintegral_iSup", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [345, 9], "def_end_pos": [345, 23]}, {"full_name": "MeasureTheory.Measure.LebesgueDecomposition.iSup_mem_measurableLE", "def_path": "Mathlib/MeasureTheory/Decomposition/Lebesgue.lean", "def_pos": [498, 9], "def_end_pos": [498, 30]}, {"full_name": "MeasureTheory.Measure.LebesgueDecomposition.iSup_monotone", "def_path": "Mathlib/MeasureTheory/Decomposition/Lebesgue.lean", "def_pos": [522, 9], "def_end_pos": [522, 22]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd\nh\u03bem : Measurable \u03be\n\u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be\nh\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be\nhle : withDensity \u03bd \u03be \u2264 \u03bc\nthis : IsFiniteMeasure (withDensity \u03bd \u03be)\nh : \u00ac(\u03bc\u2081, \u03be).1 \u27c2\u2098 \u03bd\n\u03b5 : \u211d\u22650\nh\u03b5\u2081 : 0 < \u03b5\nE : Set \u03b1\nhE\u2081 : MeasurableSet E\nhE\u2082 : 0 < \u2191\u2191\u03bd E\nhE\u2083 :\n  VectorMeasure.restrict 0 E \u2264\n    VectorMeasure.restrict (toSignedMeasure (\u03bc - withDensity \u03bd \u03be) - toSignedMeasure (\u03b5 \u2022 \u03bd)) E\nA : Set \u03b1\nhA : MeasurableSet A\n\u22a2 \u222b\u207b (a : \u03b1) in A, \u2a06 i, \u2a06 i_1, \u2a06 (_ : i_1 \u2264 i), f i_1 a \u2202\u03bd \u2264 \u2191\u2191\u03bc A", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd\nh\u03bem : Measurable \u03be\n\u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be\nh\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be\nhle : withDensity \u03bd \u03be \u2264 \u03bc\nthis : IsFiniteMeasure (withDensity \u03bd \u03be)\nh : \u00ac(\u03bc\u2081, \u03be).1 \u27c2\u2098 \u03bd\n\u03b5 : \u211d\u22650\nh\u03b5\u2081 : 0 < \u03b5\nE : Set \u03b1\nhE\u2081 : MeasurableSet E\nhE\u2082 : 0 < \u2191\u2191\u03bd E\nhE\u2083 :\n  VectorMeasure.restrict 0 E \u2264\n    VectorMeasure.restrict (toSignedMeasure (\u03bc - withDensity \u03bd \u03be) - toSignedMeasure (\u03b5 \u2022 \u03bd)) E\nA : Set \u03b1\nhA : MeasurableSet A\n\u22a2 \u2a06 n, \u222b\u207b (a : \u03b1) in A, \u2a06 k, \u2a06 (_ : k \u2264 n), f k a \u2202\u03bd \u2264 \u2191\u2191\u03bc A"}, {"tactic": "exact iSup_le fun n => (iSup_mem_measurableLE _ hf\u2081 n).2 A hA", "annotated_tactic": ["exact <a>iSup_le</a> fun n => (<a>iSup_mem_measurableLE</a> _ hf\u2081 n).2 A hA", [{"full_name": "iSup_le", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [875, 9], "def_end_pos": [875, 16]}, {"full_name": "MeasureTheory.Measure.LebesgueDecomposition.iSup_mem_measurableLE", "def_path": "Mathlib/MeasureTheory/Decomposition/Lebesgue.lean", "def_pos": [498, 9], "def_end_pos": [498, 30]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd\nh\u03bem : Measurable \u03be\n\u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be\nh\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be\nhle : withDensity \u03bd \u03be \u2264 \u03bc\nthis : IsFiniteMeasure (withDensity \u03bd \u03be)\nh : \u00ac(\u03bc\u2081, \u03be).1 \u27c2\u2098 \u03bd\n\u03b5 : \u211d\u22650\nh\u03b5\u2081 : 0 < \u03b5\nE : Set \u03b1\nhE\u2081 : MeasurableSet E\nhE\u2082 : 0 < \u2191\u2191\u03bd E\nhE\u2083 :\n  VectorMeasure.restrict 0 E \u2264\n    VectorMeasure.restrict (toSignedMeasure (\u03bc - withDensity \u03bd \u03be) - toSignedMeasure (\u03b5 \u2022 \u03bd)) E\nA : Set \u03b1\nhA : MeasurableSet A\n\u22a2 \u2a06 n, \u222b\u207b (a : \u03b1) in A, \u2a06 k, \u2a06 (_ : k \u2264 n), f k a \u2202\u03bd \u2264 \u2191\u2191\u03bc A", "state_after": "no goals"}, {"tactic": "intro A hA", "annotated_tactic": ["intro A hA", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd\nh\u03bem : Measurable \u03be\n\u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be\nh\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be\nhle : withDensity \u03bd \u03be \u2264 \u03bc\nthis : IsFiniteMeasure (withDensity \u03bd \u03be)\nh : \u00ac(\u03bc\u2081, \u03be).1 \u27c2\u2098 \u03bd\n\u03b5 : \u211d\u22650\nh\u03b5\u2081 : 0 < \u03b5\nE : Set \u03b1\nhE\u2081 : MeasurableSet E\nhE\u2082 : 0 < \u2191\u2191\u03bd E\nhE\u2083 :\n  VectorMeasure.restrict 0 E \u2264\n    VectorMeasure.restrict (toSignedMeasure (\u03bc - withDensity \u03bd \u03be) - toSignedMeasure (\u03b5 \u2022 \u03bd)) E\nh\u03bele : \u2200 (A : Set \u03b1), MeasurableSet A \u2192 \u222b\u207b (a : \u03b1) in A, \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc A\n\u22a2 \u2200 (A : Set \u03b1), MeasurableSet A \u2192 \u222b\u207b (a : \u03b1) in A \u2229 E, \u2191\u03b5 + \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc (A \u2229 E)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd\nh\u03bem : Measurable \u03be\n\u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be\nh\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be\nhle : withDensity \u03bd \u03be \u2264 \u03bc\nthis : IsFiniteMeasure (withDensity \u03bd \u03be)\nh : \u00ac(\u03bc\u2081, \u03be).1 \u27c2\u2098 \u03bd\n\u03b5 : \u211d\u22650\nh\u03b5\u2081 : 0 < \u03b5\nE : Set \u03b1\nhE\u2081 : MeasurableSet E\nhE\u2082 : 0 < \u2191\u2191\u03bd E\nhE\u2083 :\n  VectorMeasure.restrict 0 E \u2264\n    VectorMeasure.restrict (toSignedMeasure (\u03bc - withDensity \u03bd \u03be) - toSignedMeasure (\u03b5 \u2022 \u03bd)) E\nh\u03bele : \u2200 (A : Set \u03b1), MeasurableSet A \u2192 \u222b\u207b (a : \u03b1) in A, \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc A\nA : Set \u03b1\nhA : MeasurableSet A\n\u22a2 \u222b\u207b (a : \u03b1) in A \u2229 E, \u2191\u03b5 + \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc (A \u2229 E)"}, {"tactic": "have := subset_le_of_restrict_le_restrict _ _ hE\u2081 hE\u2083 (inter_subset_right A E)", "annotated_tactic": ["have := <a>subset_le_of_restrict_le_restrict</a> _ _ hE\u2081 hE\u2083 (<a>inter_subset_right</a> A E)", [{"full_name": "MeasureTheory.VectorMeasure.subset_le_of_restrict_le_restrict", "def_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "def_pos": [869, 9], "def_end_pos": [869, 42]}, {"full_name": "Set.inter_subset_right", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [969, 9], "def_end_pos": [969, 27]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd\nh\u03bem : Measurable \u03be\n\u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be\nh\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be\nhle : withDensity \u03bd \u03be \u2264 \u03bc\nthis : IsFiniteMeasure (withDensity \u03bd \u03be)\nh : \u00ac(\u03bc\u2081, \u03be).1 \u27c2\u2098 \u03bd\n\u03b5 : \u211d\u22650\nh\u03b5\u2081 : 0 < \u03b5\nE : Set \u03b1\nhE\u2081 : MeasurableSet E\nhE\u2082 : 0 < \u2191\u2191\u03bd E\nhE\u2083 :\n  VectorMeasure.restrict 0 E \u2264\n    VectorMeasure.restrict (toSignedMeasure (\u03bc - withDensity \u03bd \u03be) - toSignedMeasure (\u03b5 \u2022 \u03bd)) E\nh\u03bele : \u2200 (A : Set \u03b1), MeasurableSet A \u2192 \u222b\u207b (a : \u03b1) in A, \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc A\nA : Set \u03b1\nhA : MeasurableSet A\n\u22a2 \u222b\u207b (a : \u03b1) in A \u2229 E, \u2191\u03b5 + \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc (A \u2229 E)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd\nh\u03bem : Measurable \u03be\n\u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be\nh\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be\nhle : withDensity \u03bd \u03be \u2264 \u03bc\nthis\u271d : IsFiniteMeasure (withDensity \u03bd \u03be)\nh : \u00ac(\u03bc\u2081, \u03be).1 \u27c2\u2098 \u03bd\n\u03b5 : \u211d\u22650\nh\u03b5\u2081 : 0 < \u03b5\nE : Set \u03b1\nhE\u2081 : MeasurableSet E\nhE\u2082 : 0 < \u2191\u2191\u03bd E\nhE\u2083 :\n  VectorMeasure.restrict 0 E \u2264\n    VectorMeasure.restrict (toSignedMeasure (\u03bc - withDensity \u03bd \u03be) - toSignedMeasure (\u03b5 \u2022 \u03bd)) E\nh\u03bele : \u2200 (A : Set \u03b1), MeasurableSet A \u2192 \u222b\u207b (a : \u03b1) in A, \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc A\nA : Set \u03b1\nhA : MeasurableSet A\nthis : \u21910 (A \u2229 E) \u2264 \u2191(toSignedMeasure (\u03bc - withDensity \u03bd \u03be) - toSignedMeasure (\u03b5 \u2022 \u03bd)) (A \u2229 E)\n\u22a2 \u222b\u207b (a : \u03b1) in A \u2229 E, \u2191\u03b5 + \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc (A \u2229 E)"}, {"tactic": "rwa [zero_apply, toSignedMeasure_sub_apply (hA.inter hE\u2081),\n  Measure.sub_apply (hA.inter hE\u2081) hle,\n  ENNReal.toReal_sub_of_le _ (ne_of_lt (measure_lt_top _ _)), sub_nonneg, le_sub_iff_add_le,\n  \u2190 ENNReal.toReal_add, ENNReal.toReal_le_toReal, Measure.coe_smul, Pi.smul_apply,\n  withDensity_apply _ (hA.inter hE\u2081), show \u03b5 \u2022 \u03bd (A \u2229 E) = (\u03b5 : \u211d\u22650\u221e) * \u03bd (A \u2229 E) by rfl,\n  \u2190 set_lintegral_const, \u2190 lintegral_add_left measurable_const] at this", "annotated_tactic": ["rwa [<a>zero_apply</a>, <a>toSignedMeasure_sub_apply</a> (hA.inter hE\u2081),\n          <a>Measure.sub_apply</a> (hA.inter hE\u2081) hle,\n          <a>ENNReal.toReal_sub_of_le</a> _ (<a>ne_of_lt</a> (<a>measure_lt_top</a> _ _)), <a>sub_nonneg</a>, <a>le_sub_iff_add_le</a>,\n          \u2190 <a>ENNReal.toReal_add</a>, <a>ENNReal.toReal_le_toReal</a>, <a>Measure.coe_smul</a>, <a>Pi.smul_apply</a>,\n          <a>withDensity_apply</a> _ (hA.inter hE\u2081), show \u03b5 \u2022 \u03bd (A \u2229 E) = (\u03b5 : \u211d\u22650\u221e) * \u03bd (A \u2229 E) by rfl,\n          \u2190 <a>set_lintegral_const</a>, \u2190 <a>lintegral_add_left</a> <a>measurable_const</a>] at this", [{"full_name": "MeasureTheory.VectorMeasure.zero_apply", "def_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "def_pos": [293, 9], "def_end_pos": [293, 19]}, {"full_name": "MeasureTheory.Measure.toSignedMeasure_sub_apply", "def_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "def_pos": [514, 9], "def_end_pos": [514, 34]}, {"full_name": "MeasureTheory.Measure.sub_apply", "def_path": "Mathlib/MeasureTheory/Measure/Sub.lean", "def_pos": [71, 9], "def_end_pos": [71, 18]}, {"full_name": "ENNReal.toReal_sub_of_le", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2003, 9], "def_end_pos": [2003, 25]}, {"full_name": "ne_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [101, 9], "def_end_pos": [101, 17]}, {"full_name": "MeasureTheory.measure_lt_top", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2866, 9], "def_end_pos": [2866, 23]}, {"full_name": "sub_nonneg", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [720, 30], "def_end_pos": [720, 40]}, {"full_name": "le_sub_iff_add_le", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [740, 3], "def_end_pos": [740, 14]}, {"full_name": "ENNReal.toReal_add", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1997, 9], "def_end_pos": [1997, 19]}, {"full_name": "ENNReal.toReal_le_toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2036, 9], "def_end_pos": [2036, 25]}, {"full_name": "MeasureTheory.Measure.coe_smul", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [836, 9], "def_end_pos": [836, 17]}, {"full_name": "Pi.smul_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [116, 60], "def_end_pos": [116, 70]}, {"full_name": "MeasureTheory.withDensity_apply", "def_path": "Mathlib/MeasureTheory/Measure/WithDensity.lean", "def_pos": [39, 9], "def_end_pos": [39, 26]}, {"full_name": "MeasureTheory.set_lintegral_const", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [152, 9], "def_end_pos": [152, 28]}, {"full_name": "MeasureTheory.lintegral_add_left", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [554, 9], "def_end_pos": [554, 27]}, {"full_name": "measurable_const", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [570, 9], "def_end_pos": [570, 25]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd\nh\u03bem : Measurable \u03be\n\u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be\nh\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be\nhle : withDensity \u03bd \u03be \u2264 \u03bc\nthis\u271d : IsFiniteMeasure (withDensity \u03bd \u03be)\nh : \u00ac(\u03bc\u2081, \u03be).1 \u27c2\u2098 \u03bd\n\u03b5 : \u211d\u22650\nh\u03b5\u2081 : 0 < \u03b5\nE : Set \u03b1\nhE\u2081 : MeasurableSet E\nhE\u2082 : 0 < \u2191\u2191\u03bd E\nhE\u2083 :\n  VectorMeasure.restrict 0 E \u2264\n    VectorMeasure.restrict (toSignedMeasure (\u03bc - withDensity \u03bd \u03be) - toSignedMeasure (\u03b5 \u2022 \u03bd)) E\nh\u03bele : \u2200 (A : Set \u03b1), MeasurableSet A \u2192 \u222b\u207b (a : \u03b1) in A, \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc A\nA : Set \u03b1\nhA : MeasurableSet A\nthis : \u21910 (A \u2229 E) \u2264 \u2191(toSignedMeasure (\u03bc - withDensity \u03bd \u03be) - toSignedMeasure (\u03b5 \u2022 \u03bd)) (A \u2229 E)\n\u22a2 \u222b\u207b (a : \u03b1) in A \u2229 E, \u2191\u03b5 + \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc (A \u2229 E)", "state_after": "case ha\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd\nh\u03bem : Measurable \u03be\n\u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be\nh\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be\nhle : withDensity \u03bd \u03be \u2264 \u03bc\nthis\u271d : IsFiniteMeasure (withDensity \u03bd \u03be)\nh : \u00ac(\u03bc\u2081, \u03be).1 \u27c2\u2098 \u03bd\n\u03b5 : \u211d\u22650\nh\u03b5\u2081 : 0 < \u03b5\nE : Set \u03b1\nhE\u2081 : MeasurableSet E\nhE\u2082 : 0 < \u2191\u2191\u03bd E\nhE\u2083 :\n  VectorMeasure.restrict 0 E \u2264\n    VectorMeasure.restrict (toSignedMeasure (\u03bc - withDensity \u03bd \u03be) - toSignedMeasure (\u03b5 \u2022 \u03bd)) E\nh\u03bele : \u2200 (A : Set \u03b1), MeasurableSet A \u2192 \u222b\u207b (a : \u03b1) in A, \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc A\nA : Set \u03b1\nhA : MeasurableSet A\nthis : ENNReal.toReal (\u2191\u2191(\u03b5 \u2022 \u03bd) (A \u2229 E) + \u2191\u2191(withDensity \u03bd \u03be) (A \u2229 E)) \u2264 ENNReal.toReal (\u2191\u2191\u03bc (A \u2229 E))\n\u22a2 \u2191\u2191(\u03b5 \u2022 \u03bd) (A \u2229 E) + \u2191\u2191(withDensity \u03bd \u03be) (A \u2229 E) \u2260 \u22a4\n\ncase hb\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd\nh\u03bem : Measurable \u03be\n\u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be\nh\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be\nhle : withDensity \u03bd \u03be \u2264 \u03bc\nthis\u271d : IsFiniteMeasure (withDensity \u03bd \u03be)\nh : \u00ac(\u03bc\u2081, \u03be).1 \u27c2\u2098 \u03bd\n\u03b5 : \u211d\u22650\nh\u03b5\u2081 : 0 < \u03b5\nE : Set \u03b1\nhE\u2081 : MeasurableSet E\nhE\u2082 : 0 < \u2191\u2191\u03bd E\nhE\u2083 :\n  VectorMeasure.restrict 0 E \u2264\n    VectorMeasure.restrict (toSignedMeasure (\u03bc - withDensity \u03bd \u03be) - toSignedMeasure (\u03b5 \u2022 \u03bd)) E\nh\u03bele : \u2200 (A : Set \u03b1), MeasurableSet A \u2192 \u222b\u207b (a : \u03b1) in A, \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc A\nA : Set \u03b1\nhA : MeasurableSet A\nthis : ENNReal.toReal (\u2191\u2191(\u03b5 \u2022 \u03bd) (A \u2229 E) + \u2191\u2191(withDensity \u03bd \u03be) (A \u2229 E)) \u2264 ENNReal.toReal (\u2191\u2191\u03bc (A \u2229 E))\n\u22a2 \u2191\u2191\u03bc (A \u2229 E) \u2260 \u22a4\n\ncase ha\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd\nh\u03bem : Measurable \u03be\n\u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be\nh\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be\nhle : withDensity \u03bd \u03be \u2264 \u03bc\nthis\u271d : IsFiniteMeasure (withDensity \u03bd \u03be)\nh : \u00ac(\u03bc\u2081, \u03be).1 \u27c2\u2098 \u03bd\n\u03b5 : \u211d\u22650\nh\u03b5\u2081 : 0 < \u03b5\nE : Set \u03b1\nhE\u2081 : MeasurableSet E\nhE\u2082 : 0 < \u2191\u2191\u03bd E\nhE\u2083 :\n  VectorMeasure.restrict 0 E \u2264\n    VectorMeasure.restrict (toSignedMeasure (\u03bc - withDensity \u03bd \u03be) - toSignedMeasure (\u03b5 \u2022 \u03bd)) E\nh\u03bele : \u2200 (A : Set \u03b1), MeasurableSet A \u2192 \u222b\u207b (a : \u03b1) in A, \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc A\nA : Set \u03b1\nhA : MeasurableSet A\nthis : ENNReal.toReal (\u2191\u2191(\u03b5 \u2022 \u03bd) (A \u2229 E)) + ENNReal.toReal (\u2191\u2191(withDensity \u03bd \u03be) (A \u2229 E)) \u2264 ENNReal.toReal (\u2191\u2191\u03bc (A \u2229 E))\n\u22a2 \u2191\u2191(\u03b5 \u2022 \u03bd) (A \u2229 E) \u2260 \u22a4\n\ncase hb\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd\nh\u03bem : Measurable \u03be\n\u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be\nh\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be\nhle : withDensity \u03bd \u03be \u2264 \u03bc\nthis\u271d : IsFiniteMeasure (withDensity \u03bd \u03be)\nh : \u00ac(\u03bc\u2081, \u03be).1 \u27c2\u2098 \u03bd\n\u03b5 : \u211d\u22650\nh\u03b5\u2081 : 0 < \u03b5\nE : Set \u03b1\nhE\u2081 : MeasurableSet E\nhE\u2082 : 0 < \u2191\u2191\u03bd E\nhE\u2083 :\n  VectorMeasure.restrict 0 E \u2264\n    VectorMeasure.restrict (toSignedMeasure (\u03bc - withDensity \u03bd \u03be) - toSignedMeasure (\u03b5 \u2022 \u03bd)) E\nh\u03bele : \u2200 (A : Set \u03b1), MeasurableSet A \u2192 \u222b\u207b (a : \u03b1) in A, \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc A\nA : Set \u03b1\nhA : MeasurableSet A\nthis : ENNReal.toReal (\u2191\u2191(\u03b5 \u2022 \u03bd) (A \u2229 E)) + ENNReal.toReal (\u2191\u2191(withDensity \u03bd \u03be) (A \u2229 E)) \u2264 ENNReal.toReal (\u2191\u2191\u03bc (A \u2229 E))\n\u22a2 \u2191\u2191(withDensity \u03bd \u03be) (A \u2229 E) \u2260 \u22a4\n\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd\nh\u03bem : Measurable \u03be\n\u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be\nh\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be\nhle : withDensity \u03bd \u03be \u2264 \u03bc\nthis\u271d : IsFiniteMeasure (withDensity \u03bd \u03be)\nh : \u00ac(\u03bc\u2081, \u03be).1 \u27c2\u2098 \u03bd\n\u03b5 : \u211d\u22650\nh\u03b5\u2081 : 0 < \u03b5\nE : Set \u03b1\nhE\u2081 : MeasurableSet E\nhE\u2082 : 0 < \u2191\u2191\u03bd E\nhE\u2083 :\n  VectorMeasure.restrict 0 E \u2264\n    VectorMeasure.restrict (toSignedMeasure (\u03bc - withDensity \u03bd \u03be) - toSignedMeasure (\u03b5 \u2022 \u03bd)) E\nh\u03bele : \u2200 (A : Set \u03b1), MeasurableSet A \u2192 \u222b\u207b (a : \u03b1) in A, \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc A\nA : Set \u03b1\nhA : MeasurableSet A\nthis : 0 \u2264 ENNReal.toReal (\u2191\u2191\u03bc (A \u2229 E) - \u2191\u2191(withDensity \u03bd \u03be) (A \u2229 E)) - ENNReal.toReal (\u2191\u2191(\u03b5 \u2022 \u03bd) (A \u2229 E))\n\u22a2 \u2191\u2191(withDensity \u03bd \u03be) (A \u2229 E) \u2264 \u2191\u2191\u03bc (A \u2229 E)"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd\nh\u03bem : Measurable \u03be\n\u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be\nh\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be\nhle : withDensity \u03bd \u03be \u2264 \u03bc\nthis\u271d : IsFiniteMeasure (withDensity \u03bd \u03be)\nh : \u00ac(\u03bc\u2081, \u03be).1 \u27c2\u2098 \u03bd\n\u03b5 : \u211d\u22650\nh\u03b5\u2081 : 0 < \u03b5\nE : Set \u03b1\nhE\u2081 : MeasurableSet E\nhE\u2082 : 0 < \u2191\u2191\u03bd E\nhE\u2083 :\n  VectorMeasure.restrict 0 E \u2264\n    VectorMeasure.restrict (toSignedMeasure (\u03bc - withDensity \u03bd \u03be) - toSignedMeasure (\u03b5 \u2022 \u03bd)) E\nh\u03bele : \u2200 (A : Set \u03b1), MeasurableSet A \u2192 \u222b\u207b (a : \u03b1) in A, \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc A\nA : Set \u03b1\nhA : MeasurableSet A\nthis : \u03b5 \u2022 \u2191\u2191\u03bd (A \u2229 E) + \u222b\u207b (a : \u03b1) in A \u2229 E, \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc (A \u2229 E)\n\u22a2 \u03b5 \u2022 \u2191\u2191\u03bd (A \u2229 E) = \u2191\u03b5 * \u2191\u2191\u03bd (A \u2229 E)", "state_after": "no goals"}, {"tactic": "rw [Ne.def, ENNReal.add_eq_top, not_or]", "annotated_tactic": ["rw [<a>Ne.def</a>, <a>ENNReal.add_eq_top</a>, <a>not_or</a>]", [{"full_name": "Ne.def", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [59, 9], "def_end_pos": [59, 15]}, {"full_name": "ENNReal.add_eq_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [558, 17], "def_end_pos": [558, 27]}, {"full_name": "not_or", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [340, 9], "def_end_pos": [340, 15]}]], "state_before": "case ha\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd\nh\u03bem : Measurable \u03be\n\u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be\nh\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be\nhle : withDensity \u03bd \u03be \u2264 \u03bc\nthis\u271d : IsFiniteMeasure (withDensity \u03bd \u03be)\nh : \u00ac(\u03bc\u2081, \u03be).1 \u27c2\u2098 \u03bd\n\u03b5 : \u211d\u22650\nh\u03b5\u2081 : 0 < \u03b5\nE : Set \u03b1\nhE\u2081 : MeasurableSet E\nhE\u2082 : 0 < \u2191\u2191\u03bd E\nhE\u2083 :\n  VectorMeasure.restrict 0 E \u2264\n    VectorMeasure.restrict (toSignedMeasure (\u03bc - withDensity \u03bd \u03be) - toSignedMeasure (\u03b5 \u2022 \u03bd)) E\nh\u03bele : \u2200 (A : Set \u03b1), MeasurableSet A \u2192 \u222b\u207b (a : \u03b1) in A, \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc A\nA : Set \u03b1\nhA : MeasurableSet A\nthis : ENNReal.toReal (\u2191\u2191(\u03b5 \u2022 \u03bd) (A \u2229 E) + \u2191\u2191(withDensity \u03bd \u03be) (A \u2229 E)) \u2264 ENNReal.toReal (\u2191\u2191\u03bc (A \u2229 E))\n\u22a2 \u2191\u2191(\u03b5 \u2022 \u03bd) (A \u2229 E) + \u2191\u2191(withDensity \u03bd \u03be) (A \u2229 E) \u2260 \u22a4", "state_after": "case ha\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd\nh\u03bem : Measurable \u03be\n\u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be\nh\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be\nhle : withDensity \u03bd \u03be \u2264 \u03bc\nthis\u271d : IsFiniteMeasure (withDensity \u03bd \u03be)\nh : \u00ac(\u03bc\u2081, \u03be).1 \u27c2\u2098 \u03bd\n\u03b5 : \u211d\u22650\nh\u03b5\u2081 : 0 < \u03b5\nE : Set \u03b1\nhE\u2081 : MeasurableSet E\nhE\u2082 : 0 < \u2191\u2191\u03bd E\nhE\u2083 :\n  VectorMeasure.restrict 0 E \u2264\n    VectorMeasure.restrict (toSignedMeasure (\u03bc - withDensity \u03bd \u03be) - toSignedMeasure (\u03b5 \u2022 \u03bd)) E\nh\u03bele : \u2200 (A : Set \u03b1), MeasurableSet A \u2192 \u222b\u207b (a : \u03b1) in A, \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc A\nA : Set \u03b1\nhA : MeasurableSet A\nthis : ENNReal.toReal (\u2191\u2191(\u03b5 \u2022 \u03bd) (A \u2229 E) + \u2191\u2191(withDensity \u03bd \u03be) (A \u2229 E)) \u2264 ENNReal.toReal (\u2191\u2191\u03bc (A \u2229 E))\n\u22a2 \u00ac\u2191\u2191(\u03b5 \u2022 \u03bd) (A \u2229 E) = \u22a4 \u2227 \u00ac\u2191\u2191(withDensity \u03bd \u03be) (A \u2229 E) = \u22a4"}, {"tactic": "exact \u27e8ne_of_lt (measure_lt_top _ _), ne_of_lt (measure_lt_top _ _)\u27e9", "annotated_tactic": ["exact \u27e8<a>ne_of_lt</a> (<a>measure_lt_top</a> _ _), <a>ne_of_lt</a> (<a>measure_lt_top</a> _ _)\u27e9", [{"full_name": "ne_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [101, 9], "def_end_pos": [101, 17]}, {"full_name": "MeasureTheory.measure_lt_top", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2866, 9], "def_end_pos": [2866, 23]}, {"full_name": "ne_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [101, 9], "def_end_pos": [101, 17]}, {"full_name": "MeasureTheory.measure_lt_top", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2866, 9], "def_end_pos": [2866, 23]}]], "state_before": "case ha\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd\nh\u03bem : Measurable \u03be\n\u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be\nh\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be\nhle : withDensity \u03bd \u03be \u2264 \u03bc\nthis\u271d : IsFiniteMeasure (withDensity \u03bd \u03be)\nh : \u00ac(\u03bc\u2081, \u03be).1 \u27c2\u2098 \u03bd\n\u03b5 : \u211d\u22650\nh\u03b5\u2081 : 0 < \u03b5\nE : Set \u03b1\nhE\u2081 : MeasurableSet E\nhE\u2082 : 0 < \u2191\u2191\u03bd E\nhE\u2083 :\n  VectorMeasure.restrict 0 E \u2264\n    VectorMeasure.restrict (toSignedMeasure (\u03bc - withDensity \u03bd \u03be) - toSignedMeasure (\u03b5 \u2022 \u03bd)) E\nh\u03bele : \u2200 (A : Set \u03b1), MeasurableSet A \u2192 \u222b\u207b (a : \u03b1) in A, \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc A\nA : Set \u03b1\nhA : MeasurableSet A\nthis : ENNReal.toReal (\u2191\u2191(\u03b5 \u2022 \u03bd) (A \u2229 E) + \u2191\u2191(withDensity \u03bd \u03be) (A \u2229 E)) \u2264 ENNReal.toReal (\u2191\u2191\u03bc (A \u2229 E))\n\u22a2 \u00ac\u2191\u2191(\u03b5 \u2022 \u03bd) (A \u2229 E) = \u22a4 \u2227 \u00ac\u2191\u2191(withDensity \u03bd \u03be) (A \u2229 E) = \u22a4", "state_after": "no goals"}, {"tactic": "exact ne_of_lt (measure_lt_top _ _)", "annotated_tactic": ["exact <a>ne_of_lt</a> (<a>measure_lt_top</a> _ _)", [{"full_name": "ne_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [101, 9], "def_end_pos": [101, 17]}, {"full_name": "MeasureTheory.measure_lt_top", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2866, 9], "def_end_pos": [2866, 23]}]], "state_before": "case hb\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd\nh\u03bem : Measurable \u03be\n\u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be\nh\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be\nhle : withDensity \u03bd \u03be \u2264 \u03bc\nthis\u271d : IsFiniteMeasure (withDensity \u03bd \u03be)\nh : \u00ac(\u03bc\u2081, \u03be).1 \u27c2\u2098 \u03bd\n\u03b5 : \u211d\u22650\nh\u03b5\u2081 : 0 < \u03b5\nE : Set \u03b1\nhE\u2081 : MeasurableSet E\nhE\u2082 : 0 < \u2191\u2191\u03bd E\nhE\u2083 :\n  VectorMeasure.restrict 0 E \u2264\n    VectorMeasure.restrict (toSignedMeasure (\u03bc - withDensity \u03bd \u03be) - toSignedMeasure (\u03b5 \u2022 \u03bd)) E\nh\u03bele : \u2200 (A : Set \u03b1), MeasurableSet A \u2192 \u222b\u207b (a : \u03b1) in A, \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc A\nA : Set \u03b1\nhA : MeasurableSet A\nthis : ENNReal.toReal (\u2191\u2191(\u03b5 \u2022 \u03bd) (A \u2229 E) + \u2191\u2191(withDensity \u03bd \u03be) (A \u2229 E)) \u2264 ENNReal.toReal (\u2191\u2191\u03bc (A \u2229 E))\n\u22a2 \u2191\u2191\u03bc (A \u2229 E) \u2260 \u22a4", "state_after": "no goals"}, {"tactic": "exact ne_of_lt (measure_lt_top _ _)", "annotated_tactic": ["exact <a>ne_of_lt</a> (<a>measure_lt_top</a> _ _)", [{"full_name": "ne_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [101, 9], "def_end_pos": [101, 17]}, {"full_name": "MeasureTheory.measure_lt_top", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2866, 9], "def_end_pos": [2866, 23]}]], "state_before": "case ha\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd\nh\u03bem : Measurable \u03be\n\u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be\nh\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be\nhle : withDensity \u03bd \u03be \u2264 \u03bc\nthis\u271d : IsFiniteMeasure (withDensity \u03bd \u03be)\nh : \u00ac(\u03bc\u2081, \u03be).1 \u27c2\u2098 \u03bd\n\u03b5 : \u211d\u22650\nh\u03b5\u2081 : 0 < \u03b5\nE : Set \u03b1\nhE\u2081 : MeasurableSet E\nhE\u2082 : 0 < \u2191\u2191\u03bd E\nhE\u2083 :\n  VectorMeasure.restrict 0 E \u2264\n    VectorMeasure.restrict (toSignedMeasure (\u03bc - withDensity \u03bd \u03be) - toSignedMeasure (\u03b5 \u2022 \u03bd)) E\nh\u03bele : \u2200 (A : Set \u03b1), MeasurableSet A \u2192 \u222b\u207b (a : \u03b1) in A, \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc A\nA : Set \u03b1\nhA : MeasurableSet A\nthis : ENNReal.toReal (\u2191\u2191(\u03b5 \u2022 \u03bd) (A \u2229 E)) + ENNReal.toReal (\u2191\u2191(withDensity \u03bd \u03be) (A \u2229 E)) \u2264 ENNReal.toReal (\u2191\u2191\u03bc (A \u2229 E))\n\u22a2 \u2191\u2191(\u03b5 \u2022 \u03bd) (A \u2229 E) \u2260 \u22a4", "state_after": "no goals"}, {"tactic": "exact ne_of_lt (measure_lt_top _ _)", "annotated_tactic": ["exact <a>ne_of_lt</a> (<a>measure_lt_top</a> _ _)", [{"full_name": "ne_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [101, 9], "def_end_pos": [101, 17]}, {"full_name": "MeasureTheory.measure_lt_top", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2866, 9], "def_end_pos": [2866, 23]}]], "state_before": "case hb\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd\nh\u03bem : Measurable \u03be\n\u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be\nh\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be\nhle : withDensity \u03bd \u03be \u2264 \u03bc\nthis\u271d : IsFiniteMeasure (withDensity \u03bd \u03be)\nh : \u00ac(\u03bc\u2081, \u03be).1 \u27c2\u2098 \u03bd\n\u03b5 : \u211d\u22650\nh\u03b5\u2081 : 0 < \u03b5\nE : Set \u03b1\nhE\u2081 : MeasurableSet E\nhE\u2082 : 0 < \u2191\u2191\u03bd E\nhE\u2083 :\n  VectorMeasure.restrict 0 E \u2264\n    VectorMeasure.restrict (toSignedMeasure (\u03bc - withDensity \u03bd \u03be) - toSignedMeasure (\u03b5 \u2022 \u03bd)) E\nh\u03bele : \u2200 (A : Set \u03b1), MeasurableSet A \u2192 \u222b\u207b (a : \u03b1) in A, \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc A\nA : Set \u03b1\nhA : MeasurableSet A\nthis : ENNReal.toReal (\u2191\u2191(\u03b5 \u2022 \u03bd) (A \u2229 E)) + ENNReal.toReal (\u2191\u2191(withDensity \u03bd \u03be) (A \u2229 E)) \u2264 ENNReal.toReal (\u2191\u2191\u03bc (A \u2229 E))\n\u22a2 \u2191\u2191(withDensity \u03bd \u03be) (A \u2229 E) \u2260 \u22a4", "state_after": "no goals"}, {"tactic": "rw [withDensity_apply _ (hA.inter hE\u2081)]", "annotated_tactic": ["rw [<a>withDensity_apply</a> _ (hA.inter hE\u2081)]", [{"full_name": "MeasureTheory.withDensity_apply", "def_path": "Mathlib/MeasureTheory/Measure/WithDensity.lean", "def_pos": [39, 9], "def_end_pos": [39, 26]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd\nh\u03bem : Measurable \u03be\n\u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be\nh\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be\nhle : withDensity \u03bd \u03be \u2264 \u03bc\nthis\u271d : IsFiniteMeasure (withDensity \u03bd \u03be)\nh : \u00ac(\u03bc\u2081, \u03be).1 \u27c2\u2098 \u03bd\n\u03b5 : \u211d\u22650\nh\u03b5\u2081 : 0 < \u03b5\nE : Set \u03b1\nhE\u2081 : MeasurableSet E\nhE\u2082 : 0 < \u2191\u2191\u03bd E\nhE\u2083 :\n  VectorMeasure.restrict 0 E \u2264\n    VectorMeasure.restrict (toSignedMeasure (\u03bc - withDensity \u03bd \u03be) - toSignedMeasure (\u03b5 \u2022 \u03bd)) E\nh\u03bele : \u2200 (A : Set \u03b1), MeasurableSet A \u2192 \u222b\u207b (a : \u03b1) in A, \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc A\nA : Set \u03b1\nhA : MeasurableSet A\nthis : 0 \u2264 ENNReal.toReal (\u2191\u2191\u03bc (A \u2229 E) - \u2191\u2191(withDensity \u03bd \u03be) (A \u2229 E)) - ENNReal.toReal (\u2191\u2191(\u03b5 \u2022 \u03bd) (A \u2229 E))\n\u22a2 \u2191\u2191(withDensity \u03bd \u03be) (A \u2229 E) \u2264 \u2191\u2191\u03bc (A \u2229 E)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd\nh\u03bem : Measurable \u03be\n\u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be\nh\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be\nhle : withDensity \u03bd \u03be \u2264 \u03bc\nthis\u271d : IsFiniteMeasure (withDensity \u03bd \u03be)\nh : \u00ac(\u03bc\u2081, \u03be).1 \u27c2\u2098 \u03bd\n\u03b5 : \u211d\u22650\nh\u03b5\u2081 : 0 < \u03b5\nE : Set \u03b1\nhE\u2081 : MeasurableSet E\nhE\u2082 : 0 < \u2191\u2191\u03bd E\nhE\u2083 :\n  VectorMeasure.restrict 0 E \u2264\n    VectorMeasure.restrict (toSignedMeasure (\u03bc - withDensity \u03bd \u03be) - toSignedMeasure (\u03b5 \u2022 \u03bd)) E\nh\u03bele : \u2200 (A : Set \u03b1), MeasurableSet A \u2192 \u222b\u207b (a : \u03b1) in A, \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc A\nA : Set \u03b1\nhA : MeasurableSet A\nthis : 0 \u2264 ENNReal.toReal (\u2191\u2191\u03bc (A \u2229 E) - \u2191\u2191(withDensity \u03bd \u03be) (A \u2229 E)) - ENNReal.toReal (\u2191\u2191(\u03b5 \u2022 \u03bd) (A \u2229 E))\n\u22a2 \u222b\u207b (a : \u03b1) in A \u2229 E, \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc (A \u2229 E)"}, {"tactic": "exact h\u03bele (A \u2229 E) (hA.inter hE\u2081)", "annotated_tactic": ["exact h\u03bele (A \u2229 E) (hA.inter hE\u2081)", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd\nh\u03bem : Measurable \u03be\n\u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be\nh\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be\nhle : withDensity \u03bd \u03be \u2264 \u03bc\nthis\u271d : IsFiniteMeasure (withDensity \u03bd \u03be)\nh : \u00ac(\u03bc\u2081, \u03be).1 \u27c2\u2098 \u03bd\n\u03b5 : \u211d\u22650\nh\u03b5\u2081 : 0 < \u03b5\nE : Set \u03b1\nhE\u2081 : MeasurableSet E\nhE\u2082 : 0 < \u2191\u2191\u03bd E\nhE\u2083 :\n  VectorMeasure.restrict 0 E \u2264\n    VectorMeasure.restrict (toSignedMeasure (\u03bc - withDensity \u03bd \u03be) - toSignedMeasure (\u03b5 \u2022 \u03bd)) E\nh\u03bele : \u2200 (A : Set \u03b1), MeasurableSet A \u2192 \u222b\u207b (a : \u03b1) in A, \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc A\nA : Set \u03b1\nhA : MeasurableSet A\nthis : 0 \u2264 ENNReal.toReal (\u2191\u2191\u03bc (A \u2229 E) - \u2191\u2191(withDensity \u03bd \u03be) (A \u2229 E)) - ENNReal.toReal (\u2191\u2191(\u03b5 \u2022 \u03bd) (A \u2229 E))\n\u22a2 \u222b\u207b (a : \u03b1) in A \u2229 E, \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc (A \u2229 E)", "state_after": "no goals"}, {"tactic": "refine' \u27e8Measurable.add h\u03bem (Measurable.indicator measurable_const hE\u2081), fun A hA => _\u27e9", "annotated_tactic": ["refine' \u27e8<a>Measurable.add</a> h\u03bem (<a>Measurable.indicator</a> <a>measurable_const</a> hE\u2081), fun A hA => _\u27e9", [{"full_name": "Measurable.add", "def_path": "Mathlib/MeasureTheory/Group/Arithmetic.lean", "def_pos": [140, 3], "def_end_pos": [140, 14]}, {"full_name": "Measurable.indicator", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [353, 9], "def_end_pos": [353, 29]}, {"full_name": "measurable_const", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [570, 9], "def_end_pos": [570, 25]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd\nh\u03bem : Measurable \u03be\n\u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be\nh\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be\nhle : withDensity \u03bd \u03be \u2264 \u03bc\nthis : IsFiniteMeasure (withDensity \u03bd \u03be)\nh : \u00ac(\u03bc\u2081, \u03be).1 \u27c2\u2098 \u03bd\n\u03b5 : \u211d\u22650\nh\u03b5\u2081 : 0 < \u03b5\nE : Set \u03b1\nhE\u2081 : MeasurableSet E\nhE\u2082 : 0 < \u2191\u2191\u03bd E\nhE\u2083 :\n  VectorMeasure.restrict 0 E \u2264\n    VectorMeasure.restrict (toSignedMeasure (\u03bc - withDensity \u03bd \u03be) - toSignedMeasure (\u03b5 \u2022 \u03bd)) E\nh\u03bele : \u2200 (A : Set \u03b1), MeasurableSet A \u2192 \u222b\u207b (a : \u03b1) in A, \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc A\nh\u03b5\u2082 : \u2200 (A : Set \u03b1), MeasurableSet A \u2192 \u222b\u207b (a : \u03b1) in A \u2229 E, \u2191\u03b5 + \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc (A \u2229 E)\n\u22a2 (\u03be + indicator E fun x => \u2191\u03b5) \u2208 measurableLE \u03bd \u03bc", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd\nh\u03bem : Measurable \u03be\n\u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be\nh\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be\nhle : withDensity \u03bd \u03be \u2264 \u03bc\nthis : IsFiniteMeasure (withDensity \u03bd \u03be)\nh : \u00ac(\u03bc\u2081, \u03be).1 \u27c2\u2098 \u03bd\n\u03b5 : \u211d\u22650\nh\u03b5\u2081 : 0 < \u03b5\nE : Set \u03b1\nhE\u2081 : MeasurableSet E\nhE\u2082 : 0 < \u2191\u2191\u03bd E\nhE\u2083 :\n  VectorMeasure.restrict 0 E \u2264\n    VectorMeasure.restrict (toSignedMeasure (\u03bc - withDensity \u03bd \u03be) - toSignedMeasure (\u03b5 \u2022 \u03bd)) E\nh\u03bele : \u2200 (A : Set \u03b1), MeasurableSet A \u2192 \u222b\u207b (a : \u03b1) in A, \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc A\nh\u03b5\u2082 : \u2200 (A : Set \u03b1), MeasurableSet A \u2192 \u222b\u207b (a : \u03b1) in A \u2229 E, \u2191\u03b5 + \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc (A \u2229 E)\nA : Set \u03b1\nhA : MeasurableSet A\n\u22a2 \u222b\u207b (x : \u03b1) in A, (\u03be + indicator E fun x => \u2191\u03b5) x \u2202\u03bd \u2264 \u2191\u2191\u03bc A"}, {"tactic": "have :\n  (\u222b\u207b a in A, (\u03be + E.indicator fun _ => (\u03b5 : \u211d\u22650\u221e)) a \u2202\u03bd) =\n    (\u222b\u207b a in A \u2229 E, \u03b5 + \u03be a \u2202\u03bd) + \u222b\u207b a in A \\ E, \u03be a \u2202\u03bd := by\n  simp only [lintegral_add_left measurable_const, lintegral_add_left h\u03bem,\n    set_lintegral_const, add_assoc, lintegral_inter_add_diff _ _ hE\u2081, Pi.add_apply,\n    lintegral_indicator _ hE\u2081, restrict_apply hE\u2081]\n  rw [inter_comm, add_comm]", "annotated_tactic": ["have :\n          (\u222b\u207b a in A, (\u03be + E.indicator fun _ => (\u03b5 : \u211d\u22650\u221e)) a \u2202\u03bd) =\n            (\u222b\u207b a in A \u2229 E, \u03b5 + \u03be a \u2202\u03bd) + \u222b\u207b a in A \\ E, \u03be a \u2202\u03bd := by\n          simp only [<a>lintegral_add_left</a> <a>measurable_const</a>, <a>lintegral_add_left</a> h\u03bem,\n            <a>set_lintegral_const</a>, <a>add_assoc</a>, <a>lintegral_inter_add_diff</a> _ _ hE\u2081, <a>Pi.add_apply</a>,\n            <a>lintegral_indicator</a> _ hE\u2081, <a>restrict_apply</a> hE\u2081]\n          rw [<a>inter_comm</a>, <a>add_comm</a>]", [{"full_name": "MeasureTheory.lintegral_add_left", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [554, 9], "def_end_pos": [554, 27]}, {"full_name": "measurable_const", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [570, 9], "def_end_pos": [570, 25]}, {"full_name": "MeasureTheory.lintegral_add_left", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [554, 9], "def_end_pos": [554, 27]}, {"full_name": "MeasureTheory.set_lintegral_const", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [152, 9], "def_end_pos": [152, 28]}, {"full_name": "add_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [263, 3], "def_end_pos": [263, 14]}, {"full_name": "MeasureTheory.lintegral_inter_add_diff", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [1253, 9], "def_end_pos": [1253, 33]}, {"full_name": "Pi.add_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [82, 3], "def_end_pos": [82, 14]}, {"full_name": "MeasureTheory.lintegral_indicator", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [762, 9], "def_end_pos": [762, 28]}, {"full_name": "MeasureTheory.Measure.restrict_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1533, 9], "def_end_pos": [1533, 23]}, {"full_name": "Set.inter_comm", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [940, 9], "def_end_pos": [940, 19]}, {"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [301, 3], "def_end_pos": [301, 14]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd\nh\u03bem : Measurable \u03be\n\u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be\nh\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be\nhle : withDensity \u03bd \u03be \u2264 \u03bc\nthis : IsFiniteMeasure (withDensity \u03bd \u03be)\nh : \u00ac(\u03bc\u2081, \u03be).1 \u27c2\u2098 \u03bd\n\u03b5 : \u211d\u22650\nh\u03b5\u2081 : 0 < \u03b5\nE : Set \u03b1\nhE\u2081 : MeasurableSet E\nhE\u2082 : 0 < \u2191\u2191\u03bd E\nhE\u2083 :\n  VectorMeasure.restrict 0 E \u2264\n    VectorMeasure.restrict (toSignedMeasure (\u03bc - withDensity \u03bd \u03be) - toSignedMeasure (\u03b5 \u2022 \u03bd)) E\nh\u03bele : \u2200 (A : Set \u03b1), MeasurableSet A \u2192 \u222b\u207b (a : \u03b1) in A, \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc A\nh\u03b5\u2082 : \u2200 (A : Set \u03b1), MeasurableSet A \u2192 \u222b\u207b (a : \u03b1) in A \u2229 E, \u2191\u03b5 + \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc (A \u2229 E)\nA : Set \u03b1\nhA : MeasurableSet A\n\u22a2 \u222b\u207b (x : \u03b1) in A, (\u03be + indicator E fun x => \u2191\u03b5) x \u2202\u03bd \u2264 \u2191\u2191\u03bc A", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd\nh\u03bem : Measurable \u03be\n\u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be\nh\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be\nhle : withDensity \u03bd \u03be \u2264 \u03bc\nthis\u271d : IsFiniteMeasure (withDensity \u03bd \u03be)\nh : \u00ac(\u03bc\u2081, \u03be).1 \u27c2\u2098 \u03bd\n\u03b5 : \u211d\u22650\nh\u03b5\u2081 : 0 < \u03b5\nE : Set \u03b1\nhE\u2081 : MeasurableSet E\nhE\u2082 : 0 < \u2191\u2191\u03bd E\nhE\u2083 :\n  VectorMeasure.restrict 0 E \u2264\n    VectorMeasure.restrict (toSignedMeasure (\u03bc - withDensity \u03bd \u03be) - toSignedMeasure (\u03b5 \u2022 \u03bd)) E\nh\u03bele : \u2200 (A : Set \u03b1), MeasurableSet A \u2192 \u222b\u207b (a : \u03b1) in A, \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc A\nh\u03b5\u2082 : \u2200 (A : Set \u03b1), MeasurableSet A \u2192 \u222b\u207b (a : \u03b1) in A \u2229 E, \u2191\u03b5 + \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc (A \u2229 E)\nA : Set \u03b1\nhA : MeasurableSet A\nthis :\n  \u222b\u207b (a : \u03b1) in A, (\u03be + indicator E fun x => \u2191\u03b5) a \u2202\u03bd = \u222b\u207b (a : \u03b1) in A \u2229 E, \u2191\u03b5 + \u03be a \u2202\u03bd + \u222b\u207b (a : \u03b1) in A \\ E, \u03be a \u2202\u03bd\n\u22a2 \u222b\u207b (x : \u03b1) in A, (\u03be + indicator E fun x => \u2191\u03b5) x \u2202\u03bd \u2264 \u2191\u2191\u03bc A"}, {"tactic": "rw [this, \u2190 measure_inter_add_diff A hE\u2081]", "annotated_tactic": ["rw [this, \u2190 <a>measure_inter_add_diff</a> A hE\u2081]", [{"full_name": "MeasureTheory.measure_inter_add_diff", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [132, 9], "def_end_pos": [132, 31]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd\nh\u03bem : Measurable \u03be\n\u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be\nh\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be\nhle : withDensity \u03bd \u03be \u2264 \u03bc\nthis\u271d : IsFiniteMeasure (withDensity \u03bd \u03be)\nh : \u00ac(\u03bc\u2081, \u03be).1 \u27c2\u2098 \u03bd\n\u03b5 : \u211d\u22650\nh\u03b5\u2081 : 0 < \u03b5\nE : Set \u03b1\nhE\u2081 : MeasurableSet E\nhE\u2082 : 0 < \u2191\u2191\u03bd E\nhE\u2083 :\n  VectorMeasure.restrict 0 E \u2264\n    VectorMeasure.restrict (toSignedMeasure (\u03bc - withDensity \u03bd \u03be) - toSignedMeasure (\u03b5 \u2022 \u03bd)) E\nh\u03bele : \u2200 (A : Set \u03b1), MeasurableSet A \u2192 \u222b\u207b (a : \u03b1) in A, \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc A\nh\u03b5\u2082 : \u2200 (A : Set \u03b1), MeasurableSet A \u2192 \u222b\u207b (a : \u03b1) in A \u2229 E, \u2191\u03b5 + \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc (A \u2229 E)\nA : Set \u03b1\nhA : MeasurableSet A\nthis :\n  \u222b\u207b (a : \u03b1) in A, (\u03be + indicator E fun x => \u2191\u03b5) a \u2202\u03bd = \u222b\u207b (a : \u03b1) in A \u2229 E, \u2191\u03b5 + \u03be a \u2202\u03bd + \u222b\u207b (a : \u03b1) in A \\ E, \u03be a \u2202\u03bd\n\u22a2 \u222b\u207b (x : \u03b1) in A, (\u03be + indicator E fun x => \u2191\u03b5) x \u2202\u03bd \u2264 \u2191\u2191\u03bc A", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd\nh\u03bem : Measurable \u03be\n\u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be\nh\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be\nhle : withDensity \u03bd \u03be \u2264 \u03bc\nthis\u271d : IsFiniteMeasure (withDensity \u03bd \u03be)\nh : \u00ac(\u03bc\u2081, \u03be).1 \u27c2\u2098 \u03bd\n\u03b5 : \u211d\u22650\nh\u03b5\u2081 : 0 < \u03b5\nE : Set \u03b1\nhE\u2081 : MeasurableSet E\nhE\u2082 : 0 < \u2191\u2191\u03bd E\nhE\u2083 :\n  VectorMeasure.restrict 0 E \u2264\n    VectorMeasure.restrict (toSignedMeasure (\u03bc - withDensity \u03bd \u03be) - toSignedMeasure (\u03b5 \u2022 \u03bd)) E\nh\u03bele : \u2200 (A : Set \u03b1), MeasurableSet A \u2192 \u222b\u207b (a : \u03b1) in A, \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc A\nh\u03b5\u2082 : \u2200 (A : Set \u03b1), MeasurableSet A \u2192 \u222b\u207b (a : \u03b1) in A \u2229 E, \u2191\u03b5 + \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc (A \u2229 E)\nA : Set \u03b1\nhA : MeasurableSet A\nthis :\n  \u222b\u207b (a : \u03b1) in A, (\u03be + indicator E fun x => \u2191\u03b5) a \u2202\u03bd = \u222b\u207b (a : \u03b1) in A \u2229 E, \u2191\u03b5 + \u03be a \u2202\u03bd + \u222b\u207b (a : \u03b1) in A \\ E, \u03be a \u2202\u03bd\n\u22a2 \u222b\u207b (a : \u03b1) in A \u2229 E, \u2191\u03b5 + \u03be a \u2202\u03bd + \u222b\u207b (a : \u03b1) in A \\ E, \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc (A \u2229 E) + \u2191\u2191\u03bc (A \\ E)"}, {"tactic": "exact add_le_add (h\u03b5\u2082 A hA) (h\u03bele (A \\ E) (hA.diff hE\u2081))", "annotated_tactic": ["exact <a>add_le_add</a> (h\u03b5\u2082 A hA) (h\u03bele (A \\ E) (hA.diff hE\u2081))", [{"full_name": "add_le_add", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [205, 15], "def_end_pos": [205, 25]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd\nh\u03bem : Measurable \u03be\n\u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be\nh\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be\nhle : withDensity \u03bd \u03be \u2264 \u03bc\nthis\u271d : IsFiniteMeasure (withDensity \u03bd \u03be)\nh : \u00ac(\u03bc\u2081, \u03be).1 \u27c2\u2098 \u03bd\n\u03b5 : \u211d\u22650\nh\u03b5\u2081 : 0 < \u03b5\nE : Set \u03b1\nhE\u2081 : MeasurableSet E\nhE\u2082 : 0 < \u2191\u2191\u03bd E\nhE\u2083 :\n  VectorMeasure.restrict 0 E \u2264\n    VectorMeasure.restrict (toSignedMeasure (\u03bc - withDensity \u03bd \u03be) - toSignedMeasure (\u03b5 \u2022 \u03bd)) E\nh\u03bele : \u2200 (A : Set \u03b1), MeasurableSet A \u2192 \u222b\u207b (a : \u03b1) in A, \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc A\nh\u03b5\u2082 : \u2200 (A : Set \u03b1), MeasurableSet A \u2192 \u222b\u207b (a : \u03b1) in A \u2229 E, \u2191\u03b5 + \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc (A \u2229 E)\nA : Set \u03b1\nhA : MeasurableSet A\nthis :\n  \u222b\u207b (a : \u03b1) in A, (\u03be + indicator E fun x => \u2191\u03b5) a \u2202\u03bd = \u222b\u207b (a : \u03b1) in A \u2229 E, \u2191\u03b5 + \u03be a \u2202\u03bd + \u222b\u207b (a : \u03b1) in A \\ E, \u03be a \u2202\u03bd\n\u22a2 \u222b\u207b (a : \u03b1) in A \u2229 E, \u2191\u03b5 + \u03be a \u2202\u03bd + \u222b\u207b (a : \u03b1) in A \\ E, \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc (A \u2229 E) + \u2191\u2191\u03bc (A \\ E)", "state_after": "no goals"}, {"tactic": "simp only [lintegral_add_left measurable_const, lintegral_add_left h\u03bem,\n  set_lintegral_const, add_assoc, lintegral_inter_add_diff _ _ hE\u2081, Pi.add_apply,\n  lintegral_indicator _ hE\u2081, restrict_apply hE\u2081]", "annotated_tactic": ["simp only [<a>lintegral_add_left</a> <a>measurable_const</a>, <a>lintegral_add_left</a> h\u03bem,\n            <a>set_lintegral_const</a>, <a>add_assoc</a>, <a>lintegral_inter_add_diff</a> _ _ hE\u2081, <a>Pi.add_apply</a>,\n            <a>lintegral_indicator</a> _ hE\u2081, <a>restrict_apply</a> hE\u2081]", [{"full_name": "MeasureTheory.lintegral_add_left", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [554, 9], "def_end_pos": [554, 27]}, {"full_name": "measurable_const", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [570, 9], "def_end_pos": [570, 25]}, {"full_name": "MeasureTheory.lintegral_add_left", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [554, 9], "def_end_pos": [554, 27]}, {"full_name": "MeasureTheory.set_lintegral_const", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [152, 9], "def_end_pos": [152, 28]}, {"full_name": "add_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [263, 3], "def_end_pos": [263, 14]}, {"full_name": "MeasureTheory.lintegral_inter_add_diff", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [1253, 9], "def_end_pos": [1253, 33]}, {"full_name": "Pi.add_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [82, 3], "def_end_pos": [82, 14]}, {"full_name": "MeasureTheory.lintegral_indicator", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [762, 9], "def_end_pos": [762, 28]}, {"full_name": "MeasureTheory.Measure.restrict_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1533, 9], "def_end_pos": [1533, 23]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd\nh\u03bem : Measurable \u03be\n\u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be\nh\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be\nhle : withDensity \u03bd \u03be \u2264 \u03bc\nthis : IsFiniteMeasure (withDensity \u03bd \u03be)\nh : \u00ac(\u03bc\u2081, \u03be).1 \u27c2\u2098 \u03bd\n\u03b5 : \u211d\u22650\nh\u03b5\u2081 : 0 < \u03b5\nE : Set \u03b1\nhE\u2081 : MeasurableSet E\nhE\u2082 : 0 < \u2191\u2191\u03bd E\nhE\u2083 :\n  VectorMeasure.restrict 0 E \u2264\n    VectorMeasure.restrict (toSignedMeasure (\u03bc - withDensity \u03bd \u03be) - toSignedMeasure (\u03b5 \u2022 \u03bd)) E\nh\u03bele : \u2200 (A : Set \u03b1), MeasurableSet A \u2192 \u222b\u207b (a : \u03b1) in A, \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc A\nh\u03b5\u2082 : \u2200 (A : Set \u03b1), MeasurableSet A \u2192 \u222b\u207b (a : \u03b1) in A \u2229 E, \u2191\u03b5 + \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc (A \u2229 E)\nA : Set \u03b1\nhA : MeasurableSet A\n\u22a2 \u222b\u207b (a : \u03b1) in A, (\u03be + indicator E fun x => \u2191\u03b5) a \u2202\u03bd = \u222b\u207b (a : \u03b1) in A \u2229 E, \u2191\u03b5 + \u03be a \u2202\u03bd + \u222b\u207b (a : \u03b1) in A \\ E, \u03be a \u2202\u03bd", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd\nh\u03bem : Measurable \u03be\n\u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be\nh\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be\nhle : withDensity \u03bd \u03be \u2264 \u03bc\nthis : IsFiniteMeasure (withDensity \u03bd \u03be)\nh : \u00ac(\u03bc\u2081, \u03be).1 \u27c2\u2098 \u03bd\n\u03b5 : \u211d\u22650\nh\u03b5\u2081 : 0 < \u03b5\nE : Set \u03b1\nhE\u2081 : MeasurableSet E\nhE\u2082 : 0 < \u2191\u2191\u03bd E\nhE\u2083 :\n  VectorMeasure.restrict 0 E \u2264\n    VectorMeasure.restrict (toSignedMeasure (\u03bc - withDensity \u03bd \u03be) - toSignedMeasure (\u03b5 \u2022 \u03bd)) E\nh\u03bele : \u2200 (A : Set \u03b1), MeasurableSet A \u2192 \u222b\u207b (a : \u03b1) in A, \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc A\nh\u03b5\u2082 : \u2200 (A : Set \u03b1), MeasurableSet A \u2192 \u222b\u207b (a : \u03b1) in A \u2229 E, \u2191\u03b5 + \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc (A \u2229 E)\nA : Set \u03b1\nhA : MeasurableSet A\n\u22a2 \u222b\u207b (a : \u03b1) in A, iSup (fun n => \u2a06 k, \u2a06 (_ : k \u2264 n), f k) a \u2202\u03bd + \u2191\u03b5 * \u2191\u2191\u03bd (E \u2229 A) =\n    \u2191\u03b5 * \u2191\u2191\u03bd (A \u2229 E) + \u222b\u207b (a : \u03b1) in A, iSup (fun n => \u2a06 k, \u2a06 (_ : k \u2264 n), f k) a \u2202\u03bd"}, {"tactic": "rw [inter_comm, add_comm]", "annotated_tactic": ["rw [<a>inter_comm</a>, <a>add_comm</a>]", [{"full_name": "Set.inter_comm", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [940, 9], "def_end_pos": [940, 19]}, {"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [301, 3], "def_end_pos": [301, 14]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd\nh\u03bem : Measurable \u03be\n\u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be\nh\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be\nhle : withDensity \u03bd \u03be \u2264 \u03bc\nthis : IsFiniteMeasure (withDensity \u03bd \u03be)\nh : \u00ac(\u03bc\u2081, \u03be).1 \u27c2\u2098 \u03bd\n\u03b5 : \u211d\u22650\nh\u03b5\u2081 : 0 < \u03b5\nE : Set \u03b1\nhE\u2081 : MeasurableSet E\nhE\u2082 : 0 < \u2191\u2191\u03bd E\nhE\u2083 :\n  VectorMeasure.restrict 0 E \u2264\n    VectorMeasure.restrict (toSignedMeasure (\u03bc - withDensity \u03bd \u03be) - toSignedMeasure (\u03b5 \u2022 \u03bd)) E\nh\u03bele : \u2200 (A : Set \u03b1), MeasurableSet A \u2192 \u222b\u207b (a : \u03b1) in A, \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc A\nh\u03b5\u2082 : \u2200 (A : Set \u03b1), MeasurableSet A \u2192 \u222b\u207b (a : \u03b1) in A \u2229 E, \u2191\u03b5 + \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc (A \u2229 E)\nA : Set \u03b1\nhA : MeasurableSet A\n\u22a2 \u222b\u207b (a : \u03b1) in A, iSup (fun n => \u2a06 k, \u2a06 (_ : k \u2264 n), f k) a \u2202\u03bd + \u2191\u03b5 * \u2191\u2191\u03bd (E \u2229 A) =\n    \u2191\u03b5 * \u2191\u2191\u03bd (A \u2229 E) + \u222b\u207b (a : \u03b1) in A, iSup (fun n => \u2a06 k, \u2a06 (_ : k \u2264 n), f k) a \u2202\u03bd", "state_after": "no goals"}, {"tactic": "rw [h\u03bc\u2081]", "annotated_tactic": ["rw [h\u03bc\u2081]", []], "state_before": "case refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd\nh\u03bem : Measurable \u03be\n\u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be\nh\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be\nhle : withDensity \u03bd \u03be \u2264 \u03bc\nthis : IsFiniteMeasure (withDensity \u03bd \u03be)\n\u22a2 \u03bc = (\u03bc\u2081, \u03be).1 + withDensity \u03bd (\u03bc\u2081, \u03be).2", "state_after": "case refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd\nh\u03bem : Measurable \u03be\n\u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be\nh\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be\nhle : withDensity \u03bd \u03be \u2264 \u03bc\nthis : IsFiniteMeasure (withDensity \u03bd \u03be)\n\u22a2 \u03bc = (\u03bc - withDensity \u03bd \u03be, \u03be).1 + withDensity \u03bd (\u03bc - withDensity \u03bd \u03be, \u03be).2"}, {"tactic": "ext1 A hA", "annotated_tactic": ["ext1 A hA", []], "state_before": "case refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd\nh\u03bem : Measurable \u03be\n\u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be\nh\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be\nhle : withDensity \u03bd \u03be \u2264 \u03bc\nthis : IsFiniteMeasure (withDensity \u03bd \u03be)\n\u22a2 \u03bc = (\u03bc - withDensity \u03bd \u03be, \u03be).1 + withDensity \u03bd (\u03bc - withDensity \u03bd \u03be, \u03be).2", "state_after": "case refine'_2.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd\nh\u03bem : Measurable \u03be\n\u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be\nh\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be\nhle : withDensity \u03bd \u03be \u2264 \u03bc\nthis : IsFiniteMeasure (withDensity \u03bd \u03be)\nA : Set \u03b1\nhA : MeasurableSet A\n\u22a2 \u2191\u2191\u03bc A = \u2191\u2191((\u03bc - withDensity \u03bd \u03be, \u03be).1 + withDensity \u03bd (\u03bc - withDensity \u03bd \u03be, \u03be).2) A"}, {"tactic": "rw [Measure.coe_add, Pi.add_apply, Measure.sub_apply hA hle, add_comm,\n  add_tsub_cancel_of_le (hle A hA)]", "annotated_tactic": ["rw [<a>Measure.coe_add</a>, <a>Pi.add_apply</a>, <a>Measure.sub_apply</a> hA hle, <a>add_comm</a>,\n        <a>add_tsub_cancel_of_le</a> (hle A hA)]", [{"full_name": "MeasureTheory.Measure.coe_add", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [794, 9], "def_end_pos": [794, 16]}, {"full_name": "Pi.add_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [82, 3], "def_end_pos": [82, 14]}, {"full_name": "MeasureTheory.Measure.sub_apply", "def_path": "Mathlib/MeasureTheory/Measure/Sub.lean", "def_pos": [71, 9], "def_end_pos": [71, 18]}, {"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [301, 3], "def_end_pos": [301, 14]}, {"full_name": "add_tsub_cancel_of_le", "def_path": "Mathlib/Algebra/Order/Sub/Canonical.lean", "def_pos": [24, 9], "def_end_pos": [24, 30]}]], "state_before": "case refine'_2.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ng : \u2115 \u2192 \u211d\u22650\u221e\nh\u271d : Monotone g\nhg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc)))\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc\nhf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n\n\u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k\nh\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd\nh\u03bem : Measurable \u03be\n\u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be\nh\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be\nhle : withDensity \u03bd \u03be \u2264 \u03bc\nthis : IsFiniteMeasure (withDensity \u03bd \u03be)\nA : Set \u03b1\nhA : MeasurableSet A\n\u22a2 \u2191\u2191\u03bc A = \u2191\u2191((\u03bc - withDensity \u03bd \u03be, \u03be).1 + withDensity \u03bd (\u03bc - withDensity \u03bd \u03be, \u03be).2) A", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Rat/Lemmas.lean", "full_name": "Rat.normalize_num_den", "start": [83, 1], "end": [85, 52], "traced_tactics": [{"tactic": "have := normalize_num_den' n d z", "annotated_tactic": ["have := <a>normalize_num_den'</a> n d z", [{"full_name": "Rat.normalize_num_den'", "def_path": "lake-packages/std/Std/Data/Rat/Lemmas.lean", "def_pos": [77, 9], "def_end_pos": [77, 27]}]], "state_before": "n : Int\nd : Nat\nz : d \u2260 0\nn' : Int\nd' : Nat\nz' : d' \u2260 0\nc : Nat.Coprime (Int.natAbs n') d'\nh : normalize n d = mk' n' d'\n\u22a2 \u2203 m, m \u2260 0 \u2227 n = n' * \u2191m \u2227 d = d' * m", "state_after": "n : Int\nd : Nat\nz : d \u2260 0\nn' : Int\nd' : Nat\nz' : d' \u2260 0\nc : Nat.Coprime (Int.natAbs n') d'\nh : normalize n d = mk' n' d'\nthis : \u2203 d_1, d_1 \u2260 0 \u2227 n = (normalize n d).num * \u2191d_1 \u2227 d = (normalize n d).den * d_1\n\u22a2 \u2203 m, m \u2260 0 \u2227 n = n' * \u2191m \u2227 d = d' * m"}, {"tactic": "rwa [h] at this", "annotated_tactic": ["rwa [h] at this", []], "state_before": "n : Int\nd : Nat\nz : d \u2260 0\nn' : Int\nd' : Nat\nz' : d' \u2260 0\nc : Nat.Coprime (Int.natAbs n') d'\nh : normalize n d = mk' n' d'\nthis : \u2203 d_1, d_1 \u2260 0 \u2227 n = (normalize n d).num * \u2191d_1 \u2227 d = (normalize n d).den * d_1\n\u22a2 \u2203 m, m \u2260 0 \u2227 n = n' * \u2191m \u2227 d = d' * m", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Covering/Besicovitch.lean", "full_name": "Besicovitch.exists_closedBall_covering_tsum_measure_le", "start": [916, 1], "end": [1081, 59], "traced_tactics": [{"tactic": "obtain \u27e8u, su, u_open, \u03bcu\u27e9 : \u2203 U, U \u2287 s \u2227 IsOpen U \u2227 \u03bc U \u2264 \u03bc s + \u03b5 / 2 :=\n  Set.exists_isOpen_le_add _ _\n    (by\n      simpa only [or_false_iff, Ne.def, ENNReal.div_eq_zero_iff, ENNReal.one_ne_top] using h\u03b5)", "annotated_tactic": ["obtain \u27e8u, su, u_open, \u03bcu\u27e9 : \u2203 U, U \u2287 s \u2227 <a>IsOpen</a> U \u2227 \u03bc U \u2264 \u03bc s + \u03b5 / 2 :=\n    <a>Set.exists_isOpen_le_add</a> _ _\n      (by\n        simpa only [<a>or_false_iff</a>, <a>Ne.def</a>, <a>ENNReal.div_eq_zero_iff</a>, <a>ENNReal.one_ne_top</a>] using h\u03b5)", [{"full_name": "IsOpen", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [101, 5], "def_end_pos": [101, 11]}, {"full_name": "Set.exists_isOpen_le_add", "def_path": "Mathlib/MeasureTheory/Measure/Regular.lean", "def_pos": [264, 9], "def_end_pos": [264, 40]}, {"full_name": "or_false_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [188, 9], "def_end_pos": [188, 21]}, {"full_name": "Ne.def", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [59, 9], "def_end_pos": [59, 15]}, {"full_name": "ENNReal.div_eq_zero_iff", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1790, 17], "def_end_pos": [1790, 32]}, {"full_name": "ENNReal.one_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [340, 17], "def_end_pos": [340, 27]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\n\u22a2 \u2203 t r,\n    Set.Countable t \u2227\n      t \u2286 s \u2227\n        (\u2200 (x : \u03b1), x \u2208 t \u2192 r x \u2208 f x) \u2227\n          s \u2286 \u22c3 x \u2208 t, closedBall x (r x) \u2227 \u2211' (x : \u2191t), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\n\u22a2 \u2203 t r,\n    Set.Countable t \u2227\n      t \u2286 s \u2227\n        (\u2200 (x : \u03b1), x \u2208 t \u2192 r x \u2208 f x) \u2227\n          s \u2286 \u22c3 x \u2208 t, closedBall x (r x) \u2227 \u2211' (x : \u2191t), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5"}, {"tactic": "have : \u2200 x \u2208 s, \u2203 R > 0, ball x R \u2286 u := fun x hx =>\n  Metric.mem_nhds_iff.1 (u_open.mem_nhds (su hx))", "annotated_tactic": ["have : \u2200 x \u2208 s, \u2203 R > 0, <a>ball</a> x R \u2286 u := fun x hx =>\n    <a>Metric.mem_nhds_iff</a>.1 (u_open.mem_nhds (su hx))", [{"full_name": "Metric.ball", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [409, 5], "def_end_pos": [409, 9]}, {"full_name": "Metric.mem_nhds_iff", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [954, 9], "def_end_pos": [954, 21]}]], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\n\u22a2 \u2203 t r,\n    Set.Countable t \u2227\n      t \u2286 s \u2227\n        (\u2200 (x : \u03b1), x \u2208 t \u2192 r x \u2208 f x) \u2227\n          s \u2286 \u22c3 x \u2208 t, closedBall x (r x) \u2227 \u2211' (x : \u2191t), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nthis : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2203 R, R > 0 \u2227 ball x R \u2286 u\n\u22a2 \u2203 t r,\n    Set.Countable t \u2227\n      t \u2286 s \u2227\n        (\u2200 (x : \u03b1), x \u2208 t \u2192 r x \u2208 f x) \u2227\n          s \u2286 \u22c3 x \u2208 t, closedBall x (r x) \u2227 \u2211' (x : \u2191t), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5"}, {"tactic": "choose! R hR using this", "annotated_tactic": ["choose! R hR using this", []], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nthis : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2203 R, R > 0 \u2227 ball x R \u2286 u\n\u22a2 \u2203 t r,\n    Set.Countable t \u2227\n      t \u2286 s \u2227\n        (\u2200 (x : \u03b1), x \u2208 t \u2192 r x \u2208 f x) \u2227\n          s \u2286 \u22c3 x \u2208 t, closedBall x (r x) \u2227 \u2211' (x : \u2191t), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\n\u22a2 \u2203 t r,\n    Set.Countable t \u2227\n      t \u2286 s \u2227\n        (\u2200 (x : \u03b1), x \u2208 t \u2192 r x \u2208 f x) \u2227\n          s \u2286 \u22c3 x \u2208 t, closedBall x (r x) \u2227 \u2211' (x : \u2191t), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5"}, {"tactic": "obtain \u27e8t0, r0, t0_count, t0s, hr0, \u03bct0, t0_disj\u27e9 :\n  \u2203 (t0 : Set \u03b1) (r0 : \u03b1 \u2192 \u211d), t0.Countable \u2227 t0 \u2286 s \u2227\n    (\u2200 x \u2208 t0, r0 x \u2208 f x \u2229 Ioo 0 (R x)) \u2227 \u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0 \u2227\n      t0.PairwiseDisjoint fun x => closedBall x (r0 x) :=\n  exists_disjoint_closedBall_covering_ae \u03bc f s hf R fun x hx => (hR x hx).1", "annotated_tactic": ["obtain \u27e8t0, r0, t0_count, t0s, hr0, \u03bct0, t0_disj\u27e9 :\n    \u2203 (t0 : <a>Set</a> \u03b1) (r0 : \u03b1 \u2192 \u211d), t0.Countable \u2227 t0 \u2286 s \u2227\n      (\u2200 x \u2208 t0, r0 x \u2208 f x \u2229 <a>Ioo</a> 0 (R x)) \u2227 \u03bc (s \\ \u22c3 x \u2208 t0, <a>closedBall</a> x (r0 x)) = 0 \u2227\n        t0.PairwiseDisjoint fun x => <a>closedBall</a> x (r0 x) :=\n    <a>exists_disjoint_closedBall_covering_ae</a> \u03bc f s hf R fun x hx => (hR x hx).1", [{"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}, {"full_name": "Set.Ioo", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [44, 5], "def_end_pos": [44, 8]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "Besicovitch.exists_disjoint_closedBall_covering_ae", "def_path": "Mathlib/MeasureTheory/Covering/Besicovitch.lean", "def_pos": [853, 9], "def_end_pos": [853, 47]}]], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\n\u22a2 \u2203 t r,\n    Set.Countable t \u2227\n      t \u2286 s \u2227\n        (\u2200 (x : \u03b1), x \u2208 t \u2192 r x \u2208 f x) \u2227\n          s \u2286 \u22c3 x \u2208 t, closedBall x (r x) \u2227 \u2211' (x : \u2191t), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\n\u22a2 \u2203 t r,\n    Set.Countable t \u2227\n      t \u2286 s \u2227\n        (\u2200 (x : \u03b1), x \u2208 t \u2192 r x \u2208 f x) \u2227\n          s \u2286 \u22c3 x \u2208 t, closedBall x (r x) \u2227 \u2211' (x : \u2191t), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5"}, {"tactic": "let s' := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)", "annotated_tactic": ["let s' := s \\ \u22c3 x \u2208 t0, <a>closedBall</a> x (r0 x)", [{"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}]], "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\n\u22a2 \u2203 t r,\n    Set.Countable t \u2227\n      t \u2286 s \u2227\n        (\u2200 (x : \u03b1), x \u2208 t \u2192 r x \u2208 f x) \u2227\n          s \u2286 \u22c3 x \u2208 t, closedBall x (r x) \u2227 \u2211' (x : \u2191t), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\n\u22a2 \u2203 t r,\n    Set.Countable t \u2227\n      t \u2286 s \u2227\n        (\u2200 (x : \u03b1), x \u2208 t \u2192 r x \u2208 f x) \u2227\n          s \u2286 \u22c3 x \u2208 t, closedBall x (r x) \u2227 \u2211' (x : \u2191t), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5"}, {"tactic": "have s's : s' \u2286 s := diff_subset _ _", "annotated_tactic": ["have s's : s' \u2286 s := <a>diff_subset</a> _ _", [{"full_name": "Set.diff_subset", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1845, 9], "def_end_pos": [1845, 20]}]], "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\n\u22a2 \u2203 t r,\n    Set.Countable t \u2227\n      t \u2286 s \u2227\n        (\u2200 (x : \u03b1), x \u2208 t \u2192 r x \u2208 f x) \u2227\n          s \u2286 \u22c3 x \u2208 t, closedBall x (r x) \u2227 \u2211' (x : \u2191t), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\n\u22a2 \u2203 t r,\n    Set.Countable t \u2227\n      t \u2286 s \u2227\n        (\u2200 (x : \u03b1), x \u2208 t \u2192 r x \u2208 f x) \u2227\n          s \u2286 \u22c3 x \u2208 t, closedBall x (r x) \u2227 \u2211' (x : \u2191t), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5"}, {"tactic": "obtain \u27e8N, \u03c4, h\u03c4, H\u27e9 : \u2203 N \u03c4, 1 < \u03c4 \u2227 IsEmpty (Besicovitch.SatelliteConfig \u03b1 N \u03c4) :=\n  HasBesicovitchCovering.no_satelliteConfig", "annotated_tactic": ["obtain \u27e8N, \u03c4, h\u03c4, H\u27e9 : \u2203 N \u03c4, 1 < \u03c4 \u2227 <a>IsEmpty</a> (<a>Besicovitch.SatelliteConfig</a> \u03b1 N \u03c4) :=\n    <a>HasBesicovitchCovering.no_satelliteConfig</a>", [{"full_name": "IsEmpty", "def_path": "Mathlib/Logic/IsEmpty.lean", "def_pos": [26, 7], "def_end_pos": [26, 14]}, {"full_name": "Besicovitch.SatelliteConfig", "def_path": "Mathlib/MeasureTheory/Covering/Besicovitch.lean", "def_pos": [127, 11], "def_end_pos": [127, 38]}, {"full_name": "HasBesicovitchCovering.no_satelliteConfig", "def_path": "Mathlib/MeasureTheory/Covering/Besicovitch.lean", "def_pos": [147, 3], "def_end_pos": [147, 21]}]], "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\n\u22a2 \u2203 t r,\n    Set.Countable t \u2227\n      t \u2286 s \u2227\n        (\u2200 (x : \u03b1), x \u2208 t \u2192 r x \u2208 f x) \u2227\n          s \u2286 \u22c3 x \u2208 t, closedBall x (r x) \u2227 \u2211' (x : \u2191t), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\n\u22a2 \u2203 t r,\n    Set.Countable t \u2227\n      t \u2286 s \u2227\n        (\u2200 (x : \u03b1), x \u2208 t \u2192 r x \u2208 f x) \u2227\n          s \u2286 \u22c3 x \u2208 t, closedBall x (r x) \u2227 \u2211' (x : \u2191t), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5"}, {"tactic": "obtain \u27e8v, s'v, v_open, \u03bcv\u27e9 : \u2203 v, v \u2287 s' \u2227 IsOpen v \u2227 \u03bc v \u2264 \u03bc s' + \u03b5 / 2 / N :=\n  Set.exists_isOpen_le_add _ _\n    (by\n      simp only [h\u03b5, ENNReal.nat_ne_top, WithTop.mul_eq_top_iff, Ne.def, ENNReal.div_eq_zero_iff,\n        ENNReal.one_ne_top, not_false_iff, and_false_iff, false_and_iff, or_self_iff])", "annotated_tactic": ["obtain \u27e8v, s'v, v_open, \u03bcv\u27e9 : \u2203 v, v \u2287 s' \u2227 <a>IsOpen</a> v \u2227 \u03bc v \u2264 \u03bc s' + \u03b5 / 2 / N :=\n    <a>Set.exists_isOpen_le_add</a> _ _\n      (by\n        simp only [h\u03b5, <a>ENNReal.nat_ne_top</a>, <a>WithTop.mul_eq_top_iff</a>, <a>Ne.def</a>, <a>ENNReal.div_eq_zero_iff</a>,\n          <a>ENNReal.one_ne_top</a>, <a>not_false_iff</a>, <a>and_false_iff</a>, <a>false_and_iff</a>, <a>or_self_iff</a>])", [{"full_name": "IsOpen", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [101, 5], "def_end_pos": [101, 11]}, {"full_name": "Set.exists_isOpen_le_add", "def_path": "Mathlib/MeasureTheory/Measure/Regular.lean", "def_pos": [264, 9], "def_end_pos": [264, 40]}, {"full_name": "ENNReal.nat_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [717, 17], "def_end_pos": [717, 27]}, {"full_name": "WithTop.mul_eq_top_iff", "def_path": "Mathlib/Algebra/Order/Ring/WithTop.lean", "def_pos": [61, 9], "def_end_pos": [61, 23]}, {"full_name": "Ne.def", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [59, 9], "def_end_pos": [59, 15]}, {"full_name": "ENNReal.div_eq_zero_iff", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1790, 17], "def_end_pos": [1790, 32]}, {"full_name": "ENNReal.one_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [340, 17], "def_end_pos": [340, 27]}, {"full_name": "not_false_iff", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [82, 9], "def_end_pos": [82, 22]}, {"full_name": "and_false_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [149, 9], "def_end_pos": [149, 22]}, {"full_name": "false_and_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [151, 9], "def_end_pos": [151, 22]}, {"full_name": "or_self_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [190, 9], "def_end_pos": [190, 20]}]], "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\n\u22a2 \u2203 t r,\n    Set.Countable t \u2227\n      t \u2286 s \u2227\n        (\u2200 (x : \u03b1), x \u2208 t \u2192 r x \u2208 f x) \u2227\n          s \u2286 \u22c3 x \u2208 t, closedBall x (r x) \u2227 \u2211' (x : \u2191t), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\n\u22a2 \u2203 t r,\n    Set.Countable t \u2227\n      t \u2286 s \u2227\n        (\u2200 (x : \u03b1), x \u2208 t \u2192 r x \u2208 f x) \u2227\n          s \u2286 \u22c3 x \u2208 t, closedBall x (r x) \u2227 \u2211' (x : \u2191t), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5"}, {"tactic": "have : \u2200 x \u2208 s', \u2203 r1 \u2208 f x \u2229 Ioo (0 : \u211d) 1, closedBall x r1 \u2286 v := by\n  intro x hx\n  rcases Metric.mem_nhds_iff.1 (v_open.mem_nhds (s'v hx)) with \u27e8r, rpos, hr\u27e9\n  rcases hf x (s's hx) (min r 1) (lt_min rpos zero_lt_one) with \u27e8R', hR'\u27e9\n  exact\n    \u27e8R', \u27e8hR'.1, hR'.2.1, hR'.2.2.trans_le (min_le_right _ _)\u27e9,\n      Subset.trans (closedBall_subset_ball (hR'.2.2.trans_le (min_le_left _ _))) hr\u27e9", "annotated_tactic": ["have : \u2200 x \u2208 s', \u2203 r1 \u2208 f x \u2229 <a>Ioo</a> (0 : \u211d) 1, <a>closedBall</a> x r1 \u2286 v := by\n    intro x hx\n    rcases <a>Metric.mem_nhds_iff</a>.1 (v_open.mem_nhds (s'v hx)) with \u27e8r, rpos, hr\u27e9\n    rcases hf x (s's hx) (<a>min</a> r 1) (<a>lt_min</a> rpos <a>zero_lt_one</a>) with \u27e8R', hR'\u27e9\n    exact\n      \u27e8R', \u27e8hR'.1, hR'.2.1, hR'.2.2.<a>trans_le</a> (<a>min_le_right</a> _ _)\u27e9,\n        <a>Subset.trans</a> (<a>closedBall_subset_ball</a> (hR'.2.2.<a>trans_le</a> (<a>min_le_left</a> _ _))) hr\u27e9", [{"full_name": "Set.Ioo", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [44, 5], "def_end_pos": [44, 8]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "Metric.mem_nhds_iff", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [954, 9], "def_end_pos": [954, 21]}, {"full_name": "Min.min", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1103, 3], "def_end_pos": [1103, 6]}, {"full_name": "lt_min", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [159, 9], "def_end_pos": [159, 15]}, {"full_name": "zero_lt_one", "def_path": "Mathlib/Algebra/Order/ZeroLEOne.lean", "def_pos": [39, 15], "def_end_pos": [39, 26]}, {"full_name": "LT.lt.trans_le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [148, 7], "def_end_pos": [148, 21]}, {"full_name": "min_le_right", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [40, 9], "def_end_pos": [40, 21]}, {"full_name": "Set.Subset.trans", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [362, 9], "def_end_pos": [362, 21]}, {"full_name": "Metric.closedBall_subset_ball", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [621, 9], "def_end_pos": [621, 31]}, {"full_name": "LT.lt.trans_le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [148, 7], "def_end_pos": [148, 21]}, {"full_name": "min_le_left", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [33, 9], "def_end_pos": [33, 20]}]], "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\n\u22a2 \u2203 t r,\n    Set.Countable t \u2227\n      t \u2286 s \u2227\n        (\u2200 (x : \u03b1), x \u2208 t \u2192 r x \u2208 f x) \u2227\n          s \u2286 \u22c3 x \u2208 t, closedBall x (r x) \u2227 \u2211' (x : \u2191t), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nthis : \u2200 (x : \u03b1), x \u2208 s' \u2192 \u2203 r1, r1 \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x r1 \u2286 v\n\u22a2 \u2203 t r,\n    Set.Countable t \u2227\n      t \u2286 s \u2227\n        (\u2200 (x : \u03b1), x \u2208 t \u2192 r x \u2208 f x) \u2227\n          s \u2286 \u22c3 x \u2208 t, closedBall x (r x) \u2227 \u2211' (x : \u2191t), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5"}, {"tactic": "choose! r1 hr1 using this", "annotated_tactic": ["choose! r1 hr1 using this", []], "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nthis : \u2200 (x : \u03b1), x \u2208 s' \u2192 \u2203 r1, r1 \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x r1 \u2286 v\n\u22a2 \u2203 t r,\n    Set.Countable t \u2227\n      t \u2286 s \u2227\n        (\u2200 (x : \u03b1), x \u2208 t \u2192 r x \u2208 f x) \u2227\n          s \u2286 \u22c3 x \u2208 t, closedBall x (r x) \u2227 \u2211' (x : \u2191t), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\n\u22a2 \u2203 t r,\n    Set.Countable t \u2227\n      t \u2286 s \u2227\n        (\u2200 (x : \u03b1), x \u2208 t \u2192 r x \u2208 f x) \u2227\n          s \u2286 \u22c3 x \u2208 t, closedBall x (r x) \u2227 \u2211' (x : \u2191t), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5"}, {"tactic": "let q : BallPackage s' \u03b1 :=\n  { c := fun x => x\n    r := fun x => r1 x\n    rpos := fun x => (hr1 x.1 x.2).1.2.1\n    r_bound := 1\n    r_le := fun x => (hr1 x.1 x.2).1.2.2.le }", "annotated_tactic": ["let q : <a>BallPackage</a> s' \u03b1 :=\n    { c := fun x => x\n      r := fun x => r1 x\n      rpos := fun x => (hr1 x.1 x.2).1.2.1\n      r_bound := 1\n      r_le := fun x => (hr1 x.1 x.2).1.2.2.<a>le</a> }", [{"full_name": "Besicovitch.BallPackage", "def_path": "Mathlib/MeasureTheory/Covering/Besicovitch.lean", "def_pos": [192, 11], "def_end_pos": [192, 22]}, {"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [142, 7], "def_end_pos": [142, 15]}]], "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\n\u22a2 \u2203 t r,\n    Set.Countable t \u2227\n      t \u2286 s \u2227\n        (\u2200 (x : \u03b1), x \u2208 t \u2192 r x \u2208 f x) \u2227\n          s \u2286 \u22c3 x \u2208 t, closedBall x (r x) \u2227 \u2211' (x : \u2191t), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\n\u22a2 \u2203 t r,\n    Set.Countable t \u2227\n      t \u2286 s \u2227\n        (\u2200 (x : \u03b1), x \u2208 t \u2192 r x \u2208 f x) \u2227\n          s \u2286 \u22c3 x \u2208 t, closedBall x (r x) \u2227 \u2211' (x : \u2191t), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5"}, {"tactic": "obtain \u27e8S, S_disj, hS\u27e9 :\n  \u2203 S : Fin N \u2192 Set s',\n    (\u2200 i : Fin N, (S i).PairwiseDisjoint fun j => closedBall (q.c j) (q.r j)) \u2227\n      range q.c \u2286 \u22c3 i : Fin N, \u22c3 j \u2208 S i, ball (q.c j) (q.r j) :=\n  exist_disjoint_covering_families h\u03c4 H q", "annotated_tactic": ["obtain \u27e8S, S_disj, hS\u27e9 :\n    \u2203 S : <a>Fin</a> N \u2192 <a>Set</a> s',\n      (\u2200 i : <a>Fin</a> N, (S i).<a>PairwiseDisjoint</a> fun j => <a>closedBall</a> (q.c j) (q.r j)) \u2227\n        <a>range</a> q.c \u2286 \u22c3 i : <a>Fin</a> N, \u22c3 j \u2208 S i, <a>ball</a> (q.c j) (q.r j) :=\n    <a>exist_disjoint_covering_families</a> h\u03c4 H q", [{"full_name": "Fin", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1745, 11], "def_end_pos": [1745, 14]}, {"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}, {"full_name": "Fin", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1745, 11], "def_end_pos": [1745, 14]}, {"full_name": "Set.PairwiseDisjoint", "def_path": "Mathlib/Data/Set/Pairwise/Basic.lean", "def_pos": [242, 5], "def_end_pos": [242, 21]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "Set.range", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [668, 5], "def_end_pos": [668, 10]}, {"full_name": "Fin", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1745, 11], "def_end_pos": [1745, 14]}, {"full_name": "Metric.ball", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [409, 5], "def_end_pos": [409, 9]}, {"full_name": "Besicovitch.exist_disjoint_covering_families", "def_path": "Mathlib/MeasureTheory/Covering/Besicovitch.lean", "def_pos": [471, 9], "def_end_pos": [471, 41]}]], "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\n\u22a2 \u2203 t r,\n    Set.Countable t \u2227\n      t \u2286 s \u2227\n        (\u2200 (x : \u03b1), x \u2208 t \u2192 r x \u2208 f x) \u2227\n          s \u2286 \u22c3 x \u2208 t, closedBall x (r x) \u2227 \u2211' (x : \u2191t), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\n\u22a2 \u2203 t r,\n    Set.Countable t \u2227\n      t \u2286 s \u2227\n        (\u2200 (x : \u03b1), x \u2208 t \u2192 r x \u2208 f x) \u2227\n          s \u2286 \u22c3 x \u2208 t, closedBall x (r x) \u2227 \u2211' (x : \u2191t), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5"}, {"tactic": "have S_count : \u2200 i, (S i).Countable := by\n  intro i\n  apply (S_disj i).countable_of_nonempty_interior fun j _ => ?_\n  have : (ball (j : \u03b1) (r1 j)).Nonempty := nonempty_ball.2 (q.rpos _)\n  exact this.mono ball_subset_interior_closedBall", "annotated_tactic": ["have S_count : \u2200 i, (S i).<a>Countable</a> := by\n    intro i\n    apply (S_disj i).<a>countable_of_nonempty_interior</a> fun j _ => ?_\n    have : (<a>ball</a> (j : \u03b1) (r1 j)).<a>Nonempty</a> := <a>nonempty_ball</a>.2 (q.rpos _)\n    exact this.mono <a>ball_subset_interior_closedBall</a>", [{"full_name": "Set.Countable", "def_path": "Mathlib/Data/Set/Countable.lean", "def_pos": [31, 15], "def_end_pos": [31, 24]}, {"full_name": "Set.PairwiseDisjoint.countable_of_nonempty_interior", "def_path": "Mathlib/Topology/Bases.lean", "def_pos": [427, 9], "def_end_pos": [427, 67]}, {"full_name": "Metric.ball", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [409, 5], "def_end_pos": [409, 9]}, {"full_name": "Set.Nonempty", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [439, 15], "def_end_pos": [439, 23]}, {"full_name": "Metric.nonempty_ball", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [430, 9], "def_end_pos": [430, 22]}, {"full_name": "Metric.ball_subset_interior_closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [1920, 9], "def_end_pos": [1920, 40]}]], "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\n\u22a2 \u2203 t r,\n    Set.Countable t \u2227\n      t \u2286 s \u2227\n        (\u2200 (x : \u03b1), x \u2208 t \u2192 r x \u2208 f x) \u2227\n          s \u2286 \u22c3 x \u2208 t, closedBall x (r x) \u2227 \u2211' (x : \u2191t), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\n\u22a2 \u2203 t r,\n    Set.Countable t \u2227\n      t \u2286 s \u2227\n        (\u2200 (x : \u03b1), x \u2208 t \u2192 r x \u2208 f x) \u2227\n          s \u2286 \u22c3 x \u2208 t, closedBall x (r x) \u2227 \u2211' (x : \u2191t), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5"}, {"tactic": "let r x := if x \u2208 s' then r1 x else r0 x", "annotated_tactic": ["let r x := if x \u2208 s' then r1 x else r0 x", []], "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\n\u22a2 \u2203 t r,\n    Set.Countable t \u2227\n      t \u2286 s \u2227\n        (\u2200 (x : \u03b1), x \u2208 t \u2192 r x \u2208 f x) \u2227\n          s \u2286 \u22c3 x \u2208 t, closedBall x (r x) \u2227 \u2211' (x : \u2191t), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\n\u22a2 \u2203 t r,\n    Set.Countable t \u2227\n      t \u2286 s \u2227\n        (\u2200 (x : \u03b1), x \u2208 t \u2192 r x \u2208 f x) \u2227\n          s \u2286 \u22c3 x \u2208 t, closedBall x (r x) \u2227 \u2211' (x : \u2191t), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5"}, {"tactic": "have r_t0 : \u2200 x \u2208 t0, r x = r0 x := by\n  intro x hx\n  have : \u00acx \u2208 s' := by\n    simp only [not_exists, exists_prop, mem_iUnion, mem_closedBall, not_and, not_lt, not_le,\n      mem_diff, not_forall]\n    intro _\n    refine' \u27e8x, hx, _\u27e9\n    rw [dist_self]\n    exact (hr0 x hx).2.1.le\n  simp only [if_neg this]", "annotated_tactic": ["have r_t0 : \u2200 x \u2208 t0, r x = r0 x := by\n    intro x hx\n    have : \u00acx \u2208 s' := by\n      simp only [<a>not_exists</a>, <a>exists_prop</a>, <a>mem_iUnion</a>, <a>mem_closedBall</a>, <a>not_and</a>, <a>not_lt</a>, <a>not_le</a>,\n        <a>mem_diff</a>, <a>not_forall</a>]\n      intro _\n      refine' \u27e8x, hx, _\u27e9\n      rw [<a>dist_self</a>]\n      exact (hr0 x hx).2.1.<a>le</a>\n    simp only [<a>if_neg</a> this]", [{"full_name": "not_exists", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [422, 17], "def_end_pos": [422, 27]}, {"full_name": "exists_prop", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [485, 17], "def_end_pos": [485, 28]}, {"full_name": "Set.mem_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [201, 9], "def_end_pos": [201, 19]}, {"full_name": "Metric.mem_closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [478, 17], "def_end_pos": [478, 31]}, {"full_name": "not_and", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [316, 17], "def_end_pos": [316, 24]}, {"full_name": "not_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [368, 9], "def_end_pos": [368, 15]}, {"full_name": "not_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [373, 9], "def_end_pos": [373, 15]}, {"full_name": "Set.mem_diff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1819, 9], "def_end_pos": [1819, 17]}, {"full_name": "Classical.not_forall", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [686, 9], "def_end_pos": [686, 19]}, {"full_name": "dist_self", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [184, 9], "def_end_pos": [184, 18]}, {"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [142, 7], "def_end_pos": [142, 15]}, {"full_name": "if_neg", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [795, 9], "def_end_pos": [795, 15]}]], "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\n\u22a2 \u2203 t r,\n    Set.Countable t \u2227\n      t \u2286 s \u2227\n        (\u2200 (x : \u03b1), x \u2208 t \u2192 r x \u2208 f x) \u2227\n          s \u2286 \u22c3 x \u2208 t, closedBall x (r x) \u2227 \u2211' (x : \u2191t), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\n\u22a2 \u2203 t r,\n    Set.Countable t \u2227\n      t \u2286 s \u2227\n        (\u2200 (x : \u03b1), x \u2208 t \u2192 r x \u2208 f x) \u2227\n          s \u2286 \u22c3 x \u2208 t, closedBall x (r x) \u2227 \u2211' (x : \u2191t), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5"}, {"tactic": "refine' \u27e8t0 \u222a \u22c3 i : Fin N, ((\u2191) : s' \u2192 \u03b1) '' S i, r, _, _, _, _, _\u27e9", "annotated_tactic": ["refine' \u27e8t0 \u222a \u22c3 i : <a>Fin</a> N, ((\u2191) : s' \u2192 \u03b1) '' S i, r, _, _, _, _, _\u27e9", [{"full_name": "Fin", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1745, 11], "def_end_pos": [1745, 14]}]], "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\n\u22a2 \u2203 t r,\n    Set.Countable t \u2227\n      t \u2286 s \u2227\n        (\u2200 (x : \u03b1), x \u2208 t \u2192 r x \u2208 f x) \u2227\n          s \u2286 \u22c3 x \u2208 t, closedBall x (r x) \u2227 \u2211' (x : \u2191t), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\n\u22a2 Set.Countable (t0 \u222a \u22c3 i, Subtype.val '' S i)\n\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\n\u22a2 t0 \u222a \u22c3 i, Subtype.val '' S i \u2286 s\n\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_3\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\n\u22a2 \u2200 (x : \u03b1), x \u2208 t0 \u222a \u22c3 i, Subtype.val '' S i \u2192 r x \u2208 f x\n\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_4\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\n\u22a2 s \u2286 \u22c3 x \u2208 t0 \u222a \u22c3 i, Subtype.val '' S i, closedBall x (r x)\n\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_5\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\n\u22a2 \u2211' (x : \u2191(t0 \u222a \u22c3 i, Subtype.val '' S i)), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5"}, {"tactic": "simpa only [or_false_iff, Ne.def, ENNReal.div_eq_zero_iff, ENNReal.one_ne_top] using h\u03b5", "annotated_tactic": ["simpa only [<a>or_false_iff</a>, <a>Ne.def</a>, <a>ENNReal.div_eq_zero_iff</a>, <a>ENNReal.one_ne_top</a>] using h\u03b5", [{"full_name": "or_false_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [188, 9], "def_end_pos": [188, 21]}, {"full_name": "Ne.def", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [59, 9], "def_end_pos": [59, 15]}, {"full_name": "ENNReal.div_eq_zero_iff", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1790, 17], "def_end_pos": [1790, 32]}, {"full_name": "ENNReal.one_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [340, 17], "def_end_pos": [340, 27]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\n\u22a2 \u03b5 / 2 \u2260 0", "state_after": "no goals"}, {"tactic": "simp only [h\u03b5, ENNReal.nat_ne_top, WithTop.mul_eq_top_iff, Ne.def, ENNReal.div_eq_zero_iff,\n  ENNReal.one_ne_top, not_false_iff, and_false_iff, false_and_iff, or_self_iff]", "annotated_tactic": ["simp only [h\u03b5, <a>ENNReal.nat_ne_top</a>, <a>WithTop.mul_eq_top_iff</a>, <a>Ne.def</a>, <a>ENNReal.div_eq_zero_iff</a>,\n          <a>ENNReal.one_ne_top</a>, <a>not_false_iff</a>, <a>and_false_iff</a>, <a>false_and_iff</a>, <a>or_self_iff</a>]", [{"full_name": "ENNReal.nat_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [717, 17], "def_end_pos": [717, 27]}, {"full_name": "WithTop.mul_eq_top_iff", "def_path": "Mathlib/Algebra/Order/Ring/WithTop.lean", "def_pos": [61, 9], "def_end_pos": [61, 23]}, {"full_name": "Ne.def", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [59, 9], "def_end_pos": [59, 15]}, {"full_name": "ENNReal.div_eq_zero_iff", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1790, 17], "def_end_pos": [1790, 32]}, {"full_name": "ENNReal.one_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [340, 17], "def_end_pos": [340, 27]}, {"full_name": "not_false_iff", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [82, 9], "def_end_pos": [82, 22]}, {"full_name": "and_false_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [149, 9], "def_end_pos": [149, 22]}, {"full_name": "false_and_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [151, 9], "def_end_pos": [151, 22]}, {"full_name": "or_self_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [190, 9], "def_end_pos": [190, 20]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\n\u22a2 \u03b5 / 2 / \u2191N \u2260 0", "state_after": "no goals"}, {"tactic": "intro x hx", "annotated_tactic": ["intro x hx", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\n\u22a2 \u2200 (x : \u03b1), x \u2208 s' \u2192 \u2203 r1, r1 \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x r1 \u2286 v", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nx : \u03b1\nhx : x \u2208 s'\n\u22a2 \u2203 r1, r1 \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x r1 \u2286 v"}, {"tactic": "rcases Metric.mem_nhds_iff.1 (v_open.mem_nhds (s'v hx)) with \u27e8r, rpos, hr\u27e9", "annotated_tactic": ["rcases <a>Metric.mem_nhds_iff</a>.1 (v_open.mem_nhds (s'v hx)) with \u27e8r, rpos, hr\u27e9", [{"full_name": "Metric.mem_nhds_iff", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [954, 9], "def_end_pos": [954, 21]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nx : \u03b1\nhx : x \u2208 s'\n\u22a2 \u2203 r1, r1 \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x r1 \u2286 v", "state_after": "case intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nx : \u03b1\nhx : x \u2208 s'\nr : \u211d\nrpos : r > 0\nhr : ball x r \u2286 v\n\u22a2 \u2203 r1, r1 \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x r1 \u2286 v"}, {"tactic": "rcases hf x (s's hx) (min r 1) (lt_min rpos zero_lt_one) with \u27e8R', hR'\u27e9", "annotated_tactic": ["rcases hf x (s's hx) (<a>min</a> r 1) (<a>lt_min</a> rpos <a>zero_lt_one</a>) with \u27e8R', hR'\u27e9", [{"full_name": "Min.min", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1103, 3], "def_end_pos": [1103, 6]}, {"full_name": "lt_min", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [159, 9], "def_end_pos": [159, 15]}, {"full_name": "zero_lt_one", "def_path": "Mathlib/Algebra/Order/ZeroLEOne.lean", "def_pos": [39, 15], "def_end_pos": [39, 26]}]], "state_before": "case intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nx : \u03b1\nhx : x \u2208 s'\nr : \u211d\nrpos : r > 0\nhr : ball x r \u2286 v\n\u22a2 \u2203 r1, r1 \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x r1 \u2286 v", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nx : \u03b1\nhx : x \u2208 s'\nr : \u211d\nrpos : r > 0\nhr : ball x r \u2286 v\nR' : \u211d\nhR' : R' \u2208 f x \u2229 Ioo 0 (min r 1)\n\u22a2 \u2203 r1, r1 \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x r1 \u2286 v"}, {"tactic": "exact\n  \u27e8R', \u27e8hR'.1, hR'.2.1, hR'.2.2.trans_le (min_le_right _ _)\u27e9,\n    Subset.trans (closedBall_subset_ball (hR'.2.2.trans_le (min_le_left _ _))) hr\u27e9", "annotated_tactic": ["exact\n      \u27e8R', \u27e8hR'.1, hR'.2.1, hR'.2.2.<a>trans_le</a> (<a>min_le_right</a> _ _)\u27e9,\n        <a>Subset.trans</a> (<a>closedBall_subset_ball</a> (hR'.2.2.<a>trans_le</a> (<a>min_le_left</a> _ _))) hr\u27e9", [{"full_name": "LT.lt.trans_le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [148, 7], "def_end_pos": [148, 21]}, {"full_name": "min_le_right", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [40, 9], "def_end_pos": [40, 21]}, {"full_name": "Set.Subset.trans", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [362, 9], "def_end_pos": [362, 21]}, {"full_name": "Metric.closedBall_subset_ball", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [621, 9], "def_end_pos": [621, 31]}, {"full_name": "LT.lt.trans_le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [148, 7], "def_end_pos": [148, 21]}, {"full_name": "min_le_left", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [33, 9], "def_end_pos": [33, 20]}]], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nx : \u03b1\nhx : x \u2208 s'\nr : \u211d\nrpos : r > 0\nhr : ball x r \u2286 v\nR' : \u211d\nhR' : R' \u2208 f x \u2229 Ioo 0 (min r 1)\n\u22a2 \u2203 r1, r1 \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x r1 \u2286 v", "state_after": "no goals"}, {"tactic": "intro i", "annotated_tactic": ["intro i", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\n\u22a2 \u2200 (i : Fin N), Set.Countable (S i)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\ni : Fin N\n\u22a2 Set.Countable (S i)"}, {"tactic": "apply (S_disj i).countable_of_nonempty_interior fun j _ => ?_", "annotated_tactic": ["apply (S_disj i).<a>countable_of_nonempty_interior</a> fun j _ => ?_", [{"full_name": "Set.PairwiseDisjoint.countable_of_nonempty_interior", "def_path": "Mathlib/Topology/Bases.lean", "def_pos": [427, 9], "def_end_pos": [427, 67]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\ni : Fin N\n\u22a2 Set.Countable (S i)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\ni : Fin N\nj : \u2191s'\nx\u271d : j \u2208 S i\n\u22a2 Set.Nonempty (interior (closedBall (BallPackage.c q j) (BallPackage.r q j)))"}, {"tactic": "have : (ball (j : \u03b1) (r1 j)).Nonempty := nonempty_ball.2 (q.rpos _)", "annotated_tactic": ["have : (<a>ball</a> (j : \u03b1) (r1 j)).<a>Nonempty</a> := <a>nonempty_ball</a>.2 (q.rpos _)", [{"full_name": "Metric.ball", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [409, 5], "def_end_pos": [409, 9]}, {"full_name": "Set.Nonempty", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [439, 15], "def_end_pos": [439, 23]}, {"full_name": "Metric.nonempty_ball", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [430, 9], "def_end_pos": [430, 22]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\ni : Fin N\nj : \u2191s'\nx\u271d : j \u2208 S i\n\u22a2 Set.Nonempty (interior (closedBall (BallPackage.c q j) (BallPackage.r q j)))", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\ni : Fin N\nj : \u2191s'\nx\u271d : j \u2208 S i\nthis : Set.Nonempty (ball (\u2191j) (r1 \u2191j))\n\u22a2 Set.Nonempty (interior (closedBall (BallPackage.c q j) (BallPackage.r q j)))"}, {"tactic": "exact this.mono ball_subset_interior_closedBall", "annotated_tactic": ["exact this.mono <a>ball_subset_interior_closedBall</a>", [{"full_name": "Metric.ball_subset_interior_closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [1920, 9], "def_end_pos": [1920, 40]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\ni : Fin N\nj : \u2191s'\nx\u271d : j \u2208 S i\nthis : Set.Nonempty (ball (\u2191j) (r1 \u2191j))\n\u22a2 Set.Nonempty (interior (closedBall (BallPackage.c q j) (BallPackage.r q j)))", "state_after": "no goals"}, {"tactic": "intro x hx", "annotated_tactic": ["intro x hx", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\n\u22a2 \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nx : \u03b1\nhx : x \u2208 t0\n\u22a2 r x = r0 x"}, {"tactic": "have : \u00acx \u2208 s' := by\n  simp only [not_exists, exists_prop, mem_iUnion, mem_closedBall, not_and, not_lt, not_le,\n    mem_diff, not_forall]\n  intro _\n  refine' \u27e8x, hx, _\u27e9\n  rw [dist_self]\n  exact (hr0 x hx).2.1.le", "annotated_tactic": ["have : \u00acx \u2208 s' := by\n      simp only [<a>not_exists</a>, <a>exists_prop</a>, <a>mem_iUnion</a>, <a>mem_closedBall</a>, <a>not_and</a>, <a>not_lt</a>, <a>not_le</a>,\n        <a>mem_diff</a>, <a>not_forall</a>]\n      intro _\n      refine' \u27e8x, hx, _\u27e9\n      rw [<a>dist_self</a>]\n      exact (hr0 x hx).2.1.<a>le</a>", [{"full_name": "not_exists", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [422, 17], "def_end_pos": [422, 27]}, {"full_name": "exists_prop", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [485, 17], "def_end_pos": [485, 28]}, {"full_name": "Set.mem_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [201, 9], "def_end_pos": [201, 19]}, {"full_name": "Metric.mem_closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [478, 17], "def_end_pos": [478, 31]}, {"full_name": "not_and", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [316, 17], "def_end_pos": [316, 24]}, {"full_name": "not_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [368, 9], "def_end_pos": [368, 15]}, {"full_name": "not_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [373, 9], "def_end_pos": [373, 15]}, {"full_name": "Set.mem_diff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1819, 9], "def_end_pos": [1819, 17]}, {"full_name": "Classical.not_forall", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [686, 9], "def_end_pos": [686, 19]}, {"full_name": "dist_self", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [184, 9], "def_end_pos": [184, 18]}, {"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [142, 7], "def_end_pos": [142, 15]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nx : \u03b1\nhx : x \u2208 t0\n\u22a2 r x = r0 x", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nx : \u03b1\nhx : x \u2208 t0\nthis : \u00acx \u2208 s'\n\u22a2 r x = r0 x"}, {"tactic": "simp only [if_neg this]", "annotated_tactic": ["simp only [<a>if_neg</a> this]", [{"full_name": "if_neg", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [795, 9], "def_end_pos": [795, 15]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nx : \u03b1\nhx : x \u2208 t0\nthis : \u00acx \u2208 s'\n\u22a2 r x = r0 x", "state_after": "no goals"}, {"tactic": "simp only [not_exists, exists_prop, mem_iUnion, mem_closedBall, not_and, not_lt, not_le,\n  mem_diff, not_forall]", "annotated_tactic": ["simp only [<a>not_exists</a>, <a>exists_prop</a>, <a>mem_iUnion</a>, <a>mem_closedBall</a>, <a>not_and</a>, <a>not_lt</a>, <a>not_le</a>,\n        <a>mem_diff</a>, <a>not_forall</a>]", [{"full_name": "not_exists", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [422, 17], "def_end_pos": [422, 27]}, {"full_name": "exists_prop", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [485, 17], "def_end_pos": [485, 28]}, {"full_name": "Set.mem_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [201, 9], "def_end_pos": [201, 19]}, {"full_name": "Metric.mem_closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [478, 17], "def_end_pos": [478, 31]}, {"full_name": "not_and", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [316, 17], "def_end_pos": [316, 24]}, {"full_name": "not_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [368, 9], "def_end_pos": [368, 15]}, {"full_name": "not_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [373, 9], "def_end_pos": [373, 15]}, {"full_name": "Set.mem_diff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1819, 9], "def_end_pos": [1819, 17]}, {"full_name": "Classical.not_forall", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [686, 9], "def_end_pos": [686, 19]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nx : \u03b1\nhx : x \u2208 t0\n\u22a2 \u00acx \u2208 s'", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nx : \u03b1\nhx : x \u2208 t0\n\u22a2 x \u2208 s \u2192 \u2203 x_1, x_1 \u2208 t0 \u2227 dist x x_1 \u2264 r0 x_1"}, {"tactic": "intro _", "annotated_tactic": ["intro _", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nx : \u03b1\nhx : x \u2208 t0\n\u22a2 x \u2208 s \u2192 \u2203 x_1, x_1 \u2208 t0 \u2227 dist x x_1 \u2264 r0 x_1", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nx : \u03b1\nhx : x \u2208 t0\na\u271d : x \u2208 s\n\u22a2 \u2203 x_1, x_1 \u2208 t0 \u2227 dist x x_1 \u2264 r0 x_1"}, {"tactic": "refine' \u27e8x, hx, _\u27e9", "annotated_tactic": ["refine' \u27e8x, hx, _\u27e9", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nx : \u03b1\nhx : x \u2208 t0\na\u271d : x \u2208 s\n\u22a2 \u2203 x_1, x_1 \u2208 t0 \u2227 dist x x_1 \u2264 r0 x_1", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nx : \u03b1\nhx : x \u2208 t0\na\u271d : x \u2208 s\n\u22a2 dist x x \u2264 r0 x"}, {"tactic": "rw [dist_self]", "annotated_tactic": ["rw [<a>dist_self</a>]", [{"full_name": "dist_self", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [184, 9], "def_end_pos": [184, 18]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nx : \u03b1\nhx : x \u2208 t0\na\u271d : x \u2208 s\n\u22a2 dist x x \u2264 r0 x", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nx : \u03b1\nhx : x \u2208 t0\na\u271d : x \u2208 s\n\u22a2 0 \u2264 r0 x"}, {"tactic": "exact (hr0 x hx).2.1.le", "annotated_tactic": ["exact (hr0 x hx).2.1.<a>le</a>", [{"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [142, 7], "def_end_pos": [142, 15]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nx : \u03b1\nhx : x \u2208 t0\na\u271d : x \u2208 s\n\u22a2 0 \u2264 r0 x", "state_after": "no goals"}, {"tactic": "exact t0_count.union (countable_iUnion fun i => (S_count i).image _)", "annotated_tactic": ["exact t0_count.union (<a>countable_iUnion</a> fun i => (S_count i).<a>image</a> _)", [{"full_name": "Set.countable_iUnion", "def_path": "Mathlib/Data/Set/Countable.lean", "def_pos": [185, 9], "def_end_pos": [185, 25]}, {"full_name": "Set.Countable.image", "def_path": "Mathlib/Data/Set/Countable.lean", "def_pos": [134, 9], "def_end_pos": [134, 24]}]], "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\n\u22a2 Set.Countable (t0 \u222a \u22c3 i, Subtype.val '' S i)", "state_after": "no goals"}, {"tactic": "simp only [t0s, true_and_iff, union_subset_iff, image_subset_iff, iUnion_subset_iff]", "annotated_tactic": ["simp only [t0s, <a>true_and_iff</a>, <a>union_subset_iff</a>, <a>image_subset_iff</a>, <a>iUnion_subset_iff</a>]", [{"full_name": "true_and_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [147, 9], "def_end_pos": [147, 21]}, {"full_name": "Set.union_subset_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [841, 9], "def_end_pos": [841, 25]}, {"full_name": "Set.image_subset_iff", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [497, 9], "def_end_pos": [497, 25]}, {"full_name": "Set.iUnion_subset_iff", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [411, 9], "def_end_pos": [411, 26]}]], "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\n\u22a2 t0 \u222a \u22c3 i, Subtype.val '' S i \u2286 s", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\n\u22a2 \u2200 (i : Fin N), S i \u2286 (fun a => \u2191a) \u207b\u00b9' s"}, {"tactic": "intro i x _", "annotated_tactic": ["intro i x _", []], "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\n\u22a2 \u2200 (i : Fin N), S i \u2286 (fun a => \u2191a) \u207b\u00b9' s", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\ni : Fin N\nx : { x // x \u2208 s \\ \u22c3 x \u2208 t0, closedBall x (r0 x) }\na\u271d : x \u2208 S i\n\u22a2 x \u2208 (fun a => \u2191a) \u207b\u00b9' s"}, {"tactic": "exact s's x.2", "annotated_tactic": ["exact s's x.2", []], "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\ni : Fin N\nx : { x // x \u2208 s \\ \u22c3 x \u2208 t0, closedBall x (r0 x) }\na\u271d : x \u2208 S i\n\u22a2 x \u2208 (fun a => \u2191a) \u207b\u00b9' s", "state_after": "no goals"}, {"tactic": "intro x hx", "annotated_tactic": ["intro x hx", []], "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_3\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\n\u22a2 \u2200 (x : \u03b1), x \u2208 t0 \u222a \u22c3 i, Subtype.val '' S i \u2192 r x \u2208 f x", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_3\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nx : \u03b1\nhx : x \u2208 t0 \u222a \u22c3 i, Subtype.val '' S i\n\u22a2 r x \u2208 f x"}, {"tactic": "cases hx with\n| inl hx =>\n  rw [r_t0 x hx]\n  exact (hr0 _ hx).1\n| inr hx =>\n  have h'x : x \u2208 s' := by\n    simp only [mem_iUnion, mem_image] at hx\n    rcases hx with \u27e8i, y, _, rfl\u27e9\n    exact y.2\n  simp only [if_pos h'x, (hr1 x h'x).1.1]", "annotated_tactic": ["cases hx with\n    | <a>inl</a> hx =>\n      rw [r_t0 x hx]\n      exact (hr0 _ hx).1\n    | <a>inr</a> hx =>\n      have h'x : x \u2208 s' := by\n        simp only [<a>mem_iUnion</a>, <a>mem_image</a>] at hx\n        rcases hx with \u27e8i, y, _, rfl\u27e9\n        exact y.2\n      simp only [<a>if_pos</a> h'x, (hr1 x h'x).1.1]", [{"full_name": "Or.inl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [517, 5], "def_end_pos": [517, 8]}, {"full_name": "Or.inr", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [519, 5], "def_end_pos": [519, 8]}, {"full_name": "Set.mem_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [201, 9], "def_end_pos": [201, 19]}, {"full_name": "Set.mem_image", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [231, 9], "def_end_pos": [231, 18]}, {"full_name": "if_pos", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [790, 9], "def_end_pos": [790, 15]}]], "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_3\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nx : \u03b1\nhx : x \u2208 t0 \u222a \u22c3 i, Subtype.val '' S i\n\u22a2 r x \u2208 f x", "state_after": "no goals"}, {"tactic": "rw [r_t0 x hx]", "annotated_tactic": ["rw [r_t0 x hx]", []], "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_3.inl\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nx : \u03b1\nhx : x \u2208 t0\n\u22a2 r x \u2208 f x", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_3.inl\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nx : \u03b1\nhx : x \u2208 t0\n\u22a2 r0 x \u2208 f x"}, {"tactic": "exact (hr0 _ hx).1", "annotated_tactic": ["exact (hr0 _ hx).1", []], "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_3.inl\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nx : \u03b1\nhx : x \u2208 t0\n\u22a2 r0 x \u2208 f x", "state_after": "no goals"}, {"tactic": "have h'x : x \u2208 s' := by\n  simp only [mem_iUnion, mem_image] at hx\n  rcases hx with \u27e8i, y, _, rfl\u27e9\n  exact y.2", "annotated_tactic": ["have h'x : x \u2208 s' := by\n        simp only [<a>mem_iUnion</a>, <a>mem_image</a>] at hx\n        rcases hx with \u27e8i, y, _, rfl\u27e9\n        exact y.2", [{"full_name": "Set.mem_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [201, 9], "def_end_pos": [201, 19]}, {"full_name": "Set.mem_image", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [231, 9], "def_end_pos": [231, 18]}]], "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_3.inr\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nx : \u03b1\nhx : x \u2208 \u22c3 i, Subtype.val '' S i\n\u22a2 r x \u2208 f x", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_3.inr\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nx : \u03b1\nhx : x \u2208 \u22c3 i, Subtype.val '' S i\nh'x : x \u2208 s'\n\u22a2 r x \u2208 f x"}, {"tactic": "simp only [if_pos h'x, (hr1 x h'x).1.1]", "annotated_tactic": ["simp only [<a>if_pos</a> h'x, (hr1 x h'x).1.1]", [{"full_name": "if_pos", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [790, 9], "def_end_pos": [790, 15]}]], "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_3.inr\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nx : \u03b1\nhx : x \u2208 \u22c3 i, Subtype.val '' S i\nh'x : x \u2208 s'\n\u22a2 r x \u2208 f x", "state_after": "no goals"}, {"tactic": "simp only [mem_iUnion, mem_image] at hx", "annotated_tactic": ["simp only [<a>mem_iUnion</a>, <a>mem_image</a>] at hx", [{"full_name": "Set.mem_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [201, 9], "def_end_pos": [201, 19]}, {"full_name": "Set.mem_image", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [231, 9], "def_end_pos": [231, 18]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nx : \u03b1\nhx : x \u2208 \u22c3 i, Subtype.val '' S i\n\u22a2 x \u2208 s'", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nx : \u03b1\nhx : \u2203 i x_1, x_1 \u2208 S i \u2227 \u2191x_1 = x\n\u22a2 x \u2208 s'"}, {"tactic": "rcases hx with \u27e8i, y, _, rfl\u27e9", "annotated_tactic": ["rcases hx with \u27e8i, y, _, rfl\u27e9", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nx : \u03b1\nhx : \u2203 i x_1, x_1 \u2208 S i \u2227 \u2191x_1 = x\n\u22a2 x \u2208 s'", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\ni : Fin N\ny : { x // x \u2208 s \\ \u22c3 x \u2208 t0, closedBall x (r0 x) }\nleft\u271d : y \u2208 S i\n\u22a2 \u2191y \u2208 s'"}, {"tactic": "exact y.2", "annotated_tactic": ["exact y.2", []], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\ni : Fin N\ny : { x // x \u2208 s \\ \u22c3 x \u2208 t0, closedBall x (r0 x) }\nleft\u271d : y \u2208 S i\n\u22a2 \u2191y \u2208 s'", "state_after": "no goals"}, {"tactic": "intro x hx", "annotated_tactic": ["intro x hx", []], "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_4\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\n\u22a2 s \u2286 \u22c3 x \u2208 t0 \u222a \u22c3 i, Subtype.val '' S i, closedBall x (r x)", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_4\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nx : \u03b1\nhx : x \u2208 s\n\u22a2 x \u2208 \u22c3 x \u2208 t0 \u222a \u22c3 i, Subtype.val '' S i, closedBall x (r x)"}, {"tactic": "by_cases h'x : x \u2208 s'", "annotated_tactic": ["by_cases h'x : x \u2208 s'", []], "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_4\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nx : \u03b1\nhx : x \u2208 s\n\u22a2 x \u2208 \u22c3 x \u2208 t0 \u222a \u22c3 i, Subtype.val '' S i, closedBall x (r x)", "state_after": "case pos\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nx : \u03b1\nhx : x \u2208 s\nh'x : x \u2208 s'\n\u22a2 x \u2208 \u22c3 x \u2208 t0 \u222a \u22c3 i, Subtype.val '' S i, closedBall x (r x)\n\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nx : \u03b1\nhx : x \u2208 s\nh'x : \u00acx \u2208 s'\n\u22a2 x \u2208 \u22c3 x \u2208 t0 \u222a \u22c3 i, Subtype.val '' S i, closedBall x (r x)"}, {"tactic": "obtain \u27e8i, y, ySi, xy\u27e9 : \u2203 (i : Fin N) (y : \u21a5s'), y \u2208 S i \u2227 x \u2208 ball (y : \u03b1) (r1 y) := by\n  have A : x \u2208 range q.c := by\n    simpa only [not_exists, exists_prop, mem_iUnion, mem_closedBall, not_and, not_le,\n      mem_setOf_eq, Subtype.range_coe_subtype, mem_diff] using h'x\n  simpa only [mem_iUnion, mem_image, bex_def] using hS A", "annotated_tactic": ["obtain \u27e8i, y, ySi, xy\u27e9 : \u2203 (i : <a>Fin</a> N) (y : \u21a5s'), y \u2208 S i \u2227 x \u2208 <a>ball</a> (y : \u03b1) (r1 y) := by\n        have A : x \u2208 <a>range</a> q.c := by\n          simpa only [<a>not_exists</a>, <a>exists_prop</a>, <a>mem_iUnion</a>, <a>mem_closedBall</a>, <a>not_and</a>, <a>not_le</a>,\n            <a>mem_setOf_eq</a>, <a>Subtype.range_coe_subtype</a>, <a>mem_diff</a>] using h'x\n        simpa only [<a>mem_iUnion</a>, <a>mem_image</a>, <a>bex_def</a>] using hS A", [{"full_name": "Fin", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1745, 11], "def_end_pos": [1745, 14]}, {"full_name": "Metric.ball", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [409, 5], "def_end_pos": [409, 9]}, {"full_name": "Set.range", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [668, 5], "def_end_pos": [668, 10]}, {"full_name": "not_exists", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [422, 17], "def_end_pos": [422, 27]}, {"full_name": "exists_prop", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [485, 17], "def_end_pos": [485, 28]}, {"full_name": "Set.mem_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [201, 9], "def_end_pos": [201, 19]}, {"full_name": "Metric.mem_closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [478, 17], "def_end_pos": [478, 31]}, {"full_name": "not_and", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [316, 17], "def_end_pos": [316, 24]}, {"full_name": "not_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [373, 9], "def_end_pos": [373, 15]}, {"full_name": "Set.mem_setOf_eq", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [256, 29], "def_end_pos": [256, 41]}, {"full_name": "Subtype.range_coe_subtype", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [1425, 9], "def_end_pos": [1425, 26]}, {"full_name": "Set.mem_diff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1819, 9], "def_end_pos": [1819, 17]}, {"full_name": "Set.mem_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [201, 9], "def_end_pos": [201, 19]}, {"full_name": "Set.mem_image", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [231, 9], "def_end_pos": [231, 18]}, {"full_name": "bex_def", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [1027, 9], "def_end_pos": [1027, 16]}]], "state_before": "case pos\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nx : \u03b1\nhx : x \u2208 s\nh'x : x \u2208 s'\n\u22a2 x \u2208 \u22c3 x \u2208 t0 \u222a \u22c3 i, Subtype.val '' S i, closedBall x (r x)", "state_after": "case pos.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nx : \u03b1\nhx : x \u2208 s\nh'x : x \u2208 s'\ni : Fin N\ny : \u2191s'\nySi : y \u2208 S i\nxy : x \u2208 ball (\u2191y) (r1 \u2191y)\n\u22a2 x \u2208 \u22c3 x \u2208 t0 \u222a \u22c3 i, Subtype.val '' S i, closedBall x (r x)"}, {"tactic": "refine' mem_iUnion\u2082.2 \u27e8y, Or.inr _, _\u27e9", "annotated_tactic": ["refine' <a>mem_iUnion\u2082</a>.2 \u27e8y, <a>Or.inr</a> _, _\u27e9", [{"full_name": "Set.mem_iUnion\u2082", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [212, 9], "def_end_pos": [212, 20]}, {"full_name": "Or.inr", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [519, 5], "def_end_pos": [519, 8]}]], "state_before": "case pos.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nx : \u03b1\nhx : x \u2208 s\nh'x : x \u2208 s'\ni : Fin N\ny : \u2191s'\nySi : y \u2208 S i\nxy : x \u2208 ball (\u2191y) (r1 \u2191y)\n\u22a2 x \u2208 \u22c3 x \u2208 t0 \u222a \u22c3 i, Subtype.val '' S i, closedBall x (r x)", "state_after": "case pos.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nx : \u03b1\nhx : x \u2208 s\nh'x : x \u2208 s'\ni : Fin N\ny : \u2191s'\nySi : y \u2208 S i\nxy : x \u2208 ball (\u2191y) (r1 \u2191y)\n\u22a2 \u2191y \u2208 \u22c3 i, Subtype.val '' S i\n\ncase pos.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nx : \u03b1\nhx : x \u2208 s\nh'x : x \u2208 s'\ni : Fin N\ny : \u2191s'\nySi : y \u2208 S i\nxy : x \u2208 ball (\u2191y) (r1 \u2191y)\n\u22a2 x \u2208 closedBall (\u2191y) (r \u2191y)"}, {"tactic": "have A : x \u2208 range q.c := by\n  simpa only [not_exists, exists_prop, mem_iUnion, mem_closedBall, not_and, not_le,\n    mem_setOf_eq, Subtype.range_coe_subtype, mem_diff] using h'x", "annotated_tactic": ["have A : x \u2208 <a>range</a> q.c := by\n          simpa only [<a>not_exists</a>, <a>exists_prop</a>, <a>mem_iUnion</a>, <a>mem_closedBall</a>, <a>not_and</a>, <a>not_le</a>,\n            <a>mem_setOf_eq</a>, <a>Subtype.range_coe_subtype</a>, <a>mem_diff</a>] using h'x", [{"full_name": "Set.range", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [668, 5], "def_end_pos": [668, 10]}, {"full_name": "not_exists", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [422, 17], "def_end_pos": [422, 27]}, {"full_name": "exists_prop", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [485, 17], "def_end_pos": [485, 28]}, {"full_name": "Set.mem_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [201, 9], "def_end_pos": [201, 19]}, {"full_name": "Metric.mem_closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [478, 17], "def_end_pos": [478, 31]}, {"full_name": "not_and", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [316, 17], "def_end_pos": [316, 24]}, {"full_name": "not_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [373, 9], "def_end_pos": [373, 15]}, {"full_name": "Set.mem_setOf_eq", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [256, 29], "def_end_pos": [256, 41]}, {"full_name": "Subtype.range_coe_subtype", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [1425, 9], "def_end_pos": [1425, 26]}, {"full_name": "Set.mem_diff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1819, 9], "def_end_pos": [1819, 17]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nx : \u03b1\nhx : x \u2208 s\nh'x : x \u2208 s'\n\u22a2 \u2203 i y, y \u2208 S i \u2227 x \u2208 ball (\u2191y) (r1 \u2191y)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nx : \u03b1\nhx : x \u2208 s\nh'x : x \u2208 s'\nA : x \u2208 range q.c\n\u22a2 \u2203 i y, y \u2208 S i \u2227 x \u2208 ball (\u2191y) (r1 \u2191y)"}, {"tactic": "simpa only [mem_iUnion, mem_image, bex_def] using hS A", "annotated_tactic": ["simpa only [<a>mem_iUnion</a>, <a>mem_image</a>, <a>bex_def</a>] using hS A", [{"full_name": "Set.mem_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [201, 9], "def_end_pos": [201, 19]}, {"full_name": "Set.mem_image", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [231, 9], "def_end_pos": [231, 18]}, {"full_name": "bex_def", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [1027, 9], "def_end_pos": [1027, 16]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nx : \u03b1\nhx : x \u2208 s\nh'x : x \u2208 s'\nA : x \u2208 range q.c\n\u22a2 \u2203 i y, y \u2208 S i \u2227 x \u2208 ball (\u2191y) (r1 \u2191y)", "state_after": "no goals"}, {"tactic": "simpa only [not_exists, exists_prop, mem_iUnion, mem_closedBall, not_and, not_le,\n  mem_setOf_eq, Subtype.range_coe_subtype, mem_diff] using h'x", "annotated_tactic": ["simpa only [<a>not_exists</a>, <a>exists_prop</a>, <a>mem_iUnion</a>, <a>mem_closedBall</a>, <a>not_and</a>, <a>not_le</a>,\n            <a>mem_setOf_eq</a>, <a>Subtype.range_coe_subtype</a>, <a>mem_diff</a>] using h'x", [{"full_name": "not_exists", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [422, 17], "def_end_pos": [422, 27]}, {"full_name": "exists_prop", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [485, 17], "def_end_pos": [485, 28]}, {"full_name": "Set.mem_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [201, 9], "def_end_pos": [201, 19]}, {"full_name": "Metric.mem_closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [478, 17], "def_end_pos": [478, 31]}, {"full_name": "not_and", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [316, 17], "def_end_pos": [316, 24]}, {"full_name": "not_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [373, 9], "def_end_pos": [373, 15]}, {"full_name": "Set.mem_setOf_eq", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [256, 29], "def_end_pos": [256, 41]}, {"full_name": "Subtype.range_coe_subtype", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [1425, 9], "def_end_pos": [1425, 26]}, {"full_name": "Set.mem_diff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1819, 9], "def_end_pos": [1819, 17]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nx : \u03b1\nhx : x \u2208 s\nh'x : x \u2208 s'\n\u22a2 x \u2208 range q.c", "state_after": "no goals"}, {"tactic": "simp only [mem_iUnion, mem_image]", "annotated_tactic": ["simp only [<a>mem_iUnion</a>, <a>mem_image</a>]", [{"full_name": "Set.mem_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [201, 9], "def_end_pos": [201, 19]}, {"full_name": "Set.mem_image", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [231, 9], "def_end_pos": [231, 18]}]], "state_before": "case pos.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nx : \u03b1\nhx : x \u2208 s\nh'x : x \u2208 s'\ni : Fin N\ny : \u2191s'\nySi : y \u2208 S i\nxy : x \u2208 ball (\u2191y) (r1 \u2191y)\n\u22a2 \u2191y \u2208 \u22c3 i, Subtype.val '' S i", "state_after": "case pos.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nx : \u03b1\nhx : x \u2208 s\nh'x : x \u2208 s'\ni : Fin N\ny : \u2191s'\nySi : y \u2208 S i\nxy : x \u2208 ball (\u2191y) (r1 \u2191y)\n\u22a2 \u2203 i x, x \u2208 S i \u2227 \u2191x = \u2191y"}, {"tactic": "exact \u27e8i, y, ySi, rfl\u27e9", "annotated_tactic": ["exact \u27e8i, y, ySi, <a>rfl</a>\u27e9", [{"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "case pos.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nx : \u03b1\nhx : x \u2208 s\nh'x : x \u2208 s'\ni : Fin N\ny : \u2191s'\nySi : y \u2208 S i\nxy : x \u2208 ball (\u2191y) (r1 \u2191y)\n\u22a2 \u2203 i x, x \u2208 S i \u2227 \u2191x = \u2191y", "state_after": "no goals"}, {"tactic": "have : (y : \u03b1) \u2208 s' := y.2", "annotated_tactic": ["have : (y : \u03b1) \u2208 s' := y.2", []], "state_before": "case pos.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nx : \u03b1\nhx : x \u2208 s\nh'x : x \u2208 s'\ni : Fin N\ny : \u2191s'\nySi : y \u2208 S i\nxy : x \u2208 ball (\u2191y) (r1 \u2191y)\n\u22a2 x \u2208 closedBall (\u2191y) (r \u2191y)", "state_after": "case pos.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nx : \u03b1\nhx : x \u2208 s\nh'x : x \u2208 s'\ni : Fin N\ny : \u2191s'\nySi : y \u2208 S i\nxy : x \u2208 ball (\u2191y) (r1 \u2191y)\nthis : \u2191y \u2208 s'\n\u22a2 x \u2208 closedBall (\u2191y) (r \u2191y)"}, {"tactic": "simp only [if_pos this]", "annotated_tactic": ["simp only [<a>if_pos</a> this]", [{"full_name": "if_pos", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [790, 9], "def_end_pos": [790, 15]}]], "state_before": "case pos.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nx : \u03b1\nhx : x \u2208 s\nh'x : x \u2208 s'\ni : Fin N\ny : \u2191s'\nySi : y \u2208 S i\nxy : x \u2208 ball (\u2191y) (r1 \u2191y)\nthis : \u2191y \u2208 s'\n\u22a2 x \u2208 closedBall (\u2191y) (r \u2191y)", "state_after": "case pos.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nx : \u03b1\nhx : x \u2208 s\nh'x : x \u2208 s'\ni : Fin N\ny : \u2191s'\nySi : y \u2208 S i\nxy : x \u2208 ball (\u2191y) (r1 \u2191y)\nthis : \u2191y \u2208 s'\n\u22a2 x \u2208 closedBall (\u2191y) (r1 \u2191y)"}, {"tactic": "exact ball_subset_closedBall xy", "annotated_tactic": ["exact <a>ball_subset_closedBall</a> xy", [{"full_name": "Metric.ball_subset_closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [534, 9], "def_end_pos": [534, 31]}]], "state_before": "case pos.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nx : \u03b1\nhx : x \u2208 s\nh'x : x \u2208 s'\ni : Fin N\ny : \u2191s'\nySi : y \u2208 S i\nxy : x \u2208 ball (\u2191y) (r1 \u2191y)\nthis : \u2191y \u2208 s'\n\u22a2 x \u2208 closedBall (\u2191y) (r1 \u2191y)", "state_after": "no goals"}, {"tactic": "obtain \u27e8y, yt0, hxy\u27e9 : \u2203 y : \u03b1, y \u2208 t0 \u2227 x \u2208 closedBall y (r0 y) := by\n  simpa [hx, -mem_closedBall] using h'x", "annotated_tactic": ["obtain \u27e8y, yt0, hxy\u27e9 : \u2203 y : \u03b1, y \u2208 t0 \u2227 x \u2208 <a>closedBall</a> y (r0 y) := by\n        simpa [hx, -<a>mem_closedBall</a>] using h'x", [{"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "Metric.mem_closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [478, 17], "def_end_pos": [478, 31]}]], "state_before": "case neg\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nx : \u03b1\nhx : x \u2208 s\nh'x : \u00acx \u2208 s'\n\u22a2 x \u2208 \u22c3 x \u2208 t0 \u222a \u22c3 i, Subtype.val '' S i, closedBall x (r x)", "state_after": "case neg.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nx : \u03b1\nhx : x \u2208 s\nh'x : \u00acx \u2208 s'\ny : \u03b1\nyt0 : y \u2208 t0\nhxy : x \u2208 closedBall y (r0 y)\n\u22a2 x \u2208 \u22c3 x \u2208 t0 \u222a \u22c3 i, Subtype.val '' S i, closedBall x (r x)"}, {"tactic": "refine' mem_iUnion\u2082.2 \u27e8y, Or.inl yt0, _\u27e9", "annotated_tactic": ["refine' <a>mem_iUnion\u2082</a>.2 \u27e8y, <a>Or.inl</a> yt0, _\u27e9", [{"full_name": "Set.mem_iUnion\u2082", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [212, 9], "def_end_pos": [212, 20]}, {"full_name": "Or.inl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [517, 5], "def_end_pos": [517, 8]}]], "state_before": "case neg.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nx : \u03b1\nhx : x \u2208 s\nh'x : \u00acx \u2208 s'\ny : \u03b1\nyt0 : y \u2208 t0\nhxy : x \u2208 closedBall y (r0 y)\n\u22a2 x \u2208 \u22c3 x \u2208 t0 \u222a \u22c3 i, Subtype.val '' S i, closedBall x (r x)", "state_after": "case neg.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nx : \u03b1\nhx : x \u2208 s\nh'x : \u00acx \u2208 s'\ny : \u03b1\nyt0 : y \u2208 t0\nhxy : x \u2208 closedBall y (r0 y)\n\u22a2 x \u2208 closedBall y (r y)"}, {"tactic": "rwa [r_t0 _ yt0]", "annotated_tactic": ["rwa [r_t0 _ yt0]", []], "state_before": "case neg.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nx : \u03b1\nhx : x \u2208 s\nh'x : \u00acx \u2208 s'\ny : \u03b1\nyt0 : y \u2208 t0\nhxy : x \u2208 closedBall y (r0 y)\n\u22a2 x \u2208 closedBall y (r y)", "state_after": "no goals"}, {"tactic": "simpa [hx, -mem_closedBall] using h'x", "annotated_tactic": ["simpa [hx, -<a>mem_closedBall</a>] using h'x", [{"full_name": "Metric.mem_closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [478, 17], "def_end_pos": [478, 31]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nx : \u03b1\nhx : x \u2208 s\nh'x : \u00acx \u2208 s'\n\u22a2 \u2203 y, y \u2208 t0 \u2227 x \u2208 closedBall y (r0 y)", "state_after": "no goals"}, {"tactic": "calc\n  (\u2211' x : \u21a5(t0 \u222a \u22c3 i : Fin N, ((\u2191) : s' \u2192 \u03b1) '' S i), \u03bc (closedBall x (r x))) \u2264\n      (\u2211' x : t0, \u03bc (closedBall x (r x))) +\n        \u2211' x : \u22c3 i : Fin N, ((\u2191) : s' \u2192 \u03b1) '' S i, \u03bc (closedBall x (r x)) :=\n    ENNReal.tsum_union_le (fun x => \u03bc (closedBall x (r x))) _ _\n  _ \u2264\n      (\u2211' x : t0, \u03bc (closedBall x (r x))) +\n        \u2211 i : Fin N, \u2211' x : ((\u2191) : s' \u2192 \u03b1) '' S i, \u03bc (closedBall x (r x)) :=\n    (add_le_add le_rfl (ENNReal.tsum_iUnion_le (fun x => \u03bc (closedBall x (r x))) _))\n  _ \u2264 \u03bc s + \u03b5 / 2 + \u2211 i : Fin N, \u03b5 / 2 / N := by\n    refine' add_le_add A _\n    refine' Finset.sum_le_sum _\n    intro i _\n    exact B i\n  _ \u2264 \u03bc s + \u03b5 / 2 + \u03b5 / 2 := by\n    refine' add_le_add le_rfl _\n    simp only [Finset.card_fin, Finset.sum_const, nsmul_eq_mul, ENNReal.mul_div_le]\n  _ = \u03bc s + \u03b5 := by rw [add_assoc, ENNReal.add_halves]", "annotated_tactic": ["calc\n      (\u2211' x : \u21a5(t0 \u222a \u22c3 i : <a>Fin</a> N, ((\u2191) : s' \u2192 \u03b1) '' S i), \u03bc (<a>closedBall</a> x (r x))) \u2264\n          (\u2211' x : t0, \u03bc (<a>closedBall</a> x (r x))) +\n            \u2211' x : \u22c3 i : <a>Fin</a> N, ((\u2191) : s' \u2192 \u03b1) '' S i, \u03bc (<a>closedBall</a> x (r x)) :=\n        <a>ENNReal.tsum_union_le</a> (fun x => \u03bc (<a>closedBall</a> x (r x))) _ _\n      _ \u2264\n          (\u2211' x : t0, \u03bc (<a>closedBall</a> x (r x))) +\n            \u2211 i : <a>Fin</a> N, \u2211' x : ((\u2191) : s' \u2192 \u03b1) '' S i, \u03bc (<a>closedBall</a> x (r x)) :=\n        (<a>add_le_add</a> <a>le_rfl</a> (<a>ENNReal.tsum_iUnion_le</a> (fun x => \u03bc (<a>closedBall</a> x (r x))) _))\n      _ \u2264 \u03bc s + \u03b5 / 2 + \u2211 i : <a>Fin</a> N, \u03b5 / 2 / N := by\n        refine' <a>add_le_add</a> A _\n        refine' <a>Finset.sum_le_sum</a> _\n        intro i _\n        exact B i\n      _ \u2264 \u03bc s + \u03b5 / 2 + \u03b5 / 2 := by\n        refine' <a>add_le_add</a> <a>le_rfl</a> _\n        simp only [<a>Finset.card_fin</a>, <a>Finset.sum_const</a>, <a>nsmul_eq_mul</a>, <a>ENNReal.mul_div_le</a>]\n      _ = \u03bc s + \u03b5 := by rw [<a>add_assoc</a>, <a>ENNReal.add_halves</a>]", [{"full_name": "Fin", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1745, 11], "def_end_pos": [1745, 14]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "Fin", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1745, 11], "def_end_pos": [1745, 14]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "ENNReal.tsum_union_le", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [1018, 9], "def_end_pos": [1018, 22]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "Fin", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1745, 11], "def_end_pos": [1745, 14]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "add_le_add", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [205, 15], "def_end_pos": [205, 25]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}, {"full_name": "ENNReal.tsum_iUnion_le", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [1012, 9], "def_end_pos": [1012, 23]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "Fin", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1745, 11], "def_end_pos": [1745, 14]}, {"full_name": "add_le_add", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [205, 15], "def_end_pos": [205, 25]}, {"full_name": "Finset.sum_le_sum", "def_path": "Mathlib/Algebra/BigOperators/Order.lean", "def_pos": [111, 15], "def_end_pos": [111, 25]}, {"full_name": "add_le_add", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [205, 15], "def_end_pos": [205, 25]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}, {"full_name": "Finset.card_fin", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [322, 9], "def_end_pos": [322, 24]}, {"full_name": "Finset.sum_const", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [1440, 3], "def_end_pos": [1440, 14]}, {"full_name": "nsmul_eq_mul", "def_path": "Mathlib/Algebra/GroupPower/Lemmas.lean", "def_pos": [509, 9], "def_end_pos": [509, 21]}, {"full_name": "ENNReal.mul_div_le", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1750, 9], "def_end_pos": [1750, 19]}, {"full_name": "add_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [263, 3], "def_end_pos": [263, 14]}, {"full_name": "ENNReal.add_halves", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1781, 19], "def_end_pos": [1781, 29]}]], "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_5\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nA : \u2211' (x : \u2191t0), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nB : \u2200 (i : Fin N), \u2211' (x : \u2191(Subtype.val '' S i)), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u03b5 / 2 / \u2191N\n\u22a2 \u2211' (x : \u2191(t0 \u222a \u22c3 i, Subtype.val '' S i)), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5", "state_after": "no goals"}, {"tactic": "congr 1", "annotated_tactic": ["congr 1", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\n\u22a2 \u2211' (x : \u2191t0), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) = \u2211' (x : \u2191t0), \u2191\u2191\u03bc (closedBall (\u2191x) (r0 \u2191x))", "state_after": "case e_f\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\n\u22a2 (fun x => \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x))) = fun x => \u2191\u2191\u03bc (closedBall (\u2191x) (r0 \u2191x))"}, {"tactic": "ext x", "annotated_tactic": ["ext x", []], "state_before": "case e_f\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\n\u22a2 (fun x => \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x))) = fun x => \u2191\u2191\u03bc (closedBall (\u2191x) (r0 \u2191x))", "state_after": "case e_f.h\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nx : \u2191t0\n\u22a2 \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) = \u2191\u2191\u03bc (closedBall (\u2191x) (r0 \u2191x))"}, {"tactic": "rw [r_t0 x x.2]", "annotated_tactic": ["rw [r_t0 x x.2]", []], "state_before": "case e_f.h\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nx : \u2191t0\n\u22a2 \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) = \u2191\u2191\u03bc (closedBall (\u2191x) (r0 \u2191x))", "state_after": "no goals"}, {"tactic": "haveI : Encodable t0 := t0_count.toEncodable", "annotated_tactic": ["haveI : <a>Encodable</a> t0 := t0_count.toEncodable", [{"full_name": "Encodable", "def_path": "Mathlib/Logic/Encodable/Basic.lean", "def_pos": [45, 7], "def_end_pos": [45, 16]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\n\u22a2 \u2211' (x : \u2191t0), \u2191\u2191\u03bc (closedBall (\u2191x) (r0 \u2191x)) = \u2191\u2191\u03bc (\u22c3 x, closedBall (\u2191x) (r0 \u2191x))", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nthis : Encodable \u2191t0\n\u22a2 \u2211' (x : \u2191t0), \u2191\u2191\u03bc (closedBall (\u2191x) (r0 \u2191x)) = \u2191\u2191\u03bc (\u22c3 x, closedBall (\u2191x) (r0 \u2191x))"}, {"tactic": "rw [measure_iUnion]", "annotated_tactic": ["rw [<a>measure_iUnion</a>]", [{"full_name": "MeasureTheory.measure_iUnion", "def_path": "Mathlib/MeasureTheory/Measure/NullMeasurable.lean", "def_pos": [272, 9], "def_end_pos": [272, 23]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nthis : Encodable \u2191t0\n\u22a2 \u2211' (x : \u2191t0), \u2191\u2191\u03bc (closedBall (\u2191x) (r0 \u2191x)) = \u2191\u2191\u03bc (\u22c3 x, closedBall (\u2191x) (r0 \u2191x))", "state_after": "case hn\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nthis : Encodable \u2191t0\n\u22a2 Pairwise (Disjoint on fun x => closedBall (\u2191x) (r0 \u2191x))\n\ncase h\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nthis : Encodable \u2191t0\n\u22a2 \u2200 (i : \u2191t0), MeasurableSet (closedBall (\u2191i) (r0 \u2191i))"}, {"tactic": "exact (pairwise_subtype_iff_pairwise_set _ _).2 t0_disj", "annotated_tactic": ["exact (<a>pairwise_subtype_iff_pairwise_set</a> _ _).2 t0_disj", [{"full_name": "pairwise_subtype_iff_pairwise_set", "def_path": "Mathlib/Data/Set/Pairwise/Basic.lean", "def_pos": [221, 9], "def_end_pos": [221, 42]}]], "state_before": "case hn\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nthis : Encodable \u2191t0\n\u22a2 Pairwise (Disjoint on fun x => closedBall (\u2191x) (r0 \u2191x))", "state_after": "no goals"}, {"tactic": "exact fun i => measurableSet_closedBall", "annotated_tactic": ["exact fun i => <a>measurableSet_closedBall</a>", [{"full_name": "measurableSet_closedBall", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [1681, 9], "def_end_pos": [1681, 33]}]], "state_before": "case h\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nthis : Encodable \u2191t0\n\u22a2 \u2200 (i : \u2191t0), MeasurableSet (closedBall (\u2191i) (r0 \u2191i))", "state_after": "no goals"}, {"tactic": "apply measure_mono", "annotated_tactic": ["apply <a>measure_mono</a>", [{"full_name": "MeasureTheory.measure_mono", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [193, 9], "def_end_pos": [193, 21]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\n\u22a2 \u2191\u2191\u03bc (\u22c3 x, closedBall (\u2191x) (r0 \u2191x)) \u2264 \u2191\u2191\u03bc u", "state_after": "case h\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\n\u22a2 \u22c3 x, closedBall (\u2191x) (r0 \u2191x) \u2286 u"}, {"tactic": "simp only [SetCoe.forall, Subtype.coe_mk, iUnion_subset_iff]", "annotated_tactic": ["simp only [<a>SetCoe.forall</a>, <a>Subtype.coe_mk</a>, <a>iUnion_subset_iff</a>]", [{"full_name": "SetCoe.forall", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [185, 9], "def_end_pos": [185, 22]}, {"full_name": "Subtype.coe_mk", "def_path": "Mathlib/Data/Subtype.lean", "def_pos": [99, 9], "def_end_pos": [99, 15]}, {"full_name": "Set.iUnion_subset_iff", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [411, 9], "def_end_pos": [411, 26]}]], "state_before": "case h\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\n\u22a2 \u22c3 x, closedBall (\u2191x) (r0 \u2191x) \u2286 u", "state_after": "case h\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\n\u22a2 \u2200 (x : \u03b1), x \u2208 t0 \u2192 closedBall x (r0 x) \u2286 u"}, {"tactic": "intro x hx", "annotated_tactic": ["intro x hx", []], "state_before": "case h\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\n\u22a2 \u2200 (x : \u03b1), x \u2208 t0 \u2192 closedBall x (r0 x) \u2286 u", "state_after": "case h\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nx : \u03b1\nhx : x \u2208 t0\n\u22a2 closedBall x (r0 x) \u2286 u"}, {"tactic": "apply Subset.trans (closedBall_subset_ball (hr0 x hx).2.2) (hR x (t0s hx)).2", "annotated_tactic": ["apply <a>Subset.trans</a> (<a>closedBall_subset_ball</a> (hr0 x hx).2.2) (hR x (t0s hx)).2", [{"full_name": "Set.Subset.trans", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [362, 9], "def_end_pos": [362, 21]}, {"full_name": "Metric.closedBall_subset_ball", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [621, 9], "def_end_pos": [621, 31]}]], "state_before": "case h\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nx : \u03b1\nhx : x \u2208 t0\n\u22a2 closedBall x (r0 x) \u2286 u", "state_after": "no goals"}, {"tactic": "have : InjOn ((\u2191) : s' \u2192 \u03b1) (S i) := Subtype.val_injective.injOn _", "annotated_tactic": ["have : <a>InjOn</a> ((\u2191) : s' \u2192 \u03b1) (S i) := Subtype.val_injective.injOn _", [{"full_name": "Set.InjOn", "def_path": "Mathlib/Data/Set/Function.lean", "def_pos": [603, 5], "def_end_pos": [603, 10]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nA : \u2211' (x : \u2191t0), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\ni : Fin N\n\u22a2 \u2211' (x : \u2191(Subtype.val '' S i)), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) = \u2211' (x : \u2191(S i)), \u2191\u2191\u03bc (closedBall (\u2191\u2191x) (r \u2191\u2191x))", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nA : \u2211' (x : \u2191t0), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\ni : Fin N\nthis : InjOn Subtype.val (S i)\n\u22a2 \u2211' (x : \u2191(Subtype.val '' S i)), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) = \u2211' (x : \u2191(S i)), \u2191\u2191\u03bc (closedBall (\u2191\u2191x) (r \u2191\u2191x))"}, {"tactic": "let F : S i \u2243 ((\u2191) : s' \u2192 \u03b1) '' S i := this.bijOn_image.equiv _", "annotated_tactic": ["let F : S i \u2243 ((\u2191) : s' \u2192 \u03b1) '' S i := this.bijOn_image.equiv _", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nA : \u2211' (x : \u2191t0), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\ni : Fin N\nthis : InjOn Subtype.val (S i)\n\u22a2 \u2211' (x : \u2191(Subtype.val '' S i)), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) = \u2211' (x : \u2191(S i)), \u2191\u2191\u03bc (closedBall (\u2191\u2191x) (r \u2191\u2191x))", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nA : \u2211' (x : \u2191t0), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\ni : Fin N\nthis : InjOn Subtype.val (S i)\nF : \u2191(S i) \u2243 \u2191(Subtype.val '' S i) := BijOn.equiv Subtype.val (_ : BijOn Subtype.val (S i) (Subtype.val '' S i))\n\u22a2 \u2211' (x : \u2191(Subtype.val '' S i)), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) = \u2211' (x : \u2191(S i)), \u2191\u2191\u03bc (closedBall (\u2191\u2191x) (r \u2191\u2191x))"}, {"tactic": "exact (F.tsum_eq fun x => \u03bc (closedBall x (r x))).symm", "annotated_tactic": ["exact (F.tsum_eq fun x => \u03bc (<a>closedBall</a> x (r x))).<a>symm</a>", [{"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "Eq.symm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [310, 9], "def_end_pos": [310, 16]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nA : \u2211' (x : \u2191t0), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\ni : Fin N\nthis : InjOn Subtype.val (S i)\nF : \u2191(S i) \u2243 \u2191(Subtype.val '' S i) := BijOn.equiv Subtype.val (_ : BijOn Subtype.val (S i) (Subtype.val '' S i))\n\u22a2 \u2211' (x : \u2191(Subtype.val '' S i)), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) = \u2211' (x : \u2191(S i)), \u2191\u2191\u03bc (closedBall (\u2191\u2191x) (r \u2191\u2191x))", "state_after": "no goals"}, {"tactic": "congr 1", "annotated_tactic": ["congr 1", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nA : \u2211' (x : \u2191t0), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\ni : Fin N\n\u22a2 \u2211' (x : \u2191(S i)), \u2191\u2191\u03bc (closedBall (\u2191\u2191x) (r \u2191\u2191x)) = \u2211' (x : \u2191(S i)), \u2191\u2191\u03bc (closedBall (\u2191\u2191x) (r1 \u2191\u2191x))", "state_after": "case e_f\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nA : \u2211' (x : \u2191t0), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\ni : Fin N\n\u22a2 (fun x => \u2191\u2191\u03bc (closedBall (\u2191\u2191x) (r \u2191\u2191x))) = fun x => \u2191\u2191\u03bc (closedBall (\u2191\u2191x) (r1 \u2191\u2191x))"}, {"tactic": "ext x", "annotated_tactic": ["ext x", []], "state_before": "case e_f\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nA : \u2211' (x : \u2191t0), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\ni : Fin N\n\u22a2 (fun x => \u2191\u2191\u03bc (closedBall (\u2191\u2191x) (r \u2191\u2191x))) = fun x => \u2191\u2191\u03bc (closedBall (\u2191\u2191x) (r1 \u2191\u2191x))", "state_after": "case e_f.h\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nA : \u2211' (x : \u2191t0), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\ni : Fin N\nx : \u2191(S i)\n\u22a2 \u2191\u2191\u03bc (closedBall (\u2191\u2191x) (r \u2191\u2191x)) = \u2191\u2191\u03bc (closedBall (\u2191\u2191x) (r1 \u2191\u2191x))"}, {"tactic": "have : (x : \u03b1) \u2208 s' := x.1.2", "annotated_tactic": ["have : (x : \u03b1) \u2208 s' := x.1.2", []], "state_before": "case e_f.h\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nA : \u2211' (x : \u2191t0), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\ni : Fin N\nx : \u2191(S i)\n\u22a2 \u2191\u2191\u03bc (closedBall (\u2191\u2191x) (r \u2191\u2191x)) = \u2191\u2191\u03bc (closedBall (\u2191\u2191x) (r1 \u2191\u2191x))", "state_after": "case e_f.h\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nA : \u2211' (x : \u2191t0), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\ni : Fin N\nx : \u2191(S i)\nthis : \u2191\u2191x \u2208 s'\n\u22a2 \u2191\u2191\u03bc (closedBall (\u2191\u2191x) (r \u2191\u2191x)) = \u2191\u2191\u03bc (closedBall (\u2191\u2191x) (r1 \u2191\u2191x))"}, {"tactic": "simp only [if_pos this]", "annotated_tactic": ["simp only [<a>if_pos</a> this]", [{"full_name": "if_pos", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [790, 9], "def_end_pos": [790, 15]}]], "state_before": "case e_f.h\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nA : \u2211' (x : \u2191t0), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\ni : Fin N\nx : \u2191(S i)\nthis : \u2191\u2191x \u2208 s'\n\u22a2 \u2191\u2191\u03bc (closedBall (\u2191\u2191x) (r \u2191\u2191x)) = \u2191\u2191\u03bc (closedBall (\u2191\u2191x) (r1 \u2191\u2191x))", "state_after": "no goals"}, {"tactic": "haveI : Encodable (S i) := (S_count i).toEncodable", "annotated_tactic": ["haveI : <a>Encodable</a> (S i) := (S_count i).<a>toEncodable</a>", [{"full_name": "Encodable", "def_path": "Mathlib/Logic/Encodable/Basic.lean", "def_pos": [45, 7], "def_end_pos": [45, 16]}, {"full_name": "Set.Countable.toEncodable", "def_path": "Mathlib/Data/Set/Countable.lean", "def_pos": [62, 15], "def_end_pos": [62, 36]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nA : \u2211' (x : \u2191t0), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\ni : Fin N\n\u22a2 \u2211' (x : \u2191(S i)), \u2191\u2191\u03bc (closedBall (\u2191\u2191x) (r1 \u2191\u2191x)) = \u2191\u2191\u03bc (\u22c3 x, closedBall (\u2191\u2191x) (r1 \u2191\u2191x))", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nA : \u2211' (x : \u2191t0), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\ni : Fin N\nthis : Encodable \u2191(S i)\n\u22a2 \u2211' (x : \u2191(S i)), \u2191\u2191\u03bc (closedBall (\u2191\u2191x) (r1 \u2191\u2191x)) = \u2191\u2191\u03bc (\u22c3 x, closedBall (\u2191\u2191x) (r1 \u2191\u2191x))"}, {"tactic": "rw [measure_iUnion]", "annotated_tactic": ["rw [<a>measure_iUnion</a>]", [{"full_name": "MeasureTheory.measure_iUnion", "def_path": "Mathlib/MeasureTheory/Measure/NullMeasurable.lean", "def_pos": [272, 9], "def_end_pos": [272, 23]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nA : \u2211' (x : \u2191t0), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\ni : Fin N\nthis : Encodable \u2191(S i)\n\u22a2 \u2211' (x : \u2191(S i)), \u2191\u2191\u03bc (closedBall (\u2191\u2191x) (r1 \u2191\u2191x)) = \u2191\u2191\u03bc (\u22c3 x, closedBall (\u2191\u2191x) (r1 \u2191\u2191x))", "state_after": "case hn\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nA : \u2211' (x : \u2191t0), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\ni : Fin N\nthis : Encodable \u2191(S i)\n\u22a2 Pairwise (Disjoint on fun x => closedBall (\u2191\u2191x) (r1 \u2191\u2191x))\n\ncase h\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nA : \u2211' (x : \u2191t0), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\ni : Fin N\nthis : Encodable \u2191(S i)\n\u22a2 \u2200 (i_1 : \u2191(S i)), MeasurableSet (closedBall (\u2191\u2191i_1) (r1 \u2191\u2191i_1))"}, {"tactic": "exact (pairwise_subtype_iff_pairwise_set _ _).2 (S_disj i)", "annotated_tactic": ["exact (<a>pairwise_subtype_iff_pairwise_set</a> _ _).2 (S_disj i)", [{"full_name": "pairwise_subtype_iff_pairwise_set", "def_path": "Mathlib/Data/Set/Pairwise/Basic.lean", "def_pos": [221, 9], "def_end_pos": [221, 42]}]], "state_before": "case hn\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nA : \u2211' (x : \u2191t0), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\ni : Fin N\nthis : Encodable \u2191(S i)\n\u22a2 Pairwise (Disjoint on fun x => closedBall (\u2191\u2191x) (r1 \u2191\u2191x))", "state_after": "no goals"}, {"tactic": "exact fun i => measurableSet_closedBall", "annotated_tactic": ["exact fun i => <a>measurableSet_closedBall</a>", [{"full_name": "measurableSet_closedBall", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [1681, 9], "def_end_pos": [1681, 33]}]], "state_before": "case h\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nA : \u2211' (x : \u2191t0), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\ni : Fin N\nthis : Encodable \u2191(S i)\n\u22a2 \u2200 (i_1 : \u2191(S i)), MeasurableSet (closedBall (\u2191\u2191i_1) (r1 \u2191\u2191i_1))", "state_after": "no goals"}, {"tactic": "apply measure_mono", "annotated_tactic": ["apply <a>measure_mono</a>", [{"full_name": "MeasureTheory.measure_mono", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [193, 9], "def_end_pos": [193, 21]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nA : \u2211' (x : \u2191t0), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\ni : Fin N\n\u22a2 \u2191\u2191\u03bc (\u22c3 x, closedBall (\u2191\u2191x) (r1 \u2191\u2191x)) \u2264 \u2191\u2191\u03bc v", "state_after": "case h\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nA : \u2211' (x : \u2191t0), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\ni : Fin N\n\u22a2 \u22c3 x, closedBall (\u2191\u2191x) (r1 \u2191\u2191x) \u2286 v"}, {"tactic": "simp only [SetCoe.forall, Subtype.coe_mk, iUnion_subset_iff]", "annotated_tactic": ["simp only [<a>SetCoe.forall</a>, <a>Subtype.coe_mk</a>, <a>iUnion_subset_iff</a>]", [{"full_name": "SetCoe.forall", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [185, 9], "def_end_pos": [185, 22]}, {"full_name": "Subtype.coe_mk", "def_path": "Mathlib/Data/Subtype.lean", "def_pos": [99, 9], "def_end_pos": [99, 15]}, {"full_name": "Set.iUnion_subset_iff", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [411, 9], "def_end_pos": [411, 26]}]], "state_before": "case h\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nA : \u2211' (x : \u2191t0), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\ni : Fin N\n\u22a2 \u22c3 x, closedBall (\u2191\u2191x) (r1 \u2191\u2191x) \u2286 v", "state_after": "case h\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nA : \u2211' (x : \u2191t0), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\ni : Fin N\n\u22a2 \u2200 (x : \u03b1) (h : x \u2208 s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)), { val := x, property := h } \u2208 S i \u2192 closedBall x (r1 x) \u2286 v"}, {"tactic": "intro x xs' _", "annotated_tactic": ["intro x xs' _", []], "state_before": "case h\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nA : \u2211' (x : \u2191t0), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\ni : Fin N\n\u22a2 \u2200 (x : \u03b1) (h : x \u2208 s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)), { val := x, property := h } \u2208 S i \u2192 closedBall x (r1 x) \u2286 v", "state_after": "case h\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nA : \u2211' (x : \u2191t0), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\ni : Fin N\nx : \u03b1\nxs' : x \u2208 s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\nh\u271d : { val := x, property := xs' } \u2208 S i\n\u22a2 closedBall x (r1 x) \u2286 v"}, {"tactic": "exact (hr1 x xs').2", "annotated_tactic": ["exact (hr1 x xs').2", []], "state_before": "case h\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nA : \u2211' (x : \u2191t0), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\ni : Fin N\nx : \u03b1\nxs' : x \u2208 s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\nh\u271d : { val := x, property := xs' } \u2208 S i\n\u22a2 closedBall x (r1 x) \u2286 v", "state_after": "no goals"}, {"tactic": "have : \u03bc s' = 0 := \u03bct0", "annotated_tactic": ["have : \u03bc s' = 0 := \u03bct0", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nA : \u2211' (x : \u2191t0), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\ni : Fin N\n\u22a2 \u2191\u2191\u03bc v \u2264 \u03b5 / 2 / \u2191N", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nA : \u2211' (x : \u2191t0), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\ni : Fin N\nthis : \u2191\u2191\u03bc s' = 0\n\u22a2 \u2191\u2191\u03bc v \u2264 \u03b5 / 2 / \u2191N"}, {"tactic": "rwa [this, zero_add] at \u03bcv", "annotated_tactic": ["rwa [this, <a>zero_add</a>] at \u03bcv", [{"full_name": "zero_add", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [463, 3], "def_end_pos": [463, 14]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nA : \u2211' (x : \u2191t0), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\ni : Fin N\nthis : \u2191\u2191\u03bc s' = 0\n\u22a2 \u2191\u2191\u03bc v \u2264 \u03b5 / 2 / \u2191N", "state_after": "no goals"}, {"tactic": "refine' add_le_add A _", "annotated_tactic": ["refine' <a>add_le_add</a> A _", [{"full_name": "add_le_add", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [205, 15], "def_end_pos": [205, 25]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nA : \u2211' (x : \u2191t0), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nB : \u2200 (i : Fin N), \u2211' (x : \u2191(Subtype.val '' S i)), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u03b5 / 2 / \u2191N\n\u22a2 \u2211' (x : \u2191t0), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) +\n      \u2211 i : Fin N, \u2211' (x : \u2191(Subtype.val '' S i)), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264\n    \u2191\u2191\u03bc s + \u03b5 / 2 + \u2211 i : Fin N, \u03b5 / 2 / \u2191N", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nA : \u2211' (x : \u2191t0), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nB : \u2200 (i : Fin N), \u2211' (x : \u2191(Subtype.val '' S i)), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u03b5 / 2 / \u2191N\n\u22a2 \u2211 i : Fin N, \u2211' (x : \u2191(Subtype.val '' S i)), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2211 i : Fin N, \u03b5 / 2 / \u2191N"}, {"tactic": "refine' Finset.sum_le_sum _", "annotated_tactic": ["refine' <a>Finset.sum_le_sum</a> _", [{"full_name": "Finset.sum_le_sum", "def_path": "Mathlib/Algebra/BigOperators/Order.lean", "def_pos": [111, 15], "def_end_pos": [111, 25]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nA : \u2211' (x : \u2191t0), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nB : \u2200 (i : Fin N), \u2211' (x : \u2191(Subtype.val '' S i)), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u03b5 / 2 / \u2191N\n\u22a2 \u2211 i : Fin N, \u2211' (x : \u2191(Subtype.val '' S i)), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2211 i : Fin N, \u03b5 / 2 / \u2191N", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nA : \u2211' (x : \u2191t0), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nB : \u2200 (i : Fin N), \u2211' (x : \u2191(Subtype.val '' S i)), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u03b5 / 2 / \u2191N\n\u22a2 \u2200 (i : Fin N), i \u2208 Finset.univ \u2192 \u2211' (x : \u2191(Subtype.val '' S i)), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u03b5 / 2 / \u2191N"}, {"tactic": "intro i _", "annotated_tactic": ["intro i _", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nA : \u2211' (x : \u2191t0), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nB : \u2200 (i : Fin N), \u2211' (x : \u2191(Subtype.val '' S i)), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u03b5 / 2 / \u2191N\n\u22a2 \u2200 (i : Fin N), i \u2208 Finset.univ \u2192 \u2211' (x : \u2191(Subtype.val '' S i)), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u03b5 / 2 / \u2191N", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nA : \u2211' (x : \u2191t0), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nB : \u2200 (i : Fin N), \u2211' (x : \u2191(Subtype.val '' S i)), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u03b5 / 2 / \u2191N\ni : Fin N\na\u271d : i \u2208 Finset.univ\n\u22a2 \u2211' (x : \u2191(Subtype.val '' S i)), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u03b5 / 2 / \u2191N"}, {"tactic": "exact B i", "annotated_tactic": ["exact B i", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nA : \u2211' (x : \u2191t0), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nB : \u2200 (i : Fin N), \u2211' (x : \u2191(Subtype.val '' S i)), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u03b5 / 2 / \u2191N\ni : Fin N\na\u271d : i \u2208 Finset.univ\n\u22a2 \u2211' (x : \u2191(Subtype.val '' S i)), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u03b5 / 2 / \u2191N", "state_after": "no goals"}, {"tactic": "refine' add_le_add le_rfl _", "annotated_tactic": ["refine' <a>add_le_add</a> <a>le_rfl</a> _", [{"full_name": "add_le_add", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [205, 15], "def_end_pos": [205, 25]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nA : \u2211' (x : \u2191t0), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nB : \u2200 (i : Fin N), \u2211' (x : \u2191(Subtype.val '' S i)), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u03b5 / 2 / \u2191N\n\u22a2 \u2191\u2191\u03bc s + \u03b5 / 2 + \u2211 i : Fin N, \u03b5 / 2 / \u2191N \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 + \u03b5 / 2", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nA : \u2211' (x : \u2191t0), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nB : \u2200 (i : Fin N), \u2211' (x : \u2191(Subtype.val '' S i)), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u03b5 / 2 / \u2191N\n\u22a2 \u2211 i : Fin N, \u03b5 / 2 / \u2191N \u2264 \u03b5 / 2"}, {"tactic": "simp only [Finset.card_fin, Finset.sum_const, nsmul_eq_mul, ENNReal.mul_div_le]", "annotated_tactic": ["simp only [<a>Finset.card_fin</a>, <a>Finset.sum_const</a>, <a>nsmul_eq_mul</a>, <a>ENNReal.mul_div_le</a>]", [{"full_name": "Finset.card_fin", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [322, 9], "def_end_pos": [322, 24]}, {"full_name": "Finset.sum_const", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [1440, 3], "def_end_pos": [1440, 14]}, {"full_name": "nsmul_eq_mul", "def_path": "Mathlib/Algebra/GroupPower/Lemmas.lean", "def_pos": [509, 9], "def_end_pos": [509, 21]}, {"full_name": "ENNReal.mul_div_le", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1750, 9], "def_end_pos": [1750, 19]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nA : \u2211' (x : \u2191t0), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nB : \u2200 (i : Fin N), \u2211' (x : \u2191(Subtype.val '' S i)), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u03b5 / 2 / \u2191N\n\u22a2 \u2211 i : Fin N, \u03b5 / 2 / \u2191N \u2264 \u03b5 / 2", "state_after": "no goals"}, {"tactic": "rw [add_assoc, ENNReal.add_halves]", "annotated_tactic": ["rw [<a>add_assoc</a>, <a>ENNReal.add_halves</a>]", [{"full_name": "add_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [263, 3], "def_end_pos": [263, 14]}, {"full_name": "ENNReal.add_halves", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1781, 19], "def_end_pos": [1781, 29]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nA : \u2211' (x : \u2191t0), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nB : \u2200 (i : Fin N), \u2211' (x : \u2191(Subtype.val '' S i)), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u03b5 / 2 / \u2191N\n\u22a2 \u2191\u2191\u03bc s + \u03b5 / 2 + \u03b5 / 2 = \u2191\u2191\u03bc s + \u03b5", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "full_name": "String.drop_empty", "start": [1110, 9], "end": [1110, 88], "traced_tactics": [{"tactic": "rw [drop_eq, List.drop_nil]", "annotated_tactic": ["rw [<a>drop_eq</a>, <a>List.drop_nil</a>]", [{"full_name": "String.drop_eq", "def_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "def_pos": [1105, 9], "def_end_pos": [1105, 16]}, {"full_name": "List.drop_nil", "def_path": "lake-packages/lean4/src/lean/Init/Data/List/Basic.lean", "def_pos": [480, 17], "def_end_pos": [480, 25]}]], "state_before": "n : Nat\n\u22a2 drop \"\" n = \"\"", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Part.lean", "full_name": "Part.sdiff_get_eq", "start": [875, 1], "end": [877, 29], "traced_tactics": [{"tactic": "simp [sdiff_def]", "annotated_tactic": ["simp [<a>sdiff_def</a>]", [{"full_name": "Part.sdiff_def", "def_path": "Mathlib/Data/Part.lean", "def_pos": [709, 9], "def_end_pos": [709, 18]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : SDiff \u03b1\na b : Part \u03b1\nhab : (a \\ b).Dom\n\u22a2 get (a \\ b) hab = get a (_ : a.Dom) \\ get b (_ : b.Dom)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : SDiff \u03b1\na b : Part \u03b1\nhab : (a \\ b).Dom\n\u22a2 get (Part.bind a fun y => map (fun x => y \\ x) b) (_ : (Part.bind a fun y => map (fun x => y \\ x) b).Dom) =\n    get a (_ : a.Dom) \\ get b (_ : b.Dom)"}, {"tactic": "aesop", "annotated_tactic": ["aesop", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : SDiff \u03b1\na b : Part \u03b1\nhab : (a \\ b).Dom\n\u22a2 get (Part.bind a fun y => map (fun x => y \\ x) b) (_ : (Part.bind a fun y => map (fun x => y \\ x) b).Dom) =\n    get a (_ : a.Dom) \\ get b (_ : b.Dom)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Primrec.lean", "full_name": "Primrec.pred", "start": [268, 1], "end": [269, 29], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "full_name": "MeasureTheory.L1.SimpleFunc.setToL1S_add_left", "start": [730, 1], "end": [732, 43], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "full_name": "Int.mul_emod", "start": [474, 1], "end": [478, 40], "traced_tactics": [{"tactic": "conv => lhs; rw [\n  \u2190 emod_add_ediv a n, \u2190 emod_add_ediv' b n, Int.add_mul, Int.mul_add, Int.mul_add,\n  Int.mul_assoc, Int.mul_assoc, \u2190 Int.mul_add n _ _, add_mul_emod_self_left,\n  \u2190 Int.mul_assoc, add_mul_emod_self]", "annotated_tactic": ["conv => lhs; rw [\n    \u2190 <a>emod_add_ediv</a> a n, \u2190 <a>emod_add_ediv'</a> b n, <a>Int.add_mul</a>, <a>Int.mul_add</a>, <a>Int.mul_add</a>,\n    <a>Int.mul_assoc</a>, <a>Int.mul_assoc</a>, \u2190 <a>Int.mul_add</a> n _ _, <a>add_mul_emod_self_left</a>,\n    \u2190 <a>Int.mul_assoc</a>, <a>add_mul_emod_self</a>]", [{"full_name": "Int.emod_add_ediv", "def_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "def_pos": [299, 9], "def_end_pos": [299, 22]}, {"full_name": "Int.emod_add_ediv'", "def_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "def_pos": [399, 9], "def_end_pos": [399, 23]}, {"full_name": "Int.add_mul", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [459, 19], "def_end_pos": [459, 26]}, {"full_name": "Int.mul_add", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [446, 19], "def_end_pos": [446, 26]}, {"full_name": "Int.mul_add", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [446, 19], "def_end_pos": [446, 26]}, {"full_name": "Int.mul_assoc", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [398, 19], "def_end_pos": [398, 28]}, {"full_name": "Int.mul_assoc", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [398, 19], "def_end_pos": [398, 28]}, {"full_name": "Int.mul_add", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [446, 19], "def_end_pos": [446, 26]}, {"full_name": "Int.add_mul_emod_self_left", "def_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "def_pos": [412, 17], "def_end_pos": [412, 39]}, {"full_name": "Int.mul_assoc", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [398, 19], "def_end_pos": [398, 28]}, {"full_name": "Int.add_mul_emod_self", "def_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "def_pos": [405, 17], "def_end_pos": [405, 34]}]], "state_before": "a b n : Int\n\u22a2 a * b % n = a % n * (b % n) % n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/BinomialHeap/Basic.lean", "full_name": "Std.BinomialHeap.Imp.Heap.realSize_tail?", "start": [260, 1], "end": [264, 54], "traced_tactics": [{"tactic": "simp only [Heap.tail?]", "annotated_tactic": ["simp only [<a>Heap.tail?</a>]", [{"full_name": "Std.BinomialHeap.Imp.Heap.tail?", "def_path": "lake-packages/std/Std/Data/BinomialHeap/Basic.lean", "def_pos": [200, 15], "def_end_pos": [200, 25]}]], "state_before": "\u03b1 : Type u_1\nle : \u03b1 \u2192 \u03b1 \u2192 Bool\ns' s : Heap \u03b1\n\u22a2 tail? le s = some s' \u2192 realSize s = realSize s' + 1", "state_after": "\u03b1 : Type u_1\nle : \u03b1 \u2192 \u03b1 \u2192 Bool\ns' s : Heap \u03b1\n\u22a2 Option.map (fun x => x.snd) (deleteMin le s) = some s' \u2192 realSize s = realSize s' + 1"}, {"tactic": "intro eq", "annotated_tactic": ["intro eq", []], "state_before": "\u03b1 : Type u_1\nle : \u03b1 \u2192 \u03b1 \u2192 Bool\ns' s : Heap \u03b1\n\u22a2 Option.map (fun x => x.snd) (deleteMin le s) = some s' \u2192 realSize s = realSize s' + 1", "state_after": "\u03b1 : Type u_1\nle : \u03b1 \u2192 \u03b1 \u2192 Bool\ns' s : Heap \u03b1\neq : Option.map (fun x => x.snd) (deleteMin le s) = some s'\n\u22a2 realSize s = realSize s' + 1"}, {"tactic": "match eq\u2082 : s.deleteMin le, eq with\n| some (a, tl), rfl => exact realSize_deleteMin eq\u2082", "annotated_tactic": ["match eq\u2082 : s.deleteMin le, eq with\n  | <a>some</a> (a, tl), <a>rfl</a> => exact <a>realSize_deleteMin</a> eq\u2082", [{"full_name": "Option.some", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2143, 5], "def_end_pos": [2143, 9]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}, {"full_name": "Std.BinomialHeap.Imp.Heap.realSize_deleteMin", "def_path": "lake-packages/std/Std/Data/BinomialHeap/Basic.lean", "def_pos": [248, 9], "def_end_pos": [248, 32]}]], "state_before": "\u03b1 : Type u_1\nle : \u03b1 \u2192 \u03b1 \u2192 Bool\ns' s : Heap \u03b1\neq : Option.map (fun x => x.snd) (deleteMin le s) = some s'\n\u22a2 realSize s = realSize s' + 1", "state_after": "no goals"}, {"tactic": "exact realSize_deleteMin eq\u2082", "annotated_tactic": ["exact <a>realSize_deleteMin</a> eq\u2082", [{"full_name": "Std.BinomialHeap.Imp.Heap.realSize_deleteMin", "def_path": "lake-packages/std/Std/Data/BinomialHeap/Basic.lean", "def_pos": [248, 9], "def_end_pos": [248, 32]}]], "state_before": "\u03b1 : Type u_1\nle : \u03b1 \u2192 \u03b1 \u2192 Bool\ns' s : Heap \u03b1\neq : Option.map (fun x => x.snd) (deleteMin le s) = some s'\na : \u03b1\ntl : Heap \u03b1\neq\u2082 : deleteMin le s = some (a, tl)\n\u22a2 realSize s = realSize ((fun x => x.snd) (a, tl)) + 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Part.lean", "full_name": "Part.div_get_eq", "start": [775, 1], "end": [777, 27], "traced_tactics": [{"tactic": "simp [div_def]", "annotated_tactic": ["simp [<a>div_def</a>]", [{"full_name": "Part.div_def", "def_path": "Mathlib/Data/Part.lean", "def_pos": [704, 9], "def_end_pos": [704, 16]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : Div \u03b1\na b : Part \u03b1\nhab : (a / b).Dom\n\u22a2 get (a / b) hab = get a (_ : a.Dom) / get b (_ : b.Dom)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : Div \u03b1\na b : Part \u03b1\nhab : (a / b).Dom\n\u22a2 get (Part.bind a fun y => map (fun x => y / x) b) (_ : (Part.bind a fun y => map (fun x => y / x) b).Dom) =\n    get a (_ : a.Dom) / get b (_ : b.Dom)"}, {"tactic": "aesop", "annotated_tactic": ["aesop", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : Div \u03b1\na b : Part \u03b1\nhab : (a / b).Dom\n\u22a2 get (Part.bind a fun y => map (fun x => y / x) b) (_ : (Part.bind a fun y => map (fun x => y / x) b).Dom) =\n    get a (_ : a.Dom) / get b (_ : b.Dom)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Semiquot.lean", "full_name": "Semiquot.pure_isPure", "start": [226, 1], "end": [229, 20], "traced_tactics": [{"tactic": "rw [mem_pure] at ab ac", "annotated_tactic": ["rw [<a>mem_pure</a>] at ab ac", [{"full_name": "Semiquot.mem_pure", "def_path": "Mathlib/Data/Semiquot.lean", "def_pos": [160, 9], "def_end_pos": [160, 17]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\na b : \u03b1\nab : b \u2208 pure a\nc : \u03b1\nac : c \u2208 pure a\n\u22a2 b = c", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\na b : \u03b1\nab : b = a\nc : \u03b1\nac : c = a\n\u22a2 b = c"}, {"tactic": "rwa [\u2190ac] at ab", "annotated_tactic": ["rwa [\u2190ac] at ab", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\na b : \u03b1\nab : b = a\nc : \u03b1\nac : c = a\n\u22a2 b = c", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Card.lean", "full_name": "Finset.card_le_card_sdiff_add_card", "start": [457, 1], "end": [458, 43], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Intervals/Pi.lean", "full_name": "Set.image_mulSingle_Icc_right", "start": [251, 1], "end": [253, 31], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/FundThmCalculus.lean", "full_name": "intervalIntegral.derivWithin_integral_left", "start": [973, 1], "end": [978, 62], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Vector/MapLemmas.lean", "full_name": "Vector.mapAccumr\u2082_eq_map\u2082_of_unused_state", "start": [297, 1], "end": [300, 95], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "full_name": "List.exists_of_set'", "start": [909, 1], "end": [911, 90], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "full_name": "MeasurableSpace.measurableSpace_iSup_eq", "start": [517, 1], "end": [521, 6], "traced_tactics": [{"tactic": "ext s", "annotated_tactic": ["ext s", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u_6\ns t u : Set \u03b1\nm : \u03b9 \u2192 MeasurableSpace \u03b1\n\u22a2 \u2a06 n, m n = generateFrom {s | \u2203 n, MeasurableSet s}", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u_6\ns\u271d t u : Set \u03b1\nm : \u03b9 \u2192 MeasurableSpace \u03b1\ns : Set \u03b1\n\u22a2 MeasurableSet s \u2194 MeasurableSet s"}, {"tactic": "rw [measurableSet_iSup]", "annotated_tactic": ["rw [<a>measurableSet_iSup</a>]", [{"full_name": "MeasurableSpace.measurableSet_iSup", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [512, 9], "def_end_pos": [512, 27]}]], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u_6\ns\u271d t u : Set \u03b1\nm : \u03b9 \u2192 MeasurableSpace \u03b1\ns : Set \u03b1\n\u22a2 MeasurableSet s \u2194 MeasurableSet s", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u_6\ns\u271d t u : Set \u03b1\nm : \u03b9 \u2192 MeasurableSpace \u03b1\ns : Set \u03b1\n\u22a2 GenerateMeasurable {s | \u2203 i, MeasurableSet s} s \u2194 MeasurableSet s"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u_6\ns\u271d t u : Set \u03b1\nm : \u03b9 \u2192 MeasurableSpace \u03b1\ns : Set \u03b1\n\u22a2 GenerateMeasurable {s | \u2203 i, MeasurableSet s} s \u2194 MeasurableSet s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/Layercake.lean", "full_name": "MeasureTheory.lintegral_comp_eq_lintegral_meas_le_mul_of_measurable", "start": [195, 1], "end": [381, 39], "traced_tactics": [{"tactic": "have f_nonneg : \u2200 \u03c9, 0 \u2264 f \u03c9 := fun \u03c9 \u21a6 f_nn \u03c9", "annotated_tactic": ["have f_nonneg : \u2200 \u03c9, 0 \u2264 f \u03c9 := fun \u03c9 \u21a6 f_nn \u03c9", []], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\n\u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =\n    \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)", "state_after": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\n\u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =\n    \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)"}, {"tactic": "by_cases H1 : g =\u1d50[volume.restrict (Ioi (0 : \u211d))] 0", "annotated_tactic": ["by_cases H1 : g =\u1d50[volume.restrict (<a>Ioi</a> (0 : \u211d))] 0", [{"full_name": "Set.Ioi", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [79, 5], "def_end_pos": [79, 8]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\n\u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =\n    \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)", "state_after": "case pos\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : g =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\n\u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =\n    \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)\n\ncase neg\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\n\u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =\n    \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)"}, {"tactic": "by_cases H2 : \u2203 s > 0, 0 < \u222b t in (0)..s, g t \u2227 \u03bc {a : \u03b1 | s < f a} = \u221e", "annotated_tactic": ["by_cases H2 : \u2203 s > 0, 0 < \u222b t in (0)..s, g t \u2227 \u03bc {a : \u03b1 | s < f a} = \u221e", []], "state_before": "case neg\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\n\u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =\n    \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)", "state_after": "case pos\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2203 s, s > 0 \u2227 0 < \u222b (t : \u211d) in 0 ..s, g t \u2227 \u2191\u2191\u03bc {a | s < f a} = \u22a4\n\u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =\n    \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)\n\ncase neg\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u00ac\u2203 s, s > 0 \u2227 0 < \u222b (t : \u211d) in 0 ..s, g t \u2227 \u2191\u2191\u03bc {a | s < f a} = \u22a4\n\u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =\n    \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)"}, {"tactic": "push_neg at H2", "annotated_tactic": ["push_neg at H2", []], "state_before": "case neg\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u00ac\u2203 s, s > 0 \u2227 0 < \u222b (t : \u211d) in 0 ..s, g t \u2227 \u2191\u2191\u03bc {a | s < f a} = \u22a4\n\u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =\n    \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)", "state_after": "case neg\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\n\u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =\n    \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)"}, {"tactic": "have M_bdd : BddAbove {s : \u211d | g =\u1d50[volume.restrict (Ioc (0 : \u211d) s)] 0} := by\n  contrapose! H1\n  have : \u2200 (n : \u2115), g =\u1d50[volume.restrict (Ioc (0 : \u211d) n)] 0 := by\n    intro n\n    rcases not_bddAbove_iff.1 H1 n with \u27e8s, hs, ns\u27e9\n    exact ae_restrict_of_ae_restrict_of_subset (Ioc_subset_Ioc_right ns.le) hs\n  have Hg : g =\u1d50[volume.restrict (\u22c3 (n : \u2115), (Ioc (0 : \u211d) n))] 0 :=\n    (ae_restrict_iUnion_iff _ _).2 this\n  have : (\u22c3 (n : \u2115), (Ioc (0 : \u211d) n)) = Ioi 0 :=\n    iUnion_Ioc_eq_Ioi_self_iff.2 (fun x _ \u21a6 exists_nat_ge x)\n  rwa [this] at Hg", "annotated_tactic": ["have M_bdd : <a>BddAbove</a> {s : \u211d | g =\u1d50[volume.restrict (<a>Ioc</a> (0 : \u211d) s)] 0} := by\n    contrapose! H1\n    have : \u2200 (n : \u2115), g =\u1d50[volume.restrict (<a>Ioc</a> (0 : \u211d) n)] 0 := by\n      intro n\n      rcases <a>not_bddAbove_iff</a>.1 H1 n with \u27e8s, hs, ns\u27e9\n      exact <a>ae_restrict_of_ae_restrict_of_subset</a> (<a>Ioc_subset_Ioc_right</a> ns.le) hs\n    have Hg : g =\u1d50[volume.restrict (\u22c3 (n : \u2115), (<a>Ioc</a> (0 : \u211d) n))] 0 :=\n      (<a>ae_restrict_iUnion_iff</a> _ _).2 this\n    have : (\u22c3 (n : \u2115), (<a>Ioc</a> (0 : \u211d) n)) = <a>Ioi</a> 0 :=\n      <a>iUnion_Ioc_eq_Ioi_self_iff</a>.2 (fun x _ \u21a6 <a>exists_nat_ge</a> x)\n    rwa [this] at Hg", [{"full_name": "BddAbove", "def_path": "Mathlib/Order/Bounds/Basic.lean", "def_pos": [56, 5], "def_end_pos": [56, 13]}, {"full_name": "Set.Ioc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [69, 5], "def_end_pos": [69, 8]}, {"full_name": "Set.Ioc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [69, 5], "def_end_pos": [69, 8]}, {"full_name": "not_bddAbove_iff", "def_path": "Mathlib/Order/Bounds/Basic.lean", "def_pos": [133, 9], "def_end_pos": [133, 25]}, {"full_name": "MeasureTheory.ae_restrict_of_ae_restrict_of_subset", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2606, 9], "def_end_pos": [2606, 45]}, {"full_name": "Set.Ioc_subset_Ioc_right", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [488, 9], "def_end_pos": [488, 29]}, {"full_name": "Set.Ioc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [69, 5], "def_end_pos": [69, 8]}, {"full_name": "MeasureTheory.ae_restrict_iUnion_iff", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2514, 9], "def_end_pos": [2514, 31]}, {"full_name": "Set.Ioc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [69, 5], "def_end_pos": [69, 8]}, {"full_name": "Set.Ioi", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [79, 5], "def_end_pos": [79, 8]}, {"full_name": "Set.iUnion_Ioc_eq_Ioi_self_iff", "def_path": "Mathlib/Data/Set/Intervals/Disjoint.lean", "def_pos": [164, 9], "def_end_pos": [164, 35]}, {"full_name": "exists_nat_ge", "def_path": "Mathlib/Algebra/Order/Archimedean.lean", "def_pos": [122, 9], "def_end_pos": [122, 22]}]], "state_before": "case neg\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\n\u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =\n    \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)", "state_after": "case neg\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\n\u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =\n    \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)"}, {"tactic": "let M : \u211d := sSup {s : \u211d | g =\u1d50[volume.restrict (Ioc (0 : \u211d) s)] 0}", "annotated_tactic": ["let M : \u211d := <a>sSup</a> {s : \u211d | g =\u1d50[volume.restrict (<a>Ioc</a> (0 : \u211d) s)] 0}", [{"full_name": "SupSet.sSup", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [55, 3], "def_end_pos": [55, 7]}, {"full_name": "Set.Ioc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [69, 5], "def_end_pos": [69, 8]}]], "state_before": "case neg\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\n\u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =\n    \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)", "state_after": "case neg\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\n\u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =\n    \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)"}, {"tactic": "have zero_mem : 0 \u2208 {s : \u211d | g =\u1d50[volume.restrict (Ioc (0 : \u211d) s)] 0} := by simpa using trivial", "annotated_tactic": ["have zero_mem : 0 \u2208 {s : \u211d | g =\u1d50[volume.restrict (<a>Ioc</a> (0 : \u211d) s)] 0} := by simpa using <a>trivial</a>", [{"full_name": "Set.Ioc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [69, 5], "def_end_pos": [69, 8]}, {"full_name": "trivial", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [514, 31], "def_end_pos": [514, 38]}]], "state_before": "case neg\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\n\u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =\n    \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)", "state_after": "case neg\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\n\u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =\n    \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)"}, {"tactic": "have M_nonneg : 0 \u2264 M := le_csSup M_bdd zero_mem", "annotated_tactic": ["have M_nonneg : 0 \u2264 M := <a>le_csSup</a> M_bdd zero_mem", [{"full_name": "le_csSup", "def_path": "Mathlib/Order/ConditionallyCompleteLattice/Basic.lean", "def_pos": [457, 9], "def_end_pos": [457, 17]}]], "state_before": "case neg\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\n\u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =\n    \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)", "state_after": "case neg\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\n\u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =\n    \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)"}, {"tactic": "have hgM : g =\u1d50[volume.restrict (Ioc (0 : \u211d) M)] 0 := by\n  rw [\u2190 restrict_Ioo_eq_restrict_Ioc]\n  obtain \u27e8u, -, uM, ulim\u27e9 : \u2203 u, StrictMono u \u2227 (\u2200 (n : \u2115), u n < M) \u2227 Tendsto u atTop (\ud835\udcdd M) :=\n    exists_seq_strictMono_tendsto M\n  have I : \u2200 n, g =\u1d50[volume.restrict (Ioc (0 : \u211d) (u n))] 0 := by\n    intro n\n    obtain \u27e8s, hs, uns\u27e9 : \u2203 s, g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0 \u2227 u n < s :=\n      exists_lt_of_lt_csSup (Set.nonempty_of_mem zero_mem) (uM n)\n    exact ae_restrict_of_ae_restrict_of_subset (Ioc_subset_Ioc_right uns.le) hs\n  have : g =\u1d50[volume.restrict (\u22c3 n, Ioc (0 : \u211d) (u n))] 0 := (ae_restrict_iUnion_iff _ _).2 I\n  apply ae_restrict_of_ae_restrict_of_subset _ this\n  rintro x \u27e8x_pos, xM\u27e9\n  obtain \u27e8n, hn\u27e9 : \u2203 n, x < u n := ((tendsto_order.1 ulim).1 _ xM).exists\n  exact mem_iUnion.2 \u27e8n, \u27e8x_pos, hn.le\u27e9\u27e9", "annotated_tactic": ["have hgM : g =\u1d50[volume.restrict (<a>Ioc</a> (0 : \u211d) M)] 0 := by\n    rw [\u2190 <a>restrict_Ioo_eq_restrict_Ioc</a>]\n    obtain \u27e8u, -, uM, ulim\u27e9 : \u2203 u, <a>StrictMono</a> u \u2227 (\u2200 (n : \u2115), u n < M) \u2227 <a>Tendsto</a> u <a>atTop</a> (\ud835\udcdd M) :=\n      <a>exists_seq_strictMono_tendsto</a> M\n    have I : \u2200 n, g =\u1d50[volume.restrict (<a>Ioc</a> (0 : \u211d) (u n))] 0 := by\n      intro n\n      obtain \u27e8s, hs, uns\u27e9 : \u2203 s, g =\u1da0[<a>ae</a> (<a>Measure.restrict</a> <a>volume</a> (<a>Ioc</a> 0 s))] 0 \u2227 u n < s :=\n        <a>exists_lt_of_lt_csSup</a> (<a>Set.nonempty_of_mem</a> zero_mem) (uM n)\n      exact <a>ae_restrict_of_ae_restrict_of_subset</a> (<a>Ioc_subset_Ioc_right</a> uns.le) hs\n    have : g =\u1d50[volume.restrict (\u22c3 n, <a>Ioc</a> (0 : \u211d) (u n))] 0 := (<a>ae_restrict_iUnion_iff</a> _ _).2 I\n    apply <a>ae_restrict_of_ae_restrict_of_subset</a> _ this\n    rintro x \u27e8x_pos, xM\u27e9\n    obtain \u27e8n, hn\u27e9 : \u2203 n, x < u n := ((<a>tendsto_order</a>.1 ulim).1 _ xM).<a>exists</a>\n    exact <a>mem_iUnion</a>.2 \u27e8n, \u27e8x_pos, hn.le\u27e9\u27e9", [{"full_name": "Set.Ioc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [69, 5], "def_end_pos": [69, 8]}, {"full_name": "MeasureTheory.restrict_Ioo_eq_restrict_Ioc", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3198, 9], "def_end_pos": [3198, 37]}, {"full_name": "StrictMono", "def_path": "Mathlib/Order/Monotone/Basic.lean", "def_pos": [97, 5], "def_end_pos": [97, 15]}, {"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2939, 5], "def_end_pos": [2939, 12]}, {"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}, {"full_name": "exists_seq_strictMono_tendsto", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [2219, 9], "def_end_pos": [2219, 38]}, {"full_name": "Set.Ioc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [69, 5], "def_end_pos": [69, 8]}, {"full_name": "MeasureTheory.Measure.ae", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [365, 5], "def_end_pos": [365, 15]}, {"full_name": "MeasureTheory.Measure.restrict", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1503, 5], "def_end_pos": [1503, 13]}, {"full_name": "MeasureTheory.MeasureSpace.volume", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [663, 3], "def_end_pos": [663, 9]}, {"full_name": "Set.Ioc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [69, 5], "def_end_pos": [69, 8]}, {"full_name": "exists_lt_of_lt_csSup", "def_path": "Mathlib/Order/ConditionallyCompleteLattice/Basic.lean", "def_pos": [999, 9], "def_end_pos": [999, 30]}, {"full_name": "Set.nonempty_of_mem", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [458, 9], "def_end_pos": [458, 24]}, {"full_name": "MeasureTheory.ae_restrict_of_ae_restrict_of_subset", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2606, 9], "def_end_pos": [2606, 45]}, {"full_name": "Set.Ioc_subset_Ioc_right", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [488, 9], "def_end_pos": [488, 29]}, {"full_name": "Set.Ioc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [69, 5], "def_end_pos": [69, 8]}, {"full_name": "MeasureTheory.ae_restrict_iUnion_iff", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2514, 9], "def_end_pos": [2514, 31]}, {"full_name": "MeasureTheory.ae_restrict_of_ae_restrict_of_subset", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2606, 9], "def_end_pos": [2606, 45]}, {"full_name": "tendsto_order", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [919, 9], "def_end_pos": [919, 22]}, {"full_name": "Filter.Eventually.exists", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1308, 9], "def_end_pos": [1308, 26]}, {"full_name": "Set.mem_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [201, 9], "def_end_pos": [201, 19]}]], "state_before": "case neg\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\n\u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =\n    \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)", "state_after": "case neg\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nhgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0\n\u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =\n    \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)"}, {"tactic": "let \u03bd := \u03bc.restrict {a : \u03b1 | M < f a}", "annotated_tactic": ["let \u03bd := \u03bc.restrict {a : \u03b1 | M < f a}", []], "state_before": "case neg\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nhgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0\n\u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =\n    \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)", "state_after": "case neg\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nhgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0\n\u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a}\n\u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =\n    \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)"}, {"tactic": "have A : \u222b\u207b \u03c9, ENNReal.ofReal (\u222b t in (0)..f \u03c9, g t) \u2202\u03bc\n       = \u222b\u207b \u03c9, ENNReal.ofReal (\u222b t in (0)..f \u03c9, g t) \u2202\u03bd := by\n  have meas : MeasurableSet {a | M < f a} := measurableSet_lt measurable_const f_mble\n  have I : \u222b\u207b \u03c9 in {a | M < f a}\u1d9c, ENNReal.ofReal (\u222b t in (0).. f \u03c9, g t) \u2202\u03bc\n           = \u222b\u207b _ in {a | M < f a}\u1d9c, 0 \u2202\u03bc := by\n    apply set_lintegral_congr_fun meas.compl (eventually_of_forall (fun s hs \u21a6 ?_))\n    have : \u222b (t : \u211d) in (0)..f s, g t = \u222b (t : \u211d) in (0)..f s, 0 := by\n      simp_rw [intervalIntegral.integral_of_le (f_nonneg s)]\n      apply integral_congr_ae\n      apply ae_mono (restrict_mono ?_ le_rfl) hgM\n      apply Ioc_subset_Ioc_right\n      simpa using hs\n    simp [this]\n  simp only [lintegral_const, zero_mul] at I\n  rw [\u2190 lintegral_add_compl _ meas, I, add_zero]", "annotated_tactic": ["have A : \u222b\u207b \u03c9, <a>ENNReal.ofReal</a> (\u222b t in (0)..f \u03c9, g t) \u2202\u03bc\n         = \u222b\u207b \u03c9, <a>ENNReal.ofReal</a> (\u222b t in (0)..f \u03c9, g t) \u2202\u03bd := by\n    have meas : <a>MeasurableSet</a> {a | M < f a} := <a>measurableSet_lt</a> <a>measurable_const</a> f_mble\n    have I : \u222b\u207b \u03c9 in {a | M < f a}\u1d9c, <a>ENNReal.ofReal</a> (\u222b t in (0).. f \u03c9, g t) \u2202\u03bc\n             = \u222b\u207b _ in {a | M < f a}\u1d9c, 0 \u2202\u03bc := by\n      apply <a>set_lintegral_congr_fun</a> meas.compl (<a>eventually_of_forall</a> (fun s hs \u21a6 ?_))\n      have : \u222b (t : \u211d) in (0)..f s, g t = \u222b (t : \u211d) in (0)..f s, 0 := by\n        simp_rw [<a>intervalIntegral.integral_of_le</a> (f_nonneg s)]\n        apply <a>integral_congr_ae</a>\n        apply <a>ae_mono</a> (<a>restrict_mono</a> ?_ <a>le_rfl</a>) hgM\n        apply <a>Ioc_subset_Ioc_right</a>\n        simpa using hs\n      simp [this]\n    simp only [<a>lintegral_const</a>, <a>zero_mul</a>] at I\n    rw [\u2190 <a>lintegral_add_compl</a> _ meas, I, <a>add_zero</a>]", [{"full_name": "ENNReal.ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [172, 29], "def_end_pos": [172, 35]}, {"full_name": "ENNReal.ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [172, 29], "def_end_pos": [172, 35]}, {"full_name": "MeasurableSet", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [64, 5], "def_end_pos": [64, 18]}, {"full_name": "measurableSet_lt", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [616, 9], "def_end_pos": [616, 25]}, {"full_name": "measurable_const", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [570, 9], "def_end_pos": [570, 25]}, {"full_name": "ENNReal.ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [172, 29], "def_end_pos": [172, 35]}, {"full_name": "MeasureTheory.set_lintegral_congr_fun", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [316, 9], "def_end_pos": [316, 32]}, {"full_name": "Filter.eventually_of_forall", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1111, 9], "def_end_pos": [1111, 29]}, {"full_name": "intervalIntegral.integral_of_le", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [465, 9], "def_end_pos": [465, 23]}, {"full_name": "MeasureTheory.integral_congr_ae", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [938, 9], "def_end_pos": [938, 26]}, {"full_name": "MeasureTheory.ae_mono", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2456, 9], "def_end_pos": [2456, 16]}, {"full_name": "MeasureTheory.Measure.restrict_mono", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1550, 9], "def_end_pos": [1550, 22]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}, {"full_name": "Set.Ioc_subset_Ioc_right", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [488, 9], "def_end_pos": [488, 29]}, {"full_name": "MeasureTheory.lintegral_const", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [136, 9], "def_end_pos": [136, 24]}, {"full_name": "MulZeroClass.zero_mul", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [36, 3], "def_end_pos": [36, 11]}, {"full_name": "MeasureTheory.lintegral_add_compl", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [1258, 9], "def_end_pos": [1258, 28]}, {"full_name": "add_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [469, 3], "def_end_pos": [469, 14]}]], "state_before": "case neg\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nhgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0\n\u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a}\nthis : SigmaFinite \u03bd\n\u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =\n    \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)", "state_after": "case neg\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nhgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0\n\u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a}\nthis : SigmaFinite \u03bd\nA :\n  \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc = \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bd\n\u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =\n    \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)"}, {"tactic": "have B : \u222b\u207b t in Ioi 0, \u03bc {a : \u03b1 | t \u2264 f a} * ENNReal.ofReal (g t)\n         = \u222b\u207b t in Ioi 0, \u03bd {a : \u03b1 | t \u2264 f a} * ENNReal.ofReal (g t) := by\n  have B1 : \u222b\u207b t in Ioc 0 M, \u03bc {a : \u03b1 | t \u2264 f a} * ENNReal.ofReal (g t)\n       = \u222b\u207b t in Ioc 0 M, \u03bd {a : \u03b1 | t \u2264 f a} * ENNReal.ofReal (g t) := by\n    apply lintegral_congr_ae\n    filter_upwards [hgM] with t ht\n    simp [ht]\n  have B2 : \u222b\u207b t in Ioi M, \u03bc {a : \u03b1 | t \u2264 f a} * ENNReal.ofReal (g t)\n            = \u222b\u207b t in Ioi M, \u03bd {a : \u03b1 | t \u2264 f a} * ENNReal.ofReal (g t) := by\n    apply set_lintegral_congr_fun measurableSet_Ioi (eventually_of_forall (fun t ht \u21a6 ?_))\n    rw [Measure.restrict_apply (measurableSet_le measurable_const f_mble)]\n    congr 3\n    exact (inter_eq_left.2 (fun a ha \u21a6 (mem_Ioi.1 ht).trans_le ha)).symm\n  have I : Ioi (0 : \u211d) = Ioc (0 : \u211d) M \u222a Ioi M := (Ioc_union_Ioi_eq_Ioi M_nonneg).symm\n  have J : Disjoint (Ioc 0 M) (Ioi M) := Ioc_disjoint_Ioi le_rfl\n  rw [I, lintegral_union measurableSet_Ioi J, lintegral_union measurableSet_Ioi J, B1, B2]", "annotated_tactic": ["have B : \u222b\u207b t in <a>Ioi</a> 0, \u03bc {a : \u03b1 | t \u2264 f a} * <a>ENNReal.ofReal</a> (g t)\n           = \u222b\u207b t in <a>Ioi</a> 0, \u03bd {a : \u03b1 | t \u2264 f a} * <a>ENNReal.ofReal</a> (g t) := by\n    have B1 : \u222b\u207b t in <a>Ioc</a> 0 M, \u03bc {a : \u03b1 | t \u2264 f a} * <a>ENNReal.ofReal</a> (g t)\n         = \u222b\u207b t in <a>Ioc</a> 0 M, \u03bd {a : \u03b1 | t \u2264 f a} * <a>ENNReal.ofReal</a> (g t) := by\n      apply <a>lintegral_congr_ae</a>\n      filter_upwards [hgM] with t ht\n      simp [ht]\n    have B2 : \u222b\u207b t in <a>Ioi</a> M, \u03bc {a : \u03b1 | t \u2264 f a} * <a>ENNReal.ofReal</a> (g t)\n              = \u222b\u207b t in <a>Ioi</a> M, \u03bd {a : \u03b1 | t \u2264 f a} * <a>ENNReal.ofReal</a> (g t) := by\n      apply <a>set_lintegral_congr_fun</a> <a>measurableSet_Ioi</a> (<a>eventually_of_forall</a> (fun t ht \u21a6 ?_))\n      rw [<a>Measure.restrict_apply</a> (<a>measurableSet_le</a> <a>measurable_const</a> f_mble)]\n      congr 3\n      exact (<a>inter_eq_left</a>.2 (fun a ha \u21a6 (<a>mem_Ioi</a>.1 ht).<a>trans_le</a> ha)).<a>symm</a>\n    have I : <a>Ioi</a> (0 : \u211d) = <a>Ioc</a> (0 : \u211d) M \u222a <a>Ioi</a> M := (<a>Ioc_union_Ioi_eq_Ioi</a> M_nonneg).<a>symm</a>\n    have J : <a>Disjoint</a> (<a>Ioc</a> 0 M) (<a>Ioi</a> M) := <a>Ioc_disjoint_Ioi</a> <a>le_rfl</a>\n    rw [I, <a>lintegral_union</a> <a>measurableSet_Ioi</a> J, <a>lintegral_union</a> <a>measurableSet_Ioi</a> J, B1, B2]", [{"full_name": "Set.Ioi", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [79, 5], "def_end_pos": [79, 8]}, {"full_name": "ENNReal.ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [172, 29], "def_end_pos": [172, 35]}, {"full_name": "Set.Ioi", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [79, 5], "def_end_pos": [79, 8]}, {"full_name": "ENNReal.ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [172, 29], "def_end_pos": [172, 35]}, {"full_name": "Set.Ioc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [69, 5], "def_end_pos": [69, 8]}, {"full_name": "ENNReal.ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [172, 29], "def_end_pos": [172, 35]}, {"full_name": "Set.Ioc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [69, 5], "def_end_pos": [69, 8]}, {"full_name": "ENNReal.ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [172, 29], "def_end_pos": [172, 35]}, {"full_name": "MeasureTheory.lintegral_congr_ae", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [304, 9], "def_end_pos": [304, 27]}, {"full_name": "Set.Ioi", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [79, 5], "def_end_pos": [79, 8]}, {"full_name": "ENNReal.ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [172, 29], "def_end_pos": [172, 35]}, {"full_name": "Set.Ioi", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [79, 5], "def_end_pos": [79, 8]}, {"full_name": "ENNReal.ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [172, 29], "def_end_pos": [172, 35]}, {"full_name": "MeasureTheory.set_lintegral_congr_fun", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [316, 9], "def_end_pos": [316, 32]}, {"full_name": "measurableSet_Ioi", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [579, 9], "def_end_pos": [579, 26]}, {"full_name": "Filter.eventually_of_forall", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1111, 9], "def_end_pos": [1111, 29]}, {"full_name": "MeasureTheory.Measure.restrict_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1533, 9], "def_end_pos": [1533, 23]}, {"full_name": "measurableSet_le", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [559, 9], "def_end_pos": [559, 25]}, {"full_name": "measurable_const", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [570, 9], "def_end_pos": [570, 25]}, {"full_name": "Set.inter_eq_left", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [981, 15], "def_end_pos": [981, 28]}, {"full_name": "Set.mem_Ioi", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [151, 9], "def_end_pos": [151, 16]}, {"full_name": "LT.lt.trans_le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [148, 7], "def_end_pos": [148, 21]}, {"full_name": "Eq.symm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [310, 9], "def_end_pos": [310, 16]}, {"full_name": "Set.Ioi", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [79, 5], "def_end_pos": [79, 8]}, {"full_name": "Set.Ioc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [69, 5], "def_end_pos": [69, 8]}, {"full_name": "Set.Ioi", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [79, 5], "def_end_pos": [79, 8]}, {"full_name": "Set.Ioc_union_Ioi_eq_Ioi", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [1317, 9], "def_end_pos": [1317, 29]}, {"full_name": "Eq.symm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [310, 9], "def_end_pos": [310, 16]}, {"full_name": "Disjoint", "def_path": "Mathlib/Order/Disjoint.lean", "def_pos": [41, 5], "def_end_pos": [41, 13]}, {"full_name": "Set.Ioc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [69, 5], "def_end_pos": [69, 8]}, {"full_name": "Set.Ioi", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [79, 5], "def_end_pos": [79, 8]}, {"full_name": "Set.Ioc_disjoint_Ioi", "def_path": "Mathlib/Data/Set/Intervals/Disjoint.lean", "def_pos": [64, 9], "def_end_pos": [64, 25]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}, {"full_name": "MeasureTheory.lintegral_union", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [1243, 9], "def_end_pos": [1243, 24]}, {"full_name": "measurableSet_Ioi", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [579, 9], "def_end_pos": [579, 26]}, {"full_name": "MeasureTheory.lintegral_union", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [1243, 9], "def_end_pos": [1243, 24]}, {"full_name": "measurableSet_Ioi", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [579, 9], "def_end_pos": [579, 26]}]], "state_before": "case neg\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nhgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0\n\u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a}\nthis : SigmaFinite \u03bd\nA :\n  \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc = \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bd\n\u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =\n    \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)", "state_after": "case neg\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nhgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0\n\u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a}\nthis : SigmaFinite \u03bd\nA :\n  \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc = \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bd\nB :\n  \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) =\n    \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bd {a | t \u2264 f a} * ENNReal.ofReal (g t)\n\u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =\n    \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)"}, {"tactic": "rw [A, B]", "annotated_tactic": ["rw [A, B]", []], "state_before": "case neg\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nhgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0\n\u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a}\nthis : SigmaFinite \u03bd\nA :\n  \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc = \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bd\nB :\n  \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) =\n    \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bd {a | t \u2264 f a} * ENNReal.ofReal (g t)\n\u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =\n    \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)", "state_after": "case neg\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nhgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0\n\u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a}\nthis : SigmaFinite \u03bd\nA :\n  \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc = \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bd\nB :\n  \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) =\n    \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bd {a | t \u2264 f a} * ENNReal.ofReal (g t)\n\u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bd =\n    \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bd {a | t \u2264 f a} * ENNReal.ofReal (g t)"}, {"tactic": "exact lintegral_comp_eq_lintegral_meas_le_mul_of_measurable_of_sigmaFinite\n  \u03bd f_nn f_mble g_intble g_mble g_nn", "annotated_tactic": ["exact <a>lintegral_comp_eq_lintegral_meas_le_mul_of_measurable_of_sigmaFinite</a>\n    \u03bd f_nn f_mble g_intble g_mble g_nn", [{"full_name": "MeasureTheory.lintegral_comp_eq_lintegral_meas_le_mul_of_measurable_of_sigmaFinite", "def_path": "Mathlib/MeasureTheory/Integral/Layercake.lean", "def_pos": [105, 9], "def_end_pos": [105, 77]}]], "state_before": "case neg\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nhgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0\n\u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a}\nthis : SigmaFinite \u03bd\nA :\n  \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc = \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bd\nB :\n  \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) =\n    \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bd {a | t \u2264 f a} * ENNReal.ofReal (g t)\n\u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bd =\n    \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bd {a | t \u2264 f a} * ENNReal.ofReal (g t)", "state_after": "no goals"}, {"tactic": "have A : \u222b\u207b \u03c9, ENNReal.ofReal (\u222b t in (0)..f \u03c9, g t) \u2202\u03bc = 0 := by\n  have : \u2200 \u03c9, \u222b t in (0)..f \u03c9, g t = \u222b t in (0)..f \u03c9, 0 := by\n    intro \u03c9\n    simp_rw [intervalIntegral.integral_of_le (f_nonneg \u03c9)]\n    apply integral_congr_ae\n    exact ae_restrict_of_ae_restrict_of_subset Ioc_subset_Ioi_self H1\n  simp [this]", "annotated_tactic": ["have A : \u222b\u207b \u03c9, <a>ENNReal.ofReal</a> (\u222b t in (0)..f \u03c9, g t) \u2202\u03bc = 0 := by\n      have : \u2200 \u03c9, \u222b t in (0)..f \u03c9, g t = \u222b t in (0)..f \u03c9, 0 := by\n        intro \u03c9\n        simp_rw [<a>intervalIntegral.integral_of_le</a> (f_nonneg \u03c9)]\n        apply <a>integral_congr_ae</a>\n        exact <a>ae_restrict_of_ae_restrict_of_subset</a> <a>Ioc_subset_Ioi_self</a> H1\n      simp [this]", [{"full_name": "ENNReal.ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [172, 29], "def_end_pos": [172, 35]}, {"full_name": "intervalIntegral.integral_of_le", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [465, 9], "def_end_pos": [465, 23]}, {"full_name": "MeasureTheory.integral_congr_ae", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [938, 9], "def_end_pos": [938, 26]}, {"full_name": "MeasureTheory.ae_restrict_of_ae_restrict_of_subset", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2606, 9], "def_end_pos": [2606, 45]}, {"full_name": "Set.Ioc_subset_Ioi_self", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [526, 9], "def_end_pos": [526, 28]}]], "state_before": "case pos\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : g =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\n\u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =\n    \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)", "state_after": "case pos\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : g =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nA : \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc = 0\n\u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =\n    \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)"}, {"tactic": "have B : \u222b\u207b t in Ioi 0, \u03bc {a : \u03b1 | t \u2264 f a} * ENNReal.ofReal (g t) = 0 := by\n  have : (fun t \u21a6 \u03bc {a : \u03b1 | t \u2264 f a} * ENNReal.ofReal (g t))\n    =\u1d50[volume.restrict (Ioi (0:\u211d))] 0 := by\n      filter_upwards [H1] with t ht using by simp [ht]\n  simp [lintegral_congr_ae this]", "annotated_tactic": ["have B : \u222b\u207b t in <a>Ioi</a> 0, \u03bc {a : \u03b1 | t \u2264 f a} * <a>ENNReal.ofReal</a> (g t) = 0 := by\n      have : (fun t \u21a6 \u03bc {a : \u03b1 | t \u2264 f a} * <a>ENNReal.ofReal</a> (g t))\n        =\u1d50[volume.restrict (<a>Ioi</a> (0:\u211d))] 0 := by\n          filter_upwards [H1] with t ht using by simp [ht]\n      simp [<a>lintegral_congr_ae</a> this]", [{"full_name": "Set.Ioi", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [79, 5], "def_end_pos": [79, 8]}, {"full_name": "ENNReal.ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [172, 29], "def_end_pos": [172, 35]}, {"full_name": "ENNReal.ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [172, 29], "def_end_pos": [172, 35]}, {"full_name": "Set.Ioi", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [79, 5], "def_end_pos": [79, 8]}, {"full_name": "MeasureTheory.lintegral_congr_ae", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [304, 9], "def_end_pos": [304, 27]}]], "state_before": "case pos\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : g =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nA : \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc = 0\n\u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =\n    \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)", "state_after": "case pos\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : g =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nA : \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc = 0\nB : \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) = 0\n\u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =\n    \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)"}, {"tactic": "rw [A, B]", "annotated_tactic": ["rw [A, B]", []], "state_before": "case pos\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : g =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nA : \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc = 0\nB : \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) = 0\n\u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =\n    \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)", "state_after": "no goals"}, {"tactic": "have : \u2200 \u03c9, \u222b t in (0)..f \u03c9, g t = \u222b t in (0)..f \u03c9, 0 := by\n  intro \u03c9\n  simp_rw [intervalIntegral.integral_of_le (f_nonneg \u03c9)]\n  apply integral_congr_ae\n  exact ae_restrict_of_ae_restrict_of_subset Ioc_subset_Ioi_self H1", "annotated_tactic": ["have : \u2200 \u03c9, \u222b t in (0)..f \u03c9, g t = \u222b t in (0)..f \u03c9, 0 := by\n        intro \u03c9\n        simp_rw [<a>intervalIntegral.integral_of_le</a> (f_nonneg \u03c9)]\n        apply <a>integral_congr_ae</a>\n        exact <a>ae_restrict_of_ae_restrict_of_subset</a> <a>Ioc_subset_Ioi_self</a> H1", [{"full_name": "intervalIntegral.integral_of_le", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [465, 9], "def_end_pos": [465, 23]}, {"full_name": "MeasureTheory.integral_congr_ae", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [938, 9], "def_end_pos": [938, 26]}, {"full_name": "MeasureTheory.ae_restrict_of_ae_restrict_of_subset", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2606, 9], "def_end_pos": [2606, 45]}, {"full_name": "Set.Ioc_subset_Ioi_self", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [526, 9], "def_end_pos": [526, 28]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : g =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\n\u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc = 0", "state_after": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : g =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nthis : \u2200 (\u03c9 : \u03b1), \u222b (t : \u211d) in 0 ..f \u03c9, g t = \u222b (t : \u211d) in 0 ..f \u03c9, 0\n\u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc = 0"}, {"tactic": "simp [this]", "annotated_tactic": ["simp [this]", []], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : g =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nthis : \u2200 (\u03c9 : \u03b1), \u222b (t : \u211d) in 0 ..f \u03c9, g t = \u222b (t : \u211d) in 0 ..f \u03c9, 0\n\u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc = 0", "state_after": "no goals"}, {"tactic": "intro \u03c9", "annotated_tactic": ["intro \u03c9", []], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : g =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\n\u22a2 \u2200 (\u03c9 : \u03b1), \u222b (t : \u211d) in 0 ..f \u03c9, g t = \u222b (t : \u211d) in 0 ..f \u03c9, 0", "state_after": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : g =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\n\u03c9 : \u03b1\n\u22a2 \u222b (t : \u211d) in 0 ..f \u03c9, g t = \u222b (t : \u211d) in 0 ..f \u03c9, 0"}, {"tactic": "simp_rw [intervalIntegral.integral_of_le (f_nonneg \u03c9)]", "annotated_tactic": ["simp_rw [<a>intervalIntegral.integral_of_le</a> (f_nonneg \u03c9)]", [{"full_name": "intervalIntegral.integral_of_le", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [465, 9], "def_end_pos": [465, 23]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : g =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\n\u03c9 : \u03b1\n\u22a2 \u222b (t : \u211d) in 0 ..f \u03c9, g t = \u222b (t : \u211d) in 0 ..f \u03c9, 0", "state_after": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : g =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\n\u03c9 : \u03b1\n\u22a2 \u222b (t : \u211d) in Ioc 0 (f \u03c9), g t = \u222b (t : \u211d) in Ioc 0 (f \u03c9), 0"}, {"tactic": "apply integral_congr_ae", "annotated_tactic": ["apply <a>integral_congr_ae</a>", [{"full_name": "MeasureTheory.integral_congr_ae", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [938, 9], "def_end_pos": [938, 26]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : g =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\n\u03c9 : \u03b1\n\u22a2 \u222b (t : \u211d) in Ioc 0 (f \u03c9), g t = \u222b (t : \u211d) in Ioc 0 (f \u03c9), 0", "state_after": "case h\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : g =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\n\u03c9 : \u03b1\n\u22a2 (fun a => g a) =\u1da0[ae (Measure.restrict volume (Ioc 0 (f \u03c9)))] fun a => 0"}, {"tactic": "exact ae_restrict_of_ae_restrict_of_subset Ioc_subset_Ioi_self H1", "annotated_tactic": ["exact <a>ae_restrict_of_ae_restrict_of_subset</a> <a>Ioc_subset_Ioi_self</a> H1", [{"full_name": "MeasureTheory.ae_restrict_of_ae_restrict_of_subset", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2606, 9], "def_end_pos": [2606, 45]}, {"full_name": "Set.Ioc_subset_Ioi_self", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [526, 9], "def_end_pos": [526, 28]}]], "state_before": "case h\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : g =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\n\u03c9 : \u03b1\n\u22a2 (fun a => g a) =\u1da0[ae (Measure.restrict volume (Ioc 0 (f \u03c9)))] fun a => 0", "state_after": "no goals"}, {"tactic": "have : (fun t \u21a6 \u03bc {a : \u03b1 | t \u2264 f a} * ENNReal.ofReal (g t))\n  =\u1d50[volume.restrict (Ioi (0:\u211d))] 0 := by\n    filter_upwards [H1] with t ht using by simp [ht]", "annotated_tactic": ["have : (fun t \u21a6 \u03bc {a : \u03b1 | t \u2264 f a} * <a>ENNReal.ofReal</a> (g t))\n        =\u1d50[volume.restrict (<a>Ioi</a> (0:\u211d))] 0 := by\n          filter_upwards [H1] with t ht using by simp [ht]", [{"full_name": "ENNReal.ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [172, 29], "def_end_pos": [172, 35]}, {"full_name": "Set.Ioi", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [79, 5], "def_end_pos": [79, 8]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : g =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nA : \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc = 0\n\u22a2 \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) = 0", "state_after": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : g =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nA : \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc = 0\nthis : (fun t => \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)) =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\n\u22a2 \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) = 0"}, {"tactic": "simp [lintegral_congr_ae this]", "annotated_tactic": ["simp [<a>lintegral_congr_ae</a> this]", [{"full_name": "MeasureTheory.lintegral_congr_ae", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [304, 9], "def_end_pos": [304, 27]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : g =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nA : \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc = 0\nthis : (fun t => \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)) =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\n\u22a2 \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) = 0", "state_after": "no goals"}, {"tactic": "filter_upwards [H1] with t ht using by simp [ht]", "annotated_tactic": ["filter_upwards [H1] with t ht using by simp [ht]", []], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : g =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nA : \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc = 0\n\u22a2 (fun t => \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)) =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0", "state_after": "no goals"}, {"tactic": "simp [ht]", "annotated_tactic": ["simp [ht]", []], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : g =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nA : \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc = 0\nt : \u211d\nht : g t = OfNat.ofNat 0 t\n\u22a2 \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) = OfNat.ofNat 0 t", "state_after": "no goals"}, {"tactic": "rcases H2 with \u27e8s, s_pos, hs, h's\u27e9", "annotated_tactic": ["rcases H2 with \u27e8s, s_pos, hs, h's\u27e9", []], "state_before": "case pos\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2203 s, s > 0 \u2227 0 < \u222b (t : \u211d) in 0 ..s, g t \u2227 \u2191\u2191\u03bc {a | s < f a} = \u22a4\n\u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =\n    \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)", "state_after": "case pos.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\ns : \u211d\ns_pos : s > 0\nhs : 0 < \u222b (t : \u211d) in 0 ..s, g t\nh's : \u2191\u2191\u03bc {a | s < f a} = \u22a4\n\u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =\n    \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)"}, {"tactic": "rw [intervalIntegral.integral_of_le s_pos.le] at hs", "annotated_tactic": ["rw [<a>intervalIntegral.integral_of_le</a> s_pos.le] at hs", [{"full_name": "intervalIntegral.integral_of_le", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [465, 9], "def_end_pos": [465, 23]}]], "state_before": "case pos.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\ns : \u211d\ns_pos : s > 0\nhs : 0 < \u222b (t : \u211d) in 0 ..s, g t\nh's : \u2191\u2191\u03bc {a | s < f a} = \u22a4\n\u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =\n    \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)", "state_after": "case pos.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\ns : \u211d\ns_pos : s > 0\nhs : 0 < \u222b (x : \u211d) in Ioc 0 s, g x\nh's : \u2191\u2191\u03bc {a | s < f a} = \u22a4\n\u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =\n    \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)"}, {"tactic": "rw [A, B]", "annotated_tactic": ["rw [A, B]", []], "state_before": "case pos.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\ns : \u211d\ns_pos : s > 0\nhs : 0 < \u222b (x : \u211d) in Ioc 0 s, g x\nh's : \u2191\u2191\u03bc {a | s < f a} = \u22a4\nA : \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) = \u22a4\nB : \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc = \u22a4\n\u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =\n    \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)", "state_after": "no goals"}, {"tactic": "rw [eq_top_iff]", "annotated_tactic": ["rw [<a>eq_top_iff</a>]", [{"full_name": "eq_top_iff", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [165, 9], "def_end_pos": [165, 19]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\ns : \u211d\ns_pos : s > 0\nhs : 0 < \u222b (x : \u211d) in Ioc 0 s, g x\nh's : \u2191\u2191\u03bc {a | s < f a} = \u22a4\n\u22a2 \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) = \u22a4", "state_after": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\ns : \u211d\ns_pos : s > 0\nhs : 0 < \u222b (x : \u211d) in Ioc 0 s, g x\nh's : \u2191\u2191\u03bc {a | s < f a} = \u22a4\n\u22a2 \u22a4 \u2264 \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)"}, {"tactic": "rw [lintegral_const_mul, ENNReal.top_mul I_pos]", "annotated_tactic": ["rw [<a>lintegral_const_mul</a>, <a>ENNReal.top_mul</a> I_pos]", [{"full_name": "MeasureTheory.lintegral_const_mul", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [671, 9], "def_end_pos": [671, 28]}, {"full_name": "ENNReal.top_mul", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [587, 17], "def_end_pos": [587, 24]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\ns : \u211d\ns_pos : s > 0\nhs : 0 < \u222b (x : \u211d) in Ioc 0 s, g x\nh's : \u2191\u2191\u03bc {a | s < f a} = \u22a4\nI_pos : \u222b\u207b (a : \u211d) in Ioc 0 s, ENNReal.ofReal (g a) \u2260 0\n\u22a2 \u22a4 = \u222b\u207b (t : \u211d) in Ioc 0 s, \u22a4 * ENNReal.ofReal (g t)", "state_after": "case hf\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\ns : \u211d\ns_pos : s > 0\nhs : 0 < \u222b (x : \u211d) in Ioc 0 s, g x\nh's : \u2191\u2191\u03bc {a | s < f a} = \u22a4\nI_pos : \u222b\u207b (a : \u211d) in Ioc 0 s, ENNReal.ofReal (g a) \u2260 0\n\u22a2 Measurable fun t => ENNReal.ofReal (g t)"}, {"tactic": "exact ENNReal.measurable_ofReal.comp g_mble", "annotated_tactic": ["exact ENNReal.measurable_ofReal.comp g_mble", []], "state_before": "case hf\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\ns : \u211d\ns_pos : s > 0\nhs : 0 < \u222b (x : \u211d) in Ioc 0 s, g x\nh's : \u2191\u2191\u03bc {a | s < f a} = \u22a4\nI_pos : \u222b\u207b (a : \u211d) in Ioc 0 s, ENNReal.ofReal (g a) \u2260 0\n\u22a2 Measurable fun t => ENNReal.ofReal (g t)", "state_after": "no goals"}, {"tactic": "rw [\u2190 ofReal_integral_eq_lintegral_ofReal (g_intble s s_pos).1]", "annotated_tactic": ["rw [\u2190 <a>ofReal_integral_eq_lintegral_ofReal</a> (g_intble s s_pos).1]", [{"full_name": "MeasureTheory.ofReal_integral_eq_lintegral_ofReal", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1219, 9], "def_end_pos": [1219, 44]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\ns : \u211d\ns_pos : s > 0\nhs : 0 < \u222b (x : \u211d) in Ioc 0 s, g x\nh's : \u2191\u2191\u03bc {a | s < f a} = \u22a4\n\u22a2 \u222b\u207b (a : \u211d) in Ioc 0 s, ENNReal.ofReal (g a) \u2260 0", "state_after": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\ns : \u211d\ns_pos : s > 0\nhs : 0 < \u222b (x : \u211d) in Ioc 0 s, g x\nh's : \u2191\u2191\u03bc {a | s < f a} = \u22a4\n\u22a2 ENNReal.ofReal (\u222b (x : \u211d) in Ioc 0 s, g x) \u2260 0\n\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\ns : \u211d\ns_pos : s > 0\nhs : 0 < \u222b (x : \u211d) in Ioc 0 s, g x\nh's : \u2191\u2191\u03bc {a | s < f a} = \u22a4\n\u22a2 0 \u2264\u1da0[ae (Measure.restrict volume (Ioc 0 s))] g"}, {"tactic": "simpa only [not_lt, ge_iff_le, ne_eq, ENNReal.ofReal_eq_zero, not_le] using hs", "annotated_tactic": ["simpa only [<a>not_lt</a>, <a>ge_iff_le</a>, <a>ne_eq</a>, <a>ENNReal.ofReal_eq_zero</a>, <a>not_le</a>] using hs", [{"full_name": "not_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [368, 9], "def_end_pos": [368, 15]}, {"full_name": "ge_iff_le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [359, 9], "def_end_pos": [359, 18]}, {"full_name": "ne_eq", "def_path": "lake-packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [76, 17], "def_end_pos": [76, 22]}, {"full_name": "ENNReal.ofReal_eq_zero", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2170, 9], "def_end_pos": [2170, 23]}, {"full_name": "not_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [373, 9], "def_end_pos": [373, 15]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\ns : \u211d\ns_pos : s > 0\nhs : 0 < \u222b (x : \u211d) in Ioc 0 s, g x\nh's : \u2191\u2191\u03bc {a | s < f a} = \u22a4\n\u22a2 ENNReal.ofReal (\u222b (x : \u211d) in Ioc 0 s, g x) \u2260 0", "state_after": "no goals"}, {"tactic": "filter_upwards [ae_restrict_mem measurableSet_Ioc] with t ht using g_nn _ ht.1", "annotated_tactic": ["filter_upwards [<a>ae_restrict_mem</a> <a>measurableSet_Ioc</a>] with t ht using g_nn _ ht.1", [{"full_name": "MeasureTheory.ae_restrict_mem", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2586, 9], "def_end_pos": [2586, 24]}, {"full_name": "measurableSet_Ioc", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [589, 9], "def_end_pos": [589, 26]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\ns : \u211d\ns_pos : s > 0\nhs : 0 < \u222b (x : \u211d) in Ioc 0 s, g x\nh's : \u2191\u2191\u03bc {a | s < f a} = \u22a4\n\u22a2 0 \u2264\u1da0[ae (Measure.restrict volume (Ioc 0 s))] g", "state_after": "no goals"}, {"tactic": "apply set_lintegral_mono' measurableSet_Ioc (fun x hx \u21a6 ?_)", "annotated_tactic": ["apply <a>set_lintegral_mono'</a> <a>measurableSet_Ioc</a> (fun x hx \u21a6 ?_)", [{"full_name": "MeasureTheory.set_lintegral_mono'", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [296, 9], "def_end_pos": [296, 28]}, {"full_name": "measurableSet_Ioc", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [589, 9], "def_end_pos": [589, 26]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\ns : \u211d\ns_pos : s > 0\nhs : 0 < \u222b (x : \u211d) in Ioc 0 s, g x\nh's : \u2191\u2191\u03bc {a | s < f a} = \u22a4\n\u22a2 \u222b\u207b (t : \u211d) in Ioc 0 s, \u22a4 * ENNReal.ofReal (g t) \u2264 \u222b\u207b (t : \u211d) in Ioc 0 s, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)", "state_after": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\ns : \u211d\ns_pos : s > 0\nhs : 0 < \u222b (x : \u211d) in Ioc 0 s, g x\nh's : \u2191\u2191\u03bc {a | s < f a} = \u22a4\nx : \u211d\nhx : x \u2208 Ioc 0 s\n\u22a2 \u22a4 * ENNReal.ofReal (g x) \u2264 \u2191\u2191\u03bc {a | x \u2264 f a} * ENNReal.ofReal (g x)"}, {"tactic": "rw [\u2190 h's]", "annotated_tactic": ["rw [\u2190 h's]", []], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\ns : \u211d\ns_pos : s > 0\nhs : 0 < \u222b (x : \u211d) in Ioc 0 s, g x\nh's : \u2191\u2191\u03bc {a | s < f a} = \u22a4\nx : \u211d\nhx : x \u2208 Ioc 0 s\n\u22a2 \u22a4 * ENNReal.ofReal (g x) \u2264 \u2191\u2191\u03bc {a | x \u2264 f a} * ENNReal.ofReal (g x)", "state_after": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\ns : \u211d\ns_pos : s > 0\nhs : 0 < \u222b (x : \u211d) in Ioc 0 s, g x\nh's : \u2191\u2191\u03bc {a | s < f a} = \u22a4\nx : \u211d\nhx : x \u2208 Ioc 0 s\n\u22a2 \u2191\u2191\u03bc {a | s < f a} * ENNReal.ofReal (g x) \u2264 \u2191\u2191\u03bc {a | x \u2264 f a} * ENNReal.ofReal (g x)"}, {"tactic": "gcongr", "annotated_tactic": ["gcongr", []], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\ns : \u211d\ns_pos : s > 0\nhs : 0 < \u222b (x : \u211d) in Ioc 0 s, g x\nh's : \u2191\u2191\u03bc {a | s < f a} = \u22a4\nx : \u211d\nhx : x \u2208 Ioc 0 s\n\u22a2 \u2191\u2191\u03bc {a | s < f a} * ENNReal.ofReal (g x) \u2264 \u2191\u2191\u03bc {a | x \u2264 f a} * ENNReal.ofReal (g x)", "state_after": "case bc\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\ns : \u211d\ns_pos : s > 0\nhs : 0 < \u222b (x : \u211d) in Ioc 0 s, g x\nh's : \u2191\u2191\u03bc {a | s < f a} = \u22a4\nx : \u211d\nhx : x \u2208 Ioc 0 s\n\u22a2 \u2191\u2191\u03bc {a | s < f a} \u2264 \u2191\u2191\u03bc {a | x \u2264 f a}"}, {"tactic": "exact measure_mono (fun a ha \u21a6 hx.2.trans (le_of_lt ha))", "annotated_tactic": ["exact <a>measure_mono</a> (fun a ha \u21a6 hx.2.<a>trans</a> (<a>le_of_lt</a> ha))", [{"full_name": "MeasureTheory.measure_mono", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [193, 9], "def_end_pos": [193, 21]}, {"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [120, 7], "def_end_pos": [120, 18]}, {"full_name": "le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [110, 9], "def_end_pos": [110, 17]}]], "state_before": "case bc\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\ns : \u211d\ns_pos : s > 0\nhs : 0 < \u222b (x : \u211d) in Ioc 0 s, g x\nh's : \u2191\u2191\u03bc {a | s < f a} = \u22a4\nx : \u211d\nhx : x \u2208 Ioc 0 s\n\u22a2 \u2191\u2191\u03bc {a | s < f a} \u2264 \u2191\u2191\u03bc {a | x \u2264 f a}", "state_after": "no goals"}, {"tactic": "rw [eq_top_iff]", "annotated_tactic": ["rw [<a>eq_top_iff</a>]", [{"full_name": "eq_top_iff", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [165, 9], "def_end_pos": [165, 19]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\ns : \u211d\ns_pos : s > 0\nhs : 0 < \u222b (x : \u211d) in Ioc 0 s, g x\nh's : \u2191\u2191\u03bc {a | s < f a} = \u22a4\nA : \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) = \u22a4\n\u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc = \u22a4", "state_after": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\ns : \u211d\ns_pos : s > 0\nhs : 0 < \u222b (x : \u211d) in Ioc 0 s, g x\nh's : \u2191\u2191\u03bc {a | s < f a} = \u22a4\nA : \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) = \u22a4\n\u22a2 \u22a4 \u2264 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc"}, {"tactic": "simp only [lintegral_const, MeasurableSet.univ, Measure.restrict_apply, univ_inter,\n  h's, ne_eq, ENNReal.ofReal_eq_zero, not_le]", "annotated_tactic": ["simp only [<a>lintegral_const</a>, <a>MeasurableSet.univ</a>, <a>Measure.restrict_apply</a>, <a>univ_inter</a>,\n            h's, <a>ne_eq</a>, <a>ENNReal.ofReal_eq_zero</a>, <a>not_le</a>]", [{"full_name": "MeasureTheory.lintegral_const", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [136, 9], "def_end_pos": [136, 24]}, {"full_name": "MeasurableSet.univ", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [101, 19], "def_end_pos": [101, 37]}, {"full_name": "MeasureTheory.Measure.restrict_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1533, 9], "def_end_pos": [1533, 23]}, {"full_name": "Set.univ_inter", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1017, 9], "def_end_pos": [1017, 19]}, {"full_name": "ne_eq", "def_path": "lake-packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [76, 17], "def_end_pos": [76, 22]}, {"full_name": "ENNReal.ofReal_eq_zero", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2170, 9], "def_end_pos": [2170, 23]}, {"full_name": "not_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [373, 9], "def_end_pos": [373, 15]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\ns : \u211d\ns_pos : s > 0\nhs : 0 < \u222b (x : \u211d) in Ioc 0 s, g x\nh's : \u2191\u2191\u03bc {a | s < f a} = \u22a4\nA : \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) = \u22a4\n\u22a2 \u22a4 = \u222b\u207b (x : \u03b1) in {a | s < f a}, ENNReal.ofReal (\u222b (t : \u211d) in 0 ..s, g t) \u2202\u03bc", "state_after": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\ns : \u211d\ns_pos : s > 0\nhs : 0 < \u222b (x : \u211d) in Ioc 0 s, g x\nh's : \u2191\u2191\u03bc {a | s < f a} = \u22a4\nA : \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) = \u22a4\n\u22a2 \u22a4 = ENNReal.ofReal (\u222b (t : \u211d) in 0 ..s, g t) * \u22a4"}, {"tactic": "rw [ENNReal.mul_top]", "annotated_tactic": ["rw [<a>ENNReal.mul_top</a>]", [{"full_name": "ENNReal.mul_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [581, 17], "def_end_pos": [581, 24]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\ns : \u211d\ns_pos : s > 0\nhs : 0 < \u222b (x : \u211d) in Ioc 0 s, g x\nh's : \u2191\u2191\u03bc {a | s < f a} = \u22a4\nA : \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) = \u22a4\n\u22a2 \u22a4 = ENNReal.ofReal (\u222b (t : \u211d) in 0 ..s, g t) * \u22a4", "state_after": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\ns : \u211d\ns_pos : s > 0\nhs : 0 < \u222b (x : \u211d) in Ioc 0 s, g x\nh's : \u2191\u2191\u03bc {a | s < f a} = \u22a4\nA : \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) = \u22a4\n\u22a2 ENNReal.ofReal (\u222b (t : \u211d) in 0 ..s, g t) \u2260 0"}, {"tactic": "simpa [intervalIntegral.integral_of_le s_pos.le] using hs", "annotated_tactic": ["simpa [<a>intervalIntegral.integral_of_le</a> s_pos.le] using hs", [{"full_name": "intervalIntegral.integral_of_le", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [465, 9], "def_end_pos": [465, 23]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\ns : \u211d\ns_pos : s > 0\nhs : 0 < \u222b (x : \u211d) in Ioc 0 s, g x\nh's : \u2191\u2191\u03bc {a | s < f a} = \u22a4\nA : \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) = \u22a4\n\u22a2 ENNReal.ofReal (\u222b (t : \u211d) in 0 ..s, g t) \u2260 0", "state_after": "no goals"}, {"tactic": "apply set_lintegral_mono' (measurableSet_lt measurable_const f_mble) (fun a ha \u21a6 ?_)", "annotated_tactic": ["apply <a>set_lintegral_mono'</a> (<a>measurableSet_lt</a> <a>measurable_const</a> f_mble) (fun a ha \u21a6 ?_)", [{"full_name": "MeasureTheory.set_lintegral_mono'", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [296, 9], "def_end_pos": [296, 28]}, {"full_name": "measurableSet_lt", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [616, 9], "def_end_pos": [616, 25]}, {"full_name": "measurable_const", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [570, 9], "def_end_pos": [570, 25]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\ns : \u211d\ns_pos : s > 0\nhs : 0 < \u222b (x : \u211d) in Ioc 0 s, g x\nh's : \u2191\u2191\u03bc {a | s < f a} = \u22a4\nA : \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) = \u22a4\n\u22a2 \u222b\u207b (x : \u03b1) in {a | s < f a}, ENNReal.ofReal (\u222b (t : \u211d) in 0 ..s, g t) \u2202\u03bc \u2264\n    \u222b\u207b (\u03c9 : \u03b1) in {a | s < f a}, ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc", "state_after": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\ns : \u211d\ns_pos : s > 0\nhs : 0 < \u222b (x : \u211d) in Ioc 0 s, g x\nh's : \u2191\u2191\u03bc {a | s < f a} = \u22a4\nA : \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) = \u22a4\na : \u03b1\nha : a \u2208 {a | s < f a}\n\u22a2 ENNReal.ofReal (\u222b (t : \u211d) in 0 ..s, g t) \u2264 ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f a, g t)"}, {"tactic": "apply ENNReal.ofReal_le_ofReal", "annotated_tactic": ["apply <a>ENNReal.ofReal_le_ofReal</a>", [{"full_name": "ENNReal.ofReal_le_ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2135, 9], "def_end_pos": [2135, 25]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\ns : \u211d\ns_pos : s > 0\nhs : 0 < \u222b (x : \u211d) in Ioc 0 s, g x\nh's : \u2191\u2191\u03bc {a | s < f a} = \u22a4\nA : \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) = \u22a4\na : \u03b1\nha : a \u2208 {a | s < f a}\n\u22a2 ENNReal.ofReal (\u222b (t : \u211d) in 0 ..s, g t) \u2264 ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f a, g t)", "state_after": "case h\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\ns : \u211d\ns_pos : s > 0\nhs : 0 < \u222b (x : \u211d) in Ioc 0 s, g x\nh's : \u2191\u2191\u03bc {a | s < f a} = \u22a4\nA : \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) = \u22a4\na : \u03b1\nha : a \u2208 {a | s < f a}\n\u22a2 \u222b (t : \u211d) in 0 ..s, g t \u2264 \u222b (t : \u211d) in 0 ..f a, g t"}, {"tactic": "apply intervalIntegral.integral_mono_interval le_rfl s_pos.le (le_of_lt ha)", "annotated_tactic": ["apply <a>intervalIntegral.integral_mono_interval</a> <a>le_rfl</a> s_pos.le (<a>le_of_lt</a> ha)", [{"full_name": "intervalIntegral.integral_mono_interval", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [1401, 9], "def_end_pos": [1401, 31]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}, {"full_name": "le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [110, 9], "def_end_pos": [110, 17]}]], "state_before": "case h\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\ns : \u211d\ns_pos : s > 0\nhs : 0 < \u222b (x : \u211d) in Ioc 0 s, g x\nh's : \u2191\u2191\u03bc {a | s < f a} = \u22a4\nA : \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) = \u22a4\na : \u03b1\nha : a \u2208 {a | s < f a}\n\u22a2 \u222b (t : \u211d) in 0 ..s, g t \u2264 \u222b (t : \u211d) in 0 ..f a, g t", "state_after": "case h.hf\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\ns : \u211d\ns_pos : s > 0\nhs : 0 < \u222b (x : \u211d) in Ioc 0 s, g x\nh's : \u2191\u2191\u03bc {a | s < f a} = \u22a4\nA : \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) = \u22a4\na : \u03b1\nha : a \u2208 {a | s < f a}\n\u22a2 0 \u2264\u1da0[ae (Measure.restrict volume (Ioc 0 (f a)))] fun x => g x\n\ncase h.hfi\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\ns : \u211d\ns_pos : s > 0\nhs : 0 < \u222b (x : \u211d) in Ioc 0 s, g x\nh's : \u2191\u2191\u03bc {a | s < f a} = \u22a4\nA : \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) = \u22a4\na : \u03b1\nha : a \u2208 {a | s < f a}\n\u22a2 IntervalIntegrable (fun x => g x) volume 0 (f a)"}, {"tactic": "filter_upwards [ae_restrict_mem measurableSet_Ioc] with t ht using g_nn _ ht.1", "annotated_tactic": ["filter_upwards [<a>ae_restrict_mem</a> <a>measurableSet_Ioc</a>] with t ht using g_nn _ ht.1", [{"full_name": "MeasureTheory.ae_restrict_mem", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2586, 9], "def_end_pos": [2586, 24]}, {"full_name": "measurableSet_Ioc", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [589, 9], "def_end_pos": [589, 26]}]], "state_before": "case h.hf\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\ns : \u211d\ns_pos : s > 0\nhs : 0 < \u222b (x : \u211d) in Ioc 0 s, g x\nh's : \u2191\u2191\u03bc {a | s < f a} = \u22a4\nA : \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) = \u22a4\na : \u03b1\nha : a \u2208 {a | s < f a}\n\u22a2 0 \u2264\u1da0[ae (Measure.restrict volume (Ioc 0 (f a)))] fun x => g x", "state_after": "no goals"}, {"tactic": "exact g_intble _ (s_pos.trans ha)", "annotated_tactic": ["exact g_intble _ (s_pos.trans ha)", []], "state_before": "case h.hfi\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\ns : \u211d\ns_pos : s > 0\nhs : 0 < \u222b (x : \u211d) in Ioc 0 s, g x\nh's : \u2191\u2191\u03bc {a | s < f a} = \u22a4\nA : \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) = \u22a4\na : \u03b1\nha : a \u2208 {a | s < f a}\n\u22a2 IntervalIntegrable (fun x => g x) volume 0 (f a)", "state_after": "no goals"}, {"tactic": "contrapose! H1", "annotated_tactic": ["contrapose! H1", []], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\n\u22a2 BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}", "state_after": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nH1 : \u00acBddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\n\u22a2 g =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0"}, {"tactic": "have : \u2200 (n : \u2115), g =\u1d50[volume.restrict (Ioc (0 : \u211d) n)] 0 := by\n  intro n\n  rcases not_bddAbove_iff.1 H1 n with \u27e8s, hs, ns\u27e9\n  exact ae_restrict_of_ae_restrict_of_subset (Ioc_subset_Ioc_right ns.le) hs", "annotated_tactic": ["have : \u2200 (n : \u2115), g =\u1d50[volume.restrict (<a>Ioc</a> (0 : \u211d) n)] 0 := by\n      intro n\n      rcases <a>not_bddAbove_iff</a>.1 H1 n with \u27e8s, hs, ns\u27e9\n      exact <a>ae_restrict_of_ae_restrict_of_subset</a> (<a>Ioc_subset_Ioc_right</a> ns.le) hs", [{"full_name": "Set.Ioc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [69, 5], "def_end_pos": [69, 8]}, {"full_name": "not_bddAbove_iff", "def_path": "Mathlib/Order/Bounds/Basic.lean", "def_pos": [133, 9], "def_end_pos": [133, 25]}, {"full_name": "MeasureTheory.ae_restrict_of_ae_restrict_of_subset", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2606, 9], "def_end_pos": [2606, 45]}, {"full_name": "Set.Ioc_subset_Ioc_right", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [488, 9], "def_end_pos": [488, 29]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nH1 : \u00acBddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\n\u22a2 g =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0", "state_after": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nH1 : \u00acBddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nthis : \u2200 (n : \u2115), g =\u1da0[ae (Measure.restrict volume (Ioc 0 \u2191n))] 0\n\u22a2 g =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0"}, {"tactic": "have Hg : g =\u1d50[volume.restrict (\u22c3 (n : \u2115), (Ioc (0 : \u211d) n))] 0 :=\n  (ae_restrict_iUnion_iff _ _).2 this", "annotated_tactic": ["have Hg : g =\u1d50[volume.restrict (\u22c3 (n : \u2115), (<a>Ioc</a> (0 : \u211d) n))] 0 :=\n      (<a>ae_restrict_iUnion_iff</a> _ _).2 this", [{"full_name": "Set.Ioc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [69, 5], "def_end_pos": [69, 8]}, {"full_name": "MeasureTheory.ae_restrict_iUnion_iff", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2514, 9], "def_end_pos": [2514, 31]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nH1 : \u00acBddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nthis : \u2200 (n : \u2115), g =\u1da0[ae (Measure.restrict volume (Ioc 0 \u2191n))] 0\n\u22a2 g =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0", "state_after": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nH1 : \u00acBddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nthis : \u2200 (n : \u2115), g =\u1da0[ae (Measure.restrict volume (Ioc 0 \u2191n))] 0\nHg : g =\u1da0[ae (Measure.restrict volume (\u22c3 n, Ioc 0 \u2191n))] 0\n\u22a2 g =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0"}, {"tactic": "have : (\u22c3 (n : \u2115), (Ioc (0 : \u211d) n)) = Ioi 0 :=\n  iUnion_Ioc_eq_Ioi_self_iff.2 (fun x _ \u21a6 exists_nat_ge x)", "annotated_tactic": ["have : (\u22c3 (n : \u2115), (<a>Ioc</a> (0 : \u211d) n)) = <a>Ioi</a> 0 :=\n      <a>iUnion_Ioc_eq_Ioi_self_iff</a>.2 (fun x _ \u21a6 <a>exists_nat_ge</a> x)", [{"full_name": "Set.Ioc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [69, 5], "def_end_pos": [69, 8]}, {"full_name": "Set.Ioi", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [79, 5], "def_end_pos": [79, 8]}, {"full_name": "Set.iUnion_Ioc_eq_Ioi_self_iff", "def_path": "Mathlib/Data/Set/Intervals/Disjoint.lean", "def_pos": [164, 9], "def_end_pos": [164, 35]}, {"full_name": "exists_nat_ge", "def_path": "Mathlib/Algebra/Order/Archimedean.lean", "def_pos": [122, 9], "def_end_pos": [122, 22]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nH1 : \u00acBddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nthis : \u2200 (n : \u2115), g =\u1da0[ae (Measure.restrict volume (Ioc 0 \u2191n))] 0\nHg : g =\u1da0[ae (Measure.restrict volume (\u22c3 n, Ioc 0 \u2191n))] 0\n\u22a2 g =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0", "state_after": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nH1 : \u00acBddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nthis\u271d : \u2200 (n : \u2115), g =\u1da0[ae (Measure.restrict volume (Ioc 0 \u2191n))] 0\nHg : g =\u1da0[ae (Measure.restrict volume (\u22c3 n, Ioc 0 \u2191n))] 0\nthis : \u22c3 n, Ioc 0 \u2191n = Ioi 0\n\u22a2 g =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0"}, {"tactic": "rwa [this] at Hg", "annotated_tactic": ["rwa [this] at Hg", []], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nH1 : \u00acBddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nthis\u271d : \u2200 (n : \u2115), g =\u1da0[ae (Measure.restrict volume (Ioc 0 \u2191n))] 0\nHg : g =\u1da0[ae (Measure.restrict volume (\u22c3 n, Ioc 0 \u2191n))] 0\nthis : \u22c3 n, Ioc 0 \u2191n = Ioi 0\n\u22a2 g =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0", "state_after": "no goals"}, {"tactic": "intro n", "annotated_tactic": ["intro n", []], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nH1 : \u00acBddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\n\u22a2 \u2200 (n : \u2115), g =\u1da0[ae (Measure.restrict volume (Ioc 0 \u2191n))] 0", "state_after": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nH1 : \u00acBddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nn : \u2115\n\u22a2 g =\u1da0[ae (Measure.restrict volume (Ioc 0 \u2191n))] 0"}, {"tactic": "rcases not_bddAbove_iff.1 H1 n with \u27e8s, hs, ns\u27e9", "annotated_tactic": ["rcases <a>not_bddAbove_iff</a>.1 H1 n with \u27e8s, hs, ns\u27e9", [{"full_name": "not_bddAbove_iff", "def_path": "Mathlib/Order/Bounds/Basic.lean", "def_pos": [133, 9], "def_end_pos": [133, 25]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nH1 : \u00acBddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nn : \u2115\n\u22a2 g =\u1da0[ae (Measure.restrict volume (Ioc 0 \u2191n))] 0", "state_after": "case intro.intro\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nH1 : \u00acBddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nn : \u2115\ns : \u211d\nhs : s \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nns : \u2191n < s\n\u22a2 g =\u1da0[ae (Measure.restrict volume (Ioc 0 \u2191n))] 0"}, {"tactic": "exact ae_restrict_of_ae_restrict_of_subset (Ioc_subset_Ioc_right ns.le) hs", "annotated_tactic": ["exact <a>ae_restrict_of_ae_restrict_of_subset</a> (<a>Ioc_subset_Ioc_right</a> ns.le) hs", [{"full_name": "MeasureTheory.ae_restrict_of_ae_restrict_of_subset", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2606, 9], "def_end_pos": [2606, 45]}, {"full_name": "Set.Ioc_subset_Ioc_right", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [488, 9], "def_end_pos": [488, 29]}]], "state_before": "case intro.intro\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nH1 : \u00acBddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nn : \u2115\ns : \u211d\nhs : s \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nns : \u2191n < s\n\u22a2 g =\u1da0[ae (Measure.restrict volume (Ioc 0 \u2191n))] 0", "state_after": "no goals"}, {"tactic": "simpa using trivial", "annotated_tactic": ["simpa using <a>trivial</a>", [{"full_name": "trivial", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [514, 31], "def_end_pos": [514, 38]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\n\u22a2 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}", "state_after": "no goals"}, {"tactic": "rw [\u2190 restrict_Ioo_eq_restrict_Ioc]", "annotated_tactic": ["rw [\u2190 <a>restrict_Ioo_eq_restrict_Ioc</a>]", [{"full_name": "MeasureTheory.restrict_Ioo_eq_restrict_Ioc", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3198, 9], "def_end_pos": [3198, 37]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\n\u22a2 g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0", "state_after": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\n\u22a2 g =\u1da0[ae (Measure.restrict volume (Ioo 0 M))] 0"}, {"tactic": "obtain \u27e8u, -, uM, ulim\u27e9 : \u2203 u, StrictMono u \u2227 (\u2200 (n : \u2115), u n < M) \u2227 Tendsto u atTop (\ud835\udcdd M) :=\n  exists_seq_strictMono_tendsto M", "annotated_tactic": ["obtain \u27e8u, -, uM, ulim\u27e9 : \u2203 u, <a>StrictMono</a> u \u2227 (\u2200 (n : \u2115), u n < M) \u2227 <a>Tendsto</a> u <a>atTop</a> (\ud835\udcdd M) :=\n      <a>exists_seq_strictMono_tendsto</a> M", [{"full_name": "StrictMono", "def_path": "Mathlib/Order/Monotone/Basic.lean", "def_pos": [97, 5], "def_end_pos": [97, 15]}, {"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2939, 5], "def_end_pos": [2939, 12]}, {"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}, {"full_name": "exists_seq_strictMono_tendsto", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [2219, 9], "def_end_pos": [2219, 38]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\n\u22a2 g =\u1da0[ae (Measure.restrict volume (Ioo 0 M))] 0", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nu : \u2115 \u2192 \u211d\nuM : \u2200 (n : \u2115), u n < M\nulim : Tendsto u atTop (\ud835\udcdd M)\n\u22a2 g =\u1da0[ae (Measure.restrict volume (Ioo 0 M))] 0"}, {"tactic": "have I : \u2200 n, g =\u1d50[volume.restrict (Ioc (0 : \u211d) (u n))] 0 := by\n  intro n\n  obtain \u27e8s, hs, uns\u27e9 : \u2203 s, g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0 \u2227 u n < s :=\n    exists_lt_of_lt_csSup (Set.nonempty_of_mem zero_mem) (uM n)\n  exact ae_restrict_of_ae_restrict_of_subset (Ioc_subset_Ioc_right uns.le) hs", "annotated_tactic": ["have I : \u2200 n, g =\u1d50[volume.restrict (<a>Ioc</a> (0 : \u211d) (u n))] 0 := by\n      intro n\n      obtain \u27e8s, hs, uns\u27e9 : \u2203 s, g =\u1da0[<a>ae</a> (<a>Measure.restrict</a> <a>volume</a> (<a>Ioc</a> 0 s))] 0 \u2227 u n < s :=\n        <a>exists_lt_of_lt_csSup</a> (<a>Set.nonempty_of_mem</a> zero_mem) (uM n)\n      exact <a>ae_restrict_of_ae_restrict_of_subset</a> (<a>Ioc_subset_Ioc_right</a> uns.le) hs", [{"full_name": "Set.Ioc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [69, 5], "def_end_pos": [69, 8]}, {"full_name": "MeasureTheory.Measure.ae", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [365, 5], "def_end_pos": [365, 15]}, {"full_name": "MeasureTheory.Measure.restrict", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1503, 5], "def_end_pos": [1503, 13]}, {"full_name": "MeasureTheory.MeasureSpace.volume", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [663, 3], "def_end_pos": [663, 9]}, {"full_name": "Set.Ioc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [69, 5], "def_end_pos": [69, 8]}, {"full_name": "exists_lt_of_lt_csSup", "def_path": "Mathlib/Order/ConditionallyCompleteLattice/Basic.lean", "def_pos": [999, 9], "def_end_pos": [999, 30]}, {"full_name": "Set.nonempty_of_mem", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [458, 9], "def_end_pos": [458, 24]}, {"full_name": "MeasureTheory.ae_restrict_of_ae_restrict_of_subset", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2606, 9], "def_end_pos": [2606, 45]}, {"full_name": "Set.Ioc_subset_Ioc_right", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [488, 9], "def_end_pos": [488, 29]}]], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nu : \u2115 \u2192 \u211d\nuM : \u2200 (n : \u2115), u n < M\nulim : Tendsto u atTop (\ud835\udcdd M)\n\u22a2 g =\u1da0[ae (Measure.restrict volume (Ioo 0 M))] 0", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nu : \u2115 \u2192 \u211d\nuM : \u2200 (n : \u2115), u n < M\nulim : Tendsto u atTop (\ud835\udcdd M)\nI : \u2200 (n : \u2115), g =\u1da0[ae (Measure.restrict volume (Ioc 0 (u n)))] 0\n\u22a2 g =\u1da0[ae (Measure.restrict volume (Ioo 0 M))] 0"}, {"tactic": "have : g =\u1d50[volume.restrict (\u22c3 n, Ioc (0 : \u211d) (u n))] 0 := (ae_restrict_iUnion_iff _ _).2 I", "annotated_tactic": ["have : g =\u1d50[volume.restrict (\u22c3 n, <a>Ioc</a> (0 : \u211d) (u n))] 0 := (<a>ae_restrict_iUnion_iff</a> _ _).2 I", [{"full_name": "Set.Ioc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [69, 5], "def_end_pos": [69, 8]}, {"full_name": "MeasureTheory.ae_restrict_iUnion_iff", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2514, 9], "def_end_pos": [2514, 31]}]], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nu : \u2115 \u2192 \u211d\nuM : \u2200 (n : \u2115), u n < M\nulim : Tendsto u atTop (\ud835\udcdd M)\nI : \u2200 (n : \u2115), g =\u1da0[ae (Measure.restrict volume (Ioc 0 (u n)))] 0\n\u22a2 g =\u1da0[ae (Measure.restrict volume (Ioo 0 M))] 0", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nu : \u2115 \u2192 \u211d\nuM : \u2200 (n : \u2115), u n < M\nulim : Tendsto u atTop (\ud835\udcdd M)\nI : \u2200 (n : \u2115), g =\u1da0[ae (Measure.restrict volume (Ioc 0 (u n)))] 0\nthis : g =\u1da0[ae (Measure.restrict volume (\u22c3 n, Ioc 0 (u n)))] 0\n\u22a2 g =\u1da0[ae (Measure.restrict volume (Ioo 0 M))] 0"}, {"tactic": "apply ae_restrict_of_ae_restrict_of_subset _ this", "annotated_tactic": ["apply <a>ae_restrict_of_ae_restrict_of_subset</a> _ this", [{"full_name": "MeasureTheory.ae_restrict_of_ae_restrict_of_subset", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2606, 9], "def_end_pos": [2606, 45]}]], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nu : \u2115 \u2192 \u211d\nuM : \u2200 (n : \u2115), u n < M\nulim : Tendsto u atTop (\ud835\udcdd M)\nI : \u2200 (n : \u2115), g =\u1da0[ae (Measure.restrict volume (Ioc 0 (u n)))] 0\nthis : g =\u1da0[ae (Measure.restrict volume (\u22c3 n, Ioc 0 (u n)))] 0\n\u22a2 g =\u1da0[ae (Measure.restrict volume (Ioo 0 M))] 0", "state_after": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nu : \u2115 \u2192 \u211d\nuM : \u2200 (n : \u2115), u n < M\nulim : Tendsto u atTop (\ud835\udcdd M)\nI : \u2200 (n : \u2115), g =\u1da0[ae (Measure.restrict volume (Ioc 0 (u n)))] 0\nthis : g =\u1da0[ae (Measure.restrict volume (\u22c3 n, Ioc 0 (u n)))] 0\n\u22a2 Ioo 0 M \u2286 \u22c3 n, Ioc 0 (u n)"}, {"tactic": "rintro x \u27e8x_pos, xM\u27e9", "annotated_tactic": ["rintro x \u27e8x_pos, xM\u27e9", []], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nu : \u2115 \u2192 \u211d\nuM : \u2200 (n : \u2115), u n < M\nulim : Tendsto u atTop (\ud835\udcdd M)\nI : \u2200 (n : \u2115), g =\u1da0[ae (Measure.restrict volume (Ioc 0 (u n)))] 0\nthis : g =\u1da0[ae (Measure.restrict volume (\u22c3 n, Ioc 0 (u n)))] 0\n\u22a2 Ioo 0 M \u2286 \u22c3 n, Ioc 0 (u n)", "state_after": "case intro\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nu : \u2115 \u2192 \u211d\nuM : \u2200 (n : \u2115), u n < M\nulim : Tendsto u atTop (\ud835\udcdd M)\nI : \u2200 (n : \u2115), g =\u1da0[ae (Measure.restrict volume (Ioc 0 (u n)))] 0\nthis : g =\u1da0[ae (Measure.restrict volume (\u22c3 n, Ioc 0 (u n)))] 0\nx : \u211d\nx_pos : 0 < x\nxM : x < M\n\u22a2 x \u2208 \u22c3 n, Ioc 0 (u n)"}, {"tactic": "obtain \u27e8n, hn\u27e9 : \u2203 n, x < u n := ((tendsto_order.1 ulim).1 _ xM).exists", "annotated_tactic": ["obtain \u27e8n, hn\u27e9 : \u2203 n, x < u n := ((<a>tendsto_order</a>.1 ulim).1 _ xM).<a>exists</a>", [{"full_name": "tendsto_order", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [919, 9], "def_end_pos": [919, 22]}, {"full_name": "Filter.Eventually.exists", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1308, 9], "def_end_pos": [1308, 26]}]], "state_before": "case intro\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nu : \u2115 \u2192 \u211d\nuM : \u2200 (n : \u2115), u n < M\nulim : Tendsto u atTop (\ud835\udcdd M)\nI : \u2200 (n : \u2115), g =\u1da0[ae (Measure.restrict volume (Ioc 0 (u n)))] 0\nthis : g =\u1da0[ae (Measure.restrict volume (\u22c3 n, Ioc 0 (u n)))] 0\nx : \u211d\nx_pos : 0 < x\nxM : x < M\n\u22a2 x \u2208 \u22c3 n, Ioc 0 (u n)", "state_after": "case intro.intro\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nu : \u2115 \u2192 \u211d\nuM : \u2200 (n : \u2115), u n < M\nulim : Tendsto u atTop (\ud835\udcdd M)\nI : \u2200 (n : \u2115), g =\u1da0[ae (Measure.restrict volume (Ioc 0 (u n)))] 0\nthis : g =\u1da0[ae (Measure.restrict volume (\u22c3 n, Ioc 0 (u n)))] 0\nx : \u211d\nx_pos : 0 < x\nxM : x < M\nn : \u2115\nhn : x < u n\n\u22a2 x \u2208 \u22c3 n, Ioc 0 (u n)"}, {"tactic": "exact mem_iUnion.2 \u27e8n, \u27e8x_pos, hn.le\u27e9\u27e9", "annotated_tactic": ["exact <a>mem_iUnion</a>.2 \u27e8n, \u27e8x_pos, hn.le\u27e9\u27e9", [{"full_name": "Set.mem_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [201, 9], "def_end_pos": [201, 19]}]], "state_before": "case intro.intro\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nu : \u2115 \u2192 \u211d\nuM : \u2200 (n : \u2115), u n < M\nulim : Tendsto u atTop (\ud835\udcdd M)\nI : \u2200 (n : \u2115), g =\u1da0[ae (Measure.restrict volume (Ioc 0 (u n)))] 0\nthis : g =\u1da0[ae (Measure.restrict volume (\u22c3 n, Ioc 0 (u n)))] 0\nx : \u211d\nx_pos : 0 < x\nxM : x < M\nn : \u2115\nhn : x < u n\n\u22a2 x \u2208 \u22c3 n, Ioc 0 (u n)", "state_after": "no goals"}, {"tactic": "intro n", "annotated_tactic": ["intro n", []], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nu : \u2115 \u2192 \u211d\nuM : \u2200 (n : \u2115), u n < M\nulim : Tendsto u atTop (\ud835\udcdd M)\n\u22a2 \u2200 (n : \u2115), g =\u1da0[ae (Measure.restrict volume (Ioc 0 (u n)))] 0", "state_after": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nu : \u2115 \u2192 \u211d\nuM : \u2200 (n : \u2115), u n < M\nulim : Tendsto u atTop (\ud835\udcdd M)\nn : \u2115\n\u22a2 g =\u1da0[ae (Measure.restrict volume (Ioc 0 (u n)))] 0"}, {"tactic": "obtain \u27e8s, hs, uns\u27e9 : \u2203 s, g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0 \u2227 u n < s :=\n  exists_lt_of_lt_csSup (Set.nonempty_of_mem zero_mem) (uM n)", "annotated_tactic": ["obtain \u27e8s, hs, uns\u27e9 : \u2203 s, g =\u1da0[<a>ae</a> (<a>Measure.restrict</a> <a>volume</a> (<a>Ioc</a> 0 s))] 0 \u2227 u n < s :=\n        <a>exists_lt_of_lt_csSup</a> (<a>Set.nonempty_of_mem</a> zero_mem) (uM n)", [{"full_name": "MeasureTheory.Measure.ae", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [365, 5], "def_end_pos": [365, 15]}, {"full_name": "MeasureTheory.Measure.restrict", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1503, 5], "def_end_pos": [1503, 13]}, {"full_name": "MeasureTheory.MeasureSpace.volume", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [663, 3], "def_end_pos": [663, 9]}, {"full_name": "Set.Ioc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [69, 5], "def_end_pos": [69, 8]}, {"full_name": "exists_lt_of_lt_csSup", "def_path": "Mathlib/Order/ConditionallyCompleteLattice/Basic.lean", "def_pos": [999, 9], "def_end_pos": [999, 30]}, {"full_name": "Set.nonempty_of_mem", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [458, 9], "def_end_pos": [458, 24]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nu : \u2115 \u2192 \u211d\nuM : \u2200 (n : \u2115), u n < M\nulim : Tendsto u atTop (\ud835\udcdd M)\nn : \u2115\n\u22a2 g =\u1da0[ae (Measure.restrict volume (Ioc 0 (u n)))] 0", "state_after": "case intro.intro\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nu : \u2115 \u2192 \u211d\nuM : \u2200 (n : \u2115), u n < M\nulim : Tendsto u atTop (\ud835\udcdd M)\nn : \u2115\ns : \u211d\nhs : g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0\nuns : u n < s\n\u22a2 g =\u1da0[ae (Measure.restrict volume (Ioc 0 (u n)))] 0"}, {"tactic": "exact ae_restrict_of_ae_restrict_of_subset (Ioc_subset_Ioc_right uns.le) hs", "annotated_tactic": ["exact <a>ae_restrict_of_ae_restrict_of_subset</a> (<a>Ioc_subset_Ioc_right</a> uns.le) hs", [{"full_name": "MeasureTheory.ae_restrict_of_ae_restrict_of_subset", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2606, 9], "def_end_pos": [2606, 45]}, {"full_name": "Set.Ioc_subset_Ioc_right", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [488, 9], "def_end_pos": [488, 29]}]], "state_before": "case intro.intro\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nu : \u2115 \u2192 \u211d\nuM : \u2200 (n : \u2115), u n < M\nulim : Tendsto u atTop (\ud835\udcdd M)\nn : \u2115\ns : \u211d\nhs : g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0\nuns : u n < s\n\u22a2 g =\u1da0[ae (Measure.restrict volume (Ioc 0 (u n)))] 0", "state_after": "no goals"}, {"tactic": "obtain \u27e8u, -, uM, ulim\u27e9 : \u2203 u, StrictAnti u \u2227 (\u2200 (n : \u2115), M < u n) \u2227 Tendsto u atTop (\ud835\udcdd M) :=\n  exists_seq_strictAnti_tendsto M", "annotated_tactic": ["obtain \u27e8u, -, uM, ulim\u27e9 : \u2203 u, <a>StrictAnti</a> u \u2227 (\u2200 (n : \u2115), M < u n) \u2227 <a>Tendsto</a> u <a>atTop</a> (\ud835\udcdd M) :=\n      <a>exists_seq_strictAnti_tendsto</a> M", [{"full_name": "StrictAnti", "def_path": "Mathlib/Order/Monotone/Basic.lean", "def_pos": [102, 5], "def_end_pos": [102, 15]}, {"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2939, 5], "def_end_pos": [2939, 12]}, {"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}, {"full_name": "exists_seq_strictAnti_tendsto", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [2258, 9], "def_end_pos": [2258, 38]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nhgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0\n\u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a}\n\u22a2 SigmaFinite \u03bd", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nhgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0\n\u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a}\nu : \u2115 \u2192 \u211d\nuM : \u2200 (n : \u2115), M < u n\nulim : Tendsto u atTop (\ud835\udcdd M)\n\u22a2 SigmaFinite \u03bd"}, {"tactic": "exact \u27e8\u27e8s\u27e9\u27e9", "annotated_tactic": ["exact \u27e8\u27e8s\u27e9\u27e9", []], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nhgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0\n\u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a}\nu : \u2115 \u2192 \u211d\nuM : \u2200 (n : \u2115), M < u n\nulim : Tendsto u atTop (\ud835\udcdd M)\ns : FiniteSpanningSetsIn \u03bd univ :=\n  { set := fun n => {a | f a \u2264 M} \u222a {a | u n < f a}, set_mem := (_ : \u2115 \u2192 True),\n    finite := (_ : \u2200 (n : \u2115), \u2191\u2191\u03bd ((fun n => {a | f a \u2264 M} \u222a {a | u n < f a}) n) < \u22a4),\n    spanning := (_ : \u22c3 i, (fun n => {a | f a \u2264 M} \u222a {a | u n < f a}) i = univ) }\n\u22a2 SigmaFinite \u03bd", "state_after": "no goals"}, {"tactic": "intro n", "annotated_tactic": ["intro n", []], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nhgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0\n\u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a}\nu : \u2115 \u2192 \u211d\nuM : \u2200 (n : \u2115), M < u n\nulim : Tendsto u atTop (\ud835\udcdd M)\n\u22a2 \u2200 (i : \u2115), \u2191\u2191\u03bd ((fun n => {a | f a \u2264 M} \u222a {a | u n < f a}) i) < \u22a4", "state_after": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nhgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0\n\u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a}\nu : \u2115 \u2192 \u211d\nuM : \u2200 (n : \u2115), M < u n\nulim : Tendsto u atTop (\ud835\udcdd M)\nn : \u2115\n\u22a2 \u2191\u2191\u03bd ((fun n => {a | f a \u2264 M} \u222a {a | u n < f a}) n) < \u22a4"}, {"tactic": "have I : \u03bd {a | f a \u2264 M} = 0 := by\n  rw [Measure.restrict_apply (measurableSet_le f_mble measurable_const)]\n  convert measure_empty\n  rw [\u2190 disjoint_iff_inter_eq_empty]\n  exact disjoint_left.mpr (fun a ha \u21a6 by simpa using ha)", "annotated_tactic": ["have I : \u03bd {a | f a \u2264 M} = 0 := by\n          rw [<a>Measure.restrict_apply</a> (<a>measurableSet_le</a> f_mble <a>measurable_const</a>)]\n          convert <a>measure_empty</a>\n          rw [\u2190 <a>disjoint_iff_inter_eq_empty</a>]\n          exact disjoint_left.mpr (fun a ha \u21a6 by simpa using ha)", [{"full_name": "MeasureTheory.Measure.restrict_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1533, 9], "def_end_pos": [1533, 23]}, {"full_name": "measurableSet_le", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [559, 9], "def_end_pos": [559, 25]}, {"full_name": "measurable_const", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [570, 9], "def_end_pos": [570, 25]}, {"full_name": "MeasureTheory.measure_empty", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [185, 9], "def_end_pos": [185, 22]}, {"full_name": "Set.disjoint_iff_inter_eq_empty", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1538, 9], "def_end_pos": [1538, 36]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nhgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0\n\u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a}\nu : \u2115 \u2192 \u211d\nuM : \u2200 (n : \u2115), M < u n\nulim : Tendsto u atTop (\ud835\udcdd M)\nn : \u2115\n\u22a2 \u2191\u2191\u03bd ((fun n => {a | f a \u2264 M} \u222a {a | u n < f a}) n) < \u22a4", "state_after": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nhgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0\n\u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a}\nu : \u2115 \u2192 \u211d\nuM : \u2200 (n : \u2115), M < u n\nulim : Tendsto u atTop (\ud835\udcdd M)\nn : \u2115\nI : \u2191\u2191\u03bd {a | f a \u2264 M} = 0\n\u22a2 \u2191\u2191\u03bd ((fun n => {a | f a \u2264 M} \u222a {a | u n < f a}) n) < \u22a4"}, {"tactic": "have J : \u03bc {a | u n < f a} < \u221e := by\n  rw [lt_top_iff_ne_top]\n  apply H2 _ (M_nonneg.trans_lt (uM n))\n  by_contra H3\n  rw [not_lt, intervalIntegral.integral_of_le (M_nonneg.trans (uM n).le)] at H3\n  have g_nn_ae : \u2200\u1d50 t \u2202(volume.restrict (Ioc 0 (u n))), 0 \u2264 g t := by\n    filter_upwards [ae_restrict_mem measurableSet_Ioc] with s hs using g_nn _ hs.1\n  have Ig : \u222b (t : \u211d) in Ioc 0 (u n), g t = 0 :=\n    le_antisymm H3 (integral_nonneg_of_ae g_nn_ae)\n  have J : \u2200\u1d50 t \u2202(volume.restrict (Ioc 0 (u n))), g t = 0 :=\n    (integral_eq_zero_iff_of_nonneg_ae g_nn_ae\n      (g_intble (u n) (M_nonneg.trans_lt (uM n))).1).1 Ig\n  have : u n \u2264 M := le_csSup M_bdd J\n  exact lt_irrefl _ (this.trans_lt (uM n))", "annotated_tactic": ["have J : \u03bc {a | u n < f a} < \u221e := by\n          rw [<a>lt_top_iff_ne_top</a>]\n          apply H2 _ (M_nonneg.trans_lt (uM n))\n          by_contra H3\n          rw [<a>not_lt</a>, <a>intervalIntegral.integral_of_le</a> (M_nonneg.trans (uM n).<a>le</a>)] at H3\n          have g_nn_ae : \u2200\u1d50 t \u2202(volume.restrict (<a>Ioc</a> 0 (u n))), 0 \u2264 g t := by\n            filter_upwards [<a>ae_restrict_mem</a> <a>measurableSet_Ioc</a>] with s hs using g_nn _ hs.1\n          have Ig : \u222b (t : \u211d) in <a>Ioc</a> 0 (u n), g t = 0 :=\n            <a>le_antisymm</a> H3 (<a>integral_nonneg_of_ae</a> g_nn_ae)\n          have J : \u2200\u1d50 t \u2202(volume.restrict (<a>Ioc</a> 0 (u n))), g t = 0 :=\n            (<a>integral_eq_zero_iff_of_nonneg_ae</a> g_nn_ae\n              (g_intble (u n) (M_nonneg.trans_lt (uM n))).1).1 Ig\n          have : u n \u2264 M := <a>le_csSup</a> M_bdd J\n          exact <a>lt_irrefl</a> _ (this.trans_lt (uM n))", [{"full_name": "lt_top_iff_ne_top", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [173, 9], "def_end_pos": [173, 26]}, {"full_name": "not_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [368, 9], "def_end_pos": [368, 15]}, {"full_name": "intervalIntegral.integral_of_le", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [465, 9], "def_end_pos": [465, 23]}, {"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [142, 7], "def_end_pos": [142, 15]}, {"full_name": "Set.Ioc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [69, 5], "def_end_pos": [69, 8]}, {"full_name": "MeasureTheory.ae_restrict_mem", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2586, 9], "def_end_pos": [2586, 24]}, {"full_name": "measurableSet_Ioc", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [589, 9], "def_end_pos": [589, 26]}, {"full_name": "Set.Ioc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [69, 5], "def_end_pos": [69, 8]}, {"full_name": "le_antisymm", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [188, 9], "def_end_pos": [188, 20]}, {"full_name": "MeasureTheory.integral_nonneg_of_ae", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1201, 9], "def_end_pos": [1201, 30]}, {"full_name": "Set.Ioc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [69, 5], "def_end_pos": [69, 8]}, {"full_name": "MeasureTheory.integral_eq_zero_iff_of_nonneg_ae", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1265, 9], "def_end_pos": [1265, 42]}, {"full_name": "le_csSup", "def_path": "Mathlib/Order/ConditionallyCompleteLattice/Basic.lean", "def_pos": [457, 9], "def_end_pos": [457, 17]}, {"full_name": "lt_irrefl", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [79, 9], "def_end_pos": [79, 18]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nhgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0\n\u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a}\nu : \u2115 \u2192 \u211d\nuM : \u2200 (n : \u2115), M < u n\nulim : Tendsto u atTop (\ud835\udcdd M)\nn : \u2115\nI : \u2191\u2191\u03bd {a | f a \u2264 M} = 0\n\u22a2 \u2191\u2191\u03bd ((fun n => {a | f a \u2264 M} \u222a {a | u n < f a}) n) < \u22a4", "state_after": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nhgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0\n\u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a}\nu : \u2115 \u2192 \u211d\nuM : \u2200 (n : \u2115), M < u n\nulim : Tendsto u atTop (\ud835\udcdd M)\nn : \u2115\nI : \u2191\u2191\u03bd {a | f a \u2264 M} = 0\nJ : \u2191\u2191\u03bc {a | u n < f a} < \u22a4\n\u22a2 \u2191\u2191\u03bd ((fun n => {a | f a \u2264 M} \u222a {a | u n < f a}) n) < \u22a4"}, {"tactic": "refine lt_of_le_of_lt (measure_union_le _ _) ?_", "annotated_tactic": ["refine <a>lt_of_le_of_lt</a> (<a>measure_union_le</a> _ _) ?_", [{"full_name": "lt_of_le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [122, 9], "def_end_pos": [122, 23]}, {"full_name": "MeasureTheory.measure_union_le", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [298, 9], "def_end_pos": [298, 25]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nhgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0\n\u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a}\nu : \u2115 \u2192 \u211d\nuM : \u2200 (n : \u2115), M < u n\nulim : Tendsto u atTop (\ud835\udcdd M)\nn : \u2115\nI : \u2191\u2191\u03bd {a | f a \u2264 M} = 0\nJ : \u2191\u2191\u03bc {a | u n < f a} < \u22a4\n\u22a2 \u2191\u2191\u03bd ((fun n => {a | f a \u2264 M} \u222a {a | u n < f a}) n) < \u22a4", "state_after": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nhgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0\n\u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a}\nu : \u2115 \u2192 \u211d\nuM : \u2200 (n : \u2115), M < u n\nulim : Tendsto u atTop (\ud835\udcdd M)\nn : \u2115\nI : \u2191\u2191\u03bd {a | f a \u2264 M} = 0\nJ : \u2191\u2191\u03bc {a | u n < f a} < \u22a4\n\u22a2 \u2191\u2191\u03bd {a | f a \u2264 M} + \u2191\u2191\u03bd {a | u n < f a} < \u22a4"}, {"tactic": "rw [I, zero_add]", "annotated_tactic": ["rw [I, <a>zero_add</a>]", [{"full_name": "zero_add", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [463, 3], "def_end_pos": [463, 14]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nhgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0\n\u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a}\nu : \u2115 \u2192 \u211d\nuM : \u2200 (n : \u2115), M < u n\nulim : Tendsto u atTop (\ud835\udcdd M)\nn : \u2115\nI : \u2191\u2191\u03bd {a | f a \u2264 M} = 0\nJ : \u2191\u2191\u03bc {a | u n < f a} < \u22a4\n\u22a2 \u2191\u2191\u03bd {a | f a \u2264 M} + \u2191\u2191\u03bd {a | u n < f a} < \u22a4", "state_after": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nhgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0\n\u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a}\nu : \u2115 \u2192 \u211d\nuM : \u2200 (n : \u2115), M < u n\nulim : Tendsto u atTop (\ud835\udcdd M)\nn : \u2115\nI : \u2191\u2191\u03bd {a | f a \u2264 M} = 0\nJ : \u2191\u2191\u03bc {a | u n < f a} < \u22a4\n\u22a2 \u2191\u2191\u03bd {a | u n < f a} < \u22a4"}, {"tactic": "apply lt_of_le_of_lt _ J", "annotated_tactic": ["apply <a>lt_of_le_of_lt</a> _ J", [{"full_name": "lt_of_le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [122, 9], "def_end_pos": [122, 23]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nhgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0\n\u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a}\nu : \u2115 \u2192 \u211d\nuM : \u2200 (n : \u2115), M < u n\nulim : Tendsto u atTop (\ud835\udcdd M)\nn : \u2115\nI : \u2191\u2191\u03bd {a | f a \u2264 M} = 0\nJ : \u2191\u2191\u03bc {a | u n < f a} < \u22a4\n\u22a2 \u2191\u2191\u03bd {a | u n < f a} < \u22a4", "state_after": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nhgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0\n\u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a}\nu : \u2115 \u2192 \u211d\nuM : \u2200 (n : \u2115), M < u n\nulim : Tendsto u atTop (\ud835\udcdd M)\nn : \u2115\nI : \u2191\u2191\u03bd {a | f a \u2264 M} = 0\nJ : \u2191\u2191\u03bc {a | u n < f a} < \u22a4\n\u22a2 \u2191\u2191\u03bd {a | u n < f a} \u2264 \u2191\u2191\u03bc {a | u n < f a}"}, {"tactic": "exact restrict_le_self _ (measurableSet_lt measurable_const f_mble)", "annotated_tactic": ["exact <a>restrict_le_self</a> _ (<a>measurableSet_lt</a> <a>measurable_const</a> f_mble)", [{"full_name": "MeasureTheory.Measure.restrict_le_self", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1578, 9], "def_end_pos": [1578, 25]}, {"full_name": "measurableSet_lt", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [616, 9], "def_end_pos": [616, 25]}, {"full_name": "measurable_const", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [570, 9], "def_end_pos": [570, 25]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nhgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0\n\u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a}\nu : \u2115 \u2192 \u211d\nuM : \u2200 (n : \u2115), M < u n\nulim : Tendsto u atTop (\ud835\udcdd M)\nn : \u2115\nI : \u2191\u2191\u03bd {a | f a \u2264 M} = 0\nJ : \u2191\u2191\u03bc {a | u n < f a} < \u22a4\n\u22a2 \u2191\u2191\u03bd {a | u n < f a} \u2264 \u2191\u2191\u03bc {a | u n < f a}", "state_after": "no goals"}, {"tactic": "rw [Measure.restrict_apply (measurableSet_le f_mble measurable_const)]", "annotated_tactic": ["rw [<a>Measure.restrict_apply</a> (<a>measurableSet_le</a> f_mble <a>measurable_const</a>)]", [{"full_name": "MeasureTheory.Measure.restrict_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1533, 9], "def_end_pos": [1533, 23]}, {"full_name": "measurableSet_le", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [559, 9], "def_end_pos": [559, 25]}, {"full_name": "measurable_const", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [570, 9], "def_end_pos": [570, 25]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nhgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0\n\u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a}\nu : \u2115 \u2192 \u211d\nuM : \u2200 (n : \u2115), M < u n\nulim : Tendsto u atTop (\ud835\udcdd M)\nn : \u2115\n\u22a2 \u2191\u2191\u03bd {a | f a \u2264 M} = 0", "state_after": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nhgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0\n\u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a}\nu : \u2115 \u2192 \u211d\nuM : \u2200 (n : \u2115), M < u n\nulim : Tendsto u atTop (\ud835\udcdd M)\nn : \u2115\n\u22a2 \u2191\u2191\u03bc ({a | f a \u2264 M} \u2229 {a | M < f a}) = 0"}, {"tactic": "convert measure_empty", "annotated_tactic": ["convert <a>measure_empty</a>", [{"full_name": "MeasureTheory.measure_empty", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [185, 9], "def_end_pos": [185, 22]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nhgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0\n\u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a}\nu : \u2115 \u2192 \u211d\nuM : \u2200 (n : \u2115), M < u n\nulim : Tendsto u atTop (\ud835\udcdd M)\nn : \u2115\n\u22a2 \u2191\u2191\u03bc ({a | f a \u2264 M} \u2229 {a | M < f a}) = 0", "state_after": "case h.e'_2.h.e'_3\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nhgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0\n\u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a}\nu : \u2115 \u2192 \u211d\nuM : \u2200 (n : \u2115), M < u n\nulim : Tendsto u atTop (\ud835\udcdd M)\nn : \u2115\n\u22a2 {a | f a \u2264 M} \u2229 {a | M < f a} = \u2205"}, {"tactic": "rw [\u2190 disjoint_iff_inter_eq_empty]", "annotated_tactic": ["rw [\u2190 <a>disjoint_iff_inter_eq_empty</a>]", [{"full_name": "Set.disjoint_iff_inter_eq_empty", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1538, 9], "def_end_pos": [1538, 36]}]], "state_before": "case h.e'_2.h.e'_3\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nhgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0\n\u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a}\nu : \u2115 \u2192 \u211d\nuM : \u2200 (n : \u2115), M < u n\nulim : Tendsto u atTop (\ud835\udcdd M)\nn : \u2115\n\u22a2 {a | f a \u2264 M} \u2229 {a | M < f a} = \u2205", "state_after": "case h.e'_2.h.e'_3\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nhgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0\n\u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a}\nu : \u2115 \u2192 \u211d\nuM : \u2200 (n : \u2115), M < u n\nulim : Tendsto u atTop (\ud835\udcdd M)\nn : \u2115\n\u22a2 Disjoint {a | f a \u2264 M} {a | M < f a}"}, {"tactic": "exact disjoint_left.mpr (fun a ha \u21a6 by simpa using ha)", "annotated_tactic": ["exact disjoint_left.mpr (fun a ha \u21a6 by simpa using ha)", []], "state_before": "case h.e'_2.h.e'_3\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nhgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0\n\u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a}\nu : \u2115 \u2192 \u211d\nuM : \u2200 (n : \u2115), M < u n\nulim : Tendsto u atTop (\ud835\udcdd M)\nn : \u2115\n\u22a2 Disjoint {a | f a \u2264 M} {a | M < f a}", "state_after": "no goals"}, {"tactic": "simpa using ha", "annotated_tactic": ["simpa using ha", []], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nhgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0\n\u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a}\nu : \u2115 \u2192 \u211d\nuM : \u2200 (n : \u2115), M < u n\nulim : Tendsto u atTop (\ud835\udcdd M)\nn : \u2115\na : \u03b1\nha : a \u2208 {a | f a \u2264 M}\n\u22a2 \u00aca \u2208 {a | M < f a}", "state_after": "no goals"}, {"tactic": "rw [lt_top_iff_ne_top]", "annotated_tactic": ["rw [<a>lt_top_iff_ne_top</a>]", [{"full_name": "lt_top_iff_ne_top", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [173, 9], "def_end_pos": [173, 26]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nhgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0\n\u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a}\nu : \u2115 \u2192 \u211d\nuM : \u2200 (n : \u2115), M < u n\nulim : Tendsto u atTop (\ud835\udcdd M)\nn : \u2115\nI : \u2191\u2191\u03bd {a | f a \u2264 M} = 0\n\u22a2 \u2191\u2191\u03bc {a | u n < f a} < \u22a4", "state_after": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nhgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0\n\u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a}\nu : \u2115 \u2192 \u211d\nuM : \u2200 (n : \u2115), M < u n\nulim : Tendsto u atTop (\ud835\udcdd M)\nn : \u2115\nI : \u2191\u2191\u03bd {a | f a \u2264 M} = 0\n\u22a2 \u2191\u2191\u03bc {a | u n < f a} \u2260 \u22a4"}, {"tactic": "apply H2 _ (M_nonneg.trans_lt (uM n))", "annotated_tactic": ["apply H2 _ (M_nonneg.trans_lt (uM n))", []], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nhgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0\n\u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a}\nu : \u2115 \u2192 \u211d\nuM : \u2200 (n : \u2115), M < u n\nulim : Tendsto u atTop (\ud835\udcdd M)\nn : \u2115\nI : \u2191\u2191\u03bd {a | f a \u2264 M} = 0\n\u22a2 \u2191\u2191\u03bc {a | u n < f a} \u2260 \u22a4", "state_after": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nhgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0\n\u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a}\nu : \u2115 \u2192 \u211d\nuM : \u2200 (n : \u2115), M < u n\nulim : Tendsto u atTop (\ud835\udcdd M)\nn : \u2115\nI : \u2191\u2191\u03bd {a | f a \u2264 M} = 0\n\u22a2 0 < \u222b (t : \u211d) in 0 ..u n, g t"}, {"tactic": "by_contra H3", "annotated_tactic": ["by_contra H3", []], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nhgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0\n\u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a}\nu : \u2115 \u2192 \u211d\nuM : \u2200 (n : \u2115), M < u n\nulim : Tendsto u atTop (\ud835\udcdd M)\nn : \u2115\nI : \u2191\u2191\u03bd {a | f a \u2264 M} = 0\n\u22a2 0 < \u222b (t : \u211d) in 0 ..u n, g t", "state_after": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nhgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0\n\u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a}\nu : \u2115 \u2192 \u211d\nuM : \u2200 (n : \u2115), M < u n\nulim : Tendsto u atTop (\ud835\udcdd M)\nn : \u2115\nI : \u2191\u2191\u03bd {a | f a \u2264 M} = 0\nH3 : \u00ac0 < \u222b (t : \u211d) in 0 ..u n, g t\n\u22a2 False"}, {"tactic": "rw [not_lt, intervalIntegral.integral_of_le (M_nonneg.trans (uM n).le)] at H3", "annotated_tactic": ["rw [<a>not_lt</a>, <a>intervalIntegral.integral_of_le</a> (M_nonneg.trans (uM n).<a>le</a>)] at H3", [{"full_name": "not_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [368, 9], "def_end_pos": [368, 15]}, {"full_name": "intervalIntegral.integral_of_le", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [465, 9], "def_end_pos": [465, 23]}, {"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [142, 7], "def_end_pos": [142, 15]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nhgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0\n\u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a}\nu : \u2115 \u2192 \u211d\nuM : \u2200 (n : \u2115), M < u n\nulim : Tendsto u atTop (\ud835\udcdd M)\nn : \u2115\nI : \u2191\u2191\u03bd {a | f a \u2264 M} = 0\nH3 : \u00ac0 < \u222b (t : \u211d) in 0 ..u n, g t\n\u22a2 False", "state_after": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nhgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0\n\u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a}\nu : \u2115 \u2192 \u211d\nuM : \u2200 (n : \u2115), M < u n\nulim : Tendsto u atTop (\ud835\udcdd M)\nn : \u2115\nI : \u2191\u2191\u03bd {a | f a \u2264 M} = 0\nH3 : \u222b (x : \u211d) in Ioc 0 (u n), g x \u2264 0\n\u22a2 False"}, {"tactic": "have g_nn_ae : \u2200\u1d50 t \u2202(volume.restrict (Ioc 0 (u n))), 0 \u2264 g t := by\n  filter_upwards [ae_restrict_mem measurableSet_Ioc] with s hs using g_nn _ hs.1", "annotated_tactic": ["have g_nn_ae : \u2200\u1d50 t \u2202(volume.restrict (<a>Ioc</a> 0 (u n))), 0 \u2264 g t := by\n            filter_upwards [<a>ae_restrict_mem</a> <a>measurableSet_Ioc</a>] with s hs using g_nn _ hs.1", [{"full_name": "Set.Ioc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [69, 5], "def_end_pos": [69, 8]}, {"full_name": "MeasureTheory.ae_restrict_mem", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2586, 9], "def_end_pos": [2586, 24]}, {"full_name": "measurableSet_Ioc", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [589, 9], "def_end_pos": [589, 26]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nhgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0\n\u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a}\nu : \u2115 \u2192 \u211d\nuM : \u2200 (n : \u2115), M < u n\nulim : Tendsto u atTop (\ud835\udcdd M)\nn : \u2115\nI : \u2191\u2191\u03bd {a | f a \u2264 M} = 0\nH3 : \u222b (x : \u211d) in Ioc 0 (u n), g x \u2264 0\n\u22a2 False", "state_after": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nhgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0\n\u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a}\nu : \u2115 \u2192 \u211d\nuM : \u2200 (n : \u2115), M < u n\nulim : Tendsto u atTop (\ud835\udcdd M)\nn : \u2115\nI : \u2191\u2191\u03bd {a | f a \u2264 M} = 0\nH3 : \u222b (x : \u211d) in Ioc 0 (u n), g x \u2264 0\ng_nn_ae : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioc 0 (u n)), 0 \u2264 g t\n\u22a2 False"}, {"tactic": "have Ig : \u222b (t : \u211d) in Ioc 0 (u n), g t = 0 :=\n  le_antisymm H3 (integral_nonneg_of_ae g_nn_ae)", "annotated_tactic": ["have Ig : \u222b (t : \u211d) in <a>Ioc</a> 0 (u n), g t = 0 :=\n            <a>le_antisymm</a> H3 (<a>integral_nonneg_of_ae</a> g_nn_ae)", [{"full_name": "Set.Ioc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [69, 5], "def_end_pos": [69, 8]}, {"full_name": "le_antisymm", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [188, 9], "def_end_pos": [188, 20]}, {"full_name": "MeasureTheory.integral_nonneg_of_ae", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1201, 9], "def_end_pos": [1201, 30]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nhgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0\n\u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a}\nu : \u2115 \u2192 \u211d\nuM : \u2200 (n : \u2115), M < u n\nulim : Tendsto u atTop (\ud835\udcdd M)\nn : \u2115\nI : \u2191\u2191\u03bd {a | f a \u2264 M} = 0\nH3 : \u222b (x : \u211d) in Ioc 0 (u n), g x \u2264 0\ng_nn_ae : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioc 0 (u n)), 0 \u2264 g t\n\u22a2 False", "state_after": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nhgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0\n\u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a}\nu : \u2115 \u2192 \u211d\nuM : \u2200 (n : \u2115), M < u n\nulim : Tendsto u atTop (\ud835\udcdd M)\nn : \u2115\nI : \u2191\u2191\u03bd {a | f a \u2264 M} = 0\nH3 : \u222b (x : \u211d) in Ioc 0 (u n), g x \u2264 0\ng_nn_ae : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioc 0 (u n)), 0 \u2264 g t\nIg : \u222b (t : \u211d) in Ioc 0 (u n), g t = 0\n\u22a2 False"}, {"tactic": "have J : \u2200\u1d50 t \u2202(volume.restrict (Ioc 0 (u n))), g t = 0 :=\n  (integral_eq_zero_iff_of_nonneg_ae g_nn_ae\n    (g_intble (u n) (M_nonneg.trans_lt (uM n))).1).1 Ig", "annotated_tactic": ["have J : \u2200\u1d50 t \u2202(volume.restrict (<a>Ioc</a> 0 (u n))), g t = 0 :=\n            (<a>integral_eq_zero_iff_of_nonneg_ae</a> g_nn_ae\n              (g_intble (u n) (M_nonneg.trans_lt (uM n))).1).1 Ig", [{"full_name": "Set.Ioc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [69, 5], "def_end_pos": [69, 8]}, {"full_name": "MeasureTheory.integral_eq_zero_iff_of_nonneg_ae", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1265, 9], "def_end_pos": [1265, 42]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nhgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0\n\u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a}\nu : \u2115 \u2192 \u211d\nuM : \u2200 (n : \u2115), M < u n\nulim : Tendsto u atTop (\ud835\udcdd M)\nn : \u2115\nI : \u2191\u2191\u03bd {a | f a \u2264 M} = 0\nH3 : \u222b (x : \u211d) in Ioc 0 (u n), g x \u2264 0\ng_nn_ae : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioc 0 (u n)), 0 \u2264 g t\nIg : \u222b (t : \u211d) in Ioc 0 (u n), g t = 0\n\u22a2 False", "state_after": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nhgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0\n\u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a}\nu : \u2115 \u2192 \u211d\nuM : \u2200 (n : \u2115), M < u n\nulim : Tendsto u atTop (\ud835\udcdd M)\nn : \u2115\nI : \u2191\u2191\u03bd {a | f a \u2264 M} = 0\nH3 : \u222b (x : \u211d) in Ioc 0 (u n), g x \u2264 0\ng_nn_ae : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioc 0 (u n)), 0 \u2264 g t\nIg : \u222b (t : \u211d) in Ioc 0 (u n), g t = 0\nJ : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioc 0 (u n)), g t = 0\n\u22a2 False"}, {"tactic": "have : u n \u2264 M := le_csSup M_bdd J", "annotated_tactic": ["have : u n \u2264 M := <a>le_csSup</a> M_bdd J", [{"full_name": "le_csSup", "def_path": "Mathlib/Order/ConditionallyCompleteLattice/Basic.lean", "def_pos": [457, 9], "def_end_pos": [457, 17]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nhgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0\n\u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a}\nu : \u2115 \u2192 \u211d\nuM : \u2200 (n : \u2115), M < u n\nulim : Tendsto u atTop (\ud835\udcdd M)\nn : \u2115\nI : \u2191\u2191\u03bd {a | f a \u2264 M} = 0\nH3 : \u222b (x : \u211d) in Ioc 0 (u n), g x \u2264 0\ng_nn_ae : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioc 0 (u n)), 0 \u2264 g t\nIg : \u222b (t : \u211d) in Ioc 0 (u n), g t = 0\nJ : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioc 0 (u n)), g t = 0\n\u22a2 False", "state_after": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nhgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0\n\u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a}\nu : \u2115 \u2192 \u211d\nuM : \u2200 (n : \u2115), M < u n\nulim : Tendsto u atTop (\ud835\udcdd M)\nn : \u2115\nI : \u2191\u2191\u03bd {a | f a \u2264 M} = 0\nH3 : \u222b (x : \u211d) in Ioc 0 (u n), g x \u2264 0\ng_nn_ae : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioc 0 (u n)), 0 \u2264 g t\nIg : \u222b (t : \u211d) in Ioc 0 (u n), g t = 0\nJ : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioc 0 (u n)), g t = 0\nthis : u n \u2264 M\n\u22a2 False"}, {"tactic": "exact lt_irrefl _ (this.trans_lt (uM n))", "annotated_tactic": ["exact <a>lt_irrefl</a> _ (this.trans_lt (uM n))", [{"full_name": "lt_irrefl", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [79, 9], "def_end_pos": [79, 18]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nhgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0\n\u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a}\nu : \u2115 \u2192 \u211d\nuM : \u2200 (n : \u2115), M < u n\nulim : Tendsto u atTop (\ud835\udcdd M)\nn : \u2115\nI : \u2191\u2191\u03bd {a | f a \u2264 M} = 0\nH3 : \u222b (x : \u211d) in Ioc 0 (u n), g x \u2264 0\ng_nn_ae : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioc 0 (u n)), 0 \u2264 g t\nIg : \u222b (t : \u211d) in Ioc 0 (u n), g t = 0\nJ : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioc 0 (u n)), g t = 0\nthis : u n \u2264 M\n\u22a2 False", "state_after": "no goals"}, {"tactic": "filter_upwards [ae_restrict_mem measurableSet_Ioc] with s hs using g_nn _ hs.1", "annotated_tactic": ["filter_upwards [<a>ae_restrict_mem</a> <a>measurableSet_Ioc</a>] with s hs using g_nn _ hs.1", [{"full_name": "MeasureTheory.ae_restrict_mem", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2586, 9], "def_end_pos": [2586, 24]}, {"full_name": "measurableSet_Ioc", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [589, 9], "def_end_pos": [589, 26]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nhgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0\n\u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a}\nu : \u2115 \u2192 \u211d\nuM : \u2200 (n : \u2115), M < u n\nulim : Tendsto u atTop (\ud835\udcdd M)\nn : \u2115\nI : \u2191\u2191\u03bd {a | f a \u2264 M} = 0\nH3 : \u222b (x : \u211d) in Ioc 0 (u n), g x \u2264 0\n\u22a2 \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioc 0 (u n)), 0 \u2264 g t", "state_after": "no goals"}, {"tactic": "apply eq_univ_iff_forall.2 (fun a \u21a6 ?_)", "annotated_tactic": ["apply <a>eq_univ_iff_forall</a>.2 (fun a \u21a6 ?_)", [{"full_name": "Set.eq_univ_iff_forall", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [701, 9], "def_end_pos": [701, 27]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nhgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0\n\u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a}\nu : \u2115 \u2192 \u211d\nuM : \u2200 (n : \u2115), M < u n\nulim : Tendsto u atTop (\ud835\udcdd M)\n\u22a2 \u22c3 i, (fun n => {a | f a \u2264 M} \u222a {a | u n < f a}) i = univ", "state_after": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nhgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0\n\u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a}\nu : \u2115 \u2192 \u211d\nuM : \u2200 (n : \u2115), M < u n\nulim : Tendsto u atTop (\ud835\udcdd M)\na : \u03b1\n\u22a2 a \u2208 \u22c3 i, (fun n => {a | f a \u2264 M} \u222a {a | u n < f a}) i"}, {"tactic": "rcases le_or_lt (f a) M with ha|ha", "annotated_tactic": ["rcases <a>le_or_lt</a> (f a) M with ha|ha", [{"full_name": "le_or_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [340, 9], "def_end_pos": [340, 17]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nhgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0\n\u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a}\nu : \u2115 \u2192 \u211d\nuM : \u2200 (n : \u2115), M < u n\nulim : Tendsto u atTop (\ud835\udcdd M)\na : \u03b1\n\u22a2 a \u2208 \u22c3 i, (fun n => {a | f a \u2264 M} \u222a {a | u n < f a}) i", "state_after": "case inl\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nhgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0\n\u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a}\nu : \u2115 \u2192 \u211d\nuM : \u2200 (n : \u2115), M < u n\nulim : Tendsto u atTop (\ud835\udcdd M)\na : \u03b1\nha : f a \u2264 M\n\u22a2 a \u2208 \u22c3 i, (fun n => {a | f a \u2264 M} \u222a {a | u n < f a}) i\n\ncase inr\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nhgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0\n\u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a}\nu : \u2115 \u2192 \u211d\nuM : \u2200 (n : \u2115), M < u n\nulim : Tendsto u atTop (\ud835\udcdd M)\na : \u03b1\nha : M < f a\n\u22a2 a \u2208 \u22c3 i, (fun n => {a | f a \u2264 M} \u222a {a | u n < f a}) i"}, {"tactic": "exact mem_iUnion.2 \u27e80, Or.inl ha\u27e9", "annotated_tactic": ["exact <a>mem_iUnion</a>.2 \u27e80, <a>Or.inl</a> ha\u27e9", [{"full_name": "Set.mem_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [201, 9], "def_end_pos": [201, 19]}, {"full_name": "Or.inl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [517, 5], "def_end_pos": [517, 8]}]], "state_before": "case inl\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nhgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0\n\u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a}\nu : \u2115 \u2192 \u211d\nuM : \u2200 (n : \u2115), M < u n\nulim : Tendsto u atTop (\ud835\udcdd M)\na : \u03b1\nha : f a \u2264 M\n\u22a2 a \u2208 \u22c3 i, (fun n => {a | f a \u2264 M} \u222a {a | u n < f a}) i", "state_after": "no goals"}, {"tactic": "obtain \u27e8n, hn\u27e9 : \u2203 n, u n < f a := ((tendsto_order.1 ulim).2 _ ha).exists", "annotated_tactic": ["obtain \u27e8n, hn\u27e9 : \u2203 n, u n < f a := ((<a>tendsto_order</a>.1 ulim).2 _ ha).<a>exists</a>", [{"full_name": "tendsto_order", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [919, 9], "def_end_pos": [919, 22]}, {"full_name": "Filter.Eventually.exists", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1308, 9], "def_end_pos": [1308, 26]}]], "state_before": "case inr\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nhgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0\n\u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a}\nu : \u2115 \u2192 \u211d\nuM : \u2200 (n : \u2115), M < u n\nulim : Tendsto u atTop (\ud835\udcdd M)\na : \u03b1\nha : M < f a\n\u22a2 a \u2208 \u22c3 i, (fun n => {a | f a \u2264 M} \u222a {a | u n < f a}) i", "state_after": "case inr.intro\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nhgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0\n\u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a}\nu : \u2115 \u2192 \u211d\nuM : \u2200 (n : \u2115), M < u n\nulim : Tendsto u atTop (\ud835\udcdd M)\na : \u03b1\nha : M < f a\nn : \u2115\nhn : u n < f a\n\u22a2 a \u2208 \u22c3 i, (fun n => {a | f a \u2264 M} \u222a {a | u n < f a}) i"}, {"tactic": "exact mem_iUnion.2 \u27e8n, Or.inr hn\u27e9", "annotated_tactic": ["exact <a>mem_iUnion</a>.2 \u27e8n, <a>Or.inr</a> hn\u27e9", [{"full_name": "Set.mem_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [201, 9], "def_end_pos": [201, 19]}, {"full_name": "Or.inr", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [519, 5], "def_end_pos": [519, 8]}]], "state_before": "case inr.intro\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nhgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0\n\u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a}\nu : \u2115 \u2192 \u211d\nuM : \u2200 (n : \u2115), M < u n\nulim : Tendsto u atTop (\ud835\udcdd M)\na : \u03b1\nha : M < f a\nn : \u2115\nhn : u n < f a\n\u22a2 a \u2208 \u22c3 i, (fun n => {a | f a \u2264 M} \u222a {a | u n < f a}) i", "state_after": "no goals"}, {"tactic": "have meas : MeasurableSet {a | M < f a} := measurableSet_lt measurable_const f_mble", "annotated_tactic": ["have meas : <a>MeasurableSet</a> {a | M < f a} := <a>measurableSet_lt</a> <a>measurable_const</a> f_mble", [{"full_name": "MeasurableSet", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [64, 5], "def_end_pos": [64, 18]}, {"full_name": "measurableSet_lt", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [616, 9], "def_end_pos": [616, 25]}, {"full_name": "measurable_const", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [570, 9], "def_end_pos": [570, 25]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nhgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0\n\u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a}\nthis : SigmaFinite \u03bd\n\u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc = \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bd", "state_after": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nhgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0\n\u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a}\nthis : SigmaFinite \u03bd\nmeas : MeasurableSet {a | M < f a}\n\u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc = \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bd"}, {"tactic": "have I : \u222b\u207b \u03c9 in {a | M < f a}\u1d9c, ENNReal.ofReal (\u222b t in (0).. f \u03c9, g t) \u2202\u03bc\n         = \u222b\u207b _ in {a | M < f a}\u1d9c, 0 \u2202\u03bc := by\n  apply set_lintegral_congr_fun meas.compl (eventually_of_forall (fun s hs \u21a6 ?_))\n  have : \u222b (t : \u211d) in (0)..f s, g t = \u222b (t : \u211d) in (0)..f s, 0 := by\n    simp_rw [intervalIntegral.integral_of_le (f_nonneg s)]\n    apply integral_congr_ae\n    apply ae_mono (restrict_mono ?_ le_rfl) hgM\n    apply Ioc_subset_Ioc_right\n    simpa using hs\n  simp [this]", "annotated_tactic": ["have I : \u222b\u207b \u03c9 in {a | M < f a}\u1d9c, <a>ENNReal.ofReal</a> (\u222b t in (0).. f \u03c9, g t) \u2202\u03bc\n             = \u222b\u207b _ in {a | M < f a}\u1d9c, 0 \u2202\u03bc := by\n      apply <a>set_lintegral_congr_fun</a> meas.compl (<a>eventually_of_forall</a> (fun s hs \u21a6 ?_))\n      have : \u222b (t : \u211d) in (0)..f s, g t = \u222b (t : \u211d) in (0)..f s, 0 := by\n        simp_rw [<a>intervalIntegral.integral_of_le</a> (f_nonneg s)]\n        apply <a>integral_congr_ae</a>\n        apply <a>ae_mono</a> (<a>restrict_mono</a> ?_ <a>le_rfl</a>) hgM\n        apply <a>Ioc_subset_Ioc_right</a>\n        simpa using hs\n      simp [this]", [{"full_name": "ENNReal.ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [172, 29], "def_end_pos": [172, 35]}, {"full_name": "MeasureTheory.set_lintegral_congr_fun", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [316, 9], "def_end_pos": [316, 32]}, {"full_name": "Filter.eventually_of_forall", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1111, 9], "def_end_pos": [1111, 29]}, {"full_name": "intervalIntegral.integral_of_le", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [465, 9], "def_end_pos": [465, 23]}, {"full_name": "MeasureTheory.integral_congr_ae", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [938, 9], "def_end_pos": [938, 26]}, {"full_name": "MeasureTheory.ae_mono", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2456, 9], "def_end_pos": [2456, 16]}, {"full_name": "MeasureTheory.Measure.restrict_mono", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1550, 9], "def_end_pos": [1550, 22]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}, {"full_name": "Set.Ioc_subset_Ioc_right", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [488, 9], "def_end_pos": [488, 29]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nhgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0\n\u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a}\nthis : SigmaFinite \u03bd\nmeas : MeasurableSet {a | M < f a}\n\u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc = \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bd", "state_after": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nhgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0\n\u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a}\nthis : SigmaFinite \u03bd\nmeas : MeasurableSet {a | M < f a}\nI : \u222b\u207b (\u03c9 : \u03b1) in {a | M < f a}\u1d9c, ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc = \u222b\u207b (x : \u03b1) in {a | M < f a}\u1d9c, 0 \u2202\u03bc\n\u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc = \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bd"}, {"tactic": "simp only [lintegral_const, zero_mul] at I", "annotated_tactic": ["simp only [<a>lintegral_const</a>, <a>zero_mul</a>] at I", [{"full_name": "MeasureTheory.lintegral_const", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [136, 9], "def_end_pos": [136, 24]}, {"full_name": "MulZeroClass.zero_mul", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [36, 3], "def_end_pos": [36, 11]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nhgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0\n\u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a}\nthis : SigmaFinite \u03bd\nmeas : MeasurableSet {a | M < f a}\nI : \u222b\u207b (\u03c9 : \u03b1) in {a | M < f a}\u1d9c, ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc = \u222b\u207b (x : \u03b1) in {a | M < f a}\u1d9c, 0 \u2202\u03bc\n\u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc = \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bd", "state_after": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nhgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0\n\u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a}\nthis : SigmaFinite \u03bd\nmeas : MeasurableSet {a | M < f a}\nI :\n  \u222b\u207b (\u03c9 : \u03b1) in {a | sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} < f a}\u1d9c,\n      ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =\n    0\n\u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc = \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bd"}, {"tactic": "rw [\u2190 lintegral_add_compl _ meas, I, add_zero]", "annotated_tactic": ["rw [\u2190 <a>lintegral_add_compl</a> _ meas, I, <a>add_zero</a>]", [{"full_name": "MeasureTheory.lintegral_add_compl", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [1258, 9], "def_end_pos": [1258, 28]}, {"full_name": "add_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [469, 3], "def_end_pos": [469, 14]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nhgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0\n\u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a}\nthis : SigmaFinite \u03bd\nmeas : MeasurableSet {a | M < f a}\nI :\n  \u222b\u207b (\u03c9 : \u03b1) in {a | sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} < f a}\u1d9c,\n      ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =\n    0\n\u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc = \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bd", "state_after": "no goals"}, {"tactic": "apply set_lintegral_congr_fun meas.compl (eventually_of_forall (fun s hs \u21a6 ?_))", "annotated_tactic": ["apply <a>set_lintegral_congr_fun</a> meas.compl (<a>eventually_of_forall</a> (fun s hs \u21a6 ?_))", [{"full_name": "MeasureTheory.set_lintegral_congr_fun", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [316, 9], "def_end_pos": [316, 32]}, {"full_name": "Filter.eventually_of_forall", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1111, 9], "def_end_pos": [1111, 29]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nhgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0\n\u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a}\nthis : SigmaFinite \u03bd\nmeas : MeasurableSet {a | M < f a}\n\u22a2 \u222b\u207b (\u03c9 : \u03b1) in {a | M < f a}\u1d9c, ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc = \u222b\u207b (x : \u03b1) in {a | M < f a}\u1d9c, 0 \u2202\u03bc", "state_after": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nhgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0\n\u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a}\nthis : SigmaFinite \u03bd\nmeas : MeasurableSet {a | M < f a}\ns : \u03b1\nhs : s \u2208 {a | M < f a}\u1d9c\n\u22a2 ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f s, g t) = 0"}, {"tactic": "have : \u222b (t : \u211d) in (0)..f s, g t = \u222b (t : \u211d) in (0)..f s, 0 := by\n  simp_rw [intervalIntegral.integral_of_le (f_nonneg s)]\n  apply integral_congr_ae\n  apply ae_mono (restrict_mono ?_ le_rfl) hgM\n  apply Ioc_subset_Ioc_right\n  simpa using hs", "annotated_tactic": ["have : \u222b (t : \u211d) in (0)..f s, g t = \u222b (t : \u211d) in (0)..f s, 0 := by\n        simp_rw [<a>intervalIntegral.integral_of_le</a> (f_nonneg s)]\n        apply <a>integral_congr_ae</a>\n        apply <a>ae_mono</a> (<a>restrict_mono</a> ?_ <a>le_rfl</a>) hgM\n        apply <a>Ioc_subset_Ioc_right</a>\n        simpa using hs", [{"full_name": "intervalIntegral.integral_of_le", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [465, 9], "def_end_pos": [465, 23]}, {"full_name": "MeasureTheory.integral_congr_ae", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [938, 9], "def_end_pos": [938, 26]}, {"full_name": "MeasureTheory.ae_mono", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2456, 9], "def_end_pos": [2456, 16]}, {"full_name": "MeasureTheory.Measure.restrict_mono", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1550, 9], "def_end_pos": [1550, 22]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}, {"full_name": "Set.Ioc_subset_Ioc_right", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [488, 9], "def_end_pos": [488, 29]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nhgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0\n\u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a}\nthis : SigmaFinite \u03bd\nmeas : MeasurableSet {a | M < f a}\ns : \u03b1\nhs : s \u2208 {a | M < f a}\u1d9c\n\u22a2 ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f s, g t) = 0", "state_after": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nhgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0\n\u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a}\nthis\u271d : SigmaFinite \u03bd\nmeas : MeasurableSet {a | M < f a}\ns : \u03b1\nhs : s \u2208 {a | M < f a}\u1d9c\nthis : \u222b (t : \u211d) in 0 ..f s, g t = \u222b (t : \u211d) in 0 ..f s, 0\n\u22a2 ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f s, g t) = 0"}, {"tactic": "simp [this]", "annotated_tactic": ["simp [this]", []], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nhgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0\n\u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a}\nthis\u271d : SigmaFinite \u03bd\nmeas : MeasurableSet {a | M < f a}\ns : \u03b1\nhs : s \u2208 {a | M < f a}\u1d9c\nthis : \u222b (t : \u211d) in 0 ..f s, g t = \u222b (t : \u211d) in 0 ..f s, 0\n\u22a2 ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f s, g t) = 0", "state_after": "no goals"}, {"tactic": "simp_rw [intervalIntegral.integral_of_le (f_nonneg s)]", "annotated_tactic": ["simp_rw [<a>intervalIntegral.integral_of_le</a> (f_nonneg s)]", [{"full_name": "intervalIntegral.integral_of_le", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [465, 9], "def_end_pos": [465, 23]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nhgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0\n\u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a}\nthis : SigmaFinite \u03bd\nmeas : MeasurableSet {a | M < f a}\ns : \u03b1\nhs : s \u2208 {a | M < f a}\u1d9c\n\u22a2 \u222b (t : \u211d) in 0 ..f s, g t = \u222b (t : \u211d) in 0 ..f s, 0", "state_after": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nhgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0\n\u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a}\nthis : SigmaFinite \u03bd\nmeas : MeasurableSet {a | M < f a}\ns : \u03b1\nhs : s \u2208 {a | M < f a}\u1d9c\n\u22a2 \u222b (t : \u211d) in Ioc 0 (f s), g t = \u222b (t : \u211d) in Ioc 0 (f s), 0"}, {"tactic": "apply integral_congr_ae", "annotated_tactic": ["apply <a>integral_congr_ae</a>", [{"full_name": "MeasureTheory.integral_congr_ae", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [938, 9], "def_end_pos": [938, 26]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nhgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0\n\u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a}\nthis : SigmaFinite \u03bd\nmeas : MeasurableSet {a | M < f a}\ns : \u03b1\nhs : s \u2208 {a | M < f a}\u1d9c\n\u22a2 \u222b (t : \u211d) in Ioc 0 (f s), g t = \u222b (t : \u211d) in Ioc 0 (f s), 0", "state_after": "case h\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nhgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0\n\u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a}\nthis : SigmaFinite \u03bd\nmeas : MeasurableSet {a | M < f a}\ns : \u03b1\nhs : s \u2208 {a | M < f a}\u1d9c\n\u22a2 (fun a => g a) =\u1da0[ae (Measure.restrict volume (Ioc 0 (f s)))] fun a => 0"}, {"tactic": "apply ae_mono (restrict_mono ?_ le_rfl) hgM", "annotated_tactic": ["apply <a>ae_mono</a> (<a>restrict_mono</a> ?_ <a>le_rfl</a>) hgM", [{"full_name": "MeasureTheory.ae_mono", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2456, 9], "def_end_pos": [2456, 16]}, {"full_name": "MeasureTheory.Measure.restrict_mono", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1550, 9], "def_end_pos": [1550, 22]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}]], "state_before": "case h\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nhgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0\n\u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a}\nthis : SigmaFinite \u03bd\nmeas : MeasurableSet {a | M < f a}\ns : \u03b1\nhs : s \u2208 {a | M < f a}\u1d9c\n\u22a2 (fun a => g a) =\u1da0[ae (Measure.restrict volume (Ioc 0 (f s)))] fun a => 0", "state_after": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nhgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0\n\u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a}\nthis : SigmaFinite \u03bd\nmeas : MeasurableSet {a | M < f a}\ns : \u03b1\nhs : s \u2208 {a | M < f a}\u1d9c\n\u22a2 Ioc 0 (f s) \u2286 Ioc 0 M"}, {"tactic": "apply Ioc_subset_Ioc_right", "annotated_tactic": ["apply <a>Ioc_subset_Ioc_right</a>", [{"full_name": "Set.Ioc_subset_Ioc_right", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [488, 9], "def_end_pos": [488, 29]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nhgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0\n\u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a}\nthis : SigmaFinite \u03bd\nmeas : MeasurableSet {a | M < f a}\ns : \u03b1\nhs : s \u2208 {a | M < f a}\u1d9c\n\u22a2 Ioc 0 (f s) \u2286 Ioc 0 M", "state_after": "case h\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nhgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0\n\u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a}\nthis : SigmaFinite \u03bd\nmeas : MeasurableSet {a | M < f a}\ns : \u03b1\nhs : s \u2208 {a | M < f a}\u1d9c\n\u22a2 f s \u2264 M"}, {"tactic": "simpa using hs", "annotated_tactic": ["simpa using hs", []], "state_before": "case h\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nhgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0\n\u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a}\nthis : SigmaFinite \u03bd\nmeas : MeasurableSet {a | M < f a}\ns : \u03b1\nhs : s \u2208 {a | M < f a}\u1d9c\n\u22a2 f s \u2264 M", "state_after": "no goals"}, {"tactic": "have B1 : \u222b\u207b t in Ioc 0 M, \u03bc {a : \u03b1 | t \u2264 f a} * ENNReal.ofReal (g t)\n     = \u222b\u207b t in Ioc 0 M, \u03bd {a : \u03b1 | t \u2264 f a} * ENNReal.ofReal (g t) := by\n  apply lintegral_congr_ae\n  filter_upwards [hgM] with t ht\n  simp [ht]", "annotated_tactic": ["have B1 : \u222b\u207b t in <a>Ioc</a> 0 M, \u03bc {a : \u03b1 | t \u2264 f a} * <a>ENNReal.ofReal</a> (g t)\n         = \u222b\u207b t in <a>Ioc</a> 0 M, \u03bd {a : \u03b1 | t \u2264 f a} * <a>ENNReal.ofReal</a> (g t) := by\n      apply <a>lintegral_congr_ae</a>\n      filter_upwards [hgM] with t ht\n      simp [ht]", [{"full_name": "Set.Ioc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [69, 5], "def_end_pos": [69, 8]}, {"full_name": "ENNReal.ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [172, 29], "def_end_pos": [172, 35]}, {"full_name": "Set.Ioc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [69, 5], "def_end_pos": [69, 8]}, {"full_name": "ENNReal.ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [172, 29], "def_end_pos": [172, 35]}, {"full_name": "MeasureTheory.lintegral_congr_ae", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [304, 9], "def_end_pos": [304, 27]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nhgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0\n\u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a}\nthis : SigmaFinite \u03bd\nA :\n  \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc = \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bd\n\u22a2 \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) =\n    \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bd {a | t \u2264 f a} * ENNReal.ofReal (g t)", "state_after": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nhgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0\n\u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a}\nthis : SigmaFinite \u03bd\nA :\n  \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc = \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bd\nB1 :\n  \u222b\u207b (t : \u211d) in Ioc 0 M, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) =\n    \u222b\u207b (t : \u211d) in Ioc 0 M, \u2191\u2191\u03bd {a | t \u2264 f a} * ENNReal.ofReal (g t)\n\u22a2 \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) =\n    \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bd {a | t \u2264 f a} * ENNReal.ofReal (g t)"}, {"tactic": "have B2 : \u222b\u207b t in Ioi M, \u03bc {a : \u03b1 | t \u2264 f a} * ENNReal.ofReal (g t)\n          = \u222b\u207b t in Ioi M, \u03bd {a : \u03b1 | t \u2264 f a} * ENNReal.ofReal (g t) := by\n  apply set_lintegral_congr_fun measurableSet_Ioi (eventually_of_forall (fun t ht \u21a6 ?_))\n  rw [Measure.restrict_apply (measurableSet_le measurable_const f_mble)]\n  congr 3\n  exact (inter_eq_left.2 (fun a ha \u21a6 (mem_Ioi.1 ht).trans_le ha)).symm", "annotated_tactic": ["have B2 : \u222b\u207b t in <a>Ioi</a> M, \u03bc {a : \u03b1 | t \u2264 f a} * <a>ENNReal.ofReal</a> (g t)\n              = \u222b\u207b t in <a>Ioi</a> M, \u03bd {a : \u03b1 | t \u2264 f a} * <a>ENNReal.ofReal</a> (g t) := by\n      apply <a>set_lintegral_congr_fun</a> <a>measurableSet_Ioi</a> (<a>eventually_of_forall</a> (fun t ht \u21a6 ?_))\n      rw [<a>Measure.restrict_apply</a> (<a>measurableSet_le</a> <a>measurable_const</a> f_mble)]\n      congr 3\n      exact (<a>inter_eq_left</a>.2 (fun a ha \u21a6 (<a>mem_Ioi</a>.1 ht).<a>trans_le</a> ha)).<a>symm</a>", [{"full_name": "Set.Ioi", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [79, 5], "def_end_pos": [79, 8]}, {"full_name": "ENNReal.ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [172, 29], "def_end_pos": [172, 35]}, {"full_name": "Set.Ioi", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [79, 5], "def_end_pos": [79, 8]}, {"full_name": "ENNReal.ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [172, 29], "def_end_pos": [172, 35]}, {"full_name": "MeasureTheory.set_lintegral_congr_fun", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [316, 9], "def_end_pos": [316, 32]}, {"full_name": "measurableSet_Ioi", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [579, 9], "def_end_pos": [579, 26]}, {"full_name": "Filter.eventually_of_forall", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1111, 9], "def_end_pos": [1111, 29]}, {"full_name": "MeasureTheory.Measure.restrict_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1533, 9], "def_end_pos": [1533, 23]}, {"full_name": "measurableSet_le", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [559, 9], "def_end_pos": [559, 25]}, {"full_name": "measurable_const", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [570, 9], "def_end_pos": [570, 25]}, {"full_name": "Set.inter_eq_left", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [981, 15], "def_end_pos": [981, 28]}, {"full_name": "Set.mem_Ioi", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [151, 9], "def_end_pos": [151, 16]}, {"full_name": "LT.lt.trans_le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [148, 7], "def_end_pos": [148, 21]}, {"full_name": "Eq.symm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [310, 9], "def_end_pos": [310, 16]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nhgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0\n\u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a}\nthis : SigmaFinite \u03bd\nA :\n  \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc = \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bd\nB1 :\n  \u222b\u207b (t : \u211d) in Ioc 0 M, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) =\n    \u222b\u207b (t : \u211d) in Ioc 0 M, \u2191\u2191\u03bd {a | t \u2264 f a} * ENNReal.ofReal (g t)\n\u22a2 \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) =\n    \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bd {a | t \u2264 f a} * ENNReal.ofReal (g t)", "state_after": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nhgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0\n\u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a}\nthis : SigmaFinite \u03bd\nA :\n  \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc = \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bd\nB1 :\n  \u222b\u207b (t : \u211d) in Ioc 0 M, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) =\n    \u222b\u207b (t : \u211d) in Ioc 0 M, \u2191\u2191\u03bd {a | t \u2264 f a} * ENNReal.ofReal (g t)\nB2 :\n  \u222b\u207b (t : \u211d) in Ioi M, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) =\n    \u222b\u207b (t : \u211d) in Ioi M, \u2191\u2191\u03bd {a | t \u2264 f a} * ENNReal.ofReal (g t)\n\u22a2 \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) =\n    \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bd {a | t \u2264 f a} * ENNReal.ofReal (g t)"}, {"tactic": "have I : Ioi (0 : \u211d) = Ioc (0 : \u211d) M \u222a Ioi M := (Ioc_union_Ioi_eq_Ioi M_nonneg).symm", "annotated_tactic": ["have I : <a>Ioi</a> (0 : \u211d) = <a>Ioc</a> (0 : \u211d) M \u222a <a>Ioi</a> M := (<a>Ioc_union_Ioi_eq_Ioi</a> M_nonneg).<a>symm</a>", [{"full_name": "Set.Ioi", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [79, 5], "def_end_pos": [79, 8]}, {"full_name": "Set.Ioc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [69, 5], "def_end_pos": [69, 8]}, {"full_name": "Set.Ioi", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [79, 5], "def_end_pos": [79, 8]}, {"full_name": "Set.Ioc_union_Ioi_eq_Ioi", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [1317, 9], "def_end_pos": [1317, 29]}, {"full_name": "Eq.symm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [310, 9], "def_end_pos": [310, 16]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nhgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0\n\u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a}\nthis : SigmaFinite \u03bd\nA :\n  \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc = \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bd\nB1 :\n  \u222b\u207b (t : \u211d) in Ioc 0 M, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) =\n    \u222b\u207b (t : \u211d) in Ioc 0 M, \u2191\u2191\u03bd {a | t \u2264 f a} * ENNReal.ofReal (g t)\nB2 :\n  \u222b\u207b (t : \u211d) in Ioi M, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) =\n    \u222b\u207b (t : \u211d) in Ioi M, \u2191\u2191\u03bd {a | t \u2264 f a} * ENNReal.ofReal (g t)\n\u22a2 \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) =\n    \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bd {a | t \u2264 f a} * ENNReal.ofReal (g t)", "state_after": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nhgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0\n\u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a}\nthis : SigmaFinite \u03bd\nA :\n  \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc = \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bd\nB1 :\n  \u222b\u207b (t : \u211d) in Ioc 0 M, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) =\n    \u222b\u207b (t : \u211d) in Ioc 0 M, \u2191\u2191\u03bd {a | t \u2264 f a} * ENNReal.ofReal (g t)\nB2 :\n  \u222b\u207b (t : \u211d) in Ioi M, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) =\n    \u222b\u207b (t : \u211d) in Ioi M, \u2191\u2191\u03bd {a | t \u2264 f a} * ENNReal.ofReal (g t)\nI : Ioi 0 = Ioc 0 M \u222a Ioi M\n\u22a2 \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) =\n    \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bd {a | t \u2264 f a} * ENNReal.ofReal (g t)"}, {"tactic": "have J : Disjoint (Ioc 0 M) (Ioi M) := Ioc_disjoint_Ioi le_rfl", "annotated_tactic": ["have J : <a>Disjoint</a> (<a>Ioc</a> 0 M) (<a>Ioi</a> M) := <a>Ioc_disjoint_Ioi</a> <a>le_rfl</a>", [{"full_name": "Disjoint", "def_path": "Mathlib/Order/Disjoint.lean", "def_pos": [41, 5], "def_end_pos": [41, 13]}, {"full_name": "Set.Ioc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [69, 5], "def_end_pos": [69, 8]}, {"full_name": "Set.Ioi", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [79, 5], "def_end_pos": [79, 8]}, {"full_name": "Set.Ioc_disjoint_Ioi", "def_path": "Mathlib/Data/Set/Intervals/Disjoint.lean", "def_pos": [64, 9], "def_end_pos": [64, 25]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nhgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0\n\u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a}\nthis : SigmaFinite \u03bd\nA :\n  \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc = \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bd\nB1 :\n  \u222b\u207b (t : \u211d) in Ioc 0 M, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) =\n    \u222b\u207b (t : \u211d) in Ioc 0 M, \u2191\u2191\u03bd {a | t \u2264 f a} * ENNReal.ofReal (g t)\nB2 :\n  \u222b\u207b (t : \u211d) in Ioi M, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) =\n    \u222b\u207b (t : \u211d) in Ioi M, \u2191\u2191\u03bd {a | t \u2264 f a} * ENNReal.ofReal (g t)\nI : Ioi 0 = Ioc 0 M \u222a Ioi M\n\u22a2 \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) =\n    \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bd {a | t \u2264 f a} * ENNReal.ofReal (g t)", "state_after": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nhgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0\n\u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a}\nthis : SigmaFinite \u03bd\nA :\n  \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc = \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bd\nB1 :\n  \u222b\u207b (t : \u211d) in Ioc 0 M, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) =\n    \u222b\u207b (t : \u211d) in Ioc 0 M, \u2191\u2191\u03bd {a | t \u2264 f a} * ENNReal.ofReal (g t)\nB2 :\n  \u222b\u207b (t : \u211d) in Ioi M, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) =\n    \u222b\u207b (t : \u211d) in Ioi M, \u2191\u2191\u03bd {a | t \u2264 f a} * ENNReal.ofReal (g t)\nI : Ioi 0 = Ioc 0 M \u222a Ioi M\nJ : Disjoint (Ioc 0 M) (Ioi M)\n\u22a2 \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) =\n    \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bd {a | t \u2264 f a} * ENNReal.ofReal (g t)"}, {"tactic": "rw [I, lintegral_union measurableSet_Ioi J, lintegral_union measurableSet_Ioi J, B1, B2]", "annotated_tactic": ["rw [I, <a>lintegral_union</a> <a>measurableSet_Ioi</a> J, <a>lintegral_union</a> <a>measurableSet_Ioi</a> J, B1, B2]", [{"full_name": "MeasureTheory.lintegral_union", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [1243, 9], "def_end_pos": [1243, 24]}, {"full_name": "measurableSet_Ioi", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [579, 9], "def_end_pos": [579, 26]}, {"full_name": "MeasureTheory.lintegral_union", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [1243, 9], "def_end_pos": [1243, 24]}, {"full_name": "measurableSet_Ioi", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [579, 9], "def_end_pos": [579, 26]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nhgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0\n\u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a}\nthis : SigmaFinite \u03bd\nA :\n  \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc = \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bd\nB1 :\n  \u222b\u207b (t : \u211d) in Ioc 0 M, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) =\n    \u222b\u207b (t : \u211d) in Ioc 0 M, \u2191\u2191\u03bd {a | t \u2264 f a} * ENNReal.ofReal (g t)\nB2 :\n  \u222b\u207b (t : \u211d) in Ioi M, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) =\n    \u222b\u207b (t : \u211d) in Ioi M, \u2191\u2191\u03bd {a | t \u2264 f a} * ENNReal.ofReal (g t)\nI : Ioi 0 = Ioc 0 M \u222a Ioi M\nJ : Disjoint (Ioc 0 M) (Ioi M)\n\u22a2 \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) =\n    \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bd {a | t \u2264 f a} * ENNReal.ofReal (g t)", "state_after": "no goals"}, {"tactic": "apply lintegral_congr_ae", "annotated_tactic": ["apply <a>lintegral_congr_ae</a>", [{"full_name": "MeasureTheory.lintegral_congr_ae", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [304, 9], "def_end_pos": [304, 27]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nhgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0\n\u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a}\nthis : SigmaFinite \u03bd\nA :\n  \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc = \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bd\n\u22a2 \u222b\u207b (t : \u211d) in Ioc 0 M, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) =\n    \u222b\u207b (t : \u211d) in Ioc 0 M, \u2191\u2191\u03bd {a | t \u2264 f a} * ENNReal.ofReal (g t)", "state_after": "case h\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nhgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0\n\u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a}\nthis : SigmaFinite \u03bd\nA :\n  \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc = \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bd\n\u22a2 (fun a => \u2191\u2191\u03bc {a_1 | a \u2264 f a_1} * ENNReal.ofReal (g a)) =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] fun a =>\n    \u2191\u2191\u03bd {a_1 | a \u2264 f a_1} * ENNReal.ofReal (g a)"}, {"tactic": "filter_upwards [hgM] with t ht", "annotated_tactic": ["filter_upwards [hgM] with t ht", []], "state_before": "case h\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nhgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0\n\u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a}\nthis : SigmaFinite \u03bd\nA :\n  \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc = \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bd\n\u22a2 (fun a => \u2191\u2191\u03bc {a_1 | a \u2264 f a_1} * ENNReal.ofReal (g a)) =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] fun a =>\n    \u2191\u2191\u03bd {a_1 | a \u2264 f a_1} * ENNReal.ofReal (g a)", "state_after": "case h\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nhgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0\n\u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a}\nthis : SigmaFinite \u03bd\nA :\n  \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc = \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bd\nt : \u211d\nht : g t = OfNat.ofNat 0 t\n\u22a2 \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) = \u2191\u2191\u03bd {a | t \u2264 f a} * ENNReal.ofReal (g t)"}, {"tactic": "simp [ht]", "annotated_tactic": ["simp [ht]", []], "state_before": "case h\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nhgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0\n\u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a}\nthis : SigmaFinite \u03bd\nA :\n  \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc = \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bd\nt : \u211d\nht : g t = OfNat.ofNat 0 t\n\u22a2 \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) = \u2191\u2191\u03bd {a | t \u2264 f a} * ENNReal.ofReal (g t)", "state_after": "no goals"}, {"tactic": "apply set_lintegral_congr_fun measurableSet_Ioi (eventually_of_forall (fun t ht \u21a6 ?_))", "annotated_tactic": ["apply <a>set_lintegral_congr_fun</a> <a>measurableSet_Ioi</a> (<a>eventually_of_forall</a> (fun t ht \u21a6 ?_))", [{"full_name": "MeasureTheory.set_lintegral_congr_fun", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [316, 9], "def_end_pos": [316, 32]}, {"full_name": "measurableSet_Ioi", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [579, 9], "def_end_pos": [579, 26]}, {"full_name": "Filter.eventually_of_forall", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1111, 9], "def_end_pos": [1111, 29]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nhgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0\n\u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a}\nthis : SigmaFinite \u03bd\nA :\n  \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc = \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bd\nB1 :\n  \u222b\u207b (t : \u211d) in Ioc 0 M, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) =\n    \u222b\u207b (t : \u211d) in Ioc 0 M, \u2191\u2191\u03bd {a | t \u2264 f a} * ENNReal.ofReal (g t)\n\u22a2 \u222b\u207b (t : \u211d) in Ioi M, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) =\n    \u222b\u207b (t : \u211d) in Ioi M, \u2191\u2191\u03bd {a | t \u2264 f a} * ENNReal.ofReal (g t)", "state_after": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nhgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0\n\u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a}\nthis : SigmaFinite \u03bd\nA :\n  \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc = \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bd\nB1 :\n  \u222b\u207b (t : \u211d) in Ioc 0 M, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) =\n    \u222b\u207b (t : \u211d) in Ioc 0 M, \u2191\u2191\u03bd {a | t \u2264 f a} * ENNReal.ofReal (g t)\nt : \u211d\nht : t \u2208 Ioi M\n\u22a2 \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) = \u2191\u2191\u03bd {a | t \u2264 f a} * ENNReal.ofReal (g t)"}, {"tactic": "rw [Measure.restrict_apply (measurableSet_le measurable_const f_mble)]", "annotated_tactic": ["rw [<a>Measure.restrict_apply</a> (<a>measurableSet_le</a> <a>measurable_const</a> f_mble)]", [{"full_name": "MeasureTheory.Measure.restrict_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1533, 9], "def_end_pos": [1533, 23]}, {"full_name": "measurableSet_le", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [559, 9], "def_end_pos": [559, 25]}, {"full_name": "measurable_const", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [570, 9], "def_end_pos": [570, 25]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nhgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0\n\u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a}\nthis : SigmaFinite \u03bd\nA :\n  \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc = \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bd\nB1 :\n  \u222b\u207b (t : \u211d) in Ioc 0 M, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) =\n    \u222b\u207b (t : \u211d) in Ioc 0 M, \u2191\u2191\u03bd {a | t \u2264 f a} * ENNReal.ofReal (g t)\nt : \u211d\nht : t \u2208 Ioi M\n\u22a2 \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) = \u2191\u2191\u03bd {a | t \u2264 f a} * ENNReal.ofReal (g t)", "state_after": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nhgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0\n\u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a}\nthis : SigmaFinite \u03bd\nA :\n  \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc = \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bd\nB1 :\n  \u222b\u207b (t : \u211d) in Ioc 0 M, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) =\n    \u222b\u207b (t : \u211d) in Ioc 0 M, \u2191\u2191\u03bd {a | t \u2264 f a} * ENNReal.ofReal (g t)\nt : \u211d\nht : t \u2208 Ioi M\n\u22a2 \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) = \u2191\u2191\u03bc ({a | t \u2264 f a} \u2229 {a | M < f a}) * ENNReal.ofReal (g t)"}, {"tactic": "congr 3", "annotated_tactic": ["congr 3", []], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nhgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0\n\u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a}\nthis : SigmaFinite \u03bd\nA :\n  \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc = \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bd\nB1 :\n  \u222b\u207b (t : \u211d) in Ioc 0 M, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) =\n    \u222b\u207b (t : \u211d) in Ioc 0 M, \u2191\u2191\u03bd {a | t \u2264 f a} * ENNReal.ofReal (g t)\nt : \u211d\nht : t \u2208 Ioi M\n\u22a2 \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) = \u2191\u2191\u03bc ({a | t \u2264 f a} \u2229 {a | M < f a}) * ENNReal.ofReal (g t)", "state_after": "case e_a.e_a\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nhgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0\n\u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a}\nthis : SigmaFinite \u03bd\nA :\n  \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc = \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bd\nB1 :\n  \u222b\u207b (t : \u211d) in Ioc 0 M, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) =\n    \u222b\u207b (t : \u211d) in Ioc 0 M, \u2191\u2191\u03bd {a | t \u2264 f a} * ENNReal.ofReal (g t)\nt : \u211d\nht : t \u2208 Ioi M\n\u22a2 {a | t \u2264 f a} = {a | t \u2264 f a} \u2229 {a | M < f a}"}, {"tactic": "exact (inter_eq_left.2 (fun a ha \u21a6 (mem_Ioi.1 ht).trans_le ha)).symm", "annotated_tactic": ["exact (<a>inter_eq_left</a>.2 (fun a ha \u21a6 (<a>mem_Ioi</a>.1 ht).<a>trans_le</a> ha)).<a>symm</a>", [{"full_name": "Set.inter_eq_left", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [981, 15], "def_end_pos": [981, 28]}, {"full_name": "Set.mem_Ioi", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [151, 9], "def_end_pos": [151, 16]}, {"full_name": "LT.lt.trans_le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [148, 7], "def_end_pos": [148, 21]}, {"full_name": "Eq.symm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [310, 9], "def_end_pos": [310, 16]}]], "state_before": "case e_a.e_a\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nf_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9\nH1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0\nH2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4\nM_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nzero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0}\nM_nonneg : 0 \u2264 M\nhgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0\n\u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a}\nthis : SigmaFinite \u03bd\nA :\n  \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc = \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bd\nB1 :\n  \u222b\u207b (t : \u211d) in Ioc 0 M, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) =\n    \u222b\u207b (t : \u211d) in Ioc 0 M, \u2191\u2191\u03bd {a | t \u2264 f a} * ENNReal.ofReal (g t)\nt : \u211d\nht : t \u2208 Ioi M\n\u22a2 {a | t \u2264 f a} = {a | t \u2264 f a} \u2229 {a | M < f a}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "full_name": "MeasureTheory.Measure.NullMeasurableSet.subtype_coe", "start": [1417, 1], "end": [1420, 65], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/FiniteMeasure.lean", "full_name": "MeasureTheory.FiniteMeasure.mass_zero_iff", "start": [184, 1], "end": [188, 44], "traced_tactics": [{"tactic": "refine' \u27e8fun \u03bc_mass => _, fun h\u03bc => by simp only [h\u03bc, zero_mass]\u27e9", "annotated_tactic": ["refine' \u27e8fun \u03bc_mass => _, fun h\u03bc => by simp only [h\u03bc, <a>zero_mass</a>]\u27e9", [{"full_name": "MeasureTheory.FiniteMeasure.zero_mass", "def_path": "Mathlib/MeasureTheory/Measure/FiniteMeasure.lean", "def_pos": [179, 9], "def_end_pos": [179, 18]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\n\u22a2 mass \u03bc = 0 \u2194 \u03bc = 0", "state_after": "\u03a9 : Type u_1\ninst\u271d : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\n\u03bc_mass : mass \u03bc = 0\n\u22a2 \u03bc = 0"}, {"tactic": "apply toMeasure_injective", "annotated_tactic": ["apply <a>toMeasure_injective</a>", [{"full_name": "MeasureTheory.FiniteMeasure.toMeasure_injective", "def_path": "Mathlib/MeasureTheory/Measure/FiniteMeasure.lean", "def_pos": [151, 9], "def_end_pos": [151, 28]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\n\u03bc_mass : mass \u03bc = 0\n\u22a2 \u03bc = 0", "state_after": "case a\n\u03a9 : Type u_1\ninst\u271d : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\n\u03bc_mass : mass \u03bc = 0\n\u22a2 \u2191\u03bc = \u21910"}, {"tactic": "apply Measure.measure_univ_eq_zero.mp", "annotated_tactic": ["apply Measure.measure_univ_eq_zero.mp", []], "state_before": "case a\n\u03a9 : Type u_1\ninst\u271d : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\n\u03bc_mass : mass \u03bc = 0\n\u22a2 \u2191\u03bc = \u21910", "state_after": "case a\n\u03a9 : Type u_1\ninst\u271d : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\n\u03bc_mass : mass \u03bc = 0\n\u22a2 \u2191\u2191\u2191\u03bc univ = 0"}, {"tactic": "rwa [\u2190 ennreal_mass, ENNReal.coe_eq_zero]", "annotated_tactic": ["rwa [\u2190 <a>ennreal_mass</a>, <a>ENNReal.coe_eq_zero</a>]", [{"full_name": "MeasureTheory.FiniteMeasure.ennreal_mass", "def_path": "Mathlib/MeasureTheory/Measure/FiniteMeasure.lean", "def_pos": [171, 9], "def_end_pos": [171, 21]}, {"full_name": "ENNReal.coe_eq_zero", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [368, 28], "def_end_pos": [368, 39]}]], "state_before": "case a\n\u03a9 : Type u_1\ninst\u271d : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\n\u03bc_mass : mass \u03bc = 0\n\u22a2 \u2191\u2191\u2191\u03bc univ = 0", "state_after": "no goals"}, {"tactic": "simp only [h\u03bc, zero_mass]", "annotated_tactic": ["simp only [h\u03bc, <a>zero_mass</a>]", [{"full_name": "MeasureTheory.FiniteMeasure.zero_mass", "def_path": "Mathlib/MeasureTheory/Measure/FiniteMeasure.lean", "def_pos": [179, 9], "def_end_pos": [179, 18]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\nh\u03bc : \u03bc = 0\n\u22a2 mass \u03bc = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "full_name": "intervalIntegral.integral_interval_sub_interval_comm", "start": [943, 1], "end": [948, 70], "traced_tactics": [{"tactic": "simp only [sub_eq_add_neg, \u2190 integral_symm,\n  integral_interval_add_interval_comm hab hcd.symm (hac.trans hcd)]", "annotated_tactic": ["simp only [<a>sub_eq_add_neg</a>, \u2190 <a>integral_symm</a>,\n    <a>integral_interval_add_interval_comm</a> hab hcd.symm (hac.trans hcd)]", [{"full_name": "sub_eq_add_neg", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [975, 3], "def_end_pos": [975, 14]}, {"full_name": "intervalIntegral.integral_symm", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [474, 9], "def_end_pos": [474, 22]}, {"full_name": "intervalIntegral.integral_interval_add_interval_comm", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [935, 9], "def_end_pos": [935, 44]}]], "state_before": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\na b c d : \u211d\nf g : \u211d \u2192 E\n\u03bc : Measure \u211d\nhab : IntervalIntegrable f \u03bc a b\nhcd : IntervalIntegrable f \u03bc c d\nhac : IntervalIntegrable f \u03bc a c\n\u22a2 \u222b (x : \u211d) in a..b, f x \u2202\u03bc - \u222b (x : \u211d) in c..d, f x \u2202\u03bc = \u222b (x : \u211d) in a..c, f x \u2202\u03bc - \u222b (x : \u211d) in b..d, f x \u2202\u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/AssocList.lean", "full_name": "Std.AssocList.erase_toList", "start": [182, 9], "end": [183, 72], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Int/ModEq.lean", "full_name": "Int.ModEq.cancel_right_div_gcd", "start": [223, 1], "end": [233, 88], "traced_tactics": [{"tactic": "letI d := gcd m c", "annotated_tactic": ["letI d := <a>gcd</a> m c", [{"full_name": "Int.gcd", "def_path": "lake-packages/std/Std/Data/Int/Basic.lean", "def_pos": [120, 5], "def_end_pos": [120, 8]}]], "state_before": "m n a b c d : \u2124\nhm : 0 < m\nh : a * c \u2261 b * c [ZMOD m]\n\u22a2 a \u2261 b [ZMOD m / \u2191(gcd m c)]", "state_after": "m n a b c d\u271d : \u2124\nhm : 0 < m\nh : a * c \u2261 b * c [ZMOD m]\nd : \u2115 := gcd m c\n\u22a2 a \u2261 b [ZMOD m / \u2191(gcd m c)]"}, {"tactic": "have hmd := gcd_dvd_left m c", "annotated_tactic": ["have hmd := <a>gcd_dvd_left</a> m c", [{"full_name": "Int.gcd_dvd_left", "def_path": "Mathlib/Data/Int/GCD.lean", "def_pos": [247, 9], "def_end_pos": [247, 21]}]], "state_before": "m n a b c d\u271d : \u2124\nhm : 0 < m\nh : a * c \u2261 b * c [ZMOD m]\nd : \u2115 := gcd m c\n\u22a2 a \u2261 b [ZMOD m / \u2191(gcd m c)]", "state_after": "m n a b c d\u271d : \u2124\nhm : 0 < m\nh : a * c \u2261 b * c [ZMOD m]\nd : \u2115 := gcd m c\nhmd : \u2191(gcd m c) \u2223 m\n\u22a2 a \u2261 b [ZMOD m / \u2191(gcd m c)]"}, {"tactic": "have hcd := gcd_dvd_right m c", "annotated_tactic": ["have hcd := <a>gcd_dvd_right</a> m c", [{"full_name": "Int.gcd_dvd_right", "def_path": "Mathlib/Data/Int/GCD.lean", "def_pos": [251, 9], "def_end_pos": [251, 22]}]], "state_before": "m n a b c d\u271d : \u2124\nhm : 0 < m\nh : a * c \u2261 b * c [ZMOD m]\nd : \u2115 := gcd m c\nhmd : \u2191(gcd m c) \u2223 m\n\u22a2 a \u2261 b [ZMOD m / \u2191(gcd m c)]", "state_after": "m n a b c d\u271d : \u2124\nhm : 0 < m\nh : a * c \u2261 b * c [ZMOD m]\nd : \u2115 := gcd m c\nhmd : \u2191(gcd m c) \u2223 m\nhcd : \u2191(gcd m c) \u2223 c\n\u22a2 a \u2261 b [ZMOD m / \u2191(gcd m c)]"}, {"tactic": "rw [modEq_iff_dvd] at h \u22a2", "annotated_tactic": ["rw [<a>modEq_iff_dvd</a>] at h \u22a2", [{"full_name": "Int.modEq_iff_dvd", "def_path": "Mathlib/Data/Int/ModEq.lean", "def_pos": [94, 9], "def_end_pos": [94, 22]}]], "state_before": "m n a b c d\u271d : \u2124\nhm : 0 < m\nh : a * c \u2261 b * c [ZMOD m]\nd : \u2115 := gcd m c\nhmd : \u2191(gcd m c) \u2223 m\nhcd : \u2191(gcd m c) \u2223 c\n\u22a2 a \u2261 b [ZMOD m / \u2191(gcd m c)]", "state_after": "m n a b c d\u271d : \u2124\nhm : 0 < m\nh : m \u2223 b * c - a * c\nd : \u2115 := gcd m c\nhmd : \u2191(gcd m c) \u2223 m\nhcd : \u2191(gcd m c) \u2223 c\n\u22a2 m / \u2191(gcd m c) \u2223 b - a"}, {"tactic": "refine Int.dvd_of_dvd_mul_right_of_gcd_one (?_ : m / d \u2223 c / d * (b - a)) ?_", "annotated_tactic": ["refine <a>Int.dvd_of_dvd_mul_right_of_gcd_one</a> (?_ : m / d \u2223 c / d * (b - a)) ?_", [{"full_name": "Int.dvd_of_dvd_mul_right_of_gcd_one", "def_path": "Mathlib/Data/Int/GCD.lean", "def_pos": [424, 9], "def_end_pos": [424, 40]}]], "state_before": "m n a b c d\u271d : \u2124\nhm : 0 < m\nh : m \u2223 b * c - a * c\nd : \u2115 := gcd m c\nhmd : \u2191(gcd m c) \u2223 m\nhcd : \u2191(gcd m c) \u2223 c\n\u22a2 m / \u2191(gcd m c) \u2223 b - a", "state_after": "case refine_1\nm n a b c d\u271d : \u2124\nhm : 0 < m\nh : m \u2223 b * c - a * c\nd : \u2115 := gcd m c\nhmd : \u2191(gcd m c) \u2223 m\nhcd : \u2191(gcd m c) \u2223 c\n\u22a2 m / \u2191d \u2223 c / \u2191d * (b - a)\n\ncase refine_2\nm n a b c d\u271d : \u2124\nhm : 0 < m\nh : m \u2223 b * c - a * c\nd : \u2115 := gcd m c\nhmd : \u2191(gcd m c) \u2223 m\nhcd : \u2191(gcd m c) \u2223 c\n\u22a2 gcd (m / \u2191(gcd m c)) (c / \u2191d) = 1"}, {"tactic": "rw [mul_comm, \u2190 Int.mul_ediv_assoc (b - a) hcd, sub_mul]", "annotated_tactic": ["rw [<a>mul_comm</a>, \u2190 <a>Int.mul_ediv_assoc</a> (b - a) hcd, <a>sub_mul</a>]", [{"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [302, 9], "def_end_pos": [302, 17]}, {"full_name": "Int.mul_ediv_assoc", "def_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "def_pos": [718, 19], "def_end_pos": [718, 33]}, {"full_name": "sub_mul", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [372, 7], "def_end_pos": [372, 14]}]], "state_before": "case refine_1\nm n a b c d\u271d : \u2124\nhm : 0 < m\nh : m \u2223 b * c - a * c\nd : \u2115 := gcd m c\nhmd : \u2191(gcd m c) \u2223 m\nhcd : \u2191(gcd m c) \u2223 c\n\u22a2 m / \u2191d \u2223 c / \u2191d * (b - a)", "state_after": "case refine_1\nm n a b c d\u271d : \u2124\nhm : 0 < m\nh : m \u2223 b * c - a * c\nd : \u2115 := gcd m c\nhmd : \u2191(gcd m c) \u2223 m\nhcd : \u2191(gcd m c) \u2223 c\n\u22a2 m / \u2191d \u2223 (b * c - a * c) / \u2191(gcd m c)"}, {"tactic": "exact Int.ediv_dvd_ediv hmd h", "annotated_tactic": ["exact <a>Int.ediv_dvd_ediv</a> hmd h", [{"full_name": "Int.ediv_dvd_ediv", "def_path": "Mathlib/Data/Int/Order/Basic.lean", "def_pos": [350, 9], "def_end_pos": [350, 22]}]], "state_before": "case refine_1\nm n a b c d\u271d : \u2124\nhm : 0 < m\nh : m \u2223 b * c - a * c\nd : \u2115 := gcd m c\nhmd : \u2191(gcd m c) \u2223 m\nhcd : \u2191(gcd m c) \u2223 c\n\u22a2 m / \u2191d \u2223 (b * c - a * c) / \u2191(gcd m c)", "state_after": "no goals"}, {"tactic": "rw [gcd_div hmd hcd, natAbs_ofNat, Nat.div_self (gcd_pos_of_ne_zero_left c hm.ne')]", "annotated_tactic": ["rw [<a>gcd_div</a> hmd hcd, <a>natAbs_ofNat</a>, <a>Nat.div_self</a> (<a>gcd_pos_of_ne_zero_left</a> c hm.ne')]", [{"full_name": "Int.gcd_div", "def_path": "Mathlib/Data/Int/GCD.lean", "def_pos": [328, 9], "def_end_pos": [328, 16]}, {"full_name": "Int.natAbs_ofNat", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [138, 17], "def_end_pos": [138, 29]}, {"full_name": "Nat.div_self", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [629, 19], "def_end_pos": [629, 27]}, {"full_name": "Int.gcd_pos_of_ne_zero_left", "def_path": "Mathlib/Data/Int/GCD.lean", "def_pos": [312, 9], "def_end_pos": [312, 32]}]], "state_before": "case refine_2\nm n a b c d\u271d : \u2124\nhm : 0 < m\nh : m \u2223 b * c - a * c\nd : \u2115 := gcd m c\nhmd : \u2191(gcd m c) \u2223 m\nhcd : \u2191(gcd m c) \u2223 c\n\u22a2 gcd (m / \u2191(gcd m c)) (c / \u2191d) = 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Kernel/Basic.lean", "full_name": "ProbabilityTheory.kernel.sum_fintype", "start": [261, 1], "end": [263, 73], "traced_tactics": [{"tactic": "ext a s hs", "annotated_tactic": ["ext a s hs", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : Fintype \u03b9\n\u03ba : \u03b9 \u2192 { x // x \u2208 kernel \u03b1 \u03b2 }\n\u22a2 kernel.sum \u03ba = \u2211 i : \u03b9, \u03ba i", "state_after": "case h.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : Fintype \u03b9\n\u03ba : \u03b9 \u2192 { x // x \u2208 kernel \u03b1 \u03b2 }\na : \u03b1\ns : Set \u03b2\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(\u2191(kernel.sum \u03ba) a) s = \u2191\u2191(\u2191(\u2211 i : \u03b9, \u03ba i) a) s"}, {"tactic": "simp only [sum_apply' \u03ba a hs, finset_sum_apply' _ \u03ba a s, tsum_fintype]", "annotated_tactic": ["simp only [<a>sum_apply'</a> \u03ba a hs, <a>finset_sum_apply'</a> _ \u03ba a s, <a>tsum_fintype</a>]", [{"full_name": "ProbabilityTheory.kernel.sum_apply'", "def_path": "Mathlib/Probability/Kernel/Basic.lean", "def_pos": [244, 9], "def_end_pos": [244, 19]}, {"full_name": "ProbabilityTheory.kernel.finset_sum_apply'", "def_path": "Mathlib/Probability/Kernel/Basic.lean", "def_pos": [107, 9], "def_end_pos": [107, 26]}, {"full_name": "tsum_fintype", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/Basic.lean", "def_pos": [503, 9], "def_end_pos": [503, 21]}]], "state_before": "case h.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : Fintype \u03b9\n\u03ba : \u03b9 \u2192 { x // x \u2208 kernel \u03b1 \u03b2 }\na : \u03b1\ns : Set \u03b2\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(\u2191(kernel.sum \u03ba) a) s = \u2191\u2191(\u2191(\u2211 i : \u03b9, \u03ba i) a) s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "full_name": "Set.nsmul_univ", "start": [1002, 1], "end": [1005, 83], "traced_tactics": [{"tactic": "rw [succ_nsmul, nsmul_univ n.succ_ne_zero, univ_add_univ]", "annotated_tactic": ["rw [<a>succ_nsmul</a>, nsmul_univ n.succ_ne_zero, <a>univ_add_univ</a>]", [{"full_name": "succ_nsmul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [644, 15], "def_end_pos": [644, 25]}, {"full_name": "Set.univ_add_univ", "def_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "def_pos": [994, 3], "def_end_pos": [994, 14]}]], "state_before": "F : Type u_1\n\u03b1\u271d : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b9 : Monoid \u03b1\u271d\ns t : Set \u03b1\u271d\na : \u03b1\u271d\nm n\u271d : \u2115\n\u03b1 : Type u_5\ninst\u271d : AddMonoid \u03b1\nn : \u2115\nx\u271d : n + 2 \u2260 0\n\u22a2 (n + 2) \u2022 univ = univ", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "full_name": "String.Pos.addChar_right_comm", "start": [127, 1], "end": [130, 27], "traced_tactics": [{"tactic": "apply ext", "annotated_tactic": ["apply <a>ext</a>", [{"full_name": "String.Pos.ext", "def_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "def_pos": [105, 16], "def_end_pos": [105, 19]}]], "state_before": "p : Pos\nc\u2081 c\u2082 : Char\n\u22a2 p + c\u2081 + c\u2082 = p + c\u2082 + c\u2081", "state_after": "case h\np : Pos\nc\u2081 c\u2082 : Char\n\u22a2 (p + c\u2081 + c\u2082).byteIdx = (p + c\u2082 + c\u2081).byteIdx"}, {"tactic": "repeat rw [pos_add_char]", "annotated_tactic": ["repeat rw [<a>pos_add_char</a>]", [{"full_name": "String.pos_add_char", "def_path": "lake-packages/lean4/src/lean/Init/Data/String/Basic.lean", "def_pos": [135, 17], "def_end_pos": [135, 29]}]], "state_before": "case h\np : Pos\nc\u2081 c\u2082 : Char\n\u22a2 (p + c\u2081 + c\u2082).byteIdx = (p + c\u2082 + c\u2081).byteIdx", "state_after": "case h\np : Pos\nc\u2081 c\u2082 : Char\n\u22a2 p.byteIdx + csize c\u2081 + csize c\u2082 = p.byteIdx + csize c\u2082 + csize c\u2081"}, {"tactic": "apply Nat.add_right_comm", "annotated_tactic": ["apply <a>Nat.add_right_comm</a>", [{"full_name": "Nat.add_right_comm", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [145, 19], "def_end_pos": [145, 33]}]], "state_before": "case h\np : Pos\nc\u2081 c\u2082 : Char\n\u22a2 p.byteIdx + csize c\u2081 + csize c\u2082 = p.byteIdx + csize c\u2082 + csize c\u2081", "state_after": "no goals"}, {"tactic": "rw [pos_add_char]", "annotated_tactic": ["rw [<a>pos_add_char</a>]", [{"full_name": "String.pos_add_char", "def_path": "lake-packages/lean4/src/lean/Init/Data/String/Basic.lean", "def_pos": [135, 17], "def_end_pos": [135, 29]}]], "state_before": "case h\np : Pos\nc\u2081 c\u2082 : Char\n\u22a2 p.byteIdx + csize c\u2081 + csize c\u2082 = (p + c\u2082).byteIdx + csize c\u2081", "state_after": "case h\np : Pos\nc\u2081 c\u2082 : Char\n\u22a2 p.byteIdx + csize c\u2081 + csize c\u2082 = p.byteIdx + csize c\u2082 + csize c\u2081"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/StrongLaw.lean", "full_name": "ProbabilityTheory.tsum_prob_mem_Ioi_lt_top", "start": [314, 1], "end": [336, 44], "traced_tactics": [{"tactic": "suffices : \u2200 K : \u2115, \u2211 j in range K, \u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioi (j : \u211d)} \u2264 ENNReal.ofReal (\ud835\udd3c[X] + 1)", "annotated_tactic": ["suffices : \u2200 K : \u2115, \u2211 j in <a>range</a> K, \u2119 {\u03c9 | X \u03c9 \u2208 <a>Set.Ioi</a> (j : \u211d)} \u2264 <a>ENNReal.ofReal</a> (\ud835\udd3c[X] + 1)", [{"full_name": "Finset.range", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3027, 5], "def_end_pos": [3027, 10]}, {"full_name": "Set.Ioi", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [79, 5], "def_end_pos": [79, 8]}, {"full_name": "ENNReal.ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [172, 29], "def_end_pos": [172, 35]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\n\u22a2 \u2211' (j : \u2115), \u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioi \u2191j} < \u22a4", "state_after": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nthis : \u2200 (K : \u2115), \u2211 j in range K, \u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioi \u2191j} \u2264 ENNReal.ofReal ((\u222b (a : \u03a9), X a) + 1)\n\u22a2 \u2211' (j : \u2115), \u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioi \u2191j} < \u22a4\n\ncase this\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\n\u22a2 \u2200 (K : \u2115), \u2211 j in range K, \u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioi \u2191j} \u2264 ENNReal.ofReal ((\u222b (a : \u03a9), X a) + 1)"}, {"tactic": "exact (le_of_tendsto_of_tendsto (ENNReal.tendsto_nat_tsum _) tendsto_const_nhds\n  (eventually_of_forall this)).trans_lt ENNReal.ofReal_lt_top", "annotated_tactic": ["exact (<a>le_of_tendsto_of_tendsto</a> (<a>ENNReal.tendsto_nat_tsum</a> _) <a>tendsto_const_nhds</a>\n    (<a>eventually_of_forall</a> this)).<a>trans_lt</a> <a>ENNReal.ofReal_lt_top</a>", [{"full_name": "le_of_tendsto_of_tendsto", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [223, 9], "def_end_pos": [223, 33]}, {"full_name": "ENNReal.tendsto_nat_tsum", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [929, 9], "def_end_pos": [929, 25]}, {"full_name": "tendsto_const_nhds", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [1049, 9], "def_end_pos": [1049, 27]}, {"full_name": "Filter.eventually_of_forall", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1111, 9], "def_end_pos": [1111, 29]}, {"full_name": "LE.le.trans_lt", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [124, 7], "def_end_pos": [124, 21]}, {"full_name": "ENNReal.ofReal_lt_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [314, 17], "def_end_pos": [314, 30]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nthis : \u2200 (K : \u2115), \u2211 j in range K, \u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioi \u2191j} \u2264 ENNReal.ofReal ((\u222b (a : \u03a9), X a) + 1)\n\u22a2 \u2211' (j : \u2115), \u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioi \u2191j} < \u22a4\n\ncase this\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\n\u22a2 \u2200 (K : \u2115), \u2211 j in range K, \u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioi \u2191j} \u2264 ENNReal.ofReal ((\u222b (a : \u03a9), X a) + 1)", "state_after": "case this\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\n\u22a2 \u2200 (K : \u2115), \u2211 j in range K, \u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioi \u2191j} \u2264 ENNReal.ofReal ((\u222b (a : \u03a9), X a) + 1)"}, {"tactic": "intro K", "annotated_tactic": ["intro K", []], "state_before": "case this\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\n\u22a2 \u2200 (K : \u2115), \u2211 j in range K, \u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioi \u2191j} \u2264 ENNReal.ofReal ((\u222b (a : \u03a9), X a) + 1)", "state_after": "case this\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\n\u22a2 \u2211 j in range K, \u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioi \u2191j} \u2264 ENNReal.ofReal ((\u222b (a : \u03a9), X a) + 1)"}, {"tactic": "apply le_of_tendsto_of_tendsto A tendsto_const_nhds", "annotated_tactic": ["apply <a>le_of_tendsto_of_tendsto</a> A <a>tendsto_const_nhds</a>", [{"full_name": "le_of_tendsto_of_tendsto", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [223, 9], "def_end_pos": [223, 33]}, {"full_name": "tendsto_const_nhds", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [1049, 9], "def_end_pos": [1049, 27]}]], "state_before": "case this\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nA :\n  Tendsto (fun N => \u2211 j in range K, \u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioc \u2191j \u2191N}) atTop\n    (\ud835\udcdd (\u2211 j in range K, \u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioi \u2191j}))\n\u22a2 \u2211 j in range K, \u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioi \u2191j} \u2264 ENNReal.ofReal ((\u222b (a : \u03a9), X a) + 1)", "state_after": "case this\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nA :\n  Tendsto (fun N => \u2211 j in range K, \u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioc \u2191j \u2191N}) atTop\n    (\ud835\udcdd (\u2211 j in range K, \u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioi \u2191j}))\n\u22a2 (fun N => \u2211 j in range K, \u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioc \u2191j \u2191N}) \u2264\u1da0[atTop] fun x => ENNReal.ofReal ((\u222b (a : \u03a9), X a) + 1)"}, {"tactic": "filter_upwards [Ici_mem_atTop K] with N hN", "annotated_tactic": ["filter_upwards [<a>Ici_mem_atTop</a> K] with N hN", [{"full_name": "Filter.Ici_mem_atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [57, 9], "def_end_pos": [57, 22]}]], "state_before": "case this\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nA :\n  Tendsto (fun N => \u2211 j in range K, \u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioc \u2191j \u2191N}) atTop\n    (\ud835\udcdd (\u2211 j in range K, \u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioi \u2191j}))\n\u22a2 (fun N => \u2211 j in range K, \u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioc \u2191j \u2191N}) \u2264\u1da0[atTop] fun x => ENNReal.ofReal ((\u222b (a : \u03a9), X a) + 1)", "state_after": "case h\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nA :\n  Tendsto (fun N => \u2211 j in range K, \u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioc \u2191j \u2191N}) atTop\n    (\ud835\udcdd (\u2211 j in range K, \u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioi \u2191j}))\nN : \u2115\nhN : N \u2208 Set.Ici K\n\u22a2 \u2211 j in range K, \u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioc \u2191j \u2191N} \u2264 ENNReal.ofReal ((\u222b (a : \u03a9), X a) + 1)"}, {"tactic": "exact sum_prob_mem_Ioc_le hint hnonneg hN", "annotated_tactic": ["exact <a>sum_prob_mem_Ioc_le</a> hint hnonneg hN", [{"full_name": "ProbabilityTheory.sum_prob_mem_Ioc_le", "def_path": "Mathlib/Probability/StrongLaw.lean", "def_pos": [232, 9], "def_end_pos": [232, 28]}]], "state_before": "case h\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nA :\n  Tendsto (fun N => \u2211 j in range K, \u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioc \u2191j \u2191N}) atTop\n    (\ud835\udcdd (\u2211 j in range K, \u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioi \u2191j}))\nN : \u2115\nhN : N \u2208 Set.Ici K\n\u22a2 \u2211 j in range K, \u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioc \u2191j \u2191N} \u2264 ENNReal.ofReal ((\u222b (a : \u03a9), X a) + 1)", "state_after": "no goals"}, {"tactic": "refine' tendsto_finset_sum _ fun i _ => _", "annotated_tactic": ["refine' <a>tendsto_finset_sum</a> _ fun i _ => _", [{"full_name": "tendsto_finset_sum", "def_path": "Mathlib/Topology/Algebra/Monoid.lean", "def_pos": [736, 3], "def_end_pos": [736, 14]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\n\u22a2 Tendsto (fun N => \u2211 j in range K, \u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioc \u2191j \u2191N}) atTop\n    (\ud835\udcdd (\u2211 j in range K, \u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioi \u2191j}))", "state_after": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK i : \u2115\nx\u271d : i \u2208 range K\n\u22a2 Tendsto (fun N => \u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioc \u2191i \u2191N}) atTop (\ud835\udcdd (\u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioi \u2191i}))"}, {"tactic": "rw [this]", "annotated_tactic": ["rw [this]", []], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK i : \u2115\nx\u271d : i \u2208 range K\nthis : {\u03c9 | X \u03c9 \u2208 Set.Ioi \u2191i} = \u22c3 N, {\u03c9 | X \u03c9 \u2208 Set.Ioc \u2191i \u2191N}\n\u22a2 Tendsto (fun N => \u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioc \u2191i \u2191N}) atTop (\ud835\udcdd (\u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioi \u2191i}))", "state_after": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK i : \u2115\nx\u271d : i \u2208 range K\nthis : {\u03c9 | X \u03c9 \u2208 Set.Ioi \u2191i} = \u22c3 N, {\u03c9 | X \u03c9 \u2208 Set.Ioc \u2191i \u2191N}\n\u22a2 Tendsto (fun N => \u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioc \u2191i \u2191N}) atTop (\ud835\udcdd (\u2191\u2191\u2119 (\u22c3 N, {\u03c9 | X \u03c9 \u2208 Set.Ioc \u2191i \u2191N})))"}, {"tactic": "apply tendsto_measure_iUnion", "annotated_tactic": ["apply <a>tendsto_measure_iUnion</a>", [{"full_name": "MeasureTheory.tendsto_measure_iUnion", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [530, 9], "def_end_pos": [530, 31]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK i : \u2115\nx\u271d : i \u2208 range K\nthis : {\u03c9 | X \u03c9 \u2208 Set.Ioi \u2191i} = \u22c3 N, {\u03c9 | X \u03c9 \u2208 Set.Ioc \u2191i \u2191N}\n\u22a2 Tendsto (fun N => \u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioc \u2191i \u2191N}) atTop (\ud835\udcdd (\u2191\u2191\u2119 (\u22c3 N, {\u03c9 | X \u03c9 \u2208 Set.Ioc \u2191i \u2191N})))", "state_after": "case hm\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK i : \u2115\nx\u271d : i \u2208 range K\nthis : {\u03c9 | X \u03c9 \u2208 Set.Ioi \u2191i} = \u22c3 N, {\u03c9 | X \u03c9 \u2208 Set.Ioc \u2191i \u2191N}\n\u22a2 Monotone fun N => {\u03c9 | X \u03c9 \u2208 Set.Ioc \u2191i \u2191N}"}, {"tactic": "intro m n hmn x hx", "annotated_tactic": ["intro m n hmn x hx", []], "state_before": "case hm\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK i : \u2115\nx\u271d : i \u2208 range K\nthis : {\u03c9 | X \u03c9 \u2208 Set.Ioi \u2191i} = \u22c3 N, {\u03c9 | X \u03c9 \u2208 Set.Ioc \u2191i \u2191N}\n\u22a2 Monotone fun N => {\u03c9 | X \u03c9 \u2208 Set.Ioc \u2191i \u2191N}", "state_after": "case hm\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK i : \u2115\nx\u271d : i \u2208 range K\nthis : {\u03c9 | X \u03c9 \u2208 Set.Ioi \u2191i} = \u22c3 N, {\u03c9 | X \u03c9 \u2208 Set.Ioc \u2191i \u2191N}\nm n : \u2115\nhmn : m \u2264 n\nx : \u03a9\nhx : x \u2208 (fun N => {\u03c9 | X \u03c9 \u2208 Set.Ioc \u2191i \u2191N}) m\n\u22a2 x \u2208 (fun N => {\u03c9 | X \u03c9 \u2208 Set.Ioc \u2191i \u2191N}) n"}, {"tactic": "exact \u27e8hx.1, hx.2.trans (Nat.cast_le.2 hmn)\u27e9", "annotated_tactic": ["exact \u27e8hx.1, hx.2.<a>trans</a> (<a>Nat.cast_le</a>.2 hmn)\u27e9", [{"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [120, 7], "def_end_pos": [120, 18]}, {"full_name": "Nat.cast_le", "def_path": "Mathlib/Data/Nat/Cast/Order.lean", "def_pos": [91, 9], "def_end_pos": [91, 16]}]], "state_before": "case hm\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK i : \u2115\nx\u271d : i \u2208 range K\nthis : {\u03c9 | X \u03c9 \u2208 Set.Ioi \u2191i} = \u22c3 N, {\u03c9 | X \u03c9 \u2208 Set.Ioc \u2191i \u2191N}\nm n : \u2115\nhmn : m \u2264 n\nx : \u03a9\nhx : x \u2208 (fun N => {\u03c9 | X \u03c9 \u2208 Set.Ioc \u2191i \u2191N}) m\n\u22a2 x \u2208 (fun N => {\u03c9 | X \u03c9 \u2208 Set.Ioc \u2191i \u2191N}) n", "state_after": "no goals"}, {"tactic": "apply Set.Subset.antisymm _ _", "annotated_tactic": ["apply <a>Set.Subset.antisymm</a> _ _", [{"full_name": "Set.Subset.antisymm", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [370, 9], "def_end_pos": [370, 24]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK i : \u2115\nx\u271d : i \u2208 range K\n\u22a2 {\u03c9 | X \u03c9 \u2208 Set.Ioi \u2191i} = \u22c3 N, {\u03c9 | X \u03c9 \u2208 Set.Ioc \u2191i \u2191N}", "state_after": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK i : \u2115\nx\u271d : i \u2208 range K\n\u22a2 {\u03c9 | X \u03c9 \u2208 Set.Ioi \u2191i} \u2286 \u22c3 N, {\u03c9 | X \u03c9 \u2208 Set.Ioc \u2191i \u2191N}\n\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK i : \u2115\nx\u271d : i \u2208 range K\n\u22a2 \u22c3 N, {\u03c9 | X \u03c9 \u2208 Set.Ioc \u2191i \u2191N} \u2286 {\u03c9 | X \u03c9 \u2208 Set.Ioi \u2191i}"}, {"tactic": "intro \u03c9 h\u03c9", "annotated_tactic": ["intro \u03c9 h\u03c9", []], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK i : \u2115\nx\u271d : i \u2208 range K\n\u22a2 {\u03c9 | X \u03c9 \u2208 Set.Ioi \u2191i} \u2286 \u22c3 N, {\u03c9 | X \u03c9 \u2208 Set.Ioc \u2191i \u2191N}", "state_after": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK i : \u2115\nx\u271d : i \u2208 range K\n\u03c9 : \u03a9\nh\u03c9 : \u03c9 \u2208 {\u03c9 | X \u03c9 \u2208 Set.Ioi \u2191i}\n\u22a2 \u03c9 \u2208 \u22c3 N, {\u03c9 | X \u03c9 \u2208 Set.Ioc \u2191i \u2191N}"}, {"tactic": "obtain \u27e8N, hN\u27e9 : \u2203 N : \u2115, X \u03c9 \u2264 N := exists_nat_ge (X \u03c9)", "annotated_tactic": ["obtain \u27e8N, hN\u27e9 : \u2203 N : \u2115, X \u03c9 \u2264 N := <a>exists_nat_ge</a> (X \u03c9)", [{"full_name": "exists_nat_ge", "def_path": "Mathlib/Algebra/Order/Archimedean.lean", "def_pos": [122, 9], "def_end_pos": [122, 22]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK i : \u2115\nx\u271d : i \u2208 range K\n\u03c9 : \u03a9\nh\u03c9 : \u03c9 \u2208 {\u03c9 | X \u03c9 \u2208 Set.Ioi \u2191i}\n\u22a2 \u03c9 \u2208 \u22c3 N, {\u03c9 | X \u03c9 \u2208 Set.Ioc \u2191i \u2191N}", "state_after": "case intro\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK i : \u2115\nx\u271d : i \u2208 range K\n\u03c9 : \u03a9\nh\u03c9 : \u03c9 \u2208 {\u03c9 | X \u03c9 \u2208 Set.Ioi \u2191i}\nN : \u2115\nhN : X \u03c9 \u2264 \u2191N\n\u22a2 \u03c9 \u2208 \u22c3 N, {\u03c9 | X \u03c9 \u2208 Set.Ioc \u2191i \u2191N}"}, {"tactic": "exact Set.mem_iUnion.2 \u27e8N, h\u03c9, hN\u27e9", "annotated_tactic": ["exact <a>Set.mem_iUnion</a>.2 \u27e8N, h\u03c9, hN\u27e9", [{"full_name": "Set.mem_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [201, 9], "def_end_pos": [201, 19]}]], "state_before": "case intro\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK i : \u2115\nx\u271d : i \u2208 range K\n\u03c9 : \u03a9\nh\u03c9 : \u03c9 \u2208 {\u03c9 | X \u03c9 \u2208 Set.Ioi \u2191i}\nN : \u2115\nhN : X \u03c9 \u2264 \u2191N\n\u22a2 \u03c9 \u2208 \u22c3 N, {\u03c9 | X \u03c9 \u2208 Set.Ioc \u2191i \u2191N}", "state_after": "no goals"}, {"tactic": "simp (config := {contextual := true}) only [Set.mem_Ioc, Set.mem_Ioi,\n  Set.iUnion_subset_iff, Set.setOf_subset_setOf, imp_true_iff]", "annotated_tactic": ["simp (config := {contextual := <a>true</a>}) only [<a>Set.mem_Ioc</a>, <a>Set.mem_Ioi</a>,\n          <a>Set.iUnion_subset_iff</a>, <a>Set.setOf_subset_setOf</a>, <a>imp_true_iff</a>]", [{"full_name": "Bool.true", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [549, 5], "def_end_pos": [549, 9]}, {"full_name": "Set.mem_Ioc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [141, 9], "def_end_pos": [141, 16]}, {"full_name": "Set.mem_Ioi", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [151, 9], "def_end_pos": [151, 16]}, {"full_name": "Set.iUnion_subset_iff", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [411, 9], "def_end_pos": [411, 26]}, {"full_name": "Set.setOf_subset_setOf", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [296, 9], "def_end_pos": [296, 27]}, {"full_name": "imp_true_iff", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [116, 9], "def_end_pos": [116, 21]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK i : \u2115\nx\u271d : i \u2208 range K\n\u22a2 \u22c3 N, {\u03c9 | X \u03c9 \u2208 Set.Ioc \u2191i \u2191N} \u2286 {\u03c9 | X \u03c9 \u2208 Set.Ioi \u2191i}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Option/Basic.lean", "full_name": "Option.orElse_eq_none'", "start": [400, 1], "end": [401, 29], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "full_name": "MeasureTheory.SimpleFunc.measure_preimage_lt_top_of_mem\u2112p", "start": [307, 1], "end": [333, 38], "traced_tactics": [{"tactic": "have hp_pos_real : 0 < p.toReal := ENNReal.toReal_pos hp_pos hp_ne_top", "annotated_tactic": ["have hp_pos_real : 0 < p.toReal := <a>ENNReal.toReal_pos</a> hp_pos hp_ne_top", [{"full_name": "ENNReal.toReal_pos", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2131, 9], "def_end_pos": [2131, 19]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\nhp_pos : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nf : \u03b1 \u2192\u209b E\nhf : Mem\u2112p (\u2191f) p\ny : E\nhy_ne : y \u2260 0\n\u22a2 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\nhp_pos : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nf : \u03b1 \u2192\u209b E\nhf : Mem\u2112p (\u2191f) p\ny : E\nhy_ne : y \u2260 0\nhp_pos_real : 0 < ENNReal.toReal p\n\u22a2 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4"}, {"tactic": "have hf_snorm := Mem\u2112p.snorm_lt_top hf", "annotated_tactic": ["have hf_snorm := <a>Mem\u2112p.snorm_lt_top</a> hf", [{"full_name": "MeasureTheory.Mem\u2112p.snorm_lt_top", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [135, 9], "def_end_pos": [135, 27]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\nhp_pos : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nf : \u03b1 \u2192\u209b E\nhf : Mem\u2112p (\u2191f) p\ny : E\nhy_ne : y \u2260 0\nhp_pos_real : 0 < ENNReal.toReal p\n\u22a2 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\nhp_pos : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nf : \u03b1 \u2192\u209b E\nhf : Mem\u2112p (\u2191f) p\ny : E\nhy_ne : y \u2260 0\nhp_pos_real : 0 < ENNReal.toReal p\nhf_snorm : snorm (\u2191f) p \u03bc < \u22a4\n\u22a2 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4"}, {"tactic": "rw [snorm_eq_snorm' hp_pos hp_ne_top, f.snorm'_eq, \u2190\n  @ENNReal.lt_rpow_one_div_iff _ _ (1 / p.toReal) (by simp [hp_pos_real]),\n  @ENNReal.top_rpow_of_pos (1 / (1 / p.toReal)) (by simp [hp_pos_real]),\n  ENNReal.sum_lt_top_iff] at hf_snorm", "annotated_tactic": ["rw [<a>snorm_eq_snorm'</a> hp_pos hp_ne_top, f.snorm'_eq, \u2190\n    @<a>ENNReal.lt_rpow_one_div_iff</a> _ _ (1 / p.toReal) (by simp [hp_pos_real]),\n    @<a>ENNReal.top_rpow_of_pos</a> (1 / (1 / p.toReal)) (by simp [hp_pos_real]),\n    <a>ENNReal.sum_lt_top_iff</a>] at hf_snorm", [{"full_name": "MeasureTheory.snorm_eq_snorm'", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [88, 9], "def_end_pos": [88, 24]}, {"full_name": "ENNReal.lt_rpow_one_div_iff", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [664, 9], "def_end_pos": [664, 28]}, {"full_name": "ENNReal.top_rpow_of_pos", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [392, 9], "def_end_pos": [392, 24]}, {"full_name": "ENNReal.sum_lt_top_iff", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1263, 9], "def_end_pos": [1263, 23]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\nhp_pos : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nf : \u03b1 \u2192\u209b E\nhf : Mem\u2112p (\u2191f) p\ny : E\nhy_ne : y \u2260 0\nhp_pos_real : 0 < ENNReal.toReal p\nhf_snorm : snorm (\u2191f) p \u03bc < \u22a4\n\u22a2 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\nhp_pos : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nf : \u03b1 \u2192\u209b E\nhf : Mem\u2112p (\u2191f) p\ny : E\nhy_ne : y \u2260 0\nhp_pos_real : 0 < ENNReal.toReal p\nhf_snorm : \u2200 (a : E), a \u2208 SimpleFunc.range f \u2192 \u2191\u2016a\u2016\u208a ^ ENNReal.toReal p * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {a}) < \u22a4\n\u22a2 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4"}, {"tactic": "by_cases hyf : y \u2208 f.range", "annotated_tactic": ["by_cases hyf : y \u2208 f.range", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\nhp_pos : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nf : \u03b1 \u2192\u209b E\nhf : Mem\u2112p (\u2191f) p\ny : E\nhy_ne : y \u2260 0\nhp_pos_real : 0 < ENNReal.toReal p\nhf_snorm : \u2200 (a : E), a \u2208 SimpleFunc.range f \u2192 \u2191\u2016a\u2016\u208a ^ ENNReal.toReal p * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {a}) < \u22a4\n\u22a2 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\nhp_pos : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nf : \u03b1 \u2192\u209b E\nhf : Mem\u2112p (\u2191f) p\ny : E\nhy_ne : y \u2260 0\nhp_pos_real : 0 < ENNReal.toReal p\nhf_snorm : \u2200 (a : E), a \u2208 SimpleFunc.range f \u2192 \u2191\u2016a\u2016\u208a ^ ENNReal.toReal p * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {a}) < \u22a4\nhyf : y \u2208 SimpleFunc.range f\n\u22a2 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\nhp_pos : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nf : \u03b1 \u2192\u209b E\nhf : Mem\u2112p (\u2191f) p\ny : E\nhy_ne : y \u2260 0\nhp_pos_real : 0 < ENNReal.toReal p\nhf_snorm : \u2200 (a : E), a \u2208 SimpleFunc.range f \u2192 \u2191\u2016a\u2016\u208a ^ ENNReal.toReal p * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {a}) < \u22a4\nhyf : \u00acy \u2208 SimpleFunc.range f\n\u22a2 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4"}, {"tactic": "swap", "annotated_tactic": ["swap", []], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\nhp_pos : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nf : \u03b1 \u2192\u209b E\nhf : Mem\u2112p (\u2191f) p\ny : E\nhy_ne : y \u2260 0\nhp_pos_real : 0 < ENNReal.toReal p\nhf_snorm : \u2200 (a : E), a \u2208 SimpleFunc.range f \u2192 \u2191\u2016a\u2016\u208a ^ ENNReal.toReal p * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {a}) < \u22a4\nhyf : y \u2208 SimpleFunc.range f\n\u22a2 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\nhp_pos : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nf : \u03b1 \u2192\u209b E\nhf : Mem\u2112p (\u2191f) p\ny : E\nhy_ne : y \u2260 0\nhp_pos_real : 0 < ENNReal.toReal p\nhf_snorm : \u2200 (a : E), a \u2208 SimpleFunc.range f \u2192 \u2191\u2016a\u2016\u208a ^ ENNReal.toReal p * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {a}) < \u22a4\nhyf : \u00acy \u2208 SimpleFunc.range f\n\u22a2 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4", "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\nhp_pos : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nf : \u03b1 \u2192\u209b E\nhf : Mem\u2112p (\u2191f) p\ny : E\nhy_ne : y \u2260 0\nhp_pos_real : 0 < ENNReal.toReal p\nhf_snorm : \u2200 (a : E), a \u2208 SimpleFunc.range f \u2192 \u2191\u2016a\u2016\u208a ^ ENNReal.toReal p * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {a}) < \u22a4\nhyf : \u00acy \u2208 SimpleFunc.range f\n\u22a2 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4\n\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\nhp_pos : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nf : \u03b1 \u2192\u209b E\nhf : Mem\u2112p (\u2191f) p\ny : E\nhy_ne : y \u2260 0\nhp_pos_real : 0 < ENNReal.toReal p\nhf_snorm : \u2200 (a : E), a \u2208 SimpleFunc.range f \u2192 \u2191\u2016a\u2016\u208a ^ ENNReal.toReal p * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {a}) < \u22a4\nhyf : y \u2208 SimpleFunc.range f\n\u22a2 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4"}, {"tactic": "specialize hf_snorm y hyf", "annotated_tactic": ["specialize hf_snorm y hyf", []], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\nhp_pos : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nf : \u03b1 \u2192\u209b E\nhf : Mem\u2112p (\u2191f) p\ny : E\nhy_ne : y \u2260 0\nhp_pos_real : 0 < ENNReal.toReal p\nhf_snorm : \u2200 (a : E), a \u2208 SimpleFunc.range f \u2192 \u2191\u2016a\u2016\u208a ^ ENNReal.toReal p * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {a}) < \u22a4\nhyf : y \u2208 SimpleFunc.range f\n\u22a2 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\nhp_pos : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nf : \u03b1 \u2192\u209b E\nhf : Mem\u2112p (\u2191f) p\ny : E\nhy_ne : y \u2260 0\nhp_pos_real : 0 < ENNReal.toReal p\nhyf : y \u2208 SimpleFunc.range f\nhf_snorm : \u2191\u2016y\u2016\u208a ^ ENNReal.toReal p * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4\n\u22a2 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4"}, {"tactic": "rw [ENNReal.mul_lt_top_iff] at hf_snorm", "annotated_tactic": ["rw [<a>ENNReal.mul_lt_top_iff</a>] at hf_snorm", [{"full_name": "ENNReal.mul_lt_top_iff", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [626, 9], "def_end_pos": [626, 23]}]], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\nhp_pos : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nf : \u03b1 \u2192\u209b E\nhf : Mem\u2112p (\u2191f) p\ny : E\nhy_ne : y \u2260 0\nhp_pos_real : 0 < ENNReal.toReal p\nhyf : y \u2208 SimpleFunc.range f\nhf_snorm : \u2191\u2016y\u2016\u208a ^ ENNReal.toReal p * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4\n\u22a2 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\nhp_pos : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nf : \u03b1 \u2192\u209b E\nhf : Mem\u2112p (\u2191f) p\ny : E\nhy_ne : y \u2260 0\nhp_pos_real : 0 < ENNReal.toReal p\nhyf : y \u2208 SimpleFunc.range f\nhf_snorm : \u2191\u2016y\u2016\u208a ^ ENNReal.toReal p < \u22a4 \u2227 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4 \u2228 \u2191\u2016y\u2016\u208a ^ ENNReal.toReal p = 0 \u2228 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) = 0\n\u22a2 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4"}, {"tactic": "cases hf_snorm with\n| inl hf_snorm => exact hf_snorm.2\n| inr hf_snorm =>\n  cases hf_snorm with\n  | inl hf_snorm =>\n    refine' absurd _ hy_ne\n    simpa [hp_pos_real] using hf_snorm\n  | inr hf_snorm => simp [hf_snorm]", "annotated_tactic": ["cases hf_snorm with\n  | <a>inl</a> hf_snorm => exact hf_snorm.2\n  | <a>inr</a> hf_snorm =>\n    cases hf_snorm with\n    | <a>inl</a> hf_snorm =>\n      refine' <a>absurd</a> _ hy_ne\n      simpa [hp_pos_real] using hf_snorm\n    | <a>inr</a> hf_snorm => simp [hf_snorm]", [{"full_name": "Or.inl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [517, 5], "def_end_pos": [517, 8]}, {"full_name": "Or.inr", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [519, 5], "def_end_pos": [519, 8]}, {"full_name": "Or.inl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [517, 5], "def_end_pos": [517, 8]}, {"full_name": "absurd", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [233, 21], "def_end_pos": [233, 27]}, {"full_name": "Or.inr", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [519, 5], "def_end_pos": [519, 8]}]], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\nhp_pos : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nf : \u03b1 \u2192\u209b E\nhf : Mem\u2112p (\u2191f) p\ny : E\nhy_ne : y \u2260 0\nhp_pos_real : 0 < ENNReal.toReal p\nhyf : y \u2208 SimpleFunc.range f\nhf_snorm : \u2191\u2016y\u2016\u208a ^ ENNReal.toReal p < \u22a4 \u2227 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4 \u2228 \u2191\u2016y\u2016\u208a ^ ENNReal.toReal p = 0 \u2228 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) = 0\n\u22a2 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4", "state_after": "no goals"}, {"tactic": "simp [hp_pos_real]", "annotated_tactic": ["simp [hp_pos_real]", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\nhp_pos : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nf : \u03b1 \u2192\u209b E\nhf : Mem\u2112p (\u2191f) p\ny : E\nhy_ne : y \u2260 0\nhp_pos_real : 0 < ENNReal.toReal p\nhf_snorm : (\u2211 y in SimpleFunc.range f, \u2191\u2016y\u2016\u208a ^ ENNReal.toReal p * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y})) ^ (1 / ENNReal.toReal p) < \u22a4\n\u22a2 0 < 1 / ENNReal.toReal p", "state_after": "no goals"}, {"tactic": "simp [hp_pos_real]", "annotated_tactic": ["simp [hp_pos_real]", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\nhp_pos : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nf : \u03b1 \u2192\u209b E\nhf : Mem\u2112p (\u2191f) p\ny : E\nhy_ne : y \u2260 0\nhp_pos_real : 0 < ENNReal.toReal p\nhf_snorm : \u2211 y in SimpleFunc.range f, \u2191\u2016y\u2016\u208a ^ ENNReal.toReal p * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4 ^ (1 / (1 / ENNReal.toReal p))\n\u22a2 0 < 1 / (1 / ENNReal.toReal p)", "state_after": "no goals"}, {"tactic": "suffices h_empty : f \u207b\u00b9' {y} = \u2205", "annotated_tactic": ["suffices h_empty : f \u207b\u00b9' {y} = \u2205", []], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\nhp_pos : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nf : \u03b1 \u2192\u209b E\nhf : Mem\u2112p (\u2191f) p\ny : E\nhy_ne : y \u2260 0\nhp_pos_real : 0 < ENNReal.toReal p\nhf_snorm : \u2200 (a : E), a \u2208 SimpleFunc.range f \u2192 \u2191\u2016a\u2016\u208a ^ ENNReal.toReal p * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {a}) < \u22a4\nhyf : \u00acy \u2208 SimpleFunc.range f\n\u22a2 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4", "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\nhp_pos : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nf : \u03b1 \u2192\u209b E\nhf : Mem\u2112p (\u2191f) p\ny : E\nhy_ne : y \u2260 0\nhp_pos_real : 0 < ENNReal.toReal p\nhf_snorm : \u2200 (a : E), a \u2208 SimpleFunc.range f \u2192 \u2191\u2016a\u2016\u208a ^ ENNReal.toReal p * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {a}) < \u22a4\nhyf : \u00acy \u2208 SimpleFunc.range f\nh_empty : \u2191f \u207b\u00b9' {y} = \u2205\n\u22a2 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4\n\ncase h_empty\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\nhp_pos : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nf : \u03b1 \u2192\u209b E\nhf : Mem\u2112p (\u2191f) p\ny : E\nhy_ne : y \u2260 0\nhp_pos_real : 0 < ENNReal.toReal p\nhf_snorm : \u2200 (a : E), a \u2208 SimpleFunc.range f \u2192 \u2191\u2016a\u2016\u208a ^ ENNReal.toReal p * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {a}) < \u22a4\nhyf : \u00acy \u2208 SimpleFunc.range f\n\u22a2 \u2191f \u207b\u00b9' {y} = \u2205"}, {"tactic": "ext1 x", "annotated_tactic": ["ext1 x", []], "state_before": "case h_empty\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\nhp_pos : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nf : \u03b1 \u2192\u209b E\nhf : Mem\u2112p (\u2191f) p\ny : E\nhy_ne : y \u2260 0\nhp_pos_real : 0 < ENNReal.toReal p\nhf_snorm : \u2200 (a : E), a \u2208 SimpleFunc.range f \u2192 \u2191\u2016a\u2016\u208a ^ ENNReal.toReal p * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {a}) < \u22a4\nhyf : \u00acy \u2208 SimpleFunc.range f\n\u22a2 \u2191f \u207b\u00b9' {y} = \u2205", "state_after": "case h_empty.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\nhp_pos : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nf : \u03b1 \u2192\u209b E\nhf : Mem\u2112p (\u2191f) p\ny : E\nhy_ne : y \u2260 0\nhp_pos_real : 0 < ENNReal.toReal p\nhf_snorm : \u2200 (a : E), a \u2208 SimpleFunc.range f \u2192 \u2191\u2016a\u2016\u208a ^ ENNReal.toReal p * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {a}) < \u22a4\nhyf : \u00acy \u2208 SimpleFunc.range f\nx : \u03b1\n\u22a2 x \u2208 \u2191f \u207b\u00b9' {y} \u2194 x \u2208 \u2205"}, {"tactic": "rw [Set.mem_preimage, Set.mem_singleton_iff, mem_empty_iff_false, iff_false_iff]", "annotated_tactic": ["rw [<a>Set.mem_preimage</a>, <a>Set.mem_singleton_iff</a>, <a>mem_empty_iff_false</a>, <a>iff_false_iff</a>]", [{"full_name": "Set.mem_preimage", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [64, 9], "def_end_pos": [64, 21]}, {"full_name": "Set.mem_singleton_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1273, 9], "def_end_pos": [1273, 26]}, {"full_name": "Set.mem_empty_iff_false", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [562, 9], "def_end_pos": [562, 28]}, {"full_name": "iff_false_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [201, 9], "def_end_pos": [201, 22]}]], "state_before": "case h_empty.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\nhp_pos : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nf : \u03b1 \u2192\u209b E\nhf : Mem\u2112p (\u2191f) p\ny : E\nhy_ne : y \u2260 0\nhp_pos_real : 0 < ENNReal.toReal p\nhf_snorm : \u2200 (a : E), a \u2208 SimpleFunc.range f \u2192 \u2191\u2016a\u2016\u208a ^ ENNReal.toReal p * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {a}) < \u22a4\nhyf : \u00acy \u2208 SimpleFunc.range f\nx : \u03b1\n\u22a2 x \u2208 \u2191f \u207b\u00b9' {y} \u2194 x \u2208 \u2205", "state_after": "case h_empty.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\nhp_pos : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nf : \u03b1 \u2192\u209b E\nhf : Mem\u2112p (\u2191f) p\ny : E\nhy_ne : y \u2260 0\nhp_pos_real : 0 < ENNReal.toReal p\nhf_snorm : \u2200 (a : E), a \u2208 SimpleFunc.range f \u2192 \u2191\u2016a\u2016\u208a ^ ENNReal.toReal p * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {a}) < \u22a4\nhyf : \u00acy \u2208 SimpleFunc.range f\nx : \u03b1\n\u22a2 \u00ac\u2191f x = y"}, {"tactic": "refine' fun hxy => hyf _", "annotated_tactic": ["refine' fun hxy => hyf _", []], "state_before": "case h_empty.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\nhp_pos : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nf : \u03b1 \u2192\u209b E\nhf : Mem\u2112p (\u2191f) p\ny : E\nhy_ne : y \u2260 0\nhp_pos_real : 0 < ENNReal.toReal p\nhf_snorm : \u2200 (a : E), a \u2208 SimpleFunc.range f \u2192 \u2191\u2016a\u2016\u208a ^ ENNReal.toReal p * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {a}) < \u22a4\nhyf : \u00acy \u2208 SimpleFunc.range f\nx : \u03b1\n\u22a2 \u00ac\u2191f x = y", "state_after": "case h_empty.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\nhp_pos : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nf : \u03b1 \u2192\u209b E\nhf : Mem\u2112p (\u2191f) p\ny : E\nhy_ne : y \u2260 0\nhp_pos_real : 0 < ENNReal.toReal p\nhf_snorm : \u2200 (a : E), a \u2208 SimpleFunc.range f \u2192 \u2191\u2016a\u2016\u208a ^ ENNReal.toReal p * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {a}) < \u22a4\nhyf : \u00acy \u2208 SimpleFunc.range f\nx : \u03b1\nhxy : \u2191f x = y\n\u22a2 y \u2208 SimpleFunc.range f"}, {"tactic": "rw [mem_range, Set.mem_range]", "annotated_tactic": ["rw [<a>mem_range</a>, <a>Set.mem_range</a>]", [{"full_name": "MeasureTheory.SimpleFunc.mem_range", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [108, 9], "def_end_pos": [108, 18]}, {"full_name": "Set.mem_range", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [673, 9], "def_end_pos": [673, 18]}]], "state_before": "case h_empty.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\nhp_pos : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nf : \u03b1 \u2192\u209b E\nhf : Mem\u2112p (\u2191f) p\ny : E\nhy_ne : y \u2260 0\nhp_pos_real : 0 < ENNReal.toReal p\nhf_snorm : \u2200 (a : E), a \u2208 SimpleFunc.range f \u2192 \u2191\u2016a\u2016\u208a ^ ENNReal.toReal p * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {a}) < \u22a4\nhyf : \u00acy \u2208 SimpleFunc.range f\nx : \u03b1\nhxy : \u2191f x = y\n\u22a2 y \u2208 SimpleFunc.range f", "state_after": "case h_empty.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\nhp_pos : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nf : \u03b1 \u2192\u209b E\nhf : Mem\u2112p (\u2191f) p\ny : E\nhy_ne : y \u2260 0\nhp_pos_real : 0 < ENNReal.toReal p\nhf_snorm : \u2200 (a : E), a \u2208 SimpleFunc.range f \u2192 \u2191\u2016a\u2016\u208a ^ ENNReal.toReal p * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {a}) < \u22a4\nhyf : \u00acy \u2208 SimpleFunc.range f\nx : \u03b1\nhxy : \u2191f x = y\n\u22a2 \u2203 y_1, \u2191f y_1 = y"}, {"tactic": "exact \u27e8x, hxy\u27e9", "annotated_tactic": ["exact \u27e8x, hxy\u27e9", []], "state_before": "case h_empty.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\nhp_pos : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nf : \u03b1 \u2192\u209b E\nhf : Mem\u2112p (\u2191f) p\ny : E\nhy_ne : y \u2260 0\nhp_pos_real : 0 < ENNReal.toReal p\nhf_snorm : \u2200 (a : E), a \u2208 SimpleFunc.range f \u2192 \u2191\u2016a\u2016\u208a ^ ENNReal.toReal p * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {a}) < \u22a4\nhyf : \u00acy \u2208 SimpleFunc.range f\nx : \u03b1\nhxy : \u2191f x = y\n\u22a2 \u2203 y_1, \u2191f y_1 = y", "state_after": "no goals"}, {"tactic": "rw [h_empty, measure_empty]", "annotated_tactic": ["rw [h_empty, <a>measure_empty</a>]", [{"full_name": "MeasureTheory.measure_empty", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [185, 9], "def_end_pos": [185, 22]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\nhp_pos : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nf : \u03b1 \u2192\u209b E\nhf : Mem\u2112p (\u2191f) p\ny : E\nhy_ne : y \u2260 0\nhp_pos_real : 0 < ENNReal.toReal p\nhf_snorm : \u2200 (a : E), a \u2208 SimpleFunc.range f \u2192 \u2191\u2016a\u2016\u208a ^ ENNReal.toReal p * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {a}) < \u22a4\nhyf : \u00acy \u2208 SimpleFunc.range f\nh_empty : \u2191f \u207b\u00b9' {y} = \u2205\n\u22a2 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4", "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\nhp_pos : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nf : \u03b1 \u2192\u209b E\nhf : Mem\u2112p (\u2191f) p\ny : E\nhy_ne : y \u2260 0\nhp_pos_real : 0 < ENNReal.toReal p\nhf_snorm : \u2200 (a : E), a \u2208 SimpleFunc.range f \u2192 \u2191\u2016a\u2016\u208a ^ ENNReal.toReal p * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {a}) < \u22a4\nhyf : \u00acy \u2208 SimpleFunc.range f\nh_empty : \u2191f \u207b\u00b9' {y} = \u2205\n\u22a2 0 < \u22a4"}, {"tactic": "exact ENNReal.coe_lt_top", "annotated_tactic": ["exact <a>ENNReal.coe_lt_top</a>", [{"full_name": "ENNReal.coe_lt_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [308, 17], "def_end_pos": [308, 27]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\nhp_pos : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nf : \u03b1 \u2192\u209b E\nhf : Mem\u2112p (\u2191f) p\ny : E\nhy_ne : y \u2260 0\nhp_pos_real : 0 < ENNReal.toReal p\nhf_snorm : \u2200 (a : E), a \u2208 SimpleFunc.range f \u2192 \u2191\u2016a\u2016\u208a ^ ENNReal.toReal p * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {a}) < \u22a4\nhyf : \u00acy \u2208 SimpleFunc.range f\nh_empty : \u2191f \u207b\u00b9' {y} = \u2205\n\u22a2 0 < \u22a4", "state_after": "no goals"}, {"tactic": "exact hf_snorm.2", "annotated_tactic": ["exact hf_snorm.2", []], "state_before": "case pos.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\nhp_pos : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nf : \u03b1 \u2192\u209b E\nhf : Mem\u2112p (\u2191f) p\ny : E\nhy_ne : y \u2260 0\nhp_pos_real : 0 < ENNReal.toReal p\nhyf : y \u2208 SimpleFunc.range f\nhf_snorm : \u2191\u2016y\u2016\u208a ^ ENNReal.toReal p < \u22a4 \u2227 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4\n\u22a2 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4", "state_after": "no goals"}, {"tactic": "cases hf_snorm with\n| inl hf_snorm =>\n  refine' absurd _ hy_ne\n  simpa [hp_pos_real] using hf_snorm\n| inr hf_snorm => simp [hf_snorm]", "annotated_tactic": ["cases hf_snorm with\n    | <a>inl</a> hf_snorm =>\n      refine' <a>absurd</a> _ hy_ne\n      simpa [hp_pos_real] using hf_snorm\n    | <a>inr</a> hf_snorm => simp [hf_snorm]", [{"full_name": "Or.inl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [517, 5], "def_end_pos": [517, 8]}, {"full_name": "absurd", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [233, 21], "def_end_pos": [233, 27]}, {"full_name": "Or.inr", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [519, 5], "def_end_pos": [519, 8]}]], "state_before": "case pos.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\nhp_pos : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nf : \u03b1 \u2192\u209b E\nhf : Mem\u2112p (\u2191f) p\ny : E\nhy_ne : y \u2260 0\nhp_pos_real : 0 < ENNReal.toReal p\nhyf : y \u2208 SimpleFunc.range f\nhf_snorm : \u2191\u2016y\u2016\u208a ^ ENNReal.toReal p = 0 \u2228 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) = 0\n\u22a2 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4", "state_after": "no goals"}, {"tactic": "refine' absurd _ hy_ne", "annotated_tactic": ["refine' <a>absurd</a> _ hy_ne", [{"full_name": "absurd", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [233, 21], "def_end_pos": [233, 27]}]], "state_before": "case pos.inr.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\nhp_pos : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nf : \u03b1 \u2192\u209b E\nhf : Mem\u2112p (\u2191f) p\ny : E\nhy_ne : y \u2260 0\nhp_pos_real : 0 < ENNReal.toReal p\nhyf : y \u2208 SimpleFunc.range f\nhf_snorm : \u2191\u2016y\u2016\u208a ^ ENNReal.toReal p = 0\n\u22a2 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4", "state_after": "case pos.inr.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\nhp_pos : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nf : \u03b1 \u2192\u209b E\nhf : Mem\u2112p (\u2191f) p\ny : E\nhy_ne : y \u2260 0\nhp_pos_real : 0 < ENNReal.toReal p\nhyf : y \u2208 SimpleFunc.range f\nhf_snorm : \u2191\u2016y\u2016\u208a ^ ENNReal.toReal p = 0\n\u22a2 y = 0"}, {"tactic": "simpa [hp_pos_real] using hf_snorm", "annotated_tactic": ["simpa [hp_pos_real] using hf_snorm", []], "state_before": "case pos.inr.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\nhp_pos : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nf : \u03b1 \u2192\u209b E\nhf : Mem\u2112p (\u2191f) p\ny : E\nhy_ne : y \u2260 0\nhp_pos_real : 0 < ENNReal.toReal p\nhyf : y \u2208 SimpleFunc.range f\nhf_snorm : \u2191\u2016y\u2016\u208a ^ ENNReal.toReal p = 0\n\u22a2 y = 0", "state_after": "no goals"}, {"tactic": "simp [hf_snorm]", "annotated_tactic": ["simp [hf_snorm]", []], "state_before": "case pos.inr.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\nhp_pos : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nf : \u03b1 \u2192\u209b E\nhf : Mem\u2112p (\u2191f) p\ny : E\nhy_ne : y \u2260 0\nhp_pos_real : 0 < ENNReal.toReal p\nhyf : y \u2208 SimpleFunc.range f\nhf_snorm : \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) = 0\n\u22a2 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "full_name": "MeasureTheory.FinStronglyMeasurable.inf", "start": [1124, 11], "end": [1130, 88], "traced_tactics": [{"tactic": "refine'\n  \u27e8fun n => hf.approx n \u2293 hg.approx n, fun n => _, fun x =>\n    (hf.tendsto_approx x).inf_right_nhds (hg.tendsto_approx x)\u27e9", "annotated_tactic": ["refine'\n    \u27e8fun n => hf.approx n \u2293 hg.approx n, fun n => _, fun x =>\n      (hf.tendsto_approx x).<a>inf_right_nhds</a> (hg.tendsto_approx x)\u27e9", [{"full_name": "Filter.Tendsto.inf_right_nhds", "def_path": "Mathlib/Topology/Order/Lattice.lean", "def_pos": [126, 9], "def_end_pos": [126, 38]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u2074 : Countable \u03b9\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : SemilatticeInf \u03b2\ninst\u271d : ContinuousInf \u03b2\nhf : FinStronglyMeasurable f \u03bc\nhg : FinStronglyMeasurable g \u03bc\n\u22a2 FinStronglyMeasurable (f \u2293 g) \u03bc", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u2074 : Countable \u03b9\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : SemilatticeInf \u03b2\ninst\u271d : ContinuousInf \u03b2\nhf : FinStronglyMeasurable f \u03bc\nhg : FinStronglyMeasurable g \u03bc\nn : \u2115\n\u22a2 \u2191\u2191\u03bc (support \u2191((fun n => FinStronglyMeasurable.approx hf n \u2293 FinStronglyMeasurable.approx hg n) n)) < \u22a4"}, {"tactic": "refine' (measure_mono (support_inf _ _)).trans_lt _", "annotated_tactic": ["refine' (<a>measure_mono</a> (<a>support_inf</a> _ _)).<a>trans_lt</a> _", [{"full_name": "MeasureTheory.measure_mono", "def_path": 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Measure \u03b1\nf g : \u03b1 \u2192 \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : SemilatticeInf \u03b2\ninst\u271d : ContinuousInf \u03b2\nhf : FinStronglyMeasurable f \u03bc\nhg : FinStronglyMeasurable g \u03bc\nn : \u2115\n\u22a2 \u2191\u2191\u03bc\n      ((support fun x => \u2191(FinStronglyMeasurable.approx hf n) x) \u222a\n        support fun x => \u2191(FinStronglyMeasurable.approx hg n) x) <\n    \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "full_name": "MeasureTheory.SimpleFunc.finset_sup_apply", "start": [715, 1], "end": [719, 59], "traced_tactics": [{"tactic": "refine' Finset.induction_on s rfl _", "annotated_tactic": ["refine' <a>Finset.induction_on</a> s <a>rfl</a> _", [{"full_name": "Finset.induction_on", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1251, 19], "def_end_pos": [1251, 31]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : MeasurableSpace \u03b1\nK : Type u_5\ninst\u271d\u00b9 : SemilatticeSup \u03b2\ninst\u271d : OrderBot \u03b2\nf : \u03b3 \u2192 \u03b1 \u2192\u209b \u03b2\ns : Finset \u03b3\na : \u03b1\n\u22a2 \u2191(Finset.sup s f) a = Finset.sup s fun c => \u2191(f c) a", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : MeasurableSpace \u03b1\nK : Type u_5\ninst\u271d\u00b9 : SemilatticeSup \u03b2\ninst\u271d : OrderBot \u03b2\nf : \u03b3 \u2192 \u03b1 \u2192\u209b \u03b2\ns : Finset \u03b3\na : \u03b1\n\u22a2 \u2200 \u2983a_1 : \u03b3\u2984 {s : Finset \u03b3},\n    \u00aca_1 \u2208 s \u2192\n      (\u2191(Finset.sup s f) a = Finset.sup s fun c => \u2191(f c) a) \u2192\n        \u2191(Finset.sup (insert a_1 s) f) a = Finset.sup (insert a_1 s) fun c => \u2191(f c) a"}, {"tactic": "intro a s _ ih", "annotated_tactic": ["intro a s _ ih", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : MeasurableSpace \u03b1\nK : Type u_5\ninst\u271d\u00b9 : SemilatticeSup \u03b2\ninst\u271d : OrderBot \u03b2\nf : \u03b3 \u2192 \u03b1 \u2192\u209b \u03b2\ns : Finset \u03b3\na : \u03b1\n\u22a2 \u2200 \u2983a_1 : \u03b3\u2984 {s : Finset \u03b3},\n    \u00aca_1 \u2208 s \u2192\n      (\u2191(Finset.sup s f) a = Finset.sup s fun c => \u2191(f c) a) \u2192\n        \u2191(Finset.sup (insert a_1 s) f) a = Finset.sup (insert a_1 s) fun c => \u2191(f c) a", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : MeasurableSpace \u03b1\nK : Type u_5\ninst\u271d\u00b9 : SemilatticeSup \u03b2\ninst\u271d : OrderBot \u03b2\nf : \u03b3 \u2192 \u03b1 \u2192\u209b \u03b2\ns\u271d : Finset \u03b3\na\u271d\u00b9 : \u03b1\na : \u03b3\ns : Finset \u03b3\na\u271d : \u00aca \u2208 s\nih : \u2191(Finset.sup s f) a\u271d\u00b9 = Finset.sup s fun c => \u2191(f c) a\u271d\u00b9\n\u22a2 \u2191(Finset.sup (insert a s) f) a\u271d\u00b9 = Finset.sup (insert a s) fun c => \u2191(f c) a\u271d\u00b9"}, {"tactic": "rw [Finset.sup_insert, Finset.sup_insert, sup_apply, ih]", "annotated_tactic": ["rw [<a>Finset.sup_insert</a>, <a>Finset.sup_insert</a>, <a>sup_apply</a>, ih]", [{"full_name": "Finset.sup_insert", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [59, 9], "def_end_pos": [59, 19]}, {"full_name": "Finset.sup_insert", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [59, 9], "def_end_pos": [59, 19]}, {"full_name": "MeasureTheory.SimpleFunc.sup_apply", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [520, 9], "def_end_pos": [520, 18]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : MeasurableSpace \u03b1\nK : Type u_5\ninst\u271d\u00b9 : SemilatticeSup \u03b2\ninst\u271d : OrderBot \u03b2\nf : \u03b3 \u2192 \u03b1 \u2192\u209b \u03b2\ns\u271d : Finset \u03b3\na\u271d\u00b9 : \u03b1\na : \u03b3\ns : Finset \u03b3\na\u271d : \u00aca \u2208 s\nih : \u2191(Finset.sup s f) a\u271d\u00b9 = Finset.sup s fun c => \u2191(f c) a\u271d\u00b9\n\u22a2 \u2191(Finset.sup (insert a s) f) a\u271d\u00b9 = Finset.sup (insert a s) fun c => \u2191(f c) a\u271d\u00b9", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Stieltjes.lean", "full_name": "StieltjesFunction.measure_Icc", "start": [385, 1], "end": [392, 57], "traced_tactics": [{"tactic": "rcases le_or_lt a b with (hab | hab)", "annotated_tactic": ["rcases <a>le_or_lt</a> a b with (hab | hab)", [{"full_name": "le_or_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [340, 9], "def_end_pos": [340, 17]}]], "state_before": "f : StieltjesFunction\na b : \u211d\n\u22a2 \u2191\u2191(StieltjesFunction.measure f) (Icc a b) = ofReal (\u2191f b - leftLim (\u2191f) a)", "state_after": "case inl\nf : StieltjesFunction\na b : \u211d\nhab : a \u2264 b\n\u22a2 \u2191\u2191(StieltjesFunction.measure f) (Icc a b) = ofReal (\u2191f b - leftLim (\u2191f) a)\n\ncase inr\nf : StieltjesFunction\na b : \u211d\nhab : b < a\n\u22a2 \u2191\u2191(StieltjesFunction.measure f) (Icc a b) = ofReal (\u2191f b - leftLim (\u2191f) a)"}, {"tactic": "have A : Disjoint {a} (Ioc a b) := by simp", "annotated_tactic": ["have A : <a>Disjoint</a> {a} (<a>Ioc</a> a b) := by simp", [{"full_name": "Disjoint", "def_path": "Mathlib/Order/Disjoint.lean", "def_pos": [41, 5], "def_end_pos": [41, 13]}, {"full_name": "Set.Ioc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [69, 5], "def_end_pos": [69, 8]}]], "state_before": "case inl\nf : StieltjesFunction\na b : \u211d\nhab : a \u2264 b\n\u22a2 \u2191\u2191(StieltjesFunction.measure f) (Icc a b) = ofReal (\u2191f b - leftLim (\u2191f) a)", "state_after": "case inl\nf : StieltjesFunction\na b : \u211d\nhab : a \u2264 b\nA : Disjoint {a} (Ioc a b)\n\u22a2 \u2191\u2191(StieltjesFunction.measure f) (Icc a b) = ofReal (\u2191f b - leftLim (\u2191f) a)"}, {"tactic": "simp [\u2190 Icc_union_Ioc_eq_Icc le_rfl hab, -singleton_union, \u2190 ENNReal.ofReal_add,\n  f.mono.leftLim_le, measure_union A measurableSet_Ioc, f.mono hab]", "annotated_tactic": ["simp [\u2190 <a>Icc_union_Ioc_eq_Icc</a> <a>le_rfl</a> hab, -<a>singleton_union</a>, \u2190 <a>ENNReal.ofReal_add</a>,\n      f.mono.leftLim_le, <a>measure_union</a> A <a>measurableSet_Ioc</a>, f.mono hab]", [{"full_name": "Set.Icc_union_Ioc_eq_Icc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [1606, 9], "def_end_pos": [1606, 29]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}, {"full_name": "Set.singleton_union", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1342, 9], "def_end_pos": [1342, 24]}, {"full_name": "ENNReal.ofReal_add", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2025, 9], "def_end_pos": [2025, 19]}, {"full_name": "MeasureTheory.measure_union", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [124, 9], "def_end_pos": [124, 22]}, {"full_name": "measurableSet_Ioc", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [589, 9], "def_end_pos": [589, 26]}]], "state_before": "case inl\nf : StieltjesFunction\na b : \u211d\nhab : a \u2264 b\nA : Disjoint {a} (Ioc a b)\n\u22a2 \u2191\u2191(StieltjesFunction.measure f) (Icc a b) = ofReal (\u2191f b - leftLim (\u2191f) a)", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "f : StieltjesFunction\na b : \u211d\nhab : a \u2264 b\n\u22a2 Disjoint {a} (Ioc a b)", "state_after": "no goals"}, {"tactic": "simp only [hab, measure_empty, Icc_eq_empty, not_le]", "annotated_tactic": ["simp only [hab, <a>measure_empty</a>, <a>Icc_eq_empty</a>, <a>not_le</a>]", [{"full_name": "MeasureTheory.measure_empty", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [185, 9], "def_end_pos": [185, 22]}, {"full_name": "Set.Icc_eq_empty", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [358, 9], "def_end_pos": [358, 21]}, {"full_name": "not_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [373, 9], "def_end_pos": [373, 15]}]], "state_before": "case inr\nf : StieltjesFunction\na b : \u211d\nhab : b < a\n\u22a2 \u2191\u2191(StieltjesFunction.measure f) (Icc a b) = ofReal (\u2191f b - leftLim (\u2191f) a)", "state_after": "case inr\nf : StieltjesFunction\na b : \u211d\nhab : b < a\n\u22a2 0 = ofReal (\u2191f b - leftLim (\u2191f) a)"}, {"tactic": "symm", "annotated_tactic": ["symm", []], "state_before": "case inr\nf : StieltjesFunction\na b : \u211d\nhab : b < a\n\u22a2 0 = ofReal (\u2191f b - leftLim (\u2191f) a)", "state_after": "case inr\nf : StieltjesFunction\na b : \u211d\nhab : b < a\n\u22a2 ofReal (\u2191f b - leftLim (\u2191f) a) = 0"}, {"tactic": "simp [ENNReal.ofReal_eq_zero, f.mono.le_leftLim hab]", "annotated_tactic": ["simp [<a>ENNReal.ofReal_eq_zero</a>, f.mono.le_leftLim hab]", [{"full_name": "ENNReal.ofReal_eq_zero", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2170, 9], "def_end_pos": [2170, 23]}]], "state_before": "case inr\nf : StieltjesFunction\na b : \u211d\nhab : b < a\n\u22a2 ofReal (\u2191f b - leftLim (\u2191f) a) = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/AEMeasurable.lean", "full_name": "aemeasurable_uIoc_iff", "start": [252, 1], "end": [255, 48], "traced_tactics": [{"tactic": "rw [uIoc_eq_union, aemeasurable_union_iff]", "annotated_tactic": ["rw [<a>uIoc_eq_union</a>, <a>aemeasurable_union_iff</a>]", [{"full_name": "Set.uIoc_eq_union", "def_path": "Mathlib/Data/Set/Intervals/UnorderedInterval.lean", "def_pos": [293, 7], "def_end_pos": [293, 20]}, {"full_name": "aemeasurable_union_iff", "def_path": "Mathlib/MeasureTheory/Measure/AEMeasurable.lean", "def_pos": [146, 9], "def_end_pos": [146, 38]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\nR : Type u_6\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b3\ninst\u271d\u00b9 : MeasurableSpace \u03b4\nf\u271d g : \u03b1 \u2192 \u03b2\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : LinearOrder \u03b1\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\n\u22a2 AEMeasurable f \u2194 AEMeasurable f \u2227 AEMeasurable f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Prod.lean", "full_name": "Set.prod_eq_iff_eq", "start": [450, 1], "end": [454, 6], "traced_tactics": [{"tactic": "simp_rw [prod_eq_prod_iff, ht.ne_empty, and_true_iff, or_iff_left_iff_imp,\n  or_false_iff]", "annotated_tactic": ["simp_rw [<a>prod_eq_prod_iff</a>, ht.ne_empty, <a>and_true_iff</a>, <a>or_iff_left_iff_imp</a>,\n    <a>or_false_iff</a>]", [{"full_name": "Set.prod_eq_prod_iff", "def_path": "Mathlib/Data/Set/Prod.lean", "def_pos": [436, 9], "def_end_pos": [436, 25]}, {"full_name": "and_true_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [145, 9], "def_end_pos": [145, 21]}, {"full_name": "or_iff_left_iff_imp", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [295, 17], "def_end_pos": [295, 36]}, {"full_name": "or_false_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [188, 9], "def_end_pos": [188, 21]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\nht : Set.Nonempty t\n\u22a2 s \u00d7\u02e2 t = s\u2081 \u00d7\u02e2 t \u2194 s = s\u2081", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\nht : Set.Nonempty t\n\u22a2 s = \u2205 \u2227 s\u2081 = \u2205 \u2192 s = s\u2081"}, {"tactic": "rintro \u27e8rfl, rfl\u27e9", "annotated_tactic": ["rintro \u27e8rfl, rfl\u27e9", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\nht : Set.Nonempty t\n\u22a2 s = \u2205 \u2227 s\u2081 = \u2205 \u2192 s = s\u2081", "state_after": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\nht : Set.Nonempty t\n\u22a2 \u2205 = \u2205"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\nht : Set.Nonempty t\n\u22a2 \u2205 = \u2205", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Num/Lemmas.lean", "full_name": "Num.pred_succ", "start": [1236, 1], "end": [1239, 89], "traced_tactics": [{"tactic": "rw [PosNum.pred'_succ']", "annotated_tactic": ["rw [<a>PosNum.pred'_succ'</a>]", [{"full_name": "PosNum.pred'_succ'", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [547, 9], "def_end_pos": [547, 20]}]], "state_before": "\u03b1 : Type u_1\np : PosNum\n\u22a2 toZNumNeg (PosNum.pred' (succ' (pos p))) = ZNum.neg p", "state_after": "\u03b1 : Type u_1\np : PosNum\n\u22a2 toZNumNeg (pos p) = ZNum.neg p"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\u03b1 : Type u_1\np : PosNum\n\u22a2 toZNumNeg (pos p) = ZNum.neg p", "state_after": "no goals"}, {"tactic": "rw [ZNum.pred, \u2190 toZNum_succ, Num.succ, PosNum.succ'_pred', toZNum]", "annotated_tactic": ["rw [<a>ZNum.pred</a>, \u2190 <a>toZNum_succ</a>, <a>Num.succ</a>, <a>PosNum.succ'_pred'</a>, <a>toZNum</a>]", [{"full_name": "ZNum.pred", "def_path": "Mathlib/Data/Num/Basic.lean", "def_pos": [368, 5], "def_end_pos": [368, 9]}, {"full_name": "Num.toZNum_succ", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [1225, 9], "def_end_pos": [1225, 20]}, {"full_name": "Num.succ", "def_path": "Mathlib/Data/Num/Basic.lean", "def_pos": [243, 5], "def_end_pos": [243, 9]}, {"full_name": "PosNum.succ'_pred'", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [552, 9], "def_end_pos": [552, 20]}, {"full_name": "Num.toZNum", "def_path": "Mathlib/Data/Num/Basic.lean", "def_pos": [321, 5], "def_end_pos": [321, 11]}]], "state_before": "\u03b1 : Type u_1\np : PosNum\n\u22a2 ZNum.succ (ZNum.pred (ZNum.pos p)) = ZNum.pos p", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Nat/Init/Lemmas.lean", "full_name": "Nat.max_eq_max", "start": [29, 11], "end": [29, 70], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "full_name": "MeasureTheory.SimpleFunc.pair_preimage_singleton", "start": [412, 1], "end": [415, 30], "traced_tactics": [{"tactic": "rw [\u2190 singleton_prod_singleton]", "annotated_tactic": ["rw [\u2190 <a>singleton_prod_singleton</a>]", [{"full_name": "Set.singleton_prod_singleton", "def_path": "Mathlib/Data/Set/Prod.lean", "def_pos": [148, 9], "def_end_pos": [148, 33]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192\u209b \u03b2\ng : \u03b1 \u2192\u209b \u03b3\nb : \u03b2\nc : \u03b3\n\u22a2 \u2191(pair f g) \u207b\u00b9' {(b, c)} = \u2191f \u207b\u00b9' {b} \u2229 \u2191g \u207b\u00b9' {c}", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192\u209b \u03b2\ng : \u03b1 \u2192\u209b \u03b3\nb : \u03b2\nc : \u03b3\n\u22a2 \u2191(pair f g) \u207b\u00b9' {b} \u00d7\u02e2 {c} = \u2191f \u207b\u00b9' {b} \u2229 \u2191g \u207b\u00b9' {c}"}, {"tactic": "exact pair_preimage _ _ _ _", "annotated_tactic": ["exact <a>pair_preimage</a> _ _ _ _", [{"full_name": "MeasureTheory.SimpleFunc.pair_preimage", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [406, 9], "def_end_pos": [406, 22]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192\u209b \u03b2\ng : \u03b1 \u2192\u209b \u03b3\nb : \u03b2\nc : \u03b3\n\u22a2 \u2191(pair f g) \u207b\u00b9' {b} \u00d7\u02e2 {c} = \u2191f \u207b\u00b9' {b} \u2229 \u2191g \u207b\u00b9' {c}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Real.lean", "full_name": "MeasureTheory.ae_bdd_condexp_of_ae_bdd", "start": [146, 1], "end": [181, 15], "traced_tactics": [{"tactic": "by_cases hnm : m \u2264 m0", "annotated_tactic": ["by_cases hnm : m \u2264 m0", []], "state_before": "\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nR : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhbdd : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |f x| \u2264 \u2191R\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |(\u03bc[f|m]) x| \u2264 \u2191R", "state_after": "case pos\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nR : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhbdd : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |f x| \u2264 \u2191R\nhnm : m \u2264 m0\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |(\u03bc[f|m]) x| \u2264 \u2191R\n\ncase neg\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nR : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhbdd : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |f x| \u2264 \u2191R\nhnm : \u00acm \u2264 m0\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |(\u03bc[f|m]) x| \u2264 \u2191R"}, {"tactic": "swap", "annotated_tactic": ["swap", []], "state_before": "case pos\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nR : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhbdd : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |f x| \u2264 \u2191R\nhnm : m \u2264 m0\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |(\u03bc[f|m]) x| \u2264 \u2191R\n\ncase neg\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nR : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhbdd : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |f x| \u2264 \u2191R\nhnm : \u00acm \u2264 m0\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |(\u03bc[f|m]) x| \u2264 \u2191R", "state_after": "case neg\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nR : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhbdd : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |f x| \u2264 \u2191R\nhnm : \u00acm \u2264 m0\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |(\u03bc[f|m]) x| \u2264 \u2191R\n\ncase pos\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nR : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhbdd : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |f x| \u2264 \u2191R\nhnm : m \u2264 m0\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |(\u03bc[f|m]) x| \u2264 \u2191R"}, {"tactic": "by_cases hfint : Integrable f \u03bc", "annotated_tactic": ["by_cases hfint : <a>Integrable</a> f \u03bc", [{"full_name": "MeasureTheory.Integrable", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [442, 5], "def_end_pos": [442, 15]}]], "state_before": "case pos\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nR : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhbdd : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |f x| \u2264 \u2191R\nhnm : m \u2264 m0\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |(\u03bc[f|m]) x| \u2264 \u2191R", "state_after": "case pos\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nR : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhbdd : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |f x| \u2264 \u2191R\nhnm : m \u2264 m0\nhfint : Integrable f\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |(\u03bc[f|m]) x| \u2264 \u2191R\n\ncase neg\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nR : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhbdd : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |f x| \u2264 \u2191R\nhnm : m \u2264 m0\nhfint : \u00acIntegrable f\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |(\u03bc[f|m]) x| \u2264 \u2191R"}, {"tactic": "swap", "annotated_tactic": ["swap", []], "state_before": "case pos\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nR : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhbdd : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |f x| \u2264 \u2191R\nhnm : m \u2264 m0\nhfint : Integrable f\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |(\u03bc[f|m]) x| \u2264 \u2191R\n\ncase neg\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nR : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhbdd : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |f x| \u2264 \u2191R\nhnm : m \u2264 m0\nhfint : \u00acIntegrable f\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |(\u03bc[f|m]) x| \u2264 \u2191R", "state_after": "case neg\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nR : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhbdd : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |f x| \u2264 \u2191R\nhnm : m \u2264 m0\nhfint : \u00acIntegrable f\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |(\u03bc[f|m]) x| \u2264 \u2191R\n\ncase pos\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nR : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhbdd : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |f x| \u2264 \u2191R\nhnm : m \u2264 m0\nhfint : Integrable f\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |(\u03bc[f|m]) x| \u2264 \u2191R"}, {"tactic": "by_contra h", "annotated_tactic": ["by_contra h", []], "state_before": "case pos\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nR : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhbdd : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |f x| \u2264 \u2191R\nhnm : m \u2264 m0\nhfint : Integrable f\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |(\u03bc[f|m]) x| \u2264 \u2191R", "state_after": "case pos\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nR : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhbdd : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |f x| \u2264 \u2191R\nhnm : m \u2264 m0\nhfint : Integrable f\nh : \u00ac\u2200\u1d50 (x : \u03b1) \u2202\u03bc, |(\u03bc[f|m]) x| \u2264 \u2191R\n\u22a2 False"}, {"tactic": "change \u03bc _ \u2260 0 at h", "annotated_tactic": ["change \u03bc _ \u2260 0 at h", []], "state_before": "case pos\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nR : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhbdd : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |f x| \u2264 \u2191R\nhnm : m \u2264 m0\nhfint : Integrable f\nh : \u00ac\u2200\u1d50 (x : \u03b1) \u2202\u03bc, |(\u03bc[f|m]) x| \u2264 \u2191R\n\u22a2 False", "state_after": "case pos\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nR : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhbdd : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |f x| \u2264 \u2191R\nhnm : m \u2264 m0\nhfint : Integrable f\nh : \u2191\u2191\u03bc {x | (fun x => |(\u03bc[f|m]) x| \u2264 \u2191R) x}\u1d9c \u2260 0\n\u22a2 False"}, {"tactic": "simp only [\u2190 zero_lt_iff, Set.compl_def, Set.mem_setOf_eq, not_le] at h", "annotated_tactic": ["simp only [\u2190 <a>zero_lt_iff</a>, <a>Set.compl_def</a>, <a>Set.mem_setOf_eq</a>, <a>not_le</a>] at h", [{"full_name": "zero_lt_iff", "def_path": "Mathlib/Algebra/Order/WithZero.lean", "def_pos": [106, 9], "def_end_pos": [106, 20]}, {"full_name": "Set.compl_def", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1641, 9], "def_end_pos": [1641, 18]}, {"full_name": "Set.mem_setOf_eq", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [256, 29], "def_end_pos": [256, 41]}, {"full_name": "not_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [373, 9], "def_end_pos": [373, 15]}]], "state_before": "case pos\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nR : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhbdd : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |f x| \u2264 \u2191R\nhnm : m \u2264 m0\nhfint : Integrable f\nh : \u2191\u2191\u03bc {x | (fun x => |(\u03bc[f|m]) x| \u2264 \u2191R) x}\u1d9c \u2260 0\n\u22a2 False", "state_after": "case pos\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nR : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhbdd : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |f x| \u2264 \u2191R\nhnm : m \u2264 m0\nhfint : Integrable f\nh : 0 < \u2191\u2191\u03bc {x | \u2191R < |(\u03bc[f|m]) x|}\n\u22a2 False"}, {"tactic": "suffices (\u03bc {x | \u2191R < |(\u03bc[f|m]) x|}).toReal * \u2191R < (\u03bc {x | \u2191R < |(\u03bc[f|m]) x|}).toReal * \u2191R by\n  exact this.ne rfl", "annotated_tactic": ["suffices (\u03bc {x | \u2191R < |(\u03bc[f|m]) x|}).<a>toReal</a> * \u2191R < (\u03bc {x | \u2191R < |(\u03bc[f|m]) x|}).<a>toReal</a> * \u2191R by\n    exact this.ne <a>rfl</a>", [{"full_name": "ENNReal.toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [168, 15], "def_end_pos": [168, 21]}, {"full_name": "ENNReal.toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [168, 15], "def_end_pos": [168, 21]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "case pos\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nR : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhbdd : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |f x| \u2264 \u2191R\nhnm : m \u2264 m0\nhfint : Integrable f\nh : 0 < \u2191\u2191\u03bc {x | \u2191R < |(\u03bc[f|m]) x|}\n\u22a2 False", "state_after": "case pos\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nR : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhbdd : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |f x| \u2264 \u2191R\nhnm : m \u2264 m0\nhfint : Integrable f\nh : 0 < \u2191\u2191\u03bc {x | \u2191R < |(\u03bc[f|m]) x|}\n\u22a2 ENNReal.toReal (\u2191\u2191\u03bc {x | \u2191R < |(\u03bc[f|m]) x|}) * \u2191R < ENNReal.toReal (\u2191\u2191\u03bc {x | \u2191R < |(\u03bc[f|m]) x|}) * \u2191R"}, {"tactic": "refine' lt_of_lt_of_le (set_integral_gt_gt R.coe_nonneg _ _ h.ne.symm) _", "annotated_tactic": ["refine' <a>lt_of_lt_of_le</a> (<a>set_integral_gt_gt</a> R.coe_nonneg _ _ h.ne.symm) _", [{"full_name": "lt_of_lt_of_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [115, 9], "def_end_pos": [115, 23]}, {"full_name": "MeasureTheory.set_integral_gt_gt", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [597, 9], "def_end_pos": [597, 27]}]], "state_before": "case pos\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nR : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhbdd : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |f x| \u2264 \u2191R\nhnm : m \u2264 m0\nhfint : Integrable f\nh : 0 < \u2191\u2191\u03bc {x | \u2191R < |(\u03bc[f|m]) x|}\n\u22a2 ENNReal.toReal (\u2191\u2191\u03bc {x | \u2191R < |(\u03bc[f|m]) x|}) * \u2191R < ENNReal.toReal (\u2191\u2191\u03bc {x | \u2191R < |(\u03bc[f|m]) x|}) * \u2191R", "state_after": "case pos.refine'_1\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nR : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhbdd : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |f x| \u2264 \u2191R\nhnm : m \u2264 m0\nhfint : Integrable f\nh : 0 < \u2191\u2191\u03bc {x | \u2191R < |(\u03bc[f|m]) x|}\n\u22a2 Measurable fun x => |(\u03bc[f|m]) x|\n\ncase pos.refine'_2\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nR : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhbdd : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |f x| \u2264 \u2191R\nhnm : m \u2264 m0\nhfint : Integrable f\nh : 0 < \u2191\u2191\u03bc {x | \u2191R < |(\u03bc[f|m]) x|}\n\u22a2 IntegrableOn (fun x => |(\u03bc[f|m]) x|) {x | \u2191R < |(\u03bc[f|m]) x|}\n\ncase pos.refine'_3\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nR : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhbdd : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |f x| \u2264 \u2191R\nhnm : m \u2264 m0\nhfint : Integrable f\nh : 0 < \u2191\u2191\u03bc {x | \u2191R < |(\u03bc[f|m]) x|}\n\u22a2 \u222b (x : \u03b1) in {x | \u2191R < |(\u03bc[f|m]) x|}, |(\u03bc[f|m]) x| \u2202\u03bc \u2264 ENNReal.toReal (\u2191\u2191\u03bc {x | \u2191R < |(\u03bc[f|m]) x|}) * \u2191R"}, {"tactic": "refine' (set_integral_abs_condexp_le _ _).trans _", "annotated_tactic": ["refine' (<a>set_integral_abs_condexp_le</a> _ _).<a>trans</a> _", [{"full_name": "MeasureTheory.set_integral_abs_condexp_le", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Real.lean", "def_pos": [117, 9], "def_end_pos": [117, 36]}, {"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [120, 7], "def_end_pos": [120, 18]}]], "state_before": "case pos.refine'_3\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nR : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhbdd : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |f x| \u2264 \u2191R\nhnm : m \u2264 m0\nhfint : Integrable f\nh : 0 < \u2191\u2191\u03bc {x | \u2191R < |(\u03bc[f|m]) x|}\n\u22a2 \u222b (x : \u03b1) in {x | \u2191R < |(\u03bc[f|m]) x|}, |(\u03bc[f|m]) x| \u2202\u03bc \u2264 ENNReal.toReal (\u2191\u2191\u03bc {x | \u2191R < |(\u03bc[f|m]) x|}) * \u2191R", "state_after": "case pos.refine'_3.refine'_1\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nR : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhbdd : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |f x| \u2264 \u2191R\nhnm : m \u2264 m0\nhfint : Integrable f\nh : 0 < \u2191\u2191\u03bc {x | \u2191R < |(\u03bc[f|m]) x|}\n\u22a2 MeasurableSet {x | \u2191R < |(\u03bc[f|m]) x|}\n\ncase pos.refine'_3.refine'_2\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nR : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhbdd : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |f x| \u2264 \u2191R\nhnm : m \u2264 m0\nhfint : Integrable f\nh : 0 < \u2191\u2191\u03bc {x | \u2191R < |(\u03bc[f|m]) x|}\n\u22a2 \u222b (x : \u03b1) in {x | \u2191R < |(\u03bc[f|m]) x|}, |f x| \u2202\u03bc \u2264 ENNReal.toReal (\u2191\u2191\u03bc {x | \u2191R < |(\u03bc[f|m]) x|}) * \u2191R"}, {"tactic": "simp only [\u2190 smul_eq_mul, \u2190 set_integral_const, NNReal.val_eq_coe, IsROrC.ofReal_real_eq_id,\n  id.def]", "annotated_tactic": ["simp only [\u2190 <a>smul_eq_mul</a>, \u2190 <a>set_integral_const</a>, <a>NNReal.val_eq_coe</a>, <a>IsROrC.ofReal_real_eq_id</a>,\n    <a>id.def</a>]", [{"full_name": "smul_eq_mul", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [93, 9], "def_end_pos": [93, 20]}, {"full_name": "MeasureTheory.set_integral_const", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [474, 9], "def_end_pos": [474, 27]}, {"full_name": "NNReal.val_eq_coe", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [87, 9], "def_end_pos": [87, 19]}, {"full_name": "IsROrC.ofReal_real_eq_id", "def_path": "Mathlib/Data/IsROrC/Basic.lean", "def_pos": [948, 9], "def_end_pos": [948, 26]}, {"full_name": "id.def", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [527, 9], "def_end_pos": [527, 15]}]], "state_before": "case pos.refine'_3.refine'_2\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nR : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhbdd : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |f x| \u2264 \u2191R\nhnm : m \u2264 m0\nhfint : Integrable f\nh : 0 < \u2191\u2191\u03bc {x | \u2191R < |(\u03bc[f|m]) x|}\n\u22a2 \u222b (x : \u03b1) in {x | \u2191R < |(\u03bc[f|m]) x|}, |f x| \u2202\u03bc \u2264 ENNReal.toReal (\u2191\u2191\u03bc {x | \u2191R < |(\u03bc[f|m]) x|}) * \u2191R", "state_after": "case pos.refine'_3.refine'_2\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nR : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhbdd : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |f x| \u2264 \u2191R\nhnm : m \u2264 m0\nhfint : Integrable f\nh : 0 < \u2191\u2191\u03bc {x | \u2191R < |(\u03bc[f|m]) x|}\n\u22a2 \u222b (x : \u03b1) in {x | \u2191R < |(\u03bc[f|m]) x|}, |f x| \u2202\u03bc \u2264 \u222b (x : \u03b1) in {x | \u2191R < |(\u03bc[f|m]) x|}, \u2191R \u2202\u03bc"}, {"tactic": "refine' set_integral_mono_ae hfint.abs.integrableOn _ _", "annotated_tactic": ["refine' <a>set_integral_mono_ae</a> hfint.abs.integrableOn _ _", [{"full_name": "MeasureTheory.set_integral_mono_ae", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [717, 9], "def_end_pos": [717, 29]}]], "state_before": "case pos.refine'_3.refine'_2\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nR : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhbdd : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |f x| \u2264 \u2191R\nhnm : m \u2264 m0\nhfint : Integrable f\nh : 0 < \u2191\u2191\u03bc {x | \u2191R < |(\u03bc[f|m]) x|}\n\u22a2 \u222b (x : \u03b1) in {x | \u2191R < |(\u03bc[f|m]) x|}, |f x| \u2202\u03bc \u2264 \u222b (x : \u03b1) in {x | \u2191R < |(\u03bc[f|m]) x|}, \u2191R \u2202\u03bc", "state_after": "case pos.refine'_3.refine'_2.refine'_1\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nR : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhbdd : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |f x| \u2264 \u2191R\nhnm : m \u2264 m0\nhfint : Integrable f\nh : 0 < \u2191\u2191\u03bc {x | \u2191R < |(\u03bc[f|m]) x|}\n\u22a2 IntegrableOn (fun x => \u2191R) {x | \u2191R < |(\u03bc[f|m]) x|}\n\ncase pos.refine'_3.refine'_2.refine'_2\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nR : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhbdd : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |f x| \u2264 \u2191R\nhnm : m \u2264 m0\nhfint : Integrable f\nh : 0 < \u2191\u2191\u03bc {x | \u2191R < |(\u03bc[f|m]) x|}\n\u22a2 (fun x => |f x|) \u2264\u1d50[\u03bc] fun x => \u2191R"}, {"tactic": "simp_rw [condexp_of_not_le hnm, Pi.zero_apply, abs_zero]", "annotated_tactic": ["simp_rw [<a>condexp_of_not_le</a> hnm, <a>Pi.zero_apply</a>, <a>abs_zero</a>]", [{"full_name": "MeasureTheory.condexp_of_not_le", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean", "def_pos": [106, 9], "def_end_pos": [106, 26]}, {"full_name": "Pi.zero_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [46, 3], "def_end_pos": [46, 14]}, {"full_name": "abs_zero", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [128, 9], "def_end_pos": [128, 17]}]], "state_before": "case neg\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nR : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhbdd : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |f x| \u2264 \u2191R\nhnm : \u00acm \u2264 m0\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |(\u03bc[f|m]) x| \u2264 \u2191R", "state_after": "case neg\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nR : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhbdd : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |f x| \u2264 \u2191R\nhnm : \u00acm \u2264 m0\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, 0 \u2264 \u2191R"}, {"tactic": "refine' eventually_of_forall fun _ => R.coe_nonneg", "annotated_tactic": ["refine' <a>eventually_of_forall</a> fun _ => R.coe_nonneg", [{"full_name": "Filter.eventually_of_forall", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1111, 9], "def_end_pos": [1111, 29]}]], "state_before": "case neg\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nR : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhbdd : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |f x| \u2264 \u2191R\nhnm : \u00acm \u2264 m0\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, 0 \u2264 \u2191R", "state_after": "no goals"}, {"tactic": "simp_rw [condexp_undef hfint]", "annotated_tactic": ["simp_rw [<a>condexp_undef</a> hfint]", [{"full_name": "MeasureTheory.condexp_undef", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean", "def_pos": [159, 9], "def_end_pos": [159, 22]}]], "state_before": "case neg\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nR : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhbdd : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |f x| \u2264 \u2191R\nhnm : m \u2264 m0\nhfint : \u00acIntegrable f\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |(\u03bc[f|m]) x| \u2264 \u2191R", "state_after": "case neg\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nR : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhbdd : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |f x| \u2264 \u2191R\nhnm : m \u2264 m0\nhfint : \u00acIntegrable f\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |OfNat.ofNat 0 x| \u2264 \u2191R"}, {"tactic": "filter_upwards [hbdd] with x hx", "annotated_tactic": ["filter_upwards [hbdd] with x hx", []], "state_before": "case neg\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nR : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhbdd : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |f x| \u2264 \u2191R\nhnm : m \u2264 m0\nhfint : \u00acIntegrable f\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |OfNat.ofNat 0 x| \u2264 \u2191R", "state_after": "case h\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nR : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhbdd : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |f x| \u2264 \u2191R\nhnm : m \u2264 m0\nhfint : \u00acIntegrable f\nx : \u03b1\nhx : |f x| \u2264 \u2191R\n\u22a2 |OfNat.ofNat 0 x| \u2264 \u2191R"}, {"tactic": "rw [Pi.zero_apply, abs_zero]", "annotated_tactic": ["rw [<a>Pi.zero_apply</a>, <a>abs_zero</a>]", [{"full_name": "Pi.zero_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [46, 3], "def_end_pos": [46, 14]}, {"full_name": "abs_zero", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [128, 9], "def_end_pos": [128, 17]}]], "state_before": "case h\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nR : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhbdd : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |f x| \u2264 \u2191R\nhnm : m \u2264 m0\nhfint : \u00acIntegrable f\nx : \u03b1\nhx : |f x| \u2264 \u2191R\n\u22a2 |OfNat.ofNat 0 x| \u2264 \u2191R", "state_after": "case h\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nR : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhbdd : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |f x| \u2264 \u2191R\nhnm : m \u2264 m0\nhfint : \u00acIntegrable f\nx : \u03b1\nhx : |f x| \u2264 \u2191R\n\u22a2 0 \u2264 \u2191R"}, {"tactic": "exact (abs_nonneg _).trans hx", "annotated_tactic": ["exact (<a>abs_nonneg</a> _).<a>trans</a> hx", [{"full_name": "abs_nonneg", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [169, 9], "def_end_pos": [169, 19]}, {"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [120, 7], "def_end_pos": [120, 18]}]], "state_before": "case h\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nR : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhbdd : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |f x| \u2264 \u2191R\nhnm : m \u2264 m0\nhfint : \u00acIntegrable f\nx : \u03b1\nhx : |f x| \u2264 \u2191R\n\u22a2 0 \u2264 \u2191R", "state_after": "no goals"}, {"tactic": "exact this.ne rfl", "annotated_tactic": ["exact this.ne <a>rfl</a>", [{"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nR : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhbdd : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |f x| \u2264 \u2191R\nhnm : m \u2264 m0\nhfint : Integrable f\nh : 0 < \u2191\u2191\u03bc {x | \u2191R < |(\u03bc[f|m]) x|}\nthis : ENNReal.toReal (\u2191\u2191\u03bc {x | \u2191R < |(\u03bc[f|m]) x|}) * \u2191R < ENNReal.toReal (\u2191\u2191\u03bc {x | \u2191R < |(\u03bc[f|m]) x|}) * \u2191R\n\u22a2 False", "state_after": "no goals"}, {"tactic": "simp_rw [\u2190 Real.norm_eq_abs]", "annotated_tactic": ["simp_rw [\u2190 <a>Real.norm_eq_abs</a>]", [{"full_name": "Real.norm_eq_abs", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [1761, 9], "def_end_pos": [1761, 20]}]], "state_before": "case pos.refine'_1\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nR : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhbdd : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |f x| \u2264 \u2191R\nhnm : m \u2264 m0\nhfint : Integrable f\nh : 0 < \u2191\u2191\u03bc {x | \u2191R < |(\u03bc[f|m]) x|}\n\u22a2 Measurable fun x => |(\u03bc[f|m]) x|", "state_after": "case pos.refine'_1\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nR : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhbdd : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |f x| \u2264 \u2191R\nhnm : m \u2264 m0\nhfint : Integrable f\nh : 0 < \u2191\u2191\u03bc {x | \u2191R < |(\u03bc[f|m]) x|}\n\u22a2 Measurable fun x => \u2016(\u03bc[f|m]) x\u2016"}, {"tactic": "exact (stronglyMeasurable_condexp.mono hnm).measurable.norm", "annotated_tactic": ["exact (stronglyMeasurable_condexp.mono hnm).measurable.norm", []], "state_before": "case pos.refine'_1\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nR : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhbdd : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |f x| \u2264 \u2191R\nhnm : m \u2264 m0\nhfint : Integrable f\nh : 0 < \u2191\u2191\u03bc {x | \u2191R < |(\u03bc[f|m]) x|}\n\u22a2 Measurable fun x => \u2016(\u03bc[f|m]) x\u2016", "state_after": "no goals"}, {"tactic": "exact integrable_condexp.abs.integrableOn", "annotated_tactic": ["exact integrable_condexp.abs.integrableOn", []], "state_before": "case pos.refine'_2\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nR : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhbdd : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |f x| \u2264 \u2191R\nhnm : m \u2264 m0\nhfint : Integrable f\nh : 0 < \u2191\u2191\u03bc {x | \u2191R < |(\u03bc[f|m]) x|}\n\u22a2 IntegrableOn (fun x => |(\u03bc[f|m]) x|) {x | \u2191R < |(\u03bc[f|m]) x|}", "state_after": "no goals"}, {"tactic": "simp_rw [\u2190 Real.norm_eq_abs]", "annotated_tactic": ["simp_rw [\u2190 <a>Real.norm_eq_abs</a>]", [{"full_name": "Real.norm_eq_abs", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [1761, 9], "def_end_pos": [1761, 20]}]], "state_before": "case pos.refine'_3.refine'_1\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nR : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhbdd : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |f x| \u2264 \u2191R\nhnm : m \u2264 m0\nhfint : Integrable f\nh : 0 < \u2191\u2191\u03bc {x | \u2191R < |(\u03bc[f|m]) x|}\n\u22a2 MeasurableSet {x | \u2191R < |(\u03bc[f|m]) x|}", "state_after": "case pos.refine'_3.refine'_1\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nR : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhbdd : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |f x| \u2264 \u2191R\nhnm : m \u2264 m0\nhfint : Integrable f\nh : 0 < \u2191\u2191\u03bc {x | \u2191R < |(\u03bc[f|m]) x|}\n\u22a2 MeasurableSet {x | \u2191R < \u2016(\u03bc[f|m]) x\u2016}"}, {"tactic": "exact @measurableSet_lt _ _ _ _ _ m _ _ _ _ _ measurable_const\n  stronglyMeasurable_condexp.norm.measurable", "annotated_tactic": ["exact @<a>measurableSet_lt</a> _ _ _ _ _ m _ _ _ _ _ <a>measurable_const</a>\n      stronglyMeasurable_condexp.norm.measurable", [{"full_name": "measurableSet_lt", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [616, 9], "def_end_pos": [616, 25]}, {"full_name": "measurable_const", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [570, 9], "def_end_pos": [570, 25]}]], "state_before": "case pos.refine'_3.refine'_1\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nR : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhbdd : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |f x| \u2264 \u2191R\nhnm : m \u2264 m0\nhfint : Integrable f\nh : 0 < \u2191\u2191\u03bc {x | \u2191R < |(\u03bc[f|m]) x|}\n\u22a2 MeasurableSet {x | \u2191R < \u2016(\u03bc[f|m]) x\u2016}", "state_after": "no goals"}, {"tactic": "refine' \u27e8aestronglyMeasurable_const, lt_of_le_of_lt _\n  (integrable_condexp.integrableOn : IntegrableOn (\u03bc[f|m]) {x | \u2191R < |(\u03bc[f|m]) x|} \u03bc).2\u27e9", "annotated_tactic": ["refine' \u27e8<a>aestronglyMeasurable_const</a>, <a>lt_of_le_of_lt</a> _\n      (integrable_condexp.integrableOn : <a>IntegrableOn</a> (\u03bc[f|m]) {x | \u2191R < |(\u03bc[f|m]) x|} \u03bc).2\u27e9", [{"full_name": "MeasureTheory.aestronglyMeasurable_const", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1155, 9], "def_end_pos": [1155, 35]}, {"full_name": "lt_of_le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [122, 9], "def_end_pos": [122, 23]}, {"full_name": "MeasureTheory.IntegrableOn", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [90, 5], "def_end_pos": [90, 17]}]], "state_before": "case pos.refine'_3.refine'_2.refine'_1\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nR : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhbdd : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |f x| \u2264 \u2191R\nhnm : m \u2264 m0\nhfint : Integrable f\nh : 0 < \u2191\u2191\u03bc {x | \u2191R < |(\u03bc[f|m]) x|}\n\u22a2 IntegrableOn (fun x => \u2191R) {x | \u2191R < |(\u03bc[f|m]) x|}", "state_after": "case pos.refine'_3.refine'_2.refine'_1\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nR : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhbdd : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |f x| \u2264 \u2191R\nhnm : m \u2264 m0\nhfint : Integrable f\nh : 0 < \u2191\u2191\u03bc {x | \u2191R < |(\u03bc[f|m]) x|}\n\u22a2 \u222b\u207b (a : \u03b1) in {x | \u2191R < |(\u03bc[f|m]) x|}, \u2191\u2016(fun x => \u2191R) a\u2016\u208a \u2202\u03bc \u2264\n    \u222b\u207b (a : \u03b1) in {x | \u2191R < |(\u03bc[f|m]) x|}, \u2191\u2016(\u03bc[f|m]) a\u2016\u208a \u2202\u03bc"}, {"tactic": "refine' set_lintegral_mono (Measurable.nnnorm _).coe_nnreal_ennreal\n  (stronglyMeasurable_condexp.mono hnm).measurable.nnnorm.coe_nnreal_ennreal fun x hx => _", "annotated_tactic": ["refine' <a>set_lintegral_mono</a> (<a>Measurable.nnnorm</a> _).<a>coe_nnreal_ennreal</a>\n      (stronglyMeasurable_condexp.mono hnm).measurable.nnnorm.coe_nnreal_ennreal fun x hx => _", [{"full_name": "MeasureTheory.set_lintegral_mono", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [287, 9], "def_end_pos": [287, 27]}, {"full_name": "Measurable.nnnorm", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [2261, 9], "def_end_pos": [2261, 26]}, {"full_name": "Measurable.coe_nnreal_ennreal", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [1993, 9], "def_end_pos": [1993, 38]}]], "state_before": "case pos.refine'_3.refine'_2.refine'_1\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nR : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhbdd : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |f x| \u2264 \u2191R\nhnm : m \u2264 m0\nhfint : Integrable f\nh : 0 < \u2191\u2191\u03bc {x | \u2191R < |(\u03bc[f|m]) x|}\n\u22a2 \u222b\u207b (a : \u03b1) in {x | \u2191R < |(\u03bc[f|m]) x|}, \u2191\u2016(fun x => \u2191R) a\u2016\u208a \u2202\u03bc \u2264\n    \u222b\u207b (a : \u03b1) in {x | \u2191R < |(\u03bc[f|m]) x|}, \u2191\u2016(\u03bc[f|m]) a\u2016\u208a \u2202\u03bc", "state_after": "case pos.refine'_3.refine'_2.refine'_1.refine'_1\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nR : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhbdd : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |f x| \u2264 \u2191R\nhnm : m \u2264 m0\nhfint : Integrable f\nh : 0 < \u2191\u2191\u03bc {x | \u2191R < |(\u03bc[f|m]) x|}\n\u22a2 Measurable fun a => (fun x => \u2191R) a\n\ncase pos.refine'_3.refine'_2.refine'_1.refine'_2\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nR : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhbdd : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |f x| \u2264 \u2191R\nhnm : m \u2264 m0\nhfint : Integrable f\nh : 0 < \u2191\u2191\u03bc {x | \u2191R < |(\u03bc[f|m]) x|}\nx : \u03b1\nhx : x \u2208 {x | \u2191R < |(\u03bc[f|m]) x|}\n\u22a2 \u2191\u2016(fun x => \u2191R) x\u2016\u208a \u2264 \u2191\u2016(\u03bc[f|m]) x\u2016\u208a"}, {"tactic": "exact measurable_const", "annotated_tactic": ["exact <a>measurable_const</a>", [{"full_name": "measurable_const", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [570, 9], "def_end_pos": [570, 25]}]], "state_before": "case pos.refine'_3.refine'_2.refine'_1.refine'_1\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nR : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhbdd : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |f x| \u2264 \u2191R\nhnm : m \u2264 m0\nhfint : Integrable f\nh : 0 < \u2191\u2191\u03bc {x | \u2191R < |(\u03bc[f|m]) x|}\n\u22a2 Measurable fun a => (fun x => \u2191R) a", "state_after": "no goals"}, {"tactic": "rw [ENNReal.coe_le_coe, Real.nnnorm_of_nonneg R.coe_nonneg]", "annotated_tactic": ["rw [<a>ENNReal.coe_le_coe</a>, <a>Real.nnnorm_of_nonneg</a> R.coe_nonneg]", [{"full_name": "ENNReal.coe_le_coe", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [349, 28], "def_end_pos": [349, 38]}, {"full_name": "Real.nnnorm_of_nonneg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [1800, 9], "def_end_pos": [1800, 25]}]], "state_before": "case pos.refine'_3.refine'_2.refine'_1.refine'_2\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nR : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhbdd : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |f x| \u2264 \u2191R\nhnm : m \u2264 m0\nhfint : Integrable f\nh : 0 < \u2191\u2191\u03bc {x | \u2191R < |(\u03bc[f|m]) x|}\nx : \u03b1\nhx : x \u2208 {x | \u2191R < |(\u03bc[f|m]) x|}\n\u22a2 \u2191\u2016(fun x => \u2191R) x\u2016\u208a \u2264 \u2191\u2016(\u03bc[f|m]) x\u2016\u208a", "state_after": "case pos.refine'_3.refine'_2.refine'_1.refine'_2\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nR : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhbdd : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |f x| \u2264 \u2191R\nhnm : m \u2264 m0\nhfint : Integrable f\nh : 0 < \u2191\u2191\u03bc {x | \u2191R < |(\u03bc[f|m]) x|}\nx : \u03b1\nhx : x \u2208 {x | \u2191R < |(\u03bc[f|m]) x|}\n\u22a2 { val := \u2191R, property := (_ : 0 \u2264 \u2191R) } \u2264 \u2016(\u03bc[f|m]) x\u2016\u208a"}, {"tactic": "exact Subtype.mk_le_mk.2 (le_of_lt hx)", "annotated_tactic": ["exact <a>Subtype.mk_le_mk</a>.2 (<a>le_of_lt</a> hx)", [{"full_name": "Subtype.mk_le_mk", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [1150, 9], "def_end_pos": [1150, 17]}, {"full_name": "le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [110, 9], "def_end_pos": [110, 17]}]], "state_before": "case pos.refine'_3.refine'_2.refine'_1.refine'_2\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nR : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhbdd : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |f x| \u2264 \u2191R\nhnm : m \u2264 m0\nhfint : Integrable f\nh : 0 < \u2191\u2191\u03bc {x | \u2191R < |(\u03bc[f|m]) x|}\nx : \u03b1\nhx : x \u2208 {x | \u2191R < |(\u03bc[f|m]) x|}\n\u22a2 { val := \u2191R, property := (_ : 0 \u2264 \u2191R) } \u2264 \u2016(\u03bc[f|m]) x\u2016\u208a", "state_after": "no goals"}, {"tactic": "exact hbdd", "annotated_tactic": ["exact hbdd", []], "state_before": "case pos.refine'_3.refine'_2.refine'_2\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nR : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhbdd : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |f x| \u2264 \u2191R\nhnm : m \u2264 m0\nhfint : Integrable f\nh : 0 < \u2191\u2191\u03bc {x | \u2191R < |(\u03bc[f|m]) x|}\n\u22a2 (fun x => |f x|) \u2264\u1d50[\u03bc] fun x => \u2191R", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/TypeVec.lean", "full_name": "TypeVec.splitFun_inj", "start": [229, 1], "end": [231, 66], "traced_tactics": [{"tactic": "rw [\u2190 dropFun_splitFun f g, H, \u2190 lastFun_splitFun f g, H]", "annotated_tactic": ["rw [\u2190 <a>dropFun_splitFun</a> f g, H, \u2190 <a>lastFun_splitFun</a> f g, H]", [{"full_name": "TypeVec.dropFun_splitFun", "def_path": "Mathlib/Data/TypeVec.lean", "def_pos": [181, 9], "def_end_pos": [181, 25]}, {"full_name": "TypeVec.lastFun_splitFun", "def_path": "Mathlib/Data/TypeVec.lean", "def_pos": [207, 9], "def_end_pos": [207, 25]}]], "state_before": "n : \u2115\n\u03b1 : TypeVec.{u_1} (n + 1)\n\u03b1' : TypeVec.{u_2} (n + 1)\nf f' : drop \u03b1 \u27f9 drop \u03b1'\ng g' : last \u03b1 \u2192 last \u03b1'\nH : splitFun f g = splitFun f' g'\n\u22a2 f = f' \u2227 g = g'", "state_after": "n : \u2115\n\u03b1 : TypeVec.{u_1} (n + 1)\n\u03b1' : TypeVec.{u_2} (n + 1)\nf f' : drop \u03b1 \u27f9 drop \u03b1'\ng g' : last \u03b1 \u2192 last \u03b1'\nH : splitFun f g = splitFun f' g'\n\u22a2 dropFun (splitFun f' g') = f' \u2227 lastFun (splitFun f' g') = g'"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "n : \u2115\n\u03b1 : TypeVec.{u_1} (n + 1)\n\u03b1' : TypeVec.{u_2} (n + 1)\nf f' : drop \u03b1 \u27f9 drop \u03b1'\ng g' : last \u03b1 \u2192 last \u03b1'\nH : splitFun f g = splitFun f' g'\n\u22a2 dropFun (splitFun f' g') = f' \u2227 lastFun (splitFun f' g') = g'", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Basic.lean", "full_name": "Set.ite_inter", "start": [2338, 1], "end": [2339, 33], "traced_tactics": [{"tactic": "rw [ite_inter_inter, ite_same]", "annotated_tactic": ["rw [<a>ite_inter_inter</a>, <a>ite_same</a>]", [{"full_name": "Set.ite_inter_inter", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [2331, 9], "def_end_pos": [2331, 24]}, {"full_name": "Set.ite_same", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [2290, 9], "def_end_pos": [2290, 17]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\na b : \u03b1\ns\u271d s\u2081\u271d s\u2082\u271d t\u271d t\u2081 t\u2082 u t s\u2081 s\u2082 s : Set \u03b1\n\u22a2 Set.ite t (s\u2081 \u2229 s) (s\u2082 \u2229 s) = Set.ite t s\u2081 s\u2082 \u2229 s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Num/Lemmas.lean", "full_name": "ZNum.of_nat_toZNumNeg", "start": [1548, 1], "end": [1548, 100], "traced_tactics": [{"tactic": "rw [\u2190 of_nat_toZNum, Num.zneg_toZNum]", "annotated_tactic": ["rw [\u2190 <a>of_nat_toZNum</a>, <a>Num.zneg_toZNum</a>]", [{"full_name": "ZNum.of_nat_toZNum", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [1534, 9], "def_end_pos": [1534, 22]}, {"full_name": "Num.zneg_toZNum", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [773, 9], "def_end_pos": [773, 20]}]], "state_before": "\u03b1 : Type u_1\nn : \u2115\n\u22a2 Num.toZNumNeg \u2191n = -\u2191n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Unique.lean", "full_name": "MeasureTheory.Lp.ae_eq_zero_of_forall_set_integral_eq_zero'", "start": [72, 1], "end": [89, 27], "traced_tactics": [{"tactic": "let f_meas : lpMeas E' \ud835\udd5c m p \u03bc := \u27e8f, hf_meas\u27e9", "annotated_tactic": ["let f_meas : <a>lpMeas</a> E' \ud835\udd5c m p \u03bc := \u27e8f, hf_meas\u27e9", [{"full_name": "MeasureTheory.lpMeas", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean", "def_pos": [222, 5], "def_end_pos": [222, 11]}]], "state_before": "\u03b1 : Type u_1\nE' : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2075 : CompleteSpace E'\ninst\u271d\u2074 : NormedSpace \u211d E'\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nhm : m \u2264 m0\nf : { x // x \u2208 Lp E' p }\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn (\u2191\u2191f) s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc = 0\nhf_meas : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\n\u22a2 \u2191\u2191f =\u1d50[\u03bc] 0", "state_after": "\u03b1 : Type u_1\nE' : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2075 : CompleteSpace E'\ninst\u271d\u2074 : NormedSpace \u211d E'\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nhm : m \u2264 m0\nf : { x // x \u2208 Lp E' p }\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn (\u2191\u2191f) s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc = 0\nhf_meas : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\nf_meas : { x // x \u2208 lpMeas E' \ud835\udd5c m p \u03bc } := { val := f, property := hf_meas }\n\u22a2 \u2191\u2191f =\u1d50[\u03bc] 0"}, {"tactic": "have hf_f_meas : f =\u1d50[\u03bc] f_meas := by simp only [Subtype.coe_mk]; rfl", "annotated_tactic": ["have hf_f_meas : f =\u1d50[\u03bc] f_meas := by simp only [<a>Subtype.coe_mk</a>]; rfl", [{"full_name": "Subtype.coe_mk", "def_path": "Mathlib/Data/Subtype.lean", "def_pos": [99, 9], "def_end_pos": [99, 15]}]], "state_before": "\u03b1 : Type u_1\nE' : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2075 : CompleteSpace E'\ninst\u271d\u2074 : NormedSpace \u211d E'\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nhm : m \u2264 m0\nf : { x // x \u2208 Lp E' p }\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn (\u2191\u2191f) s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc = 0\nhf_meas : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\nf_meas : { x // x \u2208 lpMeas E' \ud835\udd5c m p \u03bc } := { val := f, property := hf_meas }\n\u22a2 \u2191\u2191f =\u1d50[\u03bc] 0", "state_after": "\u03b1 : Type u_1\nE' : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2075 : CompleteSpace E'\ninst\u271d\u2074 : NormedSpace \u211d E'\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nhm : m \u2264 m0\nf : { x // x \u2208 Lp E' p }\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn (\u2191\u2191f) s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc = 0\nhf_meas : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\nf_meas : { x // x \u2208 lpMeas E' \ud835\udd5c m p \u03bc } := { val := f, property := hf_meas }\nhf_f_meas : \u2191\u2191f =\u1d50[\u03bc] \u2191\u2191\u2191f_meas\n\u22a2 \u2191\u2191f =\u1d50[\u03bc] 0"}, {"tactic": "refine' hf_f_meas.trans _", "annotated_tactic": ["refine' hf_f_meas.trans _", []], "state_before": "\u03b1 : Type u_1\nE' : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2075 : CompleteSpace E'\ninst\u271d\u2074 : NormedSpace \u211d E'\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nhm : m \u2264 m0\nf : { x // x \u2208 Lp E' p }\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn (\u2191\u2191f) s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc = 0\nhf_meas : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\nf_meas : { x // x \u2208 lpMeas E' \ud835\udd5c m p \u03bc } := { val := f, property := hf_meas }\nhf_f_meas : \u2191\u2191f =\u1d50[\u03bc] \u2191\u2191\u2191f_meas\n\u22a2 \u2191\u2191f =\u1d50[\u03bc] 0", "state_after": "\u03b1 : Type u_1\nE' : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2075 : CompleteSpace E'\ninst\u271d\u2074 : NormedSpace \u211d E'\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nhm : m \u2264 m0\nf : { x // x \u2208 Lp E' p }\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn (\u2191\u2191f) s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc = 0\nhf_meas : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\nf_meas : { x // x \u2208 lpMeas E' \ud835\udd5c m p \u03bc } := { val := f, property := hf_meas }\nhf_f_meas : \u2191\u2191f =\u1d50[\u03bc] \u2191\u2191\u2191f_meas\n\u22a2 \u2191\u2191\u2191f_meas =\u1d50[\u03bc] 0"}, {"tactic": "refine' lpMeas.ae_eq_zero_of_forall_set_integral_eq_zero hm f_meas hp_ne_zero hp_ne_top _ _", "annotated_tactic": ["refine' <a>lpMeas.ae_eq_zero_of_forall_set_integral_eq_zero</a> hm f_meas hp_ne_zero hp_ne_top _ _", [{"full_name": "MeasureTheory.lpMeas.ae_eq_zero_of_forall_set_integral_eq_zero", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Unique.lean", "def_pos": [49, 9], "def_end_pos": [49, 57]}]], "state_before": "\u03b1 : Type u_1\nE' : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2075 : CompleteSpace E'\ninst\u271d\u2074 : NormedSpace \u211d E'\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nhm : m \u2264 m0\nf : { x // x \u2208 Lp E' p }\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn (\u2191\u2191f) s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc = 0\nhf_meas : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\nf_meas : { x // x \u2208 lpMeas E' \ud835\udd5c m p \u03bc } := { val := f, property := hf_meas }\nhf_f_meas : \u2191\u2191f =\u1d50[\u03bc] \u2191\u2191\u2191f_meas\n\u22a2 \u2191\u2191\u2191f_meas =\u1d50[\u03bc] 0", "state_after": "case refine'_1\n\u03b1 : Type u_1\nE' : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2075 : CompleteSpace E'\ninst\u271d\u2074 : NormedSpace \u211d E'\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nhm : m \u2264 m0\nf : { x // x \u2208 Lp E' p }\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn (\u2191\u2191f) s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc = 0\nhf_meas : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\nf_meas : { x // x \u2208 lpMeas E' \ud835\udd5c m p \u03bc } := { val := f, property := hf_meas }\nhf_f_meas : \u2191\u2191f =\u1d50[\u03bc] \u2191\u2191\u2191f_meas\n\u22a2 \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn (\u2191\u2191\u2191f_meas) s\n\ncase refine'_2\n\u03b1 : Type u_1\nE' : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2075 : CompleteSpace E'\ninst\u271d\u2074 : NormedSpace \u211d E'\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nhm : m \u2264 m0\nf : { x // x \u2208 Lp E' p }\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn (\u2191\u2191f) s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc = 0\nhf_meas : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\nf_meas : { x // x \u2208 lpMeas E' \ud835\udd5c m p \u03bc } := { val := f, property := hf_meas }\nhf_f_meas : \u2191\u2191f =\u1d50[\u03bc] \u2191\u2191\u2191f_meas\n\u22a2 \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191\u2191f_meas x \u2202\u03bc = 0"}, {"tactic": "simp only [Subtype.coe_mk]", "annotated_tactic": ["simp only [<a>Subtype.coe_mk</a>]", [{"full_name": "Subtype.coe_mk", "def_path": "Mathlib/Data/Subtype.lean", "def_pos": [99, 9], "def_end_pos": [99, 15]}]], "state_before": "\u03b1 : Type u_1\nE' : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2075 : CompleteSpace E'\ninst\u271d\u2074 : NormedSpace \u211d E'\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nhm : m \u2264 m0\nf : { x // x \u2208 Lp E' p }\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn (\u2191\u2191f) s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc = 0\nhf_meas : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\nf_meas : { x // x \u2208 lpMeas E' \ud835\udd5c m p \u03bc } := { val := f, property := hf_meas }\n\u22a2 \u2191\u2191f =\u1d50[\u03bc] \u2191\u2191\u2191f_meas", "state_after": "\u03b1 : Type u_1\nE' : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2075 : CompleteSpace E'\ninst\u271d\u2074 : NormedSpace \u211d E'\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nhm : m \u2264 m0\nf : { x // x \u2208 Lp E' p }\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn (\u2191\u2191f) s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc = 0\nhf_meas : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\nf_meas : { x // x \u2208 lpMeas E' \ud835\udd5c m p \u03bc } := { val := f, property := hf_meas }\n\u22a2 \u2191\u2191f =\u1d50[\u03bc] \u2191\u2191f"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\u03b1 : Type u_1\nE' : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2075 : CompleteSpace E'\ninst\u271d\u2074 : NormedSpace \u211d E'\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nhm : m \u2264 m0\nf : { x // x \u2208 Lp E' p }\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn (\u2191\u2191f) s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc = 0\nhf_meas : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\nf_meas : { x // x \u2208 lpMeas E' \ud835\udd5c m p \u03bc } := { val := f, property := hf_meas }\n\u22a2 \u2191\u2191f =\u1d50[\u03bc] \u2191\u2191f", "state_after": "no goals"}, {"tactic": "intro s hs h\u03bcs", "annotated_tactic": ["intro s hs h\u03bcs", []], "state_before": "case refine'_1\n\u03b1 : Type u_1\nE' : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2075 : CompleteSpace E'\ninst\u271d\u2074 : NormedSpace \u211d E'\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nhm : m \u2264 m0\nf : { x // x \u2208 Lp E' p }\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn (\u2191\u2191f) s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc = 0\nhf_meas : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\nf_meas : { x // x \u2208 lpMeas E' \ud835\udd5c m p \u03bc } := { val := f, property := hf_meas }\nhf_f_meas : \u2191\u2191f =\u1d50[\u03bc] \u2191\u2191\u2191f_meas\n\u22a2 \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn (\u2191\u2191\u2191f_meas) s", "state_after": "case refine'_1\n\u03b1 : Type u_1\nE' : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2075 : CompleteSpace E'\ninst\u271d\u2074 : NormedSpace \u211d E'\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nhm : m \u2264 m0\nf : { x // x \u2208 Lp E' p }\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn (\u2191\u2191f) s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc = 0\nhf_meas : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\nf_meas : { x // x \u2208 lpMeas E' \ud835\udd5c m p \u03bc } := { val := f, property := hf_meas }\nhf_f_meas : \u2191\u2191f =\u1d50[\u03bc] \u2191\u2191\u2191f_meas\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 IntegrableOn (\u2191\u2191\u2191f_meas) s"}, {"tactic": "have hfg_restrict : f =\u1d50[\u03bc.restrict s] f_meas := ae_restrict_of_ae hf_f_meas", "annotated_tactic": ["have hfg_restrict : f =\u1d50[\u03bc.restrict s] f_meas := <a>ae_restrict_of_ae</a> hf_f_meas", [{"full_name": "MeasureTheory.ae_restrict_of_ae", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2596, 9], "def_end_pos": [2596, 26]}]], "state_before": "case refine'_1\n\u03b1 : Type u_1\nE' : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2075 : CompleteSpace E'\ninst\u271d\u2074 : NormedSpace \u211d E'\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nhm : m \u2264 m0\nf : { x // x \u2208 Lp E' p }\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn (\u2191\u2191f) s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc = 0\nhf_meas : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\nf_meas : { x // x \u2208 lpMeas E' \ud835\udd5c m p \u03bc } := { val := f, property := hf_meas }\nhf_f_meas : \u2191\u2191f =\u1d50[\u03bc] \u2191\u2191\u2191f_meas\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 IntegrableOn (\u2191\u2191\u2191f_meas) s", "state_after": "case refine'_1\n\u03b1 : Type u_1\nE' : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2075 : CompleteSpace E'\ninst\u271d\u2074 : NormedSpace \u211d E'\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nhm : m \u2264 m0\nf : { x // x \u2208 Lp E' p }\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn (\u2191\u2191f) s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc = 0\nhf_meas : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\nf_meas : { x // x \u2208 lpMeas E' \ud835\udd5c m p \u03bc } := { val := f, property := hf_meas }\nhf_f_meas : \u2191\u2191f =\u1d50[\u03bc] \u2191\u2191\u2191f_meas\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\nhfg_restrict : \u2191\u2191f =\u1d50[Measure.restrict \u03bc s] \u2191\u2191\u2191f_meas\n\u22a2 IntegrableOn (\u2191\u2191\u2191f_meas) s"}, {"tactic": "rw [IntegrableOn, integrable_congr hfg_restrict.symm]", "annotated_tactic": ["rw [<a>IntegrableOn</a>, <a>integrable_congr</a> hfg_restrict.symm]", [{"full_name": "MeasureTheory.IntegrableOn", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [90, 5], "def_end_pos": [90, 17]}, {"full_name": "MeasureTheory.integrable_congr", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [496, 9], "def_end_pos": [496, 25]}]], "state_before": "case refine'_1\n\u03b1 : Type u_1\nE' : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2075 : CompleteSpace E'\ninst\u271d\u2074 : NormedSpace \u211d E'\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nhm : m \u2264 m0\nf : { x // x \u2208 Lp E' p }\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn (\u2191\u2191f) s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc = 0\nhf_meas : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\nf_meas : { x // x \u2208 lpMeas E' \ud835\udd5c m p \u03bc } := { val := f, property := hf_meas }\nhf_f_meas : \u2191\u2191f =\u1d50[\u03bc] \u2191\u2191\u2191f_meas\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\nhfg_restrict : \u2191\u2191f =\u1d50[Measure.restrict \u03bc s] \u2191\u2191\u2191f_meas\n\u22a2 IntegrableOn (\u2191\u2191\u2191f_meas) s", "state_after": "case refine'_1\n\u03b1 : Type u_1\nE' : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2075 : CompleteSpace E'\ninst\u271d\u2074 : NormedSpace \u211d E'\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nhm : m \u2264 m0\nf : { x // x \u2208 Lp E' p }\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn (\u2191\u2191f) s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc = 0\nhf_meas : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\nf_meas : { x // x \u2208 lpMeas E' \ud835\udd5c m p \u03bc } := { val := f, property := hf_meas }\nhf_f_meas : \u2191\u2191f =\u1d50[\u03bc] \u2191\u2191\u2191f_meas\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\nhfg_restrict : \u2191\u2191f =\u1d50[Measure.restrict \u03bc s] \u2191\u2191\u2191f_meas\n\u22a2 Integrable \u2191\u2191f"}, {"tactic": "exact hf_int_finite s hs h\u03bcs", "annotated_tactic": ["exact hf_int_finite s hs h\u03bcs", []], "state_before": "case refine'_1\n\u03b1 : Type u_1\nE' : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2075 : CompleteSpace E'\ninst\u271d\u2074 : NormedSpace \u211d E'\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nhm : m \u2264 m0\nf : { x // x \u2208 Lp E' p }\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn (\u2191\u2191f) s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc = 0\nhf_meas : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\nf_meas : { x // x \u2208 lpMeas E' \ud835\udd5c m p \u03bc } := { val := f, property := hf_meas }\nhf_f_meas : \u2191\u2191f =\u1d50[\u03bc] \u2191\u2191\u2191f_meas\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\nhfg_restrict : \u2191\u2191f =\u1d50[Measure.restrict \u03bc s] \u2191\u2191\u2191f_meas\n\u22a2 Integrable \u2191\u2191f", "state_after": "no goals"}, {"tactic": "intro s hs h\u03bcs", "annotated_tactic": ["intro s hs h\u03bcs", []], "state_before": "case refine'_2\n\u03b1 : Type u_1\nE' : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2075 : CompleteSpace E'\ninst\u271d\u2074 : NormedSpace \u211d E'\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nhm : m \u2264 m0\nf : { x // x \u2208 Lp E' p }\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn (\u2191\u2191f) s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc = 0\nhf_meas : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\nf_meas : { x // x \u2208 lpMeas E' \ud835\udd5c m p \u03bc } := { val := f, property := hf_meas }\nhf_f_meas : \u2191\u2191f =\u1d50[\u03bc] \u2191\u2191\u2191f_meas\n\u22a2 \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191\u2191f_meas x \u2202\u03bc = 0", "state_after": "case refine'_2\n\u03b1 : Type u_1\nE' : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2075 : CompleteSpace E'\ninst\u271d\u2074 : NormedSpace \u211d E'\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nhm : m \u2264 m0\nf : { x // x \u2208 Lp E' p }\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn (\u2191\u2191f) s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc = 0\nhf_meas : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\nf_meas : { x // x \u2208 lpMeas E' \ud835\udd5c m p \u03bc } := { val := f, property := hf_meas }\nhf_f_meas : \u2191\u2191f =\u1d50[\u03bc] \u2191\u2191\u2191f_meas\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u222b (x : \u03b1) in s, \u2191\u2191\u2191f_meas x \u2202\u03bc = 0"}, {"tactic": "have hfg_restrict : f =\u1d50[\u03bc.restrict s] f_meas := ae_restrict_of_ae hf_f_meas", "annotated_tactic": ["have hfg_restrict : f =\u1d50[\u03bc.restrict s] f_meas := <a>ae_restrict_of_ae</a> hf_f_meas", [{"full_name": "MeasureTheory.ae_restrict_of_ae", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2596, 9], "def_end_pos": [2596, 26]}]], "state_before": "case refine'_2\n\u03b1 : Type u_1\nE' : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2075 : CompleteSpace E'\ninst\u271d\u2074 : NormedSpace \u211d E'\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nhm : m \u2264 m0\nf : { x // x \u2208 Lp E' p }\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn (\u2191\u2191f) s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc = 0\nhf_meas : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\nf_meas : { x // x \u2208 lpMeas E' \ud835\udd5c m p \u03bc } := { val := f, property := hf_meas }\nhf_f_meas : \u2191\u2191f =\u1d50[\u03bc] \u2191\u2191\u2191f_meas\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u222b (x : \u03b1) in s, \u2191\u2191\u2191f_meas x \u2202\u03bc = 0", "state_after": "case refine'_2\n\u03b1 : Type u_1\nE' : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2075 : CompleteSpace E'\ninst\u271d\u2074 : NormedSpace \u211d E'\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nhm : m \u2264 m0\nf : { x // x \u2208 Lp E' p }\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn (\u2191\u2191f) s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc = 0\nhf_meas : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\nf_meas : { x // x \u2208 lpMeas E' \ud835\udd5c m p \u03bc } := { val := f, property := hf_meas }\nhf_f_meas : \u2191\u2191f =\u1d50[\u03bc] \u2191\u2191\u2191f_meas\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\nhfg_restrict : \u2191\u2191f =\u1d50[Measure.restrict \u03bc s] \u2191\u2191\u2191f_meas\n\u22a2 \u222b (x : \u03b1) in s, \u2191\u2191\u2191f_meas x \u2202\u03bc = 0"}, {"tactic": "rw [integral_congr_ae hfg_restrict.symm]", "annotated_tactic": ["rw [<a>integral_congr_ae</a> hfg_restrict.symm]", [{"full_name": "MeasureTheory.integral_congr_ae", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [938, 9], "def_end_pos": [938, 26]}]], "state_before": "case refine'_2\n\u03b1 : Type u_1\nE' : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2075 : CompleteSpace E'\ninst\u271d\u2074 : NormedSpace \u211d E'\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nhm : m \u2264 m0\nf : { x // x \u2208 Lp E' p }\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn (\u2191\u2191f) s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc = 0\nhf_meas : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\nf_meas : { x // x \u2208 lpMeas E' \ud835\udd5c m p \u03bc } := { val := f, property := hf_meas }\nhf_f_meas : \u2191\u2191f =\u1d50[\u03bc] \u2191\u2191\u2191f_meas\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\nhfg_restrict : \u2191\u2191f =\u1d50[Measure.restrict \u03bc s] \u2191\u2191\u2191f_meas\n\u22a2 \u222b (x : \u03b1) in s, \u2191\u2191\u2191f_meas x \u2202\u03bc = 0", "state_after": "case refine'_2\n\u03b1 : Type u_1\nE' : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2075 : CompleteSpace E'\ninst\u271d\u2074 : NormedSpace \u211d E'\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nhm : m \u2264 m0\nf : { x // x \u2208 Lp E' p }\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn (\u2191\u2191f) s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc = 0\nhf_meas : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\nf_meas : { x // x \u2208 lpMeas E' \ud835\udd5c m p \u03bc } := { val := f, property := hf_meas }\nhf_f_meas : \u2191\u2191f =\u1d50[\u03bc] \u2191\u2191\u2191f_meas\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\nhfg_restrict : \u2191\u2191f =\u1d50[Measure.restrict \u03bc s] \u2191\u2191\u2191f_meas\n\u22a2 \u222b (a : \u03b1) in s, \u2191\u2191f a \u2202\u03bc = 0"}, {"tactic": "exact hf_zero s hs h\u03bcs", "annotated_tactic": ["exact hf_zero s hs h\u03bcs", []], "state_before": "case refine'_2\n\u03b1 : Type u_1\nE' : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2075 : CompleteSpace E'\ninst\u271d\u2074 : NormedSpace \u211d E'\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nhm : m \u2264 m0\nf : { x // x \u2208 Lp E' p }\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn (\u2191\u2191f) s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc = 0\nhf_meas : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\nf_meas : { x // x \u2208 lpMeas E' \ud835\udd5c m p \u03bc } := { val := f, property := hf_meas }\nhf_f_meas : \u2191\u2191f =\u1d50[\u03bc] \u2191\u2191\u2191f_meas\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\nhfg_restrict : \u2191\u2191f =\u1d50[Measure.restrict \u03bc s] \u2191\u2191\u2191f_meas\n\u22a2 \u222b (a : \u03b1) in s, \u2191\u2191f a \u2202\u03bc = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/MeanInequalities.lean", "full_name": "ENNReal.lintegral_Lp_add_le", "start": [346, 1], "end": [371, 64], "traced_tactics": [{"tactic": "have hp_pos : 0 < p := lt_of_lt_of_le zero_lt_one hp1", "annotated_tactic": ["have hp_pos : 0 < p := <a>lt_of_lt_of_le</a> <a>zero_lt_one</a> hp1", [{"full_name": "lt_of_lt_of_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [115, 9], "def_end_pos": [115, 23]}, {"full_name": "zero_lt_one", "def_path": "Mathlib/Algebra/Order/ZeroLEOne.lean", "def_pos": [39, 15], "def_end_pos": [39, 26]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp1 : 1 \u2264 p\n\u22a2 (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)", "state_after": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp1 : 1 \u2264 p\nhp_pos : 0 < p\n\u22a2 (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)"}, {"tactic": "by_cases hf_top : \u222b\u207b a, f a ^ p \u2202\u03bc = \u22a4", "annotated_tactic": ["by_cases hf_top : \u222b\u207b a, f a ^ p \u2202\u03bc = \u22a4", []], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp1 : 1 \u2264 p\nhp_pos : 0 < p\n\u22a2 (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)", "state_after": "case pos\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp1 : 1 \u2264 p\nhp_pos : 0 < p\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4\n\u22a2 (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)\n\ncase neg\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp1 : 1 \u2264 p\nhp_pos : 0 < p\nhf_top : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4\n\u22a2 (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)"}, {"tactic": "by_cases hg_top : \u222b\u207b a, g a ^ p \u2202\u03bc = \u22a4", "annotated_tactic": ["by_cases hg_top : \u222b\u207b a, g a ^ p \u2202\u03bc = \u22a4", []], "state_before": "case neg\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp1 : 1 \u2264 p\nhp_pos : 0 < p\nhf_top : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4\n\u22a2 (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)", "state_after": "case pos\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp1 : 1 \u2264 p\nhp_pos : 0 < p\nhf_top : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc = \u22a4\n\u22a2 (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)\n\ncase neg\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp1 : 1 \u2264 p\nhp_pos : 0 < p\nhf_top : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4\nhg_top : \u00ac\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc = \u22a4\n\u22a2 (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)"}, {"tactic": "by_cases h1 : p = 1", "annotated_tactic": ["by_cases h1 : p = 1", []], "state_before": "case neg\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp1 : 1 \u2264 p\nhp_pos : 0 < p\nhf_top : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4\nhg_top : \u00ac\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc = \u22a4\n\u22a2 (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)", "state_after": "case pos\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp1 : 1 \u2264 p\nhp_pos : 0 < p\nhf_top : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4\nhg_top : \u00ac\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc = \u22a4\nh1 : p = 1\n\u22a2 (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)\n\ncase neg\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp1 : 1 \u2264 p\nhp_pos : 0 < p\nhf_top : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4\nhg_top : \u00ac\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc = \u22a4\nh1 : \u00acp = 1\n\u22a2 (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)"}, {"tactic": "have hp1_lt : 1 < p := by\n  refine' lt_of_le_of_ne hp1 _\n  symm\n  exact h1", "annotated_tactic": ["have hp1_lt : 1 < p := by\n    refine' <a>lt_of_le_of_ne</a> hp1 _\n    symm\n    exact h1", [{"full_name": "lt_of_le_of_ne", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [196, 9], "def_end_pos": [196, 23]}]], "state_before": "case neg\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp1 : 1 \u2264 p\nhp_pos : 0 < p\nhf_top : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4\nhg_top : \u00ac\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc = \u22a4\nh1 : \u00acp = 1\n\u22a2 (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)", "state_after": "case neg\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp1 : 1 \u2264 p\nhp_pos : 0 < p\nhf_top : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4\nhg_top : \u00ac\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc = \u22a4\nh1 : \u00acp = 1\nhp1_lt : 1 < p\n\u22a2 (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)"}, {"tactic": "have hpq := Real.isConjugateExponent_conjugateExponent hp1_lt", "annotated_tactic": ["have hpq := <a>Real.isConjugateExponent_conjugateExponent</a> hp1_lt", [{"full_name": "Real.isConjugateExponent_conjugateExponent", "def_path": "Mathlib/Data/Real/ConjugateExponents.lean", "def_pos": [123, 9], "def_end_pos": [123, 46]}]], "state_before": "case neg\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp1 : 1 \u2264 p\nhp_pos : 0 < p\nhf_top : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4\nhg_top : \u00ac\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc = \u22a4\nh1 : \u00acp = 1\nhp1_lt : 1 < p\n\u22a2 (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)", "state_after": "case neg\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp1 : 1 \u2264 p\nhp_pos : 0 < p\nhf_top : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4\nhg_top : \u00ac\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc = \u22a4\nh1 : \u00acp = 1\nhp1_lt : 1 < p\nhpq : Real.IsConjugateExponent p (Real.conjugateExponent p)\n\u22a2 (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)"}, {"tactic": "by_cases h0 : (\u222b\u207b a, (f + g) a ^ p \u2202\u03bc) = 0", "annotated_tactic": ["by_cases h0 : (\u222b\u207b a, (f + g) a ^ p \u2202\u03bc) = 0", []], "state_before": "case neg\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp1 : 1 \u2264 p\nhp_pos : 0 < p\nhf_top : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4\nhg_top : \u00ac\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc = \u22a4\nh1 : \u00acp = 1\nhp1_lt : 1 < p\nhpq : Real.IsConjugateExponent p (Real.conjugateExponent p)\n\u22a2 (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)", "state_after": "case pos\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp1 : 1 \u2264 p\nhp_pos : 0 < p\nhf_top : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4\nhg_top : \u00ac\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc = \u22a4\nh1 : \u00acp = 1\nhp1_lt : 1 < p\nhpq : Real.IsConjugateExponent p (Real.conjugateExponent p)\nh0 : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc = 0\n\u22a2 (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)\n\ncase neg\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp1 : 1 \u2264 p\nhp_pos : 0 < p\nhf_top : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4\nhg_top : \u00ac\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc = \u22a4\nh1 : \u00acp = 1\nhp1_lt : 1 < p\nhpq : Real.IsConjugateExponent p (Real.conjugateExponent p)\nh0 : \u00ac\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc = 0\n\u22a2 (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)"}, {"tactic": "have htop : (\u222b\u207b a, (f + g) a ^ p \u2202\u03bc) \u2260 \u22a4 := by\n  rw [\u2190 Ne.def] at hf_top hg_top\n  rw [\u2190 lt_top_iff_ne_top] at hf_top hg_top \u22a2\n  exact lintegral_rpow_add_lt_top_of_lintegral_rpow_lt_top hf hf_top hg_top hp1", "annotated_tactic": ["have htop : (\u222b\u207b a, (f + g) a ^ p \u2202\u03bc) \u2260 \u22a4 := by\n    rw [\u2190 <a>Ne.def</a>] at hf_top hg_top\n    rw [\u2190 <a>lt_top_iff_ne_top</a>] at hf_top hg_top \u22a2\n    exact <a>lintegral_rpow_add_lt_top_of_lintegral_rpow_lt_top</a> hf hf_top hg_top hp1", [{"full_name": "Ne.def", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [59, 9], "def_end_pos": [59, 15]}, {"full_name": "lt_top_iff_ne_top", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [173, 9], "def_end_pos": [173, 26]}, {"full_name": "ENNReal.lintegral_rpow_add_lt_top_of_lintegral_rpow_lt_top", "def_path": "Mathlib/MeasureTheory/Integral/MeanInequalities.lean", "def_pos": [178, 9], "def_end_pos": [178, 59]}]], "state_before": "case neg\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp1 : 1 \u2264 p\nhp_pos : 0 < p\nhf_top : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4\nhg_top : \u00ac\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc = \u22a4\nh1 : \u00acp = 1\nhp1_lt : 1 < p\nhpq : Real.IsConjugateExponent p (Real.conjugateExponent p)\nh0 : \u00ac\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc = 0\n\u22a2 (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)", "state_after": "case neg\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp1 : 1 \u2264 p\nhp_pos : 0 < p\nhf_top : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4\nhg_top : \u00ac\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc = \u22a4\nh1 : \u00acp = 1\nhp1_lt : 1 < p\nhpq : Real.IsConjugateExponent p (Real.conjugateExponent p)\nh0 : \u00ac\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc = 0\nhtop : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 \u22a4\n\u22a2 (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)"}, {"tactic": "exact lintegral_Lp_add_le_aux hpq hf hf_top hg hg_top h0 htop", "annotated_tactic": ["exact <a>lintegral_Lp_add_le_aux</a> hpq hf hf_top hg hg_top h0 htop", [{"full_name": "_private.Mathlib.MeasureTheory.Integral.MeanInequalities.0.ENNReal.lintegral_Lp_add_le_aux", "def_path": "Mathlib/MeasureTheory/Integral/MeanInequalities.lean", "def_pos": [312, 17], "def_end_pos": [312, 40]}]], "state_before": "case neg\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp1 : 1 \u2264 p\nhp_pos : 0 < p\nhf_top : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4\nhg_top : \u00ac\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc = \u22a4\nh1 : \u00acp = 1\nhp1_lt : 1 < p\nhpq : Real.IsConjugateExponent p (Real.conjugateExponent p)\nh0 : \u00ac\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc = 0\nhtop : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 \u22a4\n\u22a2 (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)", "state_after": "no goals"}, {"tactic": "simp [hf_top, hp_pos]", "annotated_tactic": ["simp [hf_top, hp_pos]", []], "state_before": "case pos\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp1 : 1 \u2264 p\nhp_pos : 0 < p\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4\n\u22a2 (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)", "state_after": "no goals"}, {"tactic": "simp [hg_top, hp_pos]", "annotated_tactic": ["simp [hg_top, hp_pos]", []], "state_before": "case pos\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp1 : 1 \u2264 p\nhp_pos : 0 < p\nhf_top : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc = \u22a4\n\u22a2 (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)", "state_after": "no goals"}, {"tactic": "refine' le_of_eq _", "annotated_tactic": ["refine' <a>le_of_eq</a> _", [{"full_name": "le_of_eq", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [72, 9], "def_end_pos": [72, 17]}]], "state_before": "case pos\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp1 : 1 \u2264 p\nhp_pos : 0 < p\nhf_top : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4\nhg_top : \u00ac\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc = \u22a4\nh1 : p = 1\n\u22a2 (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)", "state_after": "case pos\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp1 : 1 \u2264 p\nhp_pos : 0 < p\nhf_top : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4\nhg_top : \u00ac\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc = \u22a4\nh1 : p = 1\n\u22a2 (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) = (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)"}, {"tactic": "simp_rw [h1, one_div_one, ENNReal.rpow_one]", "annotated_tactic": ["simp_rw [h1, <a>one_div_one</a>, <a>ENNReal.rpow_one</a>]", [{"full_name": "one_div_one", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [345, 9], "def_end_pos": [345, 20]}, {"full_name": "ENNReal.rpow_one", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [450, 9], "def_end_pos": [450, 17]}]], "state_before": "case pos\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp1 : 1 \u2264 p\nhp_pos : 0 < p\nhf_top : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4\nhg_top : \u00ac\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc = \u22a4\nh1 : p = 1\n\u22a2 (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) = (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)", "state_after": "case pos\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp1 : 1 \u2264 p\nhp_pos : 0 < p\nhf_top : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4\nhg_top : \u00ac\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc = \u22a4\nh1 : p = 1\n\u22a2 \u222b\u207b (a : \u03b1), (f + g) a \u2202\u03bc = \u222b\u207b (a : \u03b1), f a \u2202\u03bc + \u222b\u207b (a : \u03b1), g a \u2202\u03bc"}, {"tactic": "exact lintegral_add_left' hf _", "annotated_tactic": ["exact <a>lintegral_add_left'</a> hf _", [{"full_name": "MeasureTheory.lintegral_add_left'", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [566, 9], "def_end_pos": [566, 28]}]], "state_before": "case pos\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp1 : 1 \u2264 p\nhp_pos : 0 < p\nhf_top : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4\nhg_top : \u00ac\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc = \u22a4\nh1 : p = 1\n\u22a2 \u222b\u207b (a : \u03b1), (f + g) a \u2202\u03bc = \u222b\u207b (a : \u03b1), f a \u2202\u03bc + \u222b\u207b (a : \u03b1), g a \u2202\u03bc", "state_after": "no goals"}, {"tactic": "refine' lt_of_le_of_ne hp1 _", "annotated_tactic": ["refine' <a>lt_of_le_of_ne</a> hp1 _", [{"full_name": "lt_of_le_of_ne", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [196, 9], "def_end_pos": [196, 23]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp1 : 1 \u2264 p\nhp_pos : 0 < p\nhf_top : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4\nhg_top : \u00ac\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc = \u22a4\nh1 : \u00acp = 1\n\u22a2 1 < p", "state_after": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp1 : 1 \u2264 p\nhp_pos : 0 < p\nhf_top : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4\nhg_top : \u00ac\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc = \u22a4\nh1 : \u00acp = 1\n\u22a2 1 \u2260 p"}, {"tactic": "symm", "annotated_tactic": ["symm", []], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp1 : 1 \u2264 p\nhp_pos : 0 < p\nhf_top : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4\nhg_top : \u00ac\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc = \u22a4\nh1 : \u00acp = 1\n\u22a2 1 \u2260 p", "state_after": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp1 : 1 \u2264 p\nhp_pos : 0 < p\nhf_top : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4\nhg_top : \u00ac\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc = \u22a4\nh1 : \u00acp = 1\n\u22a2 p \u2260 1"}, {"tactic": "exact h1", "annotated_tactic": ["exact h1", []], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp1 : 1 \u2264 p\nhp_pos : 0 < p\nhf_top : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4\nhg_top : \u00ac\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc = \u22a4\nh1 : \u00acp = 1\n\u22a2 p \u2260 1", "state_after": "no goals"}, {"tactic": "rw [h0, @ENNReal.zero_rpow_of_pos (1 / p) (by simp [lt_of_lt_of_le zero_lt_one hp1])]", "annotated_tactic": ["rw [h0, @<a>ENNReal.zero_rpow_of_pos</a> (1 / p) (by simp [<a>lt_of_lt_of_le</a> <a>zero_lt_one</a> hp1])]", [{"full_name": "ENNReal.zero_rpow_of_pos", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [401, 9], "def_end_pos": [401, 25]}, {"full_name": "lt_of_lt_of_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [115, 9], "def_end_pos": [115, 23]}, {"full_name": "zero_lt_one", "def_path": "Mathlib/Algebra/Order/ZeroLEOne.lean", "def_pos": [39, 15], "def_end_pos": [39, 26]}]], "state_before": "case pos\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp1 : 1 \u2264 p\nhp_pos : 0 < p\nhf_top : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4\nhg_top : \u00ac\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc = \u22a4\nh1 : \u00acp = 1\nhp1_lt : 1 < p\nhpq : Real.IsConjugateExponent p (Real.conjugateExponent p)\nh0 : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc = 0\n\u22a2 (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)", "state_after": "case pos\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp1 : 1 \u2264 p\nhp_pos : 0 < p\nhf_top : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4\nhg_top : \u00ac\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc = \u22a4\nh1 : \u00acp = 1\nhp1_lt : 1 < p\nhpq : Real.IsConjugateExponent p (Real.conjugateExponent p)\nh0 : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc = 0\n\u22a2 0 \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)"}, {"tactic": "exact zero_le _", "annotated_tactic": ["exact <a>zero_le</a> _", [{"full_name": "zero_le", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [217, 30], "def_end_pos": [217, 37]}]], "state_before": "case pos\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp1 : 1 \u2264 p\nhp_pos : 0 < p\nhf_top : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4\nhg_top : \u00ac\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc = \u22a4\nh1 : \u00acp = 1\nhp1_lt : 1 < p\nhpq : Real.IsConjugateExponent p (Real.conjugateExponent p)\nh0 : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc = 0\n\u22a2 0 \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)", "state_after": "no goals"}, {"tactic": "simp [lt_of_lt_of_le zero_lt_one hp1]", "annotated_tactic": ["simp [<a>lt_of_lt_of_le</a> <a>zero_lt_one</a> hp1]", [{"full_name": "lt_of_lt_of_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [115, 9], "def_end_pos": [115, 23]}, {"full_name": "zero_lt_one", "def_path": "Mathlib/Algebra/Order/ZeroLEOne.lean", "def_pos": [39, 15], "def_end_pos": [39, 26]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp1 : 1 \u2264 p\nhp_pos : 0 < p\nhf_top : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4\nhg_top : \u00ac\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc = \u22a4\nh1 : \u00acp = 1\nhp1_lt : 1 < p\nhpq : Real.IsConjugateExponent p (Real.conjugateExponent p)\nh0 : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc = 0\n\u22a2 0 < 1 / p", "state_after": "no goals"}, {"tactic": "rw [\u2190 Ne.def] at hf_top hg_top", "annotated_tactic": ["rw [\u2190 <a>Ne.def</a>] at hf_top hg_top", [{"full_name": "Ne.def", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [59, 9], "def_end_pos": [59, 15]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp1 : 1 \u2264 p\nhp_pos : 0 < p\nhf_top : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4\nhg_top : \u00ac\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc = \u22a4\nh1 : \u00acp = 1\nhp1_lt : 1 < p\nhpq : Real.IsConjugateExponent p (Real.conjugateExponent p)\nh0 : \u00ac\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc = 0\n\u22a2 \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 \u22a4", "state_after": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp1 : 1 \u2264 p\nhp_pos : 0 < p\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc \u2260 \u22a4\nh1 : \u00acp = 1\nhp1_lt : 1 < p\nhpq : Real.IsConjugateExponent p (Real.conjugateExponent p)\nh0 : \u00ac\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc = 0\n\u22a2 \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 \u22a4"}, {"tactic": "rw [\u2190 lt_top_iff_ne_top] at hf_top hg_top \u22a2", "annotated_tactic": ["rw [\u2190 <a>lt_top_iff_ne_top</a>] at hf_top hg_top \u22a2", [{"full_name": "lt_top_iff_ne_top", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [173, 9], "def_end_pos": [173, 26]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp1 : 1 \u2264 p\nhp_pos : 0 < p\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc \u2260 \u22a4\nh1 : \u00acp = 1\nhp1_lt : 1 < p\nhpq : Real.IsConjugateExponent p (Real.conjugateExponent p)\nh0 : \u00ac\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc = 0\n\u22a2 \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 \u22a4", "state_after": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp1 : 1 \u2264 p\nhp_pos : 0 < p\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc < \u22a4\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc < \u22a4\nh1 : \u00acp = 1\nhp1_lt : 1 < p\nhpq : Real.IsConjugateExponent p (Real.conjugateExponent p)\nh0 : \u00ac\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc = 0\n\u22a2 \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc < \u22a4"}, {"tactic": "exact lintegral_rpow_add_lt_top_of_lintegral_rpow_lt_top hf hf_top hg_top hp1", "annotated_tactic": ["exact <a>lintegral_rpow_add_lt_top_of_lintegral_rpow_lt_top</a> hf hf_top hg_top hp1", [{"full_name": "ENNReal.lintegral_rpow_add_lt_top_of_lintegral_rpow_lt_top", "def_path": "Mathlib/MeasureTheory/Integral/MeanInequalities.lean", "def_pos": [178, 9], "def_end_pos": [178, 59]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp1 : 1 \u2264 p\nhp_pos : 0 < p\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc < \u22a4\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc < \u22a4\nh1 : \u00acp = 1\nhp1_lt : 1 < p\nhpq : Real.IsConjugateExponent p (Real.conjugateExponent p)\nh0 : \u00ac\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc = 0\n\u22a2 \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc < \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/RBMap/Lemmas.lean", "full_name": "Std.RBNode.lowerBound?_le", "start": [260, 1], "end": [261, 25], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/WithDensity.lean", "full_name": "MeasureTheory.lintegral_withDensity_eq_lintegral_mul_non_measurable", "start": [345, 1], "end": [368, 64], "traced_tactics": [{"tactic": "refine' le_antisymm (lintegral_withDensity_le_lintegral_mul \u03bc f_meas g) _", "annotated_tactic": ["refine' <a>le_antisymm</a> (<a>lintegral_withDensity_le_lintegral_mul</a> \u03bc f_meas g) _", [{"full_name": "le_antisymm", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [188, 9], "def_end_pos": [188, 20]}, {"full_name": "MeasureTheory.lintegral_withDensity_le_lintegral_mul", "def_path": "Mathlib/MeasureTheory/Measure/WithDensity.lean", "def_pos": [336, 9], "def_end_pos": [336, 47]}]], "state_before": "\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nf_meas : Measurable f\nhf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\n\u22a2 \u222b\u207b (a : \u03b1), g a \u2202withDensity \u03bc f = \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc", "state_after": "\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nf_meas : Measurable f\nhf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\n\u22a2 \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), g a \u2202withDensity \u03bc f"}, {"tactic": "rw [\u2190 iSup_lintegral_measurable_le_eq_lintegral, \u2190 iSup_lintegral_measurable_le_eq_lintegral]", "annotated_tactic": ["rw [\u2190 <a>iSup_lintegral_measurable_le_eq_lintegral</a>, \u2190 <a>iSup_lintegral_measurable_le_eq_lintegral</a>]", [{"full_name": "MeasureTheory.iSup_lintegral_measurable_le_eq_lintegral", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [112, 9], "def_end_pos": [112, 50]}, {"full_name": "MeasureTheory.iSup_lintegral_measurable_le_eq_lintegral", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [112, 9], "def_end_pos": [112, 50]}]], "state_before": "\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nf_meas : Measurable f\nhf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\n\u22a2 \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), g a \u2202withDensity \u03bc f", "state_after": "\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nf_meas : Measurable f\nhf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\n\u22a2 \u2a06 g_1, \u2a06 (_ : Measurable g_1), \u2a06 (_ : g_1 \u2264 fun a => (f * g) a), \u222b\u207b (a : \u03b1), g_1 a \u2202\u03bc \u2264\n    \u2a06 g_1, \u2a06 (_ : Measurable g_1), \u2a06 (_ : g_1 \u2264 fun a => g a), \u222b\u207b (a : \u03b1), g_1 a \u2202withDensity \u03bc f"}, {"tactic": "refine' iSup\u2082_le fun i i_meas => iSup_le fun hi => _", "annotated_tactic": ["refine' <a>iSup\u2082_le</a> fun i i_meas => <a>iSup_le</a> fun hi => _", [{"full_name": "iSup\u2082_le", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [883, 9], "def_end_pos": [883, 17]}, {"full_name": "iSup_le", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [875, 9], "def_end_pos": [875, 16]}]], "state_before": "\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nf_meas : Measurable f\nhf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\n\u22a2 \u2a06 g_1, \u2a06 (_ : Measurable g_1), \u2a06 (_ : g_1 \u2264 fun a => (f * g) a), \u222b\u207b (a : \u03b1), g_1 a \u2202\u03bc \u2264\n    \u2a06 g_1, \u2a06 (_ : Measurable g_1), \u2a06 (_ : g_1 \u2264 fun a => g a), \u222b\u207b (a : \u03b1), g_1 a \u2202withDensity \u03bc f", "state_after": "\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nf_meas : Measurable f\nhf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4\ng i : \u03b1 \u2192 \u211d\u22650\u221e\ni_meas : Measurable i\nhi : i \u2264 fun a => (f * g) a\n\u22a2 \u222b\u207b (a : \u03b1), i a \u2202\u03bc \u2264 \u2a06 g_1, \u2a06 (_ : Measurable g_1), \u2a06 (_ : g_1 \u2264 fun a => g a), \u222b\u207b (a : \u03b1), g_1 a \u2202withDensity \u03bc f"}, {"tactic": "have A : (fun x => (f x)\u207b\u00b9 * i x) \u2264 g := by\n  intro x\n  dsimp\n  rw [mul_comm, \u2190 div_eq_mul_inv]\n  exact div_le_of_le_mul' (hi x)", "annotated_tactic": ["have A : (fun x => (f x)\u207b\u00b9 * i x) \u2264 g := by\n    intro x\n    dsimp\n    rw [<a>mul_comm</a>, \u2190 <a>div_eq_mul_inv</a>]\n    exact <a>div_le_of_le_mul'</a> (hi x)", [{"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [302, 9], "def_end_pos": [302, 17]}, {"full_name": "div_eq_mul_inv", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [977, 9], "def_end_pos": [977, 23]}, {"full_name": "ENNReal.div_le_of_le_mul'", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1643, 9], "def_end_pos": [1643, 26]}]], "state_before": "\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nf_meas : Measurable f\nhf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4\ng i : \u03b1 \u2192 \u211d\u22650\u221e\ni_meas : Measurable i\nhi : i \u2264 fun a => (f * g) a\n\u22a2 \u222b\u207b (a : \u03b1), i a \u2202\u03bc \u2264 \u2a06 g_1, \u2a06 (_ : Measurable g_1), \u2a06 (_ : g_1 \u2264 fun a => g a), \u222b\u207b (a : \u03b1), g_1 a \u2202withDensity \u03bc f", "state_after": "\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nf_meas : Measurable f\nhf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4\ng i : \u03b1 \u2192 \u211d\u22650\u221e\ni_meas : Measurable i\nhi : i \u2264 fun a => (f * g) a\nA : (fun x => (f x)\u207b\u00b9 * i x) \u2264 g\n\u22a2 \u222b\u207b (a : \u03b1), i a \u2202\u03bc \u2264 \u2a06 g_1, \u2a06 (_ : Measurable g_1), \u2a06 (_ : g_1 \u2264 fun a => g a), \u222b\u207b (a : \u03b1), g_1 a \u2202withDensity \u03bc f"}, {"tactic": "refine' le_iSup_of_le (fun x => (f x)\u207b\u00b9 * i x) (le_iSup_of_le (f_meas.inv.mul i_meas) _)", "annotated_tactic": ["refine' <a>le_iSup_of_le</a> (fun x => (f x)\u207b\u00b9 * i x) (<a>le_iSup_of_le</a> (f_meas.inv.mul i_meas) _)", [{"full_name": "le_iSup_of_le", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [849, 9], "def_end_pos": [849, 22]}, {"full_name": "le_iSup_of_le", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [849, 9], "def_end_pos": [849, 22]}]], "state_before": "\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nf_meas : Measurable f\nhf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4\ng i : \u03b1 \u2192 \u211d\u22650\u221e\ni_meas : Measurable i\nhi : i \u2264 fun a => (f * g) a\nA : (fun x => (f x)\u207b\u00b9 * i x) \u2264 g\n\u22a2 \u222b\u207b (a : \u03b1), i a \u2202\u03bc \u2264 \u2a06 g_1, \u2a06 (_ : Measurable g_1), \u2a06 (_ : g_1 \u2264 fun a => g a), \u222b\u207b (a : \u03b1), g_1 a \u2202withDensity \u03bc f", "state_after": "\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nf_meas : Measurable f\nhf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4\ng i : \u03b1 \u2192 \u211d\u22650\u221e\ni_meas : Measurable i\nhi : i \u2264 fun a => (f * g) a\nA : (fun x => (f x)\u207b\u00b9 * i x) \u2264 g\n\u22a2 \u222b\u207b (a : \u03b1), i a \u2202\u03bc \u2264\n    \u2a06 (_ : (fun x => (f x)\u207b\u00b9 * i x) \u2264 fun a => g a), \u222b\u207b (a : \u03b1), (fun x => (f x)\u207b\u00b9 * i x) a \u2202withDensity \u03bc f"}, {"tactic": "refine' le_iSup_of_le A _", "annotated_tactic": ["refine' <a>le_iSup_of_le</a> A _", [{"full_name": "le_iSup_of_le", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [849, 9], "def_end_pos": [849, 22]}]], "state_before": "\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nf_meas : Measurable f\nhf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4\ng i : \u03b1 \u2192 \u211d\u22650\u221e\ni_meas : Measurable i\nhi : i \u2264 fun a => (f * g) a\nA : (fun x => (f x)\u207b\u00b9 * i x) \u2264 g\n\u22a2 \u222b\u207b (a : \u03b1), i a \u2202\u03bc \u2264\n    \u2a06 (_ : (fun x => (f x)\u207b\u00b9 * i x) \u2264 fun a => g a), \u222b\u207b (a : \u03b1), (fun x => (f x)\u207b\u00b9 * i x) a \u2202withDensity \u03bc f", "state_after": "\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nf_meas : Measurable f\nhf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4\ng i : \u03b1 \u2192 \u211d\u22650\u221e\ni_meas : Measurable i\nhi : i \u2264 fun a => (f * g) a\nA : (fun x => (f x)\u207b\u00b9 * i x) \u2264 g\n\u22a2 \u222b\u207b (a : \u03b1), i a \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), (fun x => (f x)\u207b\u00b9 * i x) a \u2202withDensity \u03bc f"}, {"tactic": "rw [lintegral_withDensity_eq_lintegral_mul _ f_meas (f_meas.inv.mul i_meas)]", "annotated_tactic": ["rw [<a>lintegral_withDensity_eq_lintegral_mul</a> _ f_meas (f_meas.inv.mul i_meas)]", [{"full_name": "MeasureTheory.lintegral_withDensity_eq_lintegral_mul", "def_path": "Mathlib/MeasureTheory/Measure/WithDensity.lean", "def_pos": [275, 9], "def_end_pos": [275, 47]}]], "state_before": "\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nf_meas : Measurable f\nhf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4\ng i : \u03b1 \u2192 \u211d\u22650\u221e\ni_meas : Measurable i\nhi : i \u2264 fun a => (f * g) a\nA : (fun x => (f x)\u207b\u00b9 * i x) \u2264 g\n\u22a2 \u222b\u207b (a : \u03b1), i a \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), (fun x => (f x)\u207b\u00b9 * i x) a \u2202withDensity \u03bc f", "state_after": "\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nf_meas : Measurable f\nhf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4\ng i : \u03b1 \u2192 \u211d\u22650\u221e\ni_meas : Measurable i\nhi : i \u2264 fun a => (f * g) a\nA : (fun x => (f x)\u207b\u00b9 * i x) \u2264 g\n\u22a2 \u222b\u207b (a : \u03b1), i a \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), (f * fun a => (f a)\u207b\u00b9 * i a) a \u2202\u03bc"}, {"tactic": "apply lintegral_mono_ae", "annotated_tactic": ["apply <a>lintegral_mono_ae</a>", [{"full_name": "MeasureTheory.lintegral_mono_ae", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [265, 9], "def_end_pos": [265, 26]}]], "state_before": "\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nf_meas : Measurable f\nhf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4\ng i : \u03b1 \u2192 \u211d\u22650\u221e\ni_meas : Measurable i\nhi : i \u2264 fun a => (f * g) a\nA : (fun x => (f x)\u207b\u00b9 * i x) \u2264 g\n\u22a2 \u222b\u207b (a : \u03b1), i a \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), (f * fun a => (f a)\u207b\u00b9 * i a) a \u2202\u03bc", "state_after": "case h\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nf_meas : Measurable f\nhf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4\ng i : \u03b1 \u2192 \u211d\u22650\u221e\ni_meas : Measurable i\nhi : i \u2264 fun a => (f * g) a\nA : (fun x => (f x)\u207b\u00b9 * i x) \u2264 g\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, i a \u2264 (f * fun a => (f a)\u207b\u00b9 * i a) a"}, {"tactic": "filter_upwards [hf]", "annotated_tactic": ["filter_upwards [hf]", []], "state_before": "case h\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nf_meas : Measurable f\nhf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4\ng i : \u03b1 \u2192 \u211d\u22650\u221e\ni_meas : Measurable i\nhi : i \u2264 fun a => (f * g) a\nA : (fun x => (f x)\u207b\u00b9 * i x) \u2264 g\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, i a \u2264 (f * fun a => (f a)\u207b\u00b9 * i a) a", "state_after": "case h\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nf_meas : Measurable f\nhf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4\ng i : \u03b1 \u2192 \u211d\u22650\u221e\ni_meas : Measurable i\nhi : i \u2264 fun a => (f * g) a\nA : (fun x => (f x)\u207b\u00b9 * i x) \u2264 g\n\u22a2 \u2200 (a : \u03b1), f a < \u22a4 \u2192 i a \u2264 (f * fun a => (f a)\u207b\u00b9 * i a) a"}, {"tactic": "intro x h'x", "annotated_tactic": ["intro x h'x", []], "state_before": "case h\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nf_meas : Measurable f\nhf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4\ng i : \u03b1 \u2192 \u211d\u22650\u221e\ni_meas : Measurable i\nhi : i \u2264 fun a => (f * g) a\nA : (fun x => (f x)\u207b\u00b9 * i x) \u2264 g\n\u22a2 \u2200 (a : \u03b1), f a < \u22a4 \u2192 i a \u2264 (f * fun a => (f a)\u207b\u00b9 * i a) a", "state_after": "case h\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nf_meas : Measurable f\nhf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4\ng i : \u03b1 \u2192 \u211d\u22650\u221e\ni_meas : Measurable i\nhi : i \u2264 fun a => (f * g) a\nA : (fun x => (f x)\u207b\u00b9 * i x) \u2264 g\nx : \u03b1\nh'x : f x < \u22a4\n\u22a2 i x \u2264 (f * fun a => (f a)\u207b\u00b9 * i a) x"}, {"tactic": "rcases eq_or_ne (f x) 0 with (hx | hx)", "annotated_tactic": ["rcases <a>eq_or_ne</a> (f x) 0 with (hx | hx)", [{"full_name": "eq_or_ne", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [209, 9], "def_end_pos": [209, 17]}]], "state_before": "case h\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nf_meas : Measurable f\nhf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4\ng i : \u03b1 \u2192 \u211d\u22650\u221e\ni_meas : Measurable i\nhi : i \u2264 fun a => (f * g) a\nA : (fun x => (f x)\u207b\u00b9 * i x) \u2264 g\nx : \u03b1\nh'x : f x < \u22a4\n\u22a2 i x \u2264 (f * fun a => (f a)\u207b\u00b9 * i a) x", "state_after": "case h.inl\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nf_meas : Measurable f\nhf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4\ng i : \u03b1 \u2192 \u211d\u22650\u221e\ni_meas : Measurable i\nhi : i \u2264 fun a => (f * g) a\nA : (fun x => (f x)\u207b\u00b9 * i x) \u2264 g\nx : \u03b1\nh'x : f x < \u22a4\nhx : f x = 0\n\u22a2 i x \u2264 (f * fun a => (f a)\u207b\u00b9 * i a) x\n\ncase h.inr\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nf_meas : Measurable f\nhf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4\ng i : \u03b1 \u2192 \u211d\u22650\u221e\ni_meas : Measurable i\nhi : i \u2264 fun a => (f * g) a\nA : (fun x => (f x)\u207b\u00b9 * i x) \u2264 g\nx : \u03b1\nh'x : f x < \u22a4\nhx : f x \u2260 0\n\u22a2 i x \u2264 (f * fun a => (f a)\u207b\u00b9 * i a) x"}, {"tactic": "intro x", "annotated_tactic": ["intro x", []], "state_before": "\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nf_meas : Measurable f\nhf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4\ng i : \u03b1 \u2192 \u211d\u22650\u221e\ni_meas : Measurable i\nhi : i \u2264 fun a => (f * g) a\n\u22a2 (fun x => (f x)\u207b\u00b9 * i x) \u2264 g", "state_after": "\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nf_meas : Measurable f\nhf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4\ng i : \u03b1 \u2192 \u211d\u22650\u221e\ni_meas : Measurable i\nhi : i \u2264 fun a => (f * g) a\nx : \u03b1\n\u22a2 (fun x => (f x)\u207b\u00b9 * i x) x \u2264 g x"}, {"tactic": "dsimp", "annotated_tactic": ["dsimp", []], "state_before": "\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nf_meas : Measurable f\nhf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4\ng i : \u03b1 \u2192 \u211d\u22650\u221e\ni_meas : Measurable i\nhi : i \u2264 fun a => (f * g) a\nx : \u03b1\n\u22a2 (fun x => (f x)\u207b\u00b9 * i x) x \u2264 g x", "state_after": "\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nf_meas : Measurable f\nhf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4\ng i : \u03b1 \u2192 \u211d\u22650\u221e\ni_meas : Measurable i\nhi : i \u2264 fun a => (f * g) a\nx : \u03b1\n\u22a2 (f x)\u207b\u00b9 * i x \u2264 g x"}, {"tactic": "rw [mul_comm, \u2190 div_eq_mul_inv]", "annotated_tactic": ["rw [<a>mul_comm</a>, \u2190 <a>div_eq_mul_inv</a>]", [{"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [302, 9], "def_end_pos": [302, 17]}, {"full_name": "div_eq_mul_inv", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [977, 9], "def_end_pos": [977, 23]}]], "state_before": "\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nf_meas : Measurable f\nhf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4\ng i : \u03b1 \u2192 \u211d\u22650\u221e\ni_meas : Measurable i\nhi : i \u2264 fun a => (f * g) a\nx : \u03b1\n\u22a2 (f x)\u207b\u00b9 * i x \u2264 g x", "state_after": "\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nf_meas : Measurable f\nhf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4\ng i : \u03b1 \u2192 \u211d\u22650\u221e\ni_meas : Measurable i\nhi : i \u2264 fun a => (f * g) a\nx : \u03b1\n\u22a2 i x / f x \u2264 g x"}, {"tactic": "exact div_le_of_le_mul' (hi x)", "annotated_tactic": ["exact <a>div_le_of_le_mul'</a> (hi x)", [{"full_name": "ENNReal.div_le_of_le_mul'", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1643, 9], "def_end_pos": [1643, 26]}]], "state_before": "\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nf_meas : Measurable f\nhf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4\ng i : \u03b1 \u2192 \u211d\u22650\u221e\ni_meas : Measurable i\nhi : i \u2264 fun a => (f * g) a\nx : \u03b1\n\u22a2 i x / f x \u2264 g x", "state_after": "no goals"}, {"tactic": "have := hi x", "annotated_tactic": ["have := hi x", []], "state_before": "case h.inl\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nf_meas : Measurable f\nhf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4\ng i : \u03b1 \u2192 \u211d\u22650\u221e\ni_meas : Measurable i\nhi : i \u2264 fun a => (f * g) a\nA : (fun x => (f x)\u207b\u00b9 * i x) \u2264 g\nx : \u03b1\nh'x : f x < \u22a4\nhx : f x = 0\n\u22a2 i x \u2264 (f * fun a => (f a)\u207b\u00b9 * i a) x", "state_after": "case h.inl\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nf_meas : Measurable f\nhf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4\ng i : \u03b1 \u2192 \u211d\u22650\u221e\ni_meas : Measurable i\nhi : i \u2264 fun a => (f * g) a\nA : (fun x => (f x)\u207b\u00b9 * i x) \u2264 g\nx : \u03b1\nh'x : f x < \u22a4\nhx : f x = 0\nthis : i x \u2264 (fun a => (f * g) a) x\n\u22a2 i x \u2264 (f * fun a => (f a)\u207b\u00b9 * i a) x"}, {"tactic": "simp only [hx, zero_mul, Pi.mul_apply, nonpos_iff_eq_zero] at this", "annotated_tactic": ["simp only [hx, <a>zero_mul</a>, <a>Pi.mul_apply</a>, <a>nonpos_iff_eq_zero</a>] at this", [{"full_name": "MulZeroClass.zero_mul", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [36, 3], "def_end_pos": [36, 11]}, {"full_name": "Pi.mul_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [83, 9], "def_end_pos": [83, 18]}, {"full_name": "nonpos_iff_eq_zero", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [237, 3], "def_end_pos": [237, 14]}]], "state_before": "case h.inl\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nf_meas : Measurable f\nhf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4\ng i : \u03b1 \u2192 \u211d\u22650\u221e\ni_meas : Measurable i\nhi : i \u2264 fun a => (f * g) a\nA : (fun x => (f x)\u207b\u00b9 * i x) \u2264 g\nx : \u03b1\nh'x : f x < \u22a4\nhx : f x = 0\nthis : i x \u2264 (fun a => (f * g) a) x\n\u22a2 i x \u2264 (f * fun a => (f a)\u207b\u00b9 * i a) x", "state_after": "case h.inl\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nf_meas : Measurable f\nhf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4\ng i : \u03b1 \u2192 \u211d\u22650\u221e\ni_meas : Measurable i\nhi : i \u2264 fun a => (f * g) a\nA : (fun x => (f x)\u207b\u00b9 * i x) \u2264 g\nx : \u03b1\nh'x : f x < \u22a4\nhx : f x = 0\nthis : i x = 0\n\u22a2 i x \u2264 (f * fun a => (f a)\u207b\u00b9 * i a) x"}, {"tactic": "simp [this]", "annotated_tactic": ["simp [this]", []], "state_before": "case h.inl\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nf_meas : Measurable f\nhf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4\ng i : \u03b1 \u2192 \u211d\u22650\u221e\ni_meas : Measurable i\nhi : i \u2264 fun a => (f * g) a\nA : (fun x => (f x)\u207b\u00b9 * i x) \u2264 g\nx : \u03b1\nh'x : f x < \u22a4\nhx : f x = 0\nthis : i x = 0\n\u22a2 i x \u2264 (f * fun a => (f a)\u207b\u00b9 * i a) x", "state_after": "no goals"}, {"tactic": "apply le_of_eq _", "annotated_tactic": ["apply <a>le_of_eq</a> _", [{"full_name": "le_of_eq", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [72, 9], "def_end_pos": [72, 17]}]], "state_before": "case h.inr\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nf_meas : Measurable f\nhf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4\ng i : \u03b1 \u2192 \u211d\u22650\u221e\ni_meas : Measurable i\nhi : i \u2264 fun a => (f * g) a\nA : (fun x => (f x)\u207b\u00b9 * i x) \u2264 g\nx : \u03b1\nh'x : f x < \u22a4\nhx : f x \u2260 0\n\u22a2 i x \u2264 (f * fun a => (f a)\u207b\u00b9 * i a) x", "state_after": "\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nf_meas : Measurable f\nhf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4\ng i : \u03b1 \u2192 \u211d\u22650\u221e\ni_meas : Measurable i\nhi : i \u2264 fun a => (f * g) a\nA : (fun x => (f x)\u207b\u00b9 * i x) \u2264 g\nx : \u03b1\nh'x : f x < \u22a4\nhx : f x \u2260 0\n\u22a2 i x = (f * fun a => (f a)\u207b\u00b9 * i a) x"}, {"tactic": "dsimp", "annotated_tactic": ["dsimp", []], "state_before": "\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nf_meas : Measurable f\nhf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4\ng i : \u03b1 \u2192 \u211d\u22650\u221e\ni_meas : Measurable i\nhi : i \u2264 fun a => (f * g) a\nA : (fun x => (f x)\u207b\u00b9 * i x) \u2264 g\nx : \u03b1\nh'x : f x < \u22a4\nhx : f x \u2260 0\n\u22a2 i x = (f * fun a => (f a)\u207b\u00b9 * i a) x", "state_after": "\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nf_meas : Measurable f\nhf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4\ng i : \u03b1 \u2192 \u211d\u22650\u221e\ni_meas : Measurable i\nhi : i \u2264 fun a => (f * g) a\nA : (fun x => (f x)\u207b\u00b9 * i x) \u2264 g\nx : \u03b1\nh'x : f x < \u22a4\nhx : f x \u2260 0\n\u22a2 i x = f x * ((f x)\u207b\u00b9 * i x)"}, {"tactic": "rw [\u2190 mul_assoc, ENNReal.mul_inv_cancel hx h'x.ne, one_mul]", "annotated_tactic": ["rw [\u2190 <a>mul_assoc</a>, <a>ENNReal.mul_inv_cancel</a> hx h'x.ne, <a>one_mul</a>]", [{"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [264, 9], "def_end_pos": [264, 18]}, {"full_name": "ENNReal.mul_inv_cancel", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1418, 19], "def_end_pos": [1418, 33]}, {"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [464, 9], "def_end_pos": [464, 16]}]], "state_before": "\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nf_meas : Measurable f\nhf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4\ng i : \u03b1 \u2192 \u211d\u22650\u221e\ni_meas : Measurable i\nhi : i \u2264 fun a => (f * g) a\nA : (fun x => (f x)\u207b\u00b9 * i x) \u2264 g\nx : \u03b1\nh'x : f x < \u22a4\nhx : f x \u2260 0\n\u22a2 i x = f x * ((f x)\u207b\u00b9 * i x)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean", "full_name": "Real.volume_val", "start": [75, 1], "end": [76, 32], "traced_tactics": [{"tactic": "simp [volume_eq_stieltjes_id]", "annotated_tactic": ["simp [<a>volume_eq_stieltjes_id</a>]", [{"full_name": "Real.volume_eq_stieltjes_id", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean", "def_pos": [56, 9], "def_end_pos": [56, 31]}]], "state_before": "\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\ns : Set \u211d\n\u22a2 \u2191\u2191volume s = \u2191\u2191(StieltjesFunction.measure StieltjesFunction.id) s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "full_name": "MeasureTheory.SimpleFunc.tendsto_approxOn_range_Lp", "start": [189, 1], "end": [197, 57], "traced_tactics": [{"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace E\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\nq : \u211d\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace E\nf : \u03b2 \u2192 E\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\n\u03bc : Measure \u03b2\nfmeas : Measurable f\ninst\u271d : SeparableSpace \u2191(Set.range f \u222a {0})\nhf : Mem\u2112p f p\nn : \u2115\n\u22a2 0 \u2208 Set.range f \u222a {0}", "state_after": "no goals"}, {"tactic": "simpa only [Lp.tendsto_Lp_iff_tendsto_\u2112p''] using\n  tendsto_approxOn_range_Lp_snorm hp_ne_top fmeas hf.2", "annotated_tactic": ["simpa only [<a>Lp.tendsto_Lp_iff_tendsto_\u2112p''</a>] using\n    <a>tendsto_approxOn_range_Lp_snorm</a> hp_ne_top fmeas hf.2", [{"full_name": "MeasureTheory.Lp.tendsto_Lp_iff_tendsto_\u2112p''", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [1389, 9], "def_end_pos": [1389, 36]}, {"full_name": "MeasureTheory.SimpleFunc.tendsto_approxOn_range_Lp_snorm", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "def_pos": [170, 9], "def_end_pos": [170, 40]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace E\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\nq : \u211d\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace E\nf : \u03b2 \u2192 E\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\n\u03bc : Measure \u03b2\nfmeas : Measurable f\ninst\u271d : SeparableSpace \u2191(Set.range f \u222a {0})\nhf : Mem\u2112p f p\n\u22a2 Tendsto\n    (fun n =>\n      Mem\u2112p.toLp \u2191(approxOn f fmeas (Set.range f \u222a {0}) 0 (_ : 0 \u2208 Set.range f \u222a {0}) n)\n        (_ : Mem\u2112p (\u2191(approxOn f fmeas (Set.range f \u222a {0}) 0 (_ : 0 \u2208 Set.range f \u222a {0}) n)) p))\n    atTop (\ud835\udcdd (Mem\u2112p.toLp f hf))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Density.lean", "full_name": "Real.hasPDF_iff", "start": [279, 1], "end": [283, 76], "traced_tactics": [{"tactic": "by_cases hX : Measurable X", "annotated_tactic": ["by_cases hX : <a>Measurable</a> X", [{"full_name": "Measurable", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [535, 5], "def_end_pos": [535, 15]}]], "state_before": "\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b9 : MeasurableSpace E\nm : MeasurableSpace \u03a9\n\u2119 : Measure \u03a9\n\u03bc : Measure E\ninst\u271d : IsFiniteMeasure \u2119\nX : \u03a9 \u2192 \u211d\n\u22a2 HasPDF X \u2119 \u2194 Measurable X \u2227 map X \u2119 \u226a volume", "state_after": "case pos\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b9 : MeasurableSpace E\nm : MeasurableSpace \u03a9\n\u2119 : Measure \u03a9\n\u03bc : Measure E\ninst\u271d : IsFiniteMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhX : Measurable X\n\u22a2 HasPDF X \u2119 \u2194 Measurable X \u2227 map X \u2119 \u226a volume\n\ncase neg\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b9 : MeasurableSpace E\nm : MeasurableSpace \u03a9\n\u2119 : Measure \u03a9\n\u03bc : Measure E\ninst\u271d : IsFiniteMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhX : \u00acMeasurable X\n\u22a2 HasPDF X \u2119 \u2194 Measurable X \u2227 map X \u2119 \u226a volume"}, {"tactic": "rw [Real.hasPDF_iff_of_measurable hX, iff_and_self]", "annotated_tactic": ["rw [<a>Real.hasPDF_iff_of_measurable</a> hX, <a>iff_and_self</a>]", [{"full_name": "Real.hasPDF_iff_of_measurable", "def_path": "Mathlib/Probability/Density.lean", "def_pos": [273, 16], "def_end_pos": [273, 52]}, {"full_name": "iff_and_self", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [217, 17], "def_end_pos": [217, 29]}]], "state_before": "case pos\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b9 : MeasurableSpace E\nm : MeasurableSpace \u03a9\n\u2119 : Measure \u03a9\n\u03bc : Measure E\ninst\u271d : IsFiniteMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhX : Measurable X\n\u22a2 HasPDF X \u2119 \u2194 Measurable X \u2227 map X \u2119 \u226a volume", "state_after": "case pos\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b9 : MeasurableSpace E\nm : MeasurableSpace \u03a9\n\u2119 : Measure \u03a9\n\u03bc : Measure E\ninst\u271d : IsFiniteMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhX : Measurable X\n\u22a2 map X \u2119 \u226a volume \u2192 Measurable X"}, {"tactic": "exact fun _ => hX", "annotated_tactic": ["exact fun _ => hX", []], "state_before": "case pos\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b9 : MeasurableSpace E\nm : MeasurableSpace \u03a9\n\u2119 : Measure \u03a9\n\u03bc : Measure E\ninst\u271d : IsFiniteMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhX : Measurable X\n\u22a2 map X \u2119 \u226a volume \u2192 Measurable X", "state_after": "no goals"}, {"tactic": "exact \u27e8fun h => False.elim (hX h.pdf'.1), fun h => False.elim (hX h.1)\u27e9", "annotated_tactic": ["exact \u27e8fun h => <a>False.elim</a> (hX h.pdf'.1), fun h => <a>False.elim</a> (hX h.1)\u27e9", [{"full_name": "False.elim", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [223, 21], "def_end_pos": [223, 31]}, {"full_name": "False.elim", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [223, 21], "def_end_pos": [223, 31]}]], "state_before": "case neg\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b9 : MeasurableSpace E\nm : MeasurableSpace \u03a9\n\u2119 : Measure \u03a9\n\u03bc : Measure E\ninst\u271d : IsFiniteMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhX : \u00acMeasurable X\n\u22a2 HasPDF X \u2119 \u2194 Measurable X \u2227 map X \u2119 \u226a volume", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Lattice.lean", "full_name": "Finset.sup_inf", "start": [594, 1], "end": [596, 31], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "full_name": "intervalIntegral.continuousOn_primitive_Icc", "start": [1208, 1], "end": [1214, 37], "traced_tactics": [{"tactic": "have aux : (fun x => \u222b t in Icc a x, f t \u2202\u03bc) = fun x => \u222b t in Ioc a x, f t \u2202\u03bc := by\n  ext x\n  exact integral_Icc_eq_integral_Ioc", "annotated_tactic": ["have aux : (fun x => \u222b t in <a>Icc</a> a x, f t \u2202\u03bc) = fun x => \u222b t in <a>Ioc</a> a x, f t \u2202\u03bc := by\n    ext x\n    exact <a>integral_Icc_eq_integral_Ioc</a>", [{"full_name": "Set.Icc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [59, 5], "def_end_pos": [59, 8]}, {"full_name": "Set.Ioc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [69, 5], "def_end_pos": [69, 8]}, {"full_name": "MeasureTheory.integral_Icc_eq_integral_Ioc", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [675, 9], "def_end_pos": [675, 37]}]], "state_before": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : NormedSpace \u211d E\na b b\u2080 b\u2081 b\u2082 : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 E\ninst\u271d : NoAtoms \u03bc\nh_int : IntegrableOn f (Icc a b)\n\u22a2 ContinuousOn (fun x => \u222b (t : \u211d) in Icc a x, f t \u2202\u03bc) (Icc a b)", "state_after": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : NormedSpace \u211d E\na b b\u2080 b\u2081 b\u2082 : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 E\ninst\u271d : NoAtoms \u03bc\nh_int : IntegrableOn f (Icc a b)\naux : (fun x => \u222b (t : \u211d) in Icc a x, f t \u2202\u03bc) = fun x => \u222b (t : \u211d) in Ioc a x, f t \u2202\u03bc\n\u22a2 ContinuousOn (fun x => \u222b (t : \u211d) in Icc a x, f t \u2202\u03bc) (Icc a b)"}, {"tactic": "rw [aux]", "annotated_tactic": ["rw [aux]", []], "state_before": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : NormedSpace \u211d E\na b b\u2080 b\u2081 b\u2082 : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 E\ninst\u271d : NoAtoms \u03bc\nh_int : IntegrableOn f (Icc a b)\naux : (fun x => \u222b (t : \u211d) in Icc a x, f t \u2202\u03bc) = fun x => \u222b (t : \u211d) in Ioc a x, f t \u2202\u03bc\n\u22a2 ContinuousOn (fun x => \u222b (t : \u211d) in Icc a x, f t \u2202\u03bc) (Icc a b)", "state_after": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : NormedSpace \u211d E\na b b\u2080 b\u2081 b\u2082 : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 E\ninst\u271d : NoAtoms \u03bc\nh_int : IntegrableOn f (Icc a b)\naux : (fun x => \u222b (t : \u211d) in Icc a x, f t \u2202\u03bc) = fun x => \u222b (t : \u211d) in Ioc a x, f t \u2202\u03bc\n\u22a2 ContinuousOn (fun x => \u222b (t : \u211d) in Ioc a x, f t \u2202\u03bc) (Icc a b)"}, {"tactic": "exact continuousOn_primitive h_int", "annotated_tactic": ["exact <a>continuousOn_primitive</a> h_int", [{"full_name": "intervalIntegral.continuousOn_primitive", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [1193, 9], "def_end_pos": [1193, 31]}]], "state_before": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : NormedSpace \u211d E\na b b\u2080 b\u2081 b\u2082 : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 E\ninst\u271d : NoAtoms \u03bc\nh_int : IntegrableOn f (Icc a b)\naux : (fun x => \u222b (t : \u211d) in Icc a x, f t \u2202\u03bc) = fun x => \u222b (t : \u211d) in Ioc a x, f t \u2202\u03bc\n\u22a2 ContinuousOn (fun x => \u222b (t : \u211d) in Ioc a x, f t \u2202\u03bc) (Icc a b)", "state_after": "no goals"}, {"tactic": "ext x", "annotated_tactic": ["ext x", []], "state_before": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : NormedSpace \u211d E\na b b\u2080 b\u2081 b\u2082 : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 E\ninst\u271d : NoAtoms \u03bc\nh_int : IntegrableOn f (Icc a b)\n\u22a2 (fun x => \u222b (t : \u211d) in Icc a x, f t \u2202\u03bc) = fun x => \u222b (t : \u211d) in Ioc a x, f t \u2202\u03bc", "state_after": "case h\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : NormedSpace \u211d E\na b b\u2080 b\u2081 b\u2082 : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 E\ninst\u271d : NoAtoms \u03bc\nh_int : IntegrableOn f (Icc a b)\nx : \u211d\n\u22a2 \u222b (t : \u211d) in Icc a x, f t \u2202\u03bc = \u222b (t : \u211d) in Ioc a x, f t \u2202\u03bc"}, {"tactic": "exact integral_Icc_eq_integral_Ioc", "annotated_tactic": ["exact <a>integral_Icc_eq_integral_Ioc</a>", [{"full_name": "MeasureTheory.integral_Icc_eq_integral_Ioc", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [675, 9], "def_end_pos": [675, 37]}]], "state_before": "case h\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : NormedSpace \u211d E\na b b\u2080 b\u2081 b\u2082 : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 E\ninst\u271d : NoAtoms \u03bc\nh_int : IntegrableOn f (Icc a b)\nx : \u211d\n\u22a2 \u222b (t : \u211d) in Icc a x, f t \u2202\u03bc = \u222b (t : \u211d) in Ioc a x, f t \u2202\u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean", "full_name": "MeasureTheory.isClosed_aeStronglyMeasurable'", "start": [510, 1], "end": [512, 60], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Kernel/CondCdf.lean", "full_name": "ProbabilityTheory.measure_condCdf_univ", "start": [971, 1], "end": [973, 97], "traced_tactics": [{"tactic": "rw [\u2190 ENNReal.ofReal_one, \u2190 sub_zero (1 : \u211d)]", "annotated_tactic": ["rw [\u2190 <a>ENNReal.ofReal_one</a>, \u2190 <a>sub_zero</a> (1 : \u211d)]", [{"full_name": "ENNReal.ofReal_one", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [248, 17], "def_end_pos": [248, 27]}, {"full_name": "sub_zero", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [339, 3], "def_end_pos": [339, 14]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\n\u22a2 \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 a)) univ = 1", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\n\u22a2 \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 a)) univ = ENNReal.ofReal (1 - 0)"}, {"tactic": "exact StieltjesFunction.measure_univ _ (tendsto_condCdf_atBot \u03c1 a) (tendsto_condCdf_atTop \u03c1 a)", "annotated_tactic": ["exact <a>StieltjesFunction.measure_univ</a> _ (<a>tendsto_condCdf_atBot</a> \u03c1 a) (<a>tendsto_condCdf_atTop</a> \u03c1 a)", [{"full_name": "StieltjesFunction.measure_univ", "def_path": "Mathlib/MeasureTheory/Measure/Stieltjes.lean", "def_pos": [442, 9], "def_end_pos": [442, 21]}, {"full_name": "ProbabilityTheory.tendsto_condCdf_atBot", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [803, 9], "def_end_pos": [803, 30]}, {"full_name": "ProbabilityTheory.tendsto_condCdf_atTop", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [822, 9], "def_end_pos": [822, 30]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\n\u22a2 \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 a)) univ = ENNReal.ofReal (1 - 0)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Reduce.lean", "full_name": "ManyOneDegree.add_le", "start": [488, 11], "end": [492, 51], "traced_tactics": [{"tactic": "induction d\u2081 using ManyOneDegree.ind_on", "annotated_tactic": ["induction d\u2081 using <a>ManyOneDegree.ind_on</a>", [{"full_name": "ManyOneDegree.ind_on", "def_path": "Mathlib/Computability/Reduce.lean", "def_pos": [375, 19], "def_end_pos": [375, 25]}]], "state_before": "\u03b1 : Type u\ninst\u271d\u2075 : Primcodable \u03b1\ninst\u271d\u2074 : Inhabited \u03b1\n\u03b2 : Type v\ninst\u271d\u00b3 : Primcodable \u03b2\ninst\u271d\u00b2 : Inhabited \u03b2\n\u03b3 : Type w\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Inhabited \u03b3\nd\u2081 d\u2082 d\u2083 : ManyOneDegree\n\u22a2 d\u2081 + d\u2082 \u2264 d\u2083 \u2194 d\u2081 \u2264 d\u2083 \u2227 d\u2082 \u2264 d\u2083", "state_after": "case h\n\u03b1 : Type u\ninst\u271d\u2075 : Primcodable \u03b1\ninst\u271d\u2074 : Inhabited \u03b1\n\u03b2 : Type v\ninst\u271d\u00b3 : Primcodable \u03b2\ninst\u271d\u00b2 : Inhabited \u03b2\n\u03b3 : Type w\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Inhabited \u03b3\nd\u2082 d\u2083 : ManyOneDegree\np\u271d : Set \u2115\n\u22a2 of p\u271d + d\u2082 \u2264 d\u2083 \u2194 of p\u271d \u2264 d\u2083 \u2227 d\u2082 \u2264 d\u2083"}, {"tactic": "induction d\u2082 using ManyOneDegree.ind_on", "annotated_tactic": ["induction d\u2082 using <a>ManyOneDegree.ind_on</a>", [{"full_name": "ManyOneDegree.ind_on", "def_path": "Mathlib/Computability/Reduce.lean", "def_pos": [375, 19], "def_end_pos": [375, 25]}]], "state_before": "case h\n\u03b1 : Type u\ninst\u271d\u2075 : Primcodable \u03b1\ninst\u271d\u2074 : Inhabited \u03b1\n\u03b2 : Type v\ninst\u271d\u00b3 : Primcodable \u03b2\ninst\u271d\u00b2 : Inhabited \u03b2\n\u03b3 : Type w\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Inhabited \u03b3\nd\u2082 d\u2083 : ManyOneDegree\np\u271d : Set \u2115\n\u22a2 of p\u271d + d\u2082 \u2264 d\u2083 \u2194 of p\u271d \u2264 d\u2083 \u2227 d\u2082 \u2264 d\u2083", "state_after": "case h.h\n\u03b1 : Type u\ninst\u271d\u2075 : Primcodable \u03b1\ninst\u271d\u2074 : Inhabited \u03b1\n\u03b2 : Type v\ninst\u271d\u00b3 : Primcodable \u03b2\ninst\u271d\u00b2 : Inhabited \u03b2\n\u03b3 : Type w\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Inhabited \u03b3\nd\u2083 : ManyOneDegree\np\u271d\u00b9 p\u271d : Set \u2115\n\u22a2 of p\u271d\u00b9 + of p\u271d \u2264 d\u2083 \u2194 of p\u271d\u00b9 \u2264 d\u2083 \u2227 of p\u271d \u2264 d\u2083"}, {"tactic": "induction d\u2083 using ManyOneDegree.ind_on", "annotated_tactic": ["induction d\u2083 using <a>ManyOneDegree.ind_on</a>", [{"full_name": "ManyOneDegree.ind_on", "def_path": "Mathlib/Computability/Reduce.lean", "def_pos": [375, 19], "def_end_pos": [375, 25]}]], "state_before": "case h.h\n\u03b1 : Type u\ninst\u271d\u2075 : Primcodable \u03b1\ninst\u271d\u2074 : Inhabited \u03b1\n\u03b2 : Type v\ninst\u271d\u00b3 : Primcodable \u03b2\ninst\u271d\u00b2 : Inhabited \u03b2\n\u03b3 : Type w\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Inhabited \u03b3\nd\u2083 : ManyOneDegree\np\u271d\u00b9 p\u271d : Set \u2115\n\u22a2 of p\u271d\u00b9 + of p\u271d \u2264 d\u2083 \u2194 of p\u271d\u00b9 \u2264 d\u2083 \u2227 of p\u271d \u2264 d\u2083", "state_after": "case h.h.h\n\u03b1 : Type u\ninst\u271d\u2075 : Primcodable \u03b1\ninst\u271d\u2074 : Inhabited \u03b1\n\u03b2 : Type v\ninst\u271d\u00b3 : Primcodable \u03b2\ninst\u271d\u00b2 : Inhabited \u03b2\n\u03b3 : Type w\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Inhabited \u03b3\np\u271d\u00b2 p\u271d\u00b9 p\u271d : Set \u2115\n\u22a2 of p\u271d\u00b2 + of p\u271d\u00b9 \u2264 of p\u271d \u2194 of p\u271d\u00b2 \u2264 of p\u271d \u2227 of p\u271d\u00b9 \u2264 of p\u271d"}, {"tactic": "simpa only [\u2190 add_of, of_le_of] using disjoin_le", "annotated_tactic": ["simpa only [\u2190 <a>add_of</a>, <a>of_le_of</a>] using <a>disjoin_le</a>", [{"full_name": "ManyOneDegree.add_of", "def_path": "Mathlib/Computability/Reduce.lean", "def_pos": [477, 9], "def_end_pos": [477, 15]}, {"full_name": "ManyOneDegree.of_le_of", "def_path": "Mathlib/Computability/Reduce.lean", "def_pos": [436, 9], "def_end_pos": [436, 17]}, {"full_name": "disjoin_le", "def_path": "Mathlib/Computability/Reduce.lean", "def_pos": [316, 9], "def_end_pos": [316, 19]}]], "state_before": "case h.h.h\n\u03b1 : Type u\ninst\u271d\u2075 : Primcodable \u03b1\ninst\u271d\u2074 : Inhabited \u03b1\n\u03b2 : Type v\ninst\u271d\u00b3 : Primcodable \u03b2\ninst\u271d\u00b2 : Inhabited \u03b2\n\u03b3 : Type w\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Inhabited \u03b3\np\u271d\u00b2 p\u271d\u00b9 p\u271d : Set \u2115\n\u22a2 of p\u271d\u00b2 + of p\u271d\u00b9 \u2264 of p\u271d \u2194 of p\u271d\u00b2 \u2264 of p\u271d \u2227 of p\u271d\u00b9 \u2264 of p\u271d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/NoncommProd.lean", "full_name": "Finset.noncommProd_commute", "start": [335, 1], "end": [341, 15], "traced_tactics": [{"tactic": "apply Multiset.noncommProd_commute", "annotated_tactic": ["apply <a>Multiset.noncommProd_commute</a>", [{"full_name": "Multiset.noncommProd_commute", "def_path": "Mathlib/Data/Finset/NoncommProd.lean", "def_pos": [213, 9], "def_end_pos": [213, 28]}]], "state_before": "F : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b2\ninst\u271d : Monoid \u03b3\ns : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\ncomm : Set.Pairwise \u2191s fun a b => Commute (f a) (f b)\ny : \u03b2\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 Commute y (f x)\n\u22a2 Commute y (noncommProd s f comm)", "state_after": "case h\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b2\ninst\u271d : Monoid \u03b3\ns : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\ncomm : Set.Pairwise \u2191s fun a b => Commute (f a) (f b)\ny : \u03b2\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 Commute y (f x)\n\u22a2 \u2200 (x : \u03b2), x \u2208 Multiset.map f s.val \u2192 Commute y x"}, {"tactic": "intro y", "annotated_tactic": ["intro y", []], "state_before": "case h\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b2\ninst\u271d : Monoid \u03b3\ns : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\ncomm : Set.Pairwise \u2191s fun a b => Commute (f a) (f b)\ny : \u03b2\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 Commute y (f x)\n\u22a2 \u2200 (x : \u03b2), x \u2208 Multiset.map f s.val \u2192 Commute y x", "state_after": "case h\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b2\ninst\u271d : Monoid \u03b3\ns : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\ncomm : Set.Pairwise \u2191s fun a b => Commute (f a) (f b)\ny\u271d : \u03b2\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 Commute y\u271d (f x)\ny : \u03b2\n\u22a2 y \u2208 Multiset.map f s.val \u2192 Commute y\u271d y"}, {"tactic": "rw [Multiset.mem_map]", "annotated_tactic": ["rw [<a>Multiset.mem_map</a>]", [{"full_name": "Multiset.mem_map", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [1228, 9], "def_end_pos": [1228, 16]}]], "state_before": "case h\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b2\ninst\u271d : Monoid \u03b3\ns : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\ncomm : Set.Pairwise \u2191s fun a b => Commute (f a) (f b)\ny\u271d : \u03b2\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 Commute y\u271d (f x)\ny : \u03b2\n\u22a2 y \u2208 Multiset.map f s.val \u2192 Commute y\u271d y", "state_after": "case h\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b2\ninst\u271d : Monoid \u03b3\ns : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\ncomm : Set.Pairwise \u2191s fun a b => Commute (f a) (f b)\ny\u271d : \u03b2\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 Commute y\u271d (f x)\ny : \u03b2\n\u22a2 (\u2203 a, a \u2208 s.val \u2227 f a = y) \u2192 Commute y\u271d y"}, {"tactic": "rintro \u27e8x, \u27e8hx, rfl\u27e9\u27e9", "annotated_tactic": ["rintro \u27e8x, \u27e8hx, rfl\u27e9\u27e9", []], "state_before": "case h\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b2\ninst\u271d : Monoid \u03b3\ns : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\ncomm : Set.Pairwise \u2191s fun a b => Commute (f a) (f b)\ny\u271d : \u03b2\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 Commute y\u271d (f x)\ny : \u03b2\n\u22a2 (\u2203 a, a \u2208 s.val \u2227 f a = y) \u2192 Commute y\u271d y", "state_after": "case h.intro.intro\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b2\ninst\u271d : Monoid \u03b3\ns : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\ncomm : Set.Pairwise \u2191s fun a b => Commute (f a) (f b)\ny : \u03b2\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 Commute y (f x)\nx : \u03b1\nhx : x \u2208 s.val\n\u22a2 Commute y (f x)"}, {"tactic": "exact h x hx", "annotated_tactic": ["exact h x hx", []], "state_before": "case h.intro.intro\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b2\ninst\u271d : Monoid \u03b3\ns : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\ncomm : Set.Pairwise \u2191s fun a b => Commute (f a) (f b)\ny : \u03b2\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 Commute y (f x)\nx : \u03b1\nhx : x \u2208 s.val\n\u22a2 Commute y (f x)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Constructions/Polish.lean", "full_name": "MeasureTheory.measurablySeparable_range_of_disjoint", "start": [397, 1], "end": [506, 14], "traced_tactics": [{"tactic": "by_contra hfg", "annotated_tactic": ["by_contra hfg", []], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\n\u22a2 MeasurablySeparable (range f) (range g)", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\n\u22a2 False"}, {"tactic": "have I : \u2200 n x y, \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n    \u2203 x' y', x' \u2208 cylinder x n \u2227 y' \u2208 cylinder y n \u2227\n    \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1)) := by\n  intro n x y\n  contrapose!\n  intro H\n  rw [\u2190 iUnion_cylinder_update x n, \u2190 iUnion_cylinder_update y n, image_iUnion, image_iUnion]\n  refine' MeasurablySeparable.iUnion fun i j => _\n  exact H _ _ (update_mem_cylinder _ _ _) (update_mem_cylinder _ _ _)", "annotated_tactic": ["have I : \u2200 n x y, \u00ac<a>MeasurablySeparable</a> (f '' <a>cylinder</a> x n) (g '' <a>cylinder</a> y n) \u2192\n      \u2203 x' y', x' \u2208 <a>cylinder</a> x n \u2227 y' \u2208 <a>cylinder</a> y n \u2227\n      \u00ac<a>MeasurablySeparable</a> (f '' <a>cylinder</a> x' (n + 1)) (g '' <a>cylinder</a> y' (n + 1)) := by\n    intro n x y\n    contrapose!\n    intro H\n    rw [\u2190 <a>iUnion_cylinder_update</a> x n, \u2190 <a>iUnion_cylinder_update</a> y n, <a>image_iUnion</a>, <a>image_iUnion</a>]\n    refine' <a>MeasurablySeparable.iUnion</a> fun i j => _\n    exact H _ _ (<a>update_mem_cylinder</a> _ _ _) (<a>update_mem_cylinder</a> _ _ _)", [{"full_name": "MeasureTheory.MeasurablySeparable", "def_path": "Mathlib/MeasureTheory/Constructions/Polish.lean", "def_pos": [375, 5], "def_end_pos": [375, 24]}, {"full_name": "PiNat.cylinder", "def_path": "Mathlib/Topology/MetricSpace/PiNat.lean", "def_pos": [107, 5], "def_end_pos": [107, 13]}, {"full_name": "PiNat.cylinder", "def_path": "Mathlib/Topology/MetricSpace/PiNat.lean", "def_pos": [107, 5], "def_end_pos": [107, 13]}, {"full_name": "PiNat.cylinder", "def_path": "Mathlib/Topology/MetricSpace/PiNat.lean", "def_pos": [107, 5], "def_end_pos": [107, 13]}, {"full_name": "PiNat.cylinder", "def_path": "Mathlib/Topology/MetricSpace/PiNat.lean", "def_pos": [107, 5], "def_end_pos": [107, 13]}, {"full_name": "MeasureTheory.MeasurablySeparable", "def_path": "Mathlib/MeasureTheory/Constructions/Polish.lean", "def_pos": [375, 5], "def_end_pos": [375, 24]}, {"full_name": "PiNat.cylinder", "def_path": "Mathlib/Topology/MetricSpace/PiNat.lean", "def_pos": [107, 5], "def_end_pos": [107, 13]}, {"full_name": "PiNat.cylinder", "def_path": "Mathlib/Topology/MetricSpace/PiNat.lean", "def_pos": [107, 5], "def_end_pos": [107, 13]}, {"full_name": "PiNat.iUnion_cylinder_update", "def_path": "Mathlib/Topology/MetricSpace/PiNat.lean", "def_pos": [174, 9], "def_end_pos": [174, 31]}, {"full_name": "PiNat.iUnion_cylinder_update", "def_path": "Mathlib/Topology/MetricSpace/PiNat.lean", "def_pos": [174, 9], "def_end_pos": [174, 31]}, {"full_name": "Set.image_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [1791, 9], "def_end_pos": [1791, 21]}, {"full_name": "Set.image_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [1791, 9], "def_end_pos": [1791, 21]}, {"full_name": "MeasureTheory.MeasurablySeparable.iUnion", "def_path": "Mathlib/MeasureTheory/Constructions/Polish.lean", "def_pos": [379, 9], "def_end_pos": [379, 35]}, {"full_name": "PiNat.update_mem_cylinder", "def_path": "Mathlib/Topology/MetricSpace/PiNat.lean", "def_pos": [188, 9], "def_end_pos": [188, 28]}, {"full_name": "PiNat.update_mem_cylinder", "def_path": "Mathlib/Topology/MetricSpace/PiNat.lean", "def_pos": [188, 9], "def_end_pos": [188, 28]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\n\u22a2 False", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\n\u22a2 False"}, {"tactic": "let A :=\n  { p : \u2115 \u00d7 (\u2115 \u2192 \u2115) \u00d7 (\u2115 \u2192 \u2115) //\n    \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }", "annotated_tactic": ["let A :=\n    { p : \u2115 \u00d7 (\u2115 \u2192 \u2115) \u00d7 (\u2115 \u2192 \u2115) //\n      \u00ac<a>MeasurablySeparable</a> (f '' <a>cylinder</a> p.2.1 p.1) (g '' <a>cylinder</a> p.2.2 p.1) }", [{"full_name": "MeasureTheory.MeasurablySeparable", "def_path": "Mathlib/MeasureTheory/Constructions/Polish.lean", "def_pos": [375, 5], "def_end_pos": [375, 24]}, {"full_name": "PiNat.cylinder", "def_path": "Mathlib/Topology/MetricSpace/PiNat.lean", "def_pos": [107, 5], "def_end_pos": [107, 13]}, {"full_name": "PiNat.cylinder", "def_path": "Mathlib/Topology/MetricSpace/PiNat.lean", "def_pos": [107, 5], "def_end_pos": [107, 13]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\n\u22a2 False", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\n\u22a2 False"}, {"tactic": "have : \u2200 p : A, \u2203 q : A,\n    q.1.1 = p.1.1 + 1 \u2227 q.1.2.1 \u2208 cylinder p.1.2.1 p.1.1 \u2227 q.1.2.2 \u2208 cylinder p.1.2.2 p.1.1 := by\n  rintro \u27e8\u27e8n, x, y\u27e9, hp\u27e9\n  rcases I n x y hp with \u27e8x', y', hx', hy', h'\u27e9\n  exact \u27e8\u27e8\u27e8n + 1, x', y'\u27e9, h'\u27e9, rfl, hx', hy'\u27e9", "annotated_tactic": ["have : \u2200 p : A, \u2203 q : A,\n      q.1.1 = p.1.1 + 1 \u2227 q.1.2.1 \u2208 <a>cylinder</a> p.1.2.1 p.1.1 \u2227 q.1.2.2 \u2208 <a>cylinder</a> p.1.2.2 p.1.1 := by\n    rintro \u27e8\u27e8n, x, y\u27e9, hp\u27e9\n    rcases I n x y hp with \u27e8x', y', hx', hy', h'\u27e9\n    exact \u27e8\u27e8\u27e8n + 1, x', y'\u27e9, h'\u27e9, <a>rfl</a>, hx', hy'\u27e9", [{"full_name": "PiNat.cylinder", "def_path": "Mathlib/Topology/MetricSpace/PiNat.lean", "def_pos": [107, 5], "def_end_pos": [107, 13]}, {"full_name": "PiNat.cylinder", "def_path": "Mathlib/Topology/MetricSpace/PiNat.lean", "def_pos": [107, 5], "def_end_pos": [107, 13]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\n\u22a2 False", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nthis : \u2200 (p : A), \u2203 q, (\u2191q).1 = (\u2191p).1 + 1 \u2227 (\u2191q).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1 \u2227 (\u2191q).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\n\u22a2 False"}, {"tactic": "choose F hFn hFx hFy using this", "annotated_tactic": ["choose F hFn hFx hFy using this", []], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nthis : \u2200 (p : A), \u2203 q, (\u2191q).1 = (\u2191p).1 + 1 \u2227 (\u2191q).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1 \u2227 (\u2191q).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\n\u22a2 False", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\n\u22a2 False"}, {"tactic": "let p0 : A := \u27e8\u27e80, fun _ => 0, fun _ => 0\u27e9, by simp [hfg]\u27e9", "annotated_tactic": ["let p0 : A := \u27e8\u27e80, fun _ => 0, fun _ => 0\u27e9, by simp [hfg]\u27e9", []], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\n\u22a2 False", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\n\u22a2 False"}, {"tactic": "let p : \u2115 \u2192 A := fun n => F^[n] p0", "annotated_tactic": ["let p : \u2115 \u2192 A := fun n => F^[n] p0", []], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\n\u22a2 False", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\n\u22a2 False"}, {"tactic": "have prec : \u2200 n, p (n + 1) = F (p n) := fun n => by simp only [iterate_succ', Function.comp]", "annotated_tactic": ["have prec : \u2200 n, p (n + 1) = F (p n) := fun n => by simp only [<a>iterate_succ'</a>, <a>Function.comp</a>]", [{"full_name": "Function.iterate_succ'", "def_path": "Mathlib/Logic/Function/Iterate.lean", "def_pos": [186, 9], "def_end_pos": [186, 22]}, {"full_name": "Function.comp", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [52, 15], "def_end_pos": [52, 28]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\n\u22a2 False", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\n\u22a2 False"}, {"tactic": "set x : \u2115 \u2192 \u2115 := fun n => (p (n + 1)).1.2.1 n with hx", "annotated_tactic": ["set x : \u2115 \u2192 \u2115 := fun n => (p (n + 1)).1.2.1 n with hx", []], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m\nIy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m\n\u22a2 False", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m\nIy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m\nx : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.1 n\nhx : x = fun n => (\u2191(p (n + 1))).2.1 n\n\u22a2 False"}, {"tactic": "set y : \u2115 \u2192 \u2115 := fun n => (p (n + 1)).1.2.2 n with hy", "annotated_tactic": ["set y : \u2115 \u2192 \u2115 := fun n => (p (n + 1)).1.2.2 n with hy", []], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m\nIy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m\nx : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.1 n\nhx : x = fun n => (\u2191(p (n + 1))).2.1 n\n\u22a2 False", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m\nIy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m\nx : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.1 n\nhx : x = fun n => (\u2191(p (n + 1))).2.1 n\ny : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.2 n\nhy : y = fun n => (\u2191(p (n + 1))).2.2 n\n\u22a2 False"}, {"tactic": "obtain \u27e8u, v, u_open, v_open, xu, yv, huv\u27e9 :\n  \u2203 u v : Set \u03b1, IsOpen u \u2227 IsOpen v \u2227 f x \u2208 u \u2227 g y \u2208 v \u2227 Disjoint u v := by\n  apply t2_separation\n  exact disjoint_iff_forall_ne.1 h (mem_range_self _) (mem_range_self _)", "annotated_tactic": ["obtain \u27e8u, v, u_open, v_open, xu, yv, huv\u27e9 :\n    \u2203 u v : <a>Set</a> \u03b1, <a>IsOpen</a> u \u2227 <a>IsOpen</a> v \u2227 f x \u2208 u \u2227 g y \u2208 v \u2227 <a>Disjoint</a> u v := by\n    apply <a>t2_separation</a>\n    exact <a>disjoint_iff_forall_ne</a>.1 h (<a>mem_range_self</a> _) (<a>mem_range_self</a> _)", [{"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}, {"full_name": "IsOpen", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [101, 5], "def_end_pos": [101, 11]}, {"full_name": "IsOpen", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [101, 5], "def_end_pos": [101, 11]}, {"full_name": "Disjoint", "def_path": "Mathlib/Order/Disjoint.lean", "def_pos": [41, 5], "def_end_pos": [41, 13]}, {"full_name": "t2_separation", "def_path": "Mathlib/Topology/Separation.lean", "def_pos": [906, 9], "def_end_pos": [906, 22]}, {"full_name": "Set.disjoint_iff_forall_ne", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1567, 7], "def_end_pos": [1567, 29]}, {"full_name": "Set.mem_range_self", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [680, 9], "def_end_pos": [680, 23]}, {"full_name": "Set.mem_range_self", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [680, 9], "def_end_pos": [680, 23]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m\nIy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m\nx : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.1 n\nhx : x = fun n => (\u2191(p (n + 1))).2.1 n\ny : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.2 n\nhy : y = fun n => (\u2191(p (n + 1))).2.2 n\nM : \u2200 (n : \u2115), \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\n\u22a2 False", "state_after": "case intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m\nIy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m\nx : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.1 n\nhx : x = fun n => (\u2191(p (n + 1))).2.1 n\ny : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.2 n\nhy : y = fun n => (\u2191(p (n + 1))).2.2 n\nM : \u2200 (n : \u2115), \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\nu v : Set \u03b1\nu_open : IsOpen u\nv_open : IsOpen v\nxu : f x \u2208 u\nyv : g y \u2208 v\nhuv : Disjoint u v\n\u22a2 False"}, {"tactic": "letI : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat", "annotated_tactic": ["letI : <a>MetricSpace</a> (\u2115 \u2192 \u2115) := <a>metricSpaceNatNat</a>", [{"full_name": "MetricSpace", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [2892, 7], "def_end_pos": [2892, 18]}, {"full_name": "PiNat.metricSpaceNatNat", "def_path": "Mathlib/Topology/MetricSpace/PiNat.lean", "def_pos": [462, 5], "def_end_pos": [462, 22]}]], "state_before": "case intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m\nIy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m\nx : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.1 n\nhx : x = fun n => (\u2191(p (n + 1))).2.1 n\ny : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.2 n\nhy : y = fun n => (\u2191(p (n + 1))).2.2 n\nM : \u2200 (n : \u2115), \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\nu v : Set \u03b1\nu_open : IsOpen u\nv_open : IsOpen v\nxu : f x \u2208 u\nyv : g y \u2208 v\nhuv : Disjoint u v\n\u22a2 False", "state_after": "case intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m\nIy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m\nx : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.1 n\nhx : x = fun n => (\u2191(p (n + 1))).2.1 n\ny : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.2 n\nhy : y = fun n => (\u2191(p (n + 1))).2.2 n\nM : \u2200 (n : \u2115), \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\nu v : Set \u03b1\nu_open : IsOpen u\nv_open : IsOpen v\nxu : f x \u2208 u\nyv : g y \u2208 v\nhuv : Disjoint u v\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\n\u22a2 False"}, {"tactic": "obtain \u27e8\u03b5x, \u03b5xpos, h\u03b5x\u27e9 : \u2203 (\u03b5x : \u211d), \u03b5x > 0 \u2227 Metric.ball x \u03b5x \u2286 f \u207b\u00b9' u := by\n  apply Metric.mem_nhds_iff.1\n  exact hf.continuousAt.preimage_mem_nhds (u_open.mem_nhds xu)", "annotated_tactic": ["obtain \u27e8\u03b5x, \u03b5xpos, h\u03b5x\u27e9 : \u2203 (\u03b5x : \u211d), \u03b5x > 0 \u2227 <a>Metric.ball</a> x \u03b5x \u2286 f \u207b\u00b9' u := by\n    apply <a>Metric.mem_nhds_iff</a>.1\n    exact hf.continuousAt.preimage_mem_nhds (u_open.mem_nhds xu)", [{"full_name": "Metric.ball", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [409, 5], "def_end_pos": [409, 9]}, {"full_name": "Metric.mem_nhds_iff", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [954, 9], "def_end_pos": [954, 21]}]], "state_before": "case intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m\nIy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m\nx : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.1 n\nhx : x = fun n => (\u2191(p (n + 1))).2.1 n\ny : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.2 n\nhy : y = fun n => (\u2191(p (n + 1))).2.2 n\nM : \u2200 (n : \u2115), \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\nu v : Set \u03b1\nu_open : IsOpen u\nv_open : IsOpen v\nxu : f x \u2208 u\nyv : g y \u2208 v\nhuv : Disjoint u v\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\n\u22a2 False", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m\nIy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m\nx : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.1 n\nhx : x = fun n => (\u2191(p (n + 1))).2.1 n\ny : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.2 n\nhy : y = fun n => (\u2191(p (n + 1))).2.2 n\nM : \u2200 (n : \u2115), \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\nu v : Set \u03b1\nu_open : IsOpen u\nv_open : IsOpen v\nxu : f x \u2208 u\nyv : g y \u2208 v\nhuv : Disjoint u v\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\n\u03b5x : \u211d\n\u03b5xpos : \u03b5x > 0\nh\u03b5x : ball x \u03b5x \u2286 f \u207b\u00b9' u\n\u22a2 False"}, {"tactic": "obtain \u27e8\u03b5y, \u03b5ypos, h\u03b5y\u27e9 : \u2203 (\u03b5y : \u211d), \u03b5y > 0 \u2227 Metric.ball y \u03b5y \u2286 g \u207b\u00b9' v := by\n  apply Metric.mem_nhds_iff.1\n  exact hg.continuousAt.preimage_mem_nhds (v_open.mem_nhds yv)", "annotated_tactic": ["obtain \u27e8\u03b5y, \u03b5ypos, h\u03b5y\u27e9 : \u2203 (\u03b5y : \u211d), \u03b5y > 0 \u2227 <a>Metric.ball</a> y \u03b5y \u2286 g \u207b\u00b9' v := by\n    apply <a>Metric.mem_nhds_iff</a>.1\n    exact hg.continuousAt.preimage_mem_nhds (v_open.mem_nhds yv)", [{"full_name": "Metric.ball", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [409, 5], "def_end_pos": [409, 9]}, {"full_name": "Metric.mem_nhds_iff", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [954, 9], "def_end_pos": [954, 21]}]], "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m\nIy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m\nx : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.1 n\nhx : x = fun n => (\u2191(p (n + 1))).2.1 n\ny : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.2 n\nhy : y = fun n => (\u2191(p (n + 1))).2.2 n\nM : \u2200 (n : \u2115), \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\nu v : Set \u03b1\nu_open : IsOpen u\nv_open : IsOpen v\nxu : f x \u2208 u\nyv : g y \u2208 v\nhuv : Disjoint u v\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\n\u03b5x : \u211d\n\u03b5xpos : \u03b5x > 0\nh\u03b5x : ball x \u03b5x \u2286 f \u207b\u00b9' u\n\u22a2 False", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m\nIy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m\nx : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.1 n\nhx : x = fun n => (\u2191(p (n + 1))).2.1 n\ny : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.2 n\nhy : y = fun n => (\u2191(p (n + 1))).2.2 n\nM : \u2200 (n : \u2115), \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\nu v : Set \u03b1\nu_open : IsOpen u\nv_open : IsOpen v\nxu : f x \u2208 u\nyv : g y \u2208 v\nhuv : Disjoint u v\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\n\u03b5x : \u211d\n\u03b5xpos : \u03b5x > 0\nh\u03b5x : ball x \u03b5x \u2286 f \u207b\u00b9' u\n\u03b5y : \u211d\n\u03b5ypos : \u03b5y > 0\nh\u03b5y : ball y \u03b5y \u2286 g \u207b\u00b9' v\n\u22a2 False"}, {"tactic": "obtain \u27e8n, hn\u27e9 : \u2203 n : \u2115, (1 / 2 : \u211d) ^ n < min \u03b5x \u03b5y :=\n  exists_pow_lt_of_lt_one (lt_min \u03b5xpos \u03b5ypos) (by norm_num)", "annotated_tactic": ["obtain \u27e8n, hn\u27e9 : \u2203 n : \u2115, (1 / 2 : \u211d) ^ n < <a>min</a> \u03b5x \u03b5y :=\n    <a>exists_pow_lt_of_lt_one</a> (<a>lt_min</a> \u03b5xpos \u03b5ypos) (by norm_num)", [{"full_name": "Min.min", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1103, 3], "def_end_pos": [1103, 6]}, {"full_name": "exists_pow_lt_of_lt_one", "def_path": "Mathlib/Algebra/Order/Archimedean.lean", "def_pos": [232, 9], "def_end_pos": [232, 32]}, {"full_name": "lt_min", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [159, 9], "def_end_pos": [159, 15]}]], "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m\nIy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m\nx : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.1 n\nhx : x = fun n => (\u2191(p (n + 1))).2.1 n\ny : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.2 n\nhy : y = fun n => (\u2191(p (n + 1))).2.2 n\nM : \u2200 (n : \u2115), \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\nu v : Set \u03b1\nu_open : IsOpen u\nv_open : IsOpen v\nxu : f x \u2208 u\nyv : g y \u2208 v\nhuv : Disjoint u v\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\n\u03b5x : \u211d\n\u03b5xpos : \u03b5x > 0\nh\u03b5x : ball x \u03b5x \u2286 f \u207b\u00b9' u\n\u03b5y : \u211d\n\u03b5ypos : \u03b5y > 0\nh\u03b5y : ball y \u03b5y \u2286 g \u207b\u00b9' v\n\u22a2 False", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m\nIy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m\nx : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.1 n\nhx : x = fun n => (\u2191(p (n + 1))).2.1 n\ny : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.2 n\nhy : y = fun n => (\u2191(p (n + 1))).2.2 n\nM : \u2200 (n : \u2115), \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\nu v : Set \u03b1\nu_open : IsOpen u\nv_open : IsOpen v\nxu : f x \u2208 u\nyv : g y \u2208 v\nhuv : Disjoint u v\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\n\u03b5x : \u211d\n\u03b5xpos : \u03b5x > 0\nh\u03b5x : ball x \u03b5x \u2286 f \u207b\u00b9' u\n\u03b5y : \u211d\n\u03b5ypos : \u03b5y > 0\nh\u03b5y : ball y \u03b5y \u2286 g \u207b\u00b9' v\nn : \u2115\nhn : (1 / 2) ^ n < min \u03b5x \u03b5y\n\u22a2 False"}, {"tactic": "exact M n B", "annotated_tactic": ["exact M n B", []], "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m\nIy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m\nx : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.1 n\nhx : x = fun n => (\u2191(p (n + 1))).2.1 n\ny : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.2 n\nhy : y = fun n => (\u2191(p (n + 1))).2.2 n\nM : \u2200 (n : \u2115), \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\nu v : Set \u03b1\nu_open : IsOpen u\nv_open : IsOpen v\nxu : f x \u2208 u\nyv : g y \u2208 v\nhuv : Disjoint u v\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\n\u03b5x : \u211d\n\u03b5xpos : \u03b5x > 0\nh\u03b5x : ball x \u03b5x \u2286 f \u207b\u00b9' u\n\u03b5y : \u211d\n\u03b5ypos : \u03b5y > 0\nh\u03b5y : ball y \u03b5y \u2286 g \u207b\u00b9' v\nn : \u2115\nhn : (1 / 2) ^ n < min \u03b5x \u03b5y\nB : MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\n\u22a2 False", "state_after": "no goals"}, {"tactic": "intro n x y", "annotated_tactic": ["intro n x y", []], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\n\u22a2 \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nn : \u2115\nx y : \u2115 \u2192 \u2115\n\u22a2 \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n    \u2203 x' y',\n      x' \u2208 cylinder x n \u2227 y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))"}, {"tactic": "contrapose!", "annotated_tactic": ["contrapose!", []], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nn : \u2115\nx y : \u2115 \u2192 \u2115\n\u22a2 \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n    \u2203 x' y',\n      x' \u2208 cylinder x n \u2227 y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nn : \u2115\nx y : \u2115 \u2192 \u2115\n\u22a2 (\u2200 (x' y' : \u2115 \u2192 \u2115),\n      x' \u2208 cylinder x n \u2192\n        y' \u2208 cylinder y n \u2192 MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))) \u2192\n    MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)"}, {"tactic": "intro H", "annotated_tactic": ["intro H", []], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nn : \u2115\nx y : \u2115 \u2192 \u2115\n\u22a2 (\u2200 (x' y' : \u2115 \u2192 \u2115),\n      x' \u2208 cylinder x n \u2192\n        y' \u2208 cylinder y n \u2192 MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))) \u2192\n    MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nn : \u2115\nx y : \u2115 \u2192 \u2115\nH :\n  \u2200 (x' y' : \u2115 \u2192 \u2115),\n    x' \u2208 cylinder x n \u2192 y' \u2208 cylinder y n \u2192 MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\n\u22a2 MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)"}, {"tactic": "rw [\u2190 iUnion_cylinder_update x n, \u2190 iUnion_cylinder_update y n, image_iUnion, image_iUnion]", "annotated_tactic": ["rw [\u2190 <a>iUnion_cylinder_update</a> x n, \u2190 <a>iUnion_cylinder_update</a> y n, <a>image_iUnion</a>, <a>image_iUnion</a>]", [{"full_name": "PiNat.iUnion_cylinder_update", "def_path": "Mathlib/Topology/MetricSpace/PiNat.lean", "def_pos": [174, 9], "def_end_pos": [174, 31]}, {"full_name": "PiNat.iUnion_cylinder_update", "def_path": "Mathlib/Topology/MetricSpace/PiNat.lean", "def_pos": [174, 9], "def_end_pos": [174, 31]}, {"full_name": "Set.image_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [1791, 9], "def_end_pos": [1791, 21]}, {"full_name": "Set.image_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [1791, 9], "def_end_pos": [1791, 21]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nn : \u2115\nx y : \u2115 \u2192 \u2115\nH :\n  \u2200 (x' y' : \u2115 \u2192 \u2115),\n    x' \u2208 cylinder x n \u2192 y' \u2208 cylinder y n \u2192 MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\n\u22a2 MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nn : \u2115\nx y : \u2115 \u2192 \u2115\nH :\n  \u2200 (x' y' : \u2115 \u2192 \u2115),\n    x' \u2208 cylinder x n \u2192 y' \u2208 cylinder y n \u2192 MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\n\u22a2 MeasurablySeparable (\u22c3 i, f '' cylinder (update x n i) (n + 1)) (\u22c3 i, g '' cylinder (update y n i) (n + 1))"}, {"tactic": "refine' MeasurablySeparable.iUnion fun i j => _", "annotated_tactic": ["refine' <a>MeasurablySeparable.iUnion</a> fun i j => _", [{"full_name": "MeasureTheory.MeasurablySeparable.iUnion", "def_path": "Mathlib/MeasureTheory/Constructions/Polish.lean", "def_pos": [379, 9], "def_end_pos": [379, 35]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nn : \u2115\nx y : \u2115 \u2192 \u2115\nH :\n  \u2200 (x' y' : \u2115 \u2192 \u2115),\n    x' \u2208 cylinder x n \u2192 y' \u2208 cylinder y n \u2192 MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\n\u22a2 MeasurablySeparable (\u22c3 i, f '' cylinder (update x n i) (n + 1)) (\u22c3 i, g '' cylinder (update y n i) (n + 1))", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nn : \u2115\nx y : \u2115 \u2192 \u2115\nH :\n  \u2200 (x' y' : \u2115 \u2192 \u2115),\n    x' \u2208 cylinder x n \u2192 y' \u2208 cylinder y n \u2192 MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\ni j : \u2115\n\u22a2 MeasurablySeparable (f '' cylinder (update x n i) (n + 1)) (g '' cylinder (update y n j) (n + 1))"}, {"tactic": "exact H _ _ (update_mem_cylinder _ _ _) (update_mem_cylinder _ _ _)", "annotated_tactic": ["exact H _ _ (<a>update_mem_cylinder</a> _ _ _) (<a>update_mem_cylinder</a> _ _ _)", [{"full_name": "PiNat.update_mem_cylinder", "def_path": "Mathlib/Topology/MetricSpace/PiNat.lean", "def_pos": [188, 9], "def_end_pos": [188, 28]}, {"full_name": "PiNat.update_mem_cylinder", "def_path": "Mathlib/Topology/MetricSpace/PiNat.lean", "def_pos": [188, 9], "def_end_pos": [188, 28]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nn : \u2115\nx y : \u2115 \u2192 \u2115\nH :\n  \u2200 (x' y' : \u2115 \u2192 \u2115),\n    x' \u2208 cylinder x n \u2192 y' \u2208 cylinder y n \u2192 MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\ni j : \u2115\n\u22a2 MeasurablySeparable (f '' cylinder (update x n i) (n + 1)) (g '' cylinder (update y n j) (n + 1))", "state_after": "no goals"}, {"tactic": "rintro \u27e8\u27e8n, x, y\u27e9, hp\u27e9", "annotated_tactic": ["rintro \u27e8\u27e8n, x, y\u27e9, hp\u27e9", []], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\n\u22a2 \u2200 (p : A), \u2203 q, (\u2191q).1 = (\u2191p).1 + 1 \u2227 (\u2191q).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1 \u2227 (\u2191q).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1", "state_after": "case mk.mk.mk\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nn : \u2115\nx y : \u2115 \u2192 \u2115\nhp : \u00acMeasurablySeparable (f '' cylinder (n, x, y).2.1 (n, x, y).1) (g '' cylinder (n, x, y).2.2 (n, x, y).1)\n\u22a2 \u2203 q,\n    (\u2191q).1 = (\u2191{ val := (n, x, y), property := hp }).1 + 1 \u2227\n      (\u2191q).2.1 \u2208 cylinder (\u2191{ val := (n, x, y), property := hp }).2.1 (\u2191{ val := (n, x, y), property := hp }).1 \u2227\n        (\u2191q).2.2 \u2208 cylinder (\u2191{ val := (n, x, y), property := hp }).2.2 (\u2191{ val := (n, x, y), property := hp }).1"}, {"tactic": "rcases I n x y hp with \u27e8x', y', hx', hy', h'\u27e9", "annotated_tactic": ["rcases I n x y hp with \u27e8x', y', hx', hy', h'\u27e9", []], "state_before": "case mk.mk.mk\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nn : \u2115\nx y : \u2115 \u2192 \u2115\nhp : \u00acMeasurablySeparable (f '' cylinder (n, x, y).2.1 (n, x, y).1) (g '' cylinder (n, x, y).2.2 (n, x, y).1)\n\u22a2 \u2203 q,\n    (\u2191q).1 = (\u2191{ val := (n, x, y), property := hp }).1 + 1 \u2227\n      (\u2191q).2.1 \u2208 cylinder (\u2191{ val := (n, x, y), property := hp }).2.1 (\u2191{ val := (n, x, y), property := hp }).1 \u2227\n        (\u2191q).2.2 \u2208 cylinder (\u2191{ val := (n, x, y), property := hp }).2.2 (\u2191{ val := (n, x, y), property := hp }).1", "state_after": "case mk.mk.mk.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nn : \u2115\nx y : \u2115 \u2192 \u2115\nhp : \u00acMeasurablySeparable (f '' cylinder (n, x, y).2.1 (n, x, y).1) (g '' cylinder (n, x, y).2.2 (n, x, y).1)\nx' y' : \u2115 \u2192 \u2115\nhx' : x' \u2208 cylinder x n\nhy' : y' \u2208 cylinder y n\nh' : \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\n\u22a2 \u2203 q,\n    (\u2191q).1 = (\u2191{ val := (n, x, y), property := hp }).1 + 1 \u2227\n      (\u2191q).2.1 \u2208 cylinder (\u2191{ val := (n, x, y), property := hp }).2.1 (\u2191{ val := (n, x, y), property := hp }).1 \u2227\n        (\u2191q).2.2 \u2208 cylinder (\u2191{ val := (n, x, y), property := hp }).2.2 (\u2191{ val := (n, x, y), property := hp }).1"}, {"tactic": "exact \u27e8\u27e8\u27e8n + 1, x', y'\u27e9, h'\u27e9, rfl, hx', hy'\u27e9", "annotated_tactic": ["exact \u27e8\u27e8\u27e8n + 1, x', y'\u27e9, h'\u27e9, <a>rfl</a>, hx', hy'\u27e9", [{"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "case mk.mk.mk.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nn : \u2115\nx y : \u2115 \u2192 \u2115\nhp : \u00acMeasurablySeparable (f '' cylinder (n, x, y).2.1 (n, x, y).1) (g '' cylinder (n, x, y).2.2 (n, x, y).1)\nx' y' : \u2115 \u2192 \u2115\nhx' : x' \u2208 cylinder x n\nhy' : y' \u2208 cylinder y n\nh' : \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\n\u22a2 \u2203 q,\n    (\u2191q).1 = (\u2191{ val := (n, x, y), property := hp }).1 + 1 \u2227\n      (\u2191q).2.1 \u2208 cylinder (\u2191{ val := (n, x, y), property := hp }).2.1 (\u2191{ val := (n, x, y), property := hp }).1 \u2227\n        (\u2191q).2.2 \u2208 cylinder (\u2191{ val := (n, x, y), property := hp }).2.2 (\u2191{ val := (n, x, y), property := hp }).1", "state_after": "no goals"}, {"tactic": "simp [hfg]", "annotated_tactic": ["simp [hfg]", []], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\n\u22a2 \u00acMeasurablySeparable (f '' cylinder (0, fun x => 0, fun x => 0).2.1 (0, fun x => 0, fun x => 0).1)\n      (g '' cylinder (0, fun x => 0, fun x => 0).2.2 (0, fun x => 0, fun x => 0).1)", "state_after": "no goals"}, {"tactic": "simp only [iterate_succ', Function.comp]", "annotated_tactic": ["simp only [<a>iterate_succ'</a>, <a>Function.comp</a>]", [{"full_name": "Function.iterate_succ'", "def_path": "Mathlib/Logic/Function/Iterate.lean", "def_pos": [186, 9], "def_end_pos": [186, 22]}, {"full_name": "Function.comp", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [52, 15], "def_end_pos": [52, 28]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nn : \u2115\n\u22a2 p (n + 1) = F (p n)", "state_after": "no goals"}, {"tactic": "intro n", "annotated_tactic": ["intro n", []], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\n\u22a2 \u2200 (n : \u2115), (\u2191(p n)).1 = n", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\nn : \u2115\n\u22a2 (\u2191(p n)).1 = n"}, {"tactic": "induction' n with n IH", "annotated_tactic": ["induction' n with n IH", []], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\nn : \u2115\n\u22a2 (\u2191(p n)).1 = n", "state_after": "case zero\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\n\u22a2 (\u2191(p Nat.zero)).1 = Nat.zero\n\ncase succ\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\nn : \u2115\nIH : (\u2191(p n)).1 = n\n\u22a2 (\u2191(p (Nat.succ n))).1 = Nat.succ n"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case zero\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\n\u22a2 (\u2191(p Nat.zero)).1 = Nat.zero", "state_after": "no goals"}, {"tactic": "simp only [prec, hFn, IH]", "annotated_tactic": ["simp only [prec, hFn, IH]", []], "state_before": "case succ\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\nn : \u2115\nIH : (\u2191(p n)).1 = n\n\u22a2 (\u2191(p (Nat.succ n))).1 = Nat.succ n", "state_after": "no goals"}, {"tactic": "intro m", "annotated_tactic": ["intro m", []], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n\n\u22a2 \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n\nm : \u2115\n\u22a2 \u2200 (n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m"}, {"tactic": "apply Nat.le_induction", "annotated_tactic": ["apply <a>Nat.le_induction</a>", [{"full_name": "Nat.le_induction", "def_path": "Mathlib/Data/Nat/Basic.lean", "def_pos": [509, 9], "def_end_pos": [509, 21]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n\nm : \u2115\n\u22a2 \u2200 (n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m", "state_after": "case base\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n\nm : \u2115\n\u22a2 (\u2191(p (m + 1))).2.1 m = (\u2191(p (m + 1))).2.1 m\n\ncase succ\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n\nm : \u2115\n\u22a2 \u2200 (n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m \u2192 (\u2191(p (n + 1))).2.1 m = (\u2191(p (m + 1))).2.1 m"}, {"tactic": "intro n hmn IH", "annotated_tactic": ["intro n hmn IH", []], "state_before": "case succ\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n\nm : \u2115\n\u22a2 \u2200 (n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m \u2192 (\u2191(p (n + 1))).2.1 m = (\u2191(p (m + 1))).2.1 m", "state_after": "case succ\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n\nm n : \u2115\nhmn : m + 1 \u2264 n\nIH : (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m\n\u22a2 (\u2191(p (n + 1))).2.1 m = (\u2191(p (m + 1))).2.1 m"}, {"tactic": "have I : (F (p n)).val.snd.fst m = (p n).val.snd.fst m := by\n  apply hFx (p n) m\n  rw [pn_fst]\n  exact hmn", "annotated_tactic": ["have I : (F (p n)).val.snd.fst m = (p n).val.snd.fst m := by\n      apply hFx (p n) m\n      rw [pn_fst]\n      exact hmn", []], "state_before": "case succ\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n\nm n : \u2115\nhmn : m + 1 \u2264 n\nIH : (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m\n\u22a2 (\u2191(p (n + 1))).2.1 m = (\u2191(p (m + 1))).2.1 m", "state_after": "case succ\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI\u271d :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n\nm n : \u2115\nhmn : m + 1 \u2264 n\nIH : (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m\nI : (\u2191(F (p n))).2.1 m = (\u2191(p n)).2.1 m\n\u22a2 (\u2191(p (n + 1))).2.1 m = (\u2191(p (m + 1))).2.1 m"}, {"tactic": "rw [prec, I, IH]", "annotated_tactic": ["rw [prec, I, IH]", []], "state_before": "case succ\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI\u271d :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n\nm n : \u2115\nhmn : m + 1 \u2264 n\nIH : (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m\nI : (\u2191(F (p n))).2.1 m = (\u2191(p n)).2.1 m\n\u22a2 (\u2191(p (n + 1))).2.1 m = (\u2191(p (m + 1))).2.1 m", "state_after": "no goals"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case base\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n\nm : \u2115\n\u22a2 (\u2191(p (m + 1))).2.1 m = (\u2191(p (m + 1))).2.1 m", "state_after": "no goals"}, {"tactic": "apply hFx (p n) m", "annotated_tactic": ["apply hFx (p n) m", []], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n\nm n : \u2115\nhmn : m + 1 \u2264 n\nIH : (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m\n\u22a2 (\u2191(F (p n))).2.1 m = (\u2191(p n)).2.1 m", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n\nm n : \u2115\nhmn : m + 1 \u2264 n\nIH : (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m\n\u22a2 m < (\u2191(p n)).1"}, {"tactic": "rw [pn_fst]", "annotated_tactic": ["rw [pn_fst]", []], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n\nm n : \u2115\nhmn : m + 1 \u2264 n\nIH : (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m\n\u22a2 m < (\u2191(p n)).1", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n\nm n : \u2115\nhmn : m + 1 \u2264 n\nIH : (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m\n\u22a2 m < n"}, {"tactic": "exact hmn", "annotated_tactic": ["exact hmn", []], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n\nm n : \u2115\nhmn : m + 1 \u2264 n\nIH : (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m\n\u22a2 m < n", "state_after": "no goals"}, {"tactic": "intro m", "annotated_tactic": ["intro m", []], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m\n\u22a2 \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m\nm : \u2115\n\u22a2 \u2200 (n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m"}, {"tactic": "apply Nat.le_induction", "annotated_tactic": ["apply <a>Nat.le_induction</a>", [{"full_name": "Nat.le_induction", "def_path": "Mathlib/Data/Nat/Basic.lean", "def_pos": [509, 9], "def_end_pos": [509, 21]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m\nm : \u2115\n\u22a2 \u2200 (n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m", "state_after": "case base\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m\nm : \u2115\n\u22a2 (\u2191(p (m + 1))).2.2 m = (\u2191(p (m + 1))).2.2 m\n\ncase succ\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m\nm : \u2115\n\u22a2 \u2200 (n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m \u2192 (\u2191(p (n + 1))).2.2 m = (\u2191(p (m + 1))).2.2 m"}, {"tactic": "intro n hmn IH", "annotated_tactic": ["intro n hmn IH", []], "state_before": "case succ\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m\nm : \u2115\n\u22a2 \u2200 (n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m \u2192 (\u2191(p (n + 1))).2.2 m = (\u2191(p (m + 1))).2.2 m", "state_after": "case succ\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m\nm n : \u2115\nhmn : m + 1 \u2264 n\nIH : (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m\n\u22a2 (\u2191(p (n + 1))).2.2 m = (\u2191(p (m + 1))).2.2 m"}, {"tactic": "have I : (F (p n)).val.snd.snd m = (p n).val.snd.snd m := by\n  apply hFy (p n) m\n  rw [pn_fst]\n  exact hmn", "annotated_tactic": ["have I : (F (p n)).val.snd.snd m = (p n).val.snd.snd m := by\n      apply hFy (p n) m\n      rw [pn_fst]\n      exact hmn", []], "state_before": "case succ\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m\nm n : \u2115\nhmn : m + 1 \u2264 n\nIH : (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m\n\u22a2 (\u2191(p (n + 1))).2.2 m = (\u2191(p (m + 1))).2.2 m", "state_after": "case succ\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI\u271d :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m\nm n : \u2115\nhmn : m + 1 \u2264 n\nIH : (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m\nI : (\u2191(F (p n))).2.2 m = (\u2191(p n)).2.2 m\n\u22a2 (\u2191(p (n + 1))).2.2 m = (\u2191(p (m + 1))).2.2 m"}, {"tactic": "rw [prec, I, IH]", "annotated_tactic": ["rw [prec, I, IH]", []], "state_before": "case succ\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI\u271d :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m\nm n : \u2115\nhmn : m + 1 \u2264 n\nIH : (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m\nI : (\u2191(F (p n))).2.2 m = (\u2191(p n)).2.2 m\n\u22a2 (\u2191(p (n + 1))).2.2 m = (\u2191(p (m + 1))).2.2 m", "state_after": "no goals"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case base\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m\nm : \u2115\n\u22a2 (\u2191(p (m + 1))).2.2 m = (\u2191(p (m + 1))).2.2 m", "state_after": "no goals"}, {"tactic": "apply hFy (p n) m", "annotated_tactic": ["apply hFy (p n) m", []], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m\nm n : \u2115\nhmn : m + 1 \u2264 n\nIH : (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m\n\u22a2 (\u2191(F (p n))).2.2 m = (\u2191(p n)).2.2 m", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m\nm n : \u2115\nhmn : m + 1 \u2264 n\nIH : (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m\n\u22a2 m < (\u2191(p n)).1"}, {"tactic": "rw [pn_fst]", "annotated_tactic": ["rw [pn_fst]", []], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m\nm n : \u2115\nhmn : m + 1 \u2264 n\nIH : (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m\n\u22a2 m < (\u2191(p n)).1", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m\nm n : \u2115\nhmn : m + 1 \u2264 n\nIH : (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m\n\u22a2 m < n"}, {"tactic": "exact hmn", "annotated_tactic": ["exact hmn", []], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m\nm n : \u2115\nhmn : m + 1 \u2264 n\nIH : (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m\n\u22a2 m < n", "state_after": "no goals"}, {"tactic": "intro n", "annotated_tactic": ["intro n", []], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m\nIy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m\nx : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.1 n\nhx : x = fun n => (\u2191(p (n + 1))).2.1 n\ny : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.2 n\nhy : y = fun n => (\u2191(p (n + 1))).2.2 n\n\u22a2 \u2200 (n : \u2115), \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m\nIy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m\nx : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.1 n\nhx : x = fun n => (\u2191(p (n + 1))).2.1 n\ny : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.2 n\nhy : y = fun n => (\u2191(p (n + 1))).2.2 n\nn : \u2115\n\u22a2 \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)"}, {"tactic": "convert (p n).2 using 3", "annotated_tactic": ["convert (p n).2 using 3", []], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m\nIy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m\nx : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.1 n\nhx : x = fun n => (\u2191(p (n + 1))).2.1 n\ny : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.2 n\nhy : y = fun n => (\u2191(p (n + 1))).2.2 n\nn : \u2115\n\u22a2 \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)", "state_after": "case h.e'_1.h.e'_3.h.e'_4\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m\nIy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m\nx : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.1 n\nhx : x = fun n => (\u2191(p (n + 1))).2.1 n\ny : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.2 n\nhy : y = fun n => (\u2191(p (n + 1))).2.2 n\nn : \u2115\n\u22a2 cylinder x n = cylinder (\u2191(p n)).2.1 (\u2191(p n)).1\n\ncase h.e'_1.h.e'_4.h.e'_4\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m\nIy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m\nx : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.1 n\nhx : x = fun n => (\u2191(p (n + 1))).2.1 n\ny : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.2 n\nhy : y = fun n => (\u2191(p (n + 1))).2.2 n\nn : \u2115\n\u22a2 cylinder y n = cylinder (\u2191(p n)).2.2 (\u2191(p n)).1"}, {"tactic": "rw [pn_fst, \u2190 mem_cylinder_iff_eq, mem_cylinder_iff]", "annotated_tactic": ["rw [pn_fst, \u2190 <a>mem_cylinder_iff_eq</a>, <a>mem_cylinder_iff</a>]", [{"full_name": "PiNat.mem_cylinder_iff_eq", "def_path": "Mathlib/Topology/MetricSpace/PiNat.lean", "def_pos": [133, 9], "def_end_pos": [133, 28]}, {"full_name": "PiNat.mem_cylinder_iff", "def_path": "Mathlib/Topology/MetricSpace/PiNat.lean", "def_pos": [126, 9], "def_end_pos": [126, 25]}]], "state_before": "case h.e'_1.h.e'_3.h.e'_4\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m\nIy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m\nx : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.1 n\nhx : x = fun n => (\u2191(p (n + 1))).2.1 n\ny : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.2 n\nhy : y = fun n => (\u2191(p (n + 1))).2.2 n\nn : \u2115\n\u22a2 cylinder x n = cylinder (\u2191(p n)).2.1 (\u2191(p n)).1", "state_after": "case h.e'_1.h.e'_3.h.e'_4\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m\nIy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m\nx : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.1 n\nhx : x = fun n => (\u2191(p (n + 1))).2.1 n\ny : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.2 n\nhy : y = fun n => (\u2191(p (n + 1))).2.2 n\nn : \u2115\n\u22a2 \u2200 (i : \u2115), i < n \u2192 x i = (\u2191(p n)).2.1 i"}, {"tactic": "intro i hi", "annotated_tactic": ["intro i hi", []], "state_before": "case h.e'_1.h.e'_3.h.e'_4\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m\nIy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m\nx : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.1 n\nhx : x = fun n => (\u2191(p (n + 1))).2.1 n\ny : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.2 n\nhy : y = fun n => (\u2191(p (n + 1))).2.2 n\nn : \u2115\n\u22a2 \u2200 (i : \u2115), i < n \u2192 x i = (\u2191(p n)).2.1 i", "state_after": "case h.e'_1.h.e'_3.h.e'_4\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m\nIy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m\nx : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.1 n\nhx : x = fun n => (\u2191(p (n + 1))).2.1 n\ny : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.2 n\nhy : y = fun n => (\u2191(p (n + 1))).2.2 n\nn i : \u2115\nhi : i < n\n\u22a2 x i = (\u2191(p n)).2.1 i"}, {"tactic": "rw [hx]", "annotated_tactic": ["rw [hx]", []], "state_before": "case h.e'_1.h.e'_3.h.e'_4\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m\nIy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m\nx : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.1 n\nhx : x = fun n => (\u2191(p (n + 1))).2.1 n\ny : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.2 n\nhy : y = fun n => (\u2191(p (n + 1))).2.2 n\nn i : \u2115\nhi : i < n\n\u22a2 x i = (\u2191(p n)).2.1 i", "state_after": "case h.e'_1.h.e'_3.h.e'_4\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m\nIy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m\nx : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.1 n\nhx : x = fun n => (\u2191(p (n + 1))).2.1 n\ny : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.2 n\nhy : y = fun n => (\u2191(p (n + 1))).2.2 n\nn i : \u2115\nhi : i < n\n\u22a2 (fun n => (\u2191(p (n + 1))).2.1 n) i = (\u2191(p n)).2.1 i"}, {"tactic": "exact (Ix i n hi).symm", "annotated_tactic": ["exact (Ix i n hi).<a>symm</a>", [{"full_name": "Eq.symm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [310, 9], "def_end_pos": [310, 16]}]], "state_before": "case h.e'_1.h.e'_3.h.e'_4\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m\nIy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m\nx : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.1 n\nhx : x = fun n => (\u2191(p (n + 1))).2.1 n\ny : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.2 n\nhy : y = fun n => (\u2191(p (n + 1))).2.2 n\nn i : \u2115\nhi : i < n\n\u22a2 (fun n => (\u2191(p (n + 1))).2.1 n) i = (\u2191(p n)).2.1 i", "state_after": "no goals"}, {"tactic": "rw [pn_fst, \u2190 mem_cylinder_iff_eq, mem_cylinder_iff]", "annotated_tactic": ["rw [pn_fst, \u2190 <a>mem_cylinder_iff_eq</a>, <a>mem_cylinder_iff</a>]", [{"full_name": "PiNat.mem_cylinder_iff_eq", "def_path": "Mathlib/Topology/MetricSpace/PiNat.lean", "def_pos": [133, 9], "def_end_pos": [133, 28]}, {"full_name": "PiNat.mem_cylinder_iff", "def_path": "Mathlib/Topology/MetricSpace/PiNat.lean", "def_pos": [126, 9], "def_end_pos": [126, 25]}]], "state_before": "case h.e'_1.h.e'_4.h.e'_4\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m\nIy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m\nx : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.1 n\nhx : x = fun n => (\u2191(p (n + 1))).2.1 n\ny : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.2 n\nhy : y = fun n => (\u2191(p (n + 1))).2.2 n\nn : \u2115\n\u22a2 cylinder y n = cylinder (\u2191(p n)).2.2 (\u2191(p n)).1", "state_after": "case h.e'_1.h.e'_4.h.e'_4\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m\nIy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m\nx : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.1 n\nhx : x = fun n => (\u2191(p (n + 1))).2.1 n\ny : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.2 n\nhy : y = fun n => (\u2191(p (n + 1))).2.2 n\nn : \u2115\n\u22a2 \u2200 (i : \u2115), i < n \u2192 y i = (\u2191(p n)).2.2 i"}, {"tactic": "intro i hi", "annotated_tactic": ["intro i hi", []], "state_before": "case h.e'_1.h.e'_4.h.e'_4\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m\nIy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m\nx : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.1 n\nhx : x = fun n => (\u2191(p (n + 1))).2.1 n\ny : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.2 n\nhy : y = fun n => (\u2191(p (n + 1))).2.2 n\nn : \u2115\n\u22a2 \u2200 (i : \u2115), i < n \u2192 y i = (\u2191(p n)).2.2 i", "state_after": "case h.e'_1.h.e'_4.h.e'_4\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m\nIy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m\nx : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.1 n\nhx : x = fun n => (\u2191(p (n + 1))).2.1 n\ny : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.2 n\nhy : y = fun n => (\u2191(p (n + 1))).2.2 n\nn i : \u2115\nhi : i < n\n\u22a2 y i = (\u2191(p n)).2.2 i"}, {"tactic": "rw [hy]", "annotated_tactic": ["rw [hy]", []], "state_before": "case h.e'_1.h.e'_4.h.e'_4\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m\nIy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m\nx : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.1 n\nhx : x = fun n => (\u2191(p (n + 1))).2.1 n\ny : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.2 n\nhy : y = fun n => (\u2191(p (n + 1))).2.2 n\nn i : \u2115\nhi : i < n\n\u22a2 y i = (\u2191(p n)).2.2 i", "state_after": "case h.e'_1.h.e'_4.h.e'_4\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m\nIy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m\nx : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.1 n\nhx : x = fun n => (\u2191(p (n + 1))).2.1 n\ny : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.2 n\nhy : y = fun n => (\u2191(p (n + 1))).2.2 n\nn i : \u2115\nhi : i < n\n\u22a2 (fun n => (\u2191(p (n + 1))).2.2 n) i = (\u2191(p n)).2.2 i"}, {"tactic": "exact (Iy i n hi).symm", "annotated_tactic": ["exact (Iy i n hi).<a>symm</a>", [{"full_name": "Eq.symm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [310, 9], "def_end_pos": [310, 16]}]], "state_before": "case h.e'_1.h.e'_4.h.e'_4\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m\nIy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m\nx : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.1 n\nhx : x = fun n => (\u2191(p (n + 1))).2.1 n\ny : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.2 n\nhy : y = fun n => (\u2191(p (n + 1))).2.2 n\nn i : \u2115\nhi : i < n\n\u22a2 (fun n => (\u2191(p (n + 1))).2.2 n) i = (\u2191(p n)).2.2 i", "state_after": "no goals"}, {"tactic": "apply t2_separation", "annotated_tactic": ["apply <a>t2_separation</a>", [{"full_name": "t2_separation", "def_path": "Mathlib/Topology/Separation.lean", "def_pos": [906, 9], "def_end_pos": [906, 22]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m\nIy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m\nx : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.1 n\nhx : x = fun n => (\u2191(p (n + 1))).2.1 n\ny : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.2 n\nhy : y = fun n => (\u2191(p (n + 1))).2.2 n\nM : \u2200 (n : \u2115), \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\n\u22a2 \u2203 u v, IsOpen u \u2227 IsOpen v \u2227 f x \u2208 u \u2227 g y \u2208 v \u2227 Disjoint u v", "state_after": "case h\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m\nIy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m\nx : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.1 n\nhx : x = fun n => (\u2191(p (n + 1))).2.1 n\ny : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.2 n\nhy : y = fun n => (\u2191(p (n + 1))).2.2 n\nM : \u2200 (n : \u2115), \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\n\u22a2 f x \u2260 g y"}, {"tactic": "exact disjoint_iff_forall_ne.1 h (mem_range_self _) (mem_range_self _)", "annotated_tactic": ["exact <a>disjoint_iff_forall_ne</a>.1 h (<a>mem_range_self</a> _) (<a>mem_range_self</a> _)", [{"full_name": "Set.disjoint_iff_forall_ne", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1567, 7], "def_end_pos": [1567, 29]}, {"full_name": "Set.mem_range_self", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [680, 9], "def_end_pos": [680, 23]}, {"full_name": "Set.mem_range_self", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [680, 9], "def_end_pos": [680, 23]}]], "state_before": "case h\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m\nIy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m\nx : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.1 n\nhx : x = fun n => (\u2191(p (n + 1))).2.1 n\ny : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.2 n\nhy : y = fun n => (\u2191(p (n + 1))).2.2 n\nM : \u2200 (n : \u2115), \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\n\u22a2 f x \u2260 g y", "state_after": "no goals"}, {"tactic": "apply Metric.mem_nhds_iff.1", "annotated_tactic": ["apply <a>Metric.mem_nhds_iff</a>.1", [{"full_name": "Metric.mem_nhds_iff", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [954, 9], "def_end_pos": [954, 21]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m\nIy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m\nx : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.1 n\nhx : x = fun n => (\u2191(p (n + 1))).2.1 n\ny : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.2 n\nhy : y = fun n => (\u2191(p (n + 1))).2.2 n\nM : \u2200 (n : \u2115), \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\nu v : Set \u03b1\nu_open : IsOpen u\nv_open : IsOpen v\nxu : f x \u2208 u\nyv : g y \u2208 v\nhuv : Disjoint u v\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\n\u22a2 \u2203 \u03b5x, \u03b5x > 0 \u2227 ball x \u03b5x \u2286 f \u207b\u00b9' u", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m\nIy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m\nx : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.1 n\nhx : x = fun n => (\u2191(p (n + 1))).2.1 n\ny : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.2 n\nhy : y = fun n => (\u2191(p (n + 1))).2.2 n\nM : \u2200 (n : \u2115), \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\nu v : Set \u03b1\nu_open : IsOpen u\nv_open : IsOpen v\nxu : f x \u2208 u\nyv : g y \u2208 v\nhuv : Disjoint u v\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\n\u22a2 f \u207b\u00b9' u \u2208 \ud835\udcdd x"}, {"tactic": "exact hf.continuousAt.preimage_mem_nhds (u_open.mem_nhds xu)", "annotated_tactic": ["exact hf.continuousAt.preimage_mem_nhds (u_open.mem_nhds xu)", []], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m\nIy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m\nx : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.1 n\nhx : x = fun n => (\u2191(p (n + 1))).2.1 n\ny : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.2 n\nhy : y = fun n => (\u2191(p (n + 1))).2.2 n\nM : \u2200 (n : \u2115), \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\nu v : Set \u03b1\nu_open : IsOpen u\nv_open : IsOpen v\nxu : f x \u2208 u\nyv : g y \u2208 v\nhuv : Disjoint u v\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\n\u22a2 f \u207b\u00b9' u \u2208 \ud835\udcdd x", "state_after": "no goals"}, {"tactic": "apply Metric.mem_nhds_iff.1", "annotated_tactic": ["apply <a>Metric.mem_nhds_iff</a>.1", [{"full_name": "Metric.mem_nhds_iff", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [954, 9], "def_end_pos": [954, 21]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m\nIy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m\nx : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.1 n\nhx : x = fun n => (\u2191(p (n + 1))).2.1 n\ny : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.2 n\nhy : y = fun n => (\u2191(p (n + 1))).2.2 n\nM : \u2200 (n : \u2115), \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\nu v : Set \u03b1\nu_open : IsOpen u\nv_open : IsOpen v\nxu : f x \u2208 u\nyv : g y \u2208 v\nhuv : Disjoint u v\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\n\u03b5x : \u211d\n\u03b5xpos : \u03b5x > 0\nh\u03b5x : ball x \u03b5x \u2286 f \u207b\u00b9' u\n\u22a2 \u2203 \u03b5y, \u03b5y > 0 \u2227 ball y \u03b5y \u2286 g \u207b\u00b9' v", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m\nIy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m\nx : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.1 n\nhx : x = fun n => (\u2191(p (n + 1))).2.1 n\ny : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.2 n\nhy : y = fun n => (\u2191(p (n + 1))).2.2 n\nM : \u2200 (n : \u2115), \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\nu v : Set \u03b1\nu_open : IsOpen u\nv_open : IsOpen v\nxu : f x \u2208 u\nyv : g y \u2208 v\nhuv : Disjoint u v\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\n\u03b5x : \u211d\n\u03b5xpos : \u03b5x > 0\nh\u03b5x : ball x \u03b5x \u2286 f \u207b\u00b9' u\n\u22a2 g \u207b\u00b9' v \u2208 \ud835\udcdd y"}, {"tactic": "exact hg.continuousAt.preimage_mem_nhds (v_open.mem_nhds yv)", "annotated_tactic": ["exact hg.continuousAt.preimage_mem_nhds (v_open.mem_nhds yv)", []], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m\nIy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m\nx : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.1 n\nhx : x = fun n => (\u2191(p (n + 1))).2.1 n\ny : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.2 n\nhy : y = fun n => (\u2191(p (n + 1))).2.2 n\nM : \u2200 (n : \u2115), \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\nu v : Set \u03b1\nu_open : IsOpen u\nv_open : IsOpen v\nxu : f x \u2208 u\nyv : g y \u2208 v\nhuv : Disjoint u v\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\n\u03b5x : \u211d\n\u03b5xpos : \u03b5x > 0\nh\u03b5x : ball x \u03b5x \u2286 f \u207b\u00b9' u\n\u22a2 g \u207b\u00b9' v \u2208 \ud835\udcdd y", "state_after": "no goals"}, {"tactic": "norm_num", "annotated_tactic": ["norm_num", []], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m\nIy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m\nx : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.1 n\nhx : x = fun n => (\u2191(p (n + 1))).2.1 n\ny : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.2 n\nhy : y = fun n => (\u2191(p (n + 1))).2.2 n\nM : \u2200 (n : \u2115), \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\nu v : Set \u03b1\nu_open : IsOpen u\nv_open : IsOpen v\nxu : f x \u2208 u\nyv : g y \u2208 v\nhuv : Disjoint u v\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\n\u03b5x : \u211d\n\u03b5xpos : \u03b5x > 0\nh\u03b5x : ball x \u03b5x \u2286 f \u207b\u00b9' u\n\u03b5y : \u211d\n\u03b5ypos : \u03b5y > 0\nh\u03b5y : ball y \u03b5y \u2286 g \u207b\u00b9' v\n\u22a2 1 / 2 < 1", "state_after": "no goals"}, {"tactic": "refine' \u27e8u, _, _, u_open.measurableSet\u27e9", "annotated_tactic": ["refine' \u27e8u, _, _, u_open.measurableSet\u27e9", []], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m\nIy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m\nx : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.1 n\nhx : x = fun n => (\u2191(p (n + 1))).2.1 n\ny : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.2 n\nhy : y = fun n => (\u2191(p (n + 1))).2.2 n\nM : \u2200 (n : \u2115), \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\nu v : Set \u03b1\nu_open : IsOpen u\nv_open : IsOpen v\nxu : f x \u2208 u\nyv : g y \u2208 v\nhuv : Disjoint u v\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\n\u03b5x : \u211d\n\u03b5xpos : \u03b5x > 0\nh\u03b5x : ball x \u03b5x \u2286 f \u207b\u00b9' u\n\u03b5y : \u211d\n\u03b5ypos : \u03b5y > 0\nh\u03b5y : ball y \u03b5y \u2286 g \u207b\u00b9' v\nn : \u2115\nhn : (1 / 2) ^ n < min \u03b5x \u03b5y\n\u22a2 MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)", "state_after": "case refine'_1\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m\nIy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m\nx : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.1 n\nhx : x = fun n => (\u2191(p (n + 1))).2.1 n\ny : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.2 n\nhy : y = fun n => (\u2191(p (n + 1))).2.2 n\nM : \u2200 (n : \u2115), \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\nu v : Set \u03b1\nu_open : IsOpen u\nv_open : IsOpen v\nxu : f x \u2208 u\nyv : g y \u2208 v\nhuv : Disjoint u v\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\n\u03b5x : \u211d\n\u03b5xpos : \u03b5x > 0\nh\u03b5x : ball x \u03b5x \u2286 f \u207b\u00b9' u\n\u03b5y : \u211d\n\u03b5ypos : \u03b5y > 0\nh\u03b5y : ball y \u03b5y \u2286 g \u207b\u00b9' v\nn : \u2115\nhn : (1 / 2) ^ n < min \u03b5x \u03b5y\n\u22a2 f '' cylinder x n \u2286 u\n\ncase refine'_2\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m\nIy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m\nx : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.1 n\nhx : x = fun n => (\u2191(p (n + 1))).2.1 n\ny : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.2 n\nhy : y = fun n => (\u2191(p (n + 1))).2.2 n\nM : \u2200 (n : \u2115), \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\nu v : Set \u03b1\nu_open : IsOpen u\nv_open : IsOpen v\nxu : f x \u2208 u\nyv : g y \u2208 v\nhuv : Disjoint u v\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\n\u03b5x : \u211d\n\u03b5xpos : \u03b5x > 0\nh\u03b5x : ball x \u03b5x \u2286 f \u207b\u00b9' u\n\u03b5y : \u211d\n\u03b5ypos : \u03b5y > 0\nh\u03b5y : ball y \u03b5y \u2286 g \u207b\u00b9' v\nn : \u2115\nhn : (1 / 2) ^ n < min \u03b5x \u03b5y\n\u22a2 Disjoint (g '' cylinder y n) u"}, {"tactic": "rw [image_subset_iff]", "annotated_tactic": ["rw [<a>image_subset_iff</a>]", [{"full_name": "Set.image_subset_iff", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [497, 9], "def_end_pos": [497, 25]}]], "state_before": "case refine'_1\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m\nIy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m\nx : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.1 n\nhx : x = fun n => (\u2191(p (n + 1))).2.1 n\ny : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.2 n\nhy : y = fun n => (\u2191(p (n + 1))).2.2 n\nM : \u2200 (n : \u2115), \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\nu v : Set \u03b1\nu_open : IsOpen u\nv_open : IsOpen v\nxu : f x \u2208 u\nyv : g y \u2208 v\nhuv : Disjoint u v\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\n\u03b5x : \u211d\n\u03b5xpos : \u03b5x > 0\nh\u03b5x : ball x \u03b5x \u2286 f \u207b\u00b9' u\n\u03b5y : \u211d\n\u03b5ypos : \u03b5y > 0\nh\u03b5y : ball y \u03b5y \u2286 g \u207b\u00b9' v\nn : \u2115\nhn : (1 / 2) ^ n < min \u03b5x \u03b5y\n\u22a2 f '' cylinder x n \u2286 u", "state_after": "case refine'_1\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m\nIy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m\nx : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.1 n\nhx : x = fun n => (\u2191(p (n + 1))).2.1 n\ny : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.2 n\nhy : y = fun n => (\u2191(p (n + 1))).2.2 n\nM : \u2200 (n : \u2115), \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\nu v : Set \u03b1\nu_open : IsOpen u\nv_open : IsOpen v\nxu : f x \u2208 u\nyv : g y \u2208 v\nhuv : Disjoint u v\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\n\u03b5x : \u211d\n\u03b5xpos : \u03b5x > 0\nh\u03b5x : ball x \u03b5x \u2286 f \u207b\u00b9' u\n\u03b5y : \u211d\n\u03b5ypos : \u03b5y > 0\nh\u03b5y : ball y \u03b5y \u2286 g \u207b\u00b9' v\nn : \u2115\nhn : (1 / 2) ^ n < min \u03b5x \u03b5y\n\u22a2 cylinder x n \u2286 f \u207b\u00b9' u"}, {"tactic": "apply Subset.trans _ h\u03b5x", "annotated_tactic": ["apply <a>Subset.trans</a> _ h\u03b5x", [{"full_name": "Set.Subset.trans", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [362, 9], "def_end_pos": [362, 21]}]], "state_before": "case refine'_1\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m\nIy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m\nx : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.1 n\nhx : x = fun n => (\u2191(p (n + 1))).2.1 n\ny : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.2 n\nhy : y = fun n => (\u2191(p (n + 1))).2.2 n\nM : \u2200 (n : \u2115), \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\nu v : Set \u03b1\nu_open : IsOpen u\nv_open : IsOpen v\nxu : f x \u2208 u\nyv : g y \u2208 v\nhuv : Disjoint u v\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\n\u03b5x : \u211d\n\u03b5xpos : \u03b5x > 0\nh\u03b5x : ball x \u03b5x \u2286 f \u207b\u00b9' u\n\u03b5y : \u211d\n\u03b5ypos : \u03b5y > 0\nh\u03b5y : ball y \u03b5y \u2286 g \u207b\u00b9' v\nn : \u2115\nhn : (1 / 2) ^ n < min \u03b5x \u03b5y\n\u22a2 cylinder x n \u2286 f \u207b\u00b9' u", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m\nIy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m\nx : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.1 n\nhx : x = fun n => (\u2191(p (n + 1))).2.1 n\ny : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.2 n\nhy : y = fun n => (\u2191(p (n + 1))).2.2 n\nM : \u2200 (n : \u2115), \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\nu v : Set \u03b1\nu_open : IsOpen u\nv_open : IsOpen v\nxu : f x \u2208 u\nyv : g y \u2208 v\nhuv : Disjoint u v\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\n\u03b5x : \u211d\n\u03b5xpos : \u03b5x > 0\nh\u03b5x : ball x \u03b5x \u2286 f \u207b\u00b9' u\n\u03b5y : \u211d\n\u03b5ypos : \u03b5y > 0\nh\u03b5y : ball y \u03b5y \u2286 g \u207b\u00b9' v\nn : \u2115\nhn : (1 / 2) ^ n < min \u03b5x \u03b5y\n\u22a2 cylinder x n \u2286 ball x \u03b5x"}, {"tactic": "intro z hz", "annotated_tactic": ["intro z hz", []], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m\nIy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m\nx : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.1 n\nhx : x = fun n => (\u2191(p (n + 1))).2.1 n\ny : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.2 n\nhy : y = fun n => (\u2191(p (n + 1))).2.2 n\nM : \u2200 (n : \u2115), \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\nu v : Set \u03b1\nu_open : IsOpen u\nv_open : IsOpen v\nxu : f x \u2208 u\nyv : g y \u2208 v\nhuv : Disjoint u v\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\n\u03b5x : \u211d\n\u03b5xpos : \u03b5x > 0\nh\u03b5x : ball x \u03b5x \u2286 f \u207b\u00b9' u\n\u03b5y : \u211d\n\u03b5ypos : \u03b5y > 0\nh\u03b5y : ball y \u03b5y \u2286 g \u207b\u00b9' v\nn : \u2115\nhn : (1 / 2) ^ n < min \u03b5x \u03b5y\n\u22a2 cylinder x n \u2286 ball x \u03b5x", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m\nIy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m\nx : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.1 n\nhx : x = fun n => (\u2191(p (n + 1))).2.1 n\ny : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.2 n\nhy : y = fun n => (\u2191(p (n + 1))).2.2 n\nM : \u2200 (n : \u2115), \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\nu v : Set \u03b1\nu_open : IsOpen u\nv_open : IsOpen v\nxu : f x \u2208 u\nyv : g y \u2208 v\nhuv : Disjoint u v\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\n\u03b5x : \u211d\n\u03b5xpos : \u03b5x > 0\nh\u03b5x : ball x \u03b5x \u2286 f \u207b\u00b9' u\n\u03b5y : \u211d\n\u03b5ypos : \u03b5y > 0\nh\u03b5y : ball y \u03b5y \u2286 g \u207b\u00b9' v\nn : \u2115\nhn : (1 / 2) ^ n < min \u03b5x \u03b5y\nz : \u2115 \u2192 \u2115\nhz : z \u2208 cylinder x n\n\u22a2 z \u2208 ball x \u03b5x"}, {"tactic": "rw [mem_cylinder_iff_dist_le] at hz", "annotated_tactic": ["rw [<a>mem_cylinder_iff_dist_le</a>] at hz", [{"full_name": "PiNat.mem_cylinder_iff_dist_le", "def_path": "Mathlib/Topology/MetricSpace/PiNat.lean", "def_pos": [314, 9], "def_end_pos": [314, 33]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m\nIy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m\nx : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.1 n\nhx : x = fun n => (\u2191(p (n + 1))).2.1 n\ny : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.2 n\nhy : y = fun n => (\u2191(p (n + 1))).2.2 n\nM : \u2200 (n : \u2115), \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\nu v : Set \u03b1\nu_open : IsOpen u\nv_open : IsOpen v\nxu : f x \u2208 u\nyv : g y \u2208 v\nhuv : Disjoint u v\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\n\u03b5x : \u211d\n\u03b5xpos : \u03b5x > 0\nh\u03b5x : ball x \u03b5x \u2286 f \u207b\u00b9' u\n\u03b5y : \u211d\n\u03b5ypos : \u03b5y > 0\nh\u03b5y : ball y \u03b5y \u2286 g \u207b\u00b9' v\nn : \u2115\nhn : (1 / 2) ^ n < min \u03b5x \u03b5y\nz : \u2115 \u2192 \u2115\nhz : z \u2208 cylinder x n\n\u22a2 z \u2208 ball x \u03b5x", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m\nIy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m\nx : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.1 n\nhx : x = fun n => (\u2191(p (n + 1))).2.1 n\ny : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.2 n\nhy : y = fun n => (\u2191(p (n + 1))).2.2 n\nM : \u2200 (n : \u2115), \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\nu v : Set \u03b1\nu_open : IsOpen u\nv_open : IsOpen v\nxu : f x \u2208 u\nyv : g y \u2208 v\nhuv : Disjoint u v\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\n\u03b5x : \u211d\n\u03b5xpos : \u03b5x > 0\nh\u03b5x : ball x \u03b5x \u2286 f \u207b\u00b9' u\n\u03b5y : \u211d\n\u03b5ypos : \u03b5y > 0\nh\u03b5y : ball y \u03b5y \u2286 g \u207b\u00b9' v\nn : \u2115\nhn : (1 / 2) ^ n < min \u03b5x \u03b5y\nz : \u2115 \u2192 \u2115\nhz : dist z x \u2264 (1 / 2) ^ n\n\u22a2 z \u2208 ball x \u03b5x"}, {"tactic": "exact hz.trans_lt (hn.trans_le (min_le_left _ _))", "annotated_tactic": ["exact hz.trans_lt (hn.trans_le (<a>min_le_left</a> _ _))", [{"full_name": "min_le_left", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [33, 9], "def_end_pos": [33, 20]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m\nIy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m\nx : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.1 n\nhx : x = fun n => (\u2191(p (n + 1))).2.1 n\ny : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.2 n\nhy : y = fun n => (\u2191(p (n + 1))).2.2 n\nM : \u2200 (n : \u2115), \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\nu v : Set \u03b1\nu_open : IsOpen u\nv_open : IsOpen v\nxu : f x \u2208 u\nyv : g y \u2208 v\nhuv : Disjoint u v\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\n\u03b5x : \u211d\n\u03b5xpos : \u03b5x > 0\nh\u03b5x : ball x \u03b5x \u2286 f \u207b\u00b9' u\n\u03b5y : \u211d\n\u03b5ypos : \u03b5y > 0\nh\u03b5y : ball y \u03b5y \u2286 g \u207b\u00b9' v\nn : \u2115\nhn : (1 / 2) ^ n < min \u03b5x \u03b5y\nz : \u2115 \u2192 \u2115\nhz : dist z x \u2264 (1 / 2) ^ n\n\u22a2 z \u2208 ball x \u03b5x", "state_after": "no goals"}, {"tactic": "refine' Disjoint.mono_left _ huv.symm", "annotated_tactic": ["refine' <a>Disjoint.mono_left</a> _ huv.symm", [{"full_name": "Disjoint.mono_left", "def_path": "Mathlib/Order/Disjoint.lean", "def_pos": [70, 9], "def_end_pos": [70, 27]}]], "state_before": "case refine'_2\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m\nIy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m\nx : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.1 n\nhx : x = fun n => (\u2191(p (n + 1))).2.1 n\ny : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.2 n\nhy : y = fun n => (\u2191(p (n + 1))).2.2 n\nM : \u2200 (n : \u2115), \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\nu v : Set \u03b1\nu_open : IsOpen u\nv_open : IsOpen v\nxu : f x \u2208 u\nyv : g y \u2208 v\nhuv : Disjoint u v\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\n\u03b5x : \u211d\n\u03b5xpos : \u03b5x > 0\nh\u03b5x : ball x \u03b5x \u2286 f \u207b\u00b9' u\n\u03b5y : \u211d\n\u03b5ypos : \u03b5y > 0\nh\u03b5y : ball y \u03b5y \u2286 g \u207b\u00b9' v\nn : \u2115\nhn : (1 / 2) ^ n < min \u03b5x \u03b5y\n\u22a2 Disjoint (g '' cylinder y n) u", "state_after": "case refine'_2\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m\nIy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m\nx : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.1 n\nhx : x = fun n => (\u2191(p (n + 1))).2.1 n\ny : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.2 n\nhy : y = fun n => (\u2191(p (n + 1))).2.2 n\nM : \u2200 (n : \u2115), \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\nu v : Set \u03b1\nu_open : IsOpen u\nv_open : IsOpen v\nxu : f x \u2208 u\nyv : g y \u2208 v\nhuv : Disjoint u v\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\n\u03b5x : \u211d\n\u03b5xpos : \u03b5x > 0\nh\u03b5x : ball x \u03b5x \u2286 f \u207b\u00b9' u\n\u03b5y : \u211d\n\u03b5ypos : \u03b5y > 0\nh\u03b5y : ball y \u03b5y \u2286 g \u207b\u00b9' v\nn : \u2115\nhn : (1 / 2) ^ n < min \u03b5x \u03b5y\n\u22a2 g '' cylinder y n \u2264 v"}, {"tactic": "change g '' cylinder y n \u2286 v", "annotated_tactic": ["change g '' <a>cylinder</a> y n \u2286 v", [{"full_name": "PiNat.cylinder", "def_path": "Mathlib/Topology/MetricSpace/PiNat.lean", "def_pos": [107, 5], "def_end_pos": [107, 13]}]], "state_before": "case refine'_2\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m\nIy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m\nx : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.1 n\nhx : x = fun n => (\u2191(p (n + 1))).2.1 n\ny : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.2 n\nhy : y = fun n => (\u2191(p (n + 1))).2.2 n\nM : \u2200 (n : \u2115), \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\nu v : Set \u03b1\nu_open : IsOpen u\nv_open : IsOpen v\nxu : f x \u2208 u\nyv : g y \u2208 v\nhuv : Disjoint u v\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\n\u03b5x : \u211d\n\u03b5xpos : \u03b5x > 0\nh\u03b5x : ball x \u03b5x \u2286 f \u207b\u00b9' u\n\u03b5y : \u211d\n\u03b5ypos : \u03b5y > 0\nh\u03b5y : ball y \u03b5y \u2286 g \u207b\u00b9' v\nn : \u2115\nhn : (1 / 2) ^ n < min \u03b5x \u03b5y\n\u22a2 g '' cylinder y n \u2264 v", "state_after": "case refine'_2\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m\nIy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m\nx : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.1 n\nhx : x = fun n => (\u2191(p (n + 1))).2.1 n\ny : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.2 n\nhy : y = fun n => (\u2191(p (n + 1))).2.2 n\nM : \u2200 (n : \u2115), \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\nu v : Set \u03b1\nu_open : IsOpen u\nv_open : IsOpen v\nxu : f x \u2208 u\nyv : g y \u2208 v\nhuv : Disjoint u v\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\n\u03b5x : \u211d\n\u03b5xpos : \u03b5x > 0\nh\u03b5x : ball x \u03b5x \u2286 f \u207b\u00b9' u\n\u03b5y : \u211d\n\u03b5ypos : \u03b5y > 0\nh\u03b5y : ball y \u03b5y \u2286 g \u207b\u00b9' v\nn : \u2115\nhn : (1 / 2) ^ n < min \u03b5x \u03b5y\n\u22a2 g '' cylinder y n \u2286 v"}, {"tactic": "rw [image_subset_iff]", "annotated_tactic": ["rw [<a>image_subset_iff</a>]", [{"full_name": "Set.image_subset_iff", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [497, 9], "def_end_pos": [497, 25]}]], "state_before": "case refine'_2\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m\nIy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m\nx : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.1 n\nhx : x = fun n => (\u2191(p (n + 1))).2.1 n\ny : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.2 n\nhy : y = fun n => (\u2191(p (n + 1))).2.2 n\nM : \u2200 (n : \u2115), \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\nu v : Set \u03b1\nu_open : IsOpen u\nv_open : IsOpen v\nxu : f x \u2208 u\nyv : g y \u2208 v\nhuv : Disjoint u v\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\n\u03b5x : \u211d\n\u03b5xpos : \u03b5x > 0\nh\u03b5x : ball x \u03b5x \u2286 f \u207b\u00b9' u\n\u03b5y : \u211d\n\u03b5ypos : \u03b5y > 0\nh\u03b5y : ball y \u03b5y \u2286 g \u207b\u00b9' v\nn : \u2115\nhn : (1 / 2) ^ n < min \u03b5x \u03b5y\n\u22a2 g '' cylinder y n \u2286 v", "state_after": "case refine'_2\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m\nIy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m\nx : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.1 n\nhx : x = fun n => (\u2191(p (n + 1))).2.1 n\ny : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.2 n\nhy : y = fun n => (\u2191(p (n + 1))).2.2 n\nM : \u2200 (n : \u2115), \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\nu v : Set \u03b1\nu_open : IsOpen u\nv_open : IsOpen v\nxu : f x \u2208 u\nyv : g y \u2208 v\nhuv : Disjoint u v\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\n\u03b5x : \u211d\n\u03b5xpos : \u03b5x > 0\nh\u03b5x : ball x \u03b5x \u2286 f \u207b\u00b9' u\n\u03b5y : \u211d\n\u03b5ypos : \u03b5y > 0\nh\u03b5y : ball y \u03b5y \u2286 g \u207b\u00b9' v\nn : \u2115\nhn : (1 / 2) ^ n < min \u03b5x \u03b5y\n\u22a2 cylinder y n \u2286 g \u207b\u00b9' v"}, {"tactic": "apply Subset.trans _ h\u03b5y", "annotated_tactic": ["apply <a>Subset.trans</a> _ h\u03b5y", [{"full_name": "Set.Subset.trans", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [362, 9], "def_end_pos": [362, 21]}]], "state_before": "case refine'_2\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m\nIy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m\nx : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.1 n\nhx : x = fun n => (\u2191(p (n + 1))).2.1 n\ny : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.2 n\nhy : y = fun n => (\u2191(p (n + 1))).2.2 n\nM : \u2200 (n : \u2115), \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\nu v : Set \u03b1\nu_open : IsOpen u\nv_open : IsOpen v\nxu : f x \u2208 u\nyv : g y \u2208 v\nhuv : Disjoint u v\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\n\u03b5x : \u211d\n\u03b5xpos : \u03b5x > 0\nh\u03b5x : ball x \u03b5x \u2286 f \u207b\u00b9' u\n\u03b5y : \u211d\n\u03b5ypos : \u03b5y > 0\nh\u03b5y : ball y \u03b5y \u2286 g \u207b\u00b9' v\nn : \u2115\nhn : (1 / 2) ^ n < min \u03b5x \u03b5y\n\u22a2 cylinder y n \u2286 g \u207b\u00b9' v", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m\nIy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m\nx : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.1 n\nhx : x = fun n => (\u2191(p (n + 1))).2.1 n\ny : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.2 n\nhy : y = fun n => (\u2191(p (n + 1))).2.2 n\nM : \u2200 (n : \u2115), \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\nu v : Set \u03b1\nu_open : IsOpen u\nv_open : IsOpen v\nxu : f x \u2208 u\nyv : g y \u2208 v\nhuv : Disjoint u v\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\n\u03b5x : \u211d\n\u03b5xpos : \u03b5x > 0\nh\u03b5x : ball x \u03b5x \u2286 f \u207b\u00b9' u\n\u03b5y : \u211d\n\u03b5ypos : \u03b5y > 0\nh\u03b5y : ball y \u03b5y \u2286 g \u207b\u00b9' v\nn : \u2115\nhn : (1 / 2) ^ n < min \u03b5x \u03b5y\n\u22a2 cylinder y n \u2286 ball y \u03b5y"}, {"tactic": "intro z hz", "annotated_tactic": ["intro z hz", []], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m\nIy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m\nx : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.1 n\nhx : x = fun n => (\u2191(p (n + 1))).2.1 n\ny : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.2 n\nhy : y = fun n => (\u2191(p (n + 1))).2.2 n\nM : \u2200 (n : \u2115), \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\nu v : Set \u03b1\nu_open : IsOpen u\nv_open : IsOpen v\nxu : f x \u2208 u\nyv : g y \u2208 v\nhuv : Disjoint u v\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\n\u03b5x : \u211d\n\u03b5xpos : \u03b5x > 0\nh\u03b5x : ball x \u03b5x \u2286 f \u207b\u00b9' u\n\u03b5y : \u211d\n\u03b5ypos : \u03b5y > 0\nh\u03b5y : ball y \u03b5y \u2286 g \u207b\u00b9' v\nn : \u2115\nhn : (1 / 2) ^ n < min \u03b5x \u03b5y\n\u22a2 cylinder y n \u2286 ball y \u03b5y", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m\nIy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m\nx : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.1 n\nhx : x = fun n => (\u2191(p (n + 1))).2.1 n\ny : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.2 n\nhy : y = fun n => (\u2191(p (n + 1))).2.2 n\nM : \u2200 (n : \u2115), \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\nu v : Set \u03b1\nu_open : IsOpen u\nv_open : IsOpen v\nxu : f x \u2208 u\nyv : g y \u2208 v\nhuv : Disjoint u v\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\n\u03b5x : \u211d\n\u03b5xpos : \u03b5x > 0\nh\u03b5x : ball x \u03b5x \u2286 f \u207b\u00b9' u\n\u03b5y : \u211d\n\u03b5ypos : \u03b5y > 0\nh\u03b5y : ball y \u03b5y \u2286 g \u207b\u00b9' v\nn : \u2115\nhn : (1 / 2) ^ n < min \u03b5x \u03b5y\nz : \u2115 \u2192 \u2115\nhz : z \u2208 cylinder y n\n\u22a2 z \u2208 ball y \u03b5y"}, {"tactic": "rw [mem_cylinder_iff_dist_le] at hz", "annotated_tactic": ["rw [<a>mem_cylinder_iff_dist_le</a>] at hz", [{"full_name": "PiNat.mem_cylinder_iff_dist_le", "def_path": "Mathlib/Topology/MetricSpace/PiNat.lean", "def_pos": [314, 9], "def_end_pos": [314, 33]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m\nIy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m\nx : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.1 n\nhx : x = fun n => (\u2191(p (n + 1))).2.1 n\ny : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.2 n\nhy : y = fun n => (\u2191(p (n + 1))).2.2 n\nM : \u2200 (n : \u2115), \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\nu v : Set \u03b1\nu_open : IsOpen u\nv_open : IsOpen v\nxu : f x \u2208 u\nyv : g y \u2208 v\nhuv : Disjoint u v\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\n\u03b5x : \u211d\n\u03b5xpos : \u03b5x > 0\nh\u03b5x : ball x \u03b5x \u2286 f \u207b\u00b9' u\n\u03b5y : \u211d\n\u03b5ypos : \u03b5y > 0\nh\u03b5y : ball y \u03b5y \u2286 g \u207b\u00b9' v\nn : \u2115\nhn : (1 / 2) ^ n < min \u03b5x \u03b5y\nz : \u2115 \u2192 \u2115\nhz : z \u2208 cylinder y n\n\u22a2 z \u2208 ball y \u03b5y", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m\nIy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m\nx : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.1 n\nhx : x = fun n => (\u2191(p (n + 1))).2.1 n\ny : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.2 n\nhy : y = fun n => (\u2191(p (n + 1))).2.2 n\nM : \u2200 (n : \u2115), \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\nu v : Set \u03b1\nu_open : IsOpen u\nv_open : IsOpen v\nxu : f x \u2208 u\nyv : g y \u2208 v\nhuv : Disjoint u v\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\n\u03b5x : \u211d\n\u03b5xpos : \u03b5x > 0\nh\u03b5x : ball x \u03b5x \u2286 f \u207b\u00b9' u\n\u03b5y : \u211d\n\u03b5ypos : \u03b5y > 0\nh\u03b5y : ball y \u03b5y \u2286 g \u207b\u00b9' v\nn : \u2115\nhn : (1 / 2) ^ n < min \u03b5x \u03b5y\nz : \u2115 \u2192 \u2115\nhz : dist z y \u2264 (1 / 2) ^ n\n\u22a2 z \u2208 ball y \u03b5y"}, {"tactic": "exact hz.trans_lt (hn.trans_le (min_le_right _ _))", "annotated_tactic": ["exact hz.trans_lt (hn.trans_le (<a>min_le_right</a> _ _))", [{"full_name": "min_le_right", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [40, 9], "def_end_pos": [40, 21]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1\nhFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1\nhFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m\nIy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m\nx : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.1 n\nhx : x = fun n => (\u2191(p (n + 1))).2.1 n\ny : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.2 n\nhy : y = fun n => (\u2191(p (n + 1))).2.2 n\nM : \u2200 (n : \u2115), \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\nu v : Set \u03b1\nu_open : IsOpen u\nv_open : IsOpen v\nxu : f x \u2208 u\nyv : g y \u2208 v\nhuv : Disjoint u v\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\n\u03b5x : \u211d\n\u03b5xpos : \u03b5x > 0\nh\u03b5x : ball x \u03b5x \u2286 f \u207b\u00b9' u\n\u03b5y : \u211d\n\u03b5ypos : \u03b5y > 0\nh\u03b5y : ball y \u03b5y \u2286 g \u207b\u00b9' v\nn : \u2115\nhn : (1 / 2) ^ n < min \u03b5x \u03b5y\nz : \u2115 \u2192 \u2115\nhz : dist z y \u2264 (1 / 2) ^ n\n\u22a2 z \u2208 ball y \u03b5y", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/ConvergenceInMeasure.lean", "full_name": "MeasureTheory.tendstoInMeasure_of_tendsto_snorm_of_stronglyMeasurable", "start": [278, 1], "end": [299, 45], "traced_tactics": [{"tactic": "intro \u03b5 h\u03b5", "annotated_tactic": ["intro \u03b5 h\u03b5", []], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\nE : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup E\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhf : \u2200 (n : \u03b9), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nl : Filter \u03b9\nhfg : Tendsto (fun n => snorm (f n - g) p \u03bc) l (\ud835\udcdd 0)\n\u22a2 TendstoInMeasure \u03bc f l g", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\nE : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup E\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhf : \u2200 (n : \u03b9), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nl : Filter \u03b9\nhfg : Tendsto (fun n => snorm (f n - g) p \u03bc) l (\ud835\udcdd 0)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u22a2 Tendsto (fun i => \u2191\u2191\u03bc {x | \u03b5 \u2264 dist (f i x) (g x)}) l (\ud835\udcdd 0)"}, {"tactic": "replace hfg := ENNReal.Tendsto.const_mul\n  (Tendsto.ennrpow_const p.toReal hfg) (Or.inr <| @ENNReal.ofReal_ne_top (1 / \u03b5 ^ p.toReal))", "annotated_tactic": ["replace hfg := <a>ENNReal.Tendsto.const_mul</a>\n    (<a>Tendsto.ennrpow_const</a> p.toReal hfg) (<a>Or.inr</a> <| @<a>ENNReal.ofReal_ne_top</a> (1 / \u03b5 ^ p.toReal))", [{"full_name": "ENNReal.Tendsto.const_mul", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [373, 19], "def_end_pos": [373, 36]}, {"full_name": "Filter.Tendsto.ennrpow_const", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Continuity.lean", "def_pos": [534, 9], "def_end_pos": [534, 37]}, {"full_name": "Or.inr", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [519, 5], "def_end_pos": [519, 8]}, {"full_name": "ENNReal.ofReal_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [311, 17], "def_end_pos": [311, 30]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\nE : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup E\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhf : \u2200 (n : \u03b9), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nl : Filter \u03b9\nhfg : Tendsto (fun n => snorm (f n - g) p \u03bc) l (\ud835\udcdd 0)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u22a2 Tendsto (fun i => \u2191\u2191\u03bc {x | \u03b5 \u2264 dist (f i x) (g x)}) l (\ud835\udcdd 0)", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\nE : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup E\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhf : \u2200 (n : \u03b9), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nl : Filter \u03b9\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhfg :\n  Tendsto (fun b => ENNReal.ofReal (1 / \u03b5 ^ ENNReal.toReal p) * snorm (f b - g) p \u03bc ^ ENNReal.toReal p) l\n    (\ud835\udcdd (ENNReal.ofReal (1 / \u03b5 ^ ENNReal.toReal p) * 0 ^ ENNReal.toReal p))\n\u22a2 Tendsto (fun i => \u2191\u2191\u03bc {x | \u03b5 \u2264 dist (f i x) (g x)}) l (\ud835\udcdd 0)"}, {"tactic": "simp only [mul_zero,\n  ENNReal.zero_rpow_of_pos (ENNReal.toReal_pos hp_ne_zero hp_ne_top)] at hfg", "annotated_tactic": ["simp only [<a>mul_zero</a>,\n    <a>ENNReal.zero_rpow_of_pos</a> (<a>ENNReal.toReal_pos</a> hp_ne_zero hp_ne_top)] at hfg", [{"full_name": "MulZeroClass.mul_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [38, 3], "def_end_pos": [38, 11]}, {"full_name": "ENNReal.zero_rpow_of_pos", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [401, 9], "def_end_pos": [401, 25]}, {"full_name": "ENNReal.toReal_pos", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2131, 9], "def_end_pos": [2131, 19]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\nE : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup E\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhf : \u2200 (n : \u03b9), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nl : Filter \u03b9\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhfg :\n  Tendsto (fun b => ENNReal.ofReal (1 / \u03b5 ^ ENNReal.toReal p) * snorm (f b - g) p \u03bc ^ ENNReal.toReal p) l\n    (\ud835\udcdd (ENNReal.ofReal (1 / \u03b5 ^ ENNReal.toReal p) * 0 ^ ENNReal.toReal p))\n\u22a2 Tendsto (fun i => \u2191\u2191\u03bc {x | \u03b5 \u2264 dist (f i x) (g x)}) l (\ud835\udcdd 0)", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\nE : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup E\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhf : \u2200 (n : \u03b9), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nl : Filter \u03b9\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhfg : Tendsto (fun b => ENNReal.ofReal (1 / \u03b5 ^ ENNReal.toReal p) * snorm (f b - g) p \u03bc ^ ENNReal.toReal p) l (\ud835\udcdd 0)\n\u22a2 Tendsto (fun i => \u2191\u2191\u03bc {x | \u03b5 \u2264 dist (f i x) (g x)}) l (\ud835\udcdd 0)"}, {"tactic": "rw [ENNReal.tendsto_nhds_zero] at hfg \u22a2", "annotated_tactic": ["rw [<a>ENNReal.tendsto_nhds_zero</a>] at hfg \u22a2", [{"full_name": "ENNReal.tendsto_nhds_zero", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [279, 19], "def_end_pos": [279, 36]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\nE : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup E\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhf : \u2200 (n : \u03b9), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nl : Filter \u03b9\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhfg : Tendsto (fun b => ENNReal.ofReal (1 / \u03b5 ^ ENNReal.toReal p) * snorm (f b - g) p \u03bc ^ ENNReal.toReal p) l (\ud835\udcdd 0)\n\u22a2 Tendsto (fun i => \u2191\u2191\u03bc {x | \u03b5 \u2264 dist (f i x) (g x)}) l (\ud835\udcdd 0)", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\nE : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup E\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhf : \u2200 (n : \u03b9), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nl : Filter \u03b9\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhfg :\n  \u2200 (\u03b5_1 : \u211d\u22650\u221e),\n    \u03b5_1 > 0 \u2192 \u2200\u1da0 (x : \u03b9) in l, ENNReal.ofReal (1 / \u03b5 ^ ENNReal.toReal p) * snorm (f x - g) p \u03bc ^ ENNReal.toReal p \u2264 \u03b5_1\n\u22a2 \u2200 (\u03b5_1 : \u211d\u22650\u221e), \u03b5_1 > 0 \u2192 \u2200\u1da0 (x : \u03b9) in l, \u2191\u2191\u03bc {x_1 | \u03b5 \u2264 dist (f x x_1) (g x_1)} \u2264 \u03b5_1"}, {"tactic": "intro \u03b4 h\u03b4", "annotated_tactic": ["intro \u03b4 h\u03b4", []], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\nE : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup E\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhf : \u2200 (n : \u03b9), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nl : Filter \u03b9\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhfg :\n  \u2200 (\u03b5_1 : \u211d\u22650\u221e),\n    \u03b5_1 > 0 \u2192 \u2200\u1da0 (x : \u03b9) in l, ENNReal.ofReal (1 / \u03b5 ^ ENNReal.toReal p) * snorm (f x - g) p \u03bc ^ ENNReal.toReal p \u2264 \u03b5_1\n\u22a2 \u2200 (\u03b5_1 : \u211d\u22650\u221e), \u03b5_1 > 0 \u2192 \u2200\u1da0 (x : \u03b9) in l, \u2191\u2191\u03bc {x_1 | \u03b5 \u2264 dist (f x x_1) (g x_1)} \u2264 \u03b5_1", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\nE : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup E\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhf : \u2200 (n : \u03b9), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nl : Filter \u03b9\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhfg :\n  \u2200 (\u03b5_1 : \u211d\u22650\u221e),\n    \u03b5_1 > 0 \u2192 \u2200\u1da0 (x : \u03b9) in l, ENNReal.ofReal (1 / \u03b5 ^ ENNReal.toReal p) * snorm (f x - g) p \u03bc ^ ENNReal.toReal p \u2264 \u03b5_1\n\u03b4 : \u211d\u22650\u221e\nh\u03b4 : \u03b4 > 0\n\u22a2 \u2200\u1da0 (x : \u03b9) in l, \u2191\u2191\u03bc {x_1 | \u03b5 \u2264 dist (f x x_1) (g x_1)} \u2264 \u03b4"}, {"tactic": "refine' (hfg \u03b4 h\u03b4).mono fun n hn => _", "annotated_tactic": ["refine' (hfg \u03b4 h\u03b4).<a>mono</a> fun n hn => _", [{"full_name": "Filter.Eventually.mono", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1140, 9], "def_end_pos": [1140, 24]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\nE : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup E\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhf : \u2200 (n : \u03b9), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nl : Filter \u03b9\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhfg :\n  \u2200 (\u03b5_1 : \u211d\u22650\u221e),\n    \u03b5_1 > 0 \u2192 \u2200\u1da0 (x : \u03b9) in l, ENNReal.ofReal (1 / \u03b5 ^ ENNReal.toReal p) * snorm (f x - g) p \u03bc ^ ENNReal.toReal p \u2264 \u03b5_1\n\u03b4 : \u211d\u22650\u221e\nh\u03b4 : \u03b4 > 0\n\u22a2 \u2200\u1da0 (x : \u03b9) in l, \u2191\u2191\u03bc {x_1 | \u03b5 \u2264 dist (f x x_1) (g x_1)} \u2264 \u03b4", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\nE : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup E\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhf : \u2200 (n : \u03b9), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nl : Filter \u03b9\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhfg :\n  \u2200 (\u03b5_1 : \u211d\u22650\u221e),\n    \u03b5_1 > 0 \u2192 \u2200\u1da0 (x : \u03b9) in l, ENNReal.ofReal (1 / \u03b5 ^ ENNReal.toReal p) * snorm (f x - g) p \u03bc ^ ENNReal.toReal p \u2264 \u03b5_1\n\u03b4 : \u211d\u22650\u221e\nh\u03b4 : \u03b4 > 0\nn : \u03b9\nhn : ENNReal.ofReal (1 / \u03b5 ^ ENNReal.toReal p) * snorm (f n - g) p \u03bc ^ ENNReal.toReal p \u2264 \u03b4\n\u22a2 \u2191\u2191\u03bc {x | \u03b5 \u2264 dist (f n x) (g x)} \u2264 \u03b4"}, {"tactic": "refine' le_trans _ hn", "annotated_tactic": ["refine' <a>le_trans</a> _ hn", [{"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\nE : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup E\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhf : \u2200 (n : \u03b9), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nl : Filter \u03b9\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhfg :\n  \u2200 (\u03b5_1 : \u211d\u22650\u221e),\n    \u03b5_1 > 0 \u2192 \u2200\u1da0 (x : \u03b9) in l, ENNReal.ofReal (1 / \u03b5 ^ ENNReal.toReal p) * snorm (f x - g) p \u03bc ^ ENNReal.toReal p \u2264 \u03b5_1\n\u03b4 : \u211d\u22650\u221e\nh\u03b4 : \u03b4 > 0\nn : \u03b9\nhn : ENNReal.ofReal (1 / \u03b5 ^ ENNReal.toReal p) * snorm (f n - g) p \u03bc ^ ENNReal.toReal p \u2264 \u03b4\n\u22a2 \u2191\u2191\u03bc {x | \u03b5 \u2264 dist (f n x) (g x)} \u2264 \u03b4", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\nE : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup E\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhf : \u2200 (n : \u03b9), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nl : Filter \u03b9\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhfg :\n  \u2200 (\u03b5_1 : \u211d\u22650\u221e),\n    \u03b5_1 > 0 \u2192 \u2200\u1da0 (x : \u03b9) in l, ENNReal.ofReal (1 / \u03b5 ^ ENNReal.toReal p) * snorm (f x - g) p \u03bc ^ ENNReal.toReal p \u2264 \u03b5_1\n\u03b4 : \u211d\u22650\u221e\nh\u03b4 : \u03b4 > 0\nn : \u03b9\nhn : ENNReal.ofReal (1 / \u03b5 ^ ENNReal.toReal p) * snorm (f n - g) p \u03bc ^ ENNReal.toReal p \u2264 \u03b4\n\u22a2 \u2191\u2191\u03bc {x | \u03b5 \u2264 dist (f n x) (g x)} \u2264 ENNReal.ofReal (1 / \u03b5 ^ ENNReal.toReal p) * snorm (f n - g) p \u03bc ^ ENNReal.toReal p"}, {"tactic": "rw [ENNReal.ofReal_div_of_pos (Real.rpow_pos_of_pos h\u03b5 _), ENNReal.ofReal_one, mul_comm,\n  mul_one_div, ENNReal.le_div_iff_mul_le _ (Or.inl ENNReal.ofReal_ne_top), mul_comm]", "annotated_tactic": ["rw [<a>ENNReal.ofReal_div_of_pos</a> (<a>Real.rpow_pos_of_pos</a> h\u03b5 _), <a>ENNReal.ofReal_one</a>, <a>mul_comm</a>,\n    <a>mul_one_div</a>, <a>ENNReal.le_div_iff_mul_le</a> _ (<a>Or.inl</a> <a>ENNReal.ofReal_ne_top</a>), <a>mul_comm</a>]", [{"full_name": "ENNReal.ofReal_div_of_pos", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2248, 9], "def_end_pos": [2248, 26]}, {"full_name": "Real.rpow_pos_of_pos", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [92, 9], "def_end_pos": [92, 24]}, {"full_name": "ENNReal.ofReal_one", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [248, 17], "def_end_pos": [248, 27]}, {"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [302, 9], "def_end_pos": [302, 17]}, {"full_name": "mul_one_div", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [300, 9], "def_end_pos": [300, 20]}, {"full_name": "ENNReal.le_div_iff_mul_le", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1611, 19], "def_end_pos": [1611, 36]}, {"full_name": "Or.inl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [517, 5], "def_end_pos": [517, 8]}, {"full_name": "ENNReal.ofReal_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [311, 17], "def_end_pos": [311, 30]}, {"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [302, 9], "def_end_pos": [302, 17]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\nE : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup E\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhf : \u2200 (n : \u03b9), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nl : Filter \u03b9\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhfg :\n  \u2200 (\u03b5_1 : \u211d\u22650\u221e),\n    \u03b5_1 > 0 \u2192 \u2200\u1da0 (x : \u03b9) in l, ENNReal.ofReal (1 / \u03b5 ^ ENNReal.toReal p) * snorm (f x - g) p \u03bc ^ ENNReal.toReal p \u2264 \u03b5_1\n\u03b4 : \u211d\u22650\u221e\nh\u03b4 : \u03b4 > 0\nn : \u03b9\nhn : ENNReal.ofReal (1 / \u03b5 ^ ENNReal.toReal p) * snorm (f n - g) p \u03bc ^ ENNReal.toReal p \u2264 \u03b4\n\u22a2 \u2191\u2191\u03bc {x | \u03b5 \u2264 dist (f n x) (g x)} \u2264 ENNReal.ofReal (1 / \u03b5 ^ ENNReal.toReal p) * snorm (f n - g) p \u03bc ^ ENNReal.toReal p", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\nE : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup E\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhf : \u2200 (n : \u03b9), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nl : Filter \u03b9\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhfg :\n  \u2200 (\u03b5_1 : \u211d\u22650\u221e),\n    \u03b5_1 > 0 \u2192 \u2200\u1da0 (x : \u03b9) in l, ENNReal.ofReal (1 / \u03b5 ^ ENNReal.toReal p) * snorm (f x - g) p \u03bc ^ ENNReal.toReal p \u2264 \u03b5_1\n\u03b4 : \u211d\u22650\u221e\nh\u03b4 : \u03b4 > 0\nn : \u03b9\nhn : ENNReal.ofReal (1 / \u03b5 ^ ENNReal.toReal p) * snorm (f n - g) p \u03bc ^ ENNReal.toReal p \u2264 \u03b4\n\u22a2 ENNReal.ofReal (\u03b5 ^ ENNReal.toReal p) * \u2191\u2191\u03bc {x | \u03b5 \u2264 dist (f n x) (g x)} \u2264 snorm (f n - g) p \u03bc ^ ENNReal.toReal p\n\n\u03b1 : Type u_1\n\u03b9 : Type u_2\nE : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup E\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhf : \u2200 (n : \u03b9), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nl : Filter \u03b9\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhfg :\n  \u2200 (\u03b5_1 : \u211d\u22650\u221e),\n    \u03b5_1 > 0 \u2192 \u2200\u1da0 (x : \u03b9) in l, ENNReal.ofReal (1 / \u03b5 ^ ENNReal.toReal p) * snorm (f x - g) p \u03bc ^ ENNReal.toReal p \u2264 \u03b5_1\n\u03b4 : \u211d\u22650\u221e\nh\u03b4 : \u03b4 > 0\nn : \u03b9\nhn : ENNReal.ofReal (1 / \u03b5 ^ ENNReal.toReal p) * snorm (f n - g) p \u03bc ^ ENNReal.toReal p \u2264 \u03b4\n\u22a2 ENNReal.ofReal (\u03b5 ^ ENNReal.toReal p) \u2260 0 \u2228 snorm (f n - g) p \u03bc ^ ENNReal.toReal p \u2260 0"}, {"tactic": "rw [\u2190 ENNReal.ofReal_rpow_of_pos h\u03b5]", "annotated_tactic": ["rw [\u2190 <a>ENNReal.ofReal_rpow_of_pos</a> h\u03b5]", [{"full_name": "ENNReal.ofReal_rpow_of_pos", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [816, 9], "def_end_pos": [816, 27]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\nE : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup E\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhf : \u2200 (n : \u03b9), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nl : Filter \u03b9\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhfg :\n  \u2200 (\u03b5_1 : \u211d\u22650\u221e),\n    \u03b5_1 > 0 \u2192 \u2200\u1da0 (x : \u03b9) in l, ENNReal.ofReal (1 / \u03b5 ^ ENNReal.toReal p) * snorm (f x - g) p \u03bc ^ ENNReal.toReal p \u2264 \u03b5_1\n\u03b4 : \u211d\u22650\u221e\nh\u03b4 : \u03b4 > 0\nn : \u03b9\nhn : ENNReal.ofReal (1 / \u03b5 ^ ENNReal.toReal p) * snorm (f n - g) p \u03bc ^ ENNReal.toReal p \u2264 \u03b4\n\u22a2 ENNReal.ofReal (\u03b5 ^ ENNReal.toReal p) * \u2191\u2191\u03bc {x | \u03b5 \u2264 dist (f n x) (g x)} \u2264 snorm (f n - g) p \u03bc ^ ENNReal.toReal p", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\nE : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup E\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhf : \u2200 (n : \u03b9), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nl : Filter \u03b9\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhfg :\n  \u2200 (\u03b5_1 : \u211d\u22650\u221e),\n    \u03b5_1 > 0 \u2192 \u2200\u1da0 (x : \u03b9) in l, ENNReal.ofReal (1 / \u03b5 ^ ENNReal.toReal p) * snorm (f x - g) p \u03bc ^ ENNReal.toReal p \u2264 \u03b5_1\n\u03b4 : \u211d\u22650\u221e\nh\u03b4 : \u03b4 > 0\nn : \u03b9\nhn : ENNReal.ofReal (1 / \u03b5 ^ ENNReal.toReal p) * snorm (f n - g) p \u03bc ^ ENNReal.toReal p \u2264 \u03b4\n\u22a2 ENNReal.ofReal \u03b5 ^ ENNReal.toReal p * \u2191\u2191\u03bc {x | \u03b5 \u2264 dist (f n x) (g x)} \u2264 snorm (f n - g) p \u03bc ^ ENNReal.toReal p"}, {"tactic": "convert mul_meas_ge_le_pow_snorm' \u03bc hp_ne_zero hp_ne_top ((hf n).sub hg).aestronglyMeasurable\n    (ENNReal.ofReal \u03b5)", "annotated_tactic": ["convert <a>mul_meas_ge_le_pow_snorm'</a> \u03bc hp_ne_zero hp_ne_top ((hf n).<a>sub</a> hg).<a>aestronglyMeasurable</a>\n        (<a>ENNReal.ofReal</a> \u03b5)", [{"full_name": "MeasureTheory.mul_meas_ge_le_pow_snorm'", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [1164, 9], "def_end_pos": [1164, 34]}, {"full_name": "MeasureTheory.StronglyMeasurable.sub", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [436, 3], "def_end_pos": [436, 14]}, {"full_name": "MeasureTheory.StronglyMeasurable.aestronglyMeasurable", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [110, 19], "def_end_pos": [110, 58]}, {"full_name": "ENNReal.ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [172, 29], "def_end_pos": [172, 35]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\nE : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup E\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhf : \u2200 (n : \u03b9), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nl : Filter \u03b9\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhfg :\n  \u2200 (\u03b5_1 : \u211d\u22650\u221e),\n    \u03b5_1 > 0 \u2192 \u2200\u1da0 (x : \u03b9) in l, ENNReal.ofReal (1 / \u03b5 ^ ENNReal.toReal p) * snorm (f x - g) p \u03bc ^ ENNReal.toReal p \u2264 \u03b5_1\n\u03b4 : \u211d\u22650\u221e\nh\u03b4 : \u03b4 > 0\nn : \u03b9\nhn : ENNReal.ofReal (1 / \u03b5 ^ ENNReal.toReal p) * snorm (f n - g) p \u03bc ^ ENNReal.toReal p \u2264 \u03b4\n\u22a2 ENNReal.ofReal \u03b5 ^ ENNReal.toReal p * \u2191\u2191\u03bc {x | \u03b5 \u2264 dist (f n x) (g x)} \u2264 snorm (f n - g) p \u03bc ^ ENNReal.toReal p", "state_after": "case h.e'_3.h.e'_6.h.e'_3.h.e'_2.h.a\n\u03b1 : Type u_1\n\u03b9 : Type u_2\nE : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup E\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhf : \u2200 (n : \u03b9), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nl : Filter \u03b9\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhfg :\n  \u2200 (\u03b5_1 : \u211d\u22650\u221e),\n    \u03b5_1 > 0 \u2192 \u2200\u1da0 (x : \u03b9) in l, ENNReal.ofReal (1 / \u03b5 ^ ENNReal.toReal p) * snorm (f x - g) p \u03bc ^ ENNReal.toReal p \u2264 \u03b5_1\n\u03b4 : \u211d\u22650\u221e\nh\u03b4 : \u03b4 > 0\nn : \u03b9\nhn : ENNReal.ofReal (1 / \u03b5 ^ ENNReal.toReal p) * snorm (f n - g) p \u03bc ^ ENNReal.toReal p \u2264 \u03b4\nx\u271d : \u03b1\n\u22a2 \u03b5 \u2264 dist (f n x\u271d) (g x\u271d) \u2194 ENNReal.ofReal \u03b5 \u2264 \u2191\u2016(f n - g) x\u271d\u2016\u208a"}, {"tactic": "rw [dist_eq_norm, \u2190 ENNReal.ofReal_le_ofReal_iff (norm_nonneg _), ofReal_norm_eq_coe_nnnorm]", "annotated_tactic": ["rw [<a>dist_eq_norm</a>, \u2190 <a>ENNReal.ofReal_le_ofReal_iff</a> (<a>norm_nonneg</a> _), <a>ofReal_norm_eq_coe_nnnorm</a>]", [{"full_name": "dist_eq_norm", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [383, 7], "def_end_pos": [383, 19]}, {"full_name": "ENNReal.ofReal_le_ofReal_iff", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2145, 9], "def_end_pos": [2145, 29]}, {"full_name": "norm_nonneg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [500, 30], "def_end_pos": [500, 41]}, {"full_name": "ofReal_norm_eq_coe_nnnorm", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [999, 15], "def_end_pos": [999, 40]}]], "state_before": "case h.e'_3.h.e'_6.h.e'_3.h.e'_2.h.a\n\u03b1 : Type u_1\n\u03b9 : Type u_2\nE : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup E\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhf : \u2200 (n : \u03b9), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nl : Filter \u03b9\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhfg :\n  \u2200 (\u03b5_1 : \u211d\u22650\u221e),\n    \u03b5_1 > 0 \u2192 \u2200\u1da0 (x : \u03b9) in l, ENNReal.ofReal (1 / \u03b5 ^ ENNReal.toReal p) * snorm (f x - g) p \u03bc ^ ENNReal.toReal p \u2264 \u03b5_1\n\u03b4 : \u211d\u22650\u221e\nh\u03b4 : \u03b4 > 0\nn : \u03b9\nhn : ENNReal.ofReal (1 / \u03b5 ^ ENNReal.toReal p) * snorm (f n - g) p \u03bc ^ ENNReal.toReal p \u2264 \u03b4\nx\u271d : \u03b1\n\u22a2 \u03b5 \u2264 dist (f n x\u271d) (g x\u271d) \u2194 ENNReal.ofReal \u03b5 \u2264 \u2191\u2016(f n - g) x\u271d\u2016\u208a", "state_after": "case h.e'_3.h.e'_6.h.e'_3.h.e'_2.h.a\n\u03b1 : Type u_1\n\u03b9 : Type u_2\nE : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup E\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhf : \u2200 (n : \u03b9), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nl : Filter \u03b9\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhfg :\n  \u2200 (\u03b5_1 : \u211d\u22650\u221e),\n    \u03b5_1 > 0 \u2192 \u2200\u1da0 (x : \u03b9) in l, ENNReal.ofReal (1 / \u03b5 ^ ENNReal.toReal p) * snorm (f x - g) p \u03bc ^ ENNReal.toReal p \u2264 \u03b5_1\n\u03b4 : \u211d\u22650\u221e\nh\u03b4 : \u03b4 > 0\nn : \u03b9\nhn : ENNReal.ofReal (1 / \u03b5 ^ ENNReal.toReal p) * snorm (f n - g) p \u03bc ^ ENNReal.toReal p \u2264 \u03b4\nx\u271d : \u03b1\n\u22a2 ENNReal.ofReal \u03b5 \u2264 \u2191\u2016f n x\u271d - g x\u271d\u2016\u208a \u2194 ENNReal.ofReal \u03b5 \u2264 \u2191\u2016(f n - g) x\u271d\u2016\u208a"}, {"tactic": "exact Iff.rfl", "annotated_tactic": ["exact <a>Iff.rfl</a>", [{"full_name": "Iff.rfl", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [663, 19], "def_end_pos": [663, 26]}]], "state_before": "case h.e'_3.h.e'_6.h.e'_3.h.e'_2.h.a\n\u03b1 : Type u_1\n\u03b9 : Type u_2\nE : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup E\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhf : \u2200 (n : \u03b9), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nl : Filter \u03b9\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhfg :\n  \u2200 (\u03b5_1 : \u211d\u22650\u221e),\n    \u03b5_1 > 0 \u2192 \u2200\u1da0 (x : \u03b9) in l, ENNReal.ofReal (1 / \u03b5 ^ ENNReal.toReal p) * snorm (f x - g) p \u03bc ^ ENNReal.toReal p \u2264 \u03b5_1\n\u03b4 : \u211d\u22650\u221e\nh\u03b4 : \u03b4 > 0\nn : \u03b9\nhn : ENNReal.ofReal (1 / \u03b5 ^ ENNReal.toReal p) * snorm (f n - g) p \u03bc ^ ENNReal.toReal p \u2264 \u03b4\nx\u271d : \u03b1\n\u22a2 ENNReal.ofReal \u03b5 \u2264 \u2191\u2016f n x\u271d - g x\u271d\u2016\u208a \u2194 ENNReal.ofReal \u03b5 \u2264 \u2191\u2016(f n - g) x\u271d\u2016\u208a", "state_after": "no goals"}, {"tactic": "rw [Ne, ENNReal.ofReal_eq_zero, not_le]", "annotated_tactic": ["rw [<a>Ne</a>, <a>ENNReal.ofReal_eq_zero</a>, <a>not_le</a>]", [{"full_name": "Ne", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [560, 18], "def_end_pos": [560, 20]}, {"full_name": "ENNReal.ofReal_eq_zero", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2170, 9], "def_end_pos": [2170, 23]}, {"full_name": "not_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [373, 9], "def_end_pos": [373, 15]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\nE : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup E\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhf : \u2200 (n : \u03b9), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nl : Filter \u03b9\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhfg :\n  \u2200 (\u03b5_1 : \u211d\u22650\u221e),\n    \u03b5_1 > 0 \u2192 \u2200\u1da0 (x : \u03b9) in l, ENNReal.ofReal (1 / \u03b5 ^ ENNReal.toReal p) * snorm (f x - g) p \u03bc ^ ENNReal.toReal p \u2264 \u03b5_1\n\u03b4 : \u211d\u22650\u221e\nh\u03b4 : \u03b4 > 0\nn : \u03b9\nhn : ENNReal.ofReal (1 / \u03b5 ^ ENNReal.toReal p) * snorm (f n - g) p \u03bc ^ ENNReal.toReal p \u2264 \u03b4\n\u22a2 ENNReal.ofReal (\u03b5 ^ ENNReal.toReal p) \u2260 0 \u2228 snorm (f n - g) p \u03bc ^ ENNReal.toReal p \u2260 0", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\nE : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup E\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhf : \u2200 (n : \u03b9), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nl : Filter \u03b9\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhfg :\n  \u2200 (\u03b5_1 : \u211d\u22650\u221e),\n    \u03b5_1 > 0 \u2192 \u2200\u1da0 (x : \u03b9) in l, ENNReal.ofReal (1 / \u03b5 ^ ENNReal.toReal p) * snorm (f x - g) p \u03bc ^ ENNReal.toReal p \u2264 \u03b5_1\n\u03b4 : \u211d\u22650\u221e\nh\u03b4 : \u03b4 > 0\nn : \u03b9\nhn : ENNReal.ofReal (1 / \u03b5 ^ ENNReal.toReal p) * snorm (f n - g) p \u03bc ^ ENNReal.toReal p \u2264 \u03b4\n\u22a2 0 < \u03b5 ^ ENNReal.toReal p \u2228 snorm (f n - g) p \u03bc ^ ENNReal.toReal p \u2260 0"}, {"tactic": "exact Or.inl (Real.rpow_pos_of_pos h\u03b5 _)", "annotated_tactic": ["exact <a>Or.inl</a> (<a>Real.rpow_pos_of_pos</a> h\u03b5 _)", [{"full_name": "Or.inl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [517, 5], "def_end_pos": [517, 8]}, {"full_name": "Real.rpow_pos_of_pos", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [92, 9], "def_end_pos": [92, 24]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\nE : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup E\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhf : \u2200 (n : \u03b9), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nl : Filter \u03b9\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhfg :\n  \u2200 (\u03b5_1 : \u211d\u22650\u221e),\n    \u03b5_1 > 0 \u2192 \u2200\u1da0 (x : \u03b9) in l, ENNReal.ofReal (1 / \u03b5 ^ ENNReal.toReal p) * snorm (f x - g) p \u03bc ^ ENNReal.toReal p \u2264 \u03b5_1\n\u03b4 : \u211d\u22650\u221e\nh\u03b4 : \u03b4 > 0\nn : \u03b9\nhn : ENNReal.ofReal (1 / \u03b5 ^ ENNReal.toReal p) * snorm (f n - g) p \u03bc ^ ENNReal.toReal p \u2264 \u03b4\n\u22a2 0 < \u03b5 ^ ENNReal.toReal p \u2228 snorm (f n - g) p \u03bc ^ ENNReal.toReal p \u2260 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Basic.lean", "full_name": "Set.insert_union_distrib", "start": [1228, 1], "end": [1229, 34], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Pointwise/SMul.lean", "full_name": "Set.image2_vsub", "start": [602, 1], "end": [603, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/ZMod/Basic.lean", "full_name": "ZMod.castHom_bijective", "start": [420, 1], "end": [428, 31], "traced_tactics": [{"tactic": "haveI : NeZero n :=\n  \u27e8by\n    intro hn\n    rw [hn] at h\n    exact (Fintype.card_eq_zero_iff.mp h).elim' 0\u27e9", "annotated_tactic": ["haveI : <a>NeZero</a> n :=\n    \u27e8by\n      intro hn\n      rw [hn] at h\n      exact (Fintype.card_eq_zero_iff.mp h).<a>elim'</a> 0\u27e9", [{"full_name": "NeZero", "def_path": "Mathlib/Algebra/NeZero.lean", "def_pos": [25, 7], "def_end_pos": [25, 13]}, {"full_name": "IsEmpty.elim'", "def_path": "Mathlib/Logic/IsEmpty.lean", "def_pos": [110, 15], "def_end_pos": [110, 20]}]], "state_before": "n : \u2115\nR : Type u_1\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : CharP R n\ninst\u271d : Fintype R\nh : Fintype.card R = n\n\u22a2 Bijective \u2191(castHom (_ : n \u2223 n) R)", "state_after": "n : \u2115\nR : Type u_1\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : CharP R n\ninst\u271d : Fintype R\nh : Fintype.card R = n\nthis : NeZero n\n\u22a2 Bijective \u2191(castHom (_ : n \u2223 n) R)"}, {"tactic": "rw [Fintype.bijective_iff_injective_and_card, ZMod.card, h, eq_self_iff_true, and_true_iff]", "annotated_tactic": ["rw [<a>Fintype.bijective_iff_injective_and_card</a>, <a>ZMod.card</a>, h, <a>eq_self_iff_true</a>, <a>and_true_iff</a>]", [{"full_name": "Fintype.bijective_iff_injective_and_card", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [688, 9], "def_end_pos": [688, 41]}, {"full_name": "ZMod.card", "def_path": "Mathlib/Data/ZMod/Defs.lean", "def_pos": [121, 9], "def_end_pos": [121, 13]}, {"full_name": "eq_self_iff_true", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [86, 9], "def_end_pos": [86, 25]}, {"full_name": "and_true_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [145, 9], "def_end_pos": [145, 21]}]], "state_before": "n : \u2115\nR : Type u_1\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : CharP R n\ninst\u271d : Fintype R\nh : Fintype.card R = n\nthis : NeZero n\n\u22a2 Bijective \u2191(castHom (_ : n \u2223 n) R)", "state_after": "n : \u2115\nR : Type u_1\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : CharP R n\ninst\u271d : Fintype R\nh : Fintype.card R = n\nthis : NeZero n\n\u22a2 Injective \u2191(castHom (_ : n \u2223 n) R)"}, {"tactic": "apply ZMod.castHom_injective", "annotated_tactic": ["apply <a>ZMod.castHom_injective</a>", [{"full_name": "ZMod.castHom_injective", "def_path": "Mathlib/Data/ZMod/Basic.lean", "def_pos": [412, 9], "def_end_pos": [412, 26]}]], "state_before": "n : \u2115\nR : Type u_1\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : CharP R n\ninst\u271d : Fintype R\nh : Fintype.card R = n\nthis : NeZero n\n\u22a2 Injective \u2191(castHom (_ : n \u2223 n) R)", "state_after": "no goals"}, {"tactic": "intro hn", "annotated_tactic": ["intro hn", []], "state_before": "n : \u2115\nR : Type u_1\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : CharP R n\ninst\u271d : Fintype R\nh : Fintype.card R = n\n\u22a2 n \u2260 0", "state_after": "n : \u2115\nR : Type u_1\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : CharP R n\ninst\u271d : Fintype R\nh : Fintype.card R = n\nhn : n = 0\n\u22a2 False"}, {"tactic": "rw [hn] at h", "annotated_tactic": ["rw [hn] at h", []], "state_before": "n : \u2115\nR : Type u_1\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : CharP R n\ninst\u271d : Fintype R\nh : Fintype.card R = n\nhn : n = 0\n\u22a2 False", "state_after": "n : \u2115\nR : Type u_1\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : CharP R n\ninst\u271d : Fintype R\nh : Fintype.card R = 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NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : NormedAddCommGroup F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc T' C'\nc : \u211d\nh_smul : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T' s = c \u2022 T s\nf : { x // x \u2208 Lp E 1 }\n\u22a2 ContinuousLinearMap.comp (c \u2022 setToL1 hT) (coeToLp \u03b1 E \u211d) = setToL1SCLM \u03b1 E \u03bc hT'"}, {"tactic": "ext1 f", "annotated_tactic": ["ext1 f", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : NormedAddCommGroup F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc T' C'\nc : \u211d\nh_smul : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T' s = c \u2022 T s\nf : { x // x \u2208 Lp E 1 }\n\u22a2 ContinuousLinearMap.comp (c \u2022 setToL1 hT) (coeToLp \u03b1 E \u211d) = setToL1SCLM \u03b1 E \u03bc hT'", "state_after": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : NormedAddCommGroup F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc T' C'\nc : \u211d\nh_smul : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T' s = c \u2022 T s\nf\u271d : { x // x \u2208 Lp E 1 }\nf : { x // x \u2208 simpleFunc E 1 \u03bc }\n\u22a2 \u2191(ContinuousLinearMap.comp (c \u2022 setToL1 hT) (coeToLp \u03b1 E \u211d)) f = \u2191(setToL1SCLM \u03b1 E \u03bc hT') f"}, {"tactic": "suffices c \u2022 setToL1 hT f = setToL1SCLM \u03b1 E \u03bc hT' f by rw [\u2190 this]; congr", "annotated_tactic": ["suffices c \u2022 <a>setToL1</a> hT f = <a>setToL1SCLM</a> \u03b1 E \u03bc hT' f by rw [\u2190 this]; congr", [{"full_name": "MeasureTheory.L1.setToL1", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [1019, 5], "def_end_pos": [1019, 12]}, {"full_name": "MeasureTheory.L1.SimpleFunc.setToL1SCLM", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [877, 5], "def_end_pos": [877, 16]}]], "state_before": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : NormedAddCommGroup F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc T' C'\nc : \u211d\nh_smul : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T' s = c \u2022 T s\nf\u271d : { x // x \u2208 Lp E 1 }\nf : { x // x \u2208 simpleFunc E 1 \u03bc }\n\u22a2 \u2191(ContinuousLinearMap.comp (c \u2022 setToL1 hT) (coeToLp \u03b1 E \u211d)) f = \u2191(setToL1SCLM \u03b1 E \u03bc hT') f", "state_after": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : NormedAddCommGroup F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc T' C'\nc : \u211d\nh_smul : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T' s = c \u2022 T s\nf\u271d : { x // x \u2208 Lp E 1 }\nf : { x // x \u2208 simpleFunc E 1 \u03bc }\n\u22a2 c \u2022 \u2191(setToL1 hT) \u2191f = \u2191(setToL1SCLM \u03b1 E \u03bc hT') f"}, {"tactic": "rw [setToL1_eq_setToL1SCLM, setToL1SCLM_smul_left' c hT hT' h_smul]", "annotated_tactic": ["rw [<a>setToL1_eq_setToL1SCLM</a>, <a>setToL1SCLM_smul_left'</a> c hT hT' h_smul]", [{"full_name": "MeasureTheory.L1.setToL1_eq_setToL1SCLM", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [1024, 9], "def_end_pos": [1024, 31]}, {"full_name": "MeasureTheory.L1.SimpleFunc.setToL1SCLM_smul_left'", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [937, 9], "def_end_pos": [937, 31]}]], "state_before": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : NormedAddCommGroup F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc T' C'\nc : \u211d\nh_smul : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T' s = c \u2022 T s\nf\u271d : { x // x \u2208 Lp E 1 }\nf : { x // x \u2208 simpleFunc E 1 \u03bc }\n\u22a2 c \u2022 \u2191(setToL1 hT) \u2191f = \u2191(setToL1SCLM \u03b1 E \u03bc hT') f", "state_after": "no goals"}, {"tactic": "rw [this, ContinuousLinearMap.smul_apply]", "annotated_tactic": ["rw [this, <a>ContinuousLinearMap.smul_apply</a>]", [{"full_name": "ContinuousLinearMap.smul_apply", "def_path": "Mathlib/Topology/Algebra/Module/Basic.lean", "def_pos": [602, 9], "def_end_pos": [602, 19]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : NormedAddCommGroup F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc T' C'\nc : \u211d\nh_smul : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T' s = c \u2022 T s\nf : { x // x \u2208 Lp E 1 }\nthis : setToL1 hT' = c \u2022 setToL1 hT\n\u22a2 \u2191(setToL1 hT') f = c \u2022 \u2191(setToL1 hT) f", "state_after": "no goals"}, {"tactic": "rw [\u2190 this]", "annotated_tactic": ["rw [\u2190 this]", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : NormedAddCommGroup F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc T' C'\nc : \u211d\nh_smul : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T' s = c \u2022 T s\nf\u271d : { x // x \u2208 Lp E 1 }\nf : { x // x \u2208 simpleFunc E 1 \u03bc }\nthis : c \u2022 \u2191(setToL1 hT) \u2191f = \u2191(setToL1SCLM \u03b1 E \u03bc hT') f\n\u22a2 \u2191(ContinuousLinearMap.comp (c \u2022 setToL1 hT) (coeToLp \u03b1 E \u211d)) f = \u2191(setToL1SCLM \u03b1 E \u03bc hT') f", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : NormedAddCommGroup F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc T' C'\nc : \u211d\nh_smul : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T' s = c \u2022 T s\nf\u271d : { x // x \u2208 Lp E 1 }\nf : { x // x \u2208 simpleFunc E 1 \u03bc }\nthis : c \u2022 \u2191(setToL1 hT) \u2191f = \u2191(setToL1SCLM \u03b1 E \u03bc hT') f\n\u22a2 \u2191(ContinuousLinearMap.comp (c \u2022 setToL1 hT) (coeToLp \u03b1 E \u211d)) f = c \u2022 \u2191(setToL1 hT) \u2191f"}, {"tactic": "congr", "annotated_tactic": ["congr", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : NormedAddCommGroup F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc T' C'\nc : \u211d\nh_smul : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T' s = c \u2022 T s\nf\u271d : { x // x \u2208 Lp E 1 }\nf : { x // x \u2208 simpleFunc E 1 \u03bc }\nthis : c \u2022 \u2191(setToL1 hT) \u2191f = \u2191(setToL1SCLM \u03b1 E \u03bc hT') f\n\u22a2 \u2191(ContinuousLinearMap.comp (c \u2022 setToL1 hT) (coeToLp \u03b1 E \u211d)) f = c \u2022 \u2191(setToL1 hT) \u2191f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Kernel/Disintegration.lean", "full_name": "ProbabilityTheory.exists_cond_kernel", "start": [265, 1], "end": [343, 14], "traced_tactics": [{"tactic": "obtain \u27e8f, hf\u27e9 := exists_measurableEmbedding_real \u03a9", "annotated_tactic": ["obtain \u27e8f, hf\u27e9 := <a>exists_measurableEmbedding_real</a> \u03a9", [{"full_name": "MeasureTheory.exists_measurableEmbedding_real", "def_path": "Mathlib/MeasureTheory/Constructions/Polish.lean", "def_pos": [1072, 9], "def_end_pos": [1072, 40]}]], "state_before": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\n\u22a2 \u2203 \u03b7 _h, kernel.const \u03b3 \u03c1 = kernel.const \u03b3 (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft \u03b3 \u03b7", "state_after": "case intro\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u22a2 \u2203 \u03b7 _h, kernel.const \u03b3 \u03c1 = kernel.const \u03b3 (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft \u03b3 \u03b7"}, {"tactic": "let \u03c1' : Measure (\u03b1 \u00d7 \u211d) := \u03c1.map (Prod.map id f)", "annotated_tactic": ["let \u03c1' : <a>Measure</a> (\u03b1 \u00d7 \u211d) := \u03c1.map (<a>Prod.map</a> <a>id</a> f)", [{"full_name": "MeasureTheory.Measure", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [74, 11], "def_end_pos": [74, 18]}, {"full_name": "Prod.map", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [1043, 5], "def_end_pos": [1043, 13]}, {"full_name": "id", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [33, 15], "def_end_pos": [33, 17]}]], "state_before": "case intro\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u22a2 \u2203 \u03b7 _h, kernel.const \u03b3 \u03c1 = kernel.const \u03b3 (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft \u03b3 \u03b7", "state_after": "case intro\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u22a2 \u2203 \u03b7 _h, kernel.const \u03b3 \u03c1 = kernel.const \u03b3 (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft \u03b3 \u03b7"}, {"tactic": "let \u03c1_set := (toMeasurable \u03c1.fst {a | condKernelReal \u03c1' a (range f) = 1}\u1d9c)\u1d9c", "annotated_tactic": ["let \u03c1_set := (<a>toMeasurable</a> \u03c1.fst {a | <a>condKernelReal</a> \u03c1' a (<a>range</a> f) = 1}\u1d9c)\u1d9c", [{"full_name": "MeasureTheory.toMeasurable", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [626, 17], "def_end_pos": [626, 29]}, {"full_name": "ProbabilityTheory.condKernelReal", "def_path": "Mathlib/Probability/Kernel/Disintegration.lean", "def_pos": [65, 19], "def_end_pos": [65, 33]}, {"full_name": "Set.range", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [668, 5], "def_end_pos": [668, 10]}]], "state_before": "case intro\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u22a2 \u2203 \u03b7 _h, kernel.const \u03b3 \u03c1 = kernel.const \u03b3 (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft \u03b3 \u03b7", "state_after": "case intro\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\n\u22a2 \u2203 \u03b7 _h, kernel.const \u03b3 \u03c1 = kernel.const \u03b3 (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft \u03b3 \u03b7"}, {"tactic": "have hm : MeasurableSet \u03c1_set := (measurableSet_toMeasurable _ _).compl", "annotated_tactic": ["have hm : <a>MeasurableSet</a> \u03c1_set := (<a>measurableSet_toMeasurable</a> _ _).<a>compl</a>", [{"full_name": "MeasurableSet", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [64, 5], "def_end_pos": [64, 18]}, {"full_name": "MeasureTheory.measurableSet_toMeasurable", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [645, 9], "def_end_pos": [645, 35]}, {"full_name": "MeasurableSet.compl", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [87, 19], "def_end_pos": [87, 38]}]], "state_before": "case intro\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\n\u22a2 \u2203 \u03b7 _h, kernel.const \u03b3 \u03c1 = kernel.const \u03b3 (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft \u03b3 \u03b7", "state_after": "case intro\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\n\u22a2 \u2203 \u03b7 _h, kernel.const \u03b3 \u03c1 = kernel.const \u03b3 (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft \u03b3 \u03b7"}, {"tactic": "have h_prod_embed : MeasurableEmbedding (Prod.map (id : \u03b1 \u2192 \u03b1) f) :=\n  MeasurableEmbedding.id.prod_mk hf", "annotated_tactic": ["have h_prod_embed : <a>MeasurableEmbedding</a> (<a>Prod.map</a> (<a>id</a> : \u03b1 \u2192 \u03b1) f) :=\n    MeasurableEmbedding.id.prod_mk hf", [{"full_name": "MeasurableEmbedding", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [1178, 11], "def_end_pos": [1178, 30]}, {"full_name": "Prod.map", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [1043, 5], "def_end_pos": [1043, 13]}, {"full_name": "id", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [33, 15], "def_end_pos": [33, 17]}]], "state_before": "case intro\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\n\u22a2 \u2203 \u03b7 _h, kernel.const \u03b3 \u03c1 = kernel.const \u03b3 (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft \u03b3 \u03b7", "state_after": "case intro\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\n\u22a2 \u2203 \u03b7 _h, kernel.const \u03b3 \u03c1 = kernel.const \u03b3 (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft \u03b3 \u03b7"}, {"tactic": "have h_fst : \u03c1'.fst = \u03c1.fst := by\n  ext1 u hu\n  rw [Measure.fst_apply hu, Measure.fst_apply hu,\n    Measure.map_apply h_prod_embed.measurable (measurable_fst hu)]\n  rfl", "annotated_tactic": ["have h_fst : \u03c1'.fst = \u03c1.fst := by\n    ext1 u hu\n    rw [<a>Measure.fst_apply</a> hu, <a>Measure.fst_apply</a> hu,\n      <a>Measure.map_apply</a> h_prod_embed.measurable (<a>measurable_fst</a> hu)]\n    rfl", [{"full_name": "MeasureTheory.Measure.fst_apply", "def_path": "Mathlib/MeasureTheory/Constructions/Prod/Basic.lean", "def_pos": [914, 9], "def_end_pos": [914, 18]}, {"full_name": "MeasureTheory.Measure.fst_apply", "def_path": "Mathlib/MeasureTheory/Constructions/Prod/Basic.lean", "def_pos": [914, 9], "def_end_pos": [914, 18]}, {"full_name": "MeasureTheory.Measure.map_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1236, 9], "def_end_pos": [1236, 18]}, {"full_name": "measurable_fst", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [692, 9], "def_end_pos": [692, 23]}]], "state_before": "case intro\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\n\u22a2 \u2203 \u03b7 _h, kernel.const \u03b3 \u03c1 = kernel.const \u03b3 (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft \u03b3 \u03b7", "state_after": "case intro\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\n\u22a2 \u2203 \u03b7 _h, kernel.const \u03b3 \u03c1 = kernel.const \u03b3 (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft \u03b3 \u03b7"}, {"tactic": "intro a ha", "annotated_tactic": ["intro a ha", []], "state_before": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\n\u22a2 \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1", "state_after": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\na : \u03b1\nha : a \u2208 \u03c1_set\n\u22a2 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1"}, {"tactic": "rw [mem_compl_iff] at ha", "annotated_tactic": ["rw [<a>mem_compl_iff</a>] at ha", [{"full_name": "Set.mem_compl_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1658, 9], "def_end_pos": [1658, 22]}]], "state_before": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\na : \u03b1\nha : a \u2208 \u03c1_set\n\u22a2 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1", "state_after": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\na : \u03b1\nha : \u00aca \u2208 toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c\n\u22a2 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1"}, {"tactic": "have h_ss := subset_toMeasurable \u03c1.fst {a : \u03b1 | condKernelReal \u03c1' a (range f) = 1}\u1d9c", "annotated_tactic": ["have h_ss := <a>subset_toMeasurable</a> \u03c1.fst {a : \u03b1 | <a>condKernelReal</a> \u03c1' a (<a>range</a> f) = 1}\u1d9c", [{"full_name": "MeasureTheory.subset_toMeasurable", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [633, 9], "def_end_pos": [633, 28]}, {"full_name": "ProbabilityTheory.condKernelReal", "def_path": "Mathlib/Probability/Kernel/Disintegration.lean", "def_pos": [65, 19], "def_end_pos": [65, 33]}, {"full_name": "Set.range", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [668, 5], "def_end_pos": [668, 10]}]], "state_before": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\na : \u03b1\nha : \u00aca \u2208 toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c\n\u22a2 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1", "state_after": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\na : \u03b1\nha : \u00aca \u2208 toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c\nh_ss :\n  {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c \u2286\n    toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c\n\u22a2 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1"}, {"tactic": "suffices ha' : a \u2209 {a : \u03b1 | condKernelReal \u03c1' a (range f) = 1}\u1d9c", "annotated_tactic": ["suffices ha' : a \u2209 {a : \u03b1 | <a>condKernelReal</a> \u03c1' a (<a>range</a> f) = 1}\u1d9c", [{"full_name": "ProbabilityTheory.condKernelReal", "def_path": "Mathlib/Probability/Kernel/Disintegration.lean", "def_pos": [65, 19], "def_end_pos": [65, 33]}, {"full_name": "Set.range", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [668, 5], "def_end_pos": [668, 10]}]], "state_before": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\na : \u03b1\nha : \u00aca \u2208 toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c\nh_ss :\n  {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c \u2286\n    toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c\n\u22a2 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1", "state_after": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\na : \u03b1\nha : \u00aca \u2208 toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c\nh_ss :\n  {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c \u2286\n    toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c\nha' : \u00aca \u2208 {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c\n\u22a2 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\n\ncase ha'\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\na : \u03b1\nha : \u00aca \u2208 toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c\nh_ss :\n  {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c \u2286\n    toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c\n\u22a2 \u00aca \u2208 {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c"}, {"tactic": "exact not_mem_subset h_ss ha", "annotated_tactic": ["exact <a>not_mem_subset</a> h_ss ha", [{"full_name": "Set.not_mem_subset", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [387, 9], "def_end_pos": [387, 23]}]], "state_before": "case ha'\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\na : \u03b1\nha : \u00aca \u2208 toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c\nh_ss :\n  {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c \u2286\n    toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c\n\u22a2 \u00aca \u2208 {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c", "state_after": "no goals"}, {"tactic": "rwa [not_mem_compl_iff] at ha'", "annotated_tactic": ["rwa [<a>not_mem_compl_iff</a>] at ha'", [{"full_name": "Set.not_mem_compl_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1662, 9], "def_end_pos": [1662, 26]}]], "state_before": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\na : \u03b1\nha : \u00aca \u2208 toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c\nh_ss :\n  {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c \u2286\n    toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c\nha' : \u00aca \u2208 {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c\n\u22a2 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1", "state_after": "no goals"}, {"tactic": "ext1 u hu", "annotated_tactic": ["ext1 u hu", []], "state_before": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\n\u22a2 Measure.fst \u03c1' = Measure.fst \u03c1", "state_after": "case h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nu : Set \u03b1\nhu : MeasurableSet u\n\u22a2 \u2191\u2191(Measure.fst \u03c1') u = \u2191\u2191(Measure.fst \u03c1) u"}, {"tactic": "rw [Measure.fst_apply hu, Measure.fst_apply hu,\n  Measure.map_apply h_prod_embed.measurable (measurable_fst hu)]", "annotated_tactic": ["rw [<a>Measure.fst_apply</a> hu, <a>Measure.fst_apply</a> hu,\n      <a>Measure.map_apply</a> h_prod_embed.measurable (<a>measurable_fst</a> hu)]", [{"full_name": "MeasureTheory.Measure.fst_apply", "def_path": "Mathlib/MeasureTheory/Constructions/Prod/Basic.lean", "def_pos": [914, 9], "def_end_pos": [914, 18]}, {"full_name": "MeasureTheory.Measure.fst_apply", "def_path": "Mathlib/MeasureTheory/Constructions/Prod/Basic.lean", "def_pos": [914, 9], "def_end_pos": [914, 18]}, {"full_name": "MeasureTheory.Measure.map_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1236, 9], "def_end_pos": [1236, 18]}, {"full_name": "measurable_fst", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [692, 9], "def_end_pos": [692, 23]}]], "state_before": "case h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nu : Set \u03b1\nhu : MeasurableSet u\n\u22a2 \u2191\u2191(Measure.fst \u03c1') u = \u2191\u2191(Measure.fst \u03c1) u", "state_after": "case h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nu : Set \u03b1\nhu : MeasurableSet u\n\u22a2 \u2191\u2191\u03c1 (Prod.map id f \u207b\u00b9' (Prod.fst \u207b\u00b9' u)) = \u2191\u2191\u03c1 (Prod.fst \u207b\u00b9' u)"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nu : Set \u03b1\nhu : MeasurableSet u\n\u22a2 \u2191\u2191\u03c1 (Prod.map id f \u207b\u00b9' (Prod.fst \u207b\u00b9' u)) = \u2191\u2191\u03c1 (Prod.fst \u207b\u00b9' u)", "state_after": "no goals"}, {"tactic": "rw [ae_iff]", "annotated_tactic": ["rw [<a>ae_iff</a>]", [{"full_name": "MeasureTheory.ae_iff", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [388, 9], "def_end_pos": [388, 15]}]], "state_before": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set", "state_after": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\n\u22a2 \u2191\u2191(Measure.fst \u03c1) {a | \u00aca \u2208 \u03c1_set} = 0"}, {"tactic": "simp only [not_mem_compl_iff, setOf_mem_eq, measure_toMeasurable]", "annotated_tactic": ["simp only [<a>not_mem_compl_iff</a>, <a>setOf_mem_eq</a>, <a>measure_toMeasurable</a>]", [{"full_name": "Set.not_mem_compl_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1662, 9], "def_end_pos": [1662, 26]}, {"full_name": "Set.setOf_mem_eq", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [275, 9], "def_end_pos": [275, 21]}, {"full_name": "MeasureTheory.measure_toMeasurable", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [653, 9], "def_end_pos": [653, 29]}]], "state_before": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\n\u22a2 \u2191\u2191(Measure.fst \u03c1) {a | \u00aca \u2208 \u03c1_set} = 0", "state_after": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\n\u22a2 \u2191\u2191(Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal (Measure.map (Prod.map id f) \u03c1)) a) (range f) = 1}\u1d9c = 0"}, {"tactic": "change \u03c1.fst {a : \u03b1 | a \u2209 {a' : \u03b1 | condKernelReal \u03c1' a' (range f) = 1}} = 0", "annotated_tactic": ["change \u03c1.fst {a : \u03b1 | a \u2209 {a' : \u03b1 | <a>condKernelReal</a> \u03c1' a' (<a>range</a> f) = 1}} = 0", [{"full_name": "ProbabilityTheory.condKernelReal", "def_path": "Mathlib/Probability/Kernel/Disintegration.lean", "def_pos": [65, 19], "def_end_pos": [65, 33]}, {"full_name": "Set.range", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [668, 5], "def_end_pos": [668, 10]}]], "state_before": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\n\u22a2 \u2191\u2191(Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal (Measure.map (Prod.map id f) \u03c1)) a) (range f) = 1}\u1d9c = 0", "state_after": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\n\u22a2 \u2191\u2191(Measure.fst \u03c1) {a | \u00aca \u2208 {a' | \u2191\u2191(\u2191(condKernelReal \u03c1') a') (range f) = 1}} = 0"}, {"tactic": "rw [\u2190 ae_iff, \u2190 h_fst]", "annotated_tactic": ["rw [\u2190 <a>ae_iff</a>, \u2190 h_fst]", [{"full_name": "MeasureTheory.ae_iff", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [388, 9], "def_end_pos": [388, 15]}]], "state_before": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\n\u22a2 \u2191\u2191(Measure.fst \u03c1) {a | \u00aca \u2208 {a' | \u2191\u2191(\u2191(condKernelReal \u03c1') a') (range f) = 1}} = 0", "state_after": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1', a \u2208 {a' | \u2191\u2191(\u2191(condKernelReal \u03c1') a') (range f) = 1}"}, {"tactic": "refine' ae_condKernelReal_eq_one \u03c1' hf.measurableSet_range _", "annotated_tactic": ["refine' <a>ae_condKernelReal_eq_one</a> \u03c1' hf.measurableSet_range _", [{"full_name": "ProbabilityTheory.ae_condKernelReal_eq_one", "def_path": "Mathlib/Probability/Kernel/Disintegration.lean", "def_pos": [232, 9], "def_end_pos": [232, 33]}]], "state_before": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1', a \u2208 {a' | \u2191\u2191(\u2191(condKernelReal \u03c1') a') (range f) = 1}", "state_after": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\n\u22a2 \u2191\u2191\u03c1' {x | x.2 \u2208 (range f)\u1d9c} = 0"}, {"tactic": "rw [Measure.map_apply h_prod_embed.measurable]", "annotated_tactic": ["rw [<a>Measure.map_apply</a> h_prod_embed.measurable]", [{"full_name": "MeasureTheory.Measure.map_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1236, 9], "def_end_pos": [1236, 18]}]], "state_before": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\n\u22a2 \u2191\u2191\u03c1' {x | x.2 \u2208 (range f)\u1d9c} = 0", "state_after": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\n\u22a2 \u2191\u2191\u03c1 (Prod.map id f \u207b\u00b9' {x | x.2 \u2208 (range f)\u1d9c}) = 0\n\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\n\u22a2 MeasurableSet {x | x.2 \u2208 (range f)\u1d9c}"}, {"tactic": "swap", "annotated_tactic": ["swap", []], "state_before": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\n\u22a2 \u2191\u2191\u03c1 (Prod.map id f \u207b\u00b9' {x | x.2 \u2208 (range f)\u1d9c}) = 0\n\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\n\u22a2 MeasurableSet {x | x.2 \u2208 (range f)\u1d9c}", "state_after": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\n\u22a2 MeasurableSet {x | x.2 \u2208 (range f)\u1d9c}\n\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\n\u22a2 \u2191\u2191\u03c1 (Prod.map id f \u207b\u00b9' {x | x.2 \u2208 (range f)\u1d9c}) = 0"}, {"tactic": "convert measure_empty (\u03b1 := \u03b1 \u00d7 \u03a9)", "annotated_tactic": ["convert <a>measure_empty</a> (\u03b1 := \u03b1 \u00d7 \u03a9)", [{"full_name": "MeasureTheory.measure_empty", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [185, 9], "def_end_pos": [185, 22]}]], "state_before": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\n\u22a2 \u2191\u2191\u03c1 (Prod.map id f \u207b\u00b9' {x | x.2 \u2208 (range f)\u1d9c}) = 0", "state_after": "case h.e'_2.h.e'_3\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\n\u22a2 Prod.map id f \u207b\u00b9' {x | x.2 \u2208 (range f)\u1d9c} = \u2205"}, {"tactic": "ext1 x", "annotated_tactic": ["ext1 x", []], "state_before": "case h.e'_2.h.e'_3\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\n\u22a2 Prod.map id f \u207b\u00b9' {x | x.2 \u2208 (range f)\u1d9c} = \u2205", "state_after": "case h.e'_2.h.e'_3.h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\nx : \u03b1 \u00d7 \u03a9\n\u22a2 x \u2208 Prod.map id f \u207b\u00b9' {x | x.2 \u2208 (range f)\u1d9c} \u2194 x \u2208 \u2205"}, {"tactic": "simp only [mem_compl_iff, mem_range, preimage_setOf_eq, Prod_map, mem_setOf_eq,\n  mem_empty_iff_false, iff_false_iff, Classical.not_not, exists_apply_eq_apply]", "annotated_tactic": ["simp only [<a>mem_compl_iff</a>, <a>mem_range</a>, <a>preimage_setOf_eq</a>, <a>Prod_map</a>, <a>mem_setOf_eq</a>,\n      <a>mem_empty_iff_false</a>, <a>iff_false_iff</a>, <a>Classical.not_not</a>, <a>exists_apply_eq_apply</a>]", [{"full_name": "Set.mem_compl_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1658, 9], "def_end_pos": [1658, 22]}, {"full_name": "Set.mem_range", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [673, 9], "def_end_pos": [673, 18]}, {"full_name": "Set.preimage_setOf_eq", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [118, 9], "def_end_pos": [118, 26]}, {"full_name": "Prod_map", "def_path": "Mathlib/Data/Prod/Basic.lean", "def_pos": [25, 9], "def_end_pos": [25, 17]}, {"full_name": "Set.mem_setOf_eq", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [256, 29], "def_end_pos": [256, 41]}, {"full_name": "Set.mem_empty_iff_false", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [562, 9], "def_end_pos": [562, 28]}, {"full_name": "iff_false_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [201, 9], "def_end_pos": [201, 22]}, {"full_name": "Classical.not_not", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [683, 24], "def_end_pos": [683, 31]}, {"full_name": "exists_apply_eq_apply", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [488, 17], "def_end_pos": [488, 38]}]], "state_before": "case h.e'_2.h.e'_3.h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\nx : \u03b1 \u00d7 \u03a9\n\u22a2 x \u2208 Prod.map id f \u207b\u00b9' {x | x.2 \u2208 (range f)\u1d9c} \u2194 x \u2208 \u2205", "state_after": "no goals"}, {"tactic": "exact measurable_snd hf.measurableSet_range.compl", "annotated_tactic": ["exact <a>measurable_snd</a> hf.measurableSet_range.compl", [{"full_name": "measurable_snd", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [698, 9], "def_end_pos": [698, 23]}]], "state_before": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\n\u22a2 MeasurableSet {x | x.2 \u2208 (range f)\u1d9c}", "state_after": "no goals"}, {"tactic": "obtain \u27e8x\u2080, hx\u2080\u27e9 : \u2203 x, x \u2208 range f := range_nonempty _", "annotated_tactic": ["obtain \u27e8x\u2080, hx\u2080\u27e9 : \u2203 x, x \u2208 <a>range</a> f := <a>range_nonempty</a> _", [{"full_name": "Set.range", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [668, 5], "def_end_pos": [668, 10]}, {"full_name": "Set.range_nonempty", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [771, 9], "def_end_pos": [771, 23]}]], "state_before": "case intro\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\nh_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set\n\u22a2 \u2203 \u03b7 _h, kernel.const \u03b3 \u03c1 = kernel.const \u03b3 (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft \u03b3 \u03b7", "state_after": "case intro.intro\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\nh_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set\nx\u2080 : \u211d\nhx\u2080 : x\u2080 \u2208 range f\n\u22a2 \u2203 \u03b7 _h, kernel.const \u03b3 \u03c1 = kernel.const \u03b3 (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft \u03b3 \u03b7"}, {"tactic": "let \u03b7' :=\n  kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun _ => x\u2080) measurable_const)", "annotated_tactic": ["let \u03b7' :=\n    <a>kernel.piecewise</a> hm (<a>condKernelReal</a> \u03c1') (<a>kernel.deterministic</a> (fun _ => x\u2080) <a>measurable_const</a>)", [{"full_name": "ProbabilityTheory.kernel.piecewise", "def_path": "Mathlib/Probability/Kernel/Basic.lean", "def_pos": [623, 5], "def_end_pos": [623, 14]}, {"full_name": "ProbabilityTheory.condKernelReal", "def_path": "Mathlib/Probability/Kernel/Disintegration.lean", "def_pos": [65, 19], "def_end_pos": [65, 33]}, {"full_name": "ProbabilityTheory.kernel.deterministic", "def_path": "Mathlib/Probability/Kernel/Basic.lean", "def_pos": [357, 19], "def_end_pos": [357, 32]}, {"full_name": "measurable_const", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [570, 9], "def_end_pos": [570, 25]}]], "state_before": "case intro.intro\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\nh_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set\nx\u2080 : \u211d\nhx\u2080 : x\u2080 \u2208 range f\n\u22a2 \u2203 \u03b7 _h, kernel.const \u03b3 \u03c1 = kernel.const \u03b3 (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft \u03b3 \u03b7", "state_after": "case intro.intro\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\nh_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set\nx\u2080 : \u211d\nhx\u2080 : x\u2080 \u2208 range f\n\u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=\n  kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080))\n\u22a2 \u2203 \u03b7 _h, kernel.const \u03b3 \u03c1 = kernel.const \u03b3 (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft \u03b3 \u03b7"}, {"tactic": "refine' \u27e8kernel.comapRight \u03b7' hf, _, _\u27e9", "annotated_tactic": ["refine' \u27e8<a>kernel.comapRight</a> \u03b7' hf, _, _\u27e9", [{"full_name": "ProbabilityTheory.kernel.comapRight", "def_path": "Mathlib/Probability/Kernel/Basic.lean", "def_pos": [560, 19], "def_end_pos": [560, 29]}]], "state_before": "case intro.intro\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\nh_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set\nx\u2080 : \u211d\nhx\u2080 : x\u2080 \u2208 range f\n\u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=\n  kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080))\n\u22a2 \u2203 \u03b7 _h, kernel.const \u03b3 \u03c1 = kernel.const \u03b3 (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft \u03b3 \u03b7", "state_after": "case intro.intro.refine'_1\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\nh_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set\nx\u2080 : \u211d\nhx\u2080 : x\u2080 \u2208 range f\n\u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=\n  kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080))\n\u22a2 IsMarkovKernel (kernel.comapRight \u03b7' hf)\n\ncase intro.intro.refine'_2\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\nh_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set\nx\u2080 : \u211d\nhx\u2080 : x\u2080 \u2208 range f\n\u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=\n  kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080))\n\u22a2 kernel.const \u03b3 \u03c1 = kernel.const \u03b3 (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft \u03b3 (kernel.comapRight \u03b7' hf)"}, {"tactic": "have : kernel.const \u03b3 \u03c1 = kernel.comapRight (kernel.const \u03b3 \u03c1') h_prod_embed := by\n  ext c t ht : 2\n  rw [kernel.const_apply, kernel.comapRight_apply' _ _ _ ht, kernel.const_apply,\n    Measure.map_apply h_prod_embed.measurable (h_prod_embed.measurableSet_image.mpr ht)]\n  congr with x : 1\n  rw [\u2190 @Prod.mk.eta _ _ x]\n  simp only [id.def, mem_preimage, Prod.map_mk, mem_image, Prod.mk.inj_iff, Prod.exists]\n  refine' \u27e8fun h => \u27e8x.1, x.2, h, rfl, rfl\u27e9, _\u27e9\n  rintro \u27e8a, b, h_mem, rfl, hf_eq\u27e9\n  rwa [hf.injective hf_eq] at h_mem", "annotated_tactic": ["have : <a>kernel.const</a> \u03b3 \u03c1 = <a>kernel.comapRight</a> (<a>kernel.const</a> \u03b3 \u03c1') h_prod_embed := by\n    ext c t ht : 2\n    rw [<a>kernel.const_apply</a>, <a>kernel.comapRight_apply'</a> _ _ _ ht, <a>kernel.const_apply</a>,\n      <a>Measure.map_apply</a> h_prod_embed.measurable (h_prod_embed.measurableSet_image.mpr ht)]\n    congr with x : 1\n    rw [\u2190 @<a>Prod.mk.eta</a> _ _ x]\n    simp only [<a>id.def</a>, <a>mem_preimage</a>, <a>Prod.map_mk</a>, <a>mem_image</a>, <a>Prod.mk.inj_iff</a>, <a>Prod.exists</a>]\n    refine' \u27e8fun h => \u27e8x.1, x.2, h, <a>rfl</a>, <a>rfl</a>\u27e9, _\u27e9\n    rintro \u27e8a, b, h_mem, rfl, hf_eq\u27e9\n    rwa [hf.injective hf_eq] at h_mem", [{"full_name": "ProbabilityTheory.kernel.const", "def_path": "Mathlib/Probability/Kernel/Basic.lean", "def_pos": [439, 5], "def_end_pos": [439, 10]}, {"full_name": "ProbabilityTheory.kernel.comapRight", "def_path": "Mathlib/Probability/Kernel/Basic.lean", "def_pos": [560, 19], "def_end_pos": [560, 29]}, {"full_name": "ProbabilityTheory.kernel.const", "def_path": "Mathlib/Probability/Kernel/Basic.lean", "def_pos": [439, 5], "def_end_pos": [439, 10]}, {"full_name": "ProbabilityTheory.kernel.const_apply", "def_path": "Mathlib/Probability/Kernel/Basic.lean", "def_pos": [445, 9], "def_end_pos": [445, 20]}, {"full_name": "ProbabilityTheory.kernel.comapRight_apply'", "def_path": "Mathlib/Probability/Kernel/Basic.lean", "def_pos": [577, 9], "def_end_pos": [577, 26]}, {"full_name": "ProbabilityTheory.kernel.const_apply", "def_path": "Mathlib/Probability/Kernel/Basic.lean", "def_pos": [445, 9], "def_end_pos": [445, 20]}, {"full_name": "MeasureTheory.Measure.map_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1236, 9], "def_end_pos": [1236, 18]}, {"full_name": "Prod.mk.eta", "def_path": "Mathlib/Data/Prod/Basic.lean", "def_pos": [32, 9], "def_end_pos": [32, 15]}, {"full_name": "id.def", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [527, 9], "def_end_pos": [527, 15]}, {"full_name": "Set.mem_preimage", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [64, 9], "def_end_pos": [64, 21]}, {"full_name": "Prod.map_mk", "def_path": "Mathlib/Data/Prod/Basic.lean", "def_pos": [64, 9], "def_end_pos": [64, 15]}, {"full_name": "Set.mem_image", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [231, 9], "def_end_pos": [231, 18]}, {"full_name": "Prod.mk.inj_iff", "def_path": "Mathlib/Data/Prod/Basic.lean", "def_pos": [103, 9], "def_end_pos": [103, 19]}, {"full_name": "Prod.exists", "def_path": "Mathlib/Data/Prod/Basic.lean", "def_pos": [41, 9], "def_end_pos": [41, 17]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "case intro.intro.refine'_2\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\nh_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set\nx\u2080 : \u211d\nhx\u2080 : x\u2080 \u2208 range f\n\u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=\n  kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080))\n\u22a2 kernel.const \u03b3 \u03c1 = kernel.const \u03b3 (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft \u03b3 (kernel.comapRight \u03b7' hf)", "state_after": "case intro.intro.refine'_2\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\nh_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set\nx\u2080 : \u211d\nhx\u2080 : x\u2080 \u2208 range f\n\u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=\n  kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080))\nthis : kernel.const \u03b3 \u03c1 = kernel.comapRight (kernel.const \u03b3 \u03c1') h_prod_embed\n\u22a2 kernel.const \u03b3 \u03c1 = kernel.const \u03b3 (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft \u03b3 (kernel.comapRight \u03b7' hf)"}, {"tactic": "rw [this, kernel.const_eq_compProd_real _ \u03c1']", "annotated_tactic": ["rw [this, <a>kernel.const_eq_compProd_real</a> _ \u03c1']", [{"full_name": "ProbabilityTheory.kernel.const_eq_compProd_real", "def_path": "Mathlib/Probability/Kernel/Disintegration.lean", "def_pos": [211, 9], "def_end_pos": [211, 38]}]], "state_before": "case intro.intro.refine'_2\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\nh_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set\nx\u2080 : \u211d\nhx\u2080 : x\u2080 \u2208 range f\n\u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=\n  kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080))\nthis : kernel.const \u03b3 \u03c1 = kernel.comapRight (kernel.const \u03b3 \u03c1') h_prod_embed\n\u22a2 kernel.const \u03b3 \u03c1 = kernel.const \u03b3 (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft \u03b3 (kernel.comapRight \u03b7' hf)", "state_after": "case intro.intro.refine'_2\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\nh_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set\nx\u2080 : \u211d\nhx\u2080 : x\u2080 \u2208 range f\n\u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=\n  kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080))\nthis : kernel.const \u03b3 \u03c1 = kernel.comapRight (kernel.const \u03b3 \u03c1') h_prod_embed\n\u22a2 kernel.comapRight (kernel.const \u03b3 (Measure.fst \u03c1') \u2297\u2096 kernel.prodMkLeft \u03b3 (condKernelReal \u03c1')) h_prod_embed =\n    kernel.const \u03b3 (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft \u03b3 (kernel.comapRight \u03b7' hf)"}, {"tactic": "ext c t ht : 2", "annotated_tactic": ["ext c t ht : 2", []], "state_before": "case intro.intro.refine'_2\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\nh_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set\nx\u2080 : \u211d\nhx\u2080 : x\u2080 \u2208 range f\n\u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=\n  kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080))\nthis : kernel.const \u03b3 \u03c1 = kernel.comapRight (kernel.const \u03b3 \u03c1') h_prod_embed\n\u22a2 kernel.comapRight (kernel.const \u03b3 (Measure.fst \u03c1') \u2297\u2096 kernel.prodMkLeft \u03b3 (condKernelReal \u03c1')) h_prod_embed =\n    kernel.const \u03b3 (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft \u03b3 (kernel.comapRight \u03b7' hf)", "state_after": "case intro.intro.refine'_2.h.h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\nh_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set\nx\u2080 : \u211d\nhx\u2080 : x\u2080 \u2208 range f\n\u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=\n  kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080))\nthis : kernel.const \u03b3 \u03c1 = kernel.comapRight (kernel.const \u03b3 \u03c1') h_prod_embed\nc : \u03b3\nt : Set (\u03b1 \u00d7 \u03a9)\nht : MeasurableSet t\n\u22a2 \u2191\u2191(\u2191(kernel.comapRight (kernel.const \u03b3 (Measure.fst \u03c1') \u2297\u2096 kernel.prodMkLeft \u03b3 (condKernelReal \u03c1')) h_prod_embed) c)\n      t =\n    \u2191\u2191(\u2191(kernel.const \u03b3 (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft \u03b3 (kernel.comapRight \u03b7' hf)) c) t"}, {"tactic": "rw [kernel.comapRight_apply' _ _ _ ht,\n  kernel.compProd_apply _ _ _ (h_prod_embed.measurableSet_image.mpr ht), kernel.const_apply,\n  h_fst, kernel.compProd_apply _ _ _ ht, kernel.const_apply]", "annotated_tactic": ["rw [<a>kernel.comapRight_apply'</a> _ _ _ ht,\n    <a>kernel.compProd_apply</a> _ _ _ (h_prod_embed.measurableSet_image.mpr ht), <a>kernel.const_apply</a>,\n    h_fst, <a>kernel.compProd_apply</a> _ _ _ ht, <a>kernel.const_apply</a>]", [{"full_name": "ProbabilityTheory.kernel.comapRight_apply'", "def_path": "Mathlib/Probability/Kernel/Basic.lean", "def_pos": [577, 9], "def_end_pos": [577, 26]}, {"full_name": "ProbabilityTheory.kernel.compProd_apply", "def_path": "Mathlib/Probability/Kernel/Composition.lean", "def_pos": [242, 9], "def_end_pos": [242, 23]}, {"full_name": "ProbabilityTheory.kernel.const_apply", "def_path": "Mathlib/Probability/Kernel/Basic.lean", "def_pos": [445, 9], "def_end_pos": [445, 20]}, {"full_name": "ProbabilityTheory.kernel.compProd_apply", "def_path": "Mathlib/Probability/Kernel/Composition.lean", "def_pos": [242, 9], "def_end_pos": [242, 23]}, {"full_name": "ProbabilityTheory.kernel.const_apply", "def_path": "Mathlib/Probability/Kernel/Basic.lean", "def_pos": [445, 9], "def_end_pos": [445, 20]}]], "state_before": "case intro.intro.refine'_2.h.h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\nh_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set\nx\u2080 : \u211d\nhx\u2080 : x\u2080 \u2208 range f\n\u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=\n  kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080))\nthis : kernel.const \u03b3 \u03c1 = kernel.comapRight (kernel.const \u03b3 \u03c1') h_prod_embed\nc : \u03b3\nt : Set (\u03b1 \u00d7 \u03a9)\nht : MeasurableSet t\n\u22a2 \u2191\u2191(\u2191(kernel.comapRight (kernel.const \u03b3 (Measure.fst \u03c1') \u2297\u2096 kernel.prodMkLeft \u03b3 (condKernelReal \u03c1')) h_prod_embed) c)\n      t =\n    \u2191\u2191(\u2191(kernel.const \u03b3 (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft \u03b3 (kernel.comapRight \u03b7' hf)) c) t", "state_after": "case intro.intro.refine'_2.h.h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\nh_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set\nx\u2080 : \u211d\nhx\u2080 : x\u2080 \u2208 range f\n\u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=\n  kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080))\nthis : kernel.const \u03b3 \u03c1 = kernel.comapRight (kernel.const \u03b3 \u03c1') h_prod_embed\nc : \u03b3\nt : Set (\u03b1 \u00d7 \u03a9)\nht : MeasurableSet t\n\u22a2 \u222b\u207b (b : \u03b1), \u2191\u2191(\u2191(kernel.prodMkLeft \u03b3 (condKernelReal \u03c1')) (c, b)) {c | (b, c) \u2208 Prod.map id f '' t} \u2202Measure.fst \u03c1 =\n    \u222b\u207b (b : \u03b1), \u2191\u2191(\u2191(kernel.prodMkLeft \u03b3 (kernel.comapRight \u03b7' hf)) (c, b)) {c | (b, c) \u2208 t} \u2202Measure.fst \u03c1"}, {"tactic": "refine' lintegral_congr_ae _", "annotated_tactic": ["refine' <a>lintegral_congr_ae</a> _", [{"full_name": "MeasureTheory.lintegral_congr_ae", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [304, 9], "def_end_pos": [304, 27]}]], "state_before": "case intro.intro.refine'_2.h.h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\nh_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set\nx\u2080 : \u211d\nhx\u2080 : x\u2080 \u2208 range f\n\u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=\n  kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080))\nthis : kernel.const \u03b3 \u03c1 = kernel.comapRight (kernel.const \u03b3 \u03c1') h_prod_embed\nc : \u03b3\nt : Set (\u03b1 \u00d7 \u03a9)\nht : MeasurableSet t\n\u22a2 \u222b\u207b (b : \u03b1), \u2191\u2191(\u2191(kernel.prodMkLeft \u03b3 (condKernelReal \u03c1')) (c, b)) {c | (b, c) \u2208 Prod.map id f '' t} \u2202Measure.fst \u03c1 =\n    \u222b\u207b (b : \u03b1), \u2191\u2191(\u2191(kernel.prodMkLeft \u03b3 (kernel.comapRight \u03b7' hf)) (c, b)) {c | (b, c) \u2208 t} \u2202Measure.fst \u03c1", "state_after": "case intro.intro.refine'_2.h.h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\nh_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set\nx\u2080 : \u211d\nhx\u2080 : x\u2080 \u2208 range f\n\u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=\n  kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080))\nthis : kernel.const \u03b3 \u03c1 = kernel.comapRight (kernel.const \u03b3 \u03c1') h_prod_embed\nc : \u03b3\nt : Set (\u03b1 \u00d7 \u03a9)\nht : MeasurableSet t\n\u22a2 (fun b => \u2191\u2191(\u2191(kernel.prodMkLeft \u03b3 (condKernelReal \u03c1')) (c, b)) {c | (b, c) \u2208 Prod.map id f '' t}) =\u1d50[Measure.fst \u03c1]\n    fun b => \u2191\u2191(\u2191(kernel.prodMkLeft \u03b3 (kernel.comapRight \u03b7' hf)) (c, b)) {c | (b, c) \u2208 t}"}, {"tactic": "filter_upwards [h_ae] with a ha", "annotated_tactic": ["filter_upwards [h_ae] with a ha", []], "state_before": "case intro.intro.refine'_2.h.h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\nh_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set\nx\u2080 : \u211d\nhx\u2080 : x\u2080 \u2208 range f\n\u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=\n  kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080))\nthis : kernel.const \u03b3 \u03c1 = kernel.comapRight (kernel.const \u03b3 \u03c1') h_prod_embed\nc : \u03b3\nt : Set (\u03b1 \u00d7 \u03a9)\nht : MeasurableSet t\n\u22a2 (fun b => \u2191\u2191(\u2191(kernel.prodMkLeft \u03b3 (condKernelReal \u03c1')) (c, b)) {c | (b, c) \u2208 Prod.map id f '' t}) =\u1d50[Measure.fst \u03c1]\n    fun b => \u2191\u2191(\u2191(kernel.prodMkLeft \u03b3 (kernel.comapRight \u03b7' hf)) (c, b)) {c | (b, c) \u2208 t}", "state_after": "case h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\nh_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set\nx\u2080 : \u211d\nhx\u2080 : x\u2080 \u2208 range f\n\u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=\n  kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080))\nthis : kernel.const \u03b3 \u03c1 = kernel.comapRight (kernel.const \u03b3 \u03c1') h_prod_embed\nc : \u03b3\nt : Set (\u03b1 \u00d7 \u03a9)\nht : MeasurableSet t\na : \u03b1\nha : a \u2208 \u03c1_set\n\u22a2 \u2191\u2191(\u2191(kernel.prodMkLeft \u03b3 (condKernelReal \u03c1')) (c, a)) {c | (a, c) \u2208 Prod.map id f '' t} =\n    \u2191\u2191(\u2191(kernel.prodMkLeft \u03b3 (kernel.comapRight \u03b7' hf)) (c, a)) {c | (a, c) \u2208 t}"}, {"tactic": "rw [kernel.prodMkLeft_apply', kernel.prodMkLeft_apply', kernel.comapRight_apply']", "annotated_tactic": ["rw [<a>kernel.prodMkLeft_apply'</a>, <a>kernel.prodMkLeft_apply'</a>, <a>kernel.comapRight_apply'</a>]", [{"full_name": "ProbabilityTheory.kernel.prodMkLeft_apply'", "def_path": "Mathlib/Probability/Kernel/Composition.lean", "def_pos": [689, 9], "def_end_pos": [689, 26]}, {"full_name": "ProbabilityTheory.kernel.prodMkLeft_apply'", "def_path": "Mathlib/Probability/Kernel/Composition.lean", "def_pos": [689, 9], "def_end_pos": [689, 26]}, {"full_name": "ProbabilityTheory.kernel.comapRight_apply'", "def_path": "Mathlib/Probability/Kernel/Basic.lean", "def_pos": [577, 9], "def_end_pos": [577, 26]}]], "state_before": "case h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\nh_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set\nx\u2080 : \u211d\nhx\u2080 : x\u2080 \u2208 range f\n\u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=\n  kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080))\nthis : kernel.const \u03b3 \u03c1 = kernel.comapRight (kernel.const \u03b3 \u03c1') h_prod_embed\nc : \u03b3\nt : Set (\u03b1 \u00d7 \u03a9)\nht : MeasurableSet t\na : \u03b1\nha : a \u2208 \u03c1_set\n\u22a2 \u2191\u2191(\u2191(kernel.prodMkLeft \u03b3 (condKernelReal \u03c1')) (c, a)) {c | (a, c) \u2208 Prod.map id f '' t} =\n    \u2191\u2191(\u2191(kernel.prodMkLeft \u03b3 (kernel.comapRight \u03b7' hf)) (c, a)) {c | (a, c) \u2208 t}", "state_after": "case h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\nh_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set\nx\u2080 : \u211d\nhx\u2080 : x\u2080 \u2208 range f\n\u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=\n  kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080))\nthis : kernel.const \u03b3 \u03c1 = kernel.comapRight (kernel.const \u03b3 \u03c1') h_prod_embed\nc : \u03b3\nt : Set (\u03b1 \u00d7 \u03a9)\nht : MeasurableSet t\na : \u03b1\nha : a \u2208 \u03c1_set\n\u22a2 \u2191\u2191(\u2191(condKernelReal \u03c1') (c, a).2) {c | (a, c) \u2208 Prod.map id f '' t} = \u2191\u2191(\u2191\u03b7' (c, a).2) (f '' {c | (a, c) \u2208 t})\n\ncase h.ht\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\nh_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set\nx\u2080 : \u211d\nhx\u2080 : x\u2080 \u2208 range f\n\u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=\n  kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080))\nthis : kernel.const \u03b3 \u03c1 = kernel.comapRight (kernel.const \u03b3 \u03c1') h_prod_embed\nc : \u03b3\nt : Set (\u03b1 \u00d7 \u03a9)\nht : MeasurableSet t\na : \u03b1\nha : a \u2208 \u03c1_set\n\u22a2 MeasurableSet {c | (a, c) \u2208 t}"}, {"tactic": "swap", "annotated_tactic": ["swap", []], "state_before": "case h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\nh_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set\nx\u2080 : \u211d\nhx\u2080 : x\u2080 \u2208 range f\n\u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=\n  kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080))\nthis : kernel.const \u03b3 \u03c1 = kernel.comapRight (kernel.const \u03b3 \u03c1') h_prod_embed\nc : \u03b3\nt : Set (\u03b1 \u00d7 \u03a9)\nht : MeasurableSet t\na : \u03b1\nha : a \u2208 \u03c1_set\n\u22a2 \u2191\u2191(\u2191(condKernelReal \u03c1') (c, a).2) {c | (a, c) \u2208 Prod.map id f '' t} = \u2191\u2191(\u2191\u03b7' (c, a).2) (f '' {c | (a, c) \u2208 t})\n\ncase h.ht\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\nh_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set\nx\u2080 : \u211d\nhx\u2080 : x\u2080 \u2208 range f\n\u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=\n  kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080))\nthis : kernel.const \u03b3 \u03c1 = kernel.comapRight (kernel.const \u03b3 \u03c1') h_prod_embed\nc : \u03b3\nt : Set (\u03b1 \u00d7 \u03a9)\nht : MeasurableSet t\na : \u03b1\nha : a \u2208 \u03c1_set\n\u22a2 MeasurableSet {c | (a, c) \u2208 t}", "state_after": "case h.ht\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\nh_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set\nx\u2080 : \u211d\nhx\u2080 : x\u2080 \u2208 range f\n\u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=\n  kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080))\nthis : kernel.const \u03b3 \u03c1 = kernel.comapRight (kernel.const \u03b3 \u03c1') h_prod_embed\nc : \u03b3\nt : Set (\u03b1 \u00d7 \u03a9)\nht : MeasurableSet t\na : \u03b1\nha : a \u2208 \u03c1_set\n\u22a2 MeasurableSet {c | (a, c) \u2208 t}\n\ncase h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\nh_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set\nx\u2080 : \u211d\nhx\u2080 : x\u2080 \u2208 range f\n\u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=\n  kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080))\nthis : kernel.const \u03b3 \u03c1 = kernel.comapRight (kernel.const \u03b3 \u03c1') h_prod_embed\nc : \u03b3\nt : Set (\u03b1 \u00d7 \u03a9)\nht : MeasurableSet t\na : \u03b1\nha : a \u2208 \u03c1_set\n\u22a2 \u2191\u2191(\u2191(condKernelReal \u03c1') (c, a).2) {c | (a, c) \u2208 Prod.map id f '' t} = \u2191\u2191(\u2191\u03b7' (c, a).2) (f '' {c | (a, c) \u2208 t})"}, {"tactic": "have h2 : condKernelReal \u03c1' (c, a).snd = \u03b7' (c, a).snd := by\n  rw [kernel.piecewise_apply, if_pos ha]", "annotated_tactic": ["have h2 : <a>condKernelReal</a> \u03c1' (c, a).<a>snd</a> = \u03b7' (c, a).<a>snd</a> := by\n    rw [<a>kernel.piecewise_apply</a>, <a>if_pos</a> ha]", [{"full_name": "ProbabilityTheory.condKernelReal", "def_path": "Mathlib/Probability/Kernel/Disintegration.lean", "def_pos": [65, 19], "def_end_pos": [65, 33]}, {"full_name": "Prod.snd", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [470, 3], "def_end_pos": [470, 6]}, {"full_name": "Prod.snd", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [470, 3], "def_end_pos": [470, 6]}, {"full_name": "ProbabilityTheory.kernel.piecewise_apply", "def_path": "Mathlib/Probability/Kernel/Basic.lean", "def_pos": [628, 9], "def_end_pos": [628, 24]}, {"full_name": "if_pos", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [790, 9], "def_end_pos": [790, 15]}]], "state_before": "case h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\nh_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set\nx\u2080 : \u211d\nhx\u2080 : x\u2080 \u2208 range f\n\u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=\n  kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080))\nthis : kernel.const \u03b3 \u03c1 = kernel.comapRight (kernel.const \u03b3 \u03c1') h_prod_embed\nc : \u03b3\nt : Set (\u03b1 \u00d7 \u03a9)\nht : MeasurableSet t\na : \u03b1\nha : a \u2208 \u03c1_set\nh1 : {c | (a, c) \u2208 Prod.map id f '' t} = f '' {c | (a, c) \u2208 t}\n\u22a2 \u2191\u2191(\u2191(condKernelReal \u03c1') (c, a).2) {c | (a, c) \u2208 Prod.map id f '' t} = \u2191\u2191(\u2191\u03b7' (c, a).2) (f '' {c | (a, c) \u2208 t})", "state_after": "case h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\nh_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set\nx\u2080 : \u211d\nhx\u2080 : x\u2080 \u2208 range f\n\u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=\n  kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080))\nthis : kernel.const \u03b3 \u03c1 = kernel.comapRight (kernel.const \u03b3 \u03c1') h_prod_embed\nc : \u03b3\nt : Set (\u03b1 \u00d7 \u03a9)\nht : MeasurableSet t\na : \u03b1\nha : a \u2208 \u03c1_set\nh1 : {c | (a, c) \u2208 Prod.map id f '' t} = f '' {c | (a, c) \u2208 t}\nh2 : \u2191(condKernelReal \u03c1') (c, a).2 = \u2191\u03b7' (c, a).2\n\u22a2 \u2191\u2191(\u2191(condKernelReal \u03c1') (c, a).2) {c | (a, c) \u2208 Prod.map id f '' t} = \u2191\u2191(\u2191\u03b7' (c, a).2) (f '' {c | (a, c) \u2208 t})"}, {"tactic": "rw [h1, h2]", "annotated_tactic": ["rw [h1, h2]", []], "state_before": "case h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\nh_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set\nx\u2080 : \u211d\nhx\u2080 : x\u2080 \u2208 range f\n\u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=\n  kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080))\nthis : kernel.const \u03b3 \u03c1 = kernel.comapRight (kernel.const \u03b3 \u03c1') h_prod_embed\nc : \u03b3\nt : Set (\u03b1 \u00d7 \u03a9)\nht : MeasurableSet t\na : \u03b1\nha : a \u2208 \u03c1_set\nh1 : {c | (a, c) \u2208 Prod.map id f '' t} = f '' {c | (a, c) \u2208 t}\nh2 : \u2191(condKernelReal \u03c1') (c, a).2 = \u2191\u03b7' (c, a).2\n\u22a2 \u2191\u2191(\u2191(condKernelReal \u03c1') (c, a).2) {c | (a, c) \u2208 Prod.map id f '' t} = \u2191\u2191(\u2191\u03b7' (c, a).2) (f '' {c | (a, c) \u2208 t})", "state_after": "no goals"}, {"tactic": "refine' kernel.IsMarkovKernel.comapRight _ _ fun a => _", "annotated_tactic": ["refine' <a>kernel.IsMarkovKernel.comapRight</a> _ _ fun a => _", [{"full_name": "ProbabilityTheory.kernel.IsMarkovKernel.comapRight", "def_path": "Mathlib/Probability/Kernel/Basic.lean", "def_pos": [583, 9], "def_end_pos": [583, 34]}]], "state_before": "case intro.intro.refine'_1\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\nh_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set\nx\u2080 : \u211d\nhx\u2080 : x\u2080 \u2208 range f\n\u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=\n  kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080))\n\u22a2 IsMarkovKernel (kernel.comapRight \u03b7' hf)", "state_after": "case intro.intro.refine'_1\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\nh_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set\nx\u2080 : \u211d\nhx\u2080 : x\u2080 \u2208 range f\n\u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=\n  kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080))\na : \u03b1\n\u22a2 \u2191\u2191(\u2191\u03b7' a) (range f) = 1"}, {"tactic": "rw [kernel.piecewise_apply']", "annotated_tactic": ["rw [<a>kernel.piecewise_apply'</a>]", [{"full_name": "ProbabilityTheory.kernel.piecewise_apply'", "def_path": "Mathlib/Probability/Kernel/Basic.lean", "def_pos": [632, 9], "def_end_pos": [632, 25]}]], "state_before": "case intro.intro.refine'_1\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\nh_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set\nx\u2080 : \u211d\nhx\u2080 : x\u2080 \u2208 range f\n\u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=\n  kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080))\na : \u03b1\n\u22a2 \u2191\u2191(\u2191\u03b7' a) (range f) = 1", "state_after": "case intro.intro.refine'_1\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\nh_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set\nx\u2080 : \u211d\nhx\u2080 : x\u2080 \u2208 range f\n\u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=\n  kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080))\na : \u03b1\n\u22a2 (if a \u2208 \u03c1_set then \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f)\n    else \u2191\u2191(\u2191(kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080)) a) (range f)) =\n    1"}, {"tactic": "split_ifs with h_mem", "annotated_tactic": ["split_ifs with h_mem", []], "state_before": "case intro.intro.refine'_1\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\nh_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set\nx\u2080 : \u211d\nhx\u2080 : x\u2080 \u2208 range f\n\u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=\n  kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080))\na : \u03b1\n\u22a2 (if a \u2208 \u03c1_set then \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f)\n    else \u2191\u2191(\u2191(kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080)) a) (range f)) =\n    1", "state_after": "case pos\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\nh_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set\nx\u2080 : \u211d\nhx\u2080 : x\u2080 \u2208 range f\n\u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=\n  kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080))\na : \u03b1\nh_mem : a \u2208 \u03c1_set\n\u22a2 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\n\ncase neg\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\nh_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set\nx\u2080 : \u211d\nhx\u2080 : x\u2080 \u2208 range f\n\u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=\n  kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080))\na : \u03b1\nh_mem : \u00aca \u2208 \u03c1_set\n\u22a2 \u2191\u2191(\u2191(kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080)) a) (range f) = 1"}, {"tactic": "exact h_eq_one_of_mem _ h_mem", "annotated_tactic": ["exact h_eq_one_of_mem _ h_mem", []], "state_before": "case pos\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\nh_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set\nx\u2080 : \u211d\nhx\u2080 : x\u2080 \u2208 range f\n\u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=\n  kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080))\na : \u03b1\nh_mem : a \u2208 \u03c1_set\n\u22a2 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1", "state_after": "no goals"}, {"tactic": "rw [kernel.deterministic_apply' _ _ hf.measurableSet_range, Set.indicator_apply, if_pos hx\u2080]", "annotated_tactic": ["rw [<a>kernel.deterministic_apply'</a> _ _ hf.measurableSet_range, <a>Set.indicator_apply</a>, <a>if_pos</a> hx\u2080]", [{"full_name": "ProbabilityTheory.kernel.deterministic_apply'", "def_path": "Mathlib/Probability/Kernel/Basic.lean", "def_pos": [370, 9], "def_end_pos": [370, 29]}, {"full_name": "Set.indicator_apply", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [59, 3], "def_end_pos": [59, 14]}, {"full_name": "if_pos", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [790, 9], "def_end_pos": [790, 15]}]], "state_before": "case neg\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\nh_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set\nx\u2080 : \u211d\nhx\u2080 : x\u2080 \u2208 range f\n\u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=\n  kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080))\na : \u03b1\nh_mem : \u00aca \u2208 \u03c1_set\n\u22a2 \u2191\u2191(\u2191(kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080)) a) (range f) = 1", "state_after": "no goals"}, {"tactic": "ext c t ht : 2", "annotated_tactic": ["ext c t ht : 2", []], "state_before": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\nh_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set\nx\u2080 : \u211d\nhx\u2080 : x\u2080 \u2208 range f\n\u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=\n  kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080))\n\u22a2 kernel.const \u03b3 \u03c1 = kernel.comapRight (kernel.const \u03b3 \u03c1') h_prod_embed", "state_after": "case h.h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\nh_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set\nx\u2080 : \u211d\nhx\u2080 : x\u2080 \u2208 range f\n\u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=\n  kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080))\nc : \u03b3\nt : Set (\u03b1 \u00d7 \u03a9)\nht : MeasurableSet t\n\u22a2 \u2191\u2191(\u2191(kernel.const \u03b3 \u03c1) c) t = \u2191\u2191(\u2191(kernel.comapRight (kernel.const \u03b3 \u03c1') h_prod_embed) c) t"}, {"tactic": "rw [kernel.const_apply, kernel.comapRight_apply' _ _ _ ht, kernel.const_apply,\n  Measure.map_apply h_prod_embed.measurable (h_prod_embed.measurableSet_image.mpr ht)]", "annotated_tactic": ["rw [<a>kernel.const_apply</a>, <a>kernel.comapRight_apply'</a> _ _ _ ht, <a>kernel.const_apply</a>,\n      <a>Measure.map_apply</a> h_prod_embed.measurable (h_prod_embed.measurableSet_image.mpr ht)]", [{"full_name": "ProbabilityTheory.kernel.const_apply", "def_path": "Mathlib/Probability/Kernel/Basic.lean", "def_pos": [445, 9], "def_end_pos": [445, 20]}, {"full_name": "ProbabilityTheory.kernel.comapRight_apply'", "def_path": "Mathlib/Probability/Kernel/Basic.lean", "def_pos": [577, 9], "def_end_pos": [577, 26]}, {"full_name": "ProbabilityTheory.kernel.const_apply", "def_path": "Mathlib/Probability/Kernel/Basic.lean", "def_pos": [445, 9], "def_end_pos": [445, 20]}, {"full_name": "MeasureTheory.Measure.map_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1236, 9], "def_end_pos": [1236, 18]}]], "state_before": "case h.h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\nh_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set\nx\u2080 : \u211d\nhx\u2080 : x\u2080 \u2208 range f\n\u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=\n  kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080))\nc : \u03b3\nt : Set (\u03b1 \u00d7 \u03a9)\nht : MeasurableSet t\n\u22a2 \u2191\u2191(\u2191(kernel.const \u03b3 \u03c1) c) t = \u2191\u2191(\u2191(kernel.comapRight (kernel.const \u03b3 \u03c1') h_prod_embed) c) t", "state_after": "case h.h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\nh_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set\nx\u2080 : \u211d\nhx\u2080 : x\u2080 \u2208 range f\n\u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=\n  kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080))\nc : \u03b3\nt : Set (\u03b1 \u00d7 \u03a9)\nht : MeasurableSet t\n\u22a2 \u2191\u2191\u03c1 t = \u2191\u2191\u03c1 (Prod.map id f \u207b\u00b9' (Prod.map id f '' t))"}, {"tactic": "congr with x : 1", "annotated_tactic": ["congr with x : 1", []], "state_before": "case h.h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\nh_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set\nx\u2080 : \u211d\nhx\u2080 : x\u2080 \u2208 range f\n\u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=\n  kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080))\nc : \u03b3\nt : Set (\u03b1 \u00d7 \u03a9)\nht : MeasurableSet t\n\u22a2 \u2191\u2191\u03c1 t = \u2191\u2191\u03c1 (Prod.map id f \u207b\u00b9' (Prod.map id f '' t))", "state_after": "case h.h.e_a.h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\nh_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set\nx\u2080 : \u211d\nhx\u2080 : x\u2080 \u2208 range f\n\u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=\n  kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080))\nc : \u03b3\nt : Set (\u03b1 \u00d7 \u03a9)\nht : MeasurableSet t\nx : \u03b1 \u00d7 \u03a9\n\u22a2 x \u2208 t \u2194 x \u2208 Prod.map id f \u207b\u00b9' (Prod.map id f '' t)"}, {"tactic": "rw [\u2190 @Prod.mk.eta _ _ x]", "annotated_tactic": ["rw [\u2190 @<a>Prod.mk.eta</a> _ _ x]", [{"full_name": "Prod.mk.eta", "def_path": "Mathlib/Data/Prod/Basic.lean", "def_pos": [32, 9], "def_end_pos": [32, 15]}]], "state_before": "case h.h.e_a.h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\nh_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set\nx\u2080 : \u211d\nhx\u2080 : x\u2080 \u2208 range f\n\u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=\n  kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080))\nc : \u03b3\nt : Set (\u03b1 \u00d7 \u03a9)\nht : MeasurableSet t\nx : \u03b1 \u00d7 \u03a9\n\u22a2 x \u2208 t \u2194 x \u2208 Prod.map id f \u207b\u00b9' (Prod.map id f '' t)", "state_after": "case h.h.e_a.h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\nh_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set\nx\u2080 : \u211d\nhx\u2080 : x\u2080 \u2208 range f\n\u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=\n  kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080))\nc : \u03b3\nt : Set (\u03b1 \u00d7 \u03a9)\nht : MeasurableSet t\nx : \u03b1 \u00d7 \u03a9\n\u22a2 (x.1, x.2) \u2208 t \u2194 (x.1, x.2) \u2208 Prod.map id f \u207b\u00b9' (Prod.map id f '' t)"}, {"tactic": "simp only [id.def, mem_preimage, Prod.map_mk, mem_image, Prod.mk.inj_iff, Prod.exists]", "annotated_tactic": ["simp only [<a>id.def</a>, <a>mem_preimage</a>, <a>Prod.map_mk</a>, <a>mem_image</a>, <a>Prod.mk.inj_iff</a>, <a>Prod.exists</a>]", [{"full_name": "id.def", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [527, 9], "def_end_pos": [527, 15]}, {"full_name": "Set.mem_preimage", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [64, 9], "def_end_pos": [64, 21]}, {"full_name": "Prod.map_mk", "def_path": "Mathlib/Data/Prod/Basic.lean", "def_pos": [64, 9], "def_end_pos": [64, 15]}, {"full_name": "Set.mem_image", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [231, 9], "def_end_pos": [231, 18]}, {"full_name": "Prod.mk.inj_iff", "def_path": "Mathlib/Data/Prod/Basic.lean", "def_pos": [103, 9], "def_end_pos": [103, 19]}, {"full_name": "Prod.exists", "def_path": "Mathlib/Data/Prod/Basic.lean", "def_pos": [41, 9], "def_end_pos": [41, 17]}]], "state_before": "case h.h.e_a.h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\nh_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set\nx\u2080 : \u211d\nhx\u2080 : x\u2080 \u2208 range f\n\u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=\n  kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080))\nc : \u03b3\nt : Set (\u03b1 \u00d7 \u03a9)\nht : MeasurableSet t\nx : \u03b1 \u00d7 \u03a9\n\u22a2 (x.1, x.2) \u2208 t \u2194 (x.1, x.2) \u2208 Prod.map id f \u207b\u00b9' (Prod.map id f '' t)", "state_after": "case h.h.e_a.h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\nh_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set\nx\u2080 : \u211d\nhx\u2080 : x\u2080 \u2208 range f\n\u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=\n  kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080))\nc : \u03b3\nt : Set (\u03b1 \u00d7 \u03a9)\nht : MeasurableSet t\nx : \u03b1 \u00d7 \u03a9\n\u22a2 (x.1, x.2) \u2208 t \u2194 \u2203 a b, (a, b) \u2208 t \u2227 a = x.1 \u2227 f b = f x.2"}, {"tactic": "refine' \u27e8fun h => \u27e8x.1, x.2, h, rfl, rfl\u27e9, _\u27e9", "annotated_tactic": ["refine' \u27e8fun h => \u27e8x.1, x.2, h, <a>rfl</a>, <a>rfl</a>\u27e9, _\u27e9", [{"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "case h.h.e_a.h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\nh_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set\nx\u2080 : \u211d\nhx\u2080 : x\u2080 \u2208 range f\n\u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=\n  kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080))\nc : \u03b3\nt : Set (\u03b1 \u00d7 \u03a9)\nht : MeasurableSet t\nx : \u03b1 \u00d7 \u03a9\n\u22a2 (x.1, x.2) \u2208 t \u2194 \u2203 a b, (a, b) \u2208 t \u2227 a = x.1 \u2227 f b = f x.2", "state_after": "case h.h.e_a.h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\nh_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set\nx\u2080 : \u211d\nhx\u2080 : x\u2080 \u2208 range f\n\u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=\n  kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080))\nc : \u03b3\nt : Set (\u03b1 \u00d7 \u03a9)\nht : MeasurableSet t\nx : \u03b1 \u00d7 \u03a9\n\u22a2 (\u2203 a b, (a, b) \u2208 t \u2227 a = x.1 \u2227 f b = f x.2) \u2192 (x.1, x.2) \u2208 t"}, {"tactic": "rintro \u27e8a, b, h_mem, rfl, hf_eq\u27e9", "annotated_tactic": ["rintro \u27e8a, b, h_mem, rfl, hf_eq\u27e9", []], "state_before": "case h.h.e_a.h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\nh_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set\nx\u2080 : \u211d\nhx\u2080 : x\u2080 \u2208 range f\n\u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=\n  kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080))\nc : \u03b3\nt : Set (\u03b1 \u00d7 \u03a9)\nht : MeasurableSet t\nx : \u03b1 \u00d7 \u03a9\n\u22a2 (\u2203 a b, (a, b) \u2208 t \u2227 a = x.1 \u2227 f b = f x.2) \u2192 (x.1, x.2) \u2208 t", "state_after": "case h.h.e_a.h.intro.intro.intro.intro\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\nh_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set\nx\u2080 : \u211d\nhx\u2080 : x\u2080 \u2208 range f\n\u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=\n  kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080))\nc : \u03b3\nt : Set (\u03b1 \u00d7 \u03a9)\nht : MeasurableSet t\nx : \u03b1 \u00d7 \u03a9\nb : \u03a9\nhf_eq : f b = f x.2\nh_mem : (x.1, b) \u2208 t\n\u22a2 (x.1, x.2) \u2208 t"}, {"tactic": "rwa [hf.injective hf_eq] at h_mem", "annotated_tactic": ["rwa [hf.injective hf_eq] at h_mem", []], "state_before": "case h.h.e_a.h.intro.intro.intro.intro\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\nh_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set\nx\u2080 : \u211d\nhx\u2080 : x\u2080 \u2208 range f\n\u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=\n  kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080))\nc : \u03b3\nt : Set (\u03b1 \u00d7 \u03a9)\nht : MeasurableSet t\nx : \u03b1 \u00d7 \u03a9\nb : \u03a9\nhf_eq : f b = f x.2\nh_mem : (x.1, b) \u2208 t\n\u22a2 (x.1, x.2) \u2208 t", "state_after": "no goals"}, {"tactic": "exact measurable_prod_mk_left ht", "annotated_tactic": ["exact <a>measurable_prod_mk_left</a> ht", [{"full_name": "measurable_prod_mk_left", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [736, 9], "def_end_pos": [736, 32]}]], "state_before": "case h.ht\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\nh_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set\nx\u2080 : \u211d\nhx\u2080 : x\u2080 \u2208 range f\n\u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=\n  kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080))\nthis : kernel.const \u03b3 \u03c1 = kernel.comapRight (kernel.const \u03b3 \u03c1') h_prod_embed\nc : \u03b3\nt : Set (\u03b1 \u00d7 \u03a9)\nht : MeasurableSet t\na : \u03b1\nha : a \u2208 \u03c1_set\n\u22a2 MeasurableSet {c | (a, c) \u2208 t}", "state_after": "no goals"}, {"tactic": "ext1 x", "annotated_tactic": ["ext1 x", []], "state_before": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\nh_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set\nx\u2080 : \u211d\nhx\u2080 : x\u2080 \u2208 range f\n\u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=\n  kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080))\nthis : kernel.const \u03b3 \u03c1 = kernel.comapRight (kernel.const \u03b3 \u03c1') h_prod_embed\nc : \u03b3\nt : Set (\u03b1 \u00d7 \u03a9)\nht : MeasurableSet t\na : \u03b1\nha : a \u2208 \u03c1_set\n\u22a2 {c | (a, c) \u2208 Prod.map id f '' t} = f '' {c | (a, c) \u2208 t}", "state_after": "case h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\nh_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set\nx\u2080 : \u211d\nhx\u2080 : x\u2080 \u2208 range f\n\u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=\n  kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080))\nthis : kernel.const \u03b3 \u03c1 = kernel.comapRight (kernel.const \u03b3 \u03c1') h_prod_embed\nc : \u03b3\nt : Set (\u03b1 \u00d7 \u03a9)\nht : MeasurableSet t\na : \u03b1\nha : a \u2208 \u03c1_set\nx : \u211d\n\u22a2 x \u2208 {c | (a, c) \u2208 Prod.map id f '' t} \u2194 x \u2208 f '' {c | (a, c) \u2208 t}"}, {"tactic": "simp only [Prod_map, id.def, mem_image, Prod.mk.inj_iff, Prod.exists, mem_setOf_eq]", "annotated_tactic": ["simp only [<a>Prod_map</a>, <a>id.def</a>, <a>mem_image</a>, <a>Prod.mk.inj_iff</a>, <a>Prod.exists</a>, <a>mem_setOf_eq</a>]", [{"full_name": "Prod_map", "def_path": "Mathlib/Data/Prod/Basic.lean", "def_pos": [25, 9], "def_end_pos": [25, 17]}, {"full_name": "id.def", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [527, 9], "def_end_pos": [527, 15]}, {"full_name": "Set.mem_image", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [231, 9], "def_end_pos": [231, 18]}, {"full_name": "Prod.mk.inj_iff", "def_path": "Mathlib/Data/Prod/Basic.lean", "def_pos": [103, 9], "def_end_pos": [103, 19]}, {"full_name": "Prod.exists", "def_path": "Mathlib/Data/Prod/Basic.lean", "def_pos": [41, 9], "def_end_pos": [41, 17]}, {"full_name": "Set.mem_setOf_eq", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [256, 29], "def_end_pos": [256, 41]}]], "state_before": "case h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\nh_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set\nx\u2080 : \u211d\nhx\u2080 : x\u2080 \u2208 range f\n\u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=\n  kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080))\nthis : kernel.const \u03b3 \u03c1 = kernel.comapRight (kernel.const \u03b3 \u03c1') h_prod_embed\nc : \u03b3\nt : Set (\u03b1 \u00d7 \u03a9)\nht : MeasurableSet t\na : \u03b1\nha : a \u2208 \u03c1_set\nx : \u211d\n\u22a2 x \u2208 {c | (a, c) \u2208 Prod.map id f '' t} \u2194 x \u2208 f '' {c | (a, c) \u2208 t}", "state_after": "case h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\nh_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set\nx\u2080 : \u211d\nhx\u2080 : x\u2080 \u2208 range f\n\u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=\n  kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080))\nthis : kernel.const \u03b3 \u03c1 = kernel.comapRight (kernel.const \u03b3 \u03c1') h_prod_embed\nc : \u03b3\nt : Set (\u03b1 \u00d7 \u03a9)\nht : MeasurableSet t\na : \u03b1\nha : a \u2208 \u03c1_set\nx : \u211d\n\u22a2 (\u2203 a_1 b, (a_1, b) \u2208 t \u2227 a_1 = a \u2227 f b = x) \u2194 \u2203 x_1, (a, x_1) \u2208 t \u2227 f x_1 = x"}, {"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "case h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\nh_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set\nx\u2080 : \u211d\nhx\u2080 : x\u2080 \u2208 range f\n\u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=\n  kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080))\nthis : kernel.const \u03b3 \u03c1 = kernel.comapRight (kernel.const \u03b3 \u03c1') h_prod_embed\nc : \u03b3\nt : Set (\u03b1 \u00d7 \u03a9)\nht : MeasurableSet t\na : \u03b1\nha : a \u2208 \u03c1_set\nx : \u211d\n\u22a2 (\u2203 a_1 b, (a_1, b) \u2208 t \u2227 a_1 = a \u2227 f b = x) \u2194 \u2203 x_1, (a, x_1) \u2208 t \u2227 f x_1 = x", "state_after": "case h.mp\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\nh_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set\nx\u2080 : \u211d\nhx\u2080 : x\u2080 \u2208 range f\n\u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=\n  kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080))\nthis : kernel.const \u03b3 \u03c1 = kernel.comapRight (kernel.const \u03b3 \u03c1') h_prod_embed\nc : \u03b3\nt : Set (\u03b1 \u00d7 \u03a9)\nht : MeasurableSet t\na : \u03b1\nha : a \u2208 \u03c1_set\nx : \u211d\n\u22a2 (\u2203 a_1 b, (a_1, b) \u2208 t \u2227 a_1 = a \u2227 f b = x) \u2192 \u2203 x_1, (a, x_1) \u2208 t \u2227 f x_1 = x\n\ncase h.mpr\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\nh_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set\nx\u2080 : \u211d\nhx\u2080 : x\u2080 \u2208 range f\n\u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=\n  kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080))\nthis : kernel.const \u03b3 \u03c1 = kernel.comapRight (kernel.const \u03b3 \u03c1') h_prod_embed\nc : \u03b3\nt : Set (\u03b1 \u00d7 \u03a9)\nht : MeasurableSet t\na : \u03b1\nha : a \u2208 \u03c1_set\nx : \u211d\n\u22a2 (\u2203 x_1, (a, x_1) \u2208 t \u2227 f x_1 = x) \u2192 \u2203 a_2 b, (a_2, b) \u2208 t \u2227 a_2 = a \u2227 f b = x"}, {"tactic": "rintro \u27e8a', b, h_mem, rfl, hf_eq\u27e9", "annotated_tactic": ["rintro \u27e8a', b, h_mem, rfl, hf_eq\u27e9", []], "state_before": "case h.mp\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\nh_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set\nx\u2080 : \u211d\nhx\u2080 : x\u2080 \u2208 range f\n\u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=\n  kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080))\nthis : kernel.const \u03b3 \u03c1 = kernel.comapRight (kernel.const \u03b3 \u03c1') h_prod_embed\nc : \u03b3\nt : Set (\u03b1 \u00d7 \u03a9)\nht : MeasurableSet t\na : \u03b1\nha : a \u2208 \u03c1_set\nx : \u211d\n\u22a2 (\u2203 a_1 b, (a_1, b) \u2208 t \u2227 a_1 = a \u2227 f b = x) \u2192 \u2203 x_1, (a, x_1) \u2208 t \u2227 f x_1 = x", "state_after": "case h.mp.intro.intro.intro.intro\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\nh_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set\nx\u2080 : \u211d\nhx\u2080 : x\u2080 \u2208 range f\n\u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=\n  kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080))\nthis : kernel.const \u03b3 \u03c1 = kernel.comapRight (kernel.const \u03b3 \u03c1') h_prod_embed\nc : \u03b3\nt : Set (\u03b1 \u00d7 \u03a9)\nht : MeasurableSet t\nx : \u211d\na' : \u03b1\nb : \u03a9\nh_mem : (a', b) \u2208 t\nhf_eq : f b = x\nha : a' \u2208 \u03c1_set\n\u22a2 \u2203 x_1, (a', x_1) \u2208 t \u2227 f x_1 = x"}, {"tactic": "exact \u27e8b, h_mem, hf_eq\u27e9", "annotated_tactic": ["exact \u27e8b, h_mem, hf_eq\u27e9", []], "state_before": "case h.mp.intro.intro.intro.intro\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\nh_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set\nx\u2080 : \u211d\nhx\u2080 : x\u2080 \u2208 range f\n\u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=\n  kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080))\nthis : kernel.const \u03b3 \u03c1 = kernel.comapRight (kernel.const \u03b3 \u03c1') h_prod_embed\nc : \u03b3\nt : Set (\u03b1 \u00d7 \u03a9)\nht : MeasurableSet t\nx : \u211d\na' : \u03b1\nb : \u03a9\nh_mem : (a', b) \u2208 t\nhf_eq : f b = x\nha : a' \u2208 \u03c1_set\n\u22a2 \u2203 x_1, (a', x_1) \u2208 t \u2227 f x_1 = x", "state_after": "no goals"}, {"tactic": "rintro \u27e8b, h_mem, hf_eq\u27e9", "annotated_tactic": ["rintro \u27e8b, h_mem, hf_eq\u27e9", []], "state_before": "case h.mpr\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\nh_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set\nx\u2080 : \u211d\nhx\u2080 : x\u2080 \u2208 range f\n\u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=\n  kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080))\nthis : kernel.const \u03b3 \u03c1 = kernel.comapRight (kernel.const \u03b3 \u03c1') h_prod_embed\nc : \u03b3\nt : Set (\u03b1 \u00d7 \u03a9)\nht : MeasurableSet t\na : \u03b1\nha : a \u2208 \u03c1_set\nx : \u211d\n\u22a2 (\u2203 x_1, (a, x_1) \u2208 t \u2227 f x_1 = x) \u2192 \u2203 a_2 b, (a_2, b) \u2208 t \u2227 a_2 = a \u2227 f b = x", "state_after": "case h.mpr.intro.intro\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\nh_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set\nx\u2080 : \u211d\nhx\u2080 : x\u2080 \u2208 range f\n\u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=\n  kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080))\nthis : kernel.const \u03b3 \u03c1 = kernel.comapRight (kernel.const \u03b3 \u03c1') h_prod_embed\nc : \u03b3\nt : Set (\u03b1 \u00d7 \u03a9)\nht : MeasurableSet t\na : \u03b1\nha : a \u2208 \u03c1_set\nx : \u211d\nb : \u03a9\nh_mem : (a, b) \u2208 t\nhf_eq : f b = x\n\u22a2 \u2203 a_1 b, (a_1, b) \u2208 t \u2227 a_1 = a \u2227 f b = x"}, {"tactic": "exact \u27e8a, b, h_mem, rfl, hf_eq\u27e9", "annotated_tactic": ["exact \u27e8a, b, h_mem, <a>rfl</a>, hf_eq\u27e9", [{"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "case h.mpr.intro.intro\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\nh_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set\nx\u2080 : \u211d\nhx\u2080 : x\u2080 \u2208 range f\n\u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=\n  kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080))\nthis : kernel.const \u03b3 \u03c1 = kernel.comapRight (kernel.const \u03b3 \u03c1') h_prod_embed\nc : \u03b3\nt : Set (\u03b1 \u00d7 \u03a9)\nht : MeasurableSet t\na : \u03b1\nha : a \u2208 \u03c1_set\nx : \u211d\nb : \u03a9\nh_mem : (a, b) \u2208 t\nhf_eq : f b = x\n\u22a2 \u2203 a_1 b, (a_1, b) \u2208 t \u2227 a_1 = a \u2227 f b = x", "state_after": "no goals"}, {"tactic": "rw [kernel.piecewise_apply, if_pos ha]", "annotated_tactic": ["rw [<a>kernel.piecewise_apply</a>, <a>if_pos</a> ha]", [{"full_name": "ProbabilityTheory.kernel.piecewise_apply", "def_path": "Mathlib/Probability/Kernel/Basic.lean", "def_pos": [628, 9], "def_end_pos": [628, 24]}, {"full_name": "if_pos", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [790, 9], "def_end_pos": [790, 15]}]], "state_before": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\nh_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set\nx\u2080 : \u211d\nhx\u2080 : x\u2080 \u2208 range f\n\u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=\n  kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080))\nthis : kernel.const \u03b3 \u03c1 = kernel.comapRight (kernel.const \u03b3 \u03c1') h_prod_embed\nc : \u03b3\nt : Set (\u03b1 \u00d7 \u03a9)\nht : MeasurableSet t\na : \u03b1\nha : a \u2208 \u03c1_set\nh1 : {c | (a, c) \u2208 Prod.map id f '' t} = f '' {c | (a, c) \u2208 t}\n\u22a2 \u2191(condKernelReal \u03c1') (c, a).2 = \u2191\u03b7' (c, a).2", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Covering/BesicovitchVectorSpace.lean", "full_name": "Besicovitch.le_multiplicity_of_\u03b4_of_fin", "start": [310, 1], "end": [331, 23], "traced_tactics": [{"tactic": "classical\nhave finj : Function.Injective f := by\n  intro i j hij\n  by_contra h\n  have : 1 - good\u03b4 E \u2264 \u2016f i - f j\u2016 := h' i j h\n  simp only [hij, norm_zero, sub_self] at this\n  linarith [good\u03b4_lt_one E]\nlet s := Finset.image f Finset.univ\nhave s_card : s.card = n := by rw [Finset.card_image_of_injective _ finj]; exact Finset.card_fin n\nhave hs : \u2200 c \u2208 s, \u2016c\u2016 \u2264 2 := by\n  simp only [h, forall_apply_eq_imp_iff, forall_const, forall_exists_index, Finset.mem_univ,\n    Finset.mem_image, imp_true_iff, true_and]\nhave h's : \u2200 c \u2208 s, \u2200 d \u2208 s, c \u2260 d \u2192 1 - good\u03b4 E \u2264 \u2016c - d\u2016 := by\n  simp only [forall_apply_eq_imp_iff, forall_exists_index, Finset.mem_univ, Finset.mem_image,\n    Ne.def, exists_true_left, forall_apply_eq_imp_iff, forall_true_left, true_and]\n  intro i j hij\n  have : i \u2260 j := fun h => by rw [h] at hij; exact hij rfl\n  exact h' i j this\nhave : s.card \u2264 multiplicity E := card_le_multiplicity_of_\u03b4 hs h's\nrwa [s_card] at this", "annotated_tactic": ["classical\n  have finj : <a>Function.Injective</a> f := by\n    intro i j hij\n    by_contra h\n    have : 1 - <a>good\u03b4</a> E \u2264 \u2016f i - f j\u2016 := h' i j h\n    simp only [hij, <a>norm_zero</a>, <a>sub_self</a>] at this\n    linarith [<a>good\u03b4_lt_one</a> E]\n  let s := <a>Finset.image</a> f <a>Finset.univ</a>\n  have s_card : s.card = n := by rw [<a>Finset.card_image_of_injective</a> _ finj]; exact <a>Finset.card_fin</a> n\n  have hs : \u2200 c \u2208 s, \u2016c\u2016 \u2264 2 := by\n    simp only [h, <a>forall_apply_eq_imp_iff</a>, <a>forall_const</a>, <a>forall_exists_index</a>, <a>Finset.mem_univ</a>,\n      <a>Finset.mem_image</a>, <a>imp_true_iff</a>, <a>true_and</a>]\n  have h's : \u2200 c \u2208 s, \u2200 d \u2208 s, c \u2260 d \u2192 1 - <a>good\u03b4</a> E \u2264 \u2016c - d\u2016 := by\n    simp only [<a>forall_apply_eq_imp_iff</a>, <a>forall_exists_index</a>, <a>Finset.mem_univ</a>, <a>Finset.mem_image</a>,\n      <a>Ne.def</a>, <a>exists_true_left</a>, <a>forall_apply_eq_imp_iff</a>, <a>forall_true_left</a>, <a>true_and</a>]\n    intro i j hij\n    have : i \u2260 j := fun h => by rw [h] at hij; exact hij <a>rfl</a>\n    exact h' i j this\n  have : s.card \u2264 <a>multiplicity</a> E := <a>card_le_multiplicity_of_\u03b4</a> hs h's\n  rwa [s_card] at this", [{"full_name": "Function.Injective", "def_path": "Mathlib/Init/Function.lean", "def_pos": [109, 5], "def_end_pos": [109, 14]}, {"full_name": "Besicovitch.good\u03b4", "def_path": "Mathlib/MeasureTheory/Covering/BesicovitchVectorSpace.lean", "def_pos": [284, 5], "def_end_pos": [284, 10]}, {"full_name": "norm_zero", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [528, 30], "def_end_pos": [528, 39]}, {"full_name": "sub_self", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [734, 30], "def_end_pos": [734, 38]}, {"full_name": "Besicovitch.good\u03b4_lt_one", "def_path": "Mathlib/MeasureTheory/Covering/BesicovitchVectorSpace.lean", "def_pos": [288, 9], "def_end_pos": [288, 21]}, {"full_name": "Finset.image", "def_path": "Mathlib/Data/Finset/Image.lean", "def_pos": [313, 5], "def_end_pos": [313, 10]}, {"full_name": "Finset.univ", "def_path": "Mathlib/Data/Fintype/Basic.lean", "def_pos": [67, 5], "def_end_pos": [67, 9]}, {"full_name": "Finset.card_image_of_injective", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [250, 9], "def_end_pos": [250, 32]}, {"full_name": "Finset.card_fin", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [322, 9], "def_end_pos": [322, 24]}, {"full_name": "forall_apply_eq_imp_iff", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [499, 17], "def_end_pos": [499, 40]}, {"full_name": "forall_const", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [435, 17], "def_end_pos": [435, 29]}, {"full_name": "forall_exists_index", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [356, 17], "def_end_pos": [356, 36]}, {"full_name": "Finset.mem_univ", "def_path": "Mathlib/Data/Fintype/Basic.lean", "def_pos": [72, 9], "def_end_pos": [72, 17]}, {"full_name": "Finset.mem_image", "def_path": "Mathlib/Data/Finset/Image.lean", "def_pos": [330, 9], "def_end_pos": [330, 18]}, {"full_name": "imp_true_iff", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [116, 9], "def_end_pos": [116, 21]}, {"full_name": "true_and", "def_path": "lake-packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [84, 17], "def_end_pos": [84, 25]}, {"full_name": "Besicovitch.good\u03b4", "def_path": "Mathlib/MeasureTheory/Covering/BesicovitchVectorSpace.lean", "def_pos": [284, 5], "def_end_pos": [284, 10]}, {"full_name": "forall_apply_eq_imp_iff", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [499, 17], "def_end_pos": [499, 40]}, {"full_name": "forall_exists_index", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [356, 17], "def_end_pos": [356, 36]}, {"full_name": "Finset.mem_univ", "def_path": "Mathlib/Data/Fintype/Basic.lean", "def_pos": [72, 9], "def_end_pos": [72, 17]}, {"full_name": "Finset.mem_image", "def_path": "Mathlib/Data/Finset/Image.lean", "def_pos": [330, 9], "def_end_pos": [330, 18]}, {"full_name": "Ne.def", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [59, 9], "def_end_pos": [59, 15]}, {"full_name": "exists_true_left", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [912, 17], "def_end_pos": [912, 33]}, {"full_name": "forall_apply_eq_imp_iff", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [499, 17], "def_end_pos": [499, 40]}, {"full_name": "forall_true_left", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [931, 17], "def_end_pos": [931, 33]}, {"full_name": "true_and", "def_path": "lake-packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [84, 17], "def_end_pos": [84, 25]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}, {"full_name": "Besicovitch.multiplicity", "def_path": "Mathlib/MeasureTheory/Covering/BesicovitchVectorSpace.lean", "def_pos": [131, 5], "def_end_pos": [131, 17]}, {"full_name": "Besicovitch.card_le_multiplicity_of_\u03b4", "def_path": "Mathlib/MeasureTheory/Covering/BesicovitchVectorSpace.lean", "def_pos": [305, 9], "def_end_pos": [305, 34]}]], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nn : \u2115\nf : Fin n \u2192 E\nh : \u2200 (i : Fin n), \u2016f i\u2016 \u2264 2\nh' : \u2200 (i j : Fin n), i \u2260 j \u2192 1 - good\u03b4 E \u2264 \u2016f i - f j\u2016\n\u22a2 n \u2264 multiplicity E", "state_after": "no goals"}, {"tactic": "have finj : Function.Injective f := by\n  intro i j hij\n  by_contra h\n  have : 1 - good\u03b4 E \u2264 \u2016f i - f j\u2016 := h' i j h\n  simp only [hij, norm_zero, sub_self] at this\n  linarith [good\u03b4_lt_one E]", "annotated_tactic": ["have finj : <a>Function.Injective</a> f := by\n    intro i j hij\n    by_contra h\n    have : 1 - <a>good\u03b4</a> E \u2264 \u2016f i - f j\u2016 := h' i j h\n    simp only [hij, <a>norm_zero</a>, <a>sub_self</a>] at this\n    linarith [<a>good\u03b4_lt_one</a> E]", [{"full_name": "Function.Injective", "def_path": "Mathlib/Init/Function.lean", "def_pos": [109, 5], "def_end_pos": [109, 14]}, {"full_name": "Besicovitch.good\u03b4", "def_path": "Mathlib/MeasureTheory/Covering/BesicovitchVectorSpace.lean", "def_pos": [284, 5], "def_end_pos": [284, 10]}, {"full_name": "norm_zero", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [528, 30], "def_end_pos": [528, 39]}, {"full_name": "sub_self", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [734, 30], "def_end_pos": [734, 38]}, {"full_name": "Besicovitch.good\u03b4_lt_one", "def_path": "Mathlib/MeasureTheory/Covering/BesicovitchVectorSpace.lean", "def_pos": [288, 9], "def_end_pos": [288, 21]}]], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nn : \u2115\nf : Fin n \u2192 E\nh : \u2200 (i : Fin n), \u2016f i\u2016 \u2264 2\nh' : \u2200 (i j : Fin n), i \u2260 j \u2192 1 - good\u03b4 E \u2264 \u2016f i - f j\u2016\n\u22a2 n \u2264 multiplicity E", "state_after": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nn : \u2115\nf : Fin n \u2192 E\nh : \u2200 (i : Fin n), \u2016f i\u2016 \u2264 2\nh' : \u2200 (i j : Fin n), i \u2260 j \u2192 1 - good\u03b4 E \u2264 \u2016f i - f j\u2016\nfinj : Function.Injective f\n\u22a2 n \u2264 multiplicity E"}, {"tactic": "let s := Finset.image f Finset.univ", "annotated_tactic": ["let s := <a>Finset.image</a> f <a>Finset.univ</a>", [{"full_name": "Finset.image", "def_path": "Mathlib/Data/Finset/Image.lean", "def_pos": [313, 5], "def_end_pos": [313, 10]}, {"full_name": "Finset.univ", "def_path": "Mathlib/Data/Fintype/Basic.lean", "def_pos": [67, 5], "def_end_pos": [67, 9]}]], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nn : \u2115\nf : Fin n \u2192 E\nh : \u2200 (i : Fin n), \u2016f i\u2016 \u2264 2\nh' : \u2200 (i j : Fin n), i \u2260 j \u2192 1 - good\u03b4 E \u2264 \u2016f i - f j\u2016\nfinj : Function.Injective f\n\u22a2 n \u2264 multiplicity E", "state_after": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nn : \u2115\nf : Fin n \u2192 E\nh : \u2200 (i : Fin n), \u2016f i\u2016 \u2264 2\nh' : \u2200 (i j : Fin n), i \u2260 j \u2192 1 - good\u03b4 E \u2264 \u2016f i - f j\u2016\nfinj : Function.Injective f\ns : Finset E := Finset.image f Finset.univ\n\u22a2 n \u2264 multiplicity E"}, {"tactic": "have s_card : s.card = n := by rw [Finset.card_image_of_injective _ finj]; exact Finset.card_fin n", "annotated_tactic": ["have s_card : s.card = n := by rw [<a>Finset.card_image_of_injective</a> _ finj]; exact <a>Finset.card_fin</a> n", [{"full_name": "Finset.card_image_of_injective", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [250, 9], "def_end_pos": [250, 32]}, {"full_name": "Finset.card_fin", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [322, 9], "def_end_pos": [322, 24]}]], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nn : \u2115\nf : Fin n \u2192 E\nh : \u2200 (i : Fin n), \u2016f i\u2016 \u2264 2\nh' : \u2200 (i j : Fin n), i \u2260 j \u2192 1 - good\u03b4 E \u2264 \u2016f i - f j\u2016\nfinj : Function.Injective f\ns : Finset E := Finset.image f Finset.univ\n\u22a2 n \u2264 multiplicity E", "state_after": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nn : \u2115\nf : Fin n \u2192 E\nh : \u2200 (i : Fin n), \u2016f i\u2016 \u2264 2\nh' : \u2200 (i j : Fin n), i \u2260 j \u2192 1 - good\u03b4 E \u2264 \u2016f i - f j\u2016\nfinj : Function.Injective f\ns : Finset E := Finset.image f Finset.univ\ns_card : Finset.card s = n\n\u22a2 n \u2264 multiplicity E"}, {"tactic": "have hs : \u2200 c \u2208 s, \u2016c\u2016 \u2264 2 := by\n  simp only [h, forall_apply_eq_imp_iff, forall_const, forall_exists_index, Finset.mem_univ,\n    Finset.mem_image, imp_true_iff, true_and]", "annotated_tactic": ["have hs : \u2200 c \u2208 s, \u2016c\u2016 \u2264 2 := by\n    simp only [h, <a>forall_apply_eq_imp_iff</a>, <a>forall_const</a>, <a>forall_exists_index</a>, <a>Finset.mem_univ</a>,\n      <a>Finset.mem_image</a>, <a>imp_true_iff</a>, <a>true_and</a>]", [{"full_name": "forall_apply_eq_imp_iff", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [499, 17], "def_end_pos": [499, 40]}, {"full_name": "forall_const", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [435, 17], "def_end_pos": [435, 29]}, {"full_name": "forall_exists_index", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [356, 17], "def_end_pos": [356, 36]}, {"full_name": "Finset.mem_univ", "def_path": "Mathlib/Data/Fintype/Basic.lean", "def_pos": [72, 9], "def_end_pos": [72, 17]}, {"full_name": "Finset.mem_image", "def_path": "Mathlib/Data/Finset/Image.lean", "def_pos": [330, 9], "def_end_pos": [330, 18]}, {"full_name": "imp_true_iff", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [116, 9], "def_end_pos": [116, 21]}, {"full_name": "true_and", "def_path": "lake-packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [84, 17], "def_end_pos": [84, 25]}]], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nn : \u2115\nf : Fin n \u2192 E\nh : \u2200 (i : Fin n), \u2016f i\u2016 \u2264 2\nh' : \u2200 (i j : Fin n), i \u2260 j \u2192 1 - good\u03b4 E \u2264 \u2016f i - f j\u2016\nfinj : Function.Injective f\ns : Finset E := Finset.image f Finset.univ\ns_card : Finset.card s = n\n\u22a2 n \u2264 multiplicity E", "state_after": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nn : \u2115\nf : Fin n \u2192 E\nh : \u2200 (i : Fin n), \u2016f i\u2016 \u2264 2\nh' : \u2200 (i j : Fin n), i \u2260 j \u2192 1 - good\u03b4 E \u2264 \u2016f i - f j\u2016\nfinj : Function.Injective f\ns : Finset E := Finset.image f Finset.univ\ns_card : Finset.card s = n\nhs : \u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2\n\u22a2 n \u2264 multiplicity E"}, {"tactic": "have h's : \u2200 c \u2208 s, \u2200 d \u2208 s, c \u2260 d \u2192 1 - good\u03b4 E \u2264 \u2016c - d\u2016 := by\n  simp only [forall_apply_eq_imp_iff, forall_exists_index, Finset.mem_univ, Finset.mem_image,\n    Ne.def, exists_true_left, forall_apply_eq_imp_iff, forall_true_left, true_and]\n  intro i j hij\n  have : i \u2260 j := fun h => by rw [h] at hij; exact hij rfl\n  exact h' i j this", "annotated_tactic": ["have h's : \u2200 c \u2208 s, \u2200 d \u2208 s, c \u2260 d \u2192 1 - <a>good\u03b4</a> E \u2264 \u2016c - d\u2016 := by\n    simp only [<a>forall_apply_eq_imp_iff</a>, <a>forall_exists_index</a>, <a>Finset.mem_univ</a>, <a>Finset.mem_image</a>,\n      <a>Ne.def</a>, <a>exists_true_left</a>, <a>forall_apply_eq_imp_iff</a>, <a>forall_true_left</a>, <a>true_and</a>]\n    intro i j hij\n    have : i \u2260 j := fun h => by rw [h] at hij; exact hij <a>rfl</a>\n    exact h' i j this", [{"full_name": "Besicovitch.good\u03b4", "def_path": "Mathlib/MeasureTheory/Covering/BesicovitchVectorSpace.lean", "def_pos": [284, 5], "def_end_pos": [284, 10]}, {"full_name": "forall_apply_eq_imp_iff", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [499, 17], "def_end_pos": [499, 40]}, {"full_name": "forall_exists_index", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [356, 17], "def_end_pos": [356, 36]}, {"full_name": "Finset.mem_univ", "def_path": "Mathlib/Data/Fintype/Basic.lean", "def_pos": [72, 9], "def_end_pos": [72, 17]}, {"full_name": "Finset.mem_image", "def_path": "Mathlib/Data/Finset/Image.lean", "def_pos": [330, 9], "def_end_pos": [330, 18]}, {"full_name": "Ne.def", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [59, 9], "def_end_pos": [59, 15]}, {"full_name": "exists_true_left", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [912, 17], "def_end_pos": [912, 33]}, {"full_name": "forall_apply_eq_imp_iff", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [499, 17], "def_end_pos": [499, 40]}, {"full_name": "forall_true_left", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [931, 17], "def_end_pos": [931, 33]}, {"full_name": "true_and", "def_path": "lake-packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [84, 17], "def_end_pos": [84, 25]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nn : \u2115\nf : Fin n \u2192 E\nh : \u2200 (i : Fin n), \u2016f i\u2016 \u2264 2\nh' : \u2200 (i j : Fin n), i \u2260 j \u2192 1 - good\u03b4 E \u2264 \u2016f i - f j\u2016\nfinj : Function.Injective f\ns : Finset E := Finset.image f Finset.univ\ns_card : Finset.card s = n\nhs : \u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2\n\u22a2 n \u2264 multiplicity E", "state_after": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nn : \u2115\nf : Fin n \u2192 E\nh : \u2200 (i : Fin n), \u2016f i\u2016 \u2264 2\nh' : \u2200 (i j : Fin n), i \u2260 j \u2192 1 - good\u03b4 E \u2264 \u2016f i - f j\u2016\nfinj : Function.Injective f\ns : Finset E := Finset.image f Finset.univ\ns_card : Finset.card s = n\nhs : \u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2\nh's : \u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - good\u03b4 E \u2264 \u2016c - d\u2016\n\u22a2 n \u2264 multiplicity E"}, {"tactic": "have : s.card \u2264 multiplicity E := card_le_multiplicity_of_\u03b4 hs h's", "annotated_tactic": ["have : s.card \u2264 <a>multiplicity</a> E := <a>card_le_multiplicity_of_\u03b4</a> hs h's", [{"full_name": "Besicovitch.multiplicity", "def_path": "Mathlib/MeasureTheory/Covering/BesicovitchVectorSpace.lean", "def_pos": [131, 5], "def_end_pos": [131, 17]}, {"full_name": "Besicovitch.card_le_multiplicity_of_\u03b4", "def_path": "Mathlib/MeasureTheory/Covering/BesicovitchVectorSpace.lean", "def_pos": [305, 9], "def_end_pos": [305, 34]}]], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nn : \u2115\nf : Fin n \u2192 E\nh : \u2200 (i : Fin n), \u2016f i\u2016 \u2264 2\nh' : \u2200 (i j : Fin n), i \u2260 j \u2192 1 - good\u03b4 E \u2264 \u2016f i - f j\u2016\nfinj : Function.Injective f\ns : Finset E := Finset.image f Finset.univ\ns_card : Finset.card s = n\nhs : \u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2\nh's : \u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - good\u03b4 E \u2264 \u2016c - d\u2016\n\u22a2 n \u2264 multiplicity E", "state_after": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nn : \u2115\nf : Fin n \u2192 E\nh : \u2200 (i : Fin n), \u2016f i\u2016 \u2264 2\nh' : \u2200 (i j : Fin n), i \u2260 j \u2192 1 - good\u03b4 E \u2264 \u2016f i - f j\u2016\nfinj : Function.Injective f\ns : Finset E := Finset.image f Finset.univ\ns_card : Finset.card s = n\nhs : \u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2\nh's : \u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - good\u03b4 E \u2264 \u2016c - d\u2016\nthis : Finset.card s \u2264 multiplicity E\n\u22a2 n \u2264 multiplicity E"}, {"tactic": "rwa [s_card] at this", "annotated_tactic": ["rwa [s_card] at this", []], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nn : \u2115\nf : Fin n \u2192 E\nh : \u2200 (i : Fin n), \u2016f i\u2016 \u2264 2\nh' : \u2200 (i j : Fin n), i \u2260 j \u2192 1 - good\u03b4 E \u2264 \u2016f i - f j\u2016\nfinj : Function.Injective f\ns : Finset E := Finset.image f Finset.univ\ns_card : Finset.card s = n\nhs : \u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2\nh's : \u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - good\u03b4 E \u2264 \u2016c - d\u2016\nthis : Finset.card s \u2264 multiplicity E\n\u22a2 n \u2264 multiplicity E", "state_after": "no goals"}, {"tactic": "intro i j hij", "annotated_tactic": ["intro i j hij", []], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nn : \u2115\nf : Fin n \u2192 E\nh : \u2200 (i : Fin n), \u2016f i\u2016 \u2264 2\nh' : \u2200 (i j : Fin n), i \u2260 j \u2192 1 - good\u03b4 E \u2264 \u2016f i - f j\u2016\n\u22a2 Function.Injective f", "state_after": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nn : \u2115\nf : Fin n \u2192 E\nh : \u2200 (i : Fin n), \u2016f i\u2016 \u2264 2\nh' : \u2200 (i j : Fin n), i \u2260 j \u2192 1 - good\u03b4 E \u2264 \u2016f i - f j\u2016\ni j : Fin n\nhij : f i = f j\n\u22a2 i = j"}, {"tactic": "by_contra h", "annotated_tactic": ["by_contra h", []], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nn : \u2115\nf : Fin n \u2192 E\nh : \u2200 (i : Fin n), \u2016f i\u2016 \u2264 2\nh' : \u2200 (i j : Fin n), i \u2260 j \u2192 1 - good\u03b4 E \u2264 \u2016f i - f j\u2016\ni j : Fin n\nhij : f i = f j\n\u22a2 i = j", "state_after": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nn : \u2115\nf : Fin n \u2192 E\nh\u271d : \u2200 (i : Fin n), \u2016f i\u2016 \u2264 2\nh' : \u2200 (i j : Fin n), i \u2260 j \u2192 1 - good\u03b4 E \u2264 \u2016f i - f j\u2016\ni j : Fin n\nhij : f i = f j\nh : \u00aci = j\n\u22a2 False"}, {"tactic": "have : 1 - good\u03b4 E \u2264 \u2016f i - f j\u2016 := h' i j h", "annotated_tactic": ["have : 1 - <a>good\u03b4</a> E \u2264 \u2016f i - f j\u2016 := h' i j h", [{"full_name": "Besicovitch.good\u03b4", "def_path": "Mathlib/MeasureTheory/Covering/BesicovitchVectorSpace.lean", "def_pos": [284, 5], "def_end_pos": [284, 10]}]], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nn : \u2115\nf : Fin n \u2192 E\nh\u271d : \u2200 (i : Fin n), \u2016f i\u2016 \u2264 2\nh' : \u2200 (i j : Fin n), i \u2260 j \u2192 1 - good\u03b4 E \u2264 \u2016f i - f j\u2016\ni j : Fin n\nhij : f i = f j\nh : \u00aci = j\n\u22a2 False", "state_after": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nn : \u2115\nf : Fin n \u2192 E\nh\u271d : \u2200 (i : Fin n), \u2016f i\u2016 \u2264 2\nh' : \u2200 (i j : Fin n), i \u2260 j \u2192 1 - good\u03b4 E \u2264 \u2016f i - f j\u2016\ni j : Fin n\nhij : f i = f j\nh : \u00aci = j\nthis : 1 - good\u03b4 E \u2264 \u2016f i - f j\u2016\n\u22a2 False"}, {"tactic": "simp only [hij, norm_zero, sub_self] at this", "annotated_tactic": ["simp only [hij, <a>norm_zero</a>, <a>sub_self</a>] at this", [{"full_name": "norm_zero", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [528, 30], "def_end_pos": [528, 39]}, {"full_name": "sub_self", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [734, 30], "def_end_pos": [734, 38]}]], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nn : \u2115\nf : Fin n \u2192 E\nh\u271d : \u2200 (i : Fin n), \u2016f i\u2016 \u2264 2\nh' : \u2200 (i j : Fin n), i \u2260 j \u2192 1 - good\u03b4 E \u2264 \u2016f i - f j\u2016\ni j : Fin n\nhij : f i = f j\nh : \u00aci = j\nthis : 1 - good\u03b4 E \u2264 \u2016f i - f j\u2016\n\u22a2 False", "state_after": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nn : \u2115\nf : Fin n \u2192 E\nh\u271d : \u2200 (i : Fin n), \u2016f i\u2016 \u2264 2\nh' : \u2200 (i j : Fin n), i \u2260 j \u2192 1 - good\u03b4 E \u2264 \u2016f i - f j\u2016\ni j : Fin n\nhij : f i = f j\nh : \u00aci = j\nthis : 1 - good\u03b4 E \u2264 0\n\u22a2 False"}, {"tactic": "linarith [good\u03b4_lt_one E]", "annotated_tactic": ["linarith [<a>good\u03b4_lt_one</a> E]", [{"full_name": "Besicovitch.good\u03b4_lt_one", "def_path": "Mathlib/MeasureTheory/Covering/BesicovitchVectorSpace.lean", "def_pos": [288, 9], "def_end_pos": [288, 21]}]], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nn : \u2115\nf : Fin n \u2192 E\nh\u271d : \u2200 (i : Fin n), \u2016f i\u2016 \u2264 2\nh' : \u2200 (i j : Fin n), i \u2260 j \u2192 1 - good\u03b4 E \u2264 \u2016f i - f j\u2016\ni j : Fin n\nhij : f i = f j\nh : \u00aci = j\nthis : 1 - good\u03b4 E \u2264 0\n\u22a2 False", "state_after": "no goals"}, {"tactic": "rw [Finset.card_image_of_injective _ finj]", "annotated_tactic": ["rw [<a>Finset.card_image_of_injective</a> _ finj]", [{"full_name": "Finset.card_image_of_injective", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [250, 9], "def_end_pos": [250, 32]}]], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nn : \u2115\nf : Fin n \u2192 E\nh : \u2200 (i : Fin n), \u2016f i\u2016 \u2264 2\nh' : \u2200 (i j : Fin n), i \u2260 j \u2192 1 - good\u03b4 E \u2264 \u2016f i - f j\u2016\nfinj : Function.Injective f\ns : Finset E := Finset.image f Finset.univ\n\u22a2 Finset.card s = n", "state_after": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nn : \u2115\nf : Fin n \u2192 E\nh : \u2200 (i : Fin n), \u2016f i\u2016 \u2264 2\nh' : \u2200 (i j : Fin n), i \u2260 j \u2192 1 - good\u03b4 E \u2264 \u2016f i - f j\u2016\nfinj : Function.Injective f\ns : Finset E := Finset.image f Finset.univ\n\u22a2 Finset.card Finset.univ = n"}, {"tactic": "exact Finset.card_fin n", "annotated_tactic": ["exact <a>Finset.card_fin</a> n", [{"full_name": "Finset.card_fin", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [322, 9], "def_end_pos": [322, 24]}]], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nn : \u2115\nf : Fin n \u2192 E\nh : \u2200 (i : Fin n), \u2016f i\u2016 \u2264 2\nh' : \u2200 (i j : Fin n), i \u2260 j \u2192 1 - good\u03b4 E \u2264 \u2016f i - f j\u2016\nfinj : Function.Injective f\ns : Finset E := Finset.image f Finset.univ\n\u22a2 Finset.card Finset.univ = n", "state_after": "no goals"}, {"tactic": "simp only [h, forall_apply_eq_imp_iff, forall_const, forall_exists_index, Finset.mem_univ,\n  Finset.mem_image, imp_true_iff, true_and]", "annotated_tactic": ["simp only [h, <a>forall_apply_eq_imp_iff</a>, <a>forall_const</a>, <a>forall_exists_index</a>, <a>Finset.mem_univ</a>,\n      <a>Finset.mem_image</a>, <a>imp_true_iff</a>, <a>true_and</a>]", [{"full_name": "forall_apply_eq_imp_iff", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [499, 17], "def_end_pos": [499, 40]}, {"full_name": "forall_const", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [435, 17], "def_end_pos": [435, 29]}, {"full_name": "forall_exists_index", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [356, 17], "def_end_pos": [356, 36]}, {"full_name": "Finset.mem_univ", "def_path": "Mathlib/Data/Fintype/Basic.lean", "def_pos": [72, 9], "def_end_pos": [72, 17]}, {"full_name": "Finset.mem_image", "def_path": "Mathlib/Data/Finset/Image.lean", "def_pos": [330, 9], "def_end_pos": [330, 18]}, {"full_name": "imp_true_iff", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [116, 9], "def_end_pos": [116, 21]}, {"full_name": "true_and", "def_path": "lake-packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [84, 17], "def_end_pos": [84, 25]}]], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nn : \u2115\nf : Fin n \u2192 E\nh : \u2200 (i : Fin n), \u2016f i\u2016 \u2264 2\nh' : \u2200 (i j : Fin n), i \u2260 j \u2192 1 - good\u03b4 E \u2264 \u2016f i - f j\u2016\nfinj : Function.Injective f\ns : Finset E := Finset.image f Finset.univ\ns_card : Finset.card s = n\n\u22a2 \u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2", "state_after": "no goals"}, {"tactic": "simp only [forall_apply_eq_imp_iff, forall_exists_index, Finset.mem_univ, Finset.mem_image,\n  Ne.def, exists_true_left, forall_apply_eq_imp_iff, forall_true_left, true_and]", "annotated_tactic": ["simp only [<a>forall_apply_eq_imp_iff</a>, <a>forall_exists_index</a>, <a>Finset.mem_univ</a>, <a>Finset.mem_image</a>,\n      <a>Ne.def</a>, <a>exists_true_left</a>, <a>forall_apply_eq_imp_iff</a>, <a>forall_true_left</a>, <a>true_and</a>]", [{"full_name": "forall_apply_eq_imp_iff", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [499, 17], "def_end_pos": [499, 40]}, {"full_name": "forall_exists_index", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [356, 17], "def_end_pos": [356, 36]}, {"full_name": "Finset.mem_univ", "def_path": "Mathlib/Data/Fintype/Basic.lean", "def_pos": [72, 9], "def_end_pos": [72, 17]}, {"full_name": "Finset.mem_image", "def_path": "Mathlib/Data/Finset/Image.lean", "def_pos": [330, 9], "def_end_pos": [330, 18]}, {"full_name": "Ne.def", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [59, 9], "def_end_pos": [59, 15]}, {"full_name": "exists_true_left", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [912, 17], "def_end_pos": [912, 33]}, {"full_name": "forall_apply_eq_imp_iff", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [499, 17], "def_end_pos": [499, 40]}, {"full_name": "forall_true_left", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [931, 17], "def_end_pos": [931, 33]}, {"full_name": "true_and", "def_path": "lake-packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [84, 17], "def_end_pos": [84, 25]}]], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nn : \u2115\nf : Fin n \u2192 E\nh : \u2200 (i : Fin n), \u2016f i\u2016 \u2264 2\nh' : \u2200 (i j : Fin n), i \u2260 j \u2192 1 - good\u03b4 E \u2264 \u2016f i - f j\u2016\nfinj : Function.Injective f\ns : Finset E := Finset.image f Finset.univ\ns_card : Finset.card s = n\nhs : \u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2\n\u22a2 \u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - good\u03b4 E \u2264 \u2016c - d\u2016", "state_after": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nn : \u2115\nf : Fin n \u2192 E\nh : \u2200 (i : Fin n), \u2016f i\u2016 \u2264 2\nh' : \u2200 (i j : Fin n), i \u2260 j \u2192 1 - good\u03b4 E \u2264 \u2016f i - f j\u2016\nfinj : Function.Injective f\ns : Finset E := Finset.image f Finset.univ\ns_card : Finset.card s = n\nhs : \u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2\n\u22a2 \u2200 (a a_1 : Fin n), \u00acf a = f a_1 \u2192 1 - good\u03b4 E \u2264 \u2016f a - f a_1\u2016"}, {"tactic": "intro i j hij", "annotated_tactic": ["intro i j hij", []], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nn : \u2115\nf : Fin n \u2192 E\nh : \u2200 (i : Fin n), \u2016f i\u2016 \u2264 2\nh' : \u2200 (i j : Fin n), i \u2260 j \u2192 1 - good\u03b4 E \u2264 \u2016f i - f j\u2016\nfinj : Function.Injective f\ns : Finset E := Finset.image f Finset.univ\ns_card : Finset.card s = n\nhs : \u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2\n\u22a2 \u2200 (a a_1 : Fin n), \u00acf a = f a_1 \u2192 1 - good\u03b4 E \u2264 \u2016f a - f a_1\u2016", "state_after": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nn : \u2115\nf : Fin n \u2192 E\nh : \u2200 (i : Fin n), \u2016f i\u2016 \u2264 2\nh' : \u2200 (i j : Fin n), i \u2260 j \u2192 1 - good\u03b4 E \u2264 \u2016f i - f j\u2016\nfinj : Function.Injective f\ns : Finset E := Finset.image f Finset.univ\ns_card : Finset.card s = n\nhs : \u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2\ni j : Fin n\nhij : \u00acf i = f j\n\u22a2 1 - good\u03b4 E \u2264 \u2016f i - f j\u2016"}, {"tactic": "have : i \u2260 j := fun h => by rw [h] at hij; exact hij rfl", "annotated_tactic": ["have : i \u2260 j := fun h => by rw [h] at hij; exact hij <a>rfl</a>", [{"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nn : \u2115\nf : Fin n \u2192 E\nh : \u2200 (i : Fin n), \u2016f i\u2016 \u2264 2\nh' : \u2200 (i j : Fin n), i \u2260 j \u2192 1 - good\u03b4 E \u2264 \u2016f i - f j\u2016\nfinj : Function.Injective f\ns : Finset E := Finset.image f Finset.univ\ns_card : Finset.card s = n\nhs : \u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2\ni j : Fin n\nhij : \u00acf i = f j\n\u22a2 1 - good\u03b4 E \u2264 \u2016f i - f j\u2016", "state_after": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nn : \u2115\nf : Fin n \u2192 E\nh : \u2200 (i : Fin n), \u2016f i\u2016 \u2264 2\nh' : \u2200 (i j : Fin n), i \u2260 j \u2192 1 - good\u03b4 E \u2264 \u2016f i - f j\u2016\nfinj : Function.Injective f\ns : Finset E := Finset.image f Finset.univ\ns_card : Finset.card s = n\nhs : \u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2\ni j : Fin n\nhij : \u00acf i = f j\nthis : i \u2260 j\n\u22a2 1 - good\u03b4 E \u2264 \u2016f i - f j\u2016"}, {"tactic": "exact h' i j this", "annotated_tactic": ["exact h' i j this", []], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nn : \u2115\nf : Fin n \u2192 E\nh : \u2200 (i : Fin n), \u2016f i\u2016 \u2264 2\nh' : \u2200 (i j : Fin n), i \u2260 j \u2192 1 - good\u03b4 E \u2264 \u2016f i - f j\u2016\nfinj : Function.Injective f\ns : Finset E := Finset.image f Finset.univ\ns_card : Finset.card s = n\nhs : \u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2\ni j : Fin n\nhij : \u00acf i = f j\nthis : i \u2260 j\n\u22a2 1 - good\u03b4 E \u2264 \u2016f i - f j\u2016", "state_after": "no goals"}, {"tactic": "rw [h] at hij", "annotated_tactic": ["rw [h] at hij", []], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nn : \u2115\nf : Fin n \u2192 E\nh\u271d : \u2200 (i : Fin n), \u2016f i\u2016 \u2264 2\nh' : \u2200 (i j : Fin n), i \u2260 j \u2192 1 - good\u03b4 E \u2264 \u2016f i - f j\u2016\nfinj : Function.Injective f\ns : Finset E := Finset.image f Finset.univ\ns_card : Finset.card s = n\nhs : \u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2\ni j : Fin n\nhij : \u00acf i = f j\nh : i = j\n\u22a2 False", "state_after": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nn : \u2115\nf : Fin n \u2192 E\nh\u271d : \u2200 (i : Fin n), \u2016f i\u2016 \u2264 2\nh' : \u2200 (i j : Fin n), i \u2260 j \u2192 1 - good\u03b4 E \u2264 \u2016f i - f j\u2016\nfinj : Function.Injective f\ns : Finset E := Finset.image f Finset.univ\ns_card : Finset.card s = n\nhs : \u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2\ni j : Fin n\nhij : \u00acf j = f j\nh : i = j\n\u22a2 False"}, {"tactic": "exact hij rfl", "annotated_tactic": ["exact hij <a>rfl</a>", [{"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nn : \u2115\nf : Fin n \u2192 E\nh\u271d : \u2200 (i : Fin n), \u2016f i\u2016 \u2264 2\nh' : \u2200 (i j : Fin n), i \u2260 j \u2192 1 - good\u03b4 E \u2264 \u2016f i - f j\u2016\nfinj : Function.Injective f\ns : Finset E := Finset.image f Finset.univ\ns_card : Finset.card s = n\nhs : \u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2\ni j : Fin n\nhij : \u00acf j = f j\nh : i = j\n\u22a2 False", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Part.lean", "full_name": "Part.mem_mk_iff", "start": [110, 1], "end": [111, 10], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Logic.lean", "full_name": "iff_true_right", "start": [65, 1], "end": [65, 83], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Setoid/Partition.lean", "full_name": "Setoid.eq_of_mem_eqv_class", "start": [47, 1], "end": [49, 30], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "full_name": "MeasureTheory.Lp.simpleFunc.norm_toSimpleFunc", "start": [678, 1], "end": [680, 82], "traced_tactics": [{"tactic": "simpa [toLp_toSimpleFunc] using norm_toLp (toSimpleFunc f) (simpleFunc.mem\u2112p f)", "annotated_tactic": ["simpa [<a>toLp_toSimpleFunc</a>] using <a>norm_toLp</a> (<a>toSimpleFunc</a> f) (<a>simpleFunc.mem\u2112p</a> f)", [{"full_name": "MeasureTheory.Lp.simpleFunc.toLp_toSimpleFunc", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "def_pos": [626, 9], "def_end_pos": [626, 26]}, {"full_name": "MeasureTheory.Lp.simpleFunc.norm_toLp", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "def_pos": [577, 16], "def_end_pos": [577, 25]}, {"full_name": "MeasureTheory.Lp.simpleFunc.toSimpleFunc", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "def_pos": [587, 5], "def_end_pos": [587, 17]}, {"full_name": "MeasureTheory.Lp.simpleFunc.mem\u2112p", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "def_pos": [622, 19], "def_end_pos": [622, 24]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : NormedRing \ud835\udd5c\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : BoundedSMul \ud835\udd5c E\ninst\u271d : Fact (1 \u2264 p)\nf : { x // x \u2208 simpleFunc E p \u03bc }\n\u22a2 \u2016f\u2016 = ENNReal.toReal (snorm (\u2191(toSimpleFunc f)) p \u03bc)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "full_name": "Int.toNat_sub_toNat_neg", "start": [1401, 9], "end": [1404, 29], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean", "full_name": "MeasureTheory.condexpInd_smul'", "start": [302, 1], "end": [304, 25], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Intervals/Pi.lean", "full_name": "Set.pi_univ_Ioc_update_union", "start": [335, 1], "end": [340, 50], "traced_tactics": [{"tactic": "simp_rw [pi_univ_Ioc_update_left hm.1, pi_univ_Ioc_update_right hm.2, \u2190 union_inter_distrib_right,\n  \u2190 setOf_or, le_or_lt, setOf_true, univ_inter]", "annotated_tactic": ["simp_rw [<a>pi_univ_Ioc_update_left</a> hm.1, <a>pi_univ_Ioc_update_right</a> hm.2, \u2190 <a>union_inter_distrib_right</a>,\n    \u2190 <a>setOf_or</a>, <a>le_or_lt</a>, <a>setOf_true</a>, <a>univ_inter</a>]", [{"full_name": "Set.pi_univ_Ioc_update_left", "def_path": "Mathlib/Data/Set/Intervals/Pi.lean", "def_pos": [90, 9], "def_end_pos": [90, 32]}, {"full_name": "Set.pi_univ_Ioc_update_right", "def_path": "Mathlib/Data/Set/Intervals/Pi.lean", "def_pos": [101, 9], "def_end_pos": [101, 33]}, {"full_name": "Set.union_inter_distrib_right", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1067, 9], "def_end_pos": [1067, 34]}, {"full_name": "Set.setOf_or", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [304, 9], "def_end_pos": [304, 17]}, {"full_name": "le_or_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [340, 9], "def_end_pos": [340, 17]}, {"full_name": "Set.setOf_true", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [669, 9], "def_end_pos": [669, 19]}, {"full_name": "Set.univ_inter", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1017, 9], "def_end_pos": [1017, 19]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 LinearOrder (\u03b1 i)\nx y : (i : \u03b9) \u2192 \u03b1 i\ni\u2080 : \u03b9\nm : \u03b1 i\u2080\nhm : m \u2208 Icc (x i\u2080) (y i\u2080)\n\u22a2 ((pi univ fun i => Ioc (x i) (update y i\u2080 m i)) \u222a pi univ fun i => Ioc (update x i\u2080 m i) (y i)) =\n    pi univ fun i => Ioc (x i) (y i)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "full_name": "MeasureTheory.set_lintegral_lt_top_of_isCompact", "start": [1553, 1], "end": [1556, 76], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "full_name": "List.mem_iff_get?", "start": [626, 1], "end": [627, 51], "traced_tactics": [{"tactic": "simp [get?_eq_some, Fin.exists_iff, mem_iff_get]", "annotated_tactic": ["simp [<a>get?_eq_some</a>, <a>Fin.exists_iff</a>, <a>mem_iff_get</a>]", [{"full_name": "List.get?_eq_some", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [600, 9], "def_end_pos": [600, 21]}, {"full_name": "List.Fin.exists_iff", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [623, 9], "def_end_pos": [623, 23]}, {"full_name": "List.mem_iff_get", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [619, 9], "def_end_pos": [619, 20]}]], "state_before": "\u03b1 : Type u_1\na : \u03b1\nl : List \u03b1\n\u22a2 a \u2208 l \u2194 \u2203 n, get? l n = some a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "full_name": "MeasureTheory.snorm_mono_measure", "start": [612, 1], "end": [618, 56], "traced_tactics": [{"tactic": "by_cases hp0 : p = 0", "annotated_tactic": ["by_cases hp0 : p = 0", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nh\u03bc\u03bd : \u03bd \u2264 \u03bc\n\u22a2 snorm f p \u03bd \u2264 snorm f p \u03bc", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nh\u03bc\u03bd : \u03bd \u2264 \u03bc\nhp0 : p = 0\n\u22a2 snorm f p \u03bd \u2264 snorm f p \u03bc\n\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nh\u03bc\u03bd : \u03bd \u2264 \u03bc\nhp0 : \u00acp = 0\n\u22a2 snorm f p \u03bd \u2264 snorm f p \u03bc"}, {"tactic": "by_cases hp_top : p = \u221e", "annotated_tactic": ["by_cases hp_top : p = \u221e", []], "state_before": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nh\u03bc\u03bd : \u03bd \u2264 \u03bc\nhp0 : \u00acp = 0\n\u22a2 snorm f p \u03bd \u2264 snorm f p \u03bc", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nh\u03bc\u03bd : \u03bd \u2264 \u03bc\nhp0 : \u00acp = 0\nhp_top : p = \u22a4\n\u22a2 snorm f p \u03bd \u2264 snorm f p \u03bc\n\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nh\u03bc\u03bd : \u03bd \u2264 \u03bc\nhp0 : \u00acp = 0\nhp_top : \u00acp = \u22a4\n\u22a2 snorm f p \u03bd \u2264 snorm f p \u03bc"}, {"tactic": "simp_rw [snorm_eq_snorm' hp0 hp_top]", "annotated_tactic": ["simp_rw [<a>snorm_eq_snorm'</a> hp0 hp_top]", [{"full_name": "MeasureTheory.snorm_eq_snorm'", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [88, 9], "def_end_pos": [88, 24]}]], "state_before": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nh\u03bc\u03bd : \u03bd \u2264 \u03bc\nhp0 : \u00acp = 0\nhp_top : \u00acp = \u22a4\n\u22a2 snorm f p \u03bd \u2264 snorm f p \u03bc", "state_after": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nh\u03bc\u03bd : \u03bd \u2264 \u03bc\nhp0 : \u00acp = 0\nhp_top : \u00acp = \u22a4\n\u22a2 snorm' f (ENNReal.toReal p) \u03bd \u2264 snorm' f (ENNReal.toReal p) \u03bc"}, {"tactic": "exact snorm'_mono_measure f h\u03bc\u03bd ENNReal.toReal_nonneg", "annotated_tactic": ["exact <a>snorm'_mono_measure</a> f h\u03bc\u03bd <a>ENNReal.toReal_nonneg</a>", [{"full_name": "MeasureTheory.snorm'_mono_measure", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [597, 9], "def_end_pos": [597, 28]}, {"full_name": "ENNReal.toReal_nonneg", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [221, 17], "def_end_pos": [221, 30]}]], "state_before": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nh\u03bc\u03bd : \u03bd \u2264 \u03bc\nhp0 : \u00acp = 0\nhp_top : \u00acp = \u22a4\n\u22a2 snorm' f (ENNReal.toReal p) \u03bd \u2264 snorm' f (ENNReal.toReal p) \u03bc", "state_after": "no goals"}, {"tactic": "simp [hp0]", "annotated_tactic": ["simp [hp0]", []], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nh\u03bc\u03bd : \u03bd \u2264 \u03bc\nhp0 : p = 0\n\u22a2 snorm f p \u03bd \u2264 snorm f p \u03bc", "state_after": "no goals"}, {"tactic": "simp [hp_top, snormEssSup_mono_measure f (Measure.absolutelyContinuous_of_le h\u03bc\u03bd)]", "annotated_tactic": ["simp [hp_top, <a>snormEssSup_mono_measure</a> f (<a>Measure.absolutelyContinuous_of_le</a> h\u03bc\u03bd)]", [{"full_name": "MeasureTheory.snormEssSup_mono_measure", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [606, 9], "def_end_pos": [606, 33]}, {"full_name": "MeasureTheory.Measure.absolutelyContinuous_of_le", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2119, 9], "def_end_pos": [2119, 35]}]], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nh\u03bc\u03bd : \u03bd \u2264 \u03bc\nhp0 : \u00acp = 0\nhp_top : p = \u22a4\n\u22a2 snorm f p \u03bd \u2264 snorm f p \u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Int/Basic.lean", "full_name": "Int.succ_neg_nat_succ", "start": [186, 1], "end": [186, 79], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "full_name": "MeasureTheory.L1.setToL1_congr_left'", "start": [1065, 1], "end": [1074, 50], "traced_tactics": [{"tactic": "suffices setToL1 hT = setToL1 hT' by rw [this]", "annotated_tactic": ["suffices <a>setToL1</a> hT = <a>setToL1</a> hT' by rw [this]", [{"full_name": "MeasureTheory.L1.setToL1", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [1019, 5], "def_end_pos": [1019, 12]}, {"full_name": "MeasureTheory.L1.setToL1", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [1019, 5], "def_end_pos": [1019, 12]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : NormedAddCommGroup F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nT\u271d T'\u271d T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC\u271d C'\u271d C'' : \u211d\nT T' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc T' C'\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T s = T' s\nf : { x // x \u2208 Lp E 1 }\n\u22a2 \u2191(setToL1 hT) f = \u2191(setToL1 hT') f", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : NormedAddCommGroup F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nT\u271d T'\u271d T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC\u271d C'\u271d C'' : \u211d\nT T' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc T' C'\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T s = T' s\nf : { x // x \u2208 Lp E 1 }\n\u22a2 setToL1 hT = setToL1 hT'"}, {"tactic": "refine' ContinuousLinearMap.extend_unique (setToL1SCLM \u03b1 E \u03bc hT) _ _ _ _ _", "annotated_tactic": ["refine' <a>ContinuousLinearMap.extend_unique</a> (<a>setToL1SCLM</a> \u03b1 E \u03bc hT) _ _ _ _ _", [{"full_name": "ContinuousLinearMap.extend_unique", "def_path": "Mathlib/Analysis/NormedSpace/OperatorNorm.lean", "def_pos": [1746, 9], "def_end_pos": [1746, 22]}, {"full_name": "MeasureTheory.L1.SimpleFunc.setToL1SCLM", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [877, 5], "def_end_pos": [877, 16]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : NormedAddCommGroup F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nT\u271d T'\u271d T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC\u271d C'\u271d C'' : \u211d\nT T' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc T' C'\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T s = T' s\nf : { x // x \u2208 Lp E 1 }\n\u22a2 setToL1 hT = setToL1 hT'", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : NormedAddCommGroup F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nT\u271d T'\u271d T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC\u271d C'\u271d C'' : \u211d\nT T' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc T' C'\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T s = T' s\nf : { x // x \u2208 Lp E 1 }\n\u22a2 ContinuousLinearMap.comp (setToL1 hT') (coeToLp \u03b1 E \u211d) = setToL1SCLM \u03b1 E \u03bc hT"}, {"tactic": "ext1 f", "annotated_tactic": ["ext1 f", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : NormedAddCommGroup F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nT\u271d T'\u271d T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC\u271d C'\u271d C'' : \u211d\nT T' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc T' C'\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T s = T' s\nf : { x // x \u2208 Lp E 1 }\n\u22a2 ContinuousLinearMap.comp (setToL1 hT') (coeToLp \u03b1 E \u211d) = setToL1SCLM \u03b1 E \u03bc hT", "state_after": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : NormedAddCommGroup F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nT\u271d T'\u271d T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC\u271d C'\u271d C'' : \u211d\nT T' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc T' C'\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T s = T' s\nf\u271d : { x // x \u2208 Lp E 1 }\nf : { x // x \u2208 simpleFunc E 1 \u03bc }\n\u22a2 \u2191(ContinuousLinearMap.comp (setToL1 hT') (coeToLp \u03b1 E \u211d)) f = \u2191(setToL1SCLM \u03b1 E \u03bc hT) f"}, {"tactic": "suffices setToL1 hT' f = setToL1SCLM \u03b1 E \u03bc hT f by rw [\u2190 this]; rfl", "annotated_tactic": ["suffices <a>setToL1</a> hT' f = <a>setToL1SCLM</a> \u03b1 E \u03bc hT f by rw [\u2190 this]; rfl", [{"full_name": "MeasureTheory.L1.setToL1", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [1019, 5], "def_end_pos": [1019, 12]}, {"full_name": "MeasureTheory.L1.SimpleFunc.setToL1SCLM", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [877, 5], "def_end_pos": [877, 16]}]], "state_before": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : NormedAddCommGroup F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nT\u271d T'\u271d T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC\u271d C'\u271d C'' : \u211d\nT T' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc T' C'\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T s = T' s\nf\u271d : { x // x \u2208 Lp E 1 }\nf : { x // x \u2208 simpleFunc E 1 \u03bc }\n\u22a2 \u2191(ContinuousLinearMap.comp (setToL1 hT') (coeToLp \u03b1 E \u211d)) f = \u2191(setToL1SCLM \u03b1 E \u03bc hT) f", "state_after": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : NormedAddCommGroup F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nT\u271d T'\u271d T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC\u271d C'\u271d C'' : \u211d\nT T' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc T' C'\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T s = T' s\nf\u271d : { x // x \u2208 Lp E 1 }\nf : { x // x \u2208 simpleFunc E 1 \u03bc }\n\u22a2 \u2191(setToL1 hT') \u2191f = \u2191(setToL1SCLM \u03b1 E \u03bc hT) f"}, {"tactic": "rw [setToL1_eq_setToL1SCLM]", "annotated_tactic": ["rw [<a>setToL1_eq_setToL1SCLM</a>]", [{"full_name": "MeasureTheory.L1.setToL1_eq_setToL1SCLM", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [1024, 9], "def_end_pos": [1024, 31]}]], "state_before": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : NormedAddCommGroup F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nT\u271d T'\u271d T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC\u271d C'\u271d C'' : \u211d\nT T' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc T' C'\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T s = T' s\nf\u271d : { x // x \u2208 Lp E 1 }\nf : { x // x \u2208 simpleFunc E 1 \u03bc }\n\u22a2 \u2191(setToL1 hT') \u2191f = \u2191(setToL1SCLM \u03b1 E \u03bc hT) f", "state_after": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : NormedAddCommGroup F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nT\u271d T'\u271d T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC\u271d C'\u271d C'' : \u211d\nT T' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc T' C'\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T s = T' s\nf\u271d : { x // x \u2208 Lp E 1 }\nf : { x // x \u2208 simpleFunc E 1 \u03bc }\n\u22a2 \u2191(setToL1SCLM \u03b1 E \u03bc hT') f = \u2191(setToL1SCLM \u03b1 E \u03bc hT) f"}, {"tactic": "exact (setToL1SCLM_congr_left' hT hT' h f).symm", "annotated_tactic": ["exact (<a>setToL1SCLM_congr_left'</a> hT hT' h f).<a>symm</a>", [{"full_name": "MeasureTheory.L1.SimpleFunc.setToL1SCLM_congr_left'", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [907, 9], "def_end_pos": [907, 32]}, {"full_name": "Eq.symm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [310, 9], "def_end_pos": [310, 16]}]], "state_before": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : NormedAddCommGroup F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nT\u271d T'\u271d T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC\u271d C'\u271d C'' : \u211d\nT T' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc T' C'\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T s = T' s\nf\u271d : { x // x \u2208 Lp E 1 }\nf : { x // x \u2208 simpleFunc E 1 \u03bc }\n\u22a2 \u2191(setToL1SCLM \u03b1 E \u03bc hT') f = \u2191(setToL1SCLM \u03b1 E \u03bc hT) f", "state_after": "no goals"}, {"tactic": "rw [this]", "annotated_tactic": ["rw [this]", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : NormedAddCommGroup F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nT\u271d T'\u271d T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC\u271d C'\u271d C'' : \u211d\nT T' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc T' C'\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T s = T' s\nf : { x // x \u2208 Lp E 1 }\nthis : setToL1 hT = setToL1 hT'\n\u22a2 \u2191(setToL1 hT) f = \u2191(setToL1 hT') f", "state_after": "no goals"}, {"tactic": "rw [\u2190 this]", "annotated_tactic": ["rw [\u2190 this]", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : NormedAddCommGroup F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nT\u271d T'\u271d T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC\u271d C'\u271d C'' : \u211d\nT T' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc T' C'\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T s = T' s\nf\u271d : { x // x \u2208 Lp E 1 }\nf : { x // x \u2208 simpleFunc E 1 \u03bc }\nthis : \u2191(setToL1 hT') \u2191f = \u2191(setToL1SCLM \u03b1 E \u03bc hT) f\n\u22a2 \u2191(ContinuousLinearMap.comp (setToL1 hT') (coeToLp \u03b1 E \u211d)) f = \u2191(setToL1SCLM \u03b1 E \u03bc hT) f", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : NormedAddCommGroup F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nT\u271d T'\u271d T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC\u271d C'\u271d C'' : \u211d\nT T' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc T' C'\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T s = T' s\nf\u271d : { x // x \u2208 Lp E 1 }\nf : { x // x \u2208 simpleFunc E 1 \u03bc }\nthis : \u2191(setToL1 hT') \u2191f = \u2191(setToL1SCLM \u03b1 E \u03bc hT) f\n\u22a2 \u2191(ContinuousLinearMap.comp (setToL1 hT') (coeToLp \u03b1 E \u211d)) f = \u2191(setToL1 hT') \u2191f"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : NormedAddCommGroup F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nT\u271d T'\u271d T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC\u271d C'\u271d C'' : \u211d\nT T' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc T' C'\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T s = T' s\nf\u271d : { x // x \u2208 Lp E 1 }\nf : { x // x \u2208 simpleFunc E 1 \u03bc }\nthis : \u2191(setToL1 hT') \u2191f = \u2191(setToL1SCLM \u03b1 E \u03bc hT) f\n\u22a2 \u2191(ContinuousLinearMap.comp (setToL1 hT') (coeToLp \u03b1 E \u211d)) f = \u2191(setToL1 hT') \u2191f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "full_name": "MeasureTheory.Lp.continuous_posPart", "start": [1268, 1], "end": [1269, 38], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/TuringMachine.lean", "full_name": "Turing.TM2to1.trNormal_run", "start": [2526, 1], "end": [2528, 18], "traced_tactics": [{"tactic": "cases s <;> rfl", "annotated_tactic": ["cases s <;> rfl", []], "state_before": "K : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nk : K\ns : StAct k\nq : Stmt\u2082\n\u22a2 trNormal (stRun s q) = goto fun x x => go k s q", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "full_name": "Int.mul_self_le_mul_self", "start": [1236, 11], "end": [1237, 47], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/TuringMachine.lean", "full_name": "Turing.reaches_total", "start": [760, 1], "end": [762, 77], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/LocallyIntegrable.lean", "full_name": "MeasureTheory.locallyIntegrable_finset_sum", "start": [325, 1], "end": [327, 75], "traced_tactics": [{"tactic": "simpa only [\u2190 Finset.sum_apply] using locallyIntegrable_finset_sum' s hf", "annotated_tactic": ["simpa only [\u2190 <a>Finset.sum_apply</a>] using <a>locallyIntegrable_finset_sum'</a> s hf", [{"full_name": "Finset.sum_apply", "def_path": "Mathlib/Algebra/BigOperators/Pi.lean", "def_pos": [41, 3], "def_end_pos": [41, 14]}, {"full_name": "MeasureTheory.locallyIntegrable_finset_sum'", "def_path": "Mathlib/MeasureTheory/Function/LocallyIntegrable.lean", "def_pos": [320, 9], "def_end_pos": [320, 38]}]], "state_before": "X : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormedAddCommGroup E\nf\u271d g : X \u2192 E\n\u03bc : Measure X\ns\u271d : Set X\n\u03b9 : Type u_5\ns : Finset \u03b9\nf : \u03b9 \u2192 X \u2192 E\nhf : \u2200 (i : \u03b9), i \u2208 s \u2192 LocallyIntegrable (f i)\n\u22a2 LocallyIntegrable fun a => \u2211 i in s, f i a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Sigma.lean", "full_name": "Finset.disjiUnion_map_sigma_mk", "start": [85, 1], "end": [88, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Constructions/Prod/Basic.lean", "full_name": "generateFrom_eq_prod", "start": [134, 1], "end": [137, 48], "traced_tactics": [{"tactic": "rw [\u2190 hC, \u2190 hD, generateFrom_prod_eq h2C h2D]", "annotated_tactic": ["rw [\u2190 hC, \u2190 hD, <a>generateFrom_prod_eq</a> h2C h2D]", [{"full_name": "generateFrom_prod_eq", "def_path": "Mathlib/MeasureTheory/Constructions/Prod/Basic.lean", "def_pos": [104, 9], "def_end_pos": [104, 29]}]], "state_before": "\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1'\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b2'\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d : NormedAddCommGroup E\nC : Set (Set \u03b1)\nD : Set (Set \u03b2)\nhC : generateFrom C = inst\u271d\u2075\nhD : generateFrom D = inst\u271d\u00b3\nh2C : IsCountablySpanning C\nh2D : IsCountablySpanning D\n\u22a2 generateFrom (image2 (fun x x_1 => x \u00d7\u02e2 x_1) C D) = Prod.instMeasurableSpace", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "full_name": "MeasureTheory.Integrable.smul_of_top_right", "start": [1088, 1], "end": [1091, 38], "traced_tactics": [{"tactic": "rw [\u2190 mem\u2112p_one_iff_integrable] at hf \u22a2", "annotated_tactic": ["rw [\u2190 <a>mem\u2112p_one_iff_integrable</a>] at hf \u22a2", [{"full_name": "MeasureTheory.mem\u2112p_one_iff_integrable", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [453, 9], "def_end_pos": [453, 33]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b4\ninst\u271d\u2074 : NormedAddCommGroup \u03b2\ninst\u271d\u00b3 : NormedAddCommGroup \u03b3\n\ud835\udd5c : Type u_5\ninst\u271d\u00b2 : NormedRing \ud835\udd5c\ninst\u271d\u00b9 : Module \ud835\udd5c \u03b2\ninst\u271d : BoundedSMul \ud835\udd5c \u03b2\nf : \u03b1 \u2192 \u03b2\n\u03c6 : \u03b1 \u2192 \ud835\udd5c\nhf : Integrable f\nh\u03c6 : Mem\u2112p \u03c6 \u22a4\n\u22a2 Integrable (\u03c6 \u2022 f)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b4\ninst\u271d\u2074 : NormedAddCommGroup \u03b2\ninst\u271d\u00b3 : NormedAddCommGroup \u03b3\n\ud835\udd5c : Type u_5\ninst\u271d\u00b2 : NormedRing \ud835\udd5c\ninst\u271d\u00b9 : Module \ud835\udd5c \u03b2\ninst\u271d : BoundedSMul \ud835\udd5c \u03b2\nf : \u03b1 \u2192 \u03b2\n\u03c6 : \u03b1 \u2192 \ud835\udd5c\nhf : Mem\u2112p f 1\nh\u03c6 : Mem\u2112p \u03c6 \u22a4\n\u22a2 Mem\u2112p (\u03c6 \u2022 f) 1"}, {"tactic": "exact Mem\u2112p.smul_of_top_right hf h\u03c6", "annotated_tactic": ["exact <a>Mem\u2112p.smul_of_top_right</a> hf h\u03c6", [{"full_name": "MeasureTheory.Mem\u2112p.smul_of_top_right", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [1538, 9], "def_end_pos": [1538, 32]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b4\ninst\u271d\u2074 : NormedAddCommGroup \u03b2\ninst\u271d\u00b3 : NormedAddCommGroup \u03b3\n\ud835\udd5c : Type u_5\ninst\u271d\u00b2 : NormedRing \ud835\udd5c\ninst\u271d\u00b9 : Module \ud835\udd5c \u03b2\ninst\u271d : BoundedSMul \ud835\udd5c \u03b2\nf : \u03b1 \u2192 \u03b2\n\u03c6 : \u03b1 \u2192 \ud835\udd5c\nhf : Mem\u2112p f 1\nh\u03c6 : Mem\u2112p \u03c6 \u22a4\n\u22a2 Mem\u2112p (\u03c6 \u2022 f) 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "full_name": "MeasureTheory.snorm_sub_le", "start": [888, 1], "end": [890, 62], "traced_tactics": [{"tactic": "simpa [LpAddConst_of_one_le hp] using snorm_sub_le' hf hg p", "annotated_tactic": ["simpa [<a>LpAddConst_of_one_le</a> hp] using <a>snorm_sub_le'</a> hf hg p", [{"full_name": "MeasureTheory.LpAddConst_of_one_le", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [820, 9], "def_end_pos": [820, 29]}, {"full_name": "MeasureTheory.snorm_sub_le'", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [883, 9], "def_end_pos": [883, 22]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf g : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhg : AEStronglyMeasurable g \u03bc\nhp : 1 \u2264 p\n\u22a2 snorm (f - g) p \u03bc \u2264 snorm f p \u03bc + snorm g p \u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "full_name": "MeasurableEmbedding.measurable_rangeSplitting", "start": [1224, 1], "end": [1228, 57], "traced_tactics": [{"tactic": "rwa [preimage_rangeSplitting hf.injective,\n  \u2190 (subtype_coe hf.measurableSet_range).measurableSet_image, \u2190 image_comp,\n  coe_comp_rangeFactorization, hf.measurableSet_image]", "annotated_tactic": ["rwa [<a>preimage_rangeSplitting</a> hf.injective,\n    \u2190 (<a>subtype_coe</a> hf.measurableSet_range).<a>measurableSet_image</a>, \u2190 <a>image_comp</a>,\n    <a>coe_comp_rangeFactorization</a>, hf.measurableSet_image]", [{"full_name": "Set.preimage_rangeSplitting", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [1193, 9], "def_end_pos": [1193, 32]}, {"full_name": "MeasurableEmbedding.subtype_coe", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [1208, 9], "def_end_pos": [1208, 20]}, {"full_name": "MeasurableEmbedding.measurableSet_image", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [1192, 9], "def_end_pos": [1192, 28]}, {"full_name": "Set.image_comp", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [293, 9], "def_end_pos": [293, 19]}, {"full_name": 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"3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/UniformIntegrable.lean", "full_name": "MeasureTheory.unifIntegrable_of_tendsto_Lp", "start": [593, 1], "end": [601, 82], "traced_tactics": [{"tactic": "have : f = (fun _ => g) + fun n => f n - g := by ext1 n; simp", "annotated_tactic": ["have : f = (fun _ => g) + fun n => f n - g := by ext1 n; simp", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nhf : \u2200 (n : \u2115), Mem\u2112p (f n) p\nhg : Mem\u2112p g p\nhfg : Tendsto (fun n => snorm (f n - g) p \u03bc) atTop (\ud835\udcdd 0)\n\u22a2 UnifIntegrable f p \u03bc", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure 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"\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nhf : \u2200 (n : \u2115), Mem\u2112p (f n) p\nhg : Mem\u2112p g p\nhfg : Tendsto (fun n => snorm (f n - g) p \u03bc) atTop (\ud835\udcdd 0)\nthis : f = (fun x => g) + fun n => f n - g\n\u22a2 UnifIntegrable ((fun x => g) + fun n => f n - g) p \u03bc"}, {"tactic": "refine' UnifIntegrable.add _ _ hp (fun _ => hg.aestronglyMeasurable)\n    fun n => (hf n).1.sub hg.aestronglyMeasurable", "annotated_tactic": ["refine' <a>UnifIntegrable.add</a> _ _ hp (fun _ => hg.aestronglyMeasurable)\n      fun n => (hf n).1.<a>sub</a> hg.aestronglyMeasurable", [{"full_name": "MeasureTheory.UnifIntegrable.add", "def_path": "Mathlib/MeasureTheory/Function/UniformIntegrable.lean", "def_pos": [108, 19], "def_end_pos": [108, 22]}, {"full_name": "MeasureTheory.AEStronglyMeasurable.sub", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1322, 3], "def_end_pos": [1322, 14]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nhf : \u2200 (n : \u2115), Mem\u2112p (f n) p\nhg : Mem\u2112p g p\nhfg : Tendsto (fun n => snorm (f n - g) p \u03bc) atTop (\ud835\udcdd 0)\nthis : f = (fun x => g) + fun n => f n - g\n\u22a2 UnifIntegrable ((fun x => g) + fun n => f n - g) p \u03bc", "state_after": "case refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nhf : \u2200 (n : \u2115), Mem\u2112p (f n) p\nhg : Mem\u2112p g p\nhfg : Tendsto (fun n => snorm (f n - g) p \u03bc) atTop (\ud835\udcdd 0)\nthis : f = (fun x => g) + fun n => f n - g\n\u22a2 UnifIntegrable (fun x => g) p \u03bc\n\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nhf : \u2200 (n : \u2115), Mem\u2112p (f n) p\nhg : Mem\u2112p g p\nhfg : Tendsto (fun n => snorm (f n - g) p \u03bc) atTop (\ud835\udcdd 0)\nthis : f = (fun x => g) + fun n => f n - g\n\u22a2 UnifIntegrable (fun n => f n - g) p \u03bc"}, {"tactic": "ext1 n", "annotated_tactic": ["ext1 n", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nhf : \u2200 (n : \u2115), Mem\u2112p (f n) p\nhg : Mem\u2112p g p\nhfg : Tendsto (fun n => snorm (f n - g) p \u03bc) atTop (\ud835\udcdd 0)\n\u22a2 f = (fun x => g) + fun n => f n - g", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nhf : \u2200 (n : \u2115), Mem\u2112p (f n) p\nhg : Mem\u2112p g p\nhfg : Tendsto (fun n => snorm (f n - g) p \u03bc) atTop (\ud835\udcdd 0)\nn : \u2115\n\u22a2 f n = ((fun x => g) + fun n => f n - g) n"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nhf : \u2200 (n : \u2115), Mem\u2112p (f n) p\nhg : Mem\u2112p g p\nhfg : Tendsto (fun n => snorm (f n - g) p \u03bc) atTop (\ud835\udcdd 0)\nn : \u2115\n\u22a2 f n = ((fun x => g) + fun n => f n - g) n", "state_after": "no goals"}, {"tactic": "exact unifIntegrable_const \u03bc hp hp' hg", "annotated_tactic": ["exact <a>unifIntegrable_const</a> \u03bc hp hp' hg", [{"full_name": "MeasureTheory.unifIntegrable_const", "def_path": "Mathlib/MeasureTheory/Function/UniformIntegrable.lean", "def_pos": [402, 9], "def_end_pos": [402, 29]}]], "state_before": "case refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nhf : \u2200 (n : \u2115), Mem\u2112p (f n) p\nhg : Mem\u2112p g p\nhfg : Tendsto (fun n => snorm (f n - g) p \u03bc) atTop (\ud835\udcdd 0)\nthis : f = (fun x => g) + fun n => f n - g\n\u22a2 UnifIntegrable (fun x => g) p \u03bc", "state_after": "no goals"}, {"tactic": "exact unifIntegrable_of_tendsto_Lp_zero \u03bc hp hp' (fun n => (hf n).sub hg) hfg", "annotated_tactic": ["exact <a>unifIntegrable_of_tendsto_Lp_zero</a> \u03bc hp hp' (fun n => (hf n).<a>sub</a> hg) hfg", [{"full_name": "MeasureTheory.unifIntegrable_of_tendsto_Lp_zero", "def_path": "Mathlib/MeasureTheory/Function/UniformIntegrable.lean", "def_pos": [577, 9], "def_end_pos": [577, 42]}, {"full_name": "MeasureTheory.Mem\u2112p.sub", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [1238, 9], "def_end_pos": [1238, 18]}]], "state_before": "case refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nhf : \u2200 (n : \u2115), Mem\u2112p (f n) p\nhg : Mem\u2112p g p\nhfg : Tendsto (fun n => snorm (f n - g) p \u03bc) atTop (\ud835\udcdd 0)\nthis : f = (fun x => g) + fun n => f n - g\n\u22a2 UnifIntegrable (fun n => f n - g) p \u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Partrec.lean", "full_name": "Nat.Partrec.prec'", "start": [208, 1], "end": [212, 35], "traced_tactics": [{"tactic": "simp [Seq.seq]", "annotated_tactic": ["simp [<a>Seq.seq</a>]", [{"full_name": "Seq.seq", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2698, 3], "def_end_pos": [2698, 6]}]], "state_before": "f g h : \u2115 \u2192. \u2115\nhf : Partrec f\nhg : Partrec g\nhh : Partrec h\na s : \u2115\n\u22a2 (s \u2208\n      (Seq.seq (Nat.pair <$> Part.some a) fun x => f a) >>=\n        unpaired fun a n =>\n          Nat.rec (g a)\n            (fun y IH => do\n              let i \u2190 IH\n              h (Nat.pair a (Nat.pair y i)))\n            n) \u2194\n    s \u2208\n      Part.bind (f a) fun n =>\n        Nat.rec (g a)\n          (fun y IH => do\n            let i \u2190 IH\n            h (Nat.pair a (Nat.pair y i)))\n          n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Independence/Basic.lean", "full_name": "ProbabilityTheory.IndepSets.inter", "start": [300, 1], "end": [302, 31], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "full_name": "String.revPosOf_eq", "start": [332, 1], "end": [332, 69], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/List/Basic.lean", "full_name": "List.sections_eq_sectionsTR", "start": [971, 10], "end": [977, 40], "traced_tactics": [{"tactic": "funext \u03b1 L", "annotated_tactic": ["funext \u03b1 L", []], "state_before": "\u22a2 @sections = @sectionsTR", "state_after": "case h.h\n\u03b1 : Type u_1\nL : List (List \u03b1)\n\u22a2 sections L = sectionsTR L"}, {"tactic": "simp [sectionsTR]", "annotated_tactic": ["simp [<a>sectionsTR</a>]", [{"full_name": "List.sectionsTR", "def_path": "lake-packages/std/Std/Data/List/Basic.lean", "def_pos": [954, 5], "def_end_pos": [954, 15]}]], "state_before": "case h.h\n\u03b1 : Type u_1\nL : List (List \u03b1)\n\u22a2 sections L = sectionsTR L", "state_after": "case h.h\n\u03b1 : Type u_1\nL : List (List \u03b1)\n\u22a2 sections L = bif any L isEmpty then [] else (foldr sectionsTR.go #[[]] L).data"}, {"tactic": "cases e : L.any isEmpty <;> simp [sections_eq_nil_of_isEmpty, *]", "annotated_tactic": ["cases e : L.any <a>isEmpty</a> <;> simp [<a>sections_eq_nil_of_isEmpty</a>, *]", [{"full_name": "List.isEmpty", "def_path": "lake-packages/lean4/src/lean/Init/Data/List/Basic.lean", "def_pos": [143, 5], "def_end_pos": [143, 12]}, {"full_name": "List.sections_eq_nil_of_isEmpty", "def_path": "lake-packages/std/Std/Data/List/Basic.lean", "def_pos": [964, 9], "def_end_pos": [964, 35]}]], "state_before": "case h.h\n\u03b1 : Type u_1\nL : List (List \u03b1)\n\u22a2 sections L = bif any L isEmpty then [] else (foldr sectionsTR.go #[[]] L).data", "state_after": "case h.h.false\n\u03b1 : Type u_1\nL : List (List \u03b1)\ne : any L isEmpty = false\n\u22a2 sections L = (foldr sectionsTR.go #[[]] L).data"}, {"tactic": "clear e", "annotated_tactic": ["clear e", []], "state_before": "case h.h.false\n\u03b1 : Type u_1\nL : List (List \u03b1)\ne : any L isEmpty = false\n\u22a2 sections L = (foldr sectionsTR.go #[[]] L).data", "state_after": "case h.h.false\n\u03b1 : Type u_1\nL : List (List \u03b1)\n\u22a2 sections L = (foldr sectionsTR.go #[[]] L).data"}, {"tactic": "induction L with | nil => rfl | cons l L IH => ?_", "annotated_tactic": ["induction L with | <a>nil</a> => rfl | <a>cons</a> l L IH => ?_", [{"full_name": "List.nil", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2184, 5], "def_end_pos": [2184, 8]}, {"full_name": "List.cons", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2187, 5], "def_end_pos": [2187, 9]}]], "state_before": "case h.h.false\n\u03b1 : Type u_1\nL : List (List \u03b1)\n\u22a2 sections L = (foldr sectionsTR.go #[[]] L).data", "state_after": "case h.h.false.cons\n\u03b1 : Type u_1\nl : List \u03b1\nL : List (List \u03b1)\nIH : sections L = (foldr sectionsTR.go #[[]] L).data\n\u22a2 sections (l :: L) = (foldr sectionsTR.go #[[]] (l :: L)).data"}, {"tactic": "simp [IH, sectionsTR.go]", "annotated_tactic": ["simp [IH, <a>sectionsTR.go</a>]", [{"full_name": "List.sectionsTR.go", "def_path": "lake-packages/std/Std/Data/List/Basic.lean", "def_pos": [959, 3], "def_end_pos": [959, 5]}]], "state_before": "case h.h.false.cons\n\u03b1 : Type u_1\nl : List \u03b1\nL : List (List \u03b1)\nIH : sections L = (foldr sectionsTR.go #[[]] L).data\n\u22a2 sections (l :: L) = (foldr sectionsTR.go #[[]] (l :: L)).data", "state_after": "case h.h.false.cons\n\u03b1 : Type u_1\nl : List \u03b1\nL : List (List \u03b1)\nIH : sections L = (foldr sectionsTR.go #[[]] L).data\n\u22a2 (List.bind\n      (foldr\n          (fun l acc =>\n            Array.foldl (fun acc' l' => foldl (fun acc' a => Array.push acc' (a :: l')) acc' l) #[] acc 0\n              (Array.size acc))\n          #[[]] L).data\n      fun s => map (fun a => a :: s) l) =\n    (Array.foldl (fun acc' l' => foldl (fun acc' a => Array.push acc' (a :: l')) acc' l) #[]\n        (foldr\n          (fun l acc =>\n            Array.foldl (fun acc' l' => foldl (fun acc' a => Array.push acc' (a :: l')) acc' l) #[] acc 0\n              (Array.size acc))\n          #[[]] L)\n        0\n        (Array.size\n          (foldr\n            (fun l acc =>\n              Array.foldl (fun acc' l' => foldl (fun acc' a => Array.push acc' (a :: l')) acc' l) #[] acc 0\n                (Array.size acc))\n            #[[]] L))).data"}, {"tactic": "rw [Array.foldl_eq_foldl_data, Array.foldl_data_eq_bind]", "annotated_tactic": ["rw [<a>Array.foldl_eq_foldl_data</a>, <a>Array.foldl_data_eq_bind</a>]", [{"full_name": "Array.foldl_eq_foldl_data", "def_path": "lake-packages/std/Std/Data/Array/Init/Lemmas.lean", "def_pos": [47, 9], "def_end_pos": [47, 28]}, {"full_name": "Array.foldl_data_eq_bind", "def_path": "lake-packages/std/Std/Data/Array/Init/Lemmas.lean", "def_pos": [219, 9], "def_end_pos": [219, 27]}]], "state_before": "case h.h.false.cons\n\u03b1 : Type u_1\nl : List \u03b1\nL : List (List \u03b1)\nIH : sections L = (foldr sectionsTR.go #[[]] L).data\n\u22a2 (List.bind\n      (foldr\n          (fun l acc =>\n            Array.foldl (fun acc' l' => foldl (fun acc' a => Array.push acc' (a :: l')) acc' l) #[] acc 0\n              (Array.size acc))\n          #[[]] L).data\n      fun s => map (fun a => a :: s) l) =\n    (Array.foldl (fun acc' l' => foldl (fun acc' a => Array.push acc' (a :: l')) acc' l) #[]\n        (foldr\n          (fun l acc =>\n            Array.foldl (fun acc' l' => foldl (fun acc' a => Array.push acc' (a :: l')) acc' l) #[] acc 0\n              (Array.size acc))\n          #[[]] L)\n        0\n        (Array.size\n          (foldr\n            (fun l acc =>\n              Array.foldl (fun acc' l' => foldl (fun acc' a => Array.push acc' (a :: l')) acc' l) #[] acc 0\n                (Array.size acc))\n            #[[]] L))).data", "state_after": "case h.h.false.cons\n\u03b1 : Type u_1\nl : List \u03b1\nL : List (List \u03b1)\nIH : sections L = (foldr sectionsTR.go #[[]] L).data\n\u22a2 (List.bind\n      (foldr\n          (fun l acc =>\n            Array.foldl (fun acc' l' => foldl (fun acc' a => Array.push acc' (a :: l')) acc' l) #[] acc 0\n              (Array.size acc))\n          #[[]] L).data\n      fun s => map (fun a => a :: s) l) =\n    #[].data ++\n      List.bind\n        (foldr\n            (fun l acc =>\n              Array.foldl (fun acc' l' => foldl (fun acc' a => Array.push acc' (a :: l')) acc' l) #[] acc 0\n                (Array.size acc))\n            #[[]] L).data\n        ?h.h.false.cons.G\n\ncase h.h.false.cons.G\n\u03b1 : Type u_1\nl : List \u03b1\nL : List (List \u03b1)\nIH : sections L = (foldr sectionsTR.go #[[]] L).data\n\u22a2 List \u03b1 \u2192 List (List \u03b1)\n\ncase h.h.false.cons.H\n\u03b1 : Type u_1\nl : List \u03b1\nL : List (List \u03b1)\nIH : sections L = (foldr sectionsTR.go #[[]] L).data\n\u22a2 \u2200 (acc : Array (List \u03b1)) (a : List \u03b1),\n    (foldl (fun acc' a_1 => Array.push acc' (a_1 :: a)) acc l).data = acc.data ++ ?h.h.false.cons.G a"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case h.h.false.cons\n\u03b1 : Type u_1\nl : List \u03b1\nL : List (List \u03b1)\nIH : sections L = (foldr sectionsTR.go #[[]] L).data\n\u22a2 (List.bind\n      (foldr\n          (fun l acc =>\n            Array.foldl (fun acc' l' => foldl (fun acc' a => Array.push acc' (a :: l')) acc' l) #[] acc 0\n              (Array.size acc))\n          #[[]] L).data\n      fun s => map (fun a => a :: s) l) =\n    #[].data ++\n      List.bind\n        (foldr\n            (fun l acc =>\n              Array.foldl (fun acc' l' => foldl (fun acc' a => Array.push acc' (a :: l')) acc' l) #[] acc 0\n                (Array.size acc))\n            #[[]] L).data\n        ?h.h.false.cons.G\n\ncase h.h.false.cons.G\n\u03b1 : Type u_1\nl : List \u03b1\nL : List (List \u03b1)\nIH : sections L = (foldr sectionsTR.go #[[]] L).data\n\u22a2 List \u03b1 \u2192 List (List \u03b1)\n\ncase h.h.false.cons.H\n\u03b1 : Type u_1\nl : List \u03b1\nL : List (List \u03b1)\nIH : sections L = (foldr sectionsTR.go #[[]] L).data\n\u22a2 \u2200 (acc : Array (List \u03b1)) (a : List \u03b1),\n    (foldl (fun acc' a_1 => Array.push acc' (a_1 :: a)) acc l).data = acc.data ++ ?h.h.false.cons.G a", "state_after": "case h.h.false.cons.H\n\u03b1 : Type u_1\nl : List \u03b1\nL : List (List \u03b1)\nIH : sections L = (foldr sectionsTR.go #[[]] L).data\n\u22a2 \u2200 (acc : Array (List \u03b1)) (a : List \u03b1),\n    (foldl (fun acc' a_1 => Array.push acc' (a_1 :: a)) acc l).data = acc.data ++ map (fun a_1 => a_1 :: a) l"}, {"tactic": "intros", "annotated_tactic": ["intros", []], "state_before": "case h.h.false.cons.H\n\u03b1 : Type u_1\nl : List \u03b1\nL : List (List \u03b1)\nIH : sections L = (foldr sectionsTR.go #[[]] L).data\n\u22a2 \u2200 (acc : Array (List \u03b1)) (a : List \u03b1),\n    (foldl (fun acc' a_1 => Array.push acc' (a_1 :: a)) acc l).data = acc.data ++ map (fun a_1 => a_1 :: a) l", "state_after": "case h.h.false.cons.H\n\u03b1 : Type u_1\nl : List \u03b1\nL : List (List \u03b1)\nIH : sections L = (foldr sectionsTR.go #[[]] L).data\nacc\u271d : Array (List \u03b1)\na\u271d : List \u03b1\n\u22a2 (foldl (fun acc' a => Array.push acc' (a :: a\u271d)) acc\u271d l).data = acc\u271d.data ++ map (fun a => a :: a\u271d) l"}, {"tactic": "apply Array.foldl_data_eq_map", "annotated_tactic": ["apply <a>Array.foldl_data_eq_map</a>", [{"full_name": "Array.foldl_data_eq_map", "def_path": "lake-packages/std/Std/Data/Array/Init/Lemmas.lean", "def_pos": [225, 9], "def_end_pos": [225, 26]}]], "state_before": "case h.h.false.cons.H\n\u03b1 : Type u_1\nl : List \u03b1\nL : List (List \u03b1)\nIH : sections L = (foldr sectionsTR.go #[[]] L).data\nacc\u271d : Array (List \u03b1)\na\u271d : List \u03b1\n\u22a2 (foldl (fun acc' a => Array.push acc' (a :: a\u271d)) acc\u271d l).data = acc\u271d.data ++ map (fun a => a :: a\u271d) l", "state_after": "no goals"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case h.h.false.nil\n\u03b1 : Type u_1\n\u22a2 sections [] = (foldr sectionsTR.go #[[]] []).data", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Interval.lean", "full_name": "Finset.card_Iio_finset", "start": [129, 1], "end": [130, 89], "traced_tactics": [{"tactic": "rw [Iio_eq_ssubsets, ssubsets, card_erase_of_mem (mem_powerset_self _), card_powerset]", "annotated_tactic": ["rw [<a>Iio_eq_ssubsets</a>, <a>ssubsets</a>, <a>card_erase_of_mem</a> (<a>mem_powerset_self</a> _), <a>card_powerset</a>]", [{"full_name": "Finset.Iio_eq_ssubsets", "def_path": "Mathlib/Data/Finset/Interval.lean", "def_pos": [74, 9], "def_end_pos": [74, 24]}, {"full_name": "Finset.ssubsets", "def_path": "Mathlib/Data/Finset/Powerset.lean", "def_pos": [150, 5], "def_end_pos": [150, 13]}, {"full_name": "Finset.card_erase_of_mem", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [135, 9], "def_end_pos": [135, 26]}, {"full_name": "Finset.mem_powerset_self", "def_path": "Mathlib/Data/Finset/Powerset.lean", "def_pos": [53, 9], "def_end_pos": [53, 26]}, {"full_name": "Finset.card_powerset", "def_path": "Mathlib/Data/Finset/Powerset.lean", "def_pos": [88, 9], "def_end_pos": [88, 22]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\n\u22a2 card (Iio s) = 2 ^ card s - 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Lattice.lean", "full_name": "Finset.set_biInter_insert_update", "start": [2161, 1], "end": [2163, 26], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Constructions/Pi.lean", "full_name": "MeasureTheory.volume_preserving_pi_empty", "start": [874, 1], "end": [877, 45], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Lattice.lean", "full_name": "Finset.inf_of_mem", "start": [924, 1], "end": [926, 30], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Constructions/Prod/Integral.lean", "full_name": "MeasureTheory.integral_prod", "start": [455, 1], "end": [475, 48], "traced_tactics": [{"tactic": "by_cases hE : CompleteSpace E", "annotated_tactic": ["by_cases hE : <a>CompleteSpace</a> E", [{"full_name": "CompleteSpace", "def_path": "Mathlib/Topology/UniformSpace/Cauchy.lean", "def_pos": [397, 7], "def_end_pos": [397, 20]}]], "state_before": "\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1'\ninst\u271d\u2078 : MeasurableSpace \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2'\ninst\u271d\u2076 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b9 : NormedAddCommGroup E'\ninst\u271d : NormedSpace \u211d E'\nf : \u03b1 \u00d7 \u03b2 \u2192 E\nhf : Integrable f\n\u22a2 \u222b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b (x : \u03b1), \u222b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1'\ninst\u271d\u2078 : MeasurableSpace \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2'\ninst\u271d\u2076 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b9 : NormedAddCommGroup E'\ninst\u271d : NormedSpace \u211d E'\nf : \u03b1 \u00d7 \u03b2 \u2192 E\nhf : Integrable f\nhE : CompleteSpace E\n\u22a2 \u222b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b (x : \u03b1), \u222b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc\n\ncase neg\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1'\ninst\u271d\u2078 : MeasurableSpace \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2'\ninst\u271d\u2076 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b9 : NormedAddCommGroup E'\ninst\u271d : NormedSpace \u211d E'\nf : \u03b1 \u00d7 \u03b2 \u2192 E\nhf : Integrable f\nhE : \u00acCompleteSpace E\n\u22a2 \u222b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b (x : \u03b1), \u222b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc"}, {"tactic": "swap", "annotated_tactic": ["swap", []], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1'\ninst\u271d\u2078 : MeasurableSpace \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2'\ninst\u271d\u2076 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b9 : NormedAddCommGroup E'\ninst\u271d : NormedSpace \u211d E'\nf : \u03b1 \u00d7 \u03b2 \u2192 E\nhf : Integrable f\nhE : CompleteSpace E\n\u22a2 \u222b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b (x : \u03b1), \u222b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc\n\ncase neg\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1'\ninst\u271d\u2078 : MeasurableSpace \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2'\ninst\u271d\u2076 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b9 : NormedAddCommGroup E'\ninst\u271d : NormedSpace \u211d E'\nf : \u03b1 \u00d7 \u03b2 \u2192 E\nhf : Integrable f\nhE : \u00acCompleteSpace E\n\u22a2 \u222b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b (x : \u03b1), \u222b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc", "state_after": "case neg\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1'\ninst\u271d\u2078 : MeasurableSpace \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2'\ninst\u271d\u2076 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b9 : NormedAddCommGroup E'\ninst\u271d : NormedSpace \u211d E'\nf : \u03b1 \u00d7 \u03b2 \u2192 E\nhf : Integrable f\nhE : \u00acCompleteSpace E\n\u22a2 \u222b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b (x : \u03b1), \u222b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc\n\ncase pos\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1'\ninst\u271d\u2078 : MeasurableSpace \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2'\ninst\u271d\u2076 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b9 : NormedAddCommGroup E'\ninst\u271d : NormedSpace \u211d E'\nf : \u03b1 \u00d7 \u03b2 \u2192 E\nhf : Integrable f\nhE : CompleteSpace E\n\u22a2 \u222b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b (x : \u03b1), \u222b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc"}, {"tactic": "revert f", "annotated_tactic": ["revert f", []], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1'\ninst\u271d\u2078 : MeasurableSpace \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2'\ninst\u271d\u2076 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b9 : NormedAddCommGroup E'\ninst\u271d : NormedSpace \u211d E'\nf : \u03b1 \u00d7 \u03b2 \u2192 E\nhf : Integrable f\nhE : CompleteSpace E\n\u22a2 \u222b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b (x : \u03b1), \u222b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1'\ninst\u271d\u2078 : MeasurableSpace \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2'\ninst\u271d\u2076 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b9 : NormedAddCommGroup E'\ninst\u271d : NormedSpace \u211d E'\nhE : CompleteSpace E\n\u22a2 \u2200 (f : \u03b1 \u00d7 \u03b2 \u2192 E), Integrable f \u2192 \u222b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b (x : \u03b1), \u222b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc"}, {"tactic": "apply Integrable.induction", "annotated_tactic": ["apply <a>Integrable.induction</a>", [{"full_name": "MeasureTheory.Integrable.induction", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "def_pos": [1058, 9], "def_end_pos": [1058, 29]}]], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1'\ninst\u271d\u2078 : MeasurableSpace \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2'\ninst\u271d\u2076 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b9 : NormedAddCommGroup E'\ninst\u271d : NormedSpace \u211d E'\nhE : CompleteSpace E\n\u22a2 \u2200 (f : \u03b1 \u00d7 \u03b2 \u2192 E), Integrable f \u2192 \u222b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b (x : \u03b1), \u222b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc", "state_after": "case pos.h_ind\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1'\ninst\u271d\u2078 : MeasurableSpace \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2'\ninst\u271d\u2076 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b9 : NormedAddCommGroup E'\ninst\u271d : NormedSpace \u211d E'\nhE : CompleteSpace E\n\u22a2 \u2200 (c : E) \u2983s : Set (\u03b1 \u00d7 \u03b2)\u2984,\n    MeasurableSet s \u2192\n      \u2191\u2191(Measure.prod \u03bc \u03bd) s < \u22a4 \u2192\n        \u222b (z : \u03b1 \u00d7 \u03b2), indicator s (fun x => c) z \u2202Measure.prod \u03bc \u03bd =\n          \u222b (x : \u03b1), \u222b (y : \u03b2), indicator s (fun x => c) (x, y) \u2202\u03bd \u2202\u03bc\n\ncase pos.h_add\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1'\ninst\u271d\u2078 : MeasurableSpace \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2'\ninst\u271d\u2076 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b9 : NormedAddCommGroup E'\ninst\u271d : NormedSpace \u211d E'\nhE : CompleteSpace E\n\u22a2 \u2200 \u2983f g : \u03b1 \u00d7 \u03b2 \u2192 E\u2984,\n    Disjoint (support f) (support g) \u2192\n      Integrable f \u2192\n        Integrable g \u2192\n          \u222b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b (x : \u03b1), \u222b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc \u2192\n            \u222b (z : \u03b1 \u00d7 \u03b2), g z \u2202Measure.prod \u03bc \u03bd = \u222b (x : \u03b1), \u222b (y : \u03b2), g (x, y) \u2202\u03bd \u2202\u03bc \u2192\n              \u222b (z : \u03b1 \u00d7 \u03b2), (f + g) z \u2202Measure.prod \u03bc \u03bd = \u222b (x : \u03b1), \u222b (y : \u03b2), (f + g) (x, y) \u2202\u03bd \u2202\u03bc\n\ncase pos.h_closed\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1'\ninst\u271d\u2078 : MeasurableSpace \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2'\ninst\u271d\u2076 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b9 : NormedAddCommGroup E'\ninst\u271d : NormedSpace \u211d E'\nhE : CompleteSpace E\n\u22a2 IsClosed {f | \u222b (z : \u03b1 \u00d7 \u03b2), \u2191\u2191f z \u2202Measure.prod \u03bc \u03bd = \u222b (x : \u03b1), \u222b (y : \u03b2), \u2191\u2191f (x, y) \u2202\u03bd \u2202\u03bc}\n\ncase pos.h_ae\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1'\ninst\u271d\u2078 : MeasurableSpace \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2'\ninst\u271d\u2076 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b9 : NormedAddCommGroup E'\ninst\u271d : NormedSpace \u211d E'\nhE : CompleteSpace E\n\u22a2 \u2200 \u2983f g : \u03b1 \u00d7 \u03b2 \u2192 E\u2984,\n    f =\u1da0[ae (Measure.prod \u03bc \u03bd)] g \u2192\n      Integrable f \u2192\n        \u222b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b (x : \u03b1), \u222b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc \u2192\n          \u222b (z : \u03b1 \u00d7 \u03b2), g z \u2202Measure.prod \u03bc \u03bd = \u222b (x : \u03b1), \u222b (y : \u03b2), g (x, y) \u2202\u03bd \u2202\u03bc"}, {"tactic": "simp only [integral, dif_neg hE]", "annotated_tactic": ["simp only [<a>integral</a>, <a>dif_neg</a> hE]", [{"full_name": "MeasureTheory.integral", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [791, 17], "def_end_pos": [791, 25]}, {"full_name": "dif_neg", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [812, 9], "def_end_pos": [812, 16]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1'\ninst\u271d\u2078 : MeasurableSpace \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2'\ninst\u271d\u2076 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b9 : NormedAddCommGroup E'\ninst\u271d : NormedSpace \u211d E'\nf : \u03b1 \u00d7 \u03b2 \u2192 E\nhf : Integrable f\nhE : \u00acCompleteSpace E\n\u22a2 \u222b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b (x : \u03b1), \u222b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc", "state_after": "no goals"}, {"tactic": "intro c s hs h2s", "annotated_tactic": ["intro c s hs h2s", []], "state_before": "case pos.h_ind\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1'\ninst\u271d\u2078 : MeasurableSpace \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2'\ninst\u271d\u2076 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b9 : NormedAddCommGroup E'\ninst\u271d : NormedSpace \u211d E'\nhE : CompleteSpace E\n\u22a2 \u2200 (c : E) \u2983s : Set (\u03b1 \u00d7 \u03b2)\u2984,\n    MeasurableSet s \u2192\n      \u2191\u2191(Measure.prod \u03bc \u03bd) s < \u22a4 \u2192\n        \u222b (z : \u03b1 \u00d7 \u03b2), indicator s (fun x => c) z \u2202Measure.prod \u03bc \u03bd =\n          \u222b (x : \u03b1), \u222b (y : \u03b2), indicator s (fun x => c) (x, y) \u2202\u03bd \u2202\u03bc", "state_after": "case pos.h_ind\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1'\ninst\u271d\u2078 : MeasurableSpace \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2'\ninst\u271d\u2076 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b9 : NormedAddCommGroup E'\ninst\u271d : NormedSpace \u211d E'\nhE : CompleteSpace E\nc : E\ns : Set (\u03b1 \u00d7 \u03b2)\nhs : MeasurableSet s\nh2s : \u2191\u2191(Measure.prod \u03bc \u03bd) s < \u22a4\n\u22a2 \u222b (z : \u03b1 \u00d7 \u03b2), indicator s (fun x => c) z \u2202Measure.prod \u03bc \u03bd =\n    \u222b (x : \u03b1), \u222b (y : \u03b2), indicator s (fun x => c) (x, y) \u2202\u03bd \u2202\u03bc"}, {"tactic": "simp_rw [integral_indicator hs, \u2190 indicator_comp_right, Function.comp,\n  integral_indicator (measurable_prod_mk_left hs), set_integral_const, integral_smul_const,\n  integral_toReal (measurable_measure_prod_mk_left hs).aemeasurable\n    (ae_measure_lt_top hs h2s.ne)]", "annotated_tactic": ["simp_rw [<a>integral_indicator</a> hs, \u2190 <a>indicator_comp_right</a>, <a>Function.comp</a>,\n      <a>integral_indicator</a> (<a>measurable_prod_mk_left</a> hs), <a>set_integral_const</a>, <a>integral_smul_const</a>,\n      <a>integral_toReal</a> (<a>measurable_measure_prod_mk_left</a> hs).<a>aemeasurable</a>\n        (<a>ae_measure_lt_top</a> hs h2s.ne)]", [{"full_name": "MeasureTheory.integral_indicator", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [169, 9], "def_end_pos": [169, 27]}, {"full_name": "Set.indicator_comp_right", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [245, 3], "def_end_pos": [245, 14]}, {"full_name": "Function.comp", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [52, 15], "def_end_pos": [52, 28]}, {"full_name": "MeasureTheory.integral_indicator", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [169, 9], "def_end_pos": [169, 27]}, {"full_name": "measurable_prod_mk_left", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [736, 9], "def_end_pos": [736, 32]}, {"full_name": "MeasureTheory.set_integral_const", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [474, 9], "def_end_pos": [474, 27]}, {"full_name": "integral_smul_const", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [1257, 9], "def_end_pos": [1257, 28]}, {"full_name": "MeasureTheory.integral_toReal", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1230, 9], "def_end_pos": [1230, 24]}, {"full_name": "measurable_measure_prod_mk_left", "def_path": "Mathlib/MeasureTheory/Constructions/Prod/Basic.lean", "def_pos": [179, 9], "def_end_pos": [179, 40]}, {"full_name": "Measurable.aemeasurable", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [713, 9], "def_end_pos": [713, 32]}, {"full_name": "MeasureTheory.Measure.ae_measure_lt_top", "def_path": "Mathlib/MeasureTheory/Constructions/Prod/Basic.lean", "def_pos": [420, 9], "def_end_pos": [420, 26]}]], "state_before": "case pos.h_ind\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1'\ninst\u271d\u2078 : MeasurableSpace \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2'\ninst\u271d\u2076 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b9 : NormedAddCommGroup E'\ninst\u271d : NormedSpace \u211d E'\nhE : CompleteSpace E\nc : E\ns : Set (\u03b1 \u00d7 \u03b2)\nhs : MeasurableSet s\nh2s : \u2191\u2191(Measure.prod \u03bc \u03bd) s < \u22a4\n\u22a2 \u222b (z : \u03b1 \u00d7 \u03b2), indicator s (fun x => c) z \u2202Measure.prod \u03bc \u03bd =\n    \u222b (x : \u03b1), \u222b (y : \u03b2), indicator s (fun x => c) (x, y) \u2202\u03bd \u2202\u03bc", "state_after": "case pos.h_ind\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1'\ninst\u271d\u2078 : MeasurableSpace \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2'\ninst\u271d\u2076 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b9 : NormedAddCommGroup E'\ninst\u271d : NormedSpace \u211d E'\nhE : CompleteSpace E\nc : E\ns : Set (\u03b1 \u00d7 \u03b2)\nhs : MeasurableSet s\nh2s : \u2191\u2191(Measure.prod \u03bc \u03bd) s < \u22a4\n\u22a2 ENNReal.toReal (\u2191\u2191(Measure.prod \u03bc \u03bd) s) \u2022 c = ENNReal.toReal (\u222b\u207b (a : \u03b1), \u2191\u2191\u03bd (Prod.mk a \u207b\u00b9' s) \u2202\u03bc) \u2022 c"}, {"tactic": "rw [prod_apply hs]", "annotated_tactic": ["rw [<a>prod_apply</a> hs]", [{"full_name": "MeasureTheory.Measure.prod_apply", "def_path": "Mathlib/MeasureTheory/Constructions/Prod/Basic.lean", "def_pos": [307, 9], "def_end_pos": [307, 19]}]], "state_before": "case pos.h_ind\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1'\ninst\u271d\u2078 : MeasurableSpace \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2'\ninst\u271d\u2076 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b9 : NormedAddCommGroup E'\ninst\u271d : NormedSpace \u211d E'\nhE : CompleteSpace E\nc : E\ns : Set (\u03b1 \u00d7 \u03b2)\nhs : MeasurableSet s\nh2s : \u2191\u2191(Measure.prod \u03bc \u03bd) s < \u22a4\n\u22a2 ENNReal.toReal (\u2191\u2191(Measure.prod \u03bc \u03bd) s) \u2022 c = ENNReal.toReal (\u222b\u207b (a : \u03b1), \u2191\u2191\u03bd (Prod.mk a \u207b\u00b9' s) \u2202\u03bc) \u2022 c", "state_after": "no goals"}, {"tactic": "rintro f g - i_f i_g hf hg", "annotated_tactic": ["rintro f g - i_f i_g hf hg", []], "state_before": "case pos.h_add\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1'\ninst\u271d\u2078 : MeasurableSpace \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2'\ninst\u271d\u2076 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b9 : NormedAddCommGroup E'\ninst\u271d : NormedSpace \u211d E'\nhE : CompleteSpace E\n\u22a2 \u2200 \u2983f g : \u03b1 \u00d7 \u03b2 \u2192 E\u2984,\n    Disjoint (support f) (support g) \u2192\n      Integrable f \u2192\n        Integrable g \u2192\n          \u222b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b (x : \u03b1), \u222b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc \u2192\n            \u222b (z : \u03b1 \u00d7 \u03b2), g z \u2202Measure.prod \u03bc \u03bd = \u222b (x : \u03b1), \u222b (y : \u03b2), g (x, y) \u2202\u03bd \u2202\u03bc \u2192\n              \u222b (z : \u03b1 \u00d7 \u03b2), (f + g) z \u2202Measure.prod \u03bc \u03bd = \u222b (x : \u03b1), \u222b (y : \u03b2), (f + g) (x, y) \u2202\u03bd \u2202\u03bc", "state_after": "case pos.h_add\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1'\ninst\u271d\u2078 : MeasurableSpace \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2'\ninst\u271d\u2076 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b9 : NormedAddCommGroup E'\ninst\u271d : NormedSpace \u211d E'\nhE : CompleteSpace E\nf g : \u03b1 \u00d7 \u03b2 \u2192 E\ni_f : Integrable f\ni_g : Integrable g\nhf : \u222b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b (x : \u03b1), \u222b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc\nhg : \u222b (z : \u03b1 \u00d7 \u03b2), g z \u2202Measure.prod \u03bc \u03bd = \u222b (x : \u03b1), \u222b (y : \u03b2), g (x, y) \u2202\u03bd \u2202\u03bc\n\u22a2 \u222b (z : \u03b1 \u00d7 \u03b2), (f + g) z \u2202Measure.prod \u03bc \u03bd = \u222b (x : \u03b1), \u222b (y : \u03b2), (f + g) (x, y) \u2202\u03bd \u2202\u03bc"}, {"tactic": "simp_rw [integral_add' i_f i_g, integral_integral_add' i_f i_g, hf, hg]", "annotated_tactic": ["simp_rw [<a>integral_add'</a> i_f i_g, <a>integral_integral_add'</a> i_f i_g, hf, hg]", [{"full_name": "MeasureTheory.integral_add'", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [876, 9], "def_end_pos": [876, 22]}, {"full_name": "MeasureTheory.integral_integral_add'", "def_path": "Mathlib/MeasureTheory/Constructions/Prod/Integral.lean", "def_pos": [398, 9], "def_end_pos": [398, 31]}]], "state_before": "case pos.h_add\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1'\ninst\u271d\u2078 : MeasurableSpace \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2'\ninst\u271d\u2076 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b9 : NormedAddCommGroup E'\ninst\u271d : NormedSpace \u211d E'\nhE : CompleteSpace E\nf g : \u03b1 \u00d7 \u03b2 \u2192 E\ni_f : Integrable f\ni_g : Integrable g\nhf : \u222b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b (x : \u03b1), \u222b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc\nhg : \u222b (z : \u03b1 \u00d7 \u03b2), g z \u2202Measure.prod \u03bc \u03bd = \u222b (x : \u03b1), \u222b (y : \u03b2), g (x, y) \u2202\u03bd \u2202\u03bc\n\u22a2 \u222b (z : \u03b1 \u00d7 \u03b2), (f + g) z \u2202Measure.prod \u03bc \u03bd = \u222b (x : \u03b1), \u222b (y : \u03b2), (f + g) (x, y) \u2202\u03bd \u2202\u03bc", "state_after": "no goals"}, {"tactic": "exact isClosed_eq continuous_integral continuous_integral_integral", "annotated_tactic": ["exact <a>isClosed_eq</a> <a>continuous_integral</a> <a>continuous_integral_integral</a>", [{"full_name": "isClosed_eq", "def_path": "Mathlib/Topology/Separation.lean", "def_pos": [1217, 9], "def_end_pos": [1217, 20]}, {"full_name": "MeasureTheory.continuous_integral", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [957, 9], "def_end_pos": [957, 28]}, {"full_name": "MeasureTheory.continuous_integral_integral", "def_path": "Mathlib/MeasureTheory/Constructions/Prod/Integral.lean", "def_pos": [421, 9], "def_end_pos": [421, 37]}]], "state_before": "case pos.h_closed\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1'\ninst\u271d\u2078 : MeasurableSpace \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2'\ninst\u271d\u2076 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b9 : NormedAddCommGroup E'\ninst\u271d : NormedSpace \u211d E'\nhE : CompleteSpace E\n\u22a2 IsClosed {f | \u222b (z : \u03b1 \u00d7 \u03b2), \u2191\u2191f z \u2202Measure.prod \u03bc \u03bd = \u222b (x : \u03b1), \u222b (y : \u03b2), \u2191\u2191f (x, y) \u2202\u03bd \u2202\u03bc}", "state_after": "no goals"}, {"tactic": "rintro f g hfg - hf", "annotated_tactic": ["rintro f g hfg - hf", []], "state_before": "case pos.h_ae\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1'\ninst\u271d\u2078 : MeasurableSpace \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2'\ninst\u271d\u2076 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b9 : NormedAddCommGroup E'\ninst\u271d : NormedSpace \u211d E'\nhE : CompleteSpace E\n\u22a2 \u2200 \u2983f g : \u03b1 \u00d7 \u03b2 \u2192 E\u2984,\n    f =\u1da0[ae (Measure.prod \u03bc \u03bd)] g \u2192\n      Integrable f \u2192\n        \u222b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b (x : \u03b1), \u222b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc \u2192\n          \u222b (z : \u03b1 \u00d7 \u03b2), g z \u2202Measure.prod \u03bc \u03bd = \u222b (x : \u03b1), \u222b (y : \u03b2), g (x, y) \u2202\u03bd \u2202\u03bc", "state_after": "case pos.h_ae\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1'\ninst\u271d\u2078 : MeasurableSpace \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2'\ninst\u271d\u2076 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b9 : NormedAddCommGroup E'\ninst\u271d : NormedSpace \u211d E'\nhE : CompleteSpace E\nf g : \u03b1 \u00d7 \u03b2 \u2192 E\nhfg : f =\u1da0[ae (Measure.prod \u03bc \u03bd)] g\nhf : \u222b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b (x : \u03b1), \u222b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc\n\u22a2 \u222b (z : \u03b1 \u00d7 \u03b2), g z \u2202Measure.prod \u03bc \u03bd = \u222b (x : \u03b1), \u222b (y : \u03b2), g (x, y) \u2202\u03bd \u2202\u03bc"}, {"tactic": "convert hf using 1", "annotated_tactic": ["convert hf using 1", []], "state_before": "case pos.h_ae\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1'\ninst\u271d\u2078 : MeasurableSpace \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2'\ninst\u271d\u2076 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b9 : NormedAddCommGroup E'\ninst\u271d : NormedSpace \u211d E'\nhE : CompleteSpace E\nf g : \u03b1 \u00d7 \u03b2 \u2192 E\nhfg : f =\u1da0[ae (Measure.prod \u03bc \u03bd)] g\nhf : \u222b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b (x : \u03b1), \u222b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc\n\u22a2 \u222b (z : \u03b1 \u00d7 \u03b2), g z \u2202Measure.prod \u03bc \u03bd = \u222b (x : \u03b1), \u222b (y : \u03b2), g (x, y) \u2202\u03bd \u2202\u03bc", "state_after": "case h.e'_2\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1'\ninst\u271d\u2078 : MeasurableSpace \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2'\ninst\u271d\u2076 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b9 : NormedAddCommGroup E'\ninst\u271d : NormedSpace \u211d E'\nhE : CompleteSpace E\nf g : \u03b1 \u00d7 \u03b2 \u2192 E\nhfg : f =\u1da0[ae (Measure.prod \u03bc \u03bd)] g\nhf : \u222b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b (x : \u03b1), \u222b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc\n\u22a2 \u222b (z : \u03b1 \u00d7 \u03b2), g z \u2202Measure.prod \u03bc \u03bd = \u222b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd\n\ncase h.e'_3\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1'\ninst\u271d\u2078 : MeasurableSpace \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2'\ninst\u271d\u2076 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b9 : NormedAddCommGroup E'\ninst\u271d : NormedSpace \u211d E'\nhE : CompleteSpace E\nf g : \u03b1 \u00d7 \u03b2 \u2192 E\nhfg : f =\u1da0[ae (Measure.prod \u03bc \u03bd)] g\nhf : \u222b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b (x : \u03b1), \u222b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc\n\u22a2 \u222b (x : \u03b1), \u222b (y : \u03b2), g (x, y) \u2202\u03bd \u2202\u03bc = \u222b (x : \u03b1), \u222b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc"}, {"tactic": "exact integral_congr_ae hfg.symm", "annotated_tactic": ["exact <a>integral_congr_ae</a> hfg.symm", [{"full_name": "MeasureTheory.integral_congr_ae", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [938, 9], "def_end_pos": [938, 26]}]], "state_before": "case h.e'_2\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1'\ninst\u271d\u2078 : MeasurableSpace \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2'\ninst\u271d\u2076 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b9 : NormedAddCommGroup E'\ninst\u271d : NormedSpace \u211d E'\nhE : CompleteSpace E\nf g : \u03b1 \u00d7 \u03b2 \u2192 E\nhfg : f =\u1da0[ae (Measure.prod \u03bc \u03bd)] g\nhf : \u222b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b (x : \u03b1), \u222b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc\n\u22a2 \u222b (z : \u03b1 \u00d7 \u03b2), g z \u2202Measure.prod \u03bc \u03bd = \u222b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd", "state_after": "no goals"}, {"tactic": "refine' integral_congr_ae _", "annotated_tactic": ["refine' <a>integral_congr_ae</a> _", [{"full_name": "MeasureTheory.integral_congr_ae", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [938, 9], "def_end_pos": [938, 26]}]], "state_before": "case h.e'_3\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1'\ninst\u271d\u2078 : MeasurableSpace \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2'\ninst\u271d\u2076 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b9 : NormedAddCommGroup E'\ninst\u271d : NormedSpace \u211d E'\nhE : CompleteSpace E\nf g : \u03b1 \u00d7 \u03b2 \u2192 E\nhfg : f =\u1da0[ae (Measure.prod \u03bc \u03bd)] g\nhf : \u222b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b (x : \u03b1), \u222b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc\n\u22a2 \u222b (x : \u03b1), \u222b (y : \u03b2), g (x, y) \u2202\u03bd \u2202\u03bc = \u222b (x : \u03b1), \u222b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc", "state_after": "case h.e'_3\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1'\ninst\u271d\u2078 : MeasurableSpace \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2'\ninst\u271d\u2076 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b9 : NormedAddCommGroup E'\ninst\u271d : NormedSpace \u211d E'\nhE : CompleteSpace E\nf g : \u03b1 \u00d7 \u03b2 \u2192 E\nhfg : f =\u1da0[ae (Measure.prod \u03bc \u03bd)] g\nhf : \u222b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b (x : \u03b1), \u222b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc\n\u22a2 (fun x => \u222b (y : \u03b2), g (x, y) \u2202\u03bd) =\u1da0[ae \u03bc] fun x => \u222b (y : \u03b2), f (x, y) \u2202\u03bd"}, {"tactic": "refine' (ae_ae_of_ae_prod hfg).mp _", "annotated_tactic": ["refine' (<a>ae_ae_of_ae_prod</a> hfg).<a>mp</a> _", [{"full_name": "MeasureTheory.Measure.ae_ae_of_ae_prod", "def_path": "Mathlib/MeasureTheory/Constructions/Prod/Basic.lean", "def_pos": [456, 9], "def_end_pos": [456, 25]}, {"full_name": "Filter.Eventually.mp", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1135, 9], "def_end_pos": [1135, 22]}]], "state_before": "case h.e'_3\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1'\ninst\u271d\u2078 : MeasurableSpace \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2'\ninst\u271d\u2076 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b9 : NormedAddCommGroup E'\ninst\u271d : NormedSpace \u211d E'\nhE : CompleteSpace E\nf g : \u03b1 \u00d7 \u03b2 \u2192 E\nhfg : f =\u1da0[ae (Measure.prod \u03bc \u03bd)] g\nhf : \u222b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b (x : \u03b1), \u222b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc\n\u22a2 (fun x => \u222b (y : \u03b2), g (x, y) \u2202\u03bd) =\u1da0[ae \u03bc] fun x => \u222b (y : \u03b2), f (x, y) \u2202\u03bd", "state_after": "case h.e'_3\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1'\ninst\u271d\u2078 : MeasurableSpace \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2'\ninst\u271d\u2076 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b9 : NormedAddCommGroup E'\ninst\u271d : NormedSpace \u211d E'\nhE : CompleteSpace E\nf g : \u03b1 \u00d7 \u03b2 \u2192 E\nhfg : f =\u1da0[ae (Measure.prod \u03bc \u03bd)] g\nhf : \u222b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b (x : \u03b1), \u222b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    (\u2200\u1d50 (y : \u03b2) \u2202\u03bd, f (x, y) = g (x, y)) \u2192 (fun x => \u222b (y : \u03b2), g (x, y) \u2202\u03bd) x = (fun x => \u222b (y : \u03b2), f (x, y) \u2202\u03bd) x"}, {"tactic": "apply eventually_of_forall", "annotated_tactic": ["apply <a>eventually_of_forall</a>", [{"full_name": "Filter.eventually_of_forall", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1111, 9], "def_end_pos": [1111, 29]}]], "state_before": "case h.e'_3\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1'\ninst\u271d\u2078 : MeasurableSpace \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2'\ninst\u271d\u2076 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b9 : NormedAddCommGroup E'\ninst\u271d : NormedSpace \u211d E'\nhE : CompleteSpace E\nf g : \u03b1 \u00d7 \u03b2 \u2192 E\nhfg : f =\u1da0[ae (Measure.prod \u03bc \u03bd)] g\nhf : \u222b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b (x : \u03b1), \u222b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    (\u2200\u1d50 (y : \u03b2) \u2202\u03bd, f (x, y) = g (x, y)) \u2192 (fun x => \u222b (y : \u03b2), g (x, y) \u2202\u03bd) x = (fun x => \u222b (y : \u03b2), f (x, y) \u2202\u03bd) x", "state_after": "case h.e'_3.hp\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1'\ninst\u271d\u2078 : MeasurableSpace \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2'\ninst\u271d\u2076 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b9 : NormedAddCommGroup E'\ninst\u271d : NormedSpace \u211d E'\nhE : CompleteSpace E\nf g : \u03b1 \u00d7 \u03b2 \u2192 E\nhfg : f =\u1da0[ae (Measure.prod \u03bc \u03bd)] g\nhf : \u222b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b (x : \u03b1), \u222b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc\n\u22a2 \u2200 (x : \u03b1),\n    (\u2200\u1d50 (y : \u03b2) \u2202\u03bd, f (x, y) = g (x, y)) \u2192 (fun x => \u222b (y : \u03b2), g (x, y) \u2202\u03bd) x = (fun x => \u222b (y : \u03b2), f (x, y) \u2202\u03bd) x"}, {"tactic": "intro x hfgx", "annotated_tactic": ["intro x hfgx", []], "state_before": "case h.e'_3.hp\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1'\ninst\u271d\u2078 : MeasurableSpace \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2'\ninst\u271d\u2076 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b9 : NormedAddCommGroup E'\ninst\u271d : NormedSpace \u211d E'\nhE : CompleteSpace E\nf g : \u03b1 \u00d7 \u03b2 \u2192 E\nhfg : f =\u1da0[ae (Measure.prod \u03bc \u03bd)] g\nhf : \u222b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b (x : \u03b1), \u222b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc\n\u22a2 \u2200 (x : \u03b1),\n    (\u2200\u1d50 (y : \u03b2) \u2202\u03bd, f (x, y) = g (x, y)) \u2192 (fun x => \u222b (y : \u03b2), g (x, y) \u2202\u03bd) x = (fun x => \u222b (y : \u03b2), f (x, y) \u2202\u03bd) x", "state_after": "case h.e'_3.hp\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1'\ninst\u271d\u2078 : MeasurableSpace \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2'\ninst\u271d\u2076 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b9 : NormedAddCommGroup E'\ninst\u271d : NormedSpace \u211d E'\nhE : CompleteSpace E\nf g : \u03b1 \u00d7 \u03b2 \u2192 E\nhfg : f =\u1da0[ae (Measure.prod \u03bc \u03bd)] g\nhf : \u222b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b (x : \u03b1), \u222b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc\nx : \u03b1\nhfgx : \u2200\u1d50 (y : \u03b2) \u2202\u03bd, f (x, y) = g (x, y)\n\u22a2 (fun x => \u222b (y : \u03b2), g (x, y) \u2202\u03bd) x = (fun x => \u222b (y : \u03b2), f (x, y) \u2202\u03bd) x"}, {"tactic": "exact integral_congr_ae (ae_eq_symm hfgx)", "annotated_tactic": ["exact <a>integral_congr_ae</a> (<a>ae_eq_symm</a> hfgx)", [{"full_name": "MeasureTheory.integral_congr_ae", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [938, 9], "def_end_pos": [938, 26]}, {"full_name": "MeasureTheory.ae_eq_symm", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [440, 9], "def_end_pos": [440, 19]}]], "state_before": "case h.e'_3.hp\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1'\ninst\u271d\u2078 : MeasurableSpace \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2'\ninst\u271d\u2076 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b9 : NormedAddCommGroup E'\ninst\u271d : NormedSpace \u211d E'\nhE : CompleteSpace E\nf g : \u03b1 \u00d7 \u03b2 \u2192 E\nhfg : f =\u1da0[ae (Measure.prod \u03bc \u03bd)] g\nhf : \u222b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b (x : \u03b1), \u222b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc\nx : \u03b1\nhfgx : \u2200\u1d50 (y : \u03b2) \u2202\u03bd, f (x, y) = g (x, y)\n\u22a2 (fun x => \u222b (y : \u03b2), g (x, y) \u2202\u03bd) x = (fun x => \u222b (y : \u03b2), f (x, y) \u2202\u03bd) x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Card.lean", "full_name": "Set.map_eq_of_subset", "start": [716, 1], "end": [718, 49], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/LocallyIntegrable.lean", "full_name": "MeasureTheory.locallyIntegrable_map_homeomorph", "start": [292, 1], "end": [304, 58], "traced_tactics": [{"tactic": "refine' \u27e8fun h x => _, fun h x => _\u27e9", "annotated_tactic": ["refine' \u27e8fun h x => _, fun h x => _\u27e9", []], "state_before": "X : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2076 : MeasurableSpace X\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : MeasurableSpace Y\ninst\u271d\u00b3 : TopologicalSpace Y\ninst\u271d\u00b2 : NormedAddCommGroup E\nf\u271d g : X \u2192 E\n\u03bc\u271d : Measure X\ns : Set X\ninst\u271d\u00b9 : BorelSpace X\ninst\u271d : BorelSpace Y\ne : X \u2243\u209c Y\nf : Y \u2192 E\n\u03bc : Measure X\n\u22a2 LocallyIntegrable f \u2194 LocallyIntegrable (f \u2218 \u2191e)", "state_after": "case refine'_1\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2076 : MeasurableSpace X\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : MeasurableSpace Y\ninst\u271d\u00b3 : TopologicalSpace Y\ninst\u271d\u00b2 : NormedAddCommGroup E\nf\u271d g : X \u2192 E\n\u03bc\u271d : Measure X\ns : Set X\ninst\u271d\u00b9 : BorelSpace X\ninst\u271d : BorelSpace Y\ne : X \u2243\u209c Y\nf : Y \u2192 E\n\u03bc : Measure X\nh : LocallyIntegrable f\nx : X\n\u22a2 IntegrableAtFilter (f \u2218 \u2191e) (\ud835\udcdd x)\n\ncase refine'_2\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2076 : MeasurableSpace X\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : MeasurableSpace Y\ninst\u271d\u00b3 : TopologicalSpace Y\ninst\u271d\u00b2 : NormedAddCommGroup E\nf\u271d g : X \u2192 E\n\u03bc\u271d : Measure X\ns : Set X\ninst\u271d\u00b9 : BorelSpace X\ninst\u271d : BorelSpace Y\ne : X \u2243\u209c Y\nf : Y \u2192 E\n\u03bc : Measure X\nh : LocallyIntegrable (f \u2218 \u2191e)\nx : Y\n\u22a2 IntegrableAtFilter f (\ud835\udcdd x)"}, {"tactic": "rcases h (e x) with \u27e8U, hU, h'U\u27e9", "annotated_tactic": ["rcases h (e x) with \u27e8U, hU, h'U\u27e9", []], "state_before": "case refine'_1\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2076 : MeasurableSpace X\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : MeasurableSpace Y\ninst\u271d\u00b3 : TopologicalSpace Y\ninst\u271d\u00b2 : NormedAddCommGroup E\nf\u271d g : X \u2192 E\n\u03bc\u271d : Measure X\ns : Set X\ninst\u271d\u00b9 : BorelSpace X\ninst\u271d : BorelSpace Y\ne : X \u2243\u209c Y\nf : Y \u2192 E\n\u03bc : Measure X\nh : LocallyIntegrable f\nx : X\n\u22a2 IntegrableAtFilter (f \u2218 \u2191e) (\ud835\udcdd x)", "state_after": "case refine'_1.intro.intro\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2076 : MeasurableSpace X\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : MeasurableSpace Y\ninst\u271d\u00b3 : TopologicalSpace Y\ninst\u271d\u00b2 : NormedAddCommGroup E\nf\u271d g : X \u2192 E\n\u03bc\u271d : Measure X\ns : Set X\ninst\u271d\u00b9 : BorelSpace X\ninst\u271d : BorelSpace Y\ne : X \u2243\u209c Y\nf : Y \u2192 E\n\u03bc : Measure X\nh : LocallyIntegrable f\nx : X\nU : Set Y\nhU : U \u2208 \ud835\udcdd (\u2191e x)\nh'U : IntegrableOn f U\n\u22a2 IntegrableAtFilter (f \u2218 \u2191e) (\ud835\udcdd x)"}, {"tactic": "refine' \u27e8e \u207b\u00b9' U, e.continuous.continuousAt.preimage_mem_nhds hU, _\u27e9", "annotated_tactic": ["refine' \u27e8e \u207b\u00b9' U, e.continuous.continuousAt.preimage_mem_nhds hU, _\u27e9", []], "state_before": "case refine'_1.intro.intro\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2076 : MeasurableSpace X\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : MeasurableSpace Y\ninst\u271d\u00b3 : TopologicalSpace Y\ninst\u271d\u00b2 : NormedAddCommGroup E\nf\u271d g : X \u2192 E\n\u03bc\u271d : Measure X\ns : Set X\ninst\u271d\u00b9 : BorelSpace X\ninst\u271d : BorelSpace Y\ne : X \u2243\u209c Y\nf : Y \u2192 E\n\u03bc : Measure X\nh : LocallyIntegrable f\nx : X\nU : Set Y\nhU : U \u2208 \ud835\udcdd (\u2191e x)\nh'U : IntegrableOn f U\n\u22a2 IntegrableAtFilter (f \u2218 \u2191e) (\ud835\udcdd x)", "state_after": "case refine'_1.intro.intro\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2076 : MeasurableSpace X\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : MeasurableSpace Y\ninst\u271d\u00b3 : TopologicalSpace Y\ninst\u271d\u00b2 : NormedAddCommGroup E\nf\u271d g : X \u2192 E\n\u03bc\u271d : Measure X\ns : Set X\ninst\u271d\u00b9 : BorelSpace X\ninst\u271d : BorelSpace Y\ne : X \u2243\u209c Y\nf : Y \u2192 E\n\u03bc : Measure X\nh : LocallyIntegrable f\nx : X\nU : Set Y\nhU : U \u2208 \ud835\udcdd (\u2191e x)\nh'U : IntegrableOn f U\n\u22a2 IntegrableOn (f \u2218 \u2191e) (\u2191e \u207b\u00b9' U)"}, {"tactic": "exact (integrableOn_map_equiv e.toMeasurableEquiv).1 h'U", "annotated_tactic": ["exact (<a>integrableOn_map_equiv</a> e.toMeasurableEquiv).1 h'U", [{"full_name": "MeasureTheory.integrableOn_map_equiv", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [240, 9], "def_end_pos": [240, 31]}]], "state_before": "case refine'_1.intro.intro\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2076 : MeasurableSpace X\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : MeasurableSpace Y\ninst\u271d\u00b3 : TopologicalSpace Y\ninst\u271d\u00b2 : NormedAddCommGroup E\nf\u271d g : X \u2192 E\n\u03bc\u271d : Measure X\ns : Set X\ninst\u271d\u00b9 : BorelSpace X\ninst\u271d : BorelSpace Y\ne : X \u2243\u209c Y\nf : Y \u2192 E\n\u03bc : Measure X\nh : LocallyIntegrable f\nx : X\nU : Set Y\nhU : U \u2208 \ud835\udcdd (\u2191e x)\nh'U : IntegrableOn f U\n\u22a2 IntegrableOn (f \u2218 \u2191e) (\u2191e \u207b\u00b9' U)", "state_after": "no goals"}, {"tactic": "rcases h (e.symm x) with \u27e8U, hU, h'U\u27e9", "annotated_tactic": ["rcases h (e.symm x) with \u27e8U, hU, h'U\u27e9", []], "state_before": "case refine'_2\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2076 : MeasurableSpace X\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : MeasurableSpace Y\ninst\u271d\u00b3 : TopologicalSpace Y\ninst\u271d\u00b2 : NormedAddCommGroup E\nf\u271d g : X \u2192 E\n\u03bc\u271d : Measure X\ns : Set X\ninst\u271d\u00b9 : BorelSpace X\ninst\u271d : BorelSpace Y\ne : X \u2243\u209c Y\nf : Y \u2192 E\n\u03bc : Measure X\nh : LocallyIntegrable (f \u2218 \u2191e)\nx : Y\n\u22a2 IntegrableAtFilter f (\ud835\udcdd x)", "state_after": "case refine'_2.intro.intro\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2076 : MeasurableSpace X\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : MeasurableSpace Y\ninst\u271d\u00b3 : TopologicalSpace Y\ninst\u271d\u00b2 : NormedAddCommGroup E\nf\u271d g : X \u2192 E\n\u03bc\u271d : Measure X\ns : Set X\ninst\u271d\u00b9 : BorelSpace X\ninst\u271d : BorelSpace Y\ne : X \u2243\u209c Y\nf : Y \u2192 E\n\u03bc : Measure X\nh : LocallyIntegrable (f \u2218 \u2191e)\nx : Y\nU : Set X\nhU : U \u2208 \ud835\udcdd (\u2191(Homeomorph.symm e) x)\nh'U : IntegrableOn (f \u2218 \u2191e) U\n\u22a2 IntegrableAtFilter f (\ud835\udcdd x)"}, {"tactic": "refine' \u27e8e.symm \u207b\u00b9' U, e.symm.continuous.continuousAt.preimage_mem_nhds hU, _\u27e9", "annotated_tactic": ["refine' \u27e8e.symm \u207b\u00b9' U, e.symm.continuous.continuousAt.preimage_mem_nhds hU, _\u27e9", []], "state_before": "case refine'_2.intro.intro\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2076 : MeasurableSpace X\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : MeasurableSpace Y\ninst\u271d\u00b3 : TopologicalSpace Y\ninst\u271d\u00b2 : NormedAddCommGroup E\nf\u271d g : X \u2192 E\n\u03bc\u271d : Measure X\ns : Set X\ninst\u271d\u00b9 : BorelSpace X\ninst\u271d : BorelSpace Y\ne : X \u2243\u209c Y\nf : Y \u2192 E\n\u03bc : Measure X\nh : LocallyIntegrable (f \u2218 \u2191e)\nx : Y\nU : Set X\nhU : U \u2208 \ud835\udcdd (\u2191(Homeomorph.symm e) x)\nh'U : IntegrableOn (f \u2218 \u2191e) U\n\u22a2 IntegrableAtFilter f (\ud835\udcdd x)", "state_after": "case refine'_2.intro.intro\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2076 : MeasurableSpace X\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : MeasurableSpace Y\ninst\u271d\u00b3 : TopologicalSpace Y\ninst\u271d\u00b2 : NormedAddCommGroup E\nf\u271d g : X \u2192 E\n\u03bc\u271d : Measure X\ns : Set X\ninst\u271d\u00b9 : BorelSpace X\ninst\u271d : BorelSpace Y\ne : X \u2243\u209c Y\nf : Y \u2192 E\n\u03bc : Measure X\nh : LocallyIntegrable (f \u2218 \u2191e)\nx : Y\nU : Set X\nhU : U \u2208 \ud835\udcdd (\u2191(Homeomorph.symm e) x)\nh'U : IntegrableOn (f \u2218 \u2191e) U\n\u22a2 IntegrableOn f (\u2191(Homeomorph.symm e) \u207b\u00b9' U)"}, {"tactic": "apply (integrableOn_map_equiv e.toMeasurableEquiv).2", "annotated_tactic": ["apply (<a>integrableOn_map_equiv</a> e.toMeasurableEquiv).2", [{"full_name": "MeasureTheory.integrableOn_map_equiv", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [240, 9], "def_end_pos": [240, 31]}]], "state_before": "case refine'_2.intro.intro\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2076 : MeasurableSpace X\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : MeasurableSpace Y\ninst\u271d\u00b3 : TopologicalSpace Y\ninst\u271d\u00b2 : NormedAddCommGroup E\nf\u271d g : X \u2192 E\n\u03bc\u271d : Measure X\ns : Set X\ninst\u271d\u00b9 : BorelSpace X\ninst\u271d : BorelSpace Y\ne : X \u2243\u209c Y\nf : Y \u2192 E\n\u03bc : Measure X\nh : LocallyIntegrable (f \u2218 \u2191e)\nx : Y\nU : Set X\nhU : U \u2208 \ud835\udcdd (\u2191(Homeomorph.symm e) x)\nh'U : IntegrableOn (f \u2218 \u2191e) U\n\u22a2 IntegrableOn f (\u2191(Homeomorph.symm e) \u207b\u00b9' U)", "state_after": "case refine'_2.intro.intro\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2076 : MeasurableSpace X\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : MeasurableSpace Y\ninst\u271d\u00b3 : TopologicalSpace Y\ninst\u271d\u00b2 : NormedAddCommGroup E\nf\u271d g : X \u2192 E\n\u03bc\u271d : Measure X\ns : Set X\ninst\u271d\u00b9 : BorelSpace X\ninst\u271d : BorelSpace Y\ne : X \u2243\u209c Y\nf : Y \u2192 E\n\u03bc : Measure X\nh : LocallyIntegrable (f \u2218 \u2191e)\nx : Y\nU : Set X\nhU : U \u2208 \ud835\udcdd (\u2191(Homeomorph.symm e) x)\nh'U : IntegrableOn (f \u2218 \u2191e) U\n\u22a2 IntegrableOn (f \u2218 \u2191(Homeomorph.toMeasurableEquiv e))\n    (\u2191(Homeomorph.toMeasurableEquiv e) \u207b\u00b9' (\u2191(Homeomorph.symm e) \u207b\u00b9' U))"}, {"tactic": "simp only [Homeomorph.toMeasurableEquiv_coe]", "annotated_tactic": ["simp only [<a>Homeomorph.toMeasurableEquiv_coe</a>]", [{"full_name": "Homeomorph.toMeasurableEquiv_coe", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [993, 9], "def_end_pos": [993, 41]}]], "state_before": "case refine'_2.intro.intro\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2076 : MeasurableSpace X\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : MeasurableSpace Y\ninst\u271d\u00b3 : TopologicalSpace Y\ninst\u271d\u00b2 : NormedAddCommGroup E\nf\u271d g : X \u2192 E\n\u03bc\u271d : Measure X\ns : Set X\ninst\u271d\u00b9 : BorelSpace X\ninst\u271d : BorelSpace Y\ne : X \u2243\u209c Y\nf : Y \u2192 E\n\u03bc : Measure X\nh : LocallyIntegrable (f \u2218 \u2191e)\nx : Y\nU : Set X\nhU : U \u2208 \ud835\udcdd (\u2191(Homeomorph.symm e) x)\nh'U : IntegrableOn (f \u2218 \u2191e) U\n\u22a2 IntegrableOn (f \u2218 \u2191(Homeomorph.toMeasurableEquiv e))\n    (\u2191(Homeomorph.toMeasurableEquiv e) \u207b\u00b9' (\u2191(Homeomorph.symm e) \u207b\u00b9' U))", "state_after": "case refine'_2.intro.intro\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2076 : MeasurableSpace X\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : MeasurableSpace Y\ninst\u271d\u00b3 : TopologicalSpace Y\ninst\u271d\u00b2 : NormedAddCommGroup E\nf\u271d g : X \u2192 E\n\u03bc\u271d : Measure X\ns : Set X\ninst\u271d\u00b9 : BorelSpace X\ninst\u271d : BorelSpace Y\ne : X \u2243\u209c Y\nf : Y \u2192 E\n\u03bc : Measure X\nh : LocallyIntegrable (f \u2218 \u2191e)\nx : Y\nU : Set X\nhU : U \u2208 \ud835\udcdd (\u2191(Homeomorph.symm e) x)\nh'U : IntegrableOn (f \u2218 \u2191e) U\n\u22a2 IntegrableOn (f \u2218 \u2191e) (\u2191e \u207b\u00b9' (\u2191(Homeomorph.symm e) \u207b\u00b9' U))"}, {"tactic": "convert h'U", "annotated_tactic": ["convert h'U", []], "state_before": "case refine'_2.intro.intro\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2076 : MeasurableSpace X\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : MeasurableSpace Y\ninst\u271d\u00b3 : TopologicalSpace Y\ninst\u271d\u00b2 : NormedAddCommGroup E\nf\u271d g : X \u2192 E\n\u03bc\u271d : Measure X\ns : Set X\ninst\u271d\u00b9 : BorelSpace X\ninst\u271d : BorelSpace Y\ne : X \u2243\u209c Y\nf : Y \u2192 E\n\u03bc : Measure X\nh : LocallyIntegrable (f \u2218 \u2191e)\nx : Y\nU : Set X\nhU : U \u2208 \ud835\udcdd (\u2191(Homeomorph.symm e) x)\nh'U : IntegrableOn (f \u2218 \u2191e) U\n\u22a2 IntegrableOn (f \u2218 \u2191e) (\u2191e \u207b\u00b9' (\u2191(Homeomorph.symm e) \u207b\u00b9' U))", "state_after": "case h.e'_6\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2076 : MeasurableSpace X\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : MeasurableSpace Y\ninst\u271d\u00b3 : TopologicalSpace Y\ninst\u271d\u00b2 : NormedAddCommGroup E\nf\u271d g : X \u2192 E\n\u03bc\u271d : Measure X\ns : Set X\ninst\u271d\u00b9 : BorelSpace X\ninst\u271d : BorelSpace Y\ne : X \u2243\u209c Y\nf : Y \u2192 E\n\u03bc : Measure X\nh : LocallyIntegrable (f \u2218 \u2191e)\nx : Y\nU : Set X\nhU : U \u2208 \ud835\udcdd (\u2191(Homeomorph.symm e) x)\nh'U : IntegrableOn (f \u2218 \u2191e) U\n\u22a2 \u2191e \u207b\u00b9' (\u2191(Homeomorph.symm e) \u207b\u00b9' U) = U"}, {"tactic": "ext x", "annotated_tactic": ["ext x", []], "state_before": "case h.e'_6\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2076 : MeasurableSpace X\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : MeasurableSpace Y\ninst\u271d\u00b3 : TopologicalSpace Y\ninst\u271d\u00b2 : NormedAddCommGroup E\nf\u271d g : X \u2192 E\n\u03bc\u271d : Measure X\ns : Set X\ninst\u271d\u00b9 : BorelSpace X\ninst\u271d : BorelSpace Y\ne : X \u2243\u209c Y\nf : Y \u2192 E\n\u03bc : Measure X\nh : LocallyIntegrable (f \u2218 \u2191e)\nx : Y\nU : Set X\nhU : U \u2208 \ud835\udcdd (\u2191(Homeomorph.symm e) x)\nh'U : IntegrableOn (f \u2218 \u2191e) U\n\u22a2 \u2191e \u207b\u00b9' (\u2191(Homeomorph.symm e) \u207b\u00b9' U) = U", "state_after": "case h.e'_6.h\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2076 : MeasurableSpace X\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : MeasurableSpace Y\ninst\u271d\u00b3 : TopologicalSpace Y\ninst\u271d\u00b2 : NormedAddCommGroup E\nf\u271d g : X \u2192 E\n\u03bc\u271d : Measure X\ns : Set X\ninst\u271d\u00b9 : BorelSpace X\ninst\u271d : BorelSpace Y\ne : X \u2243\u209c Y\nf : Y \u2192 E\n\u03bc : Measure X\nh : LocallyIntegrable (f \u2218 \u2191e)\nx\u271d : Y\nU : Set X\nhU : U \u2208 \ud835\udcdd (\u2191(Homeomorph.symm e) x\u271d)\nh'U : IntegrableOn (f \u2218 \u2191e) U\nx : X\n\u22a2 x \u2208 \u2191e \u207b\u00b9' (\u2191(Homeomorph.symm e) \u207b\u00b9' U) \u2194 x \u2208 U"}, {"tactic": "simp only [mem_preimage, Homeomorph.symm_apply_apply]", "annotated_tactic": ["simp only [<a>mem_preimage</a>, <a>Homeomorph.symm_apply_apply</a>]", [{"full_name": "Set.mem_preimage", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [64, 9], "def_end_pos": [64, 21]}, {"full_name": "Homeomorph.symm_apply_apply", "def_path": "Mathlib/Topology/Homeomorph.lean", "def_pos": [156, 9], "def_end_pos": [156, 25]}]], "state_before": "case h.e'_6.h\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2076 : MeasurableSpace X\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : MeasurableSpace Y\ninst\u271d\u00b3 : TopologicalSpace Y\ninst\u271d\u00b2 : NormedAddCommGroup E\nf\u271d g : X \u2192 E\n\u03bc\u271d : Measure X\ns : Set X\ninst\u271d\u00b9 : BorelSpace X\ninst\u271d : BorelSpace Y\ne : X \u2243\u209c Y\nf : Y \u2192 E\n\u03bc : Measure X\nh : LocallyIntegrable (f \u2218 \u2191e)\nx\u271d : Y\nU : Set X\nhU : U \u2208 \ud835\udcdd (\u2191(Homeomorph.symm e) x\u271d)\nh'U : IntegrableOn (f \u2218 \u2191e) U\nx : X\n\u22a2 x \u2208 \u2191e \u207b\u00b9' (\u2191(Homeomorph.symm e) \u207b\u00b9' U) \u2194 x \u2208 U", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/PiSystem.lean", "full_name": "MeasurableSpace.DynkinSystem.generate_inter", "start": [713, 1], "end": [725, 13], "traced_tactics": [{"tactic": "rwa [inter_comm]", "annotated_tactic": ["rwa [<a>inter_comm</a>]", [{"full_name": "Set.inter_comm", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [940, 9], "def_end_pos": [940, 19]}]], "state_before": "\u03b1 : Type u_1\nd : DynkinSystem \u03b1\ns : Set (Set \u03b1)\nhs : IsPiSystem s\nt\u2081 t\u2082 : Set \u03b1\nht\u2081 : Has (generate s) t\u2081\nht\u2082 : Has (generate s) t\u2082\ns\u2081 : Set \u03b1\nhs\u2081 : s\u2081 \u2208 s\nthis\u271d\u00b9 : Has (generate s) s\u2081\nthis\u271d : generate s \u2264 restrictOn (generate s) this\u271d\u00b9\nthis : Has (generate s) (t\u2082 \u2229 s\u2081)\n\u22a2 Has (generate s) (s\u2081 \u2229 t\u2082)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "full_name": "MeasureTheory.OuterMeasure.ofFunction_eq_sSup", "start": [725, 1], "end": [726, 58], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Finite.lean", "full_name": "Set.finite_preimage_inl_and_inr", "start": [994, 1], "end": [997, 95], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/ProbabilityMassFunction/Constructions.lean", "full_name": "PMF.filter_apply_eq_zero_of_not_mem", "start": [278, 1], "end": [279, 88], "traced_tactics": [{"tactic": "rw [filter_apply, Set.indicator_apply_eq_zero.mpr fun ha' => absurd ha' ha, zero_mul]", "annotated_tactic": ["rw [<a>filter_apply</a>, Set.indicator_apply_eq_zero.mpr fun ha' => <a>absurd</a> ha' ha, <a>zero_mul</a>]", [{"full_name": "PMF.filter_apply", "def_path": "Mathlib/Probability/ProbabilityMassFunction/Constructions.lean", "def_pos": [273, 9], "def_end_pos": [273, 21]}, {"full_name": "absurd", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [233, 21], "def_end_pos": [233, 27]}, {"full_name": "MulZeroClass.zero_mul", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [36, 3], "def_end_pos": [36, 11]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\ns : Set \u03b1\nh : \u2203 a, a \u2208 s \u2227 a \u2208 support p\na : \u03b1\nha : \u00aca \u2208 s\n\u22a2 \u2191(filter p s h) a = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/ProbabilityMeasure.lean", "full_name": 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tail? le s = some s' \u2192 size s = size s' + 1", "state_after": "\u03b1 : Type u_1\nle : \u03b1 \u2192 \u03b1 \u2192 Bool\ns' s : Heap \u03b1\nh : NoSibling s\n\u22a2 Option.map (fun x => x.snd) (deleteMin le s) = some s' \u2192 size s = size s' + 1"}, {"tactic": "intro eq", "annotated_tactic": ["intro eq", []], "state_before": "\u03b1 : Type u_1\nle : \u03b1 \u2192 \u03b1 \u2192 Bool\ns' s : Heap \u03b1\nh : NoSibling s\n\u22a2 Option.map (fun x => x.snd) (deleteMin le s) = some s' \u2192 size s = size s' + 1", "state_after": "\u03b1 : Type u_1\nle : \u03b1 \u2192 \u03b1 \u2192 Bool\ns' s : Heap \u03b1\nh : NoSibling s\neq : Option.map (fun x => x.snd) (deleteMin le s) = some s'\n\u22a2 size s = size s' + 1"}, {"tactic": "match eq\u2082 : s.deleteMin le, eq with\n| some (a, tl), rfl => exact size_deleteMin h eq\u2082", "annotated_tactic": ["match eq\u2082 : s.deleteMin le, eq with\n  | <a>some</a> (a, tl), <a>rfl</a> => exact <a>size_deleteMin</a> h eq\u2082", [{"full_name": 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"https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Int/Lemmas.lean", "full_name": "Int.div2_bit", "start": [122, 1], "end": [128, 11], "traced_tactics": [{"tactic": "rw [bit_val, div2_val, add_comm, Int.add_mul_ediv_left, (_ : (_ / 2 : \u2124) = 0), zero_add]", "annotated_tactic": ["rw [<a>bit_val</a>, <a>div2_val</a>, <a>add_comm</a>, <a>Int.add_mul_ediv_left</a>, (_ : (_ / 2 : \u2124) = 0), <a>zero_add</a>]", [{"full_name": "Int.bit_val", "def_path": "Mathlib/Data/Int/Bitwise.lean", "def_pos": [130, 9], "def_end_pos": [130, 16]}, {"full_name": "Int.div2_val", "def_path": "Mathlib/Data/Int/Bitwise.lean", "def_pos": [111, 9], "def_end_pos": [111, 17]}, {"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [301, 3], "def_end_pos": [301, 14]}, {"full_name": "Int.add_mul_ediv_left", "def_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "def_pos": [172, 9], "def_end_pos": 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\u2260 0"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case false\na b : \u2124\nn\u271d : \u2115\nn : \u2124\n\u22a2 (bif false then 1 else 0) / 2 = 0", "state_after": "no goals"}, {"tactic": "show ofNat _ = _", "annotated_tactic": ["show <a>ofNat</a> _ = _", [{"full_name": "Int.ofNat", "def_path": "lake-packages/lean4/src/lean/Init/Data/Int/Basic.lean", "def_pos": [42, 5], "def_end_pos": [42, 10]}]], "state_before": "case true\na b : \u2124\nn\u271d : \u2115\nn : \u2124\n\u22a2 (bif true then 1 else 0) / 2 = 0", "state_after": "case true\na b : \u2124\nn\u271d : \u2115\nn : \u2124\n\u22a2 ofNat (1 / 2) = 0"}, {"tactic": "rw [Nat.div_eq_of_lt] <;> simp", "annotated_tactic": ["rw [<a>Nat.div_eq_of_lt</a>] <;> simp", [{"full_name": "Nat.div_eq_of_lt", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [590, 9], "def_end_pos": [590, 21]}]], "state_before": "case true\na b : \u2124\nn\u271d : \u2115\nn : \u2124\n\u22a2 ofNat (1 / 2) = 0", "state_after": "no goals"}, {"tactic": "decide", "annotated_tactic": ["decide", []], "state_before": "case H\na b\u271d : \u2124\nn\u271d : \u2115\nb : Bool\nn : \u2124\n\u22a2 2 \u2260 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Intervals/Monoid.lean", "full_name": "Set.image_add_const_Ioc", "start": [99, 1], "end": [100, 31], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/ConvergenceInMeasure.lean", "full_name": "MeasureTheory.ExistsSeqTendstoAe.seqTendstoAeSeq_succ", "start": [166, 1], "end": [169, 23], "traced_tactics": [{"tactic": "rw [seqTendstoAeSeq]", "annotated_tactic": ["rw [<a>seqTendstoAeSeq</a>]", [{"full_name": "MeasureTheory.ExistsSeqTendstoAe.seqTendstoAeSeq", "def_path": "Mathlib/MeasureTheory/Function/ConvergenceInMeasure.lean", "def_pos": [161, 19], "def_end_pos": [161, 34]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\nE : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : MetricSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\nhfg : TendstoInMeasure \u03bc f atTop g\nn : \u2115\n\u22a2 seqTendstoAeSeq hfg (n + 1) = max (seqTendstoAeSeqAux hfg (n + 1)) (seqTendstoAeSeq hfg n + 1)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/PImage.lean", "full_name": "Finset.pimage_inter", "start": [120, 1], "end": [121, 68], "traced_tactics": [{"tactic": "simp only [\u2190 coe_subset, coe_pimage, coe_inter, PFun.image_inter]", "annotated_tactic": ["simp only [\u2190 <a>coe_subset</a>, <a>coe_pimage</a>, <a>coe_inter</a>, <a>PFun.image_inter</a>]", [{"full_name": 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"3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Num/Lemmas.lean", "full_name": "ZNum.of_to_int", "start": [1540, 1], "end": [1540, 87], "traced_tactics": [{"tactic": "rw [\u2190 ofInt'_eq, of_to_int']", "annotated_tactic": ["rw [\u2190 <a>ofInt'_eq</a>, <a>of_to_int'</a>]", [{"full_name": "ZNum.ofInt'_eq", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [1524, 9], "def_end_pos": [1524, 18]}, {"full_name": "ZNum.of_to_int'", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [1352, 9], "def_end_pos": [1352, 19]}]], "state_before": "\u03b1 : Type u_1\nn : ZNum\n\u22a2 \u2191\u2191n = n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Sups.lean", "full_name": "upperClosure_sups", "start": [420, 1], "end": [429, 51], "traced_tactics": [{"tactic": "ext a", "annotated_tactic": ["ext a", []], "state_before": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : SemilatticeSup \u03b1\ns t : Set \u03b1\n\u22a2 upperClosure (s \u22bb t) = upperClosure s \u2294 upperClosure t", "state_after": "case a.h\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : SemilatticeSup \u03b1\ns t : Set \u03b1\na : \u03b1\n\u22a2 a \u2208 \u2191(upperClosure (s \u22bb t)) \u2194 a \u2208 \u2191(upperClosure s \u2294 upperClosure t)"}, {"tactic": "simp only [SetLike.mem_coe, mem_upperClosure, Set.mem_sups, exists_and_left, exists_prop,\n  UpperSet.coe_sup, Set.mem_inter_iff]", "annotated_tactic": ["simp only [<a>SetLike.mem_coe</a>, <a>mem_upperClosure</a>, <a>Set.mem_sups</a>, <a>exists_and_left</a>, <a>exists_prop</a>,\n    <a>UpperSet.coe_sup</a>, <a>Set.mem_inter_iff</a>]", [{"full_name": "SetLike.mem_coe", "def_path": "Mathlib/Data/SetLike/Basic.lean", "def_pos": [163, 9], "def_end_pos": [163, 16]}, {"full_name": "mem_upperClosure", "def_path": "Mathlib/Order/UpperLower/Basic.lean", "def_pos": [1374, 9], "def_end_pos": [1374, 25]}, {"full_name": "Set.mem_sups", "def_path": "Mathlib/Data/Set/Sups.lean", "def_pos": [74, 9], "def_end_pos": [74, 17]}, {"full_name": "exists_and_left", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [465, 17], "def_end_pos": [465, 32]}, {"full_name": "exists_prop", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [485, 17], "def_end_pos": [485, 28]}, {"full_name": "UpperSet.coe_sup", "def_path": "Mathlib/Order/UpperLower/Basic.lean", "def_pos": [576, 9], "def_end_pos": [576, 16]}, {"full_name": "Set.mem_inter_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [909, 9], "def_end_pos": [909, 22]}]], "state_before": "case a.h\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : SemilatticeSup \u03b1\ns t : Set \u03b1\na : \u03b1\n\u22a2 a \u2208 \u2191(upperClosure (s \u22bb t)) \u2194 a \u2208 \u2191(upperClosure s \u2294 upperClosure t)", "state_after": "case a.h\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : SemilatticeSup \u03b1\ns t : Set \u03b1\na : \u03b1\n\u22a2 (\u2203 a_1, (\u2203 a, a \u2208 s \u2227 \u2203 b, b \u2208 t \u2227 a \u2294 b = a_1) \u2227 a_1 \u2264 a) \u2194 (\u2203 a_1, a_1 \u2208 s \u2227 a_1 \u2264 a) \u2227 \u2203 a_1, a_1 \u2208 t \u2227 a_1 \u2264 a"}, {"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "case a.h\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : SemilatticeSup \u03b1\ns t : Set \u03b1\na : \u03b1\n\u22a2 (\u2203 a_1, (\u2203 a, a \u2208 s \u2227 \u2203 b, b \u2208 t \u2227 a \u2294 b = a_1) \u2227 a_1 \u2264 a) \u2194 (\u2203 a_1, a_1 \u2208 s \u2227 a_1 \u2264 a) \u2227 \u2203 a_1, a_1 \u2208 t \u2227 a_1 \u2264 a", "state_after": "case a.h.mp\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : SemilatticeSup \u03b1\ns t : Set \u03b1\na : \u03b1\n\u22a2 (\u2203 a_1, (\u2203 a, a \u2208 s \u2227 \u2203 b, b \u2208 t \u2227 a \u2294 b = a_1) \u2227 a_1 \u2264 a) \u2192 (\u2203 a_2, a_2 \u2208 s \u2227 a_2 \u2264 a) \u2227 \u2203 a_2, a_2 \u2208 t \u2227 a_2 \u2264 a\n\ncase a.h.mpr\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : SemilatticeSup \u03b1\ns t : Set \u03b1\na : \u03b1\n\u22a2 ((\u2203 a_1, a_1 \u2208 s \u2227 a_1 \u2264 a) \u2227 \u2203 a_1, a_1 \u2208 t \u2227 a_1 \u2264 a) \u2192 \u2203 a_2, (\u2203 a, a \u2208 s \u2227 \u2203 b, b \u2208 t \u2227 a \u2294 b = a_2) \u2227 a_2 \u2264 a"}, {"tactic": "rintro \u27e8_, \u27e8b, hb, c, hc, rfl\u27e9, ha\u27e9", "annotated_tactic": ["rintro \u27e8_, \u27e8b, hb, c, hc, rfl\u27e9, ha\u27e9", []], "state_before": "case a.h.mp\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : SemilatticeSup \u03b1\ns t : Set \u03b1\na : \u03b1\n\u22a2 (\u2203 a_1, (\u2203 a, a \u2208 s \u2227 \u2203 b, b \u2208 t \u2227 a \u2294 b = a_1) \u2227 a_1 \u2264 a) \u2192 (\u2203 a_2, a_2 \u2208 s \u2227 a_2 \u2264 a) \u2227 \u2203 a_2, a_2 \u2208 t \u2227 a_2 \u2264 a", "state_after": "case a.h.mp.intro.intro.intro.intro.intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : SemilatticeSup \u03b1\ns t : Set \u03b1\na b : \u03b1\nhb : b \u2208 s\nc : \u03b1\nhc : c \u2208 t\nha : b \u2294 c \u2264 a\n\u22a2 (\u2203 a_1, a_1 \u2208 s \u2227 a_1 \u2264 a) \u2227 \u2203 a_1, a_1 \u2208 t \u2227 a_1 \u2264 a"}, {"tactic": "exact \u27e8\u27e8b, hb, le_sup_left.trans ha\u27e9, c, hc, le_sup_right.trans ha\u27e9", "annotated_tactic": ["exact \u27e8\u27e8b, hb, le_sup_left.trans ha\u27e9, c, hc, le_sup_right.trans ha\u27e9", []], "state_before": "case a.h.mp.intro.intro.intro.intro.intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : SemilatticeSup \u03b1\ns t : Set \u03b1\na b : \u03b1\nhb : b \u2208 s\nc : \u03b1\nhc : c \u2208 t\nha : b \u2294 c \u2264 a\n\u22a2 (\u2203 a_1, a_1 \u2208 s \u2227 a_1 \u2264 a) \u2227 \u2203 a_1, a_1 \u2208 t \u2227 a_1 \u2264 a", "state_after": "no goals"}, {"tactic": "rintro \u27e8\u27e8b, hb, hab\u27e9, c, hc, hac\u27e9", "annotated_tactic": ["rintro \u27e8\u27e8b, hb, hab\u27e9, c, hc, hac\u27e9", []], "state_before": "case a.h.mpr\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : SemilatticeSup \u03b1\ns t : Set \u03b1\na : \u03b1\n\u22a2 ((\u2203 a_1, a_1 \u2208 s \u2227 a_1 \u2264 a) \u2227 \u2203 a_1, a_1 \u2208 t \u2227 a_1 \u2264 a) \u2192 \u2203 a_2, (\u2203 a, a \u2208 s \u2227 \u2203 b, b \u2208 t \u2227 a \u2294 b = a_2) \u2227 a_2 \u2264 a", "state_after": "case a.h.mpr.intro.intro.intro.intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : SemilatticeSup \u03b1\ns t : Set \u03b1\na b : \u03b1\nhb : b \u2208 s\nhab : b \u2264 a\nc : \u03b1\nhc : c \u2208 t\nhac : c \u2264 a\n\u22a2 \u2203 a_1, (\u2203 a, a \u2208 s \u2227 \u2203 b, b \u2208 t \u2227 a \u2294 b = a_1) \u2227 a_1 \u2264 a"}, {"tactic": "exact \u27e8_, \u27e8b, hb, c, hc, rfl\u27e9, sup_le hab hac\u27e9", "annotated_tactic": ["exact \u27e8_, \u27e8b, hb, c, hc, <a>rfl</a>\u27e9, <a>sup_le</a> hab hac\u27e9", [{"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}, {"full_name": "sup_le", "def_path": "Mathlib/Order/Lattice.lean", "def_pos": [167, 9], "def_end_pos": [167, 15]}]], "state_before": "case a.h.mpr.intro.intro.intro.intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : SemilatticeSup \u03b1\ns t : Set \u03b1\na b : \u03b1\nhb : b \u2208 s\nhab : b \u2264 a\nc : \u03b1\nhc : c \u2208 t\nhac : c \u2264 a\n\u22a2 \u2203 a_1, (\u2203 a, a \u2208 s \u2227 \u2203 b, b \u2208 t \u2227 a \u2294 b = a_1) \u2227 a_1 \u2264 a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/TMToPartrec.lean", "full_name": "Turing.PartrecToTM2.tr_pred", "start": [1124, 1], "end": [1129, 75], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/Expand.lean", "full_name": "MvPolynomial.expand_bind\u2081", "start": [73, 1], "end": [75, 46], "traced_tactics": [{"tactic": "rw [\u2190 AlgHom.comp_apply, expand_comp_bind\u2081]", "annotated_tactic": ["rw [\u2190 <a>AlgHom.comp_apply</a>, <a>expand_comp_bind\u2081</a>]", [{"full_name": "AlgHom.comp_apply", "def_path": "Mathlib/Algebra/Algebra/Hom.lean", "def_pos": [337, 9], "def_end_pos": [337, 19]}, {"full_name": "MvPolynomial.expand_comp_bind\u2081", "def_path": "Mathlib/Data/MvPolynomial/Expand.lean", "def_pos": [66, 9], "def_end_pos": [66, 26]}]], "state_before": "\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : CommSemiring S\np : \u2115\nf : \u03c3 \u2192 MvPolynomial \u03c4 R\n\u03c6 : MvPolynomial \u03c3 R\n\u22a2 \u2191(expand p) (\u2191(bind\u2081 f) \u03c6) = \u2191(bind\u2081 fun i => \u2191(expand p) (f i)) \u03c6", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Hausdorff.lean", "full_name": "MeasureTheory.hausdorffMeasure_prod_real", "start": [1067, 1], "end": [1070, 64], "traced_tactics": [{"tactic": "rw [\u2190 (volume_preserving_piFinTwo fun _ => \u211d).map_eq,\n  \u2190 (hausdorffMeasure_measurePreserving_piFinTwo (fun _ => \u211d) _).map_eq,\n  \u2190 hausdorffMeasure_pi_real, Fintype.card_fin, Nat.cast_two]", "annotated_tactic": ["rw [\u2190 (<a>volume_preserving_piFinTwo</a> fun _ => \u211d).<a>map_eq</a>,\n    \u2190 (<a>hausdorffMeasure_measurePreserving_piFinTwo</a> (fun _ => \u211d) _).<a>map_eq</a>,\n    \u2190 <a>hausdorffMeasure_pi_real</a>, <a>Fintype.card_fin</a>, <a>Nat.cast_two</a>]", [{"full_name": "MeasureTheory.volume_preserving_piFinTwo", "def_path": "Mathlib/MeasureTheory/Constructions/Pi.lean", "def_pos": [839, 9], "def_end_pos": [839, 35]}, {"full_name": "MeasureTheory.MeasurePreserving.map_eq", "def_path": "Mathlib/Dynamics/Ergodic/MeasurePreserving.lean", "def_pos": [45, 13], "def_end_pos": [45, 19]}, {"full_name": "MeasureTheory.hausdorffMeasure_measurePreserving_piFinTwo", "def_path": "Mathlib/MeasureTheory/Measure/Hausdorff.lean", "def_pos": [1050, 9], "def_end_pos": [1050, 52]}, {"full_name": "MeasureTheory.MeasurePreserving.map_eq", "def_path": "Mathlib/Dynamics/Ergodic/MeasurePreserving.lean", "def_pos": [45, 13], "def_end_pos": [45, 19]}, {"full_name": "MeasureTheory.hausdorffMeasure_pi_real", "def_path": "Mathlib/MeasureTheory/Measure/Hausdorff.lean", "def_pos": [950, 9], "def_end_pos": [950, 33]}, {"full_name": "Fintype.card_fin", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [308, 9], "def_end_pos": [308, 25]}, {"full_name": "Nat.cast_two", "def_path": "Mathlib/Data/Nat/Cast/Defs.lean", "def_pos": [193, 9], "def_end_pos": [193, 17]}]], "state_before": "\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2075 : EMetricSpace X\ninst\u271d\u2074 : EMetricSpace Y\ninst\u271d\u00b3 : MeasurableSpace X\ninst\u271d\u00b2 : BorelSpace X\ninst\u271d\u00b9 : MeasurableSpace Y\ninst\u271d : BorelSpace Y\n\u22a2 \u03bcH[2] = volume", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "full_name": "MeasurableEmbedding.stronglyMeasurable_extend", "start": [775, 1], "end": [786, 27], "traced_tactics": [{"tactic": "refine' \u27e8fun n => SimpleFunc.extend (hf.approx n) g hg (hg'.approx n), _\u27e9", "annotated_tactic": ["refine' \u27e8fun n => <a>SimpleFunc.extend</a> (hf.approx n) g hg (hg'.approx n), _\u27e9", [{"full_name": "MeasureTheory.SimpleFunc.extend", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [349, 5], "def_end_pos": [349, 11]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : Countable \u03b9\nf\u271d g\u271d f : \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b3\ng' : \u03b3 \u2192 \u03b2\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d : TopologicalSpace \u03b2\nhg : MeasurableEmbedding g\nhf : StronglyMeasurable f\nhg' : StronglyMeasurable g'\n\u22a2 StronglyMeasurable (Function.extend g f g')", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : Countable \u03b9\nf\u271d g\u271d f : \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b3\ng' : \u03b3 \u2192 \u03b2\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d : TopologicalSpace \u03b2\nhg : MeasurableEmbedding g\nhf : StronglyMeasurable f\nhg' : StronglyMeasurable g'\n\u22a2 \u2200 (x : \u03b3),\n    Tendsto\n      (fun n =>\n        \u2191((fun n => SimpleFunc.extend (StronglyMeasurable.approx hf n) g hg (StronglyMeasurable.approx hg' n)) n) x)\n      atTop (\ud835\udcdd (Function.extend g f g' x))"}, {"tactic": "intro x", "annotated_tactic": ["intro x", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : Countable \u03b9\nf\u271d g\u271d f : \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b3\ng' : \u03b3 \u2192 \u03b2\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d : TopologicalSpace \u03b2\nhg : MeasurableEmbedding g\nhf : StronglyMeasurable f\nhg' : StronglyMeasurable g'\n\u22a2 \u2200 (x : \u03b3),\n    Tendsto\n      (fun n =>\n        \u2191((fun n => SimpleFunc.extend (StronglyMeasurable.approx hf n) g hg (StronglyMeasurable.approx hg' n)) n) x)\n      atTop (\ud835\udcdd (Function.extend g f g' x))", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : Countable \u03b9\nf\u271d g\u271d f : \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b3\ng' : \u03b3 \u2192 \u03b2\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d : TopologicalSpace \u03b2\nhg : MeasurableEmbedding g\nhf : StronglyMeasurable f\nhg' : StronglyMeasurable g'\nx : \u03b3\n\u22a2 Tendsto\n    (fun n =>\n      \u2191((fun n => SimpleFunc.extend (StronglyMeasurable.approx hf n) g hg (StronglyMeasurable.approx hg' n)) n) x)\n    atTop (\ud835\udcdd (Function.extend g f g' x))"}, {"tactic": "by_cases hx : \u2203 y, g y = x", "annotated_tactic": ["by_cases hx : \u2203 y, g y = x", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : Countable \u03b9\nf\u271d g\u271d f : \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b3\ng' : \u03b3 \u2192 \u03b2\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d : TopologicalSpace \u03b2\nhg : MeasurableEmbedding g\nhf : StronglyMeasurable f\nhg' : StronglyMeasurable g'\nx : \u03b3\n\u22a2 Tendsto\n    (fun n =>\n      \u2191((fun n => SimpleFunc.extend (StronglyMeasurable.approx hf n) g hg (StronglyMeasurable.approx hg' n)) n) x)\n    atTop (\ud835\udcdd (Function.extend g f g' x))", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : Countable \u03b9\nf\u271d g\u271d f : \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b3\ng' : \u03b3 \u2192 \u03b2\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d : TopologicalSpace \u03b2\nhg : MeasurableEmbedding g\nhf : StronglyMeasurable f\nhg' : StronglyMeasurable g'\nx : \u03b3\nhx : \u2203 y, g y = x\n\u22a2 Tendsto\n    (fun n =>\n      \u2191((fun n => SimpleFunc.extend (StronglyMeasurable.approx hf n) g hg (StronglyMeasurable.approx hg' n)) n) x)\n    atTop (\ud835\udcdd (Function.extend g f g' x))\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : Countable \u03b9\nf\u271d g\u271d f : \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b3\ng' : \u03b3 \u2192 \u03b2\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d : TopologicalSpace \u03b2\nhg : MeasurableEmbedding g\nhf : StronglyMeasurable f\nhg' : StronglyMeasurable g'\nx : \u03b3\nhx : \u00ac\u2203 y, g y = x\n\u22a2 Tendsto\n    (fun n =>\n      \u2191((fun n => SimpleFunc.extend (StronglyMeasurable.approx hf n) g hg (StronglyMeasurable.approx hg' n)) n) x)\n    atTop (\ud835\udcdd (Function.extend g f g' x))"}, {"tactic": "rcases hx with \u27e8y, rfl\u27e9", "annotated_tactic": ["rcases hx with \u27e8y, rfl\u27e9", []], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : Countable \u03b9\nf\u271d g\u271d f : \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b3\ng' : \u03b3 \u2192 \u03b2\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d : TopologicalSpace \u03b2\nhg : MeasurableEmbedding g\nhf : StronglyMeasurable f\nhg' : StronglyMeasurable g'\nx : \u03b3\nhx : \u2203 y, g y = x\n\u22a2 Tendsto\n    (fun n =>\n      \u2191((fun n => SimpleFunc.extend (StronglyMeasurable.approx hf n) g hg (StronglyMeasurable.approx hg' n)) n) x)\n    atTop (\ud835\udcdd (Function.extend g f g' x))", "state_after": "case pos.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : Countable \u03b9\nf\u271d g\u271d f : \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b3\ng' : \u03b3 \u2192 \u03b2\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d : TopologicalSpace \u03b2\nhg : MeasurableEmbedding g\nhf : StronglyMeasurable f\nhg' : StronglyMeasurable g'\ny : \u03b1\n\u22a2 Tendsto\n    (fun n =>\n      \u2191((fun n => SimpleFunc.extend (StronglyMeasurable.approx hf n) g hg (StronglyMeasurable.approx hg' n)) n) (g y))\n    atTop (\ud835\udcdd (Function.extend g f g' (g y)))"}, {"tactic": "simpa only [SimpleFunc.extend_apply, hg.injective, Injective.extend_apply] using\n  hf.tendsto_approx y", "annotated_tactic": ["simpa only [<a>SimpleFunc.extend_apply</a>, hg.injective, <a>Injective.extend_apply</a>] using\n      hf.tendsto_approx y", [{"full_name": "MeasureTheory.SimpleFunc.extend_apply", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [361, 9], "def_end_pos": [361, 21]}, {"full_name": "Function.Injective.extend_apply", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [737, 9], "def_end_pos": [737, 31]}]], "state_before": "case pos.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : Countable \u03b9\nf\u271d g\u271d f : \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b3\ng' : \u03b3 \u2192 \u03b2\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d : TopologicalSpace \u03b2\nhg : MeasurableEmbedding g\nhf : StronglyMeasurable f\nhg' : StronglyMeasurable g'\ny : \u03b1\n\u22a2 Tendsto\n    (fun n =>\n      \u2191((fun n => SimpleFunc.extend (StronglyMeasurable.approx hf n) g hg (StronglyMeasurable.approx hg' n)) n) (g y))\n    atTop (\ud835\udcdd (Function.extend g f g' (g y)))", "state_after": "no goals"}, {"tactic": "simpa only [hx, SimpleFunc.extend_apply', not_false_iff, extend_apply'] using\n  hg'.tendsto_approx x", "annotated_tactic": ["simpa only [hx, <a>SimpleFunc.extend_apply'</a>, <a>not_false_iff</a>, <a>extend_apply'</a>] using\n      hg'.tendsto_approx x", [{"full_name": "MeasureTheory.SimpleFunc.extend_apply'", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [367, 9], "def_end_pos": [367, 22]}, {"full_name": "not_false_iff", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [82, 9], "def_end_pos": [82, 22]}, {"full_name": "Function.extend_apply'", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [743, 9], "def_end_pos": [743, 22]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : Countable \u03b9\nf\u271d g\u271d f : \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b3\ng' : \u03b3 \u2192 \u03b2\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d : TopologicalSpace \u03b2\nhg : MeasurableEmbedding g\nhf : StronglyMeasurable f\nhg' : StronglyMeasurable g'\nx : \u03b3\nhx : \u00ac\u2203 y, g y = x\n\u22a2 Tendsto\n    (fun n =>\n      \u2191((fun n => SimpleFunc.extend (StronglyMeasurable.approx hf n) g hg (StronglyMeasurable.approx hg' n)) n) x)\n    atTop (\ud835\udcdd (Function.extend g f g' x))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/ProbabilityMeasure.lean", "full_name": "MeasureTheory.ProbabilityMeasure.tendsto_nhds_iff_toFiniteMeasure_tendsto_nhds", "start": [272, 1], "end": [275, 59], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Sym.lean", "full_name": "Finset.diag_mem_sym2_iff", "start": [105, 1], "end": [105, 92], "traced_tactics": [{"tactic": "simp [diag_mem_sym2_mem_iff]", "annotated_tactic": ["simp [<a>diag_mem_sym2_mem_iff</a>]", [{"full_name": "Finset.diag_mem_sym2_mem_iff", "def_path": "Mathlib/Data/Finset/Sym.lean", "def_pos": [101, 9], "def_end_pos": [101, 30]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na b : \u03b1\nm : Sym2 \u03b1\n\u22a2 Sym2.diag a \u2208 Finset.sym2 s \u2194 a \u2208 s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Kernel/CondDistrib.lean", "full_name": "MeasureTheory.Integrable.comp_snd_map_prod_mk", "start": [312, 1], "end": [322, 64], "traced_tactics": [{"tactic": "by_cases hX : AEMeasurable X \u03bc", "annotated_tactic": ["by_cases hX : <a>AEMeasurable</a> X \u03bc", [{"full_name": "AEMeasurable", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [708, 5], "def_end_pos": [708, 17]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9\u271d : Type u_3\nF : Type u_4\ninst\u271d\u2076 : TopologicalSpace \u03a9\u271d\ninst\u271d\u2075 : MeasurableSpace \u03a9\u271d\ninst\u271d\u2074 : PolishSpace \u03a9\u271d\ninst\u271d\u00b3 : BorelSpace \u03a9\u271d\ninst\u271d\u00b2 : Nonempty \u03a9\u271d\ninst\u271d\u00b9 : NormedAddCommGroup F\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\u271d\nX\u271d : \u03b1 \u2192 \u03b2\nY : \u03b1 \u2192 \u03a9\u271d\nm\u03b2 : MeasurableSpace \u03b2\ns : Set \u03a9\u271d\nt : Set \u03b2\nf\u271d : \u03b2 \u00d7 \u03a9\u271d \u2192 F\n\u03a9 : Type u_5\nm\u03a9 : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u03b2\n\u03bc : Measure \u03a9\nf : \u03a9 \u2192 F\nhf_int : Integrable f\n\u22a2 Integrable fun x => f x.2", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9\u271d : Type u_3\nF : Type u_4\ninst\u271d\u2076 : TopologicalSpace \u03a9\u271d\ninst\u271d\u2075 : MeasurableSpace \u03a9\u271d\ninst\u271d\u2074 : PolishSpace \u03a9\u271d\ninst\u271d\u00b3 : BorelSpace \u03a9\u271d\ninst\u271d\u00b2 : Nonempty \u03a9\u271d\ninst\u271d\u00b9 : NormedAddCommGroup F\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\u271d\nX\u271d : \u03b1 \u2192 \u03b2\nY : \u03b1 \u2192 \u03a9\u271d\nm\u03b2 : MeasurableSpace \u03b2\ns : Set \u03a9\u271d\nt : Set \u03b2\nf\u271d : \u03b2 \u00d7 \u03a9\u271d \u2192 F\n\u03a9 : Type u_5\nm\u03a9 : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u03b2\n\u03bc : Measure \u03a9\nf : \u03a9 \u2192 F\nhf_int : Integrable f\nhX : AEMeasurable X\n\u22a2 Integrable fun x => f x.2\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9\u271d : Type u_3\nF : Type u_4\ninst\u271d\u2076 : TopologicalSpace \u03a9\u271d\ninst\u271d\u2075 : MeasurableSpace \u03a9\u271d\ninst\u271d\u2074 : PolishSpace \u03a9\u271d\ninst\u271d\u00b3 : BorelSpace \u03a9\u271d\ninst\u271d\u00b2 : Nonempty \u03a9\u271d\ninst\u271d\u00b9 : NormedAddCommGroup F\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\u271d\nX\u271d : \u03b1 \u2192 \u03b2\nY : \u03b1 \u2192 \u03a9\u271d\nm\u03b2 : MeasurableSpace \u03b2\ns : Set \u03a9\u271d\nt : Set \u03b2\nf\u271d : \u03b2 \u00d7 \u03a9\u271d \u2192 F\n\u03a9 : Type u_5\nm\u03a9 : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u03b2\n\u03bc : Measure \u03a9\nf : \u03a9 \u2192 F\nhf_int : Integrable f\nhX : \u00acAEMeasurable X\n\u22a2 Integrable fun x => f x.2"}, {"tactic": "have hf := hf_int.1.comp_snd_map_prod_mk X (m\u03a9 := m\u03a9) (m\u03b2 := m\u03b2)", "annotated_tactic": ["have hf := hf_int.1.<a>comp_snd_map_prod_mk</a> X (m\u03a9 := m\u03a9) (m\u03b2 := m\u03b2)", [{"full_name": "MeasureTheory.AEStronglyMeasurable.comp_snd_map_prod_mk", "def_path": "Mathlib/Probability/Kernel/CondDistrib.lean", "def_pos": [291, 9], "def_end_pos": [291, 71]}]], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9\u271d : Type u_3\nF : Type u_4\ninst\u271d\u2076 : TopologicalSpace \u03a9\u271d\ninst\u271d\u2075 : MeasurableSpace \u03a9\u271d\ninst\u271d\u2074 : PolishSpace \u03a9\u271d\ninst\u271d\u00b3 : BorelSpace \u03a9\u271d\ninst\u271d\u00b2 : Nonempty \u03a9\u271d\ninst\u271d\u00b9 : NormedAddCommGroup F\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\u271d\nX\u271d : \u03b1 \u2192 \u03b2\nY : \u03b1 \u2192 \u03a9\u271d\nm\u03b2 : MeasurableSpace \u03b2\ns : Set \u03a9\u271d\nt : Set \u03b2\nf\u271d : \u03b2 \u00d7 \u03a9\u271d \u2192 F\n\u03a9 : Type u_5\nm\u03a9 : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u03b2\n\u03bc : Measure \u03a9\nf : \u03a9 \u2192 F\nhf_int : Integrable f\nhX : AEMeasurable X\n\u22a2 Integrable fun x => f x.2", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9\u271d : Type u_3\nF : Type u_4\ninst\u271d\u2076 : TopologicalSpace \u03a9\u271d\ninst\u271d\u2075 : MeasurableSpace \u03a9\u271d\ninst\u271d\u2074 : PolishSpace \u03a9\u271d\ninst\u271d\u00b3 : BorelSpace \u03a9\u271d\ninst\u271d\u00b2 : Nonempty \u03a9\u271d\ninst\u271d\u00b9 : NormedAddCommGroup F\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\u271d\nX\u271d : \u03b1 \u2192 \u03b2\nY : \u03b1 \u2192 \u03a9\u271d\nm\u03b2 : MeasurableSpace \u03b2\ns : Set \u03a9\u271d\nt : Set \u03b2\nf\u271d : \u03b2 \u00d7 \u03a9\u271d \u2192 F\n\u03a9 : Type u_5\nm\u03a9 : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u03b2\n\u03bc : Measure \u03a9\nf : \u03a9 \u2192 F\nhf_int : Integrable f\nhX : AEMeasurable X\nhf : AEStronglyMeasurable (fun x => f x.2) (Measure.map (fun \u03c9 => (X \u03c9, \u03c9)) \u03bc)\n\u22a2 Integrable fun x => f x.2"}, {"tactic": "refine' \u27e8hf, _\u27e9", "annotated_tactic": ["refine' \u27e8hf, _\u27e9", []], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9\u271d : Type u_3\nF : Type u_4\ninst\u271d\u2076 : TopologicalSpace \u03a9\u271d\ninst\u271d\u2075 : MeasurableSpace \u03a9\u271d\ninst\u271d\u2074 : PolishSpace \u03a9\u271d\ninst\u271d\u00b3 : BorelSpace \u03a9\u271d\ninst\u271d\u00b2 : Nonempty \u03a9\u271d\ninst\u271d\u00b9 : NormedAddCommGroup F\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\u271d\nX\u271d : \u03b1 \u2192 \u03b2\nY : \u03b1 \u2192 \u03a9\u271d\nm\u03b2 : MeasurableSpace \u03b2\ns : Set \u03a9\u271d\nt : Set \u03b2\nf\u271d : \u03b2 \u00d7 \u03a9\u271d \u2192 F\n\u03a9 : Type u_5\nm\u03a9 : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u03b2\n\u03bc : Measure \u03a9\nf : \u03a9 \u2192 F\nhf_int : Integrable f\nhX : AEMeasurable X\nhf : AEStronglyMeasurable (fun x => f x.2) (Measure.map (fun \u03c9 => (X \u03c9, \u03c9)) \u03bc)\n\u22a2 Integrable fun x => f x.2", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9\u271d : Type u_3\nF : Type u_4\ninst\u271d\u2076 : TopologicalSpace \u03a9\u271d\ninst\u271d\u2075 : MeasurableSpace \u03a9\u271d\ninst\u271d\u2074 : PolishSpace \u03a9\u271d\ninst\u271d\u00b3 : BorelSpace \u03a9\u271d\ninst\u271d\u00b2 : Nonempty \u03a9\u271d\ninst\u271d\u00b9 : NormedAddCommGroup F\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\u271d\nX\u271d : \u03b1 \u2192 \u03b2\nY : \u03b1 \u2192 \u03a9\u271d\nm\u03b2 : MeasurableSpace \u03b2\ns : Set \u03a9\u271d\nt : Set \u03b2\nf\u271d : \u03b2 \u00d7 \u03a9\u271d \u2192 F\n\u03a9 : Type u_5\nm\u03a9 : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u03b2\n\u03bc : Measure \u03a9\nf : \u03a9 \u2192 F\nhf_int : Integrable f\nhX : AEMeasurable X\nhf : AEStronglyMeasurable (fun x => f x.2) (Measure.map (fun \u03c9 => (X \u03c9, \u03c9)) \u03bc)\n\u22a2 HasFiniteIntegral fun x => f x.2"}, {"tactic": "rw [HasFiniteIntegral, lintegral_map' hf.ennnorm (hX.prod_mk aemeasurable_id)]", "annotated_tactic": ["rw [<a>HasFiniteIntegral</a>, <a>lintegral_map'</a> hf.ennnorm (hX.prod_mk <a>aemeasurable_id</a>)]", [{"full_name": "MeasureTheory.HasFiniteIntegral", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [106, 5], "def_end_pos": [106, 22]}, {"full_name": "MeasureTheory.lintegral_map'", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [1289, 9], "def_end_pos": [1289, 23]}, {"full_name": "aemeasurable_id", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [751, 9], "def_end_pos": [751, 24]}]], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9\u271d : Type u_3\nF : Type u_4\ninst\u271d\u2076 : TopologicalSpace \u03a9\u271d\ninst\u271d\u2075 : MeasurableSpace \u03a9\u271d\ninst\u271d\u2074 : PolishSpace \u03a9\u271d\ninst\u271d\u00b3 : BorelSpace \u03a9\u271d\ninst\u271d\u00b2 : Nonempty \u03a9\u271d\ninst\u271d\u00b9 : NormedAddCommGroup F\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\u271d\nX\u271d : \u03b1 \u2192 \u03b2\nY : \u03b1 \u2192 \u03a9\u271d\nm\u03b2 : MeasurableSpace \u03b2\ns : Set \u03a9\u271d\nt : Set \u03b2\nf\u271d : \u03b2 \u00d7 \u03a9\u271d \u2192 F\n\u03a9 : Type u_5\nm\u03a9 : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u03b2\n\u03bc : Measure \u03a9\nf : \u03a9 \u2192 F\nhf_int : Integrable f\nhX : AEMeasurable X\nhf : AEStronglyMeasurable (fun x => f x.2) (Measure.map (fun \u03c9 => (X \u03c9, \u03c9)) \u03bc)\n\u22a2 HasFiniteIntegral fun x => f x.2", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9\u271d : Type u_3\nF : Type u_4\ninst\u271d\u2076 : TopologicalSpace \u03a9\u271d\ninst\u271d\u2075 : MeasurableSpace \u03a9\u271d\ninst\u271d\u2074 : PolishSpace \u03a9\u271d\ninst\u271d\u00b3 : BorelSpace \u03a9\u271d\ninst\u271d\u00b2 : Nonempty \u03a9\u271d\ninst\u271d\u00b9 : NormedAddCommGroup F\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\u271d\nX\u271d : \u03b1 \u2192 \u03b2\nY : \u03b1 \u2192 \u03a9\u271d\nm\u03b2 : MeasurableSpace \u03b2\ns : Set \u03a9\u271d\nt : Set \u03b2\nf\u271d : \u03b2 \u00d7 \u03a9\u271d \u2192 F\n\u03a9 : Type u_5\nm\u03a9 : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u03b2\n\u03bc : Measure \u03a9\nf : \u03a9 \u2192 F\nhf_int : Integrable f\nhX : AEMeasurable X\nhf : AEStronglyMeasurable (fun x => f x.2) (Measure.map (fun \u03c9 => (X \u03c9, \u03c9)) \u03bc)\n\u22a2 \u222b\u207b (a : \u03a9), \u2191\u2016f (X a, a).2\u2016\u208a \u2202\u03bc < \u22a4"}, {"tactic": "exact hf_int.2", "annotated_tactic": ["exact hf_int.2", []], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9\u271d : Type u_3\nF : Type u_4\ninst\u271d\u2076 : TopologicalSpace \u03a9\u271d\ninst\u271d\u2075 : MeasurableSpace \u03a9\u271d\ninst\u271d\u2074 : PolishSpace \u03a9\u271d\ninst\u271d\u00b3 : BorelSpace \u03a9\u271d\ninst\u271d\u00b2 : Nonempty \u03a9\u271d\ninst\u271d\u00b9 : NormedAddCommGroup F\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\u271d\nX\u271d : \u03b1 \u2192 \u03b2\nY : \u03b1 \u2192 \u03a9\u271d\nm\u03b2 : MeasurableSpace \u03b2\ns : Set \u03a9\u271d\nt : Set \u03b2\nf\u271d : \u03b2 \u00d7 \u03a9\u271d \u2192 F\n\u03a9 : Type u_5\nm\u03a9 : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u03b2\n\u03bc : Measure \u03a9\nf : \u03a9 \u2192 F\nhf_int : Integrable f\nhX : AEMeasurable X\nhf : AEStronglyMeasurable (fun x => f x.2) (Measure.map (fun \u03c9 => (X \u03c9, \u03c9)) \u03bc)\n\u22a2 \u222b\u207b (a : \u03a9), \u2191\u2016f (X a, a).2\u2016\u208a \u2202\u03bc < \u22a4", "state_after": "no goals"}, {"tactic": "rw [Measure.map_of_not_aemeasurable]", "annotated_tactic": ["rw [<a>Measure.map_of_not_aemeasurable</a>]", [{"full_name": "MeasureTheory.Measure.map_of_not_aemeasurable", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1187, 9], "def_end_pos": [1187, 32]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9\u271d : Type u_3\nF : Type u_4\ninst\u271d\u2076 : TopologicalSpace \u03a9\u271d\ninst\u271d\u2075 : MeasurableSpace \u03a9\u271d\ninst\u271d\u2074 : PolishSpace \u03a9\u271d\ninst\u271d\u00b3 : BorelSpace \u03a9\u271d\ninst\u271d\u00b2 : Nonempty \u03a9\u271d\ninst\u271d\u00b9 : NormedAddCommGroup F\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\u271d\nX\u271d : \u03b1 \u2192 \u03b2\nY : \u03b1 \u2192 \u03a9\u271d\nm\u03b2 : MeasurableSpace \u03b2\ns : Set \u03a9\u271d\nt : Set \u03b2\nf\u271d : \u03b2 \u00d7 \u03a9\u271d \u2192 F\n\u03a9 : Type u_5\nm\u03a9 : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u03b2\n\u03bc : Measure \u03a9\nf : \u03a9 \u2192 F\nhf_int : Integrable f\nhX : \u00acAEMeasurable X\n\u22a2 Integrable fun x => f x.2", "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9\u271d : Type u_3\nF : Type u_4\ninst\u271d\u2076 : TopologicalSpace \u03a9\u271d\ninst\u271d\u2075 : MeasurableSpace \u03a9\u271d\ninst\u271d\u2074 : PolishSpace \u03a9\u271d\ninst\u271d\u00b3 : BorelSpace \u03a9\u271d\ninst\u271d\u00b2 : Nonempty \u03a9\u271d\ninst\u271d\u00b9 : NormedAddCommGroup F\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\u271d\nX\u271d : \u03b1 \u2192 \u03b2\nY : \u03b1 \u2192 \u03a9\u271d\nm\u03b2 : MeasurableSpace \u03b2\ns : Set \u03a9\u271d\nt : Set \u03b2\nf\u271d : \u03b2 \u00d7 \u03a9\u271d \u2192 F\n\u03a9 : Type u_5\nm\u03a9 : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u03b2\n\u03bc : Measure \u03a9\nf : \u03a9 \u2192 F\nhf_int : Integrable f\nhX : \u00acAEMeasurable X\n\u22a2 Integrable fun x => f x.2\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9\u271d : Type u_3\nF : Type u_4\ninst\u271d\u2076 : TopologicalSpace \u03a9\u271d\ninst\u271d\u2075 : MeasurableSpace \u03a9\u271d\ninst\u271d\u2074 : PolishSpace \u03a9\u271d\ninst\u271d\u00b3 : BorelSpace \u03a9\u271d\ninst\u271d\u00b2 : Nonempty \u03a9\u271d\ninst\u271d\u00b9 : NormedAddCommGroup F\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\u271d\nX\u271d : \u03b1 \u2192 \u03b2\nY : \u03b1 \u2192 \u03a9\u271d\nm\u03b2 : MeasurableSpace \u03b2\ns : Set \u03a9\u271d\nt : Set \u03b2\nf\u271d : \u03b2 \u00d7 \u03a9\u271d \u2192 F\n\u03a9 : Type u_5\nm\u03a9 : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u03b2\n\u03bc : Measure \u03a9\nf : \u03a9 \u2192 F\nhf_int : Integrable f\nhX : \u00acAEMeasurable X\n\u22a2 \u00acAEMeasurable fun \u03c9 => (X \u03c9, \u03c9)"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9\u271d : Type u_3\nF : Type u_4\ninst\u271d\u2076 : TopologicalSpace \u03a9\u271d\ninst\u271d\u2075 : MeasurableSpace \u03a9\u271d\ninst\u271d\u2074 : PolishSpace \u03a9\u271d\ninst\u271d\u00b3 : BorelSpace \u03a9\u271d\ninst\u271d\u00b2 : Nonempty \u03a9\u271d\ninst\u271d\u00b9 : NormedAddCommGroup F\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\u271d\nX\u271d : \u03b1 \u2192 \u03b2\nY : \u03b1 \u2192 \u03a9\u271d\nm\u03b2 : MeasurableSpace \u03b2\ns : Set \u03a9\u271d\nt : Set \u03b2\nf\u271d : \u03b2 \u00d7 \u03a9\u271d \u2192 F\n\u03a9 : Type u_5\nm\u03a9 : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u03b2\n\u03bc : Measure \u03a9\nf : \u03a9 \u2192 F\nhf_int : Integrable f\nhX : \u00acAEMeasurable X\n\u22a2 Integrable fun x => f x.2", "state_after": "no goals"}, {"tactic": "contrapose! hX", "annotated_tactic": ["contrapose! hX", []], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9\u271d : Type u_3\nF : Type u_4\ninst\u271d\u2076 : TopologicalSpace \u03a9\u271d\ninst\u271d\u2075 : MeasurableSpace \u03a9\u271d\ninst\u271d\u2074 : PolishSpace \u03a9\u271d\ninst\u271d\u00b3 : BorelSpace \u03a9\u271d\ninst\u271d\u00b2 : Nonempty \u03a9\u271d\ninst\u271d\u00b9 : NormedAddCommGroup F\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\u271d\nX\u271d : \u03b1 \u2192 \u03b2\nY : \u03b1 \u2192 \u03a9\u271d\nm\u03b2 : MeasurableSpace \u03b2\ns : Set \u03a9\u271d\nt : Set \u03b2\nf\u271d : \u03b2 \u00d7 \u03a9\u271d \u2192 F\n\u03a9 : Type u_5\nm\u03a9 : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u03b2\n\u03bc : Measure \u03a9\nf : \u03a9 \u2192 F\nhf_int : Integrable f\nhX : \u00acAEMeasurable X\n\u22a2 \u00acAEMeasurable fun \u03c9 => (X \u03c9, \u03c9)", "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9\u271d : Type u_3\nF : Type u_4\ninst\u271d\u2076 : TopologicalSpace \u03a9\u271d\ninst\u271d\u2075 : MeasurableSpace \u03a9\u271d\ninst\u271d\u2074 : PolishSpace \u03a9\u271d\ninst\u271d\u00b3 : BorelSpace \u03a9\u271d\ninst\u271d\u00b2 : Nonempty \u03a9\u271d\ninst\u271d\u00b9 : NormedAddCommGroup F\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\u271d\nX\u271d : \u03b1 \u2192 \u03b2\nY : \u03b1 \u2192 \u03a9\u271d\nm\u03b2 : MeasurableSpace \u03b2\ns : Set \u03a9\u271d\nt : Set \u03b2\nf\u271d : \u03b2 \u00d7 \u03a9\u271d \u2192 F\n\u03a9 : Type u_5\nm\u03a9 : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u03b2\n\u03bc : Measure \u03a9\nf : \u03a9 \u2192 F\nhf_int : Integrable f\nhX : AEMeasurable fun \u03c9 => (X \u03c9, \u03c9)\n\u22a2 AEMeasurable X"}, {"tactic": "exact measurable_fst.comp_aemeasurable hX", "annotated_tactic": ["exact measurable_fst.comp_aemeasurable hX", []], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9\u271d : Type u_3\nF : Type u_4\ninst\u271d\u2076 : TopologicalSpace \u03a9\u271d\ninst\u271d\u2075 : MeasurableSpace \u03a9\u271d\ninst\u271d\u2074 : PolishSpace \u03a9\u271d\ninst\u271d\u00b3 : BorelSpace \u03a9\u271d\ninst\u271d\u00b2 : Nonempty \u03a9\u271d\ninst\u271d\u00b9 : NormedAddCommGroup F\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\u271d\nX\u271d : \u03b1 \u2192 \u03b2\nY : \u03b1 \u2192 \u03a9\u271d\nm\u03b2 : MeasurableSpace \u03b2\ns : Set \u03a9\u271d\nt : Set \u03b2\nf\u271d : \u03b2 \u00d7 \u03a9\u271d \u2192 F\n\u03a9 : Type u_5\nm\u03a9 : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u03b2\n\u03bc : Measure \u03a9\nf : \u03a9 \u2192 F\nhf_int : Integrable f\nhX : AEMeasurable fun \u03c9 => (X \u03c9, \u03c9)\n\u22a2 AEMeasurable X", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Decomposition/Lebesgue.lean", "full_name": "MeasureTheory.Measure.rnDeriv_withDensity", "start": [369, 1], "end": [372, 55], "traced_tactics": [{"tactic": "rw [zero_add]", "annotated_tactic": ["rw [<a>zero_add</a>]", [{"full_name": "zero_add", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [463, 3], "def_end_pos": [463, 14]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd\u271d \u03bd : Measure \u03b1\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u22a2 withDensity \u03bd f = 0 + withDensity \u03bd f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Kernel/Disintegration.lean", "full_name": "ProbabilityTheory.kernel.const_eq_compProd_real", "start": [211, 1], "end": [217, 41], "traced_tactics": [{"tactic": "ext a s hs : 2", "annotated_tactic": ["ext a s hs : 2", []], "state_before": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1\u271d : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d\u00b2 : IsFiniteMeasure \u03c1\u271d\n\u03b3 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\n\u22a2 const \u03b3 \u03c1 = const \u03b3 (Measure.fst \u03c1) \u2297\u2096 prodMkLeft \u03b3 (condKernelReal \u03c1)", "state_after": "case h.h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1\u271d : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d\u00b2 : IsFiniteMeasure \u03c1\u271d\n\u03b3 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\na : \u03b3\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(\u2191(const \u03b3 \u03c1) a) s = \u2191\u2191(\u2191(const \u03b3 (Measure.fst \u03c1) \u2297\u2096 prodMkLeft \u03b3 (condKernelReal \u03c1)) a) s"}, {"tactic": "rw [kernel.compProd_apply _ _ _ hs, kernel.const_apply, kernel.const_apply]", "annotated_tactic": ["rw [<a>kernel.compProd_apply</a> _ _ _ hs, <a>kernel.const_apply</a>, <a>kernel.const_apply</a>]", [{"full_name": "ProbabilityTheory.kernel.compProd_apply", "def_path": "Mathlib/Probability/Kernel/Composition.lean", "def_pos": [242, 9], "def_end_pos": [242, 23]}, {"full_name": "ProbabilityTheory.kernel.const_apply", "def_path": "Mathlib/Probability/Kernel/Basic.lean", "def_pos": [445, 9], "def_end_pos": [445, 20]}, {"full_name": "ProbabilityTheory.kernel.const_apply", "def_path": "Mathlib/Probability/Kernel/Basic.lean", "def_pos": [445, 9], "def_end_pos": [445, 20]}]], "state_before": "case h.h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1\u271d : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d\u00b2 : IsFiniteMeasure \u03c1\u271d\n\u03b3 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\na : \u03b3\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(\u2191(const \u03b3 \u03c1) a) s = \u2191\u2191(\u2191(const \u03b3 (Measure.fst \u03c1) \u2297\u2096 prodMkLeft \u03b3 (condKernelReal \u03c1)) a) s", "state_after": "case h.h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1\u271d : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d\u00b2 : IsFiniteMeasure \u03c1\u271d\n\u03b3 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\na : \u03b3\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\n\u22a2 \u2191\u2191\u03c1 s = \u222b\u207b (b : \u03b1), \u2191\u2191(\u2191(prodMkLeft \u03b3 (condKernelReal \u03c1)) (a, b)) {c | (b, c) \u2208 s} \u2202Measure.fst \u03c1"}, {"tactic": "simp_rw [kernel.prodMkLeft_apply]", "annotated_tactic": ["simp_rw [<a>kernel.prodMkLeft_apply</a>]", [{"full_name": "ProbabilityTheory.kernel.prodMkLeft_apply", "def_path": "Mathlib/Probability/Kernel/Composition.lean", "def_pos": [685, 9], "def_end_pos": [685, 25]}]], "state_before": "case h.h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1\u271d : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d\u00b2 : IsFiniteMeasure \u03c1\u271d\n\u03b3 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\na : \u03b3\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\n\u22a2 \u2191\u2191\u03c1 s = \u222b\u207b (b : \u03b1), \u2191\u2191(\u2191(prodMkLeft \u03b3 (condKernelReal \u03c1)) (a, b)) {c | (b, c) \u2208 s} \u2202Measure.fst \u03c1", "state_after": "case h.h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1\u271d : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d\u00b2 : IsFiniteMeasure \u03c1\u271d\n\u03b3 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\na : \u03b3\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\n\u22a2 \u2191\u2191\u03c1 s = \u222b\u207b (b : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) b) {c | (b, c) \u2208 s} \u2202Measure.fst \u03c1"}, {"tactic": "rw [lintegral_condKernelReal_mem \u03c1 hs]", "annotated_tactic": ["rw [<a>lintegral_condKernelReal_mem</a> \u03c1 hs]", [{"full_name": "ProbabilityTheory.lintegral_condKernelReal_mem", "def_path": "Mathlib/Probability/Kernel/Disintegration.lean", "def_pos": [136, 9], "def_end_pos": [136, 37]}]], "state_before": "case h.h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1\u271d : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d\u00b2 : IsFiniteMeasure \u03c1\u271d\n\u03b3 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\na : \u03b3\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\n\u22a2 \u2191\u2191\u03c1 s = \u222b\u207b (b : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) b) {c | (b, c) \u2208 s} \u2202Measure.fst \u03c1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/LpOrder.lean", "full_name": "MeasureTheory.Lp.coeFn_abs", "start": [101, 1], "end": [102, 22], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Int/ModEq.lean", "full_name": "Int.mod_mul_right_mod", "start": [321, 1], "end": [322, 33], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "full_name": "MeasureTheory.ae_bdd_liminf_atTop_of_snorm_bdd", "start": [1658, 1], "end": [1684, 68], "traced_tactics": [{"tactic": "by_cases hp' : p = \u221e", "annotated_tactic": ["by_cases hp' : p = \u221e", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : OpensMeasurableSpace E\nR : \u211d\u22650\np : \u211d\u22650\u221e\nhp : p \u2260 0\nf : \u2115 \u2192 \u03b1 \u2192 E\nhfmeas : \u2200 (n : \u2115), Measurable (f n)\nhbdd : \u2200 (n : \u2115), snorm (f n) p \u03bc \u2264 \u2191R\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, liminf (fun n => \u2191\u2016f n x\u2016\u208a) atTop < \u22a4", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : OpensMeasurableSpace E\nR : \u211d\u22650\np : \u211d\u22650\u221e\nhp : p \u2260 0\nf : \u2115 \u2192 \u03b1 \u2192 E\nhfmeas : \u2200 (n : \u2115), Measurable (f n)\nhbdd : \u2200 (n : \u2115), snorm (f n) p \u03bc \u2264 \u2191R\nhp' : p = \u22a4\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, liminf (fun n => \u2191\u2016f n x\u2016\u208a) atTop < \u22a4\n\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : OpensMeasurableSpace E\nR : \u211d\u22650\np : \u211d\u22650\u221e\nhp : p \u2260 0\nf : \u2115 \u2192 \u03b1 \u2192 E\nhfmeas : \u2200 (n : \u2115), Measurable (f n)\nhbdd : \u2200 (n : \u2115), snorm (f n) p \u03bc \u2264 \u2191R\nhp' : \u00acp = \u22a4\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, liminf (fun n => \u2191\u2016f n x\u2016\u208a) atTop < \u22a4"}, {"tactic": "filter_upwards [ae_bdd_liminf_atTop_rpow_of_snorm_bdd hfmeas hbdd] with x hx", "annotated_tactic": ["filter_upwards [<a>ae_bdd_liminf_atTop_rpow_of_snorm_bdd</a> hfmeas hbdd] with x hx", [{"full_name": "MeasureTheory.ae_bdd_liminf_atTop_rpow_of_snorm_bdd", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [1634, 9], "def_end_pos": [1634, 46]}]], "state_before": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : OpensMeasurableSpace E\nR : \u211d\u22650\np : \u211d\u22650\u221e\nhp : p \u2260 0\nf : \u2115 \u2192 \u03b1 \u2192 E\nhfmeas : \u2200 (n : \u2115), Measurable (f n)\nhbdd : \u2200 (n : \u2115), snorm (f n) p \u03bc \u2264 \u2191R\nhp' : \u00acp = \u22a4\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, liminf (fun n => \u2191\u2016f n x\u2016\u208a) atTop < \u22a4", "state_after": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : OpensMeasurableSpace E\nR : \u211d\u22650\np : \u211d\u22650\u221e\nhp : p \u2260 0\nf : \u2115 \u2192 \u03b1 \u2192 E\nhfmeas : \u2200 (n : \u2115), Measurable (f n)\nhbdd : \u2200 (n : \u2115), snorm (f n) p \u03bc \u2264 \u2191R\nhp' : \u00acp = \u22a4\nx : \u03b1\nhx : liminf (fun n => \u2191\u2016f n x\u2016\u208a ^ ENNReal.toReal p) atTop < \u22a4\n\u22a2 liminf (fun n => \u2191\u2016f n x\u2016\u208a) atTop < \u22a4"}, {"tactic": "have hppos : 0 < p.toReal := ENNReal.toReal_pos hp hp'", "annotated_tactic": ["have hppos : 0 < p.toReal := <a>ENNReal.toReal_pos</a> hp hp'", [{"full_name": "ENNReal.toReal_pos", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2131, 9], "def_end_pos": [2131, 19]}]], "state_before": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : OpensMeasurableSpace E\nR : \u211d\u22650\np : \u211d\u22650\u221e\nhp : p \u2260 0\nf : \u2115 \u2192 \u03b1 \u2192 E\nhfmeas : \u2200 (n : \u2115), Measurable (f n)\nhbdd : \u2200 (n : \u2115), snorm (f n) p \u03bc \u2264 \u2191R\nhp' : \u00acp = \u22a4\nx : \u03b1\nhx : liminf (fun n => \u2191\u2016f n x\u2016\u208a ^ ENNReal.toReal p) atTop < \u22a4\n\u22a2 liminf (fun n => \u2191\u2016f n x\u2016\u208a) atTop < \u22a4", "state_after": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : OpensMeasurableSpace E\nR : \u211d\u22650\np : \u211d\u22650\u221e\nhp : p \u2260 0\nf : \u2115 \u2192 \u03b1 \u2192 E\nhfmeas : \u2200 (n : \u2115), Measurable (f n)\nhbdd : \u2200 (n : \u2115), snorm (f n) p \u03bc \u2264 \u2191R\nhp' : \u00acp = \u22a4\nx : \u03b1\nhx : liminf (fun n => \u2191\u2016f n x\u2016\u208a ^ ENNReal.toReal p) atTop < \u22a4\nhppos : 0 < ENNReal.toReal p\n\u22a2 liminf (fun n => \u2191\u2016f n x\u2016\u208a) atTop < \u22a4"}, {"tactic": "have :\n  liminf (fun n => (\u2016f n x\u2016\u208a : \u211d\u22650\u221e) ^ p.toReal) atTop =\n    liminf (fun n => (\u2016f n x\u2016\u208a : \u211d\u22650\u221e)) atTop ^ p.toReal := by\n  change\n    liminf (fun n => ENNReal.orderIsoRpow p.toReal hppos (\u2016f n x\u2016\u208a : \u211d\u22650\u221e)) atTop =\n      ENNReal.orderIsoRpow p.toReal hppos (liminf (fun n => (\u2016f n x\u2016\u208a : \u211d\u22650\u221e)) atTop)\n  refine' (OrderIso.liminf_apply (ENNReal.orderIsoRpow p.toReal _) _ _ _ _).symm <;>\n    isBoundedDefault", "annotated_tactic": ["have :\n    <a>liminf</a> (fun n => (\u2016f n x\u2016\u208a : \u211d\u22650\u221e) ^ p.toReal) <a>atTop</a> =\n      <a>liminf</a> (fun n => (\u2016f n x\u2016\u208a : \u211d\u22650\u221e)) <a>atTop</a> ^ p.toReal := by\n    change\n      <a>liminf</a> (fun n => <a>ENNReal.orderIsoRpow</a> p.toReal hppos (\u2016f n x\u2016\u208a : \u211d\u22650\u221e)) <a>atTop</a> =\n        <a>ENNReal.orderIsoRpow</a> p.toReal hppos (<a>liminf</a> (fun n => (\u2016f n x\u2016\u208a : \u211d\u22650\u221e)) <a>atTop</a>)\n    refine' (<a>OrderIso.liminf_apply</a> (<a>ENNReal.orderIsoRpow</a> p.toReal _) _ _ _ _).<a>symm</a> <;>\n      isBoundedDefault", [{"full_name": "Filter.liminf", "def_path": "Mathlib/Order/LiminfLimsup.lean", "def_pos": [426, 5], "def_end_pos": [426, 11]}, {"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}, {"full_name": "Filter.liminf", "def_path": "Mathlib/Order/LiminfLimsup.lean", "def_pos": [426, 5], "def_end_pos": [426, 11]}, {"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}, {"full_name": "Filter.liminf", "def_path": "Mathlib/Order/LiminfLimsup.lean", "def_pos": [426, 5], "def_end_pos": [426, 11]}, {"full_name": "ENNReal.orderIsoRpow", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [629, 5], "def_end_pos": [629, 17]}, {"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}, {"full_name": "ENNReal.orderIsoRpow", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [629, 5], "def_end_pos": [629, 17]}, {"full_name": "Filter.liminf", "def_path": "Mathlib/Order/LiminfLimsup.lean", "def_pos": [426, 5], "def_end_pos": [426, 11]}, {"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}, {"full_name": "OrderIso.liminf_apply", "def_path": "Mathlib/Order/LiminfLimsup.lean", "def_pos": [1468, 9], "def_end_pos": [1468, 30]}, {"full_name": "ENNReal.orderIsoRpow", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [629, 5], "def_end_pos": [629, 17]}, {"full_name": "Eq.symm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [310, 9], "def_end_pos": [310, 16]}]], "state_before": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : OpensMeasurableSpace E\nR : \u211d\u22650\np : \u211d\u22650\u221e\nhp : p \u2260 0\nf : \u2115 \u2192 \u03b1 \u2192 E\nhfmeas : \u2200 (n : \u2115), Measurable (f n)\nhbdd : \u2200 (n : \u2115), snorm (f n) p \u03bc \u2264 \u2191R\nhp' : \u00acp = \u22a4\nx : \u03b1\nhx : liminf (fun n => \u2191\u2016f n x\u2016\u208a ^ ENNReal.toReal p) atTop < \u22a4\nhppos : 0 < ENNReal.toReal p\n\u22a2 liminf (fun n => \u2191\u2016f n x\u2016\u208a) atTop < \u22a4", "state_after": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : OpensMeasurableSpace E\nR : \u211d\u22650\np : \u211d\u22650\u221e\nhp : p \u2260 0\nf : \u2115 \u2192 \u03b1 \u2192 E\nhfmeas : \u2200 (n : \u2115), Measurable (f n)\nhbdd : \u2200 (n : \u2115), snorm (f n) p \u03bc \u2264 \u2191R\nhp' : \u00acp = \u22a4\nx : \u03b1\nhx : liminf (fun n => \u2191\u2016f n x\u2016\u208a ^ ENNReal.toReal p) atTop < \u22a4\nhppos : 0 < ENNReal.toReal p\nthis : liminf (fun n => \u2191\u2016f n x\u2016\u208a ^ ENNReal.toReal p) atTop = liminf (fun n => \u2191\u2016f n x\u2016\u208a) atTop ^ ENNReal.toReal p\n\u22a2 liminf (fun n => \u2191\u2016f n x\u2016\u208a) atTop < \u22a4"}, {"tactic": "rw [this] at hx", "annotated_tactic": ["rw [this] at hx", []], "state_before": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : OpensMeasurableSpace E\nR : \u211d\u22650\np : \u211d\u22650\u221e\nhp : p \u2260 0\nf : \u2115 \u2192 \u03b1 \u2192 E\nhfmeas : \u2200 (n : \u2115), Measurable (f n)\nhbdd : \u2200 (n : \u2115), snorm (f n) p \u03bc \u2264 \u2191R\nhp' : \u00acp = \u22a4\nx : \u03b1\nhx : liminf (fun n => \u2191\u2016f n x\u2016\u208a ^ ENNReal.toReal p) atTop < \u22a4\nhppos : 0 < ENNReal.toReal p\nthis : liminf (fun n => \u2191\u2016f n x\u2016\u208a ^ ENNReal.toReal p) atTop = liminf (fun n => \u2191\u2016f n x\u2016\u208a) atTop ^ ENNReal.toReal p\n\u22a2 liminf (fun n => \u2191\u2016f n x\u2016\u208a) atTop < \u22a4", "state_after": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : OpensMeasurableSpace E\nR : \u211d\u22650\np : \u211d\u22650\u221e\nhp : p \u2260 0\nf : \u2115 \u2192 \u03b1 \u2192 E\nhfmeas : \u2200 (n : \u2115), Measurable (f n)\nhbdd : \u2200 (n : \u2115), snorm (f n) p \u03bc \u2264 \u2191R\nhp' : \u00acp = \u22a4\nx : \u03b1\nhx : liminf (fun n => \u2191\u2016f n x\u2016\u208a) atTop ^ ENNReal.toReal p < \u22a4\nhppos : 0 < ENNReal.toReal p\nthis : liminf (fun n => \u2191\u2016f n x\u2016\u208a ^ ENNReal.toReal p) atTop = liminf (fun n => \u2191\u2016f n x\u2016\u208a) atTop ^ ENNReal.toReal p\n\u22a2 liminf (fun n => \u2191\u2016f n x\u2016\u208a) atTop < \u22a4"}, {"tactic": "rw [\u2190 ENNReal.rpow_one (liminf (fun n => \u2016f n x\u2016\u208a) atTop), \u2190 mul_inv_cancel hppos.ne.symm,\n  ENNReal.rpow_mul]", "annotated_tactic": ["rw [\u2190 <a>ENNReal.rpow_one</a> (<a>liminf</a> (fun n => \u2016f n x\u2016\u208a) <a>atTop</a>), \u2190 <a>mul_inv_cancel</a> hppos.ne.symm,\n    <a>ENNReal.rpow_mul</a>]", [{"full_name": "ENNReal.rpow_one", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [450, 9], "def_end_pos": [450, 17]}, {"full_name": "Filter.liminf", "def_path": "Mathlib/Order/LiminfLimsup.lean", "def_pos": [426, 5], "def_end_pos": [426, 11]}, {"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}, {"full_name": "mul_inv_cancel", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [194, 15], "def_end_pos": [194, 29]}, {"full_name": "ENNReal.rpow_mul", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [532, 9], "def_end_pos": [532, 17]}]], "state_before": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : OpensMeasurableSpace E\nR : \u211d\u22650\np : \u211d\u22650\u221e\nhp : p \u2260 0\nf : \u2115 \u2192 \u03b1 \u2192 E\nhfmeas : \u2200 (n : \u2115), Measurable (f n)\nhbdd : \u2200 (n : \u2115), snorm (f n) p \u03bc \u2264 \u2191R\nhp' : \u00acp = \u22a4\nx : \u03b1\nhx : liminf (fun n => \u2191\u2016f n x\u2016\u208a) atTop ^ ENNReal.toReal p < \u22a4\nhppos : 0 < ENNReal.toReal p\nthis : liminf (fun n => \u2191\u2016f n x\u2016\u208a ^ ENNReal.toReal p) atTop = liminf (fun n => \u2191\u2016f n x\u2016\u208a) atTop ^ ENNReal.toReal p\n\u22a2 liminf (fun n => \u2191\u2016f n x\u2016\u208a) atTop < \u22a4", "state_after": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : OpensMeasurableSpace E\nR : \u211d\u22650\np : \u211d\u22650\u221e\nhp : p \u2260 0\nf : \u2115 \u2192 \u03b1 \u2192 E\nhfmeas : \u2200 (n : \u2115), Measurable (f n)\nhbdd : \u2200 (n : \u2115), snorm (f n) p \u03bc \u2264 \u2191R\nhp' : \u00acp = \u22a4\nx : \u03b1\nhx : liminf (fun n => \u2191\u2016f n x\u2016\u208a) atTop ^ ENNReal.toReal p < \u22a4\nhppos : 0 < ENNReal.toReal p\nthis : liminf (fun n => \u2191\u2016f n x\u2016\u208a ^ ENNReal.toReal p) atTop = liminf (fun n => \u2191\u2016f n x\u2016\u208a) atTop ^ ENNReal.toReal p\n\u22a2 (liminf (fun n => \u2191\u2016f n x\u2016\u208a) atTop ^ ENNReal.toReal p) ^ (ENNReal.toReal p)\u207b\u00b9 < \u22a4"}, {"tactic": "exact ENNReal.rpow_lt_top_of_nonneg (inv_nonneg.2 hppos.le) hx.ne", "annotated_tactic": ["exact <a>ENNReal.rpow_lt_top_of_nonneg</a> (<a>inv_nonneg</a>.2 hppos.le) hx.ne", [{"full_name": "ENNReal.rpow_lt_top_of_nonneg", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [503, 9], "def_end_pos": [503, 30]}, {"full_name": "inv_nonneg", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [58, 9], "def_end_pos": [58, 19]}]], "state_before": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : OpensMeasurableSpace E\nR : \u211d\u22650\np : \u211d\u22650\u221e\nhp : p \u2260 0\nf : \u2115 \u2192 \u03b1 \u2192 E\nhfmeas : \u2200 (n : \u2115), Measurable (f n)\nhbdd : \u2200 (n : \u2115), snorm (f n) p \u03bc \u2264 \u2191R\nhp' : \u00acp = \u22a4\nx : \u03b1\nhx : liminf (fun n => \u2191\u2016f n x\u2016\u208a) atTop ^ ENNReal.toReal p < \u22a4\nhppos : 0 < ENNReal.toReal p\nthis : liminf (fun n => \u2191\u2016f n x\u2016\u208a ^ ENNReal.toReal p) atTop = liminf (fun n => \u2191\u2016f n x\u2016\u208a) atTop ^ ENNReal.toReal p\n\u22a2 (liminf (fun n => \u2191\u2016f n x\u2016\u208a) atTop ^ ENNReal.toReal p) ^ (ENNReal.toReal p)\u207b\u00b9 < \u22a4", "state_after": "no goals"}, {"tactic": "subst hp'", "annotated_tactic": ["subst hp'", []], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : OpensMeasurableSpace E\nR : \u211d\u22650\np : \u211d\u22650\u221e\nhp : p \u2260 0\nf : \u2115 \u2192 \u03b1 \u2192 E\nhfmeas : \u2200 (n : \u2115), Measurable (f n)\nhbdd : \u2200 (n : \u2115), snorm (f n) p \u03bc \u2264 \u2191R\nhp' : p = \u22a4\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, liminf (fun n => \u2191\u2016f n x\u2016\u208a) atTop < \u22a4", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : OpensMeasurableSpace E\nR : \u211d\u22650\nf : \u2115 \u2192 \u03b1 \u2192 E\nhfmeas : \u2200 (n : \u2115), Measurable (f n)\nhp : \u22a4 \u2260 0\nhbdd : \u2200 (n : \u2115), snorm (f n) \u22a4 \u03bc \u2264 \u2191R\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, liminf (fun n => \u2191\u2016f n x\u2016\u208a) atTop < \u22a4"}, {"tactic": "simp_rw [snorm_exponent_top] at hbdd", "annotated_tactic": ["simp_rw [<a>snorm_exponent_top</a>] at hbdd", [{"full_name": "MeasureTheory.snorm_exponent_top", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [103, 9], "def_end_pos": [103, 27]}]], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : OpensMeasurableSpace E\nR : \u211d\u22650\nf : \u2115 \u2192 \u03b1 \u2192 E\nhfmeas : \u2200 (n : \u2115), Measurable (f n)\nhp : \u22a4 \u2260 0\nhbdd : \u2200 (n : \u2115), snorm (f n) \u22a4 \u03bc \u2264 \u2191R\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, liminf (fun n => \u2191\u2016f n x\u2016\u208a) atTop < \u22a4", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : OpensMeasurableSpace E\nR : \u211d\u22650\nf : \u2115 \u2192 \u03b1 \u2192 E\nhfmeas : \u2200 (n : \u2115), Measurable (f n)\nhp : \u22a4 \u2260 0\nhbdd : \u2200 (n : \u2115), snormEssSup (f n) \u03bc \u2264 \u2191R\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, liminf (fun n => \u2191\u2016f n x\u2016\u208a) atTop < \u22a4"}, {"tactic": "have : \u2200 n, \u2200\u1d50 x \u2202\u03bc, (\u2016f n x\u2016\u208a : \u211d\u22650\u221e) < R + 1 := fun n =>\n  ae_lt_of_essSup_lt\n    (lt_of_le_of_lt (hbdd n) <| ENNReal.lt_add_right ENNReal.coe_ne_top one_ne_zero)", "annotated_tactic": ["have : \u2200 n, \u2200\u1d50 x \u2202\u03bc, (\u2016f n x\u2016\u208a : \u211d\u22650\u221e) < R + 1 := fun n =>\n      <a>ae_lt_of_essSup_lt</a>\n        (<a>lt_of_le_of_lt</a> (hbdd n) <| <a>ENNReal.lt_add_right</a> <a>ENNReal.coe_ne_top</a> <a>one_ne_zero</a>)", [{"full_name": "ae_lt_of_essSup_lt", "def_path": "Mathlib/MeasureTheory/Function/EssSup.lean", "def_pos": [98, 9], "def_end_pos": [98, 27]}, {"full_name": "lt_of_le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [122, 9], "def_end_pos": [122, 23]}, {"full_name": "ENNReal.lt_add_right", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [829, 9], "def_end_pos": [829, 21]}, {"full_name": "ENNReal.coe_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [302, 17], "def_end_pos": [302, 27]}, {"full_name": "one_ne_zero", "def_path": "Mathlib/Algebra/NeZero.lean", "def_pos": [55, 15], "def_end_pos": [55, 26]}]], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : OpensMeasurableSpace E\nR : \u211d\u22650\nf : \u2115 \u2192 \u03b1 \u2192 E\nhfmeas : \u2200 (n : \u2115), Measurable (f n)\nhp : \u22a4 \u2260 0\nhbdd : \u2200 (n : \u2115), snormEssSup (f n) \u03bc \u2264 \u2191R\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, liminf (fun n => \u2191\u2016f n x\u2016\u208a) atTop < \u22a4", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : OpensMeasurableSpace E\nR : \u211d\u22650\nf : \u2115 \u2192 \u03b1 \u2192 E\nhfmeas : \u2200 (n : \u2115), Measurable (f n)\nhp : \u22a4 \u2260 0\nhbdd : \u2200 (n : \u2115), snormEssSup (f n) \u03bc \u2264 \u2191R\nthis : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2191\u2016f n x\u2016\u208a < \u2191R + 1\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, liminf (fun n => \u2191\u2016f n x\u2016\u208a) atTop < \u22a4"}, {"tactic": "rw [\u2190 ae_all_iff] at this", "annotated_tactic": ["rw [\u2190 <a>ae_all_iff</a>] at this", [{"full_name": "MeasureTheory.ae_all_iff", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [422, 9], "def_end_pos": [422, 19]}]], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : OpensMeasurableSpace E\nR : \u211d\u22650\nf : \u2115 \u2192 \u03b1 \u2192 E\nhfmeas : \u2200 (n : \u2115), Measurable (f n)\nhp : \u22a4 \u2260 0\nhbdd : \u2200 (n : \u2115), snormEssSup (f n) \u03bc \u2264 \u2191R\nthis : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2191\u2016f n x\u2016\u208a < \u2191R + 1\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, liminf (fun n => \u2191\u2016f n x\u2016\u208a) atTop < \u22a4", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : OpensMeasurableSpace E\nR : \u211d\u22650\nf : \u2115 \u2192 \u03b1 \u2192 E\nhfmeas : \u2200 (n : \u2115), Measurable (f n)\nhp : \u22a4 \u2260 0\nhbdd : \u2200 (n : \u2115), snormEssSup (f n) \u03bc \u2264 \u2191R\nthis : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (i : \u2115), \u2191\u2016f i a\u2016\u208a < \u2191R + 1\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, liminf (fun n => \u2191\u2016f n x\u2016\u208a) atTop < \u22a4"}, {"tactic": "filter_upwards [this] with x hx using lt_of_le_of_lt\n    (liminf_le_of_frequently_le' <| frequently_of_forall fun n => (hx n).le)\n    (ENNReal.add_lt_top.2 \u27e8ENNReal.coe_lt_top, ENNReal.one_lt_top\u27e9)", "annotated_tactic": ["filter_upwards [this] with x hx using <a>lt_of_le_of_lt</a>\n        (<a>liminf_le_of_frequently_le'</a> <| <a>frequently_of_forall</a> fun n => (hx n).<a>le</a>)\n        (<a>ENNReal.add_lt_top</a>.2 \u27e8<a>ENNReal.coe_lt_top</a>, <a>ENNReal.one_lt_top</a>\u27e9)", [{"full_name": "lt_of_le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [122, 9], "def_end_pos": [122, 23]}, {"full_name": "Filter.liminf_le_of_frequently_le'", "def_path": "Mathlib/Order/LiminfLimsup.lean", "def_pos": [914, 9], "def_end_pos": [914, 36]}, {"full_name": "Filter.frequently_of_forall", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1271, 9], "def_end_pos": [1271, 29]}, {"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [142, 7], "def_end_pos": [142, 15]}, {"full_name": "ENNReal.add_lt_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [561, 17], "def_end_pos": [561, 27]}, {"full_name": "ENNReal.coe_lt_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [308, 17], "def_end_pos": [308, 27]}, {"full_name": "ENNReal.one_lt_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [723, 17], "def_end_pos": [723, 27]}]], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : OpensMeasurableSpace E\nR : \u211d\u22650\nf : \u2115 \u2192 \u03b1 \u2192 E\nhfmeas : \u2200 (n : \u2115), Measurable (f n)\nhp : \u22a4 \u2260 0\nhbdd : \u2200 (n : \u2115), snormEssSup (f n) \u03bc \u2264 \u2191R\nthis : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (i : \u2115), \u2191\u2016f i a\u2016\u208a < \u2191R + 1\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, liminf (fun n => \u2191\u2016f n x\u2016\u208a) atTop < \u22a4", "state_after": "no goals"}, {"tactic": "change\n  liminf (fun n => ENNReal.orderIsoRpow p.toReal hppos (\u2016f n x\u2016\u208a : \u211d\u22650\u221e)) atTop =\n    ENNReal.orderIsoRpow p.toReal hppos (liminf (fun n => (\u2016f n x\u2016\u208a : \u211d\u22650\u221e)) atTop)", "annotated_tactic": ["change\n      <a>liminf</a> (fun n => <a>ENNReal.orderIsoRpow</a> p.toReal hppos (\u2016f n x\u2016\u208a : \u211d\u22650\u221e)) <a>atTop</a> =\n        <a>ENNReal.orderIsoRpow</a> p.toReal hppos (<a>liminf</a> (fun n => (\u2016f n x\u2016\u208a : \u211d\u22650\u221e)) <a>atTop</a>)", [{"full_name": "Filter.liminf", "def_path": "Mathlib/Order/LiminfLimsup.lean", "def_pos": [426, 5], "def_end_pos": [426, 11]}, {"full_name": "ENNReal.orderIsoRpow", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [629, 5], "def_end_pos": [629, 17]}, {"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}, {"full_name": "ENNReal.orderIsoRpow", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [629, 5], "def_end_pos": [629, 17]}, {"full_name": "Filter.liminf", "def_path": "Mathlib/Order/LiminfLimsup.lean", "def_pos": [426, 5], "def_end_pos": [426, 11]}, {"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : OpensMeasurableSpace E\nR : \u211d\u22650\np : \u211d\u22650\u221e\nhp : p \u2260 0\nf : \u2115 \u2192 \u03b1 \u2192 E\nhfmeas : \u2200 (n : \u2115), Measurable (f n)\nhbdd : \u2200 (n : \u2115), snorm (f n) p \u03bc \u2264 \u2191R\nhp' : \u00acp = \u22a4\nx : \u03b1\nhx : liminf (fun n => \u2191\u2016f n x\u2016\u208a ^ ENNReal.toReal p) atTop < \u22a4\nhppos : 0 < ENNReal.toReal p\n\u22a2 liminf (fun n => \u2191\u2016f n x\u2016\u208a ^ ENNReal.toReal p) atTop = liminf (fun n => \u2191\u2016f n x\u2016\u208a) atTop ^ ENNReal.toReal p", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : OpensMeasurableSpace E\nR : \u211d\u22650\np : \u211d\u22650\u221e\nhp : p \u2260 0\nf : \u2115 \u2192 \u03b1 \u2192 E\nhfmeas : \u2200 (n : \u2115), Measurable (f n)\nhbdd : \u2200 (n : \u2115), snorm (f n) p \u03bc \u2264 \u2191R\nhp' : \u00acp = \u22a4\nx : \u03b1\nhx : liminf (fun n => \u2191\u2016f n x\u2016\u208a ^ ENNReal.toReal p) atTop < \u22a4\nhppos : 0 < ENNReal.toReal p\n\u22a2 liminf (fun n => \u2191(ENNReal.orderIsoRpow (ENNReal.toReal p) hppos) \u2191\u2016f n x\u2016\u208a) atTop =\n    \u2191(ENNReal.orderIsoRpow (ENNReal.toReal p) hppos) (liminf (fun n => \u2191\u2016f n x\u2016\u208a) atTop)"}, {"tactic": "refine' (OrderIso.liminf_apply (ENNReal.orderIsoRpow p.toReal _) _ _ _ _).symm <;>\n  isBoundedDefault", "annotated_tactic": ["refine' (<a>OrderIso.liminf_apply</a> (<a>ENNReal.orderIsoRpow</a> p.toReal _) _ _ _ _).<a>symm</a> <;>\n      isBoundedDefault", [{"full_name": "OrderIso.liminf_apply", "def_path": "Mathlib/Order/LiminfLimsup.lean", "def_pos": [1468, 9], "def_end_pos": [1468, 30]}, {"full_name": "ENNReal.orderIsoRpow", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [629, 5], "def_end_pos": [629, 17]}, {"full_name": "Eq.symm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [310, 9], "def_end_pos": [310, 16]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : OpensMeasurableSpace E\nR : \u211d\u22650\np : \u211d\u22650\u221e\nhp : p \u2260 0\nf : \u2115 \u2192 \u03b1 \u2192 E\nhfmeas : \u2200 (n : \u2115), Measurable (f n)\nhbdd : \u2200 (n : \u2115), snorm (f n) p \u03bc \u2264 \u2191R\nhp' : \u00acp = \u22a4\nx : \u03b1\nhx : liminf (fun n => \u2191\u2016f n x\u2016\u208a ^ ENNReal.toReal p) atTop < \u22a4\nhppos : 0 < ENNReal.toReal p\n\u22a2 liminf (fun n => \u2191(ENNReal.orderIsoRpow (ENNReal.toReal p) hppos) \u2191\u2016f n x\u2016\u208a) atTop =\n    \u2191(ENNReal.orderIsoRpow (ENNReal.toReal p) hppos) (liminf (fun n => \u2191\u2016f n x\u2016\u208a) atTop)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/ProbabilityMassFunction/Integrals.lean", "full_name": "PMF.integral_eq_sum", "start": [43, 1], "end": [47, 70], "traced_tactics": [{"tactic": "rw [integral_fintype _ (integrable_of_fintype _ f)]", "annotated_tactic": ["rw [<a>integral_fintype</a> _ (<a>integrable_of_fintype</a> _ f)]", [{"full_name": "MeasureTheory.integral_fintype", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1821, 9], "def_end_pos": [1821, 25]}, {"full_name": "MeasureTheory.integrable_of_fintype", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [511, 9], "def_end_pos": [511, 30]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MeasurableSpace \u03b1\ninst\u271d\u2074 : MeasurableSingletonClass \u03b1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : Fintype \u03b1\np : PMF \u03b1\nf : \u03b1 \u2192 E\n\u22a2 \u222b (a : \u03b1), f a \u2202toMeasure p = \u2211 a : \u03b1, ENNReal.toReal (\u2191p a) \u2022 f a", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2075 : MeasurableSpace \u03b1\ninst\u271d\u2074 : MeasurableSingletonClass \u03b1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : Fintype \u03b1\np : PMF \u03b1\nf : \u03b1 \u2192 E\n\u22a2 \u2211 x : \u03b1, ENNReal.toReal (\u2191\u2191(toMeasure p) {x}) \u2022 f x = \u2211 a : \u03b1, ENNReal.toReal (\u2191p a) \u2022 f a"}, {"tactic": "congr with x", "annotated_tactic": ["congr with x", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MeasurableSpace \u03b1\ninst\u271d\u2074 : MeasurableSingletonClass \u03b1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : Fintype \u03b1\np : PMF \u03b1\nf : \u03b1 \u2192 E\n\u22a2 \u2211 x : \u03b1, ENNReal.toReal (\u2191\u2191(toMeasure p) {x}) \u2022 f x = \u2211 a : \u03b1, ENNReal.toReal (\u2191p a) \u2022 f a", "state_after": "case e_f.h\n\u03b1 : Type u_1\ninst\u271d\u2075 : MeasurableSpace \u03b1\ninst\u271d\u2074 : MeasurableSingletonClass \u03b1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : Fintype \u03b1\np : PMF \u03b1\nf : \u03b1 \u2192 E\nx : \u03b1\n\u22a2 ENNReal.toReal (\u2191\u2191(toMeasure p) {x}) \u2022 f x = ENNReal.toReal (\u2191p x) \u2022 f x"}, {"tactic": "congr", "annotated_tactic": ["congr", []], "state_before": "case e_f.h\n\u03b1 : Type u_1\ninst\u271d\u2075 : MeasurableSpace \u03b1\ninst\u271d\u2074 : MeasurableSingletonClass \u03b1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : Fintype \u03b1\np : PMF \u03b1\nf : \u03b1 \u2192 E\nx : \u03b1\n\u22a2 ENNReal.toReal (\u2191\u2191(toMeasure p) {x}) \u2022 f x = ENNReal.toReal (\u2191p x) \u2022 f x", "state_after": "case e_f.h.e_a.e_a\n\u03b1 : Type u_1\ninst\u271d\u2075 : MeasurableSpace \u03b1\ninst\u271d\u2074 : MeasurableSingletonClass \u03b1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : Fintype \u03b1\np : PMF \u03b1\nf : \u03b1 \u2192 E\nx : \u03b1\n\u22a2 \u2191\u2191(toMeasure p) {x} = \u2191p x"}, {"tactic": "exact PMF.toMeasure_apply_singleton p x (MeasurableSet.singleton _)", "annotated_tactic": ["exact <a>PMF.toMeasure_apply_singleton</a> p x (<a>MeasurableSet.singleton</a> _)", [{"full_name": "PMF.toMeasure_apply_singleton", "def_path": "Mathlib/Probability/ProbabilityMassFunction/Basic.lean", "def_pos": [264, 9], "def_end_pos": [264, 34]}, {"full_name": "MeasurableSet.singleton", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [275, 7], "def_end_pos": [275, 30]}]], "state_before": "case e_f.h.e_a.e_a\n\u03b1 : Type u_1\ninst\u271d\u2075 : MeasurableSpace \u03b1\ninst\u271d\u2074 : MeasurableSingletonClass \u03b1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : Fintype \u03b1\np : PMF \u03b1\nf : \u03b1 \u2192 E\nx : \u03b1\n\u22a2 \u2191\u2191(toMeasure p) {x} = \u2191p x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "full_name": "BoundedContinuousFunction.range_toLpHom", "start": [1765, 1], "end": [1771, 60], "traced_tactics": [{"tactic": "symm", "annotated_tactic": ["symm", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\ninst\u271d\u00b2 : SecondCountableTopologyEither \u03b1 E\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : Fact (1 \u2264 p)\n\u22a2 NormedAddGroupHom.range (toLpHom p \u03bc) = Lp.boundedContinuousFunction E p \u03bc", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\ninst\u271d\u00b2 : SecondCountableTopologyEither \u03b1 E\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : Fact (1 \u2264 p)\n\u22a2 Lp.boundedContinuousFunction E p \u03bc = NormedAddGroupHom.range (toLpHom p \u03bc)"}, {"tactic": "convert AddMonoidHom.addSubgroupOf_range_eq_of_le\n    ((ContinuousMap.toAEEqFunAddHom \u03bc).comp (toContinuousMapAddHom \u03b1 E))\n    (by rintro - \u27e8f, rfl\u27e9; exact mem_Lp f : _ \u2264 Lp E p \u03bc)", "annotated_tactic": ["convert <a>AddMonoidHom.addSubgroupOf_range_eq_of_le</a>\n      ((<a>ContinuousMap.toAEEqFunAddHom</a> \u03bc).<a>comp</a> (<a>toContinuousMapAddHom</a> \u03b1 E))\n      (by rintro - \u27e8f, rfl\u27e9; exact <a>mem_Lp</a> f : _ \u2264 Lp E p \u03bc)", [{"full_name": "AddMonoidHom.addSubgroupOf_range_eq_of_le", "def_path": "Mathlib/GroupTheory/Subgroup/Basic.lean", "def_pos": [2735, 3], "def_end_pos": [2735, 14]}, {"full_name": "ContinuousMap.toAEEqFunAddHom", "def_path": "Mathlib/MeasureTheory/Function/AEEqFun.lean", "def_pos": [998, 3], "def_end_pos": [998, 14]}, {"full_name": "AddMonoidHom.comp", "def_path": "Mathlib/Algebra/Hom/Group/Defs.lean", "def_pos": [1046, 3], "def_end_pos": [1046, 14]}, {"full_name": "BoundedContinuousFunction.toContinuousMapAddHom", "def_path": "Mathlib/Topology/ContinuousFunction/Bounded.lean", "def_pos": [745, 5], "def_end_pos": [745, 26]}, {"full_name": "BoundedContinuousFunction.mem_Lp", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [1729, 9], "def_end_pos": [1729, 15]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\ninst\u271d\u00b2 : SecondCountableTopologyEither \u03b1 E\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : Fact (1 \u2264 p)\n\u22a2 Lp.boundedContinuousFunction E p \u03bc = NormedAddGroupHom.range (toLpHom p \u03bc)", "state_after": "no goals"}, {"tactic": "rintro - \u27e8f, rfl\u27e9", "annotated_tactic": ["rintro - \u27e8f, rfl\u27e9", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\ninst\u271d\u00b2 : SecondCountableTopologyEither \u03b1 E\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : Fact (1 \u2264 p)\n\u22a2 AddMonoidHom.range (AddMonoidHom.comp (ContinuousMap.toAEEqFunAddHom \u03bc) (toContinuousMapAddHom \u03b1 E)) \u2264 Lp E p", "state_after": "case intro\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\ninst\u271d\u00b2 : SecondCountableTopologyEither \u03b1 E\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : Fact (1 \u2264 p)\nf : \u03b1 \u2192\u1d47 E\n\u22a2 \u2191(AddMonoidHom.comp (ContinuousMap.toAEEqFunAddHom \u03bc) (toContinuousMapAddHom \u03b1 E)) f \u2208 Lp E p"}, {"tactic": "exact mem_Lp f", "annotated_tactic": ["exact <a>mem_Lp</a> f", [{"full_name": "BoundedContinuousFunction.mem_Lp", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [1729, 9], "def_end_pos": [1729, 15]}]], "state_before": "case intro\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\ninst\u271d\u00b2 : SecondCountableTopologyEither \u03b1 E\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : Fact (1 \u2264 p)\nf : \u03b1 \u2192\u1d47 E\n\u22a2 \u2191(AddMonoidHom.comp (ContinuousMap.toAEEqFunAddHom \u03bc) (toContinuousMapAddHom \u03b1 E)) f \u2208 Lp E p", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/PFun.lean", "full_name": "PFun.lift_injective", "start": [153, 1], "end": [154, 55], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "full_name": "MeasureTheory.L1.SimpleFunc.setToL1SCLM_const", "start": [954, 1], "end": [958, 62], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/FundThmCalculus.lean", "full_name": "intervalIntegral.integrableOn_deriv_right_of_nonneg", "start": [1254, 1], "end": [1285, 63], "traced_tactics": [{"tactic": "by_cases hab : a < b", "annotated_tactic": ["by_cases hab : a < b", []], "state_before": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf\u271d : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\nf f' : \u211d \u2192 E\na b : \u211d\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\ng'pos : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 0 \u2264 g' x\n\u22a2 IntegrableOn g' (Ioc a b)", "state_after": "case pos\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf\u271d : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\nf f' : \u211d \u2192 E\na b : \u211d\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\ng'pos : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 0 \u2264 g' x\nhab : a < b\n\u22a2 IntegrableOn g' (Ioc a b)\n\ncase neg\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf\u271d : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\nf f' : \u211d \u2192 E\na b : \u211d\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\ng'pos : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 0 \u2264 g' x\nhab : \u00aca < b\n\u22a2 IntegrableOn g' (Ioc a b)"}, {"tactic": "swap", "annotated_tactic": ["swap", []], "state_before": "case pos\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf\u271d : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\nf f' : \u211d \u2192 E\na b : \u211d\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\ng'pos : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 0 \u2264 g' x\nhab : a < b\n\u22a2 IntegrableOn g' (Ioc a b)\n\ncase neg\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf\u271d : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\nf f' : \u211d \u2192 E\na b : \u211d\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\ng'pos : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 0 \u2264 g' x\nhab : \u00aca < b\n\u22a2 IntegrableOn g' (Ioc a b)", "state_after": "case neg\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf\u271d : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\nf f' : \u211d \u2192 E\na b : \u211d\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\ng'pos : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 0 \u2264 g' x\nhab : \u00aca < b\n\u22a2 IntegrableOn g' (Ioc a b)\n\ncase pos\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf\u271d : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\nf f' : \u211d \u2192 E\na b : \u211d\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\ng'pos : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 0 \u2264 g' x\nhab : a < b\n\u22a2 IntegrableOn g' (Ioc a b)"}, {"tactic": "rw [integrableOn_Ioc_iff_integrableOn_Ioo]", "annotated_tactic": ["rw [<a>integrableOn_Ioc_iff_integrableOn_Ioo</a>]", [{"full_name": "integrableOn_Ioc_iff_integrableOn_Ioo", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [713, 9], "def_end_pos": [713, 46]}]], "state_before": "case pos\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf\u271d : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\nf f' : \u211d \u2192 E\na b : \u211d\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\ng'pos : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 0 \u2264 g' x\nhab : a < b\n\u22a2 IntegrableOn g' (Ioc a b)", "state_after": "case pos\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf\u271d : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\nf f' : \u211d \u2192 E\na b : \u211d\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\ng'pos : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 0 \u2264 g' x\nhab : a < b\n\u22a2 IntegrableOn g' (Ioo a b)"}, {"tactic": "have meas_g' : AEMeasurable g' (volume.restrict (Ioo a b)) := by\n  apply (aemeasurable_derivWithin_Ioi g _).congr\n  refine' (ae_restrict_mem measurableSet_Ioo).mono fun x hx => _\n  exact (hderiv x hx).derivWithin (uniqueDiffWithinAt_Ioi _)", "annotated_tactic": ["have meas_g' : <a>AEMeasurable</a> g' (volume.restrict (<a>Ioo</a> a b)) := by\n    apply (<a>aemeasurable_derivWithin_Ioi</a> g _).<a>congr</a>\n    refine' (<a>ae_restrict_mem</a> <a>measurableSet_Ioo</a>).<a>mono</a> fun x hx => _\n    exact (hderiv x hx).<a>derivWithin</a> (<a>uniqueDiffWithinAt_Ioi</a> _)", [{"full_name": "AEMeasurable", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [708, 5], "def_end_pos": [708, 17]}, {"full_name": "Set.Ioo", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [44, 5], "def_end_pos": [44, 8]}, {"full_name": "aemeasurable_derivWithin_Ioi", "def_path": "Mathlib/Analysis/Calculus/FDeriv/Measurable.lean", "def_pos": [840, 9], "def_end_pos": [840, 37]}, {"full_name": "AEMeasurable.congr", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [735, 9], "def_end_pos": [735, 14]}, {"full_name": "MeasureTheory.ae_restrict_mem", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2586, 9], "def_end_pos": [2586, 24]}, {"full_name": "measurableSet_Ioo", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [584, 9], "def_end_pos": [584, 26]}, {"full_name": "Filter.Eventually.mono", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1140, 9], "def_end_pos": [1140, 24]}, {"full_name": "HasDerivWithinAt.derivWithin", "def_path": "Mathlib/Analysis/Calculus/Deriv/Basic.lean", "def_pos": [463, 9], "def_end_pos": [463, 37]}, {"full_name": "uniqueDiffWithinAt_Ioi", "def_path": "Mathlib/Analysis/Calculus/TangentCone.lean", "def_pos": [439, 9], "def_end_pos": [439, 31]}]], "state_before": "case pos\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf\u271d : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\nf f' : \u211d \u2192 E\na b : \u211d\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\ng'pos : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 0 \u2264 g' x\nhab : a < b\n\u22a2 IntegrableOn g' (Ioo a b)", "state_after": "case pos\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf\u271d : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\nf f' : \u211d \u2192 E\na b : \u211d\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\ng'pos : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 0 \u2264 g' x\nhab : a < b\nmeas_g' : AEMeasurable g'\n\u22a2 IntegrableOn g' (Ioo a b)"}, {"tactic": "suffices H : (\u222b\u207b x in Ioo a b, \u2016g' x\u2016\u208a) \u2264 ENNReal.ofReal (g b - g a) from\n  \u27e8meas_g'.aestronglyMeasurable, H.trans_lt ENNReal.ofReal_lt_top\u27e9", "annotated_tactic": ["suffices H : (\u222b\u207b x in <a>Ioo</a> a b, \u2016g' x\u2016\u208a) \u2264 <a>ENNReal.ofReal</a> (g b - g a) from\n    \u27e8meas_g'.aestronglyMeasurable, H.trans_lt <a>ENNReal.ofReal_lt_top</a>\u27e9", [{"full_name": "Set.Ioo", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [44, 5], "def_end_pos": [44, 8]}, {"full_name": "ENNReal.ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [172, 29], "def_end_pos": [172, 35]}, {"full_name": "ENNReal.ofReal_lt_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [314, 17], "def_end_pos": [314, 30]}]], "state_before": "case pos\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf\u271d : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\nf f' : \u211d \u2192 E\na b : \u211d\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\ng'pos : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 0 \u2264 g' x\nhab : a < b\nmeas_g' : AEMeasurable g'\n\u22a2 IntegrableOn g' (Ioo a b)", "state_after": "case pos\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf\u271d : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\nf f' : \u211d \u2192 E\na b : \u211d\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\ng'pos : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 0 \u2264 g' x\nhab : a < b\nmeas_g' : AEMeasurable g'\n\u22a2 \u222b\u207b (x : \u211d) in Ioo a b, \u2191\u2016g' x\u2016\u208a \u2264 ENNReal.ofReal (g b - g a)"}, {"tactic": "by_contra' H", "annotated_tactic": ["by_contra' H", []], "state_before": "case pos\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf\u271d : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\nf f' : \u211d \u2192 E\na b : \u211d\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\ng'pos : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 0 \u2264 g' x\nhab : a < b\nmeas_g' : AEMeasurable g'\n\u22a2 \u222b\u207b (x : \u211d) in Ioo a b, \u2191\u2016g' x\u2016\u208a \u2264 ENNReal.ofReal (g b - g a)", "state_after": "case pos\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf\u271d : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\nf f' : \u211d \u2192 E\na b : \u211d\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\ng'pos : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 0 \u2264 g' x\nhab : a < b\nmeas_g' : AEMeasurable g'\nH : ENNReal.ofReal (g b - g a) < \u222b\u207b (x : \u211d) in Ioo a b, \u2191\u2016g' x\u2016\u208a\n\u22a2 False"}, {"tactic": "obtain \u27e8f, fle, fint, hf\u27e9 :\n  \u2203 f : SimpleFunc \u211d \u211d\u22650,\n    (\u2200 x, f x \u2264 \u2016g' x\u2016\u208a) \u2227\n      (\u222b\u207b x : \u211d in Ioo a b, f x) < \u221e \u2227 ENNReal.ofReal (g b - g a) < \u222b\u207b x : \u211d in Ioo a b, f x :=\n  exists_lt_lintegral_simpleFunc_of_lt_lintegral H", "annotated_tactic": ["obtain \u27e8f, fle, fint, hf\u27e9 :\n    \u2203 f : <a>SimpleFunc</a> \u211d \u211d\u22650,\n      (\u2200 x, f x \u2264 \u2016g' x\u2016\u208a) \u2227\n        (\u222b\u207b x : \u211d in <a>Ioo</a> a b, f x) < \u221e \u2227 <a>ENNReal.ofReal</a> (g b - g a) < \u222b\u207b x : \u211d in <a>Ioo</a> a b, f x :=\n    <a>exists_lt_lintegral_simpleFunc_of_lt_lintegral</a> H", [{"full_name": "MeasureTheory.SimpleFunc", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [44, 11], "def_end_pos": [44, 21]}, {"full_name": "Set.Ioo", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [44, 5], "def_end_pos": [44, 8]}, {"full_name": "ENNReal.ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [172, 29], "def_end_pos": [172, 35]}, {"full_name": "Set.Ioo", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [44, 5], "def_end_pos": [44, 8]}, {"full_name": "MeasureTheory.exists_lt_lintegral_simpleFunc_of_lt_lintegral", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [1760, 9], "def_end_pos": [1760, 55]}]], "state_before": "case pos\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf\u271d : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\nf f' : \u211d \u2192 E\na b : \u211d\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\ng'pos : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 0 \u2264 g' x\nhab : a < b\nmeas_g' : AEMeasurable g'\nH : ENNReal.ofReal (g b - g a) < \u222b\u207b (x : \u211d) in Ioo a b, \u2191\u2016g' x\u2016\u208a\n\u22a2 False", "state_after": "case pos.intro.intro.intro\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf\u271d\u00b9 : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\nf\u271d f' : \u211d \u2192 E\na b : \u211d\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\ng'pos : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 0 \u2264 g' x\nhab : a < b\nmeas_g' : AEMeasurable g'\nH : ENNReal.ofReal (g b - g a) < \u222b\u207b (x : \u211d) in Ioo a b, \u2191\u2016g' x\u2016\u208a\nf : SimpleFunc \u211d \u211d\u22650\nfle : \u2200 (x : \u211d), \u2191f x \u2264 \u2016g' x\u2016\u208a\nfint : \u222b\u207b (x : \u211d) in Ioo a b, \u2191(\u2191f x) < \u22a4\nhf : ENNReal.ofReal (g b - g a) < \u222b\u207b (x : \u211d) in Ioo a b, \u2191(\u2191f x)\n\u22a2 False"}, {"tactic": "let F : \u211d \u2192 \u211d := (\u2191) \u2218 f", "annotated_tactic": ["let F : \u211d \u2192 \u211d := (\u2191) \u2218 f", []], "state_before": "case pos.intro.intro.intro\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf\u271d\u00b9 : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\nf\u271d f' : \u211d \u2192 E\na b : \u211d\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\ng'pos : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 0 \u2264 g' x\nhab : a < b\nmeas_g' : AEMeasurable g'\nH : ENNReal.ofReal (g b - g a) < \u222b\u207b (x : \u211d) in Ioo a b, \u2191\u2016g' x\u2016\u208a\nf : SimpleFunc \u211d \u211d\u22650\nfle : \u2200 (x : \u211d), \u2191f x \u2264 \u2016g' x\u2016\u208a\nfint : \u222b\u207b (x : \u211d) in Ioo a b, \u2191(\u2191f x) < \u22a4\nhf : ENNReal.ofReal (g b - g a) < \u222b\u207b (x : \u211d) in Ioo a b, \u2191(\u2191f x)\n\u22a2 False", "state_after": "case pos.intro.intro.intro\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF\u271d : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf\u271d\u00b9 : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\nf\u271d f' : \u211d \u2192 E\na b : \u211d\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\ng'pos : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 0 \u2264 g' x\nhab : a < b\nmeas_g' : AEMeasurable g'\nH : ENNReal.ofReal (g b - g a) < \u222b\u207b (x : \u211d) in Ioo a b, \u2191\u2016g' x\u2016\u208a\nf : SimpleFunc \u211d \u211d\u22650\nfle : \u2200 (x : \u211d), \u2191f x \u2264 \u2016g' x\u2016\u208a\nfint : \u222b\u207b (x : \u211d) in Ioo a b, \u2191(\u2191f x) < \u22a4\nhf : ENNReal.ofReal (g b - g a) < \u222b\u207b (x : \u211d) in Ioo a b, \u2191(\u2191f x)\nF : \u211d \u2192 \u211d := NNReal.toReal \u2218 \u2191f\n\u22a2 False"}, {"tactic": "have intF : IntegrableOn F (Ioo a b) := by\n  refine' \u27e8f.measurable.coe_nnreal_real.aestronglyMeasurable, _\u27e9\n  simpa only [HasFiniteIntegral, comp_apply, NNReal.nnnorm_eq] using fint", "annotated_tactic": ["have intF : <a>IntegrableOn</a> F (<a>Ioo</a> a b) := by\n    refine' \u27e8f.measurable.coe_nnreal_real.aestronglyMeasurable, _\u27e9\n    simpa only [<a>HasFiniteIntegral</a>, <a>comp_apply</a>, <a>NNReal.nnnorm_eq</a>] using fint", [{"full_name": "MeasureTheory.IntegrableOn", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [90, 5], "def_end_pos": [90, 17]}, {"full_name": "Set.Ioo", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [44, 5], "def_end_pos": [44, 8]}, {"full_name": "MeasureTheory.HasFiniteIntegral", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [106, 5], "def_end_pos": [106, 22]}, {"full_name": "Function.comp_apply", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [33, 17], "def_end_pos": [33, 36]}, {"full_name": "NNReal.nnnorm_eq", "def_path": "Mathlib/Analysis/Normed/Field/Basic.lean", "def_pos": [830, 9], "def_end_pos": [830, 18]}]], "state_before": "case pos.intro.intro.intro\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF\u271d : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf\u271d\u00b9 : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\nf\u271d f' : \u211d \u2192 E\na b : \u211d\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\ng'pos : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 0 \u2264 g' x\nhab : a < b\nmeas_g' : AEMeasurable g'\nH : ENNReal.ofReal (g b - g a) < \u222b\u207b (x : \u211d) in Ioo a b, \u2191\u2016g' x\u2016\u208a\nf : SimpleFunc \u211d \u211d\u22650\nfle : \u2200 (x : \u211d), \u2191f x \u2264 \u2016g' x\u2016\u208a\nfint : \u222b\u207b (x : \u211d) in Ioo a b, \u2191(\u2191f x) < \u22a4\nhf : ENNReal.ofReal (g b - g a) < \u222b\u207b (x : \u211d) in Ioo a b, \u2191(\u2191f x)\nF : \u211d \u2192 \u211d := NNReal.toReal \u2218 \u2191f\n\u22a2 False", "state_after": "case pos.intro.intro.intro\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF\u271d : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf\u271d\u00b9 : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\nf\u271d f' : \u211d \u2192 E\na b : \u211d\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\ng'pos : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 0 \u2264 g' x\nhab : a < b\nmeas_g' : AEMeasurable g'\nH : ENNReal.ofReal (g b - g a) < \u222b\u207b (x : \u211d) in Ioo a b, \u2191\u2016g' x\u2016\u208a\nf : SimpleFunc \u211d \u211d\u22650\nfle : \u2200 (x : \u211d), \u2191f x \u2264 \u2016g' x\u2016\u208a\nfint : \u222b\u207b (x : \u211d) in Ioo a b, \u2191(\u2191f x) < \u22a4\nhf : ENNReal.ofReal (g b - g a) < \u222b\u207b (x : \u211d) in Ioo a b, \u2191(\u2191f x)\nF : \u211d \u2192 \u211d := NNReal.toReal \u2218 \u2191f\nintF : IntegrableOn F (Ioo a b)\n\u22a2 False"}, {"tactic": "have A : \u222b\u207b x : \u211d in Ioo a b, f x = ENNReal.ofReal (\u222b x in Ioo a b, F x) :=\n  lintegral_coe_eq_integral _ intF", "annotated_tactic": ["have A : \u222b\u207b x : \u211d in <a>Ioo</a> a b, f x = <a>ENNReal.ofReal</a> (\u222b x in <a>Ioo</a> a b, F x) :=\n    <a>lintegral_coe_eq_integral</a> _ intF", [{"full_name": "Set.Ioo", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [44, 5], "def_end_pos": [44, 8]}, {"full_name": "ENNReal.ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [172, 29], "def_end_pos": [172, 35]}, {"full_name": "Set.Ioo", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [44, 5], "def_end_pos": [44, 8]}, {"full_name": "MeasureTheory.lintegral_coe_eq_integral", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1208, 9], "def_end_pos": [1208, 34]}]], "state_before": "case pos.intro.intro.intro\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF\u271d : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf\u271d\u00b9 : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\nf\u271d f' : \u211d \u2192 E\na b : \u211d\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\ng'pos : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 0 \u2264 g' x\nhab : a < b\nmeas_g' : AEMeasurable g'\nH : ENNReal.ofReal (g b - g a) < \u222b\u207b (x : \u211d) in Ioo a b, \u2191\u2016g' x\u2016\u208a\nf : SimpleFunc \u211d \u211d\u22650\nfle : \u2200 (x : \u211d), \u2191f x \u2264 \u2016g' x\u2016\u208a\nfint : \u222b\u207b (x : \u211d) in Ioo a b, \u2191(\u2191f x) < \u22a4\nhf : ENNReal.ofReal (g b - g a) < \u222b\u207b (x : \u211d) in Ioo a b, \u2191(\u2191f x)\nF : \u211d \u2192 \u211d := NNReal.toReal \u2218 \u2191f\nintF : IntegrableOn F (Ioo a b)\n\u22a2 False", "state_after": "case pos.intro.intro.intro\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF\u271d : Type u_4\nA\u271d : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf\u271d\u00b9 : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\nf\u271d f' : \u211d \u2192 E\na b : \u211d\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\ng'pos : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 0 \u2264 g' x\nhab : a < b\nmeas_g' : AEMeasurable g'\nH : ENNReal.ofReal (g b - g a) < \u222b\u207b (x : \u211d) in Ioo a b, \u2191\u2016g' x\u2016\u208a\nf : SimpleFunc \u211d \u211d\u22650\nfle : \u2200 (x : \u211d), \u2191f x \u2264 \u2016g' x\u2016\u208a\nfint : \u222b\u207b (x : \u211d) in Ioo a b, \u2191(\u2191f x) < \u22a4\nhf : ENNReal.ofReal (g b - g a) < \u222b\u207b (x : \u211d) in Ioo a b, \u2191(\u2191f x)\nF : \u211d \u2192 \u211d := NNReal.toReal \u2218 \u2191f\nintF : IntegrableOn F (Ioo a b)\nA : \u222b\u207b (x : \u211d) in Ioo a b, \u2191(\u2191f x) = ENNReal.ofReal (\u222b (x : \u211d) in Ioo a b, F x)\n\u22a2 False"}, {"tactic": "rw [A] at hf", "annotated_tactic": ["rw [A] at hf", []], "state_before": "case pos.intro.intro.intro\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF\u271d : Type u_4\nA\u271d : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf\u271d\u00b9 : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\nf\u271d f' : \u211d \u2192 E\na b : \u211d\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\ng'pos : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 0 \u2264 g' x\nhab : a < b\nmeas_g' : AEMeasurable g'\nH : ENNReal.ofReal (g b - g a) < \u222b\u207b (x : \u211d) in Ioo a b, \u2191\u2016g' x\u2016\u208a\nf : SimpleFunc \u211d \u211d\u22650\nfle : \u2200 (x : \u211d), \u2191f x \u2264 \u2016g' x\u2016\u208a\nfint : \u222b\u207b (x : \u211d) in Ioo a b, \u2191(\u2191f x) < \u22a4\nhf : ENNReal.ofReal (g b - g a) < \u222b\u207b (x : \u211d) in Ioo a b, \u2191(\u2191f x)\nF : \u211d \u2192 \u211d := NNReal.toReal \u2218 \u2191f\nintF : IntegrableOn F (Ioo a b)\nA : \u222b\u207b (x : \u211d) in Ioo a b, \u2191(\u2191f x) = ENNReal.ofReal (\u222b (x : \u211d) in Ioo a b, F x)\n\u22a2 False", "state_after": "case pos.intro.intro.intro\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF\u271d : Type u_4\nA\u271d : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf\u271d\u00b9 : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\nf\u271d f' : \u211d \u2192 E\na b : \u211d\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\ng'pos : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 0 \u2264 g' x\nhab : a < b\nmeas_g' : AEMeasurable g'\nH : ENNReal.ofReal (g b - g a) < \u222b\u207b (x : \u211d) in Ioo a b, \u2191\u2016g' x\u2016\u208a\nf : SimpleFunc \u211d \u211d\u22650\nfle : \u2200 (x : \u211d), \u2191f x \u2264 \u2016g' x\u2016\u208a\nfint : \u222b\u207b (x : \u211d) in Ioo a b, \u2191(\u2191f x) < \u22a4\nF : \u211d \u2192 \u211d := NNReal.toReal \u2218 \u2191f\nhf : ENNReal.ofReal (g b - g a) < ENNReal.ofReal (\u222b (x : \u211d) in Ioo a b, F x)\nintF : IntegrableOn F (Ioo a b)\nA : \u222b\u207b (x : \u211d) in Ioo a b, \u2191(\u2191f x) = ENNReal.ofReal (\u222b (x : \u211d) in Ioo a b, F x)\n\u22a2 False"}, {"tactic": "exact lt_irrefl _ (hf.trans_le (ENNReal.ofReal_le_ofReal B))", "annotated_tactic": ["exact <a>lt_irrefl</a> _ (hf.trans_le (<a>ENNReal.ofReal_le_ofReal</a> B))", [{"full_name": "lt_irrefl", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [79, 9], "def_end_pos": [79, 18]}, {"full_name": "ENNReal.ofReal_le_ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2135, 9], "def_end_pos": [2135, 25]}]], "state_before": "case pos.intro.intro.intro\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF\u271d : Type u_4\nA\u271d : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf\u271d\u00b9 : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\nf\u271d f' : \u211d \u2192 E\na b : \u211d\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\ng'pos : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 0 \u2264 g' x\nhab : a < b\nmeas_g' : AEMeasurable g'\nH : ENNReal.ofReal (g b - g a) < \u222b\u207b (x : \u211d) in Ioo a b, \u2191\u2016g' x\u2016\u208a\nf : SimpleFunc \u211d \u211d\u22650\nfle : \u2200 (x : \u211d), \u2191f x \u2264 \u2016g' x\u2016\u208a\nfint : \u222b\u207b (x : \u211d) in Ioo a b, \u2191(\u2191f x) < \u22a4\nF : \u211d \u2192 \u211d := NNReal.toReal \u2218 \u2191f\nhf : ENNReal.ofReal (g b - g a) < ENNReal.ofReal (\u222b (x : \u211d) in Ioo a b, F x)\nintF : IntegrableOn F (Ioo a b)\nA : \u222b\u207b (x : \u211d) in Ioo a b, \u2191(\u2191f x) = ENNReal.ofReal (\u222b (x : \u211d) in Ioo a b, F x)\nB : \u222b (x : \u211d) in Ioo a b, F x \u2264 g b - g a\n\u22a2 False", "state_after": "no goals"}, {"tactic": "simp [Ioc_eq_empty hab]", "annotated_tactic": ["simp [<a>Ioc_eq_empty</a> hab]", [{"full_name": "Set.Ioc_eq_empty", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [368, 9], "def_end_pos": [368, 21]}]], "state_before": "case neg\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf\u271d : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\nf f' : \u211d \u2192 E\na b : \u211d\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\ng'pos : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 0 \u2264 g' x\nhab : \u00aca < b\n\u22a2 IntegrableOn g' (Ioc a b)", "state_after": "no goals"}, {"tactic": "apply (aemeasurable_derivWithin_Ioi g _).congr", "annotated_tactic": ["apply (<a>aemeasurable_derivWithin_Ioi</a> g _).<a>congr</a>", [{"full_name": "aemeasurable_derivWithin_Ioi", "def_path": "Mathlib/Analysis/Calculus/FDeriv/Measurable.lean", "def_pos": [840, 9], "def_end_pos": [840, 37]}, {"full_name": "AEMeasurable.congr", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [735, 9], "def_end_pos": [735, 14]}]], "state_before": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf\u271d : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\nf f' : \u211d \u2192 E\na b : \u211d\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\ng'pos : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 0 \u2264 g' x\nhab : a < b\n\u22a2 AEMeasurable g'", "state_after": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf\u271d : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\nf f' : \u211d \u2192 E\na b : \u211d\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\ng'pos : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 0 \u2264 g' x\nhab : a < b\n\u22a2 (fun x => derivWithin g (Ioi x) x) =\u1d50[Measure.restrict volume (Ioo a b)] g'"}, {"tactic": "refine' (ae_restrict_mem measurableSet_Ioo).mono fun x hx => _", "annotated_tactic": ["refine' (<a>ae_restrict_mem</a> <a>measurableSet_Ioo</a>).<a>mono</a> fun x hx => _", [{"full_name": "MeasureTheory.ae_restrict_mem", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2586, 9], "def_end_pos": [2586, 24]}, {"full_name": "measurableSet_Ioo", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [584, 9], "def_end_pos": [584, 26]}, {"full_name": "Filter.Eventually.mono", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1140, 9], "def_end_pos": [1140, 24]}]], "state_before": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf\u271d : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\nf f' : \u211d \u2192 E\na b : \u211d\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\ng'pos : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 0 \u2264 g' x\nhab : a < b\n\u22a2 (fun x => derivWithin g (Ioi x) x) =\u1d50[Measure.restrict volume (Ioo a b)] g'", "state_after": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf\u271d : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\nf f' : \u211d \u2192 E\na b : \u211d\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\ng'pos : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 0 \u2264 g' x\nhab : a < b\nx : \u211d\nhx : x \u2208 Ioo a b\n\u22a2 (fun x => derivWithin g (Ioi x) x) x = g' x"}, {"tactic": "exact (hderiv x hx).derivWithin (uniqueDiffWithinAt_Ioi _)", "annotated_tactic": ["exact (hderiv x hx).<a>derivWithin</a> (<a>uniqueDiffWithinAt_Ioi</a> _)", [{"full_name": "HasDerivWithinAt.derivWithin", "def_path": "Mathlib/Analysis/Calculus/Deriv/Basic.lean", "def_pos": [463, 9], "def_end_pos": [463, 37]}, {"full_name": "uniqueDiffWithinAt_Ioi", "def_path": "Mathlib/Analysis/Calculus/TangentCone.lean", "def_pos": [439, 9], "def_end_pos": [439, 31]}]], "state_before": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf\u271d : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\nf f' : \u211d \u2192 E\na b : \u211d\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\ng'pos : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 0 \u2264 g' x\nhab : a < b\nx : \u211d\nhx : x \u2208 Ioo a b\n\u22a2 (fun x => derivWithin g (Ioi x) x) x = g' x", "state_after": "no goals"}, {"tactic": "refine' \u27e8f.measurable.coe_nnreal_real.aestronglyMeasurable, _\u27e9", "annotated_tactic": ["refine' \u27e8f.measurable.coe_nnreal_real.aestronglyMeasurable, _\u27e9", []], "state_before": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF\u271d : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf\u271d\u00b9 : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\nf\u271d f' : \u211d \u2192 E\na b : \u211d\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\ng'pos : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 0 \u2264 g' x\nhab : a < b\nmeas_g' : AEMeasurable g'\nH : ENNReal.ofReal (g b - g a) < \u222b\u207b (x : \u211d) in Ioo a b, \u2191\u2016g' x\u2016\u208a\nf : SimpleFunc \u211d \u211d\u22650\nfle : \u2200 (x : \u211d), \u2191f x \u2264 \u2016g' x\u2016\u208a\nfint : \u222b\u207b (x : \u211d) in Ioo a b, \u2191(\u2191f x) < \u22a4\nhf : ENNReal.ofReal (g b - g a) < \u222b\u207b (x : \u211d) in Ioo a b, \u2191(\u2191f x)\nF : \u211d \u2192 \u211d := NNReal.toReal \u2218 \u2191f\n\u22a2 IntegrableOn F (Ioo a b)", "state_after": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF\u271d : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf\u271d\u00b9 : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\nf\u271d f' : \u211d \u2192 E\na b : \u211d\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\ng'pos : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 0 \u2264 g' x\nhab : a < b\nmeas_g' : AEMeasurable g'\nH : ENNReal.ofReal (g b - g a) < \u222b\u207b (x : \u211d) in Ioo a b, \u2191\u2016g' x\u2016\u208a\nf : SimpleFunc \u211d \u211d\u22650\nfle : \u2200 (x : \u211d), \u2191f x \u2264 \u2016g' x\u2016\u208a\nfint : \u222b\u207b (x : \u211d) in Ioo a b, \u2191(\u2191f x) < \u22a4\nhf : ENNReal.ofReal (g b - g a) < \u222b\u207b (x : \u211d) in Ioo a b, \u2191(\u2191f x)\nF : \u211d \u2192 \u211d := NNReal.toReal \u2218 \u2191f\n\u22a2 HasFiniteIntegral F"}, {"tactic": "simpa only [HasFiniteIntegral, comp_apply, NNReal.nnnorm_eq] using fint", "annotated_tactic": ["simpa only [<a>HasFiniteIntegral</a>, <a>comp_apply</a>, <a>NNReal.nnnorm_eq</a>] using fint", [{"full_name": "MeasureTheory.HasFiniteIntegral", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [106, 5], "def_end_pos": [106, 22]}, {"full_name": "Function.comp_apply", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [33, 17], "def_end_pos": [33, 36]}, {"full_name": "NNReal.nnnorm_eq", "def_path": "Mathlib/Analysis/Normed/Field/Basic.lean", "def_pos": [830, 9], "def_end_pos": [830, 18]}]], "state_before": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF\u271d : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf\u271d\u00b9 : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\nf\u271d f' : \u211d \u2192 E\na b : \u211d\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\ng'pos : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 0 \u2264 g' x\nhab : a < b\nmeas_g' : AEMeasurable g'\nH : ENNReal.ofReal (g b - g a) < \u222b\u207b (x : \u211d) in Ioo a b, \u2191\u2016g' x\u2016\u208a\nf : SimpleFunc \u211d \u211d\u22650\nfle : \u2200 (x : \u211d), \u2191f x \u2264 \u2016g' x\u2016\u208a\nfint : \u222b\u207b (x : \u211d) in Ioo a b, \u2191(\u2191f x) < \u22a4\nhf : ENNReal.ofReal (g b - g a) < \u222b\u207b (x : \u211d) in Ioo a b, \u2191(\u2191f x)\nF : \u211d \u2192 \u211d := NNReal.toReal \u2218 \u2191f\n\u22a2 HasFiniteIntegral F", "state_after": "no goals"}, {"tactic": "rw [\u2190 integral_Ioc_eq_integral_Ioo, \u2190 intervalIntegral.integral_of_le hab.le]", "annotated_tactic": ["rw [\u2190 <a>integral_Ioc_eq_integral_Ioo</a>, \u2190 <a>intervalIntegral.integral_of_le</a> hab.le]", [{"full_name": "MeasureTheory.integral_Ioc_eq_integral_Ioo", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [683, 9], "def_end_pos": [683, 37]}, {"full_name": "intervalIntegral.integral_of_le", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [465, 9], "def_end_pos": [465, 23]}]], "state_before": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF\u271d : Type u_4\nA\u271d : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf\u271d\u00b9 : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\nf\u271d f' : \u211d \u2192 E\na b : \u211d\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\ng'pos : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 0 \u2264 g' x\nhab : a < b\nmeas_g' : AEMeasurable g'\nH : ENNReal.ofReal (g b - g a) < \u222b\u207b (x : \u211d) in Ioo a b, \u2191\u2016g' x\u2016\u208a\nf : SimpleFunc \u211d \u211d\u22650\nfle : \u2200 (x : \u211d), \u2191f x \u2264 \u2016g' x\u2016\u208a\nfint : \u222b\u207b (x : \u211d) in Ioo a b, \u2191(\u2191f x) < \u22a4\nF : \u211d \u2192 \u211d := NNReal.toReal \u2218 \u2191f\nhf : ENNReal.ofReal (g b - g a) < ENNReal.ofReal (\u222b (x : \u211d) in Ioo a b, F x)\nintF : IntegrableOn F (Ioo a b)\nA : \u222b\u207b (x : \u211d) in Ioo a b, \u2191(\u2191f x) = ENNReal.ofReal (\u222b (x : \u211d) in Ioo a b, F x)\n\u22a2 \u222b (x : \u211d) in Ioo a b, F x \u2264 g b - g a", "state_after": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF\u271d : Type u_4\nA\u271d : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf\u271d\u00b9 : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\nf\u271d f' : \u211d \u2192 E\na b : \u211d\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\ng'pos : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 0 \u2264 g' x\nhab : a < b\nmeas_g' : AEMeasurable g'\nH : ENNReal.ofReal (g b - g a) < \u222b\u207b (x : \u211d) in Ioo a b, \u2191\u2016g' x\u2016\u208a\nf : SimpleFunc \u211d \u211d\u22650\nfle : \u2200 (x : \u211d), \u2191f x \u2264 \u2016g' x\u2016\u208a\nfint : \u222b\u207b (x : \u211d) in Ioo a b, \u2191(\u2191f x) < \u22a4\nF : \u211d \u2192 \u211d := NNReal.toReal \u2218 \u2191f\nhf : ENNReal.ofReal (g b - g a) < ENNReal.ofReal (\u222b (x : \u211d) in Ioo a b, F x)\nintF : IntegrableOn F (Ioo a b)\nA : \u222b\u207b (x : \u211d) in Ioo a b, \u2191(\u2191f x) = ENNReal.ofReal (\u222b (x : \u211d) in Ioo a b, F x)\n\u22a2 \u222b (x : \u211d) in a..b, F x \u2264 g b - g a"}, {"tactic": "refine integral_le_sub_of_hasDeriv_right_of_le hab.le hcont hderiv ?_ fun x hx => ?_", "annotated_tactic": ["refine <a>integral_le_sub_of_hasDeriv_right_of_le</a> hab.le hcont hderiv ?_ fun x hx => ?_", [{"full_name": "intervalIntegral.integral_le_sub_of_hasDeriv_right_of_le", "def_path": "Mathlib/MeasureTheory/Integral/FundThmCalculus.lean", "def_pos": [1148, 9], "def_end_pos": [1148, 48]}]], "state_before": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF\u271d : Type u_4\nA\u271d : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf\u271d\u00b9 : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\nf\u271d f' : \u211d \u2192 E\na b : \u211d\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\ng'pos : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 0 \u2264 g' x\nhab : a < b\nmeas_g' : AEMeasurable g'\nH : ENNReal.ofReal (g b - g a) < \u222b\u207b (x : \u211d) in Ioo a b, \u2191\u2016g' x\u2016\u208a\nf : SimpleFunc \u211d \u211d\u22650\nfle : \u2200 (x : \u211d), \u2191f x \u2264 \u2016g' x\u2016\u208a\nfint : \u222b\u207b (x : \u211d) in Ioo a b, \u2191(\u2191f x) < \u22a4\nF : \u211d \u2192 \u211d := NNReal.toReal \u2218 \u2191f\nhf : ENNReal.ofReal (g b - g a) < ENNReal.ofReal (\u222b (x : \u211d) in Ioo a b, F x)\nintF : IntegrableOn F (Ioo a b)\nA : \u222b\u207b (x : \u211d) in Ioo a b, \u2191(\u2191f x) = ENNReal.ofReal (\u222b (x : \u211d) in Ioo a b, F x)\n\u22a2 \u222b (x : \u211d) in a..b, F x \u2264 g b - g a", "state_after": "case refine_1\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF\u271d : Type u_4\nA\u271d : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf\u271d\u00b9 : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\nf\u271d f' : \u211d \u2192 E\na b : \u211d\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\ng'pos : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 0 \u2264 g' x\nhab : a < b\nmeas_g' : AEMeasurable g'\nH : ENNReal.ofReal (g b - g a) < \u222b\u207b (x : \u211d) in Ioo a b, \u2191\u2016g' x\u2016\u208a\nf : SimpleFunc \u211d \u211d\u22650\nfle : \u2200 (x : \u211d), \u2191f x \u2264 \u2016g' x\u2016\u208a\nfint : \u222b\u207b (x : \u211d) in Ioo a b, \u2191(\u2191f x) < \u22a4\nF : \u211d \u2192 \u211d := NNReal.toReal \u2218 \u2191f\nhf : ENNReal.ofReal (g b - g a) < ENNReal.ofReal (\u222b (x : \u211d) in Ioo a b, F x)\nintF : IntegrableOn F (Ioo a b)\nA : \u222b\u207b (x : \u211d) in Ioo a b, \u2191(\u2191f x) = ENNReal.ofReal (\u222b (x : \u211d) in Ioo a b, F x)\n\u22a2 IntegrableOn (fun x => F x) (Icc a b)\n\ncase refine_2\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF\u271d : Type u_4\nA\u271d : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf\u271d\u00b9 : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\nf\u271d f' : \u211d \u2192 E\na b : \u211d\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\ng'pos : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 0 \u2264 g' x\nhab : a < b\nmeas_g' : AEMeasurable g'\nH : ENNReal.ofReal (g b - g a) < \u222b\u207b (x : \u211d) in Ioo a b, \u2191\u2016g' x\u2016\u208a\nf : SimpleFunc \u211d \u211d\u22650\nfle : \u2200 (x : \u211d), \u2191f x \u2264 \u2016g' x\u2016\u208a\nfint : \u222b\u207b (x : \u211d) in Ioo a b, \u2191(\u2191f x) < \u22a4\nF : \u211d \u2192 \u211d := NNReal.toReal \u2218 \u2191f\nhf : ENNReal.ofReal (g b - g a) < ENNReal.ofReal (\u222b (x : \u211d) in Ioo a b, F x)\nintF : IntegrableOn F (Ioo a b)\nA : \u222b\u207b (x : \u211d) in Ioo a b, \u2191(\u2191f x) = ENNReal.ofReal (\u222b (x : \u211d) in Ioo a b, F x)\nx : \u211d\nhx : x \u2208 Ioo a b\n\u22a2 F x \u2264 g' x"}, {"tactic": "rwa [integrableOn_Icc_iff_integrableOn_Ioo]", "annotated_tactic": ["rwa [<a>integrableOn_Icc_iff_integrableOn_Ioo</a>]", [{"full_name": "integrableOn_Icc_iff_integrableOn_Ioo", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [718, 9], "def_end_pos": [718, 46]}]], "state_before": "case refine_1\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF\u271d : Type u_4\nA\u271d : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf\u271d\u00b9 : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\nf\u271d f' : \u211d \u2192 E\na b : \u211d\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\ng'pos : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 0 \u2264 g' x\nhab : a < b\nmeas_g' : AEMeasurable g'\nH : ENNReal.ofReal (g b - g a) < \u222b\u207b (x : \u211d) in Ioo a b, \u2191\u2016g' x\u2016\u208a\nf : SimpleFunc \u211d \u211d\u22650\nfle : \u2200 (x : \u211d), \u2191f x \u2264 \u2016g' x\u2016\u208a\nfint : \u222b\u207b (x : \u211d) in Ioo a b, \u2191(\u2191f x) < \u22a4\nF : \u211d \u2192 \u211d := NNReal.toReal \u2218 \u2191f\nhf : ENNReal.ofReal (g b - g a) < ENNReal.ofReal (\u222b (x : \u211d) in Ioo a b, F x)\nintF : IntegrableOn F (Ioo a b)\nA : \u222b\u207b (x : \u211d) in Ioo a b, \u2191(\u2191f x) = ENNReal.ofReal (\u222b (x : \u211d) in Ioo a b, F x)\n\u22a2 IntegrableOn (fun x => F x) (Icc a b)", "state_after": "no goals"}, {"tactic": "convert NNReal.coe_le_coe.2 (fle x)", "annotated_tactic": ["convert <a>NNReal.coe_le_coe</a>.2 (fle x)", [{"full_name": "NNReal.coe_le_coe", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [366, 19], "def_end_pos": [366, 29]}]], "state_before": "case refine_2\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF\u271d : Type u_4\nA\u271d : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf\u271d\u00b9 : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\nf\u271d f' : \u211d \u2192 E\na b : \u211d\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\ng'pos : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 0 \u2264 g' x\nhab : a < b\nmeas_g' : AEMeasurable g'\nH : ENNReal.ofReal (g b - g a) < \u222b\u207b (x : \u211d) in Ioo a b, \u2191\u2016g' x\u2016\u208a\nf : SimpleFunc \u211d \u211d\u22650\nfle : \u2200 (x : \u211d), \u2191f x \u2264 \u2016g' x\u2016\u208a\nfint : \u222b\u207b (x : \u211d) in Ioo a b, \u2191(\u2191f x) < \u22a4\nF : \u211d \u2192 \u211d := NNReal.toReal \u2218 \u2191f\nhf : ENNReal.ofReal (g b - g a) < ENNReal.ofReal (\u222b (x : \u211d) in Ioo a b, F x)\nintF : IntegrableOn F (Ioo a b)\nA : \u222b\u207b (x : \u211d) in Ioo a b, \u2191(\u2191f x) = ENNReal.ofReal (\u222b (x : \u211d) in Ioo a b, F x)\nx : \u211d\nhx : x \u2208 Ioo a b\n\u22a2 F x \u2264 g' x", "state_after": "case h.e'_4\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF\u271d : Type u_4\nA\u271d : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf\u271d\u00b9 : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\nf\u271d f' : \u211d \u2192 E\na b : \u211d\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\ng'pos : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 0 \u2264 g' x\nhab : a < b\nmeas_g' : AEMeasurable g'\nH : ENNReal.ofReal (g b - g a) < \u222b\u207b (x : \u211d) in Ioo a b, \u2191\u2016g' x\u2016\u208a\nf : SimpleFunc \u211d \u211d\u22650\nfle : \u2200 (x : \u211d), \u2191f x \u2264 \u2016g' x\u2016\u208a\nfint : \u222b\u207b (x : \u211d) in Ioo a b, \u2191(\u2191f x) < \u22a4\nF : \u211d \u2192 \u211d := NNReal.toReal \u2218 \u2191f\nhf : ENNReal.ofReal (g b - g a) < ENNReal.ofReal (\u222b (x : \u211d) in Ioo a b, F x)\nintF : IntegrableOn F (Ioo a b)\nA : \u222b\u207b (x : \u211d) in Ioo a b, \u2191(\u2191f x) = ENNReal.ofReal (\u222b (x : \u211d) in Ioo a b, F x)\nx : \u211d\nhx : x \u2208 Ioo a b\n\u22a2 g' x = \u2191\u2016g' x\u2016\u208a"}, {"tactic": "simp only [Real.norm_of_nonneg (g'pos x hx), coe_nnnorm]", "annotated_tactic": ["simp only [<a>Real.norm_of_nonneg</a> (g'pos x hx), <a>coe_nnnorm</a>]", [{"full_name": "Real.norm_of_nonneg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [1768, 9], "def_end_pos": [1768, 23]}, {"full_name": "coe_nnnorm", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [905, 41], "def_end_pos": [905, 51]}]], "state_before": "case h.e'_4\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF\u271d : Type u_4\nA\u271d : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf\u271d\u00b9 : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\nf\u271d f' : \u211d \u2192 E\na b : \u211d\nhcont : ContinuousOn g (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\ng'pos : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 0 \u2264 g' x\nhab : a < b\nmeas_g' : AEMeasurable g'\nH : ENNReal.ofReal (g b - g a) < \u222b\u207b (x : \u211d) in Ioo a b, \u2191\u2016g' x\u2016\u208a\nf : SimpleFunc \u211d \u211d\u22650\nfle : \u2200 (x : \u211d), \u2191f x \u2264 \u2016g' x\u2016\u208a\nfint : \u222b\u207b (x : \u211d) in Ioo a b, \u2191(\u2191f x) < \u22a4\nF : \u211d \u2192 \u211d := NNReal.toReal \u2218 \u2191f\nhf : ENNReal.ofReal (g b - g a) < ENNReal.ofReal (\u222b (x : \u211d) in Ioo a b, F x)\nintF : IntegrableOn F (Ioo a b)\nA : \u222b\u207b (x : \u211d) in Ioo a b, \u2191(\u2191f x) = ENNReal.ofReal (\u222b (x : \u211d) in Ioo a b, F x)\nx : \u211d\nhx : x \u2208 Ioo a b\n\u22a2 g' x = \u2191\u2016g' x\u2016\u208a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "full_name": "pi_le_borel_pi", "start": [1061, 1], "end": [1067, 88], "traced_tactics": [{"tactic": "have : \u2039\u2200 i, MeasurableSpace (\u03c0 i)\u203a = fun i => borel (\u03c0 i) :=\n  funext fun i => BorelSpace.measurable_eq", "annotated_tactic": ["have : \u2039\u2200 i, MeasurableSpace (\u03c0 i)\u203a = fun i => <a>borel</a> (\u03c0 i) :=\n    <a>funext</a> fun i => <a>BorelSpace.measurable_eq</a>", [{"full_name": "borel", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [59, 5], "def_end_pos": [59, 10]}, {"full_name": "funext", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [1555, 9], "def_end_pos": [1555, 15]}, {"full_name": "BorelSpace.measurable_eq", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [204, 3], "def_end_pos": [204, 16]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9\u271d : Sort y\ns t u : Set \u03b1\ninst\u271d\u00b9\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u2070 : BorelSpace \u03b1\ninst\u271d\u2079 : TopologicalSpace \u03b2\ninst\u271d\u2078 : MeasurableSpace \u03b2\ninst\u271d\u2077 : BorelSpace \u03b2\ninst\u271d\u2076 : TopologicalSpace \u03b3\ninst\u271d\u2075 : MeasurableSpace \u03b3\ninst\u271d\u2074 : BorelSpace \u03b3\ninst\u271d\u00b3 : MeasurableSpace \u03b4\n\u03b9 : Type u_6\n\u03c0 : \u03b9 \u2192 Type u_7\ninst\u271d\u00b2 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 MeasurableSpace (\u03c0 i)\ninst\u271d : \u2200 (i : \u03b9), BorelSpace (\u03c0 i)\n\u22a2 MeasurableSpace.pi \u2264 borel ((i : \u03b9) \u2192 \u03c0 i)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9\u271d : Sort y\ns t u : Set \u03b1\ninst\u271d\u00b9\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u2070 : BorelSpace \u03b1\ninst\u271d\u2079 : TopologicalSpace \u03b2\ninst\u271d\u2078 : MeasurableSpace \u03b2\ninst\u271d\u2077 : BorelSpace \u03b2\ninst\u271d\u2076 : TopologicalSpace \u03b3\ninst\u271d\u2075 : MeasurableSpace \u03b3\ninst\u271d\u2074 : BorelSpace \u03b3\ninst\u271d\u00b3 : MeasurableSpace \u03b4\n\u03b9 : Type u_6\n\u03c0 : \u03b9 \u2192 Type u_7\ninst\u271d\u00b2 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 MeasurableSpace (\u03c0 i)\ninst\u271d : \u2200 (i : \u03b9), BorelSpace (\u03c0 i)\nthis : inst\u271d\u00b9 = fun i => borel (\u03c0 i)\n\u22a2 MeasurableSpace.pi \u2264 borel ((i : \u03b9) \u2192 \u03c0 i)"}, {"tactic": "exact iSup_le fun i => comap_le_iff_le_map.2 <| (continuous_apply i).borel_measurable", "annotated_tactic": ["exact <a>iSup_le</a> fun i => <a>comap_le_iff_le_map</a>.2 <| (<a>continuous_apply</a> i).<a>borel_measurable</a>", [{"full_name": "iSup_le", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [875, 9], "def_end_pos": [875, 16]}, {"full_name": "MeasurableSpace.comap_le_iff_le_map", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [127, 9], "def_end_pos": [127, 28]}, {"full_name": "continuous_apply", "def_path": "Mathlib/Topology/Constructions.lean", "def_pos": [1208, 9], "def_end_pos": [1208, 25]}, {"full_name": "Continuous.borel_measurable", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [186, 9], "def_end_pos": [186, 36]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9\u271d : Sort y\ns t u : Set \u03b1\ninst\u271d\u00b9\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u2070 : BorelSpace \u03b1\ninst\u271d\u2079 : TopologicalSpace \u03b2\ninst\u271d\u2078 : MeasurableSpace \u03b2\ninst\u271d\u2077 : BorelSpace \u03b2\ninst\u271d\u2076 : TopologicalSpace \u03b3\ninst\u271d\u2075 : MeasurableSpace \u03b3\ninst\u271d\u2074 : BorelSpace \u03b3\ninst\u271d\u00b3 : MeasurableSpace \u03b4\n\u03b9 : Type u_6\n\u03c0 : \u03b9 \u2192 Type u_7\ninst\u271d\u00b2 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 MeasurableSpace (\u03c0 i)\ninst\u271d : \u2200 (i : \u03b9), BorelSpace (\u03c0 i)\nthis : inst\u271d\u00b9 = fun i => borel (\u03c0 i)\n\u22a2 MeasurableSpace.pi \u2264 borel ((i : \u03b9) \u2192 \u03c0 i)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "full_name": "Int.sub_one_lt_of_le", "start": [1261, 1], "end": [1262, 51], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Prod/TProd.lean", "full_name": "List.TProd.ext", "start": [116, 1], "end": [123, 47], "traced_tactics": [{"tactic": "apply Prod.ext", "annotated_tactic": ["apply <a>Prod.ext</a>", [{"full_name": "Prod.ext", "def_path": "Mathlib/Data/Prod/Basic.lean", "def_pos": [129, 9], "def_end_pos": [129, 12]}]], "state_before": "\u03b9 : Type u\n\u03b1 : \u03b9 \u2192 Type v\ni\u271d j : \u03b9\nl : List \u03b9\nf : (i : \u03b9) \u2192 \u03b1 i\ninst\u271d : DecidableEq \u03b9\ni : \u03b9\nis : List \u03b9\nhl : Nodup (i :: is)\nv w : TProd \u03b1 (i :: is)\nhvw : \u2200 (i_1 : \u03b9) (hi : i_1 \u2208 i :: is), TProd.elim v hi = TProd.elim w hi\n\u22a2 v = w", "state_after": "case h\u2081\n\u03b9 : Type u\n\u03b1 : \u03b9 \u2192 Type v\ni\u271d j : \u03b9\nl : List \u03b9\nf : (i : \u03b9) \u2192 \u03b1 i\ninst\u271d : DecidableEq \u03b9\ni : \u03b9\nis : List \u03b9\nhl : Nodup (i :: is)\nv w : TProd \u03b1 (i :: is)\nhvw : \u2200 (i_1 : \u03b9) (hi : i_1 \u2208 i :: is), TProd.elim v hi = TProd.elim w hi\n\u22a2 v.1 = w.1\n\ncase h\u2082\n\u03b9 : Type u\n\u03b1 : \u03b9 \u2192 Type v\ni\u271d j : \u03b9\nl : List \u03b9\nf : (i : \u03b9) \u2192 \u03b1 i\ninst\u271d : DecidableEq \u03b9\ni : \u03b9\nis : List \u03b9\nhl : Nodup (i :: is)\nv w : TProd \u03b1 (i :: is)\nhvw : \u2200 (i_1 : \u03b9) (hi : i_1 \u2208 i :: is), TProd.elim v hi = TProd.elim w hi\n\u22a2 v.2 = w.2"}, {"tactic": "rw [\u2190 elim_self v, hvw, elim_self]", "annotated_tactic": ["rw [\u2190 <a>elim_self</a> v, hvw, <a>elim_self</a>]", [{"full_name": "List.TProd.elim_self", "def_path": "Mathlib/Data/Prod/TProd.lean", "def_pos": [90, 9], "def_end_pos": [90, 18]}, {"full_name": "List.TProd.elim_self", "def_path": "Mathlib/Data/Prod/TProd.lean", "def_pos": [90, 9], "def_end_pos": [90, 18]}]], "state_before": "case h\u2081\n\u03b9 : Type u\n\u03b1 : \u03b9 \u2192 Type v\ni\u271d j : \u03b9\nl : List \u03b9\nf : (i : \u03b9) \u2192 \u03b1 i\ninst\u271d : DecidableEq \u03b9\ni : \u03b9\nis : List \u03b9\nhl : Nodup (i :: is)\nv w : TProd \u03b1 (i :: is)\nhvw : \u2200 (i_1 : \u03b9) (hi : i_1 \u2208 i :: is), TProd.elim v hi = TProd.elim w hi\n\u22a2 v.1 = w.1\n\ncase h\u2082\n\u03b9 : Type u\n\u03b1 : \u03b9 \u2192 Type v\ni\u271d j : \u03b9\nl : List \u03b9\nf : (i : \u03b9) \u2192 \u03b1 i\ninst\u271d : DecidableEq \u03b9\ni : \u03b9\nis : List \u03b9\nhl : Nodup (i :: is)\nv w : TProd \u03b1 (i :: is)\nhvw : \u2200 (i_1 : \u03b9) (hi : i_1 \u2208 i :: is), TProd.elim v hi = TProd.elim w hi\n\u22a2 v.2 = w.2", "state_after": "case h\u2082\n\u03b9 : Type u\n\u03b1 : \u03b9 \u2192 Type v\ni\u271d j : \u03b9\nl : List \u03b9\nf : (i : \u03b9) \u2192 \u03b1 i\ninst\u271d : DecidableEq \u03b9\ni : \u03b9\nis : List \u03b9\nhl : Nodup (i :: is)\nv w : TProd \u03b1 (i :: is)\nhvw : \u2200 (i_1 : \u03b9) (hi : i_1 \u2208 i :: is), TProd.elim v hi = TProd.elim w hi\n\u22a2 v.2 = w.2"}, {"tactic": "refine' ext (nodup_cons.mp hl).2 fun j hj => _", "annotated_tactic": ["refine' ext (nodup_cons.mp hl).2 fun j hj => _", []], "state_before": "case h\u2082\n\u03b9 : Type u\n\u03b1 : \u03b9 \u2192 Type v\ni\u271d j : \u03b9\nl : List \u03b9\nf : (i : \u03b9) \u2192 \u03b1 i\ninst\u271d : DecidableEq \u03b9\ni : \u03b9\nis : List \u03b9\nhl : Nodup (i :: is)\nv w : TProd \u03b1 (i :: is)\nhvw : \u2200 (i_1 : \u03b9) (hi : i_1 \u2208 i :: is), TProd.elim v hi = TProd.elim w hi\n\u22a2 v.2 = w.2", "state_after": "case h\u2082\n\u03b9 : Type u\n\u03b1 : \u03b9 \u2192 Type v\ni\u271d j\u271d : \u03b9\nl : List \u03b9\nf : (i : \u03b9) \u2192 \u03b1 i\ninst\u271d : DecidableEq \u03b9\ni : \u03b9\nis : List \u03b9\nhl : Nodup (i :: is)\nv w : TProd \u03b1 (i :: is)\nhvw : \u2200 (i_1 : \u03b9) (hi : i_1 \u2208 i :: is), TProd.elim v hi = TProd.elim w hi\nj : \u03b9\nhj : j \u2208 is\n\u22a2 TProd.elim v.2 hj = TProd.elim w.2 hj"}, {"tactic": "rw [\u2190 elim_of_mem hl, hvw, elim_of_mem hl]", "annotated_tactic": ["rw [\u2190 <a>elim_of_mem</a> hl, hvw, <a>elim_of_mem</a> hl]", [{"full_name": "List.TProd.elim_of_mem", "def_path": "Mathlib/Data/Prod/TProd.lean", "def_pos": [99, 9], "def_end_pos": [99, 20]}, {"full_name": "List.TProd.elim_of_mem", "def_path": "Mathlib/Data/Prod/TProd.lean", "def_pos": [99, 9], "def_end_pos": [99, 20]}]], "state_before": "case h\u2082\n\u03b9 : Type u\n\u03b1 : \u03b9 \u2192 Type v\ni\u271d j\u271d : \u03b9\nl : List \u03b9\nf : (i : \u03b9) \u2192 \u03b1 i\ninst\u271d : DecidableEq \u03b9\ni : \u03b9\nis : List \u03b9\nhl : Nodup (i :: is)\nv w : TProd \u03b1 (i :: is)\nhvw : \u2200 (i_1 : \u03b9) (hi : i_1 \u2208 i :: is), TProd.elim v hi = TProd.elim w hi\nj : \u03b9\nhj : j \u2208 is\n\u22a2 TProd.elim v.2 hj = TProd.elim w.2 hj", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Intervals/WithBotTop.lean", "full_name": "WithTop.image_coe_Iic", "start": [102, 1], "end": [104, 69], "traced_tactics": [{"tactic": "rw [\u2190 preimage_coe_Iic, image_preimage_eq_inter_range, range_coe,\n  inter_eq_self_of_subset_left (Iic_subset_Iio.2 <| coe_lt_top a)]", "annotated_tactic": ["rw [\u2190 <a>preimage_coe_Iic</a>, <a>image_preimage_eq_inter_range</a>, <a>range_coe</a>,\n    <a>inter_eq_self_of_subset_left</a> (<a>Iic_subset_Iio</a>.2 <| <a>coe_lt_top</a> a)]", [{"full_name": "WithTop.preimage_coe_Iic", "def_path": "Mathlib/Data/Set/Intervals/WithBotTop.lean", "def_pos": [54, 9], "def_end_pos": [54, 25]}, {"full_name": "Set.image_preimage_eq_inter_range", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [796, 9], "def_end_pos": [796, 38]}, {"full_name": "WithTop.range_coe", "def_path": "Mathlib/Data/Set/Intervals/WithBotTop.lean", "def_pos": [32, 9], "def_end_pos": [32, 18]}, {"full_name": "Set.inter_eq_self_of_subset_left", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [987, 9], "def_end_pos": [987, 37]}, {"full_name": "Set.Iic_subset_Iio", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [427, 9], "def_end_pos": [427, 23]}, {"full_name": "WithTop.coe_lt_top", "def_path": "Mathlib/Order/WithBot.lean", "def_pos": [1096, 9], "def_end_pos": [1096, 19]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : PartialOrder \u03b1\na b : \u03b1\n\u22a2 some '' Iic a = Iic \u2191a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/UniformIntegrable.lean", "full_name": "MeasureTheory.uniformIntegrable_subsingleton", "start": [780, 1], "end": [782, 44], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Setoid/Basic.lean", "full_name": "Quot.subsingleton_iff", "start": [458, 1], "end": [464, 45], "traced_tactics": [{"tactic": "simp only [_root_.subsingleton_iff, _root_.eq_top_iff, Pi.le_def, Pi.top_apply, forall_const]", "annotated_tactic": ["simp only [<a>_root_.subsingleton_iff</a>, <a>_root_.eq_top_iff</a>, <a>Pi.le_def</a>, <a>Pi.top_apply</a>, <a>forall_const</a>]", [{"full_name": "subsingleton_iff", "def_path": "Mathlib/Logic/Nontrivial/Defs.lean", "def_pos": [75, 9], "def_end_pos": [75, 25]}, {"full_name": "eq_top_iff", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [165, 9], "def_end_pos": [165, 19]}, {"full_name": "Pi.le_def", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [814, 9], "def_end_pos": [814, 18]}, {"full_name": "Pi.top_apply", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [649, 9], "def_end_pos": [649, 18]}, {"full_name": "forall_const", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [435, 17], "def_end_pos": [435, 29]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\n\u22a2 Subsingleton (Quot r) \u2194 EqvGen r = \u22a4", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\n\u22a2 (\u2200 (x y : Quot r), x = y) \u2194 \u2200 (i i_1 : \u03b1), \u22a4 \u2264 EqvGen r i i_1"}, {"tactic": "refine' (surjective_quot_mk _).forall.trans (forall_congr' fun a => _)", "annotated_tactic": ["refine' (<a>surjective_quot_mk</a> _).forall.trans (<a>forall_congr'</a> fun a => _)", [{"full_name": "surjective_quot_mk", "def_path": "Mathlib/Data/Quot.lean", "def_pos": [345, 9], "def_end_pos": [345, 27]}, {"full_name": "forall_congr'", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [376, 9], "def_end_pos": [376, 22]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\n\u22a2 (\u2200 (x y : Quot r), x = y) \u2194 \u2200 (i i_1 : \u03b1), \u22a4 \u2264 EqvGen r i i_1", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\n\u22a2 (\u2200 (y : Quot r), mk r a = y) \u2194 \u2200 (i : \u03b1), \u22a4 \u2264 EqvGen r a i"}, {"tactic": "refine' (surjective_quot_mk _).forall.trans (forall_congr' fun b => _)", "annotated_tactic": ["refine' (<a>surjective_quot_mk</a> _).forall.trans (<a>forall_congr'</a> fun b => _)", [{"full_name": "surjective_quot_mk", "def_path": "Mathlib/Data/Quot.lean", "def_pos": [345, 9], "def_end_pos": [345, 27]}, {"full_name": "forall_congr'", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [376, 9], "def_end_pos": [376, 22]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\n\u22a2 (\u2200 (y : Quot r), mk r a = y) \u2194 \u2200 (i : \u03b1), \u22a4 \u2264 EqvGen r a i", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : \u03b1\n\u22a2 mk r a = mk r b \u2194 \u22a4 \u2264 EqvGen r a b"}, {"tactic": "rw [Quot.eq]", "annotated_tactic": ["rw [<a>Quot.eq</a>]", [{"full_name": "Quot.eq", "def_path": "Mathlib/Data/Quot.lean", "def_pos": [293, 9], "def_end_pos": [293, 16]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : \u03b1\n\u22a2 mk r a = mk r b \u2194 \u22a4 \u2264 EqvGen r a b", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : \u03b1\n\u22a2 EqvGen r a b \u2194 \u22a4 \u2264 EqvGen r a b"}, {"tactic": "simp only [forall_const, le_Prop_eq, OrderTop.toTop, Pi.orderTop, Pi.top_apply,\n           Prop.top_eq_true, true_implies]", "annotated_tactic": ["simp only [<a>forall_const</a>, <a>le_Prop_eq</a>, OrderTop.toTop, <a>Pi.orderTop</a>, <a>Pi.top_apply</a>,\n             <a>Prop.top_eq_true</a>, <a>true_implies</a>]", [{"full_name": "forall_const", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [435, 17], "def_end_pos": [435, 29]}, {"full_name": "le_Prop_eq", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [1419, 9], "def_end_pos": [1419, 19]}, {"full_name": "Pi.orderTop", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [657, 10], "def_end_pos": [657, 18]}, {"full_name": "Pi.top_apply", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [649, 9], "def_end_pos": [649, 18]}, {"full_name": "Prop.top_eq_true", "def_path": "Mathlib/Order/PropInstances.lean", "def_pos": [47, 9], "def_end_pos": [47, 25]}, {"full_name": "true_implies", "def_path": "lake-packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [99, 17], "def_end_pos": [99, 29]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : \u03b1\n\u22a2 EqvGen r a b \u2194 \u22a4 \u2264 EqvGen r a b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/AEEqOfIntegral.lean", "full_name": "MeasureTheory.AEMeasurable.ae_eq_of_forall_set_lintegral_eq", "start": [606, 1], "end": [627, 53], "traced_tactics": [{"tactic": "refine'\n  ENNReal.eventuallyEq_of_toReal_eventuallyEq (ae_lt_top' hf hfi).ne_of_lt\n    (ae_lt_top' hg hgi).ne_of_lt\n    (Integrable.ae_eq_of_forall_set_integral_eq _ _\n      (integrable_toReal_of_lintegral_ne_top hf hfi)\n      (integrable_toReal_of_lintegral_ne_top hg hgi) fun s hs hs' => _)", "annotated_tactic": ["refine'\n    <a>ENNReal.eventuallyEq_of_toReal_eventuallyEq</a> (<a>ae_lt_top'</a> hf hfi).<a>ne_of_lt</a>\n      (<a>ae_lt_top'</a> hg hgi).<a>ne_of_lt</a>\n      (<a>Integrable.ae_eq_of_forall_set_integral_eq</a> _ _\n        (<a>integrable_toReal_of_lintegral_ne_top</a> hf hfi)\n        (<a>integrable_toReal_of_lintegral_ne_top</a> hg hgi) fun s hs hs' => _)", [{"full_name": "ENNReal.eventuallyEq_of_toReal_eventuallyEq", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [120, 9], "def_end_pos": [120, 44]}, {"full_name": "MeasureTheory.ae_lt_top'", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [1535, 9], "def_end_pos": [1535, 19]}, {"full_name": "Filter.Eventually.ne_of_lt", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1718, 9], "def_end_pos": [1718, 28]}, {"full_name": "MeasureTheory.ae_lt_top'", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [1535, 9], "def_end_pos": [1535, 19]}, {"full_name": "Filter.Eventually.ne_of_lt", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1718, 9], "def_end_pos": [1718, 28]}, {"full_name": "MeasureTheory.Integrable.ae_eq_of_forall_set_integral_eq", "def_path": "Mathlib/MeasureTheory/Function/AEEqOfIntegral.lean", "def_pos": [533, 9], "def_end_pos": [533, 51]}, {"full_name": "MeasureTheory.integrable_toReal_of_lintegral_ne_top", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [1044, 9], "def_end_pos": [1044, 46]}, {"full_name": "MeasureTheory.integrable_toReal_of_lintegral_ne_top", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [1044, 9], "def_end_pos": [1044, 46]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhfi : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\nhgi : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\nhfg : \u2200 \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc = \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\n\u22a2 f =\u1d50[\u03bc] g", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhfi : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\nhgi : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\nhfg : \u2200 \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc = \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nhs' : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u222b (x : \u03b1) in s, ENNReal.toReal (f x) \u2202\u03bc = \u222b (x : \u03b1) in s, ENNReal.toReal (g x) \u2202\u03bc"}, {"tactic": "rw [integral_eq_lintegral_of_nonneg_ae, integral_eq_lintegral_of_nonneg_ae]", "annotated_tactic": ["rw [<a>integral_eq_lintegral_of_nonneg_ae</a>, <a>integral_eq_lintegral_of_nonneg_ae</a>]", [{"full_name": "MeasureTheory.integral_eq_lintegral_of_nonneg_ae", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1158, 9], "def_end_pos": [1158, 43]}, {"full_name": "MeasureTheory.integral_eq_lintegral_of_nonneg_ae", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1158, 9], "def_end_pos": [1158, 43]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhfi : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\nhgi : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\nhfg : \u2200 \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc = \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nhs' : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u222b (x : \u03b1) in s, ENNReal.toReal (f x) \u2202\u03bc = \u222b (x : \u03b1) in s, ENNReal.toReal (g x) \u2202\u03bc", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhfi : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\nhgi : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\nhfg : \u2200 \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc = \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nhs' : \u2191\u2191\u03bc s < \u22a4\n\u22a2 ENNReal.toReal (\u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (ENNReal.toReal (f a)) \u2202\u03bc) =\n    ENNReal.toReal (\u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc)\n\ncase hf\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhfi : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\nhgi : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\nhfg : \u2200 \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc = \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nhs' : \u2191\u2191\u03bc s < \u22a4\n\u22a2 0 \u2264\u1d50[Measure.restrict \u03bc s] fun x => ENNReal.toReal (g x)\n\ncase hfm\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhfi : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\nhgi : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\nhfg : \u2200 \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc = \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nhs' : \u2191\u2191\u03bc s < \u22a4\n\u22a2 AEStronglyMeasurable (fun x => ENNReal.toReal (g x)) (Measure.restrict \u03bc s)\n\ncase hf\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhfi : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\nhgi : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\nhfg : \u2200 \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc = \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nhs' : \u2191\u2191\u03bc s < \u22a4\n\u22a2 0 \u2264\u1d50[Measure.restrict \u03bc s] fun x => ENNReal.toReal (f x)\n\ncase hfm\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhfi : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\nhgi : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\nhfg : \u2200 \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc = \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nhs' : \u2191\u2191\u03bc s < \u22a4\n\u22a2 AEStronglyMeasurable (fun x => ENNReal.toReal (f x)) (Measure.restrict \u03bc s)"}, {"tactic": "exacts [ae_of_all _ fun x => ENNReal.toReal_nonneg,\n  hg.ennreal_toReal.restrict.aestronglyMeasurable, ae_of_all _ fun x => ENNReal.toReal_nonneg,\n  hf.ennreal_toReal.restrict.aestronglyMeasurable]", "annotated_tactic": ["exacts [<a>ae_of_all</a> _ fun x => <a>ENNReal.toReal_nonneg</a>,\n    hg.ennreal_toReal.restrict.aestronglyMeasurable, <a>ae_of_all</a> _ fun x => <a>ENNReal.toReal_nonneg</a>,\n    hf.ennreal_toReal.restrict.aestronglyMeasurable]", [{"full_name": "MeasureTheory.ae_of_all", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [407, 9], "def_end_pos": [407, 18]}, {"full_name": "ENNReal.toReal_nonneg", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [221, 17], "def_end_pos": [221, 30]}, {"full_name": "MeasureTheory.ae_of_all", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [407, 9], "def_end_pos": [407, 18]}, {"full_name": "ENNReal.toReal_nonneg", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [221, 17], "def_end_pos": [221, 30]}]], "state_before": "case hf\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhfi : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\nhgi : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\nhfg : \u2200 \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc = \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nhs' : \u2191\u2191\u03bc s < \u22a4\n\u22a2 0 \u2264\u1d50[Measure.restrict \u03bc s] fun x => ENNReal.toReal (g x)\n\ncase hfm\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhfi : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\nhgi : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\nhfg : \u2200 \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc = \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nhs' : \u2191\u2191\u03bc s < \u22a4\n\u22a2 AEStronglyMeasurable (fun x => ENNReal.toReal (g x)) (Measure.restrict \u03bc s)\n\ncase hf\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhfi : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\nhgi : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\nhfg : \u2200 \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc = \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nhs' : \u2191\u2191\u03bc s < \u22a4\n\u22a2 0 \u2264\u1d50[Measure.restrict \u03bc s] fun x => ENNReal.toReal (f x)\n\ncase hfm\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhfi : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\nhgi : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\nhfg : \u2200 \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc = \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nhs' : \u2191\u2191\u03bc s < \u22a4\n\u22a2 AEStronglyMeasurable (fun x => ENNReal.toReal (f x)) (Measure.restrict \u03bc s)", "state_after": "no goals"}, {"tactic": "congr 1", "annotated_tactic": ["congr 1", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhfi : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\nhgi : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\nhfg : \u2200 \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc = \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nhs' : \u2191\u2191\u03bc s < \u22a4\n\u22a2 ENNReal.toReal (\u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (ENNReal.toReal (f a)) \u2202\u03bc) =\n    ENNReal.toReal (\u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc)", "state_after": "case e_a\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhfi : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\nhgi : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\nhfg : \u2200 \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc = \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nhs' : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (ENNReal.toReal (f a)) \u2202\u03bc = \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc"}, {"tactic": "rw [lintegral_congr_ae (ofReal_toReal_ae_eq _), lintegral_congr_ae (ofReal_toReal_ae_eq _)]", "annotated_tactic": ["rw [<a>lintegral_congr_ae</a> (<a>ofReal_toReal_ae_eq</a> _), <a>lintegral_congr_ae</a> (<a>ofReal_toReal_ae_eq</a> _)]", [{"full_name": "MeasureTheory.lintegral_congr_ae", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [304, 9], "def_end_pos": [304, 27]}, {"full_name": "MeasureTheory.ofReal_toReal_ae_eq", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [882, 9], "def_end_pos": [882, 28]}, {"full_name": "MeasureTheory.lintegral_congr_ae", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [304, 9], "def_end_pos": [304, 27]}, {"full_name": "MeasureTheory.ofReal_toReal_ae_eq", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [882, 9], "def_end_pos": [882, 28]}]], "state_before": "case e_a\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhfi : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\nhgi : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\nhfg : \u2200 \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc = \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nhs' : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (ENNReal.toReal (f a)) \u2202\u03bc = \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc", "state_after": "case e_a\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhfi : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\nhgi : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\nhfg : \u2200 \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc = \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nhs' : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u222b\u207b (a : \u03b1) in s, f a \u2202\u03bc = \u222b\u207b (a : \u03b1) in s, g a \u2202\u03bc\n\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhfi : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\nhgi : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\nhfg : \u2200 \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc = \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nhs' : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc s, g x < \u22a4\n\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhfi : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\nhgi : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\nhfg : \u2200 \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc = \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nhs' : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc s, f x < \u22a4"}, {"tactic": "exact hfg hs hs'", "annotated_tactic": ["exact hfg hs hs'", []], "state_before": "case e_a\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhfi : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\nhgi : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\nhfg : \u2200 \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc = \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nhs' : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u222b\u207b (a : \u03b1) in s, f a \u2202\u03bc = \u222b\u207b (a : \u03b1) in s, g a \u2202\u03bc", "state_after": "no goals"}, {"tactic": "refine' ae_lt_top' hg.restrict (ne_of_lt (lt_of_le_of_lt _ hgi.lt_top))", "annotated_tactic": ["refine' <a>ae_lt_top'</a> hg.restrict (<a>ne_of_lt</a> (<a>lt_of_le_of_lt</a> _ hgi.lt_top))", [{"full_name": "MeasureTheory.ae_lt_top'", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [1535, 9], "def_end_pos": [1535, 19]}, {"full_name": "ne_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [101, 9], "def_end_pos": [101, 17]}, {"full_name": "lt_of_le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [122, 9], "def_end_pos": [122, 23]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhfi : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\nhgi : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\nhfg : \u2200 \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc = \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nhs' : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc s, g x < \u22a4", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhfi : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\nhgi : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\nhfg : \u2200 \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc = \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nhs' : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), g x \u2202\u03bc"}, {"tactic": "exact @set_lintegral_univ \u03b1 _ \u03bc g \u25b8 lintegral_mono_set (Set.subset_univ _)", "annotated_tactic": ["exact @<a>set_lintegral_univ</a> \u03b1 _ \u03bc g \u25b8 <a>lintegral_mono_set</a> (<a>Set.subset_univ</a> _)", [{"full_name": "MeasureTheory.set_lintegral_univ", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [645, 9], "def_end_pos": [645, 27]}, {"full_name": "MeasureTheory.lintegral_mono_set", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [121, 9], "def_end_pos": [121, 27]}, {"full_name": "Set.subset_univ", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [691, 9], "def_end_pos": [691, 20]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhfi : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\nhgi : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\nhfg : \u2200 \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc = \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nhs' : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), g x \u2202\u03bc", "state_after": "no goals"}, {"tactic": "refine' ae_lt_top' hf.restrict (ne_of_lt (lt_of_le_of_lt _ hfi.lt_top))", "annotated_tactic": ["refine' <a>ae_lt_top'</a> hf.restrict (<a>ne_of_lt</a> (<a>lt_of_le_of_lt</a> _ hfi.lt_top))", [{"full_name": "MeasureTheory.ae_lt_top'", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [1535, 9], "def_end_pos": [1535, 19]}, {"full_name": "ne_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [101, 9], "def_end_pos": [101, 17]}, {"full_name": "lt_of_le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [122, 9], "def_end_pos": [122, 23]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhfi : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\nhgi : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\nhfg : \u2200 \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc = \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nhs' : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc s, f x < \u22a4", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhfi : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\nhgi : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\nhfg : \u2200 \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc = \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nhs' : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), f x \u2202\u03bc"}, {"tactic": "exact @set_lintegral_univ \u03b1 _ \u03bc f \u25b8 lintegral_mono_set (Set.subset_univ _)", "annotated_tactic": ["exact @<a>set_lintegral_univ</a> \u03b1 _ \u03bc f \u25b8 <a>lintegral_mono_set</a> (<a>Set.subset_univ</a> _)", [{"full_name": "MeasureTheory.set_lintegral_univ", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [645, 9], "def_end_pos": [645, 27]}, {"full_name": "MeasureTheory.lintegral_mono_set", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [121, 9], "def_end_pos": [121, 27]}, {"full_name": "Set.subset_univ", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [691, 9], "def_end_pos": [691, 20]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhfi : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\nhgi : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\nhfg : \u2200 \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc = \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nhs' : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), f x \u2202\u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Constructions/Prod/Integral.lean", "full_name": "MeasureTheory.integral_integral_add", "start": [389, 1], "end": [393, 61], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "full_name": "List.ne_nil_of_drop_ne_nil", "start": [805, 1], "end": [806, 29], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/List/Init/Lemmas.lean", "full_name": "List.lookup_cons_self", "start": [325, 9], "end": [326, 21], "traced_tactics": [{"tactic": "simp [lookup_cons]", "annotated_tactic": ["simp [<a>lookup_cons</a>]", [{"full_name": "List.lookup_cons", "def_path": "lake-packages/std/Std/Data/List/Init/Lemmas.lean", "def_pos": [322, 9], "def_end_pos": [322, 20]}]], "state_before": "\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\nb : \u03b2\u271d\nes : List (\u03b1 \u00d7 \u03b2\u271d)\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : LawfulBEq \u03b1\nk : \u03b1\n\u22a2 lookup k ((k, b) :: es) = some b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Rat/Basic.lean", "full_name": "Rat.sub.aux", "start": [236, 1], "end": [242, 13], "traced_tactics": [{"tactic": "have := add.aux a (-b) hg had hbd", "annotated_tactic": ["have := <a>add.aux</a> a (-b) hg had hbd", [{"full_name": "Rat.add.aux", "def_path": "lake-packages/std/Std/Data/Rat/Basic.lean", "def_pos": [181, 9], "def_end_pos": [181, 16]}]], "state_before": "a b : Rat\ng ad bd : Nat\nhg : g = Nat.gcd a.den b.den\nhad : ad = a.den / g\nhbd : bd = b.den / g\n\u22a2 let den := ad * b.den;\n  let num := a.num * \u2191bd - b.num * \u2191ad;\n  Nat.gcd (Int.natAbs num) g = Nat.gcd (Int.natAbs num) den", "state_after": "a b : Rat\ng ad bd : Nat\nhg : g = Nat.gcd a.den b.den\nhad : ad = a.den / g\nhbd : bd = b.den / g\nthis :\n  let den := ad * (-b).den;\n  let num := a.num * \u2191bd + (-b).num * \u2191ad;\n  Nat.gcd (Int.natAbs num) g = Nat.gcd (Int.natAbs num) den\n\u22a2 let den := ad * b.den;\n  let num := a.num * \u2191bd - b.num * \u2191ad;\n  Nat.gcd (Int.natAbs num) g = Nat.gcd (Int.natAbs num) den"}, {"tactic": "simp only [show (-b).num = -b.num from rfl, Int.neg_mul] at this", "annotated_tactic": ["simp only [show (-b).<a>num</a> = -b.num from <a>rfl</a>, <a>Int.neg_mul</a>] at this", [{"full_name": "Rat.num", "def_path": "lake-packages/std/Std/Data/Rat/Basic.lean", "def_pos": [21, 3], "def_end_pos": [21, 6]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}, {"full_name": "Int.neg_mul", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [468, 33], "def_end_pos": [468, 40]}]], "state_before": "a b : Rat\ng ad bd : Nat\nhg : g = Nat.gcd a.den b.den\nhad : ad = a.den / g\nhbd : bd = b.den / g\nthis :\n  let den := ad * (-b).den;\n  let num := a.num * \u2191bd + (-b).num * \u2191ad;\n  Nat.gcd (Int.natAbs num) g = Nat.gcd (Int.natAbs num) den\n\u22a2 let den := ad * b.den;\n  let num := a.num * \u2191bd - b.num * \u2191ad;\n  Nat.gcd (Int.natAbs num) g = Nat.gcd (Int.natAbs num) den", "state_after": "a b : Rat\ng ad bd : Nat\nhg : g = Nat.gcd a.den b.den\nhad : ad = a.den / g\nhbd : bd = b.den / g\nthis :\n  Nat.gcd (Int.natAbs (a.num * \u2191bd + -(b.num * \u2191ad))) g =\n    Nat.gcd (Int.natAbs (a.num * \u2191bd + -(b.num * \u2191ad))) (ad * (-b).den)\n\u22a2 let den := ad * b.den;\n  let num := a.num * \u2191bd - b.num * \u2191ad;\n  Nat.gcd (Int.natAbs num) g = Nat.gcd (Int.natAbs num) den"}, {"tactic": "exact this", "annotated_tactic": ["exact this", []], "state_before": "a b : Rat\ng ad bd : Nat\nhg : g = Nat.gcd a.den b.den\nhad : ad = a.den / g\nhbd : bd = b.den / g\nthis :\n  Nat.gcd (Int.natAbs (a.num * \u2191bd + -(b.num * \u2191ad))) g =\n    Nat.gcd (Int.natAbs (a.num * \u2191bd + -(b.num * \u2191ad))) (ad * (-b).den)\n\u22a2 let den := ad * b.den;\n  let num := a.num * \u2191bd - b.num * \u2191ad;\n  Nat.gcd (Int.natAbs num) g = Nat.gcd (Int.natAbs num) den", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "full_name": "MeasureTheory.VectorMeasure.of_iUnion_nonpos", "start": [227, 1], "end": [230, 60], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Reduce.lean", "full_name": "ManyOneDegree.le_antisymm", "start": [443, 9], "end": [447, 65], "traced_tactics": [{"tactic": "induction d\u2081 using ManyOneDegree.ind_on", "annotated_tactic": ["induction d\u2081 using <a>ManyOneDegree.ind_on</a>", [{"full_name": "ManyOneDegree.ind_on", "def_path": "Mathlib/Computability/Reduce.lean", "def_pos": [375, 19], "def_end_pos": [375, 25]}]], "state_before": "\u03b1 : Type u\ninst\u271d\u2075 : Primcodable \u03b1\ninst\u271d\u2074 : Inhabited \u03b1\n\u03b2 : Type v\ninst\u271d\u00b3 : Primcodable \u03b2\ninst\u271d\u00b2 : Inhabited \u03b2\n\u03b3 : Type w\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Inhabited \u03b3\nd\u2081 d\u2082 : ManyOneDegree\n\u22a2 d\u2081 \u2264 d\u2082 \u2192 d\u2082 \u2264 d\u2081 \u2192 d\u2081 = d\u2082", "state_after": "case h\n\u03b1 : Type u\ninst\u271d\u2075 : Primcodable \u03b1\ninst\u271d\u2074 : Inhabited \u03b1\n\u03b2 : Type v\ninst\u271d\u00b3 : Primcodable \u03b2\ninst\u271d\u00b2 : Inhabited \u03b2\n\u03b3 : Type w\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Inhabited \u03b3\nd\u2082 : ManyOneDegree\np\u271d : Set \u2115\n\u22a2 of p\u271d \u2264 d\u2082 \u2192 d\u2082 \u2264 of p\u271d \u2192 of p\u271d = d\u2082"}, {"tactic": "induction d\u2082 using ManyOneDegree.ind_on", "annotated_tactic": ["induction d\u2082 using <a>ManyOneDegree.ind_on</a>", [{"full_name": "ManyOneDegree.ind_on", "def_path": "Mathlib/Computability/Reduce.lean", "def_pos": [375, 19], "def_end_pos": [375, 25]}]], "state_before": "case h\n\u03b1 : Type u\ninst\u271d\u2075 : Primcodable \u03b1\ninst\u271d\u2074 : Inhabited \u03b1\n\u03b2 : Type v\ninst\u271d\u00b3 : Primcodable \u03b2\ninst\u271d\u00b2 : Inhabited \u03b2\n\u03b3 : Type w\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Inhabited \u03b3\nd\u2082 : ManyOneDegree\np\u271d : Set \u2115\n\u22a2 of p\u271d \u2264 d\u2082 \u2192 d\u2082 \u2264 of p\u271d \u2192 of p\u271d = d\u2082", "state_after": "case h.h\n\u03b1 : Type u\ninst\u271d\u2075 : Primcodable \u03b1\ninst\u271d\u2074 : Inhabited \u03b1\n\u03b2 : Type v\ninst\u271d\u00b3 : Primcodable \u03b2\ninst\u271d\u00b2 : Inhabited \u03b2\n\u03b3 : Type w\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Inhabited \u03b3\np\u271d\u00b9 p\u271d : Set \u2115\n\u22a2 of p\u271d\u00b9 \u2264 of p\u271d \u2192 of p\u271d \u2264 of p\u271d\u00b9 \u2192 of p\u271d\u00b9 = of p\u271d"}, {"tactic": "intro hp hq", "annotated_tactic": ["intro hp hq", []], "state_before": "case h.h\n\u03b1 : Type u\ninst\u271d\u2075 : Primcodable \u03b1\ninst\u271d\u2074 : Inhabited \u03b1\n\u03b2 : Type v\ninst\u271d\u00b3 : Primcodable \u03b2\ninst\u271d\u00b2 : Inhabited \u03b2\n\u03b3 : Type w\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Inhabited \u03b3\np\u271d\u00b9 p\u271d : Set \u2115\n\u22a2 of p\u271d\u00b9 \u2264 of p\u271d \u2192 of p\u271d \u2264 of p\u271d\u00b9 \u2192 of p\u271d\u00b9 = of p\u271d", "state_after": "case h.h\n\u03b1 : Type u\ninst\u271d\u2075 : Primcodable \u03b1\ninst\u271d\u2074 : Inhabited \u03b1\n\u03b2 : Type v\ninst\u271d\u00b3 : Primcodable \u03b2\ninst\u271d\u00b2 : Inhabited \u03b2\n\u03b3 : Type w\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Inhabited \u03b3\np\u271d\u00b9 p\u271d : Set \u2115\nhp : of p\u271d\u00b9 \u2264 of p\u271d\nhq : of p\u271d \u2264 of p\u271d\u00b9\n\u22a2 of p\u271d\u00b9 = of p\u271d"}, {"tactic": "simp_all only [ManyOneEquiv, of_le_of, of_eq_of, true_and_iff]", "annotated_tactic": ["simp_all only [<a>ManyOneEquiv</a>, <a>of_le_of</a>, <a>of_eq_of</a>, <a>true_and_iff</a>]", [{"full_name": "ManyOneEquiv", "def_path": "Mathlib/Computability/Reduce.lean", "def_pos": [150, 5], "def_end_pos": [150, 17]}, {"full_name": "ManyOneDegree.of_le_of", "def_path": "Mathlib/Computability/Reduce.lean", "def_pos": [436, 9], "def_end_pos": [436, 17]}, {"full_name": "ManyOneDegree.of_eq_of", "def_path": "Mathlib/Computability/Reduce.lean", "def_pos": [416, 9], "def_end_pos": [416, 17]}, {"full_name": "true_and_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [147, 9], "def_end_pos": [147, 21]}]], "state_before": "case h.h\n\u03b1 : Type u\ninst\u271d\u2075 : Primcodable \u03b1\ninst\u271d\u2074 : Inhabited \u03b1\n\u03b2 : Type v\ninst\u271d\u00b3 : Primcodable \u03b2\ninst\u271d\u00b2 : Inhabited \u03b2\n\u03b3 : Type w\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Inhabited \u03b3\np\u271d\u00b9 p\u271d : Set \u2115\nhp : of p\u271d\u00b9 \u2264 of p\u271d\nhq : of p\u271d \u2264 of p\u271d\u00b9\n\u22a2 of p\u271d\u00b9 = of p\u271d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/LocallyFinite.lean", "full_name": "Finset.uIcc_subset_uIcc_iff_le", "start": [1060, 1], "end": [1062, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Fold.lean", "full_name": "Finset.fold_const", "start": [84, 1], "end": [92, 16], "traced_tactics": [{"tactic": "induction' s using Finset.induction_on with x s hx IH generalizing hd", "annotated_tactic": ["induction' s using <a>Finset.induction_on</a> with x s hx IH generalizing hd", [{"full_name": "Finset.induction_on", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1251, 19], "def_end_pos": [1251, 31]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nop : \u03b2 \u2192 \u03b2 \u2192 \u03b2\nhc : IsCommutative \u03b2 op\nha : IsAssociative \u03b2 op\nf : \u03b1 \u2192 \u03b2\nb : \u03b2\ns : Finset \u03b1\na : \u03b1\nhd : Decidable (s = \u2205)\nc : \u03b2\nh : op c (op b c) = op b c\n\u22a2 fold op b (fun x => c) s = if s = \u2205 then b else op b c", "state_after": "case empty\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nop : \u03b2 \u2192 \u03b2 \u2192 \u03b2\nhc : IsCommutative \u03b2 op\nha : IsAssociative \u03b2 op\nf : \u03b1 \u2192 \u03b2\nb : \u03b2\ns : Finset \u03b1\na : \u03b1\nhd\u271d : Decidable (s = \u2205)\nc : \u03b2\nh : op c (op b c) = op b c\nhd : Decidable (\u2205 = \u2205)\n\u22a2 fold op b (fun x => c) \u2205 = if \u2205 = \u2205 then b else op b c\n\ncase insert\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nop : \u03b2 \u2192 \u03b2 \u2192 \u03b2\nhc : IsCommutative \u03b2 op\nha : IsAssociative \u03b2 op\nf : \u03b1 \u2192 \u03b2\nb : \u03b2\ns\u271d : Finset \u03b1\na : \u03b1\nhd\u271d : Decidable (s\u271d = \u2205)\nc : \u03b2\nh : op c (op b c) = op b c\nx : \u03b1\ns : Finset \u03b1\nhx : \u00acx \u2208 s\nIH : \u2200 [hd : Decidable (s = \u2205)], fold op b (fun x => c) s = if s = \u2205 then b else op b c\nhd : Decidable (insert x s = \u2205)\n\u22a2 fold op b (fun x => c) (insert x s) = if insert x s = \u2205 then b else op b c"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case empty\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nop : \u03b2 \u2192 \u03b2 \u2192 \u03b2\nhc : IsCommutative \u03b2 op\nha : IsAssociative \u03b2 op\nf : \u03b1 \u2192 \u03b2\nb : \u03b2\ns : Finset \u03b1\na : \u03b1\nhd\u271d : Decidable (s = \u2205)\nc : \u03b2\nh : op c (op b c) = op b c\nhd : Decidable (\u2205 = \u2205)\n\u22a2 fold op b (fun x => c) \u2205 = if \u2205 = \u2205 then b else op b c", "state_after": "no goals"}, {"tactic": "simp only [Finset.fold_insert hx, IH, if_false, Finset.insert_ne_empty]", "annotated_tactic": ["simp only [<a>Finset.fold_insert</a> hx, IH, <a>if_false</a>, <a>Finset.insert_ne_empty</a>]", [{"full_name": "Finset.fold_insert", "def_path": "Mathlib/Data/Finset/Fold.lean", "def_pos": [52, 9], "def_end_pos": [52, 20]}, {"full_name": "if_false", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [729, 17], "def_end_pos": [729, 25]}, {"full_name": "Finset.insert_ne_empty", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1174, 9], "def_end_pos": [1174, 24]}]], "state_before": "case insert\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nop : \u03b2 \u2192 \u03b2 \u2192 \u03b2\nhc : IsCommutative \u03b2 op\nha : IsAssociative \u03b2 op\nf : \u03b1 \u2192 \u03b2\nb : \u03b2\ns\u271d : Finset \u03b1\na : \u03b1\nhd\u271d : Decidable (s\u271d = \u2205)\nc : \u03b2\nh : op c (op b c) = op b c\nx : \u03b1\ns : Finset \u03b1\nhx : \u00acx \u2208 s\nIH : \u2200 [hd : Decidable (s = \u2205)], fold op b (fun x => c) s = if s = \u2205 then b else op b c\nhd : Decidable (insert x s = \u2205)\n\u22a2 fold op b (fun x => c) (insert x s) = if insert x s = \u2205 then b else op b c", "state_after": "case insert\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nop : \u03b2 \u2192 \u03b2 \u2192 \u03b2\nhc : IsCommutative \u03b2 op\nha : IsAssociative \u03b2 op\nf : \u03b1 \u2192 \u03b2\nb : \u03b2\ns\u271d : Finset \u03b1\na : \u03b1\nhd\u271d : Decidable (s\u271d = \u2205)\nc : \u03b2\nh : op c (op b c) = op b c\nx : \u03b1\ns : Finset \u03b1\nhx : \u00acx \u2208 s\nIH : \u2200 [hd : Decidable (s = \u2205)], fold op b (fun x => c) s = if s = \u2205 then b else op b c\nhd : Decidable (insert x s = \u2205)\n\u22a2 op c (if s = \u2205 then b else op b c) = op b c"}, {"tactic": "split_ifs", "annotated_tactic": ["split_ifs", []], "state_before": "case insert\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nop : \u03b2 \u2192 \u03b2 \u2192 \u03b2\nhc : IsCommutative \u03b2 op\nha : IsAssociative \u03b2 op\nf : \u03b1 \u2192 \u03b2\nb : \u03b2\ns\u271d : Finset \u03b1\na : \u03b1\nhd\u271d : Decidable (s\u271d = \u2205)\nc : \u03b2\nh : op c (op b c) = op b c\nx : \u03b1\ns : Finset \u03b1\nhx : \u00acx \u2208 s\nIH : \u2200 [hd : Decidable (s = \u2205)], fold op b (fun x => c) s = if s = \u2205 then b else op b c\nhd : Decidable (insert x s = \u2205)\n\u22a2 op c (if s = \u2205 then b else op b c) = op b c", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nop : \u03b2 \u2192 \u03b2 \u2192 \u03b2\nhc : IsCommutative \u03b2 op\nha : IsAssociative \u03b2 op\nf : \u03b1 \u2192 \u03b2\nb : \u03b2\ns\u271d : Finset \u03b1\na : \u03b1\nhd\u271d : Decidable (s\u271d = \u2205)\nc : \u03b2\nh : op c (op b c) = op b c\nx : \u03b1\ns : Finset \u03b1\nhx : \u00acx \u2208 s\nIH : \u2200 [hd : Decidable (s = \u2205)], fold op b (fun x => c) s = if s = \u2205 then b else op b c\nhd : Decidable (insert x s = \u2205)\nh\u271d : s = \u2205\n\u22a2 op c b = op b c\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nop : \u03b2 \u2192 \u03b2 \u2192 \u03b2\nhc : IsCommutative \u03b2 op\nha : IsAssociative \u03b2 op\nf : \u03b1 \u2192 \u03b2\nb : \u03b2\ns\u271d : Finset \u03b1\na : \u03b1\nhd\u271d : Decidable (s\u271d = \u2205)\nc : \u03b2\nh : op c (op b c) = op b c\nx : \u03b1\ns : Finset \u03b1\nhx : \u00acx \u2208 s\nIH : \u2200 [hd : Decidable (s = \u2205)], fold op b (fun x => c) s = if s = \u2205 then b else op b c\nhd : Decidable (insert x s = \u2205)\nh\u271d : \u00acs = \u2205\n\u22a2 op c (op b c) = op b c"}, {"tactic": "rw [hc.comm]", "annotated_tactic": ["rw [hc.comm]", []], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nop : \u03b2 \u2192 \u03b2 \u2192 \u03b2\nhc : IsCommutative \u03b2 op\nha : IsAssociative \u03b2 op\nf : \u03b1 \u2192 \u03b2\nb : \u03b2\ns\u271d : Finset \u03b1\na : \u03b1\nhd\u271d : Decidable (s\u271d = \u2205)\nc : \u03b2\nh : op c (op b c) = op b c\nx : \u03b1\ns : Finset \u03b1\nhx : \u00acx \u2208 s\nIH : \u2200 [hd : Decidable (s = \u2205)], fold op b (fun x => c) s = if s = \u2205 then b else op b c\nhd : Decidable (insert x s = \u2205)\nh\u271d : s = \u2205\n\u22a2 op c b = op b c", "state_after": "no goals"}, {"tactic": "exact h", "annotated_tactic": ["exact h", []], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nop : \u03b2 \u2192 \u03b2 \u2192 \u03b2\nhc : IsCommutative \u03b2 op\nha : IsAssociative \u03b2 op\nf : \u03b1 \u2192 \u03b2\nb : \u03b2\ns\u271d : Finset \u03b1\na : \u03b1\nhd\u271d : Decidable (s\u271d = \u2205)\nc : \u03b2\nh : op c (op b c) = op b c\nx : \u03b1\ns : Finset \u03b1\nhx : \u00acx \u2208 s\nIH : \u2200 [hd : Decidable (s = \u2205)], fold op b (fun x => c) s = if s = \u2205 then b else op b c\nhd : Decidable (insert x s = \u2205)\nh\u271d : \u00acs = \u2205\n\u22a2 op c (op b c) = op b c", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/MeanInequalities.lean", "full_name": "ENNReal.lintegral_mul_le_one_of_lintegral_rpow_eq_one", "start": [61, 1], "end": [74, 43], "traced_tactics": [{"tactic": "simp only [div_eq_mul_inv]", "annotated_tactic": ["simp only [<a>div_eq_mul_inv</a>]", [{"full_name": "div_eq_mul_inv", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [977, 9], "def_end_pos": [977, 23]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_norm : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 1\nhg_norm : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc = 1\n\u22a2 \u222b\u207b (a : \u03b1), f a ^ p / ENNReal.ofReal p + g a ^ q / ENNReal.ofReal q \u2202\u03bc = 1", "state_after": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_norm : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 1\nhg_norm : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc = 1\n\u22a2 \u222b\u207b (a : \u03b1), f a ^ p * (ENNReal.ofReal p)\u207b\u00b9 + g a ^ q * (ENNReal.ofReal q)\u207b\u00b9 \u2202\u03bc = 1"}, {"tactic": "rw [lintegral_add_left']", "annotated_tactic": ["rw [<a>lintegral_add_left'</a>]", [{"full_name": "MeasureTheory.lintegral_add_left'", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [566, 9], "def_end_pos": [566, 28]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_norm : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 1\nhg_norm : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc = 1\n\u22a2 \u222b\u207b (a : \u03b1), f a ^ p * (ENNReal.ofReal p)\u207b\u00b9 + g a ^ q * (ENNReal.ofReal q)\u207b\u00b9 \u2202\u03bc = 1", "state_after": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_norm : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 1\nhg_norm : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc = 1\n\u22a2 \u222b\u207b (a : \u03b1), f a ^ p * (ENNReal.ofReal p)\u207b\u00b9 \u2202\u03bc + \u222b\u207b (a : \u03b1), g a ^ q * (ENNReal.ofReal q)\u207b\u00b9 \u2202\u03bc = 1\n\ncase hf\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_norm : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 1\nhg_norm : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc = 1\n\u22a2 AEMeasurable fun a => f a ^ p * (ENNReal.ofReal p)\u207b\u00b9"}, {"tactic": "rw [lintegral_mul_const'' _ (hf.pow_const p), lintegral_mul_const', hf_norm, hg_norm, \u2190\n  div_eq_mul_inv, \u2190 div_eq_mul_inv, hpq.inv_add_inv_conj_ennreal]", "annotated_tactic": ["rw [<a>lintegral_mul_const''</a> _ (hf.pow_const p), <a>lintegral_mul_const'</a>, hf_norm, hg_norm, \u2190\n          <a>div_eq_mul_inv</a>, \u2190 <a>div_eq_mul_inv</a>, hpq.inv_add_inv_conj_ennreal]", [{"full_name": "MeasureTheory.lintegral_mul_const''", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [729, 9], "def_end_pos": [729, 30]}, {"full_name": "MeasureTheory.lintegral_mul_const'", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [737, 9], "def_end_pos": [737, 29]}, {"full_name": "div_eq_mul_inv", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [977, 9], "def_end_pos": [977, 23]}, {"full_name": "div_eq_mul_inv", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [977, 9], "def_end_pos": [977, 23]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_norm : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 1\nhg_norm : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc = 1\n\u22a2 \u222b\u207b (a : \u03b1), f a ^ p * (ENNReal.ofReal p)\u207b\u00b9 \u2202\u03bc + \u222b\u207b (a : \u03b1), g a ^ q * (ENNReal.ofReal q)\u207b\u00b9 \u2202\u03bc = 1", "state_after": "case hr\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_norm : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 1\nhg_norm : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc = 1\n\u22a2 (ENNReal.ofReal q)\u207b\u00b9 \u2260 \u22a4"}, {"tactic": "simp [hpq.symm.pos]", "annotated_tactic": ["simp [hpq.symm.pos]", []], "state_before": "case hr\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_norm : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 1\nhg_norm : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc = 1\n\u22a2 (ENNReal.ofReal q)\u207b\u00b9 \u2260 \u22a4", "state_after": "no goals"}, {"tactic": "exact (hf.pow_const _).mul_const _", "annotated_tactic": ["exact (hf.pow_const _).<a>mul_const</a> _", [{"full_name": "AEMeasurable.mul_const", "def_path": "Mathlib/MeasureTheory/Group/Arithmetic.lean", "def_pos": [127, 9], "def_end_pos": [127, 31]}]], "state_before": "case hf\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_norm : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 1\nhg_norm : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc = 1\n\u22a2 AEMeasurable fun a => f a ^ p * (ENNReal.ofReal p)\u207b\u00b9", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Part.lean", "full_name": "Part.eq_get_iff_mem", "start": [241, 1], "end": [242, 35], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finmap.lean", "full_name": "Finmap.insert_singleton_eq", "start": [539, 1], "end": [540, 75], "traced_tactics": [{"tactic": "simp only [singleton, Finmap.insert_toFinmap, AList.insert_singleton_eq]", "annotated_tactic": ["simp only [<a>singleton</a>, <a>Finmap.insert_toFinmap</a>, <a>AList.insert_singleton_eq</a>]", [{"full_name": "Finmap.singleton", "def_path": "Mathlib/Data/Finmap.lean", "def_pos": [237, 5], "def_end_pos": [237, 14]}, {"full_name": "Finmap.insert_toFinmap", "def_path": "Mathlib/Data/Finmap.lean", "def_pos": [480, 9], "def_end_pos": [480, 24]}, {"full_name": "AList.insert_singleton_eq", "def_path": "Mathlib/Data/List/AList.lean", "def_pos": [338, 9], "def_end_pos": [338, 28]}]], "state_before": "\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nb b' : \u03b2 a\n\u22a2 insert a b (singleton a b') = singleton a b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "full_name": "MeasureTheory.set_lintegral_mono'", "start": [296, 1], "end": [298, 46], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/LocallyFinite.lean", "full_name": "Finset.image_add_left_Ioo", "start": [1152, 1], "end": [1153, 78], "traced_tactics": [{"tactic": "rw [\u2190 map_add_left_Ioo, map_eq_image, addLeftEmbedding, Embedding.coeFn_mk]", "annotated_tactic": ["rw [\u2190 <a>map_add_left_Ioo</a>, <a>map_eq_image</a>, <a>addLeftEmbedding</a>, <a>Embedding.coeFn_mk</a>]", [{"full_name": "Finset.map_add_left_Ioo", "def_path": "Mathlib/Data/Finset/LocallyFinite.lean", "def_pos": [1121, 9], "def_end_pos": [1121, 25]}, {"full_name": "Finset.map_eq_image", "def_path": "Mathlib/Data/Finset/Image.lean", "def_pos": [342, 9], "def_end_pos": [342, 21]}, {"full_name": "addLeftEmbedding", "def_path": "Mathlib/Algebra/Hom/Embedding.lean", "def_pos": [23, 3], "def_end_pos": [23, 14]}, {"full_name": "Function.Embedding.coeFn_mk", "def_path": "Mathlib/Logic/Embedding/Basic.lean", "def_pos": [115, 9], "def_end_pos": [115, 17]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : OrderedCancelAddCommMonoid \u03b1\ninst\u271d\u00b2 : ExistsAddOfLE \u03b1\ninst\u271d\u00b9 : LocallyFiniteOrder \u03b1\ninst\u271d : DecidableEq \u03b1\na b c : \u03b1\n\u22a2 image ((fun x x_1 => x + x_1) c) (Ioo a b) = Ioo (c + a) (c + b)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/ZMod/Basic.lean", "full_name": "ZMod.addOrderOf_one", "start": [104, 1], "end": [105, 53], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/PairingHeap.lean", "full_name": "Std.PairingHeapImp.Heap.noSibling_tail?", "start": [108, 1], "end": [112, 55], "traced_tactics": [{"tactic": "simp only [Heap.tail?]", "annotated_tactic": ["simp only [<a>Heap.tail?</a>]", [{"full_name": "Std.PairingHeapImp.Heap.tail?", "def_path": "lake-packages/std/Std/Data/PairingHeap.lean", "def_pos": [71, 15], "def_end_pos": [71, 25]}]], "state_before": "\u03b1 : Type u_1\nle : \u03b1 \u2192 \u03b1 \u2192 Bool\ns' s : Heap \u03b1\n\u22a2 tail? le s = some s' \u2192 NoSibling s'", "state_after": "\u03b1 : Type u_1\nle : \u03b1 \u2192 \u03b1 \u2192 Bool\ns' s : Heap \u03b1\n\u22a2 Option.map (fun x => x.snd) (deleteMin le s) = some s' \u2192 NoSibling s'"}, {"tactic": "intro eq", "annotated_tactic": ["intro eq", []], "state_before": "\u03b1 : Type u_1\nle : \u03b1 \u2192 \u03b1 \u2192 Bool\ns' s : Heap \u03b1\n\u22a2 Option.map (fun x => x.snd) (deleteMin le s) = some s' \u2192 NoSibling s'", "state_after": "\u03b1 : Type u_1\nle : \u03b1 \u2192 \u03b1 \u2192 Bool\ns' s : Heap \u03b1\neq : Option.map (fun x => x.snd) (deleteMin le s) = some s'\n\u22a2 NoSibling s'"}, {"tactic": "match eq\u2082 : s.deleteMin le, eq with\n| some (a, tl), rfl => exact noSibling_deleteMin eq\u2082", "annotated_tactic": ["match eq\u2082 : s.deleteMin le, eq with\n  | <a>some</a> (a, tl), <a>rfl</a> => exact <a>noSibling_deleteMin</a> eq\u2082", [{"full_name": "Option.some", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2143, 5], "def_end_pos": [2143, 9]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}, {"full_name": "Std.PairingHeapImp.Heap.noSibling_deleteMin", "def_path": "lake-packages/std/Std/Data/PairingHeap.lean", "def_pos": [104, 9], "def_end_pos": [104, 33]}]], "state_before": "\u03b1 : Type u_1\nle : \u03b1 \u2192 \u03b1 \u2192 Bool\ns' s : Heap \u03b1\neq : Option.map (fun x => x.snd) (deleteMin le s) = some s'\n\u22a2 NoSibling s'", "state_after": "no goals"}, {"tactic": "exact noSibling_deleteMin eq\u2082", "annotated_tactic": ["exact <a>noSibling_deleteMin</a> eq\u2082", [{"full_name": "Std.PairingHeapImp.Heap.noSibling_deleteMin", "def_path": "lake-packages/std/Std/Data/PairingHeap.lean", "def_pos": [104, 9], "def_end_pos": [104, 33]}]], "state_before": "\u03b1 : Type u_1\nle : \u03b1 \u2192 \u03b1 \u2192 Bool\ns' s : Heap \u03b1\neq : Option.map (fun x => x.snd) (deleteMin le s) = some s'\na : \u03b1\ntl : Heap \u03b1\neq\u2082 : deleteMin le s = some (a, tl)\n\u22a2 NoSibling ((fun x => x.snd) (a, tl))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Analysis/Topology.lean", "full_name": "Ctop.Realizer.is_basis", "start": [127, 11], "end": [129, 61], "traced_tactics": [{"tactic": "have := toTopsp_isTopologicalBasis F.F", "annotated_tactic": ["have := <a>toTopsp_isTopologicalBasis</a> F.F", [{"full_name": "Ctop.toTopsp_isTopologicalBasis", "def_path": "Mathlib/Data/Analysis/Topology.lean", "def_pos": [89, 9], "def_end_pos": [89, 35]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03c3 : Type u_3\n\u03c4 : Type u_4\nT : TopologicalSpace \u03b1\nF : Realizer \u03b1\n\u22a2 TopologicalSpace.IsTopologicalBasis (range F.F.f)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03c3 : Type u_3\n\u03c4 : Type u_4\nT : TopologicalSpace \u03b1\nF : Realizer \u03b1\nthis : TopologicalSpace.IsTopologicalBasis (range F.F.f)\n\u22a2 TopologicalSpace.IsTopologicalBasis (range F.F.f)"}, {"tactic": "rwa [F.eq] at this", "annotated_tactic": ["rwa [F.eq] at this", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03c3 : Type u_3\n\u03c4 : Type u_4\nT : TopologicalSpace \u03b1\nF : Realizer \u03b1\nthis : TopologicalSpace.IsTopologicalBasis (range F.F.f)\n\u22a2 TopologicalSpace.IsTopologicalBasis (range F.F.f)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/PFun.lean", "full_name": "PFun.mem_restrict", "start": [180, 1], "end": [181, 63], "traced_tactics": [{"tactic": "simp [restrict]", "annotated_tactic": ["simp [<a>restrict</a>]", [{"full_name": "PFun.restrict", "def_path": "Mathlib/Data/PFun.lean", "def_pos": [175, 5], "def_end_pos": [175, 13]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf : \u03b1 \u2192. \u03b2\ns : Set \u03b1\nh : s \u2286 Dom f\na : \u03b1\nb : \u03b2\n\u22a2 b \u2208 restrict f h a \u2194 a \u2208 s \u2227 b \u2208 f a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "full_name": "MeasureTheory.Lp.simpleFunc.coeFn_nonneg", "start": [833, 1], "end": [834, 68], "traced_tactics": [{"tactic": "rw [\u2190 Subtype.coe_le_coe, Lp.coeFn_nonneg, AddSubmonoid.coe_zero]", "annotated_tactic": ["rw [\u2190 <a>Subtype.coe_le_coe</a>, <a>Lp.coeFn_nonneg</a>, <a>AddSubmonoid.coe_zero</a>]", [{"full_name": "Subtype.coe_le_coe", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [1162, 9], "def_end_pos": [1162, 19]}, {"full_name": "MeasureTheory.Lp.coeFn_nonneg", "def_path": "Mathlib/MeasureTheory/Function/LpOrder.lean", "def_pos": [46, 9], "def_end_pos": [46, 21]}, {"full_name": "AddSubmonoid.coe_zero", "def_path": "Mathlib/GroupTheory/Submonoid/Operations.lean", "def_pos": [683, 3], "def_end_pos": [683, 14]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nf : { x // x \u2208 simpleFunc G p \u03bc }\n\u22a2 0 \u2264\u1d50[\u03bc] \u2191\u2191\u2191f \u2194 0 \u2264 f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Intervals/Infinite.lean", "full_name": "Set.Ioo.infinite", "start": [40, 1], "end": [41, 54], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Num/Lemmas.lean", "full_name": "ZNum.gcd_to_nat", "start": [1761, 1], "end": [1762, 45], "traced_tactics": [{"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "a b : ZNum\n\u22a2 Nat.gcd \u2191(abs a) \u2191(abs b) = Int.gcd \u2191a \u2191b", "state_after": "a b : ZNum\n\u22a2 Nat.gcd (natAbs \u2191a) (natAbs \u2191b) = Int.gcd \u2191a \u2191b"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "a b : ZNum\n\u22a2 Nat.gcd (natAbs \u2191a) (natAbs \u2191b) = Int.gcd \u2191a \u2191b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Rat/Lemmas.lean", "full_name": "Rat.mk_eq_divInt", "start": [124, 1], "end": [125, 21], "traced_tactics": [{"tactic": "simp [mk_eq_mkRat]", "annotated_tactic": ["simp [<a>mk_eq_mkRat</a>]", [{"full_name": "Rat.mk_eq_mkRat", "def_path": "lake-packages/std/Std/Data/Rat/Lemmas.lean", "def_pos": [99, 9], "def_end_pos": [99, 20]}]], "state_before": "num : Int\nden : Nat\nnz : den \u2260 0\nc : Nat.Coprime (Int.natAbs num) den\n\u22a2 mk' num den = num /. \u2191den", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "full_name": "MeasurableSpace.CountablyGenerated.sup", "start": [1866, 1], "end": [1870, 74], "traced_tactics": [{"tactic": "rcases h\u2081 with \u27e8\u27e8b\u2081, hb\u2081c, rfl\u27e9\u27e9", "annotated_tactic": ["rcases h\u2081 with \u27e8\u27e8b\u2081, hb\u2081c, rfl\u27e9\u27e9", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u\u03b9\ns t u : Set \u03b1\nm\u2081 m\u2082 : MeasurableSpace \u03b2\nh\u2081 : CountablyGenerated \u03b2\nh\u2082 : CountablyGenerated \u03b2\n\u22a2 CountablyGenerated \u03b2", "state_after": "case mk.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u\u03b9\ns t u : Set \u03b1\nm\u2082 : MeasurableSpace \u03b2\nh\u2082 : CountablyGenerated \u03b2\nb\u2081 : Set (Set \u03b2)\nhb\u2081c : Set.Countable b\u2081\n\u22a2 CountablyGenerated \u03b2"}, {"tactic": "rcases h\u2082 with \u27e8\u27e8b\u2082, hb\u2082c, rfl\u27e9\u27e9", "annotated_tactic": ["rcases h\u2082 with \u27e8\u27e8b\u2082, hb\u2082c, rfl\u27e9\u27e9", []], "state_before": "case mk.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u\u03b9\ns t u : Set \u03b1\nm\u2082 : MeasurableSpace \u03b2\nh\u2082 : CountablyGenerated \u03b2\nb\u2081 : Set (Set \u03b2)\nhb\u2081c : Set.Countable b\u2081\n\u22a2 CountablyGenerated \u03b2", "state_after": "case mk.intro.intro.mk.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u\u03b9\ns t u : Set \u03b1\nb\u2081 : Set (Set \u03b2)\nhb\u2081c : Set.Countable b\u2081\nb\u2082 : Set (Set \u03b2)\nhb\u2082c : Set.Countable b\u2082\n\u22a2 CountablyGenerated \u03b2"}, {"tactic": "exact @mk _ (_ \u2294 _) \u27e8_, hb\u2081c.union hb\u2082c, generateFrom_sup_generateFrom\u27e9", "annotated_tactic": ["exact @<a>mk</a> _ (_ \u2294 _) \u27e8_, hb\u2081c.union hb\u2082c, <a>generateFrom_sup_generateFrom</a>\u27e9", [{"full_name": "MeasurableSpace.CountablyGenerated.mk", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [1853, 26], "def_end_pos": [1853, 49]}, {"full_name": "MeasurableSpace.generateFrom_sup_generateFrom", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [442, 9], "def_end_pos": [442, 38]}]], "state_before": "case mk.intro.intro.mk.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u\u03b9\ns t u : Set \u03b1\nb\u2081 : Set (Set \u03b2)\nhb\u2081c : Set.Countable b\u2081\nb\u2082 : Set (Set \u03b2)\nhb\u2082c : Set.Countable b\u2082\n\u22a2 CountablyGenerated \u03b2", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Lattice.lean", "full_name": "List.foldr_inf_eq_inf_toFinset", "start": [448, 1], "end": [452, 6], "traced_tactics": [{"tactic": "rw [\u2190 coe_fold_r, \u2190 Multiset.fold_dedup_idem, inf_def, \u2190 List.toFinset_coe, toFinset_val,\n  Multiset.map_id]", "annotated_tactic": ["rw [\u2190 <a>coe_fold_r</a>, \u2190 <a>Multiset.fold_dedup_idem</a>, <a>inf_def</a>, \u2190 <a>List.toFinset_coe</a>, <a>toFinset_val</a>,\n    <a>Multiset.map_id</a>]", [{"full_name": "Multiset.coe_fold_r", "def_path": "Mathlib/Data/Multiset/Fold.lean", "def_pos": [41, 9], "def_end_pos": [41, 19]}, {"full_name": "Multiset.fold_dedup_idem", "def_path": "Mathlib/Data/Multiset/Fold.lean", "def_pos": [115, 9], "def_end_pos": [115, 24]}, {"full_name": "Finset.inf_def", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [313, 9], "def_end_pos": [313, 16]}, {"full_name": "List.toFinset_coe", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3310, 9], "def_end_pos": [3310, 21]}, {"full_name": "Multiset.toFinset_val", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3186, 9], "def_end_pos": [3186, 21]}, {"full_name": "Multiset.map_id", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [1287, 9], "def_end_pos": [1287, 15]}]], "state_before": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Type u_5\n\u03ba : Type u_6\ninst\u271d\u00b2 : SemilatticeInf \u03b1\ninst\u271d\u00b9 : OrderTop \u03b1\ns s\u2081 s\u2082 : Finset \u03b2\nf g : \u03b2 \u2192 \u03b1\na : \u03b1\ninst\u271d : DecidableEq \u03b1\nl : List \u03b1\n\u22a2 List.foldr (fun x x_1 => x \u2293 x_1) \u22a4 l = inf (List.toFinset l) id", "state_after": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Type u_5\n\u03ba : Type u_6\ninst\u271d\u00b2 : SemilatticeInf \u03b1\ninst\u271d\u00b9 : OrderTop \u03b1\ns s\u2081 s\u2082 : Finset \u03b2\nf g : \u03b2 \u2192 \u03b1\na : \u03b1\ninst\u271d : DecidableEq \u03b1\nl : List \u03b1\n\u22a2 Multiset.fold (fun x x_1 => x \u2293 x_1) \u22a4 (dedup \u2191l) = Multiset.inf (dedup \u2191l)"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Type u_5\n\u03ba : Type u_6\ninst\u271d\u00b2 : SemilatticeInf \u03b1\ninst\u271d\u00b9 : OrderTop \u03b1\ns s\u2081 s\u2082 : Finset \u03b2\nf g : \u03b2 \u2192 \u03b1\na : \u03b1\ninst\u271d : DecidableEq \u03b1\nl : List \u03b1\n\u22a2 Multiset.fold (fun x x_1 => x \u2293 x_1) \u22a4 (dedup \u2191l) = Multiset.inf (dedup \u2191l)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/RBMap/Lemmas.lean", "full_name": "Std.RBSet.mem_congr", "start": [626, 1], "end": [627, 21], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Independence/Basic.lean", "full_name": "ProbabilityTheory.iIndepFun.indepFun_prod", "start": [592, 1], "end": [596, 64], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Kernel/Composition.lean", "full_name": "ProbabilityTheory.kernel.compProd_restrict", "start": [328, 1], "end": [347, 32], "traced_tactics": [{"tactic": "ext a u hu", "annotated_tactic": ["ext a u hu", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03b3 : Type u_4\nm\u03b3 : MeasurableSpace \u03b3\ns\u271d : Set (\u03b2 \u00d7 \u03b3)\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d : IsSFiniteKernel \u03b7\na : \u03b1\ns : Set \u03b2\nt : Set \u03b3\nhs : MeasurableSet s\nht : MeasurableSet t\n\u22a2 kernel.restrict \u03ba hs \u2297\u2096 kernel.restrict \u03b7 ht = kernel.restrict (\u03ba \u2297\u2096 \u03b7) (_ : MeasurableSet (s \u00d7\u02e2 t))", "state_after": "case h.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03b3 : Type u_4\nm\u03b3 : MeasurableSpace \u03b3\ns\u271d : Set (\u03b2 \u00d7 \u03b3)\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d : IsSFiniteKernel \u03b7\na\u271d : \u03b1\ns : Set \u03b2\nt : Set \u03b3\nhs : MeasurableSet s\nht : MeasurableSet t\na : \u03b1\nu : Set (\u03b2 \u00d7 \u03b3)\nhu : MeasurableSet u\n\u22a2 \u2191\u2191(\u2191(kernel.restrict \u03ba hs \u2297\u2096 kernel.restrict \u03b7 ht) a) u =\n    \u2191\u2191(\u2191(kernel.restrict (\u03ba \u2297\u2096 \u03b7) (_ : MeasurableSet (s \u00d7\u02e2 t))) a) u"}, {"tactic": "rw [compProd_apply _ _ _ hu, restrict_apply' _ _ _ hu,\n  compProd_apply _ _ _ (hu.inter (hs.prod ht))]", "annotated_tactic": ["rw [<a>compProd_apply</a> _ _ _ hu, <a>restrict_apply'</a> _ _ _ hu,\n    <a>compProd_apply</a> _ _ _ (hu.inter (hs.prod ht))]", [{"full_name": "ProbabilityTheory.kernel.compProd_apply", "def_path": "Mathlib/Probability/Kernel/Composition.lean", "def_pos": [242, 9], "def_end_pos": [242, 23]}, {"full_name": "ProbabilityTheory.kernel.restrict_apply'", "def_path": "Mathlib/Probability/Kernel/Basic.lean", "def_pos": [509, 9], "def_end_pos": [509, 24]}, {"full_name": "ProbabilityTheory.kernel.compProd_apply", "def_path": "Mathlib/Probability/Kernel/Composition.lean", "def_pos": [242, 9], "def_end_pos": [242, 23]}]], "state_before": "case h.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03b3 : Type u_4\nm\u03b3 : MeasurableSpace \u03b3\ns\u271d : Set (\u03b2 \u00d7 \u03b3)\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d : IsSFiniteKernel \u03b7\na\u271d : \u03b1\ns : Set \u03b2\nt : Set \u03b3\nhs : MeasurableSet s\nht : MeasurableSet t\na : \u03b1\nu : Set (\u03b2 \u00d7 \u03b3)\nhu : MeasurableSet u\n\u22a2 \u2191\u2191(\u2191(kernel.restrict \u03ba hs \u2297\u2096 kernel.restrict \u03b7 ht) a) u =\n    \u2191\u2191(\u2191(kernel.restrict (\u03ba \u2297\u2096 \u03b7) (_ : MeasurableSet (s \u00d7\u02e2 t))) a) u", "state_after": "case h.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03b3 : Type u_4\nm\u03b3 : MeasurableSpace \u03b3\ns\u271d : Set (\u03b2 \u00d7 \u03b3)\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d : IsSFiniteKernel \u03b7\na\u271d : \u03b1\ns : Set \u03b2\nt : Set \u03b3\nhs : MeasurableSet s\nht : MeasurableSet t\na : \u03b1\nu : Set (\u03b2 \u00d7 \u03b3)\nhu : MeasurableSet u\n\u22a2 \u222b\u207b (b : \u03b2), \u2191\u2191(\u2191(kernel.restrict \u03b7 ht) (a, b)) {c | (b, c) \u2208 u} \u2202\u2191(kernel.restrict \u03ba hs) a =\n    \u222b\u207b (b : \u03b2), \u2191\u2191(\u2191\u03b7 (a, b)) {c | (b, c) \u2208 u \u2229 s \u00d7\u02e2 t} \u2202\u2191\u03ba a"}, {"tactic": "simp only [kernel.restrict_apply, Measure.restrict_apply' ht, Set.mem_inter_iff,\n  Set.prod_mk_mem_set_prod_eq]", "annotated_tactic": ["simp only [<a>kernel.restrict_apply</a>, <a>Measure.restrict_apply'</a> ht, <a>Set.mem_inter_iff</a>,\n    <a>Set.prod_mk_mem_set_prod_eq</a>]", [{"full_name": "ProbabilityTheory.kernel.restrict_apply", "def_path": "Mathlib/Probability/Kernel/Basic.lean", "def_pos": [504, 9], "def_end_pos": [504, 23]}, {"full_name": "MeasureTheory.Measure.restrict_apply'", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1567, 9], "def_end_pos": [1567, 24]}, {"full_name": "Set.mem_inter_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [909, 9], "def_end_pos": [909, 22]}, {"full_name": "Set.prod_mk_mem_set_prod_eq", "def_path": "Mathlib/Data/Set/Prod.lean", "def_pos": [62, 9], "def_end_pos": [62, 32]}]], "state_before": "case h.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03b3 : Type u_4\nm\u03b3 : MeasurableSpace \u03b3\ns\u271d : Set (\u03b2 \u00d7 \u03b3)\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d : IsSFiniteKernel \u03b7\na\u271d : \u03b1\ns : Set \u03b2\nt : Set \u03b3\nhs : MeasurableSet s\nht : MeasurableSet t\na : \u03b1\nu : Set (\u03b2 \u00d7 \u03b3)\nhu : MeasurableSet u\n\u22a2 \u222b\u207b (b : \u03b2), \u2191\u2191(\u2191(kernel.restrict \u03b7 ht) (a, b)) {c | (b, c) \u2208 u} \u2202\u2191(kernel.restrict \u03ba hs) a =\n    \u222b\u207b (b : \u03b2), \u2191\u2191(\u2191\u03b7 (a, b)) {c | (b, c) \u2208 u \u2229 s \u00d7\u02e2 t} \u2202\u2191\u03ba a", "state_after": "case h.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03b3 : Type u_4\nm\u03b3 : MeasurableSpace \u03b3\ns\u271d : Set (\u03b2 \u00d7 \u03b3)\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d : IsSFiniteKernel \u03b7\na\u271d : \u03b1\ns : Set \u03b2\nt : Set \u03b3\nhs : MeasurableSet s\nht : MeasurableSet t\na : \u03b1\nu : Set (\u03b2 \u00d7 \u03b3)\nhu : MeasurableSet u\n\u22a2 \u222b\u207b (b : \u03b2) in s, \u2191\u2191(\u2191\u03b7 (a, b)) ({c | (b, c) \u2208 u} \u2229 t) \u2202\u2191\u03ba a =\n    \u222b\u207b (b : \u03b2), \u2191\u2191(\u2191\u03b7 (a, b)) {c | (b, c) \u2208 u \u2227 b \u2208 s \u2227 c \u2208 t} \u2202\u2191\u03ba a"}, {"tactic": "simp_rw [this]", "annotated_tactic": ["simp_rw [this]", []], "state_before": "case h.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03b3 : Type u_4\nm\u03b3 : MeasurableSpace \u03b3\ns\u271d : Set (\u03b2 \u00d7 \u03b3)\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d : IsSFiniteKernel \u03b7\na\u271d : \u03b1\ns : Set \u03b2\nt : Set \u03b3\nhs : MeasurableSet s\nht : MeasurableSet t\na : \u03b1\nu : Set (\u03b2 \u00d7 \u03b3)\nhu : MeasurableSet u\nthis :\n  \u2200 (b : \u03b2),\n    \u2191\u2191(\u2191\u03b7 (a, b)) {c | (b, c) \u2208 u \u2227 b \u2208 s \u2227 c \u2208 t} = Set.indicator s (fun b => \u2191\u2191(\u2191\u03b7 (a, b)) ({c | (b, c) \u2208 u} \u2229 t)) b\n\u22a2 \u222b\u207b (b : \u03b2) in s, \u2191\u2191(\u2191\u03b7 (a, b)) ({c | (b, c) \u2208 u} \u2229 t) \u2202\u2191\u03ba a =\n    \u222b\u207b (b : \u03b2), \u2191\u2191(\u2191\u03b7 (a, b)) {c | (b, c) \u2208 u \u2227 b \u2208 s \u2227 c \u2208 t} \u2202\u2191\u03ba a", "state_after": "case h.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03b3 : Type u_4\nm\u03b3 : MeasurableSpace \u03b3\ns\u271d : Set (\u03b2 \u00d7 \u03b3)\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d : IsSFiniteKernel \u03b7\na\u271d : \u03b1\ns : Set \u03b2\nt : Set \u03b3\nhs : MeasurableSet s\nht : MeasurableSet t\na : \u03b1\nu : Set (\u03b2 \u00d7 \u03b3)\nhu : MeasurableSet u\nthis :\n  \u2200 (b : \u03b2),\n    \u2191\u2191(\u2191\u03b7 (a, b)) {c | (b, c) \u2208 u \u2227 b \u2208 s \u2227 c \u2208 t} = Set.indicator s (fun b => \u2191\u2191(\u2191\u03b7 (a, b)) ({c | (b, c) \u2208 u} \u2229 t)) b\n\u22a2 \u222b\u207b (b : \u03b2) in s, \u2191\u2191(\u2191\u03b7 (a, b)) ({c | (b, c) \u2208 u} \u2229 t) \u2202\u2191\u03ba a =\n    \u222b\u207b (b : \u03b2), Set.indicator s (fun b => \u2191\u2191(\u2191\u03b7 (a, b)) ({c | (b, c) \u2208 u} \u2229 t)) b \u2202\u2191\u03ba a"}, {"tactic": "rw [lintegral_indicator _ hs]", "annotated_tactic": ["rw [<a>lintegral_indicator</a> _ hs]", [{"full_name": "MeasureTheory.lintegral_indicator", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [762, 9], "def_end_pos": [762, 28]}]], "state_before": "case h.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03b3 : Type u_4\nm\u03b3 : MeasurableSpace \u03b3\ns\u271d : Set (\u03b2 \u00d7 \u03b3)\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d : IsSFiniteKernel \u03b7\na\u271d : \u03b1\ns : Set \u03b2\nt : Set \u03b3\nhs : MeasurableSet s\nht : MeasurableSet t\na : \u03b1\nu : Set (\u03b2 \u00d7 \u03b3)\nhu : MeasurableSet u\nthis :\n  \u2200 (b : \u03b2),\n    \u2191\u2191(\u2191\u03b7 (a, b)) {c | (b, c) \u2208 u \u2227 b \u2208 s \u2227 c \u2208 t} = Set.indicator s (fun b => \u2191\u2191(\u2191\u03b7 (a, b)) ({c | (b, c) \u2208 u} \u2229 t)) b\n\u22a2 \u222b\u207b (b : \u03b2) in s, \u2191\u2191(\u2191\u03b7 (a, b)) ({c | (b, c) \u2208 u} \u2229 t) \u2202\u2191\u03ba a =\n    \u222b\u207b (b : \u03b2), Set.indicator s (fun b => \u2191\u2191(\u2191\u03b7 (a, b)) ({c | (b, c) \u2208 u} \u2229 t)) b \u2202\u2191\u03ba a", "state_after": "no goals"}, {"tactic": "intro b", "annotated_tactic": ["intro b", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03b3 : Type u_4\nm\u03b3 : MeasurableSpace \u03b3\ns\u271d : Set (\u03b2 \u00d7 \u03b3)\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d : IsSFiniteKernel \u03b7\na\u271d : \u03b1\ns : Set \u03b2\nt : Set \u03b3\nhs : MeasurableSet s\nht : MeasurableSet t\na : \u03b1\nu : Set (\u03b2 \u00d7 \u03b3)\nhu : MeasurableSet u\n\u22a2 \u2200 (b : \u03b2),\n    \u2191\u2191(\u2191\u03b7 (a, b)) {c | (b, c) \u2208 u \u2227 b \u2208 s \u2227 c \u2208 t} = Set.indicator s (fun b => \u2191\u2191(\u2191\u03b7 (a, b)) ({c | (b, c) \u2208 u} \u2229 t)) b", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03b3 : Type u_4\nm\u03b3 : MeasurableSpace \u03b3\ns\u271d : Set (\u03b2 \u00d7 \u03b3)\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d : IsSFiniteKernel \u03b7\na\u271d : \u03b1\ns : Set \u03b2\nt : Set \u03b3\nhs : MeasurableSet s\nht : MeasurableSet t\na : \u03b1\nu : Set (\u03b2 \u00d7 \u03b3)\nhu : MeasurableSet u\nb : \u03b2\n\u22a2 \u2191\u2191(\u2191\u03b7 (a, b)) {c | (b, c) \u2208 u \u2227 b \u2208 s \u2227 c \u2208 t} = Set.indicator s (fun b => \u2191\u2191(\u2191\u03b7 (a, b)) ({c | (b, c) \u2208 u} \u2229 t)) b"}, {"tactic": "rw [Set.indicator_apply]", "annotated_tactic": ["rw [<a>Set.indicator_apply</a>]", [{"full_name": "Set.indicator_apply", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [59, 3], "def_end_pos": [59, 14]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03b3 : Type u_4\nm\u03b3 : MeasurableSpace \u03b3\ns\u271d : Set (\u03b2 \u00d7 \u03b3)\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d : IsSFiniteKernel \u03b7\na\u271d : \u03b1\ns : Set \u03b2\nt : Set \u03b3\nhs : MeasurableSet s\nht : MeasurableSet t\na : \u03b1\nu : Set (\u03b2 \u00d7 \u03b3)\nhu : MeasurableSet u\nb : \u03b2\n\u22a2 \u2191\u2191(\u2191\u03b7 (a, b)) {c | (b, c) \u2208 u \u2227 b \u2208 s \u2227 c \u2208 t} = Set.indicator s (fun b => \u2191\u2191(\u2191\u03b7 (a, b)) ({c | (b, c) \u2208 u} \u2229 t)) b", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03b3 : Type u_4\nm\u03b3 : MeasurableSpace \u03b3\ns\u271d : Set (\u03b2 \u00d7 \u03b3)\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d : IsSFiniteKernel \u03b7\na\u271d : \u03b1\ns : Set \u03b2\nt : Set \u03b3\nhs : MeasurableSet s\nht : MeasurableSet t\na : \u03b1\nu : Set (\u03b2 \u00d7 \u03b3)\nhu : MeasurableSet u\nb : \u03b2\n\u22a2 \u2191\u2191(\u2191\u03b7 (a, b)) {c | (b, c) \u2208 u \u2227 b \u2208 s \u2227 c \u2208 t} = if b \u2208 s then \u2191\u2191(\u2191\u03b7 (a, b)) ({c | (b, c) \u2208 u} \u2229 t) else 0"}, {"tactic": "split_ifs with h", "annotated_tactic": ["split_ifs with h", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03b3 : Type u_4\nm\u03b3 : MeasurableSpace \u03b3\ns\u271d : Set (\u03b2 \u00d7 \u03b3)\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d : IsSFiniteKernel \u03b7\na\u271d : \u03b1\ns : Set \u03b2\nt : Set \u03b3\nhs : MeasurableSet s\nht : MeasurableSet t\na : \u03b1\nu : Set (\u03b2 \u00d7 \u03b3)\nhu : MeasurableSet u\nb : \u03b2\n\u22a2 \u2191\u2191(\u2191\u03b7 (a, b)) {c | (b, c) \u2208 u \u2227 b \u2208 s \u2227 c \u2208 t} = if b \u2208 s then \u2191\u2191(\u2191\u03b7 (a, b)) ({c | (b, c) \u2208 u} \u2229 t) else 0", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03b3 : Type u_4\nm\u03b3 : MeasurableSpace \u03b3\ns\u271d : Set (\u03b2 \u00d7 \u03b3)\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d : IsSFiniteKernel \u03b7\na\u271d : \u03b1\ns : Set \u03b2\nt : Set \u03b3\nhs : MeasurableSet s\nht : MeasurableSet t\na : \u03b1\nu : Set (\u03b2 \u00d7 \u03b3)\nhu : MeasurableSet u\nb : \u03b2\nh : b \u2208 s\n\u22a2 \u2191\u2191(\u2191\u03b7 (a, b)) {c | (b, c) \u2208 u \u2227 b \u2208 s \u2227 c \u2208 t} = \u2191\u2191(\u2191\u03b7 (a, b)) ({c | (b, c) \u2208 u} \u2229 t)\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03b3 : Type u_4\nm\u03b3 : MeasurableSpace \u03b3\ns\u271d : Set (\u03b2 \u00d7 \u03b3)\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d : IsSFiniteKernel \u03b7\na\u271d : \u03b1\ns : Set \u03b2\nt : Set \u03b3\nhs : MeasurableSet s\nht : MeasurableSet t\na : \u03b1\nu : Set (\u03b2 \u00d7 \u03b3)\nhu : MeasurableSet u\nb : \u03b2\nh : \u00acb \u2208 s\n\u22a2 \u2191\u2191(\u2191\u03b7 (a, b)) {c | (b, c) \u2208 u \u2227 b \u2208 s \u2227 c \u2208 t} = 0"}, {"tactic": "simp only [h, true_and_iff]", "annotated_tactic": ["simp only [h, <a>true_and_iff</a>]", [{"full_name": "true_and_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [147, 9], "def_end_pos": [147, 21]}]], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03b3 : Type u_4\nm\u03b3 : MeasurableSpace \u03b3\ns\u271d : Set (\u03b2 \u00d7 \u03b3)\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d : IsSFiniteKernel \u03b7\na\u271d : \u03b1\ns : Set \u03b2\nt : Set \u03b3\nhs : MeasurableSet s\nht : MeasurableSet t\na : \u03b1\nu : Set (\u03b2 \u00d7 \u03b3)\nhu : MeasurableSet u\nb : \u03b2\nh : b \u2208 s\n\u22a2 \u2191\u2191(\u2191\u03b7 (a, b)) {c | (b, c) \u2208 u \u2227 b \u2208 s \u2227 c \u2208 t} = \u2191\u2191(\u2191\u03b7 (a, b)) ({c | (b, c) \u2208 u} \u2229 t)", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03b3 : Type u_4\nm\u03b3 : MeasurableSpace \u03b3\ns\u271d : Set (\u03b2 \u00d7 \u03b3)\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d : IsSFiniteKernel \u03b7\na\u271d : \u03b1\ns : Set \u03b2\nt : Set \u03b3\nhs : MeasurableSet s\nht : MeasurableSet t\na : \u03b1\nu : Set (\u03b2 \u00d7 \u03b3)\nhu : MeasurableSet u\nb : \u03b2\nh : b \u2208 s\n\u22a2 \u2191\u2191(\u2191\u03b7 (a, b)) {c | (b, c) \u2208 u \u2227 c \u2208 t} = \u2191\u2191(\u2191\u03b7 (a, b)) ({c | (b, c) \u2208 u} \u2229 t)"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03b3 : Type u_4\nm\u03b3 : MeasurableSpace \u03b3\ns\u271d : Set (\u03b2 \u00d7 \u03b3)\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d : IsSFiniteKernel \u03b7\na\u271d : \u03b1\ns : Set \u03b2\nt : Set \u03b3\nhs : MeasurableSet s\nht : MeasurableSet t\na : \u03b1\nu : Set (\u03b2 \u00d7 \u03b3)\nhu : MeasurableSet u\nb : \u03b2\nh : b \u2208 s\n\u22a2 \u2191\u2191(\u2191\u03b7 (a, b)) {c | (b, c) \u2208 u \u2227 c \u2208 t} = \u2191\u2191(\u2191\u03b7 (a, b)) ({c | (b, c) \u2208 u} \u2229 t)", "state_after": "no goals"}, {"tactic": "simp only [h, false_and_iff, and_false_iff, Set.setOf_false, measure_empty]", "annotated_tactic": ["simp only [h, <a>false_and_iff</a>, <a>and_false_iff</a>, <a>Set.setOf_false</a>, <a>measure_empty</a>]", [{"full_name": "false_and_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [151, 9], "def_end_pos": [151, 22]}, {"full_name": "and_false_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [149, 9], "def_end_pos": [149, 22]}, {"full_name": "Set.setOf_false", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [567, 9], "def_end_pos": [567, 20]}, {"full_name": "MeasureTheory.measure_empty", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [185, 9], "def_end_pos": [185, 22]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03b3 : Type u_4\nm\u03b3 : MeasurableSpace \u03b3\ns\u271d : Set (\u03b2 \u00d7 \u03b3)\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d : IsSFiniteKernel \u03b7\na\u271d : \u03b1\ns : Set \u03b2\nt : Set \u03b3\nhs : MeasurableSet s\nht : MeasurableSet t\na : \u03b1\nu : Set (\u03b2 \u00d7 \u03b3)\nhu : MeasurableSet u\nb : \u03b2\nh : \u00acb \u2208 s\n\u22a2 \u2191\u2191(\u2191\u03b7 (a, b)) {c | (b, c) \u2208 u \u2227 b \u2208 s \u2227 c \u2208 t} = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/Average.lean", "full_name": "MeasureTheory.ofReal_setAverage", "start": [445, 1], "end": [447, 36], "traced_tactics": [{"tactic": "simpa using ofReal_average hf hf\u2080", "annotated_tactic": ["simpa using <a>ofReal_average</a> hf hf\u2080", [{"full_name": "MeasureTheory.ofReal_average", "def_path": "Mathlib/MeasureTheory/Integral/Average.lean", "def_pos": [436, 9], "def_end_pos": [436, 23]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\nf : \u03b1 \u2192 \u211d\nhf : IntegrableOn f s\nhf\u2080 : 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] f\n\u22a2 ENNReal.ofReal (\u2a0d (x : \u03b1) in s, f x \u2202\u03bc) = (\u222b\u207b (x : \u03b1) in s, ENNReal.ofReal (f x) \u2202\u03bc) / \u2191\u2191\u03bc s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/ZMod/Basic.lean", "full_name": "ZMod.mul_inv_cancel_aux", "start": [1170, 9], "end": [1174, 82], "traced_tactics": [{"tactic": "obtain \u27e8k, rfl\u27e9 := nat_cast_zmod_surjective a", "annotated_tactic": ["obtain \u27e8k, rfl\u27e9 := <a>nat_cast_zmod_surjective</a> a", [{"full_name": "ZMod.nat_cast_zmod_surjective", "def_path": "Mathlib/Data/ZMod/Basic.lean", "def_pos": [205, 9], "def_end_pos": [205, 33]}]], "state_before": "n a\u271d p : \u2115\ninst\u271d : Fact (Nat.Prime p)\na : ZMod p\nh : a \u2260 0\n\u22a2 a * a\u207b\u00b9 = 1", "state_after": "case intro\nn a p : \u2115\ninst\u271d : Fact (Nat.Prime p)\nk : \u2115\nh : \u2191k \u2260 0\n\u22a2 \u2191k * (\u2191k)\u207b\u00b9 = 1"}, {"tactic": "apply coe_mul_inv_eq_one", "annotated_tactic": ["apply <a>coe_mul_inv_eq_one</a>", [{"full_name": "ZMod.coe_mul_inv_eq_one", "def_path": "Mathlib/Data/ZMod/Basic.lean", "def_pos": [726, 9], "def_end_pos": [726, 27]}]], "state_before": "case intro\nn a p : \u2115\ninst\u271d : Fact (Nat.Prime p)\nk : \u2115\nh : \u2191k \u2260 0\n\u22a2 \u2191k * (\u2191k)\u207b\u00b9 = 1", "state_after": "case intro.h\nn a p : \u2115\ninst\u271d : Fact (Nat.Prime p)\nk : \u2115\nh : \u2191k \u2260 0\n\u22a2 Nat.Coprime k p"}, {"tactic": "apply Nat.Coprime.symm", "annotated_tactic": ["apply <a>Nat.Coprime.symm</a>", [{"full_name": "Nat.Coprime.symm", "def_path": "lake-packages/std/Std/Data/Nat/Gcd.lean", "def_pos": [259, 9], "def_end_pos": [259, 21]}]], "state_before": "case intro.h\nn a p : \u2115\ninst\u271d : Fact (Nat.Prime p)\nk : \u2115\nh : \u2191k \u2260 0\n\u22a2 Nat.Coprime k p", "state_after": "case intro.h.a\nn a p : \u2115\ninst\u271d : Fact (Nat.Prime p)\nk : \u2115\nh : \u2191k \u2260 0\n\u22a2 Nat.Coprime p k"}, {"tactic": "rwa [Nat.Prime.coprime_iff_not_dvd Fact.out, \u2190 CharP.cast_eq_zero_iff (ZMod p)]", "annotated_tactic": ["rwa [<a>Nat.Prime.coprime_iff_not_dvd</a> <a>Fact.out</a>, \u2190 <a>CharP.cast_eq_zero_iff</a> (<a>ZMod</a> p)]", [{"full_name": "Nat.Prime.coprime_iff_not_dvd", "def_path": "Mathlib/Data/Nat/Prime.lean", "def_pos": [541, 9], "def_end_pos": [541, 34]}, {"full_name": "Fact.out", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [118, 3], "def_end_pos": [118, 6]}, {"full_name": "CharP.cast_eq_zero_iff", "def_path": "Mathlib/Algebra/CharP/Basic.lean", "def_pos": [112, 9], "def_end_pos": [112, 31]}, {"full_name": "ZMod", "def_path": "Mathlib/Data/ZMod/Defs.lean", "def_pos": [94, 5], "def_end_pos": [94, 9]}]], "state_before": "case intro.h.a\nn a p : \u2115\ninst\u271d : Fact (Nat.Prime p)\nk : \u2115\nh : \u2191k \u2260 0\n\u22a2 Nat.Coprime p k", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Array/Lemmas.lean", "full_name": "Array.swap_def", "start": [115, 1], "end": [117, 28], "traced_tactics": [{"tactic": "simp [j.2]", "annotated_tactic": ["simp [j.2]", []], "state_before": "\u03b1 : Type ?u.17701\na : Array \u03b1\ni j : Fin (size a)\n\u22a2 j.val < size (set a i (get a j))", "state_after": "no goals"}, {"tactic": "simp [swap, fin_cast_val]", "annotated_tactic": ["simp [<a>swap</a>, <a>fin_cast_val</a>]", [{"full_name": "Array.swap", "def_path": "lake-packages/lean4/src/lean/Init/Data/Array/Basic.lean", "def_pos": [75, 5], "def_end_pos": [75, 9]}, {"full_name": "_private.\u00ablake-packages\u00bb.std.Std.Data.Array.Lemmas.0.Array.fin_cast_val", "def_path": "lake-packages/std/Std/Data/Array/Lemmas.lean", "def_pos": [113, 17], "def_end_pos": [113, 29]}]], "state_before": "\u03b1 : Type u_1\na : Array \u03b1\ni j : Fin (size a)\n\u22a2 swap a i j = set (set a i (get a j)) { val := j.val, isLt := (_ : j.val < size (set a i a[j.val])) } (get a i)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Haar/OfBasis.lean", "full_name": "parallelepiped_eq_convexHull", "start": [133, 1], "end": [135, 75], "traced_tactics": [{"tactic": "simp_rw [convexHull_sum, convexHull_pair, parallelepiped_eq_sum_segment]", "annotated_tactic": ["simp_rw [<a>convexHull_sum</a>, <a>convexHull_pair</a>, <a>parallelepiped_eq_sum_segment</a>]", [{"full_name": "convexHull_sum", "def_path": "Mathlib/Analysis/Convex/Combination.lean", "def_pos": [484, 9], "def_end_pos": [484, 23]}, {"full_name": "convexHull_pair", "def_path": "Mathlib/Analysis/Convex/Hull.lean", "def_pos": [132, 9], "def_end_pos": [132, 24]}, {"full_name": "parallelepiped_eq_sum_segment", "def_path": "Mathlib/MeasureTheory/Measure/Haar/OfBasis.lean", "def_pos": [114, 9], "def_end_pos": [114, 38]}]], "state_before": "\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2075 : Fintype \u03b9\ninst\u271d\u2074 : Fintype \u03b9'\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \u211d E\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \u211d F\nv : \u03b9 \u2192 E\n\u22a2 parallelepiped v = \u2191(convexHull \u211d) (\u2211 i : \u03b9, {0, v i})", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Constructions/Polish.lean", "full_name": "Measurable.measurable_comp_iff_of_surjective", "start": [609, 1], "end": [611, 92], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Lattice.lean", "full_name": "Finset.inf_eq_sInf_image", "start": [748, 1], "end": [750, 33], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "Finset.mem_sdiff", "start": [2077, 1], "end": [2078, 23], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "full_name": "Set.univ_pow", "start": [1009, 1], "end": [1012, 79], "traced_tactics": [{"tactic": "rw [pow_succ, univ_pow n.succ_ne_zero, univ_mul_univ]", "annotated_tactic": ["rw [<a>pow_succ</a>, univ_pow n.succ_ne_zero, <a>univ_mul_univ</a>]", [{"full_name": "pow_succ", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [645, 9], "def_end_pos": [645, 17]}, {"full_name": "Set.univ_mul_univ", "def_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "def_pos": [995, 9], "def_end_pos": [995, 22]}]], "state_before": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : Monoid \u03b1\ns t : Set \u03b1\na : \u03b1\nm n\u271d n : \u2115\nx\u271d : n + 2 \u2260 0\n\u22a2 univ ^ (n + 2) = univ", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Lattice.lean", "full_name": "Finset.set_biInter_option_toFinset", "start": [2114, 1], "end": [2116, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/Jacobian.lean", "full_name": "MeasureTheory.lintegral_image_eq_lintegral_abs_det_fderiv_mul", "start": [1170, 1], "end": [1181, 60], "traced_tactics": [{"tactic": "rw [\u2190 restrict_map_withDensity_abs_det_fderiv_eq_addHaar \u03bc hs hf' hf,\n  (measurableEmbedding_of_fderivWithin hs hf' hf).lintegral_map]", "annotated_tactic": ["rw [\u2190 <a>restrict_map_withDensity_abs_det_fderiv_eq_addHaar</a> \u03bc hs hf' hf,\n    (<a>measurableEmbedding_of_fderivWithin</a> hs hf' hf).<a>lintegral_map</a>]", [{"full_name": "MeasureTheory.restrict_map_withDensity_abs_det_fderiv_eq_addHaar", "def_path": "Mathlib/MeasureTheory/Function/Jacobian.lean", "def_pos": [1137, 9], "def_end_pos": [1137, 59]}, {"full_name": "MeasureTheory.measurableEmbedding_of_fderivWithin", "def_path": "Mathlib/MeasureTheory/Function/Jacobian.lean", "def_pos": [786, 9], "def_end_pos": [786, 44]}, {"full_name": "MeasurableEmbedding.lintegral_map", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [1332, 9], "def_end_pos": [1332, 49]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\ng : E \u2192 \u211d\u22650\u221e\n\u22a2 \u222b\u207b (x : E) in f '' s, g x \u2202\u03bc = \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| * g (f x) \u2202\u03bc", "state_after": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\ng : E \u2192 \u211d\u22650\u221e\n\u22a2 \u222b\u207b (a : \u2191s),\n      g\n        (Set.restrict s f\n          a) \u2202Measure.comap Subtype.val (withDensity \u03bc fun x => ENNReal.ofReal |ContinuousLinearMap.det (f' x)|) =\n    \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| * g (f x) \u2202\u03bc"}, {"tactic": "have : \u2200 x : s, g (s.restrict f x) = (g \u2218 f) x := fun x => rfl", "annotated_tactic": ["have : \u2200 x : s, g (s.restrict f x) = (g \u2218 f) x := fun x => <a>rfl</a>", [{"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\ng : E \u2192 \u211d\u22650\u221e\n\u22a2 \u222b\u207b (a : \u2191s),\n      g\n        (Set.restrict s f\n          a) \u2202Measure.comap Subtype.val (withDensity \u03bc fun x => ENNReal.ofReal |ContinuousLinearMap.det (f' x)|) =\n    \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| * g (f x) \u2202\u03bc", "state_after": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\ng : E \u2192 \u211d\u22650\u221e\nthis : \u2200 (x : \u2191s), g (Set.restrict s f x) = (g \u2218 f) \u2191x\n\u22a2 \u222b\u207b (a : \u2191s),\n      g\n        (Set.restrict s f\n          a) \u2202Measure.comap Subtype.val (withDensity \u03bc fun x => ENNReal.ofReal |ContinuousLinearMap.det (f' x)|) =\n    \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| * g (f x) \u2202\u03bc"}, {"tactic": "simp only [this]", "annotated_tactic": ["simp only [this]", []], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\ng : E \u2192 \u211d\u22650\u221e\nthis : \u2200 (x : \u2191s), g (Set.restrict s f x) = (g \u2218 f) \u2191x\n\u22a2 \u222b\u207b (a : \u2191s),\n      g\n        (Set.restrict s f\n          a) \u2202Measure.comap Subtype.val (withDensity \u03bc fun x => ENNReal.ofReal |ContinuousLinearMap.det (f' x)|) =\n    \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| * g (f x) \u2202\u03bc", "state_after": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\ng : E \u2192 \u211d\u22650\u221e\nthis : \u2200 (x : \u2191s), g (Set.restrict s f x) = (g \u2218 f) \u2191x\n\u22a2 \u222b\u207b (a : \u2191s),\n      (g \u2218 f) \u2191a \u2202Measure.comap Subtype.val (withDensity \u03bc fun x => ENNReal.ofReal |ContinuousLinearMap.det (f' x)|) =\n    \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| * g (f x) \u2202\u03bc"}, {"tactic": "rw [\u2190 (MeasurableEmbedding.subtype_coe hs).lintegral_map, map_comap_subtype_coe hs,\n  set_lintegral_withDensity_eq_set_lintegral_mul_non_measurable\u2080 _ _ _ hs]", "annotated_tactic": ["rw [\u2190 (<a>MeasurableEmbedding.subtype_coe</a> hs).<a>lintegral_map</a>, <a>map_comap_subtype_coe</a> hs,\n    <a>set_lintegral_withDensity_eq_set_lintegral_mul_non_measurable\u2080</a> _ _ _ hs]", [{"full_name": "MeasurableEmbedding.subtype_coe", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [1208, 9], "def_end_pos": [1208, 20]}, {"full_name": "MeasurableEmbedding.lintegral_map", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [1332, 9], "def_end_pos": [1332, 49]}, {"full_name": "map_comap_subtype_coe", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [4159, 9], "def_end_pos": [4159, 30]}, {"full_name": "MeasureTheory.set_lintegral_withDensity_eq_set_lintegral_mul_non_measurable\u2080", "def_path": "Mathlib/MeasureTheory/Measure/WithDensity.lean", "def_pos": [397, 9], "def_end_pos": [397, 71]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\ng : E \u2192 \u211d\u22650\u221e\nthis : \u2200 (x : \u2191s), g (Set.restrict s f x) = (g \u2218 f) \u2191x\n\u22a2 \u222b\u207b (a : \u2191s),\n      (g \u2218 f) \u2191a \u2202Measure.comap Subtype.val (withDensity \u03bc fun x => ENNReal.ofReal |ContinuousLinearMap.det (f' x)|) =\n    \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| * g (f x) \u2202\u03bc", "state_after": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\ng : E \u2192 \u211d\u22650\u221e\nthis : \u2200 (x : \u2191s), g (Set.restrict s f x) = (g \u2218 f) \u2191x\n\u22a2 \u222b\u207b (a : E) in s, ((fun x => ENNReal.ofReal |ContinuousLinearMap.det (f' x)|) * fun a => (g \u2218 f) a) a \u2202\u03bc =\n    \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| * g (f x) \u2202\u03bc\n\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\ng : E \u2192 \u211d\u22650\u221e\nthis : \u2200 (x : \u2191s), g (Set.restrict s f x) = (g \u2218 f) \u2191x\n\u22a2 \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| < \u22a4\n\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\ng : E \u2192 \u211d\u22650\u221e\nthis : \u2200 (x : \u2191s), g (Set.restrict s f x) = (g \u2218 f) \u2191x\n\u22a2 AEMeasurable fun x => ENNReal.ofReal |ContinuousLinearMap.det (f' x)|"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\ng : E \u2192 \u211d\u22650\u221e\nthis : \u2200 (x : \u2191s), g (Set.restrict s f x) = (g \u2218 f) \u2191x\n\u22a2 \u222b\u207b (a : E) in s, ((fun x => ENNReal.ofReal |ContinuousLinearMap.det (f' x)|) * fun a => (g \u2218 f) a) a \u2202\u03bc =\n    \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| * g (f x) \u2202\u03bc", "state_after": "no goals"}, {"tactic": "simp only [eventually_true, ENNReal.ofReal_lt_top]", "annotated_tactic": ["simp only [<a>eventually_true</a>, <a>ENNReal.ofReal_lt_top</a>]", [{"full_name": "Filter.eventually_true", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1108, 17], "def_end_pos": [1108, 32]}, {"full_name": "ENNReal.ofReal_lt_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [314, 17], "def_end_pos": [314, 30]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\ng : E \u2192 \u211d\u22650\u221e\nthis : \u2200 (x : \u2191s), g (Set.restrict s f x) = (g \u2218 f) \u2191x\n\u22a2 \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| < \u22a4", "state_after": "no goals"}, {"tactic": "exact aemeasurable_ofReal_abs_det_fderivWithin \u03bc hs hf'", "annotated_tactic": ["exact <a>aemeasurable_ofReal_abs_det_fderivWithin</a> \u03bc hs hf'", [{"full_name": "MeasureTheory.aemeasurable_ofReal_abs_det_fderivWithin", "def_path": "Mathlib/MeasureTheory/Function/Jacobian.lean", "def_pos": [758, 9], "def_end_pos": [758, 49]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\ng : E \u2192 \u211d\u22650\u221e\nthis : \u2200 (x : \u2191s), g (Set.restrict s f x) = (g \u2218 f) \u2191x\n\u22a2 AEMeasurable fun x => ENNReal.ofReal |ContinuousLinearMap.det (f' x)|", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/Basic.lean", "full_name": "MvPolynomial.eval_mem", "start": [1708, 1], "end": [1710, 18], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/MulAntidiagonal.lean", "full_name": "Set.MulAntidiagonal.finite_of_isPwo", "start": [119, 1], "end": [129, 64], "traced_tactics": [{"tactic": "refine' not_infinite.1 fun h => _", "annotated_tactic": ["refine' <a>not_infinite</a>.1 fun h => _", [{"full_name": "Set.not_infinite", "def_path": "Mathlib/Data/Set/Finite.lean", "def_pos": [147, 9], "def_end_pos": [147, 21]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : OrderedCancelCommMonoid \u03b1\ns t : Set \u03b1\na\u271d : \u03b1\nx y : \u2191(mulAntidiagonal s t a\u271d)\nhs : IsPwo s\nht : IsPwo t\na : \u03b1\n\u22a2 Set.Finite (mulAntidiagonal s t a)", "state_after": "\u03b1 : Type u_1\ninst\u271d : OrderedCancelCommMonoid \u03b1\ns t : Set \u03b1\na\u271d : \u03b1\nx y : \u2191(mulAntidiagonal s t a\u271d)\nhs : IsPwo s\nht : IsPwo t\na : \u03b1\nh : Set.Infinite (mulAntidiagonal s t a)\n\u22a2 False"}, {"tactic": "obtain \u27e8g, hg\u27e9 :=\n  h1.exists_monotone_subseq (fun n => h.natEmbedding _ n) fun n => (h.natEmbedding _ n).2", "annotated_tactic": ["obtain \u27e8g, hg\u27e9 :=\n    h1.exists_monotone_subseq (fun n => h.natEmbedding _ n) fun n => (h.natEmbedding _ n).2", []], "state_before": "\u03b1 : Type u_1\ninst\u271d : OrderedCancelCommMonoid \u03b1\ns t : Set \u03b1\na\u271d : \u03b1\nx y : \u2191(mulAntidiagonal s t a\u271d)\nhs : IsPwo s\nht : IsPwo t\na : \u03b1\nh : Set.Infinite (mulAntidiagonal s t a)\nh1 : PartiallyWellOrderedOn (mulAntidiagonal s t a) (Prod.fst \u207b\u00b9'o fun x x_1 => x \u2264 x_1)\nh2 : PartiallyWellOrderedOn (mulAntidiagonal s t a) (Prod.snd \u207b\u00b9'o fun x x_1 => x \u2264 x_1)\n\u22a2 False", "state_after": "case intro\n\u03b1 : Type u_1\ninst\u271d : OrderedCancelCommMonoid \u03b1\ns t : Set \u03b1\na\u271d : \u03b1\nx y : \u2191(mulAntidiagonal s t a\u271d)\nhs : IsPwo s\nht : IsPwo t\na : \u03b1\nh : Set.Infinite (mulAntidiagonal s t a)\nh1 : PartiallyWellOrderedOn (mulAntidiagonal s t a) (Prod.fst \u207b\u00b9'o fun x x_1 => x \u2264 x_1)\nh2 : PartiallyWellOrderedOn (mulAntidiagonal s t a) (Prod.snd \u207b\u00b9'o fun x x_1 => x \u2264 x_1)\ng : \u2115 \u21aao \u2115\nhg :\n  \u2200 (m n : \u2115),\n    m \u2264 n \u2192\n      (Prod.fst \u207b\u00b9'o fun x x_1 => x \u2264 x_1) \u2191(\u2191(Infinite.natEmbedding (mulAntidiagonal s t a) h) (\u2191g m))\n        \u2191(\u2191(Infinite.natEmbedding (mulAntidiagonal s t a) h) (\u2191g n))\n\u22a2 False"}, {"tactic": "obtain \u27e8m, n, mn, h2'\u27e9 := h2 (fun x => (h.natEmbedding _) (g x)) fun n => (h.natEmbedding _ _).2", "annotated_tactic": ["obtain \u27e8m, n, mn, h2'\u27e9 := h2 (fun x => (h.natEmbedding _) (g x)) fun n => (h.natEmbedding _ _).2", []], "state_before": "case intro\n\u03b1 : Type u_1\ninst\u271d : OrderedCancelCommMonoid \u03b1\ns t : Set \u03b1\na\u271d : \u03b1\nx y : \u2191(mulAntidiagonal s t a\u271d)\nhs : IsPwo s\nht : IsPwo t\na : \u03b1\nh : Set.Infinite (mulAntidiagonal s t a)\nh1 : PartiallyWellOrderedOn (mulAntidiagonal s t a) (Prod.fst \u207b\u00b9'o fun x x_1 => x \u2264 x_1)\nh2 : PartiallyWellOrderedOn (mulAntidiagonal s t a) (Prod.snd \u207b\u00b9'o fun x x_1 => x \u2264 x_1)\ng : \u2115 \u21aao \u2115\nhg :\n  \u2200 (m n : \u2115),\n    m \u2264 n \u2192\n      (Prod.fst \u207b\u00b9'o fun x x_1 => x \u2264 x_1) \u2191(\u2191(Infinite.natEmbedding (mulAntidiagonal s t a) h) (\u2191g m))\n        \u2191(\u2191(Infinite.natEmbedding (mulAntidiagonal s t a) h) (\u2191g n))\n\u22a2 False", "state_after": "case intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : OrderedCancelCommMonoid \u03b1\ns t : Set \u03b1\na\u271d : \u03b1\nx y : \u2191(mulAntidiagonal s t a\u271d)\nhs : IsPwo s\nht : IsPwo t\na : \u03b1\nh : Set.Infinite (mulAntidiagonal s t a)\nh1 : PartiallyWellOrderedOn (mulAntidiagonal s t a) (Prod.fst \u207b\u00b9'o fun x x_1 => x \u2264 x_1)\nh2 : PartiallyWellOrderedOn (mulAntidiagonal s t a) (Prod.snd \u207b\u00b9'o fun x x_1 => x \u2264 x_1)\ng : \u2115 \u21aao \u2115\nhg :\n  \u2200 (m n : \u2115),\n    m \u2264 n \u2192\n      (Prod.fst \u207b\u00b9'o fun x x_1 => x \u2264 x_1) \u2191(\u2191(Infinite.natEmbedding (mulAntidiagonal s t a) h) (\u2191g m))\n        \u2191(\u2191(Infinite.natEmbedding (mulAntidiagonal s t a) h) (\u2191g n))\nm n : \u2115\nmn : m < n\nh2' :\n  (Prod.snd \u207b\u00b9'o fun x x_1 => x \u2264 x_1) \u2191(\u2191(Infinite.natEmbedding (mulAntidiagonal s t a) h) (\u2191g m))\n    \u2191(\u2191(Infinite.natEmbedding (mulAntidiagonal s t a) h) (\u2191g n))\n\u22a2 False"}, {"tactic": "refine' mn.ne (g.injective <| (h.natEmbedding _).injective _)", "annotated_tactic": ["refine' mn.ne (g.injective <| (h.natEmbedding _).<a>injective</a> _)", [{"full_name": "Function.Embedding.injective", "def_path": "Mathlib/Logic/Embedding/Basic.lean", "def_pos": [124, 19], "def_end_pos": [124, 28]}]], "state_before": "case intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : OrderedCancelCommMonoid \u03b1\ns t : Set \u03b1\na\u271d : \u03b1\nx y : \u2191(mulAntidiagonal s t a\u271d)\nhs : IsPwo s\nht : IsPwo t\na : \u03b1\nh : Set.Infinite (mulAntidiagonal s t a)\nh1 : PartiallyWellOrderedOn (mulAntidiagonal s t a) (Prod.fst \u207b\u00b9'o fun x x_1 => x \u2264 x_1)\nh2 : PartiallyWellOrderedOn (mulAntidiagonal s t a) (Prod.snd \u207b\u00b9'o fun x x_1 => x \u2264 x_1)\ng : \u2115 \u21aao \u2115\nhg :\n  \u2200 (m n : \u2115),\n    m \u2264 n \u2192\n      (Prod.fst \u207b\u00b9'o fun x x_1 => x \u2264 x_1) \u2191(\u2191(Infinite.natEmbedding (mulAntidiagonal s t a) h) (\u2191g m))\n        \u2191(\u2191(Infinite.natEmbedding (mulAntidiagonal s t a) h) (\u2191g n))\nm n : \u2115\nmn : m < n\nh2' :\n  (Prod.snd \u207b\u00b9'o fun x x_1 => x \u2264 x_1) \u2191(\u2191(Infinite.natEmbedding (mulAntidiagonal s t a) h) (\u2191g m))\n    \u2191(\u2191(Infinite.natEmbedding (mulAntidiagonal s t a) h) (\u2191g n))\n\u22a2 False", "state_after": "case intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : OrderedCancelCommMonoid \u03b1\ns t : Set \u03b1\na\u271d : \u03b1\nx y : \u2191(mulAntidiagonal s t a\u271d)\nhs : IsPwo s\nht : IsPwo t\na : \u03b1\nh : Set.Infinite (mulAntidiagonal s t a)\nh1 : PartiallyWellOrderedOn (mulAntidiagonal s t a) (Prod.fst \u207b\u00b9'o fun x x_1 => x \u2264 x_1)\nh2 : PartiallyWellOrderedOn (mulAntidiagonal s t a) (Prod.snd \u207b\u00b9'o fun x x_1 => x \u2264 x_1)\ng : \u2115 \u21aao \u2115\nhg :\n  \u2200 (m n : \u2115),\n    m \u2264 n \u2192\n      (Prod.fst \u207b\u00b9'o fun x x_1 => x \u2264 x_1) \u2191(\u2191(Infinite.natEmbedding (mulAntidiagonal s t a) h) (\u2191g m))\n        \u2191(\u2191(Infinite.natEmbedding (mulAntidiagonal s t a) h) (\u2191g n))\nm n : \u2115\nmn : m < n\nh2' :\n  (Prod.snd \u207b\u00b9'o fun x x_1 => x \u2264 x_1) \u2191(\u2191(Infinite.natEmbedding (mulAntidiagonal s t a) h) (\u2191g m))\n    \u2191(\u2191(Infinite.natEmbedding (mulAntidiagonal s t a) h) (\u2191g n))\n\u22a2 \u2191(Infinite.natEmbedding (mulAntidiagonal s t a) h) (\u2191g m) = \u2191(Infinite.natEmbedding (mulAntidiagonal s t a) h) (\u2191g n)"}, {"tactic": "exact eq_of_fst_le_fst_of_snd_le_snd _ _ _ (hg _ _ mn.le) h2'", "annotated_tactic": ["exact <a>eq_of_fst_le_fst_of_snd_le_snd</a> _ _ _ (hg _ _ mn.le) h2'", [{"full_name": "Set.MulAntidiagonal.eq_of_fst_le_fst_of_snd_le_snd", "def_path": "Mathlib/Data/Set/MulAntidiagonal.lean", "def_pos": [107, 9], "def_end_pos": [107, 39]}]], "state_before": "case intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : OrderedCancelCommMonoid \u03b1\ns t : Set \u03b1\na\u271d : \u03b1\nx y : \u2191(mulAntidiagonal s t a\u271d)\nhs : IsPwo s\nht : IsPwo t\na : \u03b1\nh : Set.Infinite (mulAntidiagonal s t a)\nh1 : PartiallyWellOrderedOn (mulAntidiagonal s t a) (Prod.fst \u207b\u00b9'o fun x x_1 => x \u2264 x_1)\nh2 : PartiallyWellOrderedOn (mulAntidiagonal s t a) (Prod.snd \u207b\u00b9'o fun x x_1 => x \u2264 x_1)\ng : \u2115 \u21aao \u2115\nhg :\n  \u2200 (m n : \u2115),\n    m \u2264 n \u2192\n      (Prod.fst \u207b\u00b9'o fun x x_1 => x \u2264 x_1) \u2191(\u2191(Infinite.natEmbedding (mulAntidiagonal s t a) h) (\u2191g m))\n        \u2191(\u2191(Infinite.natEmbedding (mulAntidiagonal s t a) h) (\u2191g n))\nm n : \u2115\nmn : m < n\nh2' :\n  (Prod.snd \u207b\u00b9'o fun x x_1 => x \u2264 x_1) \u2191(\u2191(Infinite.natEmbedding (mulAntidiagonal s t a) h) (\u2191g m))\n    \u2191(\u2191(Infinite.natEmbedding (mulAntidiagonal s t a) h) (\u2191g n))\n\u22a2 \u2191(Infinite.natEmbedding (mulAntidiagonal s t a) h) (\u2191g m) = \u2191(Infinite.natEmbedding (mulAntidiagonal s t a) h) (\u2191g n)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "full_name": "MeasureTheory.AEEqFun.integrable_iff_mem_L1", "start": [1288, 1], "end": [1289, 75], "traced_tactics": [{"tactic": "rw [\u2190 integrable_coeFn, \u2190 mem\u2112p_one_iff_integrable, Lp.mem_Lp_iff_mem\u2112p]", "annotated_tactic": ["rw [\u2190 <a>integrable_coeFn</a>, \u2190 <a>mem\u2112p_one_iff_integrable</a>, <a>Lp.mem_Lp_iff_mem\u2112p</a>]", [{"full_name": "MeasureTheory.AEEqFun.integrable_coeFn", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [1270, 9], "def_end_pos": [1270, 25]}, {"full_name": "MeasureTheory.mem\u2112p_one_iff_integrable", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [453, 9], "def_end_pos": [453, 33]}, {"full_name": "MeasureTheory.Lp.mem_Lp_iff_mem\u2112p", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [174, 9], "def_end_pos": [174, 25]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : NormedAddCommGroup \u03b3\nf : \u03b1 \u2192\u2098[\u03bc] \u03b2\n\u22a2 Integrable f \u2194 f \u2208 Lp \u03b2 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "full_name": "MeasureTheory.L1.SimpleFunc.setToL1S_nonneg", "start": [839, 1], "end": [849, 96], "traced_tactics": [{"tactic": "simp_rw [setToL1S]", "annotated_tactic": ["simp_rw [<a>setToL1S</a>]", [{"full_name": "MeasureTheory.L1.SimpleFunc.setToL1S", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [685, 5], "def_end_pos": [685, 13]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : NormedAddCommGroup F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : NormedField \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nG'' : Type u_7\nG' : Type u_8\ninst\u271d\u00b3 : NormedLatticeAddCommGroup G'\ninst\u271d\u00b2 : NormedSpace \u211d G'\ninst\u271d\u00b9 : NormedLatticeAddCommGroup G''\ninst\u271d : NormedSpace \u211d G''\nT : Set \u03b1 \u2192 G'' \u2192L[\u211d] G'\nh_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s = 0 \u2192 T s = 0\nh_add : FinMeasAdditive \u03bc T\nhT_nonneg : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 (x : G''), 0 \u2264 x \u2192 0 \u2264 \u2191(T s) x\nf : { x // x \u2208 simpleFunc G'' 1 \u03bc }\nhf : 0 \u2264 f\n\u22a2 0 \u2264 setToL1S T f", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : NormedAddCommGroup F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : NormedField \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nG'' : Type u_7\nG' : Type u_8\ninst\u271d\u00b3 : NormedLatticeAddCommGroup G'\ninst\u271d\u00b2 : NormedSpace \u211d G'\ninst\u271d\u00b9 : NormedLatticeAddCommGroup G''\ninst\u271d : NormedSpace \u211d G''\nT : Set \u03b1 \u2192 G'' \u2192L[\u211d] G'\nh_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s = 0 \u2192 T s = 0\nh_add : FinMeasAdditive \u03bc T\nhT_nonneg : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 (x : G''), 0 \u2264 x \u2192 0 \u2264 \u2191(T s) x\nf : { x // x \u2208 simpleFunc G'' 1 \u03bc }\nhf : 0 \u2264 f\n\u22a2 0 \u2264 SimpleFunc.setToSimpleFunc T (toSimpleFunc f)"}, {"tactic": "obtain \u27e8f', hf', hff'\u27e9 : \u2203 f' : \u03b1 \u2192\u209b G'', 0 \u2264 f' \u2227 simpleFunc.toSimpleFunc f =\u1d50[\u03bc] f' := by\n  obtain \u27e8f'', hf'', hff''\u27e9 := exists_simpleFunc_nonneg_ae_eq hf\n  exact \u27e8f'', hf'', (Lp.simpleFunc.toSimpleFunc_eq_toFun f).trans hff''\u27e9", "annotated_tactic": ["obtain \u27e8f', hf', hff'\u27e9 : \u2203 f' : \u03b1 \u2192\u209b G'', 0 \u2264 f' \u2227 <a>simpleFunc.toSimpleFunc</a> f =\u1d50[\u03bc] f' := by\n    obtain \u27e8f'', hf'', hff''\u27e9 := <a>exists_simpleFunc_nonneg_ae_eq</a> hf\n    exact \u27e8f'', hf'', (<a>Lp.simpleFunc.toSimpleFunc_eq_toFun</a> f).<a>trans</a> hff''\u27e9", [{"full_name": "MeasureTheory.Lp.simpleFunc.toSimpleFunc", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "def_pos": [587, 5], "def_end_pos": [587, 17]}, {"full_name": "MeasureTheory.Lp.simpleFunc.exists_simpleFunc_nonneg_ae_eq", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "def_pos": [837, 9], "def_end_pos": [837, 39]}, {"full_name": "MeasureTheory.Lp.simpleFunc.toSimpleFunc_eq_toFun", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "def_pos": [614, 9], "def_end_pos": [614, 30]}, {"full_name": "Filter.EventuallyEq.trans", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1503, 9], "def_end_pos": [1503, 27]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : NormedAddCommGroup F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : NormedField \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nG'' : Type u_7\nG' : Type u_8\ninst\u271d\u00b3 : NormedLatticeAddCommGroup G'\ninst\u271d\u00b2 : NormedSpace \u211d G'\ninst\u271d\u00b9 : NormedLatticeAddCommGroup G''\ninst\u271d : NormedSpace \u211d G''\nT : Set \u03b1 \u2192 G'' \u2192L[\u211d] G'\nh_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s = 0 \u2192 T s = 0\nh_add : FinMeasAdditive \u03bc T\nhT_nonneg : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 (x : G''), 0 \u2264 x \u2192 0 \u2264 \u2191(T s) x\nf : { x // x \u2208 simpleFunc G'' 1 \u03bc }\nhf : 0 \u2264 f\n\u22a2 0 \u2264 SimpleFunc.setToSimpleFunc T (toSimpleFunc f)", "state_after": "case intro.intro\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : NormedAddCommGroup F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : NormedField \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nG'' : Type u_7\nG' : Type u_8\ninst\u271d\u00b3 : NormedLatticeAddCommGroup G'\ninst\u271d\u00b2 : NormedSpace \u211d G'\ninst\u271d\u00b9 : NormedLatticeAddCommGroup G''\ninst\u271d : NormedSpace \u211d G''\nT : Set \u03b1 \u2192 G'' \u2192L[\u211d] G'\nh_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s = 0 \u2192 T s = 0\nh_add : FinMeasAdditive \u03bc T\nhT_nonneg : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 (x : G''), 0 \u2264 x \u2192 0 \u2264 \u2191(T s) x\nf : { x // x \u2208 simpleFunc G'' 1 \u03bc }\nhf : 0 \u2264 f\nf' : \u03b1 \u2192\u209b G''\nhf' : 0 \u2264 f'\nhff' : \u2191(toSimpleFunc f) =\u1d50[\u03bc] \u2191f'\n\u22a2 0 \u2264 SimpleFunc.setToSimpleFunc T (toSimpleFunc f)"}, {"tactic": "rw [SimpleFunc.setToSimpleFunc_congr _ h_zero h_add (SimpleFunc.integrable _) hff']", "annotated_tactic": ["rw [<a>SimpleFunc.setToSimpleFunc_congr</a> _ h_zero h_add (<a>SimpleFunc.integrable</a> _) hff']", [{"full_name": "MeasureTheory.SimpleFunc.setToSimpleFunc_congr", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [380, 9], "def_end_pos": [380, 30]}, {"full_name": "MeasureTheory.L1.SimpleFunc.integrable", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "def_pos": [1040, 19], "def_end_pos": [1040, 43]}]], "state_before": "case intro.intro\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : NormedAddCommGroup F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : NormedField \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nG'' : Type u_7\nG' : Type u_8\ninst\u271d\u00b3 : NormedLatticeAddCommGroup G'\ninst\u271d\u00b2 : NormedSpace \u211d G'\ninst\u271d\u00b9 : NormedLatticeAddCommGroup G''\ninst\u271d : NormedSpace \u211d G''\nT : Set \u03b1 \u2192 G'' \u2192L[\u211d] G'\nh_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s = 0 \u2192 T s = 0\nh_add : FinMeasAdditive \u03bc T\nhT_nonneg : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 (x : G''), 0 \u2264 x \u2192 0 \u2264 \u2191(T s) x\nf : { x // x \u2208 simpleFunc G'' 1 \u03bc }\nhf : 0 \u2264 f\nf' : \u03b1 \u2192\u209b G''\nhf' : 0 \u2264 f'\nhff' : \u2191(toSimpleFunc f) =\u1d50[\u03bc] \u2191f'\n\u22a2 0 \u2264 SimpleFunc.setToSimpleFunc T (toSimpleFunc f)", "state_after": "case intro.intro\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : NormedAddCommGroup F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : NormedField \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nG'' : Type u_7\nG' : Type u_8\ninst\u271d\u00b3 : NormedLatticeAddCommGroup G'\ninst\u271d\u00b2 : NormedSpace \u211d G'\ninst\u271d\u00b9 : NormedLatticeAddCommGroup G''\ninst\u271d : NormedSpace \u211d G''\nT : Set \u03b1 \u2192 G'' \u2192L[\u211d] G'\nh_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s = 0 \u2192 T s = 0\nh_add : FinMeasAdditive \u03bc T\nhT_nonneg : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 (x : G''), 0 \u2264 x \u2192 0 \u2264 \u2191(T s) x\nf : { x // x \u2208 simpleFunc G'' 1 \u03bc }\nhf : 0 \u2264 f\nf' : \u03b1 \u2192\u209b G''\nhf' : 0 \u2264 f'\nhff' : \u2191(toSimpleFunc f) =\u1d50[\u03bc] \u2191f'\n\u22a2 0 \u2264 SimpleFunc.setToSimpleFunc (fun s => T s) f'"}, {"tactic": "exact\n  SimpleFunc.setToSimpleFunc_nonneg' T hT_nonneg _ hf' ((SimpleFunc.integrable f).congr hff')", "annotated_tactic": ["exact\n    <a>SimpleFunc.setToSimpleFunc_nonneg'</a> T hT_nonneg _ hf' ((<a>SimpleFunc.integrable</a> f).<a>congr</a> hff')", [{"full_name": "MeasureTheory.SimpleFunc.setToSimpleFunc_nonneg'", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [540, 9], "def_end_pos": [540, 32]}, {"full_name": "MeasureTheory.L1.SimpleFunc.integrable", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "def_pos": [1040, 19], "def_end_pos": [1040, 43]}, {"full_name": "MeasureTheory.Integrable.congr", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [492, 9], "def_end_pos": [492, 25]}]], "state_before": "case intro.intro\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : NormedAddCommGroup F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : NormedField \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nG'' : Type u_7\nG' : Type u_8\ninst\u271d\u00b3 : NormedLatticeAddCommGroup G'\ninst\u271d\u00b2 : NormedSpace \u211d G'\ninst\u271d\u00b9 : NormedLatticeAddCommGroup G''\ninst\u271d : NormedSpace \u211d G''\nT : Set \u03b1 \u2192 G'' \u2192L[\u211d] G'\nh_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s = 0 \u2192 T s = 0\nh_add : FinMeasAdditive \u03bc T\nhT_nonneg : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 (x : G''), 0 \u2264 x \u2192 0 \u2264 \u2191(T s) x\nf : { x // x \u2208 simpleFunc G'' 1 \u03bc }\nhf : 0 \u2264 f\nf' : \u03b1 \u2192\u209b G''\nhf' : 0 \u2264 f'\nhff' : \u2191(toSimpleFunc f) =\u1d50[\u03bc] \u2191f'\n\u22a2 0 \u2264 SimpleFunc.setToSimpleFunc (fun s => T s) f'", "state_after": "no goals"}, {"tactic": "obtain \u27e8f'', hf'', hff''\u27e9 := exists_simpleFunc_nonneg_ae_eq hf", "annotated_tactic": ["obtain \u27e8f'', hf'', hff''\u27e9 := <a>exists_simpleFunc_nonneg_ae_eq</a> hf", [{"full_name": "MeasureTheory.Lp.simpleFunc.exists_simpleFunc_nonneg_ae_eq", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "def_pos": [837, 9], "def_end_pos": [837, 39]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : NormedAddCommGroup F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : NormedField \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nG'' : Type u_7\nG' : Type u_8\ninst\u271d\u00b3 : NormedLatticeAddCommGroup G'\ninst\u271d\u00b2 : NormedSpace \u211d G'\ninst\u271d\u00b9 : NormedLatticeAddCommGroup G''\ninst\u271d : NormedSpace \u211d G''\nT : Set \u03b1 \u2192 G'' \u2192L[\u211d] G'\nh_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s = 0 \u2192 T s = 0\nh_add : FinMeasAdditive \u03bc T\nhT_nonneg : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 (x : G''), 0 \u2264 x \u2192 0 \u2264 \u2191(T s) x\nf : { x // x \u2208 simpleFunc G'' 1 \u03bc }\nhf : 0 \u2264 f\n\u22a2 \u2203 f', 0 \u2264 f' \u2227 \u2191(toSimpleFunc f) =\u1d50[\u03bc] \u2191f'", "state_after": "case intro.intro\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : NormedAddCommGroup F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : NormedField \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nG'' : Type u_7\nG' : Type u_8\ninst\u271d\u00b3 : NormedLatticeAddCommGroup G'\ninst\u271d\u00b2 : NormedSpace \u211d G'\ninst\u271d\u00b9 : NormedLatticeAddCommGroup G''\ninst\u271d : NormedSpace \u211d G''\nT : Set \u03b1 \u2192 G'' \u2192L[\u211d] G'\nh_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s = 0 \u2192 T s = 0\nh_add : FinMeasAdditive \u03bc T\nhT_nonneg : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 (x : G''), 0 \u2264 x \u2192 0 \u2264 \u2191(T s) x\nf : { x // x \u2208 simpleFunc G'' 1 \u03bc }\nhf : 0 \u2264 f\nf'' : \u03b1 \u2192\u209b G''\nhf'' : 0 \u2264 f''\nhff'' : \u2191\u2191\u2191f =\u1d50[\u03bc] \u2191f''\n\u22a2 \u2203 f', 0 \u2264 f' \u2227 \u2191(toSimpleFunc f) =\u1d50[\u03bc] \u2191f'"}, {"tactic": "exact \u27e8f'', hf'', (Lp.simpleFunc.toSimpleFunc_eq_toFun f).trans hff''\u27e9", "annotated_tactic": ["exact \u27e8f'', hf'', (<a>Lp.simpleFunc.toSimpleFunc_eq_toFun</a> f).<a>trans</a> hff''\u27e9", [{"full_name": "MeasureTheory.Lp.simpleFunc.toSimpleFunc_eq_toFun", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "def_pos": [614, 9], "def_end_pos": [614, 30]}, {"full_name": "Filter.EventuallyEq.trans", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1503, 9], "def_end_pos": [1503, 27]}]], "state_before": "case intro.intro\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : NormedAddCommGroup F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : NormedField \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nG'' : Type u_7\nG' : Type u_8\ninst\u271d\u00b3 : NormedLatticeAddCommGroup G'\ninst\u271d\u00b2 : NormedSpace \u211d G'\ninst\u271d\u00b9 : NormedLatticeAddCommGroup G''\ninst\u271d : NormedSpace \u211d G''\nT : Set \u03b1 \u2192 G'' \u2192L[\u211d] G'\nh_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s = 0 \u2192 T s = 0\nh_add : FinMeasAdditive \u03bc T\nhT_nonneg : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 (x : G''), 0 \u2264 x \u2192 0 \u2264 \u2191(T s) x\nf : { x // x \u2208 simpleFunc G'' 1 \u03bc }\nhf : 0 \u2264 f\nf'' : \u03b1 \u2192\u209b G''\nhf'' : 0 \u2264 f''\nhff'' : \u2191\u2191\u2191f =\u1d50[\u03bc] \u2191f''\n\u22a2 \u2203 f', 0 \u2264 f' \u2227 \u2191(toSimpleFunc f) =\u1d50[\u03bc] \u2191f'", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Constructions/Polish.lean", "full_name": "Measurable.measurableSet_preimage_iff_of_surjective", "start": [547, 1], "end": [555, 39], "traced_tactics": [{"tactic": "refine \u27e8fun h => ?_, fun h => hf h\u27e9", "annotated_tactic": ["refine \u27e8fun h => ?_, fun h => hf h\u27e9", []], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\nX : Type u_3\nY : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u2077 : MeasurableSpace X\ninst\u271d\u2076 : StandardBorelSpace X\ninst\u271d\u2075 : TopologicalSpace Y\ninst\u271d\u2074 : T2Space Y\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : OpensMeasurableSpace Y\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : SecondCountableTopology Y\nf : X \u2192 Y\nhf : Measurable f\nhsurj : Surjective f\ns : Set Y\n\u22a2 MeasurableSet (f \u207b\u00b9' s) \u2194 MeasurableSet s", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\nX : Type u_3\nY : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u2077 : MeasurableSpace X\ninst\u271d\u2076 : StandardBorelSpace X\ninst\u271d\u2075 : TopologicalSpace Y\ninst\u271d\u2074 : T2Space Y\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : OpensMeasurableSpace Y\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : SecondCountableTopology Y\nf : X \u2192 Y\nhf : Measurable f\nhsurj : Surjective f\ns : Set Y\nh : MeasurableSet (f \u207b\u00b9' s)\n\u22a2 MeasurableSet s"}, {"tactic": "apply AnalyticSet.measurableSet_of_compl", "annotated_tactic": ["apply <a>AnalyticSet.measurableSet_of_compl</a>", [{"full_name": "MeasureTheory.AnalyticSet.measurableSet_of_compl", "def_path": "Mathlib/MeasureTheory/Constructions/Polish.lean", "def_pos": [524, 9], "def_end_pos": [524, 43]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\nX : Type u_3\nY : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u2077 : MeasurableSpace X\ninst\u271d\u2076 : StandardBorelSpace X\ninst\u271d\u2075 : TopologicalSpace Y\ninst\u271d\u2074 : T2Space Y\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : OpensMeasurableSpace Y\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : SecondCountableTopology Y\nf : X \u2192 Y\nhf : Measurable f\nhsurj : Surjective f\ns : Set Y\nh : MeasurableSet (f \u207b\u00b9' s)\n\u22a2 MeasurableSet s", "state_after": "case hs\n\u03b1 : Type u_1\n\u03b9 : Type u_2\nX : Type u_3\nY : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u2077 : MeasurableSpace X\ninst\u271d\u2076 : StandardBorelSpace X\ninst\u271d\u2075 : TopologicalSpace Y\ninst\u271d\u2074 : T2Space Y\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : OpensMeasurableSpace Y\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : SecondCountableTopology Y\nf : X \u2192 Y\nhf : Measurable f\nhsurj : Surjective f\ns : Set Y\nh : MeasurableSet (f \u207b\u00b9' s)\n\u22a2 AnalyticSet s\n\ncase hsc\n\u03b1 : Type u_1\n\u03b9 : Type u_2\nX : Type u_3\nY : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u2077 : MeasurableSpace X\ninst\u271d\u2076 : StandardBorelSpace X\ninst\u271d\u2075 : TopologicalSpace Y\ninst\u271d\u2074 : T2Space Y\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : OpensMeasurableSpace Y\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : SecondCountableTopology Y\nf : X \u2192 Y\nhf : Measurable f\nhsurj : Surjective f\ns : Set Y\nh : MeasurableSet (f \u207b\u00b9' s)\n\u22a2 AnalyticSet s\u1d9c"}, {"tactic": "rw [\u2190 image_preimage_eq s hsurj]", "annotated_tactic": ["rw [\u2190 <a>image_preimage_eq</a> s hsurj]", [{"full_name": "Set.image_preimage_eq", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [515, 9], "def_end_pos": [515, 26]}]], "state_before": "case hs\n\u03b1 : Type u_1\n\u03b9 : Type u_2\nX : Type u_3\nY : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u2077 : MeasurableSpace X\ninst\u271d\u2076 : StandardBorelSpace X\ninst\u271d\u2075 : TopologicalSpace Y\ninst\u271d\u2074 : T2Space Y\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : OpensMeasurableSpace Y\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : SecondCountableTopology Y\nf : X \u2192 Y\nhf : Measurable f\nhsurj : Surjective f\ns : Set Y\nh : MeasurableSet (f \u207b\u00b9' s)\n\u22a2 AnalyticSet s", "state_after": "case hs\n\u03b1 : Type u_1\n\u03b9 : Type u_2\nX : Type u_3\nY : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u2077 : MeasurableSpace X\ninst\u271d\u2076 : StandardBorelSpace X\ninst\u271d\u2075 : TopologicalSpace Y\ninst\u271d\u2074 : T2Space Y\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : OpensMeasurableSpace Y\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : SecondCountableTopology Y\nf : X \u2192 Y\nhf : Measurable f\nhsurj : Surjective f\ns : Set Y\nh : MeasurableSet (f \u207b\u00b9' s)\n\u22a2 AnalyticSet (f '' (f \u207b\u00b9' s))"}, {"tactic": "exact h.analyticSet_image hf", "annotated_tactic": ["exact h.analyticSet_image hf", []], "state_before": "case hs\n\u03b1 : Type u_1\n\u03b9 : Type u_2\nX : Type u_3\nY : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u2077 : MeasurableSpace X\ninst\u271d\u2076 : StandardBorelSpace X\ninst\u271d\u2075 : TopologicalSpace Y\ninst\u271d\u2074 : T2Space Y\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : OpensMeasurableSpace Y\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : SecondCountableTopology Y\nf : X \u2192 Y\nhf : Measurable f\nhsurj : Surjective f\ns : Set Y\nh : MeasurableSet (f \u207b\u00b9' s)\n\u22a2 AnalyticSet (f '' (f \u207b\u00b9' s))", "state_after": "no goals"}, {"tactic": "rw [\u2190 image_preimage_eq s\u1d9c hsurj]", "annotated_tactic": ["rw [\u2190 <a>image_preimage_eq</a> s\u1d9c hsurj]", [{"full_name": "Set.image_preimage_eq", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [515, 9], "def_end_pos": [515, 26]}]], "state_before": "case hsc\n\u03b1 : Type u_1\n\u03b9 : Type u_2\nX : Type u_3\nY : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u2077 : MeasurableSpace X\ninst\u271d\u2076 : StandardBorelSpace X\ninst\u271d\u2075 : TopologicalSpace Y\ninst\u271d\u2074 : T2Space Y\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : OpensMeasurableSpace Y\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : SecondCountableTopology Y\nf : X \u2192 Y\nhf : Measurable f\nhsurj : Surjective f\ns : Set Y\nh : MeasurableSet (f \u207b\u00b9' s)\n\u22a2 AnalyticSet s\u1d9c", "state_after": "case hsc\n\u03b1 : Type u_1\n\u03b9 : Type u_2\nX : Type u_3\nY : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u2077 : MeasurableSpace X\ninst\u271d\u2076 : StandardBorelSpace X\ninst\u271d\u2075 : TopologicalSpace Y\ninst\u271d\u2074 : T2Space Y\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : OpensMeasurableSpace Y\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : SecondCountableTopology Y\nf : X \u2192 Y\nhf : Measurable f\nhsurj : Surjective f\ns : Set Y\nh : MeasurableSet (f \u207b\u00b9' s)\n\u22a2 AnalyticSet (f '' (f \u207b\u00b9' s\u1d9c))"}, {"tactic": "exact h.compl.analyticSet_image hf", "annotated_tactic": ["exact h.compl.analyticSet_image hf", []], "state_before": "case hsc\n\u03b1 : Type u_1\n\u03b9 : Type u_2\nX : Type u_3\nY : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u2077 : MeasurableSpace X\ninst\u271d\u2076 : StandardBorelSpace X\ninst\u271d\u2075 : TopologicalSpace Y\ninst\u271d\u2074 : T2Space Y\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : OpensMeasurableSpace Y\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : SecondCountableTopology Y\nf : X \u2192 Y\nhf : Measurable f\nhsurj : Surjective f\ns : Set Y\nh : MeasurableSet (f \u207b\u00b9' s)\n\u22a2 AnalyticSet (f '' (f \u207b\u00b9' s\u1d9c))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "full_name": "MeasureTheory.setToFun_congr_smul_measure", "start": [1675, 1], "end": [1685, 67], "traced_tactics": [{"tactic": "by_cases hc0 : c = 0", "annotated_tactic": ["by_cases hc0 : c = 0", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nc : \u211d\u22650\u221e\nhc_ne_top : c \u2260 \u22a4\nhT : DominatedFinMeasAdditive \u03bc T C\nhT_smul : DominatedFinMeasAdditive (c \u2022 \u03bc) T C'\nf : \u03b1 \u2192 E\n\u22a2 setToFun \u03bc T hT f = setToFun (c \u2022 \u03bc) T hT_smul f", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nc : \u211d\u22650\u221e\nhc_ne_top : c \u2260 \u22a4\nhT : DominatedFinMeasAdditive \u03bc T C\nhT_smul : DominatedFinMeasAdditive (c \u2022 \u03bc) T C'\nf : \u03b1 \u2192 E\nhc0 : c = 0\n\u22a2 setToFun \u03bc T hT f = setToFun (c \u2022 \u03bc) T hT_smul f\n\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nc : \u211d\u22650\u221e\nhc_ne_top : c \u2260 \u22a4\nhT : DominatedFinMeasAdditive \u03bc T C\nhT_smul : DominatedFinMeasAdditive (c \u2022 \u03bc) T C'\nf : \u03b1 \u2192 E\nhc0 : \u00acc = 0\n\u22a2 setToFun \u03bc T hT f = setToFun (c \u2022 \u03bc) T hT_smul f"}, {"tactic": "refine' setToFun_congr_measure c\u207b\u00b9 c _ hc_ne_top (le_of_eq _) le_rfl hT hT_smul f", "annotated_tactic": ["refine' <a>setToFun_congr_measure</a> c\u207b\u00b9 c _ hc_ne_top (<a>le_of_eq</a> _) <a>le_rfl</a> hT hT_smul f", [{"full_name": "MeasureTheory.setToFun_congr_measure", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [1634, 9], "def_end_pos": [1634, 31]}, {"full_name": "le_of_eq", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [72, 9], "def_end_pos": [72, 17]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}]], "state_before": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nc : \u211d\u22650\u221e\nhc_ne_top : c \u2260 \u22a4\nhT : DominatedFinMeasAdditive \u03bc T C\nhT_smul : DominatedFinMeasAdditive (c \u2022 \u03bc) T C'\nf : \u03b1 \u2192 E\nhc0 : \u00acc = 0\n\u22a2 setToFun \u03bc T hT f = setToFun (c \u2022 \u03bc) T hT_smul f", "state_after": "case neg.refine'_1\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nc : \u211d\u22650\u221e\nhc_ne_top : c \u2260 \u22a4\nhT : DominatedFinMeasAdditive \u03bc T C\nhT_smul : DominatedFinMeasAdditive (c \u2022 \u03bc) T C'\nf : \u03b1 \u2192 E\nhc0 : \u00acc = 0\n\u22a2 c\u207b\u00b9 \u2260 \u22a4\n\ncase neg.refine'_2\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nc : \u211d\u22650\u221e\nhc_ne_top : c \u2260 \u22a4\nhT : DominatedFinMeasAdditive \u03bc T C\nhT_smul : DominatedFinMeasAdditive (c \u2022 \u03bc) T C'\nf : \u03b1 \u2192 E\nhc0 : \u00acc = 0\n\u22a2 \u03bc = c\u207b\u00b9 \u2022 c \u2022 \u03bc"}, {"tactic": "simp [hc0] at hT_smul", "annotated_tactic": ["simp [hc0] at hT_smul", []], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nc : \u211d\u22650\u221e\nhc_ne_top : c \u2260 \u22a4\nhT : DominatedFinMeasAdditive \u03bc T C\nhT_smul : DominatedFinMeasAdditive (c \u2022 \u03bc) T C'\nf : \u03b1 \u2192 E\nhc0 : c = 0\n\u22a2 setToFun \u03bc T hT f = setToFun (c \u2022 \u03bc) T hT_smul f", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nc : \u211d\u22650\u221e\nhc_ne_top : c \u2260 \u22a4\nhT : DominatedFinMeasAdditive \u03bc T C\nhT_smul\u271d : DominatedFinMeasAdditive (c \u2022 \u03bc) T C'\nf : \u03b1 \u2192 E\nhc0 : c = 0\nhT_smul : DominatedFinMeasAdditive 0 T C'\n\u22a2 setToFun \u03bc T hT f = setToFun (c \u2022 \u03bc) T hT_smul\u271d f"}, {"tactic": "have h : \u2200 s, MeasurableSet s \u2192 \u03bc s < \u221e \u2192 T s = 0 := fun s hs _ => hT_smul.eq_zero hs", "annotated_tactic": ["have h : \u2200 s, <a>MeasurableSet</a> s \u2192 \u03bc s < \u221e \u2192 T s = 0 := fun s hs _ => hT_smul.eq_zero hs", [{"full_name": "MeasurableSet", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [64, 5], "def_end_pos": [64, 18]}]], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nc : \u211d\u22650\u221e\nhc_ne_top : c \u2260 \u22a4\nhT : DominatedFinMeasAdditive \u03bc T C\nhT_smul\u271d : DominatedFinMeasAdditive (c \u2022 \u03bc) T C'\nf : \u03b1 \u2192 E\nhc0 : c = 0\nhT_smul : DominatedFinMeasAdditive 0 T C'\n\u22a2 setToFun \u03bc T hT f = setToFun (c \u2022 \u03bc) T hT_smul\u271d f", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nc : \u211d\u22650\u221e\nhc_ne_top : c \u2260 \u22a4\nhT : DominatedFinMeasAdditive \u03bc T C\nhT_smul\u271d : DominatedFinMeasAdditive (c \u2022 \u03bc) T C'\nf : \u03b1 \u2192 E\nhc0 : c = 0\nhT_smul : DominatedFinMeasAdditive 0 T C'\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T s = 0\n\u22a2 setToFun \u03bc T hT f = setToFun (c \u2022 \u03bc) T hT_smul\u271d f"}, {"tactic": "rw [setToFun_zero_left' _ h, setToFun_measure_zero]", "annotated_tactic": ["rw [<a>setToFun_zero_left'</a> _ h, <a>setToFun_measure_zero</a>]", [{"full_name": "MeasureTheory.setToFun_zero_left'", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [1358, 9], "def_end_pos": [1358, 28]}, {"full_name": "MeasureTheory.setToFun_measure_zero", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [1427, 9], "def_end_pos": [1427, 30]}]], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nc : \u211d\u22650\u221e\nhc_ne_top : c \u2260 \u22a4\nhT : DominatedFinMeasAdditive \u03bc T C\nhT_smul\u271d : DominatedFinMeasAdditive (c \u2022 \u03bc) T C'\nf : \u03b1 \u2192 E\nhc0 : c = 0\nhT_smul : DominatedFinMeasAdditive 0 T C'\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T s = 0\n\u22a2 setToFun \u03bc T hT f = setToFun (c \u2022 \u03bc) T hT_smul\u271d f", "state_after": "case pos.h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nc : \u211d\u22650\u221e\nhc_ne_top : c \u2260 \u22a4\nhT : DominatedFinMeasAdditive \u03bc T C\nhT_smul\u271d : DominatedFinMeasAdditive (c \u2022 \u03bc) T C'\nf : \u03b1 \u2192 E\nhc0 : c = 0\nhT_smul : DominatedFinMeasAdditive 0 T C'\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T s = 0\n\u22a2 c \u2022 \u03bc = 0"}, {"tactic": "simp [hc0]", "annotated_tactic": ["simp [hc0]", []], "state_before": "case pos.h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nc : \u211d\u22650\u221e\nhc_ne_top : c \u2260 \u22a4\nhT : DominatedFinMeasAdditive \u03bc T C\nhT_smul\u271d : DominatedFinMeasAdditive (c \u2022 \u03bc) T C'\nf : \u03b1 \u2192 E\nhc0 : c = 0\nhT_smul : DominatedFinMeasAdditive 0 T C'\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T s = 0\n\u22a2 c \u2022 \u03bc = 0", "state_after": "no goals"}, {"tactic": "simp [hc0]", "annotated_tactic": ["simp [hc0]", []], "state_before": "case neg.refine'_1\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nc : \u211d\u22650\u221e\nhc_ne_top : c \u2260 \u22a4\nhT : DominatedFinMeasAdditive \u03bc T C\nhT_smul : DominatedFinMeasAdditive (c \u2022 \u03bc) T C'\nf : \u03b1 \u2192 E\nhc0 : \u00acc = 0\n\u22a2 c\u207b\u00b9 \u2260 \u22a4", "state_after": "no goals"}, {"tactic": "rw [smul_smul, ENNReal.inv_mul_cancel hc0 hc_ne_top, one_smul]", "annotated_tactic": ["rw [<a>smul_smul</a>, <a>ENNReal.inv_mul_cancel</a> hc0 hc_ne_top, <a>one_smul</a>]", [{"full_name": "smul_smul", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [484, 9], "def_end_pos": [484, 18]}, {"full_name": "ENNReal.inv_mul_cancel", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1424, 19], "def_end_pos": [1424, 33]}, {"full_name": "one_smul", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [492, 9], "def_end_pos": [492, 17]}]], "state_before": "case neg.refine'_2\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nc : \u211d\u22650\u221e\nhc_ne_top : c \u2260 \u22a4\nhT : DominatedFinMeasAdditive \u03bc T C\nhT_smul : DominatedFinMeasAdditive (c \u2022 \u03bc) T C'\nf : \u03b1 \u2192 E\nhc0 : \u00acc = 0\n\u22a2 \u03bc = c\u207b\u00b9 \u2022 c \u2022 \u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "List.toFinset_surjective", "start": [3343, 1], "end": [3345, 11], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "full_name": "MeasureTheory.snormEssSup_indicator_eq_snormEssSup_restrict", "start": [642, 1], "end": [654, 18], "traced_tactics": [{"tactic": "simp_rw [snormEssSup, nnnorm_indicator_eq_indicator_nnnorm, ENNReal.coe_indicator]", "annotated_tactic": ["simp_rw [<a>snormEssSup</a>, <a>nnnorm_indicator_eq_indicator_nnnorm</a>, <a>ENNReal.coe_indicator</a>]", [{"full_name": "MeasureTheory.snormEssSup", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [78, 5], "def_end_pos": [78, 16]}, {"full_name": "nnnorm_indicator_eq_indicator_nnnorm", "def_path": "Mathlib/Analysis/NormedSpace/IndicatorFunction.lean", "def_pos": [29, 9], "def_end_pos": [29, 45]}, {"full_name": "ENNReal.coe_indicator", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [548, 9], "def_end_pos": [548, 22]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nc : E\nf\u271d : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f\u271d \u03bc\ns : Set \u03b1\nf : \u03b1 \u2192 F\nhs : MeasurableSet s\n\u22a2 snormEssSup (Set.indicator s f) \u03bc = snormEssSup f (Measure.restrict \u03bc s)", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nc : E\nf\u271d : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f\u271d \u03bc\ns : Set \u03b1\nf : \u03b1 \u2192 F\nhs : MeasurableSet s\n\u22a2 essSup (fun x => Set.indicator s (fun x => \u2191\u2016f x\u2016\u208a) x) \u03bc = essSup (fun x => \u2191\u2016f x\u2016\u208a) (Measure.restrict \u03bc s)"}, {"tactic": "by_cases hs_null : \u03bc s = 0", "annotated_tactic": ["by_cases hs_null : \u03bc s = 0", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nc : E\nf\u271d : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f\u271d \u03bc\ns : Set \u03b1\nf : \u03b1 \u2192 F\nhs : MeasurableSet s\n\u22a2 essSup (fun x => Set.indicator s (fun x => \u2191\u2016f x\u2016\u208a) x) \u03bc = essSup (fun x => \u2191\u2016f x\u2016\u208a) (Measure.restrict \u03bc s)", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nc : E\nf\u271d : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f\u271d \u03bc\ns : Set \u03b1\nf : \u03b1 \u2192 F\nhs : MeasurableSet s\nhs_null : \u2191\u2191\u03bc s = 0\n\u22a2 essSup (fun x => Set.indicator s (fun x => \u2191\u2016f x\u2016\u208a) x) \u03bc = essSup (fun x => \u2191\u2016f x\u2016\u208a) (Measure.restrict \u03bc s)\n\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nc : E\nf\u271d : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f\u271d \u03bc\ns : Set \u03b1\nf : \u03b1 \u2192 F\nhs : MeasurableSet s\nhs_null : \u00ac\u2191\u2191\u03bc s = 0\n\u22a2 essSup (fun x => Set.indicator s (fun x => \u2191\u2016f x\u2016\u208a) x) \u03bc = essSup (fun x => \u2191\u2016f x\u2016\u208a) (Measure.restrict \u03bc s)"}, {"tactic": "rw [essSup_indicator_eq_essSup_restrict (eventually_of_forall fun x => ?_) hs hs_null]", "annotated_tactic": ["rw [<a>essSup_indicator_eq_essSup_restrict</a> (<a>eventually_of_forall</a> fun x => ?_) hs hs_null]", [{"full_name": "essSup_indicator_eq_essSup_restrict", "def_path": "Mathlib/MeasureTheory/Function/EssSup.lean", "def_pos": [268, 9], "def_end_pos": [268, 44]}, {"full_name": "Filter.eventually_of_forall", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1111, 9], "def_end_pos": [1111, 29]}]], "state_before": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nc : E\nf\u271d : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f\u271d \u03bc\ns : Set \u03b1\nf : \u03b1 \u2192 F\nhs : MeasurableSet s\nhs_null : \u00ac\u2191\u2191\u03bc s = 0\n\u22a2 essSup (fun x => Set.indicator s (fun x => \u2191\u2016f x\u2016\u208a) x) \u03bc = essSup (fun x => \u2191\u2016f x\u2016\u208a) (Measure.restrict \u03bc s)", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nc : E\nf\u271d : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f\u271d \u03bc\ns : Set \u03b1\nf : \u03b1 \u2192 F\nhs : MeasurableSet s\nhs_null : \u00ac\u2191\u2191\u03bc s = 0\nx : \u03b1\n\u22a2 OfNat.ofNat 0 x \u2264 \u2191\u2016f x\u2016\u208a"}, {"tactic": "rw [Pi.zero_apply]", "annotated_tactic": ["rw [<a>Pi.zero_apply</a>]", [{"full_name": "Pi.zero_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [46, 3], "def_end_pos": [46, 14]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nc : E\nf\u271d : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f\u271d \u03bc\ns : Set \u03b1\nf : \u03b1 \u2192 F\nhs : MeasurableSet s\nhs_null : \u00ac\u2191\u2191\u03bc s = 0\nx : \u03b1\n\u22a2 OfNat.ofNat 0 x \u2264 \u2191\u2016f x\u2016\u208a", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nc : E\nf\u271d : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f\u271d \u03bc\ns : Set \u03b1\nf : \u03b1 \u2192 F\nhs : MeasurableSet s\nhs_null : \u00ac\u2191\u2191\u03bc s = 0\nx : \u03b1\n\u22a2 0 \u2264 \u2191\u2016f x\u2016\u208a"}, {"tactic": "exact zero_le _", "annotated_tactic": ["exact <a>zero_le</a> _", [{"full_name": "zero_le", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [217, 30], "def_end_pos": [217, 37]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nc : E\nf\u271d : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f\u271d \u03bc\ns : Set \u03b1\nf : \u03b1 \u2192 F\nhs : MeasurableSet s\nhs_null : \u00ac\u2191\u2191\u03bc s = 0\nx : \u03b1\n\u22a2 0 \u2264 \u2191\u2016f x\u2016\u208a", "state_after": "no goals"}, {"tactic": "rw [Measure.restrict_zero_set hs_null]", "annotated_tactic": ["rw [<a>Measure.restrict_zero_set</a> hs_null]", [{"full_name": "MeasureTheory.Measure.restrict_zero_set", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1691, 9], "def_end_pos": [1691, 26]}]], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nc : E\nf\u271d : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f\u271d \u03bc\ns : Set \u03b1\nf : \u03b1 \u2192 F\nhs : MeasurableSet s\nhs_null : \u2191\u2191\u03bc s = 0\n\u22a2 essSup (fun x => Set.indicator s (fun x => \u2191\u2016f x\u2016\u208a) x) \u03bc = essSup (fun x => \u2191\u2016f x\u2016\u208a) (Measure.restrict \u03bc s)", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nc : E\nf\u271d : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f\u271d \u03bc\ns : Set \u03b1\nf : \u03b1 \u2192 F\nhs : MeasurableSet s\nhs_null : \u2191\u2191\u03bc s = 0\n\u22a2 essSup (fun x => Set.indicator s (fun x => \u2191\u2016f x\u2016\u208a) x) \u03bc = essSup (fun x => \u2191\u2016f x\u2016\u208a) 0"}, {"tactic": "simp only [essSup_measure_zero, ENNReal.essSup_eq_zero_iff, ENNReal.bot_eq_zero]", "annotated_tactic": ["simp only [<a>essSup_measure_zero</a>, <a>ENNReal.essSup_eq_zero_iff</a>, <a>ENNReal.bot_eq_zero</a>]", [{"full_name": "essSup_measure_zero", "def_path": "Mathlib/MeasureTheory/Function/EssSup.lean", "def_pos": [145, 9], "def_end_pos": [145, 28]}, {"full_name": "ENNReal.essSup_eq_zero_iff", "def_path": "Mathlib/MeasureTheory/Function/EssSup.lean", "def_pos": [309, 9], "def_end_pos": [309, 27]}, {"full_name": "ENNReal.bot_eq_zero", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [682, 9], "def_end_pos": [682, 20]}]], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nc : E\nf\u271d : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f\u271d \u03bc\ns : Set \u03b1\nf : \u03b1 \u2192 F\nhs : MeasurableSet s\nhs_null : \u2191\u2191\u03bc s = 0\n\u22a2 essSup (fun x => Set.indicator s (fun x => \u2191\u2016f x\u2016\u208a) x) \u03bc = essSup (fun x => \u2191\u2016f x\u2016\u208a) 0", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nc : E\nf\u271d : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f\u271d \u03bc\ns : Set \u03b1\nf : \u03b1 \u2192 F\nhs : MeasurableSet s\nhs_null : \u2191\u2191\u03bc s = 0\n\u22a2 (fun x => Set.indicator s (fun x => \u2191\u2016f x\u2016\u208a) x) =\u1d50[\u03bc] 0"}, {"tactic": "have hs_empty : s =\u1d50[\u03bc] (\u2205 : Set \u03b1) := by rw [ae_eq_set]; simpa using hs_null", "annotated_tactic": ["have hs_empty : s =\u1d50[\u03bc] (\u2205 : <a>Set</a> \u03b1) := by rw [<a>ae_eq_set</a>]; simpa using hs_null", [{"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}, {"full_name": "MeasureTheory.ae_eq_set", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [497, 9], "def_end_pos": [497, 18]}]], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nc : E\nf\u271d : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f\u271d \u03bc\ns : Set \u03b1\nf : \u03b1 \u2192 F\nhs : MeasurableSet s\nhs_null : \u2191\u2191\u03bc s = 0\n\u22a2 (fun x => Set.indicator s (fun x => \u2191\u2016f x\u2016\u208a) x) =\u1d50[\u03bc] 0", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nc : E\nf\u271d : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f\u271d \u03bc\ns : Set \u03b1\nf : \u03b1 \u2192 F\nhs : MeasurableSet s\nhs_null : \u2191\u2191\u03bc s = 0\nhs_empty : s =\u1d50[\u03bc] \u2205\n\u22a2 (fun x => Set.indicator s (fun x => \u2191\u2016f x\u2016\u208a) x) =\u1d50[\u03bc] 0"}, {"tactic": "refine' (indicator_ae_eq_of_ae_eq_set hs_empty).trans _", "annotated_tactic": ["refine' (<a>indicator_ae_eq_of_ae_eq_set</a> hs_empty).<a>trans</a> _", [{"full_name": "indicator_ae_eq_of_ae_eq_set", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [4512, 9], "def_end_pos": [4512, 37]}, {"full_name": "Filter.EventuallyEq.trans", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1503, 9], "def_end_pos": [1503, 27]}]], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nc : E\nf\u271d : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f\u271d \u03bc\ns : Set \u03b1\nf : \u03b1 \u2192 F\nhs : MeasurableSet s\nhs_null : \u2191\u2191\u03bc s = 0\nhs_empty : s =\u1d50[\u03bc] \u2205\n\u22a2 (fun x => Set.indicator s (fun x => \u2191\u2016f x\u2016\u208a) x) =\u1d50[\u03bc] 0", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nc : E\nf\u271d : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f\u271d \u03bc\ns : Set \u03b1\nf : \u03b1 \u2192 F\nhs : MeasurableSet s\nhs_null : \u2191\u2191\u03bc s = 0\nhs_empty : s =\u1d50[\u03bc] \u2205\n\u22a2 (Set.indicator \u2205 fun x => \u2191\u2016f x\u2016\u208a) =\u1d50[\u03bc] 0"}, {"tactic": "rw [Set.indicator_empty]", "annotated_tactic": ["rw [<a>Set.indicator_empty</a>]", [{"full_name": "Set.indicator_empty", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [193, 3], "def_end_pos": [193, 14]}]], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nc : E\nf\u271d : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f\u271d \u03bc\ns : Set \u03b1\nf : \u03b1 \u2192 F\nhs : MeasurableSet s\nhs_null : \u2191\u2191\u03bc s = 0\nhs_empty : s =\u1d50[\u03bc] \u2205\n\u22a2 (Set.indicator \u2205 fun x => \u2191\u2016f x\u2016\u208a) =\u1d50[\u03bc] 0", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nc : E\nf\u271d : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f\u271d \u03bc\ns : Set \u03b1\nf : \u03b1 \u2192 F\nhs : MeasurableSet s\nhs_null : \u2191\u2191\u03bc s = 0\nhs_empty : s =\u1d50[\u03bc] \u2205\n\u22a2 (fun x => 0) =\u1d50[\u03bc] 0"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nc : E\nf\u271d : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f\u271d \u03bc\ns : Set \u03b1\nf : \u03b1 \u2192 F\nhs : MeasurableSet s\nhs_null : \u2191\u2191\u03bc s = 0\nhs_empty : s =\u1d50[\u03bc] \u2205\n\u22a2 (fun x => 0) =\u1d50[\u03bc] 0", "state_after": "no goals"}, {"tactic": "rw [ae_eq_set]", "annotated_tactic": ["rw [<a>ae_eq_set</a>]", [{"full_name": "MeasureTheory.ae_eq_set", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [497, 9], "def_end_pos": [497, 18]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nc : E\nf\u271d : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f\u271d \u03bc\ns : Set \u03b1\nf : \u03b1 \u2192 F\nhs : MeasurableSet s\nhs_null : \u2191\u2191\u03bc s = 0\n\u22a2 s =\u1d50[\u03bc] \u2205", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nc : E\nf\u271d : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f\u271d \u03bc\ns : Set \u03b1\nf : \u03b1 \u2192 F\nhs : MeasurableSet s\nhs_null : \u2191\u2191\u03bc s = 0\n\u22a2 \u2191\u2191\u03bc (s \\ \u2205) = 0 \u2227 \u2191\u2191\u03bc (\u2205 \\ s) = 0"}, {"tactic": "simpa using hs_null", "annotated_tactic": ["simpa using hs_null", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nc : E\nf\u271d : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f\u271d \u03bc\ns : Set \u03b1\nf : \u03b1 \u2192 F\nhs : MeasurableSet s\nhs_null : \u2191\u2191\u03bc s = 0\n\u22a2 \u2191\u2191\u03bc (s \\ \u2205) = 0 \u2227 \u2191\u2191\u03bc (\u2205 \\ s) = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Variance.lean", "full_name": "ProbabilityTheory.evariance_lt_top_iff_mem\u2112p", "start": [94, 1], "end": [99, 28], "traced_tactics": [{"tactic": "refine' \u27e8_, MeasureTheory.Mem\u2112p.evariance_lt_top\u27e9", "annotated_tactic": ["refine' \u27e8_, <a>MeasureTheory.Mem\u2112p.evariance_lt_top</a>\u27e9", [{"full_name": "MeasureTheory.Mem\u2112p.evariance_lt_top", "def_path": "Mathlib/Probability/Variance.lean", "def_pos": [67, 9], "def_end_pos": [67, 52]}]], "state_before": "\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d : IsFiniteMeasure \u03bc\nhX : AEStronglyMeasurable X \u03bc\n\u22a2 evariance X \u03bc < \u22a4 \u2194 Mem\u2112p X 2", "state_after": "\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d : IsFiniteMeasure \u03bc\nhX : AEStronglyMeasurable X \u03bc\n\u22a2 evariance X \u03bc < \u22a4 \u2192 Mem\u2112p X 2"}, {"tactic": "contrapose", "annotated_tactic": ["contrapose", []], "state_before": "\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d : IsFiniteMeasure \u03bc\nhX : AEStronglyMeasurable X \u03bc\n\u22a2 evariance X \u03bc < \u22a4 \u2192 Mem\u2112p X 2", "state_after": "\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d : IsFiniteMeasure \u03bc\nhX : AEStronglyMeasurable X \u03bc\n\u22a2 \u00acMem\u2112p X 2 \u2192 \u00acevariance X \u03bc < \u22a4"}, {"tactic": "rw [not_lt, top_le_iff]", "annotated_tactic": ["rw [<a>not_lt</a>, <a>top_le_iff</a>]", [{"full_name": "not_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [368, 9], "def_end_pos": [368, 15]}, {"full_name": "top_le_iff", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [157, 9], "def_end_pos": [157, 19]}]], "state_before": "\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d : IsFiniteMeasure \u03bc\nhX : AEStronglyMeasurable X \u03bc\n\u22a2 \u00acMem\u2112p X 2 \u2192 \u00acevariance X \u03bc < \u22a4", "state_after": "\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d : IsFiniteMeasure \u03bc\nhX : AEStronglyMeasurable X \u03bc\n\u22a2 \u00acMem\u2112p X 2 \u2192 evariance X \u03bc = \u22a4"}, {"tactic": "exact evariance_eq_top hX", "annotated_tactic": ["exact <a>evariance_eq_top</a> hX", [{"full_name": "ProbabilityTheory.evariance_eq_top", "def_path": "Mathlib/Probability/Variance.lean", "def_pos": [77, 9], "def_end_pos": [77, 25]}]], "state_before": "\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d : IsFiniteMeasure \u03bc\nhX : AEStronglyMeasurable X \u03bc\n\u22a2 \u00acMem\u2112p X 2 \u2192 evariance X \u03bc = \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL2.lean", "full_name": "MeasureTheory.condexpL2_indicator_ae_eq_smul", "start": [308, 1], "end": [318, 45], "traced_tactics": [{"tactic": "rw [indicatorConstLp_eq_toSpanSingleton_compLp hs h\u03bcs x]", "annotated_tactic": ["rw [<a>indicatorConstLp_eq_toSpanSingleton_compLp</a> hs h\u03bcs x]", [{"full_name": "MeasureTheory.indicatorConstLp_eq_toSpanSingleton_compLp", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [1204, 9], "def_end_pos": [1204, 51]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2075 : CompleteSpace E\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b3 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b2 : CompleteSpace E'\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup G\ninst\u271d\u2077 : NormedAddCommGroup G'\ninst\u271d\u2076 : NormedSpace \u211d G'\ninst\u271d\u2075 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2074 : IsROrC \ud835\udd5c'\ninst\u271d\u00b3 : NormedAddCommGroup E''\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b9 : CompleteSpace E''\ninst\u271d : NormedSpace \u211d E''\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx : E'\n\u22a2 \u2191\u2191\u2191(\u2191(condexpL2 E' \ud835\udd5c hm) (indicatorConstLp 2 hs h\u03bcs x)) =\u1d50[\u03bc] fun a =>\n    \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)) a \u2022 x", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2075 : CompleteSpace E\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b3 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b2 : CompleteSpace E'\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup G\ninst\u271d\u2077 : NormedAddCommGroup G'\ninst\u271d\u2076 : NormedSpace \u211d G'\ninst\u271d\u2075 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2074 : IsROrC \ud835\udd5c'\ninst\u271d\u00b3 : NormedAddCommGroup E''\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b9 : CompleteSpace E''\ninst\u271d : NormedSpace \u211d E''\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx : E'\n\u22a2 \u2191\u2191\u2191(\u2191(condexpL2 E' \ud835\udd5c hm) (compLp (toSpanSingleton \u211d x) (indicatorConstLp 2 hs h\u03bcs 1))) =\u1d50[\u03bc] fun a =>\n    \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)) a \u2022 x"}, {"tactic": "have h_comp :=\n  condexpL2_comp_continuousLinearMap \u211d \ud835\udd5c hm (toSpanSingleton \u211d x)\n    (indicatorConstLp 2 hs h\u03bcs (1 : \u211d))", "annotated_tactic": ["have h_comp :=\n    <a>condexpL2_comp_continuousLinearMap</a> \u211d \ud835\udd5c hm (<a>toSpanSingleton</a> \u211d x)\n      (<a>indicatorConstLp</a> 2 hs h\u03bcs (1 : \u211d))", [{"full_name": "MeasureTheory.condexpL2_comp_continuousLinearMap", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL2.lean", "def_pos": [281, 9], "def_end_pos": [281, 43]}, {"full_name": "ContinuousLinearMap.toSpanSingleton", "def_path": "Mathlib/Topology/Algebra/Module/Basic.lean", "def_pos": [1242, 5], "def_end_pos": [1242, 20]}, {"full_name": "MeasureTheory.indicatorConstLp", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [744, 5], "def_end_pos": [744, 21]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2075 : CompleteSpace E\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b3 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b2 : CompleteSpace E'\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup G\ninst\u271d\u2077 : NormedAddCommGroup G'\ninst\u271d\u2076 : NormedSpace \u211d G'\ninst\u271d\u2075 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2074 : IsROrC \ud835\udd5c'\ninst\u271d\u00b3 : NormedAddCommGroup E''\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b9 : CompleteSpace E''\ninst\u271d : NormedSpace \u211d E''\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx : E'\n\u22a2 \u2191\u2191\u2191(\u2191(condexpL2 E' \ud835\udd5c hm) (compLp (toSpanSingleton \u211d x) (indicatorConstLp 2 hs h\u03bcs 1))) =\u1d50[\u03bc] fun a =>\n    \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)) a \u2022 x", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2075 : CompleteSpace E\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b3 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b2 : CompleteSpace E'\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup G\ninst\u271d\u2077 : NormedAddCommGroup G'\ninst\u271d\u2076 : NormedSpace \u211d G'\ninst\u271d\u2075 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2074 : IsROrC \ud835\udd5c'\ninst\u271d\u00b3 : NormedAddCommGroup E''\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b9 : CompleteSpace E''\ninst\u271d : NormedSpace \u211d E''\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx : E'\nh_comp :\n  \u2191\u2191\u2191(\u2191(condexpL2 E' \ud835\udd5c hm) (compLp (toSpanSingleton \u211d x) (indicatorConstLp 2 hs h\u03bcs 1))) =\u1d50[\u03bc]\n    \u2191\u2191(compLp (toSpanSingleton \u211d x) \u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)))\n\u22a2 \u2191\u2191\u2191(\u2191(condexpL2 E' \ud835\udd5c hm) (compLp (toSpanSingleton \u211d x) (indicatorConstLp 2 hs h\u03bcs 1))) =\u1d50[\u03bc] fun a =>\n    \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)) a \u2022 x"}, {"tactic": "refine' h_comp.trans _", "annotated_tactic": ["refine' h_comp.trans _", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2075 : CompleteSpace E\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b3 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b2 : CompleteSpace E'\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup G\ninst\u271d\u2077 : NormedAddCommGroup G'\ninst\u271d\u2076 : NormedSpace \u211d G'\ninst\u271d\u2075 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2074 : IsROrC \ud835\udd5c'\ninst\u271d\u00b3 : NormedAddCommGroup E''\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b9 : CompleteSpace E''\ninst\u271d : NormedSpace \u211d E''\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx : E'\nh_comp :\n  \u2191\u2191\u2191(\u2191(condexpL2 E' \ud835\udd5c hm) (compLp (toSpanSingleton \u211d x) (indicatorConstLp 2 hs h\u03bcs 1))) =\u1d50[\u03bc]\n    \u2191\u2191(compLp (toSpanSingleton \u211d x) \u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)))\n\u22a2 \u2191\u2191\u2191(\u2191(condexpL2 E' \ud835\udd5c hm) (compLp (toSpanSingleton \u211d x) (indicatorConstLp 2 hs h\u03bcs 1))) =\u1d50[\u03bc] fun a =>\n    \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)) a \u2022 x", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2075 : CompleteSpace E\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b3 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b2 : CompleteSpace E'\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup G\ninst\u271d\u2077 : NormedAddCommGroup G'\ninst\u271d\u2076 : NormedSpace \u211d G'\ninst\u271d\u2075 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2074 : IsROrC \ud835\udd5c'\ninst\u271d\u00b3 : NormedAddCommGroup E''\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b9 : CompleteSpace E''\ninst\u271d : NormedSpace \u211d E''\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx : E'\nh_comp :\n  \u2191\u2191\u2191(\u2191(condexpL2 E' \ud835\udd5c hm) (compLp (toSpanSingleton \u211d x) (indicatorConstLp 2 hs h\u03bcs 1))) =\u1d50[\u03bc]\n    \u2191\u2191(compLp (toSpanSingleton \u211d x) \u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)))\n\u22a2 \u2191\u2191(compLp (toSpanSingleton \u211d x) \u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) =\u1d50[\u03bc] fun a =>\n    \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)) a \u2022 x"}, {"tactic": "exact (toSpanSingleton \u211d x).coeFn_compLp _", "annotated_tactic": ["exact (<a>toSpanSingleton</a> \u211d x).<a>coeFn_compLp</a> _", [{"full_name": "ContinuousLinearMap.toSpanSingleton", "def_path": "Mathlib/Topology/Algebra/Module/Basic.lean", "def_pos": [1242, 5], "def_end_pos": [1242, 20]}, {"full_name": "ContinuousLinearMap.coeFn_compLp", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [1090, 9], "def_end_pos": [1090, 21]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2075 : CompleteSpace E\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b3 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b2 : CompleteSpace E'\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup G\ninst\u271d\u2077 : NormedAddCommGroup G'\ninst\u271d\u2076 : NormedSpace \u211d G'\ninst\u271d\u2075 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2074 : IsROrC \ud835\udd5c'\ninst\u271d\u00b3 : NormedAddCommGroup E''\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b9 : CompleteSpace E''\ninst\u271d : NormedSpace \u211d E''\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx : E'\nh_comp :\n  \u2191\u2191\u2191(\u2191(condexpL2 E' \ud835\udd5c hm) (compLp (toSpanSingleton \u211d x) (indicatorConstLp 2 hs h\u03bcs 1))) =\u1d50[\u03bc]\n    \u2191\u2191(compLp (toSpanSingleton \u211d x) \u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)))\n\u22a2 \u2191\u2191(compLp (toSpanSingleton \u211d x) \u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) =\u1d50[\u03bc] fun a =>\n    \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)) a \u2022 x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/Rename.lean", "full_name": "MvPolynomial.renameEquiv_refl", "start": [164, 1], "end": [165, 25], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Group/Measure.lean", "full_name": "MeasureTheory.Measure.isHaarMeasure_of_isCompact_nonempty_interior", "start": [779, 1], "end": [784, 83], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Constructions/Prod/Basic.lean", "full_name": "measurable_measure_prod_mk_left", "start": [179, 1], "end": [185, 50], "traced_tactics": [{"tactic": "have : \u2200 x, MeasurableSet (Prod.mk x \u207b\u00b9' s) := fun x => measurable_prod_mk_left hs", "annotated_tactic": ["have : \u2200 x, <a>MeasurableSet</a> (<a>Prod.mk</a> x \u207b\u00b9' s) := fun x => <a>measurable_prod_mk_left</a> hs", [{"full_name": "MeasurableSet", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [64, 5], "def_end_pos": [64, 18]}, {"full_name": "Prod.mk", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [466, 16], "def_end_pos": [466, 41]}, {"full_name": "measurable_prod_mk_left", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [736, 9], "def_end_pos": [736, 32]}]], "state_before": "\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\ns : Set (\u03b1 \u00d7 \u03b2)\nhs : MeasurableSet s\n\u22a2 Measurable fun x => \u2191\u2191\u03bd (Prod.mk x \u207b\u00b9' s)", "state_after": "\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\ns : Set (\u03b1 \u00d7 \u03b2)\nhs : MeasurableSet s\nthis : \u2200 (x : \u03b1), MeasurableSet (Prod.mk x \u207b\u00b9' s)\n\u22a2 Measurable fun x => \u2191\u2191\u03bd (Prod.mk x \u207b\u00b9' s)"}, {"tactic": "simp only [\u2190 @iSup_restrict_spanningSets _ _ \u03bd, this]", "annotated_tactic": ["simp only [\u2190 @<a>iSup_restrict_spanningSets</a> _ _ \u03bd, this]", [{"full_name": "MeasureTheory.Measure.iSup_restrict_spanningSets", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3385, 9], "def_end_pos": [3385, 35]}]], "state_before": "\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\ns : Set (\u03b1 \u00d7 \u03b2)\nhs : MeasurableSet s\nthis : \u2200 (x : \u03b1), MeasurableSet (Prod.mk x \u207b\u00b9' s)\n\u22a2 Measurable fun x => \u2191\u2191\u03bd (Prod.mk x \u207b\u00b9' s)", "state_after": "\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\ns : Set (\u03b1 \u00d7 \u03b2)\nhs : MeasurableSet s\nthis : \u2200 (x : \u03b1), MeasurableSet (Prod.mk x \u207b\u00b9' s)\n\u22a2 Measurable fun x => \u2a06 i, \u2191\u2191(Measure.restrict \u03bd (spanningSets \u03bd i)) (Prod.mk x \u207b\u00b9' s)"}, {"tactic": "apply measurable_iSup", "annotated_tactic": ["apply <a>measurable_iSup</a>", [{"full_name": "measurable_iSup", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [1360, 9], "def_end_pos": [1360, 24]}]], "state_before": "\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\ns : Set (\u03b1 \u00d7 \u03b2)\nhs : MeasurableSet s\nthis : \u2200 (x : \u03b1), MeasurableSet (Prod.mk x \u207b\u00b9' s)\n\u22a2 Measurable fun x => \u2a06 i, \u2191\u2191(Measure.restrict \u03bd (spanningSets \u03bd i)) (Prod.mk x \u207b\u00b9' s)", "state_after": "case hf\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\ns : Set (\u03b1 \u00d7 \u03b2)\nhs : MeasurableSet s\nthis : \u2200 (x : \u03b1), MeasurableSet (Prod.mk x \u207b\u00b9' s)\n\u22a2 \u2200 (i : \u2115), Measurable fun b => \u2191\u2191(Measure.restrict \u03bd (spanningSets \u03bd i)) (Prod.mk b \u207b\u00b9' s)"}, {"tactic": "intro i", "annotated_tactic": ["intro i", []], "state_before": "case hf\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\ns : Set (\u03b1 \u00d7 \u03b2)\nhs : MeasurableSet s\nthis : \u2200 (x : \u03b1), MeasurableSet (Prod.mk x \u207b\u00b9' s)\n\u22a2 \u2200 (i : \u2115), Measurable fun b => \u2191\u2191(Measure.restrict \u03bd (spanningSets \u03bd i)) (Prod.mk b \u207b\u00b9' s)", "state_after": "case hf\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\ns : Set (\u03b1 \u00d7 \u03b2)\nhs : MeasurableSet s\nthis : \u2200 (x : \u03b1), MeasurableSet (Prod.mk x \u207b\u00b9' s)\ni : \u2115\n\u22a2 Measurable fun b => \u2191\u2191(Measure.restrict \u03bd (spanningSets \u03bd i)) (Prod.mk b \u207b\u00b9' s)"}, {"tactic": "haveI := Fact.mk (measure_spanningSets_lt_top \u03bd i)", "annotated_tactic": ["haveI := <a>Fact.mk</a> (<a>measure_spanningSets_lt_top</a> \u03bd i)", [{"full_name": "Fact.mk", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [115, 12], "def_end_pos": [115, 22]}, {"full_name": "MeasureTheory.measure_spanningSets_lt_top", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3329, 9], "def_end_pos": [3329, 36]}]], "state_before": "case hf\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\ns : Set (\u03b1 \u00d7 \u03b2)\nhs : MeasurableSet s\nthis : \u2200 (x : \u03b1), MeasurableSet (Prod.mk x \u207b\u00b9' s)\ni : \u2115\n\u22a2 Measurable fun b => \u2191\u2191(Measure.restrict \u03bd (spanningSets \u03bd i)) (Prod.mk b \u207b\u00b9' s)", "state_after": "case hf\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\ns : Set (\u03b1 \u00d7 \u03b2)\nhs : MeasurableSet s\nthis\u271d : \u2200 (x : \u03b1), MeasurableSet (Prod.mk x \u207b\u00b9' s)\ni : \u2115\nthis : Fact (\u2191\u2191\u03bd (spanningSets \u03bd i) < \u22a4)\n\u22a2 Measurable fun b => \u2191\u2191(Measure.restrict \u03bd (spanningSets \u03bd i)) (Prod.mk b \u207b\u00b9' s)"}, {"tactic": "exact measurable_measure_prod_mk_left_finite hs", "annotated_tactic": ["exact <a>measurable_measure_prod_mk_left_finite</a> hs", [{"full_name": "measurable_measure_prod_mk_left_finite", "def_path": "Mathlib/MeasureTheory/Constructions/Prod/Basic.lean", "def_pos": [157, 9], "def_end_pos": [157, 47]}]], "state_before": "case hf\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\ns : Set (\u03b1 \u00d7 \u03b2)\nhs : MeasurableSet s\nthis\u271d : \u2200 (x : \u03b1), MeasurableSet (Prod.mk x \u207b\u00b9' s)\ni : \u2115\nthis : Fact (\u2191\u2191\u03bd (spanningSets \u03bd i) < \u22a4)\n\u22a2 Measurable fun b => \u2191\u2191(Measure.restrict \u03bd (spanningSets \u03bd i)) (Prod.mk b \u207b\u00b9' s)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean", "full_name": "MeasureTheory.integrableOn_Ioi_comp_rpow_iff", "start": [902, 1], "end": [924, 75], "traced_tactics": [{"tactic": "let S := Ioi (0 : \u211d)", "annotated_tactic": ["let S := <a>Ioi</a> (0 : \u211d)", [{"full_name": "Set.Ioi", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [79, 5], "def_end_pos": [79, 8]}]], "state_before": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\np : \u211d\nhp : p \u2260 0\n\u22a2 IntegrableOn (fun x => (|p| * x ^ (p - 1)) \u2022 f (x ^ p)) (Ioi 0) \u2194 IntegrableOn f (Ioi 0)", "state_after": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\np : \u211d\nhp : p \u2260 0\nS : Set \u211d := Ioi 0\n\u22a2 IntegrableOn (fun x => (|p| * x ^ (p - 1)) \u2022 f (x ^ p)) (Ioi 0) \u2194 IntegrableOn f (Ioi 0)"}, {"tactic": "have a1 : \u2200 x : \u211d, x \u2208 S \u2192 HasDerivWithinAt (fun t : \u211d => t ^ p) (p * x ^ (p - 1)) S x :=\n  fun x hx => (hasDerivAt_rpow_const (Or.inl (mem_Ioi.mp hx).ne')).hasDerivWithinAt", "annotated_tactic": ["have a1 : \u2200 x : \u211d, x \u2208 S \u2192 <a>HasDerivWithinAt</a> (fun t : \u211d => t ^ p) (p * x ^ (p - 1)) S x :=\n    fun x hx => (<a>hasDerivAt_rpow_const</a> (<a>Or.inl</a> (mem_Ioi.mp hx).<a>ne'</a>)).<a>hasDerivWithinAt</a>", [{"full_name": "HasDerivWithinAt", "def_path": "Mathlib/Analysis/Calculus/Deriv/Basic.lean", "def_pos": [115, 5], "def_end_pos": [115, 21]}, {"full_name": "Real.hasDerivAt_rpow_const", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Deriv.lean", "def_pos": [350, 9], "def_end_pos": [350, 30]}, {"full_name": "Or.inl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [517, 5], "def_end_pos": [517, 8]}, {"full_name": "LT.lt.ne'", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [328, 9], "def_end_pos": [328, 12]}, {"full_name": "HasDerivAt.hasDerivWithinAt", "def_path": "Mathlib/Analysis/Calculus/Deriv/Basic.lean", "def_pos": [388, 9], "def_end_pos": [388, 36]}]], "state_before": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\np : \u211d\nhp : p \u2260 0\nS : Set \u211d := Ioi 0\n\u22a2 IntegrableOn (fun x => (|p| * x ^ (p - 1)) \u2022 f (x ^ p)) (Ioi 0) \u2194 IntegrableOn f (Ioi 0)", "state_after": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\np : \u211d\nhp : p \u2260 0\nS : Set \u211d := Ioi 0\na1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x\n\u22a2 IntegrableOn (fun x => (|p| * x ^ (p - 1)) \u2022 f (x ^ p)) (Ioi 0) \u2194 IntegrableOn f (Ioi 0)"}, {"tactic": "have := integrableOn_image_iff_integrableOn_abs_deriv_smul measurableSet_Ioi a1 a2 f", "annotated_tactic": ["have := <a>integrableOn_image_iff_integrableOn_abs_deriv_smul</a> <a>measurableSet_Ioi</a> a1 a2 f", [{"full_name": "MeasureTheory.integrableOn_image_iff_integrableOn_abs_deriv_smul", "def_path": "Mathlib/MeasureTheory/Function/Jacobian.lean", "def_pos": [1237, 9], "def_end_pos": [1237, 59]}, {"full_name": "measurableSet_Ioi", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [579, 9], "def_end_pos": [579, 26]}]], "state_before": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\np : \u211d\nhp : p \u2260 0\nS : Set \u211d := Ioi 0\na1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x\na2 : InjOn (fun x => x ^ p) S\na3 : (fun t => t ^ p) '' S = S\n\u22a2 IntegrableOn (fun x => (|p| * x ^ (p - 1)) \u2022 f (x ^ p)) (Ioi 0) \u2194 IntegrableOn f (Ioi 0)", "state_after": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\np : \u211d\nhp : p \u2260 0\nS : Set \u211d := Ioi 0\na1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x\na2 : InjOn (fun x => x ^ p) S\na3 : (fun t => t ^ p) '' S = S\nthis : IntegrableOn f ((fun t => t ^ p) '' Ioi 0) \u2194 IntegrableOn (fun x => |p * x ^ (p - 1)| \u2022 f (x ^ p)) (Ioi 0)\n\u22a2 IntegrableOn (fun x => (|p| * x ^ (p - 1)) \u2022 f (x ^ p)) (Ioi 0) \u2194 IntegrableOn f (Ioi 0)"}, {"tactic": "rw [a3] at this", "annotated_tactic": ["rw [a3] at this", []], "state_before": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\np : \u211d\nhp : p \u2260 0\nS : Set \u211d := Ioi 0\na1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x\na2 : InjOn (fun x => x ^ p) S\na3 : (fun t => t ^ p) '' S = S\nthis : IntegrableOn f ((fun t => t ^ p) '' Ioi 0) \u2194 IntegrableOn (fun x => |p * x ^ (p - 1)| \u2022 f (x ^ p)) (Ioi 0)\n\u22a2 IntegrableOn (fun x => (|p| * x ^ (p - 1)) \u2022 f (x ^ p)) (Ioi 0) \u2194 IntegrableOn f (Ioi 0)", "state_after": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\np : \u211d\nhp : p \u2260 0\nS : Set \u211d := Ioi 0\na1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x\na2 : InjOn (fun x => x ^ p) S\na3 : (fun t => t ^ p) '' S = S\nthis : IntegrableOn f S \u2194 IntegrableOn (fun x => |p * x ^ (p - 1)| \u2022 f (x ^ p)) (Ioi 0)\n\u22a2 IntegrableOn (fun x => (|p| * x ^ (p - 1)) \u2022 f (x ^ p)) (Ioi 0) \u2194 IntegrableOn f (Ioi 0)"}, {"tactic": "rw [this]", "annotated_tactic": ["rw [this]", []], "state_before": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\np : \u211d\nhp : p \u2260 0\nS : Set \u211d := Ioi 0\na1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x\na2 : InjOn (fun x => x ^ p) S\na3 : (fun t => t ^ p) '' S = S\nthis : IntegrableOn f S \u2194 IntegrableOn (fun x => |p * x ^ (p - 1)| \u2022 f (x ^ p)) (Ioi 0)\n\u22a2 IntegrableOn (fun x => (|p| * x ^ (p - 1)) \u2022 f (x ^ p)) (Ioi 0) \u2194 IntegrableOn f (Ioi 0)", "state_after": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\np : \u211d\nhp : p \u2260 0\nS : Set \u211d := Ioi 0\na1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x\na2 : InjOn (fun x => x ^ p) S\na3 : (fun t => t ^ p) '' S = S\nthis : IntegrableOn f S \u2194 IntegrableOn (fun x => |p * x ^ (p - 1)| \u2022 f (x ^ p)) (Ioi 0)\n\u22a2 IntegrableOn (fun x => (|p| * x ^ (p - 1)) \u2022 f (x ^ p)) (Ioi 0) \u2194\n    IntegrableOn (fun x => |p * x ^ (p - 1)| \u2022 f (x ^ p)) (Ioi 0)"}, {"tactic": "refine' integrableOn_congr_fun (fun x hx => _) measurableSet_Ioi", "annotated_tactic": ["refine' <a>integrableOn_congr_fun</a> (fun x hx => _) <a>measurableSet_Ioi</a>", [{"full_name": "MeasureTheory.integrableOn_congr_fun", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [158, 9], "def_end_pos": [158, 31]}, {"full_name": "measurableSet_Ioi", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [579, 9], "def_end_pos": [579, 26]}]], "state_before": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\np : \u211d\nhp : p \u2260 0\nS : Set \u211d := Ioi 0\na1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x\na2 : InjOn (fun x => x ^ p) S\na3 : (fun t => t ^ p) '' S = S\nthis : IntegrableOn f S \u2194 IntegrableOn (fun x => |p * x ^ (p - 1)| \u2022 f (x ^ p)) (Ioi 0)\n\u22a2 IntegrableOn (fun x => (|p| * x ^ (p - 1)) \u2022 f (x ^ p)) (Ioi 0) \u2194\n    IntegrableOn (fun x => |p * x ^ (p - 1)| \u2022 f (x ^ p)) (Ioi 0)", "state_after": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\np : \u211d\nhp : p \u2260 0\nS : Set \u211d := Ioi 0\na1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x\na2 : InjOn (fun x => x ^ p) S\na3 : (fun t => t ^ p) '' S = S\nthis : IntegrableOn f S \u2194 IntegrableOn (fun x => |p * x ^ (p - 1)| \u2022 f (x ^ p)) (Ioi 0)\nx : \u211d\nhx : x \u2208 Ioi 0\n\u22a2 (|p| * x ^ (p - 1)) \u2022 f (x ^ p) = |p * x ^ (p - 1)| \u2022 f (x ^ p)"}, {"tactic": "simp_rw [abs_mul, abs_of_nonneg (rpow_nonneg_of_nonneg (le_of_lt hx) _)]", "annotated_tactic": ["simp_rw [<a>abs_mul</a>, <a>abs_of_nonneg</a> (<a>rpow_nonneg_of_nonneg</a> (<a>le_of_lt</a> hx) _)]", [{"full_name": "abs_mul", "def_path": "Mathlib/Algebra/Order/Ring/Abs.lean", "def_pos": [33, 9], "def_end_pos": [33, 16]}, {"full_name": "abs_of_nonneg", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [107, 9], "def_end_pos": [107, 22]}, {"full_name": "Real.rpow_nonneg_of_nonneg", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [141, 9], "def_end_pos": [141, 30]}, {"full_name": "le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [110, 9], "def_end_pos": [110, 17]}]], "state_before": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\np : \u211d\nhp : p \u2260 0\nS : Set \u211d := Ioi 0\na1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x\na2 : InjOn (fun x => x ^ p) S\na3 : (fun t => t ^ p) '' S = S\nthis : IntegrableOn f S \u2194 IntegrableOn (fun x => |p * x ^ (p - 1)| \u2022 f (x ^ p)) (Ioi 0)\nx : \u211d\nhx : x \u2208 Ioi 0\n\u22a2 (|p| * x ^ (p - 1)) \u2022 f (x ^ p) = |p * x ^ (p - 1)| \u2022 f (x ^ p)", "state_after": "no goals"}, {"tactic": "rcases lt_or_gt_of_ne hp with (h | h)", "annotated_tactic": ["rcases <a>lt_or_gt_of_ne</a> hp with (h | h)", [{"full_name": "lt_or_gt_of_ne", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [352, 9], "def_end_pos": [352, 23]}]], "state_before": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\np : \u211d\nhp : p \u2260 0\nS : Set \u211d := Ioi 0\na1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x\n\u22a2 InjOn (fun x => x ^ p) S", "state_after": "case inl\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\np : \u211d\nhp : p \u2260 0\nS : Set \u211d := Ioi 0\na1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x\nh : p < 0\n\u22a2 InjOn (fun x => x ^ p) S\n\ncase inr\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\np : \u211d\nhp : p \u2260 0\nS : Set \u211d := Ioi 0\na1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x\nh : p > 0\n\u22a2 InjOn (fun x => x ^ p) S"}, {"tactic": "exact StrictMonoOn.injOn fun x hx y _hy hxy => rpow_lt_rpow (mem_Ioi.mp hx).le hxy h", "annotated_tactic": ["exact <a>StrictMonoOn.injOn</a> fun x hx y _hy hxy => <a>rpow_lt_rpow</a> (mem_Ioi.mp hx).<a>le</a> hxy h", [{"full_name": "StrictMonoOn.injOn", "def_path": "Mathlib/Data/Set/Function.lean", "def_pos": [1579, 9], "def_end_pos": [1579, 27]}, {"full_name": "Real.rpow_lt_rpow", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [420, 9], "def_end_pos": [420, 21]}, {"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [142, 7], "def_end_pos": [142, 15]}]], "state_before": "case inr\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\np : \u211d\nhp : p \u2260 0\nS : Set \u211d := Ioi 0\na1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x\nh : p > 0\n\u22a2 InjOn (fun x => x ^ p) S", "state_after": "no goals"}, {"tactic": "apply StrictAntiOn.injOn", "annotated_tactic": ["apply <a>StrictAntiOn.injOn</a>", [{"full_name": "StrictAntiOn.injOn", "def_path": "Mathlib/Data/Set/Function.lean", "def_pos": [1584, 9], "def_end_pos": [1584, 27]}]], "state_before": "case inl\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\np : \u211d\nhp : p \u2260 0\nS : Set \u211d := Ioi 0\na1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x\nh : p < 0\n\u22a2 InjOn (fun x => x ^ p) S", "state_after": "case inl.H\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\np : \u211d\nhp : p \u2260 0\nS : Set \u211d := Ioi 0\na1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x\nh : p < 0\n\u22a2 StrictAntiOn (fun x => x ^ p) S"}, {"tactic": "intro x hx y hy hxy", "annotated_tactic": ["intro x hx y hy hxy", []], "state_before": "case inl.H\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\np : \u211d\nhp : p \u2260 0\nS : Set \u211d := Ioi 0\na1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x\nh : p < 0\n\u22a2 StrictAntiOn (fun x => x ^ p) S", "state_after": "case inl.H\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\np : \u211d\nhp : p \u2260 0\nS : Set \u211d := Ioi 0\na1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x\nh : p < 0\nx : \u211d\nhx : x \u2208 S\ny : \u211d\nhy : y \u2208 S\nhxy : x < y\n\u22a2 (fun x => x ^ p) y < (fun x => x ^ p) x"}, {"tactic": "rw [\u2190 inv_lt_inv (rpow_pos_of_pos hx p) (rpow_pos_of_pos hy p), \u2190 rpow_neg (le_of_lt hx), \u2190\n  rpow_neg (le_of_lt hy)]", "annotated_tactic": ["rw [\u2190 <a>inv_lt_inv</a> (<a>rpow_pos_of_pos</a> hx p) (<a>rpow_pos_of_pos</a> hy p), \u2190 <a>rpow_neg</a> (<a>le_of_lt</a> hx), \u2190\n        <a>rpow_neg</a> (<a>le_of_lt</a> hy)]", [{"full_name": "inv_lt_inv", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [275, 9], "def_end_pos": [275, 19]}, {"full_name": "Real.rpow_pos_of_pos", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [92, 9], "def_end_pos": [92, 24]}, {"full_name": "Real.rpow_pos_of_pos", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [92, 9], "def_end_pos": [92, 24]}, {"full_name": "Real.rpow_neg", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [221, 9], "def_end_pos": [221, 17]}, {"full_name": "le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [110, 9], "def_end_pos": [110, 17]}, {"full_name": "Real.rpow_neg", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [221, 9], "def_end_pos": [221, 17]}, {"full_name": "le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [110, 9], "def_end_pos": [110, 17]}]], "state_before": "case inl.H\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\np : \u211d\nhp : p \u2260 0\nS : Set \u211d := Ioi 0\na1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x\nh : p < 0\nx : \u211d\nhx : x \u2208 S\ny : \u211d\nhy : y \u2208 S\nhxy : x < y\n\u22a2 (fun x => x ^ p) y < (fun x => x ^ p) x", "state_after": "case inl.H\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\np : \u211d\nhp : p \u2260 0\nS : Set \u211d := Ioi 0\na1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x\nh : p < 0\nx : \u211d\nhx : x \u2208 S\ny : \u211d\nhy : y \u2208 S\nhxy : x < y\n\u22a2 x ^ (-p) < y ^ (-p)"}, {"tactic": "exact rpow_lt_rpow (le_of_lt hx) hxy (neg_pos.mpr h)", "annotated_tactic": ["exact <a>rpow_lt_rpow</a> (<a>le_of_lt</a> hx) hxy (neg_pos.mpr h)", [{"full_name": "Real.rpow_lt_rpow", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [420, 9], "def_end_pos": [420, 21]}, {"full_name": "le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [110, 9], "def_end_pos": [110, 17]}]], "state_before": "case inl.H\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\np : \u211d\nhp : p \u2260 0\nS : Set \u211d := Ioi 0\na1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x\nh : p < 0\nx : \u211d\nhx : x \u2208 S\ny : \u211d\nhy : y \u2208 S\nhxy : x < y\n\u22a2 x ^ (-p) < y ^ (-p)", "state_after": "no goals"}, {"tactic": "ext1 x", "annotated_tactic": ["ext1 x", []], "state_before": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\np : \u211d\nhp : p \u2260 0\nS : Set \u211d := Ioi 0\na1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x\na2 : InjOn (fun x => x ^ p) S\n\u22a2 (fun t => t ^ p) '' S = S", "state_after": "case h\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\np : \u211d\nhp : p \u2260 0\nS : Set \u211d := Ioi 0\na1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x\na2 : InjOn (fun x => x ^ p) S\nx : \u211d\n\u22a2 x \u2208 (fun t => t ^ p) '' S \u2194 x \u2208 S"}, {"tactic": "rw [mem_image]", "annotated_tactic": ["rw [<a>mem_image</a>]", [{"full_name": "Set.mem_image", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [231, 9], "def_end_pos": [231, 18]}]], "state_before": "case h\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\np : \u211d\nhp : p \u2260 0\nS : Set \u211d := Ioi 0\na1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x\na2 : InjOn (fun x => x ^ p) S\nx : \u211d\n\u22a2 x \u2208 (fun t => t ^ p) '' S \u2194 x \u2208 S", "state_after": "case h\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\np : \u211d\nhp : p \u2260 0\nS : Set \u211d := Ioi 0\na1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x\na2 : InjOn (fun x => x ^ p) S\nx : \u211d\n\u22a2 (\u2203 x_1, x_1 \u2208 S \u2227 x_1 ^ p = x) \u2194 x \u2208 S"}, {"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "case h\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\np : \u211d\nhp : p \u2260 0\nS : Set \u211d := Ioi 0\na1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x\na2 : InjOn (fun x => x ^ p) S\nx : \u211d\n\u22a2 (\u2203 x_1, x_1 \u2208 S \u2227 x_1 ^ p = x) \u2194 x \u2208 S", "state_after": "case h.mp\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\np : \u211d\nhp : p \u2260 0\nS : Set \u211d := Ioi 0\na1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x\na2 : InjOn (fun x => x ^ p) S\nx : \u211d\n\u22a2 (\u2203 x_1, x_1 \u2208 S \u2227 x_1 ^ p = x) \u2192 x \u2208 S\n\ncase h.mpr\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\np : \u211d\nhp : p \u2260 0\nS : Set \u211d := Ioi 0\na1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x\na2 : InjOn (fun x => x ^ p) S\nx : \u211d\n\u22a2 x \u2208 S \u2192 \u2203 x_1, x_1 \u2208 S \u2227 x_1 ^ p = x"}, {"tactic": "rintro \u27e8y, hy, rfl\u27e9", "annotated_tactic": ["rintro \u27e8y, hy, rfl\u27e9", []], "state_before": "case h.mp\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\np : \u211d\nhp : p \u2260 0\nS : Set \u211d := Ioi 0\na1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x\na2 : InjOn (fun x => x ^ p) S\nx : \u211d\n\u22a2 (\u2203 x_1, x_1 \u2208 S \u2227 x_1 ^ p = x) \u2192 x \u2208 S", "state_after": "case h.mp.intro.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\np : \u211d\nhp : p \u2260 0\nS : Set \u211d := Ioi 0\na1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x\na2 : InjOn (fun x => x ^ p) S\ny : \u211d\nhy : y \u2208 S\n\u22a2 y ^ p \u2208 S"}, {"tactic": "exact rpow_pos_of_pos hy p", "annotated_tactic": ["exact <a>rpow_pos_of_pos</a> hy p", [{"full_name": "Real.rpow_pos_of_pos", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [92, 9], "def_end_pos": [92, 24]}]], "state_before": "case h.mp.intro.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\np : \u211d\nhp : p \u2260 0\nS : Set \u211d := Ioi 0\na1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x\na2 : InjOn (fun x => x ^ p) S\ny : \u211d\nhy : y \u2208 S\n\u22a2 y ^ p \u2208 S", "state_after": "no goals"}, {"tactic": "intro hx", "annotated_tactic": ["intro hx", []], "state_before": "case h.mpr\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\np : \u211d\nhp : p \u2260 0\nS : Set \u211d := Ioi 0\na1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x\na2 : InjOn (fun x => x ^ p) S\nx : \u211d\n\u22a2 x \u2208 S \u2192 \u2203 x_1, x_1 \u2208 S \u2227 x_1 ^ p = x", "state_after": "case h.mpr\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\np : \u211d\nhp : p \u2260 0\nS : Set \u211d := Ioi 0\na1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x\na2 : InjOn (fun x => x ^ p) S\nx : \u211d\nhx : x \u2208 S\n\u22a2 \u2203 x_1, x_1 \u2208 S \u2227 x_1 ^ p = x"}, {"tactic": "refine' \u27e8x ^ (1 / p), rpow_pos_of_pos hx _, _\u27e9", "annotated_tactic": ["refine' \u27e8x ^ (1 / p), <a>rpow_pos_of_pos</a> hx _, _\u27e9", [{"full_name": "Real.rpow_pos_of_pos", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [92, 9], "def_end_pos": [92, 24]}]], "state_before": "case h.mpr\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\np : \u211d\nhp : p \u2260 0\nS : Set \u211d := Ioi 0\na1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x\na2 : InjOn (fun x => x ^ p) S\nx : \u211d\nhx : x \u2208 S\n\u22a2 \u2203 x_1, x_1 \u2208 S \u2227 x_1 ^ p = x", "state_after": "case h.mpr\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\np : \u211d\nhp : p \u2260 0\nS : Set \u211d := Ioi 0\na1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x\na2 : InjOn (fun x => x ^ p) S\nx : \u211d\nhx : x \u2208 S\n\u22a2 (x ^ (1 / p)) ^ p = x"}, {"tactic": "rw [\u2190 rpow_mul (le_of_lt hx), one_div_mul_cancel hp, rpow_one]", "annotated_tactic": ["rw [\u2190 <a>rpow_mul</a> (<a>le_of_lt</a> hx), <a>one_div_mul_cancel</a> hp, <a>rpow_one</a>]", [{"full_name": "Real.rpow_mul", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [317, 9], "def_end_pos": [317, 17]}, {"full_name": "le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [110, 9], "def_end_pos": [110, 17]}, {"full_name": "one_div_mul_cancel", "def_path": "Mathlib/Algebra/GroupWithZero/Units/Lemmas.lean", "def_pos": [79, 9], "def_end_pos": [79, 27]}, {"full_name": "Real.rpow_one", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [126, 9], "def_end_pos": [126, 17]}]], "state_before": "case h.mpr\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\np : \u211d\nhp : p \u2260 0\nS : Set \u211d := Ioi 0\na1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x\na2 : InjOn (fun x => x ^ p) S\nx : \u211d\nhx : x \u2208 S\n\u22a2 (x ^ (1 / p)) ^ p = x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Covering/Besicovitch.lean", "full_name": "Besicovitch.exist_disjoint_covering_families", "start": [471, 1], "end": [530, 56], "traced_tactics": [{"tactic": "cases isEmpty_or_nonempty \u03b2", "annotated_tactic": ["cases <a>isEmpty_or_nonempty</a> \u03b2", [{"full_name": "isEmpty_or_nonempty", "def_path": "Mathlib/Logic/IsEmpty.lean", "def_pos": [207, 9], "def_end_pos": [207, 28]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\n\u03b2 : Type u\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nq : BallPackage \u03b2 \u03b1\n\u22a2 \u2203 s,\n    (\u2200 (i : Fin N), PairwiseDisjoint (s i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)) \u2227\n      range q.c \u2286 \u22c3 i, \u22c3 j \u2208 s i, ball (BallPackage.c q j) (BallPackage.r q j)", "state_after": "case inl\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\n\u03b2 : Type u\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nq : BallPackage \u03b2 \u03b1\nh\u271d : IsEmpty \u03b2\n\u22a2 \u2203 s,\n    (\u2200 (i : Fin N), PairwiseDisjoint (s i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)) \u2227\n      range q.c \u2286 \u22c3 i, \u22c3 j \u2208 s i, ball (BallPackage.c q j) (BallPackage.r q j)\n\ncase inr\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\n\u03b2 : Type u\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nq : BallPackage \u03b2 \u03b1\nh\u271d : Nonempty \u03b2\n\u22a2 \u2203 s,\n    (\u2200 (i : Fin N), PairwiseDisjoint (s i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)) \u2227\n      range q.c \u2286 \u22c3 i, \u22c3 j \u2208 s i, ball (BallPackage.c q j) (BallPackage.r q j)"}, {"tactic": "let p : TauPackage \u03b2 \u03b1 :=\n  { q with\n    \u03c4\n    one_lt_tau := h\u03c4 }", "annotated_tactic": ["let p : <a>TauPackage</a> \u03b2 \u03b1 :=\n    { <a>q</a> with\n      \u03c4\n      one_lt_tau := h\u03c4 }", [{"full_name": "Besicovitch.TauPackage", "def_path": "Mathlib/MeasureTheory/Covering/Besicovitch.lean", "def_pos": [222, 11], "def_end_pos": [222, 21]}, {"full_name": "Besicovitch.BallPackage.r_le", "def_path": "Mathlib/MeasureTheory/Covering/Besicovitch.lean", "def_pos": [197, 3], "def_end_pos": [197, 7]}]], "state_before": "case inr\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\n\u03b2 : Type u\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nq : BallPackage \u03b2 \u03b1\nh\u271d : Nonempty \u03b2\n\u22a2 \u2203 s,\n    (\u2200 (i : Fin N), PairwiseDisjoint (s i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)) \u2227\n      range q.c \u2286 \u22c3 i, \u22c3 j \u2208 s i, ball (BallPackage.c q j) (BallPackage.r q j)", "state_after": "case inr\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\n\u03b2 : Type u\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nq : BallPackage \u03b2 \u03b1\nh\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1 :=\n  {\n    toBallPackage :=\n      { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n        r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n    \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\n\u22a2 \u2203 s,\n    (\u2200 (i : Fin N), PairwiseDisjoint (s i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)) \u2227\n      range q.c \u2286 \u22c3 i, \u22c3 j \u2208 s i, ball (BallPackage.c q j) (BallPackage.r q j)"}, {"tactic": "let s := fun i : Fin N =>\n  \u22c3 (k : Ordinal.{u}) (_ : k < p.lastStep) (_ : p.color k = i), ({p.index k} : Set \u03b2)", "annotated_tactic": ["let s := fun i : <a>Fin</a> N =>\n    \u22c3 (k : <a>Ordinal</a>.{u}) (_ : k < p.lastStep) (_ : p.color k = i), ({p.index k} : <a>Set</a> \u03b2)", [{"full_name": "Fin", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1745, 11], "def_end_pos": [1745, 14]}, {"full_name": "Ordinal", "def_path": "Mathlib/SetTheory/Ordinal/Basic.lean", "def_pos": [152, 5], "def_end_pos": [152, 12]}, {"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}]], "state_before": "case inr\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\n\u03b2 : Type u\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nq : BallPackage \u03b2 \u03b1\nh\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1 :=\n  {\n    toBallPackage :=\n      { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n        r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n    \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\n\u22a2 \u2203 s,\n    (\u2200 (i : Fin N), PairwiseDisjoint (s i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)) \u2227\n      range q.c \u2286 \u22c3 i, \u22c3 j \u2208 s i, ball (BallPackage.c q j) (BallPackage.r q j)", "state_after": "case inr\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\n\u03b2 : Type u\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nq : BallPackage \u03b2 \u03b1\nh\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1 :=\n  {\n    toBallPackage :=\n      { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n        r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n    \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\ns : Fin N \u2192 Set \u03b2 := fun i => \u22c3 k, \u22c3 (_ : k < lastStep p), \u22c3 (_ : color p k = \u2191i), {index p k}\n\u22a2 \u2203 s,\n    (\u2200 (i : Fin N), PairwiseDisjoint (s i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)) \u2227\n      range q.c \u2286 \u22c3 i, \u22c3 j \u2208 s i, ball (BallPackage.c q j) (BallPackage.r q j)"}, {"tactic": "refine' \u27e8s, fun i => _, _\u27e9", "annotated_tactic": ["refine' \u27e8s, fun i => _, _\u27e9", []], "state_before": "case inr\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\n\u03b2 : Type u\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nq : BallPackage \u03b2 \u03b1\nh\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1 :=\n  {\n    toBallPackage :=\n      { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n        r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n    \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\ns : Fin N \u2192 Set \u03b2 := fun i => \u22c3 k, \u22c3 (_ : k < lastStep p), \u22c3 (_ : color p k = \u2191i), {index p k}\n\u22a2 \u2203 s,\n    (\u2200 (i : Fin N), PairwiseDisjoint (s i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)) \u2227\n      range q.c \u2286 \u22c3 i, \u22c3 j \u2208 s i, ball (BallPackage.c q j) (BallPackage.r q j)", "state_after": "case inr.refine'_1\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\n\u03b2 : Type u\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nq : BallPackage \u03b2 \u03b1\nh\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1 :=\n  {\n    toBallPackage :=\n      { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n        r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n    \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\ns : Fin N \u2192 Set \u03b2 := fun i => \u22c3 k, \u22c3 (_ : k < lastStep p), \u22c3 (_ : color p k = \u2191i), {index p k}\ni : Fin N\n\u22a2 PairwiseDisjoint (s i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\n\ncase inr.refine'_2\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\n\u03b2 : Type u\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nq : BallPackage \u03b2 \u03b1\nh\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1 :=\n  {\n    toBallPackage :=\n      { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n        r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n    \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\ns : Fin N \u2192 Set \u03b2 := fun i => \u22c3 k, \u22c3 (_ : k < lastStep p), \u22c3 (_ : color p k = \u2191i), {index p k}\n\u22a2 range q.c \u2286 \u22c3 i, \u22c3 j \u2208 s i, ball (BallPackage.c q j) (BallPackage.r q j)"}, {"tactic": "refine' \u27e8fun _ => \u2205, fun _ => pairwiseDisjoint_empty, _\u27e9", "annotated_tactic": ["refine' \u27e8fun _ => \u2205, fun _ => <a>pairwiseDisjoint_empty</a>, _\u27e9", [{"full_name": "Set.pairwiseDisjoint_empty", "def_path": "Mathlib/Data/Set/Pairwise/Basic.lean", "def_pos": [259, 9], "def_end_pos": [259, 31]}]], "state_before": "case inl\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\n\u03b2 : Type u\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nq : BallPackage \u03b2 \u03b1\nh\u271d : IsEmpty \u03b2\n\u22a2 \u2203 s,\n    (\u2200 (i : Fin N), PairwiseDisjoint (s i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)) \u2227\n      range q.c \u2286 \u22c3 i, \u22c3 j \u2208 s i, ball (BallPackage.c q j) (BallPackage.r q j)", "state_after": "case inl\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\n\u03b2 : Type u\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nq : BallPackage \u03b2 \u03b1\nh\u271d : IsEmpty \u03b2\n\u22a2 range q.c \u2286 \u22c3 i, \u22c3 j \u2208 (fun x => \u2205) i, ball (BallPackage.c q j) (BallPackage.r q j)"}, {"tactic": "rw [\u2190 image_univ, eq_empty_of_isEmpty (univ : Set \u03b2)]", "annotated_tactic": ["rw [\u2190 <a>image_univ</a>, <a>eq_empty_of_isEmpty</a> (<a>univ</a> : <a>Set</a> \u03b2)]", [{"full_name": "Set.image_univ", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [718, 9], "def_end_pos": [718, 19]}, {"full_name": "Set.eq_empty_of_isEmpty", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [594, 9], "def_end_pos": [594, 28]}, {"full_name": "Set.univ", "def_path": "Mathlib/Init/Set.lean", "def_pos": [90, 5], "def_end_pos": [90, 9]}, {"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}]], "state_before": "case inl\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\n\u03b2 : Type u\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nq : BallPackage \u03b2 \u03b1\nh\u271d : IsEmpty \u03b2\n\u22a2 range q.c \u2286 \u22c3 i, \u22c3 j \u2208 (fun x => \u2205) i, ball (BallPackage.c q j) (BallPackage.r q j)", "state_after": "case inl\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\n\u03b2 : Type u\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nq : BallPackage \u03b2 \u03b1\nh\u271d : IsEmpty \u03b2\n\u22a2 q.c '' \u2205 \u2286 \u22c3 i, \u22c3 j \u2208 (fun x => \u2205) i, ball (BallPackage.c q j) (BallPackage.r q j)"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case inl\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\n\u03b2 : Type u\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nq : BallPackage \u03b2 \u03b1\nh\u271d : IsEmpty \u03b2\n\u22a2 q.c '' \u2205 \u2286 \u22c3 i, \u22c3 j \u2208 (fun x => \u2205) i, ball (BallPackage.c q j) (BallPackage.r q j)", "state_after": "no goals"}, {"tactic": "intro x hx y hy x_ne_y", "annotated_tactic": ["intro x hx y hy x_ne_y", []], "state_before": "case inr.refine'_1\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\n\u03b2 : Type u\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nq : BallPackage \u03b2 \u03b1\nh\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1 :=\n  {\n    toBallPackage :=\n      { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n        r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n    \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\ns : Fin N \u2192 Set \u03b2 := fun i => \u22c3 k, \u22c3 (_ : k < lastStep p), \u22c3 (_ : color p k = \u2191i), {index p k}\ni : Fin N\n\u22a2 PairwiseDisjoint (s i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)", "state_after": "case inr.refine'_1\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\n\u03b2 : Type u\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nq : BallPackage \u03b2 \u03b1\nh\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1 :=\n  {\n    toBallPackage :=\n      { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n        r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n    \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\ns : Fin N \u2192 Set \u03b2 := fun i => \u22c3 k, \u22c3 (_ : k < lastStep p), \u22c3 (_ : color p k = \u2191i), {index p k}\ni : Fin N\nx : \u03b2\nhx : x \u2208 s i\ny : \u03b2\nhy : y \u2208 s i\nx_ne_y : x \u2260 y\n\u22a2 (Disjoint on fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)) x y"}, {"tactic": "obtain \u27e8jx, jx_lt, jxi, rfl\u27e9 :\n  \u2203 jx : Ordinal, jx < p.lastStep \u2227 p.color jx = i \u2227 x = p.index jx := by\n  simpa only [exists_prop, mem_iUnion, mem_singleton_iff] using hx", "annotated_tactic": ["obtain \u27e8jx, jx_lt, jxi, rfl\u27e9 :\n      \u2203 jx : <a>Ordinal</a>, jx < p.lastStep \u2227 p.color jx = i \u2227 x = p.index jx := by\n      simpa only [<a>exists_prop</a>, <a>mem_iUnion</a>, <a>mem_singleton_iff</a>] using hx", [{"full_name": "Ordinal", "def_path": "Mathlib/SetTheory/Ordinal/Basic.lean", "def_pos": [152, 5], "def_end_pos": [152, 12]}, {"full_name": "exists_prop", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [485, 17], "def_end_pos": [485, 28]}, {"full_name": "Set.mem_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [201, 9], "def_end_pos": [201, 19]}, {"full_name": "Set.mem_singleton_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1273, 9], "def_end_pos": [1273, 26]}]], "state_before": "case inr.refine'_1\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\n\u03b2 : Type u\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nq : BallPackage \u03b2 \u03b1\nh\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1 :=\n  {\n    toBallPackage :=\n      { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n        r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n    \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\ns : Fin N \u2192 Set \u03b2 := fun i => \u22c3 k, \u22c3 (_ : k < lastStep p), \u22c3 (_ : color p k = \u2191i), {index p k}\ni : Fin N\nx : \u03b2\nhx : x \u2208 s i\ny : \u03b2\nhy : y \u2208 s i\nx_ne_y : x \u2260 y\n\u22a2 (Disjoint on fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)) x y", "state_after": "case inr.refine'_1.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\n\u03b2 : Type u\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nq : BallPackage \u03b2 \u03b1\nh\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1 :=\n  {\n    toBallPackage :=\n      { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n        r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n    \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\ns : Fin N \u2192 Set \u03b2 := fun i => \u22c3 k, \u22c3 (_ : k < lastStep p), \u22c3 (_ : color p k = \u2191i), {index p k}\ni : Fin N\ny : \u03b2\nhy : y \u2208 s i\njx : Ordinal.{u}\njx_lt : jx < lastStep p\njxi : color p jx = \u2191i\nhx : index p jx \u2208 s i\nx_ne_y : index p jx \u2260 y\n\u22a2 (Disjoint on fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)) (index p jx) y"}, {"tactic": "obtain \u27e8jy, jy_lt, jyi, rfl\u27e9 :\n  \u2203 jy : Ordinal, jy < p.lastStep \u2227 p.color jy = i \u2227 y = p.index jy := by\n  simpa only [exists_prop, mem_iUnion, mem_singleton_iff] using hy", "annotated_tactic": ["obtain \u27e8jy, jy_lt, jyi, rfl\u27e9 :\n      \u2203 jy : <a>Ordinal</a>, jy < p.lastStep \u2227 p.color jy = i \u2227 y = p.index jy := by\n      simpa only [<a>exists_prop</a>, <a>mem_iUnion</a>, <a>mem_singleton_iff</a>] using hy", [{"full_name": "Ordinal", "def_path": "Mathlib/SetTheory/Ordinal/Basic.lean", "def_pos": [152, 5], "def_end_pos": [152, 12]}, {"full_name": "exists_prop", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [485, 17], "def_end_pos": [485, 28]}, {"full_name": "Set.mem_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [201, 9], "def_end_pos": [201, 19]}, {"full_name": "Set.mem_singleton_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1273, 9], "def_end_pos": [1273, 26]}]], "state_before": "case inr.refine'_1.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\n\u03b2 : Type u\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nq : BallPackage \u03b2 \u03b1\nh\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1 :=\n  {\n    toBallPackage :=\n      { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n        r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n    \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\ns : Fin N \u2192 Set \u03b2 := fun i => \u22c3 k, \u22c3 (_ : k < lastStep p), \u22c3 (_ : color p k = \u2191i), {index p k}\ni : Fin N\ny : \u03b2\nhy : y \u2208 s i\njx : Ordinal.{u}\njx_lt : jx < lastStep p\njxi : color p jx = \u2191i\nhx : index p jx \u2208 s i\nx_ne_y : index p jx \u2260 y\n\u22a2 (Disjoint on fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)) (index p jx) y", "state_after": "case inr.refine'_1.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\n\u03b2 : Type u\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nq : BallPackage \u03b2 \u03b1\nh\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1 :=\n  {\n    toBallPackage :=\n      { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n        r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n    \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\ns : Fin N \u2192 Set \u03b2 := fun i => \u22c3 k, \u22c3 (_ : k < lastStep p), \u22c3 (_ : color p k = \u2191i), {index p k}\ni : Fin N\njx : Ordinal.{u}\njx_lt : jx < lastStep p\njxi : color p jx = \u2191i\nhx : index p jx \u2208 s i\njy : Ordinal.{u}\njy_lt : jy < lastStep p\njyi : color p jy = \u2191i\nhy : index p jy \u2208 s i\nx_ne_y : index p jx \u2260 index p jy\n\u22a2 (Disjoint on fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)) (index p jx) (index p jy)"}, {"tactic": "wlog jxy : jx \u2264 jy generalizing jx jy", "annotated_tactic": ["wlog jxy : jx \u2264 jy generalizing jx jy", []], "state_before": "case inr.refine'_1.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\n\u03b2 : Type u\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nq : BallPackage \u03b2 \u03b1\nh\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1 :=\n  {\n    toBallPackage :=\n      { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n        r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n    \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\ns : Fin N \u2192 Set \u03b2 := fun i => \u22c3 k, \u22c3 (_ : k < lastStep p), \u22c3 (_ : color p k = \u2191i), {index p k}\ni : Fin N\njx : Ordinal.{u}\njx_lt : jx < lastStep p\njxi : color p jx = \u2191i\nhx : index p jx \u2208 s i\njy : Ordinal.{u}\njy_lt : jy < lastStep p\njyi : color p jy = \u2191i\nhy : index p jy \u2208 s i\nx_ne_y : index p jx \u2260 index p jy\n\u22a2 (Disjoint on fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)) (index p jx) (index p jy)", "state_after": "case inr.refine'_1.intro.intro.intro.intro.intro.intro.inr\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\n\u03b2 : Type u\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nq : BallPackage \u03b2 \u03b1\nh\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1 :=\n  {\n    toBallPackage :=\n      { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n        r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n    \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\ns : Fin N \u2192 Set \u03b2 := fun i => \u22c3 k, \u22c3 (_ : k < lastStep p), \u22c3 (_ : color p k = \u2191i), {index p k}\ni : Fin N\njx : Ordinal.{u}\njx_lt : jx < lastStep p\njxi : color p jx = \u2191i\nhx : index p jx \u2208 s i\njy : Ordinal.{u}\njy_lt : jy < lastStep p\njyi : color p jy = \u2191i\nhy : index p jy \u2208 s i\nx_ne_y : index p jx \u2260 index p jy\nthis :\n  \u2200 (jx : Ordinal.{u}),\n    jx < lastStep p \u2192\n      color p jx = \u2191i \u2192\n        index p jx \u2208 s i \u2192\n          \u2200 (jy : Ordinal.{u}),\n            jy < lastStep p \u2192\n              color p jy = \u2191i \u2192\n                index p jy \u2208 s i \u2192\n                  index p jx \u2260 index p jy \u2192\n                    jx \u2264 jy \u2192\n                      (Disjoint on fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)) (index p jx)\n                        (index p jy)\njxy : \u00acjx \u2264 jy\n\u22a2 (Disjoint on fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)) (index p jx) (index p jy)\n\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\n\u03b2 : Type u\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nq : BallPackage \u03b2 \u03b1\nh\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1 :=\n  {\n    toBallPackage :=\n      { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n        r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n    \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\ns : Fin N \u2192 Set \u03b2 := fun i => \u22c3 k, \u22c3 (_ : k < lastStep p), \u22c3 (_ : color p k = \u2191i), {index p k}\ni : Fin N\njx : Ordinal.{u}\njx_lt : jx < lastStep p\njxi : color p jx = \u2191i\nhx : index p jx \u2208 s i\njy : Ordinal.{u}\njy_lt : jy < lastStep p\njyi : color p jy = \u2191i\nhy : index p jy \u2208 s i\nx_ne_y : index p jx \u2260 index p jy\njxy : jx \u2264 jy\n\u22a2 (Disjoint on fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)) (index p jx) (index p jy)"}, {"tactic": "replace jxy : jx < jy", "annotated_tactic": ["replace jxy : jx < jy", []], "state_before": "\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\n\u03b2 : Type u\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nq : BallPackage \u03b2 \u03b1\nh\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1 :=\n  {\n    toBallPackage :=\n      { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n        r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n    \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\ns : Fin N \u2192 Set \u03b2 := fun i => \u22c3 k, \u22c3 (_ : k < lastStep p), \u22c3 (_ : color p k = \u2191i), {index p k}\ni : Fin N\njx : Ordinal.{u}\njx_lt : jx < lastStep p\njxi : color p jx = \u2191i\nhx : index p jx \u2208 s i\njy : Ordinal.{u}\njy_lt : jy < lastStep p\njyi : color p jy = \u2191i\nhy : index p jy \u2208 s i\nx_ne_y : index p jx \u2260 index p jy\njxy : jx \u2264 jy\n\u22a2 (Disjoint on fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)) (index p jx) (index p jy)", "state_after": "case jxy\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\n\u03b2 : Type u\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nq : BallPackage \u03b2 \u03b1\nh\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1 :=\n  {\n    toBallPackage :=\n      { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n        r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n    \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\ns : Fin N \u2192 Set \u03b2 := fun i => \u22c3 k, \u22c3 (_ : k < lastStep p), \u22c3 (_ : color p k = \u2191i), {index p k}\ni : Fin N\njx : Ordinal.{u}\njx_lt : jx < lastStep p\njxi : color p jx = \u2191i\nhx : index p jx \u2208 s i\njy : Ordinal.{u}\njy_lt : jy < lastStep p\njyi : color p jy = \u2191i\nhy : index p jy \u2208 s i\nx_ne_y : index p jx \u2260 index p jy\njxy : jx \u2264 jy\n\u22a2 jx < jy\n\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\n\u03b2 : Type u\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nq : BallPackage \u03b2 \u03b1\nh\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1 :=\n  {\n    toBallPackage :=\n      { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n        r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n    \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\ns : Fin N \u2192 Set \u03b2 := fun i => \u22c3 k, \u22c3 (_ : k < lastStep p), \u22c3 (_ : color p k = \u2191i), {index p k}\ni : Fin N\njx : Ordinal.{u}\njx_lt : jx < lastStep p\njxi : color p jx = \u2191i\nhx : index p jx \u2208 s i\njy : Ordinal.{u}\njy_lt : jy < lastStep p\njyi : color p jy = \u2191i\nhy : index p jy \u2208 s i\nx_ne_y : index p jx \u2260 index p jy\njxy : jx < jy\n\u22a2 (Disjoint on fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)) (index p jx) (index p jy)"}, {"tactic": "let A : Set \u2115 :=\n  \u22c3 (j : { j // j < jy })\n    (_ : (closedBall (p.c (p.index j)) (p.r (p.index j)) \u2229\n      closedBall (p.c (p.index jy)) (p.r (p.index jy))).Nonempty),\n    {p.color j}", "annotated_tactic": ["let A : <a>Set</a> \u2115 :=\n      \u22c3 (j : { j // j < jy })\n        (_ : (<a>closedBall</a> (p.c (p.index j)) (p.r (p.index j)) \u2229\n          <a>closedBall</a> (p.c (p.index jy)) (p.r (p.index jy))).<a>Nonempty</a>),\n        {p.color j}", [{"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "Set.Nonempty", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [439, 15], "def_end_pos": [439, 23]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\n\u03b2 : Type u\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nq : BallPackage \u03b2 \u03b1\nh\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1 :=\n  {\n    toBallPackage :=\n      { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n        r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n    \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\ns : Fin N \u2192 Set \u03b2 := fun i => \u22c3 k, \u22c3 (_ : k < lastStep p), \u22c3 (_ : color p k = \u2191i), {index p k}\ni : Fin N\njx : Ordinal.{u}\njx_lt : jx < lastStep p\njxi : color p jx = \u2191i\nhx : index p jx \u2208 s i\njy : Ordinal.{u}\njy_lt : jy < lastStep p\njyi : color p jy = \u2191i\nhy : index p jy \u2208 s i\nx_ne_y : index p jx \u2260 index p jy\njxy : jx < jy\n\u22a2 (Disjoint on fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)) (index p jx) (index p jy)", "state_after": "\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\n\u03b2 : Type u\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nq : BallPackage \u03b2 \u03b1\nh\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1 :=\n  {\n    toBallPackage :=\n      { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n        r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n    \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\ns : Fin N \u2192 Set \u03b2 := fun i => \u22c3 k, \u22c3 (_ : k < lastStep p), \u22c3 (_ : color p k = \u2191i), {index p k}\ni : Fin N\njx : Ordinal.{u}\njx_lt : jx < lastStep p\njxi : color p jx = \u2191i\nhx : index p jx \u2208 s i\njy : Ordinal.{u}\njy_lt : jy < lastStep p\njyi : color p jy = \u2191i\nhy : index p jy \u2208 s i\nx_ne_y : index p jx \u2260 index p jy\njxy : jx < jy\nA : Set \u2115 :=\n  \u22c3 j,\n    \u22c3 (_ :\n      Set.Nonempty\n        (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n          closedBall (BallPackage.c p.toBallPackage (index p jy)) (BallPackage.r p.toBallPackage (index p jy)))),\n      {color p \u2191j}\n\u22a2 (Disjoint on fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)) (index p jx) (index p jy)"}, {"tactic": "have color_j : p.color jy = sInf (univ \\ A) := by rw [TauPackage.color]", "annotated_tactic": ["have color_j : p.color jy = <a>sInf</a> (<a>univ</a> \\ A) := by rw [<a>TauPackage.color</a>]", [{"full_name": "InfSet.sInf", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [62, 3], "def_end_pos": [62, 7]}, {"full_name": "Set.univ", "def_path": "Mathlib/Init/Set.lean", "def_pos": [90, 5], "def_end_pos": [90, 9]}, {"full_name": "Besicovitch.TauPackage.color", "def_path": "Mathlib/MeasureTheory/Covering/Besicovitch.lean", "def_pos": [275, 19], "def_end_pos": [275, 24]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\n\u03b2 : Type u\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nq : BallPackage \u03b2 \u03b1\nh\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1 :=\n  {\n    toBallPackage :=\n      { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n        r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n    \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\ns : Fin N \u2192 Set \u03b2 := fun i => \u22c3 k, \u22c3 (_ : k < lastStep p), \u22c3 (_ : color p k = \u2191i), {index p k}\ni : Fin N\njx : Ordinal.{u}\njx_lt : jx < lastStep p\njxi : color p jx = \u2191i\nhx : index p jx \u2208 s i\njy : Ordinal.{u}\njy_lt : jy < lastStep p\njyi : color p jy = \u2191i\nhy : index p jy \u2208 s i\nx_ne_y : index p jx \u2260 index p jy\njxy : jx < jy\nA : Set \u2115 :=\n  \u22c3 j,\n    \u22c3 (_ :\n      Set.Nonempty\n        (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n          closedBall (BallPackage.c p.toBallPackage (index p jy)) (BallPackage.r p.toBallPackage (index p jy)))),\n      {color p \u2191j}\n\u22a2 (Disjoint on fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)) (index p jx) (index p jy)", "state_after": "\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\n\u03b2 : Type u\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nq : BallPackage \u03b2 \u03b1\nh\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1 :=\n  {\n    toBallPackage :=\n      { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n        r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n    \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\ns : Fin N \u2192 Set \u03b2 := fun i => \u22c3 k, \u22c3 (_ : k < lastStep p), \u22c3 (_ : color p k = \u2191i), {index p k}\ni : Fin N\njx : Ordinal.{u}\njx_lt : jx < lastStep p\njxi : color p jx = \u2191i\nhx : index p jx \u2208 s i\njy : Ordinal.{u}\njy_lt : jy < lastStep p\njyi : color p jy = \u2191i\nhy : index p jy \u2208 s i\nx_ne_y : index p jx \u2260 index p jy\njxy : jx < jy\nA : Set \u2115 :=\n  \u22c3 j,\n    \u22c3 (_ :\n      Set.Nonempty\n        (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n          closedBall (BallPackage.c p.toBallPackage (index p jy)) (BallPackage.r p.toBallPackage (index p jy)))),\n      {color p \u2191j}\ncolor_j : color p jy = sInf (univ \\ A)\n\u22a2 (Disjoint on fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)) (index p jx) (index p jy)"}, {"tactic": "have h : p.color jy \u2208 univ \\ A := by\n  rw [color_j]\n  apply csInf_mem\n  refine' \u27e8N, _\u27e9\n  simp only [not_exists, true_and_iff, exists_prop, mem_iUnion, mem_singleton_iff, not_and,\n    mem_univ, mem_diff, Subtype.exists, Subtype.coe_mk]\n  intro k hk _\n  exact (p.color_lt (hk.trans jy_lt) hN).ne'", "annotated_tactic": ["have h : p.color jy \u2208 <a>univ</a> \\ A := by\n      rw [color_j]\n      apply <a>csInf_mem</a>\n      refine' \u27e8N, _\u27e9\n      simp only [<a>not_exists</a>, <a>true_and_iff</a>, <a>exists_prop</a>, <a>mem_iUnion</a>, <a>mem_singleton_iff</a>, <a>not_and</a>,\n        <a>mem_univ</a>, <a>mem_diff</a>, <a>Subtype.exists</a>, <a>Subtype.coe_mk</a>]\n      intro k hk _\n      exact (p.color_lt (hk.trans jy_lt) hN).<a>ne'</a>", [{"full_name": "Set.univ", "def_path": "Mathlib/Init/Set.lean", "def_pos": [90, 5], "def_end_pos": [90, 9]}, {"full_name": "csInf_mem", "def_path": "Mathlib/Order/ConditionallyCompleteLattice/Basic.lean", "def_pos": [1152, 9], "def_end_pos": [1152, 18]}, {"full_name": "not_exists", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [422, 17], "def_end_pos": [422, 27]}, {"full_name": "true_and_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [147, 9], "def_end_pos": [147, 21]}, {"full_name": "exists_prop", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [485, 17], "def_end_pos": [485, 28]}, {"full_name": "Set.mem_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [201, 9], "def_end_pos": [201, 19]}, {"full_name": "Set.mem_singleton_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1273, 9], "def_end_pos": [1273, 26]}, {"full_name": "not_and", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [316, 17], "def_end_pos": [316, 24]}, {"full_name": "Set.mem_univ", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [676, 9], "def_end_pos": [676, 17]}, {"full_name": "Set.mem_diff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1819, 9], "def_end_pos": [1819, 17]}, {"full_name": "Subtype.exists", "def_path": "Mathlib/Data/Subtype.lean", "def_pos": [54, 19], "def_end_pos": [54, 27]}, {"full_name": "Subtype.coe_mk", "def_path": "Mathlib/Data/Subtype.lean", "def_pos": [99, 9], "def_end_pos": [99, 15]}, {"full_name": "LT.lt.ne'", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [328, 9], "def_end_pos": [328, 12]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\n\u03b2 : Type u\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nq : BallPackage \u03b2 \u03b1\nh\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1 :=\n  {\n    toBallPackage :=\n      { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n        r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n    \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\ns : Fin N \u2192 Set \u03b2 := fun i => \u22c3 k, \u22c3 (_ : k < lastStep p), \u22c3 (_ : color p k = \u2191i), {index p k}\ni : Fin N\njx : Ordinal.{u}\njx_lt : jx < lastStep p\njxi : color p jx = \u2191i\nhx : index p jx \u2208 s i\njy : Ordinal.{u}\njy_lt : jy < lastStep p\njyi : color p jy = \u2191i\nhy : index p jy \u2208 s i\nx_ne_y : index p jx \u2260 index p jy\njxy : jx < jy\nA : Set \u2115 :=\n  \u22c3 j,\n    \u22c3 (_ :\n      Set.Nonempty\n        (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n          closedBall (BallPackage.c p.toBallPackage (index p jy)) (BallPackage.r p.toBallPackage (index p jy)))),\n      {color p \u2191j}\ncolor_j : color p jy = sInf (univ \\ A)\n\u22a2 (Disjoint on fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)) (index p jx) (index p jy)", "state_after": "\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\n\u03b2 : Type u\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nq : BallPackage \u03b2 \u03b1\nh\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1 :=\n  {\n    toBallPackage :=\n      { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n        r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n    \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\ns : Fin N \u2192 Set \u03b2 := fun i => \u22c3 k, \u22c3 (_ : k < lastStep p), \u22c3 (_ : color p k = \u2191i), {index p k}\ni : Fin N\njx : Ordinal.{u}\njx_lt : jx < lastStep p\njxi : color p jx = \u2191i\nhx : index p jx \u2208 s i\njy : Ordinal.{u}\njy_lt : jy < lastStep p\njyi : color p jy = \u2191i\nhy : index p jy \u2208 s i\nx_ne_y : index p jx \u2260 index p jy\njxy : jx < jy\nA : Set \u2115 :=\n  \u22c3 j,\n    \u22c3 (_ :\n      Set.Nonempty\n        (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n          closedBall (BallPackage.c p.toBallPackage (index p jy)) (BallPackage.r p.toBallPackage (index p jy)))),\n      {color p \u2191j}\ncolor_j : color p jy = sInf (univ \\ A)\nh : color p jy \u2208 univ \\ A\n\u22a2 (Disjoint on fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)) (index p jx) (index p jy)"}, {"tactic": "simp only [not_exists, true_and_iff, exists_prop, mem_iUnion, mem_singleton_iff, not_and,\n  mem_univ, mem_diff, Subtype.exists, Subtype.coe_mk] at h", "annotated_tactic": ["simp only [<a>not_exists</a>, <a>true_and_iff</a>, <a>exists_prop</a>, <a>mem_iUnion</a>, <a>mem_singleton_iff</a>, <a>not_and</a>,\n      <a>mem_univ</a>, <a>mem_diff</a>, <a>Subtype.exists</a>, <a>Subtype.coe_mk</a>] at h", [{"full_name": "not_exists", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [422, 17], "def_end_pos": [422, 27]}, {"full_name": "true_and_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [147, 9], "def_end_pos": [147, 21]}, {"full_name": "exists_prop", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [485, 17], "def_end_pos": [485, 28]}, {"full_name": "Set.mem_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [201, 9], "def_end_pos": [201, 19]}, {"full_name": "Set.mem_singleton_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1273, 9], "def_end_pos": [1273, 26]}, {"full_name": "not_and", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [316, 17], "def_end_pos": [316, 24]}, {"full_name": "Set.mem_univ", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [676, 9], "def_end_pos": [676, 17]}, {"full_name": "Set.mem_diff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1819, 9], "def_end_pos": [1819, 17]}, {"full_name": "Subtype.exists", "def_path": "Mathlib/Data/Subtype.lean", "def_pos": [54, 19], "def_end_pos": [54, 27]}, {"full_name": "Subtype.coe_mk", "def_path": "Mathlib/Data/Subtype.lean", "def_pos": [99, 9], "def_end_pos": [99, 15]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\n\u03b2 : Type u\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nq : BallPackage \u03b2 \u03b1\nh\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1 :=\n  {\n    toBallPackage :=\n      { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n        r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n    \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\ns : Fin N \u2192 Set \u03b2 := fun i => \u22c3 k, \u22c3 (_ : k < lastStep p), \u22c3 (_ : color p k = \u2191i), {index p k}\ni : Fin N\njx : Ordinal.{u}\njx_lt : jx < lastStep p\njxi : color p jx = \u2191i\nhx : index p jx \u2208 s i\njy : Ordinal.{u}\njy_lt : jy < lastStep p\njyi : color p jy = \u2191i\nhy : index p jy \u2208 s i\nx_ne_y : index p jx \u2260 index p jy\njxy : jx < jy\nA : Set \u2115 :=\n  \u22c3 j,\n    \u22c3 (_ :\n      Set.Nonempty\n        (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n          closedBall (BallPackage.c p.toBallPackage (index p jy)) (BallPackage.r p.toBallPackage (index p jy)))),\n      {color p \u2191j}\ncolor_j : color p jy = sInf (univ \\ A)\nh : color p jy \u2208 univ \\ A\n\u22a2 (Disjoint on fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)) (index p jx) (index p jy)", "state_after": "\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\n\u03b2 : Type u\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nq : BallPackage \u03b2 \u03b1\nh\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1 :=\n  {\n    toBallPackage :=\n      { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n        r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n    \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\ns : Fin N \u2192 Set \u03b2 := fun i => \u22c3 k, \u22c3 (_ : k < lastStep p), \u22c3 (_ : color p k = \u2191i), {index p k}\ni : Fin N\njx : Ordinal.{u}\njx_lt : jx < lastStep p\njxi : color p jx = \u2191i\nhx : index p jx \u2208 s i\njy : Ordinal.{u}\njy_lt : jy < lastStep p\njyi : color p jy = \u2191i\nhy : index p jy \u2208 s i\nx_ne_y : index p jx \u2260 index p jy\njxy : jx < jy\nA : Set \u2115 :=\n  \u22c3 j,\n    \u22c3 (_ :\n      Set.Nonempty\n        (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n          closedBall (BallPackage.c p.toBallPackage (index p jy)) (BallPackage.r p.toBallPackage (index p jy)))),\n      {color p \u2191j}\ncolor_j : color p jy = sInf (univ \\ A)\nh :\n  \u2200 (x : Ordinal.{u}),\n    x < jy \u2192\n      Set.Nonempty\n          (closedBall\n              (BallPackage.c q\n                (index\n                  {\n                    toBallPackage :=\n                      { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n                        r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n                    \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\n                  x))\n              (BallPackage.r q\n                (index\n                  {\n                    toBallPackage :=\n                      { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n                        r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n                    \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\n                  x)) \u2229\n            closedBall\n              (BallPackage.c q\n                (index\n                  {\n                    toBallPackage :=\n                      { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n                        r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n                    \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\n                  jy))\n              (BallPackage.r q\n                (index\n                  {\n                    toBallPackage :=\n                      { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n                        r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n                    \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\n                  jy))) \u2192\n        \u00accolor\n              {\n                toBallPackage :=\n                  { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n                    r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n                \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\n              jy =\n            color\n              {\n                toBallPackage :=\n                  { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n                    r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n                \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\n              x\n\u22a2 (Disjoint on fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)) (index p jx) (index p jy)"}, {"tactic": "specialize h jx jxy", "annotated_tactic": ["specialize h jx jxy", []], "state_before": "\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\n\u03b2 : Type u\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nq : BallPackage \u03b2 \u03b1\nh\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1 :=\n  {\n    toBallPackage :=\n      { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n        r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n    \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\ns : Fin N \u2192 Set \u03b2 := fun i => \u22c3 k, \u22c3 (_ : k < lastStep p), \u22c3 (_ : color p k = \u2191i), {index p k}\ni : Fin N\njx : Ordinal.{u}\njx_lt : jx < lastStep p\njxi : color p jx = \u2191i\nhx : index p jx \u2208 s i\njy : Ordinal.{u}\njy_lt : jy < lastStep p\njyi : color p jy = \u2191i\nhy : index p jy \u2208 s i\nx_ne_y : index p jx \u2260 index p jy\njxy : jx < jy\nA : Set \u2115 :=\n  \u22c3 j,\n    \u22c3 (_ :\n      Set.Nonempty\n        (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n          closedBall (BallPackage.c p.toBallPackage (index p jy)) (BallPackage.r p.toBallPackage (index p jy)))),\n      {color p \u2191j}\ncolor_j : color p jy = sInf (univ \\ A)\nh :\n  \u2200 (x : Ordinal.{u}),\n    x < jy \u2192\n      Set.Nonempty\n          (closedBall\n              (BallPackage.c q\n                (index\n                  {\n                    toBallPackage :=\n                      { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n                        r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n                    \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\n                  x))\n              (BallPackage.r q\n                (index\n                  {\n                    toBallPackage :=\n                      { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n                        r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n                    \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\n                  x)) \u2229\n            closedBall\n              (BallPackage.c q\n                (index\n                  {\n                    toBallPackage :=\n                      { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n                        r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n                    \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\n                  jy))\n              (BallPackage.r q\n                (index\n                  {\n                    toBallPackage :=\n                      { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n                        r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n                    \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\n                  jy))) \u2192\n        \u00accolor\n              {\n                toBallPackage :=\n                  { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n                    r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n                \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\n              jy =\n            color\n              {\n                toBallPackage :=\n                  { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n                    r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n                \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\n              x\n\u22a2 (Disjoint on fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)) (index p jx) (index p jy)", "state_after": "\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\n\u03b2 : Type u\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nq : BallPackage \u03b2 \u03b1\nh\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1 :=\n  {\n    toBallPackage :=\n      { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n        r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n    \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\ns : Fin N \u2192 Set \u03b2 := fun i => \u22c3 k, \u22c3 (_ : k < lastStep p), \u22c3 (_ : color p k = \u2191i), {index p k}\ni : Fin N\njx : Ordinal.{u}\njx_lt : jx < lastStep p\njxi : color p jx = \u2191i\nhx : index p jx \u2208 s i\njy : Ordinal.{u}\njy_lt : jy < lastStep p\njyi : color p jy = \u2191i\nhy : index p jy \u2208 s i\nx_ne_y : index p jx \u2260 index p jy\njxy : jx < jy\nA : Set \u2115 :=\n  \u22c3 j,\n    \u22c3 (_ :\n      Set.Nonempty\n        (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n          closedBall (BallPackage.c p.toBallPackage (index p jy)) (BallPackage.r p.toBallPackage (index p jy)))),\n      {color p \u2191j}\ncolor_j : color p jy = sInf (univ \\ A)\nh :\n  Set.Nonempty\n      (closedBall\n          (BallPackage.c q\n            (index\n              {\n                toBallPackage :=\n                  { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n                    r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n                \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\n              jx))\n          (BallPackage.r q\n            (index\n              {\n                toBallPackage :=\n                  { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n                    r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n                \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\n              jx)) \u2229\n        closedBall\n          (BallPackage.c q\n            (index\n              {\n                toBallPackage :=\n                  { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n                    r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n                \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\n              jy))\n          (BallPackage.r q\n            (index\n              {\n                toBallPackage :=\n                  { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n                    r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n                \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\n              jy))) \u2192\n    \u00accolor\n          {\n            toBallPackage :=\n              { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n                r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n            \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\n          jy =\n        color\n          {\n            toBallPackage :=\n              { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n                r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n            \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\n          jx\n\u22a2 (Disjoint on fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)) (index p jx) (index p jy)"}, {"tactic": "contrapose! h", "annotated_tactic": ["contrapose! h", []], "state_before": "\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\n\u03b2 : Type u\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nq : BallPackage \u03b2 \u03b1\nh\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1 :=\n  {\n    toBallPackage :=\n      { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n        r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n    \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\ns : Fin N \u2192 Set \u03b2 := fun i => \u22c3 k, \u22c3 (_ : k < lastStep p), \u22c3 (_ : color p k = \u2191i), {index p k}\ni : Fin N\njx : Ordinal.{u}\njx_lt : jx < lastStep p\njxi : color p jx = \u2191i\nhx : index p jx \u2208 s i\njy : Ordinal.{u}\njy_lt : jy < lastStep p\njyi : color p jy = \u2191i\nhy : index p jy \u2208 s i\nx_ne_y : index p jx \u2260 index p jy\njxy : jx < jy\nA : Set \u2115 :=\n  \u22c3 j,\n    \u22c3 (_ :\n      Set.Nonempty\n        (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n          closedBall (BallPackage.c p.toBallPackage (index p jy)) (BallPackage.r p.toBallPackage (index p jy)))),\n      {color p \u2191j}\ncolor_j : color p jy = sInf (univ \\ A)\nh :\n  Set.Nonempty\n      (closedBall\n          (BallPackage.c q\n            (index\n              {\n                toBallPackage :=\n                  { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n                    r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n                \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\n              jx))\n          (BallPackage.r q\n            (index\n              {\n                toBallPackage :=\n                  { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n                    r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n                \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\n              jx)) \u2229\n        closedBall\n          (BallPackage.c q\n            (index\n              {\n                toBallPackage :=\n                  { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n                    r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n                \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\n              jy))\n          (BallPackage.r q\n            (index\n              {\n                toBallPackage :=\n                  { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n                    r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n                \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\n              jy))) \u2192\n    \u00accolor\n          {\n            toBallPackage :=\n              { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n                r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n            \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\n          jy =\n        color\n          {\n            toBallPackage :=\n              { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n                r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n            \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\n          jx\n\u22a2 (Disjoint on fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)) (index p jx) (index p jy)", "state_after": "\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\n\u03b2 : Type u\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nq : BallPackage \u03b2 \u03b1\nh\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1 :=\n  {\n    toBallPackage :=\n      { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n        r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n    \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\ns : Fin N \u2192 Set \u03b2 := fun i => \u22c3 k, \u22c3 (_ : k < lastStep p), \u22c3 (_ : color p k = \u2191i), {index p k}\ni : Fin N\njx : Ordinal.{u}\njx_lt : jx < lastStep p\njxi : color p jx = \u2191i\nhx : index p jx \u2208 s i\njy : Ordinal.{u}\njy_lt : jy < lastStep p\njyi : color p jy = \u2191i\nhy : index p jy \u2208 s i\nx_ne_y : index p jx \u2260 index p jy\njxy : jx < jy\nA : Set \u2115 :=\n  \u22c3 j,\n    \u22c3 (_ :\n      Set.Nonempty\n        (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n          closedBall (BallPackage.c p.toBallPackage (index p jy)) (BallPackage.r p.toBallPackage (index p jy)))),\n      {color p \u2191j}\ncolor_j : color p jy = sInf (univ \\ A)\nh : \u00ac(Disjoint on fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)) (index p jx) (index p jy)\n\u22a2 Set.Nonempty\n      (closedBall\n          (BallPackage.c q\n            (index\n              {\n                toBallPackage :=\n                  { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n                    r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n                \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\n              jx))\n          (BallPackage.r q\n            (index\n              {\n                toBallPackage :=\n                  { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n                    r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n                \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\n              jx)) \u2229\n        closedBall\n          (BallPackage.c q\n            (index\n              {\n                toBallPackage :=\n                  { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n                    r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n                \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\n              jy))\n          (BallPackage.r q\n            (index\n              {\n                toBallPackage :=\n                  { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n                    r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n                \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\n              jy))) \u2227\n    color\n        {\n          toBallPackage :=\n            { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n              r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n          \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\n        jy =\n      color\n        {\n          toBallPackage :=\n            { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n              r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n          \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\n        jx"}, {"tactic": "simpa only [jxi, jyi, and_true_iff, eq_self_iff_true, \u2190 not_disjoint_iff_nonempty_inter] using h", "annotated_tactic": ["simpa only [jxi, jyi, <a>and_true_iff</a>, <a>eq_self_iff_true</a>, \u2190 <a>not_disjoint_iff_nonempty_inter</a>] using h", [{"full_name": "and_true_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [145, 9], "def_end_pos": [145, 21]}, {"full_name": "eq_self_iff_true", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [86, 9], "def_end_pos": [86, 25]}, {"full_name": "Set.not_disjoint_iff_nonempty_inter", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1557, 7], "def_end_pos": [1557, 38]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\n\u03b2 : Type u\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nq : BallPackage \u03b2 \u03b1\nh\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1 :=\n  {\n    toBallPackage :=\n      { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n        r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n    \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\ns : Fin N \u2192 Set \u03b2 := fun i => \u22c3 k, \u22c3 (_ : k < lastStep p), \u22c3 (_ : color p k = \u2191i), {index p k}\ni : Fin N\njx : Ordinal.{u}\njx_lt : jx < lastStep p\njxi : color p jx = \u2191i\nhx : index p jx \u2208 s i\njy : Ordinal.{u}\njy_lt : jy < lastStep p\njyi : color p jy = \u2191i\nhy : index p jy \u2208 s i\nx_ne_y : index p jx \u2260 index p jy\njxy : jx < jy\nA : Set \u2115 :=\n  \u22c3 j,\n    \u22c3 (_ :\n      Set.Nonempty\n        (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n          closedBall (BallPackage.c p.toBallPackage (index p jy)) (BallPackage.r p.toBallPackage (index p jy)))),\n      {color p \u2191j}\ncolor_j : color p jy = sInf (univ \\ A)\nh : \u00ac(Disjoint on fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)) (index p jx) (index p jy)\n\u22a2 Set.Nonempty\n      (closedBall\n          (BallPackage.c q\n            (index\n              {\n                toBallPackage :=\n                  { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n                    r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n                \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\n              jx))\n          (BallPackage.r q\n            (index\n              {\n                toBallPackage :=\n                  { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n                    r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n                \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\n              jx)) \u2229\n        closedBall\n          (BallPackage.c q\n            (index\n              {\n                toBallPackage :=\n                  { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n                    r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n                \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\n              jy))\n          (BallPackage.r q\n            (index\n              {\n                toBallPackage :=\n                  { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n                    r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n                \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\n              jy))) \u2227\n    color\n        {\n          toBallPackage :=\n            { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n              r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n          \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\n        jy =\n      color\n        {\n          toBallPackage :=\n            { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n              r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n          \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\n        jx", "state_after": "no goals"}, {"tactic": "simpa only [exists_prop, mem_iUnion, mem_singleton_iff] using hx", "annotated_tactic": ["simpa only [<a>exists_prop</a>, <a>mem_iUnion</a>, <a>mem_singleton_iff</a>] using hx", [{"full_name": "exists_prop", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [485, 17], "def_end_pos": [485, 28]}, {"full_name": "Set.mem_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [201, 9], "def_end_pos": [201, 19]}, {"full_name": "Set.mem_singleton_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1273, 9], "def_end_pos": [1273, 26]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\n\u03b2 : Type u\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nq : BallPackage \u03b2 \u03b1\nh\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1 :=\n  {\n    toBallPackage :=\n      { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n        r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n    \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\ns : Fin N \u2192 Set \u03b2 := fun i => \u22c3 k, \u22c3 (_ : k < lastStep p), \u22c3 (_ : color p k = \u2191i), {index p k}\ni : Fin N\nx : \u03b2\nhx : x \u2208 s i\ny : \u03b2\nhy : y \u2208 s i\nx_ne_y : x \u2260 y\n\u22a2 \u2203 jx, jx < lastStep p \u2227 color p jx = \u2191i \u2227 x = index p jx", "state_after": "no goals"}, {"tactic": "simpa only [exists_prop, mem_iUnion, mem_singleton_iff] using hy", "annotated_tactic": ["simpa only [<a>exists_prop</a>, <a>mem_iUnion</a>, <a>mem_singleton_iff</a>] using hy", [{"full_name": "exists_prop", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [485, 17], "def_end_pos": [485, 28]}, {"full_name": "Set.mem_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [201, 9], "def_end_pos": [201, 19]}, {"full_name": "Set.mem_singleton_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1273, 9], "def_end_pos": [1273, 26]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\n\u03b2 : Type u\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nq : BallPackage \u03b2 \u03b1\nh\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1 :=\n  {\n    toBallPackage :=\n      { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n        r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n    \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\ns : Fin N \u2192 Set \u03b2 := fun i => \u22c3 k, \u22c3 (_ : k < lastStep p), \u22c3 (_ : color p k = \u2191i), {index p k}\ni : Fin N\ny : \u03b2\nhy : y \u2208 s i\njx : Ordinal.{u}\njx_lt : jx < lastStep p\njxi : color p jx = \u2191i\nhx : index p jx \u2208 s i\nx_ne_y : index p jx \u2260 y\n\u22a2 \u2203 jy, jy < lastStep p \u2227 color p jy = \u2191i \u2227 y = index p jy", "state_after": "no goals"}, {"tactic": "exact (this jy jy_lt jyi hy jx jx_lt jxi hx x_ne_y.symm (le_of_not_le jxy)).symm", "annotated_tactic": ["exact (this jy jy_lt jyi hy jx jx_lt jxi hx x_ne_y.symm (<a>le_of_not_le</a> jxy)).<a>symm</a>", [{"full_name": "le_of_not_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [303, 9], "def_end_pos": [303, 21]}, {"full_name": "Disjoint.symm", "def_path": "Mathlib/Order/Disjoint.lean", "def_pos": [50, 9], "def_end_pos": [50, 22]}]], "state_before": "case inr.refine'_1.intro.intro.intro.intro.intro.intro.inr\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\n\u03b2 : Type u\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nq : BallPackage \u03b2 \u03b1\nh\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1 :=\n  {\n    toBallPackage :=\n      { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n        r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n    \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\ns : Fin N \u2192 Set \u03b2 := fun i => \u22c3 k, \u22c3 (_ : k < lastStep p), \u22c3 (_ : color p k = \u2191i), {index p k}\ni : Fin N\njx : Ordinal.{u}\njx_lt : jx < lastStep p\njxi : color p jx = \u2191i\nhx : index p jx \u2208 s i\njy : Ordinal.{u}\njy_lt : jy < lastStep p\njyi : color p jy = \u2191i\nhy : index p jy \u2208 s i\nx_ne_y : index p jx \u2260 index p jy\nthis :\n  \u2200 (jx : Ordinal.{u}),\n    jx < lastStep p \u2192\n      color p jx = \u2191i \u2192\n        index p jx \u2208 s i \u2192\n          \u2200 (jy : Ordinal.{u}),\n            jy < lastStep p \u2192\n              color p jy = \u2191i \u2192\n                index p jy \u2208 s i \u2192\n                  index p jx \u2260 index p jy \u2192\n                    jx \u2264 jy \u2192\n                      (Disjoint on fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)) (index p jx)\n                        (index p jy)\njxy : \u00acjx \u2264 jy\n\u22a2 (Disjoint on fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)) (index p jx) (index p jy)", "state_after": "no goals"}, {"tactic": "rcases lt_or_eq_of_le jxy with (H | rfl)", "annotated_tactic": ["rcases <a>lt_or_eq_of_le</a> jxy with (H | rfl)", [{"full_name": "lt_or_eq_of_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [228, 9], "def_end_pos": [228, 23]}]], "state_before": "case jxy\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\n\u03b2 : Type u\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nq : BallPackage \u03b2 \u03b1\nh\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1 :=\n  {\n    toBallPackage :=\n      { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n        r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n    \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\ns : Fin N \u2192 Set \u03b2 := fun i => \u22c3 k, \u22c3 (_ : k < lastStep p), \u22c3 (_ : color p k = \u2191i), {index p k}\ni : Fin N\njx : Ordinal.{u}\njx_lt : jx < lastStep p\njxi : color p jx = \u2191i\nhx : index p jx \u2208 s i\njy : Ordinal.{u}\njy_lt : jy < lastStep p\njyi : color p jy = \u2191i\nhy : index p jy \u2208 s i\nx_ne_y : index p jx \u2260 index p jy\njxy : jx \u2264 jy\n\u22a2 jx < jy", "state_after": "case jxy.inl\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\n\u03b2 : Type u\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nq : BallPackage \u03b2 \u03b1\nh\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1 :=\n  {\n    toBallPackage :=\n      { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n        r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n    \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\ns : Fin N \u2192 Set \u03b2 := fun i => \u22c3 k, \u22c3 (_ : k < lastStep p), \u22c3 (_ : color p k = \u2191i), {index p k}\ni : Fin N\njx : Ordinal.{u}\njx_lt : jx < lastStep p\njxi : color p jx = \u2191i\nhx : index p jx \u2208 s i\njy : Ordinal.{u}\njy_lt : jy < lastStep p\njyi : color p jy = \u2191i\nhy : index p jy \u2208 s i\nx_ne_y : index p jx \u2260 index p jy\njxy : jx \u2264 jy\nH : jx < jy\n\u22a2 jx < jy\n\ncase jxy.inr\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\n\u03b2 : Type u\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nq : BallPackage \u03b2 \u03b1\nh\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1 :=\n  {\n    toBallPackage :=\n      { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n        r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n    \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\ns : Fin N \u2192 Set \u03b2 := fun i => \u22c3 k, \u22c3 (_ : k < lastStep p), \u22c3 (_ : color p k = \u2191i), {index p k}\ni : Fin N\njx : Ordinal.{u}\njx_lt : jx < lastStep p\njxi : color p jx = \u2191i\nhx : index p jx \u2208 s i\njy_lt : jx < lastStep p\njyi : color p jx = \u2191i\nhy : index p jx \u2208 s i\nx_ne_y : index p jx \u2260 index p jx\njxy : jx \u2264 jx\n\u22a2 jx < jx"}, {"tactic": "rw [TauPackage.color]", "annotated_tactic": ["rw [<a>TauPackage.color</a>]", [{"full_name": "Besicovitch.TauPackage.color", "def_path": "Mathlib/MeasureTheory/Covering/Besicovitch.lean", "def_pos": [275, 19], "def_end_pos": [275, 24]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\n\u03b2 : Type u\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nq : BallPackage \u03b2 \u03b1\nh\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1 :=\n  {\n    toBallPackage :=\n      { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n        r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n    \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\ns : Fin N \u2192 Set \u03b2 := fun i => \u22c3 k, \u22c3 (_ : k < lastStep p), \u22c3 (_ : color p k = \u2191i), {index p k}\ni : Fin N\njx : Ordinal.{u}\njx_lt : jx < lastStep p\njxi : color p jx = \u2191i\nhx : index p jx \u2208 s i\njy : Ordinal.{u}\njy_lt : jy < lastStep p\njyi : color p jy = \u2191i\nhy : index p jy \u2208 s i\nx_ne_y : index p jx \u2260 index p jy\njxy : jx < jy\nA : Set \u2115 :=\n  \u22c3 j,\n    \u22c3 (_ :\n      Set.Nonempty\n        (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n          closedBall (BallPackage.c p.toBallPackage (index p jy)) (BallPackage.r p.toBallPackage (index p jy)))),\n      {color p \u2191j}\n\u22a2 color p jy = sInf (univ \\ A)", "state_after": "no goals"}, {"tactic": "rw [color_j]", "annotated_tactic": ["rw [color_j]", []], "state_before": "\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\n\u03b2 : Type u\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nq : BallPackage \u03b2 \u03b1\nh\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1 :=\n  {\n    toBallPackage :=\n      { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n        r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n    \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\ns : Fin N \u2192 Set \u03b2 := fun i => \u22c3 k, \u22c3 (_ : k < lastStep p), \u22c3 (_ : color p k = \u2191i), {index p k}\ni : Fin N\njx : Ordinal.{u}\njx_lt : jx < lastStep p\njxi : color p jx = \u2191i\nhx : index p jx \u2208 s i\njy : Ordinal.{u}\njy_lt : jy < lastStep p\njyi : color p jy = \u2191i\nhy : index p jy \u2208 s i\nx_ne_y : index p jx \u2260 index p jy\njxy : jx < jy\nA : Set \u2115 :=\n  \u22c3 j,\n    \u22c3 (_ :\n      Set.Nonempty\n        (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n          closedBall (BallPackage.c p.toBallPackage (index p jy)) (BallPackage.r p.toBallPackage (index p jy)))),\n      {color p \u2191j}\ncolor_j : color p jy = sInf (univ \\ A)\n\u22a2 color p jy \u2208 univ \\ A", "state_after": "\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\n\u03b2 : Type u\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nq : BallPackage \u03b2 \u03b1\nh\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1 :=\n  {\n    toBallPackage :=\n      { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n        r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n    \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\ns : Fin N \u2192 Set \u03b2 := fun i => \u22c3 k, \u22c3 (_ : k < lastStep p), \u22c3 (_ : color p k = \u2191i), {index p k}\ni : Fin N\njx : Ordinal.{u}\njx_lt : jx < lastStep p\njxi : color p jx = \u2191i\nhx : index p jx \u2208 s i\njy : Ordinal.{u}\njy_lt : jy < lastStep p\njyi : color p jy = \u2191i\nhy : index p jy \u2208 s i\nx_ne_y : index p jx \u2260 index p jy\njxy : jx < jy\nA : Set \u2115 :=\n  \u22c3 j,\n    \u22c3 (_ :\n      Set.Nonempty\n        (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n          closedBall (BallPackage.c p.toBallPackage (index p jy)) (BallPackage.r p.toBallPackage (index p jy)))),\n      {color p \u2191j}\ncolor_j : color p jy = sInf (univ \\ A)\n\u22a2 sInf (univ \\ A) \u2208 univ \\ A"}, {"tactic": "apply csInf_mem", "annotated_tactic": ["apply <a>csInf_mem</a>", [{"full_name": "csInf_mem", "def_path": "Mathlib/Order/ConditionallyCompleteLattice/Basic.lean", "def_pos": [1152, 9], "def_end_pos": [1152, 18]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\n\u03b2 : Type u\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nq : BallPackage \u03b2 \u03b1\nh\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1 :=\n  {\n    toBallPackage :=\n      { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n        r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n    \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\ns : Fin N \u2192 Set \u03b2 := fun i => \u22c3 k, \u22c3 (_ : k < lastStep p), \u22c3 (_ : color p k = \u2191i), {index p k}\ni : Fin N\njx : Ordinal.{u}\njx_lt : jx < lastStep p\njxi : color p jx = \u2191i\nhx : index p jx \u2208 s i\njy : Ordinal.{u}\njy_lt : jy < lastStep p\njyi : color p jy = \u2191i\nhy : index p jy \u2208 s i\nx_ne_y : index p jx \u2260 index p jy\njxy : jx < jy\nA : Set \u2115 :=\n  \u22c3 j,\n    \u22c3 (_ :\n      Set.Nonempty\n        (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n          closedBall (BallPackage.c p.toBallPackage (index p jy)) (BallPackage.r p.toBallPackage (index p jy)))),\n      {color p \u2191j}\ncolor_j : color p jy = sInf (univ \\ A)\n\u22a2 sInf (univ \\ A) \u2208 univ \\ A", "state_after": "case hs\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\n\u03b2 : Type u\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nq : BallPackage \u03b2 \u03b1\nh\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1 :=\n  {\n    toBallPackage :=\n      { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n        r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n    \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\ns : Fin N \u2192 Set \u03b2 := fun i => \u22c3 k, \u22c3 (_ : k < lastStep p), \u22c3 (_ : color p k = \u2191i), {index p k}\ni : Fin N\njx : Ordinal.{u}\njx_lt : jx < lastStep p\njxi : color p jx = \u2191i\nhx : index p jx \u2208 s i\njy : Ordinal.{u}\njy_lt : jy < lastStep p\njyi : color p jy = \u2191i\nhy : index p jy \u2208 s i\nx_ne_y : index p jx \u2260 index p jy\njxy : jx < jy\nA : Set \u2115 :=\n  \u22c3 j,\n    \u22c3 (_ :\n      Set.Nonempty\n        (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n          closedBall (BallPackage.c p.toBallPackage (index p jy)) (BallPackage.r p.toBallPackage (index p jy)))),\n      {color p \u2191j}\ncolor_j : color p jy = sInf (univ \\ A)\n\u22a2 Set.Nonempty (univ \\ A)"}, {"tactic": "refine' \u27e8N, _\u27e9", "annotated_tactic": ["refine' \u27e8N, _\u27e9", []], "state_before": "case hs\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\n\u03b2 : Type u\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nq : BallPackage \u03b2 \u03b1\nh\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1 :=\n  {\n    toBallPackage :=\n      { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n        r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n    \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\ns : Fin N \u2192 Set \u03b2 := fun i => \u22c3 k, \u22c3 (_ : k < lastStep p), \u22c3 (_ : color p k = \u2191i), {index p k}\ni : Fin N\njx : Ordinal.{u}\njx_lt : jx < lastStep p\njxi : color p jx = \u2191i\nhx : index p jx \u2208 s i\njy : Ordinal.{u}\njy_lt : jy < lastStep p\njyi : color p jy = \u2191i\nhy : index p jy \u2208 s i\nx_ne_y : index p jx \u2260 index p jy\njxy : jx < jy\nA : Set \u2115 :=\n  \u22c3 j,\n    \u22c3 (_ :\n      Set.Nonempty\n        (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n          closedBall (BallPackage.c p.toBallPackage (index p jy)) (BallPackage.r p.toBallPackage (index p jy)))),\n      {color p \u2191j}\ncolor_j : color p jy = sInf (univ \\ A)\n\u22a2 Set.Nonempty (univ \\ A)", "state_after": "case hs\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\n\u03b2 : Type u\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nq : BallPackage \u03b2 \u03b1\nh\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1 :=\n  {\n    toBallPackage :=\n      { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n        r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n    \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\ns : Fin N \u2192 Set \u03b2 := fun i => \u22c3 k, \u22c3 (_ : k < lastStep p), \u22c3 (_ : color p k = \u2191i), {index p k}\ni : Fin N\njx : Ordinal.{u}\njx_lt : jx < lastStep p\njxi : color p jx = \u2191i\nhx : index p jx \u2208 s i\njy : Ordinal.{u}\njy_lt : jy < lastStep p\njyi : color p jy = \u2191i\nhy : index p jy \u2208 s i\nx_ne_y : index p jx \u2260 index p jy\njxy : jx < jy\nA : Set \u2115 :=\n  \u22c3 j,\n    \u22c3 (_ :\n      Set.Nonempty\n        (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n          closedBall (BallPackage.c p.toBallPackage (index p jy)) (BallPackage.r p.toBallPackage (index p jy)))),\n      {color p \u2191j}\ncolor_j : color p jy = sInf (univ \\ A)\n\u22a2 N \u2208 univ \\ A"}, {"tactic": "simp only [not_exists, true_and_iff, exists_prop, mem_iUnion, mem_singleton_iff, not_and,\n  mem_univ, mem_diff, Subtype.exists, Subtype.coe_mk]", "annotated_tactic": ["simp only [<a>not_exists</a>, <a>true_and_iff</a>, <a>exists_prop</a>, <a>mem_iUnion</a>, <a>mem_singleton_iff</a>, <a>not_and</a>,\n        <a>mem_univ</a>, <a>mem_diff</a>, <a>Subtype.exists</a>, <a>Subtype.coe_mk</a>]", [{"full_name": "not_exists", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [422, 17], "def_end_pos": [422, 27]}, {"full_name": "true_and_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [147, 9], "def_end_pos": [147, 21]}, {"full_name": "exists_prop", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [485, 17], "def_end_pos": [485, 28]}, {"full_name": "Set.mem_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [201, 9], "def_end_pos": [201, 19]}, {"full_name": "Set.mem_singleton_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1273, 9], "def_end_pos": [1273, 26]}, {"full_name": "not_and", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [316, 17], "def_end_pos": [316, 24]}, {"full_name": "Set.mem_univ", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [676, 9], "def_end_pos": [676, 17]}, {"full_name": "Set.mem_diff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1819, 9], "def_end_pos": [1819, 17]}, {"full_name": "Subtype.exists", "def_path": "Mathlib/Data/Subtype.lean", "def_pos": [54, 19], "def_end_pos": [54, 27]}, {"full_name": "Subtype.coe_mk", "def_path": "Mathlib/Data/Subtype.lean", "def_pos": [99, 9], "def_end_pos": [99, 15]}]], "state_before": "case hs\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\n\u03b2 : Type u\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nq : BallPackage \u03b2 \u03b1\nh\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1 :=\n  {\n    toBallPackage :=\n      { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n        r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n    \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\ns : Fin N \u2192 Set \u03b2 := fun i => \u22c3 k, \u22c3 (_ : k < lastStep p), \u22c3 (_ : color p k = \u2191i), {index p k}\ni : Fin N\njx : Ordinal.{u}\njx_lt : jx < lastStep p\njxi : color p jx = \u2191i\nhx : index p jx \u2208 s i\njy : Ordinal.{u}\njy_lt : jy < lastStep p\njyi : color p jy = \u2191i\nhy : index p jy \u2208 s i\nx_ne_y : index p jx \u2260 index p jy\njxy : jx < jy\nA : Set \u2115 :=\n  \u22c3 j,\n    \u22c3 (_ :\n      Set.Nonempty\n        (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n          closedBall (BallPackage.c p.toBallPackage (index p jy)) (BallPackage.r p.toBallPackage (index p jy)))),\n      {color p \u2191j}\ncolor_j : color p jy = sInf (univ \\ A)\n\u22a2 N \u2208 univ \\ A", "state_after": "case hs\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\n\u03b2 : Type u\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nq : BallPackage \u03b2 \u03b1\nh\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1 :=\n  {\n    toBallPackage :=\n      { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n        r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n    \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\ns : Fin N \u2192 Set \u03b2 := fun i => \u22c3 k, \u22c3 (_ : k < lastStep p), \u22c3 (_ : color p k = \u2191i), {index p k}\ni : Fin N\njx : Ordinal.{u}\njx_lt : jx < lastStep p\njxi : color p jx = \u2191i\nhx : index p jx \u2208 s i\njy : Ordinal.{u}\njy_lt : jy < lastStep p\njyi : color p jy = \u2191i\nhy : index p jy \u2208 s i\nx_ne_y : index p jx \u2260 index p jy\njxy : jx < jy\nA : Set \u2115 :=\n  \u22c3 j,\n    \u22c3 (_ :\n      Set.Nonempty\n        (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n          closedBall (BallPackage.c p.toBallPackage (index p jy)) (BallPackage.r p.toBallPackage (index p jy)))),\n      {color p \u2191j}\ncolor_j : color p jy = sInf (univ \\ A)\n\u22a2 \u2200 (x : Ordinal.{u}),\n    x < jy \u2192\n      Set.Nonempty\n          (closedBall\n              (BallPackage.c q\n                (index\n                  {\n                    toBallPackage :=\n                      { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n                        r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n                    \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\n                  x))\n              (BallPackage.r q\n                (index\n                  {\n                    toBallPackage :=\n                      { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n                        r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n                    \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\n                  x)) \u2229\n            closedBall\n              (BallPackage.c q\n                (index\n                  {\n                    toBallPackage :=\n                      { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n                        r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n                    \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\n                  jy))\n              (BallPackage.r q\n                (index\n                  {\n                    toBallPackage :=\n                      { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n                        r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n                    \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\n                  jy))) \u2192\n        \u00acN =\n            color\n              {\n                toBallPackage :=\n                  { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n                    r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n                \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\n              x"}, {"tactic": "intro k hk _", "annotated_tactic": ["intro k hk _", []], "state_before": "case hs\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\n\u03b2 : Type u\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nq : BallPackage \u03b2 \u03b1\nh\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1 :=\n  {\n    toBallPackage :=\n      { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n        r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n    \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\ns : Fin N \u2192 Set \u03b2 := fun i => \u22c3 k, \u22c3 (_ : k < lastStep p), \u22c3 (_ : color p k = \u2191i), {index p k}\ni : Fin N\njx : Ordinal.{u}\njx_lt : jx < lastStep p\njxi : color p jx = \u2191i\nhx : index p jx \u2208 s i\njy : Ordinal.{u}\njy_lt : jy < lastStep p\njyi : color p jy = \u2191i\nhy : index p jy \u2208 s i\nx_ne_y : index p jx \u2260 index p jy\njxy : jx < jy\nA : Set \u2115 :=\n  \u22c3 j,\n    \u22c3 (_ :\n      Set.Nonempty\n        (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n          closedBall (BallPackage.c p.toBallPackage (index p jy)) (BallPackage.r p.toBallPackage (index p jy)))),\n      {color p \u2191j}\ncolor_j : color p jy = sInf (univ \\ A)\n\u22a2 \u2200 (x : Ordinal.{u}),\n    x < jy \u2192\n      Set.Nonempty\n          (closedBall\n              (BallPackage.c q\n                (index\n                  {\n                    toBallPackage :=\n                      { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n                        r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n                    \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\n                  x))\n              (BallPackage.r q\n                (index\n                  {\n                    toBallPackage :=\n                      { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n                        r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n                    \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\n                  x)) \u2229\n            closedBall\n              (BallPackage.c q\n                (index\n                  {\n                    toBallPackage :=\n                      { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n                        r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n                    \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\n                  jy))\n              (BallPackage.r q\n                (index\n                  {\n                    toBallPackage :=\n                      { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n                        r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n                    \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\n                  jy))) \u2192\n        \u00acN =\n            color\n              {\n                toBallPackage :=\n                  { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n                    r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n                \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\n              x", "state_after": "case hs\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\n\u03b2 : Type u\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nq : BallPackage \u03b2 \u03b1\nh\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1 :=\n  {\n    toBallPackage :=\n      { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n        r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n    \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\ns : Fin N \u2192 Set \u03b2 := fun i => \u22c3 k, \u22c3 (_ : k < lastStep p), \u22c3 (_ : color p k = \u2191i), {index p k}\ni : Fin N\njx : Ordinal.{u}\njx_lt : jx < lastStep p\njxi : color p jx = \u2191i\nhx : index p jx \u2208 s i\njy : Ordinal.{u}\njy_lt : jy < lastStep p\njyi : color p jy = \u2191i\nhy : index p jy \u2208 s i\nx_ne_y : index p jx \u2260 index p jy\njxy : jx < jy\nA : Set \u2115 :=\n  \u22c3 j,\n    \u22c3 (_ :\n      Set.Nonempty\n        (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n          closedBall (BallPackage.c p.toBallPackage (index p jy)) (BallPackage.r p.toBallPackage (index p jy)))),\n      {color p \u2191j}\ncolor_j : color p jy = sInf (univ \\ A)\nk : Ordinal.{u}\nhk : k < jy\na\u271d :\n  Set.Nonempty\n    (closedBall\n        (BallPackage.c q\n          (index\n            {\n              toBallPackage :=\n                { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n                  r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n              \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\n            k))\n        (BallPackage.r q\n          (index\n            {\n              toBallPackage :=\n                { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n                  r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n              \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\n            k)) \u2229\n      closedBall\n        (BallPackage.c q\n          (index\n            {\n              toBallPackage :=\n                { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n                  r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n              \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\n            jy))\n        (BallPackage.r q\n          (index\n            {\n              toBallPackage :=\n                { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n                  r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n              \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\n            jy)))\n\u22a2 \u00acN =\n      color\n        {\n          toBallPackage :=\n            { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n              r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n          \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\n        k"}, {"tactic": "exact (p.color_lt (hk.trans jy_lt) hN).ne'", "annotated_tactic": ["exact (p.color_lt (hk.trans jy_lt) hN).<a>ne'</a>", [{"full_name": "LT.lt.ne'", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [328, 9], "def_end_pos": [328, 12]}]], "state_before": "case hs\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\n\u03b2 : Type u\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nq : BallPackage \u03b2 \u03b1\nh\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1 :=\n  {\n    toBallPackage :=\n      { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n        r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n    \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\ns : Fin N \u2192 Set \u03b2 := fun i => \u22c3 k, \u22c3 (_ : k < lastStep p), \u22c3 (_ : color p k = \u2191i), {index p k}\ni : Fin N\njx : Ordinal.{u}\njx_lt : jx < lastStep p\njxi : color p jx = \u2191i\nhx : index p jx \u2208 s i\njy : Ordinal.{u}\njy_lt : jy < lastStep p\njyi : color p jy = \u2191i\nhy : index p jy \u2208 s i\nx_ne_y : index p jx \u2260 index p jy\njxy : jx < jy\nA : Set \u2115 :=\n  \u22c3 j,\n    \u22c3 (_ :\n      Set.Nonempty\n        (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n          closedBall (BallPackage.c p.toBallPackage (index p jy)) (BallPackage.r p.toBallPackage (index p jy)))),\n      {color p \u2191j}\ncolor_j : color p jy = sInf (univ \\ A)\nk : Ordinal.{u}\nhk : k < jy\na\u271d :\n  Set.Nonempty\n    (closedBall\n        (BallPackage.c q\n          (index\n            {\n              toBallPackage :=\n                { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n                  r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n              \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\n            k))\n        (BallPackage.r q\n          (index\n            {\n              toBallPackage :=\n                { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n                  r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n              \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\n            k)) \u2229\n      closedBall\n        (BallPackage.c q\n          (index\n            {\n              toBallPackage :=\n                { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n                  r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n              \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\n            jy))\n        (BallPackage.r q\n          (index\n            {\n              toBallPackage :=\n                { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n                  r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n              \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\n            jy)))\n\u22a2 \u00acN =\n      color\n        {\n          toBallPackage :=\n            { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n              r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n          \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\n        k", "state_after": "no goals"}, {"tactic": "refine' range_subset_iff.2 fun b => _", "annotated_tactic": ["refine' <a>range_subset_iff</a>.2 fun b => _", [{"full_name": "Set.range_subset_iff", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [749, 9], "def_end_pos": [749, 25]}]], "state_before": "case inr.refine'_2\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\n\u03b2 : Type u\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nq : BallPackage \u03b2 \u03b1\nh\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1 :=\n  {\n    toBallPackage :=\n      { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n        r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n    \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\ns : Fin N \u2192 Set \u03b2 := fun i => \u22c3 k, \u22c3 (_ : k < lastStep p), \u22c3 (_ : color p k = \u2191i), {index p k}\n\u22a2 range q.c \u2286 \u22c3 i, \u22c3 j \u2208 s i, ball (BallPackage.c q j) (BallPackage.r q j)", "state_after": "case inr.refine'_2\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\n\u03b2 : Type u\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nq : BallPackage \u03b2 \u03b1\nh\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1 :=\n  {\n    toBallPackage :=\n      { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n        r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n    \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\ns : Fin N \u2192 Set \u03b2 := fun i => \u22c3 k, \u22c3 (_ : k < lastStep p), \u22c3 (_ : color p k = \u2191i), {index p k}\nb : \u03b2\n\u22a2 BallPackage.c q b \u2208 \u22c3 i, \u22c3 j \u2208 s i, ball (BallPackage.c q j) (BallPackage.r q j)"}, {"tactic": "obtain \u27e8a, ha\u27e9 :\n  \u2203 a : Ordinal, a < p.lastStep \u2227 dist (p.c b) (p.c (p.index a)) < p.r (p.index a) := by\n  simpa only [iUnionUpTo, exists_prop, mem_iUnion, mem_ball, Subtype.exists,\n    Subtype.coe_mk] using p.mem_iUnionUpTo_lastStep b", "annotated_tactic": ["obtain \u27e8a, ha\u27e9 :\n      \u2203 a : <a>Ordinal</a>, a < p.lastStep \u2227 <a>dist</a> (p.c b) (p.c (p.index a)) < p.r (p.index a) := by\n      simpa only [<a>iUnionUpTo</a>, <a>exists_prop</a>, <a>mem_iUnion</a>, <a>mem_ball</a>, <a>Subtype.exists</a>,\n        <a>Subtype.coe_mk</a>] using p.mem_iUnionUpTo_lastStep b", [{"full_name": "Ordinal", "def_path": "Mathlib/SetTheory/Ordinal/Basic.lean", "def_pos": [152, 5], "def_end_pos": [152, 12]}, {"full_name": "Dist.dist", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [94, 3], "def_end_pos": [94, 7]}, {"full_name": "Besicovitch.TauPackage.iUnionUpTo", "def_path": "Mathlib/MeasureTheory/Covering/Besicovitch.lean", "def_pos": [257, 5], "def_end_pos": [257, 15]}, {"full_name": "exists_prop", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [485, 17], "def_end_pos": [485, 28]}, {"full_name": "Set.mem_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [201, 9], "def_end_pos": [201, 19]}, {"full_name": "Metric.mem_ball", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [414, 9], "def_end_pos": [414, 17]}, {"full_name": "Subtype.exists", "def_path": "Mathlib/Data/Subtype.lean", "def_pos": [54, 19], "def_end_pos": [54, 27]}, {"full_name": "Subtype.coe_mk", "def_path": "Mathlib/Data/Subtype.lean", "def_pos": [99, 9], "def_end_pos": [99, 15]}]], "state_before": "case inr.refine'_2\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\n\u03b2 : Type u\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nq : BallPackage \u03b2 \u03b1\nh\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1 :=\n  {\n    toBallPackage :=\n      { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n        r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n    \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\ns : Fin N \u2192 Set \u03b2 := fun i => \u22c3 k, \u22c3 (_ : k < lastStep p), \u22c3 (_ : color p k = \u2191i), {index p k}\nb : \u03b2\n\u22a2 BallPackage.c q b \u2208 \u22c3 i, \u22c3 j \u2208 s i, ball (BallPackage.c q j) (BallPackage.r q j)", "state_after": "case inr.refine'_2.intro\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\n\u03b2 : Type u\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nq : BallPackage \u03b2 \u03b1\nh\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1 :=\n  {\n    toBallPackage :=\n      { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n        r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n    \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\ns : Fin N \u2192 Set \u03b2 := fun i => \u22c3 k, \u22c3 (_ : k < lastStep p), \u22c3 (_ : color p k = \u2191i), {index p k}\nb : \u03b2\na : Ordinal.{u}\nha :\n  a < lastStep p \u2227\n    dist (BallPackage.c p.toBallPackage b) (BallPackage.c p.toBallPackage (index p a)) <\n      BallPackage.r p.toBallPackage (index p a)\n\u22a2 BallPackage.c q b \u2208 \u22c3 i, \u22c3 j \u2208 s i, ball (BallPackage.c q j) (BallPackage.r q j)"}, {"tactic": "simp only [exists_prop, mem_iUnion, mem_ball, mem_singleton_iff, biUnion_and', exists_eq_left,\n  iUnion_exists, exists_and_left]", "annotated_tactic": ["simp only [<a>exists_prop</a>, <a>mem_iUnion</a>, <a>mem_ball</a>, <a>mem_singleton_iff</a>, <a>biUnion_and'</a>, <a>exists_eq_left</a>,\n      <a>iUnion_exists</a>, <a>exists_and_left</a>]", [{"full_name": "exists_prop", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [485, 17], "def_end_pos": [485, 28]}, {"full_name": "Set.mem_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [201, 9], "def_end_pos": [201, 19]}, {"full_name": "Metric.mem_ball", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [414, 9], "def_end_pos": [414, 17]}, {"full_name": "Set.mem_singleton_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1273, 9], "def_end_pos": [1273, 26]}, {"full_name": "Set.biUnion_and'", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [928, 9], "def_end_pos": [928, 21]}, {"full_name": "exists_eq_left", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [459, 17], "def_end_pos": [459, 31]}, {"full_name": "Set.iUnion_exists", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [804, 9], "def_end_pos": [804, 22]}, {"full_name": "exists_and_left", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [465, 17], "def_end_pos": [465, 32]}]], "state_before": "case inr.refine'_2.intro\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\n\u03b2 : Type u\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nq : BallPackage \u03b2 \u03b1\nh\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1 :=\n  {\n    toBallPackage :=\n      { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n        r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n    \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\ns : Fin N \u2192 Set \u03b2 := fun i => \u22c3 k, \u22c3 (_ : k < lastStep p), \u22c3 (_ : color p k = \u2191i), {index p k}\nb : \u03b2\na : Ordinal.{u}\nha :\n  a < lastStep p \u2227\n    dist (BallPackage.c p.toBallPackage b) (BallPackage.c p.toBallPackage (index p a)) <\n      BallPackage.r p.toBallPackage (index p a)\n\u22a2 BallPackage.c q b \u2208 \u22c3 i, \u22c3 j \u2208 s i, ball (BallPackage.c q j) (BallPackage.r q j)", "state_after": "case inr.refine'_2.intro\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\n\u03b2 : Type u\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nq : BallPackage \u03b2 \u03b1\nh\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1 :=\n  {\n    toBallPackage :=\n      { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n        r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n    \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\ns : Fin N \u2192 Set \u03b2 := fun i => \u22c3 k, \u22c3 (_ : k < lastStep p), \u22c3 (_ : color p k = \u2191i), {index p k}\nb : \u03b2\na : Ordinal.{u}\nha :\n  a < lastStep p \u2227\n    dist (BallPackage.c p.toBallPackage b) (BallPackage.c p.toBallPackage (index p a)) <\n      BallPackage.r p.toBallPackage (index p a)\n\u22a2 \u2203 i i_1,\n    color\n          {\n            toBallPackage :=\n              { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n                r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n            \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\n          i_1 =\n        \u2191i \u2227\n      i_1 <\n          lastStep\n            {\n              toBallPackage :=\n                { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n                  r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n              \u03c4 := \u03c4, one_lt_tau := h\u03c4 } \u2227\n        dist (BallPackage.c q b)\n            (BallPackage.c q\n              (index\n                {\n                  toBallPackage :=\n                    { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n                      r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n                  \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\n                i_1)) <\n          BallPackage.r q\n            (index\n              {\n                toBallPackage :=\n                  { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n                    r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n                \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\n              i_1)"}, {"tactic": "exact \u27e8\u27e8p.color a, p.color_lt ha.1 hN\u27e9, a, rfl, ha\u27e9", "annotated_tactic": ["exact \u27e8\u27e8p.color a, p.color_lt ha.1 hN\u27e9, a, <a>rfl</a>, ha\u27e9", [{"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "case inr.refine'_2.intro\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\n\u03b2 : Type u\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nq : BallPackage \u03b2 \u03b1\nh\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1 :=\n  {\n    toBallPackage :=\n      { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n        r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n    \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\ns : Fin N \u2192 Set \u03b2 := fun i => \u22c3 k, \u22c3 (_ : k < lastStep p), \u22c3 (_ : color p k = \u2191i), {index p k}\nb : \u03b2\na : Ordinal.{u}\nha :\n  a < lastStep p \u2227\n    dist (BallPackage.c p.toBallPackage b) (BallPackage.c p.toBallPackage (index p a)) <\n      BallPackage.r p.toBallPackage (index p a)\n\u22a2 \u2203 i i_1,\n    color\n          {\n            toBallPackage :=\n              { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n                r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n            \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\n          i_1 =\n        \u2191i \u2227\n      i_1 <\n          lastStep\n            {\n              toBallPackage :=\n                { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n                  r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n              \u03c4 := \u03c4, one_lt_tau := h\u03c4 } \u2227\n        dist (BallPackage.c q b)\n            (BallPackage.c q\n              (index\n                {\n                  toBallPackage :=\n                    { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n                      r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n                  \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\n                i_1)) <\n          BallPackage.r q\n            (index\n              {\n                toBallPackage :=\n                  { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n                    r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n                \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\n              i_1)", "state_after": "no goals"}, {"tactic": "simpa only [iUnionUpTo, exists_prop, mem_iUnion, mem_ball, Subtype.exists,\n  Subtype.coe_mk] using p.mem_iUnionUpTo_lastStep b", "annotated_tactic": ["simpa only [<a>iUnionUpTo</a>, <a>exists_prop</a>, <a>mem_iUnion</a>, <a>mem_ball</a>, <a>Subtype.exists</a>,\n        <a>Subtype.coe_mk</a>] using p.mem_iUnionUpTo_lastStep b", [{"full_name": "Besicovitch.TauPackage.iUnionUpTo", "def_path": "Mathlib/MeasureTheory/Covering/Besicovitch.lean", "def_pos": [257, 5], "def_end_pos": [257, 15]}, {"full_name": "exists_prop", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [485, 17], "def_end_pos": [485, 28]}, {"full_name": "Set.mem_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [201, 9], "def_end_pos": [201, 19]}, {"full_name": "Metric.mem_ball", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [414, 9], "def_end_pos": [414, 17]}, {"full_name": "Subtype.exists", "def_path": "Mathlib/Data/Subtype.lean", "def_pos": [54, 19], "def_end_pos": [54, 27]}, {"full_name": "Subtype.coe_mk", "def_path": "Mathlib/Data/Subtype.lean", "def_pos": [99, 9], "def_end_pos": [99, 15]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\n\u03b2 : Type u\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nq : BallPackage \u03b2 \u03b1\nh\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1 :=\n  {\n    toBallPackage :=\n      { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n        r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n    \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\ns : Fin N \u2192 Set \u03b2 := fun i => \u22c3 k, \u22c3 (_ : k < lastStep p), \u22c3 (_ : color p k = \u2191i), {index p k}\nb : \u03b2\n\u22a2 \u2203 a,\n    a < lastStep p \u2227\n      dist (BallPackage.c p.toBallPackage b) (BallPackage.c p.toBallPackage (index p a)) <\n        BallPackage.r p.toBallPackage (index p a)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Intervals/Infinite.lean", "full_name": "Set.Ioc.infinite", "start": [60, 1], "end": [61, 39], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/TMToPartrec.lean", "full_name": "Turing.PartrecToTM2.contSupp_supports", "start": [2056, 1], "end": [2073, 84], "traced_tactics": [{"tactic": "induction k", "annotated_tactic": ["induction k", []], "state_before": "S : Finset \u039b'\nk : Cont'\nH : contSupp k \u2286 S\n\u22a2 Supports (contSupp k) S", "state_after": "case halt\nS : Finset \u039b'\nH : contSupp Cont'.halt \u2286 S\n\u22a2 Supports (contSupp Cont'.halt) S\n\ncase cons\u2081\nS : Finset \u039b'\na\u271d\u00b9 : Code\na\u271d : Cont'\na_ih\u271d : contSupp a\u271d \u2286 S \u2192 Supports (contSupp a\u271d) S\nH : contSupp (Cont'.cons\u2081 a\u271d\u00b9 a\u271d) \u2286 S\n\u22a2 Supports (contSupp (Cont'.cons\u2081 a\u271d\u00b9 a\u271d)) S\n\ncase cons\u2082\nS : Finset \u039b'\na\u271d : Cont'\na_ih\u271d : contSupp a\u271d \u2286 S \u2192 Supports (contSupp a\u271d) S\nH : contSupp (Cont'.cons\u2082 a\u271d) \u2286 S\n\u22a2 Supports (contSupp (Cont'.cons\u2082 a\u271d)) S\n\ncase comp\nS : Finset \u039b'\na\u271d\u00b9 : Code\na\u271d : Cont'\na_ih\u271d : contSupp a\u271d \u2286 S \u2192 Supports (contSupp a\u271d) S\nH : contSupp (Cont'.comp a\u271d\u00b9 a\u271d) \u2286 S\n\u22a2 Supports (contSupp (Cont'.comp a\u271d\u00b9 a\u271d)) S\n\ncase fix\nS : Finset \u039b'\na\u271d\u00b9 : Code\na\u271d : Cont'\na_ih\u271d : contSupp a\u271d \u2286 S \u2192 Supports (contSupp a\u271d) S\nH : contSupp (Cont'.fix a\u271d\u00b9 a\u271d) \u2286 S\n\u22a2 Supports (contSupp (Cont'.fix a\u271d\u00b9 a\u271d)) S"}, {"tactic": "case cons\u2081 f k IH =>\n  have H\u2081 := H; rw [contSupp_cons\u2081] at H\u2081; have H\u2082 := Finset.union_subset_right H\u2081\n  refine' trStmts\u2081_supports' (trNormal_supports H\u2082) H\u2081 fun h => _\n  refine' supports_union.2 \u27e8codeSupp'_supports H\u2082, _\u27e9\n  simp only [codeSupp, contSupp_cons\u2082, Finset.union_subset_iff] at H\u2082\n  exact trStmts\u2081_supports' (head_supports H\u2082.2.2) (Finset.union_subset_right h) IH", "annotated_tactic": ["case cons\u2081 f k IH =>\n    have H\u2081 := H; rw [<a>contSupp_cons\u2081</a>] at H\u2081; have H\u2082 := <a>Finset.union_subset_right</a> H\u2081\n    refine' <a>trStmts\u2081_supports'</a> (<a>trNormal_supports</a> H\u2082) H\u2081 fun h => _\n    refine' <a>supports_union</a>.2 \u27e8<a>codeSupp'_supports</a> H\u2082, _\u27e9\n    simp only [<a>codeSupp</a>, <a>contSupp_cons\u2082</a>, <a>Finset.union_subset_iff</a>] at H\u2082\n    exact <a>trStmts\u2081_supports'</a> (<a>head_supports</a> H\u2082.2.2) (<a>Finset.union_subset_right</a> h) IH", [{"full_name": "Turing.PartrecToTM2.contSupp_cons\u2081", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1878, 9], "def_end_pos": [1878, 23]}, {"full_name": "Finset.union_subset_right", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1452, 9], "def_end_pos": [1452, 27]}, {"full_name": "Turing.PartrecToTM2.trStmts\u2081_supports'", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1993, 9], "def_end_pos": [1993, 27]}, {"full_name": "Turing.PartrecToTM2.trNormal_supports", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1999, 9], "def_end_pos": [1999, 26]}, {"full_name": "Turing.PartrecToTM2.supports_union", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1940, 9], "def_end_pos": [1940, 23]}, {"full_name": "Turing.PartrecToTM2.codeSupp'_supports", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [2016, 9], "def_end_pos": [2016, 27]}, {"full_name": "Turing.PartrecToTM2.codeSupp", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1823, 5], "def_end_pos": [1823, 13]}, {"full_name": "Turing.PartrecToTM2.contSupp_cons\u2082", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1889, 9], "def_end_pos": [1889, 23]}, {"full_name": "Finset.union_subset_iff", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1814, 9], "def_end_pos": [1814, 25]}, {"full_name": "Turing.PartrecToTM2.trStmts\u2081_supports'", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1993, 9], "def_end_pos": [1993, 27]}, {"full_name": "Turing.PartrecToTM2.head_supports", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1949, 9], "def_end_pos": [1949, 22]}, {"full_name": "Finset.union_subset_right", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1452, 9], "def_end_pos": [1452, 27]}]], "state_before": "case cons\u2081\nS : Finset \u039b'\na\u271d\u00b9 : Code\na\u271d : Cont'\na_ih\u271d : contSupp a\u271d \u2286 S \u2192 Supports (contSupp a\u271d) S\nH : contSupp (Cont'.cons\u2081 a\u271d\u00b9 a\u271d) \u2286 S\n\u22a2 Supports (contSupp (Cont'.cons\u2081 a\u271d\u00b9 a\u271d)) S\n\ncase cons\u2082\nS : Finset \u039b'\na\u271d : Cont'\na_ih\u271d : contSupp a\u271d \u2286 S \u2192 Supports (contSupp a\u271d) S\nH : contSupp (Cont'.cons\u2082 a\u271d) \u2286 S\n\u22a2 Supports (contSupp (Cont'.cons\u2082 a\u271d)) S\n\ncase comp\nS : Finset \u039b'\na\u271d\u00b9 : Code\na\u271d : Cont'\na_ih\u271d : contSupp a\u271d \u2286 S \u2192 Supports (contSupp a\u271d) S\nH : contSupp (Cont'.comp a\u271d\u00b9 a\u271d) \u2286 S\n\u22a2 Supports (contSupp (Cont'.comp a\u271d\u00b9 a\u271d)) S\n\ncase fix\nS : Finset \u039b'\na\u271d\u00b9 : Code\na\u271d : Cont'\na_ih\u271d : contSupp a\u271d \u2286 S \u2192 Supports (contSupp a\u271d) S\nH : contSupp (Cont'.fix a\u271d\u00b9 a\u271d) \u2286 S\n\u22a2 Supports (contSupp (Cont'.fix a\u271d\u00b9 a\u271d)) S", "state_after": "case cons\u2082\nS : Finset \u039b'\na\u271d : Cont'\na_ih\u271d : contSupp a\u271d \u2286 S \u2192 Supports (contSupp a\u271d) S\nH : contSupp (Cont'.cons\u2082 a\u271d) \u2286 S\n\u22a2 Supports (contSupp (Cont'.cons\u2082 a\u271d)) S\n\ncase comp\nS : Finset \u039b'\na\u271d\u00b9 : Code\na\u271d : Cont'\na_ih\u271d : contSupp a\u271d \u2286 S \u2192 Supports (contSupp a\u271d) S\nH : contSupp (Cont'.comp a\u271d\u00b9 a\u271d) \u2286 S\n\u22a2 Supports (contSupp (Cont'.comp a\u271d\u00b9 a\u271d)) S\n\ncase fix\nS : Finset \u039b'\na\u271d\u00b9 : Code\na\u271d : Cont'\na_ih\u271d : contSupp a\u271d \u2286 S \u2192 Supports (contSupp a\u271d) S\nH : contSupp (Cont'.fix a\u271d\u00b9 a\u271d) \u2286 S\n\u22a2 Supports (contSupp (Cont'.fix a\u271d\u00b9 a\u271d)) S"}, {"tactic": "case cons\u2082 k IH =>\n  have H' := H; rw [contSupp_cons\u2082] at H'\n  exact trStmts\u2081_supports' (head_supports <| Finset.union_subset_right H') H' IH", "annotated_tactic": ["case cons\u2082 k IH =>\n    have H' := H; rw [<a>contSupp_cons\u2082</a>] at H'\n    exact <a>trStmts\u2081_supports'</a> (<a>head_supports</a> <| <a>Finset.union_subset_right</a> H') H' IH", [{"full_name": "Turing.PartrecToTM2.contSupp_cons\u2082", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1889, 9], "def_end_pos": [1889, 23]}, {"full_name": "Turing.PartrecToTM2.trStmts\u2081_supports'", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1993, 9], "def_end_pos": [1993, 27]}, {"full_name": "Turing.PartrecToTM2.head_supports", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1949, 9], "def_end_pos": [1949, 22]}, {"full_name": "Finset.union_subset_right", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1452, 9], "def_end_pos": [1452, 27]}]], "state_before": "case cons\u2082\nS : Finset \u039b'\na\u271d : Cont'\na_ih\u271d : contSupp a\u271d \u2286 S \u2192 Supports (contSupp a\u271d) S\nH : contSupp (Cont'.cons\u2082 a\u271d) \u2286 S\n\u22a2 Supports (contSupp (Cont'.cons\u2082 a\u271d)) S\n\ncase comp\nS : Finset \u039b'\na\u271d\u00b9 : Code\na\u271d : Cont'\na_ih\u271d : contSupp a\u271d \u2286 S \u2192 Supports (contSupp a\u271d) S\nH : contSupp (Cont'.comp a\u271d\u00b9 a\u271d) \u2286 S\n\u22a2 Supports (contSupp (Cont'.comp a\u271d\u00b9 a\u271d)) S\n\ncase fix\nS : Finset \u039b'\na\u271d\u00b9 : Code\na\u271d : Cont'\na_ih\u271d : contSupp a\u271d \u2286 S \u2192 Supports (contSupp a\u271d) S\nH : contSupp (Cont'.fix a\u271d\u00b9 a\u271d) \u2286 S\n\u22a2 Supports (contSupp (Cont'.fix a\u271d\u00b9 a\u271d)) S", "state_after": "case comp\nS : Finset \u039b'\na\u271d\u00b9 : Code\na\u271d : Cont'\na_ih\u271d : contSupp a\u271d \u2286 S \u2192 Supports (contSupp a\u271d) S\nH : contSupp (Cont'.comp a\u271d\u00b9 a\u271d) \u2286 S\n\u22a2 Supports (contSupp (Cont'.comp a\u271d\u00b9 a\u271d)) S\n\ncase fix\nS : Finset \u039b'\na\u271d\u00b9 : Code\na\u271d : Cont'\na_ih\u271d : contSupp a\u271d \u2286 S \u2192 Supports (contSupp a\u271d) S\nH : contSupp (Cont'.fix a\u271d\u00b9 a\u271d) \u2286 S\n\u22a2 Supports (contSupp (Cont'.fix a\u271d\u00b9 a\u271d)) S"}, {"tactic": "case comp f k IH =>\n  have H' := H; rw [contSupp_comp] at H'; have H\u2082 := Finset.union_subset_right H'\n  exact supports_union.2 \u27e8codeSupp'_supports H', IH H\u2082\u27e9", "annotated_tactic": ["case comp f k IH =>\n    have H' := H; rw [<a>contSupp_comp</a>] at H'; have H\u2082 := <a>Finset.union_subset_right</a> H'\n    exact <a>supports_union</a>.2 \u27e8<a>codeSupp'_supports</a> H', IH H\u2082\u27e9", [{"full_name": "Turing.PartrecToTM2.contSupp_comp", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1895, 9], "def_end_pos": [1895, 22]}, {"full_name": "Finset.union_subset_right", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1452, 9], "def_end_pos": [1452, 27]}, {"full_name": "Turing.PartrecToTM2.supports_union", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1940, 9], "def_end_pos": [1940, 23]}, {"full_name": "Turing.PartrecToTM2.codeSupp'_supports", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [2016, 9], "def_end_pos": [2016, 27]}]], "state_before": "case comp\nS : Finset \u039b'\na\u271d\u00b9 : Code\na\u271d : Cont'\na_ih\u271d : contSupp a\u271d \u2286 S \u2192 Supports (contSupp a\u271d) S\nH : contSupp (Cont'.comp a\u271d\u00b9 a\u271d) \u2286 S\n\u22a2 Supports (contSupp (Cont'.comp a\u271d\u00b9 a\u271d)) S\n\ncase fix\nS : Finset \u039b'\na\u271d\u00b9 : Code\na\u271d : Cont'\na_ih\u271d : contSupp a\u271d \u2286 S \u2192 Supports (contSupp a\u271d) S\nH : contSupp (Cont'.fix a\u271d\u00b9 a\u271d) \u2286 S\n\u22a2 Supports (contSupp (Cont'.fix a\u271d\u00b9 a\u271d)) S", "state_after": "case fix\nS : Finset \u039b'\na\u271d\u00b9 : Code\na\u271d : Cont'\na_ih\u271d : contSupp a\u271d \u2286 S \u2192 Supports (contSupp a\u271d) S\nH : contSupp (Cont'.fix a\u271d\u00b9 a\u271d) \u2286 S\n\u22a2 Supports (contSupp (Cont'.fix a\u271d\u00b9 a\u271d)) S"}, {"tactic": "case fix f k IH =>\n  rw [contSupp] at H\n  exact supports_union.2 \u27e8codeSupp'_supports H, IH (Finset.union_subset_right H)\u27e9", "annotated_tactic": ["case fix f k IH =>\n    rw [<a>contSupp</a>] at H\n    exact <a>supports_union</a>.2 \u27e8<a>codeSupp'_supports</a> H, IH (<a>Finset.union_subset_right</a> H)\u27e9", [{"full_name": "Turing.PartrecToTM2.contSupp", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1806, 5], "def_end_pos": [1806, 13]}, {"full_name": "Turing.PartrecToTM2.supports_union", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1940, 9], "def_end_pos": [1940, 23]}, {"full_name": "Turing.PartrecToTM2.codeSupp'_supports", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [2016, 9], "def_end_pos": [2016, 27]}, {"full_name": "Finset.union_subset_right", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1452, 9], "def_end_pos": [1452, 27]}]], "state_before": "case fix\nS : Finset \u039b'\na\u271d\u00b9 : Code\na\u271d : Cont'\na_ih\u271d : contSupp a\u271d \u2286 S \u2192 Supports (contSupp a\u271d) S\nH : contSupp (Cont'.fix a\u271d\u00b9 a\u271d) \u2286 S\n\u22a2 Supports (contSupp (Cont'.fix a\u271d\u00b9 a\u271d)) S", "state_after": "no goals"}, {"tactic": "simp [contSupp_halt, Supports]", "annotated_tactic": ["simp [<a>contSupp_halt</a>, <a>Supports</a>]", [{"full_name": "Turing.PartrecToTM2.contSupp_halt", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1905, 9], "def_end_pos": [1905, 22]}, {"full_name": "Turing.PartrecToTM2.Supports", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1929, 5], "def_end_pos": [1929, 13]}]], "state_before": "case halt\nS : Finset \u039b'\nH : contSupp Cont'.halt \u2286 S\n\u22a2 Supports (contSupp Cont'.halt) S", "state_after": "no goals"}, {"tactic": "have H\u2081 := H", "annotated_tactic": ["have H\u2081 := H", []], "state_before": "S : Finset \u039b'\nf : Code\nk : Cont'\nIH : contSupp k \u2286 S \u2192 Supports (contSupp k) S\nH : contSupp (Cont'.cons\u2081 f k) \u2286 S\n\u22a2 Supports (contSupp (Cont'.cons\u2081 f k)) S", "state_after": "S : Finset \u039b'\nf : Code\nk : Cont'\nIH : contSupp k \u2286 S \u2192 Supports (contSupp k) S\nH H\u2081 : contSupp (Cont'.cons\u2081 f k) \u2286 S\n\u22a2 Supports (contSupp (Cont'.cons\u2081 f k)) S"}, {"tactic": "rw [contSupp_cons\u2081] at H\u2081", "annotated_tactic": ["rw [<a>contSupp_cons\u2081</a>] at H\u2081", [{"full_name": "Turing.PartrecToTM2.contSupp_cons\u2081", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1878, 9], "def_end_pos": [1878, 23]}]], "state_before": "S : Finset \u039b'\nf : Code\nk : Cont'\nIH : contSupp k \u2286 S \u2192 Supports (contSupp k) S\nH H\u2081 : contSupp (Cont'.cons\u2081 f k) \u2286 S\n\u22a2 Supports (contSupp (Cont'.cons\u2081 f k)) S", "state_after": "S : Finset \u039b'\nf : Code\nk : Cont'\nIH : contSupp k \u2286 S \u2192 Supports (contSupp k) S\nH : contSupp (Cont'.cons\u2081 f k) \u2286 S\nH\u2081 :\n  trStmts\u2081\n        (move\u2082 (fun x => false) main aux\n          (move\u2082 (fun s => decide (s = \u0393'.cons\u2097)) stack main\n            (move\u2082 (fun x => false) aux stack (trNormal f (Cont'.cons\u2082 k))))) \u222a\n      codeSupp f (Cont'.cons\u2082 k) \u2286\n    S\n\u22a2 Supports (contSupp (Cont'.cons\u2081 f k)) S"}, {"tactic": "have H\u2082 := Finset.union_subset_right H\u2081", "annotated_tactic": ["have H\u2082 := <a>Finset.union_subset_right</a> H\u2081", [{"full_name": "Finset.union_subset_right", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1452, 9], "def_end_pos": [1452, 27]}]], "state_before": "S : Finset \u039b'\nf : Code\nk : Cont'\nIH : contSupp k \u2286 S \u2192 Supports (contSupp k) S\nH : contSupp (Cont'.cons\u2081 f k) \u2286 S\nH\u2081 :\n  trStmts\u2081\n        (move\u2082 (fun x => false) main aux\n          (move\u2082 (fun s => decide (s = \u0393'.cons\u2097)) stack main\n            (move\u2082 (fun x => false) aux stack (trNormal f (Cont'.cons\u2082 k))))) \u222a\n      codeSupp f (Cont'.cons\u2082 k) \u2286\n    S\n\u22a2 Supports (contSupp (Cont'.cons\u2081 f k)) S", "state_after": "S : Finset \u039b'\nf : Code\nk : Cont'\nIH : contSupp k \u2286 S \u2192 Supports (contSupp k) S\nH : contSupp (Cont'.cons\u2081 f k) \u2286 S\nH\u2081 :\n  trStmts\u2081\n        (move\u2082 (fun x => false) main aux\n          (move\u2082 (fun s => decide (s = \u0393'.cons\u2097)) stack main\n            (move\u2082 (fun x => false) aux stack (trNormal f (Cont'.cons\u2082 k))))) \u222a\n      codeSupp f (Cont'.cons\u2082 k) \u2286\n    S\nH\u2082 : codeSupp f (Cont'.cons\u2082 k) \u2286 S\n\u22a2 Supports (contSupp (Cont'.cons\u2081 f k)) S"}, {"tactic": "refine' trStmts\u2081_supports' (trNormal_supports H\u2082) H\u2081 fun h => _", "annotated_tactic": ["refine' <a>trStmts\u2081_supports'</a> (<a>trNormal_supports</a> H\u2082) H\u2081 fun h => _", [{"full_name": "Turing.PartrecToTM2.trStmts\u2081_supports'", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1993, 9], "def_end_pos": [1993, 27]}, {"full_name": "Turing.PartrecToTM2.trNormal_supports", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1999, 9], "def_end_pos": [1999, 26]}]], "state_before": "S : Finset \u039b'\nf : Code\nk : Cont'\nIH : contSupp k \u2286 S \u2192 Supports (contSupp k) S\nH : contSupp (Cont'.cons\u2081 f k) \u2286 S\nH\u2081 :\n  trStmts\u2081\n        (move\u2082 (fun x => false) main aux\n          (move\u2082 (fun s => decide (s = \u0393'.cons\u2097)) stack main\n            (move\u2082 (fun x => false) aux stack (trNormal f (Cont'.cons\u2082 k))))) \u222a\n      codeSupp f (Cont'.cons\u2082 k) \u2286\n    S\nH\u2082 : codeSupp f (Cont'.cons\u2082 k) \u2286 S\n\u22a2 Supports (contSupp (Cont'.cons\u2081 f k)) S", "state_after": "S : Finset \u039b'\nf : Code\nk : Cont'\nIH : contSupp k \u2286 S \u2192 Supports (contSupp k) S\nH : contSupp (Cont'.cons\u2081 f k) \u2286 S\nH\u2081 :\n  trStmts\u2081\n        (move\u2082 (fun x => false) main aux\n          (move\u2082 (fun s => decide (s = \u0393'.cons\u2097)) stack main\n            (move\u2082 (fun x => false) aux stack (trNormal f (Cont'.cons\u2082 k))))) \u222a\n      codeSupp f (Cont'.cons\u2082 k) \u2286\n    S\nH\u2082 : codeSupp f (Cont'.cons\u2082 k) \u2286 S\nh : codeSupp' f (Cont'.cons\u2082 k) \u222a (trStmts\u2081 (head stack (\u039b'.ret k)) \u222a contSupp k) \u2286 S\n\u22a2 Supports (codeSupp' f (Cont'.cons\u2082 k) \u222a (trStmts\u2081 (head stack (\u039b'.ret k)) \u222a contSupp k)) S"}, {"tactic": "refine' supports_union.2 \u27e8codeSupp'_supports H\u2082, _\u27e9", "annotated_tactic": ["refine' <a>supports_union</a>.2 \u27e8<a>codeSupp'_supports</a> H\u2082, _\u27e9", [{"full_name": "Turing.PartrecToTM2.supports_union", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1940, 9], "def_end_pos": [1940, 23]}, {"full_name": "Turing.PartrecToTM2.codeSupp'_supports", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [2016, 9], "def_end_pos": [2016, 27]}]], "state_before": "S : Finset \u039b'\nf : Code\nk : Cont'\nIH : contSupp k \u2286 S \u2192 Supports (contSupp k) S\nH : contSupp (Cont'.cons\u2081 f k) \u2286 S\nH\u2081 :\n  trStmts\u2081\n        (move\u2082 (fun x => false) main aux\n          (move\u2082 (fun s => decide (s = \u0393'.cons\u2097)) stack main\n            (move\u2082 (fun x => false) aux stack (trNormal f (Cont'.cons\u2082 k))))) \u222a\n      codeSupp f (Cont'.cons\u2082 k) \u2286\n    S\nH\u2082 : codeSupp f (Cont'.cons\u2082 k) \u2286 S\nh : codeSupp' f (Cont'.cons\u2082 k) \u222a (trStmts\u2081 (head stack (\u039b'.ret k)) \u222a contSupp k) \u2286 S\n\u22a2 Supports (codeSupp' f (Cont'.cons\u2082 k) \u222a (trStmts\u2081 (head stack (\u039b'.ret k)) \u222a contSupp k)) S", "state_after": "S : Finset \u039b'\nf : Code\nk : Cont'\nIH : contSupp k \u2286 S \u2192 Supports (contSupp k) S\nH : contSupp (Cont'.cons\u2081 f k) \u2286 S\nH\u2081 :\n  trStmts\u2081\n        (move\u2082 (fun x => false) main aux\n          (move\u2082 (fun s => decide (s = \u0393'.cons\u2097)) stack main\n            (move\u2082 (fun x => false) aux stack (trNormal f (Cont'.cons\u2082 k))))) \u222a\n      codeSupp f (Cont'.cons\u2082 k) \u2286\n    S\nH\u2082 : codeSupp f (Cont'.cons\u2082 k) \u2286 S\nh : codeSupp' f (Cont'.cons\u2082 k) \u222a (trStmts\u2081 (head stack (\u039b'.ret k)) \u222a contSupp k) \u2286 S\n\u22a2 Supports (trStmts\u2081 (head stack (\u039b'.ret k)) \u222a contSupp k) S"}, {"tactic": "simp only [codeSupp, contSupp_cons\u2082, Finset.union_subset_iff] at H\u2082", "annotated_tactic": ["simp only [<a>codeSupp</a>, <a>contSupp_cons\u2082</a>, <a>Finset.union_subset_iff</a>] at H\u2082", [{"full_name": "Turing.PartrecToTM2.codeSupp", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1823, 5], "def_end_pos": [1823, 13]}, {"full_name": "Turing.PartrecToTM2.contSupp_cons\u2082", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1889, 9], "def_end_pos": [1889, 23]}, {"full_name": "Finset.union_subset_iff", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1814, 9], "def_end_pos": [1814, 25]}]], "state_before": "S : Finset \u039b'\nf : Code\nk : Cont'\nIH : contSupp k \u2286 S \u2192 Supports (contSupp k) S\nH : contSupp (Cont'.cons\u2081 f k) \u2286 S\nH\u2081 :\n  trStmts\u2081\n        (move\u2082 (fun x => false) main aux\n          (move\u2082 (fun s => decide (s = \u0393'.cons\u2097)) stack main\n            (move\u2082 (fun x => false) aux stack (trNormal f (Cont'.cons\u2082 k))))) \u222a\n      codeSupp f (Cont'.cons\u2082 k) \u2286\n    S\nH\u2082 : codeSupp f (Cont'.cons\u2082 k) \u2286 S\nh : codeSupp' f (Cont'.cons\u2082 k) \u222a (trStmts\u2081 (head stack (\u039b'.ret k)) \u222a contSupp k) \u2286 S\n\u22a2 Supports (trStmts\u2081 (head stack (\u039b'.ret k)) \u222a contSupp k) S", "state_after": "S : Finset \u039b'\nf : Code\nk : Cont'\nIH : contSupp k \u2286 S \u2192 Supports (contSupp k) S\nH : contSupp (Cont'.cons\u2081 f k) \u2286 S\nH\u2081 :\n  trStmts\u2081\n        (move\u2082 (fun x => false) main aux\n          (move\u2082 (fun s => decide (s = \u0393'.cons\u2097)) stack main\n            (move\u2082 (fun x => false) aux stack (trNormal f (Cont'.cons\u2082 k))))) \u222a\n      codeSupp f (Cont'.cons\u2082 k) \u2286\n    S\nh : codeSupp' f (Cont'.cons\u2082 k) \u222a (trStmts\u2081 (head stack (\u039b'.ret k)) \u222a contSupp k) \u2286 S\nH\u2082 : codeSupp' f (Cont'.cons\u2082 k) \u2286 S \u2227 trStmts\u2081 (head stack (\u039b'.ret k)) \u2286 S \u2227 contSupp k \u2286 S\n\u22a2 Supports (trStmts\u2081 (head stack (\u039b'.ret k)) \u222a contSupp k) S"}, {"tactic": "exact trStmts\u2081_supports' (head_supports H\u2082.2.2) (Finset.union_subset_right h) IH", "annotated_tactic": ["exact <a>trStmts\u2081_supports'</a> (<a>head_supports</a> H\u2082.2.2) (<a>Finset.union_subset_right</a> h) IH", [{"full_name": "Turing.PartrecToTM2.trStmts\u2081_supports'", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1993, 9], "def_end_pos": [1993, 27]}, {"full_name": "Turing.PartrecToTM2.head_supports", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1949, 9], "def_end_pos": [1949, 22]}, {"full_name": "Finset.union_subset_right", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1452, 9], "def_end_pos": [1452, 27]}]], "state_before": "S : Finset \u039b'\nf : Code\nk : Cont'\nIH : contSupp k \u2286 S \u2192 Supports (contSupp k) S\nH : contSupp (Cont'.cons\u2081 f k) \u2286 S\nH\u2081 :\n  trStmts\u2081\n        (move\u2082 (fun x => false) main aux\n          (move\u2082 (fun s => decide (s = \u0393'.cons\u2097)) stack main\n            (move\u2082 (fun x => false) aux stack (trNormal f (Cont'.cons\u2082 k))))) \u222a\n      codeSupp f (Cont'.cons\u2082 k) \u2286\n    S\nh : codeSupp' f (Cont'.cons\u2082 k) \u222a (trStmts\u2081 (head stack (\u039b'.ret k)) \u222a contSupp k) \u2286 S\nH\u2082 : codeSupp' f (Cont'.cons\u2082 k) \u2286 S \u2227 trStmts\u2081 (head stack (\u039b'.ret k)) \u2286 S \u2227 contSupp k \u2286 S\n\u22a2 Supports (trStmts\u2081 (head stack (\u039b'.ret k)) \u222a contSupp k) S", "state_after": "no goals"}, {"tactic": "have H' := H", "annotated_tactic": ["have H' := H", []], "state_before": "S : Finset \u039b'\nk : Cont'\nIH : contSupp k \u2286 S \u2192 Supports (contSupp k) S\nH : contSupp (Cont'.cons\u2082 k) \u2286 S\n\u22a2 Supports (contSupp (Cont'.cons\u2082 k)) S", "state_after": "S : Finset \u039b'\nk : Cont'\nIH : contSupp k \u2286 S \u2192 Supports (contSupp k) S\nH H' : contSupp (Cont'.cons\u2082 k) \u2286 S\n\u22a2 Supports (contSupp (Cont'.cons\u2082 k)) S"}, {"tactic": "rw [contSupp_cons\u2082] at H'", "annotated_tactic": ["rw [<a>contSupp_cons\u2082</a>] at H'", [{"full_name": "Turing.PartrecToTM2.contSupp_cons\u2082", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1889, 9], "def_end_pos": [1889, 23]}]], "state_before": "S : Finset \u039b'\nk : Cont'\nIH : contSupp k \u2286 S \u2192 Supports (contSupp k) S\nH H' : contSupp (Cont'.cons\u2082 k) \u2286 S\n\u22a2 Supports (contSupp (Cont'.cons\u2082 k)) S", "state_after": "S : Finset \u039b'\nk : Cont'\nIH : contSupp k \u2286 S \u2192 Supports (contSupp k) S\nH : contSupp (Cont'.cons\u2082 k) \u2286 S\nH' : trStmts\u2081 (head stack (\u039b'.ret k)) \u222a contSupp k \u2286 S\n\u22a2 Supports (contSupp (Cont'.cons\u2082 k)) S"}, {"tactic": "exact trStmts\u2081_supports' (head_supports <| Finset.union_subset_right H') H' IH", "annotated_tactic": ["exact <a>trStmts\u2081_supports'</a> (<a>head_supports</a> <| <a>Finset.union_subset_right</a> H') H' IH", [{"full_name": "Turing.PartrecToTM2.trStmts\u2081_supports'", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1993, 9], "def_end_pos": [1993, 27]}, {"full_name": "Turing.PartrecToTM2.head_supports", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1949, 9], "def_end_pos": [1949, 22]}, {"full_name": "Finset.union_subset_right", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1452, 9], "def_end_pos": [1452, 27]}]], "state_before": "S : Finset \u039b'\nk : Cont'\nIH : contSupp k \u2286 S \u2192 Supports (contSupp k) S\nH : contSupp (Cont'.cons\u2082 k) \u2286 S\nH' : trStmts\u2081 (head stack (\u039b'.ret k)) \u222a contSupp k \u2286 S\n\u22a2 Supports (contSupp (Cont'.cons\u2082 k)) S", "state_after": "no goals"}, {"tactic": "have H' := H", "annotated_tactic": ["have H' := H", []], "state_before": "S : Finset \u039b'\nf : Code\nk : Cont'\nIH : contSupp k \u2286 S \u2192 Supports (contSupp k) S\nH : contSupp (Cont'.comp f k) \u2286 S\n\u22a2 Supports (contSupp (Cont'.comp f k)) S", "state_after": "S : Finset \u039b'\nf : Code\nk : Cont'\nIH : contSupp k \u2286 S \u2192 Supports (contSupp k) S\nH H' : contSupp (Cont'.comp f k) \u2286 S\n\u22a2 Supports (contSupp (Cont'.comp f k)) S"}, {"tactic": "rw [contSupp_comp] at H'", "annotated_tactic": ["rw [<a>contSupp_comp</a>] at H'", [{"full_name": "Turing.PartrecToTM2.contSupp_comp", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1895, 9], "def_end_pos": [1895, 22]}]], "state_before": "S : Finset \u039b'\nf : Code\nk : Cont'\nIH : contSupp k \u2286 S \u2192 Supports (contSupp k) S\nH H' : contSupp (Cont'.comp f k) \u2286 S\n\u22a2 Supports (contSupp (Cont'.comp f k)) S", "state_after": "S : Finset \u039b'\nf : Code\nk : Cont'\nIH : contSupp k \u2286 S \u2192 Supports (contSupp k) S\nH : contSupp (Cont'.comp f k) \u2286 S\nH' : codeSupp f k \u2286 S\n\u22a2 Supports (contSupp (Cont'.comp f k)) S"}, {"tactic": "have H\u2082 := Finset.union_subset_right H'", "annotated_tactic": ["have H\u2082 := <a>Finset.union_subset_right</a> H'", [{"full_name": "Finset.union_subset_right", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1452, 9], "def_end_pos": [1452, 27]}]], "state_before": "S : Finset \u039b'\nf : Code\nk : Cont'\nIH : contSupp k \u2286 S \u2192 Supports (contSupp k) S\nH : contSupp (Cont'.comp f k) \u2286 S\nH' : codeSupp f k \u2286 S\n\u22a2 Supports (contSupp (Cont'.comp f k)) S", "state_after": "S : Finset \u039b'\nf : Code\nk : Cont'\nIH : contSupp k \u2286 S \u2192 Supports (contSupp k) S\nH : contSupp (Cont'.comp f k) \u2286 S\nH' : codeSupp f k \u2286 S\nH\u2082 : contSupp k \u2286 S\n\u22a2 Supports (contSupp (Cont'.comp f k)) S"}, {"tactic": "exact supports_union.2 \u27e8codeSupp'_supports H', IH H\u2082\u27e9", "annotated_tactic": ["exact <a>supports_union</a>.2 \u27e8<a>codeSupp'_supports</a> H', IH H\u2082\u27e9", [{"full_name": "Turing.PartrecToTM2.supports_union", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1940, 9], "def_end_pos": [1940, 23]}, {"full_name": "Turing.PartrecToTM2.codeSupp'_supports", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [2016, 9], "def_end_pos": [2016, 27]}]], "state_before": "S : Finset \u039b'\nf : Code\nk : Cont'\nIH : contSupp k \u2286 S \u2192 Supports (contSupp k) S\nH : contSupp (Cont'.comp f k) \u2286 S\nH' : codeSupp f k \u2286 S\nH\u2082 : contSupp k \u2286 S\n\u22a2 Supports (contSupp (Cont'.comp f k)) S", "state_after": "no goals"}, {"tactic": "rw [contSupp] at H", "annotated_tactic": ["rw [<a>contSupp</a>] at H", [{"full_name": "Turing.PartrecToTM2.contSupp", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1806, 5], "def_end_pos": [1806, 13]}]], "state_before": "S : Finset \u039b'\nf : Code\nk : Cont'\nIH : contSupp k \u2286 S \u2192 Supports (contSupp k) S\nH : contSupp (Cont'.fix f k) \u2286 S\n\u22a2 Supports (contSupp (Cont'.fix f k)) S", "state_after": "S : Finset \u039b'\nf : Code\nk : Cont'\nIH : contSupp k \u2286 S \u2192 Supports (contSupp k) S\nH : codeSupp' (Code.fix f) k \u222a contSupp k \u2286 S\n\u22a2 Supports (contSupp (Cont'.fix f k)) S"}, {"tactic": "exact supports_union.2 \u27e8codeSupp'_supports H, IH (Finset.union_subset_right H)\u27e9", "annotated_tactic": ["exact <a>supports_union</a>.2 \u27e8<a>codeSupp'_supports</a> H, IH (<a>Finset.union_subset_right</a> H)\u27e9", [{"full_name": "Turing.PartrecToTM2.supports_union", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1940, 9], "def_end_pos": [1940, 23]}, {"full_name": "Turing.PartrecToTM2.codeSupp'_supports", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [2016, 9], "def_end_pos": [2016, 27]}, {"full_name": "Finset.union_subset_right", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1452, 9], "def_end_pos": [1452, 27]}]], "state_before": "S : Finset \u039b'\nf : Code\nk : Cont'\nIH : contSupp k \u2286 S \u2192 Supports (contSupp k) S\nH : codeSupp' (Code.fix f) k \u222a contSupp k \u2286 S\n\u22a2 Supports (contSupp (Cont'.fix f k)) S", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "Finset.disjoint_erase_comm", "start": [2268, 1], "end": [2269, 42], "traced_tactics": [{"tactic": "simp_rw [erase_eq, disjoint_sdiff_comm]", "annotated_tactic": ["simp_rw [<a>erase_eq</a>, <a>disjoint_sdiff_comm</a>]", [{"full_name": "Finset.erase_eq", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2264, 9], "def_end_pos": [2264, 17]}, {"full_name": "disjoint_sdiff_comm", "def_path": "Mathlib/Order/BooleanAlgebra.lean", "def_pos": [458, 9], "def_end_pos": [458, 28]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns t u v : Finset \u03b1\na b : \u03b1\n\u22a2 _root_.Disjoint (erase s a) t \u2194 _root_.Disjoint s (erase t a)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/Jacobian.lean", "full_name": "MeasureTheory.map_withDensity_abs_det_fderiv_eq_addHaar", "start": [1118, 1], "end": [1127, 26], "traced_tactics": [{"tactic": "apply Measure.ext fun t ht => ?_", "annotated_tactic": ["apply <a>Measure.ext</a> fun t ht => ?_", [{"full_name": "MeasureTheory.Measure.ext", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [135, 9], "def_end_pos": [135, 12]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nh'f : Measurable f\n\u22a2 Measure.map f (withDensity (Measure.restrict \u03bc s) fun x => ENNReal.ofReal |ContinuousLinearMap.det (f' x)|) =\n    Measure.restrict \u03bc (f '' s)", "state_after": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nh'f : Measurable f\nt : Set E\nht : MeasurableSet t\n\u22a2 \u2191\u2191(Measure.map f (withDensity (Measure.restrict \u03bc s) fun x => ENNReal.ofReal |ContinuousLinearMap.det (f' x)|)) t =\n    \u2191\u2191(Measure.restrict \u03bc (f '' s)) t"}, {"tactic": "rw [map_apply h'f ht, withDensity_apply _ (h'f ht), Measure.restrict_apply ht,\n  restrict_restrict (h'f ht),\n  lintegral_abs_det_fderiv_eq_addHaar_image \u03bc ((h'f ht).inter hs)\n    (fun x hx => (hf' x hx.2).mono (inter_subset_right _ _)) (hf.mono (inter_subset_right _ _)),\n  image_preimage_inter]", "annotated_tactic": ["rw [<a>map_apply</a> h'f ht, <a>withDensity_apply</a> _ (h'f ht), <a>Measure.restrict_apply</a> ht,\n    <a>restrict_restrict</a> (h'f ht),\n    <a>lintegral_abs_det_fderiv_eq_addHaar_image</a> \u03bc ((h'f ht).<a>inter</a> hs)\n      (fun x hx => (hf' x hx.2).<a>mono</a> (<a>inter_subset_right</a> _ _)) (hf.mono (<a>inter_subset_right</a> _ _)),\n    <a>image_preimage_inter</a>]", [{"full_name": "MeasureTheory.Measure.map_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1236, 9], "def_end_pos": [1236, 18]}, {"full_name": "MeasureTheory.withDensity_apply", "def_path": "Mathlib/MeasureTheory/Measure/WithDensity.lean", "def_pos": [39, 9], "def_end_pos": [39, 26]}, {"full_name": "MeasureTheory.Measure.restrict_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1533, 9], "def_end_pos": [1533, 23]}, {"full_name": "MeasureTheory.Measure.restrict_restrict", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1645, 9], "def_end_pos": [1645, 26]}, {"full_name": "MeasureTheory.lintegral_abs_det_fderiv_eq_addHaar_image", "def_path": "Mathlib/MeasureTheory/Function/Jacobian.lean", "def_pos": [1104, 9], "def_end_pos": [1104, 50]}, {"full_name": "MeasurableSet.inter", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [198, 19], "def_end_pos": [198, 38]}, {"full_name": "HasFDerivWithinAt.mono", "def_path": "Mathlib/Analysis/Calculus/FDeriv/Basic.lean", "def_pos": [380, 16], "def_end_pos": [380, 38]}, {"full_name": "Set.inter_subset_right", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [969, 9], "def_end_pos": [969, 27]}, {"full_name": "Set.inter_subset_right", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [969, 9], "def_end_pos": [969, 27]}, {"full_name": "Set.image_preimage_inter", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [538, 9], "def_end_pos": [538, 29]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nh'f : Measurable f\nt : Set E\nht : MeasurableSet t\n\u22a2 \u2191\u2191(Measure.map f (withDensity (Measure.restrict \u03bc s) fun x => ENNReal.ofReal |ContinuousLinearMap.det (f' x)|)) t =\n    \u2191\u2191(Measure.restrict \u03bc (f '' s)) t", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Lattice.lean", "full_name": "Finset.min'_lt_of_mem_erase_min'", "start": [1583, 1], "end": [1585, 46], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Covering/Besicovitch.lean", "full_name": "Besicovitch.exist_finset_disjoint_balls_large_measure", "start": [546, 1], "end": [670, 32], "traced_tactics": [{"tactic": "rcases le_or_lt (\u03bc s) 0 with (h\u03bcs | h\u03bcs)", "annotated_tactic": ["rcases <a>le_or_lt</a> (\u03bc s) 0 with (h\u03bcs | h\u03bcs)", [{"full_name": "le_or_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [340, 9], "def_end_pos": [340, 17]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\n\u22a2 \u2203 t,\n    \u2191t \u2286 s \u2227\n      \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s \u2227 PairwiseDisjoint \u2191t fun x => closedBall x (r x)", "state_after": "case inl\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : \u2191\u2191\u03bc s \u2264 0\n\u22a2 \u2203 t,\n    \u2191t \u2286 s \u2227\n      \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s \u2227 PairwiseDisjoint \u2191t fun x => closedBall x (r x)\n\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\n\u22a2 \u2203 t,\n    \u2191t \u2286 s \u2227\n      \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s \u2227 PairwiseDisjoint \u2191t fun x => closedBall x (r x)"}, {"tactic": "cases isEmpty_or_nonempty \u03b1", "annotated_tactic": ["cases <a>isEmpty_or_nonempty</a> \u03b1", [{"full_name": "isEmpty_or_nonempty", "def_path": "Mathlib/Logic/IsEmpty.lean", "def_pos": [207, 9], "def_end_pos": [207, 28]}]], "state_before": "case inr\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\n\u22a2 \u2203 t,\n    \u2191t \u2286 s \u2227\n      \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s \u2227 PairwiseDisjoint \u2191t fun x => closedBall x (r x)", "state_after": "case inr.inl\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : IsEmpty \u03b1\n\u22a2 \u2203 t,\n    \u2191t \u2286 s \u2227\n      \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s \u2227 PairwiseDisjoint \u2191t fun x => closedBall x (r x)\n\ncase inr.inr\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\n\u22a2 \u2203 t,\n    \u2191t \u2286 s \u2227\n      \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s \u2227 PairwiseDisjoint \u2191t fun x => closedBall x (r x)"}, {"tactic": "have Npos : N \u2260 0 := by\n  rintro rfl\n  inhabit \u03b1\n  exact not_isEmpty_of_nonempty _ hN", "annotated_tactic": ["have Npos : N \u2260 0 := by\n    rintro rfl\n    inhabit \u03b1\n    exact <a>not_isEmpty_of_nonempty</a> _ hN", [{"full_name": "not_isEmpty_of_nonempty", "def_path": "Mathlib/Logic/IsEmpty.lean", "def_pos": [212, 9], "def_end_pos": [212, 32]}]], "state_before": "case inr.inr\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\n\u22a2 \u2203 t,\n    \u2191t \u2286 s \u2227\n      \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s \u2227 PairwiseDisjoint \u2191t fun x => closedBall x (r x)", "state_after": "case inr.inr\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\n\u22a2 \u2203 t,\n    \u2191t \u2286 s \u2227\n      \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s \u2227 PairwiseDisjoint \u2191t fun x => closedBall x (r x)"}, {"tactic": "obtain \u27e8o, so, omeas, \u03bco\u27e9 : \u2203 o : Set \u03b1, s \u2286 o \u2227 MeasurableSet o \u2227 \u03bc o = \u03bc s :=\n  exists_measurable_superset \u03bc s", "annotated_tactic": ["obtain \u27e8o, so, omeas, \u03bco\u27e9 : \u2203 o : <a>Set</a> \u03b1, s \u2286 o \u2227 <a>MeasurableSet</a> o \u2227 \u03bc o = \u03bc s :=\n    <a>exists_measurable_superset</a> \u03bc s", [{"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}, {"full_name": "MeasurableSet", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [64, 5], "def_end_pos": [64, 18]}, {"full_name": "MeasureTheory.exists_measurable_superset", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [212, 9], "def_end_pos": [212, 35]}]], "state_before": "case inr.inr\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\n\u22a2 \u2203 t,\n    \u2191t \u2286 s \u2227\n      \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s \u2227 PairwiseDisjoint \u2191t fun x => closedBall x (r x)", "state_after": "case inr.inr.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\n\u22a2 \u2203 t,\n    \u2191t \u2286 s \u2227\n      \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s \u2227 PairwiseDisjoint \u2191t fun x => closedBall x (r x)"}, {"tactic": "let a : BallPackage s \u03b1 :=\n  { c := fun x => x\n    r := fun x => r x\n    rpos := fun x => rpos x x.2\n    r_bound := 1\n    r_le := fun x => rle x x.2 }", "annotated_tactic": ["let a : <a>BallPackage</a> s \u03b1 :=\n    { c := fun x => x\n      r := fun x => r x\n      rpos := fun x => rpos x x.2\n      r_bound := 1\n      r_le := fun x => rle x x.2 }", [{"full_name": "Besicovitch.BallPackage", "def_path": "Mathlib/MeasureTheory/Covering/Besicovitch.lean", "def_pos": [192, 11], "def_end_pos": [192, 22]}]], "state_before": "case inr.inr.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\n\u22a2 \u2203 t,\n    \u2191t \u2286 s \u2227\n      \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s \u2227 PairwiseDisjoint \u2191t fun x => closedBall x (r x)", "state_after": "case inr.inr.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\n\u22a2 \u2203 t,\n    \u2191t \u2286 s \u2227\n      \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s \u2227 PairwiseDisjoint \u2191t fun x => closedBall x (r x)"}, {"tactic": "rcases exist_disjoint_covering_families h\u03c4 hN a with \u27e8u, hu, hu'\u27e9", "annotated_tactic": ["rcases <a>exist_disjoint_covering_families</a> h\u03c4 hN a with \u27e8u, hu, hu'\u27e9", [{"full_name": "Besicovitch.exist_disjoint_covering_families", "def_path": "Mathlib/MeasureTheory/Covering/Besicovitch.lean", "def_pos": [471, 9], "def_end_pos": [471, 41]}]], "state_before": "case inr.inr.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\n\u22a2 \u2203 t,\n    \u2191t \u2286 s \u2227\n      \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s \u2227 PairwiseDisjoint \u2191t fun x => closedBall x (r x)", "state_after": "case inr.inr.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\n\u22a2 \u2203 t,\n    \u2191t \u2286 s \u2227\n      \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s \u2227 PairwiseDisjoint \u2191t fun x => closedBall x (r x)"}, {"tactic": "have u_count : \u2200 i, (u i).Countable := by\n  intro i\n  refine' (hu i).countable_of_nonempty_interior fun j _ => _\n  have : (ball (j : \u03b1) (r j)).Nonempty := nonempty_ball.2 (a.rpos _)\n  exact this.mono ball_subset_interior_closedBall", "annotated_tactic": ["have u_count : \u2200 i, (u i).<a>Countable</a> := by\n    intro i\n    refine' (hu i).<a>countable_of_nonempty_interior</a> fun j _ => _\n    have : (<a>ball</a> (j : \u03b1) (r j)).<a>Nonempty</a> := <a>nonempty_ball</a>.2 (a.rpos _)\n    exact this.mono <a>ball_subset_interior_closedBall</a>", [{"full_name": "Set.Countable", "def_path": "Mathlib/Data/Set/Countable.lean", "def_pos": [31, 15], "def_end_pos": [31, 24]}, {"full_name": "Set.PairwiseDisjoint.countable_of_nonempty_interior", "def_path": "Mathlib/Topology/Bases.lean", "def_pos": [427, 9], "def_end_pos": [427, 67]}, {"full_name": "Metric.ball", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [409, 5], "def_end_pos": [409, 9]}, {"full_name": "Set.Nonempty", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [439, 15], "def_end_pos": [439, 23]}, {"full_name": "Metric.nonempty_ball", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [430, 9], "def_end_pos": [430, 22]}, {"full_name": "Metric.ball_subset_interior_closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [1920, 9], "def_end_pos": [1920, 40]}]], "state_before": "case inr.inr.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\n\u22a2 \u2203 t,\n    \u2191t \u2286 s \u2227\n      \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s \u2227 PairwiseDisjoint \u2191t fun x => closedBall x (r x)", "state_after": "case inr.inr.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\n\u22a2 \u2203 t,\n    \u2191t \u2286 s \u2227\n      \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s \u2227 PairwiseDisjoint \u2191t fun x => closedBall x (r x)"}, {"tactic": "let v : Fin N \u2192 Set \u03b1 := fun i => \u22c3 (x : s) (_ : x \u2208 u i), closedBall x (r x)", "annotated_tactic": ["let v : <a>Fin</a> N \u2192 <a>Set</a> \u03b1 := fun i => \u22c3 (x : s) (_ : x \u2208 u i), <a>closedBall</a> x (r x)", [{"full_name": "Fin", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1745, 11], "def_end_pos": [1745, 14]}, {"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}]], "state_before": "case inr.inr.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\n\u22a2 \u2203 t,\n    \u2191t \u2286 s \u2227\n      \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s \u2227 PairwiseDisjoint \u2191t fun x => closedBall x (r x)", "state_after": "case inr.inr.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\n\u22a2 \u2203 t,\n    \u2191t \u2286 s \u2227\n      \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s \u2227 PairwiseDisjoint \u2191t fun x => closedBall x (r x)"}, {"tactic": "have A : s = \u22c3 i : Fin N, s \u2229 v i := by\n  refine' Subset.antisymm _ (iUnion_subset fun i => inter_subset_left _ _)\n  intro x hx\n  obtain \u27e8i, y, hxy, h'\u27e9 :\n      \u2203 (i : Fin N) (i_1 : \u21a5s), i_1 \u2208 u i \u2227 x \u2208 ball (\u2191i_1) (r \u2191i_1) := by\n    have : x \u2208 range a.c := by simpa only [Subtype.range_coe_subtype, setOf_mem_eq]\n    simpa only [mem_iUnion, bex_def] using hu' this\n  refine' mem_iUnion.2 \u27e8i, \u27e8hx, _\u27e9\u27e9\n  simp only [exists_prop, mem_iUnion, SetCoe.exists, exists_and_right, Subtype.coe_mk]\n  exact \u27e8y, \u27e8y.2, by simpa only [Subtype.coe_eta]\u27e9, ball_subset_closedBall h'\u27e9", "annotated_tactic": ["have A : s = \u22c3 i : <a>Fin</a> N, s \u2229 v i := by\n    refine' <a>Subset.antisymm</a> _ (<a>iUnion_subset</a> fun i => <a>inter_subset_left</a> _ _)\n    intro x hx\n    obtain \u27e8i, y, hxy, h'\u27e9 :\n        \u2203 (i : <a>Fin</a> N) (i_1 : \u21a5s), i_1 \u2208 u i \u2227 x \u2208 <a>ball</a> (\u2191i_1) (r \u2191i_1) := by\n      have : x \u2208 <a>range</a> a.c := by simpa only [<a>Subtype.range_coe_subtype</a>, <a>setOf_mem_eq</a>]\n      simpa only [<a>mem_iUnion</a>, <a>bex_def</a>] using hu' this\n    refine' <a>mem_iUnion</a>.2 \u27e8i, \u27e8hx, _\u27e9\u27e9\n    simp only [<a>exists_prop</a>, <a>mem_iUnion</a>, <a>SetCoe.exists</a>, <a>exists_and_right</a>, <a>Subtype.coe_mk</a>]\n    exact \u27e8y, \u27e8y.2, by simpa only [<a>Subtype.coe_eta</a>]\u27e9, <a>ball_subset_closedBall</a> h'\u27e9", [{"full_name": "Fin", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1745, 11], "def_end_pos": [1745, 14]}, {"full_name": "Set.Subset.antisymm", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [370, 9], "def_end_pos": [370, 24]}, {"full_name": "Set.iUnion_subset", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [390, 9], "def_end_pos": [390, 22]}, {"full_name": "Set.inter_subset_left", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [965, 9], "def_end_pos": [965, 26]}, {"full_name": "Fin", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1745, 11], "def_end_pos": [1745, 14]}, {"full_name": "Metric.ball", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [409, 5], "def_end_pos": [409, 9]}, {"full_name": "Set.range", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [668, 5], "def_end_pos": [668, 10]}, {"full_name": "Subtype.range_coe_subtype", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [1425, 9], "def_end_pos": [1425, 26]}, {"full_name": "Set.setOf_mem_eq", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [275, 9], "def_end_pos": [275, 21]}, {"full_name": "Set.mem_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [201, 9], "def_end_pos": [201, 19]}, {"full_name": "bex_def", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [1027, 9], "def_end_pos": [1027, 16]}, {"full_name": "Set.mem_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [201, 9], "def_end_pos": [201, 19]}, {"full_name": "exists_prop", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [485, 17], "def_end_pos": [485, 28]}, {"full_name": "Set.mem_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [201, 9], "def_end_pos": [201, 19]}, {"full_name": "SetCoe.exists", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [190, 9], "def_end_pos": [190, 22]}, {"full_name": "exists_and_right", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [468, 17], "def_end_pos": [468, 33]}, {"full_name": "Subtype.coe_mk", "def_path": "Mathlib/Data/Subtype.lean", "def_pos": [99, 9], "def_end_pos": [99, 15]}, {"full_name": "Subtype.coe_eta", "def_path": "Mathlib/Data/Subtype.lean", "def_pos": [95, 9], "def_end_pos": [95, 16]}, {"full_name": "Metric.ball_subset_closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [534, 9], "def_end_pos": [534, 31]}]], "state_before": "case inr.inr.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\n\u22a2 \u2203 t,\n    \u2191t \u2286 s \u2227\n      \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s \u2227 PairwiseDisjoint \u2191t fun x => closedBall x (r x)", "state_after": "case inr.inr.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 i, s \u2229 v i\n\u22a2 \u2203 t,\n    \u2191t \u2286 s \u2227\n      \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s \u2227 PairwiseDisjoint \u2191t fun x => closedBall x (r x)"}, {"tactic": "obtain \u27e8i, -, hi\u27e9 : \u2203 (i : Fin N), i \u2208 Finset.univ \u2227 \u03bc s / N \u2264 \u03bc (s \u2229 v i) := by\n  apply ENNReal.exists_le_of_sum_le _ S\n  exact \u27e8\u27e80, bot_lt_iff_ne_bot.2 Npos\u27e9, Finset.mem_univ _\u27e9", "annotated_tactic": ["obtain \u27e8i, -, hi\u27e9 : \u2203 (i : <a>Fin</a> N), i \u2208 <a>Finset.univ</a> \u2227 \u03bc s / N \u2264 \u03bc (s \u2229 v i) := by\n    apply <a>ENNReal.exists_le_of_sum_le</a> _ S\n    exact \u27e8\u27e80, <a>bot_lt_iff_ne_bot</a>.2 Npos\u27e9, <a>Finset.mem_univ</a> _\u27e9", [{"full_name": "Fin", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1745, 11], "def_end_pos": [1745, 14]}, {"full_name": "Finset.univ", "def_path": "Mathlib/Data/Fintype/Basic.lean", "def_pos": [67, 5], "def_end_pos": [67, 9]}, {"full_name": "ENNReal.exists_le_of_sum_le", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1310, 9], "def_end_pos": [1310, 28]}, {"full_name": "bot_lt_iff_ne_bot", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [371, 9], "def_end_pos": [371, 26]}, {"full_name": "Finset.mem_univ", "def_path": "Mathlib/Data/Fintype/Basic.lean", "def_pos": [72, 9], "def_end_pos": [72, 17]}]], "state_before": "case inr.inr.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 i, s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\n\u22a2 \u2203 t,\n    \u2191t \u2286 s \u2227\n      \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s \u2227 PairwiseDisjoint \u2191t fun x => closedBall x (r x)", "state_after": "case inr.inr.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 i, s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / \u2191N \u2264 \u2191\u2191\u03bc (s \u2229 v i)\n\u22a2 \u2203 t,\n    \u2191t \u2286 s \u2227\n      \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s \u2227 PairwiseDisjoint \u2191t fun x => closedBall x (r x)"}, {"tactic": "replace hi : \u03bc s / (N + 1) < \u03bc (s \u2229 v i)", "annotated_tactic": ["replace hi : \u03bc s / (N + 1) < \u03bc (s \u2229 v i)", []], "state_before": "case inr.inr.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 i, s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / \u2191N \u2264 \u2191\u2191\u03bc (s \u2229 v i)\n\u22a2 \u2203 t,\n    \u2191t \u2286 s \u2227\n      \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s \u2227 PairwiseDisjoint \u2191t fun x => closedBall x (r x)", "state_after": "case hi\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 i, s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / \u2191N \u2264 \u2191\u2191\u03bc (s \u2229 v i)\n\u22a2 \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\n\ncase inr.inr.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 i, s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\n\u22a2 \u2203 t,\n    \u2191t \u2286 s \u2227\n      \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s \u2227 PairwiseDisjoint \u2191t fun x => closedBall x (r x)"}, {"tactic": "obtain \u27e8w, hw\u27e9 :\n  \u2203 w : Finset (u i), \u03bc s / (N + 1) <\n    \u2211 x : u i in w, \u03bc (o \u2229 closedBall (x : \u03b1) (r (x : \u03b1))) := by\n  have C : HasSum (fun x : u i => \u03bc (o \u2229 closedBall x (r x))) (\u03bc (o \u2229 v i)) := by\n    rw [B]; exact ENNReal.summable.hasSum\n  have : \u03bc s / (N + 1) < \u03bc (o \u2229 v i) := hi.trans_le (measure_mono (inter_subset_inter_left _ so))\n  exact ((tendsto_order.1 C).1 _ this).exists", "annotated_tactic": ["obtain \u27e8w, hw\u27e9 :\n    \u2203 w : <a>Finset</a> (u i), \u03bc s / (N + 1) <\n      \u2211 x : u i in w, \u03bc (o \u2229 <a>closedBall</a> (x : \u03b1) (r (x : \u03b1))) := by\n    have C : <a>HasSum</a> (fun x : u i => \u03bc (o \u2229 <a>closedBall</a> x (r x))) (\u03bc (o \u2229 v i)) := by\n      rw [B]; exact ENNReal.summable.hasSum\n    have : \u03bc s / (N + 1) < \u03bc (o \u2229 v i) := hi.trans_le (<a>measure_mono</a> (<a>inter_subset_inter_left</a> _ so))\n    exact ((<a>tendsto_order</a>.1 C).1 _ this).<a>exists</a>", [{"full_name": "Finset", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [138, 11], "def_end_pos": [138, 17]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "HasSum", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/Basic.lean", "def_pos": [57, 5], "def_end_pos": [57, 11]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "MeasureTheory.measure_mono", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [193, 9], "def_end_pos": [193, 21]}, {"full_name": "Set.inter_subset_inter_left", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1027, 9], "def_end_pos": [1027, 32]}, {"full_name": "tendsto_order", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [919, 9], "def_end_pos": [919, 22]}, {"full_name": "Filter.Eventually.exists", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1308, 9], "def_end_pos": [1308, 26]}]], "state_before": "case inr.inr.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 i, s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\n\u22a2 \u2203 t,\n    \u2191t \u2286 s \u2227\n      \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s \u2227 PairwiseDisjoint \u2191t fun x => closedBall x (r x)", "state_after": "case inr.inr.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 i, s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nw : Finset \u2191(u i)\nhw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\n\u22a2 \u2203 t,\n    \u2191t \u2286 s \u2227\n      \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s \u2227 PairwiseDisjoint \u2191t fun x => closedBall x (r x)"}, {"tactic": "refine' \u27e8Finset.image (fun x : u i => x) w, _, _, _\u27e9", "annotated_tactic": ["refine' \u27e8<a>Finset.image</a> (fun x : u i => x) w, _, _, _\u27e9", [{"full_name": "Finset.image", "def_path": "Mathlib/Data/Finset/Image.lean", "def_pos": [313, 5], "def_end_pos": [313, 10]}]], "state_before": "case inr.inr.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 i, s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nw : Finset \u2191(u i)\nhw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\n\u22a2 \u2203 t,\n    \u2191t \u2286 s \u2227\n      \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s \u2227 PairwiseDisjoint \u2191t fun x => closedBall x (r x)", "state_after": "case inr.inr.intro.intro.intro.intro.intro.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 i, s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nw : Finset \u2191(u i)\nhw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\n\u22a2 \u2191(Finset.image (fun x => \u2191\u2191x) w) \u2286 s\n\ncase inr.inr.intro.intro.intro.intro.intro.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 i, s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nw : Finset \u2191(u i)\nhw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\n\u22a2 \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 Finset.image (fun x => \u2191\u2191x) w, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s\n\ncase inr.inr.intro.intro.intro.intro.intro.intro.intro.intro.refine'_3\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 i, s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nw : Finset \u2191(u i)\nhw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\n\u22a2 PairwiseDisjoint \u2191(Finset.image (fun x => \u2191\u2191x) w) fun x => closedBall x (r x)"}, {"tactic": "have : \u03bc s = 0 := le_bot_iff.1 h\u03bcs", "annotated_tactic": ["have : \u03bc s = 0 := <a>le_bot_iff</a>.1 h\u03bcs", [{"full_name": "le_bot_iff", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [355, 9], "def_end_pos": [355, 19]}]], "state_before": "case inl\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : \u2191\u2191\u03bc s \u2264 0\n\u22a2 \u2203 t,\n    \u2191t \u2286 s \u2227\n      \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s \u2227 PairwiseDisjoint \u2191t fun x => closedBall x (r x)", "state_after": "case inl\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : \u2191\u2191\u03bc s \u2264 0\nthis : \u2191\u2191\u03bc s = 0\n\u22a2 \u2203 t,\n    \u2191t \u2286 s \u2227\n      \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s \u2227 PairwiseDisjoint \u2191t fun x => closedBall x (r x)"}, {"tactic": "refine' \u27e8\u2205, by simp only [Finset.coe_empty, empty_subset], _, _\u27e9", "annotated_tactic": ["refine' \u27e8\u2205, by simp only [<a>Finset.coe_empty</a>, <a>empty_subset</a>], _, _\u27e9", [{"full_name": "Finset.coe_empty", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [618, 9], "def_end_pos": [618, 18]}, {"full_name": "Set.empty_subset", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [574, 9], "def_end_pos": [574, 21]}]], "state_before": "case inl\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : \u2191\u2191\u03bc s \u2264 0\nthis : \u2191\u2191\u03bc s = 0\n\u22a2 \u2203 t,\n    \u2191t \u2286 s \u2227\n      \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s \u2227 PairwiseDisjoint \u2191t fun x => closedBall x (r x)", "state_after": "case inl.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : \u2191\u2191\u03bc s \u2264 0\nthis : \u2191\u2191\u03bc s = 0\n\u22a2 \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 \u2205, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s\n\ncase inl.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : \u2191\u2191\u03bc s \u2264 0\nthis : \u2191\u2191\u03bc s = 0\n\u22a2 PairwiseDisjoint \u2191\u2205 fun x => closedBall x (r x)"}, {"tactic": "simp only [Finset.coe_empty, empty_subset]", "annotated_tactic": ["simp only [<a>Finset.coe_empty</a>, <a>empty_subset</a>]", [{"full_name": "Finset.coe_empty", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [618, 9], "def_end_pos": [618, 18]}, {"full_name": "Set.empty_subset", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [574, 9], "def_end_pos": [574, 21]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : \u2191\u2191\u03bc s \u2264 0\nthis : \u2191\u2191\u03bc s = 0\n\u22a2 \u2191\u2205 \u2286 s", "state_after": "no goals"}, {"tactic": "simp only [this, Finset.not_mem_empty, diff_empty, iUnion_false, iUnion_empty,\n  nonpos_iff_eq_zero, mul_zero]", "annotated_tactic": ["simp only [this, <a>Finset.not_mem_empty</a>, <a>diff_empty</a>, <a>iUnion_false</a>, <a>iUnion_empty</a>,\n        <a>nonpos_iff_eq_zero</a>, <a>mul_zero</a>]", [{"full_name": "Finset.not_mem_empty", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [548, 9], "def_end_pos": [548, 22]}, {"full_name": "Set.diff_empty", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1930, 9], "def_end_pos": [1930, 19]}, {"full_name": "Set.iUnion_false", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [783, 9], "def_end_pos": [783, 21]}, {"full_name": "Set.iUnion_empty", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [810, 9], "def_end_pos": [810, 21]}, {"full_name": "nonpos_iff_eq_zero", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [237, 3], "def_end_pos": [237, 14]}, {"full_name": "MulZeroClass.mul_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [38, 3], "def_end_pos": [38, 11]}]], "state_before": "case inl.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : \u2191\u2191\u03bc s \u2264 0\nthis : \u2191\u2191\u03bc s = 0\n\u22a2 \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 \u2205, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s", "state_after": "no goals"}, {"tactic": "simp only [Finset.coe_empty, pairwiseDisjoint_empty]", "annotated_tactic": ["simp only [<a>Finset.coe_empty</a>, <a>pairwiseDisjoint_empty</a>]", [{"full_name": "Finset.coe_empty", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [618, 9], "def_end_pos": [618, 18]}, {"full_name": "Set.pairwiseDisjoint_empty", "def_path": "Mathlib/Data/Set/Pairwise/Basic.lean", "def_pos": [259, 9], "def_end_pos": [259, 31]}]], "state_before": "case inl.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : \u2191\u2191\u03bc s \u2264 0\nthis : \u2191\u2191\u03bc s = 0\n\u22a2 PairwiseDisjoint \u2191\u2205 fun x => closedBall x (r x)", "state_after": "no goals"}, {"tactic": "simp only [eq_empty_of_isEmpty s, measure_empty] at h\u03bcs", "annotated_tactic": ["simp only [<a>eq_empty_of_isEmpty</a> s, <a>measure_empty</a>] at h\u03bcs", [{"full_name": "Set.eq_empty_of_isEmpty", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [594, 9], "def_end_pos": [594, 28]}, {"full_name": "MeasureTheory.measure_empty", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [185, 9], "def_end_pos": [185, 22]}]], "state_before": "case inr.inl\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : IsEmpty \u03b1\n\u22a2 \u2203 t,\n    \u2191t \u2286 s \u2227\n      \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s \u2227 PairwiseDisjoint \u2191t fun x => closedBall x (r x)", "state_after": "case inr.inl\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u271d : IsEmpty \u03b1\nh\u03bcs : 0 < 0\n\u22a2 \u2203 t,\n    \u2191t \u2286 s \u2227\n      \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s \u2227 PairwiseDisjoint \u2191t fun x => closedBall x (r x)"}, {"tactic": "exact (lt_irrefl _ h\u03bcs).elim", "annotated_tactic": ["exact (<a>lt_irrefl</a> _ h\u03bcs).<a>elim</a>", [{"full_name": "lt_irrefl", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [79, 9], "def_end_pos": [79, 18]}, {"full_name": "False.elim", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [223, 21], "def_end_pos": [223, 31]}]], "state_before": "case inr.inl\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u271d : IsEmpty \u03b1\nh\u03bcs : 0 < 0\n\u22a2 \u2203 t,\n    \u2191t \u2286 s \u2227\n      \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s \u2227 PairwiseDisjoint \u2191t fun x => closedBall x (r x)", "state_after": "no goals"}, {"tactic": "rintro rfl", "annotated_tactic": ["rintro rfl", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\n\u22a2 N \u2260 0", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nhN : IsEmpty (SatelliteConfig \u03b1 0 \u03c4)\n\u22a2 False"}, {"tactic": "inhabit \u03b1", "annotated_tactic": ["inhabit \u03b1", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nhN : IsEmpty (SatelliteConfig \u03b1 0 \u03c4)\n\u22a2 False", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nhN : IsEmpty (SatelliteConfig \u03b1 0 \u03c4)\ninhabited_h : Inhabited \u03b1\n\u22a2 False"}, {"tactic": "exact not_isEmpty_of_nonempty _ hN", "annotated_tactic": ["exact <a>not_isEmpty_of_nonempty</a> _ hN", [{"full_name": "not_isEmpty_of_nonempty", "def_path": "Mathlib/Logic/IsEmpty.lean", "def_pos": [212, 9], "def_end_pos": [212, 32]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nhN : IsEmpty (SatelliteConfig \u03b1 0 \u03c4)\ninhabited_h : Inhabited \u03b1\n\u22a2 False", "state_after": "no goals"}, {"tactic": "intro i", "annotated_tactic": ["intro i", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\n\u22a2 \u2200 (i : Fin N), Set.Countable (u i)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\ni : Fin N\n\u22a2 Set.Countable (u i)"}, {"tactic": "refine' (hu i).countable_of_nonempty_interior fun j _ => _", "annotated_tactic": ["refine' (hu i).<a>countable_of_nonempty_interior</a> fun j _ => _", [{"full_name": "Set.PairwiseDisjoint.countable_of_nonempty_interior", "def_path": "Mathlib/Topology/Bases.lean", "def_pos": [427, 9], "def_end_pos": [427, 67]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\ni : Fin N\n\u22a2 Set.Countable (u i)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\ni : Fin N\nj : \u2191s\nx\u271d : j \u2208 u i\n\u22a2 Set.Nonempty (interior (closedBall (BallPackage.c a j) (BallPackage.r a j)))"}, {"tactic": "have : (ball (j : \u03b1) (r j)).Nonempty := nonempty_ball.2 (a.rpos _)", "annotated_tactic": ["have : (<a>ball</a> (j : \u03b1) (r j)).<a>Nonempty</a> := <a>nonempty_ball</a>.2 (a.rpos _)", [{"full_name": "Metric.ball", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [409, 5], "def_end_pos": [409, 9]}, {"full_name": "Set.Nonempty", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [439, 15], "def_end_pos": [439, 23]}, {"full_name": "Metric.nonempty_ball", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [430, 9], "def_end_pos": [430, 22]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\ni : Fin N\nj : \u2191s\nx\u271d : j \u2208 u i\n\u22a2 Set.Nonempty (interior (closedBall (BallPackage.c a j) (BallPackage.r a j)))", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\ni : Fin N\nj : \u2191s\nx\u271d : j \u2208 u i\nthis : Set.Nonempty (ball (\u2191j) (r \u2191j))\n\u22a2 Set.Nonempty (interior (closedBall (BallPackage.c a j) (BallPackage.r a j)))"}, {"tactic": "exact this.mono ball_subset_interior_closedBall", "annotated_tactic": ["exact this.mono <a>ball_subset_interior_closedBall</a>", [{"full_name": "Metric.ball_subset_interior_closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [1920, 9], "def_end_pos": [1920, 40]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\ni : Fin N\nj : \u2191s\nx\u271d : j \u2208 u i\nthis : Set.Nonempty (ball (\u2191j) (r \u2191j))\n\u22a2 Set.Nonempty (interior (closedBall (BallPackage.c a j) (BallPackage.r a j)))", "state_after": "no goals"}, {"tactic": "refine' Subset.antisymm _ (iUnion_subset fun i => inter_subset_left _ _)", "annotated_tactic": ["refine' <a>Subset.antisymm</a> _ (<a>iUnion_subset</a> fun i => <a>inter_subset_left</a> _ _)", [{"full_name": "Set.Subset.antisymm", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [370, 9], "def_end_pos": [370, 24]}, {"full_name": "Set.iUnion_subset", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [390, 9], "def_end_pos": [390, 22]}, {"full_name": "Set.inter_subset_left", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [965, 9], "def_end_pos": [965, 26]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\n\u22a2 s = \u22c3 i, s \u2229 v i", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\n\u22a2 s \u2286 \u22c3 i, s \u2229 v i"}, {"tactic": "intro x hx", "annotated_tactic": ["intro x hx", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\n\u22a2 s \u2286 \u22c3 i, s \u2229 v i", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nx : \u03b1\nhx : x \u2208 s\n\u22a2 x \u2208 \u22c3 i, s \u2229 v i"}, {"tactic": "obtain \u27e8i, y, hxy, h'\u27e9 :\n    \u2203 (i : Fin N) (i_1 : \u21a5s), i_1 \u2208 u i \u2227 x \u2208 ball (\u2191i_1) (r \u2191i_1) := by\n  have : x \u2208 range a.c := by simpa only [Subtype.range_coe_subtype, setOf_mem_eq]\n  simpa only [mem_iUnion, bex_def] using hu' this", "annotated_tactic": ["obtain \u27e8i, y, hxy, h'\u27e9 :\n        \u2203 (i : <a>Fin</a> N) (i_1 : \u21a5s), i_1 \u2208 u i \u2227 x \u2208 <a>ball</a> (\u2191i_1) (r \u2191i_1) := by\n      have : x \u2208 <a>range</a> a.c := by simpa only [<a>Subtype.range_coe_subtype</a>, <a>setOf_mem_eq</a>]\n      simpa only [<a>mem_iUnion</a>, <a>bex_def</a>] using hu' this", [{"full_name": "Fin", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1745, 11], "def_end_pos": [1745, 14]}, {"full_name": "Metric.ball", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [409, 5], "def_end_pos": [409, 9]}, {"full_name": "Set.range", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [668, 5], "def_end_pos": [668, 10]}, {"full_name": "Subtype.range_coe_subtype", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [1425, 9], "def_end_pos": [1425, 26]}, {"full_name": "Set.setOf_mem_eq", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [275, 9], "def_end_pos": [275, 21]}, {"full_name": "Set.mem_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [201, 9], "def_end_pos": [201, 19]}, {"full_name": "bex_def", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [1027, 9], "def_end_pos": [1027, 16]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nx : \u03b1\nhx : x \u2208 s\n\u22a2 x \u2208 \u22c3 i, s \u2229 v i", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nx : \u03b1\nhx : x \u2208 s\ni : Fin N\ny : \u2191s\nhxy : y \u2208 u i\nh' : x \u2208 ball (\u2191y) (r \u2191y)\n\u22a2 x \u2208 \u22c3 i, s \u2229 v i"}, {"tactic": "refine' mem_iUnion.2 \u27e8i, \u27e8hx, _\u27e9\u27e9", "annotated_tactic": ["refine' <a>mem_iUnion</a>.2 \u27e8i, \u27e8hx, _\u27e9\u27e9", [{"full_name": "Set.mem_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [201, 9], "def_end_pos": [201, 19]}]], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nx : \u03b1\nhx : x \u2208 s\ni : Fin N\ny : \u2191s\nhxy : y \u2208 u i\nh' : x \u2208 ball (\u2191y) (r \u2191y)\n\u22a2 x \u2208 \u22c3 i, s \u2229 v i", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nx : \u03b1\nhx : x \u2208 s\ni : Fin N\ny : \u2191s\nhxy : y \u2208 u i\nh' : x \u2208 ball (\u2191y) (r \u2191y)\n\u22a2 x \u2208 v i"}, {"tactic": "simp only [exists_prop, mem_iUnion, SetCoe.exists, exists_and_right, Subtype.coe_mk]", "annotated_tactic": ["simp only [<a>exists_prop</a>, <a>mem_iUnion</a>, <a>SetCoe.exists</a>, <a>exists_and_right</a>, <a>Subtype.coe_mk</a>]", [{"full_name": "exists_prop", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [485, 17], "def_end_pos": [485, 28]}, {"full_name": "Set.mem_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [201, 9], "def_end_pos": [201, 19]}, {"full_name": "SetCoe.exists", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [190, 9], "def_end_pos": [190, 22]}, {"full_name": "exists_and_right", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [468, 17], "def_end_pos": [468, 33]}, {"full_name": "Subtype.coe_mk", "def_path": "Mathlib/Data/Subtype.lean", "def_pos": [99, 9], "def_end_pos": [99, 15]}]], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nx : \u03b1\nhx : x \u2208 s\ni : Fin N\ny : \u2191s\nhxy : y \u2208 u i\nh' : x \u2208 ball (\u2191y) (r \u2191y)\n\u22a2 x \u2208 v i", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nx : \u03b1\nhx : x \u2208 s\ni : Fin N\ny : \u2191s\nhxy : y \u2208 u i\nh' : x \u2208 ball (\u2191y) (r \u2191y)\n\u22a2 \u2203 x_1, (\u2203 x, { val := x_1, property := (_ : x_1 \u2208 s) } \u2208 u i) \u2227 x \u2208 closedBall x_1 (r x_1)"}, {"tactic": "exact \u27e8y, \u27e8y.2, by simpa only [Subtype.coe_eta]\u27e9, ball_subset_closedBall h'\u27e9", "annotated_tactic": ["exact \u27e8y, \u27e8y.2, by simpa only [<a>Subtype.coe_eta</a>]\u27e9, <a>ball_subset_closedBall</a> h'\u27e9", [{"full_name": "Subtype.coe_eta", "def_path": "Mathlib/Data/Subtype.lean", "def_pos": [95, 9], "def_end_pos": [95, 16]}, {"full_name": "Metric.ball_subset_closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [534, 9], "def_end_pos": [534, 31]}]], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nx : \u03b1\nhx : x \u2208 s\ni : Fin N\ny : \u2191s\nhxy : y \u2208 u i\nh' : x \u2208 ball (\u2191y) (r \u2191y)\n\u22a2 \u2203 x_1, (\u2203 x, { val := x_1, property := (_ : x_1 \u2208 s) } \u2208 u i) \u2227 x \u2208 closedBall x_1 (r x_1)", "state_after": "no goals"}, {"tactic": "have : x \u2208 range a.c := by simpa only [Subtype.range_coe_subtype, setOf_mem_eq]", "annotated_tactic": ["have : x \u2208 <a>range</a> a.c := by simpa only [<a>Subtype.range_coe_subtype</a>, <a>setOf_mem_eq</a>]", [{"full_name": "Set.range", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [668, 5], "def_end_pos": [668, 10]}, {"full_name": "Subtype.range_coe_subtype", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [1425, 9], "def_end_pos": [1425, 26]}, {"full_name": "Set.setOf_mem_eq", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [275, 9], "def_end_pos": [275, 21]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nx : \u03b1\nhx : x \u2208 s\n\u22a2 \u2203 i i_1, i_1 \u2208 u i \u2227 x \u2208 ball (\u2191i_1) (r \u2191i_1)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nx : \u03b1\nhx : x \u2208 s\nthis : x \u2208 range a.c\n\u22a2 \u2203 i i_1, i_1 \u2208 u i \u2227 x \u2208 ball (\u2191i_1) (r \u2191i_1)"}, {"tactic": "simpa only [mem_iUnion, bex_def] using hu' this", "annotated_tactic": ["simpa only [<a>mem_iUnion</a>, <a>bex_def</a>] using hu' this", [{"full_name": "Set.mem_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [201, 9], "def_end_pos": [201, 19]}, {"full_name": "bex_def", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [1027, 9], "def_end_pos": [1027, 16]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nx : \u03b1\nhx : x \u2208 s\nthis : x \u2208 range a.c\n\u22a2 \u2203 i i_1, i_1 \u2208 u i \u2227 x \u2208 ball (\u2191i_1) (r \u2191i_1)", "state_after": "no goals"}, {"tactic": "simpa only [Subtype.range_coe_subtype, setOf_mem_eq]", "annotated_tactic": ["simpa only [<a>Subtype.range_coe_subtype</a>, <a>setOf_mem_eq</a>]", [{"full_name": "Subtype.range_coe_subtype", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [1425, 9], "def_end_pos": [1425, 26]}, {"full_name": "Set.setOf_mem_eq", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [275, 9], "def_end_pos": [275, 21]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nx : \u03b1\nhx : x \u2208 s\n\u22a2 x \u2208 range a.c", "state_after": "no goals"}, {"tactic": "simpa only [Subtype.coe_eta]", "annotated_tactic": ["simpa only [<a>Subtype.coe_eta</a>]", [{"full_name": "Subtype.coe_eta", "def_path": "Mathlib/Data/Subtype.lean", "def_pos": [95, 9], "def_end_pos": [95, 16]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nx : \u03b1\nhx : x \u2208 s\ni : Fin N\ny : \u2191s\nhxy : y \u2208 u i\nh' : x \u2208 ball (\u2191y) (r \u2191y)\n\u22a2 { val := \u2191y, property := (_ : \u2191y \u2208 s) } \u2208 u i", "state_after": "no goals"}, {"tactic": "simp only [Finset.card_fin, Finset.sum_const, nsmul_eq_mul]", "annotated_tactic": ["simp only [<a>Finset.card_fin</a>, <a>Finset.sum_const</a>, <a>nsmul_eq_mul</a>]", [{"full_name": "Finset.card_fin", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [322, 9], "def_end_pos": [322, 24]}, {"full_name": "Finset.sum_const", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [1440, 3], "def_end_pos": [1440, 14]}, {"full_name": "nsmul_eq_mul", "def_path": "Mathlib/Algebra/GroupPower/Lemmas.lean", "def_pos": [509, 9], "def_end_pos": [509, 21]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 i, s \u2229 v i\n\u22a2 \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N = \u2191\u2191\u03bc s", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 i, s \u2229 v i\n\u22a2 \u2191N * (\u2191\u2191\u03bc s / \u2191N) = \u2191\u2191\u03bc s"}, {"tactic": "rw [ENNReal.mul_div_cancel']", "annotated_tactic": ["rw [<a>ENNReal.mul_div_cancel'</a>]", [{"full_name": "ENNReal.mul_div_cancel'", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1432, 19], "def_end_pos": [1432, 34]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 i, s \u2229 v i\n\u22a2 \u2191N * (\u2191\u2191\u03bc s / \u2191N) = \u2191\u2191\u03bc s", "state_after": "case h0\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 i, s \u2229 v i\n\u22a2 \u2191N \u2260 0\n\ncase hI\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 i, s \u2229 v i\n\u22a2 \u2191N \u2260 \u22a4"}, {"tactic": "simp only [Npos, Ne.def, Nat.cast_eq_zero, not_false_iff]", "annotated_tactic": ["simp only [Npos, <a>Ne.def</a>, <a>Nat.cast_eq_zero</a>, <a>not_false_iff</a>]", [{"full_name": "Ne.def", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [59, 9], "def_end_pos": [59, 15]}, {"full_name": "Nat.cast_eq_zero", "def_path": "Mathlib/Algebra/CharZero/Defs.lean", "def_pos": [80, 9], "def_end_pos": [80, 21]}, {"full_name": "not_false_iff", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [82, 9], "def_end_pos": [82, 22]}]], "state_before": "case h0\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 i, s \u2229 v i\n\u22a2 \u2191N \u2260 0", "state_after": "no goals"}, {"tactic": "exact ENNReal.nat_ne_top _", "annotated_tactic": ["exact <a>ENNReal.nat_ne_top</a> _", [{"full_name": "ENNReal.nat_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [717, 17], "def_end_pos": [717, 27]}]], "state_before": "case hI\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 i, s \u2229 v i\n\u22a2 \u2191N \u2260 \u22a4", "state_after": "no goals"}, {"tactic": "conv_lhs => rw [A]", "annotated_tactic": ["conv_lhs => rw [A]", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 i, s \u2229 v i\n\u22a2 \u2191\u2191\u03bc s \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 i, s \u2229 v i\n\u22a2 \u2191\u2191\u03bc (\u22c3 i, s \u2229 v i) \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)"}, {"tactic": "apply measure_iUnion_fintype_le", "annotated_tactic": ["apply <a>measure_iUnion_fintype_le</a>", [{"full_name": "MeasureTheory.measure_iUnion_fintype_le", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [257, 9], "def_end_pos": [257, 34]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 i, s \u2229 v i\n\u22a2 \u2191\u2191\u03bc (\u22c3 i, s \u2229 v i) \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)", "state_after": "no goals"}, {"tactic": "apply ENNReal.exists_le_of_sum_le _ S", "annotated_tactic": ["apply <a>ENNReal.exists_le_of_sum_le</a> _ S", [{"full_name": "ENNReal.exists_le_of_sum_le", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1310, 9], "def_end_pos": [1310, 28]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 i, s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\n\u22a2 \u2203 i, i \u2208 Finset.univ \u2227 \u2191\u2191\u03bc s / \u2191N \u2264 \u2191\u2191\u03bc (s \u2229 v i)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 i, s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\n\u22a2 Finset.Nonempty Finset.univ"}, {"tactic": "exact \u27e8\u27e80, bot_lt_iff_ne_bot.2 Npos\u27e9, Finset.mem_univ _\u27e9", "annotated_tactic": ["exact \u27e8\u27e80, <a>bot_lt_iff_ne_bot</a>.2 Npos\u27e9, <a>Finset.mem_univ</a> _\u27e9", [{"full_name": "bot_lt_iff_ne_bot", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [371, 9], "def_end_pos": [371, 26]}, {"full_name": "Finset.mem_univ", "def_path": "Mathlib/Data/Fintype/Basic.lean", "def_pos": [72, 9], "def_end_pos": [72, 17]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 i, s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\n\u22a2 Finset.Nonempty Finset.univ", "state_after": "no goals"}, {"tactic": "apply lt_of_lt_of_le _ hi", "annotated_tactic": ["apply <a>lt_of_lt_of_le</a> _ hi", [{"full_name": "lt_of_lt_of_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [115, 9], "def_end_pos": [115, 23]}]], "state_before": "case hi\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 i, s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / \u2191N \u2264 \u2191\u2191\u03bc (s \u2229 v i)\n\u22a2 \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 i, s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / \u2191N \u2264 \u2191\u2191\u03bc (s \u2229 v i)\n\u22a2 \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc s / \u2191N"}, {"tactic": "apply (ENNReal.mul_lt_mul_left h\u03bcs.ne' (measure_lt_top \u03bc s).ne).2", "annotated_tactic": ["apply (<a>ENNReal.mul_lt_mul_left</a> h\u03bcs.ne' (<a>measure_lt_top</a> \u03bc s).<a>ne</a>).2", [{"full_name": "ENNReal.mul_lt_mul_left", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1075, 9], "def_end_pos": [1075, 24]}, {"full_name": "MeasureTheory.measure_lt_top", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2866, 9], "def_end_pos": [2866, 23]}, {"full_name": "LT.lt.ne", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [152, 7], "def_end_pos": [152, 15]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 i, s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / \u2191N \u2264 \u2191\u2191\u03bc (s \u2229 v i)\n\u22a2 \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc s / \u2191N", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 i, s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / \u2191N \u2264 \u2191\u2191\u03bc (s \u2229 v i)\n\u22a2 (\u2191N + 1)\u207b\u00b9 < (\u2191N)\u207b\u00b9"}, {"tactic": "rw [ENNReal.inv_lt_inv]", "annotated_tactic": ["rw [<a>ENNReal.inv_lt_inv</a>]", [{"full_name": "ENNReal.inv_lt_inv", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1528, 19], "def_end_pos": [1528, 29]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 i, s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / \u2191N \u2264 \u2191\u2191\u03bc (s \u2229 v i)\n\u22a2 (\u2191N + 1)\u207b\u00b9 < (\u2191N)\u207b\u00b9", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 i, s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / \u2191N \u2264 \u2191\u2191\u03bc (s \u2229 v i)\n\u22a2 \u2191N < \u2191N + 1"}, {"tactic": "conv_lhs => rw [\u2190 add_zero (N : \u211d\u22650\u221e)]", "annotated_tactic": ["conv_lhs => rw [\u2190 <a>add_zero</a> (N : \u211d\u22650\u221e)]", [{"full_name": "add_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [469, 3], "def_end_pos": [469, 14]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 i, s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / \u2191N \u2264 \u2191\u2191\u03bc (s \u2229 v i)\n\u22a2 \u2191N < \u2191N + 1", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 i, s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / \u2191N \u2264 \u2191\u2191\u03bc (s \u2229 v i)\n\u22a2 \u2191N + 0 < \u2191N + 1"}, {"tactic": "exact ENNReal.add_lt_add_left (ENNReal.nat_ne_top N) zero_lt_one", "annotated_tactic": ["exact <a>ENNReal.add_lt_add_left</a> (<a>ENNReal.nat_ne_top</a> N) <a>zero_lt_one</a>", [{"full_name": "ENNReal.add_lt_add_left", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [793, 29], "def_end_pos": [793, 44]}, {"full_name": "ENNReal.nat_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [717, 17], "def_end_pos": [717, 27]}, {"full_name": "zero_lt_one", "def_path": "Mathlib/Algebra/Order/ZeroLEOne.lean", "def_pos": [39, 15], "def_end_pos": [39, 26]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 i, s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / \u2191N \u2264 \u2191\u2191\u03bc (s \u2229 v i)\n\u22a2 \u2191N + 0 < \u2191N + 1", "state_after": "no goals"}, {"tactic": "have : o \u2229 v i = \u22c3 (x : s) (_ : x \u2208 u i), o \u2229 closedBall x (r x) := by simp only [inter_iUnion]", "annotated_tactic": ["have : o \u2229 v i = \u22c3 (x : s) (_ : x \u2208 u i), o \u2229 <a>closedBall</a> x (r x) := by simp only [<a>inter_iUnion</a>]", [{"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "Set.inter_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [635, 9], "def_end_pos": [635, 21]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 i, s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\n\u22a2 \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 i, s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nthis : o \u2229 v i = \u22c3 x \u2208 u i, o \u2229 closedBall (\u2191x) (r \u2191x)\n\u22a2 \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))"}, {"tactic": "rw [this, measure_biUnion (u_count i)]", "annotated_tactic": ["rw [this, <a>measure_biUnion</a> (u_count i)]", [{"full_name": "MeasureTheory.measure_biUnion", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [172, 9], "def_end_pos": [172, 24]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 i, s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nthis : o \u2229 v i = \u22c3 x \u2208 u i, o \u2229 closedBall (\u2191x) (r \u2191x)\n\u22a2 \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))", "state_after": "case hd\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 i, s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nthis : o \u2229 v i = \u22c3 x \u2208 u i, o \u2229 closedBall (\u2191x) (r \u2191x)\n\u22a2 PairwiseDisjoint (u i) fun x => o \u2229 closedBall (\u2191x) (r \u2191x)\n\ncase h\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 i, s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nthis : o \u2229 v i = \u22c3 x \u2208 u i, o \u2229 closedBall (\u2191x) (r \u2191x)\n\u22a2 \u2200 (b : \u2191s), b \u2208 u i \u2192 MeasurableSet (o \u2229 closedBall (\u2191b) (r \u2191b))"}, {"tactic": "simp only [inter_iUnion]", "annotated_tactic": ["simp only [<a>inter_iUnion</a>]", [{"full_name": "Set.inter_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [635, 9], "def_end_pos": [635, 21]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 i, s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\n\u22a2 o \u2229 v i = \u22c3 x \u2208 u i, o \u2229 closedBall (\u2191x) (r \u2191x)", "state_after": "no goals"}, {"tactic": "exact (hu i).mono fun k => inter_subset_right _ _", "annotated_tactic": ["exact (hu i).<a>mono</a> fun k => <a>inter_subset_right</a> _ _", [{"full_name": "Set.PairwiseDisjoint.mono", "def_path": "Mathlib/Data/Set/Pairwise/Basic.lean", "def_pos": [254, 9], "def_end_pos": [254, 30]}, {"full_name": "Set.inter_subset_right", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [969, 9], "def_end_pos": [969, 27]}]], "state_before": "case hd\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 i, s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nthis : o \u2229 v i = \u22c3 x \u2208 u i, o \u2229 closedBall (\u2191x) (r \u2191x)\n\u22a2 PairwiseDisjoint (u i) fun x => o \u2229 closedBall (\u2191x) (r \u2191x)", "state_after": "no goals"}, {"tactic": "exact fun b _ => omeas.inter measurableSet_closedBall", "annotated_tactic": ["exact fun b _ => omeas.inter <a>measurableSet_closedBall</a>", [{"full_name": "measurableSet_closedBall", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [1681, 9], "def_end_pos": [1681, 33]}]], "state_before": "case h\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 i, s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nthis : o \u2229 v i = \u22c3 x \u2208 u i, o \u2229 closedBall (\u2191x) (r \u2191x)\n\u22a2 \u2200 (b : \u2191s), b \u2208 u i \u2192 MeasurableSet (o \u2229 closedBall (\u2191b) (r \u2191b))", "state_after": "no goals"}, {"tactic": "have C : HasSum (fun x : u i => \u03bc (o \u2229 closedBall x (r x))) (\u03bc (o \u2229 v i)) := by\n  rw [B]; exact ENNReal.summable.hasSum", "annotated_tactic": ["have C : <a>HasSum</a> (fun x : u i => \u03bc (o \u2229 <a>closedBall</a> x (r x))) (\u03bc (o \u2229 v i)) := by\n      rw [B]; exact ENNReal.summable.hasSum", [{"full_name": "HasSum", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/Basic.lean", "def_pos": [57, 5], "def_end_pos": [57, 11]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 i, s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\n\u22a2 \u2203 w, \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 i, s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nC : HasSum (fun x => \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))) (\u2191\u2191\u03bc (o \u2229 v i))\n\u22a2 \u2203 w, \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))"}, {"tactic": "have : \u03bc s / (N + 1) < \u03bc (o \u2229 v i) := hi.trans_le (measure_mono (inter_subset_inter_left _ so))", "annotated_tactic": ["have : \u03bc s / (N + 1) < \u03bc (o \u2229 v i) := hi.trans_le (<a>measure_mono</a> (<a>inter_subset_inter_left</a> _ so))", [{"full_name": "MeasureTheory.measure_mono", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [193, 9], "def_end_pos": [193, 21]}, {"full_name": "Set.inter_subset_inter_left", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1027, 9], "def_end_pos": [1027, 32]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 i, s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nC : HasSum (fun x => \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))) (\u2191\u2191\u03bc (o \u2229 v i))\n\u22a2 \u2203 w, \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 i, s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nC : HasSum (fun x => \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))) (\u2191\u2191\u03bc (o \u2229 v i))\nthis : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (o \u2229 v i)\n\u22a2 \u2203 w, \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))"}, {"tactic": "exact ((tendsto_order.1 C).1 _ this).exists", "annotated_tactic": ["exact ((<a>tendsto_order</a>.1 C).1 _ this).<a>exists</a>", [{"full_name": "tendsto_order", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [919, 9], "def_end_pos": [919, 22]}, {"full_name": "Filter.Eventually.exists", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1308, 9], "def_end_pos": [1308, 26]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 i, s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nC : HasSum (fun x => \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))) (\u2191\u2191\u03bc (o \u2229 v i))\nthis : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (o \u2229 v i)\n\u22a2 \u2203 w, \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))", "state_after": "no goals"}, {"tactic": "rw [B]", "annotated_tactic": ["rw [B]", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 i, s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\n\u22a2 HasSum (fun x => \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))) (\u2191\u2191\u03bc (o \u2229 v i))", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 i, s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\n\u22a2 HasSum (fun x => \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))) (\u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x)))"}, {"tactic": "exact ENNReal.summable.hasSum", "annotated_tactic": ["exact ENNReal.summable.hasSum", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 i, s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\n\u22a2 HasSum (fun x => \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))) (\u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x)))", "state_after": "no goals"}, {"tactic": "simp only [image_subset_iff, Finset.coe_image]", "annotated_tactic": ["simp only [<a>image_subset_iff</a>, <a>Finset.coe_image</a>]", [{"full_name": "Set.image_subset_iff", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [497, 9], "def_end_pos": [497, 25]}, {"full_name": "Finset.coe_image", "def_path": "Mathlib/Data/Finset/Image.lean", "def_pos": [392, 9], "def_end_pos": [392, 18]}]], "state_before": "case inr.inr.intro.intro.intro.intro.intro.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 i, s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nw : Finset \u2191(u i)\nhw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\n\u22a2 \u2191(Finset.image (fun x => \u2191\u2191x) w) \u2286 s", "state_after": "case inr.inr.intro.intro.intro.intro.intro.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 i, s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nw : Finset \u2191(u i)\nhw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\n\u22a2 \u2191w \u2286 (fun x => \u2191\u2191x) \u207b\u00b9' s"}, {"tactic": "intro y _", "annotated_tactic": ["intro y _", []], "state_before": "case inr.inr.intro.intro.intro.intro.intro.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 i, s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nw : Finset \u2191(u i)\nhw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\n\u22a2 \u2191w \u2286 (fun x => \u2191\u2191x) \u207b\u00b9' s", "state_after": "case inr.inr.intro.intro.intro.intro.intro.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 i, s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nw : Finset \u2191(u i)\nhw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\ny : \u2191(u i)\na\u271d : y \u2208 \u2191w\n\u22a2 y \u2208 (fun x => \u2191\u2191x) \u207b\u00b9' s"}, {"tactic": "simp only [Subtype.coe_prop, mem_preimage]", "annotated_tactic": ["simp only [<a>Subtype.coe_prop</a>, <a>mem_preimage</a>]", [{"full_name": "Subtype.coe_prop", "def_path": "Mathlib/Data/Subtype.lean", "def_pos": [262, 9], "def_end_pos": [262, 17]}, {"full_name": "Set.mem_preimage", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [64, 9], "def_end_pos": [64, 21]}]], "state_before": "case inr.inr.intro.intro.intro.intro.intro.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 i, s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nw : Finset \u2191(u i)\nhw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\ny : \u2191(u i)\na\u271d : y \u2208 \u2191w\n\u22a2 y \u2208 (fun x => \u2191\u2191x) \u207b\u00b9' s", "state_after": "no goals"}, {"tactic": "suffices H : \u03bc (o \\ \u22c3 x \u2208 w, closedBall (\u2191x) (r \u2191x)) \u2264 N / (N + 1) * \u03bc s", "annotated_tactic": ["suffices H : \u03bc (o \\ \u22c3 x \u2208 w, <a>closedBall</a> (\u2191x) (r \u2191x)) \u2264 N / (N + 1) * \u03bc s", [{"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}]], "state_before": "case inr.inr.intro.intro.intro.intro.intro.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 i, s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nw : Finset \u2191(u i)\nhw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\n\u22a2 \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 Finset.image (fun x => \u2191\u2191x) w, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s", "state_after": "case inr.inr.intro.intro.intro.intro.intro.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 i, s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nw : Finset \u2191(u i)\nhw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nH : \u2191\u2191\u03bc (o \\ \u22c3 x \u2208 w, closedBall (\u2191\u2191x) (r \u2191\u2191x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s\n\u22a2 \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 Finset.image (fun x => \u2191\u2191x) w, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s\n\ncase H\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 i, s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nw : Finset \u2191(u i)\nhw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\n\u22a2 \u2191\u2191\u03bc (o \\ \u22c3 x \u2208 w, closedBall (\u2191\u2191x) (r \u2191\u2191x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s"}, {"tactic": "rw [\u2190 diff_inter_self_eq_diff,\n  measure_diff_le_iff_le_add _ (inter_subset_right _ _) (measure_lt_top \u03bc _).ne]", "annotated_tactic": ["rw [\u2190 <a>diff_inter_self_eq_diff</a>,\n      <a>measure_diff_le_iff_le_add</a> _ (<a>inter_subset_right</a> _ _) (<a>measure_lt_top</a> \u03bc _).<a>ne</a>]", [{"full_name": "Set.diff_inter_self_eq_diff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [2058, 9], "def_end_pos": [2058, 32]}, {"full_name": "MeasureTheory.measure_diff_le_iff_le_add", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [271, 9], "def_end_pos": [271, 35]}, {"full_name": "Set.inter_subset_right", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [969, 9], "def_end_pos": [969, 27]}, {"full_name": "MeasureTheory.measure_lt_top", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2866, 9], "def_end_pos": [2866, 23]}, {"full_name": "LT.lt.ne", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [152, 7], "def_end_pos": [152, 15]}]], "state_before": "case H\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 i, s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nw : Finset \u2191(u i)\nhw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\n\u22a2 \u2191\u2191\u03bc (o \\ \u22c3 x \u2208 w, closedBall (\u2191\u2191x) (r \u2191\u2191x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s", "state_after": "case H\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 i, s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nw : Finset \u2191(u i)\nhw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\n\u22a2 \u2191\u2191\u03bc o \u2264 \u2191\u2191\u03bc ((\u22c3 x \u2208 w, closedBall (\u2191\u2191x) (r \u2191\u2191x)) \u2229 o) + \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s\n\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 i, s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nw : Finset \u2191(u i)\nhw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\n\u22a2 MeasurableSet ((\u22c3 x \u2208 w, closedBall (\u2191\u2191x) (r \u2191\u2191x)) \u2229 o)"}, {"tactic": "swap", "annotated_tactic": ["swap", []], "state_before": "case H\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 i, s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nw : Finset \u2191(u i)\nhw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\n\u22a2 \u2191\u2191\u03bc o \u2264 \u2191\u2191\u03bc ((\u22c3 x \u2208 w, closedBall (\u2191\u2191x) (r \u2191\u2191x)) \u2229 o) + \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s\n\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 i, s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nw : Finset \u2191(u i)\nhw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\n\u22a2 MeasurableSet ((\u22c3 x \u2208 w, closedBall (\u2191\u2191x) (r \u2191\u2191x)) \u2229 o)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 i, s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nw : Finset \u2191(u i)\nhw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\n\u22a2 MeasurableSet ((\u22c3 x \u2208 w, closedBall (\u2191\u2191x) (r \u2191\u2191x)) \u2229 o)\n\ncase H\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 i, s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nw : Finset \u2191(u i)\nhw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\n\u22a2 \u2191\u2191\u03bc o \u2264 \u2191\u2191\u03bc ((\u22c3 x \u2208 w, closedBall (\u2191\u2191x) (r \u2191\u2191x)) \u2229 o) + \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s"}, {"tactic": "rw [Finset.set_biUnion_finset_image]", "annotated_tactic": ["rw [<a>Finset.set_biUnion_finset_image</a>]", [{"full_name": "Finset.set_biUnion_finset_image", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [2146, 9], "def_end_pos": [2146, 33]}]], "state_before": "case inr.inr.intro.intro.intro.intro.intro.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 i, s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nw : Finset \u2191(u i)\nhw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nH : \u2191\u2191\u03bc (o \\ \u22c3 x \u2208 w, closedBall (\u2191\u2191x) (r \u2191\u2191x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s\n\u22a2 \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 Finset.image (fun x => \u2191\u2191x) w, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s", "state_after": "case inr.inr.intro.intro.intro.intro.intro.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 i, s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nw : Finset \u2191(u i)\nhw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nH : \u2191\u2191\u03bc (o \\ \u22c3 x \u2208 w, closedBall (\u2191\u2191x) (r \u2191\u2191x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s\n\u22a2 \u2191\u2191\u03bc (s \\ \u22c3 y \u2208 w, closedBall (\u2191\u2191y) (r \u2191\u2191y)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s"}, {"tactic": "exact le_trans (measure_mono (diff_subset_diff so (Subset.refl _))) H", "annotated_tactic": ["exact <a>le_trans</a> (<a>measure_mono</a> (<a>diff_subset_diff</a> so (<a>Subset.refl</a> _))) H", [{"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}, {"full_name": "MeasureTheory.measure_mono", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [193, 9], "def_end_pos": [193, 21]}, {"full_name": "Set.diff_subset_diff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1904, 9], "def_end_pos": [1904, 25]}, {"full_name": "Set.Subset.refl", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [354, 9], "def_end_pos": [354, 20]}]], "state_before": "case inr.inr.intro.intro.intro.intro.intro.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 i, s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nw : Finset \u2191(u i)\nhw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nH : \u2191\u2191\u03bc (o \\ \u22c3 x \u2208 w, closedBall (\u2191\u2191x) (r \u2191\u2191x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s\n\u22a2 \u2191\u2191\u03bc (s \\ \u22c3 y \u2208 w, closedBall (\u2191\u2191y) (r \u2191\u2191y)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s", "state_after": "no goals"}, {"tactic": "apply MeasurableSet.inter _ omeas", "annotated_tactic": ["apply <a>MeasurableSet.inter</a> _ omeas", [{"full_name": "MeasurableSet.inter", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [198, 19], "def_end_pos": [198, 38]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 i, s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nw : Finset \u2191(u i)\nhw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\n\u22a2 MeasurableSet ((\u22c3 x \u2208 w, closedBall (\u2191\u2191x) (r \u2191\u2191x)) \u2229 o)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 i, s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nw : Finset \u2191(u i)\nhw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\n\u22a2 MeasurableSet (\u22c3 x \u2208 w, closedBall (\u2191\u2191x) (r \u2191\u2191x))"}, {"tactic": "haveI : Encodable (u i) := (u_count i).toEncodable", "annotated_tactic": ["haveI : <a>Encodable</a> (u i) := (u_count i).<a>toEncodable</a>", [{"full_name": "Encodable", "def_path": "Mathlib/Logic/Encodable/Basic.lean", "def_pos": [45, 7], "def_end_pos": [45, 16]}, {"full_name": "Set.Countable.toEncodable", "def_path": "Mathlib/Data/Set/Countable.lean", "def_pos": [62, 15], "def_end_pos": [62, 36]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 i, s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nw : Finset \u2191(u i)\nhw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\n\u22a2 MeasurableSet (\u22c3 x \u2208 w, closedBall (\u2191\u2191x) (r \u2191\u2191x))", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 i, s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nw : Finset \u2191(u i)\nhw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nthis : Encodable \u2191(u i)\n\u22a2 MeasurableSet (\u22c3 x \u2208 w, closedBall (\u2191\u2191x) (r \u2191\u2191x))"}, {"tactic": "exact MeasurableSet.iUnion fun b => MeasurableSet.iUnion fun _ => measurableSet_closedBall", "annotated_tactic": ["exact <a>MeasurableSet.iUnion</a> fun b => <a>MeasurableSet.iUnion</a> fun _ => <a>measurableSet_closedBall</a>", [{"full_name": "MeasurableSet.iUnion", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [115, 19], "def_end_pos": [115, 39]}, {"full_name": "MeasurableSet.iUnion", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [115, 19], "def_end_pos": [115, 39]}, {"full_name": "measurableSet_closedBall", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [1681, 9], "def_end_pos": [1681, 33]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 i, s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nw : Finset \u2191(u i)\nhw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nthis : Encodable \u2191(u i)\n\u22a2 MeasurableSet (\u22c3 x \u2208 w, closedBall (\u2191\u2191x) (r \u2191\u2191x))", "state_after": "no goals"}, {"tactic": "rw [\u03bco, \u2190 add_mul, ENNReal.div_add_div_same, add_comm, ENNReal.div_self, one_mul] <;> simp", "annotated_tactic": ["rw [\u03bco, \u2190 <a>add_mul</a>, <a>ENNReal.div_add_div_same</a>, <a>add_comm</a>, <a>ENNReal.div_self</a>, <a>one_mul</a>] <;> simp", [{"full_name": "add_mul", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [91, 7], "def_end_pos": [91, 14]}, {"full_name": "ENNReal.div_add_div_same", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1742, 19], "def_end_pos": [1742, 35]}, {"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [301, 3], "def_end_pos": [301, 14]}, {"full_name": "ENNReal.div_self", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1746, 19], "def_end_pos": [1746, 27]}, {"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [464, 9], "def_end_pos": [464, 16]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 i, s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nw : Finset \u2191(u i)\nhw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\n\u22a2 \u2191\u2191\u03bc o = 1 / (\u2191N + 1) * \u2191\u2191\u03bc s + \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s", "state_after": "no goals"}, {"tactic": "refine' add_le_add _ le_rfl", "annotated_tactic": ["refine' <a>add_le_add</a> _ <a>le_rfl</a>", [{"full_name": "add_le_add", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [205, 15], "def_end_pos": [205, 25]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 i, s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nw : Finset \u2191(u i)\nhw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\n\u22a2 1 / (\u2191N + 1) * \u2191\u2191\u03bc s + \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc ((\u22c3 x \u2208 w, closedBall (\u2191\u2191x) (r \u2191\u2191x)) \u2229 o) + \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 i, s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nw : Finset \u2191(u i)\nhw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\n\u22a2 1 / (\u2191N + 1) * \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc ((\u22c3 x \u2208 w, closedBall (\u2191\u2191x) (r \u2191\u2191x)) \u2229 o)"}, {"tactic": "rw [div_eq_mul_inv, one_mul, mul_comm, \u2190 div_eq_mul_inv]", "annotated_tactic": ["rw [<a>div_eq_mul_inv</a>, <a>one_mul</a>, <a>mul_comm</a>, \u2190 <a>div_eq_mul_inv</a>]", [{"full_name": "div_eq_mul_inv", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [977, 9], "def_end_pos": [977, 23]}, {"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [464, 9], "def_end_pos": [464, 16]}, {"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [302, 9], "def_end_pos": [302, 17]}, {"full_name": "div_eq_mul_inv", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [977, 9], "def_end_pos": [977, 23]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 i, s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nw : Finset \u2191(u i)\nhw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\n\u22a2 1 / (\u2191N + 1) * \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc ((\u22c3 x \u2208 w, closedBall (\u2191\u2191x) (r \u2191\u2191x)) \u2229 o)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 i, s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nw : Finset \u2191(u i)\nhw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\n\u22a2 \u2191\u2191\u03bc s / (\u2191N + 1) \u2264 \u2191\u2191\u03bc ((\u22c3 x \u2208 w, closedBall (\u2191\u2191x) (r \u2191\u2191x)) \u2229 o)"}, {"tactic": "apply hw.le.trans (le_of_eq _)", "annotated_tactic": ["apply hw.le.trans (<a>le_of_eq</a> _)", [{"full_name": "le_of_eq", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [72, 9], "def_end_pos": [72, 17]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 i, s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nw : Finset \u2191(u i)\nhw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\n\u22a2 \u2191\u2191\u03bc s / (\u2191N + 1) \u2264 \u2191\u2191\u03bc ((\u22c3 x \u2208 w, closedBall (\u2191\u2191x) (r \u2191\u2191x)) \u2229 o)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 i, s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nw : Finset \u2191(u i)\nhw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\n\u22a2 \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x)) = \u2191\u2191\u03bc ((\u22c3 x \u2208 w, closedBall (\u2191\u2191x) (r \u2191\u2191x)) \u2229 o)"}, {"tactic": "rw [\u2190 Finset.set_biUnion_coe, inter_comm _ o, inter_iUnion\u2082, Finset.set_biUnion_coe,\n  measure_biUnion_finset]", "annotated_tactic": ["rw [\u2190 <a>Finset.set_biUnion_coe</a>, <a>inter_comm</a> _ o, <a>inter_iUnion\u2082</a>, <a>Finset.set_biUnion_coe</a>,\n          <a>measure_biUnion_finset</a>]", [{"full_name": "Finset.set_biUnion_coe", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [2087, 9], "def_end_pos": [2087, 24]}, {"full_name": "Set.inter_comm", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [940, 9], "def_end_pos": [940, 19]}, {"full_name": "Set.inter_iUnion\u2082", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [1121, 9], "def_end_pos": [1121, 22]}, {"full_name": "Finset.set_biUnion_coe", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [2087, 9], "def_end_pos": [2087, 24]}, {"full_name": "MeasureTheory.measure_biUnion_finset", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [194, 9], "def_end_pos": [194, 31]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 i, s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nw : Finset \u2191(u i)\nhw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\n\u22a2 \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x)) = \u2191\u2191\u03bc ((\u22c3 x \u2208 w, closedBall (\u2191\u2191x) (r \u2191\u2191x)) \u2229 o)", "state_after": "case hd\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 i, s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nw : Finset \u2191(u i)\nhw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\n\u22a2 PairwiseDisjoint \u2191w fun x => o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x)\n\ncase hm\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 i, s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nw : Finset \u2191(u i)\nhw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\n\u22a2 \u2200 (b : \u2191(u i)), b \u2208 w \u2192 MeasurableSet (o \u2229 closedBall (\u2191\u2191b) (r \u2191\u2191b))"}, {"tactic": "have : (w : Set (u i)).PairwiseDisjoint\n    fun b : u i => closedBall (b : \u03b1) (r (b : \u03b1)) := by\n  intro k _ l _ hkl; exact hu i k.2 l.2 (Subtype.val_injective.ne hkl)", "annotated_tactic": ["have : (w : <a>Set</a> (u i)).<a>PairwiseDisjoint</a>\n              fun b : u i => <a>closedBall</a> (b : \u03b1) (r (b : \u03b1)) := by\n            intro k _ l _ hkl; exact hu i k.2 l.2 (Subtype.val_injective.ne hkl)", [{"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}, {"full_name": "Set.PairwiseDisjoint", "def_path": "Mathlib/Data/Set/Pairwise/Basic.lean", "def_pos": [242, 5], "def_end_pos": [242, 21]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}]], "state_before": "case hd\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 i, s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nw : Finset \u2191(u i)\nhw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\n\u22a2 PairwiseDisjoint \u2191w fun x => o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x)", "state_after": "case hd\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 i, s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nw : Finset \u2191(u i)\nhw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nthis : PairwiseDisjoint \u2191w fun b => closedBall (\u2191\u2191b) (r \u2191\u2191b)\n\u22a2 PairwiseDisjoint \u2191w fun x => o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x)"}, {"tactic": "exact this.mono fun k => inter_subset_right _ _", "annotated_tactic": ["exact this.mono fun k => <a>inter_subset_right</a> _ _", [{"full_name": "Set.inter_subset_right", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [969, 9], "def_end_pos": [969, 27]}]], "state_before": "case hd\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 i, s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nw : Finset \u2191(u i)\nhw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nthis : PairwiseDisjoint \u2191w fun b => closedBall (\u2191\u2191b) (r \u2191\u2191b)\n\u22a2 PairwiseDisjoint \u2191w fun x => o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x)", "state_after": "no goals"}, {"tactic": "intro k _ l _ hkl", "annotated_tactic": ["intro k _ l _ hkl", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 i, s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nw : Finset \u2191(u i)\nhw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\n\u22a2 PairwiseDisjoint \u2191w fun b => closedBall (\u2191\u2191b) (r \u2191\u2191b)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 i, s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nw : Finset \u2191(u i)\nhw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nk : \u2191(u i)\na\u271d\u00b9 : k \u2208 \u2191w\nl : \u2191(u i)\na\u271d : l \u2208 \u2191w\nhkl : k \u2260 l\n\u22a2 (Disjoint on fun b => closedBall (\u2191\u2191b) (r \u2191\u2191b)) k l"}, {"tactic": "exact hu i k.2 l.2 (Subtype.val_injective.ne hkl)", "annotated_tactic": ["exact hu i k.2 l.2 (Subtype.val_injective.ne hkl)", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 i, s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nw : Finset \u2191(u i)\nhw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nk : \u2191(u i)\na\u271d\u00b9 : k \u2208 \u2191w\nl : \u2191(u i)\na\u271d : l \u2208 \u2191w\nhkl : k \u2260 l\n\u22a2 (Disjoint on fun b => closedBall (\u2191\u2191b) (r \u2191\u2191b)) k l", "state_after": "no goals"}, {"tactic": "intro b _", "annotated_tactic": ["intro b _", []], "state_before": "case hm\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 i, s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nw : Finset \u2191(u i)\nhw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\n\u22a2 \u2200 (b : \u2191(u i)), b \u2208 w \u2192 MeasurableSet (o \u2229 closedBall (\u2191\u2191b) (r \u2191\u2191b))", "state_after": "case hm\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 i, s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nw : Finset \u2191(u i)\nhw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nb : \u2191(u i)\na\u271d : b \u2208 w\n\u22a2 MeasurableSet (o \u2229 closedBall (\u2191\u2191b) (r \u2191\u2191b))"}, {"tactic": "apply omeas.inter measurableSet_closedBall", "annotated_tactic": ["apply omeas.inter <a>measurableSet_closedBall</a>", [{"full_name": "measurableSet_closedBall", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [1681, 9], "def_end_pos": [1681, 33]}]], "state_before": "case hm\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 i, s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nw : Finset \u2191(u i)\nhw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nb : \u2191(u i)\na\u271d : b \u2208 w\n\u22a2 MeasurableSet (o \u2229 closedBall (\u2191\u2191b) (r \u2191\u2191b))", "state_after": "no goals"}, {"tactic": "intro k hk l hl hkl", "annotated_tactic": ["intro k hk l hl hkl", []], "state_before": "case inr.inr.intro.intro.intro.intro.intro.intro.intro.intro.refine'_3\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 i, s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nw : Finset \u2191(u i)\nhw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\n\u22a2 PairwiseDisjoint \u2191(Finset.image (fun x => \u2191\u2191x) w) fun x => closedBall x (r x)", "state_after": "case inr.inr.intro.intro.intro.intro.intro.intro.intro.intro.refine'_3\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 i, s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nw : Finset \u2191(u i)\nhw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nk : \u03b1\nhk : k \u2208 \u2191(Finset.image (fun x => \u2191\u2191x) w)\nl : \u03b1\nhl : l \u2208 \u2191(Finset.image (fun x => \u2191\u2191x) w)\nhkl : k \u2260 l\n\u22a2 (Disjoint on fun x => closedBall x (r x)) k l"}, {"tactic": "obtain \u27e8k', _, rfl\u27e9 : \u2203 k' : u i, k' \u2208 w \u2227 \u2191k' = k := by\n  simpa only [mem_image, Finset.mem_coe, Finset.coe_image] using hk", "annotated_tactic": ["obtain \u27e8k', _, rfl\u27e9 : \u2203 k' : u i, k' \u2208 w \u2227 \u2191k' = k := by\n      simpa only [<a>mem_image</a>, <a>Finset.mem_coe</a>, <a>Finset.coe_image</a>] using hk", [{"full_name": "Set.mem_image", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [231, 9], "def_end_pos": [231, 18]}, {"full_name": "Finset.mem_coe", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [208, 9], "def_end_pos": [208, 16]}, {"full_name": "Finset.coe_image", "def_path": "Mathlib/Data/Finset/Image.lean", "def_pos": [392, 9], "def_end_pos": [392, 18]}]], "state_before": "case inr.inr.intro.intro.intro.intro.intro.intro.intro.intro.refine'_3\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 i, s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nw : Finset \u2191(u i)\nhw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nk : \u03b1\nhk : k \u2208 \u2191(Finset.image (fun x => \u2191\u2191x) w)\nl : \u03b1\nhl : l \u2208 \u2191(Finset.image (fun x => \u2191\u2191x) w)\nhkl : k \u2260 l\n\u22a2 (Disjoint on fun x => closedBall x (r x)) k l", "state_after": "case inr.inr.intro.intro.intro.intro.intro.intro.intro.intro.refine'_3.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 i, s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nw : Finset \u2191(u i)\nhw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nl : \u03b1\nhl : l \u2208 \u2191(Finset.image (fun x => \u2191\u2191x) w)\nk' : \u2191(u i)\nleft\u271d : k' \u2208 w\nhk : \u2191\u2191k' \u2208 \u2191(Finset.image (fun x => \u2191\u2191x) w)\nhkl : \u2191\u2191k' \u2260 l\n\u22a2 (Disjoint on fun x => closedBall x (r x)) (\u2191\u2191k') l"}, {"tactic": "obtain \u27e8l', _, rfl\u27e9 : \u2203 l' : u i, l' \u2208 w \u2227 \u2191l' = l := by\n  simpa only [mem_image, Finset.mem_coe, Finset.coe_image] using hl", "annotated_tactic": ["obtain \u27e8l', _, rfl\u27e9 : \u2203 l' : u i, l' \u2208 w \u2227 \u2191l' = l := by\n      simpa only [<a>mem_image</a>, <a>Finset.mem_coe</a>, <a>Finset.coe_image</a>] using hl", [{"full_name": "Set.mem_image", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [231, 9], "def_end_pos": [231, 18]}, {"full_name": "Finset.mem_coe", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [208, 9], "def_end_pos": [208, 16]}, {"full_name": "Finset.coe_image", "def_path": "Mathlib/Data/Finset/Image.lean", "def_pos": [392, 9], "def_end_pos": [392, 18]}]], "state_before": "case inr.inr.intro.intro.intro.intro.intro.intro.intro.intro.refine'_3.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 i, s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nw : Finset \u2191(u i)\nhw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nl : \u03b1\nhl : l \u2208 \u2191(Finset.image (fun x => \u2191\u2191x) w)\nk' : \u2191(u i)\nleft\u271d : k' \u2208 w\nhk : \u2191\u2191k' \u2208 \u2191(Finset.image (fun x => \u2191\u2191x) w)\nhkl : \u2191\u2191k' \u2260 l\n\u22a2 (Disjoint on fun x => closedBall x (r x)) (\u2191\u2191k') l", "state_after": "case inr.inr.intro.intro.intro.intro.intro.intro.intro.intro.refine'_3.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 i, s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nw : Finset \u2191(u i)\nhw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nk' : \u2191(u i)\nleft\u271d\u00b9 : k' \u2208 w\nhk : \u2191\u2191k' \u2208 \u2191(Finset.image (fun x => \u2191\u2191x) w)\nl' : \u2191(u i)\nleft\u271d : l' \u2208 w\nhl : \u2191\u2191l' \u2208 \u2191(Finset.image (fun x => \u2191\u2191x) w)\nhkl : \u2191\u2191k' \u2260 \u2191\u2191l'\n\u22a2 (Disjoint on fun x => closedBall x (r x)) \u2191\u2191k' \u2191\u2191l'"}, {"tactic": "have k'nel' : (k' : s) \u2260 l' := by intro h; rw [h] at hkl; exact hkl rfl", "annotated_tactic": ["have k'nel' : (k' : s) \u2260 l' := by intro h; rw [h] at hkl; exact hkl <a>rfl</a>", [{"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "case inr.inr.intro.intro.intro.intro.intro.intro.intro.intro.refine'_3.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 i, s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nw : Finset \u2191(u i)\nhw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nk' : \u2191(u i)\nleft\u271d\u00b9 : k' \u2208 w\nhk : \u2191\u2191k' \u2208 \u2191(Finset.image (fun x => \u2191\u2191x) w)\nl' : \u2191(u i)\nleft\u271d : l' \u2208 w\nhl : \u2191\u2191l' \u2208 \u2191(Finset.image (fun x => \u2191\u2191x) w)\nhkl : \u2191\u2191k' \u2260 \u2191\u2191l'\n\u22a2 (Disjoint on fun x => closedBall x (r x)) \u2191\u2191k' \u2191\u2191l'", "state_after": "case inr.inr.intro.intro.intro.intro.intro.intro.intro.intro.refine'_3.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 i, s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nw : Finset \u2191(u i)\nhw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nk' : \u2191(u i)\nleft\u271d\u00b9 : k' \u2208 w\nhk : \u2191\u2191k' \u2208 \u2191(Finset.image (fun x => \u2191\u2191x) w)\nl' : \u2191(u i)\nleft\u271d : l' \u2208 w\nhl : \u2191\u2191l' \u2208 \u2191(Finset.image (fun x => \u2191\u2191x) w)\nhkl : \u2191\u2191k' \u2260 \u2191\u2191l'\nk'nel' : \u2191k' \u2260 \u2191l'\n\u22a2 (Disjoint on fun x => closedBall x (r x)) \u2191\u2191k' \u2191\u2191l'"}, {"tactic": "exact hu i k'.2 l'.2 k'nel'", "annotated_tactic": ["exact hu i k'.2 l'.2 k'nel'", []], "state_before": "case inr.inr.intro.intro.intro.intro.intro.intro.intro.intro.refine'_3.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 i, s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nw : Finset \u2191(u i)\nhw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nk' : \u2191(u i)\nleft\u271d\u00b9 : k' \u2208 w\nhk : \u2191\u2191k' \u2208 \u2191(Finset.image (fun x => \u2191\u2191x) w)\nl' : \u2191(u i)\nleft\u271d : l' \u2208 w\nhl : \u2191\u2191l' \u2208 \u2191(Finset.image (fun x => \u2191\u2191x) w)\nhkl : \u2191\u2191k' \u2260 \u2191\u2191l'\nk'nel' : \u2191k' \u2260 \u2191l'\n\u22a2 (Disjoint on fun x => closedBall x (r x)) \u2191\u2191k' \u2191\u2191l'", "state_after": "no goals"}, {"tactic": "simpa only [mem_image, Finset.mem_coe, Finset.coe_image] using hk", "annotated_tactic": ["simpa only [<a>mem_image</a>, <a>Finset.mem_coe</a>, <a>Finset.coe_image</a>] using hk", [{"full_name": "Set.mem_image", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [231, 9], "def_end_pos": [231, 18]}, {"full_name": "Finset.mem_coe", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [208, 9], "def_end_pos": [208, 16]}, {"full_name": "Finset.coe_image", "def_path": "Mathlib/Data/Finset/Image.lean", "def_pos": [392, 9], "def_end_pos": [392, 18]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 i, s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nw : Finset \u2191(u i)\nhw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nk : \u03b1\nhk : k \u2208 \u2191(Finset.image (fun x => \u2191\u2191x) w)\nl : \u03b1\nhl : l \u2208 \u2191(Finset.image (fun x => \u2191\u2191x) w)\nhkl : k \u2260 l\n\u22a2 \u2203 k', k' \u2208 w \u2227 \u2191\u2191k' = k", "state_after": "no goals"}, {"tactic": "simpa only [mem_image, Finset.mem_coe, Finset.coe_image] using hl", "annotated_tactic": ["simpa only [<a>mem_image</a>, <a>Finset.mem_coe</a>, <a>Finset.coe_image</a>] using hl", [{"full_name": "Set.mem_image", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [231, 9], "def_end_pos": [231, 18]}, {"full_name": "Finset.mem_coe", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [208, 9], "def_end_pos": [208, 16]}, {"full_name": "Finset.coe_image", "def_path": "Mathlib/Data/Finset/Image.lean", "def_pos": [392, 9], "def_end_pos": [392, 18]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 i, s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nw : Finset \u2191(u i)\nhw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nl : \u03b1\nhl : l \u2208 \u2191(Finset.image (fun x => \u2191\u2191x) w)\nk' : \u2191(u i)\nleft\u271d : k' \u2208 w\nhk : \u2191\u2191k' \u2208 \u2191(Finset.image (fun x => \u2191\u2191x) w)\nhkl : \u2191\u2191k' \u2260 l\n\u22a2 \u2203 l', l' \u2208 w \u2227 \u2191\u2191l' = l", "state_after": "no goals"}, {"tactic": "intro h", "annotated_tactic": ["intro h", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 i, s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nw : Finset \u2191(u i)\nhw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nk' : \u2191(u i)\nleft\u271d\u00b9 : k' \u2208 w\nhk : \u2191\u2191k' \u2208 \u2191(Finset.image (fun x => \u2191\u2191x) w)\nl' : \u2191(u i)\nleft\u271d : l' \u2208 w\nhl : \u2191\u2191l' \u2208 \u2191(Finset.image (fun x => \u2191\u2191x) w)\nhkl : \u2191\u2191k' \u2260 \u2191\u2191l'\n\u22a2 \u2191k' \u2260 \u2191l'", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 i, s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nw : Finset \u2191(u i)\nhw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nk' : \u2191(u i)\nleft\u271d\u00b9 : k' \u2208 w\nhk : \u2191\u2191k' \u2208 \u2191(Finset.image (fun x => \u2191\u2191x) w)\nl' : \u2191(u i)\nleft\u271d : l' \u2208 w\nhl : \u2191\u2191l' \u2208 \u2191(Finset.image (fun x => \u2191\u2191x) w)\nhkl : \u2191\u2191k' \u2260 \u2191\u2191l'\nh : \u2191k' = \u2191l'\n\u22a2 False"}, {"tactic": "rw [h] at hkl", "annotated_tactic": ["rw [h] at hkl", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 i, s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nw : Finset \u2191(u i)\nhw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nk' : \u2191(u i)\nleft\u271d\u00b9 : k' \u2208 w\nhk : \u2191\u2191k' \u2208 \u2191(Finset.image (fun x => \u2191\u2191x) w)\nl' : \u2191(u i)\nleft\u271d : l' \u2208 w\nhl : \u2191\u2191l' \u2208 \u2191(Finset.image (fun x => \u2191\u2191x) w)\nhkl : \u2191\u2191k' \u2260 \u2191\u2191l'\nh : \u2191k' = \u2191l'\n\u22a2 False", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 i, s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nw : Finset \u2191(u i)\nhw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nk' : \u2191(u i)\nleft\u271d\u00b9 : k' \u2208 w\nhk : \u2191\u2191k' \u2208 \u2191(Finset.image (fun x => \u2191\u2191x) w)\nl' : \u2191(u i)\nleft\u271d : l' \u2208 w\nhl : \u2191\u2191l' \u2208 \u2191(Finset.image (fun x => \u2191\u2191x) w)\nhkl : \u2191\u2191l' \u2260 \u2191\u2191l'\nh : \u2191k' = \u2191l'\n\u22a2 False"}, {"tactic": "exact hkl rfl", "annotated_tactic": ["exact hkl <a>rfl</a>", [{"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n  { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n    r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 i, s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nw : Finset \u2191(u i)\nhw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nk' : \u2191(u i)\nleft\u271d\u00b9 : k' \u2208 w\nhk : \u2191\u2191k' \u2208 \u2191(Finset.image (fun x => \u2191\u2191x) w)\nl' : \u2191(u i)\nleft\u271d : l' \u2208 w\nhl : \u2191\u2191l' \u2208 \u2191(Finset.image (fun x => \u2191\u2191x) w)\nhkl : \u2191\u2191l' \u2260 \u2191\u2191l'\nh : \u2191k' = \u2191l'\n\u22a2 False", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "full_name": "String.utf8GetAux_of_valid", "start": [212, 1], "end": [221, 49], "traced_tactics": [{"tactic": "match cs, cs' with\n| [], [] => rfl\n| [], c::cs' => simp [\u2190 hp, utf8GetAux]\n| c::cs, cs' =>\n  simp [utf8GetAux, -List.headD_eq_head?]; rw [if_neg]\n  case hnc => simp [\u2190 hp, Pos.ext_iff]; exact ne_self_add_add_csize\n  refine utf8GetAux_of_valid cs cs' ?_\n  simpa [Nat.add_assoc, Nat.add_comm] using hp", "annotated_tactic": ["match cs, cs' with\n  | [], [] => rfl\n  | [], c::cs' => simp [\u2190 hp, <a>utf8GetAux</a>]\n  | c::cs, cs' =>\n    simp [<a>utf8GetAux</a>, -<a>List.headD_eq_head?</a>]; rw [<a>if_neg</a>]\n    case hnc => simp [\u2190 hp, <a>Pos.ext_iff</a>]; exact <a>ne_self_add_add_csize</a>\n    refine utf8GetAux_of_valid cs cs' ?_\n    simpa [<a>Nat.add_assoc</a>, <a>Nat.add_comm</a>] using hp", [{"full_name": "String.utf8GetAux", "def_path": "lake-packages/lean4/src/lean/Init/Data/String/Basic.lean", "def_pos": [47, 5], "def_end_pos": [47, 15]}, {"full_name": "String.utf8GetAux", "def_path": "lake-packages/lean4/src/lean/Init/Data/String/Basic.lean", "def_pos": [47, 5], "def_end_pos": [47, 15]}, {"full_name": "List.headD_eq_head?", "def_path": "lake-packages/std/Std/Data/List/Basic.lean", "def_pos": [592, 17], "def_end_pos": [592, 31]}, {"full_name": "if_neg", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [795, 9], "def_end_pos": [795, 15]}, {"full_name": "String.Pos.ext_iff", "def_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "def_pos": [108, 9], "def_end_pos": [108, 16]}, {"full_name": "_private.\u00ablake-packages\u00bb.std.Std.Data.String.Lemmas.0.String.ne_self_add_add_csize", "def_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "def_pos": [50, 17], "def_end_pos": [50, 38]}, {"full_name": "Nat.add_assoc", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [138, 19], "def_end_pos": [138, 28]}, {"full_name": "Nat.add_comm", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [131, 19], "def_end_pos": [131, 27]}]], "state_before": "cs cs' : List Char\ni p : Nat\nhp : i + utf8Len cs = p\n\u22a2 utf8GetAux (cs ++ cs') { byteIdx := i } { byteIdx := p } = List.headD cs' default", "state_after": "no goals"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "cs cs' : List Char\ni p : Nat\nhp : i + utf8Len [] = p\n\u22a2 utf8GetAux ([] ++ []) { byteIdx := i } { byteIdx := p } = List.headD [] default", "state_after": "no goals"}, {"tactic": "simp [\u2190 hp, utf8GetAux]", "annotated_tactic": ["simp [\u2190 hp, <a>utf8GetAux</a>]", [{"full_name": "String.utf8GetAux", "def_path": "lake-packages/lean4/src/lean/Init/Data/String/Basic.lean", "def_pos": [47, 5], "def_end_pos": [47, 15]}]], "state_before": "cs cs'\u271d : List Char\ni p : Nat\nc : Char\ncs' : List Char\nhp : i + utf8Len [] = p\n\u22a2 utf8GetAux ([] ++ c :: cs') { byteIdx := i } { byteIdx := p } = List.headD (c :: cs') default", "state_after": "no goals"}, {"tactic": "simp [utf8GetAux, -List.headD_eq_head?]", "annotated_tactic": ["simp [<a>utf8GetAux</a>, -<a>List.headD_eq_head?</a>]", [{"full_name": "String.utf8GetAux", "def_path": "lake-packages/lean4/src/lean/Init/Data/String/Basic.lean", "def_pos": [47, 5], "def_end_pos": [47, 15]}, {"full_name": "List.headD_eq_head?", "def_path": "lake-packages/std/Std/Data/List/Basic.lean", "def_pos": [592, 17], "def_end_pos": [592, 31]}]], "state_before": "cs\u271d cs'\u271d : List Char\ni p : Nat\nc : Char\ncs cs' : List Char\nhp : i + utf8Len (c :: cs) = p\n\u22a2 utf8GetAux (c :: cs ++ cs') { byteIdx := i } { byteIdx := p } = List.headD cs' default", "state_after": "cs\u271d cs'\u271d : List Char\ni p : Nat\nc : Char\ncs cs' : List Char\nhp : i + utf8Len (c :: cs) = p\n\u22a2 (if { byteIdx := i } = { byteIdx := p } then c else utf8GetAux (cs ++ cs') ({ byteIdx := i } + c) { byteIdx := p }) =\n    List.headD cs' default"}, {"tactic": "rw [if_neg]", "annotated_tactic": ["rw [<a>if_neg</a>]", [{"full_name": "if_neg", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [795, 9], "def_end_pos": [795, 15]}]], "state_before": "cs\u271d cs'\u271d : List Char\ni p : Nat\nc : Char\ncs cs' : List Char\nhp : i + utf8Len (c :: cs) = p\n\u22a2 (if { byteIdx := i } = { byteIdx := p } then c else utf8GetAux (cs ++ cs') ({ byteIdx := i } + c) { byteIdx := p }) =\n    List.headD cs' default", "state_after": "cs\u271d cs'\u271d : List Char\ni p : Nat\nc : Char\ncs cs' : List Char\nhp : i + utf8Len (c :: cs) = p\n\u22a2 utf8GetAux (cs ++ cs') ({ byteIdx := i } + c) { byteIdx := p } = List.headD cs' default\n\ncase hnc\ncs\u271d cs'\u271d : List Char\ni p : Nat\nc : Char\ncs cs' : List Char\nhp : i + utf8Len (c :: cs) = p\n\u22a2 \u00ac{ byteIdx := i } = { byteIdx := p }"}, {"tactic": "case hnc => simp [\u2190 hp, Pos.ext_iff]; exact ne_self_add_add_csize", "annotated_tactic": ["case hnc => simp [\u2190 hp, <a>Pos.ext_iff</a>]; exact <a>ne_self_add_add_csize</a>", [{"full_name": "String.Pos.ext_iff", "def_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "def_pos": [108, 9], "def_end_pos": [108, 16]}, {"full_name": "_private.\u00ablake-packages\u00bb.std.Std.Data.String.Lemmas.0.String.ne_self_add_add_csize", "def_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "def_pos": [50, 17], "def_end_pos": [50, 38]}]], "state_before": "cs\u271d cs'\u271d : List Char\ni p : Nat\nc : Char\ncs cs' : List Char\nhp : i + utf8Len (c :: cs) = p\n\u22a2 utf8GetAux (cs ++ cs') ({ byteIdx := i } + c) { byteIdx := p } = List.headD cs' default\n\ncase hnc\ncs\u271d cs'\u271d : List Char\ni p : Nat\nc : Char\ncs cs' : List Char\nhp : i + utf8Len (c :: cs) = p\n\u22a2 \u00ac{ byteIdx := i } = { byteIdx := p }", "state_after": "cs\u271d cs'\u271d : List Char\ni p : Nat\nc : Char\ncs cs' : List Char\nhp : i + utf8Len (c :: cs) = p\n\u22a2 utf8GetAux (cs ++ cs') ({ byteIdx := i } + c) { byteIdx := p } = List.headD cs' default"}, {"tactic": "refine utf8GetAux_of_valid cs cs' ?_", "annotated_tactic": ["refine utf8GetAux_of_valid cs cs' ?_", []], "state_before": "cs\u271d cs'\u271d : List Char\ni p : Nat\nc : Char\ncs cs' : List Char\nhp : i + utf8Len (c :: cs) = p\n\u22a2 utf8GetAux (cs ++ cs') ({ byteIdx := i } + c) { byteIdx := p } = List.headD cs' default", "state_after": "cs\u271d cs'\u271d : List Char\ni p : Nat\nc : Char\ncs cs' : List Char\nhp : i + utf8Len (c :: cs) = p\n\u22a2 { byteIdx := i }.byteIdx + csize c + utf8Len cs = p"}, {"tactic": "simpa [Nat.add_assoc, Nat.add_comm] using hp", "annotated_tactic": ["simpa [<a>Nat.add_assoc</a>, <a>Nat.add_comm</a>] using hp", [{"full_name": "Nat.add_assoc", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [138, 19], "def_end_pos": [138, 28]}, {"full_name": "Nat.add_comm", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [131, 19], "def_end_pos": [131, 27]}]], "state_before": "cs\u271d cs'\u271d : List Char\ni p : Nat\nc : Char\ncs cs' : List Char\nhp : i + utf8Len (c :: cs) = p\n\u22a2 { byteIdx := i }.byteIdx + csize c + utf8Len cs = p", "state_after": "no goals"}, {"tactic": "simp [\u2190 hp, Pos.ext_iff]", "annotated_tactic": ["simp [\u2190 hp, <a>Pos.ext_iff</a>]", [{"full_name": "String.Pos.ext_iff", "def_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "def_pos": [108, 9], "def_end_pos": [108, 16]}]], "state_before": "cs\u271d cs'\u271d : List Char\ni p : Nat\nc : Char\ncs cs' : List Char\nhp : i + utf8Len (c :: cs) = p\n\u22a2 \u00ac{ byteIdx := i } = { byteIdx := p }", "state_after": "cs\u271d cs'\u271d : List Char\ni p : Nat\nc : Char\ncs cs' : List Char\nhp : i + utf8Len (c :: cs) = p\n\u22a2 \u00aci = i + (utf8Len cs + csize c)"}, {"tactic": "exact ne_self_add_add_csize", "annotated_tactic": ["exact <a>ne_self_add_add_csize</a>", [{"full_name": "_private.\u00ablake-packages\u00bb.std.Std.Data.String.Lemmas.0.String.ne_self_add_add_csize", "def_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "def_pos": [50, 17], "def_end_pos": [50, 38]}]], "state_before": "cs\u271d cs'\u271d : List Char\ni p : Nat\nc : Char\ncs cs' : List Char\nhp : i + utf8Len (c :: cs) = p\n\u22a2 \u00aci = i + (utf8Len cs + csize c)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/RBMap/Alter.lean", "full_name": "Std.RBNode.Path.zoom_insert", "start": [169, 1], "end": [176, 99], "traced_tactics": [{"tactic": "have \u27e8_, _, ht', hp'\u27e9 := ht.zoom .root H", "annotated_tactic": ["have \u27e8_, _, ht', hp'\u27e9 := ht.zoom .root H", []], "state_before": "\u03b1 : Type u_1\nc : RBColor\nn : Nat\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nt' : RBNode \u03b1\nv : \u03b1\npath : Path \u03b1\nt : RBNode \u03b1\nht : Balanced t c n\nH : zoom (cmp v) t root = (t', path)\n\u22a2 setBlack (insert path t' v) = setBlack (RBNode.insert cmp t v)", "state_after": "\u03b1 : Type u_1\nc : RBColor\nn : Nat\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nt' : RBNode \u03b1\nv : \u03b1\npath : Path \u03b1\nt : RBNode \u03b1\nht : Balanced t c n\nH : zoom (cmp v) t root = (t', path)\nw\u271d\u00b9 : RBColor\nw\u271d : Nat\nht' : Balanced t' w\u271d\u00b9 w\u271d\nhp' : Path.Balanced c n path w\u271d\u00b9 w\u271d\n\u22a2 setBlack (insert path t' v) = setBlack (RBNode.insert cmp t v)"}, {"tactic": "cases ht' with simp [insert]\n| nil => simp [insertNew_eq_insert H, setBlack_idem]\n| red hl hr => rw [\u2190 ins_eq_fill hp' (.red hl hr), insert_setBlack]; exact (zoom_ins H).symm\n| black hl hr => rw [\u2190 ins_eq_fill hp' (.black hl hr), insert_setBlack]; exact (zoom_ins H).symm", "annotated_tactic": ["cases ht' with simp [<a>insert</a>]\n  | <a>nil</a> => simp [<a>insertNew_eq_insert</a> H, <a>setBlack_idem</a>]\n  | <a>red</a> hl hr => rw [\u2190 <a>ins_eq_fill</a> hp' (.red hl hr), <a>insert_setBlack</a>]; exact (<a>zoom_ins</a> H).<a>symm</a>\n  | <a>black</a> hl hr => rw [\u2190 <a>ins_eq_fill</a> hp' (.black hl hr), <a>insert_setBlack</a>]; exact (<a>zoom_ins</a> H).<a>symm</a>", [{"full_name": "Std.RBNode.Path.insert", "def_path": "lake-packages/std/Std/Data/RBMap/Basic.lean", "def_pos": [481, 5], "def_end_pos": [481, 16]}, {"full_name": "Std.RBNode.Balanced.nil", "def_path": "lake-packages/std/Std/Data/RBMap/Basic.lean", "def_pos": [572, 15], "def_end_pos": [572, 18]}, {"full_name": "Std.RBNode.Path.insertNew_eq_insert", "def_path": "lake-packages/std/Std/Data/RBMap/Alter.lean", "def_pos": [68, 9], "def_end_pos": [68, 28]}, {"full_name": "Std.RBNode.setBlack_idem", "def_path": "lake-packages/std/Std/Data/RBMap/WF.lean", "def_pos": [86, 9], "def_end_pos": [86, 22]}, {"full_name": "Std.RBNode.Balanced.red", "def_path": "lake-packages/std/Std/Data/RBMap/Basic.lean", "def_pos": [575, 15], "def_end_pos": [575, 18]}, {"full_name": "Std.RBNode.Path.ins_eq_fill", "def_path": "lake-packages/std/Std/Data/RBMap/Alter.lean", "def_pos": [137, 9], "def_end_pos": [137, 20]}, {"full_name": "Std.RBNode.insert_setBlack", "def_path": "lake-packages/std/Std/Data/RBMap/WF.lean", "def_pos": [88, 9], "def_end_pos": [88, 24]}, {"full_name": "Std.RBNode.Path.zoom_ins", "def_path": "lake-packages/std/Std/Data/RBMap/Alter.lean", "def_pos": [57, 9], "def_end_pos": [57, 17]}, {"full_name": "Eq.symm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [310, 9], "def_end_pos": [310, 16]}, {"full_name": "Std.RBNode.Balanced.black", "def_path": "lake-packages/std/Std/Data/RBMap/Basic.lean", "def_pos": [578, 15], "def_end_pos": [578, 20]}, {"full_name": "Std.RBNode.Path.ins_eq_fill", "def_path": "lake-packages/std/Std/Data/RBMap/Alter.lean", "def_pos": [137, 9], "def_end_pos": [137, 20]}, {"full_name": "Std.RBNode.insert_setBlack", "def_path": "lake-packages/std/Std/Data/RBMap/WF.lean", "def_pos": [88, 9], "def_end_pos": [88, 24]}, {"full_name": "Std.RBNode.Path.zoom_ins", "def_path": "lake-packages/std/Std/Data/RBMap/Alter.lean", "def_pos": [57, 9], "def_end_pos": [57, 17]}, {"full_name": "Eq.symm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [310, 9], "def_end_pos": [310, 16]}]], "state_before": "\u03b1 : Type u_1\nc : RBColor\nn : Nat\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nt' : RBNode \u03b1\nv : \u03b1\npath : Path \u03b1\nt : RBNode \u03b1\nht : Balanced t c n\nH : zoom (cmp v) t root = (t', path)\nw\u271d\u00b9 : RBColor\nw\u271d : Nat\nht' : Balanced t' w\u271d\u00b9 w\u271d\nhp' : Path.Balanced c n path w\u271d\u00b9 w\u271d\n\u22a2 setBlack (insert path t' v) = setBlack (RBNode.insert cmp t v)", "state_after": "no goals"}, {"tactic": "simp [insertNew_eq_insert H, setBlack_idem]", "annotated_tactic": ["simp [<a>insertNew_eq_insert</a> H, <a>setBlack_idem</a>]", [{"full_name": "Std.RBNode.Path.insertNew_eq_insert", "def_path": "lake-packages/std/Std/Data/RBMap/Alter.lean", "def_pos": [68, 9], "def_end_pos": [68, 28]}, {"full_name": "Std.RBNode.setBlack_idem", "def_path": "lake-packages/std/Std/Data/RBMap/WF.lean", "def_pos": [86, 9], "def_end_pos": [86, 22]}]], "state_before": "case nil\n\u03b1 : Type u_1\nc : RBColor\nn : Nat\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nv : \u03b1\npath : Path \u03b1\nt : RBNode \u03b1\nht : Balanced t c n\nH : zoom (cmp v) t root = (nil, path)\nhp' : Path.Balanced c n path black 0\n\u22a2 setBlack (insertNew path v) = setBlack (RBNode.insert cmp t v)", "state_after": "no goals"}, {"tactic": "rw [\u2190 ins_eq_fill hp' (.red hl hr), insert_setBlack]", "annotated_tactic": ["rw [\u2190 <a>ins_eq_fill</a> hp' (.red hl hr), <a>insert_setBlack</a>]", [{"full_name": "Std.RBNode.Path.ins_eq_fill", "def_path": "lake-packages/std/Std/Data/RBMap/Alter.lean", "def_pos": [137, 9], "def_end_pos": [137, 20]}, {"full_name": "Std.RBNode.insert_setBlack", "def_path": "lake-packages/std/Std/Data/RBMap/WF.lean", "def_pos": [88, 9], "def_end_pos": [88, 24]}]], "state_before": "case red\n\u03b1 : Type u_1\nc : RBColor\nn : Nat\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nv : \u03b1\npath : Path \u03b1\nt : RBNode \u03b1\nht : Balanced t c n\nw\u271d : Nat\nx\u271d y\u271d : RBNode \u03b1\nv\u271d : \u03b1\nH : zoom (cmp v) t root = (node red x\u271d v\u271d y\u271d, path)\nhp' : Path.Balanced c n path red w\u271d\nhl : Balanced x\u271d black w\u271d\nhr : Balanced y\u271d black w\u271d\n\u22a2 setBlack (fill path (node red x\u271d v y\u271d)) = setBlack (RBNode.insert cmp t v)", "state_after": "case red\n\u03b1 : Type u_1\nc : RBColor\nn : Nat\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nv : \u03b1\npath : Path \u03b1\nt : RBNode \u03b1\nht : Balanced t c n\nw\u271d : Nat\nx\u271d y\u271d : RBNode \u03b1\nv\u271d : \u03b1\nH : zoom (cmp v) t root = (node red x\u271d v\u271d y\u271d, path)\nhp' : Path.Balanced c n path red w\u271d\nhl : Balanced x\u271d black w\u271d\nhr : Balanced y\u271d black w\u271d\n\u22a2 ins path (node red x\u271d v y\u271d) = setBlack (RBNode.ins cmp v t)"}, {"tactic": "exact (zoom_ins H).symm", "annotated_tactic": ["exact (<a>zoom_ins</a> H).<a>symm</a>", [{"full_name": "Std.RBNode.Path.zoom_ins", "def_path": "lake-packages/std/Std/Data/RBMap/Alter.lean", "def_pos": [57, 9], "def_end_pos": [57, 17]}, {"full_name": "Eq.symm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [310, 9], "def_end_pos": [310, 16]}]], "state_before": "case red\n\u03b1 : Type u_1\nc : RBColor\nn : Nat\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nv : \u03b1\npath : Path \u03b1\nt : RBNode \u03b1\nht : Balanced t c n\nw\u271d : Nat\nx\u271d y\u271d : RBNode \u03b1\nv\u271d : \u03b1\nH : zoom (cmp v) t root = (node red x\u271d v\u271d y\u271d, path)\nhp' : Path.Balanced c n path red w\u271d\nhl : Balanced x\u271d black w\u271d\nhr : Balanced y\u271d black w\u271d\n\u22a2 ins path (node red x\u271d v y\u271d) = setBlack (RBNode.ins cmp v t)", "state_after": "no goals"}, {"tactic": "rw [\u2190 ins_eq_fill hp' (.black hl hr), insert_setBlack]", "annotated_tactic": ["rw [\u2190 <a>ins_eq_fill</a> hp' (.black hl hr), <a>insert_setBlack</a>]", [{"full_name": "Std.RBNode.Path.ins_eq_fill", "def_path": "lake-packages/std/Std/Data/RBMap/Alter.lean", "def_pos": [137, 9], "def_end_pos": [137, 20]}, {"full_name": "Std.RBNode.insert_setBlack", "def_path": "lake-packages/std/Std/Data/RBMap/WF.lean", "def_pos": [88, 9], "def_end_pos": [88, 24]}]], "state_before": "case black\n\u03b1 : Type u_1\nc : RBColor\nn : Nat\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nv : \u03b1\npath : Path \u03b1\nt : RBNode \u03b1\nht : Balanced t c n\nx\u271d : RBNode \u03b1\nc\u2081\u271d : RBColor\nn\u271d : Nat\ny\u271d : RBNode \u03b1\nc\u2082\u271d : RBColor\nv\u271d : \u03b1\nhl : Balanced x\u271d c\u2081\u271d n\u271d\nhr : Balanced y\u271d c\u2082\u271d n\u271d\nH : zoom (cmp v) t root = (node black x\u271d v\u271d y\u271d, path)\nhp' : Path.Balanced c n path black (n\u271d + 1)\n\u22a2 setBlack (fill path (node black x\u271d v y\u271d)) = setBlack (RBNode.insert cmp t v)", "state_after": "case black\n\u03b1 : Type u_1\nc : RBColor\nn : Nat\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nv : \u03b1\npath : Path \u03b1\nt : RBNode \u03b1\nht : Balanced t c n\nx\u271d : RBNode \u03b1\nc\u2081\u271d : RBColor\nn\u271d : Nat\ny\u271d : RBNode \u03b1\nc\u2082\u271d : RBColor\nv\u271d : \u03b1\nhl : Balanced x\u271d c\u2081\u271d n\u271d\nhr : Balanced y\u271d c\u2082\u271d n\u271d\nH : zoom (cmp v) t root = (node black x\u271d v\u271d y\u271d, path)\nhp' : Path.Balanced c n path black (n\u271d + 1)\n\u22a2 ins path (node black x\u271d v y\u271d) = setBlack (RBNode.ins cmp v t)"}, {"tactic": "exact (zoom_ins H).symm", "annotated_tactic": ["exact (<a>zoom_ins</a> H).<a>symm</a>", [{"full_name": "Std.RBNode.Path.zoom_ins", "def_path": "lake-packages/std/Std/Data/RBMap/Alter.lean", "def_pos": [57, 9], "def_end_pos": [57, 17]}, {"full_name": "Eq.symm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [310, 9], "def_end_pos": [310, 16]}]], "state_before": "case black\n\u03b1 : Type u_1\nc : RBColor\nn : Nat\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nv : \u03b1\npath : Path \u03b1\nt : RBNode \u03b1\nht : Balanced t c n\nx\u271d : RBNode \u03b1\nc\u2081\u271d : RBColor\nn\u271d : Nat\ny\u271d : RBNode \u03b1\nc\u2082\u271d : RBColor\nv\u271d : \u03b1\nhl : Balanced x\u271d c\u2081\u271d n\u271d\nhr : Balanced y\u271d c\u2082\u271d n\u271d\nH : zoom (cmp v) t root = (node black x\u271d v\u271d y\u271d, path)\nhp' : Path.Balanced c n path black (n\u271d + 1)\n\u22a2 ins path (node black x\u271d v y\u271d) = setBlack (RBNode.ins cmp v t)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/ProbabilityMeasure.lean", "full_name": "ProbabilityMeasure.toFiniteMeasure_normalize_eq_self", "start": [383, 1], "end": [389, 8], "traced_tactics": [{"tactic": "apply ProbabilityMeasure.eq_of_forall_apply_eq", "annotated_tactic": ["apply <a>ProbabilityMeasure.eq_of_forall_apply_eq</a>", [{"full_name": "MeasureTheory.ProbabilityMeasure.eq_of_forall_apply_eq", "def_path": "Mathlib/MeasureTheory/Measure/ProbabilityMeasure.lean", "def_pos": [201, 9], "def_end_pos": [201, 30]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d : Nonempty \u03a9\nm0\u271d : MeasurableSpace \u03a9\n\u03bc\u271d : FiniteMeasure \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : ProbabilityMeasure \u03a9\n\u22a2 normalize (ProbabilityMeasure.toFiniteMeasure \u03bc) = \u03bc", "state_after": "case h\n\u03a9 : Type u_1\ninst\u271d : Nonempty \u03a9\nm0\u271d : MeasurableSpace \u03a9\n\u03bc\u271d : FiniteMeasure \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : ProbabilityMeasure \u03a9\n\u22a2 \u2200 (s : Set \u03a9),\n    MeasurableSet s \u2192\n      (fun s => ENNReal.toNNReal (\u2191\u2191\u2191(normalize (ProbabilityMeasure.toFiniteMeasure \u03bc)) s)) s =\n        (fun s => ENNReal.toNNReal (\u2191\u2191\u2191\u03bc s)) s"}, {"tactic": "intro s _s_mble", "annotated_tactic": ["intro s _s_mble", []], "state_before": "case h\n\u03a9 : Type u_1\ninst\u271d : Nonempty \u03a9\nm0\u271d : MeasurableSpace \u03a9\n\u03bc\u271d : FiniteMeasure \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : ProbabilityMeasure \u03a9\n\u22a2 \u2200 (s : Set \u03a9),\n    MeasurableSet s \u2192\n      (fun s => ENNReal.toNNReal (\u2191\u2191\u2191(normalize (ProbabilityMeasure.toFiniteMeasure \u03bc)) s)) s =\n        (fun s => ENNReal.toNNReal (\u2191\u2191\u2191\u03bc s)) s", "state_after": "case h\n\u03a9 : Type u_1\ninst\u271d : Nonempty \u03a9\nm0\u271d : MeasurableSpace \u03a9\n\u03bc\u271d : FiniteMeasure \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : ProbabilityMeasure \u03a9\ns : Set \u03a9\n_s_mble : MeasurableSet s\n\u22a2 (fun s => ENNReal.toNNReal (\u2191\u2191\u2191(normalize (ProbabilityMeasure.toFiniteMeasure \u03bc)) s)) s =\n    (fun s => ENNReal.toNNReal (\u2191\u2191\u2191\u03bc s)) s"}, {"tactic": "rw [\u03bc.toFiniteMeasure.normalize_eq_of_nonzero \u03bc.toFiniteMeasure_nonzero s]", "annotated_tactic": ["rw [\u03bc.toFiniteMeasure.normalize_eq_of_nonzero \u03bc.toFiniteMeasure_nonzero s]", []], "state_before": "case h\n\u03a9 : Type u_1\ninst\u271d : Nonempty \u03a9\nm0\u271d : MeasurableSpace \u03a9\n\u03bc\u271d : FiniteMeasure \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : ProbabilityMeasure \u03a9\ns : Set \u03a9\n_s_mble : MeasurableSet s\n\u22a2 (fun s => ENNReal.toNNReal (\u2191\u2191\u2191(normalize (ProbabilityMeasure.toFiniteMeasure \u03bc)) s)) s =\n    (fun s => ENNReal.toNNReal (\u2191\u2191\u2191\u03bc s)) s", "state_after": "case h\n\u03a9 : Type u_1\ninst\u271d : Nonempty \u03a9\nm0\u271d : MeasurableSpace \u03a9\n\u03bc\u271d : FiniteMeasure \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : ProbabilityMeasure \u03a9\ns : Set \u03a9\n_s_mble : MeasurableSet s\n\u22a2 (mass (ProbabilityMeasure.toFiniteMeasure \u03bc))\u207b\u00b9 *\n      (fun s => ENNReal.toNNReal (\u2191\u2191\u2191(ProbabilityMeasure.toFiniteMeasure \u03bc) s)) s =\n    (fun s => ENNReal.toNNReal (\u2191\u2191\u2191\u03bc s)) s"}, {"tactic": "simp only [ProbabilityMeasure.mass_toFiniteMeasure, inv_one, one_mul]", "annotated_tactic": ["simp only [<a>ProbabilityMeasure.mass_toFiniteMeasure</a>, <a>inv_one</a>, <a>one_mul</a>]", [{"full_name": "MeasureTheory.ProbabilityMeasure.mass_toFiniteMeasure", "def_path": "Mathlib/MeasureTheory/Measure/ProbabilityMeasure.lean", "def_pos": [208, 9], "def_end_pos": [208, 29]}, {"full_name": "inv_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [1015, 9], "def_end_pos": [1015, 16]}, {"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [464, 9], "def_end_pos": [464, 16]}]], "state_before": "case h\n\u03a9 : Type u_1\ninst\u271d : Nonempty \u03a9\nm0\u271d : MeasurableSpace \u03a9\n\u03bc\u271d : FiniteMeasure \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : ProbabilityMeasure \u03a9\ns : Set \u03a9\n_s_mble : MeasurableSet s\n\u22a2 (mass (ProbabilityMeasure.toFiniteMeasure \u03bc))\u207b\u00b9 *\n      (fun s => ENNReal.toNNReal (\u2191\u2191\u2191(ProbabilityMeasure.toFiniteMeasure \u03bc) s)) s =\n    (fun s => ENNReal.toNNReal (\u2191\u2191\u2191\u03bc s)) s", "state_after": "case h\n\u03a9 : Type u_1\ninst\u271d : Nonempty \u03a9\nm0\u271d : MeasurableSpace \u03a9\n\u03bc\u271d : FiniteMeasure \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : ProbabilityMeasure \u03a9\ns : Set \u03a9\n_s_mble : MeasurableSet s\n\u22a2 ENNReal.toNNReal (\u2191\u2191\u2191(ProbabilityMeasure.toFiniteMeasure \u03bc) s) = ENNReal.toNNReal (\u2191\u2191\u2191\u03bc s)"}, {"tactic": "congr", "annotated_tactic": ["congr", []], "state_before": "case h\n\u03a9 : Type u_1\ninst\u271d : Nonempty \u03a9\nm0\u271d : MeasurableSpace \u03a9\n\u03bc\u271d : FiniteMeasure \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : ProbabilityMeasure \u03a9\ns : Set \u03a9\n_s_mble : MeasurableSet s\n\u22a2 ENNReal.toNNReal (\u2191\u2191\u2191(ProbabilityMeasure.toFiniteMeasure \u03bc) s) = ENNReal.toNNReal (\u2191\u2191\u2191\u03bc s)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Pointwise.lean", "full_name": "Finset.op_smul_finset_mul_eq_mul_smul_finset", "start": [1864, 1], "end": [1866, 77], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "full_name": "Int.sign_eq_one_iff_pos", "start": [1290, 1], "end": [1291, 43], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "full_name": "MeasureTheory.Mem\u2112p.im", "start": [1619, 1], "end": [1625, 60], "traced_tactics": [{"tactic": "have : \u2200 x, \u2016IsROrC.im (f x)\u2016 \u2264 1 * \u2016f x\u2016 := by\n  intro x\n  rw [one_mul]\n  exact IsROrC.norm_im_le_norm (f x)", "annotated_tactic": ["have : \u2200 x, \u2016<a>IsROrC.im</a> (f x)\u2016 \u2264 1 * \u2016f x\u2016 := by\n    intro x\n    rw [<a>one_mul</a>]\n    exact <a>IsROrC.norm_im_le_norm</a> (f x)", [{"full_name": "IsROrC.im", "def_path": "Mathlib/Data/IsROrC/Basic.lean", "def_pos": [59, 3], "def_end_pos": [59, 5]}, {"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [464, 9], "def_end_pos": [464, 16]}, {"full_name": "IsROrC.norm_im_le_norm", "def_path": "Mathlib/Data/IsROrC/Basic.lean", "def_pos": [735, 9], "def_end_pos": [735, 24]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\n\ud835\udd5c : Type u_5\ninst\u271d : IsROrC \ud835\udd5c\nf : \u03b1 \u2192 \ud835\udd5c\nhf : Mem\u2112p f p\n\u22a2 Mem\u2112p (fun x => \u2191IsROrC.im (f x)) p", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\n\ud835\udd5c : Type u_5\ninst\u271d : IsROrC \ud835\udd5c\nf : \u03b1 \u2192 \ud835\udd5c\nhf : Mem\u2112p f p\nthis : \u2200 (x : \u03b1), \u2016\u2191IsROrC.im (f x)\u2016 \u2264 1 * \u2016f x\u2016\n\u22a2 Mem\u2112p (fun x => \u2191IsROrC.im (f x)) p"}, {"tactic": "refine' hf.of_le_mul _ (eventually_of_forall this)", "annotated_tactic": ["refine' hf.of_le_mul _ (<a>eventually_of_forall</a> this)", [{"full_name": "Filter.eventually_of_forall", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1111, 9], "def_end_pos": [1111, 29]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\n\ud835\udd5c : Type u_5\ninst\u271d : IsROrC \ud835\udd5c\nf : \u03b1 \u2192 \ud835\udd5c\nhf : Mem\u2112p f p\nthis : \u2200 (x : \u03b1), \u2016\u2191IsROrC.im (f x)\u2016 \u2264 1 * \u2016f x\u2016\n\u22a2 Mem\u2112p (fun x => \u2191IsROrC.im (f x)) p", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\n\ud835\udd5c : Type u_5\ninst\u271d : IsROrC \ud835\udd5c\nf : \u03b1 \u2192 \ud835\udd5c\nhf : Mem\u2112p f p\nthis : \u2200 (x : \u03b1), \u2016\u2191IsROrC.im (f x)\u2016 \u2264 1 * \u2016f x\u2016\n\u22a2 AEStronglyMeasurable (fun x => \u2191IsROrC.im (f x)) \u03bc"}, {"tactic": "exact IsROrC.continuous_im.comp_aestronglyMeasurable hf.1", "annotated_tactic": ["exact IsROrC.continuous_im.comp_aestronglyMeasurable hf.1", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\n\ud835\udd5c : Type u_5\ninst\u271d : IsROrC \ud835\udd5c\nf : \u03b1 \u2192 \ud835\udd5c\nhf : Mem\u2112p f p\nthis : \u2200 (x : \u03b1), \u2016\u2191IsROrC.im (f x)\u2016 \u2264 1 * \u2016f x\u2016\n\u22a2 AEStronglyMeasurable (fun x => \u2191IsROrC.im (f x)) \u03bc", "state_after": "no goals"}, {"tactic": "intro x", "annotated_tactic": ["intro x", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\n\ud835\udd5c : Type u_5\ninst\u271d : IsROrC \ud835\udd5c\nf : \u03b1 \u2192 \ud835\udd5c\nhf : Mem\u2112p f p\n\u22a2 \u2200 (x : \u03b1), \u2016\u2191IsROrC.im (f x)\u2016 \u2264 1 * \u2016f x\u2016", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\n\ud835\udd5c : Type u_5\ninst\u271d : IsROrC \ud835\udd5c\nf : \u03b1 \u2192 \ud835\udd5c\nhf : Mem\u2112p f p\nx : \u03b1\n\u22a2 \u2016\u2191IsROrC.im (f x)\u2016 \u2264 1 * \u2016f x\u2016"}, {"tactic": "rw [one_mul]", "annotated_tactic": ["rw [<a>one_mul</a>]", [{"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [464, 9], "def_end_pos": [464, 16]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\n\ud835\udd5c : Type u_5\ninst\u271d : IsROrC \ud835\udd5c\nf : \u03b1 \u2192 \ud835\udd5c\nhf : Mem\u2112p f p\nx : \u03b1\n\u22a2 \u2016\u2191IsROrC.im (f x)\u2016 \u2264 1 * \u2016f x\u2016", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\n\ud835\udd5c : Type u_5\ninst\u271d : IsROrC \ud835\udd5c\nf : \u03b1 \u2192 \ud835\udd5c\nhf : Mem\u2112p f p\nx : \u03b1\n\u22a2 \u2016\u2191IsROrC.im (f x)\u2016 \u2264 \u2016f x\u2016"}, {"tactic": "exact IsROrC.norm_im_le_norm (f x)", "annotated_tactic": ["exact <a>IsROrC.norm_im_le_norm</a> (f x)", [{"full_name": "IsROrC.norm_im_le_norm", "def_path": "Mathlib/Data/IsROrC/Basic.lean", "def_pos": [735, 9], "def_end_pos": [735, 24]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\n\ud835\udd5c : Type u_5\ninst\u271d : IsROrC \ud835\udd5c\nf : \u03b1 \u2192 \ud835\udd5c\nhf : Mem\u2112p f p\nx : \u03b1\n\u22a2 \u2016\u2191IsROrC.im (f x)\u2016 \u2264 \u2016f x\u2016", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Basic.lean", "full_name": "Set.inclusion_eq_id", "start": [2800, 1], "end": [2801, 24], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Halting.lean", "full_name": "Nat.Partrec'.bind", "start": [324, 11], "end": [329, 52], "traced_tactics": [{"tactic": "refine' Fin.cases _ (fun i => _) i <;> simp [*]", "annotated_tactic": ["refine' <a>Fin.cases</a> _ (fun i => _) i <;> simp [*]", [{"full_name": "Fin.cases", "def_path": "lake-packages/std/Std/Data/Fin/Lemmas.lean", "def_pos": [613, 21], "def_end_pos": [613, 26]}]], "state_before": "n : \u2115\nf : Vector \u2115 n \u2192. \u2115\ng : Vector \u2115 (n + 1) \u2192. \u2115\nhf : Partrec' f\nhg : Partrec' g\ni : Fin (n + 1)\n\u22a2 Partrec' ((fun i => Fin.cases f (fun i v => \u2191(some (Vector.get v i))) i) i)", "state_after": "case refine'_2\nn : \u2115\nf : Vector \u2115 n \u2192. \u2115\ng : Vector \u2115 (n + 1) \u2192. \u2115\nhf : Partrec' f\nhg : Partrec' g\ni\u271d : Fin (n + 1)\ni : Fin n\n\u22a2 Partrec' fun v => Part.some (Vector.get v i)"}, {"tactic": "exact prim (Nat.Primrec'.get _)", "annotated_tactic": ["exact <a>prim</a> (<a>Nat.Primrec'.get</a> _)", [{"full_name": "Nat.Partrec'.prim", "def_path": "Mathlib/Computability/Halting.lean", "def_pos": [278, 5], "def_end_pos": [278, 9]}, {"full_name": "Nat.Primrec'.get", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [1364, 5], "def_end_pos": [1364, 8]}]], "state_before": "case refine'_2\nn : \u2115\nf : Vector \u2115 n \u2192. \u2115\ng : Vector \u2115 (n + 1) \u2192. \u2115\nhf : Partrec' f\nhg : Partrec' g\ni\u271d : Fin (n + 1)\ni : Fin n\n\u22a2 Partrec' fun v => Part.some (Vector.get v i)", "state_after": "no goals"}, {"tactic": "simp [mOfFn, Part.bind_assoc, pure]", "annotated_tactic": ["simp [<a>mOfFn</a>, <a>Part.bind_assoc</a>, <a>pure</a>]", [{"full_name": "Vector.mOfFn", "def_path": "Mathlib/Data/Vector/Basic.lean", "def_pos": [408, 5], "def_end_pos": [408, 10]}, {"full_name": "Part.bind_assoc", "def_path": "Mathlib/Data/Part.lean", "def_pos": [540, 9], "def_end_pos": [540, 19]}, {"full_name": "Pure.pure", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2670, 3], "def_end_pos": [2670, 7]}]], "state_before": "n : \u2115\nf : Vector \u2115 n \u2192. \u2115\ng : Vector \u2115 (n + 1) \u2192. \u2115\nhf : Partrec' f\nhg : Partrec' g\nv : Vector \u2115 n\n\u22a2 (mOfFn fun i => Fin.cases f (fun i v => \u2191(some (Vector.get v i))) i v) >>= g = Part.bind (f v) fun a => g (a ::\u1d65 v)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Haar/NormedSpace.lean", "full_name": "MeasureTheory.Measure.integral_comp_div", "start": [127, 1], "end": [128, 34], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "full_name": 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\u03b1\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t\u271d : Set \u03b1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t u : Set \u03b1\nhs : MeasurableSet s\nh's : s \u2286 u\nh't : t \u2286 u\nh : \u2191\u2191\u03bc u < \u2191\u2191\u03bc t + \u2191\u2191\u03bc s\n\u22a2 Set.Nonempty (s \u2229 t)"}, {"tactic": "rw [inter_comm]", "annotated_tactic": ["rw [<a>inter_comm</a>]", [{"full_name": "Set.inter_comm", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [940, 9], "def_end_pos": [940, 19]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t\u271d : Set \u03b1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t u : Set \u03b1\nhs : MeasurableSet s\nh's : s \u2286 u\nh't : t \u2286 u\nh : \u2191\u2191\u03bc u < \u2191\u2191\u03bc t + \u2191\u2191\u03bc s\n\u22a2 Set.Nonempty (s \u2229 t)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t\u271d : Set \u03b1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t u : Set \u03b1\nhs : MeasurableSet s\nh's : s \u2286 u\nh't : t \u2286 u\nh : \u2191\u2191\u03bc u < \u2191\u2191\u03bc t + \u2191\u2191\u03bc s\n\u22a2 Set.Nonempty (t \u2229 s)"}, {"tactic": "exact nonempty_inter_of_measure_lt_add \u03bc hs h't h's h", "annotated_tactic": ["exact <a>nonempty_inter_of_measure_lt_add</a> \u03bc hs h't h's h", [{"full_name": "MeasureTheory.nonempty_inter_of_measure_lt_add", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [440, 9], "def_end_pos": [440, 41]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t\u271d : Set \u03b1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t u : Set \u03b1\nhs : MeasurableSet s\nh's : s \u2286 u\nh't : t \u2286 u\nh : \u2191\u2191\u03bc u < \u2191\u2191\u03bc t + \u2191\u2191\u03bc s\n\u22a2 Set.Nonempty (t \u2229 s)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/Floor.lean", "full_name": "Nat.measurable_ceil", "start": [82, 1], "end": [84, 98], "traced_tactics": [{"tactic": "cases' eq_or_ne \u2308n\u2309\u208a 0 with h h <;> simp_all [h, Nat.preimage_ceil_of_ne_zero, -ceil_eq_zero]", "annotated_tactic": ["cases' <a>eq_or_ne</a> \u2308n\u2309\u208a 0 with h h <;> simp_all [h, <a>Nat.preimage_ceil_of_ne_zero</a>, -<a>ceil_eq_zero</a>]", [{"full_name": "eq_or_ne", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [209, 9], "def_end_pos": [209, 17]}, {"full_name": "Nat.preimage_ceil_of_ne_zero", "def_path": "Mathlib/Algebra/Order/Floor.lean", "def_pos": [370, 9], "def_end_pos": [370, 33]}, {"full_name": "Nat.ceil_eq_zero", "def_path": "Mathlib/Algebra/Order/Floor.lean", "def_pos": [329, 9], "def_end_pos": [329, 21]}]], "state_before": "\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : LinearOrderedSemiring R\ninst\u271d\u2074 : FloorSemiring R\ninst\u271d\u00b3 : TopologicalSpace R\ninst\u271d\u00b2 : OrderTopology R\ninst\u271d\u00b9 : MeasurableSpace R\ninst\u271d : OpensMeasurableSpace R\nf : \u03b1 \u2192 R\nn : R\n\u22a2 MeasurableSet (ceil \u207b\u00b9' {\u2308n\u2309\u208a})", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/UniformIntegrable.lean", "full_name": "MeasureTheory.unifIntegrable_of", "start": [700, 1], "end": [722, 94], "traced_tactics": [{"tactic": "set g : \u03b9 \u2192 \u03b1 \u2192 \u03b2 := fun i => (hf i).choose", "annotated_tactic": ["set g : \u03b9 \u2192 \u03b1 \u2192 \u03b2 := fun i => (hf i).<a>choose</a>", [{"full_name": "Exists.choose", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [442, 32], "def_end_pos": [442, 45]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\nh : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\n\u22a2 UnifIntegrable f p \u03bc", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng\u271d : \u03b1 \u2192 \u03b2\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\nh : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\ng : \u03b9 \u2192 \u03b1 \u2192 \u03b2 := fun i => Exists.choose (_ : AEStronglyMeasurable (f i) \u03bc)\n\u22a2 UnifIntegrable f p \u03bc"}, {"tactic": "refine'\n  (unifIntegrable_of' \u03bc hp hp' (fun i => (Exists.choose_spec <| hf i).1) fun \u03b5 h\u03b5 => _).ae_eq\n    fun i => (Exists.choose_spec <| hf i).2.symm", "annotated_tactic": ["refine'\n    (<a>unifIntegrable_of'</a> \u03bc hp hp' (fun i => (<a>Exists.choose_spec</a> <| hf i).1) fun \u03b5 h\u03b5 => _).<a>ae_eq</a>\n      fun i => (<a>Exists.choose_spec</a> <| hf i).2.<a>symm</a>", [{"full_name": "MeasureTheory.unifIntegrable_of'", "def_path": "Mathlib/MeasureTheory/Function/UniformIntegrable.lean", "def_pos": [635, 9], "def_end_pos": [635, 27]}, {"full_name": "Exists.choose_spec", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [445, 9], "def_end_pos": [445, 27]}, {"full_name": "MeasureTheory.UnifIntegrable.ae_eq", "def_path": "Mathlib/MeasureTheory/Function/UniformIntegrable.lean", "def_pos": [137, 19], "def_end_pos": [137, 24]}, {"full_name": "Exists.choose_spec", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [445, 9], "def_end_pos": [445, 27]}, {"full_name": "Filter.EventuallyEq.symm", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1498, 9], "def_end_pos": [1498, 26]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng\u271d : \u03b1 \u2192 \u03b2\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\nh : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\ng : \u03b9 \u2192 \u03b1 \u2192 \u03b2 := fun i => Exists.choose (_ : AEStronglyMeasurable (f i) \u03bc)\n\u22a2 UnifIntegrable f p \u03bc", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng\u271d : \u03b1 \u2192 \u03b2\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\nh : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\ng : \u03b9 \u2192 \u03b1 \u2192 \u03b2 := fun i => Exists.choose (_ : AEStronglyMeasurable (f i) \u03bc)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u22a2 \u2203 C,\n    0 < C \u2227\n      \u2200 (i : \u03b9),\n        snorm\n            (indicator {x | C \u2264 \u2016Exists.choose (_ : AEStronglyMeasurable (f i) \u03bc) x\u2016\u208a}\n              (Exists.choose (_ : AEStronglyMeasurable (f i) \u03bc)))\n            p \u03bc \u2264\n          ENNReal.ofReal \u03b5"}, {"tactic": "obtain \u27e8C, hC\u27e9 := h \u03b5 h\u03b5", "annotated_tactic": ["obtain \u27e8C, hC\u27e9 := h \u03b5 h\u03b5", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng\u271d : \u03b1 \u2192 \u03b2\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\nh : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\ng : \u03b9 \u2192 \u03b1 \u2192 \u03b2 := fun i => Exists.choose (_ : AEStronglyMeasurable (f i) \u03bc)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u22a2 \u2203 C,\n    0 < C \u2227\n      \u2200 (i : \u03b9),\n        snorm\n            (indicator {x | C \u2264 \u2016Exists.choose (_ : AEStronglyMeasurable (f i) \u03bc) x\u2016\u208a}\n              (Exists.choose (_ : AEStronglyMeasurable (f i) \u03bc)))\n            p \u03bc \u2264\n          ENNReal.ofReal \u03b5", "state_after": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng\u271d : \u03b1 \u2192 \u03b2\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\nh : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\ng : \u03b9 \u2192 \u03b1 \u2192 \u03b2 := fun i => Exists.choose (_ : AEStronglyMeasurable (f i) \u03bc)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nC : \u211d\u22650\nhC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\n\u22a2 \u2203 C,\n    0 < C \u2227\n      \u2200 (i : \u03b9),\n        snorm\n            (indicator {x | C \u2264 \u2016Exists.choose (_ : AEStronglyMeasurable (f i) \u03bc) x\u2016\u208a}\n              (Exists.choose (_ : AEStronglyMeasurable (f i) \u03bc)))\n            p \u03bc \u2264\n          ENNReal.ofReal \u03b5"}, {"tactic": "refine' \u27e8max C 1, lt_max_of_lt_right one_pos, fun i => le_trans (snorm_mono fun x => _) (hCg i)\u27e9", "annotated_tactic": ["refine' \u27e8<a>max</a> C 1, <a>lt_max_of_lt_right</a> <a>one_pos</a>, fun i => <a>le_trans</a> (<a>snorm_mono</a> fun x => _) (hCg i)\u27e9", [{"full_name": "Max.max", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1090, 3], "def_end_pos": [1090, 6]}, {"full_name": "lt_max_of_lt_right", "def_path": "Mathlib/Order/MinMax.lean", "def_pos": [102, 9], "def_end_pos": [102, 27]}, {"full_name": "one_pos", "def_path": "Mathlib/Algebra/Order/ZeroLEOne.lean", "def_pos": [50, 7], "def_end_pos": [50, 14]}, {"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}, {"full_name": "MeasureTheory.snorm_mono", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [434, 9], "def_end_pos": [434, 19]}]], "state_before": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng\u271d : \u03b1 \u2192 \u03b2\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\nh : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\ng : \u03b9 \u2192 \u03b1 \u2192 \u03b2 := fun i => Exists.choose (_ : AEStronglyMeasurable (f i) \u03bc)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nC : \u211d\u22650\nhC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nhCg : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016g i x\u2016\u208a} (g i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\n\u22a2 \u2203 C,\n    0 < C \u2227\n      \u2200 (i : \u03b9),\n        snorm\n            (indicator {x | C \u2264 \u2016Exists.choose (_ : AEStronglyMeasurable (f i) \u03bc) x\u2016\u208a}\n              (Exists.choose (_ : AEStronglyMeasurable (f i) \u03bc)))\n            p \u03bc \u2264\n          ENNReal.ofReal \u03b5", "state_after": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng\u271d : \u03b1 \u2192 \u03b2\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\nh : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\ng : \u03b9 \u2192 \u03b1 \u2192 \u03b2 := fun i => Exists.choose (_ : AEStronglyMeasurable (f i) \u03bc)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nC : \u211d\u22650\nhC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nhCg : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016g i x\u2016\u208a} (g i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\ni : \u03b9\nx : \u03b1\n\u22a2 \u2016indicator {x | max C 1 \u2264 \u2016Exists.choose (_ : AEStronglyMeasurable (f i) \u03bc) x\u2016\u208a}\n        (Exists.choose (_ : AEStronglyMeasurable (f i) \u03bc)) x\u2016 \u2264\n    \u2016indicator {x | C \u2264 \u2016g i x\u2016\u208a} (g i) x\u2016"}, {"tactic": "rw [norm_indicator_eq_indicator_norm, norm_indicator_eq_indicator_norm]", "annotated_tactic": ["rw [<a>norm_indicator_eq_indicator_norm</a>, <a>norm_indicator_eq_indicator_norm</a>]", [{"full_name": "norm_indicator_eq_indicator_norm", "def_path": "Mathlib/Analysis/NormedSpace/IndicatorFunction.lean", "def_pos": [25, 9], "def_end_pos": [25, 41]}, {"full_name": "norm_indicator_eq_indicator_norm", "def_path": "Mathlib/Analysis/NormedSpace/IndicatorFunction.lean", "def_pos": [25, 9], "def_end_pos": [25, 41]}]], "state_before": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng\u271d : \u03b1 \u2192 \u03b2\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\nh : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\ng : \u03b9 \u2192 \u03b1 \u2192 \u03b2 := fun i => Exists.choose (_ : AEStronglyMeasurable (f i) \u03bc)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nC : \u211d\u22650\nhC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nhCg : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016g i x\u2016\u208a} (g i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\ni : \u03b9\nx : \u03b1\n\u22a2 \u2016indicator {x | max C 1 \u2264 \u2016Exists.choose (_ : AEStronglyMeasurable (f i) \u03bc) x\u2016\u208a}\n        (Exists.choose (_ : AEStronglyMeasurable (f i) \u03bc)) x\u2016 \u2264\n    \u2016indicator {x | C \u2264 \u2016g i x\u2016\u208a} (g i) x\u2016", "state_after": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng\u271d : \u03b1 \u2192 \u03b2\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\nh : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\ng : \u03b9 \u2192 \u03b1 \u2192 \u03b2 := fun i => Exists.choose (_ : AEStronglyMeasurable (f i) \u03bc)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nC : \u211d\u22650\nhC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nhCg : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016g i x\u2016\u208a} (g i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\ni : \u03b9\nx : \u03b1\n\u22a2 indicator {x | max C 1 \u2264 \u2016Exists.choose (_ : AEStronglyMeasurable (f i) \u03bc) x\u2016\u208a}\n      (fun a => \u2016Exists.choose (_ : AEStronglyMeasurable (f i) \u03bc) a\u2016) x \u2264\n    indicator {x | C \u2264 \u2016g i x\u2016\u208a} (fun a => \u2016g i a\u2016) x"}, {"tactic": "exact Set.indicator_le_indicator_of_subset\n  (fun x hx => Set.mem_setOf_eq \u25b8 le_trans (le_max_left _ _) hx) (fun _ => norm_nonneg _) _", "annotated_tactic": ["exact <a>Set.indicator_le_indicator_of_subset</a>\n    (fun x hx => <a>Set.mem_setOf_eq</a> \u25b8 <a>le_trans</a> (<a>le_max_left</a> _ _) hx) (fun _ => <a>norm_nonneg</a> _) _", [{"full_name": "Set.indicator_le_indicator_of_subset", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [849, 3], "def_end_pos": [849, 14]}, {"full_name": "Set.mem_setOf_eq", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [256, 29], "def_end_pos": [256, 41]}, {"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}, {"full_name": "le_max_left", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [54, 9], "def_end_pos": [54, 20]}, {"full_name": "norm_nonneg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [500, 30], "def_end_pos": [500, 41]}]], "state_before": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng\u271d : \u03b1 \u2192 \u03b2\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\nh : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\ng : \u03b9 \u2192 \u03b1 \u2192 \u03b2 := fun i => Exists.choose (_ : AEStronglyMeasurable (f i) \u03bc)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nC : \u211d\u22650\nhC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nhCg : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016g i x\u2016\u208a} (g i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\ni : \u03b9\nx : \u03b1\n\u22a2 indicator {x | max C 1 \u2264 \u2016Exists.choose (_ : AEStronglyMeasurable (f i) \u03bc) x\u2016\u208a}\n      (fun a => \u2016Exists.choose (_ : AEStronglyMeasurable (f i) \u03bc) a\u2016) x \u2264\n    indicator {x | C \u2264 \u2016g i x\u2016\u208a} (fun a => \u2016g i a\u2016) x", "state_after": "no goals"}, {"tactic": "intro i", "annotated_tactic": ["intro i", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng\u271d : \u03b1 \u2192 \u03b2\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\nh : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\ng : \u03b9 \u2192 \u03b1 \u2192 \u03b2 := fun i => Exists.choose (_ : AEStronglyMeasurable (f i) \u03bc)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nC : \u211d\u22650\nhC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\n\u22a2 \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016g i x\u2016\u208a} (g i)) p \u03bc \u2264 ENNReal.ofReal \u03b5", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng\u271d : \u03b1 \u2192 \u03b2\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\nh : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\ng : \u03b9 \u2192 \u03b1 \u2192 \u03b2 := fun i => Exists.choose (_ : AEStronglyMeasurable (f i) \u03bc)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nC : \u211d\u22650\nhC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\ni : \u03b9\n\u22a2 snorm (indicator {x | C \u2264 \u2016g i x\u2016\u208a} (g i)) p \u03bc \u2264 ENNReal.ofReal \u03b5"}, {"tactic": "refine' le_trans (le_of_eq <| snorm_congr_ae _) (hC i)", "annotated_tactic": ["refine' <a>le_trans</a> (<a>le_of_eq</a> <| <a>snorm_congr_ae</a> _) (hC i)", [{"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}, {"full_name": "le_of_eq", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [72, 9], "def_end_pos": [72, 17]}, {"full_name": "MeasureTheory.snorm_congr_ae", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [549, 9], "def_end_pos": [549, 23]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng\u271d : \u03b1 \u2192 \u03b2\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\nh : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\ng : \u03b9 \u2192 \u03b1 \u2192 \u03b2 := fun i => Exists.choose (_ : AEStronglyMeasurable (f i) \u03bc)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nC : \u211d\u22650\nhC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\ni : \u03b9\n\u22a2 snorm (indicator {x | C \u2264 \u2016g i x\u2016\u208a} (g i)) p \u03bc \u2264 ENNReal.ofReal \u03b5", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng\u271d : \u03b1 \u2192 \u03b2\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\nh : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\ng : \u03b9 \u2192 \u03b1 \u2192 \u03b2 := fun i => Exists.choose (_ : AEStronglyMeasurable (f i) \u03bc)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nC : \u211d\u22650\nhC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\ni : \u03b9\n\u22a2 indicator {x | C \u2264 \u2016g i x\u2016\u208a} (g i) =\u1d50[\u03bc] indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)"}, {"tactic": "filter_upwards [(Exists.choose_spec <| hf i).2] with x hx", "annotated_tactic": ["filter_upwards [(<a>Exists.choose_spec</a> <| hf i).2] with x hx", [{"full_name": "Exists.choose_spec", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [445, 9], "def_end_pos": [445, 27]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng\u271d : \u03b1 \u2192 \u03b2\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\nh : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\ng : \u03b9 \u2192 \u03b1 \u2192 \u03b2 := fun i => Exists.choose (_ : AEStronglyMeasurable (f i) \u03bc)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nC : \u211d\u22650\nhC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\ni : \u03b9\n\u22a2 indicator {x | C \u2264 \u2016g i x\u2016\u208a} (g i) =\u1d50[\u03bc] indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng\u271d : \u03b1 \u2192 \u03b2\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\nh : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\ng : \u03b9 \u2192 \u03b1 \u2192 \u03b2 := fun i => Exists.choose (_ : AEStronglyMeasurable (f i) \u03bc)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nC : \u211d\u22650\nhC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\ni : \u03b9\nx : \u03b1\nhx : f i x = Exists.choose (_ : AEStronglyMeasurable (f i) \u03bc) x\n\u22a2 indicator {x | C \u2264 \u2016g i x\u2016\u208a} (g i) x = indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i) x"}, {"tactic": "by_cases hfx : x \u2208 { x | C \u2264 \u2016f i x\u2016\u208a }", "annotated_tactic": ["by_cases hfx : x \u2208 { x | C \u2264 \u2016f i x\u2016\u208a }", []], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng\u271d : \u03b1 \u2192 \u03b2\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\nh : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\ng : \u03b9 \u2192 \u03b1 \u2192 \u03b2 := fun i => Exists.choose (_ : AEStronglyMeasurable (f i) \u03bc)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nC : \u211d\u22650\nhC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\ni : \u03b9\nx : \u03b1\nhx : f i x = Exists.choose (_ : AEStronglyMeasurable (f i) \u03bc) x\n\u22a2 indicator {x | C \u2264 \u2016g i x\u2016\u208a} (g i) x = indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i) x", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng\u271d : \u03b1 \u2192 \u03b2\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\nh : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\ng : \u03b9 \u2192 \u03b1 \u2192 \u03b2 := fun i => Exists.choose (_ : AEStronglyMeasurable (f i) \u03bc)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nC : \u211d\u22650\nhC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\ni : \u03b9\nx : \u03b1\nhx : f i x = Exists.choose (_ : AEStronglyMeasurable (f i) \u03bc) x\nhfx : x \u2208 {x | C \u2264 \u2016f i x\u2016\u208a}\n\u22a2 indicator {x | C \u2264 \u2016g i x\u2016\u208a} (g i) x = indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i) x\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng\u271d : \u03b1 \u2192 \u03b2\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\nh : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\ng : \u03b9 \u2192 \u03b1 \u2192 \u03b2 := fun i => Exists.choose (_ : AEStronglyMeasurable (f i) \u03bc)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nC : \u211d\u22650\nhC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\ni : \u03b9\nx : \u03b1\nhx : f i x = Exists.choose (_ : AEStronglyMeasurable (f i) \u03bc) x\nhfx : \u00acx \u2208 {x | C \u2264 \u2016f i x\u2016\u208a}\n\u22a2 indicator {x | C \u2264 \u2016g i x\u2016\u208a} (g i) x = indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i) x"}, {"tactic": "rw [Set.indicator_of_mem hfx, Set.indicator_of_mem, hx]", "annotated_tactic": ["rw [<a>Set.indicator_of_mem</a> hfx, <a>Set.indicator_of_mem</a>, hx]", [{"full_name": "Set.indicator_of_mem", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [67, 3], "def_end_pos": [67, 14]}, {"full_name": "Set.indicator_of_mem", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [67, 3], "def_end_pos": [67, 14]}]], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng\u271d : \u03b1 \u2192 \u03b2\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\nh : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\ng : \u03b9 \u2192 \u03b1 \u2192 \u03b2 := fun i => Exists.choose (_ : AEStronglyMeasurable (f i) \u03bc)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nC : \u211d\u22650\nhC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\ni : \u03b9\nx : \u03b1\nhx : f i x = Exists.choose (_ : AEStronglyMeasurable (f i) \u03bc) x\nhfx : x \u2208 {x | C \u2264 \u2016f i x\u2016\u208a}\n\u22a2 indicator {x | C \u2264 \u2016g i x\u2016\u208a} (g i) x = indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i) x", "state_after": "case pos.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng\u271d : \u03b1 \u2192 \u03b2\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\nh : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\ng : \u03b9 \u2192 \u03b1 \u2192 \u03b2 := fun i => Exists.choose (_ : AEStronglyMeasurable (f i) \u03bc)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nC : \u211d\u22650\nhC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\ni : \u03b9\nx : \u03b1\nhx : f i x = Exists.choose (_ : AEStronglyMeasurable (f i) \u03bc) x\nhfx : x \u2208 {x | C \u2264 \u2016f i x\u2016\u208a}\n\u22a2 x \u2208 {x | C \u2264 \u2016g i x\u2016\u208a}"}, {"tactic": "rwa [Set.mem_setOf, hx] at hfx", "annotated_tactic": ["rwa [<a>Set.mem_setOf</a>, hx] at hfx", [{"full_name": "Set.mem_setOf", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [259, 9], "def_end_pos": [259, 18]}]], "state_before": "case pos.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng\u271d : \u03b1 \u2192 \u03b2\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\nh : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\ng : \u03b9 \u2192 \u03b1 \u2192 \u03b2 := fun i => Exists.choose (_ : AEStronglyMeasurable (f i) \u03bc)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nC : \u211d\u22650\nhC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\ni : \u03b9\nx : \u03b1\nhx : f i x = Exists.choose (_ : AEStronglyMeasurable (f i) \u03bc) x\nhfx : x \u2208 {x | C \u2264 \u2016f i x\u2016\u208a}\n\u22a2 x \u2208 {x | C \u2264 \u2016g i x\u2016\u208a}", "state_after": "no goals"}, {"tactic": "rw [Set.indicator_of_not_mem hfx, Set.indicator_of_not_mem]", "annotated_tactic": ["rw [<a>Set.indicator_of_not_mem</a> hfx, <a>Set.indicator_of_not_mem</a>]", [{"full_name": "Set.indicator_of_not_mem", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [73, 3], "def_end_pos": [73, 14]}, {"full_name": "Set.indicator_of_not_mem", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [73, 3], "def_end_pos": [73, 14]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng\u271d : \u03b1 \u2192 \u03b2\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\nh : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\ng : \u03b9 \u2192 \u03b1 \u2192 \u03b2 := fun i => Exists.choose (_ : AEStronglyMeasurable (f i) \u03bc)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nC : \u211d\u22650\nhC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\ni : \u03b9\nx : \u03b1\nhx : f i x = Exists.choose (_ : AEStronglyMeasurable (f i) \u03bc) x\nhfx : \u00acx \u2208 {x | C \u2264 \u2016f i x\u2016\u208a}\n\u22a2 indicator {x | C \u2264 \u2016g i x\u2016\u208a} (g i) x = indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i) x", "state_after": "case neg.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng\u271d : \u03b1 \u2192 \u03b2\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\nh : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\ng : \u03b9 \u2192 \u03b1 \u2192 \u03b2 := fun i => Exists.choose (_ : AEStronglyMeasurable (f i) \u03bc)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nC : \u211d\u22650\nhC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\ni : \u03b9\nx : \u03b1\nhx : f i x = Exists.choose (_ : AEStronglyMeasurable (f i) \u03bc) x\nhfx : \u00acx \u2208 {x | C \u2264 \u2016f i x\u2016\u208a}\n\u22a2 \u00acx \u2208 {x | C \u2264 \u2016g i x\u2016\u208a}"}, {"tactic": "rwa [Set.mem_setOf, hx] at hfx", "annotated_tactic": ["rwa [<a>Set.mem_setOf</a>, hx] at hfx", [{"full_name": "Set.mem_setOf", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [259, 9], "def_end_pos": [259, 18]}]], "state_before": "case neg.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng\u271d : \u03b1 \u2192 \u03b2\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\nh : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\ng : \u03b9 \u2192 \u03b1 \u2192 \u03b2 := fun i => Exists.choose (_ : AEStronglyMeasurable (f i) \u03bc)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nC : \u211d\u22650\nhC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\ni : \u03b9\nx : \u03b1\nhx : f i x = Exists.choose (_ : AEStronglyMeasurable (f i) \u03bc) x\nhfx : \u00acx \u2208 {x | C \u2264 \u2016f i x\u2016\u208a}\n\u22a2 \u00acx \u2208 {x | C \u2264 \u2016g i x\u2016\u208a}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/BinomialHeap/Basic.lean", "full_name": "Std.BinomialHeap.Imp.Heap.realSize_tail", "start": [266, 1], "end": [270, 50], "traced_tactics": [{"tactic": "simp only [Heap.tail]", "annotated_tactic": ["simp only [<a>Heap.tail</a>]", [{"full_name": "Std.BinomialHeap.Imp.Heap.tail", "def_path": "lake-packages/std/Std/Data/BinomialHeap/Basic.lean", "def_pos": [204, 15], "def_end_pos": [204, 24]}]], "state_before": "\u03b1 : Type u_1\nle : \u03b1 \u2192 \u03b1 \u2192 Bool\ns : Heap \u03b1\n\u22a2 realSize (tail le s) = realSize s - 1", "state_after": "\u03b1 : Type u_1\nle : \u03b1 \u2192 \u03b1 \u2192 Bool\ns : Heap \u03b1\n\u22a2 realSize (Option.getD (tail? le s) nil) = realSize s - 1"}, {"tactic": "match eq : s.tail? le with\n| none => cases s with cases eq | nil => rfl\n| some tl => simp [Heap.realSize_tail? eq]; rfl", "annotated_tactic": ["match eq : s.tail? le with\n  | <a>none</a> => cases s with cases eq | <a>nil</a> => rfl\n  | <a>some</a> tl => simp [<a>Heap.realSize_tail?</a> eq]; rfl", [{"full_name": "Option.none", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2141, 5], "def_end_pos": [2141, 9]}, {"full_name": "Std.BinomialHeap.Imp.Heap.nil", "def_path": "lake-packages/std/Std/Data/BinomialHeap/Basic.lean", "def_pos": [74, 5], "def_end_pos": [74, 8]}, {"full_name": "Option.some", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2143, 5], "def_end_pos": [2143, 9]}, {"full_name": "Std.BinomialHeap.Imp.Heap.realSize_tail?", "def_path": "lake-packages/std/Std/Data/BinomialHeap/Basic.lean", "def_pos": [260, 9], "def_end_pos": [260, 28]}]], "state_before": "\u03b1 : Type u_1\nle : \u03b1 \u2192 \u03b1 \u2192 Bool\ns : Heap \u03b1\n\u22a2 realSize (Option.getD (tail? le s) nil) = realSize s - 1", "state_after": "no goals"}, {"tactic": "cases s with cases eq | nil => rfl", "annotated_tactic": ["cases s with cases eq | <a>nil</a> => rfl", [{"full_name": "Std.BinomialHeap.Imp.Heap.nil", "def_path": "lake-packages/std/Std/Data/BinomialHeap/Basic.lean", "def_pos": [74, 5], "def_end_pos": [74, 8]}]], "state_before": "\u03b1 : Type u_1\nle : \u03b1 \u2192 \u03b1 \u2192 Bool\ns : Heap \u03b1\neq : tail? le s = none\n\u22a2 realSize (Option.getD none nil) = realSize s - 1", "state_after": "no goals"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case nil.refl\n\u03b1 : Type u_1\nle : \u03b1 \u2192 \u03b1 \u2192 Bool\n\u22a2 realSize (Option.getD none nil) = realSize nil - 1", "state_after": "no goals"}, {"tactic": "simp [Heap.realSize_tail? eq]", "annotated_tactic": ["simp [<a>Heap.realSize_tail?</a> eq]", [{"full_name": "Std.BinomialHeap.Imp.Heap.realSize_tail?", "def_path": "lake-packages/std/Std/Data/BinomialHeap/Basic.lean", "def_pos": [260, 9], "def_end_pos": [260, 28]}]], "state_before": "\u03b1 : Type u_1\nle : \u03b1 \u2192 \u03b1 \u2192 Bool\ns tl : Heap \u03b1\neq : tail? le s = some tl\n\u22a2 realSize (Option.getD (some tl) nil) = realSize s - 1", "state_after": "\u03b1 : Type u_1\nle : \u03b1 \u2192 \u03b1 \u2192 Bool\ns tl : Heap \u03b1\neq : tail? le s = some tl\n\u22a2 realSize (Option.getD (some tl) nil) = realSize tl"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\u03b1 : Type u_1\nle : \u03b1 \u2192 \u03b1 \u2192 Bool\ns tl : Heap \u03b1\neq : tail? le s = some tl\n\u22a2 realSize (Option.getD (some tl) nil) = realSize tl", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Image.lean", "full_name": "Set.compl_image_set_of", "start": [556, 1], "end": [557, 26], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Martingale/Convergence.lean", "full_name": "MeasureTheory.Submartingale.upcrossings_ae_lt_top'", "start": [157, 1], "end": [185, 74], "traced_tactics": [{"tactic": "refine' ae_lt_top (hf.adapted.measurable_upcrossings hab) _", "annotated_tactic": ["refine' <a>ae_lt_top</a> (hf.adapted.measurable_upcrossings hab) _", [{"full_name": "MeasureTheory.ae_lt_top", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [1522, 9], "def_end_pos": [1522, 18]}]], "state_before": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200 (n : \u2115), snorm (f n) 1 \u03bc \u2264 \u2191R\nhab : a < b\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, upcrossings a b f \u03c9 < \u22a4", "state_after": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200 (n : \u2115), snorm (f n) 1 \u03bc \u2264 \u2191R\nhab : a < b\n\u22a2 \u222b\u207b (x : \u03a9), upcrossings a b f x \u2202\u03bc \u2260 \u22a4"}, {"tactic": "have := hf.mul_lintegral_upcrossings_le_lintegral_pos_part a b", "annotated_tactic": ["have := hf.mul_lintegral_upcrossings_le_lintegral_pos_part a b", []], "state_before": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200 (n : \u2115), snorm (f n) 1 \u03bc \u2264 \u2191R\nhab : a < b\n\u22a2 \u222b\u207b (x : \u03a9), upcrossings a b f x \u2202\u03bc \u2260 \u22a4", "state_after": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200 (n : \u2115), snorm (f n) 1 \u03bc \u2264 \u2191R\nhab : a < b\nthis : ENNReal.ofReal (b - a) * \u222b\u207b (\u03c9 : \u03a9), upcrossings a b f \u03c9 \u2202\u03bc \u2264 \u2a06 N, \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc\n\u22a2 \u222b\u207b (x : \u03a9), upcrossings a b f x \u2202\u03bc \u2260 \u22a4"}, {"tactic": "rw [mul_comm, \u2190 ENNReal.le_div_iff_mul_le] at this", "annotated_tactic": ["rw [<a>mul_comm</a>, \u2190 <a>ENNReal.le_div_iff_mul_le</a>] at this", [{"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [302, 9], "def_end_pos": [302, 17]}, {"full_name": "ENNReal.le_div_iff_mul_le", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1611, 19], "def_end_pos": [1611, 36]}]], "state_before": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200 (n : \u2115), snorm (f n) 1 \u03bc \u2264 \u2191R\nhab : a < b\nthis : ENNReal.ofReal (b - a) * \u222b\u207b (\u03c9 : \u03a9), upcrossings a b f \u03c9 \u2202\u03bc \u2264 \u2a06 N, \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc\n\u22a2 \u222b\u207b (x : \u03a9), upcrossings a b f x \u2202\u03bc \u2260 \u22a4", "state_after": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200 (n : \u2115), snorm (f n) 1 \u03bc \u2264 \u2191R\nhab : a < b\nthis : \u222b\u207b (\u03c9 : \u03a9), upcrossings a b f \u03c9 \u2202\u03bc \u2264 (\u2a06 N, \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc) / ENNReal.ofReal (b - a)\n\u22a2 \u222b\u207b (x : \u03a9), upcrossings a b f x \u2202\u03bc \u2260 \u22a4\n\ncase h0\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200 (n : \u2115), snorm (f n) 1 \u03bc \u2264 \u2191R\nhab : a < b\nthis : (\u222b\u207b (\u03c9 : \u03a9), upcrossings a b f \u03c9 \u2202\u03bc) * ENNReal.ofReal (b - a) \u2264 \u2a06 N, \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc\n\u22a2 ENNReal.ofReal (b - a) \u2260 0 \u2228 \u2a06 N, \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc \u2260 0\n\ncase ht\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200 (n : \u2115), snorm (f n) 1 \u03bc \u2264 \u2191R\nhab : a < b\nthis : (\u222b\u207b (\u03c9 : \u03a9), upcrossings a b f \u03c9 \u2202\u03bc) * ENNReal.ofReal (b - a) \u2264 \u2a06 N, \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc\n\u22a2 ENNReal.ofReal (b - a) \u2260 \u22a4 \u2228 \u2a06 N, \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc \u2260 \u22a4"}, {"tactic": "refine' (lt_of_le_of_lt this (ENNReal.div_lt_top _ _)).ne", "annotated_tactic": ["refine' (<a>lt_of_le_of_lt</a> this (<a>ENNReal.div_lt_top</a> _ _)).<a>ne</a>", [{"full_name": "lt_of_le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [122, 9], "def_end_pos": [122, 23]}, {"full_name": "ENNReal.div_lt_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1460, 9], "def_end_pos": [1460, 19]}, {"full_name": "LT.lt.ne", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [152, 7], "def_end_pos": [152, 15]}]], "state_before": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200 (n : \u2115), snorm (f n) 1 \u03bc \u2264 \u2191R\nhab : a < b\nthis : \u222b\u207b (\u03c9 : \u03a9), upcrossings a b f \u03c9 \u2202\u03bc \u2264 (\u2a06 N, \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc) / ENNReal.ofReal (b - a)\n\u22a2 \u222b\u207b (x : \u03a9), upcrossings a b f x \u2202\u03bc \u2260 \u22a4", "state_after": "case refine'_1\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200 (n : \u2115), snorm (f n) 1 \u03bc \u2264 \u2191R\nhab : a < b\nthis : \u222b\u207b (\u03c9 : \u03a9), upcrossings a b f \u03c9 \u2202\u03bc \u2264 (\u2a06 N, \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc) / ENNReal.ofReal (b - a)\n\u22a2 \u2a06 N, \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc \u2260 \u22a4\n\ncase refine'_2\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200 (n : \u2115), snorm (f n) 1 \u03bc \u2264 \u2191R\nhab : a < b\nthis : \u222b\u207b (\u03c9 : \u03a9), upcrossings a b f \u03c9 \u2202\u03bc \u2264 (\u2a06 N, \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc) / ENNReal.ofReal (b - a)\n\u22a2 ENNReal.ofReal (b - a) \u2260 0"}, {"tactic": "refine' ne_of_lt (iSup_lt_iff.2 \u27e8R + \u2016a\u2016\u208a * \u03bc Set.univ, ENNReal.add_lt_top.2\n  \u27e8ENNReal.coe_lt_top, ENNReal.mul_lt_top ENNReal.coe_lt_top.ne (measure_ne_top _ _)\u27e9,\n  fun n => le_trans _ (hR' n)\u27e9)", "annotated_tactic": ["refine' <a>ne_of_lt</a> (<a>iSup_lt_iff</a>.2 \u27e8R + \u2016a\u2016\u208a * \u03bc <a>Set.univ</a>, <a>ENNReal.add_lt_top</a>.2\n        \u27e8<a>ENNReal.coe_lt_top</a>, <a>ENNReal.mul_lt_top</a> ENNReal.coe_lt_top.ne (<a>measure_ne_top</a> _ _)\u27e9,\n        fun n => <a>le_trans</a> _ (hR' n)\u27e9)", [{"full_name": "ne_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [101, 9], "def_end_pos": [101, 17]}, {"full_name": "iSup_lt_iff", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [981, 9], "def_end_pos": [981, 20]}, {"full_name": "Set.univ", "def_path": "Mathlib/Init/Set.lean", "def_pos": [90, 5], "def_end_pos": [90, 9]}, {"full_name": "ENNReal.add_lt_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [561, 17], "def_end_pos": [561, 27]}, {"full_name": "ENNReal.coe_lt_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [308, 17], "def_end_pos": [308, 27]}, {"full_name": "ENNReal.mul_lt_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [612, 9], "def_end_pos": [612, 19]}, {"full_name": "MeasureTheory.measure_ne_top", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2875, 9], "def_end_pos": [2875, 23]}, {"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}]], "state_before": "case refine'_1\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200 (n : \u2115), snorm (f n) 1 \u03bc \u2264 \u2191R\nhab : a < b\nthis : \u222b\u207b (\u03c9 : \u03a9), upcrossings a b f \u03c9 \u2202\u03bc \u2264 (\u2a06 N, \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc) / ENNReal.ofReal (b - a)\nhR' : \u2200 (n : \u2115), \u222b\u207b (\u03c9 : \u03a9), \u2191\u2016f n \u03c9 - a\u2016\u208a \u2202\u03bc \u2264 \u2191R + \u2191\u2016a\u2016\u208a * \u2191\u2191\u03bc Set.univ\n\u22a2 \u2a06 N, \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc \u2260 \u22a4", "state_after": "case refine'_1\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200 (n : \u2115), snorm (f n) 1 \u03bc \u2264 \u2191R\nhab : a < b\nthis : \u222b\u207b (\u03c9 : \u03a9), upcrossings a b f \u03c9 \u2202\u03bc \u2264 (\u2a06 N, \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc) / ENNReal.ofReal (b - a)\nhR' : \u2200 (n : \u2115), \u222b\u207b (\u03c9 : \u03a9), \u2191\u2016f n \u03c9 - a\u2016\u208a \u2202\u03bc \u2264 \u2191R + \u2191\u2016a\u2016\u208a * \u2191\u2191\u03bc Set.univ\nn : \u2115\n\u22a2 \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f n \u03c9 - a)\u207a \u2202\u03bc \u2264 \u222b\u207b (\u03c9 : \u03a9), \u2191\u2016f n \u03c9 - a\u2016\u208a \u2202\u03bc"}, {"tactic": "refine' lintegral_mono fun \u03c9 => _", "annotated_tactic": ["refine' <a>lintegral_mono</a> fun \u03c9 => _", [{"full_name": "MeasureTheory.lintegral_mono", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [99, 9], "def_end_pos": [99, 23]}]], "state_before": "case refine'_1\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200 (n : \u2115), snorm (f n) 1 \u03bc \u2264 \u2191R\nhab : a < b\nthis : \u222b\u207b (\u03c9 : \u03a9), upcrossings a b f \u03c9 \u2202\u03bc \u2264 (\u2a06 N, \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc) / ENNReal.ofReal (b - a)\nhR' : \u2200 (n : \u2115), \u222b\u207b (\u03c9 : \u03a9), \u2191\u2016f n \u03c9 - a\u2016\u208a \u2202\u03bc \u2264 \u2191R + \u2191\u2016a\u2016\u208a * \u2191\u2191\u03bc Set.univ\nn : \u2115\n\u22a2 \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f n \u03c9 - a)\u207a \u2202\u03bc \u2264 \u222b\u207b (\u03c9 : \u03a9), \u2191\u2016f n \u03c9 - a\u2016\u208a \u2202\u03bc", "state_after": "case refine'_1\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200 (n : \u2115), snorm (f n) 1 \u03bc \u2264 \u2191R\nhab : a < b\nthis : \u222b\u207b (\u03c9 : \u03a9), upcrossings a b f \u03c9 \u2202\u03bc \u2264 (\u2a06 N, \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc) / ENNReal.ofReal (b - a)\nhR' : \u2200 (n : \u2115), \u222b\u207b (\u03c9 : \u03a9), \u2191\u2016f n \u03c9 - a\u2016\u208a \u2202\u03bc \u2264 \u2191R + \u2191\u2016a\u2016\u208a * \u2191\u2191\u03bc Set.univ\nn : \u2115\n\u03c9 : \u03a9\n\u22a2 ENNReal.ofReal (f n \u03c9 - a)\u207a \u2264 \u2191\u2016f n \u03c9 - a\u2016\u208a"}, {"tactic": "rw [ENNReal.ofReal_le_iff_le_toReal, ENNReal.coe_toReal, coe_nnnorm]", "annotated_tactic": ["rw [<a>ENNReal.ofReal_le_iff_le_toReal</a>, <a>ENNReal.coe_toReal</a>, <a>coe_nnnorm</a>]", [{"full_name": "ENNReal.ofReal_le_iff_le_toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2189, 9], "def_end_pos": [2189, 32]}, {"full_name": "ENNReal.coe_toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [236, 17], "def_end_pos": [236, 27]}, {"full_name": "coe_nnnorm", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [905, 41], "def_end_pos": [905, 51]}]], "state_before": "case refine'_1\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200 (n : \u2115), snorm (f n) 1 \u03bc \u2264 \u2191R\nhab : a < b\nthis : \u222b\u207b (\u03c9 : \u03a9), upcrossings a b f \u03c9 \u2202\u03bc \u2264 (\u2a06 N, \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc) / ENNReal.ofReal (b - a)\nhR' : \u2200 (n : \u2115), \u222b\u207b (\u03c9 : \u03a9), \u2191\u2016f n \u03c9 - a\u2016\u208a \u2202\u03bc \u2264 \u2191R + \u2191\u2016a\u2016\u208a * \u2191\u2191\u03bc Set.univ\nn : \u2115\n\u03c9 : \u03a9\n\u22a2 ENNReal.ofReal (f n \u03c9 - a)\u207a \u2264 \u2191\u2016f n \u03c9 - a\u2016\u208a", "state_after": "case refine'_1\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200 (n : \u2115), snorm (f n) 1 \u03bc \u2264 \u2191R\nhab : a < b\nthis : \u222b\u207b (\u03c9 : \u03a9), upcrossings a b f \u03c9 \u2202\u03bc \u2264 (\u2a06 N, \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc) / ENNReal.ofReal (b - a)\nhR' : \u2200 (n : \u2115), \u222b\u207b (\u03c9 : \u03a9), \u2191\u2016f n \u03c9 - a\u2016\u208a \u2202\u03bc \u2264 \u2191R + \u2191\u2016a\u2016\u208a * \u2191\u2191\u03bc Set.univ\nn : \u2115\n\u03c9 : \u03a9\n\u22a2 (f n \u03c9 - a)\u207a \u2264 \u2016f n \u03c9 - a\u2016\n\ncase refine'_1\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200 (n : \u2115), snorm (f n) 1 \u03bc \u2264 \u2191R\nhab : a < b\nthis : \u222b\u207b (\u03c9 : \u03a9), upcrossings a b f \u03c9 \u2202\u03bc \u2264 (\u2a06 N, \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc) / ENNReal.ofReal (b - a)\nhR' : \u2200 (n : \u2115), \u222b\u207b (\u03c9 : \u03a9), \u2191\u2016f n \u03c9 - a\u2016\u208a \u2202\u03bc \u2264 \u2191R + \u2191\u2016a\u2016\u208a * \u2191\u2191\u03bc Set.univ\nn : \u2115\n\u03c9 : \u03a9\n\u22a2 \u2191\u2016f n \u03c9 - a\u2016\u208a \u2260 \u22a4"}, {"tactic": "by_cases hnonneg : 0 \u2264 f n \u03c9 - a", "annotated_tactic": ["by_cases hnonneg : 0 \u2264 f n \u03c9 - a", []], "state_before": "case refine'_1\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200 (n : \u2115), snorm (f n) 1 \u03bc \u2264 \u2191R\nhab : a < b\nthis : \u222b\u207b (\u03c9 : \u03a9), upcrossings a b f \u03c9 \u2202\u03bc \u2264 (\u2a06 N, \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc) / ENNReal.ofReal (b - a)\nhR' : \u2200 (n : \u2115), \u222b\u207b (\u03c9 : \u03a9), \u2191\u2016f n \u03c9 - a\u2016\u208a \u2202\u03bc \u2264 \u2191R + \u2191\u2016a\u2016\u208a * \u2191\u2191\u03bc Set.univ\nn : \u2115\n\u03c9 : \u03a9\n\u22a2 (f n \u03c9 - a)\u207a \u2264 \u2016f n \u03c9 - a\u2016\n\ncase refine'_1\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200 (n : \u2115), snorm (f n) 1 \u03bc \u2264 \u2191R\nhab : a < b\nthis : \u222b\u207b (\u03c9 : \u03a9), upcrossings a b f \u03c9 \u2202\u03bc \u2264 (\u2a06 N, \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc) / ENNReal.ofReal (b - a)\nhR' : \u2200 (n : \u2115), \u222b\u207b (\u03c9 : \u03a9), \u2191\u2016f n \u03c9 - a\u2016\u208a \u2202\u03bc \u2264 \u2191R + \u2191\u2016a\u2016\u208a * \u2191\u2191\u03bc Set.univ\nn : \u2115\n\u03c9 : \u03a9\n\u22a2 \u2191\u2016f n \u03c9 - a\u2016\u208a \u2260 \u22a4", "state_after": "case pos\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200 (n : \u2115), snorm (f n) 1 \u03bc \u2264 \u2191R\nhab : a < b\nthis : \u222b\u207b (\u03c9 : \u03a9), upcrossings a b f \u03c9 \u2202\u03bc \u2264 (\u2a06 N, \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc) / ENNReal.ofReal (b - a)\nhR' : \u2200 (n : \u2115), \u222b\u207b (\u03c9 : \u03a9), \u2191\u2016f n \u03c9 - a\u2016\u208a \u2202\u03bc \u2264 \u2191R + \u2191\u2016a\u2016\u208a * \u2191\u2191\u03bc Set.univ\nn : \u2115\n\u03c9 : \u03a9\nhnonneg : 0 \u2264 f n \u03c9 - a\n\u22a2 (f n \u03c9 - a)\u207a \u2264 \u2016f n \u03c9 - a\u2016\n\ncase neg\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200 (n : \u2115), snorm (f n) 1 \u03bc \u2264 \u2191R\nhab : a < b\nthis : \u222b\u207b (\u03c9 : \u03a9), upcrossings a b f \u03c9 \u2202\u03bc \u2264 (\u2a06 N, \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc) / ENNReal.ofReal (b - a)\nhR' : \u2200 (n : \u2115), \u222b\u207b (\u03c9 : \u03a9), \u2191\u2016f n \u03c9 - a\u2016\u208a \u2202\u03bc \u2264 \u2191R + \u2191\u2016a\u2016\u208a * \u2191\u2191\u03bc Set.univ\nn : \u2115\n\u03c9 : \u03a9\nhnonneg : \u00ac0 \u2264 f n \u03c9 - a\n\u22a2 (f n \u03c9 - a)\u207a \u2264 \u2016f n \u03c9 - a\u2016\n\ncase refine'_1\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200 (n : \u2115), snorm (f n) 1 \u03bc \u2264 \u2191R\nhab : a < b\nthis : \u222b\u207b (\u03c9 : \u03a9), upcrossings a b f \u03c9 \u2202\u03bc \u2264 (\u2a06 N, \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc) / ENNReal.ofReal (b - a)\nhR' : \u2200 (n : \u2115), \u222b\u207b (\u03c9 : \u03a9), \u2191\u2016f n \u03c9 - a\u2016\u208a \u2202\u03bc \u2264 \u2191R + \u2191\u2016a\u2016\u208a * \u2191\u2191\u03bc Set.univ\nn : \u2115\n\u03c9 : \u03a9\n\u22a2 \u2191\u2016f n \u03c9 - a\u2016\u208a \u2260 \u22a4"}, {"tactic": "simp_rw [snorm_one_eq_lintegral_nnnorm] at hbdd", "annotated_tactic": ["simp_rw [<a>snorm_one_eq_lintegral_nnnorm</a>] at hbdd", [{"full_name": "MeasureTheory.snorm_one_eq_lintegral_nnnorm", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [97, 9], "def_end_pos": [97, 38]}]], "state_before": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200 (n : \u2115), snorm (f n) 1 \u03bc \u2264 \u2191R\nhab : a < b\nthis : \u222b\u207b (\u03c9 : \u03a9), upcrossings a b f \u03c9 \u2202\u03bc \u2264 (\u2a06 N, \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc) / ENNReal.ofReal (b - a)\n\u22a2 \u2200 (n : \u2115), \u222b\u207b (\u03c9 : \u03a9), \u2191\u2016f n \u03c9 - a\u2016\u208a \u2202\u03bc \u2264 \u2191R + \u2191\u2016a\u2016\u208a * \u2191\u2191\u03bc Set.univ", "state_after": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhab : a < b\nthis : \u222b\u207b (\u03c9 : \u03a9), upcrossings a b f \u03c9 \u2202\u03bc \u2264 (\u2a06 N, \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc) / ENNReal.ofReal (b - a)\nhbdd : \u2200 (n : \u2115), \u222b\u207b (x : \u03a9), \u2191\u2016f n x\u2016\u208a \u2202\u03bc \u2264 \u2191R\n\u22a2 \u2200 (n : \u2115), \u222b\u207b (\u03c9 : \u03a9), \u2191\u2016f n \u03c9 - a\u2016\u208a \u2202\u03bc \u2264 \u2191R + \u2191\u2016a\u2016\u208a * \u2191\u2191\u03bc Set.univ"}, {"tactic": "intro n", "annotated_tactic": ["intro n", []], "state_before": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhab : a < b\nthis : \u222b\u207b (\u03c9 : \u03a9), upcrossings a b f \u03c9 \u2202\u03bc \u2264 (\u2a06 N, \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc) / ENNReal.ofReal (b - a)\nhbdd : \u2200 (n : \u2115), \u222b\u207b (x : \u03a9), \u2191\u2016f n x\u2016\u208a \u2202\u03bc \u2264 \u2191R\n\u22a2 \u2200 (n : \u2115), \u222b\u207b (\u03c9 : \u03a9), \u2191\u2016f n \u03c9 - a\u2016\u208a \u2202\u03bc \u2264 \u2191R + \u2191\u2016a\u2016\u208a * \u2191\u2191\u03bc Set.univ", "state_after": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhab : a < b\nthis : \u222b\u207b (\u03c9 : \u03a9), upcrossings a b f \u03c9 \u2202\u03bc \u2264 (\u2a06 N, \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc) / ENNReal.ofReal (b - a)\nhbdd : \u2200 (n : \u2115), \u222b\u207b (x : \u03a9), \u2191\u2016f n x\u2016\u208a \u2202\u03bc \u2264 \u2191R\nn : \u2115\n\u22a2 \u222b\u207b (\u03c9 : \u03a9), \u2191\u2016f n \u03c9 - a\u2016\u208a \u2202\u03bc \u2264 \u2191R + \u2191\u2016a\u2016\u208a * \u2191\u2191\u03bc Set.univ"}, {"tactic": "refine' (lintegral_mono _ : \u222b\u207b \u03c9, \u2016f n \u03c9 - a\u2016\u208a \u2202\u03bc \u2264 \u222b\u207b \u03c9, \u2016f n \u03c9\u2016\u208a + \u2016a\u2016\u208a \u2202\u03bc).trans _", "annotated_tactic": ["refine' (<a>lintegral_mono</a> _ : \u222b\u207b \u03c9, \u2016f n \u03c9 - a\u2016\u208a \u2202\u03bc \u2264 \u222b\u207b \u03c9, \u2016f n \u03c9\u2016\u208a + \u2016a\u2016\u208a \u2202\u03bc).<a>trans</a> _", [{"full_name": "MeasureTheory.lintegral_mono", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [99, 9], "def_end_pos": [99, 23]}, {"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [120, 7], "def_end_pos": [120, 18]}]], "state_before": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhab : a < b\nthis : \u222b\u207b (\u03c9 : \u03a9), upcrossings a b f \u03c9 \u2202\u03bc \u2264 (\u2a06 N, \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc) / ENNReal.ofReal (b - a)\nhbdd : \u2200 (n : \u2115), \u222b\u207b (x : \u03a9), \u2191\u2016f n x\u2016\u208a \u2202\u03bc \u2264 \u2191R\nn : \u2115\n\u22a2 \u222b\u207b (\u03c9 : \u03a9), \u2191\u2016f n \u03c9 - a\u2016\u208a \u2202\u03bc \u2264 \u2191R + \u2191\u2016a\u2016\u208a * \u2191\u2191\u03bc Set.univ", "state_after": "case refine'_1\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhab : a < b\nthis : \u222b\u207b (\u03c9 : \u03a9), upcrossings a b f \u03c9 \u2202\u03bc \u2264 (\u2a06 N, \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc) / ENNReal.ofReal (b - a)\nhbdd : \u2200 (n : \u2115), \u222b\u207b (x : \u03a9), \u2191\u2016f n x\u2016\u208a \u2202\u03bc \u2264 \u2191R\nn : \u2115\n\u22a2 (fun \u03c9 => \u2191\u2016f n \u03c9 - a\u2016\u208a) \u2264 fun \u03c9 => \u2191\u2016f n \u03c9\u2016\u208a + \u2191\u2016a\u2016\u208a\n\ncase refine'_2\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhab : a < b\nthis : \u222b\u207b (\u03c9 : \u03a9), upcrossings a b f \u03c9 \u2202\u03bc \u2264 (\u2a06 N, \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc) / ENNReal.ofReal (b - a)\nhbdd : \u2200 (n : \u2115), \u222b\u207b (x : \u03a9), \u2191\u2016f n x\u2016\u208a \u2202\u03bc \u2264 \u2191R\nn : \u2115\n\u22a2 \u222b\u207b (a_1 : \u03a9), \u2191\u2016f n a_1\u2016\u208a + \u2191\u2016a\u2016\u208a \u2202\u03bc \u2264 \u2191R + \u2191\u2016a\u2016\u208a * \u2191\u2191\u03bc Set.univ"}, {"tactic": "intro \u03c9", "annotated_tactic": ["intro \u03c9", []], "state_before": "case refine'_1\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhab : a < b\nthis : \u222b\u207b (\u03c9 : \u03a9), upcrossings a b f \u03c9 \u2202\u03bc \u2264 (\u2a06 N, \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc) / ENNReal.ofReal (b - a)\nhbdd : \u2200 (n : \u2115), \u222b\u207b (x : \u03a9), \u2191\u2016f n x\u2016\u208a \u2202\u03bc \u2264 \u2191R\nn : \u2115\n\u22a2 (fun \u03c9 => \u2191\u2016f n \u03c9 - a\u2016\u208a) \u2264 fun \u03c9 => \u2191\u2016f n \u03c9\u2016\u208a + \u2191\u2016a\u2016\u208a", "state_after": "case refine'_1\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhab : a < b\nthis : \u222b\u207b (\u03c9 : \u03a9), upcrossings a b f \u03c9 \u2202\u03bc \u2264 (\u2a06 N, \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc) / ENNReal.ofReal (b - a)\nhbdd : \u2200 (n : \u2115), \u222b\u207b (x : \u03a9), \u2191\u2016f n x\u2016\u208a \u2202\u03bc \u2264 \u2191R\nn : \u2115\n\u03c9 : \u03a9\n\u22a2 (fun \u03c9 => \u2191\u2016f n \u03c9 - a\u2016\u208a) \u03c9 \u2264 (fun \u03c9 => \u2191\u2016f n \u03c9\u2016\u208a + \u2191\u2016a\u2016\u208a) \u03c9"}, {"tactic": "simp_rw [sub_eq_add_neg, \u2190 nnnorm_neg a, \u2190 ENNReal.coe_add, ENNReal.coe_le_coe]", "annotated_tactic": ["simp_rw [<a>sub_eq_add_neg</a>, \u2190 <a>nnnorm_neg</a> a, \u2190 <a>ENNReal.coe_add</a>, <a>ENNReal.coe_le_coe</a>]", [{"full_name": "sub_eq_add_neg", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [975, 3], "def_end_pos": [975, 14]}, {"full_name": "nnnorm_neg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [952, 30], "def_end_pos": [952, 40]}, {"full_name": "ENNReal.coe_add", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [386, 28], "def_end_pos": [386, 35]}, {"full_name": "ENNReal.coe_le_coe", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [349, 28], "def_end_pos": [349, 38]}]], "state_before": "case refine'_1\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhab : a < b\nthis : \u222b\u207b (\u03c9 : \u03a9), upcrossings a b f \u03c9 \u2202\u03bc \u2264 (\u2a06 N, \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc) / ENNReal.ofReal (b - a)\nhbdd : \u2200 (n : \u2115), \u222b\u207b (x : \u03a9), \u2191\u2016f n x\u2016\u208a \u2202\u03bc \u2264 \u2191R\nn : \u2115\n\u03c9 : \u03a9\n\u22a2 (fun \u03c9 => \u2191\u2016f n \u03c9 - a\u2016\u208a) \u03c9 \u2264 (fun \u03c9 => \u2191\u2016f n \u03c9\u2016\u208a + \u2191\u2016a\u2016\u208a) \u03c9", "state_after": "case refine'_1\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhab : a < b\nthis : \u222b\u207b (\u03c9 : \u03a9), upcrossings a b f \u03c9 \u2202\u03bc \u2264 (\u2a06 N, \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc) / ENNReal.ofReal (b - a)\nhbdd : \u2200 (n : \u2115), \u222b\u207b (x : \u03a9), \u2191\u2016f n x\u2016\u208a \u2202\u03bc \u2264 \u2191R\nn : \u2115\n\u03c9 : \u03a9\n\u22a2 \u2016f n \u03c9 + -a\u2016\u208a \u2264 \u2016f n \u03c9\u2016\u208a + \u2016-a\u2016\u208a"}, {"tactic": "exact nnnorm_add_le _ _", "annotated_tactic": ["exact <a>nnnorm_add_le</a> _ _", [{"full_name": "nnnorm_add_le", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [946, 15], "def_end_pos": [946, 28]}]], "state_before": "case refine'_1\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhab : a < b\nthis : \u222b\u207b (\u03c9 : \u03a9), upcrossings a b f \u03c9 \u2202\u03bc \u2264 (\u2a06 N, \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc) / ENNReal.ofReal (b - a)\nhbdd : \u2200 (n : \u2115), \u222b\u207b (x : \u03a9), \u2191\u2016f n x\u2016\u208a \u2202\u03bc \u2264 \u2191R\nn : \u2115\n\u03c9 : \u03a9\n\u22a2 \u2016f n \u03c9 + -a\u2016\u208a \u2264 \u2016f n \u03c9\u2016\u208a + \u2016-a\u2016\u208a", "state_after": "no goals"}, {"tactic": "simp_rw [lintegral_add_right _ measurable_const, lintegral_const]", "annotated_tactic": ["simp_rw [<a>lintegral_add_right</a> _ <a>measurable_const</a>, <a>lintegral_const</a>]", [{"full_name": "MeasureTheory.lintegral_add_right", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [581, 9], "def_end_pos": [581, 28]}, {"full_name": "measurable_const", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [570, 9], "def_end_pos": [570, 25]}, {"full_name": "MeasureTheory.lintegral_const", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [136, 9], "def_end_pos": [136, 24]}]], "state_before": "case refine'_2\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhab : a < b\nthis : \u222b\u207b (\u03c9 : \u03a9), upcrossings a b f \u03c9 \u2202\u03bc \u2264 (\u2a06 N, \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc) / ENNReal.ofReal (b - a)\nhbdd : \u2200 (n : \u2115), \u222b\u207b (x : \u03a9), \u2191\u2016f n x\u2016\u208a \u2202\u03bc \u2264 \u2191R\nn : \u2115\n\u22a2 \u222b\u207b (a_1 : \u03a9), \u2191\u2016f n a_1\u2016\u208a + \u2191\u2016a\u2016\u208a \u2202\u03bc \u2264 \u2191R + \u2191\u2016a\u2016\u208a * \u2191\u2191\u03bc Set.univ", "state_after": "case refine'_2\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhab : a < b\nthis : \u222b\u207b (\u03c9 : \u03a9), upcrossings a b f \u03c9 \u2202\u03bc \u2264 (\u2a06 N, \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc) / ENNReal.ofReal (b - a)\nhbdd : \u2200 (n : \u2115), \u222b\u207b (x : \u03a9), \u2191\u2016f n x\u2016\u208a \u2202\u03bc \u2264 \u2191R\nn : \u2115\n\u22a2 \u222b\u207b (a : \u03a9), \u2191\u2016f n a\u2016\u208a \u2202\u03bc + \u2191\u2016a\u2016\u208a * \u2191\u2191\u03bc Set.univ \u2264 \u2191R + \u2191\u2016a\u2016\u208a * \u2191\u2191\u03bc Set.univ"}, {"tactic": "exact add_le_add (hbdd _) le_rfl", "annotated_tactic": ["exact <a>add_le_add</a> (hbdd _) <a>le_rfl</a>", [{"full_name": "add_le_add", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [205, 15], "def_end_pos": [205, 25]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}]], "state_before": "case refine'_2\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhab : a < b\nthis : \u222b\u207b (\u03c9 : \u03a9), upcrossings a b f \u03c9 \u2202\u03bc \u2264 (\u2a06 N, \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc) / ENNReal.ofReal (b - a)\nhbdd : \u2200 (n : \u2115), \u222b\u207b (x : \u03a9), \u2191\u2016f n x\u2016\u208a \u2202\u03bc \u2264 \u2191R\nn : \u2115\n\u22a2 \u222b\u207b (a : \u03a9), \u2191\u2016f n a\u2016\u208a \u2202\u03bc + \u2191\u2016a\u2016\u208a * \u2191\u2191\u03bc Set.univ \u2264 \u2191R + \u2191\u2016a\u2016\u208a * \u2191\u2191\u03bc Set.univ", "state_after": "no goals"}, {"tactic": "rw [LatticeOrderedGroup.pos_of_nonneg _ hnonneg, Real.norm_eq_abs,\n  abs_of_nonneg hnonneg]", "annotated_tactic": ["rw [<a>LatticeOrderedGroup.pos_of_nonneg</a> _ hnonneg, <a>Real.norm_eq_abs</a>,\n          <a>abs_of_nonneg</a> hnonneg]", [{"full_name": "LatticeOrderedGroup.pos_of_nonneg", "def_path": "Mathlib/Algebra/Order/LatticeGroup.lean", "def_pos": [299, 3], "def_end_pos": [299, 14]}, {"full_name": "Real.norm_eq_abs", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [1761, 9], "def_end_pos": [1761, 20]}, {"full_name": "abs_of_nonneg", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [107, 9], "def_end_pos": [107, 22]}]], "state_before": "case pos\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200 (n : \u2115), snorm (f n) 1 \u03bc \u2264 \u2191R\nhab : a < b\nthis : \u222b\u207b (\u03c9 : \u03a9), upcrossings a b f \u03c9 \u2202\u03bc \u2264 (\u2a06 N, \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc) / ENNReal.ofReal (b - a)\nhR' : \u2200 (n : \u2115), \u222b\u207b (\u03c9 : \u03a9), \u2191\u2016f n \u03c9 - a\u2016\u208a \u2202\u03bc \u2264 \u2191R + \u2191\u2016a\u2016\u208a * \u2191\u2191\u03bc Set.univ\nn : \u2115\n\u03c9 : \u03a9\nhnonneg : 0 \u2264 f n \u03c9 - a\n\u22a2 (f n \u03c9 - a)\u207a \u2264 \u2016f n \u03c9 - a\u2016", "state_after": "no goals"}, {"tactic": "rw [LatticeOrderedGroup.pos_of_nonpos _ (not_le.1 hnonneg).le]", "annotated_tactic": ["rw [<a>LatticeOrderedGroup.pos_of_nonpos</a> _ (<a>not_le</a>.1 hnonneg).<a>le</a>]", [{"full_name": "LatticeOrderedGroup.pos_of_nonpos", "def_path": "Mathlib/Algebra/Order/LatticeGroup.lean", "def_pos": [308, 3], "def_end_pos": [308, 14]}, {"full_name": "not_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [373, 9], "def_end_pos": [373, 15]}, {"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [142, 7], "def_end_pos": [142, 15]}]], "state_before": "case neg\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200 (n : \u2115), snorm (f n) 1 \u03bc \u2264 \u2191R\nhab : a < b\nthis : \u222b\u207b (\u03c9 : \u03a9), upcrossings a b f \u03c9 \u2202\u03bc \u2264 (\u2a06 N, \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc) / ENNReal.ofReal (b - a)\nhR' : \u2200 (n : \u2115), \u222b\u207b (\u03c9 : \u03a9), \u2191\u2016f n \u03c9 - a\u2016\u208a \u2202\u03bc \u2264 \u2191R + \u2191\u2016a\u2016\u208a * \u2191\u2191\u03bc Set.univ\nn : \u2115\n\u03c9 : \u03a9\nhnonneg : \u00ac0 \u2264 f n \u03c9 - a\n\u22a2 (f n \u03c9 - a)\u207a \u2264 \u2016f n \u03c9 - a\u2016", "state_after": "case neg\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200 (n : \u2115), snorm (f n) 1 \u03bc \u2264 \u2191R\nhab : a < b\nthis : \u222b\u207b (\u03c9 : \u03a9), upcrossings a b f \u03c9 \u2202\u03bc \u2264 (\u2a06 N, \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc) / ENNReal.ofReal (b - a)\nhR' : \u2200 (n : \u2115), \u222b\u207b (\u03c9 : \u03a9), \u2191\u2016f n \u03c9 - a\u2016\u208a \u2202\u03bc \u2264 \u2191R + \u2191\u2016a\u2016\u208a * \u2191\u2191\u03bc Set.univ\nn : \u2115\n\u03c9 : \u03a9\nhnonneg : \u00ac0 \u2264 f n \u03c9 - a\n\u22a2 0 \u2264 \u2016f n \u03c9 - a\u2016"}, {"tactic": "exact norm_nonneg _", "annotated_tactic": ["exact <a>norm_nonneg</a> _", [{"full_name": "norm_nonneg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [500, 30], "def_end_pos": [500, 41]}]], "state_before": "case neg\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200 (n : \u2115), snorm (f n) 1 \u03bc \u2264 \u2191R\nhab : a < b\nthis : \u222b\u207b (\u03c9 : \u03a9), upcrossings a b f \u03c9 \u2202\u03bc \u2264 (\u2a06 N, \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc) / ENNReal.ofReal (b - a)\nhR' : \u2200 (n : \u2115), \u222b\u207b (\u03c9 : \u03a9), \u2191\u2016f n \u03c9 - a\u2016\u208a \u2202\u03bc \u2264 \u2191R + \u2191\u2016a\u2016\u208a * \u2191\u2191\u03bc Set.univ\nn : \u2115\n\u03c9 : \u03a9\nhnonneg : \u00ac0 \u2264 f n \u03c9 - a\n\u22a2 0 \u2264 \u2016f n \u03c9 - a\u2016", "state_after": "no goals"}, {"tactic": "simp only [Ne.def, ENNReal.coe_ne_top, not_false_iff]", "annotated_tactic": ["simp only [<a>Ne.def</a>, <a>ENNReal.coe_ne_top</a>, <a>not_false_iff</a>]", [{"full_name": "Ne.def", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [59, 9], "def_end_pos": [59, 15]}, {"full_name": "ENNReal.coe_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [302, 17], "def_end_pos": [302, 27]}, {"full_name": "not_false_iff", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [82, 9], "def_end_pos": [82, 22]}]], "state_before": "case refine'_1\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200 (n : \u2115), snorm (f n) 1 \u03bc \u2264 \u2191R\nhab : a < b\nthis : \u222b\u207b (\u03c9 : \u03a9), upcrossings a b f \u03c9 \u2202\u03bc \u2264 (\u2a06 N, \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc) / ENNReal.ofReal (b - a)\nhR' : \u2200 (n : \u2115), \u222b\u207b (\u03c9 : \u03a9), \u2191\u2016f n \u03c9 - a\u2016\u208a \u2202\u03bc \u2264 \u2191R + \u2191\u2016a\u2016\u208a * \u2191\u2191\u03bc Set.univ\nn : \u2115\n\u03c9 : \u03a9\n\u22a2 \u2191\u2016f n \u03c9 - a\u2016\u208a \u2260 \u22a4", "state_after": "no goals"}, {"tactic": "simp only [hab, Ne.def, ENNReal.ofReal_eq_zero, sub_nonpos, not_le]", "annotated_tactic": ["simp only [hab, <a>Ne.def</a>, <a>ENNReal.ofReal_eq_zero</a>, <a>sub_nonpos</a>, <a>not_le</a>]", [{"full_name": "Ne.def", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [59, 9], "def_end_pos": [59, 15]}, {"full_name": "ENNReal.ofReal_eq_zero", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2170, 9], "def_end_pos": [2170, 23]}, {"full_name": "sub_nonpos", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [730, 30], "def_end_pos": [730, 40]}, {"full_name": "not_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [373, 9], "def_end_pos": [373, 15]}]], "state_before": "case refine'_2\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200 (n : \u2115), snorm (f n) 1 \u03bc \u2264 \u2191R\nhab : a < b\nthis : \u222b\u207b (\u03c9 : \u03a9), upcrossings a b f \u03c9 \u2202\u03bc \u2264 (\u2a06 N, \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc) / ENNReal.ofReal (b - a)\n\u22a2 ENNReal.ofReal (b - a) \u2260 0", "state_after": "no goals"}, {"tactic": "simp only [hab, Ne.def, ENNReal.ofReal_eq_zero, sub_nonpos, not_le, true_or_iff]", "annotated_tactic": ["simp only [hab, <a>Ne.def</a>, <a>ENNReal.ofReal_eq_zero</a>, <a>sub_nonpos</a>, <a>not_le</a>, <a>true_or_iff</a>]", [{"full_name": "Ne.def", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [59, 9], "def_end_pos": [59, 15]}, {"full_name": "ENNReal.ofReal_eq_zero", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2170, 9], "def_end_pos": [2170, 23]}, {"full_name": "sub_nonpos", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [730, 30], "def_end_pos": [730, 40]}, {"full_name": "not_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [373, 9], "def_end_pos": [373, 15]}, {"full_name": "true_or_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [182, 9], "def_end_pos": [182, 20]}]], "state_before": "case h0\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200 (n : \u2115), snorm (f n) 1 \u03bc \u2264 \u2191R\nhab : a < b\nthis : (\u222b\u207b (\u03c9 : \u03a9), upcrossings a b f \u03c9 \u2202\u03bc) * ENNReal.ofReal (b - a) \u2264 \u2a06 N, \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc\n\u22a2 ENNReal.ofReal (b - a) \u2260 0 \u2228 \u2a06 N, \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc \u2260 0", "state_after": "no goals"}, {"tactic": "simp only [Ne.def, ENNReal.ofReal_ne_top, not_false_iff, true_or_iff]", "annotated_tactic": ["simp only [<a>Ne.def</a>, <a>ENNReal.ofReal_ne_top</a>, <a>not_false_iff</a>, <a>true_or_iff</a>]", [{"full_name": "Ne.def", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [59, 9], "def_end_pos": [59, 15]}, {"full_name": "ENNReal.ofReal_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [311, 17], "def_end_pos": [311, 30]}, {"full_name": "not_false_iff", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [82, 9], "def_end_pos": [82, 22]}, {"full_name": "true_or_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [182, 9], "def_end_pos": [182, 20]}]], "state_before": "case ht\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200 (n : \u2115), snorm (f n) 1 \u03bc \u2264 \u2191R\nhab : a < b\nthis : (\u222b\u207b (\u03c9 : \u03a9), upcrossings a b f \u03c9 \u2202\u03bc) * ENNReal.ofReal (b - a) \u2264 \u2a06 N, \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc\n\u22a2 ENNReal.ofReal (b - a) \u2260 \u22a4 \u2228 \u2a06 N, \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc \u2260 \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Regular.lean", "full_name": "MeasureTheory.Measure.Regular.map", "start": [531, 11], "end": [538, 69], "traced_tactics": [{"tactic": "haveI := OuterRegular.map f \u03bc", "annotated_tactic": ["haveI := <a>OuterRegular.map</a> f \u03bc", [{"full_name": "MeasureTheory.Measure.OuterRegular.map", "def_path": "Mathlib/MeasureTheory/Measure/Regular.lean", "def_pos": [280, 19], "def_end_pos": [280, 22]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : OpensMeasurableSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : BorelSpace \u03b2\ninst\u271d : Regular \u03bc\nf : \u03b1 \u2243\u209c \u03b2\n\u22a2 Regular (map (\u2191f) \u03bc)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : OpensMeasurableSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : BorelSpace \u03b2\ninst\u271d : Regular \u03bc\nf : \u03b1 \u2243\u209c \u03b2\nthis : OuterRegular (map (\u2191f) \u03bc)\n\u22a2 Regular (map (\u2191f) \u03bc)"}, {"tactic": "haveI := IsFiniteMeasureOnCompacts.map \u03bc f", "annotated_tactic": ["haveI := <a>IsFiniteMeasureOnCompacts.map</a> \u03bc f", [{"full_name": "IsFiniteMeasureOnCompacts.map", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [1541, 19], "def_end_pos": [1541, 48]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : OpensMeasurableSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : BorelSpace \u03b2\ninst\u271d : Regular \u03bc\nf : \u03b1 \u2243\u209c \u03b2\nthis : OuterRegular (map (\u2191f) \u03bc)\n\u22a2 Regular (map (\u2191f) \u03bc)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : OpensMeasurableSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : BorelSpace \u03b2\ninst\u271d : Regular \u03bc\nf : \u03b1 \u2243\u209c \u03b2\nthis\u271d : OuterRegular (map (\u2191f) \u03bc)\nthis : IsFiniteMeasureOnCompacts (map (\u2191f) \u03bc)\n\u22a2 Regular (map (\u2191f) \u03bc)"}, {"tactic": "exact\n  \u27e8Regular.innerRegular.map f.toEquiv f.measurable.aemeasurable\n      (fun U hU => hU.preimage f.continuous) (fun K hK => hK.image f.continuous)\n      (fun K hK => hK.measurableSet) fun U hU => hU.measurableSet\u27e9", "annotated_tactic": ["exact\n    \u27e8Regular.innerRegular.map f.toEquiv f.measurable.aemeasurable\n        (fun U hU => hU.preimage f.continuous) (fun K hK => hK.image f.continuous)\n        (fun K hK => hK.measurableSet) fun U hU => hU.measurableSet\u27e9", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : OpensMeasurableSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : BorelSpace \u03b2\ninst\u271d : Regular \u03bc\nf : \u03b1 \u2243\u209c \u03b2\nthis\u271d : OuterRegular (map (\u2191f) \u03bc)\nthis : IsFiniteMeasureOnCompacts (map (\u2191f) \u03bc)\n\u22a2 Regular (map (\u2191f) \u03bc)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Portmanteau.lean", "full_name": "MeasureTheory.ProbabilityMeasure.tendsto_measure_of_null_frontier_of_tendsto'", "start": [406, 1], "end": [413, 54], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/HashMap/WF.lean", "full_name": "Std.HashMap.Imp.WF_iff", "start": [292, 1], "end": [294, 39], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Reduce.lean", "full_name": "OneOneReducible.disjoin_right", "start": [303, 1], "end": [305, 80], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "full_name": "ContinuousAt.stronglyMeasurableAtFilter", "start": [610, 1], "end": [613, 77], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/Jacobian.lean", "full_name": "MeasureTheory.restrict_map_withDensity_abs_det_fderiv_eq_addHaar", "start": [1137, 1], "end": [1160, 18], "traced_tactics": [{"tactic": "obtain \u27e8u, u_meas, uf\u27e9 : \u2203 u, Measurable u \u2227 EqOn u f s := by\n  classical\n  refine' \u27e8piecewise s f 0, _, piecewise_eqOn _ _ _\u27e9\n  refine' ContinuousOn.measurable_piecewise _ continuous_zero.continuousOn hs\n  have : DifferentiableOn \u211d f s := fun x hx => (hf' x hx).differentiableWithinAt\n  exact this.continuousOn", "annotated_tactic": ["obtain \u27e8u, u_meas, uf\u27e9 : \u2203 u, <a>Measurable</a> u \u2227 <a>EqOn</a> u f s := by\n    classical\n    refine' \u27e8<a>piecewise</a> s f 0, _, <a>piecewise_eqOn</a> _ _ _\u27e9\n    refine' <a>ContinuousOn.measurable_piecewise</a> _ continuous_zero.continuousOn hs\n    have : <a>DifferentiableOn</a> \u211d f s := fun x hx => (hf' x hx).<a>differentiableWithinAt</a>\n    exact this.continuousOn", [{"full_name": "Measurable", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [535, 5], "def_end_pos": [535, 15]}, {"full_name": "Set.EqOn", "def_path": "Mathlib/Data/Set/Function.lean", "def_pos": [180, 5], "def_end_pos": [180, 9]}, {"full_name": "Set.piecewise", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [993, 5], "def_end_pos": [993, 18]}, {"full_name": "Set.piecewise_eqOn", "def_path": "Mathlib/Data/Set/Function.lean", "def_pos": [1438, 9], "def_end_pos": [1438, 23]}, {"full_name": "ContinuousOn.measurable_piecewise", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [907, 9], "def_end_pos": [907, 42]}, {"full_name": "DifferentiableOn", "def_path": "Mathlib/Analysis/Calculus/FDeriv/Basic.lean", "def_pos": [197, 5], "def_end_pos": [197, 21]}, {"full_name": "HasFDerivWithinAt.differentiableWithinAt", "def_path": "Mathlib/Analysis/Calculus/FDeriv/Basic.lean", "def_pos": [394, 9], "def_end_pos": [394, 49]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u22a2 Measure.map (Set.restrict s f)\n      (Measure.comap Subtype.val (withDensity \u03bc fun x => ENNReal.ofReal |ContinuousLinearMap.det (f' x)|)) =\n    Measure.restrict \u03bc (f '' s)", "state_after": "case intro.intro\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nu : E \u2192 E\nu_meas : Measurable u\nuf : EqOn u f s\n\u22a2 Measure.map (Set.restrict s f)\n      (Measure.comap Subtype.val (withDensity \u03bc fun x => ENNReal.ofReal |ContinuousLinearMap.det (f' x)|)) =\n    Measure.restrict \u03bc (f '' s)"}, {"tactic": "have u' : \u2200 x \u2208 s, HasFDerivWithinAt u (f' x) s x := fun x hx =>\n  (hf' x hx).congr (fun y hy => uf hy) (uf hx)", "annotated_tactic": ["have u' : \u2200 x \u2208 s, <a>HasFDerivWithinAt</a> u (f' x) s x := fun x hx =>\n    (hf' x hx).<a>congr</a> (fun y hy => uf hy) (uf hx)", [{"full_name": "HasFDerivWithinAt", "def_path": "Mathlib/Analysis/Calculus/FDeriv/Basic.lean", "def_pos": [150, 5], "def_end_pos": [150, 22]}, {"full_name": "HasFDerivWithinAt.congr", "def_path": "Mathlib/Analysis/Calculus/FDeriv/Basic.lean", "def_pos": [959, 9], "def_end_pos": [959, 32]}]], "state_before": "case intro.intro\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nu : E \u2192 E\nu_meas : Measurable u\nuf : EqOn u f s\n\u22a2 Measure.map (Set.restrict s f)\n      (Measure.comap Subtype.val (withDensity \u03bc fun x => ENNReal.ofReal |ContinuousLinearMap.det (f' x)|)) =\n    Measure.restrict \u03bc (f '' s)", "state_after": "case intro.intro\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nu : E \u2192 E\nu_meas : Measurable u\nuf : EqOn u f s\nu' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt u (f' x) s x\n\u22a2 Measure.map (Set.restrict s f)\n      (Measure.comap Subtype.val (withDensity \u03bc fun x => ENNReal.ofReal |ContinuousLinearMap.det (f' x)|)) =\n    Measure.restrict \u03bc (f '' s)"}, {"tactic": "set F : s \u2192 E := u \u2218 (\u2191) with hF", "annotated_tactic": ["set F : s \u2192 E := u \u2218 (\u2191) with hF", []], "state_before": "case intro.intro\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nu : E \u2192 E\nu_meas : Measurable u\nuf : EqOn u f s\nu' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt u (f' x) s x\n\u22a2 Measure.map (Set.restrict s f)\n      (Measure.comap Subtype.val (withDensity \u03bc fun x => ENNReal.ofReal |ContinuousLinearMap.det (f' x)|)) =\n    Measure.restrict \u03bc (f '' s)", "state_after": "case intro.intro\nE : Type u_1\nF\u271d : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\u271d\ninst\u271d\u00b3 : NormedSpace \u211d F\u271d\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nu : E \u2192 E\nu_meas : Measurable u\nuf : EqOn u f s\nu' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt u (f' x) s x\nF : \u2191s \u2192 E := u \u2218 Subtype.val\nhF : F = u \u2218 Subtype.val\n\u22a2 Measure.map (Set.restrict s f)\n      (Measure.comap Subtype.val (withDensity \u03bc fun x => ENNReal.ofReal |ContinuousLinearMap.det (f' x)|)) =\n    Measure.restrict \u03bc (f '' s)"}, {"tactic": "have A :\n  Measure.map F (comap (\u2191) (\u03bc.withDensity fun x => ENNReal.ofReal |(f' x).det|)) =\n    \u03bc.restrict (u '' s) := by\n  rw [hF, \u2190 Measure.map_map u_meas measurable_subtype_coe, map_comap_subtype_coe hs,\n    restrict_withDensity hs]\n  exact map_withDensity_abs_det_fderiv_eq_addHaar \u03bc hs u' (hf.congr uf.symm) u_meas", "annotated_tactic": ["have A :\n    <a>Measure.map</a> F (<a>comap</a> (\u2191) (\u03bc.withDensity fun x => <a>ENNReal.ofReal</a> |(f' x).<a>det</a>|)) =\n      \u03bc.restrict (u '' s) := by\n    rw [hF, \u2190 <a>Measure.map_map</a> u_meas <a>measurable_subtype_coe</a>, <a>map_comap_subtype_coe</a> hs,\n      <a>restrict_withDensity</a> hs]\n    exact <a>map_withDensity_abs_det_fderiv_eq_addHaar</a> \u03bc hs u' (hf.congr uf.symm) u_meas", [{"full_name": "MeasureTheory.Measure.map", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1163, 17], "def_end_pos": [1163, 20]}, {"full_name": "MeasureTheory.Measure.comap", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1313, 5], "def_end_pos": [1313, 10]}, {"full_name": "ENNReal.ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [172, 29], "def_end_pos": [172, 35]}, {"full_name": "ContinuousLinearMap.det", "def_path": "Mathlib/Topology/Algebra/Module/Determinant.lean", "def_pos": [22, 19], "def_end_pos": [22, 22]}, {"full_name": "MeasureTheory.Measure.map_map", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1258, 9], "def_end_pos": [1258, 16]}, {"full_name": "measurable_subtype_coe", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [577, 9], "def_end_pos": [577, 31]}, {"full_name": "map_comap_subtype_coe", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [4159, 9], "def_end_pos": [4159, 30]}, {"full_name": "MeasureTheory.restrict_withDensity", "def_path": "Mathlib/MeasureTheory/Measure/WithDensity.lean", "def_pos": [176, 9], "def_end_pos": [176, 29]}, {"full_name": "MeasureTheory.map_withDensity_abs_det_fderiv_eq_addHaar", "def_path": "Mathlib/MeasureTheory/Function/Jacobian.lean", "def_pos": [1118, 9], "def_end_pos": [1118, 50]}]], "state_before": "case intro.intro\nE : Type u_1\nF\u271d : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\u271d\ninst\u271d\u00b3 : NormedSpace \u211d F\u271d\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nu : E \u2192 E\nu_meas : Measurable u\nuf : EqOn u f s\nu' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt u (f' x) s x\nF : \u2191s \u2192 E := u \u2218 Subtype.val\nhF : F = u \u2218 Subtype.val\n\u22a2 Measure.map (Set.restrict s f)\n      (Measure.comap Subtype.val (withDensity \u03bc fun x => ENNReal.ofReal |ContinuousLinearMap.det (f' x)|)) =\n    Measure.restrict \u03bc (f '' s)", "state_after": "case intro.intro\nE : Type u_1\nF\u271d : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\u271d\ninst\u271d\u00b3 : NormedSpace \u211d F\u271d\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nu : E \u2192 E\nu_meas : Measurable u\nuf : EqOn u f s\nu' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt u (f' x) s x\nF : \u2191s \u2192 E := u \u2218 Subtype.val\nhF : F = u \u2218 Subtype.val\nA :\n  Measure.map F (Measure.comap Subtype.val (withDensity \u03bc fun x => ENNReal.ofReal |ContinuousLinearMap.det (f' x)|)) =\n    Measure.restrict \u03bc (u '' s)\n\u22a2 Measure.map (Set.restrict s f)\n      (Measure.comap Subtype.val (withDensity \u03bc fun x => ENNReal.ofReal |ContinuousLinearMap.det (f' x)|)) =\n    Measure.restrict \u03bc (f '' s)"}, {"tactic": "rw [uf.image_eq] at A", "annotated_tactic": ["rw [uf.image_eq] at A", []], "state_before": "case intro.intro\nE : Type u_1\nF\u271d : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\u271d\ninst\u271d\u00b3 : NormedSpace \u211d F\u271d\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nu : E \u2192 E\nu_meas : Measurable u\nuf : EqOn u f s\nu' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt u (f' x) s x\nF : \u2191s \u2192 E := u \u2218 Subtype.val\nhF : F = u \u2218 Subtype.val\nA :\n  Measure.map F (Measure.comap Subtype.val (withDensity \u03bc fun x => ENNReal.ofReal |ContinuousLinearMap.det (f' x)|)) =\n    Measure.restrict \u03bc (u '' s)\n\u22a2 Measure.map (Set.restrict s f)\n      (Measure.comap Subtype.val (withDensity \u03bc fun x => ENNReal.ofReal |ContinuousLinearMap.det (f' x)|)) =\n    Measure.restrict \u03bc (f '' s)", "state_after": "case intro.intro\nE : Type u_1\nF\u271d : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\u271d\ninst\u271d\u00b3 : NormedSpace \u211d F\u271d\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nu : E \u2192 E\nu_meas : Measurable u\nuf : EqOn u f s\nu' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt u (f' x) s x\nF : \u2191s \u2192 E := u \u2218 Subtype.val\nhF : F = u \u2218 Subtype.val\nA :\n  Measure.map F (Measure.comap Subtype.val (withDensity \u03bc fun x => ENNReal.ofReal |ContinuousLinearMap.det (f' x)|)) =\n    Measure.restrict \u03bc (f '' s)\n\u22a2 Measure.map (Set.restrict s f)\n      (Measure.comap Subtype.val (withDensity \u03bc fun x => ENNReal.ofReal |ContinuousLinearMap.det (f' x)|)) =\n    Measure.restrict \u03bc (f '' s)"}, {"tactic": "have : F = s.restrict f := by\n  ext x\n  exact uf x.2", "annotated_tactic": ["have : F = s.restrict f := by\n    ext x\n    exact uf x.2", []], "state_before": "case intro.intro\nE : Type u_1\nF\u271d : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\u271d\ninst\u271d\u00b3 : NormedSpace \u211d F\u271d\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nu : E \u2192 E\nu_meas : Measurable u\nuf : EqOn u f s\nu' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt u (f' x) s x\nF : \u2191s \u2192 E := u \u2218 Subtype.val\nhF : F = u \u2218 Subtype.val\nA :\n  Measure.map F (Measure.comap Subtype.val (withDensity \u03bc fun x => ENNReal.ofReal |ContinuousLinearMap.det (f' x)|)) =\n    Measure.restrict \u03bc (f '' s)\n\u22a2 Measure.map (Set.restrict s f)\n      (Measure.comap Subtype.val (withDensity \u03bc fun x => ENNReal.ofReal |ContinuousLinearMap.det (f' x)|)) =\n    Measure.restrict \u03bc (f '' s)", "state_after": "case intro.intro\nE : Type u_1\nF\u271d : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\u271d\ninst\u271d\u00b3 : NormedSpace \u211d F\u271d\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nu : E \u2192 E\nu_meas : Measurable u\nuf : EqOn u f s\nu' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt u (f' x) s x\nF : \u2191s \u2192 E := u \u2218 Subtype.val\nhF : F = u \u2218 Subtype.val\nA :\n  Measure.map F (Measure.comap Subtype.val (withDensity \u03bc fun x => ENNReal.ofReal |ContinuousLinearMap.det (f' x)|)) =\n    Measure.restrict \u03bc (f '' s)\nthis : F = Set.restrict s f\n\u22a2 Measure.map (Set.restrict s f)\n      (Measure.comap Subtype.val (withDensity \u03bc fun x => ENNReal.ofReal |ContinuousLinearMap.det (f' x)|)) =\n    Measure.restrict \u03bc (f '' s)"}, {"tactic": "rwa [this] at A", "annotated_tactic": ["rwa [this] at A", []], "state_before": "case intro.intro\nE : Type u_1\nF\u271d : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\u271d\ninst\u271d\u00b3 : NormedSpace \u211d F\u271d\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nu : E \u2192 E\nu_meas : Measurable u\nuf : EqOn u f s\nu' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt u (f' x) s x\nF : \u2191s \u2192 E := u \u2218 Subtype.val\nhF : F = u \u2218 Subtype.val\nA :\n  Measure.map F (Measure.comap Subtype.val (withDensity \u03bc fun x => ENNReal.ofReal |ContinuousLinearMap.det (f' x)|)) =\n    Measure.restrict \u03bc (f '' s)\nthis : F = Set.restrict s f\n\u22a2 Measure.map (Set.restrict s f)\n      (Measure.comap Subtype.val (withDensity \u03bc fun x => ENNReal.ofReal |ContinuousLinearMap.det (f' x)|)) =\n    Measure.restrict \u03bc (f '' s)", "state_after": "no goals"}, {"tactic": "classical\nrefine' \u27e8piecewise s f 0, _, piecewise_eqOn _ _ _\u27e9\nrefine' ContinuousOn.measurable_piecewise _ continuous_zero.continuousOn hs\nhave : DifferentiableOn \u211d f s := fun x hx => (hf' x hx).differentiableWithinAt\nexact this.continuousOn", "annotated_tactic": ["classical\n    refine' \u27e8<a>piecewise</a> s f 0, _, <a>piecewise_eqOn</a> _ _ _\u27e9\n    refine' <a>ContinuousOn.measurable_piecewise</a> _ continuous_zero.continuousOn hs\n    have : <a>DifferentiableOn</a> \u211d f s := fun x hx => (hf' x hx).<a>differentiableWithinAt</a>\n    exact this.continuousOn", [{"full_name": "Set.piecewise", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [993, 5], "def_end_pos": [993, 18]}, {"full_name": "Set.piecewise_eqOn", "def_path": "Mathlib/Data/Set/Function.lean", "def_pos": [1438, 9], "def_end_pos": [1438, 23]}, {"full_name": "ContinuousOn.measurable_piecewise", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [907, 9], "def_end_pos": [907, 42]}, {"full_name": "DifferentiableOn", "def_path": "Mathlib/Analysis/Calculus/FDeriv/Basic.lean", "def_pos": [197, 5], "def_end_pos": [197, 21]}, {"full_name": "HasFDerivWithinAt.differentiableWithinAt", "def_path": "Mathlib/Analysis/Calculus/FDeriv/Basic.lean", "def_pos": [394, 9], "def_end_pos": [394, 49]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u22a2 \u2203 u, Measurable u \u2227 EqOn u f s", "state_after": "no goals"}, {"tactic": "refine' \u27e8piecewise s f 0, _, piecewise_eqOn _ _ _\u27e9", "annotated_tactic": ["refine' \u27e8<a>piecewise</a> s f 0, _, <a>piecewise_eqOn</a> _ _ _\u27e9", [{"full_name": "Set.piecewise", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [993, 5], "def_end_pos": [993, 18]}, {"full_name": "Set.piecewise_eqOn", "def_path": "Mathlib/Data/Set/Function.lean", "def_pos": [1438, 9], "def_end_pos": [1438, 23]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u22a2 \u2203 u, Measurable u \u2227 EqOn u f s", "state_after": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u22a2 Measurable (piecewise s f 0)"}, {"tactic": "refine' ContinuousOn.measurable_piecewise _ continuous_zero.continuousOn hs", "annotated_tactic": ["refine' <a>ContinuousOn.measurable_piecewise</a> _ continuous_zero.continuousOn hs", [{"full_name": "ContinuousOn.measurable_piecewise", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [907, 9], "def_end_pos": [907, 42]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u22a2 Measurable (piecewise s f 0)", "state_after": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u22a2 ContinuousOn f s"}, {"tactic": "have : DifferentiableOn \u211d f s := fun x hx => (hf' x hx).differentiableWithinAt", "annotated_tactic": ["have : <a>DifferentiableOn</a> \u211d f s := fun x hx => (hf' x hx).<a>differentiableWithinAt</a>", [{"full_name": "DifferentiableOn", "def_path": "Mathlib/Analysis/Calculus/FDeriv/Basic.lean", "def_pos": [197, 5], "def_end_pos": [197, 21]}, {"full_name": "HasFDerivWithinAt.differentiableWithinAt", "def_path": "Mathlib/Analysis/Calculus/FDeriv/Basic.lean", "def_pos": [394, 9], "def_end_pos": [394, 49]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u22a2 ContinuousOn f s", "state_after": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nthis : DifferentiableOn \u211d f s\n\u22a2 ContinuousOn f s"}, {"tactic": "exact this.continuousOn", "annotated_tactic": ["exact this.continuousOn", []], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nthis : DifferentiableOn \u211d f s\n\u22a2 ContinuousOn f s", "state_after": "no goals"}, {"tactic": "rw [hF, \u2190 Measure.map_map u_meas measurable_subtype_coe, map_comap_subtype_coe hs,\n  restrict_withDensity hs]", "annotated_tactic": ["rw [hF, \u2190 <a>Measure.map_map</a> u_meas <a>measurable_subtype_coe</a>, <a>map_comap_subtype_coe</a> hs,\n      <a>restrict_withDensity</a> hs]", [{"full_name": "MeasureTheory.Measure.map_map", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1258, 9], "def_end_pos": [1258, 16]}, {"full_name": "measurable_subtype_coe", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [577, 9], "def_end_pos": [577, 31]}, {"full_name": "map_comap_subtype_coe", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [4159, 9], "def_end_pos": [4159, 30]}, {"full_name": "MeasureTheory.restrict_withDensity", "def_path": "Mathlib/MeasureTheory/Measure/WithDensity.lean", "def_pos": [176, 9], "def_end_pos": [176, 29]}]], "state_before": "E : Type u_1\nF\u271d : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\u271d\ninst\u271d\u00b3 : NormedSpace \u211d F\u271d\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nu : E \u2192 E\nu_meas : Measurable u\nuf : EqOn u f s\nu' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt u (f' x) s x\nF : \u2191s \u2192 E := u \u2218 Subtype.val\nhF : F = u \u2218 Subtype.val\n\u22a2 Measure.map F (Measure.comap Subtype.val (withDensity \u03bc fun x => ENNReal.ofReal |ContinuousLinearMap.det (f' x)|)) =\n    Measure.restrict \u03bc (u '' s)", "state_after": "E : Type u_1\nF\u271d : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\u271d\ninst\u271d\u00b3 : NormedSpace \u211d F\u271d\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nu : E \u2192 E\nu_meas : Measurable u\nuf : EqOn u f s\nu' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt u (f' x) s x\nF : \u2191s \u2192 E := u \u2218 Subtype.val\nhF : F = u \u2218 Subtype.val\n\u22a2 Measure.map u (withDensity (Measure.restrict \u03bc s) fun x => ENNReal.ofReal |ContinuousLinearMap.det (f' x)|) =\n    Measure.restrict \u03bc (u '' s)"}, {"tactic": "exact map_withDensity_abs_det_fderiv_eq_addHaar \u03bc hs u' (hf.congr uf.symm) u_meas", "annotated_tactic": ["exact <a>map_withDensity_abs_det_fderiv_eq_addHaar</a> \u03bc hs u' (hf.congr uf.symm) u_meas", [{"full_name": "MeasureTheory.map_withDensity_abs_det_fderiv_eq_addHaar", "def_path": "Mathlib/MeasureTheory/Function/Jacobian.lean", "def_pos": [1118, 9], "def_end_pos": [1118, 50]}]], "state_before": "E : Type u_1\nF\u271d : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\u271d\ninst\u271d\u00b3 : NormedSpace \u211d F\u271d\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nu : E \u2192 E\nu_meas : Measurable u\nuf : EqOn u f s\nu' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt u (f' x) s x\nF : \u2191s \u2192 E := u \u2218 Subtype.val\nhF : F = u \u2218 Subtype.val\n\u22a2 Measure.map u (withDensity (Measure.restrict \u03bc s) fun x => ENNReal.ofReal |ContinuousLinearMap.det (f' x)|) =\n    Measure.restrict \u03bc (u '' s)", "state_after": "no goals"}, {"tactic": "ext x", "annotated_tactic": ["ext x", []], "state_before": "E : Type u_1\nF\u271d : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\u271d\ninst\u271d\u00b3 : NormedSpace \u211d F\u271d\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nu : E \u2192 E\nu_meas : Measurable u\nuf : EqOn u f s\nu' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt u (f' x) s x\nF : \u2191s \u2192 E := u \u2218 Subtype.val\nhF : F = u \u2218 Subtype.val\nA :\n  Measure.map F (Measure.comap Subtype.val (withDensity \u03bc fun x => ENNReal.ofReal |ContinuousLinearMap.det (f' x)|)) =\n    Measure.restrict \u03bc (f '' s)\n\u22a2 F = Set.restrict s f", "state_after": "case h\nE : Type u_1\nF\u271d : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\u271d\ninst\u271d\u00b3 : NormedSpace \u211d F\u271d\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nu : E \u2192 E\nu_meas : Measurable u\nuf : EqOn u f s\nu' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt u (f' x) s x\nF : \u2191s \u2192 E := u \u2218 Subtype.val\nhF : F = u \u2218 Subtype.val\nA :\n  Measure.map F (Measure.comap Subtype.val (withDensity \u03bc fun x => ENNReal.ofReal |ContinuousLinearMap.det (f' x)|)) =\n    Measure.restrict \u03bc (f '' s)\nx : \u2191s\n\u22a2 F x = Set.restrict s f x"}, {"tactic": "exact uf x.2", "annotated_tactic": ["exact uf x.2", []], "state_before": "case h\nE : Type u_1\nF\u271d : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\u271d\ninst\u271d\u00b3 : NormedSpace \u211d F\u271d\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nu : E \u2192 E\nu_meas : Measurable u\nuf : EqOn u f s\nu' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt u (f' x) s x\nF : \u2191s \u2192 E := u \u2218 Subtype.val\nhF : F = u \u2218 Subtype.val\nA :\n  Measure.map F (Measure.comap Subtype.val (withDensity \u03bc fun x => ENNReal.ofReal |ContinuousLinearMap.det (f' x)|)) =\n    Measure.restrict \u03bc (f '' s)\nx : \u2191s\n\u22a2 F x = Set.restrict s f x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Decomposition/SignedHahn.lean", "full_name": "MeasureTheory.SignedMeasure.restrictNonposSeq_lt", "start": [179, 9], "end": [183, 33], "traced_tactics": [{"tactic": "rw [restrictNonposSeq_succ]", "annotated_tactic": ["rw [<a>restrictNonposSeq_succ</a>]", [{"full_name": "_private.Mathlib.MeasureTheory.Decomposition.SignedHahn.0.MeasureTheory.SignedMeasure.restrictNonposSeq_succ", "def_path": "Mathlib/MeasureTheory/Decomposition/SignedHahn.lean", "def_pos": [172, 17], "def_end_pos": [172, 39]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : TopologicalSpace M\ninst\u271d : OrderedAddCommMonoid M\ns : SignedMeasure \u03b1\ni j : Set \u03b1\nn : \u2115\nhn :\n  \u00acrestrict s (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) \u2264\n      restrict 0 (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)\n\u22a2 1 /\n      (\u2191(MeasureTheory.SignedMeasure.findExistsOneDivLT s\n            (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)) +\n        1) <\n    \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i (Nat.succ n))", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : TopologicalSpace M\ninst\u271d : OrderedAddCommMonoid M\ns : SignedMeasure \u03b1\ni j : Set \u03b1\nn : \u2115\nhn :\n  \u00acrestrict s (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) \u2264\n      restrict 0 (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)\n\u22a2 1 /\n      (\u2191(MeasureTheory.SignedMeasure.findExistsOneDivLT s\n            (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)) +\n        1) <\n    \u2191s\n      (MeasureTheory.SignedMeasure.someExistsOneDivLT s\n        (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k))"}, {"tactic": "apply someExistsOneDivLT_lt hn", "annotated_tactic": ["apply <a>someExistsOneDivLT_lt</a> hn", [{"full_name": "_private.Mathlib.MeasureTheory.Decomposition.SignedHahn.0.MeasureTheory.SignedMeasure.someExistsOneDivLT_lt", "def_path": "Mathlib/MeasureTheory/Decomposition/SignedHahn.lean", "def_pos": [152, 17], "def_end_pos": [152, 38]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : TopologicalSpace M\ninst\u271d : OrderedAddCommMonoid M\ns : SignedMeasure \u03b1\ni j : Set \u03b1\nn : \u2115\nhn :\n  \u00acrestrict s (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) \u2264\n      restrict 0 (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)\n\u22a2 1 /\n      (\u2191(MeasureTheory.SignedMeasure.findExistsOneDivLT s\n            (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)) +\n        1) <\n    \u2191s\n      (MeasureTheory.SignedMeasure.someExistsOneDivLT s\n        (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/Basic.lean", "full_name": "MvPolynomial.is_id", "start": [481, 1], "end": [483, 38], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/QPF/Multivariate/Constructions/Cofix.lean", "full_name": "MvQPF.Cofix.abs_repr", "start": [432, 1], "end": [448, 6], "traced_tactics": [{"tactic": "let R := fun x y : Cofix F \u03b1 => abs (repr y) = x", "annotated_tactic": ["let R := fun x y : <a>Cofix</a> F \u03b1 => <a>abs</a> (<a>repr</a> y) = x", [{"full_name": "MvQPF.Cofix", "def_path": "Mathlib/Data/QPF/Multivariate/Constructions/Cofix.lean", "def_pos": [91, 5], "def_end_pos": [91, 10]}, {"full_name": "MvQPF.Cofix.abs", "def_path": "Mathlib/Data/QPF/Multivariate/Constructions/Cofix.lean", "def_pos": [154, 5], "def_end_pos": [154, 14]}, {"full_name": "MvQPF.Cofix.repr", "def_path": "Mathlib/Data/QPF/Multivariate/Constructions/Cofix.lean", "def_pos": [159, 5], "def_end_pos": [159, 15]}]], "state_before": "n : \u2115\nF\u271d : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\u271d\nq\u271d : MvQPF F\u271d\nF : TypeVec.{u} (n + 1) \u2192 Type u\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nx : Cofix F \u03b1\n\u22a2 Quot.mk Mcongr (repr x) = x", "state_after": "n : \u2115\nF\u271d : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\u271d\nq\u271d : MvQPF F\u271d\nF : TypeVec.{u} (n + 1) \u2192 Type u\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nx : Cofix F \u03b1\nR : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => abs (repr y) = x\n\u22a2 Quot.mk Mcongr (repr x) = x"}, {"tactic": "refine' Cofix.bisim\u2082 R _ _ _ rfl", "annotated_tactic": ["refine' <a>Cofix.bisim\u2082</a> R _ _ _ <a>rfl</a>", [{"full_name": "MvQPF.Cofix.bisim\u2082", "def_path": "Mathlib/Data/QPF/Multivariate/Constructions/Cofix.lean", "def_pos": [305, 9], "def_end_pos": [305, 21]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "n : \u2115\nF\u271d : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\u271d\nq\u271d : MvQPF F\u271d\nF : TypeVec.{u} (n + 1) \u2192 Type u\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nx : Cofix F \u03b1\nR : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => abs (repr y) = x\n\u22a2 Quot.mk Mcongr (repr x) = x", "state_after": "n : \u2115\nF\u271d : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\u271d\nq\u271d : MvQPF F\u271d\nF : TypeVec.{u} (n + 1) \u2192 Type u\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nx : Cofix F \u03b1\nR : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => abs (repr y) = x\n\u22a2 \u2200 (x y : Cofix F \u03b1), R x y \u2192 LiftR' (RelLast' \u03b1 R) (dest x) (dest y)"}, {"tactic": "clear x", "annotated_tactic": ["clear x", []], "state_before": "n : \u2115\nF\u271d : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\u271d\nq\u271d : MvQPF F\u271d\nF : TypeVec.{u} (n + 1) \u2192 Type u\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nx : Cofix F \u03b1\nR : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => abs (repr y) = x\n\u22a2 \u2200 (x y : Cofix F \u03b1), R x y \u2192 LiftR' (RelLast' \u03b1 R) (dest x) (dest y)", "state_after": "n : \u2115\nF\u271d : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\u271d\nq\u271d : MvQPF F\u271d\nF : TypeVec.{u} (n + 1) \u2192 Type u\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nR : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => abs (repr y) = x\n\u22a2 \u2200 (x y : Cofix F \u03b1), R x y \u2192 LiftR' (RelLast' \u03b1 R) (dest x) (dest y)"}, {"tactic": "rintro x y h", "annotated_tactic": ["rintro x y h", []], "state_before": "n : \u2115\nF\u271d : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\u271d\nq\u271d : MvQPF F\u271d\nF : TypeVec.{u} (n + 1) \u2192 Type u\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nR : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => abs (repr y) = x\n\u22a2 \u2200 (x y : Cofix F \u03b1), R x y \u2192 LiftR' (RelLast' \u03b1 R) (dest x) (dest y)", "state_after": "n : \u2115\nF\u271d : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\u271d\nq\u271d : MvQPF F\u271d\nF : TypeVec.{u} (n + 1) \u2192 Type u\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nR : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => abs (repr y) = x\nx y : Cofix F \u03b1\nh : R x y\n\u22a2 LiftR' (RelLast' \u03b1 R) (dest x) (dest y)"}, {"tactic": "subst h", "annotated_tactic": ["subst h", []], "state_before": "n : \u2115\nF\u271d : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\u271d\nq\u271d : MvQPF F\u271d\nF : TypeVec.{u} (n + 1) \u2192 Type u\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nR : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => abs (repr y) = x\nx y : Cofix F \u03b1\nh : R x y\n\u22a2 LiftR' (RelLast' \u03b1 R) (dest x) (dest y)", "state_after": "n : \u2115\nF\u271d : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\u271d\nq\u271d : MvQPF F\u271d\nF : TypeVec.{u} (n + 1) \u2192 Type u\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nR : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => abs (repr y) = x\ny : Cofix F \u03b1\n\u22a2 LiftR' (RelLast' \u03b1 R) (dest (abs (repr y))) (dest y)"}, {"tactic": "dsimp [Cofix.dest, Cofix.abs]", "annotated_tactic": ["dsimp [<a>Cofix.dest</a>, <a>Cofix.abs</a>]", [{"full_name": "MvQPF.Cofix.dest", "def_path": "Mathlib/Data/QPF/Multivariate/Constructions/Cofix.lean", "def_pos": [138, 5], "def_end_pos": [138, 15]}, {"full_name": "MvQPF.Cofix.abs", "def_path": "Mathlib/Data/QPF/Multivariate/Constructions/Cofix.lean", "def_pos": [154, 5], "def_end_pos": [154, 14]}]], "state_before": "n : \u2115\nF\u271d : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\u271d\nq\u271d : MvQPF F\u271d\nF : TypeVec.{u} (n + 1) \u2192 Type u\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nR : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => abs (repr y) = x\ny : Cofix F \u03b1\n\u22a2 LiftR' (RelLast' \u03b1 R) (dest (abs (repr y))) (dest y)", "state_after": "n : \u2115\nF\u271d : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\u271d\nq\u271d : MvQPF F\u271d\nF : TypeVec.{u} (n + 1) \u2192 Type u\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nR : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => abs (repr y) = x\ny : Cofix F \u03b1\n\u22a2 LiftR' (RelLast' \u03b1 fun x y => Quot.mk Mcongr (repr y) = x)\n    ((TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) (repr y)))\n    (Quot.lift (fun x => (TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) x))\n      (_ :\n        \u2200 (x y : M (P F) \u03b1),\n          Mcongr x y \u2192\n            (fun x => (TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) x)) x =\n              (fun x => (TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) x)) y)\n      y)"}, {"tactic": "induction y using Quot.ind", "annotated_tactic": ["induction y using <a>Quot.ind</a>", [{"full_name": "Quot.ind", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [414, 14], "def_end_pos": [414, 22]}]], "state_before": "n : \u2115\nF\u271d : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\u271d\nq\u271d : MvQPF F\u271d\nF : TypeVec.{u} (n + 1) \u2192 Type u\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nR : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => abs (repr y) = x\ny : Cofix F \u03b1\n\u22a2 LiftR' (RelLast' \u03b1 fun x y => Quot.mk Mcongr (repr y) = x)\n    ((TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) (repr y)))\n    (Quot.lift (fun x => (TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) x))\n      (_ :\n        \u2200 (x y : M (P F) \u03b1),\n          Mcongr x y \u2192\n            (fun x => (TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) x)) x =\n              (fun x => (TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) x)) y)\n      y)", "state_after": "case mk\nn : \u2115\nF\u271d : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\u271d\nq\u271d : MvQPF F\u271d\nF : TypeVec.{u} (n + 1) \u2192 Type u\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nR : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => abs (repr y) = x\na\u271d : M (P F) \u03b1\n\u22a2 LiftR' (RelLast' \u03b1 fun x y => Quot.mk Mcongr (repr y) = x)\n    ((TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) (repr (Quot.mk Mcongr a\u271d))))\n    (Quot.lift (fun x => (TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) x))\n      (_ :\n        \u2200 (x y : M (P F) \u03b1),\n          Mcongr x y \u2192\n            (fun x => (TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) x)) x =\n              (fun x => (TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) x)) y)\n      (Quot.mk Mcongr a\u271d))"}, {"tactic": "simp only [Cofix.repr, M.dest_corec, abs_map, MvQPF.abs_repr, Function.comp]", "annotated_tactic": ["simp only [<a>Cofix.repr</a>, <a>M.dest_corec</a>, <a>abs_map</a>, <a>MvQPF.abs_repr</a>, <a>Function.comp</a>]", [{"full_name": "MvQPF.Cofix.repr", "def_path": "Mathlib/Data/QPF/Multivariate/Constructions/Cofix.lean", "def_pos": [159, 5], "def_end_pos": [159, 15]}, {"full_name": "MvPFunctor.M.dest_corec", "def_path": "Mathlib/Data/PFunctor/Multivariate/M.lean", "def_pos": [213, 9], "def_end_pos": [213, 21]}, {"full_name": "MvQPF.abs_map", "def_path": "Mathlib/Data/QPF/Multivariate/Basic.lean", "def_pos": [90, 3], "def_end_pos": [90, 10]}, {"full_name": "MvQPF.abs_repr", "def_path": "Mathlib/Data/QPF/Multivariate/Basic.lean", "def_pos": [89, 3], "def_end_pos": [89, 11]}, {"full_name": "Function.comp", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [52, 15], "def_end_pos": [52, 28]}]], "state_before": "case mk\nn : \u2115\nF\u271d : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\u271d\nq\u271d : MvQPF F\u271d\nF : TypeVec.{u} (n + 1) \u2192 Type u\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nR : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => abs (repr y) = x\na\u271d : M (P F) \u03b1\n\u22a2 LiftR' (RelLast' \u03b1 fun x y => Quot.mk Mcongr (repr y) = x)\n    ((TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) (repr (Quot.mk Mcongr a\u271d))))\n    (Quot.lift (fun x => (TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) x))\n      (_ :\n        \u2200 (x y : M (P F) \u03b1),\n          Mcongr x y \u2192\n            (fun x => (TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) x)) x =\n              (fun x => (TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) x)) y)\n      (Quot.mk Mcongr a\u271d))", "state_after": "case mk\nn : \u2115\nF\u271d : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\u271d\nq\u271d : MvQPF F\u271d\nF : TypeVec.{u} (n + 1) \u2192 Type u\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nR : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => abs (repr y) = x\na\u271d : M (P F) \u03b1\n\u22a2 LiftR' (RelLast' \u03b1 fun x y => Quot.mk Mcongr (M.corec (P F) (fun x => MvQPF.repr (dest x)) y) = x)\n    ((TypeVec.id ::: Quot.mk Mcongr) <$$>\n      (TypeVec.id ::: M.corec (P F) fun x => MvQPF.repr (dest x)) <$$> dest (Quot.mk Mcongr a\u271d))\n    ((TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) a\u271d))"}, {"tactic": "conv =>\n  congr\n  rfl\n  rw [Cofix.dest]", "annotated_tactic": ["conv =>\n    congr\n    rfl\n    rw [<a>Cofix.dest</a>]", [{"full_name": "MvQPF.Cofix.dest", "def_path": "Mathlib/Data/QPF/Multivariate/Constructions/Cofix.lean", "def_pos": [138, 5], "def_end_pos": [138, 15]}]], "state_before": "case mk\nn : \u2115\nF\u271d : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\u271d\nq\u271d : MvQPF F\u271d\nF : TypeVec.{u} (n + 1) \u2192 Type u\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nR : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => abs (repr y) = x\na\u271d : M (P F) \u03b1\n\u22a2 LiftR' (RelLast' \u03b1 fun x y => Quot.mk Mcongr (M.corec (P F) (fun x => MvQPF.repr (dest x)) y) = x)\n    ((TypeVec.id ::: Quot.mk Mcongr) <$$>\n      (TypeVec.id ::: M.corec (P F) fun x => MvQPF.repr (dest x)) <$$> dest (Quot.mk Mcongr a\u271d))\n    ((TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) a\u271d))", "state_after": "case mk\nn : \u2115\nF\u271d : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\u271d\nq\u271d : MvQPF F\u271d\nF : TypeVec.{u} (n + 1) \u2192 Type u\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nR : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => abs (repr y) = x\na\u271d : M (P F) \u03b1\n\u22a2 LiftR' (RelLast' \u03b1 fun x y => Quot.mk Mcongr (M.corec (P F) (fun x => MvQPF.repr (dest x)) y) = x)\n    ((TypeVec.id ::: Quot.mk Mcongr) <$$>\n      (TypeVec.id :::\n          M.corec (P F) fun x =>\n            MvQPF.repr\n              (Quot.lift (fun x => (TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) x))\n                (_ :\n                  \u2200 (x y : M (P F) \u03b1),\n                    Mcongr x y \u2192\n                      (fun x => (TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) x)) x =\n                        (fun x => (TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) x)) y)\n                x)) <$$>\n        Quot.lift (fun x => (TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) x))\n          (_ :\n            \u2200 (x y : M (P F) \u03b1),\n              Mcongr x y \u2192\n                (fun x => (TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) x)) x =\n                  (fun x => (TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) x)) y)\n          (Quot.mk Mcongr a\u271d))\n    ((TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) a\u271d))"}, {"tactic": "rw [MvFunctor.map_map, MvFunctor.map_map, \u2190appendFun_comp_id, \u2190appendFun_comp_id]", "annotated_tactic": ["rw [<a>MvFunctor.map_map</a>, <a>MvFunctor.map_map</a>, \u2190<a>appendFun_comp_id</a>, \u2190<a>appendFun_comp_id</a>]", [{"full_name": "MvFunctor.map_map", "def_path": "Mathlib/Control/Functor/Multivariate.lean", "def_pos": [115, 9], "def_end_pos": [115, 16]}, {"full_name": "MvFunctor.map_map", "def_path": "Mathlib/Control/Functor/Multivariate.lean", "def_pos": [115, 9], "def_end_pos": [115, 16]}, {"full_name": "TypeVec.appendFun_comp_id", "def_path": "Mathlib/Data/TypeVec.lean", "def_pos": [270, 9], "def_end_pos": [270, 26]}, {"full_name": "TypeVec.appendFun_comp_id", "def_path": "Mathlib/Data/TypeVec.lean", "def_pos": [270, 9], "def_end_pos": [270, 26]}]], "state_before": "case mk\nn : \u2115\nF\u271d : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\u271d\nq\u271d : MvQPF F\u271d\nF : TypeVec.{u} (n + 1) \u2192 Type u\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nR : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => abs (repr y) = x\na\u271d : M (P F) \u03b1\n\u22a2 LiftR' (RelLast' \u03b1 fun x y => Quot.mk Mcongr (M.corec (P F) (fun x => MvQPF.repr (dest x)) y) = x)\n    ((TypeVec.id ::: Quot.mk Mcongr) <$$>\n      (TypeVec.id :::\n          M.corec (P F) fun x =>\n            MvQPF.repr\n              (Quot.lift (fun x => (TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) x))\n                (_ :\n                  \u2200 (x y : M (P F) \u03b1),\n                    Mcongr x y \u2192\n                      (fun x => (TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) x)) x =\n                        (fun x => (TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) x)) y)\n                x)) <$$>\n        Quot.lift (fun x => (TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) x))\n          (_ :\n            \u2200 (x y : M (P F) \u03b1),\n              Mcongr x y \u2192\n                (fun x => (TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) x)) x =\n                  (fun x => (TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) x)) y)\n          (Quot.mk Mcongr a\u271d))\n    ((TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) a\u271d))", "state_after": "case mk\nn : \u2115\nF\u271d : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\u271d\nq\u271d : MvQPF F\u271d\nF : TypeVec.{u} (n + 1) \u2192 Type u\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nR : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => abs (repr y) = x\na\u271d : M (P F) \u03b1\n\u22a2 LiftR' (RelLast' \u03b1 fun x y => Quot.mk Mcongr (M.corec (P F) (fun x => MvQPF.repr (dest x)) y) = x)\n    ((TypeVec.id :::\n        (Quot.mk Mcongr \u2218\n            M.corec (P F) fun x =>\n              MvQPF.repr\n                (Quot.lift (fun x => (TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) x))\n                  (_ :\n                    \u2200 (x y : M (P F) \u03b1),\n                      Mcongr x y \u2192\n                        (fun x => (TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) x)) x =\n                          (fun x => (TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) x)) y)\n                  x)) \u2218\n          Quot.mk Mcongr) <$$>\n      MvQPF.abs (M.dest (P F) a\u271d))\n    ((TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) a\u271d))"}, {"tactic": "apply liftR_map_last", "annotated_tactic": ["apply <a>liftR_map_last</a>", [{"full_name": "MvQPF.liftR_map_last", "def_path": "Mathlib/Data/QPF/Multivariate/Constructions/Cofix.lean", "def_pos": [390, 9], "def_end_pos": [390, 23]}]], "state_before": "case mk\nn : \u2115\nF\u271d : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\u271d\nq\u271d : MvQPF F\u271d\nF : TypeVec.{u} (n + 1) \u2192 Type u\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nR : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => abs (repr y) = x\na\u271d : M (P F) \u03b1\n\u22a2 LiftR' (RelLast' \u03b1 fun x y => Quot.mk Mcongr (M.corec (P F) (fun x => MvQPF.repr (dest x)) y) = x)\n    ((TypeVec.id :::\n        (Quot.mk Mcongr \u2218\n            M.corec (P F) fun x =>\n              MvQPF.repr\n                (Quot.lift (fun x => (TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) x))\n                  (_ :\n                    \u2200 (x y : M (P F) \u03b1),\n                      Mcongr x y \u2192\n                        (fun x => (TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) x)) x =\n                          (fun x => (TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) x)) y)\n                  x)) \u2218\n          Quot.mk Mcongr) <$$>\n      MvQPF.abs (M.dest (P F) a\u271d))\n    ((TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) a\u271d))", "state_after": "case mk.hh\nn : \u2115\nF\u271d : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\u271d\nq\u271d : MvQPF F\u271d\nF : TypeVec.{u} (n + 1) \u2192 Type u\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nR : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => abs (repr y) = x\na\u271d : M (P F) \u03b1\n\u22a2 \u2200 (x : M (P F) \u03b1),\n    Quot.mk Mcongr (M.corec (P F) (fun x => MvQPF.repr (dest x)) (Quot.mk Mcongr x)) =\n      ((Quot.mk Mcongr \u2218\n            M.corec (P F) fun x =>\n              MvQPF.repr\n                (Quot.lift (fun x => (TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) x))\n                  (_ :\n                    \u2200 (x y : M (P F) \u03b1),\n                      Mcongr x y \u2192\n                        (fun x => (TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) x)) x =\n                          (fun x => (TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) x)) y)\n                  x)) \u2218\n          Quot.mk Mcongr)\n        x"}, {"tactic": "intros", "annotated_tactic": ["intros", []], "state_before": "case mk.hh\nn : \u2115\nF\u271d : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\u271d\nq\u271d : MvQPF F\u271d\nF : TypeVec.{u} (n + 1) \u2192 Type u\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nR : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => abs (repr y) = x\na\u271d : M (P F) \u03b1\n\u22a2 \u2200 (x : M (P F) \u03b1),\n    Quot.mk Mcongr (M.corec (P F) (fun x => MvQPF.repr (dest x)) (Quot.mk Mcongr x)) =\n      ((Quot.mk Mcongr \u2218\n            M.corec (P F) fun x =>\n              MvQPF.repr\n                (Quot.lift (fun x => (TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) x))\n                  (_ :\n                    \u2200 (x y : M (P F) \u03b1),\n                      Mcongr x y \u2192\n                        (fun x => (TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) x)) x =\n                          (fun x => (TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) x)) y)\n                  x)) \u2218\n          Quot.mk Mcongr)\n        x", "state_after": "case mk.hh\nn : \u2115\nF\u271d : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\u271d\nq\u271d : MvQPF F\u271d\nF : TypeVec.{u} (n + 1) \u2192 Type u\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nR : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => abs (repr y) = x\na\u271d x\u271d : M (P F) \u03b1\n\u22a2 Quot.mk Mcongr (M.corec (P F) (fun x => MvQPF.repr (dest x)) (Quot.mk Mcongr x\u271d)) =\n    ((Quot.mk Mcongr \u2218\n          M.corec (P F) fun x =>\n            MvQPF.repr\n              (Quot.lift (fun x => (TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) x))\n                (_ :\n                  \u2200 (x y : M (P F) \u03b1),\n                    Mcongr x y \u2192\n                      (fun x => (TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) x)) x =\n                        (fun x => (TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) x)) y)\n                x)) \u2218\n        Quot.mk Mcongr)\n      x\u271d"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case mk.hh\nn : \u2115\nF\u271d : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\u271d\nq\u271d : MvQPF F\u271d\nF : TypeVec.{u} (n + 1) \u2192 Type u\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nR : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => abs (repr y) = x\na\u271d x\u271d : M (P F) \u03b1\n\u22a2 Quot.mk Mcongr (M.corec (P F) (fun x => MvQPF.repr (dest x)) (Quot.mk Mcongr x\u271d)) =\n    ((Quot.mk Mcongr \u2218\n          M.corec (P F) fun x =>\n            MvQPF.repr\n              (Quot.lift (fun x => (TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) x))\n                (_ :\n                  \u2200 (x y : M (P F) \u03b1),\n                    Mcongr x y \u2192\n                      (fun x => (TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) x)) x =\n                        (fun x => (TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) x)) y)\n                x)) \u2218\n        Quot.mk Mcongr)\n      x\u271d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Covering/DensityTheorem.lean", "full_name": "IsUnifLocDoublingMeasure.closedBall_mem_vitaliFamily_of_dist_le_mul", "start": [71, 1], "end": [109, 54], "traced_tactics": [{"tactic": "let R := scalingScaleOf \u03bc (max (4 * K + 3) 3)", "annotated_tactic": ["let R := <a>scalingScaleOf</a> \u03bc (<a>max</a> (4 * K + 3) 3)", [{"full_name": "IsUnifLocDoublingMeasure.scalingScaleOf", "def_path": "Mathlib/MeasureTheory/Measure/Doubling.lean", "def_pos": [152, 5], "def_end_pos": [152, 19]}, {"full_name": "Max.max", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1090, 3], "def_end_pos": [1090, 6]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : IsUnifLocDoublingMeasure \u03bc\ninst\u271d\u00b2 : SecondCountableTopology \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nK : \u211d\nx y : \u03b1\nr : \u211d\nh : dist x y \u2264 K * r\nrpos : 0 < r\n\u22a2 closedBall y r \u2208 VitaliFamily.setsAt (vitaliFamily \u03bc K) x", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : IsUnifLocDoublingMeasure \u03bc\ninst\u271d\u00b2 : SecondCountableTopology \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nK : \u211d\nx y : \u03b1\nr : \u211d\nh : dist x y \u2264 K * r\nrpos : 0 < r\nR : \u211d := scalingScaleOf \u03bc (max (4 * K + 3) 3)\n\u22a2 closedBall y r \u2208 VitaliFamily.setsAt (vitaliFamily \u03bc K) x"}, {"tactic": "simp only [vitaliFamily, VitaliFamily.enlarge, Vitali.vitaliFamily, mem_union, mem_setOf_eq,\n  isClosed_ball, true_and_iff, (nonempty_ball.2 rpos).mono ball_subset_interior_closedBall,\n  measurableSet_closedBall]", "annotated_tactic": ["simp only [<a>vitaliFamily</a>, <a>VitaliFamily.enlarge</a>, <a>Vitali.vitaliFamily</a>, <a>mem_union</a>, <a>mem_setOf_eq</a>,\n    <a>isClosed_ball</a>, <a>true_and_iff</a>, (<a>nonempty_ball</a>.2 rpos).<a>mono</a> <a>ball_subset_interior_closedBall</a>,\n    <a>measurableSet_closedBall</a>]", [{"full_name": "IsUnifLocDoublingMeasure.vitaliFamily", "def_path": "Mathlib/MeasureTheory/Covering/DensityTheorem.lean", "def_pos": [51, 17], "def_end_pos": [51, 29]}, {"full_name": "VitaliFamily.enlarge", "def_path": "Mathlib/MeasureTheory/Covering/VitaliFamily.lean", "def_pos": [186, 5], "def_end_pos": [186, 12]}, {"full_name": "Vitali.vitaliFamily", "def_path": "Mathlib/MeasureTheory/Covering/Vitali.lean", "def_pos": [402, 15], "def_end_pos": [402, 27]}, {"full_name": "Set.mem_union", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [767, 9], "def_end_pos": [767, 18]}, {"full_name": "Set.mem_setOf_eq", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [256, 29], "def_end_pos": [256, 41]}, {"full_name": "Metric.isClosed_ball", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [1890, 9], "def_end_pos": [1890, 22]}, {"full_name": "true_and_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [147, 9], "def_end_pos": [147, 21]}, {"full_name": "Metric.nonempty_ball", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [430, 9], "def_end_pos": [430, 22]}, {"full_name": "Set.Nonempty.mono", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [476, 9], "def_end_pos": [476, 22]}, {"full_name": "Metric.ball_subset_interior_closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [1920, 9], "def_end_pos": [1920, 40]}, {"full_name": "measurableSet_closedBall", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [1681, 9], "def_end_pos": [1681, 33]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : IsUnifLocDoublingMeasure \u03bc\ninst\u271d\u00b2 : SecondCountableTopology \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nK : \u211d\nx y : \u03b1\nr : \u211d\nh : dist x y \u2264 K * r\nrpos : 0 < r\nR : \u211d := scalingScaleOf \u03bc (max (4 * K + 3) 3)\n\u22a2 closedBall y r \u2208 VitaliFamily.setsAt (vitaliFamily \u03bc K) x", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : IsUnifLocDoublingMeasure \u03bc\ninst\u271d\u00b2 : SecondCountableTopology \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nK : \u211d\nx y : \u03b1\nr : \u211d\nh : dist x y \u2264 K * r\nrpos : 0 < r\nR : \u211d := scalingScaleOf \u03bc (max (4 * K + 3) 3)\n\u22a2 (\u2203 r_1,\n      closedBall y r \u2286 closedBall x r_1 \u2227\n        \u2191\u2191\u03bc (closedBall x (3 * r_1)) \u2264 \u2191(scalingConstantOf \u03bc (max (4 * K + 3) 3)) * \u2191\u2191\u03bc (closedBall y r)) \u2228\n    \u00acclosedBall y r \u2286 closedBall x (scalingScaleOf \u03bc (max (4 * K + 3) 3) / 4)"}, {"tactic": "by_cases H : closedBall y r \u2286 closedBall x (R / 4)", "annotated_tactic": ["by_cases H : <a>closedBall</a> y r \u2286 <a>closedBall</a> x (R / 4)", [{"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : IsUnifLocDoublingMeasure \u03bc\ninst\u271d\u00b2 : SecondCountableTopology \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nK : \u211d\nx y : \u03b1\nr : \u211d\nh : dist x y \u2264 K * r\nrpos : 0 < r\nR : \u211d := scalingScaleOf \u03bc (max (4 * K + 3) 3)\n\u22a2 (\u2203 r_1,\n      closedBall y r \u2286 closedBall x r_1 \u2227\n        \u2191\u2191\u03bc (closedBall x (3 * r_1)) \u2264 \u2191(scalingConstantOf \u03bc (max (4 * K + 3) 3)) * \u2191\u2191\u03bc (closedBall y r)) \u2228\n    \u00acclosedBall y r \u2286 closedBall x (scalingScaleOf \u03bc (max (4 * K + 3) 3) / 4)", "state_after": "case pos\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : IsUnifLocDoublingMeasure \u03bc\ninst\u271d\u00b2 : SecondCountableTopology \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nK : \u211d\nx y : \u03b1\nr : \u211d\nh : dist x y \u2264 K * r\nrpos : 0 < r\nR : \u211d := scalingScaleOf \u03bc (max (4 * K + 3) 3)\nH : closedBall y r \u2286 closedBall x (R / 4)\n\u22a2 (\u2203 r_1,\n      closedBall y r \u2286 closedBall x r_1 \u2227\n        \u2191\u2191\u03bc (closedBall x (3 * r_1)) \u2264 \u2191(scalingConstantOf \u03bc (max (4 * K + 3) 3)) * \u2191\u2191\u03bc (closedBall y r)) \u2228\n    \u00acclosedBall y r \u2286 closedBall x (scalingScaleOf \u03bc (max (4 * K + 3) 3) / 4)\n\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : IsUnifLocDoublingMeasure \u03bc\ninst\u271d\u00b2 : SecondCountableTopology \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nK : \u211d\nx y : \u03b1\nr : \u211d\nh : dist x y \u2264 K * r\nrpos : 0 < r\nR : \u211d := scalingScaleOf \u03bc (max (4 * K + 3) 3)\nH : \u00acclosedBall y r \u2286 closedBall x (R / 4)\n\u22a2 (\u2203 r_1,\n      closedBall y r \u2286 closedBall x r_1 \u2227\n        \u2191\u2191\u03bc (closedBall x (3 * r_1)) \u2264 \u2191(scalingConstantOf \u03bc (max (4 * K + 3) 3)) * \u2191\u2191\u03bc (closedBall y r)) \u2228\n    \u00acclosedBall y r \u2286 closedBall x (scalingScaleOf \u03bc (max (4 * K + 3) 3) / 4)"}, {"tactic": "swap", "annotated_tactic": ["swap", []], "state_before": "case pos\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : IsUnifLocDoublingMeasure \u03bc\ninst\u271d\u00b2 : SecondCountableTopology \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nK : \u211d\nx y : \u03b1\nr : \u211d\nh : dist x y \u2264 K * r\nrpos : 0 < r\nR : \u211d := scalingScaleOf \u03bc (max (4 * K + 3) 3)\nH : closedBall y r \u2286 closedBall x (R / 4)\n\u22a2 (\u2203 r_1,\n      closedBall y r \u2286 closedBall x r_1 \u2227\n        \u2191\u2191\u03bc (closedBall x (3 * r_1)) \u2264 \u2191(scalingConstantOf \u03bc (max (4 * K + 3) 3)) * \u2191\u2191\u03bc (closedBall y r)) \u2228\n    \u00acclosedBall y r \u2286 closedBall x (scalingScaleOf \u03bc (max (4 * K + 3) 3) / 4)\n\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : IsUnifLocDoublingMeasure \u03bc\ninst\u271d\u00b2 : SecondCountableTopology \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nK : \u211d\nx y : \u03b1\nr : \u211d\nh : dist x y \u2264 K * r\nrpos : 0 < r\nR : \u211d := scalingScaleOf \u03bc (max (4 * K + 3) 3)\nH : \u00acclosedBall y r \u2286 closedBall x (R / 4)\n\u22a2 (\u2203 r_1,\n      closedBall y r \u2286 closedBall x r_1 \u2227\n        \u2191\u2191\u03bc (closedBall x (3 * r_1)) \u2264 \u2191(scalingConstantOf \u03bc (max (4 * K + 3) 3)) * \u2191\u2191\u03bc (closedBall y r)) \u2228\n    \u00acclosedBall y r \u2286 closedBall x (scalingScaleOf \u03bc (max (4 * K + 3) 3) / 4)", "state_after": "case neg\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : IsUnifLocDoublingMeasure \u03bc\ninst\u271d\u00b2 : SecondCountableTopology \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nK : \u211d\nx y : \u03b1\nr : \u211d\nh : dist x y \u2264 K * r\nrpos : 0 < r\nR : \u211d := scalingScaleOf \u03bc (max (4 * K + 3) 3)\nH : \u00acclosedBall y r \u2286 closedBall x (R / 4)\n\u22a2 (\u2203 r_1,\n      closedBall y r \u2286 closedBall x r_1 \u2227\n        \u2191\u2191\u03bc (closedBall x (3 * r_1)) \u2264 \u2191(scalingConstantOf \u03bc (max (4 * K + 3) 3)) * \u2191\u2191\u03bc (closedBall y r)) \u2228\n    \u00acclosedBall y r \u2286 closedBall x (scalingScaleOf \u03bc (max (4 * K + 3) 3) / 4)\n\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : IsUnifLocDoublingMeasure \u03bc\ninst\u271d\u00b2 : SecondCountableTopology \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nK : \u211d\nx y : \u03b1\nr : \u211d\nh : dist x y \u2264 K * r\nrpos : 0 < r\nR : \u211d := scalingScaleOf \u03bc (max (4 * K + 3) 3)\nH : closedBall y r \u2286 closedBall x (R / 4)\n\u22a2 (\u2203 r_1,\n      closedBall y r \u2286 closedBall x r_1 \u2227\n        \u2191\u2191\u03bc (closedBall x (3 * r_1)) \u2264 \u2191(scalingConstantOf \u03bc (max (4 * K + 3) 3)) * \u2191\u2191\u03bc (closedBall y r)) \u2228\n    \u00acclosedBall y r \u2286 closedBall x (scalingScaleOf \u03bc (max (4 * K + 3) 3) / 4)"}, {"tactic": "left", "annotated_tactic": ["left", []], "state_before": "case pos\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : IsUnifLocDoublingMeasure \u03bc\ninst\u271d\u00b2 : SecondCountableTopology \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nK : \u211d\nx y : \u03b1\nr : \u211d\nh : dist x y \u2264 K * r\nrpos : 0 < r\nR : \u211d := scalingScaleOf \u03bc (max (4 * K + 3) 3)\nH : closedBall y r \u2286 closedBall x (R / 4)\n\u22a2 (\u2203 r_1,\n      closedBall y r \u2286 closedBall x r_1 \u2227\n        \u2191\u2191\u03bc (closedBall x (3 * r_1)) \u2264 \u2191(scalingConstantOf \u03bc (max (4 * K + 3) 3)) * \u2191\u2191\u03bc (closedBall y r)) \u2228\n    \u00acclosedBall y r \u2286 closedBall x (scalingScaleOf \u03bc (max (4 * K + 3) 3) / 4)", "state_after": "case pos.h\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : IsUnifLocDoublingMeasure \u03bc\ninst\u271d\u00b2 : SecondCountableTopology \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nK : \u211d\nx y : \u03b1\nr : \u211d\nh : dist x y \u2264 K * r\nrpos : 0 < r\nR : \u211d := scalingScaleOf \u03bc (max (4 * K + 3) 3)\nH : closedBall y r \u2286 closedBall x (R / 4)\n\u22a2 \u2203 r_1,\n    closedBall y r \u2286 closedBall x r_1 \u2227\n      \u2191\u2191\u03bc (closedBall x (3 * r_1)) \u2264 \u2191(scalingConstantOf \u03bc (max (4 * K + 3) 3)) * \u2191\u2191\u03bc (closedBall y r)"}, {"tactic": "rcases le_or_lt r R with (hr | hr)", "annotated_tactic": ["rcases <a>le_or_lt</a> r R with (hr | hr)", [{"full_name": "le_or_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [340, 9], "def_end_pos": [340, 17]}]], "state_before": "case pos.h\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : IsUnifLocDoublingMeasure \u03bc\ninst\u271d\u00b2 : SecondCountableTopology \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nK : \u211d\nx y : \u03b1\nr : \u211d\nh : dist x y \u2264 K * r\nrpos : 0 < r\nR : \u211d := scalingScaleOf \u03bc (max (4 * K + 3) 3)\nH : closedBall y r \u2286 closedBall x (R / 4)\n\u22a2 \u2203 r_1,\n    closedBall y r \u2286 closedBall x r_1 \u2227\n      \u2191\u2191\u03bc (closedBall x (3 * r_1)) \u2264 \u2191(scalingConstantOf \u03bc (max (4 * K + 3) 3)) * \u2191\u2191\u03bc (closedBall y r)", "state_after": "case pos.h.inl\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : IsUnifLocDoublingMeasure \u03bc\ninst\u271d\u00b2 : SecondCountableTopology \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nK : \u211d\nx y : \u03b1\nr : \u211d\nh : dist x y \u2264 K * r\nrpos : 0 < r\nR : \u211d := scalingScaleOf \u03bc (max (4 * K + 3) 3)\nH : closedBall y r \u2286 closedBall x (R / 4)\nhr : r \u2264 R\n\u22a2 \u2203 r_1,\n    closedBall y r \u2286 closedBall x r_1 \u2227\n      \u2191\u2191\u03bc (closedBall x (3 * r_1)) \u2264 \u2191(scalingConstantOf \u03bc (max (4 * K + 3) 3)) * \u2191\u2191\u03bc (closedBall y r)\n\ncase pos.h.inr\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : IsUnifLocDoublingMeasure \u03bc\ninst\u271d\u00b2 : SecondCountableTopology \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nK : \u211d\nx y : \u03b1\nr : \u211d\nh : dist x y \u2264 K * r\nrpos : 0 < r\nR : \u211d := scalingScaleOf \u03bc (max (4 * K + 3) 3)\nH : closedBall y r \u2286 closedBall x (R / 4)\nhr : R < r\n\u22a2 \u2203 r_1,\n    closedBall y r \u2286 closedBall x r_1 \u2227\n      \u2191\u2191\u03bc (closedBall x (3 * r_1)) \u2264 \u2191(scalingConstantOf \u03bc (max (4 * K + 3) 3)) * \u2191\u2191\u03bc (closedBall y r)"}, {"tactic": "exact Or.inr H", "annotated_tactic": ["exact <a>Or.inr</a> H", [{"full_name": "Or.inr", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [519, 5], "def_end_pos": [519, 8]}]], "state_before": "case neg\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : IsUnifLocDoublingMeasure \u03bc\ninst\u271d\u00b2 : SecondCountableTopology \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nK : \u211d\nx y : \u03b1\nr : \u211d\nh : dist x y \u2264 K * r\nrpos : 0 < r\nR : \u211d := scalingScaleOf \u03bc (max (4 * K + 3) 3)\nH : \u00acclosedBall y r \u2286 closedBall x (R / 4)\n\u22a2 (\u2203 r_1,\n      closedBall y r \u2286 closedBall x r_1 \u2227\n        \u2191\u2191\u03bc (closedBall x (3 * r_1)) \u2264 \u2191(scalingConstantOf \u03bc (max (4 * K + 3) 3)) * \u2191\u2191\u03bc (closedBall y r)) \u2228\n    \u00acclosedBall y r \u2286 closedBall x (scalingScaleOf \u03bc (max (4 * K + 3) 3) / 4)", "state_after": "no goals"}, {"tactic": "refine' \u27e8(K + 1) * r, _\u27e9", "annotated_tactic": ["refine' \u27e8(K + 1) * r, _\u27e9", []], "state_before": "case pos.h.inl\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : IsUnifLocDoublingMeasure \u03bc\ninst\u271d\u00b2 : SecondCountableTopology \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nK : \u211d\nx y : \u03b1\nr : \u211d\nh : dist x y \u2264 K * r\nrpos : 0 < r\nR : \u211d := scalingScaleOf \u03bc (max (4 * K + 3) 3)\nH : closedBall y r \u2286 closedBall x (R / 4)\nhr : r \u2264 R\n\u22a2 \u2203 r_1,\n    closedBall y r \u2286 closedBall x r_1 \u2227\n      \u2191\u2191\u03bc (closedBall x (3 * r_1)) \u2264 \u2191(scalingConstantOf \u03bc (max (4 * K + 3) 3)) * \u2191\u2191\u03bc (closedBall y r)", "state_after": "case pos.h.inl\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : IsUnifLocDoublingMeasure \u03bc\ninst\u271d\u00b2 : SecondCountableTopology \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nK : \u211d\nx y : \u03b1\nr : \u211d\nh : dist x y \u2264 K * r\nrpos : 0 < r\nR : \u211d := scalingScaleOf \u03bc (max (4 * K + 3) 3)\nH : closedBall y r \u2286 closedBall x (R / 4)\nhr : r \u2264 R\n\u22a2 closedBall y r \u2286 closedBall x ((K + 1) * r) \u2227\n    \u2191\u2191\u03bc (closedBall x (3 * ((K + 1) * r))) \u2264 \u2191(scalingConstantOf \u03bc (max (4 * K + 3) 3)) * \u2191\u2191\u03bc (closedBall y r)"}, {"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "case pos.h.inl\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : IsUnifLocDoublingMeasure \u03bc\ninst\u271d\u00b2 : SecondCountableTopology \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nK : \u211d\nx y : \u03b1\nr : \u211d\nh : dist x y \u2264 K * r\nrpos : 0 < r\nR : \u211d := scalingScaleOf \u03bc (max (4 * K + 3) 3)\nH : closedBall y r \u2286 closedBall x (R / 4)\nhr : r \u2264 R\n\u22a2 closedBall y r \u2286 closedBall x ((K + 1) * r) \u2227\n    \u2191\u2191\u03bc (closedBall x (3 * ((K + 1) * r))) \u2264 \u2191(scalingConstantOf \u03bc (max (4 * K + 3) 3)) * \u2191\u2191\u03bc (closedBall y r)", "state_after": "case pos.h.inl.left\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : IsUnifLocDoublingMeasure \u03bc\ninst\u271d\u00b2 : SecondCountableTopology \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nK : \u211d\nx y : \u03b1\nr : \u211d\nh : dist x y \u2264 K * r\nrpos : 0 < r\nR : \u211d := scalingScaleOf \u03bc (max (4 * K + 3) 3)\nH : closedBall y r \u2286 closedBall x (R / 4)\nhr : r \u2264 R\n\u22a2 closedBall y r \u2286 closedBall x ((K + 1) * r)\n\ncase pos.h.inl.right\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : IsUnifLocDoublingMeasure \u03bc\ninst\u271d\u00b2 : SecondCountableTopology \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nK : \u211d\nx y : \u03b1\nr : \u211d\nh : dist x y \u2264 K * r\nrpos : 0 < r\nR : \u211d := scalingScaleOf \u03bc (max (4 * K + 3) 3)\nH : closedBall y r \u2286 closedBall x (R / 4)\nhr : r \u2264 R\n\u22a2 \u2191\u2191\u03bc (closedBall x (3 * ((K + 1) * r))) \u2264 \u2191(scalingConstantOf \u03bc (max (4 * K + 3) 3)) * \u2191\u2191\u03bc (closedBall y r)"}, {"tactic": "apply closedBall_subset_closedBall'", "annotated_tactic": ["apply <a>closedBall_subset_closedBall'</a>", [{"full_name": "Metric.closedBall_subset_closedBall'", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [613, 9], "def_end_pos": [613, 38]}]], "state_before": "case pos.h.inl.left\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : IsUnifLocDoublingMeasure \u03bc\ninst\u271d\u00b2 : SecondCountableTopology \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nK : \u211d\nx y : \u03b1\nr : \u211d\nh : dist x y \u2264 K * r\nrpos : 0 < r\nR : \u211d := scalingScaleOf \u03bc (max (4 * K + 3) 3)\nH : closedBall y r \u2286 closedBall x (R / 4)\nhr : r \u2264 R\n\u22a2 closedBall y r \u2286 closedBall x ((K + 1) * r)", "state_after": "case pos.h.inl.left.h\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : IsUnifLocDoublingMeasure \u03bc\ninst\u271d\u00b2 : SecondCountableTopology \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nK : \u211d\nx y : \u03b1\nr : \u211d\nh : dist x y \u2264 K * r\nrpos : 0 < r\nR : \u211d := scalingScaleOf \u03bc (max (4 * K + 3) 3)\nH : closedBall y r \u2286 closedBall x (R / 4)\nhr : r \u2264 R\n\u22a2 r + dist y x \u2264 (K + 1) * r"}, {"tactic": "rw [dist_comm]", "annotated_tactic": ["rw [<a>dist_comm</a>]", [{"full_name": "dist_comm", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [188, 9], "def_end_pos": [188, 18]}]], "state_before": "case pos.h.inl.left.h\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : IsUnifLocDoublingMeasure \u03bc\ninst\u271d\u00b2 : SecondCountableTopology \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nK : \u211d\nx y : \u03b1\nr : \u211d\nh : dist x y \u2264 K * r\nrpos : 0 < r\nR : \u211d := scalingScaleOf \u03bc (max (4 * K + 3) 3)\nH : closedBall y r \u2286 closedBall x (R / 4)\nhr : r \u2264 R\n\u22a2 r + dist y x \u2264 (K + 1) * r", "state_after": "case pos.h.inl.left.h\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : IsUnifLocDoublingMeasure \u03bc\ninst\u271d\u00b2 : SecondCountableTopology \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nK : \u211d\nx y : \u03b1\nr : \u211d\nh : dist x y \u2264 K * r\nrpos : 0 < r\nR : \u211d := scalingScaleOf \u03bc (max (4 * K + 3) 3)\nH : closedBall y r \u2286 closedBall x (R / 4)\nhr : r \u2264 R\n\u22a2 r + dist x y \u2264 (K + 1) * r"}, {"tactic": "linarith", "annotated_tactic": ["linarith", []], "state_before": "case pos.h.inl.left.h\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : IsUnifLocDoublingMeasure \u03bc\ninst\u271d\u00b2 : SecondCountableTopology \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nK : \u211d\nx y : \u03b1\nr : \u211d\nh : dist x y \u2264 K * r\nrpos : 0 < r\nR : \u211d := scalingScaleOf \u03bc (max (4 * K + 3) 3)\nH : closedBall y r \u2286 closedBall x (R / 4)\nhr : r \u2264 R\n\u22a2 r + dist x y \u2264 (K + 1) * r", "state_after": "no goals"}, {"tactic": "have I1 : closedBall x (3 * ((K + 1) * r)) \u2286 closedBall y ((4 * K + 3) * r) := by\n  apply closedBall_subset_closedBall'\n  linarith", "annotated_tactic": ["have I1 : <a>closedBall</a> x (3 * ((K + 1) * r)) \u2286 <a>closedBall</a> y ((4 * K + 3) * r) := by\n        apply <a>closedBall_subset_closedBall'</a>\n        linarith", [{"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "Metric.closedBall_subset_closedBall'", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [613, 9], "def_end_pos": [613, 38]}]], "state_before": "case pos.h.inl.right\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : IsUnifLocDoublingMeasure \u03bc\ninst\u271d\u00b2 : SecondCountableTopology \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nK : \u211d\nx y : \u03b1\nr : \u211d\nh : dist x y \u2264 K * r\nrpos : 0 < r\nR : \u211d := scalingScaleOf \u03bc (max (4 * K + 3) 3)\nH : closedBall y r \u2286 closedBall x (R / 4)\nhr : r \u2264 R\n\u22a2 \u2191\u2191\u03bc (closedBall x (3 * ((K + 1) * r))) \u2264 \u2191(scalingConstantOf \u03bc (max (4 * K + 3) 3)) * \u2191\u2191\u03bc (closedBall y r)", "state_after": "case pos.h.inl.right\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : IsUnifLocDoublingMeasure \u03bc\ninst\u271d\u00b2 : SecondCountableTopology \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nK : \u211d\nx y : \u03b1\nr : \u211d\nh : dist x y \u2264 K * r\nrpos : 0 < r\nR : \u211d := scalingScaleOf \u03bc (max (4 * K + 3) 3)\nH : closedBall y r \u2286 closedBall x (R / 4)\nhr : r \u2264 R\nI1 : closedBall x (3 * ((K + 1) * r)) \u2286 closedBall y ((4 * K + 3) * r)\n\u22a2 \u2191\u2191\u03bc (closedBall x (3 * ((K + 1) * r))) \u2264 \u2191(scalingConstantOf \u03bc (max (4 * K + 3) 3)) * \u2191\u2191\u03bc (closedBall y r)"}, {"tactic": "have I2 : closedBall y ((4 * K + 3) * r) \u2286 closedBall y (max (4 * K + 3) 3 * r) := by\n  apply closedBall_subset_closedBall\n  exact mul_le_mul_of_nonneg_right (le_max_left _ _) rpos.le", "annotated_tactic": ["have I2 : <a>closedBall</a> y ((4 * K + 3) * r) \u2286 <a>closedBall</a> y (<a>max</a> (4 * K + 3) 3 * r) := by\n        apply <a>closedBall_subset_closedBall</a>\n        exact <a>mul_le_mul_of_nonneg_right</a> (<a>le_max_left</a> _ _) rpos.le", [{"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "Max.max", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1090, 3], "def_end_pos": [1090, 6]}, {"full_name": "Metric.closedBall_subset_closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [609, 9], "def_end_pos": [609, 37]}, {"full_name": "mul_le_mul_of_nonneg_right", "def_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "def_pos": [156, 9], "def_end_pos": [156, 35]}, {"full_name": "le_max_left", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [54, 9], "def_end_pos": [54, 20]}]], "state_before": "case pos.h.inl.right\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : IsUnifLocDoublingMeasure \u03bc\ninst\u271d\u00b2 : SecondCountableTopology \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nK : \u211d\nx y : \u03b1\nr : \u211d\nh : dist x y \u2264 K * r\nrpos : 0 < r\nR : \u211d := scalingScaleOf \u03bc (max (4 * K + 3) 3)\nH : closedBall y r \u2286 closedBall x (R / 4)\nhr : r \u2264 R\nI1 : closedBall x (3 * ((K + 1) * r)) \u2286 closedBall y ((4 * K + 3) * r)\n\u22a2 \u2191\u2191\u03bc (closedBall x (3 * ((K + 1) * r))) \u2264 \u2191(scalingConstantOf \u03bc (max (4 * K + 3) 3)) * \u2191\u2191\u03bc (closedBall y r)", "state_after": "case pos.h.inl.right\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : IsUnifLocDoublingMeasure \u03bc\ninst\u271d\u00b2 : SecondCountableTopology \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nK : \u211d\nx y : \u03b1\nr : \u211d\nh : dist x y \u2264 K * r\nrpos : 0 < r\nR : \u211d := scalingScaleOf \u03bc (max (4 * K + 3) 3)\nH : closedBall y r \u2286 closedBall x (R / 4)\nhr : r \u2264 R\nI1 : closedBall x (3 * ((K + 1) * r)) \u2286 closedBall y ((4 * K + 3) * r)\nI2 : closedBall y ((4 * K + 3) * r) \u2286 closedBall y (max (4 * K + 3) 3 * r)\n\u22a2 \u2191\u2191\u03bc (closedBall x (3 * ((K + 1) * r))) \u2264 \u2191(scalingConstantOf \u03bc (max (4 * K + 3) 3)) * \u2191\u2191\u03bc (closedBall y r)"}, {"tactic": "apply (measure_mono (I1.trans I2)).trans", "annotated_tactic": ["apply (<a>measure_mono</a> (I1.trans I2)).<a>trans</a>", [{"full_name": "MeasureTheory.measure_mono", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [193, 9], "def_end_pos": [193, 21]}, {"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [120, 7], "def_end_pos": [120, 18]}]], "state_before": "case pos.h.inl.right\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : IsUnifLocDoublingMeasure \u03bc\ninst\u271d\u00b2 : SecondCountableTopology \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nK : \u211d\nx y : \u03b1\nr : \u211d\nh : dist x y \u2264 K * r\nrpos : 0 < r\nR : \u211d := scalingScaleOf \u03bc (max (4 * K + 3) 3)\nH : closedBall y r \u2286 closedBall x (R / 4)\nhr : r \u2264 R\nI1 : closedBall x (3 * ((K + 1) * r)) \u2286 closedBall y ((4 * K + 3) * r)\nI2 : closedBall y ((4 * K + 3) * r) \u2286 closedBall y (max (4 * K + 3) 3 * r)\n\u22a2 \u2191\u2191\u03bc (closedBall x (3 * ((K + 1) * r))) \u2264 \u2191(scalingConstantOf \u03bc (max (4 * K + 3) 3)) * \u2191\u2191\u03bc (closedBall y r)", "state_after": "case pos.h.inl.right\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : IsUnifLocDoublingMeasure \u03bc\ninst\u271d\u00b2 : SecondCountableTopology \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nK : \u211d\nx y : \u03b1\nr : \u211d\nh : dist x y \u2264 K * r\nrpos : 0 < r\nR : \u211d := scalingScaleOf \u03bc (max (4 * K + 3) 3)\nH : closedBall y r \u2286 closedBall x (R / 4)\nhr : r \u2264 R\nI1 : closedBall x (3 * ((K + 1) * r)) \u2286 closedBall y ((4 * K + 3) * r)\nI2 : closedBall y ((4 * K + 3) * r) \u2286 closedBall y (max (4 * K + 3) 3 * r)\n\u22a2 \u2191\u2191\u03bc (closedBall y (max (4 * K + 3) 3 * r)) \u2264 \u2191(scalingConstantOf \u03bc (max (4 * K + 3) 3)) * \u2191\u2191\u03bc (closedBall y r)"}, {"tactic": "exact measure_mul_le_scalingConstantOf_mul _\n  \u27e8zero_lt_three.trans_le (le_max_right _ _), le_rfl\u27e9 hr", "annotated_tactic": ["exact <a>measure_mul_le_scalingConstantOf_mul</a> _\n        \u27e8zero_lt_three.trans_le (<a>le_max_right</a> _ _), <a>le_rfl</a>\u27e9 hr", [{"full_name": "IsUnifLocDoublingMeasure.measure_mul_le_scalingConstantOf_mul", "def_path": "Mathlib/MeasureTheory/Measure/Doubling.lean", "def_pos": [160, 9], "def_end_pos": [160, 45]}, {"full_name": "le_max_right", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [61, 9], "def_end_pos": [61, 21]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}]], "state_before": "case pos.h.inl.right\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : IsUnifLocDoublingMeasure \u03bc\ninst\u271d\u00b2 : SecondCountableTopology \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nK : \u211d\nx y : \u03b1\nr : \u211d\nh : dist x y \u2264 K * r\nrpos : 0 < r\nR : \u211d := scalingScaleOf \u03bc (max (4 * K + 3) 3)\nH : closedBall y r \u2286 closedBall x (R / 4)\nhr : r \u2264 R\nI1 : closedBall x (3 * ((K + 1) * r)) \u2286 closedBall y ((4 * K + 3) * r)\nI2 : closedBall y ((4 * K + 3) * r) \u2286 closedBall y (max (4 * K + 3) 3 * r)\n\u22a2 \u2191\u2191\u03bc (closedBall y (max (4 * K + 3) 3 * r)) \u2264 \u2191(scalingConstantOf \u03bc (max (4 * K + 3) 3)) * \u2191\u2191\u03bc (closedBall y r)", "state_after": "no goals"}, {"tactic": "apply closedBall_subset_closedBall'", "annotated_tactic": ["apply <a>closedBall_subset_closedBall'</a>", [{"full_name": "Metric.closedBall_subset_closedBall'", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [613, 9], "def_end_pos": [613, 38]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : IsUnifLocDoublingMeasure \u03bc\ninst\u271d\u00b2 : SecondCountableTopology \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nK : \u211d\nx y : \u03b1\nr : \u211d\nh : dist x y \u2264 K * r\nrpos : 0 < r\nR : \u211d := scalingScaleOf \u03bc (max (4 * K + 3) 3)\nH : closedBall y r \u2286 closedBall x (R / 4)\nhr : r \u2264 R\n\u22a2 closedBall x (3 * ((K + 1) * r)) \u2286 closedBall y ((4 * K + 3) * r)", "state_after": "case h\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : IsUnifLocDoublingMeasure \u03bc\ninst\u271d\u00b2 : SecondCountableTopology \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nK : \u211d\nx y : \u03b1\nr : \u211d\nh : dist x y \u2264 K * r\nrpos : 0 < r\nR : \u211d := scalingScaleOf \u03bc (max (4 * K + 3) 3)\nH : closedBall y r \u2286 closedBall x (R / 4)\nhr : r \u2264 R\n\u22a2 3 * ((K + 1) * r) + dist x y \u2264 (4 * K + 3) * r"}, {"tactic": "linarith", "annotated_tactic": ["linarith", []], "state_before": "case h\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : IsUnifLocDoublingMeasure \u03bc\ninst\u271d\u00b2 : SecondCountableTopology \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nK : \u211d\nx y : \u03b1\nr : \u211d\nh : dist x y \u2264 K * r\nrpos : 0 < r\nR : \u211d := scalingScaleOf \u03bc (max (4 * K + 3) 3)\nH : closedBall y r \u2286 closedBall x (R / 4)\nhr : r \u2264 R\n\u22a2 3 * ((K + 1) * r) + dist x y \u2264 (4 * K + 3) * r", "state_after": "no goals"}, {"tactic": "apply closedBall_subset_closedBall", "annotated_tactic": ["apply <a>closedBall_subset_closedBall</a>", [{"full_name": "Metric.closedBall_subset_closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [609, 9], "def_end_pos": [609, 37]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : IsUnifLocDoublingMeasure \u03bc\ninst\u271d\u00b2 : SecondCountableTopology \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nK : \u211d\nx y : \u03b1\nr : \u211d\nh : dist x y \u2264 K * r\nrpos : 0 < r\nR : \u211d := scalingScaleOf \u03bc (max (4 * K + 3) 3)\nH : closedBall y r \u2286 closedBall x (R / 4)\nhr : r \u2264 R\nI1 : closedBall x (3 * ((K + 1) * r)) \u2286 closedBall y ((4 * K + 3) * r)\n\u22a2 closedBall y ((4 * K + 3) * r) \u2286 closedBall y (max (4 * K + 3) 3 * r)", "state_after": "case h\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : IsUnifLocDoublingMeasure \u03bc\ninst\u271d\u00b2 : SecondCountableTopology \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nK : \u211d\nx y : \u03b1\nr : \u211d\nh : dist x y \u2264 K * r\nrpos : 0 < r\nR : \u211d := scalingScaleOf \u03bc (max (4 * K + 3) 3)\nH : closedBall y r \u2286 closedBall x (R / 4)\nhr : r \u2264 R\nI1 : closedBall x (3 * ((K + 1) * r)) \u2286 closedBall y ((4 * K + 3) * r)\n\u22a2 (4 * K + 3) * r \u2264 max (4 * K + 3) 3 * r"}, {"tactic": "exact mul_le_mul_of_nonneg_right (le_max_left _ _) rpos.le", "annotated_tactic": ["exact <a>mul_le_mul_of_nonneg_right</a> (<a>le_max_left</a> _ _) rpos.le", [{"full_name": "mul_le_mul_of_nonneg_right", "def_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "def_pos": [156, 9], "def_end_pos": [156, 35]}, {"full_name": "le_max_left", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [54, 9], "def_end_pos": [54, 20]}]], "state_before": "case h\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : IsUnifLocDoublingMeasure \u03bc\ninst\u271d\u00b2 : SecondCountableTopology \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nK : \u211d\nx y : \u03b1\nr : \u211d\nh : dist x y \u2264 K * r\nrpos : 0 < r\nR : \u211d := scalingScaleOf \u03bc (max (4 * K + 3) 3)\nH : closedBall y r \u2286 closedBall x (R / 4)\nhr : r \u2264 R\nI1 : closedBall x (3 * ((K + 1) * r)) \u2286 closedBall y ((4 * K + 3) * r)\n\u22a2 (4 * K + 3) * r \u2264 max (4 * K + 3) 3 * r", "state_after": "no goals"}, {"tactic": "refine' \u27e8R / 4, H, _\u27e9", "annotated_tactic": ["refine' \u27e8R / 4, H, _\u27e9", []], "state_before": "case pos.h.inr\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : IsUnifLocDoublingMeasure \u03bc\ninst\u271d\u00b2 : SecondCountableTopology \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nK : \u211d\nx y : \u03b1\nr : \u211d\nh : dist x y \u2264 K * r\nrpos : 0 < r\nR : \u211d := scalingScaleOf \u03bc (max (4 * K + 3) 3)\nH : closedBall y r \u2286 closedBall x (R / 4)\nhr : R < r\n\u22a2 \u2203 r_1,\n    closedBall y r \u2286 closedBall x r_1 \u2227\n      \u2191\u2191\u03bc (closedBall x (3 * r_1)) \u2264 \u2191(scalingConstantOf \u03bc (max (4 * K + 3) 3)) * \u2191\u2191\u03bc (closedBall y r)", "state_after": "case pos.h.inr\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : IsUnifLocDoublingMeasure \u03bc\ninst\u271d\u00b2 : SecondCountableTopology \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nK : \u211d\nx y : \u03b1\nr : \u211d\nh : dist x y \u2264 K * r\nrpos : 0 < r\nR : \u211d := scalingScaleOf \u03bc (max (4 * K + 3) 3)\nH : closedBall y r \u2286 closedBall x (R / 4)\nhr : R < r\n\u22a2 \u2191\u2191\u03bc (closedBall x (3 * (R / 4))) \u2264 \u2191(scalingConstantOf \u03bc (max (4 * K + 3) 3)) * \u2191\u2191\u03bc (closedBall y r)"}, {"tactic": "have : closedBall x (3 * (R / 4)) \u2286 closedBall y r := by\n  apply closedBall_subset_closedBall'\n  have A : y \u2208 closedBall y r := mem_closedBall_self rpos.le\n  have B := mem_closedBall'.1 (H A)\n  linarith", "annotated_tactic": ["have : <a>closedBall</a> x (3 * (R / 4)) \u2286 <a>closedBall</a> y r := by\n      apply <a>closedBall_subset_closedBall'</a>\n      have A : y \u2208 <a>closedBall</a> y r := <a>mem_closedBall_self</a> rpos.le\n      have B := <a>mem_closedBall'</a>.1 (H A)\n      linarith", [{"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "Metric.closedBall_subset_closedBall'", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [613, 9], "def_end_pos": [613, 38]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "Metric.mem_closedBall_self", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [515, 9], "def_end_pos": [515, 28]}, {"full_name": "Metric.mem_closedBall'", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [481, 9], "def_end_pos": [481, 24]}]], "state_before": "case pos.h.inr\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : IsUnifLocDoublingMeasure \u03bc\ninst\u271d\u00b2 : SecondCountableTopology \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nK : \u211d\nx y : \u03b1\nr : \u211d\nh : dist x y \u2264 K * r\nrpos : 0 < r\nR : \u211d := scalingScaleOf \u03bc (max (4 * K + 3) 3)\nH : closedBall y r \u2286 closedBall x (R / 4)\nhr : R < r\n\u22a2 \u2191\u2191\u03bc (closedBall x (3 * (R / 4))) \u2264 \u2191(scalingConstantOf \u03bc (max (4 * K + 3) 3)) * \u2191\u2191\u03bc (closedBall y r)", "state_after": "case pos.h.inr\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : IsUnifLocDoublingMeasure \u03bc\ninst\u271d\u00b2 : SecondCountableTopology \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nK : \u211d\nx y : \u03b1\nr : \u211d\nh : dist x y \u2264 K * r\nrpos : 0 < r\nR : \u211d := scalingScaleOf \u03bc (max (4 * K + 3) 3)\nH : closedBall y r \u2286 closedBall x (R / 4)\nhr : R < r\nthis : closedBall x (3 * (R / 4)) \u2286 closedBall y r\n\u22a2 \u2191\u2191\u03bc (closedBall x (3 * (R / 4))) \u2264 \u2191(scalingConstantOf \u03bc (max (4 * K + 3) 3)) * \u2191\u2191\u03bc (closedBall y r)"}, {"tactic": "apply (measure_mono this).trans _", "annotated_tactic": ["apply (<a>measure_mono</a> this).<a>trans</a> _", [{"full_name": "MeasureTheory.measure_mono", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [193, 9], "def_end_pos": [193, 21]}, {"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [120, 7], "def_end_pos": [120, 18]}]], "state_before": "case pos.h.inr\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : IsUnifLocDoublingMeasure \u03bc\ninst\u271d\u00b2 : SecondCountableTopology \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nK : \u211d\nx y : \u03b1\nr : \u211d\nh : dist x y \u2264 K * r\nrpos : 0 < r\nR : \u211d := scalingScaleOf \u03bc (max (4 * K + 3) 3)\nH : closedBall y r \u2286 closedBall x (R / 4)\nhr : R < r\nthis : closedBall x (3 * (R / 4)) \u2286 closedBall y r\n\u22a2 \u2191\u2191\u03bc (closedBall x (3 * (R / 4))) \u2264 \u2191(scalingConstantOf \u03bc (max (4 * K + 3) 3)) * \u2191\u2191\u03bc (closedBall y r)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : IsUnifLocDoublingMeasure \u03bc\ninst\u271d\u00b2 : SecondCountableTopology \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nK : \u211d\nx y : \u03b1\nr : \u211d\nh : dist x y \u2264 K * r\nrpos : 0 < r\nR : \u211d := scalingScaleOf \u03bc (max (4 * K + 3) 3)\nH : closedBall y r \u2286 closedBall x (R / 4)\nhr : R < r\nthis : closedBall x (3 * (R / 4)) \u2286 closedBall y r\n\u22a2 \u2191\u2191\u03bc (closedBall y r) \u2264 \u2191(scalingConstantOf \u03bc (max (4 * K + 3) 3)) * \u2191\u2191\u03bc (closedBall y r)"}, {"tactic": "refine' le_mul_of_one_le_left (zero_le _) _", "annotated_tactic": ["refine' <a>le_mul_of_one_le_left</a> (<a>zero_le</a> _) _", [{"full_name": "le_mul_of_one_le_left", "def_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "def_pos": [670, 9], "def_end_pos": [670, 30]}, {"full_name": "zero_le", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [217, 30], "def_end_pos": [217, 37]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : IsUnifLocDoublingMeasure \u03bc\ninst\u271d\u00b2 : SecondCountableTopology \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nK : \u211d\nx y : \u03b1\nr : \u211d\nh : dist x y \u2264 K * r\nrpos : 0 < r\nR : \u211d := scalingScaleOf \u03bc (max (4 * K + 3) 3)\nH : closedBall y r \u2286 closedBall x (R / 4)\nhr : R < r\nthis : closedBall x (3 * (R / 4)) \u2286 closedBall y r\n\u22a2 \u2191\u2191\u03bc (closedBall y r) \u2264 \u2191(scalingConstantOf \u03bc (max (4 * K + 3) 3)) * \u2191\u2191\u03bc (closedBall y r)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : IsUnifLocDoublingMeasure \u03bc\ninst\u271d\u00b2 : SecondCountableTopology \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nK : \u211d\nx y : \u03b1\nr : \u211d\nh : dist x y \u2264 K * r\nrpos : 0 < r\nR : \u211d := scalingScaleOf \u03bc (max (4 * K + 3) 3)\nH : closedBall y r \u2286 closedBall x (R / 4)\nhr : R < r\nthis : closedBall x (3 * (R / 4)) \u2286 closedBall y r\n\u22a2 1 \u2264 \u2191(scalingConstantOf \u03bc (max (4 * K + 3) 3))"}, {"tactic": "exact ENNReal.one_le_coe_iff.2 (le_max_right _ _)", "annotated_tactic": ["exact <a>ENNReal.one_le_coe_iff</a>.2 (<a>le_max_right</a> _ _)", [{"full_name": "ENNReal.one_le_coe_iff", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [691, 9], "def_end_pos": [691, 23]}, {"full_name": "le_max_right", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [61, 9], "def_end_pos": [61, 21]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : IsUnifLocDoublingMeasure \u03bc\ninst\u271d\u00b2 : SecondCountableTopology \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nK : \u211d\nx y : \u03b1\nr : \u211d\nh : dist x y \u2264 K * r\nrpos : 0 < r\nR : \u211d := scalingScaleOf \u03bc (max (4 * K + 3) 3)\nH : closedBall y r \u2286 closedBall x (R / 4)\nhr : R < r\nthis : closedBall x (3 * (R / 4)) \u2286 closedBall y r\n\u22a2 1 \u2264 \u2191(scalingConstantOf \u03bc (max (4 * K + 3) 3))", "state_after": "no goals"}, {"tactic": "apply closedBall_subset_closedBall'", "annotated_tactic": ["apply <a>closedBall_subset_closedBall'</a>", [{"full_name": "Metric.closedBall_subset_closedBall'", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [613, 9], "def_end_pos": [613, 38]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : IsUnifLocDoublingMeasure \u03bc\ninst\u271d\u00b2 : SecondCountableTopology \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nK : \u211d\nx y : \u03b1\nr : \u211d\nh : dist x y \u2264 K * r\nrpos : 0 < r\nR : \u211d := scalingScaleOf \u03bc (max (4 * K + 3) 3)\nH : closedBall y r \u2286 closedBall x (R / 4)\nhr : R < r\n\u22a2 closedBall x (3 * (R / 4)) \u2286 closedBall y r", "state_after": "case h\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : IsUnifLocDoublingMeasure \u03bc\ninst\u271d\u00b2 : SecondCountableTopology \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nK : \u211d\nx y : \u03b1\nr : \u211d\nh : dist x y \u2264 K * r\nrpos : 0 < r\nR : \u211d := scalingScaleOf \u03bc (max (4 * K + 3) 3)\nH : closedBall y r \u2286 closedBall x (R / 4)\nhr : R < r\n\u22a2 3 * (R / 4) + dist x y \u2264 r"}, {"tactic": "have A : y \u2208 closedBall y r := mem_closedBall_self rpos.le", "annotated_tactic": ["have A : y \u2208 <a>closedBall</a> y r := <a>mem_closedBall_self</a> rpos.le", [{"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "Metric.mem_closedBall_self", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [515, 9], "def_end_pos": [515, 28]}]], "state_before": "case h\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : IsUnifLocDoublingMeasure \u03bc\ninst\u271d\u00b2 : SecondCountableTopology \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nK : \u211d\nx y : \u03b1\nr : \u211d\nh : dist x y \u2264 K * r\nrpos : 0 < r\nR : \u211d := scalingScaleOf \u03bc (max (4 * K + 3) 3)\nH : closedBall y r \u2286 closedBall x (R / 4)\nhr : R < r\n\u22a2 3 * (R / 4) + dist x y \u2264 r", "state_after": "case h\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : IsUnifLocDoublingMeasure \u03bc\ninst\u271d\u00b2 : SecondCountableTopology \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nK : \u211d\nx y : \u03b1\nr : \u211d\nh : dist x y \u2264 K * r\nrpos : 0 < r\nR : \u211d := scalingScaleOf \u03bc (max (4 * K + 3) 3)\nH : closedBall y r \u2286 closedBall x (R / 4)\nhr : R < r\nA : y \u2208 closedBall y r\n\u22a2 3 * (R / 4) + dist x y \u2264 r"}, {"tactic": "have B := mem_closedBall'.1 (H A)", "annotated_tactic": ["have B := <a>mem_closedBall'</a>.1 (H A)", [{"full_name": "Metric.mem_closedBall'", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [481, 9], "def_end_pos": [481, 24]}]], "state_before": "case h\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : IsUnifLocDoublingMeasure \u03bc\ninst\u271d\u00b2 : SecondCountableTopology \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nK : \u211d\nx y : \u03b1\nr : \u211d\nh : dist x y \u2264 K * r\nrpos : 0 < r\nR : \u211d := scalingScaleOf \u03bc (max (4 * K + 3) 3)\nH : closedBall y r \u2286 closedBall x (R / 4)\nhr : R < r\nA : y \u2208 closedBall y r\n\u22a2 3 * (R / 4) + dist x y \u2264 r", "state_after": "case h\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : IsUnifLocDoublingMeasure \u03bc\ninst\u271d\u00b2 : SecondCountableTopology \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nK : \u211d\nx y : \u03b1\nr : \u211d\nh : dist x y \u2264 K * r\nrpos : 0 < r\nR : \u211d := scalingScaleOf \u03bc (max (4 * K + 3) 3)\nH : closedBall y r \u2286 closedBall x (R / 4)\nhr : R < r\nA : y \u2208 closedBall y r\nB : dist x y \u2264 R / 4\n\u22a2 3 * (R / 4) + dist x y \u2264 r"}, {"tactic": "linarith", "annotated_tactic": ["linarith", []], "state_before": "case h\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : IsUnifLocDoublingMeasure \u03bc\ninst\u271d\u00b2 : SecondCountableTopology \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nK : \u211d\nx y : \u03b1\nr : \u211d\nh : dist x y \u2264 K * r\nrpos : 0 < r\nR : \u211d := scalingScaleOf \u03bc (max (4 * K + 3) 3)\nH : closedBall y r \u2286 closedBall x (R / 4)\nhr : R < r\nA : y \u2208 closedBall y r\nB : dist x y \u2264 R / 4\n\u22a2 3 * (R / 4) + dist x y \u2264 r", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/Basic.lean", "full_name": "MvPolynomial.aevalTower_comp_algebraMap", "start": [1646, 1], "end": [1648, 24], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/StrongLaw.lean", "full_name": "ProbabilityTheory.strong_law_aux7", "start": [628, 1], "end": [640, 13], "traced_tactics": [{"tactic": "obtain \u27e8c, -, cone, clim\u27e9 :\n    \u2203 c : \u2115 \u2192 \u211d, StrictAnti c \u2227 (\u2200 n : \u2115, 1 < c n) \u2227 Tendsto c atTop (\ud835\udcdd 1) :=\n  exists_seq_strictAnti_tendsto (1 : \u211d)", "annotated_tactic": ["obtain \u27e8c, -, cone, clim\u27e9 :\n      \u2203 c : \u2115 \u2192 \u211d, <a>StrictAnti</a> c \u2227 (\u2200 n : \u2115, 1 < c n) \u2227 <a>Tendsto</a> c <a>atTop</a> (\ud835\udcdd 1) :=\n    <a>exists_seq_strictAnti_tendsto</a> (1 : \u211d)", [{"full_name": "StrictAnti", "def_path": "Mathlib/Order/Monotone/Basic.lean", "def_pos": [102, 5], "def_end_pos": [102, 15]}, {"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2939, 5], "def_end_pos": [2939, 12]}, {"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}, {"full_name": "exists_seq_strictAnti_tendsto", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [2258, 9], "def_end_pos": [2258, 38]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9), Tendsto (fun n => (\u2211 i in range n, X i \u03c9) / \u2191n) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a))", "state_after": "case intro.intro.intro\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u2115 \u2192 \u211d\ncone : \u2200 (n : \u2115), 1 < c n\nclim : Tendsto c atTop (\ud835\udcdd 1)\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9), Tendsto (fun n => (\u2211 i in range n, X i \u03c9) / \u2191n) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a))"}, {"tactic": "have : \u2200 k, \u2200\u1d50 \u03c9,\n    Tendsto (fun n : \u2115 => (\u2211 i in range \u230ac k ^ n\u230b\u208a, X i \u03c9) / \u230ac k ^ n\u230b\u208a) atTop (\ud835\udcdd \ud835\udd3c[X 0]) :=\n  fun k => strong_law_aux6 X hint hindep hident hnonneg (cone k)", "annotated_tactic": ["have : \u2200 k, \u2200\u1d50 \u03c9,\n      <a>Tendsto</a> (fun n : \u2115 => (\u2211 i in <a>range</a> \u230ac k ^ n\u230b\u208a, X i \u03c9) / \u230ac k ^ n\u230b\u208a) <a>atTop</a> (\ud835\udcdd \ud835\udd3c[X 0]) :=\n    fun k => <a>strong_law_aux6</a> X hint hindep hident hnonneg (cone k)", [{"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2939, 5], "def_end_pos": [2939, 12]}, {"full_name": "Finset.range", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3027, 5], "def_end_pos": [3027, 10]}, {"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}, {"full_name": "ProbabilityTheory.strong_law_aux6", "def_path": "Mathlib/Probability/StrongLaw.lean", "def_pos": [604, 9], "def_end_pos": [604, 24]}]], "state_before": "case intro.intro.intro\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u2115 \u2192 \u211d\ncone : \u2200 (n : \u2115), 1 < c n\nclim : Tendsto c atTop (\ud835\udcdd 1)\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9), Tendsto (fun n => (\u2211 i in range n, X i \u03c9) / \u2191n) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a))", "state_after": "case intro.intro.intro\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u2115 \u2192 \u211d\ncone : \u2200 (n : \u2115), 1 < c n\nclim : Tendsto c atTop (\ud835\udcdd 1)\nthis :\n  \u2200 (k : \u2115), \u2200\u1d50 (\u03c9 : \u03a9), Tendsto (fun n => (\u2211 i in range \u230ac k ^ n\u230b\u208a, X i \u03c9) / \u2191\u230ac k ^ n\u230b\u208a) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a))\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9), Tendsto (fun n => (\u2211 i in range n, X i \u03c9) / \u2191n) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a))"}, {"tactic": "filter_upwards [ae_all_iff.2 this] with \u03c9 h\u03c9", "annotated_tactic": ["filter_upwards [<a>ae_all_iff</a>.2 this] with \u03c9 h\u03c9", [{"full_name": "MeasureTheory.ae_all_iff", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [422, 9], "def_end_pos": [422, 19]}]], "state_before": "case intro.intro.intro\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u2115 \u2192 \u211d\ncone : \u2200 (n : \u2115), 1 < c n\nclim : Tendsto c atTop (\ud835\udcdd 1)\nthis :\n  \u2200 (k : \u2115), \u2200\u1d50 (\u03c9 : \u03a9), Tendsto (fun n => (\u2211 i in range \u230ac k ^ n\u230b\u208a, X i \u03c9) / \u2191\u230ac k ^ n\u230b\u208a) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a))\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9), Tendsto (fun n => (\u2211 i in range n, X i \u03c9) / \u2191n) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a))", "state_after": "case h\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u2115 \u2192 \u211d\ncone : \u2200 (n : \u2115), 1 < c n\nclim : Tendsto c atTop (\ud835\udcdd 1)\nthis :\n  \u2200 (k : \u2115), \u2200\u1d50 (\u03c9 : \u03a9), Tendsto (fun n => (\u2211 i in range \u230ac k ^ n\u230b\u208a, X i \u03c9) / \u2191\u230ac k ^ n\u230b\u208a) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a))\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (i : \u2115), Tendsto (fun n => (\u2211 i in range \u230ac i ^ n\u230b\u208a, X i \u03c9) / \u2191\u230ac i ^ n\u230b\u208a) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a))\n\u22a2 Tendsto (fun n => (\u2211 i in range n, X i \u03c9) / \u2191n) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a))"}, {"tactic": "apply tendsto_div_of_monotone_of_tendsto_div_floor_pow _ _ _ c cone clim _", "annotated_tactic": ["apply <a>tendsto_div_of_monotone_of_tendsto_div_floor_pow</a> _ _ _ c cone clim _", [{"full_name": "tendsto_div_of_monotone_of_tendsto_div_floor_pow", "def_path": "Mathlib/Analysis/SpecificLimits/FloorPow.lean", "def_pos": [209, 9], "def_end_pos": [209, 57]}]], "state_before": "case h\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u2115 \u2192 \u211d\ncone : \u2200 (n : \u2115), 1 < c n\nclim : Tendsto c atTop (\ud835\udcdd 1)\nthis :\n  \u2200 (k : \u2115), \u2200\u1d50 (\u03c9 : \u03a9), Tendsto (fun n => (\u2211 i in range \u230ac k ^ n\u230b\u208a, X i \u03c9) / \u2191\u230ac k ^ n\u230b\u208a) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a))\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (i : \u2115), Tendsto (fun n => (\u2211 i in range \u230ac i ^ n\u230b\u208a, X i \u03c9) / \u2191\u230ac i ^ n\u230b\u208a) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a))\n\u22a2 Tendsto (fun n => (\u2211 i in range n, X i \u03c9) / \u2191n) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a))", "state_after": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u2115 \u2192 \u211d\ncone : \u2200 (n : \u2115), 1 < c n\nclim : Tendsto c atTop (\ud835\udcdd 1)\nthis :\n  \u2200 (k : \u2115), \u2200\u1d50 (\u03c9 : \u03a9), Tendsto (fun n => (\u2211 i in range \u230ac k ^ n\u230b\u208a, X i \u03c9) / \u2191\u230ac k ^ n\u230b\u208a) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a))\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (i : \u2115), Tendsto (fun n => (\u2211 i in range \u230ac i ^ n\u230b\u208a, X i \u03c9) / \u2191\u230ac i ^ n\u230b\u208a) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a))\n\u22a2 Monotone fun n => \u2211 i in range n, X i \u03c9\n\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u2115 \u2192 \u211d\ncone : \u2200 (n : \u2115), 1 < c n\nclim : Tendsto c atTop (\ud835\udcdd 1)\nthis :\n  \u2200 (k : \u2115), \u2200\u1d50 (\u03c9 : \u03a9), Tendsto (fun n => (\u2211 i in range \u230ac k ^ n\u230b\u208a, X i \u03c9) / \u2191\u230ac k ^ n\u230b\u208a) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a))\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (i : \u2115), Tendsto (fun n => (\u2211 i in range \u230ac i ^ n\u230b\u208a, X i \u03c9) / \u2191\u230ac i ^ n\u230b\u208a) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a))\n\u22a2 \u2200 (k : \u2115), Tendsto (fun n => (\u2211 i in range \u230ac k ^ n\u230b\u208a, X i \u03c9) / \u2191\u230ac k ^ n\u230b\u208a) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a))"}, {"tactic": "intro m n hmn", "annotated_tactic": ["intro m n hmn", []], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u2115 \u2192 \u211d\ncone : \u2200 (n : \u2115), 1 < c n\nclim : Tendsto c atTop (\ud835\udcdd 1)\nthis :\n  \u2200 (k : \u2115), \u2200\u1d50 (\u03c9 : \u03a9), Tendsto (fun n => (\u2211 i in range \u230ac k ^ n\u230b\u208a, X i \u03c9) / \u2191\u230ac k ^ n\u230b\u208a) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a))\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (i : \u2115), Tendsto (fun n => (\u2211 i in range \u230ac i ^ n\u230b\u208a, X i \u03c9) / \u2191\u230ac i ^ n\u230b\u208a) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a))\n\u22a2 Monotone fun n => \u2211 i in range n, X i \u03c9", "state_after": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u2115 \u2192 \u211d\ncone : \u2200 (n : \u2115), 1 < c n\nclim : Tendsto c atTop (\ud835\udcdd 1)\nthis :\n  \u2200 (k : \u2115), \u2200\u1d50 (\u03c9 : \u03a9), Tendsto (fun n => (\u2211 i in range \u230ac k ^ n\u230b\u208a, X i \u03c9) / \u2191\u230ac k ^ n\u230b\u208a) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a))\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (i : \u2115), Tendsto (fun n => (\u2211 i in range \u230ac i ^ n\u230b\u208a, X i \u03c9) / \u2191\u230ac i ^ n\u230b\u208a) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a))\nm n : \u2115\nhmn : m \u2264 n\n\u22a2 (fun n => \u2211 i in range n, X i \u03c9) m \u2264 (fun n => \u2211 i in range n, X i \u03c9) n"}, {"tactic": "exact sum_le_sum_of_subset_of_nonneg (range_mono hmn) fun i _ _ => hnonneg i \u03c9", "annotated_tactic": ["exact <a>sum_le_sum_of_subset_of_nonneg</a> (<a>range_mono</a> hmn) fun i _ _ => hnonneg i \u03c9", [{"full_name": "Finset.sum_le_sum_of_subset_of_nonneg", "def_path": "Mathlib/Algebra/BigOperators/Order.lean", "def_pos": [156, 15], "def_end_pos": [156, 45]}, {"full_name": "Finset.range_mono", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3079, 9], "def_end_pos": [3079, 19]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u2115 \u2192 \u211d\ncone : \u2200 (n : \u2115), 1 < c n\nclim : Tendsto c atTop (\ud835\udcdd 1)\nthis :\n  \u2200 (k : \u2115), \u2200\u1d50 (\u03c9 : \u03a9), Tendsto (fun n => (\u2211 i in range \u230ac k ^ n\u230b\u208a, X i \u03c9) / \u2191\u230ac k ^ n\u230b\u208a) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a))\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (i : \u2115), Tendsto (fun n => (\u2211 i in range \u230ac i ^ n\u230b\u208a, X i \u03c9) / \u2191\u230ac i ^ n\u230b\u208a) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a))\nm n : \u2115\nhmn : m \u2264 n\n\u22a2 (fun n => \u2211 i in range n, X i \u03c9) m \u2264 (fun n => \u2211 i in range n, X i \u03c9) n", "state_after": "no goals"}, {"tactic": "exact h\u03c9", "annotated_tactic": ["exact h\u03c9", []], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u2115 \u2192 \u211d\ncone : \u2200 (n : \u2115), 1 < c n\nclim : Tendsto c atTop (\ud835\udcdd 1)\nthis :\n  \u2200 (k : \u2115), \u2200\u1d50 (\u03c9 : \u03a9), Tendsto (fun n => (\u2211 i in range \u230ac k ^ n\u230b\u208a, X i \u03c9) / \u2191\u230ac k ^ n\u230b\u208a) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a))\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (i : \u2115), Tendsto (fun n => (\u2211 i in range \u230ac i ^ n\u230b\u208a, X i \u03c9) / \u2191\u230ac i ^ n\u230b\u208a) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a))\n\u22a2 \u2200 (k : \u2115), Tendsto (fun n => (\u2211 i in range \u230ac k ^ n\u230b\u208a, X i \u03c9) / \u2191\u230ac k ^ n\u230b\u208a) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/GiryMonad.lean", "full_name": "MeasureTheory.Measure.join_map_dirac", "start": [227, 1], "end": [228, 13], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "full_name": "map_restrict_ae_le_map_indicator_ae", "start": [4473, 1], "end": [4480, 101], "traced_tactics": [{"tactic": "intro t", "annotated_tactic": ["intro t", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nf : \u03b1 \u2192 \u03b2\ninst\u271d : Zero \u03b2\nhs : MeasurableSet s\n\u22a2 Filter.map f (ae (Measure.restrict \u03bc s)) \u2264 Filter.map (indicator s f) (ae \u03bc)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\nf : \u03b1 \u2192 \u03b2\ninst\u271d : Zero \u03b2\nhs : MeasurableSet s\nt : Set \u03b2\n\u22a2 t \u2208 Filter.map (indicator s f) (ae \u03bc) \u2192 t \u2208 Filter.map f (ae (Measure.restrict \u03bc s))"}, {"tactic": "by_cases ht : (0 : \u03b2) \u2208 t", "annotated_tactic": ["by_cases ht : (0 : \u03b2) \u2208 t", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\nf : \u03b1 \u2192 \u03b2\ninst\u271d : Zero \u03b2\nhs : MeasurableSet s\nt : Set \u03b2\n\u22a2 t \u2208 Filter.map (indicator s f) (ae \u03bc) \u2192 t \u2208 Filter.map f (ae (Measure.restrict \u03bc s))", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\nf : \u03b1 \u2192 \u03b2\ninst\u271d : Zero \u03b2\nhs : MeasurableSet s\nt : Set \u03b2\nht : 0 \u2208 t\n\u22a2 t \u2208 Filter.map (indicator s f) (ae \u03bc) \u2192 t \u2208 Filter.map f (ae (Measure.restrict \u03bc s))\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\nf : \u03b1 \u2192 \u03b2\ninst\u271d : Zero \u03b2\nhs : MeasurableSet s\nt : Set \u03b2\nht : \u00ac0 \u2208 t\n\u22a2 t \u2208 Filter.map (indicator s f) (ae \u03bc) \u2192 t \u2208 Filter.map f (ae (Measure.restrict \u03bc s))"}, {"tactic": "rw [mem_map_indicator_ae_iff_of_zero_nmem ht, mem_map_restrict_ae_iff hs]", "annotated_tactic": ["rw [<a>mem_map_indicator_ae_iff_of_zero_nmem</a> ht, <a>mem_map_restrict_ae_iff</a> hs]", [{"full_name": "mem_map_indicator_ae_iff_of_zero_nmem", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [4466, 9], "def_end_pos": [4466, 46]}, {"full_name": "MeasureTheory.mem_map_restrict_ae_iff", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2624, 9], "def_end_pos": [2624, 32]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\nf : \u03b1 \u2192 \u03b2\ninst\u271d : Zero \u03b2\nhs : MeasurableSet s\nt : Set \u03b2\nht : \u00ac0 \u2208 t\n\u22a2 t \u2208 Filter.map (indicator s f) (ae \u03bc) \u2192 t \u2208 Filter.map f (ae (Measure.restrict \u03bc s))", "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\nf : \u03b1 \u2192 \u03b2\ninst\u271d : Zero \u03b2\nhs : MeasurableSet s\nt : Set \u03b2\nht : \u00ac0 \u2208 t\n\u22a2 \u2191\u2191\u03bc ((f \u207b\u00b9' t)\u1d9c \u222a s\u1d9c) = 0 \u2192 \u2191\u2191\u03bc ((f \u207b\u00b9' t)\u1d9c \u2229 s) = 0"}, {"tactic": "exact fun h => measure_mono_null ((Set.inter_subset_left _ _).trans (Set.subset_union_left _ _)) h", "annotated_tactic": ["exact fun h => <a>measure_mono_null</a> ((<a>Set.inter_subset_left</a> _ _).<a>trans</a> (<a>Set.subset_union_left</a> _ _)) h", [{"full_name": "MeasureTheory.measure_mono_null", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [197, 9], "def_end_pos": [197, 26]}, {"full_name": "Set.inter_subset_left", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [965, 9], "def_end_pos": [965, 26]}, {"full_name": "HasSubset.Subset.trans", "def_path": "Mathlib/Order/RelClasses.lean", "def_pos": [664, 7], "def_end_pos": [664, 29]}, {"full_name": "Set.subset_union_left", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [829, 9], "def_end_pos": [829, 26]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\nf : \u03b1 \u2192 \u03b2\ninst\u271d : Zero \u03b2\nhs : MeasurableSet s\nt : Set \u03b2\nht : \u00ac0 \u2208 t\n\u22a2 \u2191\u2191\u03bc ((f \u207b\u00b9' t)\u1d9c \u222a s\u1d9c) = 0 \u2192 \u2191\u2191\u03bc ((f \u207b\u00b9' t)\u1d9c \u2229 s) = 0", "state_after": "no goals"}, {"tactic": "rw [mem_map_indicator_ae_iff_mem_map_restrict_ae_of_zero_mem ht hs]", "annotated_tactic": ["rw [<a>mem_map_indicator_ae_iff_mem_map_restrict_ae_of_zero_mem</a> ht hs]", [{"full_name": "mem_map_indicator_ae_iff_mem_map_restrict_ae_of_zero_mem", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [4454, 9], "def_end_pos": [4454, 65]}]], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\nf : \u03b1 \u2192 \u03b2\ninst\u271d : Zero \u03b2\nhs : MeasurableSet s\nt : Set \u03b2\nht : 0 \u2208 t\n\u22a2 t \u2208 Filter.map (indicator s f) (ae \u03bc) \u2192 t \u2208 Filter.map f (ae (Measure.restrict \u03bc s))", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\nf : \u03b1 \u2192 \u03b2\ninst\u271d : Zero \u03b2\nhs : MeasurableSet s\nt : Set \u03b2\nht : 0 \u2208 t\n\u22a2 t \u2208 Filter.map f (ae (Measure.restrict \u03bc s)) \u2192 t \u2208 Filter.map f (ae (Measure.restrict \u03bc s))"}, {"tactic": "exact id", "annotated_tactic": ["exact <a>id</a>", [{"full_name": "id", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [33, 15], "def_end_pos": [33, 17]}]], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\nf : \u03b1 \u2192 \u03b2\ninst\u271d : Zero \u03b2\nhs : MeasurableSet s\nt : Set \u03b2\nht : 0 \u2208 t\n\u22a2 t \u2208 Filter.map f (ae (Measure.restrict \u03bc s)) \u2192 t \u2208 Filter.map f (ae (Measure.restrict \u03bc s))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Analysis/Filter.lean", "full_name": "Filter.Realizer.tendsto_iff", "start": [342, 1], "end": [344, 94], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "full_name": "MeasureTheory.VectorMeasure.of_union", "start": [185, 1], "end": [188, 72], "traced_tactics": [{"tactic": "rw [Set.union_eq_iUnion, of_disjoint_iUnion, tsum_fintype, Fintype.sum_bool, cond, cond]", "annotated_tactic": ["rw [<a>Set.union_eq_iUnion</a>, <a>of_disjoint_iUnion</a>, <a>tsum_fintype</a>, <a>Fintype.sum_bool</a>, <a>cond</a>, <a>cond</a>]", [{"full_name": "Set.union_eq_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [1478, 9], "def_end_pos": [1478, 24]}, {"full_name": "MeasureTheory.VectorMeasure.of_disjoint_iUnion", "def_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "def_pos": [180, 9], "def_end_pos": [180, 27]}, {"full_name": "tsum_fintype", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/Basic.lean", "def_pos": [503, 9], "def_end_pos": [503, 21]}, {"full_name": "Fintype.sum_bool", "def_path": "Mathlib/Data/Fintype/BigOperators.lean", "def_pos": [38, 3], "def_end_pos": [38, 14]}, {"full_name": "cond", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [971, 21], "def_end_pos": [971, 25]}, {"full_name": "cond", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [971, 21], "def_end_pos": [971, 25]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : TopologicalSpace M\ninst\u271d : T2Space M\nv : VectorMeasure \u03b1 M\nf : \u2115 \u2192 Set \u03b1\nA B : Set \u03b1\nh : Disjoint A B\nhA : MeasurableSet A\nhB : MeasurableSet B\n\u22a2 \u2191v (A \u222a B) = \u2191v A + \u2191v B", "state_after": "case hf\u2081\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : TopologicalSpace M\ninst\u271d : T2Space M\nv : VectorMeasure \u03b1 M\nf : \u2115 \u2192 Set \u03b1\nA B : Set \u03b1\nh : Disjoint A B\nhA : MeasurableSet A\nhB : MeasurableSet B\n\u22a2 \u2200 (i : Bool), MeasurableSet (bif i then A else B)\n\ncase hf\u2082\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : TopologicalSpace M\ninst\u271d : T2Space M\nv : VectorMeasure \u03b1 M\nf : \u2115 \u2192 Set \u03b1\nA B : Set \u03b1\nh : Disjoint A B\nhA : MeasurableSet A\nhB : MeasurableSet B\n\u22a2 Pairwise (Disjoint on fun b => bif b then A else B)"}, {"tactic": "exacts [fun b => Bool.casesOn b hB hA, pairwise_disjoint_on_bool.2 h]", "annotated_tactic": ["exacts [fun b => <a>Bool.casesOn</a> b hB hA, <a>pairwise_disjoint_on_bool</a>.2 h]", [{"full_name": "Bool.casesOn", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [545, 11], "def_end_pos": [545, 15]}, {"full_name": "pairwise_disjoint_on_bool", "def_path": "Mathlib/Data/Set/Pairwise/Basic.lean", "def_pos": [45, 9], "def_end_pos": [45, 34]}]], "state_before": "case hf\u2081\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : TopologicalSpace M\ninst\u271d : T2Space M\nv : VectorMeasure \u03b1 M\nf : \u2115 \u2192 Set \u03b1\nA B : Set \u03b1\nh : Disjoint A B\nhA : MeasurableSet A\nhB : MeasurableSet B\n\u22a2 \u2200 (i : Bool), MeasurableSet (bif i then A else B)\n\ncase hf\u2082\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : TopologicalSpace M\ninst\u271d : T2Space M\nv : VectorMeasure \u03b1 M\nf : \u2115 \u2192 Set \u03b1\nA B : Set \u03b1\nh : Disjoint A B\nhA : MeasurableSet A\nhB : MeasurableSet B\n\u22a2 Pairwise (Disjoint on fun b => bif b then A else B)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean", "full_name": "MeasureTheory.condexpIndL1Fin_smul", "start": [105, 1], "end": [113, 40], "traced_tactics": [{"tactic": "ext1", "annotated_tactic": ["ext1", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : CompleteSpace F'\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \u211d G'\ninst\u271d\u00b2 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nc : \u211d\nx : G\n\u22a2 condexpIndL1Fin hm hs h\u03bcs (c \u2022 x) = c \u2022 condexpIndL1Fin hm hs h\u03bcs x", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : CompleteSpace F'\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \u211d G'\ninst\u271d\u00b2 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nc : \u211d\nx : G\n\u22a2 \u2191\u2191(condexpIndL1Fin hm hs h\u03bcs (c \u2022 x)) =\u1d50[\u03bc] \u2191\u2191(c \u2022 condexpIndL1Fin hm hs h\u03bcs x)"}, {"tactic": "refine' (Mem\u2112p.coeFn_toLp q).trans _", "annotated_tactic": ["refine' (<a>Mem\u2112p.coeFn_toLp</a> <a>q</a>).<a>trans</a> _", [{"full_name": "MeasureTheory.Mem\u2112p.coeFn_toLp", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [119, 9], "def_end_pos": [119, 19]}, {"full_name": "_private.Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1.0.MeasureTheory.q", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean", "def_pos": [88, 17], "def_end_pos": [88, 18]}, {"full_name": "Filter.EventuallyEq.trans", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1503, 9], "def_end_pos": [1503, 27]}]], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : CompleteSpace F'\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \u211d G'\ninst\u271d\u00b2 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nc : \u211d\nx : G\n\u22a2 \u2191\u2191(condexpIndL1Fin hm hs h\u03bcs (c \u2022 x)) =\u1d50[\u03bc] \u2191\u2191(c \u2022 condexpIndL1Fin hm hs h\u03bcs x)", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : CompleteSpace F'\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \u211d G'\ninst\u271d\u00b2 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nc : \u211d\nx : G\n\u22a2 \u2191\u2191(condexpIndSMul hm hs h\u03bcs (c \u2022 x)) =\u1d50[\u03bc] \u2191\u2191(c \u2022 condexpIndL1Fin hm hs h\u03bcs x)"}, {"tactic": "refine' EventuallyEq.trans _ (Lp.coeFn_smul _ _).symm", "annotated_tactic": ["refine' <a>EventuallyEq.trans</a> _ (<a>Lp.coeFn_smul</a> _ _).<a>symm</a>", [{"full_name": "Filter.EventuallyEq.trans", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1503, 9], "def_end_pos": [1503, 27]}, {"full_name": "MeasureTheory.Lp.coeFn_smul", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [498, 9], "def_end_pos": [498, 19]}, {"full_name": "Filter.EventuallyEq.symm", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1498, 9], "def_end_pos": [1498, 26]}]], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : CompleteSpace F'\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \u211d G'\ninst\u271d\u00b2 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nc : \u211d\nx : G\n\u22a2 \u2191\u2191(condexpIndSMul hm hs h\u03bcs (c \u2022 x)) =\u1d50[\u03bc] \u2191\u2191(c \u2022 condexpIndL1Fin hm hs h\u03bcs x)", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : CompleteSpace F'\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \u211d G'\ninst\u271d\u00b2 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nc : \u211d\nx : G\n\u22a2 \u2191\u2191(condexpIndSMul hm hs h\u03bcs (c \u2022 x)) =\u1d50[\u03bc] c \u2022 \u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x)"}, {"tactic": "rw [condexpIndSMul_smul hs h\u03bcs c x]", "annotated_tactic": ["rw [<a>condexpIndSMul_smul</a> hs h\u03bcs c x]", [{"full_name": "MeasureTheory.condexpIndSMul_smul", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL2.lean", "def_pos": [407, 9], "def_end_pos": [407, 28]}]], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : CompleteSpace F'\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \u211d G'\ninst\u271d\u00b2 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nc : \u211d\nx : G\n\u22a2 \u2191\u2191(condexpIndSMul hm hs h\u03bcs (c \u2022 x)) =\u1d50[\u03bc] c \u2022 \u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x)", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : CompleteSpace F'\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \u211d G'\ninst\u271d\u00b2 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nc : \u211d\nx : G\n\u22a2 \u2191\u2191(c \u2022 condexpIndSMul hm hs h\u03bcs x) =\u1d50[\u03bc] c \u2022 \u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x)"}, {"tactic": "refine' (Lp.coeFn_smul _ _).trans _", "annotated_tactic": ["refine' (<a>Lp.coeFn_smul</a> _ _).<a>trans</a> _", [{"full_name": "MeasureTheory.Lp.coeFn_smul", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [498, 9], "def_end_pos": [498, 19]}, {"full_name": "Filter.EventuallyEq.trans", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1503, 9], "def_end_pos": [1503, 27]}]], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : CompleteSpace F'\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \u211d G'\ninst\u271d\u00b2 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nc : \u211d\nx : G\n\u22a2 \u2191\u2191(c \u2022 condexpIndSMul hm hs h\u03bcs x) =\u1d50[\u03bc] c \u2022 \u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x)", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : CompleteSpace F'\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \u211d G'\ninst\u271d\u00b2 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nc : \u211d\nx : G\n\u22a2 c \u2022 \u2191\u2191(condexpIndSMul hm hs h\u03bcs x) =\u1d50[\u03bc] c \u2022 \u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x)"}, {"tactic": "refine' (condexpIndL1Fin_ae_eq_condexpIndSMul hm hs h\u03bcs x).mono fun y hy => _", "annotated_tactic": ["refine' (<a>condexpIndL1Fin_ae_eq_condexpIndSMul</a> hm hs h\u03bcs x).<a>mono</a> fun y hy => _", [{"full_name": "MeasureTheory.condexpIndL1Fin_ae_eq_condexpIndSMul", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean", "def_pos": [78, 9], "def_end_pos": [78, 45]}, {"full_name": "Filter.Eventually.mono", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1140, 9], "def_end_pos": [1140, 24]}]], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : CompleteSpace F'\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \u211d G'\ninst\u271d\u00b2 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nc : \u211d\nx : G\n\u22a2 c \u2022 \u2191\u2191(condexpIndSMul hm hs h\u03bcs x) =\u1d50[\u03bc] c \u2022 \u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x)", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : CompleteSpace F'\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \u211d G'\ninst\u271d\u00b2 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nc : \u211d\nx : G\ny : \u03b1\nhy : \u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) y = \u2191\u2191(condexpIndSMul hm hs h\u03bcs x) y\n\u22a2 (c \u2022 \u2191\u2191(condexpIndSMul hm hs h\u03bcs x)) y = (c \u2022 \u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x)) y"}, {"tactic": "rw [Pi.smul_apply, Pi.smul_apply, hy]", "annotated_tactic": ["rw [<a>Pi.smul_apply</a>, <a>Pi.smul_apply</a>, hy]", [{"full_name": "Pi.smul_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [116, 60], "def_end_pos": [116, 70]}, {"full_name": "Pi.smul_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [116, 60], "def_end_pos": [116, 70]}]], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : CompleteSpace F'\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \u211d G'\ninst\u271d\u00b2 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nc : \u211d\nx : G\ny : \u03b1\nhy : \u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) y = \u2191\u2191(condexpIndSMul hm hs h\u03bcs x) y\n\u22a2 (c \u2022 \u2191\u2191(condexpIndSMul hm hs h\u03bcs x)) y = (c \u2022 \u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x)) y", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "full_name": "String.get'_eq", "start": [322, 9], "end": [322, 81], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Subtype.lean", "full_name": "exists_subtype_mk_eq_iff", "start": [145, 1], "end": [147, 69], "traced_tactics": [{"tactic": "simp only [@eq_comm _ b, exists_eq_subtype_mk_iff, @eq_comm _ _ a]", "annotated_tactic": ["simp only [@<a>eq_comm</a> _ b, <a>exists_eq_subtype_mk_iff</a>, @<a>eq_comm</a> _ _ a]", [{"full_name": "eq_comm", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [104, 9], "def_end_pos": [104, 16]}, {"full_name": "exists_eq_subtype_mk_iff", "def_path": "Mathlib/Data/Subtype.lean", "def_pos": [137, 9], "def_end_pos": [137, 40]}, {"full_name": "eq_comm", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [104, 9], "def_end_pos": [104, 16]}]], "state_before": "\u03b1 : Sort u_1\n\u03b2 : Sort u_2\n\u03b3 : Sort u_3\np q : \u03b1 \u2192 Prop\na : Subtype p\nb : \u03b1\n\u22a2 (\u2203 h, { val := b, property := h } = a) \u2194 b = \u2191a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/RBMap/Lemmas.lean", "full_name": "Std.RBSet.mem_insert", "start": [711, 1], "end": [717, 62], "traced_tactics": [{"tactic": "refine \u27e8fun h => ?_, fun | .inl h => mem_insert_of_mem _ h | .inr h => mem_insert_of_eq _ h\u27e9", "annotated_tactic": ["refine \u27e8fun h => ?_, fun | .inl h => <a>mem_insert_of_mem</a> _ h | .inr h => <a>mem_insert_of_eq</a> _ h\u27e9", [{"full_name": "Std.RBSet.mem_insert_of_mem", "def_path": "lake-packages/std/Std/Data/RBMap/Lemmas.lean", "def_pos": [702, 9], "def_end_pos": [702, 26]}, {"full_name": "Std.RBSet.mem_insert_of_eq", "def_path": "lake-packages/std/Std/Data/RBMap/Lemmas.lean", "def_pos": [688, 9], "def_end_pos": [688, 25]}]], "state_before": "\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nv' v : \u03b1\ninst\u271d : TransCmp cmp\nt : RBSet \u03b1 cmp\n\u22a2 v' \u2208 insert t v \u2194 v' \u2208 t \u2228 cmp v v' = Ordering.eq", "state_after": "\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nv' v : \u03b1\ninst\u271d : TransCmp cmp\nt : RBSet \u03b1 cmp\nh : v' \u2208 insert t v\n\u22a2 v' \u2208 t \u2228 cmp v v' = Ordering.eq"}, {"tactic": "let \u27e8_, h\u2081, h\u2082\u27e9 := mem_iff_mem_toList.1 h", "annotated_tactic": ["let \u27e8_, h\u2081, h\u2082\u27e9 := <a>mem_iff_mem_toList</a>.1 h", [{"full_name": "Std.RBSet.mem_iff_mem_toList", "def_path": "lake-packages/std/Std/Data/RBMap/Lemmas.lean", "def_pos": [629, 9], "def_end_pos": [629, 27]}]], "state_before": "\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nv' v : \u03b1\ninst\u271d : TransCmp cmp\nt : RBSet \u03b1 cmp\nh : v' \u2208 insert t v\n\u22a2 v' \u2208 t \u2228 cmp v v' = Ordering.eq", "state_after": "\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nv' v : \u03b1\ninst\u271d : TransCmp cmp\nt : RBSet \u03b1 cmp\nh : v' \u2208 insert t v\nw\u271d : \u03b1\nh\u2081 : w\u271d \u2208 toList (insert t v)\nh\u2082 : cmp v' w\u271d = Ordering.eq\n\u22a2 v' \u2208 t \u2228 cmp v v' = Ordering.eq"}, {"tactic": "match mem_toList_insert.1 h\u2081 with\n| .inl \u27e8h\u2083, _\u27e9 => exact .inl <| mem_iff_mem_toList.2 \u27e8_, h\u2083, h\u2082\u27e9\n| .inr rfl => exact .inr <| OrientedCmp.cmp_eq_eq_symm.1 h\u2082", "annotated_tactic": ["match <a>mem_toList_insert</a>.1 h\u2081 with\n  | .inl \u27e8h\u2083, _\u27e9 => exact .inl <| <a>mem_iff_mem_toList</a>.2 \u27e8_, h\u2083, h\u2082\u27e9\n  | .inr <a>rfl</a> => exact .inr <| <a>OrientedCmp.cmp_eq_eq_symm</a>.1 h\u2082", [{"full_name": "Std.RBSet.mem_toList_insert", "def_path": "lake-packages/std/Std/Data/RBMap/Lemmas.lean", "def_pos": [706, 9], "def_end_pos": [706, 26]}, {"full_name": "Std.RBSet.mem_iff_mem_toList", "def_path": "lake-packages/std/Std/Data/RBMap/Lemmas.lean", "def_pos": [629, 9], "def_end_pos": [629, 27]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}, {"full_name": "Std.OrientedCmp.cmp_eq_eq_symm", "def_path": "lake-packages/std/Std/Classes/Order.lean", "def_pos": [34, 9], "def_end_pos": [34, 23]}]], "state_before": "\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nv' v : \u03b1\ninst\u271d : TransCmp cmp\nt : RBSet \u03b1 cmp\nh : v' \u2208 insert t v\nw\u271d : \u03b1\nh\u2081 : w\u271d \u2208 toList (insert t v)\nh\u2082 : cmp v' w\u271d = Ordering.eq\n\u22a2 v' \u2208 t \u2228 cmp v v' = Ordering.eq", "state_after": "no goals"}, {"tactic": "exact .inl <| mem_iff_mem_toList.2 \u27e8_, h\u2083, h\u2082\u27e9", "annotated_tactic": ["exact .inl <| <a>mem_iff_mem_toList</a>.2 \u27e8_, h\u2083, h\u2082\u27e9", [{"full_name": "Std.RBSet.mem_iff_mem_toList", "def_path": "lake-packages/std/Std/Data/RBMap/Lemmas.lean", "def_pos": [629, 9], "def_end_pos": [629, 27]}]], "state_before": "\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nv' : \u03b1\ninst\u271d : TransCmp cmp\nt : RBSet \u03b1 cmp\nw\u271d : \u03b1\nh\u2082 : cmp v' w\u271d = Ordering.eq\nv : \u03b1\nh\u2083 : w\u271d \u2208 toList t\nright\u271d : find? t v \u2260 some w\u271d\nh : v' \u2208 insert t v\nh\u2081 : w\u271d \u2208 toList (insert t v)\n\u22a2 v' \u2208 t \u2228 cmp v v' = Ordering.eq", "state_after": "no goals"}, {"tactic": "exact .inr <| OrientedCmp.cmp_eq_eq_symm.1 h\u2082", "annotated_tactic": ["exact .inr <| <a>OrientedCmp.cmp_eq_eq_symm</a>.1 h\u2082", [{"full_name": "Std.OrientedCmp.cmp_eq_eq_symm", "def_path": "lake-packages/std/Std/Classes/Order.lean", "def_pos": [34, 9], "def_end_pos": [34, 23]}]], "state_before": "\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nv' : \u03b1\ninst\u271d : TransCmp cmp\nt : RBSet \u03b1 cmp\nw\u271d : \u03b1\nh\u2082 : cmp v' w\u271d = Ordering.eq\nh : v' \u2208 insert t w\u271d\nh\u2081 : w\u271d \u2208 toList (insert t w\u271d)\n\u22a2 v' \u2208 t \u2228 cmp w\u271d v' = Ordering.eq", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "full_name": "Substring.ValidFor.atEnd", "start": [920, 1], "end": [921, 77], "traced_tactics": [{"tactic": "simp [Substring.atEnd, Pos.ext_iff, Nat.add_left_cancel_iff]", "annotated_tactic": ["simp [<a>Substring.atEnd</a>, <a>Pos.ext_iff</a>, <a>Nat.add_left_cancel_iff</a>]", [{"full_name": "Substring.atEnd", "def_path": "lake-packages/lean4/src/lean/Init/Data/String/Basic.lean", "def_pos": [587, 15], "def_end_pos": [587, 20]}, {"full_name": "String.Pos.ext_iff", "def_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "def_pos": [108, 9], "def_end_pos": [108, 16]}, {"full_name": "Nat.add_left_cancel_iff", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [290, 19], "def_end_pos": [290, 38]}]], "state_before": "l m r : List Char\np : Nat\n\u22a2 Substring.atEnd\n        { str := { data := l ++ m ++ r }, startPos := { byteIdx := utf8Len l },\n          stopPos := { byteIdx := utf8Len l + utf8Len m } }\n        { byteIdx := p } =\n      true \u2194\n    p = utf8Len m", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "full_name": "MeasureTheory.L1.SimpleFunc.setToL1SCLM_add_left'", "start": [925, 1], "end": [929, 38], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/LocallyFinite.lean", "full_name": "Finset.Iic_eq_cons_Iio", "start": [746, 1], "end": [747, 44], "traced_tactics": [{"tactic": "classical rw [cons_eq_insert, Iio_insert]", "annotated_tactic": ["classical rw [<a>cons_eq_insert</a>, <a>Iio_insert</a>]", [{"full_name": "Finset.cons_eq_insert", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1108, 9], "def_end_pos": [1108, 23]}, {"full_name": "Finset.Iio_insert", "def_path": "Mathlib/Data/Finset/LocallyFinite.lean", "def_pos": [734, 9], "def_end_pos": [734, 19]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : LocallyFiniteOrderBot \u03b1\nb : \u03b1\n\u22a2 Iic b = cons b (Iio b) (_ : \u00acb \u2208 Iio b)", "state_after": "no goals"}, {"tactic": "rw [cons_eq_insert, Iio_insert]", "annotated_tactic": ["rw [<a>cons_eq_insert</a>, <a>Iio_insert</a>]", [{"full_name": "Finset.cons_eq_insert", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1108, 9], "def_end_pos": [1108, 23]}, {"full_name": "Finset.Iio_insert", "def_path": "Mathlib/Data/Finset/LocallyFinite.lean", "def_pos": [734, 9], "def_end_pos": [734, 19]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : LocallyFiniteOrderBot \u03b1\nb : \u03b1\n\u22a2 Iic b = cons b (Iio b) (_ : \u00acb \u2208 Iio b)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/AssocList.lean", "full_name": "Std.AssocList.findEntry?_eq", "start": [111, 9], "end": [112, 68], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "full_name": "String.extract.go\u2081_add_right_cancel", "start": [417, 1], "end": [428, 13], "traced_tactics": [{"tactic": "apply utf8InductionOn s \u27e8i\u27e9 \u27e8b\u27e9 (motive := fun s i =>\n  go\u2081 s \u27e8i.byteIdx + n\u27e9 \u27e8b + n\u27e9 \u27e8e + n\u27e9 = go\u2081 s i \u27e8b\u27e9 \u27e8e\u27e9) <;>\nsimp [go\u2081]", "annotated_tactic": ["apply <a>utf8InductionOn</a> s \u27e8i\u27e9 \u27e8b\u27e9 (motive := fun s i =>\n    <a>go\u2081</a> s \u27e8i.byteIdx + n\u27e9 \u27e8b + n\u27e9 \u27e8e + n\u27e9 = <a>go\u2081</a> s i \u27e8b\u27e9 \u27e8e\u27e9) <;>\n  simp [<a>go\u2081</a>]", [{"full_name": "String.utf8InductionOn", "def_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "def_pos": [185, 5], "def_end_pos": [185, 20]}, {"full_name": "String.extract.go\u2081", "def_path": "lake-packages/lean4/src/lean/Init/Data/String/Basic.lean", "def_pos": [223, 3], "def_end_pos": [223, 6]}, {"full_name": "String.extract.go\u2081", "def_path": "lake-packages/lean4/src/lean/Init/Data/String/Basic.lean", "def_pos": [223, 3], "def_end_pos": [223, 6]}, {"full_name": "String.extract.go\u2081", "def_path": "lake-packages/lean4/src/lean/Init/Data/String/Basic.lean", "def_pos": [223, 3], "def_end_pos": [223, 6]}]], "state_before": "s : List Char\ni b e n : Nat\n\u22a2 go\u2081 s { byteIdx := i + n } { byteIdx := b + n } { byteIdx := e + n } =\n    go\u2081 s { byteIdx := i } { byteIdx := b } { byteIdx := e }", "state_after": "case eq\ns : List Char\ni b e n : Nat\n\u22a2 \u2200 (c : Char) (cs : List Char),\n    go\u2082 (c :: cs) { byteIdx := b + n } { byteIdx := e + n } = go\u2082 (c :: cs) { byteIdx := b } { byteIdx := e }\n\ncase ind\ns : List Char\ni b e n : Nat\n\u22a2 \u2200 (c : Char) (cs : List Char) (i : Pos),\n    \u00aci = { byteIdx := b } \u2192\n      go\u2081 cs { byteIdx := i.byteIdx + csize c + n } { byteIdx := b + n } { byteIdx := e + n } =\n          go\u2081 cs (i + c) { byteIdx := b } { byteIdx := e } \u2192\n        (if { byteIdx := i.byteIdx + n } = { byteIdx := b + n } then\n            go\u2082 (c :: cs) { byteIdx := i.byteIdx + n } { byteIdx := e + n }\n          else go\u2081 cs ({ byteIdx := i.byteIdx + n } + c) { byteIdx := b + n } { byteIdx := e + n }) =\n          if i = { byteIdx := b } then go\u2082 (c :: cs) i { byteIdx := e }\n          else go\u2081 cs (i + c) { byteIdx := b } { byteIdx := e }"}, {"tactic": "intro c cs", "annotated_tactic": ["intro c cs", []], "state_before": "case eq\ns : List Char\ni b e n : Nat\n\u22a2 \u2200 (c : Char) (cs : List Char),\n    go\u2082 (c :: cs) { byteIdx := b + n } { byteIdx := e + n } = go\u2082 (c :: cs) { byteIdx := b } { byteIdx := e }", "state_after": "case eq\ns : List Char\ni b e n : Nat\nc : Char\ncs : List Char\n\u22a2 go\u2082 (c :: cs) { byteIdx := b + n } { byteIdx := e + n } = go\u2082 (c :: cs) { byteIdx := b } { byteIdx := e }"}, {"tactic": "apply go\u2082_add_right_cancel", "annotated_tactic": ["apply <a>go\u2082_add_right_cancel</a>", [{"full_name": "String.extract.go\u2082_add_right_cancel", "def_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "def_pos": [400, 9], "def_end_pos": [400, 37]}]], "state_before": "case eq\ns : List Char\ni b e n : Nat\nc : Char\ncs : List Char\n\u22a2 go\u2082 (c :: cs) { byteIdx := b + n } { byteIdx := e + n } = go\u2082 (c :: cs) { byteIdx := b } { byteIdx := e }", "state_after": "no goals"}, {"tactic": "intro c cs \u27e8i\u27e9 h ih", "annotated_tactic": ["intro c cs \u27e8i\u27e9 h ih", []], "state_before": "case ind\ns : List Char\ni b e n : Nat\n\u22a2 \u2200 (c : Char) (cs : List Char) (i : Pos),\n    \u00aci = { byteIdx := b } \u2192\n      go\u2081 cs { byteIdx := i.byteIdx + csize c + n } { byteIdx := b + n } { byteIdx := e + n } =\n          go\u2081 cs (i + c) { byteIdx := b } { byteIdx := e } \u2192\n        (if { byteIdx := i.byteIdx + n } = { byteIdx := b + n } then\n            go\u2082 (c :: cs) { byteIdx := i.byteIdx + n } { byteIdx := e + n }\n          else go\u2081 cs ({ byteIdx := i.byteIdx + n } + c) { byteIdx := b + n } { byteIdx := e + n }) =\n          if i = { byteIdx := b } then go\u2082 (c :: cs) i { byteIdx := e }\n          else go\u2081 cs (i + c) { byteIdx := b } { byteIdx := e }", "state_after": "case ind\ns : List Char\ni\u271d b e n : Nat\nc : Char\ncs : List Char\ni : Nat\nh : \u00ac{ byteIdx := i } = { byteIdx := b }\nih :\n  go\u2081 cs { byteIdx := { byteIdx := i }.byteIdx + csize c + n } { byteIdx := b + n } { byteIdx := e + n } =\n    go\u2081 cs ({ byteIdx := i } + c) { byteIdx := b } { byteIdx := e }\n\u22a2 (if { byteIdx := { byteIdx := i }.byteIdx + n } = { byteIdx := b + n } then\n      go\u2082 (c :: cs) { byteIdx := { byteIdx := i }.byteIdx + n } { byteIdx := e + n }\n    else go\u2081 cs ({ byteIdx := { byteIdx := i }.byteIdx + n } + c) { byteIdx := b + n } { byteIdx := e + n }) =\n    if { byteIdx := i } = { byteIdx := b } then go\u2082 (c :: cs) { byteIdx := i } { byteIdx := e }\n    else go\u2081 cs ({ byteIdx := i } + c) { byteIdx := b } { byteIdx := e }"}, {"tactic": "simp [Pos.ext_iff, Pos.addChar_eq] at h ih \u22a2", "annotated_tactic": ["simp [<a>Pos.ext_iff</a>, <a>Pos.addChar_eq</a>] at h ih \u22a2", [{"full_name": "String.Pos.ext_iff", "def_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "def_pos": [108, 9], "def_end_pos": [108, 16]}, {"full_name": "String.Pos.addChar_eq", "def_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "def_pos": [120, 9], "def_end_pos": [120, 19]}]], "state_before": "case ind\ns : List Char\ni\u271d b e n : Nat\nc : Char\ncs : List Char\ni : Nat\nh : \u00ac{ byteIdx := i } = { byteIdx := b }\nih :\n  go\u2081 cs { byteIdx := { byteIdx := i }.byteIdx + csize c + n } { byteIdx := b + n } { byteIdx := e + n } =\n    go\u2081 cs ({ byteIdx := i } + c) { byteIdx := b } { byteIdx := e }\n\u22a2 (if { byteIdx := { byteIdx := i }.byteIdx + n } = { byteIdx := b + n } then\n      go\u2082 (c :: cs) { byteIdx := { byteIdx := i }.byteIdx + n } { byteIdx := e + n }\n    else go\u2081 cs ({ byteIdx := { byteIdx := i }.byteIdx + n } + c) { byteIdx := b + n } { byteIdx := e + n }) =\n    if { byteIdx := i } = { byteIdx := b } then go\u2082 (c :: cs) { byteIdx := i } { byteIdx := e }\n    else go\u2081 cs ({ byteIdx := i } + c) { byteIdx := b } { byteIdx := e }", "state_after": "case ind\ns : List Char\ni\u271d b e n : Nat\nc : Char\ncs : List Char\ni : Nat\nih :\n  go\u2081 cs { byteIdx := i + csize c + n } { byteIdx := b + n } { byteIdx := e + n } =\n    go\u2081 cs { byteIdx := i + csize c } { byteIdx := b } { byteIdx := e }\nh : \u00aci = b\n\u22a2 (if i + n = b + n then go\u2082 (c :: cs) { byteIdx := i + n } { byteIdx := e + n }\n    else go\u2081 cs { byteIdx := i + n + csize c } { byteIdx := b + n } { byteIdx := e + n }) =\n    if i = b then go\u2082 (c :: cs) { byteIdx := i } { byteIdx := e }\n    else go\u2081 cs { byteIdx := i + csize c } { byteIdx := b } { byteIdx := e }"}, {"tactic": "simp [Nat.add_right_cancel_iff, h]", "annotated_tactic": ["simp [<a>Nat.add_right_cancel_iff</a>, h]", [{"full_name": "Nat.add_right_cancel_iff", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [293, 19], "def_end_pos": [293, 39]}]], "state_before": "case ind\ns : List Char\ni\u271d b e n : Nat\nc : Char\ncs : List Char\ni : Nat\nih :\n  go\u2081 cs { byteIdx := i + csize c + n } { byteIdx := b + n } { byteIdx := e + n } =\n    go\u2081 cs { byteIdx := i + csize c } { byteIdx := b } { byteIdx := e }\nh : \u00aci = b\n\u22a2 (if i + n = b + n then go\u2082 (c :: cs) { byteIdx := i + n } { byteIdx := e + n }\n    else go\u2081 cs { byteIdx := i + n + csize c } { byteIdx := b + n } { byteIdx := e + n }) =\n    if i = b then go\u2082 (c :: cs) { byteIdx := i } { byteIdx := e }\n    else go\u2081 cs { byteIdx := i + csize c } { byteIdx := b } { byteIdx := e }", "state_after": "case ind\ns : List Char\ni\u271d b e n : Nat\nc : Char\ncs : List Char\ni : Nat\nih :\n  go\u2081 cs { byteIdx := i + csize c + n } { byteIdx := b + n } { byteIdx := e + n } =\n    go\u2081 cs { byteIdx := i + csize c } { byteIdx := b } { byteIdx := e }\nh : \u00aci = b\n\u22a2 go\u2081 cs { byteIdx := i + n + csize c } { byteIdx := b + n } { byteIdx := e + n } =\n    go\u2081 cs { byteIdx := i + csize c } { byteIdx := b } { byteIdx := e }"}, {"tactic": "rw [Nat.add_right_comm]", "annotated_tactic": ["rw [<a>Nat.add_right_comm</a>]", [{"full_name": "Nat.add_right_comm", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [145, 19], "def_end_pos": [145, 33]}]], "state_before": "case ind\ns : List Char\ni\u271d b e n : Nat\nc : Char\ncs : List Char\ni : Nat\nih :\n  go\u2081 cs { byteIdx := i + csize c + n } { byteIdx := b + n } { byteIdx := e + n } =\n    go\u2081 cs { byteIdx := i + csize c } { byteIdx := b } { byteIdx := e }\nh : \u00aci = b\n\u22a2 go\u2081 cs { byteIdx := i + n + csize c } { byteIdx := b + n } { byteIdx := e + n } =\n    go\u2081 cs { byteIdx := i + csize c } { byteIdx := b } { byteIdx := e }", "state_after": "case ind\ns : List Char\ni\u271d b e n : Nat\nc : Char\ncs : List Char\ni : Nat\nih :\n  go\u2081 cs { byteIdx := i + csize c + n } { byteIdx := b + n } { byteIdx := e + n } =\n    go\u2081 cs { byteIdx := i + csize c } { byteIdx := b } { byteIdx := e }\nh : \u00aci = b\n\u22a2 go\u2081 cs { byteIdx := i + csize c + n } { byteIdx := b + n } { byteIdx := e + n } =\n    go\u2081 cs { byteIdx := i + csize c } { byteIdx := b } { byteIdx := e }"}, {"tactic": "exact ih", "annotated_tactic": ["exact ih", []], "state_before": "case ind\ns : List Char\ni\u271d b e n : Nat\nc : Char\ncs : List Char\ni : Nat\nih :\n  go\u2081 cs { byteIdx := i + csize c + n } { byteIdx := b + n } { byteIdx := e + n } =\n    go\u2081 cs { byteIdx := i + csize c } { byteIdx := b } { byteIdx := e }\nh : \u00aci = b\n\u22a2 go\u2081 cs { byteIdx := i + csize c + n } { byteIdx := b + n } { byteIdx := e + n } =\n    go\u2081 cs { byteIdx := i + csize c } { byteIdx := b } { byteIdx := e }", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "full_name": "MeasureTheory.Measure.countable_meas_level_set_pos", "start": [3525, 1], "end": [3528, 54], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/CircleTransform.lean", "full_name": "Complex.circleTransformDeriv_bound", "start": [143, 1], "end": [174, 13], "traced_tactics": [{"tactic": "obtain \u27e8r, hr, hrx\u27e9 := exists_lt_mem_ball_of_mem_ball hx", "annotated_tactic": ["obtain \u27e8r, hr, hrx\u27e9 := <a>exists_lt_mem_ball_of_mem_ball</a> hx", [{"full_name": "Metric.exists_lt_mem_ball_of_mem_ball", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [447, 9], "def_end_pos": [447, 39]}]], "state_before": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR : \u211d\nhR : 0 < R\nz x : \u2102\nf : \u2102 \u2192 \u2102\nhx : x \u2208 ball z R\nhf : ContinuousOn f (sphere z R)\n\u22a2 \u2203 B \u03b5, 0 < \u03b5 \u2227 ball x \u03b5 \u2286 ball z R \u2227 \u2200 (t : \u211d) (y : \u2102), y \u2208 ball x \u03b5 \u2192 \u2016circleTransformDeriv R z y f t\u2016 \u2264 B", "state_after": "case intro.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR : \u211d\nhR : 0 < R\nz x : \u2102\nf : \u2102 \u2192 \u2102\nhx : x \u2208 ball z R\nhf : ContinuousOn f (sphere z R)\nr : \u211d\nhr : r < R\nhrx : x \u2208 ball z r\n\u22a2 \u2203 B \u03b5, 0 < \u03b5 \u2227 ball x \u03b5 \u2286 ball z R \u2227 \u2200 (t : \u211d) (y : \u2102), y \u2208 ball x \u03b5 \u2192 \u2016circleTransformDeriv R z y f t\u2016 \u2264 B"}, {"tactic": "obtain \u27e8\u03b5', h\u03b5', H\u27e9 := exists_ball_subset_ball hrx", "annotated_tactic": ["obtain \u27e8\u03b5', h\u03b5', H\u27e9 := <a>exists_ball_subset_ball</a> hrx", [{"full_name": "Metric.exists_ball_subset_ball", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [680, 9], "def_end_pos": [680, 32]}]], "state_before": "case intro.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR : \u211d\nhR : 0 < R\nz x : \u2102\nf : \u2102 \u2192 \u2102\nhx : x \u2208 ball z R\nhf : ContinuousOn f (sphere z R)\nr : \u211d\nhr : r < R\nhrx : x \u2208 ball z r\n\u22a2 \u2203 B \u03b5, 0 < \u03b5 \u2227 ball x \u03b5 \u2286 ball z R \u2227 \u2200 (t : \u211d) (y : \u2102), y \u2208 ball x \u03b5 \u2192 \u2016circleTransformDeriv R z y f t\u2016 \u2264 B", "state_after": "case intro.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR : \u211d\nhR : 0 < R\nz x : \u2102\nf : \u2102 \u2192 \u2102\nhx : x \u2208 ball z R\nhf : ContinuousOn f (sphere z R)\nr : \u211d\nhr : r < R\nhrx : x \u2208 ball z r\n\u03b5' : \u211d\nh\u03b5' : \u03b5' > 0\nH : ball x \u03b5' \u2286 ball z r\n\u22a2 \u2203 B \u03b5, 0 < \u03b5 \u2227 ball x \u03b5 \u2286 ball z R \u2227 \u2200 (t : \u211d) (y : \u2102), y \u2208 ball x \u03b5 \u2192 \u2016circleTransformDeriv R z y f t\u2016 \u2264 B"}, {"tactic": "obtain \u27e8\u27e8\u27e8a, b\u27e9, \u27e8ha, hb\u27e9\u27e9, hab\u27e9 :=\n  abs_circleTransformBoundingFunction_le hr (pos_of_mem_ball hrx).le z", "annotated_tactic": ["obtain \u27e8\u27e8\u27e8a, b\u27e9, \u27e8ha, hb\u27e9\u27e9, hab\u27e9 :=\n    <a>abs_circleTransformBoundingFunction_le</a> hr (<a>pos_of_mem_ball</a> hrx).<a>le</a> z", [{"full_name": "Complex.abs_circleTransformBoundingFunction_le", "def_path": "Mathlib/MeasureTheory/Integral/CircleTransform.lean", "def_pos": [128, 9], "def_end_pos": [128, 47]}, {"full_name": "Metric.pos_of_mem_ball", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [421, 9], "def_end_pos": [421, 24]}, {"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [142, 7], "def_end_pos": [142, 15]}]], "state_before": "case intro.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR : \u211d\nhR : 0 < R\nz x : \u2102\nf : \u2102 \u2192 \u2102\nhx : x \u2208 ball z R\nhf : ContinuousOn f (sphere z R)\nr : \u211d\nhr : r < R\nhrx : x \u2208 ball z r\n\u03b5' : \u211d\nh\u03b5' : \u03b5' > 0\nH : ball x \u03b5' \u2286 ball z r\n\u22a2 \u2203 B \u03b5, 0 < \u03b5 \u2227 ball x \u03b5 \u2286 ball z R \u2227 \u2200 (t : \u211d) (y : \u2102), y \u2208 ball x \u03b5 \u2192 \u2016circleTransformDeriv R z y f t\u2016 \u2264 B", "state_after": "case intro.intro.intro.intro.intro.mk.mk.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR : \u211d\nhR : 0 < R\nz x : \u2102\nf : \u2102 \u2192 \u2102\nhx : x \u2208 ball z R\nhf : ContinuousOn f (sphere z R)\nr : \u211d\nhr : r < R\nhrx : x \u2208 ball z r\n\u03b5' : \u211d\nh\u03b5' : \u03b5' > 0\nH : ball x \u03b5' \u2286 ball z r\na : \u2102\nb : \u211d\nha : (a, b).1 \u2208 closedBall z r\nhb : (a, b).2 \u2208 [[0, 2 * \u03c0]]\nhab :\n  \u2200 (y : \u2191(closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])),\n    \u2191abs (circleTransformBoundingFunction R z \u2191y) \u2264\n      \u2191abs\n        (circleTransformBoundingFunction R z\n          \u2191{ val := (a, b), property := (_ : (a, b).1 \u2208 closedBall z r \u2227 (a, b).2 \u2208 [[0, 2 * \u03c0]]) })\n\u22a2 \u2203 B \u03b5, 0 < \u03b5 \u2227 ball x \u03b5 \u2286 ball z R \u2227 \u2200 (t : \u211d) (y : \u2102), y \u2208 ball x \u03b5 \u2192 \u2016circleTransformDeriv R z y f t\u2016 \u2264 B"}, {"tactic": "let V : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b8 w => circleTransformDeriv R z w (fun _ => 1) \u03b8", "annotated_tactic": ["let V : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b8 w => <a>circleTransformDeriv</a> R z w (fun _ => 1) \u03b8", [{"full_name": "Complex.circleTransformDeriv", "def_path": "Mathlib/MeasureTheory/Integral/CircleTransform.lean", "def_pos": [46, 5], "def_end_pos": [46, 25]}]], "state_before": "case intro.intro.intro.intro.intro.mk.mk.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR : \u211d\nhR : 0 < R\nz x : \u2102\nf : \u2102 \u2192 \u2102\nhx : x \u2208 ball z R\nhf : ContinuousOn f (sphere z R)\nr : \u211d\nhr : r < R\nhrx : x \u2208 ball z r\n\u03b5' : \u211d\nh\u03b5' : \u03b5' > 0\nH : ball x \u03b5' \u2286 ball z r\na : \u2102\nb : \u211d\nha : (a, b).1 \u2208 closedBall z r\nhb : (a, b).2 \u2208 [[0, 2 * \u03c0]]\nhab :\n  \u2200 (y : \u2191(closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])),\n    \u2191abs (circleTransformBoundingFunction R z \u2191y) \u2264\n      \u2191abs\n        (circleTransformBoundingFunction R z\n          \u2191{ val := (a, b), property := (_ : (a, b).1 \u2208 closedBall z r \u2227 (a, b).2 \u2208 [[0, 2 * \u03c0]]) })\n\u22a2 \u2203 B \u03b5, 0 < \u03b5 \u2227 ball x \u03b5 \u2286 ball z R \u2227 \u2200 (t : \u211d) (y : \u2102), y \u2208 ball x \u03b5 \u2192 \u2016circleTransformDeriv R z y f t\u2016 \u2264 B", "state_after": "case intro.intro.intro.intro.intro.mk.mk.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR : \u211d\nhR : 0 < R\nz x : \u2102\nf : \u2102 \u2192 \u2102\nhx : x \u2208 ball z R\nhf : ContinuousOn f (sphere z R)\nr : \u211d\nhr : r < R\nhrx : x \u2208 ball z r\n\u03b5' : \u211d\nh\u03b5' : \u03b5' > 0\nH : ball x \u03b5' \u2286 ball z r\na : \u2102\nb : \u211d\nha : (a, b).1 \u2208 closedBall z r\nhb : (a, b).2 \u2208 [[0, 2 * \u03c0]]\nhab :\n  \u2200 (y : \u2191(closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])),\n    \u2191abs (circleTransformBoundingFunction R z \u2191y) \u2264\n      \u2191abs\n        (circleTransformBoundingFunction R z\n          \u2191{ val := (a, b), property := (_ : (a, b).1 \u2208 closedBall z r \u2227 (a, b).2 \u2208 [[0, 2 * \u03c0]]) })\nV : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b8 w => circleTransformDeriv R z w (fun x => 1) \u03b8\n\u22a2 \u2203 B \u03b5, 0 < \u03b5 \u2227 ball x \u03b5 \u2286 ball z R \u2227 \u2200 (t : \u211d) (y : \u2102), y \u2208 ball x \u03b5 \u2192 \u2016circleTransformDeriv R z y f t\u2016 \u2264 B"}, {"tactic": "have funccomp : ContinuousOn (fun r => abs (f r)) (sphere z R) := by\n  have cabs : ContinuousOn abs \u22a4 := by apply continuous_abs.continuousOn\n  apply cabs.comp hf; rw [MapsTo]; tauto", "annotated_tactic": ["have funccomp : <a>ContinuousOn</a> (fun r => <a>abs</a> (f r)) (<a>sphere</a> z R) := by\n    have cabs : <a>ContinuousOn</a> <a>abs</a> \u22a4 := by apply continuous_abs.continuousOn\n    apply cabs.comp hf; rw [<a>MapsTo</a>]; tauto", [{"full_name": "ContinuousOn", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [532, 5], "def_end_pos": [532, 17]}, {"full_name": "Complex.abs", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [966, 19], "def_end_pos": [966, 37]}, {"full_name": "Metric.sphere", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [485, 5], "def_end_pos": [485, 11]}, {"full_name": "ContinuousOn", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [532, 5], "def_end_pos": [532, 17]}, {"full_name": "Complex.abs", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [966, 19], "def_end_pos": [966, 37]}, {"full_name": "Set.MapsTo", "def_path": "Mathlib/Data/Set/Function.lean", "def_pos": [348, 5], "def_end_pos": [348, 11]}]], "state_before": "case intro.intro.intro.intro.intro.mk.mk.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR : \u211d\nhR : 0 < R\nz x : \u2102\nf : \u2102 \u2192 \u2102\nhx : x \u2208 ball z R\nhf : ContinuousOn f (sphere z R)\nr : \u211d\nhr : r < R\nhrx : x \u2208 ball z r\n\u03b5' : \u211d\nh\u03b5' : \u03b5' > 0\nH : ball x \u03b5' \u2286 ball z r\na : \u2102\nb : \u211d\nha : (a, b).1 \u2208 closedBall z r\nhb : (a, b).2 \u2208 [[0, 2 * \u03c0]]\nhab :\n  \u2200 (y : \u2191(closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])),\n    \u2191abs (circleTransformBoundingFunction R z \u2191y) \u2264\n      \u2191abs\n        (circleTransformBoundingFunction R z\n          \u2191{ val := (a, b), property := (_ : (a, b).1 \u2208 closedBall z r \u2227 (a, b).2 \u2208 [[0, 2 * \u03c0]]) })\nV : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b8 w => circleTransformDeriv R z w (fun x => 1) \u03b8\n\u22a2 \u2203 B \u03b5, 0 < \u03b5 \u2227 ball x \u03b5 \u2286 ball z R \u2227 \u2200 (t : \u211d) (y : \u2102), y \u2208 ball x \u03b5 \u2192 \u2016circleTransformDeriv R z y f t\u2016 \u2264 B", "state_after": "case intro.intro.intro.intro.intro.mk.mk.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR : \u211d\nhR : 0 < R\nz x : \u2102\nf : \u2102 \u2192 \u2102\nhx : x \u2208 ball z R\nhf : ContinuousOn f (sphere z R)\nr : \u211d\nhr : r < R\nhrx : x \u2208 ball z r\n\u03b5' : \u211d\nh\u03b5' : \u03b5' > 0\nH : ball x \u03b5' \u2286 ball z r\na : \u2102\nb : \u211d\nha : (a, b).1 \u2208 closedBall z r\nhb : (a, b).2 \u2208 [[0, 2 * \u03c0]]\nhab :\n  \u2200 (y : \u2191(closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])),\n    \u2191abs (circleTransformBoundingFunction R z \u2191y) \u2264\n      \u2191abs\n        (circleTransformBoundingFunction R z\n          \u2191{ val := (a, b), property := (_ : (a, b).1 \u2208 closedBall z r \u2227 (a, b).2 \u2208 [[0, 2 * \u03c0]]) })\nV : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b8 w => circleTransformDeriv R z w (fun x => 1) \u03b8\nfunccomp : ContinuousOn (fun r => \u2191abs (f r)) (sphere z R)\n\u22a2 \u2203 B \u03b5, 0 < \u03b5 \u2227 ball x \u03b5 \u2286 ball z R \u2227 \u2200 (t : \u211d) (y : \u2102), y \u2208 ball x \u03b5 \u2192 \u2016circleTransformDeriv R z y f t\u2016 \u2264 B"}, {"tactic": "have sbou :=\n  IsCompact.exists_isMaxOn (isCompact_sphere z R) (NormedSpace.sphere_nonempty.2 hR.le) funccomp", "annotated_tactic": ["have sbou :=\n    <a>IsCompact.exists_isMaxOn</a> (<a>isCompact_sphere</a> z R) (<a>NormedSpace.sphere_nonempty</a>.2 hR.le) funccomp", [{"full_name": "IsCompact.exists_isMaxOn", "def_path": "Mathlib/Topology/Algebra/Order/Compact.lean", "def_pos": [212, 9], "def_end_pos": [212, 33]}, {"full_name": "isCompact_sphere", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [2205, 9], "def_end_pos": [2205, 25]}, {"full_name": "NormedSpace.sphere_nonempty", "def_path": "Mathlib/Analysis/NormedSpace/Pointwise.lean", "def_pos": [411, 9], "def_end_pos": [411, 36]}]], "state_before": "case intro.intro.intro.intro.intro.mk.mk.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR : \u211d\nhR : 0 < R\nz x : \u2102\nf : \u2102 \u2192 \u2102\nhx : x \u2208 ball z R\nhf : ContinuousOn f (sphere z R)\nr : \u211d\nhr : r < R\nhrx : x \u2208 ball z r\n\u03b5' : \u211d\nh\u03b5' : \u03b5' > 0\nH : ball x \u03b5' \u2286 ball z r\na : \u2102\nb : \u211d\nha : (a, b).1 \u2208 closedBall z r\nhb : (a, b).2 \u2208 [[0, 2 * \u03c0]]\nhab :\n  \u2200 (y : \u2191(closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])),\n    \u2191abs (circleTransformBoundingFunction R z \u2191y) \u2264\n      \u2191abs\n        (circleTransformBoundingFunction R z\n          \u2191{ val := (a, b), property := (_ : (a, b).1 \u2208 closedBall z r \u2227 (a, b).2 \u2208 [[0, 2 * \u03c0]]) })\nV : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b8 w => circleTransformDeriv R z w (fun x => 1) \u03b8\nfunccomp : ContinuousOn (fun r => \u2191abs (f r)) (sphere z R)\n\u22a2 \u2203 B \u03b5, 0 < \u03b5 \u2227 ball x \u03b5 \u2286 ball z R \u2227 \u2200 (t : \u211d) (y : \u2102), y \u2208 ball x \u03b5 \u2192 \u2016circleTransformDeriv R z y f t\u2016 \u2264 B", "state_after": "case intro.intro.intro.intro.intro.mk.mk.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR : \u211d\nhR : 0 < R\nz x : \u2102\nf : \u2102 \u2192 \u2102\nhx : x \u2208 ball z R\nhf : ContinuousOn f (sphere z R)\nr : \u211d\nhr : r < R\nhrx : x \u2208 ball z r\n\u03b5' : \u211d\nh\u03b5' : \u03b5' > 0\nH : ball x \u03b5' \u2286 ball z r\na : \u2102\nb : \u211d\nha : (a, b).1 \u2208 closedBall z r\nhb : (a, b).2 \u2208 [[0, 2 * \u03c0]]\nhab :\n  \u2200 (y : \u2191(closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])),\n    \u2191abs (circleTransformBoundingFunction R z \u2191y) \u2264\n      \u2191abs\n        (circleTransformBoundingFunction R z\n          \u2191{ val := (a, b), property := (_ : (a, b).1 \u2208 closedBall z r \u2227 (a, b).2 \u2208 [[0, 2 * \u03c0]]) })\nV : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b8 w => circleTransformDeriv R z w (fun x => 1) \u03b8\nfunccomp : ContinuousOn (fun r => \u2191abs (f r)) (sphere z R)\nsbou : \u2203 x, x \u2208 sphere z R \u2227 IsMaxOn (fun r => \u2191abs (f r)) (sphere z R) x\n\u22a2 \u2203 B \u03b5, 0 < \u03b5 \u2227 ball x \u03b5 \u2286 ball z R \u2227 \u2200 (t : \u211d) (y : \u2102), y \u2208 ball x \u03b5 \u2192 \u2016circleTransformDeriv R z y f t\u2016 \u2264 B"}, {"tactic": "obtain \u27e8X, HX, HX2\u27e9 := sbou", "annotated_tactic": ["obtain \u27e8X, HX, HX2\u27e9 := sbou", []], "state_before": "case intro.intro.intro.intro.intro.mk.mk.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR : \u211d\nhR : 0 < R\nz x : \u2102\nf : \u2102 \u2192 \u2102\nhx : x \u2208 ball z R\nhf : ContinuousOn f (sphere z R)\nr : \u211d\nhr : r < R\nhrx : x \u2208 ball z r\n\u03b5' : \u211d\nh\u03b5' : \u03b5' > 0\nH : ball x \u03b5' \u2286 ball z r\na : \u2102\nb : \u211d\nha : (a, b).1 \u2208 closedBall z r\nhb : (a, b).2 \u2208 [[0, 2 * \u03c0]]\nhab :\n  \u2200 (y : \u2191(closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])),\n    \u2191abs (circleTransformBoundingFunction R z \u2191y) \u2264\n      \u2191abs\n        (circleTransformBoundingFunction R z\n          \u2191{ val := (a, b), property := (_ : (a, b).1 \u2208 closedBall z r \u2227 (a, b).2 \u2208 [[0, 2 * \u03c0]]) })\nV : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b8 w => circleTransformDeriv R z w (fun x => 1) \u03b8\nfunccomp : ContinuousOn (fun r => \u2191abs (f r)) (sphere z R)\nsbou : \u2203 x, x \u2208 sphere z R \u2227 IsMaxOn (fun r => \u2191abs (f r)) (sphere z R) x\n\u22a2 \u2203 B \u03b5, 0 < \u03b5 \u2227 ball x \u03b5 \u2286 ball z R \u2227 \u2200 (t : \u211d) (y : \u2102), y \u2208 ball x \u03b5 \u2192 \u2016circleTransformDeriv R z y f t\u2016 \u2264 B", "state_after": "case intro.intro.intro.intro.intro.mk.mk.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR : \u211d\nhR : 0 < R\nz x : \u2102\nf : \u2102 \u2192 \u2102\nhx : x \u2208 ball z R\nhf : ContinuousOn f (sphere z R)\nr : \u211d\nhr : r < R\nhrx : x \u2208 ball z r\n\u03b5' : \u211d\nh\u03b5' : \u03b5' > 0\nH : ball x \u03b5' \u2286 ball z r\na : \u2102\nb : \u211d\nha : (a, b).1 \u2208 closedBall z r\nhb : (a, b).2 \u2208 [[0, 2 * \u03c0]]\nhab :\n  \u2200 (y : \u2191(closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])),\n    \u2191abs (circleTransformBoundingFunction R z \u2191y) \u2264\n      \u2191abs\n        (circleTransformBoundingFunction R z\n          \u2191{ val := (a, b), property := (_ : (a, b).1 \u2208 closedBall z r \u2227 (a, b).2 \u2208 [[0, 2 * \u03c0]]) })\nV : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b8 w => circleTransformDeriv R z w (fun x => 1) \u03b8\nfunccomp : ContinuousOn (fun r => \u2191abs (f r)) (sphere z R)\nX : \u2102\nHX : X \u2208 sphere z R\nHX2 : IsMaxOn (fun r => \u2191abs (f r)) (sphere z R) X\n\u22a2 \u2203 B \u03b5, 0 < \u03b5 \u2227 ball x \u03b5 \u2286 ball z R \u2227 \u2200 (t : \u211d) (y : \u2102), y \u2208 ball x \u03b5 \u2192 \u2016circleTransformDeriv R z y f t\u2016 \u2264 B"}, {"tactic": "refine' \u27e8abs (V b a) * abs (f X), \u03b5', h\u03b5', Subset.trans H (ball_subset_ball hr.le), _\u27e9", "annotated_tactic": ["refine' \u27e8<a>abs</a> (V b a) * <a>abs</a> (f X), \u03b5', h\u03b5', <a>Subset.trans</a> H (<a>ball_subset_ball</a> hr.le), _\u27e9", [{"full_name": "Complex.abs", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [966, 19], "def_end_pos": [966, 37]}, {"full_name": "Complex.abs", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [966, 19], "def_end_pos": [966, 37]}, {"full_name": "Set.Subset.trans", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [362, 9], "def_end_pos": [362, 21]}, {"full_name": "Metric.ball_subset_ball", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [594, 9], "def_end_pos": [594, 25]}]], "state_before": "case intro.intro.intro.intro.intro.mk.mk.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR : \u211d\nhR : 0 < R\nz x : \u2102\nf : \u2102 \u2192 \u2102\nhx : x \u2208 ball z R\nhf : ContinuousOn f (sphere z R)\nr : \u211d\nhr : r < R\nhrx : x \u2208 ball z r\n\u03b5' : \u211d\nh\u03b5' : \u03b5' > 0\nH : ball x \u03b5' \u2286 ball z r\na : \u2102\nb : \u211d\nha : (a, b).1 \u2208 closedBall z r\nhb : (a, b).2 \u2208 [[0, 2 * \u03c0]]\nhab :\n  \u2200 (y : \u2191(closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])),\n    \u2191abs (circleTransformBoundingFunction R z \u2191y) \u2264\n      \u2191abs\n        (circleTransformBoundingFunction R z\n          \u2191{ val := (a, b), property := (_ : (a, b).1 \u2208 closedBall z r \u2227 (a, b).2 \u2208 [[0, 2 * \u03c0]]) })\nV : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b8 w => circleTransformDeriv R z w (fun x => 1) \u03b8\nfunccomp : ContinuousOn (fun r => \u2191abs (f r)) (sphere z R)\nX : \u2102\nHX : X \u2208 sphere z R\nHX2 : IsMaxOn (fun r => \u2191abs (f r)) (sphere z R) X\n\u22a2 \u2203 B \u03b5, 0 < \u03b5 \u2227 ball x \u03b5 \u2286 ball z R \u2227 \u2200 (t : \u211d) (y : \u2102), y \u2208 ball x \u03b5 \u2192 \u2016circleTransformDeriv R z y f t\u2016 \u2264 B", "state_after": "case intro.intro.intro.intro.intro.mk.mk.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR : \u211d\nhR : 0 < R\nz x : \u2102\nf : \u2102 \u2192 \u2102\nhx : x \u2208 ball z R\nhf : ContinuousOn f (sphere z R)\nr : \u211d\nhr : r < R\nhrx : x \u2208 ball z r\n\u03b5' : \u211d\nh\u03b5' : \u03b5' > 0\nH : ball x \u03b5' \u2286 ball z r\na : \u2102\nb : \u211d\nha : (a, b).1 \u2208 closedBall z r\nhb : (a, b).2 \u2208 [[0, 2 * \u03c0]]\nhab :\n  \u2200 (y : \u2191(closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])),\n    \u2191abs (circleTransformBoundingFunction R z \u2191y) \u2264\n      \u2191abs\n        (circleTransformBoundingFunction R z\n          \u2191{ val := (a, b), property := (_ : (a, b).1 \u2208 closedBall z r \u2227 (a, b).2 \u2208 [[0, 2 * \u03c0]]) })\nV : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b8 w => circleTransformDeriv R z w (fun x => 1) \u03b8\nfunccomp : ContinuousOn (fun r => \u2191abs (f r)) (sphere z R)\nX : \u2102\nHX : X \u2208 sphere z R\nHX2 : IsMaxOn (fun r => \u2191abs (f r)) (sphere z R) X\n\u22a2 \u2200 (t : \u211d) (y : \u2102), y \u2208 ball x \u03b5' \u2192 \u2016circleTransformDeriv R z y f t\u2016 \u2264 \u2191abs (V b a) * \u2191abs (f X)"}, {"tactic": "intro y v hv", "annotated_tactic": ["intro y v hv", []], "state_before": "case intro.intro.intro.intro.intro.mk.mk.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR : \u211d\nhR : 0 < R\nz x : \u2102\nf : \u2102 \u2192 \u2102\nhx : x \u2208 ball z R\nhf : ContinuousOn f (sphere z R)\nr : \u211d\nhr : r < R\nhrx : x \u2208 ball z r\n\u03b5' : \u211d\nh\u03b5' : \u03b5' > 0\nH : ball x \u03b5' \u2286 ball z r\na : \u2102\nb : \u211d\nha : (a, b).1 \u2208 closedBall z r\nhb : (a, b).2 \u2208 [[0, 2 * \u03c0]]\nhab :\n  \u2200 (y : \u2191(closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])),\n    \u2191abs (circleTransformBoundingFunction R z \u2191y) \u2264\n      \u2191abs\n        (circleTransformBoundingFunction R z\n          \u2191{ val := (a, b), property := (_ : (a, b).1 \u2208 closedBall z r \u2227 (a, b).2 \u2208 [[0, 2 * \u03c0]]) })\nV : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b8 w => circleTransformDeriv R z w (fun x => 1) \u03b8\nfunccomp : ContinuousOn (fun r => \u2191abs (f r)) (sphere z R)\nX : \u2102\nHX : X \u2208 sphere z R\nHX2 : IsMaxOn (fun r => \u2191abs (f r)) (sphere z R) X\n\u22a2 \u2200 (t : \u211d) (y : \u2102), y \u2208 ball x \u03b5' \u2192 \u2016circleTransformDeriv R z y f t\u2016 \u2264 \u2191abs (V b a) * \u2191abs (f X)", "state_after": "case intro.intro.intro.intro.intro.mk.mk.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR : \u211d\nhR : 0 < R\nz x : \u2102\nf : \u2102 \u2192 \u2102\nhx : x \u2208 ball z R\nhf : ContinuousOn f (sphere z R)\nr : \u211d\nhr : r < R\nhrx : x \u2208 ball z r\n\u03b5' : \u211d\nh\u03b5' : \u03b5' > 0\nH : ball x \u03b5' \u2286 ball z r\na : \u2102\nb : \u211d\nha : (a, b).1 \u2208 closedBall z r\nhb : (a, b).2 \u2208 [[0, 2 * \u03c0]]\nhab :\n  \u2200 (y : \u2191(closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])),\n    \u2191abs (circleTransformBoundingFunction R z \u2191y) \u2264\n      \u2191abs\n        (circleTransformBoundingFunction R z\n          \u2191{ val := (a, b), property := (_ : (a, b).1 \u2208 closedBall z r \u2227 (a, b).2 \u2208 [[0, 2 * \u03c0]]) })\nV : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b8 w => circleTransformDeriv R z w (fun x => 1) \u03b8\nfunccomp : ContinuousOn (fun r => \u2191abs (f r)) (sphere z R)\nX : \u2102\nHX : X \u2208 sphere z R\nHX2 : IsMaxOn (fun r => \u2191abs (f r)) (sphere z R) X\ny : \u211d\nv : \u2102\nhv : v \u2208 ball x \u03b5'\n\u22a2 \u2016circleTransformDeriv R z v f y\u2016 \u2264 \u2191abs (V b a) * \u2191abs (f X)"}, {"tactic": "obtain \u27e8y1, hy1, hfun\u27e9 :=\n  Periodic.exists_mem_Ico\u2080 (circleTransformDeriv_periodic R z v f) Real.two_pi_pos y", "annotated_tactic": ["obtain \u27e8y1, hy1, hfun\u27e9 :=\n    <a>Periodic.exists_mem_Ico\u2080</a> (<a>circleTransformDeriv_periodic</a> R z v f) <a>Real.two_pi_pos</a> y", [{"full_name": "Function.Periodic.exists_mem_Ico\u2080", "def_path": "Mathlib/Algebra/Periodic.lean", "def_pos": [284, 9], "def_end_pos": [284, 33]}, {"full_name": "Complex.circleTransformDeriv_periodic", "def_path": "Mathlib/MeasureTheory/Integral/CircleTransform.lean", "def_pos": [50, 9], "def_end_pos": [50, 38]}, {"full_name": "Real.two_pi_pos", "def_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean", "def_pos": [186, 9], "def_end_pos": [186, 19]}]], "state_before": "case intro.intro.intro.intro.intro.mk.mk.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR : \u211d\nhR : 0 < R\nz x : \u2102\nf : \u2102 \u2192 \u2102\nhx : x \u2208 ball z R\nhf : ContinuousOn f (sphere z R)\nr : \u211d\nhr : r < R\nhrx : x \u2208 ball z r\n\u03b5' : \u211d\nh\u03b5' : \u03b5' > 0\nH : ball x \u03b5' \u2286 ball z r\na : \u2102\nb : \u211d\nha : (a, b).1 \u2208 closedBall z r\nhb : (a, b).2 \u2208 [[0, 2 * \u03c0]]\nhab :\n  \u2200 (y : \u2191(closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])),\n    \u2191abs (circleTransformBoundingFunction R z \u2191y) \u2264\n      \u2191abs\n        (circleTransformBoundingFunction R z\n          \u2191{ val := (a, b), property := (_ : (a, b).1 \u2208 closedBall z r \u2227 (a, b).2 \u2208 [[0, 2 * \u03c0]]) })\nV : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b8 w => circleTransformDeriv R z w (fun x => 1) \u03b8\nfunccomp : ContinuousOn (fun r => \u2191abs (f r)) (sphere z R)\nX : \u2102\nHX : X \u2208 sphere z R\nHX2 : IsMaxOn (fun r => \u2191abs (f r)) (sphere z R) X\ny : \u211d\nv : \u2102\nhv : v \u2208 ball x \u03b5'\n\u22a2 \u2016circleTransformDeriv R z v f y\u2016 \u2264 \u2191abs (V b a) * \u2191abs (f X)", "state_after": "case intro.intro.intro.intro.intro.mk.mk.intro.intro.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR : \u211d\nhR : 0 < R\nz x : \u2102\nf : \u2102 \u2192 \u2102\nhx : x \u2208 ball z R\nhf : ContinuousOn f (sphere z R)\nr : \u211d\nhr : r < R\nhrx : x \u2208 ball z r\n\u03b5' : \u211d\nh\u03b5' : \u03b5' > 0\nH : ball x \u03b5' \u2286 ball z r\na : \u2102\nb : \u211d\nha : (a, b).1 \u2208 closedBall z r\nhb : (a, b).2 \u2208 [[0, 2 * \u03c0]]\nhab :\n  \u2200 (y : \u2191(closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])),\n    \u2191abs (circleTransformBoundingFunction R z \u2191y) \u2264\n      \u2191abs\n        (circleTransformBoundingFunction R z\n          \u2191{ val := (a, b), property := (_ : (a, b).1 \u2208 closedBall z r \u2227 (a, b).2 \u2208 [[0, 2 * \u03c0]]) })\nV : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b8 w => circleTransformDeriv R z w (fun x => 1) \u03b8\nfunccomp : ContinuousOn (fun r => \u2191abs (f r)) (sphere z R)\nX : \u2102\nHX : X \u2208 sphere z R\nHX2 : IsMaxOn (fun r => \u2191abs (f r)) (sphere z R) X\ny : \u211d\nv : \u2102\nhv : v \u2208 ball x \u03b5'\ny1 : \u211d\nhy1 : y1 \u2208 Ico 0 (2 * \u03c0)\nhfun : circleTransformDeriv R z v f y = circleTransformDeriv R z v f y1\n\u22a2 \u2016circleTransformDeriv R z v f y\u2016 \u2264 \u2191abs (V b a) * \u2191abs (f X)"}, {"tactic": "have hy2 : y1 \u2208 [[0, 2 * \u03c0]] := by\n  convert Ico_subset_Icc_self hy1 using 1\n  simp [uIcc_of_le Real.two_pi_pos.le]", "annotated_tactic": ["have hy2 : y1 \u2208 [[0, 2 * \u03c0]] := by\n    convert <a>Ico_subset_Icc_self</a> hy1 using 1\n    simp [<a>uIcc_of_le</a> Real.two_pi_pos.le]", [{"full_name": "Set.Ico_subset_Icc_self", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [510, 9], "def_end_pos": [510, 28]}, {"full_name": "Set.uIcc_of_le", "def_path": "Mathlib/Data/Set/Intervals/UnorderedInterval.lean", "def_pos": [69, 7], "def_end_pos": [69, 17]}]], "state_before": "case intro.intro.intro.intro.intro.mk.mk.intro.intro.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR : \u211d\nhR : 0 < R\nz x : \u2102\nf : \u2102 \u2192 \u2102\nhx : x \u2208 ball z R\nhf : ContinuousOn f (sphere z R)\nr : \u211d\nhr : r < R\nhrx : x \u2208 ball z r\n\u03b5' : \u211d\nh\u03b5' : \u03b5' > 0\nH : ball x \u03b5' \u2286 ball z r\na : \u2102\nb : \u211d\nha : (a, b).1 \u2208 closedBall z r\nhb : (a, b).2 \u2208 [[0, 2 * \u03c0]]\nhab :\n  \u2200 (y : \u2191(closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])),\n    \u2191abs (circleTransformBoundingFunction R z \u2191y) \u2264\n      \u2191abs\n        (circleTransformBoundingFunction R z\n          \u2191{ val := (a, b), property := (_ : (a, b).1 \u2208 closedBall z r \u2227 (a, b).2 \u2208 [[0, 2 * \u03c0]]) })\nV : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b8 w => circleTransformDeriv R z w (fun x => 1) \u03b8\nfunccomp : ContinuousOn (fun r => \u2191abs (f r)) (sphere z R)\nX : \u2102\nHX : X \u2208 sphere z R\nHX2 : IsMaxOn (fun r => \u2191abs (f r)) (sphere z R) X\ny : \u211d\nv : \u2102\nhv : v \u2208 ball x \u03b5'\ny1 : \u211d\nhy1 : y1 \u2208 Ico 0 (2 * \u03c0)\nhfun : circleTransformDeriv R z v f y = circleTransformDeriv R z v f y1\n\u22a2 \u2016circleTransformDeriv R z v f y\u2016 \u2264 \u2191abs (V b a) * \u2191abs (f X)", "state_after": "case intro.intro.intro.intro.intro.mk.mk.intro.intro.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR : \u211d\nhR : 0 < R\nz x : \u2102\nf : \u2102 \u2192 \u2102\nhx : x \u2208 ball z R\nhf : ContinuousOn f (sphere z R)\nr : \u211d\nhr : r < R\nhrx : x \u2208 ball z r\n\u03b5' : \u211d\nh\u03b5' : \u03b5' > 0\nH : ball x \u03b5' \u2286 ball z r\na : \u2102\nb : \u211d\nha : (a, b).1 \u2208 closedBall z r\nhb : (a, b).2 \u2208 [[0, 2 * \u03c0]]\nhab :\n  \u2200 (y : \u2191(closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])),\n    \u2191abs (circleTransformBoundingFunction R z \u2191y) \u2264\n      \u2191abs\n        (circleTransformBoundingFunction R z\n          \u2191{ val := (a, b), property := (_ : (a, b).1 \u2208 closedBall z r \u2227 (a, b).2 \u2208 [[0, 2 * \u03c0]]) })\nV : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b8 w => circleTransformDeriv R z w (fun x => 1) \u03b8\nfunccomp : ContinuousOn (fun r => \u2191abs (f r)) (sphere z R)\nX : \u2102\nHX : X \u2208 sphere z R\nHX2 : IsMaxOn (fun r => \u2191abs (f r)) (sphere z R) X\ny : \u211d\nv : \u2102\nhv : v \u2208 ball x \u03b5'\ny1 : \u211d\nhy1 : y1 \u2208 Ico 0 (2 * \u03c0)\nhfun : circleTransformDeriv R z v f y = circleTransformDeriv R z v f y1\nhy2 : y1 \u2208 [[0, 2 * \u03c0]]\n\u22a2 \u2016circleTransformDeriv R z v f y\u2016 \u2264 \u2191abs (V b a) * \u2191abs (f X)"}, {"tactic": "simp only [IsMaxOn, IsMaxFilter, eventually_principal, mem_sphere_iff_norm, norm_eq_abs] at HX2", "annotated_tactic": ["simp only [<a>IsMaxOn</a>, <a>IsMaxFilter</a>, <a>eventually_principal</a>, <a>mem_sphere_iff_norm</a>, <a>norm_eq_abs</a>] at HX2", [{"full_name": "IsMaxOn", "def_path": "Mathlib/Order/Filter/Extr.lean", "def_pos": [116, 5], "def_end_pos": [116, 12]}, {"full_name": "IsMaxFilter", "def_path": "Mathlib/Order/Filter/Extr.lean", "def_pos": [101, 5], "def_end_pos": [101, 16]}, {"full_name": "Filter.eventually_principal", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1241, 9], "def_end_pos": [1241, 29]}, {"full_name": "mem_sphere_iff_norm", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [709, 35], "def_end_pos": [709, 54]}, {"full_name": "Complex.norm_eq_abs", "def_path": "Mathlib/Analysis/Complex/Basic.lean", "def_pos": [51, 9], "def_end_pos": [51, 20]}]], "state_before": "case intro.intro.intro.intro.intro.mk.mk.intro.intro.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR : \u211d\nhR : 0 < R\nz x : \u2102\nf : \u2102 \u2192 \u2102\nhx : x \u2208 ball z R\nhf : ContinuousOn f (sphere z R)\nr : \u211d\nhr : r < R\nhrx : x \u2208 ball z r\n\u03b5' : \u211d\nh\u03b5' : \u03b5' > 0\nH : ball x \u03b5' \u2286 ball z r\na : \u2102\nb : \u211d\nha : (a, b).1 \u2208 closedBall z r\nhb : (a, b).2 \u2208 [[0, 2 * \u03c0]]\nhab :\n  \u2200 (y : \u2191(closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])),\n    \u2191abs (circleTransformBoundingFunction R z \u2191y) \u2264\n      \u2191abs\n        (circleTransformBoundingFunction R z\n          \u2191{ val := (a, b), property := (_ : (a, b).1 \u2208 closedBall z r \u2227 (a, b).2 \u2208 [[0, 2 * \u03c0]]) })\nV : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b8 w => circleTransformDeriv R z w (fun x => 1) \u03b8\nfunccomp : ContinuousOn (fun r => \u2191abs (f r)) (sphere z R)\nX : \u2102\nHX : X \u2208 sphere z R\nHX2 : IsMaxOn (fun r => \u2191abs (f r)) (sphere z R) X\ny : \u211d\nv : \u2102\nhv : v \u2208 ball x \u03b5'\ny1 : \u211d\nhy1 : y1 \u2208 Ico 0 (2 * \u03c0)\nhfun : circleTransformDeriv R z v f y = circleTransformDeriv R z v f y1\nhy2 : y1 \u2208 [[0, 2 * \u03c0]]\n\u22a2 \u2016circleTransformDeriv R z v f y\u2016 \u2264 \u2191abs (V b a) * \u2191abs (f X)", "state_after": "case intro.intro.intro.intro.intro.mk.mk.intro.intro.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR : \u211d\nhR : 0 < R\nz x : \u2102\nf : \u2102 \u2192 \u2102\nhx : x \u2208 ball z R\nhf : ContinuousOn f (sphere z R)\nr : \u211d\nhr : r < R\nhrx : x \u2208 ball z r\n\u03b5' : \u211d\nh\u03b5' : \u03b5' > 0\nH : ball x \u03b5' \u2286 ball z r\na : \u2102\nb : \u211d\nha : (a, b).1 \u2208 closedBall z r\nhb : (a, b).2 \u2208 [[0, 2 * \u03c0]]\nhab :\n  \u2200 (y : \u2191(closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])),\n    \u2191abs (circleTransformBoundingFunction R z \u2191y) \u2264\n      \u2191abs\n        (circleTransformBoundingFunction R z\n          \u2191{ val := (a, b), property := (_ : (a, b).1 \u2208 closedBall z r \u2227 (a, b).2 \u2208 [[0, 2 * \u03c0]]) })\nV : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b8 w => circleTransformDeriv R z w (fun x => 1) \u03b8\nfunccomp : ContinuousOn (fun r => \u2191abs (f r)) (sphere z R)\nX : \u2102\nHX : X \u2208 sphere z R\ny : \u211d\nv : \u2102\nhv : v \u2208 ball x \u03b5'\ny1 : \u211d\nhy1 : y1 \u2208 Ico 0 (2 * \u03c0)\nhfun : circleTransformDeriv R z v f y = circleTransformDeriv R z v f y1\nhy2 : y1 \u2208 [[0, 2 * \u03c0]]\nHX2 : \u2200 (x : \u2102), \u2191abs (x - z) = R \u2192 \u2191abs (f x) \u2264 \u2191abs (f X)\n\u22a2 \u2016circleTransformDeriv R z v f y\u2016 \u2264 \u2191abs (V b a) * \u2191abs (f X)"}, {"tactic": "have := mul_le_mul (hab \u27e8\u27e8v, y1\u27e9, \u27e8ball_subset_closedBall (H hv), hy2\u27e9\u27e9)\n  (HX2 (circleMap z R y1) (circleMap_mem_sphere z hR.le y1)) (Complex.abs.nonneg _)\n  (Complex.abs.nonneg _)", "annotated_tactic": ["have := <a>mul_le_mul</a> (hab \u27e8\u27e8v, y1\u27e9, \u27e8<a>ball_subset_closedBall</a> (H hv), hy2\u27e9\u27e9)\n    (HX2 (<a>circleMap</a> z R y1) (<a>circleMap_mem_sphere</a> z hR.le y1)) (Complex.abs.nonneg _)\n    (Complex.abs.nonneg _)", [{"full_name": "mul_le_mul", "def_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "def_pos": [414, 9], "def_end_pos": [414, 19]}, {"full_name": "Metric.ball_subset_closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [534, 9], "def_end_pos": [534, 31]}, {"full_name": "circleMap", "def_path": "Mathlib/MeasureTheory/Integral/CircleIntegral.lean", "def_pos": [89, 5], "def_end_pos": [89, 14]}, {"full_name": "circleMap_mem_sphere", "def_path": "Mathlib/MeasureTheory/Integral/CircleIntegral.lean", "def_pos": [122, 9], "def_end_pos": [122, 29]}]], "state_before": "case intro.intro.intro.intro.intro.mk.mk.intro.intro.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR : \u211d\nhR : 0 < R\nz x : \u2102\nf : \u2102 \u2192 \u2102\nhx : x \u2208 ball z R\nhf : ContinuousOn f (sphere z R)\nr : \u211d\nhr : r < R\nhrx : x \u2208 ball z r\n\u03b5' : \u211d\nh\u03b5' : \u03b5' > 0\nH : ball x \u03b5' \u2286 ball z r\na : \u2102\nb : \u211d\nha : (a, b).1 \u2208 closedBall z r\nhb : (a, b).2 \u2208 [[0, 2 * \u03c0]]\nhab :\n  \u2200 (y : \u2191(closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])),\n    \u2191abs (circleTransformBoundingFunction R z \u2191y) \u2264\n      \u2191abs\n        (circleTransformBoundingFunction R z\n          \u2191{ val := (a, b), property := (_ : (a, b).1 \u2208 closedBall z r \u2227 (a, b).2 \u2208 [[0, 2 * \u03c0]]) })\nV : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b8 w => circleTransformDeriv R z w (fun x => 1) \u03b8\nfunccomp : ContinuousOn (fun r => \u2191abs (f r)) (sphere z R)\nX : \u2102\nHX : X \u2208 sphere z R\ny : \u211d\nv : \u2102\nhv : v \u2208 ball x \u03b5'\ny1 : \u211d\nhy1 : y1 \u2208 Ico 0 (2 * \u03c0)\nhfun : circleTransformDeriv R z v f y = circleTransformDeriv R z v f y1\nhy2 : y1 \u2208 [[0, 2 * \u03c0]]\nHX2 : \u2200 (x : \u2102), \u2191abs (x - z) = R \u2192 \u2191abs (f x) \u2264 \u2191abs (f X)\n\u22a2 \u2016circleTransformDeriv R z v f y\u2016 \u2264 \u2191abs (V b a) * \u2191abs (f X)", "state_after": "case intro.intro.intro.intro.intro.mk.mk.intro.intro.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR : \u211d\nhR : 0 < R\nz x : \u2102\nf : \u2102 \u2192 \u2102\nhx : x \u2208 ball z R\nhf : ContinuousOn f (sphere z R)\nr : \u211d\nhr : r < R\nhrx : x \u2208 ball z r\n\u03b5' : \u211d\nh\u03b5' : \u03b5' > 0\nH : ball x \u03b5' \u2286 ball z r\na : \u2102\nb : \u211d\nha : (a, b).1 \u2208 closedBall z r\nhb : (a, b).2 \u2208 [[0, 2 * \u03c0]]\nhab :\n  \u2200 (y : \u2191(closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])),\n    \u2191abs (circleTransformBoundingFunction R z \u2191y) \u2264\n      \u2191abs\n        (circleTransformBoundingFunction R z\n          \u2191{ val := (a, b), property := (_ : (a, b).1 \u2208 closedBall z r \u2227 (a, b).2 \u2208 [[0, 2 * \u03c0]]) })\nV : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b8 w => circleTransformDeriv R z w (fun x => 1) \u03b8\nfunccomp : ContinuousOn (fun r => \u2191abs (f r)) (sphere z R)\nX : \u2102\nHX : X \u2208 sphere z R\ny : \u211d\nv : \u2102\nhv : v \u2208 ball x \u03b5'\ny1 : \u211d\nhy1 : y1 \u2208 Ico 0 (2 * \u03c0)\nhfun : circleTransformDeriv R z v f y = circleTransformDeriv R z v f y1\nhy2 : y1 \u2208 [[0, 2 * \u03c0]]\nHX2 : \u2200 (x : \u2102), \u2191abs (x - z) = R \u2192 \u2191abs (f x) \u2264 \u2191abs (f X)\nthis :\n  \u2191abs\n        (circleTransformBoundingFunction R z\n          \u2191{ val := (v, y1), property := (_ : (v, y1).1 \u2208 closedBall z r \u2227 (v, y1).2 \u2208 [[0, 2 * \u03c0]]) }) *\n      \u2191abs (f (circleMap z R y1)) \u2264\n    \u2191abs\n        (circleTransformBoundingFunction R z\n          \u2191{ val := (a, b), property := (_ : (a, b).1 \u2208 closedBall z r \u2227 (a, b).2 \u2208 [[0, 2 * \u03c0]]) }) *\n      \u2191abs (f X)\n\u22a2 \u2016circleTransformDeriv R z v f y\u2016 \u2264 \u2191abs (V b a) * \u2191abs (f X)"}, {"tactic": "simp_rw [hfun]", "annotated_tactic": ["simp_rw [hfun]", []], "state_before": "case intro.intro.intro.intro.intro.mk.mk.intro.intro.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR : \u211d\nhR : 0 < R\nz x : \u2102\nf : \u2102 \u2192 \u2102\nhx : x \u2208 ball z R\nhf : ContinuousOn f (sphere z R)\nr : \u211d\nhr : r < R\nhrx : x \u2208 ball z r\n\u03b5' : \u211d\nh\u03b5' : \u03b5' > 0\nH : ball x \u03b5' \u2286 ball z r\na : \u2102\nb : \u211d\nha : (a, b).1 \u2208 closedBall z r\nhb : (a, b).2 \u2208 [[0, 2 * \u03c0]]\nhab :\n  \u2200 (y : \u2191(closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])),\n    \u2191abs (circleTransformBoundingFunction R z \u2191y) \u2264\n      \u2191abs\n        (circleTransformBoundingFunction R z\n          \u2191{ val := (a, b), property := (_ : (a, b).1 \u2208 closedBall z r \u2227 (a, b).2 \u2208 [[0, 2 * \u03c0]]) })\nV : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b8 w => circleTransformDeriv R z w (fun x => 1) \u03b8\nfunccomp : ContinuousOn (fun r => \u2191abs (f r)) (sphere z R)\nX : \u2102\nHX : X \u2208 sphere z R\ny : \u211d\nv : \u2102\nhv : v \u2208 ball x \u03b5'\ny1 : \u211d\nhy1 : y1 \u2208 Ico 0 (2 * \u03c0)\nhfun : circleTransformDeriv R z v f y = circleTransformDeriv R z v f y1\nhy2 : y1 \u2208 [[0, 2 * \u03c0]]\nHX2 : \u2200 (x : \u2102), \u2191abs (x - z) = R \u2192 \u2191abs (f x) \u2264 \u2191abs (f X)\nthis :\n  \u2191abs\n        (circleTransformBoundingFunction R z\n          \u2191{ val := (v, y1), property := (_ : (v, y1).1 \u2208 closedBall z r \u2227 (v, y1).2 \u2208 [[0, 2 * \u03c0]]) }) *\n      \u2191abs (f (circleMap z R y1)) \u2264\n    \u2191abs\n        (circleTransformBoundingFunction R z\n          \u2191{ val := (a, b), property := (_ : (a, b).1 \u2208 closedBall z r \u2227 (a, b).2 \u2208 [[0, 2 * \u03c0]]) }) *\n      \u2191abs (f X)\n\u22a2 \u2016circleTransformDeriv R z v f y\u2016 \u2264 \u2191abs (V b a) * \u2191abs (f X)", "state_after": "case intro.intro.intro.intro.intro.mk.mk.intro.intro.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR : \u211d\nhR : 0 < R\nz x : \u2102\nf : \u2102 \u2192 \u2102\nhx : x \u2208 ball z R\nhf : ContinuousOn f (sphere z R)\nr : \u211d\nhr : r < R\nhrx : x \u2208 ball z r\n\u03b5' : \u211d\nh\u03b5' : \u03b5' > 0\nH : ball x \u03b5' \u2286 ball z r\na : \u2102\nb : \u211d\nha : (a, b).1 \u2208 closedBall z r\nhb : (a, b).2 \u2208 [[0, 2 * \u03c0]]\nhab :\n  \u2200 (y : \u2191(closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])),\n    \u2191abs (circleTransformBoundingFunction R z \u2191y) \u2264\n      \u2191abs\n        (circleTransformBoundingFunction R z\n          \u2191{ val := (a, b), property := (_ : (a, b).1 \u2208 closedBall z r \u2227 (a, b).2 \u2208 [[0, 2 * \u03c0]]) })\nV : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b8 w => circleTransformDeriv R z w (fun x => 1) \u03b8\nfunccomp : ContinuousOn (fun r => \u2191abs (f r)) (sphere z R)\nX : \u2102\nHX : X \u2208 sphere z R\ny : \u211d\nv : \u2102\nhv : v \u2208 ball x \u03b5'\ny1 : \u211d\nhy1 : y1 \u2208 Ico 0 (2 * \u03c0)\nhfun : circleTransformDeriv R z v f y = circleTransformDeriv R z v f y1\nhy2 : y1 \u2208 [[0, 2 * \u03c0]]\nHX2 : \u2200 (x : \u2102), \u2191abs (x - z) = R \u2192 \u2191abs (f x) \u2264 \u2191abs (f X)\nthis :\n  \u2191abs\n        (circleTransformBoundingFunction R z\n          \u2191{ val := (v, y1), property := (_ : (v, y1).1 \u2208 closedBall z r \u2227 (v, y1).2 \u2208 [[0, 2 * \u03c0]]) }) *\n      \u2191abs (f (circleMap z R y1)) \u2264\n    \u2191abs\n        (circleTransformBoundingFunction R z\n          \u2191{ val := (a, b), property := (_ : (a, b).1 \u2208 closedBall z r \u2227 (a, b).2 \u2208 [[0, 2 * \u03c0]]) }) *\n      \u2191abs (f X)\n\u22a2 \u2016circleTransformDeriv R z v f y1\u2016 \u2264 \u2191abs (circleTransformDeriv R z a (fun x => 1) b) * \u2191abs (f X)"}, {"tactic": "simp only [circleTransformBoundingFunction, circleTransformDeriv, norm_eq_abs,\n  Algebra.id.smul_eq_mul, deriv_circleMap, map_mul, abs_circleMap_zero, abs_I, mul_one, \u2190\n  mul_assoc, mul_inv_rev, inv_I, abs_neg, abs_inv, abs_ofReal, one_mul, abs_two, abs_pow,\n  mem_ball, gt_iff_lt, Subtype.coe_mk, SetCoe.forall, mem_prod, mem_closedBall, and_imp,\n  Prod.forall, NormedSpace.sphere_nonempty, mem_sphere_iff_norm] at *", "annotated_tactic": ["simp only [<a>circleTransformBoundingFunction</a>, <a>circleTransformDeriv</a>, <a>norm_eq_abs</a>,\n    <a>Algebra.id.smul_eq_mul</a>, <a>deriv_circleMap</a>, <a>map_mul</a>, <a>abs_circleMap_zero</a>, <a>abs_I</a>, <a>mul_one</a>, \u2190\n    <a>mul_assoc</a>, <a>mul_inv_rev</a>, <a>inv_I</a>, <a>abs_neg</a>, <a>abs_inv</a>, <a>abs_ofReal</a>, <a>one_mul</a>, <a>abs_two</a>, <a>abs_pow</a>,\n    <a>mem_ball</a>, <a>gt_iff_lt</a>, <a>Subtype.coe_mk</a>, <a>SetCoe.forall</a>, <a>mem_prod</a>, <a>mem_closedBall</a>, <a>and_imp</a>,\n    <a>Prod.forall</a>, <a>NormedSpace.sphere_nonempty</a>, <a>mem_sphere_iff_norm</a>] at *", [{"full_name": "Complex.circleTransformBoundingFunction", "def_path": "Mathlib/MeasureTheory/Integral/CircleTransform.lean", "def_pos": [96, 5], "def_end_pos": [96, 36]}, {"full_name": "Complex.circleTransformDeriv", "def_path": "Mathlib/MeasureTheory/Integral/CircleTransform.lean", "def_pos": [46, 5], "def_end_pos": [46, 25]}, {"full_name": "Complex.norm_eq_abs", "def_path": "Mathlib/Analysis/Complex/Basic.lean", "def_pos": [51, 9], "def_end_pos": [51, 20]}, {"full_name": "Algebra.id.smul_eq_mul", "def_path": "Mathlib/Algebra/Algebra/Basic.lean", "def_pos": [453, 9], "def_end_pos": [453, 20]}, {"full_name": "deriv_circleMap", "def_path": "Mathlib/MeasureTheory/Integral/CircleIntegral.lean", "def_pos": [195, 9], "def_end_pos": [195, 24]}, {"full_name": "map_mul", "def_path": "Mathlib/Algebra/Hom/Group/Defs.lean", "def_pos": [299, 9], "def_end_pos": [299, 16]}, {"full_name": "abs_circleMap_zero", "def_path": "Mathlib/MeasureTheory/Integral/CircleIntegral.lean", "def_pos": [116, 9], "def_end_pos": [116, 27]}, {"full_name": "Complex.abs_I", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [1018, 9], "def_end_pos": [1018, 14]}, {"full_name": "mul_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [470, 9], "def_end_pos": [470, 16]}, {"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [264, 9], "def_end_pos": [264, 18]}, {"full_name": "mul_inv_rev", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [1050, 9], "def_end_pos": [1050, 20]}, {"full_name": "Complex.inv_I", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [879, 9], "def_end_pos": [879, 14]}, {"full_name": "abs_neg", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [84, 9], "def_end_pos": [84, 16]}, {"full_name": "abs_inv", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [998, 9], "def_end_pos": [998, 16]}, {"full_name": "Complex.abs_ofReal", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [985, 9], "def_end_pos": [985, 19]}, {"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [464, 9], "def_end_pos": [464, 16]}, {"full_name": "Complex.abs_two", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [1023, 9], "def_end_pos": [1023, 16]}, {"full_name": "Complex.abs_pow", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [1050, 9], "def_end_pos": [1050, 16]}, {"full_name": "Metric.mem_ball", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [414, 9], "def_end_pos": [414, 17]}, {"full_name": "gt_iff_lt", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [366, 9], "def_end_pos": [366, 18]}, {"full_name": "Subtype.coe_mk", "def_path": "Mathlib/Data/Subtype.lean", "def_pos": [99, 9], "def_end_pos": [99, 15]}, {"full_name": "SetCoe.forall", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [185, 9], "def_end_pos": [185, 22]}, {"full_name": "Set.mem_prod", "def_path": "Mathlib/Data/Set/Prod.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}, {"full_name": "Metric.mem_closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [478, 17], "def_end_pos": [478, 31]}, {"full_name": "and_imp", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [313, 17], "def_end_pos": [313, 24]}, {"full_name": "Prod.forall", "def_path": "Mathlib/Data/Prod/Basic.lean", "def_pos": [36, 9], "def_end_pos": [36, 17]}, {"full_name": "NormedSpace.sphere_nonempty", "def_path": "Mathlib/Analysis/NormedSpace/Pointwise.lean", "def_pos": [411, 9], "def_end_pos": [411, 36]}, {"full_name": "mem_sphere_iff_norm", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [709, 35], "def_end_pos": [709, 54]}]], "state_before": "case intro.intro.intro.intro.intro.mk.mk.intro.intro.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR : \u211d\nhR : 0 < R\nz x : \u2102\nf : \u2102 \u2192 \u2102\nhx : x \u2208 ball z R\nhf : ContinuousOn f (sphere z R)\nr : \u211d\nhr : r < R\nhrx : x \u2208 ball z r\n\u03b5' : \u211d\nh\u03b5' : \u03b5' > 0\nH : ball x \u03b5' \u2286 ball z r\na : \u2102\nb : \u211d\nha : (a, b).1 \u2208 closedBall z r\nhb : (a, b).2 \u2208 [[0, 2 * \u03c0]]\nhab :\n  \u2200 (y : \u2191(closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])),\n    \u2191abs (circleTransformBoundingFunction R z \u2191y) \u2264\n      \u2191abs\n        (circleTransformBoundingFunction R z\n          \u2191{ val := (a, b), property := (_ : (a, b).1 \u2208 closedBall z r \u2227 (a, b).2 \u2208 [[0, 2 * \u03c0]]) })\nV : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b8 w => circleTransformDeriv R z w (fun x => 1) \u03b8\nfunccomp : ContinuousOn (fun r => \u2191abs (f r)) (sphere z R)\nX : \u2102\nHX : X \u2208 sphere z R\ny : \u211d\nv : \u2102\nhv : v \u2208 ball x \u03b5'\ny1 : \u211d\nhy1 : y1 \u2208 Ico 0 (2 * \u03c0)\nhfun : circleTransformDeriv R z v f y = circleTransformDeriv R z v f y1\nhy2 : y1 \u2208 [[0, 2 * \u03c0]]\nHX2 : \u2200 (x : \u2102), \u2191abs (x - z) = R \u2192 \u2191abs (f x) \u2264 \u2191abs (f X)\nthis :\n  \u2191abs\n        (circleTransformBoundingFunction R z\n          \u2191{ val := (v, y1), property := (_ : (v, y1).1 \u2208 closedBall z r \u2227 (v, y1).2 \u2208 [[0, 2 * \u03c0]]) }) *\n      \u2191abs (f (circleMap z R y1)) \u2264\n    \u2191abs\n        (circleTransformBoundingFunction R z\n          \u2191{ val := (a, b), property := (_ : (a, b).1 \u2208 closedBall z r \u2227 (a, b).2 \u2208 [[0, 2 * \u03c0]]) }) *\n      \u2191abs (f X)\n\u22a2 \u2016circleTransformDeriv R z v f y1\u2016 \u2264 \u2191abs (circleTransformDeriv R z a (fun x => 1) b) * \u2191abs (f X)", "state_after": "case intro.intro.intro.intro.intro.mk.mk.intro.intro.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR : \u211d\nhR : 0 < R\nz x : \u2102\nf : \u2102 \u2192 \u2102\nhf : ContinuousOn f (sphere z R)\nr : \u211d\nhr : r < R\n\u03b5' : \u211d\nH : ball x \u03b5' \u2286 ball z r\na : \u2102\nb : \u211d\nha : (a, b).1 \u2208 closedBall z r\nhb : (a, b).2 \u2208 [[0, 2 * \u03c0]]\nV : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b8 w => circleTransformDeriv R z w (fun x => 1) \u03b8\nfunccomp : ContinuousOn (fun r => \u2191abs (f r)) (sphere z R)\nX : \u2102\ny : \u211d\nv : \u2102\nhv : v \u2208 ball x \u03b5'\ny1 : \u211d\nhy1 : y1 \u2208 Ico 0 (2 * \u03c0)\nhy2 : y1 \u2208 [[0, 2 * \u03c0]]\nHX2 : \u2200 (x : \u2102), \u2191abs (x - z) = R \u2192 \u2191abs (f x) \u2264 \u2191abs (f X)\nhx : dist x z < R\nhrx : dist x z < r\nh\u03b5' : 0 < \u03b5'\nhab :\n  \u2200 (a_1 : \u2102) (b_1 : \u211d),\n    dist a_1 z \u2264 r \u2192\n      b_1 \u2208 [[0, 2 * \u03c0]] \u2192\n        \u2191abs (-I) * \u2191abs (\u2191\u03c0)\u207b\u00b9 * \u2191abs 2\u207b\u00b9 * |R| * \u2191abs ((circleMap z R b_1 - a_1) ^ 2)\u207b\u00b9 \u2264\n          \u2191abs (-I) * \u2191abs (\u2191\u03c0)\u207b\u00b9 * \u2191abs 2\u207b\u00b9 * |R| * \u2191abs ((circleMap z R b - a) ^ 2)\u207b\u00b9\nHX : \u2191abs (X - z) = R\nhfun :\n  -I * (\u2191\u03c0)\u207b\u00b9 * 2\u207b\u00b9 * circleMap 0 R y * I * ((circleMap z R y - v) ^ 2)\u207b\u00b9 * f (circleMap z R y) =\n    -I * (\u2191\u03c0)\u207b\u00b9 * 2\u207b\u00b9 * circleMap 0 R y1 * I * ((circleMap z R y1 - v) ^ 2)\u207b\u00b9 * f (circleMap z R y1)\nthis :\n  \u2191abs (-I) * \u2191abs (\u2191\u03c0)\u207b\u00b9 * \u2191abs 2\u207b\u00b9 * |R| * \u2191abs ((circleMap z R y1 - v) ^ 2)\u207b\u00b9 * \u2191abs (f (circleMap z R y1)) \u2264\n    \u2191abs (-I) * \u2191abs (\u2191\u03c0)\u207b\u00b9 * \u2191abs 2\u207b\u00b9 * |R| * \u2191abs ((circleMap z R b - a) ^ 2)\u207b\u00b9 * \u2191abs (f X)\n\u22a2 \u2191abs (-I) * \u2191abs (\u2191\u03c0)\u207b\u00b9 * \u2191abs 2\u207b\u00b9 * |R| * \u2191abs ((circleMap z R y1 - v) ^ 2)\u207b\u00b9 * \u2191abs (f (circleMap z R y1)) \u2264\n    \u2191abs (-I) * \u2191abs (\u2191\u03c0)\u207b\u00b9 * \u2191abs 2\u207b\u00b9 * |R| * \u2191abs ((circleMap z R b - a) ^ 2)\u207b\u00b9 * \u2191abs (f X)"}, {"tactic": "exact this", "annotated_tactic": ["exact this", []], "state_before": "case intro.intro.intro.intro.intro.mk.mk.intro.intro.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR : \u211d\nhR : 0 < R\nz x : \u2102\nf : \u2102 \u2192 \u2102\nhf : ContinuousOn f (sphere z R)\nr : \u211d\nhr : r < R\n\u03b5' : \u211d\nH : ball x \u03b5' \u2286 ball z r\na : \u2102\nb : \u211d\nha : (a, b).1 \u2208 closedBall z r\nhb : (a, b).2 \u2208 [[0, 2 * \u03c0]]\nV : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b8 w => circleTransformDeriv R z w (fun x => 1) \u03b8\nfunccomp : ContinuousOn (fun r => \u2191abs (f r)) (sphere z R)\nX : \u2102\ny : \u211d\nv : \u2102\nhv : v \u2208 ball x \u03b5'\ny1 : \u211d\nhy1 : y1 \u2208 Ico 0 (2 * \u03c0)\nhy2 : y1 \u2208 [[0, 2 * \u03c0]]\nHX2 : \u2200 (x : \u2102), \u2191abs (x - z) = R \u2192 \u2191abs (f x) \u2264 \u2191abs (f X)\nhx : dist x z < R\nhrx : dist x z < r\nh\u03b5' : 0 < \u03b5'\nhab :\n  \u2200 (a_1 : \u2102) (b_1 : \u211d),\n    dist a_1 z \u2264 r \u2192\n      b_1 \u2208 [[0, 2 * \u03c0]] \u2192\n        \u2191abs (-I) * \u2191abs (\u2191\u03c0)\u207b\u00b9 * \u2191abs 2\u207b\u00b9 * |R| * \u2191abs ((circleMap z R b_1 - a_1) ^ 2)\u207b\u00b9 \u2264\n          \u2191abs (-I) * \u2191abs (\u2191\u03c0)\u207b\u00b9 * \u2191abs 2\u207b\u00b9 * |R| * \u2191abs ((circleMap z R b - a) ^ 2)\u207b\u00b9\nHX : \u2191abs (X - z) = R\nhfun :\n  -I * (\u2191\u03c0)\u207b\u00b9 * 2\u207b\u00b9 * circleMap 0 R y * I * ((circleMap z R y - v) ^ 2)\u207b\u00b9 * f (circleMap z R y) =\n    -I * (\u2191\u03c0)\u207b\u00b9 * 2\u207b\u00b9 * circleMap 0 R y1 * I * ((circleMap z R y1 - v) ^ 2)\u207b\u00b9 * f (circleMap z R y1)\nthis :\n  \u2191abs (-I) * \u2191abs (\u2191\u03c0)\u207b\u00b9 * \u2191abs 2\u207b\u00b9 * |R| * \u2191abs ((circleMap z R y1 - v) ^ 2)\u207b\u00b9 * \u2191abs (f (circleMap z R y1)) \u2264\n    \u2191abs (-I) * \u2191abs (\u2191\u03c0)\u207b\u00b9 * \u2191abs 2\u207b\u00b9 * |R| * \u2191abs ((circleMap z R b - a) ^ 2)\u207b\u00b9 * \u2191abs (f X)\n\u22a2 \u2191abs (-I) * \u2191abs (\u2191\u03c0)\u207b\u00b9 * \u2191abs 2\u207b\u00b9 * |R| * \u2191abs ((circleMap z R y1 - v) ^ 2)\u207b\u00b9 * \u2191abs (f (circleMap z R y1)) \u2264\n    \u2191abs (-I) * \u2191abs (\u2191\u03c0)\u207b\u00b9 * \u2191abs 2\u207b\u00b9 * |R| * \u2191abs ((circleMap z R b - a) ^ 2)\u207b\u00b9 * \u2191abs (f X)", "state_after": "no goals"}, {"tactic": "have cabs : ContinuousOn abs \u22a4 := by apply continuous_abs.continuousOn", "annotated_tactic": ["have cabs : <a>ContinuousOn</a> <a>abs</a> \u22a4 := by apply continuous_abs.continuousOn", [{"full_name": "ContinuousOn", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [532, 5], "def_end_pos": [532, 17]}, {"full_name": "Complex.abs", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [966, 19], "def_end_pos": [966, 37]}]], "state_before": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR : \u211d\nhR : 0 < R\nz x : \u2102\nf : \u2102 \u2192 \u2102\nhx : x \u2208 ball z R\nhf : ContinuousOn f (sphere z R)\nr : \u211d\nhr : r < R\nhrx : x \u2208 ball z r\n\u03b5' : \u211d\nh\u03b5' : \u03b5' > 0\nH : ball x \u03b5' \u2286 ball z r\na : \u2102\nb : \u211d\nha : (a, b).1 \u2208 closedBall z r\nhb : (a, b).2 \u2208 [[0, 2 * \u03c0]]\nhab :\n  \u2200 (y : \u2191(closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])),\n    \u2191abs (circleTransformBoundingFunction R z \u2191y) \u2264\n      \u2191abs\n        (circleTransformBoundingFunction R z\n          \u2191{ val := (a, b), property := (_ : (a, b).1 \u2208 closedBall z r \u2227 (a, b).2 \u2208 [[0, 2 * \u03c0]]) })\nV : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b8 w => circleTransformDeriv R z w (fun x => 1) \u03b8\n\u22a2 ContinuousOn (fun r => \u2191abs (f r)) (sphere z R)", "state_after": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR : \u211d\nhR : 0 < R\nz x : \u2102\nf : \u2102 \u2192 \u2102\nhx : x \u2208 ball z R\nhf : ContinuousOn f (sphere z R)\nr : \u211d\nhr : r < R\nhrx : x \u2208 ball z r\n\u03b5' : \u211d\nh\u03b5' : \u03b5' > 0\nH : ball x \u03b5' \u2286 ball z r\na : \u2102\nb : \u211d\nha : (a, b).1 \u2208 closedBall z r\nhb : (a, b).2 \u2208 [[0, 2 * \u03c0]]\nhab :\n  \u2200 (y : \u2191(closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])),\n    \u2191abs (circleTransformBoundingFunction R z \u2191y) \u2264\n      \u2191abs\n        (circleTransformBoundingFunction R z\n          \u2191{ val := (a, b), property := (_ : (a, b).1 \u2208 closedBall z r \u2227 (a, b).2 \u2208 [[0, 2 * \u03c0]]) })\nV : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b8 w => circleTransformDeriv R z w (fun x => 1) \u03b8\ncabs : ContinuousOn \u2191abs \u22a4\n\u22a2 ContinuousOn (fun r => \u2191abs (f r)) (sphere z R)"}, {"tactic": "apply cabs.comp hf", "annotated_tactic": ["apply cabs.comp hf", []], "state_before": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR : \u211d\nhR : 0 < R\nz x : \u2102\nf : \u2102 \u2192 \u2102\nhx : x \u2208 ball z R\nhf : ContinuousOn f (sphere z R)\nr : \u211d\nhr : r < R\nhrx : x \u2208 ball z r\n\u03b5' : \u211d\nh\u03b5' : \u03b5' > 0\nH : ball x \u03b5' \u2286 ball z r\na : \u2102\nb : \u211d\nha : (a, b).1 \u2208 closedBall z r\nhb : (a, b).2 \u2208 [[0, 2 * \u03c0]]\nhab :\n  \u2200 (y : \u2191(closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])),\n    \u2191abs (circleTransformBoundingFunction R z \u2191y) \u2264\n      \u2191abs\n        (circleTransformBoundingFunction R z\n          \u2191{ val := (a, b), property := (_ : (a, b).1 \u2208 closedBall z r \u2227 (a, b).2 \u2208 [[0, 2 * \u03c0]]) })\nV : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b8 w => circleTransformDeriv R z w (fun x => 1) \u03b8\ncabs : ContinuousOn \u2191abs \u22a4\n\u22a2 ContinuousOn (fun r => \u2191abs (f r)) (sphere z R)", "state_after": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR : \u211d\nhR : 0 < R\nz x : \u2102\nf : \u2102 \u2192 \u2102\nhx : x \u2208 ball z R\nhf : ContinuousOn f (sphere z R)\nr : \u211d\nhr : r < R\nhrx : x \u2208 ball z r\n\u03b5' : \u211d\nh\u03b5' : \u03b5' > 0\nH : ball x \u03b5' \u2286 ball z r\na : \u2102\nb : \u211d\nha : (a, b).1 \u2208 closedBall z r\nhb : (a, b).2 \u2208 [[0, 2 * \u03c0]]\nhab :\n  \u2200 (y : \u2191(closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])),\n    \u2191abs (circleTransformBoundingFunction R z \u2191y) \u2264\n      \u2191abs\n        (circleTransformBoundingFunction R z\n          \u2191{ val := (a, b), property := (_ : (a, b).1 \u2208 closedBall z r \u2227 (a, b).2 \u2208 [[0, 2 * \u03c0]]) })\nV : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b8 w => circleTransformDeriv R z w (fun x => 1) \u03b8\ncabs : ContinuousOn \u2191abs \u22a4\n\u22a2 MapsTo f (sphere z R) \u22a4"}, {"tactic": "rw [MapsTo]", "annotated_tactic": ["rw [<a>MapsTo</a>]", [{"full_name": "Set.MapsTo", "def_path": "Mathlib/Data/Set/Function.lean", "def_pos": [348, 5], "def_end_pos": [348, 11]}]], "state_before": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR : \u211d\nhR : 0 < R\nz x : \u2102\nf : \u2102 \u2192 \u2102\nhx : x \u2208 ball z R\nhf : ContinuousOn f (sphere z R)\nr : \u211d\nhr : r < R\nhrx : x \u2208 ball z r\n\u03b5' : \u211d\nh\u03b5' : \u03b5' > 0\nH : ball x \u03b5' \u2286 ball z r\na : \u2102\nb : \u211d\nha : (a, b).1 \u2208 closedBall z r\nhb : (a, b).2 \u2208 [[0, 2 * \u03c0]]\nhab :\n  \u2200 (y : \u2191(closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])),\n    \u2191abs (circleTransformBoundingFunction R z \u2191y) \u2264\n      \u2191abs\n        (circleTransformBoundingFunction R z\n          \u2191{ val := (a, b), property := (_ : (a, b).1 \u2208 closedBall z r \u2227 (a, b).2 \u2208 [[0, 2 * \u03c0]]) })\nV : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b8 w => circleTransformDeriv R z w (fun x => 1) \u03b8\ncabs : ContinuousOn \u2191abs \u22a4\n\u22a2 MapsTo f (sphere z R) \u22a4", "state_after": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR : \u211d\nhR : 0 < R\nz x : \u2102\nf : \u2102 \u2192 \u2102\nhx : x \u2208 ball z R\nhf : ContinuousOn f (sphere z R)\nr : \u211d\nhr : r < R\nhrx : x \u2208 ball z r\n\u03b5' : \u211d\nh\u03b5' : \u03b5' > 0\nH : ball x \u03b5' \u2286 ball z r\na : \u2102\nb : \u211d\nha : (a, b).1 \u2208 closedBall z r\nhb : (a, b).2 \u2208 [[0, 2 * \u03c0]]\nhab :\n  \u2200 (y : \u2191(closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])),\n    \u2191abs (circleTransformBoundingFunction R z \u2191y) \u2264\n      \u2191abs\n        (circleTransformBoundingFunction R z\n          \u2191{ val := (a, b), property := (_ : (a, b).1 \u2208 closedBall z r \u2227 (a, b).2 \u2208 [[0, 2 * \u03c0]]) })\nV : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b8 w => circleTransformDeriv R z w (fun x => 1) \u03b8\ncabs : ContinuousOn \u2191abs \u22a4\n\u22a2 \u2200 \u2983x : \u2102\u2984, x \u2208 sphere z R \u2192 f x \u2208 \u22a4"}, {"tactic": "tauto", "annotated_tactic": ["tauto", []], "state_before": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR : \u211d\nhR : 0 < R\nz x : \u2102\nf : \u2102 \u2192 \u2102\nhx : x \u2208 ball z R\nhf : ContinuousOn f (sphere z R)\nr : \u211d\nhr : r < R\nhrx : x \u2208 ball z r\n\u03b5' : \u211d\nh\u03b5' : \u03b5' > 0\nH : ball x \u03b5' \u2286 ball z r\na : \u2102\nb : \u211d\nha : (a, b).1 \u2208 closedBall z r\nhb : (a, b).2 \u2208 [[0, 2 * \u03c0]]\nhab :\n  \u2200 (y : \u2191(closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])),\n    \u2191abs (circleTransformBoundingFunction R z \u2191y) \u2264\n      \u2191abs\n        (circleTransformBoundingFunction R z\n          \u2191{ val := (a, b), property := (_ : (a, b).1 \u2208 closedBall z r \u2227 (a, b).2 \u2208 [[0, 2 * \u03c0]]) })\nV : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b8 w => circleTransformDeriv R z w (fun x => 1) \u03b8\ncabs : ContinuousOn \u2191abs \u22a4\n\u22a2 \u2200 \u2983x : \u2102\u2984, x \u2208 sphere z R \u2192 f x \u2208 \u22a4", "state_after": "no goals"}, {"tactic": "apply continuous_abs.continuousOn", "annotated_tactic": ["apply continuous_abs.continuousOn", []], "state_before": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR : \u211d\nhR : 0 < R\nz x : \u2102\nf : \u2102 \u2192 \u2102\nhx : x \u2208 ball z R\nhf : ContinuousOn f (sphere z R)\nr : \u211d\nhr : r < R\nhrx : x \u2208 ball z r\n\u03b5' : \u211d\nh\u03b5' : \u03b5' > 0\nH : ball x \u03b5' \u2286 ball z r\na : \u2102\nb : \u211d\nha : (a, b).1 \u2208 closedBall z r\nhb : (a, b).2 \u2208 [[0, 2 * \u03c0]]\nhab :\n  \u2200 (y : \u2191(closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])),\n    \u2191abs (circleTransformBoundingFunction R z \u2191y) \u2264\n      \u2191abs\n        (circleTransformBoundingFunction R z\n          \u2191{ val := (a, b), property := (_ : (a, b).1 \u2208 closedBall z r \u2227 (a, b).2 \u2208 [[0, 2 * \u03c0]]) })\nV : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b8 w => circleTransformDeriv R z w (fun x => 1) \u03b8\n\u22a2 ContinuousOn \u2191abs \u22a4", "state_after": "no goals"}, {"tactic": "convert Ico_subset_Icc_self hy1 using 1", "annotated_tactic": ["convert <a>Ico_subset_Icc_self</a> hy1 using 1", [{"full_name": "Set.Ico_subset_Icc_self", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [510, 9], "def_end_pos": [510, 28]}]], "state_before": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR : \u211d\nhR : 0 < R\nz x : \u2102\nf : \u2102 \u2192 \u2102\nhx : x \u2208 ball z R\nhf : ContinuousOn f (sphere z R)\nr : \u211d\nhr : r < R\nhrx : x \u2208 ball z r\n\u03b5' : \u211d\nh\u03b5' : \u03b5' > 0\nH : ball x \u03b5' \u2286 ball z r\na : \u2102\nb : \u211d\nha : (a, b).1 \u2208 closedBall z r\nhb : (a, b).2 \u2208 [[0, 2 * \u03c0]]\nhab :\n  \u2200 (y : \u2191(closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])),\n    \u2191abs (circleTransformBoundingFunction R z \u2191y) \u2264\n      \u2191abs\n        (circleTransformBoundingFunction R z\n          \u2191{ val := (a, b), property := (_ : (a, b).1 \u2208 closedBall z r \u2227 (a, b).2 \u2208 [[0, 2 * \u03c0]]) })\nV : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b8 w => circleTransformDeriv R z w (fun x => 1) \u03b8\nfunccomp : ContinuousOn (fun r => \u2191abs (f r)) (sphere z R)\nX : \u2102\nHX : X \u2208 sphere z R\nHX2 : IsMaxOn (fun r => \u2191abs (f r)) (sphere z R) X\ny : \u211d\nv : \u2102\nhv : v \u2208 ball x \u03b5'\ny1 : \u211d\nhy1 : y1 \u2208 Ico 0 (2 * \u03c0)\nhfun : circleTransformDeriv R z v f y = circleTransformDeriv R z v f y1\n\u22a2 y1 \u2208 [[0, 2 * \u03c0]]", "state_after": "case h.e'_5\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR : \u211d\nhR : 0 < R\nz x : \u2102\nf : \u2102 \u2192 \u2102\nhx : x \u2208 ball z R\nhf : ContinuousOn f (sphere z R)\nr : \u211d\nhr : r < R\nhrx : x \u2208 ball z r\n\u03b5' : \u211d\nh\u03b5' : \u03b5' > 0\nH : ball x \u03b5' \u2286 ball z r\na : \u2102\nb : \u211d\nha : (a, b).1 \u2208 closedBall z r\nhb : (a, b).2 \u2208 [[0, 2 * \u03c0]]\nhab :\n  \u2200 (y : \u2191(closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])),\n    \u2191abs (circleTransformBoundingFunction R z \u2191y) \u2264\n      \u2191abs\n        (circleTransformBoundingFunction R z\n          \u2191{ val := (a, b), property := (_ : (a, b).1 \u2208 closedBall z r \u2227 (a, b).2 \u2208 [[0, 2 * \u03c0]]) })\nV : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b8 w => circleTransformDeriv R z w (fun x => 1) \u03b8\nfunccomp : ContinuousOn (fun r => \u2191abs (f r)) (sphere z R)\nX : \u2102\nHX : X \u2208 sphere z R\nHX2 : IsMaxOn (fun r => \u2191abs (f r)) (sphere z R) X\ny : \u211d\nv : \u2102\nhv : v \u2208 ball x \u03b5'\ny1 : \u211d\nhy1 : y1 \u2208 Ico 0 (2 * \u03c0)\nhfun : circleTransformDeriv R z v f y = circleTransformDeriv R z v f y1\n\u22a2 [[0, 2 * \u03c0]] = Icc 0 (2 * \u03c0)"}, {"tactic": "simp [uIcc_of_le Real.two_pi_pos.le]", "annotated_tactic": ["simp [<a>uIcc_of_le</a> Real.two_pi_pos.le]", [{"full_name": "Set.uIcc_of_le", "def_path": "Mathlib/Data/Set/Intervals/UnorderedInterval.lean", "def_pos": [69, 7], "def_end_pos": [69, 17]}]], "state_before": "case h.e'_5\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR : \u211d\nhR : 0 < R\nz x : \u2102\nf : \u2102 \u2192 \u2102\nhx : x \u2208 ball z R\nhf : ContinuousOn f (sphere z R)\nr : \u211d\nhr : r < R\nhrx : x \u2208 ball z r\n\u03b5' : \u211d\nh\u03b5' : \u03b5' > 0\nH : ball x \u03b5' \u2286 ball z r\na : \u2102\nb : \u211d\nha : (a, b).1 \u2208 closedBall z r\nhb : (a, b).2 \u2208 [[0, 2 * \u03c0]]\nhab :\n  \u2200 (y : \u2191(closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])),\n    \u2191abs (circleTransformBoundingFunction R z \u2191y) \u2264\n      \u2191abs\n        (circleTransformBoundingFunction R z\n          \u2191{ val := (a, b), property := (_ : (a, b).1 \u2208 closedBall z r \u2227 (a, b).2 \u2208 [[0, 2 * \u03c0]]) })\nV : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b8 w => circleTransformDeriv R z w (fun x => 1) \u03b8\nfunccomp : ContinuousOn (fun r => \u2191abs (f r)) (sphere z R)\nX : \u2102\nHX : X \u2208 sphere z R\nHX2 : IsMaxOn (fun r => \u2191abs (f r)) (sphere z R) X\ny : \u211d\nv : \u2102\nhv : v \u2208 ball x \u03b5'\ny1 : \u211d\nhy1 : y1 \u2208 Ico 0 (2 * \u03c0)\nhfun : circleTransformDeriv R z v f y = circleTransformDeriv R z v f y1\n\u22a2 [[0, 2 * \u03c0]] = Icc 0 (2 * \u03c0)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "full_name": "snd_integral", "start": [1244, 1], "end": [1247, 46], "traced_tactics": [{"tactic": "rw [\u2190 Prod.fst_swap, swap_integral]", "annotated_tactic": ["rw [\u2190 <a>Prod.fst_swap</a>, <a>swap_integral</a>]", [{"full_name": "Prod.fst_swap", "def_path": "Mathlib/Data/Prod/Basic.lean", "def_pos": [172, 9], "def_end_pos": [172, 17]}, {"full_name": "swap_integral", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [1233, 7], "def_end_pos": [1233, 20]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2078 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2077 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace E\nf : \u03b1 \u2192 E \u00d7 F\nhf : Integrable f\n\u22a2 (\u222b (x : \u03b1), f x \u2202\u03bc).2 = \u222b (x : \u03b1), (f x).2 \u2202\u03bc", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2078 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2077 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace E\nf : \u03b1 \u2192 E \u00d7 F\nhf : Integrable f\n\u22a2 (\u222b (x : \u03b1), Prod.swap (f x) \u2202\u03bc).1 = \u222b (x : \u03b1), (f x).2 \u2202\u03bc"}, {"tactic": "exact fst_integral <| hf.snd.prod_mk hf.fst", "annotated_tactic": ["exact <a>fst_integral</a> <| hf.snd.prod_mk hf.fst", [{"full_name": "fst_integral", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [1236, 9], "def_end_pos": [1236, 21]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2078 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2077 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace E\nf : \u03b1 \u2192 E \u00d7 F\nhf : Integrable f\n\u22a2 (\u222b (x : \u03b1), Prod.swap (f x) \u2202\u03bc).1 = \u222b (x : \u03b1), (f x).2 \u2202\u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "full_name": "MeasureTheory.Mem\u2112p.ofReal", "start": [1110, 1], "end": [1112, 41], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/BinomialHeap/Basic.lean", "full_name": "Std.BinomialHeap.Imp.Heap.WF.tail", "start": [467, 1], "end": [471, 34], "traced_tactics": [{"tactic": "simp only [Heap.tail]", "annotated_tactic": ["simp only [<a>Heap.tail</a>]", [{"full_name": "Std.BinomialHeap.Imp.Heap.tail", "def_path": "lake-packages/std/Std/Data/BinomialHeap/Basic.lean", "def_pos": [204, 15], "def_end_pos": [204, 24]}]], "state_before": "\u03b1 : Type u_1\ns : Heap \u03b1\nle : \u03b1 \u2192 \u03b1 \u2192 Bool\nn : Nat\nhwf : WF le n s\n\u22a2 WF le 0 (Heap.tail le s)", "state_after": "\u03b1 : Type u_1\ns : Heap \u03b1\nle : \u03b1 \u2192 \u03b1 \u2192 Bool\nn : Nat\nhwf : WF le n s\n\u22a2 WF le 0 (Option.getD (Heap.tail? le s) Heap.nil)"}, {"tactic": "match eq : s.tail? le with\n| none => exact Heap.WF.nil\n| some tl => exact hwf.tail? eq", "annotated_tactic": ["match eq : s.tail? le with\n  | <a>none</a> => exact <a>Heap.WF.nil</a>\n  | <a>some</a> tl => exact hwf.tail? eq", [{"full_name": "Option.none", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2141, 5], "def_end_pos": [2141, 9]}, {"full_name": "Std.BinomialHeap.Imp.Heap.WF.nil", "def_path": "lake-packages/std/Std/Data/BinomialHeap/Basic.lean", "def_pos": [347, 9], "def_end_pos": [347, 20]}, {"full_name": "Option.some", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2143, 5], "def_end_pos": [2143, 9]}]], "state_before": "\u03b1 : Type u_1\ns : Heap \u03b1\nle : \u03b1 \u2192 \u03b1 \u2192 Bool\nn : Nat\nhwf : WF le n s\n\u22a2 WF le 0 (Option.getD (Heap.tail? le s) Heap.nil)", "state_after": "no goals"}, {"tactic": "exact Heap.WF.nil", "annotated_tactic": ["exact <a>Heap.WF.nil</a>", [{"full_name": "Std.BinomialHeap.Imp.Heap.WF.nil", "def_path": "lake-packages/std/Std/Data/BinomialHeap/Basic.lean", "def_pos": [347, 9], "def_end_pos": [347, 20]}]], "state_before": "\u03b1 : Type u_1\ns : Heap \u03b1\nle : \u03b1 \u2192 \u03b1 \u2192 Bool\nn : Nat\nhwf : WF le n s\neq : Heap.tail? le s = none\n\u22a2 WF le 0 (Option.getD none Heap.nil)", "state_after": "no goals"}, {"tactic": "exact hwf.tail? eq", "annotated_tactic": ["exact hwf.tail? eq", []], "state_before": "\u03b1 : Type u_1\ns : Heap \u03b1\nle : \u03b1 \u2192 \u03b1 \u2192 Bool\nn : Nat\nhwf : WF le n s\ntl : Heap \u03b1\neq : Heap.tail? le s = some tl\n\u22a2 WF le 0 (Option.getD (some tl) Heap.nil)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Classes/LawfulMonad.lean", "full_name": "SatisfiesM_ReaderT_eq", "start": [195, 9], "end": [197, 94], "traced_tactics": [{"tactic": "exact \u27e8fun eq _ => eq \u25b8 rfl, funext\u27e9", "annotated_tactic": ["exact \u27e8fun eq _ => eq \u25b8 <a>rfl</a>, <a>funext</a>\u27e9", [{"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}, {"full_name": "funext", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [1555, 9], "def_end_pos": [1555, 15]}]], "state_before": "m : Type u_1 \u2192 Type u_2\n\u03c1 \u03b1\u271d : Type u_1\np : \u03b1\u271d \u2192 Prop\nx : ReaderT \u03c1 m \u03b1\u271d\ninst\u271d : Monad m\na : ReaderT \u03c1 m { a // p a }\n\u22a2 Subtype.val <$> a = x \u2194 \u2200 (x_1 : \u03c1), Subtype.val <$> a x_1 = x x_1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "full_name": "Nat.strongRecOn_eq", "start": [43, 1], "end": [45, 22], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/ZMod/Basic.lean", "full_name": "ZMod.ringHom_rightInverse", "start": [1223, 1], "end": [1225, 21], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Subtype.lean", "full_name": "Subtype.equiv_iff", "start": [229, 1], "end": [230, 10], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Basic.lean", "full_name": "Set.not_monotoneOn_not_antitoneOn_iff_exists_lt_lt", "start": [2754, 1], "end": [2759, 72], "traced_tactics": [{"tactic": "simp [monotoneOn_iff_monotone, antitoneOn_iff_antitone, and_assoc, exists_and_left,\n  not_monotone_not_antitone_iff_exists_lt_lt, @and_left_comm (_ \u2208 s)]", "annotated_tactic": ["simp [<a>monotoneOn_iff_monotone</a>, <a>antitoneOn_iff_antitone</a>, <a>and_assoc</a>, <a>exists_and_left</a>,\n    <a>not_monotone_not_antitone_iff_exists_lt_lt</a>, @<a>and_left_comm</a> (_ \u2208 s)]", [{"full_name": "Set.monotoneOn_iff_monotone", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [2675, 9], "def_end_pos": [2675, 32]}, {"full_name": "Set.antitoneOn_iff_antitone", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [2680, 9], "def_end_pos": [2680, 32]}, {"full_name": "and_assoc", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [177, 9], "def_end_pos": [177, 18]}, {"full_name": "exists_and_left", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [465, 17], "def_end_pos": [465, 32]}, {"full_name": "not_monotone_not_antitone_iff_exists_lt_lt", "def_path": "Mathlib/Order/Monotone/Basic.lean", "def_pos": [959, 7], "def_end_pos": [959, 49]}, {"full_name": "and_left_comm", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [180, 9], "def_end_pos": [180, 22]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\na b : \u03b1\ns s\u2081 s\u2082 t t\u2081 t\u2082 u : Set \u03b1\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\n\u22a2 \u00acMonotoneOn f s \u2227 \u00acAntitoneOn f s \u2194 \u2203 a x b x c x, a < b \u2227 b < c \u2227 (f a < f b \u2227 f c < f b \u2228 f b < f a \u2227 f b < f c)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Intervals/Monotone.lean", "full_name": "strictAntiOn_Ici_of_lt_pred", "start": [266, 1], "end": [268, 68], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Image.lean", "full_name": "Set.mem_image_elim_on", "start": [267, 1], "end": [269, 23], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/UniformIntegrable.lean", "full_name": "MeasureTheory.UniformIntegrable.ae_eq", "start": [747, 1], "end": [752, 13], "traced_tactics": [{"tactic": "obtain \u27e8hfm, hunif, C, hC\u27e9 := hf", "annotated_tactic": ["obtain \u27e8hfm, hunif, C, hC\u27e9 := hf", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf g : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhf : UniformIntegrable f p \u03bc\nhfg : \u2200 (n : \u03b9), f n =\u1d50[\u03bc] g n\n\u22a2 UniformIntegrable g p \u03bc", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf g : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhfg : \u2200 (n : \u03b9), f n =\u1d50[\u03bc] g n\nhfm : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\nhunif : UnifIntegrable f p \u03bc\nC : \u211d\u22650\nhC : \u2200 (i : \u03b9), snorm (f i) p \u03bc \u2264 \u2191C\n\u22a2 UniformIntegrable g p \u03bc"}, {"tactic": "refine' \u27e8fun i => (hfm i).congr (hfg i), (unifIntegrable_congr_ae hfg).1 hunif, C, fun i => _\u27e9", "annotated_tactic": ["refine' \u27e8fun i => (hfm i).<a>congr</a> (hfg i), (<a>unifIntegrable_congr_ae</a> hfg).1 hunif, C, fun i => _\u27e9", [{"full_name": "MeasureTheory.AEStronglyMeasurable.congr", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1228, 9], "def_end_pos": [1228, 14]}, {"full_name": "MeasureTheory.unifIntegrable_congr_ae", "def_path": "Mathlib/MeasureTheory/Function/UniformIntegrable.lean", "def_pos": [153, 9], "def_end_pos": [153, 32]}]], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf g : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhfg : \u2200 (n : \u03b9), f n =\u1d50[\u03bc] g n\nhfm : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\nhunif : UnifIntegrable f p \u03bc\nC : \u211d\u22650\nhC : \u2200 (i : \u03b9), snorm (f i) p \u03bc \u2264 \u2191C\n\u22a2 UniformIntegrable g p \u03bc", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf g : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhfg : \u2200 (n : \u03b9), f n =\u1d50[\u03bc] g n\nhfm : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\nhunif : UnifIntegrable f p \u03bc\nC : \u211d\u22650\nhC : \u2200 (i : \u03b9), snorm (f i) p \u03bc \u2264 \u2191C\ni : \u03b9\n\u22a2 snorm (g i) p \u03bc \u2264 \u2191C"}, {"tactic": "rw [\u2190 snorm_congr_ae (hfg i)]", "annotated_tactic": ["rw [\u2190 <a>snorm_congr_ae</a> (hfg i)]", [{"full_name": "MeasureTheory.snorm_congr_ae", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [549, 9], "def_end_pos": [549, 23]}]], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf g : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhfg : \u2200 (n : \u03b9), f n =\u1d50[\u03bc] g n\nhfm : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\nhunif : UnifIntegrable f p \u03bc\nC : \u211d\u22650\nhC : \u2200 (i : \u03b9), snorm (f i) p \u03bc \u2264 \u2191C\ni : \u03b9\n\u22a2 snorm (g i) p \u03bc \u2264 \u2191C", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf g : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhfg : \u2200 (n : \u03b9), f n =\u1d50[\u03bc] g n\nhfm : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\nhunif : UnifIntegrable f p \u03bc\nC : \u211d\u22650\nhC : \u2200 (i : \u03b9), snorm (f i) p \u03bc \u2264 \u2191C\ni : \u03b9\n\u22a2 snorm (f i) p \u03bc \u2264 \u2191C"}, {"tactic": "exact hC i", "annotated_tactic": ["exact hC i", []], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf g : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhfg : \u2200 (n : \u03b9), f n =\u1d50[\u03bc] g n\nhfm : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\nhunif : UnifIntegrable f p \u03bc\nC : \u211d\u22650\nhC : \u2200 (i : \u03b9), snorm (f i) p \u03bc \u2264 \u2191C\ni : \u03b9\n\u22a2 snorm (f i) p \u03bc \u2264 \u2191C", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/LocallyFinite.lean", "full_name": "Finset.uIcc_injective_right", "start": [1014, 1], "end": [1016, 83], "traced_tactics": [{"tactic": "rw [ext_iff] at h", "annotated_tactic": ["rw [<a>ext_iff</a>] at h", [{"full_name": "Finset.ext_iff", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [234, 9], "def_end_pos": [234, 16]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : DistribLattice \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na\u271d a\u2081 a\u2082 b\u271d b\u2081 b\u2082 c\u271d x a b c : \u03b1\nh : (fun b => [[b, a]]) b = (fun b => [[b, a]]) c\n\u22a2 b = c", "state_after": "\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : DistribLattice \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na\u271d a\u2081 a\u2082 b\u271d b\u2081 b\u2082 c\u271d x a b c : \u03b1\nh : \u2200 (a_1 : \u03b1), a_1 \u2208 (fun b => [[b, a]]) b \u2194 a_1 \u2208 (fun b => [[b, a]]) c\n\u22a2 b = c"}, {"tactic": "exact eq_of_mem_uIcc_of_mem_uIcc ((h _).1 left_mem_uIcc) ((h _).2 left_mem_uIcc)", "annotated_tactic": ["exact <a>eq_of_mem_uIcc_of_mem_uIcc</a> ((h _).1 <a>left_mem_uIcc</a>) ((h _).2 <a>left_mem_uIcc</a>)", [{"full_name": "Finset.eq_of_mem_uIcc_of_mem_uIcc", "def_path": "Mathlib/Data/Finset/LocallyFinite.lean", "def_pos": [1004, 9], "def_end_pos": [1004, 35]}, {"full_name": "Finset.left_mem_uIcc", "def_path": "Mathlib/Data/Finset/LocallyFinite.lean", "def_pos": [952, 9], "def_end_pos": [952, 22]}, {"full_name": "Finset.left_mem_uIcc", "def_path": "Mathlib/Data/Finset/LocallyFinite.lean", "def_pos": [952, 9], "def_end_pos": [952, 22]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : DistribLattice \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na\u271d a\u2081 a\u2082 b\u271d b\u2081 b\u2082 c\u271d x a b c : \u03b1\nh : \u2200 (a_1 : \u03b1), a_1 \u2208 (fun b => [[b, a]]) b \u2194 a_1 \u2208 (fun b => [[b, a]]) c\n\u22a2 b = c", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "full_name": "String.utf8Len_reverse", "start": [71, 9], "end": [71, 96], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/QPF/Univariate/Basic.lean", "full_name": "QPF.Cofix.bisim_rel", "start": [467, 1], "end": [484, 19], "traced_tactics": [{"tactic": "let r' (x y) := x = y \u2228 r x y", "annotated_tactic": ["let r' (x y) := x = y \u2228 r x y", []], "state_before": "F : Type u \u2192 Type u\ninst\u271d : Functor F\nq : QPF F\nr : Cofix F \u2192 Cofix F \u2192 Prop\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\n\u22a2 \u2200 (x y : Cofix F), r x y \u2192 x = y", "state_after": "F : Type u \u2192 Type u\ninst\u271d : Functor F\nq : QPF F\nr : Cofix F \u2192 Cofix F \u2192 Prop\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nr' : Cofix F \u2192 Cofix F \u2192 Prop := fun x y => x = y \u2228 r x y\n\u22a2 \u2200 (x y : Cofix F), r x y \u2192 x = y"}, {"tactic": "intro x y rxy", "annotated_tactic": ["intro x y rxy", []], "state_before": "F : Type u \u2192 Type u\ninst\u271d : Functor F\nq : QPF F\nr : Cofix F \u2192 Cofix F \u2192 Prop\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nr' : Cofix F \u2192 Cofix F \u2192 Prop := fun x y => x = y \u2228 r x y\n\u22a2 \u2200 (x y : Cofix F), r x y \u2192 x = y", "state_after": "F : Type u \u2192 Type u\ninst\u271d : Functor F\nq : QPF F\nr : Cofix F \u2192 Cofix F \u2192 Prop\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nr' : Cofix F \u2192 Cofix F \u2192 Prop := fun x y => x = y \u2228 r x y\nx y : Cofix F\nrxy : r x y\n\u22a2 x = y"}, {"tactic": "apply Cofix.bisim_aux r'", "annotated_tactic": ["apply <a>Cofix.bisim_aux</a> r'", [{"full_name": "_private.Mathlib.Data.QPF.Univariate.Basic.0.QPF.Cofix.bisim_aux", "def_path": "Mathlib/Data/QPF/Univariate/Basic.lean", "def_pos": [431, 17], "def_end_pos": [431, 32]}]], "state_before": "F : Type u \u2192 Type u\ninst\u271d : Functor F\nq : QPF F\nr : Cofix F \u2192 Cofix F \u2192 Prop\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nr' : Cofix F \u2192 Cofix F \u2192 Prop := fun x y => x = y \u2228 r x y\nx y : Cofix F\nrxy : r x y\n\u22a2 x = y", "state_after": "case h'\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : QPF F\nr : Cofix F \u2192 Cofix F \u2192 Prop\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nr' : Cofix F \u2192 Cofix F \u2192 Prop := fun x y => x = y \u2228 r x y\nx y : Cofix F\nrxy : r x y\n\u22a2 \u2200 (x : Cofix F), r' x x\n\ncase h\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : QPF F\nr : Cofix F \u2192 Cofix F \u2192 Prop\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nr' : Cofix F \u2192 Cofix F \u2192 Prop := fun x y => x = y \u2228 r x y\nx y : Cofix F\nrxy : r x y\n\u22a2 \u2200 (x y : Cofix F), r' x y \u2192 Quot.mk r' <$> dest x = Quot.mk r' <$> dest y\n\ncase a\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : QPF F\nr : Cofix F \u2192 Cofix F \u2192 Prop\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nr' : Cofix F \u2192 Cofix F \u2192 Prop := fun x y => x = y \u2228 r x y\nx y : Cofix F\nrxy : r x y\n\u22a2 r' x y"}, {"tactic": "right", "annotated_tactic": ["right", []], "state_before": "case a\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : QPF F\nr : Cofix F \u2192 Cofix F \u2192 Prop\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nr' : Cofix F \u2192 Cofix F \u2192 Prop := fun x y => x = y \u2228 r x y\nx y : Cofix F\nrxy : r x y\n\u22a2 r' x y", "state_after": "case a.h\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : QPF F\nr : Cofix F \u2192 Cofix F \u2192 Prop\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nr' : Cofix F \u2192 Cofix F \u2192 Prop := fun x y => x = y \u2228 r x y\nx y : Cofix F\nrxy : r x y\n\u22a2 r x y"}, {"tactic": "exact rxy", "annotated_tactic": ["exact rxy", []], "state_before": "case a.h\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : QPF F\nr : Cofix F \u2192 Cofix F \u2192 Prop\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nr' : Cofix F \u2192 Cofix F \u2192 Prop := fun x y => x = y \u2228 r x y\nx y : Cofix F\nrxy : r x y\n\u22a2 r x y", "state_after": "no goals"}, {"tactic": "intro x", "annotated_tactic": ["intro x", []], "state_before": "case h'\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : QPF F\nr : Cofix F \u2192 Cofix F \u2192 Prop\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nr' : Cofix F \u2192 Cofix F \u2192 Prop := fun x y => x = y \u2228 r x y\nx y : Cofix F\nrxy : r x y\n\u22a2 \u2200 (x : Cofix F), r' x x", "state_after": "case h'\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : QPF F\nr : Cofix F \u2192 Cofix F \u2192 Prop\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nr' : Cofix F \u2192 Cofix F \u2192 Prop := fun x y => x = y \u2228 r x y\nx\u271d y : Cofix F\nrxy : r x\u271d y\nx : Cofix F\n\u22a2 r' x x"}, {"tactic": "left", "annotated_tactic": ["left", []], "state_before": "case h'\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : QPF F\nr : Cofix F \u2192 Cofix F \u2192 Prop\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nr' : Cofix F \u2192 Cofix F \u2192 Prop := fun x y => x = y \u2228 r x y\nx\u271d y : Cofix F\nrxy : r x\u271d y\nx : Cofix F\n\u22a2 r' x x", "state_after": "case h'.h\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : QPF F\nr : Cofix F \u2192 Cofix F \u2192 Prop\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nr' : Cofix F \u2192 Cofix F \u2192 Prop := fun x y => x = y \u2228 r x y\nx\u271d y : Cofix F\nrxy : r x\u271d y\nx : Cofix F\n\u22a2 x = x"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case h'.h\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : QPF F\nr : Cofix F \u2192 Cofix F \u2192 Prop\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nr' : Cofix F \u2192 Cofix F \u2192 Prop := fun x y => x = y \u2228 r x y\nx\u271d y : Cofix F\nrxy : r x\u271d y\nx : Cofix F\n\u22a2 x = x", "state_after": "no goals"}, {"tactic": "intro x y r'xy", "annotated_tactic": ["intro x y r'xy", []], "state_before": "case h\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : QPF F\nr : Cofix F \u2192 Cofix F \u2192 Prop\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nr' : Cofix F \u2192 Cofix F \u2192 Prop := fun x y => x = y \u2228 r x y\nx y : Cofix F\nrxy : r x y\n\u22a2 \u2200 (x y : Cofix F), r' x y \u2192 Quot.mk r' <$> dest x = Quot.mk r' <$> dest y", "state_after": "case h\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : QPF F\nr : Cofix F \u2192 Cofix F \u2192 Prop\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nr' : Cofix F \u2192 Cofix F \u2192 Prop := fun x y => x = y \u2228 r x y\nx\u271d y\u271d : Cofix F\nrxy : r x\u271d y\u271d\nx y : Cofix F\nr'xy : r' x y\n\u22a2 Quot.mk r' <$> dest x = Quot.mk r' <$> dest y"}, {"tactic": "cases' r'xy with r'xy r'xy", "annotated_tactic": ["cases' r'xy with r'xy r'xy", []], "state_before": "case h\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : QPF F\nr : Cofix F \u2192 Cofix F \u2192 Prop\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nr' : Cofix F \u2192 Cofix F \u2192 Prop := fun x y => x = y \u2228 r x y\nx\u271d y\u271d : Cofix F\nrxy : r x\u271d y\u271d\nx y : Cofix F\nr'xy : r' x y\n\u22a2 Quot.mk r' <$> dest x = Quot.mk r' <$> dest y", "state_after": "case h.inl\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : QPF F\nr : Cofix F \u2192 Cofix F \u2192 Prop\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nr' : Cofix F \u2192 Cofix F \u2192 Prop := fun x y => x = y \u2228 r x y\nx\u271d y\u271d : Cofix F\nrxy : r x\u271d y\u271d\nx y : Cofix F\nr'xy : x = y\n\u22a2 Quot.mk r' <$> dest x = Quot.mk r' <$> dest y\n\ncase h.inr\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : QPF F\nr : Cofix F \u2192 Cofix F \u2192 Prop\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nr' : Cofix F \u2192 Cofix F \u2192 Prop := fun x y => x = y \u2228 r x y\nx\u271d y\u271d : Cofix F\nrxy : r x\u271d y\u271d\nx y : Cofix F\nr'xy : r x y\n\u22a2 Quot.mk r' <$> dest x = Quot.mk r' <$> dest y"}, {"tactic": "have : \u2200 x y, r x y \u2192 r' x y := fun x y h => Or.inr h", "annotated_tactic": ["have : \u2200 x y, r x y \u2192 r' x y := fun x y h => <a>Or.inr</a> h", [{"full_name": "Or.inr", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [519, 5], "def_end_pos": [519, 8]}]], "state_before": "case h.inr\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : QPF F\nr : Cofix F \u2192 Cofix F \u2192 Prop\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nr' : Cofix F \u2192 Cofix F \u2192 Prop := fun x y => x = y \u2228 r x y\nx\u271d y\u271d : Cofix F\nrxy : r x\u271d y\u271d\nx y : Cofix F\nr'xy : r x y\n\u22a2 Quot.mk r' <$> dest x = Quot.mk r' <$> dest y", "state_after": "case h.inr\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : QPF F\nr : Cofix F \u2192 Cofix F \u2192 Prop\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nr' : Cofix F \u2192 Cofix F \u2192 Prop := fun x y => x = y \u2228 r x y\nx\u271d y\u271d : Cofix F\nrxy : r x\u271d y\u271d\nx y : Cofix F\nr'xy : r x y\nthis : \u2200 (x y : Cofix F), r x y \u2192 r' x y\n\u22a2 Quot.mk r' <$> dest x = Quot.mk r' <$> dest y"}, {"tactic": "rw [\u2190 Quot.factor_mk_eq _ _ this]", "annotated_tactic": ["rw [\u2190 <a>Quot.factor_mk_eq</a> _ _ this]", [{"full_name": "Quot.factor_mk_eq", "def_path": "Mathlib/Data/Quot.lean", "def_pos": [90, 9], "def_end_pos": [90, 21]}]], "state_before": "case h.inr\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : QPF F\nr : Cofix F \u2192 Cofix F \u2192 Prop\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nr' : Cofix F \u2192 Cofix F \u2192 Prop := fun x y => x = y \u2228 r x y\nx\u271d y\u271d : Cofix F\nrxy : r x\u271d y\u271d\nx y : Cofix F\nr'xy : r x y\nthis : \u2200 (x y : Cofix F), r x y \u2192 r' x y\n\u22a2 Quot.mk r' <$> dest x = Quot.mk r' <$> dest y", "state_after": "case h.inr\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : QPF F\nr : Cofix F \u2192 Cofix F \u2192 Prop\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nr' : Cofix F \u2192 Cofix F \u2192 Prop := fun x y => x = y \u2228 r x y\nx\u271d y\u271d : Cofix F\nrxy : r x\u271d y\u271d\nx y : Cofix F\nr'xy : r x y\nthis : \u2200 (x y : Cofix F), r x y \u2192 r' x y\n\u22a2 (Quot.factor (fun x y => r x y) (fun x y => r' x y) this \u2218 Quot.mk fun x y => r x y) <$> dest x =\n    (Quot.factor (fun x y => r x y) (fun x y => r' x y) this \u2218 Quot.mk fun x y => r x y) <$> dest y"}, {"tactic": "dsimp", "annotated_tactic": ["dsimp", []], "state_before": "case h.inr\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : QPF F\nr : Cofix F \u2192 Cofix F \u2192 Prop\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nr' : Cofix F \u2192 Cofix F \u2192 Prop := fun x y => x = y \u2228 r x y\nx\u271d y\u271d : Cofix F\nrxy : r x\u271d y\u271d\nx y : Cofix F\nr'xy : r x y\nthis : \u2200 (x y : Cofix F), r x y \u2192 r' x y\n\u22a2 (Quot.factor (fun x y => r x y) (fun x y => r' x y) this \u2218 Quot.mk fun x y => r x y) <$> dest x =\n    (Quot.factor (fun x y => r x y) (fun x y => r' x y) this \u2218 Quot.mk fun x y => r x y) <$> dest y", "state_after": "case h.inr\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : QPF F\nr : Cofix F \u2192 Cofix F \u2192 Prop\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nr' : Cofix F \u2192 Cofix F \u2192 Prop := fun x y => x = y \u2228 r x y\nx\u271d y\u271d : Cofix F\nrxy : r x\u271d y\u271d\nx y : Cofix F\nr'xy : r x y\nthis : \u2200 (x y : Cofix F), r x y \u2192 r' x y\n\u22a2 (Quot.factor (fun x y => r x y) (fun x y => x = y \u2228 r x y) this \u2218 Quot.mk fun x y => r x y) <$> dest x =\n    (Quot.factor (fun x y => r x y) (fun x y => x = y \u2228 r x y) this \u2218 Quot.mk fun x y => r x y) <$> dest y"}, {"tactic": "rw [@comp_map _ _ q _ _ _ (Quot.mk r), @comp_map _ _ q _ _ _ (Quot.mk r)]", "annotated_tactic": ["rw [@<a>comp_map</a> _ _ q _ _ _ (<a>Quot.mk</a> r), @<a>comp_map</a> _ _ q _ _ _ (<a>Quot.mk</a> r)]", [{"full_name": "QPF.comp_map", "def_path": "Mathlib/Data/QPF/Univariate/Basic.lean", "def_pos": [78, 9], "def_end_pos": [78, 17]}, {"full_name": "Quot.mk", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [406, 14], "def_end_pos": [406, 21]}, {"full_name": "QPF.comp_map", "def_path": "Mathlib/Data/QPF/Univariate/Basic.lean", "def_pos": [78, 9], "def_end_pos": [78, 17]}, {"full_name": "Quot.mk", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [406, 14], "def_end_pos": [406, 21]}]], "state_before": "case h.inr\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : QPF F\nr : Cofix F \u2192 Cofix F \u2192 Prop\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nr' : Cofix F \u2192 Cofix F \u2192 Prop := fun x y => x = y \u2228 r x y\nx\u271d y\u271d : Cofix F\nrxy : r x\u271d y\u271d\nx y : Cofix F\nr'xy : r x y\nthis : \u2200 (x y : Cofix F), r x y \u2192 r' x y\n\u22a2 (Quot.factor (fun x y => r x y) (fun x y => x = y \u2228 r x y) this \u2218 Quot.mk fun x y => r x y) <$> dest x =\n    (Quot.factor (fun x y => r x y) (fun x y => x = y \u2228 r x y) this \u2218 Quot.mk fun x y => r x y) <$> dest y", "state_after": "case h.inr\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : QPF F\nr : Cofix F \u2192 Cofix F \u2192 Prop\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nr' : Cofix F \u2192 Cofix F \u2192 Prop := fun x y => x = y \u2228 r x y\nx\u271d y\u271d : Cofix F\nrxy : r x\u271d y\u271d\nx y : Cofix F\nr'xy : r x y\nthis : \u2200 (x y : Cofix F), r x y \u2192 r' x y\n\u22a2 Quot.factor (fun x y => r x y) (fun x y => x = y \u2228 r x y) this <$> Quot.mk r <$> dest x =\n    Quot.factor (fun x y => r x y) (fun x y => x = y \u2228 r x y) this <$> Quot.mk r <$> dest y"}, {"tactic": "rw [h _ _ r'xy]", "annotated_tactic": ["rw [h _ _ r'xy]", []], "state_before": "case h.inr\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : QPF F\nr : Cofix F \u2192 Cofix F \u2192 Prop\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nr' : Cofix F \u2192 Cofix F \u2192 Prop := fun x y => x = y \u2228 r x y\nx\u271d y\u271d : Cofix F\nrxy : r x\u271d y\u271d\nx y : Cofix F\nr'xy : r x y\nthis : \u2200 (x y : Cofix F), r x y \u2192 r' x y\n\u22a2 Quot.factor (fun x y => r x y) (fun x y => x = y \u2228 r x y) this <$> Quot.mk r <$> dest x =\n    Quot.factor (fun x y => r x y) (fun x y => x = y \u2228 r x y) this <$> Quot.mk r <$> dest y", "state_after": "no goals"}, {"tactic": "rw [r'xy]", "annotated_tactic": ["rw [r'xy]", []], "state_before": "case h.inl\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : QPF F\nr : Cofix F \u2192 Cofix F \u2192 Prop\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nr' : Cofix F \u2192 Cofix F \u2192 Prop := fun x y => x = y \u2228 r x y\nx\u271d y\u271d : Cofix F\nrxy : r x\u271d y\u271d\nx y : Cofix F\nr'xy : x = y\n\u22a2 Quot.mk r' <$> dest x = Quot.mk r' <$> dest y", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/BorelCantelli.lean", "full_name": "ProbabilityTheory.iIndepSet.condexp_indicator_filtrationOfSet_ae_eq", "start": [58, 1], "end": [64, 87], "traced_tactics": [{"tactic": "rw [Filtration.filtrationOfSet_eq_natural (\u03b2 := \u211d) hsm]", "annotated_tactic": ["rw [<a>Filtration.filtrationOfSet_eq_natural</a> (\u03b2 := \u211d) hsm]", [{"full_name": "MeasureTheory.Filtration.filtrationOfSet_eq_natural", "def_path": "Mathlib/Probability/Process/Filtration.lean", "def_pos": [281, 9], "def_end_pos": [281, 35]}]], "state_before": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b9\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ni j : \u03b9\ns : \u03b9 \u2192 Set \u03a9\nhsm : \u2200 (n : \u03b9), MeasurableSet (s n)\nhs : iIndepSet s\nhij : i < j\n\u22a2 \u03bc[Set.indicator (s j) fun x => 1|\u2191(filtrationOfSet hsm) i] =\u1d50[\u03bc] fun x => ENNReal.toReal (\u2191\u2191\u03bc (s j))", "state_after": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b9\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ni j : \u03b9\ns : \u03b9 \u2192 Set \u03a9\nhsm : \u2200 (n : \u03b9), MeasurableSet (s n)\nhs : iIndepSet s\nhij : i < j\n\u22a2 \u03bc[Set.indicator (s j) fun x =>\n        1|\u2191(Filtration.natural (fun i => Set.indicator (s i) fun x => 1)\n            (_ : \u2200 (i : \u03b9), StronglyMeasurable (Set.indicator (s i) 1)))\n        i] =\u1d50[\u03bc]\n    fun x => ENNReal.toReal (\u2191\u2191\u03bc (s j))"}, {"tactic": "refine' (iIndepFun.condexp_natural_ae_eq_of_lt _ hs.iIndepFun_indicator hij).trans _", "annotated_tactic": ["refine' (<a>iIndepFun.condexp_natural_ae_eq_of_lt</a> _ hs.iIndepFun_indicator hij).<a>trans</a> _", [{"full_name": "ProbabilityTheory.iIndepFun.condexp_natural_ae_eq_of_lt", "def_path": "Mathlib/Probability/BorelCantelli.lean", "def_pos": [50, 9], "def_end_pos": [50, 46]}, {"full_name": "Filter.EventuallyEq.trans", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1503, 9], "def_end_pos": [1503, 27]}]], "state_before": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b9\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ni j : \u03b9\ns : \u03b9 \u2192 Set \u03a9\nhsm : \u2200 (n : \u03b9), MeasurableSet (s n)\nhs : iIndepSet s\nhij : i < j\n\u22a2 \u03bc[Set.indicator (s j) fun x =>\n        1|\u2191(Filtration.natural (fun i => Set.indicator (s i) fun x => 1)\n            (_ : \u2200 (i : \u03b9), StronglyMeasurable (Set.indicator (s i) 1)))\n        i] =\u1d50[\u03bc]\n    fun x => ENNReal.toReal (\u2191\u2191\u03bc (s j))", "state_after": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b9\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ni j : \u03b9\ns : \u03b9 \u2192 Set \u03a9\nhsm : \u2200 (n : \u03b9), MeasurableSet (s n)\nhs : iIndepSet s\nhij : i < j\n\u22a2 (fun x => \u222b (x : \u03a9), Set.indicator (s j) (fun _\u03c9 => 1) x \u2202\u03bc) =\u1d50[\u03bc] fun x => ENNReal.toReal (\u2191\u2191\u03bc (s j))"}, {"tactic": "simp only [integral_indicator_const _ (hsm _), Algebra.id.smul_eq_mul, mul_one]", "annotated_tactic": ["simp only [<a>integral_indicator_const</a> _ (hsm _), <a>Algebra.id.smul_eq_mul</a>, <a>mul_one</a>]", [{"full_name": "MeasureTheory.integral_indicator_const", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [479, 9], "def_end_pos": [479, 33]}, {"full_name": "Algebra.id.smul_eq_mul", "def_path": "Mathlib/Algebra/Algebra/Basic.lean", "def_pos": [453, 9], "def_end_pos": [453, 20]}, {"full_name": "mul_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [470, 9], "def_end_pos": [470, 16]}]], "state_before": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b9\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ni j : \u03b9\ns : \u03b9 \u2192 Set \u03a9\nhsm : \u2200 (n : \u03b9), MeasurableSet (s n)\nhs : iIndepSet s\nhij : i < j\n\u22a2 (fun x => \u222b (x : \u03a9), Set.indicator (s j) (fun _\u03c9 => 1) x \u2202\u03bc) =\u1d50[\u03bc] fun x => ENNReal.toReal (\u2191\u2191\u03bc (s j))", "state_after": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b9\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ni j : \u03b9\ns : \u03b9 \u2192 Set \u03a9\nhsm : \u2200 (n : \u03b9), MeasurableSet (s n)\nhs : iIndepSet s\nhij : i < j\n\u22a2 (fun x => ENNReal.toReal (\u2191\u2191\u03bc (s j))) =\u1d50[\u03bc] fun x => ENNReal.toReal (\u2191\u2191\u03bc (s j))"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b9\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ni j : \u03b9\ns : \u03b9 \u2192 Set \u03a9\nhsm : \u2200 (n : \u03b9), MeasurableSet (s n)\nhs : iIndepSet s\nhij : i < j\n\u22a2 (fun x => ENNReal.toReal (\u2191\u2191\u03bc (s j))) =\u1d50[\u03bc] fun x => ENNReal.toReal (\u2191\u2191\u03bc (s j))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Process/Adapted.lean", "full_name": "MeasureTheory.progMeasurable_of_tendsto", "start": [201, 1], "end": [203, 47], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Decomposition/Jordan.lean", "full_name": "MeasureTheory.SignedMeasure.toJordanDecomposition_smul_real_nonneg", "start": [467, 9], "end": [471, 6], "traced_tactics": [{"tactic": "lift r to \u211d\u22650 using hr", "annotated_tactic": ["lift r to \u211d\u22650 using hr", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\nr : \u211d\nhr : 0 \u2264 r\n\u22a2 toJordanDecomposition (r \u2022 s) = r \u2022 toJordanDecomposition s", "state_after": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\nr : \u211d\u22650\n\u22a2 toJordanDecomposition (\u2191r \u2022 s) = \u2191r \u2022 toJordanDecomposition s"}, {"tactic": "rw [JordanDecomposition.coe_smul, \u2190 toJordanDecomposition_smul]", "annotated_tactic": ["rw [<a>JordanDecomposition.coe_smul</a>, \u2190 <a>toJordanDecomposition_smul</a>]", [{"full_name": "MeasureTheory.JordanDecomposition.coe_smul", "def_path": "Mathlib/MeasureTheory/Decomposition/Jordan.lean", "def_pos": [135, 9], "def_end_pos": [135, 17]}, {"full_name": "MeasureTheory.SignedMeasure.toJordanDecomposition_smul", "def_path": "Mathlib/MeasureTheory/Decomposition/Jordan.lean", "def_pos": [461, 9], "def_end_pos": [461, 35]}]], "state_before": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\nr : \u211d\u22650\n\u22a2 toJordanDecomposition (\u2191r \u2022 s) = \u2191r \u2022 toJordanDecomposition s", "state_after": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\nr : \u211d\u22650\n\u22a2 toJordanDecomposition (\u2191r \u2022 s) = toJordanDecomposition (r \u2022 s)"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\nr : \u211d\u22650\n\u22a2 toJordanDecomposition (\u2191r \u2022 s) = toJordanDecomposition (r \u2022 s)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Control/ForInStep/Lemmas.lean", "full_name": "ForInStep.bind_done", "start": [10, 9], "end": [11, 63], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Int/Bitwise.lean", "full_name": "Int.bitwise_xor", "start": [274, 1], "end": [289, 32], "traced_tactics": [{"tactic": "funext m n", "annotated_tactic": ["funext m n", []], "state_before": "\u22a2 bitwise xor = Int.xor", "state_after": "case h.h\nm n : \u2124\n\u22a2 bitwise xor m n = Int.xor m n"}, {"tactic": "cases' m with m m <;> cases' n with n n <;> try {rfl}\n  <;> simp only [bitwise, natBitwise, Bool.not_false, Bool.or_true, Bool.bne_eq_xor,\n    cond_false, cond_true, lor, Nat.ldiff, Bool.and_true, negSucc.injEq, Bool.false_xor,\n    Bool.true_xor, Bool.and_false, Nat.land, Bool.not_true, ldiff,\n    HOr.hOr, OrOp.or, Nat.lor, Int.xor, HXor.hXor, Xor.xor, Nat.xor]", "annotated_tactic": ["cases' m with m m <;> cases' n with n n <;> try {rfl}\n    <;> simp only [<a>bitwise</a>, <a>natBitwise</a>, <a>Bool.not_false</a>, <a>Bool.or_true</a>, <a>Bool.bne_eq_xor</a>,\n      <a>cond_false</a>, <a>cond_true</a>, <a>lor</a>, <a>Nat.ldiff</a>, <a>Bool.and_true</a>, negSucc.injEq, <a>Bool.false_xor</a>,\n      <a>Bool.true_xor</a>, <a>Bool.and_false</a>, <a>Nat.land</a>, <a>Bool.not_true</a>, <a>ldiff</a>,\n      <a>HOr.hOr</a>, <a>OrOp.or</a>, <a>Nat.lor</a>, <a>Int.xor</a>, <a>HXor.hXor</a>, <a>Xor.xor</a>, <a>Nat.xor</a>]", [{"full_name": "Int.bitwise", "def_path": "Mathlib/Init/Data/Int/Bitwise.lean", "def_pos": [55, 5], "def_end_pos": [55, 12]}, {"full_name": "Int.natBitwise", "def_path": "Mathlib/Init/Data/Int/Bitwise.lean", "def_pos": [49, 5], "def_end_pos": [49, 15]}, {"full_name": "Bool.not_false", "def_path": "lake-packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [125, 17], "def_end_pos": [125, 31]}, {"full_name": "Bool.or_true", "def_path": "lake-packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [103, 17], "def_end_pos": [103, 29]}, {"full_name": "Bool.bne_eq_xor", "def_path": "Mathlib/Data/Bool/Basic.lean", "def_pos": [262, 9], "def_end_pos": [262, 19]}, {"full_name": "cond_false", "def_path": "lake-packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [146, 17], "def_end_pos": [146, 27]}, {"full_name": "cond_true", "def_path": "lake-packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [145, 17], "def_end_pos": [145, 26]}, {"full_name": "Int.lor", "def_path": "Mathlib/Init/Data/Int/Bitwise.lean", "def_pos": [69, 5], "def_end_pos": [69, 8]}, {"full_name": "Nat.ldiff", "def_path": "Mathlib/Init/Data/Nat/Bitwise.lean", "def_pos": [261, 5], "def_end_pos": [261, 10]}, {"full_name": "Bool.and_true", "def_path": "lake-packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [111, 17], "def_end_pos": [111, 30]}, {"full_name": "Bool.false_xor", "def_path": "Mathlib/Init/Data/Bool/Lemmas.lean", "def_pos": [65, 9], "def_end_pos": [65, 18]}, {"full_name": "Bool.true_xor", "def_path": "Mathlib/Init/Data/Bool/Lemmas.lean", "def_pos": [62, 9], "def_end_pos": [62, 17]}, {"full_name": "Bool.and_false", "def_path": "lake-packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [110, 17], "def_end_pos": [110, 31]}, {"full_name": "Nat.land", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Bitwise.lean", "def_pos": [34, 5], "def_end_pos": [34, 9]}, {"full_name": "Bool.not_true", "def_path": "lake-packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [124, 17], "def_end_pos": [124, 30]}, {"full_name": "Int.ldiff", "def_path": "Mathlib/Init/Data/Int/Bitwise.lean", "def_pos": [88, 5], "def_end_pos": [88, 10]}, {"full_name": "HOr.hOr", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1243, 3], "def_end_pos": [1243, 6]}, {"full_name": "OrOp.or", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1347, 3], "def_end_pos": [1347, 5]}, {"full_name": "Nat.lor", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Bitwise.lean", "def_pos": [36, 5], "def_end_pos": [36, 8]}, {"full_name": "Int.xor", "def_path": "Mathlib/Init/Data/Int/Bitwise.lean", "def_pos": [97, 15], "def_end_pos": [97, 18]}, {"full_name": "HXor.hXor", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1237, 3], "def_end_pos": [1237, 7]}, {"full_name": "Xor.xor", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1339, 3], "def_end_pos": [1339, 6]}, {"full_name": "Nat.xor", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Bitwise.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}]], "state_before": "case h.h\nm n : \u2124\n\u22a2 bitwise xor m n = Int.xor m n", "state_after": "case h.h.ofNat.negSucc\nm n : \u2115\n\u22a2 Nat.bitwise (fun x y => !xor x !y) m n = Nat.bitwise xor m n\n\ncase h.h.negSucc.ofNat\nm n : \u2115\n\u22a2 Nat.bitwise (fun x y => !xor (!x) y) m n = Nat.bitwise xor m n\n\ncase h.h.negSucc.negSucc\nm n : \u2115\n\u22a2 \u2191(Nat.bitwise (fun x y => xor (!x) !y) m n) = \u2191(Nat.bitwise xor m n)"}, {"tactic": "congr", "annotated_tactic": ["congr", []], "state_before": "case h.h.ofNat.negSucc\nm n : \u2115\n\u22a2 Nat.bitwise (fun x y => !xor x !y) m n = Nat.bitwise xor m n", "state_after": "case h.h.ofNat.negSucc.e_f\nm n : \u2115\n\u22a2 (fun x y => !xor x !y) = xor"}, {"tactic": "funext x y", "annotated_tactic": ["funext x y", []], "state_before": "case h.h.ofNat.negSucc.e_f\nm n : \u2115\n\u22a2 (fun x y => !xor x !y) = xor", "state_after": "case h.h.ofNat.negSucc.e_f.h.h\nm n : \u2115\nx y : Bool\n\u22a2 (!xor x !y) = xor x y"}, {"tactic": "cases x <;> cases y <;> rfl", "annotated_tactic": ["cases x <;> cases y <;> rfl", []], "state_before": "case h.h.ofNat.negSucc.e_f.h.h\nm n : \u2115\nx y : Bool\n\u22a2 (!xor x !y) = xor x y", "state_after": "no goals"}, {"tactic": "congr", "annotated_tactic": ["congr", []], "state_before": "case h.h.negSucc.ofNat\nm n : \u2115\n\u22a2 Nat.bitwise (fun x y => !xor (!x) y) m n = Nat.bitwise xor m n", "state_after": "case h.h.negSucc.ofNat.e_f\nm n : \u2115\n\u22a2 (fun x y => !xor (!x) y) = xor"}, {"tactic": "funext x y", "annotated_tactic": ["funext x y", []], "state_before": "case h.h.negSucc.ofNat.e_f\nm n : \u2115\n\u22a2 (fun x y => !xor (!x) y) = xor", "state_after": "case h.h.negSucc.ofNat.e_f.h.h\nm n : \u2115\nx y : Bool\n\u22a2 (!xor (!x) y) = xor x y"}, {"tactic": "cases x <;> cases y <;> rfl", "annotated_tactic": ["cases x <;> cases y <;> rfl", []], "state_before": "case h.h.negSucc.ofNat.e_f.h.h\nm n : \u2115\nx y : Bool\n\u22a2 (!xor (!x) y) = xor x y", "state_after": "no goals"}, {"tactic": "congr", "annotated_tactic": ["congr", []], "state_before": "case h.h.negSucc.negSucc\nm n : \u2115\n\u22a2 \u2191(Nat.bitwise (fun x y => xor (!x) !y) m n) = \u2191(Nat.bitwise xor m n)", "state_after": "case h.h.negSucc.negSucc.e_a.e_f\nm n : \u2115\n\u22a2 (fun x y => xor (!x) !y) = xor"}, {"tactic": "funext x y", "annotated_tactic": ["funext x y", []], "state_before": "case h.h.negSucc.negSucc.e_a.e_f\nm n : \u2115\n\u22a2 (fun x y => xor (!x) !y) = xor", "state_after": "case h.h.negSucc.negSucc.e_a.e_f.h.h\nm n : \u2115\nx y : Bool\n\u22a2 (xor (!x) !y) = xor x y"}, {"tactic": "cases x <;> cases y <;> rfl", "annotated_tactic": ["cases x <;> cases y <;> rfl", []], "state_before": "case h.h.negSucc.negSucc.e_a.e_f.h.h\nm n : \u2115\nx y : Bool\n\u22a2 (xor (!x) !y) = xor x y", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "full_name": "MeasureTheory.Measure.restrict_toOuterMeasure_eq_toOuterMeasure_restrict", "start": [1515, 1], "end": [1518, 71], "traced_tactics": [{"tactic": "simp_rw [restrict, restrict\u2097, liftLinear, LinearMap.coe_mk, AddHom.coe_mk,\n  toMeasure_toOuterMeasure, OuterMeasure.restrict_trim h, \u03bc.trimmed]", "annotated_tactic": ["simp_rw [<a>restrict</a>, <a>restrict\u2097</a>, <a>liftLinear</a>, <a>LinearMap.coe_mk</a>, <a>AddHom.coe_mk</a>,\n    <a>toMeasure_toOuterMeasure</a>, <a>OuterMeasure.restrict_trim</a> h, \u03bc.trimmed]", [{"full_name": "MeasureTheory.Measure.restrict", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1503, 5], "def_end_pos": [1503, 13]}, {"full_name": "MeasureTheory.Measure.restrict\u2097", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1495, 5], "def_end_pos": [1495, 14]}, {"full_name": "MeasureTheory.Measure.liftLinear", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1121, 5], "def_end_pos": [1121, 15]}, {"full_name": "LinearMap.coe_mk", "def_path": "Mathlib/Algebra/Module/LinearMap.lean", "def_pos": [251, 9], "def_end_pos": [251, 15]}, {"full_name": "AddHom.coe_mk", "def_path": "Mathlib/Algebra/Hom/Group/Defs.lean", "def_pos": [584, 3], "def_end_pos": [584, 14]}, {"full_name": "MeasureTheory.toMeasure_toOuterMeasure", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [665, 9], "def_end_pos": [665, 33]}, {"full_name": "MeasureTheory.OuterMeasure.restrict_trim", "def_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "def_pos": [1775, 9], "def_end_pos": [1775, 22]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nh : MeasurableSet s\n\u22a2 \u2191(restrict \u03bc s) = \u2191(OuterMeasure.restrict s) \u2191\u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/TuringMachine.lean", "full_name": "Turing.tr_reaches", "start": [895, 1], "end": [900, 38], "traced_tactics": [{"tactic": "rcases reflTransGen_iff_eq_or_transGen.1 ab with (rfl | ab)", "annotated_tactic": ["rcases <a>reflTransGen_iff_eq_or_transGen</a>.1 ab with (rfl | ab)", [{"full_name": "Relation.reflTransGen_iff_eq_or_transGen", "def_path": "Mathlib/Logic/Relation.lean", "def_pos": [535, 9], "def_end_pos": [535, 40]}]], "state_before": "\u03c3\u2081 : Type u_1\n\u03c3\u2082 : Type u_2\nf\u2081 : \u03c3\u2081 \u2192 Option \u03c3\u2081\nf\u2082 : \u03c3\u2082 \u2192 Option \u03c3\u2082\ntr : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop\nH : Respects f\u2081 f\u2082 tr\na\u2081 : \u03c3\u2081\na\u2082 : \u03c3\u2082\naa : tr a\u2081 a\u2082\nb\u2081 : \u03c3\u2081\nab : Reaches f\u2081 a\u2081 b\u2081\n\u22a2 \u2203 b\u2082, tr b\u2081 b\u2082 \u2227 Reaches f\u2082 a\u2082 b\u2082", "state_after": "case inl\n\u03c3\u2081 : Type u_1\n\u03c3\u2082 : Type u_2\nf\u2081 : \u03c3\u2081 \u2192 Option \u03c3\u2081\nf\u2082 : \u03c3\u2082 \u2192 Option \u03c3\u2082\ntr : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop\nH : Respects f\u2081 f\u2082 tr\na\u2082 : \u03c3\u2082\nb\u2081 : \u03c3\u2081\naa : tr b\u2081 a\u2082\nab : Reaches f\u2081 b\u2081 b\u2081\n\u22a2 \u2203 b\u2082, tr b\u2081 b\u2082 \u2227 Reaches f\u2082 a\u2082 b\u2082\n\ncase inr\n\u03c3\u2081 : Type u_1\n\u03c3\u2082 : Type u_2\nf\u2081 : \u03c3\u2081 \u2192 Option \u03c3\u2081\nf\u2082 : \u03c3\u2082 \u2192 Option \u03c3\u2082\ntr : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop\nH : Respects f\u2081 f\u2082 tr\na\u2081 : \u03c3\u2081\na\u2082 : \u03c3\u2082\naa : tr a\u2081 a\u2082\nb\u2081 : \u03c3\u2081\nab\u271d : Reaches f\u2081 a\u2081 b\u2081\nab : TransGen (fun a b => b \u2208 f\u2081 a) a\u2081 b\u2081\n\u22a2 \u2203 b\u2082, tr b\u2081 b\u2082 \u2227 Reaches f\u2082 a\u2082 b\u2082"}, {"tactic": "exact \u27e8_, aa, ReflTransGen.refl\u27e9", "annotated_tactic": ["exact \u27e8_, aa, <a>ReflTransGen.refl</a>\u27e9", [{"full_name": "Relation.ReflTransGen.refl", "def_path": "Mathlib/Logic/Relation.lean", "def_pos": [223, 5], "def_end_pos": [223, 9]}]], "state_before": "case inl\n\u03c3\u2081 : Type u_1\n\u03c3\u2082 : Type u_2\nf\u2081 : \u03c3\u2081 \u2192 Option \u03c3\u2081\nf\u2082 : \u03c3\u2082 \u2192 Option \u03c3\u2082\ntr : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop\nH : Respects f\u2081 f\u2082 tr\na\u2082 : \u03c3\u2082\nb\u2081 : \u03c3\u2081\naa : tr b\u2081 a\u2082\nab : Reaches f\u2081 b\u2081 b\u2081\n\u22a2 \u2203 b\u2082, tr b\u2081 b\u2082 \u2227 Reaches f\u2082 a\u2082 b\u2082", "state_after": "no goals"}, {"tactic": "have \u27e8b\u2082, bb, h\u27e9 := tr_reaches\u2081 H aa ab", "annotated_tactic": ["have \u27e8b\u2082, bb, h\u27e9 := <a>tr_reaches\u2081</a> H aa ab", [{"full_name": "Turing.tr_reaches\u2081", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [883, 9], "def_end_pos": [883, 20]}]], "state_before": "case inr\n\u03c3\u2081 : Type u_1\n\u03c3\u2082 : Type u_2\nf\u2081 : \u03c3\u2081 \u2192 Option \u03c3\u2081\nf\u2082 : \u03c3\u2082 \u2192 Option \u03c3\u2082\ntr : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop\nH : Respects f\u2081 f\u2082 tr\na\u2081 : \u03c3\u2081\na\u2082 : \u03c3\u2082\naa : tr a\u2081 a\u2082\nb\u2081 : \u03c3\u2081\nab\u271d : Reaches f\u2081 a\u2081 b\u2081\nab : TransGen (fun a b => b \u2208 f\u2081 a) a\u2081 b\u2081\n\u22a2 \u2203 b\u2082, tr b\u2081 b\u2082 \u2227 Reaches f\u2082 a\u2082 b\u2082", "state_after": "case inr\n\u03c3\u2081 : Type u_1\n\u03c3\u2082 : Type u_2\nf\u2081 : \u03c3\u2081 \u2192 Option \u03c3\u2081\nf\u2082 : \u03c3\u2082 \u2192 Option \u03c3\u2082\ntr : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop\nH : Respects f\u2081 f\u2082 tr\na\u2081 : \u03c3\u2081\na\u2082 : \u03c3\u2082\naa : tr a\u2081 a\u2082\nb\u2081 : \u03c3\u2081\nab\u271d : Reaches f\u2081 a\u2081 b\u2081\nab : TransGen (fun a b => b \u2208 f\u2081 a) a\u2081 b\u2081\nb\u2082 : \u03c3\u2082\nbb : tr b\u2081 b\u2082\nh : Reaches\u2081 f\u2082 a\u2082 b\u2082\n\u22a2 \u2203 b\u2082, tr b\u2081 b\u2082 \u2227 Reaches f\u2082 a\u2082 b\u2082"}, {"tactic": "exact \u27e8b\u2082, bb, h.to_reflTransGen\u27e9", "annotated_tactic": ["exact \u27e8b\u2082, bb, h.to_reflTransGen\u27e9", []], "state_before": "case inr\n\u03c3\u2081 : Type u_1\n\u03c3\u2082 : Type u_2\nf\u2081 : \u03c3\u2081 \u2192 Option \u03c3\u2081\nf\u2082 : \u03c3\u2082 \u2192 Option \u03c3\u2082\ntr : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop\nH : Respects f\u2081 f\u2082 tr\na\u2081 : \u03c3\u2081\na\u2082 : \u03c3\u2082\naa : tr a\u2081 a\u2082\nb\u2081 : \u03c3\u2081\nab\u271d : Reaches f\u2081 a\u2081 b\u2081\nab : TransGen (fun a b => b \u2208 f\u2081 a) a\u2081 b\u2081\nb\u2082 : \u03c3\u2082\nbb : tr b\u2081 b\u2082\nh : Reaches\u2081 f\u2082 a\u2082 b\u2082\n\u22a2 \u2203 b\u2082, tr b\u2081 b\u2082 \u2227 Reaches f\u2082 a\u2082 b\u2082", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Primrec.lean", "full_name": "Nat.Primrec'.sqrt", "start": [1487, 1], "end": [1505, 101], "traced_tactics": [{"tactic": "suffices H : \u2200 n : \u2115, n.sqrt = n.rec 0 fun x y => if x.succ < y.succ * y.succ then y else y.succ", "annotated_tactic": ["suffices H : \u2200 n : \u2115, n.sqrt = n.rec 0 fun x y => if x.succ < y.succ * y.succ then y else y.succ", []], "state_before": "\u22a2 Primrec' fun v => Nat.sqrt (Vector.head v)", "state_after": "H : \u2200 (n : \u2115), Nat.sqrt n = Nat.rec 0 (fun x y => if Nat.succ x < Nat.succ y * Nat.succ y then y else Nat.succ y) n\n\u22a2 Primrec' fun v => Nat.sqrt (Vector.head v)\n\ncase H\n\n\u22a2 \u2200 (n : \u2115), Nat.sqrt n = Nat.rec 0 (fun x y => if Nat.succ x < Nat.succ y * Nat.succ y then y else Nat.succ y) n"}, {"tactic": "introv", "annotated_tactic": ["introv", []], "state_before": "case H\n\n\u22a2 \u2200 (n : \u2115), Nat.sqrt n = Nat.rec 0 (fun x y => if Nat.succ x < Nat.succ y * Nat.succ y then y else Nat.succ y) n", "state_after": "case H\nn : \u2115\n\u22a2 Nat.sqrt n = Nat.rec 0 (fun x y => if Nat.succ x < Nat.succ y * Nat.succ y then y else Nat.succ y) n"}, {"tactic": "symm", "annotated_tactic": ["symm", []], "state_before": "case H\nn : \u2115\n\u22a2 Nat.sqrt n = Nat.rec 0 (fun x y => if Nat.succ x < Nat.succ y * Nat.succ y then y else Nat.succ y) n", "state_after": "case H\nn : \u2115\n\u22a2 Nat.rec 0 (fun x y => if Nat.succ x < Nat.succ y * Nat.succ y then y else Nat.succ y) n = Nat.sqrt n"}, {"tactic": "induction' n with n IH", "annotated_tactic": ["induction' n with n IH", []], "state_before": "case H\nn : \u2115\n\u22a2 Nat.rec 0 (fun x y => if Nat.succ x < Nat.succ y * Nat.succ y then y else Nat.succ y) n = Nat.sqrt n", "state_after": "case H.zero\n\n\u22a2 Nat.rec 0 (fun x y => if Nat.succ x < Nat.succ y * Nat.succ y then y else Nat.succ y) Nat.zero = Nat.sqrt Nat.zero\n\ncase H.succ\nn : \u2115\nIH : Nat.rec 0 (fun x y => if Nat.succ x < Nat.succ y * Nat.succ y then y else Nat.succ y) n = Nat.sqrt n\n\u22a2 Nat.rec 0 (fun x y => if Nat.succ x < Nat.succ y * Nat.succ y then y else Nat.succ y) (Nat.succ n) =\n    Nat.sqrt (Nat.succ n)"}, {"tactic": "dsimp", "annotated_tactic": ["dsimp", []], "state_before": "case H.succ\nn : \u2115\nIH : Nat.rec 0 (fun x y => if Nat.succ x < Nat.succ y * Nat.succ y then y else Nat.succ y) n = Nat.sqrt n\n\u22a2 Nat.rec 0 (fun x y => if Nat.succ x < Nat.succ y * Nat.succ y then y else Nat.succ y) (Nat.succ n) =\n    Nat.sqrt (Nat.succ n)", "state_after": "case H.succ\nn : \u2115\nIH : Nat.rec 0 (fun x y => if Nat.succ x < Nat.succ y * Nat.succ y then y else Nat.succ y) n = Nat.sqrt n\n\u22a2 (if\n        Nat.succ n <\n          Nat.succ (Nat.rec 0 (fun x y => if Nat.succ x < Nat.succ y * Nat.succ y then y else Nat.succ y) n) *\n            Nat.succ (Nat.rec 0 (fun x y => if Nat.succ x < Nat.succ y * Nat.succ y then y else Nat.succ y) n) then\n      Nat.rec 0 (fun x y => if Nat.succ x < Nat.succ y * Nat.succ y then y else Nat.succ y) n\n    else Nat.succ (Nat.rec 0 (fun x y => if Nat.succ x < Nat.succ y * Nat.succ y then y else Nat.succ y) n)) =\n    Nat.sqrt (Nat.succ n)"}, {"tactic": "rw [IH]", "annotated_tactic": ["rw [IH]", []], "state_before": "case H.succ\nn : \u2115\nIH : Nat.rec 0 (fun x y => if Nat.succ x < Nat.succ y * Nat.succ y then y else Nat.succ y) n = Nat.sqrt n\n\u22a2 (if\n        Nat.succ n <\n          Nat.succ (Nat.rec 0 (fun x y => if Nat.succ x < Nat.succ y * Nat.succ y then y else Nat.succ y) n) *\n            Nat.succ (Nat.rec 0 (fun x y => if Nat.succ x < Nat.succ y * Nat.succ y then y else Nat.succ y) n) then\n      Nat.rec 0 (fun x y => if Nat.succ x < Nat.succ y * Nat.succ y then y else Nat.succ y) n\n    else Nat.succ (Nat.rec 0 (fun x y => if Nat.succ x < Nat.succ y * Nat.succ y then y else Nat.succ y) n)) =\n    Nat.sqrt (Nat.succ n)", "state_after": "case H.succ\nn : \u2115\nIH : Nat.rec 0 (fun x y => if Nat.succ x < Nat.succ y * Nat.succ y then y else Nat.succ y) n = Nat.sqrt n\n\u22a2 (if Nat.succ n < Nat.succ (Nat.sqrt n) * Nat.succ (Nat.sqrt n) then Nat.sqrt n else Nat.succ (Nat.sqrt n)) =\n    Nat.sqrt (Nat.succ n)"}, {"tactic": "split_ifs with h", "annotated_tactic": ["split_ifs with h", []], "state_before": "case H.succ\nn : \u2115\nIH : Nat.rec 0 (fun x y => if Nat.succ x < Nat.succ y * Nat.succ y then y else Nat.succ y) n = Nat.sqrt n\n\u22a2 (if Nat.succ n < Nat.succ (Nat.sqrt n) * Nat.succ (Nat.sqrt n) then Nat.sqrt n else Nat.succ (Nat.sqrt n)) =\n    Nat.sqrt (Nat.succ n)", "state_after": "case pos\nn : \u2115\nIH : Nat.rec 0 (fun x y => if Nat.succ x < Nat.succ y * Nat.succ y then y else Nat.succ y) n = Nat.sqrt n\nh : Nat.succ n < Nat.succ (Nat.sqrt n) * Nat.succ (Nat.sqrt n)\n\u22a2 Nat.sqrt n = Nat.sqrt (Nat.succ n)\n\ncase neg\nn : \u2115\nIH : Nat.rec 0 (fun x y => if Nat.succ x < Nat.succ y * Nat.succ y then y else Nat.succ y) n = Nat.sqrt n\nh : \u00acNat.succ n < Nat.succ (Nat.sqrt n) * Nat.succ (Nat.sqrt n)\n\u22a2 Nat.succ (Nat.sqrt n) = Nat.sqrt (Nat.succ n)"}, {"tactic": "simp [H]", "annotated_tactic": ["simp [H]", []], "state_before": "H : \u2200 (n : \u2115), Nat.sqrt n = Nat.rec 0 (fun x y => if Nat.succ x < Nat.succ y * Nat.succ y then y else Nat.succ y) n\n\u22a2 Primrec' fun v => Nat.sqrt (Vector.head v)", "state_after": "H : \u2200 (n : \u2115), Nat.sqrt n = Nat.rec 0 (fun x y => if Nat.succ x < Nat.succ y * Nat.succ y then y else Nat.succ y) n\n\u22a2 Primrec' fun v =>\n    Nat.rec 0 (fun x y => if Nat.succ x < Nat.succ y * Nat.succ y then y else Nat.succ y) (Vector.head v)"}, {"tactic": "have :=\n  @prec' 1 _ _\n    (fun v => by\n      have x := v.head; have y := v.tail.head;\n        exact if x.succ < y.succ * y.succ then y else y.succ)\n    head (const 0) ?_", "annotated_tactic": ["have :=\n      @<a>prec'</a> 1 _ _\n        (fun v => by\n          have x := v.head; have y := v.tail.head;\n            exact if x.succ < y.succ * y.succ then y else y.succ)\n        <a>head</a> (<a>const</a> 0) ?_", [{"full_name": "Nat.Primrec'.prec'", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [1443, 9], "def_end_pos": [1443, 14]}, {"full_name": "Nat.Primrec'.head", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [1404, 9], "def_end_pos": [1404, 13]}, {"full_name": "Nat.Primrec'.const", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [1399, 9], "def_end_pos": [1399, 14]}]], "state_before": "H : \u2200 (n : \u2115), Nat.sqrt n = Nat.rec 0 (fun x y => if Nat.succ x < Nat.succ y * Nat.succ y then y else Nat.succ y) n\n\u22a2 Primrec' fun v =>\n    Nat.rec 0 (fun x y => if Nat.succ x < Nat.succ y * Nat.succ y then y else Nat.succ y) (Vector.head v)", "state_after": "case refine_2\nH : \u2200 (n : \u2115), Nat.sqrt n = Nat.rec 0 (fun x y => if Nat.succ x < Nat.succ y * Nat.succ y then y else Nat.succ y) n\nthis :\n  Primrec' fun v =>\n    Nat.rec 0\n      (fun y IH =>\n        (fun v =>\n            let_fun x := Vector.head v;\n            let_fun y := Vector.head (Vector.tail v);\n            if Nat.succ x < Nat.succ y * Nat.succ y then y else Nat.succ y)\n          (y ::\u1d65 IH ::\u1d65 v))\n      (Vector.head v)\n\u22a2 Primrec' fun v =>\n    Nat.rec 0 (fun x y => if Nat.succ x < Nat.succ y * Nat.succ y then y else Nat.succ y) (Vector.head v)\n\ncase refine_1\nH : \u2200 (n : \u2115), Nat.sqrt n = Nat.rec 0 (fun x y => if Nat.succ x < Nat.succ y * Nat.succ y then y else Nat.succ y) n\n\u22a2 Primrec' fun v =>\n    let_fun x := Vector.head v;\n    let_fun y := Vector.head (Vector.tail v);\n    if Nat.succ x < Nat.succ y * Nat.succ y then y else Nat.succ y"}, {"tactic": "have x1 : @Primrec' 3 fun v => v.head.succ := succ.comp\u2081 _ head", "annotated_tactic": ["have x1 : @<a>Primrec'</a> 3 fun v => v.head.succ := succ.comp\u2081 _ <a>head</a>", [{"full_name": "Nat.Primrec'", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [1361, 11], "def_end_pos": [1361, 19]}, {"full_name": "Nat.Primrec'.head", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [1404, 9], "def_end_pos": [1404, 13]}]], "state_before": "case refine_1\nH : \u2200 (n : \u2115), Nat.sqrt n = Nat.rec 0 (fun x y => if Nat.succ x < Nat.succ y * Nat.succ y then y else Nat.succ y) n\n\u22a2 Primrec' fun v =>\n    let_fun x := Vector.head v;\n    let_fun y := Vector.head (Vector.tail v);\n    if Nat.succ x < Nat.succ y * Nat.succ y then y else Nat.succ y", "state_after": "case refine_1\nH : \u2200 (n : \u2115), Nat.sqrt n = Nat.rec 0 (fun x y => if Nat.succ x < Nat.succ y * Nat.succ y then y else Nat.succ y) n\nx1 : Primrec' fun v => Nat.succ (Vector.head v)\n\u22a2 Primrec' fun v =>\n    let_fun x := Vector.head v;\n    let_fun y := Vector.head (Vector.tail v);\n    if Nat.succ x < Nat.succ y * Nat.succ y then y else Nat.succ y"}, {"tactic": "have y1 : @Primrec' 3 fun v => v.tail.head.succ := succ.comp\u2081 _ (tail head)", "annotated_tactic": ["have y1 : @<a>Primrec'</a> 3 fun v => v.tail.head.succ := succ.comp\u2081 _ (<a>tail</a> <a>head</a>)", [{"full_name": "Nat.Primrec'", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [1361, 11], "def_end_pos": [1361, 19]}, {"full_name": "Nat.Primrec'.tail", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [1408, 9], "def_end_pos": [1408, 13]}, {"full_name": "Nat.Primrec'.head", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [1404, 9], "def_end_pos": [1404, 13]}]], "state_before": "case refine_1\nH : \u2200 (n : \u2115), Nat.sqrt n = Nat.rec 0 (fun x y => if Nat.succ x < Nat.succ y * Nat.succ y then y else Nat.succ y) n\nx1 : Primrec' fun v => Nat.succ (Vector.head v)\n\u22a2 Primrec' fun v =>\n    let_fun x := Vector.head v;\n    let_fun y := Vector.head (Vector.tail v);\n    if Nat.succ x < Nat.succ y * Nat.succ y then y else Nat.succ y", "state_after": "case refine_1\nH : \u2200 (n : \u2115), Nat.sqrt n = Nat.rec 0 (fun x y => if Nat.succ x < Nat.succ y * Nat.succ y then y else Nat.succ y) n\nx1 : Primrec' fun v => Nat.succ (Vector.head v)\ny1 : Primrec' fun v => Nat.succ (Vector.head (Vector.tail v))\n\u22a2 Primrec' fun v =>\n    let_fun x := Vector.head v;\n    let_fun y := Vector.head (Vector.tail v);\n    if Nat.succ x < Nat.succ y * Nat.succ y then y else Nat.succ y"}, {"tactic": "exact if_lt x1 (mul.comp\u2082 _ y1 y1) (tail head) y1", "annotated_tactic": ["exact <a>if_lt</a> x1 (mul.comp\u2082 _ y1 y1) (<a>tail</a> <a>head</a>) y1", [{"full_name": "Nat.Primrec'.if_lt", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [1468, 9], "def_end_pos": [1468, 14]}, {"full_name": "Nat.Primrec'.tail", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [1408, 9], "def_end_pos": [1408, 13]}, {"full_name": "Nat.Primrec'.head", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [1404, 9], "def_end_pos": [1404, 13]}]], "state_before": "case refine_1\nH : \u2200 (n : \u2115), Nat.sqrt n = Nat.rec 0 (fun x y => if Nat.succ x < Nat.succ y * Nat.succ y then y else Nat.succ y) n\nx1 : Primrec' fun v => Nat.succ (Vector.head v)\ny1 : Primrec' fun v => Nat.succ (Vector.head (Vector.tail v))\n\u22a2 Primrec' fun v =>\n    let_fun x := Vector.head v;\n    let_fun y := Vector.head (Vector.tail v);\n    if Nat.succ x < Nat.succ y * Nat.succ y then y else Nat.succ y", "state_after": "no goals"}, {"tactic": "have x := v.head", "annotated_tactic": ["have x := v.head", []], "state_before": "H : \u2200 (n : \u2115), Nat.sqrt n = Nat.rec 0 (fun x y => if Nat.succ x < Nat.succ y * Nat.succ y then y else Nat.succ y) n\nv : Vector \u2115 (1 + 2)\n\u22a2 \u2115", "state_after": "H : \u2200 (n : \u2115), Nat.sqrt n = Nat.rec 0 (fun x y => if Nat.succ x < Nat.succ y * Nat.succ y then y else Nat.succ y) n\nv : Vector \u2115 (1 + 2)\nx : \u2115\n\u22a2 \u2115"}, {"tactic": "have y := v.tail.head", "annotated_tactic": ["have y := v.tail.head", []], "state_before": "H : \u2200 (n : \u2115), Nat.sqrt n = Nat.rec 0 (fun x y => if Nat.succ x < Nat.succ y * Nat.succ y then y else Nat.succ y) n\nv : Vector \u2115 (1 + 2)\nx : \u2115\n\u22a2 \u2115", "state_after": "H : \u2200 (n : \u2115), Nat.sqrt n = Nat.rec 0 (fun x y => if Nat.succ x < Nat.succ y * Nat.succ y then y else Nat.succ y) n\nv : Vector \u2115 (1 + 2)\nx y : \u2115\n\u22a2 \u2115"}, {"tactic": "exact if x.succ < y.succ * y.succ then y else y.succ", "annotated_tactic": ["exact if x.succ < y.succ * y.succ then y else y.succ", []], "state_before": "H : \u2200 (n : \u2115), Nat.sqrt n = Nat.rec 0 (fun x y => if Nat.succ x < Nat.succ y * Nat.succ y then y else Nat.succ y) n\nv : Vector \u2115 (1 + 2)\nx y : \u2115\n\u22a2 \u2115", "state_after": "no goals"}, {"tactic": "exact this", "annotated_tactic": ["exact this", []], "state_before": "case refine_2\nH : \u2200 (n : \u2115), Nat.sqrt n = Nat.rec 0 (fun x y => if Nat.succ x < Nat.succ y * Nat.succ y then y else Nat.succ y) n\nthis :\n  Primrec' fun v =>\n    Nat.rec 0\n      (fun y IH =>\n        (fun v =>\n            let_fun x := Vector.head v;\n            let_fun y := Vector.head (Vector.tail v);\n            if Nat.succ x < Nat.succ y * Nat.succ y then y else Nat.succ y)\n          (y ::\u1d65 IH ::\u1d65 v))\n      (Vector.head v)\n\u22a2 Primrec' fun v =>\n    Nat.rec 0 (fun x y => if Nat.succ x < Nat.succ y * Nat.succ y then y else Nat.succ y) (Vector.head v)", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case H.zero\n\n\u22a2 Nat.rec 0 (fun x y => if Nat.succ x < Nat.succ y * Nat.succ y then y else Nat.succ y) Nat.zero = Nat.sqrt Nat.zero", "state_after": "no goals"}, {"tactic": "exact le_antisymm (Nat.sqrt_le_sqrt (Nat.le_succ _)) (Nat.lt_succ_iff.1 <| Nat.sqrt_lt.2 h)", "annotated_tactic": ["exact <a>le_antisymm</a> (<a>Nat.sqrt_le_sqrt</a> (<a>Nat.le_succ</a> _)) (<a>Nat.lt_succ_iff</a>.1 <| <a>Nat.sqrt_lt</a>.2 h)", [{"full_name": "le_antisymm", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [188, 9], "def_end_pos": [188, 20]}, {"full_name": "Nat.sqrt_le_sqrt", "def_path": "Mathlib/Data/Nat/Sqrt.lean", "def_pos": [104, 9], "def_end_pos": [104, 21]}, {"full_name": "Nat.le_succ", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1599, 9], "def_end_pos": [1599, 20]}, {"full_name": "Nat.lt_succ_iff", "def_path": "Mathlib/Data/Nat/Basic.lean", "def_pos": [207, 9], "def_end_pos": [207, 20]}, {"full_name": "Nat.sqrt_lt", "def_path": "Mathlib/Data/Nat/Sqrt.lean", "def_pos": [92, 9], "def_end_pos": [92, 16]}]], "state_before": "case pos\nn : \u2115\nIH : Nat.rec 0 (fun x y => if Nat.succ x < Nat.succ y * Nat.succ y then y else Nat.succ y) n = Nat.sqrt n\nh : Nat.succ n < Nat.succ (Nat.sqrt n) * Nat.succ (Nat.sqrt n)\n\u22a2 Nat.sqrt n = Nat.sqrt (Nat.succ n)", "state_after": "no goals"}, {"tactic": "exact\n  Nat.eq_sqrt.2 \u27e8not_lt.1 h, Nat.sqrt_lt.1 <| Nat.lt_succ_iff.2 <| Nat.sqrt_succ_le_succ_sqrt _\u27e9", "annotated_tactic": ["exact\n      <a>Nat.eq_sqrt</a>.2 \u27e8<a>not_lt</a>.1 h, <a>Nat.sqrt_lt</a>.1 <| <a>Nat.lt_succ_iff</a>.2 <| <a>Nat.sqrt_succ_le_succ_sqrt</a> _\u27e9", [{"full_name": "Nat.eq_sqrt", "def_path": "Mathlib/Data/Nat/Sqrt.lean", "def_pos": [118, 9], "def_end_pos": [118, 16]}, {"full_name": "not_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [368, 9], "def_end_pos": [368, 15]}, {"full_name": "Nat.sqrt_lt", "def_path": "Mathlib/Data/Nat/Sqrt.lean", "def_pos": [92, 9], "def_end_pos": [92, 16]}, {"full_name": "Nat.lt_succ_iff", "def_path": "Mathlib/Data/Nat/Basic.lean", "def_pos": [207, 9], "def_end_pos": [207, 20]}, {"full_name": "Nat.sqrt_succ_le_succ_sqrt", "def_path": "Mathlib/Data/Nat/Sqrt.lean", "def_pos": [165, 9], "def_end_pos": [165, 31]}]], "state_before": "case neg\nn : \u2115\nIH : Nat.rec 0 (fun x y => if Nat.succ x < Nat.succ y * Nat.succ y then y else Nat.succ y) n = Nat.sqrt n\nh : \u00acNat.succ n < Nat.succ (Nat.sqrt n) * Nat.succ (Nat.sqrt n)\n\u22a2 Nat.succ (Nat.sqrt n) = Nat.sqrt (Nat.succ n)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/TMToPartrec.lean", "full_name": "Turing.PartrecToTM2.codeSupp_fix", "start": [1871, 1], "end": [1874, 28], "traced_tactics": [{"tactic": "simp [codeSupp, codeSupp', contSupp, Finset.union_assoc, Finset.union_left_comm,\n  Finset.union_left_idem]", "annotated_tactic": ["simp [<a>codeSupp</a>, <a>codeSupp'</a>, <a>contSupp</a>, <a>Finset.union_assoc</a>, <a>Finset.union_left_comm</a>,\n    <a>Finset.union_left_idem</a>]", [{"full_name": "Turing.PartrecToTM2.codeSupp", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1823, 5], "def_end_pos": [1823, 13]}, {"full_name": "Turing.PartrecToTM2.codeSupp'", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1777, 5], "def_end_pos": [1777, 14]}, {"full_name": "Turing.PartrecToTM2.contSupp", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1806, 5], "def_end_pos": [1806, 13]}, {"full_name": "Finset.union_assoc", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1433, 9], "def_end_pos": [1433, 20]}, {"full_name": "Finset.union_left_comm", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1456, 9], "def_end_pos": [1456, 24]}, {"full_name": "Finset.union_left_idem", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1752, 9], "def_end_pos": [1752, 24]}]], "state_before": "f : Code\nk : Cont'\n\u22a2 codeSupp (Code.fix f) k = trStmts\u2081 (trNormal (Code.fix f) k) \u222a codeSupp f (Cont'.fix f k)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Hausdorff.lean", "full_name": "MeasureTheory.OuterMeasure.IsMetric.borel_le_caratheodory", "start": [160, 1], "end": [228, 34], "traced_tactics": [{"tactic": "rw [borel_eq_generateFrom_isClosed]", "annotated_tactic": ["rw [<a>borel_eq_generateFrom_isClosed</a>]", [{"full_name": "borel_eq_generateFrom_isClosed", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [108, 9], "def_end_pos": [108, 39]}]], "state_before": "\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\n\u22a2 borel X \u2264 OuterMeasure.caratheodory \u03bc", "state_after": "\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\n\u22a2 MeasurableSpace.generateFrom {s | IsClosed s} \u2264 OuterMeasure.caratheodory \u03bc"}, {"tactic": "refine' MeasurableSpace.generateFrom_le fun t ht => \u03bc.isCaratheodory_iff_le.2 fun s => _", "annotated_tactic": ["refine' <a>MeasurableSpace.generateFrom_le</a> fun t ht => \u03bc.isCaratheodory_iff_le.2 fun s => _", [{"full_name": "MeasurableSpace.generateFrom_le", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [384, 9], "def_end_pos": [384, 24]}]], "state_before": "\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\n\u22a2 MeasurableSpace.generateFrom {s | IsClosed s} \u2264 OuterMeasure.caratheodory \u03bc", "state_after": "\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\n\u22a2 \u2191\u03bc (s \u2229 t) + \u2191\u03bc (s \\ t) \u2264 \u2191\u03bc s"}, {"tactic": "set S : \u2115 \u2192 Set X := fun n => {x \u2208 s | (\u2191n)\u207b\u00b9 \u2264 infEdist x t}", "annotated_tactic": ["set S : \u2115 \u2192 <a>Set</a> X := fun n => {x \u2208 s | (\u2191n)\u207b\u00b9 \u2264 <a>infEdist</a> x t}", [{"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}, {"full_name": "EMetric.infEdist", "def_path": "Mathlib/Topology/MetricSpace/HausdorffDistance.lean", "def_pos": [52, 5], "def_end_pos": [52, 13]}]], "state_before": "\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\n\u22a2 \u2191\u03bc (s \u2229 t) + \u2191\u03bc (s \\ t) \u2264 \u2191\u03bc s", "state_after": "\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\n\u22a2 \u2191\u03bc (s \u2229 t) + \u2191\u03bc (s \\ t) \u2264 \u2191\u03bc s"}, {"tactic": "have n0 : \u2200 {n : \u2115}, (n\u207b\u00b9 : \u211d\u22650\u221e) \u2260 0 := fun {n} => ENNReal.inv_ne_zero.2 (ENNReal.nat_ne_top _)", "annotated_tactic": ["have n0 : \u2200 {n : \u2115}, (n\u207b\u00b9 : \u211d\u22650\u221e) \u2260 0 := fun {n} => <a>ENNReal.inv_ne_zero</a>.2 (<a>ENNReal.nat_ne_top</a> _)", [{"full_name": "ENNReal.inv_ne_zero", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1469, 19], "def_end_pos": [1469, 30]}, {"full_name": "ENNReal.nat_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [717, 17], "def_end_pos": [717, 27]}]], "state_before": "\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\n\u22a2 \u2191\u03bc (s \u2229 t) + \u2191\u03bc (s \\ t) \u2264 \u2191\u03bc s", "state_after": "\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\n\u22a2 \u2191\u03bc (s \u2229 t) + \u2191\u03bc (s \\ t) \u2264 \u2191\u03bc s"}, {"tactic": "have Ssep : \u2200 n, IsMetricSeparated (S n) t := fun n =>\n  \u27e8n\u207b\u00b9, n0, fun x hx y hy => hx.2.trans <| infEdist_le_edist_of_mem hy\u27e9", "annotated_tactic": ["have Ssep : \u2200 n, <a>IsMetricSeparated</a> (S n) t := fun n =>\n    \u27e8n\u207b\u00b9, n0, fun x hx y hy => hx.2.<a>trans</a> <| <a>infEdist_le_edist_of_mem</a> hy\u27e9", [{"full_name": "IsMetricSeparated", "def_path": "Mathlib/Topology/MetricSpace/MetricSeparated.lean", "def_pos": [27, 5], "def_end_pos": [27, 22]}, {"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [120, 7], "def_end_pos": [120, 18]}, {"full_name": "EMetric.infEdist_le_edist_of_mem", "def_path": "Mathlib/Topology/MetricSpace/HausdorffDistance.lean", "def_pos": [83, 9], "def_end_pos": [83, 33]}]], "state_before": "\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\n\u22a2 \u2191\u03bc (s \u2229 t) + \u2191\u03bc (s \\ t) \u2264 \u2191\u03bc s", "state_after": "\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\n\u22a2 \u2191\u03bc (s \u2229 t) + \u2191\u03bc (s \\ t) \u2264 \u2191\u03bc s"}, {"tactic": "have Ssep' : \u2200 n, IsMetricSeparated (S n) (s \u2229 t) := fun n =>\n  (Ssep n).mono Subset.rfl (inter_subset_right _ _)", "annotated_tactic": ["have Ssep' : \u2200 n, <a>IsMetricSeparated</a> (S n) (s \u2229 t) := fun n =>\n    (Ssep n).<a>mono</a> <a>Subset.rfl</a> (<a>inter_subset_right</a> _ _)", [{"full_name": "IsMetricSeparated", "def_path": "Mathlib/Topology/MetricSpace/MetricSeparated.lean", "def_pos": [27, 5], "def_end_pos": [27, 22]}, {"full_name": "IsMetricSeparated.mono", "def_path": "Mathlib/Topology/MetricSpace/MetricSeparated.lean", "def_pos": [65, 9], "def_end_pos": [65, 13]}, {"full_name": "Set.Subset.rfl", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [357, 9], "def_end_pos": [357, 19]}, {"full_name": "Set.inter_subset_right", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [969, 9], "def_end_pos": [969, 27]}]], "state_before": "\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\n\u22a2 \u2191\u03bc (s \u2229 t) + \u2191\u03bc (s \\ t) \u2264 \u2191\u03bc s", "state_after": "\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\n\u22a2 \u2191\u03bc (s \u2229 t) + \u2191\u03bc (s \\ t) \u2264 \u2191\u03bc s"}, {"tactic": "have S_sub : \u2200 n, S n \u2286 s \\ t := fun n =>\n  subset_inter (inter_subset_left _ _) (Ssep n).subset_compl_right", "annotated_tactic": ["have S_sub : \u2200 n, S n \u2286 s \\ t := fun n =>\n    <a>subset_inter</a> (<a>inter_subset_left</a> _ _) (Ssep n).<a>subset_compl_right</a>", [{"full_name": "Set.subset_inter", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [972, 9], "def_end_pos": [972, 21]}, {"full_name": "Set.inter_subset_left", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [965, 9], "def_end_pos": [965, 26]}, {"full_name": "IsMetricSeparated.subset_compl_right", "def_path": "Mathlib/Topology/MetricSpace/MetricSeparated.lean", "def_pos": [60, 9], "def_end_pos": [60, 27]}]], "state_before": "\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\n\u22a2 \u2191\u03bc (s \u2229 t) + \u2191\u03bc (s \\ t) \u2264 \u2191\u03bc s", "state_after": "\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\n\u22a2 \u2191\u03bc (s \u2229 t) + \u2191\u03bc (s \\ t) \u2264 \u2191\u03bc s"}, {"tactic": "have hSs : \u2200 n, \u03bc (s \u2229 t) + \u03bc (S n) \u2264 \u03bc s := fun n =>\n  calc\n    \u03bc (s \u2229 t) + \u03bc (S n) = \u03bc (s \u2229 t \u222a S n) := Eq.symm <| hm _ _ <| (Ssep' n).symm\n    _ \u2264 \u03bc (s \u2229 t \u222a s \\ t) := \u03bc.mono <| union_subset_union_right _ <| S_sub n\n    _ = \u03bc s := by rw [inter_union_diff]", "annotated_tactic": ["have hSs : \u2200 n, \u03bc (s \u2229 t) + \u03bc (S n) \u2264 \u03bc s := fun n =>\n    calc\n      \u03bc (s \u2229 t) + \u03bc (S n) = \u03bc (s \u2229 t \u222a S n) := <a>Eq.symm</a> <| hm _ _ <| (Ssep' n).<a>symm</a>\n      _ \u2264 \u03bc (s \u2229 t \u222a s \\ t) := \u03bc.mono <| <a>union_subset_union_right</a> _ <| S_sub n\n      _ = \u03bc s := by rw [<a>inter_union_diff</a>]", [{"full_name": "Eq.symm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [310, 9], "def_end_pos": [310, 16]}, {"full_name": "IsMetricSeparated.symm", "def_path": "Mathlib/Topology/MetricSpace/MetricSeparated.lean", "def_pos": [36, 9], "def_end_pos": [36, 13]}, {"full_name": "Set.union_subset_union_right", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [856, 9], "def_end_pos": [856, 33]}, {"full_name": "Set.inter_union_diff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1889, 9], "def_end_pos": [1889, 25]}]], "state_before": "\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\n\u22a2 \u2191\u03bc (s \u2229 t) + \u2191\u03bc (s \\ t) \u2264 \u2191\u03bc s", "state_after": "\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\n\u22a2 \u2191\u03bc (s \u2229 t) + \u2191\u03bc (s \\ t) \u2264 \u2191\u03bc s"}, {"tactic": "have iUnion_S : \u22c3 n, S n = s \\ t := by\n  refine' Subset.antisymm (iUnion_subset S_sub) _\n  rintro x \u27e8hxs, hxt\u27e9\n  rw [mem_iff_infEdist_zero_of_closed ht] at hxt\n  rcases ENNReal.exists_inv_nat_lt hxt with \u27e8n, hn\u27e9\n  exact mem_iUnion.2 \u27e8n, hxs, hn.le\u27e9", "annotated_tactic": ["have iUnion_S : \u22c3 n, S n = s \\ t := by\n    refine' <a>Subset.antisymm</a> (<a>iUnion_subset</a> S_sub) _\n    rintro x \u27e8hxs, hxt\u27e9\n    rw [<a>mem_iff_infEdist_zero_of_closed</a> ht] at hxt\n    rcases <a>ENNReal.exists_inv_nat_lt</a> hxt with \u27e8n, hn\u27e9\n    exact <a>mem_iUnion</a>.2 \u27e8n, hxs, hn.le\u27e9", [{"full_name": "Set.Subset.antisymm", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [370, 9], "def_end_pos": [370, 24]}, {"full_name": "Set.iUnion_subset", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [390, 9], "def_end_pos": [390, 22]}, {"full_name": "EMetric.mem_iff_infEdist_zero_of_closed", "def_path": "Mathlib/Topology/MetricSpace/HausdorffDistance.lean", "def_pos": [159, 9], "def_end_pos": [159, 40]}, {"full_name": "ENNReal.exists_inv_nat_lt", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1870, 9], "def_end_pos": [1870, 26]}, {"full_name": "Set.mem_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [201, 9], "def_end_pos": [201, 19]}]], "state_before": "\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\n\u22a2 \u2191\u03bc (s \u2229 t) + \u2191\u03bc (s \\ t) \u2264 \u2191\u03bc s", "state_after": "\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\niUnion_S : \u22c3 n, S n = s \\ t\n\u22a2 \u2191\u03bc (s \u2229 t) + \u2191\u03bc (s \\ t) \u2264 \u2191\u03bc s"}, {"tactic": "by_cases htop : \u03bc (s \\ t) = \u221e", "annotated_tactic": ["by_cases htop : \u03bc (s \\ t) = \u221e", []], "state_before": "\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\niUnion_S : \u22c3 n, S n = s \\ t\n\u22a2 \u2191\u03bc (s \u2229 t) + \u2191\u03bc (s \\ t) \u2264 \u2191\u03bc s", "state_after": "case pos\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\niUnion_S : \u22c3 n, S n = s \\ t\nhtop : \u2191\u03bc (s \\ t) = \u22a4\n\u22a2 \u2191\u03bc (s \u2229 t) + \u2191\u03bc (s \\ t) \u2264 \u2191\u03bc s\n\ncase neg\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\niUnion_S : \u22c3 n, S n = s \\ t\nhtop : \u00ac\u2191\u03bc (s \\ t) = \u22a4\n\u22a2 \u2191\u03bc (s \u2229 t) + \u2191\u03bc (s \\ t) \u2264 \u2191\u03bc s"}, {"tactic": "suffices : \u03bc (\u22c3 n, S n) \u2264 \u2a06 n, \u03bc (S n)", "annotated_tactic": ["suffices : \u03bc (\u22c3 n, S n) \u2264 \u2a06 n, \u03bc (S n)", []], "state_before": "case neg\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\niUnion_S : \u22c3 n, S n = s \\ t\nhtop : \u00ac\u2191\u03bc (s \\ t) = \u22a4\n\u22a2 \u2191\u03bc (s \u2229 t) + \u2191\u03bc (s \\ t) \u2264 \u2191\u03bc s", "state_after": "case neg\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\niUnion_S : \u22c3 n, S n = s \\ t\nhtop : \u00ac\u2191\u03bc (s \\ t) = \u22a4\nthis : \u2191\u03bc (\u22c3 n, S n) \u2264 \u2a06 n, \u2191\u03bc (S n)\n\u22a2 \u2191\u03bc (s \u2229 t) + \u2191\u03bc (s \\ t) \u2264 \u2191\u03bc s\n\ncase this\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\niUnion_S : \u22c3 n, S n = s \\ t\nhtop : \u00ac\u2191\u03bc (s \\ t) = \u22a4\n\u22a2 \u2191\u03bc (\u22c3 n, S n) \u2264 \u2a06 n, \u2191\u03bc (S n)"}, {"tactic": "calc\n  \u03bc (s \u2229 t) + \u03bc (s \\ t) = \u03bc (s \u2229 t) + \u03bc (\u22c3 n, S n) := by rw [iUnion_S]\n  _ \u2264 \u03bc (s \u2229 t) + \u2a06 n, \u03bc (S n) := (add_le_add le_rfl this)\n  _ = \u2a06 n, \u03bc (s \u2229 t) + \u03bc (S n) := ENNReal.add_iSup\n  _ \u2264 \u03bc s := iSup_le hSs", "annotated_tactic": ["calc\n    \u03bc (s \u2229 t) + \u03bc (s \\ t) = \u03bc (s \u2229 t) + \u03bc (\u22c3 n, S n) := by rw [iUnion_S]\n    _ \u2264 \u03bc (s \u2229 t) + \u2a06 n, \u03bc (S n) := (<a>add_le_add</a> <a>le_rfl</a> this)\n    _ = \u2a06 n, \u03bc (s \u2229 t) + \u03bc (S n) := <a>ENNReal.add_iSup</a>\n    _ \u2264 \u03bc s := <a>iSup_le</a> hSs", [{"full_name": "add_le_add", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [205, 15], "def_end_pos": [205, 25]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}, {"full_name": "ENNReal.add_iSup", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [590, 9], "def_end_pos": [590, 17]}, {"full_name": "iSup_le", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [875, 9], "def_end_pos": [875, 16]}]], "state_before": "case neg\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\niUnion_S : \u22c3 n, S n = s \\ t\nhtop : \u00ac\u2191\u03bc (s \\ t) = \u22a4\nthis : \u2191\u03bc (\u22c3 n, S n) \u2264 \u2a06 n, \u2191\u03bc (S n)\n\u22a2 \u2191\u03bc (s \u2229 t) + \u2191\u03bc (s \\ t) \u2264 \u2191\u03bc s\n\ncase this\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\niUnion_S : \u22c3 n, S n = s \\ t\nhtop : \u00ac\u2191\u03bc (s \\ t) = \u22a4\n\u22a2 \u2191\u03bc (\u22c3 n, S n) \u2264 \u2a06 n, \u2191\u03bc (S n)", "state_after": "case this\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\niUnion_S : \u22c3 n, S n = s \\ t\nhtop : \u00ac\u2191\u03bc (s \\ t) = \u22a4\n\u22a2 \u2191\u03bc (\u22c3 n, S n) \u2264 \u2a06 n, \u2191\u03bc (S n)"}, {"tactic": "have : \u2200 n, S n \u2286 S (n + 1) := fun n x hx =>\n  \u27e8hx.1, le_trans (ENNReal.inv_le_inv.2 <| Nat.cast_le.2 n.le_succ) hx.2\u27e9", "annotated_tactic": ["have : \u2200 n, S n \u2286 S (n + 1) := fun n x hx =>\n    \u27e8hx.1, <a>le_trans</a> (<a>ENNReal.inv_le_inv</a>.2 <| <a>Nat.cast_le</a>.2 n.le_succ) hx.2\u27e9", [{"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}, {"full_name": "ENNReal.inv_le_inv", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1541, 19], "def_end_pos": [1541, 29]}, {"full_name": "Nat.cast_le", "def_path": "Mathlib/Data/Nat/Cast/Order.lean", "def_pos": [91, 9], "def_end_pos": [91, 16]}]], "state_before": "case this\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\niUnion_S : \u22c3 n, S n = s \\ t\nhtop : \u00ac\u2191\u03bc (s \\ t) = \u22a4\n\u22a2 \u2191\u03bc (\u22c3 n, S n) \u2264 \u2a06 n, \u2191\u03bc (S n)", "state_after": "case this\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\niUnion_S : \u22c3 n, S n = s \\ t\nhtop : \u00ac\u2191\u03bc (s \\ t) = \u22a4\nthis : \u2200 (n : \u2115), S n \u2286 S (n + 1)\n\u22a2 \u2191\u03bc (\u22c3 n, S n) \u2264 \u2a06 n, \u2191\u03bc (S n)"}, {"tactic": "rw [inter_union_diff]", "annotated_tactic": ["rw [<a>inter_union_diff</a>]", [{"full_name": "Set.inter_union_diff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1889, 9], "def_end_pos": [1889, 25]}]], "state_before": "\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nn : \u2115\n\u22a2 \u2191\u03bc (s \u2229 t \u222a s \\ t) = \u2191\u03bc s", "state_after": "no goals"}, {"tactic": "refine' Subset.antisymm (iUnion_subset S_sub) _", "annotated_tactic": ["refine' <a>Subset.antisymm</a> (<a>iUnion_subset</a> S_sub) _", [{"full_name": "Set.Subset.antisymm", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [370, 9], "def_end_pos": [370, 24]}, {"full_name": "Set.iUnion_subset", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [390, 9], "def_end_pos": [390, 22]}]], "state_before": "\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\n\u22a2 \u22c3 n, S n = s \\ t", "state_after": "\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\n\u22a2 s \\ t \u2286 \u22c3 n, S n"}, {"tactic": "rintro x \u27e8hxs, hxt\u27e9", "annotated_tactic": ["rintro x \u27e8hxs, hxt\u27e9", []], "state_before": "\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\n\u22a2 s \\ t \u2286 \u22c3 n, S n", "state_after": "case intro\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\nx : X\nhxs : x \u2208 s\nhxt : \u00acx \u2208 t\n\u22a2 x \u2208 \u22c3 n, S n"}, {"tactic": "rw [mem_iff_infEdist_zero_of_closed ht] at hxt", "annotated_tactic": ["rw [<a>mem_iff_infEdist_zero_of_closed</a> ht] at hxt", [{"full_name": "EMetric.mem_iff_infEdist_zero_of_closed", "def_path": "Mathlib/Topology/MetricSpace/HausdorffDistance.lean", "def_pos": [159, 9], "def_end_pos": [159, 40]}]], "state_before": "case intro\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\nx : X\nhxs : x \u2208 s\nhxt : \u00acx \u2208 t\n\u22a2 x \u2208 \u22c3 n, S n", "state_after": "case intro\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\nx : X\nhxs : x \u2208 s\nhxt : \u00acinfEdist x t = 0\n\u22a2 x \u2208 \u22c3 n, S n"}, {"tactic": "rcases ENNReal.exists_inv_nat_lt hxt with \u27e8n, hn\u27e9", "annotated_tactic": ["rcases <a>ENNReal.exists_inv_nat_lt</a> hxt with \u27e8n, hn\u27e9", [{"full_name": "ENNReal.exists_inv_nat_lt", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1870, 9], "def_end_pos": [1870, 26]}]], "state_before": "case intro\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\nx : X\nhxs : x \u2208 s\nhxt : \u00acinfEdist x t = 0\n\u22a2 x \u2208 \u22c3 n, S n", "state_after": "case intro.intro\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\nx : X\nhxs : x \u2208 s\nhxt : \u00acinfEdist x t = 0\nn : \u2115\nhn : (\u2191n)\u207b\u00b9 < infEdist x t\n\u22a2 x \u2208 \u22c3 n, S n"}, {"tactic": "exact mem_iUnion.2 \u27e8n, hxs, hn.le\u27e9", "annotated_tactic": ["exact <a>mem_iUnion</a>.2 \u27e8n, hxs, hn.le\u27e9", [{"full_name": "Set.mem_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [201, 9], "def_end_pos": [201, 19]}]], "state_before": "case intro.intro\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\nx : X\nhxs : x \u2208 s\nhxt : \u00acinfEdist x t = 0\nn : \u2115\nhn : (\u2191n)\u207b\u00b9 < infEdist x t\n\u22a2 x \u2208 \u22c3 n, S n", "state_after": "no goals"}, {"tactic": "rw [htop, add_top, \u2190 htop]", "annotated_tactic": ["rw [htop, <a>add_top</a>, \u2190 htop]", [{"full_name": "add_top", "def_path": "Mathlib/Algebra/Order/Monoid/Defs.lean", "def_pos": [110, 9], "def_end_pos": [110, 16]}]], "state_before": "case pos\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\niUnion_S : \u22c3 n, S n = s \\ t\nhtop : \u2191\u03bc (s \\ t) = \u22a4\n\u22a2 \u2191\u03bc (s \u2229 t) + \u2191\u03bc (s \\ t) \u2264 \u2191\u03bc s", "state_after": "case pos\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\niUnion_S : \u22c3 n, S n = s \\ t\nhtop : \u2191\u03bc (s \\ t) = \u22a4\n\u22a2 \u2191\u03bc (s \\ t) \u2264 \u2191\u03bc s"}, {"tactic": "exact \u03bc.mono (diff_subset _ _)", "annotated_tactic": ["exact \u03bc.mono (<a>diff_subset</a> _ _)", [{"full_name": "Set.diff_subset", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1845, 9], "def_end_pos": [1845, 20]}]], "state_before": "case pos\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\niUnion_S : \u22c3 n, S n = s \\ t\nhtop : \u2191\u03bc (s \\ t) = \u22a4\n\u22a2 \u2191\u03bc (s \\ t) \u2264 \u2191\u03bc s", "state_after": "no goals"}, {"tactic": "rw [iUnion_S]", "annotated_tactic": ["rw [iUnion_S]", []], "state_before": "\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\niUnion_S : \u22c3 n, S n = s \\ t\nhtop : \u00ac\u2191\u03bc (s \\ t) = \u22a4\nthis : \u2191\u03bc (\u22c3 n, S n) \u2264 \u2a06 n, \u2191\u03bc (S n)\n\u22a2 \u2191\u03bc (s \u2229 t) + \u2191\u03bc (s \\ t) = \u2191\u03bc (s \u2229 t) + \u2191\u03bc (\u22c3 n, S n)", "state_after": "no goals"}, {"tactic": "refine' (\u03bc.iUnion_nat_of_monotone_of_tsum_ne_top this _).le", "annotated_tactic": ["refine' (\u03bc.iUnion_nat_of_monotone_of_tsum_ne_top this _).<a>le</a>", [{"full_name": "Eq.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [159, 7], "def_end_pos": [159, 12]}]], "state_before": "case this\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\niUnion_S : \u22c3 n, S n = s \\ t\nhtop : \u00ac\u2191\u03bc (s \\ t) = \u22a4\nthis : \u2200 (n : \u2115), S n \u2286 S (n + 1)\n\u22a2 \u2191\u03bc (\u22c3 n, S n) \u2264 \u2a06 n, \u2191\u03bc (S n)", "state_after": "case this\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\niUnion_S : \u22c3 n, S n = s \\ t\nhtop : \u00ac\u2191\u03bc (s \\ t) = \u22a4\nthis : \u2200 (n : \u2115), S n \u2286 S (n + 1)\n\u22a2 \u2211' (k : \u2115), \u2191\u03bc (S (k + 1) \\ S k) \u2260 \u22a4"}, {"tactic": "clear this", "annotated_tactic": ["clear this", []], "state_before": "case this\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\niUnion_S : \u22c3 n, S n = s \\ t\nhtop : \u00ac\u2191\u03bc (s \\ t) = \u22a4\nthis : \u2200 (n : \u2115), S n \u2286 S (n + 1)\n\u22a2 \u2211' (k : \u2115), \u2191\u03bc (S (k + 1) \\ S k) \u2260 \u22a4", "state_after": "case this\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\niUnion_S : \u22c3 n, S n = s \\ t\nhtop : \u00ac\u2191\u03bc (s \\ t) = \u22a4\n\u22a2 \u2211' (k : \u2115), \u2191\u03bc (S (k + 1) \\ S k) \u2260 \u22a4"}, {"tactic": "rw [\u2190 tsum_even_add_odd ENNReal.summable ENNReal.summable, ENNReal.add_ne_top]", "annotated_tactic": ["rw [\u2190 <a>tsum_even_add_odd</a> <a>ENNReal.summable</a> <a>ENNReal.summable</a>, <a>ENNReal.add_ne_top</a>]", [{"full_name": "tsum_even_add_odd", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/Basic.lean", "def_pos": [793, 9], "def_end_pos": [793, 26]}, {"full_name": "ENNReal.summable", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [777, 19], "def_end_pos": [777, 27]}, {"full_name": "ENNReal.summable", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [777, 19], "def_end_pos": [777, 27]}, {"full_name": "ENNReal.add_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [574, 9], "def_end_pos": [574, 19]}]], "state_before": "case this\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\niUnion_S : \u22c3 n, S n = s \\ t\nhtop : \u00ac\u2191\u03bc (s \\ t) = \u22a4\n\u22a2 \u2211' (k : \u2115), \u2191\u03bc (S (k + 1) \\ S k) \u2260 \u22a4", "state_after": "case this\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\niUnion_S : \u22c3 n, S n = s \\ t\nhtop : \u00ac\u2191\u03bc (s \\ t) = \u22a4\n\u22a2 \u2211' (k : \u2115), \u2191\u03bc (S (2 * k + 1) \\ S (2 * k)) \u2260 \u22a4 \u2227 \u2211' (k : \u2115), \u2191\u03bc (S (2 * k + 1 + 1) \\ S (2 * k + 1)) \u2260 \u22a4"}, {"tactic": "suffices : \u2200 a, (\u2211' k : \u2115, \u03bc (S (2 * k + 1 + a) \\ S (2 * k + a))) \u2260 \u221e", "annotated_tactic": ["suffices : \u2200 a, (\u2211' k : \u2115, \u03bc (S (2 * k + 1 + a) \\ S (2 * k + a))) \u2260 \u221e", []], "state_before": "case this\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\niUnion_S : \u22c3 n, S n = s \\ t\nhtop : \u00ac\u2191\u03bc (s \\ t) = \u22a4\n\u22a2 \u2211' (k : \u2115), \u2191\u03bc (S (2 * k + 1) \\ S (2 * k)) \u2260 \u22a4 \u2227 \u2211' (k : \u2115), \u2191\u03bc (S (2 * k + 1 + 1) \\ S (2 * k + 1)) \u2260 \u22a4", "state_after": "case this\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\niUnion_S : \u22c3 n, S n = s \\ t\nhtop : \u00ac\u2191\u03bc (s \\ t) = \u22a4\nthis : \u2200 (a : \u2115), \u2211' (k : \u2115), \u2191\u03bc (S (2 * k + 1 + a) \\ S (2 * k + a)) \u2260 \u22a4\n\u22a2 \u2211' (k : \u2115), \u2191\u03bc (S (2 * k + 1) \\ S (2 * k)) \u2260 \u22a4 \u2227 \u2211' (k : \u2115), \u2191\u03bc (S (2 * k + 1 + 1) \\ S (2 * k + 1)) \u2260 \u22a4\n\ncase this\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\niUnion_S : \u22c3 n, S n = s \\ t\nhtop : \u00ac\u2191\u03bc (s \\ t) = \u22a4\n\u22a2 \u2200 (a : \u2115), \u2211' (k : \u2115), \u2191\u03bc (S (2 * k + 1 + a) \\ S (2 * k + a)) \u2260 \u22a4"}, {"tactic": "exact \u27e8by simpa using this 0, by simpa using this 1\u27e9", "annotated_tactic": ["exact \u27e8by simpa using this 0, by simpa using this 1\u27e9", []], "state_before": "case this\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\niUnion_S : \u22c3 n, S n = s \\ t\nhtop : \u00ac\u2191\u03bc (s \\ t) = \u22a4\nthis : \u2200 (a : \u2115), \u2211' (k : \u2115), \u2191\u03bc (S (2 * k + 1 + a) \\ S (2 * k + a)) \u2260 \u22a4\n\u22a2 \u2211' (k : \u2115), \u2191\u03bc (S (2 * k + 1) \\ S (2 * k)) \u2260 \u22a4 \u2227 \u2211' (k : \u2115), \u2191\u03bc (S (2 * k + 1 + 1) \\ S (2 * k + 1)) \u2260 \u22a4\n\ncase this\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\niUnion_S : \u22c3 n, S n = s \\ t\nhtop : \u00ac\u2191\u03bc (s \\ t) = \u22a4\n\u22a2 \u2200 (a : \u2115), \u2211' (k : \u2115), \u2191\u03bc (S (2 * k + 1 + a) \\ S (2 * k + a)) \u2260 \u22a4", "state_after": "case this\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\niUnion_S : \u22c3 n, S n = s \\ t\nhtop : \u00ac\u2191\u03bc (s \\ t) = \u22a4\n\u22a2 \u2200 (a : \u2115), \u2211' (k : \u2115), \u2191\u03bc (S (2 * k + 1 + a) \\ S (2 * k + a)) \u2260 \u22a4"}, {"tactic": "refine' fun r => ne_top_of_le_ne_top htop _", "annotated_tactic": ["refine' fun r => <a>ne_top_of_le_ne_top</a> htop _", [{"full_name": "ne_top_of_le_ne_top", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [194, 9], "def_end_pos": [194, 28]}]], "state_before": "case this\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\niUnion_S : \u22c3 n, S n = s \\ t\nhtop : \u00ac\u2191\u03bc (s \\ t) = \u22a4\n\u22a2 \u2200 (a : \u2115), \u2211' (k : \u2115), \u2191\u03bc (S (2 * k + 1 + a) \\ S (2 * k + a)) \u2260 \u22a4", "state_after": "case this\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\niUnion_S : \u22c3 n, S n = s \\ t\nhtop : \u00ac\u2191\u03bc (s \\ t) = \u22a4\nr : \u2115\n\u22a2 \u2211' (k : \u2115), \u2191\u03bc (S (2 * k + 1 + r) \\ S (2 * k + r)) \u2264 \u2191\u03bc (s \\ t)"}, {"tactic": "rw [\u2190 iUnion_S, ENNReal.tsum_eq_iSup_nat, iSup_le_iff]", "annotated_tactic": ["rw [\u2190 iUnion_S, <a>ENNReal.tsum_eq_iSup_nat</a>, <a>iSup_le_iff</a>]", [{"full_name": "ENNReal.tsum_eq_iSup_nat", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [843, 19], "def_end_pos": [843, 35]}, {"full_name": "iSup_le_iff", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [964, 9], "def_end_pos": [964, 20]}]], "state_before": "case this\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\niUnion_S : \u22c3 n, S n = s \\ t\nhtop : \u00ac\u2191\u03bc (s \\ t) = \u22a4\nr : \u2115\n\u22a2 \u2211' (k : \u2115), \u2191\u03bc (S (2 * k + 1 + r) \\ S (2 * k + r)) \u2264 \u2191\u03bc (s \\ t)", "state_after": "case this\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\niUnion_S : \u22c3 n, S n = s \\ t\nhtop : \u00ac\u2191\u03bc (s \\ t) = \u22a4\nr : \u2115\n\u22a2 \u2200 (i : \u2115), \u2211 a in Finset.range i, \u2191\u03bc (S (2 * a + 1 + r) \\ S (2 * a + r)) \u2264 \u2191\u03bc (\u22c3 n, S n)"}, {"tactic": "intro n", "annotated_tactic": ["intro n", []], "state_before": "case this\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\niUnion_S : \u22c3 n, S n = s \\ t\nhtop : \u00ac\u2191\u03bc (s \\ t) = \u22a4\nr : \u2115\n\u22a2 \u2200 (i : \u2115), \u2211 a in Finset.range i, \u2191\u03bc (S (2 * a + 1 + r) \\ S (2 * a + r)) \u2264 \u2191\u03bc (\u22c3 n, S n)", "state_after": "case this\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\niUnion_S : \u22c3 n, S n = s \\ t\nhtop : \u00ac\u2191\u03bc (s \\ t) = \u22a4\nr n : \u2115\n\u22a2 \u2211 a in Finset.range n, \u2191\u03bc (S (2 * a + 1 + r) \\ S (2 * a + r)) \u2264 \u2191\u03bc (\u22c3 n, S n)"}, {"tactic": "rw [\u2190 hm.finset_iUnion_of_pairwise_separated]", "annotated_tactic": ["rw [\u2190 hm.finset_iUnion_of_pairwise_separated]", []], "state_before": "case this\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\niUnion_S : \u22c3 n, S n = s \\ t\nhtop : \u00ac\u2191\u03bc (s \\ t) = \u22a4\nr n : \u2115\n\u22a2 \u2211 a in Finset.range n, \u2191\u03bc (S (2 * a + 1 + r) \\ S (2 * a + r)) \u2264 \u2191\u03bc (\u22c3 n, S n)", "state_after": "case this\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\niUnion_S : \u22c3 n, S n = s \\ t\nhtop : \u00ac\u2191\u03bc (s \\ t) = \u22a4\nr n : \u2115\n\u22a2 \u2191\u03bc (\u22c3 i \u2208 Finset.range n, S (2 * i + 1 + r) \\ S (2 * i + r)) \u2264 \u2191\u03bc (\u22c3 n, S n)\n\ncase this\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\niUnion_S : \u22c3 n, S n = s \\ t\nhtop : \u00ac\u2191\u03bc (s \\ t) = \u22a4\nr n : \u2115\n\u22a2 \u2200 (i : \u2115),\n    i \u2208 Finset.range n \u2192\n      \u2200 (j : \u2115),\n        j \u2208 Finset.range n \u2192\n          i \u2260 j \u2192 IsMetricSeparated (S (2 * i + 1 + r) \\ S (2 * i + r)) (S (2 * j + 1 + r) \\ S (2 * j + r))"}, {"tactic": "suffices : \u2200 i j, i < j \u2192 IsMetricSeparated (S (2 * i + 1 + r)) (s \\ S (2 * j + r))", "annotated_tactic": ["suffices : \u2200 i j, i < j \u2192 <a>IsMetricSeparated</a> (S (2 * i + 1 + r)) (s \\ S (2 * j + r))", [{"full_name": "IsMetricSeparated", "def_path": "Mathlib/Topology/MetricSpace/MetricSeparated.lean", "def_pos": [27, 5], "def_end_pos": [27, 22]}]], "state_before": "case this\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\niUnion_S : \u22c3 n, S n = s \\ t\nhtop : \u00ac\u2191\u03bc (s \\ t) = \u22a4\nr n : \u2115\n\u22a2 \u2200 (i : \u2115),\n    i \u2208 Finset.range n \u2192\n      \u2200 (j : \u2115),\n        j \u2208 Finset.range n \u2192\n          i \u2260 j \u2192 IsMetricSeparated (S (2 * i + 1 + r) \\ S (2 * i + r)) (S (2 * j + 1 + r) \\ S (2 * j + r))", "state_after": "case this\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\niUnion_S : \u22c3 n, S n = s \\ t\nhtop : \u00ac\u2191\u03bc (s \\ t) = \u22a4\nr n : \u2115\nthis : \u2200 (i j : \u2115), i < j \u2192 IsMetricSeparated (S (2 * i + 1 + r)) (s \\ S (2 * j + r))\n\u22a2 \u2200 (i : \u2115),\n    i \u2208 Finset.range n \u2192\n      \u2200 (j : \u2115),\n        j \u2208 Finset.range n \u2192\n          i \u2260 j \u2192 IsMetricSeparated (S (2 * i + 1 + r) \\ S (2 * i + r)) (S (2 * j + 1 + r) \\ S (2 * j + r))\n\ncase this\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\niUnion_S : \u22c3 n, S n = s \\ t\nhtop : \u00ac\u2191\u03bc (s \\ t) = \u22a4\nr n : \u2115\n\u22a2 \u2200 (i j : \u2115), i < j \u2192 IsMetricSeparated (S (2 * i + 1 + r)) (s \\ S (2 * j + r))"}, {"tactic": "exact fun i _ j _ hij =>\n  hij.lt_or_lt.elim\n    (fun h => (this i j h).mono (inter_subset_left _ _) fun x hx => by exact \u27e8hx.1.1, hx.2\u27e9)\n    fun h => (this j i h).symm.mono (fun x hx => by exact \u27e8hx.1.1, hx.2\u27e9) (inter_subset_left _ _)", "annotated_tactic": ["exact fun i _ j _ hij =>\n    hij.lt_or_lt.elim\n      (fun h => (this i j h).<a>mono</a> (<a>inter_subset_left</a> _ _) fun x hx => by exact \u27e8hx.1.1, hx.2\u27e9)\n      fun h => (this j i h).symm.mono (fun x hx => by exact \u27e8hx.1.1, hx.2\u27e9) (<a>inter_subset_left</a> _ _)", [{"full_name": "IsMetricSeparated.mono", "def_path": "Mathlib/Topology/MetricSpace/MetricSeparated.lean", "def_pos": [65, 9], "def_end_pos": [65, 13]}, {"full_name": "Set.inter_subset_left", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [965, 9], "def_end_pos": [965, 26]}, {"full_name": "Set.inter_subset_left", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [965, 9], "def_end_pos": [965, 26]}]], "state_before": "case this\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\niUnion_S : \u22c3 n, S n = s \\ t\nhtop : \u00ac\u2191\u03bc (s \\ t) = \u22a4\nr n : \u2115\nthis : \u2200 (i j : \u2115), i < j \u2192 IsMetricSeparated (S (2 * i + 1 + r)) (s \\ S (2 * j + r))\n\u22a2 \u2200 (i : \u2115),\n    i \u2208 Finset.range n \u2192\n      \u2200 (j : \u2115),\n        j \u2208 Finset.range n \u2192\n          i \u2260 j \u2192 IsMetricSeparated (S (2 * i + 1 + r) \\ S (2 * i + r)) (S (2 * j + 1 + r) \\ S (2 * j + r))\n\ncase this\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\niUnion_S : \u22c3 n, S n = s \\ t\nhtop : \u00ac\u2191\u03bc (s \\ t) = \u22a4\nr n : \u2115\n\u22a2 \u2200 (i j : \u2115), i < j \u2192 IsMetricSeparated (S (2 * i + 1 + r)) (s \\ S (2 * j + r))", "state_after": "case this\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\niUnion_S : \u22c3 n, S n = s \\ t\nhtop : \u00ac\u2191\u03bc (s \\ t) = \u22a4\nr n : \u2115\n\u22a2 \u2200 (i j : \u2115), i < j \u2192 IsMetricSeparated (S (2 * i + 1 + r)) (s \\ S (2 * j + r))"}, {"tactic": "intro i j hj", "annotated_tactic": ["intro i j hj", []], "state_before": "case this\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\niUnion_S : \u22c3 n, S n = s \\ t\nhtop : \u00ac\u2191\u03bc (s \\ t) = \u22a4\nr n : \u2115\n\u22a2 \u2200 (i j : \u2115), i < j \u2192 IsMetricSeparated (S (2 * i + 1 + r)) (s \\ S (2 * j + r))", "state_after": "case this\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\niUnion_S : \u22c3 n, S n = s \\ t\nhtop : \u00ac\u2191\u03bc (s \\ t) = \u22a4\nr n i j : \u2115\nhj : i < j\n\u22a2 IsMetricSeparated (S (2 * i + 1 + r)) (s \\ S (2 * j + r))"}, {"tactic": "have A : ((\u2191(2 * j + r))\u207b\u00b9 : \u211d\u22650\u221e) < (\u2191(2 * i + 1 + r))\u207b\u00b9 := by\n  rw [ENNReal.inv_lt_inv, Nat.cast_lt]; linarith", "annotated_tactic": ["have A : ((\u2191(2 * j + r))\u207b\u00b9 : \u211d\u22650\u221e) < (\u2191(2 * i + 1 + r))\u207b\u00b9 := by\n    rw [<a>ENNReal.inv_lt_inv</a>, <a>Nat.cast_lt</a>]; linarith", [{"full_name": "ENNReal.inv_lt_inv", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1528, 19], "def_end_pos": [1528, 29]}, {"full_name": "Nat.cast_lt", "def_path": "Mathlib/Data/Nat/Cast/Order.lean", "def_pos": [96, 9], "def_end_pos": [96, 16]}]], "state_before": "case this\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\niUnion_S : \u22c3 n, S n = s \\ t\nhtop : \u00ac\u2191\u03bc (s \\ t) = \u22a4\nr n i j : \u2115\nhj : i < j\n\u22a2 IsMetricSeparated (S (2 * i + 1 + r)) (s \\ S (2 * j + r))", "state_after": "case this\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\niUnion_S : \u22c3 n, S n = s \\ t\nhtop : \u00ac\u2191\u03bc (s \\ t) = \u22a4\nr n i j : \u2115\nhj : i < j\nA : (\u2191(2 * j + r))\u207b\u00b9 < (\u2191(2 * i + 1 + r))\u207b\u00b9\n\u22a2 IsMetricSeparated (S (2 * i + 1 + r)) (s \\ S (2 * j + r))"}, {"tactic": "refine' \u27e8(\u2191(2 * i + 1 + r))\u207b\u00b9 - (\u2191(2 * j + r))\u207b\u00b9, by simpa using A, fun x hx y hy => _\u27e9", "annotated_tactic": ["refine' \u27e8(\u2191(2 * i + 1 + r))\u207b\u00b9 - (\u2191(2 * j + r))\u207b\u00b9, by simpa using A, fun x hx y hy => _\u27e9", []], "state_before": "case this\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\niUnion_S : \u22c3 n, S n = s \\ t\nhtop : \u00ac\u2191\u03bc (s \\ t) = \u22a4\nr n i j : \u2115\nhj : i < j\nA : (\u2191(2 * j + r))\u207b\u00b9 < (\u2191(2 * i + 1 + r))\u207b\u00b9\n\u22a2 IsMetricSeparated (S (2 * i + 1 + r)) (s \\ S (2 * j + r))", "state_after": "case this\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\niUnion_S : \u22c3 n, S n = s \\ t\nhtop : \u00ac\u2191\u03bc (s \\ t) = \u22a4\nr n i j : \u2115\nhj : i < j\nA : (\u2191(2 * j + r))\u207b\u00b9 < (\u2191(2 * i + 1 + r))\u207b\u00b9\nx : X\nhx : x \u2208 S (2 * i + 1 + r)\ny : X\nhy : y \u2208 s \\ S (2 * j + r)\n\u22a2 (\u2191(2 * i + 1 + r))\u207b\u00b9 - (\u2191(2 * j + r))\u207b\u00b9 \u2264 edist x y"}, {"tactic": "have : infEdist y t < (\u2191(2 * j + r))\u207b\u00b9 := not_le.1 fun hle => hy.2 \u27e8hy.1, hle\u27e9", "annotated_tactic": ["have : <a>infEdist</a> y t < (\u2191(2 * j + r))\u207b\u00b9 := <a>not_le</a>.1 fun hle => hy.2 \u27e8hy.1, hle\u27e9", [{"full_name": "EMetric.infEdist", "def_path": "Mathlib/Topology/MetricSpace/HausdorffDistance.lean", "def_pos": [52, 5], "def_end_pos": [52, 13]}, {"full_name": "not_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [373, 9], "def_end_pos": [373, 15]}]], "state_before": "case this\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\niUnion_S : \u22c3 n, S n = s \\ t\nhtop : \u00ac\u2191\u03bc (s \\ t) = \u22a4\nr n i j : \u2115\nhj : i < j\nA : (\u2191(2 * j + r))\u207b\u00b9 < (\u2191(2 * i + 1 + r))\u207b\u00b9\nx : X\nhx : x \u2208 S (2 * i + 1 + r)\ny : X\nhy : y \u2208 s \\ S (2 * j + r)\n\u22a2 (\u2191(2 * i + 1 + r))\u207b\u00b9 - (\u2191(2 * j + r))\u207b\u00b9 \u2264 edist x y", "state_after": "case this\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\niUnion_S : \u22c3 n, S n = s \\ t\nhtop : \u00ac\u2191\u03bc (s \\ t) = \u22a4\nr n i j : \u2115\nhj : i < j\nA : (\u2191(2 * j + r))\u207b\u00b9 < (\u2191(2 * i + 1 + r))\u207b\u00b9\nx : X\nhx : x \u2208 S (2 * i + 1 + r)\ny : X\nhy : y \u2208 s \\ S (2 * j + r)\nthis : infEdist y t < (\u2191(2 * j + r))\u207b\u00b9\n\u22a2 (\u2191(2 * i + 1 + r))\u207b\u00b9 - (\u2191(2 * j + r))\u207b\u00b9 \u2264 edist x y"}, {"tactic": "rcases infEdist_lt_iff.mp this with \u27e8z, hzt, hyz\u27e9", "annotated_tactic": ["rcases infEdist_lt_iff.mp this with \u27e8z, hzt, hyz\u27e9", []], "state_before": "case this\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\niUnion_S : \u22c3 n, S n = s \\ t\nhtop : \u00ac\u2191\u03bc (s \\ t) = \u22a4\nr n i j : \u2115\nhj : i < j\nA : (\u2191(2 * j + r))\u207b\u00b9 < (\u2191(2 * i + 1 + r))\u207b\u00b9\nx : X\nhx : x \u2208 S (2 * i + 1 + r)\ny : X\nhy : y \u2208 s \\ S (2 * j + r)\nthis : infEdist y t < (\u2191(2 * j + r))\u207b\u00b9\n\u22a2 (\u2191(2 * i + 1 + r))\u207b\u00b9 - (\u2191(2 * j + r))\u207b\u00b9 \u2264 edist x y", "state_after": "case this.intro.intro\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\niUnion_S : \u22c3 n, S n = s \\ t\nhtop : \u00ac\u2191\u03bc (s \\ t) = \u22a4\nr n i j : \u2115\nhj : i < j\nA : (\u2191(2 * j + r))\u207b\u00b9 < (\u2191(2 * i + 1 + r))\u207b\u00b9\nx : X\nhx : x \u2208 S (2 * i + 1 + r)\ny : X\nhy : y \u2208 s \\ S (2 * j + r)\nthis : infEdist y t < (\u2191(2 * j + r))\u207b\u00b9\nz : X\nhzt : z \u2208 t\nhyz : edist y z < (\u2191(2 * j + r))\u207b\u00b9\n\u22a2 (\u2191(2 * i + 1 + r))\u207b\u00b9 - (\u2191(2 * j + r))\u207b\u00b9 \u2264 edist x y"}, {"tactic": "have hxz : (\u2191(2 * i + 1 + r))\u207b\u00b9 \u2264 edist x z := le_infEdist.1 hx.2 _ hzt", "annotated_tactic": ["have hxz : (\u2191(2 * i + 1 + r))\u207b\u00b9 \u2264 <a>edist</a> x z := <a>le_infEdist</a>.1 hx.2 _ hzt", [{"full_name": "EDist.edist", "def_path": "Mathlib/Topology/EMetricSpace/Basic.lean", "def_pos": [48, 3], "def_end_pos": [48, 8]}, {"full_name": "EMetric.le_infEdist", "def_path": "Mathlib/Topology/MetricSpace/HausdorffDistance.lean", "def_pos": [61, 9], "def_end_pos": [61, 20]}]], "state_before": "case this.intro.intro\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\niUnion_S : \u22c3 n, S n = s \\ t\nhtop : \u00ac\u2191\u03bc (s \\ t) = \u22a4\nr n i j : \u2115\nhj : i < j\nA : (\u2191(2 * j + r))\u207b\u00b9 < (\u2191(2 * i + 1 + r))\u207b\u00b9\nx : X\nhx : x \u2208 S (2 * i + 1 + r)\ny : X\nhy : y \u2208 s \\ S (2 * j + r)\nthis : infEdist y t < (\u2191(2 * j + r))\u207b\u00b9\nz : X\nhzt : z \u2208 t\nhyz : edist y z < (\u2191(2 * j + r))\u207b\u00b9\n\u22a2 (\u2191(2 * i + 1 + r))\u207b\u00b9 - (\u2191(2 * j + r))\u207b\u00b9 \u2264 edist x y", "state_after": "case this.intro.intro\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\niUnion_S : \u22c3 n, S n = s \\ t\nhtop : \u00ac\u2191\u03bc (s \\ t) = \u22a4\nr n i j : \u2115\nhj : i < j\nA : (\u2191(2 * j + r))\u207b\u00b9 < (\u2191(2 * i + 1 + r))\u207b\u00b9\nx : X\nhx : x \u2208 S (2 * i + 1 + r)\ny : X\nhy : y \u2208 s \\ S (2 * j + r)\nthis : infEdist y t < (\u2191(2 * j + r))\u207b\u00b9\nz : X\nhzt : z \u2208 t\nhyz : edist y z < (\u2191(2 * j + r))\u207b\u00b9\nhxz : (\u2191(2 * i + 1 + r))\u207b\u00b9 \u2264 edist x z\n\u22a2 (\u2191(2 * i + 1 + r))\u207b\u00b9 - (\u2191(2 * j + r))\u207b\u00b9 \u2264 edist x y"}, {"tactic": "apply ENNReal.le_of_add_le_add_right hyz.ne_top", "annotated_tactic": ["apply <a>ENNReal.le_of_add_le_add_right</a> hyz.ne_top", [{"full_name": "ENNReal.le_of_add_le_add_right", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [789, 19], "def_end_pos": [789, 41]}]], "state_before": "case this.intro.intro\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\niUnion_S : \u22c3 n, S n = s \\ t\nhtop : \u00ac\u2191\u03bc (s \\ t) = \u22a4\nr n i j : \u2115\nhj : i < j\nA : (\u2191(2 * j + r))\u207b\u00b9 < (\u2191(2 * i + 1 + r))\u207b\u00b9\nx : X\nhx : x \u2208 S (2 * i + 1 + r)\ny : X\nhy : y \u2208 s \\ S (2 * j + r)\nthis : infEdist y t < (\u2191(2 * j + r))\u207b\u00b9\nz : X\nhzt : z \u2208 t\nhyz : edist y z < (\u2191(2 * j + r))\u207b\u00b9\nhxz : (\u2191(2 * i + 1 + r))\u207b\u00b9 \u2264 edist x z\n\u22a2 (\u2191(2 * i + 1 + r))\u207b\u00b9 - (\u2191(2 * j + r))\u207b\u00b9 \u2264 edist x y", "state_after": "case this.intro.intro\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\niUnion_S : \u22c3 n, S n = s \\ t\nhtop : \u00ac\u2191\u03bc (s \\ t) = \u22a4\nr n i j : \u2115\nhj : i < j\nA : (\u2191(2 * j + r))\u207b\u00b9 < (\u2191(2 * i + 1 + r))\u207b\u00b9\nx : X\nhx : x \u2208 S (2 * i + 1 + r)\ny : X\nhy : y \u2208 s \\ S (2 * j + r)\nthis : infEdist y t < (\u2191(2 * j + r))\u207b\u00b9\nz : X\nhzt : z \u2208 t\nhyz : edist y z < (\u2191(2 * j + r))\u207b\u00b9\nhxz : (\u2191(2 * i + 1 + r))\u207b\u00b9 \u2264 edist x z\n\u22a2 (\u2191(2 * i + 1 + r))\u207b\u00b9 - (\u2191(2 * j + r))\u207b\u00b9 + edist y z \u2264 edist x y + edist y z"}, {"tactic": "refine' le_trans _ (edist_triangle _ _ _)", "annotated_tactic": ["refine' <a>le_trans</a> _ (<a>edist_triangle</a> _ _ _)", [{"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}, {"full_name": "PseudoEMetricSpace.edist_triangle", "def_path": "Mathlib/Topology/EMetricSpace/Basic.lean", "def_pos": [80, 3], "def_end_pos": [80, 17]}]], "state_before": "case this.intro.intro\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\niUnion_S : \u22c3 n, S n = s \\ t\nhtop : \u00ac\u2191\u03bc (s \\ t) = \u22a4\nr n i j : \u2115\nhj : i < j\nA : (\u2191(2 * j + r))\u207b\u00b9 < (\u2191(2 * i + 1 + r))\u207b\u00b9\nx : X\nhx : x \u2208 S (2 * i + 1 + r)\ny : X\nhy : y \u2208 s \\ S (2 * j + r)\nthis : infEdist y t < (\u2191(2 * j + r))\u207b\u00b9\nz : X\nhzt : z \u2208 t\nhyz : edist y z < (\u2191(2 * j + r))\u207b\u00b9\nhxz : (\u2191(2 * i + 1 + r))\u207b\u00b9 \u2264 edist x z\n\u22a2 (\u2191(2 * i + 1 + r))\u207b\u00b9 - (\u2191(2 * j + r))\u207b\u00b9 + edist y z \u2264 edist x y + edist y z", "state_after": "case this.intro.intro\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\niUnion_S : \u22c3 n, S n = s \\ t\nhtop : \u00ac\u2191\u03bc (s \\ t) = \u22a4\nr n i j : \u2115\nhj : i < j\nA : (\u2191(2 * j + r))\u207b\u00b9 < (\u2191(2 * i + 1 + r))\u207b\u00b9\nx : X\nhx : x \u2208 S (2 * i + 1 + r)\ny : X\nhy : y \u2208 s \\ S (2 * j + r)\nthis : infEdist y t < (\u2191(2 * j + r))\u207b\u00b9\nz : X\nhzt : z \u2208 t\nhyz : edist y z < (\u2191(2 * j + r))\u207b\u00b9\nhxz : (\u2191(2 * i + 1 + r))\u207b\u00b9 \u2264 edist x z\n\u22a2 (\u2191(2 * i + 1 + r))\u207b\u00b9 - (\u2191(2 * j + r))\u207b\u00b9 + edist y z \u2264 edist x z"}, {"tactic": "refine' (add_le_add le_rfl hyz.le).trans (Eq.trans_le _ hxz)", "annotated_tactic": ["refine' (<a>add_le_add</a> <a>le_rfl</a> hyz.le).<a>trans</a> (<a>Eq.trans_le</a> _ hxz)", [{"full_name": "add_le_add", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [205, 15], "def_end_pos": [205, 25]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}, {"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [120, 7], "def_end_pos": [120, 18]}, {"full_name": "Eq.trans_le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [219, 7], "def_end_pos": [219, 18]}]], "state_before": "case this.intro.intro\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\niUnion_S : \u22c3 n, S n = s \\ t\nhtop : \u00ac\u2191\u03bc (s \\ t) = \u22a4\nr n i j : \u2115\nhj : i < j\nA : (\u2191(2 * j + r))\u207b\u00b9 < (\u2191(2 * i + 1 + r))\u207b\u00b9\nx : X\nhx : x \u2208 S (2 * i + 1 + r)\ny : X\nhy : y \u2208 s \\ S (2 * j + r)\nthis : infEdist y t < (\u2191(2 * j + r))\u207b\u00b9\nz : X\nhzt : z \u2208 t\nhyz : edist y z < (\u2191(2 * j + r))\u207b\u00b9\nhxz : (\u2191(2 * i + 1 + r))\u207b\u00b9 \u2264 edist x z\n\u22a2 (\u2191(2 * i + 1 + r))\u207b\u00b9 - (\u2191(2 * j + r))\u207b\u00b9 + edist y z \u2264 edist x z", "state_after": "case this.intro.intro\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\niUnion_S : \u22c3 n, S n = s \\ t\nhtop : \u00ac\u2191\u03bc (s \\ t) = \u22a4\nr n i j : \u2115\nhj : i < j\nA : (\u2191(2 * j + r))\u207b\u00b9 < (\u2191(2 * i + 1 + r))\u207b\u00b9\nx : X\nhx : x \u2208 S (2 * i + 1 + r)\ny : X\nhy : y \u2208 s \\ S (2 * j + r)\nthis : infEdist y t < (\u2191(2 * j + r))\u207b\u00b9\nz : X\nhzt : z \u2208 t\nhyz : edist y z < (\u2191(2 * j + r))\u207b\u00b9\nhxz : (\u2191(2 * i + 1 + r))\u207b\u00b9 \u2264 edist x z\n\u22a2 (\u2191(2 * i + 1 + r))\u207b\u00b9 - (\u2191(2 * j + r))\u207b\u00b9 + (\u2191(2 * j + r))\u207b\u00b9 = (\u2191(2 * i + 1 + r))\u207b\u00b9"}, {"tactic": "rw [tsub_add_cancel_of_le A.le]", "annotated_tactic": ["rw [<a>tsub_add_cancel_of_le</a> A.le]", [{"full_name": "tsub_add_cancel_of_le", "def_path": "Mathlib/Algebra/Order/Sub/Canonical.lean", "def_pos": [30, 9], "def_end_pos": [30, 30]}]], "state_before": "case this.intro.intro\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\niUnion_S : \u22c3 n, S n = s \\ t\nhtop : \u00ac\u2191\u03bc (s \\ t) = \u22a4\nr n i j : \u2115\nhj : i < j\nA : (\u2191(2 * j + r))\u207b\u00b9 < (\u2191(2 * i + 1 + r))\u207b\u00b9\nx : X\nhx : x \u2208 S (2 * i + 1 + r)\ny : X\nhy : y \u2208 s \\ S (2 * j + r)\nthis : infEdist y t < (\u2191(2 * j + r))\u207b\u00b9\nz : X\nhzt : z \u2208 t\nhyz : edist y z < (\u2191(2 * j + r))\u207b\u00b9\nhxz : (\u2191(2 * i + 1 + r))\u207b\u00b9 \u2264 edist x z\n\u22a2 (\u2191(2 * i + 1 + r))\u207b\u00b9 - (\u2191(2 * j + r))\u207b\u00b9 + (\u2191(2 * j + r))\u207b\u00b9 = (\u2191(2 * i + 1 + r))\u207b\u00b9", "state_after": "no goals"}, {"tactic": "simpa using this 0", "annotated_tactic": ["simpa using this 0", []], "state_before": "\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\niUnion_S : \u22c3 n, S n = s \\ t\nhtop : \u00ac\u2191\u03bc (s \\ t) = \u22a4\nthis : \u2200 (a : \u2115), \u2211' (k : \u2115), \u2191\u03bc (S (2 * k + 1 + a) \\ S (2 * k + a)) \u2260 \u22a4\n\u22a2 \u2211' (k : \u2115), \u2191\u03bc (S (2 * k + 1) \\ S (2 * k)) \u2260 \u22a4", "state_after": "no goals"}, {"tactic": "simpa using this 1", "annotated_tactic": ["simpa using this 1", []], "state_before": "\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\niUnion_S : \u22c3 n, S n = s \\ t\nhtop : \u00ac\u2191\u03bc (s \\ t) = \u22a4\nthis : \u2200 (a : \u2115), \u2211' (k : \u2115), \u2191\u03bc (S (2 * k + 1 + a) \\ S (2 * k + a)) \u2260 \u22a4\n\u22a2 \u2211' (k : \u2115), \u2191\u03bc (S (2 * k + 1 + 1) \\ S (2 * k + 1)) \u2260 \u22a4", "state_after": "no goals"}, {"tactic": "exact \u03bc.mono (iUnion_subset fun i => iUnion_subset fun _ x hx => mem_iUnion.2 \u27e8_, hx.1\u27e9)", "annotated_tactic": ["exact \u03bc.mono (<a>iUnion_subset</a> fun i => <a>iUnion_subset</a> fun _ x hx => <a>mem_iUnion</a>.2 \u27e8_, hx.1\u27e9)", [{"full_name": "Set.iUnion_subset", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [390, 9], "def_end_pos": [390, 22]}, {"full_name": "Set.iUnion_subset", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [390, 9], "def_end_pos": [390, 22]}, {"full_name": "Set.mem_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [201, 9], "def_end_pos": [201, 19]}]], "state_before": "case this\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\niUnion_S : \u22c3 n, S n = s \\ t\nhtop : \u00ac\u2191\u03bc (s \\ t) = \u22a4\nr n : \u2115\n\u22a2 \u2191\u03bc (\u22c3 i \u2208 Finset.range n, S (2 * i + 1 + r) \\ S (2 * i + r)) \u2264 \u2191\u03bc (\u22c3 n, S n)", "state_after": "no goals"}, {"tactic": "exact \u27e8hx.1.1, hx.2\u27e9", "annotated_tactic": ["exact \u27e8hx.1.1, hx.2\u27e9", []], "state_before": "\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\niUnion_S : \u22c3 n, S n = s \\ t\nhtop : \u00ac\u2191\u03bc (s \\ t) = \u22a4\nr n : \u2115\nthis : \u2200 (i j : \u2115), i < j \u2192 IsMetricSeparated (S (2 * i + 1 + r)) (s \\ S (2 * j + r))\ni : \u2115\nx\u271d\u00b9 : i \u2208 Finset.range n\nj : \u2115\nx\u271d : j \u2208 Finset.range n\nhij : i \u2260 j\nh : i < j\nx : X\nhx : x \u2208 S (2 * j + 1 + r) \\ S (2 * j + r)\n\u22a2 x \u2208 s \\ S (2 * j + r)", "state_after": "no goals"}, {"tactic": "exact \u27e8hx.1.1, hx.2\u27e9", "annotated_tactic": ["exact \u27e8hx.1.1, hx.2\u27e9", []], "state_before": "\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\niUnion_S : \u22c3 n, S n = s \\ t\nhtop : \u00ac\u2191\u03bc (s \\ t) = \u22a4\nr n : \u2115\nthis : \u2200 (i j : \u2115), i < j \u2192 IsMetricSeparated (S (2 * i + 1 + r)) (s \\ S (2 * j + r))\ni : \u2115\nx\u271d\u00b9 : i \u2208 Finset.range n\nj : \u2115\nx\u271d : j \u2208 Finset.range n\nhij : i \u2260 j\nh : j < i\nx : X\nhx : x \u2208 S (2 * i + 1 + r) \\ S (2 * i + r)\n\u22a2 x \u2208 s \\ S (2 * i + r)", "state_after": "no goals"}, {"tactic": "rw [ENNReal.inv_lt_inv, Nat.cast_lt]", "annotated_tactic": ["rw [<a>ENNReal.inv_lt_inv</a>, <a>Nat.cast_lt</a>]", [{"full_name": "ENNReal.inv_lt_inv", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1528, 19], "def_end_pos": [1528, 29]}, {"full_name": "Nat.cast_lt", "def_path": "Mathlib/Data/Nat/Cast/Order.lean", "def_pos": [96, 9], "def_end_pos": [96, 16]}]], "state_before": "\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\niUnion_S : \u22c3 n, S n = s \\ t\nhtop : \u00ac\u2191\u03bc (s \\ t) = \u22a4\nr n i j : \u2115\nhj : i < j\n\u22a2 (\u2191(2 * j + r))\u207b\u00b9 < (\u2191(2 * i + 1 + r))\u207b\u00b9", "state_after": "\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\niUnion_S : \u22c3 n, S n = s \\ t\nhtop : \u00ac\u2191\u03bc (s \\ t) = \u22a4\nr n i j : \u2115\nhj : i < j\n\u22a2 2 * i + 1 + r < 2 * j + r"}, {"tactic": "linarith", "annotated_tactic": ["linarith", []], "state_before": "\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\niUnion_S : \u22c3 n, S n = s \\ t\nhtop : \u00ac\u2191\u03bc (s \\ t) = \u22a4\nr n i j : \u2115\nhj : i < j\n\u22a2 2 * i + 1 + r < 2 * j + r", "state_after": "no goals"}, {"tactic": "simpa using A", "annotated_tactic": ["simpa using A", []], "state_before": "\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\niUnion_S : \u22c3 n, S n = s \\ t\nhtop : \u00ac\u2191\u03bc (s \\ t) = \u22a4\nr n i j : \u2115\nhj : i < j\nA : (\u2191(2 * j + r))\u207b\u00b9 < (\u2191(2 * i + 1 + r))\u207b\u00b9\n\u22a2 (\u2191(2 * i + 1 + r))\u207b\u00b9 - (\u2191(2 * j + r))\u207b\u00b9 \u2260 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/ProbabilityMassFunction/Binomial.lean", "full_name": "PMF.binomial_one_eq_bernoulli", "start": [51, 1], "end": [53, 58], "traced_tactics": [{"tactic": "ext i", "annotated_tactic": ["ext i", []], "state_before": "p : \u211d\u22650\u221e\nh : p \u2264 1\n\u22a2 binomial p h 1 = map (fun x => bif x then 1 else 0) (bernoulli p h)", "state_after": "case h\np : \u211d\u22650\u221e\nh : p \u2264 1\ni : Fin (1 + 1)\n\u22a2 \u2191(binomial p h 1) i = \u2191(map (fun x => bif x then 1 else 0) (bernoulli p h)) i"}, {"tactic": "fin_cases i <;> simp [tsum_bool, binomial_apply]", "annotated_tactic": ["fin_cases i <;> simp [<a>tsum_bool</a>, <a>binomial_apply</a>]", [{"full_name": "tsum_bool", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/Basic.lean", "def_pos": [516, 9], "def_end_pos": [516, 18]}, {"full_name": "PMF.binomial_apply", "def_path": "Mathlib/Probability/ProbabilityMassFunction/Binomial.lean", "def_pos": [37, 9], "def_end_pos": [37, 23]}]], "state_before": "case h\np : \u211d\u22650\u221e\nh : p \u2264 1\ni : Fin (1 + 1)\n\u22a2 \u2191(binomial p h 1) i = \u2191(map (fun x => bif x then 1 else 0) (bernoulli p h)) i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "full_name": "MeasureTheory.set_integral_nonpos_le", "start": [804, 1], "end": [810, 61], "traced_tactics": [{"tactic": "rw [\u2190 integral_indicator hs, \u2190\n  integral_indicator (hf.measurableSet_le stronglyMeasurable_const)]", "annotated_tactic": ["rw [\u2190 <a>integral_indicator</a> hs, \u2190\n    <a>integral_indicator</a> (hf.measurableSet_le <a>stronglyMeasurable_const</a>)]", [{"full_name": "MeasureTheory.integral_indicator", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [169, 9], "def_end_pos": [169, 27]}, {"full_name": "MeasureTheory.integral_indicator", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [169, 9], "def_end_pos": [169, 27]}, {"full_name": "MeasureTheory.stronglyMeasurable_const", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [143, 9], "def_end_pos": [143, 33]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\ns\u271d s : Set \u03b1\nhs : MeasurableSet s\nhf : StronglyMeasurable f\nhfi : Integrable f\n\u22a2 \u222b (x : \u03b1) in {y | f y \u2264 0}, f x \u2202\u03bc \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\ns\u271d s : Set \u03b1\nhs : MeasurableSet s\nhf : StronglyMeasurable f\nhfi : Integrable f\n\u22a2 \u222b (x : \u03b1), indicator {a | f a \u2264 0} (fun x => f x) x \u2202\u03bc \u2264 \u222b (x : \u03b1), indicator s (fun x => f x) x \u2202\u03bc"}, {"tactic": "exact\n  integral_mono (hfi.indicator (hf.measurableSet_le stronglyMeasurable_const))\n    (hfi.indicator hs) (indicator_nonpos_le_indicator s f)", "annotated_tactic": ["exact\n    <a>integral_mono</a> (hfi.indicator (hf.measurableSet_le <a>stronglyMeasurable_const</a>))\n      (hfi.indicator hs) (<a>indicator_nonpos_le_indicator</a> s f)", [{"full_name": "MeasureTheory.integral_mono", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1351, 9], "def_end_pos": [1351, 22]}, {"full_name": "MeasureTheory.stronglyMeasurable_const", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [143, 9], "def_end_pos": [143, 33]}, {"full_name": "Set.indicator_nonpos_le_indicator", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [934, 9], "def_end_pos": [934, 38]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\ns\u271d s : Set \u03b1\nhs : MeasurableSet s\nhf : StronglyMeasurable f\nhfi : Integrable f\n\u22a2 \u222b (x : \u03b1), indicator {a | f a \u2264 0} (fun x => f x) x \u2202\u03bc \u2264 \u222b (x : \u03b1), indicator s (fun x => f x) x \u2202\u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/ZMod/Basic.lean", "full_name": "ZMod.addOrderOf_coe", "start": [111, 1], "end": [115, 86], "traced_tactics": [{"tactic": "cases' a with a", "annotated_tactic": ["cases' a with a", []], "state_before": "a n : \u2115\nn0 : n \u2260 0\n\u22a2 addOrderOf \u2191a = n / Nat.gcd n a", "state_after": "case zero\nn : \u2115\nn0 : n \u2260 0\n\u22a2 addOrderOf \u2191Nat.zero = n / Nat.gcd n Nat.zero\n\ncase succ\nn : \u2115\nn0 : n \u2260 0\na : \u2115\n\u22a2 addOrderOf \u2191(Nat.succ a) = n / Nat.gcd n (Nat.succ a)"}, {"tactic": "simp only [Nat.zero_eq, Nat.cast_zero, addOrderOf_zero, Nat.gcd_zero_right, Nat.pos_of_ne_zero n0,\n  Nat.div_self]", "annotated_tactic": ["simp only [<a>Nat.zero_eq</a>, <a>Nat.cast_zero</a>, <a>addOrderOf_zero</a>, <a>Nat.gcd_zero_right</a>, <a>Nat.pos_of_ne_zero</a> n0,\n    <a>Nat.div_self</a>]", [{"full_name": "Nat.zero_eq", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [83, 17], "def_end_pos": [83, 24]}, {"full_name": "Nat.cast_zero", "def_path": "Mathlib/Data/Nat/Cast/Defs.lean", "def_pos": [114, 9], "def_end_pos": [114, 18]}, {"full_name": "addOrderOf_zero", "def_path": "Mathlib/GroupTheory/OrderOfElement.lean", "def_pos": [226, 3], "def_end_pos": [226, 14]}, {"full_name": "Nat.gcd_zero_right", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Gcd.lean", "def_pos": [30, 17], "def_end_pos": [30, 31]}, {"full_name": "Nat.pos_of_ne_zero", "def_path": "lake-packages/std/Std/Data/Nat/Init/Lemmas.lean", "def_pos": [25, 19], "def_end_pos": [25, 33]}, {"full_name": "Nat.div_self", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [629, 19], "def_end_pos": [629, 27]}]], "state_before": "case zero\nn : \u2115\nn0 : n \u2260 0\n\u22a2 addOrderOf \u2191Nat.zero = n / Nat.gcd n Nat.zero\n\ncase succ\nn : \u2115\nn0 : n \u2260 0\na : \u2115\n\u22a2 addOrderOf \u2191(Nat.succ a) = n / Nat.gcd n (Nat.succ a)", "state_after": "case succ\nn : \u2115\nn0 : n \u2260 0\na : \u2115\n\u22a2 addOrderOf \u2191(Nat.succ a) = n / Nat.gcd n (Nat.succ a)"}, {"tactic": "rw [\u2190 Nat.smul_one_eq_coe, addOrderOf_nsmul' _ a.succ_ne_zero, ZMod.addOrderOf_one]", "annotated_tactic": ["rw [\u2190 <a>Nat.smul_one_eq_coe</a>, <a>addOrderOf_nsmul'</a> _ a.succ_ne_zero, <a>ZMod.addOrderOf_one</a>]", [{"full_name": "Nat.smul_one_eq_coe", "def_path": "Mathlib/Algebra/Module/Basic.lean", "def_pos": [758, 9], "def_end_pos": [758, 28]}, {"full_name": "addOrderOf_nsmul'", "def_path": "Mathlib/GroupTheory/OrderOfElement.lean", "def_pos": [360, 3], "def_end_pos": [360, 14]}, {"full_name": "ZMod.addOrderOf_one", "def_path": "Mathlib/Data/ZMod/Basic.lean", "def_pos": [104, 9], "def_end_pos": [104, 23]}]], "state_before": "case succ\nn : \u2115\nn0 : n \u2260 0\na : \u2115\n\u22a2 addOrderOf \u2191(Nat.succ a) = n / Nat.gcd n (Nat.succ a)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "full_name": "MeasureTheory.inducedOuterMeasure_eq_extend", "start": [1577, 1], "end": [1579, 85], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Function.lean", "full_name": "Set.InjOn.cancel_left", "start": [741, 1], "end": [743, 54], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "full_name": "IsUnit.integrable_smul_iff", "start": [1076, 1], "end": [1078, 85], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "full_name": "measure_closure_of_null_frontier", "start": [500, 1], "end": [502, 51], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Lattice.lean", "full_name": "Finset.min_singleton", "start": [1348, 1], "end": [1350, 19], "traced_tactics": [{"tactic": "rw [\u2190 insert_emptyc_eq]", "annotated_tactic": ["rw [\u2190 <a>insert_emptyc_eq</a>]", [{"full_name": "IsLawfulSingleton.insert_emptyc_eq", "def_path": "lake-packages/std/Std/Classes/SetNotation.lean", "def_pos": [116, 3], "def_end_pos": [116, 19]}]], "state_before": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Type u_5\n\u03ba : Type u_6\ninst\u271d : LinearOrder \u03b1\na : \u03b1\n\u22a2 Finset.min {a} = \u2191a", "state_after": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Type u_5\n\u03ba : Type u_6\ninst\u271d : LinearOrder \u03b1\na : \u03b1\n\u22a2 Finset.min (insert a \u2205) = \u2191a"}, {"tactic": "exact min_insert", "annotated_tactic": ["exact <a>min_insert</a>", [{"full_name": "Finset.min_insert", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [1343, 9], "def_end_pos": [1343, 19]}]], "state_before": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Type u_5\n\u03ba : Type u_6\ninst\u271d : LinearOrder \u03b1\na : \u03b1\n\u22a2 Finset.min (insert a \u2205) = \u2191a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean", "full_name": "MeasureTheory.Measure.tendsto_addHaar_inter_smul_zero_of_density_zero", "start": [715, 1], "end": [772, 34], "traced_tactics": [{"tactic": "refine' tendsto_order.2 \u27e8fun a' ha' => (ENNReal.not_lt_zero ha').elim, fun \u03b5 (\u03b5pos : 0 < \u03b5) => _\u27e9", "annotated_tactic": ["refine' <a>tendsto_order</a>.2 \u27e8fun a' ha' => (<a>ENNReal.not_lt_zero</a> ha').<a>elim</a>, fun \u03b5 (\u03b5pos : 0 < \u03b5) => _\u27e9", [{"full_name": "tendsto_order", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [919, 9], "def_end_pos": [919, 22]}, {"full_name": "ENNReal.not_lt_zero", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [782, 9], "def_end_pos": [782, 20]}, {"full_name": "False.elim", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [223, 21], "def_end_pos": [223, 31]}]], "state_before": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt : Set E\nht : MeasurableSet t\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\n\u22a2 Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)", "state_after": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt : Set E\nht : MeasurableSet t\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\n\u22a2 \u2200\u1da0 (b : \u211d) in \ud835\udcdd[Ioi 0] 0, \u2191\u2191\u03bc (s \u2229 ({x} + b \u2022 t)) / \u2191\u2191\u03bc ({x} + b \u2022 t) < \u03b5"}, {"tactic": "rcases eq_or_ne (\u03bc t) 0 with (h't | h't)", "annotated_tactic": ["rcases <a>eq_or_ne</a> (\u03bc t) 0 with (h't | h't)", [{"full_name": "eq_or_ne", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [209, 9], "def_end_pos": [209, 17]}]], "state_before": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt : Set E\nht : MeasurableSet t\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\n\u22a2 \u2200\u1da0 (b : \u211d) in \ud835\udcdd[Ioi 0] 0, \u2191\u2191\u03bc (s \u2229 ({x} + b \u2022 t)) / \u2191\u2191\u03bc ({x} + b \u2022 t) < \u03b5", "state_after": "case inl\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt : Set E\nht : MeasurableSet t\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nh't : \u2191\u2191\u03bc t = 0\n\u22a2 \u2200\u1da0 (b : \u211d) in \ud835\udcdd[Ioi 0] 0, \u2191\u2191\u03bc (s \u2229 ({x} + b \u2022 t)) / \u2191\u2191\u03bc ({x} + b \u2022 t) < \u03b5\n\ncase inr\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt : Set E\nht : MeasurableSet t\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nh't : \u2191\u2191\u03bc t \u2260 0\n\u22a2 \u2200\u1da0 (b : \u211d) in \ud835\udcdd[Ioi 0] 0, \u2191\u2191\u03bc (s \u2229 ({x} + b \u2022 t)) / \u2191\u2191\u03bc ({x} + b \u2022 t) < \u03b5"}, {"tactic": "obtain \u27e8n, npos, hn\u27e9 : \u2203 n : \u2115, 0 < n \u2227 \u03bc (t \\ closedBall 0 n) < \u03b5 / 2 * \u03bc t := by\n  have A :\n    Tendsto (fun n : \u2115 => \u03bc (t \\ closedBall 0 n)) atTop\n      (\ud835\udcdd (\u03bc (\u22c2 n : \u2115, t \\ closedBall 0 n))) := by\n    have N : \u2203 n : \u2115, \u03bc (t \\ closedBall 0 n) \u2260 \u221e :=\n      \u27e80, ((measure_mono (diff_subset t _)).trans_lt h''t.lt_top).ne\u27e9\n    refine' tendsto_measure_iInter (fun n \u21a6 ht.diff measurableSet_closedBall) (fun m n hmn \u21a6 _) N\n    exact diff_subset_diff Subset.rfl (closedBall_subset_closedBall (Nat.cast_le.2 hmn))\n  have : \u22c2 n : \u2115, t \\ closedBall 0 n = \u2205 := by\n    simp_rw [diff_eq, \u2190 inter_iInter, iInter_eq_compl_iUnion_compl, compl_compl,\n      iUnion_closedBall_nat, compl_univ, inter_empty]\n  simp only [this, measure_empty] at A\n  have I : 0 < \u03b5 / 2 * \u03bc t := ENNReal.mul_pos (ENNReal.half_pos \u03b5pos.ne').ne' h't\n  exact (Eventually.and (Ioi_mem_atTop 0) ((tendsto_order.1 A).2 _ I)).exists", "annotated_tactic": ["obtain \u27e8n, npos, hn\u27e9 : \u2203 n : \u2115, 0 < n \u2227 \u03bc (t \\ <a>closedBall</a> 0 n) < \u03b5 / 2 * \u03bc t := by\n    have A :\n      <a>Tendsto</a> (fun n : \u2115 => \u03bc (t \\ <a>closedBall</a> 0 n)) <a>atTop</a>\n        (\ud835\udcdd (\u03bc (\u22c2 n : \u2115, t \\ <a>closedBall</a> 0 n))) := by\n      have N : \u2203 n : \u2115, \u03bc (t \\ <a>closedBall</a> 0 n) \u2260 \u221e :=\n        \u27e80, ((<a>measure_mono</a> (<a>diff_subset</a> t _)).<a>trans_lt</a> h''t.lt_top).<a>ne</a>\u27e9\n      refine' <a>tendsto_measure_iInter</a> (fun n \u21a6 ht.diff <a>measurableSet_closedBall</a>) (fun m n hmn \u21a6 _) N\n      exact <a>diff_subset_diff</a> <a>Subset.rfl</a> (<a>closedBall_subset_closedBall</a> (<a>Nat.cast_le</a>.2 hmn))\n    have : \u22c2 n : \u2115, t \\ <a>closedBall</a> 0 n = \u2205 := by\n      simp_rw [<a>diff_eq</a>, \u2190 <a>inter_iInter</a>, <a>iInter_eq_compl_iUnion_compl</a>, <a>compl_compl</a>,\n        <a>iUnion_closedBall_nat</a>, <a>compl_univ</a>, <a>inter_empty</a>]\n    simp only [this, <a>measure_empty</a>] at A\n    have I : 0 < \u03b5 / 2 * \u03bc t := <a>ENNReal.mul_pos</a> (<a>ENNReal.half_pos</a> \u03b5pos.ne').<a>ne'</a> h't\n    exact (<a>Eventually.and</a> (<a>Ioi_mem_atTop</a> 0) ((<a>tendsto_order</a>.1 A).2 _ I)).<a>exists</a>", [{"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2939, 5], "def_end_pos": [2939, 12]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "MeasureTheory.measure_mono", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [193, 9], "def_end_pos": [193, 21]}, {"full_name": "Set.diff_subset", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1845, 9], "def_end_pos": [1845, 20]}, {"full_name": "LE.le.trans_lt", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [124, 7], "def_end_pos": [124, 21]}, {"full_name": "LT.lt.ne", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [152, 7], "def_end_pos": [152, 15]}, {"full_name": "MeasureTheory.tendsto_measure_iInter", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [538, 9], "def_end_pos": [538, 31]}, {"full_name": "measurableSet_closedBall", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [1681, 9], "def_end_pos": [1681, 33]}, {"full_name": "Set.diff_subset_diff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1904, 9], "def_end_pos": [1904, 25]}, {"full_name": "Set.Subset.rfl", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [357, 9], "def_end_pos": [357, 19]}, {"full_name": "Metric.closedBall_subset_closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [609, 9], "def_end_pos": [609, 37]}, {"full_name": "Nat.cast_le", "def_path": "Mathlib/Data/Nat/Cast/Order.lean", "def_pos": [91, 9], "def_end_pos": [91, 16]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "Set.diff_eq", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1814, 9], "def_end_pos": [1814, 16]}, {"full_name": "Set.inter_iInter", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [661, 9], "def_end_pos": [661, 21]}, {"full_name": "Set.iInter_eq_compl_iUnion_compl", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [631, 9], "def_end_pos": [631, 37]}, {"full_name": "compl_compl", "def_path": "Mathlib/Order/BooleanAlgebra.lean", "def_pos": [634, 9], "def_end_pos": [634, 20]}, {"full_name": "Metric.iUnion_closedBall_nat", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [663, 9], "def_end_pos": [663, 30]}, {"full_name": "Set.compl_univ", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1691, 9], "def_end_pos": [1691, 19]}, {"full_name": "Set.inter_empty", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [931, 9], "def_end_pos": [931, 20]}, {"full_name": "MeasureTheory.measure_empty", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [185, 9], "def_end_pos": [185, 22]}, {"full_name": "ENNReal.mul_pos", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [645, 9], "def_end_pos": [645, 16]}, {"full_name": "ENNReal.half_pos", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1796, 19], "def_end_pos": [1796, 27]}, {"full_name": "LT.lt.ne'", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [328, 9], "def_end_pos": [328, 12]}, {"full_name": "Filter.Eventually.and", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1103, 19], "def_end_pos": [1103, 33]}, {"full_name": "Filter.Ioi_mem_atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [61, 9], "def_end_pos": [61, 22]}, {"full_name": "tendsto_order", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [919, 9], "def_end_pos": [919, 22]}, {"full_name": "Filter.Eventually.exists", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1308, 9], "def_end_pos": [1308, 26]}]], "state_before": "case inr\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt : Set E\nht : MeasurableSet t\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nh't : \u2191\u2191\u03bc t \u2260 0\n\u22a2 \u2200\u1da0 (b : \u211d) in \ud835\udcdd[Ioi 0] 0, \u2191\u2191\u03bc (s \u2229 ({x} + b \u2022 t)) / \u2191\u2191\u03bc ({x} + b \u2022 t) < \u03b5", "state_after": "case inr.intro.intro\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt : Set E\nht : MeasurableSet t\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nh't : \u2191\u2191\u03bc t \u2260 0\nn : \u2115\nnpos : 0 < n\nhn : \u2191\u2191\u03bc (t \\ closedBall 0 \u2191n) < \u03b5 / 2 * \u2191\u2191\u03bc t\n\u22a2 \u2200\u1da0 (b : \u211d) in \ud835\udcdd[Ioi 0] 0, \u2191\u2191\u03bc (s \u2229 ({x} + b \u2022 t)) / \u2191\u2191\u03bc ({x} + b \u2022 t) < \u03b5"}, {"tactic": "have L :\n  Tendsto (fun r : \u211d => \u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 n))) / \u03bc ({x} + r \u2022 t)) (\ud835\udcdd[>] 0)\n    (\ud835\udcdd 0) :=\n  tendsto_addHaar_inter_smul_zero_of_density_zero_aux2 \u03bc s x h _ t h't n (Nat.cast_pos.2 npos)\n    (inter_subset_right _ _)", "annotated_tactic": ["have L :\n    <a>Tendsto</a> (fun r : \u211d => \u03bc (s \u2229 ({x} + r \u2022 (t \u2229 <a>closedBall</a> 0 n))) / \u03bc ({x} + r \u2022 t)) (\ud835\udcdd[>] 0)\n      (\ud835\udcdd 0) :=\n    <a>tendsto_addHaar_inter_smul_zero_of_density_zero_aux2</a> \u03bc s x h _ t h't n (<a>Nat.cast_pos</a>.2 npos)\n      (<a>inter_subset_right</a> _ _)", [{"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2939, 5], "def_end_pos": [2939, 12]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "MeasureTheory.Measure.tendsto_addHaar_inter_smul_zero_of_density_zero_aux2", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean", "def_pos": [680, 9], "def_end_pos": [680, 61]}, {"full_name": "Nat.cast_pos", "def_path": "Mathlib/Data/Nat/Cast/Order.lean", "def_pos": [72, 9], "def_end_pos": [72, 17]}, {"full_name": "Set.inter_subset_right", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [969, 9], "def_end_pos": [969, 27]}]], "state_before": "case inr.intro.intro\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt : Set E\nht : MeasurableSet t\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nh't : \u2191\u2191\u03bc t \u2260 0\nn : \u2115\nnpos : 0 < n\nhn : \u2191\u2191\u03bc (t \\ closedBall 0 \u2191n) < \u03b5 / 2 * \u2191\u2191\u03bc t\n\u22a2 \u2200\u1da0 (b : \u211d) in \ud835\udcdd[Ioi 0] 0, \u2191\u2191\u03bc (s \u2229 ({x} + b \u2022 t)) / \u2191\u2191\u03bc ({x} + b \u2022 t) < \u03b5", "state_after": "case inr.intro.intro\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt : Set E\nht : MeasurableSet t\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nh't : \u2191\u2191\u03bc t \u2260 0\nn : \u2115\nnpos : 0 < n\nhn : \u2191\u2191\u03bc (t \\ closedBall 0 \u2191n) < \u03b5 / 2 * \u2191\u2191\u03bc t\nL : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 \u2191n))) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\n\u22a2 \u2200\u1da0 (b : \u211d) in \ud835\udcdd[Ioi 0] 0, \u2191\u2191\u03bc (s \u2229 ({x} + b \u2022 t)) / \u2191\u2191\u03bc ({x} + b \u2022 t) < \u03b5"}, {"tactic": "filter_upwards [(tendsto_order.1 L).2 _ (ENNReal.half_pos \u03b5pos.ne'), self_mem_nhdsWithin]", "annotated_tactic": ["filter_upwards [(<a>tendsto_order</a>.1 L).2 _ (<a>ENNReal.half_pos</a> \u03b5pos.ne'), <a>self_mem_nhdsWithin</a>]", [{"full_name": "tendsto_order", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [919, 9], "def_end_pos": [919, 22]}, {"full_name": "ENNReal.half_pos", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1796, 19], "def_end_pos": [1796, 27]}, {"full_name": "self_mem_nhdsWithin", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [151, 9], "def_end_pos": [151, 28]}]], "state_before": "case inr.intro.intro\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt : Set E\nht : MeasurableSet t\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nh't : \u2191\u2191\u03bc t \u2260 0\nn : \u2115\nnpos : 0 < n\nhn : \u2191\u2191\u03bc (t \\ closedBall 0 \u2191n) < \u03b5 / 2 * \u2191\u2191\u03bc t\nL : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 \u2191n))) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\n\u22a2 \u2200\u1da0 (b : \u211d) in \ud835\udcdd[Ioi 0] 0, \u2191\u2191\u03bc (s \u2229 ({x} + b \u2022 t)) / \u2191\u2191\u03bc ({x} + b \u2022 t) < \u03b5", "state_after": "case h\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt : Set E\nht : MeasurableSet t\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nh't : \u2191\u2191\u03bc t \u2260 0\nn : \u2115\nnpos : 0 < n\nhn : \u2191\u2191\u03bc (t \\ closedBall 0 \u2191n) < \u03b5 / 2 * \u2191\u2191\u03bc t\nL : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 \u2191n))) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\n\u22a2 \u2200 (a : \u211d),\n    \u2191\u2191\u03bc (s \u2229 ({x} + a \u2022 (t \u2229 closedBall 0 \u2191n))) / \u2191\u2191\u03bc ({x} + a \u2022 t) < \u03b5 / 2 \u2192\n      a \u2208 Ioi 0 \u2192 \u2191\u2191\u03bc (s \u2229 ({x} + a \u2022 t)) / \u2191\u2191\u03bc ({x} + a \u2022 t) < \u03b5"}, {"tactic": "rintro r hr (rpos : 0 < r)", "annotated_tactic": ["rintro r hr (rpos : 0 < r)", []], "state_before": "case h\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt : Set E\nht : MeasurableSet t\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nh't : \u2191\u2191\u03bc t \u2260 0\nn : \u2115\nnpos : 0 < n\nhn : \u2191\u2191\u03bc (t \\ closedBall 0 \u2191n) < \u03b5 / 2 * \u2191\u2191\u03bc t\nL : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 \u2191n))) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\n\u22a2 \u2200 (a : \u211d),\n    \u2191\u2191\u03bc (s \u2229 ({x} + a \u2022 (t \u2229 closedBall 0 \u2191n))) / \u2191\u2191\u03bc ({x} + a \u2022 t) < \u03b5 / 2 \u2192\n      a \u2208 Ioi 0 \u2192 \u2191\u2191\u03bc (s \u2229 ({x} + a \u2022 t)) / \u2191\u2191\u03bc ({x} + a \u2022 t) < \u03b5", "state_after": "case h\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt : Set E\nht : MeasurableSet t\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nh't : \u2191\u2191\u03bc t \u2260 0\nn : \u2115\nnpos : 0 < n\nhn : \u2191\u2191\u03bc (t \\ closedBall 0 \u2191n) < \u03b5 / 2 * \u2191\u2191\u03bc t\nL : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 \u2191n))) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nr : \u211d\nhr : \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 \u2191n))) / \u2191\u2191\u03bc ({x} + r \u2022 t) < \u03b5 / 2\nrpos : 0 < r\n\u22a2 \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t) < \u03b5"}, {"tactic": "have I :\n  \u03bc (s \u2229 ({x} + r \u2022 t)) \u2264\n    \u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 n))) + \u03bc ({x} + r \u2022 (t \\ closedBall 0 n)) :=\n  calc\n    \u03bc (s \u2229 ({x} + r \u2022 t)) =\n        \u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 n)) \u222a s \u2229 ({x} + r \u2022 (t \\ closedBall 0 n))) :=\n      by rw [\u2190 inter_union_distrib_left, \u2190 add_union, \u2190 smul_set_union, inter_union_diff]\n    _ \u2264 \u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 n))) + \u03bc (s \u2229 ({x} + r \u2022 (t \\ closedBall 0 n))) :=\n      (measure_union_le _ _)\n    _ \u2264 \u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 n))) + \u03bc ({x} + r \u2022 (t \\ closedBall 0 n)) :=\n      add_le_add le_rfl (measure_mono (inter_subset_right _ _))", "annotated_tactic": ["have I :\n    \u03bc (s \u2229 ({x} + r \u2022 t)) \u2264\n      \u03bc (s \u2229 ({x} + r \u2022 (t \u2229 <a>closedBall</a> 0 n))) + \u03bc ({x} + r \u2022 (t \\ <a>closedBall</a> 0 n)) :=\n    calc\n      \u03bc (s \u2229 ({x} + r \u2022 t)) =\n          \u03bc (s \u2229 ({x} + r \u2022 (t \u2229 <a>closedBall</a> 0 n)) \u222a s \u2229 ({x} + r \u2022 (t \\ <a>closedBall</a> 0 n))) :=\n        by rw [\u2190 <a>inter_union_distrib_left</a>, \u2190 <a>add_union</a>, \u2190 <a>smul_set_union</a>, <a>inter_union_diff</a>]\n      _ \u2264 \u03bc (s \u2229 ({x} + r \u2022 (t \u2229 <a>closedBall</a> 0 n))) + \u03bc (s \u2229 ({x} + r \u2022 (t \\ <a>closedBall</a> 0 n))) :=\n        (<a>measure_union_le</a> _ _)\n      _ \u2264 \u03bc (s \u2229 ({x} + r \u2022 (t \u2229 <a>closedBall</a> 0 n))) + \u03bc ({x} + r \u2022 (t \\ <a>closedBall</a> 0 n)) :=\n        <a>add_le_add</a> <a>le_rfl</a> (<a>measure_mono</a> (<a>inter_subset_right</a> _ _))", [{"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "Set.inter_union_distrib_left", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1059, 9], "def_end_pos": [1059, 33]}, {"full_name": "Set.add_union", "def_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "def_pos": [445, 3], "def_end_pos": [445, 14]}, {"full_name": "Set.smul_set_union", "def_path": "Mathlib/Data/Set/Pointwise/SMul.lean", "def_pos": [366, 9], "def_end_pos": [366, 23]}, {"full_name": "Set.inter_union_diff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1889, 9], "def_end_pos": [1889, 25]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "MeasureTheory.measure_union_le", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [298, 9], "def_end_pos": [298, 25]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "add_le_add", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [205, 15], "def_end_pos": [205, 25]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}, {"full_name": "MeasureTheory.measure_mono", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [193, 9], "def_end_pos": [193, 21]}, {"full_name": "Set.inter_subset_right", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [969, 9], "def_end_pos": [969, 27]}]], "state_before": "case h\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt : Set E\nht : MeasurableSet t\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nh't : \u2191\u2191\u03bc t \u2260 0\nn : \u2115\nnpos : 0 < n\nhn : \u2191\u2191\u03bc (t \\ closedBall 0 \u2191n) < \u03b5 / 2 * \u2191\u2191\u03bc t\nL : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 \u2191n))) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nr : \u211d\nhr : \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 \u2191n))) / \u2191\u2191\u03bc ({x} + r \u2022 t) < \u03b5 / 2\nrpos : 0 < r\n\u22a2 \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t) < \u03b5", "state_after": "case h\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt : Set E\nht : MeasurableSet t\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nh't : \u2191\u2191\u03bc t \u2260 0\nn : \u2115\nnpos : 0 < n\nhn : \u2191\u2191\u03bc (t \\ closedBall 0 \u2191n) < \u03b5 / 2 * \u2191\u2191\u03bc t\nL : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 \u2191n))) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nr : \u211d\nhr : \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 \u2191n))) / \u2191\u2191\u03bc ({x} + r \u2022 t) < \u03b5 / 2\nrpos : 0 < r\nI : \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) \u2264 \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 \u2191n))) + \u2191\u2191\u03bc ({x} + r \u2022 (t \\ closedBall 0 \u2191n))\n\u22a2 \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t) < \u03b5"}, {"tactic": "calc\n  \u03bc (s \u2229 ({x} + r \u2022 t)) / \u03bc ({x} + r \u2022 t) \u2264\n      (\u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 n))) + \u03bc ({x} + r \u2022 (t \\ closedBall 0 n))) /\n        \u03bc ({x} + r \u2022 t) :=\n    mul_le_mul_right' I _\n  _ < \u03b5 / 2 + \u03b5 / 2 := by\n    rw [ENNReal.add_div]\n    apply ENNReal.add_lt_add hr _\n    rwa [addHaar_singleton_add_smul_div_singleton_add_smul \u03bc rpos.ne',\n      ENNReal.div_lt_iff (Or.inl h't) (Or.inl h''t)]\n  _ = \u03b5 := ENNReal.add_halves _", "annotated_tactic": ["calc\n    \u03bc (s \u2229 ({x} + r \u2022 t)) / \u03bc ({x} + r \u2022 t) \u2264\n        (\u03bc (s \u2229 ({x} + r \u2022 (t \u2229 <a>closedBall</a> 0 n))) + \u03bc ({x} + r \u2022 (t \\ <a>closedBall</a> 0 n))) /\n          \u03bc ({x} + r \u2022 t) :=\n      <a>mul_le_mul_right'</a> I _\n    _ < \u03b5 / 2 + \u03b5 / 2 := by\n      rw [<a>ENNReal.add_div</a>]\n      apply <a>ENNReal.add_lt_add</a> hr _\n      rwa [<a>addHaar_singleton_add_smul_div_singleton_add_smul</a> \u03bc rpos.ne',\n        <a>ENNReal.div_lt_iff</a> (<a>Or.inl</a> h't) (<a>Or.inl</a> h''t)]\n    _ = \u03b5 := <a>ENNReal.add_halves</a> _", [{"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "mul_le_mul_right'", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [67, 9], "def_end_pos": [67, 26]}, {"full_name": "ENNReal.add_div", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1738, 19], "def_end_pos": [1738, 26]}, {"full_name": "ENNReal.add_lt_add", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [931, 19], "def_end_pos": [931, 29]}, {"full_name": "MeasureTheory.Measure.addHaar_singleton_add_smul_div_singleton_add_smul", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean", "def_pos": [529, 9], "def_end_pos": [529, 58]}, {"full_name": "ENNReal.div_lt_iff", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1658, 19], "def_end_pos": [1658, 29]}, {"full_name": "Or.inl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [517, 5], "def_end_pos": [517, 8]}, {"full_name": "Or.inl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [517, 5], "def_end_pos": [517, 8]}, {"full_name": "ENNReal.add_halves", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1781, 19], "def_end_pos": [1781, 29]}]], "state_before": "case h\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt : Set E\nht : MeasurableSet t\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nh't : \u2191\u2191\u03bc t \u2260 0\nn : \u2115\nnpos : 0 < n\nhn : \u2191\u2191\u03bc (t \\ closedBall 0 \u2191n) < \u03b5 / 2 * \u2191\u2191\u03bc t\nL : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 \u2191n))) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nr : \u211d\nhr : \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 \u2191n))) / \u2191\u2191\u03bc ({x} + r \u2022 t) < \u03b5 / 2\nrpos : 0 < r\nI : \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) \u2264 \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 \u2191n))) + \u2191\u2191\u03bc ({x} + r \u2022 (t \\ closedBall 0 \u2191n))\n\u22a2 \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t) < \u03b5", "state_after": "no goals"}, {"tactic": "apply eventually_of_forall fun r => ?_", "annotated_tactic": ["apply <a>eventually_of_forall</a> fun r => ?_", [{"full_name": "Filter.eventually_of_forall", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1111, 9], "def_end_pos": [1111, 29]}]], "state_before": "case inl\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt : Set E\nht : MeasurableSet t\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nh't : \u2191\u2191\u03bc t = 0\n\u22a2 \u2200\u1da0 (b : \u211d) in \ud835\udcdd[Ioi 0] 0, \u2191\u2191\u03bc (s \u2229 ({x} + b \u2022 t)) / \u2191\u2191\u03bc ({x} + b \u2022 t) < \u03b5", "state_after": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt : Set E\nht : MeasurableSet t\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nh't : \u2191\u2191\u03bc t = 0\nr : \u211d\n\u22a2 \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t) < \u03b5"}, {"tactic": "suffices H : \u03bc (s \u2229 ({x} + r \u2022 t)) = 0", "annotated_tactic": ["suffices H : \u03bc (s \u2229 ({x} + r \u2022 t)) = 0", []], "state_before": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt : Set E\nht : MeasurableSet t\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nh't : \u2191\u2191\u03bc t = 0\nr : \u211d\n\u22a2 \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t) < \u03b5", "state_after": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt : Set E\nht : MeasurableSet t\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nh't : \u2191\u2191\u03bc t = 0\nr : \u211d\nH : \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) = 0\n\u22a2 \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t) < \u03b5\n\ncase H\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt : Set E\nht : MeasurableSet t\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nh't : \u2191\u2191\u03bc t = 0\nr : \u211d\n\u22a2 \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) = 0"}, {"tactic": "apply le_antisymm _ (zero_le _)", "annotated_tactic": ["apply <a>le_antisymm</a> _ (<a>zero_le</a> _)", [{"full_name": "le_antisymm", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [188, 9], "def_end_pos": [188, 20]}, {"full_name": "zero_le", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [217, 30], "def_end_pos": [217, 37]}]], "state_before": "case H\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt : Set E\nht : MeasurableSet t\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nh't : \u2191\u2191\u03bc t = 0\nr : \u211d\n\u22a2 \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) = 0", "state_after": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt : Set E\nht : MeasurableSet t\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nh't : \u2191\u2191\u03bc t = 0\nr : \u211d\n\u22a2 \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) \u2264 0"}, {"tactic": "calc\n  \u03bc (s \u2229 ({x} + r \u2022 t)) \u2264 \u03bc ({x} + r \u2022 t) := measure_mono (inter_subset_right _ _)\n  _ = 0 := by\n    simp only [h't, addHaar_smul, image_add_left, measure_preimage_add, singleton_add,\n      mul_zero]", "annotated_tactic": ["calc\n      \u03bc (s \u2229 ({x} + r \u2022 t)) \u2264 \u03bc ({x} + r \u2022 t) := <a>measure_mono</a> (<a>inter_subset_right</a> _ _)\n      _ = 0 := by\n        simp only [h't, <a>addHaar_smul</a>, <a>image_add_left</a>, <a>measure_preimage_add</a>, <a>singleton_add</a>,\n          <a>mul_zero</a>]", [{"full_name": "MeasureTheory.measure_mono", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [193, 9], "def_end_pos": [193, 21]}, {"full_name": "Set.inter_subset_right", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [969, 9], "def_end_pos": [969, 27]}, {"full_name": "MeasureTheory.Measure.addHaar_smul", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean", "def_pos": [371, 9], "def_end_pos": [371, 21]}, {"full_name": "Set.image_add_left", "def_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "def_pos": [1198, 3], "def_end_pos": [1198, 14]}, {"full_name": "MeasureTheory.measure_preimage_add", "def_path": "Mathlib/MeasureTheory/Group/Measure.lean", "def_pos": [317, 3], "def_end_pos": [317, 14]}, {"full_name": "Set.singleton_add", "def_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "def_pos": [402, 3], "def_end_pos": [402, 14]}, {"full_name": "MulZeroClass.mul_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [38, 3], "def_end_pos": [38, 11]}]], "state_before": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt : Set E\nht : MeasurableSet t\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nh't : \u2191\u2191\u03bc t = 0\nr : \u211d\n\u22a2 \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) \u2264 0", "state_after": "no goals"}, {"tactic": "rw [H]", "annotated_tactic": ["rw [H]", []], "state_before": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt : Set E\nht : MeasurableSet t\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nh't : \u2191\u2191\u03bc t = 0\nr : \u211d\nH : \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) = 0\n\u22a2 \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t) < \u03b5", "state_after": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt : Set E\nht : MeasurableSet t\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nh't : \u2191\u2191\u03bc t = 0\nr : \u211d\nH : \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) = 0\n\u22a2 0 / \u2191\u2191\u03bc ({x} + r \u2022 t) < \u03b5"}, {"tactic": "simpa only [ENNReal.zero_div] using \u03b5pos", "annotated_tactic": ["simpa only [<a>ENNReal.zero_div</a>] using \u03b5pos", [{"full_name": "ENNReal.zero_div", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1604, 27], "def_end_pos": [1604, 35]}]], "state_before": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt : Set E\nht : MeasurableSet t\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nh't : \u2191\u2191\u03bc t = 0\nr : \u211d\nH : \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) = 0\n\u22a2 0 / \u2191\u2191\u03bc ({x} + r \u2022 t) < \u03b5", "state_after": "no goals"}, {"tactic": "simp only [h't, addHaar_smul, image_add_left, measure_preimage_add, singleton_add,\n  mul_zero]", "annotated_tactic": ["simp only [h't, <a>addHaar_smul</a>, <a>image_add_left</a>, <a>measure_preimage_add</a>, <a>singleton_add</a>,\n          <a>mul_zero</a>]", [{"full_name": "MeasureTheory.Measure.addHaar_smul", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean", "def_pos": [371, 9], "def_end_pos": [371, 21]}, {"full_name": "Set.image_add_left", "def_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "def_pos": [1198, 3], "def_end_pos": [1198, 14]}, {"full_name": "MeasureTheory.measure_preimage_add", "def_path": "Mathlib/MeasureTheory/Group/Measure.lean", "def_pos": [317, 3], "def_end_pos": [317, 14]}, {"full_name": "Set.singleton_add", "def_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "def_pos": [402, 3], "def_end_pos": [402, 14]}, {"full_name": "MulZeroClass.mul_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [38, 3], "def_end_pos": [38, 11]}]], "state_before": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt : Set E\nht : MeasurableSet t\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nh't : \u2191\u2191\u03bc t = 0\nr : \u211d\n\u22a2 \u2191\u2191\u03bc ({x} + r \u2022 t) = 0", "state_after": "no goals"}, {"tactic": "have A :\n  Tendsto (fun n : \u2115 => \u03bc (t \\ closedBall 0 n)) atTop\n    (\ud835\udcdd (\u03bc (\u22c2 n : \u2115, t \\ closedBall 0 n))) := by\n  have N : \u2203 n : \u2115, \u03bc (t \\ closedBall 0 n) \u2260 \u221e :=\n    \u27e80, ((measure_mono (diff_subset t _)).trans_lt h''t.lt_top).ne\u27e9\n  refine' tendsto_measure_iInter (fun n \u21a6 ht.diff measurableSet_closedBall) (fun m n hmn \u21a6 _) N\n  exact diff_subset_diff Subset.rfl (closedBall_subset_closedBall (Nat.cast_le.2 hmn))", "annotated_tactic": ["have A :\n      <a>Tendsto</a> (fun n : \u2115 => \u03bc (t \\ <a>closedBall</a> 0 n)) <a>atTop</a>\n        (\ud835\udcdd (\u03bc (\u22c2 n : \u2115, t \\ <a>closedBall</a> 0 n))) := by\n      have N : \u2203 n : \u2115, \u03bc (t \\ <a>closedBall</a> 0 n) \u2260 \u221e :=\n        \u27e80, ((<a>measure_mono</a> (<a>diff_subset</a> t _)).<a>trans_lt</a> h''t.lt_top).<a>ne</a>\u27e9\n      refine' <a>tendsto_measure_iInter</a> (fun n \u21a6 ht.diff <a>measurableSet_closedBall</a>) (fun m n hmn \u21a6 _) N\n      exact <a>diff_subset_diff</a> <a>Subset.rfl</a> (<a>closedBall_subset_closedBall</a> (<a>Nat.cast_le</a>.2 hmn))", [{"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2939, 5], "def_end_pos": [2939, 12]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "MeasureTheory.measure_mono", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [193, 9], "def_end_pos": [193, 21]}, {"full_name": "Set.diff_subset", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1845, 9], "def_end_pos": [1845, 20]}, {"full_name": "LE.le.trans_lt", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [124, 7], "def_end_pos": [124, 21]}, {"full_name": "LT.lt.ne", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [152, 7], "def_end_pos": [152, 15]}, {"full_name": "MeasureTheory.tendsto_measure_iInter", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [538, 9], "def_end_pos": [538, 31]}, {"full_name": "measurableSet_closedBall", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [1681, 9], "def_end_pos": [1681, 33]}, {"full_name": "Set.diff_subset_diff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1904, 9], "def_end_pos": [1904, 25]}, {"full_name": "Set.Subset.rfl", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [357, 9], "def_end_pos": [357, 19]}, {"full_name": "Metric.closedBall_subset_closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [609, 9], "def_end_pos": [609, 37]}, {"full_name": "Nat.cast_le", "def_path": "Mathlib/Data/Nat/Cast/Order.lean", "def_pos": [91, 9], "def_end_pos": [91, 16]}]], "state_before": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt : Set E\nht : MeasurableSet t\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nh't : \u2191\u2191\u03bc t \u2260 0\n\u22a2 \u2203 n, 0 < n \u2227 \u2191\u2191\u03bc (t \\ closedBall 0 \u2191n) < \u03b5 / 2 * \u2191\u2191\u03bc t", "state_after": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt : Set E\nht : MeasurableSet t\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nh't : \u2191\u2191\u03bc t \u2260 0\nA : Tendsto (fun n => \u2191\u2191\u03bc (t \\ closedBall 0 \u2191n)) atTop (\ud835\udcdd (\u2191\u2191\u03bc (\u22c2 n, t \\ closedBall 0 \u2191n)))\n\u22a2 \u2203 n, 0 < n \u2227 \u2191\u2191\u03bc (t \\ closedBall 0 \u2191n) < \u03b5 / 2 * \u2191\u2191\u03bc t"}, {"tactic": "have : \u22c2 n : \u2115, t \\ closedBall 0 n = \u2205 := by\n  simp_rw [diff_eq, \u2190 inter_iInter, iInter_eq_compl_iUnion_compl, compl_compl,\n    iUnion_closedBall_nat, compl_univ, inter_empty]", "annotated_tactic": ["have : \u22c2 n : \u2115, t \\ <a>closedBall</a> 0 n = \u2205 := by\n      simp_rw [<a>diff_eq</a>, \u2190 <a>inter_iInter</a>, <a>iInter_eq_compl_iUnion_compl</a>, <a>compl_compl</a>,\n        <a>iUnion_closedBall_nat</a>, <a>compl_univ</a>, <a>inter_empty</a>]", [{"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "Set.diff_eq", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1814, 9], "def_end_pos": [1814, 16]}, {"full_name": "Set.inter_iInter", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [661, 9], "def_end_pos": [661, 21]}, {"full_name": "Set.iInter_eq_compl_iUnion_compl", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [631, 9], "def_end_pos": [631, 37]}, {"full_name": "compl_compl", "def_path": "Mathlib/Order/BooleanAlgebra.lean", "def_pos": [634, 9], "def_end_pos": [634, 20]}, {"full_name": "Metric.iUnion_closedBall_nat", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [663, 9], "def_end_pos": [663, 30]}, {"full_name": "Set.compl_univ", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1691, 9], "def_end_pos": [1691, 19]}, {"full_name": "Set.inter_empty", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [931, 9], "def_end_pos": [931, 20]}]], "state_before": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt : Set E\nht : MeasurableSet t\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nh't : \u2191\u2191\u03bc t \u2260 0\nA : Tendsto (fun n => \u2191\u2191\u03bc (t \\ closedBall 0 \u2191n)) atTop (\ud835\udcdd (\u2191\u2191\u03bc (\u22c2 n, t \\ closedBall 0 \u2191n)))\n\u22a2 \u2203 n, 0 < n \u2227 \u2191\u2191\u03bc (t \\ closedBall 0 \u2191n) < \u03b5 / 2 * \u2191\u2191\u03bc t", "state_after": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt : Set E\nht : MeasurableSet t\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nh't : \u2191\u2191\u03bc t \u2260 0\nA : Tendsto (fun n => \u2191\u2191\u03bc (t \\ closedBall 0 \u2191n)) atTop (\ud835\udcdd (\u2191\u2191\u03bc (\u22c2 n, t \\ closedBall 0 \u2191n)))\nthis : \u22c2 n, t \\ closedBall 0 \u2191n = \u2205\n\u22a2 \u2203 n, 0 < n \u2227 \u2191\u2191\u03bc (t \\ closedBall 0 \u2191n) < \u03b5 / 2 * \u2191\u2191\u03bc t"}, {"tactic": "simp only [this, measure_empty] at A", "annotated_tactic": ["simp only [this, <a>measure_empty</a>] at A", [{"full_name": "MeasureTheory.measure_empty", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [185, 9], "def_end_pos": [185, 22]}]], "state_before": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt : Set E\nht : MeasurableSet t\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nh't : \u2191\u2191\u03bc t \u2260 0\nA : Tendsto (fun n => \u2191\u2191\u03bc (t \\ closedBall 0 \u2191n)) atTop (\ud835\udcdd (\u2191\u2191\u03bc (\u22c2 n, t \\ closedBall 0 \u2191n)))\nthis : \u22c2 n, t \\ closedBall 0 \u2191n = \u2205\n\u22a2 \u2203 n, 0 < n \u2227 \u2191\u2191\u03bc (t \\ closedBall 0 \u2191n) < \u03b5 / 2 * \u2191\u2191\u03bc t", "state_after": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt : Set E\nht : MeasurableSet t\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nh't : \u2191\u2191\u03bc t \u2260 0\nthis : \u22c2 n, t \\ closedBall 0 \u2191n = \u2205\nA : Tendsto (fun n => \u2191\u2191\u03bc (t \\ closedBall 0 \u2191n)) atTop (\ud835\udcdd 0)\n\u22a2 \u2203 n, 0 < n \u2227 \u2191\u2191\u03bc (t \\ closedBall 0 \u2191n) < \u03b5 / 2 * \u2191\u2191\u03bc t"}, {"tactic": "have I : 0 < \u03b5 / 2 * \u03bc t := ENNReal.mul_pos (ENNReal.half_pos \u03b5pos.ne').ne' h't", "annotated_tactic": ["have I : 0 < \u03b5 / 2 * \u03bc t := <a>ENNReal.mul_pos</a> (<a>ENNReal.half_pos</a> \u03b5pos.ne').<a>ne'</a> h't", [{"full_name": "ENNReal.mul_pos", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [645, 9], "def_end_pos": [645, 16]}, {"full_name": "ENNReal.half_pos", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1796, 19], "def_end_pos": [1796, 27]}, {"full_name": "LT.lt.ne'", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [328, 9], "def_end_pos": [328, 12]}]], "state_before": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt : Set E\nht : MeasurableSet t\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nh't : \u2191\u2191\u03bc t \u2260 0\nthis : \u22c2 n, t \\ closedBall 0 \u2191n = \u2205\nA : Tendsto (fun n => \u2191\u2191\u03bc (t \\ closedBall 0 \u2191n)) atTop (\ud835\udcdd 0)\n\u22a2 \u2203 n, 0 < n \u2227 \u2191\u2191\u03bc (t \\ closedBall 0 \u2191n) < \u03b5 / 2 * \u2191\u2191\u03bc t", "state_after": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt : Set E\nht : MeasurableSet t\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nh't : \u2191\u2191\u03bc t \u2260 0\nthis : \u22c2 n, t \\ closedBall 0 \u2191n = \u2205\nA : Tendsto (fun n => \u2191\u2191\u03bc (t \\ closedBall 0 \u2191n)) atTop (\ud835\udcdd 0)\nI : 0 < \u03b5 / 2 * \u2191\u2191\u03bc t\n\u22a2 \u2203 n, 0 < n \u2227 \u2191\u2191\u03bc (t \\ closedBall 0 \u2191n) < \u03b5 / 2 * \u2191\u2191\u03bc t"}, {"tactic": "exact (Eventually.and (Ioi_mem_atTop 0) ((tendsto_order.1 A).2 _ I)).exists", "annotated_tactic": ["exact (<a>Eventually.and</a> (<a>Ioi_mem_atTop</a> 0) ((<a>tendsto_order</a>.1 A).2 _ I)).<a>exists</a>", [{"full_name": "Filter.Eventually.and", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1103, 19], "def_end_pos": [1103, 33]}, {"full_name": "Filter.Ioi_mem_atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [61, 9], "def_end_pos": [61, 22]}, {"full_name": "tendsto_order", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [919, 9], "def_end_pos": [919, 22]}, {"full_name": "Filter.Eventually.exists", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1308, 9], "def_end_pos": [1308, 26]}]], "state_before": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt : Set E\nht : MeasurableSet t\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nh't : \u2191\u2191\u03bc t \u2260 0\nthis : \u22c2 n, t \\ closedBall 0 \u2191n = \u2205\nA : Tendsto (fun n => \u2191\u2191\u03bc (t \\ closedBall 0 \u2191n)) atTop (\ud835\udcdd 0)\nI : 0 < \u03b5 / 2 * \u2191\u2191\u03bc t\n\u22a2 \u2203 n, 0 < n \u2227 \u2191\u2191\u03bc (t \\ closedBall 0 \u2191n) < \u03b5 / 2 * \u2191\u2191\u03bc t", "state_after": "no goals"}, {"tactic": "have N : \u2203 n : \u2115, \u03bc (t \\ closedBall 0 n) \u2260 \u221e :=\n  \u27e80, ((measure_mono (diff_subset t _)).trans_lt h''t.lt_top).ne\u27e9", "annotated_tactic": ["have N : \u2203 n : \u2115, \u03bc (t \\ <a>closedBall</a> 0 n) \u2260 \u221e :=\n        \u27e80, ((<a>measure_mono</a> (<a>diff_subset</a> t _)).<a>trans_lt</a> h''t.lt_top).<a>ne</a>\u27e9", [{"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "MeasureTheory.measure_mono", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [193, 9], "def_end_pos": [193, 21]}, {"full_name": "Set.diff_subset", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1845, 9], "def_end_pos": [1845, 20]}, {"full_name": "LE.le.trans_lt", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [124, 7], "def_end_pos": [124, 21]}, {"full_name": "LT.lt.ne", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [152, 7], "def_end_pos": [152, 15]}]], "state_before": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt : Set E\nht : MeasurableSet t\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nh't : \u2191\u2191\u03bc t \u2260 0\n\u22a2 Tendsto (fun n => \u2191\u2191\u03bc (t \\ closedBall 0 \u2191n)) atTop (\ud835\udcdd (\u2191\u2191\u03bc (\u22c2 n, t \\ closedBall 0 \u2191n)))", "state_after": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt : Set E\nht : MeasurableSet t\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nh't : \u2191\u2191\u03bc t \u2260 0\nN : \u2203 n, \u2191\u2191\u03bc (t \\ closedBall 0 \u2191n) \u2260 \u22a4\n\u22a2 Tendsto (fun n => \u2191\u2191\u03bc (t \\ closedBall 0 \u2191n)) atTop (\ud835\udcdd (\u2191\u2191\u03bc (\u22c2 n, t \\ closedBall 0 \u2191n)))"}, {"tactic": "refine' tendsto_measure_iInter (fun n \u21a6 ht.diff measurableSet_closedBall) (fun m n hmn \u21a6 _) N", "annotated_tactic": ["refine' <a>tendsto_measure_iInter</a> (fun n \u21a6 ht.diff <a>measurableSet_closedBall</a>) (fun m n hmn \u21a6 _) N", [{"full_name": "MeasureTheory.tendsto_measure_iInter", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [538, 9], "def_end_pos": [538, 31]}, {"full_name": "measurableSet_closedBall", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [1681, 9], "def_end_pos": [1681, 33]}]], "state_before": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt : Set E\nht : MeasurableSet t\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nh't : \u2191\u2191\u03bc t \u2260 0\nN : \u2203 n, \u2191\u2191\u03bc (t \\ closedBall 0 \u2191n) \u2260 \u22a4\n\u22a2 Tendsto (fun n => \u2191\u2191\u03bc (t \\ closedBall 0 \u2191n)) atTop (\ud835\udcdd (\u2191\u2191\u03bc (\u22c2 n, t \\ closedBall 0 \u2191n)))", "state_after": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt : Set E\nht : MeasurableSet t\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nh't : \u2191\u2191\u03bc t \u2260 0\nN : \u2203 n, \u2191\u2191\u03bc (t \\ closedBall 0 \u2191n) \u2260 \u22a4\nm n : \u2115\nhmn : m \u2264 n\n\u22a2 t \\ closedBall 0 \u2191n \u2264 t \\ closedBall 0 \u2191m"}, {"tactic": "exact diff_subset_diff Subset.rfl (closedBall_subset_closedBall (Nat.cast_le.2 hmn))", "annotated_tactic": ["exact <a>diff_subset_diff</a> <a>Subset.rfl</a> (<a>closedBall_subset_closedBall</a> (<a>Nat.cast_le</a>.2 hmn))", [{"full_name": "Set.diff_subset_diff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1904, 9], "def_end_pos": [1904, 25]}, {"full_name": "Set.Subset.rfl", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [357, 9], "def_end_pos": [357, 19]}, {"full_name": "Metric.closedBall_subset_closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [609, 9], "def_end_pos": [609, 37]}, {"full_name": "Nat.cast_le", "def_path": "Mathlib/Data/Nat/Cast/Order.lean", "def_pos": [91, 9], "def_end_pos": [91, 16]}]], "state_before": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt : Set E\nht : MeasurableSet t\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nh't : \u2191\u2191\u03bc t \u2260 0\nN : \u2203 n, \u2191\u2191\u03bc (t \\ closedBall 0 \u2191n) \u2260 \u22a4\nm n : \u2115\nhmn : m \u2264 n\n\u22a2 t \\ closedBall 0 \u2191n \u2264 t \\ closedBall 0 \u2191m", "state_after": "no goals"}, {"tactic": "simp_rw [diff_eq, \u2190 inter_iInter, iInter_eq_compl_iUnion_compl, compl_compl,\n  iUnion_closedBall_nat, compl_univ, inter_empty]", "annotated_tactic": ["simp_rw [<a>diff_eq</a>, \u2190 <a>inter_iInter</a>, <a>iInter_eq_compl_iUnion_compl</a>, <a>compl_compl</a>,\n        <a>iUnion_closedBall_nat</a>, <a>compl_univ</a>, <a>inter_empty</a>]", [{"full_name": "Set.diff_eq", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1814, 9], "def_end_pos": [1814, 16]}, {"full_name": "Set.inter_iInter", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [661, 9], "def_end_pos": [661, 21]}, {"full_name": "Set.iInter_eq_compl_iUnion_compl", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [631, 9], "def_end_pos": [631, 37]}, {"full_name": "compl_compl", "def_path": "Mathlib/Order/BooleanAlgebra.lean", "def_pos": [634, 9], "def_end_pos": [634, 20]}, {"full_name": "Metric.iUnion_closedBall_nat", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [663, 9], "def_end_pos": [663, 30]}, {"full_name": "Set.compl_univ", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1691, 9], "def_end_pos": [1691, 19]}, {"full_name": "Set.inter_empty", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [931, 9], "def_end_pos": [931, 20]}]], "state_before": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt : Set E\nht : MeasurableSet t\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nh't : \u2191\u2191\u03bc t \u2260 0\nA : Tendsto (fun n => \u2191\u2191\u03bc (t \\ closedBall 0 \u2191n)) atTop (\ud835\udcdd (\u2191\u2191\u03bc (\u22c2 n, t \\ closedBall 0 \u2191n)))\n\u22a2 \u22c2 n, t \\ closedBall 0 \u2191n = \u2205", "state_after": "no goals"}, {"tactic": "rw [\u2190 inter_union_distrib_left, \u2190 add_union, \u2190 smul_set_union, inter_union_diff]", "annotated_tactic": ["rw [\u2190 <a>inter_union_distrib_left</a>, \u2190 <a>add_union</a>, \u2190 <a>smul_set_union</a>, <a>inter_union_diff</a>]", [{"full_name": "Set.inter_union_distrib_left", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1059, 9], "def_end_pos": [1059, 33]}, {"full_name": "Set.add_union", "def_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "def_pos": [445, 3], "def_end_pos": [445, 14]}, {"full_name": "Set.smul_set_union", "def_path": "Mathlib/Data/Set/Pointwise/SMul.lean", "def_pos": [366, 9], "def_end_pos": [366, 23]}, {"full_name": "Set.inter_union_diff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1889, 9], "def_end_pos": [1889, 25]}]], "state_before": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt : Set E\nht : MeasurableSet t\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nh't : \u2191\u2191\u03bc t \u2260 0\nn : \u2115\nnpos : 0 < n\nhn : \u2191\u2191\u03bc (t \\ closedBall 0 \u2191n) < \u03b5 / 2 * \u2191\u2191\u03bc t\nL : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 \u2191n))) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nr : \u211d\nhr : \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 \u2191n))) / \u2191\u2191\u03bc ({x} + r \u2022 t) < \u03b5 / 2\nrpos : 0 < r\n\u22a2 \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) = \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 \u2191n)) \u222a s \u2229 ({x} + r \u2022 (t \\ closedBall 0 \u2191n)))", "state_after": "no goals"}, {"tactic": "rw [ENNReal.add_div]", "annotated_tactic": ["rw [<a>ENNReal.add_div</a>]", [{"full_name": "ENNReal.add_div", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1738, 19], "def_end_pos": [1738, 26]}]], "state_before": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt : Set E\nht : MeasurableSet t\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nh't : \u2191\u2191\u03bc t \u2260 0\nn : \u2115\nnpos : 0 < n\nhn : \u2191\u2191\u03bc (t \\ closedBall 0 \u2191n) < \u03b5 / 2 * \u2191\u2191\u03bc t\nL : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 \u2191n))) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nr : \u211d\nhr : \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 \u2191n))) / \u2191\u2191\u03bc ({x} + r \u2022 t) < \u03b5 / 2\nrpos : 0 < r\nI : \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) \u2264 \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 \u2191n))) + \u2191\u2191\u03bc ({x} + r \u2022 (t \\ closedBall 0 \u2191n))\n\u22a2 (\u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 \u2191n))) + \u2191\u2191\u03bc ({x} + r \u2022 (t \\ closedBall 0 \u2191n))) / \u2191\u2191\u03bc ({x} + r \u2022 t) <\n    \u03b5 / 2 + \u03b5 / 2", "state_after": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt : Set E\nht : MeasurableSet t\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nh't : \u2191\u2191\u03bc t \u2260 0\nn : \u2115\nnpos : 0 < n\nhn : \u2191\u2191\u03bc (t \\ closedBall 0 \u2191n) < \u03b5 / 2 * \u2191\u2191\u03bc t\nL : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 \u2191n))) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nr : \u211d\nhr : \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 \u2191n))) / \u2191\u2191\u03bc ({x} + r \u2022 t) < \u03b5 / 2\nrpos : 0 < r\nI : \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) \u2264 \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 \u2191n))) + \u2191\u2191\u03bc ({x} + r \u2022 (t \\ closedBall 0 \u2191n))\n\u22a2 \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 \u2191n))) / \u2191\u2191\u03bc ({x} + r \u2022 t) +\n      \u2191\u2191\u03bc ({x} + r \u2022 (t \\ closedBall 0 \u2191n)) / \u2191\u2191\u03bc ({x} + r \u2022 t) <\n    \u03b5 / 2 + \u03b5 / 2"}, {"tactic": "apply ENNReal.add_lt_add hr _", "annotated_tactic": ["apply <a>ENNReal.add_lt_add</a> hr _", [{"full_name": "ENNReal.add_lt_add", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [931, 19], "def_end_pos": [931, 29]}]], "state_before": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt : Set E\nht : MeasurableSet t\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nh't : \u2191\u2191\u03bc t \u2260 0\nn : \u2115\nnpos : 0 < n\nhn : \u2191\u2191\u03bc (t \\ closedBall 0 \u2191n) < \u03b5 / 2 * \u2191\u2191\u03bc t\nL : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 \u2191n))) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nr : \u211d\nhr : \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 \u2191n))) / \u2191\u2191\u03bc ({x} + r \u2022 t) < \u03b5 / 2\nrpos : 0 < r\nI : \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) \u2264 \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 \u2191n))) + \u2191\u2191\u03bc ({x} + r \u2022 (t \\ closedBall 0 \u2191n))\n\u22a2 \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 \u2191n))) / \u2191\u2191\u03bc ({x} + r \u2022 t) +\n      \u2191\u2191\u03bc ({x} + r \u2022 (t \\ closedBall 0 \u2191n)) / \u2191\u2191\u03bc ({x} + r \u2022 t) <\n    \u03b5 / 2 + \u03b5 / 2", "state_after": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt : Set E\nht : MeasurableSet t\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nh't : \u2191\u2191\u03bc t \u2260 0\nn : \u2115\nnpos : 0 < n\nhn : \u2191\u2191\u03bc (t \\ closedBall 0 \u2191n) < \u03b5 / 2 * \u2191\u2191\u03bc t\nL : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 \u2191n))) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nr : \u211d\nhr : \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 \u2191n))) / \u2191\u2191\u03bc ({x} + r \u2022 t) < \u03b5 / 2\nrpos : 0 < r\nI : \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) \u2264 \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 \u2191n))) + \u2191\u2191\u03bc ({x} + r \u2022 (t \\ closedBall 0 \u2191n))\n\u22a2 \u2191\u2191\u03bc ({x} + r \u2022 (t \\ closedBall 0 \u2191n)) / \u2191\u2191\u03bc ({x} + r \u2022 t) < \u03b5 / 2"}, {"tactic": "rwa [addHaar_singleton_add_smul_div_singleton_add_smul \u03bc rpos.ne',\n  ENNReal.div_lt_iff (Or.inl h't) (Or.inl h''t)]", "annotated_tactic": ["rwa [<a>addHaar_singleton_add_smul_div_singleton_add_smul</a> \u03bc rpos.ne',\n        <a>ENNReal.div_lt_iff</a> (<a>Or.inl</a> h't) (<a>Or.inl</a> h''t)]", [{"full_name": "MeasureTheory.Measure.addHaar_singleton_add_smul_div_singleton_add_smul", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean", "def_pos": [529, 9], "def_end_pos": [529, 58]}, {"full_name": "ENNReal.div_lt_iff", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1658, 19], "def_end_pos": [1658, 29]}, {"full_name": "Or.inl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [517, 5], "def_end_pos": [517, 8]}, {"full_name": "Or.inl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [517, 5], "def_end_pos": [517, 8]}]], "state_before": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt : Set E\nht : MeasurableSet t\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nh't : \u2191\u2191\u03bc t \u2260 0\nn : \u2115\nnpos : 0 < n\nhn : \u2191\u2191\u03bc (t \\ closedBall 0 \u2191n) < \u03b5 / 2 * \u2191\u2191\u03bc t\nL : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 \u2191n))) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nr : \u211d\nhr : \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 \u2191n))) / \u2191\u2191\u03bc ({x} + r \u2022 t) < \u03b5 / 2\nrpos : 0 < r\nI : \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) \u2264 \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 \u2191n))) + \u2191\u2191\u03bc ({x} + r \u2022 (t \\ closedBall 0 \u2191n))\n\u22a2 \u2191\u2191\u03bc ({x} + r \u2022 (t \\ closedBall 0 \u2191n)) / \u2191\u2191\u03bc ({x} + r \u2022 t) < \u03b5 / 2", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": 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"state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Basic.lean", "full_name": "Set.inter_diff_distrib_right", "start": [1634, 1], "end": [1635, 32], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Rat/Lemmas.lean", "full_name": "Rat.divInt_mul_left", "start": [148, 1], "end": [150, 84], "traced_tactics": [{"tactic": "if d0 : d = 0 then simp [d0] else\nsimp [divInt_eq_iff (Int.mul_ne_zero a0 d0) d0, Int.mul_assoc, Int.mul_left_comm]", "annotated_tactic": ["if d0 : d = 0 then simp [d0] else\n  simp [<a>divInt_eq_iff</a> (<a>Int.mul_ne_zero</a> a0 d0) d0, <a>Int.mul_assoc</a>, <a>Int.mul_left_comm</a>]", [{"full_name": "Rat.divInt_eq_iff", "def_path": "lake-packages/std/Std/Data/Rat/Lemmas.lean", "def_pos": [141, 9], "def_end_pos": [141, 22]}, 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(a * d) = n /. d", "state_after": "no goals"}, {"tactic": "simp [d0]", "annotated_tactic": ["simp [d0]", []], "state_before": "n d a : Int\na0 : a \u2260 0\nd0 : d = 0\n\u22a2 a * n /. (a * d) = n /. d", "state_after": "no goals"}, {"tactic": "simp [divInt_eq_iff (Int.mul_ne_zero a0 d0) d0, Int.mul_assoc, Int.mul_left_comm]", "annotated_tactic": ["simp [<a>divInt_eq_iff</a> (<a>Int.mul_ne_zero</a> a0 d0) d0, <a>Int.mul_assoc</a>, <a>Int.mul_left_comm</a>]", [{"full_name": "Rat.divInt_eq_iff", "def_path": "lake-packages/std/Std/Data/Rat/Lemmas.lean", "def_pos": [141, 9], "def_end_pos": [141, 22]}, {"full_name": "Int.mul_ne_zero", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [1309, 19], "def_end_pos": [1309, 30]}, {"full_name": "Int.mul_assoc", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [398, 19], "def_end_pos": [398, 28]}, {"full_name": "Int.mul_left_comm", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [401, 19], "def_end_pos": [401, 32]}]], "state_before": "n d a : Int\na0 : a \u2260 0\nd0 : \u00acd = 0\n\u22a2 a * n /. (a * d) = n /. d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "full_name": "Int.mul_pos_of_neg_of_neg", "start": [1232, 11], "end": [1234, 29], "traced_tactics": [{"tactic": "have : 0 * b < a * b := Int.mul_lt_mul_of_neg_right ha hb", "annotated_tactic": ["have : 0 * b < a * b := <a>Int.mul_lt_mul_of_neg_right</a> ha hb", [{"full_name": "Int.mul_lt_mul_of_neg_right", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [1225, 19], "def_end_pos": [1225, 42]}]], "state_before": "a b : Int\nha : a < 0\nhb : b < 0\n\u22a2 0 < a * b", "state_after": "a b : Int\nha : a < 0\nhb : b < 0\nthis : 0 * b < a * b\n\u22a2 0 < a * b"}, {"tactic": "rwa [Int.zero_mul] at this", "annotated_tactic": ["rwa [<a>Int.zero_mul</a>] at this", [{"full_name": "Int.zero_mul", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [409, 27], "def_end_pos": [409, 35]}]], "state_before": "a b : Int\nha : a < 0\nhb : b < 0\nthis : 0 * b < a * b\n\u22a2 0 < a * b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "full_name": "measure_le_lintegral_thickenedIndicator", "start": [1360, 1], "end": [1365, 91], "traced_tactics": [{"tactic": "convert measure_le_lintegral_thickenedIndicatorAux \u03bc E_mble \u03b4", "annotated_tactic": ["convert <a>measure_le_lintegral_thickenedIndicatorAux</a> \u03bc E_mble \u03b4", [{"full_name": "measure_le_lintegral_thickenedIndicatorAux", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [1351, 9], "def_end_pos": [1351, 51]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nE\u271d : Type u_3\nF : Type u_4\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup E\u271d\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03bc : Measure \u03b1\nE : Set \u03b1\nE_mble : MeasurableSet E\n\u03b4 : \u211d\n\u03b4_pos : 0 < \u03b4\n\u22a2 \u2191\u2191\u03bc E \u2264 \u222b\u207b (a : \u03b1), \u2191(\u2191(thickenedIndicator \u03b4_pos E) a) \u2202\u03bc", "state_after": "case h.e'_4.h.e'_4.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE\u271d : Type u_3\nF : Type u_4\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup E\u271d\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03bc : Measure \u03b1\nE : Set \u03b1\nE_mble : MeasurableSet E\n\u03b4 : \u211d\n\u03b4_pos : 0 < \u03b4\nx\u271d : \u03b1\n\u22a2 \u2191(\u2191(thickenedIndicator \u03b4_pos E) x\u271d) = thickenedIndicatorAux \u03b4 E x\u271d"}, {"tactic": "dsimp", "annotated_tactic": ["dsimp", []], "state_before": "case h.e'_4.h.e'_4.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE\u271d : Type u_3\nF : Type u_4\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup E\u271d\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03bc : Measure \u03b1\nE : Set \u03b1\nE_mble : MeasurableSet E\n\u03b4 : \u211d\n\u03b4_pos : 0 < \u03b4\nx\u271d : \u03b1\n\u22a2 \u2191(\u2191(thickenedIndicator \u03b4_pos E) x\u271d) = thickenedIndicatorAux \u03b4 E x\u271d", "state_after": "case h.e'_4.h.e'_4.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE\u271d : Type u_3\nF : Type u_4\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup E\u271d\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03bc : Measure \u03b1\nE : Set \u03b1\nE_mble : MeasurableSet E\n\u03b4 : \u211d\n\u03b4_pos : 0 < \u03b4\nx\u271d : \u03b1\n\u22a2 \u2191(ENNReal.toNNReal (thickenedIndicatorAux \u03b4 E x\u271d)) = thickenedIndicatorAux \u03b4 E x\u271d"}, {"tactic": "simp only [thickenedIndicatorAux_lt_top.ne, ENNReal.coe_toNNReal, Ne.def, not_false_iff]", "annotated_tactic": ["simp only [thickenedIndicatorAux_lt_top.ne, <a>ENNReal.coe_toNNReal</a>, <a>Ne.def</a>, <a>not_false_iff</a>]", [{"full_name": "ENNReal.coe_toNNReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [180, 9], "def_end_pos": [180, 21]}, {"full_name": "Ne.def", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [59, 9], "def_end_pos": [59, 15]}, {"full_name": "not_false_iff", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [82, 9], "def_end_pos": [82, 22]}]], "state_before": "case h.e'_4.h.e'_4.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE\u271d : Type u_3\nF : Type u_4\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup E\u271d\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03bc : Measure \u03b1\nE : Set \u03b1\nE_mble : MeasurableSet E\n\u03b4 : \u211d\n\u03b4_pos : 0 < \u03b4\nx\u271d : \u03b1\n\u22a2 \u2191(ENNReal.toNNReal (thickenedIndicatorAux \u03b4 E x\u271d)) = thickenedIndicatorAux \u03b4 E x\u271d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "full_name": "MeasureTheory.lintegral_iInf_ae", "start": [982, 1], "end": [1010, 74], "traced_tactics": [{"tactic": "induction' n with n ih", "annotated_tactic": ["induction' n with n ih", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nh_meas : \u2200 (n : \u2115), Measurable (f n)\nh_mono\u271d : \u2200 (n : \u2115), f (Nat.succ n) \u2264\u1d50[\u03bc] f n\nh_fin : \u222b\u207b (a : \u03b1), f 0 a \u2202\u03bc \u2260 \u22a4\nfn_le_f0 : \u222b\u207b (a : \u03b1), \u2a05 n, f n a \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), f 0 a \u2202\u03bc\nfn_le_f0' : \u2a05 n, \u222b\u207b (a : \u03b1), f n a \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), f 0 a \u2202\u03bc\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (n : \u2115), f (Nat.succ n) a \u2264 f n a\nn : \u2115\na : \u03b1\nh : \u2200 (n : \u2115), f (Nat.succ n) a \u2264 f n a\n\u22a2 f n a \u2264 f 0 a", "state_after": "case zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nh_meas : \u2200 (n : \u2115), Measurable (f n)\nh_mono\u271d : \u2200 (n : \u2115), f (Nat.succ n) \u2264\u1d50[\u03bc] f n\nh_fin : \u222b\u207b (a : \u03b1), f 0 a \u2202\u03bc \u2260 \u22a4\nfn_le_f0 : \u222b\u207b (a : \u03b1), \u2a05 n, f n a \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), f 0 a \u2202\u03bc\nfn_le_f0' : \u2a05 n, \u222b\u207b (a : \u03b1), f n a \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), f 0 a \u2202\u03bc\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (n : \u2115), f (Nat.succ n) a \u2264 f n a\na : \u03b1\nh : \u2200 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a \u2264 f 0 a", "state_after": "no goals"}, {"tactic": "exact le_trans (h n) ih", "annotated_tactic": ["exact <a>le_trans</a> (h n) ih", [{"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}]], "state_before": "case succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nh_meas : \u2200 (n : \u2115), Measurable (f n)\nh_mono\u271d : \u2200 (n : \u2115), f (Nat.succ n) \u2264\u1d50[\u03bc] f n\nh_fin : \u222b\u207b (a : \u03b1), f 0 a \u2202\u03bc \u2260 \u22a4\nfn_le_f0 : \u222b\u207b (a : \u03b1), \u2a05 n, f n a \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), f 0 a \u2202\u03bc\nfn_le_f0' : \u2a05 n, \u222b\u207b (a : \u03b1), f n a \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), f 0 a \u2202\u03bc\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (n : \u2115), f (Nat.succ n) a 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"MeasureTheory.det_one_smulRight", "def_path": "Mathlib/MeasureTheory/Function/Jacobian.lean", "def_pos": [1222, 9], "def_end_pos": [1222, 26]}, {"full_name": "MeasureTheory.integrableOn_image_iff_integrableOn_abs_det_fderiv_smul", "def_path": "Mathlib/MeasureTheory/Function/Jacobian.lean", "def_pos": [1188, 9], "def_end_pos": [1188, 64]}, {"full_name": "MeasureTheory.MeasureSpace.volume", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [663, 3], "def_end_pos": [663, 9]}, {"full_name": "HasDerivWithinAt.hasFDerivWithinAt", "def_path": "Mathlib/Analysis/Calculus/Deriv/Basic.lean", "def_pos": [189, 9], "def_end_pos": [189, 43]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] 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"\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03c3 : Type u_5\ninst\u271d\u2074 : Primcodable \u03b1\ninst\u271d\u00b3 : Primcodable \u03b2\ninst\u271d\u00b2 : Primcodable \u03b3\ninst\u271d\u00b9 : Primcodable \u03b4\ninst\u271d : Primcodable \u03c3\np : \u2115 \u00d7 \u2115\n\u22a2 (Nat.casesOn ((fun x x_1 => x - x_1) p.1 p.2) true fun b => false) =\n    (fun a b => decide ((fun x x_1 => x \u2264 x_1) a b)) p.1 p.2", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03c3 : Type u_5\ninst\u271d\u2074 : Primcodable \u03b1\ninst\u271d\u00b3 : Primcodable \u03b2\ninst\u271d\u00b2 : Primcodable \u03b3\ninst\u271d\u00b9 : Primcodable \u03b4\ninst\u271d : Primcodable \u03c3\np : \u2115 \u00d7 \u2115\n\u22a2 Nat.rec true (fun n n_ih => false) (p.1 - p.2) = decide (p.1 \u2264 p.2)"}, {"tactic": "cases' e : p.1 - p.2 with n", "annotated_tactic": ["cases' e : p.1 - p.2 with n", []], "state_before": "\u03b1 : Type 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\u03b2\ninst\u271d\u00b2 : Primcodable \u03b3\ninst\u271d\u00b9 : Primcodable \u03b4\ninst\u271d : Primcodable \u03c3\np : \u2115 \u00d7 \u2115\nn : \u2115\ne : p.1 - p.2 = Nat.succ n\n\u22a2 Nat.rec true (fun n n_ih => false) (Nat.succ n) = decide (p.1 \u2264 p.2)"}, {"tactic": "simp [tsub_eq_zero_iff_le.1 e]", "annotated_tactic": ["simp [<a>tsub_eq_zero_iff_le</a>.1 e]", [{"full_name": "tsub_eq_zero_iff_le", "def_path": "Mathlib/Algebra/Order/Sub/Canonical.lean", "def_pos": [324, 9], "def_end_pos": [324, 28]}]], "state_before": "case zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03c3 : Type u_5\ninst\u271d\u2074 : Primcodable \u03b1\ninst\u271d\u00b3 : Primcodable \u03b2\ninst\u271d\u00b2 : Primcodable \u03b3\ninst\u271d\u00b9 : Primcodable \u03b4\ninst\u271d : Primcodable \u03c3\np : \u2115 \u00d7 \u2115\ne : p.1 - p.2 = Nat.zero\n\u22a2 Nat.rec true (fun n n_ih => false) Nat.zero = decide (p.1 \u2264 p.2)", "state_after": "no goals"}, {"tactic": "simp [not_le.2 (Nat.lt_of_sub_eq_succ e)]", "annotated_tactic": ["simp [<a>not_le</a>.2 (<a>Nat.lt_of_sub_eq_succ</a> e)]", [{"full_name": "not_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [373, 9], "def_end_pos": [373, 15]}, {"full_name": "Nat.lt_of_sub_eq_succ", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [391, 19], "def_end_pos": [391, 36]}]], "state_before": "case succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03c3 : Type u_5\ninst\u271d\u2074 : Primcodable \u03b1\ninst\u271d\u00b3 : Primcodable \u03b2\ninst\u271d\u00b2 : Primcodable \u03b3\ninst\u271d\u00b9 : Primcodable \u03b4\ninst\u271d : Primcodable \u03c3\np : \u2115 \u00d7 \u2115\nn : \u2115\ne : p.1 - p.2 = Nat.succ n\n\u22a2 Nat.rec true (fun n n_ih => false) (Nat.succ n) = decide (p.1 \u2264 p.2)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/TuringMachine.lean", "full_name": "Turing.BlankRel.refl", "start": [122, 1], "end": [123, 31], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Pointwise.lean", "full_name": "Finset.mul_subset_iff_right", "start": [1852, 1], "end": [1853, 26], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/EssSup.lean", "full_name": "essSup_mono_measure", "start": [192, 1], "end": [194, 29], "traced_tactics": [{"tactic": "refine' limsup_le_limsup_of_le (Measure.ae_le_iff_absolutelyContinuous.mpr h\u03bc\u03bd) _ _", "annotated_tactic": ["refine' <a>limsup_le_limsup_of_le</a> (Measure.ae_le_iff_absolutelyContinuous.mpr h\u03bc\u03bd) _ _", [{"full_name": "Filter.limsup_le_limsup_of_le", "def_path": "Mathlib/Order/LiminfLimsup.lean", "def_pos": [610, 9], "def_end_pos": [610, 31]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : CompleteLattice \u03b2\nf : \u03b1 \u2192 \u03b2\nh\u03bc\u03bd : \u03bd \u226a \u03bc\n\u22a2 essSup f \u03bd \u2264 essSup f \u03bc", "state_after": "case refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : CompleteLattice \u03b2\nf : \u03b1 \u2192 \u03b2\nh\u03bc\u03bd : \u03bd \u226a \u03bc\n\u22a2 IsCoboundedUnder (fun x x_1 => x \u2264 x_1) (Measure.ae \u03bd) f\n\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : CompleteLattice \u03b2\nf : \u03b1 \u2192 \u03b2\nh\u03bc\u03bd : \u03bd \u226a \u03bc\n\u22a2 IsBoundedUnder (fun x x_1 => x \u2264 x_1) (Measure.ae \u03bc) f"}, {"tactic": "all_goals isBoundedDefault", "annotated_tactic": ["all_goals isBoundedDefault", []], "state_before": "case refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : CompleteLattice \u03b2\nf : \u03b1 \u2192 \u03b2\nh\u03bc\u03bd : \u03bd \u226a \u03bc\n\u22a2 IsCoboundedUnder (fun x x_1 => x \u2264 x_1) (Measure.ae \u03bd) f\n\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : CompleteLattice \u03b2\nf : \u03b1 \u2192 \u03b2\nh\u03bc\u03bd : \u03bd \u226a \u03bc\n\u22a2 IsBoundedUnder (fun x x_1 => x \u2264 x_1) (Measure.ae \u03bc) f", "state_after": "no goals"}, {"tactic": "isBoundedDefault", "annotated_tactic": ["isBoundedDefault", []], "state_before": "case refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : CompleteLattice \u03b2\nf : \u03b1 \u2192 \u03b2\nh\u03bc\u03bd : \u03bd \u226a \u03bc\n\u22a2 IsBoundedUnder (fun x x_1 => x \u2264 x_1) (Measure.ae \u03bc) f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Interval.lean", "full_name": "Finset.card_Iic_finset", "start": [125, 1], "end": [125, 94], "traced_tactics": [{"tactic": "rw [Iic_eq_powerset, card_powerset]", "annotated_tactic": ["rw [<a>Iic_eq_powerset</a>, <a>card_powerset</a>]", [{"full_name": "Finset.Iic_eq_powerset", "def_path": "Mathlib/Data/Finset/Interval.lean", "def_pos": [70, 9], "def_end_pos": [70, 24]}, {"full_name": "Finset.card_powerset", "def_path": "Mathlib/Data/Finset/Powerset.lean", "def_pos": [88, 9], "def_end_pos": [88, 22]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\n\u22a2 card (Iic s) = 2 ^ card s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "Finset.inter_insert_of_not_mem", "start": [1701, 1], "end": [1702, 92], "traced_tactics": [{"tactic": "rw [inter_comm, insert_inter_of_not_mem h, inter_comm]", "annotated_tactic": ["rw [<a>inter_comm</a>, <a>insert_inter_of_not_mem</a> h, <a>inter_comm</a>]", [{"full_name": "Finset.inter_comm", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1642, 9], "def_end_pos": [1642, 19]}, {"full_name": "Finset.insert_inter_of_not_mem", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1693, 9], "def_end_pos": [1693, 32]}, {"full_name": "Finset.inter_comm", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1642, 9], "def_end_pos": [1642, 19]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns s\u2081\u271d s\u2082\u271d t t\u2081 t\u2082 u v : Finset \u03b1\na\u271d b : \u03b1\ns\u2081 s\u2082 : Finset \u03b1\na : \u03b1\nh : \u00aca \u2208 s\u2081\n\u22a2 s\u2081 \u2229 insert a s\u2082 = s\u2081 \u2229 s\u2082", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/UnionFind.lean", "full_name": "UFModel.Models.size_eq", "start": [123, 1], "end": [124, 32], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "full_name": "MeasureTheory.VectorMeasure.measurable_of_not_zero_le_restrict", "start": [1006, 1], "end": [1007, 54], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/Layercake.lean", "full_name": "MeasureTheory.lintegral_rpow_eq_lintegral_meas_le_mul", "start": [469, 1], "end": [492, 37], "traced_tactics": [{"tactic": "have one_lt_p : -1 < p - 1 := by linarith", "annotated_tactic": ["have one_lt_p : -1 < p - 1 := by linarith", []], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nf_mble : AEMeasurable f\np : \u211d\np_pos : 0 < p\n\u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9 ^ p) \u2202\u03bc =\n    ENNReal.ofReal p * \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (t ^ (p - 1))", "state_after": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nf_mble : AEMeasurable f\np : \u211d\np_pos : 0 < p\none_lt_p : -1 < p - 1\n\u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9 ^ p) \u2202\u03bc =\n    ENNReal.ofReal p * \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (t ^ (p - 1))"}, {"tactic": "have obs : \u2200 x : \u211d, (\u222b t : \u211d in (0)..x, t ^ (p - 1)) = x ^ p / p := by\n  intro x\n  rw [integral_rpow (Or.inl one_lt_p)]\n  simp [Real.zero_rpow p_pos.ne.symm]", "annotated_tactic": ["have obs : \u2200 x : \u211d, (\u222b t : \u211d in (0)..x, t ^ (p - 1)) = x ^ p / p := by\n    intro x\n    rw [<a>integral_rpow</a> (<a>Or.inl</a> one_lt_p)]\n    simp [<a>Real.zero_rpow</a> p_pos.ne.symm]", [{"full_name": "integral_rpow", "def_path": "Mathlib/Analysis/SpecialFunctions/Integrals.lean", "def_pos": [348, 9], "def_end_pos": [348, 22]}, {"full_name": "Or.inl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [517, 5], "def_end_pos": [517, 8]}, {"full_name": "Real.zero_rpow", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [103, 9], "def_end_pos": [103, 18]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nf_mble : AEMeasurable f\np : \u211d\np_pos : 0 < p\none_lt_p : -1 < p - 1\n\u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9 ^ p) \u2202\u03bc =\n    ENNReal.ofReal p * \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (t ^ (p - 1))", "state_after": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nf_mble : AEMeasurable f\np : \u211d\np_pos : 0 < p\none_lt_p : -1 < p - 1\nobs : \u2200 (x : \u211d), \u222b (t : \u211d) in 0 ..x, t ^ (p - 1) = x ^ p / p\n\u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9 ^ p) \u2202\u03bc =\n    ENNReal.ofReal p * \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (t ^ (p - 1))"}, {"tactic": "set g := fun t : \u211d => t ^ (p - 1)", "annotated_tactic": ["set g := fun t : \u211d => t ^ (p - 1)", []], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nf_mble : AEMeasurable f\np : \u211d\np_pos : 0 < p\none_lt_p : -1 < p - 1\nobs : \u2200 (x : \u211d), \u222b (t : \u211d) in 0 ..x, t ^ (p - 1) = x ^ p / p\n\u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9 ^ p) \u2202\u03bc =\n    ENNReal.ofReal p * \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (t ^ (p - 1))", "state_after": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng\u271d : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nf_mble : AEMeasurable f\np : \u211d\np_pos : 0 < p\none_lt_p : -1 < p - 1\ng : \u211d \u2192 \u211d := fun t => t ^ (p - 1)\nobs : \u2200 (x : \u211d), intervalIntegral g 0 x volume = x ^ p / p\n\u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9 ^ p) \u2202\u03bc =\n    ENNReal.ofReal p * \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (t ^ (p - 1))"}, {"tactic": "have g_nn : \u2200\u1d50 t \u2202volume.restrict (Ioi (0 : \u211d)), 0 \u2264 g t := by\n  filter_upwards [self_mem_ae_restrict (measurableSet_Ioi : MeasurableSet (Ioi (0 : \u211d)))]\n  intro t t_pos\n  exact Real.rpow_nonneg_of_nonneg (mem_Ioi.mp t_pos).le (p - 1)", "annotated_tactic": ["have g_nn : \u2200\u1d50 t \u2202volume.restrict (<a>Ioi</a> (0 : \u211d)), 0 \u2264 g t := by\n    filter_upwards [<a>self_mem_ae_restrict</a> (<a>measurableSet_Ioi</a> : <a>MeasurableSet</a> (<a>Ioi</a> (0 : \u211d)))]\n    intro t t_pos\n    exact <a>Real.rpow_nonneg_of_nonneg</a> (mem_Ioi.mp t_pos).<a>le</a> (p - 1)", [{"full_name": "Set.Ioi", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [79, 5], "def_end_pos": [79, 8]}, {"full_name": "MeasureTheory.self_mem_ae_restrict", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2681, 9], "def_end_pos": [2681, 29]}, {"full_name": "measurableSet_Ioi", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [579, 9], "def_end_pos": [579, 26]}, {"full_name": "MeasurableSet", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [64, 5], "def_end_pos": [64, 18]}, {"full_name": "Set.Ioi", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [79, 5], "def_end_pos": [79, 8]}, {"full_name": "Real.rpow_nonneg_of_nonneg", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [141, 9], "def_end_pos": [141, 30]}, {"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [142, 7], "def_end_pos": [142, 15]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng\u271d : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nf_mble : AEMeasurable f\np : \u211d\np_pos : 0 < p\none_lt_p : -1 < p - 1\ng : \u211d \u2192 \u211d := fun t => t ^ (p - 1)\nobs : \u2200 (x : \u211d), intervalIntegral g 0 x volume = x ^ p / p\n\u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9 ^ p) \u2202\u03bc =\n    ENNReal.ofReal p * \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (t ^ (p - 1))", "state_after": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng\u271d : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nf_mble : AEMeasurable f\np : \u211d\np_pos : 0 < p\none_lt_p : -1 < p - 1\ng : \u211d \u2192 \u211d := fun t => t ^ (p - 1)\nobs : \u2200 (x : \u211d), intervalIntegral g 0 x volume = x ^ p / p\ng_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t\n\u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9 ^ p) \u2202\u03bc =\n    ENNReal.ofReal p * \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (t ^ (p - 1))"}, {"tactic": "have g_intble : \u2200 t > 0, IntervalIntegrable g volume 0 t := fun _ _ =>\n  intervalIntegral.intervalIntegrable_rpow' one_lt_p", "annotated_tactic": ["have g_intble : \u2200 t > 0, <a>IntervalIntegrable</a> g <a>volume</a> 0 t := fun _ _ =>\n    <a>intervalIntegral.intervalIntegrable_rpow'</a> one_lt_p", [{"full_name": "IntervalIntegrable", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [70, 5], "def_end_pos": [70, 23]}, {"full_name": "MeasureTheory.MeasureSpace.volume", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [663, 3], "def_end_pos": [663, 9]}, {"full_name": "intervalIntegral.intervalIntegrable_rpow'", "def_path": "Mathlib/Analysis/SpecialFunctions/Integrals.lean", "def_pos": [73, 9], "def_end_pos": [73, 33]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng\u271d : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nf_mble : AEMeasurable f\np : \u211d\np_pos : 0 < p\none_lt_p : -1 < p - 1\ng : \u211d \u2192 \u211d := fun t => t ^ (p - 1)\nobs : \u2200 (x : \u211d), intervalIntegral g 0 x volume = x ^ p / p\ng_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t\n\u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9 ^ p) \u2202\u03bc =\n    ENNReal.ofReal p * \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (t ^ (p - 1))", "state_after": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng\u271d : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nf_mble : AEMeasurable f\np : \u211d\np_pos : 0 < p\none_lt_p : -1 < p - 1\ng : \u211d \u2192 \u211d := fun t => t ^ (p - 1)\nobs : \u2200 (x : \u211d), intervalIntegral g 0 x volume = x ^ p / p\ng_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\n\u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9 ^ p) \u2202\u03bc =\n    ENNReal.ofReal p * \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (t ^ (p - 1))"}, {"tactic": "have key := lintegral_comp_eq_lintegral_meas_le_mul \u03bc f_nn f_mble g_intble g_nn", "annotated_tactic": ["have key := <a>lintegral_comp_eq_lintegral_meas_le_mul</a> \u03bc f_nn f_mble g_intble g_nn", [{"full_name": "MeasureTheory.lintegral_comp_eq_lintegral_meas_le_mul", "def_path": "Mathlib/MeasureTheory/Integral/Layercake.lean", "def_pos": [395, 9], "def_end_pos": [395, 48]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng\u271d : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nf_mble : AEMeasurable f\np : \u211d\np_pos : 0 < p\none_lt_p : -1 < p - 1\ng : \u211d \u2192 \u211d := fun t => t ^ (p - 1)\nobs : \u2200 (x : \u211d), intervalIntegral g 0 x volume = x ^ p / p\ng_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\n\u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9 ^ p) \u2202\u03bc =\n    ENNReal.ofReal p * \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (t ^ (p - 1))", "state_after": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng\u271d : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nf_mble : AEMeasurable f\np : \u211d\np_pos : 0 < p\none_lt_p : -1 < p - 1\ng : \u211d \u2192 \u211d := fun t => t ^ (p - 1)\nobs : \u2200 (x : \u211d), intervalIntegral g 0 x volume = x ^ p / p\ng_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\nkey :\n  \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =\n    \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)\n\u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9 ^ p) \u2202\u03bc =\n    ENNReal.ofReal p * \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (t ^ (p - 1))"}, {"tactic": "rw [\u2190 key, \u2190 lintegral_const_mul'' (ENNReal.ofReal p)] <;> simp_rw [obs]", "annotated_tactic": ["rw [\u2190 key, \u2190 <a>lintegral_const_mul''</a> (<a>ENNReal.ofReal</a> p)] <;> simp_rw [obs]", [{"full_name": "MeasureTheory.lintegral_const_mul''", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [691, 9], "def_end_pos": [691, 30]}, {"full_name": "ENNReal.ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [172, 29], "def_end_pos": [172, 35]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng\u271d : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nf_mble : AEMeasurable f\np : \u211d\np_pos : 0 < p\none_lt_p : -1 < p - 1\ng : \u211d \u2192 \u211d := fun t => t ^ (p - 1)\nobs : \u2200 (x : \u211d), intervalIntegral g 0 x volume = x ^ p / p\ng_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\nkey :\n  \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =\n    \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)\n\u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9 ^ p) \u2202\u03bc =\n    ENNReal.ofReal p * \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (t ^ (p - 1))", "state_after": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng\u271d : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nf_mble : AEMeasurable f\np : \u211d\np_pos : 0 < p\none_lt_p : -1 < p - 1\ng : \u211d \u2192 \u211d := fun t => t ^ (p - 1)\nobs : \u2200 (x : \u211d), intervalIntegral g 0 x volume = x ^ p / p\ng_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\nkey :\n  \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =\n    \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)\n\u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9 ^ p) \u2202\u03bc = \u222b\u207b (a : \u03b1), ENNReal.ofReal p * ENNReal.ofReal (f a ^ p / p) \u2202\u03bc\n\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng\u271d : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nf_mble : AEMeasurable f\np : \u211d\np_pos : 0 < p\none_lt_p : -1 < p - 1\ng : \u211d \u2192 \u211d := fun t => t ^ (p - 1)\nobs : \u2200 (x : \u211d), intervalIntegral g 0 x volume = x ^ p / p\ng_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\nkey :\n  \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =\n    \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)\n\u22a2 AEMeasurable fun \u03c9 => ENNReal.ofReal (f \u03c9 ^ p / p)"}, {"tactic": "linarith", "annotated_tactic": ["linarith", []], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nf_mble : AEMeasurable f\np : \u211d\np_pos : 0 < p\n\u22a2 -1 < p - 1", "state_after": "no goals"}, {"tactic": "intro x", "annotated_tactic": ["intro x", []], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nf_mble : AEMeasurable f\np : \u211d\np_pos : 0 < p\none_lt_p : -1 < p - 1\n\u22a2 \u2200 (x : \u211d), \u222b (t : \u211d) in 0 ..x, t ^ (p - 1) = x ^ p / p", "state_after": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nf_mble : AEMeasurable f\np : \u211d\np_pos : 0 < p\none_lt_p : -1 < p - 1\nx : \u211d\n\u22a2 \u222b (t : \u211d) in 0 ..x, t ^ (p - 1) = x ^ p / p"}, {"tactic": "rw [integral_rpow (Or.inl one_lt_p)]", "annotated_tactic": ["rw [<a>integral_rpow</a> (<a>Or.inl</a> one_lt_p)]", [{"full_name": "integral_rpow", "def_path": "Mathlib/Analysis/SpecialFunctions/Integrals.lean", "def_pos": [348, 9], "def_end_pos": [348, 22]}, {"full_name": "Or.inl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [517, 5], "def_end_pos": [517, 8]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nf_mble : AEMeasurable f\np : \u211d\np_pos : 0 < p\none_lt_p : -1 < p - 1\nx : \u211d\n\u22a2 \u222b (t : \u211d) in 0 ..x, t ^ (p - 1) = x ^ p / p", "state_after": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nf_mble : AEMeasurable f\np : \u211d\np_pos : 0 < p\none_lt_p : -1 < p - 1\nx : \u211d\n\u22a2 (x ^ (p - 1 + 1) - 0 ^ (p - 1 + 1)) / (p - 1 + 1) = x ^ p / p"}, {"tactic": "simp [Real.zero_rpow p_pos.ne.symm]", "annotated_tactic": ["simp [<a>Real.zero_rpow</a> p_pos.ne.symm]", [{"full_name": "Real.zero_rpow", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [103, 9], "def_end_pos": [103, 18]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nf_mble : AEMeasurable f\np : \u211d\np_pos : 0 < p\none_lt_p : -1 < p - 1\nx : \u211d\n\u22a2 (x ^ (p - 1 + 1) - 0 ^ (p - 1 + 1)) / (p - 1 + 1) = x ^ p / p", "state_after": "no goals"}, {"tactic": "filter_upwards [self_mem_ae_restrict (measurableSet_Ioi : MeasurableSet (Ioi (0 : \u211d)))]", "annotated_tactic": ["filter_upwards [<a>self_mem_ae_restrict</a> (<a>measurableSet_Ioi</a> : <a>MeasurableSet</a> (<a>Ioi</a> (0 : \u211d)))]", [{"full_name": "MeasureTheory.self_mem_ae_restrict", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2681, 9], "def_end_pos": [2681, 29]}, {"full_name": "measurableSet_Ioi", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [579, 9], "def_end_pos": [579, 26]}, {"full_name": "MeasurableSet", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [64, 5], "def_end_pos": [64, 18]}, {"full_name": "Set.Ioi", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [79, 5], "def_end_pos": [79, 8]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng\u271d : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nf_mble : AEMeasurable f\np : \u211d\np_pos : 0 < p\none_lt_p : -1 < p - 1\ng : \u211d \u2192 \u211d := fun t => t ^ (p - 1)\nobs : \u2200 (x : \u211d), intervalIntegral g 0 x volume = x ^ p / p\n\u22a2 \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t", "state_after": "case h\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng\u271d : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nf_mble : AEMeasurable f\np : \u211d\np_pos : 0 < p\none_lt_p : -1 < p - 1\ng : \u211d \u2192 \u211d := fun t => t ^ (p - 1)\nobs : \u2200 (x : \u211d), intervalIntegral g 0 x volume = x ^ p / p\n\u22a2 \u2200 (a : \u211d), a \u2208 Ioi 0 \u2192 0 \u2264 g a"}, {"tactic": "intro t t_pos", "annotated_tactic": ["intro t t_pos", []], "state_before": "case h\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng\u271d : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nf_mble : AEMeasurable f\np : \u211d\np_pos : 0 < p\none_lt_p : -1 < p - 1\ng : \u211d \u2192 \u211d := fun t => t ^ (p - 1)\nobs : \u2200 (x : \u211d), intervalIntegral g 0 x volume = x ^ p / p\n\u22a2 \u2200 (a : \u211d), a \u2208 Ioi 0 \u2192 0 \u2264 g a", "state_after": "case h\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng\u271d : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nf_mble : AEMeasurable f\np : \u211d\np_pos : 0 < p\none_lt_p : -1 < p - 1\ng : \u211d \u2192 \u211d := fun t => t ^ (p - 1)\nobs : \u2200 (x : \u211d), intervalIntegral g 0 x volume = x ^ p / p\nt : \u211d\nt_pos : t \u2208 Ioi 0\n\u22a2 0 \u2264 g t"}, {"tactic": "exact Real.rpow_nonneg_of_nonneg (mem_Ioi.mp t_pos).le (p - 1)", "annotated_tactic": ["exact <a>Real.rpow_nonneg_of_nonneg</a> (mem_Ioi.mp t_pos).<a>le</a> (p - 1)", [{"full_name": "Real.rpow_nonneg_of_nonneg", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [141, 9], "def_end_pos": [141, 30]}, {"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [142, 7], "def_end_pos": [142, 15]}]], "state_before": "case h\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng\u271d : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nf_mble : AEMeasurable f\np : \u211d\np_pos : 0 < p\none_lt_p : -1 < p - 1\ng : \u211d \u2192 \u211d := fun t => t ^ (p - 1)\nobs : \u2200 (x : \u211d), intervalIntegral g 0 x volume = x ^ p / p\nt : \u211d\nt_pos : t \u2208 Ioi 0\n\u22a2 0 \u2264 g t", "state_after": "no goals"}, {"tactic": "congr with \u03c9", "annotated_tactic": ["congr with \u03c9", []], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng\u271d : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nf_mble : AEMeasurable f\np : \u211d\np_pos : 0 < p\none_lt_p : -1 < p - 1\ng : \u211d \u2192 \u211d := fun t => t ^ (p - 1)\nobs : \u2200 (x : \u211d), intervalIntegral g 0 x volume = x ^ p / p\ng_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\nkey :\n  \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =\n    \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)\n\u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9 ^ p) \u2202\u03bc = \u222b\u207b (a : \u03b1), ENNReal.ofReal p * ENNReal.ofReal (f a ^ p / p) \u2202\u03bc", "state_after": "case e_f.h\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng\u271d : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nf_mble : AEMeasurable f\np : \u211d\np_pos : 0 < p\none_lt_p : -1 < p - 1\ng : \u211d \u2192 \u211d := fun t => t ^ (p - 1)\nobs : \u2200 (x : \u211d), intervalIntegral g 0 x volume = x ^ p / p\ng_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\nkey :\n  \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =\n    \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)\n\u03c9 : \u03b1\n\u22a2 ENNReal.ofReal (f \u03c9 ^ p) = ENNReal.ofReal p * ENNReal.ofReal (f \u03c9 ^ p / p)"}, {"tactic": "rw [\u2190 ENNReal.ofReal_mul p_pos.le, mul_div_cancel' (f \u03c9 ^ p) p_pos.ne.symm]", "annotated_tactic": ["rw [\u2190 <a>ENNReal.ofReal_mul</a> p_pos.le, <a>mul_div_cancel'</a> (f \u03c9 ^ p) p_pos.ne.symm]", [{"full_name": "ENNReal.ofReal_mul", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2225, 9], "def_end_pos": [2225, 19]}, {"full_name": "mul_div_cancel'", "def_path": "Mathlib/Algebra/GroupWithZero/Units/Lemmas.lean", "def_pos": [173, 9], "def_end_pos": [173, 24]}]], "state_before": "case e_f.h\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng\u271d : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nf_mble : AEMeasurable f\np : \u211d\np_pos : 0 < p\none_lt_p : -1 < p - 1\ng : \u211d \u2192 \u211d := fun t => t ^ (p - 1)\nobs : \u2200 (x : \u211d), intervalIntegral g 0 x volume = x ^ p / p\ng_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\nkey :\n  \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =\n    \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)\n\u03c9 : \u03b1\n\u22a2 ENNReal.ofReal (f \u03c9 ^ p) = ENNReal.ofReal p * ENNReal.ofReal (f \u03c9 ^ p / p)", "state_after": "no goals"}, {"tactic": "have aux := (@measurable_const \u211d \u03b1 (by infer_instance) (by infer_instance) p).aemeasurable\n              (\u03bc := \u03bc)", "annotated_tactic": ["have aux := (@<a>measurable_const</a> \u211d \u03b1 (by infer_instance) (by infer_instance) p).<a>aemeasurable</a>\n                  (\u03bc := \u03bc)", [{"full_name": "measurable_const", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [570, 9], "def_end_pos": [570, 25]}, {"full_name": "Measurable.aemeasurable", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [713, 9], "def_end_pos": [713, 32]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng\u271d : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nf_mble : AEMeasurable f\np : \u211d\np_pos : 0 < p\none_lt_p : -1 < p - 1\ng : \u211d \u2192 \u211d := fun t => t ^ (p - 1)\nobs : \u2200 (x : \u211d), intervalIntegral g 0 x volume = x ^ p / p\ng_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\nkey :\n  \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =\n    \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)\n\u22a2 AEMeasurable fun \u03c9 => ENNReal.ofReal (f \u03c9 ^ p / p)", "state_after": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng\u271d : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nf_mble : AEMeasurable f\np : \u211d\np_pos : 0 < p\none_lt_p : -1 < p - 1\ng : \u211d \u2192 \u211d := fun t => t ^ (p - 1)\nobs : \u2200 (x : \u211d), intervalIntegral g 0 x volume = x ^ p / p\ng_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\nkey :\n  \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =\n    \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)\naux : AEMeasurable fun x => p\n\u22a2 AEMeasurable fun \u03c9 => ENNReal.ofReal (f \u03c9 ^ p / p)"}, {"tactic": "exact (Measurable.ennreal_ofReal (hf := measurable_id)).comp_aemeasurable\n  ((f_mble.pow aux).div_const p)", "annotated_tactic": ["exact (<a>Measurable.ennreal_ofReal</a> (hf := <a>measurable_id</a>)).<a>comp_aemeasurable</a>\n      ((f_mble.pow aux).<a>div_const</a> p)", [{"full_name": "Measurable.ennreal_ofReal", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [2005, 9], "def_end_pos": [2005, 34]}, {"full_name": "measurable_id", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [550, 9], "def_end_pos": [550, 22]}, {"full_name": "Measurable.comp_aemeasurable", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [760, 9], "def_end_pos": [760, 37]}, {"full_name": "AEMeasurable.div_const", "def_path": "Mathlib/MeasureTheory/Group/Arithmetic.lean", "def_pos": [320, 9], "def_end_pos": [320, 31]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng\u271d : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nf_mble : AEMeasurable f\np : \u211d\np_pos : 0 < p\none_lt_p : -1 < p - 1\ng : \u211d \u2192 \u211d := fun t => t ^ (p - 1)\nobs : \u2200 (x : \u211d), intervalIntegral g 0 x volume = x ^ p / p\ng_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\nkey :\n  \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =\n    \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)\naux : AEMeasurable fun x => p\n\u22a2 AEMeasurable fun \u03c9 => ENNReal.ofReal (f \u03c9 ^ p / p)", "state_after": "no goals"}, {"tactic": "infer_instance", "annotated_tactic": ["infer_instance", []], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng\u271d : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nf_mble : AEMeasurable f\np : \u211d\np_pos : 0 < p\none_lt_p : -1 < p - 1\ng : \u211d \u2192 \u211d := fun t => t ^ (p - 1)\nobs : \u2200 (x : \u211d), intervalIntegral g 0 x volume = x ^ p / p\ng_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\nkey :\n  \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =\n    \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)\n\u22a2 MeasurableSpace \u211d", "state_after": "no goals"}, {"tactic": "infer_instance", "annotated_tactic": ["infer_instance", []], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng\u271d : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nf_mble : AEMeasurable f\np : \u211d\np_pos : 0 < p\none_lt_p : -1 < p - 1\ng : \u211d \u2192 \u211d := fun t => t ^ (p - 1)\nobs : \u2200 (x : \u211d), intervalIntegral g 0 x volume = x ^ p / p\ng_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\nkey :\n  \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =\n    \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)\n\u22a2 MeasurableSpace \u03b1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/Division.lean", "full_name": "MvPolynomial.divMonomial_monomial", "start": [94, 1], "end": [95, 31], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "full_name": "MeasureTheory.Lp.coeFn_sub", "start": [236, 1], "end": [237, 24], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "full_name": "MeasureTheory.Lp.simpleFunc.toLp_sub", "start": [565, 1], "end": [567, 53], "traced_tactics": [{"tactic": "simp only [sub_eq_add_neg, \u2190 toLp_neg, \u2190 toLp_add]", "annotated_tactic": ["simp only [<a>sub_eq_add_neg</a>, \u2190 <a>toLp_neg</a>, \u2190 <a>toLp_add</a>]", [{"full_name": "sub_eq_add_neg", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [975, 3], "def_end_pos": [975, 14]}, {"full_name": "MeasureTheory.Lp.simpleFunc.toLp_neg", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "def_pos": [561, 9], "def_end_pos": [561, 17]}, {"full_name": "MeasureTheory.Lp.simpleFunc.toLp_add", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "def_pos": [556, 9], "def_end_pos": [556, 17]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192\u209b E\nhf : Mem\u2112p (\u2191f) p\nhg : Mem\u2112p (\u2191g) p\n\u22a2 toLp (f - g) (_ : Mem\u2112p (\u2191f - fun a => \u2191g a) p) = toLp f hf - toLp g hg", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "full_name": "MeasureTheory.weightedSMul_union", "start": [222, 1], "end": [225, 57], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "full_name": "MeasureTheory.set_integral_congr\u2080", "start": [82, 1], "end": [84, 54], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/Periodic.lean", "full_name": "AddCircle.measurePreserving_mk", "start": [86, 11], "end": [93, 91], "traced_tactics": [{"tactic": "apply MeasurePreservingQuotientAddGroup.mk'", "annotated_tactic": ["apply <a>MeasurePreservingQuotientAddGroup.mk'</a>", [{"full_name": "MeasurePreservingQuotientAddGroup.mk'", "def_path": "Mathlib/MeasureTheory/Measure/Haar/Quotient.lean", "def_pos": [145, 15], "def_end_pos": [145, 52]}]], "state_before": "T : \u211d\nhT : Fact (0 < T)\nt : \u211d\n\u22a2 MeasurePreserving QuotientAddGroup.mk", "state_after": "case h\ud835\udcd5\nT : \u211d\nhT : Fact (0 < T)\nt : \u211d\n\u22a2 IsAddFundamentalDomain { x // x \u2208 AddSubgroup.op (zmultiples T) } (Ioc t (t + T))\n\ncase h\ud835\udcd5_finite\nT : \u211d\nhT : Fact (0 < T)\nt : \u211d\n\u22a2 \u2191\u2191volume (Ioc t (t + T)) < \u22a4\n\ncase h\nT : \u211d\nhT : Fact (0 < T)\nt : \u211d\n\u22a2 \u2191\u2191volume (Ioc t (t + T) \u2229 \u2191(QuotientAddGroup.mk' (zmultiples T)) \u207b\u00b9' \u2191\u22a4) = \u2191(Real.toNNReal T)"}, {"tactic": "exact isAddFundamentalDomain_Ioc' hT.out t", "annotated_tactic": ["exact <a>isAddFundamentalDomain_Ioc'</a> hT.out t", [{"full_name": "isAddFundamentalDomain_Ioc'", "def_path": "Mathlib/MeasureTheory/Integral/Periodic.lean", "def_pos": [49, 9], "def_end_pos": [49, 36]}]], "state_before": "case h\ud835\udcd5\nT : \u211d\nhT : Fact (0 < T)\nt : \u211d\n\u22a2 IsAddFundamentalDomain { x // x \u2208 AddSubgroup.op (zmultiples T) } (Ioc t (t + T))", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case h\ud835\udcd5_finite\nT : \u211d\nhT : Fact (0 < T)\nt : \u211d\n\u22a2 \u2191\u2191volume (Ioc t (t + T)) < \u22a4", "state_after": "no goals"}, {"tactic": "haveI : CompactSpace (\u211d \u29f8 zmultiples T) := inferInstanceAs (CompactSpace (AddCircle T))", "annotated_tactic": ["haveI : <a>CompactSpace</a> (\u211d \u29f8 <a>zmultiples</a> T) := <a>inferInstanceAs</a> (<a>CompactSpace</a> (<a>AddCircle</a> T))", [{"full_name": "CompactSpace", "def_path": "Mathlib/Topology/Compactness/Compact.lean", "def_pos": [669, 7], "def_end_pos": [669, 19]}, {"full_name": "AddSubgroup.zmultiples", "def_path": "Mathlib/GroupTheory/Subgroup/ZPowers.lean", "def_pos": [85, 5], "def_end_pos": [85, 15]}, {"full_name": "inferInstanceAs", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [100, 8], "def_end_pos": [100, 23]}, {"full_name": "CompactSpace", "def_path": "Mathlib/Topology/Compactness/Compact.lean", "def_pos": [669, 7], "def_end_pos": [669, 19]}, {"full_name": "AddCircle", "def_path": "Mathlib/Topology/Instances/AddCircle.lean", "def_pos": [124, 8], "def_end_pos": [124, 17]}]], "state_before": "case h\nT : \u211d\nhT : Fact (0 < T)\nt : \u211d\n\u22a2 \u2191\u2191volume (Ioc t (t + T) \u2229 \u2191(QuotientAddGroup.mk' (zmultiples T)) \u207b\u00b9' \u2191\u22a4) = \u2191(Real.toNNReal T)", "state_after": "case h\nT : \u211d\nhT : Fact (0 < T)\nt : \u211d\nthis : CompactSpace (\u211d \u29f8 zmultiples T)\n\u22a2 \u2191\u2191volume (Ioc t (t + T) \u2229 \u2191(QuotientAddGroup.mk' (zmultiples T)) \u207b\u00b9' \u2191\u22a4) = \u2191(Real.toNNReal T)"}, {"tactic": "simp [\u2190 ENNReal.ofReal_coe_nnreal, Real.coe_toNNReal T hT.out.le, -Real.coe_toNNReal']", "annotated_tactic": ["simp [\u2190 <a>ENNReal.ofReal_coe_nnreal</a>, <a>Real.coe_toNNReal</a> T hT.out.le, -<a>Real.coe_toNNReal'</a>]", [{"full_name": "ENNReal.ofReal_coe_nnreal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [212, 17], "def_end_pos": [212, 34]}, {"full_name": "Real.coe_toNNReal", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [122, 9], "def_end_pos": [122, 33]}, {"full_name": "Real.coe_toNNReal'", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [615, 9], "def_end_pos": [615, 22]}]], "state_before": "case h\nT : \u211d\nhT : Fact (0 < T)\nt : \u211d\nthis : CompactSpace (\u211d \u29f8 zmultiples T)\n\u22a2 \u2191\u2191volume (Ioc t (t + T) \u2229 \u2191(QuotientAddGroup.mk' (zmultiples T)) \u207b\u00b9' \u2191\u22a4) = \u2191(Real.toNNReal T)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/Comap.lean", "full_name": "MvPolynomial.comap_comp", "start": [77, 1], "end": [80, 31], "traced_tactics": [{"tactic": "funext x", "annotated_tactic": ["funext x", []], "state_before": "\u03c3 : Type u_1\n\u03c4 : Type u_2\n\u03c5 : Type u_3\nR : Type u_4\ninst\u271d : CommSemiring R\nf : MvPolynomial \u03c3 R \u2192\u2090[R] MvPolynomial \u03c4 R\ng : MvPolynomial \u03c4 R \u2192\u2090[R] MvPolynomial \u03c5 R\n\u22a2 comap (AlgHom.comp g f) = comap f \u2218 comap g", "state_after": "case h\n\u03c3 : Type u_1\n\u03c4 : Type u_2\n\u03c5 : Type u_3\nR : Type u_4\ninst\u271d : CommSemiring R\nf : MvPolynomial \u03c3 R \u2192\u2090[R] MvPolynomial \u03c4 R\ng : MvPolynomial \u03c4 R \u2192\u2090[R] MvPolynomial \u03c5 R\nx : \u03c5 \u2192 R\n\u22a2 comap (AlgHom.comp g f) x = (comap f \u2218 comap g) x"}, {"tactic": "exact comap_comp_apply _ _ _", "annotated_tactic": ["exact <a>comap_comp_apply</a> _ _ _", [{"full_name": "MvPolynomial.comap_comp_apply", "def_path": "Mathlib/Data/MvPolynomial/Comap.lean", "def_pos": [62, 9], "def_end_pos": [62, 25]}]], "state_before": "case h\n\u03c3 : Type u_1\n\u03c4 : Type u_2\n\u03c5 : Type u_3\nR : Type u_4\ninst\u271d : CommSemiring R\nf : MvPolynomial \u03c3 R \u2192\u2090[R] MvPolynomial \u03c4 R\ng : MvPolynomial \u03c4 R \u2192\u2090[R] MvPolynomial \u03c5 R\nx : \u03c5 \u2192 R\n\u22a2 comap (AlgHom.comp g f) x = (comap f \u2218 comap g) x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean", "full_name": "MeasureTheory.stronglyMeasurable_condexp", "start": [179, 1], "end": [189, 34], "traced_tactics": [{"tactic": "by_cases hm : m \u2264 m0", "annotated_tactic": ["by_cases hm : m \u2264 m0", []], "state_before": "\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\n\u22a2 StronglyMeasurable (\u03bc[f|m])", "state_after": "case pos\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nhm : m \u2264 m0\n\u22a2 StronglyMeasurable (\u03bc[f|m])\n\ncase neg\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nhm : \u00acm \u2264 m0\n\u22a2 StronglyMeasurable (\u03bc[f|m])"}, {"tactic": "swap", "annotated_tactic": ["swap", []], "state_before": "case pos\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nhm : m \u2264 m0\n\u22a2 StronglyMeasurable (\u03bc[f|m])\n\ncase neg\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nhm : \u00acm \u2264 m0\n\u22a2 StronglyMeasurable (\u03bc[f|m])", "state_after": "case neg\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nhm : \u00acm \u2264 m0\n\u22a2 StronglyMeasurable (\u03bc[f|m])\n\ncase pos\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nhm : m \u2264 m0\n\u22a2 StronglyMeasurable (\u03bc[f|m])"}, {"tactic": "by_cases h\u03bcm : SigmaFinite (\u03bc.trim hm)", "annotated_tactic": ["by_cases h\u03bcm : <a>SigmaFinite</a> (\u03bc.trim hm)", [{"full_name": "MeasureTheory.SigmaFinite", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3289, 7], "def_end_pos": [3289, 18]}]], "state_before": "case pos\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nhm : m \u2264 m0\n\u22a2 StronglyMeasurable (\u03bc[f|m])", "state_after": "case pos\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nhm : m \u2264 m0\nh\u03bcm : SigmaFinite (Measure.trim \u03bc hm)\n\u22a2 StronglyMeasurable (\u03bc[f|m])\n\ncase neg\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nhm : m \u2264 m0\nh\u03bcm : \u00acSigmaFinite (Measure.trim \u03bc hm)\n\u22a2 StronglyMeasurable (\u03bc[f|m])"}, {"tactic": "swap", "annotated_tactic": ["swap", []], "state_before": "case pos\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nhm : m \u2264 m0\nh\u03bcm : SigmaFinite (Measure.trim \u03bc hm)\n\u22a2 StronglyMeasurable (\u03bc[f|m])\n\ncase neg\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nhm : m \u2264 m0\nh\u03bcm : \u00acSigmaFinite (Measure.trim \u03bc hm)\n\u22a2 StronglyMeasurable (\u03bc[f|m])", "state_after": "case neg\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nhm : m \u2264 m0\nh\u03bcm : \u00acSigmaFinite (Measure.trim \u03bc hm)\n\u22a2 StronglyMeasurable (\u03bc[f|m])\n\ncase pos\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nhm : m \u2264 m0\nh\u03bcm : SigmaFinite (Measure.trim \u03bc hm)\n\u22a2 StronglyMeasurable (\u03bc[f|m])"}, {"tactic": "haveI : SigmaFinite (\u03bc.trim hm) := h\u03bcm", "annotated_tactic": ["haveI : <a>SigmaFinite</a> (\u03bc.trim hm) := h\u03bcm", [{"full_name": "MeasureTheory.SigmaFinite", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3289, 7], "def_end_pos": [3289, 18]}]], "state_before": "case pos\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nhm : m \u2264 m0\nh\u03bcm : SigmaFinite (Measure.trim \u03bc hm)\n\u22a2 StronglyMeasurable (\u03bc[f|m])", "state_after": "case pos\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nhm : m \u2264 m0\nh\u03bcm this : SigmaFinite (Measure.trim \u03bc hm)\n\u22a2 StronglyMeasurable (\u03bc[f|m])"}, {"tactic": "rw [condexp_of_sigmaFinite hm]", "annotated_tactic": ["rw [<a>condexp_of_sigmaFinite</a> hm]", [{"full_name": "MeasureTheory.condexp_of_sigmaFinite", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean", "def_pos": [113, 9], "def_end_pos": [113, 31]}]], "state_before": "case pos\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nhm : m \u2264 m0\nh\u03bcm this : SigmaFinite (Measure.trim \u03bc hm)\n\u22a2 StronglyMeasurable (\u03bc[f|m])", "state_after": "case pos\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nhm : m \u2264 m0\nh\u03bcm this : SigmaFinite (Measure.trim \u03bc hm)\n\u22a2 StronglyMeasurable\n    (if Integrable f then\n      if StronglyMeasurable f then f\n      else AEStronglyMeasurable'.mk \u2191\u2191(condexpL1 hm \u03bc f) (_ : AEStronglyMeasurable' m (\u2191\u2191(condexpL1 hm \u03bc f)) \u03bc)\n    else 0)"}, {"tactic": "split_ifs with hfi hfm", "annotated_tactic": ["split_ifs with hfi hfm", []], "state_before": "case pos\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nhm : m \u2264 m0\nh\u03bcm this : SigmaFinite (Measure.trim \u03bc hm)\n\u22a2 StronglyMeasurable\n    (if Integrable f then\n      if StronglyMeasurable f then f\n      else AEStronglyMeasurable'.mk \u2191\u2191(condexpL1 hm \u03bc f) (_ : AEStronglyMeasurable' m (\u2191\u2191(condexpL1 hm \u03bc f)) \u03bc)\n    else 0)", "state_after": "case pos\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nhm : m \u2264 m0\nh\u03bcm this : SigmaFinite (Measure.trim \u03bc hm)\nhfi : Integrable f\nhfm : StronglyMeasurable f\n\u22a2 StronglyMeasurable f\n\ncase neg\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nhm : m \u2264 m0\nh\u03bcm this : SigmaFinite (Measure.trim \u03bc hm)\nhfi : Integrable f\nhfm : \u00acStronglyMeasurable f\n\u22a2 StronglyMeasurable\n    (AEStronglyMeasurable'.mk \u2191\u2191(condexpL1 hm \u03bc f) (_ : AEStronglyMeasurable' m (\u2191\u2191(condexpL1 hm \u03bc f)) \u03bc))\n\ncase neg\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nhm : m \u2264 m0\nh\u03bcm this : SigmaFinite (Measure.trim \u03bc hm)\nhfi : \u00acIntegrable f\n\u22a2 StronglyMeasurable 0"}, {"tactic": "rw [condexp_of_not_le hm]", "annotated_tactic": ["rw [<a>condexp_of_not_le</a> hm]", [{"full_name": "MeasureTheory.condexp_of_not_le", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean", "def_pos": [106, 9], "def_end_pos": [106, 26]}]], "state_before": "case neg\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nhm : \u00acm \u2264 m0\n\u22a2 StronglyMeasurable (\u03bc[f|m])", "state_after": "case neg\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nhm : \u00acm \u2264 m0\n\u22a2 StronglyMeasurable 0"}, {"tactic": "exact stronglyMeasurable_zero", "annotated_tactic": ["exact <a>stronglyMeasurable_zero</a>", [{"full_name": "MeasureTheory.stronglyMeasurable_zero", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [148, 3], "def_end_pos": [148, 14]}]], "state_before": "case neg\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nhm : \u00acm \u2264 m0\n\u22a2 StronglyMeasurable 0", "state_after": "no goals"}, {"tactic": "rw [condexp_of_not_sigmaFinite hm h\u03bcm]", "annotated_tactic": ["rw [<a>condexp_of_not_sigmaFinite</a> hm h\u03bcm]", [{"full_name": "MeasureTheory.condexp_of_not_sigmaFinite", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean", "def_pos": [109, 9], "def_end_pos": [109, 35]}]], "state_before": "case neg\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nhm : m \u2264 m0\nh\u03bcm : \u00acSigmaFinite (Measure.trim \u03bc hm)\n\u22a2 StronglyMeasurable (\u03bc[f|m])", "state_after": "case neg\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nhm : m \u2264 m0\nh\u03bcm : \u00acSigmaFinite (Measure.trim \u03bc hm)\n\u22a2 StronglyMeasurable 0"}, {"tactic": "exact stronglyMeasurable_zero", "annotated_tactic": ["exact <a>stronglyMeasurable_zero</a>", [{"full_name": "MeasureTheory.stronglyMeasurable_zero", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [148, 3], "def_end_pos": [148, 14]}]], "state_before": "case neg\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nhm : m \u2264 m0\nh\u03bcm : \u00acSigmaFinite (Measure.trim \u03bc hm)\n\u22a2 StronglyMeasurable 0", "state_after": "no goals"}, {"tactic": "exact hfm", "annotated_tactic": ["exact hfm", []], "state_before": "case pos\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nhm : m \u2264 m0\nh\u03bcm this : SigmaFinite (Measure.trim \u03bc hm)\nhfi : Integrable f\nhfm : StronglyMeasurable f\n\u22a2 StronglyMeasurable f", "state_after": "no goals"}, {"tactic": "exact AEStronglyMeasurable'.stronglyMeasurable_mk _", "annotated_tactic": ["exact <a>AEStronglyMeasurable'.stronglyMeasurable_mk</a> _", [{"full_name": "MeasureTheory.AEStronglyMeasurable'.stronglyMeasurable_mk", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean", "def_pos": [114, 9], "def_end_pos": [114, 30]}]], "state_before": "case neg\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nhm : m \u2264 m0\nh\u03bcm this : SigmaFinite (Measure.trim \u03bc hm)\nhfi : Integrable f\nhfm : \u00acStronglyMeasurable f\n\u22a2 StronglyMeasurable\n    (AEStronglyMeasurable'.mk \u2191\u2191(condexpL1 hm \u03bc f) (_ : AEStronglyMeasurable' m (\u2191\u2191(condexpL1 hm \u03bc f)) \u03bc))", "state_after": "no goals"}, {"tactic": "exact stronglyMeasurable_zero", "annotated_tactic": ["exact <a>stronglyMeasurable_zero</a>", [{"full_name": "MeasureTheory.stronglyMeasurable_zero", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [148, 3], "def_end_pos": [148, 14]}]], "state_before": "case neg\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nhm : m \u2264 m0\nh\u03bcm this : SigmaFinite (Measure.trim \u03bc hm)\nhfi : \u00acIntegrable f\n\u22a2 StronglyMeasurable 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean", "full_name": "MeasureTheory.condexpL1_of_aestronglyMeasurable'", "start": [584, 1], "end": [589, 73], "traced_tactics": [{"tactic": "rw [condexpL1_eq hfi]", "annotated_tactic": ["rw [<a>condexpL1_eq</a> hfi]", [{"full_name": "MeasureTheory.condexpL1_eq", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean", "def_pos": [522, 9], "def_end_pos": [522, 21]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup F'\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2076 : NormedSpace \u211d F'\ninst\u271d\u2075 : CompleteSpace F'\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedAddCommGroup G'\ninst\u271d\u00b2 : NormedSpace \u211d G'\ninst\u271d\u00b9 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nhfm : AEStronglyMeasurable' m f \u03bc\nhfi : Integrable f\n\u22a2 \u2191\u2191(condexpL1 hm \u03bc f) =\u1d50[\u03bc] f", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup F'\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2076 : NormedSpace \u211d F'\ninst\u271d\u2075 : CompleteSpace F'\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedAddCommGroup G'\ninst\u271d\u00b2 : NormedSpace \u211d G'\ninst\u271d\u00b9 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nhfm : AEStronglyMeasurable' m f \u03bc\nhfi : Integrable f\n\u22a2 \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) (Integrable.toL1 f hfi)) =\u1d50[\u03bc] f"}, {"tactic": "refine' EventuallyEq.trans _ (Integrable.coeFn_toL1 hfi)", "annotated_tactic": ["refine' <a>EventuallyEq.trans</a> _ (<a>Integrable.coeFn_toL1</a> hfi)", [{"full_name": "Filter.EventuallyEq.trans", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1503, 9], "def_end_pos": [1503, 27]}, {"full_name": "MeasureTheory.Integrable.coeFn_toL1", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [1411, 9], "def_end_pos": [1411, 19]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup F'\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2076 : NormedSpace \u211d F'\ninst\u271d\u2075 : CompleteSpace F'\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedAddCommGroup G'\ninst\u271d\u00b2 : NormedSpace \u211d G'\ninst\u271d\u00b9 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nhfm : AEStronglyMeasurable' m f \u03bc\nhfi : Integrable f\n\u22a2 \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) (Integrable.toL1 f hfi)) =\u1d50[\u03bc] f", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup F'\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2076 : NormedSpace \u211d F'\ninst\u271d\u2075 : CompleteSpace F'\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedAddCommGroup G'\ninst\u271d\u00b2 : NormedSpace \u211d G'\ninst\u271d\u00b9 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nhfm : AEStronglyMeasurable' m f \u03bc\nhfi : Integrable f\n\u22a2 \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) (Integrable.toL1 f hfi)) =\u1d50[\u03bc] \u2191\u2191(Integrable.toL1 f hfi)"}, {"tactic": "rw [condexpL1Clm_of_aestronglyMeasurable']", "annotated_tactic": ["rw [<a>condexpL1Clm_of_aestronglyMeasurable'</a>]", [{"full_name": "MeasureTheory.condexpL1Clm_of_aestronglyMeasurable'", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean", "def_pos": [507, 9], "def_end_pos": [507, 46]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup F'\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2076 : NormedSpace \u211d F'\ninst\u271d\u2075 : CompleteSpace F'\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedAddCommGroup G'\ninst\u271d\u00b2 : NormedSpace \u211d G'\ninst\u271d\u00b9 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nhfm : AEStronglyMeasurable' m f \u03bc\nhfi : Integrable f\n\u22a2 \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) (Integrable.toL1 f hfi)) =\u1d50[\u03bc] \u2191\u2191(Integrable.toL1 f hfi)", "state_after": "case hfm\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup F'\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2076 : NormedSpace \u211d F'\ninst\u271d\u2075 : CompleteSpace F'\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedAddCommGroup G'\ninst\u271d\u00b2 : NormedSpace \u211d G'\ninst\u271d\u00b9 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nhfm : AEStronglyMeasurable' m f \u03bc\nhfi : Integrable f\n\u22a2 AEStronglyMeasurable' m (\u2191\u2191(Integrable.toL1 f hfi)) \u03bc"}, {"tactic": "exact AEStronglyMeasurable'.congr hfm (Integrable.coeFn_toL1 hfi).symm", "annotated_tactic": ["exact <a>AEStronglyMeasurable'.congr</a> hfm (<a>Integrable.coeFn_toL1</a> hfi).<a>symm</a>", [{"full_name": "MeasureTheory.AEStronglyMeasurable'.congr", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean", "def_pos": [59, 9], "def_end_pos": [59, 14]}, {"full_name": "MeasureTheory.Integrable.coeFn_toL1", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [1411, 9], "def_end_pos": [1411, 19]}, {"full_name": "Filter.EventuallyEq.symm", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1498, 9], "def_end_pos": [1498, 26]}]], "state_before": "case hfm\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup F'\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2076 : NormedSpace \u211d F'\ninst\u271d\u2075 : CompleteSpace F'\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedAddCommGroup G'\ninst\u271d\u00b2 : NormedSpace \u211d G'\ninst\u271d\u00b9 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nhfm : AEStronglyMeasurable' m f \u03bc\nhfi : Integrable f\n\u22a2 AEStronglyMeasurable' m (\u2191\u2191(Integrable.toL1 f hfi)) \u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Independence/Basic.lean", "full_name": "ProbabilityTheory.indep_iSup_of_monotone", "start": [447, 1], "end": [451, 54], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Halting.lean", "full_name": "Nat.Partrec'.part_iff\u2082", "start": [422, 1], "end": [427, 66], "traced_tactics": [{"tactic": "simp only [head_cons, tail_cons]", "annotated_tactic": ["simp only [<a>head_cons</a>, <a>tail_cons</a>]", [{"full_name": "Vector.head_cons", "def_path": "Mathlib/Data/Vector.lean", "def_pos": [64, 9], "def_end_pos": [64, 18]}, {"full_name": "Vector.tail_cons", "def_path": "Mathlib/Data/Vector.lean", "def_pos": [75, 9], "def_end_pos": [75, 18]}]], "state_before": "f : \u2115 \u2192 \u2115 \u2192. \u2115\nh : _root_.Partrec fun v => f (Vector.head v) (Vector.head (Vector.tail v))\nv : \u2115 \u00d7 \u2115\n\u22a2 f (Vector.head (v.1 ::\u1d65 v.2 ::\u1d65 nil)) (Vector.head (Vector.tail (v.1 ::\u1d65 v.2 ::\u1d65 nil))) = f v.1 v.2", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Group/GeometryOfNumbers.lean", "full_name": "MeasureTheory.exists_ne_zero_mem_lattice_of_measure_mul_two_pow_lt_measure", "start": [64, 1], "end": [83, 53], "traced_tactics": [{"tactic": "have h_vol : \u03bc F < \u03bc ((2\u207b\u00b9 : \u211d) \u2022 s) := by\n  rw [addHaar_smul_of_nonneg \u03bc (by norm_num : 0 \u2264 (2 : \u211d)\u207b\u00b9) s, \u2190\n    mul_lt_mul_right (pow_ne_zero (finrank \u211d E) (two_ne_zero' _)) (pow_ne_top two_ne_top),\n    mul_right_comm, ofReal_pow (by norm_num : 0 \u2264 (2 : \u211d)\u207b\u00b9), \u2190 ofReal_inv_of_pos zero_lt_two]\n  norm_num\n  rwa [\u2190 mul_pow, ENNReal.inv_mul_cancel two_ne_zero two_ne_top, one_pow, one_mul]", "annotated_tactic": ["have h_vol : \u03bc F < \u03bc ((2\u207b\u00b9 : \u211d) \u2022 s) := by\n    rw [<a>addHaar_smul_of_nonneg</a> \u03bc (by norm_num : 0 \u2264 (2 : \u211d)\u207b\u00b9) s, \u2190\n      <a>mul_lt_mul_right</a> (<a>pow_ne_zero</a> (<a>finrank</a> \u211d E) (<a>two_ne_zero'</a> _)) (<a>pow_ne_top</a> <a>two_ne_top</a>),\n      <a>mul_right_comm</a>, <a>ofReal_pow</a> (by norm_num : 0 \u2264 (2 : \u211d)\u207b\u00b9), \u2190 <a>ofReal_inv_of_pos</a> <a>zero_lt_two</a>]\n    norm_num\n    rwa [\u2190 <a>mul_pow</a>, <a>ENNReal.inv_mul_cancel</a> <a>two_ne_zero</a> <a>two_ne_top</a>, <a>one_pow</a>, <a>one_mul</a>]", [{"full_name": "MeasureTheory.Measure.addHaar_smul_of_nonneg", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean", "def_pos": [387, 9], "def_end_pos": [387, 31]}, {"full_name": "ENNReal.mul_lt_mul_right", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1080, 9], "def_end_pos": [1080, 25]}, {"full_name": "pow_ne_zero", "def_path": "Mathlib/Algebra/GroupPower/Ring.lean", "def_pos": [84, 9], "def_end_pos": [84, 20]}, {"full_name": "FiniteDimensional.finrank", "def_path": "Mathlib/LinearAlgebra/Finrank.lean", "def_pos": [58, 19], "def_end_pos": [58, 26]}, {"full_name": "two_ne_zero'", "def_path": "Mathlib/Algebra/NeZero.lean", "def_pos": [81, 7], "def_end_pos": [81, 19]}, {"full_name": "ENNReal.pow_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [662, 9], "def_end_pos": [662, 19]}, {"full_name": "ENNReal.two_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [431, 9], "def_end_pos": [431, 19]}, {"full_name": "mul_right_comm", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [102, 9], "def_end_pos": [102, 23]}, {"full_name": "ENNReal.ofReal_pow", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2235, 9], "def_end_pos": [2235, 19]}, {"full_name": "ENNReal.ofReal_inv_of_pos", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2243, 9], "def_end_pos": [2243, 26]}, {"full_name": "zero_lt_two", "def_path": "Mathlib/Algebra/Order/Monoid/NatCast.lean", "def_pos": [71, 15], "def_end_pos": [71, 26]}, {"full_name": "mul_pow", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [257, 9], "def_end_pos": [257, 16]}, {"full_name": "ENNReal.inv_mul_cancel", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1424, 19], "def_end_pos": [1424, 33]}, {"full_name": "two_ne_zero", "def_path": "Mathlib/Algebra/NeZero.lean", "def_pos": [62, 7], "def_end_pos": [62, 18]}, {"full_name": "ENNReal.two_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [431, 9], "def_end_pos": [431, 19]}, {"full_name": "one_pow", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [90, 9], "def_end_pos": [90, 16]}, {"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [464, 9], "def_end_pos": [464, 16]}]], "state_before": "E : Type u_1\nL\u271d : Type u_2\ninst\u271d\u2076 : MeasurableSpace E\n\u03bc : Measure E\nF s : Set E\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : BorelSpace E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bc\nL : AddSubgroup E\ninst\u271d : Countable { x // x \u2208 L }\nfund : IsAddFundamentalDomain { x // x \u2208 L } F\nh : \u2191\u2191\u03bc F * 2 ^ finrank \u211d E < \u2191\u2191\u03bc s\nh_symm : \u2200 (x : E), x \u2208 s \u2192 -x \u2208 s\nh_conv : Convex \u211d s\n\u22a2 \u2203 x x_1, \u2191x \u2208 s", "state_after": "E : Type u_1\nL\u271d : Type u_2\ninst\u271d\u2076 : MeasurableSpace E\n\u03bc : Measure E\nF s : Set E\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : BorelSpace E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bc\nL : AddSubgroup E\ninst\u271d : Countable { x // x \u2208 L }\nfund : IsAddFundamentalDomain { x // x \u2208 L } F\nh : \u2191\u2191\u03bc F * 2 ^ finrank \u211d E < \u2191\u2191\u03bc s\nh_symm : \u2200 (x : E), x \u2208 s \u2192 -x \u2208 s\nh_conv : Convex \u211d s\nh_vol : \u2191\u2191\u03bc F < \u2191\u2191\u03bc (2\u207b\u00b9 \u2022 s)\n\u22a2 \u2203 x x_1, \u2191x \u2208 s"}, {"tactic": "obtain \u27e8x, y, hxy, h\u27e9 :=\n  exists_pair_mem_lattice_not_disjoint_vadd fund ((h_conv.smul _).nullMeasurableSet _) h_vol", "annotated_tactic": ["obtain \u27e8x, y, hxy, h\u27e9 :=\n    <a>exists_pair_mem_lattice_not_disjoint_vadd</a> fund ((h_conv.smul _).<a>nullMeasurableSet</a> _) h_vol", [{"full_name": "MeasureTheory.exists_pair_mem_lattice_not_disjoint_vadd", "def_path": "Mathlib/MeasureTheory/Group/GeometryOfNumbers.lean", "def_pos": [50, 9], "def_end_pos": [50, 50]}, {"full_name": "Convex.nullMeasurableSet", "def_path": "Mathlib/Analysis/Convex/Measure.lean", "def_pos": [85, 19], "def_end_pos": [85, 36]}]], "state_before": "E : Type u_1\nL\u271d : Type u_2\ninst\u271d\u2076 : MeasurableSpace E\n\u03bc : Measure E\nF s : Set E\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : BorelSpace E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bc\nL : AddSubgroup E\ninst\u271d : Countable { x // x \u2208 L }\nfund : IsAddFundamentalDomain { x // x \u2208 L } F\nh : \u2191\u2191\u03bc F * 2 ^ finrank \u211d E < \u2191\u2191\u03bc s\nh_symm : \u2200 (x : E), x \u2208 s \u2192 -x \u2208 s\nh_conv : Convex \u211d s\nh_vol : \u2191\u2191\u03bc F < \u2191\u2191\u03bc (2\u207b\u00b9 \u2022 s)\n\u22a2 \u2203 x x_1, \u2191x \u2208 s", "state_after": "case intro.intro.intro\nE : Type u_1\nL\u271d : Type u_2\ninst\u271d\u2076 : MeasurableSpace E\n\u03bc : Measure E\nF s : Set E\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : BorelSpace E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bc\nL : AddSubgroup E\ninst\u271d : Countable { x // x \u2208 L }\nfund : IsAddFundamentalDomain { x // x \u2208 L } F\nh\u271d : \u2191\u2191\u03bc F * 2 ^ finrank \u211d E < \u2191\u2191\u03bc s\nh_symm : \u2200 (x : E), x \u2208 s \u2192 -x \u2208 s\nh_conv : Convex \u211d s\nh_vol : \u2191\u2191\u03bc F < \u2191\u2191\u03bc (2\u207b\u00b9 \u2022 s)\nx y : { x // x \u2208 L }\nhxy : x \u2260 y\nh : \u00acDisjoint (x +\u1d65 2\u207b\u00b9 \u2022 s) (y +\u1d65 2\u207b\u00b9 \u2022 s)\n\u22a2 \u2203 x x_1, \u2191x \u2208 s"}, {"tactic": "obtain \u27e8_, \u27e8v, hv, rfl\u27e9, w, hw, hvw\u27e9 := Set.not_disjoint_iff.mp h", "annotated_tactic": ["obtain \u27e8_, \u27e8v, hv, rfl\u27e9, w, hw, hvw\u27e9 := Set.not_disjoint_iff.mp h", []], "state_before": "case intro.intro.intro\nE : Type u_1\nL\u271d : Type u_2\ninst\u271d\u2076 : MeasurableSpace E\n\u03bc : Measure E\nF s : Set E\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : BorelSpace E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bc\nL : AddSubgroup E\ninst\u271d : Countable { x // x \u2208 L }\nfund : IsAddFundamentalDomain { x // x \u2208 L } F\nh\u271d : \u2191\u2191\u03bc F * 2 ^ finrank \u211d E < \u2191\u2191\u03bc s\nh_symm : \u2200 (x : E), x \u2208 s \u2192 -x \u2208 s\nh_conv : Convex \u211d s\nh_vol : \u2191\u2191\u03bc F < \u2191\u2191\u03bc (2\u207b\u00b9 \u2022 s)\nx y : { x // x \u2208 L }\nhxy : x \u2260 y\nh : \u00acDisjoint (x +\u1d65 2\u207b\u00b9 \u2022 s) (y +\u1d65 2\u207b\u00b9 \u2022 s)\n\u22a2 \u2203 x x_1, \u2191x \u2208 s", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro\nE : Type u_1\nL\u271d : Type u_2\ninst\u271d\u2076 : MeasurableSpace E\n\u03bc : Measure E\nF s : Set E\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : BorelSpace E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bc\nL : AddSubgroup E\ninst\u271d : Countable { x // x \u2208 L }\nfund : IsAddFundamentalDomain { x // x \u2208 L } F\nh\u271d : \u2191\u2191\u03bc F * 2 ^ finrank \u211d E < \u2191\u2191\u03bc s\nh_symm : \u2200 (x : E), x \u2208 s \u2192 -x \u2208 s\nh_conv : Convex \u211d s\nh_vol : \u2191\u2191\u03bc F < \u2191\u2191\u03bc (2\u207b\u00b9 \u2022 s)\nx y : { x // x \u2208 L }\nhxy : x \u2260 y\nh : \u00acDisjoint (x +\u1d65 2\u207b\u00b9 \u2022 s) (y +\u1d65 2\u207b\u00b9 \u2022 s)\nv : E\nhv : v \u2208 2\u207b\u00b9 \u2022 s\nw : E\nhw : w \u2208 2\u207b\u00b9 \u2022 s\nhvw : (fun x => y +\u1d65 x) w = (fun x_1 => x +\u1d65 x_1) v\n\u22a2 \u2203 x x_1, \u2191x \u2208 s"}, {"tactic": "refine' \u27e8x - y, sub_ne_zero.2 hxy, _\u27e9", "annotated_tactic": ["refine' \u27e8x - y, <a>sub_ne_zero</a>.2 hxy, _\u27e9", [{"full_name": "sub_ne_zero", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [813, 3], "def_end_pos": [813, 14]}]], "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro\nE : Type u_1\nL\u271d : Type u_2\ninst\u271d\u2076 : MeasurableSpace E\n\u03bc : Measure E\nF s : Set E\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : BorelSpace E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bc\nL : AddSubgroup E\ninst\u271d : Countable { x // x \u2208 L }\nfund : IsAddFundamentalDomain { x // x \u2208 L } F\nh\u271d : \u2191\u2191\u03bc F * 2 ^ finrank \u211d E < \u2191\u2191\u03bc s\nh_symm : \u2200 (x : E), x \u2208 s \u2192 -x \u2208 s\nh_conv : Convex \u211d s\nh_vol : \u2191\u2191\u03bc F < \u2191\u2191\u03bc (2\u207b\u00b9 \u2022 s)\nx y : { x // x \u2208 L }\nhxy : x \u2260 y\nh : \u00acDisjoint (x +\u1d65 2\u207b\u00b9 \u2022 s) (y +\u1d65 2\u207b\u00b9 \u2022 s)\nv : E\nhv : v \u2208 2\u207b\u00b9 \u2022 s\nw : E\nhw : w \u2208 2\u207b\u00b9 \u2022 s\nhvw : (fun x => y +\u1d65 x) w = (fun x_1 => x +\u1d65 x_1) v\n\u22a2 \u2203 x x_1, \u2191x \u2208 s", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro\nE : Type u_1\nL\u271d : Type u_2\ninst\u271d\u2076 : MeasurableSpace E\n\u03bc : Measure E\nF s : Set E\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : BorelSpace E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bc\nL : AddSubgroup E\ninst\u271d : Countable { x // x \u2208 L }\nfund : IsAddFundamentalDomain { x // x \u2208 L } F\nh\u271d : \u2191\u2191\u03bc F * 2 ^ finrank \u211d E < \u2191\u2191\u03bc s\nh_symm : \u2200 (x : E), x \u2208 s \u2192 -x \u2208 s\nh_conv : Convex \u211d s\nh_vol : \u2191\u2191\u03bc F < \u2191\u2191\u03bc (2\u207b\u00b9 \u2022 s)\nx y : { x // x \u2208 L }\nhxy : x \u2260 y\nh : \u00acDisjoint (x +\u1d65 2\u207b\u00b9 \u2022 s) (y +\u1d65 2\u207b\u00b9 \u2022 s)\nv : E\nhv : v \u2208 2\u207b\u00b9 \u2022 s\nw : E\nhw : w \u2208 2\u207b\u00b9 \u2022 s\nhvw : (fun x => y +\u1d65 x) w = (fun x_1 => x +\u1d65 x_1) v\n\u22a2 \u2191(x - y) \u2208 s"}, {"tactic": "rw [Set.mem_inv_smul_set_iff\u2080 (two_ne_zero' \u211d)] at hv hw", "annotated_tactic": ["rw [<a>Set.mem_inv_smul_set_iff\u2080</a> (<a>two_ne_zero'</a> \u211d)] at hv hw", [{"full_name": "Set.mem_inv_smul_set_iff\u2080", "def_path": "Mathlib/Data/Set/Pointwise/SMul.lean", "def_pos": [1023, 9], "def_end_pos": [1023, 30]}, {"full_name": "two_ne_zero'", "def_path": "Mathlib/Algebra/NeZero.lean", "def_pos": [81, 7], "def_end_pos": [81, 19]}]], "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro\nE : Type u_1\nL\u271d : Type u_2\ninst\u271d\u2076 : MeasurableSpace E\n\u03bc : Measure E\nF s : Set E\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : BorelSpace E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bc\nL : AddSubgroup E\ninst\u271d : Countable { x // x \u2208 L }\nfund : IsAddFundamentalDomain { x // x \u2208 L } F\nh\u271d : \u2191\u2191\u03bc F * 2 ^ finrank \u211d E < \u2191\u2191\u03bc s\nh_symm : \u2200 (x : E), x \u2208 s \u2192 -x \u2208 s\nh_conv : Convex \u211d s\nh_vol : \u2191\u2191\u03bc F < \u2191\u2191\u03bc (2\u207b\u00b9 \u2022 s)\nx y : { x // x \u2208 L }\nhxy : x \u2260 y\nh : \u00acDisjoint (x +\u1d65 2\u207b\u00b9 \u2022 s) (y +\u1d65 2\u207b\u00b9 \u2022 s)\nv : E\nhv : v \u2208 2\u207b\u00b9 \u2022 s\nw : E\nhw : w \u2208 2\u207b\u00b9 \u2022 s\nhvw : (fun x => y +\u1d65 x) w = (fun x_1 => x +\u1d65 x_1) v\n\u22a2 \u2191(x - y) \u2208 s", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro\nE : Type u_1\nL\u271d : Type u_2\ninst\u271d\u2076 : MeasurableSpace E\n\u03bc : Measure E\nF s : Set E\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : BorelSpace E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bc\nL : AddSubgroup E\ninst\u271d : Countable { x // x \u2208 L }\nfund : IsAddFundamentalDomain { x // x \u2208 L } F\nh\u271d : \u2191\u2191\u03bc F * 2 ^ finrank \u211d E < \u2191\u2191\u03bc s\nh_symm : \u2200 (x : E), x \u2208 s \u2192 -x \u2208 s\nh_conv : Convex \u211d s\nh_vol : \u2191\u2191\u03bc F < \u2191\u2191\u03bc (2\u207b\u00b9 \u2022 s)\nx y : { x // x \u2208 L }\nhxy : x \u2260 y\nh : \u00acDisjoint (x +\u1d65 2\u207b\u00b9 \u2022 s) (y +\u1d65 2\u207b\u00b9 \u2022 s)\nv : E\nhv : 2 \u2022 v \u2208 s\nw : E\nhw : 2 \u2022 w \u2208 s\nhvw : (fun x => y +\u1d65 x) w = (fun x_1 => x +\u1d65 x_1) v\n\u22a2 \u2191(x - y) \u2208 s"}, {"tactic": "simp_rw [AddSubgroup.vadd_def, vadd_eq_add, add_comm _ w, \u2190 sub_eq_sub_iff_add_eq_add, \u2190\n  AddSubgroup.coe_sub] at hvw", "annotated_tactic": ["simp_rw [<a>AddSubgroup.vadd_def</a>, <a>vadd_eq_add</a>, <a>add_comm</a> _ w, \u2190 <a>sub_eq_sub_iff_add_eq_add</a>, \u2190\n    <a>AddSubgroup.coe_sub</a>] at hvw", [{"full_name": "AddSubgroup.vadd_def", "def_path": "Mathlib/GroupTheory/Subgroup/Actions.lean", "def_pos": [32, 3], "def_end_pos": [32, 14]}, {"full_name": "vadd_eq_add", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [92, 3], "def_end_pos": [92, 14]}, {"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [301, 3], "def_end_pos": [301, 14]}, {"full_name": "sub_eq_sub_iff_add_eq_add", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [1003, 3], "def_end_pos": [1003, 14]}, {"full_name": "AddSubgroup.coe_sub", "def_path": "Mathlib/GroupTheory/Subgroup/Basic.lean", "def_pos": [718, 3], "def_end_pos": [718, 14]}]], "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro\nE : Type u_1\nL\u271d : Type u_2\ninst\u271d\u2076 : MeasurableSpace E\n\u03bc : Measure E\nF s : Set E\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : BorelSpace E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bc\nL : AddSubgroup E\ninst\u271d : Countable { x // x \u2208 L }\nfund : IsAddFundamentalDomain { x // x \u2208 L } F\nh\u271d : \u2191\u2191\u03bc F * 2 ^ finrank \u211d E < \u2191\u2191\u03bc s\nh_symm : \u2200 (x : E), x \u2208 s \u2192 -x \u2208 s\nh_conv : Convex \u211d s\nh_vol : \u2191\u2191\u03bc F < \u2191\u2191\u03bc (2\u207b\u00b9 \u2022 s)\nx y : { x // x \u2208 L }\nhxy : x \u2260 y\nh : \u00acDisjoint (x +\u1d65 2\u207b\u00b9 \u2022 s) (y +\u1d65 2\u207b\u00b9 \u2022 s)\nv : E\nhv : 2 \u2022 v \u2208 s\nw : E\nhw : 2 \u2022 w \u2208 s\nhvw : (fun x => y +\u1d65 x) w = (fun x_1 => x +\u1d65 x_1) v\n\u22a2 \u2191(x - y) \u2208 s", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro\nE : Type u_1\nL\u271d : Type u_2\ninst\u271d\u2076 : MeasurableSpace E\n\u03bc : Measure E\nF s : Set E\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : BorelSpace E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bc\nL : AddSubgroup E\ninst\u271d : Countable { x // x \u2208 L }\nfund : IsAddFundamentalDomain { x // x \u2208 L } F\nh\u271d : \u2191\u2191\u03bc F * 2 ^ finrank \u211d E < \u2191\u2191\u03bc s\nh_symm : \u2200 (x : E), x \u2208 s \u2192 -x \u2208 s\nh_conv : Convex \u211d s\nh_vol : \u2191\u2191\u03bc F < \u2191\u2191\u03bc (2\u207b\u00b9 \u2022 s)\nx y : { x // x \u2208 L }\nhxy : x \u2260 y\nh : \u00acDisjoint (x +\u1d65 2\u207b\u00b9 \u2022 s) (y +\u1d65 2\u207b\u00b9 \u2022 s)\nv : E\nhv : 2 \u2022 v \u2208 s\nw : E\nhw : 2 \u2022 w \u2208 s\nhvw : w - v = \u2191(x - y)\n\u22a2 \u2191(x - y) \u2208 s"}, {"tactic": "rw [\u2190 hvw, \u2190 inv_smul_smul\u2080 (two_ne_zero' \u211d) (_ - _), smul_sub, sub_eq_add_neg, smul_add]", "annotated_tactic": ["rw [\u2190 hvw, \u2190 <a>inv_smul_smul\u2080</a> (<a>two_ne_zero'</a> \u211d) (_ - _), <a>smul_sub</a>, <a>sub_eq_add_neg</a>, <a>smul_add</a>]", [{"full_name": "inv_smul_smul\u2080", "def_path": "Mathlib/GroupTheory/GroupAction/Group.lean", "def_pos": [192, 9], "def_end_pos": [192, 23]}, {"full_name": "two_ne_zero'", "def_path": "Mathlib/Algebra/NeZero.lean", "def_pos": [81, 7], "def_end_pos": [81, 19]}, {"full_name": "smul_sub", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [988, 9], "def_end_pos": [988, 17]}, {"full_name": "sub_eq_add_neg", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [975, 3], "def_end_pos": [975, 14]}, {"full_name": "smul_add", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [807, 9], "def_end_pos": [807, 17]}]], "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro\nE : Type u_1\nL\u271d : Type u_2\ninst\u271d\u2076 : MeasurableSpace E\n\u03bc : Measure E\nF s : Set E\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : BorelSpace E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bc\nL : AddSubgroup E\ninst\u271d : Countable { x // x \u2208 L }\nfund : IsAddFundamentalDomain { x // x \u2208 L } F\nh\u271d : \u2191\u2191\u03bc F * 2 ^ finrank \u211d E < \u2191\u2191\u03bc s\nh_symm : \u2200 (x : E), x \u2208 s \u2192 -x \u2208 s\nh_conv : Convex \u211d s\nh_vol : \u2191\u2191\u03bc F < \u2191\u2191\u03bc (2\u207b\u00b9 \u2022 s)\nx y : { x // x \u2208 L }\nhxy : x \u2260 y\nh : \u00acDisjoint (x +\u1d65 2\u207b\u00b9 \u2022 s) (y +\u1d65 2\u207b\u00b9 \u2022 s)\nv : E\nhv : 2 \u2022 v \u2208 s\nw : E\nhw : 2 \u2022 w \u2208 s\nhvw : w - v = \u2191(x - y)\n\u22a2 \u2191(x - y) \u2208 s", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro\nE : Type u_1\nL\u271d : Type u_2\ninst\u271d\u2076 : MeasurableSpace E\n\u03bc : Measure E\nF s : Set E\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : BorelSpace E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bc\nL : AddSubgroup E\ninst\u271d : Countable { x // x \u2208 L }\nfund : IsAddFundamentalDomain { x // x \u2208 L } F\nh\u271d : \u2191\u2191\u03bc F * 2 ^ finrank \u211d E < \u2191\u2191\u03bc s\nh_symm : \u2200 (x : E), x \u2208 s \u2192 -x \u2208 s\nh_conv : Convex \u211d s\nh_vol : \u2191\u2191\u03bc F < \u2191\u2191\u03bc (2\u207b\u00b9 \u2022 s)\nx y : { x // x \u2208 L }\nhxy : x \u2260 y\nh : \u00acDisjoint (x +\u1d65 2\u207b\u00b9 \u2022 s) (y +\u1d65 2\u207b\u00b9 \u2022 s)\nv : E\nhv : 2 \u2022 v \u2208 s\nw : E\nhw : 2 \u2022 w \u2208 s\nhvw : w - v = \u2191(x - y)\n\u22a2 2\u207b\u00b9 \u2022 2 \u2022 w + 2\u207b\u00b9 \u2022 -(2 \u2022 v) \u2208 s"}, {"tactic": "refine' h_conv hw (h_symm _ hv) _ _ _ <;> norm_num", "annotated_tactic": ["refine' h_conv hw (h_symm _ hv) _ _ _ <;> norm_num", []], "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro\nE : Type u_1\nL\u271d : Type u_2\ninst\u271d\u2076 : MeasurableSpace E\n\u03bc : Measure E\nF s : Set E\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : BorelSpace E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bc\nL : AddSubgroup E\ninst\u271d : Countable { x // x \u2208 L }\nfund : IsAddFundamentalDomain { x // x \u2208 L } F\nh\u271d : \u2191\u2191\u03bc F * 2 ^ finrank \u211d E < \u2191\u2191\u03bc s\nh_symm : \u2200 (x : E), x \u2208 s \u2192 -x \u2208 s\nh_conv : Convex \u211d s\nh_vol : \u2191\u2191\u03bc F < \u2191\u2191\u03bc (2\u207b\u00b9 \u2022 s)\nx y : { x // x \u2208 L }\nhxy : x \u2260 y\nh : \u00acDisjoint (x +\u1d65 2\u207b\u00b9 \u2022 s) (y +\u1d65 2\u207b\u00b9 \u2022 s)\nv : E\nhv : 2 \u2022 v \u2208 s\nw : E\nhw : 2 \u2022 w \u2208 s\nhvw : w - v = \u2191(x - y)\n\u22a2 2\u207b\u00b9 \u2022 2 \u2022 w + 2\u207b\u00b9 \u2022 -(2 \u2022 v) \u2208 s", "state_after": "no goals"}, {"tactic": "rw [addHaar_smul_of_nonneg \u03bc (by norm_num : 0 \u2264 (2 : \u211d)\u207b\u00b9) s, \u2190\n  mul_lt_mul_right (pow_ne_zero (finrank \u211d E) (two_ne_zero' _)) (pow_ne_top two_ne_top),\n  mul_right_comm, ofReal_pow (by norm_num : 0 \u2264 (2 : \u211d)\u207b\u00b9), \u2190 ofReal_inv_of_pos zero_lt_two]", "annotated_tactic": ["rw [<a>addHaar_smul_of_nonneg</a> \u03bc (by norm_num : 0 \u2264 (2 : \u211d)\u207b\u00b9) s, \u2190\n      <a>mul_lt_mul_right</a> (<a>pow_ne_zero</a> (<a>finrank</a> \u211d E) (<a>two_ne_zero'</a> _)) (<a>pow_ne_top</a> <a>two_ne_top</a>),\n      <a>mul_right_comm</a>, <a>ofReal_pow</a> (by norm_num : 0 \u2264 (2 : \u211d)\u207b\u00b9), \u2190 <a>ofReal_inv_of_pos</a> <a>zero_lt_two</a>]", [{"full_name": "MeasureTheory.Measure.addHaar_smul_of_nonneg", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean", "def_pos": [387, 9], "def_end_pos": [387, 31]}, {"full_name": "ENNReal.mul_lt_mul_right", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1080, 9], "def_end_pos": [1080, 25]}, {"full_name": "pow_ne_zero", "def_path": "Mathlib/Algebra/GroupPower/Ring.lean", "def_pos": [84, 9], "def_end_pos": [84, 20]}, {"full_name": "FiniteDimensional.finrank", "def_path": "Mathlib/LinearAlgebra/Finrank.lean", "def_pos": [58, 19], "def_end_pos": [58, 26]}, {"full_name": "two_ne_zero'", "def_path": "Mathlib/Algebra/NeZero.lean", "def_pos": [81, 7], "def_end_pos": [81, 19]}, {"full_name": "ENNReal.pow_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [662, 9], "def_end_pos": [662, 19]}, {"full_name": "ENNReal.two_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [431, 9], "def_end_pos": [431, 19]}, {"full_name": "mul_right_comm", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [102, 9], "def_end_pos": [102, 23]}, {"full_name": "ENNReal.ofReal_pow", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2235, 9], "def_end_pos": [2235, 19]}, {"full_name": "ENNReal.ofReal_inv_of_pos", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2243, 9], "def_end_pos": [2243, 26]}, {"full_name": "zero_lt_two", "def_path": "Mathlib/Algebra/Order/Monoid/NatCast.lean", "def_pos": [71, 15], "def_end_pos": [71, 26]}]], "state_before": "E : Type u_1\nL\u271d : Type u_2\ninst\u271d\u2076 : MeasurableSpace E\n\u03bc : Measure E\nF s : Set E\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : BorelSpace E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bc\nL : AddSubgroup E\ninst\u271d : Countable { x // x \u2208 L }\nfund : IsAddFundamentalDomain { x // x \u2208 L } F\nh : \u2191\u2191\u03bc F * 2 ^ finrank \u211d E < \u2191\u2191\u03bc s\nh_symm : \u2200 (x : E), x \u2208 s \u2192 -x \u2208 s\nh_conv : Convex \u211d s\n\u22a2 \u2191\u2191\u03bc F < \u2191\u2191\u03bc (2\u207b\u00b9 \u2022 s)", "state_after": "E : Type u_1\nL\u271d : Type u_2\ninst\u271d\u2076 : MeasurableSpace E\n\u03bc : Measure E\nF s : Set E\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : BorelSpace E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bc\nL : AddSubgroup E\ninst\u271d : Countable { x // x \u2208 L }\nfund : IsAddFundamentalDomain { x // x \u2208 L } F\nh : \u2191\u2191\u03bc F * 2 ^ finrank \u211d E < \u2191\u2191\u03bc s\nh_symm : \u2200 (x : E), x \u2208 s \u2192 -x \u2208 s\nh_conv : Convex \u211d s\n\u22a2 \u2191\u2191\u03bc F * 2 ^ finrank \u211d E < (ENNReal.ofReal 2)\u207b\u00b9 ^ finrank \u211d E * 2 ^ finrank \u211d E * \u2191\u2191\u03bc s"}, {"tactic": "norm_num", "annotated_tactic": ["norm_num", []], "state_before": "E : Type u_1\nL\u271d : Type u_2\ninst\u271d\u2076 : MeasurableSpace E\n\u03bc : Measure E\nF s : Set E\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : BorelSpace E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bc\nL : AddSubgroup E\ninst\u271d : Countable { x // x \u2208 L }\nfund : IsAddFundamentalDomain { x // x \u2208 L } F\nh : \u2191\u2191\u03bc F * 2 ^ finrank \u211d E < \u2191\u2191\u03bc s\nh_symm : \u2200 (x : E), x \u2208 s \u2192 -x \u2208 s\nh_conv : Convex \u211d s\n\u22a2 \u2191\u2191\u03bc F * 2 ^ finrank \u211d E < (ENNReal.ofReal 2)\u207b\u00b9 ^ finrank \u211d E * 2 ^ finrank \u211d E * \u2191\u2191\u03bc s", "state_after": "E : Type u_1\nL\u271d : Type u_2\ninst\u271d\u2076 : MeasurableSpace E\n\u03bc : Measure E\nF s : Set E\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : BorelSpace E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bc\nL : AddSubgroup E\ninst\u271d : Countable { x // x \u2208 L }\nfund : IsAddFundamentalDomain { x // x \u2208 L } F\nh : \u2191\u2191\u03bc F * 2 ^ finrank \u211d E < \u2191\u2191\u03bc s\nh_symm : \u2200 (x : E), x \u2208 s \u2192 -x \u2208 s\nh_conv : Convex \u211d s\n\u22a2 \u2191\u2191\u03bc F * 2 ^ finrank \u211d E < 2\u207b\u00b9 ^ finrank \u211d E * 2 ^ finrank \u211d E * \u2191\u2191\u03bc s"}, {"tactic": "rwa [\u2190 mul_pow, ENNReal.inv_mul_cancel two_ne_zero two_ne_top, one_pow, one_mul]", "annotated_tactic": ["rwa [\u2190 <a>mul_pow</a>, <a>ENNReal.inv_mul_cancel</a> <a>two_ne_zero</a> <a>two_ne_top</a>, <a>one_pow</a>, <a>one_mul</a>]", [{"full_name": "mul_pow", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [257, 9], "def_end_pos": [257, 16]}, {"full_name": "ENNReal.inv_mul_cancel", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1424, 19], 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finrank \u211d E < \u2191\u2191\u03bc s\nh_symm : \u2200 (x : E), x \u2208 s \u2192 -x \u2208 s\nh_conv : Convex \u211d s\n\u22a2 \u2191\u2191\u03bc F * 2 ^ finrank \u211d E < 2\u207b\u00b9 ^ finrank \u211d E * 2 ^ finrank \u211d E * \u2191\u2191\u03bc s", "state_after": "no goals"}, {"tactic": "norm_num", "annotated_tactic": ["norm_num", []], "state_before": "E : Type u_1\nL\u271d : Type u_2\ninst\u271d\u2076 : MeasurableSpace E\n\u03bc : Measure E\nF s : Set E\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : BorelSpace E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bc\nL : AddSubgroup E\ninst\u271d : Countable { x // x \u2208 L }\nfund : IsAddFundamentalDomain { x // x \u2208 L } F\nh : \u2191\u2191\u03bc F * 2 ^ finrank \u211d E < \u2191\u2191\u03bc s\nh_symm : \u2200 (x : E), x \u2208 s \u2192 -x \u2208 s\nh_conv : Convex \u211d s\n\u22a2 0 \u2264 2\u207b\u00b9", "state_after": "no 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CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf\u271d\u00b9 : \u211d \u2192 E\ng'\u271d g\u271d \u03c6 : \u211d \u2192 \u211d\nf\u271d f'\u271d : \u211d \u2192 E\na b : \u211d\nf f' g g' : \u211d \u2192 \u211d\nhf : ContinuousOn f [[a, b]]\nhff' : \u2200 (x : \u211d), x \u2208 Ioo (min a b) (max a b) \u2192 HasDerivWithinAt f (f' x) (Ioi x) x\nhf' : ContinuousOn f' [[a, b]]\nhg : ContinuousOn g [[f a, f b]]\nhgg' : \u2200 (x : \u211d), x \u2208 Ioo (min (f a) (f b)) (max (f a) (f b)) \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\nhg' : ContinuousOn g' (f '' [[a, b]])\n\u22a2 \u222b (x : \u211d) in a..b, (g' \u2218 f) x * f' x = (g \u2218 f) b - (g \u2218 f) a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Card.lean", "full_name": "Set.encard_tsub_one_le_encard_diff_singleton", "start": [252, 1], "end": [254, 61], "traced_tactics": [{"tactic": "rw [\u2190encard_singleton x]", "annotated_tactic": ["rw [\u2190<a>encard_singleton</a> x]", [{"full_name": "Set.encard_singleton", "def_path": "Mathlib/Data/Set/Card.lean", "def_pos": [109, 17], "def_end_pos": [109, 33]}]], "state_before": "\u03b1 : Type u_1\ns\u271d t s : Set \u03b1\nx : \u03b1\n\u22a2 encard s - 1 \u2264 encard (s \\ {x})", "state_after": "\u03b1 : Type u_1\ns\u271d t s : Set \u03b1\nx : \u03b1\n\u22a2 encard s - encard {x} \u2264 encard (s \\ {x})"}, {"tactic": "apply tsub_encard_le_encard_diff", "annotated_tactic": ["apply <a>tsub_encard_le_encard_diff</a>", [{"full_name": "Set.tsub_encard_le_encard_diff", "def_path": "Mathlib/Data/Set/Card.lean", "def_pos": [227, 9], "def_end_pos": [227, 35]}]], "state_before": "\u03b1 : Type u_1\ns\u271d t s : Set \u03b1\nx : \u03b1\n\u22a2 encard s - encard {x} \u2264 encard (s \\ {x})", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", 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"Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [193, 9], "def_end_pos": [193, 21]}, {"full_name": "Function.support_sup", "def_path": "Mathlib/Algebra/Support.lean", "def_pos": [153, 3], "def_end_pos": [153, 14]}, {"full_name": "LE.le.trans_lt", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [124, 7], "def_end_pos": [124, 21]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u2074 : Countable \u03b9\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : SemilatticeSup \u03b2\ninst\u271d : ContinuousSup \u03b2\nhf : FinStronglyMeasurable f \u03bc\nhg : FinStronglyMeasurable g \u03bc\nn : \u2115\n\u22a2 \u2191\u2191\u03bc (support \u2191((fun n => FinStronglyMeasurable.approx hf n \u2294 FinStronglyMeasurable.approx hg n) n)) < \u22a4", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type 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Measure \u03b1\nf g : \u03b1 \u2192 \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : SemilatticeSup \u03b2\ninst\u271d : ContinuousSup \u03b2\nhf : FinStronglyMeasurable f \u03bc\nhg : FinStronglyMeasurable g \u03bc\nn : \u2115\n\u22a2 \u2191\u2191\u03bc\n      ((support fun x => \u2191(FinStronglyMeasurable.approx hf n) x) \u222a\n        support fun x => \u2191(FinStronglyMeasurable.approx hg n) x) <\n    \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/TuringMachine.lean", "full_name": "Turing.TM1to1.tr_respects", "start": [1849, 1], "end": [1888, 30], "traced_tactics": [{"tactic": "obtain \u27e8L, R, rfl\u27e9 := T.exists_mk'", "annotated_tactic": ["obtain \u27e8L, R, rfl\u27e9 := T.exists_mk'", []], "state_before": "\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nl\u2081 : Option \u039b\nv : \u03c3\nT : Tape \u0393\n\u22a2 FRespects (step (tr enc dec M)) (fun c\u2081 => trCfg enc enc\u2080 c\u2081) (trCfg enc enc\u2080 { l := l\u2081, var := v, Tape := T })\n    (step M { l := l\u2081, var := v, Tape := T })", "state_after": "case intro.intro\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nl\u2081 : Option \u039b\nv : \u03c3\nL R : ListBlank \u0393\n\u22a2 FRespects (step (tr enc dec M)) (fun c\u2081 => trCfg enc enc\u2080 c\u2081)\n    (trCfg enc enc\u2080 { l := l\u2081, var := v, Tape := Tape.mk' L R }) (step M { l := l\u2081, var := v, Tape := Tape.mk' L R })"}, {"tactic": "cases' l\u2081 with l\u2081", "annotated_tactic": ["cases' l\u2081 with l\u2081", []], "state_before": "case intro.intro\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nl\u2081 : Option \u039b\nv : \u03c3\nL R : ListBlank \u0393\n\u22a2 FRespects (step (tr enc dec M)) (fun c\u2081 => trCfg enc enc\u2080 c\u2081)\n    (trCfg enc enc\u2080 { l := l\u2081, var := v, Tape := Tape.mk' L R }) (step M { l := l\u2081, var := v, Tape := Tape.mk' L R })", "state_after": "case intro.intro.none\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv : \u03c3\nL R : ListBlank \u0393\n\u22a2 FRespects (step (tr enc dec M)) (fun c\u2081 => trCfg enc enc\u2080 c\u2081)\n    (trCfg enc enc\u2080 { l := none, var := v, Tape := Tape.mk' L R })\n    (step M { l := none, var := v, Tape := Tape.mk' L R })\n\ncase intro.intro.some\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv : \u03c3\nL R : ListBlank \u0393\nl\u2081 : \u039b\n\u22a2 FRespects (step (tr enc dec M)) (fun c\u2081 => trCfg enc enc\u2080 c\u2081)\n    (trCfg enc enc\u2080 { l := some l\u2081, var := v, Tape := Tape.mk' L R })\n    (step M { l := some l\u2081, var := v, Tape := Tape.mk' L R })"}, {"tactic": "suffices \u2200 q R, Reaches (step (tr enc dec M)) (stepAux (trNormal dec q) v (trTape' enc0 L R))\n    (trCfg enc enc0 (stepAux q v (Tape.mk' L R))) by\n  refine' TransGen.head' rfl _\n  rw [trTape_mk']\n  exact this _ R", "annotated_tactic": ["suffices \u2200 q R, <a>Reaches</a> (<a>step</a> (<a>tr</a> enc dec M)) (<a>stepAux</a> (<a>trNormal</a> dec q) v (<a>trTape'</a> enc0 L R))\n        (<a>trCfg</a> enc enc0 (<a>stepAux</a> q v (<a>Tape.mk'</a> L R))) by\n      refine' <a>TransGen.head'</a> <a>rfl</a> _\n      rw [<a>trTape_mk'</a>]\n      exact this _ R", [{"full_name": "Turing.Reaches", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [744, 5], "def_end_pos": [744, 12]}, {"full_name": "Turing.TM1.step", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1291, 5], "def_end_pos": [1291, 9]}, {"full_name": "Turing.TM1to1.tr", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1755, 5], "def_end_pos": [1755, 7]}, {"full_name": "Turing.TM1.stepAux", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1281, 5], "def_end_pos": [1281, 12]}, {"full_name": "Turing.TM1to1.trNormal", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1687, 5], "def_end_pos": [1687, 13]}, {"full_name": "Turing.TM1to1.trTape'", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1734, 5], "def_end_pos": [1734, 12]}, {"full_name": "Turing.TM1to1.trCfg", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1761, 5], "def_end_pos": [1761, 10]}, {"full_name": "Turing.TM1.stepAux", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1281, 5], "def_end_pos": [1281, 12]}, {"full_name": "Turing.Tape.mk'", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [545, 5], "def_end_pos": [545, 13]}, {"full_name": "Relation.TransGen.head'", "def_path": "Mathlib/Logic/Relation.lean", "def_pos": [375, 9], "def_end_pos": [375, 14]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}, {"full_name": "Turing.TM1to1.trTape_mk'", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1746, 9], "def_end_pos": [1746, 19]}]], "state_before": "case intro.intro.some\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv : \u03c3\nL R : ListBlank \u0393\nl\u2081 : \u039b\n\u22a2 FRespects (step (tr enc dec M)) (fun c\u2081 => trCfg enc enc\u2080 c\u2081)\n    (trCfg enc enc\u2080 { l := some l\u2081, var := v, Tape := Tape.mk' L R })\n    (step M { l := some l\u2081, var := v, Tape := Tape.mk' L R })", "state_after": "case intro.intro.some\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv : \u03c3\nL R : ListBlank \u0393\nl\u2081 : \u039b\n\u22a2 \u2200 (q : Stmt\u2081) (R : ListBlank \u0393),\n    Reaches (step (tr enc dec M)) (stepAux (trNormal dec q) v (trTape' enc0 L R))\n      (trCfg enc enc0 (stepAux q v (Tape.mk' L R)))"}, {"tactic": "clear R l\u2081", "annotated_tactic": ["clear R l\u2081", []], "state_before": "case intro.intro.some\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv : \u03c3\nL R : ListBlank \u0393\nl\u2081 : \u039b\n\u22a2 \u2200 (q : Stmt\u2081) (R : ListBlank \u0393),\n    Reaches (step (tr enc dec M)) (stepAux (trNormal dec q) v (trTape' enc0 L R))\n      (trCfg enc enc0 (stepAux q v (Tape.mk' L R)))", "state_after": "case intro.intro.some\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv : \u03c3\nL : ListBlank \u0393\n\u22a2 \u2200 (q : Stmt\u2081) (R : ListBlank \u0393),\n    Reaches (step (tr enc dec M)) (stepAux (trNormal dec q) v (trTape' enc0 L R))\n      (trCfg enc enc0 (stepAux q v (Tape.mk' L R)))"}, {"tactic": "intro q R", "annotated_tactic": ["intro q R", []], "state_before": "case intro.intro.some\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv : \u03c3\nL : ListBlank \u0393\n\u22a2 \u2200 (q : Stmt\u2081) (R : ListBlank \u0393),\n    Reaches (step (tr enc dec M)) (stepAux (trNormal dec q) v (trTape' enc0 L R))\n      (trCfg enc enc0 (stepAux q v (Tape.mk' L R)))", "state_after": "case intro.intro.some\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv : \u03c3\nL : ListBlank \u0393\nq : Stmt\u2081\nR : ListBlank \u0393\n\u22a2 Reaches (step (tr enc dec M)) (stepAux (trNormal dec q) v (trTape' enc0 L R))\n    (trCfg enc enc0 (stepAux q v (Tape.mk' L R)))"}, {"tactic": "induction' q generalizing v L R", "annotated_tactic": ["induction' q generalizing v L R", []], "state_before": "case intro.intro.some\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv : \u03c3\nL : ListBlank \u0393\nq : Stmt\u2081\nR : ListBlank \u0393\n\u22a2 Reaches (step (tr enc dec M)) (stepAux (trNormal dec q) v (trTape' enc0 L R))\n    (trCfg enc enc0 (stepAux q v (Tape.mk' L R)))", "state_after": "case intro.intro.some.move\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nL\u271d R\u271d : ListBlank \u0393\na\u271d\u00b9 : Dir\na\u271d : Stmt\u2081\na_ih\u271d :\n  \u2200 (v : \u03c3) (L R : ListBlank \u0393),\n    Reaches (step (tr enc dec M)) (stepAux (trNormal dec a\u271d) v (trTape' enc0 L R))\n      (trCfg enc enc0 (stepAux a\u271d v (Tape.mk' L R)))\nv : \u03c3\nL R : ListBlank \u0393\n\u22a2 Reaches (step (tr enc dec M)) (stepAux (trNormal dec (Stmt.move a\u271d\u00b9 a\u271d)) v (trTape' enc0 L R))\n    (trCfg enc enc0 (stepAux (Stmt.move a\u271d\u00b9 a\u271d) v (Tape.mk' L R)))\n\ncase intro.intro.some.write\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nL\u271d R\u271d : ListBlank \u0393\na\u271d\u00b9 : \u0393 \u2192 \u03c3 \u2192 \u0393\na\u271d : Stmt\u2081\na_ih\u271d :\n  \u2200 (v : \u03c3) (L R : ListBlank \u0393),\n    Reaches (step (tr enc dec M)) (stepAux (trNormal dec a\u271d) v (trTape' enc0 L R))\n      (trCfg enc enc0 (stepAux a\u271d v (Tape.mk' L R)))\nv : \u03c3\nL R : ListBlank \u0393\n\u22a2 Reaches (step (tr enc dec M)) (stepAux (trNormal dec (Stmt.write a\u271d\u00b9 a\u271d)) v (trTape' enc0 L R))\n    (trCfg enc enc0 (stepAux (Stmt.write a\u271d\u00b9 a\u271d) v (Tape.mk' L R)))\n\ncase intro.intro.some.load\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nL\u271d R\u271d : ListBlank \u0393\na\u271d\u00b9 : \u0393 \u2192 \u03c3 \u2192 \u03c3\na\u271d : Stmt\u2081\na_ih\u271d :\n  \u2200 (v : \u03c3) (L R : ListBlank \u0393),\n    Reaches (step (tr enc dec M)) (stepAux (trNormal dec a\u271d) v (trTape' enc0 L R))\n      (trCfg enc enc0 (stepAux a\u271d v (Tape.mk' L R)))\nv : \u03c3\nL R : ListBlank \u0393\n\u22a2 Reaches (step (tr enc dec M)) (stepAux (trNormal dec (Stmt.load a\u271d\u00b9 a\u271d)) v (trTape' enc0 L R))\n    (trCfg enc enc0 (stepAux (Stmt.load a\u271d\u00b9 a\u271d) v (Tape.mk' L R)))\n\ncase intro.intro.some.branch\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nL\u271d R\u271d : ListBlank \u0393\na\u271d\u00b2 : \u0393 \u2192 \u03c3 \u2192 Bool\na\u271d\u00b9 a\u271d : Stmt\u2081\na_ih\u271d\u00b9 :\n  \u2200 (v : \u03c3) (L R : ListBlank \u0393),\n    Reaches (step (tr enc dec M)) (stepAux (trNormal dec a\u271d\u00b9) v (trTape' enc0 L R))\n      (trCfg enc enc0 (stepAux a\u271d\u00b9 v (Tape.mk' L R)))\na_ih\u271d :\n  \u2200 (v : \u03c3) (L R : ListBlank \u0393),\n    Reaches (step (tr enc dec M)) (stepAux (trNormal dec a\u271d) v (trTape' enc0 L R))\n      (trCfg enc enc0 (stepAux a\u271d v (Tape.mk' L R)))\nv : \u03c3\nL R : ListBlank \u0393\n\u22a2 Reaches (step (tr enc dec M)) (stepAux (trNormal dec (Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d)) v (trTape' enc0 L R))\n    (trCfg enc enc0 (stepAux (Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) v (Tape.mk' L R)))\n\ncase intro.intro.some.goto\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nL\u271d R\u271d : ListBlank \u0393\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u039b\nv : \u03c3\nL R : ListBlank \u0393\n\u22a2 Reaches (step (tr enc dec M)) (stepAux (trNormal dec (Stmt.goto a\u271d)) v (trTape' enc0 L R))\n    (trCfg enc enc0 (stepAux (Stmt.goto a\u271d) v (Tape.mk' L R)))\n\ncase intro.intro.some.halt\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nL\u271d R\u271d : ListBlank \u0393\nv : \u03c3\nL R : ListBlank \u0393\n\u22a2 Reaches (step (tr enc dec M)) (stepAux (trNormal dec Stmt.halt) v (trTape' enc0 L R))\n    (trCfg enc enc0 (stepAux Stmt.halt v (Tape.mk' L R)))"}, {"tactic": "case move d q IH =>\n  cases d <;>\n      simp only [trNormal, iterate, stepAux_move, stepAux, ListBlank.head_cons,\n        Tape.move_left_mk', ListBlank.cons_head_tail, ListBlank.tail_cons,\n        trTape'_move_left enc0, trTape'_move_right enc0] <;>\n    apply IH", "annotated_tactic": ["case move d q IH =>\n      cases d <;>\n          simp only [<a>trNormal</a>, <a>iterate</a>, <a>stepAux_move</a>, <a>stepAux</a>, <a>ListBlank.head_cons</a>,\n            <a>Tape.move_left_mk'</a>, <a>ListBlank.cons_head_tail</a>, <a>ListBlank.tail_cons</a>,\n            <a>trTape'_move_left</a> enc0, <a>trTape'_move_right</a> enc0] <;>\n        apply IH", [{"full_name": "Turing.TM1to1.trNormal", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1687, 5], "def_end_pos": [1687, 13]}, {"full_name": "Nat.iterate", "def_path": "Mathlib/Logic/Function/Iterate.lean", "def_pos": [38, 5], "def_end_pos": [38, 16]}, {"full_name": "Turing.TM1to1.stepAux_move", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1697, 9], "def_end_pos": [1697, 21]}, {"full_name": "Turing.TM1.stepAux", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1281, 5], "def_end_pos": [1281, 12]}, {"full_name": "Turing.ListBlank.head_cons", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [253, 9], "def_end_pos": [253, 28]}, {"full_name": "Turing.Tape.move_left_mk'", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [581, 9], "def_end_pos": [581, 27]}, {"full_name": "Turing.ListBlank.cons_head_tail", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [265, 9], "def_end_pos": [265, 33]}, {"full_name": "Turing.ListBlank.tail_cons", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [258, 9], "def_end_pos": [258, 28]}, {"full_name": "Turing.TM1to1.trTape'_move_left", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1767, 9], "def_end_pos": [1767, 26]}, {"full_name": "Turing.TM1to1.trTape'_move_right", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1785, 9], "def_end_pos": [1785, 27]}]], "state_before": "case intro.intro.some.move\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nL\u271d R\u271d : ListBlank \u0393\na\u271d\u00b9 : Dir\na\u271d : Stmt\u2081\na_ih\u271d :\n  \u2200 (v : \u03c3) (L R : ListBlank \u0393),\n    Reaches (step (tr enc dec M)) (stepAux (trNormal dec a\u271d) v (trTape' enc0 L R))\n      (trCfg enc enc0 (stepAux a\u271d v (Tape.mk' L R)))\nv : \u03c3\nL R : ListBlank \u0393\n\u22a2 Reaches (step (tr enc dec M)) (stepAux (trNormal dec (Stmt.move a\u271d\u00b9 a\u271d)) v (trTape' enc0 L R))\n    (trCfg enc enc0 (stepAux (Stmt.move a\u271d\u00b9 a\u271d) v (Tape.mk' L R)))\n\ncase intro.intro.some.write\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nL\u271d R\u271d : ListBlank \u0393\na\u271d\u00b9 : \u0393 \u2192 \u03c3 \u2192 \u0393\na\u271d : Stmt\u2081\na_ih\u271d :\n  \u2200 (v : \u03c3) (L R : ListBlank \u0393),\n    Reaches (step (tr enc dec M)) (stepAux (trNormal dec a\u271d) v (trTape' enc0 L R))\n      (trCfg enc enc0 (stepAux a\u271d v (Tape.mk' L R)))\nv : \u03c3\nL R : ListBlank \u0393\n\u22a2 Reaches (step (tr enc dec M)) (stepAux (trNormal dec (Stmt.write a\u271d\u00b9 a\u271d)) v (trTape' enc0 L R))\n    (trCfg enc enc0 (stepAux (Stmt.write a\u271d\u00b9 a\u271d) v (Tape.mk' L R)))\n\ncase intro.intro.some.load\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nL\u271d R\u271d : ListBlank \u0393\na\u271d\u00b9 : \u0393 \u2192 \u03c3 \u2192 \u03c3\na\u271d : Stmt\u2081\na_ih\u271d :\n  \u2200 (v : \u03c3) (L R : ListBlank \u0393),\n    Reaches (step (tr enc dec M)) (stepAux (trNormal dec a\u271d) v (trTape' enc0 L R))\n      (trCfg enc enc0 (stepAux a\u271d v (Tape.mk' L R)))\nv : \u03c3\nL R : ListBlank \u0393\n\u22a2 Reaches (step (tr enc dec M)) (stepAux (trNormal dec (Stmt.load a\u271d\u00b9 a\u271d)) v (trTape' enc0 L R))\n    (trCfg enc enc0 (stepAux (Stmt.load a\u271d\u00b9 a\u271d) v (Tape.mk' L R)))\n\ncase intro.intro.some.branch\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nL\u271d R\u271d : ListBlank \u0393\na\u271d\u00b2 : \u0393 \u2192 \u03c3 \u2192 Bool\na\u271d\u00b9 a\u271d : Stmt\u2081\na_ih\u271d\u00b9 :\n  \u2200 (v : \u03c3) (L R : ListBlank \u0393),\n    Reaches (step (tr enc dec M)) (stepAux (trNormal dec a\u271d\u00b9) v (trTape' enc0 L R))\n      (trCfg enc enc0 (stepAux a\u271d\u00b9 v (Tape.mk' L R)))\na_ih\u271d :\n  \u2200 (v : \u03c3) (L R : ListBlank \u0393),\n    Reaches (step (tr enc dec M)) (stepAux (trNormal dec a\u271d) v (trTape' enc0 L R))\n      (trCfg enc enc0 (stepAux a\u271d v (Tape.mk' L R)))\nv : \u03c3\nL R : ListBlank \u0393\n\u22a2 Reaches (step (tr enc dec M)) (stepAux (trNormal dec (Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d)) v (trTape' enc0 L R))\n    (trCfg enc enc0 (stepAux (Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) v (Tape.mk' L R)))\n\ncase intro.intro.some.goto\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nL\u271d R\u271d : ListBlank \u0393\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u039b\nv : \u03c3\nL R : ListBlank \u0393\n\u22a2 Reaches (step (tr enc dec M)) (stepAux (trNormal dec (Stmt.goto a\u271d)) v (trTape' enc0 L R))\n    (trCfg enc enc0 (stepAux (Stmt.goto a\u271d) v (Tape.mk' L R)))\n\ncase intro.intro.some.halt\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nL\u271d R\u271d : ListBlank \u0393\nv : \u03c3\nL R : ListBlank \u0393\n\u22a2 Reaches (step (tr enc dec M)) (stepAux (trNormal dec Stmt.halt) v (trTape' enc0 L R))\n    (trCfg enc enc0 (stepAux Stmt.halt v (Tape.mk' L R)))", "state_after": "case intro.intro.some.write\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nL\u271d R\u271d : ListBlank \u0393\na\u271d\u00b9 : \u0393 \u2192 \u03c3 \u2192 \u0393\na\u271d : Stmt\u2081\na_ih\u271d :\n  \u2200 (v : \u03c3) (L R : ListBlank \u0393),\n    Reaches (step (tr enc dec M)) (stepAux (trNormal dec a\u271d) v (trTape' enc0 L R))\n      (trCfg enc enc0 (stepAux a\u271d v (Tape.mk' L R)))\nv : \u03c3\nL R : ListBlank \u0393\n\u22a2 Reaches (step (tr enc dec M)) (stepAux (trNormal dec (Stmt.write a\u271d\u00b9 a\u271d)) v (trTape' enc0 L R))\n    (trCfg enc enc0 (stepAux (Stmt.write a\u271d\u00b9 a\u271d) v (Tape.mk' L R)))\n\ncase intro.intro.some.load\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nL\u271d R\u271d : ListBlank \u0393\na\u271d\u00b9 : \u0393 \u2192 \u03c3 \u2192 \u03c3\na\u271d : Stmt\u2081\na_ih\u271d :\n  \u2200 (v : \u03c3) (L R : ListBlank \u0393),\n    Reaches (step (tr enc dec M)) (stepAux (trNormal dec a\u271d) v (trTape' enc0 L R))\n      (trCfg enc enc0 (stepAux a\u271d v (Tape.mk' L R)))\nv : \u03c3\nL R : ListBlank \u0393\n\u22a2 Reaches (step (tr enc dec M)) (stepAux (trNormal dec (Stmt.load a\u271d\u00b9 a\u271d)) v (trTape' enc0 L R))\n    (trCfg enc enc0 (stepAux (Stmt.load a\u271d\u00b9 a\u271d) v (Tape.mk' L R)))\n\ncase intro.intro.some.branch\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nL\u271d R\u271d : ListBlank \u0393\na\u271d\u00b2 : \u0393 \u2192 \u03c3 \u2192 Bool\na\u271d\u00b9 a\u271d : Stmt\u2081\na_ih\u271d\u00b9 :\n  \u2200 (v : \u03c3) (L R : ListBlank \u0393),\n    Reaches (step (tr enc dec M)) (stepAux (trNormal dec a\u271d\u00b9) v (trTape' enc0 L R))\n      (trCfg enc enc0 (stepAux a\u271d\u00b9 v (Tape.mk' L R)))\na_ih\u271d :\n  \u2200 (v : \u03c3) (L R : ListBlank \u0393),\n    Reaches (step (tr enc dec M)) (stepAux (trNormal dec a\u271d) v (trTape' enc0 L R))\n      (trCfg enc enc0 (stepAux a\u271d v (Tape.mk' L R)))\nv : \u03c3\nL R : ListBlank \u0393\n\u22a2 Reaches (step (tr enc dec M)) (stepAux (trNormal dec (Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d)) v (trTape' enc0 L R))\n    (trCfg enc enc0 (stepAux (Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) v (Tape.mk' L R)))\n\ncase intro.intro.some.goto\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nL\u271d R\u271d : ListBlank \u0393\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u039b\nv : \u03c3\nL R : ListBlank \u0393\n\u22a2 Reaches (step (tr enc dec M)) (stepAux (trNormal dec (Stmt.goto a\u271d)) v (trTape' enc0 L R))\n    (trCfg enc enc0 (stepAux (Stmt.goto a\u271d) v (Tape.mk' L R)))\n\ncase intro.intro.some.halt\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nL\u271d R\u271d : ListBlank \u0393\nv : \u03c3\nL R : ListBlank \u0393\n\u22a2 Reaches (step (tr enc dec M)) (stepAux (trNormal dec Stmt.halt) v (trTape' enc0 L R))\n    (trCfg enc enc0 (stepAux Stmt.halt v (Tape.mk' L R)))"}, {"tactic": "case write f q IH =>\n  simp only [trNormal, stepAux_read dec enc0 encdec, stepAux]\n  refine' ReflTransGen.head rfl _\n  obtain \u27e8a, R, rfl\u27e9 := R.exists_cons\n  rw [tr, Tape.mk'_head, stepAux_write, ListBlank.head_cons, stepAux_move,\n    trTape'_move_left enc0, ListBlank.head_cons, ListBlank.tail_cons, Tape.write_mk']\n  apply IH", "annotated_tactic": ["case write f q IH =>\n      simp only [<a>trNormal</a>, <a>stepAux_read</a> dec enc0 encdec, <a>stepAux</a>]\n      refine' <a>ReflTransGen.head</a> <a>rfl</a> _\n      obtain \u27e8a, R, rfl\u27e9 := R.exists_cons\n      rw [<a>tr</a>, <a>Tape.mk'_head</a>, <a>stepAux_write</a>, <a>ListBlank.head_cons</a>, <a>stepAux_move</a>,\n        <a>trTape'_move_left</a> enc0, <a>ListBlank.head_cons</a>, <a>ListBlank.tail_cons</a>, <a>Tape.write_mk'</a>]\n      apply IH", [{"full_name": "Turing.TM1to1.trNormal", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1687, 5], "def_end_pos": [1687, 13]}, {"full_name": "Turing.TM1to1.stepAux_read", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1818, 9], "def_end_pos": [1818, 21]}, {"full_name": "Turing.TM1.stepAux", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1281, 5], "def_end_pos": [1281, 12]}, {"full_name": "Relation.ReflTransGen.head", "def_path": "Mathlib/Logic/Relation.lean", "def_pos": [280, 9], "def_end_pos": [280, 13]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}, {"full_name": "Turing.TM1to1.tr", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1755, 5], "def_end_pos": [1755, 7]}, {"full_name": "Turing.Tape.mk'_head", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [555, 9], "def_end_pos": [555, 22]}, {"full_name": "Turing.TM1to1.stepAux_write", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1797, 9], "def_end_pos": [1797, 22]}, {"full_name": "Turing.ListBlank.head_cons", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [253, 9], "def_end_pos": [253, 28]}, {"full_name": "Turing.TM1to1.stepAux_move", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1697, 9], "def_end_pos": [1697, 21]}, {"full_name": "Turing.TM1to1.trTape'_move_left", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1767, 9], "def_end_pos": [1767, 26]}, {"full_name": "Turing.ListBlank.head_cons", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [253, 9], "def_end_pos": [253, 28]}, {"full_name": "Turing.ListBlank.tail_cons", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [258, 9], "def_end_pos": [258, 28]}, {"full_name": "Turing.Tape.write_mk'", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [675, 9], "def_end_pos": [675, 23]}]], "state_before": "case intro.intro.some.write\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nL\u271d R\u271d : ListBlank \u0393\na\u271d\u00b9 : \u0393 \u2192 \u03c3 \u2192 \u0393\na\u271d : Stmt\u2081\na_ih\u271d :\n  \u2200 (v : \u03c3) (L R : ListBlank \u0393),\n    Reaches (step (tr enc dec M)) (stepAux (trNormal dec a\u271d) v (trTape' enc0 L R))\n      (trCfg enc enc0 (stepAux a\u271d v (Tape.mk' L R)))\nv : \u03c3\nL R : ListBlank \u0393\n\u22a2 Reaches (step (tr enc dec M)) (stepAux (trNormal dec (Stmt.write a\u271d\u00b9 a\u271d)) v (trTape' enc0 L R))\n    (trCfg enc enc0 (stepAux (Stmt.write a\u271d\u00b9 a\u271d) v (Tape.mk' L R)))\n\ncase intro.intro.some.load\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nL\u271d R\u271d : ListBlank \u0393\na\u271d\u00b9 : \u0393 \u2192 \u03c3 \u2192 \u03c3\na\u271d : Stmt\u2081\na_ih\u271d :\n  \u2200 (v : \u03c3) (L R : ListBlank \u0393),\n    Reaches (step (tr enc dec M)) (stepAux (trNormal dec a\u271d) v (trTape' enc0 L R))\n      (trCfg enc enc0 (stepAux a\u271d v (Tape.mk' L R)))\nv : \u03c3\nL R : ListBlank \u0393\n\u22a2 Reaches (step (tr enc dec M)) (stepAux (trNormal dec (Stmt.load a\u271d\u00b9 a\u271d)) v (trTape' enc0 L R))\n    (trCfg enc enc0 (stepAux (Stmt.load a\u271d\u00b9 a\u271d) v (Tape.mk' L R)))\n\ncase intro.intro.some.branch\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nL\u271d R\u271d : ListBlank \u0393\na\u271d\u00b2 : \u0393 \u2192 \u03c3 \u2192 Bool\na\u271d\u00b9 a\u271d : Stmt\u2081\na_ih\u271d\u00b9 :\n  \u2200 (v : \u03c3) (L R : ListBlank \u0393),\n    Reaches (step (tr enc dec M)) (stepAux (trNormal dec a\u271d\u00b9) v (trTape' enc0 L R))\n      (trCfg enc enc0 (stepAux a\u271d\u00b9 v (Tape.mk' L R)))\na_ih\u271d :\n  \u2200 (v : \u03c3) (L R : ListBlank \u0393),\n    Reaches (step (tr enc dec M)) (stepAux (trNormal dec a\u271d) v (trTape' enc0 L R))\n      (trCfg enc enc0 (stepAux a\u271d v (Tape.mk' L R)))\nv : \u03c3\nL R : ListBlank \u0393\n\u22a2 Reaches (step (tr enc dec M)) (stepAux (trNormal dec (Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d)) v (trTape' enc0 L R))\n    (trCfg enc enc0 (stepAux (Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) v (Tape.mk' L R)))\n\ncase intro.intro.some.goto\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nL\u271d R\u271d : ListBlank \u0393\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u039b\nv : \u03c3\nL R : ListBlank \u0393\n\u22a2 Reaches (step (tr enc dec M)) (stepAux (trNormal dec (Stmt.goto a\u271d)) v (trTape' enc0 L R))\n    (trCfg enc enc0 (stepAux (Stmt.goto a\u271d) v (Tape.mk' L R)))\n\ncase intro.intro.some.halt\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nL\u271d R\u271d : ListBlank \u0393\nv : \u03c3\nL R : ListBlank \u0393\n\u22a2 Reaches (step (tr enc dec M)) (stepAux (trNormal dec Stmt.halt) v (trTape' enc0 L R))\n    (trCfg enc enc0 (stepAux Stmt.halt v (Tape.mk' L R)))", "state_after": "case intro.intro.some.load\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nL\u271d R\u271d : ListBlank \u0393\na\u271d\u00b9 : \u0393 \u2192 \u03c3 \u2192 \u03c3\na\u271d : Stmt\u2081\na_ih\u271d :\n  \u2200 (v : \u03c3) (L R : ListBlank \u0393),\n    Reaches (step (tr enc dec M)) (stepAux (trNormal dec a\u271d) v (trTape' enc0 L R))\n      (trCfg enc enc0 (stepAux a\u271d v (Tape.mk' L R)))\nv : \u03c3\nL R : ListBlank \u0393\n\u22a2 Reaches (step (tr enc dec M)) (stepAux (trNormal dec (Stmt.load a\u271d\u00b9 a\u271d)) v (trTape' enc0 L R))\n    (trCfg enc enc0 (stepAux (Stmt.load a\u271d\u00b9 a\u271d) v (Tape.mk' L R)))\n\ncase intro.intro.some.branch\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nL\u271d R\u271d : ListBlank \u0393\na\u271d\u00b2 : \u0393 \u2192 \u03c3 \u2192 Bool\na\u271d\u00b9 a\u271d : Stmt\u2081\na_ih\u271d\u00b9 :\n  \u2200 (v : \u03c3) (L R : ListBlank \u0393),\n    Reaches (step (tr enc dec M)) (stepAux (trNormal dec a\u271d\u00b9) v (trTape' enc0 L R))\n      (trCfg enc enc0 (stepAux a\u271d\u00b9 v (Tape.mk' L R)))\na_ih\u271d :\n  \u2200 (v : \u03c3) (L R : ListBlank \u0393),\n    Reaches (step (tr enc dec M)) (stepAux (trNormal dec a\u271d) v (trTape' enc0 L R))\n      (trCfg enc enc0 (stepAux a\u271d v (Tape.mk' L R)))\nv : \u03c3\nL R : ListBlank \u0393\n\u22a2 Reaches (step (tr enc dec M)) (stepAux (trNormal dec (Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d)) v (trTape' enc0 L R))\n    (trCfg enc enc0 (stepAux (Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) v (Tape.mk' L R)))\n\ncase intro.intro.some.goto\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nL\u271d R\u271d : ListBlank \u0393\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u039b\nv : \u03c3\nL R : ListBlank \u0393\n\u22a2 Reaches (step (tr enc dec M)) (stepAux (trNormal dec (Stmt.goto a\u271d)) v (trTape' enc0 L R))\n    (trCfg enc enc0 (stepAux (Stmt.goto a\u271d) v (Tape.mk' L R)))\n\ncase intro.intro.some.halt\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nL\u271d R\u271d : ListBlank \u0393\nv : \u03c3\nL R : ListBlank \u0393\n\u22a2 Reaches (step (tr enc dec M)) (stepAux (trNormal dec Stmt.halt) v (trTape' enc0 L R))\n    (trCfg enc enc0 (stepAux Stmt.halt v (Tape.mk' L R)))"}, {"tactic": "case load a q IH =>\n  simp only [trNormal, stepAux_read dec enc0 encdec]\n  apply IH", "annotated_tactic": ["case load a q IH =>\n      simp only [<a>trNormal</a>, <a>stepAux_read</a> dec enc0 encdec]\n      apply IH", [{"full_name": "Turing.TM1to1.trNormal", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1687, 5], "def_end_pos": [1687, 13]}, {"full_name": "Turing.TM1to1.stepAux_read", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1818, 9], "def_end_pos": [1818, 21]}]], "state_before": "case intro.intro.some.load\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nL\u271d R\u271d : ListBlank \u0393\na\u271d\u00b9 : \u0393 \u2192 \u03c3 \u2192 \u03c3\na\u271d : Stmt\u2081\na_ih\u271d :\n  \u2200 (v : \u03c3) (L R : ListBlank \u0393),\n    Reaches (step (tr enc dec M)) (stepAux (trNormal dec a\u271d) v (trTape' enc0 L R))\n      (trCfg enc enc0 (stepAux a\u271d v (Tape.mk' L R)))\nv : \u03c3\nL R : ListBlank \u0393\n\u22a2 Reaches (step (tr enc dec M)) (stepAux (trNormal dec (Stmt.load a\u271d\u00b9 a\u271d)) v (trTape' enc0 L R))\n    (trCfg enc enc0 (stepAux (Stmt.load a\u271d\u00b9 a\u271d) v (Tape.mk' L R)))\n\ncase intro.intro.some.branch\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nL\u271d R\u271d : ListBlank \u0393\na\u271d\u00b2 : \u0393 \u2192 \u03c3 \u2192 Bool\na\u271d\u00b9 a\u271d : Stmt\u2081\na_ih\u271d\u00b9 :\n  \u2200 (v : \u03c3) (L R : ListBlank \u0393),\n    Reaches (step (tr enc dec M)) (stepAux (trNormal dec a\u271d\u00b9) v (trTape' enc0 L R))\n      (trCfg enc enc0 (stepAux a\u271d\u00b9 v (Tape.mk' L R)))\na_ih\u271d :\n  \u2200 (v : \u03c3) (L R : ListBlank \u0393),\n    Reaches (step (tr enc dec M)) (stepAux (trNormal dec a\u271d) v (trTape' enc0 L R))\n      (trCfg enc enc0 (stepAux a\u271d v (Tape.mk' L R)))\nv : \u03c3\nL R : ListBlank \u0393\n\u22a2 Reaches (step (tr enc dec M)) (stepAux (trNormal dec (Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d)) v (trTape' enc0 L R))\n    (trCfg enc enc0 (stepAux (Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) v (Tape.mk' L R)))\n\ncase intro.intro.some.goto\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nL\u271d R\u271d : ListBlank \u0393\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u039b\nv : \u03c3\nL R : ListBlank \u0393\n\u22a2 Reaches (step (tr enc dec M)) (stepAux (trNormal dec (Stmt.goto a\u271d)) v (trTape' enc0 L R))\n    (trCfg enc enc0 (stepAux (Stmt.goto a\u271d) v (Tape.mk' L R)))\n\ncase intro.intro.some.halt\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nL\u271d R\u271d : ListBlank \u0393\nv : \u03c3\nL R : ListBlank \u0393\n\u22a2 Reaches (step (tr enc dec M)) (stepAux (trNormal dec Stmt.halt) v (trTape' enc0 L R))\n    (trCfg enc enc0 (stepAux Stmt.halt v (Tape.mk' L R)))", "state_after": "case intro.intro.some.branch\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nL\u271d R\u271d : ListBlank \u0393\na\u271d\u00b2 : \u0393 \u2192 \u03c3 \u2192 Bool\na\u271d\u00b9 a\u271d : Stmt\u2081\na_ih\u271d\u00b9 :\n  \u2200 (v : \u03c3) (L R : ListBlank \u0393),\n    Reaches (step (tr enc dec M)) (stepAux (trNormal dec a\u271d\u00b9) v (trTape' enc0 L R))\n      (trCfg enc enc0 (stepAux a\u271d\u00b9 v (Tape.mk' L R)))\na_ih\u271d :\n  \u2200 (v : \u03c3) (L R : ListBlank \u0393),\n    Reaches (step (tr enc dec M)) (stepAux (trNormal dec a\u271d) v (trTape' enc0 L R))\n      (trCfg enc enc0 (stepAux a\u271d v (Tape.mk' L R)))\nv : \u03c3\nL R : ListBlank \u0393\n\u22a2 Reaches (step (tr enc dec M)) (stepAux (trNormal dec (Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d)) v (trTape' enc0 L R))\n    (trCfg enc enc0 (stepAux (Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) v (Tape.mk' L R)))\n\ncase intro.intro.some.goto\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nL\u271d R\u271d : ListBlank \u0393\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u039b\nv : \u03c3\nL R : ListBlank \u0393\n\u22a2 Reaches (step (tr enc dec M)) (stepAux (trNormal dec (Stmt.goto a\u271d)) v (trTape' enc0 L R))\n    (trCfg enc enc0 (stepAux (Stmt.goto a\u271d) v (Tape.mk' L R)))\n\ncase intro.intro.some.halt\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nL\u271d R\u271d : ListBlank \u0393\nv : \u03c3\nL R : ListBlank \u0393\n\u22a2 Reaches (step (tr enc dec M)) (stepAux (trNormal dec Stmt.halt) v (trTape' enc0 L R))\n    (trCfg enc enc0 (stepAux Stmt.halt v (Tape.mk' L R)))"}, {"tactic": "case branch p q\u2081 q\u2082 IH\u2081 IH\u2082 =>\n  simp only [trNormal, stepAux_read dec enc0 encdec, stepAux]\n  cases p R.head v <;> [apply IH\u2082; apply IH\u2081]", "annotated_tactic": ["case branch p q\u2081 q\u2082 IH\u2081 IH\u2082 =>\n      simp only [<a>trNormal</a>, <a>stepAux_read</a> dec enc0 encdec, <a>stepAux</a>]\n      cases p R.head v <;> [apply IH\u2082; apply IH\u2081]", [{"full_name": "Turing.TM1to1.trNormal", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1687, 5], "def_end_pos": [1687, 13]}, {"full_name": "Turing.TM1to1.stepAux_read", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1818, 9], "def_end_pos": [1818, 21]}, {"full_name": "Turing.TM1.stepAux", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1281, 5], "def_end_pos": [1281, 12]}]], "state_before": "case intro.intro.some.branch\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nL\u271d R\u271d : ListBlank \u0393\na\u271d\u00b2 : \u0393 \u2192 \u03c3 \u2192 Bool\na\u271d\u00b9 a\u271d : Stmt\u2081\na_ih\u271d\u00b9 :\n  \u2200 (v : \u03c3) (L R : ListBlank \u0393),\n    Reaches (step (tr enc dec M)) (stepAux (trNormal dec a\u271d\u00b9) v (trTape' enc0 L R))\n      (trCfg enc enc0 (stepAux a\u271d\u00b9 v (Tape.mk' L R)))\na_ih\u271d :\n  \u2200 (v : \u03c3) (L R : ListBlank \u0393),\n    Reaches (step (tr enc dec M)) (stepAux (trNormal dec a\u271d) v (trTape' enc0 L R))\n      (trCfg enc enc0 (stepAux a\u271d v (Tape.mk' L R)))\nv : \u03c3\nL R : ListBlank \u0393\n\u22a2 Reaches (step (tr enc dec M)) (stepAux (trNormal dec (Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d)) v (trTape' enc0 L R))\n    (trCfg enc enc0 (stepAux (Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) v (Tape.mk' L R)))\n\ncase intro.intro.some.goto\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nL\u271d R\u271d : ListBlank \u0393\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u039b\nv : \u03c3\nL R : ListBlank \u0393\n\u22a2 Reaches (step (tr enc dec M)) (stepAux (trNormal dec (Stmt.goto a\u271d)) v (trTape' enc0 L R))\n    (trCfg enc enc0 (stepAux (Stmt.goto a\u271d) v (Tape.mk' L R)))\n\ncase intro.intro.some.halt\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nL\u271d R\u271d : ListBlank \u0393\nv : \u03c3\nL R : ListBlank \u0393\n\u22a2 Reaches (step (tr enc dec M)) (stepAux (trNormal dec Stmt.halt) v (trTape' enc0 L R))\n    (trCfg enc enc0 (stepAux Stmt.halt v (Tape.mk' L R)))", "state_after": "case intro.intro.some.goto\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nL\u271d R\u271d : ListBlank \u0393\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u039b\nv : \u03c3\nL R : ListBlank \u0393\n\u22a2 Reaches (step (tr enc dec M)) (stepAux (trNormal dec (Stmt.goto a\u271d)) v (trTape' enc0 L R))\n    (trCfg enc enc0 (stepAux (Stmt.goto a\u271d) v (Tape.mk' L R)))\n\ncase intro.intro.some.halt\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nL\u271d R\u271d : ListBlank \u0393\nv : \u03c3\nL R : ListBlank \u0393\n\u22a2 Reaches (step (tr enc dec M)) (stepAux (trNormal dec Stmt.halt) v (trTape' enc0 L R))\n    (trCfg enc enc0 (stepAux Stmt.halt v (Tape.mk' L R)))"}, {"tactic": "case goto l =>\n  simp only [trNormal, stepAux_read dec enc0 encdec, stepAux, trCfg, trTape_mk']\n  apply ReflTransGen.refl", "annotated_tactic": ["case goto l =>\n      simp only [<a>trNormal</a>, <a>stepAux_read</a> dec enc0 encdec, <a>stepAux</a>, <a>trCfg</a>, <a>trTape_mk'</a>]\n      apply <a>ReflTransGen.refl</a>", [{"full_name": "Turing.TM1to1.trNormal", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1687, 5], "def_end_pos": [1687, 13]}, {"full_name": "Turing.TM1to1.stepAux_read", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1818, 9], "def_end_pos": [1818, 21]}, {"full_name": "Turing.TM1.stepAux", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1281, 5], "def_end_pos": [1281, 12]}, {"full_name": "Turing.TM1to1.trCfg", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1761, 5], "def_end_pos": [1761, 10]}, {"full_name": "Turing.TM1to1.trTape_mk'", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1746, 9], "def_end_pos": [1746, 19]}, {"full_name": "Relation.ReflTransGen.refl", "def_path": "Mathlib/Logic/Relation.lean", "def_pos": [223, 5], "def_end_pos": [223, 9]}]], "state_before": "case intro.intro.some.goto\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nL\u271d R\u271d : ListBlank \u0393\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u039b\nv : \u03c3\nL R : ListBlank \u0393\n\u22a2 Reaches (step (tr enc dec M)) (stepAux (trNormal dec (Stmt.goto a\u271d)) v (trTape' enc0 L R))\n    (trCfg enc enc0 (stepAux (Stmt.goto a\u271d) v (Tape.mk' L R)))\n\ncase intro.intro.some.halt\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nL\u271d R\u271d : ListBlank \u0393\nv : \u03c3\nL R : ListBlank \u0393\n\u22a2 Reaches (step (tr enc dec M)) (stepAux (trNormal dec Stmt.halt) v (trTape' enc0 L R))\n    (trCfg enc enc0 (stepAux Stmt.halt v (Tape.mk' L R)))", "state_after": "case intro.intro.some.halt\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nL\u271d R\u271d : ListBlank \u0393\nv : \u03c3\nL R : ListBlank \u0393\n\u22a2 Reaches (step (tr enc dec M)) (stepAux (trNormal dec Stmt.halt) v (trTape' enc0 L R))\n    (trCfg enc enc0 (stepAux Stmt.halt v (Tape.mk' L R)))"}, {"tactic": "case halt =>\n  simp only [trNormal, stepAux, trCfg, stepAux_move, trTape'_move_left enc0,\n    trTape'_move_right enc0, trTape_mk']\n  apply ReflTransGen.refl", "annotated_tactic": ["case halt =>\n      simp only [<a>trNormal</a>, <a>stepAux</a>, <a>trCfg</a>, <a>stepAux_move</a>, <a>trTape'_move_left</a> enc0,\n        <a>trTape'_move_right</a> enc0, <a>trTape_mk'</a>]\n      apply <a>ReflTransGen.refl</a>", [{"full_name": "Turing.TM1to1.trNormal", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1687, 5], "def_end_pos": [1687, 13]}, {"full_name": "Turing.TM1.stepAux", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1281, 5], "def_end_pos": [1281, 12]}, {"full_name": "Turing.TM1to1.trCfg", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1761, 5], "def_end_pos": [1761, 10]}, {"full_name": "Turing.TM1to1.stepAux_move", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1697, 9], "def_end_pos": [1697, 21]}, {"full_name": "Turing.TM1to1.trTape'_move_left", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1767, 9], "def_end_pos": [1767, 26]}, {"full_name": "Turing.TM1to1.trTape'_move_right", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1785, 9], "def_end_pos": [1785, 27]}, {"full_name": "Turing.TM1to1.trTape_mk'", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1746, 9], "def_end_pos": [1746, 19]}, {"full_name": "Relation.ReflTransGen.refl", "def_path": "Mathlib/Logic/Relation.lean", "def_pos": [223, 5], "def_end_pos": [223, 9]}]], "state_before": "case intro.intro.some.halt\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nL\u271d R\u271d : ListBlank \u0393\nv : \u03c3\nL R : ListBlank \u0393\n\u22a2 Reaches (step (tr enc dec M)) (stepAux (trNormal dec Stmt.halt) v (trTape' enc0 L R))\n    (trCfg enc enc0 (stepAux Stmt.halt v (Tape.mk' L R)))", "state_after": "no goals"}, {"tactic": "exact rfl", "annotated_tactic": ["exact <a>rfl</a>", [{"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "case intro.intro.none\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv : \u03c3\nL R : ListBlank \u0393\n\u22a2 FRespects (step (tr enc dec M)) (fun c\u2081 => trCfg enc enc\u2080 c\u2081)\n    (trCfg enc enc\u2080 { l := none, var := v, Tape := Tape.mk' L R })\n    (step M { l := none, var := v, Tape := Tape.mk' L R })", "state_after": "no goals"}, {"tactic": "refine' TransGen.head' rfl _", "annotated_tactic": ["refine' <a>TransGen.head'</a> <a>rfl</a> _", [{"full_name": "Relation.TransGen.head'", "def_path": "Mathlib/Logic/Relation.lean", "def_pos": [375, 9], "def_end_pos": [375, 14]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv : \u03c3\nL R : ListBlank \u0393\nl\u2081 : \u039b\nthis :\n  \u2200 (q : Stmt\u2081) (R : ListBlank \u0393),\n    Reaches (step (tr enc dec M)) (stepAux (trNormal dec q) v (trTape' enc0 L R))\n      (trCfg enc enc0 (stepAux q v (Tape.mk' L R)))\n\u22a2 FRespects (step (tr enc dec M)) (fun c\u2081 => trCfg enc enc\u2080 c\u2081)\n    (trCfg enc enc\u2080 { l := some l\u2081, var := v, Tape := Tape.mk' L R })\n    (step M { l := some l\u2081, var := v, Tape := Tape.mk' L R })", "state_after": "\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv : \u03c3\nL R : ListBlank \u0393\nl\u2081 : \u039b\nthis :\n  \u2200 (q : Stmt\u2081) (R : ListBlank \u0393),\n    Reaches (step (tr enc dec M)) (stepAux (trNormal dec q) v (trTape' enc0 L R))\n      (trCfg enc enc0 (stepAux q v (Tape.mk' L R)))\n\u22a2 ReflTransGen (fun a b => b \u2208 step (tr enc dec M) a)\n    (stepAux (tr enc dec M (\u039b'.normal l\u2081)) v (trTape enc\u2080 (Tape.mk' L R)))\n    ((fun c\u2081 => trCfg enc enc\u2080 c\u2081) (stepAux (M l\u2081) v (Tape.mk' L R)))"}, {"tactic": "rw [trTape_mk']", "annotated_tactic": ["rw [<a>trTape_mk'</a>]", [{"full_name": "Turing.TM1to1.trTape_mk'", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1746, 9], "def_end_pos": [1746, 19]}]], "state_before": "\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv : \u03c3\nL R : ListBlank \u0393\nl\u2081 : \u039b\nthis :\n  \u2200 (q : Stmt\u2081) (R : ListBlank \u0393),\n    Reaches (step (tr enc dec M)) (stepAux (trNormal dec q) v (trTape' enc0 L R))\n      (trCfg enc enc0 (stepAux q v (Tape.mk' L R)))\n\u22a2 ReflTransGen (fun a b => b \u2208 step (tr enc dec M) a)\n    (stepAux (tr enc dec M (\u039b'.normal l\u2081)) v (trTape enc\u2080 (Tape.mk' L R)))\n    ((fun c\u2081 => trCfg enc enc\u2080 c\u2081) (stepAux (M l\u2081) v (Tape.mk' L R)))", "state_after": "\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv : \u03c3\nL R : ListBlank \u0393\nl\u2081 : \u039b\nthis :\n  \u2200 (q : Stmt\u2081) (R : ListBlank \u0393),\n    Reaches (step (tr enc dec M)) (stepAux (trNormal dec q) v (trTape' enc0 L R))\n      (trCfg enc enc0 (stepAux q v (Tape.mk' L R)))\n\u22a2 ReflTransGen (fun a b => b \u2208 step (tr enc dec M) a) (stepAux (tr enc dec M (\u039b'.normal l\u2081)) v (trTape' enc\u2080 L R))\n    ((fun c\u2081 => trCfg enc enc\u2080 c\u2081) (stepAux (M l\u2081) v (Tape.mk' L R)))"}, {"tactic": "exact this _ R", "annotated_tactic": ["exact this _ R", []], "state_before": "\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv : \u03c3\nL R : ListBlank \u0393\nl\u2081 : \u039b\nthis :\n  \u2200 (q : Stmt\u2081) (R : ListBlank \u0393),\n    Reaches (step (tr enc dec M)) (stepAux (trNormal dec q) v (trTape' enc0 L R))\n      (trCfg enc enc0 (stepAux q v (Tape.mk' L R)))\n\u22a2 ReflTransGen (fun a b => b \u2208 step (tr enc dec M) a) (stepAux (tr enc dec M (\u039b'.normal l\u2081)) v (trTape' enc\u2080 L R))\n    ((fun c\u2081 => trCfg enc enc\u2080 c\u2081) (stepAux (M l\u2081) v (Tape.mk' L R)))", "state_after": "no goals"}, {"tactic": "cases d <;>\n    simp only [trNormal, iterate, stepAux_move, stepAux, ListBlank.head_cons,\n      Tape.move_left_mk', ListBlank.cons_head_tail, ListBlank.tail_cons,\n      trTape'_move_left enc0, trTape'_move_right enc0] <;>\n  apply IH", "annotated_tactic": ["cases d <;>\n          simp only [<a>trNormal</a>, <a>iterate</a>, <a>stepAux_move</a>, <a>stepAux</a>, <a>ListBlank.head_cons</a>,\n            <a>Tape.move_left_mk'</a>, <a>ListBlank.cons_head_tail</a>, <a>ListBlank.tail_cons</a>,\n            <a>trTape'_move_left</a> enc0, <a>trTape'_move_right</a> enc0] <;>\n        apply IH", [{"full_name": "Turing.TM1to1.trNormal", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1687, 5], "def_end_pos": [1687, 13]}, {"full_name": "Nat.iterate", "def_path": "Mathlib/Logic/Function/Iterate.lean", "def_pos": [38, 5], "def_end_pos": [38, 16]}, {"full_name": "Turing.TM1to1.stepAux_move", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1697, 9], "def_end_pos": [1697, 21]}, {"full_name": "Turing.TM1.stepAux", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1281, 5], "def_end_pos": [1281, 12]}, {"full_name": "Turing.ListBlank.head_cons", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [253, 9], "def_end_pos": [253, 28]}, {"full_name": "Turing.Tape.move_left_mk'", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [581, 9], "def_end_pos": [581, 27]}, {"full_name": "Turing.ListBlank.cons_head_tail", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [265, 9], "def_end_pos": [265, 33]}, {"full_name": "Turing.ListBlank.tail_cons", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [258, 9], "def_end_pos": [258, 28]}, {"full_name": "Turing.TM1to1.trTape'_move_left", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1767, 9], "def_end_pos": [1767, 26]}, {"full_name": "Turing.TM1to1.trTape'_move_right", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1785, 9], "def_end_pos": [1785, 27]}]], "state_before": "\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nL\u271d R\u271d : ListBlank \u0393\nd : Dir\nq : Stmt\u2081\nIH :\n  \u2200 (v : \u03c3) (L R : ListBlank \u0393),\n    Reaches (step (tr enc dec M)) (stepAux (trNormal dec q) v (trTape' enc0 L R))\n      (trCfg enc enc0 (stepAux q v (Tape.mk' L R)))\nv : \u03c3\nL R : ListBlank \u0393\n\u22a2 Reaches (step (tr enc dec M)) (stepAux (trNormal dec (Stmt.move d q)) v (trTape' enc0 L R))\n    (trCfg enc enc0 (stepAux (Stmt.move d q) v (Tape.mk' L R)))", "state_after": "no goals"}, {"tactic": "simp only [trNormal, stepAux_read dec enc0 encdec, stepAux]", "annotated_tactic": ["simp only [<a>trNormal</a>, <a>stepAux_read</a> dec enc0 encdec, <a>stepAux</a>]", [{"full_name": "Turing.TM1to1.trNormal", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1687, 5], "def_end_pos": [1687, 13]}, {"full_name": "Turing.TM1to1.stepAux_read", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1818, 9], "def_end_pos": [1818, 21]}, {"full_name": "Turing.TM1.stepAux", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1281, 5], "def_end_pos": [1281, 12]}]], "state_before": "\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nL\u271d R\u271d : ListBlank \u0393\nf : \u0393 \u2192 \u03c3 \u2192 \u0393\nq : Stmt\u2081\nIH :\n  \u2200 (v : \u03c3) (L R : ListBlank \u0393),\n    Reaches (step (tr enc dec M)) (stepAux (trNormal dec q) v (trTape' enc0 L R))\n      (trCfg enc enc0 (stepAux q v (Tape.mk' L R)))\nv : \u03c3\nL R : ListBlank \u0393\n\u22a2 Reaches (step (tr enc dec M)) (stepAux (trNormal dec (Stmt.write f q)) v (trTape' enc0 L R))\n    (trCfg enc enc0 (stepAux (Stmt.write f q) v (Tape.mk' L R)))", "state_after": "\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nL\u271d R\u271d : ListBlank \u0393\nf : \u0393 \u2192 \u03c3 \u2192 \u0393\nq : Stmt\u2081\nIH :\n  \u2200 (v : \u03c3) (L R : ListBlank \u0393),\n    Reaches (step (tr enc dec M)) (stepAux (trNormal dec q) v (trTape' enc0 L R))\n      (trCfg enc enc0 (stepAux q v (Tape.mk' L R)))\nv : \u03c3\nL R : ListBlank \u0393\n\u22a2 Reaches (step (tr enc dec M)) { l := some (\u039b'.write (f (ListBlank.head R) v) q), var := v, Tape := trTape' enc0 L R }\n    (trCfg enc enc0 (stepAux q v (Tape.write (f (Tape.mk' L R).head v) (Tape.mk' L R))))"}, {"tactic": "refine' ReflTransGen.head rfl _", "annotated_tactic": ["refine' <a>ReflTransGen.head</a> <a>rfl</a> _", [{"full_name": "Relation.ReflTransGen.head", "def_path": "Mathlib/Logic/Relation.lean", "def_pos": [280, 9], "def_end_pos": [280, 13]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nL\u271d R\u271d : ListBlank \u0393\nf : \u0393 \u2192 \u03c3 \u2192 \u0393\nq : Stmt\u2081\nIH :\n  \u2200 (v : \u03c3) (L R : ListBlank \u0393),\n    Reaches (step (tr enc dec M)) (stepAux (trNormal dec q) v (trTape' enc0 L R))\n      (trCfg enc enc0 (stepAux q v (Tape.mk' L R)))\nv : \u03c3\nL R : ListBlank \u0393\n\u22a2 Reaches (step (tr enc dec M)) { l := some (\u039b'.write (f (ListBlank.head R) v) q), var := v, Tape := trTape' enc0 L R }\n    (trCfg enc enc0 (stepAux q v (Tape.write (f (Tape.mk' L R).head v) (Tape.mk' L R))))", "state_after": "\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nL\u271d R\u271d : ListBlank \u0393\nf : \u0393 \u2192 \u03c3 \u2192 \u0393\nq : Stmt\u2081\nIH :\n  \u2200 (v : \u03c3) (L R : ListBlank \u0393),\n    Reaches (step (tr enc dec M)) (stepAux (trNormal dec q) v (trTape' enc0 L R))\n      (trCfg enc enc0 (stepAux q v (Tape.mk' L R)))\nv : \u03c3\nL R : ListBlank \u0393\n\u22a2 ReflTransGen (fun a b => b \u2208 step (tr enc dec M) a)\n    (stepAux (tr enc dec M (\u039b'.write (f (ListBlank.head R) v) q)) v (trTape' enc0 L R))\n    (trCfg enc enc0 (stepAux q v (Tape.write (f (Tape.mk' L R).head v) (Tape.mk' L R))))"}, {"tactic": "obtain \u27e8a, R, rfl\u27e9 := R.exists_cons", "annotated_tactic": ["obtain \u27e8a, R, rfl\u27e9 := R.exists_cons", []], "state_before": "\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nL\u271d R\u271d : ListBlank \u0393\nf : \u0393 \u2192 \u03c3 \u2192 \u0393\nq : Stmt\u2081\nIH :\n  \u2200 (v : \u03c3) (L R : ListBlank \u0393),\n    Reaches (step (tr enc dec M)) (stepAux (trNormal dec q) v (trTape' enc0 L R))\n      (trCfg enc enc0 (stepAux q v (Tape.mk' L R)))\nv : \u03c3\nL R : ListBlank \u0393\n\u22a2 ReflTransGen (fun a b => b \u2208 step (tr enc dec M) a)\n    (stepAux (tr enc dec M (\u039b'.write (f (ListBlank.head R) v) q)) v (trTape' enc0 L R))\n    (trCfg enc enc0 (stepAux q v (Tape.write (f (Tape.mk' L R).head v) (Tape.mk' L R))))", "state_after": "case intro.intro\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nL\u271d R\u271d : ListBlank \u0393\nf : \u0393 \u2192 \u03c3 \u2192 \u0393\nq : Stmt\u2081\nIH :\n  \u2200 (v : \u03c3) (L R : ListBlank \u0393),\n    Reaches (step (tr enc dec M)) (stepAux (trNormal dec q) v (trTape' enc0 L R))\n      (trCfg enc enc0 (stepAux q v (Tape.mk' L R)))\nv : \u03c3\nL : ListBlank \u0393\na : \u0393\nR : ListBlank \u0393\n\u22a2 ReflTransGen (fun a b => b \u2208 step (tr enc dec M) a)\n    (stepAux (tr enc dec M (\u039b'.write (f (ListBlank.head (ListBlank.cons a R)) v) q)) v\n      (trTape' enc0 L (ListBlank.cons a R)))\n    (trCfg enc enc0\n      (stepAux q v (Tape.write (f (Tape.mk' L (ListBlank.cons a R)).head v) (Tape.mk' L (ListBlank.cons a R)))))"}, {"tactic": "rw [tr, Tape.mk'_head, stepAux_write, ListBlank.head_cons, stepAux_move,\n  trTape'_move_left enc0, ListBlank.head_cons, ListBlank.tail_cons, Tape.write_mk']", "annotated_tactic": ["rw [<a>tr</a>, <a>Tape.mk'_head</a>, <a>stepAux_write</a>, <a>ListBlank.head_cons</a>, <a>stepAux_move</a>,\n        <a>trTape'_move_left</a> enc0, <a>ListBlank.head_cons</a>, <a>ListBlank.tail_cons</a>, <a>Tape.write_mk'</a>]", [{"full_name": "Turing.TM1to1.tr", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1755, 5], "def_end_pos": [1755, 7]}, {"full_name": "Turing.Tape.mk'_head", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [555, 9], "def_end_pos": [555, 22]}, {"full_name": "Turing.TM1to1.stepAux_write", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1797, 9], "def_end_pos": [1797, 22]}, {"full_name": "Turing.ListBlank.head_cons", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [253, 9], "def_end_pos": [253, 28]}, {"full_name": "Turing.TM1to1.stepAux_move", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1697, 9], "def_end_pos": [1697, 21]}, {"full_name": "Turing.TM1to1.trTape'_move_left", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1767, 9], "def_end_pos": [1767, 26]}, {"full_name": "Turing.ListBlank.head_cons", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [253, 9], "def_end_pos": [253, 28]}, {"full_name": "Turing.ListBlank.tail_cons", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [258, 9], "def_end_pos": [258, 28]}, {"full_name": "Turing.Tape.write_mk'", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [675, 9], "def_end_pos": [675, 23]}]], "state_before": "case intro.intro\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nL\u271d R\u271d : ListBlank \u0393\nf : \u0393 \u2192 \u03c3 \u2192 \u0393\nq : Stmt\u2081\nIH :\n  \u2200 (v : \u03c3) (L R : ListBlank \u0393),\n    Reaches (step (tr enc dec M)) (stepAux (trNormal dec q) v (trTape' enc0 L R))\n      (trCfg enc enc0 (stepAux q v (Tape.mk' L R)))\nv : \u03c3\nL : ListBlank \u0393\na : \u0393\nR : ListBlank \u0393\n\u22a2 ReflTransGen (fun a b => b \u2208 step (tr enc dec M) a)\n    (stepAux (tr enc dec M (\u039b'.write (f (ListBlank.head (ListBlank.cons a R)) v) q)) v\n      (trTape' enc0 L (ListBlank.cons a R)))\n    (trCfg enc enc0\n      (stepAux q v (Tape.write (f (Tape.mk' L (ListBlank.cons a R)).head v) (Tape.mk' L (ListBlank.cons a R)))))", "state_after": "case intro.intro\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nL\u271d R\u271d : ListBlank \u0393\nf : \u0393 \u2192 \u03c3 \u2192 \u0393\nq : Stmt\u2081\nIH :\n  \u2200 (v : \u03c3) (L R : ListBlank \u0393),\n    Reaches (step (tr enc dec M)) (stepAux (trNormal dec q) v (trTape' enc0 L R))\n      (trCfg enc enc0 (stepAux q v (Tape.mk' L R)))\nv : \u03c3\nL : ListBlank \u0393\na : \u0393\nR : ListBlank \u0393\n\u22a2 ReflTransGen (fun a b => b \u2208 step (tr enc dec M) a)\n    (stepAux (trNormal dec q) v (trTape' enc0 L (ListBlank.cons (f a v) R)))\n    (trCfg enc enc0 (stepAux q v (Tape.mk' L (ListBlank.cons (f a v) R))))"}, {"tactic": "apply IH", "annotated_tactic": ["apply IH", []], "state_before": "case intro.intro\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nL\u271d R\u271d : ListBlank \u0393\nf : \u0393 \u2192 \u03c3 \u2192 \u0393\nq : Stmt\u2081\nIH :\n  \u2200 (v : \u03c3) (L R : ListBlank \u0393),\n    Reaches (step (tr enc dec M)) (stepAux (trNormal dec q) v (trTape' enc0 L R))\n      (trCfg enc enc0 (stepAux q v (Tape.mk' L R)))\nv : \u03c3\nL : ListBlank \u0393\na : \u0393\nR : ListBlank \u0393\n\u22a2 ReflTransGen (fun a b => b \u2208 step (tr enc dec M) a)\n    (stepAux (trNormal dec q) v (trTape' enc0 L (ListBlank.cons (f a v) R)))\n    (trCfg enc enc0 (stepAux q v (Tape.mk' L (ListBlank.cons (f a v) R))))", "state_after": "no goals"}, {"tactic": "simp only [trNormal, stepAux_read dec enc0 encdec]", "annotated_tactic": ["simp only [<a>trNormal</a>, <a>stepAux_read</a> dec enc0 encdec]", [{"full_name": "Turing.TM1to1.trNormal", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1687, 5], "def_end_pos": [1687, 13]}, {"full_name": "Turing.TM1to1.stepAux_read", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1818, 9], "def_end_pos": [1818, 21]}]], "state_before": "\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nL\u271d R\u271d : ListBlank \u0393\na : \u0393 \u2192 \u03c3 \u2192 \u03c3\nq : Stmt\u2081\nIH :\n  \u2200 (v : \u03c3) (L R : ListBlank \u0393),\n    Reaches (step (tr enc dec M)) (stepAux (trNormal dec q) v (trTape' enc0 L R))\n      (trCfg enc enc0 (stepAux q v (Tape.mk' L R)))\nv : \u03c3\nL R : ListBlank \u0393\n\u22a2 Reaches (step (tr enc dec M)) (stepAux (trNormal dec (Stmt.load a q)) v (trTape' enc0 L R))\n    (trCfg enc enc0 (stepAux (Stmt.load a q) v (Tape.mk' L R)))", "state_after": "\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nL\u271d R\u271d : ListBlank \u0393\na : \u0393 \u2192 \u03c3 \u2192 \u03c3\nq : Stmt\u2081\nIH :\n  \u2200 (v : \u03c3) (L R : ListBlank \u0393),\n    Reaches (step (tr enc dec M)) (stepAux (trNormal dec q) v (trTape' enc0 L R))\n      (trCfg enc enc0 (stepAux q v (Tape.mk' L R)))\nv : \u03c3\nL R : ListBlank \u0393\n\u22a2 Reaches (step (tr enc dec M))\n    (stepAux (Stmt.load (fun x s => a (ListBlank.head R) s) (trNormal dec q)) v (trTape' enc0 L R))\n    (trCfg enc enc0 (stepAux (Stmt.load a q) v (Tape.mk' L R)))"}, {"tactic": "apply IH", "annotated_tactic": ["apply IH", []], "state_before": "\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nL\u271d R\u271d : ListBlank \u0393\na : \u0393 \u2192 \u03c3 \u2192 \u03c3\nq : Stmt\u2081\nIH :\n  \u2200 (v : \u03c3) (L R : ListBlank \u0393),\n    Reaches (step (tr enc dec M)) (stepAux (trNormal dec q) v (trTape' enc0 L R))\n      (trCfg enc enc0 (stepAux q v (Tape.mk' L R)))\nv : \u03c3\nL R : ListBlank \u0393\n\u22a2 Reaches (step (tr enc dec M))\n    (stepAux (Stmt.load (fun x s => a (ListBlank.head R) s) (trNormal dec q)) v (trTape' enc0 L R))\n    (trCfg enc enc0 (stepAux (Stmt.load a q) v (Tape.mk' L R)))", "state_after": "no goals"}, {"tactic": "simp only [trNormal, stepAux_read dec enc0 encdec, stepAux]", "annotated_tactic": ["simp only [<a>trNormal</a>, <a>stepAux_read</a> dec enc0 encdec, <a>stepAux</a>]", [{"full_name": "Turing.TM1to1.trNormal", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1687, 5], "def_end_pos": [1687, 13]}, {"full_name": "Turing.TM1to1.stepAux_read", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1818, 9], "def_end_pos": [1818, 21]}, {"full_name": "Turing.TM1.stepAux", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1281, 5], "def_end_pos": [1281, 12]}]], "state_before": "\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nL\u271d R\u271d : ListBlank \u0393\np : \u0393 \u2192 \u03c3 \u2192 Bool\nq\u2081 q\u2082 : Stmt\u2081\nIH\u2081 :\n  \u2200 (v : \u03c3) (L R : ListBlank \u0393),\n    Reaches (step (tr enc dec M)) (stepAux (trNormal dec q\u2081) v (trTape' enc0 L R))\n      (trCfg enc enc0 (stepAux q\u2081 v (Tape.mk' L R)))\nIH\u2082 :\n  \u2200 (v : \u03c3) (L R : ListBlank \u0393),\n    Reaches (step (tr enc dec M)) (stepAux (trNormal dec q\u2082) v (trTape' enc0 L R))\n      (trCfg enc enc0 (stepAux q\u2082 v (Tape.mk' L R)))\nv : \u03c3\nL R : ListBlank \u0393\n\u22a2 Reaches (step (tr enc dec M)) (stepAux (trNormal dec (Stmt.branch p q\u2081 q\u2082)) v (trTape' enc0 L R))\n    (trCfg enc enc0 (stepAux (Stmt.branch p q\u2081 q\u2082) v (Tape.mk' L R)))", "state_after": "\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nL\u271d R\u271d : ListBlank \u0393\np : \u0393 \u2192 \u03c3 \u2192 Bool\nq\u2081 q\u2082 : Stmt\u2081\nIH\u2081 :\n  \u2200 (v : \u03c3) (L R : ListBlank \u0393),\n    Reaches (step (tr enc dec M)) (stepAux (trNormal dec q\u2081) v (trTape' enc0 L R))\n      (trCfg enc enc0 (stepAux q\u2081 v (Tape.mk' L R)))\nIH\u2082 :\n  \u2200 (v : \u03c3) (L R : ListBlank \u0393),\n    Reaches (step (tr enc dec M)) (stepAux (trNormal dec q\u2082) v (trTape' enc0 L R))\n      (trCfg enc enc0 (stepAux q\u2082 v (Tape.mk' L R)))\nv : \u03c3\nL R : ListBlank 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\u0393 \u2192 \u03c3 \u2192 Bool\nq\u2081 q\u2082 : Stmt\u2081\nIH\u2081 :\n  \u2200 (v : \u03c3) (L R : ListBlank \u0393),\n    Reaches (step (tr enc dec M)) (stepAux (trNormal dec q\u2081) v (trTape' enc0 L R))\n      (trCfg enc enc0 (stepAux q\u2081 v (Tape.mk' L R)))\nIH\u2082 :\n  \u2200 (v : \u03c3) (L R : ListBlank \u0393),\n    Reaches (step (tr enc dec M)) (stepAux (trNormal dec q\u2082) v (trTape' enc0 L R))\n      (trCfg enc enc0 (stepAux q\u2082 v (Tape.mk' L R)))\nv : \u03c3\nL R : ListBlank \u0393\n\u22a2 Reaches (step (tr enc dec M))\n    (bif p (ListBlank.head R) v then stepAux (trNormal dec q\u2081) v (trTape' enc0 L R)\n    else stepAux (trNormal dec q\u2082) v (trTape' enc0 L R))\n    (trCfg enc enc0 (bif p (Tape.mk' L R).head v then stepAux q\u2081 v (Tape.mk' L R) else stepAux q\u2082 v (Tape.mk' L R)))", "state_after": "no goals"}, {"tactic": "simp only [trNormal, stepAux_read dec enc0 encdec, stepAux, trCfg, trTape_mk']", "annotated_tactic": ["simp only 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\u03c9}\n\u22a2 MeasurableSet {\u03c9 | \u03c4 \u03c9 \u2264 \u03c0 \u03c9}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Sigma.lean", "full_name": "Finset.mem_sigma", "start": [49, 1], "end": [50, 21], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Intervals/Group.lean", "full_name": "Set.pairwise_disjoint_Ioo_int_cast", "start": [257, 1], "end": [258, 76], "traced_tactics": [{"tactic": "simpa only [zero_add] using pairwise_disjoint_Ioo_add_int_cast (0 : \u03b1)", "annotated_tactic": ["simpa only [<a>zero_add</a>] using <a>pairwise_disjoint_Ioo_add_int_cast</a> (0 : \u03b1)", [{"full_name": "zero_add", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [463, 3], "def_end_pos": [463, 14]}, {"full_name": "Set.pairwise_disjoint_Ioo_add_int_cast", "def_path": "Mathlib/Data/Set/Intervals/Group.lean", "def_pos": [245, 9], "def_end_pos": [245, 43]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : OrderedRing \u03b1\na : \u03b1\n\u22a2 Pairwise (Disjoint on fun n => Ioo (\u2191n) (\u2191n + 1))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "full_name": "MeasureTheory.setToFun_zero", "start": [1346, 1], "end": [1347, 99], "traced_tactics": [{"tactic": "erw [setToFun_eq hT (integrable_zero _ _ _), Integrable.toL1_zero, ContinuousLinearMap.map_zero]", "annotated_tactic": ["erw [<a>setToFun_eq</a> hT (<a>integrable_zero</a> _ _ _), <a>Integrable.toL1_zero</a>, <a>ContinuousLinearMap.map_zero</a>]", [{"full_name": "MeasureTheory.setToFun_eq", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [1276, 9], "def_end_pos": [1276, 20]}, {"full_name": "MeasureTheory.integrable_zero", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [662, 9], "def_end_pos": [662, 24]}, {"full_name": "MeasureTheory.Integrable.toL1_zero", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [1416, 9], "def_end_pos": [1416, 18]}, {"full_name": "ContinuousLinearMap.map_zero", "def_path": "Mathlib/Topology/Algebra/Module/Basic.lean", "def_pos": [506, 19], "def_end_pos": [506, 27]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\n\u22a2 setToFun \u03bc T hT 0 = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "full_name": "Int.sign_eq_div_abs", "start": [835, 11], "end": [838, 37], "traced_tactics": [{"tactic": "simp [az]", "annotated_tactic": ["simp [az]", []], "state_before": "a : Int\naz : a = 0\n\u22a2 sign a = div a \u2191(natAbs a)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Holor.lean", "full_name": "Holor.cprankMax_mul", "start": [350, 1], "end": [358, 37], "traced_tactics": [{"tactic": "simp [mul_zero x, CPRankMax.zero]", "annotated_tactic": ["simp [<a>mul_zero</a> x, <a>CPRankMax.zero</a>]", [{"full_name": "Holor.mul_zero", "def_path": "Mathlib/Data/Holor.lean", "def_pos": [222, 16], "def_end_pos": [222, 24]}, {"full_name": "Holor.CPRankMax.zero", "def_path": "Mathlib/Data/Holor.lean", "def_pos": [319, 5], "def_end_pos": [319, 9]}]], "state_before": "\u03b1 : Type\nd : \u2115\nds ds\u2081 ds\u2082 ds\u2083 : List \u2115\ninst\u271d : Ring \u03b1\nx : Holor \u03b1 [d]\n\u22a2 CPRankMax 0 (x \u2297 0)", "state_after": "no goals"}, {"tactic": "rw [mul_left_distrib]", "annotated_tactic": ["rw [<a>mul_left_distrib</a>]", [{"full_name": "Holor.mul_left_distrib", "def_path": "Mathlib/Data/Holor.lean", "def_pos": [208, 9], "def_end_pos": [208, 25]}]], "state_before": "\u03b1 : Type\nd : \u2115\nds ds\u2081 ds\u2082 ds\u2083 : List \u2115\ninst\u271d : Ring \u03b1\nn : \u2115\nx : Holor \u03b1 [d]\ny\u2081 y\u2082 : Holor \u03b1 ds\nhy\u2081 : CPRankMax1 y\u2081\nhy\u2082 : CPRankMax n y\u2082\n\u22a2 CPRankMax (n + 1) (x \u2297 (y\u2081 + y\u2082))", "state_after": "\u03b1 : Type\nd : \u2115\nds ds\u2081 ds\u2082 ds\u2083 : List \u2115\ninst\u271d : Ring \u03b1\nn : \u2115\nx : Holor \u03b1 [d]\ny\u2081 y\u2082 : Holor \u03b1 ds\nhy\u2081 : CPRankMax1 y\u2081\nhy\u2082 : CPRankMax n y\u2082\n\u22a2 CPRankMax (n + 1) (x \u2297 y\u2081 + x \u2297 y\u2082)"}, {"tactic": "rw [Nat.add_comm]", "annotated_tactic": ["rw [<a>Nat.add_comm</a>]", [{"full_name": "Nat.add_comm", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [131, 19], "def_end_pos": [131, 27]}]], "state_before": "\u03b1 : Type\nd : \u2115\nds ds\u2081 ds\u2082 ds\u2083 : List \u2115\ninst\u271d : Ring \u03b1\nn : \u2115\nx : Holor \u03b1 [d]\ny\u2081 y\u2082 : Holor \u03b1 ds\nhy\u2081 : CPRankMax1 y\u2081\nhy\u2082 : CPRankMax n y\u2082\n\u22a2 CPRankMax (n + 1) (x \u2297 y\u2081 + x \u2297 y\u2082)", "state_after": "\u03b1 : Type\nd : \u2115\nds ds\u2081 ds\u2082 ds\u2083 : List \u2115\ninst\u271d : Ring \u03b1\nn : \u2115\nx : Holor \u03b1 [d]\ny\u2081 y\u2082 : Holor \u03b1 ds\nhy\u2081 : CPRankMax1 y\u2081\nhy\u2082 : CPRankMax n y\u2082\n\u22a2 CPRankMax (1 + n) (x \u2297 y\u2081 + x \u2297 y\u2082)"}, {"tactic": "apply cprankMax_add", "annotated_tactic": ["apply <a>cprankMax_add</a>", [{"full_name": "Holor.cprankMax_add", "def_path": "Mathlib/Data/Holor.lean", "def_pos": [335, 9], "def_end_pos": [335, 22]}]], "state_before": "\u03b1 : Type\nd : \u2115\nds ds\u2081 ds\u2082 ds\u2083 : List \u2115\ninst\u271d : Ring \u03b1\nn : \u2115\nx : Holor \u03b1 [d]\ny\u2081 y\u2082 : Holor \u03b1 ds\nhy\u2081 : CPRankMax1 y\u2081\nhy\u2082 : CPRankMax n y\u2082\n\u22a2 CPRankMax (1 + n) (x \u2297 y\u2081 + x \u2297 y\u2082)", "state_after": "case a\n\u03b1 : Type\nd : \u2115\nds ds\u2081 ds\u2082 ds\u2083 : List \u2115\ninst\u271d : Ring \u03b1\nn : \u2115\nx : Holor \u03b1 [d]\ny\u2081 y\u2082 : Holor \u03b1 ds\nhy\u2081 : CPRankMax1 y\u2081\nhy\u2082 : CPRankMax n y\u2082\n\u22a2 CPRankMax 1 (x \u2297 y\u2081)\n\ncase a\n\u03b1 : Type\nd : \u2115\nds ds\u2081 ds\u2082 ds\u2083 : List \u2115\ninst\u271d : Ring \u03b1\nn : \u2115\nx : Holor \u03b1 [d]\ny\u2081 y\u2082 : Holor \u03b1 ds\nhy\u2081 : CPRankMax1 y\u2081\nhy\u2082 : CPRankMax n y\u2082\n\u22a2 CPRankMax n (x \u2297 y\u2082)"}, {"tactic": "exact cprankMax_1 (CPRankMax1.cons _ _ hy\u2081)", "annotated_tactic": ["exact <a>cprankMax_1</a> (<a>CPRankMax1.cons</a> _ _ hy\u2081)", [{"full_name": "Holor.cprankMax_1", "def_path": "Mathlib/Data/Holor.lean", "def_pos": [329, 9], "def_end_pos": [329, 20]}, {"full_name": "Holor.CPRankMax1.cons", "def_path": "Mathlib/Data/Holor.lean", "def_pos": [313, 5], "def_end_pos": [313, 9]}]], "state_before": "case a\n\u03b1 : Type\nd : \u2115\nds ds\u2081 ds\u2082 ds\u2083 : List \u2115\ninst\u271d : Ring \u03b1\nn : \u2115\nx : Holor \u03b1 [d]\ny\u2081 y\u2082 : Holor \u03b1 ds\nhy\u2081 : CPRankMax1 y\u2081\nhy\u2082 : CPRankMax n y\u2082\n\u22a2 CPRankMax 1 (x \u2297 y\u2081)", "state_after": "no goals"}, {"tactic": "exact cprankMax_mul _ x y\u2082 hy\u2082", "annotated_tactic": ["exact cprankMax_mul _ x y\u2082 hy\u2082", []], "state_before": "case a\n\u03b1 : Type\nd : \u2115\nds ds\u2081 ds\u2082 ds\u2083 : List \u2115\ninst\u271d : Ring \u03b1\nn : \u2115\nx : Holor \u03b1 [d]\ny\u2081 y\u2082 : Holor \u03b1 ds\nhy\u2081 : CPRankMax1 y\u2081\nhy\u2082 : CPRankMax n y\u2082\n\u22a2 CPRankMax n (x \u2297 y\u2082)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Independence/Basic.lean", "full_name": "ProbabilityTheory.indep_bot_left", "start": [238, 1], "end": [239, 73], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Constructions/Prod/Integral.lean", "full_name": "MeasureTheory.integral_integral_add'", "start": [398, 1], "end": [401, 30], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/Average.lean", "full_name": "MeasureTheory.measure_mul_laverage", "start": [100, 1], "end": [104, 95], "traced_tactics": [{"tactic": "cases' eq_or_ne \u03bc 0 with h\u03bc h\u03bc", "annotated_tactic": ["cases' <a>eq_or_ne</a> \u03bc 0 with h\u03bc h\u03bc", [{"full_name": "eq_or_ne", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [209, 9], "def_end_pos": [209, 17]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\nf\u271d g : \u03b1 \u2192 \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u22a2 \u2191\u2191\u03bc univ * \u2a0d\u207b (x : \u03b1), f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc", "state_after": "case inl\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\nf\u271d g : \u03b1 \u2192 \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh\u03bc : \u03bc = 0\n\u22a2 \u2191\u2191\u03bc univ * \u2a0d\u207b (x : \u03b1), f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc\n\ncase inr\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\nf\u271d g : \u03b1 \u2192 \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh\u03bc : \u03bc \u2260 0\n\u22a2 \u2191\u2191\u03bc univ * \u2a0d\u207b (x : \u03b1), f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc"}, {"tactic": "rw [h\u03bc, lintegral_zero_measure, laverage_zero_measure, mul_zero]", "annotated_tactic": ["rw [h\u03bc, <a>lintegral_zero_measure</a>, <a>laverage_zero_measure</a>, <a>mul_zero</a>]", [{"full_name": "MeasureTheory.lintegral_zero_measure", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [632, 9], "def_end_pos": [632, 31]}, {"full_name": "MeasureTheory.laverage_zero_measure", "def_path": "Mathlib/MeasureTheory/Integral/Average.lean", "def_pos": [85, 9], "def_end_pos": [85, 30]}, {"full_name": "MulZeroClass.mul_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [38, 3], "def_end_pos": [38, 11]}]], "state_before": "case inl\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\nf\u271d g : \u03b1 \u2192 \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh\u03bc : \u03bc = 0\n\u22a2 \u2191\u2191\u03bc univ * \u2a0d\u207b (x : \u03b1), f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc", "state_after": "no goals"}, {"tactic": "rw [laverage_eq, ENNReal.mul_div_cancel' (measure_univ_ne_zero.2 h\u03bc) (measure_ne_top _ _)]", "annotated_tactic": ["rw [<a>laverage_eq</a>, <a>ENNReal.mul_div_cancel'</a> (<a>measure_univ_ne_zero</a>.2 h\u03bc) (<a>measure_ne_top</a> _ _)]", [{"full_name": "MeasureTheory.laverage_eq", "def_path": "Mathlib/MeasureTheory/Integral/Average.lean", "def_pos": [91, 9], "def_end_pos": [91, 20]}, {"full_name": "ENNReal.mul_div_cancel'", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1432, 19], "def_end_pos": [1432, 34]}, {"full_name": "MeasureTheory.Measure.measure_univ_ne_zero", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1105, 9], "def_end_pos": [1105, 29]}, {"full_name": "MeasureTheory.measure_ne_top", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2875, 9], "def_end_pos": [2875, 23]}]], "state_before": "case inr\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\nf\u271d g : \u03b1 \u2192 \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh\u03bc : \u03bc \u2260 0\n\u22a2 \u2191\u2191\u03bc univ * \u2a0d\u207b (x : \u03b1), f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Sum.lean", "full_name": "Finset.disjSum_strictMono_left", "start": [104, 1], "end": [105, 75], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Independence/Basic.lean", "full_name": "ProbabilityTheory.IndepSets.union_iff", "start": [283, 1], "end": [285, 29], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "full_name": "List.length_filterMap_le", "start": [1345, 1], "end": [1348, 25], "traced_tactics": [{"tactic": "rw [\u2190 length_map _ some, map_filterMap_some_eq_filter_map_is_some, \u2190 length_map _ f]", "annotated_tactic": ["rw [\u2190 <a>length_map</a> _ <a>some</a>, <a>map_filterMap_some_eq_filter_map_is_some</a>, \u2190 <a>length_map</a> _ f]", [{"full_name": "List.length_map", "def_path": "lake-packages/lean4/src/lean/Init/Data/List/Basic.lean", "def_pos": [795, 17], "def_end_pos": [795, 27]}, {"full_name": "Option.some", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2143, 5], "def_end_pos": [2143, 9]}, {"full_name": "List.map_filterMap_some_eq_filter_map_is_some", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [1327, 9], "def_end_pos": [1327, 49]}, {"full_name": "List.length_map", "def_path": "lake-packages/lean4/src/lean/Init/Data/List/Basic.lean", "def_pos": [795, 17], "def_end_pos": [795, 27]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b1 \u2192 Option \u03b2\nl : List \u03b1\n\u22a2 length (filterMap f l) \u2264 length l", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b1 \u2192 Option \u03b2\nl : List \u03b1\n\u22a2 length (filter (fun b => Option.isSome b) (map f l)) \u2264 length (map f l)"}, {"tactic": "apply length_filter_le", "annotated_tactic": ["apply <a>length_filter_le</a>", [{"full_name": "List.length_filter_le", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [1342, 9], "def_end_pos": [1342, 25]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b1 \u2192 Option \u03b2\nl : List \u03b1\n\u22a2 length (filter (fun b => Option.isSome b) (map f l)) \u2264 length (map f l)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Constructions/Pi.lean", "full_name": "MeasureTheory.Measure.pi_eq", "start": [380, 1], "end": [384, 46], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/VitaliCaratheodory.lean", "full_name": "MeasureTheory.exists_lt_lowerSemicontinuous_integral_lt", "start": [457, 1], "end": [530, 55], "traced_tactics": [{"tactic": "let \u03b4 : \u211d\u22650 := \u27e8\u03b5 / 2, (half_pos \u03b5pos).le\u27e9", "annotated_tactic": ["let \u03b4 : \u211d\u22650 := \u27e8\u03b5 / 2, (<a>half_pos</a> \u03b5pos).<a>le</a>\u27e9", [{"full_name": "half_pos", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [504, 9], "def_end_pos": [504, 17]}, {"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [142, 7], "def_end_pos": [142, 15]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227\n      LowerSemicontinuous g \u2227\n        (Integrable fun x => EReal.toReal (g x)) \u2227\n          (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4) \u2227 \u222b (x : \u03b1), EReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), f x \u2202\u03bc + \u03b5", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227\n      LowerSemicontinuous g \u2227\n        (Integrable fun x => EReal.toReal (g x)) \u2227\n          (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4) \u2227 \u222b (x : \u03b1), EReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), f x \u2202\u03bc + \u03b5"}, {"tactic": "have \u03b4pos : 0 < \u03b4 := half_pos \u03b5pos", "annotated_tactic": ["have \u03b4pos : 0 < \u03b4 := <a>half_pos</a> \u03b5pos", [{"full_name": "half_pos", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [504, 9], "def_end_pos": [504, 17]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227\n      LowerSemicontinuous g \u2227\n        (Integrable fun x => EReal.toReal (g x)) \u2227\n          (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4) \u2227 \u222b (x : \u03b1), EReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), f x \u2202\u03bc + \u03b5", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227\n      LowerSemicontinuous g \u2227\n        (Integrable fun x => EReal.toReal (g x)) \u2227\n          (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4) \u2227 \u222b (x : \u03b1), EReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), f x \u2202\u03bc + \u03b5"}, {"tactic": "let fp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)", "annotated_tactic": ["let fp : \u03b1 \u2192 \u211d\u22650 := fun x => <a>Real.toNNReal</a> (f x)", [{"full_name": "Real.toNNReal", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [118, 19], "def_end_pos": [118, 39]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227\n      LowerSemicontinuous g \u2227\n        (Integrable fun x => EReal.toReal (g x)) \u2227\n          (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4) \u2227 \u222b (x : \u03b1), EReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), f x \u2202\u03bc + \u03b5", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227\n      LowerSemicontinuous g \u2227\n        (Integrable fun x => EReal.toReal (g x)) \u2227\n          (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4) \u2227 \u222b (x : \u03b1), EReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), f x \u2202\u03bc + \u03b5"}, {"tactic": "have int_fp : Integrable (fun x => (fp x : \u211d)) \u03bc := hf.real_toNNReal", "annotated_tactic": ["have int_fp : <a>Integrable</a> (fun x => (fp x : \u211d)) \u03bc := hf.real_toNNReal", [{"full_name": "MeasureTheory.Integrable", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [442, 5], "def_end_pos": [442, 15]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227\n      LowerSemicontinuous g \u2227\n        (Integrable fun x => EReal.toReal (g x)) \u2227\n          (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4) \u2227 \u222b (x : \u03b1), EReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), f x \u2202\u03bc + \u03b5", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227\n      LowerSemicontinuous g \u2227\n        (Integrable fun x => EReal.toReal (g x)) \u2227\n          (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4) \u2227 \u222b (x : \u03b1), EReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), f x \u2202\u03bc + \u03b5"}, {"tactic": "rcases exists_lt_lowerSemicontinuous_integral_gt_nnreal fp int_fp \u03b4pos with\n  \u27e8gp, fp_lt_gp, gpcont, gp_lt_top, gp_integrable, gpint\u27e9", "annotated_tactic": ["rcases <a>exists_lt_lowerSemicontinuous_integral_gt_nnreal</a> fp int_fp \u03b4pos with\n    \u27e8gp, fp_lt_gp, gpcont, gp_lt_top, gp_integrable, gpint\u27e9", [{"full_name": "MeasureTheory.exists_lt_lowerSemicontinuous_integral_gt_nnreal", "def_path": "Mathlib/MeasureTheory/Integral/VitaliCaratheodory.lean", "def_pos": [270, 9], "def_end_pos": [270, 57]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227\n      LowerSemicontinuous g \u2227\n        (Integrable fun x => EReal.toReal (g x)) \u2227\n          (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4) \u2227 \u222b (x : \u03b1), EReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), f x \u2202\u03bc + \u03b5", "state_after": "case intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227\n      LowerSemicontinuous g \u2227\n        (Integrable fun x => EReal.toReal (g x)) \u2227\n          (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4) \u2227 \u222b (x : \u03b1), EReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), f x \u2202\u03bc + \u03b5"}, {"tactic": "let fm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)", "annotated_tactic": ["let fm : \u03b1 \u2192 \u211d\u22650 := fun x => <a>Real.toNNReal</a> (-f x)", [{"full_name": "Real.toNNReal", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [118, 19], "def_end_pos": [118, 39]}]], "state_before": "case intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227\n      LowerSemicontinuous g \u2227\n        (Integrable fun x => EReal.toReal (g x)) \u2227\n          (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4) \u2227 \u222b (x : \u03b1), EReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), f x \u2202\u03bc + \u03b5", "state_after": "case intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227\n      LowerSemicontinuous g \u2227\n        (Integrable fun x => EReal.toReal (g x)) \u2227\n          (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4) \u2227 \u222b (x : \u03b1), EReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), f x \u2202\u03bc + \u03b5"}, {"tactic": "have int_fm : Integrable (fun x => (fm x : \u211d)) \u03bc := hf.neg.real_toNNReal", "annotated_tactic": ["have int_fm : <a>Integrable</a> (fun x => (fm x : \u211d)) \u03bc := hf.neg.real_toNNReal", [{"full_name": "MeasureTheory.Integrable", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [442, 5], "def_end_pos": [442, 15]}]], "state_before": "case intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227\n      LowerSemicontinuous g \u2227\n        (Integrable fun x => EReal.toReal (g x)) \u2227\n          (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4) \u2227 \u222b (x : \u03b1), EReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), f x \u2202\u03bc + \u03b5", "state_after": "case intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227\n      LowerSemicontinuous g \u2227\n        (Integrable fun x => EReal.toReal (g x)) \u2227\n          (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4) \u2227 \u222b (x : \u03b1), EReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), f x \u2202\u03bc + \u03b5"}, {"tactic": "rcases exists_upperSemicontinuous_le_integral_le fm int_fm \u03b4pos with\n  \u27e8gm, gm_le_fm, gmcont, gm_integrable, gmint\u27e9", "annotated_tactic": ["rcases <a>exists_upperSemicontinuous_le_integral_le</a> fm int_fm \u03b4pos with\n    \u27e8gm, gm_le_fm, gmcont, gm_integrable, gmint\u27e9", [{"full_name": "MeasureTheory.exists_upperSemicontinuous_le_integral_le", "def_path": "Mathlib/MeasureTheory/Integral/VitaliCaratheodory.lean", "def_pos": [422, 9], "def_end_pos": [422, 50]}]], "state_before": "case intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227\n      LowerSemicontinuous g \u2227\n        (Integrable fun x => EReal.toReal (g x)) \u2227\n          (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4) \u2227 \u222b (x : \u03b1), EReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), f x \u2202\u03bc + \u03b5", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227\n      LowerSemicontinuous g \u2227\n        (Integrable fun x => EReal.toReal (g x)) \u2227\n          (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4) \u2227 \u222b (x : \u03b1), EReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), f x \u2202\u03bc + \u03b5"}, {"tactic": "let g : \u03b1 \u2192 EReal := fun x => (gp x : EReal) - gm x", "annotated_tactic": ["let g : \u03b1 \u2192 <a>EReal</a> := fun x => (gp x : <a>EReal</a>) - gm x", [{"full_name": "EReal", "def_path": "Mathlib/Data/Real/EReal.lean", "def_pos": [57, 5], "def_end_pos": [57, 10]}, {"full_name": "EReal", "def_path": "Mathlib/Data/Real/EReal.lean", "def_pos": [57, 5], "def_end_pos": [57, 10]}]], "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227\n      LowerSemicontinuous g \u2227\n        (Integrable fun x => EReal.toReal (g x)) \u2227\n          (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4) \u2227 \u222b (x : \u03b1), EReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), f x \u2202\u03bc + \u03b5", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227\n      LowerSemicontinuous g \u2227\n        (Integrable fun x => EReal.toReal (g x)) \u2227\n          (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4) \u2227 \u222b (x : \u03b1), EReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), f x \u2202\u03bc + \u03b5"}, {"tactic": "have ae_g : \u2200\u1d50 x \u2202\u03bc, (g x).toReal = (gp x : EReal).toReal - (gm x : EReal).toReal := by\n  filter_upwards [gp_lt_top] with _ hx\n  rw [EReal.toReal_sub] <;> simp [hx.ne]", "annotated_tactic": ["have ae_g : \u2200\u1d50 x \u2202\u03bc, (g x).<a>toReal</a> = (gp x : <a>EReal</a>).<a>toReal</a> - (gm x : <a>EReal</a>).<a>toReal</a> := by\n    filter_upwards [gp_lt_top] with _ hx\n    rw [<a>EReal.toReal_sub</a>] <;> simp [hx.ne]", [{"full_name": "EReal.toReal", "def_path": "Mathlib/Data/Real/EReal.lean", "def_pos": [254, 5], "def_end_pos": [254, 11]}, {"full_name": "EReal", "def_path": "Mathlib/Data/Real/EReal.lean", "def_pos": [57, 5], "def_end_pos": [57, 10]}, {"full_name": "EReal.toReal", "def_path": "Mathlib/Data/Real/EReal.lean", "def_pos": [254, 5], "def_end_pos": [254, 11]}, {"full_name": "EReal", "def_path": "Mathlib/Data/Real/EReal.lean", "def_pos": [57, 5], "def_end_pos": [57, 10]}, {"full_name": "EReal.toReal", "def_path": "Mathlib/Data/Real/EReal.lean", "def_pos": [254, 5], "def_end_pos": [254, 11]}, {"full_name": "EReal.toReal_sub", "def_path": "Mathlib/Data/Real/EReal.lean", "def_pos": [894, 9], "def_end_pos": [894, 19]}]], "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227\n      LowerSemicontinuous g \u2227\n        (Integrable fun x => EReal.toReal (g x)) \u2227\n          (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4) \u2227 \u222b (x : \u03b1), EReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), f x \u2202\u03bc + \u03b5", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227\n      LowerSemicontinuous g \u2227\n        (Integrable fun x => EReal.toReal (g x)) \u2227\n          (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4) \u2227 \u222b (x : \u03b1), EReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), f x \u2202\u03bc + \u03b5"}, {"tactic": "refine' \u27e8g, ?lt, ?lsc, ?int, ?aelt, ?intlt\u27e9", "annotated_tactic": ["refine' \u27e8g, ?lt, ?lsc, ?int, ?aelt, ?intlt\u27e9", []], "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227\n      LowerSemicontinuous g \u2227\n        (Integrable fun x => EReal.toReal (g x)) \u2227\n          (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4) \u2227 \u222b (x : \u03b1), EReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), f x \u2202\u03bc + \u03b5", "state_after": "case lt\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\n\u22a2 \u2200 (x : \u03b1), \u2191(f x) < g x\n\ncase lsc\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\n\u22a2 LowerSemicontinuous g\n\ncase int\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\n\u22a2 Integrable fun x => EReal.toReal (g x)\n\ncase aelt\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\n\ncase intlt\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\n\u22a2 \u222b (x : \u03b1), EReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), f x \u2202\u03bc + \u03b5"}, {"tactic": "case int =>\n  show Integrable (fun x => EReal.toReal (g x)) \u03bc\n  rw [integrable_congr ae_g]\n  convert gp_integrable.sub gm_integrable\n  simp", "annotated_tactic": ["case int =>\n    show <a>Integrable</a> (fun x => <a>EReal.toReal</a> (g x)) \u03bc\n    rw [<a>integrable_congr</a> ae_g]\n    convert gp_integrable.sub gm_integrable\n    simp", [{"full_name": "MeasureTheory.Integrable", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [442, 5], "def_end_pos": [442, 15]}, {"full_name": "EReal.toReal", "def_path": "Mathlib/Data/Real/EReal.lean", "def_pos": [254, 5], "def_end_pos": [254, 11]}, {"full_name": "MeasureTheory.integrable_congr", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [496, 9], "def_end_pos": [496, 25]}]], "state_before": "case lt\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\n\u22a2 \u2200 (x : \u03b1), \u2191(f x) < g x\n\ncase lsc\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\n\u22a2 LowerSemicontinuous g\n\ncase int\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\n\u22a2 Integrable fun x => EReal.toReal (g x)\n\ncase aelt\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\n\ncase intlt\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\n\u22a2 \u222b (x : \u03b1), EReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), f x \u2202\u03bc + \u03b5", "state_after": "case lt\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\n\u22a2 \u2200 (x : \u03b1), \u2191(f x) < g x\n\ncase lsc\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\n\u22a2 LowerSemicontinuous g\n\ncase aelt\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\n\ncase intlt\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\n\u22a2 \u222b (x : \u03b1), EReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), f x \u2202\u03bc + \u03b5"}, {"tactic": "case intlt =>\n  show (\u222b x : \u03b1, (g x).toReal \u2202\u03bc) < (\u222b x : \u03b1, f x \u2202\u03bc) + \u03b5;\n  exact\n    calc\n      (\u222b x : \u03b1, (g x).toReal \u2202\u03bc) = \u222b x : \u03b1, EReal.toReal (gp x) - EReal.toReal (gm x) \u2202\u03bc :=\n        integral_congr_ae ae_g\n      _ = (\u222b x : \u03b1, EReal.toReal (gp x) \u2202\u03bc) - \u222b x : \u03b1, \u2191(gm x) \u2202\u03bc := by\n        simp only [EReal.toReal_coe_ennreal, ENNReal.coe_toReal]\n        exact integral_sub gp_integrable gm_integrable\n      _ < (\u222b x : \u03b1, \u2191(fp x) \u2202\u03bc) + \u2191\u03b4 - \u222b x : \u03b1, \u2191(gm x) \u2202\u03bc := by\n        apply sub_lt_sub_right\n        convert gpint\n        simp only [EReal.toReal_coe_ennreal]\n      _ \u2264 (\u222b x : \u03b1, \u2191(fp x) \u2202\u03bc) + \u2191\u03b4 - ((\u222b x : \u03b1, \u2191(fm x) \u2202\u03bc) - \u03b4) := (sub_le_sub_left gmint _)\n      _ = (\u222b x : \u03b1, f x \u2202\u03bc) + 2 * \u03b4 := by\n        simp_rw [integral_eq_integral_pos_part_sub_integral_neg_part hf]; ring\n      _ = (\u222b x : \u03b1, f x \u2202\u03bc) + \u03b5 := by congr 1; field_simp [mul_comm]", "annotated_tactic": ["case intlt =>\n    show (\u222b x : \u03b1, (g x).<a>toReal</a> \u2202\u03bc) < (\u222b x : \u03b1, f x \u2202\u03bc) + \u03b5;\n    exact\n      calc\n        (\u222b x : \u03b1, (g x).<a>toReal</a> \u2202\u03bc) = \u222b x : \u03b1, <a>EReal.toReal</a> (gp x) - <a>EReal.toReal</a> (gm x) \u2202\u03bc :=\n          <a>integral_congr_ae</a> ae_g\n        _ = (\u222b x : \u03b1, <a>EReal.toReal</a> (gp x) \u2202\u03bc) - \u222b x : \u03b1, \u2191(gm x) \u2202\u03bc := by\n          simp only [<a>EReal.toReal_coe_ennreal</a>, <a>ENNReal.coe_toReal</a>]\n          exact <a>integral_sub</a> gp_integrable gm_integrable\n        _ < (\u222b x : \u03b1, \u2191(fp x) \u2202\u03bc) + \u2191\u03b4 - \u222b x : \u03b1, \u2191(gm x) \u2202\u03bc := by\n          apply <a>sub_lt_sub_right</a>\n          convert gpint\n          simp only [<a>EReal.toReal_coe_ennreal</a>]\n        _ \u2264 (\u222b x : \u03b1, \u2191(fp x) \u2202\u03bc) + \u2191\u03b4 - ((\u222b x : \u03b1, \u2191(fm x) \u2202\u03bc) - \u03b4) := (<a>sub_le_sub_left</a> gmint _)\n        _ = (\u222b x : \u03b1, f x \u2202\u03bc) + 2 * \u03b4 := by\n          simp_rw [<a>integral_eq_integral_pos_part_sub_integral_neg_part</a> hf]; ring\n        _ = (\u222b x : \u03b1, f x \u2202\u03bc) + \u03b5 := by congr 1; field_simp [<a>mul_comm</a>]", [{"full_name": "EReal.toReal", "def_path": "Mathlib/Data/Real/EReal.lean", "def_pos": [254, 5], "def_end_pos": [254, 11]}, {"full_name": "EReal.toReal", "def_path": "Mathlib/Data/Real/EReal.lean", "def_pos": [254, 5], "def_end_pos": [254, 11]}, {"full_name": "EReal.toReal", "def_path": "Mathlib/Data/Real/EReal.lean", "def_pos": [254, 5], "def_end_pos": [254, 11]}, {"full_name": "EReal.toReal", "def_path": "Mathlib/Data/Real/EReal.lean", "def_pos": [254, 5], "def_end_pos": [254, 11]}, {"full_name": "MeasureTheory.integral_congr_ae", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [938, 9], "def_end_pos": [938, 26]}, {"full_name": "EReal.toReal", "def_path": "Mathlib/Data/Real/EReal.lean", "def_pos": [254, 5], "def_end_pos": [254, 11]}, {"full_name": "EReal.toReal_coe_ennreal", "def_path": "Mathlib/Data/Real/EReal.lean", "def_pos": [451, 9], "def_end_pos": [451, 27]}, {"full_name": "ENNReal.coe_toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [236, 17], "def_end_pos": [236, 27]}, {"full_name": "MeasureTheory.integral_sub", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [901, 9], "def_end_pos": [901, 21]}, {"full_name": "sub_lt_sub_right", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [877, 15], "def_end_pos": [877, 31]}, {"full_name": "EReal.toReal_coe_ennreal", "def_path": "Mathlib/Data/Real/EReal.lean", "def_pos": [451, 9], "def_end_pos": [451, 27]}, {"full_name": "sub_le_sub_left", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [779, 15], "def_end_pos": [779, 30]}, {"full_name": "MeasureTheory.integral_eq_integral_pos_part_sub_integral_neg_part", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1194, 9], "def_end_pos": [1194, 60]}, {"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [302, 9], "def_end_pos": [302, 17]}]], "state_before": "case lt\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\n\u22a2 \u2200 (x : \u03b1), \u2191(f x) < g x\n\ncase lsc\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\n\u22a2 LowerSemicontinuous g\n\ncase aelt\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\n\ncase intlt\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\n\u22a2 \u222b (x : \u03b1), EReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), f x \u2202\u03bc + \u03b5", "state_after": "case lt\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\n\u22a2 \u2200 (x : \u03b1), \u2191(f x) < g x\n\ncase lsc\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\n\u22a2 LowerSemicontinuous g\n\ncase aelt\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4"}, {"tactic": "case aelt =>\n  show \u2200\u1d50 x : \u03b1 \u2202\u03bc, g x < \u22a4\n  filter_upwards [gp_lt_top] with ?_ hx\n  simp only [sub_eq_add_neg, Ne.def, (EReal.add_lt_top _ _).ne, lt_top_iff_ne_top,\n    lt_top_iff_ne_top.1 hx, EReal.coe_ennreal_eq_top_iff, not_false_iff, EReal.neg_eq_top_iff,\n    EReal.coe_ennreal_ne_bot]", "annotated_tactic": ["case aelt =>\n    show \u2200\u1d50 x : \u03b1 \u2202\u03bc, g x < \u22a4\n    filter_upwards [gp_lt_top] with ?_ hx\n    simp only [<a>sub_eq_add_neg</a>, <a>Ne.def</a>, (<a>EReal.add_lt_top</a> _ _).<a>ne</a>, <a>lt_top_iff_ne_top</a>,\n      <a>lt_top_iff_ne_top</a>.1 hx, <a>EReal.coe_ennreal_eq_top_iff</a>, <a>not_false_iff</a>, <a>EReal.neg_eq_top_iff</a>,\n      <a>EReal.coe_ennreal_ne_bot</a>]", [{"full_name": "sub_eq_add_neg", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [975, 3], "def_end_pos": [975, 14]}, {"full_name": "Ne.def", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [59, 9], "def_end_pos": [59, 15]}, {"full_name": "EReal.add_lt_top", "def_path": "Mathlib/Data/Real/EReal.lean", "def_pos": [721, 9], "def_end_pos": [721, 19]}, {"full_name": "LT.lt.ne", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [152, 7], "def_end_pos": [152, 15]}, {"full_name": "lt_top_iff_ne_top", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [173, 9], "def_end_pos": [173, 26]}, {"full_name": "lt_top_iff_ne_top", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [173, 9], "def_end_pos": [173, 26]}, {"full_name": "EReal.coe_ennreal_eq_top_iff", "def_path": "Mathlib/Data/Real/EReal.lean", "def_pos": [489, 9], "def_end_pos": [489, 31]}, {"full_name": "not_false_iff", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [82, 9], "def_end_pos": [82, 22]}, {"full_name": "EReal.neg_eq_top_iff", "def_path": "Mathlib/Data/Real/EReal.lean", "def_pos": [786, 9], "def_end_pos": [786, 23]}, {"full_name": "EReal.coe_ennreal_ne_bot", "def_path": "Mathlib/Data/Real/EReal.lean", "def_pos": [562, 9], "def_end_pos": [562, 27]}]], "state_before": "case lt\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\n\u22a2 \u2200 (x : \u03b1), \u2191(f x) < g x\n\ncase lsc\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\n\u22a2 LowerSemicontinuous g\n\ncase aelt\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4", "state_after": "case lt\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\n\u22a2 \u2200 (x : \u03b1), \u2191(f x) < g x\n\ncase lsc\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\n\u22a2 LowerSemicontinuous g"}, {"tactic": "filter_upwards [gp_lt_top] with _ hx", "annotated_tactic": ["filter_upwards [gp_lt_top] with _ hx", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)", "state_after": "case h\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\na\u271d : \u03b1\nhx : gp a\u271d < \u22a4\n\u22a2 EReal.toReal (g a\u271d) = EReal.toReal \u2191(gp a\u271d) - EReal.toReal \u2191\u2191(gm a\u271d)"}, {"tactic": "rw [EReal.toReal_sub] <;> simp [hx.ne]", "annotated_tactic": ["rw [<a>EReal.toReal_sub</a>] <;> simp [hx.ne]", [{"full_name": "EReal.toReal_sub", "def_path": "Mathlib/Data/Real/EReal.lean", "def_pos": [894, 9], "def_end_pos": [894, 19]}]], "state_before": "case h\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\na\u271d : \u03b1\nhx : gp a\u271d < \u22a4\n\u22a2 EReal.toReal (g a\u271d) = EReal.toReal \u2191(gp a\u271d) - EReal.toReal \u2191\u2191(gm a\u271d)", "state_after": "no goals"}, {"tactic": "rw [integrable_congr ae_g]", "annotated_tactic": ["rw [<a>integrable_congr</a> ae_g]", [{"full_name": "MeasureTheory.integrable_congr", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [496, 9], "def_end_pos": [496, 25]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\n\u22a2 Integrable fun x => EReal.toReal (g x)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\n\u22a2 Integrable fun x => EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)"}, {"tactic": "convert gp_integrable.sub gm_integrable", "annotated_tactic": ["convert gp_integrable.sub gm_integrable", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\n\u22a2 Integrable fun x => EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)", "state_after": "case h.e'_5.h\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\nx\u271d : \u03b1\n\u22a2 EReal.toReal \u2191(gp x\u271d) - EReal.toReal \u2191\u2191(gm x\u271d) = ((fun x => ENNReal.toReal (gp x)) - fun x => \u2191(gm x)) x\u271d"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case h.e'_5.h\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\nx\u271d : \u03b1\n\u22a2 EReal.toReal \u2191(gp x\u271d) - EReal.toReal \u2191\u2191(gm x\u271d) = ((fun x => ENNReal.toReal (gp x)) - fun x => \u2191(gm x)) x\u271d", "state_after": "no goals"}, {"tactic": "exact\n  calc\n    (\u222b x : \u03b1, (g x).toReal \u2202\u03bc) = \u222b x : \u03b1, EReal.toReal (gp x) - EReal.toReal (gm x) \u2202\u03bc :=\n      integral_congr_ae ae_g\n    _ = (\u222b x : \u03b1, EReal.toReal (gp x) \u2202\u03bc) - \u222b x : \u03b1, \u2191(gm x) \u2202\u03bc := by\n      simp only [EReal.toReal_coe_ennreal, ENNReal.coe_toReal]\n      exact integral_sub gp_integrable gm_integrable\n    _ < (\u222b x : \u03b1, \u2191(fp x) \u2202\u03bc) + \u2191\u03b4 - \u222b x : \u03b1, \u2191(gm x) \u2202\u03bc := by\n      apply sub_lt_sub_right\n      convert gpint\n      simp only [EReal.toReal_coe_ennreal]\n    _ \u2264 (\u222b x : \u03b1, \u2191(fp x) \u2202\u03bc) + \u2191\u03b4 - ((\u222b x : \u03b1, \u2191(fm x) \u2202\u03bc) - \u03b4) := (sub_le_sub_left gmint _)\n    _ = (\u222b x : \u03b1, f x \u2202\u03bc) + 2 * \u03b4 := by\n      simp_rw [integral_eq_integral_pos_part_sub_integral_neg_part hf]; ring\n    _ = (\u222b x : \u03b1, f x \u2202\u03bc) + \u03b5 := by congr 1; field_simp [mul_comm]", "annotated_tactic": ["exact\n      calc\n        (\u222b x : \u03b1, (g x).<a>toReal</a> \u2202\u03bc) = \u222b x : \u03b1, <a>EReal.toReal</a> (gp x) - <a>EReal.toReal</a> (gm x) \u2202\u03bc :=\n          <a>integral_congr_ae</a> ae_g\n        _ = (\u222b x : \u03b1, <a>EReal.toReal</a> (gp x) \u2202\u03bc) - \u222b x : \u03b1, \u2191(gm x) \u2202\u03bc := by\n          simp only [<a>EReal.toReal_coe_ennreal</a>, <a>ENNReal.coe_toReal</a>]\n          exact <a>integral_sub</a> gp_integrable gm_integrable\n        _ < (\u222b x : \u03b1, \u2191(fp x) \u2202\u03bc) + \u2191\u03b4 - \u222b x : \u03b1, \u2191(gm x) \u2202\u03bc := by\n          apply <a>sub_lt_sub_right</a>\n          convert gpint\n          simp only [<a>EReal.toReal_coe_ennreal</a>]\n        _ \u2264 (\u222b x : \u03b1, \u2191(fp x) \u2202\u03bc) + \u2191\u03b4 - ((\u222b x : \u03b1, \u2191(fm x) \u2202\u03bc) - \u03b4) := (<a>sub_le_sub_left</a> gmint _)\n        _ = (\u222b x : \u03b1, f x \u2202\u03bc) + 2 * \u03b4 := by\n          simp_rw [<a>integral_eq_integral_pos_part_sub_integral_neg_part</a> hf]; ring\n        _ = (\u222b x : \u03b1, f x \u2202\u03bc) + \u03b5 := by congr 1; field_simp [<a>mul_comm</a>]", [{"full_name": "EReal.toReal", "def_path": "Mathlib/Data/Real/EReal.lean", "def_pos": [254, 5], "def_end_pos": [254, 11]}, {"full_name": "EReal.toReal", "def_path": "Mathlib/Data/Real/EReal.lean", "def_pos": [254, 5], "def_end_pos": [254, 11]}, {"full_name": "EReal.toReal", "def_path": "Mathlib/Data/Real/EReal.lean", "def_pos": [254, 5], "def_end_pos": [254, 11]}, {"full_name": "MeasureTheory.integral_congr_ae", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [938, 9], "def_end_pos": [938, 26]}, {"full_name": "EReal.toReal", "def_path": "Mathlib/Data/Real/EReal.lean", "def_pos": [254, 5], "def_end_pos": [254, 11]}, {"full_name": "EReal.toReal_coe_ennreal", "def_path": "Mathlib/Data/Real/EReal.lean", "def_pos": [451, 9], "def_end_pos": [451, 27]}, {"full_name": "ENNReal.coe_toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [236, 17], "def_end_pos": [236, 27]}, {"full_name": "MeasureTheory.integral_sub", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [901, 9], "def_end_pos": [901, 21]}, {"full_name": "sub_lt_sub_right", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [877, 15], "def_end_pos": [877, 31]}, {"full_name": "EReal.toReal_coe_ennreal", "def_path": "Mathlib/Data/Real/EReal.lean", "def_pos": [451, 9], "def_end_pos": [451, 27]}, {"full_name": "sub_le_sub_left", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [779, 15], "def_end_pos": [779, 30]}, {"full_name": "MeasureTheory.integral_eq_integral_pos_part_sub_integral_neg_part", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1194, 9], "def_end_pos": [1194, 60]}, {"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [302, 9], "def_end_pos": [302, 17]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\n\u22a2 \u222b (x : \u03b1), EReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), f x \u2202\u03bc + \u03b5", "state_after": "no goals"}, {"tactic": "simp only [EReal.toReal_coe_ennreal, ENNReal.coe_toReal]", "annotated_tactic": ["simp only [<a>EReal.toReal_coe_ennreal</a>, <a>ENNReal.coe_toReal</a>]", [{"full_name": "EReal.toReal_coe_ennreal", "def_path": "Mathlib/Data/Real/EReal.lean", "def_pos": [451, 9], "def_end_pos": [451, 27]}, {"full_name": "ENNReal.coe_toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [236, 17], "def_end_pos": [236, 27]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\n\u22a2 \u222b (x : \u03b1), EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x) \u2202\u03bc =\n    \u222b (x : \u03b1), EReal.toReal \u2191(gp x) \u2202\u03bc - \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\n\u22a2 \u222b (x : \u03b1), ENNReal.toReal (gp x) - \u2191(gm x) \u2202\u03bc = \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc - \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc"}, {"tactic": "exact integral_sub gp_integrable gm_integrable", "annotated_tactic": ["exact <a>integral_sub</a> gp_integrable gm_integrable", [{"full_name": "MeasureTheory.integral_sub", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [901, 9], "def_end_pos": [901, 21]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\n\u22a2 \u222b (x : \u03b1), ENNReal.toReal (gp x) - \u2191(gm x) \u2202\u03bc = \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc - \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc", "state_after": "no goals"}, {"tactic": "apply sub_lt_sub_right", "annotated_tactic": ["apply <a>sub_lt_sub_right</a>", [{"full_name": "sub_lt_sub_right", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [877, 15], "def_end_pos": [877, 31]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\n\u22a2 \u222b (x : \u03b1), EReal.toReal \u2191(gp x) \u2202\u03bc - \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + \u2191\u03b4 - \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc", "state_after": "case h\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\n\u22a2 \u222b (x : \u03b1), EReal.toReal \u2191(gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + \u2191\u03b4"}, {"tactic": "convert gpint", "annotated_tactic": ["convert gpint", []], "state_before": "case h\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\n\u22a2 \u222b (x : \u03b1), EReal.toReal \u2191(gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + \u2191\u03b4", "state_after": "case h.e'_3.h.e'_7.h\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\nx\u271d : \u03b1\n\u22a2 EReal.toReal \u2191(gp x\u271d) = ENNReal.toReal (gp x\u271d)"}, {"tactic": "simp only [EReal.toReal_coe_ennreal]", "annotated_tactic": ["simp only [<a>EReal.toReal_coe_ennreal</a>]", [{"full_name": "EReal.toReal_coe_ennreal", "def_path": "Mathlib/Data/Real/EReal.lean", "def_pos": [451, 9], "def_end_pos": [451, 27]}]], "state_before": "case h.e'_3.h.e'_7.h\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\nx\u271d : \u03b1\n\u22a2 EReal.toReal \u2191(gp x\u271d) = ENNReal.toReal (gp x\u271d)", "state_after": "no goals"}, {"tactic": "simp_rw [integral_eq_integral_pos_part_sub_integral_neg_part hf]", "annotated_tactic": ["simp_rw [<a>integral_eq_integral_pos_part_sub_integral_neg_part</a> hf]", [{"full_name": "MeasureTheory.integral_eq_integral_pos_part_sub_integral_neg_part", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1194, 9], "def_end_pos": [1194, 60]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\n\u22a2 \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + \u2191\u03b4 - (\u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - \u2191\u03b4) = \u222b (x : \u03b1), f x \u2202\u03bc + 2 * \u2191\u03b4", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\n\u22a2 \u222b (x : \u03b1), \u2191(Real.toNNReal (f x)) \u2202\u03bc + \u2191{ val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) } -\n      (\u222b (x : \u03b1), \u2191(Real.toNNReal (-f x)) \u2202\u03bc - \u2191{ val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }) =\n    \u222b (x : \u03b1), \u2191(Real.toNNReal (f x)) \u2202\u03bc - \u222b (x : \u03b1), \u2191(Real.toNNReal (-f x)) \u2202\u03bc +\n      2 * \u2191{ val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }"}, {"tactic": "ring", "annotated_tactic": ["ring", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\n\u22a2 \u222b (x : \u03b1), \u2191(Real.toNNReal (f x)) \u2202\u03bc + \u2191{ val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) } -\n      (\u222b (x : \u03b1), \u2191(Real.toNNReal (-f x)) \u2202\u03bc - \u2191{ val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }) =\n    \u222b (x : \u03b1), \u2191(Real.toNNReal (f x)) \u2202\u03bc - \u222b (x : \u03b1), \u2191(Real.toNNReal (-f x)) \u2202\u03bc +\n      2 * \u2191{ val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }", "state_after": "no goals"}, {"tactic": "congr 1", "annotated_tactic": ["congr 1", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\n\u22a2 \u222b (x : \u03b1), f x \u2202\u03bc + 2 * \u2191\u03b4 = \u222b (x : \u03b1), f x \u2202\u03bc + \u03b5", "state_after": "case e_a\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\n\u22a2 2 * \u2191\u03b4 = \u03b5"}, {"tactic": "field_simp [mul_comm]", "annotated_tactic": ["field_simp [<a>mul_comm</a>]", [{"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [302, 9], "def_end_pos": [302, 17]}]], "state_before": "case e_a\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\n\u22a2 2 * \u2191\u03b4 = \u03b5", "state_after": "no goals"}, {"tactic": "filter_upwards [gp_lt_top] with ?_ hx", "annotated_tactic": ["filter_upwards [gp_lt_top] with ?_ hx", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4", "state_after": "case h\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\nx\u271d : \u03b1\nhx : gp x\u271d < \u22a4\n\u22a2 g x\u271d < \u22a4"}, {"tactic": "simp only [sub_eq_add_neg, Ne.def, (EReal.add_lt_top _ _).ne, lt_top_iff_ne_top,\n  lt_top_iff_ne_top.1 hx, EReal.coe_ennreal_eq_top_iff, not_false_iff, EReal.neg_eq_top_iff,\n  EReal.coe_ennreal_ne_bot]", "annotated_tactic": ["simp only [<a>sub_eq_add_neg</a>, <a>Ne.def</a>, (<a>EReal.add_lt_top</a> _ _).<a>ne</a>, <a>lt_top_iff_ne_top</a>,\n      <a>lt_top_iff_ne_top</a>.1 hx, <a>EReal.coe_ennreal_eq_top_iff</a>, <a>not_false_iff</a>, <a>EReal.neg_eq_top_iff</a>,\n      <a>EReal.coe_ennreal_ne_bot</a>]", [{"full_name": "sub_eq_add_neg", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [975, 3], "def_end_pos": [975, 14]}, {"full_name": "Ne.def", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [59, 9], "def_end_pos": [59, 15]}, {"full_name": "EReal.add_lt_top", "def_path": "Mathlib/Data/Real/EReal.lean", "def_pos": [721, 9], "def_end_pos": [721, 19]}, {"full_name": "LT.lt.ne", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [152, 7], "def_end_pos": [152, 15]}, {"full_name": "lt_top_iff_ne_top", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [173, 9], "def_end_pos": [173, 26]}, {"full_name": "lt_top_iff_ne_top", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [173, 9], "def_end_pos": [173, 26]}, {"full_name": "EReal.coe_ennreal_eq_top_iff", "def_path": "Mathlib/Data/Real/EReal.lean", "def_pos": [489, 9], "def_end_pos": [489, 31]}, {"full_name": "not_false_iff", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [82, 9], "def_end_pos": [82, 22]}, {"full_name": "EReal.neg_eq_top_iff", "def_path": "Mathlib/Data/Real/EReal.lean", "def_pos": [786, 9], "def_end_pos": [786, 23]}, {"full_name": "EReal.coe_ennreal_ne_bot", "def_path": "Mathlib/Data/Real/EReal.lean", "def_pos": [562, 9], "def_end_pos": [562, 27]}]], "state_before": "case h\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\nx\u271d : \u03b1\nhx : gp x\u271d < \u22a4\n\u22a2 g x\u271d < \u22a4", "state_after": "no goals"}, {"tactic": "intro x", "annotated_tactic": ["intro x", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\n\u22a2 \u2200 (x : \u03b1), \u2191(f x) < g x", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\nx : \u03b1\n\u22a2 \u2191(f x) < g x"}, {"tactic": "rw [EReal.coe_real_ereal_eq_coe_toNNReal_sub_coe_toNNReal (f x)]", "annotated_tactic": ["rw [<a>EReal.coe_real_ereal_eq_coe_toNNReal_sub_coe_toNNReal</a> (f x)]", [{"full_name": "EReal.coe_real_ereal_eq_coe_toNNReal_sub_coe_toNNReal", "def_path": "Mathlib/Data/Real/EReal.lean", "def_pos": [882, 9], "def_end_pos": [882, 56]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\nx : \u03b1\n\u22a2 \u2191(f x) < g x", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\nx : \u03b1\n\u22a2 \u2191\u2191(Real.toNNReal (f x)) - \u2191\u2191(Real.toNNReal (-f x)) < g x"}, {"tactic": "refine' EReal.sub_lt_sub_of_lt_of_le _ _ _ _", "annotated_tactic": ["refine' <a>EReal.sub_lt_sub_of_lt_of_le</a> _ _ _ _", [{"full_name": "EReal.sub_lt_sub_of_lt_of_le", "def_path": "Mathlib/Data/Real/EReal.lean", "def_pos": [877, 9], "def_end_pos": [877, 31]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\nx : \u03b1\n\u22a2 \u2191\u2191(Real.toNNReal (f x)) - \u2191\u2191(Real.toNNReal (-f x)) < g x", "state_after": "case refine'_1\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\nx : \u03b1\n\u22a2 \u2191\u2191(Real.toNNReal (f x)) < \u2191(gp x)\n\ncase refine'_2\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\nx : \u03b1\n\u22a2 \u2191\u2191(gm x) \u2264 \u2191\u2191(Real.toNNReal (-f x))\n\ncase refine'_3\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\nx : \u03b1\n\u22a2 \u2191\u2191(gm x) \u2260 \u22a5\n\ncase refine'_4\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\nx : \u03b1\n\u22a2 \u2191\u2191(Real.toNNReal (-f x)) \u2260 \u22a4"}, {"tactic": "simp only [EReal.coe_ennreal_lt_coe_ennreal_iff]", "annotated_tactic": ["simp only [<a>EReal.coe_ennreal_lt_coe_ennreal_iff</a>]", [{"full_name": "EReal.coe_ennreal_lt_coe_ennreal_iff", "def_path": "Mathlib/Data/Real/EReal.lean", "def_pos": [506, 9], "def_end_pos": [506, 39]}]], "state_before": "case refine'_1\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\nx : \u03b1\n\u22a2 \u2191\u2191(Real.toNNReal (f x)) < \u2191(gp x)", "state_after": "case refine'_1\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\nx : \u03b1\n\u22a2 \u2191(Real.toNNReal (f x)) < gp x"}, {"tactic": "exact fp_lt_gp x", "annotated_tactic": ["exact fp_lt_gp x", []], "state_before": "case refine'_1\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\nx : \u03b1\n\u22a2 \u2191(Real.toNNReal (f x)) < gp x", "state_after": "no goals"}, {"tactic": "simp only [ENNReal.coe_le_coe, EReal.coe_ennreal_le_coe_ennreal_iff]", "annotated_tactic": ["simp only [<a>ENNReal.coe_le_coe</a>, <a>EReal.coe_ennreal_le_coe_ennreal_iff</a>]", [{"full_name": "ENNReal.coe_le_coe", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [349, 28], "def_end_pos": [349, 38]}, {"full_name": "EReal.coe_ennreal_le_coe_ennreal_iff", "def_path": "Mathlib/Data/Real/EReal.lean", "def_pos": [501, 9], "def_end_pos": [501, 39]}]], "state_before": "case refine'_2\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\nx : \u03b1\n\u22a2 \u2191\u2191(gm x) \u2264 \u2191\u2191(Real.toNNReal (-f x))", "state_after": "case refine'_2\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\nx : \u03b1\n\u22a2 gm x \u2264 Real.toNNReal (-f x)"}, {"tactic": "exact gm_le_fm x", "annotated_tactic": ["exact gm_le_fm x", []], "state_before": "case refine'_2\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\nx : \u03b1\n\u22a2 gm x \u2264 Real.toNNReal (-f x)", "state_after": "no goals"}, {"tactic": "simp only [EReal.coe_ennreal_ne_bot, Ne.def, not_false_iff]", "annotated_tactic": ["simp only [<a>EReal.coe_ennreal_ne_bot</a>, <a>Ne.def</a>, <a>not_false_iff</a>]", [{"full_name": "EReal.coe_ennreal_ne_bot", "def_path": "Mathlib/Data/Real/EReal.lean", "def_pos": [562, 9], "def_end_pos": [562, 27]}, {"full_name": "Ne.def", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [59, 9], "def_end_pos": [59, 15]}, {"full_name": "not_false_iff", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [82, 9], "def_end_pos": [82, 22]}]], "state_before": "case refine'_3\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\nx : \u03b1\n\u22a2 \u2191\u2191(gm x) \u2260 \u22a5", "state_after": "no goals"}, {"tactic": "simp only [EReal.coe_nnreal_ne_top, Ne.def, not_false_iff]", "annotated_tactic": ["simp only [<a>EReal.coe_nnreal_ne_top</a>, <a>Ne.def</a>, <a>not_false_iff</a>]", [{"full_name": "EReal.coe_nnreal_ne_top", "def_path": "Mathlib/Data/Real/EReal.lean", "def_pos": [493, 9], "def_end_pos": [493, 26]}, {"full_name": "Ne.def", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [59, 9], "def_end_pos": [59, 15]}, {"full_name": "not_false_iff", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [82, 9], "def_end_pos": [82, 22]}]], "state_before": "case refine'_4\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\nx : \u03b1\n\u22a2 \u2191\u2191(Real.toNNReal (-f x)) \u2260 \u22a4", "state_after": "no goals"}, {"tactic": "apply LowerSemicontinuous.add'", "annotated_tactic": ["apply <a>LowerSemicontinuous.add'</a>", [{"full_name": "LowerSemicontinuous.add'", "def_path": "Mathlib/Topology/Semicontinuous.lean", "def_pos": [470, 9], "def_end_pos": [470, 33]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\n\u22a2 LowerSemicontinuous g", "state_after": "case hf\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\n\u22a2 LowerSemicontinuous fun z => \u2191(gp z)\n\ncase hg\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\n\u22a2 LowerSemicontinuous fun z => -\u2191\u2191(gm z)\n\ncase hcont\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\n\u22a2 \u2200 (x : \u03b1), ContinuousAt (fun p => p.1 + p.2) (\u2191(gp x), -\u2191\u2191(gm x))"}, {"tactic": "exact continuous_coe_ennreal_ereal.comp_lowerSemicontinuous gpcont fun x y hxy =>\n    EReal.coe_ennreal_le_coe_ennreal_iff.2 hxy", "annotated_tactic": ["exact continuous_coe_ennreal_ereal.comp_lowerSemicontinuous gpcont fun x y hxy =>\n          <a>EReal.coe_ennreal_le_coe_ennreal_iff</a>.2 hxy", [{"full_name": "EReal.coe_ennreal_le_coe_ennreal_iff", "def_path": "Mathlib/Data/Real/EReal.lean", "def_pos": [501, 9], "def_end_pos": [501, 39]}]], "state_before": "case hf\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\n\u22a2 LowerSemicontinuous fun z => \u2191(gp z)", "state_after": "no goals"}, {"tactic": "apply continuous_neg.comp_upperSemicontinuous_antitone _ fun x y hxy =>\n    EReal.neg_le_neg_iff.2 hxy", "annotated_tactic": ["apply continuous_neg.comp_upperSemicontinuous_antitone _ fun x y hxy =>\n          <a>EReal.neg_le_neg_iff</a>.2 hxy", [{"full_name": "EReal.neg_le_neg_iff", "def_path": "Mathlib/Data/Real/EReal.lean", "def_pos": [805, 17], "def_end_pos": [805, 31]}]], "state_before": "case hg\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\n\u22a2 LowerSemicontinuous fun z => -\u2191\u2191(gm z)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\n\u22a2 UpperSemicontinuous fun z => \u2191\u2191(gm z)"}, {"tactic": "apply continuous_coe_ennreal_ereal.comp_upperSemicontinuous _ fun x y hxy =>\n    EReal.coe_ennreal_le_coe_ennreal_iff.2 hxy", "annotated_tactic": ["apply continuous_coe_ennreal_ereal.comp_upperSemicontinuous _ fun x y hxy =>\n          <a>EReal.coe_ennreal_le_coe_ennreal_iff</a>.2 hxy", [{"full_name": "EReal.coe_ennreal_le_coe_ennreal_iff", "def_path": "Mathlib/Data/Real/EReal.lean", "def_pos": [501, 9], "def_end_pos": [501, 39]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\n\u22a2 UpperSemicontinuous fun z => \u2191\u2191(gm z)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\n\u22a2 UpperSemicontinuous fun z => \u2191(gm z)"}, {"tactic": "exact ENNReal.continuous_coe.comp_upperSemicontinuous gmcont fun x y hxy =>\n    ENNReal.coe_le_coe.2 hxy", "annotated_tactic": ["exact ENNReal.continuous_coe.comp_upperSemicontinuous gmcont fun x y hxy =>\n          <a>ENNReal.coe_le_coe</a>.2 hxy", [{"full_name": "ENNReal.coe_le_coe", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [349, 28], "def_end_pos": [349, 38]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\n\u22a2 UpperSemicontinuous fun z => \u2191(gm z)", "state_after": "no goals"}, {"tactic": "intro x", "annotated_tactic": ["intro x", []], "state_before": "case hcont\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\n\u22a2 \u2200 (x : \u03b1), ContinuousAt (fun p => p.1 + p.2) (\u2191(gp x), -\u2191\u2191(gm x))", "state_after": "case hcont\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\nx : \u03b1\n\u22a2 ContinuousAt (fun p => p.1 + p.2) (\u2191(gp x), -\u2191\u2191(gm x))"}, {"tactic": "exact EReal.continuousAt_add (by simp) (by simp)", "annotated_tactic": ["exact <a>EReal.continuousAt_add</a> (by simp) (by simp)", [{"full_name": "EReal.continuousAt_add", "def_path": "Mathlib/Topology/Instances/EReal.lean", "def_pos": [225, 9], "def_end_pos": [225, 25]}]], "state_before": "case hcont\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\nx : \u03b1\n\u22a2 ContinuousAt (fun p => p.1 + p.2) (\u2191(gp x), -\u2191\u2191(gm x))", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\nx : \u03b1\n\u22a2 (\u2191(gp x), -\u2191\u2191(gm x)).1 \u2260 \u22a4 \u2228 (\u2191(gp x), -\u2191\u2191(gm x)).2 \u2260 \u22a5", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\nx : \u03b1\n\u22a2 (\u2191(gp x), -\u2191\u2191(gm x)).1 \u2260 \u22a5 \u2228 (\u2191(gp x), -\u2191\u2191(gm x)).2 \u2260 \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Constructions/Pi.lean", "full_name": "MeasureTheory.measurePreserving_finTwoArrow", "start": [852, 1], "end": [856, 42], "traced_tactics": [{"tactic": "simpa only [Matrix.vec_single_eq_const, Matrix.vecCons_const] using\n  measurePreserving_finTwoArrow_vec \u03bc \u03bc", "annotated_tactic": ["simpa only [<a>Matrix.vec_single_eq_const</a>, <a>Matrix.vecCons_const</a>] using\n    <a>measurePreserving_finTwoArrow_vec</a> \u03bc \u03bc", [{"full_name": "Matrix.vec_single_eq_const", "def_path": "Mathlib/Data/Fin/VecNotation.lean", "def_pos": [198, 9], "def_end_pos": [198, 28]}, {"full_name": "Matrix.vecCons_const", "def_path": "Mathlib/Data/Fin/VecNotation.lean", "def_pos": [194, 9], "def_end_pos": [194, 22]}, {"full_name": "MeasureTheory.measurePreserving_finTwoArrow_vec", "def_path": "Mathlib/MeasureTheory/Constructions/Pi.lean", "def_pos": [845, 9], "def_end_pos": [845, 42]}]], "state_before": "\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1\u271d : \u03b9 \u2192 Type u_3\ninst\u271d\u00b3 : Fintype \u03b9\nm\u271d\u00b9 : (i : \u03b9) \u2192 OuterMeasure (\u03b1\u271d i)\nm\u271d : (i : \u03b9) \u2192 MeasurableSpace (\u03b1\u271d i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (\u03b1\u271d i)\ninst\u271d\u00b2 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\ninst\u271d\u00b9 : Fintype \u03b9'\n\u03b1 : Type u\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\n\u22a2 MeasurePreserving \u2191MeasurableEquiv.finTwoArrow", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "full_name": "MeasureTheory.Measure.map_add", "start": [1178, 1], "end": [1179, 81], "traced_tactics": [{"tactic": "simp [\u2190 map\u2097_apply_of_measurable hf]", "annotated_tactic": ["simp [\u2190 <a>map\u2097_apply_of_measurable</a> hf]", [{"full_name": "MeasureTheory.Measure.map\u2097_apply_of_measurable", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1171, 9], "def_end_pos": [1171, 33]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd\u271d \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : Measurable f\n\u22a2 map f (\u03bc + \u03bd) = map f \u03bc + map f \u03bd", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/List/Init/Lemmas.lean", "full_name": "List.elem_cons_self", "start": [316, 9], "end": [317, 19], "traced_tactics": [{"tactic": "simp [elem_cons]", "annotated_tactic": ["simp [<a>elem_cons</a>]", [{"full_name": "List.elem_cons", "def_path": "lake-packages/std/Std/Data/List/Init/Lemmas.lean", "def_pos": [313, 9], "def_end_pos": [313, 18]}]], "state_before": "\u03b1 : Type u_1\nas : List \u03b1\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : LawfulBEq \u03b1\na : \u03b1\n\u22a2 elem a (a :: as) = true", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Array/Init/Lemmas.lean", "full_name": "Array.foldr_push'", "start": [89, 9], "end": [90, 98], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/UniformIntegrable.lean", "full_name": "MeasureTheory.Mem\u2112p.integral_indicator_norm_ge_le", "start": [176, 1], "end": [208, 6], "traced_tactics": [{"tactic": "have htendsto :\n    \u2200\u1d50 x \u2202\u03bc, Tendsto (fun M : \u2115 => { x | (M : \u211d) \u2264 \u2016f x\u2016\u208a }.indicator f x) atTop (\ud835\udcdd 0) :=\n  univ_mem' (id fun x => tendsto_indicator_ge f x)", "annotated_tactic": ["have htendsto :\n      \u2200\u1d50 x \u2202\u03bc, <a>Tendsto</a> (fun M : \u2115 => { x | (M : \u211d) \u2264 \u2016f x\u2016\u208a }.<a>indicator</a> f x) <a>atTop</a> (\ud835\udcdd 0) :=\n    <a>univ_mem'</a> (<a>id</a> fun x => <a>tendsto_indicator_ge</a> f x)", [{"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2939, 5], "def_end_pos": [2939, 12]}, {"full_name": "Set.indicator", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [46, 3], "def_end_pos": [46, 14]}, {"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}, {"full_name": "Filter.univ_mem'", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 18]}, {"full_name": "id", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [33, 15], "def_end_pos": [33, 17]}, {"full_name": "MeasureTheory.tendsto_indicator_ge", "def_path": "Mathlib/MeasureTheory/Function/UniformIntegrable.lean", "def_pos": [158, 9], "def_end_pos": [158, 29]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u03b2\nhf : Mem\u2112p f 1\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u22a2 \u2203 M, \u222b\u207b (x : \u03b1), \u2191\u2016Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f x\u2016\u208a \u2202\u03bc \u2264 ENNReal.ofReal \u03b5", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u03b2\nhf : Mem\u2112p f 1\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhtendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun M => Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f x) atTop (\ud835\udcdd 0)\n\u22a2 \u2203 M, \u222b\u207b (x : \u03b1), \u2191\u2016Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f x\u2016\u208a \u2202\u03bc \u2264 ENNReal.ofReal \u03b5"}, {"tactic": "have hmeas : \u2200 M : \u2115, AEStronglyMeasurable ({ x | (M : \u211d) \u2264 \u2016f x\u2016\u208a }.indicator f) \u03bc := by\n  intro M\n  apply hf.1.indicator\n  apply StronglyMeasurable.measurableSet_le stronglyMeasurable_const\n    hmeas.nnnorm.measurable.coe_nnreal_real.stronglyMeasurable", "annotated_tactic": ["have hmeas : \u2200 M : \u2115, <a>AEStronglyMeasurable</a> ({ x | (M : \u211d) \u2264 \u2016f x\u2016\u208a }.<a>indicator</a> f) \u03bc := by\n    intro M\n    apply hf.1.<a>indicator</a>\n    apply <a>StronglyMeasurable.measurableSet_le</a> <a>stronglyMeasurable_const</a>\n      hmeas.nnnorm.measurable.coe_nnreal_real.stronglyMeasurable", [{"full_name": "MeasureTheory.AEStronglyMeasurable", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [93, 5], "def_end_pos": [93, 25]}, {"full_name": "Set.indicator", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [46, 3], "def_end_pos": [46, 14]}, {"full_name": "MeasureTheory.AEStronglyMeasurable.indicator", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1524, 19], "def_end_pos": [1524, 28]}, {"full_name": "MeasureTheory.StronglyMeasurable.measurableSet_le", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [876, 9], "def_end_pos": [876, 25]}, {"full_name": "MeasureTheory.stronglyMeasurable_const", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [143, 9], "def_end_pos": [143, 33]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u03b2\nhf : Mem\u2112p f 1\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhtendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun M => Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f x) atTop (\ud835\udcdd 0)\n\u22a2 \u2203 M, \u222b\u207b (x : \u03b1), \u2191\u2016Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f x\u2016\u208a \u2202\u03bc \u2264 ENNReal.ofReal \u03b5", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u03b2\nhf : Mem\u2112p f 1\nhmeas\u271d : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhtendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun M => Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f x) atTop (\ud835\udcdd 0)\nhmeas : \u2200 (M : \u2115), AEStronglyMeasurable (Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f) \u03bc\n\u22a2 \u2203 M, \u222b\u207b (x : \u03b1), \u2191\u2016Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f x\u2016\u208a \u2202\u03bc \u2264 ENNReal.ofReal \u03b5"}, {"tactic": "have hbound : HasFiniteIntegral (fun x => \u2016f x\u2016) \u03bc := by\n  rw [mem\u2112p_one_iff_integrable] at hf\n  exact hf.norm.2", "annotated_tactic": ["have hbound : <a>HasFiniteIntegral</a> (fun x => \u2016f x\u2016) \u03bc := by\n    rw [<a>mem\u2112p_one_iff_integrable</a>] at hf\n    exact hf.norm.2", [{"full_name": "MeasureTheory.HasFiniteIntegral", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [106, 5], "def_end_pos": [106, 22]}, {"full_name": "MeasureTheory.mem\u2112p_one_iff_integrable", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [453, 9], "def_end_pos": [453, 33]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u03b2\nhf : Mem\u2112p f 1\nhmeas\u271d : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhtendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun M => Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f x) atTop (\ud835\udcdd 0)\nhmeas : \u2200 (M : \u2115), AEStronglyMeasurable (Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f) \u03bc\n\u22a2 \u2203 M, \u222b\u207b (x : \u03b1), \u2191\u2016Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f x\u2016\u208a \u2202\u03bc \u2264 ENNReal.ofReal \u03b5", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u03b2\nhf : Mem\u2112p f 1\nhmeas\u271d : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhtendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun M => Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f x) atTop (\ud835\udcdd 0)\nhmeas : \u2200 (M : \u2115), AEStronglyMeasurable (Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f) \u03bc\nhbound : HasFiniteIntegral fun x => \u2016f x\u2016\n\u22a2 \u2203 M, \u222b\u207b (x : \u03b1), \u2191\u2016Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f x\u2016\u208a \u2202\u03bc \u2264 ENNReal.ofReal \u03b5"}, {"tactic": "rw [ENNReal.tendsto_atTop_zero] at this", "annotated_tactic": ["rw [<a>ENNReal.tendsto_atTop_zero</a>] at this", [{"full_name": "ENNReal.tendsto_atTop_zero", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [298, 19], "def_end_pos": [298, 37]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u03b2\nhf : Mem\u2112p f 1\nhmeas\u271d : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhtendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun M => Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f x) atTop (\ud835\udcdd 0)\nhmeas : \u2200 (M : \u2115), AEStronglyMeasurable (Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f) \u03bc\nhbound : HasFiniteIntegral fun x => \u2016f x\u2016\nthis : Tendsto (fun n => \u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016Set.indicator {x | \u2191n \u2264 \u2016f x\u2016\u208a} f a - 0\u2016 \u2202\u03bc) atTop (\ud835\udcdd 0)\n\u22a2 \u2203 M, \u222b\u207b (x : \u03b1), \u2191\u2016Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f x\u2016\u208a \u2202\u03bc \u2264 ENNReal.ofReal \u03b5", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u03b2\nhf : Mem\u2112p f 1\nhmeas\u271d : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhtendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun M => Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f x) atTop (\ud835\udcdd 0)\nhmeas : \u2200 (M : \u2115), AEStronglyMeasurable (Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f) \u03bc\nhbound : HasFiniteIntegral fun x => \u2016f x\u2016\nthis :\n  \u2200 (\u03b5 : \u211d\u22650\u221e),\n    \u03b5 > 0 \u2192 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 \u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016Set.indicator {x | \u2191n \u2264 \u2016f x\u2016\u208a} f a - 0\u2016 \u2202\u03bc \u2264 \u03b5\n\u22a2 \u2203 M, \u222b\u207b (x : \u03b1), \u2191\u2016Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f x\u2016\u208a \u2202\u03bc \u2264 ENNReal.ofReal \u03b5"}, {"tactic": "obtain \u27e8M, hM\u27e9 := this (ENNReal.ofReal \u03b5) (ENNReal.ofReal_pos.2 h\u03b5)", "annotated_tactic": ["obtain \u27e8M, hM\u27e9 := this (<a>ENNReal.ofReal</a> \u03b5) (<a>ENNReal.ofReal_pos</a>.2 h\u03b5)", [{"full_name": "ENNReal.ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [172, 29], "def_end_pos": [172, 35]}, {"full_name": "ENNReal.ofReal_pos", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2166, 9], "def_end_pos": [2166, 19]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u03b2\nhf : Mem\u2112p f 1\nhmeas\u271d : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhtendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun M => Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f x) atTop (\ud835\udcdd 0)\nhmeas : \u2200 (M : \u2115), AEStronglyMeasurable (Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f) \u03bc\nhbound : HasFiniteIntegral fun x => \u2016f x\u2016\nthis :\n  \u2200 (\u03b5 : \u211d\u22650\u221e),\n    \u03b5 > 0 \u2192 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 \u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016Set.indicator {x | \u2191n \u2264 \u2016f x\u2016\u208a} f a - 0\u2016 \u2202\u03bc \u2264 \u03b5\n\u22a2 \u2203 M, \u222b\u207b (x : \u03b1), \u2191\u2016Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f x\u2016\u208a \u2202\u03bc \u2264 ENNReal.ofReal \u03b5", "state_after": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u03b2\nhf : Mem\u2112p f 1\nhmeas\u271d : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhtendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun M => Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f x) atTop (\ud835\udcdd 0)\nhmeas : \u2200 (M : \u2115), AEStronglyMeasurable (Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f) \u03bc\nhbound : HasFiniteIntegral fun x => \u2016f x\u2016\nthis :\n  \u2200 (\u03b5 : \u211d\u22650\u221e),\n    \u03b5 > 0 \u2192 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 \u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016Set.indicator {x | \u2191n \u2264 \u2016f x\u2016\u208a} f a - 0\u2016 \u2202\u03bc \u2264 \u03b5\nM : \u2115\nhM : \u2200 (n : \u2115), n \u2265 M \u2192 \u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016Set.indicator {x | \u2191n \u2264 \u2016f x\u2016\u208a} f a - 0\u2016 \u2202\u03bc \u2264 ENNReal.ofReal \u03b5\n\u22a2 \u2203 M, \u222b\u207b (x : \u03b1), \u2191\u2016Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f x\u2016\u208a \u2202\u03bc \u2264 ENNReal.ofReal \u03b5"}, {"tactic": "simp only [true_and_iff, ge_iff_le, zero_tsub, zero_le, sub_zero, zero_add, coe_nnnorm,\n  Set.mem_Icc] at hM", "annotated_tactic": ["simp only [<a>true_and_iff</a>, <a>ge_iff_le</a>, <a>zero_tsub</a>, <a>zero_le</a>, <a>sub_zero</a>, <a>zero_add</a>, <a>coe_nnnorm</a>,\n    <a>Set.mem_Icc</a>] at hM", [{"full_name": "true_and_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [147, 9], "def_end_pos": [147, 21]}, {"full_name": "ge_iff_le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [359, 9], "def_end_pos": [359, 18]}, {"full_name": "zero_tsub", "def_path": "Mathlib/Algebra/Order/Sub/Canonical.lean", "def_pos": [341, 9], "def_end_pos": [341, 18]}, {"full_name": "zero_le", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [217, 30], "def_end_pos": [217, 37]}, {"full_name": "sub_zero", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [339, 3], "def_end_pos": [339, 14]}, {"full_name": "zero_add", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [463, 3], "def_end_pos": [463, 14]}, {"full_name": "coe_nnnorm", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [905, 41], "def_end_pos": [905, 51]}, {"full_name": "Set.mem_Icc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [131, 9], "def_end_pos": [131, 16]}]], "state_before": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u03b2\nhf : Mem\u2112p f 1\nhmeas\u271d : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhtendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun M => Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f x) atTop (\ud835\udcdd 0)\nhmeas : \u2200 (M : \u2115), AEStronglyMeasurable (Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f) \u03bc\nhbound : HasFiniteIntegral fun x => \u2016f x\u2016\nthis :\n  \u2200 (\u03b5 : \u211d\u22650\u221e),\n    \u03b5 > 0 \u2192 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 \u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016Set.indicator {x | \u2191n \u2264 \u2016f x\u2016\u208a} f a - 0\u2016 \u2202\u03bc \u2264 \u03b5\nM : \u2115\nhM : \u2200 (n : \u2115), n \u2265 M \u2192 \u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016Set.indicator {x | \u2191n \u2264 \u2016f x\u2016\u208a} f a - 0\u2016 \u2202\u03bc \u2264 ENNReal.ofReal \u03b5\n\u22a2 \u2203 M, \u222b\u207b (x : \u03b1), \u2191\u2016Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f x\u2016\u208a \u2202\u03bc \u2264 ENNReal.ofReal \u03b5", "state_after": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u03b2\nhf : Mem\u2112p f 1\nhmeas\u271d : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhtendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun M => Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f x) atTop (\ud835\udcdd 0)\nhmeas : \u2200 (M : \u2115), AEStronglyMeasurable (Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f) \u03bc\nhbound : HasFiniteIntegral fun x => \u2016f x\u2016\nthis :\n  \u2200 (\u03b5 : \u211d\u22650\u221e),\n    \u03b5 > 0 \u2192 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 \u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016Set.indicator {x | \u2191n \u2264 \u2016f x\u2016\u208a} f a - 0\u2016 \u2202\u03bc \u2264 \u03b5\nM : \u2115\nhM : \u2200 (n : \u2115), M \u2264 n \u2192 \u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016Set.indicator {x | \u2191n \u2264 \u2016f x\u2016\u208a} f a\u2016 \u2202\u03bc \u2264 ENNReal.ofReal \u03b5\n\u22a2 \u2203 M, \u222b\u207b (x : \u03b1), \u2191\u2016Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f x\u2016\u208a \u2202\u03bc \u2264 ENNReal.ofReal \u03b5"}, {"tactic": "refine' \u27e8M, _\u27e9", "annotated_tactic": ["refine' \u27e8M, _\u27e9", []], "state_before": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u03b2\nhf : Mem\u2112p f 1\nhmeas\u271d : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhtendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun M => Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f x) atTop (\ud835\udcdd 0)\nhmeas : \u2200 (M : \u2115), AEStronglyMeasurable (Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f) \u03bc\nhbound : HasFiniteIntegral fun x => \u2016f x\u2016\nthis :\n  \u2200 (\u03b5 : \u211d\u22650\u221e),\n    \u03b5 > 0 \u2192 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 \u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016Set.indicator {x | \u2191n \u2264 \u2016f x\u2016\u208a} f a - 0\u2016 \u2202\u03bc \u2264 \u03b5\nM : \u2115\nhM : \u2200 (n : \u2115), M \u2264 n \u2192 \u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016Set.indicator {x | \u2191n \u2264 \u2016f x\u2016\u208a} f a\u2016 \u2202\u03bc \u2264 ENNReal.ofReal \u03b5\n\u22a2 \u2203 M, \u222b\u207b (x : \u03b1), \u2191\u2016Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f x\u2016\u208a \u2202\u03bc \u2264 ENNReal.ofReal \u03b5", "state_after": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u03b2\nhf : Mem\u2112p f 1\nhmeas\u271d : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhtendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun M => Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f x) atTop (\ud835\udcdd 0)\nhmeas : \u2200 (M : \u2115), AEStronglyMeasurable (Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f) \u03bc\nhbound : HasFiniteIntegral fun x => \u2016f x\u2016\nthis :\n  \u2200 (\u03b5 : \u211d\u22650\u221e),\n    \u03b5 > 0 \u2192 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 \u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016Set.indicator {x | \u2191n \u2264 \u2016f x\u2016\u208a} f a - 0\u2016 \u2202\u03bc \u2264 \u03b5\nM : \u2115\nhM : \u2200 (n : \u2115), M \u2264 n \u2192 \u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016Set.indicator {x | \u2191n \u2264 \u2016f x\u2016\u208a} f a\u2016 \u2202\u03bc \u2264 ENNReal.ofReal \u03b5\n\u22a2 \u222b\u207b (x : \u03b1), \u2191\u2016Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f x\u2016\u208a \u2202\u03bc \u2264 ENNReal.ofReal \u03b5"}, {"tactic": "convert hM M le_rfl", "annotated_tactic": ["convert hM M <a>le_rfl</a>", [{"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}]], "state_before": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u03b2\nhf : Mem\u2112p f 1\nhmeas\u271d : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhtendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun M => Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f x) atTop (\ud835\udcdd 0)\nhmeas : \u2200 (M : \u2115), AEStronglyMeasurable (Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f) \u03bc\nhbound : HasFiniteIntegral fun x => \u2016f x\u2016\nthis :\n  \u2200 (\u03b5 : \u211d\u22650\u221e),\n    \u03b5 > 0 \u2192 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 \u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016Set.indicator {x | \u2191n \u2264 \u2016f x\u2016\u208a} f a - 0\u2016 \u2202\u03bc \u2264 \u03b5\nM : \u2115\nhM : \u2200 (n : \u2115), M \u2264 n \u2192 \u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016Set.indicator {x | \u2191n \u2264 \u2016f x\u2016\u208a} f a\u2016 \u2202\u03bc \u2264 ENNReal.ofReal \u03b5\n\u22a2 \u222b\u207b (x : \u03b1), \u2191\u2016Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f x\u2016\u208a \u2202\u03bc \u2264 ENNReal.ofReal \u03b5", "state_after": "case h.e'_3.h.e'_4.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u03b2\nhf : Mem\u2112p f 1\nhmeas\u271d : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhtendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun M => Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f x) atTop (\ud835\udcdd 0)\nhmeas : \u2200 (M : \u2115), AEStronglyMeasurable (Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f) \u03bc\nhbound : HasFiniteIntegral fun x => \u2016f x\u2016\nthis :\n  \u2200 (\u03b5 : \u211d\u22650\u221e),\n    \u03b5 > 0 \u2192 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 \u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016Set.indicator {x | \u2191n \u2264 \u2016f x\u2016\u208a} f a - 0\u2016 \u2202\u03bc \u2264 \u03b5\nM : \u2115\nhM : \u2200 (n : \u2115), M \u2264 n \u2192 \u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016Set.indicator {x | \u2191n \u2264 \u2016f x\u2016\u208a} f a\u2016 \u2202\u03bc \u2264 ENNReal.ofReal \u03b5\nx\u271d : \u03b1\n\u22a2 \u2191\u2016Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f x\u271d\u2016\u208a = ENNReal.ofReal \u2016Set.indicator {x | \u2191M \u2264 \u2016f x\u2016\u208a} f x\u271d\u2016"}, {"tactic": "simp only [coe_nnnorm, ENNReal.ofReal_eq_coe_nnreal (norm_nonneg _)]", "annotated_tactic": ["simp only [<a>coe_nnnorm</a>, <a>ENNReal.ofReal_eq_coe_nnreal</a> (<a>norm_nonneg</a> _)]", [{"full_name": "coe_nnnorm", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [905, 41], "def_end_pos": [905, 51]}, {"full_name": "ENNReal.ofReal_eq_coe_nnreal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [207, 9], "def_end_pos": [207, 29]}, {"full_name": "norm_nonneg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [500, 30], "def_end_pos": [500, 41]}]], "state_before": "case h.e'_3.h.e'_4.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u03b2\nhf : Mem\u2112p f 1\nhmeas\u271d : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhtendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun M => Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f x) atTop (\ud835\udcdd 0)\nhmeas : \u2200 (M : \u2115), AEStronglyMeasurable (Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f) \u03bc\nhbound : HasFiniteIntegral fun x => \u2016f x\u2016\nthis :\n  \u2200 (\u03b5 : \u211d\u22650\u221e),\n    \u03b5 > 0 \u2192 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 \u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016Set.indicator {x | \u2191n \u2264 \u2016f x\u2016\u208a} f a - 0\u2016 \u2202\u03bc \u2264 \u03b5\nM : \u2115\nhM : \u2200 (n : \u2115), M \u2264 n \u2192 \u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016Set.indicator {x | \u2191n \u2264 \u2016f x\u2016\u208a} f a\u2016 \u2202\u03bc \u2264 ENNReal.ofReal \u03b5\nx\u271d : \u03b1\n\u22a2 \u2191\u2016Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f x\u271d\u2016\u208a = ENNReal.ofReal \u2016Set.indicator {x | \u2191M \u2264 \u2016f x\u2016\u208a} f x\u271d\u2016", "state_after": "case h.e'_3.h.e'_4.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u03b2\nhf : Mem\u2112p f 1\nhmeas\u271d : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhtendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun M => Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f x) atTop (\ud835\udcdd 0)\nhmeas : \u2200 (M : \u2115), AEStronglyMeasurable (Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f) \u03bc\nhbound : HasFiniteIntegral fun x => \u2016f x\u2016\nthis :\n  \u2200 (\u03b5 : \u211d\u22650\u221e),\n    \u03b5 > 0 \u2192 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 \u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016Set.indicator {x | \u2191n \u2264 \u2016f x\u2016\u208a} f a - 0\u2016 \u2202\u03bc \u2264 \u03b5\nM : \u2115\nhM : \u2200 (n : \u2115), M \u2264 n \u2192 \u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016Set.indicator {x | \u2191n \u2264 \u2016f x\u2016\u208a} f a\u2016 \u2202\u03bc \u2264 ENNReal.ofReal \u03b5\nx\u271d : \u03b1\n\u22a2 \u2191\u2016Set.indicator {x | \u2191M \u2264 \u2016f x\u2016} f x\u271d\u2016\u208a =\n    \u2191{ val := \u2016Set.indicator {x | \u2191M \u2264 \u2016f x\u2016\u208a} f x\u271d\u2016, property := (_ : 0 \u2264 \u2016Set.indicator {x | \u2191M \u2264 \u2016f x\u2016\u208a} f x\u271d\u2016) }"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case h.e'_3.h.e'_4.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u03b2\nhf : Mem\u2112p f 1\nhmeas\u271d : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhtendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun M => Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f x) atTop (\ud835\udcdd 0)\nhmeas : \u2200 (M : \u2115), AEStronglyMeasurable (Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f) \u03bc\nhbound : HasFiniteIntegral fun x => \u2016f x\u2016\nthis :\n  \u2200 (\u03b5 : \u211d\u22650\u221e),\n    \u03b5 > 0 \u2192 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 \u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016Set.indicator {x | \u2191n \u2264 \u2016f x\u2016\u208a} f a - 0\u2016 \u2202\u03bc \u2264 \u03b5\nM : \u2115\nhM : \u2200 (n : \u2115), M \u2264 n \u2192 \u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016Set.indicator {x | \u2191n \u2264 \u2016f x\u2016\u208a} f a\u2016 \u2202\u03bc \u2264 ENNReal.ofReal \u03b5\nx\u271d : \u03b1\n\u22a2 \u2191\u2016Set.indicator {x | \u2191M \u2264 \u2016f x\u2016} f x\u271d\u2016\u208a =\n    \u2191{ val := \u2016Set.indicator {x | \u2191M \u2264 \u2016f x\u2016\u208a} f x\u271d\u2016, property := (_ : 0 \u2264 \u2016Set.indicator {x | \u2191M \u2264 \u2016f x\u2016\u208a} f x\u271d\u2016) }", "state_after": "no goals"}, {"tactic": "intro M", "annotated_tactic": ["intro M", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u03b2\nhf : Mem\u2112p f 1\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhtendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun M => Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f x) atTop (\ud835\udcdd 0)\n\u22a2 \u2200 (M : \u2115), AEStronglyMeasurable (Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f) \u03bc", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u03b2\nhf : Mem\u2112p f 1\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhtendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun M => Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f x) atTop (\ud835\udcdd 0)\nM : \u2115\n\u22a2 AEStronglyMeasurable (Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f) \u03bc"}, {"tactic": "apply hf.1.indicator", "annotated_tactic": ["apply hf.1.<a>indicator</a>", [{"full_name": "MeasureTheory.AEStronglyMeasurable.indicator", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1524, 19], "def_end_pos": [1524, 28]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u03b2\nhf : Mem\u2112p f 1\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhtendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun M => Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f x) atTop (\ud835\udcdd 0)\nM : \u2115\n\u22a2 AEStronglyMeasurable (Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f) \u03bc", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u03b2\nhf : Mem\u2112p f 1\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhtendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun M => Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f x) atTop (\ud835\udcdd 0)\nM : \u2115\n\u22a2 MeasurableSet {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a}"}, {"tactic": "apply StronglyMeasurable.measurableSet_le stronglyMeasurable_const\n  hmeas.nnnorm.measurable.coe_nnreal_real.stronglyMeasurable", "annotated_tactic": ["apply <a>StronglyMeasurable.measurableSet_le</a> <a>stronglyMeasurable_const</a>\n      hmeas.nnnorm.measurable.coe_nnreal_real.stronglyMeasurable", [{"full_name": "MeasureTheory.StronglyMeasurable.measurableSet_le", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [876, 9], "def_end_pos": [876, 25]}, {"full_name": "MeasureTheory.stronglyMeasurable_const", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [143, 9], "def_end_pos": [143, 33]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u03b2\nhf : Mem\u2112p f 1\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhtendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun M => Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f x) atTop (\ud835\udcdd 0)\nM : \u2115\n\u22a2 MeasurableSet {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a}", "state_after": "no goals"}, {"tactic": "rw [mem\u2112p_one_iff_integrable] at hf", "annotated_tactic": ["rw [<a>mem\u2112p_one_iff_integrable</a>] at hf", [{"full_name": "MeasureTheory.mem\u2112p_one_iff_integrable", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [453, 9], "def_end_pos": [453, 33]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u03b2\nhf : Mem\u2112p f 1\nhmeas\u271d : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhtendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun M => Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f x) atTop (\ud835\udcdd 0)\nhmeas : \u2200 (M : \u2115), AEStronglyMeasurable (Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f) \u03bc\n\u22a2 HasFiniteIntegral fun x => \u2016f x\u2016", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u03b2\nhf : Integrable f\nhmeas\u271d : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhtendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun M => Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f x) atTop (\ud835\udcdd 0)\nhmeas : \u2200 (M : \u2115), AEStronglyMeasurable (Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f) \u03bc\n\u22a2 HasFiniteIntegral fun x => \u2016f x\u2016"}, {"tactic": "exact hf.norm.2", "annotated_tactic": ["exact hf.norm.2", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u03b2\nhf : Integrable f\nhmeas\u271d : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhtendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun M => Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f x) atTop (\ud835\udcdd 0)\nhmeas : \u2200 (M : \u2115), AEStronglyMeasurable (Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f) \u03bc\n\u22a2 HasFiniteIntegral fun x => \u2016f x\u2016", "state_after": "no goals"}, {"tactic": "refine' tendsto_lintegral_norm_of_dominated_convergence hmeas hbound _ htendsto", "annotated_tactic": ["refine' <a>tendsto_lintegral_norm_of_dominated_convergence</a> hmeas hbound _ htendsto", [{"full_name": "MeasureTheory.tendsto_lintegral_norm_of_dominated_convergence", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [327, 9], "def_end_pos": [327, 56]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u03b2\nhf : Mem\u2112p f 1\nhmeas\u271d : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhtendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun M => Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f x) atTop (\ud835\udcdd 0)\nhmeas : \u2200 (M : \u2115), AEStronglyMeasurable (Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f) \u03bc\nhbound : HasFiniteIntegral fun x => \u2016f x\u2016\n\u22a2 Tendsto (fun n => \u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016Set.indicator {x | \u2191n \u2264 \u2016f x\u2016\u208a} f a - 0\u2016 \u2202\u03bc) atTop (\ud835\udcdd 0)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u03b2\nhf : Mem\u2112p f 1\nhmeas\u271d : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhtendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun M => Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f x) atTop (\ud835\udcdd 0)\nhmeas : \u2200 (M : \u2115), AEStronglyMeasurable (Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f) \u03bc\nhbound : HasFiniteIntegral fun x => \u2016f x\u2016\n\u22a2 \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016Set.indicator {x | \u2191n \u2264 \u2016f x\u2016\u208a} f a\u2016 \u2264 \u2016f a\u2016"}, {"tactic": "refine' fun n => univ_mem' (id fun x => _)", "annotated_tactic": ["refine' fun n => <a>univ_mem'</a> (<a>id</a> fun x => _)", [{"full_name": "Filter.univ_mem'", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 18]}, {"full_name": "id", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [33, 15], "def_end_pos": [33, 17]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u03b2\nhf : Mem\u2112p f 1\nhmeas\u271d : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhtendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun M => Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f x) atTop (\ud835\udcdd 0)\nhmeas : \u2200 (M : \u2115), AEStronglyMeasurable (Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f) \u03bc\nhbound : HasFiniteIntegral fun x => \u2016f x\u2016\n\u22a2 \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016Set.indicator {x | \u2191n \u2264 \u2016f x\u2016\u208a} f a\u2016 \u2264 \u2016f a\u2016", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u03b2\nhf : Mem\u2112p f 1\nhmeas\u271d : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhtendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun M => Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f x) atTop (\ud835\udcdd 0)\nhmeas : \u2200 (M : \u2115), AEStronglyMeasurable (Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f) \u03bc\nhbound : HasFiniteIntegral fun x => \u2016f x\u2016\nn : \u2115\nx : \u03b1\n\u22a2 x \u2208 {x | (fun a => \u2016Set.indicator {x | \u2191n \u2264 \u2016f x\u2016\u208a} f a\u2016 \u2264 \u2016f a\u2016) x}"}, {"tactic": "by_cases hx : (n : \u211d) \u2264 \u2016f x\u2016", "annotated_tactic": ["by_cases hx : (n : \u211d) \u2264 \u2016f x\u2016", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u03b2\nhf : Mem\u2112p f 1\nhmeas\u271d : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhtendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun M => Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f x) atTop (\ud835\udcdd 0)\nhmeas : \u2200 (M : \u2115), AEStronglyMeasurable (Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f) \u03bc\nhbound : HasFiniteIntegral fun x => \u2016f x\u2016\nn : \u2115\nx : \u03b1\n\u22a2 x \u2208 {x | (fun a => \u2016Set.indicator {x | \u2191n \u2264 \u2016f x\u2016\u208a} f a\u2016 \u2264 \u2016f a\u2016) x}", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u03b2\nhf : Mem\u2112p f 1\nhmeas\u271d : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhtendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun M => Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f x) atTop (\ud835\udcdd 0)\nhmeas : \u2200 (M : \u2115), AEStronglyMeasurable (Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f) \u03bc\nhbound : HasFiniteIntegral fun x => \u2016f x\u2016\nn : \u2115\nx : \u03b1\nhx : \u2191n \u2264 \u2016f x\u2016\n\u22a2 x \u2208 {x | (fun a => \u2016Set.indicator {x | \u2191n \u2264 \u2016f x\u2016\u208a} f a\u2016 \u2264 \u2016f a\u2016) x}\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u03b2\nhf : Mem\u2112p f 1\nhmeas\u271d : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhtendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun M => Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f x) atTop (\ud835\udcdd 0)\nhmeas : \u2200 (M : \u2115), AEStronglyMeasurable (Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f) \u03bc\nhbound : HasFiniteIntegral fun x => \u2016f x\u2016\nn : \u2115\nx : \u03b1\nhx : \u00ac\u2191n \u2264 \u2016f x\u2016\n\u22a2 x \u2208 {x | (fun a => \u2016Set.indicator {x | \u2191n \u2264 \u2016f x\u2016\u208a} f a\u2016 \u2264 \u2016f a\u2016) x}"}, {"tactic": "dsimp", "annotated_tactic": ["dsimp", []], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u03b2\nhf : Mem\u2112p f 1\nhmeas\u271d : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhtendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun M => Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f x) atTop (\ud835\udcdd 0)\nhmeas : \u2200 (M : \u2115), AEStronglyMeasurable (Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f) \u03bc\nhbound : HasFiniteIntegral fun x => \u2016f x\u2016\nn : \u2115\nx : \u03b1\nhx : \u2191n \u2264 \u2016f x\u2016\n\u22a2 x \u2208 {x | (fun a => \u2016Set.indicator {x | \u2191n \u2264 \u2016f x\u2016\u208a} f a\u2016 \u2264 \u2016f a\u2016) x}", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u03b2\nhf : Mem\u2112p f 1\nhmeas\u271d : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhtendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun M => Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f x) atTop (\ud835\udcdd 0)\nhmeas : \u2200 (M : \u2115), AEStronglyMeasurable (Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f) \u03bc\nhbound : HasFiniteIntegral fun x => \u2016f x\u2016\nn : \u2115\nx : \u03b1\nhx : \u2191n \u2264 \u2016f x\u2016\n\u22a2 \u2016Set.indicator {x | \u2191n \u2264 \u2016f x\u2016\u208a} f x\u2016 \u2264 \u2016f x\u2016"}, {"tactic": "rwa [Set.indicator_of_mem]", "annotated_tactic": ["rwa [<a>Set.indicator_of_mem</a>]", [{"full_name": "Set.indicator_of_mem", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [67, 3], "def_end_pos": [67, 14]}]], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u03b2\nhf : Mem\u2112p f 1\nhmeas\u271d : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhtendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun M => Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f x) atTop (\ud835\udcdd 0)\nhmeas : \u2200 (M : \u2115), AEStronglyMeasurable (Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f) \u03bc\nhbound : HasFiniteIntegral fun x => \u2016f x\u2016\nn : \u2115\nx : \u03b1\nhx : \u2191n \u2264 \u2016f x\u2016\n\u22a2 \u2016Set.indicator {x | \u2191n \u2264 \u2016f x\u2016\u208a} f x\u2016 \u2264 \u2016f x\u2016", "state_after": "no goals"}, {"tactic": "dsimp", "annotated_tactic": ["dsimp", []], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u03b2\nhf : Mem\u2112p f 1\nhmeas\u271d : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhtendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun M => Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f x) atTop (\ud835\udcdd 0)\nhmeas : \u2200 (M : \u2115), AEStronglyMeasurable (Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f) \u03bc\nhbound : HasFiniteIntegral fun x => \u2016f x\u2016\nn : \u2115\nx : \u03b1\nhx : \u00ac\u2191n \u2264 \u2016f x\u2016\n\u22a2 x \u2208 {x | (fun a => \u2016Set.indicator {x | \u2191n \u2264 \u2016f x\u2016\u208a} f a\u2016 \u2264 \u2016f a\u2016) x}", "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u03b2\nhf : Mem\u2112p f 1\nhmeas\u271d : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhtendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun M => Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f x) atTop (\ud835\udcdd 0)\nhmeas : \u2200 (M : \u2115), AEStronglyMeasurable (Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f) \u03bc\nhbound : HasFiniteIntegral fun x => \u2016f x\u2016\nn : \u2115\nx : \u03b1\nhx : \u00ac\u2191n \u2264 \u2016f x\u2016\n\u22a2 \u2016Set.indicator {x | \u2191n \u2264 \u2016f x\u2016\u208a} f x\u2016 \u2264 \u2016f x\u2016"}, {"tactic": "rw [Set.indicator_of_not_mem, norm_zero]", "annotated_tactic": ["rw [<a>Set.indicator_of_not_mem</a>, <a>norm_zero</a>]", [{"full_name": "Set.indicator_of_not_mem", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [73, 3], "def_end_pos": [73, 14]}, {"full_name": "norm_zero", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [528, 30], "def_end_pos": [528, 39]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u03b2\nhf : Mem\u2112p f 1\nhmeas\u271d : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhtendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun M => Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f x) atTop (\ud835\udcdd 0)\nhmeas : \u2200 (M : \u2115), AEStronglyMeasurable (Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f) \u03bc\nhbound : HasFiniteIntegral fun x => \u2016f x\u2016\nn : \u2115\nx : \u03b1\nhx : \u00ac\u2191n \u2264 \u2016f x\u2016\n\u22a2 \u2016Set.indicator {x | \u2191n \u2264 \u2016f x\u2016\u208a} f x\u2016 \u2264 \u2016f x\u2016", "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u03b2\nhf : Mem\u2112p f 1\nhmeas\u271d : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhtendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun M => Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f x) atTop (\ud835\udcdd 0)\nhmeas : \u2200 (M : \u2115), AEStronglyMeasurable (Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f) \u03bc\nhbound : HasFiniteIntegral fun x => \u2016f x\u2016\nn : \u2115\nx : \u03b1\nhx : \u00ac\u2191n \u2264 \u2016f x\u2016\n\u22a2 0 \u2264 \u2016f x\u2016\n\ncase neg.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u03b2\nhf : Mem\u2112p f 1\nhmeas\u271d : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhtendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun M => Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f x) atTop (\ud835\udcdd 0)\nhmeas : \u2200 (M : \u2115), AEStronglyMeasurable (Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f) \u03bc\nhbound : HasFiniteIntegral fun x => \u2016f x\u2016\nn : \u2115\nx : \u03b1\nhx : \u00ac\u2191n \u2264 \u2016f x\u2016\n\u22a2 \u00acx \u2208 {x | \u2191n \u2264 \u2016f x\u2016\u208a}"}, {"tactic": "exact norm_nonneg _", "annotated_tactic": ["exact <a>norm_nonneg</a> _", [{"full_name": "norm_nonneg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [500, 30], "def_end_pos": [500, 41]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u03b2\nhf : Mem\u2112p f 1\nhmeas\u271d : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhtendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun M => Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f x) atTop (\ud835\udcdd 0)\nhmeas : \u2200 (M : \u2115), AEStronglyMeasurable (Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f) \u03bc\nhbound : HasFiniteIntegral fun x => \u2016f x\u2016\nn : \u2115\nx : \u03b1\nhx : \u00ac\u2191n \u2264 \u2016f x\u2016\n\u22a2 0 \u2264 \u2016f x\u2016", "state_after": "no goals"}, {"tactic": "assumption", "annotated_tactic": ["assumption", []], "state_before": "case neg.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u03b2\nhf : Mem\u2112p f 1\nhmeas\u271d : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhtendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun M => Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f x) atTop (\ud835\udcdd 0)\nhmeas : \u2200 (M : \u2115), AEStronglyMeasurable (Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f) \u03bc\nhbound : HasFiniteIntegral fun x => \u2016f x\u2016\nn : \u2115\nx : \u03b1\nhx : \u00ac\u2191n \u2264 \u2016f x\u2016\n\u22a2 \u00acx \u2208 {x | \u2191n \u2264 \u2016f x\u2016\u208a}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "full_name": "Int.natAbs_div", "start": [559, 9], "end": [564, 95], "traced_tactics": [{"tactic": "rw [Int.div_neg, natAbs_neg, natAbs_neg]", "annotated_tactic": ["rw [<a>Int.div_neg</a>, <a>natAbs_neg</a>, <a>natAbs_neg</a>]", [{"full_name": "Int.div_neg", "def_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "def_pos": [66, 27], "def_end_pos": [66, 34]}, {"full_name": "Int.natAbs_neg", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [157, 17], "def_end_pos": [157, 27]}, {"full_name": "Int.natAbs_neg", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [157, 17], "def_end_pos": [157, 27]}]], "state_before": "a b : Int\nw\u271d\u00b9 w\u271d : Nat\n\u22a2 natAbs (div (\u2191w\u271d\u00b9) (-\u2191w\u271d)) = Nat.div (natAbs \u2191w\u271d\u00b9) (natAbs (-\u2191w\u271d))", "state_after": "a b : Int\nw\u271d\u00b9 w\u271d : Nat\n\u22a2 natAbs (div \u2191w\u271d\u00b9 \u2191w\u271d) = Nat.div (natAbs \u2191w\u271d\u00b9) (natAbs \u2191w\u271d)"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "a b : Int\nw\u271d\u00b9 w\u271d : Nat\n\u22a2 natAbs (div \u2191w\u271d\u00b9 \u2191w\u271d) = Nat.div (natAbs \u2191w\u271d\u00b9) (natAbs \u2191w\u271d)", "state_after": "no goals"}, {"tactic": "rw [Int.neg_div, natAbs_neg, natAbs_neg]", "annotated_tactic": ["rw [<a>Int.neg_div</a>, <a>natAbs_neg</a>, <a>natAbs_neg</a>]", [{"full_name": "Int.neg_div", "def_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "def_pos": [84, 27], "def_end_pos": [84, 34]}, {"full_name": "Int.natAbs_neg", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [157, 17], "def_end_pos": [157, 27]}, {"full_name": "Int.natAbs_neg", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [157, 17], "def_end_pos": [157, 27]}]], "state_before": "a b : Int\nw\u271d\u00b9 w\u271d : Nat\n\u22a2 natAbs (div (-\u2191w\u271d\u00b9) \u2191w\u271d) = Nat.div (natAbs (-\u2191w\u271d\u00b9)) (natAbs \u2191w\u271d)", "state_after": "a b : Int\nw\u271d\u00b9 w\u271d : Nat\n\u22a2 natAbs (div \u2191w\u271d\u00b9 \u2191w\u271d) = Nat.div (natAbs \u2191w\u271d\u00b9) (natAbs \u2191w\u271d)"}, {"tactic": "rw [Int.neg_div_neg, natAbs_neg, natAbs_neg]", "annotated_tactic": ["rw [<a>Int.neg_div_neg</a>, <a>natAbs_neg</a>, <a>natAbs_neg</a>]", [{"full_name": "Int.neg_div_neg", "def_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "def_pos": [89, 19], "def_end_pos": [89, 30]}, {"full_name": "Int.natAbs_neg", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [157, 17], "def_end_pos": [157, 27]}, {"full_name": "Int.natAbs_neg", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [157, 17], "def_end_pos": [157, 27]}]], "state_before": "a b : Int\nw\u271d\u00b9 w\u271d : Nat\n\u22a2 natAbs (div (-\u2191w\u271d\u00b9) (-\u2191w\u271d)) = Nat.div (natAbs (-\u2191w\u271d\u00b9)) (natAbs (-\u2191w\u271d))", "state_after": "a b : Int\nw\u271d\u00b9 w\u271d : Nat\n\u22a2 natAbs (div \u2191w\u271d\u00b9 \u2191w\u271d) = Nat.div (natAbs \u2191w\u271d\u00b9) (natAbs \u2191w\u271d)"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Variance.lean", "full_name": "ProbabilityTheory.variance_le_expectation_sq", "start": [223, 1], "end": [242, 88], "traced_tactics": [{"tactic": "by_cases hX : Mem\u2112p X 2", "annotated_tactic": ["by_cases hX : <a>Mem\u2112p</a> X 2", [{"full_name": "MeasureTheory.Mem\u2112p", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [108, 5], "def_end_pos": [108, 10]}]], "state_before": "\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhm : AEStronglyMeasurable X \u2119\n\u22a2 variance X \u2119 \u2264 \u222b (a : \u03a9), (X ^ 2) a", "state_after": "case pos\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhm : AEStronglyMeasurable X \u2119\nhX : Mem\u2112p X 2\n\u22a2 variance X \u2119 \u2264 \u222b (a : \u03a9), (X ^ 2) a\n\ncase neg\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhm : AEStronglyMeasurable X \u2119\nhX : \u00acMem\u2112p X 2\n\u22a2 variance X \u2119 \u2264 \u222b (a : \u03a9), (X ^ 2) a"}, {"tactic": "rw [variance, evariance_eq_lintegral_ofReal, \u2190 integral_eq_lintegral_of_nonneg_ae]", "annotated_tactic": ["rw [<a>variance</a>, <a>evariance_eq_lintegral_ofReal</a>, \u2190 <a>integral_eq_lintegral_of_nonneg_ae</a>]", [{"full_name": "ProbabilityTheory.variance", "def_path": "Mathlib/Probability/Variance.lean", "def_pos": [61, 5], "def_end_pos": [61, 13]}, {"full_name": "ProbabilityTheory.evariance_eq_lintegral_ofReal", "def_path": "Mathlib/Probability/Variance.lean", "def_pos": [108, 9], "def_end_pos": [108, 38]}, {"full_name": "MeasureTheory.integral_eq_lintegral_of_nonneg_ae", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1158, 9], "def_end_pos": [1158, 43]}]], "state_before": "case neg\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhm : AEStronglyMeasurable X \u2119\nhX : \u00acMem\u2112p X 2\n\u22a2 variance X \u2119 \u2264 \u222b (a : \u03a9), (X ^ 2) a", "state_after": "case neg\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhm : AEStronglyMeasurable X \u2119\nhX : \u00acMem\u2112p X 2\n\u22a2 \u222b (a : \u03a9), (X a - \u222b (x : \u03a9), X x) ^ 2 \u2264 \u222b (a : \u03a9), (X ^ 2) a\n\ncase neg.hf\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhm : AEStronglyMeasurable X \u2119\nhX : \u00acMem\u2112p X 2\n\u22a2 0 \u2264\u1d50[\u2119] fun \u03c9 => (X \u03c9 - \u222b (x : \u03a9), X x) ^ 2\n\ncase neg.hfm\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhm : AEStronglyMeasurable X \u2119\nhX : \u00acMem\u2112p X 2\n\u22a2 AEStronglyMeasurable (fun \u03c9 => (X \u03c9 - \u222b (x : \u03a9), X x) ^ 2) \u2119"}, {"tactic": "by_cases hint : Integrable X", "annotated_tactic": ["by_cases hint : <a>Integrable</a> X", [{"full_name": "MeasureTheory.Integrable", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [442, 5], "def_end_pos": [442, 15]}]], "state_before": "case neg\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhm : AEStronglyMeasurable X \u2119\nhX : \u00acMem\u2112p X 2\n\u22a2 \u222b (a : \u03a9), (X a - \u222b (x : \u03a9), X x) ^ 2 \u2264 \u222b (a : \u03a9), (X ^ 2) a\n\ncase neg.hf\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhm : AEStronglyMeasurable X \u2119\nhX : \u00acMem\u2112p X 2\n\u22a2 0 \u2264\u1d50[\u2119] fun \u03c9 => (X \u03c9 - \u222b (x : \u03a9), X x) ^ 2\n\ncase neg.hfm\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhm : AEStronglyMeasurable X \u2119\nhX : \u00acMem\u2112p X 2\n\u22a2 AEStronglyMeasurable (fun \u03c9 => (X \u03c9 - \u222b (x : \u03a9), X x) ^ 2) \u2119", "state_after": "case pos\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhm : AEStronglyMeasurable X \u2119\nhX : \u00acMem\u2112p X 2\nhint : Integrable X\n\u22a2 \u222b (a : \u03a9), (X a - \u222b (x : \u03a9), X x) ^ 2 \u2264 \u222b (a : \u03a9), (X ^ 2) a\n\ncase neg\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhm : AEStronglyMeasurable X \u2119\nhX : \u00acMem\u2112p X 2\nhint : \u00acIntegrable X\n\u22a2 \u222b (a : \u03a9), (X a - \u222b (x : \u03a9), X x) ^ 2 \u2264 \u222b (a : \u03a9), (X ^ 2) a\n\ncase neg.hf\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhm : AEStronglyMeasurable X \u2119\nhX : \u00acMem\u2112p X 2\n\u22a2 0 \u2264\u1d50[\u2119] fun \u03c9 => (X \u03c9 - \u222b (x : \u03a9), X x) ^ 2\n\ncase neg.hfm\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhm : AEStronglyMeasurable X \u2119\nhX : \u00acMem\u2112p X 2\n\u22a2 AEStronglyMeasurable (fun \u03c9 => (X \u03c9 - \u222b (x : \u03a9), X x) ^ 2) \u2119"}, {"tactic": "swap", "annotated_tactic": ["swap", []], "state_before": "case pos\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhm : AEStronglyMeasurable X \u2119\nhX : \u00acMem\u2112p X 2\nhint : Integrable X\n\u22a2 \u222b (a : \u03a9), (X a - \u222b (x : \u03a9), X x) ^ 2 \u2264 \u222b (a : \u03a9), (X ^ 2) a\n\ncase neg\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhm : AEStronglyMeasurable X \u2119\nhX : \u00acMem\u2112p X 2\nhint : \u00acIntegrable X\n\u22a2 \u222b (a : \u03a9), (X a - \u222b (x : \u03a9), X x) ^ 2 \u2264 \u222b (a : \u03a9), (X ^ 2) a\n\ncase neg.hf\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhm : AEStronglyMeasurable X \u2119\nhX : \u00acMem\u2112p X 2\n\u22a2 0 \u2264\u1d50[\u2119] fun \u03c9 => (X \u03c9 - \u222b (x : \u03a9), X x) ^ 2\n\ncase neg.hfm\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhm : AEStronglyMeasurable X \u2119\nhX : \u00acMem\u2112p X 2\n\u22a2 AEStronglyMeasurable (fun \u03c9 => (X \u03c9 - \u222b (x : \u03a9), X x) ^ 2) \u2119", "state_after": "case neg\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhm : AEStronglyMeasurable X \u2119\nhX : \u00acMem\u2112p X 2\nhint : \u00acIntegrable X\n\u22a2 \u222b (a : \u03a9), (X a - \u222b (x : \u03a9), X x) ^ 2 \u2264 \u222b (a : \u03a9), (X ^ 2) a\n\ncase pos\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhm : AEStronglyMeasurable X \u2119\nhX : \u00acMem\u2112p X 2\nhint : Integrable X\n\u22a2 \u222b (a : \u03a9), (X a - \u222b (x : \u03a9), X x) ^ 2 \u2264 \u222b (a : \u03a9), (X ^ 2) a\n\ncase neg.hf\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhm : AEStronglyMeasurable X \u2119\nhX : \u00acMem\u2112p X 2\n\u22a2 0 \u2264\u1d50[\u2119] fun \u03c9 => (X \u03c9 - \u222b (x : \u03a9), X x) ^ 2\n\ncase neg.hfm\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhm : AEStronglyMeasurable X \u2119\nhX : \u00acMem\u2112p X 2\n\u22a2 AEStronglyMeasurable (fun \u03c9 => (X \u03c9 - \u222b (x : \u03a9), X x) ^ 2) \u2119"}, {"tactic": "rw [variance_def' hX]", "annotated_tactic": ["rw [<a>variance_def'</a> hX]", [{"full_name": "ProbabilityTheory.variance_def'", "def_path": "Mathlib/Probability/Variance.lean", "def_pos": [210, 9], "def_end_pos": [210, 22]}]], "state_before": "case pos\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhm : AEStronglyMeasurable X \u2119\nhX : Mem\u2112p X 2\n\u22a2 variance X \u2119 \u2264 \u222b (a : \u03a9), (X ^ 2) a", "state_after": "case pos\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhm : AEStronglyMeasurable X \u2119\nhX : Mem\u2112p X 2\n\u22a2 (\u222b (a : \u03a9), (X ^ 2) a) - (\u222b (a : \u03a9), X a) ^ 2 \u2264 \u222b (a : \u03a9), (X ^ 2) a"}, {"tactic": "simp only [sq_nonneg, sub_le_self_iff]", "annotated_tactic": ["simp only [<a>sq_nonneg</a>, <a>sub_le_self_iff</a>]", [{"full_name": "sq_nonneg", "def_path": "Mathlib/Algebra/GroupPower/Order.lean", "def_pos": [645, 9], "def_end_pos": [645, 18]}, {"full_name": "sub_le_self_iff", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [359, 3], "def_end_pos": [359, 14]}]], "state_before": "case pos\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhm : AEStronglyMeasurable X \u2119\nhX : Mem\u2112p X 2\n\u22a2 (\u222b (a : \u03a9), (X ^ 2) a) - (\u222b (a : \u03a9), X a) ^ 2 \u2264 \u222b (a : \u03a9), (X ^ 2) a", "state_after": "no goals"}, {"tactic": "simp only [integral_undef hint, Pi.pow_apply, Pi.sub_apply, sub_zero]", "annotated_tactic": ["simp only [<a>integral_undef</a> hint, <a>Pi.pow_apply</a>, <a>Pi.sub_apply</a>, <a>sub_zero</a>]", [{"full_name": "MeasureTheory.integral_undef", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [836, 9], "def_end_pos": [836, 23]}, {"full_name": "Pi.pow_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [117, 9], "def_end_pos": [117, 18]}, {"full_name": "Pi.sub_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [200, 3], "def_end_pos": [200, 14]}, {"full_name": "sub_zero", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [339, 3], "def_end_pos": [339, 14]}]], "state_before": "case neg\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhm : AEStronglyMeasurable X \u2119\nhX : \u00acMem\u2112p X 2\nhint : \u00acIntegrable X\n\u22a2 \u222b (a : \u03a9), (X a - \u222b (x : \u03a9), X x) ^ 2 \u2264 \u222b (a : \u03a9), (X ^ 2) a", "state_after": "case neg\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhm : AEStronglyMeasurable X \u2119\nhX : \u00acMem\u2112p X 2\nhint : \u00acIntegrable X\n\u22a2 \u222b (a : \u03a9), X a ^ 2 \u2264 \u222b (a : \u03a9), X a ^ 2"}, {"tactic": "exact le_rfl", "annotated_tactic": ["exact <a>le_rfl</a>", [{"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}]], "state_before": "case neg\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhm : AEStronglyMeasurable X \u2119\nhX : \u00acMem\u2112p X 2\nhint : \u00acIntegrable X\n\u22a2 \u222b (a : \u03a9), X a ^ 2 \u2264 \u222b (a : \u03a9), X a ^ 2", "state_after": "no goals"}, {"tactic": "rw [integral_undef]", "annotated_tactic": ["rw [<a>integral_undef</a>]", [{"full_name": "MeasureTheory.integral_undef", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [836, 9], "def_end_pos": [836, 23]}]], "state_before": "case pos\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhm : AEStronglyMeasurable X \u2119\nhX : \u00acMem\u2112p X 2\nhint : Integrable X\n\u22a2 \u222b (a : \u03a9), (X a - \u222b (x : \u03a9), X x) ^ 2 \u2264 \u222b (a : \u03a9), (X ^ 2) a", "state_after": "case pos\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhm : AEStronglyMeasurable X \u2119\nhX : \u00acMem\u2112p X 2\nhint : Integrable X\n\u22a2 0 \u2264 \u222b (a : \u03a9), (X ^ 2) a\n\ncase pos\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhm : AEStronglyMeasurable X \u2119\nhX : \u00acMem\u2112p X 2\nhint : Integrable X\n\u22a2 \u00acIntegrable fun a => (X a - \u222b (x : \u03a9), X x) ^ 2"}, {"tactic": "exact integral_nonneg fun a => sq_nonneg _", "annotated_tactic": ["exact <a>integral_nonneg</a> fun a => <a>sq_nonneg</a> _", [{"full_name": "MeasureTheory.integral_nonneg", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1251, 9], "def_end_pos": [1251, 24]}, {"full_name": "sq_nonneg", "def_path": "Mathlib/Algebra/GroupPower/Order.lean", "def_pos": [645, 9], "def_end_pos": [645, 18]}]], "state_before": "case pos\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhm : AEStronglyMeasurable X \u2119\nhX : \u00acMem\u2112p X 2\nhint : Integrable X\n\u22a2 0 \u2264 \u222b (a : \u03a9), (X ^ 2) a", "state_after": "no goals"}, {"tactic": "intro h", "annotated_tactic": ["intro h", []], "state_before": "case pos\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhm : AEStronglyMeasurable X \u2119\nhX : \u00acMem\u2112p X 2\nhint : Integrable X\n\u22a2 \u00acIntegrable fun a => (X a - \u222b (x : \u03a9), X x) ^ 2", "state_after": "case pos\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhm : AEStronglyMeasurable X \u2119\nhX : \u00acMem\u2112p X 2\nhint : Integrable X\nh : Integrable fun a => (X a - \u222b (x : \u03a9), X x) ^ 2\n\u22a2 False"}, {"tactic": "have A : Mem\u2112p (X - fun \u03c9 : \u03a9 => \ud835\udd3c[X]) 2 \u2119 :=\n  (mem\u2112p_two_iff_integrable_sq (hint.aestronglyMeasurable.sub aestronglyMeasurable_const)).2 h", "annotated_tactic": ["have A : <a>Mem\u2112p</a> (X - fun \u03c9 : \u03a9 => \ud835\udd3c[X]) 2 \u2119 :=\n        (<a>mem\u2112p_two_iff_integrable_sq</a> (hint.aestronglyMeasurable.sub <a>aestronglyMeasurable_const</a>)).2 h", [{"full_name": "MeasureTheory.Mem\u2112p", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [108, 5], "def_end_pos": [108, 10]}, {"full_name": "MeasureTheory.mem\u2112p_two_iff_integrable_sq", "def_path": "Mathlib/MeasureTheory/Function/L2Space.lean", "def_pos": [56, 9], "def_end_pos": [56, 36]}, {"full_name": "MeasureTheory.aestronglyMeasurable_const", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1155, 9], "def_end_pos": [1155, 35]}]], "state_before": "case pos\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhm : AEStronglyMeasurable X \u2119\nhX : \u00acMem\u2112p X 2\nhint : Integrable X\nh : Integrable fun a => (X a - \u222b (x : \u03a9), X x) ^ 2\n\u22a2 False", "state_after": "case pos\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhm : AEStronglyMeasurable X \u2119\nhX : \u00acMem\u2112p X 2\nhint : Integrable X\nh : Integrable fun a => (X a - \u222b (x : \u03a9), X x) ^ 2\nA : Mem\u2112p (X - fun \u03c9 => \u222b (a : \u03a9), X a) 2\n\u22a2 False"}, {"tactic": "have B : Mem\u2112p (fun _ : \u03a9 => \ud835\udd3c[X]) 2 \u2119 := mem\u2112p_const _", "annotated_tactic": ["have B : <a>Mem\u2112p</a> (fun _ : \u03a9 => \ud835\udd3c[X]) 2 \u2119 := <a>mem\u2112p_const</a> _", [{"full_name": "MeasureTheory.Mem\u2112p", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [108, 5], "def_end_pos": [108, 10]}, {"full_name": "MeasureTheory.mem\u2112p_const", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [344, 9], "def_end_pos": [344, 20]}]], "state_before": "case pos\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhm : AEStronglyMeasurable X \u2119\nhX : \u00acMem\u2112p X 2\nhint : Integrable X\nh : Integrable fun a => (X a - \u222b (x : \u03a9), X x) ^ 2\nA : Mem\u2112p (X - fun \u03c9 => \u222b (a : \u03a9), X a) 2\n\u22a2 False", "state_after": "case pos\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhm : AEStronglyMeasurable X \u2119\nhX : \u00acMem\u2112p X 2\nhint : Integrable X\nh : Integrable fun a => (X a - \u222b (x : \u03a9), X x) ^ 2\nA : Mem\u2112p (X - fun \u03c9 => \u222b (a : \u03a9), X a) 2\nB : Mem\u2112p (fun x => \u222b (a : \u03a9), X a) 2\n\u22a2 False"}, {"tactic": "apply hX", "annotated_tactic": ["apply hX", []], "state_before": "case pos\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhm : AEStronglyMeasurable X \u2119\nhX : \u00acMem\u2112p X 2\nhint : Integrable X\nh : Integrable fun a => (X a - \u222b (x : \u03a9), X x) ^ 2\nA : Mem\u2112p (X - fun \u03c9 => \u222b (a : \u03a9), X a) 2\nB : Mem\u2112p (fun x => \u222b (a : \u03a9), X a) 2\n\u22a2 False", "state_after": "case pos\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhm : AEStronglyMeasurable X \u2119\nhX : \u00acMem\u2112p X 2\nhint : Integrable X\nh : Integrable fun a => (X a - \u222b (x : \u03a9), X x) ^ 2\nA : Mem\u2112p (X - fun \u03c9 => \u222b (a : \u03a9), X a) 2\nB : Mem\u2112p (fun x => \u222b (a : \u03a9), X a) 2\n\u22a2 Mem\u2112p X 2"}, {"tactic": "convert A.add B", "annotated_tactic": ["convert A.add B", []], "state_before": "case pos\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhm : AEStronglyMeasurable X \u2119\nhX : \u00acMem\u2112p X 2\nhint : Integrable X\nh : Integrable fun a => (X a - \u222b (x : \u03a9), X x) ^ 2\nA : Mem\u2112p (X - fun \u03c9 => \u222b (a : \u03a9), X a) 2\nB : Mem\u2112p (fun x => \u222b (a : \u03a9), X a) 2\n\u22a2 Mem\u2112p X 2", "state_after": "case h.e'_5\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhm : AEStronglyMeasurable X \u2119\nhX : \u00acMem\u2112p X 2\nhint : Integrable X\nh : Integrable fun a => (X a - \u222b (x : \u03a9), X x) ^ 2\nA : Mem\u2112p (X - fun \u03c9 => \u222b (a : \u03a9), X a) 2\nB : Mem\u2112p (fun x => \u222b (a : \u03a9), X a) 2\n\u22a2 X = (X - fun \u03c9 => \u222b (a : \u03a9), X a) + fun x => \u222b (a : \u03a9), X a"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case h.e'_5\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhm : AEStronglyMeasurable X \u2119\nhX : \u00acMem\u2112p X 2\nhint : Integrable X\nh : Integrable fun a => (X a - \u222b (x : \u03a9), X x) ^ 2\nA : Mem\u2112p (X - fun \u03c9 => \u222b (a : \u03a9), X a) 2\nB : Mem\u2112p (fun x => \u222b (a : \u03a9), X a) 2\n\u22a2 X = (X - fun \u03c9 => \u222b (a : \u03a9), X a) + fun x => \u222b (a : \u03a9), X a", "state_after": "no goals"}, {"tactic": "exact @ae_of_all _ (_) _ _ fun x => sq_nonneg _", "annotated_tactic": ["exact @<a>ae_of_all</a> _ (_) _ _ fun x => <a>sq_nonneg</a> _", [{"full_name": "MeasureTheory.ae_of_all", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [407, 9], "def_end_pos": [407, 18]}, {"full_name": "sq_nonneg", "def_path": "Mathlib/Algebra/GroupPower/Order.lean", "def_pos": [645, 9], "def_end_pos": [645, 18]}]], "state_before": "case neg.hf\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhm : AEStronglyMeasurable X \u2119\nhX : \u00acMem\u2112p X 2\n\u22a2 0 \u2264\u1d50[\u2119] fun \u03c9 => (X \u03c9 - \u222b (x : \u03a9), X x) ^ 2", "state_after": "no goals"}, {"tactic": "exact (AEMeasurable.pow_const (hm.aemeasurable.sub_const _) _).aestronglyMeasurable", "annotated_tactic": ["exact (<a>AEMeasurable.pow_const</a> (hm.aemeasurable.sub_const _) _).<a>aestronglyMeasurable</a>", [{"full_name": "AEMeasurable.pow_const", "def_path": "Mathlib/MeasureTheory/Group/Arithmetic.lean", "def_pos": [233, 9], "def_end_pos": [233, 31]}, {"full_name": "AEMeasurable.aestronglyMeasurable", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1451, 9], "def_end_pos": [1451, 49]}]], "state_before": "case neg.hfm\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhm : AEStronglyMeasurable X \u2119\nhX : \u00acMem\u2112p X 2\n\u22a2 AEStronglyMeasurable (fun \u03c9 => (X \u03c9 - \u222b (x : \u03a9), X x) ^ 2) \u2119", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Hausdorff.lean", "full_name": "MeasureTheory.Measure.le_hausdorffMeasure", "start": [585, 1], "end": [587, 25], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Content.lean", "full_name": "MeasureTheory.Content.innerContent_pos_of_is_mul_left_invariant", "start": [223, 1], "end": [236, 95], "traced_tactics": [{"tactic": "have : (interior (U : Set G)).Nonempty", "annotated_tactic": ["have : (<a>interior</a> (U : <a>Set</a> G)).<a>Nonempty</a>", [{"full_name": "interior", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [288, 5], "def_end_pos": [288, 13]}, {"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}, {"full_name": "Set.Nonempty", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [439, 15], "def_end_pos": [439, 23]}]], "state_before": "G : Type w\ninst\u271d\u00b3 : TopologicalSpace G\n\u03bc : Content G\ninst\u271d\u00b2 : T2Space G\ninst\u271d\u00b9 : Group G\ninst\u271d : TopologicalGroup G\nh3 :\n  \u2200 (g : G) {K : Compacts G},\n    (fun s => \u2191(toFun \u03bc s)) (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) =\n      (fun s => \u2191(toFun \u03bc s)) K\nK : Compacts G\nhK : (fun s => \u2191(toFun \u03bc s)) K \u2260 0\nU : Opens G\nhU : Set.Nonempty \u2191U\n\u22a2 0 < innerContent \u03bc U", "state_after": "case this\nG : Type w\ninst\u271d\u00b3 : TopologicalSpace G\n\u03bc : Content G\ninst\u271d\u00b2 : T2Space G\ninst\u271d\u00b9 : Group G\ninst\u271d : TopologicalGroup G\nh3 :\n  \u2200 (g : G) {K : Compacts G},\n    (fun s => \u2191(toFun \u03bc s)) (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) =\n      (fun s => \u2191(toFun \u03bc s)) K\nK : Compacts G\nhK : (fun s => \u2191(toFun \u03bc s)) K \u2260 0\nU : Opens G\nhU : Set.Nonempty \u2191U\n\u22a2 Set.Nonempty (interior \u2191U)\n\nG : Type w\ninst\u271d\u00b3 : TopologicalSpace G\n\u03bc : Content G\ninst\u271d\u00b2 : T2Space G\ninst\u271d\u00b9 : Group G\ninst\u271d : TopologicalGroup G\nh3 :\n  \u2200 (g : G) {K : Compacts G},\n    (fun s => \u2191(toFun \u03bc s)) (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) =\n      (fun s => \u2191(toFun \u03bc s)) K\nK : Compacts G\nhK : (fun s => \u2191(toFun \u03bc s)) K \u2260 0\nU : Opens G\nhU : Set.Nonempty \u2191U\nthis : Set.Nonempty (interior \u2191U)\n\u22a2 0 < innerContent \u03bc U"}, {"tactic": "rwa [U.isOpen.interior_eq]", "annotated_tactic": ["rwa [U.isOpen.interior_eq]", []], "state_before": "case this\nG : Type w\ninst\u271d\u00b3 : TopologicalSpace G\n\u03bc : Content G\ninst\u271d\u00b2 : T2Space G\ninst\u271d\u00b9 : Group G\ninst\u271d : TopologicalGroup G\nh3 :\n  \u2200 (g : G) {K : Compacts G},\n    (fun s => \u2191(toFun \u03bc s)) (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) =\n      (fun s => \u2191(toFun \u03bc s)) K\nK : Compacts G\nhK : (fun s => \u2191(toFun \u03bc s)) K \u2260 0\nU : Opens G\nhU : Set.Nonempty \u2191U\n\u22a2 Set.Nonempty (interior \u2191U)\n\nG : Type w\ninst\u271d\u00b3 : TopologicalSpace G\n\u03bc : Content G\ninst\u271d\u00b2 : T2Space G\ninst\u271d\u00b9 : Group G\ninst\u271d : TopologicalGroup G\nh3 :\n  \u2200 (g : G) {K : Compacts G},\n    (fun s => \u2191(toFun \u03bc s)) (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) =\n      (fun s => \u2191(toFun \u03bc s)) K\nK : Compacts G\nhK : (fun s => \u2191(toFun \u03bc s)) K \u2260 0\nU : Opens G\nhU : Set.Nonempty \u2191U\nthis : Set.Nonempty (interior \u2191U)\n\u22a2 0 < innerContent \u03bc U", "state_after": "G : Type w\ninst\u271d\u00b3 : TopologicalSpace G\n\u03bc : Content G\ninst\u271d\u00b2 : T2Space G\ninst\u271d\u00b9 : Group G\ninst\u271d : TopologicalGroup G\nh3 :\n  \u2200 (g : G) {K : Compacts G},\n    (fun s => \u2191(toFun \u03bc s)) (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) =\n      (fun s => \u2191(toFun \u03bc s)) K\nK : Compacts G\nhK : (fun s => \u2191(toFun \u03bc s)) K \u2260 0\nU : Opens G\nhU : Set.Nonempty \u2191U\nthis : Set.Nonempty (interior \u2191U)\n\u22a2 0 < innerContent \u03bc U"}, {"tactic": "rcases compact_covered_by_mul_left_translates K.2 this with \u27e8s, hs\u27e9", "annotated_tactic": ["rcases <a>compact_covered_by_mul_left_translates</a> K.2 this with \u27e8s, hs\u27e9", [{"full_name": "compact_covered_by_mul_left_translates", "def_path": "Mathlib/Topology/Algebra/Group/Basic.lean", "def_pos": [1667, 9], "def_end_pos": [1667, 47]}]], "state_before": "G : Type w\ninst\u271d\u00b3 : TopologicalSpace G\n\u03bc : Content G\ninst\u271d\u00b2 : T2Space G\ninst\u271d\u00b9 : Group G\ninst\u271d : TopologicalGroup G\nh3 :\n  \u2200 (g : G) {K : Compacts G},\n    (fun s => \u2191(toFun \u03bc s)) (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) =\n      (fun s => \u2191(toFun \u03bc s)) K\nK : Compacts G\nhK : (fun s => \u2191(toFun \u03bc s)) K \u2260 0\nU : Opens G\nhU : Set.Nonempty \u2191U\nthis : Set.Nonempty (interior \u2191U)\n\u22a2 0 < innerContent \u03bc U", "state_after": "case intro\nG : Type w\ninst\u271d\u00b3 : TopologicalSpace G\n\u03bc : Content G\ninst\u271d\u00b2 : T2Space G\ninst\u271d\u00b9 : Group G\ninst\u271d : TopologicalGroup G\nh3 :\n  \u2200 (g : G) {K : Compacts G},\n    (fun s => \u2191(toFun \u03bc s)) (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) =\n      (fun s => \u2191(toFun \u03bc s)) K\nK : Compacts G\nhK : (fun s => \u2191(toFun \u03bc s)) K \u2260 0\nU : Opens G\nhU : Set.Nonempty \u2191U\nthis : Set.Nonempty (interior \u2191U)\ns : Finset G\nhs : K.carrier \u2286 \u22c3 g \u2208 s, (fun h => g * h) \u207b\u00b9' \u2191U\n\u22a2 0 < innerContent \u03bc U"}, {"tactic": "suffices \u03bc K \u2264 s.card * \u03bc.innerContent U by\n  exact (ENNReal.mul_pos_iff.mp <| hK.bot_lt.trans_le this).2", "annotated_tactic": ["suffices \u03bc K \u2264 s.card * \u03bc.innerContent U by\n    exact (ENNReal.mul_pos_iff.mp <| hK.bot_lt.trans_le this).2", []], "state_before": "case intro\nG : Type w\ninst\u271d\u00b3 : TopologicalSpace G\n\u03bc : Content G\ninst\u271d\u00b2 : T2Space G\ninst\u271d\u00b9 : Group G\ninst\u271d : TopologicalGroup G\nh3 :\n  \u2200 (g : G) {K : Compacts G},\n    (fun s => \u2191(toFun \u03bc s)) (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) =\n      (fun s => \u2191(toFun \u03bc s)) K\nK : Compacts G\nhK : (fun s => \u2191(toFun \u03bc s)) K \u2260 0\nU : Opens G\nhU : Set.Nonempty \u2191U\nthis : Set.Nonempty (interior \u2191U)\ns : Finset G\nhs : K.carrier \u2286 \u22c3 g \u2208 s, (fun h => g * h) \u207b\u00b9' \u2191U\n\u22a2 0 < innerContent \u03bc U", "state_after": "case intro\nG : Type w\ninst\u271d\u00b3 : TopologicalSpace G\n\u03bc : Content G\ninst\u271d\u00b2 : T2Space G\ninst\u271d\u00b9 : Group G\ninst\u271d : TopologicalGroup G\nh3 :\n  \u2200 (g : G) {K : Compacts G},\n    (fun s => \u2191(toFun \u03bc s)) (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) =\n      (fun s => \u2191(toFun \u03bc s)) K\nK : Compacts G\nhK : (fun s => \u2191(toFun \u03bc s)) K \u2260 0\nU : Opens G\nhU : Set.Nonempty \u2191U\nthis : Set.Nonempty (interior \u2191U)\ns : Finset G\nhs : K.carrier \u2286 \u22c3 g \u2208 s, (fun h => g * h) \u207b\u00b9' \u2191U\n\u22a2 (fun s => \u2191(toFun \u03bc s)) K \u2264 \u2191(Finset.card s) * innerContent \u03bc U"}, {"tactic": "have : (K : Set G) \u2286 \u2191(\u2a06 g \u2208 s, Opens.comap (Homeomorph.mulLeft g).toContinuousMap U) := by\n  simpa only [Opens.iSup_def, Opens.coe_comap, Subtype.coe_mk]", "annotated_tactic": ["have : (K : <a>Set</a> G) \u2286 \u2191(\u2a06 g \u2208 s, <a>Opens.comap</a> (<a>Homeomorph.mulLeft</a> g).<a>toContinuousMap</a> U) := by\n    simpa only [<a>Opens.iSup_def</a>, <a>Opens.coe_comap</a>, <a>Subtype.coe_mk</a>]", [{"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}, {"full_name": "TopologicalSpace.Opens.comap", "def_path": "Mathlib/Topology/Sets/Opens.lean", "def_pos": [360, 5], "def_end_pos": [360, 10]}, {"full_name": "Homeomorph.mulLeft", "def_path": "Mathlib/Topology/Algebra/Group/Basic.lean", "def_pos": [56, 15], "def_end_pos": [56, 33]}, {"full_name": "Homeomorph.toContinuousMap", "def_path": "Mathlib/Topology/ContinuousFunction/Basic.lean", "def_pos": [480, 5], "def_end_pos": [480, 20]}, {"full_name": "TopologicalSpace.Opens.iSup_def", "def_path": "Mathlib/Topology/Sets/Opens.lean", "def_pos": [236, 9], "def_end_pos": [236, 17]}, {"full_name": "TopologicalSpace.Opens.coe_comap", "def_path": "Mathlib/Topology/Sets/Opens.lean", "def_pos": [377, 9], "def_end_pos": [377, 18]}, {"full_name": "Subtype.coe_mk", "def_path": "Mathlib/Data/Subtype.lean", "def_pos": [99, 9], "def_end_pos": [99, 15]}]], "state_before": "case intro\nG : Type w\ninst\u271d\u00b3 : TopologicalSpace G\n\u03bc : Content G\ninst\u271d\u00b2 : T2Space G\ninst\u271d\u00b9 : Group G\ninst\u271d : TopologicalGroup G\nh3 :\n  \u2200 (g : G) {K : Compacts G},\n    (fun s => \u2191(toFun \u03bc s)) (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) =\n      (fun s => \u2191(toFun \u03bc s)) K\nK : Compacts G\nhK : (fun s => \u2191(toFun \u03bc s)) K \u2260 0\nU : Opens G\nhU : Set.Nonempty \u2191U\nthis : Set.Nonempty (interior \u2191U)\ns : Finset G\nhs : K.carrier \u2286 \u22c3 g \u2208 s, (fun h => g * h) \u207b\u00b9' \u2191U\n\u22a2 (fun s => \u2191(toFun \u03bc s)) K \u2264 \u2191(Finset.card s) * innerContent \u03bc U", "state_after": "case intro\nG : Type w\ninst\u271d\u00b3 : TopologicalSpace G\n\u03bc : Content G\ninst\u271d\u00b2 : T2Space G\ninst\u271d\u00b9 : Group G\ninst\u271d : TopologicalGroup G\nh3 :\n  \u2200 (g : G) {K : Compacts G},\n    (fun s => \u2191(toFun \u03bc s)) (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) =\n      (fun s => \u2191(toFun \u03bc s)) K\nK : Compacts G\nhK : (fun s => \u2191(toFun \u03bc s)) K \u2260 0\nU : Opens G\nhU : Set.Nonempty \u2191U\nthis\u271d : Set.Nonempty (interior \u2191U)\ns : Finset G\nhs : K.carrier \u2286 \u22c3 g \u2208 s, (fun h => g * h) \u207b\u00b9' \u2191U\nthis : \u2191K \u2286 \u2191(\u2a06 g \u2208 s, \u2191(Opens.comap (Homeomorph.toContinuousMap (Homeomorph.mulLeft g))) U)\n\u22a2 (fun s => \u2191(toFun \u03bc s)) K \u2264 \u2191(Finset.card s) * innerContent \u03bc U"}, {"tactic": "refine' (\u03bc.le_innerContent _ _ this).trans _", "annotated_tactic": ["refine' (\u03bc.le_innerContent _ _ this).<a>trans</a> _", [{"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [120, 7], "def_end_pos": [120, 18]}]], "state_before": "case intro\nG : Type w\ninst\u271d\u00b3 : TopologicalSpace G\n\u03bc : Content G\ninst\u271d\u00b2 : T2Space G\ninst\u271d\u00b9 : Group G\ninst\u271d : TopologicalGroup G\nh3 :\n  \u2200 (g : G) {K : Compacts G},\n    (fun s => \u2191(toFun \u03bc s)) (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) =\n      (fun s => \u2191(toFun \u03bc s)) K\nK : Compacts G\nhK : (fun s => \u2191(toFun \u03bc s)) K \u2260 0\nU : Opens G\nhU : Set.Nonempty \u2191U\nthis\u271d : Set.Nonempty (interior \u2191U)\ns : Finset G\nhs : K.carrier \u2286 \u22c3 g \u2208 s, (fun h => g * h) \u207b\u00b9' \u2191U\nthis : \u2191K \u2286 \u2191(\u2a06 g \u2208 s, \u2191(Opens.comap (Homeomorph.toContinuousMap (Homeomorph.mulLeft g))) U)\n\u22a2 (fun s => \u2191(toFun \u03bc s)) K \u2264 \u2191(Finset.card s) * innerContent \u03bc U", "state_after": "case intro\nG : Type w\ninst\u271d\u00b3 : TopologicalSpace G\n\u03bc : Content G\ninst\u271d\u00b2 : T2Space G\ninst\u271d\u00b9 : Group G\ninst\u271d : TopologicalGroup G\nh3 :\n  \u2200 (g : G) {K : Compacts G},\n    (fun s => \u2191(toFun \u03bc s)) (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) =\n      (fun s => \u2191(toFun \u03bc s)) K\nK : Compacts G\nhK : (fun s => \u2191(toFun \u03bc s)) K \u2260 0\nU : Opens G\nhU : Set.Nonempty \u2191U\nthis\u271d : Set.Nonempty (interior \u2191U)\ns : Finset G\nhs : K.carrier \u2286 \u22c3 g \u2208 s, (fun h => g * h) \u207b\u00b9' \u2191U\nthis : \u2191K \u2286 \u2191(\u2a06 g \u2208 s, \u2191(Opens.comap (Homeomorph.toContinuousMap (Homeomorph.mulLeft g))) U)\n\u22a2 innerContent \u03bc (\u2a06 g \u2208 s, \u2191(Opens.comap (Homeomorph.toContinuousMap (Homeomorph.mulLeft g))) U) \u2264\n    \u2191(Finset.card s) * innerContent \u03bc U"}, {"tactic": "simp only [\u03bc.is_mul_left_invariant_innerContent h3, Finset.sum_const, nsmul_eq_mul, le_refl]", "annotated_tactic": ["simp only [\u03bc.is_mul_left_invariant_innerContent h3, <a>Finset.sum_const</a>, <a>nsmul_eq_mul</a>, <a>le_refl</a>]", [{"full_name": "Finset.sum_const", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [1440, 3], "def_end_pos": [1440, 14]}, {"full_name": "nsmul_eq_mul", "def_path": "Mathlib/Algebra/GroupPower/Lemmas.lean", "def_pos": [509, 9], "def_end_pos": [509, 21]}, {"full_name": "le_refl", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [50, 9], "def_end_pos": [50, 16]}]], "state_before": "case intro\nG : Type w\ninst\u271d\u00b3 : TopologicalSpace G\n\u03bc : Content G\ninst\u271d\u00b2 : T2Space G\ninst\u271d\u00b9 : Group G\ninst\u271d : TopologicalGroup G\nh3 :\n  \u2200 (g : G) {K : Compacts G},\n    (fun s => \u2191(toFun \u03bc s)) (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) =\n      (fun s => \u2191(toFun \u03bc s)) K\nK : Compacts G\nhK : (fun s => \u2191(toFun \u03bc s)) K \u2260 0\nU : Opens G\nhU : Set.Nonempty \u2191U\nthis\u271d : Set.Nonempty (interior \u2191U)\ns : Finset G\nhs : K.carrier \u2286 \u22c3 g \u2208 s, (fun h => g * h) \u207b\u00b9' \u2191U\nthis : \u2191K \u2286 \u2191(\u2a06 g \u2208 s, \u2191(Opens.comap (Homeomorph.toContinuousMap (Homeomorph.mulLeft g))) U)\n\u22a2 (Finset.sum s fun d => innerContent \u03bc (\u2191(Opens.comap (Homeomorph.toContinuousMap (Homeomorph.mulLeft d))) U)) \u2264\n    \u2191(Finset.card s) * innerContent \u03bc U", "state_after": "no goals"}, {"tactic": "exact (ENNReal.mul_pos_iff.mp <| hK.bot_lt.trans_le this).2", "annotated_tactic": ["exact (ENNReal.mul_pos_iff.mp <| hK.bot_lt.trans_le this).2", []], "state_before": "G : Type w\ninst\u271d\u00b3 : TopologicalSpace G\n\u03bc : Content G\ninst\u271d\u00b2 : T2Space G\ninst\u271d\u00b9 : Group G\ninst\u271d : TopologicalGroup G\nh3 :\n  \u2200 (g : G) {K : Compacts G},\n    (fun s => \u2191(toFun \u03bc s)) (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) =\n      (fun s => \u2191(toFun \u03bc s)) K\nK : Compacts G\nhK : (fun s => \u2191(toFun \u03bc s)) K \u2260 0\nU : Opens G\nhU : Set.Nonempty \u2191U\nthis\u271d : Set.Nonempty (interior \u2191U)\ns : Finset G\nhs : K.carrier \u2286 \u22c3 g \u2208 s, (fun h => g * h) \u207b\u00b9' \u2191U\nthis : (fun s => \u2191(toFun \u03bc s)) K \u2264 \u2191(Finset.card s) * innerContent \u03bc U\n\u22a2 0 < innerContent \u03bc U", "state_after": "no goals"}, {"tactic": "simpa only [Opens.iSup_def, Opens.coe_comap, Subtype.coe_mk]", "annotated_tactic": ["simpa only [<a>Opens.iSup_def</a>, <a>Opens.coe_comap</a>, <a>Subtype.coe_mk</a>]", [{"full_name": "TopologicalSpace.Opens.iSup_def", "def_path": "Mathlib/Topology/Sets/Opens.lean", "def_pos": [236, 9], "def_end_pos": [236, 17]}, {"full_name": "TopologicalSpace.Opens.coe_comap", "def_path": "Mathlib/Topology/Sets/Opens.lean", "def_pos": [377, 9], "def_end_pos": [377, 18]}, {"full_name": "Subtype.coe_mk", "def_path": "Mathlib/Data/Subtype.lean", "def_pos": [99, 9], "def_end_pos": [99, 15]}]], "state_before": "G : Type w\ninst\u271d\u00b3 : TopologicalSpace G\n\u03bc : Content G\ninst\u271d\u00b2 : T2Space G\ninst\u271d\u00b9 : Group G\ninst\u271d : TopologicalGroup G\nh3 :\n  \u2200 (g : G) {K : Compacts G},\n    (fun s => \u2191(toFun \u03bc s)) (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) =\n      (fun s => \u2191(toFun \u03bc s)) K\nK : Compacts G\nhK : (fun s => \u2191(toFun \u03bc s)) K \u2260 0\nU : Opens G\nhU : Set.Nonempty \u2191U\nthis : Set.Nonempty (interior \u2191U)\ns : Finset G\nhs : K.carrier \u2286 \u22c3 g \u2208 s, (fun h => g * h) \u207b\u00b9' \u2191U\n\u22a2 \u2191K \u2286 \u2191(\u2a06 g \u2208 s, \u2191(Opens.comap (Homeomorph.toContinuousMap (Homeomorph.mulLeft g))) U)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "full_name": "Nat.le_log2", "start": [842, 1], "end": [851, 78], "traced_tactics": [{"tactic": "simp [show 1 \u2264 n from Nat.pos_of_ne_zero h]", "annotated_tactic": ["simp [show 1 \u2264 n from <a>Nat.pos_of_ne_zero</a> h]", [{"full_name": "Nat.pos_of_ne_zero", "def_path": "lake-packages/std/Std/Data/Nat/Init/Lemmas.lean", "def_pos": [25, 19], "def_end_pos": [25, 33]}]], "state_before": "n k : Nat\nh : n \u2260 0\n\u22a2 0 \u2264 log2 n \u2194 2 ^ 0 \u2264 n", "state_after": "no goals"}, {"tactic": "rw [log2]", "annotated_tactic": ["rw [<a>log2</a>]", [{"full_name": "Nat.log2", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Log2.lean", "def_pos": [27, 5], "def_end_pos": [27, 9]}]], "state_before": "n k\u271d : Nat\nh : n \u2260 0\nk : Nat\n\u22a2 k + 1 \u2264 log2 n \u2194 2 ^ (k + 1) \u2264 n", "state_after": "n k\u271d : Nat\nh : n \u2260 0\nk : Nat\n\u22a2 (k + 1 \u2264 if n \u2265 2 then log2 (n / 2) + 1 else 0) \u2194 2 ^ (k + 1) \u2264 n"}, {"tactic": "split", "annotated_tactic": ["split", []], "state_before": "n k\u271d : Nat\nh : n \u2260 0\nk : Nat\n\u22a2 (k + 1 \u2264 if n \u2265 2 then log2 (n / 2) + 1 else 0) \u2194 2 ^ (k + 1) \u2264 n", "state_after": "case inl\nn k\u271d : Nat\nh : n \u2260 0\nk : Nat\nh\u271d : n \u2265 2\n\u22a2 k + 1 \u2264 log2 (n / 2) + 1 \u2194 2 ^ (k + 1) \u2264 n\n\ncase inr\nn k\u271d : Nat\nh : n \u2260 0\nk : Nat\nh\u271d : \u00acn \u2265 2\n\u22a2 k + 1 \u2264 0 \u2194 2 ^ (k + 1) \u2264 n"}, {"tactic": "have n0 : 0 < n / 2 := (Nat.le_div_iff_mul_le (by decide)).2 \u2039_\u203a", "annotated_tactic": ["have n0 : 0 < n / 2 := (<a>Nat.le_div_iff_mul_le</a> (by decide)).2 \u2039_\u203a", [{"full_name": "Nat.le_div_iff_mul_le", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [565, 9], "def_end_pos": [565, 26]}]], "state_before": "case inl\nn k\u271d : Nat\nh : n \u2260 0\nk : Nat\nh\u271d : n \u2265 2\n\u22a2 k + 1 \u2264 log2 (n / 2) + 1 \u2194 2 ^ (k + 1) \u2264 n", "state_after": "case inl\nn k\u271d : Nat\nh : n \u2260 0\nk : Nat\nh\u271d : n \u2265 2\nn0 : 0 < n / 2\n\u22a2 k + 1 \u2264 log2 (n / 2) + 1 \u2194 2 ^ (k + 1) \u2264 n"}, {"tactic": "simp [Nat.add_le_add_iff_right, le_log2 (Nat.ne_of_gt n0), le_div_iff_mul_le, Nat.pow_succ]", "annotated_tactic": ["simp [<a>Nat.add_le_add_iff_right</a>, le_log2 (<a>Nat.ne_of_gt</a> n0), <a>le_div_iff_mul_le</a>, <a>Nat.pow_succ</a>]", [{"full_name": "Nat.add_le_add_iff_right", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [299, 19], "def_end_pos": [299, 39]}, {"full_name": "Nat.ne_of_gt", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [136, 9], "def_end_pos": [136, 17]}, {"full_name": "Nat.le_div_iff_mul_le", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [565, 9], "def_end_pos": [565, 26]}, {"full_name": "Nat.pow_succ", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [473, 9], "def_end_pos": [473, 17]}]], "state_before": "case inl\nn k\u271d : Nat\nh : n \u2260 0\nk : Nat\nh\u271d : n \u2265 2\nn0 : 0 < n / 2\n\u22a2 k + 1 \u2264 log2 (n / 2) + 1 \u2194 2 ^ (k + 1) \u2264 n", "state_after": "no goals"}, {"tactic": "decide", "annotated_tactic": ["decide", []], "state_before": "n k\u271d : Nat\nh : n \u2260 0\nk : Nat\nh\u271d : n \u2265 2\n\u22a2 0 < 2", "state_after": "no goals"}, {"tactic": "simp only [le_zero_eq, succ_ne_zero, false_iff]", "annotated_tactic": ["simp only [<a>le_zero_eq</a>, <a>succ_ne_zero</a>, <a>false_iff</a>]", [{"full_name": "Nat.le_zero_eq", "def_path": "lake-packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [154, 17], "def_end_pos": [154, 31]}, {"full_name": "Nat.succ_ne_zero", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [432, 9], "def_end_pos": [432, 21]}, {"full_name": "false_iff", "def_path": "lake-packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [96, 17], "def_end_pos": [96, 26]}]], "state_before": "case inr\nn k\u271d : Nat\nh : n \u2260 0\nk : Nat\nh\u271d : \u00acn \u2265 2\n\u22a2 k + 1 \u2264 0 \u2194 2 ^ (k + 1) \u2264 n", "state_after": "case inr\nn k\u271d : Nat\nh : n \u2260 0\nk : Nat\nh\u271d : \u00acn \u2265 2\n\u22a2 \u00ac2 ^ (k + 1) \u2264 n"}, {"tactic": "refine mt (Nat.le_trans ?_) \u2039_\u203a", "annotated_tactic": ["refine <a>mt</a> (<a>Nat.le_trans</a> ?_) \u2039_\u203a", [{"full_name": "mt", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [516, 9], "def_end_pos": [516, 11]}, {"full_name": "Nat.le_trans", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1592, 19], "def_end_pos": [1592, 31]}]], "state_before": "case inr\nn k\u271d : Nat\nh : n \u2260 0\nk : Nat\nh\u271d : \u00acn \u2265 2\n\u22a2 \u00ac2 ^ (k + 1) \u2264 n", "state_after": "case inr\nn k\u271d : Nat\nh : n \u2260 0\nk : Nat\nh\u271d : \u00acn \u2265 2\n\u22a2 2 \u2264 2 ^ (k + 1)"}, {"tactic": "exact Nat.pow_le_pow_of_le_right (Nat.succ_pos 1) (Nat.le_add_left 1 k)", "annotated_tactic": ["exact <a>Nat.pow_le_pow_of_le_right</a> (<a>Nat.succ_pos</a> 1) (<a>Nat.le_add_left</a> 1 k)", [{"full_name": "Nat.pow_le_pow_of_le_right", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [482, 9], "def_end_pos": [482, 31]}, {"full_name": "Nat.succ_pos", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1608, 9], "def_end_pos": [1608, 21]}, {"full_name": "Nat.le_add_left", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [344, 9], "def_end_pos": [344, 20]}]], "state_before": "case inr\nn k\u271d : Nat\nh : n \u2260 0\nk : Nat\nh\u271d : \u00acn \u2265 2\n\u22a2 2 \u2264 2 ^ (k + 1)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/NAry.lean", "full_name": "Finset.image\u2082_insert_left", "start": [180, 1], "end": [184, 29], "traced_tactics": [{"tactic": "push_cast", "annotated_tactic": ["push_cast", []], "state_before": "\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\ninst\u271d\u2078 : DecidableEq \u03b1'\ninst\u271d\u2077 : DecidableEq \u03b2'\ninst\u271d\u2076 : DecidableEq \u03b3\ninst\u271d\u2075 : DecidableEq \u03b3'\ninst\u271d\u2074 : DecidableEq \u03b4\ninst\u271d\u00b3 : DecidableEq \u03b4'\ninst\u271d\u00b2 : DecidableEq \u03b5\ninst\u271d\u00b9 : DecidableEq \u03b5'\nf f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Finset \u03b1\nt t' : Finset \u03b2\nu u' : Finset \u03b3\na a' : \u03b1\nb b' : \u03b2\nc : \u03b3\ninst\u271d : DecidableEq \u03b1\n\u22a2 \u2191(image\u2082 f (insert a s) t) = \u2191(image (fun b => f a b) t \u222a image\u2082 f s t)", "state_after": "\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\ninst\u271d\u2078 : DecidableEq \u03b1'\ninst\u271d\u2077 : DecidableEq \u03b2'\ninst\u271d\u2076 : DecidableEq \u03b3\ninst\u271d\u2075 : DecidableEq \u03b3'\ninst\u271d\u2074 : DecidableEq \u03b4\ninst\u271d\u00b3 : DecidableEq \u03b4'\ninst\u271d\u00b2 : DecidableEq \u03b5\ninst\u271d\u00b9 : DecidableEq \u03b5'\nf f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Finset \u03b1\nt t' : Finset \u03b2\nu u' : Finset \u03b3\na a' : \u03b1\nb b' : \u03b2\nc : \u03b3\ninst\u271d : DecidableEq \u03b1\n\u22a2 image2 f (insert a \u2191s) \u2191t = (fun b => f a b) '' \u2191t \u222a image2 f \u2191s \u2191t"}, {"tactic": "exact image2_insert_left", "annotated_tactic": ["exact <a>image2_insert_left</a>", [{"full_name": "Set.image2_insert_left", "def_path": "Mathlib/Data/Set/NAry.lean", "def_pos": [208, 9], "def_end_pos": [208, 27]}]], "state_before": "\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\ninst\u271d\u2078 : DecidableEq \u03b1'\ninst\u271d\u2077 : DecidableEq \u03b2'\ninst\u271d\u2076 : DecidableEq \u03b3\ninst\u271d\u2075 : DecidableEq \u03b3'\ninst\u271d\u2074 : DecidableEq \u03b4\ninst\u271d\u00b3 : DecidableEq \u03b4'\ninst\u271d\u00b2 : DecidableEq \u03b5\ninst\u271d\u00b9 : DecidableEq \u03b5'\nf f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Finset \u03b1\nt t' : Finset \u03b2\nu u' : Finset \u03b3\na a' : \u03b1\nb b' : \u03b2\nc : \u03b3\ninst\u271d : DecidableEq \u03b1\n\u22a2 image2 f (insert a \u2191s) \u2191t = (fun b => f a b) '' \u2191t \u222a image2 f \u2191s \u2191t", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/TuringMachine.lean", "full_name": "Turing.TM2to1.tr_eval_dom", "start": [2746, 1], "end": [2748, 54], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Finite.lean", "full_name": "Set.Finite.pi", "start": [1014, 1], "end": [1020, 30], "traced_tactics": [{"tactic": "cases _root_.nonempty_fintype \u03b4", "annotated_tactic": ["cases <a>_root_.nonempty_fintype</a> \u03b4", [{"full_name": "nonempty_fintype", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [442, 9], "def_end_pos": [442, 25]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\n\u03b3 : Type x\n\u03b4 : Type u_1\ninst\u271d : Finite \u03b4\n\u03ba : \u03b4 \u2192 Type u_2\nt : (d : \u03b4) \u2192 Set (\u03ba d)\nht : \u2200 (d : \u03b4), Set.Finite (t d)\n\u22a2 Set.Finite (Set.pi univ t)", "state_after": "case intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\n\u03b3 : Type x\n\u03b4 : Type u_1\ninst\u271d : Finite \u03b4\n\u03ba : \u03b4 \u2192 Type u_2\nt : (d : \u03b4) \u2192 Set (\u03ba d)\nht : \u2200 (d : \u03b4), Set.Finite (t d)\nval\u271d : Fintype \u03b4\n\u22a2 Set.Finite (Set.pi univ t)"}, {"tactic": "lift t to \u2200 d, Finset (\u03ba d) using ht", "annotated_tactic": ["lift t to \u2200 d, <a>Finset</a> (\u03ba d) using ht", [{"full_name": "Finset", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [138, 11], "def_end_pos": [138, 17]}]], "state_before": "case intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\n\u03b3 : Type x\n\u03b4 : Type u_1\ninst\u271d : Finite \u03b4\n\u03ba : \u03b4 \u2192 Type u_2\nt : (d : \u03b4) \u2192 Set (\u03ba d)\nht : \u2200 (d : \u03b4), Set.Finite (t d)\nval\u271d : Fintype \u03b4\n\u22a2 Set.Finite (Set.pi univ t)", "state_after": "case intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\n\u03b3 : Type x\n\u03b4 : Type u_1\ninst\u271d : Finite \u03b4\n\u03ba : \u03b4 \u2192 Type u_2\nval\u271d : Fintype \u03b4\nt : (i : \u03b4) \u2192 Finset (\u03ba i)\n\u22a2 Set.Finite (Set.pi univ fun i => \u2191(t i))"}, {"tactic": "classical\n  rw [\u2190 Fintype.coe_piFinset]\n  apply Finset.finite_toSet", "annotated_tactic": ["classical\n    rw [\u2190 <a>Fintype.coe_piFinset</a>]\n    apply <a>Finset.finite_toSet</a>", [{"full_name": "Fintype.coe_piFinset", "def_path": "Mathlib/Data/Fintype/Pi.lean", "def_pos": [46, 9], "def_end_pos": [46, 21]}, {"full_name": "Finset.finite_toSet", "def_path": "Mathlib/Data/Set/Finite.lean", "def_pos": [545, 9], "def_end_pos": [545, 21]}]], "state_before": "case intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\n\u03b3 : Type x\n\u03b4 : Type u_1\ninst\u271d : Finite \u03b4\n\u03ba : \u03b4 \u2192 Type u_2\nval\u271d : Fintype \u03b4\nt : (i : \u03b4) \u2192 Finset (\u03ba i)\n\u22a2 Set.Finite (Set.pi univ fun i => \u2191(t i))", "state_after": "no goals"}, {"tactic": "rw [\u2190 Fintype.coe_piFinset]", "annotated_tactic": ["rw [\u2190 <a>Fintype.coe_piFinset</a>]", [{"full_name": "Fintype.coe_piFinset", "def_path": "Mathlib/Data/Fintype/Pi.lean", "def_pos": [46, 9], "def_end_pos": [46, 21]}]], "state_before": "case intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\n\u03b3 : Type x\n\u03b4 : Type u_1\ninst\u271d : Finite \u03b4\n\u03ba : \u03b4 \u2192 Type u_2\nval\u271d : Fintype \u03b4\nt : (i : \u03b4) \u2192 Finset (\u03ba i)\n\u22a2 Set.Finite (Set.pi univ fun i => \u2191(t i))", "state_after": "case intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\n\u03b3 : Type x\n\u03b4 : Type u_1\ninst\u271d : Finite \u03b4\n\u03ba : \u03b4 \u2192 Type u_2\nval\u271d : Fintype \u03b4\nt : (i : \u03b4) \u2192 Finset (\u03ba i)\n\u22a2 Set.Finite \u2191(Fintype.piFinset fun i => t i)"}, {"tactic": "apply Finset.finite_toSet", "annotated_tactic": ["apply <a>Finset.finite_toSet</a>", [{"full_name": "Finset.finite_toSet", "def_path": "Mathlib/Data/Set/Finite.lean", "def_pos": [545, 9], "def_end_pos": [545, 21]}]], "state_before": "case intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\n\u03b3 : Type x\n\u03b4 : Type u_1\ninst\u271d : Finite \u03b4\n\u03ba : \u03b4 \u2192 Type u_2\nval\u271d : Fintype \u03b4\nt : (i : \u03b4) \u2192 Finset (\u03ba i)\n\u22a2 Set.Finite \u2191(Fintype.piFinset fun i => t i)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "full_name": "MeasureTheory.Lp.mem_Lp_of_ae_le", "start": [408, 1], "end": [410, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/ProbabilityMassFunction/Basic.lean", "full_name": "PMF.toMeasure_apply_eq_zero_iff", "start": [269, 1], "end": [271, 88], "traced_tactics": [{"tactic": "rw [toMeasure_apply_eq_toOuterMeasure_apply p s hs, toOuterMeasure_apply_eq_zero_iff]", "annotated_tactic": ["rw [<a>toMeasure_apply_eq_toOuterMeasure_apply</a> p s hs, <a>toOuterMeasure_apply_eq_zero_iff</a>]", [{"full_name": "PMF.toMeasure_apply_eq_toOuterMeasure_apply", "def_path": "Mathlib/Probability/ProbabilityMassFunction/Basic.lean", "def_pos": [255, 9], "def_end_pos": [255, 48]}, {"full_name": "PMF.toOuterMeasure_apply_eq_zero_iff", "def_path": "Mathlib/Probability/ProbabilityMassFunction/Basic.lean", "def_pos": [196, 9], "def_end_pos": [196, 41]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b1\np : PMF \u03b1\ns t : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(toMeasure p) s = 0 \u2194 Disjoint (support p) s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean", "full_name": "MeasureTheory.condexp_congr_ae", "start": [192, 1], "end": [200, 43], "traced_tactics": [{"tactic": "by_cases hm : m \u2264 m0", "annotated_tactic": ["by_cases hm : m \u2264 m0", []], "state_before": "\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nh : f =\u1d50[\u03bc] g\n\u22a2 \u03bc[f|m] =\u1d50[\u03bc] \u03bc[g|m]", "state_after": "case pos\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nh : f =\u1d50[\u03bc] g\nhm : m \u2264 m0\n\u22a2 \u03bc[f|m] =\u1d50[\u03bc] \u03bc[g|m]\n\ncase neg\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nh : f =\u1d50[\u03bc] g\nhm : \u00acm \u2264 m0\n\u22a2 \u03bc[f|m] =\u1d50[\u03bc] \u03bc[g|m]"}, {"tactic": "swap", "annotated_tactic": ["swap", []], "state_before": "case pos\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nh : f =\u1d50[\u03bc] g\nhm : m \u2264 m0\n\u22a2 \u03bc[f|m] =\u1d50[\u03bc] \u03bc[g|m]\n\ncase neg\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nh : f =\u1d50[\u03bc] g\nhm : \u00acm \u2264 m0\n\u22a2 \u03bc[f|m] =\u1d50[\u03bc] \u03bc[g|m]", "state_after": "case neg\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nh : f =\u1d50[\u03bc] g\nhm : \u00acm \u2264 m0\n\u22a2 \u03bc[f|m] =\u1d50[\u03bc] \u03bc[g|m]\n\ncase pos\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nh : f =\u1d50[\u03bc] g\nhm : m \u2264 m0\n\u22a2 \u03bc[f|m] =\u1d50[\u03bc] \u03bc[g|m]"}, {"tactic": "by_cases h\u03bcm : SigmaFinite (\u03bc.trim hm)", "annotated_tactic": ["by_cases h\u03bcm : <a>SigmaFinite</a> (\u03bc.trim hm)", [{"full_name": "MeasureTheory.SigmaFinite", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3289, 7], "def_end_pos": [3289, 18]}]], "state_before": "case pos\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nh : f =\u1d50[\u03bc] g\nhm : m \u2264 m0\n\u22a2 \u03bc[f|m] =\u1d50[\u03bc] \u03bc[g|m]", "state_after": "case pos\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nh : f =\u1d50[\u03bc] g\nhm : m \u2264 m0\nh\u03bcm : SigmaFinite (Measure.trim \u03bc hm)\n\u22a2 \u03bc[f|m] =\u1d50[\u03bc] \u03bc[g|m]\n\ncase neg\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nh : f =\u1d50[\u03bc] g\nhm : m \u2264 m0\nh\u03bcm : \u00acSigmaFinite (Measure.trim \u03bc hm)\n\u22a2 \u03bc[f|m] =\u1d50[\u03bc] \u03bc[g|m]"}, {"tactic": "swap", "annotated_tactic": ["swap", []], "state_before": "case pos\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nh : f =\u1d50[\u03bc] g\nhm : m \u2264 m0\nh\u03bcm : SigmaFinite (Measure.trim \u03bc hm)\n\u22a2 \u03bc[f|m] =\u1d50[\u03bc] \u03bc[g|m]\n\ncase neg\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nh : f =\u1d50[\u03bc] g\nhm : m \u2264 m0\nh\u03bcm : \u00acSigmaFinite (Measure.trim \u03bc hm)\n\u22a2 \u03bc[f|m] =\u1d50[\u03bc] \u03bc[g|m]", "state_after": "case neg\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nh : f =\u1d50[\u03bc] g\nhm : m \u2264 m0\nh\u03bcm : \u00acSigmaFinite (Measure.trim \u03bc hm)\n\u22a2 \u03bc[f|m] =\u1d50[\u03bc] \u03bc[g|m]\n\ncase pos\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nh : f =\u1d50[\u03bc] g\nhm : m \u2264 m0\nh\u03bcm : SigmaFinite (Measure.trim \u03bc hm)\n\u22a2 \u03bc[f|m] =\u1d50[\u03bc] \u03bc[g|m]"}, {"tactic": "haveI : SigmaFinite (\u03bc.trim hm) := h\u03bcm", "annotated_tactic": ["haveI : <a>SigmaFinite</a> (\u03bc.trim hm) := h\u03bcm", [{"full_name": "MeasureTheory.SigmaFinite", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3289, 7], "def_end_pos": [3289, 18]}]], "state_before": "case pos\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nh : f =\u1d50[\u03bc] g\nhm : m \u2264 m0\nh\u03bcm : SigmaFinite (Measure.trim \u03bc hm)\n\u22a2 \u03bc[f|m] =\u1d50[\u03bc] \u03bc[g|m]", "state_after": "case pos\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nh : f =\u1d50[\u03bc] g\nhm : m \u2264 m0\nh\u03bcm this : SigmaFinite (Measure.trim \u03bc hm)\n\u22a2 \u03bc[f|m] =\u1d50[\u03bc] \u03bc[g|m]"}, {"tactic": "exact (condexp_ae_eq_condexpL1 hm f).trans\n  (Filter.EventuallyEq.trans (by rw [condexpL1_congr_ae hm h])\n    (condexp_ae_eq_condexpL1 hm g).symm)", "annotated_tactic": ["exact (<a>condexp_ae_eq_condexpL1</a> hm f).<a>trans</a>\n    (<a>Filter.EventuallyEq.trans</a> (by rw [<a>condexpL1_congr_ae</a> hm h])\n      (<a>condexp_ae_eq_condexpL1</a> hm g).<a>symm</a>)", [{"full_name": "MeasureTheory.condexp_ae_eq_condexpL1", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean", "def_pos": [136, 9], "def_end_pos": [136, 32]}, {"full_name": "Filter.EventuallyEq.trans", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1503, 9], "def_end_pos": [1503, 27]}, {"full_name": "Filter.EventuallyEq.trans", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1503, 9], "def_end_pos": [1503, 27]}, {"full_name": "MeasureTheory.condexpL1_congr_ae", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean", "def_pos": [546, 9], "def_end_pos": [546, 27]}, {"full_name": "MeasureTheory.condexp_ae_eq_condexpL1", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean", "def_pos": [136, 9], "def_end_pos": [136, 32]}, {"full_name": "Filter.EventuallyEq.symm", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1498, 9], "def_end_pos": [1498, 26]}]], "state_before": "case pos\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nh : f =\u1d50[\u03bc] g\nhm : m \u2264 m0\nh\u03bcm this : SigmaFinite (Measure.trim \u03bc hm)\n\u22a2 \u03bc[f|m] =\u1d50[\u03bc] \u03bc[g|m]", "state_after": "no goals"}, {"tactic": "simp_rw [condexp_of_not_le hm]", "annotated_tactic": ["simp_rw [<a>condexp_of_not_le</a> hm]", [{"full_name": "MeasureTheory.condexp_of_not_le", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean", "def_pos": [106, 9], "def_end_pos": [106, 26]}]], "state_before": "case neg\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nh : f =\u1d50[\u03bc] g\nhm : \u00acm \u2264 m0\n\u22a2 \u03bc[f|m] =\u1d50[\u03bc] \u03bc[g|m]", "state_after": "case neg\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nh : f =\u1d50[\u03bc] g\nhm : \u00acm \u2264 m0\n\u22a2 0 =\u1d50[\u03bc] 0"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case neg\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nh : f =\u1d50[\u03bc] g\nhm : \u00acm \u2264 m0\n\u22a2 0 =\u1d50[\u03bc] 0", "state_after": "no goals"}, {"tactic": "simp_rw [condexp_of_not_sigmaFinite hm h\u03bcm]", "annotated_tactic": ["simp_rw [<a>condexp_of_not_sigmaFinite</a> hm h\u03bcm]", [{"full_name": "MeasureTheory.condexp_of_not_sigmaFinite", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean", "def_pos": [109, 9], "def_end_pos": [109, 35]}]], "state_before": "case neg\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nh : f =\u1d50[\u03bc] g\nhm : m \u2264 m0\nh\u03bcm : \u00acSigmaFinite (Measure.trim \u03bc hm)\n\u22a2 \u03bc[f|m] =\u1d50[\u03bc] \u03bc[g|m]", "state_after": "case neg\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nh : f =\u1d50[\u03bc] g\nhm : m \u2264 m0\nh\u03bcm : \u00acSigmaFinite (Measure.trim \u03bc hm)\n\u22a2 0 =\u1d50[\u03bc] 0"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case neg\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nh : f =\u1d50[\u03bc] g\nhm : m \u2264 m0\nh\u03bcm : \u00acSigmaFinite (Measure.trim \u03bc hm)\n\u22a2 0 =\u1d50[\u03bc] 0", "state_after": "no goals"}, {"tactic": "rw [condexpL1_congr_ae hm h]", "annotated_tactic": ["rw [<a>condexpL1_congr_ae</a> hm h]", [{"full_name": "MeasureTheory.condexpL1_congr_ae", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean", "def_pos": [546, 9], "def_end_pos": [546, 27]}]], "state_before": "\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nh : f =\u1d50[\u03bc] g\nhm : m \u2264 m0\nh\u03bcm this : SigmaFinite (Measure.trim \u03bc hm)\n\u22a2 \u2191\u2191(condexpL1 hm \u03bc f) =\u1d50[\u03bc] \u2191\u2191(condexpL1 hm \u03bc g)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "full_name": "MeasureTheory.ae_restrict_uIoc_iff", "start": [2554, 1], "end": [2557, 46], "traced_tactics": [{"tactic": "rw [ae_restrict_uIoc_eq, eventually_sup]", "annotated_tactic": ["rw [<a>ae_restrict_uIoc_eq</a>, <a>eventually_sup</a>]", [{"full_name": "MeasureTheory.ae_restrict_uIoc_eq", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2548, 9], "def_end_pos": [2548, 28]}, {"full_name": "Filter.eventually_sup", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1223, 9], "def_end_pos": [1223, 23]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nP : \u03b1 \u2192 Prop\n\u22a2 (\u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc (\u0399 a b), P x) \u2194\n    (\u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc (Ioc a b), P x) \u2227 \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc (Ioc b a), P x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "full_name": "MeasureTheory.Measure.ae_eq_image_of_ae_eq_comap", "start": [1354, 1], "end": [1369, 62], "traced_tactics": [{"tactic": "rw [EventuallyEq, ae_iff] at hst \u22a2", "annotated_tactic": ["rw [<a>EventuallyEq</a>, <a>ae_iff</a>] at hst \u22a2", [{"full_name": "Filter.EventuallyEq", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1438, 5], "def_end_pos": [1438, 17]}, {"full_name": "MeasureTheory.ae_iff", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [388, 9], "def_end_pos": [388, 15]}]], "state_before": "\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\n\u03b2 : Type u_8\ninst\u271d : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\n\u03bc : Measure \u03b2\nhfi : Injective f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 NullMeasurableSet (f '' s)\ns t : Set \u03b1\nhst : s =\u1da0[ae (comap f \u03bc)] t\n\u22a2 f '' s =\u1da0[ae \u03bc] f '' t", "state_after": "\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\n\u03b2 : Type u_8\ninst\u271d : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\n\u03bc : Measure \u03b2\nhfi : Injective f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 NullMeasurableSet (f '' s)\ns t : Set \u03b1\nhst : \u2191\u2191(comap f \u03bc) {a | \u00acs a = t a} = 0\n\u22a2 \u2191\u2191\u03bc {a | \u00ac(f '' s) a = (f '' t) a} = 0"}, {"tactic": "have h_eq_\u03b1 : { a : \u03b1 | \u00acs a = t a } = s \\ t \u222a t \\ s := by\n  ext1 x\n  simp only [eq_iff_iff, mem_setOf_eq, mem_union, mem_diff]\n  tauto", "annotated_tactic": ["have h_eq_\u03b1 : { a : \u03b1 | \u00acs a = t a } = s \\ t \u222a t \\ s := by\n    ext1 x\n    simp only [<a>eq_iff_iff</a>, <a>mem_setOf_eq</a>, <a>mem_union</a>, <a>mem_diff</a>]\n    tauto", [{"full_name": "eq_iff_iff", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [53, 17], "def_end_pos": [53, 27]}, {"full_name": "Set.mem_setOf_eq", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [256, 29], "def_end_pos": [256, 41]}, {"full_name": "Set.mem_union", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [767, 9], "def_end_pos": [767, 18]}, {"full_name": "Set.mem_diff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1819, 9], "def_end_pos": [1819, 17]}]], "state_before": "\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\n\u03b2 : Type u_8\ninst\u271d : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\n\u03bc : Measure \u03b2\nhfi : Injective f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 NullMeasurableSet (f '' s)\ns t : Set \u03b1\nhst : \u2191\u2191(comap f \u03bc) {a | \u00acs a = t a} = 0\n\u22a2 \u2191\u2191\u03bc {a | \u00ac(f '' s) a = (f '' t) a} = 0", "state_after": "\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\n\u03b2 : Type u_8\ninst\u271d : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\n\u03bc : Measure \u03b2\nhfi : Injective f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 NullMeasurableSet (f '' s)\ns t : Set \u03b1\nhst : \u2191\u2191(comap f \u03bc) {a | \u00acs a = t a} = 0\nh_eq_\u03b1 : {a | \u00acs a = t a} = s \\ t \u222a t \\ s\n\u22a2 \u2191\u2191\u03bc {a | \u00ac(f '' s) a = (f '' t) a} = 0"}, {"tactic": "have h_eq_\u03b2 : { a : \u03b2 | \u00ac(f '' s) a = (f '' t) a } = f '' s \\ f '' t \u222a f '' t \\ f '' s := by\n  ext1 x\n  simp only [eq_iff_iff, mem_setOf_eq, mem_union, mem_diff]\n  tauto", "annotated_tactic": ["have h_eq_\u03b2 : { a : \u03b2 | \u00ac(f '' s) a = (f '' t) a } = f '' s \\ f '' t \u222a f '' t \\ f '' s := by\n    ext1 x\n    simp only [<a>eq_iff_iff</a>, <a>mem_setOf_eq</a>, <a>mem_union</a>, <a>mem_diff</a>]\n    tauto", [{"full_name": "eq_iff_iff", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [53, 17], "def_end_pos": [53, 27]}, {"full_name": "Set.mem_setOf_eq", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [256, 29], "def_end_pos": [256, 41]}, {"full_name": "Set.mem_union", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [767, 9], "def_end_pos": [767, 18]}, {"full_name": "Set.mem_diff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1819, 9], "def_end_pos": [1819, 17]}]], "state_before": "\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\n\u03b2 : Type u_8\ninst\u271d : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\n\u03bc : Measure \u03b2\nhfi : Injective f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 NullMeasurableSet (f '' s)\ns t : Set \u03b1\nhst : \u2191\u2191(comap f \u03bc) {a | \u00acs a = t a} = 0\nh_eq_\u03b1 : {a | \u00acs a = t a} = s \\ t \u222a t \\ s\n\u22a2 \u2191\u2191\u03bc {a | \u00ac(f '' s) a = (f '' t) a} = 0", "state_after": "\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\n\u03b2 : Type u_8\ninst\u271d : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\n\u03bc : Measure \u03b2\nhfi : Injective f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 NullMeasurableSet (f '' s)\ns t : Set \u03b1\nhst : \u2191\u2191(comap f \u03bc) {a | \u00acs a = t a} = 0\nh_eq_\u03b1 : {a | \u00acs a = t a} = s \\ t \u222a t \\ s\nh_eq_\u03b2 : {a | \u00ac(f '' s) a = (f '' t) a} = f '' s \\ f '' t \u222a f '' t \\ f '' s\n\u22a2 \u2191\u2191\u03bc {a | \u00ac(f '' s) a = (f '' t) a} = 0"}, {"tactic": "rw [\u2190 Set.image_diff hfi, \u2190 Set.image_diff hfi, \u2190 Set.image_union] at h_eq_\u03b2", "annotated_tactic": ["rw [\u2190 <a>Set.image_diff</a> hfi, \u2190 <a>Set.image_diff</a> hfi, \u2190 <a>Set.image_union</a>] at h_eq_\u03b2", [{"full_name": "Set.image_diff", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [462, 9], "def_end_pos": [462, 19]}, {"full_name": "Set.image_diff", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [462, 9], "def_end_pos": [462, 19]}, {"full_name": "Set.image_union", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [330, 9], "def_end_pos": [330, 20]}]], "state_before": "\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\n\u03b2 : Type u_8\ninst\u271d : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\n\u03bc : Measure \u03b2\nhfi : Injective f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 NullMeasurableSet (f '' s)\ns t : Set \u03b1\nhst : \u2191\u2191(comap f \u03bc) {a | \u00acs a = t a} = 0\nh_eq_\u03b1 : {a | \u00acs a = t a} = s \\ t \u222a t \\ s\nh_eq_\u03b2 : {a | \u00ac(f '' s) a = (f '' t) a} = f '' s \\ f '' t \u222a f '' t \\ f '' s\n\u22a2 \u2191\u2191\u03bc {a | \u00ac(f '' s) a = (f '' t) a} = 0", "state_after": "\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\n\u03b2 : Type u_8\ninst\u271d : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\n\u03bc : Measure \u03b2\nhfi : Injective f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 NullMeasurableSet (f '' s)\ns t : Set \u03b1\nhst : \u2191\u2191(comap f \u03bc) {a | \u00acs a = t a} = 0\nh_eq_\u03b1 : {a | \u00acs a = t a} = s \\ t \u222a t \\ s\nh_eq_\u03b2 : {a | \u00ac(f '' s) a = (f '' t) a} = f '' (s \\ t \u222a t \\ s)\n\u22a2 \u2191\u2191\u03bc {a | \u00ac(f '' s) a = (f '' t) a} = 0"}, {"tactic": "rw [h_eq_\u03b2]", "annotated_tactic": ["rw [h_eq_\u03b2]", []], "state_before": "\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\n\u03b2 : Type u_8\ninst\u271d : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\n\u03bc : Measure \u03b2\nhfi : Injective f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 NullMeasurableSet (f '' s)\ns t : Set \u03b1\nhst : \u2191\u2191(comap f \u03bc) {a | \u00acs a = t a} = 0\nh_eq_\u03b1 : {a | \u00acs a = t a} = s \\ t \u222a t \\ s\nh_eq_\u03b2 : {a | \u00ac(f '' s) a = (f '' t) a} = f '' (s \\ t \u222a t \\ s)\n\u22a2 \u2191\u2191\u03bc {a | \u00ac(f '' s) a = (f '' t) a} = 0", "state_after": "\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\n\u03b2 : Type u_8\ninst\u271d : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\n\u03bc : Measure \u03b2\nhfi : Injective f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 NullMeasurableSet (f '' s)\ns t : Set \u03b1\nhst : \u2191\u2191(comap f \u03bc) {a | \u00acs a = t a} = 0\nh_eq_\u03b1 : {a | \u00acs a = t a} = s \\ t \u222a t \\ s\nh_eq_\u03b2 : {a | \u00ac(f '' s) a = (f '' t) a} = f '' (s \\ t \u222a t \\ s)\n\u22a2 \u2191\u2191\u03bc (f '' (s \\ t \u222a t \\ s)) = 0"}, {"tactic": "rw [h_eq_\u03b1] at hst", "annotated_tactic": ["rw [h_eq_\u03b1] at hst", []], "state_before": "\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\n\u03b2 : Type u_8\ninst\u271d : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\n\u03bc : Measure \u03b2\nhfi : Injective f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 NullMeasurableSet (f '' s)\ns t : Set \u03b1\nhst : \u2191\u2191(comap f \u03bc) {a | \u00acs a = t a} = 0\nh_eq_\u03b1 : {a | \u00acs a = t a} = s \\ t \u222a t \\ s\nh_eq_\u03b2 : {a | \u00ac(f '' s) a = (f '' t) a} = f '' (s \\ t \u222a t \\ s)\n\u22a2 \u2191\u2191\u03bc (f '' (s \\ t \u222a t \\ s)) = 0", "state_after": "\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\n\u03b2 : Type u_8\ninst\u271d : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\n\u03bc : Measure \u03b2\nhfi : Injective f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 NullMeasurableSet (f '' s)\ns t : Set \u03b1\nhst : \u2191\u2191(comap f \u03bc) (s \\ t \u222a t \\ s) = 0\nh_eq_\u03b1 : {a | \u00acs a = t a} = s \\ t \u222a t \\ s\nh_eq_\u03b2 : {a | \u00ac(f '' s) a = (f '' t) a} = f '' (s \\ t \u222a t \\ s)\n\u22a2 \u2191\u2191\u03bc (f '' (s \\ t \u222a t \\ s)) = 0"}, {"tactic": "exact measure_image_eq_zero_of_comap_eq_zero f \u03bc hfi hf hst", "annotated_tactic": ["exact <a>measure_image_eq_zero_of_comap_eq_zero</a> f \u03bc hfi hf hst", [{"full_name": "MeasureTheory.Measure.measure_image_eq_zero_of_comap_eq_zero", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1347, 9], "def_end_pos": [1347, 47]}]], "state_before": "\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\n\u03b2 : Type u_8\ninst\u271d : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\n\u03bc : Measure \u03b2\nhfi : Injective f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 NullMeasurableSet (f '' s)\ns t : Set \u03b1\nhst : \u2191\u2191(comap f \u03bc) (s \\ t \u222a t \\ s) = 0\nh_eq_\u03b1 : {a | \u00acs a = t a} = s \\ t \u222a t \\ s\nh_eq_\u03b2 : {a | \u00ac(f '' s) a = (f '' t) a} = f '' (s \\ t \u222a t \\ s)\n\u22a2 \u2191\u2191\u03bc (f '' (s \\ t \u222a t \\ s)) = 0", "state_after": "no goals"}, {"tactic": "ext1 x", "annotated_tactic": ["ext1 x", []], "state_before": "\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\n\u03b2 : Type u_8\ninst\u271d : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\n\u03bc : Measure \u03b2\nhfi : Injective f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 NullMeasurableSet (f '' s)\ns t : Set \u03b1\nhst : \u2191\u2191(comap f \u03bc) {a | \u00acs a = t a} = 0\n\u22a2 {a | \u00acs a = t a} = s \\ t \u222a t \\ s", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\n\u03b2 : Type u_8\ninst\u271d : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\n\u03bc : Measure \u03b2\nhfi : Injective f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 NullMeasurableSet (f '' s)\ns t : Set \u03b1\nhst : \u2191\u2191(comap f \u03bc) {a | \u00acs a = t a} = 0\nx : \u03b1\n\u22a2 x \u2208 {a | \u00acs a = t a} \u2194 x \u2208 s \\ t \u222a t \\ s"}, {"tactic": "simp only [eq_iff_iff, mem_setOf_eq, mem_union, mem_diff]", "annotated_tactic": ["simp only [<a>eq_iff_iff</a>, <a>mem_setOf_eq</a>, <a>mem_union</a>, <a>mem_diff</a>]", [{"full_name": "eq_iff_iff", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [53, 17], "def_end_pos": [53, 27]}, {"full_name": "Set.mem_setOf_eq", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [256, 29], "def_end_pos": [256, 41]}, {"full_name": "Set.mem_union", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [767, 9], "def_end_pos": [767, 18]}, {"full_name": "Set.mem_diff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1819, 9], "def_end_pos": [1819, 17]}]], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\n\u03b2 : Type u_8\ninst\u271d : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\n\u03bc : Measure \u03b2\nhfi : Injective f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 NullMeasurableSet (f '' s)\ns t : Set \u03b1\nhst : \u2191\u2191(comap f \u03bc) {a | \u00acs a = t a} = 0\nx : \u03b1\n\u22a2 x \u2208 {a | \u00acs a = t a} \u2194 x \u2208 s \\ t \u222a t \\ s", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\n\u03b2 : Type u_8\ninst\u271d : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\n\u03bc : Measure \u03b2\nhfi : Injective f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 NullMeasurableSet (f '' s)\ns t : Set \u03b1\nhst : \u2191\u2191(comap f \u03bc) {a | \u00acs a = t a} = 0\nx : \u03b1\n\u22a2 \u00ac(s x \u2194 t x) \u2194 x \u2208 s \u2227 \u00acx \u2208 t \u2228 x \u2208 t \u2227 \u00acx \u2208 s"}, {"tactic": "tauto", "annotated_tactic": ["tauto", []], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\n\u03b2 : Type u_8\ninst\u271d : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\n\u03bc : Measure \u03b2\nhfi : Injective f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 NullMeasurableSet (f '' s)\ns t : Set \u03b1\nhst : \u2191\u2191(comap f \u03bc) {a | \u00acs a = t a} = 0\nx : \u03b1\n\u22a2 \u00ac(s x \u2194 t x) \u2194 x \u2208 s \u2227 \u00acx \u2208 t \u2228 x \u2208 t \u2227 \u00acx \u2208 s", "state_after": "no goals"}, {"tactic": "ext1 x", "annotated_tactic": ["ext1 x", []], "state_before": "\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\n\u03b2 : Type u_8\ninst\u271d : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\n\u03bc : Measure \u03b2\nhfi : Injective f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 NullMeasurableSet (f '' s)\ns t : Set \u03b1\nhst : \u2191\u2191(comap f \u03bc) {a | \u00acs a = t a} = 0\nh_eq_\u03b1 : {a | \u00acs a = t a} = s \\ t \u222a t \\ s\n\u22a2 {a | \u00ac(f '' s) a = (f '' t) a} = f '' s \\ f '' t \u222a f '' t \\ f '' s", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\n\u03b2 : Type u_8\ninst\u271d : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\n\u03bc : Measure \u03b2\nhfi : Injective f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 NullMeasurableSet (f '' s)\ns t : Set \u03b1\nhst : \u2191\u2191(comap f \u03bc) {a | \u00acs a = t a} = 0\nh_eq_\u03b1 : {a | \u00acs a = t a} = s \\ t \u222a t \\ s\nx : \u03b2\n\u22a2 x \u2208 {a | \u00ac(f '' s) a = (f '' t) a} \u2194 x \u2208 f '' s \\ f '' t \u222a f '' t \\ f '' s"}, {"tactic": "simp only [eq_iff_iff, mem_setOf_eq, mem_union, mem_diff]", "annotated_tactic": ["simp only [<a>eq_iff_iff</a>, <a>mem_setOf_eq</a>, <a>mem_union</a>, <a>mem_diff</a>]", [{"full_name": "eq_iff_iff", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [53, 17], "def_end_pos": [53, 27]}, {"full_name": "Set.mem_setOf_eq", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [256, 29], "def_end_pos": [256, 41]}, {"full_name": "Set.mem_union", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [767, 9], "def_end_pos": [767, 18]}, {"full_name": "Set.mem_diff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1819, 9], "def_end_pos": [1819, 17]}]], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\n\u03b2 : Type u_8\ninst\u271d : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\n\u03bc : Measure \u03b2\nhfi : Injective f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 NullMeasurableSet (f '' s)\ns t : Set \u03b1\nhst : \u2191\u2191(comap f \u03bc) {a | \u00acs a = t a} = 0\nh_eq_\u03b1 : {a | \u00acs a = t a} = s \\ t \u222a t \\ s\nx : \u03b2\n\u22a2 x \u2208 {a | \u00ac(f '' s) a = (f '' t) a} \u2194 x \u2208 f '' s \\ f '' t \u222a f '' t \\ f '' s", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\n\u03b2 : Type u_8\ninst\u271d : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\n\u03bc : Measure \u03b2\nhfi : Injective f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 NullMeasurableSet (f '' s)\ns t : Set \u03b1\nhst : \u2191\u2191(comap f \u03bc) {a | \u00acs a = t a} = 0\nh_eq_\u03b1 : {a | \u00acs a = t a} = s \\ t \u222a t \\ s\nx : \u03b2\n\u22a2 \u00ac((f '' s) x \u2194 (f '' t) x) \u2194 x \u2208 f '' s \u2227 \u00acx \u2208 f '' t \u2228 x \u2208 f '' t \u2227 \u00acx \u2208 f '' s"}, {"tactic": "tauto", "annotated_tactic": ["tauto", []], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\n\u03b2 : Type u_8\ninst\u271d : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\n\u03bc : Measure \u03b2\nhfi : Injective f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 NullMeasurableSet (f '' s)\ns t : Set \u03b1\nhst : \u2191\u2191(comap f \u03bc) {a | \u00acs a = t a} = 0\nh_eq_\u03b1 : {a | \u00acs a = t a} = s \\ t \u222a t \\ s\nx : \u03b2\n\u22a2 \u00ac((f '' s) x \u2194 (f '' t) x) \u2194 x \u2208 f '' s \u2227 \u00acx \u2208 f '' t \u2228 x \u2208 f '' t \u2227 \u00acx \u2208 f '' s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/NullMeasurable.lean", "full_name": "MeasureTheory.exists_subordinate_pairwise_disjoint", "start": [258, 1], "end": [269, 86], "traced_tactics": [{"tactic": "choose t ht_sub htm ht_eq using fun i => exists_measurable_subset_ae_eq (h i)", "annotated_tactic": ["choose t ht_sub htm ht_eq using fun i => <a>exists_measurable_subset_ae_eq</a> (h i)", [{"full_name": "MeasureTheory.NullMeasurableSet.exists_measurable_subset_ae_eq", "def_path": "Mathlib/MeasureTheory/Measure/NullMeasurable.lean", "def_pos": [245, 9], "def_end_pos": [245, 39]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d : Countable \u03b9\ns : \u03b9 \u2192 Set \u03b1\nh : \u2200 (i : \u03b9), NullMeasurableSet (s i)\nhd : Pairwise (AEDisjoint \u03bc on s)\n\u22a2 \u2203 t, (\u2200 (i : \u03b9), t i \u2286 s i) \u2227 (\u2200 (i : \u03b9), s i =\u1d50[\u03bc] t i) \u2227 (\u2200 (i : \u03b9), MeasurableSet (t i)) \u2227 Pairwise (Disjoint on t)", "state_after": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t\u271d : Set \u03b1\ninst\u271d : Countable \u03b9\ns : \u03b9 \u2192 Set \u03b1\nh : \u2200 (i : \u03b9), NullMeasurableSet (s i)\nhd : Pairwise (AEDisjoint \u03bc on s)\nt : \u03b9 \u2192 Set \u03b1\nht_sub : \u2200 (i : \u03b9), t i \u2286 s i\nhtm : \u2200 (i : \u03b9), MeasurableSet (t i)\nht_eq : \u2200 (i : \u03b9), t i =\u1d50[\u03bc] s i\n\u22a2 \u2203 t, (\u2200 (i : \u03b9), t i \u2286 s i) \u2227 (\u2200 (i : \u03b9), s i =\u1d50[\u03bc] t i) \u2227 (\u2200 (i : \u03b9), MeasurableSet (t i)) \u2227 Pairwise (Disjoint on t)"}, {"tactic": "rcases exists_null_pairwise_disjoint_diff hd with \u27e8u, hum, hu\u2080, hud\u27e9", "annotated_tactic": ["rcases <a>exists_null_pairwise_disjoint_diff</a> hd with \u27e8u, hum, hu\u2080, hud\u27e9", [{"full_name": "MeasureTheory.exists_null_pairwise_disjoint_diff", "def_path": "Mathlib/MeasureTheory/Measure/AEDisjoint.lean", "def_pos": [34, 9], "def_end_pos": [34, 43]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t\u271d : Set \u03b1\ninst\u271d : Countable \u03b9\ns : \u03b9 \u2192 Set \u03b1\nh : \u2200 (i : \u03b9), NullMeasurableSet (s i)\nhd : Pairwise (AEDisjoint \u03bc on s)\nt : \u03b9 \u2192 Set \u03b1\nht_sub : \u2200 (i : \u03b9), t i \u2286 s i\nhtm : \u2200 (i : \u03b9), MeasurableSet (t i)\nht_eq : \u2200 (i : \u03b9), t i =\u1d50[\u03bc] s i\n\u22a2 \u2203 t, (\u2200 (i : \u03b9), t i \u2286 s i) \u2227 (\u2200 (i : \u03b9), s i =\u1d50[\u03bc] t i) \u2227 (\u2200 (i : \u03b9), MeasurableSet (t i)) \u2227 Pairwise (Disjoint on t)", "state_after": "case intro.intro.intro\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t\u271d : Set \u03b1\ninst\u271d : Countable \u03b9\ns : \u03b9 \u2192 Set \u03b1\nh : \u2200 (i : \u03b9), NullMeasurableSet (s i)\nhd : Pairwise (AEDisjoint \u03bc on s)\nt : \u03b9 \u2192 Set \u03b1\nht_sub : \u2200 (i : \u03b9), t i \u2286 s i\nhtm : \u2200 (i : \u03b9), MeasurableSet (t i)\nht_eq : \u2200 (i : \u03b9), t i =\u1d50[\u03bc] s i\nu : \u03b9 \u2192 Set \u03b1\nhum : \u2200 (i : \u03b9), MeasurableSet (u i)\nhu\u2080 : \u2200 (i : \u03b9), \u2191\u2191\u03bc (u i) = 0\nhud : Pairwise (Disjoint on fun i => s i \\ u i)\n\u22a2 \u2203 t, (\u2200 (i : \u03b9), t i \u2286 s i) \u2227 (\u2200 (i : \u03b9), s i =\u1d50[\u03bc] t i) \u2227 (\u2200 (i : \u03b9), MeasurableSet (t i)) \u2227 Pairwise (Disjoint on t)"}, {"tactic": "exact\n  \u27e8fun i => t i \\ u i, fun i => (diff_subset _ _).trans (ht_sub _), fun i =>\n    (ht_eq _).symm.trans (diff_null_ae_eq_self (hu\u2080 i)).symm, fun i => (htm i).diff (hum i),\n    hud.mono fun i j h =>\n      h.mono (diff_subset_diff_left (ht_sub i)) (diff_subset_diff_left (ht_sub j))\u27e9", "annotated_tactic": ["exact\n    \u27e8fun i => t i \\ u i, fun i => (<a>diff_subset</a> _ _).<a>trans</a> (ht_sub _), fun i =>\n      (ht_eq _).symm.trans (<a>diff_null_ae_eq_self</a> (hu\u2080 i)).<a>symm</a>, fun i => (htm i).<a>diff</a> (hum i),\n      hud.mono fun i j h =>\n        h.mono (<a>diff_subset_diff_left</a> (ht_sub i)) (<a>diff_subset_diff_left</a> (ht_sub j))\u27e9", [{"full_name": "Set.diff_subset", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1845, 9], "def_end_pos": [1845, 20]}, {"full_name": "HasSubset.Subset.trans", "def_path": "Mathlib/Order/RelClasses.lean", "def_pos": [664, 7], "def_end_pos": [664, 29]}, {"full_name": "MeasureTheory.diff_null_ae_eq_self", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [493, 9], "def_end_pos": [493, 29]}, {"full_name": "Filter.EventuallyEq.symm", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1498, 9], "def_end_pos": [1498, 26]}, {"full_name": "MeasurableSet.diff", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [205, 19], "def_end_pos": [205, 37]}, {"full_name": "Set.diff_subset_diff_left", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1908, 9], "def_end_pos": [1908, 30]}, {"full_name": "Set.diff_subset_diff_left", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1908, 9], "def_end_pos": [1908, 30]}]], "state_before": "case intro.intro.intro\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t\u271d : Set \u03b1\ninst\u271d : Countable \u03b9\ns : \u03b9 \u2192 Set \u03b1\nh : \u2200 (i : \u03b9), NullMeasurableSet (s i)\nhd : Pairwise (AEDisjoint \u03bc on s)\nt : \u03b9 \u2192 Set \u03b1\nht_sub : \u2200 (i : \u03b9), t i \u2286 s i\nhtm : \u2200 (i : \u03b9), MeasurableSet (t i)\nht_eq : \u2200 (i : \u03b9), t i =\u1d50[\u03bc] s i\nu : \u03b9 \u2192 Set \u03b1\nhum : \u2200 (i : \u03b9), MeasurableSet (u i)\nhu\u2080 : \u2200 (i : \u03b9), \u2191\u2191\u03bc (u i) = 0\nhud : Pairwise (Disjoint on fun i => s i \\ u i)\n\u22a2 \u2203 t, (\u2200 (i : \u03b9), t i \u2286 s i) \u2227 (\u2200 (i : \u03b9), s i =\u1d50[\u03bc] t i) \u2227 (\u2200 (i : \u03b9), MeasurableSet (t i)) \u2227 Pairwise (Disjoint on t)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Process/Adapted.lean", "full_name": "MeasureTheory.ProgMeasurable.finset_prod", "start": [164, 11], "end": [167, 90], "traced_tactics": [{"tactic": "convert ProgMeasurable.finset_prod' h using 1", "annotated_tactic": ["convert <a>ProgMeasurable.finset_prod'</a> h using 1", [{"full_name": "MeasureTheory.ProgMeasurable.finset_prod'", "def_path": "Mathlib/Probability/Process/Adapted.lean", "def_pos": [156, 19], "def_end_pos": [156, 31]}]], "state_before": "\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : Preorder \u03b9\nu v : \u03b9 \u2192 \u03a9 \u2192 \u03b2\nf : Filtration \u03b9 m\ninst\u271d\u00b2 : MeasurableSpace \u03b9\n\u03b3 : Type u_4\ninst\u271d\u00b9 : CommMonoid \u03b2\ninst\u271d : ContinuousMul \u03b2\nU : \u03b3 \u2192 \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Finset \u03b3\nh : \u2200 (c : \u03b3), c \u2208 s \u2192 ProgMeasurable f (U c)\n\u22a2 ProgMeasurable f fun i a => \u220f c in s, U c i a", "state_after": "case h.e'_9\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : Preorder \u03b9\nu v : \u03b9 \u2192 \u03a9 \u2192 \u03b2\nf : Filtration \u03b9 m\ninst\u271d\u00b2 : MeasurableSpace \u03b9\n\u03b3 : Type u_4\ninst\u271d\u00b9 : CommMonoid \u03b2\ninst\u271d : ContinuousMul \u03b2\nU : \u03b3 \u2192 \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Finset \u03b3\nh : \u2200 (c : \u03b3), c \u2208 s \u2192 ProgMeasurable f (U c)\n\u22a2 (fun i a => \u220f c in s, U c i a) = \u220f c in s, U c"}, {"tactic": "ext (i a)", "annotated_tactic": ["ext (i a)", []], "state_before": "case h.e'_9\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : Preorder \u03b9\nu v : \u03b9 \u2192 \u03a9 \u2192 \u03b2\nf : Filtration \u03b9 m\ninst\u271d\u00b2 : MeasurableSpace \u03b9\n\u03b3 : Type u_4\ninst\u271d\u00b9 : CommMonoid \u03b2\ninst\u271d : ContinuousMul \u03b2\nU : \u03b3 \u2192 \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Finset \u03b3\nh : \u2200 (c : \u03b3), c \u2208 s \u2192 ProgMeasurable f (U c)\n\u22a2 (fun i a => \u220f c in s, U c i a) = \u220f c in s, U c", "state_after": "case h.e'_9.h.h\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : Preorder \u03b9\nu v : \u03b9 \u2192 \u03a9 \u2192 \u03b2\nf : Filtration \u03b9 m\ninst\u271d\u00b2 : MeasurableSpace \u03b9\n\u03b3 : Type u_4\ninst\u271d\u00b9 : CommMonoid \u03b2\ninst\u271d : ContinuousMul \u03b2\nU : \u03b3 \u2192 \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Finset \u03b3\nh : \u2200 (c : \u03b3), c \u2208 s \u2192 ProgMeasurable f (U c)\ni : \u03b9\na : \u03a9\n\u22a2 \u220f c in s, U c i a = Finset.prod s (fun c => U c) i a"}, {"tactic": "simp only [Finset.prod_apply]", "annotated_tactic": ["simp only [<a>Finset.prod_apply</a>]", [{"full_name": "Finset.prod_apply", "def_path": "Mathlib/Algebra/BigOperators/Pi.lean", "def_pos": [42, 9], "def_end_pos": [42, 26]}]], "state_before": "case h.e'_9.h.h\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : Preorder \u03b9\nu v : \u03b9 \u2192 \u03a9 \u2192 \u03b2\nf : Filtration \u03b9 m\ninst\u271d\u00b2 : MeasurableSpace \u03b9\n\u03b3 : Type u_4\ninst\u271d\u00b9 : CommMonoid \u03b2\ninst\u271d : ContinuousMul \u03b2\nU : \u03b3 \u2192 \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Finset \u03b3\nh : \u2200 (c : \u03b3), c \u2208 s \u2192 ProgMeasurable f (U c)\ni : \u03b9\na : \u03a9\n\u22a2 \u220f c in s, U c i a = Finset.prod s (fun c => U c) i a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "full_name": "intervalIntegral.integral_add_adjacent_intervals", "start": [900, 1], "end": [903, 90], "traced_tactics": [{"tactic": "rw [\u2190 add_neg_eq_zero, \u2190 integral_symm, integral_add_adjacent_intervals_cancel hab hbc]", "annotated_tactic": ["rw [\u2190 <a>add_neg_eq_zero</a>, \u2190 <a>integral_symm</a>, <a>integral_add_adjacent_intervals_cancel</a> hab hbc]", [{"full_name": "add_neg_eq_zero", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [702, 3], "def_end_pos": [702, 14]}, {"full_name": "intervalIntegral.integral_symm", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [474, 9], "def_end_pos": [474, 22]}, {"full_name": "intervalIntegral.integral_add_adjacent_intervals_cancel", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [886, 9], "def_end_pos": [886, 47]}]], "state_before": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\na b c d : \u211d\nf g : \u211d \u2192 E\n\u03bc : Measure \u211d\nhab : IntervalIntegrable f \u03bc a b\nhbc : IntervalIntegrable f \u03bc b c\n\u22a2 \u222b (x : \u211d) in a..b, f x \u2202\u03bc + \u222b (x : \u211d) in b..c, f x \u2202\u03bc = \u222b (x : \u211d) in a..c, f x \u2202\u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "full_name": "intervalIntegral.norm_integral_eq_norm_integral_Ioc", "start": [531, 1], "end": [533, 64], "traced_tactics": [{"tactic": "rw [\u2190 norm_integral_min_max, integral_of_le min_le_max, uIoc]", "annotated_tactic": ["rw [\u2190 <a>norm_integral_min_max</a>, <a>integral_of_le</a> <a>min_le_max</a>, <a>uIoc</a>]", [{"full_name": "intervalIntegral.norm_integral_min_max", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [526, 9], "def_end_pos": [526, 30]}, {"full_name": "intervalIntegral.integral_of_le", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [465, 9], "def_end_pos": [465, 23]}, {"full_name": "min_le_max", "def_path": "Mathlib/Order/MinMax.lean", "def_pos": [138, 9], "def_end_pos": [138, 19]}, {"full_name": "Set.uIoc", "def_path": "Mathlib/Data/Set/Intervals/UnorderedInterval.lean", "def_pos": [279, 5], "def_end_pos": [279, 9]}]], "state_before": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\na b : \u211d\nf\u271d g : \u211d \u2192 E\n\u03bc : Measure \u211d\nf : \u211d \u2192 E\n\u22a2 \u2016\u222b (x : \u211d) in a..b, f x \u2202\u03bc\u2016 = \u2016\u222b (x : \u211d) in \u0399 a b, f x \u2202\u03bc\u2016", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/QPF/Multivariate/Constructions/Cofix.lean", "full_name": "MvQPF.Cofix.bisim\u2082", "start": [305, 1], "end": [308, 76], "traced_tactics": [{"tactic": "intros", "annotated_tactic": ["intros", []], "state_before": "n : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 LiftR' (RelLast' \u03b1 r) (dest x) (dest y)\n\u22a2 \u2200 (x y : Cofix F \u03b1), r x y \u2192 LiftR (fun {i} => RelLast \u03b1 r) (dest x) (dest y)", "state_after": "n : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 LiftR' (RelLast' \u03b1 r) (dest x) (dest y)\nx\u271d y\u271d : Cofix F \u03b1\na\u271d : r x\u271d y\u271d\n\u22a2 LiftR (fun {i} => RelLast \u03b1 r) (dest x\u271d) (dest y\u271d)"}, {"tactic": "rw [\u2190 LiftR_RelLast_iff]", "annotated_tactic": ["rw [\u2190 <a>LiftR_RelLast_iff</a>]", [{"full_name": "MvFunctor.LiftR_RelLast_iff", "def_path": "Mathlib/Control/Functor/Multivariate.lean", "def_pos": [224, 9], "def_end_pos": [224, 26]}]], "state_before": "n : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 LiftR' (RelLast' \u03b1 r) (dest x) (dest y)\nx\u271d y\u271d : Cofix F \u03b1\na\u271d : r x\u271d y\u271d\n\u22a2 LiftR (fun {i} => RelLast \u03b1 r) (dest x\u271d) (dest y\u271d)", "state_after": "n : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 LiftR' (RelLast' \u03b1 r) (dest x) (dest y)\nx\u271d y\u271d : Cofix F \u03b1\na\u271d : r x\u271d y\u271d\n\u22a2 LiftR' (RelLast' \u03b1 r) (dest x\u271d) (dest y\u271d)"}, {"tactic": "apply h", "annotated_tactic": ["apply h", []], "state_before": "n : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 LiftR' (RelLast' \u03b1 r) (dest x) (dest y)\nx\u271d y\u271d : Cofix F \u03b1\na\u271d : r x\u271d y\u271d\n\u22a2 LiftR' (RelLast' \u03b1 r) (dest x\u271d) (dest y\u271d)", "state_after": "case a\nn : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 LiftR' (RelLast' \u03b1 r) (dest x) (dest y)\nx\u271d y\u271d : Cofix F \u03b1\na\u271d : r x\u271d y\u271d\n\u22a2 r x\u271d y\u271d"}, {"tactic": "assumption", "annotated_tactic": ["assumption", []], "state_before": "case a\nn : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 LiftR' (RelLast' \u03b1 r) (dest x) (dest y)\nx\u271d y\u271d : Cofix F \u03b1\na\u271d : r x\u271d y\u271d\n\u22a2 r x\u271d y\u271d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "full_name": "MeasureTheory.DominatedFinMeasAdditive.zero", "start": [196, 1], "end": [200, 36], "traced_tactics": [{"tactic": "refine' \u27e8FinMeasAdditive.zero, fun s _ _ => _\u27e9", "annotated_tactic": ["refine' \u27e8<a>FinMeasAdditive.zero</a>, fun s _ _ => _\u27e9", [{"full_name": "MeasureTheory.FinMeasAdditive.zero", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [103, 9], "def_end_pos": [103, 13]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\n\u03b2 : Type u_7\ninst\u271d : SeminormedAddCommGroup \u03b2\nT T' : Set \u03b1 \u2192 \u03b2\nC C' : \u211d\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhC : 0 \u2264 C\n\u22a2 DominatedFinMeasAdditive \u03bc 0 C", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\n\u03b2 : Type u_7\ninst\u271d : SeminormedAddCommGroup \u03b2\nT T' : Set \u03b1 \u2192 \u03b2\nC C' : \u211d\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhC : 0 \u2264 C\ns : Set \u03b1\nx\u271d\u00b9 : MeasurableSet s\nx\u271d : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u2016OfNat.ofNat 0 s\u2016 \u2264 C * ENNReal.toReal (\u2191\u2191\u03bc s)"}, {"tactic": "rw [Pi.zero_apply, norm_zero]", "annotated_tactic": ["rw [<a>Pi.zero_apply</a>, <a>norm_zero</a>]", [{"full_name": "Pi.zero_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [46, 3], "def_end_pos": [46, 14]}, {"full_name": "norm_zero", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [528, 30], "def_end_pos": [528, 39]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\n\u03b2 : Type u_7\ninst\u271d : SeminormedAddCommGroup \u03b2\nT T' : Set \u03b1 \u2192 \u03b2\nC C' : \u211d\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhC : 0 \u2264 C\ns : Set \u03b1\nx\u271d\u00b9 : MeasurableSet s\nx\u271d : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u2016OfNat.ofNat 0 s\u2016 \u2264 C * ENNReal.toReal (\u2191\u2191\u03bc s)", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\n\u03b2 : Type u_7\ninst\u271d : SeminormedAddCommGroup \u03b2\nT T' : Set \u03b1 \u2192 \u03b2\nC C' : \u211d\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhC : 0 \u2264 C\ns : Set \u03b1\nx\u271d\u00b9 : MeasurableSet s\nx\u271d : \u2191\u2191\u03bc s < \u22a4\n\u22a2 0 \u2264 C * ENNReal.toReal (\u2191\u2191\u03bc s)"}, {"tactic": "exact mul_nonneg hC toReal_nonneg", "annotated_tactic": ["exact <a>mul_nonneg</a> hC <a>toReal_nonneg</a>", [{"full_name": "mul_nonneg", "def_path": 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\u2191\u2191(\u2191\u03ba a) s = \u2191\u2191(\u2191\u03b7 a) s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/ProbabilityMassFunction/Basic.lean", "full_name": "PMF.toOuterMeasure_mono", "start": [221, 1], "end": [223, 94], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/NoncommProd.lean", "full_name": "Finset.noncommProd_insert_of_not_mem", "start": [276, 1], "end": [287, 59], "traced_tactics": [{"tactic": "convert noncommProd_lemma _ f comm using 3", "annotated_tactic": ["convert <a>noncommProd_lemma</a> _ f comm using 3", [{"full_name": "Finset.noncommProd_lemma", "def_path": "Mathlib/Data/Finset/NoncommProd.lean", "def_pos": [230, 9], "def_end_pos": [230, 26]}]], "state_before": "F : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b2 : Monoid \u03b2\ninst\u271d\u00b9 : Monoid \u03b3\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\na : \u03b1\nf : \u03b1 \u2192 \u03b2\ncomm : Set.Pairwise \u2191(insert a s) fun a b => Commute (f a) (f b)\nha : \u00aca \u2208 s\n\u22a2 Set.Pairwise {x | x \u2208 f a ::\u2098 Multiset.map f s.val} Commute", "state_after": "case h.e'_2.h.e'_2.h.a\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b2 : Monoid \u03b2\ninst\u271d\u00b9 : Monoid \u03b3\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\na : \u03b1\nf : \u03b1 \u2192 \u03b2\ncomm : Set.Pairwise \u2191(insert a s) fun a b => Commute (f a) (f b)\nha : \u00aca \u2208 s\nx\u271d : \u03b2\n\u22a2 x\u271d \u2208 f a ::\u2098 Multiset.map f s.val \u2194 x\u271d \u2208 Multiset.map f (insert a s).val"}, {"tactic": "simp [@eq_comm _ (f a)]", "annotated_tactic": ["simp [@<a>eq_comm</a> _ (f a)]", [{"full_name": "eq_comm", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [104, 9], "def_end_pos": [104, 16]}]], "state_before": "case h.e'_2.h.e'_2.h.a\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b2 : Monoid \u03b2\ninst\u271d\u00b9 : Monoid \u03b3\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\na : \u03b1\nf : \u03b1 \u2192 \u03b2\ncomm : Set.Pairwise \u2191(insert a s) fun a b => Commute (f a) (f b)\nha : \u00aca \u2208 s\nx\u271d : \u03b2\n\u22a2 x\u271d \u2208 f a ::\u2098 Multiset.map f s.val \u2194 x\u271d \u2208 Multiset.map f (insert a s).val", "state_after": "no goals"}, {"tactic": "rw [Multiset.noncommProd_cons, noncommProd]", "annotated_tactic": ["rw [<a>Multiset.noncommProd_cons</a>, <a>noncommProd</a>]", [{"full_name": "Multiset.noncommProd_cons", "def_path": "Mathlib/Data/Finset/NoncommProd.lean", "def_pos": [142, 9], "def_end_pos": [142, 25]}, {"full_name": "Finset.noncommProd", "def_path": "Mathlib/Data/Finset/NoncommProd.lean", "def_pos": [242, 5], "def_end_pos": [242, 16]}]], "state_before": "F : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b2 : Monoid \u03b2\ninst\u271d\u00b9 : Monoid \u03b3\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\na : \u03b1\nf : \u03b1 \u2192 \u03b2\ncomm : Set.Pairwise \u2191(insert a s) fun a b => Commute (f a) (f b)\nha : \u00aca \u2208 s\n\u22a2 Multiset.noncommProd (f a ::\u2098 Multiset.map f s.val)\n      (_ : Set.Pairwise {x | x \u2208 f a ::\u2098 Multiset.map f s.val} Commute) =\n    f a * noncommProd s f (_ : Set.Pairwise \u2191s fun a b => Commute (f a) (f b))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "full_name": "integral_pair", "start": [1250, 1], "end": [1254, 51], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Hausdorff.lean", "full_name": "MeasureTheory.hausdorffMeasure_measurePreserving_funUnique", "start": [1044, 1], "end": [1047, 69], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "full_name": "MeasureTheory.Lp.norm_neg", "start": [363, 1], "end": [364, 43], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Martingale/Centering.lean", "full_name": "MeasureTheory.martingalePart_bdd_difference", "start": [174, 1], "end": [183, 57], "traced_tactics": [{"tactic": "filter_upwards [hbdd, predictablePart_bdd_difference \u2131 hbdd] with \u03c9 h\u03c9\u2081 h\u03c9\u2082 i", "annotated_tactic": ["filter_upwards [hbdd, <a>predictablePart_bdd_difference</a> \u2131 hbdd] with \u03c9 h\u03c9\u2081 h\u03c9\u2082 i", [{"full_name": "MeasureTheory.predictablePart_bdd_difference", "def_path": "Mathlib/Probability/Martingale/Centering.lean", "def_pos": [167, 9], "def_end_pos": [167, 39]}]], "state_before": "\u03a9 : Type u_1\nE : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf\u271d : \u2115 \u2192 \u03a9 \u2192 E\n\u2131\u271d : Filtration \u2115 m0\nn : \u2115\nR : \u211d\u22650\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u2131 : Filtration \u2115 m0\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |martingalePart f \u2131 \u03bc (i + 1) \u03c9 - martingalePart f \u2131 \u03bc i \u03c9| \u2264 \u2191(2 * R)", "state_after": "case h\n\u03a9 : Type u_1\nE : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf\u271d : \u2115 \u2192 \u03a9 \u2192 E\n\u2131\u271d : Filtration \u2115 m0\nn : \u2115\nR : \u211d\u22650\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u2131 : Filtration \u2115 m0\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\n\u03c9 : \u03a9\nh\u03c9\u2081 : \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\nh\u03c9\u2082 : \u2200 (i : \u2115), |predictablePart f \u2131 \u03bc (i + 1) \u03c9 - predictablePart f \u2131 \u03bc i \u03c9| \u2264 \u2191R\ni : \u2115\n\u22a2 |martingalePart f \u2131 \u03bc (i + 1) \u03c9 - martingalePart f \u2131 \u03bc i \u03c9| \u2264 \u2191(2 * R)"}, {"tactic": "simp only [two_mul, martingalePart, Pi.sub_apply]", "annotated_tactic": ["simp only [<a>two_mul</a>, <a>martingalePart</a>, <a>Pi.sub_apply</a>]", [{"full_name": "two_mul", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [177, 9], "def_end_pos": [177, 16]}, {"full_name": "MeasureTheory.martingalePart", "def_path": "Mathlib/Probability/Martingale/Centering.lean", "def_pos": [66, 19], "def_end_pos": [66, 33]}, {"full_name": "Pi.sub_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [200, 3], "def_end_pos": [200, 14]}]], "state_before": "case h\n\u03a9 : Type u_1\nE : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf\u271d : \u2115 \u2192 \u03a9 \u2192 E\n\u2131\u271d : Filtration \u2115 m0\nn : \u2115\nR : \u211d\u22650\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u2131 : Filtration \u2115 m0\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\n\u03c9 : \u03a9\nh\u03c9\u2081 : \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\nh\u03c9\u2082 : \u2200 (i : \u2115), |predictablePart f \u2131 \u03bc (i + 1) \u03c9 - predictablePart f \u2131 \u03bc i \u03c9| \u2264 \u2191R\ni : \u2115\n\u22a2 |martingalePart f \u2131 \u03bc (i + 1) \u03c9 - martingalePart f \u2131 \u03bc i \u03c9| \u2264 \u2191(2 * R)", "state_after": "case h\n\u03a9 : Type u_1\nE : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf\u271d : \u2115 \u2192 \u03a9 \u2192 E\n\u2131\u271d : Filtration \u2115 m0\nn : \u2115\nR : \u211d\u22650\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u2131 : Filtration \u2115 m0\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\n\u03c9 : \u03a9\nh\u03c9\u2081 : \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\nh\u03c9\u2082 : \u2200 (i : \u2115), |predictablePart f \u2131 \u03bc (i + 1) \u03c9 - predictablePart f \u2131 \u03bc i \u03c9| \u2264 \u2191R\ni : \u2115\n\u22a2 |f (i + 1) \u03c9 - predictablePart f \u2131 \u03bc (i + 1) \u03c9 - (f i \u03c9 - predictablePart f \u2131 \u03bc i \u03c9)| \u2264 \u2191(R + R)"}, {"tactic": "have : |f (i + 1) \u03c9 - predictablePart f \u2131 \u03bc (i + 1) \u03c9 - (f i \u03c9 - predictablePart f \u2131 \u03bc i \u03c9)| =\n    |f (i + 1) \u03c9 - f i \u03c9 - (predictablePart f \u2131 \u03bc (i + 1) \u03c9 - predictablePart f \u2131 \u03bc i \u03c9)| := by\n  ring_nf", "annotated_tactic": ["have : |f (i + 1) \u03c9 - <a>predictablePart</a> f \u2131 \u03bc (i + 1) \u03c9 - (f i \u03c9 - <a>predictablePart</a> f \u2131 \u03bc i \u03c9)| =\n      |f (i + 1) \u03c9 - f i \u03c9 - (<a>predictablePart</a> f \u2131 \u03bc (i + 1) \u03c9 - <a>predictablePart</a> f \u2131 \u03bc i \u03c9)| := by\n    ring_nf", [{"full_name": "MeasureTheory.predictablePart", "def_path": "Mathlib/Probability/Martingale/Centering.lean", "def_pos": [45, 19], "def_end_pos": [45, 34]}, {"full_name": "MeasureTheory.predictablePart", "def_path": "Mathlib/Probability/Martingale/Centering.lean", "def_pos": [45, 19], "def_end_pos": [45, 34]}, {"full_name": "MeasureTheory.predictablePart", "def_path": "Mathlib/Probability/Martingale/Centering.lean", "def_pos": [45, 19], "def_end_pos": [45, 34]}, {"full_name": "MeasureTheory.predictablePart", "def_path": "Mathlib/Probability/Martingale/Centering.lean", "def_pos": [45, 19], "def_end_pos": [45, 34]}]], "state_before": "case h\n\u03a9 : Type u_1\nE : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf\u271d : \u2115 \u2192 \u03a9 \u2192 E\n\u2131\u271d : Filtration \u2115 m0\nn : \u2115\nR : \u211d\u22650\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u2131 : Filtration \u2115 m0\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\n\u03c9 : \u03a9\nh\u03c9\u2081 : \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\nh\u03c9\u2082 : \u2200 (i : \u2115), |predictablePart f \u2131 \u03bc (i + 1) \u03c9 - predictablePart f \u2131 \u03bc i \u03c9| \u2264 \u2191R\ni : \u2115\n\u22a2 |f (i + 1) \u03c9 - predictablePart f \u2131 \u03bc (i + 1) \u03c9 - (f i \u03c9 - predictablePart f \u2131 \u03bc i \u03c9)| \u2264 \u2191(R + R)", "state_after": "case h\n\u03a9 : Type u_1\nE : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf\u271d : \u2115 \u2192 \u03a9 \u2192 E\n\u2131\u271d : Filtration \u2115 m0\nn : \u2115\nR : \u211d\u22650\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u2131 : Filtration \u2115 m0\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\n\u03c9 : \u03a9\nh\u03c9\u2081 : \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\nh\u03c9\u2082 : \u2200 (i : \u2115), |predictablePart f \u2131 \u03bc (i + 1) \u03c9 - predictablePart f \u2131 \u03bc i \u03c9| \u2264 \u2191R\ni : \u2115\nthis :\n  |f (i + 1) \u03c9 - predictablePart f \u2131 \u03bc (i + 1) \u03c9 - (f i \u03c9 - predictablePart f \u2131 \u03bc i \u03c9)| =\n    |f (i + 1) \u03c9 - f i \u03c9 - (predictablePart f \u2131 \u03bc (i + 1) \u03c9 - predictablePart f \u2131 \u03bc i \u03c9)|\n\u22a2 |f (i + 1) \u03c9 - predictablePart f \u2131 \u03bc (i + 1) \u03c9 - (f i \u03c9 - predictablePart f \u2131 \u03bc i \u03c9)| \u2264 \u2191(R + R)"}, {"tactic": "rw [this]", "annotated_tactic": ["rw [this]", []], "state_before": "case h\n\u03a9 : Type u_1\nE : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf\u271d : \u2115 \u2192 \u03a9 \u2192 E\n\u2131\u271d : Filtration \u2115 m0\nn : \u2115\nR : \u211d\u22650\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u2131 : Filtration \u2115 m0\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\n\u03c9 : \u03a9\nh\u03c9\u2081 : \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\nh\u03c9\u2082 : \u2200 (i : \u2115), |predictablePart f \u2131 \u03bc (i + 1) \u03c9 - predictablePart f \u2131 \u03bc i \u03c9| \u2264 \u2191R\ni : \u2115\nthis :\n  |f (i + 1) \u03c9 - predictablePart f \u2131 \u03bc (i + 1) \u03c9 - (f i \u03c9 - predictablePart f \u2131 \u03bc i \u03c9)| =\n    |f (i + 1) \u03c9 - f i \u03c9 - (predictablePart f \u2131 \u03bc (i + 1) \u03c9 - predictablePart f \u2131 \u03bc i \u03c9)|\n\u22a2 |f (i + 1) \u03c9 - predictablePart f \u2131 \u03bc (i + 1) \u03c9 - (f i \u03c9 - predictablePart f \u2131 \u03bc i \u03c9)| \u2264 \u2191(R + R)", "state_after": "case h\n\u03a9 : Type u_1\nE : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf\u271d : \u2115 \u2192 \u03a9 \u2192 E\n\u2131\u271d : Filtration \u2115 m0\nn : \u2115\nR : \u211d\u22650\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u2131 : Filtration \u2115 m0\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\n\u03c9 : \u03a9\nh\u03c9\u2081 : \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\nh\u03c9\u2082 : \u2200 (i : \u2115), |predictablePart f \u2131 \u03bc (i + 1) \u03c9 - predictablePart f \u2131 \u03bc i \u03c9| \u2264 \u2191R\ni : \u2115\nthis :\n  |f (i + 1) \u03c9 - predictablePart f \u2131 \u03bc (i + 1) \u03c9 - (f i \u03c9 - predictablePart f \u2131 \u03bc i \u03c9)| =\n    |f (i + 1) \u03c9 - f i \u03c9 - (predictablePart f \u2131 \u03bc (i + 1) \u03c9 - predictablePart f \u2131 \u03bc i \u03c9)|\n\u22a2 |f (i + 1) \u03c9 - f i \u03c9 - (predictablePart f \u2131 \u03bc (i + 1) \u03c9 - predictablePart f \u2131 \u03bc i \u03c9)| \u2264 \u2191(R + R)"}, {"tactic": "exact (abs_sub _ _).trans (add_le_add (h\u03c9\u2081 i) (h\u03c9\u2082 i))", "annotated_tactic": ["exact (<a>abs_sub</a> _ _).<a>trans</a> (<a>add_le_add</a> (h\u03c9\u2081 i) (h\u03c9\u2082 i))", [{"full_name": "abs_sub", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [276, 9], "def_end_pos": [276, 16]}, {"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [120, 7], "def_end_pos": [120, 18]}, {"full_name": "add_le_add", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [205, 15], "def_end_pos": [205, 25]}]], "state_before": "case h\n\u03a9 : Type u_1\nE : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf\u271d : \u2115 \u2192 \u03a9 \u2192 E\n\u2131\u271d : Filtration \u2115 m0\nn : \u2115\nR : \u211d\u22650\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u2131 : Filtration \u2115 m0\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\n\u03c9 : \u03a9\nh\u03c9\u2081 : \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\nh\u03c9\u2082 : \u2200 (i : \u2115), |predictablePart f \u2131 \u03bc (i + 1) \u03c9 - predictablePart f \u2131 \u03bc i \u03c9| \u2264 \u2191R\ni : \u2115\nthis :\n  |f (i + 1) \u03c9 - predictablePart f \u2131 \u03bc (i + 1) \u03c9 - (f i \u03c9 - predictablePart f \u2131 \u03bc i \u03c9)| =\n    |f (i + 1) \u03c9 - f i \u03c9 - (predictablePart f \u2131 \u03bc (i + 1) \u03c9 - predictablePart f \u2131 \u03bc i \u03c9)|\n\u22a2 |f (i + 1) \u03c9 - f i \u03c9 - (predictablePart f \u2131 \u03bc (i + 1) \u03c9 - predictablePart f \u2131 \u03bc i \u03c9)| \u2264 \u2191(R + R)", "state_after": "no goals"}, {"tactic": "ring_nf", "annotated_tactic": ["ring_nf", []], "state_before": "\u03a9 : Type u_1\nE : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf\u271d : \u2115 \u2192 \u03a9 \u2192 E\n\u2131\u271d : Filtration \u2115 m0\nn : \u2115\nR : \u211d\u22650\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u2131 : Filtration \u2115 m0\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\n\u03c9 : \u03a9\nh\u03c9\u2081 : \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\nh\u03c9\u2082 : \u2200 (i : \u2115), |predictablePart f \u2131 \u03bc (i + 1) \u03c9 - predictablePart f \u2131 \u03bc i \u03c9| \u2264 \u2191R\ni : \u2115\n\u22a2 |f (i + 1) \u03c9 - predictablePart f \u2131 \u03bc (i + 1) \u03c9 - (f i \u03c9 - predictablePart f \u2131 \u03bc i \u03c9)| =\n    |f (i + 1) \u03c9 - f i \u03c9 - (predictablePart f \u2131 \u03bc (i + 1) \u03c9 - predictablePart f \u2131 \u03bc i \u03c9)|", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/ProbabilityMassFunction/Constructions.lean", "full_name": "PMF.map_const", "start": [86, 1], "end": [87, 61], "traced_tactics": [{"tactic": "simp only [map, Function.comp, bind_const, Function.const]", "annotated_tactic": ["simp only [<a>map</a>, <a>Function.comp</a>, <a>bind_const</a>, <a>Function.const</a>]", [{"full_name": "PMF.map", "def_path": "Mathlib/Probability/ProbabilityMassFunction/Constructions.lean", "def_pos": [41, 5], "def_end_pos": [41, 8]}, {"full_name": "Function.comp", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [52, 15], "def_end_pos": [52, 28]}, {"full_name": "PMF.bind_const", "def_path": "Mathlib/Probability/ProbabilityMassFunction/Monad.lean", "def_pos": [146, 9], "def_end_pos": [146, 19]}, {"full_name": "Function.const", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [66, 15], "def_end_pos": [66, 29]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2\np : PMF \u03b1\nb : \u03b2\n\u22a2 map (Function.const \u03b1 b) p = pure b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Intervals/Group.lean", "full_name": "Set.pairwise_disjoint_Ico_int_cast", "start": [253, 1], "end": [254, 76], "traced_tactics": [{"tactic": "simpa only [zero_add] using pairwise_disjoint_Ico_add_int_cast (0 : \u03b1)", "annotated_tactic": ["simpa only [<a>zero_add</a>] using <a>pairwise_disjoint_Ico_add_int_cast</a> (0 : \u03b1)", [{"full_name": "zero_add", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [463, 3], "def_end_pos": [463, 14]}, {"full_name": "Set.pairwise_disjoint_Ico_add_int_cast", "def_path": "Mathlib/Data/Set/Intervals/Group.lean", "def_pos": [239, 9], "def_end_pos": [239, 43]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : OrderedRing \u03b1\na : \u03b1\n\u22a2 Pairwise (Disjoint on fun n => Ico (\u2191n) (\u2191n + 1))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "full_name": "MeasureTheory.hasFiniteIntegral_add_measure", "start": [218, 1], "end": [220, 91], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean", "full_name": "MeasureTheory.Lp.induction_stronglyMeasurable", "start": [621, 1], "end": [675, 61], "traced_tactics": [{"tactic": "intro f hf", "annotated_tactic": ["intro f hf", []], "state_before": "\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n  \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n    P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n    StronglyMeasurable f \u2192\n      StronglyMeasurable g \u2192\n        Disjoint (Function.support f) (Function.support g) \u2192\n          P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\n\u22a2 \u2200 (f : { x // x \u2208 Lp F p }), AEStronglyMeasurable' m (\u2191\u2191f) \u03bc \u2192 P f", "state_after": "\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n  \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n    P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n    StronglyMeasurable f \u2192\n      StronglyMeasurable g \u2192\n        Disjoint (Function.support f) (Function.support g) \u2192\n          P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf : { x // x \u2208 Lp F p }\nhf : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\n\u22a2 P f"}, {"tactic": "suffices h_add_ae :\n  \u2200 \u2983f g\u2984, \u2200 hf : Mem\u2112p f p \u03bc, \u2200 hg : Mem\u2112p g p \u03bc, \u2200 _ : AEStronglyMeasurable' m f \u03bc,\n    \u2200 _ : AEStronglyMeasurable' m g \u03bc, Disjoint (Function.support f) (Function.support g) \u2192\n      P (hf.toLp f) \u2192 P (hg.toLp g) \u2192 P (hf.toLp f + hg.toLp g)", "annotated_tactic": ["suffices h_add_ae :\n    \u2200 \u2983f g\u2984, \u2200 hf : <a>Mem\u2112p</a> f p \u03bc, \u2200 hg : <a>Mem\u2112p</a> g p \u03bc, \u2200 _ : <a>AEStronglyMeasurable'</a> m f \u03bc,\n      \u2200 _ : <a>AEStronglyMeasurable'</a> m g \u03bc, <a>Disjoint</a> (<a>Function.support</a> f) (<a>Function.support</a> g) \u2192\n        P (hf.toLp f) \u2192 P (hg.toLp g) \u2192 P (hf.toLp f + hg.toLp g)", [{"full_name": "MeasureTheory.Mem\u2112p", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [108, 5], "def_end_pos": [108, 10]}, {"full_name": "MeasureTheory.Mem\u2112p", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [108, 5], "def_end_pos": [108, 10]}, {"full_name": "MeasureTheory.AEStronglyMeasurable'", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean", "def_pos": [49, 5], "def_end_pos": [49, 26]}, {"full_name": "MeasureTheory.AEStronglyMeasurable'", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean", "def_pos": [49, 5], "def_end_pos": [49, 26]}, {"full_name": "Disjoint", "def_path": "Mathlib/Order/Disjoint.lean", "def_pos": [41, 5], "def_end_pos": [41, 13]}, {"full_name": "Function.support", "def_path": "Mathlib/Algebra/Support.lean", "def_pos": [37, 5], "def_end_pos": [37, 12]}, {"full_name": "Function.support", "def_path": "Mathlib/Algebra/Support.lean", "def_pos": [37, 5], "def_end_pos": [37, 12]}]], "state_before": "\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n  \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n    P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n    StronglyMeasurable f \u2192\n      StronglyMeasurable g \u2192\n        Disjoint (Function.support f) (Function.support g) \u2192\n          P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf : { x // x \u2208 Lp F p }\nhf : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\n\u22a2 P f", "state_after": "\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n  \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n    P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n    StronglyMeasurable f \u2192\n      StronglyMeasurable g \u2192\n        Disjoint (Function.support f) (Function.support g) \u2192\n          P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf : { x // x \u2208 Lp F p }\nhf : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\nh_add_ae :\n  \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n    AEStronglyMeasurable' m f \u03bc \u2192\n      AEStronglyMeasurable' m g \u03bc \u2192\n        Disjoint (Function.support f) (Function.support g) \u2192\n          P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\n\u22a2 P f\n\ncase h_add_ae\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n  \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n    P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n    StronglyMeasurable f \u2192\n      StronglyMeasurable g \u2192\n        Disjoint (Function.support f) (Function.support g) \u2192\n          P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf : { x // x \u2208 Lp F p }\nhf : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\n\u22a2 \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n    AEStronglyMeasurable' m f \u03bc \u2192\n      AEStronglyMeasurable' m g \u03bc \u2192\n        Disjoint (Function.support f) (Function.support g) \u2192\n          P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)"}, {"tactic": "exact Lp.induction_stronglyMeasurable_aux hm hp_ne_top h_ind h_add_ae h_closed f hf", "annotated_tactic": ["exact <a>Lp.induction_stronglyMeasurable_aux</a> hm hp_ne_top h_ind h_add_ae h_closed f hf", [{"full_name": "MeasureTheory.Lp.induction_stronglyMeasurable_aux", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean", "def_pos": [568, 9], "def_end_pos": [568, 44]}]], "state_before": "\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n  \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n    P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n    StronglyMeasurable f \u2192\n      StronglyMeasurable g \u2192\n        Disjoint (Function.support f) (Function.support g) \u2192\n          P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf : { x // x \u2208 Lp F p }\nhf : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\nh_add_ae :\n  \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n    AEStronglyMeasurable' m f \u03bc \u2192\n      AEStronglyMeasurable' m g \u03bc \u2192\n        Disjoint (Function.support f) (Function.support g) \u2192\n          P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\n\u22a2 P f\n\ncase h_add_ae\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n  \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n    P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n    StronglyMeasurable f \u2192\n      StronglyMeasurable g \u2192\n        Disjoint (Function.support f) (Function.support g) \u2192\n          P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf : { x // x \u2208 Lp F p }\nhf : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\n\u22a2 \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n    AEStronglyMeasurable' m f \u03bc \u2192\n      AEStronglyMeasurable' m g \u03bc \u2192\n        Disjoint (Function.support f) (Function.support g) \u2192\n          P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)", "state_after": "case h_add_ae\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n  \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n    P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n    StronglyMeasurable f \u2192\n      StronglyMeasurable g \u2192\n        Disjoint (Function.support f) (Function.support g) \u2192\n          P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf : { x // x \u2208 Lp F p }\nhf : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\n\u22a2 \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n    AEStronglyMeasurable' m f \u03bc \u2192\n      AEStronglyMeasurable' m g \u03bc \u2192\n        Disjoint (Function.support f) (Function.support g) \u2192\n          P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)"}, {"tactic": "intro f g hf hg hfm hgm h_disj hPf hPg", "annotated_tactic": ["intro f g hf hg hfm hgm h_disj hPf hPg", []], "state_before": "case h_add_ae\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n  \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n    P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n    StronglyMeasurable f \u2192\n      StronglyMeasurable g \u2192\n        Disjoint (Function.support f) (Function.support g) \u2192\n          P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf : { x // x \u2208 Lp F p }\nhf : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\n\u22a2 \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n    AEStronglyMeasurable' m f \u03bc \u2192\n      AEStronglyMeasurable' m g \u03bc \u2192\n        Disjoint (Function.support f) (Function.support g) \u2192\n          P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)", "state_after": "case h_add_ae\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n  \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n    P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n    StronglyMeasurable f \u2192\n      StronglyMeasurable g \u2192\n        Disjoint (Function.support f) (Function.support g) \u2192\n          P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nh_disj : Disjoint (Function.support f) (Function.support g)\nhPf : P (Mem\u2112p.toLp f hf)\nhPg : P (Mem\u2112p.toLp g hg)\n\u22a2 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)"}, {"tactic": "let s_f : Set \u03b1 := Function.support (hfm.mk f)", "annotated_tactic": ["let s_f : <a>Set</a> \u03b1 := <a>Function.support</a> (hfm.mk f)", [{"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}, {"full_name": "Function.support", "def_path": "Mathlib/Algebra/Support.lean", "def_pos": [37, 5], "def_end_pos": [37, 12]}]], "state_before": "case h_add_ae\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n  \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n    P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n    StronglyMeasurable f \u2192\n      StronglyMeasurable g \u2192\n        Disjoint (Function.support f) (Function.support g) \u2192\n          P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nh_disj : Disjoint (Function.support f) (Function.support g)\nhPf : P (Mem\u2112p.toLp f hf)\nhPg : P (Mem\u2112p.toLp g hg)\n\u22a2 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)", "state_after": "case h_add_ae\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n  \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n    P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n    StronglyMeasurable f \u2192\n      StronglyMeasurable g \u2192\n        Disjoint (Function.support f) (Function.support g) \u2192\n          P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nh_disj : Disjoint (Function.support f) (Function.support g)\nhPf : P (Mem\u2112p.toLp f hf)\nhPg : P (Mem\u2112p.toLp g hg)\ns_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm)\n\u22a2 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)"}, {"tactic": "have hs_f : MeasurableSet[m] s_f := hfm.stronglyMeasurable_mk.measurableSet_support", "annotated_tactic": ["have hs_f : MeasurableSet[m] s_f := hfm.stronglyMeasurable_mk.measurableSet_support", []], "state_before": "case h_add_ae\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n  \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n    P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n    StronglyMeasurable f \u2192\n      StronglyMeasurable g \u2192\n        Disjoint (Function.support f) (Function.support g) \u2192\n          P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nh_disj : Disjoint (Function.support f) (Function.support g)\nhPf : P (Mem\u2112p.toLp f hf)\nhPg : P (Mem\u2112p.toLp g hg)\ns_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm)\n\u22a2 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)", "state_after": "case h_add_ae\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n  \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n    P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n    StronglyMeasurable f \u2192\n      StronglyMeasurable g \u2192\n        Disjoint (Function.support f) (Function.support g) \u2192\n          P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nh_disj : Disjoint (Function.support f) (Function.support g)\nhPf : P (Mem\u2112p.toLp f hf)\nhPg : P (Mem\u2112p.toLp g hg)\ns_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm)\nhs_f : MeasurableSet s_f\n\u22a2 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)"}, {"tactic": "have hs_f_eq : s_f =\u1d50[\u03bc] Function.support f := hfm.ae_eq_mk.symm.support", "annotated_tactic": ["have hs_f_eq : s_f =\u1d50[\u03bc] <a>Function.support</a> f := hfm.ae_eq_mk.symm.support", [{"full_name": "Function.support", "def_path": "Mathlib/Algebra/Support.lean", "def_pos": [37, 5], "def_end_pos": [37, 12]}]], "state_before": "case h_add_ae\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n  \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n    P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n    StronglyMeasurable f \u2192\n      StronglyMeasurable g \u2192\n        Disjoint (Function.support f) (Function.support g) \u2192\n          P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nh_disj : Disjoint (Function.support f) (Function.support g)\nhPf : P (Mem\u2112p.toLp f hf)\nhPg : P (Mem\u2112p.toLp g hg)\ns_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm)\nhs_f : MeasurableSet s_f\n\u22a2 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)", "state_after": "case h_add_ae\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n  \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n    P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n    StronglyMeasurable f \u2192\n      StronglyMeasurable g \u2192\n        Disjoint (Function.support f) (Function.support g) \u2192\n          P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nh_disj : Disjoint (Function.support f) (Function.support g)\nhPf : P (Mem\u2112p.toLp f hf)\nhPg : P (Mem\u2112p.toLp g hg)\ns_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm)\nhs_f : MeasurableSet s_f\nhs_f_eq : s_f =\u1d50[\u03bc] Function.support f\n\u22a2 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)"}, {"tactic": "let s_g : Set \u03b1 := Function.support (hgm.mk g)", "annotated_tactic": ["let s_g : <a>Set</a> \u03b1 := <a>Function.support</a> (hgm.mk g)", [{"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}, {"full_name": "Function.support", "def_path": "Mathlib/Algebra/Support.lean", "def_pos": [37, 5], "def_end_pos": [37, 12]}]], "state_before": "case h_add_ae\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n  \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n    P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n    StronglyMeasurable f \u2192\n      StronglyMeasurable g \u2192\n        Disjoint (Function.support f) (Function.support g) \u2192\n          P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nh_disj : Disjoint (Function.support f) (Function.support g)\nhPf : P (Mem\u2112p.toLp f hf)\nhPg : P (Mem\u2112p.toLp g hg)\ns_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm)\nhs_f : MeasurableSet s_f\nhs_f_eq : s_f =\u1d50[\u03bc] Function.support f\n\u22a2 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)", "state_after": "case h_add_ae\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n  \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n    P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n    StronglyMeasurable f \u2192\n      StronglyMeasurable g \u2192\n        Disjoint (Function.support f) (Function.support g) \u2192\n          P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nh_disj : Disjoint (Function.support f) (Function.support g)\nhPf : P (Mem\u2112p.toLp f hf)\nhPg : P (Mem\u2112p.toLp g hg)\ns_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm)\nhs_f : MeasurableSet s_f\nhs_f_eq : s_f =\u1d50[\u03bc] Function.support f\ns_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm)\n\u22a2 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)"}, {"tactic": "have hs_g : MeasurableSet[m] s_g := hgm.stronglyMeasurable_mk.measurableSet_support", "annotated_tactic": ["have hs_g : MeasurableSet[m] s_g := hgm.stronglyMeasurable_mk.measurableSet_support", []], "state_before": "case h_add_ae\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n  \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n    P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n    StronglyMeasurable f \u2192\n      StronglyMeasurable g \u2192\n        Disjoint (Function.support f) (Function.support g) \u2192\n          P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nh_disj : Disjoint (Function.support f) (Function.support g)\nhPf : P (Mem\u2112p.toLp f hf)\nhPg : P (Mem\u2112p.toLp g hg)\ns_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm)\nhs_f : MeasurableSet s_f\nhs_f_eq : s_f =\u1d50[\u03bc] Function.support f\ns_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm)\n\u22a2 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)", "state_after": "case h_add_ae\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n  \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n    P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n    StronglyMeasurable f \u2192\n      StronglyMeasurable g \u2192\n        Disjoint (Function.support f) (Function.support g) \u2192\n          P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nh_disj : Disjoint (Function.support f) (Function.support g)\nhPf : P (Mem\u2112p.toLp f hf)\nhPg : P (Mem\u2112p.toLp g hg)\ns_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm)\nhs_f : MeasurableSet s_f\nhs_f_eq : s_f =\u1d50[\u03bc] Function.support f\ns_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm)\nhs_g : MeasurableSet s_g\n\u22a2 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)"}, {"tactic": "have hs_g_eq : s_g =\u1d50[\u03bc] Function.support g := hgm.ae_eq_mk.symm.support", "annotated_tactic": ["have hs_g_eq : s_g =\u1d50[\u03bc] <a>Function.support</a> g := hgm.ae_eq_mk.symm.support", [{"full_name": "Function.support", "def_path": "Mathlib/Algebra/Support.lean", "def_pos": [37, 5], "def_end_pos": [37, 12]}]], "state_before": "case h_add_ae\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n  \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n    P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n    StronglyMeasurable f \u2192\n      StronglyMeasurable g \u2192\n        Disjoint (Function.support f) (Function.support g) \u2192\n          P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nh_disj : Disjoint (Function.support f) (Function.support g)\nhPf : P (Mem\u2112p.toLp f hf)\nhPg : P (Mem\u2112p.toLp g hg)\ns_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm)\nhs_f : MeasurableSet s_f\nhs_f_eq : s_f =\u1d50[\u03bc] Function.support f\ns_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm)\nhs_g : MeasurableSet s_g\n\u22a2 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)", "state_after": "case h_add_ae\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n  \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n    P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n    StronglyMeasurable f \u2192\n      StronglyMeasurable g \u2192\n        Disjoint (Function.support f) (Function.support g) \u2192\n          P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nh_disj : Disjoint (Function.support f) (Function.support g)\nhPf : P (Mem\u2112p.toLp f hf)\nhPg : P (Mem\u2112p.toLp g hg)\ns_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm)\nhs_f : MeasurableSet s_f\nhs_f_eq : s_f =\u1d50[\u03bc] Function.support f\ns_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm)\nhs_g : MeasurableSet s_g\nhs_g_eq : s_g =\u1d50[\u03bc] Function.support g\n\u22a2 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)"}, {"tactic": "have h_inter_empty : (s_f \u2229 s_g : Set \u03b1) =\u1d50[\u03bc] (\u2205 : Set \u03b1) := by\n  refine' (hs_f_eq.inter hs_g_eq).trans _\n  suffices Function.support f \u2229 Function.support g = \u2205 by rw [this]\n  exact Set.disjoint_iff_inter_eq_empty.mp h_disj", "annotated_tactic": ["have h_inter_empty : (s_f \u2229 s_g : <a>Set</a> \u03b1) =\u1d50[\u03bc] (\u2205 : <a>Set</a> \u03b1) := by\n    refine' (hs_f_eq.inter hs_g_eq).<a>trans</a> _\n    suffices <a>Function.support</a> f \u2229 <a>Function.support</a> g = \u2205 by rw [this]\n    exact Set.disjoint_iff_inter_eq_empty.mp h_disj", [{"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}, {"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}, {"full_name": "Filter.EventuallyEq.trans", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1503, 9], "def_end_pos": [1503, 27]}, {"full_name": "Function.support", "def_path": "Mathlib/Algebra/Support.lean", "def_pos": [37, 5], "def_end_pos": [37, 12]}, {"full_name": "Function.support", "def_path": "Mathlib/Algebra/Support.lean", "def_pos": [37, 5], "def_end_pos": [37, 12]}]], "state_before": "case h_add_ae\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n  \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n    P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n    StronglyMeasurable f \u2192\n      StronglyMeasurable g \u2192\n        Disjoint (Function.support f) (Function.support g) \u2192\n          P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nh_disj : Disjoint (Function.support f) (Function.support g)\nhPf : P (Mem\u2112p.toLp f hf)\nhPg : P (Mem\u2112p.toLp g hg)\ns_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm)\nhs_f : MeasurableSet s_f\nhs_f_eq : s_f =\u1d50[\u03bc] Function.support f\ns_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm)\nhs_g : MeasurableSet s_g\nhs_g_eq : s_g =\u1d50[\u03bc] Function.support g\n\u22a2 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)", "state_after": "case h_add_ae\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n  \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n    P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n    StronglyMeasurable f \u2192\n      StronglyMeasurable g \u2192\n        Disjoint (Function.support f) (Function.support g) \u2192\n          P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nh_disj : Disjoint (Function.support f) (Function.support g)\nhPf : P (Mem\u2112p.toLp f hf)\nhPg : P (Mem\u2112p.toLp g hg)\ns_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm)\nhs_f : MeasurableSet s_f\nhs_f_eq : s_f =\u1d50[\u03bc] Function.support f\ns_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm)\nhs_g : MeasurableSet s_g\nhs_g_eq : s_g =\u1d50[\u03bc] Function.support g\nh_inter_empty : s_f \u2229 s_g =\u1d50[\u03bc] \u2205\n\u22a2 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)"}, {"tactic": "let f' := (s_f \\ s_g).indicator (hfm.mk f)", "annotated_tactic": ["let f' := (s_f \\ s_g).<a>indicator</a> (hfm.mk f)", [{"full_name": "Set.indicator", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [46, 3], "def_end_pos": [46, 14]}]], "state_before": "case h_add_ae\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n  \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n    P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n    StronglyMeasurable f \u2192\n      StronglyMeasurable g \u2192\n        Disjoint (Function.support f) (Function.support g) \u2192\n          P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nh_disj : Disjoint (Function.support f) (Function.support g)\nhPf : P (Mem\u2112p.toLp f hf)\nhPg : P (Mem\u2112p.toLp g hg)\ns_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm)\nhs_f : MeasurableSet s_f\nhs_f_eq : s_f =\u1d50[\u03bc] Function.support f\ns_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm)\nhs_g : MeasurableSet s_g\nhs_g_eq : s_g =\u1d50[\u03bc] Function.support g\nh_inter_empty : s_f \u2229 s_g =\u1d50[\u03bc] \u2205\n\u22a2 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)", "state_after": "case h_add_ae\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n  \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n    P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n    StronglyMeasurable f \u2192\n      StronglyMeasurable g \u2192\n        Disjoint (Function.support f) (Function.support g) \u2192\n          P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nh_disj : Disjoint (Function.support f) (Function.support g)\nhPf : P (Mem\u2112p.toLp f hf)\nhPg : P (Mem\u2112p.toLp g hg)\ns_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm)\nhs_f : MeasurableSet s_f\nhs_f_eq : s_f =\u1d50[\u03bc] Function.support f\ns_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm)\nhs_g : MeasurableSet s_g\nhs_g_eq : s_g =\u1d50[\u03bc] Function.support g\nh_inter_empty : s_f \u2229 s_g =\u1d50[\u03bc] \u2205\nf' : \u03b1 \u2192 F := Set.indicator (s_f \\ s_g) (AEStronglyMeasurable'.mk f hfm)\n\u22a2 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)"}, {"tactic": "have hff' : f =\u1d50[\u03bc] f' := by\n  have : s_f \\ s_g =\u1d50[\u03bc] s_f := by\n    rw [\u2190 Set.diff_inter_self_eq_diff, Set.inter_comm]\n    refine' ((ae_eq_refl s_f).diff h_inter_empty).trans _\n    rw [Set.diff_empty]\n  refine' ((indicator_ae_eq_of_ae_eq_set this).trans _).symm\n  rw [Set.indicator_support]\n  exact hfm.ae_eq_mk.symm", "annotated_tactic": ["have hff' : f =\u1d50[\u03bc] f' := by\n    have : s_f \\ s_g =\u1d50[\u03bc] s_f := by\n      rw [\u2190 <a>Set.diff_inter_self_eq_diff</a>, <a>Set.inter_comm</a>]\n      refine' ((<a>ae_eq_refl</a> s_f).<a>diff</a> h_inter_empty).<a>trans</a> _\n      rw [<a>Set.diff_empty</a>]\n    refine' ((<a>indicator_ae_eq_of_ae_eq_set</a> this).<a>trans</a> _).<a>symm</a>\n    rw [<a>Set.indicator_support</a>]\n    exact hfm.ae_eq_mk.symm", [{"full_name": "Set.diff_inter_self_eq_diff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [2058, 9], "def_end_pos": [2058, 32]}, {"full_name": "Set.inter_comm", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [940, 9], "def_end_pos": [940, 19]}, {"full_name": "MeasureTheory.ae_eq_refl", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [436, 9], "def_end_pos": [436, 19]}, {"full_name": "Filter.EventuallyEq.diff", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1595, 9], "def_end_pos": [1595, 26]}, {"full_name": "Filter.EventuallyEq.trans", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1503, 9], "def_end_pos": [1503, 27]}, {"full_name": "Set.diff_empty", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1930, 9], "def_end_pos": [1930, 19]}, {"full_name": "indicator_ae_eq_of_ae_eq_set", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [4512, 9], "def_end_pos": [4512, 37]}, {"full_name": "Filter.EventuallyEq.trans", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1503, 9], "def_end_pos": [1503, 27]}, {"full_name": "Filter.EventuallyEq.symm", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1498, 9], "def_end_pos": [1498, 26]}, {"full_name": "Set.indicator_support", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [163, 3], "def_end_pos": [163, 14]}]], "state_before": "case h_add_ae\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n  \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n    P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n    StronglyMeasurable f \u2192\n      StronglyMeasurable g \u2192\n        Disjoint (Function.support f) (Function.support g) \u2192\n          P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nh_disj : Disjoint (Function.support f) (Function.support g)\nhPf : P (Mem\u2112p.toLp f hf)\nhPg : P (Mem\u2112p.toLp g hg)\ns_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm)\nhs_f : MeasurableSet s_f\nhs_f_eq : s_f =\u1d50[\u03bc] Function.support f\ns_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm)\nhs_g : MeasurableSet s_g\nhs_g_eq : s_g =\u1d50[\u03bc] Function.support g\nh_inter_empty : s_f \u2229 s_g =\u1d50[\u03bc] \u2205\nf' : \u03b1 \u2192 F := Set.indicator (s_f \\ s_g) (AEStronglyMeasurable'.mk f hfm)\n\u22a2 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)", "state_after": "case h_add_ae\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n  \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n    P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n    StronglyMeasurable f \u2192\n      StronglyMeasurable g \u2192\n        Disjoint (Function.support f) (Function.support g) \u2192\n          P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nh_disj : Disjoint (Function.support f) (Function.support g)\nhPf : P (Mem\u2112p.toLp f hf)\nhPg : P (Mem\u2112p.toLp g hg)\ns_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm)\nhs_f : MeasurableSet s_f\nhs_f_eq : s_f =\u1d50[\u03bc] Function.support f\ns_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm)\nhs_g : MeasurableSet s_g\nhs_g_eq : s_g =\u1d50[\u03bc] Function.support g\nh_inter_empty : s_f \u2229 s_g =\u1d50[\u03bc] \u2205\nf' : \u03b1 \u2192 F := Set.indicator (s_f \\ s_g) (AEStronglyMeasurable'.mk f hfm)\nhff' : f =\u1d50[\u03bc] f'\n\u22a2 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)"}, {"tactic": "have hf'_meas : StronglyMeasurable[m] f' := hfm.stronglyMeasurable_mk.indicator (hs_f.diff hs_g)", "annotated_tactic": ["have hf'_meas : StronglyMeasurable[m] f' := hfm.stronglyMeasurable_mk.indicator (hs_f.diff hs_g)", []], "state_before": "case h_add_ae\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n  \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n    P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n    StronglyMeasurable f \u2192\n      StronglyMeasurable g \u2192\n        Disjoint (Function.support f) (Function.support g) \u2192\n          P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nh_disj : Disjoint (Function.support f) (Function.support g)\nhPf : P (Mem\u2112p.toLp f hf)\nhPg : P (Mem\u2112p.toLp g hg)\ns_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm)\nhs_f : MeasurableSet s_f\nhs_f_eq : s_f =\u1d50[\u03bc] Function.support f\ns_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm)\nhs_g : MeasurableSet s_g\nhs_g_eq : s_g =\u1d50[\u03bc] Function.support g\nh_inter_empty : s_f \u2229 s_g =\u1d50[\u03bc] \u2205\nf' : \u03b1 \u2192 F := Set.indicator (s_f \\ s_g) (AEStronglyMeasurable'.mk f hfm)\nhff' : f =\u1d50[\u03bc] f'\n\u22a2 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)", "state_after": "case h_add_ae\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n  \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n    P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n    StronglyMeasurable f \u2192\n      StronglyMeasurable g \u2192\n        Disjoint (Function.support f) (Function.support g) \u2192\n          P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nh_disj : Disjoint (Function.support f) (Function.support g)\nhPf : P (Mem\u2112p.toLp f hf)\nhPg : P (Mem\u2112p.toLp g hg)\ns_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm)\nhs_f : MeasurableSet s_f\nhs_f_eq : s_f =\u1d50[\u03bc] Function.support f\ns_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm)\nhs_g : MeasurableSet s_g\nhs_g_eq : s_g =\u1d50[\u03bc] Function.support g\nh_inter_empty : s_f \u2229 s_g =\u1d50[\u03bc] \u2205\nf' : \u03b1 \u2192 F := Set.indicator (s_f \\ s_g) (AEStronglyMeasurable'.mk f hfm)\nhff' : f =\u1d50[\u03bc] f'\nhf'_meas : StronglyMeasurable f'\n\u22a2 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)"}, {"tactic": "have hf'_Lp : Mem\u2112p f' p \u03bc := hf.ae_eq hff'", "annotated_tactic": ["have hf'_Lp : <a>Mem\u2112p</a> f' p \u03bc := hf.ae_eq hff'", [{"full_name": "MeasureTheory.Mem\u2112p", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [108, 5], "def_end_pos": [108, 10]}]], "state_before": "case h_add_ae\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n  \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n    P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n    StronglyMeasurable f \u2192\n      StronglyMeasurable g \u2192\n        Disjoint (Function.support f) (Function.support g) \u2192\n          P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nh_disj : Disjoint (Function.support f) (Function.support g)\nhPf : P (Mem\u2112p.toLp f hf)\nhPg : P (Mem\u2112p.toLp g hg)\ns_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm)\nhs_f : MeasurableSet s_f\nhs_f_eq : s_f =\u1d50[\u03bc] Function.support f\ns_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm)\nhs_g : MeasurableSet s_g\nhs_g_eq : s_g =\u1d50[\u03bc] Function.support g\nh_inter_empty : s_f \u2229 s_g =\u1d50[\u03bc] \u2205\nf' : \u03b1 \u2192 F := Set.indicator (s_f \\ s_g) (AEStronglyMeasurable'.mk f hfm)\nhff' : f =\u1d50[\u03bc] f'\nhf'_meas : StronglyMeasurable f'\n\u22a2 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)", "state_after": "case h_add_ae\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n  \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n    P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n    StronglyMeasurable f \u2192\n      StronglyMeasurable g \u2192\n        Disjoint (Function.support f) (Function.support g) \u2192\n          P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nh_disj : Disjoint (Function.support f) (Function.support g)\nhPf : P (Mem\u2112p.toLp f hf)\nhPg : P (Mem\u2112p.toLp g hg)\ns_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm)\nhs_f : MeasurableSet s_f\nhs_f_eq : s_f =\u1d50[\u03bc] Function.support f\ns_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm)\nhs_g : MeasurableSet s_g\nhs_g_eq : s_g =\u1d50[\u03bc] Function.support g\nh_inter_empty : s_f \u2229 s_g =\u1d50[\u03bc] \u2205\nf' : \u03b1 \u2192 F := Set.indicator (s_f \\ s_g) (AEStronglyMeasurable'.mk f hfm)\nhff' : f =\u1d50[\u03bc] f'\nhf'_meas : StronglyMeasurable f'\nhf'_Lp : Mem\u2112p f' p\n\u22a2 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)"}, {"tactic": "let g' := (s_g \\ s_f).indicator (hgm.mk g)", "annotated_tactic": ["let g' := (s_g \\ s_f).<a>indicator</a> (hgm.mk g)", [{"full_name": "Set.indicator", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [46, 3], "def_end_pos": [46, 14]}]], "state_before": "case h_add_ae\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n  \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n    P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n    StronglyMeasurable f \u2192\n      StronglyMeasurable g \u2192\n        Disjoint (Function.support f) (Function.support g) \u2192\n          P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nh_disj : Disjoint (Function.support f) (Function.support g)\nhPf : P (Mem\u2112p.toLp f hf)\nhPg : P (Mem\u2112p.toLp g hg)\ns_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm)\nhs_f : MeasurableSet s_f\nhs_f_eq : s_f =\u1d50[\u03bc] Function.support f\ns_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm)\nhs_g : MeasurableSet s_g\nhs_g_eq : s_g =\u1d50[\u03bc] Function.support g\nh_inter_empty : s_f \u2229 s_g =\u1d50[\u03bc] \u2205\nf' : \u03b1 \u2192 F := Set.indicator (s_f \\ s_g) (AEStronglyMeasurable'.mk f hfm)\nhff' : f =\u1d50[\u03bc] f'\nhf'_meas : StronglyMeasurable f'\nhf'_Lp : Mem\u2112p f' p\n\u22a2 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)", "state_after": "case h_add_ae\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n  \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n    P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n    StronglyMeasurable f \u2192\n      StronglyMeasurable g \u2192\n        Disjoint (Function.support f) (Function.support g) \u2192\n          P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nh_disj : Disjoint (Function.support f) (Function.support g)\nhPf : P (Mem\u2112p.toLp f hf)\nhPg : P (Mem\u2112p.toLp g hg)\ns_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm)\nhs_f : MeasurableSet s_f\nhs_f_eq : s_f =\u1d50[\u03bc] Function.support f\ns_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm)\nhs_g : MeasurableSet s_g\nhs_g_eq : s_g =\u1d50[\u03bc] Function.support g\nh_inter_empty : s_f \u2229 s_g =\u1d50[\u03bc] \u2205\nf' : \u03b1 \u2192 F := Set.indicator (s_f \\ s_g) (AEStronglyMeasurable'.mk f hfm)\nhff' : f =\u1d50[\u03bc] f'\nhf'_meas : StronglyMeasurable f'\nhf'_Lp : Mem\u2112p f' p\ng' : \u03b1 \u2192 F := Set.indicator (s_g \\ s_f) (AEStronglyMeasurable'.mk g hgm)\n\u22a2 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)"}, {"tactic": "have hgg' : g =\u1d50[\u03bc] g' := by\n  have : s_g \\ s_f =\u1d50[\u03bc] s_g := by\n    rw [\u2190 Set.diff_inter_self_eq_diff]\n    refine' ((ae_eq_refl s_g).diff h_inter_empty).trans _\n    rw [Set.diff_empty]\n  refine' ((indicator_ae_eq_of_ae_eq_set this).trans _).symm\n  rw [Set.indicator_support]\n  exact hgm.ae_eq_mk.symm", "annotated_tactic": ["have hgg' : g =\u1d50[\u03bc] g' := by\n    have : s_g \\ s_f =\u1d50[\u03bc] s_g := by\n      rw [\u2190 <a>Set.diff_inter_self_eq_diff</a>]\n      refine' ((<a>ae_eq_refl</a> s_g).<a>diff</a> h_inter_empty).<a>trans</a> _\n      rw [<a>Set.diff_empty</a>]\n    refine' ((<a>indicator_ae_eq_of_ae_eq_set</a> this).<a>trans</a> _).<a>symm</a>\n    rw [<a>Set.indicator_support</a>]\n    exact hgm.ae_eq_mk.symm", [{"full_name": "Set.diff_inter_self_eq_diff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [2058, 9], "def_end_pos": [2058, 32]}, {"full_name": "MeasureTheory.ae_eq_refl", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [436, 9], "def_end_pos": [436, 19]}, {"full_name": "Filter.EventuallyEq.diff", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1595, 9], "def_end_pos": [1595, 26]}, {"full_name": "Filter.EventuallyEq.trans", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1503, 9], "def_end_pos": [1503, 27]}, {"full_name": "Set.diff_empty", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1930, 9], "def_end_pos": [1930, 19]}, {"full_name": "indicator_ae_eq_of_ae_eq_set", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [4512, 9], "def_end_pos": [4512, 37]}, {"full_name": "Filter.EventuallyEq.trans", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1503, 9], "def_end_pos": [1503, 27]}, {"full_name": "Filter.EventuallyEq.symm", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1498, 9], "def_end_pos": [1498, 26]}, {"full_name": "Set.indicator_support", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [163, 3], "def_end_pos": [163, 14]}]], "state_before": "case h_add_ae\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n  \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n    P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n    StronglyMeasurable f \u2192\n      StronglyMeasurable g \u2192\n        Disjoint (Function.support f) (Function.support g) \u2192\n          P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nh_disj : Disjoint (Function.support f) (Function.support g)\nhPf : P (Mem\u2112p.toLp f hf)\nhPg : P (Mem\u2112p.toLp g hg)\ns_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm)\nhs_f : MeasurableSet s_f\nhs_f_eq : s_f =\u1d50[\u03bc] Function.support f\ns_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm)\nhs_g : MeasurableSet s_g\nhs_g_eq : s_g =\u1d50[\u03bc] Function.support g\nh_inter_empty : s_f \u2229 s_g =\u1d50[\u03bc] \u2205\nf' : \u03b1 \u2192 F := Set.indicator (s_f \\ s_g) (AEStronglyMeasurable'.mk f hfm)\nhff' : f =\u1d50[\u03bc] f'\nhf'_meas : StronglyMeasurable f'\nhf'_Lp : Mem\u2112p f' p\ng' : \u03b1 \u2192 F := Set.indicator (s_g \\ s_f) (AEStronglyMeasurable'.mk g hgm)\n\u22a2 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)", "state_after": "case h_add_ae\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n  \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n    P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n    StronglyMeasurable f \u2192\n      StronglyMeasurable g \u2192\n        Disjoint (Function.support f) (Function.support g) \u2192\n          P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nh_disj : Disjoint (Function.support f) (Function.support g)\nhPf : P (Mem\u2112p.toLp f hf)\nhPg : P (Mem\u2112p.toLp g hg)\ns_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm)\nhs_f : MeasurableSet s_f\nhs_f_eq : s_f =\u1d50[\u03bc] Function.support f\ns_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm)\nhs_g : MeasurableSet s_g\nhs_g_eq : s_g =\u1d50[\u03bc] Function.support g\nh_inter_empty : s_f \u2229 s_g =\u1d50[\u03bc] \u2205\nf' : \u03b1 \u2192 F := Set.indicator (s_f \\ s_g) (AEStronglyMeasurable'.mk f hfm)\nhff' : f =\u1d50[\u03bc] f'\nhf'_meas : StronglyMeasurable f'\nhf'_Lp : Mem\u2112p f' p\ng' : \u03b1 \u2192 F := Set.indicator (s_g \\ s_f) (AEStronglyMeasurable'.mk g hgm)\nhgg' : g =\u1d50[\u03bc] g'\n\u22a2 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)"}, {"tactic": "have hg'_meas : StronglyMeasurable[m] g' := hgm.stronglyMeasurable_mk.indicator (hs_g.diff hs_f)", "annotated_tactic": ["have hg'_meas : StronglyMeasurable[m] g' := hgm.stronglyMeasurable_mk.indicator (hs_g.diff hs_f)", []], "state_before": "case h_add_ae\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n  \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n    P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n    StronglyMeasurable f \u2192\n      StronglyMeasurable g \u2192\n        Disjoint (Function.support f) (Function.support g) \u2192\n          P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nh_disj : Disjoint (Function.support f) (Function.support g)\nhPf : P (Mem\u2112p.toLp f hf)\nhPg : P (Mem\u2112p.toLp g hg)\ns_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm)\nhs_f : MeasurableSet s_f\nhs_f_eq : s_f =\u1d50[\u03bc] Function.support f\ns_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm)\nhs_g : MeasurableSet s_g\nhs_g_eq : s_g =\u1d50[\u03bc] Function.support g\nh_inter_empty : s_f \u2229 s_g =\u1d50[\u03bc] \u2205\nf' : \u03b1 \u2192 F := Set.indicator (s_f \\ s_g) (AEStronglyMeasurable'.mk f hfm)\nhff' : f =\u1d50[\u03bc] f'\nhf'_meas : StronglyMeasurable f'\nhf'_Lp : Mem\u2112p f' p\ng' : \u03b1 \u2192 F := Set.indicator (s_g \\ s_f) (AEStronglyMeasurable'.mk g hgm)\nhgg' : g =\u1d50[\u03bc] g'\n\u22a2 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)", "state_after": "case h_add_ae\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n  \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n    P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n    StronglyMeasurable f \u2192\n      StronglyMeasurable g \u2192\n        Disjoint (Function.support f) (Function.support g) \u2192\n          P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nh_disj : Disjoint (Function.support f) (Function.support g)\nhPf : P (Mem\u2112p.toLp f hf)\nhPg : P (Mem\u2112p.toLp g hg)\ns_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm)\nhs_f : MeasurableSet s_f\nhs_f_eq : s_f =\u1d50[\u03bc] Function.support f\ns_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm)\nhs_g : MeasurableSet s_g\nhs_g_eq : s_g =\u1d50[\u03bc] Function.support g\nh_inter_empty : s_f \u2229 s_g =\u1d50[\u03bc] \u2205\nf' : \u03b1 \u2192 F := Set.indicator (s_f \\ s_g) (AEStronglyMeasurable'.mk f hfm)\nhff' : f =\u1d50[\u03bc] f'\nhf'_meas : StronglyMeasurable f'\nhf'_Lp : Mem\u2112p f' p\ng' : \u03b1 \u2192 F := Set.indicator (s_g \\ s_f) (AEStronglyMeasurable'.mk g hgm)\nhgg' : g =\u1d50[\u03bc] g'\nhg'_meas : StronglyMeasurable g'\n\u22a2 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)"}, {"tactic": "have hg'_Lp : Mem\u2112p g' p \u03bc := hg.ae_eq hgg'", "annotated_tactic": ["have hg'_Lp : <a>Mem\u2112p</a> g' p \u03bc := hg.ae_eq hgg'", [{"full_name": "MeasureTheory.Mem\u2112p", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [108, 5], "def_end_pos": [108, 10]}]], "state_before": "case h_add_ae\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n  \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n    P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n    StronglyMeasurable f \u2192\n      StronglyMeasurable g \u2192\n        Disjoint (Function.support f) (Function.support g) \u2192\n          P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nh_disj : Disjoint (Function.support f) (Function.support g)\nhPf : P (Mem\u2112p.toLp f hf)\nhPg : P (Mem\u2112p.toLp g hg)\ns_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm)\nhs_f : MeasurableSet s_f\nhs_f_eq : s_f =\u1d50[\u03bc] Function.support f\ns_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm)\nhs_g : MeasurableSet s_g\nhs_g_eq : s_g =\u1d50[\u03bc] Function.support g\nh_inter_empty : s_f \u2229 s_g =\u1d50[\u03bc] \u2205\nf' : \u03b1 \u2192 F := Set.indicator (s_f \\ s_g) (AEStronglyMeasurable'.mk f hfm)\nhff' : f =\u1d50[\u03bc] f'\nhf'_meas : StronglyMeasurable f'\nhf'_Lp : Mem\u2112p f' p\ng' : \u03b1 \u2192 F := Set.indicator (s_g \\ s_f) (AEStronglyMeasurable'.mk g hgm)\nhgg' : g =\u1d50[\u03bc] g'\nhg'_meas : StronglyMeasurable g'\n\u22a2 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)", "state_after": "case h_add_ae\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n  \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n    P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n    StronglyMeasurable f \u2192\n      StronglyMeasurable g \u2192\n        Disjoint (Function.support f) (Function.support g) \u2192\n          P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nh_disj : Disjoint (Function.support f) (Function.support g)\nhPf : P (Mem\u2112p.toLp f hf)\nhPg : P (Mem\u2112p.toLp g hg)\ns_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm)\nhs_f : MeasurableSet s_f\nhs_f_eq : s_f =\u1d50[\u03bc] Function.support f\ns_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm)\nhs_g : MeasurableSet s_g\nhs_g_eq : s_g =\u1d50[\u03bc] Function.support g\nh_inter_empty : s_f \u2229 s_g =\u1d50[\u03bc] \u2205\nf' : \u03b1 \u2192 F := Set.indicator (s_f \\ s_g) (AEStronglyMeasurable'.mk f hfm)\nhff' : f =\u1d50[\u03bc] f'\nhf'_meas : StronglyMeasurable f'\nhf'_Lp : Mem\u2112p f' p\ng' : \u03b1 \u2192 F := Set.indicator (s_g \\ s_f) (AEStronglyMeasurable'.mk g hgm)\nhgg' : g =\u1d50[\u03bc] g'\nhg'_meas : StronglyMeasurable g'\nhg'_Lp : Mem\u2112p g' p\n\u22a2 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)"}, {"tactic": "have h_disj : Disjoint (Function.support f') (Function.support g') :=\n  haveI : Disjoint (s_f \\ s_g) (s_g \\ s_f) := disjoint_sdiff_sdiff\n  this.mono Set.support_indicator_subset Set.support_indicator_subset", "annotated_tactic": ["have h_disj : <a>Disjoint</a> (<a>Function.support</a> f') (<a>Function.support</a> g') :=\n    haveI : <a>Disjoint</a> (s_f \\ s_g) (s_g \\ s_f) := <a>disjoint_sdiff_sdiff</a>\n    this.mono <a>Set.support_indicator_subset</a> <a>Set.support_indicator_subset</a>", [{"full_name": "Disjoint", "def_path": "Mathlib/Order/Disjoint.lean", "def_pos": [41, 5], "def_end_pos": [41, 13]}, {"full_name": "Function.support", "def_path": "Mathlib/Algebra/Support.lean", "def_pos": [37, 5], "def_end_pos": [37, 12]}, {"full_name": "Function.support", "def_path": "Mathlib/Algebra/Support.lean", "def_pos": [37, 5], "def_end_pos": [37, 12]}, {"full_name": "Disjoint", "def_path": "Mathlib/Order/Disjoint.lean", "def_pos": [41, 5], "def_end_pos": [41, 13]}, {"full_name": "disjoint_sdiff_sdiff", "def_path": "Mathlib/Order/BooleanAlgebra.lean", "def_pos": [163, 9], "def_end_pos": [163, 29]}, {"full_name": "Set.support_indicator_subset", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [157, 3], "def_end_pos": [157, 14]}, {"full_name": "Set.support_indicator_subset", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [157, 3], "def_end_pos": [157, 14]}]], "state_before": "case h_add_ae\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n  \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n    P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n    StronglyMeasurable f \u2192\n      StronglyMeasurable g \u2192\n        Disjoint (Function.support f) (Function.support g) \u2192\n          P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nh_disj : Disjoint (Function.support f) (Function.support g)\nhPf : P (Mem\u2112p.toLp f hf)\nhPg : P (Mem\u2112p.toLp g hg)\ns_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm)\nhs_f : MeasurableSet s_f\nhs_f_eq : s_f =\u1d50[\u03bc] Function.support f\ns_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm)\nhs_g : MeasurableSet s_g\nhs_g_eq : s_g =\u1d50[\u03bc] Function.support g\nh_inter_empty : s_f \u2229 s_g =\u1d50[\u03bc] \u2205\nf' : \u03b1 \u2192 F := Set.indicator (s_f \\ s_g) (AEStronglyMeasurable'.mk f hfm)\nhff' : f =\u1d50[\u03bc] f'\nhf'_meas : StronglyMeasurable f'\nhf'_Lp : Mem\u2112p f' p\ng' : \u03b1 \u2192 F := Set.indicator (s_g \\ s_f) (AEStronglyMeasurable'.mk g hgm)\nhgg' : g =\u1d50[\u03bc] g'\nhg'_meas : StronglyMeasurable g'\nhg'_Lp : Mem\u2112p g' p\n\u22a2 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)", "state_after": "case h_add_ae\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n  \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n    P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n    StronglyMeasurable f \u2192\n      StronglyMeasurable g \u2192\n        Disjoint (Function.support f) (Function.support g) \u2192\n          P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nh_disj\u271d : Disjoint (Function.support f) (Function.support g)\nhPf : P (Mem\u2112p.toLp f hf)\nhPg : P (Mem\u2112p.toLp g hg)\ns_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm)\nhs_f : MeasurableSet s_f\nhs_f_eq : s_f =\u1d50[\u03bc] Function.support f\ns_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm)\nhs_g : MeasurableSet s_g\nhs_g_eq : s_g =\u1d50[\u03bc] Function.support g\nh_inter_empty : s_f \u2229 s_g =\u1d50[\u03bc] \u2205\nf' : \u03b1 \u2192 F := Set.indicator (s_f \\ s_g) (AEStronglyMeasurable'.mk f hfm)\nhff' : f =\u1d50[\u03bc] f'\nhf'_meas : StronglyMeasurable f'\nhf'_Lp : Mem\u2112p f' p\ng' : \u03b1 \u2192 F := Set.indicator (s_g \\ s_f) (AEStronglyMeasurable'.mk g hgm)\nhgg' : g =\u1d50[\u03bc] g'\nhg'_meas : StronglyMeasurable g'\nhg'_Lp : Mem\u2112p g' p\nh_disj : Disjoint (Function.support f') (Function.support g')\n\u22a2 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)"}, {"tactic": "rw [\u2190 Mem\u2112p.toLp_congr hf'_Lp hf hff'.symm] at hPf \u22a2", "annotated_tactic": ["rw [\u2190 <a>Mem\u2112p.toLp_congr</a> hf'_Lp hf hff'.symm] at hPf \u22a2", [{"full_name": "MeasureTheory.Mem\u2112p.toLp_congr", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [123, 9], "def_end_pos": [123, 19]}]], "state_before": "case h_add_ae\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n  \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n    P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n    StronglyMeasurable f \u2192\n      StronglyMeasurable g \u2192\n        Disjoint (Function.support f) (Function.support g) \u2192\n          P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nh_disj\u271d : Disjoint (Function.support f) (Function.support g)\nhPf : P (Mem\u2112p.toLp f hf)\nhPg : P (Mem\u2112p.toLp g hg)\ns_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm)\nhs_f : MeasurableSet s_f\nhs_f_eq : s_f =\u1d50[\u03bc] Function.support f\ns_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm)\nhs_g : MeasurableSet s_g\nhs_g_eq : s_g =\u1d50[\u03bc] Function.support g\nh_inter_empty : s_f \u2229 s_g =\u1d50[\u03bc] \u2205\nf' : \u03b1 \u2192 F := Set.indicator (s_f \\ s_g) (AEStronglyMeasurable'.mk f hfm)\nhff' : f =\u1d50[\u03bc] f'\nhf'_meas : StronglyMeasurable f'\nhf'_Lp : Mem\u2112p f' p\ng' : \u03b1 \u2192 F := Set.indicator (s_g \\ s_f) (AEStronglyMeasurable'.mk g hgm)\nhgg' : g =\u1d50[\u03bc] g'\nhg'_meas : StronglyMeasurable g'\nhg'_Lp : Mem\u2112p g' p\nh_disj : Disjoint (Function.support f') (Function.support g')\n\u22a2 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)", "state_after": "case h_add_ae\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n  \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n    P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n    StronglyMeasurable f \u2192\n      StronglyMeasurable g \u2192\n        Disjoint (Function.support f) (Function.support g) \u2192\n          P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nh_disj\u271d : Disjoint (Function.support f) (Function.support g)\nhPg : P (Mem\u2112p.toLp g hg)\ns_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm)\nhs_f : MeasurableSet s_f\nhs_f_eq : s_f =\u1d50[\u03bc] Function.support f\ns_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm)\nhs_g : MeasurableSet s_g\nhs_g_eq : s_g =\u1d50[\u03bc] Function.support g\nh_inter_empty : s_f \u2229 s_g =\u1d50[\u03bc] \u2205\nf' : \u03b1 \u2192 F := Set.indicator (s_f \\ s_g) (AEStronglyMeasurable'.mk f hfm)\nhff' : f =\u1d50[\u03bc] f'\nhf'_meas : StronglyMeasurable f'\nhf'_Lp : Mem\u2112p f' p\nhPf : P (Mem\u2112p.toLp f' hf'_Lp)\ng' : \u03b1 \u2192 F := Set.indicator (s_g \\ s_f) (AEStronglyMeasurable'.mk g hgm)\nhgg' : g =\u1d50[\u03bc] g'\nhg'_meas : StronglyMeasurable g'\nhg'_Lp : Mem\u2112p g' p\nh_disj : Disjoint (Function.support f') (Function.support g')\n\u22a2 P (Mem\u2112p.toLp f' hf'_Lp + Mem\u2112p.toLp g hg)"}, {"tactic": "rw [\u2190 Mem\u2112p.toLp_congr hg'_Lp hg hgg'.symm] at hPg \u22a2", "annotated_tactic": ["rw [\u2190 <a>Mem\u2112p.toLp_congr</a> hg'_Lp hg hgg'.symm] at hPg \u22a2", [{"full_name": "MeasureTheory.Mem\u2112p.toLp_congr", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [123, 9], "def_end_pos": [123, 19]}]], "state_before": "case h_add_ae\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n  \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n    P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n    StronglyMeasurable f \u2192\n      StronglyMeasurable g \u2192\n        Disjoint (Function.support f) (Function.support g) \u2192\n          P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nh_disj\u271d : Disjoint (Function.support f) (Function.support g)\nhPg : P (Mem\u2112p.toLp g hg)\ns_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm)\nhs_f : MeasurableSet s_f\nhs_f_eq : s_f =\u1d50[\u03bc] Function.support f\ns_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm)\nhs_g : MeasurableSet s_g\nhs_g_eq : s_g =\u1d50[\u03bc] Function.support g\nh_inter_empty : s_f \u2229 s_g =\u1d50[\u03bc] \u2205\nf' : \u03b1 \u2192 F := Set.indicator (s_f \\ s_g) (AEStronglyMeasurable'.mk f hfm)\nhff' : f =\u1d50[\u03bc] f'\nhf'_meas : StronglyMeasurable f'\nhf'_Lp : Mem\u2112p f' p\nhPf : P (Mem\u2112p.toLp f' hf'_Lp)\ng' : \u03b1 \u2192 F := Set.indicator (s_g \\ s_f) (AEStronglyMeasurable'.mk g hgm)\nhgg' : g =\u1d50[\u03bc] g'\nhg'_meas : StronglyMeasurable g'\nhg'_Lp : Mem\u2112p g' p\nh_disj : Disjoint (Function.support f') (Function.support g')\n\u22a2 P (Mem\u2112p.toLp f' hf'_Lp + Mem\u2112p.toLp g hg)", "state_after": "case h_add_ae\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n  \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n    P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n    StronglyMeasurable f \u2192\n      StronglyMeasurable g \u2192\n        Disjoint (Function.support f) (Function.support g) \u2192\n          P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nh_disj\u271d : Disjoint (Function.support f) (Function.support g)\ns_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm)\nhs_f : MeasurableSet s_f\nhs_f_eq : s_f =\u1d50[\u03bc] Function.support f\ns_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm)\nhs_g : MeasurableSet s_g\nhs_g_eq : s_g =\u1d50[\u03bc] Function.support g\nh_inter_empty : s_f \u2229 s_g =\u1d50[\u03bc] \u2205\nf' : \u03b1 \u2192 F := Set.indicator (s_f \\ s_g) (AEStronglyMeasurable'.mk f hfm)\nhff' : f =\u1d50[\u03bc] f'\nhf'_meas : StronglyMeasurable f'\nhf'_Lp : Mem\u2112p f' p\nhPf : P (Mem\u2112p.toLp f' hf'_Lp)\ng' : \u03b1 \u2192 F := Set.indicator (s_g \\ s_f) (AEStronglyMeasurable'.mk g hgm)\nhgg' : g =\u1d50[\u03bc] g'\nhg'_meas : StronglyMeasurable g'\nhg'_Lp : Mem\u2112p g' p\nhPg : P (Mem\u2112p.toLp g' hg'_Lp)\nh_disj : Disjoint (Function.support f') (Function.support g')\n\u22a2 P (Mem\u2112p.toLp f' hf'_Lp + Mem\u2112p.toLp g' hg'_Lp)"}, {"tactic": "exact h_add hf'_Lp hg'_Lp hf'_meas hg'_meas h_disj hPf hPg", "annotated_tactic": ["exact h_add hf'_Lp hg'_Lp hf'_meas hg'_meas h_disj hPf hPg", []], "state_before": "case h_add_ae\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n  \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n    P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n    StronglyMeasurable f \u2192\n      StronglyMeasurable g \u2192\n        Disjoint (Function.support f) (Function.support g) \u2192\n          P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nh_disj\u271d : Disjoint (Function.support f) (Function.support g)\ns_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm)\nhs_f : MeasurableSet s_f\nhs_f_eq : s_f =\u1d50[\u03bc] Function.support f\ns_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm)\nhs_g : MeasurableSet s_g\nhs_g_eq : s_g =\u1d50[\u03bc] Function.support g\nh_inter_empty : s_f \u2229 s_g =\u1d50[\u03bc] \u2205\nf' : \u03b1 \u2192 F := Set.indicator (s_f \\ s_g) (AEStronglyMeasurable'.mk f hfm)\nhff' : f =\u1d50[\u03bc] f'\nhf'_meas : StronglyMeasurable f'\nhf'_Lp : Mem\u2112p f' p\nhPf : P (Mem\u2112p.toLp f' hf'_Lp)\ng' : \u03b1 \u2192 F := Set.indicator (s_g \\ s_f) (AEStronglyMeasurable'.mk g hgm)\nhgg' : g =\u1d50[\u03bc] g'\nhg'_meas : StronglyMeasurable g'\nhg'_Lp : Mem\u2112p g' p\nhPg : P (Mem\u2112p.toLp g' hg'_Lp)\nh_disj : Disjoint (Function.support f') (Function.support g')\n\u22a2 P (Mem\u2112p.toLp f' hf'_Lp + Mem\u2112p.toLp g' hg'_Lp)", "state_after": "no goals"}, {"tactic": "refine' (hs_f_eq.inter hs_g_eq).trans _", "annotated_tactic": ["refine' (hs_f_eq.inter hs_g_eq).<a>trans</a> _", [{"full_name": "Filter.EventuallyEq.trans", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1503, 9], "def_end_pos": [1503, 27]}]], "state_before": "\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n  \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n    P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n    StronglyMeasurable f \u2192\n      StronglyMeasurable g \u2192\n        Disjoint (Function.support f) (Function.support g) \u2192\n          P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nh_disj : Disjoint (Function.support f) (Function.support g)\nhPf : P (Mem\u2112p.toLp f hf)\nhPg : P (Mem\u2112p.toLp g hg)\ns_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm)\nhs_f : MeasurableSet s_f\nhs_f_eq : s_f =\u1d50[\u03bc] Function.support f\ns_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm)\nhs_g : MeasurableSet s_g\nhs_g_eq : s_g =\u1d50[\u03bc] Function.support g\n\u22a2 s_f \u2229 s_g =\u1d50[\u03bc] \u2205", "state_after": "\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n  \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n    P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n    StronglyMeasurable f \u2192\n      StronglyMeasurable g \u2192\n        Disjoint (Function.support f) (Function.support g) \u2192\n          P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nh_disj : Disjoint (Function.support f) (Function.support g)\nhPf : P (Mem\u2112p.toLp f hf)\nhPg : P (Mem\u2112p.toLp g hg)\ns_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm)\nhs_f : MeasurableSet s_f\nhs_f_eq : s_f =\u1d50[\u03bc] Function.support f\ns_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm)\nhs_g : MeasurableSet s_g\nhs_g_eq : s_g =\u1d50[\u03bc] Function.support g\n\u22a2 Function.support f \u2229 Function.support g =\u1d50[\u03bc] \u2205"}, {"tactic": "suffices Function.support f \u2229 Function.support g = \u2205 by rw [this]", "annotated_tactic": ["suffices <a>Function.support</a> f \u2229 <a>Function.support</a> g = \u2205 by rw [this]", [{"full_name": "Function.support", "def_path": "Mathlib/Algebra/Support.lean", "def_pos": [37, 5], "def_end_pos": [37, 12]}, {"full_name": "Function.support", "def_path": "Mathlib/Algebra/Support.lean", "def_pos": [37, 5], "def_end_pos": [37, 12]}]], "state_before": "\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n  \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n    P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n    StronglyMeasurable f \u2192\n      StronglyMeasurable g \u2192\n        Disjoint (Function.support f) (Function.support g) \u2192\n          P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nh_disj : Disjoint (Function.support f) (Function.support g)\nhPf : P (Mem\u2112p.toLp f hf)\nhPg : P (Mem\u2112p.toLp g hg)\ns_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm)\nhs_f : MeasurableSet s_f\nhs_f_eq : s_f =\u1d50[\u03bc] Function.support f\ns_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm)\nhs_g : MeasurableSet s_g\nhs_g_eq : s_g =\u1d50[\u03bc] Function.support g\n\u22a2 Function.support f \u2229 Function.support g =\u1d50[\u03bc] \u2205", "state_after": "\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n  \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n    P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n    StronglyMeasurable f \u2192\n      StronglyMeasurable g \u2192\n        Disjoint (Function.support f) (Function.support g) \u2192\n          P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nh_disj : Disjoint (Function.support f) (Function.support g)\nhPf : P (Mem\u2112p.toLp f hf)\nhPg : P (Mem\u2112p.toLp g hg)\ns_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm)\nhs_f : MeasurableSet s_f\nhs_f_eq : s_f =\u1d50[\u03bc] Function.support f\ns_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm)\nhs_g : MeasurableSet s_g\nhs_g_eq : s_g =\u1d50[\u03bc] Function.support g\n\u22a2 Function.support f \u2229 Function.support g = \u2205"}, {"tactic": "exact Set.disjoint_iff_inter_eq_empty.mp h_disj", "annotated_tactic": ["exact Set.disjoint_iff_inter_eq_empty.mp h_disj", []], "state_before": "\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n  \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n    P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n    StronglyMeasurable f \u2192\n      StronglyMeasurable g \u2192\n        Disjoint (Function.support f) (Function.support g) \u2192\n          P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nh_disj : Disjoint (Function.support f) (Function.support g)\nhPf : P (Mem\u2112p.toLp f hf)\nhPg : P (Mem\u2112p.toLp g hg)\ns_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm)\nhs_f : MeasurableSet s_f\nhs_f_eq : s_f =\u1d50[\u03bc] Function.support f\ns_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm)\nhs_g : MeasurableSet s_g\nhs_g_eq : s_g =\u1d50[\u03bc] Function.support g\n\u22a2 Function.support f \u2229 Function.support g = \u2205", "state_after": "no goals"}, {"tactic": "rw [this]", "annotated_tactic": ["rw [this]", []], "state_before": "\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n  \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n    P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n    StronglyMeasurable f \u2192\n      StronglyMeasurable g \u2192\n        Disjoint (Function.support f) (Function.support g) \u2192\n          P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nh_disj : Disjoint (Function.support f) (Function.support g)\nhPf : P (Mem\u2112p.toLp f hf)\nhPg : P (Mem\u2112p.toLp g hg)\ns_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm)\nhs_f : MeasurableSet s_f\nhs_f_eq : s_f =\u1d50[\u03bc] Function.support f\ns_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm)\nhs_g : MeasurableSet s_g\nhs_g_eq : s_g =\u1d50[\u03bc] Function.support g\nthis : Function.support f \u2229 Function.support g = \u2205\n\u22a2 Function.support f \u2229 Function.support g =\u1d50[\u03bc] \u2205", "state_after": "no goals"}, {"tactic": "have : s_f \\ s_g =\u1d50[\u03bc] s_f := by\n  rw [\u2190 Set.diff_inter_self_eq_diff, Set.inter_comm]\n  refine' ((ae_eq_refl s_f).diff h_inter_empty).trans _\n  rw [Set.diff_empty]", "annotated_tactic": ["have : s_f \\ s_g =\u1d50[\u03bc] s_f := by\n      rw [\u2190 <a>Set.diff_inter_self_eq_diff</a>, <a>Set.inter_comm</a>]\n      refine' ((<a>ae_eq_refl</a> s_f).<a>diff</a> h_inter_empty).<a>trans</a> _\n      rw [<a>Set.diff_empty</a>]", [{"full_name": "Set.diff_inter_self_eq_diff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [2058, 9], "def_end_pos": [2058, 32]}, {"full_name": "Set.inter_comm", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [940, 9], "def_end_pos": [940, 19]}, {"full_name": "MeasureTheory.ae_eq_refl", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [436, 9], "def_end_pos": [436, 19]}, {"full_name": "Filter.EventuallyEq.diff", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1595, 9], "def_end_pos": [1595, 26]}, {"full_name": "Filter.EventuallyEq.trans", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1503, 9], "def_end_pos": [1503, 27]}, {"full_name": "Set.diff_empty", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1930, 9], "def_end_pos": [1930, 19]}]], "state_before": "\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n  \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n    P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n    StronglyMeasurable f \u2192\n      StronglyMeasurable g \u2192\n        Disjoint (Function.support f) (Function.support g) \u2192\n          P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nh_disj : Disjoint (Function.support f) (Function.support g)\nhPf : P (Mem\u2112p.toLp f hf)\nhPg : P (Mem\u2112p.toLp g hg)\ns_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm)\nhs_f : MeasurableSet s_f\nhs_f_eq : s_f =\u1d50[\u03bc] Function.support f\ns_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm)\nhs_g : MeasurableSet s_g\nhs_g_eq : s_g =\u1d50[\u03bc] Function.support g\nh_inter_empty : s_f \u2229 s_g =\u1d50[\u03bc] \u2205\nf' : \u03b1 \u2192 F := Set.indicator (s_f \\ s_g) (AEStronglyMeasurable'.mk f hfm)\n\u22a2 f =\u1d50[\u03bc] f'", "state_after": "\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n  \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n    P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n    StronglyMeasurable f \u2192\n      StronglyMeasurable g \u2192\n        Disjoint (Function.support f) (Function.support g) \u2192\n          P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nh_disj : Disjoint (Function.support f) (Function.support g)\nhPf : P (Mem\u2112p.toLp f hf)\nhPg : P (Mem\u2112p.toLp g hg)\ns_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm)\nhs_f : MeasurableSet s_f\nhs_f_eq : s_f =\u1d50[\u03bc] Function.support f\ns_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm)\nhs_g : MeasurableSet s_g\nhs_g_eq : s_g =\u1d50[\u03bc] Function.support g\nh_inter_empty : s_f \u2229 s_g =\u1d50[\u03bc] \u2205\nf' : \u03b1 \u2192 F := Set.indicator (s_f \\ s_g) (AEStronglyMeasurable'.mk f hfm)\nthis : s_f \\ s_g =\u1d50[\u03bc] s_f\n\u22a2 f =\u1d50[\u03bc] f'"}, {"tactic": "refine' ((indicator_ae_eq_of_ae_eq_set this).trans _).symm", "annotated_tactic": ["refine' ((<a>indicator_ae_eq_of_ae_eq_set</a> this).<a>trans</a> _).<a>symm</a>", [{"full_name": "indicator_ae_eq_of_ae_eq_set", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [4512, 9], "def_end_pos": [4512, 37]}, {"full_name": "Filter.EventuallyEq.trans", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1503, 9], "def_end_pos": [1503, 27]}, {"full_name": "Filter.EventuallyEq.symm", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1498, 9], "def_end_pos": [1498, 26]}]], "state_before": "\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n  \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n    P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n    StronglyMeasurable f \u2192\n      StronglyMeasurable g \u2192\n        Disjoint (Function.support f) (Function.support g) \u2192\n          P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nh_disj : Disjoint (Function.support f) (Function.support g)\nhPf : P (Mem\u2112p.toLp f hf)\nhPg : P (Mem\u2112p.toLp g hg)\ns_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm)\nhs_f : MeasurableSet s_f\nhs_f_eq : s_f =\u1d50[\u03bc] Function.support f\ns_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm)\nhs_g : MeasurableSet s_g\nhs_g_eq : s_g =\u1d50[\u03bc] Function.support g\nh_inter_empty : s_f \u2229 s_g =\u1d50[\u03bc] \u2205\nf' : \u03b1 \u2192 F := Set.indicator (s_f \\ s_g) (AEStronglyMeasurable'.mk f hfm)\nthis : s_f \\ s_g =\u1d50[\u03bc] s_f\n\u22a2 f =\u1d50[\u03bc] f'", "state_after": "\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n  \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n    P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n    StronglyMeasurable f \u2192\n      StronglyMeasurable g \u2192\n        Disjoint (Function.support f) (Function.support g) \u2192\n          P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nh_disj : Disjoint (Function.support f) (Function.support g)\nhPf : P (Mem\u2112p.toLp f hf)\nhPg : P (Mem\u2112p.toLp g hg)\ns_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm)\nhs_f : MeasurableSet s_f\nhs_f_eq : s_f =\u1d50[\u03bc] Function.support f\ns_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm)\nhs_g : MeasurableSet s_g\nhs_g_eq : s_g =\u1d50[\u03bc] Function.support g\nh_inter_empty : s_f \u2229 s_g =\u1d50[\u03bc] \u2205\nf' : \u03b1 \u2192 F := Set.indicator (s_f \\ s_g) (AEStronglyMeasurable'.mk f hfm)\nthis : s_f \\ s_g =\u1d50[\u03bc] s_f\n\u22a2 Set.indicator s_f (AEStronglyMeasurable'.mk f hfm) =\u1d50[\u03bc] f"}, {"tactic": "rw [Set.indicator_support]", "annotated_tactic": ["rw [<a>Set.indicator_support</a>]", [{"full_name": "Set.indicator_support", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [163, 3], "def_end_pos": [163, 14]}]], "state_before": "\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n  \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n    P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n    StronglyMeasurable f \u2192\n      StronglyMeasurable g \u2192\n        Disjoint (Function.support f) (Function.support g) \u2192\n          P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nh_disj : Disjoint (Function.support f) (Function.support g)\nhPf : P (Mem\u2112p.toLp f hf)\nhPg : P (Mem\u2112p.toLp g hg)\ns_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm)\nhs_f : MeasurableSet s_f\nhs_f_eq : s_f =\u1d50[\u03bc] Function.support f\ns_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm)\nhs_g : MeasurableSet s_g\nhs_g_eq : s_g =\u1d50[\u03bc] Function.support g\nh_inter_empty : s_f \u2229 s_g =\u1d50[\u03bc] \u2205\nf' : \u03b1 \u2192 F := Set.indicator (s_f \\ s_g) (AEStronglyMeasurable'.mk f hfm)\nthis : s_f \\ s_g =\u1d50[\u03bc] s_f\n\u22a2 Set.indicator s_f (AEStronglyMeasurable'.mk f hfm) =\u1d50[\u03bc] f", "state_after": "\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n  \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n    P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n    StronglyMeasurable f \u2192\n      StronglyMeasurable g \u2192\n        Disjoint (Function.support f) (Function.support g) \u2192\n          P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nh_disj : Disjoint (Function.support f) (Function.support g)\nhPf : P (Mem\u2112p.toLp f hf)\nhPg : P (Mem\u2112p.toLp g hg)\ns_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm)\nhs_f : MeasurableSet s_f\nhs_f_eq : s_f =\u1d50[\u03bc] Function.support f\ns_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm)\nhs_g : MeasurableSet s_g\nhs_g_eq : s_g =\u1d50[\u03bc] Function.support g\nh_inter_empty : s_f \u2229 s_g =\u1d50[\u03bc] \u2205\nf' : \u03b1 \u2192 F := Set.indicator (s_f \\ s_g) (AEStronglyMeasurable'.mk f hfm)\nthis : s_f \\ s_g =\u1d50[\u03bc] s_f\n\u22a2 AEStronglyMeasurable'.mk f hfm =\u1d50[\u03bc] f"}, {"tactic": "exact hfm.ae_eq_mk.symm", "annotated_tactic": ["exact hfm.ae_eq_mk.symm", []], "state_before": "\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n  \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n    P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n    StronglyMeasurable f \u2192\n      StronglyMeasurable g \u2192\n        Disjoint (Function.support f) (Function.support g) \u2192\n          P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nh_disj : Disjoint (Function.support f) (Function.support g)\nhPf : P (Mem\u2112p.toLp f hf)\nhPg : P (Mem\u2112p.toLp g hg)\ns_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm)\nhs_f : MeasurableSet s_f\nhs_f_eq : s_f =\u1d50[\u03bc] Function.support f\ns_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm)\nhs_g : MeasurableSet s_g\nhs_g_eq : s_g =\u1d50[\u03bc] Function.support g\nh_inter_empty : s_f \u2229 s_g =\u1d50[\u03bc] \u2205\nf' : \u03b1 \u2192 F := Set.indicator (s_f \\ s_g) (AEStronglyMeasurable'.mk f hfm)\nthis : s_f \\ s_g =\u1d50[\u03bc] s_f\n\u22a2 AEStronglyMeasurable'.mk f hfm =\u1d50[\u03bc] f", "state_after": "no goals"}, {"tactic": "rw [\u2190 Set.diff_inter_self_eq_diff, Set.inter_comm]", "annotated_tactic": ["rw [\u2190 <a>Set.diff_inter_self_eq_diff</a>, <a>Set.inter_comm</a>]", [{"full_name": "Set.diff_inter_self_eq_diff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [2058, 9], "def_end_pos": [2058, 32]}, {"full_name": "Set.inter_comm", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [940, 9], "def_end_pos": [940, 19]}]], "state_before": "\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n  \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n    P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n    StronglyMeasurable f \u2192\n      StronglyMeasurable g \u2192\n        Disjoint (Function.support f) (Function.support g) \u2192\n          P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nh_disj : Disjoint (Function.support f) (Function.support g)\nhPf : P (Mem\u2112p.toLp f hf)\nhPg : P (Mem\u2112p.toLp g hg)\ns_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm)\nhs_f : MeasurableSet s_f\nhs_f_eq : s_f =\u1d50[\u03bc] Function.support f\ns_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm)\nhs_g : MeasurableSet s_g\nhs_g_eq : s_g =\u1d50[\u03bc] Function.support g\nh_inter_empty : s_f \u2229 s_g =\u1d50[\u03bc] \u2205\nf' : \u03b1 \u2192 F := Set.indicator (s_f \\ s_g) (AEStronglyMeasurable'.mk f hfm)\n\u22a2 s_f \\ s_g =\u1d50[\u03bc] s_f", "state_after": "\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n  \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n    P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n    StronglyMeasurable f \u2192\n      StronglyMeasurable g \u2192\n        Disjoint (Function.support f) (Function.support g) \u2192\n          P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nh_disj : Disjoint (Function.support f) (Function.support g)\nhPf : P (Mem\u2112p.toLp f hf)\nhPg : P (Mem\u2112p.toLp g hg)\ns_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm)\nhs_f : MeasurableSet s_f\nhs_f_eq : s_f =\u1d50[\u03bc] Function.support f\ns_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm)\nhs_g : MeasurableSet s_g\nhs_g_eq : s_g =\u1d50[\u03bc] Function.support g\nh_inter_empty : s_f \u2229 s_g =\u1d50[\u03bc] \u2205\nf' : \u03b1 \u2192 F := Set.indicator (s_f \\ s_g) (AEStronglyMeasurable'.mk f hfm)\n\u22a2 s_f \\ (s_f \u2229 s_g) =\u1d50[\u03bc] s_f"}, {"tactic": "refine' ((ae_eq_refl s_f).diff h_inter_empty).trans _", "annotated_tactic": ["refine' ((<a>ae_eq_refl</a> s_f).<a>diff</a> h_inter_empty).<a>trans</a> _", [{"full_name": "MeasureTheory.ae_eq_refl", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [436, 9], "def_end_pos": [436, 19]}, {"full_name": "Filter.EventuallyEq.diff", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1595, 9], "def_end_pos": [1595, 26]}, {"full_name": "Filter.EventuallyEq.trans", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1503, 9], "def_end_pos": [1503, 27]}]], "state_before": "\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n  \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n    P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n    StronglyMeasurable f \u2192\n      StronglyMeasurable g \u2192\n        Disjoint (Function.support f) (Function.support g) \u2192\n          P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nh_disj : Disjoint (Function.support f) (Function.support g)\nhPf : P (Mem\u2112p.toLp f hf)\nhPg : P (Mem\u2112p.toLp g hg)\ns_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm)\nhs_f : MeasurableSet s_f\nhs_f_eq : s_f =\u1d50[\u03bc] Function.support f\ns_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm)\nhs_g : MeasurableSet s_g\nhs_g_eq : s_g =\u1d50[\u03bc] Function.support g\nh_inter_empty : s_f \u2229 s_g =\u1d50[\u03bc] \u2205\nf' : \u03b1 \u2192 F := Set.indicator (s_f \\ s_g) (AEStronglyMeasurable'.mk f hfm)\n\u22a2 s_f \\ (s_f \u2229 s_g) =\u1d50[\u03bc] s_f", "state_after": "\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n  \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n    P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n    StronglyMeasurable f \u2192\n      StronglyMeasurable g \u2192\n        Disjoint (Function.support f) (Function.support g) \u2192\n          P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nh_disj : Disjoint (Function.support f) (Function.support g)\nhPf : P (Mem\u2112p.toLp f hf)\nhPg : P (Mem\u2112p.toLp g hg)\ns_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm)\nhs_f : MeasurableSet s_f\nhs_f_eq : s_f =\u1d50[\u03bc] Function.support f\ns_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm)\nhs_g : MeasurableSet s_g\nhs_g_eq : s_g =\u1d50[\u03bc] Function.support g\nh_inter_empty : s_f \u2229 s_g =\u1d50[\u03bc] \u2205\nf' : \u03b1 \u2192 F := Set.indicator (s_f \\ s_g) (AEStronglyMeasurable'.mk f hfm)\n\u22a2 s_f \\ \u2205 =\u1d50[\u03bc] s_f"}, {"tactic": "rw [Set.diff_empty]", "annotated_tactic": ["rw [<a>Set.diff_empty</a>]", [{"full_name": "Set.diff_empty", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1930, 9], "def_end_pos": [1930, 19]}]], "state_before": "\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n  \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n    P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n    StronglyMeasurable f \u2192\n      StronglyMeasurable g \u2192\n        Disjoint (Function.support f) (Function.support g) \u2192\n          P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nh_disj : Disjoint (Function.support f) (Function.support g)\nhPf : P (Mem\u2112p.toLp f hf)\nhPg : P (Mem\u2112p.toLp g hg)\ns_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm)\nhs_f : MeasurableSet s_f\nhs_f_eq : s_f =\u1d50[\u03bc] Function.support f\ns_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm)\nhs_g : MeasurableSet s_g\nhs_g_eq : s_g =\u1d50[\u03bc] Function.support g\nh_inter_empty : s_f \u2229 s_g =\u1d50[\u03bc] \u2205\nf' : \u03b1 \u2192 F := Set.indicator (s_f \\ s_g) (AEStronglyMeasurable'.mk f hfm)\n\u22a2 s_f \\ \u2205 =\u1d50[\u03bc] s_f", "state_after": "no goals"}, {"tactic": "have : s_g \\ s_f =\u1d50[\u03bc] s_g := by\n  rw [\u2190 Set.diff_inter_self_eq_diff]\n  refine' ((ae_eq_refl s_g).diff h_inter_empty).trans _\n  rw [Set.diff_empty]", "annotated_tactic": ["have : s_g \\ s_f =\u1d50[\u03bc] s_g := by\n      rw [\u2190 <a>Set.diff_inter_self_eq_diff</a>]\n      refine' ((<a>ae_eq_refl</a> s_g).<a>diff</a> h_inter_empty).<a>trans</a> _\n      rw [<a>Set.diff_empty</a>]", [{"full_name": "Set.diff_inter_self_eq_diff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [2058, 9], "def_end_pos": [2058, 32]}, {"full_name": "MeasureTheory.ae_eq_refl", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [436, 9], "def_end_pos": [436, 19]}, {"full_name": "Filter.EventuallyEq.diff", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1595, 9], "def_end_pos": [1595, 26]}, {"full_name": "Filter.EventuallyEq.trans", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1503, 9], "def_end_pos": [1503, 27]}, {"full_name": "Set.diff_empty", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1930, 9], "def_end_pos": [1930, 19]}]], "state_before": "\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n  \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n    P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n    StronglyMeasurable f \u2192\n      StronglyMeasurable g \u2192\n        Disjoint (Function.support f) (Function.support g) \u2192\n          P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nh_disj : Disjoint (Function.support f) (Function.support g)\nhPf : P (Mem\u2112p.toLp f hf)\nhPg : P (Mem\u2112p.toLp g hg)\ns_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm)\nhs_f : MeasurableSet s_f\nhs_f_eq : s_f =\u1d50[\u03bc] Function.support f\ns_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm)\nhs_g : MeasurableSet s_g\nhs_g_eq : s_g =\u1d50[\u03bc] Function.support g\nh_inter_empty : s_f \u2229 s_g =\u1d50[\u03bc] \u2205\nf' : \u03b1 \u2192 F := Set.indicator (s_f \\ s_g) (AEStronglyMeasurable'.mk f hfm)\nhff' : f =\u1d50[\u03bc] f'\nhf'_meas : StronglyMeasurable f'\nhf'_Lp : Mem\u2112p f' p\ng' : \u03b1 \u2192 F := Set.indicator (s_g \\ s_f) (AEStronglyMeasurable'.mk g hgm)\n\u22a2 g =\u1d50[\u03bc] g'", "state_after": "\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n  \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n    P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n    StronglyMeasurable f \u2192\n      StronglyMeasurable g \u2192\n        Disjoint (Function.support f) (Function.support g) \u2192\n          P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nh_disj : Disjoint (Function.support f) (Function.support g)\nhPf : P (Mem\u2112p.toLp f hf)\nhPg : P (Mem\u2112p.toLp g hg)\ns_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm)\nhs_f : MeasurableSet s_f\nhs_f_eq : s_f =\u1d50[\u03bc] Function.support f\ns_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm)\nhs_g : MeasurableSet s_g\nhs_g_eq : s_g =\u1d50[\u03bc] Function.support g\nh_inter_empty : s_f \u2229 s_g =\u1d50[\u03bc] \u2205\nf' : \u03b1 \u2192 F := Set.indicator (s_f \\ s_g) (AEStronglyMeasurable'.mk f hfm)\nhff' : f =\u1d50[\u03bc] f'\nhf'_meas : StronglyMeasurable f'\nhf'_Lp : Mem\u2112p f' p\ng' : \u03b1 \u2192 F := Set.indicator (s_g \\ s_f) (AEStronglyMeasurable'.mk g hgm)\nthis : s_g \\ s_f =\u1d50[\u03bc] s_g\n\u22a2 g =\u1d50[\u03bc] g'"}, {"tactic": "refine' ((indicator_ae_eq_of_ae_eq_set this).trans _).symm", "annotated_tactic": ["refine' ((<a>indicator_ae_eq_of_ae_eq_set</a> this).<a>trans</a> _).<a>symm</a>", [{"full_name": "indicator_ae_eq_of_ae_eq_set", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [4512, 9], "def_end_pos": [4512, 37]}, {"full_name": "Filter.EventuallyEq.trans", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1503, 9], "def_end_pos": [1503, 27]}, {"full_name": "Filter.EventuallyEq.symm", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1498, 9], "def_end_pos": [1498, 26]}]], "state_before": "\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n  \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n    P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n    StronglyMeasurable f \u2192\n      StronglyMeasurable g \u2192\n        Disjoint (Function.support f) (Function.support g) \u2192\n          P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nh_disj : Disjoint (Function.support f) (Function.support g)\nhPf : P (Mem\u2112p.toLp f hf)\nhPg : P (Mem\u2112p.toLp g hg)\ns_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm)\nhs_f : MeasurableSet s_f\nhs_f_eq : s_f =\u1d50[\u03bc] Function.support f\ns_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm)\nhs_g : MeasurableSet s_g\nhs_g_eq : s_g =\u1d50[\u03bc] Function.support g\nh_inter_empty : s_f \u2229 s_g =\u1d50[\u03bc] \u2205\nf' : \u03b1 \u2192 F := Set.indicator (s_f \\ s_g) (AEStronglyMeasurable'.mk f hfm)\nhff' : f =\u1d50[\u03bc] f'\nhf'_meas : StronglyMeasurable f'\nhf'_Lp : Mem\u2112p f' p\ng' : \u03b1 \u2192 F := Set.indicator (s_g \\ s_f) (AEStronglyMeasurable'.mk g hgm)\nthis : s_g \\ s_f =\u1d50[\u03bc] s_g\n\u22a2 g =\u1d50[\u03bc] g'", "state_after": "\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n  \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n    P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n    StronglyMeasurable f \u2192\n      StronglyMeasurable g \u2192\n        Disjoint (Function.support f) (Function.support g) \u2192\n          P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nh_disj : Disjoint (Function.support f) (Function.support g)\nhPf : P (Mem\u2112p.toLp f hf)\nhPg : P (Mem\u2112p.toLp g hg)\ns_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm)\nhs_f : MeasurableSet s_f\nhs_f_eq : s_f =\u1d50[\u03bc] Function.support f\ns_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm)\nhs_g : MeasurableSet s_g\nhs_g_eq : s_g =\u1d50[\u03bc] Function.support g\nh_inter_empty : s_f \u2229 s_g =\u1d50[\u03bc] \u2205\nf' : \u03b1 \u2192 F := Set.indicator (s_f \\ s_g) (AEStronglyMeasurable'.mk f hfm)\nhff' : f =\u1d50[\u03bc] f'\nhf'_meas : StronglyMeasurable f'\nhf'_Lp : Mem\u2112p f' p\ng' : \u03b1 \u2192 F := Set.indicator (s_g \\ s_f) (AEStronglyMeasurable'.mk g hgm)\nthis : s_g \\ s_f =\u1d50[\u03bc] s_g\n\u22a2 Set.indicator s_g (AEStronglyMeasurable'.mk g hgm) =\u1d50[\u03bc] g"}, {"tactic": "rw [Set.indicator_support]", "annotated_tactic": ["rw [<a>Set.indicator_support</a>]", [{"full_name": "Set.indicator_support", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [163, 3], "def_end_pos": [163, 14]}]], "state_before": "\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n  \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n    P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n    StronglyMeasurable f \u2192\n      StronglyMeasurable g \u2192\n        Disjoint (Function.support f) (Function.support g) \u2192\n          P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nh_disj : Disjoint (Function.support f) (Function.support g)\nhPf : P (Mem\u2112p.toLp f hf)\nhPg : P (Mem\u2112p.toLp g hg)\ns_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm)\nhs_f : MeasurableSet s_f\nhs_f_eq : s_f =\u1d50[\u03bc] Function.support f\ns_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm)\nhs_g : MeasurableSet s_g\nhs_g_eq : s_g =\u1d50[\u03bc] Function.support g\nh_inter_empty : s_f \u2229 s_g =\u1d50[\u03bc] \u2205\nf' : \u03b1 \u2192 F := Set.indicator (s_f \\ s_g) (AEStronglyMeasurable'.mk f hfm)\nhff' : f =\u1d50[\u03bc] f'\nhf'_meas : StronglyMeasurable f'\nhf'_Lp : Mem\u2112p f' p\ng' : \u03b1 \u2192 F := Set.indicator (s_g \\ s_f) (AEStronglyMeasurable'.mk g hgm)\nthis : s_g \\ s_f =\u1d50[\u03bc] s_g\n\u22a2 Set.indicator s_g (AEStronglyMeasurable'.mk g hgm) =\u1d50[\u03bc] g", "state_after": "\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n  \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n    P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n    StronglyMeasurable f \u2192\n      StronglyMeasurable g \u2192\n        Disjoint (Function.support f) (Function.support g) \u2192\n          P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nh_disj : Disjoint (Function.support f) (Function.support g)\nhPf : P (Mem\u2112p.toLp f hf)\nhPg : P (Mem\u2112p.toLp g hg)\ns_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm)\nhs_f : MeasurableSet s_f\nhs_f_eq : s_f =\u1d50[\u03bc] Function.support f\ns_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm)\nhs_g : MeasurableSet s_g\nhs_g_eq : s_g =\u1d50[\u03bc] Function.support g\nh_inter_empty : s_f \u2229 s_g =\u1d50[\u03bc] \u2205\nf' : \u03b1 \u2192 F := Set.indicator (s_f \\ s_g) (AEStronglyMeasurable'.mk f hfm)\nhff' : f =\u1d50[\u03bc] f'\nhf'_meas : StronglyMeasurable f'\nhf'_Lp : Mem\u2112p f' p\ng' : \u03b1 \u2192 F := Set.indicator (s_g \\ s_f) (AEStronglyMeasurable'.mk g hgm)\nthis : s_g \\ s_f =\u1d50[\u03bc] s_g\n\u22a2 AEStronglyMeasurable'.mk g hgm =\u1d50[\u03bc] g"}, {"tactic": "exact hgm.ae_eq_mk.symm", "annotated_tactic": ["exact hgm.ae_eq_mk.symm", []], "state_before": "\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n  \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n    P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n    StronglyMeasurable f \u2192\n      StronglyMeasurable g \u2192\n        Disjoint (Function.support f) (Function.support g) \u2192\n          P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nh_disj : Disjoint (Function.support f) (Function.support g)\nhPf : P (Mem\u2112p.toLp f hf)\nhPg : P (Mem\u2112p.toLp g hg)\ns_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm)\nhs_f : MeasurableSet s_f\nhs_f_eq : s_f =\u1d50[\u03bc] Function.support f\ns_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm)\nhs_g : MeasurableSet s_g\nhs_g_eq : s_g =\u1d50[\u03bc] Function.support g\nh_inter_empty : s_f \u2229 s_g =\u1d50[\u03bc] \u2205\nf' : \u03b1 \u2192 F := Set.indicator (s_f \\ s_g) (AEStronglyMeasurable'.mk f hfm)\nhff' : f =\u1d50[\u03bc] f'\nhf'_meas : StronglyMeasurable f'\nhf'_Lp : Mem\u2112p f' p\ng' : \u03b1 \u2192 F := Set.indicator (s_g \\ s_f) (AEStronglyMeasurable'.mk g hgm)\nthis : s_g \\ s_f =\u1d50[\u03bc] s_g\n\u22a2 AEStronglyMeasurable'.mk g hgm =\u1d50[\u03bc] g", "state_after": "no goals"}, {"tactic": "rw [\u2190 Set.diff_inter_self_eq_diff]", "annotated_tactic": ["rw [\u2190 <a>Set.diff_inter_self_eq_diff</a>]", [{"full_name": "Set.diff_inter_self_eq_diff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [2058, 9], "def_end_pos": [2058, 32]}]], "state_before": "\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n  \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n    P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n    StronglyMeasurable f \u2192\n      StronglyMeasurable g \u2192\n        Disjoint (Function.support f) (Function.support g) \u2192\n          P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nh_disj : Disjoint (Function.support f) (Function.support g)\nhPf : P (Mem\u2112p.toLp f hf)\nhPg : P (Mem\u2112p.toLp g hg)\ns_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm)\nhs_f : MeasurableSet s_f\nhs_f_eq : s_f =\u1d50[\u03bc] Function.support f\ns_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm)\nhs_g : MeasurableSet s_g\nhs_g_eq : s_g =\u1d50[\u03bc] Function.support g\nh_inter_empty : s_f \u2229 s_g =\u1d50[\u03bc] \u2205\nf' : \u03b1 \u2192 F := Set.indicator (s_f \\ s_g) (AEStronglyMeasurable'.mk f hfm)\nhff' : f =\u1d50[\u03bc] f'\nhf'_meas : StronglyMeasurable f'\nhf'_Lp : Mem\u2112p f' p\ng' : \u03b1 \u2192 F := Set.indicator (s_g \\ s_f) (AEStronglyMeasurable'.mk g hgm)\n\u22a2 s_g \\ s_f =\u1d50[\u03bc] s_g", "state_after": "\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n  \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n    P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n    StronglyMeasurable f \u2192\n      StronglyMeasurable g \u2192\n        Disjoint (Function.support f) (Function.support g) \u2192\n          P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nh_disj : Disjoint (Function.support f) (Function.support g)\nhPf : P (Mem\u2112p.toLp f hf)\nhPg : P (Mem\u2112p.toLp g hg)\ns_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm)\nhs_f : MeasurableSet s_f\nhs_f_eq : s_f =\u1d50[\u03bc] Function.support f\ns_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm)\nhs_g : MeasurableSet s_g\nhs_g_eq : s_g =\u1d50[\u03bc] Function.support g\nh_inter_empty : s_f \u2229 s_g =\u1d50[\u03bc] \u2205\nf' : \u03b1 \u2192 F := Set.indicator (s_f \\ s_g) (AEStronglyMeasurable'.mk f hfm)\nhff' : f =\u1d50[\u03bc] f'\nhf'_meas : StronglyMeasurable f'\nhf'_Lp : Mem\u2112p f' p\ng' : \u03b1 \u2192 F := Set.indicator (s_g \\ s_f) (AEStronglyMeasurable'.mk g hgm)\n\u22a2 s_g \\ (s_f \u2229 s_g) =\u1d50[\u03bc] s_g"}, {"tactic": "refine' ((ae_eq_refl s_g).diff h_inter_empty).trans _", "annotated_tactic": ["refine' ((<a>ae_eq_refl</a> s_g).<a>diff</a> h_inter_empty).<a>trans</a> _", [{"full_name": "MeasureTheory.ae_eq_refl", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [436, 9], "def_end_pos": [436, 19]}, {"full_name": "Filter.EventuallyEq.diff", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1595, 9], "def_end_pos": [1595, 26]}, {"full_name": "Filter.EventuallyEq.trans", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1503, 9], "def_end_pos": [1503, 27]}]], "state_before": "\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n  \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n    P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n    StronglyMeasurable f \u2192\n      StronglyMeasurable g \u2192\n        Disjoint (Function.support f) (Function.support g) \u2192\n          P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nh_disj : Disjoint (Function.support f) (Function.support g)\nhPf : P (Mem\u2112p.toLp f hf)\nhPg : P (Mem\u2112p.toLp g hg)\ns_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm)\nhs_f : MeasurableSet s_f\nhs_f_eq : s_f =\u1d50[\u03bc] Function.support f\ns_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm)\nhs_g : MeasurableSet s_g\nhs_g_eq : s_g =\u1d50[\u03bc] Function.support g\nh_inter_empty : s_f \u2229 s_g =\u1d50[\u03bc] \u2205\nf' : \u03b1 \u2192 F := Set.indicator (s_f \\ s_g) (AEStronglyMeasurable'.mk f hfm)\nhff' : f =\u1d50[\u03bc] f'\nhf'_meas : StronglyMeasurable f'\nhf'_Lp : Mem\u2112p f' p\ng' : \u03b1 \u2192 F := Set.indicator (s_g \\ s_f) (AEStronglyMeasurable'.mk g hgm)\n\u22a2 s_g \\ (s_f \u2229 s_g) =\u1d50[\u03bc] s_g", "state_after": "\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n  \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n    P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n    StronglyMeasurable f \u2192\n      StronglyMeasurable g \u2192\n        Disjoint (Function.support f) (Function.support g) \u2192\n          P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nh_disj : Disjoint (Function.support f) (Function.support g)\nhPf : P (Mem\u2112p.toLp f hf)\nhPg : P (Mem\u2112p.toLp g hg)\ns_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm)\nhs_f : MeasurableSet s_f\nhs_f_eq : s_f =\u1d50[\u03bc] Function.support f\ns_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm)\nhs_g : MeasurableSet s_g\nhs_g_eq : s_g =\u1d50[\u03bc] Function.support g\nh_inter_empty : s_f \u2229 s_g =\u1d50[\u03bc] \u2205\nf' : \u03b1 \u2192 F := Set.indicator (s_f \\ s_g) (AEStronglyMeasurable'.mk f hfm)\nhff' : f =\u1d50[\u03bc] f'\nhf'_meas : StronglyMeasurable f'\nhf'_Lp : Mem\u2112p f' p\ng' : \u03b1 \u2192 F := Set.indicator (s_g \\ s_f) (AEStronglyMeasurable'.mk g hgm)\n\u22a2 s_g \\ \u2205 =\u1d50[\u03bc] s_g"}, {"tactic": "rw [Set.diff_empty]", "annotated_tactic": ["rw [<a>Set.diff_empty</a>]", [{"full_name": "Set.diff_empty", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1930, 9], "def_end_pos": [1930, 19]}]], "state_before": "\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n  \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n    P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n    StronglyMeasurable f \u2192\n      StronglyMeasurable g \u2192\n        Disjoint (Function.support f) (Function.support g) \u2192\n          P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nh_disj : Disjoint (Function.support f) (Function.support g)\nhPf : P (Mem\u2112p.toLp f hf)\nhPg : P (Mem\u2112p.toLp g hg)\ns_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm)\nhs_f : MeasurableSet s_f\nhs_f_eq : s_f =\u1d50[\u03bc] Function.support f\ns_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm)\nhs_g : MeasurableSet s_g\nhs_g_eq : s_g =\u1d50[\u03bc] Function.support g\nh_inter_empty : s_f \u2229 s_g =\u1d50[\u03bc] \u2205\nf' : \u03b1 \u2192 F := Set.indicator (s_f \\ s_g) (AEStronglyMeasurable'.mk f hfm)\nhff' : f =\u1d50[\u03bc] f'\nhf'_meas : StronglyMeasurable f'\nhf'_Lp : Mem\u2112p f' p\ng' : \u03b1 \u2192 F := Set.indicator (s_g \\ s_f) (AEStronglyMeasurable'.mk g hgm)\n\u22a2 s_g \\ \u2205 =\u1d50[\u03bc] s_g", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "full_name": "String.back_eq", "start": [312, 1], "end": [315, 71], "traced_tactics": [{"tactic": "match s, s.1.eq_nil_or_concat with\n| \u27e8_\u27e9, .inl rfl => rfl\n| \u27e8_\u27e9, .inr \u27e8cs, c, rfl\u27e9 => simp [back, prev_of_valid, get_of_valid]", "annotated_tactic": ["match s, s.1.<a>eq_nil_or_concat</a> with\n  | \u27e8_\u27e9, .inl <a>rfl</a> => rfl\n  | \u27e8_\u27e9, .inr \u27e8cs, c, <a>rfl</a>\u27e9 => simp [<a>back</a>, <a>prev_of_valid</a>, <a>get_of_valid</a>]", [{"full_name": "List.eq_nil_or_concat", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [357, 9], "def_end_pos": [357, 25]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}, {"full_name": "String.back", "def_path": "lake-packages/lean4/src/lean/Init/Data/String/Basic.lean", "def_pos": [107, 5], "def_end_pos": [107, 9]}, {"full_name": "String.prev_of_valid", "def_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "def_pos": [296, 9], "def_end_pos": [296, 22]}, {"full_name": "String.get_of_valid", "def_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "def_pos": [223, 9], "def_end_pos": [223, 21]}]], "state_before": "s : String\n\u22a2 back s = List.getLastD s.data default", "state_after": "no goals"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "s : String\n\u22a2 back { data := [] } = List.getLastD { data := [] }.data default", "state_after": "no goals"}, {"tactic": "simp [back, prev_of_valid, get_of_valid]", "annotated_tactic": ["simp [<a>back</a>, <a>prev_of_valid</a>, <a>get_of_valid</a>]", [{"full_name": "String.back", "def_path": "lake-packages/lean4/src/lean/Init/Data/String/Basic.lean", "def_pos": [107, 5], "def_end_pos": [107, 9]}, {"full_name": "String.prev_of_valid", "def_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "def_pos": [296, 9], "def_end_pos": [296, 22]}, {"full_name": "String.get_of_valid", "def_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "def_pos": [223, 9], "def_end_pos": [223, 21]}]], "state_before": "s : String\ncs : List Char\nc : Char\n\u22a2 back { data := cs ++ [c] } = List.getLastD { data := cs ++ [c] }.data default", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/PairingHeap.lean", "full_name": "Std.PairingHeapImp.Heap.WF.singleton", "start": [230, 1], "end": [230, 69], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/UniformIntegrable.lean", "full_name": "MeasureTheory.unifIntegrable_of'", "start": [635, 1], "end": [697, 78], "traced_tactics": [{"tactic": "have hpzero := (lt_of_lt_of_le zero_lt_one hp).ne.symm", "annotated_tactic": ["have hpzero := (<a>lt_of_lt_of_le</a> <a>zero_lt_one</a> hp).ne.symm", [{"full_name": "lt_of_lt_of_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [115, 9], "def_end_pos": [115, 23]}, {"full_name": "zero_lt_one", "def_path": "Mathlib/Algebra/Order/ZeroLEOne.lean", "def_pos": [39, 15], "def_end_pos": [39, 26]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nh : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, 0 < C \u2227 \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\n\u22a2 UnifIntegrable f p \u03bc", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nh : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, 0 < C \u2227 \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nhpzero : p \u2260 0\n\u22a2 UnifIntegrable f p \u03bc"}, {"tactic": "by_cases h\u03bc : \u03bc Set.univ = 0", "annotated_tactic": ["by_cases h\u03bc : \u03bc <a>Set.univ</a> = 0", [{"full_name": "Set.univ", "def_path": "Mathlib/Init/Set.lean", "def_pos": [90, 5], "def_end_pos": [90, 9]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nh : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, 0 < C \u2227 \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nhpzero : p \u2260 0\n\u22a2 UnifIntegrable f p \u03bc", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nh : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, 0 < C \u2227 \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nhpzero : p \u2260 0\nh\u03bc : \u2191\u2191\u03bc univ = 0\n\u22a2 UnifIntegrable f p \u03bc\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nh : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, 0 < C \u2227 \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nhpzero : p \u2260 0\nh\u03bc : \u00ac\u2191\u2191\u03bc univ = 0\n\u22a2 UnifIntegrable f p \u03bc"}, {"tactic": "intro \u03b5 h\u03b5", "annotated_tactic": ["intro \u03b5 h\u03b5", []], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nh : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, 0 < C \u2227 \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nhpzero : p \u2260 0\nh\u03bc : \u00ac\u2191\u2191\u03bc univ = 0\n\u22a2 UnifIntegrable f p \u03bc", "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nh : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, 0 < C \u2227 \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nhpzero : p \u2260 0\nh\u03bc : \u00ac\u2191\u2191\u03bc univ = 0\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u22a2 \u2203 \u03b4 x,\n    \u2200 (i : \u03b9) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5"}, {"tactic": "obtain \u27e8C, hCpos, hC\u27e9 := h (\u03b5 / 2) (half_pos h\u03b5)", "annotated_tactic": ["obtain \u27e8C, hCpos, hC\u27e9 := h (\u03b5 / 2) (<a>half_pos</a> h\u03b5)", [{"full_name": "half_pos", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [504, 9], "def_end_pos": [504, 17]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nh : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, 0 < C \u2227 \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nhpzero : p \u2260 0\nh\u03bc : \u00ac\u2191\u2191\u03bc univ = 0\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u22a2 \u2203 \u03b4 x,\n    \u2200 (i : \u03b9) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5", "state_after": "case neg.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nh : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, 0 < C \u2227 \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nhpzero : p \u2260 0\nh\u03bc : \u00ac\u2191\u2191\u03bc univ = 0\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nC : \u211d\u22650\nhCpos : 0 < C\nhC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal (\u03b5 / 2)\n\u22a2 \u2203 \u03b4 x,\n    \u2200 (i : \u03b9) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5"}, {"tactic": "refine' \u27e8(\u03b5 / (2 * C)) ^ ENNReal.toReal p,\n  Real.rpow_pos_of_pos (div_pos h\u03b5 (mul_pos two_pos (NNReal.coe_pos.2 hCpos))) _,\n  fun i s hs h\u03bcs => _\u27e9", "annotated_tactic": ["refine' \u27e8(\u03b5 / (2 * C)) ^ <a>ENNReal.toReal</a> p,\n    <a>Real.rpow_pos_of_pos</a> (<a>div_pos</a> h\u03b5 (<a>mul_pos</a> <a>two_pos</a> (<a>NNReal.coe_pos</a>.2 hCpos))) _,\n    fun i s hs h\u03bcs => _\u27e9", [{"full_name": "ENNReal.toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [168, 15], "def_end_pos": [168, 21]}, {"full_name": "Real.rpow_pos_of_pos", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [92, 9], "def_end_pos": [92, 24]}, {"full_name": "div_pos", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [89, 9], "def_end_pos": [89, 16]}, {"full_name": "mul_pos", "def_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "def_pos": [345, 7], "def_end_pos": [345, 14]}, {"full_name": "two_pos", "def_path": "Mathlib/Algebra/Order/Monoid/NatCast.lean", "def_pos": [113, 7], "def_end_pos": [113, 14]}, {"full_name": "NNReal.coe_pos", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [376, 19], "def_end_pos": [376, 26]}]], "state_before": "case neg.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nh : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, 0 < C \u2227 \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nhpzero : p \u2260 0\nh\u03bc : \u00ac\u2191\u2191\u03bc univ = 0\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nC : \u211d\u22650\nhCpos : 0 < C\nhC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal (\u03b5 / 2)\n\u22a2 \u2203 \u03b4 x,\n    \u2200 (i : \u03b9) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5", "state_after": "case neg.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nh : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, 0 < C \u2227 \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nhpzero : p \u2260 0\nh\u03bc : \u00ac\u2191\u2191\u03bc univ = 0\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nC : \u211d\u22650\nhCpos : 0 < C\nhC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal (\u03b5 / 2)\ni : \u03b9\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal ((\u03b5 / (2 * \u2191C)) ^ ENNReal.toReal p)\n\u22a2 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5"}, {"tactic": "by_cases h\u03bcs' : \u03bc s = 0", "annotated_tactic": ["by_cases h\u03bcs' : \u03bc s = 0", []], "state_before": "case neg.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nh : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, 0 < C \u2227 \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nhpzero : p \u2260 0\nh\u03bc : \u00ac\u2191\u2191\u03bc univ = 0\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nC : \u211d\u22650\nhCpos : 0 < C\nhC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal (\u03b5 / 2)\ni : \u03b9\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal ((\u03b5 / (2 * \u2191C)) ^ ENNReal.toReal p)\n\u22a2 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nh : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, 0 < C \u2227 \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nhpzero : p \u2260 0\nh\u03bc : \u00ac\u2191\u2191\u03bc univ = 0\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nC : \u211d\u22650\nhCpos : 0 < C\nhC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal (\u03b5 / 2)\ni : \u03b9\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal ((\u03b5 / (2 * \u2191C)) ^ ENNReal.toReal p)\nh\u03bcs' : \u2191\u2191\u03bc s = 0\n\u22a2 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nh : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, 0 < C \u2227 \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nhpzero : p \u2260 0\nh\u03bc : \u00ac\u2191\u2191\u03bc univ = 0\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nC : \u211d\u22650\nhCpos : 0 < C\nhC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal (\u03b5 / 2)\ni : \u03b9\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal ((\u03b5 / (2 * \u2191C)) ^ ENNReal.toReal p)\nh\u03bcs' : \u00ac\u2191\u2191\u03bc s = 0\n\u22a2 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5"}, {"tactic": "rw [Measure.measure_univ_eq_zero] at h\u03bc", "annotated_tactic": ["rw [<a>Measure.measure_univ_eq_zero</a>] at h\u03bc", [{"full_name": "MeasureTheory.Measure.measure_univ_eq_zero", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1100, 9], "def_end_pos": [1100, 29]}]], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nh : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, 0 < C \u2227 \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nhpzero : p \u2260 0\nh\u03bc : \u2191\u2191\u03bc univ = 0\n\u22a2 UnifIntegrable f p \u03bc", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nh : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, 0 < C \u2227 \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nhpzero : p \u2260 0\nh\u03bc : \u03bc = 0\n\u22a2 UnifIntegrable f p \u03bc"}, {"tactic": "exact h\u03bc.symm \u25b8 unifIntegrable_zero_meas", "annotated_tactic": ["exact h\u03bc.symm \u25b8 <a>unifIntegrable_zero_meas</a>", [{"full_name": "MeasureTheory.unifIntegrable_zero_meas", "def_path": "Mathlib/MeasureTheory/Function/UniformIntegrable.lean", "def_pos": [148, 9], "def_end_pos": [148, 33]}]], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nh : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, 0 < C \u2227 \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nhpzero : p \u2260 0\nh\u03bc : \u03bc = 0\n\u22a2 UnifIntegrable f p \u03bc", "state_after": "no goals"}, {"tactic": "rw [(snorm_eq_zero_iff ((hf i).indicator hs).aestronglyMeasurable hpzero).2\n    (indicator_meas_zero h\u03bcs')]", "annotated_tactic": ["rw [(<a>snorm_eq_zero_iff</a> ((hf i).<a>indicator</a> hs).<a>aestronglyMeasurable</a> hpzero).2\n        (<a>indicator_meas_zero</a> h\u03bcs')]", [{"full_name": "MeasureTheory.snorm_eq_zero_iff", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [764, 9], "def_end_pos": [764, 26]}, {"full_name": "MeasureTheory.StronglyMeasurable.indicator", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [825, 19], "def_end_pos": [825, 28]}, {"full_name": "MeasureTheory.StronglyMeasurable.aestronglyMeasurable", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [110, 19], "def_end_pos": [110, 58]}, {"full_name": "indicator_meas_zero", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [4516, 9], "def_end_pos": [4516, 28]}]], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nh : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, 0 < C \u2227 \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nhpzero : p \u2260 0\nh\u03bc : \u00ac\u2191\u2191\u03bc univ = 0\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nC : \u211d\u22650\nhCpos : 0 < C\nhC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal (\u03b5 / 2)\ni : \u03b9\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal ((\u03b5 / (2 * \u2191C)) ^ ENNReal.toReal p)\nh\u03bcs' : \u2191\u2191\u03bc s = 0\n\u22a2 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nh : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, 0 < C \u2227 \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nhpzero : p \u2260 0\nh\u03bc : \u00ac\u2191\u2191\u03bc univ = 0\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nC : \u211d\u22650\nhCpos : 0 < C\nhC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal (\u03b5 / 2)\ni : \u03b9\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal ((\u03b5 / (2 * \u2191C)) ^ ENNReal.toReal p)\nh\u03bcs' : \u2191\u2191\u03bc s = 0\n\u22a2 0 \u2264 ENNReal.ofReal \u03b5"}, {"tactic": "norm_num", "annotated_tactic": ["norm_num", []], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nh : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, 0 < C \u2227 \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nhpzero : p \u2260 0\nh\u03bc : \u00ac\u2191\u2191\u03bc univ = 0\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nC : \u211d\u22650\nhCpos : 0 < C\nhC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal (\u03b5 / 2)\ni : \u03b9\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal ((\u03b5 / (2 * \u2191C)) ^ ENNReal.toReal p)\nh\u03bcs' : \u2191\u2191\u03bc s = 0\n\u22a2 0 \u2264 ENNReal.ofReal \u03b5", "state_after": "no goals"}, {"tactic": "refine' le_trans (Eq.le _) (snorm_add_le\n  (StronglyMeasurable.aestronglyMeasurable\n    ((hf i).indicator (hs.inter (stronglyMeasurable_const.measurableSet_le (hf i).nnnorm))))\n  (StronglyMeasurable.aestronglyMeasurable\n    ((hf i).indicator (hs.inter ((hf i).nnnorm.measurableSet_lt stronglyMeasurable_const))))\n  hp)", "annotated_tactic": ["refine' <a>le_trans</a> (<a>Eq.le</a> _) (<a>snorm_add_le</a>\n        (<a>StronglyMeasurable.aestronglyMeasurable</a>\n          ((hf i).<a>indicator</a> (hs.inter (stronglyMeasurable_const.measurableSet_le (hf i).<a>nnnorm</a>))))\n        (<a>StronglyMeasurable.aestronglyMeasurable</a>\n          ((hf i).<a>indicator</a> (hs.inter ((hf i).nnnorm.measurableSet_lt <a>stronglyMeasurable_const</a>))))\n        hp)", [{"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}, {"full_name": "Eq.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [159, 7], "def_end_pos": [159, 12]}, {"full_name": "MeasureTheory.snorm_add_le", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [802, 9], "def_end_pos": [802, 21]}, {"full_name": "MeasureTheory.StronglyMeasurable.aestronglyMeasurable", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [110, 19], "def_end_pos": [110, 58]}, {"full_name": "MeasureTheory.StronglyMeasurable.indicator", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [825, 19], "def_end_pos": [825, 28]}, {"full_name": "MeasureTheory.StronglyMeasurable.nnnorm", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [845, 19], "def_end_pos": [845, 25]}, {"full_name": "MeasureTheory.StronglyMeasurable.aestronglyMeasurable", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [110, 19], "def_end_pos": [110, 58]}, {"full_name": "MeasureTheory.StronglyMeasurable.indicator", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [825, 19], "def_end_pos": [825, 28]}, {"full_name": "MeasureTheory.stronglyMeasurable_const", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [143, 9], "def_end_pos": [143, 33]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nh : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, 0 < C \u2227 \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nhpzero : p \u2260 0\nh\u03bc : \u00ac\u2191\u2191\u03bc univ = 0\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nC : \u211d\u22650\nhCpos : 0 < C\nhC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal (\u03b5 / 2)\ni : \u03b9\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal ((\u03b5 / (2 * \u2191C)) ^ ENNReal.toReal p)\nh\u03bcs' : \u00ac\u2191\u2191\u03bc s = 0\n\u22a2 snorm (indicator s (f i)) p \u03bc \u2264\n    snorm (indicator (s \u2229 {x | C \u2264 \u2016f i x\u2016\u208a}) (f i)) p \u03bc + snorm (indicator (s \u2229 {x | \u2016f i x\u2016\u208a < C}) (f i)) p \u03bc", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nh : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, 0 < C \u2227 \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nhpzero : p \u2260 0\nh\u03bc : \u00ac\u2191\u2191\u03bc univ = 0\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nC : \u211d\u22650\nhCpos : 0 < C\nhC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal (\u03b5 / 2)\ni : \u03b9\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal ((\u03b5 / (2 * \u2191C)) ^ ENNReal.toReal p)\nh\u03bcs' : \u00ac\u2191\u2191\u03bc s = 0\n\u22a2 snorm (indicator s (f i)) p \u03bc =\n    snorm (indicator (s \u2229 {x | C \u2264 \u2016f i x\u2016\u208a}) (f i) + indicator (s \u2229 {x | \u2016f i x\u2016\u208a < C}) (f i)) p \u03bc"}, {"tactic": "congr", "annotated_tactic": ["congr", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nh : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, 0 < C \u2227 \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nhpzero : p \u2260 0\nh\u03bc : \u00ac\u2191\u2191\u03bc univ = 0\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nC : \u211d\u22650\nhCpos : 0 < C\nhC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal (\u03b5 / 2)\ni : \u03b9\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal ((\u03b5 / (2 * \u2191C)) ^ ENNReal.toReal p)\nh\u03bcs' : \u00ac\u2191\u2191\u03bc s = 0\n\u22a2 snorm (indicator s (f i)) p \u03bc =\n    snorm (indicator (s \u2229 {x | C \u2264 \u2016f i x\u2016\u208a}) (f i) + indicator (s \u2229 {x | \u2016f i x\u2016\u208a < C}) (f i)) p \u03bc", "state_after": "case e_f\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nh : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, 0 < C \u2227 \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nhpzero : p \u2260 0\nh\u03bc : \u00ac\u2191\u2191\u03bc univ = 0\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nC : \u211d\u22650\nhCpos : 0 < C\nhC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal (\u03b5 / 2)\ni : \u03b9\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal ((\u03b5 / (2 * \u2191C)) ^ ENNReal.toReal p)\nh\u03bcs' : \u00ac\u2191\u2191\u03bc s = 0\n\u22a2 indicator s (f i) = indicator (s \u2229 {x | C \u2264 \u2016f i x\u2016\u208a}) (f i) + indicator (s \u2229 {x | \u2016f i x\u2016\u208a < C}) (f i)"}, {"tactic": "change _ = fun x => (s \u2229 { x : \u03b1 | C \u2264 \u2016f i x\u2016\u208a }).indicator (f i) x +\n  (s \u2229 { x : \u03b1 | \u2016f i x\u2016\u208a < C }).indicator (f i) x", "annotated_tactic": ["change _ = fun x => (s \u2229 { x : \u03b1 | C \u2264 \u2016f i x\u2016\u208a }).<a>indicator</a> (f i) x +\n        (s \u2229 { x : \u03b1 | \u2016f i x\u2016\u208a < C }).<a>indicator</a> (f i) x", [{"full_name": "Set.indicator", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [46, 3], "def_end_pos": [46, 14]}, {"full_name": "Set.indicator", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [46, 3], "def_end_pos": [46, 14]}]], "state_before": "case e_f\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nh : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, 0 < C \u2227 \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nhpzero : p \u2260 0\nh\u03bc : \u00ac\u2191\u2191\u03bc univ = 0\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nC : \u211d\u22650\nhCpos : 0 < C\nhC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal (\u03b5 / 2)\ni : \u03b9\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal ((\u03b5 / (2 * \u2191C)) ^ ENNReal.toReal p)\nh\u03bcs' : \u00ac\u2191\u2191\u03bc s = 0\n\u22a2 indicator s (f i) = indicator (s \u2229 {x | C \u2264 \u2016f i x\u2016\u208a}) (f i) + indicator (s \u2229 {x | \u2016f i x\u2016\u208a < C}) (f i)", "state_after": "case e_f\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nh : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, 0 < C \u2227 \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nhpzero : p \u2260 0\nh\u03bc : \u00ac\u2191\u2191\u03bc univ = 0\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nC : \u211d\u22650\nhCpos : 0 < C\nhC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal (\u03b5 / 2)\ni : \u03b9\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal ((\u03b5 / (2 * \u2191C)) ^ ENNReal.toReal p)\nh\u03bcs' : \u00ac\u2191\u2191\u03bc s = 0\n\u22a2 indicator s (f i) = fun x => indicator (s \u2229 {x | C \u2264 \u2016f i x\u2016\u208a}) (f i) x + indicator (s \u2229 {x | \u2016f i x\u2016\u208a < C}) (f i) x"}, {"tactic": "rw [\u2190 Set.indicator_union_of_disjoint]", "annotated_tactic": ["rw [\u2190 <a>Set.indicator_union_of_disjoint</a>]", [{"full_name": "Set.indicator_union_of_disjoint", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [377, 3], "def_end_pos": [377, 14]}]], "state_before": "case e_f\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nh : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, 0 < C \u2227 \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nhpzero : p \u2260 0\nh\u03bc : \u00ac\u2191\u2191\u03bc univ = 0\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nC : \u211d\u22650\nhCpos : 0 < C\nhC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal (\u03b5 / 2)\ni : \u03b9\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal ((\u03b5 / (2 * \u2191C)) ^ ENNReal.toReal p)\nh\u03bcs' : \u00ac\u2191\u2191\u03bc s = 0\n\u22a2 indicator s (f i) = fun x => indicator (s \u2229 {x | C \u2264 \u2016f i x\u2016\u208a}) (f i) x + indicator (s \u2229 {x | \u2016f i x\u2016\u208a < C}) (f i) x", "state_after": "case e_f\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nh : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, 0 < C \u2227 \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nhpzero : p \u2260 0\nh\u03bc : \u00ac\u2191\u2191\u03bc univ = 0\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nC : \u211d\u22650\nhCpos : 0 < C\nhC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal (\u03b5 / 2)\ni : \u03b9\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal ((\u03b5 / (2 * \u2191C)) ^ ENNReal.toReal p)\nh\u03bcs' : \u00ac\u2191\u2191\u03bc s = 0\n\u22a2 indicator s (f i) = indicator (s \u2229 {x | C \u2264 \u2016f i x\u2016\u208a} \u222a s \u2229 {x | \u2016f i x\u2016\u208a < C}) (f i)\n\ncase e_f.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nh : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, 0 < C \u2227 \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nhpzero : p \u2260 0\nh\u03bc : \u00ac\u2191\u2191\u03bc univ = 0\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nC : \u211d\u22650\nhCpos : 0 < C\nhC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal (\u03b5 / 2)\ni : \u03b9\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal ((\u03b5 / (2 * \u2191C)) ^ ENNReal.toReal p)\nh\u03bcs' : \u00ac\u2191\u2191\u03bc s = 0\n\u22a2 Disjoint (s \u2229 {x | C \u2264 \u2016f i x\u2016\u208a}) (s \u2229 {x | \u2016f i x\u2016\u208a < C})"}, {"tactic": "rw [\u2190 Set.inter_union_distrib_left, (by ext; simp [le_or_lt] :\n    { x : \u03b1 | C \u2264 \u2016f i x\u2016\u208a } \u222a { x : \u03b1 | \u2016f i x\u2016\u208a < C } = Set.univ),\n  Set.inter_univ]", "annotated_tactic": ["rw [\u2190 <a>Set.inter_union_distrib_left</a>, (by ext; simp [<a>le_or_lt</a>] :\n            { x : \u03b1 | C \u2264 \u2016f i x\u2016\u208a } \u222a { x : \u03b1 | \u2016f i x\u2016\u208a < C } = Set.univ),\n          <a>Set.inter_univ</a>]", [{"full_name": "Set.inter_union_distrib_left", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1059, 9], "def_end_pos": [1059, 33]}, {"full_name": "le_or_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [340, 9], "def_end_pos": [340, 17]}, {"full_name": "Set.inter_univ", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1012, 9], "def_end_pos": [1012, 19]}]], "state_before": "case e_f\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nh : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, 0 < C \u2227 \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nhpzero : p \u2260 0\nh\u03bc : \u00ac\u2191\u2191\u03bc univ = 0\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nC : \u211d\u22650\nhCpos : 0 < C\nhC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal (\u03b5 / 2)\ni : \u03b9\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal ((\u03b5 / (2 * \u2191C)) ^ ENNReal.toReal p)\nh\u03bcs' : \u00ac\u2191\u2191\u03bc s = 0\n\u22a2 indicator s (f i) = indicator (s \u2229 {x | C \u2264 \u2016f i x\u2016\u208a} \u222a s \u2229 {x | \u2016f i x\u2016\u208a < C}) (f i)", "state_after": "no goals"}, {"tactic": "ext", "annotated_tactic": ["ext", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nh : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, 0 < C \u2227 \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nhpzero : p \u2260 0\nh\u03bc : \u00ac\u2191\u2191\u03bc univ = 0\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nC : \u211d\u22650\nhCpos : 0 < C\nhC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal (\u03b5 / 2)\ni : \u03b9\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal ((\u03b5 / (2 * \u2191C)) ^ ENNReal.toReal p)\nh\u03bcs' : \u00ac\u2191\u2191\u03bc s = 0\n\u22a2 {x | C \u2264 \u2016f i x\u2016\u208a} \u222a {x | \u2016f i x\u2016\u208a < C} = univ", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nh : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, 0 < C \u2227 \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nhpzero : p \u2260 0\nh\u03bc : \u00ac\u2191\u2191\u03bc univ = 0\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nC : \u211d\u22650\nhCpos : 0 < C\nhC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal (\u03b5 / 2)\ni : \u03b9\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal ((\u03b5 / (2 * \u2191C)) ^ ENNReal.toReal p)\nh\u03bcs' : \u00ac\u2191\u2191\u03bc s = 0\nx\u271d : \u03b1\n\u22a2 x\u271d \u2208 {x | C \u2264 \u2016f i x\u2016\u208a} \u222a {x | \u2016f i x\u2016\u208a < C} \u2194 x\u271d \u2208 univ"}, {"tactic": "simp [le_or_lt]", "annotated_tactic": ["simp [<a>le_or_lt</a>]", [{"full_name": "le_or_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [340, 9], "def_end_pos": [340, 17]}]], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nh : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, 0 < C \u2227 \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nhpzero : p \u2260 0\nh\u03bc : \u00ac\u2191\u2191\u03bc univ = 0\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nC : \u211d\u22650\nhCpos : 0 < C\nhC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal (\u03b5 / 2)\ni : \u03b9\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal ((\u03b5 / (2 * \u2191C)) ^ ENNReal.toReal p)\nh\u03bcs' : \u00ac\u2191\u2191\u03bc s = 0\nx\u271d : \u03b1\n\u22a2 x\u271d \u2208 {x | C \u2264 \u2016f i x\u2016\u208a} \u222a {x | \u2016f i x\u2016\u208a < C} \u2194 x\u271d \u2208 univ", "state_after": "no goals"}, {"tactic": "refine' (Disjoint.inf_right' _ _).inf_left' _", "annotated_tactic": ["refine' (<a>Disjoint.inf_right'</a> _ _).<a>inf_left'</a> _", [{"full_name": "Disjoint.inf_right'", "def_path": "Mathlib/Order/Disjoint.lean", "def_pos": [164, 9], "def_end_pos": [164, 28]}, {"full_name": "Disjoint.inf_left'", "def_path": "Mathlib/Order/Disjoint.lean", "def_pos": [156, 9], "def_end_pos": [156, 27]}]], "state_before": "case e_f.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nh : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, 0 < C \u2227 \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nhpzero : p \u2260 0\nh\u03bc : \u00ac\u2191\u2191\u03bc univ = 0\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nC : \u211d\u22650\nhCpos : 0 < C\nhC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal (\u03b5 / 2)\ni : \u03b9\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal ((\u03b5 / (2 * \u2191C)) ^ ENNReal.toReal p)\nh\u03bcs' : \u00ac\u2191\u2191\u03bc s = 0\n\u22a2 Disjoint (s \u2229 {x | C \u2264 \u2016f i x\u2016\u208a}) (s \u2229 {x | \u2016f i x\u2016\u208a < C})", "state_after": "case e_f.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nh : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, 0 < C \u2227 \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nhpzero : p \u2260 0\nh\u03bc : \u00ac\u2191\u2191\u03bc univ = 0\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nC : \u211d\u22650\nhCpos : 0 < C\nhC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal (\u03b5 / 2)\ni : \u03b9\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal ((\u03b5 / (2 * \u2191C)) ^ ENNReal.toReal p)\nh\u03bcs' : \u00ac\u2191\u2191\u03bc s = 0\n\u22a2 Disjoint {x | C \u2264 \u2016f i x\u2016\u208a} {x | \u2016f i x\u2016\u208a < C}"}, {"tactic": "rw [disjoint_iff_inf_le]", "annotated_tactic": ["rw [<a>disjoint_iff_inf_le</a>]", [{"full_name": "disjoint_iff_inf_le", "def_path": "Mathlib/Order/Disjoint.lean", "def_pos": [122, 9], "def_end_pos": [122, 28]}]], "state_before": "case e_f.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nh : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, 0 < C \u2227 \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nhpzero : p \u2260 0\nh\u03bc : \u00ac\u2191\u2191\u03bc univ = 0\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nC : \u211d\u22650\nhCpos : 0 < C\nhC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal (\u03b5 / 2)\ni : \u03b9\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal ((\u03b5 / (2 * \u2191C)) ^ ENNReal.toReal p)\nh\u03bcs' : \u00ac\u2191\u2191\u03bc s = 0\n\u22a2 Disjoint {x | C \u2264 \u2016f i x\u2016\u208a} {x | \u2016f i x\u2016\u208a < C}", "state_after": "case e_f.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nh : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, 0 < C \u2227 \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nhpzero : p \u2260 0\nh\u03bc : \u00ac\u2191\u2191\u03bc univ = 0\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nC : \u211d\u22650\nhCpos : 0 < C\nhC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal (\u03b5 / 2)\ni : \u03b9\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal ((\u03b5 / (2 * \u2191C)) ^ ENNReal.toReal p)\nh\u03bcs' : \u00ac\u2191\u2191\u03bc s = 0\n\u22a2 {x | C \u2264 \u2016f i x\u2016\u208a} \u2293 {x | \u2016f i x\u2016\u208a < C} \u2264 \u22a5"}, {"tactic": "rintro x \u27e8hx\u2081, hx\u2082\u27e9", "annotated_tactic": ["rintro x \u27e8hx\u2081, hx\u2082\u27e9", []], "state_before": "case e_f.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nh : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, 0 < C \u2227 \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nhpzero : p \u2260 0\nh\u03bc : \u00ac\u2191\u2191\u03bc univ = 0\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nC : \u211d\u22650\nhCpos : 0 < C\nhC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal (\u03b5 / 2)\ni : \u03b9\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal ((\u03b5 / (2 * \u2191C)) ^ ENNReal.toReal p)\nh\u03bcs' : \u00ac\u2191\u2191\u03bc s = 0\n\u22a2 {x | C \u2264 \u2016f i x\u2016\u208a} \u2293 {x | \u2016f i x\u2016\u208a < C} \u2264 \u22a5", "state_after": "case e_f.h.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nh : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, 0 < C \u2227 \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nhpzero : p \u2260 0\nh\u03bc : \u00ac\u2191\u2191\u03bc univ = 0\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nC : \u211d\u22650\nhCpos : 0 < C\nhC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal (\u03b5 / 2)\ni : \u03b9\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal ((\u03b5 / (2 * \u2191C)) ^ ENNReal.toReal p)\nh\u03bcs' : \u00ac\u2191\u2191\u03bc s = 0\nx : \u03b1\nhx\u2081 : x \u2208 {x | C \u2264 \u2016f i x\u2016\u208a}\nhx\u2082 : x \u2208 {x | \u2016f i x\u2016\u208a < C}\n\u22a2 x \u2208 \u22a5"}, {"tactic": "rw [Set.mem_setOf_eq] at hx\u2081 hx\u2082", "annotated_tactic": ["rw [<a>Set.mem_setOf_eq</a>] at hx\u2081 hx\u2082", [{"full_name": "Set.mem_setOf_eq", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [256, 29], "def_end_pos": [256, 41]}]], "state_before": "case e_f.h.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nh : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, 0 < C \u2227 \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nhpzero : p \u2260 0\nh\u03bc : \u00ac\u2191\u2191\u03bc univ = 0\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nC : \u211d\u22650\nhCpos : 0 < C\nhC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal (\u03b5 / 2)\ni : \u03b9\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal ((\u03b5 / (2 * \u2191C)) ^ ENNReal.toReal p)\nh\u03bcs' : \u00ac\u2191\u2191\u03bc s = 0\nx : \u03b1\nhx\u2081 : x \u2208 {x | C \u2264 \u2016f i x\u2016\u208a}\nhx\u2082 : x \u2208 {x | \u2016f i x\u2016\u208a < C}\n\u22a2 x \u2208 \u22a5", "state_after": "case e_f.h.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nh : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, 0 < C \u2227 \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nhpzero : p \u2260 0\nh\u03bc : \u00ac\u2191\u2191\u03bc univ = 0\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nC : \u211d\u22650\nhCpos : 0 < C\nhC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal (\u03b5 / 2)\ni : \u03b9\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal ((\u03b5 / (2 * \u2191C)) ^ ENNReal.toReal p)\nh\u03bcs' : \u00ac\u2191\u2191\u03bc s = 0\nx : \u03b1\nhx\u2081 : C \u2264 \u2016f i x\u2016\u208a\nhx\u2082 : \u2016f i x\u2016\u208a < C\n\u22a2 x \u2208 \u22a5"}, {"tactic": "exact False.elim (hx\u2082.ne (eq_of_le_of_not_lt hx\u2081 (not_lt.2 hx\u2082.le)).symm)", "annotated_tactic": ["exact <a>False.elim</a> (hx\u2082.ne (<a>eq_of_le_of_not_lt</a> hx\u2081 (<a>not_lt</a>.2 hx\u2082.le)).<a>symm</a>)", [{"full_name": "False.elim", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [223, 21], "def_end_pos": [223, 31]}, {"full_name": "eq_of_le_of_not_lt", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [437, 9], "def_end_pos": [437, 27]}, {"full_name": "not_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [368, 9], "def_end_pos": [368, 15]}, {"full_name": "Eq.symm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [310, 9], "def_end_pos": [310, 16]}]], "state_before": "case e_f.h.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nh : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, 0 < C \u2227 \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nhpzero : p \u2260 0\nh\u03bc : \u00ac\u2191\u2191\u03bc univ = 0\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nC : \u211d\u22650\nhCpos : 0 < C\nhC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal (\u03b5 / 2)\ni : \u03b9\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal ((\u03b5 / (2 * \u2191C)) ^ ENNReal.toReal p)\nh\u03bcs' : \u00ac\u2191\u2191\u03bc s = 0\nx : \u03b1\nhx\u2081 : C \u2264 \u2016f i x\u2016\u208a\nhx\u2082 : \u2016f i x\u2016\u208a < C\n\u22a2 x \u2208 \u22a5", "state_after": "no goals"}, {"tactic": "refine' add_le_add\n  (snorm_mono fun x => norm_indicator_le_of_subset (Set.inter_subset_right _ _) _ _) _", "annotated_tactic": ["refine' <a>add_le_add</a>\n        (<a>snorm_mono</a> fun x => <a>norm_indicator_le_of_subset</a> (<a>Set.inter_subset_right</a> _ _) _ _) _", [{"full_name": "add_le_add", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [205, 15], "def_end_pos": [205, 25]}, {"full_name": "MeasureTheory.snorm_mono", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [434, 9], "def_end_pos": [434, 19]}, {"full_name": "norm_indicator_le_of_subset", "def_path": "Mathlib/Analysis/NormedSpace/IndicatorFunction.lean", "def_pos": [34, 9], "def_end_pos": [34, 36]}, {"full_name": "Set.inter_subset_right", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [969, 9], "def_end_pos": [969, 27]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nh : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, 0 < C \u2227 \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nhpzero : p \u2260 0\nh\u03bc : \u00ac\u2191\u2191\u03bc univ = 0\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nC : \u211d\u22650\nhCpos : 0 < C\nhC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal (\u03b5 / 2)\ni : \u03b9\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal ((\u03b5 / (2 * \u2191C)) ^ ENNReal.toReal p)\nh\u03bcs' : \u00ac\u2191\u2191\u03bc s = 0\n\u22a2 snorm (indicator (s \u2229 {x | C \u2264 \u2016f i x\u2016\u208a}) (f i)) p \u03bc + snorm (indicator (s \u2229 {x | \u2016f i x\u2016\u208a < C}) (f i)) p \u03bc \u2264\n    snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc + \u2191C * \u2191\u2191\u03bc s ^ (1 / ENNReal.toReal p)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nh : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, 0 < C \u2227 \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nhpzero : p \u2260 0\nh\u03bc : \u00ac\u2191\u2191\u03bc univ = 0\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nC : \u211d\u22650\nhCpos : 0 < C\nhC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal (\u03b5 / 2)\ni : \u03b9\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal ((\u03b5 / (2 * \u2191C)) ^ ENNReal.toReal p)\nh\u03bcs' : \u00ac\u2191\u2191\u03bc s = 0\n\u22a2 snorm (indicator (s \u2229 {x | \u2016f i x\u2016\u208a < C}) (f i)) p \u03bc \u2264 \u2191C * \u2191\u2191\u03bc s ^ (1 / ENNReal.toReal p)"}, {"tactic": "rw [\u2190 Set.indicator_indicator]", "annotated_tactic": ["rw [\u2190 <a>Set.indicator_indicator</a>]", [{"full_name": "Set.indicator_indicator", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [221, 3], "def_end_pos": [221, 14]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nh : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, 0 < C \u2227 \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nhpzero : p \u2260 0\nh\u03bc : \u00ac\u2191\u2191\u03bc univ = 0\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nC : \u211d\u22650\nhCpos : 0 < C\nhC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal (\u03b5 / 2)\ni : \u03b9\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal ((\u03b5 / (2 * \u2191C)) ^ ENNReal.toReal p)\nh\u03bcs' : \u00ac\u2191\u2191\u03bc s = 0\n\u22a2 snorm (indicator (s \u2229 {x | \u2016f i x\u2016\u208a < C}) (f i)) p \u03bc \u2264 \u2191C * \u2191\u2191\u03bc s ^ (1 / ENNReal.toReal p)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nh : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, 0 < C \u2227 \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nhpzero : p \u2260 0\nh\u03bc : \u00ac\u2191\u2191\u03bc univ = 0\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nC : \u211d\u22650\nhCpos : 0 < C\nhC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal (\u03b5 / 2)\ni : \u03b9\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal ((\u03b5 / (2 * \u2191C)) ^ ENNReal.toReal p)\nh\u03bcs' : \u00ac\u2191\u2191\u03bc s = 0\n\u22a2 snorm (indicator s (indicator {x | \u2016f i x\u2016\u208a < C} (f i))) p \u03bc \u2264 \u2191C * \u2191\u2191\u03bc s ^ (1 / ENNReal.toReal p)"}, {"tactic": "rw [snorm_indicator_eq_snorm_restrict hs]", "annotated_tactic": ["rw [<a>snorm_indicator_eq_snorm_restrict</a> hs]", [{"full_name": "MeasureTheory.snorm_indicator_eq_snorm_restrict", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [657, 9], "def_end_pos": [657, 42]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nh : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, 0 < C \u2227 \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nhpzero : p \u2260 0\nh\u03bc : \u00ac\u2191\u2191\u03bc univ = 0\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nC : \u211d\u22650\nhCpos : 0 < C\nhC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal (\u03b5 / 2)\ni : \u03b9\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal ((\u03b5 / (2 * \u2191C)) ^ ENNReal.toReal p)\nh\u03bcs' : \u00ac\u2191\u2191\u03bc s = 0\n\u22a2 snorm (indicator s (indicator {x | \u2016f i x\u2016\u208a < C} (f i))) p \u03bc \u2264 \u2191C * \u2191\u2191\u03bc s ^ (1 / ENNReal.toReal p)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nh : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, 0 < C \u2227 \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nhpzero : p \u2260 0\nh\u03bc : \u00ac\u2191\u2191\u03bc univ = 0\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nC : \u211d\u22650\nhCpos : 0 < C\nhC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal (\u03b5 / 2)\ni : \u03b9\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal ((\u03b5 / (2 * \u2191C)) ^ ENNReal.toReal p)\nh\u03bcs' : \u00ac\u2191\u2191\u03bc s = 0\n\u22a2 snorm (indicator {x | \u2016f i x\u2016\u208a < C} (f i)) p (Measure.restrict \u03bc s) \u2264 \u2191C * \u2191\u2191\u03bc s ^ (1 / ENNReal.toReal p)"}, {"tactic": "have : \u2200\u1d50 x \u2202\u03bc.restrict s, \u2016{ x : \u03b1 | \u2016f i x\u2016\u208a < C }.indicator (f i) x\u2016 \u2264 C := by\n  refine' ae_of_all _ _\n  simp_rw [norm_indicator_eq_indicator_norm]\n  exact Set.indicator_le' (fun x (hx : _ < _) => hx.le) fun _ _ => NNReal.coe_nonneg _", "annotated_tactic": ["have : \u2200\u1d50 x \u2202\u03bc.restrict s, \u2016{ x : \u03b1 | \u2016f i x\u2016\u208a < C }.<a>indicator</a> (f i) x\u2016 \u2264 C := by\n        refine' <a>ae_of_all</a> _ _\n        simp_rw [<a>norm_indicator_eq_indicator_norm</a>]\n        exact <a>Set.indicator_le'</a> (fun x (hx : _ < _) => hx.le) fun _ _ => <a>NNReal.coe_nonneg</a> _", [{"full_name": "Set.indicator", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [46, 3], "def_end_pos": [46, 14]}, {"full_name": "MeasureTheory.ae_of_all", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [407, 9], "def_end_pos": [407, 18]}, {"full_name": "norm_indicator_eq_indicator_norm", "def_path": "Mathlib/Analysis/NormedSpace/IndicatorFunction.lean", "def_pos": [25, 9], "def_end_pos": [25, 41]}, {"full_name": "Set.indicator_le'", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [794, 3], "def_end_pos": [794, 14]}, {"full_name": "NNReal.coe_nonneg", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [134, 9], "def_end_pos": [134, 19]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nh : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, 0 < C \u2227 \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nhpzero : p \u2260 0\nh\u03bc : \u00ac\u2191\u2191\u03bc univ = 0\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nC : \u211d\u22650\nhCpos : 0 < C\nhC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal (\u03b5 / 2)\ni : \u03b9\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal ((\u03b5 / (2 * \u2191C)) ^ ENNReal.toReal p)\nh\u03bcs' : \u00ac\u2191\u2191\u03bc s = 0\n\u22a2 snorm (indicator {x | \u2016f i x\u2016\u208a < C} (f i)) p (Measure.restrict \u03bc s) \u2264 \u2191C * \u2191\u2191\u03bc s ^ (1 / ENNReal.toReal p)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nh : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, 0 < C \u2227 \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nhpzero : p \u2260 0\nh\u03bc : \u00ac\u2191\u2191\u03bc univ = 0\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nC : \u211d\u22650\nhCpos : 0 < C\nhC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal (\u03b5 / 2)\ni : \u03b9\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal ((\u03b5 / (2 * \u2191C)) ^ ENNReal.toReal p)\nh\u03bcs' : \u00ac\u2191\u2191\u03bc s = 0\nthis : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc s, \u2016indicator {x | \u2016f i x\u2016\u208a < C} (f i) x\u2016 \u2264 \u2191C\n\u22a2 snorm (indicator {x | \u2016f i x\u2016\u208a < C} (f i)) p (Measure.restrict \u03bc s) \u2264 \u2191C * \u2191\u2191\u03bc s ^ (1 / ENNReal.toReal p)"}, {"tactic": "refine' le_trans (snorm_le_of_ae_bound this) _", "annotated_tactic": ["refine' <a>le_trans</a> (<a>snorm_le_of_ae_bound</a> this) _", [{"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}, {"full_name": "MeasureTheory.snorm_le_of_ae_bound", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [474, 9], "def_end_pos": [474, 29]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nh : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, 0 < C \u2227 \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nhpzero : p \u2260 0\nh\u03bc : \u00ac\u2191\u2191\u03bc univ = 0\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nC : \u211d\u22650\nhCpos : 0 < C\nhC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal (\u03b5 / 2)\ni : \u03b9\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal ((\u03b5 / (2 * \u2191C)) ^ ENNReal.toReal p)\nh\u03bcs' : \u00ac\u2191\u2191\u03bc s = 0\nthis : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc s, \u2016indicator {x | \u2016f i x\u2016\u208a < C} (f i) x\u2016 \u2264 \u2191C\n\u22a2 snorm (indicator {x | \u2016f i x\u2016\u208a < C} (f i)) p (Measure.restrict \u03bc s) \u2264 \u2191C * \u2191\u2191\u03bc s ^ (1 / ENNReal.toReal p)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nh : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, 0 < C \u2227 \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nhpzero : p \u2260 0\nh\u03bc : \u00ac\u2191\u2191\u03bc univ = 0\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nC : \u211d\u22650\nhCpos : 0 < C\nhC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal (\u03b5 / 2)\ni : \u03b9\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal ((\u03b5 / (2 * \u2191C)) ^ ENNReal.toReal p)\nh\u03bcs' : \u00ac\u2191\u2191\u03bc s = 0\nthis : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc s, \u2016indicator {x | \u2016f i x\u2016\u208a < C} (f i) x\u2016 \u2264 \u2191C\n\u22a2 \u2191\u2191(Measure.restrict \u03bc s) univ ^ (ENNReal.toReal p)\u207b\u00b9 * ENNReal.ofReal \u2191C \u2264 \u2191C * \u2191\u2191\u03bc s ^ (1 / ENNReal.toReal p)"}, {"tactic": "rw [mul_comm, Measure.restrict_apply' hs, Set.univ_inter, ENNReal.ofReal_coe_nnreal, one_div]", "annotated_tactic": ["rw [<a>mul_comm</a>, <a>Measure.restrict_apply'</a> hs, <a>Set.univ_inter</a>, <a>ENNReal.ofReal_coe_nnreal</a>, <a>one_div</a>]", [{"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [302, 9], "def_end_pos": [302, 17]}, {"full_name": "MeasureTheory.Measure.restrict_apply'", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1567, 9], "def_end_pos": [1567, 24]}, {"full_name": "Set.univ_inter", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1017, 9], "def_end_pos": [1017, 19]}, {"full_name": "ENNReal.ofReal_coe_nnreal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [212, 17], "def_end_pos": [212, 34]}, {"full_name": "one_div", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [318, 9], "def_end_pos": [318, 16]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nh : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, 0 < C \u2227 \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nhpzero : p \u2260 0\nh\u03bc : \u00ac\u2191\u2191\u03bc univ = 0\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nC : \u211d\u22650\nhCpos : 0 < C\nhC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal (\u03b5 / 2)\ni : \u03b9\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal ((\u03b5 / (2 * \u2191C)) ^ ENNReal.toReal p)\nh\u03bcs' : \u00ac\u2191\u2191\u03bc s = 0\nthis : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc s, \u2016indicator {x | \u2016f i x\u2016\u208a < C} (f i) x\u2016 \u2264 \u2191C\n\u22a2 \u2191\u2191(Measure.restrict \u03bc s) univ ^ (ENNReal.toReal p)\u207b\u00b9 * ENNReal.ofReal \u2191C \u2264 \u2191C * \u2191\u2191\u03bc s ^ (1 / ENNReal.toReal p)", "state_after": "no goals"}, {"tactic": "refine' ae_of_all _ _", "annotated_tactic": ["refine' <a>ae_of_all</a> _ _", [{"full_name": "MeasureTheory.ae_of_all", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [407, 9], "def_end_pos": [407, 18]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nh : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, 0 < C \u2227 \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nhpzero : p \u2260 0\nh\u03bc : \u00ac\u2191\u2191\u03bc univ = 0\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nC : \u211d\u22650\nhCpos : 0 < C\nhC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal (\u03b5 / 2)\ni : \u03b9\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal ((\u03b5 / (2 * \u2191C)) ^ ENNReal.toReal p)\nh\u03bcs' : \u00ac\u2191\u2191\u03bc s = 0\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc s, \u2016indicator {x | \u2016f i x\u2016\u208a < C} (f i) x\u2016 \u2264 \u2191C", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nh : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, 0 < C \u2227 \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nhpzero : p \u2260 0\nh\u03bc : \u00ac\u2191\u2191\u03bc univ = 0\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nC : \u211d\u22650\nhCpos : 0 < C\nhC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal (\u03b5 / 2)\ni : \u03b9\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal ((\u03b5 / (2 * \u2191C)) ^ ENNReal.toReal p)\nh\u03bcs' : \u00ac\u2191\u2191\u03bc s = 0\n\u22a2 \u2200 (a : \u03b1), \u2016indicator {x | \u2016f i x\u2016\u208a < C} (f i) a\u2016 \u2264 \u2191C"}, {"tactic": "simp_rw [norm_indicator_eq_indicator_norm]", "annotated_tactic": ["simp_rw [<a>norm_indicator_eq_indicator_norm</a>]", [{"full_name": "norm_indicator_eq_indicator_norm", "def_path": "Mathlib/Analysis/NormedSpace/IndicatorFunction.lean", "def_pos": [25, 9], "def_end_pos": [25, 41]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nh : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, 0 < C \u2227 \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nhpzero : p \u2260 0\nh\u03bc : \u00ac\u2191\u2191\u03bc univ = 0\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nC : \u211d\u22650\nhCpos : 0 < C\nhC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal (\u03b5 / 2)\ni : \u03b9\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal ((\u03b5 / (2 * \u2191C)) ^ ENNReal.toReal p)\nh\u03bcs' : \u00ac\u2191\u2191\u03bc s = 0\n\u22a2 \u2200 (a : \u03b1), \u2016indicator {x | \u2016f i x\u2016\u208a < C} (f i) a\u2016 \u2264 \u2191C", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nh : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, 0 < C \u2227 \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nhpzero : p \u2260 0\nh\u03bc : \u00ac\u2191\u2191\u03bc univ = 0\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nC : \u211d\u22650\nhCpos : 0 < C\nhC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal (\u03b5 / 2)\ni : \u03b9\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal ((\u03b5 / (2 * \u2191C)) ^ ENNReal.toReal p)\nh\u03bcs' : \u00ac\u2191\u2191\u03bc s = 0\n\u22a2 \u2200 (a : \u03b1), indicator {x | \u2016f i x\u2016\u208a < C} (fun a => \u2016f i a\u2016) a \u2264 \u2191C"}, {"tactic": "exact Set.indicator_le' (fun x (hx : _ < _) => hx.le) fun _ _ => NNReal.coe_nonneg _", "annotated_tactic": ["exact <a>Set.indicator_le'</a> (fun x (hx : _ < _) => hx.le) fun _ _ => <a>NNReal.coe_nonneg</a> _", [{"full_name": "Set.indicator_le'", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [794, 3], "def_end_pos": [794, 14]}, {"full_name": "NNReal.coe_nonneg", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [134, 9], "def_end_pos": [134, 19]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nh : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, 0 < C \u2227 \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nhpzero : p \u2260 0\nh\u03bc : \u00ac\u2191\u2191\u03bc univ = 0\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nC : \u211d\u22650\nhCpos : 0 < C\nhC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal (\u03b5 / 2)\ni : \u03b9\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal ((\u03b5 / (2 * \u2191C)) ^ ENNReal.toReal p)\nh\u03bcs' : \u00ac\u2191\u2191\u03bc s = 0\n\u22a2 \u2200 (a : \u03b1), indicator {x | \u2016f i x\u2016\u208a < C} (fun a => \u2016f i a\u2016) a \u2264 \u2191C", "state_after": "no goals"}, {"tactic": "refine' add_le_add (hC i) (mul_le_mul_left' _ _)", "annotated_tactic": ["refine' <a>add_le_add</a> (hC i) (<a>mul_le_mul_left'</a> _ _)", [{"full_name": "add_le_add", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [205, 15], "def_end_pos": [205, 25]}, {"full_name": "mul_le_mul_left'", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [50, 9], "def_end_pos": [50, 25]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nh : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, 0 < C \u2227 \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nhpzero : p \u2260 0\nh\u03bc : \u00ac\u2191\u2191\u03bc univ = 0\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nC : \u211d\u22650\nhCpos : 0 < C\nhC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal (\u03b5 / 2)\ni : \u03b9\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal ((\u03b5 / (2 * \u2191C)) ^ ENNReal.toReal p)\nh\u03bcs' : \u00ac\u2191\u2191\u03bc s = 0\n\u22a2 snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc + \u2191C * \u2191\u2191\u03bc s ^ (1 / ENNReal.toReal p) \u2264\n    ENNReal.ofReal (\u03b5 / 2) + \u2191C * ENNReal.ofReal (\u03b5 / (2 * \u2191C))", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nh : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, 0 < C \u2227 \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nhpzero : p \u2260 0\nh\u03bc : \u00ac\u2191\u2191\u03bc univ = 0\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nC : \u211d\u22650\nhCpos : 0 < C\nhC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal (\u03b5 / 2)\ni : \u03b9\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal ((\u03b5 / (2 * \u2191C)) ^ ENNReal.toReal p)\nh\u03bcs' : \u00ac\u2191\u2191\u03bc s = 0\n\u22a2 \u2191\u2191\u03bc s ^ (1 / ENNReal.toReal p) \u2264 ENNReal.ofReal (\u03b5 / (2 * \u2191C))"}, {"tactic": "rwa [ENNReal.rpow_one_div_le_iff (ENNReal.toReal_pos hpzero hp'),\n  ENNReal.ofReal_rpow_of_pos (div_pos h\u03b5 (mul_pos two_pos (NNReal.coe_pos.2 hCpos)))]", "annotated_tactic": ["rwa [<a>ENNReal.rpow_one_div_le_iff</a> (<a>ENNReal.toReal_pos</a> hpzero hp'),\n        <a>ENNReal.ofReal_rpow_of_pos</a> (<a>div_pos</a> h\u03b5 (<a>mul_pos</a> <a>two_pos</a> (<a>NNReal.coe_pos</a>.2 hCpos)))]", [{"full_name": "ENNReal.rpow_one_div_le_iff", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [670, 9], "def_end_pos": [670, 28]}, {"full_name": "ENNReal.toReal_pos", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2131, 9], "def_end_pos": [2131, 19]}, {"full_name": "ENNReal.ofReal_rpow_of_pos", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [816, 9], "def_end_pos": [816, 27]}, {"full_name": "div_pos", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [89, 9], "def_end_pos": [89, 16]}, {"full_name": "mul_pos", "def_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "def_pos": [345, 7], "def_end_pos": [345, 14]}, {"full_name": "two_pos", "def_path": "Mathlib/Algebra/Order/Monoid/NatCast.lean", "def_pos": [113, 7], "def_end_pos": [113, 14]}, {"full_name": "NNReal.coe_pos", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [376, 19], "def_end_pos": [376, 26]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nh : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, 0 < C \u2227 \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nhpzero : p \u2260 0\nh\u03bc : \u00ac\u2191\u2191\u03bc univ = 0\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nC : \u211d\u22650\nhCpos : 0 < C\nhC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal (\u03b5 / 2)\ni : \u03b9\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal ((\u03b5 / (2 * \u2191C)) ^ ENNReal.toReal p)\nh\u03bcs' : \u00ac\u2191\u2191\u03bc s = 0\n\u22a2 \u2191\u2191\u03bc s ^ (1 / ENNReal.toReal p) \u2264 ENNReal.ofReal (\u03b5 / (2 * \u2191C))", "state_after": "no goals"}, {"tactic": "refine' add_le_add_left _ _", "annotated_tactic": ["refine' <a>add_le_add_left</a> _ _", [{"full_name": "add_le_add_left", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [49, 15], "def_end_pos": [49, 30]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nh : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, 0 < C \u2227 \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nhpzero : p \u2260 0\nh\u03bc : \u00ac\u2191\u2191\u03bc univ = 0\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nC : \u211d\u22650\nhCpos : 0 < C\nhC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal (\u03b5 / 2)\ni : \u03b9\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal ((\u03b5 / (2 * \u2191C)) ^ ENNReal.toReal p)\nh\u03bcs' : \u00ac\u2191\u2191\u03bc s = 0\n\u22a2 ENNReal.ofReal (\u03b5 / 2) + \u2191C * ENNReal.ofReal (\u03b5 / (2 * \u2191C)) \u2264 ENNReal.ofReal (\u03b5 / 2) + ENNReal.ofReal (\u03b5 / 2)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nh : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, 0 < C \u2227 \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nhpzero : p \u2260 0\nh\u03bc : \u00ac\u2191\u2191\u03bc univ = 0\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nC : \u211d\u22650\nhCpos : 0 < C\nhC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal (\u03b5 / 2)\ni : \u03b9\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal ((\u03b5 / (2 * \u2191C)) ^ ENNReal.toReal p)\nh\u03bcs' : \u00ac\u2191\u2191\u03bc s = 0\n\u22a2 \u2191C * ENNReal.ofReal (\u03b5 / (2 * \u2191C)) \u2264 ENNReal.ofReal (\u03b5 / 2)"}, {"tactic": "rw [\u2190 ENNReal.ofReal_coe_nnreal, \u2190 ENNReal.ofReal_mul (NNReal.coe_nonneg _), \u2190 div_div,\n  mul_div_cancel' _ (NNReal.coe_pos.2 hCpos).ne.symm]", "annotated_tactic": ["rw [\u2190 <a>ENNReal.ofReal_coe_nnreal</a>, \u2190 <a>ENNReal.ofReal_mul</a> (<a>NNReal.coe_nonneg</a> _), \u2190 <a>div_div</a>,\n        <a>mul_div_cancel'</a> _ (<a>NNReal.coe_pos</a>.2 hCpos).ne.symm]", [{"full_name": "ENNReal.ofReal_coe_nnreal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [212, 17], "def_end_pos": [212, 34]}, {"full_name": "ENNReal.ofReal_mul", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2225, 9], "def_end_pos": [2225, 19]}, {"full_name": "NNReal.coe_nonneg", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [134, 9], "def_end_pos": [134, 19]}, {"full_name": "div_div", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [527, 9], "def_end_pos": [527, 16]}, {"full_name": "mul_div_cancel'", "def_path": "Mathlib/Algebra/GroupWithZero/Units/Lemmas.lean", "def_pos": [173, 9], "def_end_pos": [173, 24]}, {"full_name": "NNReal.coe_pos", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [376, 19], "def_end_pos": [376, 26]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nh : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, 0 < C \u2227 \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nhpzero : p \u2260 0\nh\u03bc : \u00ac\u2191\u2191\u03bc univ = 0\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nC : \u211d\u22650\nhCpos : 0 < C\nhC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal (\u03b5 / 2)\ni : \u03b9\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal ((\u03b5 / (2 * \u2191C)) ^ ENNReal.toReal p)\nh\u03bcs' : \u00ac\u2191\u2191\u03bc s = 0\n\u22a2 \u2191C * ENNReal.ofReal (\u03b5 / (2 * \u2191C)) \u2264 ENNReal.ofReal (\u03b5 / 2)", "state_after": "no goals"}, {"tactic": "rw [\u2190 ENNReal.ofReal_add (half_pos h\u03b5).le (half_pos h\u03b5).le, add_halves]", "annotated_tactic": ["rw [\u2190 <a>ENNReal.ofReal_add</a> (<a>half_pos</a> h\u03b5).<a>le</a> (<a>half_pos</a> h\u03b5).<a>le</a>, <a>add_halves</a>]", [{"full_name": "ENNReal.ofReal_add", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2025, 9], "def_end_pos": [2025, 19]}, {"full_name": "half_pos", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [504, 9], "def_end_pos": [504, 17]}, {"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [142, 7], "def_end_pos": [142, 15]}, {"full_name": "half_pos", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [504, 9], "def_end_pos": [504, 17]}, {"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [142, 7], "def_end_pos": [142, 15]}, {"full_name": "add_halves", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [495, 9], "def_end_pos": [495, 19]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nh : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, 0 < C \u2227 \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nhpzero : p \u2260 0\nh\u03bc : \u00ac\u2191\u2191\u03bc univ = 0\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nC : \u211d\u22650\nhCpos : 0 < C\nhC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal (\u03b5 / 2)\ni : \u03b9\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal ((\u03b5 / (2 * \u2191C)) ^ ENNReal.toReal p)\nh\u03bcs' : \u00ac\u2191\u2191\u03bc s = 0\n\u22a2 ENNReal.ofReal (\u03b5 / 2) + ENNReal.ofReal (\u03b5 / 2) \u2264 ENNReal.ofReal \u03b5", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Quot.lean", "full_name": "Quot.map\u2082_mk", "start": [156, 1], "end": [159, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/Egorov.lean", "full_name": "MeasureTheory.Egorov.exists_notConvergentSeq_lt", "start": [99, 1], "end": [108, 29], "traced_tactics": [{"tactic": "have \u27e8N, hN\u27e9 := (ENNReal.tendsto_atTop ENNReal.zero_ne_top).1\n  (measure_notConvergentSeq_tendsto_zero hf hg hsm hs hfg n) (ENNReal.ofReal (\u03b5 * 2\u207b\u00b9 ^ n)) (by\n    rw [gt_iff_lt, ENNReal.ofReal_pos]\n    exact mul_pos h\u03b5 (pow_pos (by norm_num) n))", "annotated_tactic": ["have \u27e8N, hN\u27e9 := (<a>ENNReal.tendsto_atTop</a> <a>ENNReal.zero_ne_top</a>).1\n    (<a>measure_notConvergentSeq_tendsto_zero</a> hf hg hsm hs hfg n) (<a>ENNReal.ofReal</a> (\u03b5 * 2\u207b\u00b9 ^ n)) (by\n      rw [<a>gt_iff_lt</a>, <a>ENNReal.ofReal_pos</a>]\n      exact <a>mul_pos</a> h\u03b5 (<a>pow_pos</a> (by norm_num) n))", [{"full_name": "ENNReal.tendsto_atTop", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [284, 19], "def_end_pos": [284, 32]}, {"full_name": "ENNReal.zero_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [334, 17], "def_end_pos": [334, 28]}, {"full_name": "MeasureTheory.Egorov.measure_notConvergentSeq_tendsto_zero", "def_path": "Mathlib/MeasureTheory/Function/Egorov.lean", "def_pos": [81, 9], "def_end_pos": [81, 46]}, {"full_name": "ENNReal.ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [172, 29], "def_end_pos": [172, 35]}, {"full_name": "gt_iff_lt", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [366, 9], "def_end_pos": [366, 18]}, {"full_name": "ENNReal.ofReal_pos", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2166, 9], "def_end_pos": [2166, 19]}, {"full_name": "mul_pos", "def_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "def_pos": [345, 7], "def_end_pos": [345, 14]}, {"full_name": "pow_pos", "def_path": "Mathlib/Algebra/Order/Ring/Defs.lean", "def_pos": [530, 9], "def_end_pos": [530, 16]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MetricSpace \u03b2\n\u03bc : Measure \u03b1\nn\u271d : \u2115\ni j : \u03b9\ns : Set \u03b1\n\u03b5 : \u211d\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : SemilatticeSup \u03b9\ninst\u271d\u00b9 : Nonempty \u03b9\ninst\u271d : Countable \u03b9\nh\u03b5 : 0 < \u03b5\nhf : \u2200 (n : \u03b9), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhsm : MeasurableSet s\nhs : \u2191\u2191\u03bc s \u2260 \u22a4\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\nn : \u2115\n\u22a2 \u2203 j, \u2191\u2191\u03bc (s \u2229 notConvergentSeq f g n j) \u2264 ENNReal.ofReal (\u03b5 * 2\u207b\u00b9 ^ n)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MetricSpace \u03b2\n\u03bc : Measure \u03b1\nn\u271d : \u2115\ni j : \u03b9\ns : Set \u03b1\n\u03b5 : \u211d\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : SemilatticeSup \u03b9\ninst\u271d\u00b9 : Nonempty \u03b9\ninst\u271d : Countable \u03b9\nh\u03b5 : 0 < \u03b5\nhf : \u2200 (n : \u03b9), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhsm : MeasurableSet s\nhs : \u2191\u2191\u03bc s \u2260 \u22a4\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\nn : \u2115\nN : \u03b9\nhN :\n  \u2200 (n_1 : \u03b9),\n    n_1 \u2265 N \u2192\n      \u2191\u2191\u03bc (s \u2229 notConvergentSeq (fun n => f n) g n n_1) \u2208\n        Icc (0 - ENNReal.ofReal (\u03b5 * 2\u207b\u00b9 ^ n)) (0 + ENNReal.ofReal (\u03b5 * 2\u207b\u00b9 ^ n))\n\u22a2 \u2203 j, \u2191\u2191\u03bc (s \u2229 notConvergentSeq f g n j) \u2264 ENNReal.ofReal (\u03b5 * 2\u207b\u00b9 ^ n)"}, {"tactic": "rw [zero_add] at hN", "annotated_tactic": ["rw [<a>zero_add</a>] at hN", [{"full_name": "zero_add", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [463, 3], "def_end_pos": [463, 14]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MetricSpace \u03b2\n\u03bc : Measure \u03b1\nn\u271d : \u2115\ni j : \u03b9\ns : Set \u03b1\n\u03b5 : \u211d\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : SemilatticeSup \u03b9\ninst\u271d\u00b9 : Nonempty \u03b9\ninst\u271d : Countable \u03b9\nh\u03b5 : 0 < \u03b5\nhf : \u2200 (n : \u03b9), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhsm : MeasurableSet s\nhs : \u2191\u2191\u03bc s \u2260 \u22a4\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\nn : \u2115\nN : \u03b9\nhN :\n  \u2200 (n_1 : \u03b9),\n    n_1 \u2265 N \u2192\n      \u2191\u2191\u03bc (s \u2229 notConvergentSeq (fun n => f n) g n n_1) \u2208\n        Icc (0 - ENNReal.ofReal (\u03b5 * 2\u207b\u00b9 ^ n)) (0 + ENNReal.ofReal (\u03b5 * 2\u207b\u00b9 ^ n))\n\u22a2 \u2203 j, \u2191\u2191\u03bc (s \u2229 notConvergentSeq f g n j) \u2264 ENNReal.ofReal (\u03b5 * 2\u207b\u00b9 ^ n)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MetricSpace \u03b2\n\u03bc : Measure \u03b1\nn\u271d : \u2115\ni j : \u03b9\ns : Set \u03b1\n\u03b5 : \u211d\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : SemilatticeSup \u03b9\ninst\u271d\u00b9 : Nonempty \u03b9\ninst\u271d : Countable \u03b9\nh\u03b5 : 0 < \u03b5\nhf : \u2200 (n : \u03b9), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhsm : MeasurableSet s\nhs : \u2191\u2191\u03bc s \u2260 \u22a4\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\nn : \u2115\nN : \u03b9\nhN :\n  \u2200 (n_1 : \u03b9),\n    n_1 \u2265 N \u2192\n      \u2191\u2191\u03bc (s \u2229 notConvergentSeq (fun n => f n) g n n_1) \u2208\n        Icc (0 - ENNReal.ofReal (\u03b5 * 2\u207b\u00b9 ^ n)) (ENNReal.ofReal (\u03b5 * 2\u207b\u00b9 ^ n))\n\u22a2 \u2203 j, \u2191\u2191\u03bc (s \u2229 notConvergentSeq f g n j) \u2264 ENNReal.ofReal (\u03b5 * 2\u207b\u00b9 ^ n)"}, {"tactic": "exact \u27e8N, (hN N le_rfl).2\u27e9", "annotated_tactic": ["exact \u27e8N, (hN N <a>le_rfl</a>).2\u27e9", [{"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MetricSpace \u03b2\n\u03bc : Measure \u03b1\nn\u271d : \u2115\ni j : \u03b9\ns : Set \u03b1\n\u03b5 : \u211d\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : SemilatticeSup \u03b9\ninst\u271d\u00b9 : Nonempty \u03b9\ninst\u271d : Countable \u03b9\nh\u03b5 : 0 < \u03b5\nhf : \u2200 (n : \u03b9), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhsm : MeasurableSet s\nhs : \u2191\u2191\u03bc s \u2260 \u22a4\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\nn : \u2115\nN : \u03b9\nhN :\n  \u2200 (n_1 : \u03b9),\n    n_1 \u2265 N \u2192\n      \u2191\u2191\u03bc (s \u2229 notConvergentSeq (fun n => f n) g n n_1) \u2208\n        Icc (0 - ENNReal.ofReal (\u03b5 * 2\u207b\u00b9 ^ n)) (ENNReal.ofReal (\u03b5 * 2\u207b\u00b9 ^ n))\n\u22a2 \u2203 j, \u2191\u2191\u03bc (s \u2229 notConvergentSeq f g n j) \u2264 ENNReal.ofReal (\u03b5 * 2\u207b\u00b9 ^ n)", "state_after": "no goals"}, {"tactic": "rw [gt_iff_lt, ENNReal.ofReal_pos]", "annotated_tactic": ["rw [<a>gt_iff_lt</a>, <a>ENNReal.ofReal_pos</a>]", [{"full_name": "gt_iff_lt", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [366, 9], "def_end_pos": [366, 18]}, {"full_name": "ENNReal.ofReal_pos", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2166, 9], "def_end_pos": [2166, 19]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MetricSpace \u03b2\n\u03bc : Measure \u03b1\nn\u271d : \u2115\ni j : \u03b9\ns : Set \u03b1\n\u03b5 : \u211d\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : SemilatticeSup \u03b9\ninst\u271d\u00b9 : Nonempty \u03b9\ninst\u271d : Countable \u03b9\nh\u03b5 : 0 < \u03b5\nhf : \u2200 (n : \u03b9), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhsm : MeasurableSet s\nhs : \u2191\u2191\u03bc s \u2260 \u22a4\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\nn : \u2115\n\u22a2 ENNReal.ofReal (\u03b5 * 2\u207b\u00b9 ^ n) > 0", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MetricSpace \u03b2\n\u03bc : Measure \u03b1\nn\u271d : \u2115\ni j : \u03b9\ns : Set \u03b1\n\u03b5 : \u211d\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : SemilatticeSup \u03b9\ninst\u271d\u00b9 : Nonempty \u03b9\ninst\u271d : Countable \u03b9\nh\u03b5 : 0 < \u03b5\nhf : \u2200 (n : \u03b9), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhsm : MeasurableSet s\nhs : \u2191\u2191\u03bc s \u2260 \u22a4\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\nn : \u2115\n\u22a2 0 < \u03b5 * 2\u207b\u00b9 ^ n"}, {"tactic": "exact mul_pos h\u03b5 (pow_pos (by norm_num) n)", "annotated_tactic": ["exact <a>mul_pos</a> h\u03b5 (<a>pow_pos</a> (by norm_num) n)", [{"full_name": "mul_pos", "def_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "def_pos": [345, 7], "def_end_pos": [345, 14]}, {"full_name": "pow_pos", "def_path": "Mathlib/Algebra/Order/Ring/Defs.lean", "def_pos": [530, 9], "def_end_pos": [530, 16]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MetricSpace \u03b2\n\u03bc : Measure \u03b1\nn\u271d : \u2115\ni j : \u03b9\ns : Set \u03b1\n\u03b5 : \u211d\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : SemilatticeSup \u03b9\ninst\u271d\u00b9 : Nonempty \u03b9\ninst\u271d : Countable \u03b9\nh\u03b5 : 0 < \u03b5\nhf : \u2200 (n : \u03b9), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhsm : MeasurableSet s\nhs : \u2191\u2191\u03bc s \u2260 \u22a4\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\nn : \u2115\n\u22a2 0 < \u03b5 * 2\u207b\u00b9 ^ n", "state_after": "no goals"}, {"tactic": "norm_num", "annotated_tactic": ["norm_num", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MetricSpace \u03b2\n\u03bc : Measure \u03b1\nn\u271d : \u2115\ni j : \u03b9\ns : Set \u03b1\n\u03b5 : \u211d\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : SemilatticeSup \u03b9\ninst\u271d\u00b9 : Nonempty \u03b9\ninst\u271d : Countable \u03b9\nh\u03b5 : 0 < \u03b5\nhf : \u2200 (n : \u03b9), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhsm : MeasurableSet s\nhs : \u2191\u2191\u03bc s \u2260 \u22a4\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\nn : \u2115\n\u22a2 0 < 2\u207b\u00b9", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Pointwise.lean", "full_name": "Finset.smul_mem_smul_finset_iff", "start": [1948, 1], "end": [1949, 43], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Lp.lean", "full_name": "MeasureTheory.Mem\u2112p.finStronglyMeasurable_of_stronglyMeasurable", "start": [41, 1], "end": [55, 9], "traced_tactics": [{"tactic": "borelize G", "annotated_tactic": ["borelize G", []], "state_before": "\u03b1 : Type u_1\nG : Type u_2\np : \u211d\u22650\u221e\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 G\nhf : Mem\u2112p f p\nhf_meas : StronglyMeasurable f\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\n\u22a2 FinStronglyMeasurable f \u03bc", "state_after": "\u03b1 : Type u_1\nG : Type u_2\np : \u211d\u22650\u221e\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 G\nhf : Mem\u2112p f p\nhf_meas : StronglyMeasurable f\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\n\u22a2 FinStronglyMeasurable f \u03bc"}, {"tactic": "haveI : SeparableSpace (Set.range f \u222a {0} : Set G) :=\n  hf_meas.separableSpace_range_union_singleton", "annotated_tactic": ["haveI : <a>SeparableSpace</a> (<a>Set.range</a> f \u222a {0} : <a>Set</a> G) :=\n    hf_meas.separableSpace_range_union_singleton", [{"full_name": "TopologicalSpace.SeparableSpace", "def_path": "Mathlib/Topology/Bases.lean", "def_pos": [313, 17], "def_end_pos": [313, 31]}, {"full_name": "Set.range", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [668, 5], "def_end_pos": [668, 10]}, {"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}]], "state_before": "\u03b1 : Type u_1\nG : Type u_2\np : \u211d\u22650\u221e\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 G\nhf : Mem\u2112p f p\nhf_meas : StronglyMeasurable f\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\n\u22a2 FinStronglyMeasurable f \u03bc", "state_after": "\u03b1 : Type u_1\nG : Type u_2\np : \u211d\u22650\u221e\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 G\nhf : Mem\u2112p f p\nhf_meas : StronglyMeasurable f\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nthis : SeparableSpace \u2191(Set.range f \u222a {0})\n\u22a2 FinStronglyMeasurable f \u03bc"}, {"tactic": "let fs := SimpleFunc.approxOn f hf_meas.measurable (Set.range f \u222a {0}) 0 (by simp)", "annotated_tactic": ["let fs := <a>SimpleFunc.approxOn</a> f hf_meas.measurable (<a>Set.range</a> f \u222a {0}) 0 (by simp)", [{"full_name": "MeasureTheory.SimpleFunc.approxOn", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDense.lean", "def_pos": [127, 19], "def_end_pos": [127, 27]}, {"full_name": "Set.range", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [668, 5], "def_end_pos": [668, 10]}]], "state_before": "\u03b1 : Type u_1\nG : Type u_2\np : \u211d\u22650\u221e\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 G\nhf : Mem\u2112p f p\nhf_meas : StronglyMeasurable f\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nthis : SeparableSpace \u2191(Set.range f \u222a {0})\n\u22a2 FinStronglyMeasurable f \u03bc", "state_after": "\u03b1 : Type u_1\nG : Type u_2\np : \u211d\u22650\u221e\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 G\nhf : Mem\u2112p f p\nhf_meas : StronglyMeasurable f\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nthis : SeparableSpace \u2191(Set.range f \u222a {0})\nfs : \u2115 \u2192 \u03b1 \u2192\u209b G := SimpleFunc.approxOn f (_ : Measurable f) (Set.range f \u222a {0}) 0 (_ : 0 \u2208 Set.range f \u222a {0})\n\u22a2 FinStronglyMeasurable f \u03bc"}, {"tactic": "refine' \u27e8fs, _, _\u27e9", "annotated_tactic": ["refine' \u27e8fs, _, _\u27e9", []], "state_before": "\u03b1 : Type u_1\nG : Type u_2\np : \u211d\u22650\u221e\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 G\nhf : Mem\u2112p f p\nhf_meas : StronglyMeasurable f\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nthis : SeparableSpace \u2191(Set.range f \u222a {0})\nfs : \u2115 \u2192 \u03b1 \u2192\u209b G := SimpleFunc.approxOn f (_ : Measurable f) (Set.range f \u222a {0}) 0 (_ : 0 \u2208 Set.range f \u222a {0})\n\u22a2 FinStronglyMeasurable f \u03bc", "state_after": "case refine'_1\n\u03b1 : Type u_1\nG : Type u_2\np : \u211d\u22650\u221e\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 G\nhf : Mem\u2112p f p\nhf_meas : StronglyMeasurable f\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nthis : SeparableSpace \u2191(Set.range f \u222a {0})\nfs : \u2115 \u2192 \u03b1 \u2192\u209b G := SimpleFunc.approxOn f (_ : Measurable f) (Set.range f \u222a {0}) 0 (_ : 0 \u2208 Set.range f \u222a {0})\n\u22a2 \u2200 (n : \u2115), \u2191\u2191\u03bc (support \u2191(fs n)) < \u22a4\n\ncase refine'_2\n\u03b1 : Type u_1\nG : Type u_2\np : \u211d\u22650\u221e\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 G\nhf : Mem\u2112p f p\nhf_meas : StronglyMeasurable f\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nthis : SeparableSpace \u2191(Set.range f \u222a {0})\nfs : \u2115 \u2192 \u03b1 \u2192\u209b G := SimpleFunc.approxOn f (_ : Measurable f) (Set.range f \u222a {0}) 0 (_ : 0 \u2208 Set.range f \u222a {0})\n\u22a2 \u2200 (x : \u03b1), Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x))"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03b1 : Type u_1\nG : Type u_2\np : \u211d\u22650\u221e\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 G\nhf : Mem\u2112p f p\nhf_meas : StronglyMeasurable f\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nthis : SeparableSpace \u2191(Set.range f \u222a {0})\n\u22a2 0 \u2208 Set.range f \u222a {0}", "state_after": "no goals"}, {"tactic": "have h_fs_Lp : \u2200 n, Mem\u2112p (fs n) p \u03bc :=\n  SimpleFunc.mem\u2112p_approxOn_range hf_meas.measurable hf", "annotated_tactic": ["have h_fs_Lp : \u2200 n, <a>Mem\u2112p</a> (fs n) p \u03bc :=\n      <a>SimpleFunc.mem\u2112p_approxOn_range</a> hf_meas.measurable hf", [{"full_name": "MeasureTheory.Mem\u2112p", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [108, 5], "def_end_pos": [108, 10]}, {"full_name": "MeasureTheory.SimpleFunc.mem\u2112p_approxOn_range", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "def_pos": [183, 9], "def_end_pos": [183, 29]}]], "state_before": "case refine'_1\n\u03b1 : Type u_1\nG : Type u_2\np : \u211d\u22650\u221e\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 G\nhf : Mem\u2112p f p\nhf_meas : StronglyMeasurable f\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nthis : SeparableSpace \u2191(Set.range f \u222a {0})\nfs : \u2115 \u2192 \u03b1 \u2192\u209b G := SimpleFunc.approxOn f (_ : Measurable f) (Set.range f \u222a {0}) 0 (_ : 0 \u2208 Set.range f \u222a {0})\n\u22a2 \u2200 (n : \u2115), \u2191\u2191\u03bc (support \u2191(fs n)) < \u22a4", "state_after": "case refine'_1\n\u03b1 : Type u_1\nG : Type u_2\np : \u211d\u22650\u221e\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 G\nhf : Mem\u2112p f p\nhf_meas : StronglyMeasurable f\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nthis : SeparableSpace \u2191(Set.range f \u222a {0})\nfs : \u2115 \u2192 \u03b1 \u2192\u209b G := SimpleFunc.approxOn f (_ : Measurable f) (Set.range f \u222a {0}) 0 (_ : 0 \u2208 Set.range f \u222a {0})\nh_fs_Lp : \u2200 (n : \u2115), Mem\u2112p (\u2191(fs n)) p\n\u22a2 \u2200 (n : \u2115), \u2191\u2191\u03bc (support \u2191(fs n)) < \u22a4"}, {"tactic": "exact fun n => (fs n).measure_support_lt_top_of_mem\u2112p (h_fs_Lp n) hp_ne_zero hp_ne_top", "annotated_tactic": ["exact fun n => (fs n).<a>measure_support_lt_top_of_mem\u2112p</a> (h_fs_Lp n) hp_ne_zero hp_ne_top", [{"full_name": "MeasureTheory.SimpleFunc.measure_support_lt_top_of_mem\u2112p", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "def_pos": [407, 9], "def_end_pos": [407, 40]}]], "state_before": "case refine'_1\n\u03b1 : Type u_1\nG : Type u_2\np : \u211d\u22650\u221e\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 G\nhf : Mem\u2112p f p\nhf_meas : StronglyMeasurable f\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nthis : SeparableSpace \u2191(Set.range f \u222a {0})\nfs : \u2115 \u2192 \u03b1 \u2192\u209b G := SimpleFunc.approxOn f (_ : Measurable f) (Set.range f \u222a {0}) 0 (_ : 0 \u2208 Set.range f \u222a {0})\nh_fs_Lp : \u2200 (n : \u2115), Mem\u2112p (\u2191(fs n)) p\n\u22a2 \u2200 (n : \u2115), \u2191\u2191\u03bc (support \u2191(fs n)) < \u22a4", "state_after": "no goals"}, {"tactic": "intro x", "annotated_tactic": ["intro x", []], "state_before": "case refine'_2\n\u03b1 : Type u_1\nG : Type u_2\np : \u211d\u22650\u221e\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 G\nhf : Mem\u2112p f p\nhf_meas : StronglyMeasurable f\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nthis : SeparableSpace \u2191(Set.range f \u222a {0})\nfs : \u2115 \u2192 \u03b1 \u2192\u209b G := SimpleFunc.approxOn f (_ : Measurable f) (Set.range f \u222a {0}) 0 (_ : 0 \u2208 Set.range f \u222a {0})\n\u22a2 \u2200 (x : \u03b1), Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x))", "state_after": "case refine'_2\n\u03b1 : Type u_1\nG : Type u_2\np : \u211d\u22650\u221e\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 G\nhf : Mem\u2112p f p\nhf_meas : StronglyMeasurable f\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nthis : SeparableSpace \u2191(Set.range f \u222a {0})\nfs : \u2115 \u2192 \u03b1 \u2192\u209b G := SimpleFunc.approxOn f (_ : Measurable f) (Set.range f \u222a {0}) 0 (_ : 0 \u2208 Set.range f \u222a {0})\nx : \u03b1\n\u22a2 Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x))"}, {"tactic": "apply SimpleFunc.tendsto_approxOn", "annotated_tactic": ["apply <a>SimpleFunc.tendsto_approxOn</a>", [{"full_name": "MeasureTheory.SimpleFunc.tendsto_approxOn", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDense.lean", "def_pos": [154, 9], "def_end_pos": [154, 25]}]], "state_before": "case refine'_2\n\u03b1 : Type u_1\nG : Type u_2\np : \u211d\u22650\u221e\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 G\nhf : Mem\u2112p f p\nhf_meas : StronglyMeasurable f\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nthis : SeparableSpace \u2191(Set.range f \u222a {0})\nfs : \u2115 \u2192 \u03b1 \u2192\u209b G := SimpleFunc.approxOn f (_ : Measurable f) (Set.range f \u222a {0}) 0 (_ : 0 \u2208 Set.range f \u222a {0})\nx : \u03b1\n\u22a2 Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x))", "state_after": "case refine'_2.hx\n\u03b1 : Type u_1\nG : Type u_2\np : \u211d\u22650\u221e\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 G\nhf : Mem\u2112p f p\nhf_meas : StronglyMeasurable f\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nthis : SeparableSpace \u2191(Set.range f \u222a {0})\nfs : \u2115 \u2192 \u03b1 \u2192\u209b G := SimpleFunc.approxOn f (_ : Measurable f) (Set.range f \u222a {0}) 0 (_ : 0 \u2208 Set.range f \u222a {0})\nx : \u03b1\n\u22a2 f x \u2208 closure (Set.range f \u222a {0})"}, {"tactic": "apply subset_closure", "annotated_tactic": ["apply <a>subset_closure</a>", [{"full_name": "subset_closure", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [435, 9], "def_end_pos": [435, 23]}]], "state_before": "case refine'_2.hx\n\u03b1 : Type u_1\nG : Type u_2\np : \u211d\u22650\u221e\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 G\nhf : Mem\u2112p f p\nhf_meas : StronglyMeasurable f\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nthis : SeparableSpace \u2191(Set.range f \u222a {0})\nfs : \u2115 \u2192 \u03b1 \u2192\u209b G := SimpleFunc.approxOn f (_ : Measurable f) (Set.range f \u222a {0}) 0 (_ : 0 \u2208 Set.range f \u222a {0})\nx : \u03b1\n\u22a2 f x \u2208 closure (Set.range f \u222a {0})", "state_after": "case refine'_2.hx.a\n\u03b1 : Type u_1\nG : Type u_2\np : \u211d\u22650\u221e\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 G\nhf : Mem\u2112p f p\nhf_meas : StronglyMeasurable f\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nthis : SeparableSpace \u2191(Set.range f \u222a {0})\nfs : \u2115 \u2192 \u03b1 \u2192\u209b G := SimpleFunc.approxOn f (_ : Measurable f) (Set.range f \u222a {0}) 0 (_ : 0 \u2208 Set.range f \u222a {0})\nx : \u03b1\n\u22a2 f x \u2208 Set.range f \u222a {0}"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case refine'_2.hx.a\n\u03b1 : Type u_1\nG : Type u_2\np : \u211d\u22650\u221e\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 G\nhf : Mem\u2112p f p\nhf_meas : StronglyMeasurable f\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nthis : SeparableSpace \u2191(Set.range f \u222a {0})\nfs : \u2115 \u2192 \u03b1 \u2192\u209b G := SimpleFunc.approxOn f (_ : Measurable f) (Set.range f \u222a {0}) 0 (_ : 0 \u2208 Set.range f \u222a {0})\nx : \u03b1\n\u22a2 f x \u2208 Set.range f \u222a {0}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/CommRing.lean", "full_name": "MvPolynomial.eval_sub", "start": [138, 1], "end": [139, 18], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Card.lean", "full_name": "Set.Finite.eq_of_subset_of_encard_le'", "start": [211, 1], "end": [213, 65], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Card.lean", "full_name": "Set.encard_strictMono", "start": [218, 1], "end": [219, 46], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Group/Integral.lean", "full_name": "MeasureTheory.integral_smul_eq_self", "start": [167, 1], "end": [170, 34], "traced_tactics": [{"tactic": "have h : MeasurableEmbedding fun x : \u03b1 => g \u2022 x := (MeasurableEquiv.smul g).measurableEmbedding", "annotated_tactic": ["have h : <a>MeasurableEmbedding</a> fun x : \u03b1 => g \u2022 x := (<a>MeasurableEquiv.smul</a> g).<a>measurableEmbedding</a>", [{"full_name": "MeasurableEmbedding", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [1178, 11], "def_end_pos": [1178, 30]}, {"full_name": "MeasurableEquiv.smul", "def_path": "Mathlib/MeasureTheory/Group/MeasurableEquiv.lean", "def_pos": [49, 5], "def_end_pos": [49, 9]}, {"full_name": "MeasurableEquiv.measurableEmbedding", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [1451, 19], "def_end_pos": [1451, 38]}]], "state_before": "\ud835\udd5c : Type u_1\nM : Type u_2\n\u03b1 : Type u_3\nG : Type u_4\nE : Type u_5\nF : Type u_6\ninst\u271d\u2079 : MeasurableSpace G\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : CompleteSpace E\ninst\u271d\u2075 : NormedAddCommGroup F\n\u03bc\u271d : Measure G\nf\u271d : G \u2192 E\ng\u271d : G\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MulAction G \u03b1\ninst\u271d\u00b9 : MeasurableSMul G \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SMulInvariantMeasure G \u03b1 \u03bc\nf : \u03b1 \u2192 E\ng : G\n\u22a2 \u222b (x : \u03b1), f (g \u2022 x) \u2202\u03bc = \u222b (x : \u03b1), f x \u2202\u03bc", "state_after": "\ud835\udd5c : Type u_1\nM : Type u_2\n\u03b1 : Type u_3\nG : Type u_4\nE : Type u_5\nF : Type u_6\ninst\u271d\u2079 : MeasurableSpace G\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : CompleteSpace E\ninst\u271d\u2075 : NormedAddCommGroup F\n\u03bc\u271d : Measure G\nf\u271d : G \u2192 E\ng\u271d : G\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MulAction G \u03b1\ninst\u271d\u00b9 : MeasurableSMul G \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SMulInvariantMeasure G \u03b1 \u03bc\nf : \u03b1 \u2192 E\ng : G\nh : MeasurableEmbedding fun x => g \u2022 x\n\u22a2 \u222b (x : \u03b1), f (g \u2022 x) \u2202\u03bc = \u222b (x : \u03b1), f x \u2202\u03bc"}, {"tactic": "rw [\u2190 h.integral_map, map_smul]", "annotated_tactic": ["rw [\u2190 h.integral_map, <a>map_smul</a>]", [{"full_name": "MeasureTheory.map_smul", "def_path": "Mathlib/MeasureTheory/Group/Action.lean", "def_pos": [100, 9], "def_end_pos": [100, 17]}]], "state_before": "\ud835\udd5c : Type u_1\nM : Type u_2\n\u03b1 : Type u_3\nG : Type u_4\nE : Type u_5\nF : Type u_6\ninst\u271d\u2079 : MeasurableSpace G\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : CompleteSpace E\ninst\u271d\u2075 : NormedAddCommGroup F\n\u03bc\u271d : Measure G\nf\u271d : G \u2192 E\ng\u271d : G\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MulAction G \u03b1\ninst\u271d\u00b9 : MeasurableSMul G \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SMulInvariantMeasure G \u03b1 \u03bc\nf : \u03b1 \u2192 E\ng : G\nh : MeasurableEmbedding fun x => g \u2022 x\n\u22a2 \u222b (x : \u03b1), f (g \u2022 x) \u2202\u03bc = \u222b (x : \u03b1), f x \u2202\u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Pointwise.lean", "full_name": "Finset.mem_inv_smul_finset_iff\u2080", "start": [2071, 1], "end": [2072, 67], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/ZMod/Basic.lean", "full_name": "ZMod.val_add_of_le", "start": [633, 1], "end": [636, 30], "traced_tactics": [{"tactic": "rw [val_add_val_of_le h]", "annotated_tactic": ["rw [<a>val_add_val_of_le</a> h]", [{"full_name": "ZMod.val_add_val_of_le", "def_path": "Mathlib/Data/ZMod/Basic.lean", "def_pos": [627, 9], "def_end_pos": [627, 26]}]], "state_before": "n : \u2115\ninst\u271d : NeZero n\na b : ZMod n\nh : n \u2264 val a + val b\n\u22a2 val (a + b) = val a + val b - n", "state_after": "n : \u2115\ninst\u271d : NeZero n\na b : ZMod n\nh : n \u2264 val a + val b\n\u22a2 val (a + b) = val (a + b) + n - n"}, {"tactic": "exact eq_tsub_of_add_eq rfl", "annotated_tactic": ["exact <a>eq_tsub_of_add_eq</a> <a>rfl</a>", [{"full_name": "eq_tsub_of_add_eq", "def_path": "Mathlib/Algebra/Order/Sub/Defs.lean", "def_pos": [347, 9], "def_end_pos": [347, 26]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "n : \u2115\ninst\u271d : NeZero n\na b : ZMod n\nh : n \u2264 val a + val b\n\u22a2 val (a + b) = val (a + b) + n - n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "full_name": "List.modifyNth_eq_set_get", "start": [901, 1], "end": [903, 49], "traced_tactics": [{"tactic": "rw [modifyNth_eq_set_get?, get?_eq_get h]", "annotated_tactic": ["rw [<a>modifyNth_eq_set_get?</a>, <a>get?_eq_get</a> h]", [{"full_name": "List.modifyNth_eq_set_get?", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [894, 9], "def_end_pos": [894, 30]}, {"full_name": "List.get?_eq_get", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [581, 9], "def_end_pos": [581, 20]}]], "state_before": "\u03b1 : Type u_1\nf : \u03b1 \u2192 \u03b1\nn : Nat\nl : List \u03b1\nh : n < length l\n\u22a2 modifyNth f n l = set l n (f (get l { val := n, isLt := h }))", "state_after": "\u03b1 : Type u_1\nf : \u03b1 \u2192 \u03b1\nn : Nat\nl : List \u03b1\nh : n < length l\n\u22a2 Option.getD ((fun a => set l n (f a)) <$> some (get l { val := n, isLt := h })) l =\n    set l n (f (get l { val := n, isLt := h }))"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\u03b1 : Type u_1\nf : \u03b1 \u2192 \u03b1\nn : Nat\nl : List \u03b1\nh : n < length l\n\u22a2 Option.getD ((fun a => set l n (f a)) <$> some (get l { val := n, isLt := h })) l =\n    set l n (f (get l { val := n, isLt := h }))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Martingale/Centering.lean", "full_name": "MeasureTheory.martingalePart_eq_sum", "start": [75, 1], "end": [79, 72], "traced_tactics": [{"tactic": "unfold martingalePart predictablePart", "annotated_tactic": ["unfold <a>martingalePart</a> <a>predictablePart</a>", [{"full_name": "MeasureTheory.martingalePart", "def_path": "Mathlib/Probability/Martingale/Centering.lean", "def_pos": [66, 19], "def_end_pos": [66, 33]}, {"full_name": "MeasureTheory.predictablePart", "def_path": "Mathlib/Probability/Martingale/Centering.lean", "def_pos": [45, 19], "def_end_pos": [45, 34]}]], "state_before": "\u03a9 : Type u_1\nE : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn : \u2115\n\u22a2 martingalePart f \u2131 \u03bc = fun n => f 0 + \u2211 i in Finset.range n, (f (i + 1) - f i - \u03bc[f (i + 1) - f i|\u2191\u2131 i])", "state_after": "\u03a9 : Type u_1\nE : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn : \u2115\n\u22a2 (fun n => f n - \u2211 i in Finset.range n, \u03bc[f (i + 1) - f i|\u2191\u2131 i]) = fun n =>\n    f 0 + \u2211 i in Finset.range n, (f (i + 1) - f i - \u03bc[f (i + 1) - f i|\u2191\u2131 i])"}, {"tactic": "ext1 n", "annotated_tactic": ["ext1 n", []], "state_before": "\u03a9 : Type u_1\nE : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn : \u2115\n\u22a2 (fun n => f n - \u2211 i in Finset.range n, \u03bc[f (i + 1) - f i|\u2191\u2131 i]) = fun n =>\n    f 0 + \u2211 i in Finset.range n, (f (i + 1) - f i - \u03bc[f (i + 1) - f i|\u2191\u2131 i])", "state_after": "case h\n\u03a9 : Type u_1\nE : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn\u271d n : \u2115\n\u22a2 f n - \u2211 i in Finset.range n, \u03bc[f (i + 1) - f i|\u2191\u2131 i] =\n    f 0 + \u2211 i in Finset.range n, (f (i + 1) - f i - \u03bc[f (i + 1) - f i|\u2191\u2131 i])"}, {"tactic": "rw [Finset.eq_sum_range_sub f n, \u2190 add_sub, \u2190 Finset.sum_sub_distrib]", "annotated_tactic": ["rw [<a>Finset.eq_sum_range_sub</a> f n, \u2190 <a>add_sub</a>, \u2190 <a>Finset.sum_sub_distrib</a>]", [{"full_name": "Finset.eq_sum_range_sub", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [1410, 3], "def_end_pos": [1410, 14]}, {"full_name": "add_sub", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [323, 3], "def_end_pos": [323, 14]}, {"full_name": "Finset.sum_sub_distrib", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [1820, 3], "def_end_pos": [1820, 14]}]], "state_before": "case h\n\u03a9 : Type u_1\nE : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn\u271d n : \u2115\n\u22a2 f n - \u2211 i in Finset.range n, \u03bc[f (i + 1) - f i|\u2191\u2131 i] =\n    f 0 + \u2211 i in Finset.range n, (f (i + 1) - f i - \u03bc[f (i + 1) - f i|\u2191\u2131 i])", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Haar/InnerProductSpace.lean", "full_name": "Orientation.measure_orthonormalBasis", "start": [36, 1], "end": [45, 63], "traced_tactics": [{"tactic": "have e : \u03b9 \u2243 Fin n := by\n  refine' Fintype.equivFinOfCardEq _\n  rw [\u2190 _i.out, finrank_eq_card_basis b.toBasis]", "annotated_tactic": ["have e : \u03b9 \u2243 <a>Fin</a> n := by\n    refine' <a>Fintype.equivFinOfCardEq</a> _\n    rw [\u2190 _i.out, <a>finrank_eq_card_basis</a> b.toBasis]", [{"full_name": "Fin", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1745, 11], "def_end_pos": [1745, 14]}, {"full_name": "Fintype.equivFinOfCardEq", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [176, 19], "def_end_pos": [176, 35]}, {"full_name": "FiniteDimensional.finrank_eq_card_basis", "def_path": "Mathlib/LinearAlgebra/Finrank.lean", "def_pos": [135, 9], "def_end_pos": [135, 30]}]], "state_before": "\u03b9 : Type u_1\nF : Type u_2\ninst\u271d\u2075 : Fintype \u03b9\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\ninst\u271d\u00b2 : FiniteDimensional \u211d F\ninst\u271d\u00b9 : MeasurableSpace F\ninst\u271d : BorelSpace F\nm n : \u2115\n_i : Fact (finrank \u211d F = n)\no : Orientation \u211d F (Fin n)\nb : OrthonormalBasis \u03b9 \u211d F\n\u22a2 \u2191\u2191(AlternatingMap.measure (volumeForm o)) (parallelepiped \u2191b) = 1", "state_after": "\u03b9 : Type u_1\nF : Type u_2\ninst\u271d\u2075 : Fintype \u03b9\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\ninst\u271d\u00b2 : FiniteDimensional \u211d F\ninst\u271d\u00b9 : MeasurableSpace F\ninst\u271d : BorelSpace F\nm n : \u2115\n_i : Fact (finrank \u211d F = n)\no : Orientation \u211d F (Fin n)\nb : OrthonormalBasis \u03b9 \u211d F\ne : \u03b9 \u2243 Fin n\n\u22a2 \u2191\u2191(AlternatingMap.measure (volumeForm o)) (parallelepiped \u2191b) = 1"}, {"tactic": "have A : \u21d1b = b.reindex e \u2218 e := by\n  ext x\n  simp only [OrthonormalBasis.coe_reindex, Function.comp_apply, Equiv.symm_apply_apply]", "annotated_tactic": ["have A : \u21d1b = b.reindex e \u2218 e := by\n    ext x\n    simp only [<a>OrthonormalBasis.coe_reindex</a>, <a>Function.comp_apply</a>, <a>Equiv.symm_apply_apply</a>]", [{"full_name": "OrthonormalBasis.coe_reindex", "def_path": "Mathlib/Analysis/InnerProductSpace/PiL2.lean", "def_pos": [597, 19], "def_end_pos": [597, 30]}, {"full_name": "Function.comp_apply", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [33, 17], "def_end_pos": [33, 36]}, {"full_name": "Equiv.symm_apply_apply", "def_path": "Mathlib/Logic/Equiv/Defs.lean", "def_pos": [283, 17], "def_end_pos": [283, 33]}]], "state_before": "\u03b9 : Type u_1\nF : Type u_2\ninst\u271d\u2075 : Fintype \u03b9\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\ninst\u271d\u00b2 : FiniteDimensional \u211d F\ninst\u271d\u00b9 : MeasurableSpace F\ninst\u271d : BorelSpace F\nm n : \u2115\n_i : Fact (finrank \u211d F = n)\no : Orientation \u211d F (Fin n)\nb : OrthonormalBasis \u03b9 \u211d F\ne : \u03b9 \u2243 Fin n\n\u22a2 \u2191\u2191(AlternatingMap.measure (volumeForm o)) (parallelepiped \u2191b) = 1", "state_after": "\u03b9 : Type u_1\nF : Type u_2\ninst\u271d\u2075 : Fintype \u03b9\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\ninst\u271d\u00b2 : FiniteDimensional \u211d F\ninst\u271d\u00b9 : MeasurableSpace F\ninst\u271d : BorelSpace F\nm n : \u2115\n_i : Fact (finrank \u211d F = n)\no : Orientation \u211d F (Fin n)\nb : OrthonormalBasis \u03b9 \u211d F\ne : \u03b9 \u2243 Fin n\nA : \u2191b = \u2191(OrthonormalBasis.reindex b e) \u2218 \u2191e\n\u22a2 \u2191\u2191(AlternatingMap.measure (volumeForm o)) (parallelepiped \u2191b) = 1"}, {"tactic": "rw [A, parallelepiped_comp_equiv, AlternatingMap.measure_parallelepiped,\n  o.abs_volumeForm_apply_of_orthonormal, ENNReal.ofReal_one]", "annotated_tactic": ["rw [A, <a>parallelepiped_comp_equiv</a>, <a>AlternatingMap.measure_parallelepiped</a>,\n    o.abs_volumeForm_apply_of_orthonormal, <a>ENNReal.ofReal_one</a>]", [{"full_name": "parallelepiped_comp_equiv", "def_path": "Mathlib/MeasureTheory/Measure/Haar/OfBasis.lean", "def_pos": [62, 9], "def_end_pos": [62, 34]}, {"full_name": "AlternatingMap.measure_parallelepiped", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean", "def_pos": [588, 9], "def_end_pos": [588, 53]}, {"full_name": "ENNReal.ofReal_one", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [248, 17], "def_end_pos": [248, 27]}]], "state_before": "\u03b9 : Type u_1\nF : Type u_2\ninst\u271d\u2075 : Fintype \u03b9\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\ninst\u271d\u00b2 : FiniteDimensional \u211d F\ninst\u271d\u00b9 : MeasurableSpace F\ninst\u271d : BorelSpace F\nm n : \u2115\n_i : Fact (finrank \u211d F = n)\no : Orientation \u211d F (Fin n)\nb : OrthonormalBasis \u03b9 \u211d F\ne : \u03b9 \u2243 Fin n\nA : \u2191b = \u2191(OrthonormalBasis.reindex b e) \u2218 \u2191e\n\u22a2 \u2191\u2191(AlternatingMap.measure (volumeForm o)) (parallelepiped \u2191b) = 1", "state_after": "no goals"}, {"tactic": "refine' Fintype.equivFinOfCardEq _", "annotated_tactic": ["refine' <a>Fintype.equivFinOfCardEq</a> _", [{"full_name": "Fintype.equivFinOfCardEq", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [176, 19], "def_end_pos": [176, 35]}]], "state_before": "\u03b9 : Type u_1\nF : Type u_2\ninst\u271d\u2075 : Fintype \u03b9\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\ninst\u271d\u00b2 : FiniteDimensional \u211d F\ninst\u271d\u00b9 : MeasurableSpace F\ninst\u271d : BorelSpace F\nm n : \u2115\n_i : Fact (finrank \u211d F = n)\no : Orientation \u211d F (Fin n)\nb : OrthonormalBasis \u03b9 \u211d F\n\u22a2 \u03b9 \u2243 Fin n", "state_after": "\u03b9 : Type u_1\nF : Type u_2\ninst\u271d\u2075 : Fintype \u03b9\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\ninst\u271d\u00b2 : FiniteDimensional \u211d F\ninst\u271d\u00b9 : MeasurableSpace F\ninst\u271d : BorelSpace F\nm n : \u2115\n_i : Fact (finrank \u211d F = n)\no : Orientation \u211d F (Fin n)\nb : OrthonormalBasis \u03b9 \u211d F\n\u22a2 Fintype.card \u03b9 = n"}, {"tactic": "rw [\u2190 _i.out, finrank_eq_card_basis b.toBasis]", "annotated_tactic": ["rw [\u2190 _i.out, <a>finrank_eq_card_basis</a> b.toBasis]", [{"full_name": "FiniteDimensional.finrank_eq_card_basis", "def_path": "Mathlib/LinearAlgebra/Finrank.lean", "def_pos": [135, 9], "def_end_pos": [135, 30]}]], "state_before": "\u03b9 : Type u_1\nF : Type u_2\ninst\u271d\u2075 : Fintype \u03b9\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\ninst\u271d\u00b2 : FiniteDimensional \u211d F\ninst\u271d\u00b9 : MeasurableSpace F\ninst\u271d : BorelSpace F\nm n : \u2115\n_i : Fact (finrank \u211d F = n)\no : Orientation \u211d F (Fin n)\nb : OrthonormalBasis \u03b9 \u211d F\n\u22a2 Fintype.card \u03b9 = n", "state_after": "no goals"}, {"tactic": "ext x", "annotated_tactic": ["ext x", []], "state_before": "\u03b9 : Type u_1\nF : Type u_2\ninst\u271d\u2075 : Fintype \u03b9\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\ninst\u271d\u00b2 : FiniteDimensional \u211d F\ninst\u271d\u00b9 : MeasurableSpace F\ninst\u271d : BorelSpace F\nm n : \u2115\n_i : Fact (finrank \u211d F = n)\no : Orientation \u211d F (Fin n)\nb : OrthonormalBasis \u03b9 \u211d F\ne : \u03b9 \u2243 Fin n\n\u22a2 \u2191b = \u2191(OrthonormalBasis.reindex b e) \u2218 \u2191e", "state_after": "case h\n\u03b9 : Type u_1\nF : Type u_2\ninst\u271d\u2075 : Fintype \u03b9\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\ninst\u271d\u00b2 : FiniteDimensional \u211d F\ninst\u271d\u00b9 : MeasurableSpace F\ninst\u271d : BorelSpace F\nm n : \u2115\n_i : Fact (finrank \u211d F = n)\no : Orientation \u211d F (Fin n)\nb : OrthonormalBasis \u03b9 \u211d F\ne : \u03b9 \u2243 Fin n\nx : \u03b9\n\u22a2 \u2191b x = (\u2191(OrthonormalBasis.reindex b e) \u2218 \u2191e) x"}, {"tactic": "simp only [OrthonormalBasis.coe_reindex, Function.comp_apply, Equiv.symm_apply_apply]", "annotated_tactic": ["simp only [<a>OrthonormalBasis.coe_reindex</a>, <a>Function.comp_apply</a>, <a>Equiv.symm_apply_apply</a>]", [{"full_name": "OrthonormalBasis.coe_reindex", "def_path": "Mathlib/Analysis/InnerProductSpace/PiL2.lean", "def_pos": [597, 19], "def_end_pos": [597, 30]}, {"full_name": "Function.comp_apply", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [33, 17], "def_end_pos": [33, 36]}, {"full_name": "Equiv.symm_apply_apply", "def_path": "Mathlib/Logic/Equiv/Defs.lean", "def_pos": [283, 17], "def_end_pos": [283, 33]}]], "state_before": "case h\n\u03b9 : Type u_1\nF : Type u_2\ninst\u271d\u2075 : Fintype \u03b9\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\ninst\u271d\u00b2 : FiniteDimensional \u211d F\ninst\u271d\u00b9 : MeasurableSpace F\ninst\u271d : BorelSpace F\nm n : \u2115\n_i : Fact (finrank \u211d F = n)\no : Orientation \u211d F (Fin n)\nb : OrthonormalBasis \u03b9 \u211d F\ne : \u03b9 \u2243 Fin n\nx : \u03b9\n\u22a2 \u2191b x = (\u2191(OrthonormalBasis.reindex b e) \u2218 \u2191e) x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "full_name": "MeasureTheory.integral_piecewise", "start": [204, 1], "end": [209, 56], "traced_tactics": [{"tactic": "rw [\u2190 Set.indicator_add_compl_eq_piecewise,\n  integral_add' (hf.integrable_indicator hs) (hg.integrable_indicator hs.compl),\n  integral_indicator hs, integral_indicator hs.compl]", "annotated_tactic": ["rw [\u2190 <a>Set.indicator_add_compl_eq_piecewise</a>,\n    <a>integral_add'</a> (hf.integrable_indicator hs) (hg.integrable_indicator hs.compl),\n    <a>integral_indicator</a> hs, <a>integral_indicator</a> hs.compl]", [{"full_name": "Set.indicator_add_compl_eq_piecewise", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [453, 3], "def_end_pos": [453, 14]}, {"full_name": "MeasureTheory.integral_add'", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [876, 9], "def_end_pos": [876, 22]}, {"full_name": "MeasureTheory.integral_indicator", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [169, 9], "def_end_pos": [169, 27]}, {"full_name": "MeasureTheory.integral_indicator", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [169, 9], "def_end_pos": [169, 27]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\nf g : \u03b1 \u2192 E\ns t : Set \u03b1\n\u03bc \u03bd : Measure \u03b1\nl l' : Filter \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : DecidablePred fun x => x \u2208 s\nhs : MeasurableSet s\nhf : IntegrableOn f s\nhg : IntegrableOn g s\u1d9c\n\u22a2 \u222b (x : \u03b1), piecewise s f g x \u2202\u03bc = \u222b (x : \u03b1) in s, f x \u2202\u03bc + \u222b (x : \u03b1) in s\u1d9c, g x \u2202\u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Pointwise/Interval.lean", "full_name": "Set.abs_sub_left_of_mem_uIcc", "start": [486, 1], "end": [487, 60], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/AEMeasurable.lean", "full_name": "aemeasurable_restrict_iff_comap_subtype", "start": [278, 1], "end": [280, 93], "traced_tactics": [{"tactic": "rw [\u2190 map_comap_subtype_coe hs, (MeasurableEmbedding.subtype_coe hs).aemeasurable_map_iff]", "annotated_tactic": ["rw [\u2190 <a>map_comap_subtype_coe</a> hs, (<a>MeasurableEmbedding.subtype_coe</a> hs).<a>aemeasurable_map_iff</a>]", [{"full_name": "map_comap_subtype_coe", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [4159, 9], "def_end_pos": [4159, 30]}, {"full_name": "MeasurableEmbedding.subtype_coe", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [1208, 9], "def_end_pos": [1208, 20]}, {"full_name": "MeasurableEmbedding.aemeasurable_map_iff", "def_path": "Mathlib/MeasureTheory/Measure/AEMeasurable.lean", "def_pos": [262, 9], "def_end_pos": [262, 49]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\nR : Type u_6\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\ninst\u271d : MeasurableSpace \u03b4\nf\u271d g : \u03b1 \u2192 \u03b2\n\u03bc\u271d \u03bd : Measure \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u03b2\n\u22a2 AEMeasurable f \u2194 AEMeasurable (f \u2218 Subtype.val)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Kernel/Composition.lean", "full_name": "ProbabilityTheory.kernel.compProd_apply_eq_compProdFun", "start": [216, 1], "end": [227, 6], "traced_tactics": [{"tactic": "rw [compProd, dif_pos]", "annotated_tactic": ["rw [<a>compProd</a>, <a>dif_pos</a>]", [{"full_name": "ProbabilityTheory.kernel.compProd", "def_path": "Mathlib/Probability/Kernel/Composition.lean", "def_pos": [194, 19], "def_end_pos": [194, 27]}, {"full_name": "dif_pos", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [807, 9], "def_end_pos": [807, 16]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03b3 : Type u_4\nm\u03b3 : MeasurableSpace \u03b3\ns : Set (\u03b2 \u00d7 \u03b3)\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d : IsSFiniteKernel \u03b7\na : \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(\u2191(\u03ba \u2297\u2096 \u03b7) a) s = compProdFun \u03ba \u03b7 a s", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03b3 : Type u_4\nm\u03b3 : MeasurableSpace \u03b3\ns : Set (\u03b2 \u00d7 \u03b3)\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d : IsSFiniteKernel \u03b7\na : \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(\u2191{\n                val := fun a =>\n                  Measure.ofMeasurable (fun s x => compProdFun \u03ba \u03b7 a s) (_ : compProdFun \u03ba \u03b7 a \u2205 = 0)\n                    (_ :\n                      \u2200 (f : \u2115 \u2192 Set (\u03b2 \u00d7 \u03b3)),\n                        (\u2200 (i : \u2115), MeasurableSet (f i)) \u2192\n                          Pairwise (Disjoint on f) \u2192\n                            compProdFun \u03ba \u03b7 a (\u22c3 i, f i) = \u2211' (i : \u2115), compProdFun \u03ba \u03b7 a (f i)),\n                property :=\n                  (_ :\n                    Measurable fun a =>\n                      Measure.ofMeasurable (fun s x => compProdFun \u03ba \u03b7 a s) (_ : compProdFun \u03ba \u03b7 a \u2205 = 0)\n                        (_ :\n                          \u2200 (f : \u2115 \u2192 Set (\u03b2 \u00d7 \u03b3)),\n                            (\u2200 (i : \u2115), MeasurableSet (f i)) \u2192\n                              Pairwise (Disjoint on f) \u2192\n                                compProdFun \u03ba \u03b7 a (\u22c3 i, f i) = \u2211' (i : \u2115), compProdFun \u03ba \u03b7 a (f i))) }\n            a)\n      s =\n    compProdFun \u03ba \u03b7 a s\n\ncase hc\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03b3 : Type u_4\nm\u03b3 : MeasurableSpace \u03b3\ns : Set (\u03b2 \u00d7 \u03b3)\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d : IsSFiniteKernel \u03b7\na : \u03b1\nhs : MeasurableSet s\n\u22a2 IsSFiniteKernel \u03ba \u2227 IsSFiniteKernel \u03b7"}, {"tactic": "swap", "annotated_tactic": ["swap", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03b3 : Type u_4\nm\u03b3 : MeasurableSpace \u03b3\ns : Set (\u03b2 \u00d7 \u03b3)\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d : IsSFiniteKernel \u03b7\na : \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(\u2191{\n                val := fun a =>\n                  Measure.ofMeasurable (fun s x => compProdFun \u03ba \u03b7 a s) (_ : compProdFun \u03ba \u03b7 a \u2205 = 0)\n                    (_ :\n                      \u2200 (f : \u2115 \u2192 Set (\u03b2 \u00d7 \u03b3)),\n                        (\u2200 (i : \u2115), MeasurableSet (f i)) \u2192\n                          Pairwise (Disjoint on f) \u2192\n                            compProdFun \u03ba \u03b7 a (\u22c3 i, f i) = \u2211' (i : \u2115), compProdFun \u03ba \u03b7 a (f i)),\n                property :=\n                  (_ :\n                    Measurable fun a =>\n                      Measure.ofMeasurable (fun s x => compProdFun \u03ba \u03b7 a s) (_ : compProdFun \u03ba \u03b7 a \u2205 = 0)\n                        (_ :\n                          \u2200 (f : \u2115 \u2192 Set (\u03b2 \u00d7 \u03b3)),\n                            (\u2200 (i : \u2115), MeasurableSet (f i)) \u2192\n                              Pairwise (Disjoint on f) \u2192\n                                compProdFun \u03ba \u03b7 a (\u22c3 i, f i) = \u2211' (i : \u2115), compProdFun \u03ba \u03b7 a (f i))) }\n            a)\n      s =\n    compProdFun \u03ba \u03b7 a s\n\ncase hc\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03b3 : Type u_4\nm\u03b3 : MeasurableSpace \u03b3\ns : Set (\u03b2 \u00d7 \u03b3)\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d : IsSFiniteKernel \u03b7\na : \u03b1\nhs : MeasurableSet s\n\u22a2 IsSFiniteKernel \u03ba \u2227 IsSFiniteKernel \u03b7", "state_after": "case hc\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03b3 : Type u_4\nm\u03b3 : MeasurableSpace \u03b3\ns : Set (\u03b2 \u00d7 \u03b3)\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d : IsSFiniteKernel \u03b7\na : \u03b1\nhs : MeasurableSet s\n\u22a2 IsSFiniteKernel \u03ba \u2227 IsSFiniteKernel \u03b7\n\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03b3 : Type u_4\nm\u03b3 : MeasurableSpace \u03b3\ns : Set (\u03b2 \u00d7 \u03b3)\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d : IsSFiniteKernel \u03b7\na : \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(\u2191{\n                val := fun a =>\n                  Measure.ofMeasurable (fun s x => compProdFun \u03ba \u03b7 a s) (_ : compProdFun \u03ba \u03b7 a \u2205 = 0)\n                    (_ :\n                      \u2200 (f : \u2115 \u2192 Set (\u03b2 \u00d7 \u03b3)),\n                        (\u2200 (i : \u2115), MeasurableSet (f i)) \u2192\n                          Pairwise (Disjoint on f) \u2192\n                            compProdFun \u03ba \u03b7 a (\u22c3 i, f i) = \u2211' (i : \u2115), compProdFun \u03ba \u03b7 a (f i)),\n                property :=\n                  (_ :\n                    Measurable fun a =>\n                      Measure.ofMeasurable (fun s x => compProdFun \u03ba \u03b7 a s) (_ : compProdFun \u03ba \u03b7 a \u2205 = 0)\n                        (_ :\n                          \u2200 (f : \u2115 \u2192 Set (\u03b2 \u00d7 \u03b3)),\n                            (\u2200 (i : \u2115), MeasurableSet (f i)) \u2192\n                              Pairwise (Disjoint on f) \u2192\n                                compProdFun \u03ba \u03b7 a (\u22c3 i, f i) = \u2211' (i : \u2115), compProdFun \u03ba \u03b7 a (f i))) }\n            a)\n      s =\n    compProdFun \u03ba \u03b7 a s"}, {"tactic": "change\n  Measure.ofMeasurable (fun s _ => compProdFun \u03ba \u03b7 a s) (compProdFun_empty \u03ba \u03b7 a)\n      (compProdFun_iUnion \u03ba \u03b7 a) s =\n    \u222b\u207b b, \u03b7 (a, b) {c | (b, c) \u2208 s} \u2202\u03ba a", "annotated_tactic": ["change\n    <a>Measure.ofMeasurable</a> (fun s _ => <a>compProdFun</a> \u03ba \u03b7 a s) (<a>compProdFun_empty</a> \u03ba \u03b7 a)\n        (<a>compProdFun_iUnion</a> \u03ba \u03b7 a) s =\n      \u222b\u207b b, \u03b7 (a, b) {c | (b, c) \u2208 s} \u2202\u03ba a", [{"full_name": "MeasureTheory.Measure.ofMeasurable", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [103, 5], "def_end_pos": [103, 17]}, {"full_name": "ProbabilityTheory.kernel.compProdFun", "def_path": "Mathlib/Probability/Kernel/Composition.lean", "def_pos": [88, 19], "def_end_pos": [88, 30]}, {"full_name": "ProbabilityTheory.kernel.compProdFun_empty", "def_path": "Mathlib/Probability/Kernel/Composition.lean", "def_pos": [93, 9], "def_end_pos": [93, 26]}, {"full_name": "ProbabilityTheory.kernel.compProdFun_iUnion", "def_path": "Mathlib/Probability/Kernel/Composition.lean", "def_pos": [99, 9], "def_end_pos": [99, 27]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03b3 : Type u_4\nm\u03b3 : MeasurableSpace \u03b3\ns : Set (\u03b2 \u00d7 \u03b3)\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d : IsSFiniteKernel \u03b7\na : \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(\u2191{\n                val := fun a =>\n                  Measure.ofMeasurable (fun s x => compProdFun \u03ba \u03b7 a s) (_ : compProdFun \u03ba \u03b7 a \u2205 = 0)\n                    (_ :\n                      \u2200 (f : \u2115 \u2192 Set (\u03b2 \u00d7 \u03b3)),\n                        (\u2200 (i : \u2115), MeasurableSet (f i)) \u2192\n                          Pairwise (Disjoint on f) \u2192\n                            compProdFun \u03ba \u03b7 a (\u22c3 i, f i) = \u2211' (i : \u2115), compProdFun \u03ba \u03b7 a (f i)),\n                property :=\n                  (_ :\n                    Measurable fun a =>\n                      Measure.ofMeasurable (fun s x => compProdFun \u03ba \u03b7 a s) (_ : compProdFun \u03ba \u03b7 a \u2205 = 0)\n                        (_ :\n                          \u2200 (f : \u2115 \u2192 Set (\u03b2 \u00d7 \u03b3)),\n                            (\u2200 (i : \u2115), MeasurableSet (f i)) \u2192\n                              Pairwise (Disjoint on f) \u2192\n                                compProdFun \u03ba \u03b7 a (\u22c3 i, f i) = \u2211' (i : \u2115), compProdFun \u03ba \u03b7 a (f i))) }\n            a)\n      s =\n    compProdFun \u03ba \u03b7 a s", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03b3 : Type u_4\nm\u03b3 : MeasurableSpace \u03b3\ns : Set (\u03b2 \u00d7 \u03b3)\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d : IsSFiniteKernel \u03b7\na : \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(Measure.ofMeasurable (fun s x => compProdFun \u03ba \u03b7 a s) (_ : compProdFun \u03ba \u03b7 a \u2205 = 0)\n            (_ :\n              \u2200 (f : \u2115 \u2192 Set (\u03b2 \u00d7 \u03b3)),\n                (\u2200 (i : \u2115), MeasurableSet (f i)) \u2192\n                  Pairwise (Disjoint on f) \u2192 compProdFun \u03ba \u03b7 a (\u22c3 i, f i) = \u2211' (i : \u2115), compProdFun \u03ba \u03b7 a (f i)))\n      s =\n    \u222b\u207b (b : \u03b2), \u2191\u2191(\u2191\u03b7 (a, b)) {c | (b, c) \u2208 s} \u2202\u2191\u03ba a"}, {"tactic": "rw [Measure.ofMeasurable_apply _ hs]", "annotated_tactic": ["rw [<a>Measure.ofMeasurable_apply</a> _ hs]", [{"full_name": "MeasureTheory.Measure.ofMeasurable_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [120, 9], "def_end_pos": [120, 27]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03b3 : Type u_4\nm\u03b3 : MeasurableSpace \u03b3\ns : Set (\u03b2 \u00d7 \u03b3)\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d : IsSFiniteKernel \u03b7\na : \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(Measure.ofMeasurable (fun s x => compProdFun \u03ba \u03b7 a s) (_ : compProdFun \u03ba \u03b7 a \u2205 = 0)\n            (_ :\n              \u2200 (f : \u2115 \u2192 Set (\u03b2 \u00d7 \u03b3)),\n                (\u2200 (i : \u2115), MeasurableSet (f i)) \u2192\n                  Pairwise (Disjoint on f) \u2192 compProdFun \u03ba \u03b7 a (\u22c3 i, f i) = \u2211' (i : \u2115), compProdFun \u03ba \u03b7 a (f i)))\n      s =\n    \u222b\u207b (b : \u03b2), \u2191\u2191(\u2191\u03b7 (a, b)) {c | (b, c) \u2208 s} \u2202\u2191\u03ba a", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03b3 : Type u_4\nm\u03b3 : MeasurableSpace \u03b3\ns : Set (\u03b2 \u00d7 \u03b3)\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d : IsSFiniteKernel \u03b7\na : \u03b1\nhs : MeasurableSet s\n\u22a2 compProdFun \u03ba \u03b7 a s = \u222b\u207b (b : \u03b2), \u2191\u2191(\u2191\u03b7 (a, b)) {c | (b, c) \u2208 s} \u2202\u2191\u03ba a"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03b3 : Type u_4\nm\u03b3 : MeasurableSpace \u03b3\ns : Set (\u03b2 \u00d7 \u03b3)\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d : IsSFiniteKernel \u03b7\na : \u03b1\nhs : MeasurableSet s\n\u22a2 compProdFun \u03ba \u03b7 a s = \u222b\u207b (b : \u03b2), \u2191\u2191(\u2191\u03b7 (a, b)) {c | (b, c) \u2208 s} \u2202\u2191\u03ba a", "state_after": "no goals"}, {"tactic": "constructor <;> infer_instance", "annotated_tactic": ["constructor <;> infer_instance", []], "state_before": "case hc\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03b3 : Type u_4\nm\u03b3 : MeasurableSpace \u03b3\ns : Set (\u03b2 \u00d7 \u03b3)\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d : IsSFiniteKernel \u03b7\na : \u03b1\nhs : MeasurableSet s\n\u22a2 IsSFiniteKernel \u03ba \u2227 IsSFiniteKernel \u03b7", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "full_name": "MeasureTheory.VectorMeasure.MutuallySingular.smul_right", "start": [1224, 1], "end": [1227, 91], "traced_tactics": [{"tactic": "simp only [coe_smul, Pi.smul_apply, hs\u2082 t ht, smul_zero]", "annotated_tactic": ["simp only [<a>coe_smul</a>, <a>Pi.smul_apply</a>, hs\u2082 t ht, <a>smul_zero</a>]", [{"full_name": "MeasureTheory.VectorMeasure.coe_smul", "def_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "def_pos": [269, 9], "def_end_pos": [269, 17]}, {"full_name": "Pi.smul_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [116, 60], "def_end_pos": [116, 70]}, {"full_name": "smul_zero", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [732, 9], "def_end_pos": [732, 18]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nL : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u2078 : AddCommMonoid L\ninst\u271d\u2077 : TopologicalSpace L\ninst\u271d\u2076 : AddCommMonoid M\ninst\u271d\u2075 : TopologicalSpace M\ninst\u271d\u2074 : AddCommMonoid N\ninst\u271d\u00b3 : TopologicalSpace N\nv v\u2081 v\u2082 : VectorMeasure \u03b1 M\nw w\u2081 w\u2082 : VectorMeasure \u03b1 N\nR : Type u_6\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : DistribMulAction R N\ninst\u271d : ContinuousConstSMul R N\nr : R\nh : v \u27c2\u1d65 w\ns : Set \u03b1\nhmeas : MeasurableSet s\nhs\u2081 : \u2200 (t : Set \u03b1), t \u2286 s \u2192 \u2191v t = 0\nhs\u2082 : \u2200 (t : Set \u03b1), t \u2286 s\u1d9c \u2192 \u2191w t = 0\nt : Set \u03b1\nht : t \u2286 s\u1d9c\n\u22a2 \u2191(r \u2022 w) t = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "full_name": "MeasureTheory.snorm_one_le_of_le", "start": [1940, 1], "end": [1984, 48], "traced_tactics": [{"tactic": "by_cases hr : r = 0", "annotated_tactic": ["by_cases hr : r = 0", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r\n\u22a2 snorm f 1 \u03bc \u2264 2 * \u2191\u2191\u03bc univ * \u2191r", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r\nhr : r = 0\n\u22a2 snorm f 1 \u03bc \u2264 2 * \u2191\u2191\u03bc univ * \u2191r\n\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r\nhr : \u00acr = 0\n\u22a2 snorm f 1 \u03bc \u2264 2 * \u2191\u2191\u03bc univ * \u2191r"}, {"tactic": "by_cases h\u03bc : IsFiniteMeasure \u03bc", "annotated_tactic": ["by_cases h\u03bc : <a>IsFiniteMeasure</a> \u03bc", [{"full_name": "MeasureTheory.IsFiniteMeasure", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2850, 7], "def_end_pos": [2850, 22]}]], "state_before": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r\nhr : \u00acr = 0\n\u22a2 snorm f 1 \u03bc \u2264 2 * \u2191\u2191\u03bc univ * \u2191r", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r\nhr : \u00acr = 0\nh\u03bc : IsFiniteMeasure \u03bc\n\u22a2 snorm f 1 \u03bc \u2264 2 * \u2191\u2191\u03bc univ * \u2191r\n\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r\nhr : \u00acr = 0\nh\u03bc : \u00acIsFiniteMeasure \u03bc\n\u22a2 snorm f 1 \u03bc \u2264 2 * \u2191\u2191\u03bc univ * \u2191r"}, {"tactic": "swap", "annotated_tactic": ["swap", []], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r\nhr : \u00acr = 0\nh\u03bc : IsFiniteMeasure \u03bc\n\u22a2 snorm f 1 \u03bc \u2264 2 * \u2191\u2191\u03bc univ * \u2191r\n\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r\nhr : \u00acr = 0\nh\u03bc : \u00acIsFiniteMeasure \u03bc\n\u22a2 snorm f 1 \u03bc \u2264 2 * \u2191\u2191\u03bc univ * \u2191r", "state_after": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r\nhr : \u00acr = 0\nh\u03bc : \u00acIsFiniteMeasure \u03bc\n\u22a2 snorm f 1 \u03bc \u2264 2 * \u2191\u2191\u03bc univ * \u2191r\n\ncase pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r\nhr : \u00acr = 0\nh\u03bc : IsFiniteMeasure \u03bc\n\u22a2 snorm f 1 \u03bc \u2264 2 * \u2191\u2191\u03bc univ * \u2191r"}, {"tactic": "haveI := h\u03bc", "annotated_tactic": ["haveI := h\u03bc", []], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r\nhr : \u00acr = 0\nh\u03bc : IsFiniteMeasure \u03bc\n\u22a2 snorm f 1 \u03bc \u2264 2 * \u2191\u2191\u03bc univ * \u2191r", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r\nhr : \u00acr = 0\nh\u03bc this : IsFiniteMeasure \u03bc\n\u22a2 snorm f 1 \u03bc \u2264 2 * \u2191\u2191\u03bc univ * \u2191r"}, {"tactic": "rw [integral_eq_integral_pos_part_sub_integral_neg_part hfint, sub_nonneg] at hfint'", "annotated_tactic": ["rw [<a>integral_eq_integral_pos_part_sub_integral_neg_part</a> hfint, <a>sub_nonneg</a>] at hfint'", [{"full_name": "MeasureTheory.integral_eq_integral_pos_part_sub_integral_neg_part", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1194, 9], "def_end_pos": [1194, 60]}, {"full_name": "sub_nonneg", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [720, 30], "def_end_pos": [720, 40]}]], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r\nhr : \u00acr = 0\nh\u03bc this : IsFiniteMeasure \u03bc\n\u22a2 snorm f 1 \u03bc \u2264 2 * \u2191\u2191\u03bc univ * \u2191r", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : \u222b (a : \u03b1), \u2191(Real.toNNReal (-f a)) \u2202\u03bc \u2264 \u222b (a : \u03b1), \u2191(Real.toNNReal (f a)) \u2202\u03bc\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r\nhr : \u00acr = 0\nh\u03bc this : IsFiniteMeasure \u03bc\n\u22a2 snorm f 1 \u03bc \u2264 2 * \u2191\u2191\u03bc univ * \u2191r"}, {"tactic": "have hposbdd : \u222b \u03c9, max (f \u03c9) 0 \u2202\u03bc \u2264 (\u03bc Set.univ).toReal \u2022 (r : \u211d) := by\n  rw [\u2190 integral_const]\n  refine' integral_mono_ae hfint.real_toNNReal (integrable_const (r : \u211d)) _\n  filter_upwards [hf] with \u03c9 h\u03c9 using Real.toNNReal_le_iff_le_coe.2 h\u03c9", "annotated_tactic": ["have hposbdd : \u222b \u03c9, <a>max</a> (f \u03c9) 0 \u2202\u03bc \u2264 (\u03bc <a>Set.univ</a>).<a>toReal</a> \u2022 (r : \u211d) := by\n    rw [\u2190 <a>integral_const</a>]\n    refine' <a>integral_mono_ae</a> hfint.real_toNNReal (<a>integrable_const</a> (r : \u211d)) _\n    filter_upwards [hf] with \u03c9 h\u03c9 using <a>Real.toNNReal_le_iff_le_coe</a>.2 h\u03c9", [{"full_name": "Max.max", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1090, 3], "def_end_pos": [1090, 6]}, {"full_name": "Set.univ", "def_path": "Mathlib/Init/Set.lean", "def_pos": [90, 5], "def_end_pos": [90, 9]}, {"full_name": "ENNReal.toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [168, 15], "def_end_pos": [168, 21]}, {"full_name": "MeasureTheory.integral_const", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1409, 9], "def_end_pos": [1409, 23]}, {"full_name": "MeasureTheory.integral_mono_ae", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1342, 9], "def_end_pos": [1342, 25]}, {"full_name": "MeasureTheory.integrable_const", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [506, 9], "def_end_pos": [506, 25]}, {"full_name": "Real.toNNReal_le_iff_le_coe", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [685, 9], "def_end_pos": [685, 31]}]], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : \u222b (a : \u03b1), \u2191(Real.toNNReal (-f a)) \u2202\u03bc \u2264 \u222b (a : \u03b1), \u2191(Real.toNNReal (f a)) \u2202\u03bc\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r\nhr : \u00acr = 0\nh\u03bc this : IsFiniteMeasure \u03bc\n\u22a2 snorm f 1 \u03bc \u2264 2 * \u2191\u2191\u03bc univ * \u2191r", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : \u222b (a : \u03b1), \u2191(Real.toNNReal (-f a)) \u2202\u03bc \u2264 \u222b (a : \u03b1), \u2191(Real.toNNReal (f a)) \u2202\u03bc\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r\nhr : \u00acr = 0\nh\u03bc this : IsFiniteMeasure \u03bc\nhposbdd : \u222b (\u03c9 : \u03b1), max (f \u03c9) 0 \u2202\u03bc \u2264 ENNReal.toReal (\u2191\u2191\u03bc univ) \u2022 \u2191r\n\u22a2 snorm f 1 \u03bc \u2264 2 * \u2191\u2191\u03bc univ * \u2191r"}, {"tactic": "rw [Mem\u2112p.snorm_eq_integral_rpow_norm one_ne_zero ENNReal.one_ne_top\n    (mem\u2112p_one_iff_integrable.2 hfint),\n  ENNReal.ofReal_le_iff_le_toReal\n    (ENNReal.mul_ne_top (ENNReal.mul_ne_top ENNReal.two_ne_top <| @measure_ne_top _ _ _ h\u03bc _)\n      ENNReal.coe_ne_top)]", "annotated_tactic": ["rw [<a>Mem\u2112p.snorm_eq_integral_rpow_norm</a> <a>one_ne_zero</a> <a>ENNReal.one_ne_top</a>\n      (<a>mem\u2112p_one_iff_integrable</a>.2 hfint),\n    <a>ENNReal.ofReal_le_iff_le_toReal</a>\n      (<a>ENNReal.mul_ne_top</a> (<a>ENNReal.mul_ne_top</a> <a>ENNReal.two_ne_top</a> <| @<a>measure_ne_top</a> _ _ _ h\u03bc _)\n        <a>ENNReal.coe_ne_top</a>)]", [{"full_name": "MeasureTheory.Mem\u2112p.snorm_eq_integral_rpow_norm", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1321, 9], "def_end_pos": [1321, 42]}, {"full_name": "one_ne_zero", "def_path": "Mathlib/Algebra/NeZero.lean", "def_pos": [55, 15], "def_end_pos": [55, 26]}, {"full_name": "ENNReal.one_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [340, 17], "def_end_pos": [340, 27]}, {"full_name": "MeasureTheory.mem\u2112p_one_iff_integrable", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [453, 9], "def_end_pos": [453, 33]}, {"full_name": "ENNReal.ofReal_le_iff_le_toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2189, 9], "def_end_pos": [2189, 32]}, {"full_name": "ENNReal.mul_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [615, 9], "def_end_pos": [615, 19]}, {"full_name": "ENNReal.mul_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [615, 9], "def_end_pos": [615, 19]}, {"full_name": "ENNReal.two_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [431, 9], "def_end_pos": [431, 19]}, {"full_name": "MeasureTheory.measure_ne_top", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2875, 9], "def_end_pos": [2875, 23]}, {"full_name": "ENNReal.coe_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [302, 17], "def_end_pos": [302, 27]}]], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : \u222b (a : \u03b1), \u2191(Real.toNNReal (-f a)) \u2202\u03bc \u2264 \u222b (a : \u03b1), \u2191(Real.toNNReal (f a)) \u2202\u03bc\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r\nhr : \u00acr = 0\nh\u03bc this : IsFiniteMeasure \u03bc\nhposbdd : \u222b (\u03c9 : \u03b1), max (f \u03c9) 0 \u2202\u03bc \u2264 ENNReal.toReal (\u2191\u2191\u03bc univ) \u2022 \u2191r\n\u22a2 snorm f 1 \u03bc \u2264 2 * \u2191\u2191\u03bc univ * \u2191r", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : \u222b (a : \u03b1), \u2191(Real.toNNReal (-f a)) \u2202\u03bc \u2264 \u222b (a : \u03b1), \u2191(Real.toNNReal (f a)) \u2202\u03bc\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r\nhr : \u00acr = 0\nh\u03bc this : IsFiniteMeasure \u03bc\nhposbdd : \u222b (\u03c9 : \u03b1), max (f \u03c9) 0 \u2202\u03bc \u2264 ENNReal.toReal (\u2191\u2191\u03bc univ) \u2022 \u2191r\n\u22a2 (\u222b (a : \u03b1), \u2016f a\u2016 ^ ENNReal.toReal 1 \u2202\u03bc) ^ (ENNReal.toReal 1)\u207b\u00b9 \u2264 ENNReal.toReal (2 * \u2191\u2191\u03bc univ * \u2191r)"}, {"tactic": "simp_rw [ENNReal.one_toReal, _root_.inv_one, Real.rpow_one, Real.norm_eq_abs, \u2190\n  max_zero_add_max_neg_zero_eq_abs_self, \u2190 Real.coe_toNNReal']", "annotated_tactic": ["simp_rw [<a>ENNReal.one_toReal</a>, <a>_root_.inv_one</a>, <a>Real.rpow_one</a>, <a>Real.norm_eq_abs</a>, \u2190\n    <a>max_zero_add_max_neg_zero_eq_abs_self</a>, \u2190 <a>Real.coe_toNNReal'</a>]", [{"full_name": "ENNReal.one_toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [230, 17], "def_end_pos": [230, 27]}, {"full_name": "inv_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [1015, 9], "def_end_pos": [1015, 16]}, {"full_name": "Real.rpow_one", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [126, 9], "def_end_pos": [126, 17]}, {"full_name": "Real.norm_eq_abs", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [1761, 9], "def_end_pos": [1761, 20]}, {"full_name": "max_zero_add_max_neg_zero_eq_abs_self", "def_path": "Mathlib/Algebra/Order/Ring/Abs.lean", "def_pos": [78, 9], "def_end_pos": [78, 46]}, {"full_name": "Real.coe_toNNReal'", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [615, 9], "def_end_pos": [615, 22]}]], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : \u222b (a : \u03b1), \u2191(Real.toNNReal (-f a)) \u2202\u03bc \u2264 \u222b (a : \u03b1), \u2191(Real.toNNReal (f a)) \u2202\u03bc\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r\nhr : \u00acr = 0\nh\u03bc this : IsFiniteMeasure \u03bc\nhposbdd : \u222b (\u03c9 : \u03b1), max (f \u03c9) 0 \u2202\u03bc \u2264 ENNReal.toReal (\u2191\u2191\u03bc univ) \u2022 \u2191r\n\u22a2 (\u222b (a : \u03b1), \u2016f a\u2016 ^ ENNReal.toReal 1 \u2202\u03bc) ^ (ENNReal.toReal 1)\u207b\u00b9 \u2264 ENNReal.toReal (2 * \u2191\u2191\u03bc univ * \u2191r)", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : \u222b (a : \u03b1), \u2191(Real.toNNReal (-f a)) \u2202\u03bc \u2264 \u222b (a : \u03b1), \u2191(Real.toNNReal (f a)) \u2202\u03bc\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r\nhr : \u00acr = 0\nh\u03bc this : IsFiniteMeasure \u03bc\nhposbdd : \u222b (\u03c9 : \u03b1), max (f \u03c9) 0 \u2202\u03bc \u2264 ENNReal.toReal (\u2191\u2191\u03bc univ) \u2022 \u2191r\n\u22a2 \u222b (a : \u03b1), \u2191(Real.toNNReal (f a)) + \u2191(Real.toNNReal (-f a)) \u2202\u03bc \u2264 ENNReal.toReal (2 * \u2191\u2191\u03bc univ * \u2191r)"}, {"tactic": "rw [integral_add hfint.real_toNNReal]", "annotated_tactic": ["rw [<a>integral_add</a> hfint.real_toNNReal]", [{"full_name": "MeasureTheory.integral_add", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [868, 9], "def_end_pos": [868, 21]}]], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : \u222b (a : \u03b1), \u2191(Real.toNNReal (-f a)) \u2202\u03bc \u2264 \u222b (a : \u03b1), \u2191(Real.toNNReal (f a)) \u2202\u03bc\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r\nhr : \u00acr = 0\nh\u03bc this : IsFiniteMeasure \u03bc\nhposbdd : \u222b (\u03c9 : \u03b1), max (f \u03c9) 0 \u2202\u03bc \u2264 ENNReal.toReal (\u2191\u2191\u03bc univ) \u2022 \u2191r\n\u22a2 \u222b (a : \u03b1), \u2191(Real.toNNReal (f a)) + \u2191(Real.toNNReal (-f a)) \u2202\u03bc \u2264 ENNReal.toReal (2 * \u2191\u2191\u03bc univ * \u2191r)", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : \u222b (a : \u03b1), \u2191(Real.toNNReal (-f a)) \u2202\u03bc \u2264 \u222b (a : \u03b1), \u2191(Real.toNNReal (f a)) \u2202\u03bc\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r\nhr : \u00acr = 0\nh\u03bc this : IsFiniteMeasure \u03bc\nhposbdd : \u222b (\u03c9 : \u03b1), max (f \u03c9) 0 \u2202\u03bc \u2264 ENNReal.toReal (\u2191\u2191\u03bc univ) \u2022 \u2191r\n\u22a2 \u222b (a : \u03b1), \u2191(Real.toNNReal (f a)) \u2202\u03bc + \u222b (a : \u03b1), \u2191(Real.toNNReal (-f a)) \u2202\u03bc \u2264 ENNReal.toReal (2 * \u2191\u2191\u03bc univ * \u2191r)\n\ncase pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : \u222b (a : \u03b1), \u2191(Real.toNNReal (-f a)) \u2202\u03bc \u2264 \u222b (a : \u03b1), \u2191(Real.toNNReal (f a)) \u2202\u03bc\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r\nhr : \u00acr = 0\nh\u03bc this : IsFiniteMeasure \u03bc\nhposbdd : \u222b (\u03c9 : \u03b1), max (f \u03c9) 0 \u2202\u03bc \u2264 ENNReal.toReal (\u2191\u2191\u03bc univ) \u2022 \u2191r\n\u22a2 Integrable fun a => \u2191(Real.toNNReal (-f a))"}, {"tactic": "suffices f =\u1d50[\u03bc] 0 by\n  rw [snorm_congr_ae this, snorm_zero, hr, ENNReal.coe_zero, mul_zero]", "annotated_tactic": ["suffices f =\u1d50[\u03bc] 0 by\n      rw [<a>snorm_congr_ae</a> this, <a>snorm_zero</a>, hr, <a>ENNReal.coe_zero</a>, <a>mul_zero</a>]", [{"full_name": "MeasureTheory.snorm_congr_ae", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [549, 9], "def_end_pos": [549, 23]}, {"full_name": "MeasureTheory.snorm_zero", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [201, 9], "def_end_pos": [201, 19]}, {"full_name": "ENNReal.coe_zero", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [215, 28], "def_end_pos": [215, 36]}, {"full_name": "MulZeroClass.mul_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [38, 3], "def_end_pos": [38, 11]}]], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r\nhr : r = 0\n\u22a2 snorm f 1 \u03bc \u2264 2 * \u2191\u2191\u03bc univ * \u2191r", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r\nhr : r = 0\n\u22a2 f =\u1d50[\u03bc] 0"}, {"tactic": "rw [hr] at hf", "annotated_tactic": ["rw [hr] at hf", []], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r\nhr : r = 0\n\u22a2 f =\u1d50[\u03bc] 0", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u21910\nhr : r = 0\n\u22a2 f =\u1d50[\u03bc] 0"}, {"tactic": "norm_cast at hf", "annotated_tactic": ["norm_cast at hf", []], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u21910\nhr : r = 0\n\u22a2 f =\u1d50[\u03bc] 0", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc\nhr : r = 0\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 0\n\u22a2 f =\u1d50[\u03bc] 0"}, {"tactic": "have hnegf : \u222b x, -f x \u2202\u03bc = 0 := by\n  rw [integral_neg, neg_eq_zero]\n  exact le_antisymm (integral_nonpos_of_ae hf) hfint'", "annotated_tactic": ["have hnegf : \u222b x, -f x \u2202\u03bc = 0 := by\n      rw [<a>integral_neg</a>, <a>neg_eq_zero</a>]\n      exact <a>le_antisymm</a> (<a>integral_nonpos_of_ae</a> hf) hfint'", [{"full_name": "MeasureTheory.integral_neg", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [890, 9], "def_end_pos": [890, 21]}, {"full_name": "neg_eq_zero", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [422, 3], "def_end_pos": [422, 14]}, {"full_name": "le_antisymm", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [188, 9], "def_end_pos": [188, 20]}, {"full_name": "MeasureTheory.integral_nonpos_of_ae", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1255, 9], "def_end_pos": [1255, 30]}]], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc\nhr : r = 0\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 0\n\u22a2 f =\u1d50[\u03bc] 0", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc\nhr : r = 0\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 0\nhnegf : \u222b (x : \u03b1), -f x \u2202\u03bc = 0\n\u22a2 f =\u1d50[\u03bc] 0"}, {"tactic": "have := (integral_eq_zero_iff_of_nonneg_ae ?_ hfint.neg).1 hnegf", "annotated_tactic": ["have := (<a>integral_eq_zero_iff_of_nonneg_ae</a> ?_ hfint.neg).1 hnegf", [{"full_name": "MeasureTheory.integral_eq_zero_iff_of_nonneg_ae", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1265, 9], "def_end_pos": [1265, 42]}]], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc\nhr : r = 0\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 0\nhnegf : \u222b (x : \u03b1), -f x \u2202\u03bc = 0\n\u22a2 f =\u1d50[\u03bc] 0", "state_after": "case pos.refine_2\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc\nhr : r = 0\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 0\nhnegf : \u222b (x : \u03b1), -f x \u2202\u03bc = 0\nthis : -f =\u1d50[\u03bc] 0\n\u22a2 f =\u1d50[\u03bc] 0\n\ncase pos.refine_1\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc\nhr : r = 0\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 0\nhnegf : \u222b (x : \u03b1), -f x \u2202\u03bc = 0\n\u22a2 0 \u2264\u1d50[\u03bc] -f"}, {"tactic": "rw [snorm_congr_ae this, snorm_zero, hr, ENNReal.coe_zero, mul_zero]", "annotated_tactic": ["rw [<a>snorm_congr_ae</a> this, <a>snorm_zero</a>, hr, <a>ENNReal.coe_zero</a>, <a>mul_zero</a>]", [{"full_name": "MeasureTheory.snorm_congr_ae", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [549, 9], "def_end_pos": [549, 23]}, {"full_name": "MeasureTheory.snorm_zero", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [201, 9], "def_end_pos": [201, 19]}, {"full_name": "ENNReal.coe_zero", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [215, 28], "def_end_pos": [215, 36]}, {"full_name": "MulZeroClass.mul_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [38, 3], "def_end_pos": [38, 11]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r\nhr : r = 0\nthis : f =\u1d50[\u03bc] 0\n\u22a2 snorm f 1 \u03bc \u2264 2 * \u2191\u2191\u03bc univ * \u2191r", "state_after": "no goals"}, {"tactic": "rw [integral_neg, neg_eq_zero]", "annotated_tactic": ["rw [<a>integral_neg</a>, <a>neg_eq_zero</a>]", [{"full_name": "MeasureTheory.integral_neg", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [890, 9], "def_end_pos": [890, 21]}, {"full_name": "neg_eq_zero", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [422, 3], "def_end_pos": [422, 14]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc\nhr : r = 0\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 0\n\u22a2 \u222b (x : \u03b1), -f x \u2202\u03bc = 0", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc\nhr : r = 0\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 0\n\u22a2 \u222b (a : \u03b1), f a \u2202\u03bc = 0"}, {"tactic": "exact le_antisymm (integral_nonpos_of_ae hf) hfint'", "annotated_tactic": ["exact <a>le_antisymm</a> (<a>integral_nonpos_of_ae</a> hf) hfint'", [{"full_name": "le_antisymm", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [188, 9], "def_end_pos": [188, 20]}, {"full_name": "MeasureTheory.integral_nonpos_of_ae", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1255, 9], "def_end_pos": [1255, 30]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc\nhr : r = 0\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 0\n\u22a2 \u222b (a : \u03b1), f a \u2202\u03bc = 0", "state_after": "no goals"}, {"tactic": "filter_upwards [this] with \u03c9 h\u03c9", "annotated_tactic": ["filter_upwards [this] with \u03c9 h\u03c9", []], "state_before": "case pos.refine_2\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc\nhr : r = 0\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 0\nhnegf : \u222b (x : \u03b1), -f x \u2202\u03bc = 0\nthis : -f =\u1d50[\u03bc] 0\n\u22a2 f =\u1d50[\u03bc] 0", "state_after": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc\nhr : r = 0\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 0\nhnegf : \u222b (x : \u03b1), -f x \u2202\u03bc = 0\nthis : -f =\u1d50[\u03bc] 0\n\u03c9 : \u03b1\nh\u03c9 : (-f) \u03c9 = OfNat.ofNat 0 \u03c9\n\u22a2 f \u03c9 = OfNat.ofNat 0 \u03c9"}, {"tactic": "rwa [Pi.neg_apply, Pi.zero_apply, neg_eq_zero] at h\u03c9", "annotated_tactic": ["rwa [<a>Pi.neg_apply</a>, <a>Pi.zero_apply</a>, <a>neg_eq_zero</a>] at h\u03c9", [{"full_name": "Pi.neg_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [170, 3], "def_end_pos": [170, 14]}, {"full_name": "Pi.zero_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [46, 3], "def_end_pos": [46, 14]}, {"full_name": "neg_eq_zero", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [422, 3], "def_end_pos": [422, 14]}]], "state_before": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc\nhr : r = 0\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 0\nhnegf : \u222b (x : \u03b1), -f x \u2202\u03bc = 0\nthis : -f =\u1d50[\u03bc] 0\n\u03c9 : \u03b1\nh\u03c9 : (-f) \u03c9 = OfNat.ofNat 0 \u03c9\n\u22a2 f \u03c9 = OfNat.ofNat 0 \u03c9", "state_after": "no goals"}, {"tactic": "filter_upwards [hf] with \u03c9 h\u03c9", "annotated_tactic": ["filter_upwards [hf] with \u03c9 h\u03c9", []], "state_before": "case pos.refine_1\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc\nhr : r = 0\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 0\nhnegf : \u222b (x : \u03b1), -f x \u2202\u03bc = 0\n\u22a2 0 \u2264\u1d50[\u03bc] -f", "state_after": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc\nhr : r = 0\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 0\nhnegf : \u222b (x : \u03b1), -f x \u2202\u03bc = 0\n\u03c9 : \u03b1\nh\u03c9 : f \u03c9 \u2264 0\n\u22a2 OfNat.ofNat 0 \u03c9 \u2264 (-f) \u03c9"}, {"tactic": "rwa [Pi.zero_apply, Pi.neg_apply, Right.nonneg_neg_iff]", "annotated_tactic": ["rwa [<a>Pi.zero_apply</a>, <a>Pi.neg_apply</a>, <a>Right.nonneg_neg_iff</a>]", [{"full_name": "Pi.zero_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [46, 3], "def_end_pos": [46, 14]}, {"full_name": "Pi.neg_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [170, 3], "def_end_pos": [170, 14]}, {"full_name": "Right.nonneg_neg_iff", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [222, 3], "def_end_pos": [222, 14]}]], "state_before": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc\nhr : r = 0\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 0\nhnegf : \u222b (x : \u03b1), -f x \u2202\u03bc = 0\n\u03c9 : \u03b1\nh\u03c9 : f \u03c9 \u2264 0\n\u22a2 OfNat.ofNat 0 \u03c9 \u2264 (-f) \u03c9", "state_after": "no goals"}, {"tactic": "have : \u03bc Set.univ = \u221e := by\n  by_contra h\u03bc'\n  exact h\u03bc (IsFiniteMeasure.mk <| lt_top_iff_ne_top.2 h\u03bc')", "annotated_tactic": ["have : \u03bc <a>Set.univ</a> = \u221e := by\n      by_contra h\u03bc'\n      exact h\u03bc (<a>IsFiniteMeasure.mk</a> <| <a>lt_top_iff_ne_top</a>.2 h\u03bc')", [{"full_name": "Set.univ", "def_path": "Mathlib/Init/Set.lean", "def_pos": [90, 5], "def_end_pos": [90, 9]}, {"full_name": "MeasureTheory.IsFiniteMeasure.mk", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2850, 23], "def_end_pos": [2850, 38]}, {"full_name": "lt_top_iff_ne_top", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [173, 9], "def_end_pos": [173, 26]}]], "state_before": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r\nhr : \u00acr = 0\nh\u03bc : \u00acIsFiniteMeasure \u03bc\n\u22a2 snorm f 1 \u03bc \u2264 2 * \u2191\u2191\u03bc univ * \u2191r", "state_after": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r\nhr : \u00acr = 0\nh\u03bc : \u00acIsFiniteMeasure \u03bc\nthis : \u2191\u2191\u03bc univ = \u22a4\n\u22a2 snorm f 1 \u03bc \u2264 2 * \u2191\u2191\u03bc univ * \u2191r"}, {"tactic": "rw [this, ENNReal.mul_top', if_neg, ENNReal.top_mul', if_neg]", "annotated_tactic": ["rw [this, <a>ENNReal.mul_top'</a>, <a>if_neg</a>, <a>ENNReal.top_mul'</a>, <a>if_neg</a>]", [{"full_name": "ENNReal.mul_top'", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [577, 9], "def_end_pos": [577, 17]}, {"full_name": "if_neg", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [795, 9], "def_end_pos": [795, 15]}, {"full_name": "ENNReal.top_mul'", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [583, 9], "def_end_pos": [583, 17]}, {"full_name": "if_neg", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [795, 9], "def_end_pos": [795, 15]}]], "state_before": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r\nhr : \u00acr = 0\nh\u03bc : \u00acIsFiniteMeasure \u03bc\nthis : \u2191\u2191\u03bc univ = \u22a4\n\u22a2 snorm f 1 \u03bc \u2264 2 * \u2191\u2191\u03bc univ * \u2191r", "state_after": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r\nhr : \u00acr = 0\nh\u03bc : \u00acIsFiniteMeasure \u03bc\nthis : \u2191\u2191\u03bc univ = \u22a4\n\u22a2 snorm f 1 \u03bc \u2264 \u22a4\n\ncase neg.hnc\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r\nhr : \u00acr = 0\nh\u03bc : \u00acIsFiniteMeasure \u03bc\nthis : \u2191\u2191\u03bc univ = \u22a4\n\u22a2 \u00ac\u2191r = 0\n\ncase neg.hnc\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r\nhr : \u00acr = 0\nh\u03bc : \u00acIsFiniteMeasure \u03bc\nthis : \u2191\u2191\u03bc univ = \u22a4\n\u22a2 \u00ac2 = 0"}, {"tactic": "by_contra h\u03bc'", "annotated_tactic": ["by_contra h\u03bc'", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r\nhr : \u00acr = 0\nh\u03bc : \u00acIsFiniteMeasure \u03bc\n\u22a2 \u2191\u2191\u03bc univ = \u22a4", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r\nhr : \u00acr = 0\nh\u03bc : \u00acIsFiniteMeasure \u03bc\nh\u03bc' : \u00ac\u2191\u2191\u03bc univ = \u22a4\n\u22a2 False"}, {"tactic": "exact h\u03bc (IsFiniteMeasure.mk <| lt_top_iff_ne_top.2 h\u03bc')", "annotated_tactic": ["exact h\u03bc (<a>IsFiniteMeasure.mk</a> <| <a>lt_top_iff_ne_top</a>.2 h\u03bc')", [{"full_name": "MeasureTheory.IsFiniteMeasure.mk", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2850, 23], "def_end_pos": [2850, 38]}, {"full_name": "lt_top_iff_ne_top", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [173, 9], "def_end_pos": [173, 26]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r\nhr : \u00acr = 0\nh\u03bc : \u00acIsFiniteMeasure \u03bc\nh\u03bc' : \u00ac\u2191\u2191\u03bc univ = \u22a4\n\u22a2 False", "state_after": "no goals"}, {"tactic": "exact le_top", "annotated_tactic": ["exact <a>le_top</a>", [{"full_name": "le_top", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [98, 9], "def_end_pos": [98, 15]}]], "state_before": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r\nhr : \u00acr = 0\nh\u03bc : \u00acIsFiniteMeasure \u03bc\nthis : \u2191\u2191\u03bc univ = \u22a4\n\u22a2 snorm f 1 \u03bc \u2264 \u22a4", "state_after": "no goals"}, {"tactic": "simp [hr]", "annotated_tactic": ["simp [hr]", []], "state_before": "case neg.hnc\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r\nhr : \u00acr = 0\nh\u03bc : \u00acIsFiniteMeasure \u03bc\nthis : \u2191\u2191\u03bc univ = \u22a4\n\u22a2 \u00ac\u2191r = 0", "state_after": "no goals"}, {"tactic": "norm_num", "annotated_tactic": ["norm_num", []], "state_before": "case neg.hnc\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r\nhr : \u00acr = 0\nh\u03bc : \u00acIsFiniteMeasure \u03bc\nthis : \u2191\u2191\u03bc univ = \u22a4\n\u22a2 \u00ac2 = 0", "state_after": "no goals"}, {"tactic": "rw [\u2190 integral_const]", "annotated_tactic": ["rw [\u2190 <a>integral_const</a>]", [{"full_name": "MeasureTheory.integral_const", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1409, 9], "def_end_pos": [1409, 23]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : \u222b (a : \u03b1), \u2191(Real.toNNReal (-f a)) \u2202\u03bc \u2264 \u222b (a : \u03b1), \u2191(Real.toNNReal (f a)) \u2202\u03bc\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r\nhr : \u00acr = 0\nh\u03bc this : IsFiniteMeasure \u03bc\n\u22a2 \u222b (\u03c9 : \u03b1), max (f \u03c9) 0 \u2202\u03bc \u2264 ENNReal.toReal (\u2191\u2191\u03bc univ) \u2022 \u2191r", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : \u222b (a : \u03b1), \u2191(Real.toNNReal (-f a)) \u2202\u03bc \u2264 \u222b (a : \u03b1), \u2191(Real.toNNReal (f a)) \u2202\u03bc\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r\nhr : \u00acr = 0\nh\u03bc this : IsFiniteMeasure \u03bc\n\u22a2 \u222b (\u03c9 : \u03b1), max (f \u03c9) 0 \u2202\u03bc \u2264 \u222b (x : \u03b1), \u2191r \u2202\u03bc"}, {"tactic": "refine' integral_mono_ae hfint.real_toNNReal (integrable_const (r : \u211d)) _", "annotated_tactic": ["refine' <a>integral_mono_ae</a> hfint.real_toNNReal (<a>integrable_const</a> (r : \u211d)) _", [{"full_name": "MeasureTheory.integral_mono_ae", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1342, 9], "def_end_pos": [1342, 25]}, {"full_name": "MeasureTheory.integrable_const", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [506, 9], "def_end_pos": [506, 25]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : \u222b (a : \u03b1), \u2191(Real.toNNReal (-f a)) \u2202\u03bc \u2264 \u222b (a : \u03b1), \u2191(Real.toNNReal (f a)) \u2202\u03bc\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r\nhr : \u00acr = 0\nh\u03bc this : IsFiniteMeasure \u03bc\n\u22a2 \u222b (\u03c9 : \u03b1), max (f \u03c9) 0 \u2202\u03bc \u2264 \u222b (x : \u03b1), \u2191r \u2202\u03bc", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : \u222b (a : \u03b1), \u2191(Real.toNNReal (-f a)) \u2202\u03bc \u2264 \u222b (a : \u03b1), \u2191(Real.toNNReal (f a)) \u2202\u03bc\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r\nhr : \u00acr = 0\nh\u03bc this : IsFiniteMeasure \u03bc\n\u22a2 (fun \u03c9 => max (f \u03c9) 0) \u2264\u1d50[\u03bc] fun x => \u2191r"}, {"tactic": "filter_upwards [hf] with \u03c9 h\u03c9 using Real.toNNReal_le_iff_le_coe.2 h\u03c9", "annotated_tactic": ["filter_upwards [hf] with \u03c9 h\u03c9 using <a>Real.toNNReal_le_iff_le_coe</a>.2 h\u03c9", [{"full_name": "Real.toNNReal_le_iff_le_coe", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [685, 9], "def_end_pos": [685, 31]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : \u222b (a : \u03b1), \u2191(Real.toNNReal (-f a)) \u2202\u03bc \u2264 \u222b (a : \u03b1), \u2191(Real.toNNReal (f a)) \u2202\u03bc\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r\nhr : \u00acr = 0\nh\u03bc this : IsFiniteMeasure \u03bc\n\u22a2 (fun \u03c9 => max (f \u03c9) 0) \u2264\u1d50[\u03bc] fun x => \u2191r", "state_after": "no goals"}, {"tactic": "simp only [Real.coe_toNNReal', ENNReal.toReal_mul, ENNReal.one_toReal, ENNReal.coe_toReal,\n  ge_iff_le, Left.nonneg_neg_iff, Left.neg_nonpos_iff, toReal_ofNat] at hfint' \u22a2", "annotated_tactic": ["simp only [<a>Real.coe_toNNReal'</a>, <a>ENNReal.toReal_mul</a>, <a>ENNReal.one_toReal</a>, <a>ENNReal.coe_toReal</a>,\n      <a>ge_iff_le</a>, <a>Left.nonneg_neg_iff</a>, <a>Left.neg_nonpos_iff</a>, <a>toReal_ofNat</a>] at hfint' \u22a2", [{"full_name": "Real.coe_toNNReal'", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [615, 9], "def_end_pos": [615, 22]}, {"full_name": "ENNReal.toReal_mul", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2296, 9], "def_end_pos": [2296, 19]}, {"full_name": "ENNReal.one_toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [230, 17], "def_end_pos": [230, 27]}, {"full_name": "ENNReal.coe_toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [236, 17], "def_end_pos": [236, 27]}, {"full_name": "ge_iff_le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [359, 9], "def_end_pos": [359, 18]}, {"full_name": "Left.nonneg_neg_iff", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [105, 3], "def_end_pos": [105, 14]}, {"full_name": "Left.neg_nonpos_iff", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [97, 3], "def_end_pos": [97, 14]}, {"full_name": "ENNReal.toReal_ofNat", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [736, 17], "def_end_pos": [736, 29]}]], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : \u222b (a : \u03b1), \u2191(Real.toNNReal (-f a)) \u2202\u03bc \u2264 \u222b (a : \u03b1), \u2191(Real.toNNReal (f a)) \u2202\u03bc\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r\nhr : \u00acr = 0\nh\u03bc this : IsFiniteMeasure \u03bc\nhposbdd : \u222b (\u03c9 : \u03b1), max (f \u03c9) 0 \u2202\u03bc \u2264 ENNReal.toReal (\u2191\u2191\u03bc univ) \u2022 \u2191r\n\u22a2 \u222b (a : \u03b1), \u2191(Real.toNNReal (f a)) \u2202\u03bc + \u222b (a : \u03b1), \u2191(Real.toNNReal (-f a)) \u2202\u03bc \u2264 ENNReal.toReal (2 * \u2191\u2191\u03bc univ * \u2191r)", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : \u222b (a : \u03b1), max (-f a) 0 \u2202\u03bc \u2264 \u222b (a : \u03b1), max (f a) 0 \u2202\u03bc\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r\nhr : \u00acr = 0\nh\u03bc this : IsFiniteMeasure \u03bc\nhposbdd : \u222b (\u03c9 : \u03b1), max (f \u03c9) 0 \u2202\u03bc \u2264 ENNReal.toReal (\u2191\u2191\u03bc univ) \u2022 \u2191r\n\u22a2 \u222b (a : \u03b1), max (f a) 0 \u2202\u03bc + \u222b (a : \u03b1), max (-f a) 0 \u2202\u03bc \u2264 2 * ENNReal.toReal (\u2191\u2191\u03bc univ) * \u2191r"}, {"tactic": "refine' (add_le_add_left hfint' _).trans _", "annotated_tactic": ["refine' (<a>add_le_add_left</a> hfint' _).<a>trans</a> _", [{"full_name": "add_le_add_left", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [49, 15], "def_end_pos": [49, 30]}, {"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [120, 7], "def_end_pos": [120, 18]}]], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : \u222b (a : \u03b1), max (-f a) 0 \u2202\u03bc \u2264 \u222b (a : \u03b1), max (f a) 0 \u2202\u03bc\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r\nhr : \u00acr = 0\nh\u03bc this : IsFiniteMeasure \u03bc\nhposbdd : \u222b (\u03c9 : \u03b1), max (f \u03c9) 0 \u2202\u03bc \u2264 ENNReal.toReal (\u2191\u2191\u03bc univ) \u2022 \u2191r\n\u22a2 \u222b (a : \u03b1), max (f a) 0 \u2202\u03bc + \u222b (a : \u03b1), max (-f a) 0 \u2202\u03bc \u2264 2 * ENNReal.toReal (\u2191\u2191\u03bc univ) * \u2191r", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : \u222b (a : \u03b1), max (-f a) 0 \u2202\u03bc \u2264 \u222b (a : \u03b1), max (f a) 0 \u2202\u03bc\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r\nhr : \u00acr = 0\nh\u03bc this : IsFiniteMeasure \u03bc\nhposbdd : \u222b (\u03c9 : \u03b1), max (f \u03c9) 0 \u2202\u03bc \u2264 ENNReal.toReal (\u2191\u2191\u03bc univ) \u2022 \u2191r\n\u22a2 \u222b (a : \u03b1), max (f a) 0 \u2202\u03bc + \u222b (a : \u03b1), max (f a) 0 \u2202\u03bc \u2264 2 * ENNReal.toReal (\u2191\u2191\u03bc univ) * \u2191r"}, {"tactic": "rwa [\u2190 two_mul, mul_assoc, mul_le_mul_left (two_pos : (0 : \u211d) < 2)]", "annotated_tactic": ["rwa [\u2190 <a>two_mul</a>, <a>mul_assoc</a>, <a>mul_le_mul_left</a> (<a>two_pos</a> : (0 : \u211d) < 2)]", [{"full_name": "two_mul", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [177, 9], "def_end_pos": [177, 16]}, {"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [264, 9], "def_end_pos": [264, 18]}, {"full_name": "mul_le_mul_left", "def_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "def_pos": [209, 9], "def_end_pos": [209, 24]}, {"full_name": "two_pos", "def_path": "Mathlib/Algebra/Order/Monoid/NatCast.lean", "def_pos": [113, 7], "def_end_pos": [113, 14]}]], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : \u222b (a : \u03b1), max (-f a) 0 \u2202\u03bc \u2264 \u222b (a : \u03b1), max (f a) 0 \u2202\u03bc\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r\nhr : \u00acr = 0\nh\u03bc this : IsFiniteMeasure \u03bc\nhposbdd : \u222b (\u03c9 : \u03b1), max (f \u03c9) 0 \u2202\u03bc \u2264 ENNReal.toReal (\u2191\u2191\u03bc univ) \u2022 \u2191r\n\u22a2 \u222b (a : \u03b1), max (f a) 0 \u2202\u03bc + \u222b (a : \u03b1), max (f a) 0 \u2202\u03bc \u2264 2 * ENNReal.toReal (\u2191\u2191\u03bc univ) * \u2191r", "state_after": "no goals"}, {"tactic": "exact hfint.neg.sup (integrable_zero _ _ \u03bc)", "annotated_tactic": ["exact hfint.neg.sup (<a>integrable_zero</a> _ _ \u03bc)", [{"full_name": "MeasureTheory.integrable_zero", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [662, 9], "def_end_pos": [662, 24]}]], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : \u222b (a : \u03b1), \u2191(Real.toNNReal (-f a)) \u2202\u03bc \u2264 \u222b (a : \u03b1), \u2191(Real.toNNReal (f a)) \u2202\u03bc\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r\nhr : \u00acr = 0\nh\u03bc this : IsFiniteMeasure \u03bc\nhposbdd : \u222b (\u03c9 : \u03b1), max (f \u03c9) 0 \u2202\u03bc \u2264 ENNReal.toReal (\u2191\u2191\u03bc univ) \u2022 \u2191r\n\u22a2 Integrable fun a => \u2191(Real.toNNReal (-f a))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Quot.lean", "full_name": "Quotient.hrecOn'_mk''", "start": [722, 1], "end": [725, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Kernel/CondCdf.lean", "full_name": "ProbabilityTheory.condCdfRat_le_one", "start": [616, 1], "end": [624, 31], "traced_tactics": [{"tactic": "unfold condCdfRat", "annotated_tactic": ["unfold <a>condCdfRat</a>", [{"full_name": "ProbabilityTheory.condCdfRat", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [570, 19], "def_end_pos": [570, 29]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nr : \u211a\n\u22a2 condCdfRat \u03c1 a r \u2264 1", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nr : \u211a\n\u22a2 ite (a \u2208 condCdfSet \u03c1) (fun r => ENNReal.toReal (preCdf \u03c1 r a)) (fun r => if r < 0 then 0 else 1) r \u2264 1"}, {"tactic": "split_ifs with h", "annotated_tactic": ["split_ifs with h", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nr : \u211a\n\u22a2 ite (a \u2208 condCdfSet \u03c1) (fun r => ENNReal.toReal (preCdf \u03c1 r a)) (fun r => if r < 0 then 0 else 1) r \u2264 1", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nr : \u211a\nh : a \u2208 condCdfSet \u03c1\n\u22a2 (fun r => ENNReal.toReal (preCdf \u03c1 r a)) r \u2264 1\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nr : \u211a\nh : \u00aca \u2208 condCdfSet \u03c1\n\u22a2 (fun r => if r < 0 then 0 else 1) r \u2264 1"}, {"tactic": "dsimp only", "annotated_tactic": ["dsimp only", []], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nr : \u211a\nh : \u00aca \u2208 condCdfSet \u03c1\n\u22a2 (fun r => if r < 0 then 0 else 1) r \u2264 1", "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nr : \u211a\nh : \u00aca \u2208 condCdfSet \u03c1\n\u22a2 (if r < 0 then 0 else 1) \u2264 1"}, {"tactic": "split_ifs", "annotated_tactic": ["split_ifs", []], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nr : \u211a\nh : \u00aca \u2208 condCdfSet \u03c1\n\u22a2 (if r < 0 then 0 else 1) \u2264 1", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nr : \u211a\nh : \u00aca \u2208 condCdfSet \u03c1\nh\u271d : r < 0\n\u22a2 0 \u2264 1\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nr : \u211a\nh : \u00aca \u2208 condCdfSet \u03c1\nh\u271d : \u00acr < 0\n\u22a2 1 \u2264 1"}, {"tactic": "exacts [zero_le_one, le_rfl]", "annotated_tactic": ["exacts [<a>zero_le_one</a>, <a>le_rfl</a>]", [{"full_name": "zero_le_one", "def_path": "Mathlib/Algebra/Order/ZeroLEOne.lean", "def_pos": [26, 15], "def_end_pos": [26, 26]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}]], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nr : \u211a\nh : \u00aca \u2208 condCdfSet \u03c1\nh\u271d : r < 0\n\u22a2 0 \u2264 1\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nr : \u211a\nh : \u00aca \u2208 condCdfSet \u03c1\nh\u271d : \u00acr < 0\n\u22a2 1 \u2264 1", "state_after": "no goals"}, {"tactic": "refine' ENNReal.toReal_le_of_le_ofReal zero_le_one _", "annotated_tactic": ["refine' <a>ENNReal.toReal_le_of_le_ofReal</a> <a>zero_le_one</a> _", [{"full_name": "ENNReal.toReal_le_of_le_ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2210, 9], "def_end_pos": [2210, 31]}, {"full_name": "zero_le_one", "def_path": "Mathlib/Algebra/Order/ZeroLEOne.lean", "def_pos": [26, 15], "def_end_pos": [26, 26]}]], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nr : \u211a\nh : a \u2208 condCdfSet \u03c1\n\u22a2 (fun r => ENNReal.toReal (preCdf \u03c1 r a)) r \u2264 1", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nr : \u211a\nh : a \u2208 condCdfSet \u03c1\n\u22a2 preCdf \u03c1 r a \u2264 ENNReal.ofReal 1"}, {"tactic": "rw [ENNReal.ofReal_one]", "annotated_tactic": ["rw [<a>ENNReal.ofReal_one</a>]", [{"full_name": "ENNReal.ofReal_one", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [248, 17], "def_end_pos": [248, 27]}]], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nr : \u211a\nh : a \u2208 condCdfSet \u03c1\n\u22a2 preCdf \u03c1 r a \u2264 ENNReal.ofReal 1", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nr : \u211a\nh : a \u2208 condCdfSet \u03c1\n\u22a2 preCdf \u03c1 r a \u2264 1"}, {"tactic": "exact (hasCondCdf_of_mem_condCdfSet h).le_one r", "annotated_tactic": ["exact (<a>hasCondCdf_of_mem_condCdfSet</a> h).<a>le_one</a> r", [{"full_name": "ProbabilityTheory.hasCondCdf_of_mem_condCdfSet", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [549, 9], "def_end_pos": [549, 37]}, {"full_name": "ProbabilityTheory.HasCondCdf.le_one", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [527, 3], "def_end_pos": [527, 9]}]], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nr : \u211a\nh : a \u2208 condCdfSet \u03c1\n\u22a2 preCdf \u03c1 r a \u2264 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Ackermann.lean", "full_name": "ack_add_one_sq_lt_ack_add_three", "start": [277, 1], "end": [285, 13], "traced_tactics": [{"tactic": "simpa using sq_le_two_pow_add_one_minus_three (n + 2)", "annotated_tactic": ["simpa using <a>sq_le_two_pow_add_one_minus_three</a> (n + 2)", [{"full_name": "_private.Mathlib.Computability.Ackermann.0.sq_le_two_pow_add_one_minus_three", "def_path": "Mathlib/Computability/Ackermann.lean", "def_pos": [262, 17], "def_end_pos": [262, 50]}]], "state_before": "n : \u2115\n\u22a2 (ack 0 n + 1) ^ 2 \u2264 ack (0 + 3) n", "state_after": "no goals"}, {"tactic": "rw [ack_succ_zero, ack_succ_zero]", "annotated_tactic": ["rw [<a>ack_succ_zero</a>, <a>ack_succ_zero</a>]", [{"full_name": "ack_succ_zero", "def_path": "Mathlib/Computability/Ackermann.lean", "def_pos": [74, 9], "def_end_pos": [74, 22]}, {"full_name": "ack_succ_zero", "def_path": "Mathlib/Computability/Ackermann.lean", "def_pos": [74, 9], "def_end_pos": [74, 22]}]], "state_before": "m : \u2115\n\u22a2 (ack (m + 1) 0 + 1) ^ 2 \u2264 ack (m + 1 + 3) 0", "state_after": "m : \u2115\n\u22a2 (ack m 1 + 1) ^ 2 \u2264 ack (m + 3) 1"}, {"tactic": "apply ack_add_one_sq_lt_ack_add_three", "annotated_tactic": ["apply ack_add_one_sq_lt_ack_add_three", []], "state_before": "m : \u2115\n\u22a2 (ack m 1 + 1) ^ 2 \u2264 ack (m + 3) 1", "state_after": "no goals"}, {"tactic": "rw [ack_succ_succ, ack_succ_succ]", "annotated_tactic": ["rw [<a>ack_succ_succ</a>, <a>ack_succ_succ</a>]", [{"full_name": "ack_succ_succ", "def_path": "Mathlib/Computability/Ackermann.lean", "def_pos": [78, 9], "def_end_pos": [78, 22]}, {"full_name": "ack_succ_succ", "def_path": "Mathlib/Computability/Ackermann.lean", "def_pos": [78, 9], "def_end_pos": [78, 22]}]], "state_before": "m n : \u2115\n\u22a2 (ack (m + 1) (n + 1) + 1) ^ 2 \u2264 ack (m + 1 + 3) (n + 1)", "state_after": "m n : \u2115\n\u22a2 (ack m (ack (m + 1) n) + 1) ^ 2 \u2264 ack (m + 3) (ack (m + 3 + 1) n)"}, {"tactic": "apply (ack_add_one_sq_lt_ack_add_three _ _).trans (ack_mono_right _ <| ack_mono_left _ _)", "annotated_tactic": ["apply (ack_add_one_sq_lt_ack_add_three _ _).<a>trans</a> (<a>ack_mono_right</a> _ <| <a>ack_mono_left</a> _ _)", [{"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [120, 7], "def_end_pos": [120, 18]}, {"full_name": "ack_mono_right", "def_path": "Mathlib/Computability/Ackermann.lean", "def_pos": [148, 9], "def_end_pos": [148, 23]}, {"full_name": "ack_mono_left", "def_path": "Mathlib/Computability/Ackermann.lean", "def_pos": [222, 9], "def_end_pos": [222, 22]}]], "state_before": "m n : \u2115\n\u22a2 (ack m (ack (m + 1) n) + 1) ^ 2 \u2264 ack (m + 3) (ack (m + 3 + 1) n)", "state_after": "m n : \u2115\n\u22a2 m + 1 \u2264 m + 3 + 1"}, {"tactic": "linarith", "annotated_tactic": ["linarith", []], "state_before": "m n : \u2115\n\u22a2 m + 1 \u2264 m + 3 + 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Function.lean", "full_name": "Set.restrict_eq_restrict_iff", "start": [194, 1], "end": [195, 18], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "full_name": "Nat.exists_eq_succ_of_ne_zero", "start": [238, 1], "end": [239, 50], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Independence/Basic.lean", "full_name": "ProbabilityTheory.iIndepFun.indepFun_finset_prod_of_not_mem", "start": [612, 1], "end": [617, 71], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "full_name": "MeasureTheory.Mem\u2112p.of_le_mul", "start": [1333, 1], "end": [1337, 64], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Int/Interval.lean", "full_name": "Int.card_uIcc", "start": [125, 1], "end": [132, 94], "traced_tactics": [{"tactic": "change ((\u2191) : \u2115 \u2192 \u2124) _ = ((\u2191) : \u2115 \u2192 \u2124) _", "annotated_tactic": ["change ((\u2191) : \u2115 \u2192 \u2124) _ = ((\u2191) : \u2115 \u2192 \u2124) _", []], "state_before": "a b : \u2124\n\u22a2 ofNat (card (range (toNat (a \u2294 b + 1 - a \u2293 b)))) = ofNat (natAbs (b - a) + 1)", "state_after": "a b : \u2124\n\u22a2 \u2191(card (range (toNat (a \u2294 b + 1 - a \u2293 b)))) = \u2191(natAbs (b - a) + 1)"}, {"tactic": "rw [card_range, sup_eq_max, inf_eq_min,\n  Int.toNat_of_nonneg (sub_nonneg_of_le <| le_add_one min_le_max), Int.ofNat_add,\n  Int.coe_natAbs, add_comm, add_sub_assoc, max_sub_min_eq_abs, add_comm, Int.ofNat_one]", "annotated_tactic": ["rw [<a>card_range</a>, <a>sup_eq_max</a>, <a>inf_eq_min</a>,\n        <a>Int.toNat_of_nonneg</a> (<a>sub_nonneg_of_le</a> <| <a>le_add_one</a> <a>min_le_max</a>), <a>Int.ofNat_add</a>,\n        <a>Int.coe_natAbs</a>, <a>add_comm</a>, <a>add_sub_assoc</a>, <a>max_sub_min_eq_abs</a>, <a>add_comm</a>, <a>Int.ofNat_one</a>]", [{"full_name": "Finset.card_range", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [177, 9], "def_end_pos": [177, 19]}, {"full_name": "sup_eq_max", "def_path": "Mathlib/Order/Lattice.lean", "def_pos": [835, 9], "def_end_pos": [835, 19]}, {"full_name": "inf_eq_min", "def_path": "Mathlib/Order/Lattice.lean", "def_pos": [839, 9], "def_end_pos": [839, 19]}, {"full_name": "Int.toNat_of_nonneg", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [1374, 17], "def_end_pos": [1374, 32]}, {"full_name": "sub_nonneg_of_le", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [726, 26], "def_end_pos": [726, 42]}, {"full_name": "Int.le_add_one", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [856, 9], "def_end_pos": [856, 19]}, {"full_name": "min_le_max", "def_path": "Mathlib/Order/MinMax.lean", "def_pos": [138, 9], "def_end_pos": [138, 19]}, {"full_name": "Int.ofNat_add", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [35, 9], "def_end_pos": [35, 18]}, {"full_name": "Int.coe_natAbs", "def_path": "Mathlib/Data/Int/Order/Basic.lean", "def_pos": [61, 26], "def_end_pos": [61, 36]}, {"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [301, 3], "def_end_pos": [301, 14]}, {"full_name": "add_sub_assoc", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [305, 3], "def_end_pos": [305, 14]}, {"full_name": "max_sub_min_eq_abs", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [216, 9], "def_end_pos": [216, 27]}, {"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [301, 3], "def_end_pos": [301, 14]}, {"full_name": "Int.ofNat_one", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [21, 17], "def_end_pos": [21, 26]}]], "state_before": "a b : \u2124\n\u22a2 \u2191(card (range (toNat (a \u2294 b + 1 - a \u2293 b)))) = \u2191(natAbs (b - a) + 1)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "full_name": "MeasureTheory.L1.setToL1_add_left", "start": [1077, 1], "end": [1088, 83], "traced_tactics": [{"tactic": "suffices setToL1 (hT.add hT') = setToL1 hT + setToL1 hT' by\n  rw [this, ContinuousLinearMap.add_apply]", "annotated_tactic": ["suffices <a>setToL1</a> (hT.add hT') = <a>setToL1</a> hT + <a>setToL1</a> hT' by\n    rw [this, <a>ContinuousLinearMap.add_apply</a>]", [{"full_name": "MeasureTheory.L1.setToL1", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [1019, 5], "def_end_pos": [1019, 12]}, {"full_name": "MeasureTheory.L1.setToL1", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [1019, 5], "def_end_pos": [1019, 12]}, {"full_name": "MeasureTheory.L1.setToL1", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [1019, 5], "def_end_pos": [1019, 12]}, {"full_name": "ContinuousLinearMap.add_apply", "def_path": "Mathlib/Topology/Algebra/Module/Basic.lean", "def_pos": [726, 9], "def_end_pos": [726, 18]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : NormedAddCommGroup F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc T' C'\nf : { x // x \u2208 Lp E 1 }\n\u22a2 \u2191(setToL1 (_ : DominatedFinMeasAdditive \u03bc (T + T') (C + C'))) f = \u2191(setToL1 hT) f + \u2191(setToL1 hT') f", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : NormedAddCommGroup F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc T' C'\nf : { x // x \u2208 Lp E 1 }\n\u22a2 setToL1 (_ : DominatedFinMeasAdditive \u03bc (T + T') (C + C')) = setToL1 hT + setToL1 hT'"}, {"tactic": "refine' ContinuousLinearMap.extend_unique (setToL1SCLM \u03b1 E \u03bc (hT.add hT')) _ _ _ _ _", "annotated_tactic": ["refine' <a>ContinuousLinearMap.extend_unique</a> (<a>setToL1SCLM</a> \u03b1 E \u03bc (hT.add hT')) _ _ _ _ _", [{"full_name": "ContinuousLinearMap.extend_unique", "def_path": "Mathlib/Analysis/NormedSpace/OperatorNorm.lean", "def_pos": [1746, 9], "def_end_pos": [1746, 22]}, {"full_name": "MeasureTheory.L1.SimpleFunc.setToL1SCLM", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [877, 5], "def_end_pos": [877, 16]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : NormedAddCommGroup F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc T' C'\nf : { x // x \u2208 Lp E 1 }\n\u22a2 setToL1 (_ : DominatedFinMeasAdditive \u03bc (T + T') (C + C')) = setToL1 hT + setToL1 hT'", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : NormedAddCommGroup F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc T' C'\nf : { x // x \u2208 Lp E 1 }\n\u22a2 ContinuousLinearMap.comp (setToL1 hT + setToL1 hT') (coeToLp \u03b1 E \u211d) =\n    setToL1SCLM \u03b1 E \u03bc (_ : DominatedFinMeasAdditive \u03bc (T + T') (C + C'))"}, {"tactic": "ext1 f", "annotated_tactic": ["ext1 f", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : NormedAddCommGroup F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc T' C'\nf : { x // x \u2208 Lp E 1 }\n\u22a2 ContinuousLinearMap.comp (setToL1 hT + setToL1 hT') (coeToLp \u03b1 E \u211d) =\n    setToL1SCLM \u03b1 E \u03bc (_ : DominatedFinMeasAdditive \u03bc (T + T') (C + C'))", "state_after": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : NormedAddCommGroup F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc T' C'\nf\u271d : { x // x \u2208 Lp E 1 }\nf : { x // x \u2208 simpleFunc E 1 \u03bc }\n\u22a2 \u2191(ContinuousLinearMap.comp (setToL1 hT + setToL1 hT') (coeToLp \u03b1 E \u211d)) f =\n    \u2191(setToL1SCLM \u03b1 E \u03bc (_ : DominatedFinMeasAdditive \u03bc (T + T') (C + C'))) f"}, {"tactic": "suffices setToL1 hT f + setToL1 hT' f = setToL1SCLM \u03b1 E \u03bc (hT.add hT') f by\n  rw [\u2190 this]; rfl", "annotated_tactic": ["suffices <a>setToL1</a> hT f + <a>setToL1</a> hT' f = <a>setToL1SCLM</a> \u03b1 E \u03bc (hT.add hT') f by\n    rw [\u2190 this]; rfl", [{"full_name": "MeasureTheory.L1.setToL1", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [1019, 5], "def_end_pos": [1019, 12]}, {"full_name": "MeasureTheory.L1.setToL1", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [1019, 5], "def_end_pos": [1019, 12]}, {"full_name": "MeasureTheory.L1.SimpleFunc.setToL1SCLM", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [877, 5], "def_end_pos": [877, 16]}]], "state_before": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : NormedAddCommGroup F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc T' C'\nf\u271d : { x // x \u2208 Lp E 1 }\nf : { x // x \u2208 simpleFunc E 1 \u03bc }\n\u22a2 \u2191(ContinuousLinearMap.comp (setToL1 hT + setToL1 hT') (coeToLp \u03b1 E \u211d)) f =\n    \u2191(setToL1SCLM \u03b1 E \u03bc (_ : DominatedFinMeasAdditive \u03bc (T + T') (C + C'))) f", "state_after": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : NormedAddCommGroup F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc T' C'\nf\u271d : { x // x \u2208 Lp E 1 }\nf : { x // x \u2208 simpleFunc E 1 \u03bc }\n\u22a2 \u2191(setToL1 hT) \u2191f + \u2191(setToL1 hT') \u2191f = \u2191(setToL1SCLM \u03b1 E \u03bc (_ : DominatedFinMeasAdditive \u03bc (T + T') (C + C'))) f"}, {"tactic": "rw [setToL1_eq_setToL1SCLM, setToL1_eq_setToL1SCLM, setToL1SCLM_add_left hT hT']", "annotated_tactic": ["rw [<a>setToL1_eq_setToL1SCLM</a>, <a>setToL1_eq_setToL1SCLM</a>, <a>setToL1SCLM_add_left</a> hT hT']", [{"full_name": "MeasureTheory.L1.setToL1_eq_setToL1SCLM", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [1024, 9], "def_end_pos": [1024, 31]}, {"full_name": "MeasureTheory.L1.setToL1_eq_setToL1SCLM", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [1024, 9], "def_end_pos": [1024, 31]}, {"full_name": "MeasureTheory.L1.SimpleFunc.setToL1SCLM_add_left", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [919, 9], "def_end_pos": [919, 29]}]], "state_before": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : NormedAddCommGroup F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc T' C'\nf\u271d : { x // x \u2208 Lp E 1 }\nf : { x // x \u2208 simpleFunc E 1 \u03bc }\n\u22a2 \u2191(setToL1 hT) \u2191f + \u2191(setToL1 hT') \u2191f = \u2191(setToL1SCLM \u03b1 E \u03bc (_ : DominatedFinMeasAdditive \u03bc (T + T') (C + C'))) f", "state_after": "no goals"}, {"tactic": "rw [this, ContinuousLinearMap.add_apply]", "annotated_tactic": ["rw [this, <a>ContinuousLinearMap.add_apply</a>]", [{"full_name": "ContinuousLinearMap.add_apply", "def_path": "Mathlib/Topology/Algebra/Module/Basic.lean", "def_pos": [726, 9], "def_end_pos": [726, 18]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : NormedAddCommGroup F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc T' C'\nf : { x // x \u2208 Lp E 1 }\nthis : setToL1 (_ : DominatedFinMeasAdditive \u03bc (T + T') (C + C')) = setToL1 hT + setToL1 hT'\n\u22a2 \u2191(setToL1 (_ : DominatedFinMeasAdditive \u03bc (T + T') (C + C'))) f = \u2191(setToL1 hT) f + \u2191(setToL1 hT') f", "state_after": "no goals"}, {"tactic": "rw [\u2190 this]", "annotated_tactic": ["rw [\u2190 this]", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : NormedAddCommGroup F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc T' C'\nf\u271d : { x // x \u2208 Lp E 1 }\nf : { x // x \u2208 simpleFunc E 1 \u03bc }\nthis : \u2191(setToL1 hT) \u2191f + \u2191(setToL1 hT') \u2191f = \u2191(setToL1SCLM \u03b1 E \u03bc (_ : DominatedFinMeasAdditive \u03bc (T + T') (C + C'))) f\n\u22a2 \u2191(ContinuousLinearMap.comp (setToL1 hT + setToL1 hT') (coeToLp \u03b1 E \u211d)) f =\n    \u2191(setToL1SCLM \u03b1 E \u03bc (_ : DominatedFinMeasAdditive \u03bc (T + T') (C + C'))) f", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : NormedAddCommGroup F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc T' C'\nf\u271d : { x // x \u2208 Lp E 1 }\nf : { x // x \u2208 simpleFunc E 1 \u03bc }\nthis : \u2191(setToL1 hT) \u2191f + \u2191(setToL1 hT') \u2191f = \u2191(setToL1SCLM \u03b1 E \u03bc (_ : DominatedFinMeasAdditive \u03bc (T + T') (C + C'))) f\n\u22a2 \u2191(ContinuousLinearMap.comp (setToL1 hT + setToL1 hT') (coeToLp \u03b1 E \u211d)) f = \u2191(setToL1 hT) \u2191f + \u2191(setToL1 hT') \u2191f"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : NormedAddCommGroup F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc T' C'\nf\u271d : { x // x \u2208 Lp E 1 }\nf : { x // x \u2208 simpleFunc E 1 \u03bc }\nthis : \u2191(setToL1 hT) \u2191f + \u2191(setToL1 hT') \u2191f = \u2191(setToL1SCLM \u03b1 E \u03bc (_ : DominatedFinMeasAdditive \u03bc (T + T') (C + C'))) f\n\u22a2 \u2191(ContinuousLinearMap.comp (setToL1 hT + setToL1 hT') (coeToLp \u03b1 E \u211d)) f = \u2191(setToL1 hT) \u2191f + \u2191(setToL1 hT') \u2191f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Pointwise/Finite.lean", "full_name": "Set.infinite_mul", "start": [132, 1], "end": [134, 36], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Group/Prod.lean", "full_name": "MeasureTheory.measurePreserving_mul_prod_inv", "start": [148, 1], "end": [153, 67], "traced_tactics": [{"tactic": "convert (measurePreserving_prod_inv_mul_swap \u03bd \u03bc).comp (measurePreserving_prod_mul_swap \u03bc \u03bd)\n  using 1", "annotated_tactic": ["convert (<a>measurePreserving_prod_inv_mul_swap</a> \u03bd \u03bc).<a>comp</a> (<a>measurePreserving_prod_mul_swap</a> \u03bc \u03bd)\n    using 1", [{"full_name": "MeasureTheory.measurePreserving_prod_inv_mul_swap", "def_path": "Mathlib/MeasureTheory/Group/Prod.lean", "def_pos": [137, 9], "def_end_pos": [137, 44]}, {"full_name": "MeasureTheory.MeasurePreserving.comp", "def_path": "Mathlib/Dynamics/Ergodic/MeasurePreserving.lean", "def_pos": [102, 19], "def_end_pos": [102, 23]}, {"full_name": "MeasureTheory.measurePreserving_prod_mul_swap", "def_path": "Mathlib/MeasureTheory/Group/Prod.lean", "def_pos": [98, 9], "def_end_pos": [98, 40]}]], "state_before": "G : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\n\u22a2 MeasurePreserving fun z => (z.2 * z.1, z.1\u207b\u00b9)", "state_after": "case h.e'_5\nG : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\n\u22a2 (fun z => (z.2 * z.1, z.1\u207b\u00b9)) = (fun z => (z.2, z.2\u207b\u00b9 * z.1)) \u2218 fun z => (z.2, z.2 * z.1)"}, {"tactic": "ext1 \u27e8x, y\u27e9", "annotated_tactic": ["ext1 \u27e8x, y\u27e9", []], "state_before": "case h.e'_5\nG : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\n\u22a2 (fun z => (z.2 * z.1, z.1\u207b\u00b9)) = (fun z => (z.2, z.2\u207b\u00b9 * z.1)) \u2218 fun z => (z.2, z.2 * z.1)", "state_after": "case h.e'_5.h.mk\nG : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nx y : G\n\u22a2 ((x, y).2 * (x, y).1, (x, y).1\u207b\u00b9) = ((fun z => (z.2, z.2\u207b\u00b9 * z.1)) \u2218 fun z => (z.2, z.2 * z.1)) (x, y)"}, {"tactic": "simp_rw [Function.comp_apply, mul_inv_rev, inv_mul_cancel_right]", "annotated_tactic": ["simp_rw [<a>Function.comp_apply</a>, <a>mul_inv_rev</a>, <a>inv_mul_cancel_right</a>]", [{"full_name": "Function.comp_apply", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [33, 17], "def_end_pos": [33, 36]}, {"full_name": "mul_inv_rev", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [1050, 9], "def_end_pos": [1050, 20]}, {"full_name": "inv_mul_cancel_right", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [1165, 9], "def_end_pos": [1165, 29]}]], "state_before": "case h.e'_5.h.mk\nG : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nx y : G\n\u22a2 ((x, y).2 * (x, y).1, (x, y).1\u207b\u00b9) = ((fun z => (z.2, z.2\u207b\u00b9 * z.1)) \u2218 fun z => (z.2, z.2 * z.1)) (x, y)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Kernel/Composition.lean", "full_name": "ProbabilityTheory.kernel.snd_apply'", "start": [812, 1], "end": [813, 97], "traced_tactics": [{"tactic": "rw [snd_apply, Measure.map_apply measurable_snd hs]", "annotated_tactic": ["rw [<a>snd_apply</a>, <a>Measure.map_apply</a> <a>measurable_snd</a> hs]", [{"full_name": "ProbabilityTheory.kernel.snd_apply", "def_path": "Mathlib/Probability/Kernel/Composition.lean", "def_pos": [808, 9], "def_end_pos": [808, 18]}, {"full_name": "MeasureTheory.Measure.map_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1236, 9], "def_end_pos": [1236, 18]}, {"full_name": "measurable_snd", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [698, 9], "def_end_pos": [698, 23]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03b3 : Type u_4\nm\u03b3 : MeasurableSpace \u03b3\nf : \u03b2 \u2192 \u03b3\ng : \u03b3 \u2192 \u03b1\n\u03ba : { x // x \u2208 kernel \u03b1 (\u03b2 \u00d7 \u03b3) }\na : \u03b1\ns : Set \u03b3\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(\u2191(snd \u03ba) a) s = \u2191\u2191(\u2191\u03ba a) {p | p.2 \u2208 s}", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03b3 : Type u_4\nm\u03b3 : MeasurableSpace \u03b3\nf : \u03b2 \u2192 \u03b3\ng : \u03b3 \u2192 \u03b1\n\u03ba : { x // x \u2208 kernel \u03b1 (\u03b2 \u00d7 \u03b3) }\na : \u03b1\ns : Set \u03b3\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(\u2191\u03ba a) (Prod.snd \u207b\u00b9' s) = \u2191\u2191(\u2191\u03ba a) {p | p.2 \u2208 s}"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03b3 : Type u_4\nm\u03b3 : MeasurableSpace \u03b3\nf : \u03b2 \u2192 \u03b3\ng : \u03b3 \u2192 \u03b1\n\u03ba : { x // x \u2208 kernel \u03b1 (\u03b2 \u00d7 \u03b3) }\na : \u03b1\ns : Set \u03b3\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(\u2191\u03ba a) (Prod.snd \u207b\u00b9' s) = \u2191\u2191(\u2191\u03ba a) {p | p.2 \u2208 s}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "full_name": "MeasureTheory.VectorMeasure.ext", "start": [139, 1], "end": [140, 20], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Intervals/ProjIcc.lean", "full_name": "StrictMono.strictMonoOn_IciExtend", "start": [329, 1], "end": [331, 44], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "full_name": "List.mem_of_mem_erase", "start": [1206, 1], "end": [1206, 98], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/ProbabilityMassFunction/Monad.lean", "full_name": "PMF.pure_bindOnSupport", "start": [262, 1], "end": [267, 35], "traced_tactics": [{"tactic": "refine' PMF.ext fun b => _", "annotated_tactic": ["refine' <a>PMF.ext</a> fun b => _", [{"full_name": "PMF.ext", "def_path": "Mathlib/Probability/ProbabilityMassFunction/Basic.lean", "def_pos": [55, 19], "def_end_pos": [55, 22]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\nf\u271d : (a : \u03b1) \u2192 a \u2208 support p \u2192 PMF \u03b2\na : \u03b1\nf : (a' : \u03b1) \u2192 a' \u2208 support (pure a) \u2192 PMF \u03b2\n\u22a2 bindOnSupport (pure a) f = f a (_ : a \u2208 support (pure a))", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\nf\u271d : (a : \u03b1) \u2192 a \u2208 support p \u2192 PMF \u03b2\na : \u03b1\nf : (a' : \u03b1) \u2192 a' \u2208 support (pure a) \u2192 PMF \u03b2\nb : \u03b2\n\u22a2 \u2191(bindOnSupport (pure a) f) b = \u2191(f a (_ : a \u2208 support (pure a))) b"}, {"tactic": "simp only [bindOnSupport_apply, pure_apply]", "annotated_tactic": ["simp only [<a>bindOnSupport_apply</a>, <a>pure_apply</a>]", [{"full_name": "PMF.bindOnSupport_apply", "def_path": "Mathlib/Probability/ProbabilityMassFunction/Monad.lean", "def_pos": [219, 9], "def_end_pos": [219, 28]}, {"full_name": "PMF.pure_apply", "def_path": "Mathlib/Probability/ProbabilityMassFunction/Monad.lean", "def_pos": [45, 9], "def_end_pos": [45, 19]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\nf\u271d : (a : \u03b1) \u2192 a \u2208 support p \u2192 PMF \u03b2\na : \u03b1\nf : (a' : \u03b1) \u2192 a' \u2208 support (pure a) \u2192 PMF \u03b2\nb : \u03b2\n\u22a2 \u2191(bindOnSupport (pure a) f) b = \u2191(f a (_ : a \u2208 support (pure a))) b", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\nf\u271d : (a : \u03b1) \u2192 a \u2208 support p \u2192 PMF \u03b2\na : \u03b1\nf : (a' : \u03b1) \u2192 a' \u2208 support (pure a) \u2192 PMF \u03b2\nb : \u03b2\n\u22a2 (\u2211' (a_1 : \u03b1),\n      (if a_1 = a then 1 else 0) *\n        if h : (if a_1 = a then 1 else 0) = 0 then 0 else \u2191(f a_1 (_ : \u00ac\u2191(pure a) a_1 = 0)) b) =\n    \u2191(f a (_ : a \u2208 support (pure a))) b"}, {"tactic": "refine' _root_.trans (tsum_congr fun a' => _) (tsum_ite_eq a _)", "annotated_tactic": ["refine' <a>_root_.trans</a> (<a>tsum_congr</a> fun a' => _) (<a>tsum_ite_eq</a> a _)", [{"full_name": "trans", "def_path": "Mathlib/Init/Algebra/Classes.lean", "def_pos": [308, 9], "def_end_pos": [308, 14]}, {"full_name": "tsum_congr", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/Basic.lean", "def_pos": [498, 9], "def_end_pos": [498, 19]}, {"full_name": "tsum_ite_eq", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/Basic.lean", "def_pos": [534, 9], "def_end_pos": [534, 20]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\nf\u271d : (a : \u03b1) \u2192 a \u2208 support p \u2192 PMF \u03b2\na : \u03b1\nf : (a' : \u03b1) \u2192 a' \u2208 support (pure a) \u2192 PMF \u03b2\nb : \u03b2\n\u22a2 (\u2211' (a_1 : \u03b1),\n      (if a_1 = a then 1 else 0) *\n        if h : (if a_1 = a then 1 else 0) = 0 then 0 else \u2191(f a_1 (_ : \u00ac\u2191(pure a) a_1 = 0)) b) =\n    \u2191(f a (_ : a \u2208 support (pure a))) b", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\nf\u271d : (a : \u03b1) \u2192 a \u2208 support p \u2192 PMF \u03b2\na : \u03b1\nf : (a' : \u03b1) \u2192 a' \u2208 support (pure a) \u2192 PMF \u03b2\nb : \u03b2\na' : \u03b1\n\u22a2 ((if a' = a then 1 else 0) * if h : (if a' = a then 1 else 0) = 0 then 0 else \u2191(f a' (_ : \u00ac\u2191(pure a) a' = 0)) b) =\n    if a' = a then \u2191(f a (_ : a \u2208 support (pure a))) b else 0"}, {"tactic": "by_cases h : a' = a <;> simp [h]", "annotated_tactic": ["by_cases h : a' = a <;> simp [h]", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\nf\u271d : (a : \u03b1) \u2192 a \u2208 support p \u2192 PMF \u03b2\na : \u03b1\nf : (a' : \u03b1) \u2192 a' \u2208 support (pure a) \u2192 PMF \u03b2\nb : \u03b2\na' : \u03b1\n\u22a2 ((if a' = a then 1 else 0) * if h : (if a' = a then 1 else 0) = 0 then 0 else \u2191(f a' (_ : \u00ac\u2191(pure a) a' = 0)) b) =\n    if a' = a then \u2191(f a (_ : a \u2208 support (pure a))) b else 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Countable.lean", "full_name": "Set.countable_iff_exists_surjective", "start": [104, 11], "end": [106, 83], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Function.lean", "full_name": "Set.SurjOn.union_union", "start": [815, 1], "end": [818, 55], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean", "full_name": "MeasureTheory.lpMeasSubgroupToLpTrim_add", "start": [392, 1], "end": [406, 45], "traced_tactics": [{"tactic": "ext1", "annotated_tactic": ["ext1", []], "state_before": "\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\n\u03b9 : Type u_6\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nf g : { x // x \u2208 lpMeasSubgroup F m p \u03bc }\n\u22a2 lpMeasSubgroupToLpTrim F p \u03bc hm (f + g) = lpMeasSubgroupToLpTrim F p \u03bc hm f + lpMeasSubgroupToLpTrim F p \u03bc hm g", "state_after": "case h\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\n\u03b9 : Type u_6\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nf g : { x // x \u2208 lpMeasSubgroup F m p \u03bc }\n\u22a2 \u2191\u2191(lpMeasSubgroupToLpTrim F p \u03bc hm (f + g)) =\u1d50[Measure.trim \u03bc hm]\n    \u2191\u2191(lpMeasSubgroupToLpTrim F p \u03bc hm f + lpMeasSubgroupToLpTrim F p \u03bc hm g)"}, {"tactic": "refine' EventuallyEq.trans _ (Lp.coeFn_add _ _).symm", "annotated_tactic": ["refine' <a>EventuallyEq.trans</a> _ (<a>Lp.coeFn_add</a> _ _).<a>symm</a>", [{"full_name": "Filter.EventuallyEq.trans", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1503, 9], "def_end_pos": [1503, 27]}, {"full_name": "MeasureTheory.Lp.coeFn_add", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [232, 9], "def_end_pos": [232, 18]}, {"full_name": "Filter.EventuallyEq.symm", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1498, 9], "def_end_pos": [1498, 26]}]], "state_before": "case h\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\n\u03b9 : Type u_6\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nf g : { x // x \u2208 lpMeasSubgroup F m p \u03bc }\n\u22a2 \u2191\u2191(lpMeasSubgroupToLpTrim F p \u03bc hm (f + g)) =\u1d50[Measure.trim \u03bc hm]\n    \u2191\u2191(lpMeasSubgroupToLpTrim F p \u03bc hm f + lpMeasSubgroupToLpTrim F p \u03bc hm g)", "state_after": "case h\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\n\u03b9 : Type u_6\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nf g : { x // x \u2208 lpMeasSubgroup F m p \u03bc }\n\u22a2 \u2191\u2191(lpMeasSubgroupToLpTrim F p \u03bc hm (f + g)) =\u1d50[Measure.trim \u03bc hm]\n    \u2191\u2191(lpMeasSubgroupToLpTrim F p \u03bc hm f) + \u2191\u2191(lpMeasSubgroupToLpTrim F p \u03bc hm g)"}, {"tactic": "refine' ae_eq_trim_of_stronglyMeasurable hm (Lp.stronglyMeasurable _) _ _", "annotated_tactic": ["refine' <a>ae_eq_trim_of_stronglyMeasurable</a> hm (<a>Lp.stronglyMeasurable</a> _) _ _", [{"full_name": "MeasureTheory.ae_eq_trim_of_stronglyMeasurable", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1907, 9], "def_end_pos": [1907, 41]}, {"full_name": "MeasureTheory.Lp.stronglyMeasurable", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [207, 19], "def_end_pos": [207, 37]}]], "state_before": "case h\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\n\u03b9 : Type u_6\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nf g : { x // x \u2208 lpMeasSubgroup F m p \u03bc }\n\u22a2 \u2191\u2191(lpMeasSubgroupToLpTrim F p \u03bc hm (f + g)) =\u1d50[Measure.trim \u03bc hm]\n    \u2191\u2191(lpMeasSubgroupToLpTrim F p \u03bc hm f) + \u2191\u2191(lpMeasSubgroupToLpTrim F p \u03bc hm g)", "state_after": "case h.refine'_1\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\n\u03b9 : Type u_6\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nf g : { x // x \u2208 lpMeasSubgroup F m p \u03bc }\n\u22a2 StronglyMeasurable (\u2191\u2191(lpMeasSubgroupToLpTrim F p \u03bc hm f) + \u2191\u2191(lpMeasSubgroupToLpTrim F p \u03bc hm g))\n\ncase h.refine'_2\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\n\u03b9 : Type u_6\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nf g : { x // x \u2208 lpMeasSubgroup F m p \u03bc }\n\u22a2 \u2191\u2191(lpMeasSubgroupToLpTrim F p \u03bc hm (f + g)) =\u1d50[\u03bc]\n    \u2191\u2191(lpMeasSubgroupToLpTrim F p \u03bc hm f) + \u2191\u2191(lpMeasSubgroupToLpTrim F p \u03bc hm g)"}, {"tactic": "refine' (lpMeasSubgroupToLpTrim_ae_eq hm _).trans _", "annotated_tactic": ["refine' (<a>lpMeasSubgroupToLpTrim_ae_eq</a> hm _).<a>trans</a> _", [{"full_name": "MeasureTheory.lpMeasSubgroupToLpTrim_ae_eq", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean", "def_pos": [346, 9], "def_end_pos": [346, 37]}, {"full_name": "Filter.EventuallyEq.trans", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1503, 9], "def_end_pos": [1503, 27]}]], "state_before": "case h.refine'_2\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\n\u03b9 : Type u_6\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nf g : { x // x \u2208 lpMeasSubgroup F m p \u03bc }\n\u22a2 \u2191\u2191(lpMeasSubgroupToLpTrim F p \u03bc hm (f + g)) =\u1d50[\u03bc]\n    \u2191\u2191(lpMeasSubgroupToLpTrim F p \u03bc hm f) + \u2191\u2191(lpMeasSubgroupToLpTrim F p \u03bc hm g)", "state_after": "case h.refine'_2\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\n\u03b9 : Type u_6\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nf g : { x // x \u2208 lpMeasSubgroup F m p \u03bc }\n\u22a2 \u2191\u2191\u2191(f + g) =\u1d50[\u03bc] \u2191\u2191(lpMeasSubgroupToLpTrim F p \u03bc hm f) + \u2191\u2191(lpMeasSubgroupToLpTrim F p \u03bc hm g)"}, {"tactic": "refine'\n  EventuallyEq.trans _\n    (EventuallyEq.add (lpMeasSubgroupToLpTrim_ae_eq hm f).symm\n      (lpMeasSubgroupToLpTrim_ae_eq hm g).symm)", "annotated_tactic": ["refine'\n    <a>EventuallyEq.trans</a> _\n      (<a>EventuallyEq.add</a> (<a>lpMeasSubgroupToLpTrim_ae_eq</a> hm f).<a>symm</a>\n        (<a>lpMeasSubgroupToLpTrim_ae_eq</a> hm g).<a>symm</a>)", [{"full_name": "Filter.EventuallyEq.trans", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1503, 9], "def_end_pos": [1503, 27]}, {"full_name": "Filter.EventuallyEq.add", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1530, 3], "def_end_pos": [1530, 14]}, {"full_name": "MeasureTheory.lpMeasSubgroupToLpTrim_ae_eq", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean", "def_pos": [346, 9], "def_end_pos": [346, 37]}, {"full_name": "Filter.EventuallyEq.symm", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1498, 9], "def_end_pos": [1498, 26]}, {"full_name": "MeasureTheory.lpMeasSubgroupToLpTrim_ae_eq", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean", "def_pos": [346, 9], "def_end_pos": [346, 37]}, {"full_name": "Filter.EventuallyEq.symm", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1498, 9], "def_end_pos": [1498, 26]}]], "state_before": "case h.refine'_2\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\n\u03b9 : Type u_6\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nf g : { x // x \u2208 lpMeasSubgroup F m p \u03bc }\n\u22a2 \u2191\u2191\u2191(f + g) =\u1d50[\u03bc] \u2191\u2191(lpMeasSubgroupToLpTrim F p \u03bc hm f) + \u2191\u2191(lpMeasSubgroupToLpTrim F p \u03bc hm g)", "state_after": "case h.refine'_2\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\n\u03b9 : Type u_6\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nf g : { x // x \u2208 lpMeasSubgroup F m p \u03bc }\n\u22a2 \u2191\u2191\u2191(f + g) =\u1d50[\u03bc] fun x => \u2191\u2191\u2191f x + \u2191\u2191\u2191g x"}, {"tactic": "refine' (Lp.coeFn_add _ _).trans _", "annotated_tactic": ["refine' (<a>Lp.coeFn_add</a> _ _).<a>trans</a> _", [{"full_name": "MeasureTheory.Lp.coeFn_add", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [232, 9], "def_end_pos": [232, 18]}, {"full_name": "Filter.EventuallyEq.trans", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1503, 9], "def_end_pos": [1503, 27]}]], "state_before": "case h.refine'_2\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\n\u03b9 : Type u_6\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nf g : { x // x \u2208 lpMeasSubgroup F m p \u03bc }\n\u22a2 \u2191\u2191\u2191(f + g) =\u1d50[\u03bc] fun x => \u2191\u2191\u2191f x + \u2191\u2191\u2191g x", "state_after": "case h.refine'_2\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\n\u03b9 : Type u_6\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nf g : { x // x \u2208 lpMeasSubgroup F m p \u03bc }\n\u22a2 \u2191\u2191\u2191f + \u2191\u2191\u2191g =\u1d50[\u03bc] fun x => \u2191\u2191\u2191f x + \u2191\u2191\u2191g x"}, {"tactic": "exact eventually_of_forall fun x => by rfl", "annotated_tactic": ["exact <a>eventually_of_forall</a> fun x => by rfl", [{"full_name": "Filter.eventually_of_forall", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1111, 9], "def_end_pos": [1111, 29]}]], "state_before": "case h.refine'_2\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\n\u03b9 : Type u_6\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nf g : { x // x \u2208 lpMeasSubgroup F m p \u03bc }\n\u22a2 \u2191\u2191\u2191f + \u2191\u2191\u2191g =\u1d50[\u03bc] fun x => \u2191\u2191\u2191f x + \u2191\u2191\u2191g x", "state_after": "no goals"}, {"tactic": "exact (Lp.stronglyMeasurable _).add (Lp.stronglyMeasurable _)", "annotated_tactic": ["exact (<a>Lp.stronglyMeasurable</a> _).<a>add</a> (<a>Lp.stronglyMeasurable</a> _)", [{"full_name": "MeasureTheory.Lp.stronglyMeasurable", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [207, 19], "def_end_pos": [207, 37]}, {"full_name": "MeasureTheory.StronglyMeasurable.add", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [408, 3], "def_end_pos": [408, 14]}, {"full_name": "MeasureTheory.Lp.stronglyMeasurable", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [207, 19], "def_end_pos": [207, 37]}]], "state_before": "case h.refine'_1\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\n\u03b9 : Type u_6\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nf g : { x // x \u2208 lpMeasSubgroup F m p \u03bc }\n\u22a2 StronglyMeasurable (\u2191\u2191(lpMeasSubgroupToLpTrim F p \u03bc hm f) + \u2191\u2191(lpMeasSubgroupToLpTrim F p \u03bc hm g))", "state_after": "no goals"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\n\u03b9 : Type u_6\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nf g : { x // x \u2208 lpMeasSubgroup F m p \u03bc }\nx : \u03b1\n\u22a2 (\u2191\u2191\u2191f + \u2191\u2191\u2191g) x = (fun x => \u2191\u2191\u2191f x + \u2191\u2191\u2191g x) x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/List/Init/Lemmas.lean", "full_name": "List.append_right_inj", "start": [79, 1], "end": [80, 48], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Lattice.lean", "full_name": "Finset.inf'_map", "start": [1043, 1], "end": [1045, 31], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "full_name": "MeasureTheory.exists_simpleFunc_forall_lintegral_sub_lt_of_pos", "start": [222, 1], "end": [238, 6], "traced_tactics": [{"tactic": "rw [lintegral_eq_nnreal] at h", "annotated_tactic": ["rw [<a>lintegral_eq_nnreal</a>] at h", [{"full_name": "MeasureTheory.lintegral_eq_nnreal", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [197, 9], "def_end_pos": [197, 28]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u22a2 \u2203 \u03c6,\n    (\u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x) \u2227\n      \u2200 (\u03c8 : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (\u03c8 - \u03c6)) \u03bc < \u03b5", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u2a06 \u03c6, \u2a06 (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u22a2 \u2203 \u03c6,\n    (\u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x) \u2227\n      \u2200 (\u03c8 : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (\u03c8 - \u03c6)) \u03bc < \u03b5"}, {"tactic": "have := ENNReal.lt_add_right h h\u03b5", "annotated_tactic": ["have := <a>ENNReal.lt_add_right</a> h h\u03b5", [{"full_name": "ENNReal.lt_add_right", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [829, 9], "def_end_pos": [829, 21]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u2a06 \u03c6, \u2a06 (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u22a2 \u2203 \u03c6,\n    (\u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x) \u2227\n      \u2200 (\u03c8 : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (\u03c8 - \u03c6)) \u03bc < \u03b5", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u2a06 \u03c6, \u2a06 (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nthis :\n  \u2a06 \u03c6, \u2a06 (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc <\n    (\u2a06 \u03c6, \u2a06 (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc) + \u03b5\n\u22a2 \u2203 \u03c6,\n    (\u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x) \u2227\n      \u2200 (\u03c8 : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (\u03c8 - \u03c6)) \u03bc < \u03b5"}, {"tactic": "erw [ENNReal.biSup_add] at this <;> [skip; exact \u27e80, fun x => zero_le _\u27e9]", "annotated_tactic": ["erw [<a>ENNReal.biSup_add</a>] at this <;> [skip; exact \u27e80, fun x => <a>zero_le</a> _\u27e9]", [{"full_name": "ENNReal.biSup_add", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [576, 9], "def_end_pos": [576, 18]}, {"full_name": "zero_le", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [217, 30], "def_end_pos": [217, 37]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u2a06 \u03c6, \u2a06 (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nthis :\n  \u2a06 \u03c6, \u2a06 (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc <\n    (\u2a06 \u03c6, \u2a06 (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc) + \u03b5\n\u22a2 \u2203 \u03c6,\n    (\u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x) \u2227\n      \u2200 (\u03c8 : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (\u03c8 - \u03c6)) \u03bc < \u03b5", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u2a06 \u03c6, \u2a06 (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nthis :\n  \u2a06 \u03c6, \u2a06 (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc <\n    \u2a06 i \u2208 fun \u03c6 => \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x, SimpleFunc.lintegral (SimpleFunc.map ENNReal.some i) \u03bc + \u03b5\n\u22a2 \u2203 \u03c6,\n    (\u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x) \u2227\n      \u2200 (\u03c8 : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (\u03c8 - \u03c6)) \u03bc < \u03b5"}, {"tactic": "simp_rw [lt_iSup_iff, iSup_lt_iff, iSup_le_iff] at this", "annotated_tactic": ["simp_rw [<a>lt_iSup_iff</a>, <a>iSup_lt_iff</a>, <a>iSup_le_iff</a>] at this", [{"full_name": "lt_iSup_iff", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [668, 9], "def_end_pos": [668, 20]}, {"full_name": "iSup_lt_iff", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [981, 9], "def_end_pos": [981, 20]}, {"full_name": "iSup_le_iff", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [964, 9], "def_end_pos": [964, 20]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u2a06 \u03c6, \u2a06 (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nthis :\n  \u2a06 \u03c6, \u2a06 (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc <\n    \u2a06 i \u2208 fun \u03c6 => \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x, SimpleFunc.lintegral (SimpleFunc.map ENNReal.some i) \u03bc + \u03b5\n\u22a2 \u2203 \u03c6,\n    (\u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x) \u2227\n      \u2200 (\u03c8 : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (\u03c8 - \u03c6)) \u03bc < \u03b5", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u2a06 \u03c6, \u2a06 (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nthis :\n  \u2203 i h b,\n    b < SimpleFunc.lintegral (SimpleFunc.map ENNReal.some i) \u03bc + \u03b5 \u2227\n      \u2200 (i : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191i x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some i) \u03bc \u2264 b\n\u22a2 \u2203 \u03c6,\n    (\u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x) \u2227\n      \u2200 (\u03c8 : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (\u03c8 - \u03c6)) \u03bc < \u03b5"}, {"tactic": "rcases this with \u27e8\u03c6, hle : \u2200 x, \u2191(\u03c6 x) \u2264 f x, b, hb\u03c6, hb\u27e9", "annotated_tactic": ["rcases this with \u27e8\u03c6, hle : \u2200 x, \u2191(\u03c6 x) \u2264 f x, b, hb\u03c6, hb\u27e9", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u2a06 \u03c6, \u2a06 (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nthis :\n  \u2203 i h b,\n    b < SimpleFunc.lintegral (SimpleFunc.map ENNReal.some i) \u03bc + \u03b5 \u2227\n      \u2200 (i : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191i x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some i) \u03bc \u2264 b\n\u22a2 \u2203 \u03c6,\n    (\u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x) \u2227\n      \u2200 (\u03c8 : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (\u03c8 - \u03c6)) \u03bc < \u03b5", "state_after": "case intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u2a06 \u03c6, \u2a06 (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\nhle : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x\nb : \u211d\u22650\u221e\nhb\u03c6 : b < SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc + \u03b5\nhb : \u2200 (i : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191i x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some i) \u03bc \u2264 b\n\u22a2 \u2203 \u03c6,\n    (\u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x) \u2227\n      \u2200 (\u03c8 : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (\u03c8 - \u03c6)) \u03bc < \u03b5"}, {"tactic": "refine' \u27e8\u03c6, hle, fun \u03c8 h\u03c8 => _\u27e9", "annotated_tactic": ["refine' \u27e8\u03c6, hle, fun \u03c8 h\u03c8 => _\u27e9", []], "state_before": "case intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u2a06 \u03c6, \u2a06 (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\nhle : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x\nb : \u211d\u22650\u221e\nhb\u03c6 : b < SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc + \u03b5\nhb : \u2200 (i : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191i x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some i) \u03bc \u2264 b\n\u22a2 \u2203 \u03c6,\n    (\u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x) \u2227\n      \u2200 (\u03c8 : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (\u03c8 - \u03c6)) \u03bc < \u03b5", "state_after": "case intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u2a06 \u03c6, \u2a06 (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\nhle : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x\nb : \u211d\u22650\u221e\nhb\u03c6 : b < SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc + \u03b5\nhb : \u2200 (i : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191i x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some i) \u03bc \u2264 b\n\u03c8 : \u03b1 \u2192\u209b \u211d\u22650\nh\u03c8 : \u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x\n\u22a2 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (\u03c8 - \u03c6)) \u03bc < \u03b5"}, {"tactic": "have : (map (\u2191) \u03c6).lintegral \u03bc \u2260 \u221e := ne_top_of_le_ne_top h (by exact le_iSup\u2082 (\u03b1 := \u211d\u22650\u221e) \u03c6 hle)", "annotated_tactic": ["have : (<a>map</a> (\u2191) \u03c6).<a>lintegral</a> \u03bc \u2260 \u221e := <a>ne_top_of_le_ne_top</a> h (by exact <a>le_iSup\u2082</a> (\u03b1 := \u211d\u22650\u221e) \u03c6 hle)", [{"full_name": "MeasureTheory.SimpleFunc.map", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [290, 5], "def_end_pos": [290, 8]}, {"full_name": "MeasureTheory.SimpleFunc.lintegral", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [950, 5], "def_end_pos": [950, 14]}, {"full_name": "ne_top_of_le_ne_top", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [194, 9], "def_end_pos": [194, 28]}, {"full_name": "le_iSup\u2082", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [857, 9], "def_end_pos": [857, 17]}]], "state_before": "case intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u2a06 \u03c6, \u2a06 (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\nhle : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x\nb : \u211d\u22650\u221e\nhb\u03c6 : b < SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc + \u03b5\nhb : \u2200 (i : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191i x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some i) \u03bc \u2264 b\n\u03c8 : \u03b1 \u2192\u209b \u211d\u22650\nh\u03c8 : \u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x\n\u22a2 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (\u03c8 - \u03c6)) \u03bc < \u03b5", "state_after": "case intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u2a06 \u03c6, \u2a06 (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\nhle : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x\nb : \u211d\u22650\u221e\nhb\u03c6 : b < SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc + \u03b5\nhb : \u2200 (i : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191i x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some i) \u03bc \u2264 b\n\u03c8 : \u03b1 \u2192\u209b \u211d\u22650\nh\u03c8 : \u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x\nthis : SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc \u2260 \u22a4\n\u22a2 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (\u03c8 - \u03c6)) \u03bc < \u03b5"}, {"tactic": "rw [\u2190 ENNReal.add_lt_add_iff_left this, \u2190 add_lintegral, \u2190 SimpleFunc.map_add @ENNReal.coe_add]", "annotated_tactic": ["rw [\u2190 <a>ENNReal.add_lt_add_iff_left</a> this, \u2190 <a>add_lintegral</a>, \u2190 <a>SimpleFunc.map_add</a> @<a>ENNReal.coe_add</a>]", [{"full_name": "ENNReal.add_lt_add_iff_left", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [809, 19], "def_end_pos": [809, 38]}, {"full_name": "MeasureTheory.SimpleFunc.add_lintegral", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [990, 9], "def_end_pos": [990, 22]}, {"full_name": "MeasureTheory.SimpleFunc.map_add", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [565, 3], "def_end_pos": [565, 14]}, {"full_name": "ENNReal.coe_add", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [386, 28], "def_end_pos": [386, 35]}]], "state_before": "case intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u2a06 \u03c6, \u2a06 (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\nhle : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x\nb : \u211d\u22650\u221e\nhb\u03c6 : b < SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc + \u03b5\nhb : \u2200 (i : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191i x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some i) \u03bc \u2264 b\n\u03c8 : \u03b1 \u2192\u209b \u211d\u22650\nh\u03c8 : \u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x\nthis : SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc \u2260 \u22a4\n\u22a2 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (\u03c8 - \u03c6)) \u03bc < \u03b5", "state_after": "case intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u2a06 \u03c6, \u2a06 (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\nhle : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x\nb : \u211d\u22650\u221e\nhb\u03c6 : b < SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc + \u03b5\nhb : \u2200 (i : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191i x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some i) \u03bc \u2264 b\n\u03c8 : \u03b1 \u2192\u209b \u211d\u22650\nh\u03c8 : \u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x\nthis : SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc \u2260 \u22a4\n\u22a2 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (\u03c6 + (\u03c8 - \u03c6))) \u03bc <\n    SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc + \u03b5"}, {"tactic": "refine' (hb _ fun x => le_trans _ (max_le (hle x) (h\u03c8 x))).trans_lt hb\u03c6", "annotated_tactic": ["refine' (hb _ fun x => <a>le_trans</a> _ (<a>max_le</a> (hle x) (h\u03c8 x))).<a>trans_lt</a> hb\u03c6", [{"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}, {"full_name": "max_le", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [68, 9], "def_end_pos": [68, 15]}, {"full_name": "LE.le.trans_lt", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [124, 7], "def_end_pos": [124, 21]}]], "state_before": "case intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u2a06 \u03c6, \u2a06 (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\nhle : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x\nb : \u211d\u22650\u221e\nhb\u03c6 : b < SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc + \u03b5\nhb : \u2200 (i : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191i x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some i) \u03bc \u2264 b\n\u03c8 : \u03b1 \u2192\u209b \u211d\u22650\nh\u03c8 : \u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x\nthis : SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc \u2260 \u22a4\n\u22a2 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (\u03c6 + (\u03c8 - \u03c6))) \u03bc <\n    SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc + \u03b5", "state_after": "case intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u2a06 \u03c6, \u2a06 (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\nhle : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x\nb : \u211d\u22650\u221e\nhb\u03c6 : b < SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc + \u03b5\nhb : \u2200 (i : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191i x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some i) \u03bc \u2264 b\n\u03c8 : \u03b1 \u2192\u209b \u211d\u22650\nh\u03c8 : \u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x\nthis : SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc \u2260 \u22a4\nx : \u03b1\n\u22a2 \u2191(\u2191(\u03c6 + (\u03c8 - \u03c6)) x) \u2264 max \u2191(\u2191\u03c6 x) \u2191(\u2191\u03c8 x)"}, {"tactic": "norm_cast", "annotated_tactic": ["norm_cast", []], "state_before": "case intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u2a06 \u03c6, \u2a06 (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\nhle : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x\nb : \u211d\u22650\u221e\nhb\u03c6 : b < SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc + \u03b5\nhb : \u2200 (i : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191i x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some i) \u03bc \u2264 b\n\u03c8 : \u03b1 \u2192\u209b \u211d\u22650\nh\u03c8 : \u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x\nthis : SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc \u2260 \u22a4\nx : \u03b1\n\u22a2 \u2191(\u2191(\u03c6 + (\u03c8 - \u03c6)) x) \u2264 max \u2191(\u2191\u03c6 x) \u2191(\u2191\u03c8 x)", "state_after": "case intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u2a06 \u03c6, \u2a06 (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\nhle : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x\nb : \u211d\u22650\u221e\nhb\u03c6 : b < SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc + \u03b5\nhb : \u2200 (i : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191i x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some i) \u03bc \u2264 b\n\u03c8 : \u03b1 \u2192\u209b \u211d\u22650\nh\u03c8 : \u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x\nthis : SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc \u2260 \u22a4\nx : \u03b1\n\u22a2 \u2191(\u03c6 + (\u03c8 - \u03c6)) x \u2264 max (\u2191\u03c6 x) (\u2191\u03c8 x)"}, {"tactic": "simp only [add_apply, sub_apply, add_tsub_eq_max]", "annotated_tactic": ["simp only [<a>add_apply</a>, <a>sub_apply</a>, <a>add_tsub_eq_max</a>]", [{"full_name": "MeasureTheory.SimpleFunc.add_apply", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [502, 3], "def_end_pos": [502, 14]}, {"full_name": "MeasureTheory.SimpleFunc.sub_apply", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [508, 3], "def_end_pos": [508, 14]}, {"full_name": "add_tsub_eq_max", "def_path": "Mathlib/Algebra/Order/Sub/Canonical.lean", "def_pos": [498, 9], "def_end_pos": [498, 24]}]], "state_before": "case intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u2a06 \u03c6, \u2a06 (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\nhle : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x\nb : \u211d\u22650\u221e\nhb\u03c6 : b < SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc + \u03b5\nhb : \u2200 (i : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191i x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some i) \u03bc \u2264 b\n\u03c8 : \u03b1 \u2192\u209b \u211d\u22650\nh\u03c8 : \u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x\nthis : SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc \u2260 \u22a4\nx : \u03b1\n\u22a2 \u2191(\u03c6 + (\u03c8 - \u03c6)) x \u2264 max (\u2191\u03c6 x) (\u2191\u03c8 x)", "state_after": "case intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u2a06 \u03c6, \u2a06 (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\nhle : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x\nb : \u211d\u22650\u221e\nhb\u03c6 : b < SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc + \u03b5\nhb : \u2200 (i : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191i x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some i) \u03bc \u2264 b\n\u03c8 : \u03b1 \u2192\u209b \u211d\u22650\nh\u03c8 : \u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x\nthis : SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc \u2260 \u22a4\nx : \u03b1\n\u22a2 max (\u2191\u03c6 x) (\u2191\u03c8 x) \u2264 max (\u2191\u03c6 x) (\u2191\u03c8 x)"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u2a06 \u03c6, \u2a06 (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\nhle : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x\nb : \u211d\u22650\u221e\nhb\u03c6 : b < SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc + \u03b5\nhb : \u2200 (i : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191i x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some i) \u03bc \u2264 b\n\u03c8 : \u03b1 \u2192\u209b \u211d\u22650\nh\u03c8 : \u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x\nthis : SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc \u2260 \u22a4\nx : \u03b1\n\u22a2 max (\u2191\u03c6 x) (\u2191\u03c8 x) \u2264 max (\u2191\u03c6 x) (\u2191\u03c8 x)", "state_after": "no goals"}, {"tactic": "exact le_iSup\u2082 (\u03b1 := \u211d\u22650\u221e) \u03c6 hle", "annotated_tactic": ["exact <a>le_iSup\u2082</a> (\u03b1 := \u211d\u22650\u221e) \u03c6 hle", [{"full_name": "le_iSup\u2082", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [857, 9], "def_end_pos": [857, 17]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u2a06 \u03c6, \u2a06 (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\nhle : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x\nb : \u211d\u22650\u221e\nhb\u03c6 : b < SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc + \u03b5\nhb : \u2200 (i : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191i x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some i) \u03bc \u2264 b\n\u03c8 : \u03b1 \u2192\u209b \u211d\u22650\nh\u03c8 : \u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x\n\u22a2 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc \u2264\n    \u2a06 \u03c6, \u2a06 (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "full_name": "List.eraseP_map", "start": [1133, 1], "end": [1135, 85], "traced_tactics": [{"tactic": "by_cases h : p (f b) <;> simp [h, eraseP_map f l, eraseP_cons_of_pos]", "annotated_tactic": ["by_cases h : p (f b) <;> simp [h, eraseP_map f l, <a>eraseP_cons_of_pos</a>]", [{"full_name": "List.eraseP_cons_of_pos", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [1055, 17], "def_end_pos": [1055, 35]}]], "state_before": "\u03b2 : Type u_1\n\u03b1 : Type u_2\np : \u03b1 \u2192 Bool\nf : \u03b2 \u2192 \u03b1\nb : \u03b2\nl : List \u03b2\n\u22a2 eraseP p (map f (b :: l)) = map f (eraseP (p \u2218 f) (b :: l))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Pointwise.lean", "full_name": "Finset.preimage_mul_right_one'", "start": [1241, 1], "end": [1242, 39], "traced_tactics": [{"tactic": "rw [preimage_mul_right_one, inv_inv]", "annotated_tactic": ["rw [<a>preimage_mul_right_one</a>, <a>inv_inv</a>]", [{"full_name": "Finset.preimage_mul_right_one", "def_path": "Mathlib/Data/Finset/Pointwise.lean", "def_pos": [1229, 9], "def_end_pos": [1229, 31]}, {"full_name": "inv_inv", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [800, 9], "def_end_pos": [800, 16]}]], "state_before": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : Group \u03b1\ns t : Finset \u03b1\na b : \u03b1\n\u22a2 preimage 1 (fun x => x * b\u207b\u00b9) (_ : Set.InjOn (fun x => x * b\u207b\u00b9) ((fun x => x * b\u207b\u00b9) \u207b\u00b9' \u21911)) = {b}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/ZMod/Basic.lean", "full_name": "ZMod.val_mul", "start": [643, 1], "end": [647, 22], "traced_tactics": [{"tactic": "cases n", "annotated_tactic": ["cases n", []], "state_before": "n : \u2115\na b : ZMod n\n\u22a2 val (a * b) = val a * val b % n", "state_after": "case zero\na b : ZMod Nat.zero\n\u22a2 val (a * b) = val a * val b % Nat.zero\n\ncase succ\nn\u271d : \u2115\na b : ZMod (Nat.succ n\u271d)\n\u22a2 val (a * b) = val a * val b % Nat.succ n\u271d"}, {"tactic": "rw [Nat.mod_zero]", "annotated_tactic": ["rw [<a>Nat.mod_zero</a>]", [{"full_name": "Nat.mod_zero", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Div.lean", "def_pos": [105, 17], "def_end_pos": [105, 25]}]], "state_before": "case zero\na b : ZMod Nat.zero\n\u22a2 val (a * b) = val a * val b % Nat.zero", "state_after": "case zero\na b : ZMod Nat.zero\n\u22a2 val (a * b) = val a * val b"}, {"tactic": "apply Int.natAbs_mul", "annotated_tactic": ["apply <a>Int.natAbs_mul</a>", [{"full_name": "Int.natAbs_mul", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [171, 9], "def_end_pos": [171, 19]}]], "state_before": "case zero\na b : ZMod Nat.zero\n\u22a2 val (a * b) = val a * val b", "state_after": "no goals"}, {"tactic": "apply Fin.val_mul", "annotated_tactic": ["apply <a>Fin.val_mul</a>", [{"full_name": "Fin.val_mul", "def_path": "lake-packages/std/Std/Data/Fin/Lemmas.lean", "def_pos": [712, 9], "def_end_pos": [712, 16]}]], "state_before": "case succ\nn\u271d : \u2115\na b : ZMod (Nat.succ n\u271d)\n\u22a2 val (a * b) = val a * val b % Nat.succ n\u271d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "full_name": "MeasureTheory.SimpleFunc.induction", "start": [1266, 11], "end": [1301, 71], "traced_tactics": [{"tactic": "generalize h : f.range \\ {0} = s", "annotated_tactic": ["generalize h : f.range \\ {0} = s", []], "state_before": "\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3\u271d : Type u_3\n\u03b4 : Type u_4\n\u03b1 : Type u_5\n\u03b3 : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : AddMonoid \u03b3\nP : (\u03b1 \u2192\u209b \u03b3) \u2192 Prop\nh_ind : \u2200 (c : \u03b3) {s : Set \u03b1} (hs : MeasurableSet s), P (piecewise s hs (const \u03b1 c) (const \u03b1 0))\nh_add : \u2200 \u2983f g : \u03b1 \u2192\u209b \u03b3\u2984, Disjoint (support \u2191f) (support \u2191g) \u2192 P f \u2192 P g \u2192 P (f + g)\nf : \u03b1 \u2192\u209b \u03b3\n\u22a2 P f", "state_after": "\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3\u271d : Type u_3\n\u03b4 : Type u_4\n\u03b1 : Type u_5\n\u03b3 : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : AddMonoid \u03b3\nP : (\u03b1 \u2192\u209b \u03b3) \u2192 Prop\nh_ind : \u2200 (c : \u03b3) {s : Set \u03b1} (hs : MeasurableSet s), P (piecewise s hs (const \u03b1 c) (const \u03b1 0))\nh_add : \u2200 \u2983f g : \u03b1 \u2192\u209b \u03b3\u2984, Disjoint (support \u2191f) (support \u2191g) \u2192 P f \u2192 P g \u2192 P (f + g)\nf : \u03b1 \u2192\u209b \u03b3\ns : Finset \u03b3\nh : SimpleFunc.range f \\ {0} = s\n\u22a2 P f"}, {"tactic": "rw [\u2190 Finset.coe_inj, Finset.coe_sdiff, Finset.coe_singleton, SimpleFunc.coe_range] at h", "annotated_tactic": ["rw [\u2190 <a>Finset.coe_inj</a>, <a>Finset.coe_sdiff</a>, <a>Finset.coe_singleton</a>, <a>SimpleFunc.coe_range</a>] at h", [{"full_name": "Finset.coe_inj", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [244, 9], "def_end_pos": [244, 16]}, {"full_name": "Finset.coe_sdiff", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2150, 9], "def_end_pos": [2150, 18]}, {"full_name": "Finset.coe_singleton", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [725, 9], "def_end_pos": [725, 22]}, {"full_name": "MeasureTheory.SimpleFunc.coe_range", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [117, 9], "def_end_pos": [117, 18]}]], "state_before": "\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3\u271d : Type u_3\n\u03b4 : Type u_4\n\u03b1 : Type u_5\n\u03b3 : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : AddMonoid \u03b3\nP : (\u03b1 \u2192\u209b \u03b3) \u2192 Prop\nh_ind : \u2200 (c : \u03b3) {s : Set \u03b1} (hs : MeasurableSet s), P (piecewise s hs (const \u03b1 c) (const \u03b1 0))\nh_add : \u2200 \u2983f g : \u03b1 \u2192\u209b \u03b3\u2984, Disjoint (support \u2191f) (support \u2191g) \u2192 P f \u2192 P g \u2192 P (f + g)\nf : \u03b1 \u2192\u209b \u03b3\ns : Finset \u03b3\nh : SimpleFunc.range f \\ {0} = s\n\u22a2 P f", "state_after": "\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3\u271d : Type u_3\n\u03b4 : Type u_4\n\u03b1 : Type u_5\n\u03b3 : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : AddMonoid \u03b3\nP : (\u03b1 \u2192\u209b \u03b3) \u2192 Prop\nh_ind : \u2200 (c : \u03b3) {s : Set \u03b1} (hs : MeasurableSet s), P (piecewise s hs (const \u03b1 c) (const \u03b1 0))\nh_add : \u2200 \u2983f g : \u03b1 \u2192\u209b \u03b3\u2984, Disjoint (support \u2191f) (support \u2191g) \u2192 P f \u2192 P g \u2192 P (f + g)\nf : \u03b1 \u2192\u209b \u03b3\ns : Finset \u03b3\nh : range \u2191f \\ {0} = \u2191s\n\u22a2 P f"}, {"tactic": "rw [Finset.coe_empty, diff_eq_empty, range_subset_singleton] at h", "annotated_tactic": ["rw [<a>Finset.coe_empty</a>, <a>diff_eq_empty</a>, <a>range_subset_singleton</a>] at h", [{"full_name": "Finset.coe_empty", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [618, 9], "def_end_pos": [618, 18]}, {"full_name": "Set.diff_eq_empty", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1925, 9], "def_end_pos": [1925, 22]}, {"full_name": "Set.range_subset_singleton", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [1052, 9], "def_end_pos": [1052, 31]}]], "state_before": "case empty\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3\u271d : Type u_3\n\u03b4 : Type u_4\n\u03b1 : Type u_5\n\u03b3 : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : AddMonoid \u03b3\nP : (\u03b1 \u2192\u209b \u03b3) \u2192 Prop\nh_ind : \u2200 (c : \u03b3) {s : Set \u03b1} (hs : MeasurableSet s), P (piecewise s hs (const \u03b1 c) (const \u03b1 0))\nh_add : \u2200 \u2983f g : \u03b1 \u2192\u209b \u03b3\u2984, Disjoint (support \u2191f) (support \u2191g) \u2192 P f \u2192 P g \u2192 P (f + g)\nf : \u03b1 \u2192\u209b \u03b3\nh : range \u2191f \\ {0} = \u2191\u2205\n\u22a2 P f", "state_after": "case empty\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3\u271d : Type u_3\n\u03b4 : Type u_4\n\u03b1 : Type u_5\n\u03b3 : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : AddMonoid \u03b3\nP : (\u03b1 \u2192\u209b \u03b3) \u2192 Prop\nh_ind : \u2200 (c : \u03b3) {s : Set \u03b1} (hs : MeasurableSet s), P (piecewise s hs (const \u03b1 c) (const \u03b1 0))\nh_add : \u2200 \u2983f g : \u03b1 \u2192\u209b \u03b3\u2984, Disjoint (support \u2191f) (support \u2191g) \u2192 P f \u2192 P g \u2192 P (f + g)\nf : \u03b1 \u2192\u209b \u03b3\nh : \u2191f = Function.const \u03b1 0\n\u22a2 P f"}, {"tactic": "convert h_ind 0 MeasurableSet.univ", "annotated_tactic": ["convert h_ind 0 <a>MeasurableSet.univ</a>", [{"full_name": "MeasurableSet.univ", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [101, 19], "def_end_pos": [101, 37]}]], "state_before": "case empty\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3\u271d : Type u_3\n\u03b4 : Type u_4\n\u03b1 : Type u_5\n\u03b3 : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : AddMonoid \u03b3\nP : (\u03b1 \u2192\u209b \u03b3) \u2192 Prop\nh_ind : \u2200 (c : \u03b3) {s : Set \u03b1} (hs : MeasurableSet s), P (piecewise s hs (const \u03b1 c) (const \u03b1 0))\nh_add : \u2200 \u2983f g : \u03b1 \u2192\u209b \u03b3\u2984, Disjoint (support \u2191f) (support \u2191g) \u2192 P f \u2192 P g \u2192 P (f + g)\nf : \u03b1 \u2192\u209b \u03b3\nh : \u2191f = Function.const \u03b1 0\n\u22a2 P f", "state_after": "case h.e'_1\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3\u271d : Type u_3\n\u03b4 : Type u_4\n\u03b1 : Type u_5\n\u03b3 : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : AddMonoid \u03b3\nP : (\u03b1 \u2192\u209b \u03b3) \u2192 Prop\nh_ind : \u2200 (c : \u03b3) {s : Set \u03b1} (hs : MeasurableSet s), P (piecewise s hs (const \u03b1 c) (const \u03b1 0))\nh_add : \u2200 \u2983f g : \u03b1 \u2192\u209b \u03b3\u2984, Disjoint (support \u2191f) (support \u2191g) \u2192 P f \u2192 P g \u2192 P (f + g)\nf : \u03b1 \u2192\u209b \u03b3\nh : \u2191f = Function.const \u03b1 0\n\u22a2 f = piecewise univ (_ : MeasurableSet univ) (const \u03b1 0) (const \u03b1 0)"}, {"tactic": "ext x", "annotated_tactic": ["ext x", []], "state_before": "case h.e'_1\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3\u271d : Type u_3\n\u03b4 : Type u_4\n\u03b1 : Type u_5\n\u03b3 : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : AddMonoid \u03b3\nP : (\u03b1 \u2192\u209b \u03b3) \u2192 Prop\nh_ind : \u2200 (c : \u03b3) {s : Set \u03b1} (hs : MeasurableSet s), P (piecewise s hs (const \u03b1 c) (const \u03b1 0))\nh_add : \u2200 \u2983f g : \u03b1 \u2192\u209b \u03b3\u2984, Disjoint (support \u2191f) (support \u2191g) \u2192 P f \u2192 P g \u2192 P (f + g)\nf : \u03b1 \u2192\u209b \u03b3\nh : \u2191f = Function.const \u03b1 0\n\u22a2 f = piecewise univ (_ : MeasurableSet univ) (const \u03b1 0) (const \u03b1 0)", "state_after": "case h.e'_1.H\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3\u271d : Type u_3\n\u03b4 : Type u_4\n\u03b1 : Type u_5\n\u03b3 : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : AddMonoid \u03b3\nP : (\u03b1 \u2192\u209b \u03b3) \u2192 Prop\nh_ind : \u2200 (c : \u03b3) {s : Set \u03b1} (hs : MeasurableSet s), P (piecewise s hs (const \u03b1 c) (const \u03b1 0))\nh_add : \u2200 \u2983f g : \u03b1 \u2192\u209b \u03b3\u2984, Disjoint (support \u2191f) (support \u2191g) \u2192 P f \u2192 P g \u2192 P (f + g)\nf : \u03b1 \u2192\u209b \u03b3\nh : \u2191f = Function.const \u03b1 0\nx : \u03b1\n\u22a2 \u2191f x = \u2191(piecewise univ (_ : MeasurableSet univ) (const \u03b1 0) (const \u03b1 0)) x"}, {"tactic": "simp [h]", "annotated_tactic": ["simp [h]", []], "state_before": "case h.e'_1.H\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3\u271d : Type u_3\n\u03b4 : Type u_4\n\u03b1 : Type u_5\n\u03b3 : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : AddMonoid \u03b3\nP : (\u03b1 \u2192\u209b \u03b3) \u2192 Prop\nh_ind : \u2200 (c : \u03b3) {s : Set \u03b1} (hs : MeasurableSet s), P (piecewise s hs (const \u03b1 c) (const \u03b1 0))\nh_add : \u2200 \u2983f g : \u03b1 \u2192\u209b \u03b3\u2984, Disjoint (support \u2191f) (support \u2191g) \u2192 P f \u2192 P g \u2192 P (f + g)\nf : \u03b1 \u2192\u209b \u03b3\nh : \u2191f = Function.const \u03b1 0\nx : \u03b1\n\u22a2 \u2191f x = \u2191(piecewise univ (_ : MeasurableSet univ) (const \u03b1 0) (const \u03b1 0)) x", "state_after": "no goals"}, {"tactic": "have mx := f.measurableSet_preimage {x}", "annotated_tactic": ["have mx := f.measurableSet_preimage {x}", []], "state_before": "case insert\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3\u271d : Type u_3\n\u03b4 : Type u_4\n\u03b1 : Type u_5\n\u03b3 : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : AddMonoid \u03b3\nP : (\u03b1 \u2192\u209b \u03b3) \u2192 Prop\nh_ind : \u2200 (c : \u03b3) {s : Set \u03b1} (hs : MeasurableSet s), P (piecewise s hs (const \u03b1 c) (const \u03b1 0))\nh_add : \u2200 \u2983f g : \u03b1 \u2192\u209b \u03b3\u2984, Disjoint (support \u2191f) (support \u2191g) \u2192 P f \u2192 P g \u2192 P (f + g)\nx : \u03b3\ns : Finset \u03b3\nhxs : \u00acx \u2208 s\nih : \u2200 (f : \u03b1 \u2192\u209b \u03b3), range \u2191f \\ {0} = \u2191s \u2192 P f\nf : \u03b1 \u2192\u209b \u03b3\nh : range \u2191f \\ {0} = \u2191(insert x s)\n\u22a2 P f", "state_after": "case insert\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3\u271d : Type u_3\n\u03b4 : Type u_4\n\u03b1 : Type u_5\n\u03b3 : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : AddMonoid \u03b3\nP : (\u03b1 \u2192\u209b \u03b3) \u2192 Prop\nh_ind : \u2200 (c : \u03b3) {s : Set \u03b1} (hs : MeasurableSet s), P (piecewise s hs (const \u03b1 c) (const \u03b1 0))\nh_add : \u2200 \u2983f g : \u03b1 \u2192\u209b \u03b3\u2984, Disjoint (support \u2191f) (support \u2191g) \u2192 P f \u2192 P g \u2192 P (f + g)\nx : \u03b3\ns : Finset \u03b3\nhxs : \u00acx \u2208 s\nih : \u2200 (f : \u03b1 \u2192\u209b \u03b3), range \u2191f \\ {0} = \u2191s \u2192 P f\nf : \u03b1 \u2192\u209b \u03b3\nh : range \u2191f \\ {0} = \u2191(insert x s)\nmx : MeasurableSet (\u2191f \u207b\u00b9' {x})\n\u22a2 P f"}, {"tactic": "let g := SimpleFunc.piecewise (f \u207b\u00b9' {x}) mx 0 f", "annotated_tactic": ["let g := <a>SimpleFunc.piecewise</a> (f \u207b\u00b9' {x}) mx 0 f", [{"full_name": "MeasureTheory.SimpleFunc.piecewise", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [222, 5], "def_end_pos": [222, 14]}]], "state_before": "case insert\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3\u271d : Type u_3\n\u03b4 : Type u_4\n\u03b1 : Type u_5\n\u03b3 : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : AddMonoid \u03b3\nP : (\u03b1 \u2192\u209b \u03b3) \u2192 Prop\nh_ind : \u2200 (c : \u03b3) {s : Set \u03b1} (hs : MeasurableSet s), P (piecewise s hs (const \u03b1 c) (const \u03b1 0))\nh_add : \u2200 \u2983f g : \u03b1 \u2192\u209b \u03b3\u2984, Disjoint (support \u2191f) (support \u2191g) \u2192 P f \u2192 P g \u2192 P (f + g)\nx : \u03b3\ns : Finset \u03b3\nhxs : \u00acx \u2208 s\nih : \u2200 (f : \u03b1 \u2192\u209b \u03b3), range \u2191f \\ {0} = \u2191s \u2192 P f\nf : \u03b1 \u2192\u209b \u03b3\nh : range \u2191f \\ {0} = \u2191(insert x s)\nmx : MeasurableSet (\u2191f \u207b\u00b9' {x})\n\u22a2 P f", "state_after": "case insert\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3\u271d : Type u_3\n\u03b4 : Type u_4\n\u03b1 : Type u_5\n\u03b3 : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : AddMonoid \u03b3\nP : (\u03b1 \u2192\u209b \u03b3) \u2192 Prop\nh_ind : \u2200 (c : \u03b3) {s : Set \u03b1} (hs : MeasurableSet s), P (piecewise s hs (const \u03b1 c) (const \u03b1 0))\nh_add : \u2200 \u2983f g : \u03b1 \u2192\u209b \u03b3\u2984, Disjoint (support \u2191f) (support \u2191g) \u2192 P f \u2192 P g \u2192 P (f + g)\nx : \u03b3\ns : Finset \u03b3\nhxs : \u00acx \u2208 s\nih : \u2200 (f : \u03b1 \u2192\u209b \u03b3), range \u2191f \\ {0} = \u2191s \u2192 P f\nf : \u03b1 \u2192\u209b \u03b3\nh : range \u2191f \\ {0} = \u2191(insert x s)\nmx : MeasurableSet (\u2191f \u207b\u00b9' {x})\ng : \u03b1 \u2192\u209b \u03b3 := piecewise (\u2191f \u207b\u00b9' {x}) mx 0 f\n\u22a2 P f"}, {"tactic": "convert h_add _ Pg (h_ind x mx)", "annotated_tactic": ["convert h_add _ Pg (h_ind x mx)", []], "state_before": "case insert\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3\u271d : Type u_3\n\u03b4 : Type u_4\n\u03b1 : Type u_5\n\u03b3 : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : AddMonoid \u03b3\nP : (\u03b1 \u2192\u209b \u03b3) \u2192 Prop\nh_ind : \u2200 (c : \u03b3) {s : Set \u03b1} (hs : MeasurableSet s), P (piecewise s hs (const \u03b1 c) (const \u03b1 0))\nh_add : \u2200 \u2983f g : \u03b1 \u2192\u209b \u03b3\u2984, Disjoint (support \u2191f) (support \u2191g) \u2192 P f \u2192 P g \u2192 P (f + g)\nx : \u03b3\ns : Finset \u03b3\nhxs : \u00acx \u2208 s\nih : \u2200 (f : \u03b1 \u2192\u209b \u03b3), range \u2191f \\ {0} = \u2191s \u2192 P f\nf : \u03b1 \u2192\u209b \u03b3\nh : range \u2191f \\ {0} = \u2191(insert x s)\nmx : MeasurableSet (\u2191f \u207b\u00b9' {x})\ng : \u03b1 \u2192\u209b \u03b3 := piecewise (\u2191f \u207b\u00b9' {x}) mx 0 f\nPg : P g\n\u22a2 P f", "state_after": "case h.e'_1\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3\u271d : Type u_3\n\u03b4 : Type u_4\n\u03b1 : Type u_5\n\u03b3 : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : AddMonoid \u03b3\nP : (\u03b1 \u2192\u209b \u03b3) \u2192 Prop\nh_ind : \u2200 (c : \u03b3) {s : Set \u03b1} (hs : MeasurableSet s), P (piecewise s hs (const \u03b1 c) (const \u03b1 0))\nh_add : \u2200 \u2983f g : \u03b1 \u2192\u209b \u03b3\u2984, Disjoint (support \u2191f) (support \u2191g) \u2192 P f \u2192 P g \u2192 P (f + g)\nx : \u03b3\ns : Finset \u03b3\nhxs : \u00acx \u2208 s\nih : \u2200 (f : \u03b1 \u2192\u209b \u03b3), range \u2191f \\ {0} = \u2191s \u2192 P f\nf : \u03b1 \u2192\u209b \u03b3\nh : range \u2191f \\ {0} = \u2191(insert x s)\nmx : MeasurableSet (\u2191f \u207b\u00b9' {x})\ng : \u03b1 \u2192\u209b \u03b3 := piecewise (\u2191f \u207b\u00b9' {x}) mx 0 f\nPg : P g\n\u22a2 f = g + piecewise (\u2191f \u207b\u00b9' {x}) mx (const \u03b1 x) (const \u03b1 0)\n\ncase insert\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3\u271d : Type u_3\n\u03b4 : Type u_4\n\u03b1 : Type u_5\n\u03b3 : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : AddMonoid \u03b3\nP : (\u03b1 \u2192\u209b \u03b3) \u2192 Prop\nh_ind : \u2200 (c : \u03b3) {s : Set \u03b1} (hs : MeasurableSet s), P (piecewise s hs (const \u03b1 c) (const \u03b1 0))\nh_add : \u2200 \u2983f g : \u03b1 \u2192\u209b \u03b3\u2984, Disjoint (support \u2191f) (support \u2191g) \u2192 P f \u2192 P g \u2192 P (f + g)\nx : \u03b3\ns : Finset \u03b3\nhxs : \u00acx \u2208 s\nih : \u2200 (f : \u03b1 \u2192\u209b \u03b3), range \u2191f \\ {0} = \u2191s \u2192 P f\nf : \u03b1 \u2192\u209b \u03b3\nh : range \u2191f \\ {0} = \u2191(insert x s)\nmx : MeasurableSet (\u2191f \u207b\u00b9' {x})\ng : \u03b1 \u2192\u209b \u03b3 := piecewise (\u2191f \u207b\u00b9' {x}) mx 0 f\nPg : P g\n\u22a2 Disjoint (support \u2191g) (support \u2191(piecewise (\u2191f \u207b\u00b9' {x}) mx (const \u03b1 x) (const \u03b1 0)))"}, {"tactic": "rw [disjoint_iff_inf_le]", "annotated_tactic": ["rw [<a>disjoint_iff_inf_le</a>]", [{"full_name": "disjoint_iff_inf_le", "def_path": "Mathlib/Order/Disjoint.lean", "def_pos": [122, 9], "def_end_pos": [122, 28]}]], "state_before": "case insert\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3\u271d : Type u_3\n\u03b4 : Type u_4\n\u03b1 : Type u_5\n\u03b3 : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : AddMonoid \u03b3\nP : (\u03b1 \u2192\u209b \u03b3) \u2192 Prop\nh_ind : \u2200 (c : \u03b3) {s : Set \u03b1} (hs : MeasurableSet s), P (piecewise s hs (const \u03b1 c) (const \u03b1 0))\nh_add : \u2200 \u2983f g : \u03b1 \u2192\u209b \u03b3\u2984, Disjoint (support \u2191f) (support \u2191g) \u2192 P f \u2192 P g \u2192 P (f + g)\nx : \u03b3\ns : Finset \u03b3\nhxs : \u00acx \u2208 s\nih : \u2200 (f : \u03b1 \u2192\u209b \u03b3), range \u2191f \\ {0} = \u2191s \u2192 P f\nf : \u03b1 \u2192\u209b \u03b3\nh : range \u2191f \\ {0} = \u2191(insert x s)\nmx : MeasurableSet (\u2191f \u207b\u00b9' {x})\ng : \u03b1 \u2192\u209b \u03b3 := piecewise (\u2191f \u207b\u00b9' {x}) mx 0 f\nPg : P g\n\u22a2 Disjoint (support \u2191g) (support \u2191(piecewise (\u2191f \u207b\u00b9' {x}) mx (const \u03b1 x) (const \u03b1 0)))", "state_after": "case insert\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3\u271d : Type u_3\n\u03b4 : Type u_4\n\u03b1 : Type u_5\n\u03b3 : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : AddMonoid \u03b3\nP : (\u03b1 \u2192\u209b \u03b3) \u2192 Prop\nh_ind : \u2200 (c : \u03b3) {s : Set \u03b1} (hs : MeasurableSet s), P (piecewise s hs (const \u03b1 c) (const \u03b1 0))\nh_add : \u2200 \u2983f g : \u03b1 \u2192\u209b \u03b3\u2984, Disjoint (support \u2191f) (support \u2191g) \u2192 P f \u2192 P g \u2192 P (f + g)\nx : \u03b3\ns : Finset \u03b3\nhxs : \u00acx \u2208 s\nih : \u2200 (f : \u03b1 \u2192\u209b \u03b3), range \u2191f \\ {0} = \u2191s \u2192 P f\nf : \u03b1 \u2192\u209b \u03b3\nh : range \u2191f \\ {0} = \u2191(insert x s)\nmx : MeasurableSet (\u2191f \u207b\u00b9' {x})\ng : \u03b1 \u2192\u209b \u03b3 := piecewise (\u2191f \u207b\u00b9' {x}) mx 0 f\nPg : P g\n\u22a2 support \u2191g \u2293 support \u2191(piecewise (\u2191f \u207b\u00b9' {x}) mx (const \u03b1 x) (const \u03b1 0)) \u2264 \u22a5"}, {"tactic": "rintro y", "annotated_tactic": ["rintro y", []], "state_before": "case insert\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3\u271d : Type u_3\n\u03b4 : Type u_4\n\u03b1 : Type u_5\n\u03b3 : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : AddMonoid \u03b3\nP : (\u03b1 \u2192\u209b \u03b3) \u2192 Prop\nh_ind : \u2200 (c : \u03b3) {s : Set \u03b1} (hs : MeasurableSet s), P (piecewise s hs (const \u03b1 c) (const \u03b1 0))\nh_add : \u2200 \u2983f g : \u03b1 \u2192\u209b \u03b3\u2984, Disjoint (support \u2191f) (support \u2191g) \u2192 P f \u2192 P g \u2192 P (f + g)\nx : \u03b3\ns : Finset \u03b3\nhxs : \u00acx \u2208 s\nih : \u2200 (f : \u03b1 \u2192\u209b \u03b3), range \u2191f \\ {0} = \u2191s \u2192 P f\nf : \u03b1 \u2192\u209b \u03b3\nh : range \u2191f \\ {0} = \u2191(insert x s)\nmx : MeasurableSet (\u2191f \u207b\u00b9' {x})\ng : \u03b1 \u2192\u209b \u03b3 := piecewise (\u2191f \u207b\u00b9' {x}) mx 0 f\nPg : P g\n\u22a2 support \u2191g \u2293 support \u2191(piecewise (\u2191f \u207b\u00b9' {x}) mx (const \u03b1 x) (const \u03b1 0)) \u2264 \u22a5", "state_after": "case insert\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3\u271d : Type u_3\n\u03b4 : Type u_4\n\u03b1 : Type u_5\n\u03b3 : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : AddMonoid \u03b3\nP : (\u03b1 \u2192\u209b \u03b3) \u2192 Prop\nh_ind : \u2200 (c : \u03b3) {s : Set \u03b1} (hs : MeasurableSet s), P (piecewise s hs (const \u03b1 c) (const \u03b1 0))\nh_add : \u2200 \u2983f g : \u03b1 \u2192\u209b \u03b3\u2984, Disjoint (support \u2191f) (support \u2191g) \u2192 P f \u2192 P g \u2192 P (f + g)\nx : \u03b3\ns : Finset \u03b3\nhxs : \u00acx \u2208 s\nih : \u2200 (f : \u03b1 \u2192\u209b \u03b3), range \u2191f \\ {0} = \u2191s \u2192 P f\nf : \u03b1 \u2192\u209b \u03b3\nh : range \u2191f \\ {0} = \u2191(insert x s)\nmx : MeasurableSet (\u2191f \u207b\u00b9' {x})\ng : \u03b1 \u2192\u209b \u03b3 := piecewise (\u2191f \u207b\u00b9' {x}) mx 0 f\nPg : P g\ny : \u03b1\n\u22a2 y \u2208 support \u2191g \u2293 support \u2191(piecewise (\u2191f \u207b\u00b9' {x}) mx (const \u03b1 x) (const \u03b1 0)) \u2192 y \u2208 \u22a5"}, {"tactic": "by_cases hy : y \u2208 f \u207b\u00b9' {x}", "annotated_tactic": ["by_cases hy : y \u2208 f \u207b\u00b9' {x}", []], "state_before": "case insert\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3\u271d : Type u_3\n\u03b4 : Type u_4\n\u03b1 : Type u_5\n\u03b3 : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : AddMonoid \u03b3\nP : (\u03b1 \u2192\u209b \u03b3) \u2192 Prop\nh_ind : \u2200 (c : \u03b3) {s : Set \u03b1} (hs : MeasurableSet s), P (piecewise s hs (const \u03b1 c) (const \u03b1 0))\nh_add : \u2200 \u2983f g : \u03b1 \u2192\u209b \u03b3\u2984, Disjoint (support \u2191f) (support \u2191g) \u2192 P f \u2192 P g \u2192 P (f + g)\nx : \u03b3\ns : Finset \u03b3\nhxs : \u00acx \u2208 s\nih : \u2200 (f : \u03b1 \u2192\u209b \u03b3), range \u2191f \\ {0} = \u2191s \u2192 P f\nf : \u03b1 \u2192\u209b \u03b3\nh : range \u2191f \\ {0} = \u2191(insert x s)\nmx : MeasurableSet (\u2191f \u207b\u00b9' {x})\ng : \u03b1 \u2192\u209b \u03b3 := piecewise (\u2191f \u207b\u00b9' {x}) mx 0 f\nPg : P g\ny : \u03b1\n\u22a2 y \u2208 support \u2191g \u2293 support \u2191(piecewise (\u2191f \u207b\u00b9' {x}) mx (const \u03b1 x) (const \u03b1 0)) \u2192 y \u2208 \u22a5", "state_after": "case pos\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3\u271d : Type u_3\n\u03b4 : Type u_4\n\u03b1 : Type u_5\n\u03b3 : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : AddMonoid \u03b3\nP : (\u03b1 \u2192\u209b \u03b3) \u2192 Prop\nh_ind : \u2200 (c : \u03b3) {s : Set \u03b1} (hs : MeasurableSet s), P (piecewise s hs (const \u03b1 c) (const \u03b1 0))\nh_add : \u2200 \u2983f g : \u03b1 \u2192\u209b \u03b3\u2984, Disjoint (support \u2191f) (support \u2191g) \u2192 P f \u2192 P g \u2192 P (f + g)\nx : \u03b3\ns : Finset \u03b3\nhxs : \u00acx \u2208 s\nih : \u2200 (f : \u03b1 \u2192\u209b \u03b3), range \u2191f \\ {0} = \u2191s \u2192 P f\nf : \u03b1 \u2192\u209b \u03b3\nh : range \u2191f \\ {0} = \u2191(insert x s)\nmx : MeasurableSet (\u2191f \u207b\u00b9' {x})\ng : \u03b1 \u2192\u209b \u03b3 := piecewise (\u2191f \u207b\u00b9' {x}) mx 0 f\nPg : P g\ny : \u03b1\nhy : y \u2208 \u2191f \u207b\u00b9' {x}\n\u22a2 y \u2208 support \u2191g \u2293 support \u2191(piecewise (\u2191f \u207b\u00b9' {x}) mx (const \u03b1 x) (const \u03b1 0)) \u2192 y \u2208 \u22a5\n\ncase neg\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3\u271d : Type u_3\n\u03b4 : Type u_4\n\u03b1 : Type u_5\n\u03b3 : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : AddMonoid \u03b3\nP : (\u03b1 \u2192\u209b \u03b3) \u2192 Prop\nh_ind : \u2200 (c : \u03b3) {s : Set \u03b1} (hs : MeasurableSet s), P (piecewise s hs (const \u03b1 c) (const \u03b1 0))\nh_add : \u2200 \u2983f g : \u03b1 \u2192\u209b \u03b3\u2984, Disjoint (support \u2191f) (support \u2191g) \u2192 P f \u2192 P g \u2192 P (f + g)\nx : \u03b3\ns : Finset \u03b3\nhxs : \u00acx \u2208 s\nih : \u2200 (f : \u03b1 \u2192\u209b \u03b3), range \u2191f \\ {0} = \u2191s \u2192 P f\nf : \u03b1 \u2192\u209b \u03b3\nh : range \u2191f \\ {0} = \u2191(insert x s)\nmx : MeasurableSet (\u2191f \u207b\u00b9' {x})\ng : \u03b1 \u2192\u209b \u03b3 := piecewise (\u2191f \u207b\u00b9' {x}) mx 0 f\nPg : P g\ny : \u03b1\nhy : \u00acy \u2208 \u2191f \u207b\u00b9' {x}\n\u22a2 y \u2208 support \u2191g \u2293 support \u2191(piecewise (\u2191f \u207b\u00b9' {x}) mx (const \u03b1 x) (const \u03b1 0)) \u2192 y \u2208 \u22a5"}, {"tactic": "apply ih", "annotated_tactic": ["apply ih", []], "state_before": "\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3\u271d : Type u_3\n\u03b4 : Type u_4\n\u03b1 : Type u_5\n\u03b3 : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : AddMonoid \u03b3\nP : (\u03b1 \u2192\u209b \u03b3) \u2192 Prop\nh_ind : \u2200 (c : \u03b3) {s : Set \u03b1} (hs : MeasurableSet s), P (piecewise s hs (const \u03b1 c) (const \u03b1 0))\nh_add : \u2200 \u2983f g : \u03b1 \u2192\u209b \u03b3\u2984, Disjoint (support \u2191f) (support \u2191g) \u2192 P f \u2192 P g \u2192 P (f + g)\nx : \u03b3\ns : Finset \u03b3\nhxs : \u00acx \u2208 s\nih : \u2200 (f : \u03b1 \u2192\u209b \u03b3), range \u2191f \\ {0} = \u2191s \u2192 P f\nf : \u03b1 \u2192\u209b \u03b3\nh : range \u2191f \\ {0} = \u2191(insert x s)\nmx : MeasurableSet (\u2191f \u207b\u00b9' {x})\ng : \u03b1 \u2192\u209b \u03b3 := piecewise (\u2191f \u207b\u00b9' {x}) mx 0 f\n\u22a2 P g", "state_after": "case h\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3\u271d : Type u_3\n\u03b4 : Type u_4\n\u03b1 : Type u_5\n\u03b3 : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : AddMonoid \u03b3\nP : (\u03b1 \u2192\u209b \u03b3) \u2192 Prop\nh_ind : \u2200 (c : \u03b3) {s : Set \u03b1} (hs : MeasurableSet s), P (piecewise s hs (const \u03b1 c) (const \u03b1 0))\nh_add : \u2200 \u2983f g : \u03b1 \u2192\u209b \u03b3\u2984, Disjoint (support \u2191f) (support \u2191g) \u2192 P f \u2192 P g \u2192 P (f + g)\nx : \u03b3\ns : Finset \u03b3\nhxs : \u00acx \u2208 s\nih : \u2200 (f : \u03b1 \u2192\u209b \u03b3), range \u2191f \\ {0} = \u2191s \u2192 P f\nf : \u03b1 \u2192\u209b \u03b3\nh : range \u2191f \\ {0} = \u2191(insert x s)\nmx : MeasurableSet (\u2191f \u207b\u00b9' {x})\ng : \u03b1 \u2192\u209b \u03b3 := piecewise (\u2191f \u207b\u00b9' {x}) mx 0 f\n\u22a2 range \u2191g \\ {0} = \u2191s"}, {"tactic": "simp only [SimpleFunc.coe_piecewise, range_piecewise]", "annotated_tactic": ["simp only [<a>SimpleFunc.coe_piecewise</a>, <a>range_piecewise</a>]", [{"full_name": "MeasureTheory.SimpleFunc.coe_piecewise", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [230, 9], "def_end_pos": [230, 22]}, {"full_name": "Set.range_piecewise", "def_path": "Mathlib/Data/Set/Function.lean", "def_pos": [1537, 9], "def_end_pos": [1537, 24]}]], "state_before": "case h\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3\u271d : Type u_3\n\u03b4 : Type u_4\n\u03b1 : Type u_5\n\u03b3 : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : AddMonoid \u03b3\nP : (\u03b1 \u2192\u209b \u03b3) \u2192 Prop\nh_ind : \u2200 (c : \u03b3) {s : Set \u03b1} (hs : MeasurableSet s), P (piecewise s hs (const \u03b1 c) (const \u03b1 0))\nh_add : \u2200 \u2983f g : \u03b1 \u2192\u209b \u03b3\u2984, Disjoint (support \u2191f) (support \u2191g) \u2192 P f \u2192 P g \u2192 P (f + g)\nx : \u03b3\ns : Finset \u03b3\nhxs : \u00acx \u2208 s\nih : \u2200 (f : \u03b1 \u2192\u209b \u03b3), range \u2191f \\ {0} = \u2191s \u2192 P f\nf : \u03b1 \u2192\u209b \u03b3\nh : range \u2191f \\ {0} = \u2191(insert x s)\nmx : MeasurableSet (\u2191f \u207b\u00b9' {x})\ng : \u03b1 \u2192\u209b \u03b3 := piecewise (\u2191f \u207b\u00b9' {x}) mx 0 f\n\u22a2 range \u2191g \\ {0} = \u2191s", "state_after": "case h\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3\u271d : Type u_3\n\u03b4 : Type u_4\n\u03b1 : Type u_5\n\u03b3 : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : AddMonoid \u03b3\nP : (\u03b1 \u2192\u209b \u03b3) \u2192 Prop\nh_ind : \u2200 (c : \u03b3) {s : Set \u03b1} (hs : MeasurableSet s), P (piecewise s hs (const \u03b1 c) (const \u03b1 0))\nh_add : \u2200 \u2983f g : \u03b1 \u2192\u209b \u03b3\u2984, Disjoint (support \u2191f) (support \u2191g) \u2192 P f \u2192 P g \u2192 P (f + g)\nx : \u03b3\ns : Finset \u03b3\nhxs : \u00acx \u2208 s\nih : \u2200 (f : \u03b1 \u2192\u209b \u03b3), range \u2191f \\ {0} = \u2191s \u2192 P f\nf : \u03b1 \u2192\u209b \u03b3\nh : range \u2191f \\ {0} = \u2191(insert x s)\nmx : MeasurableSet (\u2191f \u207b\u00b9' {x})\ng : \u03b1 \u2192\u209b \u03b3 := piecewise (\u2191f \u207b\u00b9' {x}) mx 0 f\n\u22a2 (\u21910 '' (\u2191f \u207b\u00b9' {x}) \u222a \u2191f '' (\u2191f \u207b\u00b9' {x})\u1d9c) \\ {0} = \u2191s"}, {"tactic": "rw [image_compl_preimage, union_diff_distrib, diff_diff_comm, h, Finset.coe_insert,\n  insert_diff_self_of_not_mem, diff_eq_empty.mpr, Set.empty_union]", "annotated_tactic": ["rw [<a>image_compl_preimage</a>, <a>union_diff_distrib</a>, <a>diff_diff_comm</a>, h, <a>Finset.coe_insert</a>,\n        <a>insert_diff_self_of_not_mem</a>, diff_eq_empty.mpr, <a>Set.empty_union</a>]", [{"full_name": "Set.image_compl_preimage", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [1056, 9], "def_end_pos": [1056, 29]}, {"full_name": "Set.union_diff_distrib", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1875, 9], "def_end_pos": [1875, 27]}, {"full_name": "Set.diff_diff_comm", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1944, 9], "def_end_pos": [1944, 23]}, {"full_name": "Finset.coe_insert", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1113, 9], "def_end_pos": [1113, 19]}, {"full_name": "Set.insert_diff_self_of_not_mem", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [2012, 9], "def_end_pos": [2012, 36]}, {"full_name": "Set.empty_union", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [782, 9], "def_end_pos": [782, 20]}]], "state_before": "case h\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3\u271d : Type u_3\n\u03b4 : Type u_4\n\u03b1 : Type u_5\n\u03b3 : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : AddMonoid \u03b3\nP : (\u03b1 \u2192\u209b \u03b3) \u2192 Prop\nh_ind : \u2200 (c : \u03b3) {s : Set \u03b1} (hs : MeasurableSet s), P (piecewise s hs (const \u03b1 c) (const \u03b1 0))\nh_add : \u2200 \u2983f g : \u03b1 \u2192\u209b \u03b3\u2984, Disjoint (support \u2191f) (support \u2191g) \u2192 P f \u2192 P g \u2192 P (f + g)\nx : \u03b3\ns : Finset \u03b3\nhxs : \u00acx \u2208 s\nih : \u2200 (f : \u03b1 \u2192\u209b \u03b3), range \u2191f \\ {0} = \u2191s \u2192 P f\nf : \u03b1 \u2192\u209b \u03b3\nh : range \u2191f \\ {0} = \u2191(insert x s)\nmx : MeasurableSet (\u2191f \u207b\u00b9' {x})\ng : \u03b1 \u2192\u209b \u03b3 := piecewise (\u2191f \u207b\u00b9' {x}) mx 0 f\n\u22a2 (\u21910 '' (\u2191f \u207b\u00b9' {x}) \u222a \u2191f '' (\u2191f \u207b\u00b9' {x})\u1d9c) \\ {0} = \u2191s", "state_after": "case h\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3\u271d : Type u_3\n\u03b4 : Type u_4\n\u03b1 : Type u_5\n\u03b3 : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : AddMonoid \u03b3\nP : (\u03b1 \u2192\u209b \u03b3) \u2192 Prop\nh_ind : \u2200 (c : \u03b3) {s : Set \u03b1} (hs : MeasurableSet s), P (piecewise s hs (const \u03b1 c) (const \u03b1 0))\nh_add : \u2200 \u2983f g : \u03b1 \u2192\u209b \u03b3\u2984, Disjoint (support \u2191f) (support \u2191g) \u2192 P f \u2192 P g \u2192 P (f + g)\nx : \u03b3\ns : Finset \u03b3\nhxs : \u00acx \u2208 s\nih : \u2200 (f : \u03b1 \u2192\u209b \u03b3), range \u2191f \\ {0} = \u2191s \u2192 P f\nf : \u03b1 \u2192\u209b \u03b3\nh : range \u2191f \\ {0} = \u2191(insert x s)\nmx : MeasurableSet (\u2191f \u207b\u00b9' {x})\ng : \u03b1 \u2192\u209b \u03b3 := piecewise (\u2191f \u207b\u00b9' {x}) mx 0 f\n\u22a2 \u21910 '' (\u2191f \u207b\u00b9' {x}) \u2286 {0}\n\ncase h\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3\u271d : Type u_3\n\u03b4 : Type u_4\n\u03b1 : Type u_5\n\u03b3 : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : AddMonoid \u03b3\nP : (\u03b1 \u2192\u209b \u03b3) \u2192 Prop\nh_ind : \u2200 (c : \u03b3) {s : Set \u03b1} (hs : MeasurableSet s), P (piecewise s hs (const \u03b1 c) (const \u03b1 0))\nh_add : \u2200 \u2983f g : \u03b1 \u2192\u209b \u03b3\u2984, Disjoint (support \u2191f) (support \u2191g) \u2192 P f \u2192 P g \u2192 P (f + g)\nx : \u03b3\ns : Finset \u03b3\nhxs : \u00acx \u2208 s\nih : \u2200 (f : \u03b1 \u2192\u209b \u03b3), range \u2191f \\ {0} = \u2191s \u2192 P f\nf : \u03b1 \u2192\u209b \u03b3\nh : range \u2191f \\ {0} = \u2191(insert x s)\nmx : MeasurableSet (\u2191f \u207b\u00b9' {x})\ng : \u03b1 \u2192\u209b \u03b3 := piecewise (\u2191f \u207b\u00b9' {x}) mx 0 f\n\u22a2 \u00acx \u2208 \u2191s"}, {"tactic": "rw [Set.image_subset_iff]", "annotated_tactic": ["rw [<a>Set.image_subset_iff</a>]", [{"full_name": "Set.image_subset_iff", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [497, 9], "def_end_pos": [497, 25]}]], "state_before": "case h\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3\u271d : Type u_3\n\u03b4 : Type u_4\n\u03b1 : Type u_5\n\u03b3 : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : AddMonoid \u03b3\nP : (\u03b1 \u2192\u209b \u03b3) \u2192 Prop\nh_ind : \u2200 (c : \u03b3) {s : Set \u03b1} (hs : MeasurableSet s), P (piecewise s hs (const \u03b1 c) (const \u03b1 0))\nh_add : \u2200 \u2983f g : \u03b1 \u2192\u209b \u03b3\u2984, Disjoint (support \u2191f) (support \u2191g) \u2192 P f \u2192 P g \u2192 P (f + g)\nx : \u03b3\ns : Finset \u03b3\nhxs : \u00acx \u2208 s\nih : \u2200 (f : \u03b1 \u2192\u209b \u03b3), range \u2191f \\ {0} = \u2191s \u2192 P f\nf : \u03b1 \u2192\u209b \u03b3\nh : range \u2191f \\ {0} = \u2191(insert x s)\nmx : MeasurableSet (\u2191f \u207b\u00b9' {x})\ng : \u03b1 \u2192\u209b \u03b3 := piecewise (\u2191f \u207b\u00b9' {x}) mx 0 f\n\u22a2 \u21910 '' (\u2191f \u207b\u00b9' {x}) \u2286 {0}", "state_after": "case h\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3\u271d : Type u_3\n\u03b4 : Type u_4\n\u03b1 : Type u_5\n\u03b3 : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : AddMonoid \u03b3\nP : (\u03b1 \u2192\u209b \u03b3) \u2192 Prop\nh_ind : \u2200 (c : \u03b3) {s : Set \u03b1} (hs : MeasurableSet s), P (piecewise s hs (const \u03b1 c) (const \u03b1 0))\nh_add : \u2200 \u2983f g : \u03b1 \u2192\u209b \u03b3\u2984, Disjoint (support \u2191f) (support \u2191g) \u2192 P f \u2192 P g \u2192 P (f + g)\nx : \u03b3\ns : Finset \u03b3\nhxs : \u00acx \u2208 s\nih : \u2200 (f : \u03b1 \u2192\u209b \u03b3), range \u2191f \\ {0} = \u2191s \u2192 P f\nf : \u03b1 \u2192\u209b \u03b3\nh : range \u2191f \\ {0} = \u2191(insert x s)\nmx : MeasurableSet (\u2191f \u207b\u00b9' {x})\ng : \u03b1 \u2192\u209b \u03b3 := piecewise (\u2191f \u207b\u00b9' {x}) mx 0 f\n\u22a2 \u2191f \u207b\u00b9' {x} \u2286 \u21910 \u207b\u00b9' {0}"}, {"tactic": "convert Set.subset_univ _", "annotated_tactic": ["convert <a>Set.subset_univ</a> _", [{"full_name": "Set.subset_univ", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [691, 9], "def_end_pos": [691, 20]}]], "state_before": "case h\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3\u271d : Type u_3\n\u03b4 : Type u_4\n\u03b1 : Type u_5\n\u03b3 : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : AddMonoid \u03b3\nP : (\u03b1 \u2192\u209b \u03b3) \u2192 Prop\nh_ind : \u2200 (c : \u03b3) {s : Set \u03b1} (hs : MeasurableSet s), P (piecewise s hs (const \u03b1 c) (const \u03b1 0))\nh_add : \u2200 \u2983f g : \u03b1 \u2192\u209b \u03b3\u2984, Disjoint (support \u2191f) (support \u2191g) \u2192 P f \u2192 P g \u2192 P (f + g)\nx : \u03b3\ns : Finset \u03b3\nhxs : \u00acx \u2208 s\nih : \u2200 (f : \u03b1 \u2192\u209b \u03b3), range \u2191f \\ {0} = \u2191s \u2192 P f\nf : \u03b1 \u2192\u209b \u03b3\nh : range \u2191f \\ {0} = \u2191(insert x s)\nmx : MeasurableSet (\u2191f \u207b\u00b9' {x})\ng : \u03b1 \u2192\u209b \u03b3 := piecewise (\u2191f \u207b\u00b9' {x}) mx 0 f\n\u22a2 \u2191f \u207b\u00b9' {x} \u2286 \u21910 \u207b\u00b9' {0}", "state_after": "case h.e'_4\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3\u271d : Type u_3\n\u03b4 : Type u_4\n\u03b1 : Type u_5\n\u03b3 : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : AddMonoid \u03b3\nP : (\u03b1 \u2192\u209b \u03b3) \u2192 Prop\nh_ind : \u2200 (c : \u03b3) {s : Set \u03b1} (hs : MeasurableSet s), P (piecewise s hs (const \u03b1 c) (const \u03b1 0))\nh_add : \u2200 \u2983f g : \u03b1 \u2192\u209b \u03b3\u2984, Disjoint (support \u2191f) (support \u2191g) \u2192 P f \u2192 P g \u2192 P (f + g)\nx : \u03b3\ns : Finset \u03b3\nhxs : \u00acx \u2208 s\nih : \u2200 (f : \u03b1 \u2192\u209b \u03b3), range \u2191f \\ {0} = \u2191s \u2192 P f\nf : \u03b1 \u2192\u209b \u03b3\nh : range \u2191f \\ {0} = \u2191(insert x s)\nmx : MeasurableSet (\u2191f \u207b\u00b9' {x})\ng : \u03b1 \u2192\u209b \u03b3 := piecewise (\u2191f \u207b\u00b9' {x}) mx 0 f\n\u22a2 \u21910 \u207b\u00b9' {0} = univ"}, {"tactic": "exact preimage_const_of_mem (mem_singleton _)", "annotated_tactic": ["exact <a>preimage_const_of_mem</a> (<a>mem_singleton</a> _)", [{"full_name": "Set.preimage_const_of_mem", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [138, 9], "def_end_pos": [138, 30]}, {"full_name": "Set.mem_singleton", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1289, 9], "def_end_pos": [1289, 22]}]], "state_before": "case h.e'_4\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3\u271d : Type u_3\n\u03b4 : Type u_4\n\u03b1 : Type u_5\n\u03b3 : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : AddMonoid \u03b3\nP : (\u03b1 \u2192\u209b \u03b3) \u2192 Prop\nh_ind : \u2200 (c : \u03b3) {s : Set \u03b1} (hs : MeasurableSet s), P (piecewise s hs (const \u03b1 c) (const \u03b1 0))\nh_add : \u2200 \u2983f g : \u03b1 \u2192\u209b \u03b3\u2984, Disjoint (support \u2191f) (support \u2191g) \u2192 P f \u2192 P g \u2192 P (f + g)\nx : \u03b3\ns : Finset \u03b3\nhxs : \u00acx \u2208 s\nih : \u2200 (f : \u03b1 \u2192\u209b \u03b3), range \u2191f \\ {0} = \u2191s \u2192 P f\nf : \u03b1 \u2192\u209b \u03b3\nh : range \u2191f \\ {0} = \u2191(insert x s)\nmx : MeasurableSet (\u2191f \u207b\u00b9' {x})\ng : \u03b1 \u2192\u209b \u03b3 := piecewise (\u2191f \u207b\u00b9' {x}) mx 0 f\n\u22a2 \u21910 \u207b\u00b9' {0} = univ", "state_after": "no goals"}, {"tactic": "rwa [Finset.mem_coe]", "annotated_tactic": ["rwa [<a>Finset.mem_coe</a>]", [{"full_name": "Finset.mem_coe", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [208, 9], "def_end_pos": [208, 16]}]], "state_before": "case h\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3\u271d : Type u_3\n\u03b4 : Type u_4\n\u03b1 : Type u_5\n\u03b3 : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : AddMonoid \u03b3\nP : (\u03b1 \u2192\u209b \u03b3) \u2192 Prop\nh_ind : \u2200 (c : \u03b3) {s : Set \u03b1} (hs : MeasurableSet s), P (piecewise s hs (const \u03b1 c) (const \u03b1 0))\nh_add : \u2200 \u2983f g : \u03b1 \u2192\u209b \u03b3\u2984, Disjoint (support \u2191f) (support \u2191g) \u2192 P f \u2192 P g \u2192 P (f + g)\nx : \u03b3\ns : Finset \u03b3\nhxs : \u00acx \u2208 s\nih : \u2200 (f : \u03b1 \u2192\u209b \u03b3), range \u2191f \\ {0} = \u2191s \u2192 P f\nf : \u03b1 \u2192\u209b \u03b3\nh : range \u2191f \\ {0} = \u2191(insert x s)\nmx : MeasurableSet (\u2191f \u207b\u00b9' {x})\ng : \u03b1 \u2192\u209b \u03b3 := piecewise (\u2191f \u207b\u00b9' {x}) mx 0 f\n\u22a2 \u00acx \u2208 \u2191s", "state_after": "no goals"}, {"tactic": "ext1 y", "annotated_tactic": ["ext1 y", []], "state_before": "case h.e'_1\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3\u271d : Type u_3\n\u03b4 : Type u_4\n\u03b1 : Type u_5\n\u03b3 : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : AddMonoid \u03b3\nP : (\u03b1 \u2192\u209b \u03b3) \u2192 Prop\nh_ind : \u2200 (c : \u03b3) {s : Set \u03b1} (hs : MeasurableSet s), P (piecewise s hs (const \u03b1 c) (const \u03b1 0))\nh_add : \u2200 \u2983f g : \u03b1 \u2192\u209b \u03b3\u2984, Disjoint (support \u2191f) (support \u2191g) \u2192 P f \u2192 P g \u2192 P (f + g)\nx : \u03b3\ns : Finset \u03b3\nhxs : \u00acx \u2208 s\nih : \u2200 (f : \u03b1 \u2192\u209b \u03b3), range \u2191f \\ {0} = \u2191s \u2192 P f\nf : \u03b1 \u2192\u209b \u03b3\nh : range \u2191f \\ {0} = \u2191(insert x s)\nmx : MeasurableSet (\u2191f \u207b\u00b9' {x})\ng : \u03b1 \u2192\u209b \u03b3 := piecewise (\u2191f \u207b\u00b9' {x}) mx 0 f\nPg : P g\n\u22a2 f = g + piecewise (\u2191f \u207b\u00b9' {x}) mx (const \u03b1 x) (const \u03b1 0)", "state_after": "case h.e'_1.H\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3\u271d : Type u_3\n\u03b4 : Type u_4\n\u03b1 : Type u_5\n\u03b3 : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : AddMonoid \u03b3\nP : (\u03b1 \u2192\u209b \u03b3) \u2192 Prop\nh_ind : \u2200 (c : \u03b3) {s : Set \u03b1} (hs : MeasurableSet s), P (piecewise s hs (const \u03b1 c) (const \u03b1 0))\nh_add : \u2200 \u2983f g : \u03b1 \u2192\u209b \u03b3\u2984, Disjoint (support \u2191f) (support \u2191g) \u2192 P f \u2192 P g \u2192 P (f + g)\nx : \u03b3\ns : Finset \u03b3\nhxs : \u00acx \u2208 s\nih : \u2200 (f : \u03b1 \u2192\u209b \u03b3), range \u2191f \\ {0} = \u2191s \u2192 P f\nf : \u03b1 \u2192\u209b \u03b3\nh : range \u2191f \\ {0} = \u2191(insert x s)\nmx : MeasurableSet (\u2191f \u207b\u00b9' {x})\ng : \u03b1 \u2192\u209b \u03b3 := piecewise (\u2191f \u207b\u00b9' {x}) mx 0 f\nPg : P g\ny : \u03b1\n\u22a2 \u2191f y = \u2191(g + piecewise (\u2191f \u207b\u00b9' {x}) mx (const \u03b1 x) (const \u03b1 0)) y"}, {"tactic": "by_cases hy : y \u2208 f \u207b\u00b9' {x}", "annotated_tactic": ["by_cases hy : y \u2208 f \u207b\u00b9' {x}", []], "state_before": "case h.e'_1.H\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3\u271d : Type u_3\n\u03b4 : Type u_4\n\u03b1 : Type u_5\n\u03b3 : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : AddMonoid \u03b3\nP : (\u03b1 \u2192\u209b \u03b3) \u2192 Prop\nh_ind : \u2200 (c : \u03b3) {s : Set \u03b1} (hs : MeasurableSet s), P (piecewise s hs (const \u03b1 c) (const \u03b1 0))\nh_add : \u2200 \u2983f g : \u03b1 \u2192\u209b \u03b3\u2984, Disjoint (support \u2191f) (support \u2191g) \u2192 P f \u2192 P g \u2192 P (f + g)\nx : \u03b3\ns : Finset \u03b3\nhxs : \u00acx \u2208 s\nih : \u2200 (f : \u03b1 \u2192\u209b \u03b3), range \u2191f \\ {0} = \u2191s \u2192 P f\nf : \u03b1 \u2192\u209b \u03b3\nh : range \u2191f \\ {0} = \u2191(insert x s)\nmx : MeasurableSet (\u2191f \u207b\u00b9' {x})\ng : \u03b1 \u2192\u209b \u03b3 := piecewise (\u2191f \u207b\u00b9' {x}) mx 0 f\nPg : P g\ny : \u03b1\n\u22a2 \u2191f y = \u2191(g + piecewise (\u2191f \u207b\u00b9' {x}) mx (const \u03b1 x) (const \u03b1 0)) y", "state_after": "case pos\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3\u271d : Type u_3\n\u03b4 : Type u_4\n\u03b1 : Type u_5\n\u03b3 : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : AddMonoid \u03b3\nP : (\u03b1 \u2192\u209b \u03b3) \u2192 Prop\nh_ind : \u2200 (c : \u03b3) {s : Set \u03b1} (hs : MeasurableSet s), P (piecewise s hs (const \u03b1 c) (const \u03b1 0))\nh_add : \u2200 \u2983f g : \u03b1 \u2192\u209b \u03b3\u2984, Disjoint (support \u2191f) (support \u2191g) \u2192 P f \u2192 P g \u2192 P (f + g)\nx : \u03b3\ns : Finset \u03b3\nhxs : \u00acx \u2208 s\nih : \u2200 (f : \u03b1 \u2192\u209b \u03b3), range \u2191f \\ {0} = \u2191s \u2192 P f\nf : \u03b1 \u2192\u209b \u03b3\nh : range \u2191f \\ {0} = \u2191(insert x s)\nmx : MeasurableSet (\u2191f \u207b\u00b9' {x})\ng : \u03b1 \u2192\u209b \u03b3 := piecewise (\u2191f \u207b\u00b9' {x}) mx 0 f\nPg : P g\ny : \u03b1\nhy : y \u2208 \u2191f \u207b\u00b9' {x}\n\u22a2 \u2191f y = \u2191(g + piecewise (\u2191f \u207b\u00b9' {x}) mx (const \u03b1 x) (const \u03b1 0)) y\n\ncase neg\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3\u271d : Type u_3\n\u03b4 : Type u_4\n\u03b1 : Type u_5\n\u03b3 : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : AddMonoid \u03b3\nP : (\u03b1 \u2192\u209b \u03b3) \u2192 Prop\nh_ind : \u2200 (c : \u03b3) {s : Set \u03b1} (hs : MeasurableSet s), P (piecewise s hs (const \u03b1 c) (const \u03b1 0))\nh_add : \u2200 \u2983f g : \u03b1 \u2192\u209b \u03b3\u2984, Disjoint (support \u2191f) (support \u2191g) \u2192 P f \u2192 P g \u2192 P (f + g)\nx : \u03b3\ns : Finset \u03b3\nhxs : \u00acx \u2208 s\nih : \u2200 (f : \u03b1 \u2192\u209b \u03b3), range \u2191f \\ {0} = \u2191s \u2192 P f\nf : \u03b1 \u2192\u209b \u03b3\nh : range \u2191f \\ {0} = \u2191(insert x s)\nmx : MeasurableSet (\u2191f \u207b\u00b9' {x})\ng : \u03b1 \u2192\u209b \u03b3 := piecewise (\u2191f \u207b\u00b9' {x}) mx 0 f\nPg : P g\ny : \u03b1\nhy : \u00acy \u2208 \u2191f \u207b\u00b9' {x}\n\u22a2 \u2191f y = \u2191(g + piecewise (\u2191f \u207b\u00b9' {x}) mx (const \u03b1 x) (const \u03b1 0)) y"}, {"tactic": "simpa [piecewise_eq_of_mem _ _ _ hy, -piecewise_eq_indicator]", "annotated_tactic": ["simpa [<a>piecewise_eq_of_mem</a> _ _ _ hy, -<a>piecewise_eq_indicator</a>]", [{"full_name": "Set.piecewise_eq_of_mem", "def_path": "Mathlib/Data/Set/Function.lean", "def_pos": [1420, 9], "def_end_pos": [1420, 28]}, {"full_name": "Set.piecewise_eq_indicator", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [52, 3], "def_end_pos": [52, 14]}]], "state_before": "case pos\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3\u271d : Type u_3\n\u03b4 : Type u_4\n\u03b1 : Type u_5\n\u03b3 : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : AddMonoid \u03b3\nP : (\u03b1 \u2192\u209b \u03b3) \u2192 Prop\nh_ind : \u2200 (c : \u03b3) {s : Set \u03b1} (hs : MeasurableSet s), P (piecewise s hs (const \u03b1 c) (const \u03b1 0))\nh_add : \u2200 \u2983f g : \u03b1 \u2192\u209b \u03b3\u2984, Disjoint (support \u2191f) (support \u2191g) \u2192 P f \u2192 P g \u2192 P (f + g)\nx : \u03b3\ns : Finset \u03b3\nhxs : \u00acx \u2208 s\nih : \u2200 (f : \u03b1 \u2192\u209b \u03b3), range \u2191f \\ {0} = \u2191s \u2192 P f\nf : \u03b1 \u2192\u209b \u03b3\nh : range \u2191f \\ {0} = \u2191(insert x s)\nmx : MeasurableSet (\u2191f \u207b\u00b9' {x})\ng : \u03b1 \u2192\u209b \u03b3 := piecewise (\u2191f \u207b\u00b9' {x}) mx 0 f\nPg : P g\ny : \u03b1\nhy : y \u2208 \u2191f \u207b\u00b9' {x}\n\u22a2 \u2191f y = \u2191(g + piecewise (\u2191f \u207b\u00b9' {x}) mx (const \u03b1 x) (const \u03b1 0)) y", "state_after": "no goals"}, {"tactic": "simp [piecewise_eq_of_not_mem _ _ _ hy, -piecewise_eq_indicator]", "annotated_tactic": ["simp [<a>piecewise_eq_of_not_mem</a> _ _ _ hy, -<a>piecewise_eq_indicator</a>]", [{"full_name": "Set.piecewise_eq_of_not_mem", "def_path": "Mathlib/Data/Set/Function.lean", "def_pos": [1425, 9], "def_end_pos": [1425, 32]}, {"full_name": "Set.piecewise_eq_indicator", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [52, 3], "def_end_pos": [52, 14]}]], "state_before": "case neg\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3\u271d : Type u_3\n\u03b4 : Type u_4\n\u03b1 : Type u_5\n\u03b3 : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : AddMonoid \u03b3\nP : (\u03b1 \u2192\u209b \u03b3) \u2192 Prop\nh_ind : \u2200 (c : \u03b3) {s : Set \u03b1} (hs : MeasurableSet s), P (piecewise s hs (const \u03b1 c) (const \u03b1 0))\nh_add : \u2200 \u2983f g : \u03b1 \u2192\u209b \u03b3\u2984, Disjoint (support \u2191f) (support \u2191g) \u2192 P f \u2192 P g \u2192 P (f + g)\nx : \u03b3\ns : Finset \u03b3\nhxs : \u00acx \u2208 s\nih : \u2200 (f : \u03b1 \u2192\u209b \u03b3), range \u2191f \\ {0} = \u2191s \u2192 P f\nf : \u03b1 \u2192\u209b \u03b3\nh : range \u2191f \\ {0} = \u2191(insert x s)\nmx : MeasurableSet (\u2191f \u207b\u00b9' {x})\ng : \u03b1 \u2192\u209b \u03b3 := piecewise (\u2191f \u207b\u00b9' {x}) mx 0 f\nPg : P g\ny : \u03b1\nhy : \u00acy \u2208 \u2191f \u207b\u00b9' {x}\n\u22a2 \u2191f y = \u2191(g + piecewise (\u2191f \u207b\u00b9' {x}) mx (const \u03b1 x) (const \u03b1 0)) y", "state_after": "no goals"}, {"tactic": "simp [piecewise_eq_of_mem _ _ _ hy, -piecewise_eq_indicator]", "annotated_tactic": ["simp [<a>piecewise_eq_of_mem</a> _ _ _ hy, -<a>piecewise_eq_indicator</a>]", [{"full_name": "Set.piecewise_eq_of_mem", "def_path": "Mathlib/Data/Set/Function.lean", "def_pos": [1420, 9], "def_end_pos": [1420, 28]}, {"full_name": "Set.piecewise_eq_indicator", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [52, 3], "def_end_pos": [52, 14]}]], "state_before": "case pos\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3\u271d : Type u_3\n\u03b4 : Type u_4\n\u03b1 : Type u_5\n\u03b3 : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : AddMonoid \u03b3\nP : (\u03b1 \u2192\u209b \u03b3) \u2192 Prop\nh_ind : \u2200 (c : \u03b3) {s : Set \u03b1} (hs : MeasurableSet s), P (piecewise s hs (const \u03b1 c) (const \u03b1 0))\nh_add : \u2200 \u2983f g : \u03b1 \u2192\u209b \u03b3\u2984, Disjoint (support \u2191f) (support \u2191g) \u2192 P f \u2192 P g \u2192 P (f + g)\nx : \u03b3\ns : Finset \u03b3\nhxs : \u00acx \u2208 s\nih : \u2200 (f : \u03b1 \u2192\u209b \u03b3), range \u2191f \\ {0} = \u2191s \u2192 P f\nf : \u03b1 \u2192\u209b \u03b3\nh : range \u2191f \\ {0} = \u2191(insert x s)\nmx : MeasurableSet (\u2191f \u207b\u00b9' {x})\ng : \u03b1 \u2192\u209b \u03b3 := piecewise (\u2191f \u207b\u00b9' {x}) mx 0 f\nPg : P g\ny : \u03b1\nhy : y \u2208 \u2191f \u207b\u00b9' {x}\n\u22a2 y \u2208 support \u2191g \u2293 support \u2191(piecewise (\u2191f \u207b\u00b9' {x}) mx (const \u03b1 x) (const \u03b1 0)) \u2192 y \u2208 \u22a5", "state_after": "no goals"}, {"tactic": "simp [piecewise_eq_of_not_mem _ _ _ hy, -piecewise_eq_indicator]", "annotated_tactic": ["simp [<a>piecewise_eq_of_not_mem</a> _ _ _ hy, -<a>piecewise_eq_indicator</a>]", [{"full_name": "Set.piecewise_eq_of_not_mem", "def_path": "Mathlib/Data/Set/Function.lean", "def_pos": [1425, 9], "def_end_pos": [1425, 32]}, {"full_name": "Set.piecewise_eq_indicator", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [52, 3], "def_end_pos": [52, 14]}]], "state_before": "case neg\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3\u271d : Type u_3\n\u03b4 : Type u_4\n\u03b1 : Type u_5\n\u03b3 : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : AddMonoid \u03b3\nP : (\u03b1 \u2192\u209b \u03b3) \u2192 Prop\nh_ind : \u2200 (c : \u03b3) {s : Set \u03b1} (hs : MeasurableSet s), P (piecewise s hs (const \u03b1 c) (const \u03b1 0))\nh_add : \u2200 \u2983f g : \u03b1 \u2192\u209b \u03b3\u2984, Disjoint (support \u2191f) (support \u2191g) \u2192 P f \u2192 P g \u2192 P (f + g)\nx : \u03b3\ns : Finset \u03b3\nhxs : \u00acx \u2208 s\nih : \u2200 (f : \u03b1 \u2192\u209b \u03b3), range \u2191f \\ {0} = \u2191s \u2192 P f\nf : \u03b1 \u2192\u209b \u03b3\nh : range \u2191f \\ {0} = \u2191(insert x s)\nmx : MeasurableSet (\u2191f \u207b\u00b9' {x})\ng : \u03b1 \u2192\u209b \u03b3 := piecewise (\u2191f \u207b\u00b9' {x}) mx 0 f\nPg : P g\ny : \u03b1\nhy : \u00acy \u2208 \u2191f \u207b\u00b9' {x}\n\u22a2 y \u2208 support \u2191g \u2293 support \u2191(piecewise (\u2191f \u207b\u00b9' {x}) mx (const \u03b1 x) (const \u03b1 0)) \u2192 y \u2208 \u22a5", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "full_name": "MeasurableSet.inter", "start": [198, 11], "end": [201, 40], "traced_tactics": [{"tactic": "rw [inter_eq_compl_compl_union_compl]", "annotated_tactic": ["rw [<a>inter_eq_compl_compl_union_compl</a>]", [{"full_name": "Set.inter_eq_compl_compl_union_compl", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1734, 9], "def_end_pos": [1734, 41]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u_6\ns t u : Set \u03b1\nm : MeasurableSpace \u03b1\ns\u2081 s\u2082 : Set \u03b1\nh\u2081 : MeasurableSet s\u2081\nh\u2082 : MeasurableSet s\u2082\n\u22a2 MeasurableSet (s\u2081 \u2229 s\u2082)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u_6\ns t u : Set \u03b1\nm : MeasurableSpace \u03b1\ns\u2081 s\u2082 : Set \u03b1\nh\u2081 : MeasurableSet s\u2081\nh\u2082 : MeasurableSet s\u2082\n\u22a2 MeasurableSet (s\u2081\u1d9c \u222a s\u2082\u1d9c)\u1d9c"}, {"tactic": "exact (h\u2081.compl.union h\u2082.compl).compl", "annotated_tactic": ["exact (h\u2081.compl.union h\u2082.compl).<a>compl</a>", [{"full_name": "MeasurableSet.compl", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [87, 19], "def_end_pos": [87, 38]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u_6\ns t u : Set \u03b1\nm : MeasurableSpace \u03b1\ns\u2081 s\u2082 : Set \u03b1\nh\u2081 : MeasurableSet s\u2081\nh\u2082 : MeasurableSet s\u2082\n\u22a2 MeasurableSet (s\u2081\u1d9c \u222a s\u2082\u1d9c)\u1d9c", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "full_name": "MeasureTheory.dist_indicatorConstLp_eq_norm", "start": [797, 1], "end": [800, 90], "traced_tactics": [{"tactic": "rw [Lp.dist_edist, edist_indicatorConstLp_eq_nnnorm, ENNReal.coe_toReal, Lp.coe_nnnorm]", "annotated_tactic": ["rw [<a>Lp.dist_edist</a>, <a>edist_indicatorConstLp_eq_nnnorm</a>, <a>ENNReal.coe_toReal</a>, <a>Lp.coe_nnnorm</a>]", [{"full_name": "MeasureTheory.Lp.dist_edist", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [304, 19], "def_end_pos": [304, 29]}, {"full_name": "MeasureTheory.edist_indicatorConstLp_eq_nnnorm", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [791, 9], "def_end_pos": [791, 41]}, {"full_name": "ENNReal.coe_toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [236, 17], "def_end_pos": [236, 27]}, {"full_name": "MeasureTheory.Lp.coe_nnnorm", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [268, 19], "def_end_pos": [268, 29]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nc : E\nt : Set \u03b1\nht : MeasurableSet t\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\n\u22a2 dist (indicatorConstLp p hs h\u03bcs c) (indicatorConstLp p ht h\u03bct c) =\n    \u2016indicatorConstLp p (_ : MeasurableSet (s \u2206 t)) (_ : \u2191\u2191\u03bc (s \u2206 t) \u2260 \u22a4) c\u2016", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Group/Measure.lean", "full_name": "MeasureTheory.Measure.measure_inv", "start": [454, 1], "end": [455, 32], "traced_tactics": [{"tactic": "rw [\u2190 inv_apply, inv_eq_self]", "annotated_tactic": ["rw [\u2190 <a>inv_apply</a>, <a>inv_eq_self</a>]", [{"full_name": "MeasureTheory.Measure.inv_apply", "def_path": "Mathlib/MeasureTheory/Group/Measure.lean", "def_pos": [442, 9], "def_end_pos": [442, 18]}, {"full_name": "MeasureTheory.Measure.inv_eq_self", "def_path": "Mathlib/MeasureTheory/Group/Measure.lean", "def_pos": [416, 9], "def_end_pos": [416, 20]}]], "state_before": "\ud835\udd5c : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : MeasurableSpace H\ninst\u271d\u00b2 : InvolutiveInv G\ninst\u271d\u00b9 : MeasurableInv G\n\u03bc : Measure G\ninst\u271d : IsInvInvariant \u03bc\nA : Set G\n\u22a2 \u2191\u2191\u03bc A\u207b\u00b9 = \u2191\u2191\u03bc A", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/Average.lean", "full_name": "MeasureTheory.setAverage_const", "start": [393, 1], "end": [395, 70], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Variance.lean", "full_name": "MeasureTheory.Mem\u2112p.variance_eq", "start": [131, 1], "end": [143, 48], "traced_tactics": [{"tactic": "rw [variance, evariance_eq_lintegral_ofReal, \u2190 ofReal_integral_eq_lintegral_ofReal,\n  ENNReal.toReal_ofReal]", "annotated_tactic": ["rw [<a>variance</a>, <a>evariance_eq_lintegral_ofReal</a>, \u2190 <a>ofReal_integral_eq_lintegral_ofReal</a>,\n    <a>ENNReal.toReal_ofReal</a>]", [{"full_name": "ProbabilityTheory.variance", "def_path": "Mathlib/Probability/Variance.lean", "def_pos": [61, 5], "def_end_pos": [61, 13]}, {"full_name": "ProbabilityTheory.evariance_eq_lintegral_ofReal", "def_path": "Mathlib/Probability/Variance.lean", "def_pos": [108, 9], "def_end_pos": [108, 38]}, {"full_name": "MeasureTheory.ofReal_integral_eq_lintegral_ofReal", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1219, 9], "def_end_pos": [1219, 44]}, {"full_name": "ENNReal.toReal_ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [191, 9], "def_end_pos": [191, 22]}]], "state_before": "\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d : IsFiniteMeasure \u03bc\nhX : Mem\u2112p X 2\n\u22a2 variance X \u03bc = \u222b (x : \u03a9), ((X - fun x => \u222b (x : \u03a9), X x \u2202\u03bc) ^ 2) x \u2202\u03bc", "state_after": "\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d : IsFiniteMeasure \u03bc\nhX : Mem\u2112p X 2\n\u22a2 \u222b (x : \u03a9), (X x - \u222b (x : \u03a9), X x \u2202\u03bc) ^ 2 \u2202\u03bc = \u222b (x : \u03a9), ((X - fun x => \u222b (x : \u03a9), X x \u2202\u03bc) ^ 2) x \u2202\u03bc\n\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d : IsFiniteMeasure \u03bc\nhX : Mem\u2112p X 2\n\u22a2 0 \u2264 \u222b (x : \u03a9), (X x - \u222b (x : \u03a9), X x \u2202\u03bc) ^ 2 \u2202\u03bc\n\ncase hfi\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d : IsFiniteMeasure \u03bc\nhX : Mem\u2112p X 2\n\u22a2 Integrable fun \u03c9 => (X \u03c9 - \u222b (x : \u03a9), X x \u2202\u03bc) ^ 2\n\ncase f_nn\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d : IsFiniteMeasure \u03bc\nhX : Mem\u2112p X 2\n\u22a2 0 \u2264\u1d50[\u03bc] fun \u03c9 => (X \u03c9 - \u222b (x : \u03a9), X x \u2202\u03bc) ^ 2"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d : IsFiniteMeasure \u03bc\nhX : Mem\u2112p X 2\n\u22a2 \u222b (x : \u03a9), (X x - \u222b (x : \u03a9), X x \u2202\u03bc) ^ 2 \u2202\u03bc = \u222b (x : \u03a9), ((X - fun x => \u222b (x : \u03a9), X x \u2202\u03bc) ^ 2) x \u2202\u03bc", "state_after": "no goals"}, {"tactic": "exact integral_nonneg fun \u03c9 => pow_two_nonneg _", "annotated_tactic": ["exact <a>integral_nonneg</a> fun \u03c9 => <a>pow_two_nonneg</a> _", [{"full_name": "MeasureTheory.integral_nonneg", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1251, 9], "def_end_pos": [1251, 24]}, {"full_name": "pow_two_nonneg", "def_path": "Mathlib/Algebra/GroupPower/Order.lean", "def_pos": [649, 7], "def_end_pos": [649, 21]}]], "state_before": "\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d : IsFiniteMeasure \u03bc\nhX : Mem\u2112p X 2\n\u22a2 0 \u2264 \u222b (x : \u03a9), (X x - \u222b (x : \u03a9), X x \u2202\u03bc) ^ 2 \u2202\u03bc", "state_after": "no goals"}, {"tactic": "convert (hX.sub <| mem\u2112p_const (\u03bc [X])).integrable_norm_rpow two_ne_zero ENNReal.two_ne_top\n  with \u03c9", "annotated_tactic": ["convert (hX.sub <| <a>mem\u2112p_const</a> (\u03bc [X])).<a>integrable_norm_rpow</a> <a>two_ne_zero</a> <a>ENNReal.two_ne_top</a>\n      with \u03c9", [{"full_name": "MeasureTheory.mem\u2112p_const", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [344, 9], "def_end_pos": [344, 20]}, {"full_name": "MeasureTheory.Mem\u2112p.integrable_norm_rpow", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [516, 9], "def_end_pos": [516, 35]}, {"full_name": "two_ne_zero", "def_path": "Mathlib/Algebra/NeZero.lean", "def_pos": [62, 7], "def_end_pos": [62, 18]}, {"full_name": "ENNReal.two_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [431, 9], "def_end_pos": [431, 19]}]], "state_before": "case hfi\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d : IsFiniteMeasure \u03bc\nhX : Mem\u2112p X 2\n\u22a2 Integrable fun \u03c9 => (X \u03c9 - \u222b (x : \u03a9), X x \u2202\u03bc) ^ 2", "state_after": "case h.e'_5.h\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d : IsFiniteMeasure \u03bc\nhX : Mem\u2112p X 2\n\u03c9 : \u03a9\n\u22a2 (X \u03c9 - \u222b (x : \u03a9), X x \u2202\u03bc) ^ 2 = \u2016(X - fun x => \u222b (x : \u03a9), X x \u2202\u03bc) \u03c9\u2016 ^ ENNReal.toReal 2"}, {"tactic": "simp only [Pi.sub_apply, Real.norm_eq_abs, coe_two, ENNReal.one_toReal,\n  Real.rpow_two, sq_abs, abs_pow]", "annotated_tactic": ["simp only [<a>Pi.sub_apply</a>, <a>Real.norm_eq_abs</a>, <a>coe_two</a>, <a>ENNReal.one_toReal</a>,\n      <a>Real.rpow_two</a>, <a>sq_abs</a>, <a>abs_pow</a>]", [{"full_name": "Pi.sub_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [200, 3], "def_end_pos": [200, 14]}, {"full_name": "Real.norm_eq_abs", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [1761, 9], "def_end_pos": [1761, 20]}, {"full_name": "_private.Mathlib.Probability.Variance.0.ProbabilityTheory.coe_two", "def_path": "Mathlib/Probability/Variance.lean", "def_pos": [49, 15], "def_end_pos": [49, 22]}, {"full_name": "ENNReal.one_toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [230, 17], "def_end_pos": [230, 27]}, {"full_name": "Real.rpow_two", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [362, 9], "def_end_pos": [362, 17]}, {"full_name": "sq_abs", "def_path": "Mathlib/Algebra/GroupPower/Order.lean", "def_pos": [680, 9], "def_end_pos": [680, 15]}, {"full_name": "abs_pow", "def_path": "Mathlib/Algebra/GroupPower/Lemmas.lean", "def_pos": [705, 9], "def_end_pos": [705, 16]}]], "state_before": "case h.e'_5.h\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d : IsFiniteMeasure \u03bc\nhX : Mem\u2112p X 2\n\u03c9 : \u03a9\n\u22a2 (X \u03c9 - \u222b (x : \u03a9), X x \u2202\u03bc) ^ 2 = \u2016(X - fun x => \u222b (x : \u03a9), X x \u2202\u03bc) \u03c9\u2016 ^ ENNReal.toReal 2", "state_after": "no goals"}, {"tactic": "exact ae_of_all _ fun \u03c9 => pow_two_nonneg _", "annotated_tactic": ["exact <a>ae_of_all</a> _ fun \u03c9 => <a>pow_two_nonneg</a> _", [{"full_name": "MeasureTheory.ae_of_all", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [407, 9], "def_end_pos": [407, 18]}, {"full_name": "pow_two_nonneg", "def_path": "Mathlib/Algebra/GroupPower/Order.lean", "def_pos": [649, 7], "def_end_pos": [649, 21]}]], "state_before": "case f_nn\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d : IsFiniteMeasure \u03bc\nhX : Mem\u2112p X 2\n\u22a2 0 \u2264\u1d50[\u03bc] fun \u03c9 => (X \u03c9 - \u222b (x : \u03a9), X x \u2202\u03bc) ^ 2", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean", "full_name": "MeasureTheory.Measure.addHaar_ball_mul_of_pos", "start": [439, 1], "end": [443, 84], "traced_tactics": [{"tactic": "have : ball (0 : E) (r * s) = r \u2022 ball (0 : E) s := by\n  simp only [_root_.smul_ball hr.ne' (0 : E) s, Real.norm_eq_abs, abs_of_nonneg hr.le, smul_zero]", "annotated_tactic": ["have : <a>ball</a> (0 : E) (r * s) = r \u2022 <a>ball</a> (0 : E) s := by\n    simp only [<a>_root_.smul_ball</a> hr.ne' (0 : E) s, <a>Real.norm_eq_abs</a>, <a>abs_of_nonneg</a> hr.le, <a>smul_zero</a>]", [{"full_name": "Metric.ball", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [409, 5], "def_end_pos": [409, 9]}, {"full_name": "Metric.ball", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [409, 5], "def_end_pos": [409, 9]}, {"full_name": "smul_ball", "def_path": "Mathlib/Analysis/NormedSpace/Pointwise.lean", "def_pos": [86, 9], "def_end_pos": [86, 18]}, {"full_name": "Real.norm_eq_abs", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [1761, 9], "def_end_pos": [1761, 20]}, {"full_name": "abs_of_nonneg", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [107, 9], "def_end_pos": [107, 22]}, {"full_name": "smul_zero", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [732, 9], "def_end_pos": [732, 18]}]], "state_before": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d : Set E\nx : E\nr : \u211d\nhr : 0 < r\ns : \u211d\n\u22a2 \u2191\u2191\u03bc (ball x (r * s)) = ENNReal.ofReal (r ^ finrank \u211d E) * \u2191\u2191\u03bc (ball 0 s)", "state_after": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d : Set E\nx : E\nr : \u211d\nhr : 0 < r\ns : \u211d\nthis : ball 0 (r * s) = r \u2022 ball 0 s\n\u22a2 \u2191\u2191\u03bc (ball x (r * s)) = ENNReal.ofReal (r ^ finrank \u211d E) * \u2191\u2191\u03bc (ball 0 s)"}, {"tactic": "simp only [this, addHaar_smul, abs_of_nonneg hr.le, addHaar_ball_center, abs_pow]", "annotated_tactic": ["simp only [this, <a>addHaar_smul</a>, <a>abs_of_nonneg</a> hr.le, <a>addHaar_ball_center</a>, <a>abs_pow</a>]", [{"full_name": "MeasureTheory.Measure.addHaar_smul", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean", "def_pos": [371, 9], "def_end_pos": [371, 21]}, {"full_name": "abs_of_nonneg", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [107, 9], "def_end_pos": [107, 22]}, {"full_name": "MeasureTheory.Measure.addHaar_ball_center", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean", "def_pos": [426, 9], "def_end_pos": [426, 28]}, {"full_name": "abs_pow", "def_path": "Mathlib/Algebra/GroupPower/Lemmas.lean", "def_pos": [705, 9], "def_end_pos": [705, 16]}]], "state_before": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d : Set E\nx : E\nr : \u211d\nhr : 0 < r\ns : \u211d\nthis : ball 0 (r * s) = r \u2022 ball 0 s\n\u22a2 \u2191\u2191\u03bc (ball x (r * s)) = ENNReal.ofReal (r ^ finrank \u211d E) * \u2191\u2191\u03bc (ball 0 s)", "state_after": "no goals"}, {"tactic": "simp only [_root_.smul_ball hr.ne' (0 : E) s, Real.norm_eq_abs, abs_of_nonneg hr.le, smul_zero]", "annotated_tactic": ["simp only [<a>_root_.smul_ball</a> hr.ne' (0 : E) s, <a>Real.norm_eq_abs</a>, <a>abs_of_nonneg</a> hr.le, <a>smul_zero</a>]", [{"full_name": "smul_ball", "def_path": "Mathlib/Analysis/NormedSpace/Pointwise.lean", "def_pos": [86, 9], "def_end_pos": [86, 18]}, {"full_name": "Real.norm_eq_abs", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [1761, 9], "def_end_pos": [1761, 20]}, {"full_name": "abs_of_nonneg", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [107, 9], "def_end_pos": [107, 22]}, {"full_name": "smul_zero", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [732, 9], "def_end_pos": [732, 18]}]], "state_before": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d : Set E\nx : E\nr : \u211d\nhr : 0 < r\ns : \u211d\n\u22a2 ball 0 (r * s) = r \u2022 ball 0 s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "Finset.mem_erase", "start": [1887, 1], "end": [1888, 20], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "full_name": "aemeasurable_biInf", "start": [1449, 1], "end": [1451, 40], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Group/Measure.lean", "full_name": "MeasureTheory.eventually_div_right_iff", "start": [375, 1], "end": [377, 66], "traced_tactics": [{"tactic": "conv_rhs => rw [Filter.Eventually, \u2190 map_div_right_ae \u03bc t]; rfl", "annotated_tactic": ["conv_rhs => rw [<a>Filter.Eventually</a>, \u2190 <a>map_div_right_ae</a> \u03bc t]; rfl", [{"full_name": "Filter.Eventually", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1072, 15], "def_end_pos": [1072, 25]}, {"full_name": "MeasureTheory.map_div_right_ae", "def_path": "Mathlib/MeasureTheory/Group/Measure.lean", "def_pos": [354, 9], "def_end_pos": [354, 25]}]], "state_before": "\ud835\udd5c : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : MeasurableSpace H\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : MeasurableMul G\n\u03bc : Measure G\ninst\u271d : IsMulRightInvariant \u03bc\nt : G\np : G \u2192 Prop\n\u22a2 (\u2200\u1d50 (x : G) \u2202\u03bc, p (x / t)) \u2194 \u2200\u1d50 (x : G) \u2202\u03bc, p x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "full_name": "MeasurableEmbedding.of_measurable_inverse", "start": [1769, 1], "end": [1771, 76], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Intervals/OrdConnectedComponent.lean", "full_name": "Set.eq_of_mem_ordConnectedSection_of_uIcc_subset", "start": [154, 1], "end": [161, 58], "traced_tactics": [{"tactic": "rcases hx with \u27e8x, rfl\u27e9", "annotated_tactic": ["rcases hx with \u27e8x, rfl\u27e9", []], "state_before": "\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\ns t : Set \u03b1\nx y z : \u03b1\nhx : x \u2208 ordConnectedSection s\nhy : y \u2208 ordConnectedSection s\nh : [[x, y]] \u2286 s\n\u22a2 x = y", "state_after": "case intro\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\ns t : Set \u03b1\ny z : \u03b1\nhy : y \u2208 ordConnectedSection s\nx : \u2191s\nh : [[ordConnectedProj s x, y]] \u2286 s\n\u22a2 ordConnectedProj s x = y"}, {"tactic": "rcases hy with \u27e8y, rfl\u27e9", "annotated_tactic": ["rcases hy with \u27e8y, rfl\u27e9", []], "state_before": "case intro\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\ns t : Set \u03b1\ny z : \u03b1\nhy : y \u2208 ordConnectedSection s\nx : \u2191s\nh : [[ordConnectedProj s x, y]] \u2286 s\n\u22a2 ordConnectedProj s x = y", "state_after": "case intro.intro\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\ns t : Set \u03b1\nz : \u03b1\nx y : \u2191s\nh : [[ordConnectedProj s x, ordConnectedProj s y]] \u2286 s\n\u22a2 ordConnectedProj s x = ordConnectedProj s y"}, {"tactic": "exact\n  ordConnectedProj_eq.2\n    (mem_ordConnectedComponent_trans\n      (mem_ordConnectedComponent_trans (ordConnectedProj_mem_ordConnectedComponent _ _) h)\n      (mem_ordConnectedComponent_ordConnectedProj _ _))", "annotated_tactic": ["exact\n    <a>ordConnectedProj_eq</a>.2\n      (<a>mem_ordConnectedComponent_trans</a>\n        (<a>mem_ordConnectedComponent_trans</a> (<a>ordConnectedProj_mem_ordConnectedComponent</a> _ _) h)\n        (<a>mem_ordConnectedComponent_ordConnectedProj</a> _ _))", [{"full_name": "Set.ordConnectedProj_eq", "def_path": "Mathlib/Data/Set/Intervals/OrdConnectedComponent.lean", "def_pos": [126, 9], "def_end_pos": [126, 28]}, {"full_name": "Set.mem_ordConnectedComponent_trans", "def_path": "Mathlib/Data/Set/Intervals/OrdConnectedComponent.lean", "def_pos": [86, 9], "def_end_pos": [86, 40]}, {"full_name": "Set.mem_ordConnectedComponent_trans", "def_path": "Mathlib/Data/Set/Intervals/OrdConnectedComponent.lean", "def_pos": [86, 9], "def_end_pos": [86, 40]}, {"full_name": "Set.ordConnectedProj_mem_ordConnectedComponent", "def_path": "Mathlib/Data/Set/Intervals/OrdConnectedComponent.lean", "def_pos": [109, 9], "def_end_pos": [109, 51]}, {"full_name": "Set.mem_ordConnectedComponent_ordConnectedProj", "def_path": "Mathlib/Data/Set/Intervals/OrdConnectedComponent.lean", "def_pos": [114, 9], "def_end_pos": [114, 51]}]], "state_before": "case intro.intro\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\ns t : Set \u03b1\nz : \u03b1\nx y : \u2191s\nh : [[ordConnectedProj s x, ordConnectedProj s y]] \u2286 s\n\u22a2 ordConnectedProj s x = ordConnectedProj s y", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Powerset.lean", "full_name": "Finset.powersetCard_sup", "start": [303, 1], "end": [318, 56], "traced_tactics": [{"tactic": "apply le_antisymm", "annotated_tactic": ["apply <a>le_antisymm</a>", [{"full_name": "le_antisymm", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [188, 9], "def_end_pos": [188, 20]}]], "state_before": "\u03b1 : Type u_1\ns t : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\nu : Finset \u03b1\nn : \u2115\nhn : n < card u\n\u22a2 sup (powersetCard (Nat.succ n) u) id = u", "state_after": "case a\n\u03b1 : Type u_1\ns t : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\nu : Finset \u03b1\nn : \u2115\nhn : n < card u\n\u22a2 sup (powersetCard (Nat.succ n) u) id \u2264 u\n\ncase a\n\u03b1 : Type u_1\ns t : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\nu : Finset \u03b1\nn : \u2115\nhn : n < card u\n\u22a2 u \u2264 sup (powersetCard (Nat.succ n) u) id"}, {"tactic": "simp_rw [Finset.sup_le_iff, mem_powersetCard]", "annotated_tactic": ["simp_rw [<a>Finset.sup_le_iff</a>, <a>mem_powersetCard</a>]", [{"full_name": "Finset.sup_le_iff", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [99, 19], "def_end_pos": [99, 29]}, {"full_name": "Finset.mem_powersetCard", "def_path": "Mathlib/Data/Finset/Powerset.lean", "def_pos": [203, 9], "def_end_pos": [203, 25]}]], "state_before": "case a\n\u03b1 : Type u_1\ns t : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\nu : Finset \u03b1\nn : \u2115\nhn : n < card u\n\u22a2 sup (powersetCard (Nat.succ n) u) id \u2264 u", "state_after": "case a\n\u03b1 : Type u_1\ns t : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\nu : Finset \u03b1\nn : \u2115\nhn : n < card u\n\u22a2 \u2200 (b : Finset \u03b1), b \u2286 u \u2227 card b = Nat.succ n \u2192 id b \u2264 u"}, {"tactic": "rintro x \u27e8h, -\u27e9", "annotated_tactic": ["rintro x \u27e8h, -\u27e9", []], "state_before": "case a\n\u03b1 : Type u_1\ns t : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\nu : Finset \u03b1\nn : \u2115\nhn : n < card u\n\u22a2 \u2200 (b : Finset \u03b1), b \u2286 u \u2227 card b = Nat.succ n \u2192 id b \u2264 u", "state_after": "case a.intro\n\u03b1 : Type u_1\ns t : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\nu : Finset \u03b1\nn : \u2115\nhn : n < card u\nx : Finset \u03b1\nh : x \u2286 u\n\u22a2 id x \u2264 u"}, {"tactic": "exact h", "annotated_tactic": ["exact h", []], "state_before": "case a.intro\n\u03b1 : Type u_1\ns t : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\nu : Finset \u03b1\nn : \u2115\nhn : n < card u\nx : Finset \u03b1\nh : x \u2286 u\n\u22a2 id x \u2264 u", "state_after": "no goals"}, {"tactic": "rw [sup_eq_biUnion, le_iff_subset, subset_iff]", "annotated_tactic": ["rw [<a>sup_eq_biUnion</a>, <a>le_iff_subset</a>, <a>subset_iff</a>]", [{"full_name": "Finset.sup_eq_biUnion", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [1876, 9], "def_end_pos": [1876, 23]}, {"full_name": "Finset.le_iff_subset", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [402, 9], "def_end_pos": [402, 22]}, {"full_name": "Finset.subset_iff", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [371, 9], "def_end_pos": [371, 19]}]], "state_before": "case a\n\u03b1 : Type u_1\ns t : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\nu : Finset \u03b1\nn : \u2115\nhn : n < card u\n\u22a2 u \u2264 sup (powersetCard (Nat.succ n) u) id", "state_after": "case a\n\u03b1 : Type u_1\ns t : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\nu : Finset \u03b1\nn : \u2115\nhn : n < card u\n\u22a2 \u2200 \u2983x : \u03b1\u2984, x \u2208 u \u2192 x \u2208 Finset.biUnion (powersetCard (Nat.succ n) u) id"}, {"tactic": "cases' (Nat.succ_le_of_lt hn).eq_or_lt with h' h'", "annotated_tactic": ["cases' (<a>Nat.succ_le_of_lt</a> hn).<a>eq_or_lt</a> with h' h'", [{"full_name": "Nat.succ_le_of_lt", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [313, 9], "def_end_pos": [313, 22]}, {"full_name": "LE.le.eq_or_lt", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [428, 7], "def_end_pos": [428, 21]}]], "state_before": "case a\n\u03b1 : Type u_1\ns t : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\nu : Finset \u03b1\nn : \u2115\nhn : n < card u\n\u22a2 \u2200 \u2983x : \u03b1\u2984, x \u2208 u \u2192 x \u2208 Finset.biUnion (powersetCard (Nat.succ n) u) id", "state_after": "case a.inl\n\u03b1 : Type u_1\ns t : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\nu : Finset \u03b1\nn : \u2115\nhn : n < card u\nh' : Nat.succ n = card u\n\u22a2 \u2200 \u2983x : \u03b1\u2984, x \u2208 u \u2192 x \u2208 Finset.biUnion (powersetCard (Nat.succ n) u) id\n\ncase a.inr\n\u03b1 : Type u_1\ns t : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\nu : Finset \u03b1\nn : \u2115\nhn : n < card u\nh' : Nat.succ n < card u\n\u22a2 \u2200 \u2983x : \u03b1\u2984, x \u2208 u \u2192 x \u2208 Finset.biUnion (powersetCard (Nat.succ n) u) id"}, {"tactic": "simp [h']", "annotated_tactic": ["simp [h']", []], "state_before": "case a.inl\n\u03b1 : Type u_1\ns t : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\nu : Finset \u03b1\nn : \u2115\nhn : n < card u\nh' : Nat.succ n = card u\n\u22a2 \u2200 \u2983x : \u03b1\u2984, x \u2208 u \u2192 x \u2208 Finset.biUnion (powersetCard (Nat.succ n) u) id", "state_after": "no goals"}, {"tactic": "intro x hx", "annotated_tactic": ["intro x hx", []], "state_before": "case a.inr\n\u03b1 : Type u_1\ns t : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\nu : Finset \u03b1\nn : \u2115\nhn : n < card u\nh' : Nat.succ n < card u\n\u22a2 \u2200 \u2983x : \u03b1\u2984, x \u2208 u \u2192 x \u2208 Finset.biUnion (powersetCard (Nat.succ n) u) id", "state_after": "case a.inr\n\u03b1 : Type u_1\ns t : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\nu : Finset \u03b1\nn : \u2115\nhn : n < card u\nh' : Nat.succ n < card u\nx : \u03b1\nhx : x \u2208 u\n\u22a2 x \u2208 Finset.biUnion (powersetCard (Nat.succ n) u) id"}, {"tactic": "simp only [mem_biUnion, exists_prop, id.def]", "annotated_tactic": ["simp only [<a>mem_biUnion</a>, <a>exists_prop</a>, <a>id.def</a>]", [{"full_name": "Finset.mem_biUnion", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3613, 9], "def_end_pos": [3613, 20]}, {"full_name": "exists_prop", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [485, 17], "def_end_pos": [485, 28]}, {"full_name": "id.def", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [527, 9], "def_end_pos": [527, 15]}]], "state_before": "case a.inr\n\u03b1 : Type u_1\ns t : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\nu : Finset \u03b1\nn : \u2115\nhn : n < card u\nh' : Nat.succ n < card u\nx : \u03b1\nhx : x \u2208 u\n\u22a2 x \u2208 Finset.biUnion (powersetCard (Nat.succ n) u) id", "state_after": "case a.inr\n\u03b1 : Type u_1\ns t : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\nu : Finset \u03b1\nn : \u2115\nhn : n < card u\nh' : Nat.succ n < card u\nx : \u03b1\nhx : x \u2208 u\n\u22a2 \u2203 a, a \u2208 powersetCard (Nat.succ n) u \u2227 x \u2208 a"}, {"tactic": "obtain \u27e8t, ht\u27e9 : \u2203 t, t \u2208 powersetCard n (u.erase x) := powersetCard_nonempty\n  (le_trans (Nat.le_pred_of_lt hn) pred_card_le_card_erase)", "annotated_tactic": ["obtain \u27e8t, ht\u27e9 : \u2203 t, t \u2208 <a>powersetCard</a> n (u.erase x) := <a>powersetCard_nonempty</a>\n        (<a>le_trans</a> (<a>Nat.le_pred_of_lt</a> hn) <a>pred_card_le_card_erase</a>)", [{"full_name": "Finset.powersetCard", "def_path": "Mathlib/Data/Finset/Powerset.lean", "def_pos": [197, 5], "def_end_pos": [197, 17]}, {"full_name": "Finset.powersetCard_nonempty", "def_path": "Mathlib/Data/Finset/Powerset.lean", "def_pos": [254, 9], "def_end_pos": [254, 30]}, {"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}, {"full_name": "Nat.le_pred_of_lt", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [268, 9], "def_end_pos": [268, 22]}, {"full_name": "Finset.pred_card_le_card_erase", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [162, 9], "def_end_pos": [162, 32]}]], "state_before": "case a.inr\n\u03b1 : Type u_1\ns t : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\nu : Finset \u03b1\nn : \u2115\nhn : n < card u\nh' : Nat.succ n < card u\nx : \u03b1\nhx : x \u2208 u\n\u22a2 \u2203 a, a \u2208 powersetCard (Nat.succ n) u \u2227 x \u2208 a", "state_after": "case a.inr.intro\n\u03b1 : Type u_1\ns t\u271d : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\nu : Finset \u03b1\nn : \u2115\nhn : n < card u\nh' : Nat.succ n < card u\nx : \u03b1\nhx : x \u2208 u\nt : Finset \u03b1\nht : t \u2208 powersetCard n (erase u x)\n\u22a2 \u2203 a, a \u2208 powersetCard (Nat.succ n) u \u2227 x \u2208 a"}, {"tactic": "refine' \u27e8insert x t, _, mem_insert_self _ _\u27e9", "annotated_tactic": ["refine' \u27e8<a>insert</a> x t, _, <a>mem_insert_self</a> _ _\u27e9", [{"full_name": "Insert.insert", "def_path": "lake-packages/std/Std/Classes/SetNotation.lean", "def_pos": [69, 3], "def_end_pos": [69, 9]}, {"full_name": "Finset.mem_insert_self", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1091, 9], "def_end_pos": [1091, 24]}]], "state_before": "case a.inr.intro\n\u03b1 : Type u_1\ns t\u271d : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\nu : Finset \u03b1\nn : \u2115\nhn : n < card u\nh' : Nat.succ n < card u\nx : \u03b1\nhx : x \u2208 u\nt : Finset \u03b1\nht : t \u2208 powersetCard n (erase u x)\n\u22a2 \u2203 a, a \u2208 powersetCard (Nat.succ n) u \u2227 x \u2208 a", "state_after": "case a.inr.intro\n\u03b1 : Type u_1\ns t\u271d : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\nu : Finset \u03b1\nn : \u2115\nhn : n < card u\nh' : Nat.succ n < card u\nx : \u03b1\nhx : x \u2208 u\nt : Finset \u03b1\nht : t \u2208 powersetCard n (erase u x)\n\u22a2 insert x t \u2208 powersetCard (Nat.succ n) u"}, {"tactic": "rw [\u2190 insert_erase hx, powersetCard_succ_insert (not_mem_erase _ _)]", "annotated_tactic": ["rw [\u2190 <a>insert_erase</a> hx, <a>powersetCard_succ_insert</a> (<a>not_mem_erase</a> _ _)]", [{"full_name": "Finset.insert_erase", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1957, 9], "def_end_pos": [1957, 21]}, {"full_name": "Finset.powersetCard_succ_insert", "def_path": "Mathlib/Data/Finset/Powerset.lean", "def_pos": [240, 9], "def_end_pos": [240, 33]}, {"full_name": "Finset.not_mem_erase", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1891, 9], "def_end_pos": [1891, 22]}]], "state_before": "case a.inr.intro\n\u03b1 : Type u_1\ns t\u271d : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\nu : Finset \u03b1\nn : \u2115\nhn : n < card u\nh' : Nat.succ n < card u\nx : \u03b1\nhx : x \u2208 u\nt : Finset \u03b1\nht : t \u2208 powersetCard n (erase u x)\n\u22a2 insert x t \u2208 powersetCard (Nat.succ n) u", "state_after": "case a.inr.intro\n\u03b1 : Type u_1\ns t\u271d : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\nu : Finset \u03b1\nn : \u2115\nhn : n < card u\nh' : Nat.succ n < card u\nx : \u03b1\nhx : x \u2208 u\nt : Finset \u03b1\nht : t \u2208 powersetCard n (erase u x)\n\u22a2 insert x t \u2208 powersetCard (Nat.succ n) (erase u x) \u222a image (insert x) (powersetCard n (erase u x))"}, {"tactic": "exact mem_union_right _ (mem_image_of_mem _ ht)", "annotated_tactic": ["exact <a>mem_union_right</a> _ (<a>mem_image_of_mem</a> _ ht)", [{"full_name": "Finset.mem_union_right", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1386, 9], "def_end_pos": [1386, 24]}, {"full_name": "Finset.mem_image_of_mem", "def_path": "Mathlib/Data/Finset/Image.lean", "def_pos": [334, 9], "def_end_pos": [334, 25]}]], "state_before": "case a.inr.intro\n\u03b1 : Type u_1\ns t\u271d : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\nu : Finset \u03b1\nn : \u2115\nhn : n < card u\nh' : Nat.succ n < card u\nx : \u03b1\nhx : x \u2208 u\nt : Finset \u03b1\nht : t \u2208 powersetCard n (erase u x)\n\u22a2 insert x t \u2208 powersetCard (Nat.succ n) (erase u x) \u222a image (insert x) (powersetCard n (erase u x))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "full_name": "MeasureTheory.Lp.simpleFunc.aestronglyMeasurable", "start": [609, 11], "end": [611, 57], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "full_name": "MeasureTheory.VectorMeasure.MutuallySingular.neg_left_iff", "start": [1249, 1], "end": [1251, 46], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/CircleIntegral.lean", "full_name": "le_radius_cauchyPowerSeries", "start": [562, 1], "end": [576, 36], "traced_tactics": [{"tactic": "refine'\n  (cauchyPowerSeries f c R).le_radius_of_bound\n    ((2 * \u03c0)\u207b\u00b9 * \u222b \u03b8 : \u211d in (0)..2 * \u03c0, \u2016f (circleMap c R \u03b8)\u2016) fun n => _", "annotated_tactic": ["refine'\n    (<a>cauchyPowerSeries</a> f c R).<a>le_radius_of_bound</a>\n      ((2 * \u03c0)\u207b\u00b9 * \u222b \u03b8 : \u211d in (0)..2 * \u03c0, \u2016f (<a>circleMap</a> c R \u03b8)\u2016) fun n => _", [{"full_name": "cauchyPowerSeries", "def_path": "Mathlib/MeasureTheory/Integral/CircleIntegral.lean", "def_pos": [526, 5], "def_end_pos": [526, 22]}, {"full_name": "FormalMultilinearSeries.le_radius_of_bound", "def_path": "Mathlib/Analysis/Analytic/Basic.lean", "def_pos": [129, 9], "def_end_pos": [129, 27]}, {"full_name": "circleMap", "def_path": "Mathlib/MeasureTheory/Integral/CircleIntegral.lean", "def_pos": [89, 5], "def_end_pos": [89, 14]}]], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf : \u2102 \u2192 E\nc : \u2102\nR : \u211d\u22650\n\u22a2 \u2191R \u2264 FormalMultilinearSeries.radius (cauchyPowerSeries f c \u2191R)", "state_after": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf : \u2102 \u2192 E\nc : \u2102\nR : \u211d\u22650\nn : \u2115\n\u22a2 \u2016cauchyPowerSeries f c (\u2191R) n\u2016 * \u2191R ^ n \u2264 (2 * \u03c0)\u207b\u00b9 * \u222b (\u03b8 : \u211d) in 0 ..2 * \u03c0, \u2016f (circleMap c (\u2191R) \u03b8)\u2016"}, {"tactic": "refine' (mul_le_mul_of_nonneg_right (norm_cauchyPowerSeries_le _ _ _ _)\n  (pow_nonneg R.coe_nonneg _)).trans _", "annotated_tactic": ["refine' (<a>mul_le_mul_of_nonneg_right</a> (<a>norm_cauchyPowerSeries_le</a> _ _ _ _)\n    (<a>pow_nonneg</a> R.coe_nonneg _)).<a>trans</a> _", [{"full_name": "mul_le_mul_of_nonneg_right", "def_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "def_pos": [156, 9], "def_end_pos": [156, 35]}, {"full_name": "norm_cauchyPowerSeries_le", "def_path": "Mathlib/MeasureTheory/Integral/CircleIntegral.lean", "def_pos": [539, 9], "def_end_pos": [539, 34]}, {"full_name": "pow_nonneg", "def_path": "Mathlib/Algebra/Order/Ring/Defs.lean", "def_pos": [244, 9], "def_end_pos": [244, 19]}, {"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [120, 7], "def_end_pos": [120, 18]}]], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf : \u2102 \u2192 E\nc : \u2102\nR : \u211d\u22650\nn : \u2115\n\u22a2 \u2016cauchyPowerSeries f c (\u2191R) n\u2016 * \u2191R ^ n \u2264 (2 * \u03c0)\u207b\u00b9 * \u222b (\u03b8 : \u211d) in 0 ..2 * \u03c0, \u2016f (circleMap c (\u2191R) \u03b8)\u2016", "state_after": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf : \u2102 \u2192 E\nc : \u2102\nR : \u211d\u22650\nn : \u2115\n\u22a2 ((2 * \u03c0)\u207b\u00b9 * \u222b (\u03b8 : \u211d) in 0 ..2 * \u03c0, \u2016f (circleMap c (\u2191R) \u03b8)\u2016) * |\u2191R|\u207b\u00b9 ^ n * \u2191R ^ n \u2264\n    (2 * \u03c0)\u207b\u00b9 * \u222b (\u03b8 : \u211d) in 0 ..2 * \u03c0, \u2016f (circleMap c (\u2191R) \u03b8)\u2016"}, {"tactic": "rw [_root_.abs_of_nonneg R.coe_nonneg]", "annotated_tactic": ["rw [<a>_root_.abs_of_nonneg</a> R.coe_nonneg]", [{"full_name": "abs_of_nonneg", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [107, 9], "def_end_pos": [107, 22]}]], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf : \u2102 \u2192 E\nc : \u2102\nR : \u211d\u22650\nn : \u2115\n\u22a2 ((2 * \u03c0)\u207b\u00b9 * \u222b (\u03b8 : \u211d) in 0 ..2 * \u03c0, \u2016f (circleMap c (\u2191R) \u03b8)\u2016) * |\u2191R|\u207b\u00b9 ^ n * \u2191R ^ n \u2264\n    (2 * \u03c0)\u207b\u00b9 * \u222b (\u03b8 : \u211d) in 0 ..2 * \u03c0, \u2016f (circleMap c (\u2191R) \u03b8)\u2016", "state_after": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf : \u2102 \u2192 E\nc : \u2102\nR : \u211d\u22650\nn : \u2115\n\u22a2 ((2 * \u03c0)\u207b\u00b9 * \u222b (\u03b8 : \u211d) in 0 ..2 * \u03c0, \u2016f (circleMap c (\u2191R) \u03b8)\u2016) * (\u2191R)\u207b\u00b9 ^ n * \u2191R ^ n \u2264\n    (2 * \u03c0)\u207b\u00b9 * \u222b (\u03b8 : \u211d) in 0 ..2 * \u03c0, \u2016f (circleMap c (\u2191R) \u03b8)\u2016"}, {"tactic": "cases' eq_or_ne (R ^ n : \u211d) 0 with hR hR", "annotated_tactic": ["cases' <a>eq_or_ne</a> (R ^ n : \u211d) 0 with hR hR", [{"full_name": "eq_or_ne", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [209, 9], "def_end_pos": [209, 17]}]], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf : \u2102 \u2192 E\nc : \u2102\nR : \u211d\u22650\nn : \u2115\n\u22a2 ((2 * \u03c0)\u207b\u00b9 * \u222b (\u03b8 : \u211d) in 0 ..2 * \u03c0, \u2016f (circleMap c (\u2191R) \u03b8)\u2016) * (\u2191R)\u207b\u00b9 ^ n * \u2191R ^ n \u2264\n    (2 * \u03c0)\u207b\u00b9 * \u222b (\u03b8 : \u211d) in 0 ..2 * \u03c0, \u2016f (circleMap c (\u2191R) \u03b8)\u2016", "state_after": "case inl\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf : \u2102 \u2192 E\nc : \u2102\nR : \u211d\u22650\nn : \u2115\nhR : \u2191(R ^ n) = 0\n\u22a2 ((2 * \u03c0)\u207b\u00b9 * \u222b (\u03b8 : \u211d) in 0 ..2 * \u03c0, \u2016f (circleMap c (\u2191R) \u03b8)\u2016) * (\u2191R)\u207b\u00b9 ^ n * \u2191R ^ n \u2264\n    (2 * \u03c0)\u207b\u00b9 * \u222b (\u03b8 : \u211d) in 0 ..2 * \u03c0, \u2016f (circleMap c (\u2191R) \u03b8)\u2016\n\ncase inr\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf : \u2102 \u2192 E\nc : \u2102\nR : \u211d\u22650\nn : \u2115\nhR : \u2191(R ^ n) \u2260 0\n\u22a2 ((2 * \u03c0)\u207b\u00b9 * \u222b (\u03b8 : \u211d) in 0 ..2 * \u03c0, \u2016f (circleMap c (\u2191R) \u03b8)\u2016) * (\u2191R)\u207b\u00b9 ^ n * \u2191R ^ n \u2264\n    (2 * \u03c0)\u207b\u00b9 * \u222b (\u03b8 : \u211d) in 0 ..2 * \u03c0, \u2016f (circleMap c (\u2191R) \u03b8)\u2016"}, {"tactic": "rw_mod_cast [hR, mul_zero]", "annotated_tactic": ["rw_mod_cast [hR, <a>mul_zero</a>]", [{"full_name": "MulZeroClass.mul_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [38, 3], "def_end_pos": [38, 11]}]], "state_before": "case inl\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf : \u2102 \u2192 E\nc : \u2102\nR : \u211d\u22650\nn : \u2115\nhR : \u2191(R ^ n) = 0\n\u22a2 ((2 * \u03c0)\u207b\u00b9 * \u222b (\u03b8 : \u211d) in 0 ..2 * \u03c0, \u2016f (circleMap c (\u2191R) \u03b8)\u2016) * (\u2191R)\u207b\u00b9 ^ n * \u2191R ^ n \u2264\n    (2 * \u03c0)\u207b\u00b9 * \u222b (\u03b8 : \u211d) in 0 ..2 * \u03c0, \u2016f (circleMap c (\u2191R) \u03b8)\u2016", "state_after": "case inl\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf : \u2102 \u2192 E\nc : \u2102\nR : \u211d\u22650\nn : \u2115\nhR : R ^ n = 0\n\u22a2 0 \u2264 (2 * \u03c0)\u207b\u00b9 * \u222b (\u03b8 : \u211d) in 0 ..2 * \u03c0, \u2016f (circleMap c (\u2191R) \u03b8)\u2016"}, {"tactic": "exact mul_nonneg (inv_nonneg.2 Real.two_pi_pos.le)\n  (intervalIntegral.integral_nonneg Real.two_pi_pos.le fun _ _ => norm_nonneg _)", "annotated_tactic": ["exact <a>mul_nonneg</a> (<a>inv_nonneg</a>.2 Real.two_pi_pos.le)\n      (<a>intervalIntegral.integral_nonneg</a> Real.two_pi_pos.le fun _ _ => <a>norm_nonneg</a> _)", [{"full_name": "mul_nonneg", "def_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "def_pos": [380, 7], "def_end_pos": [380, 17]}, {"full_name": "inv_nonneg", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [58, 9], "def_end_pos": [58, 19]}, {"full_name": "intervalIntegral.integral_nonneg", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [1368, 9], "def_end_pos": [1368, 24]}, {"full_name": "norm_nonneg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [500, 30], "def_end_pos": [500, 41]}]], "state_before": "case inl\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf : \u2102 \u2192 E\nc : \u2102\nR : \u211d\u22650\nn : \u2115\nhR : R ^ n = 0\n\u22a2 0 \u2264 (2 * \u03c0)\u207b\u00b9 * \u222b (\u03b8 : \u211d) in 0 ..2 * \u03c0, \u2016f (circleMap c (\u2191R) \u03b8)\u2016", "state_after": "no goals"}, {"tactic": "rw [inv_pow]", "annotated_tactic": ["rw [<a>inv_pow</a>]", [{"full_name": "inv_pow", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [317, 9], "def_end_pos": [317, 16]}]], "state_before": "case inr\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf : \u2102 \u2192 E\nc : \u2102\nR : \u211d\u22650\nn : \u2115\nhR : \u2191(R ^ n) \u2260 0\n\u22a2 ((2 * \u03c0)\u207b\u00b9 * \u222b (\u03b8 : \u211d) in 0 ..2 * \u03c0, \u2016f (circleMap c (\u2191R) \u03b8)\u2016) * (\u2191R)\u207b\u00b9 ^ n * \u2191R ^ n \u2264\n    (2 * \u03c0)\u207b\u00b9 * \u222b (\u03b8 : \u211d) in 0 ..2 * \u03c0, \u2016f (circleMap c (\u2191R) \u03b8)\u2016", "state_after": "case inr\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf : \u2102 \u2192 E\nc : \u2102\nR : \u211d\u22650\nn : \u2115\nhR : \u2191(R ^ n) \u2260 0\n\u22a2 ((2 * \u03c0)\u207b\u00b9 * \u222b (\u03b8 : \u211d) in 0 ..2 * \u03c0, \u2016f (circleMap c (\u2191R) \u03b8)\u2016) * (\u2191R ^ n)\u207b\u00b9 * \u2191R ^ n \u2264\n    (2 * \u03c0)\u207b\u00b9 * \u222b (\u03b8 : \u211d) in 0 ..2 * \u03c0, \u2016f (circleMap c (\u2191R) \u03b8)\u2016"}, {"tactic": "have : (R:\u211d) ^ n \u2260 0 := by norm_cast at hR \u22a2", "annotated_tactic": ["have : (R:\u211d) ^ n \u2260 0 := by norm_cast at hR \u22a2", []], "state_before": "case inr\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf : \u2102 \u2192 E\nc : \u2102\nR : \u211d\u22650\nn : \u2115\nhR : \u2191(R ^ n) \u2260 0\n\u22a2 ((2 * \u03c0)\u207b\u00b9 * \u222b (\u03b8 : \u211d) in 0 ..2 * \u03c0, \u2016f (circleMap c (\u2191R) \u03b8)\u2016) * (\u2191R ^ n)\u207b\u00b9 * \u2191R ^ n \u2264\n    (2 * \u03c0)\u207b\u00b9 * \u222b (\u03b8 : \u211d) in 0 ..2 * \u03c0, \u2016f (circleMap c (\u2191R) \u03b8)\u2016", "state_after": "case inr\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf : \u2102 \u2192 E\nc : \u2102\nR : \u211d\u22650\nn : \u2115\nhR : \u2191(R ^ n) \u2260 0\nthis : \u2191R ^ n \u2260 0\n\u22a2 ((2 * \u03c0)\u207b\u00b9 * \u222b (\u03b8 : \u211d) in 0 ..2 * \u03c0, \u2016f (circleMap c (\u2191R) \u03b8)\u2016) * (\u2191R ^ n)\u207b\u00b9 * \u2191R ^ n \u2264\n    (2 * \u03c0)\u207b\u00b9 * \u222b (\u03b8 : \u211d) in 0 ..2 * \u03c0, \u2016f (circleMap c (\u2191R) \u03b8)\u2016"}, {"tactic": "rw [inv_mul_cancel_right\u2080 this]", "annotated_tactic": ["rw [<a>inv_mul_cancel_right\u2080</a> this]", [{"full_name": "inv_mul_cancel_right\u2080", "def_path": "Mathlib/Algebra/GroupWithZero/Basic.lean", "def_pos": [235, 9], "def_end_pos": [235, 30]}]], "state_before": "case inr\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf : \u2102 \u2192 E\nc : \u2102\nR : \u211d\u22650\nn : \u2115\nhR : \u2191(R ^ n) \u2260 0\nthis : \u2191R ^ n \u2260 0\n\u22a2 ((2 * \u03c0)\u207b\u00b9 * \u222b (\u03b8 : \u211d) in 0 ..2 * \u03c0, \u2016f (circleMap c (\u2191R) \u03b8)\u2016) * (\u2191R ^ n)\u207b\u00b9 * \u2191R ^ n \u2264\n    (2 * \u03c0)\u207b\u00b9 * \u222b (\u03b8 : \u211d) in 0 ..2 * \u03c0, \u2016f (circleMap c (\u2191R) \u03b8)\u2016", "state_after": "no goals"}, {"tactic": "norm_cast at hR \u22a2", "annotated_tactic": ["norm_cast at hR \u22a2", []], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf : \u2102 \u2192 E\nc : \u2102\nR : \u211d\u22650\nn : \u2115\nhR : \u2191(R ^ n) \u2260 0\n\u22a2 \u2191R ^ n \u2260 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Integration.lean", "full_name": "ProbabilityTheory.lintegral_mul_eq_lintegral_mul_lintegral_of_indepFun", "start": [109, 1], "end": [114, 68], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "full_name": "MeasurableSpace.measurableSet_iSup", "start": [512, 1], "end": [514, 60], "traced_tactics": [{"tactic": "simp only [iSup, measurableSet_sSup, exists_range_iff]", "annotated_tactic": ["simp only [<a>iSup</a>, <a>measurableSet_sSup</a>, <a>exists_range_iff</a>]", [{"full_name": "iSup", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [78, 5], "def_end_pos": [78, 9]}, {"full_name": "MeasurableSpace.measurableSet_sSup", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [505, 9], "def_end_pos": [505, 27]}, {"full_name": "Set.exists_range_iff", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [694, 9], "def_end_pos": [694, 25]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9\u271d : Sort u_6\ns\u271d t u : Set \u03b1\n\u03b9 : Sort u_7\nm : \u03b9 \u2192 MeasurableSpace \u03b1\ns : Set \u03b1\n\u22a2 MeasurableSet s \u2194 GenerateMeasurable {s | \u2203 i, MeasurableSet s} s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/PiSystem.lean", "full_name": "MeasurableSpace.DynkinSystem.generateFrom_eq", "start": [735, 1], "end": [740, 69], "traced_tactics": [{"tactic": "rw [ofMeasurableSpace_toMeasurableSpace]", "annotated_tactic": ["rw [<a>ofMeasurableSpace_toMeasurableSpace</a>]", [{"full_name": "MeasurableSpace.DynkinSystem.ofMeasurableSpace_toMeasurableSpace", "def_path": "Mathlib/MeasureTheory/PiSystem.lean", "def_pos": [679, 9], "def_end_pos": [679, 44]}]], "state_before": "\u03b1 : Type u_1\nd : DynkinSystem \u03b1\ns : Set (Set \u03b1)\nhs : IsPiSystem s\n\u22a2 ofMeasurableSpace\n      (toMeasurableSpace (generate s)\n        (_ : \u2200 (t\u2081 t\u2082 : Set \u03b1), Has (generate s) t\u2081 \u2192 Has (generate s) t\u2082 \u2192 Has (generate s) (t\u2081 \u2229 t\u2082))) \u2264\n    ofMeasurableSpace (generateFrom s)", "state_after": "\u03b1 : Type u_1\nd : DynkinSystem \u03b1\ns : Set (Set \u03b1)\nhs : IsPiSystem s\n\u22a2 generate s \u2264 ofMeasurableSpace (generateFrom s)"}, {"tactic": "exact generate_le _ fun t ht => measurableSet_generateFrom ht", "annotated_tactic": ["exact <a>generate_le</a> _ fun t ht => <a>measurableSet_generateFrom</a> ht", [{"full_name": "MeasurableSpace.DynkinSystem.generate_le", "def_path": "Mathlib/MeasureTheory/PiSystem.lean", "def_pos": [704, 9], "def_end_pos": [704, 20]}, {"full_name": "MeasurableSpace.measurableSet_generateFrom", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [370, 9], "def_end_pos": [370, 35]}]], "state_before": "\u03b1 : Type u_1\nd : DynkinSystem \u03b1\ns : Set (Set \u03b1)\nhs : IsPiSystem s\n\u22a2 generate s \u2264 ofMeasurableSpace (generateFrom s)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/Basic.lean", "full_name": "MvPolynomial.eval\u2082_pow", "start": [1033, 1], "end": [1037, 78], "traced_tactics": [{"tactic": "rw [pow_zero, pow_zero]", "annotated_tactic": ["rw [<a>pow_zero</a>, <a>pow_zero</a>]", [{"full_name": "pow_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [639, 9], "def_end_pos": [639, 17]}, {"full_name": "pow_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [639, 9], "def_end_pos": [639, 17]}]], "state_before": "R : Type u\nS\u2081 : Type v\nS\u2082 : Type w\nS\u2083 : Type x\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : CommSemiring S\u2081\np\u271d q : MvPolynomial \u03c3 R\nf : R \u2192+* S\u2081\ng : \u03c3 \u2192 S\u2081\np : MvPolynomial \u03c3 R\n\u22a2 eval\u2082 f g (p ^ 0) = eval\u2082 f g p ^ 0", "state_after": "R : Type u\nS\u2081 : Type v\nS\u2082 : Type w\nS\u2083 : Type x\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : CommSemiring S\u2081\np\u271d q : MvPolynomial \u03c3 R\nf : R \u2192+* S\u2081\ng : \u03c3 \u2192 S\u2081\np : MvPolynomial \u03c3 R\n\u22a2 eval\u2082 f g 1 = 1"}, {"tactic": "exact eval\u2082_one _ _", "annotated_tactic": ["exact <a>eval\u2082_one</a> _ _", [{"full_name": "MvPolynomial.eval\u2082_one", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [993, 9], "def_end_pos": [993, 18]}]], "state_before": "R : Type u\nS\u2081 : Type v\nS\u2082 : Type w\nS\u2083 : Type x\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : CommSemiring S\u2081\np\u271d q : MvPolynomial \u03c3 R\nf : R \u2192+* S\u2081\ng : \u03c3 \u2192 S\u2081\np : MvPolynomial \u03c3 R\n\u22a2 eval\u2082 f g 1 = 1", "state_after": "no goals"}, {"tactic": "rw [pow_add, pow_one, pow_add, pow_one, eval\u2082_mul, eval\u2082_pow]", "annotated_tactic": ["rw [<a>pow_add</a>, <a>pow_one</a>, <a>pow_add</a>, <a>pow_one</a>, <a>eval\u2082_mul</a>, eval\u2082_pow]", [{"full_name": "pow_add", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [118, 9], "def_end_pos": [118, 16]}, {"full_name": "pow_one", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [97, 9], "def_end_pos": [97, 16]}, 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\u2191s)) \u2286 closure {y\u2080} \u222a closure (Subtype.val '' Set.range (denseSeq \u2191s))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/AssocList.lean", "full_name": "Std.AssocList.forIn_eq", "start": [215, 9], "end": [219, 38], "traced_tactics": [{"tactic": "simp [forIn, List.forIn]", "annotated_tactic": ["simp [<a>forIn</a>, <a>List.forIn</a>]", [{"full_name": "ForIn.forIn", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [248, 3], "def_end_pos": [248, 8]}, {"full_name": "List.forIn", "def_path": "lake-packages/lean4/src/lean/Init/Data/List/Control.lean", "def_pos": [141, 25], "def_end_pos": [141, 30]}]], "state_before": "m : Type u_1 \u2192 Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b4 : Type u_1\ninst\u271d : Monad m\nl : AssocList \u03b1 \u03b2\ninit : \u03b4\nf : \u03b1 \u00d7 \u03b2 \u2192 \u03b4 \u2192 m (ForInStep \u03b4)\n\u22a2 forIn l init f = forIn (toList l) init f", "state_after": "m : Type u_1 \u2192 Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b4 : Type u_1\ninst\u271d : Monad m\nl : AssocList \u03b1 \u03b2\ninit : \u03b4\nf : \u03b1 \u00d7 \u03b2 \u2192 \u03b4 \u2192 m (ForInStep \u03b4)\n\u22a2 AssocList.forIn l init f = List.forIn.loop f (toList l) init"}, {"tactic": "induction l generalizing init <;> simp [AssocList.forIn, List.forIn.loop]", "annotated_tactic": ["induction l generalizing init <;> simp [<a>AssocList.forIn</a>, <a>List.forIn.loop</a>]", [{"full_name": "Std.AssocList.forIn", "def_path": "lake-packages/std/Std/Data/AssocList.lean", "def_pos": [203, 29], "def_end_pos": [203, 34]}, {"full_name": "List.forIn.loop", "def_path": "lake-packages/lean4/src/lean/Init/Data/List/Control.lean", "def_pos": [142, 25], "def_end_pos": [142, 29]}]], "state_before": "m : Type u_1 \u2192 Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b4 : Type u_1\ninst\u271d : Monad m\nl : AssocList \u03b1 \u03b2\ninit : \u03b4\nf : \u03b1 \u00d7 \u03b2 \u2192 \u03b4 \u2192 m (ForInStep \u03b4)\n\u22a2 AssocList.forIn l init f = List.forIn.loop f (toList l) init", "state_after": "case cons\nm : Type u_1 \u2192 Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b4 : Type u_1\ninst\u271d : Monad m\nf : \u03b1 \u00d7 \u03b2 \u2192 \u03b4 \u2192 m (ForInStep \u03b4)\nkey\u271d : \u03b1\nvalue\u271d : \u03b2\ntail\u271d : AssocList \u03b1 \u03b2\ntail_ih\u271d : \u2200 (init : \u03b4), AssocList.forIn tail\u271d init f = List.forIn.loop f (toList tail\u271d) init\ninit : \u03b4\n\u22a2 (do\n      let __do_lift \u2190 f (key\u271d, value\u271d) init\n      match __do_lift with\n        | ForInStep.done d => pure d\n        | ForInStep.yield d => AssocList.forIn tail\u271d d f) =\n    do\n    let __do_lift \u2190 f (key\u271d, value\u271d) init\n    match __do_lift with\n      | ForInStep.done d => pure d\n      | ForInStep.yield b => List.forIn.loop f (toList tail\u271d) b"}, {"tactic": "congr", "annotated_tactic": ["congr", []], "state_before": "case cons\nm : Type u_1 \u2192 Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b4 : Type u_1\ninst\u271d : Monad m\nf : \u03b1 \u00d7 \u03b2 \u2192 \u03b4 \u2192 m (ForInStep \u03b4)\nkey\u271d : \u03b1\nvalue\u271d : \u03b2\ntail\u271d : AssocList \u03b1 \u03b2\ntail_ih\u271d : \u2200 (init : \u03b4), AssocList.forIn tail\u271d init f = List.forIn.loop f (toList tail\u271d) init\ninit : \u03b4\n\u22a2 (do\n      let __do_lift \u2190 f (key\u271d, value\u271d) init\n      match __do_lift with\n        | ForInStep.done d => pure d\n        | ForInStep.yield d => AssocList.forIn tail\u271d d f) =\n    do\n    let __do_lift \u2190 f (key\u271d, value\u271d) init\n    match __do_lift with\n      | ForInStep.done d => pure d\n      | ForInStep.yield b => List.forIn.loop f (toList tail\u271d) b", "state_after": "case cons.e_a\nm : Type u_1 \u2192 Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b4 : Type u_1\ninst\u271d : Monad m\nf : \u03b1 \u00d7 \u03b2 \u2192 \u03b4 \u2192 m (ForInStep \u03b4)\nkey\u271d : \u03b1\nvalue\u271d : \u03b2\ntail\u271d : AssocList \u03b1 \u03b2\ntail_ih\u271d : \u2200 (init : \u03b4), AssocList.forIn tail\u271d init f = List.forIn.loop f (toList tail\u271d) init\ninit : \u03b4\n\u22a2 (fun __do_lift =>\n      match __do_lift with\n      | ForInStep.done d => pure d\n      | ForInStep.yield d => AssocList.forIn tail\u271d d f) =\n    fun __do_lift =>\n    match __do_lift with\n    | ForInStep.done d => pure d\n    | ForInStep.yield b => List.forIn.loop f (toList tail\u271d) b"}, {"tactic": "funext a", "annotated_tactic": ["funext a", []], "state_before": "case cons.e_a\nm : Type u_1 \u2192 Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b4 : Type u_1\ninst\u271d : Monad m\nf : \u03b1 \u00d7 \u03b2 \u2192 \u03b4 \u2192 m (ForInStep \u03b4)\nkey\u271d : \u03b1\nvalue\u271d : \u03b2\ntail\u271d : AssocList \u03b1 \u03b2\ntail_ih\u271d : \u2200 (init : \u03b4), AssocList.forIn tail\u271d init f = List.forIn.loop f (toList tail\u271d) init\ninit : \u03b4\n\u22a2 (fun __do_lift =>\n      match __do_lift with\n      | ForInStep.done d => pure d\n      | ForInStep.yield d => AssocList.forIn tail\u271d d f) =\n    fun __do_lift =>\n    match __do_lift with\n    | ForInStep.done d => pure d\n    | ForInStep.yield b => List.forIn.loop f (toList tail\u271d) b", "state_after": "case cons.e_a.h\nm : Type u_1 \u2192 Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b4 : Type u_1\ninst\u271d : Monad m\nf : \u03b1 \u00d7 \u03b2 \u2192 \u03b4 \u2192 m (ForInStep \u03b4)\nkey\u271d : \u03b1\nvalue\u271d : \u03b2\ntail\u271d : AssocList \u03b1 \u03b2\ntail_ih\u271d : \u2200 (init : \u03b4), AssocList.forIn tail\u271d init f = List.forIn.loop f (toList tail\u271d) init\ninit : \u03b4\na : ForInStep \u03b4\n\u22a2 (match a with\n    | ForInStep.done d => pure d\n    | ForInStep.yield d => AssocList.forIn tail\u271d d f) =\n    match a with\n    | ForInStep.done d => pure d\n    | ForInStep.yield b => List.forIn.loop f (toList tail\u271d) b"}, {"tactic": "split <;> simp [*]", "annotated_tactic": ["split <;> simp [*]", []], "state_before": "case cons.e_a.h\nm : Type u_1 \u2192 Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b4 : Type u_1\ninst\u271d : Monad m\nf : \u03b1 \u00d7 \u03b2 \u2192 \u03b4 \u2192 m (ForInStep \u03b4)\nkey\u271d : \u03b1\nvalue\u271d : \u03b2\ntail\u271d : AssocList \u03b1 \u03b2\ntail_ih\u271d : \u2200 (init : \u03b4), AssocList.forIn tail\u271d init f = List.forIn.loop f (toList tail\u271d) init\ninit : \u03b4\na : ForInStep \u03b4\n\u22a2 (match a with\n    | ForInStep.done d => pure d\n    | ForInStep.yield d => AssocList.forIn tail\u271d d f) =\n    match a with\n    | ForInStep.done d => pure d\n    | ForInStep.yield b => List.forIn.loop f (toList tail\u271d) b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/Layercake.lean", "full_name": "MeasureTheory.lintegral_comp_eq_lintegral_meas_le_mul_of_measurable_of_sigmaFinite", "start": [105, 1], "end": [183, 28], "traced_tactics": [{"tactic": "have integrand_eq : \u2200 \u03c9,\n    ENNReal.ofReal (\u222b t in (0)..f \u03c9, g t) = (\u222b\u207b t in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)) := by\n  intro \u03c9\n  have g_ae_nn : 0 \u2264\u1d50[volume.restrict (Ioc 0 (f \u03c9))] g := by\n    filter_upwards [self_mem_ae_restrict (measurableSet_Ioc : MeasurableSet (Ioc 0 (f \u03c9)))]\n      with x hx using g_nn x hx.1\n  rw [\u2190 ofReal_integral_eq_lintegral_ofReal (g_intble' (f \u03c9) (f_nn \u03c9)).1 g_ae_nn]\n  congr\n  exact intervalIntegral.integral_of_le (f_nn \u03c9)", "annotated_tactic": ["have integrand_eq : \u2200 \u03c9,\n      <a>ENNReal.ofReal</a> (\u222b t in (0)..f \u03c9, g t) = (\u222b\u207b t in <a>Ioc</a> 0 (f \u03c9), <a>ENNReal.ofReal</a> (g t)) := by\n    intro \u03c9\n    have g_ae_nn : 0 \u2264\u1d50[volume.restrict (<a>Ioc</a> 0 (f \u03c9))] g := by\n      filter_upwards [<a>self_mem_ae_restrict</a> (<a>measurableSet_Ioc</a> : <a>MeasurableSet</a> (<a>Ioc</a> 0 (f \u03c9)))]\n        with x hx using g_nn x hx.1\n    rw [\u2190 <a>ofReal_integral_eq_lintegral_ofReal</a> (g_intble' (f \u03c9) (f_nn \u03c9)).1 g_ae_nn]\n    congr\n    exact <a>intervalIntegral.integral_of_le</a> (f_nn \u03c9)", [{"full_name": "ENNReal.ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [172, 29], "def_end_pos": [172, 35]}, {"full_name": "Set.Ioc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [69, 5], "def_end_pos": [69, 8]}, {"full_name": "ENNReal.ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [172, 29], "def_end_pos": [172, 35]}, {"full_name": "Set.Ioc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [69, 5], "def_end_pos": [69, 8]}, {"full_name": "MeasureTheory.self_mem_ae_restrict", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2681, 9], "def_end_pos": [2681, 29]}, {"full_name": "measurableSet_Ioc", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [589, 9], "def_end_pos": [589, 26]}, {"full_name": "MeasurableSet", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [64, 5], "def_end_pos": [64, 18]}, {"full_name": "Set.Ioc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [69, 5], "def_end_pos": [69, 8]}, {"full_name": "MeasureTheory.ofReal_integral_eq_lintegral_ofReal", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1219, 9], "def_end_pos": [1219, 44]}, {"full_name": "intervalIntegral.integral_of_le", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [465, 9], "def_end_pos": [465, 23]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\n\u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =\n    \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\n\u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =\n    \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)"}, {"tactic": "rw [lintegral_congr integrand_eq]", "annotated_tactic": ["rw [<a>lintegral_congr</a> integrand_eq]", [{"full_name": "MeasureTheory.lintegral_congr", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [308, 9], "def_end_pos": [308, 24]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\n\u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =\n    \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\n\u22a2 \u222b\u207b (a : \u03b1), \u222b\u207b (t : \u211d) in Ioc 0 (f a), ENNReal.ofReal (g t) \u2202\u03bc =\n    \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)"}, {"tactic": "simp_rw [\u2190 lintegral_indicator (fun t => ENNReal.ofReal (g t)) measurableSet_Ioc]", "annotated_tactic": ["simp_rw [\u2190 <a>lintegral_indicator</a> (fun t => <a>ENNReal.ofReal</a> (g t)) <a>measurableSet_Ioc</a>]", [{"full_name": "MeasureTheory.lintegral_indicator", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [762, 9], "def_end_pos": [762, 28]}, {"full_name": "ENNReal.ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [172, 29], "def_end_pos": [172, 35]}, {"full_name": "measurableSet_Ioc", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [589, 9], "def_end_pos": [589, 26]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\n\u22a2 \u222b\u207b (a : \u03b1), \u222b\u207b (t : \u211d) in Ioc 0 (f a), ENNReal.ofReal (g t) \u2202\u03bc =\n    \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\n\u22a2 \u222b\u207b (a : \u03b1), \u222b\u207b (a_1 : \u211d), indicator (Ioc 0 (f a)) (fun t => ENNReal.ofReal (g t)) a_1 \u2202\u03bc =\n    \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)"}, {"tactic": "rw [\u2190 lintegral_indicator _ measurableSet_Ioi, lintegral_lintegral_swap]", "annotated_tactic": ["rw [\u2190 <a>lintegral_indicator</a> _ <a>measurableSet_Ioi</a>, <a>lintegral_lintegral_swap</a>]", [{"full_name": "MeasureTheory.lintegral_indicator", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [762, 9], "def_end_pos": [762, 28]}, {"full_name": "measurableSet_Ioi", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [579, 9], "def_end_pos": [579, 26]}, {"full_name": "MeasureTheory.lintegral_lintegral_swap", "def_path": "Mathlib/MeasureTheory/Constructions/Prod/Basic.lean", "def_pos": [891, 9], "def_end_pos": [891, 33]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\n\u22a2 \u222b\u207b (a : \u03b1), \u222b\u207b (a_1 : \u211d), indicator (Ioc 0 (f a)) (fun t => ENNReal.ofReal (g t)) a_1 \u2202\u03bc =\n    \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\n\u22a2 \u222b\u207b (y : \u211d), \u222b\u207b (x : \u03b1), indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) y \u2202\u03bc =\n    \u222b\u207b (a : \u211d), indicator (Ioi 0) (fun t => \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)) a\n\ncase hf\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\n\u22a2 AEMeasurable (Function.uncurry fun a a_1 => indicator (Ioc 0 (f a)) (fun t => ENNReal.ofReal (g t)) a_1)"}, {"tactic": "rw [aux\u2082]", "annotated_tactic": ["rw [aux\u2082]", []], "state_before": "case hf\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\naux\u2082 :\n  (Function.uncurry fun x y => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) y) =\n    indicator {p | p.2 \u2208 Ioc 0 (f p.1)} fun p => ENNReal.ofReal (g p.2)\n\u22a2 AEMeasurable (Function.uncurry fun a a_1 => indicator (Ioc 0 (f a)) (fun t => ENNReal.ofReal (g t)) a_1)", "state_after": "case hf\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\naux\u2082 :\n  (Function.uncurry fun x y => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) y) =\n    indicator {p | p.2 \u2208 Ioc 0 (f p.1)} fun p => ENNReal.ofReal (g p.2)\n\u22a2 AEMeasurable (indicator {p | p.2 \u2208 Ioc 0 (f p.1)} fun p => ENNReal.ofReal (g p.2))"}, {"tactic": "have mble\u2080 : MeasurableSet {p : \u03b1 \u00d7 \u211d | p.snd \u2208 Ioc 0 (f p.fst)} := by\n  simpa only [mem_univ, Pi.zero_apply, gt_iff_lt, not_lt, ge_iff_le, true_and] using\n    measurableSet_region_between_oc measurable_zero f_mble  MeasurableSet.univ", "annotated_tactic": ["have mble\u2080 : <a>MeasurableSet</a> {p : \u03b1 \u00d7 \u211d | p.snd \u2208 <a>Ioc</a> 0 (f p.fst)} := by\n    simpa only [<a>mem_univ</a>, <a>Pi.zero_apply</a>, <a>gt_iff_lt</a>, <a>not_lt</a>, <a>ge_iff_le</a>, <a>true_and</a>] using\n      <a>measurableSet_region_between_oc</a> <a>measurable_zero</a> f_mble  <a>MeasurableSet.univ</a>", [{"full_name": "MeasurableSet", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [64, 5], "def_end_pos": [64, 18]}, {"full_name": "Set.Ioc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [69, 5], "def_end_pos": [69, 8]}, {"full_name": "Set.mem_univ", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [676, 9], "def_end_pos": [676, 17]}, {"full_name": "Pi.zero_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [46, 3], "def_end_pos": [46, 14]}, {"full_name": "gt_iff_lt", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [366, 9], "def_end_pos": [366, 18]}, {"full_name": "not_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [368, 9], "def_end_pos": [368, 15]}, {"full_name": "ge_iff_le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [359, 9], "def_end_pos": [359, 18]}, {"full_name": "true_and", "def_path": "lake-packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [84, 17], "def_end_pos": [84, 25]}, {"full_name": "measurableSet_region_between_oc", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean", "def_pos": [489, 9], "def_end_pos": [489, 40]}, {"full_name": "measurable_zero", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [270, 18], "def_end_pos": [270, 29]}, {"full_name": "MeasurableSet.univ", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [101, 19], "def_end_pos": [101, 37]}]], "state_before": "case hf\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\naux\u2082 :\n  (Function.uncurry fun x y => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) y) =\n    indicator {p | p.2 \u2208 Ioc 0 (f p.1)} fun p => ENNReal.ofReal (g p.2)\n\u22a2 AEMeasurable (indicator {p | p.2 \u2208 Ioc 0 (f p.1)} fun p => ENNReal.ofReal (g p.2))", "state_after": "case hf\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\naux\u2082 :\n  (Function.uncurry fun x y => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) y) =\n    indicator {p | p.2 \u2208 Ioc 0 (f p.1)} fun p => ENNReal.ofReal (g p.2)\nmble\u2080 : MeasurableSet {p | p.2 \u2208 Ioc 0 (f p.1)}\n\u22a2 AEMeasurable (indicator {p | p.2 \u2208 Ioc 0 (f p.1)} fun p => ENNReal.ofReal (g p.2))"}, {"tactic": "exact (ENNReal.measurable_ofReal.comp (g_mble.comp measurable_snd)).aemeasurable.indicator\u2080\n  mble\u2080.nullMeasurableSet", "annotated_tactic": ["exact (ENNReal.measurable_ofReal.comp (g_mble.comp <a>measurable_snd</a>)).aemeasurable.indicator\u2080\n    mble\u2080.nullMeasurableSet", [{"full_name": "measurable_snd", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [698, 9], "def_end_pos": [698, 23]}]], "state_before": "case hf\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\naux\u2082 :\n  (Function.uncurry fun x y => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) y) =\n    indicator {p | p.2 \u2208 Ioc 0 (f p.1)} fun p => ENNReal.ofReal (g p.2)\nmble\u2080 : MeasurableSet {p | p.2 \u2208 Ioc 0 (f p.1)}\n\u22a2 AEMeasurable (indicator {p | p.2 \u2208 Ioc 0 (f p.1)} fun p => ENNReal.ofReal (g p.2))", "state_after": "no goals"}, {"tactic": "intro t ht", "annotated_tactic": ["intro t ht", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\n\u22a2 \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nt : \u211d\nht : 0 \u2264 t\n\u22a2 IntervalIntegrable g volume 0 t"}, {"tactic": "cases' eq_or_lt_of_le ht with h h", "annotated_tactic": ["cases' <a>eq_or_lt_of_le</a> ht with h h", [{"full_name": "eq_or_lt_of_le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [414, 9], "def_end_pos": [414, 23]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nt : \u211d\nht : 0 \u2264 t\n\u22a2 IntervalIntegrable g volume 0 t", "state_after": "case inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nt : \u211d\nht : 0 \u2264 t\nh : 0 = t\n\u22a2 IntervalIntegrable g volume 0 t\n\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nt : \u211d\nht : 0 \u2264 t\nh : 0 < t\n\u22a2 IntervalIntegrable g volume 0 t"}, {"tactic": "simp [\u2190 h]", "annotated_tactic": ["simp [\u2190 h]", []], "state_before": "case inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nt : \u211d\nht : 0 \u2264 t\nh : 0 = t\n\u22a2 IntervalIntegrable g volume 0 t", "state_after": "no goals"}, {"tactic": "exact g_intble t h", "annotated_tactic": ["exact g_intble t h", []], "state_before": "case inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nt : \u211d\nht : 0 \u2264 t\nh : 0 < t\n\u22a2 IntervalIntegrable g volume 0 t", "state_after": "no goals"}, {"tactic": "intro \u03c9", "annotated_tactic": ["intro \u03c9", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\n\u22a2 \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\n\u03c9 : \u03b1\n\u22a2 ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)"}, {"tactic": "have g_ae_nn : 0 \u2264\u1d50[volume.restrict (Ioc 0 (f \u03c9))] g := by\n  filter_upwards [self_mem_ae_restrict (measurableSet_Ioc : MeasurableSet (Ioc 0 (f \u03c9)))]\n    with x hx using g_nn x hx.1", "annotated_tactic": ["have g_ae_nn : 0 \u2264\u1d50[volume.restrict (<a>Ioc</a> 0 (f \u03c9))] g := by\n      filter_upwards [<a>self_mem_ae_restrict</a> (<a>measurableSet_Ioc</a> : <a>MeasurableSet</a> (<a>Ioc</a> 0 (f \u03c9)))]\n        with x hx using g_nn x hx.1", [{"full_name": "Set.Ioc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [69, 5], "def_end_pos": [69, 8]}, {"full_name": "MeasureTheory.self_mem_ae_restrict", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2681, 9], "def_end_pos": [2681, 29]}, {"full_name": "measurableSet_Ioc", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [589, 9], "def_end_pos": [589, 26]}, {"full_name": "MeasurableSet", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [64, 5], "def_end_pos": [64, 18]}, {"full_name": "Set.Ioc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [69, 5], "def_end_pos": [69, 8]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\n\u03c9 : \u03b1\n\u22a2 ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\n\u03c9 : \u03b1\ng_ae_nn : 0 \u2264\u1da0[ae (Measure.restrict volume (Ioc 0 (f \u03c9)))] g\n\u22a2 ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)"}, {"tactic": "rw [\u2190 ofReal_integral_eq_lintegral_ofReal (g_intble' (f \u03c9) (f_nn \u03c9)).1 g_ae_nn]", "annotated_tactic": ["rw [\u2190 <a>ofReal_integral_eq_lintegral_ofReal</a> (g_intble' (f \u03c9) (f_nn \u03c9)).1 g_ae_nn]", [{"full_name": "MeasureTheory.ofReal_integral_eq_lintegral_ofReal", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1219, 9], "def_end_pos": [1219, 44]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\n\u03c9 : \u03b1\ng_ae_nn : 0 \u2264\u1da0[ae (Measure.restrict volume (Ioc 0 (f \u03c9)))] g\n\u22a2 ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\n\u03c9 : \u03b1\ng_ae_nn : 0 \u2264\u1da0[ae (Measure.restrict volume (Ioc 0 (f \u03c9)))] g\n\u22a2 ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = ENNReal.ofReal (\u222b (x : \u211d) in Ioc 0 (f \u03c9), g x)"}, {"tactic": "congr", "annotated_tactic": ["congr", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\n\u03c9 : \u03b1\ng_ae_nn : 0 \u2264\u1da0[ae (Measure.restrict volume (Ioc 0 (f \u03c9)))] g\n\u22a2 ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = ENNReal.ofReal (\u222b (x : \u211d) in Ioc 0 (f \u03c9), g x)", "state_after": "case e_r\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\n\u03c9 : \u03b1\ng_ae_nn : 0 \u2264\u1da0[ae (Measure.restrict volume (Ioc 0 (f \u03c9)))] g\n\u22a2 \u222b (t : \u211d) in 0 ..f \u03c9, g t = \u222b (x : \u211d) in Ioc 0 (f \u03c9), g x"}, {"tactic": "exact intervalIntegral.integral_of_le (f_nn \u03c9)", "annotated_tactic": ["exact <a>intervalIntegral.integral_of_le</a> (f_nn \u03c9)", [{"full_name": "intervalIntegral.integral_of_le", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [465, 9], "def_end_pos": [465, 23]}]], "state_before": "case e_r\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\n\u03c9 : \u03b1\ng_ae_nn : 0 \u2264\u1da0[ae (Measure.restrict volume (Ioc 0 (f \u03c9)))] g\n\u22a2 \u222b (t : \u211d) in 0 ..f \u03c9, g t = \u222b (x : \u211d) in Ioc 0 (f \u03c9), g x", "state_after": "no goals"}, {"tactic": "filter_upwards [self_mem_ae_restrict (measurableSet_Ioc : MeasurableSet (Ioc 0 (f \u03c9)))]\n  with x hx using g_nn x hx.1", "annotated_tactic": ["filter_upwards [<a>self_mem_ae_restrict</a> (<a>measurableSet_Ioc</a> : <a>MeasurableSet</a> (<a>Ioc</a> 0 (f \u03c9)))]\n        with x hx using g_nn x hx.1", [{"full_name": "MeasureTheory.self_mem_ae_restrict", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2681, 9], "def_end_pos": [2681, 29]}, {"full_name": "measurableSet_Ioc", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [589, 9], "def_end_pos": [589, 26]}, {"full_name": "MeasurableSet", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [64, 5], "def_end_pos": [64, 18]}, {"full_name": "Set.Ioc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [69, 5], "def_end_pos": [69, 8]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\n\u03c9 : \u03b1\n\u22a2 0 \u2264\u1da0[ae (Measure.restrict volume (Ioc 0 (f \u03c9)))] g", "state_after": "no goals"}, {"tactic": "apply congr_arg", "annotated_tactic": ["apply <a>congr_arg</a>", [{"full_name": "congr_arg", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [43, 7], "def_end_pos": [43, 16]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\n\u22a2 \u222b\u207b (y : \u211d), \u222b\u207b (x : \u03b1), indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) y \u2202\u03bc =\n    \u222b\u207b (a : \u211d), indicator (Ioi 0) (fun t => \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)) a", "state_after": "case h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\n\u22a2 (fun y => \u222b\u207b (x : \u03b1), indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) y \u2202\u03bc) = fun a =>\n    indicator (Ioi 0) (fun t => \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)) a"}, {"tactic": "funext s", "annotated_tactic": ["funext s", []], "state_before": "case h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\n\u22a2 (fun y => \u222b\u207b (x : \u03b1), indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) y \u2202\u03bc) = fun a =>\n    indicator (Ioi 0) (fun t => \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)) a", "state_after": "case h.h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\ns : \u211d\n\u22a2 \u222b\u207b (x : \u03b1), indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s \u2202\u03bc =\n    indicator (Ioi 0) (fun t => \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)) s"}, {"tactic": "simp_rw [aux\u2081]", "annotated_tactic": ["simp_rw [aux\u2081]", []], "state_before": "case h.h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\ns : \u211d\naux\u2081 :\n  (fun x => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s) = fun x =>\n    ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x)\n\u22a2 \u222b\u207b (x : \u03b1), indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s \u2202\u03bc =\n    indicator (Ioi 0) (fun t => \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)) s", "state_after": "case h.h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\ns : \u211d\naux\u2081 :\n  (fun x => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s) = fun x =>\n    ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x)\n\u22a2 \u222b\u207b (x : \u03b1), ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x) \u2202\u03bc =\n    indicator (Ioi 0) (fun t => \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)) s"}, {"tactic": "rw [lintegral_const_mul']", "annotated_tactic": ["rw [<a>lintegral_const_mul'</a>]", [{"full_name": "MeasureTheory.lintegral_const_mul'", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [711, 9], "def_end_pos": [711, 29]}]], "state_before": "case h.h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\ns : \u211d\naux\u2081 :\n  (fun x => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s) = fun x =>\n    ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x)\n\u22a2 \u222b\u207b (x : \u03b1), ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x) \u2202\u03bc =\n    indicator (Ioi 0) (fun t => \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)) s", "state_after": "case h.h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\ns : \u211d\naux\u2081 :\n  (fun x => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s) = fun x =>\n    ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x)\n\u22a2 ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * \u222b\u207b (a : \u03b1), indicator (Ici s) (fun x => 1) (f a) \u2202\u03bc =\n    indicator (Ioi 0) (fun t => \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)) s\n\ncase h.h.hr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\ns : \u211d\naux\u2081 :\n  (fun x => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s) = fun x =>\n    ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x)\n\u22a2 ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s \u2260 \u22a4"}, {"tactic": "swap", "annotated_tactic": ["swap", []], "state_before": "case h.h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\ns : \u211d\naux\u2081 :\n  (fun x => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s) = fun x =>\n    ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x)\n\u22a2 ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * \u222b\u207b (a : \u03b1), indicator (Ici s) (fun x => 1) (f a) \u2202\u03bc =\n    indicator (Ioi 0) (fun t => \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)) s\n\ncase h.h.hr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\ns : \u211d\naux\u2081 :\n  (fun x => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s) = fun x =>\n    ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x)\n\u22a2 ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s \u2260 \u22a4", "state_after": "case h.h.hr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\ns : \u211d\naux\u2081 :\n  (fun x => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s) = fun x =>\n    ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x)\n\u22a2 ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s \u2260 \u22a4\n\ncase h.h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\ns : \u211d\naux\u2081 :\n  (fun x => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s) = fun x =>\n    ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x)\n\u22a2 ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * \u222b\u207b (a : \u03b1), indicator (Ici s) (fun x => 1) (f a) \u2202\u03bc =\n    indicator (Ioi 0) (fun t => \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)) s"}, {"tactic": "simp_rw [show\n    (fun a => (Ici s).indicator (fun _ : \u211d => (1 : \u211d\u22650\u221e)) (f a)) = fun a =>\n      {a : \u03b1 | s \u2264 f a}.indicator (fun _ => 1) a\n    by funext a; by_cases s \u2264 f a <;> simp [h]]", "annotated_tactic": ["simp_rw [show\n        (fun a => (<a>Ici</a> s).<a>indicator</a> (fun _ : \u211d => (1 : \u211d\u22650\u221e)) (f a)) = fun a =>\n          {a : \u03b1 | s \u2264 f a}.<a>indicator</a> (fun _ => 1) a\n        by funext a; by_cases s \u2264 f a <;> simp [h]]", [{"full_name": "Set.Ici", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [74, 5], "def_end_pos": [74, 8]}, {"full_name": "Set.indicator", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [46, 3], "def_end_pos": [46, 14]}, {"full_name": "Set.indicator", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [46, 3], "def_end_pos": [46, 14]}]], "state_before": "case h.h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\ns : \u211d\naux\u2081 :\n  (fun x => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s) = fun x =>\n    ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x)\n\u22a2 ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * \u222b\u207b (a : \u03b1), indicator (Ici s) (fun x => 1) (f a) \u2202\u03bc =\n    indicator (Ioi 0) (fun t => \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)) s", "state_after": "case h.h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\ns : \u211d\naux\u2081 :\n  (fun x => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s) = fun x =>\n    ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x)\n\u22a2 ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * \u222b\u207b (a : \u03b1), indicator {a | s \u2264 f a} (fun x => 1) a \u2202\u03bc =\n    indicator (Ioi 0) (fun t => \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)) s"}, {"tactic": "rw [lintegral_indicator\u2080]", "annotated_tactic": ["rw [<a>lintegral_indicator\u2080</a>]", [{"full_name": "MeasureTheory.lintegral_indicator\u2080", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [776, 9], "def_end_pos": [776, 29]}]], "state_before": "case h.h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\ns : \u211d\naux\u2081 :\n  (fun x => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s) = fun x =>\n    ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x)\n\u22a2 ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * \u222b\u207b (a : \u03b1), indicator {a | s \u2264 f a} (fun x => 1) a \u2202\u03bc =\n    indicator (Ioi 0) (fun t => \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)) s", "state_after": "case h.h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\ns : \u211d\naux\u2081 :\n  (fun x => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s) = fun x =>\n    ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x)\n\u22a2 ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * \u222b\u207b (a : \u03b1) in {a | s \u2264 f a}, 1 \u2202\u03bc =\n    indicator (Ioi 0) (fun t => \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)) s\n\ncase h.h.hs\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\ns : \u211d\naux\u2081 :\n  (fun x => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s) = fun x =>\n    ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x)\n\u22a2 NullMeasurableSet {a | s \u2264 f a}"}, {"tactic": "swap", "annotated_tactic": ["swap", []], "state_before": "case h.h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\ns : \u211d\naux\u2081 :\n  (fun x => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s) = fun x =>\n    ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x)\n\u22a2 ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * \u222b\u207b (a : \u03b1) in {a | s \u2264 f a}, 1 \u2202\u03bc =\n    indicator (Ioi 0) (fun t => \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)) s\n\ncase h.h.hs\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\ns : \u211d\naux\u2081 :\n  (fun x => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s) = fun x =>\n    ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x)\n\u22a2 NullMeasurableSet {a | s \u2264 f a}", "state_after": "case h.h.hs\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\ns : \u211d\naux\u2081 :\n  (fun x => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s) = fun x =>\n    ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x)\n\u22a2 NullMeasurableSet {a | s \u2264 f a}\n\ncase h.h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\ns : \u211d\naux\u2081 :\n  (fun x => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s) = fun x =>\n    ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x)\n\u22a2 ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * \u222b\u207b (a : \u03b1) in {a | s \u2264 f a}, 1 \u2202\u03bc =\n    indicator (Ioi 0) (fun t => \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)) s"}, {"tactic": "rw [lintegral_one, Measure.restrict_apply MeasurableSet.univ, univ_inter, indicator_mul_left,\n  mul_assoc,\n  show\n    (Ioi 0).indicator (fun _x : \u211d => (1 : \u211d\u22650\u221e)) s * \u03bc {a : \u03b1 | s \u2264 f a} =\n      (Ioi 0).indicator (fun _x : \u211d => 1 * \u03bc {a : \u03b1 | s \u2264 f a}) s\n    by by_cases 0 < s <;> simp [h]]", "annotated_tactic": ["rw [<a>lintegral_one</a>, <a>Measure.restrict_apply</a> <a>MeasurableSet.univ</a>, <a>univ_inter</a>, <a>indicator_mul_left</a>,\n      <a>mul_assoc</a>,\n      show\n        (<a>Ioi</a> 0).<a>indicator</a> (fun _x : \u211d => (1 : \u211d\u22650\u221e)) s * \u03bc {a : \u03b1 | s \u2264 f a} =\n          (<a>Ioi</a> 0).<a>indicator</a> (fun _x : \u211d => 1 * \u03bc {a : \u03b1 | s \u2264 f a}) s\n        by by_cases 0 < s <;> simp [h]]", [{"full_name": "MeasureTheory.lintegral_one", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [149, 9], "def_end_pos": [149, 22]}, {"full_name": "MeasureTheory.Measure.restrict_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1533, 9], "def_end_pos": [1533, 23]}, {"full_name": "MeasurableSet.univ", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [101, 19], "def_end_pos": [101, 37]}, {"full_name": "Set.univ_inter", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1017, 9], "def_end_pos": [1017, 19]}, {"full_name": "Set.indicator_mul_left", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [714, 9], "def_end_pos": [714, 27]}, {"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [264, 9], "def_end_pos": [264, 18]}, {"full_name": "Set.Ioi", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [79, 5], "def_end_pos": [79, 8]}, {"full_name": "Set.indicator", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [46, 3], "def_end_pos": [46, 14]}, {"full_name": "Set.Ioi", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [79, 5], "def_end_pos": [79, 8]}, {"full_name": "Set.indicator", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [46, 3], "def_end_pos": [46, 14]}]], "state_before": "case h.h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\ns : \u211d\naux\u2081 :\n  (fun x => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s) = fun x =>\n    ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x)\n\u22a2 ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * \u222b\u207b (a : \u03b1) in {a | s \u2264 f a}, 1 \u2202\u03bc =\n    indicator (Ioi 0) (fun t => \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)) s", "state_after": "case h.h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\ns : \u211d\naux\u2081 :\n  (fun x => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s) = fun x =>\n    ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x)\n\u22a2 ENNReal.ofReal (g s) * indicator (Ioi 0) (fun _x => 1 * \u2191\u2191\u03bc {a | s \u2264 f a}) s =\n    indicator (Ioi 0) (fun t => \u2191\u2191\u03bc {a | t \u2264 f a}) s * ENNReal.ofReal (g s)"}, {"tactic": "simp_rw [mul_comm _ (ENNReal.ofReal _), one_mul]", "annotated_tactic": ["simp_rw [<a>mul_comm</a> _ (<a>ENNReal.ofReal</a> _), <a>one_mul</a>]", [{"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [302, 9], "def_end_pos": [302, 17]}, {"full_name": "ENNReal.ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [172, 29], "def_end_pos": [172, 35]}, {"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [464, 9], "def_end_pos": [464, 16]}]], "state_before": "case h.h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\ns : \u211d\naux\u2081 :\n  (fun x => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s) = fun x =>\n    ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x)\n\u22a2 ENNReal.ofReal (g s) * indicator (Ioi 0) (fun _x => 1 * \u2191\u2191\u03bc {a | s \u2264 f a}) s =\n    indicator (Ioi 0) (fun t => \u2191\u2191\u03bc {a | t \u2264 f a}) s * ENNReal.ofReal (g s)", "state_after": "case h.h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\ns : \u211d\naux\u2081 :\n  (fun x => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s) = fun x =>\n    ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x)\n\u22a2 ENNReal.ofReal (g s) * indicator (Ioi 0) (fun _x => \u2191\u2191\u03bc {a | s \u2264 f a}) s =\n    ENNReal.ofReal (g s) * indicator (Ioi 0) (fun t => \u2191\u2191\u03bc {a | t \u2264 f a}) s"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case h.h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\ns : \u211d\naux\u2081 :\n  (fun x => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s) = fun x =>\n    ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x)\n\u22a2 ENNReal.ofReal (g s) * indicator (Ioi 0) (fun _x => \u2191\u2191\u03bc {a | s \u2264 f a}) s =\n    ENNReal.ofReal (g s) * indicator (Ioi 0) (fun t => \u2191\u2191\u03bc {a | t \u2264 f a}) s", "state_after": "no goals"}, {"tactic": "funext a", "annotated_tactic": ["funext a", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\ns : \u211d\n\u22a2 (fun x => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s) = fun x =>\n    ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x)", "state_after": "case h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\ns : \u211d\na : \u03b1\n\u22a2 indicator (Ioc 0 (f a)) (fun t => ENNReal.ofReal (g t)) s =\n    ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f a)"}, {"tactic": "by_cases s \u2208 Ioc (0 : \u211d) (f a)", "annotated_tactic": ["by_cases s \u2208 <a>Ioc</a> (0 : \u211d) (f a)", [{"full_name": "Set.Ioc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [69, 5], "def_end_pos": [69, 8]}]], "state_before": "case h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\ns : \u211d\na : \u03b1\n\u22a2 indicator (Ioc 0 (f a)) (fun t => ENNReal.ofReal (g t)) s =\n    ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f a)", "state_after": "case pos\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\ns : \u211d\na : \u03b1\nh : s \u2208 Ioc 0 (f a)\n\u22a2 indicator (Ioc 0 (f a)) (fun t => ENNReal.ofReal (g t)) s =\n    ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f a)\n\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\ns : \u211d\na : \u03b1\nh : \u00acs \u2208 Ioc 0 (f a)\n\u22a2 indicator (Ioc 0 (f a)) (fun t => ENNReal.ofReal (g t)) s =\n    ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f a)"}, {"tactic": "simp only [h, show s \u2208 Ioi (0 : \u211d) from h.1, show f a \u2208 Ici s from h.2, indicator_of_mem,\n  mul_one]", "annotated_tactic": ["simp only [h, show s \u2208 <a>Ioi</a> (0 : \u211d) from h.1, show f a \u2208 <a>Ici</a> s from h.2, <a>indicator_of_mem</a>,\n          <a>mul_one</a>]", [{"full_name": "Set.Ioi", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [79, 5], "def_end_pos": [79, 8]}, {"full_name": "Set.Ici", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [74, 5], "def_end_pos": [74, 8]}, {"full_name": "Set.indicator_of_mem", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [67, 3], "def_end_pos": [67, 14]}, {"full_name": "mul_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [470, 9], "def_end_pos": [470, 16]}]], "state_before": "case pos\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\ns : \u211d\na : \u03b1\nh : s \u2208 Ioc 0 (f a)\n\u22a2 indicator (Ioc 0 (f a)) (fun t => ENNReal.ofReal (g t)) s =\n    ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f a)", "state_after": "no goals"}, {"tactic": "have h_copy := h", "annotated_tactic": ["have h_copy := h", []], "state_before": "case neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\ns : \u211d\na : \u03b1\nh : \u00acs \u2208 Ioc 0 (f a)\n\u22a2 indicator (Ioc 0 (f a)) (fun t => ENNReal.ofReal (g t)) s =\n    ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f a)", "state_after": "case neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\ns : \u211d\na : \u03b1\nh h_copy : \u00acs \u2208 Ioc 0 (f a)\n\u22a2 indicator (Ioc 0 (f a)) (fun t => ENNReal.ofReal (g t)) s =\n    ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f a)"}, {"tactic": "simp only [mem_Ioc, not_and, not_le] at h", "annotated_tactic": ["simp only [<a>mem_Ioc</a>, <a>not_and</a>, <a>not_le</a>] at h", [{"full_name": "Set.mem_Ioc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [141, 9], "def_end_pos": [141, 16]}, {"full_name": "not_and", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [316, 17], "def_end_pos": [316, 24]}, {"full_name": "not_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [373, 9], "def_end_pos": [373, 15]}]], "state_before": "case neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\ns : \u211d\na : \u03b1\nh h_copy : \u00acs \u2208 Ioc 0 (f a)\n\u22a2 indicator (Ioc 0 (f a)) (fun t => ENNReal.ofReal (g t)) s =\n    ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f a)", "state_after": "case neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\ns : \u211d\na : \u03b1\nh_copy : \u00acs \u2208 Ioc 0 (f a)\nh : 0 < s \u2192 f a < s\n\u22a2 indicator (Ioc 0 (f a)) (fun t => ENNReal.ofReal (g t)) s =\n    ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f a)"}, {"tactic": "by_cases h' : 0 < s", "annotated_tactic": ["by_cases h' : 0 < s", []], "state_before": "case neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\ns : \u211d\na : \u03b1\nh_copy : \u00acs \u2208 Ioc 0 (f a)\nh : 0 < s \u2192 f a < s\n\u22a2 indicator (Ioc 0 (f a)) (fun t => ENNReal.ofReal (g t)) s =\n    ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f a)", "state_after": "case pos\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\ns : \u211d\na : \u03b1\nh_copy : \u00acs \u2208 Ioc 0 (f a)\nh : 0 < s \u2192 f a < s\nh' : 0 < s\n\u22a2 indicator (Ioc 0 (f a)) (fun t => ENNReal.ofReal (g t)) s =\n    ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f a)\n\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\ns : \u211d\na : \u03b1\nh_copy : \u00acs \u2208 Ioc 0 (f a)\nh : 0 < s \u2192 f a < s\nh' : \u00ac0 < s\n\u22a2 indicator (Ioc 0 (f a)) (fun t => ENNReal.ofReal (g t)) s =\n    ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f a)"}, {"tactic": "simp only [h_copy, h h', indicator_of_not_mem, not_false_iff, mem_Ici, not_le, mul_zero]", "annotated_tactic": ["simp only [h_copy, h h', <a>indicator_of_not_mem</a>, <a>not_false_iff</a>, <a>mem_Ici</a>, <a>not_le</a>, <a>mul_zero</a>]", [{"full_name": "Set.indicator_of_not_mem", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [73, 3], "def_end_pos": [73, 14]}, {"full_name": "not_false_iff", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [82, 9], "def_end_pos": [82, 22]}, {"full_name": "Set.mem_Ici", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [146, 9], "def_end_pos": [146, 16]}, {"full_name": "not_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [373, 9], "def_end_pos": [373, 15]}, {"full_name": "MulZeroClass.mul_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [38, 3], "def_end_pos": [38, 11]}]], "state_before": "case pos\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\ns : \u211d\na : \u03b1\nh_copy : \u00acs \u2208 Ioc 0 (f a)\nh : 0 < s \u2192 f a < s\nh' : 0 < s\n\u22a2 indicator (Ioc 0 (f a)) (fun t => ENNReal.ofReal (g t)) s =\n    ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f a)", "state_after": "no goals"}, {"tactic": "have : s \u2209 Ioi (0 : \u211d) := h'", "annotated_tactic": ["have : s \u2209 <a>Ioi</a> (0 : \u211d) := h'", [{"full_name": "Set.Ioi", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [79, 5], "def_end_pos": [79, 8]}]], "state_before": "case neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\ns : \u211d\na : \u03b1\nh_copy : \u00acs \u2208 Ioc 0 (f a)\nh : 0 < s \u2192 f a < s\nh' : \u00ac0 < s\n\u22a2 indicator (Ioc 0 (f a)) (fun t => ENNReal.ofReal (g t)) s =\n    ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f a)", "state_after": "case neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\ns : \u211d\na : \u03b1\nh_copy : \u00acs \u2208 Ioc 0 (f a)\nh : 0 < s \u2192 f a < s\nh' : \u00ac0 < s\nthis : \u00acs \u2208 Ioi 0\n\u22a2 indicator (Ioc 0 (f a)) (fun t => ENNReal.ofReal (g t)) s =\n    ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f a)"}, {"tactic": "simp only [this, h', indicator_of_not_mem, not_false_iff, mul_zero,\n  zero_mul, mem_Ioc, false_and_iff]", "annotated_tactic": ["simp only [this, h', <a>indicator_of_not_mem</a>, <a>not_false_iff</a>, <a>mul_zero</a>,\n            <a>zero_mul</a>, <a>mem_Ioc</a>, <a>false_and_iff</a>]", [{"full_name": "Set.indicator_of_not_mem", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [73, 3], "def_end_pos": [73, 14]}, {"full_name": "not_false_iff", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [82, 9], "def_end_pos": [82, 22]}, {"full_name": "MulZeroClass.mul_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [38, 3], "def_end_pos": [38, 11]}, {"full_name": "MulZeroClass.zero_mul", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [36, 3], "def_end_pos": [36, 11]}, {"full_name": "Set.mem_Ioc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [141, 9], "def_end_pos": [141, 16]}, {"full_name": "false_and_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [151, 9], "def_end_pos": [151, 22]}]], "state_before": "case neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\ns : \u211d\na : \u03b1\nh_copy : \u00acs \u2208 Ioc 0 (f a)\nh : 0 < s \u2192 f a < s\nh' : \u00ac0 < s\nthis : \u00acs \u2208 Ioi 0\n\u22a2 indicator (Ioc 0 (f a)) (fun t => ENNReal.ofReal (g t)) s =\n    ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f a)", "state_after": "no goals"}, {"tactic": "apply ENNReal.mul_ne_top ENNReal.ofReal_ne_top", "annotated_tactic": ["apply <a>ENNReal.mul_ne_top</a> <a>ENNReal.ofReal_ne_top</a>", [{"full_name": "ENNReal.mul_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [615, 9], "def_end_pos": [615, 19]}, {"full_name": "ENNReal.ofReal_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [311, 17], "def_end_pos": [311, 30]}]], "state_before": "case h.h.hr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\ns : \u211d\naux\u2081 :\n  (fun x => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s) = fun x =>\n    ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x)\n\u22a2 ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s \u2260 \u22a4", "state_after": "case h.h.hr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\ns : \u211d\naux\u2081 :\n  (fun x => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s) = fun x =>\n    ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x)\n\u22a2 indicator (Ioi 0) (fun x => 1) s \u2260 \u22a4"}, {"tactic": "simp [h]", "annotated_tactic": ["simp [h]", []], "state_before": "case neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\ns : \u211d\naux\u2081 :\n  (fun x => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s) = fun x =>\n    ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x)\nh : \u00ac0 < s\n\u22a2 indicator (Ioi 0) (fun x => 1) s \u2260 \u22a4", "state_after": "no goals"}, {"tactic": "funext a", "annotated_tactic": ["funext a", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\ns : \u211d\naux\u2081 :\n  (fun x => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s) = fun x =>\n    ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x)\n\u22a2 (fun a => indicator (Ici s) (fun x => 1) (f a)) = fun a => indicator {a | s \u2264 f a} (fun x => 1) a", "state_after": "case h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\ns : \u211d\naux\u2081 :\n  (fun x => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s) = fun x =>\n    ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x)\na : \u03b1\n\u22a2 indicator (Ici s) (fun x => 1) (f a) = indicator {a | s \u2264 f a} (fun x => 1) a"}, {"tactic": "by_cases s \u2264 f a <;> simp [h]", "annotated_tactic": ["by_cases s \u2264 f a <;> simp [h]", []], "state_before": "case h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\ns : \u211d\naux\u2081 :\n  (fun x => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s) = fun x =>\n    ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x)\na : \u03b1\n\u22a2 indicator (Ici s) (fun x => 1) (f a) = indicator {a | s \u2264 f a} (fun x => 1) a", "state_after": "no goals"}, {"tactic": "exact f_mble.nullMeasurable measurableSet_Ici", "annotated_tactic": ["exact f_mble.nullMeasurable <a>measurableSet_Ici</a>", [{"full_name": "measurableSet_Ici", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [510, 9], "def_end_pos": [510, 26]}]], "state_before": "case h.h.hs\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\ns : \u211d\naux\u2081 :\n  (fun x => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s) = fun x =>\n    ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x)\n\u22a2 NullMeasurableSet {a | s \u2264 f a}", "state_after": "no goals"}, {"tactic": "by_cases 0 < s <;> simp [h]", "annotated_tactic": ["by_cases 0 < s <;> simp [h]", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\ns : \u211d\naux\u2081 :\n  (fun x => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s) = fun x =>\n    ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x)\n\u22a2 indicator (Ioi 0) (fun _x => 1) s * \u2191\u2191\u03bc {a | s \u2264 f a} = indicator (Ioi 0) (fun _x => 1 * \u2191\u2191\u03bc {a | s \u2264 f a}) s", "state_after": "no goals"}, {"tactic": "funext p", "annotated_tactic": ["funext p", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\n\u22a2 (Function.uncurry fun x y => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) y) =\n    indicator {p | p.2 \u2208 Ioc 0 (f p.1)} fun p => ENNReal.ofReal (g p.2)", "state_after": "case h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\np : \u03b1 \u00d7 \u211d\n\u22a2 Function.uncurry (fun x y => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) y) p =\n    indicator {p | p.2 \u2208 Ioc 0 (f p.1)} (fun p => ENNReal.ofReal (g p.2)) p"}, {"tactic": "cases p with | mk p_fst p_snd => ?_", "annotated_tactic": ["cases p with | <a>mk</a> p_fst p_snd => ?_", [{"full_name": "Prod.mk", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [466, 16], "def_end_pos": [466, 41]}]], "state_before": "case h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\np : \u03b1 \u00d7 \u211d\n\u22a2 Function.uncurry (fun x y => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) y) p =\n    indicator {p | p.2 \u2208 Ioc 0 (f p.1)} (fun p => ENNReal.ofReal (g p.2)) p", "state_after": "case h.mk\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\np_fst : \u03b1\np_snd : \u211d\n\u22a2 Function.uncurry (fun x y => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) y) (p_fst, p_snd) =\n    indicator {p | p.2 \u2208 Ioc 0 (f p.1)} (fun p => ENNReal.ofReal (g p.2)) (p_fst, p_snd)"}, {"tactic": "rw [Function.uncurry_apply_pair]", "annotated_tactic": ["rw [<a>Function.uncurry_apply_pair</a>]", [{"full_name": "Function.uncurry_apply_pair", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [817, 9], "def_end_pos": [817, 27]}]], "state_before": "case h.mk\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\np_fst : \u03b1\np_snd : \u211d\n\u22a2 Function.uncurry (fun x y => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) y) (p_fst, p_snd) =\n    indicator {p | p.2 \u2208 Ioc 0 (f p.1)} (fun p => ENNReal.ofReal (g p.2)) (p_fst, p_snd)", "state_after": "case h.mk\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\np_fst : \u03b1\np_snd : \u211d\n\u22a2 indicator (Ioc 0 (f p_fst)) (fun t => ENNReal.ofReal (g t)) p_snd =\n    indicator {p | p.2 \u2208 Ioc 0 (f p.1)} (fun p => ENNReal.ofReal (g p.2)) (p_fst, p_snd)"}, {"tactic": "by_cases p_snd \u2208 Ioc 0 (f p_fst)", "annotated_tactic": ["by_cases p_snd \u2208 <a>Ioc</a> 0 (f p_fst)", [{"full_name": "Set.Ioc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [69, 5], "def_end_pos": [69, 8]}]], "state_before": "case h.mk\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\np_fst : \u03b1\np_snd : \u211d\n\u22a2 indicator (Ioc 0 (f p_fst)) (fun t => ENNReal.ofReal (g t)) p_snd =\n    indicator {p | p.2 \u2208 Ioc 0 (f p.1)} (fun p => ENNReal.ofReal (g p.2)) (p_fst, p_snd)", "state_after": "case pos\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\np_fst : \u03b1\np_snd : \u211d\nh : p_snd \u2208 Ioc 0 (f p_fst)\n\u22a2 indicator (Ioc 0 (f p_fst)) (fun t => ENNReal.ofReal (g t)) p_snd =\n    indicator {p | p.2 \u2208 Ioc 0 (f p.1)} (fun p => ENNReal.ofReal (g p.2)) (p_fst, p_snd)\n\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\np_fst : \u03b1\np_snd : \u211d\nh : \u00acp_snd \u2208 Ioc 0 (f p_fst)\n\u22a2 indicator (Ioc 0 (f p_fst)) (fun t => ENNReal.ofReal (g t)) p_snd =\n    indicator {p | p.2 \u2208 Ioc 0 (f p.1)} (fun p => ENNReal.ofReal (g p.2)) (p_fst, p_snd)"}, {"tactic": "have h' : (p_fst, p_snd) \u2208 {p : \u03b1 \u00d7 \u211d | p.snd \u2208 Ioc 0 (f p.fst)} := h", "annotated_tactic": ["have h' : (p_fst, p_snd) \u2208 {p : \u03b1 \u00d7 \u211d | p.snd \u2208 <a>Ioc</a> 0 (f p.fst)} := h", [{"full_name": "Set.Ioc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [69, 5], "def_end_pos": [69, 8]}]], "state_before": "case pos\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\np_fst : \u03b1\np_snd : \u211d\nh : p_snd \u2208 Ioc 0 (f p_fst)\n\u22a2 indicator (Ioc 0 (f p_fst)) (fun t => ENNReal.ofReal (g t)) p_snd =\n    indicator {p | p.2 \u2208 Ioc 0 (f p.1)} (fun p => ENNReal.ofReal (g p.2)) (p_fst, p_snd)", "state_after": "case pos\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\np_fst : \u03b1\np_snd : \u211d\nh : p_snd \u2208 Ioc 0 (f p_fst)\nh' : (p_fst, p_snd) \u2208 {p | p.2 \u2208 Ioc 0 (f p.1)}\n\u22a2 indicator (Ioc 0 (f p_fst)) (fun t => ENNReal.ofReal (g t)) p_snd =\n    indicator {p | p.2 \u2208 Ioc 0 (f p.1)} (fun p => ENNReal.ofReal (g p.2)) (p_fst, p_snd)"}, {"tactic": "rw [Set.indicator_of_mem h', Set.indicator_of_mem h]", "annotated_tactic": ["rw [<a>Set.indicator_of_mem</a> h', <a>Set.indicator_of_mem</a> h]", [{"full_name": "Set.indicator_of_mem", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [67, 3], "def_end_pos": [67, 14]}, {"full_name": "Set.indicator_of_mem", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [67, 3], "def_end_pos": [67, 14]}]], "state_before": "case pos\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\np_fst : \u03b1\np_snd : \u211d\nh : p_snd \u2208 Ioc 0 (f p_fst)\nh' : (p_fst, p_snd) \u2208 {p | p.2 \u2208 Ioc 0 (f p.1)}\n\u22a2 indicator (Ioc 0 (f p_fst)) (fun t => ENNReal.ofReal (g t)) p_snd =\n    indicator {p | p.2 \u2208 Ioc 0 (f p.1)} (fun p => ENNReal.ofReal (g p.2)) (p_fst, p_snd)", "state_after": "no goals"}, {"tactic": "have h' : (p_fst, p_snd) \u2209 {p : \u03b1 \u00d7 \u211d | p.snd \u2208 Ioc 0 (f p.fst)} := h", "annotated_tactic": ["have h' : (p_fst, p_snd) \u2209 {p : \u03b1 \u00d7 \u211d | p.snd \u2208 <a>Ioc</a> 0 (f p.fst)} := h", [{"full_name": "Set.Ioc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [69, 5], "def_end_pos": [69, 8]}]], "state_before": "case neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\np_fst : \u03b1\np_snd : \u211d\nh : \u00acp_snd \u2208 Ioc 0 (f p_fst)\n\u22a2 indicator (Ioc 0 (f p_fst)) (fun t => ENNReal.ofReal (g t)) p_snd =\n    indicator {p | p.2 \u2208 Ioc 0 (f p.1)} (fun p => ENNReal.ofReal (g p.2)) (p_fst, p_snd)", "state_after": "case neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\np_fst : \u03b1\np_snd : \u211d\nh : \u00acp_snd \u2208 Ioc 0 (f p_fst)\nh' : \u00ac(p_fst, p_snd) \u2208 {p | p.2 \u2208 Ioc 0 (f p.1)}\n\u22a2 indicator (Ioc 0 (f p_fst)) (fun t => ENNReal.ofReal (g t)) p_snd =\n    indicator {p | p.2 \u2208 Ioc 0 (f p.1)} (fun p => ENNReal.ofReal (g p.2)) (p_fst, p_snd)"}, {"tactic": "rw [Set.indicator_of_not_mem h', Set.indicator_of_not_mem h]", "annotated_tactic": ["rw [<a>Set.indicator_of_not_mem</a> h', <a>Set.indicator_of_not_mem</a> h]", [{"full_name": "Set.indicator_of_not_mem", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [73, 3], "def_end_pos": [73, 14]}, {"full_name": "Set.indicator_of_not_mem", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [73, 3], "def_end_pos": [73, 14]}]], "state_before": "case neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\np_fst : \u03b1\np_snd : \u211d\nh : \u00acp_snd \u2208 Ioc 0 (f p_fst)\nh' : \u00ac(p_fst, p_snd) \u2208 {p | p.2 \u2208 Ioc 0 (f p.1)}\n\u22a2 indicator (Ioc 0 (f p_fst)) (fun t => ENNReal.ofReal (g t)) p_snd =\n    indicator {p | p.2 \u2208 Ioc 0 (f p.1)} (fun p => ENNReal.ofReal (g p.2)) (p_fst, p_snd)", "state_after": "no goals"}, {"tactic": "simpa only [mem_univ, Pi.zero_apply, gt_iff_lt, not_lt, ge_iff_le, true_and] using\n  measurableSet_region_between_oc measurable_zero f_mble  MeasurableSet.univ", "annotated_tactic": ["simpa only [<a>mem_univ</a>, <a>Pi.zero_apply</a>, <a>gt_iff_lt</a>, <a>not_lt</a>, <a>ge_iff_le</a>, <a>true_and</a>] using\n      <a>measurableSet_region_between_oc</a> <a>measurable_zero</a> f_mble  <a>MeasurableSet.univ</a>", [{"full_name": "Set.mem_univ", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [676, 9], "def_end_pos": [676, 17]}, {"full_name": "Pi.zero_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [46, 3], "def_end_pos": [46, 14]}, {"full_name": "gt_iff_lt", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [366, 9], "def_end_pos": [366, 18]}, {"full_name": "not_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [368, 9], "def_end_pos": [368, 15]}, {"full_name": "ge_iff_le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [359, 9], "def_end_pos": [359, 18]}, {"full_name": "true_and", "def_path": "lake-packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [84, 17], "def_end_pos": [84, 25]}, {"full_name": "measurableSet_region_between_oc", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean", "def_pos": [489, 9], "def_end_pos": [489, 40]}, {"full_name": "measurable_zero", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [270, 18], "def_end_pos": [270, 29]}, {"full_name": "MeasurableSet.univ", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [101, 19], "def_end_pos": [101, 37]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\naux\u2082 :\n  (Function.uncurry fun x y => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) y) =\n    indicator {p | p.2 \u2208 Ioc 0 (f p.1)} fun p => ENNReal.ofReal (g p.2)\n\u22a2 MeasurableSet {p | p.2 \u2208 Ioc 0 (f p.1)}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/List/Count.lean", "full_name": "List.count_le_count_cons", "start": [155, 1], "end": [156, 32], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/CircleIntegral.lean", "full_name": "circleIntegral.norm_integral_le_of_norm_le_const", "start": [401, 1], "end": [406, 38], "traced_tactics": [{"tactic": "rwa [this]", "annotated_tactic": ["rwa [this]", []], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf : \u2102 \u2192 E\nc : \u2102\nR C : \u211d\nhR : 0 \u2264 R\nhf : \u2200 (z : \u2102), z \u2208 sphere c R \u2192 \u2016f z\u2016 \u2264 C\nthis : |R| = R\n\u22a2 \u2200 (z : \u2102), z \u2208 sphere c |R| \u2192 \u2016f z\u2016 \u2264 C", "state_after": "no goals"}, {"tactic": "rw [this]", "annotated_tactic": ["rw [this]", []], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf : \u2102 \u2192 E\nc : \u2102\nR C : \u211d\nhR : 0 \u2264 R\nhf : \u2200 (z : \u2102), z \u2208 sphere c R \u2192 \u2016f z\u2016 \u2264 C\nthis : |R| = R\n\u22a2 2 * \u03c0 * |R| * C = 2 * \u03c0 * R * C", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/List/Basic.lean", "full_name": "List.leftpad_eq_leftpadTR", "start": [634, 10], "end": [635, 65], "traced_tactics": [{"tactic": "funext \u03b1 n a l", "annotated_tactic": ["funext \u03b1 n a l", []], "state_before": "\u22a2 @leftpad = @leftpadTR", "state_after": "case h.h.h.h\n\u03b1 : Type u_1\nn : Nat\na : \u03b1\nl : List \u03b1\n\u22a2 leftpad n a l = leftpadTR n a l"}, {"tactic": "simp [leftpad, leftpadTR, replicateTR_loop_eq]", "annotated_tactic": ["simp [<a>leftpad</a>, <a>leftpadTR</a>, <a>replicateTR_loop_eq</a>]", [{"full_name": "List.leftpad", "def_path": "lake-packages/std/Std/Data/List/Basic.lean", "def_pos": [628, 5], "def_end_pos": [628, 12]}, {"full_name": "List.leftpadTR", "def_path": "lake-packages/std/Std/Data/List/Basic.lean", "def_pos": [631, 15], "def_end_pos": [631, 24]}, {"full_name": "List.replicateTR_loop_eq", "def_path": "lake-packages/std/Std/Data/List/Basic.lean", "def_pos": [210, 9], "def_end_pos": [210, 28]}]], "state_before": "case h.h.h.h\n\u03b1 : Type u_1\nn : Nat\na : \u03b1\nl : List \u03b1\n\u22a2 leftpad n a l = leftpadTR n a l", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "full_name": "MeasureTheory.OuterMeasure.restrict_ofFunction", "start": [805, 1], "end": [811, 83], "traced_tactics": [{"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\ns : Set \u03b1\nhm : Monotone m\n\u22a2 (fun t => m (t \u2229 s)) \u2205 = 0", "state_after": "\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\ns : Set \u03b1\nhm : Monotone m\n\u22a2 m \u2205 = 0"}, {"tactic": "simp [m_empty]", "annotated_tactic": ["simp [m_empty]", []], "state_before": "\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\ns : Set \u03b1\nhm : Monotone m\n\u22a2 m \u2205 = 0", "state_after": "no goals"}, {"tactic": "rw [restrict]", "annotated_tactic": ["rw [<a>restrict</a>]", [{"full_name": "MeasureTheory.OuterMeasure.restrict", "def_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "def_pos": [556, 5], "def_end_pos": [556, 13]}]], "state_before": "\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\ns : Set \u03b1\nhm : Monotone m\n\u22a2 \u2191(restrict s) (OuterMeasure.ofFunction m m_empty) =\n    OuterMeasure.ofFunction (fun t => m (t \u2229 s)) (_ : (fun t => m (t \u2229 s)) \u2205 = 0)", "state_after": "\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\ns : Set \u03b1\nhm : Monotone m\n\u22a2 \u2191(LinearMap.comp (map Subtype.val) (comap Subtype.val)) (OuterMeasure.ofFunction m m_empty) =\n    OuterMeasure.ofFunction (fun t => m (t \u2229 s)) (_ : (fun t => m (t \u2229 s)) \u2205 = 0)"}, {"tactic": "simp only [LinearMap.comp_apply]", "annotated_tactic": ["simp only [<a>LinearMap.comp_apply</a>]", [{"full_name": "LinearMap.comp_apply", "def_path": "Mathlib/Algebra/Module/LinearMap.lean", "def_pos": [549, 9], "def_end_pos": [549, 19]}]], "state_before": "\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\ns : Set \u03b1\nhm : Monotone m\n\u22a2 \u2191(LinearMap.comp (map Subtype.val) (comap Subtype.val)) (OuterMeasure.ofFunction m m_empty) =\n    OuterMeasure.ofFunction (fun t => m (t \u2229 s)) (_ : (fun t => m (t \u2229 s)) \u2205 = 0)", "state_after": "\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\ns : Set \u03b1\nhm : Monotone m\n\u22a2 \u2191(map Subtype.val) (\u2191(comap Subtype.val) (OuterMeasure.ofFunction m m_empty)) =\n    OuterMeasure.ofFunction (fun t => m (t \u2229 s)) (_ : (fun t => m (t \u2229 s)) \u2205 = 0)"}, {"tactic": "rw [comap_ofFunction _ (Or.inl hm)]", "annotated_tactic": ["rw [<a>comap_ofFunction</a> _ (<a>Or.inl</a> hm)]", [{"full_name": "MeasureTheory.OuterMeasure.comap_ofFunction", "def_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "def_pos": [769, 9], "def_end_pos": [769, 25]}, {"full_name": "Or.inl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [517, 5], "def_end_pos": [517, 8]}]], "state_before": "\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\ns : Set \u03b1\nhm : Monotone m\n\u22a2 \u2191(map Subtype.val) (\u2191(comap Subtype.val) (OuterMeasure.ofFunction m m_empty)) =\n    OuterMeasure.ofFunction (fun t => m (t \u2229 s)) (_ : (fun t => m (t \u2229 s)) \u2205 = 0)", "state_after": "\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\ns : Set \u03b1\nhm : Monotone m\n\u22a2 \u2191(map Subtype.val)\n      (OuterMeasure.ofFunction (fun s_1 => m (Subtype.val '' s_1)) (_ : (fun s_1 => m (Subtype.val '' s_1)) \u2205 = 0)) =\n    OuterMeasure.ofFunction (fun t => m (t \u2229 s)) (_ : (fun t => m (t \u2229 s)) \u2205 = 0)"}, {"tactic": "simp only [map_ofFunction Subtype.coe_injective, Subtype.image_preimage_coe]", "annotated_tactic": ["simp only [<a>map_ofFunction</a> <a>Subtype.coe_injective</a>, <a>Subtype.image_preimage_coe</a>]", [{"full_name": "MeasureTheory.OuterMeasure.map_ofFunction", "def_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "def_pos": [792, 9], "def_end_pos": [792, 23]}, {"full_name": "Subtype.coe_injective", "def_path": "Mathlib/Data/Subtype.lean", "def_pos": [119, 9], "def_end_pos": [119, 22]}, {"full_name": "Subtype.image_preimage_coe", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [1448, 9], "def_end_pos": [1448, 27]}]], "state_before": "\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\ns : Set \u03b1\nhm : Monotone m\n\u22a2 \u2191(map Subtype.val)\n      (OuterMeasure.ofFunction (fun s_1 => m (Subtype.val '' s_1)) (_ : (fun s_1 => m (Subtype.val '' s_1)) \u2205 = 0)) =\n    OuterMeasure.ofFunction (fun t => m (t \u2229 s)) (_ : (fun t => m (t \u2229 s)) \u2205 = 0)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "full_name": "MeasureTheory.FinStronglyMeasurable.sub", "start": [1089, 11], "end": [1095, 62], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Function.lean", "full_name": "Set.bijOn_empty", "start": [938, 1], "end": [939, 54], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/List/Init/Lemmas.lean", "full_name": "List.reverseAux_eq", "start": [130, 1], "end": [131, 26], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "full_name": "MeasureTheory.setToFun_zero_left", "start": [1351, 1], "end": [1355, 31], "traced_tactics": [{"tactic": "by_cases hf : Integrable f \u03bc", "annotated_tactic": ["by_cases hf : <a>Integrable</a> f \u03bc", [{"full_name": "MeasureTheory.Integrable", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [442, 5], "def_end_pos": [442, 15]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc 0 C\n\u22a2 setToFun \u03bc 0 hT f = 0", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc 0 C\nhf : Integrable f\n\u22a2 setToFun \u03bc 0 hT f = 0\n\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc 0 C\nhf : \u00acIntegrable f\n\u22a2 setToFun \u03bc 0 hT f = 0"}, {"tactic": "rw [setToFun_eq hT hf]", "annotated_tactic": ["rw [<a>setToFun_eq</a> hT hf]", [{"full_name": "MeasureTheory.setToFun_eq", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [1276, 9], "def_end_pos": [1276, 20]}]], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc 0 C\nhf : Integrable f\n\u22a2 setToFun \u03bc 0 hT f = 0", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc 0 C\nhf : Integrable f\n\u22a2 \u2191(L1.setToL1 hT) (Integrable.toL1 f hf) = 0"}, {"tactic": "exact L1.setToL1_zero_left hT _", "annotated_tactic": ["exact <a>L1.setToL1_zero_left</a> hT _", [{"full_name": "MeasureTheory.L1.setToL1_zero_left", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [1037, 9], "def_end_pos": [1037, 26]}]], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc 0 C\nhf : Integrable f\n\u22a2 \u2191(L1.setToL1 hT) (Integrable.toL1 f hf) = 0", "state_after": "no goals"}, {"tactic": "exact setToFun_undef hT hf", "annotated_tactic": ["exact <a>setToFun_undef</a> hT hf", [{"full_name": "MeasureTheory.setToFun_undef", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [1286, 9], "def_end_pos": [1286, 23]}]], "state_before": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc 0 C\nhf : \u00acIntegrable f\n\u22a2 setToFun \u03bc 0 hT f = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/LocallyFinite.lean", "full_name": "Finset.eq_of_mem_uIcc_of_mem_uIcc'", "start": [1009, 1], "end": [1011, 40], "traced_tactics": [{"tactic": "simp_rw [mem_uIcc]", "annotated_tactic": ["simp_rw [<a>mem_uIcc</a>]", [{"full_name": "Finset.mem_uIcc", "def_path": "Mathlib/Order/LocallyFinite.lean", "def_pos": [502, 9], "def_end_pos": [502, 17]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : DistribLattice \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\n\u22a2 b \u2208 [[a, c]] \u2192 c \u2208 [[a, b]] \u2192 b = c", "state_after": "\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : DistribLattice \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\n\u22a2 a \u2293 c \u2264 b \u2227 b \u2264 a \u2294 c \u2192 a \u2293 b \u2264 c \u2227 c \u2264 a \u2294 b \u2192 b = c"}, {"tactic": "exact Set.eq_of_mem_uIcc_of_mem_uIcc'", "annotated_tactic": ["exact <a>Set.eq_of_mem_uIcc_of_mem_uIcc'</a>", [{"full_name": "Set.eq_of_mem_uIcc_of_mem_uIcc'", "def_path": "Mathlib/Data/Set/Intervals/UnorderedInterval.lean", "def_pos": [167, 7], "def_end_pos": [167, 34]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : DistribLattice \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\n\u22a2 a \u2293 c \u2264 b \u2227 b \u2264 a \u2294 c \u2192 a \u2293 b \u2264 c \u2227 c \u2264 a \u2294 b \u2192 b = c", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "full_name": "measurable_tProd_elim'", "start": [1043, 1], "end": [1045, 62], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Basic.lean", "full_name": "Set.univ_eq_true_false", "start": [2660, 1], "end": [2663, 35], "traced_tactics": [{"tactic": "rw [mem_insert_iff, mem_singleton_iff]", "annotated_tactic": ["rw [<a>mem_insert_iff</a>, <a>mem_singleton_iff</a>]", [{"full_name": "Set.mem_insert_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1147, 9], "def_end_pos": [1147, 23]}, {"full_name": "Set.mem_singleton_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1273, 9], "def_end_pos": [1273, 26]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\na b : \u03b1\ns s\u2081 s\u2082 t t\u2081 t\u2082 u : Set \u03b1\nx : Prop\n\u22a2 x \u2208 {True, False}", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\na b : \u03b1\ns s\u2081 s\u2082 t t\u2081 t\u2082 u : Set \u03b1\nx : Prop\n\u22a2 x = True \u2228 x = False"}, {"tactic": "exact Classical.propComplete x", "annotated_tactic": ["exact <a>Classical.propComplete</a> x", [{"full_name": "Classical.propComplete", "def_path": "lake-packages/lean4/src/lean/Init/Classical.lean", "def_pos": [113, 9], "def_end_pos": [113, 21]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\na b : \u03b1\ns s\u2081 s\u2082 t t\u2081 t\u2082 u : Set \u03b1\nx : Prop\n\u22a2 x = True \u2228 x = False", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Kernel/CondCdf.lean", "full_name": "ProbabilityTheory.integrable_condCdf", "start": [931, 1], "end": [945, 39], "traced_tactics": [{"tactic": "refine' integrable_of_forall_fin_meas_le _ (measure_lt_top \u03c1.fst univ) _ fun t _ _ => _", "annotated_tactic": ["refine' <a>integrable_of_forall_fin_meas_le</a> _ (<a>measure_lt_top</a> \u03c1.fst <a>univ</a>) _ fun t _ _ => _", [{"full_name": "MeasureTheory.integrable_of_forall_fin_meas_le", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [1241, 9], "def_end_pos": [1241, 41]}, {"full_name": "MeasureTheory.measure_lt_top", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2866, 9], "def_end_pos": [2866, 23]}, {"full_name": "Set.univ", "def_path": "Mathlib/Init/Set.lean", "def_pos": [90, 5], "def_end_pos": [90, 9]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\n\u22a2 Integrable fun a => \u2191(condCdf \u03c1 a) x", "state_after": "case refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\n\u22a2 AEStronglyMeasurable (fun a => \u2191(condCdf \u03c1 a) x) (Measure.fst \u03c1)\n\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\nt : Set \u03b1\nx\u271d\u00b9 : MeasurableSet t\nx\u271d : \u2191\u2191(Measure.fst \u03c1) t \u2260 \u22a4\n\u22a2 \u222b\u207b (x_1 : \u03b1) in t, \u2191\u2016\u2191(condCdf \u03c1 x_1) x\u2016\u208a \u2202Measure.fst \u03c1 \u2264 \u2191\u2191(Measure.fst \u03c1) univ"}, {"tactic": "exact (stronglyMeasurable_condCdf \u03c1 _).aestronglyMeasurable", "annotated_tactic": ["exact (<a>stronglyMeasurable_condCdf</a> \u03c1 _).<a>aestronglyMeasurable</a>", [{"full_name": "ProbabilityTheory.stronglyMeasurable_condCdf", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [926, 9], "def_end_pos": [926, 35]}, {"full_name": "MeasureTheory.StronglyMeasurable.aestronglyMeasurable", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [110, 19], "def_end_pos": [110, 58]}]], "state_before": "case refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\n\u22a2 AEStronglyMeasurable (fun a => \u2191(condCdf \u03c1 a) x) (Measure.fst \u03c1)", "state_after": "no goals"}, {"tactic": "have : \u2200 y, (\u2016condCdf \u03c1 y x\u2016\u208a : \u211d\u22650\u221e) \u2264 1 := by\n  intro y\n  rw [Real.nnnorm_of_nonneg (condCdf_nonneg _ _ _)]\n  simp only [ENNReal.coe_le_one_iff]\n  exact condCdf_le_one _ _ _", "annotated_tactic": ["have : \u2200 y, (\u2016<a>condCdf</a> \u03c1 y x\u2016\u208a : \u211d\u22650\u221e) \u2264 1 := by\n      intro y\n      rw [<a>Real.nnnorm_of_nonneg</a> (<a>condCdf_nonneg</a> _ _ _)]\n      -- Porting note: was exact_mod_cast condCdf_le_one _ _ _\n      simp only [<a>ENNReal.coe_le_one_iff</a>]\n      exact <a>condCdf_le_one</a> _ _ _", [{"full_name": "ProbabilityTheory.condCdf", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [777, 19], "def_end_pos": [777, 26]}, {"full_name": "Real.nnnorm_of_nonneg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [1800, 9], "def_end_pos": [1800, 25]}, {"full_name": "ProbabilityTheory.condCdf_nonneg", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [789, 9], "def_end_pos": [789, 23]}, {"full_name": "ENNReal.coe_le_one_iff", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [695, 9], "def_end_pos": [695, 23]}, {"full_name": "ProbabilityTheory.condCdf_le_one", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [794, 9], "def_end_pos": [794, 23]}]], "state_before": "case refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\nt : Set \u03b1\nx\u271d\u00b9 : MeasurableSet t\nx\u271d : \u2191\u2191(Measure.fst \u03c1) t \u2260 \u22a4\n\u22a2 \u222b\u207b (x_1 : \u03b1) in t, \u2191\u2016\u2191(condCdf \u03c1 x_1) x\u2016\u208a \u2202Measure.fst \u03c1 \u2264 \u2191\u2191(Measure.fst \u03c1) univ", "state_after": "case refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\nt : Set \u03b1\nx\u271d\u00b9 : MeasurableSet t\nx\u271d : \u2191\u2191(Measure.fst \u03c1) t \u2260 \u22a4\nthis : \u2200 (y : \u03b1), \u2191\u2016\u2191(condCdf \u03c1 y) x\u2016\u208a \u2264 1\n\u22a2 \u222b\u207b (x_1 : \u03b1) in t, \u2191\u2016\u2191(condCdf \u03c1 x_1) x\u2016\u208a \u2202Measure.fst \u03c1 \u2264 \u2191\u2191(Measure.fst \u03c1) univ"}, {"tactic": "refine'\n  (set_lintegral_mono (measurable_condCdf _ _).ennnorm measurable_one fun y _ => this y).trans\n    _", "annotated_tactic": ["refine'\n      (<a>set_lintegral_mono</a> (<a>measurable_condCdf</a> _ _).<a>ennnorm</a> <a>measurable_one</a> fun y _ => this y).<a>trans</a>\n        _", [{"full_name": "MeasureTheory.set_lintegral_mono", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [287, 9], "def_end_pos": [287, 27]}, {"full_name": "ProbabilityTheory.measurable_condCdf", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [854, 9], "def_end_pos": [854, 27]}, {"full_name": "Measurable.ennnorm", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [2277, 9], "def_end_pos": [2277, 27]}, {"full_name": "measurable_one", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [271, 9], "def_end_pos": [271, 23]}, {"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [120, 7], "def_end_pos": [120, 18]}]], "state_before": "case refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\nt : Set \u03b1\nx\u271d\u00b9 : MeasurableSet t\nx\u271d : \u2191\u2191(Measure.fst \u03c1) t \u2260 \u22a4\nthis : \u2200 (y : \u03b1), \u2191\u2016\u2191(condCdf \u03c1 y) x\u2016\u208a \u2264 1\n\u22a2 \u222b\u207b (x_1 : \u03b1) in t, \u2191\u2016\u2191(condCdf \u03c1 x_1) x\u2016\u208a \u2202Measure.fst \u03c1 \u2264 \u2191\u2191(Measure.fst \u03c1) univ", "state_after": "case refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\nt : Set \u03b1\nx\u271d\u00b9 : MeasurableSet t\nx\u271d : \u2191\u2191(Measure.fst \u03c1) t \u2260 \u22a4\nthis : \u2200 (y : \u03b1), \u2191\u2016\u2191(condCdf \u03c1 y) x\u2016\u208a \u2264 1\n\u22a2 \u222b\u207b (x : \u03b1) in t, OfNat.ofNat 1 x \u2202Measure.fst \u03c1 \u2264 \u2191\u2191(Measure.fst \u03c1) univ"}, {"tactic": "simp only [Pi.one_apply, lintegral_one, Measure.restrict_apply, MeasurableSet.univ, univ_inter]", "annotated_tactic": ["simp only [<a>Pi.one_apply</a>, <a>lintegral_one</a>, <a>Measure.restrict_apply</a>, <a>MeasurableSet.univ</a>, <a>univ_inter</a>]", [{"full_name": "Pi.one_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [47, 9], "def_end_pos": [47, 18]}, {"full_name": "MeasureTheory.lintegral_one", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [149, 9], "def_end_pos": [149, 22]}, {"full_name": "MeasureTheory.Measure.restrict_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1533, 9], "def_end_pos": [1533, 23]}, {"full_name": "MeasurableSet.univ", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [101, 19], "def_end_pos": [101, 37]}, {"full_name": "Set.univ_inter", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1017, 9], "def_end_pos": [1017, 19]}]], "state_before": "case refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\nt : Set \u03b1\nx\u271d\u00b9 : MeasurableSet t\nx\u271d : \u2191\u2191(Measure.fst \u03c1) t \u2260 \u22a4\nthis : \u2200 (y : \u03b1), \u2191\u2016\u2191(condCdf \u03c1 y) x\u2016\u208a \u2264 1\n\u22a2 \u222b\u207b (x : \u03b1) in t, OfNat.ofNat 1 x \u2202Measure.fst \u03c1 \u2264 \u2191\u2191(Measure.fst \u03c1) univ", "state_after": "case refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\nt : Set \u03b1\nx\u271d\u00b9 : MeasurableSet t\nx\u271d : \u2191\u2191(Measure.fst \u03c1) t \u2260 \u22a4\nthis : \u2200 (y : \u03b1), \u2191\u2016\u2191(condCdf \u03c1 y) x\u2016\u208a \u2264 1\n\u22a2 \u2191\u2191(Measure.fst \u03c1) t \u2264 \u2191\u2191(Measure.fst \u03c1) univ"}, {"tactic": "exact measure_mono (subset_univ _)", "annotated_tactic": ["exact <a>measure_mono</a> (<a>subset_univ</a> _)", [{"full_name": "MeasureTheory.measure_mono", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [193, 9], "def_end_pos": [193, 21]}, {"full_name": "Set.subset_univ", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [691, 9], "def_end_pos": [691, 20]}]], "state_before": "case refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\nt : Set \u03b1\nx\u271d\u00b9 : MeasurableSet t\nx\u271d : \u2191\u2191(Measure.fst \u03c1) t \u2260 \u22a4\nthis : \u2200 (y : \u03b1), \u2191\u2016\u2191(condCdf \u03c1 y) x\u2016\u208a \u2264 1\n\u22a2 \u2191\u2191(Measure.fst \u03c1) t \u2264 \u2191\u2191(Measure.fst \u03c1) univ", "state_after": "no goals"}, {"tactic": "intro y", "annotated_tactic": ["intro y", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\nt : Set \u03b1\nx\u271d\u00b9 : MeasurableSet t\nx\u271d : \u2191\u2191(Measure.fst \u03c1) t \u2260 \u22a4\n\u22a2 \u2200 (y : \u03b1), \u2191\u2016\u2191(condCdf \u03c1 y) x\u2016\u208a \u2264 1", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\nt : Set \u03b1\nx\u271d\u00b9 : MeasurableSet t\nx\u271d : \u2191\u2191(Measure.fst \u03c1) t \u2260 \u22a4\ny : \u03b1\n\u22a2 \u2191\u2016\u2191(condCdf \u03c1 y) x\u2016\u208a \u2264 1"}, {"tactic": "rw [Real.nnnorm_of_nonneg (condCdf_nonneg _ _ _)]", "annotated_tactic": ["rw [<a>Real.nnnorm_of_nonneg</a> (<a>condCdf_nonneg</a> _ _ _)]", [{"full_name": "Real.nnnorm_of_nonneg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [1800, 9], "def_end_pos": [1800, 25]}, {"full_name": "ProbabilityTheory.condCdf_nonneg", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [789, 9], "def_end_pos": [789, 23]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\nt : Set \u03b1\nx\u271d\u00b9 : MeasurableSet t\nx\u271d : \u2191\u2191(Measure.fst \u03c1) t \u2260 \u22a4\ny : \u03b1\n\u22a2 \u2191\u2016\u2191(condCdf \u03c1 y) x\u2016\u208a \u2264 1", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\nt : Set \u03b1\nx\u271d\u00b9 : MeasurableSet t\nx\u271d : \u2191\u2191(Measure.fst \u03c1) t \u2260 \u22a4\ny : \u03b1\n\u22a2 \u2191{ val := \u2191(condCdf \u03c1 y) x, property := (_ : 0 \u2264 \u2191(condCdf \u03c1 y) x) } \u2264 1"}, {"tactic": "simp only [ENNReal.coe_le_one_iff]", "annotated_tactic": ["simp only [<a>ENNReal.coe_le_one_iff</a>]", [{"full_name": "ENNReal.coe_le_one_iff", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [695, 9], "def_end_pos": [695, 23]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\nt : Set \u03b1\nx\u271d\u00b9 : MeasurableSet t\nx\u271d : \u2191\u2191(Measure.fst \u03c1) t \u2260 \u22a4\ny : \u03b1\n\u22a2 \u2191{ val := \u2191(condCdf \u03c1 y) x, property := (_ : 0 \u2264 \u2191(condCdf \u03c1 y) x) } \u2264 1", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\nt : Set \u03b1\nx\u271d\u00b9 : MeasurableSet t\nx\u271d : \u2191\u2191(Measure.fst \u03c1) t \u2260 \u22a4\ny : \u03b1\n\u22a2 { val := \u2191(condCdf \u03c1 y) x, property := (_ : 0 \u2264 \u2191(condCdf \u03c1 y) x) } \u2264 1"}, {"tactic": "exact condCdf_le_one _ _ _", "annotated_tactic": ["exact <a>condCdf_le_one</a> _ _ _", [{"full_name": "ProbabilityTheory.condCdf_le_one", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [794, 9], "def_end_pos": [794, 23]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\nt : Set \u03b1\nx\u271d\u00b9 : MeasurableSet t\nx\u271d : \u2191\u2191(Measure.fst \u03c1) t \u2260 \u22a4\ny : \u03b1\n\u22a2 { val := \u2191(condCdf \u03c1 y) x, property := (_ : 0 \u2264 \u2191(condCdf \u03c1 y) x) } \u2264 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Quot.lean", "full_name": "Trunc.eq", "start": [516, 11], "end": [517, 55], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "full_name": "MeasureTheory.SimpleFunc.setToSimpleFunc_indicator", "start": [604, 1], "end": [629, 7], "traced_tactics": [{"tactic": "obtain rfl | hs_empty := s.eq_empty_or_nonempty", "annotated_tactic": ["obtain rfl | hs_empty := s.eq_empty_or_nonempty", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : NormedAddCommGroup G\nm\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nT : Set \u03b1 \u2192 F \u2192L[\u211d] F'\nhT_empty : T \u2205 = 0\nm : MeasurableSpace \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nx : F\n\u22a2 setToSimpleFunc T (piecewise s hs (const \u03b1 x) (const \u03b1 0)) = \u2191(T s) x", "state_after": "case inl\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : NormedAddCommGroup G\nm\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nT : Set \u03b1 \u2192 F \u2192L[\u211d] F'\nhT_empty : T \u2205 = 0\nm : MeasurableSpace \u03b1\nx : F\nhs : MeasurableSet \u2205\n\u22a2 setToSimpleFunc T (piecewise \u2205 hs (const \u03b1 x) (const \u03b1 0)) = \u2191(T \u2205) x\n\ncase inr\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : NormedAddCommGroup G\nm\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nT : Set \u03b1 \u2192 F \u2192L[\u211d] F'\nhT_empty : T \u2205 = 0\nm : MeasurableSpace \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nx : F\nhs_empty : Set.Nonempty s\n\u22a2 setToSimpleFunc T (piecewise s hs (const \u03b1 x) (const \u03b1 0)) = \u2191(T s) x"}, {"tactic": "simp_rw [setToSimpleFunc]", "annotated_tactic": ["simp_rw [<a>setToSimpleFunc</a>]", [{"full_name": "MeasureTheory.SimpleFunc.setToSimpleFunc", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [281, 5], "def_end_pos": [281, 20]}]], "state_before": "case inr\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : NormedAddCommGroup G\nm\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nT : Set \u03b1 \u2192 F \u2192L[\u211d] F'\nhT_empty : T \u2205 = 0\nm : MeasurableSpace \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nx : F\nhs_empty : Set.Nonempty s\n\u22a2 setToSimpleFunc T (piecewise s hs (const \u03b1 x) (const \u03b1 0)) = \u2191(T s) x", "state_after": "case inr\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : NormedAddCommGroup G\nm\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nT : Set \u03b1 \u2192 F \u2192L[\u211d] F'\nhT_empty : T \u2205 = 0\nm : MeasurableSpace \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nx : F\nhs_empty : Set.Nonempty s\n\u22a2 \u2211 x_1 in SimpleFunc.range (piecewise s hs (const \u03b1 x) (const \u03b1 0)),\n      \u2191(T (\u2191(piecewise s hs (const \u03b1 x) (const \u03b1 0)) \u207b\u00b9' {x_1})) x_1 =\n    \u2191(T s) x"}, {"tactic": "obtain rfl | hs_univ := eq_or_ne s univ", "annotated_tactic": ["obtain rfl | hs_univ := <a>eq_or_ne</a> s <a>univ</a>", [{"full_name": "eq_or_ne", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [209, 9], "def_end_pos": [209, 17]}, {"full_name": "Set.univ", "def_path": "Mathlib/Init/Set.lean", "def_pos": [90, 5], "def_end_pos": [90, 9]}]], "state_before": "case inr\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : NormedAddCommGroup G\nm\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nT : Set \u03b1 \u2192 F \u2192L[\u211d] F'\nhT_empty : T \u2205 = 0\nm : MeasurableSpace \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nx : F\nhs_empty : Set.Nonempty s\n\u22a2 \u2211 x_1 in SimpleFunc.range (piecewise s hs (const \u03b1 x) (const \u03b1 0)),\n      \u2191(T (\u2191(piecewise s hs (const \u03b1 x) (const \u03b1 0)) \u207b\u00b9' {x_1})) x_1 =\n    \u2191(T s) x", "state_after": "case inr.inl\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : NormedAddCommGroup G\nm\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nT : Set \u03b1 \u2192 F \u2192L[\u211d] F'\nhT_empty : T \u2205 = 0\nm : MeasurableSpace \u03b1\nx : F\nhs : MeasurableSet Set.univ\nhs_empty : Set.Nonempty Set.univ\n\u22a2 \u2211 x_1 in SimpleFunc.range (piecewise Set.univ hs (const \u03b1 x) (const \u03b1 0)),\n      \u2191(T (\u2191(piecewise Set.univ hs (const \u03b1 x) (const \u03b1 0)) \u207b\u00b9' {x_1})) x_1 =\n    \u2191(T Set.univ) x\n\ncase inr.inr\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : NormedAddCommGroup G\nm\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nT : Set \u03b1 \u2192 F \u2192L[\u211d] F'\nhT_empty : T \u2205 = 0\nm : MeasurableSpace \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nx : F\nhs_empty : Set.Nonempty s\nhs_univ : s \u2260 Set.univ\n\u22a2 \u2211 x_1 in SimpleFunc.range (piecewise s hs (const \u03b1 x) (const \u03b1 0)),\n      \u2191(T (\u2191(piecewise s hs (const \u03b1 x) (const \u03b1 0)) \u207b\u00b9' {x_1})) x_1 =\n    \u2191(T s) x"}, {"tactic": "rw [range_indicator hs hs_empty hs_univ]", "annotated_tactic": ["rw [<a>range_indicator</a> hs hs_empty hs_univ]", [{"full_name": "MeasureTheory.SimpleFunc.range_indicator", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [261, 9], "def_end_pos": [261, 24]}]], "state_before": "case inr.inr\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : NormedAddCommGroup G\nm\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nT : Set \u03b1 \u2192 F \u2192L[\u211d] F'\nhT_empty : T \u2205 = 0\nm : MeasurableSpace \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nx : F\nhs_empty : Set.Nonempty s\nhs_univ : s \u2260 Set.univ\n\u22a2 \u2211 x_1 in SimpleFunc.range (piecewise s hs (const \u03b1 x) (const \u03b1 0)),\n      \u2191(T (\u2191(piecewise s hs (const \u03b1 x) (const \u03b1 0)) \u207b\u00b9' {x_1})) x_1 =\n    \u2191(T s) x", "state_after": "case inr.inr\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : NormedAddCommGroup G\nm\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nT : Set \u03b1 \u2192 F \u2192L[\u211d] F'\nhT_empty : T \u2205 = 0\nm : MeasurableSpace \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nx : F\nhs_empty : Set.Nonempty s\nhs_univ : s \u2260 Set.univ\n\u22a2 \u2211 x_1 in {x, 0}, \u2191(T (\u2191(piecewise s hs (const \u03b1 x) (const \u03b1 0)) \u207b\u00b9' {x_1})) x_1 = \u2191(T s) x"}, {"tactic": "by_cases hx0 : x = 0", "annotated_tactic": ["by_cases hx0 : x = 0", []], "state_before": "case inr.inr\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : NormedAddCommGroup G\nm\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nT : Set \u03b1 \u2192 F \u2192L[\u211d] F'\nhT_empty : T \u2205 = 0\nm : MeasurableSpace \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nx : F\nhs_empty : Set.Nonempty s\nhs_univ : s \u2260 Set.univ\n\u22a2 \u2211 x_1 in {x, 0}, \u2191(T (\u2191(piecewise s hs (const \u03b1 x) (const \u03b1 0)) \u207b\u00b9' {x_1})) x_1 = \u2191(T s) x", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : NormedAddCommGroup G\nm\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nT : Set \u03b1 \u2192 F \u2192L[\u211d] F'\nhT_empty : T \u2205 = 0\nm : MeasurableSpace \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nx : F\nhs_empty : Set.Nonempty s\nhs_univ : s \u2260 Set.univ\nhx0 : x = 0\n\u22a2 \u2211 x_1 in {x, 0}, \u2191(T (\u2191(piecewise s hs (const \u03b1 x) (const \u03b1 0)) \u207b\u00b9' {x_1})) x_1 = \u2191(T s) x\n\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : NormedAddCommGroup G\nm\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nT : Set \u03b1 \u2192 F \u2192L[\u211d] F'\nhT_empty : T \u2205 = 0\nm : MeasurableSpace \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nx : F\nhs_empty : Set.Nonempty s\nhs_univ : s \u2260 Set.univ\nhx0 : \u00acx = 0\n\u22a2 \u2211 x_1 in {x, 0}, \u2191(T (\u2191(piecewise s hs (const \u03b1 x) (const \u03b1 0)) \u207b\u00b9' {x_1})) x_1 = \u2191(T s) x"}, {"tactic": "rw [sum_insert]", "annotated_tactic": ["rw [<a>sum_insert</a>]", [{"full_name": "Finset.sum_insert", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [316, 3], "def_end_pos": [316, 14]}]], "state_before": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : NormedAddCommGroup G\nm\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nT : Set \u03b1 \u2192 F \u2192L[\u211d] F'\nhT_empty : T \u2205 = 0\nm : MeasurableSpace \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nx : F\nhs_empty : Set.Nonempty s\nhs_univ : s \u2260 Set.univ\nhx0 : \u00acx = 0\n\u22a2 \u2211 x_1 in {x, 0}, \u2191(T (\u2191(piecewise s hs (const \u03b1 x) (const \u03b1 0)) \u207b\u00b9' {x_1})) x_1 = \u2191(T s) x", "state_after": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : NormedAddCommGroup G\nm\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nT : Set \u03b1 \u2192 F \u2192L[\u211d] F'\nhT_empty : T \u2205 = 0\nm : MeasurableSpace \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nx : F\nhs_empty : Set.Nonempty s\nhs_univ : s \u2260 Set.univ\nhx0 : \u00acx = 0\n\u22a2 \u2191(T (\u2191(piecewise s hs (const \u03b1 x) (const \u03b1 0)) \u207b\u00b9' {x})) x +\n      \u2211 x_1 in {0}, \u2191(T (\u2191(piecewise s hs (const \u03b1 x) (const \u03b1 0)) \u207b\u00b9' {x_1})) x_1 =\n    \u2191(T s) x\n\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : NormedAddCommGroup G\nm\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nT : Set \u03b1 \u2192 F \u2192L[\u211d] F'\nhT_empty : T \u2205 = 0\nm : MeasurableSpace \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nx : F\nhs_empty : Set.Nonempty s\nhs_univ : s \u2260 Set.univ\nhx0 : \u00acx = 0\n\u22a2 \u00acx \u2208 {0}"}, {"tactic": "swap", "annotated_tactic": ["swap", []], "state_before": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : NormedAddCommGroup G\nm\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nT : Set \u03b1 \u2192 F \u2192L[\u211d] F'\nhT_empty : T \u2205 = 0\nm : MeasurableSpace \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nx : F\nhs_empty : Set.Nonempty s\nhs_univ : s \u2260 Set.univ\nhx0 : \u00acx = 0\n\u22a2 \u2191(T (\u2191(piecewise s hs (const \u03b1 x) (const \u03b1 0)) \u207b\u00b9' {x})) x +\n      \u2211 x_1 in {0}, \u2191(T (\u2191(piecewise s hs (const \u03b1 x) (const \u03b1 0)) \u207b\u00b9' {x_1})) x_1 =\n    \u2191(T s) x\n\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : NormedAddCommGroup G\nm\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nT : Set \u03b1 \u2192 F \u2192L[\u211d] F'\nhT_empty : T \u2205 = 0\nm : MeasurableSpace \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nx : F\nhs_empty : Set.Nonempty s\nhs_univ : s \u2260 Set.univ\nhx0 : \u00acx = 0\n\u22a2 \u00acx \u2208 {0}", "state_after": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : NormedAddCommGroup G\nm\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nT : Set \u03b1 \u2192 F \u2192L[\u211d] F'\nhT_empty : T \u2205 = 0\nm : MeasurableSpace \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nx : F\nhs_empty : Set.Nonempty s\nhs_univ : s \u2260 Set.univ\nhx0 : \u00acx = 0\n\u22a2 \u00acx \u2208 {0}\n\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : NormedAddCommGroup G\nm\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nT : Set \u03b1 \u2192 F \u2192L[\u211d] F'\nhT_empty : T \u2205 = 0\nm : MeasurableSpace \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nx : F\nhs_empty : Set.Nonempty s\nhs_univ : s \u2260 Set.univ\nhx0 : \u00acx = 0\n\u22a2 \u2191(T (\u2191(piecewise s hs (const \u03b1 x) (const \u03b1 0)) \u207b\u00b9' {x})) x +\n      \u2211 x_1 in {0}, \u2191(T (\u2191(piecewise s hs (const \u03b1 x) (const \u03b1 0)) \u207b\u00b9' {x_1})) x_1 =\n    \u2191(T s) x"}, {"tactic": "rw [sum_singleton, (T _).map_zero, add_zero]", "annotated_tactic": ["rw [<a>sum_singleton</a>, (T _).<a>map_zero</a>, <a>add_zero</a>]", [{"full_name": "Finset.sum_singleton", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [343, 3], "def_end_pos": [343, 14]}, {"full_name": "ContinuousLinearMap.map_zero", "def_path": "Mathlib/Topology/Algebra/Module/Basic.lean", "def_pos": [506, 19], "def_end_pos": [506, 27]}, {"full_name": "add_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [469, 3], "def_end_pos": [469, 14]}]], "state_before": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : NormedAddCommGroup G\nm\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nT : Set \u03b1 \u2192 F \u2192L[\u211d] F'\nhT_empty : T \u2205 = 0\nm : MeasurableSpace \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nx : F\nhs_empty : Set.Nonempty s\nhs_univ : s \u2260 Set.univ\nhx0 : \u00acx = 0\n\u22a2 \u2191(T (\u2191(piecewise s hs (const \u03b1 x) (const \u03b1 0)) \u207b\u00b9' {x})) x +\n      \u2211 x_1 in {0}, \u2191(T (\u2191(piecewise s hs (const \u03b1 x) (const \u03b1 0)) \u207b\u00b9' {x_1})) x_1 =\n    \u2191(T s) x", "state_after": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : NormedAddCommGroup G\nm\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nT : Set \u03b1 \u2192 F \u2192L[\u211d] F'\nhT_empty : T \u2205 = 0\nm : MeasurableSpace \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nx : F\nhs_empty : Set.Nonempty s\nhs_univ : s \u2260 Set.univ\nhx0 : \u00acx = 0\n\u22a2 \u2191(T (\u2191(piecewise s hs (const \u03b1 x) (const \u03b1 0)) \u207b\u00b9' {x})) x = \u2191(T s) x"}, {"tactic": "congr", "annotated_tactic": ["congr", []], "state_before": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : NormedAddCommGroup G\nm\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nT : Set \u03b1 \u2192 F \u2192L[\u211d] F'\nhT_empty : T \u2205 = 0\nm : MeasurableSpace \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nx : F\nhs_empty : Set.Nonempty s\nhs_univ : s \u2260 Set.univ\nhx0 : \u00acx = 0\n\u22a2 \u2191(T (\u2191(piecewise s hs (const \u03b1 x) (const \u03b1 0)) \u207b\u00b9' {x})) x = \u2191(T s) x", "state_after": "case neg.e_a.e_a\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : NormedAddCommGroup G\nm\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nT : Set \u03b1 \u2192 F \u2192L[\u211d] F'\nhT_empty : T \u2205 = 0\nm : MeasurableSpace \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nx : F\nhs_empty : Set.Nonempty s\nhs_univ : s \u2260 Set.univ\nhx0 : \u00acx = 0\n\u22a2 \u2191(piecewise s hs (const \u03b1 x) (const \u03b1 0)) \u207b\u00b9' {x} = s"}, {"tactic": "simp only [coe_piecewise, piecewise_eq_indicator, coe_const, Pi.const_zero,\n  piecewise_eq_indicator]", "annotated_tactic": ["simp only [<a>coe_piecewise</a>, <a>piecewise_eq_indicator</a>, <a>coe_const</a>, <a>Pi.const_zero</a>,\n    <a>piecewise_eq_indicator</a>]", [{"full_name": "MeasureTheory.SimpleFunc.coe_piecewise", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [230, 9], "def_end_pos": [230, 22]}, {"full_name": "Set.piecewise_eq_indicator", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [52, 3], "def_end_pos": [52, 14]}, {"full_name": "MeasureTheory.SimpleFunc.coe_const", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [158, 9], "def_end_pos": [158, 18]}, {"full_name": "Pi.const_zero", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [58, 3], "def_end_pos": [58, 14]}, {"full_name": "Set.piecewise_eq_indicator", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [52, 3], "def_end_pos": [52, 14]}]], "state_before": "case neg.e_a.e_a\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : NormedAddCommGroup G\nm\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nT : Set \u03b1 \u2192 F \u2192L[\u211d] F'\nhT_empty : T \u2205 = 0\nm : MeasurableSpace \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nx : F\nhs_empty : Set.Nonempty s\nhs_univ : s \u2260 Set.univ\nhx0 : \u00acx = 0\n\u22a2 \u2191(piecewise s hs (const \u03b1 x) (const \u03b1 0)) \u207b\u00b9' {x} = s", "state_after": "case neg.e_a.e_a\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : NormedAddCommGroup G\nm\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nT : Set \u03b1 \u2192 F \u2192L[\u211d] F'\nhT_empty : T \u2205 = 0\nm : MeasurableSpace \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nx : F\nhs_empty : Set.Nonempty s\nhs_univ : s \u2260 Set.univ\nhx0 : \u00acx = 0\n\u22a2 indicator s (Function.const \u03b1 x) \u207b\u00b9' {x} = s"}, {"tactic": "rw [indicator_preimage, \u2190 Function.const_def, preimage_const_of_mem]", "annotated_tactic": ["rw [<a>indicator_preimage</a>, \u2190 <a>Function.const_def</a>, <a>preimage_const_of_mem</a>]", [{"full_name": "Set.indicator_preimage", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [276, 3], "def_end_pos": [276, 14]}, {"full_name": "Function.const_def", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [36, 9], "def_end_pos": [36, 18]}, {"full_name": "Set.preimage_const_of_mem", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [138, 9], "def_end_pos": [138, 30]}]], "state_before": "case neg.e_a.e_a\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : NormedAddCommGroup G\nm\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nT : Set \u03b1 \u2192 F \u2192L[\u211d] F'\nhT_empty : T \u2205 = 0\nm : MeasurableSpace \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nx : F\nhs_empty : Set.Nonempty s\nhs_univ : s \u2260 Set.univ\nhx0 : \u00acx = 0\n\u22a2 indicator s (Function.const \u03b1 x) \u207b\u00b9' {x} = s", "state_after": "case neg.e_a.e_a\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : NormedAddCommGroup G\nm\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nT : Set \u03b1 \u2192 F \u2192L[\u211d] F'\nhT_empty : T \u2205 = 0\nm : MeasurableSpace \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nx : F\nhs_empty : Set.Nonempty s\nhs_univ : s \u2260 Set.univ\nhx0 : \u00acx = 0\n\u22a2 Set.ite s Set.univ (0 \u207b\u00b9' {x}) = s\n\ncase neg.e_a.e_a\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : NormedAddCommGroup G\nm\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nT : Set \u03b1 \u2192 F \u2192L[\u211d] F'\nhT_empty : T \u2205 = 0\nm : MeasurableSpace \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nx : F\nhs_empty : Set.Nonempty s\nhs_univ : s \u2260 Set.univ\nhx0 : \u00acx = 0\n\u22a2 x \u2208 {x}"}, {"tactic": "swap", "annotated_tactic": ["swap", []], "state_before": "case neg.e_a.e_a\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : NormedAddCommGroup G\nm\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nT : Set \u03b1 \u2192 F \u2192L[\u211d] F'\nhT_empty : T \u2205 = 0\nm : MeasurableSpace \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nx : F\nhs_empty : Set.Nonempty s\nhs_univ : s \u2260 Set.univ\nhx0 : \u00acx = 0\n\u22a2 Set.ite s Set.univ (0 \u207b\u00b9' {x}) = s\n\ncase neg.e_a.e_a\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : NormedAddCommGroup G\nm\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nT : Set \u03b1 \u2192 F \u2192L[\u211d] F'\nhT_empty : T \u2205 = 0\nm : MeasurableSpace \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nx : F\nhs_empty : Set.Nonempty s\nhs_univ : s \u2260 Set.univ\nhx0 : \u00acx = 0\n\u22a2 x \u2208 {x}", "state_after": "case neg.e_a.e_a\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : NormedAddCommGroup G\nm\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nT : Set \u03b1 \u2192 F \u2192L[\u211d] F'\nhT_empty : T \u2205 = 0\nm : MeasurableSpace \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nx : F\nhs_empty : Set.Nonempty s\nhs_univ : s \u2260 Set.univ\nhx0 : \u00acx = 0\n\u22a2 x \u2208 {x}\n\ncase neg.e_a.e_a\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : NormedAddCommGroup G\nm\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nT : Set \u03b1 \u2192 F \u2192L[\u211d] F'\nhT_empty : T \u2205 = 0\nm : MeasurableSpace \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nx : F\nhs_empty : Set.Nonempty s\nhs_univ : s \u2260 Set.univ\nhx0 : \u00acx = 0\n\u22a2 Set.ite s Set.univ (0 \u207b\u00b9' {x}) = s"}, {"tactic": "rw [\u2190 Pi.const_zero, \u2190 Function.const_def, preimage_const_of_not_mem]", "annotated_tactic": ["rw [\u2190 <a>Pi.const_zero</a>, \u2190 <a>Function.const_def</a>, <a>preimage_const_of_not_mem</a>]", [{"full_name": "Pi.const_zero", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [58, 3], "def_end_pos": [58, 14]}, {"full_name": "Function.const_def", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [36, 9], "def_end_pos": [36, 18]}, {"full_name": "Set.preimage_const_of_not_mem", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [143, 9], "def_end_pos": [143, 34]}]], "state_before": "case neg.e_a.e_a\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : NormedAddCommGroup G\nm\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nT : Set \u03b1 \u2192 F \u2192L[\u211d] F'\nhT_empty : T \u2205 = 0\nm : MeasurableSpace \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nx : F\nhs_empty : Set.Nonempty s\nhs_univ : s \u2260 Set.univ\nhx0 : \u00acx = 0\n\u22a2 Set.ite s Set.univ (0 \u207b\u00b9' {x}) = s", "state_after": "case neg.e_a.e_a\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : NormedAddCommGroup G\nm\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nT : Set \u03b1 \u2192 F \u2192L[\u211d] F'\nhT_empty : T \u2205 = 0\nm : MeasurableSpace \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nx : F\nhs_empty : Set.Nonempty s\nhs_univ : s \u2260 Set.univ\nhx0 : \u00acx = 0\n\u22a2 Set.ite s Set.univ \u2205 = s\n\ncase neg.e_a.e_a\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : NormedAddCommGroup G\nm\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nT : Set \u03b1 \u2192 F \u2192L[\u211d] F'\nhT_empty : T \u2205 = 0\nm : MeasurableSpace \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nx : F\nhs_empty : Set.Nonempty s\nhs_univ : s \u2260 Set.univ\nhx0 : \u00acx = 0\n\u22a2 \u00ac0 \u2208 {x}"}, {"tactic": "swap", "annotated_tactic": ["swap", []], "state_before": "case neg.e_a.e_a\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : NormedAddCommGroup G\nm\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nT : Set \u03b1 \u2192 F \u2192L[\u211d] F'\nhT_empty : T \u2205 = 0\nm : MeasurableSpace \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nx : F\nhs_empty : Set.Nonempty s\nhs_univ : s \u2260 Set.univ\nhx0 : \u00acx = 0\n\u22a2 Set.ite s Set.univ \u2205 = s\n\ncase neg.e_a.e_a\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : NormedAddCommGroup G\nm\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nT : Set \u03b1 \u2192 F \u2192L[\u211d] F'\nhT_empty : T \u2205 = 0\nm : MeasurableSpace \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nx : F\nhs_empty : Set.Nonempty s\nhs_univ : s \u2260 Set.univ\nhx0 : \u00acx = 0\n\u22a2 \u00ac0 \u2208 {x}", "state_after": "case neg.e_a.e_a\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : NormedAddCommGroup G\nm\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nT : Set \u03b1 \u2192 F \u2192L[\u211d] F'\nhT_empty : T \u2205 = 0\nm : MeasurableSpace \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nx : F\nhs_empty : Set.Nonempty s\nhs_univ : s \u2260 Set.univ\nhx0 : \u00acx = 0\n\u22a2 \u00ac0 \u2208 {x}\n\ncase neg.e_a.e_a\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : NormedAddCommGroup G\nm\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nT : Set \u03b1 \u2192 F \u2192L[\u211d] F'\nhT_empty : T \u2205 = 0\nm : MeasurableSpace \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nx : F\nhs_empty : Set.Nonempty s\nhs_univ : s \u2260 Set.univ\nhx0 : \u00acx = 0\n\u22a2 Set.ite s Set.univ \u2205 = s"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case neg.e_a.e_a\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : NormedAddCommGroup G\nm\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nT : Set \u03b1 \u2192 F \u2192L[\u211d] F'\nhT_empty : T \u2205 = 0\nm : MeasurableSpace \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nx : F\nhs_empty : Set.Nonempty s\nhs_univ : s \u2260 Set.univ\nhx0 : \u00acx = 0\n\u22a2 Set.ite s Set.univ \u2205 = s", "state_after": "no goals"}, {"tactic": "simp only [hT_empty, ContinuousLinearMap.zero_apply, piecewise_empty, const_zero,\n  setToSimpleFunc_zero_apply]", "annotated_tactic": ["simp only [hT_empty, <a>ContinuousLinearMap.zero_apply</a>, <a>piecewise_empty</a>, <a>const_zero</a>,\n      <a>setToSimpleFunc_zero_apply</a>]", [{"full_name": "ContinuousLinearMap.zero_apply", "def_path": "Mathlib/Topology/Algebra/Module/Basic.lean", "def_pos": [644, 9], "def_end_pos": [644, 19]}, {"full_name": "MeasureTheory.SimpleFunc.piecewise_empty", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [252, 9], "def_end_pos": [252, 24]}, {"full_name": "MeasureTheory.SimpleFunc.const_zero", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [457, 3], "def_end_pos": [457, 14]}, {"full_name": "MeasureTheory.SimpleFunc.setToSimpleFunc_zero_apply", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [303, 9], "def_end_pos": [303, 35]}]], "state_before": "case inl\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : NormedAddCommGroup G\nm\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nT : Set \u03b1 \u2192 F \u2192L[\u211d] F'\nhT_empty : T \u2205 = 0\nm : MeasurableSpace \u03b1\nx : F\nhs : MeasurableSet \u2205\n\u22a2 setToSimpleFunc T (piecewise \u2205 hs (const \u03b1 x) (const \u03b1 0)) = \u2191(T \u2205) x", "state_after": "no goals"}, {"tactic": "haveI h\u03b1 := hs_empty.to_type", "annotated_tactic": ["haveI h\u03b1 := hs_empty.to_type", []], "state_before": "case inr.inl\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : NormedAddCommGroup G\nm\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nT : Set \u03b1 \u2192 F \u2192L[\u211d] F'\nhT_empty : T \u2205 = 0\nm : MeasurableSpace \u03b1\nx : F\nhs : MeasurableSet Set.univ\nhs_empty : Set.Nonempty Set.univ\n\u22a2 \u2211 x_1 in SimpleFunc.range (piecewise Set.univ hs (const \u03b1 x) (const \u03b1 0)),\n      \u2191(T (\u2191(piecewise Set.univ hs (const \u03b1 x) (const \u03b1 0)) \u207b\u00b9' {x_1})) x_1 =\n    \u2191(T Set.univ) x", "state_after": "case inr.inl\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : NormedAddCommGroup G\nm\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nT : Set \u03b1 \u2192 F \u2192L[\u211d] F'\nhT_empty : T \u2205 = 0\nm : MeasurableSpace \u03b1\nx : F\nhs : MeasurableSet Set.univ\nhs_empty : Set.Nonempty Set.univ\nh\u03b1 : Nonempty \u03b1\n\u22a2 \u2211 x_1 in SimpleFunc.range (piecewise Set.univ hs (const \u03b1 x) (const \u03b1 0)),\n      \u2191(T (\u2191(piecewise Set.univ hs (const \u03b1 x) (const \u03b1 0)) \u207b\u00b9' {x_1})) x_1 =\n    \u2191(T Set.univ) x"}, {"tactic": "simp [\u2190 Function.const_def]", "annotated_tactic": ["simp [\u2190 <a>Function.const_def</a>]", [{"full_name": "Function.const_def", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [36, 9], "def_end_pos": [36, 18]}]], "state_before": "case inr.inl\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : NormedAddCommGroup G\nm\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nT : Set \u03b1 \u2192 F \u2192L[\u211d] F'\nhT_empty : T \u2205 = 0\nm : MeasurableSpace \u03b1\nx : F\nhs : MeasurableSet Set.univ\nhs_empty : Set.Nonempty Set.univ\nh\u03b1 : Nonempty \u03b1\n\u22a2 \u2211 x_1 in SimpleFunc.range (piecewise Set.univ hs (const \u03b1 x) (const \u03b1 0)),\n      \u2191(T (\u2191(piecewise Set.univ hs (const \u03b1 x) (const \u03b1 0)) \u207b\u00b9' {x_1})) x_1 =\n    \u2191(T Set.univ) x", "state_after": "no goals"}, {"tactic": "simp_rw [hx0]", "annotated_tactic": ["simp_rw [hx0]", []], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : NormedAddCommGroup G\nm\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nT : Set \u03b1 \u2192 F \u2192L[\u211d] F'\nhT_empty : T \u2205 = 0\nm : MeasurableSpace \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nx : F\nhs_empty : Set.Nonempty s\nhs_univ : s \u2260 Set.univ\nhx0 : x = 0\n\u22a2 \u2211 x_1 in {x, 0}, \u2191(T (\u2191(piecewise s hs (const \u03b1 x) (const \u03b1 0)) \u207b\u00b9' {x_1})) x_1 = \u2191(T s) x", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : NormedAddCommGroup G\nm\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nT : Set \u03b1 \u2192 F \u2192L[\u211d] F'\nhT_empty : T \u2205 = 0\nm : MeasurableSpace \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nx : F\nhs_empty : Set.Nonempty s\nhs_univ : s \u2260 Set.univ\nhx0 : x = 0\n\u22a2 \u2211 x in {0, 0}, \u2191(T (\u2191(piecewise s hs (const \u03b1 0) (const \u03b1 0)) \u207b\u00b9' {x})) x = \u2191(T s) 0"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : NormedAddCommGroup G\nm\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nT : Set \u03b1 \u2192 F \u2192L[\u211d] F'\nhT_empty : T \u2205 = 0\nm : MeasurableSpace \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nx : F\nhs_empty : Set.Nonempty s\nhs_univ : s \u2260 Set.univ\nhx0 : x = 0\n\u22a2 \u2211 x in {0, 0}, \u2191(T (\u2191(piecewise s hs (const \u03b1 0) (const \u03b1 0)) \u207b\u00b9' {x})) x = \u2191(T s) 0", "state_after": "no goals"}, {"tactic": "rw [Finset.mem_singleton]", "annotated_tactic": ["rw [<a>Finset.mem_singleton</a>]", [{"full_name": "Finset.mem_singleton", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [678, 9], "def_end_pos": [678, 22]}]], "state_before": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : NormedAddCommGroup G\nm\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nT : Set \u03b1 \u2192 F \u2192L[\u211d] F'\nhT_empty : T \u2205 = 0\nm : MeasurableSpace \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nx : F\nhs_empty : Set.Nonempty s\nhs_univ : s \u2260 Set.univ\nhx0 : \u00acx = 0\n\u22a2 \u00acx \u2208 {0}", "state_after": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : NormedAddCommGroup G\nm\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nT : Set \u03b1 \u2192 F \u2192L[\u211d] F'\nhT_empty : T \u2205 = 0\nm : MeasurableSpace \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nx : F\nhs_empty : Set.Nonempty s\nhs_univ : s \u2260 Set.univ\nhx0 : \u00acx = 0\n\u22a2 \u00acx = 0"}, {"tactic": "exact hx0", "annotated_tactic": ["exact hx0", []], "state_before": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : NormedAddCommGroup G\nm\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nT : Set \u03b1 \u2192 F \u2192L[\u211d] F'\nhT_empty : T \u2205 = 0\nm : MeasurableSpace \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nx : F\nhs_empty : Set.Nonempty s\nhs_univ : s \u2260 Set.univ\nhx0 : \u00acx = 0\n\u22a2 \u00acx = 0", "state_after": "no goals"}, {"tactic": "exact Set.mem_singleton x", "annotated_tactic": ["exact <a>Set.mem_singleton</a> x", [{"full_name": "Set.mem_singleton", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1289, 9], "def_end_pos": [1289, 22]}]], "state_before": "case neg.e_a.e_a\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : NormedAddCommGroup G\nm\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nT : Set \u03b1 \u2192 F \u2192L[\u211d] F'\nhT_empty : T \u2205 = 0\nm : MeasurableSpace \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nx : F\nhs_empty : Set.Nonempty s\nhs_univ : s \u2260 Set.univ\nhx0 : \u00acx = 0\n\u22a2 x \u2208 {x}", "state_after": "no goals"}, {"tactic": "rw [Set.mem_singleton_iff]", "annotated_tactic": ["rw [<a>Set.mem_singleton_iff</a>]", [{"full_name": "Set.mem_singleton_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1273, 9], "def_end_pos": [1273, 26]}]], "state_before": "case neg.e_a.e_a\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : NormedAddCommGroup G\nm\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nT : Set \u03b1 \u2192 F \u2192L[\u211d] F'\nhT_empty : T \u2205 = 0\nm : MeasurableSpace \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nx : F\nhs_empty : Set.Nonempty s\nhs_univ : s \u2260 Set.univ\nhx0 : \u00acx = 0\n\u22a2 \u00ac0 \u2208 {x}", "state_after": "case neg.e_a.e_a\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : NormedAddCommGroup G\nm\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nT : Set \u03b1 \u2192 F \u2192L[\u211d] F'\nhT_empty : T \u2205 = 0\nm : MeasurableSpace \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nx : F\nhs_empty : Set.Nonempty s\nhs_univ : s \u2260 Set.univ\nhx0 : \u00acx = 0\n\u22a2 \u00ac0 = x"}, {"tactic": "exact Ne.symm hx0", "annotated_tactic": ["exact <a>Ne.symm</a> hx0", [{"full_name": "Ne.symm", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [575, 9], "def_end_pos": [575, 16]}]], "state_before": "case neg.e_a.e_a\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : NormedAddCommGroup G\nm\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nT : Set \u03b1 \u2192 F \u2192L[\u211d] F'\nhT_empty : T \u2205 = 0\nm : MeasurableSpace \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nx : F\nhs_empty : Set.Nonempty s\nhs_univ : s \u2260 Set.univ\nhx0 : \u00acx = 0\n\u22a2 \u00ac0 = x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "Finset.filter_subset", "start": [2701, 1], "end": [2702, 29], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "full_name": "ContinuousMap.toLp_injective", "start": [1872, 1], "end": [1874, 101], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "Finset.erase_inter_comm", "start": [2299, 1], "end": [2300, 32], "traced_tactics": [{"tactic": "rw [erase_inter, inter_erase]", "annotated_tactic": ["rw [<a>erase_inter</a>, <a>inter_erase</a>]", [{"full_name": "Finset.erase_inter", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2287, 9], "def_end_pos": [2287, 20]}, {"full_name": "Finset.inter_erase", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2282, 9], "def_end_pos": [2282, 20]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d u v : Finset \u03b1\na\u271d b : \u03b1\ns t : Finset \u03b1\na : \u03b1\n\u22a2 erase s a \u2229 t = s \u2229 erase t a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/AEEqFun.lean", "full_name": "MeasureTheory.AEEqFun.compQuasiMeasurePreserving_eq_mk", "start": [232, 1], "end": [235, 75], "traced_tactics": [{"tactic": "rw [\u2190 compQuasiMeasurePreserving_mk g.aestronglyMeasurable hf, mk_coeFn]", "annotated_tactic": ["rw [\u2190 <a>compQuasiMeasurePreserving_mk</a> g.aestronglyMeasurable hf, <a>mk_coeFn</a>]", [{"full_name": "MeasureTheory.AEEqFun.compQuasiMeasurePreserving_mk", "def_path": "Mathlib/MeasureTheory/Function/AEEqFun.lean", "def_pos": [227, 9], "def_end_pos": [227, 38]}, {"full_name": "MeasureTheory.AEEqFun.mk_coeFn", "def_path": "Mathlib/MeasureTheory/Function/AEEqFun.lean", "def_pos": [165, 9], "def_end_pos": [165, 17]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2074 : MeasurableSpace \u03b1\n\u03bc \u03bd\u271d : Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\ninst\u271d\u00b9 : TopologicalSpace \u03b4\ninst\u271d : MeasurableSpace \u03b2\n\u03bd : Measure \u03b2\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192\u2098[\u03bd] \u03b3\nhf : QuasiMeasurePreserving f\n\u22a2 compQuasiMeasurePreserving g f hf = mk (\u2191g \u2218 f) (_ : AEStronglyMeasurable (\u2191g \u2218 f) \u03bc)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Lebesgue/Complex.lean", "full_name": "Complex.volume_preserving_equiv_pi", "start": [39, 1], "end": [45, 90], "traced_tactics": [{"tactic": "convert (measurableEquivPi.symm.measurable.measurePreserving volume).symm", "annotated_tactic": ["convert (measurableEquivPi.symm.measurable.measurePreserving <a>volume</a>).<a>symm</a>", [{"full_name": "MeasureTheory.MeasureSpace.volume", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [663, 3], "def_end_pos": [663, 9]}, {"full_name": "MeasureTheory.MeasurePreserving.symm", "def_path": "Mathlib/Dynamics/Ergodic/MeasurePreserving.lean", "def_pos": [70, 9], "def_end_pos": [70, 13]}]], "state_before": "\u22a2 MeasurePreserving \u2191measurableEquivPi", "state_after": "case h.e'_6\n\n\u22a2 volume = Measure.map (\u2191(MeasurableEquiv.symm measurableEquivPi)) volume"}, {"tactic": "rw [\u2190 addHaarMeasure_eq_volume_pi, \u2190 Basis.parallelepiped_basisFun, \u2190 Basis.addHaar,\n  measurableEquivPi, Homeomorph.toMeasurableEquiv_symm_coe,\n  ContinuousLinearEquiv.symm_toHomeomorph, ContinuousLinearEquiv.coe_toHomeomorph,\n  Basis.map_addHaar, eq_comm]", "annotated_tactic": ["rw [\u2190 <a>addHaarMeasure_eq_volume_pi</a>, \u2190 <a>Basis.parallelepiped_basisFun</a>, \u2190 <a>Basis.addHaar</a>,\n    <a>measurableEquivPi</a>, <a>Homeomorph.toMeasurableEquiv_symm_coe</a>,\n    <a>ContinuousLinearEquiv.symm_toHomeomorph</a>, <a>ContinuousLinearEquiv.coe_toHomeomorph</a>,\n    <a>Basis.map_addHaar</a>, <a>eq_comm</a>]", [{"full_name": "MeasureTheory.addHaarMeasure_eq_volume_pi", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean", "def_pos": [120, 9], "def_end_pos": [120, 36]}, {"full_name": "Basis.parallelepiped_basisFun", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean", "def_pos": [74, 9], "def_end_pos": [74, 38]}, {"full_name": "Basis.addHaar", "def_path": "Mathlib/MeasureTheory/Measure/Haar/OfBasis.lean", "def_pos": [220, 1], "def_end_pos": [223, 42]}, {"full_name": "Complex.measurableEquivPi", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/Complex.lean", "def_pos": [30, 5], "def_end_pos": [30, 22]}, {"full_name": "Homeomorph.toMeasurableEquiv_symm_coe", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [998, 9], "def_end_pos": [998, 46]}, {"full_name": "ContinuousLinearEquiv.symm_toHomeomorph", "def_path": "Mathlib/Topology/Algebra/Module/Basic.lean", "def_pos": [2028, 9], "def_end_pos": [2028, 26]}, {"full_name": "ContinuousLinearEquiv.coe_toHomeomorph", "def_path": "Mathlib/Topology/Algebra/Module/Basic.lean", "def_pos": [1903, 9], "def_end_pos": [1903, 25]}, {"full_name": "Basis.map_addHaar", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean", "def_pos": [95, 9], "def_end_pos": [95, 26]}, {"full_name": "eq_comm", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [104, 9], "def_end_pos": [104, 16]}]], "state_before": "case h.e'_6\n\n\u22a2 volume = Measure.map (\u2191(MeasurableEquiv.symm measurableEquivPi)) volume", "state_after": "case h.e'_6\n\n\u22a2 Basis.addHaar\n      (Basis.map (Pi.basisFun \u211d (Fin 2))\n        (ContinuousLinearEquiv.symm (LinearEquiv.toContinuousLinearEquiv (Basis.equivFun basisOneI))).toLinearEquiv) =\n    volume"}, {"tactic": "exact (Basis.addHaar_eq_iff _ _).mpr Complex.orthonormalBasisOneI.volume_parallelepiped", "annotated_tactic": ["exact (<a>Basis.addHaar_eq_iff</a> _ _).<a>mpr</a> Complex.orthonormalBasisOneI.volume_parallelepiped", [{"full_name": "Basis.addHaar_eq_iff", "def_path": "Mathlib/MeasureTheory/Measure/Haar/OfBasis.lean", "def_pos": [236, 9], "def_end_pos": [236, 29]}, {"full_name": "Iff.mpr", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [92, 3], "def_end_pos": [92, 6]}]], "state_before": "case h.e'_6\n\n\u22a2 Basis.addHaar\n      (Basis.map (Pi.basisFun \u211d (Fin 2))\n        (ContinuousLinearEquiv.symm (LinearEquiv.toContinuousLinearEquiv (Basis.equivFun basisOneI))).toLinearEquiv) =\n    volume", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Kernel/Basic.lean", "full_name": "ProbabilityTheory.IsFiniteKernel.bound_ne_top", "start": [137, 1], "end": [139, 37], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Fold.lean", "full_name": "Finset.fold_ite", "start": [158, 1], "end": [161, 42], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Basic.lean", "full_name": "Set.nontrivial_iff_pair_subset", "start": [2502, 1], "end": [2505, 37], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Martingale/OptionalStopping.lean", "full_name": "MeasureTheory.submartingale_of_expected_stoppedValue_mono", "start": [69, 1], "end": [80, 75], "traced_tactics": [{"tactic": "refine' submartingale_of_set_integral_le hadp hint fun i j hij s hs => _", "annotated_tactic": ["refine' <a>submartingale_of_set_integral_le</a> hadp hint fun i j hij s hs => _", [{"full_name": "MeasureTheory.submartingale_of_set_integral_le", "def_path": "Mathlib/Probability/Martingale/Basic.lean", "def_pos": [288, 9], "def_end_pos": [288, 41]}]], "state_before": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhadp : Adapted \ud835\udca2 f\nhint : \u2200 (i : \u2115), Integrable (f i)\nhf :\n  \u2200 (\u03c4 \u03c0 : \u03a9 \u2192 \u2115),\n    IsStoppingTime \ud835\udca2 \u03c4 \u2192\n      IsStoppingTime \ud835\udca2 \u03c0 \u2192\n        \u03c4 \u2264 \u03c0 \u2192 (\u2203 N, \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 N) \u2192 \u222b (x : \u03a9), stoppedValue f \u03c4 x \u2202\u03bc \u2264 \u222b (x : \u03a9), stoppedValue f \u03c0 x \u2202\u03bc\n\u22a2 Submartingale f \ud835\udca2 \u03bc", "state_after": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhadp : Adapted \ud835\udca2 f\nhint : \u2200 (i : \u2115), Integrable (f i)\nhf :\n  \u2200 (\u03c4 \u03c0 : \u03a9 \u2192 \u2115),\n    IsStoppingTime \ud835\udca2 \u03c4 \u2192\n      IsStoppingTime \ud835\udca2 \u03c0 \u2192\n        \u03c4 \u2264 \u03c0 \u2192 (\u2203 N, \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 N) \u2192 \u222b (x : \u03a9), stoppedValue f \u03c4 x \u2202\u03bc \u2264 \u222b (x : \u03a9), stoppedValue f \u03c0 x \u2202\u03bc\ni j : \u2115\nhij : i \u2264 j\ns : Set \u03a9\nhs : MeasurableSet s\n\u22a2 \u222b (\u03c9 : \u03a9) in s, f i \u03c9 \u2202\u03bc \u2264 \u222b (\u03c9 : \u03a9) in s, f j \u03c9 \u2202\u03bc"}, {"tactic": "classical\nspecialize hf (s.piecewise (fun _ => i) fun _ => j) _ (isStoppingTime_piecewise_const hij hs)\n  (isStoppingTime_const \ud835\udca2 j) (fun x => (ite_le_sup _ _ (x \u2208 s)).trans (max_eq_right hij).le)\n  \u27e8j, fun _ => le_rfl\u27e9\nrwa [stoppedValue_const, stoppedValue_piecewise_const,\n  integral_piecewise (\ud835\udca2.le _ _ hs) (hint _).integrableOn (hint _).integrableOn, \u2190\n  integral_add_compl (\ud835\udca2.le _ _ hs) (hint j), add_le_add_iff_right] at hf", "annotated_tactic": ["classical\n  specialize hf (s.piecewise (fun _ => i) fun _ => j) _ (<a>isStoppingTime_piecewise_const</a> hij hs)\n    (<a>isStoppingTime_const</a> \ud835\udca2 j) (fun x => (<a>ite_le_sup</a> _ _ (x \u2208 s)).<a>trans</a> (<a>max_eq_right</a> hij).<a>le</a>)\n    \u27e8j, fun _ => <a>le_rfl</a>\u27e9\n  rwa [<a>stoppedValue_const</a>, <a>stoppedValue_piecewise_const</a>,\n    <a>integral_piecewise</a> (\ud835\udca2.le _ _ hs) (hint _).<a>integrableOn</a> (hint _).<a>integrableOn</a>, \u2190\n    <a>integral_add_compl</a> (\ud835\udca2.le _ _ hs) (hint j), <a>add_le_add_iff_right</a>] at hf", [{"full_name": "MeasureTheory.isStoppingTime_piecewise_const", "def_path": "Mathlib/Probability/Process/Stopping.lean", "def_pos": [1140, 9], "def_end_pos": [1140, 39]}, {"full_name": "MeasureTheory.isStoppingTime_const", "def_path": "Mathlib/Probability/Process/Stopping.lean", "def_pos": [57, 9], "def_end_pos": [57, 29]}, {"full_name": "ite_le_sup", "def_path": "Mathlib/Order/Lattice.lean", "def_pos": [341, 9], "def_end_pos": [341, 19]}, {"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [120, 7], "def_end_pos": [120, 18]}, {"full_name": "max_eq_right", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [137, 9], "def_end_pos": [137, 21]}, {"full_name": "Eq.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [159, 7], "def_end_pos": [159, 12]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}, {"full_name": "MeasureTheory.stoppedValue_const", "def_path": "Mathlib/Probability/Process/Stopping.lean", "def_pos": [771, 9], "def_end_pos": [771, 27]}, {"full_name": "MeasureTheory.stoppedValue_piecewise_const", "def_path": "Mathlib/Probability/Process/Stopping.lean", "def_pos": [1146, 9], "def_end_pos": [1146, 37]}, {"full_name": "MeasureTheory.integral_piecewise", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [204, 9], "def_end_pos": [204, 27]}, {"full_name": "MeasureTheory.Integrable.integrableOn", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [163, 9], "def_end_pos": [163, 32]}, {"full_name": "MeasureTheory.Integrable.integrableOn", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [163, 9], "def_end_pos": [163, 32]}, {"full_name": "MeasureTheory.integral_add_compl", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [162, 9], "def_end_pos": [162, 27]}, {"full_name": "add_le_add_iff_right", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [90, 3], "def_end_pos": [90, 14]}]], "state_before": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhadp : Adapted \ud835\udca2 f\nhint : \u2200 (i : \u2115), Integrable (f i)\nhf :\n  \u2200 (\u03c4 \u03c0 : \u03a9 \u2192 \u2115),\n    IsStoppingTime \ud835\udca2 \u03c4 \u2192\n      IsStoppingTime \ud835\udca2 \u03c0 \u2192\n        \u03c4 \u2264 \u03c0 \u2192 (\u2203 N, \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 N) \u2192 \u222b (x : \u03a9), stoppedValue f \u03c4 x \u2202\u03bc \u2264 \u222b (x : \u03a9), stoppedValue f \u03c0 x \u2202\u03bc\ni j : \u2115\nhij : i \u2264 j\ns : Set \u03a9\nhs : MeasurableSet s\n\u22a2 \u222b (\u03c9 : \u03a9) in s, f i \u03c9 \u2202\u03bc \u2264 \u222b (\u03c9 : \u03a9) in s, f j \u03c9 \u2202\u03bc", "state_after": "no goals"}, {"tactic": "specialize hf (s.piecewise (fun _ => i) fun _ => j) _ (isStoppingTime_piecewise_const hij hs)\n  (isStoppingTime_const \ud835\udca2 j) (fun x => (ite_le_sup _ _ (x \u2208 s)).trans (max_eq_right hij).le)\n  \u27e8j, fun _ => le_rfl\u27e9", "annotated_tactic": ["specialize hf (s.piecewise (fun _ => i) fun _ => j) _ (<a>isStoppingTime_piecewise_const</a> hij hs)\n    (<a>isStoppingTime_const</a> \ud835\udca2 j) (fun x => (<a>ite_le_sup</a> _ _ (x \u2208 s)).<a>trans</a> (<a>max_eq_right</a> hij).<a>le</a>)\n    \u27e8j, fun _ => <a>le_rfl</a>\u27e9", [{"full_name": "MeasureTheory.isStoppingTime_piecewise_const", "def_path": "Mathlib/Probability/Process/Stopping.lean", "def_pos": [1140, 9], "def_end_pos": [1140, 39]}, {"full_name": "MeasureTheory.isStoppingTime_const", "def_path": "Mathlib/Probability/Process/Stopping.lean", "def_pos": [57, 9], "def_end_pos": [57, 29]}, {"full_name": "ite_le_sup", "def_path": "Mathlib/Order/Lattice.lean", "def_pos": [341, 9], "def_end_pos": [341, 19]}, {"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [120, 7], "def_end_pos": [120, 18]}, {"full_name": "max_eq_right", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [137, 9], "def_end_pos": [137, 21]}, {"full_name": "Eq.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [159, 7], "def_end_pos": [159, 12]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}]], "state_before": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhadp : Adapted \ud835\udca2 f\nhint : \u2200 (i : \u2115), Integrable (f i)\nhf :\n  \u2200 (\u03c4 \u03c0 : \u03a9 \u2192 \u2115),\n    IsStoppingTime \ud835\udca2 \u03c4 \u2192\n      IsStoppingTime \ud835\udca2 \u03c0 \u2192\n        \u03c4 \u2264 \u03c0 \u2192 (\u2203 N, \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 N) \u2192 \u222b (x : \u03a9), stoppedValue f \u03c4 x \u2202\u03bc \u2264 \u222b (x : \u03a9), stoppedValue f \u03c0 x \u2202\u03bc\ni j : \u2115\nhij : i \u2264 j\ns : Set \u03a9\nhs : MeasurableSet s\n\u22a2 \u222b (\u03c9 : \u03a9) in s, f i \u03c9 \u2202\u03bc \u2264 \u222b (\u03c9 : \u03a9) in s, f j \u03c9 \u2202\u03bc", "state_after": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhadp : Adapted \ud835\udca2 f\nhint : \u2200 (i : \u2115), Integrable (f i)\ni j : \u2115\nhij : i \u2264 j\ns : Set \u03a9\nhs : MeasurableSet s\nhf :\n  \u222b (x : \u03a9), stoppedValue f (Set.piecewise s (fun x => i) fun x => j) x \u2202\u03bc \u2264 \u222b (x : \u03a9), stoppedValue f (fun x => j) x \u2202\u03bc\n\u22a2 \u222b (\u03c9 : \u03a9) in s, f i \u03c9 \u2202\u03bc \u2264 \u222b (\u03c9 : \u03a9) in s, f j \u03c9 \u2202\u03bc"}, {"tactic": "rwa [stoppedValue_const, stoppedValue_piecewise_const,\n  integral_piecewise (\ud835\udca2.le _ _ hs) (hint _).integrableOn (hint _).integrableOn, \u2190\n  integral_add_compl (\ud835\udca2.le _ _ hs) (hint j), add_le_add_iff_right] at hf", "annotated_tactic": ["rwa [<a>stoppedValue_const</a>, <a>stoppedValue_piecewise_const</a>,\n    <a>integral_piecewise</a> (\ud835\udca2.le _ _ hs) (hint _).<a>integrableOn</a> (hint _).<a>integrableOn</a>, \u2190\n    <a>integral_add_compl</a> (\ud835\udca2.le _ _ hs) (hint j), <a>add_le_add_iff_right</a>] at hf", [{"full_name": "MeasureTheory.stoppedValue_const", "def_path": "Mathlib/Probability/Process/Stopping.lean", "def_pos": [771, 9], "def_end_pos": [771, 27]}, {"full_name": "MeasureTheory.stoppedValue_piecewise_const", "def_path": "Mathlib/Probability/Process/Stopping.lean", "def_pos": [1146, 9], "def_end_pos": [1146, 37]}, {"full_name": "MeasureTheory.integral_piecewise", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [204, 9], "def_end_pos": [204, 27]}, {"full_name": "MeasureTheory.Integrable.integrableOn", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [163, 9], "def_end_pos": [163, 32]}, {"full_name": "MeasureTheory.Integrable.integrableOn", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [163, 9], "def_end_pos": [163, 32]}, {"full_name": "MeasureTheory.integral_add_compl", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [162, 9], "def_end_pos": [162, 27]}, {"full_name": "add_le_add_iff_right", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [90, 3], "def_end_pos": [90, 14]}]], "state_before": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhadp : Adapted \ud835\udca2 f\nhint : \u2200 (i : \u2115), Integrable (f i)\ni j : \u2115\nhij : i \u2264 j\ns : Set \u03a9\nhs : MeasurableSet s\nhf :\n  \u222b (x : \u03a9), stoppedValue f (Set.piecewise s (fun x => i) fun x => j) x \u2202\u03bc \u2264 \u222b (x : \u03a9), stoppedValue f (fun x => j) x \u2202\u03bc\n\u22a2 \u222b (\u03c9 : \u03a9) in s, f i \u03c9 \u2202\u03bc \u2264 \u222b (\u03c9 : \u03a9) in s, f j \u03c9 \u2202\u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Pointwise/SMul.lean", "full_name": "Set.mul_subset_iff_right", "start": [439, 1], "end": [440, 26], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Kernel/CondCdf.lean", "full_name": "ProbabilityTheory.tendsto_condCdf_atTop", "start": [822, 1], "end": [837, 65], "traced_tactics": [{"tactic": "have h_exists : \u2200 x : \u211d, \u2203 q : \u211a, x - 1 < q \u2227 \u2191q < x := fun x => exists_rat_btwn (sub_one_lt x)", "annotated_tactic": ["have h_exists : \u2200 x : \u211d, \u2203 q : \u211a, x - 1 < q \u2227 \u2191q < x := fun x => <a>exists_rat_btwn</a> (<a>sub_one_lt</a> x)", [{"full_name": "exists_rat_btwn", "def_path": "Mathlib/Algebra/Order/Archimedean.lean", "def_pos": [263, 9], "def_end_pos": [263, 24]}, {"full_name": "sub_one_lt", "def_path": "Mathlib/Algebra/Order/Ring/Defs.lean", "def_pos": [1166, 9], "def_end_pos": [1166, 19]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\n\u22a2 Tendsto (\u2191(condCdf \u03c1 a)) atTop (\ud835\udcdd 1)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nh_exists : \u2200 (x : \u211d), \u2203 q, x - 1 < \u2191q \u2227 \u2191q < x\n\u22a2 Tendsto (\u2191(condCdf \u03c1 a)) atTop (\ud835\udcdd 1)"}, {"tactic": "let qs : \u211d \u2192 \u211a := fun x => (h_exists x).choose", "annotated_tactic": ["let qs : \u211d \u2192 \u211a := fun x => (h_exists x).<a>choose</a>", [{"full_name": "Exists.choose", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [442, 32], "def_end_pos": [442, 45]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nh_exists : \u2200 (x : \u211d), \u2203 q, x - 1 < \u2191q \u2227 \u2191q < x\n\u22a2 Tendsto (\u2191(condCdf \u03c1 a)) atTop (\ud835\udcdd 1)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nh_exists : \u2200 (x : \u211d), \u2203 q, x - 1 < \u2191q \u2227 \u2191q < x\nqs : \u211d \u2192 \u211a := fun x => Exists.choose (_ : \u2203 q, x - 1 < \u2191q \u2227 \u2191q < x)\n\u22a2 Tendsto (\u2191(condCdf \u03c1 a)) atTop (\ud835\udcdd 1)"}, {"tactic": "have hqs_tendsto : Tendsto qs atTop atTop := by\n  rw [tendsto_atTop_atTop]\n  refine' fun q => \u27e8q + 1, fun y hy => _\u27e9\n  have h_le : y - 1 \u2264 qs y := (h_exists y).choose_spec.1.le\n  rw [sub_le_iff_le_add] at h_le\n  exact_mod_cast le_of_add_le_add_right (hy.trans h_le)", "annotated_tactic": ["have hqs_tendsto : <a>Tendsto</a> qs <a>atTop</a> <a>atTop</a> := by\n    rw [<a>tendsto_atTop_atTop</a>]\n    refine' fun q => \u27e8q + 1, fun y hy => _\u27e9\n    have h_le : y - 1 \u2264 qs y := (h_exists y).<a>choose_spec</a>.1.<a>le</a>\n    rw [<a>sub_le_iff_le_add</a>] at h_le\n    exact_mod_cast <a>le_of_add_le_add_right</a> (hy.trans h_le)", [{"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2939, 5], "def_end_pos": [2939, 12]}, {"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}, {"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}, {"full_name": "Filter.tendsto_atTop_atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [1332, 9], "def_end_pos": [1332, 28]}, {"full_name": "Exists.choose_spec", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [445, 9], "def_end_pos": [445, 27]}, {"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [142, 7], "def_end_pos": [142, 15]}, {"full_name": "sub_le_iff_le_add", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [750, 3], "def_end_pos": [750, 14]}, {"full_name": "le_of_add_le_add_right", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [74, 15], "def_end_pos": [74, 37]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nh_exists : \u2200 (x : \u211d), \u2203 q, x - 1 < \u2191q \u2227 \u2191q < x\nqs : \u211d \u2192 \u211a := fun x => Exists.choose (_ : \u2203 q, x - 1 < \u2191q \u2227 \u2191q < x)\n\u22a2 Tendsto (\u2191(condCdf \u03c1 a)) atTop (\ud835\udcdd 1)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nh_exists : \u2200 (x : \u211d), \u2203 q, x - 1 < \u2191q \u2227 \u2191q < x\nqs : \u211d \u2192 \u211a := fun x => Exists.choose (_ : \u2203 q, x - 1 < \u2191q \u2227 \u2191q < x)\nhqs_tendsto : Tendsto qs atTop atTop\n\u22a2 Tendsto (\u2191(condCdf \u03c1 a)) atTop (\ud835\udcdd 1)"}, {"tactic": "refine'\n  tendsto_of_tendsto_of_tendsto_of_le_of_le ((tendsto_condCdfRat_atTop \u03c1 a).comp hqs_tendsto)\n    tendsto_const_nhds _ (condCdf_le_one \u03c1 a)", "annotated_tactic": ["refine'\n    <a>tendsto_of_tendsto_of_tendsto_of_le_of_le</a> ((<a>tendsto_condCdfRat_atTop</a> \u03c1 a).<a>comp</a> hqs_tendsto)\n      <a>tendsto_const_nhds</a> _ (<a>condCdf_le_one</a> \u03c1 a)", [{"full_name": "tendsto_of_tendsto_of_tendsto_of_le_of_le", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [955, 9], "def_end_pos": [955, 50]}, {"full_name": "ProbabilityTheory.tendsto_condCdfRat_atTop", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [642, 9], "def_end_pos": [642, 33]}, {"full_name": "Filter.Tendsto.comp", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [3032, 9], "def_end_pos": [3032, 21]}, {"full_name": "tendsto_const_nhds", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [1049, 9], "def_end_pos": [1049, 27]}, {"full_name": "ProbabilityTheory.condCdf_le_one", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [794, 9], "def_end_pos": [794, 23]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nh_exists : \u2200 (x : \u211d), \u2203 q, x - 1 < \u2191q \u2227 \u2191q < x\nqs : \u211d \u2192 \u211a := fun x => Exists.choose (_ : \u2203 q, x - 1 < \u2191q \u2227 \u2191q < x)\nhqs_tendsto : Tendsto qs atTop atTop\n\u22a2 Tendsto (\u2191(condCdf \u03c1 a)) atTop (\ud835\udcdd 1)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nh_exists : \u2200 (x : \u211d), \u2203 q, x - 1 < \u2191q \u2227 \u2191q < x\nqs : \u211d \u2192 \u211a := fun x => Exists.choose (_ : \u2203 q, x - 1 < \u2191q \u2227 \u2191q < x)\nhqs_tendsto : Tendsto qs atTop atTop\n\u22a2 condCdfRat \u03c1 a \u2218 qs \u2264 \u2191(condCdf \u03c1 a)"}, {"tactic": "intro x", "annotated_tactic": ["intro x", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nh_exists : \u2200 (x : \u211d), \u2203 q, x - 1 < \u2191q \u2227 \u2191q < x\nqs : \u211d \u2192 \u211a := fun x => Exists.choose (_ : \u2203 q, x - 1 < \u2191q \u2227 \u2191q < x)\nhqs_tendsto : Tendsto qs atTop atTop\n\u22a2 condCdfRat \u03c1 a \u2218 qs \u2264 \u2191(condCdf \u03c1 a)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nh_exists : \u2200 (x : \u211d), \u2203 q, x - 1 < \u2191q \u2227 \u2191q < x\nqs : \u211d \u2192 \u211a := fun x => Exists.choose (_ : \u2203 q, x - 1 < \u2191q \u2227 \u2191q < x)\nhqs_tendsto : Tendsto qs atTop atTop\nx : \u211d\n\u22a2 (condCdfRat \u03c1 a \u2218 qs) x \u2264 \u2191(condCdf \u03c1 a) x"}, {"tactic": "rw [Function.comp_apply, \u2190 condCdf_eq_condCdfRat]", "annotated_tactic": ["rw [<a>Function.comp_apply</a>, \u2190 <a>condCdf_eq_condCdfRat</a>]", [{"full_name": "Function.comp_apply", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [33, 17], "def_end_pos": [33, 36]}, {"full_name": "ProbabilityTheory.condCdf_eq_condCdfRat", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [783, 9], "def_end_pos": [783, 30]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nh_exists : \u2200 (x : \u211d), \u2203 q, x - 1 < \u2191q \u2227 \u2191q < x\nqs : \u211d \u2192 \u211a := fun x => Exists.choose (_ : \u2203 q, x - 1 < \u2191q \u2227 \u2191q < x)\nhqs_tendsto : Tendsto qs atTop atTop\nx : \u211d\n\u22a2 (condCdfRat \u03c1 a \u2218 qs) x \u2264 \u2191(condCdf \u03c1 a) x", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nh_exists : \u2200 (x : \u211d), \u2203 q, x - 1 < \u2191q \u2227 \u2191q < x\nqs : \u211d \u2192 \u211a := fun x => Exists.choose (_ : \u2203 q, x - 1 < \u2191q \u2227 \u2191q < x)\nhqs_tendsto : Tendsto qs atTop atTop\nx : \u211d\n\u22a2 \u2191(condCdf \u03c1 a) \u2191(qs x) \u2264 \u2191(condCdf \u03c1 a) x"}, {"tactic": "exact (condCdf \u03c1 a).mono (le_of_lt (h_exists x).choose_spec.2)", "annotated_tactic": ["exact (<a>condCdf</a> \u03c1 a).<a>mono</a> (<a>le_of_lt</a> (h_exists x).<a>choose_spec</a>.2)", [{"full_name": "ProbabilityTheory.condCdf", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [777, 19], "def_end_pos": [777, 26]}, {"full_name": "StieltjesFunction.mono", "def_path": "Mathlib/MeasureTheory/Measure/Stieltjes.lean", "def_pos": [62, 9], "def_end_pos": [62, 13]}, {"full_name": "le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [110, 9], "def_end_pos": [110, 17]}, {"full_name": "Exists.choose_spec", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [445, 9], "def_end_pos": [445, 27]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nh_exists : \u2200 (x : \u211d), \u2203 q, x - 1 < \u2191q \u2227 \u2191q < x\nqs : \u211d \u2192 \u211a := fun x => Exists.choose (_ : \u2203 q, x - 1 < \u2191q \u2227 \u2191q < x)\nhqs_tendsto : Tendsto qs atTop atTop\nx : \u211d\n\u22a2 \u2191(condCdf \u03c1 a) \u2191(qs x) \u2264 \u2191(condCdf \u03c1 a) x", "state_after": "no goals"}, {"tactic": "rw [tendsto_atTop_atTop]", "annotated_tactic": ["rw [<a>tendsto_atTop_atTop</a>]", [{"full_name": "Filter.tendsto_atTop_atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [1332, 9], "def_end_pos": [1332, 28]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nh_exists : \u2200 (x : \u211d), \u2203 q, x - 1 < \u2191q \u2227 \u2191q < x\nqs : \u211d \u2192 \u211a := fun x => Exists.choose (_ : \u2203 q, x - 1 < \u2191q \u2227 \u2191q < x)\n\u22a2 Tendsto qs atTop atTop", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nh_exists : \u2200 (x : \u211d), \u2203 q, x - 1 < \u2191q \u2227 \u2191q < x\nqs : \u211d \u2192 \u211a := fun x => Exists.choose (_ : \u2203 q, x - 1 < \u2191q \u2227 \u2191q < x)\n\u22a2 \u2200 (b : \u211a), \u2203 i, \u2200 (a : \u211d), i \u2264 a \u2192 b \u2264 qs a"}, {"tactic": "refine' fun q => \u27e8q + 1, fun y hy => _\u27e9", "annotated_tactic": ["refine' fun q => \u27e8q + 1, fun y hy => _\u27e9", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nh_exists : \u2200 (x : \u211d), \u2203 q, x - 1 < \u2191q \u2227 \u2191q < x\nqs : \u211d \u2192 \u211a := fun x => Exists.choose (_ : \u2203 q, x - 1 < \u2191q \u2227 \u2191q < x)\n\u22a2 \u2200 (b : \u211a), \u2203 i, \u2200 (a : \u211d), i \u2264 a \u2192 b \u2264 qs a", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nh_exists : \u2200 (x : \u211d), \u2203 q, x - 1 < \u2191q \u2227 \u2191q < x\nqs : \u211d \u2192 \u211a := fun x => Exists.choose (_ : \u2203 q, x - 1 < \u2191q \u2227 \u2191q < x)\nq : \u211a\ny : \u211d\nhy : \u2191q + 1 \u2264 y\n\u22a2 q \u2264 qs y"}, {"tactic": "have h_le : y - 1 \u2264 qs y := (h_exists y).choose_spec.1.le", "annotated_tactic": ["have h_le : y - 1 \u2264 qs y := (h_exists y).<a>choose_spec</a>.1.<a>le</a>", [{"full_name": "Exists.choose_spec", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [445, 9], "def_end_pos": [445, 27]}, {"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [142, 7], "def_end_pos": [142, 15]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nh_exists : \u2200 (x : \u211d), \u2203 q, x - 1 < \u2191q \u2227 \u2191q < x\nqs : \u211d \u2192 \u211a := fun x => Exists.choose (_ : \u2203 q, x - 1 < \u2191q \u2227 \u2191q < x)\nq : \u211a\ny : \u211d\nhy : \u2191q + 1 \u2264 y\n\u22a2 q \u2264 qs y", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nh_exists : \u2200 (x : \u211d), \u2203 q, x - 1 < \u2191q \u2227 \u2191q < x\nqs : \u211d \u2192 \u211a := fun x => Exists.choose (_ : \u2203 q, x - 1 < \u2191q \u2227 \u2191q < x)\nq : \u211a\ny : \u211d\nhy : \u2191q + 1 \u2264 y\nh_le : y - 1 \u2264 \u2191(qs y)\n\u22a2 q \u2264 qs y"}, {"tactic": "rw [sub_le_iff_le_add] at h_le", "annotated_tactic": ["rw [<a>sub_le_iff_le_add</a>] at h_le", [{"full_name": "sub_le_iff_le_add", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [750, 3], "def_end_pos": [750, 14]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nh_exists : \u2200 (x : \u211d), \u2203 q, x - 1 < \u2191q \u2227 \u2191q < x\nqs : \u211d \u2192 \u211a := fun x => Exists.choose (_ : \u2203 q, x - 1 < \u2191q \u2227 \u2191q < x)\nq : \u211a\ny : \u211d\nhy : \u2191q + 1 \u2264 y\nh_le : y - 1 \u2264 \u2191(qs y)\n\u22a2 q \u2264 qs y", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nh_exists : \u2200 (x : \u211d), \u2203 q, x - 1 < \u2191q \u2227 \u2191q < x\nqs : \u211d \u2192 \u211a := fun x => Exists.choose (_ : \u2203 q, x - 1 < \u2191q \u2227 \u2191q < x)\nq : \u211a\ny : \u211d\nhy : \u2191q + 1 \u2264 y\nh_le : y \u2264 \u2191(qs y) + 1\n\u22a2 q \u2264 qs y"}, {"tactic": "exact_mod_cast le_of_add_le_add_right (hy.trans h_le)", "annotated_tactic": ["exact_mod_cast <a>le_of_add_le_add_right</a> (hy.trans h_le)", [{"full_name": "le_of_add_le_add_right", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [74, 15], "def_end_pos": [74, 37]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nh_exists : \u2200 (x : \u211d), \u2203 q, x - 1 < \u2191q \u2227 \u2191q < x\nqs : \u211d \u2192 \u211a := fun x => Exists.choose (_ : \u2203 q, x - 1 < \u2191q \u2227 \u2191q < x)\nq : \u211a\ny : \u211d\nhy : \u2191q + 1 \u2264 y\nh_le : y \u2264 \u2191(qs y) + 1\n\u22a2 q \u2264 qs y", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Pointwise.lean", "full_name": "Finset.preimage_mul_left_one'", "start": [1235, 1], "end": [1236, 41], "traced_tactics": [{"tactic": "rw [preimage_mul_left_one, inv_inv]", "annotated_tactic": ["rw [<a>preimage_mul_left_one</a>, <a>inv_inv</a>]", [{"full_name": "Finset.preimage_mul_left_one", "def_path": "Mathlib/Data/Finset/Pointwise.lean", "def_pos": [1223, 9], "def_end_pos": [1223, 30]}, {"full_name": "inv_inv", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [800, 9], "def_end_pos": [800, 16]}]], "state_before": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : Group \u03b1\ns t : Finset \u03b1\na b : \u03b1\n\u22a2 preimage 1 ((fun x x_1 => x * x_1) a\u207b\u00b9)\n      (_ : Set.InjOn ((fun x x_1 => x * x_1) a\u207b\u00b9) ((fun x x_1 => x * x_1) a\u207b\u00b9 \u207b\u00b9' \u21911)) =\n    {a}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Kernel/WithDensity.lean", "full_name": "ProbabilityTheory.kernel.IsSFiniteKernel.withDensity", "start": [215, 8], "end": [223, 70], "traced_tactics": [{"tactic": "have h_eq_sum : withDensity \u03ba f = kernel.sum fun i => withDensity (seq \u03ba i) f := by\n  rw [\u2190 withDensity_kernel_sum _ _]\n  congr\n  exact (kernel_sum_seq \u03ba).symm", "annotated_tactic": ["have h_eq_sum : <a>withDensity</a> \u03ba f = <a>kernel.sum</a> fun i => <a>withDensity</a> (<a>seq</a> \u03ba i) f := by\n    rw [\u2190 <a>withDensity_kernel_sum</a> _ _]\n    congr\n    exact (<a>kernel_sum_seq</a> \u03ba).<a>symm</a>", [{"full_name": "ProbabilityTheory.kernel.withDensity", "def_path": "Mathlib/Probability/Kernel/WithDensity.lean", "def_pos": [47, 19], "def_end_pos": [47, 30]}, {"full_name": "ProbabilityTheory.kernel.sum", "def_path": "Mathlib/Probability/Kernel/Basic.lean", "def_pos": [231, 29], "def_end_pos": [231, 32]}, {"full_name": "ProbabilityTheory.kernel.withDensity", "def_path": "Mathlib/Probability/Kernel/WithDensity.lean", "def_pos": [47, 19], "def_end_pos": [47, 30]}, {"full_name": "ProbabilityTheory.kernel.seq", "def_path": "Mathlib/Probability/Kernel/Basic.lean", "def_pos": [294, 19], "def_end_pos": [294, 22]}, {"full_name": "ProbabilityTheory.kernel.withDensity_kernel_sum", "def_path": "Mathlib/Probability/Kernel/WithDensity.lean", "def_pos": [102, 9], "def_end_pos": [102, 31]}, {"full_name": "ProbabilityTheory.kernel.kernel_sum_seq", "def_path": "Mathlib/Probability/Kernel/Basic.lean", "def_pos": [298, 9], "def_end_pos": [298, 23]}, {"full_name": "Eq.symm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [310, 9], "def_end_pos": [310, 16]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\nf : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsSFiniteKernel \u03ba\nhf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4\n\u22a2 IsSFiniteKernel (kernel.withDensity \u03ba f)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\nf : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsSFiniteKernel \u03ba\nhf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4\nh_eq_sum : kernel.withDensity \u03ba f = kernel.sum fun i => kernel.withDensity (seq \u03ba i) f\n\u22a2 IsSFiniteKernel (kernel.withDensity \u03ba f)"}, {"tactic": "rw [h_eq_sum]", "annotated_tactic": ["rw [h_eq_sum]", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\nf : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsSFiniteKernel \u03ba\nhf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4\nh_eq_sum : kernel.withDensity \u03ba f = kernel.sum fun i => kernel.withDensity (seq \u03ba i) f\n\u22a2 IsSFiniteKernel (kernel.withDensity \u03ba f)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\nf : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsSFiniteKernel \u03ba\nhf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4\nh_eq_sum : kernel.withDensity \u03ba f = kernel.sum fun i => kernel.withDensity (seq \u03ba i) f\n\u22a2 IsSFiniteKernel (kernel.sum fun i => kernel.withDensity (seq \u03ba i) f)"}, {"tactic": "exact isSFiniteKernel_sum fun n =>\n  isSFiniteKernel_withDensity_of_isFiniteKernel (seq \u03ba n) hf_ne_top", "annotated_tactic": ["exact <a>isSFiniteKernel_sum</a> fun n =>\n    <a>isSFiniteKernel_withDensity_of_isFiniteKernel</a> (<a>seq</a> \u03ba n) hf_ne_top", [{"full_name": "ProbabilityTheory.kernel.isSFiniteKernel_sum", "def_path": "Mathlib/Probability/Kernel/Basic.lean", "def_pos": [343, 9], "def_end_pos": [343, 28]}, {"full_name": "ProbabilityTheory.kernel.isSFiniteKernel_withDensity_of_isFiniteKernel", "def_path": "Mathlib/Probability/Kernel/WithDensity.lean", "def_pos": [156, 9], "def_end_pos": [156, 54]}, {"full_name": "ProbabilityTheory.kernel.seq", "def_path": "Mathlib/Probability/Kernel/Basic.lean", "def_pos": [294, 19], "def_end_pos": [294, 22]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\nf : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsSFiniteKernel \u03ba\nhf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4\nh_eq_sum : kernel.withDensity \u03ba f = kernel.sum fun i => kernel.withDensity (seq \u03ba i) f\n\u22a2 IsSFiniteKernel (kernel.sum fun i => kernel.withDensity (seq \u03ba i) f)", "state_after": "no goals"}, {"tactic": "rw [\u2190 withDensity_kernel_sum _ _]", "annotated_tactic": ["rw [\u2190 <a>withDensity_kernel_sum</a> _ _]", [{"full_name": "ProbabilityTheory.kernel.withDensity_kernel_sum", "def_path": "Mathlib/Probability/Kernel/WithDensity.lean", "def_pos": [102, 9], "def_end_pos": [102, 31]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\nf : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsSFiniteKernel \u03ba\nhf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4\n\u22a2 kernel.withDensity \u03ba f = kernel.sum fun i => kernel.withDensity (seq \u03ba i) f", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\nf : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsSFiniteKernel \u03ba\nhf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4\n\u22a2 kernel.withDensity \u03ba f = kernel.withDensity (kernel.sum fun i => seq \u03ba i) f"}, {"tactic": "congr", "annotated_tactic": ["congr", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\nf : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsSFiniteKernel \u03ba\nhf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4\n\u22a2 kernel.withDensity \u03ba f = kernel.withDensity (kernel.sum fun i => seq \u03ba i) f", "state_after": "case e_\u03ba\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\nf : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsSFiniteKernel \u03ba\nhf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4\n\u22a2 \u03ba = kernel.sum fun i => seq \u03ba i"}, {"tactic": "exact (kernel_sum_seq \u03ba).symm", "annotated_tactic": ["exact (<a>kernel_sum_seq</a> \u03ba).<a>symm</a>", [{"full_name": "ProbabilityTheory.kernel.kernel_sum_seq", "def_path": "Mathlib/Probability/Kernel/Basic.lean", "def_pos": [298, 9], "def_end_pos": [298, 23]}, {"full_name": "Eq.symm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [310, 9], "def_end_pos": [310, 16]}]], "state_before": "case e_\u03ba\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\nf : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsSFiniteKernel \u03ba\nhf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4\n\u22a2 \u03ba = kernel.sum fun i => seq \u03ba i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "full_name": "ae_restrict_iff_subtype", "start": [4164, 1], "end": [4167, 83], "traced_tactics": [{"tactic": "rw [\u2190 map_comap_subtype_coe hs, (MeasurableEmbedding.subtype_coe hs).ae_map_iff]", "annotated_tactic": ["rw [\u2190 <a>map_comap_subtype_coe</a> hs, (<a>MeasurableEmbedding.subtype_coe</a> hs).<a>ae_map_iff</a>]", [{"full_name": "map_comap_subtype_coe", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [4159, 9], "def_end_pos": [4159, 30]}, {"full_name": "MeasurableEmbedding.subtype_coe", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [1208, 9], "def_end_pos": [1208, 20]}, {"full_name": "MeasurableEmbedding.ae_map_iff", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [4135, 9], "def_end_pos": [4135, 19]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\np : \u03b1 \u2192 Prop\n\u22a2 (\u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc s, p x) \u2194 \u2200\u1d50 (x : \u2191s) \u2202Measure.comap Subtype.val \u03bc, p \u2191x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/PEquiv.lean", "full_name": "PEquiv.single_trans_single", "start": [378, 1], "end": [380, 41], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Pairwise/Basic.lean", "full_name": "Set.pairwiseDisjoint_pi", "start": [371, 1], "end": [378, 92], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Int/Bitwise.lean", "full_name": "Int.bit1_ne_zero", "start": [191, 1], "end": [191, 94], "traced_tactics": [{"tactic": "simpa only [bit0_zero] using bit1_ne_bit0 m 0", "annotated_tactic": ["simpa only [<a>bit0_zero</a>] using <a>bit1_ne_bit0</a> m 0", [{"full_name": "bit0_zero", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [152, 9], "def_end_pos": [152, 18]}, {"full_name": "Int.bit1_ne_bit0", "def_path": "Mathlib/Data/Int/Bitwise.lean", "def_pos": [186, 9], "def_end_pos": [186, 21]}]], "state_before": "m : \u2124\n\u22a2 bit1 m \u2260 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/TuringMachine.lean", "full_name": "Turing.TM1.stmts_trans", "start": [1358, 1], "end": [1362, 50], "traced_tactics": [{"tactic": "simp only [stmts, Finset.mem_insertNone, Finset.mem_biUnion, Option.mem_def, Option.some.injEq,\n  forall_eq', exists_imp, and_imp]", "annotated_tactic": ["simp only [<a>stmts</a>, <a>Finset.mem_insertNone</a>, <a>Finset.mem_biUnion</a>, <a>Option.mem_def</a>, Option.some.injEq,\n    <a>forall_eq'</a>, <a>exists_imp</a>, <a>and_imp</a>]", [{"full_name": "Turing.TM1.stmts", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1354, 19], "def_end_pos": [1354, 24]}, {"full_name": "Finset.mem_insertNone", "def_path": "Mathlib/Data/Finset/Option.lean", "def_pos": [70, 9], "def_end_pos": [70, 23]}, {"full_name": "Finset.mem_biUnion", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3613, 9], "def_end_pos": [3613, 20]}, {"full_name": "Option.mem_def", "def_path": "lake-packages/std/Std/Data/Option/Basic.lean", "def_pos": [19, 17], "def_end_pos": [19, 24]}, {"full_name": "forall_eq'", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [453, 17], "def_end_pos": [453, 27]}, {"full_name": "exists_imp", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [367, 9], "def_end_pos": [367, 19]}, {"full_name": "and_imp", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [313, 17], "def_end_pos": [313, 24]}]], "state_before": "\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nM : \u039b \u2192 Stmt\u2081\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2081\nh\u2081 : q\u2081 \u2208 stmts\u2081 q\u2082\n\u22a2 some q\u2082 \u2208 stmts M S \u2192 some q\u2081 \u2208 stmts M S", "state_after": "\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nM : \u039b \u2192 Stmt\u2081\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2081\nh\u2081 : q\u2081 \u2208 stmts\u2081 q\u2082\n\u22a2 \u2200 (x : \u039b), x \u2208 S \u2192 q\u2082 \u2208 stmts\u2081 (M x) \u2192 \u2203 a, a \u2208 S \u2227 q\u2081 \u2208 stmts\u2081 (M a)"}, {"tactic": "exact fun l ls h\u2082 \u21a6 \u27e8_, ls, stmts\u2081_trans h\u2082 h\u2081\u27e9", "annotated_tactic": ["exact fun l ls h\u2082 \u21a6 \u27e8_, ls, <a>stmts\u2081_trans</a> h\u2082 h\u2081\u27e9", [{"full_name": "Turing.TM1.stmts\u2081_trans", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1322, 9], "def_end_pos": [1322, 21]}]], "state_before": "\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nM : \u039b \u2192 Stmt\u2081\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2081\nh\u2081 : q\u2081 \u2208 stmts\u2081 q\u2082\n\u22a2 \u2200 (x : \u039b), x \u2208 S \u2192 q\u2082 \u2208 stmts\u2081 (M x) \u2192 \u2203 a, a \u2208 S \u2227 q\u2081 \u2208 stmts\u2081 (M a)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Image.lean", "full_name": "Set.image_injective", "start": [1573, 1], "end": [1576, 44], "traced_tactics": [{"tactic": "refine' \u27e8fun h x x' hx => _, Injective.image_injective\u27e9", "annotated_tactic": ["refine' \u27e8fun h x x' hx => _, <a>Injective.image_injective</a>\u27e9", [{"full_name": "Function.Injective.image_injective", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [1330, 9], "def_end_pos": [1330, 34]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\n\u22a2 Injective (image f) \u2194 Injective f", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\nh : Injective (image f)\nx x' : \u03b1\nhx : f x = f x'\n\u22a2 x = x'"}, {"tactic": "rw [\u2190 singleton_eq_singleton_iff]", "annotated_tactic": ["rw [\u2190 <a>singleton_eq_singleton_iff</a>]", [{"full_name": "Set.singleton_eq_singleton_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1298, 9], "def_end_pos": [1298, 35]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\nh : Injective (image f)\nx x' : \u03b1\nhx : f x = f x'\n\u22a2 x = x'", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\nh : Injective (image f)\nx x' : \u03b1\nhx : f x = f x'\n\u22a2 {x} = {x'}"}, {"tactic": "apply h", "annotated_tactic": ["apply h", []], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\nh : Injective (image f)\nx x' : \u03b1\nhx : f x = f x'\n\u22a2 {x} = {x'}", "state_after": "case a\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\nh : Injective (image f)\nx x' : \u03b1\nhx : f x = f x'\n\u22a2 f '' {x} = f '' {x'}"}, {"tactic": "rw [image_singleton, image_singleton, hx]", "annotated_tactic": ["rw [<a>image_singleton</a>, <a>image_singleton</a>, hx]", [{"full_name": "Set.image_singleton", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [363, 9], "def_end_pos": [363, 24]}, {"full_name": "Set.image_singleton", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [363, 9], "def_end_pos": [363, 24]}]], "state_before": "case a\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\nh : Injective (image f)\nx x' : \u03b1\nhx : f x = f x'\n\u22a2 f '' {x} = f '' {x'}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Stieltjes.lean", "full_name": "StieltjesFunction.iInf_rat_gt_eq", "start": [82, 1], "end": [88, 33], "traced_tactics": [{"tactic": "rw [\u2190 iInf_Ioi_eq f x]", "annotated_tactic": ["rw [\u2190 <a>iInf_Ioi_eq</a> f x]", [{"full_name": "StieltjesFunction.iInf_Ioi_eq", "def_path": "Mathlib/MeasureTheory/Measure/Stieltjes.lean", "def_pos": [75, 9], "def_end_pos": [75, 20]}]], "state_before": "f\u271d f : StieltjesFunction\nx : \u211d\n\u22a2 \u2a05 r, \u2191f \u2191\u2191r = \u2191f x", "state_after": "f\u271d f : StieltjesFunction\nx : \u211d\n\u22a2 \u2a05 r, \u2191f \u2191\u2191r = \u2a05 r, \u2191f \u2191r"}, {"tactic": "refine' (Real.iInf_Ioi_eq_iInf_rat_gt _ _ f.mono).symm", "annotated_tactic": ["refine' (<a>Real.iInf_Ioi_eq_iInf_rat_gt</a> _ _ f.mono).<a>symm</a>", [{"full_name": "Real.iInf_Ioi_eq_iInf_rat_gt", "def_path": "Mathlib/Data/Real/Basic.lean", "def_pos": [946, 9], "def_end_pos": [946, 32]}, {"full_name": "Eq.symm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [310, 9], "def_end_pos": [310, 16]}]], "state_before": "f\u271d f : StieltjesFunction\nx : \u211d\n\u22a2 \u2a05 r, \u2191f \u2191\u2191r = \u2a05 r, \u2191f \u2191r", "state_after": "f\u271d f : StieltjesFunction\nx : \u211d\n\u22a2 BddBelow (\u2191f '' Ioi x)"}, {"tactic": "refine' \u27e8f x, fun y => _\u27e9", "annotated_tactic": ["refine' \u27e8f x, fun y => _\u27e9", []], "state_before": "f\u271d f : StieltjesFunction\nx : \u211d\n\u22a2 BddBelow (\u2191f '' Ioi x)", "state_after": "f\u271d f : StieltjesFunction\nx y : \u211d\n\u22a2 y \u2208 \u2191f '' Ioi x \u2192 \u2191f x \u2264 y"}, {"tactic": "rintro \u27e8y, hy_mem, rfl\u27e9", "annotated_tactic": ["rintro \u27e8y, hy_mem, rfl\u27e9", []], "state_before": "f\u271d f : StieltjesFunction\nx y : \u211d\n\u22a2 y \u2208 \u2191f '' Ioi x \u2192 \u2191f x \u2264 y", "state_after": "case intro.intro\nf\u271d f : StieltjesFunction\nx y : \u211d\nhy_mem : y \u2208 Ioi x\n\u22a2 \u2191f x \u2264 \u2191f y"}, {"tactic": "exact f.mono (le_of_lt hy_mem)", "annotated_tactic": ["exact f.mono (<a>le_of_lt</a> hy_mem)", [{"full_name": "le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [110, 9], "def_end_pos": [110, 17]}]], "state_before": "case intro.intro\nf\u271d f : StieltjesFunction\nx y : \u211d\nhy_mem : y \u2208 Ioi x\n\u22a2 \u2191f x \u2264 \u2191f y", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/Jacobian.lean", "full_name": "MeasureTheory.addHaar_image_le_lintegral_abs_det_fderiv", "start": [903, 1], "end": [931, 93], "traced_tactics": [{"tactic": "let u n := disjointed (spanningSets \u03bc) n", "annotated_tactic": ["let u n := <a>disjointed</a> (<a>spanningSets</a> \u03bc) n", [{"full_name": "disjointed", "def_path": "Mathlib/Order/Disjointed.lean", "def_pos": [49, 5], "def_end_pos": [49, 15]}, {"full_name": "MeasureTheory.spanningSets", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3316, 5], "def_end_pos": [3316, 17]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u22a2 \u2191\u2191\u03bc (f '' s) \u2264 \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc", "state_after": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nu : \u2115 \u2192 Set E := fun n => disjointed (spanningSets \u03bc) n\n\u22a2 \u2191\u2191\u03bc (f '' s) \u2264 \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc"}, {"tactic": "have u_meas : \u2200 n, MeasurableSet (u n) := by\n  intro n\n  apply MeasurableSet.disjointed fun i => ?_\n  exact measurable_spanningSets \u03bc i", "annotated_tactic": ["have u_meas : \u2200 n, <a>MeasurableSet</a> (u n) := by\n    intro n\n    apply <a>MeasurableSet.disjointed</a> fun i => ?_\n    exact <a>measurable_spanningSets</a> \u03bc i", [{"full_name": "MeasurableSet", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [64, 5], "def_end_pos": [64, 18]}, {"full_name": "MeasurableSet.disjointed", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [236, 19], "def_end_pos": [236, 43]}, {"full_name": "MeasureTheory.measurable_spanningSets", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3324, 9], "def_end_pos": [3324, 32]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nu : \u2115 \u2192 Set E := fun n => disjointed (spanningSets \u03bc) n\n\u22a2 \u2191\u2191\u03bc (f '' s) \u2264 \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc", "state_after": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nu : \u2115 \u2192 Set E := fun n => disjointed (spanningSets \u03bc) n\nu_meas : \u2200 (n : \u2115), MeasurableSet (u n)\n\u22a2 \u2191\u2191\u03bc (f '' s) \u2264 \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc"}, {"tactic": "have A : s = \u22c3 n, s \u2229 u n := by\n  rw [\u2190 inter_iUnion, iUnion_disjointed, iUnion_spanningSets, inter_univ]", "annotated_tactic": ["have A : s = \u22c3 n, s \u2229 u n := by\n    rw [\u2190 <a>inter_iUnion</a>, <a>iUnion_disjointed</a>, <a>iUnion_spanningSets</a>, <a>inter_univ</a>]", [{"full_name": "Set.inter_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [635, 9], "def_end_pos": [635, 21]}, {"full_name": "iUnion_disjointed", "def_path": "Mathlib/Order/Disjointed.lean", "def_pos": [165, 9], "def_end_pos": [165, 26]}, {"full_name": "MeasureTheory.iUnion_spanningSets", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3334, 9], "def_end_pos": [3334, 28]}, {"full_name": "Set.inter_univ", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1012, 9], "def_end_pos": [1012, 19]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nu : \u2115 \u2192 Set E := fun n => disjointed (spanningSets \u03bc) n\nu_meas : \u2200 (n : \u2115), MeasurableSet (u n)\n\u22a2 \u2191\u2191\u03bc (f '' s) \u2264 \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc", "state_after": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nu : \u2115 \u2192 Set E := fun n => disjointed (spanningSets \u03bc) n\nu_meas : \u2200 (n : \u2115), MeasurableSet (u n)\nA : s = \u22c3 n, s \u2229 u n\n\u22a2 \u2191\u2191\u03bc (f '' s) \u2264 \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc"}, {"tactic": "intro n", "annotated_tactic": ["intro n", []], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nu : \u2115 \u2192 Set E := fun n => disjointed (spanningSets \u03bc) n\n\u22a2 \u2200 (n : \u2115), MeasurableSet (u n)", "state_after": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nu : \u2115 \u2192 Set E := fun n => disjointed (spanningSets \u03bc) n\nn : \u2115\n\u22a2 MeasurableSet (u n)"}, {"tactic": "apply MeasurableSet.disjointed fun i => ?_", "annotated_tactic": ["apply <a>MeasurableSet.disjointed</a> fun i => ?_", [{"full_name": "MeasurableSet.disjointed", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [236, 19], "def_end_pos": [236, 43]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nu : \u2115 \u2192 Set E := fun n => disjointed (spanningSets \u03bc) n\nn : \u2115\n\u22a2 MeasurableSet (u n)", "state_after": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nu : \u2115 \u2192 Set E := fun n => disjointed (spanningSets \u03bc) n\nn i : \u2115\n\u22a2 MeasurableSet (spanningSets \u03bc i)"}, {"tactic": "exact measurable_spanningSets \u03bc i", "annotated_tactic": ["exact <a>measurable_spanningSets</a> \u03bc i", [{"full_name": "MeasureTheory.measurable_spanningSets", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3324, 9], "def_end_pos": [3324, 32]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nu : \u2115 \u2192 Set E := fun n => disjointed (spanningSets \u03bc) n\nn i : \u2115\n\u22a2 MeasurableSet (spanningSets \u03bc i)", "state_after": "no goals"}, {"tactic": "rw [\u2190 inter_iUnion, iUnion_disjointed, iUnion_spanningSets, inter_univ]", "annotated_tactic": ["rw [\u2190 <a>inter_iUnion</a>, <a>iUnion_disjointed</a>, <a>iUnion_spanningSets</a>, <a>inter_univ</a>]", [{"full_name": "Set.inter_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [635, 9], "def_end_pos": [635, 21]}, {"full_name": "iUnion_disjointed", "def_path": "Mathlib/Order/Disjointed.lean", "def_pos": [165, 9], "def_end_pos": [165, 26]}, {"full_name": "MeasureTheory.iUnion_spanningSets", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3334, 9], "def_end_pos": [3334, 28]}, {"full_name": "Set.inter_univ", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1012, 9], "def_end_pos": [1012, 19]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nu : \u2115 \u2192 Set E := fun n => disjointed (spanningSets \u03bc) n\nu_meas : \u2200 (n : \u2115), MeasurableSet (u n)\n\u22a2 s = \u22c3 n, s \u2229 u n", "state_after": "no goals"}, {"tactic": "conv_lhs => rw [A, image_iUnion]", "annotated_tactic": ["conv_lhs => rw [A, <a>image_iUnion</a>]", [{"full_name": "Set.image_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [1791, 9], "def_end_pos": [1791, 21]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nu : \u2115 \u2192 Set E := fun n => disjointed (spanningSets \u03bc) n\nu_meas : \u2200 (n : \u2115), MeasurableSet (u n)\nA : s = \u22c3 n, s \u2229 u n\n\u22a2 \u2191\u2191\u03bc (f '' s) \u2264 \u2211' (n : \u2115), \u2191\u2191\u03bc (f '' (s \u2229 u n))", "state_after": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nu : \u2115 \u2192 Set E := fun n => disjointed (spanningSets \u03bc) n\nu_meas : \u2200 (n : \u2115), MeasurableSet (u n)\nA : s = \u22c3 n, s \u2229 u n\n\u22a2 \u2191\u2191\u03bc (\u22c3 i, f '' (s \u2229 u i)) \u2264 \u2211' (n : \u2115), \u2191\u2191\u03bc (f '' (s \u2229 u n))"}, {"tactic": "exact measure_iUnion_le _", "annotated_tactic": ["exact <a>measure_iUnion_le</a> _", [{"full_name": "MeasureTheory.measure_iUnion_le", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [240, 9], "def_end_pos": [240, 26]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nu : \u2115 \u2192 Set E := fun n => disjointed (spanningSets \u03bc) n\nu_meas : \u2200 (n : \u2115), MeasurableSet (u n)\nA : s = \u22c3 n, s \u2229 u n\n\u22a2 \u2191\u2191\u03bc (\u22c3 i, f '' (s \u2229 u i)) \u2264 \u2211' (n : \u2115), \u2191\u2191\u03bc (f '' (s \u2229 u n))", "state_after": "no goals"}, {"tactic": "apply ENNReal.tsum_le_tsum fun n => ?_", "annotated_tactic": ["apply <a>ENNReal.tsum_le_tsum</a> fun n => ?_", [{"full_name": "ENNReal.tsum_le_tsum", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [827, 19], "def_end_pos": [827, 31]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nu : \u2115 \u2192 Set E := fun n => disjointed (spanningSets \u03bc) n\nu_meas : \u2200 (n : \u2115), MeasurableSet (u n)\nA : s = \u22c3 n, s \u2229 u n\n\u22a2 \u2211' (n : \u2115), \u2191\u2191\u03bc (f '' (s \u2229 u n)) \u2264\n    \u2211' (n : \u2115), \u222b\u207b (x : E) in s \u2229 u n, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc", "state_after": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nu : \u2115 \u2192 Set E := fun n => disjointed (spanningSets \u03bc) n\nu_meas : \u2200 (n : \u2115), MeasurableSet (u n)\nA : s = \u22c3 n, s \u2229 u n\nn : \u2115\n\u22a2 \u2191\u2191\u03bc (f '' (s \u2229 u n)) \u2264 \u222b\u207b (x : E) in s \u2229 u n, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc"}, {"tactic": "apply\n  addHaar_image_le_lintegral_abs_det_fderiv_aux2 \u03bc (hs.inter (u_meas n)) _ fun x hx =>\n    (hf' x hx.1).mono (inter_subset_left _ _)", "annotated_tactic": ["apply\n        <a>addHaar_image_le_lintegral_abs_det_fderiv_aux2</a> \u03bc (hs.inter (u_meas n)) _ fun x hx =>\n          (hf' x hx.1).<a>mono</a> (<a>inter_subset_left</a> _ _)", [{"full_name": "MeasureTheory.addHaar_image_le_lintegral_abs_det_fderiv_aux2", "def_path": "Mathlib/MeasureTheory/Function/Jacobian.lean", "def_pos": [885, 9], "def_end_pos": [885, 55]}, {"full_name": "HasFDerivWithinAt.mono", "def_path": "Mathlib/Analysis/Calculus/FDeriv/Basic.lean", "def_pos": [380, 16], "def_end_pos": [380, 38]}, {"full_name": "Set.inter_subset_left", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [965, 9], "def_end_pos": [965, 26]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nu : \u2115 \u2192 Set E := fun n => disjointed (spanningSets \u03bc) n\nu_meas : \u2200 (n : \u2115), MeasurableSet (u n)\nA : s = \u22c3 n, s \u2229 u n\nn : \u2115\n\u22a2 \u2191\u2191\u03bc (f '' (s \u2229 u n)) \u2264 \u222b\u207b (x : E) in s \u2229 u n, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc", "state_after": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nu : \u2115 \u2192 Set E := fun n => disjointed (spanningSets \u03bc) n\nu_meas : \u2200 (n : \u2115), MeasurableSet (u n)\nA : s = \u22c3 n, s \u2229 u n\nn : \u2115\n\u22a2 \u2191\u2191\u03bc (s \u2229 u n) \u2260 \u22a4"}, {"tactic": "have : \u03bc (u n) < \u221e :=\n  lt_of_le_of_lt (measure_mono (disjointed_subset _ _)) (measure_spanningSets_lt_top \u03bc n)", "annotated_tactic": ["have : \u03bc (u n) < \u221e :=\n        <a>lt_of_le_of_lt</a> (<a>measure_mono</a> (<a>disjointed_subset</a> _ _)) (<a>measure_spanningSets_lt_top</a> \u03bc n)", [{"full_name": "lt_of_le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [122, 9], "def_end_pos": [122, 23]}, {"full_name": "MeasureTheory.measure_mono", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [193, 9], "def_end_pos": [193, 21]}, {"full_name": "disjointed_subset", "def_path": "Mathlib/Order/Disjointed.lean", "def_pos": [161, 9], "def_end_pos": [161, 26]}, {"full_name": "MeasureTheory.measure_spanningSets_lt_top", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3329, 9], "def_end_pos": [3329, 36]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nu : \u2115 \u2192 Set E := fun n => disjointed (spanningSets \u03bc) n\nu_meas : \u2200 (n : \u2115), MeasurableSet (u n)\nA : s = \u22c3 n, s \u2229 u n\nn : \u2115\n\u22a2 \u2191\u2191\u03bc (s \u2229 u n) \u2260 \u22a4", "state_after": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nu : \u2115 \u2192 Set E := fun n => disjointed (spanningSets \u03bc) n\nu_meas : \u2200 (n : \u2115), MeasurableSet (u n)\nA : s = \u22c3 n, s \u2229 u n\nn : \u2115\nthis : \u2191\u2191\u03bc (u n) < \u22a4\n\u22a2 \u2191\u2191\u03bc (s \u2229 u n) \u2260 \u22a4"}, {"tactic": "exact ne_of_lt (lt_of_le_of_lt (measure_mono (inter_subset_right _ _)) this)", "annotated_tactic": ["exact <a>ne_of_lt</a> (<a>lt_of_le_of_lt</a> (<a>measure_mono</a> (<a>inter_subset_right</a> _ _)) this)", [{"full_name": "ne_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [101, 9], "def_end_pos": [101, 17]}, {"full_name": "lt_of_le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [122, 9], "def_end_pos": [122, 23]}, {"full_name": "MeasureTheory.measure_mono", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [193, 9], "def_end_pos": [193, 21]}, {"full_name": "Set.inter_subset_right", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [969, 9], "def_end_pos": [969, 27]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nu : \u2115 \u2192 Set E := fun n => disjointed (spanningSets \u03bc) n\nu_meas : \u2200 (n : \u2115), MeasurableSet (u n)\nA : s = \u22c3 n, s \u2229 u n\nn : \u2115\nthis : \u2191\u2191\u03bc (u n) < \u22a4\n\u22a2 \u2191\u2191\u03bc (s \u2229 u n) \u2260 \u22a4", "state_after": "no goals"}, {"tactic": "conv_rhs => rw [A]", "annotated_tactic": ["conv_rhs => rw [A]", []], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nu : \u2115 \u2192 Set E := fun n => disjointed (spanningSets \u03bc) n\nu_meas : \u2200 (n : \u2115), MeasurableSet (u n)\nA : s = \u22c3 n, s \u2229 u n\n\u22a2 \u2211' (n : \u2115), \u222b\u207b (x : E) in s \u2229 u n, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc =\n    \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc", "state_after": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nu : \u2115 \u2192 Set E := fun n => disjointed (spanningSets \u03bc) n\nu_meas : \u2200 (n : \u2115), MeasurableSet (u n)\nA : s = \u22c3 n, s \u2229 u n\n\u22a2 \u2211' (n : \u2115), \u222b\u207b (x : E) in s \u2229 u n, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc =\n    \u222b\u207b (x : E) in \u22c3 n, s \u2229 u n, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc"}, {"tactic": "rw [lintegral_iUnion]", "annotated_tactic": ["rw [<a>lintegral_iUnion</a>]", [{"full_name": "MeasureTheory.lintegral_iUnion", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [1206, 9], "def_end_pos": [1206, 25]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nu : \u2115 \u2192 Set E := fun n => disjointed (spanningSets \u03bc) n\nu_meas : \u2200 (n : \u2115), MeasurableSet (u n)\nA : s = \u22c3 n, s \u2229 u n\n\u22a2 \u2211' (n : \u2115), \u222b\u207b (x : E) in s \u2229 u n, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc =\n    \u222b\u207b (x : E) in \u22c3 n, s \u2229 u n, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc", "state_after": "case hm\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nu : \u2115 \u2192 Set E := fun n => disjointed (spanningSets \u03bc) n\nu_meas : \u2200 (n : \u2115), MeasurableSet (u n)\nA : s = \u22c3 n, s \u2229 u n\n\u22a2 \u2200 (i : \u2115), MeasurableSet (s \u2229 u i)\n\ncase hd\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nu : \u2115 \u2192 Set E := fun n => disjointed (spanningSets \u03bc) n\nu_meas : \u2200 (n : \u2115), MeasurableSet (u n)\nA : s = \u22c3 n, s \u2229 u n\n\u22a2 Pairwise (Disjoint on fun n => s \u2229 u n)"}, {"tactic": "intro n", "annotated_tactic": ["intro n", []], "state_before": "case hm\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nu : \u2115 \u2192 Set E := fun n => disjointed (spanningSets \u03bc) n\nu_meas : \u2200 (n : \u2115), MeasurableSet (u n)\nA : s = \u22c3 n, s \u2229 u n\n\u22a2 \u2200 (i : \u2115), MeasurableSet (s \u2229 u i)", "state_after": "case hm\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nu : \u2115 \u2192 Set E := fun n => disjointed (spanningSets \u03bc) n\nu_meas : \u2200 (n : \u2115), MeasurableSet (u n)\nA : s = \u22c3 n, s \u2229 u n\nn : \u2115\n\u22a2 MeasurableSet (s \u2229 u n)"}, {"tactic": "exact hs.inter (u_meas n)", "annotated_tactic": ["exact hs.inter (u_meas n)", []], "state_before": "case hm\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nu : \u2115 \u2192 Set E := fun n => disjointed (spanningSets \u03bc) n\nu_meas : \u2200 (n : \u2115), MeasurableSet (u n)\nA : s = \u22c3 n, s \u2229 u n\nn : \u2115\n\u22a2 MeasurableSet (s \u2229 u n)", "state_after": "no goals"}, {"tactic": "exact pairwise_disjoint_mono (disjoint_disjointed _) fun n => inter_subset_right _ _", "annotated_tactic": ["exact <a>pairwise_disjoint_mono</a> (<a>disjoint_disjointed</a> _) fun n => <a>inter_subset_right</a> _ _", [{"full_name": "pairwise_disjoint_mono", "def_path": "Mathlib/Data/Set/Pairwise/Basic.lean", "def_pos": [60, 9], "def_end_pos": [60, 31]}, {"full_name": "disjoint_disjointed", "def_path": "Mathlib/Order/Disjointed.lean", "def_pos": [74, 9], "def_end_pos": [74, 28]}, {"full_name": "Set.inter_subset_right", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [969, 9], "def_end_pos": [969, 27]}]], "state_before": "case hd\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nu : \u2115 \u2192 Set E := fun n => disjointed (spanningSets \u03bc) n\nu_meas : \u2200 (n : \u2115), MeasurableSet (u n)\nA : s = \u22c3 n, s \u2229 u n\n\u22a2 Pairwise (Disjoint on fun n => s \u2229 u n)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "full_name": "MeasureTheory.SimpleFunc.integral_eq_lintegral", "start": [405, 1], "end": [410, 87], "traced_tactics": [{"tactic": "have : f =\u1d50[\u03bc] f.map (ENNReal.toReal \u2218 ENNReal.ofReal) :=\n  h_pos.mono fun a h => (ENNReal.toReal_ofReal h).symm", "annotated_tactic": ["have : f =\u1d50[\u03bc] f.map (<a>ENNReal.toReal</a> \u2218 <a>ENNReal.ofReal</a>) :=\n    h_pos.mono fun a h => (<a>ENNReal.toReal_ofReal</a> h).<a>symm</a>", [{"full_name": "ENNReal.toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [168, 15], "def_end_pos": [168, 21]}, {"full_name": "ENNReal.ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [172, 29], "def_end_pos": [172, 35]}, {"full_name": "ENNReal.toReal_ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [191, 9], "def_end_pos": [191, 22]}, {"full_name": "Eq.symm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [310, 9], "def_end_pos": [310, 16]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\np : \u211d\u22650\u221e\nG : Type u_5\nF' : Type u_6\ninst\u271d\u2076 : NormedAddCommGroup G\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \u211d F'\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : SMulCommClass \u211d \ud835\udd5c E\nf : \u03b1 \u2192\u209b \u211d\nhf : Integrable \u2191f\nh_pos : 0 \u2264\u1d50[\u03bc] \u2191f\n\u22a2 integral \u03bc f = ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (\u2191f a) \u2202\u03bc)", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\np : \u211d\u22650\u221e\nG : Type u_5\nF' : Type u_6\ninst\u271d\u2076 : NormedAddCommGroup G\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \u211d F'\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : SMulCommClass \u211d \ud835\udd5c E\nf : \u03b1 \u2192\u209b \u211d\nhf : Integrable \u2191f\nh_pos : 0 \u2264\u1d50[\u03bc] \u2191f\nthis : \u2191f =\u1d50[\u03bc] \u2191(map (ENNReal.toReal \u2218 ENNReal.ofReal) f)\n\u22a2 integral \u03bc f = ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (\u2191f a) \u2202\u03bc)"}, {"tactic": "rw [\u2190 integral_eq_lintegral' hf]", "annotated_tactic": ["rw [\u2190 <a>integral_eq_lintegral'</a> hf]", [{"full_name": "MeasureTheory.SimpleFunc.integral_eq_lintegral'", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [380, 9], "def_end_pos": [380, 31]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\np : \u211d\u22650\u221e\nG : Type u_5\nF' : Type u_6\ninst\u271d\u2076 : NormedAddCommGroup G\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \u211d F'\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : SMulCommClass \u211d \ud835\udd5c E\nf : \u03b1 \u2192\u209b \u211d\nhf : Integrable \u2191f\nh_pos : 0 \u2264\u1d50[\u03bc] \u2191f\nthis : \u2191f =\u1d50[\u03bc] \u2191(map (ENNReal.toReal \u2218 ENNReal.ofReal) f)\n\u22a2 integral \u03bc f = ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (\u2191f a) \u2202\u03bc)", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\np : \u211d\u22650\u221e\nG : Type u_5\nF' : Type u_6\ninst\u271d\u2076 : NormedAddCommGroup G\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \u211d F'\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : SMulCommClass \u211d \ud835\udd5c E\nf : \u03b1 \u2192\u209b \u211d\nhf : Integrable \u2191f\nh_pos : 0 \u2264\u1d50[\u03bc] \u2191f\nthis : \u2191f =\u1d50[\u03bc] \u2191(map (ENNReal.toReal \u2218 ENNReal.ofReal) f)\n\u22a2 integral \u03bc f = integral \u03bc (map (ENNReal.toReal \u2218 ENNReal.ofReal) f)\n\ncase hg0\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\np : \u211d\u22650\u221e\nG : Type u_5\nF' : Type u_6\ninst\u271d\u2076 : NormedAddCommGroup G\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \u211d F'\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : SMulCommClass \u211d \ud835\udd5c E\nf : \u03b1 \u2192\u209b \u211d\nhf : Integrable \u2191f\nh_pos : 0 \u2264\u1d50[\u03bc] \u2191f\nthis : \u2191f =\u1d50[\u03bc] \u2191(map (ENNReal.toReal \u2218 ENNReal.ofReal) f)\n\u22a2 ENNReal.ofReal 0 = 0\n\ncase ht\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\np : \u211d\u22650\u221e\nG : Type u_5\nF' : Type u_6\ninst\u271d\u2076 : NormedAddCommGroup G\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \u211d F'\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : SMulCommClass \u211d \ud835\udd5c E\nf : \u03b1 \u2192\u209b \u211d\nhf : Integrable \u2191f\nh_pos : 0 \u2264\u1d50[\u03bc] \u2191f\nthis : \u2191f =\u1d50[\u03bc] \u2191(map (ENNReal.toReal \u2218 ENNReal.ofReal) f)\n\u22a2 \u2200 (b : \u211d), ENNReal.ofReal b \u2260 \u22a4"}, {"tactic": "exacts [integral_congr hf this, ENNReal.ofReal_zero, fun b => ENNReal.ofReal_ne_top]", "annotated_tactic": ["exacts [<a>integral_congr</a> hf this, <a>ENNReal.ofReal_zero</a>, fun b => <a>ENNReal.ofReal_ne_top</a>]", [{"full_name": "MeasureTheory.SimpleFunc.integral_congr", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [398, 9], "def_end_pos": [398, 23]}, {"full_name": "ENNReal.ofReal_zero", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [245, 17], "def_end_pos": [245, 28]}, {"full_name": "ENNReal.ofReal_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [311, 17], "def_end_pos": [311, 30]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\np : \u211d\u22650\u221e\nG : Type u_5\nF' : Type u_6\ninst\u271d\u2076 : NormedAddCommGroup G\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \u211d F'\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : SMulCommClass \u211d \ud835\udd5c E\nf : \u03b1 \u2192\u209b \u211d\nhf : Integrable \u2191f\nh_pos : 0 \u2264\u1d50[\u03bc] \u2191f\nthis : \u2191f =\u1d50[\u03bc] \u2191(map (ENNReal.toReal \u2218 ENNReal.ofReal) f)\n\u22a2 integral \u03bc f = integral \u03bc (map (ENNReal.toReal \u2218 ENNReal.ofReal) f)\n\ncase hg0\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\np : \u211d\u22650\u221e\nG : Type u_5\nF' : Type u_6\ninst\u271d\u2076 : NormedAddCommGroup G\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \u211d F'\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : SMulCommClass \u211d \ud835\udd5c E\nf : \u03b1 \u2192\u209b \u211d\nhf : Integrable \u2191f\nh_pos : 0 \u2264\u1d50[\u03bc] \u2191f\nthis : \u2191f =\u1d50[\u03bc] \u2191(map (ENNReal.toReal \u2218 ENNReal.ofReal) f)\n\u22a2 ENNReal.ofReal 0 = 0\n\ncase ht\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\np : \u211d\u22650\u221e\nG : Type u_5\nF' : Type u_6\ninst\u271d\u2076 : NormedAddCommGroup G\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \u211d F'\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : SMulCommClass \u211d \ud835\udd5c E\nf : \u03b1 \u2192\u209b \u211d\nhf : Integrable \u2191f\nh_pos : 0 \u2264\u1d50[\u03bc] \u2191f\nthis : \u2191f =\u1d50[\u03bc] \u2191(map (ENNReal.toReal \u2218 ENNReal.ofReal) f)\n\u22a2 \u2200 (b : \u211d), ENNReal.ofReal b \u2260 \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "full_name": "List.IsPrefix.filter", "start": [1910, 1], "end": [1913, 42], "traced_tactics": [{"tactic": "obtain \u27e8xs, rfl\u27e9 := h", "annotated_tactic": ["obtain \u27e8xs, rfl\u27e9 := h", []], "state_before": "\u03b1 : Type u_1\np : \u03b1 \u2192 Bool\nl\u2081 l\u2082 : List \u03b1\nh : l\u2081 <+: l\u2082\n\u22a2 List.filter p l\u2081 <+: List.filter p l\u2082", "state_after": "case intro\n\u03b1 : Type u_1\np : \u03b1 \u2192 Bool\nl\u2081 xs : List \u03b1\n\u22a2 List.filter p l\u2081 <+: List.filter p (l\u2081 ++ xs)"}, {"tactic": "rw [filter_append]", "annotated_tactic": ["rw [<a>filter_append</a>]", [{"full_name": "List.filter_append", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [1238, 17], "def_end_pos": [1238, 30]}]], "state_before": "case intro\n\u03b1 : Type u_1\np : \u03b1 \u2192 Bool\nl\u2081 xs : List \u03b1\n\u22a2 List.filter p l\u2081 <+: List.filter p (l\u2081 ++ xs)", "state_after": "case intro\n\u03b1 : Type u_1\np : \u03b1 \u2192 Bool\nl\u2081 xs : List \u03b1\n\u22a2 List.filter p l\u2081 <+: List.filter p l\u2081 ++ List.filter p xs"}, {"tactic": "apply prefix_append", "annotated_tactic": ["apply <a>prefix_append</a>", [{"full_name": "List.prefix_append", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [1733, 17], "def_end_pos": [1733, 30]}]], "state_before": "case intro\n\u03b1 : Type u_1\np : \u03b1 \u2192 Bool\nl\u2081 xs : List \u03b1\n\u22a2 List.filter p l\u2081 <+: List.filter p l\u2081 ++ List.filter p xs", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Image.lean", "full_name": "Set.isCompl_range_inl_range_inr", "start": [902, 1], "end": [907, 71], "traced_tactics": [{"tactic": "rintro y \u27e8\u27e8x\u2081, rfl\u27e9, \u27e8x\u2082, h\u27e9\u27e9", "annotated_tactic": ["rintro y \u27e8\u27e8x\u2081, rfl\u27e9, \u27e8x\u2082, h\u27e9\u27e9", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\nf : \u03b9 \u2192 \u03b1\ns t : Set \u03b1\n\u22a2 range Sum.inl \u2293 range Sum.inr \u2264 \u22a5", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\nf : \u03b9 \u2192 \u03b1\ns t : Set \u03b1\nx\u2081 : \u03b1\nx\u2082 : \u03b2\nh : Sum.inr x\u2082 = Sum.inl x\u2081\n\u22a2 Sum.inl x\u2081 \u2208 \u22a5"}, {"tactic": "exact Sum.noConfusion h", "annotated_tactic": ["exact <a>Sum.noConfusion</a> h", [{"full_name": "Sum.noConfusion", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [102, 11], "def_end_pos": [102, 14]}]], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\nf : \u03b9 \u2192 \u03b1\ns t : Set \u03b1\nx\u2081 : \u03b1\nx\u2082 : \u03b2\nh : Sum.inr x\u2082 = Sum.inl x\u2081\n\u22a2 Sum.inl x\u2081 \u2208 \u22a5", "state_after": "no goals"}, {"tactic": "rintro (x | y) - <;> [left; right] <;> exact mem_range_self _", "annotated_tactic": ["rintro (x | y) - <;> [left; right] <;> exact <a>mem_range_self</a> _", [{"full_name": "Set.mem_range_self", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [680, 9], "def_end_pos": [680, 23]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\nf : \u03b9 \u2192 \u03b1\ns t : Set \u03b1\n\u22a2 \u22a4 \u2264 range Sum.inl \u2294 range Sum.inr", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/UniformIntegrable.lean", "full_name": "MeasureTheory.Mem\u2112p.snorm_indicator_le", "start": [388, 1], "end": [398, 38], "traced_tactics": [{"tactic": "have h\u2112p := hf", "annotated_tactic": ["have h\u2112p := hf", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u03b2\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\nhf : Mem\u2112p f p\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u22a2 \u2203 \u03b4 h\u03b4, \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (Set.indicator s f) p \u03bc \u2264 ENNReal.ofReal \u03b5", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u03b2\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\nhf : Mem\u2112p f p\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nh\u2112p : Mem\u2112p f p\n\u22a2 \u2203 \u03b4 h\u03b4, \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (Set.indicator s f) p \u03bc \u2264 ENNReal.ofReal \u03b5"}, {"tactic": "obtain \u27e8\u27e8f', hf', heq\u27e9, _\u27e9 := hf", "annotated_tactic": ["obtain \u27e8\u27e8f', hf', heq\u27e9, _\u27e9 := hf", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u03b2\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\nhf : Mem\u2112p f p\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nh\u2112p : Mem\u2112p f p\n\u22a2 \u2203 \u03b4 h\u03b4, \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (Set.indicator s f) p \u03bc \u2264 ENNReal.ofReal \u03b5", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u03b2\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nh\u2112p : Mem\u2112p f p\nright\u271d : snorm f p \u03bc < \u22a4\nf' : \u03b1 \u2192 \u03b2\nhf' : StronglyMeasurable f'\nheq : f =\u1d50[\u03bc] f'\n\u22a2 \u2203 \u03b4 h\u03b4, \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (Set.indicator s f) p \u03bc \u2264 ENNReal.ofReal \u03b5"}, {"tactic": "obtain \u27e8\u03b4, h\u03b4pos, h\u03b4\u27e9 := (h\u2112p.ae_eq heq).snorm_indicator_le_of_meas \u03bc hp_one hp_top hf' h\u03b5", "annotated_tactic": ["obtain \u27e8\u03b4, h\u03b4pos, h\u03b4\u27e9 := (h\u2112p.ae_eq heq).<a>snorm_indicator_le_of_meas</a> \u03bc hp_one hp_top hf' h\u03b5", [{"full_name": "MeasureTheory.Mem\u2112p.snorm_indicator_le_of_meas", "def_path": "Mathlib/MeasureTheory/Function/UniformIntegrable.lean", "def_pos": [377, 9], "def_end_pos": [377, 41]}]], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u03b2\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nh\u2112p : Mem\u2112p f p\nright\u271d : snorm f p \u03bc < \u22a4\nf' : \u03b1 \u2192 \u03b2\nhf' : StronglyMeasurable f'\nheq : f =\u1d50[\u03bc] f'\n\u22a2 \u2203 \u03b4 h\u03b4, \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (Set.indicator s f) p \u03bc \u2264 ENNReal.ofReal \u03b5", "state_after": "case intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u03b2\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nh\u2112p : Mem\u2112p f p\nright\u271d : snorm f p \u03bc < \u22a4\nf' : \u03b1 \u2192 \u03b2\nhf' : StronglyMeasurable f'\nheq : f =\u1d50[\u03bc] f'\n\u03b4 : \u211d\nh\u03b4pos : 0 < \u03b4\nh\u03b4 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (Set.indicator s f') p \u03bc \u2264 ENNReal.ofReal \u03b5\n\u22a2 \u2203 \u03b4 h\u03b4, \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (Set.indicator s f) p \u03bc \u2264 ENNReal.ofReal \u03b5"}, {"tactic": "refine' \u27e8\u03b4, h\u03b4pos, fun s hs h\u03bcs => _\u27e9", "annotated_tactic": ["refine' \u27e8\u03b4, h\u03b4pos, fun s hs h\u03bcs => _\u27e9", []], "state_before": "case intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u03b2\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nh\u2112p : Mem\u2112p f p\nright\u271d : snorm f p \u03bc < \u22a4\nf' : \u03b1 \u2192 \u03b2\nhf' : StronglyMeasurable f'\nheq : f =\u1d50[\u03bc] f'\n\u03b4 : \u211d\nh\u03b4pos : 0 < \u03b4\nh\u03b4 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (Set.indicator s f') p \u03bc \u2264 ENNReal.ofReal \u03b5\n\u22a2 \u2203 \u03b4 h\u03b4, \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (Set.indicator s f) p \u03bc \u2264 ENNReal.ofReal \u03b5", "state_after": "case intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u03b2\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nh\u2112p : Mem\u2112p f p\nright\u271d : snorm f p \u03bc < \u22a4\nf' : \u03b1 \u2192 \u03b2\nhf' : StronglyMeasurable f'\nheq : f =\u1d50[\u03bc] f'\n\u03b4 : \u211d\nh\u03b4pos : 0 < \u03b4\nh\u03b4 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (Set.indicator s f') p \u03bc \u2264 ENNReal.ofReal \u03b5\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\n\u22a2 snorm (Set.indicator s f) p \u03bc \u2264 ENNReal.ofReal \u03b5"}, {"tactic": "convert h\u03b4 s hs h\u03bcs using 1", "annotated_tactic": ["convert h\u03b4 s hs h\u03bcs using 1", []], "state_before": "case intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u03b2\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nh\u2112p : Mem\u2112p f p\nright\u271d : snorm f p \u03bc < \u22a4\nf' : \u03b1 \u2192 \u03b2\nhf' : StronglyMeasurable f'\nheq : f =\u1d50[\u03bc] f'\n\u03b4 : \u211d\nh\u03b4pos : 0 < \u03b4\nh\u03b4 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (Set.indicator s f') p \u03bc \u2264 ENNReal.ofReal \u03b5\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\n\u22a2 snorm (Set.indicator s f) p \u03bc \u2264 ENNReal.ofReal \u03b5", "state_after": "case h.e'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u03b2\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nh\u2112p : Mem\u2112p f p\nright\u271d : snorm f p \u03bc < \u22a4\nf' : \u03b1 \u2192 \u03b2\nhf' : StronglyMeasurable f'\nheq : f =\u1d50[\u03bc] f'\n\u03b4 : \u211d\nh\u03b4pos : 0 < \u03b4\nh\u03b4 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (Set.indicator s f') p \u03bc \u2264 ENNReal.ofReal \u03b5\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\n\u22a2 snorm (Set.indicator s f) p \u03bc = snorm (Set.indicator s f') p \u03bc"}, {"tactic": "rw [snorm_indicator_eq_snorm_restrict hs, snorm_indicator_eq_snorm_restrict hs]", "annotated_tactic": ["rw [<a>snorm_indicator_eq_snorm_restrict</a> hs, <a>snorm_indicator_eq_snorm_restrict</a> hs]", [{"full_name": "MeasureTheory.snorm_indicator_eq_snorm_restrict", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [657, 9], "def_end_pos": [657, 42]}, {"full_name": "MeasureTheory.snorm_indicator_eq_snorm_restrict", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [657, 9], "def_end_pos": [657, 42]}]], "state_before": "case h.e'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u03b2\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nh\u2112p : Mem\u2112p f p\nright\u271d : snorm f p \u03bc < \u22a4\nf' : \u03b1 \u2192 \u03b2\nhf' : StronglyMeasurable f'\nheq : f =\u1d50[\u03bc] f'\n\u03b4 : \u211d\nh\u03b4pos : 0 < \u03b4\nh\u03b4 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (Set.indicator s f') p \u03bc \u2264 ENNReal.ofReal \u03b5\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\n\u22a2 snorm (Set.indicator s f) p \u03bc = snorm (Set.indicator s f') p \u03bc", "state_after": "case h.e'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u03b2\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nh\u2112p : Mem\u2112p f p\nright\u271d : snorm f p \u03bc < \u22a4\nf' : \u03b1 \u2192 \u03b2\nhf' : StronglyMeasurable f'\nheq : f =\u1d50[\u03bc] f'\n\u03b4 : \u211d\nh\u03b4pos : 0 < \u03b4\nh\u03b4 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (Set.indicator s f') p \u03bc \u2264 ENNReal.ofReal \u03b5\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\n\u22a2 snorm f p (Measure.restrict \u03bc s) = snorm f' p (Measure.restrict \u03bc s)"}, {"tactic": "refine' snorm_congr_ae heq.restrict", "annotated_tactic": ["refine' <a>snorm_congr_ae</a> heq.restrict", [{"full_name": "MeasureTheory.snorm_congr_ae", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [549, 9], "def_end_pos": [549, 23]}]], "state_before": "case h.e'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u03b2\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nh\u2112p : Mem\u2112p f p\nright\u271d : snorm f p \u03bc < \u22a4\nf' : \u03b1 \u2192 \u03b2\nhf' : StronglyMeasurable f'\nheq : f =\u1d50[\u03bc] f'\n\u03b4 : \u211d\nh\u03b4pos : 0 < \u03b4\nh\u03b4 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (Set.indicator s f') p \u03bc \u2264 ENNReal.ofReal \u03b5\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\n\u22a2 snorm f p (Measure.restrict \u03bc s) = snorm f' p (Measure.restrict \u03bc s)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/UnionFind.lean", "full_name": "UFModel.Models.push", "start": [138, 1], "end": [144, 38], "traced_tactics": [{"tactic": "intro H", "annotated_tactic": ["intro H", []], "state_before": "case g\n\u03b1 : Type u_1\narr : Array (UFNode \u03b1)\nn : \u2115\nm : UFModel n\nH : Models arr m\nk : \u2115\nhk : k = n + 1\nx : \u03b1\n\u22a2 (Agrees arr (fun x => x.rank) fun i => rank m \u2191i) \u2192\n    Agrees (Array.push arr { parent := n, value := x, rank := 0 }) (fun x => x.rank) fun i =>\n      rank (UFModel.push m k (_ : n \u2264 k)) \u2191i", "state_after": "case g\n\u03b1 : Type u_1\narr : Array (UFNode \u03b1)\nn : \u2115\nm : UFModel n\nH\u271d : Models arr m\nk : \u2115\nhk : k = n + 1\nx : \u03b1\nH : Agrees arr (fun x => x.rank) fun i => rank m \u2191i\n\u22a2 Agrees (Array.push arr { parent := n, value := x, rank := 0 }) (fun x => x.rank) fun i =>\n    rank (UFModel.push m k (_ : n \u2264 k)) \u2191i"}, {"tactic": "refine H.push _ hk _ _ (fun i h \u21a6 ?_) (fun h \u21a6 ?_) <;>\nsimp [UFModel.push, h, lt_irrefl]", "annotated_tactic": ["refine H.push _ hk _ _ (fun i h \u21a6 ?_) (fun h \u21a6 ?_) <;>\n    simp [<a>UFModel.push</a>, h, <a>lt_irrefl</a>]", [{"full_name": "UFModel.push", "def_path": "Mathlib/Data/UnionFind.lean", "def_pos": [24, 5], "def_end_pos": [24, 9]}, {"full_name": "lt_irrefl", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [79, 9], "def_end_pos": [79, 18]}]], "state_before": "case g\n\u03b1 : Type u_1\narr : Array (UFNode \u03b1)\nn : \u2115\nm : UFModel n\nH\u271d : Models arr m\nk : \u2115\nhk : k = n + 1\nx : \u03b1\nH : Agrees arr (fun x => x.rank) fun i => rank m \u2191i\n\u22a2 Agrees (Array.push arr { parent := n, value := x, rank := 0 }) (fun x => x.rank) fun i =>\n    rank (UFModel.push m k (_ : n \u2264 k)) \u2191i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/UniformIntegrable.lean", "full_name": "MeasureTheory.tendsto_Lp_of_tendsto_ae_of_meas", "start": [484, 1], "end": [550, 37], "traced_tactics": [{"tactic": "rw [ENNReal.tendsto_atTop_zero]", "annotated_tactic": ["rw [<a>ENNReal.tendsto_atTop_zero</a>]", [{"full_name": "ENNReal.tendsto_atTop_zero", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [298, 19], "def_end_pos": [298, 37]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u22a2 Tendsto (fun n => snorm (f n - g) p \u03bc) atTop (\ud835\udcdd 0)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u22a2 \u2200 (\u03b5 : \u211d\u22650\u221e), \u03b5 > 0 \u2192 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 snorm (f n - g) p \u03bc \u2264 \u03b5"}, {"tactic": "intro \u03b5 h\u03b5", "annotated_tactic": ["intro \u03b5 h\u03b5", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u22a2 \u2200 (\u03b5 : \u211d\u22650\u221e), \u03b5 > 0 \u2192 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 snorm (f n - g) p \u03bc \u2264 \u03b5", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\n\u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 snorm (f n - g) p \u03bc \u2264 \u03b5"}, {"tactic": "by_cases \u03b5 < \u221e", "annotated_tactic": ["by_cases \u03b5 < \u221e", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\n\u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 snorm (f n - g) p \u03bc \u2264 \u03b5", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\n\u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 snorm (f n - g) p \u03bc \u2264 \u03b5\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u00ac\u03b5 < \u22a4\n\u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 snorm (f n - g) p \u03bc \u2264 \u03b5"}, {"tactic": "swap", "annotated_tactic": ["swap", []], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\n\u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 snorm (f n - g) p \u03bc \u2264 \u03b5\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u00ac\u03b5 < \u22a4\n\u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 snorm (f n - g) p \u03bc \u2264 \u03b5", "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u00ac\u03b5 < \u22a4\n\u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 snorm (f n - g) p \u03bc \u2264 \u03b5\n\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\n\u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 snorm (f n - g) p \u03bc \u2264 \u03b5"}, {"tactic": "by_cases h\u03bc : \u03bc = 0", "annotated_tactic": ["by_cases h\u03bc : \u03bc = 0", []], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\n\u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 snorm (f n - g) p \u03bc \u2264 \u03b5", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u03bc = 0\n\u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 snorm (f n - g) p \u03bc \u2264 \u03b5\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\n\u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 snorm (f n - g) p \u03bc \u2264 \u03b5"}, {"tactic": "have h\u03b5' : 0 < \u03b5.toReal / 3 :=\n  div_pos (ENNReal.toReal_pos (gt_iff_lt.1 h\u03b5).ne.symm h.ne) (by norm_num)", "annotated_tactic": ["have h\u03b5' : 0 < \u03b5.toReal / 3 :=\n    <a>div_pos</a> (<a>ENNReal.toReal_pos</a> (<a>gt_iff_lt</a>.1 h\u03b5).ne.symm h.ne) (by norm_num)", [{"full_name": "div_pos", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [89, 9], "def_end_pos": [89, 16]}, {"full_name": "ENNReal.toReal_pos", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2131, 9], "def_end_pos": [2131, 19]}, {"full_name": "gt_iff_lt", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [366, 9], "def_end_pos": [366, 18]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\n\u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 snorm (f n - g) p \u03bc \u2264 \u03b5", "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\n\u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 snorm (f n - g) p \u03bc \u2264 \u03b5"}, {"tactic": "have hdivp : 0 \u2264 1 / p.toReal := by\n  refine' one_div_nonneg.2 _\n  rw [\u2190 ENNReal.zero_toReal, ENNReal.toReal_le_toReal ENNReal.zero_ne_top hp']\n  exact le_trans (zero_le _) hp", "annotated_tactic": ["have hdivp : 0 \u2264 1 / p.toReal := by\n    refine' <a>one_div_nonneg</a>.2 _\n    rw [\u2190 <a>ENNReal.zero_toReal</a>, <a>ENNReal.toReal_le_toReal</a> <a>ENNReal.zero_ne_top</a> hp']\n    exact <a>le_trans</a> (<a>zero_le</a> _) hp", [{"full_name": "one_div_nonneg", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [81, 9], "def_end_pos": [81, 23]}, {"full_name": "ENNReal.zero_toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [242, 17], "def_end_pos": [242, 28]}, {"full_name": "ENNReal.toReal_le_toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2036, 9], "def_end_pos": [2036, 25]}, {"full_name": "ENNReal.zero_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [334, 17], "def_end_pos": [334, 28]}, {"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}, {"full_name": "zero_le", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [217, 30], "def_end_pos": [217, 37]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\n\u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 snorm (f n - g) p \u03bc \u2264 \u03b5", "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\n\u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 snorm (f n - g) p \u03bc \u2264 \u03b5"}, {"tactic": "have hpow : 0 < measureUnivNNReal \u03bc ^ (1 / p.toReal) :=\n  Real.rpow_pos_of_pos (measureUnivNNReal_pos h\u03bc) _", "annotated_tactic": ["have hpow : 0 < <a>measureUnivNNReal</a> \u03bc ^ (1 / p.toReal) :=\n    <a>Real.rpow_pos_of_pos</a> (<a>measureUnivNNReal_pos</a> h\u03bc) _", [{"full_name": "MeasureTheory.measureUnivNNReal", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2897, 5], "def_end_pos": [2897, 22]}, {"full_name": "Real.rpow_pos_of_pos", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [92, 9], "def_end_pos": [92, 24]}, {"full_name": "MeasureTheory.measureUnivNNReal_pos", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2963, 9], "def_end_pos": [2963, 30]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\n\u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 snorm (f n - g) p \u03bc \u2264 \u03b5", "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 snorm (f n - g) p \u03bc \u2264 \u03b5"}, {"tactic": "obtain \u27e8\u03b4\u2081, h\u03b4\u2081, hsnorm\u2081\u27e9 := hui h\u03b5'", "annotated_tactic": ["obtain \u27e8\u03b4\u2081, h\u03b4\u2081, hsnorm\u2081\u27e9 := hui h\u03b5'", []], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 snorm (f n - g) p \u03bc \u2264 \u03b5", "state_after": "case neg.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 snorm (f n - g) p \u03bc \u2264 \u03b5"}, {"tactic": "obtain \u27e8\u03b4\u2082, h\u03b4\u2082, hsnorm\u2082\u27e9 := hg'.snorm_indicator_le \u03bc hp hp' h\u03b5'", "annotated_tactic": ["obtain \u27e8\u03b4\u2082, h\u03b4\u2082, hsnorm\u2082\u27e9 := hg'.snorm_indicator_le \u03bc hp hp' h\u03b5'", []], "state_before": "case neg.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 snorm (f n - g) p \u03bc \u2264 \u03b5", "state_after": "case neg.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 snorm (f n - g) p \u03bc \u2264 \u03b5"}, {"tactic": "obtain \u27e8t, htm, ht\u2081, ht\u2082\u27e9 := tendstoUniformlyOn_of_ae_tendsto' hf hg hfg (lt_min h\u03b4\u2081 h\u03b4\u2082)", "annotated_tactic": ["obtain \u27e8t, htm, ht\u2081, ht\u2082\u27e9 := <a>tendstoUniformlyOn_of_ae_tendsto'</a> hf hg hfg (<a>lt_min</a> h\u03b4\u2081 h\u03b4\u2082)", [{"full_name": "MeasureTheory.tendstoUniformlyOn_of_ae_tendsto'", "def_path": "Mathlib/MeasureTheory/Function/Egorov.lean", "def_pos": [213, 9], "def_end_pos": [213, 42]}, {"full_name": "lt_min", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [159, 9], "def_end_pos": [159, 15]}]], "state_before": "case neg.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 snorm (f n - g) p \u03bc \u2264 \u03b5", "state_after": "case neg.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nht\u2082 : TendstoUniformlyOn (fun n => f n) g atTop t\u1d9c\n\u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 snorm (f n - g) p \u03bc \u2264 \u03b5"}, {"tactic": "rw [Metric.tendstoUniformlyOn_iff] at ht\u2082", "annotated_tactic": ["rw [<a>Metric.tendstoUniformlyOn_iff</a>] at ht\u2082", [{"full_name": "Metric.tendstoUniformlyOn_iff", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [922, 9], "def_end_pos": [922, 31]}]], "state_before": "case neg.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nht\u2082 : TendstoUniformlyOn (fun n => f n) g atTop t\u1d9c\n\u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 snorm (f n - g) p \u03bc \u2264 \u03b5", "state_after": "case neg.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nht\u2082 : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2200\u1da0 (n : \u2115) in atTop, \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f n x) < \u03b5\n\u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 snorm (f n - g) p \u03bc \u2264 \u03b5"}, {"tactic": "specialize ht\u2082 (\u03b5.toReal / (3 * measureUnivNNReal \u03bc ^ (1 / p.toReal)))\n  (div_pos (ENNReal.toReal_pos (gt_iff_lt.1 h\u03b5).ne.symm h.ne) (mul_pos (by norm_num) hpow))", "annotated_tactic": ["specialize ht\u2082 (\u03b5.toReal / (3 * <a>measureUnivNNReal</a> \u03bc ^ (1 / p.toReal)))\n    (<a>div_pos</a> (<a>ENNReal.toReal_pos</a> (<a>gt_iff_lt</a>.1 h\u03b5).ne.symm h.ne) (<a>mul_pos</a> (by norm_num) hpow))", [{"full_name": "MeasureTheory.measureUnivNNReal", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2897, 5], "def_end_pos": [2897, 22]}, {"full_name": "div_pos", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [89, 9], "def_end_pos": [89, 16]}, {"full_name": "ENNReal.toReal_pos", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2131, 9], "def_end_pos": [2131, 19]}, {"full_name": "gt_iff_lt", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [366, 9], "def_end_pos": [366, 18]}, {"full_name": "mul_pos", "def_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "def_pos": [345, 7], "def_end_pos": [345, 14]}]], "state_before": "case neg.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nht\u2082 : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2200\u1da0 (n : \u2115) in atTop, \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f n x) < \u03b5\n\u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 snorm (f n - g) p \u03bc \u2264 \u03b5", "state_after": "case neg.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nht\u2082 :\n  \u2200\u1da0 (n : \u2115) in atTop,\n    \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f n x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\n\u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 snorm (f n - g) p \u03bc \u2264 \u03b5"}, {"tactic": "obtain \u27e8N, hN\u27e9 := eventually_atTop.1 ht\u2082", "annotated_tactic": ["obtain \u27e8N, hN\u27e9 := <a>eventually_atTop</a>.1 ht\u2082", [{"full_name": "Filter.eventually_atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [178, 9], "def_end_pos": [178, 25]}]], "state_before": "case neg.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nht\u2082 :\n  \u2200\u1da0 (n : \u2115) in atTop,\n    \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f n x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\n\u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 snorm (f n - g) p \u03bc \u2264 \u03b5", "state_after": "case neg.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nht\u2082 :\n  \u2200\u1da0 (n : \u2115) in atTop,\n    \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f n x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nN : \u2115\nhN :\n  \u2200 (b : \u2115),\n    b \u2265 N \u2192\n      \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f b x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\n\u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 snorm (f n - g) p \u03bc \u2264 \u03b5"}, {"tactic": "clear ht\u2082", "annotated_tactic": ["clear ht\u2082", []], "state_before": "case neg.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nht\u2082 :\n  \u2200\u1da0 (n : \u2115) in atTop,\n    \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f n x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nN : \u2115\nhN :\n  \u2200 (b : \u2115),\n    b \u2265 N \u2192\n      \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f b x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\n\u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 snorm (f n - g) p \u03bc \u2264 \u03b5", "state_after": "case neg.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN : \u2115\nhN :\n  \u2200 (b : \u2115),\n    b \u2265 N \u2192\n      \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f b x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\n\u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 snorm (f n - g) p \u03bc \u2264 \u03b5"}, {"tactic": "refine' \u27e8N, fun n hn => _\u27e9", "annotated_tactic": ["refine' \u27e8N, fun n hn => _\u27e9", []], "state_before": "case neg.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN : \u2115\nhN :\n  \u2200 (b : \u2115),\n    b \u2265 N \u2192\n      \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f b x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\n\u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 snorm (f n - g) p \u03bc \u2264 \u03b5", "state_after": "case neg.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN : \u2115\nhN :\n  \u2200 (b : \u2115),\n    b \u2265 N \u2192\n      \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f b x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nn : \u2115\nhn : n \u2265 N\n\u22a2 snorm (f n - g) p \u03bc \u2264 \u03b5"}, {"tactic": "rw [\u2190 t.indicator_self_add_compl (f n - g)]", "annotated_tactic": ["rw [\u2190 t.indicator_self_add_compl (f n - g)]", []], "state_before": "case neg.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN : \u2115\nhN :\n  \u2200 (b : \u2115),\n    b \u2265 N \u2192\n      \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f b x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nn : \u2115\nhn : n \u2265 N\n\u22a2 snorm (f n - g) p \u03bc \u2264 \u03b5", "state_after": "case neg.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN : \u2115\nhN :\n  \u2200 (b : \u2115),\n    b \u2265 N \u2192\n      \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f b x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nn : \u2115\nhn : n \u2265 N\n\u22a2 snorm (indicator t (f n - g) + indicator t\u1d9c (f n - g)) p \u03bc \u2264 \u03b5"}, {"tactic": "refine' le_trans (snorm_add_le (((hf n).sub hg).indicator htm).aestronglyMeasurable\n  (((hf n).sub hg).indicator htm.compl).aestronglyMeasurable hp) _", "annotated_tactic": ["refine' <a>le_trans</a> (<a>snorm_add_le</a> (((hf n).<a>sub</a> hg).<a>indicator</a> htm).<a>aestronglyMeasurable</a>\n    (((hf n).<a>sub</a> hg).<a>indicator</a> htm.compl).<a>aestronglyMeasurable</a> hp) _", [{"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}, {"full_name": "MeasureTheory.snorm_add_le", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [802, 9], "def_end_pos": [802, 21]}, {"full_name": "MeasureTheory.StronglyMeasurable.sub", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [436, 3], "def_end_pos": [436, 14]}, {"full_name": "MeasureTheory.StronglyMeasurable.indicator", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [825, 19], "def_end_pos": [825, 28]}, {"full_name": "MeasureTheory.StronglyMeasurable.aestronglyMeasurable", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [110, 19], "def_end_pos": [110, 58]}, {"full_name": "MeasureTheory.StronglyMeasurable.sub", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [436, 3], "def_end_pos": [436, 14]}, {"full_name": "MeasureTheory.StronglyMeasurable.indicator", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [825, 19], "def_end_pos": [825, 28]}, {"full_name": "MeasureTheory.StronglyMeasurable.aestronglyMeasurable", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [110, 19], "def_end_pos": [110, 58]}]], "state_before": "case neg.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN : \u2115\nhN :\n  \u2200 (b : \u2115),\n    b \u2265 N \u2192\n      \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f b x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nn : \u2115\nhn : n \u2265 N\n\u22a2 snorm (indicator t (f n - g) + indicator t\u1d9c (f n - g)) p \u03bc \u2264 \u03b5", "state_after": "case neg.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN : \u2115\nhN :\n  \u2200 (b : \u2115),\n    b \u2265 N \u2192\n      \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f b x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nn : \u2115\nhn : n \u2265 N\n\u22a2 snorm (indicator t (f n - g)) p \u03bc + snorm (indicator t\u1d9c (f n - g)) p \u03bc \u2264 \u03b5"}, {"tactic": "rw [sub_eq_add_neg, Set.indicator_add' t, Set.indicator_neg']", "annotated_tactic": ["rw [<a>sub_eq_add_neg</a>, <a>Set.indicator_add'</a> t, <a>Set.indicator_neg'</a>]", [{"full_name": "sub_eq_add_neg", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [975, 3], "def_end_pos": [975, 14]}, {"full_name": "Set.indicator_add'", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [400, 3], "def_end_pos": [400, 14]}, {"full_name": "Set.indicator_neg'", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [533, 3], "def_end_pos": [533, 14]}]], "state_before": "case neg.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN : \u2115\nhN :\n  \u2200 (b : \u2115),\n    b \u2265 N \u2192\n      \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f b x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nn : \u2115\nhn : n \u2265 N\n\u22a2 snorm (indicator t (f n - g)) p \u03bc + snorm (indicator t\u1d9c (f n - g)) p \u03bc \u2264 \u03b5", "state_after": "case neg.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN : \u2115\nhN :\n  \u2200 (b : \u2115),\n    b \u2265 N \u2192\n      \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f b x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nn : \u2115\nhn : n \u2265 N\n\u22a2 snorm (indicator t (f n) + -indicator t g) p \u03bc + snorm (indicator t\u1d9c (f n + -g)) p \u03bc \u2264 \u03b5"}, {"tactic": "refine' le_trans (add_le_add_right (snorm_add_le ((hf n).indicator htm).aestronglyMeasurable\n  (hg.indicator htm).neg.aestronglyMeasurable hp) _) _", "annotated_tactic": ["refine' <a>le_trans</a> (<a>add_le_add_right</a> (<a>snorm_add_le</a> ((hf n).<a>indicator</a> htm).<a>aestronglyMeasurable</a>\n    (hg.indicator htm).neg.aestronglyMeasurable hp) _) _", [{"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}, {"full_name": "add_le_add_right", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [66, 15], "def_end_pos": [66, 31]}, {"full_name": "MeasureTheory.snorm_add_le", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [802, 9], "def_end_pos": [802, 21]}, {"full_name": "MeasureTheory.StronglyMeasurable.indicator", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [825, 19], "def_end_pos": [825, 28]}, {"full_name": "MeasureTheory.StronglyMeasurable.aestronglyMeasurable", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [110, 19], "def_end_pos": [110, 58]}]], "state_before": "case neg.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN : \u2115\nhN :\n  \u2200 (b : \u2115),\n    b \u2265 N \u2192\n      \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f b x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nn : \u2115\nhn : n \u2265 N\n\u22a2 snorm (indicator t (f n) + -indicator t g) p \u03bc + snorm (indicator t\u1d9c (f n + -g)) p \u03bc \u2264 \u03b5", "state_after": "case neg.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN : \u2115\nhN :\n  \u2200 (b : \u2115),\n    b \u2265 N \u2192\n      \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f b x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nn : \u2115\nhn : n \u2265 N\n\u22a2 snorm (indicator t (f n)) p \u03bc + snorm (-indicator t g) p \u03bc + snorm (indicator t\u1d9c (f n + -g)) p \u03bc \u2264 \u03b5"}, {"tactic": "have hnf : snorm (t.indicator (f n)) p \u03bc \u2264 ENNReal.ofReal (\u03b5.toReal / 3) := by\n  refine' hsnorm\u2081 n t htm (le_trans ht\u2081 _)\n  rw [ENNReal.ofReal_le_ofReal_iff h\u03b4\u2081.le]\n  exact min_le_left _ _", "annotated_tactic": ["have hnf : <a>snorm</a> (t.indicator (f n)) p \u03bc \u2264 <a>ENNReal.ofReal</a> (\u03b5.toReal / 3) := by\n    refine' hsnorm\u2081 n t htm (<a>le_trans</a> ht\u2081 _)\n    rw [<a>ENNReal.ofReal_le_ofReal_iff</a> h\u03b4\u2081.le]\n    exact <a>min_le_left</a> _ _", [{"full_name": "MeasureTheory.snorm", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [84, 5], "def_end_pos": [84, 10]}, {"full_name": "ENNReal.ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [172, 29], "def_end_pos": [172, 35]}, {"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}, {"full_name": "ENNReal.ofReal_le_ofReal_iff", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2145, 9], "def_end_pos": [2145, 29]}, {"full_name": "min_le_left", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [33, 9], "def_end_pos": [33, 20]}]], "state_before": "case neg.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN : \u2115\nhN :\n  \u2200 (b : \u2115),\n    b \u2265 N \u2192\n      \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f b x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nn : \u2115\nhn : n \u2265 N\n\u22a2 snorm (indicator t (f n)) p \u03bc + snorm (-indicator t g) p \u03bc + snorm (indicator t\u1d9c (f n + -g)) p \u03bc \u2264 \u03b5", "state_after": "case neg.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN : \u2115\nhN :\n  \u2200 (b : \u2115),\n    b \u2265 N \u2192\n      \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f b x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nn : \u2115\nhn : n \u2265 N\nhnf : snorm (indicator t (f n)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u22a2 snorm (indicator t (f n)) p \u03bc + snorm (-indicator t g) p \u03bc + snorm (indicator t\u1d9c (f n + -g)) p \u03bc \u2264 \u03b5"}, {"tactic": "have hng : snorm (t.indicator g) p \u03bc \u2264 ENNReal.ofReal (\u03b5.toReal / 3) := by\n  refine' hsnorm\u2082 t htm (le_trans ht\u2081 _)\n  rw [ENNReal.ofReal_le_ofReal_iff h\u03b4\u2082.le]\n  exact min_le_right _ _", "annotated_tactic": ["have hng : <a>snorm</a> (t.indicator g) p \u03bc \u2264 <a>ENNReal.ofReal</a> (\u03b5.toReal / 3) := by\n    refine' hsnorm\u2082 t htm (<a>le_trans</a> ht\u2081 _)\n    rw [<a>ENNReal.ofReal_le_ofReal_iff</a> h\u03b4\u2082.le]\n    exact <a>min_le_right</a> _ _", [{"full_name": "MeasureTheory.snorm", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [84, 5], "def_end_pos": [84, 10]}, {"full_name": "ENNReal.ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [172, 29], "def_end_pos": [172, 35]}, {"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}, {"full_name": "ENNReal.ofReal_le_ofReal_iff", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2145, 9], "def_end_pos": [2145, 29]}, {"full_name": "min_le_right", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [40, 9], "def_end_pos": [40, 21]}]], "state_before": "case neg.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN : \u2115\nhN :\n  \u2200 (b : \u2115),\n    b \u2265 N \u2192\n      \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f b x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nn : \u2115\nhn : n \u2265 N\nhnf : snorm (indicator t (f n)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u22a2 snorm (indicator t (f n)) p \u03bc + snorm (-indicator t g) p \u03bc + snorm (indicator t\u1d9c (f n + -g)) p \u03bc \u2264 \u03b5", "state_after": "case neg.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN : \u2115\nhN :\n  \u2200 (b : \u2115),\n    b \u2265 N \u2192\n      \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f b x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nn : \u2115\nhn : n \u2265 N\nhnf : snorm (indicator t (f n)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhng : snorm (indicator t g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u22a2 snorm (indicator t (f n)) p \u03bc + snorm (-indicator t g) p \u03bc + snorm (indicator t\u1d9c (f n + -g)) p \u03bc \u2264 \u03b5"}, {"tactic": "have : ENNReal.ofReal (\u03b5.toReal / 3) = \u03b5 / 3 := by\n  rw [ENNReal.ofReal_div_of_pos (show (0 : \u211d) < 3 by norm_num), ENNReal.ofReal_toReal h.ne]\n  simp", "annotated_tactic": ["have : <a>ENNReal.ofReal</a> (\u03b5.toReal / 3) = \u03b5 / 3 := by\n    rw [<a>ENNReal.ofReal_div_of_pos</a> (show (0 : \u211d) < 3 by norm_num), <a>ENNReal.ofReal_toReal</a> h.ne]\n    simp", [{"full_name": "ENNReal.ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [172, 29], "def_end_pos": [172, 35]}, {"full_name": "ENNReal.ofReal_div_of_pos", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2248, 9], "def_end_pos": [2248, 26]}, {"full_name": "ENNReal.ofReal_toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [186, 9], "def_end_pos": [186, 22]}]], "state_before": "case neg.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN : \u2115\nhN :\n  \u2200 (b : \u2115),\n    b \u2265 N \u2192\n      \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f b x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nn : \u2115\nhn : n \u2265 N\nhnf : snorm (indicator t (f n)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhng : snorm (indicator t g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhlt : snorm (indicator t\u1d9c (f n - g)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u22a2 snorm (indicator t (f n)) p \u03bc + snorm (-indicator t g) p \u03bc + snorm (indicator t\u1d9c (f n + -g)) p \u03bc \u2264 \u03b5", "state_after": "case neg.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN : \u2115\nhN :\n  \u2200 (b : \u2115),\n    b \u2265 N \u2192\n      \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f b x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nn : \u2115\nhn : n \u2265 N\nhnf : snorm (indicator t (f n)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhng : snorm (indicator t g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhlt : snorm (indicator t\u1d9c (f n - g)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nthis : ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) = \u03b5 / 3\n\u22a2 snorm (indicator t (f n)) p \u03bc + snorm (-indicator t g) p \u03bc + snorm (indicator t\u1d9c (f n + -g)) p \u03bc \u2264 \u03b5"}, {"tactic": "rw [this] at hnf hng hlt", "annotated_tactic": ["rw [this] at hnf hng hlt", []], "state_before": "case neg.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN : \u2115\nhN :\n  \u2200 (b : \u2115),\n    b \u2265 N \u2192\n      \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f b x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nn : \u2115\nhn : n \u2265 N\nhnf : snorm (indicator t (f n)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhng : snorm (indicator t g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhlt : snorm (indicator t\u1d9c (f n - g)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nthis : ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) = \u03b5 / 3\n\u22a2 snorm (indicator t (f n)) p \u03bc + snorm (-indicator t g) p \u03bc + snorm (indicator t\u1d9c (f n + -g)) p \u03bc \u2264 \u03b5", "state_after": "case neg.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN : \u2115\nhN :\n  \u2200 (b : \u2115),\n    b \u2265 N \u2192\n      \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f b x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nn : \u2115\nhn : n \u2265 N\nhnf : snorm (indicator t (f n)) p \u03bc \u2264 \u03b5 / 3\nhng : snorm (indicator t g) p \u03bc \u2264 \u03b5 / 3\nhlt : snorm (indicator t\u1d9c (f n - g)) p \u03bc \u2264 \u03b5 / 3\nthis : ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) = \u03b5 / 3\n\u22a2 snorm (indicator t (f n)) p \u03bc + snorm (-indicator t g) p \u03bc + snorm (indicator t\u1d9c (f n + -g)) p \u03bc \u2264 \u03b5"}, {"tactic": "rw [snorm_neg, \u2190 ENNReal.add_thirds \u03b5, \u2190 sub_eq_add_neg]", "annotated_tactic": ["rw [<a>snorm_neg</a>, \u2190 <a>ENNReal.add_thirds</a> \u03b5, \u2190 <a>sub_eq_add_neg</a>]", [{"full_name": "MeasureTheory.snorm_neg", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [261, 9], "def_end_pos": [261, 18]}, {"full_name": "ENNReal.add_thirds", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1786, 9], "def_end_pos": [1786, 19]}, {"full_name": "sub_eq_add_neg", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [975, 3], "def_end_pos": [975, 14]}]], "state_before": "case neg.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN : \u2115\nhN :\n  \u2200 (b : \u2115),\n    b \u2265 N \u2192\n      \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f b x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nn : \u2115\nhn : n \u2265 N\nhnf : snorm (indicator t (f n)) p \u03bc \u2264 \u03b5 / 3\nhng : snorm (indicator t g) p \u03bc \u2264 \u03b5 / 3\nhlt : snorm (indicator t\u1d9c (f n - g)) p \u03bc \u2264 \u03b5 / 3\nthis : ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) = \u03b5 / 3\n\u22a2 snorm (indicator t (f n)) p \u03bc + snorm (-indicator t g) p \u03bc + snorm (indicator t\u1d9c (f n + -g)) p \u03bc \u2264 \u03b5", "state_after": "case neg.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN : \u2115\nhN :\n  \u2200 (b : \u2115),\n    b \u2265 N \u2192\n      \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f b x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nn : \u2115\nhn : n \u2265 N\nhnf : snorm (indicator t (f n)) p \u03bc \u2264 \u03b5 / 3\nhng : snorm (indicator t g) p \u03bc \u2264 \u03b5 / 3\nhlt : snorm (indicator t\u1d9c (f n - g)) p \u03bc \u2264 \u03b5 / 3\nthis : ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) = \u03b5 / 3\n\u22a2 snorm (indicator t (f n)) p \u03bc + snorm (indicator t g) p \u03bc + snorm (indicator t\u1d9c (f n - g)) p \u03bc \u2264 \u03b5 / 3 + \u03b5 / 3 + \u03b5 / 3"}, {"tactic": "exact add_le_add_three hnf hng hlt", "annotated_tactic": ["exact <a>add_le_add_three</a> hnf hng hlt", [{"full_name": "add_le_add_three", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [213, 3], "def_end_pos": [213, 14]}]], "state_before": "case neg.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN : \u2115\nhN :\n  \u2200 (b : \u2115),\n    b \u2265 N \u2192\n      \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f b x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nn : \u2115\nhn : n \u2265 N\nhnf : snorm (indicator t (f n)) p \u03bc \u2264 \u03b5 / 3\nhng : snorm (indicator t g) p \u03bc \u2264 \u03b5 / 3\nhlt : snorm (indicator t\u1d9c (f n - g)) p \u03bc \u2264 \u03b5 / 3\nthis : ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) = \u03b5 / 3\n\u22a2 snorm (indicator t (f n)) p \u03bc + snorm (indicator t g) p \u03bc + snorm (indicator t\u1d9c (f n - g)) p \u03bc \u2264 \u03b5 / 3 + \u03b5 / 3 + \u03b5 / 3", "state_after": "no goals"}, {"tactic": "rw [not_lt, top_le_iff] at h", "annotated_tactic": ["rw [<a>not_lt</a>, <a>top_le_iff</a>] at h", [{"full_name": "not_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [368, 9], "def_end_pos": [368, 15]}, {"full_name": "top_le_iff", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [157, 9], "def_end_pos": [157, 19]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u00ac\u03b5 < \u22a4\n\u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 snorm (f n - g) p \u03bc \u2264 \u03b5", "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 = \u22a4\n\u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 snorm (f n - g) p \u03bc \u2264 \u03b5"}, {"tactic": "exact \u27e80, fun n _ => by simp [h]\u27e9", "annotated_tactic": ["exact \u27e80, fun n _ => by simp [h]\u27e9", []], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 = \u22a4\n\u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 snorm (f n - g) p \u03bc \u2264 \u03b5", "state_after": "no goals"}, {"tactic": "simp [h]", "annotated_tactic": ["simp [h]", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 = \u22a4\nn : \u2115\nx\u271d : n \u2265 0\n\u22a2 snorm (f n - g) p \u03bc \u2264 \u03b5", "state_after": "no goals"}, {"tactic": "exact \u27e80, fun n _ => by simp [h\u03bc]\u27e9", "annotated_tactic": ["exact \u27e80, fun n _ => by simp [h\u03bc]\u27e9", []], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u03bc = 0\n\u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 snorm (f n - g) p \u03bc \u2264 \u03b5", "state_after": "no goals"}, {"tactic": "simp [h\u03bc]", "annotated_tactic": ["simp [h\u03bc]", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u03bc = 0\nn : \u2115\nx\u271d : n \u2265 0\n\u22a2 snorm (f n - g) p \u03bc \u2264 \u03b5", "state_after": "no goals"}, {"tactic": "norm_num", "annotated_tactic": ["norm_num", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\n\u22a2 0 < 3", "state_after": "no goals"}, {"tactic": "refine' one_div_nonneg.2 _", "annotated_tactic": ["refine' <a>one_div_nonneg</a>.2 _", [{"full_name": "one_div_nonneg", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [81, 9], "def_end_pos": [81, 23]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\n\u22a2 0 \u2264 1 / ENNReal.toReal p", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\n\u22a2 0 \u2264 ENNReal.toReal p"}, {"tactic": "rw [\u2190 ENNReal.zero_toReal, ENNReal.toReal_le_toReal ENNReal.zero_ne_top hp']", "annotated_tactic": ["rw [\u2190 <a>ENNReal.zero_toReal</a>, <a>ENNReal.toReal_le_toReal</a> <a>ENNReal.zero_ne_top</a> hp']", [{"full_name": "ENNReal.zero_toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [242, 17], "def_end_pos": [242, 28]}, {"full_name": "ENNReal.toReal_le_toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2036, 9], "def_end_pos": [2036, 25]}, {"full_name": "ENNReal.zero_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [334, 17], "def_end_pos": [334, 28]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\n\u22a2 0 \u2264 ENNReal.toReal p", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\n\u22a2 0 \u2264 p"}, {"tactic": "exact le_trans (zero_le _) hp", "annotated_tactic": ["exact <a>le_trans</a> (<a>zero_le</a> _) hp", [{"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}, {"full_name": "zero_le", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [217, 30], "def_end_pos": [217, 37]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\n\u22a2 0 \u2264 p", "state_after": "no goals"}, {"tactic": "norm_num", "annotated_tactic": ["norm_num", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nht\u2082 : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2200\u1da0 (n : \u2115) in atTop, \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f n x) < \u03b5\n\u22a2 0 < 3", "state_after": "no goals"}, {"tactic": "refine' hsnorm\u2081 n t htm (le_trans ht\u2081 _)", "annotated_tactic": ["refine' hsnorm\u2081 n t htm (<a>le_trans</a> ht\u2081 _)", [{"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN : \u2115\nhN :\n  \u2200 (b : \u2115),\n    b \u2265 N \u2192\n      \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f b x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nn : \u2115\nhn : n \u2265 N\n\u22a2 snorm (indicator t (f n)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN : \u2115\nhN :\n  \u2200 (b : \u2115),\n    b \u2265 N \u2192\n      \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f b x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nn : \u2115\nhn : n \u2265 N\n\u22a2 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082) \u2264 ENNReal.ofReal \u03b4\u2081"}, {"tactic": "rw [ENNReal.ofReal_le_ofReal_iff h\u03b4\u2081.le]", "annotated_tactic": ["rw [<a>ENNReal.ofReal_le_ofReal_iff</a> h\u03b4\u2081.le]", [{"full_name": "ENNReal.ofReal_le_ofReal_iff", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2145, 9], "def_end_pos": [2145, 29]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN : \u2115\nhN :\n  \u2200 (b : \u2115),\n    b \u2265 N \u2192\n      \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f b x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nn : \u2115\nhn : n \u2265 N\n\u22a2 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082) \u2264 ENNReal.ofReal \u03b4\u2081", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN : \u2115\nhN :\n  \u2200 (b : \u2115),\n    b \u2265 N \u2192\n      \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f b x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nn : \u2115\nhn : n \u2265 N\n\u22a2 min \u03b4\u2081 \u03b4\u2082 \u2264 \u03b4\u2081"}, {"tactic": "exact min_le_left _ _", "annotated_tactic": ["exact <a>min_le_left</a> _ _", [{"full_name": "min_le_left", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [33, 9], "def_end_pos": [33, 20]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN : \u2115\nhN :\n  \u2200 (b : \u2115),\n    b \u2265 N \u2192\n      \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f b x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nn : \u2115\nhn : n \u2265 N\n\u22a2 min \u03b4\u2081 \u03b4\u2082 \u2264 \u03b4\u2081", "state_after": "no goals"}, {"tactic": "refine' hsnorm\u2082 t htm (le_trans ht\u2081 _)", "annotated_tactic": ["refine' hsnorm\u2082 t htm (<a>le_trans</a> ht\u2081 _)", [{"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN : \u2115\nhN :\n  \u2200 (b : \u2115),\n    b \u2265 N \u2192\n      \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f b x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nn : \u2115\nhn : n \u2265 N\nhnf : snorm (indicator t (f n)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u22a2 snorm (indicator t g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN : \u2115\nhN :\n  \u2200 (b : \u2115),\n    b \u2265 N \u2192\n      \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f b x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nn : \u2115\nhn : n \u2265 N\nhnf : snorm (indicator t (f n)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u22a2 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082) \u2264 ENNReal.ofReal \u03b4\u2082"}, {"tactic": "rw [ENNReal.ofReal_le_ofReal_iff h\u03b4\u2082.le]", "annotated_tactic": ["rw [<a>ENNReal.ofReal_le_ofReal_iff</a> h\u03b4\u2082.le]", [{"full_name": "ENNReal.ofReal_le_ofReal_iff", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2145, 9], "def_end_pos": [2145, 29]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN : \u2115\nhN :\n  \u2200 (b : \u2115),\n    b \u2265 N \u2192\n      \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f b x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nn : \u2115\nhn : n \u2265 N\nhnf : snorm (indicator t (f n)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u22a2 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082) \u2264 ENNReal.ofReal \u03b4\u2082", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN : \u2115\nhN :\n  \u2200 (b : \u2115),\n    b \u2265 N \u2192\n      \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f b x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nn : \u2115\nhn : n \u2265 N\nhnf : snorm (indicator t (f n)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u22a2 min \u03b4\u2081 \u03b4\u2082 \u2264 \u03b4\u2082"}, {"tactic": "exact min_le_right _ _", "annotated_tactic": ["exact <a>min_le_right</a> _ _", [{"full_name": "min_le_right", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [40, 9], "def_end_pos": [40, 21]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN : \u2115\nhN :\n  \u2200 (b : \u2115),\n    b \u2265 N \u2192\n      \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f b x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nn : \u2115\nhn : n \u2265 N\nhnf : snorm (indicator t (f n)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u22a2 min \u03b4\u2081 \u03b4\u2082 \u2264 \u03b4\u2082", "state_after": "no goals"}, {"tactic": "specialize hN n hn", "annotated_tactic": ["specialize hN n hn", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN : \u2115\nhN :\n  \u2200 (b : \u2115),\n    b \u2265 N \u2192\n      \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f b x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nn : \u2115\nhn : n \u2265 N\nhnf : snorm (indicator t (f n)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhng : snorm (indicator t g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u22a2 snorm (indicator t\u1d9c (f n - g)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN n : \u2115\nhn : n \u2265 N\nhnf : snorm (indicator t (f n)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhng : snorm (indicator t g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhN : \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f n x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\n\u22a2 snorm (indicator t\u1d9c (f n - g)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)"}, {"tactic": "have : 0 \u2264 \u03b5.toReal / (3 * measureUnivNNReal \u03bc ^ (1 / p.toReal)) := by\n  rw [div_mul_eq_div_mul_one_div]\n  exact mul_nonneg h\u03b5'.le (one_div_nonneg.2 hpow.le)", "annotated_tactic": ["have : 0 \u2264 \u03b5.toReal / (3 * <a>measureUnivNNReal</a> \u03bc ^ (1 / p.toReal)) := by\n      rw [<a>div_mul_eq_div_mul_one_div</a>]\n      exact <a>mul_nonneg</a> h\u03b5'.le (<a>one_div_nonneg</a>.2 hpow.le)", [{"full_name": "MeasureTheory.measureUnivNNReal", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2897, 5], "def_end_pos": [2897, 22]}, {"full_name": "div_mul_eq_div_mul_one_div", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [567, 9], "def_end_pos": [567, 35]}, {"full_name": "mul_nonneg", "def_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "def_pos": [380, 7], "def_end_pos": [380, 17]}, {"full_name": "one_div_nonneg", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [81, 9], "def_end_pos": [81, 23]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN n : \u2115\nhn : n \u2265 N\nhnf : snorm (indicator t (f n)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhng : snorm (indicator t g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhN : \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f n x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\n\u22a2 snorm (indicator t\u1d9c (f n - g)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN n : \u2115\nhn : n \u2265 N\nhnf : snorm (indicator t (f n)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhng : snorm (indicator t g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhN : \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f n x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nthis : 0 \u2264 ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\n\u22a2 snorm (indicator t\u1d9c (f n - g)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)"}, {"tactic": "have := snorm_sub_le_of_dist_bdd \u03bc hp' htm.compl this fun x hx =>\n  (dist_comm (g x) (f n x) \u25b8 (hN x hx).le :\n    dist (f n x) (g x) \u2264 \u03b5.toReal / (3 * measureUnivNNReal \u03bc ^ (1 / p.toReal)))", "annotated_tactic": ["have := <a>snorm_sub_le_of_dist_bdd</a> \u03bc hp' htm.compl this fun x hx =>\n      (<a>dist_comm</a> (g x) (f n x) \u25b8 (hN x hx).<a>le</a> :\n        <a>dist</a> (f n x) (g x) \u2264 \u03b5.toReal / (3 * <a>measureUnivNNReal</a> \u03bc ^ (1 / p.toReal)))", [{"full_name": "MeasureTheory.snorm_sub_le_of_dist_bdd", "def_path": "Mathlib/MeasureTheory/Function/UniformIntegrable.lean", "def_pos": [465, 9], "def_end_pos": [465, 33]}, {"full_name": "dist_comm", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [188, 9], "def_end_pos": [188, 18]}, {"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [142, 7], "def_end_pos": [142, 15]}, {"full_name": "Dist.dist", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [94, 3], "def_end_pos": [94, 7]}, {"full_name": "MeasureTheory.measureUnivNNReal", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2897, 5], "def_end_pos": [2897, 22]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN n : \u2115\nhn : n \u2265 N\nhnf : snorm (indicator t (f n)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhng : snorm (indicator t g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhN : \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f n x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nthis : 0 \u2264 ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\n\u22a2 snorm (indicator t\u1d9c (f n - g)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN n : \u2115\nhn : n \u2265 N\nhnf : snorm (indicator t (f n)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhng : snorm (indicator t g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhN : \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f n x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nthis\u271d : 0 \u2264 ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nthis :\n  snorm (indicator t\u1d9c ((fun x => f n x) - fun x => g x)) p \u03bc \u2264\n    ENNReal.ofReal (ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))) *\n      \u2191\u2191\u03bc t\u1d9c ^ (1 / ENNReal.toReal p)\n\u22a2 snorm (indicator t\u1d9c (f n - g)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)"}, {"tactic": "refine' le_trans this _", "annotated_tactic": ["refine' <a>le_trans</a> this _", [{"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN n : \u2115\nhn : n \u2265 N\nhnf : snorm (indicator t (f n)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhng : snorm (indicator t g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhN : \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f n x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nthis\u271d : 0 \u2264 ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nthis :\n  snorm (indicator t\u1d9c ((fun x => f n x) - fun x => g x)) p \u03bc \u2264\n    ENNReal.ofReal (ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))) *\n      \u2191\u2191\u03bc t\u1d9c ^ (1 / ENNReal.toReal p)\n\u22a2 snorm (indicator t\u1d9c (f n - g)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN n : \u2115\nhn : n \u2265 N\nhnf : snorm (indicator t (f n)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhng : snorm (indicator t g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhN : \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f n x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nthis\u271d : 0 \u2264 ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nthis :\n  snorm (indicator t\u1d9c ((fun x => f n x) - fun x => g x)) p \u03bc \u2264\n    ENNReal.ofReal (ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))) *\n      \u2191\u2191\u03bc t\u1d9c ^ (1 / ENNReal.toReal p)\n\u22a2 ENNReal.ofReal (ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))) *\n      \u2191\u2191\u03bc t\u1d9c ^ (1 / ENNReal.toReal p) \u2264\n    ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)"}, {"tactic": "rw [div_mul_eq_div_mul_one_div, \u2190 ENNReal.ofReal_toReal (measure_lt_top \u03bc t\u1d9c).ne,\n  ENNReal.ofReal_rpow_of_nonneg ENNReal.toReal_nonneg hdivp, \u2190 ENNReal.ofReal_mul, mul_assoc]", "annotated_tactic": ["rw [<a>div_mul_eq_div_mul_one_div</a>, \u2190 <a>ENNReal.ofReal_toReal</a> (<a>measure_lt_top</a> \u03bc t\u1d9c).<a>ne</a>,\n      <a>ENNReal.ofReal_rpow_of_nonneg</a> <a>ENNReal.toReal_nonneg</a> hdivp, \u2190 <a>ENNReal.ofReal_mul</a>, <a>mul_assoc</a>]", [{"full_name": "div_mul_eq_div_mul_one_div", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [567, 9], "def_end_pos": [567, 35]}, {"full_name": "ENNReal.ofReal_toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [186, 9], "def_end_pos": [186, 22]}, {"full_name": "MeasureTheory.measure_lt_top", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2866, 9], "def_end_pos": [2866, 23]}, {"full_name": "LT.lt.ne", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [152, 7], "def_end_pos": [152, 15]}, {"full_name": "ENNReal.ofReal_rpow_of_nonneg", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [823, 9], "def_end_pos": [823, 30]}, {"full_name": "ENNReal.toReal_nonneg", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [221, 17], "def_end_pos": [221, 30]}, {"full_name": "ENNReal.ofReal_mul", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2225, 9], "def_end_pos": [2225, 19]}, {"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [264, 9], "def_end_pos": [264, 18]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN n : \u2115\nhn : n \u2265 N\nhnf : snorm (indicator t (f n)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhng : snorm (indicator t g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhN : \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f n x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nthis\u271d : 0 \u2264 ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nthis :\n  snorm (indicator t\u1d9c ((fun x => f n x) - fun x => g x)) p \u03bc \u2264\n    ENNReal.ofReal (ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))) *\n      \u2191\u2191\u03bc t\u1d9c ^ (1 / ENNReal.toReal p)\n\u22a2 ENNReal.ofReal (ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))) *\n      \u2191\u2191\u03bc t\u1d9c ^ (1 / ENNReal.toReal p) \u2264\n    ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN n : \u2115\nhn : n \u2265 N\nhnf : snorm (indicator t (f n)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhng : snorm (indicator t g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhN : \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f n x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nthis\u271d : 0 \u2264 ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nthis :\n  snorm (indicator t\u1d9c ((fun x => f n x) - fun x => g x)) p \u03bc \u2264\n    ENNReal.ofReal (ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))) *\n      \u2191\u2191\u03bc t\u1d9c ^ (1 / ENNReal.toReal p)\n\u22a2 ENNReal.ofReal\n      (ENNReal.toReal \u03b5 / 3 *\n        (1 / \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p) * ENNReal.toReal (\u2191\u2191\u03bc t\u1d9c) ^ (1 / ENNReal.toReal p))) \u2264\n    ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN n : \u2115\nhn : n \u2265 N\nhnf : snorm (indicator t (f n)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhng : snorm (indicator t g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhN : \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f n x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nthis\u271d : 0 \u2264 ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nthis :\n  snorm (indicator t\u1d9c ((fun x => f n x) - fun x => g x)) p \u03bc \u2264\n    ENNReal.ofReal (ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))) *\n      \u2191\u2191\u03bc t\u1d9c ^ (1 / ENNReal.toReal p)\n\u22a2 0 \u2264 ENNReal.toReal \u03b5 / 3 * (1 / \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))"}, {"tactic": "rw [div_mul_eq_div_mul_one_div]", "annotated_tactic": ["rw [<a>div_mul_eq_div_mul_one_div</a>]", [{"full_name": "div_mul_eq_div_mul_one_div", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [567, 9], "def_end_pos": [567, 35]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN n : \u2115\nhn : n \u2265 N\nhnf : snorm (indicator t (f n)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhng : snorm (indicator t g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhN : \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f n x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\n\u22a2 0 \u2264 ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN n : \u2115\nhn : n \u2265 N\nhnf : snorm (indicator t (f n)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhng : snorm (indicator t g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhN : \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f n x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\n\u22a2 0 \u2264 ENNReal.toReal \u03b5 / 3 * (1 / \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))"}, {"tactic": "exact mul_nonneg h\u03b5'.le (one_div_nonneg.2 hpow.le)", "annotated_tactic": ["exact <a>mul_nonneg</a> h\u03b5'.le (<a>one_div_nonneg</a>.2 hpow.le)", [{"full_name": "mul_nonneg", "def_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "def_pos": [380, 7], "def_end_pos": [380, 17]}, {"full_name": "one_div_nonneg", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [81, 9], "def_end_pos": [81, 23]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN n : \u2115\nhn : n \u2265 N\nhnf : snorm (indicator t (f n)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhng : snorm (indicator t g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhN : \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f n x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\n\u22a2 0 \u2264 ENNReal.toReal \u03b5 / 3 * (1 / \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))", "state_after": "no goals"}, {"tactic": "refine' ENNReal.ofReal_le_ofReal (mul_le_of_le_one_right h\u03b5'.le _)", "annotated_tactic": ["refine' <a>ENNReal.ofReal_le_ofReal</a> (<a>mul_le_of_le_one_right</a> h\u03b5'.le _)", [{"full_name": "ENNReal.ofReal_le_ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2135, 9], "def_end_pos": [2135, 25]}, {"full_name": "mul_le_of_le_one_right", "def_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "def_pos": [674, 9], "def_end_pos": [674, 31]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN n : \u2115\nhn : n \u2265 N\nhnf : snorm (indicator t (f n)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhng : snorm (indicator t g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhN : \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f n x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nthis\u271d : 0 \u2264 ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nthis :\n  snorm (indicator t\u1d9c ((fun x => f n x) - fun x => g x)) p \u03bc \u2264\n    ENNReal.ofReal (ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))) *\n      \u2191\u2191\u03bc t\u1d9c ^ (1 / ENNReal.toReal p)\n\u22a2 ENNReal.ofReal\n      (ENNReal.toReal \u03b5 / 3 *\n        (1 / \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p) * ENNReal.toReal (\u2191\u2191\u03bc t\u1d9c) ^ (1 / ENNReal.toReal p))) \u2264\n    ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN n : \u2115\nhn : n \u2265 N\nhnf : snorm (indicator t (f n)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhng : snorm (indicator t g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhN : \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f n x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nthis\u271d : 0 \u2264 ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nthis :\n  snorm (indicator t\u1d9c ((fun x => f n x) - fun x => g x)) p \u03bc \u2264\n    ENNReal.ofReal (ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))) *\n      \u2191\u2191\u03bc t\u1d9c ^ (1 / ENNReal.toReal p)\n\u22a2 1 / \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p) * ENNReal.toReal (\u2191\u2191\u03bc t\u1d9c) ^ (1 / ENNReal.toReal p) \u2264 1"}, {"tactic": "rw [mul_comm, mul_one_div, div_le_one]", "annotated_tactic": ["rw [<a>mul_comm</a>, <a>mul_one_div</a>, <a>div_le_one</a>]", [{"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [302, 9], "def_end_pos": [302, 17]}, {"full_name": "mul_one_div", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [300, 9], "def_end_pos": [300, 20]}, {"full_name": "div_le_one", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [425, 9], "def_end_pos": [425, 19]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN n : \u2115\nhn : n \u2265 N\nhnf : snorm (indicator t (f n)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhng : snorm (indicator t g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhN : \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f n x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nthis\u271d : 0 \u2264 ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nthis :\n  snorm (indicator t\u1d9c ((fun x => f n x) - fun x => g x)) p \u03bc \u2264\n    ENNReal.ofReal (ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))) *\n      \u2191\u2191\u03bc t\u1d9c ^ (1 / ENNReal.toReal p)\n\u22a2 1 / \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p) * ENNReal.toReal (\u2191\u2191\u03bc t\u1d9c) ^ (1 / ENNReal.toReal p) \u2264 1", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN n : \u2115\nhn : n \u2265 N\nhnf : snorm (indicator t (f n)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhng : snorm (indicator t g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhN : \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f n x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nthis\u271d : 0 \u2264 ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nthis :\n  snorm (indicator t\u1d9c ((fun x => f n x) - fun x => g x)) p \u03bc \u2264\n    ENNReal.ofReal (ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))) *\n      \u2191\u2191\u03bc t\u1d9c ^ (1 / ENNReal.toReal p)\n\u22a2 ENNReal.toReal (\u2191\u2191\u03bc t\u1d9c) ^ (1 / ENNReal.toReal p) \u2264 \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN n : \u2115\nhn : n \u2265 N\nhnf : snorm (indicator t (f n)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhng : snorm (indicator t g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhN : \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f n x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nthis\u271d : 0 \u2264 ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nthis :\n  snorm (indicator t\u1d9c ((fun x => f n x) - fun x => g x)) p \u03bc \u2264\n    ENNReal.ofReal (ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))) *\n      \u2191\u2191\u03bc t\u1d9c ^ (1 / ENNReal.toReal p)\n\u22a2 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)"}, {"tactic": "refine' Real.rpow_le_rpow ENNReal.toReal_nonneg\n  (ENNReal.toReal_le_of_le_ofReal (measureUnivNNReal_pos h\u03bc).le _) hdivp", "annotated_tactic": ["refine' <a>Real.rpow_le_rpow</a> <a>ENNReal.toReal_nonneg</a>\n          (<a>ENNReal.toReal_le_of_le_ofReal</a> (<a>measureUnivNNReal_pos</a> h\u03bc).<a>le</a> _) hdivp", [{"full_name": "Real.rpow_le_rpow", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [433, 9], "def_end_pos": [433, 21]}, {"full_name": "ENNReal.toReal_nonneg", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [221, 17], "def_end_pos": [221, 30]}, {"full_name": "ENNReal.toReal_le_of_le_ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2210, 9], "def_end_pos": [2210, 31]}, {"full_name": "MeasureTheory.measureUnivNNReal_pos", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2963, 9], "def_end_pos": [2963, 30]}, {"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [142, 7], "def_end_pos": [142, 15]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN n : \u2115\nhn : n \u2265 N\nhnf : snorm (indicator t (f n)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhng : snorm (indicator t g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhN : \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f n x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nthis\u271d : 0 \u2264 ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nthis :\n  snorm (indicator t\u1d9c ((fun x => f n x) - fun x => g x)) p \u03bc \u2264\n    ENNReal.ofReal (ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))) *\n      \u2191\u2191\u03bc t\u1d9c ^ (1 / ENNReal.toReal p)\n\u22a2 ENNReal.toReal (\u2191\u2191\u03bc t\u1d9c) ^ (1 / ENNReal.toReal p) \u2264 \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN n : \u2115\nhn : n \u2265 N\nhnf : snorm (indicator t (f n)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhng : snorm (indicator t g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhN : \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f n x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nthis\u271d : 0 \u2264 ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nthis :\n  snorm (indicator t\u1d9c ((fun x => f n x) - fun x => g x)) p \u03bc \u2264\n    ENNReal.ofReal (ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))) *\n      \u2191\u2191\u03bc t\u1d9c ^ (1 / ENNReal.toReal p)\n\u22a2 \u2191\u2191\u03bc t\u1d9c \u2264 ENNReal.ofReal \u2191(measureUnivNNReal \u03bc)"}, {"tactic": "rw [ENNReal.ofReal_coe_nnreal, coe_measureUnivNNReal]", "annotated_tactic": ["rw [<a>ENNReal.ofReal_coe_nnreal</a>, <a>coe_measureUnivNNReal</a>]", [{"full_name": "ENNReal.ofReal_coe_nnreal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [212, 17], "def_end_pos": [212, 34]}, {"full_name": "MeasureTheory.coe_measureUnivNNReal", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2902, 9], "def_end_pos": [2902, 30]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN n : \u2115\nhn : n \u2265 N\nhnf : snorm (indicator t (f n)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhng : snorm (indicator t g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhN : \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f n x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nthis\u271d : 0 \u2264 ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nthis :\n  snorm (indicator t\u1d9c ((fun x => f n x) - fun x => g x)) p \u03bc \u2264\n    ENNReal.ofReal (ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))) *\n      \u2191\u2191\u03bc t\u1d9c ^ (1 / ENNReal.toReal p)\n\u22a2 \u2191\u2191\u03bc t\u1d9c \u2264 ENNReal.ofReal \u2191(measureUnivNNReal \u03bc)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN n : \u2115\nhn : n \u2265 N\nhnf : snorm (indicator t (f n)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhng : snorm (indicator t g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhN : \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f n x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nthis\u271d : 0 \u2264 ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nthis :\n  snorm (indicator t\u1d9c ((fun x => f n x) - fun x => g x)) p \u03bc \u2264\n    ENNReal.ofReal (ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))) *\n      \u2191\u2191\u03bc t\u1d9c ^ (1 / ENNReal.toReal p)\n\u22a2 \u2191\u2191\u03bc t\u1d9c \u2264 \u2191\u2191\u03bc univ"}, {"tactic": "exact measure_mono (Set.subset_univ _)", "annotated_tactic": ["exact <a>measure_mono</a> (<a>Set.subset_univ</a> _)", [{"full_name": "MeasureTheory.measure_mono", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [193, 9], "def_end_pos": [193, 21]}, {"full_name": "Set.subset_univ", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [691, 9], "def_end_pos": [691, 20]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN n : \u2115\nhn : n \u2265 N\nhnf : snorm (indicator t (f n)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhng : snorm (indicator t g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhN : \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f n x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nthis\u271d : 0 \u2264 ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nthis :\n  snorm (indicator t\u1d9c ((fun x => f n x) - fun x => g x)) p \u03bc \u2264\n    ENNReal.ofReal (ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))) *\n      \u2191\u2191\u03bc t\u1d9c ^ (1 / ENNReal.toReal p)\n\u22a2 \u2191\u2191\u03bc t\u1d9c \u2264 \u2191\u2191\u03bc univ", "state_after": "no goals"}, {"tactic": "exact Real.rpow_pos_of_pos (measureUnivNNReal_pos h\u03bc) _", "annotated_tactic": ["exact <a>Real.rpow_pos_of_pos</a> (<a>measureUnivNNReal_pos</a> h\u03bc) _", [{"full_name": "Real.rpow_pos_of_pos", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [92, 9], "def_end_pos": [92, 24]}, {"full_name": "MeasureTheory.measureUnivNNReal_pos", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2963, 9], "def_end_pos": [2963, 30]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN n : \u2115\nhn : n \u2265 N\nhnf : snorm (indicator t (f n)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhng : snorm (indicator t g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhN : \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f n x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nthis\u271d : 0 \u2264 ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nthis :\n  snorm (indicator t\u1d9c ((fun x => f n x) - fun x => g x)) p \u03bc \u2264\n    ENNReal.ofReal (ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))) *\n      \u2191\u2191\u03bc t\u1d9c ^ (1 / ENNReal.toReal p)\n\u22a2 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)", "state_after": "no goals"}, {"tactic": "refine' mul_nonneg h\u03b5'.le (one_div_nonneg.2 hpow.le)", "annotated_tactic": ["refine' <a>mul_nonneg</a> h\u03b5'.le (<a>one_div_nonneg</a>.2 hpow.le)", [{"full_name": "mul_nonneg", "def_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "def_pos": [380, 7], "def_end_pos": [380, 17]}, {"full_name": "one_div_nonneg", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [81, 9], "def_end_pos": [81, 23]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN n : \u2115\nhn : n \u2265 N\nhnf : snorm (indicator t (f n)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhng : snorm (indicator t g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhN : \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f n x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nthis\u271d : 0 \u2264 ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nthis :\n  snorm (indicator t\u1d9c ((fun x => f n x) - fun x => g x)) p \u03bc \u2264\n    ENNReal.ofReal (ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))) *\n      \u2191\u2191\u03bc t\u1d9c ^ (1 / ENNReal.toReal p)\n\u22a2 0 \u2264 ENNReal.toReal \u03b5 / 3 * (1 / \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))", "state_after": "no goals"}, {"tactic": "rw [ENNReal.ofReal_div_of_pos (show (0 : \u211d) < 3 by norm_num), ENNReal.ofReal_toReal h.ne]", "annotated_tactic": ["rw [<a>ENNReal.ofReal_div_of_pos</a> (show (0 : \u211d) < 3 by norm_num), <a>ENNReal.ofReal_toReal</a> h.ne]", [{"full_name": "ENNReal.ofReal_div_of_pos", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2248, 9], "def_end_pos": [2248, 26]}, {"full_name": "ENNReal.ofReal_toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [186, 9], "def_end_pos": [186, 22]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN : \u2115\nhN :\n  \u2200 (b : \u2115),\n    b \u2265 N \u2192\n      \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f b x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nn : \u2115\nhn : n \u2265 N\nhnf : snorm (indicator t (f n)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhng : snorm (indicator t g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhlt : snorm (indicator t\u1d9c (f n - g)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u22a2 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) = \u03b5 / 3", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN : \u2115\nhN :\n  \u2200 (b : \u2115),\n    b \u2265 N \u2192\n      \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f b x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nn : \u2115\nhn : n \u2265 N\nhnf : snorm (indicator t (f n)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhng : snorm (indicator t g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhlt : snorm (indicator t\u1d9c (f n - g)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u22a2 \u03b5 / ENNReal.ofReal 3 = \u03b5 / 3"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN : \u2115\nhN :\n  \u2200 (b : \u2115),\n    b \u2265 N \u2192\n      \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f b x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nn : \u2115\nhn : n \u2265 N\nhnf : snorm (indicator t (f n)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhng : snorm (indicator t g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhlt : snorm (indicator t\u1d9c (f n - g)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u22a2 \u03b5 / ENNReal.ofReal 3 = \u03b5 / 3", "state_after": "no goals"}, {"tactic": "norm_num", "annotated_tactic": ["norm_num", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN : \u2115\nhN :\n  \u2200 (b : \u2115),\n    b \u2265 N \u2192\n      \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f b x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nn : \u2115\nhn : n \u2265 N\nhnf : snorm (indicator t (f n)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhng : snorm (indicator t g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhlt : snorm (indicator t\u1d9c (f n - g)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u22a2 0 < 3", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean", "full_name": "MeasureTheory.Measure.addHaar_eq_zero_of_disjoint_translates", "start": [160, 1], "end": [175, 58], "traced_tactics": [{"tactic": "suffices H : \u2200 R, \u03bc (s \u2229 closedBall 0 R) = 0", "annotated_tactic": ["suffices H : \u2200 R, \u03bc (s \u2229 <a>closedBall</a> 0 R) = 0", [{"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}]], "state_before": "E : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\ns : Set E\nu : \u2115 \u2192 E\nhu : Bornology.IsBounded (range u)\nhs : Pairwise (Disjoint on fun n => {u n} + s)\nh's : MeasurableSet s\n\u22a2 \u2191\u2191\u03bc s = 0", "state_after": "E : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\ns : Set E\nu : \u2115 \u2192 E\nhu : Bornology.IsBounded (range u)\nhs : Pairwise (Disjoint on fun n => {u n} + s)\nh's : MeasurableSet s\nH : \u2200 (R : \u211d), \u2191\u2191\u03bc (s \u2229 closedBall 0 R) = 0\n\u22a2 \u2191\u2191\u03bc s = 0\n\ncase H\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\ns : Set E\nu : \u2115 \u2192 E\nhu : Bornology.IsBounded (range u)\nhs : Pairwise (Disjoint on fun n => {u n} + s)\nh's : MeasurableSet s\n\u22a2 \u2200 (R : \u211d), \u2191\u2191\u03bc (s \u2229 closedBall 0 R) = 0"}, {"tactic": "intro R", "annotated_tactic": ["intro R", []], "state_before": "case H\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\ns : Set E\nu : \u2115 \u2192 E\nhu : Bornology.IsBounded (range u)\nhs : Pairwise (Disjoint on fun n => {u n} + s)\nh's : MeasurableSet s\n\u22a2 \u2200 (R : \u211d), \u2191\u2191\u03bc (s \u2229 closedBall 0 R) = 0", "state_after": "case H\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\ns : Set E\nu : \u2115 \u2192 E\nhu : Bornology.IsBounded (range u)\nhs : Pairwise (Disjoint on fun n => {u n} + s)\nh's : MeasurableSet s\nR : \u211d\n\u22a2 \u2191\u2191\u03bc (s \u2229 closedBall 0 R) = 0"}, {"tactic": "apply addHaar_eq_zero_of_disjoint_translates_aux \u03bc u\n  (isBounded_closedBall.subset (inter_subset_right _ _)) hu _ (h's.inter measurableSet_closedBall)", "annotated_tactic": ["apply <a>addHaar_eq_zero_of_disjoint_translates_aux</a> \u03bc u\n    (isBounded_closedBall.subset (<a>inter_subset_right</a> _ _)) hu _ (h's.inter <a>measurableSet_closedBall</a>)", [{"full_name": "MeasureTheory.Measure.addHaar_eq_zero_of_disjoint_translates_aux", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean", "def_pos": [142, 9], "def_end_pos": [142, 51]}, {"full_name": "Set.inter_subset_right", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [969, 9], "def_end_pos": [969, 27]}, {"full_name": "measurableSet_closedBall", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [1681, 9], "def_end_pos": [1681, 33]}]], "state_before": "case H\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\ns : Set E\nu : \u2115 \u2192 E\nhu : Bornology.IsBounded (range u)\nhs : Pairwise (Disjoint on fun n => {u n} + s)\nh's : MeasurableSet s\nR : \u211d\n\u22a2 \u2191\u2191\u03bc (s \u2229 closedBall 0 R) = 0", "state_after": "E : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\ns : Set E\nu : \u2115 \u2192 E\nhu : Bornology.IsBounded (range u)\nhs : Pairwise (Disjoint on fun n => {u n} + s)\nh's : MeasurableSet s\nR : \u211d\n\u22a2 Pairwise (Disjoint on fun n => {u n} + s \u2229 closedBall 0 R)"}, {"tactic": "refine pairwise_disjoint_mono hs fun n => ?_", "annotated_tactic": ["refine <a>pairwise_disjoint_mono</a> hs fun n => ?_", [{"full_name": "pairwise_disjoint_mono", "def_path": "Mathlib/Data/Set/Pairwise/Basic.lean", "def_pos": [60, 9], "def_end_pos": [60, 31]}]], "state_before": "E : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\ns : Set E\nu : \u2115 \u2192 E\nhu : Bornology.IsBounded (range u)\nhs : Pairwise (Disjoint on fun n => {u n} + s)\nh's : MeasurableSet s\nR : \u211d\n\u22a2 Pairwise (Disjoint on fun n => {u n} + s \u2229 closedBall 0 R)", "state_after": "E : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\ns : Set E\nu : \u2115 \u2192 E\nhu : Bornology.IsBounded (range u)\nhs : Pairwise (Disjoint on fun n => {u n} + s)\nh's : MeasurableSet s\nR : \u211d\nn : \u2115\n\u22a2 {u n} + s \u2229 closedBall 0 R \u2264 {u n} + s"}, {"tactic": "exact add_subset_add Subset.rfl (inter_subset_left _ _)", "annotated_tactic": ["exact <a>add_subset_add</a> <a>Subset.rfl</a> (<a>inter_subset_left</a> _ _)", [{"full_name": "Set.add_subset_add", "def_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "def_pos": [415, 3], "def_end_pos": [415, 14]}, {"full_name": "Set.Subset.rfl", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [357, 9], "def_end_pos": [357, 19]}, {"full_name": "Set.inter_subset_left", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [965, 9], "def_end_pos": [965, 26]}]], "state_before": "E : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\ns : Set E\nu : \u2115 \u2192 E\nhu : Bornology.IsBounded (range u)\nhs : Pairwise (Disjoint on fun n => {u n} + s)\nh's : MeasurableSet s\nR : \u211d\nn : \u2115\n\u22a2 {u n} + s \u2229 closedBall 0 R \u2264 {u n} + s", "state_after": "no goals"}, {"tactic": "apply le_antisymm _ (zero_le _)", "annotated_tactic": ["apply <a>le_antisymm</a> _ (<a>zero_le</a> _)", [{"full_name": "le_antisymm", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [188, 9], "def_end_pos": [188, 20]}, {"full_name": "zero_le", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [217, 30], "def_end_pos": [217, 37]}]], "state_before": "E : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\ns : Set E\nu : \u2115 \u2192 E\nhu : Bornology.IsBounded (range u)\nhs : Pairwise (Disjoint on fun n => {u n} + s)\nh's : MeasurableSet s\nH : \u2200 (R : \u211d), \u2191\u2191\u03bc (s \u2229 closedBall 0 R) = 0\n\u22a2 \u2191\u2191\u03bc s = 0", "state_after": "E : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\ns : Set E\nu : \u2115 \u2192 E\nhu : Bornology.IsBounded (range u)\nhs : Pairwise (Disjoint on fun n => {u n} + s)\nh's : MeasurableSet s\nH : \u2200 (R : \u211d), \u2191\u2191\u03bc (s \u2229 closedBall 0 R) = 0\n\u22a2 \u2191\u2191\u03bc s \u2264 0"}, {"tactic": "calc\n  \u03bc s \u2264 \u2211' n : \u2115, \u03bc (s \u2229 closedBall 0 n) := by\n    conv_lhs => rw [\u2190 iUnion_inter_closedBall_nat s 0]\n    exact measure_iUnion_le _\n  _ = 0 := by simp only [H, tsum_zero]", "annotated_tactic": ["calc\n      \u03bc s \u2264 \u2211' n : \u2115, \u03bc (s \u2229 <a>closedBall</a> 0 n) := by\n        conv_lhs => rw [\u2190 <a>iUnion_inter_closedBall_nat</a> s 0]\n        exact <a>measure_iUnion_le</a> _\n      _ = 0 := by simp only [H, <a>tsum_zero</a>]", [{"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "Metric.iUnion_inter_closedBall_nat", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [667, 9], "def_end_pos": [667, 36]}, {"full_name": "MeasureTheory.measure_iUnion_le", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [240, 9], "def_end_pos": [240, 26]}, {"full_name": "tsum_zero", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/Basic.lean", "def_pos": [489, 9], "def_end_pos": [489, 18]}]], "state_before": "E : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\ns : Set E\nu : \u2115 \u2192 E\nhu : Bornology.IsBounded (range u)\nhs : Pairwise (Disjoint on fun n => {u n} + s)\nh's : MeasurableSet s\nH : \u2200 (R : \u211d), \u2191\u2191\u03bc (s \u2229 closedBall 0 R) = 0\n\u22a2 \u2191\u2191\u03bc s \u2264 0", "state_after": "no goals"}, {"tactic": "conv_lhs => rw [\u2190 iUnion_inter_closedBall_nat s 0]", "annotated_tactic": ["conv_lhs => rw [\u2190 <a>iUnion_inter_closedBall_nat</a> s 0]", [{"full_name": "Metric.iUnion_inter_closedBall_nat", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [667, 9], "def_end_pos": [667, 36]}]], "state_before": "E : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\ns : Set E\nu : \u2115 \u2192 E\nhu : Bornology.IsBounded (range u)\nhs : Pairwise (Disjoint on fun n => {u n} + s)\nh's : MeasurableSet s\nH : \u2200 (R : \u211d), \u2191\u2191\u03bc (s \u2229 closedBall 0 R) = 0\n\u22a2 \u2191\u2191\u03bc s \u2264 \u2211' (n : \u2115), \u2191\u2191\u03bc (s \u2229 closedBall 0 \u2191n)", "state_after": "E : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\ns : Set E\nu : \u2115 \u2192 E\nhu : Bornology.IsBounded (range u)\nhs : Pairwise (Disjoint on fun n => {u n} + s)\nh's : MeasurableSet s\nH : \u2200 (R : \u211d), \u2191\u2191\u03bc (s \u2229 closedBall 0 R) = 0\n\u22a2 \u2191\u2191\u03bc (\u22c3 n, s \u2229 closedBall 0 \u2191n) \u2264 \u2211' (n : \u2115), \u2191\u2191\u03bc (s \u2229 closedBall 0 \u2191n)"}, {"tactic": "exact measure_iUnion_le _", "annotated_tactic": ["exact <a>measure_iUnion_le</a> _", [{"full_name": "MeasureTheory.measure_iUnion_le", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [240, 9], "def_end_pos": [240, 26]}]], "state_before": "E : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\ns : Set E\nu : \u2115 \u2192 E\nhu : Bornology.IsBounded (range u)\nhs : Pairwise (Disjoint on fun n => {u n} + s)\nh's : MeasurableSet s\nH : \u2200 (R : \u211d), \u2191\u2191\u03bc (s \u2229 closedBall 0 R) = 0\n\u22a2 \u2191\u2191\u03bc (\u22c3 n, s \u2229 closedBall 0 \u2191n) \u2264 \u2211' (n : \u2115), \u2191\u2191\u03bc (s \u2229 closedBall 0 \u2191n)", "state_after": "no goals"}, {"tactic": "simp only [H, tsum_zero]", "annotated_tactic": ["simp only [H, <a>tsum_zero</a>]", [{"full_name": "tsum_zero", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/Basic.lean", "def_pos": [489, 9], "def_end_pos": [489, 18]}]], "state_before": "E : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\ns : Set E\nu : \u2115 \u2192 E\nhu : Bornology.IsBounded (range u)\nhs : Pairwise (Disjoint on fun n => {u n} + s)\nh's : MeasurableSet s\nH : \u2200 (R : \u211d), \u2191\u2191\u03bc (s \u2229 closedBall 0 R) = 0\n\u22a2 \u2211' (n : \u2115), \u2191\u2191\u03bc (s \u2229 closedBall 0 \u2191n) = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Kernel/Condexp.lean", "full_name": "MeasureTheory.Integrable.norm_integral_condexpKernel", "start": [142, 1], "end": [148, 63], "traced_tactics": [{"tactic": "rw [condexpKernel]", "annotated_tactic": ["rw [<a>condexpKernel</a>]", [{"full_name": "ProbabilityTheory.condexpKernel", "def_path": "Mathlib/Probability/Kernel/Condexp.lean", "def_pos": [71, 31], "def_end_pos": [71, 44]}]], "state_before": "\u03a9 : Type u_1\nF : Type u_2\ninst\u271d\u2077 : TopologicalSpace \u03a9\nm m\u03a9 : MeasurableSpace \u03a9\ninst\u271d\u2076 : PolishSpace \u03a9\ninst\u271d\u2075 : BorelSpace \u03a9\ninst\u271d\u2074 : Nonempty \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsFiniteMeasure \u03bc\ninst\u271d\u00b2 : NormedAddCommGroup F\nf : \u03a9 \u2192 F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nhf_int : Integrable f\n\u22a2 Integrable fun \u03c9 => \u2016\u222b (y : \u03a9), f y \u2202\u2191(condexpKernel \u03bc m) \u03c9\u2016", "state_after": "\u03a9 : Type u_1\nF : Type u_2\ninst\u271d\u2077 : TopologicalSpace \u03a9\nm m\u03a9 : MeasurableSpace \u03a9\ninst\u271d\u2076 : PolishSpace \u03a9\ninst\u271d\u2075 : BorelSpace \u03a9\ninst\u271d\u2074 : Nonempty \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsFiniteMeasure \u03bc\ninst\u271d\u00b2 : NormedAddCommGroup F\nf : \u03a9 \u2192 F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nhf_int : Integrable f\n\u22a2 Integrable fun \u03c9 => \u2016\u222b (y : \u03a9), f y \u2202\u2191(kernel.comap (condDistrib id id \u03bc) id (_ : Measurable id)) \u03c9\u2016"}, {"tactic": "exact Integrable.norm_integral_condDistrib\n  (aemeasurable_id'' \u03bc (inf_le_right : m \u2293 m\u03a9 \u2264 m\u03a9)) aemeasurable_id\n  (hf_int.comp_snd_map_prod_id (inf_le_right : m \u2293 m\u03a9 \u2264 m\u03a9))", "annotated_tactic": ["exact <a>Integrable.norm_integral_condDistrib</a>\n    (<a>aemeasurable_id''</a> \u03bc (<a>inf_le_right</a> : m \u2293 m\u03a9 \u2264 m\u03a9)) <a>aemeasurable_id</a>\n    (hf_int.comp_snd_map_prod_id (<a>inf_le_right</a> : m \u2293 m\u03a9 \u2264 m\u03a9))", [{"full_name": "MeasureTheory.Integrable.norm_integral_condDistrib", "def_path": "Mathlib/Probability/Kernel/CondDistrib.lean", "def_pos": [170, 9], "def_end_pos": [170, 66]}, {"full_name": "aemeasurable_id''", "def_path": "Mathlib/MeasureTheory/Measure/AEMeasurable.lean", "def_pos": [42, 9], "def_end_pos": [42, 26]}, {"full_name": "inf_le_right", "def_path": "Mathlib/Order/Lattice.lean", "def_pos": [399, 9], "def_end_pos": [399, 21]}, {"full_name": "aemeasurable_id", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [751, 9], "def_end_pos": [751, 24]}, {"full_name": "inf_le_right", "def_path": "Mathlib/Order/Lattice.lean", "def_pos": [399, 9], "def_end_pos": [399, 21]}]], "state_before": "\u03a9 : Type u_1\nF : Type u_2\ninst\u271d\u2077 : TopologicalSpace \u03a9\nm m\u03a9 : MeasurableSpace \u03a9\ninst\u271d\u2076 : PolishSpace \u03a9\ninst\u271d\u2075 : BorelSpace \u03a9\ninst\u271d\u2074 : Nonempty \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsFiniteMeasure \u03bc\ninst\u271d\u00b2 : NormedAddCommGroup F\nf : \u03a9 \u2192 F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nhf_int : Integrable f\n\u22a2 Integrable fun \u03c9 => \u2016\u222b (y : \u03a9), f y \u2202\u2191(kernel.comap (condDistrib id id \u03bc) id (_ : Measurable id)) \u03c9\u2016", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "full_name": "List.infix_nil", "start": [1814, 9], "end": [1814, 98], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Array/Lemmas.lean", "full_name": "Array.getD_eq_get?", "start": [62, 9], "end": [63, 36], "traced_tactics": [{"tactic": "simp [get?, getD]", "annotated_tactic": ["simp [<a>get?</a>, <a>getD</a>]", [{"full_name": "Array.get?", "def_path": "lake-packages/lean4/src/lean/Init/Data/Array/Basic.lean", "def_pos": [50, 5], "def_end_pos": [50, 9]}, {"full_name": "Array.getD", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2525, 18], "def_end_pos": [2525, 28]}]], "state_before": "\u03b1 : Type u_1\na : Array \u03b1\nn : Nat\nd : \u03b1\n\u22a2 getD a n d = Option.getD (get? a n) d", "state_after": "\u03b1 : Type u_1\na : Array \u03b1\nn : Nat\nd : \u03b1\n\u22a2 (if h : n < size a then a[n] else d) = Option.getD (if h : n < size a then some a[n] else none) d"}, {"tactic": "split <;> simp", "annotated_tactic": ["split <;> simp", []], "state_before": "\u03b1 : Type u_1\na : Array \u03b1\nn : Nat\nd : \u03b1\n\u22a2 (if h : n < size a then a[n] else d) = Option.getD (if h : n < size a then some a[n] else none) d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Constructions/Pi.lean", "full_name": "MeasureTheory.volume_preserving_piEquivPiSubtypeProd", "start": [747, 1], "end": [750, 61], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/L2Space.lean", "full_name": "MeasureTheory.L2.snorm_inner_lt_top", "start": [128, 1], "end": [143, 69], "traced_tactics": [{"tactic": "have h : \u2200 x, \u2016\u27eaf x, g x\u27eb\u2016 \u2264 \u2016\u2016f x\u2016 ^ (2 : \u211d) + \u2016g x\u2016 ^ (2 : \u211d)\u2016 := by\n  intro x\n  rw [\u2190 @Nat.cast_two \u211d, Real.rpow_nat_cast, Real.rpow_nat_cast]\n  calc\n    \u2016\u27eaf x, g x\u27eb\u2016 \u2264 \u2016f x\u2016 * \u2016g x\u2016 := norm_inner_le_norm _ _\n    _ \u2264 2 * \u2016f x\u2016 * \u2016g x\u2016 :=\n      (mul_le_mul_of_nonneg_right (le_mul_of_one_le_left (norm_nonneg _) one_le_two)\n        (norm_nonneg _))\n    _ \u2264 \u2016\u2016f x\u2016 ^ 2 + \u2016g x\u2016 ^ 2\u2016 := (two_mul_le_add_sq _ _).trans (le_abs_self _)", "annotated_tactic": ["have h : \u2200 x, \u2016\u27eaf x, g x\u27eb\u2016 \u2264 \u2016\u2016f x\u2016 ^ (2 : \u211d) + \u2016g x\u2016 ^ (2 : \u211d)\u2016 := by\n    intro x\n    rw [\u2190 @<a>Nat.cast_two</a> \u211d, <a>Real.rpow_nat_cast</a>, <a>Real.rpow_nat_cast</a>]\n    calc\n      \u2016\u27eaf x, g x\u27eb\u2016 \u2264 \u2016f x\u2016 * \u2016g x\u2016 := <a>norm_inner_le_norm</a> _ _\n      _ \u2264 2 * \u2016f x\u2016 * \u2016g x\u2016 :=\n        (<a>mul_le_mul_of_nonneg_right</a> (<a>le_mul_of_one_le_left</a> (<a>norm_nonneg</a> _) <a>one_le_two</a>)\n          (<a>norm_nonneg</a> _))\n      _ \u2264 \u2016\u2016f x\u2016 ^ 2 + \u2016g x\u2016 ^ 2\u2016 := (<a>two_mul_le_add_sq</a> _ _).<a>trans</a> (<a>le_abs_self</a> _)", [{"full_name": "Nat.cast_two", "def_path": "Mathlib/Data/Nat/Cast/Defs.lean", "def_pos": [193, 9], "def_end_pos": [193, 17]}, {"full_name": "Real.rpow_nat_cast", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [357, 9], "def_end_pos": [357, 22]}, {"full_name": "Real.rpow_nat_cast", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [357, 9], "def_end_pos": [357, 22]}, {"full_name": "norm_inner_le_norm", "def_path": "Mathlib/Analysis/InnerProductSpace/Basic.lean", "def_pos": [1090, 9], "def_end_pos": [1090, 27]}, {"full_name": "mul_le_mul_of_nonneg_right", "def_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "def_pos": [156, 9], "def_end_pos": [156, 35]}, {"full_name": "le_mul_of_one_le_left", "def_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "def_pos": [670, 9], "def_end_pos": [670, 30]}, {"full_name": "norm_nonneg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [500, 30], "def_end_pos": [500, 41]}, {"full_name": "one_le_two", "def_path": "Mathlib/Algebra/Order/Monoid/NatCast.lean", "def_pos": [50, 7], "def_end_pos": [50, 17]}, {"full_name": "norm_nonneg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [500, 30], "def_end_pos": [500, 41]}, {"full_name": "two_mul_le_add_sq", "def_path": "Mathlib/Algebra/GroupPower/Order.lean", "def_pos": [755, 9], "def_end_pos": [755, 26]}, {"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [120, 7], "def_end_pos": [120, 18]}, {"full_name": "le_abs_self", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [57, 9], "def_end_pos": [57, 20]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : NormedAddCommGroup F\nf g : { x // x \u2208 Lp E 2 }\n\u22a2 snorm (fun x => inner (\u2191\u2191f x) (\u2191\u2191g x)) 1 \u03bc < \u22a4", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : NormedAddCommGroup F\nf g : { x // x \u2208 Lp E 2 }\nh : \u2200 (x : \u03b1), \u2016inner (\u2191\u2191f x) (\u2191\u2191g x)\u2016 \u2264 \u2016\u2016\u2191\u2191f x\u2016 ^ 2 + \u2016\u2191\u2191g x\u2016 ^ 2\u2016\n\u22a2 snorm (fun x => inner (\u2191\u2191f x) (\u2191\u2191g x)) 1 \u03bc < \u22a4"}, {"tactic": "refine' (snorm_mono_ae (ae_of_all _ h)).trans_lt ((snorm_add_le _ _ le_rfl).trans_lt _)", "annotated_tactic": ["refine' (<a>snorm_mono_ae</a> (<a>ae_of_all</a> _ h)).<a>trans_lt</a> ((<a>snorm_add_le</a> _ _ <a>le_rfl</a>).<a>trans_lt</a> _)", [{"full_name": "MeasureTheory.snorm_mono_ae", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [419, 9], "def_end_pos": [419, 22]}, {"full_name": "MeasureTheory.ae_of_all", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [407, 9], "def_end_pos": [407, 18]}, {"full_name": "LE.le.trans_lt", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [124, 7], "def_end_pos": [124, 21]}, {"full_name": "MeasureTheory.snorm_add_le", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [802, 9], "def_end_pos": [802, 21]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}, {"full_name": "LE.le.trans_lt", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [124, 7], "def_end_pos": [124, 21]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : NormedAddCommGroup F\nf g : { x // x \u2208 Lp E 2 }\nh : \u2200 (x : \u03b1), \u2016inner (\u2191\u2191f x) (\u2191\u2191g x)\u2016 \u2264 \u2016\u2016\u2191\u2191f x\u2016 ^ 2 + \u2016\u2191\u2191g x\u2016 ^ 2\u2016\n\u22a2 snorm (fun x => inner (\u2191\u2191f x) (\u2191\u2191g x)) 1 \u03bc < \u22a4", "state_after": "case refine'_1\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : NormedAddCommGroup F\nf g : { x // x \u2208 Lp E 2 }\nh : \u2200 (x : \u03b1), \u2016inner (\u2191\u2191f x) (\u2191\u2191g x)\u2016 \u2264 \u2016\u2016\u2191\u2191f x\u2016 ^ 2 + \u2016\u2191\u2191g x\u2016 ^ 2\u2016\n\u22a2 AEStronglyMeasurable (fun a => \u2016\u2191\u2191f a\u2016 ^ 2) \u03bc\n\ncase refine'_2\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : NormedAddCommGroup F\nf g : { x // x \u2208 Lp E 2 }\nh : \u2200 (x : \u03b1), \u2016inner (\u2191\u2191f x) (\u2191\u2191g x)\u2016 \u2264 \u2016\u2016\u2191\u2191f x\u2016 ^ 2 + \u2016\u2191\u2191g x\u2016 ^ 2\u2016\n\u22a2 AEStronglyMeasurable (fun a => \u2016\u2191\u2191g a\u2016 ^ 2) \u03bc\n\ncase refine'_3\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : NormedAddCommGroup F\nf g : { x // x \u2208 Lp E 2 }\nh : \u2200 (x : \u03b1), \u2016inner (\u2191\u2191f x) (\u2191\u2191g x)\u2016 \u2264 \u2016\u2016\u2191\u2191f x\u2016 ^ 2 + \u2016\u2191\u2191g x\u2016 ^ 2\u2016\n\u22a2 snorm (fun a => \u2016\u2191\u2191f a\u2016 ^ 2) 1 \u03bc + snorm (fun a => \u2016\u2191\u2191g a\u2016 ^ 2) 1 \u03bc < \u22a4"}, {"tactic": "rw [ENNReal.add_lt_top]", "annotated_tactic": ["rw [<a>ENNReal.add_lt_top</a>]", [{"full_name": "ENNReal.add_lt_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [561, 17], "def_end_pos": [561, 27]}]], "state_before": "case refine'_3\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : NormedAddCommGroup F\nf g : { x // x \u2208 Lp E 2 }\nh : \u2200 (x : \u03b1), \u2016inner (\u2191\u2191f x) (\u2191\u2191g x)\u2016 \u2264 \u2016\u2016\u2191\u2191f x\u2016 ^ 2 + \u2016\u2191\u2191g x\u2016 ^ 2\u2016\n\u22a2 snorm (fun a => \u2016\u2191\u2191f a\u2016 ^ 2) 1 \u03bc + snorm (fun a => \u2016\u2191\u2191g a\u2016 ^ 2) 1 \u03bc < \u22a4", "state_after": "case refine'_3\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : NormedAddCommGroup F\nf g : { x // x \u2208 Lp E 2 }\nh : \u2200 (x : \u03b1), \u2016inner (\u2191\u2191f x) (\u2191\u2191g x)\u2016 \u2264 \u2016\u2016\u2191\u2191f x\u2016 ^ 2 + \u2016\u2191\u2191g x\u2016 ^ 2\u2016\n\u22a2 snorm (fun a => \u2016\u2191\u2191f a\u2016 ^ 2) 1 \u03bc < \u22a4 \u2227 snorm (fun a => \u2016\u2191\u2191g a\u2016 ^ 2) 1 \u03bc < \u22a4"}, {"tactic": "exact \u27e8snorm_rpow_two_norm_lt_top f, snorm_rpow_two_norm_lt_top g\u27e9", "annotated_tactic": ["exact \u27e8<a>snorm_rpow_two_norm_lt_top</a> f, <a>snorm_rpow_two_norm_lt_top</a> g\u27e9", [{"full_name": "MeasureTheory.L2.snorm_rpow_two_norm_lt_top", "def_path": "Mathlib/MeasureTheory/Function/L2Space.lean", "def_pos": [122, 9], "def_end_pos": [122, 35]}, {"full_name": "MeasureTheory.L2.snorm_rpow_two_norm_lt_top", "def_path": "Mathlib/MeasureTheory/Function/L2Space.lean", "def_pos": [122, 9], "def_end_pos": [122, 35]}]], "state_before": "case refine'_3\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : NormedAddCommGroup F\nf g : { x // x \u2208 Lp E 2 }\nh : \u2200 (x : \u03b1), \u2016inner (\u2191\u2191f x) (\u2191\u2191g x)\u2016 \u2264 \u2016\u2016\u2191\u2191f x\u2016 ^ 2 + \u2016\u2191\u2191g x\u2016 ^ 2\u2016\n\u22a2 snorm (fun a => \u2016\u2191\u2191f a\u2016 ^ 2) 1 \u03bc < \u22a4 \u2227 snorm (fun a => \u2016\u2191\u2191g a\u2016 ^ 2) 1 \u03bc < \u22a4", "state_after": "no goals"}, {"tactic": "intro x", "annotated_tactic": ["intro x", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : NormedAddCommGroup F\nf g : { x // x \u2208 Lp E 2 }\n\u22a2 \u2200 (x : \u03b1), \u2016inner (\u2191\u2191f x) (\u2191\u2191g x)\u2016 \u2264 \u2016\u2016\u2191\u2191f x\u2016 ^ 2 + \u2016\u2191\u2191g x\u2016 ^ 2\u2016", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : NormedAddCommGroup F\nf g : { x // x \u2208 Lp E 2 }\nx : \u03b1\n\u22a2 \u2016inner (\u2191\u2191f x) (\u2191\u2191g x)\u2016 \u2264 \u2016\u2016\u2191\u2191f x\u2016 ^ 2 + \u2016\u2191\u2191g x\u2016 ^ 2\u2016"}, {"tactic": "calc\n  \u2016\u27eaf x, g x\u27eb\u2016 \u2264 \u2016f x\u2016 * \u2016g x\u2016 := norm_inner_le_norm _ _\n  _ \u2264 2 * \u2016f x\u2016 * \u2016g x\u2016 :=\n    (mul_le_mul_of_nonneg_right (le_mul_of_one_le_left (norm_nonneg _) one_le_two)\n      (norm_nonneg _))\n  _ \u2264 \u2016\u2016f x\u2016 ^ 2 + \u2016g x\u2016 ^ 2\u2016 := (two_mul_le_add_sq _ _).trans (le_abs_self _)", "annotated_tactic": ["calc\n      \u2016\u27eaf x, g x\u27eb\u2016 \u2264 \u2016f x\u2016 * \u2016g x\u2016 := <a>norm_inner_le_norm</a> _ _\n      _ \u2264 2 * \u2016f x\u2016 * \u2016g x\u2016 :=\n        (<a>mul_le_mul_of_nonneg_right</a> (<a>le_mul_of_one_le_left</a> (<a>norm_nonneg</a> _) <a>one_le_two</a>)\n          (<a>norm_nonneg</a> _))\n      _ \u2264 \u2016\u2016f x\u2016 ^ 2 + \u2016g x\u2016 ^ 2\u2016 := (<a>two_mul_le_add_sq</a> _ _).<a>trans</a> (<a>le_abs_self</a> _)", [{"full_name": "norm_inner_le_norm", "def_path": "Mathlib/Analysis/InnerProductSpace/Basic.lean", "def_pos": [1090, 9], "def_end_pos": [1090, 27]}, {"full_name": "mul_le_mul_of_nonneg_right", "def_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "def_pos": [156, 9], "def_end_pos": [156, 35]}, {"full_name": "le_mul_of_one_le_left", "def_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "def_pos": [670, 9], "def_end_pos": [670, 30]}, {"full_name": "norm_nonneg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [500, 30], "def_end_pos": [500, 41]}, {"full_name": "one_le_two", "def_path": "Mathlib/Algebra/Order/Monoid/NatCast.lean", "def_pos": [50, 7], "def_end_pos": [50, 17]}, {"full_name": "norm_nonneg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [500, 30], "def_end_pos": [500, 41]}, {"full_name": "two_mul_le_add_sq", "def_path": "Mathlib/Algebra/GroupPower/Order.lean", "def_pos": [755, 9], "def_end_pos": [755, 26]}, {"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [120, 7], "def_end_pos": [120, 18]}, {"full_name": "le_abs_self", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [57, 9], "def_end_pos": [57, 20]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : NormedAddCommGroup F\nf g : { x // x \u2208 Lp E 2 }\nx : \u03b1\n\u22a2 \u2016inner (\u2191\u2191f x) (\u2191\u2191g x)\u2016 \u2264 \u2016\u2016\u2191\u2191f x\u2016 ^ 2 + \u2016\u2191\u2191g x\u2016 ^ 2\u2016", "state_after": "no goals"}, {"tactic": "exact ((Lp.aestronglyMeasurable f).norm.aemeasurable.pow_const _).aestronglyMeasurable", "annotated_tactic": ["exact ((<a>Lp.aestronglyMeasurable</a> f).norm.aemeasurable.pow_const _).<a>aestronglyMeasurable</a>", [{"full_name": "MeasureTheory.Lp.aestronglyMeasurable", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [212, 19], "def_end_pos": [212, 39]}, {"full_name": "AEMeasurable.aestronglyMeasurable", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1451, 9], "def_end_pos": [1451, 49]}]], "state_before": "case refine'_1\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : NormedAddCommGroup F\nf g : { x // x \u2208 Lp E 2 }\nh : \u2200 (x : \u03b1), \u2016inner (\u2191\u2191f x) (\u2191\u2191g x)\u2016 \u2264 \u2016\u2016\u2191\u2191f x\u2016 ^ 2 + \u2016\u2191\u2191g x\u2016 ^ 2\u2016\n\u22a2 AEStronglyMeasurable (fun a => \u2016\u2191\u2191f a\u2016 ^ 2) \u03bc", "state_after": "no goals"}, {"tactic": "exact ((Lp.aestronglyMeasurable g).norm.aemeasurable.pow_const _).aestronglyMeasurable", "annotated_tactic": ["exact ((<a>Lp.aestronglyMeasurable</a> g).norm.aemeasurable.pow_const _).<a>aestronglyMeasurable</a>", [{"full_name": "MeasureTheory.Lp.aestronglyMeasurable", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [212, 19], "def_end_pos": [212, 39]}, {"full_name": "AEMeasurable.aestronglyMeasurable", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1451, 9], "def_end_pos": [1451, 49]}]], "state_before": "case refine'_2\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : NormedAddCommGroup F\nf g : { x // x \u2208 Lp E 2 }\nh : \u2200 (x : \u03b1), \u2016inner (\u2191\u2191f x) (\u2191\u2191g x)\u2016 \u2264 \u2016\u2016\u2191\u2191f x\u2016 ^ 2 + \u2016\u2191\u2191g x\u2016 ^ 2\u2016\n\u22a2 AEStronglyMeasurable (fun a => \u2016\u2191\u2191g a\u2016 ^ 2) \u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "Finset.filter_cons", "start": [2853, 1], "end": [2863, 56], "traced_tactics": [{"tactic": "split_ifs", "annotated_tactic": ["split_ifs", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np q : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : DecidablePred q\ns\u271d : Finset \u03b1\na : \u03b1\ns : Finset \u03b1\nha : \u00aca \u2208 s\n\u22a2 _root_.Disjoint (if p a then {a} else \u2205) (filter p s)", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np q : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : DecidablePred q\ns\u271d : Finset \u03b1\na : \u03b1\ns : Finset \u03b1\nha : \u00aca \u2208 s\nh\u271d : p a\n\u22a2 _root_.Disjoint {a} (filter p s)\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np q : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : DecidablePred q\ns\u271d : Finset \u03b1\na : \u03b1\ns : Finset \u03b1\nha : \u00aca \u2208 s\nh\u271d : \u00acp a\n\u22a2 _root_.Disjoint \u2205 (filter p s)"}, {"tactic": "rw [disjoint_singleton_left]", "annotated_tactic": ["rw [<a>disjoint_singleton_left</a>]", [{"full_name": "Finset.disjoint_singleton_left", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [985, 9], "def_end_pos": [985, 32]}]], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np q : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : DecidablePred q\ns\u271d : Finset \u03b1\na : \u03b1\ns : Finset \u03b1\nha : \u00aca \u2208 s\nh\u271d : p a\n\u22a2 _root_.Disjoint {a} (filter p s)", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np q : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : DecidablePred q\ns\u271d : Finset \u03b1\na : \u03b1\ns : Finset \u03b1\nha : \u00aca \u2208 s\nh\u271d : p a\n\u22a2 \u00aca \u2208 filter p s"}, {"tactic": "exact mem_filter.not.mpr <| mt And.left ha", "annotated_tactic": ["exact mem_filter.not.mpr <| <a>mt</a> <a>And.left</a> ha", [{"full_name": "mt", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [516, 9], "def_end_pos": [516, 11]}, {"full_name": "And.left", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [504, 3], "def_end_pos": [504, 7]}]], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np q : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : DecidablePred q\ns\u271d : Finset \u03b1\na : \u03b1\ns : Finset \u03b1\nha : \u00aca \u2208 s\nh\u271d : p a\n\u22a2 \u00aca \u2208 filter p s", "state_after": "no goals"}, {"tactic": "exact disjoint_empty_left _", "annotated_tactic": ["exact <a>disjoint_empty_left</a> _", [{"full_name": "Finset.disjoint_empty_left", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [975, 9], "def_end_pos": [975, 28]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np q : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : DecidablePred q\ns\u271d : Finset \u03b1\na : \u03b1\ns : Finset \u03b1\nha : \u00aca \u2208 s\nh\u271d : \u00acp a\n\u22a2 _root_.Disjoint \u2205 (filter p s)", "state_after": "no goals"}, {"tactic": "split_ifs with h", "annotated_tactic": ["split_ifs with h", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np q : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : DecidablePred q\ns\u271d : Finset \u03b1\na : \u03b1\ns : Finset \u03b1\nha : \u00aca \u2208 s\n\u22a2 filter p (cons a s ha) =\n    disjUnion (if p a then {a} else \u2205) (filter p s) (_ : _root_.Disjoint (if p a then {a} else \u2205) (filter p s))", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np q : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : DecidablePred q\ns\u271d : Finset \u03b1\na : \u03b1\ns : Finset \u03b1\nha : \u00aca \u2208 s\nh : p a\n\u22a2 filter p (cons a s ha) = disjUnion {a} (filter p s) (_ : _root_.Disjoint {a} (filter p s))\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np q : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : DecidablePred q\ns\u271d : Finset \u03b1\na : \u03b1\ns : Finset \u03b1\nha : \u00aca \u2208 s\nh : \u00acp a\n\u22a2 filter p (cons a s ha) = disjUnion \u2205 (filter p s) (_ : _root_.Disjoint \u2205 (filter p s))"}, {"tactic": "rw [filter_cons_of_pos _ _ _ ha h, singleton_disjUnion]", "annotated_tactic": ["rw [<a>filter_cons_of_pos</a> _ _ _ ha h, <a>singleton_disjUnion</a>]", [{"full_name": "Finset.filter_cons_of_pos", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2812, 9], "def_end_pos": [2812, 27]}, {"full_name": "Finset.singleton_disjUnion", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1048, 9], "def_end_pos": [1048, 28]}]], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np q : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : DecidablePred q\ns\u271d : Finset \u03b1\na : \u03b1\ns : Finset \u03b1\nha : \u00aca \u2208 s\nh : p a\n\u22a2 filter p (cons a s ha) = disjUnion {a} (filter p s) (_ : _root_.Disjoint {a} (filter p s))", "state_after": "no goals"}, {"tactic": "rw [filter_cons_of_neg _ _ _ ha h, empty_disjUnion]", "annotated_tactic": ["rw [<a>filter_cons_of_neg</a> _ _ _ ha h, <a>empty_disjUnion</a>]", [{"full_name": "Finset.filter_cons_of_neg", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2817, 9], "def_end_pos": [2817, 27]}, {"full_name": "Finset.empty_disjUnion", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1037, 9], "def_end_pos": [1037, 24]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np q : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : DecidablePred q\ns\u271d : Finset \u03b1\na : \u03b1\ns : Finset \u03b1\nha : \u00aca \u2208 s\nh : \u00acp a\n\u22a2 filter p (cons a s ha) = disjUnion \u2205 (filter p s) (_ : _root_.Disjoint \u2205 (filter p s))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Martingale/Upcrossing.lean", "full_name": "MeasureTheory.Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part", "start": [852, 1], "end": [880, 20], "traced_tactics": [{"tactic": "by_cases hab : a < b", "annotated_tactic": ["by_cases hab : a < b", []], "state_before": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na\u271d b\u271d : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\ninst\u271d : IsFiniteMeasure \u03bc\na b : \u211d\nhf : Submartingale f \u2131 \u03bc\n\u22a2 ENNReal.ofReal (b - a) * \u222b\u207b (\u03c9 : \u03a9), upcrossings a b f \u03c9 \u2202\u03bc \u2264 \u2a06 N, \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc", "state_after": "case pos\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na\u271d b\u271d : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\ninst\u271d : IsFiniteMeasure \u03bc\na b : \u211d\nhf : Submartingale f \u2131 \u03bc\nhab : a < b\n\u22a2 ENNReal.ofReal (b - a) * \u222b\u207b (\u03c9 : \u03a9), upcrossings a b f \u03c9 \u2202\u03bc \u2264 \u2a06 N, \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc\n\ncase neg\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na\u271d b\u271d : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\ninst\u271d : IsFiniteMeasure \u03bc\na b : \u211d\nhf : Submartingale f \u2131 \u03bc\nhab : \u00aca < b\n\u22a2 ENNReal.ofReal (b - a) * \u222b\u207b (\u03c9 : \u03a9), upcrossings a b f \u03c9 \u2202\u03bc \u2264 \u2a06 N, \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc"}, {"tactic": "simp_rw [upcrossings]", "annotated_tactic": ["simp_rw [<a>upcrossings</a>]", [{"full_name": "MeasureTheory.upcrossings", "def_path": "Mathlib/Probability/Martingale/Upcrossing.lean", "def_pos": [823, 19], "def_end_pos": [823, 30]}]], "state_before": "case pos\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na\u271d b\u271d : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\ninst\u271d : IsFiniteMeasure \u03bc\na b : \u211d\nhf : Submartingale f \u2131 \u03bc\nhab : a < b\n\u22a2 ENNReal.ofReal (b - a) * \u222b\u207b (\u03c9 : \u03a9), upcrossings a b f \u03c9 \u2202\u03bc \u2264 \u2a06 N, \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc", "state_after": "case pos\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na\u271d b\u271d : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\ninst\u271d : IsFiniteMeasure \u03bc\na b : \u211d\nhf : Submartingale f \u2131 \u03bc\nhab : a < b\n\u22a2 ENNReal.ofReal (b - a) * \u222b\u207b (\u03c9 : \u03a9), \u2a06 N, \u2191(upcrossingsBefore a b f N \u03c9) \u2202\u03bc \u2264\n    \u2a06 N, \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc"}, {"tactic": "rw [lintegral_iSup']", "annotated_tactic": ["rw [<a>lintegral_iSup'</a>]", [{"full_name": "MeasureTheory.lintegral_iSup'", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [416, 9], "def_end_pos": [416, 24]}]], "state_before": "case pos\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na\u271d b\u271d : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\ninst\u271d : IsFiniteMeasure \u03bc\na b : \u211d\nhf : Submartingale f \u2131 \u03bc\nhab : a < b\nthis : \u2200 (N : \u2115), \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc = ENNReal.ofReal (\u222b (\u03c9 : \u03a9), (f N \u03c9 - a)\u207a \u2202\u03bc)\n\u22a2 ENNReal.ofReal (b - a) * \u222b\u207b (\u03c9 : \u03a9), \u2a06 N, \u2191(upcrossingsBefore a b f N \u03c9) \u2202\u03bc \u2264\n    \u2a06 N, \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc", "state_after": "case pos\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na\u271d b\u271d : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\ninst\u271d : IsFiniteMeasure \u03bc\na b : \u211d\nhf : Submartingale f \u2131 \u03bc\nhab : a < b\nthis : \u2200 (N : \u2115), \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc = ENNReal.ofReal (\u222b (\u03c9 : \u03a9), (f N \u03c9 - a)\u207a \u2202\u03bc)\n\u22a2 ENNReal.ofReal (b - a) * \u2a06 n, \u222b\u207b (a_1 : \u03a9), \u2191(upcrossingsBefore a b f n a_1) \u2202\u03bc \u2264\n    \u2a06 N, \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc\n\ncase pos.hf\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na\u271d b\u271d : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\ninst\u271d : IsFiniteMeasure \u03bc\na b : \u211d\nhf : Submartingale f \u2131 \u03bc\nhab : a < b\nthis : \u2200 (N : \u2115), \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc = ENNReal.ofReal (\u222b (\u03c9 : \u03a9), (f N \u03c9 - a)\u207a \u2202\u03bc)\n\u22a2 \u2200 (n : \u2115), AEMeasurable fun \u03c9 => \u2191(upcrossingsBefore a b f n \u03c9)\n\ncase pos.h_mono\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na\u271d b\u271d : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\ninst\u271d : IsFiniteMeasure \u03bc\na b : \u211d\nhf : Submartingale f \u2131 \u03bc\nhab : a < b\nthis : \u2200 (N : \u2115), \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc = ENNReal.ofReal (\u222b (\u03c9 : \u03a9), (f N \u03c9 - a)\u207a \u2202\u03bc)\n\u22a2 \u2200\u1d50 (x : \u03a9) \u2202\u03bc, Monotone fun n => \u2191(upcrossingsBefore a b f n x)"}, {"tactic": "intro N", "annotated_tactic": ["intro N", []], "state_before": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na\u271d b\u271d : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\ninst\u271d : IsFiniteMeasure \u03bc\na b : \u211d\nhf : Submartingale f \u2131 \u03bc\nhab : a < b\n\u22a2 \u2200 (N : \u2115), \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc = ENNReal.ofReal (\u222b (\u03c9 : \u03a9), (f N \u03c9 - a)\u207a \u2202\u03bc)", "state_after": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na\u271d b\u271d : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN\u271d n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\ninst\u271d : IsFiniteMeasure \u03bc\na b : \u211d\nhf : Submartingale f \u2131 \u03bc\nhab : a < b\nN : \u2115\n\u22a2 \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc = ENNReal.ofReal (\u222b (\u03c9 : \u03a9), (f N \u03c9 - a)\u207a \u2202\u03bc)"}, {"tactic": "rw [ofReal_integral_eq_lintegral_ofReal]", "annotated_tactic": ["rw [<a>ofReal_integral_eq_lintegral_ofReal</a>]", [{"full_name": "MeasureTheory.ofReal_integral_eq_lintegral_ofReal", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1219, 9], "def_end_pos": [1219, 44]}]], "state_before": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na\u271d b\u271d : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN\u271d n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\ninst\u271d : IsFiniteMeasure \u03bc\na b : \u211d\nhf : Submartingale f \u2131 \u03bc\nhab : a < b\nN : \u2115\n\u22a2 \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc = ENNReal.ofReal (\u222b (\u03c9 : \u03a9), (f N \u03c9 - a)\u207a \u2202\u03bc)", "state_after": "case hfi\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na\u271d b\u271d : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN\u271d n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\ninst\u271d : IsFiniteMeasure \u03bc\na b : \u211d\nhf : Submartingale f \u2131 \u03bc\nhab : a < b\nN : \u2115\n\u22a2 Integrable fun \u03c9 => (f N \u03c9 - a)\u207a\n\ncase f_nn\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na\u271d b\u271d : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN\u271d n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\ninst\u271d : IsFiniteMeasure \u03bc\na b : \u211d\nhf : Submartingale f \u2131 \u03bc\nhab : a < b\nN : \u2115\n\u22a2 0 \u2264\u1d50[\u03bc] fun \u03c9 => (f N \u03c9 - a)\u207a"}, {"tactic": "exact (hf.sub_martingale (martingale_const _ _ _)).pos.integrable _", "annotated_tactic": ["exact (hf.sub_martingale (<a>martingale_const</a> _ _ _)).pos.integrable _", [{"full_name": "MeasureTheory.martingale_const", "def_path": "Mathlib/Probability/Martingale/Basic.lean", "def_pos": [70, 9], "def_end_pos": [70, 25]}]], "state_before": "case hfi\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na\u271d b\u271d : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN\u271d n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\ninst\u271d : IsFiniteMeasure \u03bc\na b : \u211d\nhf : Submartingale f \u2131 \u03bc\nhab : a < b\nN : \u2115\n\u22a2 Integrable fun \u03c9 => (f N \u03c9 - a)\u207a", "state_after": "no goals"}, {"tactic": "exact eventually_of_forall fun \u03c9 => LatticeOrderedGroup.pos_nonneg _", "annotated_tactic": ["exact <a>eventually_of_forall</a> fun \u03c9 => <a>LatticeOrderedGroup.pos_nonneg</a> _", [{"full_name": "Filter.eventually_of_forall", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1111, 9], "def_end_pos": [1111, 29]}, {"full_name": "LatticeOrderedGroup.pos_nonneg", "def_path": "Mathlib/Algebra/Order/LatticeGroup.lean", "def_pos": [200, 15], "def_end_pos": [200, 25]}]], "state_before": "case f_nn\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na\u271d b\u271d : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN\u271d n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\ninst\u271d : IsFiniteMeasure \u03bc\na b : \u211d\nhf : Submartingale f \u2131 \u03bc\nhab : a < b\nN : \u2115\n\u22a2 0 \u2264\u1d50[\u03bc] fun \u03c9 => (f N \u03c9 - a)\u207a", "state_after": "no goals"}, {"tactic": "simp_rw [this, ENNReal.mul_iSup, iSup_le_iff]", "annotated_tactic": ["simp_rw [this, <a>ENNReal.mul_iSup</a>, <a>iSup_le_iff</a>]", [{"full_name": "ENNReal.mul_iSup", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [638, 9], "def_end_pos": [638, 17]}, {"full_name": "iSup_le_iff", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [964, 9], "def_end_pos": [964, 20]}]], "state_before": "case pos\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na\u271d b\u271d : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\ninst\u271d : IsFiniteMeasure \u03bc\na b : \u211d\nhf : Submartingale f \u2131 \u03bc\nhab : a < b\nthis : \u2200 (N : \u2115), \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc = ENNReal.ofReal (\u222b (\u03c9 : \u03a9), (f N \u03c9 - a)\u207a \u2202\u03bc)\n\u22a2 ENNReal.ofReal (b - a) * \u2a06 n, \u222b\u207b (a_1 : \u03a9), \u2191(upcrossingsBefore a b f n a_1) \u2202\u03bc \u2264\n    \u2a06 N, \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc", "state_after": "case pos\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na\u271d b\u271d : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\ninst\u271d : IsFiniteMeasure \u03bc\na b : \u211d\nhf : Submartingale f \u2131 \u03bc\nhab : a < b\nthis : \u2200 (N : \u2115), \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc = ENNReal.ofReal (\u222b (\u03c9 : \u03a9), (f N \u03c9 - a)\u207a \u2202\u03bc)\n\u22a2 \u2200 (i : \u2115),\n    ENNReal.ofReal (b - a) * \u222b\u207b (a_1 : \u03a9), \u2191(upcrossingsBefore a b f i a_1) \u2202\u03bc \u2264\n      \u2a06 N, ENNReal.ofReal (\u222b (\u03c9 : \u03a9), (f N \u03c9 - a)\u207a \u2202\u03bc)"}, {"tactic": "intro N", "annotated_tactic": ["intro N", []], "state_before": "case pos\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na\u271d b\u271d : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\ninst\u271d : IsFiniteMeasure \u03bc\na b : \u211d\nhf : Submartingale f \u2131 \u03bc\nhab : a < b\nthis : \u2200 (N : \u2115), \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc = ENNReal.ofReal (\u222b (\u03c9 : \u03a9), (f N \u03c9 - a)\u207a \u2202\u03bc)\n\u22a2 \u2200 (i : \u2115),\n    ENNReal.ofReal (b - a) * \u222b\u207b (a_1 : \u03a9), \u2191(upcrossingsBefore a b f i a_1) \u2202\u03bc \u2264\n      \u2a06 N, ENNReal.ofReal (\u222b (\u03c9 : \u03a9), (f N \u03c9 - a)\u207a \u2202\u03bc)", "state_after": "case pos\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na\u271d b\u271d : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN\u271d n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\ninst\u271d : IsFiniteMeasure \u03bc\na b : \u211d\nhf : Submartingale f \u2131 \u03bc\nhab : a < b\nthis : \u2200 (N : \u2115), \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc = ENNReal.ofReal (\u222b (\u03c9 : \u03a9), (f N \u03c9 - a)\u207a \u2202\u03bc)\nN : \u2115\n\u22a2 ENNReal.ofReal (b - a) * \u222b\u207b (a_1 : \u03a9), \u2191(upcrossingsBefore a b f N a_1) \u2202\u03bc \u2264\n    \u2a06 N, ENNReal.ofReal (\u222b (\u03c9 : \u03a9), (f N \u03c9 - a)\u207a \u2202\u03bc)"}, {"tactic": "rw [(by simp :\n    \u222b\u207b \u03c9, upcrossingsBefore a b f N \u03c9 \u2202\u03bc = \u222b\u207b \u03c9, \u2191(upcrossingsBefore a b f N \u03c9 : \u211d\u22650) \u2202\u03bc),\n  lintegral_coe_eq_integral, \u2190 ENNReal.ofReal_mul (sub_pos.2 hab).le]", "annotated_tactic": ["rw [(by simp :\n          \u222b\u207b \u03c9, upcrossingsBefore a b f N \u03c9 \u2202\u03bc = \u222b\u207b \u03c9, \u2191(upcrossingsBefore a b f N \u03c9 : \u211d\u22650) \u2202\u03bc),\n        <a>lintegral_coe_eq_integral</a>, \u2190 <a>ENNReal.ofReal_mul</a> (<a>sub_pos</a>.2 hab).<a>le</a>]", [{"full_name": "MeasureTheory.lintegral_coe_eq_integral", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1208, 9], "def_end_pos": [1208, 34]}, {"full_name": "ENNReal.ofReal_mul", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2225, 9], "def_end_pos": [2225, 19]}, {"full_name": "sub_pos", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [883, 30], "def_end_pos": [883, 37]}, {"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [142, 7], "def_end_pos": [142, 15]}]], "state_before": "case pos\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na\u271d b\u271d : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN\u271d n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\ninst\u271d : IsFiniteMeasure \u03bc\na b : \u211d\nhf : Submartingale f \u2131 \u03bc\nhab : a < b\nthis : \u2200 (N : \u2115), \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc = ENNReal.ofReal (\u222b (\u03c9 : \u03a9), (f N \u03c9 - a)\u207a \u2202\u03bc)\nN : \u2115\n\u22a2 ENNReal.ofReal (b - a) * \u222b\u207b (a_1 : \u03a9), \u2191(upcrossingsBefore a b f N a_1) \u2202\u03bc \u2264\n    \u2a06 N, ENNReal.ofReal (\u222b (\u03c9 : \u03a9), (f N \u03c9 - a)\u207a \u2202\u03bc)", "state_after": "case pos\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na\u271d b\u271d : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN\u271d n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\ninst\u271d : IsFiniteMeasure \u03bc\na b : \u211d\nhf : Submartingale f \u2131 \u03bc\nhab : a < b\nthis : \u2200 (N : \u2115), \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc = ENNReal.ofReal (\u222b (\u03c9 : \u03a9), (f N \u03c9 - a)\u207a \u2202\u03bc)\nN : \u2115\n\u22a2 ENNReal.ofReal ((b - a) * \u222b (a_1 : \u03a9), \u2191\u2191(upcrossingsBefore a b f N a_1) \u2202\u03bc) \u2264\n    \u2a06 N, ENNReal.ofReal (\u222b (\u03c9 : \u03a9), (f N \u03c9 - a)\u207a \u2202\u03bc)\n\ncase pos.hfi\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na\u271d b\u271d : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN\u271d n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\ninst\u271d : IsFiniteMeasure \u03bc\na b : \u211d\nhf : Submartingale f \u2131 \u03bc\nhab : a < b\nthis : \u2200 (N : \u2115), \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc = ENNReal.ofReal (\u222b (\u03c9 : \u03a9), (f N \u03c9 - a)\u207a \u2202\u03bc)\nN : \u2115\n\u22a2 Integrable fun x => \u2191\u2191(upcrossingsBefore a b f N x)"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na\u271d b\u271d : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN\u271d n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\ninst\u271d : IsFiniteMeasure \u03bc\na b : \u211d\nhf : Submartingale f \u2131 \u03bc\nhab : a < b\nthis : \u2200 (N : \u2115), \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc = ENNReal.ofReal (\u222b (\u03c9 : \u03a9), (f N \u03c9 - a)\u207a \u2202\u03bc)\nN : \u2115\n\u22a2 \u222b\u207b (\u03c9 : \u03a9), \u2191(upcrossingsBefore a b f N \u03c9) \u2202\u03bc = \u222b\u207b (\u03c9 : \u03a9), \u2191\u2191(upcrossingsBefore a b f N \u03c9) \u2202\u03bc", "state_after": "no goals"}, {"tactic": "simp_rw [NNReal.coe_nat_cast]", "annotated_tactic": ["simp_rw [<a>NNReal.coe_nat_cast</a>]", [{"full_name": "NNReal.coe_nat_cast", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [354, 19], "def_end_pos": [354, 31]}]], "state_before": "case pos\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na\u271d b\u271d : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN\u271d n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\ninst\u271d : IsFiniteMeasure \u03bc\na b : \u211d\nhf : Submartingale f \u2131 \u03bc\nhab : a < b\nthis : \u2200 (N : \u2115), \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc = ENNReal.ofReal (\u222b (\u03c9 : \u03a9), (f N \u03c9 - a)\u207a \u2202\u03bc)\nN : \u2115\n\u22a2 ENNReal.ofReal ((b - a) * \u222b (a_1 : \u03a9), \u2191\u2191(upcrossingsBefore a b f N a_1) \u2202\u03bc) \u2264\n    \u2a06 N, ENNReal.ofReal (\u222b (\u03c9 : \u03a9), (f N \u03c9 - a)\u207a \u2202\u03bc)", "state_after": "case pos\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na\u271d b\u271d : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN\u271d n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\ninst\u271d : IsFiniteMeasure \u03bc\na b : \u211d\nhf : Submartingale f \u2131 \u03bc\nhab : a < b\nthis : \u2200 (N : \u2115), \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc = ENNReal.ofReal (\u222b (\u03c9 : \u03a9), (f N \u03c9 - a)\u207a \u2202\u03bc)\nN : \u2115\n\u22a2 ENNReal.ofReal ((b - a) * \u222b (a_1 : \u03a9), \u2191(upcrossingsBefore a b f N a_1) \u2202\u03bc) \u2264\n    \u2a06 N, ENNReal.ofReal (\u222b (\u03c9 : \u03a9), (f N \u03c9 - a)\u207a \u2202\u03bc)"}, {"tactic": "exact (ENNReal.ofReal_le_ofReal\n  (hf.mul_integral_upcrossingsBefore_le_integral_pos_part a b N)).trans\n    (le_iSup (\u03b1 := \u211d\u22650\u221e) _ N)", "annotated_tactic": ["exact (<a>ENNReal.ofReal_le_ofReal</a>\n          (hf.mul_integral_upcrossingsBefore_le_integral_pos_part a b N)).<a>trans</a>\n            (<a>le_iSup</a> (\u03b1 := \u211d\u22650\u221e) _ N)", [{"full_name": "ENNReal.ofReal_le_ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2135, 9], "def_end_pos": [2135, 25]}, {"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [120, 7], "def_end_pos": [120, 18]}, {"full_name": "le_iSup", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [810, 9], "def_end_pos": [810, 16]}]], "state_before": "case pos\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na\u271d b\u271d : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN\u271d n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\ninst\u271d : IsFiniteMeasure \u03bc\na b : \u211d\nhf : Submartingale f \u2131 \u03bc\nhab : a < b\nthis : \u2200 (N : \u2115), \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc = ENNReal.ofReal (\u222b (\u03c9 : \u03a9), (f N \u03c9 - a)\u207a \u2202\u03bc)\nN : \u2115\n\u22a2 ENNReal.ofReal ((b - a) * \u222b (a_1 : \u03a9), \u2191(upcrossingsBefore a b f N a_1) \u2202\u03bc) \u2264\n    \u2a06 N, ENNReal.ofReal (\u222b (\u03c9 : \u03a9), (f N \u03c9 - a)\u207a \u2202\u03bc)", "state_after": "no goals"}, {"tactic": "simp only [NNReal.coe_nat_cast, hf.adapted.integrable_upcrossingsBefore hab]", "annotated_tactic": ["simp only [<a>NNReal.coe_nat_cast</a>, hf.adapted.integrable_upcrossingsBefore hab]", [{"full_name": "NNReal.coe_nat_cast", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [354, 19], "def_end_pos": [354, 31]}]], "state_before": "case pos.hfi\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na\u271d b\u271d : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN\u271d n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\ninst\u271d : IsFiniteMeasure \u03bc\na b : \u211d\nhf : Submartingale f \u2131 \u03bc\nhab : a < b\nthis : \u2200 (N : \u2115), \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc = ENNReal.ofReal (\u222b (\u03c9 : \u03a9), (f N \u03c9 - a)\u207a \u2202\u03bc)\nN : \u2115\n\u22a2 Integrable fun x => \u2191\u2191(upcrossingsBefore a b f N x)", "state_after": "no goals"}, {"tactic": "exact fun n => measurable_from_top.comp_aemeasurable\n  (hf.adapted.measurable_upcrossingsBefore hab).aemeasurable", "annotated_tactic": ["exact fun n => measurable_from_top.comp_aemeasurable\n        (hf.adapted.measurable_upcrossingsBefore hab).<a>aemeasurable</a>", [{"full_name": "Measurable.aemeasurable", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [713, 9], "def_end_pos": [713, 32]}]], "state_before": "case pos.hf\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na\u271d b\u271d : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\ninst\u271d : IsFiniteMeasure \u03bc\na b : \u211d\nhf : Submartingale f \u2131 \u03bc\nhab : a < b\nthis : \u2200 (N : \u2115), \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc = ENNReal.ofReal (\u222b (\u03c9 : \u03a9), (f N \u03c9 - a)\u207a \u2202\u03bc)\n\u22a2 \u2200 (n : \u2115), AEMeasurable fun \u03c9 => \u2191(upcrossingsBefore a b f n \u03c9)", "state_after": "no goals"}, {"tactic": "refine' eventually_of_forall fun \u03c9 N M hNM => _", "annotated_tactic": ["refine' <a>eventually_of_forall</a> fun \u03c9 N M hNM => _", [{"full_name": "Filter.eventually_of_forall", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1111, 9], "def_end_pos": [1111, 29]}]], "state_before": "case pos.h_mono\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na\u271d b\u271d : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\ninst\u271d : IsFiniteMeasure \u03bc\na b : \u211d\nhf : Submartingale f \u2131 \u03bc\nhab : a < b\nthis : \u2200 (N : \u2115), \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc = ENNReal.ofReal (\u222b (\u03c9 : \u03a9), (f N \u03c9 - a)\u207a \u2202\u03bc)\n\u22a2 \u2200\u1d50 (x : \u03a9) \u2202\u03bc, Monotone fun n => \u2191(upcrossingsBefore a b f n x)", "state_after": "case pos.h_mono\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na\u271d b\u271d : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN\u271d n m : \u2115\n\u03c9\u271d : \u03a9\n\u2131 : Filtration \u2115 m0\ninst\u271d : IsFiniteMeasure \u03bc\na b : \u211d\nhf : Submartingale f \u2131 \u03bc\nhab : a < b\nthis : \u2200 (N : \u2115), \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc = ENNReal.ofReal (\u222b (\u03c9 : \u03a9), (f N \u03c9 - a)\u207a \u2202\u03bc)\n\u03c9 : \u03a9\nN M : \u2115\nhNM : N \u2264 M\n\u22a2 (fun n => \u2191(upcrossingsBefore a b f n \u03c9)) N \u2264 (fun n => \u2191(upcrossingsBefore a b f n \u03c9)) M"}, {"tactic": "rw [Nat.cast_le]", "annotated_tactic": ["rw [<a>Nat.cast_le</a>]", [{"full_name": "Nat.cast_le", "def_path": "Mathlib/Data/Nat/Cast/Order.lean", "def_pos": [91, 9], "def_end_pos": [91, 16]}]], "state_before": "case pos.h_mono\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na\u271d b\u271d : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN\u271d n m : \u2115\n\u03c9\u271d : \u03a9\n\u2131 : Filtration \u2115 m0\ninst\u271d : IsFiniteMeasure \u03bc\na b : \u211d\nhf : Submartingale f \u2131 \u03bc\nhab : a < b\nthis : \u2200 (N : \u2115), \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc = ENNReal.ofReal (\u222b (\u03c9 : \u03a9), (f N \u03c9 - a)\u207a \u2202\u03bc)\n\u03c9 : \u03a9\nN M : \u2115\nhNM : N \u2264 M\n\u22a2 (fun n => \u2191(upcrossingsBefore a b f n \u03c9)) N \u2264 (fun n => \u2191(upcrossingsBefore a b f n \u03c9)) M", "state_after": "case pos.h_mono\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na\u271d b\u271d : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN\u271d n m : \u2115\n\u03c9\u271d : \u03a9\n\u2131 : Filtration \u2115 m0\ninst\u271d : IsFiniteMeasure \u03bc\na b : \u211d\nhf : Submartingale f \u2131 \u03bc\nhab : a < b\nthis : \u2200 (N : \u2115), \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc = ENNReal.ofReal (\u222b (\u03c9 : \u03a9), (f N \u03c9 - a)\u207a \u2202\u03bc)\n\u03c9 : \u03a9\nN M : \u2115\nhNM : N \u2264 M\n\u22a2 upcrossingsBefore a b f N \u03c9 \u2264 upcrossingsBefore a b f M \u03c9"}, {"tactic": "exact upcrossingsBefore_mono hab hNM \u03c9", "annotated_tactic": ["exact <a>upcrossingsBefore_mono</a> hab hNM \u03c9", [{"full_name": "MeasureTheory.upcrossingsBefore_mono", "def_path": "Mathlib/Probability/Martingale/Upcrossing.lean", "def_pos": [543, 9], "def_end_pos": [543, 31]}]], "state_before": "case pos.h_mono\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na\u271d b\u271d : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN\u271d n m : \u2115\n\u03c9\u271d : \u03a9\n\u2131 : Filtration \u2115 m0\ninst\u271d : IsFiniteMeasure \u03bc\na b : \u211d\nhf : Submartingale f \u2131 \u03bc\nhab : a < b\nthis : \u2200 (N : \u2115), \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc = ENNReal.ofReal (\u222b (\u03c9 : \u03a9), (f N \u03c9 - a)\u207a \u2202\u03bc)\n\u03c9 : \u03a9\nN M : \u2115\nhNM : N \u2264 M\n\u22a2 upcrossingsBefore a b f N \u03c9 \u2264 upcrossingsBefore a b f M \u03c9", "state_after": "no goals"}, {"tactic": "rw [not_lt, \u2190 sub_nonpos] at hab", "annotated_tactic": ["rw [<a>not_lt</a>, \u2190 <a>sub_nonpos</a>] at hab", [{"full_name": "not_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [368, 9], "def_end_pos": [368, 15]}, {"full_name": "sub_nonpos", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [730, 30], "def_end_pos": [730, 40]}]], "state_before": "case neg\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na\u271d b\u271d : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\ninst\u271d : IsFiniteMeasure \u03bc\na b : \u211d\nhf : Submartingale f \u2131 \u03bc\nhab : \u00aca < b\n\u22a2 ENNReal.ofReal (b - a) * \u222b\u207b (\u03c9 : \u03a9), upcrossings a b f \u03c9 \u2202\u03bc \u2264 \u2a06 N, \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc", "state_after": "case neg\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na\u271d b\u271d : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\ninst\u271d : IsFiniteMeasure \u03bc\na b : \u211d\nhf : Submartingale f \u2131 \u03bc\nhab\u271d : b \u2264 a\nhab : b - a \u2264 0\n\u22a2 ENNReal.ofReal (b - a) * \u222b\u207b (\u03c9 : \u03a9), upcrossings a b f \u03c9 \u2202\u03bc \u2264 \u2a06 N, \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc"}, {"tactic": "rw [ENNReal.ofReal_of_nonpos hab, zero_mul]", "annotated_tactic": ["rw [<a>ENNReal.ofReal_of_nonpos</a> hab, <a>zero_mul</a>]", [{"full_name": "ENNReal.ofReal_of_nonpos", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2178, 11], "def_end_pos": [2178, 27]}, {"full_name": "MulZeroClass.zero_mul", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [36, 3], "def_end_pos": [36, 11]}]], "state_before": "case neg\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na\u271d b\u271d : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\ninst\u271d : IsFiniteMeasure \u03bc\na b : \u211d\nhf : Submartingale f \u2131 \u03bc\nhab\u271d : b \u2264 a\nhab : b - a \u2264 0\n\u22a2 ENNReal.ofReal (b - a) * \u222b\u207b (\u03c9 : \u03a9), upcrossings a b f \u03c9 \u2202\u03bc \u2264 \u2a06 N, \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc", "state_after": "case neg\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na\u271d b\u271d : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\ninst\u271d : IsFiniteMeasure \u03bc\na b : \u211d\nhf : Submartingale f \u2131 \u03bc\nhab\u271d : b \u2264 a\nhab : b - a \u2264 0\n\u22a2 0 \u2264 \u2a06 N, \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc"}, {"tactic": "exact zero_le _", "annotated_tactic": ["exact <a>zero_le</a> _", [{"full_name": "zero_le", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [217, 30], "def_end_pos": [217, 37]}]], "state_before": "case neg\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na\u271d b\u271d : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\ninst\u271d : IsFiniteMeasure \u03bc\na b : \u211d\nhf : Submartingale f \u2131 \u03bc\nhab\u271d : b \u2264 a\nhab : b - a \u2264 0\n\u22a2 0 \u2264 \u2a06 N, \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/HashMap/WF.lean", "full_name": "Std.HashMap.Imp.WF.mapVal", "start": [296, 1], "end": [306, 23], "traced_tactics": [{"tactic": "have \u27e8h\u2081, h\u2082\u27e9 := H.out", "annotated_tactic": ["have \u27e8h\u2081, h\u2082\u27e9 := H.out", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nH : WF m\n\u22a2 WF (Imp.mapVal f m)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nH : WF m\nh\u2081 : m.size = Buckets.size m.buckets\nh\u2082 : Buckets.WF m.buckets\n\u22a2 WF (Imp.mapVal f m)"}, {"tactic": "simp [Imp.mapVal, Buckets.mapVal, WF_iff, h\u2081]", "annotated_tactic": ["simp [<a>Imp.mapVal</a>, <a>Buckets.mapVal</a>, <a>WF_iff</a>, h\u2081]", [{"full_name": "Std.HashMap.Imp.mapVal", "def_path": "lake-packages/std/Std/Data/HashMap/Basic.lean", "def_pos": [184, 15], "def_end_pos": [184, 21]}, {"full_name": "Std.HashMap.Imp.Buckets.mapVal", "def_path": "lake-packages/std/Std/Data/HashMap/Basic.lean", "def_pos": [46, 19], "def_end_pos": [46, 25]}, {"full_name": "Std.HashMap.Imp.WF_iff", "def_path": "lake-packages/std/Std/Data/HashMap/WF.lean", "def_pos": [292, 9], "def_end_pos": [292, 15]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nH : WF m\nh\u2081 : m.size = Buckets.size m.buckets\nh\u2082 : Buckets.WF m.buckets\n\u22a2 WF (Imp.mapVal f m)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nH : WF m\nh\u2081 : m.size = Buckets.size m.buckets\nh\u2082 : Buckets.WF m.buckets\n\u22a2 Buckets.size m.buckets =\n      Buckets.size\n        { val := Array.map (AssocList.mapVal f) m.buckets.val,\n          property := (_ : 0 < Array.size (Array.map (AssocList.mapVal f) m.buckets.val)) } \u2227\n    Buckets.WF\n      { val := Array.map (AssocList.mapVal f) m.buckets.val,\n        property := (_ : 0 < Array.size (Array.map (AssocList.mapVal f) m.buckets.val)) }"}, {"tactic": "refine \u27e8?_, ?_, fun i h => ?_\u27e9", "annotated_tactic": ["refine \u27e8?_, ?_, fun i h => ?_\u27e9", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nH : WF m\nh\u2081 : m.size = Buckets.size m.buckets\nh\u2082 : Buckets.WF m.buckets\n\u22a2 Buckets.size m.buckets =\n      Buckets.size\n        { val := Array.map (AssocList.mapVal f) m.buckets.val,\n          property := (_ : 0 < Array.size (Array.map (AssocList.mapVal f) m.buckets.val)) } \u2227\n    Buckets.WF\n      { val := Array.map (AssocList.mapVal f) m.buckets.val,\n        property := (_ : 0 < Array.size (Array.map (AssocList.mapVal f) m.buckets.val)) }", "state_after": "case refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nH : WF m\nh\u2081 : m.size = Buckets.size m.buckets\nh\u2082 : Buckets.WF m.buckets\n\u22a2 Buckets.size m.buckets =\n    Buckets.size\n      { val := Array.map (AssocList.mapVal f) m.buckets.val,\n        property := (_ : 0 < Array.size (Array.map (AssocList.mapVal f) m.buckets.val)) }\n\ncase refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nH : WF m\nh\u2081 : m.size = Buckets.size m.buckets\nh\u2082 : Buckets.WF m.buckets\n\u22a2 \u2200 [inst : LawfulHashable \u03b1] [inst : PartialEquivBEq \u03b1] (bucket : AssocList \u03b1 \u03b3),\n    bucket \u2208\n        { val := Array.map (AssocList.mapVal f) m.buckets.val,\n              property := (_ : 0 < Array.size (Array.map (AssocList.mapVal f) m.buckets.val)) }.val.data \u2192\n      List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList bucket)\n\ncase refine_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nH : WF m\nh\u2081 : m.size = Buckets.size m.buckets\nh\u2082 : Buckets.WF m.buckets\ni : Nat\nh :\n  i <\n    Array.size\n      { val := Array.map (AssocList.mapVal f) m.buckets.val,\n          property := (_ : 0 < Array.size (Array.map (AssocList.mapVal f) m.buckets.val)) }.val\n\u22a2 AssocList.All\n    (fun k x =>\n      USize.toNat\n          (UInt64.toUSize (hash k) %\n            Array.size\n              { val := Array.map (AssocList.mapVal f) m.buckets.val,\n                  property := (_ : 0 < Array.size (Array.map (AssocList.mapVal f) m.buckets.val)) }.val) =\n        i)\n    { val := Array.map (AssocList.mapVal f) m.buckets.val,\n          property := (_ : 0 < Array.size (Array.map (AssocList.mapVal f) m.buckets.val)) }.val[i]"}, {"tactic": "simp [Buckets.size]", "annotated_tactic": ["simp [<a>Buckets.size</a>]", [{"full_name": "Std.HashMap.Imp.Buckets.size", "def_path": "lake-packages/std/Std/Data/HashMap/Basic.lean", "def_pos": [40, 19], "def_end_pos": [40, 23]}]], "state_before": "case refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nH : WF m\nh\u2081 : m.size = Buckets.size m.buckets\nh\u2082 : Buckets.WF m.buckets\n\u22a2 Buckets.size m.buckets =\n    Buckets.size\n      { val := Array.map (AssocList.mapVal f) m.buckets.val,\n        property := (_ : 0 < Array.size (Array.map (AssocList.mapVal f) m.buckets.val)) }", "state_after": "case refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nH : WF m\nh\u2081 : m.size = Buckets.size m.buckets\nh\u2082 : Buckets.WF m.buckets\n\u22a2 Nat.sum (List.map (fun x => List.length (AssocList.toList x)) m.buckets.val.data) =\n    Nat.sum (List.map ((fun x => List.length (AssocList.toList x)) \u2218 AssocList.mapVal f) m.buckets.val.data)"}, {"tactic": "congr", "annotated_tactic": ["congr", []], "state_before": "case refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nH : WF m\nh\u2081 : m.size = Buckets.size m.buckets\nh\u2082 : Buckets.WF m.buckets\n\u22a2 Nat.sum (List.map (fun x => List.length (AssocList.toList x)) m.buckets.val.data) =\n    Nat.sum (List.map ((fun x => List.length (AssocList.toList x)) \u2218 AssocList.mapVal f) m.buckets.val.data)", "state_after": "case refine_1.e_l.e_f\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nH : WF m\nh\u2081 : m.size = Buckets.size m.buckets\nh\u2082 : Buckets.WF m.buckets\n\u22a2 (fun x => List.length (AssocList.toList x)) = (fun x => List.length (AssocList.toList x)) \u2218 AssocList.mapVal f"}, {"tactic": "funext l", "annotated_tactic": ["funext l", []], "state_before": "case refine_1.e_l.e_f\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nH : WF m\nh\u2081 : m.size = Buckets.size m.buckets\nh\u2082 : Buckets.WF m.buckets\n\u22a2 (fun x => List.length (AssocList.toList x)) = (fun x => List.length (AssocList.toList x)) \u2218 AssocList.mapVal f", "state_after": "case refine_1.e_l.e_f.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nH : WF m\nh\u2081 : m.size = Buckets.size m.buckets\nh\u2082 : Buckets.WF m.buckets\nl : AssocList \u03b1 \u03b2\n\u22a2 List.length (AssocList.toList l) = ((fun x => List.length (AssocList.toList x)) \u2218 AssocList.mapVal f) l"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case refine_1.e_l.e_f.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nH : WF m\nh\u2081 : m.size = Buckets.size m.buckets\nh\u2082 : Buckets.WF m.buckets\nl : AssocList \u03b1 \u03b2\n\u22a2 List.length (AssocList.toList l) = ((fun x => List.length (AssocList.toList x)) \u2218 AssocList.mapVal f) l", "state_after": "no goals"}, {"tactic": "simp only [Array.map_data, List.forall_mem_map_iff]", "annotated_tactic": ["simp only [<a>Array.map_data</a>, <a>List.forall_mem_map_iff</a>]", [{"full_name": "Array.map_data", "def_path": "lake-packages/std/Std/Data/Array/Init/Lemmas.lean", "def_pos": [185, 17], "def_end_pos": [185, 25]}, {"full_name": "List.forall_mem_map_iff", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [167, 9], "def_end_pos": [167, 27]}]], "state_before": "case refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nH : WF m\nh\u2081 : m.size = Buckets.size m.buckets\nh\u2082 : Buckets.WF m.buckets\n\u22a2 \u2200 [inst : LawfulHashable \u03b1] [inst : PartialEquivBEq \u03b1] (bucket : AssocList \u03b1 \u03b3),\n    bucket \u2208\n        { val := Array.map (AssocList.mapVal f) m.buckets.val,\n              property := (_ : 0 < Array.size (Array.map (AssocList.mapVal f) m.buckets.val)) }.val.data \u2192\n      List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList bucket)", "state_after": "case refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nH : WF m\nh\u2081 : m.size = Buckets.size m.buckets\nh\u2082 : Buckets.WF m.buckets\n\u22a2 \u2200 [inst : LawfulHashable \u03b1] [inst : PartialEquivBEq \u03b1] (j : AssocList \u03b1 \u03b2),\n    j \u2208 m.buckets.val.data \u2192\n      List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList (AssocList.mapVal f j))"}, {"tactic": "simp [List.pairwise_map]", "annotated_tactic": ["simp [<a>List.pairwise_map</a>]", [{"full_name": "List.pairwise_map", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [1451, 9], "def_end_pos": [1451, 21]}]], "state_before": "case refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nH : WF m\nh\u2081 : m.size = Buckets.size m.buckets\nh\u2082 : Buckets.WF m.buckets\n\u22a2 \u2200 [inst : LawfulHashable \u03b1] [inst : PartialEquivBEq \u03b1] (j : AssocList \u03b1 \u03b2),\n    j \u2208 m.buckets.val.data \u2192\n      List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList (AssocList.mapVal f j))", "state_after": "case refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nH : WF m\nh\u2081 : m.size = Buckets.size m.buckets\nh\u2082 : Buckets.WF m.buckets\n\u22a2 \u2200 [inst : LawfulHashable \u03b1] [inst : PartialEquivBEq \u03b1] (j : AssocList \u03b1 \u03b2),\n    j \u2208 m.buckets.val.data \u2192 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList j)"}, {"tactic": "exact fun _ => h\u2082.1 _", "annotated_tactic": ["exact fun _ => h\u2082.1 _", []], "state_before": "case refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nH : WF m\nh\u2081 : m.size = Buckets.size m.buckets\nh\u2082 : Buckets.WF m.buckets\n\u22a2 \u2200 [inst : LawfulHashable \u03b1] [inst : PartialEquivBEq \u03b1] (j : AssocList \u03b1 \u03b2),\n    j \u2208 m.buckets.val.data \u2192 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList j)", "state_after": "no goals"}, {"tactic": "simp [AssocList.All] at h \u22a2", "annotated_tactic": ["simp [<a>AssocList.All</a>] at h \u22a2", [{"full_name": "Std.AssocList.All", "def_path": "lake-packages/std/Std/Data/AssocList.lean", "def_pos": [145, 5], "def_end_pos": [145, 8]}]], "state_before": "case refine_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nH : WF m\nh\u2081 : m.size = Buckets.size m.buckets\nh\u2082 : Buckets.WF m.buckets\ni : Nat\nh :\n  i <\n    Array.size\n      { val := Array.map (AssocList.mapVal f) m.buckets.val,\n          property := (_ : 0 < Array.size (Array.map (AssocList.mapVal f) m.buckets.val)) }.val\n\u22a2 AssocList.All\n    (fun k x =>\n      USize.toNat\n          (UInt64.toUSize (hash k) %\n            Array.size\n              { val := Array.map (AssocList.mapVal f) m.buckets.val,\n                  property := (_ : 0 < Array.size (Array.map (AssocList.mapVal f) m.buckets.val)) }.val) =\n        i)\n    { val := Array.map (AssocList.mapVal f) m.buckets.val,\n          property := (_ : 0 < Array.size (Array.map (AssocList.mapVal f) m.buckets.val)) }.val[i]", "state_after": "case refine_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nH : WF m\nh\u2081 : m.size = Buckets.size m.buckets\nh\u2082 : Buckets.WF m.buckets\ni : Nat\nh\u271d :\n  i <\n    Array.size\n      { val := Array.map (AssocList.mapVal f) m.buckets.val,\n          property := (_ : 0 < Array.size (Array.map (AssocList.mapVal f) m.buckets.val)) }.val\nh : i < Array.size m.buckets.val\n\u22a2 \u2200 (a : \u03b1 \u00d7 \u03b2),\n    a \u2208 AssocList.toList m.buckets.val[i] \u2192 USize.toNat (UInt64.toUSize (hash a.fst) % Array.size m.buckets.val) = i"}, {"tactic": "intro a m", "annotated_tactic": ["intro a m", []], "state_before": "case refine_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nH : WF m\nh\u2081 : m.size = Buckets.size m.buckets\nh\u2082 : Buckets.WF m.buckets\ni : Nat\nh\u271d :\n  i <\n    Array.size\n      { val := Array.map (AssocList.mapVal f) m.buckets.val,\n          property := (_ : 0 < Array.size (Array.map (AssocList.mapVal f) m.buckets.val)) }.val\nh : i < Array.size m.buckets.val\n\u22a2 \u2200 (a : \u03b1 \u00d7 \u03b2),\n    a \u2208 AssocList.toList m.buckets.val[i] \u2192 USize.toNat (UInt64.toUSize (hash a.fst) % Array.size m.buckets.val) = i", "state_after": "case refine_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm\u271d : Imp \u03b1 \u03b2\nH : WF m\u271d\nh\u2081 : m\u271d.size = Buckets.size m\u271d.buckets\nh\u2082 : Buckets.WF m\u271d.buckets\ni : Nat\nh\u271d :\n  i <\n    Array.size\n      { val := Array.map (AssocList.mapVal f) m\u271d.buckets.val,\n          property := (_ : 0 < Array.size (Array.map (AssocList.mapVal f) m\u271d.buckets.val)) }.val\nh : i < Array.size m\u271d.buckets.val\na : \u03b1 \u00d7 \u03b2\nm : a \u2208 AssocList.toList m\u271d.buckets.val[i]\n\u22a2 USize.toNat (UInt64.toUSize (hash a.fst) % Array.size m\u271d.buckets.val) = i"}, {"tactic": "apply h\u2082.2 _ _ _ m", "annotated_tactic": ["apply h\u2082.2 _ _ _ m", []], "state_before": "case refine_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm\u271d : Imp \u03b1 \u03b2\nH : WF m\u271d\nh\u2081 : m\u271d.size = Buckets.size m\u271d.buckets\nh\u2082 : Buckets.WF m\u271d.buckets\ni : Nat\nh\u271d :\n  i <\n    Array.size\n      { val := Array.map (AssocList.mapVal f) m\u271d.buckets.val,\n          property := (_ : 0 < Array.size (Array.map (AssocList.mapVal f) m\u271d.buckets.val)) }.val\nh : i < Array.size m\u271d.buckets.val\na : \u03b1 \u00d7 \u03b2\nm : a \u2208 AssocList.toList m\u271d.buckets.val[i]\n\u22a2 USize.toNat (UInt64.toUSize (hash a.fst) % Array.size m\u271d.buckets.val) = i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/WithDensity.lean", "full_name": "MeasureTheory.withDensity_indicator", "start": [150, 1], "end": [154, 28], "traced_tactics": [{"tactic": "ext1 t ht", "annotated_tactic": ["ext1 t ht", []], "state_before": "\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u22a2 withDensity \u03bc (indicator s f) = withDensity (restrict \u03bc s) f", "state_after": "case h\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nf : \u03b1 \u2192 \u211d\u22650\u221e\nt : Set \u03b1\nht : MeasurableSet t\n\u22a2 \u2191\u2191(withDensity \u03bc (indicator s f)) t = \u2191\u2191(withDensity (restrict \u03bc s) f) t"}, {"tactic": "rw [withDensity_apply _ ht, lintegral_indicator _ hs, restrict_comm hs, \u2190\n  withDensity_apply _ ht]", "annotated_tactic": ["rw [<a>withDensity_apply</a> _ ht, <a>lintegral_indicator</a> _ hs, <a>restrict_comm</a> hs, \u2190\n    <a>withDensity_apply</a> _ ht]", [{"full_name": "MeasureTheory.withDensity_apply", "def_path": "Mathlib/MeasureTheory/Measure/WithDensity.lean", "def_pos": [39, 9], "def_end_pos": [39, 26]}, {"full_name": "MeasureTheory.lintegral_indicator", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [762, 9], "def_end_pos": [762, 28]}, {"full_name": "MeasureTheory.Measure.restrict_comm", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1665, 9], "def_end_pos": [1665, 22]}, {"full_name": "MeasureTheory.withDensity_apply", "def_path": "Mathlib/MeasureTheory/Measure/WithDensity.lean", "def_pos": [39, 9], "def_end_pos": [39, 26]}]], "state_before": "case h\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nf : \u03b1 \u2192 \u211d\u22650\u221e\nt : Set \u03b1\nht : MeasurableSet t\n\u22a2 \u2191\u2191(withDensity \u03bc (indicator s f)) t = \u2191\u2191(withDensity (restrict \u03bc s) f) t", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "full_name": "borel_eq_generateFrom_of_subbasis", "start": [80, 1], "end": [95, 95], "traced_tactics": [{"tactic": "induction hu", "annotated_tactic": ["induction hu", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u\u271d : Set \u03b1\ns : Set (Set \u03b1)\nt : TopologicalSpace \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nhs : t = TopologicalSpace.generateFrom s\nu : Set \u03b1\nhu : TopologicalSpace.IsOpen u\n\u22a2 MeasurableSet u", "state_after": "case basic\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d\u00b9 t\u271d u\u271d : Set \u03b1\ns : Set (Set \u03b1)\nt : TopologicalSpace \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nhs : t = TopologicalSpace.generateFrom s\nu s\u271d : Set \u03b1\na\u271d : s\u271d \u2208 s\n\u22a2 MeasurableSet s\u271d\n\ncase univ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u\u271d : Set \u03b1\ns : Set (Set \u03b1)\nt : TopologicalSpace \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nhs : t = TopologicalSpace.generateFrom s\nu : Set \u03b1\n\u22a2 MeasurableSet univ\n\ncase inter\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d\u00b9 t\u271d\u00b9 u\u271d : Set \u03b1\ns : Set (Set \u03b1)\nt : TopologicalSpace \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nhs : t = TopologicalSpace.generateFrom s\nu s\u271d t\u271d : Set \u03b1\na\u271d\u00b9 : GenerateOpen s s\u271d\na\u271d : GenerateOpen s t\u271d\na_ih\u271d\u00b9 : MeasurableSet s\u271d\na_ih\u271d : MeasurableSet t\u271d\n\u22a2 MeasurableSet (s\u271d \u2229 t\u271d)\n\ncase sUnion\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u\u271d : Set \u03b1\ns : Set (Set \u03b1)\nt : TopologicalSpace \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nhs : t = TopologicalSpace.generateFrom s\nu : Set \u03b1\nS\u271d : Set (Set \u03b1)\na\u271d : \u2200 (s_1 : Set \u03b1), s_1 \u2208 S\u271d \u2192 GenerateOpen s s_1\na_ih\u271d : \u2200 (s_1 : Set \u03b1), s_1 \u2208 S\u271d \u2192 MeasurableSet s_1\n\u22a2 MeasurableSet (\u22c3\u2080 S\u271d)"}, {"tactic": "case basic u hu => exact GenerateMeasurable.basic u hu", "annotated_tactic": ["case basic u hu => exact <a>GenerateMeasurable.basic</a> u hu", [{"full_name": "MeasurableSpace.GenerateMeasurable.basic", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [355, 15], "def_end_pos": [355, 20]}]], "state_before": "case basic\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d\u00b9 t\u271d u\u271d : Set \u03b1\ns : Set (Set \u03b1)\nt : TopologicalSpace \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nhs : t = TopologicalSpace.generateFrom s\nu s\u271d : Set \u03b1\na\u271d : s\u271d \u2208 s\n\u22a2 MeasurableSet s\u271d\n\ncase univ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u\u271d : Set \u03b1\ns : Set (Set \u03b1)\nt : TopologicalSpace \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nhs : t = TopologicalSpace.generateFrom s\nu : Set \u03b1\n\u22a2 MeasurableSet univ\n\ncase inter\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d\u00b9 t\u271d\u00b9 u\u271d : Set \u03b1\ns : Set (Set \u03b1)\nt : TopologicalSpace \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nhs : t = TopologicalSpace.generateFrom s\nu s\u271d t\u271d : Set \u03b1\na\u271d\u00b9 : GenerateOpen s s\u271d\na\u271d : GenerateOpen s t\u271d\na_ih\u271d\u00b9 : MeasurableSet s\u271d\na_ih\u271d : MeasurableSet t\u271d\n\u22a2 MeasurableSet (s\u271d \u2229 t\u271d)\n\ncase sUnion\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u\u271d : Set \u03b1\ns : Set (Set \u03b1)\nt : TopologicalSpace \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nhs : t = TopologicalSpace.generateFrom s\nu : Set \u03b1\nS\u271d : Set (Set \u03b1)\na\u271d : \u2200 (s_1 : Set \u03b1), s_1 \u2208 S\u271d \u2192 GenerateOpen s s_1\na_ih\u271d : \u2200 (s_1 : Set \u03b1), s_1 \u2208 S\u271d \u2192 MeasurableSet s_1\n\u22a2 MeasurableSet (\u22c3\u2080 S\u271d)", "state_after": "case univ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u\u271d : Set \u03b1\ns : Set (Set \u03b1)\nt : TopologicalSpace \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nhs : t = TopologicalSpace.generateFrom s\nu : Set \u03b1\n\u22a2 MeasurableSet univ\n\ncase inter\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d\u00b9 t\u271d\u00b9 u\u271d : Set \u03b1\ns : Set (Set \u03b1)\nt : TopologicalSpace \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nhs : t = TopologicalSpace.generateFrom s\nu s\u271d t\u271d : Set \u03b1\na\u271d\u00b9 : GenerateOpen s s\u271d\na\u271d : GenerateOpen s t\u271d\na_ih\u271d\u00b9 : MeasurableSet s\u271d\na_ih\u271d : MeasurableSet t\u271d\n\u22a2 MeasurableSet (s\u271d \u2229 t\u271d)\n\ncase sUnion\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u\u271d : Set \u03b1\ns : Set (Set \u03b1)\nt : TopologicalSpace \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nhs : t = TopologicalSpace.generateFrom s\nu : Set \u03b1\nS\u271d : Set (Set \u03b1)\na\u271d : \u2200 (s_1 : Set \u03b1), s_1 \u2208 S\u271d \u2192 GenerateOpen s s_1\na_ih\u271d : \u2200 (s_1 : Set \u03b1), s_1 \u2208 S\u271d \u2192 MeasurableSet s_1\n\u22a2 MeasurableSet (\u22c3\u2080 S\u271d)"}, {"tactic": "case univ => exact @MeasurableSet.univ \u03b1 (generateFrom s)", "annotated_tactic": ["case univ => exact @<a>MeasurableSet.univ</a> \u03b1 (<a>generateFrom</a> s)", [{"full_name": "MeasurableSet.univ", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [101, 19], "def_end_pos": [101, 37]}, {"full_name": "MeasurableSpace.generateFrom", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [363, 5], "def_end_pos": [363, 17]}]], "state_before": "case univ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u\u271d : Set \u03b1\ns : Set (Set \u03b1)\nt : TopologicalSpace \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nhs : t = TopologicalSpace.generateFrom s\nu : Set \u03b1\n\u22a2 MeasurableSet univ\n\ncase inter\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d\u00b9 t\u271d\u00b9 u\u271d : Set \u03b1\ns : Set (Set \u03b1)\nt : TopologicalSpace \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nhs : t = TopologicalSpace.generateFrom s\nu s\u271d t\u271d : Set \u03b1\na\u271d\u00b9 : GenerateOpen s s\u271d\na\u271d : GenerateOpen s t\u271d\na_ih\u271d\u00b9 : MeasurableSet s\u271d\na_ih\u271d : MeasurableSet t\u271d\n\u22a2 MeasurableSet (s\u271d \u2229 t\u271d)\n\ncase sUnion\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u\u271d : Set \u03b1\ns : Set (Set \u03b1)\nt : TopologicalSpace \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nhs : t = TopologicalSpace.generateFrom s\nu : Set \u03b1\nS\u271d : Set (Set \u03b1)\na\u271d : \u2200 (s_1 : Set \u03b1), s_1 \u2208 S\u271d \u2192 GenerateOpen s s_1\na_ih\u271d : \u2200 (s_1 : Set \u03b1), s_1 \u2208 S\u271d \u2192 MeasurableSet s_1\n\u22a2 MeasurableSet (\u22c3\u2080 S\u271d)", "state_after": "case inter\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d\u00b9 t\u271d\u00b9 u\u271d : Set \u03b1\ns : Set (Set \u03b1)\nt : TopologicalSpace \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nhs : t = TopologicalSpace.generateFrom s\nu s\u271d t\u271d : Set \u03b1\na\u271d\u00b9 : GenerateOpen s s\u271d\na\u271d : GenerateOpen s t\u271d\na_ih\u271d\u00b9 : MeasurableSet s\u271d\na_ih\u271d : MeasurableSet t\u271d\n\u22a2 MeasurableSet (s\u271d \u2229 t\u271d)\n\ncase sUnion\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u\u271d : Set \u03b1\ns : Set (Set \u03b1)\nt : TopologicalSpace \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nhs : t = TopologicalSpace.generateFrom s\nu : Set \u03b1\nS\u271d : Set (Set \u03b1)\na\u271d : \u2200 (s_1 : Set \u03b1), s_1 \u2208 S\u271d \u2192 GenerateOpen s s_1\na_ih\u271d : \u2200 (s_1 : Set \u03b1), s_1 \u2208 S\u271d \u2192 MeasurableSet s_1\n\u22a2 MeasurableSet (\u22c3\u2080 S\u271d)"}, {"tactic": "case inter s\u2081 s\u2082 _ _ hs\u2081 hs\u2082 => exact @MeasurableSet.inter \u03b1 (generateFrom s) _ _ hs\u2081 hs\u2082", "annotated_tactic": ["case inter s\u2081 s\u2082 _ _ hs\u2081 hs\u2082 => exact @<a>MeasurableSet.inter</a> \u03b1 (<a>generateFrom</a> s) _ _ hs\u2081 hs\u2082", [{"full_name": "MeasurableSet.inter", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [198, 19], "def_end_pos": [198, 38]}, {"full_name": "MeasurableSpace.generateFrom", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [363, 5], "def_end_pos": [363, 17]}]], "state_before": "case inter\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d\u00b9 t\u271d\u00b9 u\u271d : Set \u03b1\ns : Set (Set \u03b1)\nt : TopologicalSpace \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nhs : t = TopologicalSpace.generateFrom s\nu s\u271d t\u271d : Set \u03b1\na\u271d\u00b9 : GenerateOpen s s\u271d\na\u271d : GenerateOpen s t\u271d\na_ih\u271d\u00b9 : MeasurableSet s\u271d\na_ih\u271d : MeasurableSet t\u271d\n\u22a2 MeasurableSet (s\u271d \u2229 t\u271d)\n\ncase sUnion\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u\u271d : Set \u03b1\ns : Set (Set \u03b1)\nt : TopologicalSpace \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nhs : t = TopologicalSpace.generateFrom s\nu : Set \u03b1\nS\u271d : Set (Set \u03b1)\na\u271d : \u2200 (s_1 : Set \u03b1), s_1 \u2208 S\u271d \u2192 GenerateOpen s s_1\na_ih\u271d : \u2200 (s_1 : Set \u03b1), s_1 \u2208 S\u271d \u2192 MeasurableSet s_1\n\u22a2 MeasurableSet (\u22c3\u2080 S\u271d)", "state_after": "case sUnion\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u\u271d : Set \u03b1\ns : Set (Set \u03b1)\nt : TopologicalSpace \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nhs : t = TopologicalSpace.generateFrom s\nu : Set \u03b1\nS\u271d : Set (Set \u03b1)\na\u271d : \u2200 (s_1 : Set \u03b1), s_1 \u2208 S\u271d \u2192 GenerateOpen s s_1\na_ih\u271d : \u2200 (s_1 : Set \u03b1), s_1 \u2208 S\u271d \u2192 MeasurableSet s_1\n\u22a2 MeasurableSet (\u22c3\u2080 S\u271d)"}, {"tactic": "case\n  sUnion f hf ih =>\n  rcases isOpen_sUnion_countable f (by rwa [hs]) with \u27e8v, hv, vf, vu\u27e9\n  rw [\u2190 vu]\n  exact @MeasurableSet.sUnion \u03b1 (generateFrom s) _ hv fun x xv => ih _ (vf xv)", "annotated_tactic": ["case\n        sUnion f hf ih =>\n        rcases <a>isOpen_sUnion_countable</a> f (by rwa [hs]) with \u27e8v, hv, vf, vu\u27e9\n        rw [\u2190 vu]\n        exact @<a>MeasurableSet.sUnion</a> \u03b1 (<a>generateFrom</a> s) _ hv fun x xv => ih _ (vf xv)", [{"full_name": "TopologicalSpace.isOpen_sUnion_countable", "def_path": "Mathlib/Topology/Bases.lean", "def_pos": [805, 9], "def_end_pos": [805, 32]}, {"full_name": "MeasurableSet.sUnion", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [147, 19], "def_end_pos": [147, 39]}, {"full_name": "MeasurableSpace.generateFrom", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [363, 5], "def_end_pos": [363, 17]}]], "state_before": "case sUnion\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u\u271d : Set \u03b1\ns : Set (Set \u03b1)\nt : TopologicalSpace \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nhs : t = TopologicalSpace.generateFrom s\nu : Set \u03b1\nS\u271d : Set (Set \u03b1)\na\u271d : \u2200 (s_1 : Set \u03b1), s_1 \u2208 S\u271d \u2192 GenerateOpen s s_1\na_ih\u271d : \u2200 (s_1 : Set \u03b1), s_1 \u2208 S\u271d \u2192 MeasurableSet s_1\n\u22a2 MeasurableSet (\u22c3\u2080 S\u271d)", "state_after": "no goals"}, {"tactic": "exact GenerateMeasurable.basic u hu", "annotated_tactic": ["exact <a>GenerateMeasurable.basic</a> u hu", [{"full_name": "MeasurableSpace.GenerateMeasurable.basic", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [355, 15], "def_end_pos": [355, 20]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u\u271d\u00b9 : Set \u03b1\ns : Set (Set \u03b1)\nt : TopologicalSpace \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nhs : t = TopologicalSpace.generateFrom s\nu\u271d u : Set \u03b1\nhu : u \u2208 s\n\u22a2 MeasurableSet u", "state_after": "no goals"}, {"tactic": "exact @MeasurableSet.univ \u03b1 (generateFrom s)", "annotated_tactic": ["exact @<a>MeasurableSet.univ</a> \u03b1 (<a>generateFrom</a> s)", [{"full_name": "MeasurableSet.univ", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [101, 19], "def_end_pos": [101, 37]}, {"full_name": "MeasurableSpace.generateFrom", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [363, 5], "def_end_pos": [363, 17]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u\u271d : Set \u03b1\ns : Set (Set \u03b1)\nt : TopologicalSpace \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nhs : t = TopologicalSpace.generateFrom s\nu : Set \u03b1\n\u22a2 MeasurableSet univ", "state_after": "no goals"}, {"tactic": "exact @MeasurableSet.inter \u03b1 (generateFrom s) _ _ hs\u2081 hs\u2082", "annotated_tactic": ["exact @<a>MeasurableSet.inter</a> \u03b1 (<a>generateFrom</a> s) _ _ hs\u2081 hs\u2082", [{"full_name": "MeasurableSet.inter", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [198, 19], "def_end_pos": [198, 38]}, {"full_name": "MeasurableSpace.generateFrom", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [363, 5], "def_end_pos": [363, 17]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u\u271d : Set \u03b1\ns : Set (Set \u03b1)\nt : TopologicalSpace \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nhs : t = TopologicalSpace.generateFrom s\nu s\u2081 s\u2082 : Set \u03b1\na\u271d\u00b9 : GenerateOpen s s\u2081\na\u271d : GenerateOpen s s\u2082\nhs\u2081 : MeasurableSet s\u2081\nhs\u2082 : MeasurableSet s\u2082\n\u22a2 MeasurableSet (s\u2081 \u2229 s\u2082)", "state_after": "no goals"}, {"tactic": "rcases isOpen_sUnion_countable f (by rwa [hs]) with \u27e8v, hv, vf, vu\u27e9", "annotated_tactic": ["rcases <a>isOpen_sUnion_countable</a> f (by rwa [hs]) with \u27e8v, hv, vf, vu\u27e9", [{"full_name": "TopologicalSpace.isOpen_sUnion_countable", "def_path": "Mathlib/Topology/Bases.lean", "def_pos": [805, 9], "def_end_pos": [805, 32]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u\u271d : Set \u03b1\ns : Set (Set \u03b1)\nt : TopologicalSpace \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nhs : t = TopologicalSpace.generateFrom s\nu : Set \u03b1\nf : Set (Set \u03b1)\nhf : \u2200 (s_1 : Set \u03b1), s_1 \u2208 f \u2192 GenerateOpen s s_1\nih : \u2200 (s_1 : Set \u03b1), s_1 \u2208 f \u2192 MeasurableSet s_1\n\u22a2 MeasurableSet (\u22c3\u2080 f)", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u\u271d : Set \u03b1\ns : Set (Set \u03b1)\nt : TopologicalSpace \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nhs : t = TopologicalSpace.generateFrom s\nu : Set \u03b1\nf : Set (Set \u03b1)\nhf : \u2200 (s_1 : Set \u03b1), s_1 \u2208 f \u2192 GenerateOpen s s_1\nih : \u2200 (s_1 : Set \u03b1), s_1 \u2208 f \u2192 MeasurableSet s_1\nv : Set (Set \u03b1)\nhv : Set.Countable v\nvf : v \u2286 f\nvu : \u22c3\u2080 v = \u22c3\u2080 f\n\u22a2 MeasurableSet (\u22c3\u2080 f)"}, {"tactic": "rw [\u2190 vu]", "annotated_tactic": ["rw [\u2190 vu]", []], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u\u271d : Set \u03b1\ns : Set (Set \u03b1)\nt : TopologicalSpace \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nhs : t = TopologicalSpace.generateFrom s\nu : Set \u03b1\nf : Set (Set \u03b1)\nhf : \u2200 (s_1 : Set \u03b1), s_1 \u2208 f \u2192 GenerateOpen s s_1\nih : \u2200 (s_1 : Set \u03b1), s_1 \u2208 f \u2192 MeasurableSet s_1\nv : Set (Set \u03b1)\nhv : Set.Countable v\nvf : v \u2286 f\nvu : \u22c3\u2080 v = \u22c3\u2080 f\n\u22a2 MeasurableSet (\u22c3\u2080 f)", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u\u271d : Set \u03b1\ns : Set (Set \u03b1)\nt : TopologicalSpace \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nhs : t = TopologicalSpace.generateFrom s\nu : Set \u03b1\nf : Set (Set \u03b1)\nhf : \u2200 (s_1 : Set \u03b1), s_1 \u2208 f \u2192 GenerateOpen s s_1\nih : \u2200 (s_1 : Set \u03b1), s_1 \u2208 f \u2192 MeasurableSet s_1\nv : Set (Set \u03b1)\nhv : Set.Countable v\nvf : v \u2286 f\nvu : \u22c3\u2080 v = \u22c3\u2080 f\n\u22a2 MeasurableSet (\u22c3\u2080 v)"}, {"tactic": "exact @MeasurableSet.sUnion \u03b1 (generateFrom s) _ hv fun x xv => ih _ (vf xv)", "annotated_tactic": ["exact @<a>MeasurableSet.sUnion</a> \u03b1 (<a>generateFrom</a> s) _ hv fun x xv => ih _ (vf xv)", [{"full_name": "MeasurableSet.sUnion", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [147, 19], "def_end_pos": [147, 39]}, {"full_name": "MeasurableSpace.generateFrom", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [363, 5], "def_end_pos": [363, 17]}]], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u\u271d : Set \u03b1\ns : Set (Set \u03b1)\nt : TopologicalSpace \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nhs : t = TopologicalSpace.generateFrom s\nu : Set \u03b1\nf : Set (Set \u03b1)\nhf : \u2200 (s_1 : Set \u03b1), s_1 \u2208 f \u2192 GenerateOpen s s_1\nih : \u2200 (s_1 : Set \u03b1), s_1 \u2208 f \u2192 MeasurableSet s_1\nv : Set (Set \u03b1)\nhv : Set.Countable v\nvf : v \u2286 f\nvu : \u22c3\u2080 v = \u22c3\u2080 f\n\u22a2 MeasurableSet (\u22c3\u2080 v)", "state_after": "no goals"}, {"tactic": "rwa [hs]", "annotated_tactic": ["rwa [hs]", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u\u271d : Set \u03b1\ns : Set (Set \u03b1)\nt : TopologicalSpace \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nhs : t = TopologicalSpace.generateFrom s\nu : Set \u03b1\nf : Set (Set \u03b1)\nhf : \u2200 (s_1 : Set \u03b1), s_1 \u2208 f \u2192 GenerateOpen s s_1\nih : \u2200 (s_1 : Set \u03b1), s_1 \u2208 f \u2192 MeasurableSet s_1\n\u22a2 \u2200 (s : Set \u03b1), s \u2208 f \u2192 IsOpen s", "state_after": "no goals"}, {"tactic": "exact GenerateOpen.basic _ hu", "annotated_tactic": ["exact <a>GenerateOpen.basic</a> _ hu", [{"full_name": "TopologicalSpace.GenerateOpen.basic", "def_path": "Mathlib/Topology/Order.lean", "def_pos": [61, 5], "def_end_pos": [61, 10]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u\u271d : Set \u03b1\ns : Set (Set \u03b1)\nt : TopologicalSpace \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nhs : t = TopologicalSpace.generateFrom s\nu : Set \u03b1\nhu : u \u2208 s\n\u22a2 TopologicalSpace.IsOpen u", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/LocallyFinite.lean", "full_name": "Finset.mem_uIcc_of_ge", "start": [966, 1], "end": [967, 41], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "Finset.insert_union_distrib", "start": [1498, 1], "end": [1500, 54], "traced_tactics": [{"tactic": "simp only [insert_union, union_insert, insert_idem]", "annotated_tactic": ["simp only [<a>insert_union</a>, <a>union_insert</a>, <a>insert_idem</a>]", [{"full_name": "Finset.insert_union", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1489, 9], "def_end_pos": [1489, 21]}, {"full_name": "Finset.union_insert", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1494, 9], "def_end_pos": [1494, 21]}, {"full_name": "Finset.insert_idem", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1164, 9], "def_end_pos": [1164, 20]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns\u271d s\u2081 s\u2082 t\u271d t\u2081 t\u2082 u v : Finset \u03b1\na\u271d b a : \u03b1\ns t : Finset \u03b1\n\u22a2 insert a (s \u222a t) = insert a s \u222a insert a t", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Kernel/CondCdf.lean", "full_name": "ProbabilityTheory.tendsto_preCdf_atTop_one", "start": [373, 1], "end": [440, 47], "traced_tactics": [{"tactic": "have h_mono := monotone_preCdf \u03c1", "annotated_tactic": ["have h_mono := <a>monotone_preCdf</a> \u03c1", [{"full_name": "ProbabilityTheory.monotone_preCdf", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [318, 9], "def_end_pos": [318, 24]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd 1)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd 1)"}, {"tactic": "have h_le_one := preCdf_le_one \u03c1", "annotated_tactic": ["have h_le_one := <a>preCdf_le_one</a> \u03c1", [{"full_name": "ProbabilityTheory.preCdf_le_one", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [349, 9], "def_end_pos": [349, 22]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd 1)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\nh_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd 1)"}, {"tactic": "classical\nlet F : \u03b1 \u2192 \u211d\u22650\u221e := fun a =>\n  if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then h.choose else 0\nhave h_tendsto_\u211a : \u2200\u1d50 a \u2202\u03c1.fst, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a)) := by\n  filter_upwards [h_exists] with a ha\n  simp_rw [dif_pos ha]\n  exact ha.choose_spec\nhave h_tendsto_\u2115 : \u2200\u1d50 a \u2202\u03c1.fst, Tendsto (fun n : \u2115 => preCdf \u03c1 n a) atTop (\ud835\udcdd (F a)) := by\n  filter_upwards [h_tendsto_\u211a] with a ha using ha.comp tendsto_nat_cast_atTop_atTop\nhave hF_ae_meas : AEMeasurable F \u03c1.fst := by\n  refine' aemeasurable_of_tendsto_metrizable_ae _ (fun n => _) h_tendsto_\u211a\n  exact measurable_preCdf.aemeasurable\nhave hF_le_one : \u2200\u1d50 a \u2202\u03c1.fst, F a \u2264 1 := by\n  filter_upwards [h_tendsto_\u211a, h_le_one] with a ha ha_le using le_of_tendsto' ha ha_le\nsuffices \u2200\u1d50 a \u2202\u03c1.fst, F a = 1 by\n  filter_upwards [h_tendsto_\u211a, this] with a ha_tendsto ha_eq\n  rwa [ha_eq] at ha_tendsto\nhave h_lintegral_eq : \u222b\u207b a, F a \u2202\u03c1.fst = \u222b\u207b _, 1 \u2202\u03c1.fst := by\n  have h_lintegral :\n    Tendsto (fun r : \u2115 => \u222b\u207b a, preCdf \u03c1 r a \u2202\u03c1.fst) atTop (\ud835\udcdd (\u222b\u207b a, F a \u2202\u03c1.fst)) := by\n    refine'\n      lintegral_tendsto_of_tendsto_of_monotone\n        (fun _ => measurable_preCdf.aemeasurable)\n        _ h_tendsto_\u2115\n    filter_upwards [h_mono] with a ha\n    refine' fun n m hnm => ha _\n    exact_mod_cast hnm\n  have h_lintegral' :\n    Tendsto (fun r : \u2115 => \u222b\u207b a, preCdf \u03c1 r a \u2202\u03c1.fst) atTop (\ud835\udcdd (\u222b\u207b _, 1 \u2202\u03c1.fst)) := by\n    rw [lintegral_one, Measure.fst_univ]\n    exact (tendsto_lintegral_preCdf_atTop \u03c1).comp tendsto_nat_cast_atTop_atTop\n  exact tendsto_nhds_unique h_lintegral h_lintegral'\nhave : \u222b\u207b a, 1 - F a \u2202\u03c1.fst = 0 := by\n  rw [lintegral_sub' hF_ae_meas _ hF_le_one, h_lintegral_eq, tsub_self]\n  calc\n    \u222b\u207b a, F a \u2202\u03c1.fst = \u222b\u207b _, 1 \u2202\u03c1.fst := h_lintegral_eq\n    _ = \u03c1.fst univ := lintegral_one\n    _ = \u03c1 univ := Measure.fst_univ\n    _ \u2260 \u221e := measure_ne_top \u03c1 _\nrw [lintegral_eq_zero_iff' (aemeasurable_const.sub hF_ae_meas)] at this\nfilter_upwards [this, hF_le_one] with ha h_one_sub_eq_zero h_le_one\nrw [Pi.zero_apply, tsub_eq_zero_iff_le] at h_one_sub_eq_zero\nexact le_antisymm h_le_one h_one_sub_eq_zero", "annotated_tactic": ["classical\n  -- let `F` be the pointwise limit of `preCdf` where it exists, and 0 elsewhere.\n  let F : \u03b1 \u2192 \u211d\u22650\u221e := fun a =>\n    if h : \u2203 l, <a>Tendsto</a> (fun r => <a>preCdf</a> \u03c1 r a) <a>atTop</a> (\ud835\udcdd l) then h.choose else 0\n  have h_tendsto_\u211a : \u2200\u1d50 a \u2202\u03c1.fst, <a>Tendsto</a> (fun r => <a>preCdf</a> \u03c1 r a) <a>atTop</a> (\ud835\udcdd (F a)) := by\n    filter_upwards [h_exists] with a ha\n    simp_rw [<a>dif_pos</a> ha]\n    exact ha.choose_spec\n  have h_tendsto_\u2115 : \u2200\u1d50 a \u2202\u03c1.fst, <a>Tendsto</a> (fun n : \u2115 => <a>preCdf</a> \u03c1 n a) <a>atTop</a> (\ud835\udcdd (F a)) := by\n    filter_upwards [h_tendsto_\u211a] with a ha using ha.comp <a>tendsto_nat_cast_atTop_atTop</a>\n  have hF_ae_meas : <a>AEMeasurable</a> F \u03c1.fst := by\n    refine' <a>aemeasurable_of_tendsto_metrizable_ae</a> _ (fun n => _) h_tendsto_\u211a\n    exact measurable_preCdf.aemeasurable\n  have hF_le_one : \u2200\u1d50 a \u2202\u03c1.fst, F a \u2264 1 := by\n    filter_upwards [h_tendsto_\u211a, h_le_one] with a ha ha_le using <a>le_of_tendsto'</a> ha ha_le\n  -- it suffices to show that the limit `F` is 1 a.e.\n  suffices \u2200\u1d50 a \u2202\u03c1.fst, F a = 1 by\n    filter_upwards [h_tendsto_\u211a, this] with a ha_tendsto ha_eq\n    rwa [ha_eq] at ha_tendsto\n  -- since `F` is at most 1, proving that its integral is the same as the integral of 1 will tell\n  -- us that `F` is 1 a.e.\n  have h_lintegral_eq : \u222b\u207b a, F a \u2202\u03c1.fst = \u222b\u207b _, 1 \u2202\u03c1.fst := by\n    have h_lintegral :\n      <a>Tendsto</a> (fun r : \u2115 => \u222b\u207b a, <a>preCdf</a> \u03c1 r a \u2202\u03c1.fst) <a>atTop</a> (\ud835\udcdd (\u222b\u207b a, F a \u2202\u03c1.fst)) := by\n      refine'\n        <a>lintegral_tendsto_of_tendsto_of_monotone</a>\n          (-- does this exist only for \u2115?\n          fun _ => measurable_preCdf.aemeasurable)\n          _ h_tendsto_\u2115\n      filter_upwards [h_mono] with a ha\n      refine' fun n m hnm => ha _\n      exact_mod_cast hnm\n    have h_lintegral' :\n      <a>Tendsto</a> (fun r : \u2115 => \u222b\u207b a, <a>preCdf</a> \u03c1 r a \u2202\u03c1.fst) <a>atTop</a> (\ud835\udcdd (\u222b\u207b _, 1 \u2202\u03c1.fst)) := by\n      rw [<a>lintegral_one</a>, <a>Measure.fst_univ</a>]\n      exact (<a>tendsto_lintegral_preCdf_atTop</a> \u03c1).<a>comp</a> <a>tendsto_nat_cast_atTop_atTop</a>\n    exact <a>tendsto_nhds_unique</a> h_lintegral h_lintegral'\n  have : \u222b\u207b a, 1 - F a \u2202\u03c1.fst = 0 := by\n    rw [<a>lintegral_sub'</a> hF_ae_meas _ hF_le_one, h_lintegral_eq, <a>tsub_self</a>]\n    calc\n      \u222b\u207b a, F a \u2202\u03c1.fst = \u222b\u207b _, 1 \u2202\u03c1.fst := h_lintegral_eq\n      _ = \u03c1.fst <a>univ</a> := <a>lintegral_one</a>\n      _ = \u03c1 <a>univ</a> := <a>Measure.fst_univ</a>\n      _ \u2260 \u221e := <a>measure_ne_top</a> \u03c1 _\n  rw [<a>lintegral_eq_zero_iff'</a> (aemeasurable_const.sub hF_ae_meas)] at this\n  filter_upwards [this, hF_le_one] with ha h_one_sub_eq_zero h_le_one\n  rw [<a>Pi.zero_apply</a>, <a>tsub_eq_zero_iff_le</a>] at h_one_sub_eq_zero\n  exact <a>le_antisymm</a> h_le_one h_one_sub_eq_zero", [{"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2939, 5], "def_end_pos": [2939, 12]}, {"full_name": "ProbabilityTheory.preCdf", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [294, 19], "def_end_pos": [294, 25]}, {"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}, {"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2939, 5], "def_end_pos": [2939, 12]}, {"full_name": "ProbabilityTheory.preCdf", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [294, 19], "def_end_pos": [294, 25]}, {"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}, {"full_name": "dif_pos", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [807, 9], "def_end_pos": [807, 16]}, {"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2939, 5], "def_end_pos": [2939, 12]}, {"full_name": "ProbabilityTheory.preCdf", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [294, 19], "def_end_pos": [294, 25]}, {"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}, {"full_name": "tendsto_nat_cast_atTop_atTop", "def_path": "Mathlib/Order/Filter/Archimedean.lean", "def_pos": [37, 9], "def_end_pos": [37, 37]}, {"full_name": "AEMeasurable", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [708, 5], "def_end_pos": [708, 17]}, {"full_name": "aemeasurable_of_tendsto_metrizable_ae", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Metrizable.lean", "def_pos": [92, 9], "def_end_pos": [92, 46]}, {"full_name": "le_of_tendsto'", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [145, 9], "def_end_pos": [145, 23]}, {"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2939, 5], "def_end_pos": [2939, 12]}, {"full_name": "ProbabilityTheory.preCdf", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [294, 19], "def_end_pos": [294, 25]}, {"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}, {"full_name": "MeasureTheory.lintegral_tendsto_of_tendsto_of_monotone", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [435, 9], "def_end_pos": [435, 49]}, {"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2939, 5], "def_end_pos": [2939, 12]}, {"full_name": "ProbabilityTheory.preCdf", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [294, 19], "def_end_pos": [294, 25]}, {"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}, {"full_name": "MeasureTheory.lintegral_one", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [149, 9], "def_end_pos": [149, 22]}, {"full_name": "MeasureTheory.Measure.fst_univ", "def_path": "Mathlib/MeasureTheory/Constructions/Prod/Basic.lean", "def_pos": [918, 9], "def_end_pos": [918, 17]}, {"full_name": "ProbabilityTheory.tendsto_lintegral_preCdf_atTop", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [360, 9], "def_end_pos": [360, 39]}, {"full_name": "Filter.Tendsto.comp", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [3032, 9], "def_end_pos": [3032, 21]}, {"full_name": "tendsto_nat_cast_atTop_atTop", "def_path": "Mathlib/Order/Filter/Archimedean.lean", "def_pos": [37, 9], "def_end_pos": [37, 37]}, {"full_name": "tendsto_nhds_unique", "def_path": "Mathlib/Topology/Separation.lean", "def_pos": [994, 9], "def_end_pos": [994, 28]}, {"full_name": "MeasureTheory.lintegral_sub'", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [926, 9], "def_end_pos": [926, 23]}, {"full_name": "tsub_self", "def_path": "Mathlib/Algebra/Order/Sub/Canonical.lean", "def_pos": [333, 9], "def_end_pos": [333, 18]}, {"full_name": "Set.univ", "def_path": "Mathlib/Init/Set.lean", "def_pos": [90, 5], "def_end_pos": [90, 9]}, {"full_name": "MeasureTheory.lintegral_one", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [149, 9], "def_end_pos": [149, 22]}, {"full_name": "Set.univ", "def_path": "Mathlib/Init/Set.lean", "def_pos": [90, 5], "def_end_pos": [90, 9]}, {"full_name": "MeasureTheory.Measure.fst_univ", "def_path": "Mathlib/MeasureTheory/Constructions/Prod/Basic.lean", "def_pos": [918, 9], "def_end_pos": [918, 17]}, {"full_name": "MeasureTheory.measure_ne_top", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2875, 9], "def_end_pos": [2875, 23]}, {"full_name": "MeasureTheory.lintegral_eq_zero_iff'", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [888, 9], "def_end_pos": [888, 31]}, {"full_name": "Pi.zero_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [46, 3], "def_end_pos": [46, 14]}, {"full_name": "tsub_eq_zero_iff_le", "def_path": "Mathlib/Algebra/Order/Sub/Canonical.lean", "def_pos": [324, 9], "def_end_pos": [324, 28]}, {"full_name": "le_antisymm", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [188, 9], "def_end_pos": [188, 20]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\nh_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd 1)", "state_after": "no goals"}, {"tactic": "filter_upwards [h_mono, h_le_one] with a ha_mono ha_le_one", "annotated_tactic": ["filter_upwards [h_mono, h_le_one] with a ha_mono ha_le_one", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\nh_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\nh_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\na : \u03b1\nha_mono : Monotone fun r => preCdf \u03c1 r a\nha_le_one : \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\n\u22a2 \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)"}, {"tactic": "have h_tendsto :\n  Tendsto (fun r => preCdf \u03c1 r a) atTop atTop \u2228\n    \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) :=\n  tendsto_of_monotone ha_mono", "annotated_tactic": ["have h_tendsto :\n      <a>Tendsto</a> (fun r => <a>preCdf</a> \u03c1 r a) <a>atTop</a> <a>atTop</a> \u2228\n        \u2203 l, <a>Tendsto</a> (fun r => <a>preCdf</a> \u03c1 r a) <a>atTop</a> (\ud835\udcdd l) :=\n      <a>tendsto_of_monotone</a> ha_mono", [{"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2939, 5], "def_end_pos": [2939, 12]}, {"full_name": "ProbabilityTheory.preCdf", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [294, 19], "def_end_pos": [294, 25]}, {"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}, {"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}, {"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2939, 5], "def_end_pos": [2939, 12]}, {"full_name": "ProbabilityTheory.preCdf", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [294, 19], "def_end_pos": [294, 25]}, {"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}, {"full_name": "tendsto_of_monotone", "def_path": "Mathlib/Topology/Algebra/Order/MonotoneConvergence.lean", "def_pos": [223, 9], "def_end_pos": [223, 28]}]], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\nh_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\na : \u03b1\nha_mono : Monotone fun r => preCdf \u03c1 r a\nha_le_one : \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\n\u22a2 \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\nh_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\na : \u03b1\nha_mono : Monotone fun r => preCdf \u03c1 r a\nha_le_one : \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\nh_tendsto : Tendsto (fun r => preCdf \u03c1 r a) atTop atTop \u2228 \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)\n\u22a2 \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)"}, {"tactic": "cases' h_tendsto with h_absurd h_tendsto", "annotated_tactic": ["cases' h_tendsto with h_absurd h_tendsto", []], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\nh_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\na : \u03b1\nha_mono : Monotone fun r => preCdf \u03c1 r a\nha_le_one : \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\nh_tendsto : Tendsto (fun r => preCdf \u03c1 r a) atTop atTop \u2228 \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)\n\u22a2 \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)", "state_after": "case h.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\nh_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\na : \u03b1\nha_mono : Monotone fun r => preCdf \u03c1 r a\nha_le_one : \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\nh_absurd : Tendsto (fun r => preCdf \u03c1 r a) atTop atTop\n\u22a2 \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)\n\ncase h.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\nh_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\na : \u03b1\nha_mono : Monotone fun r => preCdf \u03c1 r a\nha_le_one : \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\nh_tendsto : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)\n\u22a2 \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)"}, {"tactic": "rw [Monotone.tendsto_atTop_atTop_iff ha_mono] at h_absurd", "annotated_tactic": ["rw [<a>Monotone.tendsto_atTop_atTop_iff</a> ha_mono] at h_absurd", [{"full_name": "Monotone.tendsto_atTop_atTop_iff", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [1384, 7], "def_end_pos": [1384, 46]}]], "state_before": "case h.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\nh_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\na : \u03b1\nha_mono : Monotone fun r => preCdf \u03c1 r a\nha_le_one : \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\nh_absurd : Tendsto (fun r => preCdf \u03c1 r a) atTop atTop\n\u22a2 \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)", "state_after": "case h.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\nh_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\na : \u03b1\nha_mono : Monotone fun r => preCdf \u03c1 r a\nha_le_one : \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\nh_absurd : \u2200 (b : \u211d\u22650\u221e), \u2203 a_1, b \u2264 preCdf \u03c1 a_1 a\n\u22a2 \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)"}, {"tactic": "obtain \u27e8r, hr\u27e9 := h_absurd 2", "annotated_tactic": ["obtain \u27e8r, hr\u27e9 := h_absurd 2", []], "state_before": "case h.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\nh_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\na : \u03b1\nha_mono : Monotone fun r => preCdf \u03c1 r a\nha_le_one : \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\nh_absurd : \u2200 (b : \u211d\u22650\u221e), \u2203 a_1, b \u2264 preCdf \u03c1 a_1 a\n\u22a2 \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)", "state_after": "case h.inl.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\nh_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\na : \u03b1\nha_mono : Monotone fun r => preCdf \u03c1 r a\nha_le_one : \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\nh_absurd : \u2200 (b : \u211d\u22650\u221e), \u2203 a_1, b \u2264 preCdf \u03c1 a_1 a\nr : \u211a\nhr : 2 \u2264 preCdf \u03c1 r a\n\u22a2 \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)"}, {"tactic": "exact absurd (hr.trans (ha_le_one r)) ENNReal.one_lt_two.not_le", "annotated_tactic": ["exact <a>absurd</a> (hr.trans (ha_le_one r)) ENNReal.one_lt_two.not_le", [{"full_name": "absurd", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [233, 21], "def_end_pos": [233, 27]}]], "state_before": "case h.inl.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\nh_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\na : \u03b1\nha_mono : Monotone fun r => preCdf \u03c1 r a\nha_le_one : \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\nh_absurd : \u2200 (b : \u211d\u22650\u221e), \u2203 a_1, b \u2264 preCdf \u03c1 a_1 a\nr : \u211a\nhr : 2 \u2264 preCdf \u03c1 r a\n\u22a2 \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)", "state_after": "no goals"}, {"tactic": "exact h_tendsto", "annotated_tactic": ["exact h_tendsto", []], "state_before": "case h.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\nh_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\na : \u03b1\nha_mono : Monotone fun r => preCdf \u03c1 r a\nha_le_one : \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\nh_tendsto : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)\n\u22a2 \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)", "state_after": "no goals"}, {"tactic": "let F : \u03b1 \u2192 \u211d\u22650\u221e := fun a =>\n  if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then h.choose else 0", "annotated_tactic": ["let F : \u03b1 \u2192 \u211d\u22650\u221e := fun a =>\n    if h : \u2203 l, <a>Tendsto</a> (fun r => <a>preCdf</a> \u03c1 r a) <a>atTop</a> (\ud835\udcdd l) then h.choose else 0", [{"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2939, 5], "def_end_pos": [2939, 12]}, {"full_name": "ProbabilityTheory.preCdf", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [294, 19], "def_end_pos": [294, 25]}, {"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\nh_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd 1)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\nh_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd 1)"}, {"tactic": "have h_tendsto_\u211a : \u2200\u1d50 a \u2202\u03c1.fst, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a)) := by\n  filter_upwards [h_exists] with a ha\n  simp_rw [dif_pos ha]\n  exact ha.choose_spec", "annotated_tactic": ["have h_tendsto_\u211a : \u2200\u1d50 a \u2202\u03c1.fst, <a>Tendsto</a> (fun r => <a>preCdf</a> \u03c1 r a) <a>atTop</a> (\ud835\udcdd (F a)) := by\n    filter_upwards [h_exists] with a ha\n    simp_rw [<a>dif_pos</a> ha]\n    exact ha.choose_spec", [{"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2939, 5], "def_end_pos": [2939, 12]}, {"full_name": "ProbabilityTheory.preCdf", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [294, 19], "def_end_pos": [294, 25]}, {"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}, {"full_name": "dif_pos", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [807, 9], "def_end_pos": [807, 16]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\nh_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd 1)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\nh_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto_\u211a : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a))\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd 1)"}, {"tactic": "have h_tendsto_\u2115 : \u2200\u1d50 a \u2202\u03c1.fst, Tendsto (fun n : \u2115 => preCdf \u03c1 n a) atTop (\ud835\udcdd (F a)) := by\n  filter_upwards [h_tendsto_\u211a] with a ha using ha.comp tendsto_nat_cast_atTop_atTop", "annotated_tactic": ["have h_tendsto_\u2115 : \u2200\u1d50 a \u2202\u03c1.fst, <a>Tendsto</a> (fun n : \u2115 => <a>preCdf</a> \u03c1 n a) <a>atTop</a> (\ud835\udcdd (F a)) := by\n    filter_upwards [h_tendsto_\u211a] with a ha using ha.comp <a>tendsto_nat_cast_atTop_atTop</a>", [{"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2939, 5], "def_end_pos": [2939, 12]}, {"full_name": "ProbabilityTheory.preCdf", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [294, 19], "def_end_pos": [294, 25]}, {"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}, {"full_name": "tendsto_nat_cast_atTop_atTop", "def_path": "Mathlib/Order/Filter/Archimedean.lean", "def_pos": [37, 9], "def_end_pos": [37, 37]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\nh_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto_\u211a : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a))\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd 1)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\nh_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto_\u211a : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a))\nh_tendsto_\u2115 : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun n => preCdf \u03c1 (\u2191n) a) atTop (\ud835\udcdd (F a))\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd 1)"}, {"tactic": "have hF_ae_meas : AEMeasurable F \u03c1.fst := by\n  refine' aemeasurable_of_tendsto_metrizable_ae _ (fun n => _) h_tendsto_\u211a\n  exact measurable_preCdf.aemeasurable", "annotated_tactic": ["have hF_ae_meas : <a>AEMeasurable</a> F \u03c1.fst := by\n    refine' <a>aemeasurable_of_tendsto_metrizable_ae</a> _ (fun n => _) h_tendsto_\u211a\n    exact measurable_preCdf.aemeasurable", [{"full_name": "AEMeasurable", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [708, 5], "def_end_pos": [708, 17]}, {"full_name": "aemeasurable_of_tendsto_metrizable_ae", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Metrizable.lean", "def_pos": [92, 9], "def_end_pos": [92, 46]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\nh_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto_\u211a : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a))\nh_tendsto_\u2115 : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun n => preCdf \u03c1 (\u2191n) a) atTop (\ud835\udcdd (F a))\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd 1)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\nh_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto_\u211a : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a))\nh_tendsto_\u2115 : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun n => preCdf \u03c1 (\u2191n) a) atTop (\ud835\udcdd (F a))\nhF_ae_meas : AEMeasurable F\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd 1)"}, {"tactic": "have hF_le_one : \u2200\u1d50 a \u2202\u03c1.fst, F a \u2264 1 := by\n  filter_upwards [h_tendsto_\u211a, h_le_one] with a ha ha_le using le_of_tendsto' ha ha_le", "annotated_tactic": ["have hF_le_one : \u2200\u1d50 a \u2202\u03c1.fst, F a \u2264 1 := by\n    filter_upwards [h_tendsto_\u211a, h_le_one] with a ha ha_le using <a>le_of_tendsto'</a> ha ha_le", [{"full_name": "le_of_tendsto'", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [145, 9], "def_end_pos": [145, 23]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\nh_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto_\u211a : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a))\nh_tendsto_\u2115 : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun n => preCdf \u03c1 (\u2191n) a) atTop (\ud835\udcdd (F a))\nhF_ae_meas : AEMeasurable F\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd 1)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\nh_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto_\u211a : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a))\nh_tendsto_\u2115 : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun n => preCdf \u03c1 (\u2191n) a) atTop (\ud835\udcdd (F a))\nhF_ae_meas : AEMeasurable F\nhF_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, F a \u2264 1\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd 1)"}, {"tactic": "suffices \u2200\u1d50 a \u2202\u03c1.fst, F a = 1 by\n  filter_upwards [h_tendsto_\u211a, this] with a ha_tendsto ha_eq\n  rwa [ha_eq] at ha_tendsto", "annotated_tactic": ["suffices \u2200\u1d50 a \u2202\u03c1.fst, F a = 1 by\n    filter_upwards [h_tendsto_\u211a, this] with a ha_tendsto ha_eq\n    rwa [ha_eq] at ha_tendsto", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\nh_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto_\u211a : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a))\nh_tendsto_\u2115 : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun n => preCdf \u03c1 (\u2191n) a) atTop (\ud835\udcdd (F a))\nhF_ae_meas : AEMeasurable F\nhF_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, F a \u2264 1\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd 1)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\nh_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto_\u211a : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a))\nh_tendsto_\u2115 : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun n => preCdf \u03c1 (\u2191n) a) atTop (\ud835\udcdd (F a))\nhF_ae_meas : AEMeasurable F\nhF_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, F a \u2264 1\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, F a = 1"}, {"tactic": "have h_lintegral_eq : \u222b\u207b a, F a \u2202\u03c1.fst = \u222b\u207b _, 1 \u2202\u03c1.fst := by\n  have h_lintegral :\n    Tendsto (fun r : \u2115 => \u222b\u207b a, preCdf \u03c1 r a \u2202\u03c1.fst) atTop (\ud835\udcdd (\u222b\u207b a, F a \u2202\u03c1.fst)) := by\n    refine'\n      lintegral_tendsto_of_tendsto_of_monotone\n        (fun _ => measurable_preCdf.aemeasurable)\n        _ h_tendsto_\u2115\n    filter_upwards [h_mono] with a ha\n    refine' fun n m hnm => ha _\n    exact_mod_cast hnm\n  have h_lintegral' :\n    Tendsto (fun r : \u2115 => \u222b\u207b a, preCdf \u03c1 r a \u2202\u03c1.fst) atTop (\ud835\udcdd (\u222b\u207b _, 1 \u2202\u03c1.fst)) := by\n    rw [lintegral_one, Measure.fst_univ]\n    exact (tendsto_lintegral_preCdf_atTop \u03c1).comp tendsto_nat_cast_atTop_atTop\n  exact tendsto_nhds_unique h_lintegral h_lintegral'", "annotated_tactic": ["have h_lintegral_eq : \u222b\u207b a, F a \u2202\u03c1.fst = \u222b\u207b _, 1 \u2202\u03c1.fst := by\n    have h_lintegral :\n      <a>Tendsto</a> (fun r : \u2115 => \u222b\u207b a, <a>preCdf</a> \u03c1 r a \u2202\u03c1.fst) <a>atTop</a> (\ud835\udcdd (\u222b\u207b a, F a \u2202\u03c1.fst)) := by\n      refine'\n        <a>lintegral_tendsto_of_tendsto_of_monotone</a>\n          (-- does this exist only for \u2115?\n          fun _ => measurable_preCdf.aemeasurable)\n          _ h_tendsto_\u2115\n      filter_upwards [h_mono] with a ha\n      refine' fun n m hnm => ha _\n      exact_mod_cast hnm\n    have h_lintegral' :\n      <a>Tendsto</a> (fun r : \u2115 => \u222b\u207b a, <a>preCdf</a> \u03c1 r a \u2202\u03c1.fst) <a>atTop</a> (\ud835\udcdd (\u222b\u207b _, 1 \u2202\u03c1.fst)) := by\n      rw [<a>lintegral_one</a>, <a>Measure.fst_univ</a>]\n      exact (<a>tendsto_lintegral_preCdf_atTop</a> \u03c1).<a>comp</a> <a>tendsto_nat_cast_atTop_atTop</a>\n    exact <a>tendsto_nhds_unique</a> h_lintegral h_lintegral'", [{"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2939, 5], "def_end_pos": [2939, 12]}, {"full_name": "ProbabilityTheory.preCdf", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [294, 19], "def_end_pos": [294, 25]}, {"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}, {"full_name": "MeasureTheory.lintegral_tendsto_of_tendsto_of_monotone", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [435, 9], "def_end_pos": [435, 49]}, {"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2939, 5], "def_end_pos": [2939, 12]}, {"full_name": "ProbabilityTheory.preCdf", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [294, 19], "def_end_pos": [294, 25]}, {"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}, {"full_name": "MeasureTheory.lintegral_one", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [149, 9], "def_end_pos": [149, 22]}, {"full_name": "MeasureTheory.Measure.fst_univ", "def_path": "Mathlib/MeasureTheory/Constructions/Prod/Basic.lean", "def_pos": [918, 9], "def_end_pos": [918, 17]}, {"full_name": "ProbabilityTheory.tendsto_lintegral_preCdf_atTop", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [360, 9], "def_end_pos": [360, 39]}, {"full_name": "Filter.Tendsto.comp", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [3032, 9], "def_end_pos": [3032, 21]}, {"full_name": "tendsto_nat_cast_atTop_atTop", "def_path": "Mathlib/Order/Filter/Archimedean.lean", "def_pos": [37, 9], "def_end_pos": [37, 37]}, {"full_name": "tendsto_nhds_unique", "def_path": "Mathlib/Topology/Separation.lean", "def_pos": [994, 9], "def_end_pos": [994, 28]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\nh_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto_\u211a : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a))\nh_tendsto_\u2115 : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun n => preCdf \u03c1 (\u2191n) a) atTop (\ud835\udcdd (F a))\nhF_ae_meas : AEMeasurable F\nhF_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, F a \u2264 1\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, F a = 1", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\nh_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto_\u211a : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a))\nh_tendsto_\u2115 : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun n => preCdf \u03c1 (\u2191n) a) atTop (\ud835\udcdd (F a))\nhF_ae_meas : AEMeasurable F\nhF_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, F a \u2264 1\nh_lintegral_eq : \u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1 = \u222b\u207b (x : \u03b1), 1 \u2202Measure.fst \u03c1\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, F a = 1"}, {"tactic": "have : \u222b\u207b a, 1 - F a \u2202\u03c1.fst = 0 := by\n  rw [lintegral_sub' hF_ae_meas _ hF_le_one, h_lintegral_eq, tsub_self]\n  calc\n    \u222b\u207b a, F a \u2202\u03c1.fst = \u222b\u207b _, 1 \u2202\u03c1.fst := h_lintegral_eq\n    _ = \u03c1.fst univ := lintegral_one\n    _ = \u03c1 univ := Measure.fst_univ\n    _ \u2260 \u221e := measure_ne_top \u03c1 _", "annotated_tactic": ["have : \u222b\u207b a, 1 - F a \u2202\u03c1.fst = 0 := by\n    rw [<a>lintegral_sub'</a> hF_ae_meas _ hF_le_one, h_lintegral_eq, <a>tsub_self</a>]\n    calc\n      \u222b\u207b a, F a \u2202\u03c1.fst = \u222b\u207b _, 1 \u2202\u03c1.fst := h_lintegral_eq\n      _ = \u03c1.fst <a>univ</a> := <a>lintegral_one</a>\n      _ = \u03c1 <a>univ</a> := <a>Measure.fst_univ</a>\n      _ \u2260 \u221e := <a>measure_ne_top</a> \u03c1 _", [{"full_name": "MeasureTheory.lintegral_sub'", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [926, 9], "def_end_pos": [926, 23]}, {"full_name": "tsub_self", "def_path": "Mathlib/Algebra/Order/Sub/Canonical.lean", "def_pos": [333, 9], "def_end_pos": [333, 18]}, {"full_name": "Set.univ", "def_path": "Mathlib/Init/Set.lean", "def_pos": [90, 5], "def_end_pos": [90, 9]}, {"full_name": "MeasureTheory.lintegral_one", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [149, 9], "def_end_pos": [149, 22]}, {"full_name": "Set.univ", "def_path": "Mathlib/Init/Set.lean", "def_pos": [90, 5], "def_end_pos": [90, 9]}, {"full_name": "MeasureTheory.Measure.fst_univ", "def_path": "Mathlib/MeasureTheory/Constructions/Prod/Basic.lean", "def_pos": [918, 9], "def_end_pos": [918, 17]}, {"full_name": "MeasureTheory.measure_ne_top", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2875, 9], "def_end_pos": [2875, 23]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\nh_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto_\u211a : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a))\nh_tendsto_\u2115 : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun n => preCdf \u03c1 (\u2191n) a) atTop (\ud835\udcdd (F a))\nhF_ae_meas : AEMeasurable F\nhF_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, F a \u2264 1\nh_lintegral_eq : \u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1 = \u222b\u207b (x : \u03b1), 1 \u2202Measure.fst \u03c1\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, F a = 1", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\nh_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto_\u211a : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a))\nh_tendsto_\u2115 : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun n => preCdf \u03c1 (\u2191n) a) atTop (\ud835\udcdd (F a))\nhF_ae_meas : AEMeasurable F\nhF_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, F a \u2264 1\nh_lintegral_eq : \u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1 = \u222b\u207b (x : \u03b1), 1 \u2202Measure.fst \u03c1\nthis : \u222b\u207b (a : \u03b1), 1 - F a \u2202Measure.fst \u03c1 = 0\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, F a = 1"}, {"tactic": "rw [lintegral_eq_zero_iff' (aemeasurable_const.sub hF_ae_meas)] at this", "annotated_tactic": ["rw [<a>lintegral_eq_zero_iff'</a> (aemeasurable_const.sub hF_ae_meas)] at this", [{"full_name": "MeasureTheory.lintegral_eq_zero_iff'", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [888, 9], "def_end_pos": [888, 31]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\nh_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto_\u211a : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a))\nh_tendsto_\u2115 : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun n => preCdf \u03c1 (\u2191n) a) atTop (\ud835\udcdd (F a))\nhF_ae_meas : AEMeasurable F\nhF_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, F a \u2264 1\nh_lintegral_eq : \u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1 = \u222b\u207b (x : \u03b1), 1 \u2202Measure.fst \u03c1\nthis : \u222b\u207b (a : \u03b1), 1 - F a \u2202Measure.fst \u03c1 = 0\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, F a = 1", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\nh_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto_\u211a : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a))\nh_tendsto_\u2115 : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun n => preCdf \u03c1 (\u2191n) a) atTop (\ud835\udcdd (F a))\nhF_ae_meas : AEMeasurable F\nhF_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, F a \u2264 1\nh_lintegral_eq : \u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1 = \u222b\u207b (x : \u03b1), 1 \u2202Measure.fst \u03c1\nthis : (fun a => 1 - F a) =\u1d50[Measure.fst \u03c1] 0\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, F a = 1"}, {"tactic": "filter_upwards [this, hF_le_one] with ha h_one_sub_eq_zero h_le_one", "annotated_tactic": ["filter_upwards [this, hF_le_one] with ha h_one_sub_eq_zero h_le_one", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\nh_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto_\u211a : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a))\nh_tendsto_\u2115 : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun n => preCdf \u03c1 (\u2191n) a) atTop (\ud835\udcdd (F a))\nhF_ae_meas : AEMeasurable F\nhF_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, F a \u2264 1\nh_lintegral_eq : \u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1 = \u222b\u207b (x : \u03b1), 1 \u2202Measure.fst \u03c1\nthis : (fun a => 1 - F a) =\u1d50[Measure.fst \u03c1] 0\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, F a = 1", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\nh_le_one\u271d : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto_\u211a : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a))\nh_tendsto_\u2115 : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun n => preCdf \u03c1 (\u2191n) a) atTop (\ud835\udcdd (F a))\nhF_ae_meas : AEMeasurable F\nhF_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, F a \u2264 1\nh_lintegral_eq : \u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1 = \u222b\u207b (x : \u03b1), 1 \u2202Measure.fst \u03c1\nthis : (fun a => 1 - F a) =\u1d50[Measure.fst \u03c1] 0\nha : \u03b1\nh_one_sub_eq_zero : 1 - F ha = OfNat.ofNat 0 ha\nh_le_one : F ha \u2264 1\n\u22a2 F ha = 1"}, {"tactic": "rw [Pi.zero_apply, tsub_eq_zero_iff_le] at h_one_sub_eq_zero", "annotated_tactic": ["rw [<a>Pi.zero_apply</a>, <a>tsub_eq_zero_iff_le</a>] at h_one_sub_eq_zero", [{"full_name": "Pi.zero_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [46, 3], "def_end_pos": [46, 14]}, {"full_name": "tsub_eq_zero_iff_le", "def_path": "Mathlib/Algebra/Order/Sub/Canonical.lean", "def_pos": [324, 9], "def_end_pos": [324, 28]}]], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\nh_le_one\u271d : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto_\u211a : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a))\nh_tendsto_\u2115 : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun n => preCdf \u03c1 (\u2191n) a) atTop (\ud835\udcdd (F a))\nhF_ae_meas : AEMeasurable F\nhF_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, F a \u2264 1\nh_lintegral_eq : \u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1 = \u222b\u207b (x : \u03b1), 1 \u2202Measure.fst \u03c1\nthis : (fun a => 1 - F a) =\u1d50[Measure.fst \u03c1] 0\nha : \u03b1\nh_one_sub_eq_zero : 1 - F ha = OfNat.ofNat 0 ha\nh_le_one : F ha \u2264 1\n\u22a2 F ha = 1", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\nh_le_one\u271d : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto_\u211a : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a))\nh_tendsto_\u2115 : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun n => preCdf \u03c1 (\u2191n) a) atTop (\ud835\udcdd (F a))\nhF_ae_meas : AEMeasurable F\nhF_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, F a \u2264 1\nh_lintegral_eq : \u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1 = \u222b\u207b (x : \u03b1), 1 \u2202Measure.fst \u03c1\nthis : (fun a => 1 - F a) =\u1d50[Measure.fst \u03c1] 0\nha : \u03b1\nh_one_sub_eq_zero : 1 \u2264 F ha\nh_le_one : F ha \u2264 1\n\u22a2 F ha = 1"}, {"tactic": "exact le_antisymm h_le_one h_one_sub_eq_zero", "annotated_tactic": ["exact <a>le_antisymm</a> h_le_one h_one_sub_eq_zero", [{"full_name": "le_antisymm", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [188, 9], "def_end_pos": [188, 20]}]], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\nh_le_one\u271d : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto_\u211a : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a))\nh_tendsto_\u2115 : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun n => preCdf \u03c1 (\u2191n) a) atTop (\ud835\udcdd (F a))\nhF_ae_meas : AEMeasurable F\nhF_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, F a \u2264 1\nh_lintegral_eq : \u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1 = \u222b\u207b (x : \u03b1), 1 \u2202Measure.fst \u03c1\nthis : (fun a => 1 - F a) =\u1d50[Measure.fst \u03c1] 0\nha : \u03b1\nh_one_sub_eq_zero : 1 \u2264 F ha\nh_le_one : F ha \u2264 1\n\u22a2 F ha = 1", "state_after": "no goals"}, {"tactic": "filter_upwards [h_exists] with a ha", "annotated_tactic": ["filter_upwards [h_exists] with a ha", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\nh_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a))", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\nh_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0\na : \u03b1\nha : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)\n\u22a2 Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a))"}, {"tactic": "simp_rw [dif_pos ha]", "annotated_tactic": ["simp_rw [<a>dif_pos</a> ha]", [{"full_name": "dif_pos", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [807, 9], "def_end_pos": [807, 16]}]], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\nh_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0\na : \u03b1\nha : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)\n\u22a2 Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a))", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\nh_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0\na : \u03b1\nha : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)\n\u22a2 Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (Exists.choose ha))"}, {"tactic": "exact ha.choose_spec", "annotated_tactic": ["exact ha.choose_spec", []], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\nh_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0\na : \u03b1\nha : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)\n\u22a2 Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (Exists.choose ha))", "state_after": "no goals"}, {"tactic": "filter_upwards [h_tendsto_\u211a] with a ha using ha.comp tendsto_nat_cast_atTop_atTop", "annotated_tactic": ["filter_upwards [h_tendsto_\u211a] with a ha using ha.comp <a>tendsto_nat_cast_atTop_atTop</a>", [{"full_name": "tendsto_nat_cast_atTop_atTop", "def_path": "Mathlib/Order/Filter/Archimedean.lean", "def_pos": [37, 9], "def_end_pos": [37, 37]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\nh_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto_\u211a : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a))\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun n => preCdf \u03c1 (\u2191n) a) atTop (\ud835\udcdd (F a))", "state_after": "no goals"}, {"tactic": "refine' aemeasurable_of_tendsto_metrizable_ae _ (fun n => _) h_tendsto_\u211a", "annotated_tactic": ["refine' <a>aemeasurable_of_tendsto_metrizable_ae</a> _ (fun n => _) h_tendsto_\u211a", [{"full_name": "aemeasurable_of_tendsto_metrizable_ae", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Metrizable.lean", "def_pos": [92, 9], "def_end_pos": [92, 46]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\nh_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto_\u211a : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a))\nh_tendsto_\u2115 : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun n => preCdf \u03c1 (\u2191n) a) atTop (\ud835\udcdd (F a))\n\u22a2 AEMeasurable F", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\nh_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto_\u211a : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a))\nh_tendsto_\u2115 : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun n => preCdf \u03c1 (\u2191n) a) atTop (\ud835\udcdd (F a))\nn : \u211a\n\u22a2 AEMeasurable fun x => preCdf \u03c1 n x"}, {"tactic": "exact measurable_preCdf.aemeasurable", "annotated_tactic": ["exact measurable_preCdf.aemeasurable", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\nh_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto_\u211a : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a))\nh_tendsto_\u2115 : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun n => preCdf \u03c1 (\u2191n) a) atTop (\ud835\udcdd (F a))\nn : \u211a\n\u22a2 AEMeasurable fun x => preCdf \u03c1 n x", "state_after": "no goals"}, {"tactic": "filter_upwards [h_tendsto_\u211a, h_le_one] with a ha ha_le using le_of_tendsto' ha ha_le", "annotated_tactic": ["filter_upwards [h_tendsto_\u211a, h_le_one] with a ha ha_le using <a>le_of_tendsto'</a> ha ha_le", [{"full_name": "le_of_tendsto'", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [145, 9], "def_end_pos": [145, 23]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\nh_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto_\u211a : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a))\nh_tendsto_\u2115 : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun n => preCdf \u03c1 (\u2191n) a) atTop (\ud835\udcdd (F a))\nhF_ae_meas : AEMeasurable F\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, F a \u2264 1", "state_after": "no goals"}, {"tactic": "filter_upwards [h_tendsto_\u211a, this] with a ha_tendsto ha_eq", "annotated_tactic": ["filter_upwards [h_tendsto_\u211a, this] with a ha_tendsto ha_eq", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\nh_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto_\u211a : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a))\nh_tendsto_\u2115 : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun n => preCdf \u03c1 (\u2191n) a) atTop (\ud835\udcdd (F a))\nhF_ae_meas : AEMeasurable F\nhF_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, F a \u2264 1\nthis : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, F a = 1\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd 1)", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\nh_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto_\u211a : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a))\nh_tendsto_\u2115 : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun n => preCdf \u03c1 (\u2191n) a) atTop (\ud835\udcdd (F a))\nhF_ae_meas : AEMeasurable F\nhF_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, F a \u2264 1\nthis : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, F a = 1\na : \u03b1\nha_tendsto : Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a))\nha_eq : F a = 1\n\u22a2 Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd 1)"}, {"tactic": "rwa [ha_eq] at ha_tendsto", "annotated_tactic": ["rwa [ha_eq] at ha_tendsto", []], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\nh_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto_\u211a : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a))\nh_tendsto_\u2115 : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun n => preCdf \u03c1 (\u2191n) a) atTop (\ud835\udcdd (F a))\nhF_ae_meas : AEMeasurable F\nhF_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, F a \u2264 1\nthis : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, F a = 1\na : \u03b1\nha_tendsto : Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a))\nha_eq : F a = 1\n\u22a2 Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd 1)", "state_after": "no goals"}, {"tactic": "have h_lintegral :\n  Tendsto (fun r : \u2115 => \u222b\u207b a, preCdf \u03c1 r a \u2202\u03c1.fst) atTop (\ud835\udcdd (\u222b\u207b a, F a \u2202\u03c1.fst)) := by\n  refine'\n    lintegral_tendsto_of_tendsto_of_monotone\n      (fun _ => measurable_preCdf.aemeasurable)\n      _ h_tendsto_\u2115\n  filter_upwards [h_mono] with a ha\n  refine' fun n m hnm => ha _\n  exact_mod_cast hnm", "annotated_tactic": ["have h_lintegral :\n      <a>Tendsto</a> (fun r : \u2115 => \u222b\u207b a, <a>preCdf</a> \u03c1 r a \u2202\u03c1.fst) <a>atTop</a> (\ud835\udcdd (\u222b\u207b a, F a \u2202\u03c1.fst)) := by\n      refine'\n        <a>lintegral_tendsto_of_tendsto_of_monotone</a>\n          (-- does this exist only for \u2115?\n          fun _ => measurable_preCdf.aemeasurable)\n          _ h_tendsto_\u2115\n      filter_upwards [h_mono] with a ha\n      refine' fun n m hnm => ha _\n      exact_mod_cast hnm", [{"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2939, 5], "def_end_pos": [2939, 12]}, {"full_name": "ProbabilityTheory.preCdf", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [294, 19], "def_end_pos": [294, 25]}, {"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}, {"full_name": "MeasureTheory.lintegral_tendsto_of_tendsto_of_monotone", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [435, 9], "def_end_pos": [435, 49]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\nh_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto_\u211a : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a))\nh_tendsto_\u2115 : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun n => preCdf \u03c1 (\u2191n) a) atTop (\ud835\udcdd (F a))\nhF_ae_meas : AEMeasurable F\nhF_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, F a \u2264 1\n\u22a2 \u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1 = \u222b\u207b (x : \u03b1), 1 \u2202Measure.fst \u03c1", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\nh_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto_\u211a : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a))\nh_tendsto_\u2115 : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun n => preCdf \u03c1 (\u2191n) a) atTop (\ud835\udcdd (F a))\nhF_ae_meas : AEMeasurable F\nhF_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, F a \u2264 1\nh_lintegral : Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (\u2191r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd (\u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1))\n\u22a2 \u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1 = \u222b\u207b (x : \u03b1), 1 \u2202Measure.fst \u03c1"}, {"tactic": "have h_lintegral' :\n  Tendsto (fun r : \u2115 => \u222b\u207b a, preCdf \u03c1 r a \u2202\u03c1.fst) atTop (\ud835\udcdd (\u222b\u207b _, 1 \u2202\u03c1.fst)) := by\n  rw [lintegral_one, Measure.fst_univ]\n  exact (tendsto_lintegral_preCdf_atTop \u03c1).comp tendsto_nat_cast_atTop_atTop", "annotated_tactic": ["have h_lintegral' :\n      <a>Tendsto</a> (fun r : \u2115 => \u222b\u207b a, <a>preCdf</a> \u03c1 r a \u2202\u03c1.fst) <a>atTop</a> (\ud835\udcdd (\u222b\u207b _, 1 \u2202\u03c1.fst)) := by\n      rw [<a>lintegral_one</a>, <a>Measure.fst_univ</a>]\n      exact (<a>tendsto_lintegral_preCdf_atTop</a> \u03c1).<a>comp</a> <a>tendsto_nat_cast_atTop_atTop</a>", [{"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2939, 5], "def_end_pos": [2939, 12]}, {"full_name": "ProbabilityTheory.preCdf", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [294, 19], "def_end_pos": [294, 25]}, {"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}, {"full_name": "MeasureTheory.lintegral_one", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [149, 9], "def_end_pos": [149, 22]}, {"full_name": "MeasureTheory.Measure.fst_univ", "def_path": "Mathlib/MeasureTheory/Constructions/Prod/Basic.lean", "def_pos": [918, 9], "def_end_pos": [918, 17]}, {"full_name": "ProbabilityTheory.tendsto_lintegral_preCdf_atTop", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [360, 9], "def_end_pos": [360, 39]}, {"full_name": "Filter.Tendsto.comp", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [3032, 9], "def_end_pos": [3032, 21]}, {"full_name": "tendsto_nat_cast_atTop_atTop", "def_path": "Mathlib/Order/Filter/Archimedean.lean", "def_pos": [37, 9], "def_end_pos": [37, 37]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\nh_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto_\u211a : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a))\nh_tendsto_\u2115 : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun n => preCdf \u03c1 (\u2191n) a) atTop (\ud835\udcdd (F a))\nhF_ae_meas : AEMeasurable F\nhF_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, F a \u2264 1\nh_lintegral : Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (\u2191r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd (\u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1))\n\u22a2 \u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1 = \u222b\u207b (x : \u03b1), 1 \u2202Measure.fst \u03c1", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\nh_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto_\u211a : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a))\nh_tendsto_\u2115 : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun n => preCdf \u03c1 (\u2191n) a) atTop (\ud835\udcdd (F a))\nhF_ae_meas : AEMeasurable F\nhF_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, F a \u2264 1\nh_lintegral : Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (\u2191r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd (\u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1))\nh_lintegral' : Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (\u2191r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd (\u222b\u207b (x : \u03b1), 1 \u2202Measure.fst \u03c1))\n\u22a2 \u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1 = \u222b\u207b (x : \u03b1), 1 \u2202Measure.fst \u03c1"}, {"tactic": "exact tendsto_nhds_unique h_lintegral h_lintegral'", "annotated_tactic": ["exact <a>tendsto_nhds_unique</a> h_lintegral h_lintegral'", [{"full_name": "tendsto_nhds_unique", "def_path": "Mathlib/Topology/Separation.lean", "def_pos": [994, 9], "def_end_pos": [994, 28]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\nh_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto_\u211a : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a))\nh_tendsto_\u2115 : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun n => preCdf \u03c1 (\u2191n) a) atTop (\ud835\udcdd (F a))\nhF_ae_meas : AEMeasurable F\nhF_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, F a \u2264 1\nh_lintegral : Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (\u2191r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd (\u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1))\nh_lintegral' : Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (\u2191r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd (\u222b\u207b (x : \u03b1), 1 \u2202Measure.fst \u03c1))\n\u22a2 \u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1 = \u222b\u207b (x : \u03b1), 1 \u2202Measure.fst \u03c1", "state_after": "no goals"}, {"tactic": "refine'\n  lintegral_tendsto_of_tendsto_of_monotone\n    (fun _ => measurable_preCdf.aemeasurable)\n    _ h_tendsto_\u2115", "annotated_tactic": ["refine'\n        <a>lintegral_tendsto_of_tendsto_of_monotone</a>\n          (-- does this exist only for \u2115?\n          fun _ => measurable_preCdf.aemeasurable)\n          _ h_tendsto_\u2115", [{"full_name": "MeasureTheory.lintegral_tendsto_of_tendsto_of_monotone", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [435, 9], "def_end_pos": [435, 49]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\nh_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto_\u211a : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a))\nh_tendsto_\u2115 : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun n => preCdf \u03c1 (\u2191n) a) atTop (\ud835\udcdd (F a))\nhF_ae_meas : AEMeasurable F\nhF_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, F a \u2264 1\n\u22a2 Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (\u2191r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd (\u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1))", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\nh_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto_\u211a : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a))\nh_tendsto_\u2115 : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun n => preCdf \u03c1 (\u2191n) a) atTop (\ud835\udcdd (F a))\nhF_ae_meas : AEMeasurable F\nhF_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, F a \u2264 1\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1, Monotone fun n => preCdf \u03c1 (\u2191n) x"}, {"tactic": "filter_upwards [h_mono] with a ha", "annotated_tactic": ["filter_upwards [h_mono] with a ha", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\nh_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto_\u211a : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a))\nh_tendsto_\u2115 : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun n => preCdf \u03c1 (\u2191n) a) atTop (\ud835\udcdd (F a))\nhF_ae_meas : AEMeasurable F\nhF_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, F a \u2264 1\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1, Monotone fun n => preCdf \u03c1 (\u2191n) x", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\nh_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto_\u211a : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a))\nh_tendsto_\u2115 : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun n => preCdf \u03c1 (\u2191n) a) atTop (\ud835\udcdd (F a))\nhF_ae_meas : AEMeasurable F\nhF_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, F a \u2264 1\na : \u03b1\nha : Monotone fun r => preCdf \u03c1 r a\n\u22a2 Monotone fun n => preCdf \u03c1 (\u2191n) a"}, {"tactic": "refine' fun n m hnm => ha _", "annotated_tactic": ["refine' fun n m hnm => ha _", []], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\nh_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto_\u211a : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a))\nh_tendsto_\u2115 : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun n => preCdf \u03c1 (\u2191n) a) atTop (\ud835\udcdd (F a))\nhF_ae_meas : AEMeasurable F\nhF_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, F a \u2264 1\na : \u03b1\nha : Monotone fun r => preCdf \u03c1 r a\n\u22a2 Monotone fun n => preCdf \u03c1 (\u2191n) a", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\nh_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto_\u211a : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a))\nh_tendsto_\u2115 : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun n => preCdf \u03c1 (\u2191n) a) atTop (\ud835\udcdd (F a))\nhF_ae_meas : AEMeasurable F\nhF_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, F a \u2264 1\na : \u03b1\nha : Monotone fun r => preCdf \u03c1 r a\nn m : \u2115\nhnm : n \u2264 m\n\u22a2 \u2191n \u2264 \u2191m"}, {"tactic": "exact_mod_cast hnm", "annotated_tactic": ["exact_mod_cast hnm", []], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\nh_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto_\u211a : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a))\nh_tendsto_\u2115 : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun n => preCdf \u03c1 (\u2191n) a) atTop (\ud835\udcdd (F a))\nhF_ae_meas : AEMeasurable F\nhF_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, F a \u2264 1\na : \u03b1\nha : Monotone fun r => preCdf \u03c1 r a\nn m : \u2115\nhnm : n \u2264 m\n\u22a2 \u2191n \u2264 \u2191m", "state_after": "no goals"}, {"tactic": "rw [lintegral_one, Measure.fst_univ]", "annotated_tactic": ["rw [<a>lintegral_one</a>, <a>Measure.fst_univ</a>]", [{"full_name": "MeasureTheory.lintegral_one", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [149, 9], "def_end_pos": [149, 22]}, {"full_name": "MeasureTheory.Measure.fst_univ", "def_path": "Mathlib/MeasureTheory/Constructions/Prod/Basic.lean", "def_pos": [918, 9], "def_end_pos": [918, 17]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\nh_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto_\u211a : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a))\nh_tendsto_\u2115 : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun n => preCdf \u03c1 (\u2191n) a) atTop (\ud835\udcdd (F a))\nhF_ae_meas : AEMeasurable F\nhF_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, F a \u2264 1\nh_lintegral : Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (\u2191r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd (\u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1))\n\u22a2 Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (\u2191r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd (\u222b\u207b (x : \u03b1), 1 \u2202Measure.fst \u03c1))", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\nh_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto_\u211a : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a))\nh_tendsto_\u2115 : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun n => preCdf \u03c1 (\u2191n) a) atTop (\ud835\udcdd (F a))\nhF_ae_meas : AEMeasurable F\nhF_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, F a \u2264 1\nh_lintegral : Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (\u2191r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd (\u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1))\n\u22a2 Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (\u2191r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd (\u2191\u2191\u03c1 univ))"}, {"tactic": "exact (tendsto_lintegral_preCdf_atTop \u03c1).comp tendsto_nat_cast_atTop_atTop", "annotated_tactic": ["exact (<a>tendsto_lintegral_preCdf_atTop</a> \u03c1).<a>comp</a> <a>tendsto_nat_cast_atTop_atTop</a>", [{"full_name": "ProbabilityTheory.tendsto_lintegral_preCdf_atTop", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [360, 9], "def_end_pos": [360, 39]}, {"full_name": "Filter.Tendsto.comp", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [3032, 9], "def_end_pos": [3032, 21]}, {"full_name": "tendsto_nat_cast_atTop_atTop", "def_path": "Mathlib/Order/Filter/Archimedean.lean", "def_pos": [37, 9], "def_end_pos": [37, 37]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\nh_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto_\u211a : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a))\nh_tendsto_\u2115 : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun n => preCdf \u03c1 (\u2191n) a) atTop (\ud835\udcdd (F a))\nhF_ae_meas : AEMeasurable F\nhF_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, F a \u2264 1\nh_lintegral : Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (\u2191r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd (\u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1))\n\u22a2 Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (\u2191r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd (\u2191\u2191\u03c1 univ))", "state_after": "no goals"}, {"tactic": "rw [lintegral_sub' hF_ae_meas _ hF_le_one, h_lintegral_eq, tsub_self]", "annotated_tactic": ["rw [<a>lintegral_sub'</a> hF_ae_meas _ hF_le_one, h_lintegral_eq, <a>tsub_self</a>]", [{"full_name": "MeasureTheory.lintegral_sub'", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [926, 9], "def_end_pos": [926, 23]}, {"full_name": "tsub_self", "def_path": "Mathlib/Algebra/Order/Sub/Canonical.lean", "def_pos": [333, 9], "def_end_pos": [333, 18]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\nh_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto_\u211a : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a))\nh_tendsto_\u2115 : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun n => preCdf \u03c1 (\u2191n) a) atTop (\ud835\udcdd (F a))\nhF_ae_meas : AEMeasurable F\nhF_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, F a \u2264 1\nh_lintegral_eq : \u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1 = \u222b\u207b (x : \u03b1), 1 \u2202Measure.fst \u03c1\n\u22a2 \u222b\u207b (a : \u03b1), 1 - F a \u2202Measure.fst \u03c1 = 0", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\nh_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto_\u211a : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a))\nh_tendsto_\u2115 : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun n => preCdf \u03c1 (\u2191n) a) atTop (\ud835\udcdd (F a))\nhF_ae_meas : AEMeasurable F\nhF_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, F a \u2264 1\nh_lintegral_eq : \u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1 = \u222b\u207b (x : \u03b1), 1 \u2202Measure.fst \u03c1\n\u22a2 \u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1 \u2260 \u22a4"}, {"tactic": "calc\n  \u222b\u207b a, F a \u2202\u03c1.fst = \u222b\u207b _, 1 \u2202\u03c1.fst := h_lintegral_eq\n  _ = \u03c1.fst univ := lintegral_one\n  _ = \u03c1 univ := Measure.fst_univ\n  _ \u2260 \u221e := measure_ne_top \u03c1 _", "annotated_tactic": ["calc\n      \u222b\u207b a, F a \u2202\u03c1.fst = \u222b\u207b _, 1 \u2202\u03c1.fst := h_lintegral_eq\n      _ = \u03c1.fst <a>univ</a> := <a>lintegral_one</a>\n      _ = \u03c1 <a>univ</a> := <a>Measure.fst_univ</a>\n      _ \u2260 \u221e := <a>measure_ne_top</a> \u03c1 _", [{"full_name": "Set.univ", "def_path": "Mathlib/Init/Set.lean", "def_pos": [90, 5], "def_end_pos": [90, 9]}, {"full_name": "MeasureTheory.lintegral_one", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [149, 9], "def_end_pos": [149, 22]}, {"full_name": "Set.univ", "def_path": "Mathlib/Init/Set.lean", "def_pos": [90, 5], "def_end_pos": [90, 9]}, {"full_name": "MeasureTheory.Measure.fst_univ", "def_path": "Mathlib/MeasureTheory/Constructions/Prod/Basic.lean", "def_pos": [918, 9], "def_end_pos": [918, 17]}, {"full_name": "MeasureTheory.measure_ne_top", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2875, 9], "def_end_pos": [2875, 23]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\nh_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto_\u211a : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a))\nh_tendsto_\u2115 : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun n => preCdf \u03c1 (\u2191n) a) atTop (\ud835\udcdd (F a))\nhF_ae_meas : AEMeasurable F\nhF_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, F a \u2264 1\nh_lintegral_eq : \u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1 = \u222b\u207b (x : \u03b1), 1 \u2202Measure.fst \u03c1\n\u22a2 \u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1 \u2260 \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/TMToPartrec.lean", "full_name": "Turing.ToPartrec.Code.zero_eval", "start": [201, 1], "end": [201, 65], "traced_tactics": [{"tactic": "simp [zero]", "annotated_tactic": ["simp [<a>zero</a>]", [{"full_name": "Turing.ToPartrec.Code.zero", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [196, 5], "def_end_pos": [196, 9]}]], "state_before": "v : List \u2115\n\u22a2 eval zero v = pure [0]", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "full_name": "MeasureTheory.all_ae_ofReal_f_le_bound", "start": [301, 1], "end": [308, 47], "traced_tactics": [{"tactic": "have F_le_bound := all_ae_ofReal_F_le_bound h_bound", "annotated_tactic": ["have F_le_bound := <a>all_ae_ofReal_F_le_bound</a> h_bound", [{"full_name": "MeasureTheory.all_ae_ofReal_F_le_bound", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [290, 9], "def_end_pos": [290, 33]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : NormedAddCommGroup \u03b3\nF : \u2115 \u2192 \u03b1 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b2\nbound : \u03b1 \u2192 \u211d\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016F n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a))\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, ENNReal.ofReal \u2016f a\u2016 \u2264 ENNReal.ofReal (bound a)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : NormedAddCommGroup \u03b3\nF : \u2115 \u2192 \u03b1 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b2\nbound : \u03b1 \u2192 \u211d\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016F n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a))\nF_le_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, ENNReal.ofReal \u2016F n a\u2016 \u2264 ENNReal.ofReal (bound a)\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, ENNReal.ofReal \u2016f a\u2016 \u2264 ENNReal.ofReal (bound a)"}, {"tactic": "rw [\u2190 ae_all_iff] at F_le_bound", "annotated_tactic": ["rw [\u2190 <a>ae_all_iff</a>] at F_le_bound", [{"full_name": "MeasureTheory.ae_all_iff", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [422, 9], "def_end_pos": [422, 19]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : NormedAddCommGroup \u03b3\nF : \u2115 \u2192 \u03b1 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b2\nbound : \u03b1 \u2192 \u211d\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016F n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a))\nF_le_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, ENNReal.ofReal \u2016F n a\u2016 \u2264 ENNReal.ofReal (bound a)\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, ENNReal.ofReal \u2016f a\u2016 \u2264 ENNReal.ofReal (bound a)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : NormedAddCommGroup \u03b3\nF : \u2115 \u2192 \u03b1 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b2\nbound : \u03b1 \u2192 \u211d\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016F n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a))\nF_le_bound : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (i : \u2115), ENNReal.ofReal \u2016F i a\u2016 \u2264 ENNReal.ofReal (bound a)\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, ENNReal.ofReal \u2016f a\u2016 \u2264 ENNReal.ofReal (bound a)"}, {"tactic": "apply F_le_bound.mp ((all_ae_tendsto_ofReal_norm h_lim).mono _)", "annotated_tactic": ["apply F_le_bound.mp ((<a>all_ae_tendsto_ofReal_norm</a> h_lim).<a>mono</a> _)", [{"full_name": "MeasureTheory.all_ae_tendsto_ofReal_norm", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [296, 9], "def_end_pos": [296, 35]}, {"full_name": "Filter.Eventually.mono", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1140, 9], "def_end_pos": [1140, 24]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : NormedAddCommGroup \u03b3\nF : \u2115 \u2192 \u03b1 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b2\nbound : \u03b1 \u2192 \u211d\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016F n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a))\nF_le_bound : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (i : \u2115), ENNReal.ofReal \u2016F i a\u2016 \u2264 ENNReal.ofReal (bound a)\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, ENNReal.ofReal \u2016f a\u2016 \u2264 ENNReal.ofReal (bound a)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : NormedAddCommGroup \u03b3\nF : \u2115 \u2192 \u03b1 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b2\nbound : \u03b1 \u2192 \u211d\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016F n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a))\nF_le_bound : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (i : \u2115), ENNReal.ofReal \u2016F i a\u2016 \u2264 ENNReal.ofReal (bound a)\n\u22a2 \u2200 (x : \u03b1),\n    Tendsto (fun n => ENNReal.ofReal \u2016F n x\u2016) atTop (\ud835\udcdd (ENNReal.ofReal \u2016f x\u2016)) \u2192\n      (\u2200 (i : \u2115), ENNReal.ofReal \u2016F i x\u2016 \u2264 ENNReal.ofReal (bound x)) \u2192 ENNReal.ofReal \u2016f x\u2016 \u2264 ENNReal.ofReal (bound x)"}, {"tactic": "intro a tendsto_norm F_le_bound", "annotated_tactic": ["intro a tendsto_norm F_le_bound", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : NormedAddCommGroup \u03b3\nF : \u2115 \u2192 \u03b1 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b2\nbound : \u03b1 \u2192 \u211d\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016F n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a))\nF_le_bound : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (i : \u2115), ENNReal.ofReal \u2016F i a\u2016 \u2264 ENNReal.ofReal (bound a)\n\u22a2 \u2200 (x : \u03b1),\n    Tendsto (fun n => ENNReal.ofReal \u2016F n x\u2016) atTop (\ud835\udcdd (ENNReal.ofReal \u2016f x\u2016)) \u2192\n      (\u2200 (i : \u2115), ENNReal.ofReal \u2016F i x\u2016 \u2264 ENNReal.ofReal (bound x)) \u2192 ENNReal.ofReal \u2016f x\u2016 \u2264 ENNReal.ofReal (bound x)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : NormedAddCommGroup \u03b3\nF : \u2115 \u2192 \u03b1 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b2\nbound : \u03b1 \u2192 \u211d\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016F n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a))\nF_le_bound\u271d : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (i : \u2115), ENNReal.ofReal \u2016F i a\u2016 \u2264 ENNReal.ofReal (bound a)\na : \u03b1\ntendsto_norm : Tendsto (fun n => ENNReal.ofReal \u2016F n a\u2016) atTop (\ud835\udcdd (ENNReal.ofReal \u2016f a\u2016))\nF_le_bound : \u2200 (i : \u2115), ENNReal.ofReal \u2016F i a\u2016 \u2264 ENNReal.ofReal (bound a)\n\u22a2 ENNReal.ofReal \u2016f a\u2016 \u2264 ENNReal.ofReal (bound a)"}, {"tactic": "exact le_of_tendsto' tendsto_norm F_le_bound", "annotated_tactic": ["exact <a>le_of_tendsto'</a> tendsto_norm F_le_bound", [{"full_name": "le_of_tendsto'", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [145, 9], "def_end_pos": [145, 23]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : NormedAddCommGroup \u03b3\nF : \u2115 \u2192 \u03b1 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b2\nbound : \u03b1 \u2192 \u211d\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016F n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a))\nF_le_bound\u271d : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (i : \u2115), ENNReal.ofReal \u2016F i a\u2016 \u2264 ENNReal.ofReal (bound a)\na : \u03b1\ntendsto_norm : Tendsto (fun n => ENNReal.ofReal \u2016F n a\u2016) atTop (\ud835\udcdd (ENNReal.ofReal \u2016f a\u2016))\nF_le_bound : \u2200 (i : \u2115), ENNReal.ofReal \u2016F i a\u2016 \u2264 ENNReal.ofReal (bound a)\n\u22a2 ENNReal.ofReal \u2016f a\u2016 \u2264 ENNReal.ofReal (bound a)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean", "full_name": "MeasureTheory.Measure.NullMeasurableSet.const_smul", "start": [396, 1], "end": [405, 62], "traced_tactics": [{"tactic": "obtain rfl | hs' := s.eq_empty_or_nonempty", "annotated_tactic": ["obtain rfl | hs' := s.eq_empty_or_nonempty", []], "state_before": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns : Set E\nhs : NullMeasurableSet s\nr : \u211d\n\u22a2 NullMeasurableSet (r \u2022 s)", "state_after": "case inl\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nr : \u211d\nhs : NullMeasurableSet \u2205\n\u22a2 NullMeasurableSet (r \u2022 \u2205)\n\ncase inr\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns : Set E\nhs : NullMeasurableSet s\nr : \u211d\nhs' : Set.Nonempty s\n\u22a2 NullMeasurableSet (r \u2022 s)"}, {"tactic": "obtain rfl | hr := eq_or_ne r 0", "annotated_tactic": ["obtain rfl | hr := <a>eq_or_ne</a> r 0", [{"full_name": "eq_or_ne", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [209, 9], "def_end_pos": [209, 17]}]], "state_before": "case inr\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns : Set E\nhs : NullMeasurableSet s\nr : \u211d\nhs' : Set.Nonempty s\n\u22a2 NullMeasurableSet (r \u2022 s)", "state_after": "case inr.inl\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns : Set E\nhs : NullMeasurableSet s\nhs' : Set.Nonempty s\n\u22a2 NullMeasurableSet (0 \u2022 s)\n\ncase inr.inr\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns : Set E\nhs : NullMeasurableSet s\nr : \u211d\nhs' : Set.Nonempty s\nhr : r \u2260 0\n\u22a2 NullMeasurableSet (r \u2022 s)"}, {"tactic": "obtain \u27e8t, ht, hst\u27e9 := hs", "annotated_tactic": ["obtain \u27e8t, ht, hst\u27e9 := hs", []], "state_before": "case inr.inr\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns : Set E\nhs : NullMeasurableSet s\nr : \u211d\nhs' : Set.Nonempty s\nhr : r \u2260 0\n\u22a2 NullMeasurableSet (r \u2022 s)", "state_after": "case inr.inr.intro.intro\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns : Set E\nr : \u211d\nhs' : Set.Nonempty s\nhr : r \u2260 0\nt : Set E\nht : MeasurableSet t\nhst : s =\u1da0[ae \u03bc] t\n\u22a2 NullMeasurableSet (r \u2022 s)"}, {"tactic": "refine' \u27e8_, ht.const_smul_of_ne_zero hr, _\u27e9", "annotated_tactic": ["refine' \u27e8_, ht.const_smul_of_ne_zero hr, _\u27e9", []], "state_before": "case inr.inr.intro.intro\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns : Set E\nr : \u211d\nhs' : Set.Nonempty s\nhr : r \u2260 0\nt : Set E\nht : MeasurableSet t\nhst : s =\u1da0[ae \u03bc] t\n\u22a2 NullMeasurableSet (r \u2022 s)", "state_after": "case inr.inr.intro.intro\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns : Set E\nr : \u211d\nhs' : Set.Nonempty s\nhr : r \u2260 0\nt : Set E\nht : MeasurableSet t\nhst : s =\u1da0[ae \u03bc] t\n\u22a2 r \u2022 s =\u1da0[ae \u03bc] r \u2022 t"}, {"tactic": "rw [\u2190 measure_symmDiff_eq_zero_iff] at hst \u22a2", "annotated_tactic": ["rw [\u2190 <a>measure_symmDiff_eq_zero_iff</a>] at hst \u22a2", [{"full_name": "MeasureTheory.measure_symmDiff_eq_zero_iff", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [502, 9], "def_end_pos": [502, 37]}]], "state_before": "case inr.inr.intro.intro\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns : Set E\nr : \u211d\nhs' : Set.Nonempty s\nhr : r \u2260 0\nt : Set E\nht : MeasurableSet t\nhst : s =\u1da0[ae \u03bc] t\n\u22a2 r \u2022 s =\u1da0[ae \u03bc] r \u2022 t", "state_after": "case inr.inr.intro.intro\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns : Set E\nr : \u211d\nhs' : Set.Nonempty s\nhr : r \u2260 0\nt : Set E\nht : MeasurableSet t\nhst : \u2191\u2191\u03bc (s \u2206 t) = 0\n\u22a2 \u2191\u2191\u03bc ((r \u2022 s) \u2206 (r \u2022 t)) = 0"}, {"tactic": "rw [\u2190 smul_set_symmDiff\u2080 hr, addHaar_smul \u03bc, hst, mul_zero]", "annotated_tactic": ["rw [\u2190 <a>smul_set_symmDiff\u2080</a> hr, <a>addHaar_smul</a> \u03bc, hst, <a>mul_zero</a>]", [{"full_name": "Set.smul_set_symmDiff\u2080", "def_path": "Mathlib/Data/Set/Pointwise/SMul.lean", "def_pos": [1056, 9], "def_end_pos": [1056, 27]}, {"full_name": "MeasureTheory.Measure.addHaar_smul", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean", "def_pos": [371, 9], "def_end_pos": [371, 21]}, {"full_name": "MulZeroClass.mul_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [38, 3], "def_end_pos": [38, 11]}]], "state_before": "case inr.inr.intro.intro\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns : Set E\nr : \u211d\nhs' : Set.Nonempty s\nhr : r \u2260 0\nt : Set E\nht : MeasurableSet t\nhst : \u2191\u2191\u03bc (s \u2206 t) = 0\n\u22a2 \u2191\u2191\u03bc ((r \u2022 s) \u2206 (r \u2022 t)) = 0", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case inl\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nr : \u211d\nhs : NullMeasurableSet \u2205\n\u22a2 NullMeasurableSet (r \u2022 \u2205)", "state_after": "no goals"}, {"tactic": "simpa [zero_smul_set hs'] using nullMeasurableSet_singleton _", "annotated_tactic": ["simpa [<a>zero_smul_set</a> hs'] using <a>nullMeasurableSet_singleton</a> _", [{"full_name": "Set.zero_smul_set", "def_path": "Mathlib/Data/Set/Pointwise/SMul.lean", "def_pos": [812, 9], "def_end_pos": [812, 22]}, {"full_name": "MeasureTheory.nullMeasurableSet_singleton", "def_path": "Mathlib/MeasureTheory/Measure/NullMeasurable.lean", "def_pos": [347, 9], "def_end_pos": [347, 36]}]], "state_before": "case inr.inl\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns : Set E\nhs : NullMeasurableSet s\nhs' : Set.Nonempty s\n\u22a2 NullMeasurableSet (0 \u2022 s)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean", "full_name": "Real.volume_pi_Ico", "start": [245, 1], "end": [247, 68], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "Finset.ssubset_iff_exists_cons_subset", "start": [924, 1], "end": [927, 46], "traced_tactics": [{"tactic": "refine' \u27e8fun h => _, fun \u27e8a, ha, h\u27e9 => ssubset_of_ssubset_of_subset (ssubset_cons _) h\u27e9", "annotated_tactic": ["refine' \u27e8fun h => _, fun \u27e8a, ha, h\u27e9 => <a>ssubset_of_ssubset_of_subset</a> (<a>ssubset_cons</a> _) h\u27e9", [{"full_name": "Finset.ssubset_of_ssubset_of_subset", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [428, 9], "def_end_pos": [428, 37]}, {"full_name": "Finset.ssubset_cons", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [911, 9], "def_end_pos": [911, 21]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns t : Finset \u03b1\na b : \u03b1\n\u22a2 s \u2282 t \u2194 \u2203 a h, cons a s h \u2286 t", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns t : Finset \u03b1\na b : \u03b1\nh : s \u2282 t\n\u22a2 \u2203 a h, cons a s h \u2286 t"}, {"tactic": "obtain \u27e8a, hs, ht\u27e9 := not_subset.1 h.2", "annotated_tactic": ["obtain \u27e8a, hs, ht\u27e9 := <a>not_subset</a>.1 h.2", [{"full_name": "Finset.not_subset", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [389, 9], "def_end_pos": [389, 19]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns t : Finset \u03b1\na b : \u03b1\nh : s \u2282 t\n\u22a2 \u2203 a h, cons a s h \u2286 t", "state_after": "case intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns t : Finset \u03b1\na\u271d b : \u03b1\nh : s \u2282 t\na : \u03b1\nhs : a \u2208 t\nht : \u00aca \u2208 s\n\u22a2 \u2203 a h, cons a s h \u2286 t"}, {"tactic": "exact \u27e8a, ht, cons_subset.2 \u27e8hs, h.subset\u27e9\u27e9", "annotated_tactic": ["exact \u27e8a, ht, <a>cons_subset</a>.2 \u27e8hs, h.subset\u27e9\u27e9", [{"full_name": "Finset.cons_subset", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [915, 9], "def_end_pos": [915, 20]}]], "state_before": "case intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns t : Finset \u03b1\na\u271d b : \u03b1\nh : s \u2282 t\na : \u03b1\nhs : a \u2208 t\nht : \u00aca \u2208 s\n\u22a2 \u2203 a h, cons a s h \u2286 t", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/NAry.lean", "full_name": "Finset.image_image\u2082_right_anticomm", "start": [499, 1], "end": [502, 79], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Group/Measure.lean", "full_name": "MeasureTheory.measure_univ_of_isMulLeftInvariant", "start": [653, 1], "end": [694, 63], "traced_tactics": [{"tactic": "obtain \u27e8K, hK, Kclosed, K1\u27e9 : \u2203 K : Set G, IsCompact K \u2227 IsClosed K \u2227 K \u2208 \ud835\udcdd 1 :=\n  exists_isCompact_isClosed_nhds_one G", "annotated_tactic": ["obtain \u27e8K, hK, Kclosed, K1\u27e9 : \u2203 K : <a>Set</a> G, <a>IsCompact</a> K \u2227 <a>IsClosed</a> K \u2227 K \u2208 \ud835\udcdd 1 :=\n    <a>exists_isCompact_isClosed_nhds_one</a> G", [{"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}, {"full_name": "IsCompact", "def_path": "Mathlib/Topology/Compactness/Compact.lean", "def_pos": [40, 5], "def_end_pos": [40, 14]}, {"full_name": "IsClosed", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [196, 7], "def_end_pos": [196, 15]}, {"full_name": "exists_isCompact_isClosed_nhds_one", "def_path": "Mathlib/Topology/Algebra/Group/Basic.lean", "def_pos": [1747, 9], "def_end_pos": [1747, 43]}]], "state_before": "\ud835\udd5c : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u00b9\u2070 : MeasurableSpace G\ninst\u271d\u2079 : MeasurableSpace H\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : BorelSpace G\n\u03bc\u271d : Measure G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : IsMulLeftInvariant \u03bc\u271d\ninst\u271d\u00b3 : WeaklyLocallyCompactSpace G\ninst\u271d\u00b2 : NoncompactSpace G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsOpenPosMeasure \u03bc\ninst\u271d : IsMulLeftInvariant \u03bc\n\u22a2 \u2191\u2191\u03bc univ = \u22a4", "state_after": "case intro.intro.intro\n\ud835\udd5c : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u00b9\u2070 : MeasurableSpace G\ninst\u271d\u2079 : MeasurableSpace H\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : BorelSpace G\n\u03bc\u271d : Measure G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : IsMulLeftInvariant \u03bc\u271d\ninst\u271d\u00b3 : WeaklyLocallyCompactSpace G\ninst\u271d\u00b2 : NoncompactSpace G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsOpenPosMeasure \u03bc\ninst\u271d : IsMulLeftInvariant \u03bc\nK : Set G\nhK : IsCompact K\nKclosed : IsClosed K\nK1 : K \u2208 \ud835\udcdd 1\n\u22a2 \u2191\u2191\u03bc univ = \u22a4"}, {"tactic": "have K_pos : 0 < \u03bc K := measure_pos_of_nonempty_interior _ \u27e8_, mem_interior_iff_mem_nhds.2 K1\u27e9", "annotated_tactic": ["have K_pos : 0 < \u03bc K := <a>measure_pos_of_nonempty_interior</a> _ \u27e8_, <a>mem_interior_iff_mem_nhds</a>.2 K1\u27e9", [{"full_name": "MeasureTheory.Measure.measure_pos_of_nonempty_interior", "def_path": "Mathlib/MeasureTheory/Measure/OpenPos.lean", "def_pos": [62, 9], "def_end_pos": [62, 41]}, {"full_name": "mem_interior_iff_mem_nhds", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [910, 9], "def_end_pos": [910, 34]}]], "state_before": "case intro.intro.intro\n\ud835\udd5c : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u00b9\u2070 : MeasurableSpace G\ninst\u271d\u2079 : MeasurableSpace H\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : BorelSpace G\n\u03bc\u271d : Measure G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : IsMulLeftInvariant \u03bc\u271d\ninst\u271d\u00b3 : WeaklyLocallyCompactSpace G\ninst\u271d\u00b2 : NoncompactSpace G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsOpenPosMeasure \u03bc\ninst\u271d : IsMulLeftInvariant \u03bc\nK : Set G\nhK : IsCompact K\nKclosed : IsClosed K\nK1 : K \u2208 \ud835\udcdd 1\n\u22a2 \u2191\u2191\u03bc univ = \u22a4", "state_after": "case intro.intro.intro\n\ud835\udd5c : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u00b9\u2070 : MeasurableSpace G\ninst\u271d\u2079 : MeasurableSpace H\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : BorelSpace G\n\u03bc\u271d : Measure G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : IsMulLeftInvariant \u03bc\u271d\ninst\u271d\u00b3 : WeaklyLocallyCompactSpace G\ninst\u271d\u00b2 : NoncompactSpace G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsOpenPosMeasure \u03bc\ninst\u271d : IsMulLeftInvariant \u03bc\nK : Set G\nhK : IsCompact K\nKclosed : IsClosed K\nK1 : K \u2208 \ud835\udcdd 1\nK_pos : 0 < \u2191\u2191\u03bc K\n\u22a2 \u2191\u2191\u03bc univ = \u22a4"}, {"tactic": "have A : \u2200 L : Set G, IsCompact L \u2192 \u2203 g : G, Disjoint L (g \u2022 K) := fun L hL =>\n  exists_disjoint_smul_of_isCompact hL hK", "annotated_tactic": ["have A : \u2200 L : <a>Set</a> G, <a>IsCompact</a> L \u2192 \u2203 g : G, <a>Disjoint</a> L (g \u2022 K) := fun L hL =>\n    <a>exists_disjoint_smul_of_isCompact</a> hL hK", [{"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}, {"full_name": "IsCompact", "def_path": "Mathlib/Topology/Compactness/Compact.lean", "def_pos": [40, 5], "def_end_pos": [40, 14]}, {"full_name": "Disjoint", "def_path": "Mathlib/Order/Disjoint.lean", "def_pos": [41, 5], "def_end_pos": [41, 13]}, {"full_name": "exists_disjoint_smul_of_isCompact", "def_path": "Mathlib/Topology/Algebra/Group/Basic.lean", "def_pos": [1705, 9], "def_end_pos": [1705, 42]}]], "state_before": "case intro.intro.intro\n\ud835\udd5c : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u00b9\u2070 : MeasurableSpace G\ninst\u271d\u2079 : MeasurableSpace H\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : BorelSpace G\n\u03bc\u271d : Measure G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : IsMulLeftInvariant \u03bc\u271d\ninst\u271d\u00b3 : WeaklyLocallyCompactSpace G\ninst\u271d\u00b2 : NoncompactSpace G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsOpenPosMeasure \u03bc\ninst\u271d : IsMulLeftInvariant \u03bc\nK : Set G\nhK : IsCompact K\nKclosed : IsClosed K\nK1 : K \u2208 \ud835\udcdd 1\nK_pos : 0 < \u2191\u2191\u03bc K\n\u22a2 \u2191\u2191\u03bc univ = \u22a4", "state_after": "case intro.intro.intro\n\ud835\udd5c : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u00b9\u2070 : MeasurableSpace G\ninst\u271d\u2079 : MeasurableSpace H\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : BorelSpace G\n\u03bc\u271d : Measure G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : IsMulLeftInvariant \u03bc\u271d\ninst\u271d\u00b3 : WeaklyLocallyCompactSpace G\ninst\u271d\u00b2 : NoncompactSpace G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsOpenPosMeasure \u03bc\ninst\u271d : IsMulLeftInvariant \u03bc\nK : Set G\nhK : IsCompact K\nKclosed : IsClosed K\nK1 : K \u2208 \ud835\udcdd 1\nK_pos : 0 < \u2191\u2191\u03bc K\nA : \u2200 (L : Set G), IsCompact L \u2192 \u2203 g, Disjoint L (g \u2022 K)\n\u22a2 \u2191\u2191\u03bc univ = \u22a4"}, {"tactic": "choose! g hg using A", "annotated_tactic": ["choose! g hg using A", []], "state_before": "case intro.intro.intro\n\ud835\udd5c : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u00b9\u2070 : MeasurableSpace G\ninst\u271d\u2079 : MeasurableSpace H\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : BorelSpace G\n\u03bc\u271d : Measure G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : IsMulLeftInvariant \u03bc\u271d\ninst\u271d\u00b3 : WeaklyLocallyCompactSpace G\ninst\u271d\u00b2 : NoncompactSpace G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsOpenPosMeasure \u03bc\ninst\u271d : IsMulLeftInvariant \u03bc\nK : Set G\nhK : IsCompact K\nKclosed : IsClosed K\nK1 : K \u2208 \ud835\udcdd 1\nK_pos : 0 < \u2191\u2191\u03bc K\nA : \u2200 (L : Set G), IsCompact L \u2192 \u2203 g, Disjoint L (g \u2022 K)\n\u22a2 \u2191\u2191\u03bc univ = \u22a4", "state_after": "case intro.intro.intro\n\ud835\udd5c : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u00b9\u2070 : MeasurableSpace G\ninst\u271d\u2079 : MeasurableSpace H\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : BorelSpace G\n\u03bc\u271d : Measure G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : IsMulLeftInvariant \u03bc\u271d\ninst\u271d\u00b3 : WeaklyLocallyCompactSpace G\ninst\u271d\u00b2 : NoncompactSpace G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsOpenPosMeasure \u03bc\ninst\u271d : IsMulLeftInvariant \u03bc\nK : Set G\nhK : IsCompact K\nKclosed : IsClosed K\nK1 : K \u2208 \ud835\udcdd 1\nK_pos : 0 < \u2191\u2191\u03bc K\ng : Set G \u2192 G\nhg : \u2200 (L : Set G), IsCompact L \u2192 Disjoint L (g L \u2022 K)\n\u22a2 \u2191\u2191\u03bc univ = \u22a4"}, {"tactic": "set L : \u2115 \u2192 Set G := fun n => (fun T => T \u222a g T \u2022 K)^[n] K", "annotated_tactic": ["set L : \u2115 \u2192 <a>Set</a> G := fun n => (fun T => T \u222a g T \u2022 K)^[n] K", [{"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}]], "state_before": "case intro.intro.intro\n\ud835\udd5c : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u00b9\u2070 : MeasurableSpace G\ninst\u271d\u2079 : MeasurableSpace H\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : BorelSpace G\n\u03bc\u271d : Measure G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : IsMulLeftInvariant \u03bc\u271d\ninst\u271d\u00b3 : WeaklyLocallyCompactSpace G\ninst\u271d\u00b2 : NoncompactSpace G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsOpenPosMeasure \u03bc\ninst\u271d : IsMulLeftInvariant \u03bc\nK : Set G\nhK : IsCompact K\nKclosed : IsClosed K\nK1 : K \u2208 \ud835\udcdd 1\nK_pos : 0 < \u2191\u2191\u03bc K\ng : Set G \u2192 G\nhg : \u2200 (L : Set G), IsCompact L \u2192 Disjoint L (g L \u2022 K)\n\u22a2 \u2191\u2191\u03bc univ = \u22a4", "state_after": "case intro.intro.intro\n\ud835\udd5c : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u00b9\u2070 : MeasurableSpace G\ninst\u271d\u2079 : MeasurableSpace H\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : BorelSpace G\n\u03bc\u271d : Measure G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : IsMulLeftInvariant \u03bc\u271d\ninst\u271d\u00b3 : WeaklyLocallyCompactSpace G\ninst\u271d\u00b2 : NoncompactSpace G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsOpenPosMeasure \u03bc\ninst\u271d : IsMulLeftInvariant \u03bc\nK : Set G\nhK : IsCompact K\nKclosed : IsClosed K\nK1 : K \u2208 \ud835\udcdd 1\nK_pos : 0 < \u2191\u2191\u03bc K\ng : Set G \u2192 G\nhg : \u2200 (L : Set G), IsCompact L \u2192 Disjoint L (g L \u2022 K)\nL : \u2115 \u2192 Set G := fun n => (fun T => T \u222a g T \u2022 K)^[n] K\n\u22a2 \u2191\u2191\u03bc univ = \u22a4"}, {"tactic": "have N : Tendsto (fun n => \u03bc (L n)) atTop (\ud835\udcdd (\u221e * \u03bc K)) := by\n  simp_rw [M]\n  apply ENNReal.Tendsto.mul_const _ (Or.inl ENNReal.top_ne_zero)\n  exact ENNReal.tendsto_nat_nhds_top.comp (tendsto_add_atTop_nat _)", "annotated_tactic": ["have N : <a>Tendsto</a> (fun n => \u03bc (L n)) <a>atTop</a> (\ud835\udcdd (\u221e * \u03bc K)) := by\n    simp_rw [M]\n    apply <a>ENNReal.Tendsto.mul_const</a> _ (<a>Or.inl</a> <a>ENNReal.top_ne_zero</a>)\n    exact ENNReal.tendsto_nat_nhds_top.comp (<a>tendsto_add_atTop_nat</a> _)", [{"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2939, 5], "def_end_pos": [2939, 12]}, {"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}, {"full_name": "ENNReal.Tendsto.mul_const", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [379, 19], "def_end_pos": [379, 36]}, {"full_name": "Or.inl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [517, 5], "def_end_pos": [517, 8]}, {"full_name": "ENNReal.top_ne_zero", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [337, 17], "def_end_pos": [337, 28]}, {"full_name": "Filter.tendsto_add_atTop_nat", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [1707, 9], "def_end_pos": [1707, 30]}]], "state_before": "case intro.intro.intro\n\ud835\udd5c : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u00b9\u2070 : MeasurableSpace G\ninst\u271d\u2079 : MeasurableSpace H\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : BorelSpace G\n\u03bc\u271d : Measure G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : IsMulLeftInvariant \u03bc\u271d\ninst\u271d\u00b3 : WeaklyLocallyCompactSpace G\ninst\u271d\u00b2 : NoncompactSpace G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsOpenPosMeasure \u03bc\ninst\u271d : IsMulLeftInvariant \u03bc\nK : Set G\nhK : IsCompact K\nKclosed : IsClosed K\nK1 : K \u2208 \ud835\udcdd 1\nK_pos : 0 < \u2191\u2191\u03bc K\ng : Set G \u2192 G\nhg : \u2200 (L : Set G), IsCompact L \u2192 Disjoint L (g L \u2022 K)\nL : \u2115 \u2192 Set G := fun n => (fun T => T \u222a g T \u2022 K)^[n] K\nLcompact : \u2200 (n : \u2115), IsCompact (L n)\nLclosed : \u2200 (n : \u2115), IsClosed (L n)\nM : \u2200 (n : \u2115), \u2191\u2191\u03bc (L n) = \u2191(n + 1) * \u2191\u2191\u03bc K\n\u22a2 \u2191\u2191\u03bc univ = \u22a4", "state_after": "case intro.intro.intro\n\ud835\udd5c : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u00b9\u2070 : MeasurableSpace G\ninst\u271d\u2079 : MeasurableSpace H\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : BorelSpace G\n\u03bc\u271d : Measure G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : IsMulLeftInvariant \u03bc\u271d\ninst\u271d\u00b3 : WeaklyLocallyCompactSpace G\ninst\u271d\u00b2 : NoncompactSpace G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsOpenPosMeasure \u03bc\ninst\u271d : IsMulLeftInvariant \u03bc\nK : Set G\nhK : IsCompact K\nKclosed : IsClosed K\nK1 : K \u2208 \ud835\udcdd 1\nK_pos : 0 < \u2191\u2191\u03bc K\ng : Set G \u2192 G\nhg : \u2200 (L : Set G), IsCompact L \u2192 Disjoint L (g L \u2022 K)\nL : \u2115 \u2192 Set G := fun n => (fun T => T \u222a g T \u2022 K)^[n] K\nLcompact : \u2200 (n : \u2115), IsCompact (L n)\nLclosed : \u2200 (n : \u2115), IsClosed (L n)\nM : \u2200 (n : \u2115), \u2191\u2191\u03bc (L n) = \u2191(n + 1) * \u2191\u2191\u03bc K\nN : Tendsto (fun n => \u2191\u2191\u03bc (L n)) atTop (\ud835\udcdd (\u22a4 * \u2191\u2191\u03bc K))\n\u22a2 \u2191\u2191\u03bc univ = \u22a4"}, {"tactic": "simp only [ENNReal.top_mul', K_pos.ne', if_false] at N", "annotated_tactic": ["simp only [<a>ENNReal.top_mul'</a>, K_pos.ne', <a>if_false</a>] at N", [{"full_name": "ENNReal.top_mul'", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [583, 9], "def_end_pos": [583, 17]}, {"full_name": "if_false", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [729, 17], "def_end_pos": [729, 25]}]], "state_before": "case intro.intro.intro\n\ud835\udd5c : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u00b9\u2070 : MeasurableSpace G\ninst\u271d\u2079 : MeasurableSpace H\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : BorelSpace G\n\u03bc\u271d : Measure G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : IsMulLeftInvariant \u03bc\u271d\ninst\u271d\u00b3 : WeaklyLocallyCompactSpace G\ninst\u271d\u00b2 : NoncompactSpace G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsOpenPosMeasure \u03bc\ninst\u271d : IsMulLeftInvariant \u03bc\nK : Set G\nhK : IsCompact K\nKclosed : IsClosed K\nK1 : K \u2208 \ud835\udcdd 1\nK_pos : 0 < \u2191\u2191\u03bc K\ng : Set G \u2192 G\nhg : \u2200 (L : Set G), IsCompact L \u2192 Disjoint L (g L \u2022 K)\nL : \u2115 \u2192 Set G := fun n => (fun T => T \u222a g T \u2022 K)^[n] K\nLcompact : \u2200 (n : \u2115), IsCompact (L n)\nLclosed : \u2200 (n : \u2115), IsClosed (L n)\nM : \u2200 (n : \u2115), \u2191\u2191\u03bc (L n) = \u2191(n + 1) * \u2191\u2191\u03bc K\nN : Tendsto (fun n => \u2191\u2191\u03bc (L n)) atTop (\ud835\udcdd (\u22a4 * \u2191\u2191\u03bc K))\n\u22a2 \u2191\u2191\u03bc univ = \u22a4", "state_after": "case intro.intro.intro\n\ud835\udd5c : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u00b9\u2070 : MeasurableSpace G\ninst\u271d\u2079 : MeasurableSpace H\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : BorelSpace G\n\u03bc\u271d : Measure G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : IsMulLeftInvariant \u03bc\u271d\ninst\u271d\u00b3 : WeaklyLocallyCompactSpace G\ninst\u271d\u00b2 : NoncompactSpace G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsOpenPosMeasure \u03bc\ninst\u271d : IsMulLeftInvariant \u03bc\nK : Set G\nhK : IsCompact K\nKclosed : IsClosed K\nK1 : K \u2208 \ud835\udcdd 1\nK_pos : 0 < \u2191\u2191\u03bc K\ng : Set G \u2192 G\nhg : \u2200 (L : Set G), IsCompact L \u2192 Disjoint L (g L \u2022 K)\nL : \u2115 \u2192 Set G := fun n => (fun T => T \u222a g T \u2022 K)^[n] K\nLcompact : \u2200 (n : \u2115), IsCompact (L n)\nLclosed : \u2200 (n : \u2115), IsClosed (L n)\nM : \u2200 (n : \u2115), \u2191\u2191\u03bc (L n) = \u2191(n + 1) * \u2191\u2191\u03bc K\nN : Tendsto (fun n => \u2191\u2191\u03bc ((fun T => T \u222a g T \u2022 K)^[n] K)) atTop (\ud835\udcdd \u22a4)\n\u22a2 \u2191\u2191\u03bc univ = \u22a4"}, {"tactic": "apply top_le_iff.1", "annotated_tactic": ["apply <a>top_le_iff</a>.1", [{"full_name": "top_le_iff", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [157, 9], "def_end_pos": [157, 19]}]], "state_before": "case intro.intro.intro\n\ud835\udd5c : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u00b9\u2070 : MeasurableSpace G\ninst\u271d\u2079 : MeasurableSpace H\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : BorelSpace G\n\u03bc\u271d : Measure G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : IsMulLeftInvariant \u03bc\u271d\ninst\u271d\u00b3 : WeaklyLocallyCompactSpace G\ninst\u271d\u00b2 : NoncompactSpace G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsOpenPosMeasure \u03bc\ninst\u271d : IsMulLeftInvariant \u03bc\nK : Set G\nhK : IsCompact K\nKclosed : IsClosed K\nK1 : K \u2208 \ud835\udcdd 1\nK_pos : 0 < \u2191\u2191\u03bc K\ng : Set G \u2192 G\nhg : \u2200 (L : Set G), IsCompact L \u2192 Disjoint L (g L \u2022 K)\nL : \u2115 \u2192 Set G := fun n => (fun T => T \u222a g T \u2022 K)^[n] K\nLcompact : \u2200 (n : \u2115), IsCompact (L n)\nLclosed : \u2200 (n : \u2115), IsClosed (L n)\nM : \u2200 (n : \u2115), \u2191\u2191\u03bc (L n) = \u2191(n + 1) * \u2191\u2191\u03bc K\nN : Tendsto (fun n => \u2191\u2191\u03bc ((fun T => T \u222a g T \u2022 K)^[n] K)) atTop (\ud835\udcdd \u22a4)\n\u22a2 \u2191\u2191\u03bc univ = \u22a4", "state_after": "case intro.intro.intro\n\ud835\udd5c : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u00b9\u2070 : MeasurableSpace G\ninst\u271d\u2079 : MeasurableSpace H\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : BorelSpace G\n\u03bc\u271d : Measure G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : IsMulLeftInvariant \u03bc\u271d\ninst\u271d\u00b3 : WeaklyLocallyCompactSpace G\ninst\u271d\u00b2 : NoncompactSpace G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsOpenPosMeasure \u03bc\ninst\u271d : IsMulLeftInvariant \u03bc\nK : Set G\nhK : IsCompact K\nKclosed : IsClosed K\nK1 : K \u2208 \ud835\udcdd 1\nK_pos : 0 < \u2191\u2191\u03bc K\ng : Set G \u2192 G\nhg : \u2200 (L : Set G), IsCompact L \u2192 Disjoint L (g L \u2022 K)\nL : \u2115 \u2192 Set G := fun n => (fun T => T \u222a g T \u2022 K)^[n] K\nLcompact : \u2200 (n : \u2115), IsCompact (L n)\nLclosed : \u2200 (n : \u2115), IsClosed (L n)\nM : \u2200 (n : \u2115), \u2191\u2191\u03bc (L n) = \u2191(n + 1) * \u2191\u2191\u03bc K\nN : Tendsto (fun n => \u2191\u2191\u03bc ((fun T => T \u222a g T \u2022 K)^[n] K)) atTop (\ud835\udcdd \u22a4)\n\u22a2 \u22a4 \u2264 \u2191\u2191\u03bc univ"}, {"tactic": "exact le_of_tendsto' N fun n => measure_mono (subset_univ _)", "annotated_tactic": ["exact <a>le_of_tendsto'</a> N fun n => <a>measure_mono</a> (<a>subset_univ</a> _)", [{"full_name": "le_of_tendsto'", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [145, 9], "def_end_pos": [145, 23]}, {"full_name": "MeasureTheory.measure_mono", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [193, 9], "def_end_pos": [193, 21]}, {"full_name": "Set.subset_univ", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [691, 9], "def_end_pos": [691, 20]}]], "state_before": "case intro.intro.intro\n\ud835\udd5c : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u00b9\u2070 : MeasurableSpace G\ninst\u271d\u2079 : MeasurableSpace H\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : BorelSpace G\n\u03bc\u271d : Measure G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : IsMulLeftInvariant \u03bc\u271d\ninst\u271d\u00b3 : WeaklyLocallyCompactSpace G\ninst\u271d\u00b2 : NoncompactSpace G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsOpenPosMeasure \u03bc\ninst\u271d : IsMulLeftInvariant \u03bc\nK : Set G\nhK : IsCompact K\nKclosed : IsClosed K\nK1 : K \u2208 \ud835\udcdd 1\nK_pos : 0 < \u2191\u2191\u03bc K\ng : Set G \u2192 G\nhg : \u2200 (L : Set G), IsCompact L \u2192 Disjoint L (g L \u2022 K)\nL : \u2115 \u2192 Set G := fun n => (fun T => T \u222a g T \u2022 K)^[n] K\nLcompact : \u2200 (n : \u2115), IsCompact (L n)\nLclosed : \u2200 (n : \u2115), IsClosed (L n)\nM : \u2200 (n : \u2115), \u2191\u2191\u03bc (L n) = \u2191(n + 1) * \u2191\u2191\u03bc K\nN : Tendsto (fun n => \u2191\u2191\u03bc ((fun T => T \u222a g T \u2022 K)^[n] K)) atTop (\ud835\udcdd \u22a4)\n\u22a2 \u22a4 \u2264 \u2191\u2191\u03bc univ", "state_after": "no goals"}, {"tactic": "intro n", "annotated_tactic": ["intro n", []], "state_before": "\ud835\udd5c : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u00b9\u2070 : MeasurableSpace G\ninst\u271d\u2079 : MeasurableSpace H\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : BorelSpace G\n\u03bc\u271d : Measure G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : IsMulLeftInvariant \u03bc\u271d\ninst\u271d\u00b3 : WeaklyLocallyCompactSpace G\ninst\u271d\u00b2 : NoncompactSpace G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsOpenPosMeasure \u03bc\ninst\u271d : IsMulLeftInvariant \u03bc\nK : Set G\nhK : IsCompact K\nKclosed : IsClosed K\nK1 : K \u2208 \ud835\udcdd 1\nK_pos : 0 < \u2191\u2191\u03bc K\ng : Set G \u2192 G\nhg : \u2200 (L : Set G), IsCompact L \u2192 Disjoint L (g L \u2022 K)\nL : \u2115 \u2192 Set G := fun n => (fun T => T \u222a g T \u2022 K)^[n] K\n\u22a2 \u2200 (n : \u2115), IsCompact (L n)", "state_after": "\ud835\udd5c : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u00b9\u2070 : MeasurableSpace G\ninst\u271d\u2079 : MeasurableSpace H\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : BorelSpace G\n\u03bc\u271d : Measure G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : IsMulLeftInvariant \u03bc\u271d\ninst\u271d\u00b3 : WeaklyLocallyCompactSpace G\ninst\u271d\u00b2 : NoncompactSpace G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsOpenPosMeasure \u03bc\ninst\u271d : IsMulLeftInvariant \u03bc\nK : Set G\nhK : IsCompact K\nKclosed : IsClosed K\nK1 : K \u2208 \ud835\udcdd 1\nK_pos : 0 < \u2191\u2191\u03bc K\ng : Set G \u2192 G\nhg : \u2200 (L : Set G), IsCompact L \u2192 Disjoint L (g L \u2022 K)\nL : \u2115 \u2192 Set G := fun n => (fun T => T \u222a g T \u2022 K)^[n] K\nn : \u2115\n\u22a2 IsCompact (L n)"}, {"tactic": "induction' n with n IH", "annotated_tactic": ["induction' n with n IH", []], "state_before": "\ud835\udd5c : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u00b9\u2070 : MeasurableSpace G\ninst\u271d\u2079 : MeasurableSpace H\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : BorelSpace G\n\u03bc\u271d : Measure G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : IsMulLeftInvariant \u03bc\u271d\ninst\u271d\u00b3 : WeaklyLocallyCompactSpace G\ninst\u271d\u00b2 : NoncompactSpace G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsOpenPosMeasure \u03bc\ninst\u271d : IsMulLeftInvariant \u03bc\nK : Set G\nhK : IsCompact K\nKclosed : IsClosed K\nK1 : K \u2208 \ud835\udcdd 1\nK_pos : 0 < \u2191\u2191\u03bc K\ng : Set G \u2192 G\nhg : \u2200 (L : Set G), IsCompact L \u2192 Disjoint L (g L \u2022 K)\nL : \u2115 \u2192 Set G := fun n => (fun T => T \u222a g T \u2022 K)^[n] K\nn : \u2115\n\u22a2 IsCompact (L n)", "state_after": "case zero\n\ud835\udd5c : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u00b9\u2070 : MeasurableSpace G\ninst\u271d\u2079 : MeasurableSpace H\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : BorelSpace G\n\u03bc\u271d : Measure G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : IsMulLeftInvariant \u03bc\u271d\ninst\u271d\u00b3 : WeaklyLocallyCompactSpace G\ninst\u271d\u00b2 : NoncompactSpace G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsOpenPosMeasure \u03bc\ninst\u271d : IsMulLeftInvariant \u03bc\nK : Set G\nhK : IsCompact K\nKclosed : IsClosed K\nK1 : K \u2208 \ud835\udcdd 1\nK_pos : 0 < \u2191\u2191\u03bc K\ng : Set G \u2192 G\nhg : \u2200 (L : Set G), IsCompact L \u2192 Disjoint L (g L \u2022 K)\nL : \u2115 \u2192 Set G := fun n => (fun T => T \u222a g T \u2022 K)^[n] K\n\u22a2 IsCompact (L Nat.zero)\n\ncase succ\n\ud835\udd5c : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u00b9\u2070 : MeasurableSpace G\ninst\u271d\u2079 : MeasurableSpace H\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : BorelSpace G\n\u03bc\u271d : Measure G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : IsMulLeftInvariant \u03bc\u271d\ninst\u271d\u00b3 : WeaklyLocallyCompactSpace G\ninst\u271d\u00b2 : NoncompactSpace G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsOpenPosMeasure \u03bc\ninst\u271d : IsMulLeftInvariant \u03bc\nK : Set G\nhK : IsCompact K\nKclosed : IsClosed K\nK1 : K \u2208 \ud835\udcdd 1\nK_pos : 0 < \u2191\u2191\u03bc K\ng : Set G \u2192 G\nhg : \u2200 (L : Set G), IsCompact L \u2192 Disjoint L (g L \u2022 K)\nL : \u2115 \u2192 Set G := fun n => (fun T => T \u222a g T \u2022 K)^[n] K\nn : \u2115\nIH : IsCompact (L n)\n\u22a2 IsCompact (L (Nat.succ n))"}, {"tactic": "exact hK", "annotated_tactic": ["exact hK", []], "state_before": "case zero\n\ud835\udd5c : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u00b9\u2070 : MeasurableSpace G\ninst\u271d\u2079 : MeasurableSpace H\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : BorelSpace G\n\u03bc\u271d : Measure G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : IsMulLeftInvariant \u03bc\u271d\ninst\u271d\u00b3 : WeaklyLocallyCompactSpace G\ninst\u271d\u00b2 : NoncompactSpace G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsOpenPosMeasure \u03bc\ninst\u271d : IsMulLeftInvariant \u03bc\nK : Set G\nhK : IsCompact K\nKclosed : IsClosed K\nK1 : K \u2208 \ud835\udcdd 1\nK_pos : 0 < \u2191\u2191\u03bc K\ng : Set G \u2192 G\nhg : \u2200 (L : Set G), IsCompact L \u2192 Disjoint L (g L \u2022 K)\nL : \u2115 \u2192 Set G := fun n => (fun T => T \u222a g T \u2022 K)^[n] K\n\u22a2 IsCompact (L Nat.zero)", "state_after": "no goals"}, {"tactic": "simp_rw [iterate_succ']", "annotated_tactic": ["simp_rw [<a>iterate_succ'</a>]", [{"full_name": "Function.iterate_succ'", "def_path": "Mathlib/Logic/Function/Iterate.lean", "def_pos": [186, 9], "def_end_pos": [186, 22]}]], "state_before": "case succ\n\ud835\udd5c : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u00b9\u2070 : MeasurableSpace G\ninst\u271d\u2079 : MeasurableSpace H\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : BorelSpace G\n\u03bc\u271d : Measure G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : IsMulLeftInvariant \u03bc\u271d\ninst\u271d\u00b3 : WeaklyLocallyCompactSpace G\ninst\u271d\u00b2 : NoncompactSpace G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsOpenPosMeasure \u03bc\ninst\u271d : IsMulLeftInvariant \u03bc\nK : Set G\nhK : IsCompact K\nKclosed : IsClosed K\nK1 : K \u2208 \ud835\udcdd 1\nK_pos : 0 < \u2191\u2191\u03bc K\ng : Set G \u2192 G\nhg : \u2200 (L : Set G), IsCompact L \u2192 Disjoint L (g L \u2022 K)\nL : \u2115 \u2192 Set G := fun n => (fun T => T \u222a g T \u2022 K)^[n] K\nn : \u2115\nIH : IsCompact (L n)\n\u22a2 IsCompact (L (Nat.succ n))", "state_after": "case succ\n\ud835\udd5c : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u00b9\u2070 : MeasurableSpace G\ninst\u271d\u2079 : MeasurableSpace H\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : BorelSpace G\n\u03bc\u271d : Measure G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : IsMulLeftInvariant \u03bc\u271d\ninst\u271d\u00b3 : WeaklyLocallyCompactSpace G\ninst\u271d\u00b2 : NoncompactSpace G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsOpenPosMeasure \u03bc\ninst\u271d : IsMulLeftInvariant \u03bc\nK : Set G\nhK : IsCompact K\nKclosed : IsClosed K\nK1 : K \u2208 \ud835\udcdd 1\nK_pos : 0 < \u2191\u2191\u03bc K\ng : Set G \u2192 G\nhg : \u2200 (L : Set G), IsCompact L \u2192 Disjoint L (g L \u2022 K)\nL : \u2115 \u2192 Set G := fun n => (fun T => T \u222a g T \u2022 K)^[n] K\nn : \u2115\nIH : IsCompact (L n)\n\u22a2 IsCompact (((fun T => T \u222a g T \u2022 K) \u2218 (fun T => T \u222a g T \u2022 K)^[n]) K)"}, {"tactic": "apply IsCompact.union IH (hK.smul (g (L n)))", "annotated_tactic": ["apply <a>IsCompact.union</a> IH (hK.smul (g (L n)))", [{"full_name": "IsCompact.union", "def_path": "Mathlib/Topology/Compactness/Compact.lean", "def_pos": [446, 9], "def_end_pos": [446, 24]}]], "state_before": "case succ\n\ud835\udd5c : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u00b9\u2070 : MeasurableSpace G\ninst\u271d\u2079 : MeasurableSpace H\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : BorelSpace G\n\u03bc\u271d : Measure G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : IsMulLeftInvariant \u03bc\u271d\ninst\u271d\u00b3 : WeaklyLocallyCompactSpace G\ninst\u271d\u00b2 : NoncompactSpace G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsOpenPosMeasure \u03bc\ninst\u271d : IsMulLeftInvariant \u03bc\nK : Set G\nhK : IsCompact K\nKclosed : IsClosed K\nK1 : K \u2208 \ud835\udcdd 1\nK_pos : 0 < \u2191\u2191\u03bc K\ng : Set G \u2192 G\nhg : \u2200 (L : Set G), IsCompact L \u2192 Disjoint L (g L \u2022 K)\nL : \u2115 \u2192 Set G := fun n => (fun T => T \u222a g T \u2022 K)^[n] K\nn : \u2115\nIH : IsCompact (L n)\n\u22a2 IsCompact (((fun T => T \u222a g T \u2022 K) \u2218 (fun T => T \u222a g T \u2022 K)^[n]) K)", "state_after": "no goals"}, {"tactic": "intro n", "annotated_tactic": ["intro n", []], "state_before": "\ud835\udd5c : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u00b9\u2070 : MeasurableSpace G\ninst\u271d\u2079 : MeasurableSpace H\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : BorelSpace G\n\u03bc\u271d : Measure G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : IsMulLeftInvariant \u03bc\u271d\ninst\u271d\u00b3 : WeaklyLocallyCompactSpace G\ninst\u271d\u00b2 : NoncompactSpace G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsOpenPosMeasure \u03bc\ninst\u271d : IsMulLeftInvariant \u03bc\nK : Set G\nhK : IsCompact K\nKclosed : IsClosed K\nK1 : K \u2208 \ud835\udcdd 1\nK_pos : 0 < \u2191\u2191\u03bc K\ng : Set G \u2192 G\nhg : \u2200 (L : Set G), IsCompact L \u2192 Disjoint L (g L \u2022 K)\nL : \u2115 \u2192 Set G := fun n => (fun T => T \u222a g T \u2022 K)^[n] K\nLcompact : \u2200 (n : \u2115), IsCompact (L n)\n\u22a2 \u2200 (n : \u2115), IsClosed (L n)", "state_after": "\ud835\udd5c : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u00b9\u2070 : MeasurableSpace G\ninst\u271d\u2079 : MeasurableSpace H\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : BorelSpace G\n\u03bc\u271d : Measure G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : IsMulLeftInvariant \u03bc\u271d\ninst\u271d\u00b3 : WeaklyLocallyCompactSpace G\ninst\u271d\u00b2 : NoncompactSpace G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsOpenPosMeasure \u03bc\ninst\u271d : IsMulLeftInvariant \u03bc\nK : Set G\nhK : IsCompact K\nKclosed : IsClosed K\nK1 : K \u2208 \ud835\udcdd 1\nK_pos : 0 < \u2191\u2191\u03bc K\ng : Set G \u2192 G\nhg : \u2200 (L : Set G), IsCompact L \u2192 Disjoint L (g L \u2022 K)\nL : \u2115 \u2192 Set G := fun n => (fun T => T \u222a g T \u2022 K)^[n] K\nLcompact : \u2200 (n : \u2115), IsCompact (L n)\nn : \u2115\n\u22a2 IsClosed (L n)"}, {"tactic": "induction' n with n IH", "annotated_tactic": ["induction' n with n IH", []], "state_before": "\ud835\udd5c : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u00b9\u2070 : MeasurableSpace G\ninst\u271d\u2079 : MeasurableSpace H\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : BorelSpace G\n\u03bc\u271d : Measure G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : IsMulLeftInvariant \u03bc\u271d\ninst\u271d\u00b3 : WeaklyLocallyCompactSpace G\ninst\u271d\u00b2 : NoncompactSpace G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsOpenPosMeasure \u03bc\ninst\u271d : IsMulLeftInvariant \u03bc\nK : Set G\nhK : IsCompact K\nKclosed : IsClosed K\nK1 : K \u2208 \ud835\udcdd 1\nK_pos : 0 < \u2191\u2191\u03bc K\ng : Set G \u2192 G\nhg : \u2200 (L : Set G), IsCompact L \u2192 Disjoint L (g L \u2022 K)\nL : \u2115 \u2192 Set G := fun n => (fun T => T \u222a g T \u2022 K)^[n] K\nLcompact : \u2200 (n : \u2115), IsCompact (L n)\nn : \u2115\n\u22a2 IsClosed (L n)", "state_after": "case zero\n\ud835\udd5c : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u00b9\u2070 : MeasurableSpace G\ninst\u271d\u2079 : MeasurableSpace H\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : BorelSpace G\n\u03bc\u271d : Measure G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : IsMulLeftInvariant \u03bc\u271d\ninst\u271d\u00b3 : WeaklyLocallyCompactSpace G\ninst\u271d\u00b2 : NoncompactSpace G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsOpenPosMeasure \u03bc\ninst\u271d : IsMulLeftInvariant \u03bc\nK : Set G\nhK : IsCompact K\nKclosed : IsClosed K\nK1 : K \u2208 \ud835\udcdd 1\nK_pos : 0 < \u2191\u2191\u03bc K\ng : Set G \u2192 G\nhg : \u2200 (L : Set G), IsCompact L \u2192 Disjoint L (g L \u2022 K)\nL : \u2115 \u2192 Set G := fun n => (fun T => T \u222a g T \u2022 K)^[n] K\nLcompact : \u2200 (n : \u2115), IsCompact (L n)\n\u22a2 IsClosed (L Nat.zero)\n\ncase succ\n\ud835\udd5c : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u00b9\u2070 : MeasurableSpace G\ninst\u271d\u2079 : MeasurableSpace H\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : BorelSpace G\n\u03bc\u271d : Measure G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : IsMulLeftInvariant \u03bc\u271d\ninst\u271d\u00b3 : WeaklyLocallyCompactSpace G\ninst\u271d\u00b2 : NoncompactSpace G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsOpenPosMeasure \u03bc\ninst\u271d : IsMulLeftInvariant \u03bc\nK : Set G\nhK : IsCompact K\nKclosed : IsClosed K\nK1 : K \u2208 \ud835\udcdd 1\nK_pos : 0 < \u2191\u2191\u03bc K\ng : Set G \u2192 G\nhg : \u2200 (L : Set G), IsCompact L \u2192 Disjoint L (g L \u2022 K)\nL : \u2115 \u2192 Set G := fun n => (fun T => T \u222a g T \u2022 K)^[n] K\nLcompact : \u2200 (n : \u2115), IsCompact (L n)\nn : \u2115\nIH : IsClosed (L n)\n\u22a2 IsClosed (L (Nat.succ n))"}, {"tactic": "exact Kclosed", "annotated_tactic": ["exact Kclosed", []], "state_before": "case zero\n\ud835\udd5c : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u00b9\u2070 : MeasurableSpace G\ninst\u271d\u2079 : MeasurableSpace H\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : BorelSpace G\n\u03bc\u271d : Measure G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : IsMulLeftInvariant \u03bc\u271d\ninst\u271d\u00b3 : WeaklyLocallyCompactSpace G\ninst\u271d\u00b2 : NoncompactSpace G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsOpenPosMeasure \u03bc\ninst\u271d : IsMulLeftInvariant \u03bc\nK : Set G\nhK : IsCompact K\nKclosed : IsClosed K\nK1 : K \u2208 \ud835\udcdd 1\nK_pos : 0 < \u2191\u2191\u03bc K\ng : Set G \u2192 G\nhg : \u2200 (L : Set G), IsCompact L \u2192 Disjoint L (g L \u2022 K)\nL : \u2115 \u2192 Set G := fun n => (fun T => T \u222a g T \u2022 K)^[n] K\nLcompact : \u2200 (n : \u2115), IsCompact (L n)\n\u22a2 IsClosed (L Nat.zero)", "state_after": "no goals"}, {"tactic": "simp_rw [iterate_succ']", "annotated_tactic": ["simp_rw [<a>iterate_succ'</a>]", [{"full_name": "Function.iterate_succ'", "def_path": "Mathlib/Logic/Function/Iterate.lean", "def_pos": [186, 9], "def_end_pos": [186, 22]}]], "state_before": "case succ\n\ud835\udd5c : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u00b9\u2070 : MeasurableSpace G\ninst\u271d\u2079 : MeasurableSpace H\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : BorelSpace G\n\u03bc\u271d : Measure G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : IsMulLeftInvariant \u03bc\u271d\ninst\u271d\u00b3 : WeaklyLocallyCompactSpace G\ninst\u271d\u00b2 : NoncompactSpace G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsOpenPosMeasure \u03bc\ninst\u271d : IsMulLeftInvariant \u03bc\nK : Set G\nhK : IsCompact K\nKclosed : IsClosed K\nK1 : K \u2208 \ud835\udcdd 1\nK_pos : 0 < \u2191\u2191\u03bc K\ng : Set G \u2192 G\nhg : \u2200 (L : Set G), IsCompact L \u2192 Disjoint L (g L \u2022 K)\nL : \u2115 \u2192 Set G := fun n => (fun T => T \u222a g T \u2022 K)^[n] K\nLcompact : \u2200 (n : \u2115), IsCompact (L n)\nn : \u2115\nIH : IsClosed (L n)\n\u22a2 IsClosed (L (Nat.succ n))", "state_after": "case succ\n\ud835\udd5c : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u00b9\u2070 : MeasurableSpace G\ninst\u271d\u2079 : MeasurableSpace H\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : BorelSpace G\n\u03bc\u271d : Measure G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : IsMulLeftInvariant \u03bc\u271d\ninst\u271d\u00b3 : WeaklyLocallyCompactSpace G\ninst\u271d\u00b2 : NoncompactSpace G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsOpenPosMeasure \u03bc\ninst\u271d : IsMulLeftInvariant \u03bc\nK : Set G\nhK : IsCompact K\nKclosed : IsClosed K\nK1 : K \u2208 \ud835\udcdd 1\nK_pos : 0 < \u2191\u2191\u03bc K\ng : Set G \u2192 G\nhg : \u2200 (L : Set G), IsCompact L \u2192 Disjoint L (g L \u2022 K)\nL : \u2115 \u2192 Set G := fun n => (fun T => T \u222a g T \u2022 K)^[n] K\nLcompact : \u2200 (n : \u2115), IsCompact (L n)\nn : \u2115\nIH : IsClosed (L n)\n\u22a2 IsClosed (((fun T => T \u222a g T \u2022 K) \u2218 (fun T => T \u222a g T \u2022 K)^[n]) K)"}, {"tactic": "apply IsClosed.union IH (Kclosed.smul (g (L n)))", "annotated_tactic": ["apply <a>IsClosed.union</a> IH (Kclosed.smul (g (L n)))", [{"full_name": "IsClosed.union", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [218, 9], "def_end_pos": [218, 23]}]], "state_before": "case succ\n\ud835\udd5c : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u00b9\u2070 : MeasurableSpace G\ninst\u271d\u2079 : MeasurableSpace H\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : BorelSpace G\n\u03bc\u271d : Measure G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : IsMulLeftInvariant \u03bc\u271d\ninst\u271d\u00b3 : WeaklyLocallyCompactSpace G\ninst\u271d\u00b2 : NoncompactSpace G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsOpenPosMeasure \u03bc\ninst\u271d : IsMulLeftInvariant \u03bc\nK : Set G\nhK : IsCompact K\nKclosed : IsClosed K\nK1 : K \u2208 \ud835\udcdd 1\nK_pos : 0 < \u2191\u2191\u03bc K\ng : Set G \u2192 G\nhg : \u2200 (L : Set G), IsCompact L \u2192 Disjoint L (g L \u2022 K)\nL : \u2115 \u2192 Set G := fun n => (fun T => T \u222a g T \u2022 K)^[n] K\nLcompact : \u2200 (n : \u2115), IsCompact (L n)\nn : \u2115\nIH : IsClosed (L n)\n\u22a2 IsClosed (((fun T => T \u222a g T \u2022 K) \u2218 (fun T => T \u222a g T \u2022 K)^[n]) K)", "state_after": "no goals"}, {"tactic": "intro n", "annotated_tactic": ["intro n", []], "state_before": "\ud835\udd5c : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u00b9\u2070 : MeasurableSpace G\ninst\u271d\u2079 : MeasurableSpace H\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : BorelSpace G\n\u03bc\u271d : Measure G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : IsMulLeftInvariant \u03bc\u271d\ninst\u271d\u00b3 : WeaklyLocallyCompactSpace G\ninst\u271d\u00b2 : NoncompactSpace G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsOpenPosMeasure \u03bc\ninst\u271d : IsMulLeftInvariant \u03bc\nK : Set G\nhK : IsCompact K\nKclosed : IsClosed K\nK1 : K \u2208 \ud835\udcdd 1\nK_pos : 0 < \u2191\u2191\u03bc K\ng : Set G \u2192 G\nhg : \u2200 (L : Set G), IsCompact L \u2192 Disjoint L (g L \u2022 K)\nL : \u2115 \u2192 Set G := fun n => (fun T => T \u222a g T \u2022 K)^[n] K\nLcompact : \u2200 (n : \u2115), IsCompact (L n)\nLclosed : \u2200 (n : \u2115), IsClosed (L n)\n\u22a2 \u2200 (n : \u2115), \u2191\u2191\u03bc (L n) = \u2191(n + 1) * \u2191\u2191\u03bc K", "state_after": "\ud835\udd5c : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u00b9\u2070 : MeasurableSpace G\ninst\u271d\u2079 : MeasurableSpace H\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : BorelSpace G\n\u03bc\u271d : Measure G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : IsMulLeftInvariant \u03bc\u271d\ninst\u271d\u00b3 : WeaklyLocallyCompactSpace G\ninst\u271d\u00b2 : NoncompactSpace G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsOpenPosMeasure \u03bc\ninst\u271d : IsMulLeftInvariant \u03bc\nK : Set G\nhK : IsCompact K\nKclosed : IsClosed K\nK1 : K \u2208 \ud835\udcdd 1\nK_pos : 0 < \u2191\u2191\u03bc K\ng : Set G \u2192 G\nhg : \u2200 (L : Set G), IsCompact L \u2192 Disjoint L (g L \u2022 K)\nL : \u2115 \u2192 Set G := fun n => (fun T => T \u222a g T \u2022 K)^[n] K\nLcompact : \u2200 (n : \u2115), IsCompact (L n)\nLclosed : \u2200 (n : \u2115), IsClosed (L n)\nn : \u2115\n\u22a2 \u2191\u2191\u03bc (L n) = \u2191(n + 1) * \u2191\u2191\u03bc K"}, {"tactic": "induction' n with n IH", "annotated_tactic": ["induction' n with n IH", []], "state_before": "\ud835\udd5c : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u00b9\u2070 : MeasurableSpace G\ninst\u271d\u2079 : MeasurableSpace H\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : BorelSpace G\n\u03bc\u271d : Measure G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : IsMulLeftInvariant \u03bc\u271d\ninst\u271d\u00b3 : WeaklyLocallyCompactSpace G\ninst\u271d\u00b2 : NoncompactSpace G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsOpenPosMeasure \u03bc\ninst\u271d : IsMulLeftInvariant \u03bc\nK : Set G\nhK : IsCompact K\nKclosed : IsClosed K\nK1 : K \u2208 \ud835\udcdd 1\nK_pos : 0 < \u2191\u2191\u03bc K\ng : Set G \u2192 G\nhg : \u2200 (L : Set G), IsCompact L \u2192 Disjoint L (g L \u2022 K)\nL : \u2115 \u2192 Set G := fun n => (fun T => T \u222a g T \u2022 K)^[n] K\nLcompact : \u2200 (n : \u2115), IsCompact (L n)\nLclosed : \u2200 (n : \u2115), IsClosed (L n)\nn : \u2115\n\u22a2 \u2191\u2191\u03bc (L n) = \u2191(n + 1) * \u2191\u2191\u03bc K", "state_after": "case zero\n\ud835\udd5c : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u00b9\u2070 : MeasurableSpace G\ninst\u271d\u2079 : MeasurableSpace H\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : BorelSpace G\n\u03bc\u271d : Measure G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : IsMulLeftInvariant \u03bc\u271d\ninst\u271d\u00b3 : WeaklyLocallyCompactSpace G\ninst\u271d\u00b2 : NoncompactSpace G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsOpenPosMeasure \u03bc\ninst\u271d : IsMulLeftInvariant \u03bc\nK : Set G\nhK : IsCompact K\nKclosed : IsClosed K\nK1 : K \u2208 \ud835\udcdd 1\nK_pos : 0 < \u2191\u2191\u03bc K\ng : Set G \u2192 G\nhg : \u2200 (L : Set G), IsCompact L \u2192 Disjoint L (g L \u2022 K)\nL : \u2115 \u2192 Set G := fun n => (fun T => T \u222a g T \u2022 K)^[n] K\nLcompact : \u2200 (n : \u2115), IsCompact (L n)\nLclosed : \u2200 (n : \u2115), IsClosed (L n)\n\u22a2 \u2191\u2191\u03bc (L Nat.zero) = \u2191(Nat.zero + 1) * \u2191\u2191\u03bc K\n\ncase succ\n\ud835\udd5c : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u00b9\u2070 : MeasurableSpace G\ninst\u271d\u2079 : MeasurableSpace H\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : BorelSpace G\n\u03bc\u271d : Measure G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : IsMulLeftInvariant \u03bc\u271d\ninst\u271d\u00b3 : WeaklyLocallyCompactSpace G\ninst\u271d\u00b2 : NoncompactSpace G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsOpenPosMeasure \u03bc\ninst\u271d : IsMulLeftInvariant \u03bc\nK : Set G\nhK : IsCompact K\nKclosed : IsClosed K\nK1 : K \u2208 \ud835\udcdd 1\nK_pos : 0 < \u2191\u2191\u03bc K\ng : Set G \u2192 G\nhg : \u2200 (L : Set G), IsCompact L \u2192 Disjoint L (g L \u2022 K)\nL : \u2115 \u2192 Set G := fun n => (fun T => T \u222a g T \u2022 K)^[n] K\nLcompact : \u2200 (n : \u2115), IsCompact (L n)\nLclosed : \u2200 (n : \u2115), IsClosed (L n)\nn : \u2115\nIH : \u2191\u2191\u03bc (L n) = \u2191(n + 1) * \u2191\u2191\u03bc K\n\u22a2 \u2191\u2191\u03bc (L (Nat.succ n)) = \u2191(Nat.succ n + 1) * \u2191\u2191\u03bc K"}, {"tactic": "simp only [one_mul, Nat.cast_one, iterate_zero, id.def, Nat.zero_eq, Nat.zero_add]", "annotated_tactic": ["simp only [<a>one_mul</a>, <a>Nat.cast_one</a>, <a>iterate_zero</a>, <a>id.def</a>, <a>Nat.zero_eq</a>, <a>Nat.zero_add</a>]", [{"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [464, 9], "def_end_pos": [464, 16]}, {"full_name": "Nat.cast_one", "def_path": "Mathlib/Data/Nat/Cast/Defs.lean", "def_pos": [141, 9], "def_end_pos": [141, 17]}, {"full_name": "Function.iterate_zero", "def_path": "Mathlib/Logic/Function/Iterate.lean", "def_pos": [53, 9], "def_end_pos": [53, 21]}, {"full_name": "id.def", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [527, 9], "def_end_pos": [527, 15]}, {"full_name": "Nat.zero_eq", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [83, 17], "def_end_pos": [83, 24]}, {"full_name": "Nat.zero_add", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [114, 27], "def_end_pos": [114, 35]}]], "state_before": "case zero\n\ud835\udd5c : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u00b9\u2070 : MeasurableSpace G\ninst\u271d\u2079 : MeasurableSpace H\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : BorelSpace G\n\u03bc\u271d : Measure G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : IsMulLeftInvariant \u03bc\u271d\ninst\u271d\u00b3 : WeaklyLocallyCompactSpace G\ninst\u271d\u00b2 : NoncompactSpace G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsOpenPosMeasure \u03bc\ninst\u271d : IsMulLeftInvariant \u03bc\nK : Set G\nhK : IsCompact K\nKclosed : IsClosed K\nK1 : K \u2208 \ud835\udcdd 1\nK_pos : 0 < \u2191\u2191\u03bc K\ng : Set G \u2192 G\nhg : \u2200 (L : Set G), IsCompact L \u2192 Disjoint L (g L \u2022 K)\nL : \u2115 \u2192 Set G := fun n => (fun T => T \u222a g T \u2022 K)^[n] K\nLcompact : \u2200 (n : \u2115), IsCompact (L n)\nLclosed : \u2200 (n : \u2115), IsClosed (L n)\n\u22a2 \u2191\u2191\u03bc (L Nat.zero) = \u2191(Nat.zero + 1) * \u2191\u2191\u03bc K", "state_after": "no goals"}, {"tactic": "calc\n  \u03bc (L (n + 1)) = \u03bc (L n) + \u03bc (g (L n) \u2022 K) := by\n    simp_rw [iterate_succ']\n    exact measure_union' (hg _ (Lcompact _)) (Lclosed _).measurableSet\n  _ = (n + 1 + 1 : \u2115) * \u03bc K := by\n    simp only [IH, measure_smul, add_mul, Nat.cast_add, Nat.cast_one, one_mul]", "annotated_tactic": ["calc\n        \u03bc (L (n + 1)) = \u03bc (L n) + \u03bc (g (L n) \u2022 K) := by\n          simp_rw [<a>iterate_succ'</a>]\n          exact <a>measure_union'</a> (hg _ (Lcompact _)) (Lclosed _).<a>measurableSet</a>\n        _ = (n + 1 + 1 : \u2115) * \u03bc K := by\n          simp only [IH, <a>measure_smul</a>, <a>add_mul</a>, <a>Nat.cast_add</a>, <a>Nat.cast_one</a>, <a>one_mul</a>]", [{"full_name": "Function.iterate_succ'", "def_path": "Mathlib/Logic/Function/Iterate.lean", "def_pos": [186, 9], "def_end_pos": [186, 22]}, {"full_name": "MeasureTheory.measure_union'", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [128, 9], "def_end_pos": [128, 23]}, {"full_name": "IsClosed.measurableSet", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [340, 9], "def_end_pos": [340, 31]}, {"full_name": "MeasureTheory.measure_smul", "def_path": "Mathlib/MeasureTheory/Group/Action.lean", "def_pos": [215, 9], "def_end_pos": [215, 21]}, {"full_name": "add_mul", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [91, 7], "def_end_pos": [91, 14]}, {"full_name": "Nat.cast_add", "def_path": "Mathlib/Data/Nat/Cast/Defs.lean", "def_pos": [146, 9], "def_end_pos": [146, 17]}, {"full_name": "Nat.cast_one", "def_path": "Mathlib/Data/Nat/Cast/Defs.lean", "def_pos": [141, 9], "def_end_pos": [141, 17]}, {"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [464, 9], "def_end_pos": [464, 16]}]], "state_before": "case succ\n\ud835\udd5c : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u00b9\u2070 : MeasurableSpace G\ninst\u271d\u2079 : MeasurableSpace H\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : BorelSpace G\n\u03bc\u271d : Measure G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : IsMulLeftInvariant \u03bc\u271d\ninst\u271d\u00b3 : WeaklyLocallyCompactSpace G\ninst\u271d\u00b2 : NoncompactSpace G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsOpenPosMeasure \u03bc\ninst\u271d : IsMulLeftInvariant \u03bc\nK : Set G\nhK : IsCompact K\nKclosed : IsClosed K\nK1 : K \u2208 \ud835\udcdd 1\nK_pos : 0 < \u2191\u2191\u03bc K\ng : Set G \u2192 G\nhg : \u2200 (L : Set G), IsCompact L \u2192 Disjoint L (g L \u2022 K)\nL : \u2115 \u2192 Set G := fun n => (fun T => T \u222a g T \u2022 K)^[n] K\nLcompact : \u2200 (n : \u2115), IsCompact (L n)\nLclosed : \u2200 (n : \u2115), IsClosed (L n)\nn : \u2115\nIH : \u2191\u2191\u03bc (L n) = \u2191(n + 1) * \u2191\u2191\u03bc K\n\u22a2 \u2191\u2191\u03bc (L (Nat.succ n)) = \u2191(Nat.succ n + 1) * \u2191\u2191\u03bc K", "state_after": "no goals"}, {"tactic": "simp_rw [iterate_succ']", "annotated_tactic": ["simp_rw [<a>iterate_succ'</a>]", [{"full_name": "Function.iterate_succ'", "def_path": "Mathlib/Logic/Function/Iterate.lean", "def_pos": [186, 9], "def_end_pos": [186, 22]}]], "state_before": "\ud835\udd5c : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u00b9\u2070 : MeasurableSpace G\ninst\u271d\u2079 : MeasurableSpace H\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : BorelSpace G\n\u03bc\u271d : Measure G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : IsMulLeftInvariant \u03bc\u271d\ninst\u271d\u00b3 : WeaklyLocallyCompactSpace G\ninst\u271d\u00b2 : NoncompactSpace G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsOpenPosMeasure \u03bc\ninst\u271d : IsMulLeftInvariant \u03bc\nK : Set G\nhK : IsCompact K\nKclosed : IsClosed K\nK1 : K \u2208 \ud835\udcdd 1\nK_pos : 0 < \u2191\u2191\u03bc K\ng : Set G \u2192 G\nhg : \u2200 (L : Set G), IsCompact L \u2192 Disjoint L (g L \u2022 K)\nL : \u2115 \u2192 Set G := fun n => (fun T => T \u222a g T \u2022 K)^[n] K\nLcompact : \u2200 (n : \u2115), IsCompact (L n)\nLclosed : \u2200 (n : \u2115), IsClosed (L n)\nn : \u2115\nIH : \u2191\u2191\u03bc (L n) = \u2191(n + 1) * \u2191\u2191\u03bc K\n\u22a2 \u2191\u2191\u03bc (L (n + 1)) = \u2191\u2191\u03bc (L n) + \u2191\u2191\u03bc (g (L n) \u2022 K)", "state_after": "\ud835\udd5c : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u00b9\u2070 : MeasurableSpace G\ninst\u271d\u2079 : MeasurableSpace H\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : BorelSpace G\n\u03bc\u271d : Measure G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : IsMulLeftInvariant \u03bc\u271d\ninst\u271d\u00b3 : WeaklyLocallyCompactSpace G\ninst\u271d\u00b2 : NoncompactSpace G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsOpenPosMeasure \u03bc\ninst\u271d : IsMulLeftInvariant \u03bc\nK : Set G\nhK : IsCompact K\nKclosed : IsClosed K\nK1 : K \u2208 \ud835\udcdd 1\nK_pos : 0 < \u2191\u2191\u03bc K\ng : Set G \u2192 G\nhg : \u2200 (L : Set G), IsCompact L \u2192 Disjoint L (g L \u2022 K)\nL : \u2115 \u2192 Set G := fun n => (fun T => T \u222a g T \u2022 K)^[n] K\nLcompact : \u2200 (n : \u2115), IsCompact (L n)\nLclosed : \u2200 (n : \u2115), IsClosed (L n)\nn : \u2115\nIH : \u2191\u2191\u03bc (L n) = \u2191(n + 1) * \u2191\u2191\u03bc K\n\u22a2 \u2191\u2191\u03bc (((fun T => T \u222a g T \u2022 K) \u2218 (fun T => T \u222a g T \u2022 K)^[n]) K) =\n    \u2191\u2191\u03bc ((fun T => T \u222a g T \u2022 K)^[n] K) + \u2191\u2191\u03bc (g ((fun T => T \u222a g T \u2022 K)^[n] K) \u2022 K)"}, {"tactic": "exact measure_union' (hg _ (Lcompact _)) (Lclosed _).measurableSet", "annotated_tactic": ["exact <a>measure_union'</a> (hg _ (Lcompact _)) (Lclosed _).<a>measurableSet</a>", [{"full_name": "MeasureTheory.measure_union'", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [128, 9], "def_end_pos": [128, 23]}, {"full_name": "IsClosed.measurableSet", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [340, 9], "def_end_pos": [340, 31]}]], "state_before": "\ud835\udd5c : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u00b9\u2070 : MeasurableSpace G\ninst\u271d\u2079 : MeasurableSpace H\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : BorelSpace G\n\u03bc\u271d : Measure G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : IsMulLeftInvariant \u03bc\u271d\ninst\u271d\u00b3 : WeaklyLocallyCompactSpace G\ninst\u271d\u00b2 : NoncompactSpace G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsOpenPosMeasure \u03bc\ninst\u271d : IsMulLeftInvariant \u03bc\nK : Set G\nhK : IsCompact K\nKclosed : IsClosed K\nK1 : K \u2208 \ud835\udcdd 1\nK_pos : 0 < \u2191\u2191\u03bc K\ng : Set G \u2192 G\nhg : \u2200 (L : Set G), IsCompact L \u2192 Disjoint L (g L \u2022 K)\nL : \u2115 \u2192 Set G := fun n => (fun T => T \u222a g T \u2022 K)^[n] K\nLcompact : \u2200 (n : \u2115), IsCompact (L n)\nLclosed : \u2200 (n : \u2115), IsClosed (L n)\nn : \u2115\nIH : \u2191\u2191\u03bc (L n) = \u2191(n + 1) * \u2191\u2191\u03bc K\n\u22a2 \u2191\u2191\u03bc (((fun T => T \u222a g T \u2022 K) \u2218 (fun T => T \u222a g T \u2022 K)^[n]) K) =\n    \u2191\u2191\u03bc ((fun T => T \u222a g T \u2022 K)^[n] K) + \u2191\u2191\u03bc (g ((fun T => T \u222a g T \u2022 K)^[n] K) \u2022 K)", "state_after": "no goals"}, {"tactic": "simp only [IH, measure_smul, add_mul, Nat.cast_add, Nat.cast_one, one_mul]", "annotated_tactic": ["simp only [IH, <a>measure_smul</a>, <a>add_mul</a>, <a>Nat.cast_add</a>, <a>Nat.cast_one</a>, <a>one_mul</a>]", [{"full_name": "MeasureTheory.measure_smul", "def_path": "Mathlib/MeasureTheory/Group/Action.lean", "def_pos": [215, 9], "def_end_pos": [215, 21]}, {"full_name": "add_mul", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [91, 7], "def_end_pos": [91, 14]}, {"full_name": "Nat.cast_add", "def_path": "Mathlib/Data/Nat/Cast/Defs.lean", "def_pos": [146, 9], "def_end_pos": [146, 17]}, {"full_name": "Nat.cast_one", "def_path": "Mathlib/Data/Nat/Cast/Defs.lean", "def_pos": [141, 9], "def_end_pos": [141, 17]}, {"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [464, 9], "def_end_pos": [464, 16]}]], "state_before": "\ud835\udd5c : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u00b9\u2070 : MeasurableSpace G\ninst\u271d\u2079 : MeasurableSpace H\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : BorelSpace G\n\u03bc\u271d : Measure G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : IsMulLeftInvariant \u03bc\u271d\ninst\u271d\u00b3 : WeaklyLocallyCompactSpace G\ninst\u271d\u00b2 : NoncompactSpace G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsOpenPosMeasure \u03bc\ninst\u271d : IsMulLeftInvariant \u03bc\nK : Set G\nhK : IsCompact K\nKclosed : IsClosed K\nK1 : K \u2208 \ud835\udcdd 1\nK_pos : 0 < \u2191\u2191\u03bc K\ng : Set G \u2192 G\nhg : \u2200 (L : Set G), IsCompact L \u2192 Disjoint L (g L \u2022 K)\nL : \u2115 \u2192 Set G := fun n => (fun T => T \u222a g T \u2022 K)^[n] K\nLcompact : \u2200 (n : \u2115), IsCompact (L n)\nLclosed : \u2200 (n : \u2115), IsClosed (L n)\nn : \u2115\nIH : \u2191\u2191\u03bc (L n) = \u2191(n + 1) * \u2191\u2191\u03bc K\n\u22a2 \u2191\u2191\u03bc (L n) + \u2191\u2191\u03bc (g (L n) \u2022 K) = \u2191(n + 1 + 1) * \u2191\u2191\u03bc K", "state_after": "no goals"}, {"tactic": "simp_rw [M]", "annotated_tactic": ["simp_rw [M]", []], "state_before": "\ud835\udd5c : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u00b9\u2070 : MeasurableSpace G\ninst\u271d\u2079 : MeasurableSpace H\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : BorelSpace G\n\u03bc\u271d : Measure G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : IsMulLeftInvariant \u03bc\u271d\ninst\u271d\u00b3 : WeaklyLocallyCompactSpace G\ninst\u271d\u00b2 : NoncompactSpace G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsOpenPosMeasure \u03bc\ninst\u271d : IsMulLeftInvariant \u03bc\nK : Set G\nhK : IsCompact K\nKclosed : IsClosed K\nK1 : K \u2208 \ud835\udcdd 1\nK_pos : 0 < \u2191\u2191\u03bc K\ng : Set G \u2192 G\nhg : \u2200 (L : Set G), IsCompact L \u2192 Disjoint L (g L \u2022 K)\nL : \u2115 \u2192 Set G := fun n => (fun T => T \u222a g T \u2022 K)^[n] K\nLcompact : \u2200 (n : \u2115), IsCompact (L n)\nLclosed : \u2200 (n : \u2115), IsClosed (L n)\nM : \u2200 (n : \u2115), \u2191\u2191\u03bc (L n) = \u2191(n + 1) * \u2191\u2191\u03bc K\n\u22a2 Tendsto (fun n => \u2191\u2191\u03bc (L n)) atTop (\ud835\udcdd (\u22a4 * \u2191\u2191\u03bc K))", "state_after": "\ud835\udd5c : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u00b9\u2070 : MeasurableSpace G\ninst\u271d\u2079 : MeasurableSpace H\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : BorelSpace G\n\u03bc\u271d : Measure G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : IsMulLeftInvariant \u03bc\u271d\ninst\u271d\u00b3 : WeaklyLocallyCompactSpace G\ninst\u271d\u00b2 : NoncompactSpace G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsOpenPosMeasure \u03bc\ninst\u271d : IsMulLeftInvariant \u03bc\nK : Set G\nhK : IsCompact K\nKclosed : IsClosed K\nK1 : K \u2208 \ud835\udcdd 1\nK_pos : 0 < \u2191\u2191\u03bc K\ng : Set G \u2192 G\nhg : \u2200 (L : Set G), IsCompact L \u2192 Disjoint L (g L \u2022 K)\nL : \u2115 \u2192 Set G := fun n => (fun T => T \u222a g T \u2022 K)^[n] K\nLcompact : \u2200 (n : \u2115), IsCompact (L n)\nLclosed : \u2200 (n : \u2115), IsClosed (L n)\nM : \u2200 (n : \u2115), \u2191\u2191\u03bc (L n) = \u2191(n + 1) * \u2191\u2191\u03bc K\n\u22a2 Tendsto (fun n => \u2191(n + 1) * \u2191\u2191\u03bc K) atTop (\ud835\udcdd (\u22a4 * \u2191\u2191\u03bc K))"}, {"tactic": "apply ENNReal.Tendsto.mul_const _ (Or.inl ENNReal.top_ne_zero)", "annotated_tactic": ["apply <a>ENNReal.Tendsto.mul_const</a> _ (<a>Or.inl</a> <a>ENNReal.top_ne_zero</a>)", [{"full_name": "ENNReal.Tendsto.mul_const", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [379, 19], "def_end_pos": [379, 36]}, {"full_name": "Or.inl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [517, 5], "def_end_pos": [517, 8]}, {"full_name": "ENNReal.top_ne_zero", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [337, 17], "def_end_pos": [337, 28]}]], "state_before": "\ud835\udd5c : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u00b9\u2070 : MeasurableSpace G\ninst\u271d\u2079 : MeasurableSpace H\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : BorelSpace G\n\u03bc\u271d : Measure G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : IsMulLeftInvariant \u03bc\u271d\ninst\u271d\u00b3 : WeaklyLocallyCompactSpace G\ninst\u271d\u00b2 : NoncompactSpace G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsOpenPosMeasure \u03bc\ninst\u271d : IsMulLeftInvariant \u03bc\nK : Set G\nhK : IsCompact K\nKclosed : IsClosed K\nK1 : K \u2208 \ud835\udcdd 1\nK_pos : 0 < \u2191\u2191\u03bc K\ng : Set G \u2192 G\nhg : \u2200 (L : Set G), IsCompact L \u2192 Disjoint L (g L \u2022 K)\nL : \u2115 \u2192 Set G := fun n => (fun T => T \u222a g T \u2022 K)^[n] K\nLcompact : \u2200 (n : \u2115), IsCompact (L n)\nLclosed : \u2200 (n : \u2115), IsClosed (L n)\nM : \u2200 (n : \u2115), \u2191\u2191\u03bc (L n) = \u2191(n + 1) * \u2191\u2191\u03bc K\n\u22a2 Tendsto (fun n => \u2191(n + 1) * \u2191\u2191\u03bc K) atTop (\ud835\udcdd (\u22a4 * \u2191\u2191\u03bc K))", "state_after": "\ud835\udd5c : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u00b9\u2070 : MeasurableSpace G\ninst\u271d\u2079 : MeasurableSpace H\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : BorelSpace G\n\u03bc\u271d : Measure G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : IsMulLeftInvariant \u03bc\u271d\ninst\u271d\u00b3 : WeaklyLocallyCompactSpace G\ninst\u271d\u00b2 : NoncompactSpace G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsOpenPosMeasure \u03bc\ninst\u271d : IsMulLeftInvariant \u03bc\nK : Set G\nhK : IsCompact K\nKclosed : IsClosed K\nK1 : K \u2208 \ud835\udcdd 1\nK_pos : 0 < \u2191\u2191\u03bc K\ng : Set G \u2192 G\nhg : \u2200 (L : Set G), IsCompact L \u2192 Disjoint L (g L \u2022 K)\nL : \u2115 \u2192 Set G := fun n => (fun T => T \u222a g T \u2022 K)^[n] K\nLcompact : \u2200 (n : \u2115), IsCompact (L n)\nLclosed : \u2200 (n : \u2115), IsClosed (L n)\nM : \u2200 (n : \u2115), \u2191\u2191\u03bc (L n) = \u2191(n + 1) * \u2191\u2191\u03bc K\n\u22a2 Tendsto (fun x => \u2191(x + 1)) atTop (\ud835\udcdd \u22a4)"}, {"tactic": "exact ENNReal.tendsto_nat_nhds_top.comp (tendsto_add_atTop_nat _)", "annotated_tactic": ["exact ENNReal.tendsto_nat_nhds_top.comp (<a>tendsto_add_atTop_nat</a> _)", [{"full_name": "Filter.tendsto_add_atTop_nat", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [1707, 9], "def_end_pos": [1707, 30]}]], "state_before": "\ud835\udd5c : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u00b9\u2070 : MeasurableSpace G\ninst\u271d\u2079 : MeasurableSpace H\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : BorelSpace G\n\u03bc\u271d : Measure G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : IsMulLeftInvariant \u03bc\u271d\ninst\u271d\u00b3 : WeaklyLocallyCompactSpace G\ninst\u271d\u00b2 : NoncompactSpace G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsOpenPosMeasure \u03bc\ninst\u271d : IsMulLeftInvariant \u03bc\nK : Set G\nhK : IsCompact K\nKclosed : IsClosed K\nK1 : K \u2208 \ud835\udcdd 1\nK_pos : 0 < \u2191\u2191\u03bc K\ng : Set G \u2192 G\nhg : \u2200 (L : Set G), IsCompact L \u2192 Disjoint L (g L \u2022 K)\nL : \u2115 \u2192 Set G := fun n => (fun T => T \u222a g T \u2022 K)^[n] K\nLcompact : \u2200 (n : \u2115), IsCompact (L n)\nLclosed : \u2200 (n : \u2115), IsClosed (L n)\nM : \u2200 (n : \u2115), \u2191\u2191\u03bc (L n) = \u2191(n + 1) * \u2191\u2191\u03bc K\n\u22a2 Tendsto (fun x => \u2191(x + 1)) atTop (\ud835\udcdd \u22a4)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/TorusIntegral.lean", "full_name": "torusIntegral_succAbove", "start": [235, 1], "end": [256, 78], "traced_tactics": [{"tactic": "set e : \u211d \u00d7 \u211d\u207f \u2243\u1d50 \u211d\u207f\u207a\u00b9 := (MeasurableEquiv.piFinSuccAboveEquiv (fun _ => \u211d) i).symm", "annotated_tactic": ["set e : \u211d \u00d7 \u211d\u207f \u2243\u1d50 \u211d\u207f\u207a\u00b9 := (<a>MeasurableEquiv.piFinSuccAboveEquiv</a> (fun _ => \u211d) i).<a>symm</a>", [{"full_name": "MeasurableEquiv.piFinSuccAboveEquiv", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [1654, 5], "def_end_pos": [1654, 24]}, {"full_name": "MeasurableEquiv.symm", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [1325, 5], "def_end_pos": [1325, 9]}]], "state_before": "n : \u2115\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf\u271d g : (Fin n \u2192 \u2102) \u2192 E\nc\u271d : Fin n \u2192 \u2102\nR\u271d : Fin n \u2192 \u211d\nf : (Fin (n + 1) \u2192 \u2102) \u2192 E\nc : Fin (n + 1) \u2192 \u2102\nR : Fin (n + 1) \u2192 \u211d\nhf : TorusIntegrable f c R\ni : Fin (n + 1)\n\u22a2 (\u222f (x : Fin (n + 1) \u2192 \u2102) in T(c, R), f x) =\n    \u222e (x : \u2102) in C(c i, R i), \u222f (y : Fin n \u2192 \u2102) in T(c \u2218 Fin.succAbove i, R \u2218 Fin.succAbove i), f (Fin.insertNth i x y)", "state_after": "n : \u2115\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf\u271d g : (Fin n \u2192 \u2102) \u2192 E\nc\u271d : Fin n \u2192 \u2102\nR\u271d : Fin n \u2192 \u211d\nf : (Fin (n + 1) \u2192 \u2102) \u2192 E\nc : Fin (n + 1) \u2192 \u2102\nR : Fin (n + 1) \u2192 \u211d\nhf : TorusIntegrable f c R\ni : Fin (n + 1)\ne : \u211d \u00d7 (Fin n \u2192 \u211d) \u2243\u1d50 (Fin (n + 1) \u2192 \u211d) := MeasurableEquiv.symm (MeasurableEquiv.piFinSuccAboveEquiv (fun x => \u211d) i)\n\u22a2 (\u222f (x : Fin (n + 1) \u2192 \u2102) in T(c, R), f x) =\n    \u222e (x : \u2102) in C(c i, R i), \u222f (y : Fin n \u2192 \u2102) in T(c \u2218 Fin.succAbove i, R \u2218 Fin.succAbove i), f (Fin.insertNth i x y)"}, {"tactic": "have hem : MeasurePreserving e :=\n  (volume_preserving_piFinSuccAboveEquiv (fun _ : Fin (n + 1) => \u211d) i).symm _", "annotated_tactic": ["have hem : <a>MeasurePreserving</a> e :=\n    (<a>volume_preserving_piFinSuccAboveEquiv</a> (fun _ : <a>Fin</a> (n + 1) => \u211d) i).<a>symm</a> _", [{"full_name": "MeasureTheory.MeasurePreserving", "def_path": "Mathlib/Dynamics/Ergodic/MeasurePreserving.lean", "def_pos": [42, 11], "def_end_pos": [42, 28]}, {"full_name": "MeasureTheory.volume_preserving_piFinSuccAboveEquiv", "def_path": "Mathlib/MeasureTheory/Constructions/Pi.lean", "def_pos": [807, 9], "def_end_pos": [807, 46]}, {"full_name": "Fin", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1745, 11], "def_end_pos": [1745, 14]}, {"full_name": "MeasureTheory.MeasurePreserving.symm", "def_path": "Mathlib/Dynamics/Ergodic/MeasurePreserving.lean", "def_pos": [70, 9], "def_end_pos": [70, 13]}]], "state_before": "n : \u2115\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf\u271d g : (Fin n \u2192 \u2102) \u2192 E\nc\u271d : Fin n \u2192 \u2102\nR\u271d : Fin n \u2192 \u211d\nf : (Fin (n + 1) \u2192 \u2102) \u2192 E\nc : Fin (n + 1) \u2192 \u2102\nR : Fin (n + 1) \u2192 \u211d\nhf : TorusIntegrable f c R\ni : Fin (n + 1)\ne : \u211d \u00d7 (Fin n \u2192 \u211d) \u2243\u1d50 (Fin (n + 1) \u2192 \u211d) := MeasurableEquiv.symm (MeasurableEquiv.piFinSuccAboveEquiv (fun x => \u211d) i)\n\u22a2 (\u222f (x : Fin (n + 1) \u2192 \u2102) in T(c, R), f x) =\n    \u222e (x : \u2102) in C(c i, R i), \u222f (y : Fin n \u2192 \u2102) in T(c \u2218 Fin.succAbove i, R \u2218 Fin.succAbove i), f (Fin.insertNth i x y)", "state_after": "n : \u2115\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf\u271d g : (Fin n \u2192 \u2102) \u2192 E\nc\u271d : Fin n \u2192 \u2102\nR\u271d : Fin n \u2192 \u211d\nf : (Fin (n + 1) \u2192 \u2102) \u2192 E\nc : Fin (n + 1) \u2192 \u2102\nR : Fin (n + 1) \u2192 \u211d\nhf : TorusIntegrable f c R\ni : Fin (n + 1)\ne : \u211d \u00d7 (Fin n \u2192 \u211d) \u2243\u1d50 (Fin (n + 1) \u2192 \u211d) := MeasurableEquiv.symm (MeasurableEquiv.piFinSuccAboveEquiv (fun x => \u211d) i)\nhem : MeasurePreserving \u2191e\n\u22a2 (\u222f (x : Fin (n + 1) \u2192 \u2102) in T(c, R), f x) =\n    \u222e (x : \u2102) in C(c i, R i), \u222f (y : Fin n \u2192 \u2102) in T(c \u2218 Fin.succAbove i, R \u2218 Fin.succAbove i), f (Fin.insertNth i x y)"}, {"tactic": "have he\u03c0 : (e \u207b\u00b9' Icc 0 fun _ => 2 * \u03c0) = Icc 0 (2 * \u03c0) \u00d7\u02e2 Icc (0 : \u211d\u207f) fun _ => 2 * \u03c0 :=\n  ((OrderIso.piFinSuccAboveIso (fun _ => \u211d) i).symm.preimage_Icc _ _).trans (Icc_prod_eq _ _)", "annotated_tactic": ["have he\u03c0 : (e \u207b\u00b9' <a>Icc</a> 0 fun _ => 2 * \u03c0) = <a>Icc</a> 0 (2 * \u03c0) \u00d7\u02e2 <a>Icc</a> (0 : \u211d\u207f) fun _ => 2 * \u03c0 :=\n    ((<a>OrderIso.piFinSuccAboveIso</a> (fun _ => \u211d) i).symm.preimage_Icc _ _).<a>trans</a> (<a>Icc_prod_eq</a> _ _)", [{"full_name": "Set.Icc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [59, 5], "def_end_pos": [59, 8]}, {"full_name": "Set.Icc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [59, 5], "def_end_pos": [59, 8]}, {"full_name": "Set.Icc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [59, 5], "def_end_pos": [59, 8]}, {"full_name": "OrderIso.piFinSuccAboveIso", "def_path": "Mathlib/Logic/Equiv/Fin.lean", "def_pos": [303, 5], "def_end_pos": [303, 31]}, {"full_name": "Eq.trans", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [322, 9], "def_end_pos": [322, 17]}, {"full_name": "Set.Icc_prod_eq", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [1927, 9], "def_end_pos": [1927, 20]}]], "state_before": "n : \u2115\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf\u271d g : (Fin n \u2192 \u2102) \u2192 E\nc\u271d : Fin n \u2192 \u2102\nR\u271d : Fin n \u2192 \u211d\nf : (Fin (n + 1) \u2192 \u2102) \u2192 E\nc : Fin (n + 1) \u2192 \u2102\nR : Fin (n + 1) \u2192 \u211d\nhf : TorusIntegrable f c R\ni : Fin (n + 1)\ne : \u211d \u00d7 (Fin n \u2192 \u211d) \u2243\u1d50 (Fin (n + 1) \u2192 \u211d) := MeasurableEquiv.symm (MeasurableEquiv.piFinSuccAboveEquiv (fun x => \u211d) i)\nhem : MeasurePreserving \u2191e\n\u22a2 (\u222f (x : Fin (n + 1) \u2192 \u2102) in T(c, R), f x) =\n    \u222e (x : \u2102) in C(c i, R i), \u222f (y : Fin n \u2192 \u2102) in T(c \u2218 Fin.succAbove i, R \u2218 Fin.succAbove i), f (Fin.insertNth i x y)", "state_after": "n : \u2115\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf\u271d g : (Fin n \u2192 \u2102) \u2192 E\nc\u271d : Fin n \u2192 \u2102\nR\u271d : Fin n \u2192 \u211d\nf : (Fin (n + 1) \u2192 \u2102) \u2192 E\nc : Fin (n + 1) \u2192 \u2102\nR : Fin (n + 1) \u2192 \u211d\nhf : TorusIntegrable f c R\ni : Fin (n + 1)\ne : \u211d \u00d7 (Fin n \u2192 \u211d) \u2243\u1d50 (Fin (n + 1) \u2192 \u211d) := MeasurableEquiv.symm (MeasurableEquiv.piFinSuccAboveEquiv (fun x => \u211d) i)\nhem : MeasurePreserving \u2191e\nhe\u03c0 : (\u2191e \u207b\u00b9' Icc 0 fun x => 2 * \u03c0) = Icc 0 (2 * \u03c0) \u00d7\u02e2 Icc 0 fun x => 2 * \u03c0\n\u22a2 (\u222f (x : Fin (n + 1) \u2192 \u2102) in T(c, R), f x) =\n    \u222e (x : \u2102) in C(c i, R i), \u222f (y : Fin n \u2192 \u2102) in T(c \u2218 Fin.succAbove i, R \u2218 Fin.succAbove i), f (Fin.insertNth i x y)"}, {"tactic": "rw [torusIntegral, \u2190 hem.map_eq, set_integral_map_equiv, he\u03c0, Measure.volume_eq_prod,\n  set_integral_prod, circleIntegral_def_Icc]", "annotated_tactic": ["rw [<a>torusIntegral</a>, \u2190 hem.map_eq, <a>set_integral_map_equiv</a>, he\u03c0, <a>Measure.volume_eq_prod</a>,\n    <a>set_integral_prod</a>, <a>circleIntegral_def_Icc</a>]", [{"full_name": "torusIntegral", "def_path": "Mathlib/MeasureTheory/Integral/TorusIntegral.lean", "def_pos": [155, 5], "def_end_pos": [155, 18]}, {"full_name": "MeasureTheory.set_integral_map_equiv", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [543, 9], "def_end_pos": [543, 31]}, {"full_name": "MeasureTheory.Measure.volume_eq_prod", "def_path": "Mathlib/MeasureTheory/Constructions/Prod/Basic.lean", "def_pos": [300, 9], "def_end_pos": [300, 23]}, {"full_name": "MeasureTheory.set_integral_prod", "def_path": "Mathlib/MeasureTheory/Constructions/Prod/Integral.lean", "def_pos": [505, 9], "def_end_pos": [505, 26]}, {"full_name": "circleIntegral_def_Icc", "def_path": "Mathlib/MeasureTheory/Integral/CircleIntegral.lean", "def_pos": [349, 9], "def_end_pos": [349, 31]}]], "state_before": "n : \u2115\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf\u271d g : (Fin n \u2192 \u2102) \u2192 E\nc\u271d : Fin n \u2192 \u2102\nR\u271d : Fin n \u2192 \u211d\nf : (Fin (n + 1) \u2192 \u2102) \u2192 E\nc : Fin (n + 1) \u2192 \u2102\nR : Fin (n + 1) \u2192 \u211d\nhf : TorusIntegrable f c R\ni : Fin (n + 1)\ne : \u211d \u00d7 (Fin n \u2192 \u211d) \u2243\u1d50 (Fin (n + 1) \u2192 \u211d) := MeasurableEquiv.symm (MeasurableEquiv.piFinSuccAboveEquiv (fun x => \u211d) i)\nhem : MeasurePreserving \u2191e\nhe\u03c0 : (\u2191e \u207b\u00b9' Icc 0 fun x => 2 * \u03c0) = Icc 0 (2 * \u03c0) \u00d7\u02e2 Icc 0 fun x => 2 * \u03c0\n\u22a2 (\u222f (x : Fin (n + 1) \u2192 \u2102) in T(c, R), f x) =\n    \u222e (x : \u2102) in C(c i, R i), \u222f (y : Fin n \u2192 \u2102) in T(c \u2218 Fin.succAbove i, R \u2218 Fin.succAbove i), f (Fin.insertNth i x y)", "state_after": "n : \u2115\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf\u271d g : (Fin n \u2192 \u2102) \u2192 E\nc\u271d : Fin n \u2192 \u2102\nR\u271d : Fin n \u2192 \u211d\nf : (Fin (n + 1) \u2192 \u2102) \u2192 E\nc : Fin (n + 1) \u2192 \u2102\nR : Fin (n + 1) \u2192 \u211d\nhf : TorusIntegrable f c R\ni : Fin (n + 1)\ne : \u211d \u00d7 (Fin n \u2192 \u211d) \u2243\u1d50 (Fin (n + 1) \u2192 \u211d) := MeasurableEquiv.symm (MeasurableEquiv.piFinSuccAboveEquiv (fun x => \u211d) i)\nhem : MeasurePreserving \u2191e\nhe\u03c0 : (\u2191e \u207b\u00b9' Icc 0 fun x => 2 * \u03c0) = Icc 0 (2 * \u03c0) \u00d7\u02e2 Icc 0 fun x => 2 * \u03c0\n\u22a2 \u222b (x : \u211d) in Icc 0 (2 * \u03c0),\n      \u222b (y : Fin n \u2192 \u211d) in Icc 0 fun x => 2 * \u03c0,\n        (\u220f i : Fin (n + 1), \u2191(R i) * cexp (\u2191(\u2191e (x, y) i) * I) * I) \u2022 f (torusMap c R (\u2191e (x, y))) =\n    \u222b (\u03b8 : \u211d) in Icc 0 (2 * \u03c0),\n      deriv (circleMap (c i) (R i)) \u03b8 \u2022\n        \u222f (y : Fin n \u2192 \u2102) in T(c \u2218 Fin.succAbove i, R \u2218 Fin.succAbove i),\n          f (Fin.insertNth i (circleMap (c i) (R i) \u03b8) y)\n\ncase hf\nn : \u2115\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf\u271d g : (Fin n \u2192 \u2102) \u2192 E\nc\u271d : Fin n \u2192 \u2102\nR\u271d : Fin n \u2192 \u211d\nf : (Fin (n + 1) \u2192 \u2102) \u2192 E\nc : Fin (n + 1) \u2192 \u2102\nR : Fin (n + 1) \u2192 \u211d\nhf : TorusIntegrable f c R\ni : Fin (n + 1)\ne : \u211d \u00d7 (Fin n \u2192 \u211d) \u2243\u1d50 (Fin (n + 1) \u2192 \u211d) := MeasurableEquiv.symm (MeasurableEquiv.piFinSuccAboveEquiv (fun x => \u211d) i)\nhem : MeasurePreserving \u2191e\nhe\u03c0 : (\u2191e \u207b\u00b9' Icc 0 fun x => 2 * \u03c0) = Icc 0 (2 * \u03c0) \u00d7\u02e2 Icc 0 fun x => 2 * \u03c0\n\u22a2 IntegrableOn (fun x => (\u220f i : Fin (n + 1), \u2191(R i) * cexp (\u2191(\u2191e x i) * I) * I) \u2022 f (torusMap c R (\u2191e x)))\n    (Icc 0 (2 * \u03c0) \u00d7\u02e2 Icc 0 fun x => 2 * \u03c0)"}, {"tactic": "refine' set_integral_congr measurableSet_Icc fun \u03b8 _ => _", "annotated_tactic": ["refine' <a>set_integral_congr</a> <a>measurableSet_Icc</a> fun \u03b8 _ => _", [{"full_name": "MeasureTheory.set_integral_congr", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [87, 9], "def_end_pos": [87, 27]}, {"full_name": "measurableSet_Icc", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [520, 9], "def_end_pos": [520, 26]}]], "state_before": "n : \u2115\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf\u271d g : (Fin n \u2192 \u2102) \u2192 E\nc\u271d : Fin n \u2192 \u2102\nR\u271d : Fin n \u2192 \u211d\nf : (Fin (n + 1) \u2192 \u2102) \u2192 E\nc : Fin (n + 1) \u2192 \u2102\nR : Fin (n + 1) \u2192 \u211d\nhf : TorusIntegrable f c R\ni : Fin (n + 1)\ne : \u211d \u00d7 (Fin n \u2192 \u211d) \u2243\u1d50 (Fin (n + 1) \u2192 \u211d) := MeasurableEquiv.symm (MeasurableEquiv.piFinSuccAboveEquiv (fun x => \u211d) i)\nhem : MeasurePreserving \u2191e\nhe\u03c0 : (\u2191e \u207b\u00b9' Icc 0 fun x => 2 * \u03c0) = Icc 0 (2 * \u03c0) \u00d7\u02e2 Icc 0 fun x => 2 * \u03c0\n\u22a2 \u222b (x : \u211d) in Icc 0 (2 * \u03c0),\n      \u222b (y : Fin n \u2192 \u211d) in Icc 0 fun x => 2 * \u03c0,\n        (\u220f i : Fin (n + 1), \u2191(R i) * cexp (\u2191(\u2191e (x, y) i) * I) * I) \u2022 f (torusMap c R (\u2191e (x, y))) =\n    \u222b (\u03b8 : \u211d) in Icc 0 (2 * \u03c0),\n      deriv (circleMap (c i) (R i)) \u03b8 \u2022\n        \u222f (y : Fin n \u2192 \u2102) in T(c \u2218 Fin.succAbove i, R \u2218 Fin.succAbove i),\n          f (Fin.insertNth i (circleMap (c i) (R i) \u03b8) y)", "state_after": "n : \u2115\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf\u271d g : (Fin n \u2192 \u2102) \u2192 E\nc\u271d : Fin n \u2192 \u2102\nR\u271d : Fin n \u2192 \u211d\nf : (Fin (n + 1) \u2192 \u2102) \u2192 E\nc : Fin (n + 1) \u2192 \u2102\nR : Fin (n + 1) \u2192 \u211d\nhf : TorusIntegrable f c R\ni : Fin (n + 1)\ne : \u211d \u00d7 (Fin n \u2192 \u211d) \u2243\u1d50 (Fin (n + 1) \u2192 \u211d) := MeasurableEquiv.symm (MeasurableEquiv.piFinSuccAboveEquiv (fun x => \u211d) i)\nhem : MeasurePreserving \u2191e\nhe\u03c0 : (\u2191e \u207b\u00b9' Icc 0 fun x => 2 * \u03c0) = Icc 0 (2 * \u03c0) \u00d7\u02e2 Icc 0 fun x => 2 * \u03c0\n\u03b8 : \u211d\nx\u271d : \u03b8 \u2208 Icc 0 (2 * \u03c0)\n\u22a2 \u222b (y : Fin n \u2192 \u211d) in Icc 0 fun x => 2 * \u03c0,\n      (\u220f i : Fin (n + 1), \u2191(R i) * cexp (\u2191(\u2191e (\u03b8, y) i) * I) * I) \u2022 f (torusMap c R (\u2191e (\u03b8, y))) =\n    deriv (circleMap (c i) (R i)) \u03b8 \u2022\n      \u222f (y : Fin n \u2192 \u2102) in T(c \u2218 Fin.succAbove i, R \u2218 Fin.succAbove i), f (Fin.insertNth i (circleMap (c i) (R i) \u03b8) y)"}, {"tactic": "simp only [torusIntegral, \u2190 integral_smul, deriv_circleMap, i.prod_univ_succAbove _, smul_smul,\n  torusMap, circleMap_zero]", "annotated_tactic": ["simp only [<a>torusIntegral</a>, \u2190 <a>integral_smul</a>, <a>deriv_circleMap</a>, i.prod_univ_succAbove _, <a>smul_smul</a>,\n      <a>torusMap</a>, <a>circleMap_zero</a>]", [{"full_name": "torusIntegral", "def_path": "Mathlib/MeasureTheory/Integral/TorusIntegral.lean", "def_pos": [155, 5], "def_end_pos": [155, 18]}, {"full_name": "MeasureTheory.integral_smul", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [915, 9], "def_end_pos": [915, 22]}, {"full_name": "deriv_circleMap", "def_path": "Mathlib/MeasureTheory/Integral/CircleIntegral.lean", "def_pos": [195, 9], "def_end_pos": [195, 24]}, {"full_name": "smul_smul", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [484, 9], "def_end_pos": [484, 18]}, {"full_name": "torusMap", "def_path": "Mathlib/MeasureTheory/Integral/TorusIntegral.lean", "def_pos": [83, 5], "def_end_pos": [83, 13]}, {"full_name": "circleMap_zero", "def_path": "Mathlib/MeasureTheory/Integral/CircleIntegral.lean", "def_pos": [111, 9], "def_end_pos": [111, 23]}]], "state_before": "n : \u2115\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf\u271d g : (Fin n \u2192 \u2102) \u2192 E\nc\u271d : Fin n \u2192 \u2102\nR\u271d : Fin n \u2192 \u211d\nf : (Fin (n + 1) \u2192 \u2102) \u2192 E\nc : Fin (n + 1) \u2192 \u2102\nR : Fin (n + 1) \u2192 \u211d\nhf : TorusIntegrable f c R\ni : Fin (n + 1)\ne : \u211d \u00d7 (Fin n \u2192 \u211d) \u2243\u1d50 (Fin (n + 1) \u2192 \u211d) := MeasurableEquiv.symm (MeasurableEquiv.piFinSuccAboveEquiv (fun x => \u211d) i)\nhem : MeasurePreserving \u2191e\nhe\u03c0 : (\u2191e \u207b\u00b9' Icc 0 fun x => 2 * \u03c0) = Icc 0 (2 * \u03c0) \u00d7\u02e2 Icc 0 fun x => 2 * \u03c0\n\u03b8 : \u211d\nx\u271d : \u03b8 \u2208 Icc 0 (2 * \u03c0)\n\u22a2 \u222b (y : Fin n \u2192 \u211d) in Icc 0 fun x => 2 * \u03c0,\n      (\u220f i : Fin (n + 1), \u2191(R i) * cexp (\u2191(\u2191e (\u03b8, y) i) * I) * I) \u2022 f (torusMap c R (\u2191e (\u03b8, y))) =\n    deriv (circleMap (c i) (R i)) \u03b8 \u2022\n      \u222f (y : Fin n \u2192 \u2102) in T(c \u2218 Fin.succAbove i, R \u2218 Fin.succAbove i), f (Fin.insertNth i (circleMap (c i) (R i) \u03b8) y)", "state_after": "n : \u2115\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf\u271d g : (Fin n \u2192 \u2102) \u2192 E\nc\u271d : Fin n \u2192 \u2102\nR\u271d : Fin n \u2192 \u211d\nf : (Fin (n + 1) \u2192 \u2102) \u2192 E\nc : Fin (n + 1) \u2192 \u2102\nR : Fin (n + 1) \u2192 \u211d\nhf : TorusIntegrable f c R\ni : Fin (n + 1)\ne : \u211d \u00d7 (Fin n \u2192 \u211d) \u2243\u1d50 (Fin (n + 1) \u2192 \u211d) := MeasurableEquiv.symm (MeasurableEquiv.piFinSuccAboveEquiv (fun x => \u211d) i)\nhem : MeasurePreserving \u2191e\nhe\u03c0 : (\u2191e \u207b\u00b9' Icc 0 fun x => 2 * \u03c0) = Icc 0 (2 * \u03c0) \u00d7\u02e2 Icc 0 fun x => 2 * \u03c0\n\u03b8 : \u211d\nx\u271d : \u03b8 \u2208 Icc 0 (2 * \u03c0)\n\u22a2 (\u222b (y : Fin n \u2192 \u211d) in Icc 0 fun x => 2 * \u03c0,\n      (\u2191(R i) * cexp (\u2191(\u2191(MeasurableEquiv.symm (MeasurableEquiv.piFinSuccAboveEquiv (fun x => \u211d) i)) (\u03b8, y) i) * I) *\n            I *\n          \u220f i_1 : Fin n,\n            \u2191(R (Fin.succAbove i i_1)) *\n                cexp\n                  (\u2191(\u2191(MeasurableEquiv.symm (MeasurableEquiv.piFinSuccAboveEquiv (fun x => \u211d) i)) (\u03b8, y)\n                        (Fin.succAbove i i_1)) *\n                    I) *\n              I) \u2022\n        f fun i_1 =>\n          c i_1 +\n            \u2191(R i_1) *\n              cexp (\u2191(\u2191(MeasurableEquiv.symm (MeasurableEquiv.piFinSuccAboveEquiv (fun x => \u211d) i)) (\u03b8, y) i_1) * I)) =\n    \u222b (a : Fin n \u2192 \u211d) in Icc 0 fun x => 2 * \u03c0,\n      (\u2191(R i) * cexp (\u2191\u03b8 * I) * I * \u220f i_1 : Fin n, \u2191((R \u2218 Fin.succAbove i) i_1) * cexp (\u2191(a i_1) * I) * I) \u2022\n        f\n          (Fin.insertNth i (circleMap (c i) (R i) \u03b8) fun i_1 =>\n            (c \u2218 Fin.succAbove i) i_1 + \u2191((R \u2218 Fin.succAbove i) i_1) * cexp (\u2191(a i_1) * I))"}, {"tactic": "refine' set_integral_congr measurableSet_Icc fun \u0398 _ => _", "annotated_tactic": ["refine' <a>set_integral_congr</a> <a>measurableSet_Icc</a> fun \u0398 _ => _", [{"full_name": "MeasureTheory.set_integral_congr", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [87, 9], "def_end_pos": [87, 27]}, {"full_name": "measurableSet_Icc", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [520, 9], "def_end_pos": [520, 26]}]], "state_before": "n : \u2115\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf\u271d g : (Fin n \u2192 \u2102) \u2192 E\nc\u271d : Fin n \u2192 \u2102\nR\u271d : Fin n \u2192 \u211d\nf : (Fin (n + 1) \u2192 \u2102) \u2192 E\nc : Fin (n + 1) \u2192 \u2102\nR : Fin (n + 1) \u2192 \u211d\nhf : TorusIntegrable f c R\ni : Fin (n + 1)\ne : \u211d \u00d7 (Fin n \u2192 \u211d) \u2243\u1d50 (Fin (n + 1) \u2192 \u211d) := MeasurableEquiv.symm (MeasurableEquiv.piFinSuccAboveEquiv (fun x => \u211d) i)\nhem : MeasurePreserving \u2191e\nhe\u03c0 : (\u2191e \u207b\u00b9' Icc 0 fun x => 2 * \u03c0) = Icc 0 (2 * \u03c0) \u00d7\u02e2 Icc 0 fun x => 2 * \u03c0\n\u03b8 : \u211d\nx\u271d : \u03b8 \u2208 Icc 0 (2 * \u03c0)\n\u22a2 (\u222b (y : Fin n \u2192 \u211d) in Icc 0 fun x => 2 * \u03c0,\n      (\u2191(R i) * cexp (\u2191(\u2191(MeasurableEquiv.symm (MeasurableEquiv.piFinSuccAboveEquiv (fun x => \u211d) i)) (\u03b8, y) i) * I) *\n            I *\n          \u220f i_1 : Fin n,\n            \u2191(R (Fin.succAbove i i_1)) *\n                cexp\n                  (\u2191(\u2191(MeasurableEquiv.symm (MeasurableEquiv.piFinSuccAboveEquiv (fun x => \u211d) i)) (\u03b8, y)\n                        (Fin.succAbove i i_1)) *\n                    I) *\n              I) \u2022\n        f fun i_1 =>\n          c i_1 +\n            \u2191(R i_1) *\n              cexp (\u2191(\u2191(MeasurableEquiv.symm (MeasurableEquiv.piFinSuccAboveEquiv (fun x => \u211d) i)) (\u03b8, y) i_1) * I)) =\n    \u222b (a : Fin n \u2192 \u211d) in Icc 0 fun x => 2 * \u03c0,\n      (\u2191(R i) * cexp (\u2191\u03b8 * I) * I * \u220f i_1 : Fin n, \u2191((R \u2218 Fin.succAbove i) i_1) * cexp (\u2191(a i_1) * I) * I) \u2022\n        f\n          (Fin.insertNth i (circleMap (c i) (R i) \u03b8) fun i_1 =>\n            (c \u2218 Fin.succAbove i) i_1 + \u2191((R \u2218 Fin.succAbove i) i_1) * cexp (\u2191(a i_1) * I))", "state_after": "n : \u2115\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf\u271d g : (Fin n \u2192 \u2102) \u2192 E\nc\u271d : Fin n \u2192 \u2102\nR\u271d : Fin n \u2192 \u211d\nf : (Fin (n + 1) \u2192 \u2102) \u2192 E\nc : Fin (n + 1) \u2192 \u2102\nR : Fin (n + 1) \u2192 \u211d\nhf : TorusIntegrable f c R\ni : Fin (n + 1)\ne : \u211d \u00d7 (Fin n \u2192 \u211d) \u2243\u1d50 (Fin (n + 1) \u2192 \u211d) := MeasurableEquiv.symm (MeasurableEquiv.piFinSuccAboveEquiv (fun x => \u211d) i)\nhem : MeasurePreserving \u2191e\nhe\u03c0 : (\u2191e \u207b\u00b9' Icc 0 fun x => 2 * \u03c0) = Icc 0 (2 * \u03c0) \u00d7\u02e2 Icc 0 fun x => 2 * \u03c0\n\u03b8 : \u211d\nx\u271d\u00b9 : \u03b8 \u2208 Icc 0 (2 * \u03c0)\n\u0398 : Fin n \u2192 \u211d\nx\u271d : \u0398 \u2208 Icc 0 fun x => 2 * \u03c0\n\u22a2 ((\u2191(R i) * cexp (\u2191(\u2191(MeasurableEquiv.symm (MeasurableEquiv.piFinSuccAboveEquiv (fun x => \u211d) i)) (\u03b8, \u0398) i) * I) * I *\n        \u220f i_1 : Fin n,\n          \u2191(R (Fin.succAbove i i_1)) *\n              cexp\n                (\u2191(\u2191(MeasurableEquiv.symm (MeasurableEquiv.piFinSuccAboveEquiv (fun x => \u211d) i)) (\u03b8, \u0398)\n                      (Fin.succAbove i i_1)) *\n                  I) *\n            I) \u2022\n      f fun i_1 =>\n        c i_1 +\n          \u2191(R i_1) *\n            cexp (\u2191(\u2191(MeasurableEquiv.symm (MeasurableEquiv.piFinSuccAboveEquiv (fun x => \u211d) i)) (\u03b8, \u0398) i_1) * I)) =\n    (\u2191(R i) * cexp (\u2191\u03b8 * I) * I * \u220f i_1 : Fin n, \u2191((R \u2218 Fin.succAbove i) i_1) * cexp (\u2191(\u0398 i_1) * I) * I) \u2022\n      f\n        (Fin.insertNth i (circleMap (c i) (R i) \u03b8) fun i_1 =>\n          (c \u2218 Fin.succAbove i) i_1 + \u2191((R \u2218 Fin.succAbove i) i_1) * cexp (\u2191(\u0398 i_1) * I))"}, {"tactic": "congr 2", "annotated_tactic": ["congr 2", []], "state_before": "n : \u2115\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf\u271d g : (Fin n \u2192 \u2102) \u2192 E\nc\u271d : Fin n \u2192 \u2102\nR\u271d : Fin n \u2192 \u211d\nf : (Fin (n + 1) \u2192 \u2102) \u2192 E\nc : Fin (n + 1) \u2192 \u2102\nR : Fin (n + 1) \u2192 \u211d\nhf : TorusIntegrable f c R\ni : Fin (n + 1)\ne : \u211d \u00d7 (Fin n \u2192 \u211d) \u2243\u1d50 (Fin (n + 1) \u2192 \u211d) := MeasurableEquiv.symm (MeasurableEquiv.piFinSuccAboveEquiv (fun x => \u211d) i)\nhem : MeasurePreserving \u2191e\nhe\u03c0 : (\u2191e \u207b\u00b9' Icc 0 fun x => 2 * \u03c0) = Icc 0 (2 * \u03c0) \u00d7\u02e2 Icc 0 fun x => 2 * \u03c0\n\u03b8 : \u211d\nx\u271d\u00b9 : \u03b8 \u2208 Icc 0 (2 * \u03c0)\n\u0398 : Fin n \u2192 \u211d\nx\u271d : \u0398 \u2208 Icc 0 fun x => 2 * \u03c0\n\u22a2 ((\u2191(R i) * cexp (\u2191\u03b8 * I) * I * \u220f x : Fin n, \u2191(R (Fin.succAbove i x)) * cexp (\u2191(\u0398 x) * I) * I) \u2022\n      f fun i_1 => c i_1 + \u2191(R i_1) * cexp (\u2191(Fin.insertNth i \u03b8 \u0398 i_1) * I)) =\n    (\u2191(R i) * cexp (\u2191\u03b8 * I) * I * \u220f x : Fin n, \u2191(R (Fin.succAbove i x)) * cexp (\u2191(\u0398 x) * I) * I) \u2022\n      f\n        (Fin.insertNth i (circleMap (c i) (R i) \u03b8) fun i_1 =>\n          c (Fin.succAbove i i_1) + \u2191(R (Fin.succAbove i i_1)) * cexp (\u2191(\u0398 i_1) * I))", "state_after": "case e_a.e_a\nn : \u2115\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf\u271d g : (Fin n \u2192 \u2102) \u2192 E\nc\u271d : Fin n \u2192 \u2102\nR\u271d : Fin n \u2192 \u211d\nf : (Fin (n + 1) \u2192 \u2102) \u2192 E\nc : Fin (n + 1) \u2192 \u2102\nR : Fin (n + 1) \u2192 \u211d\nhf : TorusIntegrable f c R\ni : Fin (n + 1)\ne : \u211d \u00d7 (Fin n \u2192 \u211d) \u2243\u1d50 (Fin (n + 1) \u2192 \u211d) := MeasurableEquiv.symm (MeasurableEquiv.piFinSuccAboveEquiv (fun x => \u211d) i)\nhem : MeasurePreserving \u2191e\nhe\u03c0 : (\u2191e \u207b\u00b9' Icc 0 fun x => 2 * \u03c0) = Icc 0 (2 * \u03c0) \u00d7\u02e2 Icc 0 fun x => 2 * \u03c0\n\u03b8 : \u211d\nx\u271d\u00b9 : \u03b8 \u2208 Icc 0 (2 * \u03c0)\n\u0398 : Fin n \u2192 \u211d\nx\u271d : \u0398 \u2208 Icc 0 fun x => 2 * \u03c0\n\u22a2 (fun i_1 => c i_1 + \u2191(R i_1) * cexp (\u2191(Fin.insertNth i \u03b8 \u0398 i_1) * I)) =\n    Fin.insertNth i (circleMap (c i) (R i) \u03b8) fun i_1 =>\n      c (Fin.succAbove i i_1) + \u2191(R (Fin.succAbove i i_1)) * cexp (\u2191(\u0398 i_1) * I)"}, {"tactic": "simp only [funext_iff, i.forall_iff_succAbove, circleMap, Fin.insertNth_apply_same,\n  eq_self_iff_true, Fin.insertNth_apply_succAbove, imp_true_iff, and_self_iff]", "annotated_tactic": ["simp only [<a>funext_iff</a>, i.forall_iff_succAbove, <a>circleMap</a>, <a>Fin.insertNth_apply_same</a>,\n      <a>eq_self_iff_true</a>, <a>Fin.insertNth_apply_succAbove</a>, <a>imp_true_iff</a>, <a>and_self_iff</a>]", [{"full_name": "Function.funext_iff", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [82, 9], "def_end_pos": [82, 19]}, {"full_name": "circleMap", "def_path": "Mathlib/MeasureTheory/Integral/CircleIntegral.lean", "def_pos": [89, 5], "def_end_pos": [89, 14]}, {"full_name": "Fin.insertNth_apply_same", "def_path": "Mathlib/Data/Fin/Tuple/Basic.lean", "def_pos": [667, 9], "def_end_pos": [667, 29]}, {"full_name": "eq_self_iff_true", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [86, 9], "def_end_pos": [86, 25]}, {"full_name": "Fin.insertNth_apply_succAbove", "def_path": "Mathlib/Data/Fin/Tuple/Basic.lean", "def_pos": [672, 9], "def_end_pos": [672, 34]}, {"full_name": "imp_true_iff", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [116, 9], "def_end_pos": [116, 21]}, {"full_name": "and_self_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [155, 9], "def_end_pos": [155, 21]}]], "state_before": "case e_a.e_a\nn : \u2115\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf\u271d g : (Fin n \u2192 \u2102) \u2192 E\nc\u271d : Fin n \u2192 \u2102\nR\u271d : Fin n \u2192 \u211d\nf : (Fin (n + 1) \u2192 \u2102) \u2192 E\nc : Fin (n + 1) \u2192 \u2102\nR : Fin (n + 1) \u2192 \u211d\nhf : TorusIntegrable f c R\ni : Fin (n + 1)\ne : \u211d \u00d7 (Fin n \u2192 \u211d) \u2243\u1d50 (Fin (n + 1) \u2192 \u211d) := MeasurableEquiv.symm (MeasurableEquiv.piFinSuccAboveEquiv (fun x => \u211d) i)\nhem : MeasurePreserving \u2191e\nhe\u03c0 : (\u2191e \u207b\u00b9' Icc 0 fun x => 2 * \u03c0) = Icc 0 (2 * \u03c0) \u00d7\u02e2 Icc 0 fun x => 2 * \u03c0\n\u03b8 : \u211d\nx\u271d\u00b9 : \u03b8 \u2208 Icc 0 (2 * \u03c0)\n\u0398 : Fin n \u2192 \u211d\nx\u271d : \u0398 \u2208 Icc 0 fun x => 2 * \u03c0\n\u22a2 (fun i_1 => c i_1 + \u2191(R i_1) * cexp (\u2191(Fin.insertNth i \u03b8 \u0398 i_1) * I)) =\n    Fin.insertNth i (circleMap (c i) (R i) \u03b8) fun i_1 =>\n      c (Fin.succAbove i i_1) + \u2191(R (Fin.succAbove i i_1)) * cexp (\u2191(\u0398 i_1) * I)", "state_after": "no goals"}, {"tactic": "have := hf.function_integrable", "annotated_tactic": ["have := hf.function_integrable", []], "state_before": "case hf\nn : \u2115\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf\u271d g : (Fin n \u2192 \u2102) \u2192 E\nc\u271d : Fin n \u2192 \u2102\nR\u271d : Fin n \u2192 \u211d\nf : (Fin (n + 1) \u2192 \u2102) \u2192 E\nc : Fin (n + 1) \u2192 \u2102\nR : Fin (n + 1) \u2192 \u211d\nhf : TorusIntegrable f c R\ni : Fin (n + 1)\ne : \u211d \u00d7 (Fin n \u2192 \u211d) \u2243\u1d50 (Fin (n + 1) \u2192 \u211d) := MeasurableEquiv.symm (MeasurableEquiv.piFinSuccAboveEquiv (fun x => \u211d) i)\nhem : MeasurePreserving \u2191e\nhe\u03c0 : (\u2191e \u207b\u00b9' Icc 0 fun x => 2 * \u03c0) = Icc 0 (2 * \u03c0) \u00d7\u02e2 Icc 0 fun x => 2 * \u03c0\n\u22a2 IntegrableOn (fun x => (\u220f i : Fin (n + 1), \u2191(R i) * cexp (\u2191(\u2191e x i) * I) * I) \u2022 f (torusMap c R (\u2191e x)))\n    (Icc 0 (2 * \u03c0) \u00d7\u02e2 Icc 0 fun x => 2 * \u03c0)", "state_after": "case hf\nn : \u2115\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf\u271d g : (Fin n \u2192 \u2102) \u2192 E\nc\u271d : Fin n \u2192 \u2102\nR\u271d : Fin n \u2192 \u211d\nf : (Fin (n + 1) \u2192 \u2102) \u2192 E\nc : Fin (n + 1) \u2192 \u2102\nR : Fin (n + 1) \u2192 \u211d\nhf : TorusIntegrable f c R\ni : Fin (n + 1)\ne : \u211d \u00d7 (Fin n \u2192 \u211d) \u2243\u1d50 (Fin (n + 1) \u2192 \u211d) := MeasurableEquiv.symm (MeasurableEquiv.piFinSuccAboveEquiv (fun x => \u211d) i)\nhem : MeasurePreserving \u2191e\nhe\u03c0 : (\u2191e \u207b\u00b9' Icc 0 fun x => 2 * \u03c0) = Icc 0 (2 * \u03c0) \u00d7\u02e2 Icc 0 fun x => 2 * \u03c0\nthis :\n  IntegrableOn (fun \u03b8 => (\u220f i : Fin (n + 1), \u2191(R i) * cexp (\u2191(\u03b8 i) * I) * I) \u2022 f (torusMap c R \u03b8))\n    (Icc 0 fun x => 2 * \u03c0)\n\u22a2 IntegrableOn (fun x => (\u220f i : Fin (n + 1), \u2191(R i) * cexp (\u2191(\u2191e x i) * I) * I) \u2022 f (torusMap c R (\u2191e x)))\n    (Icc 0 (2 * \u03c0) \u00d7\u02e2 Icc 0 fun x => 2 * \u03c0)"}, {"tactic": "rwa [\u2190 hem.integrableOn_comp_preimage e.measurableEmbedding, he\u03c0] at this", "annotated_tactic": ["rwa [\u2190 hem.integrableOn_comp_preimage e.measurableEmbedding, he\u03c0] at this", []], "state_before": "case hf\nn : \u2115\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf\u271d g : (Fin n \u2192 \u2102) \u2192 E\nc\u271d : Fin n \u2192 \u2102\nR\u271d : Fin n \u2192 \u211d\nf : (Fin (n + 1) \u2192 \u2102) \u2192 E\nc : Fin (n + 1) \u2192 \u2102\nR : Fin (n + 1) \u2192 \u211d\nhf : TorusIntegrable f c R\ni : Fin (n + 1)\ne : \u211d \u00d7 (Fin n \u2192 \u211d) \u2243\u1d50 (Fin (n + 1) \u2192 \u211d) := MeasurableEquiv.symm (MeasurableEquiv.piFinSuccAboveEquiv (fun x => \u211d) i)\nhem : MeasurePreserving \u2191e\nhe\u03c0 : (\u2191e \u207b\u00b9' Icc 0 fun x => 2 * \u03c0) = Icc 0 (2 * \u03c0) \u00d7\u02e2 Icc 0 fun x => 2 * \u03c0\nthis :\n  IntegrableOn (fun \u03b8 => (\u220f i : Fin (n + 1), \u2191(R i) * cexp (\u2191(\u03b8 i) * I) * I) \u2022 f (torusMap c R \u03b8))\n    (Icc 0 fun x => 2 * \u03c0)\n\u22a2 IntegrableOn (fun x => (\u220f i : Fin (n + 1), \u2191(R i) * cexp (\u2191(\u2191e x i) * I) * I) \u2022 f (torusMap c R (\u2191e x)))\n    (Icc 0 (2 * \u03c0) \u00d7\u02e2 Icc 0 fun x => 2 * \u03c0)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Lattice.lean", "full_name": "Finset.inf_inf", "start": [347, 1], "end": [348, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Kernel/Disintegration.lean", "full_name": "ProbabilityTheory.set_lintegral_condKernelReal_Iic", "start": [78, 1], "end": [81, 67], "traced_tactics": [{"tactic": "simp_rw [condKernelReal_Iic]", "annotated_tactic": ["simp_rw [<a>condKernelReal_Iic</a>]", [{"full_name": "ProbabilityTheory.condKernelReal_Iic", "def_path": "Mathlib/Probability/Kernel/Disintegration.lean", "def_pos": [73, 9], "def_end_pos": [73, 27]}]], "state_before": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) (Iic x) \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic x)", "state_after": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic x)"}, {"tactic": "exact set_lintegral_condCdf \u03c1 x hs", "annotated_tactic": ["exact <a>set_lintegral_condCdf</a> \u03c1 x hs", [{"full_name": "ProbabilityTheory.set_lintegral_condCdf", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [874, 9], "def_end_pos": [874, 30]}]], "state_before": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic x)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Setoid/Partition.lean", "full_name": "IndexedPartition.index_some", "start": [394, 1], "end": [395, 42], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "full_name": "Set.coe_singletonOneHom", "start": [149, 1], "end": [150, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Kernel/Composition.lean", "full_name": "ProbabilityTheory.kernel.ae_ae_of_ae_compProd", "start": [317, 1], "end": [319, 29], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/List/Pairwise.lean", "full_name": "List.pairwise_iff_get", "start": [236, 1], "end": [247, 29], "traced_tactics": [{"tactic": "rw [pairwise_iff_forall_sublist]", "annotated_tactic": ["rw [<a>pairwise_iff_forall_sublist</a>]", [{"full_name": "List.pairwise_iff_forall_sublist", "def_path": "lake-packages/std/Std/Data/List/Pairwise.lean", "def_pos": [172, 9], "def_end_pos": [172, 36]}]], "state_before": "\u03b1\u271d : Type u_1\nR : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\nl : List \u03b1\u271d\n\u22a2 Pairwise R l \u2194 \u2200 (i j : Fin (length l)), i < j \u2192 R (get l i) (get l j)", "state_after": "\u03b1\u271d : Type u_1\nR : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\nl : List \u03b1\u271d\n\u22a2 (\u2200 {a b : \u03b1\u271d}, [a, b] <+ l \u2192 R a b) \u2194 \u2200 (i j : Fin (length l)), i < j \u2192 R (get l i) (get l j)"}, {"tactic": "constructor <;> intro h", "annotated_tactic": ["constructor <;> intro h", []], "state_before": "\u03b1\u271d : Type u_1\nR : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\nl : List \u03b1\u271d\n\u22a2 (\u2200 {a b : \u03b1\u271d}, [a, b] <+ l \u2192 R a b) \u2194 \u2200 (i j : Fin (length l)), i < j \u2192 R (get l i) (get l j)", "state_after": "case mp\n\u03b1\u271d : Type u_1\nR : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\nl : List \u03b1\u271d\nh : \u2200 {a b : \u03b1\u271d}, [a, b] <+ l \u2192 R a b\n\u22a2 \u2200 (i j : Fin (length l)), i < j \u2192 R (get l i) (get l j)\n\ncase mpr\n\u03b1\u271d : Type u_1\nR : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\nl : List \u03b1\u271d\nh : \u2200 (i j : Fin (length l)), i < j \u2192 R (get l i) (get l j)\n\u22a2 \u2200 {a b : \u03b1\u271d}, [a, b] <+ l \u2192 R a b"}, {"tactic": "intros i j h'", "annotated_tactic": ["intros i j h'", []], "state_before": "case mp\n\u03b1\u271d : Type u_1\nR : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\nl : List \u03b1\u271d\nh : \u2200 {a b : \u03b1\u271d}, [a, b] <+ l \u2192 R a b\n\u22a2 \u2200 (i j : Fin (length l)), i < j \u2192 R (get l i) (get l j)", "state_after": "case mp\n\u03b1\u271d : Type u_1\nR : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\nl : List \u03b1\u271d\nh : \u2200 {a b : \u03b1\u271d}, [a, b] <+ l \u2192 R a b\ni j : Fin (length l)\nh' : i < j\n\u22a2 R (get l i) (get l j)"}, {"tactic": "apply h", "annotated_tactic": ["apply h", []], "state_before": "case mp\n\u03b1\u271d : Type u_1\nR : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\nl : List \u03b1\u271d\nh : \u2200 {a b : \u03b1\u271d}, [a, b] <+ l \u2192 R a b\ni j : Fin (length l)\nh' : i < j\n\u22a2 R (get l i) (get l j)", "state_after": "case mp.a\n\u03b1\u271d : Type u_1\nR : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\nl : List \u03b1\u271d\nh : \u2200 {a b : \u03b1\u271d}, [a, b] <+ l \u2192 R a b\ni j : Fin (length l)\nh' : i < j\n\u22a2 [get l i, get l j] <+ l"}, {"tactic": "apply map_get_sublist (is := [i, j])", "annotated_tactic": ["apply <a>map_get_sublist</a> (is := [i, j])", [{"full_name": "List.map_get_sublist", "def_path": "lake-packages/std/Std/Data/List/Pairwise.lean", "def_pos": [204, 9], "def_end_pos": [204, 24]}]], "state_before": "case mp.a\n\u03b1\u271d : Type u_1\nR : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\nl : List \u03b1\u271d\nh : \u2200 {a b : \u03b1\u271d}, [a, b] <+ l \u2192 R a b\ni j : Fin (length l)\nh' : i < j\n\u22a2 [get l i, get l j] <+ l", "state_after": "case mp.a\n\u03b1\u271d : Type u_1\nR : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\nl : List \u03b1\u271d\nh : \u2200 {a b : \u03b1\u271d}, [a, b] <+ l \u2192 R a b\ni j : Fin (length l)\nh' : i < j\n\u22a2 Pairwise (fun x x_1 => \u2191x < \u2191x_1) [i, j]"}, {"tactic": "rw [Fin.lt_def] at h'", "annotated_tactic": ["rw [<a>Fin.lt_def</a>] at h'", [{"full_name": "Fin.lt_def", "def_path": "lake-packages/std/Std/Data/Fin/Lemmas.lean", "def_pos": [81, 9], "def_end_pos": [81, 15]}]], "state_before": "case mp.a\n\u03b1\u271d : Type u_1\nR : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\nl : List \u03b1\u271d\nh : \u2200 {a b : \u03b1\u271d}, [a, b] <+ l \u2192 R a b\ni j : Fin (length l)\nh' : i < j\n\u22a2 Pairwise (fun x x_1 => \u2191x < \u2191x_1) [i, j]", "state_after": "case mp.a\n\u03b1\u271d : Type u_1\nR : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\nl : List \u03b1\u271d\nh : \u2200 {a b : \u03b1\u271d}, [a, b] <+ l \u2192 R a b\ni j : Fin (length l)\nh' : \u2191i < \u2191j\n\u22a2 Pairwise (fun x x_1 => \u2191x < \u2191x_1) [i, j]"}, {"tactic": "simp [h']", "annotated_tactic": ["simp [h']", []], "state_before": "case mp.a\n\u03b1\u271d : Type u_1\nR : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\nl : List \u03b1\u271d\nh : \u2200 {a b : \u03b1\u271d}, [a, b] <+ l \u2192 R a b\ni j : Fin (length l)\nh' : \u2191i < \u2191j\n\u22a2 Pairwise (fun x x_1 => \u2191x < \u2191x_1) [i, j]", "state_after": "no goals"}, {"tactic": "intros a b h'", "annotated_tactic": ["intros a b h'", []], "state_before": "case mpr\n\u03b1\u271d : Type u_1\nR : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\nl : List \u03b1\u271d\nh : \u2200 (i j : Fin (length l)), i < j \u2192 R (get l i) (get l j)\n\u22a2 \u2200 {a b : \u03b1\u271d}, [a, b] <+ l \u2192 R a b", "state_after": "case mpr\n\u03b1\u271d : Type u_1\nR : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\nl : List \u03b1\u271d\nh : \u2200 (i j : Fin (length l)), i < j \u2192 R (get l i) (get l j)\na b : \u03b1\u271d\nh' : [a, b] <+ l\n\u22a2 R a b"}, {"tactic": "have \u27e8is, h', hij\u27e9 := sublist_eq_map_get h'", "annotated_tactic": ["have \u27e8is, h', hij\u27e9 := <a>sublist_eq_map_get</a> h'", [{"full_name": "List.sublist_eq_map_get", "def_path": "lake-packages/std/Std/Data/List/Pairwise.lean", "def_pos": [222, 9], "def_end_pos": [222, 27]}]], "state_before": "case mpr\n\u03b1\u271d : Type u_1\nR : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\nl : List \u03b1\u271d\nh : \u2200 (i j : Fin (length l)), i < j \u2192 R (get l i) (get l j)\na b : \u03b1\u271d\nh' : [a, b] <+ l\n\u22a2 R a b", "state_after": "case mpr\n\u03b1\u271d : Type u_1\nR : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\nl : List \u03b1\u271d\nh : \u2200 (i j : Fin (length l)), i < j \u2192 R (get l i) (get l j)\na b : \u03b1\u271d\nh'\u271d : [a, b] <+ l\nis : List (Fin (length l))\nh' : [a, b] = map (get l) is\nhij : Pairwise (fun x x_1 => x < x_1) is\n\u22a2 R a b"}, {"tactic": "rcases is with \u27e8\u27e9 | \u27e8a', \u27e8\u27e9 | \u27e8b', \u27e8\u27e9\u27e9\u27e9 <;> simp at h'", "annotated_tactic": ["rcases is with \u27e8\u27e9 | \u27e8a', \u27e8\u27e9 | \u27e8b', \u27e8\u27e9\u27e9\u27e9 <;> simp at h'", []], "state_before": "case mpr\n\u03b1\u271d : Type u_1\nR : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\nl : List \u03b1\u271d\nh : \u2200 (i j : Fin (length l)), i < j \u2192 R (get l i) (get l j)\na b : \u03b1\u271d\nh'\u271d : [a, b] <+ l\nis : List (Fin (length l))\nh' : [a, b] = map (get l) is\nhij : Pairwise (fun x x_1 => x < x_1) is\n\u22a2 R a b", "state_after": "case mpr.cons.cons.nil\n\u03b1\u271d : Type u_1\nR : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\nl : List \u03b1\u271d\nh : \u2200 (i j : Fin (length l)), i < j \u2192 R (get l i) (get l j)\na b : \u03b1\u271d\nh'\u271d : [a, b] <+ l\na' b' : Fin (length l)\nhij : Pairwise (fun x x_1 => x < x_1) [a', b']\nh' : a = get l a' \u2227 b = get l b'\n\u22a2 R a b"}, {"tactic": "rcases h' with \u27e8rfl, rfl\u27e9", "annotated_tactic": ["rcases h' with \u27e8rfl, rfl\u27e9", []], "state_before": "case mpr.cons.cons.nil\n\u03b1\u271d : Type u_1\nR : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\nl : List \u03b1\u271d\nh : \u2200 (i j : Fin (length l)), i < j \u2192 R (get l i) (get l j)\na b : \u03b1\u271d\nh'\u271d : [a, b] <+ l\na' b' : Fin (length l)\nhij : Pairwise (fun x x_1 => x < x_1) [a', b']\nh' : a = get l a' \u2227 b = get l b'\n\u22a2 R a b", "state_after": "case mpr.cons.cons.nil.intro\n\u03b1\u271d : Type u_1\nR : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\nl : List \u03b1\u271d\nh : \u2200 (i j : Fin (length l)), i < j \u2192 R (get l i) (get l j)\na' b' : Fin (length l)\nhij : Pairwise (fun x x_1 => x < x_1) [a', b']\nh' : [get l a', get l b'] <+ l\n\u22a2 R (get l a') (get l b')"}, {"tactic": "apply h", "annotated_tactic": ["apply h", []], "state_before": "case mpr.cons.cons.nil.intro\n\u03b1\u271d : Type u_1\nR : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\nl : List \u03b1\u271d\nh : \u2200 (i j : Fin (length l)), i < j \u2192 R (get l i) (get l j)\na' b' : Fin (length l)\nhij : Pairwise (fun x x_1 => x < x_1) [a', b']\nh' : [get l a', get l b'] <+ l\n\u22a2 R (get l a') (get l b')", "state_after": "case mpr.cons.cons.nil.intro._hij\n\u03b1\u271d : Type u_1\nR : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\nl : List \u03b1\u271d\nh : \u2200 (i j : Fin (length l)), i < j \u2192 R (get l i) (get l j)\na' b' : Fin (length l)\nhij : Pairwise (fun x x_1 => x < x_1) [a', b']\nh' : [get l a', get l b'] <+ l\n\u22a2 a' < b'"}, {"tactic": "simpa using hij", "annotated_tactic": ["simpa using hij", []], "state_before": "case mpr.cons.cons.nil.intro._hij\n\u03b1\u271d : Type u_1\nR : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\nl : List \u03b1\u271d\nh : \u2200 (i j : Fin (length l)), i < j \u2192 R (get l i) (get l j)\na' b' : Fin (length l)\nhij : Pairwise (fun x x_1 => x < x_1) [a', b']\nh' : [get l a', get l b'] <+ l\n\u22a2 a' < b'", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Primrec.lean", "full_name": "Primrec.vector_get'", "start": [1326, 1], "end": [1327, 16], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/MeanInequalities.lean", "full_name": "ENNReal.lintegral_mul_eq_zero_of_lintegral_rpow_eq_zero", "start": [137, 1], "end": [144, 38], "traced_tactics": [{"tactic": "rw [\u2190 @lintegral_zero_fun \u03b1 _ \u03bc]", "annotated_tactic": ["rw [\u2190 @<a>lintegral_zero_fun</a> \u03b1 _ \u03bc]", [{"full_name": "MeasureTheory.lintegral_zero_fun", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [144, 9], "def_end_pos": [144, 27]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nhp0 : 0 \u2264 p\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_zero : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 0\n\u22a2 \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc = 0", "state_after": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nhp0 : 0 \u2264 p\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_zero : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 0\n\u22a2 \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc = lintegral \u03bc 0"}, {"tactic": "refine' lintegral_congr_ae _", "annotated_tactic": ["refine' <a>lintegral_congr_ae</a> _", [{"full_name": "MeasureTheory.lintegral_congr_ae", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [304, 9], "def_end_pos": [304, 27]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nhp0 : 0 \u2264 p\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_zero : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 0\n\u22a2 \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc = lintegral \u03bc 0", "state_after": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nhp0 : 0 \u2264 p\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_zero : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 0\n\u22a2 (fun a => (f * g) a) =\u1d50[\u03bc] fun a => OfNat.ofNat 0 a"}, {"tactic": "suffices h_mul_zero : f * g =\u1d50[\u03bc] 0 * g", "annotated_tactic": ["suffices h_mul_zero : f * g =\u1d50[\u03bc] 0 * g", []], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nhp0 : 0 \u2264 p\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_zero : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 0\n\u22a2 (fun a => (f * g) a) =\u1d50[\u03bc] fun a => OfNat.ofNat 0 a", "state_after": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nhp0 : 0 \u2264 p\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_zero : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 0\nh_mul_zero : f * g =\u1d50[\u03bc] 0 * g\n\u22a2 (fun a => (f * g) a) =\u1d50[\u03bc] fun a => OfNat.ofNat 0 a\n\ncase h_mul_zero\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nhp0 : 0 \u2264 p\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_zero : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 0\n\u22a2 f * g =\u1d50[\u03bc] 0 * g"}, {"tactic": "have hf_eq_zero : f =\u1d50[\u03bc] 0 := ae_eq_zero_of_lintegral_rpow_eq_zero hp0 hf hf_zero", "annotated_tactic": ["have hf_eq_zero : f =\u1d50[\u03bc] 0 := <a>ae_eq_zero_of_lintegral_rpow_eq_zero</a> hp0 hf hf_zero", [{"full_name": "ENNReal.ae_eq_zero_of_lintegral_rpow_eq_zero", "def_path": "Mathlib/MeasureTheory/Integral/MeanInequalities.lean", "def_pos": [128, 9], "def_end_pos": [128, 45]}]], "state_before": "case h_mul_zero\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nhp0 : 0 \u2264 p\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_zero : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 0\n\u22a2 f * g =\u1d50[\u03bc] 0 * g", "state_after": "case h_mul_zero\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nhp0 : 0 \u2264 p\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_zero : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 0\nhf_eq_zero : f =\u1d50[\u03bc] 0\n\u22a2 f * g =\u1d50[\u03bc] 0 * g"}, {"tactic": "exact hf_eq_zero.mul (ae_eq_refl g)", "annotated_tactic": ["exact hf_eq_zero.mul (<a>ae_eq_refl</a> g)", [{"full_name": "MeasureTheory.ae_eq_refl", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [436, 9], "def_end_pos": [436, 19]}]], "state_before": "case h_mul_zero\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nhp0 : 0 \u2264 p\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_zero : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 0\nhf_eq_zero : f =\u1d50[\u03bc] 0\n\u22a2 f * g =\u1d50[\u03bc] 0 * g", "state_after": "no goals"}, {"tactic": "rwa [zero_mul] at h_mul_zero", "annotated_tactic": ["rwa [<a>zero_mul</a>] at h_mul_zero", [{"full_name": "MulZeroClass.zero_mul", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [36, 3], "def_end_pos": [36, 11]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nhp0 : 0 \u2264 p\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_zero : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 0\nh_mul_zero : f * g =\u1d50[\u03bc] 0 * g\n\u22a2 (fun a => (f * g) a) =\u1d50[\u03bc] fun a => OfNat.ofNat 0 a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Pointwise/Interval.lean", "full_name": "Set.preimage_mul_const_uIcc", "start": [686, 1], "end": [691, 95], "traced_tactics": [{"tactic": "simp [\u2190 Icc_min_max, h, h.le, min_div_div_right_of_nonpos, max_div_div_right_of_nonpos]", "annotated_tactic": ["simp [\u2190 <a>Icc_min_max</a>, h, h.le, <a>min_div_div_right_of_nonpos</a>, <a>max_div_div_right_of_nonpos</a>]", [{"full_name": "Set.Icc_min_max", "def_path": "Mathlib/Data/Set/Intervals/UnorderedInterval.lean", "def_pos": [220, 9], "def_end_pos": [220, 20]}, {"full_name": "min_div_div_right_of_nonpos", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [990, 9], "def_end_pos": [990, 36]}, {"full_name": "max_div_div_right_of_nonpos", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [994, 9], "def_end_pos": [994, 36]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na : \u03b1\nha : a \u2260 0\nb c : \u03b1\nh : a < 0\n\u22a2 (fun x => x * a) \u207b\u00b9' [[b, c]] = [[b / a, c / a]]", "state_after": "no goals"}, {"tactic": "simp [\u2190 Icc_min_max, ha, ha.le, min_div_div_right, max_div_div_right]", "annotated_tactic": ["simp [\u2190 <a>Icc_min_max</a>, ha, ha.le, <a>min_div_div_right</a>, <a>max_div_div_right</a>]", [{"full_name": "Set.Icc_min_max", "def_path": "Mathlib/Data/Set/Intervals/UnorderedInterval.lean", "def_pos": [220, 9], "def_end_pos": [220, 20]}, {"full_name": "min_div_div_right", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [602, 9], "def_end_pos": [602, 26]}, {"full_name": "max_div_div_right", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [606, 9], "def_end_pos": [606, 26]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na : \u03b1\nha\u271d : a \u2260 0\nb c : \u03b1\nha : 0 < a\n\u22a2 (fun x => x * a) \u207b\u00b9' [[b, c]] = [[b / a, c / a]]", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "full_name": "Measurable.iSup_Prop", "start": [1342, 1], "end": [1345, 64], "traced_tactics": [{"tactic": "convert hf", "annotated_tactic": ["convert hf", []], "state_before": "\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns t u : Set \u03b1\u271d\ninst\u271d\u00b9\u00b9 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b1\u271d\ninst\u271d\u2079 : BorelSpace \u03b1\u271d\ninst\u271d\u2078 : TopologicalSpace \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : BorelSpace \u03b2\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : MeasurableSpace \u03b3\ninst\u271d\u00b3 : BorelSpace \u03b3\ninst\u271d\u00b2 : MeasurableSpace \u03b4\n\u03b1 : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : ConditionallyCompleteLattice \u03b1\np : Prop\nf : \u03b4 \u2192 \u03b1\nhf : Measurable f\nh : p\n\u22a2 Measurable fun b => \u2a06 (_ : p), f b", "state_after": "case h.e'_5.h\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns t u : Set \u03b1\u271d\ninst\u271d\u00b9\u00b9 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b1\u271d\ninst\u271d\u2079 : BorelSpace \u03b1\u271d\ninst\u271d\u2078 : TopologicalSpace \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : BorelSpace \u03b2\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : MeasurableSpace \u03b3\ninst\u271d\u00b3 : BorelSpace \u03b3\ninst\u271d\u00b2 : MeasurableSpace \u03b4\n\u03b1 : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : ConditionallyCompleteLattice \u03b1\np : Prop\nf : \u03b4 \u2192 \u03b1\nhf : Measurable f\nh : p\nx\u271d : \u03b4\n\u22a2 \u2a06 (_ : p), f x\u271d = f x\u271d"}, {"tactic": "exact ciSup_pos h", "annotated_tactic": ["exact <a>ciSup_pos</a> h", [{"full_name": "ciSup_pos", "def_path": "Mathlib/Order/ConditionallyCompleteLattice/Basic.lean", "def_pos": [875, 9], "def_end_pos": [875, 18]}]], "state_before": "case h.e'_5.h\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns t u : Set \u03b1\u271d\ninst\u271d\u00b9\u00b9 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b1\u271d\ninst\u271d\u2079 : BorelSpace \u03b1\u271d\ninst\u271d\u2078 : TopologicalSpace \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : BorelSpace \u03b2\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : MeasurableSpace \u03b3\ninst\u271d\u00b3 : BorelSpace \u03b3\ninst\u271d\u00b2 : MeasurableSpace \u03b4\n\u03b1 : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : ConditionallyCompleteLattice \u03b1\np : Prop\nf : \u03b4 \u2192 \u03b1\nhf : Measurable f\nh : p\nx\u271d : \u03b4\n\u22a2 \u2a06 (_ : p), f x\u271d = f x\u271d", "state_after": "no goals"}, {"tactic": "convert measurable_const using 1", "annotated_tactic": ["convert <a>measurable_const</a> using 1", [{"full_name": "measurable_const", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [570, 9], "def_end_pos": [570, 25]}]], "state_before": "\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns t u : Set \u03b1\u271d\ninst\u271d\u00b9\u00b9 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b1\u271d\ninst\u271d\u2079 : BorelSpace \u03b1\u271d\ninst\u271d\u2078 : TopologicalSpace \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : BorelSpace \u03b2\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : MeasurableSpace \u03b3\ninst\u271d\u00b3 : BorelSpace \u03b3\ninst\u271d\u00b2 : MeasurableSpace \u03b4\n\u03b1 : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : ConditionallyCompleteLattice \u03b1\np : Prop\nf : \u03b4 \u2192 \u03b1\nhf : Measurable f\nh : \u00acp\n\u22a2 Measurable fun b => \u2a06 (_ : p), f b", "state_after": "case h.e'_5\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns t u : Set \u03b1\u271d\ninst\u271d\u00b9\u00b9 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b1\u271d\ninst\u271d\u2079 : BorelSpace \u03b1\u271d\ninst\u271d\u2078 : TopologicalSpace \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : BorelSpace \u03b2\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : MeasurableSpace \u03b3\ninst\u271d\u00b3 : BorelSpace \u03b3\ninst\u271d\u00b2 : MeasurableSpace \u03b4\n\u03b1 : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : ConditionallyCompleteLattice \u03b1\np : Prop\nf : \u03b4 \u2192 \u03b1\nhf : Measurable f\nh : \u00acp\n\u22a2 (fun b => \u2a06 (_ : p), f b) = fun x => ?convert_5\n\ncase convert_5\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns t u : Set \u03b1\u271d\ninst\u271d\u00b9\u00b9 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b1\u271d\ninst\u271d\u2079 : BorelSpace \u03b1\u271d\ninst\u271d\u2078 : TopologicalSpace \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : BorelSpace \u03b2\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : MeasurableSpace \u03b3\ninst\u271d\u00b3 : BorelSpace \u03b3\ninst\u271d\u00b2 : MeasurableSpace \u03b4\n\u03b1 : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : ConditionallyCompleteLattice \u03b1\np : Prop\nf : \u03b4 \u2192 \u03b1\nhf : Measurable f\nh : \u00acp\n\u22a2 \u03b1"}, {"tactic": "funext", "annotated_tactic": ["funext", []], "state_before": "case h.e'_5\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns t u : Set \u03b1\u271d\ninst\u271d\u00b9\u00b9 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b1\u271d\ninst\u271d\u2079 : BorelSpace \u03b1\u271d\ninst\u271d\u2078 : TopologicalSpace \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : BorelSpace \u03b2\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : MeasurableSpace \u03b3\ninst\u271d\u00b3 : BorelSpace \u03b3\ninst\u271d\u00b2 : MeasurableSpace \u03b4\n\u03b1 : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : ConditionallyCompleteLattice \u03b1\np : Prop\nf : \u03b4 \u2192 \u03b1\nhf : Measurable f\nh : \u00acp\n\u22a2 (fun b => \u2a06 (_ : p), f b) = fun x => ?convert_5\n\ncase convert_5\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns t u : Set \u03b1\u271d\ninst\u271d\u00b9\u00b9 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b1\u271d\ninst\u271d\u2079 : BorelSpace \u03b1\u271d\ninst\u271d\u2078 : TopologicalSpace \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : BorelSpace \u03b2\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : MeasurableSpace \u03b3\ninst\u271d\u00b3 : BorelSpace \u03b3\ninst\u271d\u00b2 : MeasurableSpace \u03b4\n\u03b1 : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : ConditionallyCompleteLattice \u03b1\np : Prop\nf : \u03b4 \u2192 \u03b1\nhf : Measurable f\nh : \u00acp\n\u22a2 \u03b1", "state_after": "case h.e'_5.h\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns t u : Set \u03b1\u271d\ninst\u271d\u00b9\u00b9 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b1\u271d\ninst\u271d\u2079 : BorelSpace \u03b1\u271d\ninst\u271d\u2078 : TopologicalSpace \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : BorelSpace \u03b2\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : MeasurableSpace \u03b3\ninst\u271d\u00b3 : BorelSpace \u03b3\ninst\u271d\u00b2 : MeasurableSpace \u03b4\n\u03b1 : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : ConditionallyCompleteLattice \u03b1\np : Prop\nf : \u03b4 \u2192 \u03b1\nhf : Measurable f\nh : \u00acp\nx\u271d : \u03b4\n\u22a2 \u2a06 (_ : p), f x\u271d = ?convert_5\n\ncase convert_5\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns t u : Set \u03b1\u271d\ninst\u271d\u00b9\u00b9 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b1\u271d\ninst\u271d\u2079 : BorelSpace \u03b1\u271d\ninst\u271d\u2078 : TopologicalSpace \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : BorelSpace \u03b2\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : MeasurableSpace \u03b3\ninst\u271d\u00b3 : BorelSpace \u03b3\ninst\u271d\u00b2 : MeasurableSpace \u03b4\n\u03b1 : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : ConditionallyCompleteLattice \u03b1\np : Prop\nf : \u03b4 \u2192 \u03b1\nhf : Measurable f\nh : \u00acp\n\u22a2 \u03b1"}, {"tactic": "exact ciSup_neg h", "annotated_tactic": ["exact <a>ciSup_neg</a> h", [{"full_name": "ciSup_neg", "def_path": "Mathlib/Order/ConditionallyCompleteLattice/Basic.lean", "def_pos": [884, 7], "def_end_pos": [884, 16]}]], "state_before": "case h.e'_5.h\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns t u : Set \u03b1\u271d\ninst\u271d\u00b9\u00b9 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b1\u271d\ninst\u271d\u2079 : BorelSpace \u03b1\u271d\ninst\u271d\u2078 : TopologicalSpace \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : BorelSpace \u03b2\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : MeasurableSpace \u03b3\ninst\u271d\u00b3 : BorelSpace \u03b3\ninst\u271d\u00b2 : MeasurableSpace \u03b4\n\u03b1 : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : ConditionallyCompleteLattice \u03b1\np : Prop\nf : \u03b4 \u2192 \u03b1\nhf : Measurable f\nh : \u00acp\nx\u271d : \u03b4\n\u22a2 \u2a06 (_ : p), f x\u271d = ?convert_5\n\ncase convert_5\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns t u : Set \u03b1\u271d\ninst\u271d\u00b9\u00b9 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b1\u271d\ninst\u271d\u2079 : BorelSpace \u03b1\u271d\ninst\u271d\u2078 : TopologicalSpace \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : BorelSpace \u03b2\ninst\u271d\u2075 : TopologicalSpace \u03b3\ninst\u271d\u2074 : MeasurableSpace \u03b3\ninst\u271d\u00b3 : BorelSpace \u03b3\ninst\u271d\u00b2 : MeasurableSpace \u03b4\n\u03b1 : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : ConditionallyCompleteLattice \u03b1\np : Prop\nf : \u03b4 \u2192 \u03b1\nhf : Measurable f\nh : \u00acp\n\u22a2 \u03b1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Fin/Lemmas.lean", "full_name": "Fin.addCases_right", "start": [701, 1], "end": [704, 91], "traced_tactics": [{"tactic": "have : \u00ac(natAdd m i : Nat) < m := Nat.not_lt.2 (le_coe_natAdd ..)", "annotated_tactic": ["have : \u00ac(<a>natAdd</a> m i : <a>Nat</a>) < m := <a>Nat.not_lt</a>.2 (<a>le_coe_natAdd</a> ..)", [{"full_name": "Fin.natAdd", "def_path": "lake-packages/std/Std/Data/Fin/Basic.lean", "def_pos": [36, 5], "def_end_pos": [36, 11]}, {"full_name": "Nat", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1038, 11], "def_end_pos": [1038, 14]}, {"full_name": "Nat.not_lt", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [150, 27], "def_end_pos": [150, 33]}, {"full_name": "Fin.le_coe_natAdd", "def_path": "lake-packages/std/Std/Data/Fin/Lemmas.lean", "def_pos": [438, 9], "def_end_pos": [438, 22]}]], "state_before": "m n : Nat\nmotive : Fin (m + n) \u2192 Sort u_1\nleft : (i : Fin m) \u2192 motive (castAdd n i)\nright : (i : Fin n) \u2192 motive (natAdd m i)\ni : Fin n\n\u22a2 addCases left right (natAdd m i) = right i", "state_after": "m n : Nat\nmotive : Fin (m + n) \u2192 Sort u_1\nleft : (i : Fin m) \u2192 motive (castAdd n i)\nright : (i : Fin n) \u2192 motive (natAdd m i)\ni : Fin n\nthis : \u00ac\u2191(natAdd m i) < m\n\u22a2 addCases left right (natAdd m i) = right i"}, {"tactic": "rw [addCases, dif_neg this]", "annotated_tactic": ["rw [<a>addCases</a>, <a>dif_neg</a> this]", [{"full_name": "Fin.addCases", "def_path": "lake-packages/std/Std/Data/Fin/Lemmas.lean", "def_pos": [690, 21], "def_end_pos": [690, 29]}, {"full_name": "dif_neg", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [812, 9], "def_end_pos": [812, 16]}]], "state_before": "m n : Nat\nmotive : Fin (m + n) \u2192 Sort u_1\nleft : (i : Fin m) \u2192 motive (castAdd n i)\nright : (i : Fin n) \u2192 motive (natAdd m i)\ni : Fin n\nthis : \u00ac\u2191(natAdd m i) < m\n\u22a2 addCases left right (natAdd m i) = right i", "state_after": "m n : Nat\nmotive : Fin (m + n) \u2192 Sort u_1\nleft : (i : Fin m) \u2192 motive (castAdd n i)\nright : (i : Fin n) \u2192 motive (natAdd m i)\ni : Fin n\nthis : \u00ac\u2191(natAdd m i) < m\n\u22a2 (_ : natAdd m (subNat m (cast (_ : m + n = n + m) (natAdd m i)) (_ : m \u2264 \u2191(natAdd m i))) = natAdd m i) \u25b8\n      right (subNat m (cast (_ : m + n = n + m) (natAdd m i)) (_ : m \u2264 \u2191(natAdd m i))) =\n    right i"}, {"tactic": "exact eq_of_heq <| (eqRec_heq _ _).trans (by congr 1; simp)", "annotated_tactic": ["exact <a>eq_of_heq</a> <| (<a>eqRec_heq</a> _ _).<a>trans</a> (by congr 1; simp)", [{"full_name": "eq_of_heq", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [451, 9], "def_end_pos": [451, 18]}, {"full_name": "eqRec_heq", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [644, 9], "def_end_pos": [644, 18]}, {"full_name": "HEq.trans", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [630, 9], "def_end_pos": [630, 18]}]], "state_before": "m n : Nat\nmotive : Fin (m + n) \u2192 Sort u_1\nleft : (i : Fin m) \u2192 motive (castAdd n i)\nright : (i : Fin n) \u2192 motive (natAdd m i)\ni : Fin n\nthis : \u00ac\u2191(natAdd m i) < m\n\u22a2 (_ : natAdd m (subNat m (cast (_ : m + n = n + m) (natAdd m i)) (_ : m \u2264 \u2191(natAdd m i))) = natAdd m i) \u25b8\n      right (subNat m (cast (_ : m + n = n + m) (natAdd m i)) (_ : m \u2264 \u2191(natAdd m i))) =\n    right i", "state_after": "no goals"}, {"tactic": "congr 1", "annotated_tactic": ["congr 1", []], "state_before": "m n : Nat\nmotive : Fin (m + n) \u2192 Sort u_1\nleft : (i : Fin m) \u2192 motive (castAdd n i)\nright : (i : Fin n) \u2192 motive (natAdd m i)\ni : Fin n\nthis : \u00ac\u2191(natAdd m i) < m\n\u22a2 HEq (right (subNat m (cast (_ : m + n = n + m) (natAdd m i)) (_ : m \u2264 \u2191(natAdd m i)))) (right i)", "state_after": "case e_1\nm n : Nat\nmotive : Fin (m + n) \u2192 Sort u_1\nleft : (i : Fin m) \u2192 motive (castAdd n i)\nright : (i : Fin n) \u2192 motive (natAdd m i)\ni : Fin n\nthis : \u00ac\u2191(natAdd m i) < m\n\u22a2 subNat m (cast (_ : m + n = n + m) (natAdd m i)) (_ : m \u2264 \u2191(natAdd m i)) = i"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case e_1\nm n : Nat\nmotive : Fin (m + n) \u2192 Sort u_1\nleft : (i : Fin m) \u2192 motive (castAdd n i)\nright : (i : Fin n) \u2192 motive (natAdd m i)\ni : Fin n\nthis : \u00ac\u2191(natAdd m i) < m\n\u22a2 subNat m (cast (_ : m + n = n + m) (natAdd m i)) (_ : m \u2264 \u2191(natAdd m i)) = i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "full_name": "String.Pos.zero_addChar_byteIdx", "start": [122, 1], "end": [123, 58], "traced_tactics": [{"tactic": "simp only [addChar_byteIdx, byteIdx_zero, Nat.zero_add]", "annotated_tactic": ["simp only [<a>addChar_byteIdx</a>, <a>byteIdx_zero</a>, <a>Nat.zero_add</a>]", [{"full_name": "String.Pos.addChar_byteIdx", "def_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "def_pos": [118, 17], "def_end_pos": [118, 32]}, {"full_name": "String.Pos.byteIdx_zero", "def_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "def_pos": [97, 17], "def_end_pos": [97, 29]}, {"full_name": "Nat.zero_add", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [114, 27], "def_end_pos": [114, 35]}]], "state_before": "c : Char\n\u22a2 (0 + c).byteIdx = csize c", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/RBMap/Alter.lean", "full_name": "Std.RBNode.Path.ins_eq_fill", "start": [137, 1], "end": [142, 90], "traced_tactics": [{"tactic": "unfold ins", "annotated_tactic": ["unfold <a>ins</a>", [{"full_name": "Std.RBNode.Path.ins", "def_path": "lake-packages/std/Std/Data/RBMap/Basic.lean", "def_pos": [463, 5], "def_end_pos": [463, 13]}]], "state_before": "\u03b1 : Type u_1\nc\u2080 : RBColor\nn\u2080 : Nat\nc : RBColor\nn\u271d : Nat\npath : Path \u03b1\nt : RBNode \u03b1\nn : Nat\nx\u271d : RBNode \u03b1\nv\u271d : \u03b1\nparent\u271d : Path \u03b1\nha : Balanced x\u271d black n\nH : Path.Balanced c\u2080 n\u2080 parent\u271d red n\nhb : Balanced t black n\n\u22a2 ins (right red x\u271d v\u271d parent\u271d) t = setBlack (fill (right red x\u271d v\u271d parent\u271d) t)", "state_after": "\u03b1 : Type u_1\nc\u2080 : RBColor\nn\u2080 : Nat\nc : RBColor\nn\u271d : Nat\npath : Path \u03b1\nt : RBNode \u03b1\nn : Nat\nx\u271d : RBNode \u03b1\nv\u271d : \u03b1\nparent\u271d : Path \u03b1\nha : Balanced x\u271d black n\nH : Path.Balanced c\u2080 n\u2080 parent\u271d red n\nhb : Balanced t black n\n\u22a2 ins parent\u271d (node red x\u271d v\u271d t) = setBlack (fill (right red x\u271d v\u271d parent\u271d) t)"}, {"tactic": "exact ins_eq_fill H (.red ha hb)", "annotated_tactic": ["exact ins_eq_fill H (.red ha hb)", []], "state_before": "\u03b1 : Type u_1\nc\u2080 : RBColor\nn\u2080 : Nat\nc : RBColor\nn\u271d : Nat\npath : Path \u03b1\nt : RBNode \u03b1\nn : Nat\nx\u271d : RBNode \u03b1\nv\u271d : \u03b1\nparent\u271d : Path \u03b1\nha : Balanced x\u271d black n\nH : Path.Balanced c\u2080 n\u2080 parent\u271d red n\nhb : Balanced t black n\n\u22a2 ins parent\u271d (node red x\u271d v\u271d t) = setBlack (fill (right red x\u271d v\u271d parent\u271d) t)", "state_after": "no goals"}, {"tactic": "rw [ins, fill, \u2190 ins_eq_fill H (.black ha hb), balance1_eq ha]", "annotated_tactic": ["rw [<a>ins</a>, <a>fill</a>, \u2190 ins_eq_fill H (.black ha hb), <a>balance1_eq</a> ha]", [{"full_name": "Std.RBNode.Path.ins", "def_path": "lake-packages/std/Std/Data/RBMap/Basic.lean", "def_pos": [463, 5], "def_end_pos": [463, 13]}, {"full_name": "Std.RBNode.Path.fill", "def_path": "lake-packages/std/Std/Data/RBMap/Basic.lean", "def_pos": [442, 5], "def_end_pos": [442, 14]}, {"full_name": "Std.RBNode.balance1_eq", "def_path": "lake-packages/std/Std/Data/RBMap/WF.lean", "def_pos": [178, 9], "def_end_pos": [178, 20]}]], "state_before": "\u03b1 : Type u_1\nc\u2080 : RBColor\nn\u2080 : Nat\nc\u271d : RBColor\nn\u271d : Nat\npath : Path \u03b1\nt : RBNode \u03b1\nn : Nat\nc : RBColor\ny\u271d : RBNode \u03b1\nc\u2082\u271d : RBColor\nparent\u271d : Path \u03b1\nv\u271d : \u03b1\nhb : Balanced y\u271d c\u2082\u271d n\nH : Path.Balanced c\u2080 n\u2080 parent\u271d black (n + 1)\nha : Balanced t c n\n\u22a2 ins (left black parent\u271d v\u271d y\u271d) t = setBlack (fill (left black parent\u271d v\u271d y\u271d) t)", "state_after": "no goals"}, {"tactic": "rw [ins, fill, \u2190 ins_eq_fill H (.black ha hb), balance2_eq hb]", "annotated_tactic": ["rw [<a>ins</a>, <a>fill</a>, \u2190 ins_eq_fill H (.black ha hb), <a>balance2_eq</a> hb]", [{"full_name": "Std.RBNode.Path.ins", "def_path": "lake-packages/std/Std/Data/RBMap/Basic.lean", "def_pos": [463, 5], "def_end_pos": [463, 13]}, {"full_name": "Std.RBNode.Path.fill", "def_path": "lake-packages/std/Std/Data/RBMap/Basic.lean", "def_pos": [442, 5], "def_end_pos": [442, 14]}, {"full_name": "Std.RBNode.balance2_eq", "def_path": "lake-packages/std/Std/Data/RBMap/WF.lean", "def_pos": [183, 9], "def_end_pos": [183, 20]}]], "state_before": "\u03b1 : Type u_1\nc\u2080 : RBColor\nn\u2080 : Nat\nc\u271d : RBColor\nn\u271d : Nat\npath : Path \u03b1\nt : RBNode \u03b1\nn : Nat\nc : RBColor\nx\u271d : RBNode \u03b1\nc\u2081\u271d : RBColor\nv\u271d : \u03b1\nparent\u271d : Path \u03b1\nha : Balanced x\u271d c\u2081\u271d n\nH : Path.Balanced c\u2080 n\u2080 parent\u271d black (n + 1)\nhb : Balanced t c n\n\u22a2 ins (right black x\u271d v\u271d parent\u271d) t = setBlack (fill (right black x\u271d v\u271d parent\u271d) t)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Pointwise.lean", "full_name": "Finset.pow_subset_pow_of_one_mem", "start": [895, 1], "end": [900, 50], "traced_tactics": [{"tactic": "apply Nat.le_induction", "annotated_tactic": ["apply <a>Nat.le_induction</a>", [{"full_name": "Nat.le_induction", "def_path": "Mathlib/Data/Nat/Basic.lean", "def_pos": [509, 9], "def_end_pos": [509, 21]}]], "state_before": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : Monoid \u03b1\ns t : Finset \u03b1\na : \u03b1\nm n : \u2115\nhs : 1 \u2208 s\n\u22a2 m \u2264 n \u2192 s ^ m \u2286 s ^ n", "state_after": "case base\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : Monoid \u03b1\ns t : Finset \u03b1\na : \u03b1\nm n : \u2115\nhs : 1 \u2208 s\n\u22a2 s ^ m \u2286 s ^ m\n\ncase succ\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : Monoid \u03b1\ns t : Finset \u03b1\na : \u03b1\nm n : \u2115\nhs : 1 \u2208 s\n\u22a2 \u2200 (n : \u2115), m \u2264 n \u2192 s ^ m \u2286 s ^ n \u2192 s ^ m \u2286 s ^ (n + 1)"}, {"tactic": "exact fun _ hn => hn", "annotated_tactic": ["exact fun _ hn => hn", []], "state_before": "case base\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : Monoid \u03b1\ns t : Finset \u03b1\na : \u03b1\nm n : \u2115\nhs : 1 \u2208 s\n\u22a2 s ^ m \u2286 s ^ m", "state_after": "no goals"}, {"tactic": "intro n _ hmn", "annotated_tactic": ["intro n _ hmn", []], "state_before": "case succ\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : Monoid \u03b1\ns t : Finset \u03b1\na : \u03b1\nm n : \u2115\nhs : 1 \u2208 s\n\u22a2 \u2200 (n : \u2115), m \u2264 n \u2192 s ^ m \u2286 s ^ n \u2192 s ^ m \u2286 s ^ (n + 1)", "state_after": "case succ\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : Monoid \u03b1\ns t : Finset \u03b1\na : \u03b1\nm n\u271d : \u2115\nhs : 1 \u2208 s\nn : \u2115\nhn\u271d : m \u2264 n\nhmn : s ^ m \u2286 s ^ n\n\u22a2 s ^ m \u2286 s ^ (n + 1)"}, {"tactic": "rw [pow_succ]", "annotated_tactic": ["rw [<a>pow_succ</a>]", [{"full_name": "pow_succ", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [645, 9], "def_end_pos": [645, 17]}]], "state_before": "case succ\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : Monoid \u03b1\ns t : Finset \u03b1\na : \u03b1\nm n\u271d : \u2115\nhs : 1 \u2208 s\nn : \u2115\nhn\u271d : m \u2264 n\nhmn : s ^ m \u2286 s ^ n\n\u22a2 s ^ m \u2286 s ^ (n + 1)", "state_after": "case succ\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : Monoid \u03b1\ns t : Finset \u03b1\na : \u03b1\nm n\u271d : \u2115\nhs : 1 \u2208 s\nn : \u2115\nhn\u271d : m \u2264 n\nhmn : s ^ m \u2286 s ^ n\n\u22a2 s ^ m \u2286 s * s ^ n"}, {"tactic": "exact hmn.trans (subset_mul_right (s ^ n) hs)", "annotated_tactic": ["exact hmn.trans (<a>subset_mul_right</a> (s ^ n) hs)", [{"full_name": "Finset.subset_mul_right", "def_path": "Mathlib/Data/Finset/Pointwise.lean", "def_pos": [798, 9], "def_end_pos": [798, 25]}]], "state_before": "case succ\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : Monoid \u03b1\ns t : Finset \u03b1\na : \u03b1\nm n\u271d : \u2115\nhs : 1 \u2208 s\nn : \u2115\nhn\u271d : m \u2264 n\nhmn : s ^ m \u2286 s ^ n\n\u22a2 s ^ m \u2286 s * s ^ n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/CondCount.lean", "full_name": "ProbabilityTheory.condCount_of_univ", "start": [134, 1], "end": [135, 43], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "full_name": "Nat.recDiagOn_zero_succ", "start": [94, 1], "end": [99, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Card.lean", "full_name": "Set.ncard_le_ncard_of_injOn", "start": [763, 1], "end": [767, 84], "traced_tactics": [{"tactic": "have hle := encard_le_encard_of_injOn hf f_inj", "annotated_tactic": ["have hle := <a>encard_le_encard_of_injOn</a> hf f_inj", [{"full_name": "Set.encard_le_encard_of_injOn", "def_path": "Mathlib/Data/Set/Card.lean", "def_pos": [413, 9], "def_end_pos": [413, 34]}]], "state_before": "\u03b1 : Type u_2\ns t\u271d : Set \u03b1\n\u03b2 : Type u_1\nt : Set \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 (a : \u03b1), a \u2208 s \u2192 f a \u2208 t\nf_inj : InjOn f s\nht : autoParam (Set.Finite t) _auto\u271d\n\u22a2 ncard s \u2264 ncard t", "state_after": "\u03b1 : Type u_2\ns t\u271d : Set \u03b1\n\u03b2 : Type u_1\nt : Set \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 (a : \u03b1), a \u2208 s \u2192 f a \u2208 t\nf_inj : InjOn f s\nht : autoParam (Set.Finite t) _auto\u271d\nhle : encard s \u2264 encard t\n\u22a2 ncard s \u2264 ncard t"}, {"tactic": "to_encard_tac", "annotated_tactic": ["to_encard_tac", []], "state_before": "\u03b1 : Type u_2\ns t\u271d : Set \u03b1\n\u03b2 : Type u_1\nt : Set \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 (a : \u03b1), a \u2208 s \u2192 f a \u2208 t\nf_inj : InjOn f s\nht : autoParam (Set.Finite t) _auto\u271d\nhle : encard s \u2264 encard t\n\u22a2 ncard s \u2264 ncard t", "state_after": "\u03b1 : Type u_2\ns t\u271d : Set \u03b1\n\u03b2 : Type u_1\nt : Set \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 (a : \u03b1), a \u2208 s \u2192 f a \u2208 t\nf_inj : InjOn f s\nht : autoParam (Set.Finite t) _auto\u271d\nhle : encard s \u2264 encard t\n\u22a2 \u2191(ncard s) \u2264 \u2191(ncard t)"}, {"tactic": "rwa [ht.cast_ncard_eq, (ht.finite_of_encard_le hle).cast_ncard_eq]", "annotated_tactic": ["rwa [ht.cast_ncard_eq, (ht.finite_of_encard_le hle).<a>cast_ncard_eq</a>]", [{"full_name": "Set.Finite.cast_ncard_eq", "def_path": "Mathlib/Data/Set/Card.lean", "def_pos": [477, 9], "def_end_pos": [477, 29]}]], "state_before": "\u03b1 : Type u_2\ns t\u271d : Set \u03b1\n\u03b2 : Type u_1\nt : Set \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 (a : \u03b1), a \u2208 s \u2192 f a \u2208 t\nf_inj : InjOn f s\nht : autoParam (Set.Finite t) _auto\u271d\nhle : encard s \u2264 encard t\n\u22a2 \u2191(ncard s) \u2264 \u2191(ncard t)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Card.lean", "full_name": "Set.ncard_le_one_iff_eq", "start": [1029, 1], "end": [1038, 62], "traced_tactics": [{"tactic": "obtain rfl | \u27e8x, hx\u27e9 := s.eq_empty_or_nonempty", "annotated_tactic": ["obtain rfl | \u27e8x, hx\u27e9 := s.eq_empty_or_nonempty", []], "state_before": "\u03b1 : Type u_1\ns t : Set \u03b1\nhs : autoParam (Set.Finite s) _auto\u271d\n\u22a2 ncard s \u2264 1 \u2194 s = \u2205 \u2228 \u2203 a, s = {a}", "state_after": "case inl\n\u03b1 : Type u_1\nt : Set \u03b1\nhs : autoParam (Set.Finite \u2205) _auto\u271d\n\u22a2 ncard \u2205 \u2264 1 \u2194 \u2205 = \u2205 \u2228 \u2203 a, \u2205 = {a}\n\ncase inr.intro\n\u03b1 : Type u_1\ns t : Set \u03b1\nhs : autoParam (Set.Finite s) _auto\u271d\nx : \u03b1\nhx : x \u2208 s\n\u22a2 ncard s \u2264 1 \u2194 s = \u2205 \u2228 \u2203 a, s = {a}"}, {"tactic": "rw [ncard_le_one_iff hs]", "annotated_tactic": ["rw [<a>ncard_le_one_iff</a> hs]", [{"full_name": "Set.ncard_le_one_iff", "def_path": "Mathlib/Data/Set/Card.lean", "def_pos": [1023, 9], "def_end_pos": [1023, 25]}]], "state_before": "case inr.intro\n\u03b1 : Type u_1\ns t : Set \u03b1\nhs : autoParam (Set.Finite s) _auto\u271d\nx : \u03b1\nhx : x \u2208 s\n\u22a2 ncard s \u2264 1 \u2194 s = \u2205 \u2228 \u2203 a, s = {a}", "state_after": "case inr.intro\n\u03b1 : Type u_1\ns t : Set \u03b1\nhs : autoParam (Set.Finite s) _auto\u271d\nx : \u03b1\nhx : x \u2208 s\n\u22a2 (\u2200 {a b : \u03b1}, a \u2208 s \u2192 b \u2208 s \u2192 a = b) \u2194 s = \u2205 \u2228 \u2203 a, s = {a}"}, {"tactic": "refine' \u27e8fun h \u21a6 Or.inr \u27e8x, (singleton_subset_iff.mpr hx).antisymm' fun y hy \u21a6 h hy hx\u27e9, _\u27e9", "annotated_tactic": ["refine' \u27e8fun h \u21a6 <a>Or.inr</a> \u27e8x, (singleton_subset_iff.mpr hx).<a>antisymm'</a> fun y hy \u21a6 h hy hx\u27e9, _\u27e9", [{"full_name": "Or.inr", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [519, 5], "def_end_pos": [519, 8]}, {"full_name": "HasSubset.Subset.antisymm'", "def_path": "Mathlib/Order/RelClasses.lean", "def_pos": [670, 7], "def_end_pos": [670, 33]}]], "state_before": "case inr.intro\n\u03b1 : Type u_1\ns t : Set \u03b1\nhs : autoParam (Set.Finite s) _auto\u271d\nx : \u03b1\nhx : x \u2208 s\n\u22a2 (\u2200 {a b : \u03b1}, a \u2208 s \u2192 b \u2208 s \u2192 a = b) \u2194 s = \u2205 \u2228 \u2203 a, s = {a}", "state_after": "case inr.intro\n\u03b1 : Type u_1\ns t : Set \u03b1\nhs : autoParam (Set.Finite s) _auto\u271d\nx : \u03b1\nhx : x \u2208 s\n\u22a2 (s = \u2205 \u2228 \u2203 a, s = {a}) \u2192 \u2200 {a b : \u03b1}, a \u2208 s \u2192 b \u2208 s \u2192 a = b"}, {"tactic": "rintro (rfl | \u27e8a, rfl\u27e9)", "annotated_tactic": ["rintro (rfl | \u27e8a, rfl\u27e9)", []], "state_before": "case inr.intro\n\u03b1 : Type u_1\ns t : Set \u03b1\nhs : autoParam (Set.Finite s) _auto\u271d\nx : \u03b1\nhx : x \u2208 s\n\u22a2 (s = \u2205 \u2228 \u2203 a, s = {a}) \u2192 \u2200 {a b : \u03b1}, a \u2208 s \u2192 b \u2208 s \u2192 a = b", "state_after": "case inr.intro.inl\n\u03b1 : Type u_1\nt : Set \u03b1\nx : \u03b1\nhs : autoParam (Set.Finite \u2205) _auto\u271d\nhx : x \u2208 \u2205\n\u22a2 \u2200 {a b : \u03b1}, a \u2208 \u2205 \u2192 b \u2208 \u2205 \u2192 a = b\n\ncase inr.intro.inr.intro\n\u03b1 : Type u_1\nt : Set \u03b1\nx a : \u03b1\nhs : autoParam (Set.Finite {a}) _auto\u271d\nhx : x \u2208 {a}\n\u22a2 \u2200 {a_1 b : \u03b1}, a_1 \u2208 {a} \u2192 b \u2208 {a} \u2192 a_1 = b"}, {"tactic": "simp_rw [mem_singleton_iff] at hx \u22a2", "annotated_tactic": ["simp_rw [<a>mem_singleton_iff</a>] at hx \u22a2", [{"full_name": "Set.mem_singleton_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1273, 9], "def_end_pos": [1273, 26]}]], "state_before": "case inr.intro.inr.intro\n\u03b1 : Type u_1\nt : Set \u03b1\nx a : \u03b1\nhs : autoParam (Set.Finite {a}) _auto\u271d\nhx : x \u2208 {a}\n\u22a2 \u2200 {a_1 b : \u03b1}, a_1 \u2208 {a} \u2192 b \u2208 {a} \u2192 a_1 = b", "state_after": "case inr.intro.inr.intro\n\u03b1 : Type u_1\nt : Set \u03b1\nx a : \u03b1\nhs : autoParam (Set.Finite {a}) _auto\u271d\nhx : x = a\n\u22a2 \u2200 {a_1 b : \u03b1}, a_1 = a \u2192 b = a \u2192 a_1 = b"}, {"tactic": "subst hx", "annotated_tactic": ["subst hx", []], "state_before": "case inr.intro.inr.intro\n\u03b1 : Type u_1\nt : Set \u03b1\nx a : \u03b1\nhs : autoParam (Set.Finite {a}) _auto\u271d\nhx : x = a\n\u22a2 \u2200 {a_1 b : \u03b1}, a_1 = a \u2192 b = a \u2192 a_1 = b", "state_after": "case inr.intro.inr.intro\n\u03b1 : Type u_1\nt : Set \u03b1\nx : \u03b1\nhs : autoParam (Set.Finite {x}) _auto\u271d\n\u22a2 \u2200 {a b : \u03b1}, a = x \u2192 b = x \u2192 a = b"}, {"tactic": "simp only [forall_eq_apply_imp_iff, imp_self, implies_true]", "annotated_tactic": ["simp only [<a>forall_eq_apply_imp_iff</a>, <a>imp_self</a>, <a>implies_true</a>]", [{"full_name": "forall_eq_apply_imp_iff", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [502, 17], "def_end_pos": [502, 40]}, {"full_name": "imp_self", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [122, 17], "def_end_pos": [122, 25]}, {"full_name": "implies_true", "def_path": "lake-packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [98, 17], "def_end_pos": [98, 29]}]], "state_before": "case inr.intro.inr.intro\n\u03b1 : Type u_1\nt : Set \u03b1\nx : \u03b1\nhs : autoParam (Set.Finite {x}) _auto\u271d\n\u22a2 \u2200 {a b : \u03b1}, a = x \u2192 b = x \u2192 a = b", "state_after": "no goals"}, {"tactic": "exact iff_of_true (by simp) (Or.inl rfl)", "annotated_tactic": ["exact <a>iff_of_true</a> (by simp) (<a>Or.inl</a> <a>rfl</a>)", [{"full_name": "iff_of_true", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [59, 9], "def_end_pos": [59, 20]}, {"full_name": "Or.inl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [517, 5], "def_end_pos": [517, 8]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "case inl\n\u03b1 : Type u_1\nt : Set \u03b1\nhs : autoParam (Set.Finite \u2205) _auto\u271d\n\u22a2 ncard \u2205 \u2264 1 \u2194 \u2205 = \u2205 \u2228 \u2203 a, \u2205 = {a}", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03b1 : Type u_1\nt : Set \u03b1\nhs : autoParam (Set.Finite \u2205) _auto\u271d\n\u22a2 ncard \u2205 \u2264 1", "state_after": "no goals"}, {"tactic": "exact (not_mem_empty _ hx).elim", "annotated_tactic": ["exact (<a>not_mem_empty</a> _ hx).<a>elim</a>", [{"full_name": "Set.not_mem_empty", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [424, 9], "def_end_pos": [424, 22]}, {"full_name": "False.elim", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [223, 21], "def_end_pos": [223, 31]}]], "state_before": "case inr.intro.inl\n\u03b1 : Type u_1\nt : Set \u03b1\nx : \u03b1\nhs : autoParam (Set.Finite \u2205) _auto\u271d\nhx : x \u2208 \u2205\n\u22a2 \u2200 {a b : \u03b1}, a \u2208 \u2205 \u2192 b \u2208 \u2205 \u2192 a = b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Content.lean", "full_name": "MeasureTheory.Content.contentRegular_exists_compact", "start": [420, 1], "end": [430, 85], "traced_tactics": [{"tactic": "by_contra hc", "annotated_tactic": ["by_contra hc", []], "state_before": "G : Type w\ninst\u271d : TopologicalSpace G\n\u03bc : Content G\nH : ContentRegular \u03bc\nK : Compacts G\n\u03b5 : \u211d\u22650\nh\u03b5 : \u03b5 \u2260 0\n\u22a2 \u2203 K', K.carrier \u2286 interior K'.carrier \u2227 (fun s => \u2191(toFun \u03bc s)) K' \u2264 (fun s => \u2191(toFun \u03bc s)) K + \u2191\u03b5", "state_after": "G : Type w\ninst\u271d : TopologicalSpace G\n\u03bc : Content G\nH : ContentRegular \u03bc\nK : Compacts G\n\u03b5 : \u211d\u22650\nh\u03b5 : \u03b5 \u2260 0\nhc : \u00ac\u2203 K', K.carrier \u2286 interior K'.carrier \u2227 (fun s => \u2191(toFun \u03bc s)) K' \u2264 (fun s => \u2191(toFun \u03bc s)) K + \u2191\u03b5\n\u22a2 False"}, {"tactic": "simp only [not_exists, not_and, not_le] at hc", "annotated_tactic": ["simp only [<a>not_exists</a>, <a>not_and</a>, <a>not_le</a>] at hc", [{"full_name": "not_exists", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [422, 17], "def_end_pos": [422, 27]}, {"full_name": "not_and", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [316, 17], "def_end_pos": [316, 24]}, {"full_name": "not_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [373, 9], "def_end_pos": [373, 15]}]], "state_before": "G : Type w\ninst\u271d : TopologicalSpace G\n\u03bc : Content G\nH : ContentRegular \u03bc\nK : Compacts G\n\u03b5 : \u211d\u22650\nh\u03b5 : \u03b5 \u2260 0\nhc : \u00ac\u2203 K', K.carrier \u2286 interior K'.carrier \u2227 (fun s => \u2191(toFun \u03bc s)) K' \u2264 (fun s => \u2191(toFun \u03bc s)) K + \u2191\u03b5\n\u22a2 False", "state_after": "G : Type w\ninst\u271d : TopologicalSpace G\n\u03bc : Content G\nH : ContentRegular \u03bc\nK : Compacts G\n\u03b5 : \u211d\u22650\nh\u03b5 : \u03b5 \u2260 0\nhc : \u2200 (x : Compacts G), K.carrier \u2286 interior x.carrier \u2192 \u2191(toFun \u03bc K) + \u2191\u03b5 < \u2191(toFun \u03bc x)\n\u22a2 False"}, {"tactic": "have lower_bound_iInf : \u03bc K + \u03b5 \u2264\n    \u2a05 (K' : TopologicalSpace.Compacts G) (_ : (K : Set G) \u2286 interior (K' : Set G)), \u03bc K' :=\n  le_iInf fun K' => le_iInf fun K'_hyp => le_of_lt (hc K' K'_hyp)", "annotated_tactic": ["have lower_bound_iInf : \u03bc K + \u03b5 \u2264\n      \u2a05 (K' : <a>TopologicalSpace.Compacts</a> G) (_ : (K : <a>Set</a> G) \u2286 <a>interior</a> (K' : <a>Set</a> G)), \u03bc K' :=\n    <a>le_iInf</a> fun K' => <a>le_iInf</a> fun K'_hyp => <a>le_of_lt</a> (hc K' K'_hyp)", [{"full_name": "TopologicalSpace.Compacts", "def_path": "Mathlib/Topology/Sets/Compacts.lean", "def_pos": [36, 11], "def_end_pos": [36, 19]}, {"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}, {"full_name": "interior", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [288, 5], "def_end_pos": [288, 13]}, {"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}, {"full_name": "le_iInf", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [879, 9], "def_end_pos": [879, 16]}, {"full_name": "le_iInf", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [879, 9], "def_end_pos": [879, 16]}, {"full_name": "le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [110, 9], "def_end_pos": [110, 17]}]], "state_before": "G : Type w\ninst\u271d : TopologicalSpace G\n\u03bc : Content G\nH : ContentRegular \u03bc\nK : Compacts G\n\u03b5 : \u211d\u22650\nh\u03b5 : \u03b5 \u2260 0\nhc : \u2200 (x : Compacts G), K.carrier \u2286 interior x.carrier \u2192 \u2191(toFun \u03bc K) + \u2191\u03b5 < \u2191(toFun \u03bc x)\n\u22a2 False", "state_after": "G : Type w\ninst\u271d : TopologicalSpace G\n\u03bc : Content G\nH : ContentRegular \u03bc\nK : Compacts G\n\u03b5 : \u211d\u22650\nh\u03b5 : \u03b5 \u2260 0\nhc : \u2200 (x : Compacts G), K.carrier \u2286 interior x.carrier \u2192 \u2191(toFun \u03bc K) + \u2191\u03b5 < \u2191(toFun \u03bc x)\nlower_bound_iInf : (fun s => \u2191(toFun \u03bc s)) K + \u2191\u03b5 \u2264 \u2a05 K', \u2a05 (_ : \u2191K \u2286 interior \u2191K'), (fun s => \u2191(toFun \u03bc s)) K'\n\u22a2 False"}, {"tactic": "rw [\u2190 H] at lower_bound_iInf", "annotated_tactic": ["rw [\u2190 H] at lower_bound_iInf", []], "state_before": "G : Type w\ninst\u271d : TopologicalSpace G\n\u03bc : Content G\nH : ContentRegular \u03bc\nK : Compacts G\n\u03b5 : \u211d\u22650\nh\u03b5 : \u03b5 \u2260 0\nhc : \u2200 (x : Compacts G), K.carrier \u2286 interior x.carrier \u2192 \u2191(toFun \u03bc K) + \u2191\u03b5 < \u2191(toFun \u03bc x)\nlower_bound_iInf : (fun s => \u2191(toFun \u03bc s)) K + \u2191\u03b5 \u2264 \u2a05 K', \u2a05 (_ : \u2191K \u2286 interior \u2191K'), (fun s => \u2191(toFun \u03bc s)) K'\n\u22a2 False", "state_after": "G : Type w\ninst\u271d : TopologicalSpace G\n\u03bc : Content G\nH : ContentRegular \u03bc\nK : Compacts G\n\u03b5 : \u211d\u22650\nh\u03b5 : \u03b5 \u2260 0\nhc : \u2200 (x : Compacts G), K.carrier \u2286 interior x.carrier \u2192 \u2191(toFun \u03bc K) + \u2191\u03b5 < \u2191(toFun \u03bc x)\nlower_bound_iInf : (fun s => \u2191(toFun \u03bc s)) K + \u2191\u03b5 \u2264 (fun s => \u2191(toFun \u03bc s)) K\n\u22a2 False"}, {"tactic": "exact (lt_self_iff_false (\u03bc K)).mp (lt_of_le_of_lt' lower_bound_iInf\n  (ENNReal.lt_add_right (ne_top_of_lt (\u03bc.lt_top K)) (ENNReal.coe_ne_zero.mpr h\u03b5)))", "annotated_tactic": ["exact (<a>lt_self_iff_false</a> (\u03bc K)).<a>mp</a> (<a>lt_of_le_of_lt'</a> lower_bound_iInf\n    (<a>ENNReal.lt_add_right</a> (<a>ne_top_of_lt</a> (\u03bc.lt_top K)) (ENNReal.coe_ne_zero.mpr h\u03b5)))", [{"full_name": "lt_self_iff_false", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [175, 9], "def_end_pos": [175, 26]}, {"full_name": "Iff.mp", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [90, 3], "def_end_pos": [90, 5]}, {"full_name": "lt_of_le_of_lt'", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [85, 9], "def_end_pos": [85, 24]}, {"full_name": "ENNReal.lt_add_right", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [829, 9], "def_end_pos": [829, 21]}, {"full_name": "ne_top_of_lt", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [122, 9], "def_end_pos": [122, 21]}]], "state_before": "G : Type w\ninst\u271d : TopologicalSpace G\n\u03bc : Content G\nH : ContentRegular \u03bc\nK : Compacts G\n\u03b5 : \u211d\u22650\nh\u03b5 : \u03b5 \u2260 0\nhc : \u2200 (x : Compacts G), K.carrier \u2286 interior x.carrier \u2192 \u2191(toFun \u03bc K) + \u2191\u03b5 < \u2191(toFun \u03bc x)\nlower_bound_iInf : (fun s => \u2191(toFun \u03bc s)) K + \u2191\u03b5 \u2264 (fun s => \u2191(toFun \u03bc s)) K\n\u22a2 False", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Pointwise.lean", "full_name": "Finset.mem_smul_finset", "start": [1593, 1], "end": [1594, 98], "traced_tactics": [{"tactic": "simp only [Finset.smul_finset_def, and_assoc, mem_image, exists_prop, Prod.exists, mem_product]", "annotated_tactic": ["simp only [<a>Finset.smul_finset_def</a>, <a>and_assoc</a>, <a>mem_image</a>, <a>exists_prop</a>, <a>Prod.exists</a>, <a>mem_product</a>]", [{"full_name": "Finset.smul_finset_def", "def_path": "Mathlib/Data/Finset/Pointwise.lean", "def_pos": [1581, 9], "def_end_pos": [1581, 24]}, {"full_name": "and_assoc", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [177, 9], "def_end_pos": [177, 18]}, {"full_name": "Finset.mem_image", "def_path": "Mathlib/Data/Finset/Image.lean", "def_pos": [330, 9], "def_end_pos": [330, 18]}, {"full_name": "exists_prop", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [485, 17], "def_end_pos": [485, 28]}, {"full_name": "Prod.exists", "def_path": "Mathlib/Data/Prod/Basic.lean", "def_pos": [41, 9], "def_end_pos": [41, 17]}, {"full_name": "Finset.mem_product", "def_path": "Mathlib/Data/Finset/Prod.lean", "def_pos": [53, 9], "def_end_pos": [53, 20]}]], "state_before": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : SMul \u03b1 \u03b2\ns s\u2081 s\u2082 t u : Finset \u03b2\na : \u03b1\nb x : \u03b2\n\u22a2 x \u2208 a \u2022 s \u2194 \u2203 y, y \u2208 s \u2227 a \u2022 y = x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Finite.lean", "full_name": "Set.eq_of_subset_of_card_le", "start": [1255, 1], "end": [1257, 97], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "full_name": "MeasureTheory.DominatedFinMeasAdditive.add", "start": [217, 1], "end": [221, 78], "traced_tactics": [{"tactic": "refine' \u27e8hT.1.add hT'.1, fun s hs h\u03bcs => _\u27e9", "annotated_tactic": ["refine' \u27e8hT.1.<a>add</a> hT'.1, fun s hs h\u03bcs => _\u27e9", [{"full_name": "MeasureTheory.FinMeasAdditive.add", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [106, 9], "def_end_pos": [106, 12]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_7\ninst\u271d : SeminormedAddCommGroup \u03b2\nT T' : Set \u03b1 \u2192 \u03b2\nC C' : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc T' C'\n\u22a2 DominatedFinMeasAdditive \u03bc (T + T') (C + C')", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_7\ninst\u271d : SeminormedAddCommGroup \u03b2\nT T' : Set \u03b1 \u2192 \u03b2\nC C' : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc T' C'\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u2016(T + T') s\u2016 \u2264 (C + C') * ENNReal.toReal (\u2191\u2191\u03bc s)"}, {"tactic": "rw [Pi.add_apply, add_mul]", "annotated_tactic": ["rw [<a>Pi.add_apply</a>, <a>add_mul</a>]", [{"full_name": "Pi.add_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [82, 3], "def_end_pos": [82, 14]}, {"full_name": "add_mul", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [91, 7], "def_end_pos": [91, 14]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_7\ninst\u271d : SeminormedAddCommGroup \u03b2\nT T' : Set \u03b1 \u2192 \u03b2\nC C' : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc T' C'\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u2016(T + T') s\u2016 \u2264 (C + C') * ENNReal.toReal (\u2191\u2191\u03bc s)", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_7\ninst\u271d : SeminormedAddCommGroup \u03b2\nT T' : Set \u03b1 \u2192 \u03b2\nC C' : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc T' C'\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u2016T s + T' s\u2016 \u2264 C * ENNReal.toReal (\u2191\u2191\u03bc s) + C' * ENNReal.toReal (\u2191\u2191\u03bc s)"}, {"tactic": "exact (norm_add_le _ _).trans (add_le_add (hT.2 s hs h\u03bcs) (hT'.2 s hs h\u03bcs))", "annotated_tactic": ["exact (<a>norm_add_le</a> _ _).<a>trans</a> (<a>add_le_add</a> (hT.2 s hs h\u03bcs) (hT'.2 s hs h\u03bcs))", [{"full_name": "norm_add_le", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [482, 15], "def_end_pos": [482, 26]}, {"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [120, 7], "def_end_pos": [120, 18]}, {"full_name": "add_le_add", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [205, 15], "def_end_pos": [205, 25]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_7\ninst\u271d : SeminormedAddCommGroup \u03b2\nT T' : Set \u03b1 \u2192 \u03b2\nC C' : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc T' C'\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u2016T s + T' s\u2016 \u2264 C * ENNReal.toReal (\u2191\u2191\u03bc s) + C' * ENNReal.toReal (\u2191\u2191\u03bc s)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Group/FundamentalDomain.lean", "full_name": "MeasureTheory.IsFundamentalDomain.exists_ne_one_smul_eq", "start": [498, 1], "end": [507, 38], "traced_tactics": [{"tactic": "contrapose! ht", "annotated_tactic": ["contrapose! ht", []], "state_before": "G : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : Group H\ninst\u271d\u00b9\u2070 : MulAction G \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MulAction H \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : MeasurableSMul G \u03b1\ninst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b2 : Countable G\n\u03bd : Measure \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhs : IsFundamentalDomain G s\nhtm : NullMeasurableSet t\nht : \u2191\u2191\u03bc s < \u2191\u2191\u03bc t\n\u22a2 \u2203 x, x \u2208 t \u2227 \u2203 y, y \u2208 t \u2227 \u2203 g, g \u2260 1 \u2227 g \u2022 x = y", "state_after": "G : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : Group H\ninst\u271d\u00b9\u2070 : MulAction G \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MulAction H \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : MeasurableSMul G \u03b1\ninst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b2 : Countable G\n\u03bd : Measure \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhs : IsFundamentalDomain G s\nhtm : NullMeasurableSet t\nht : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2200 (y : \u03b1), y \u2208 t \u2192 \u2200 (g : G), g \u2260 1 \u2192 g \u2022 x \u2260 y\n\u22a2 \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc s"}, {"tactic": "refine' hs.measure_le_of_pairwise_disjoint htm (Pairwise.aedisjoint fun g\u2081 g\u2082 hne => _)", "annotated_tactic": ["refine' hs.measure_le_of_pairwise_disjoint htm (<a>Pairwise.aedisjoint</a> fun g\u2081 g\u2082 hne => _)", [{"full_name": "Pairwise.aedisjoint", "def_path": "Mathlib/MeasureTheory/Measure/AEDisjoint.lean", "def_pos": [69, 19], "def_end_pos": [69, 45]}]], "state_before": "G : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : Group H\ninst\u271d\u00b9\u2070 : MulAction G \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MulAction H \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : MeasurableSMul G \u03b1\ninst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b2 : Countable G\n\u03bd : Measure \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhs : IsFundamentalDomain G s\nhtm : NullMeasurableSet t\nht : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2200 (y : \u03b1), y \u2208 t \u2192 \u2200 (g : G), g \u2260 1 \u2192 g \u2022 x \u2260 y\n\u22a2 \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc s", "state_after": "G : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : Group H\ninst\u271d\u00b9\u2070 : MulAction G \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MulAction H \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : MeasurableSMul G \u03b1\ninst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b2 : Countable G\n\u03bd : Measure \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhs : IsFundamentalDomain G s\nhtm : NullMeasurableSet t\nht : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2200 (y : \u03b1), y \u2208 t \u2192 \u2200 (g : G), g \u2260 1 \u2192 g \u2022 x \u2260 y\ng\u2081 g\u2082 : G\nhne : g\u2081 \u2260 g\u2082\n\u22a2 (Disjoint on fun g => g \u2022 t \u2229 s) g\u2081 g\u2082"}, {"tactic": "dsimp [Function.onFun]", "annotated_tactic": ["dsimp [<a>Function.onFun</a>]", [{"full_name": "Function.onFun", "def_path": "Mathlib/Init/Function.lean", "def_pos": [49, 5], "def_end_pos": [49, 10]}]], "state_before": "G : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : Group H\ninst\u271d\u00b9\u2070 : MulAction G \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MulAction H \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : MeasurableSMul G \u03b1\ninst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b2 : Countable G\n\u03bd : Measure \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhs : IsFundamentalDomain G s\nhtm : NullMeasurableSet t\nht : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2200 (y : \u03b1), y \u2208 t \u2192 \u2200 (g : G), g \u2260 1 \u2192 g \u2022 x \u2260 y\ng\u2081 g\u2082 : G\nhne : g\u2081 \u2260 g\u2082\n\u22a2 (Disjoint on fun g => g \u2022 t \u2229 s) g\u2081 g\u2082", "state_after": "G : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : Group H\ninst\u271d\u00b9\u2070 : MulAction G \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MulAction H \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : MeasurableSMul G \u03b1\ninst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b2 : Countable G\n\u03bd : Measure \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhs : IsFundamentalDomain G s\nhtm : NullMeasurableSet t\nht : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2200 (y : \u03b1), y \u2208 t \u2192 \u2200 (g : G), g \u2260 1 \u2192 g \u2022 x \u2260 y\ng\u2081 g\u2082 : G\nhne : g\u2081 \u2260 g\u2082\n\u22a2 Disjoint (g\u2081 \u2022 t \u2229 s) (g\u2082 \u2022 t \u2229 s)"}, {"tactic": "refine' (Disjoint.inf_left _ _).inf_right _", "annotated_tactic": ["refine' (<a>Disjoint.inf_left</a> _ _).<a>inf_right</a> _", [{"full_name": "Disjoint.inf_left", "def_path": "Mathlib/Order/Disjoint.lean", "def_pos": [152, 9], "def_end_pos": [152, 26]}, {"full_name": "Disjoint.inf_right", "def_path": "Mathlib/Order/Disjoint.lean", "def_pos": [160, 9], "def_end_pos": [160, 27]}]], "state_before": "G : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : Group H\ninst\u271d\u00b9\u2070 : MulAction G \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MulAction H \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : MeasurableSMul G \u03b1\ninst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b2 : Countable G\n\u03bd : Measure \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhs : IsFundamentalDomain G s\nhtm : NullMeasurableSet t\nht : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2200 (y : \u03b1), y \u2208 t \u2192 \u2200 (g : G), g \u2260 1 \u2192 g \u2022 x \u2260 y\ng\u2081 g\u2082 : G\nhne : g\u2081 \u2260 g\u2082\n\u22a2 Disjoint (g\u2081 \u2022 t \u2229 s) (g\u2082 \u2022 t \u2229 s)", "state_after": "G : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : Group H\ninst\u271d\u00b9\u2070 : MulAction G \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MulAction H \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : MeasurableSMul G \u03b1\ninst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b2 : Countable G\n\u03bd : Measure \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhs : IsFundamentalDomain G s\nhtm : NullMeasurableSet t\nht : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2200 (y : \u03b1), y \u2208 t \u2192 \u2200 (g : G), g \u2260 1 \u2192 g \u2022 x \u2260 y\ng\u2081 g\u2082 : G\nhne : g\u2081 \u2260 g\u2082\n\u22a2 Disjoint (g\u2081 \u2022 t) (g\u2082 \u2022 t)"}, {"tactic": "rw [Set.disjoint_left]", "annotated_tactic": ["rw [<a>Set.disjoint_left</a>]", [{"full_name": "Set.disjoint_left", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1546, 9], "def_end_pos": [1546, 22]}]], "state_before": "G : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : Group H\ninst\u271d\u00b9\u2070 : MulAction G \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MulAction H \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : MeasurableSMul G \u03b1\ninst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b2 : Countable G\n\u03bd : Measure \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhs : IsFundamentalDomain G s\nhtm : NullMeasurableSet t\nht : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2200 (y : \u03b1), y \u2208 t \u2192 \u2200 (g : G), g \u2260 1 \u2192 g \u2022 x \u2260 y\ng\u2081 g\u2082 : G\nhne : g\u2081 \u2260 g\u2082\n\u22a2 Disjoint (g\u2081 \u2022 t) (g\u2082 \u2022 t)", "state_after": "G : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : Group H\ninst\u271d\u00b9\u2070 : MulAction G \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MulAction H \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : MeasurableSMul G \u03b1\ninst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b2 : Countable G\n\u03bd : Measure \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhs : IsFundamentalDomain G s\nhtm : NullMeasurableSet t\nht : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2200 (y : \u03b1), y \u2208 t \u2192 \u2200 (g : G), g \u2260 1 \u2192 g \u2022 x \u2260 y\ng\u2081 g\u2082 : G\nhne : g\u2081 \u2260 g\u2082\n\u22a2 \u2200 \u2983a : \u03b1\u2984, a \u2208 g\u2081 \u2022 t \u2192 \u00aca \u2208 g\u2082 \u2022 t"}, {"tactic": "rintro _ \u27e8x, hx, rfl\u27e9 \u27e8y, hy, hxy : g\u2082 \u2022 y = g\u2081 \u2022 x\u27e9", "annotated_tactic": ["rintro _ \u27e8x, hx, rfl\u27e9 \u27e8y, hy, hxy : g\u2082 \u2022 y = g\u2081 \u2022 x\u27e9", []], "state_before": "G : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : Group H\ninst\u271d\u00b9\u2070 : MulAction G \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MulAction H \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : MeasurableSMul G \u03b1\ninst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b2 : Countable G\n\u03bd : Measure \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhs : IsFundamentalDomain G s\nhtm : NullMeasurableSet t\nht : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2200 (y : \u03b1), y \u2208 t \u2192 \u2200 (g : G), g \u2260 1 \u2192 g \u2022 x \u2260 y\ng\u2081 g\u2082 : G\nhne : g\u2081 \u2260 g\u2082\n\u22a2 \u2200 \u2983a : \u03b1\u2984, a \u2208 g\u2081 \u2022 t \u2192 \u00aca \u2208 g\u2082 \u2022 t", "state_after": "case intro.intro.intro.intro\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : Group H\ninst\u271d\u00b9\u2070 : MulAction G \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MulAction H \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : MeasurableSMul G \u03b1\ninst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b2 : Countable G\n\u03bd : Measure \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhs : IsFundamentalDomain G s\nhtm : NullMeasurableSet t\nht : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2200 (y : \u03b1), y \u2208 t \u2192 \u2200 (g : G), g \u2260 1 \u2192 g \u2022 x \u2260 y\ng\u2081 g\u2082 : G\nhne : g\u2081 \u2260 g\u2082\nx : \u03b1\nhx : x \u2208 t\ny : \u03b1\nhy : y \u2208 t\nhxy : g\u2082 \u2022 y = g\u2081 \u2022 x\n\u22a2 False"}, {"tactic": "refine' ht x hx y hy (g\u2082\u207b\u00b9 * g\u2081) (mt inv_mul_eq_one.1 hne.symm) _", "annotated_tactic": ["refine' ht x hx y hy (g\u2082\u207b\u00b9 * g\u2081) (<a>mt</a> <a>inv_mul_eq_one</a>.1 hne.symm) _", [{"full_name": "mt", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [516, 9], "def_end_pos": [516, 11]}, {"full_name": "inv_mul_eq_one", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [708, 9], "def_end_pos": [708, 23]}]], "state_before": "case intro.intro.intro.intro\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : Group H\ninst\u271d\u00b9\u2070 : MulAction G \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MulAction H \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : MeasurableSMul G \u03b1\ninst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b2 : Countable G\n\u03bd : Measure \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhs : IsFundamentalDomain G s\nhtm : NullMeasurableSet t\nht : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2200 (y : \u03b1), y \u2208 t \u2192 \u2200 (g : G), g \u2260 1 \u2192 g \u2022 x \u2260 y\ng\u2081 g\u2082 : G\nhne : g\u2081 \u2260 g\u2082\nx : \u03b1\nhx : x \u2208 t\ny : \u03b1\nhy : y \u2208 t\nhxy : g\u2082 \u2022 y = g\u2081 \u2022 x\n\u22a2 False", "state_after": "case intro.intro.intro.intro\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : Group H\ninst\u271d\u00b9\u2070 : MulAction G \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MulAction H \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : MeasurableSMul G \u03b1\ninst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b2 : Countable G\n\u03bd : Measure \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhs : IsFundamentalDomain G s\nhtm : NullMeasurableSet t\nht : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2200 (y : \u03b1), y \u2208 t \u2192 \u2200 (g : G), g \u2260 1 \u2192 g \u2022 x \u2260 y\ng\u2081 g\u2082 : G\nhne : g\u2081 \u2260 g\u2082\nx : \u03b1\nhx : x \u2208 t\ny : \u03b1\nhy : y \u2208 t\nhxy : g\u2082 \u2022 y = g\u2081 \u2022 x\n\u22a2 (g\u2082\u207b\u00b9 * g\u2081) \u2022 x = y"}, {"tactic": "rw [mul_smul, \u2190 hxy, inv_smul_smul]", "annotated_tactic": ["rw [<a>mul_smul</a>, \u2190 hxy, <a>inv_smul_smul</a>]", [{"full_name": "MulAction.mul_smul", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [112, 3], "def_end_pos": [112, 11]}, {"full_name": "inv_smul_smul", "def_path": "Mathlib/GroupTheory/GroupAction/Group.lean", "def_pos": [36, 9], "def_end_pos": [36, 22]}]], "state_before": "case intro.intro.intro.intro\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : Group H\ninst\u271d\u00b9\u2070 : MulAction G \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MulAction H \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : MeasurableSMul G \u03b1\ninst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b2 : Countable G\n\u03bd : Measure \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhs : IsFundamentalDomain G s\nhtm : NullMeasurableSet t\nht : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2200 (y : \u03b1), y \u2208 t \u2192 \u2200 (g : G), g \u2260 1 \u2192 g \u2022 x \u2260 y\ng\u2081 g\u2082 : G\nhne : g\u2081 \u2260 g\u2082\nx : \u03b1\nhx : x \u2208 t\ny : \u03b1\nhy : y \u2208 t\nhxy : g\u2082 \u2022 y = g\u2081 \u2022 x\n\u22a2 (g\u2082\u207b\u00b9 * g\u2081) \u2022 x = y", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Pointwise/Interval.lean", "full_name": "Set.image_mul_const_uIcc", "start": [707, 1], "end": [713, 59], "traced_tactics": [{"tactic": "simp [ha]", "annotated_tactic": ["simp [ha]", []], "state_before": "\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na\u271d a b c : \u03b1\nha : a = 0\n\u22a2 (fun x => x * a) '' [[b, c]] = [[b * a, c * a]]", "state_after": "no goals"}, {"tactic": "simp only [div_eq_mul_inv]", "annotated_tactic": ["simp only [<a>div_eq_mul_inv</a>]", [{"full_name": "div_eq_mul_inv", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [977, 9], "def_end_pos": [977, 23]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na\u271d a b c : \u03b1\nha : \u00aca = 0\n\u22a2 (fun x => x * a\u207b\u00b9) \u207b\u00b9' [[b, c]] = (fun x => x / a) \u207b\u00b9' [[b, c]]", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/ProbabilityMassFunction/Basic.lean", "full_name": "PMF.apply_eq_one_iff", "start": [114, 1], "end": [133, 97], "traced_tactics": [{"tactic": "refine' \u27e8fun h => Set.Subset.antisymm (fun a' ha' => by_contra fun ha => _)\n  fun a' ha' => ha'.symm \u25b8 (p.mem_support_iff a).2 fun ha => zero_ne_one <| ha.symm.trans h,\n  fun h => _root_.trans (symm <| tsum_eq_single a\n    fun a' ha' => (p.apply_eq_zero_iff a').2 (h.symm \u25b8 ha')) p.tsum_coe\u27e9", "annotated_tactic": ["refine' \u27e8fun h => <a>Set.Subset.antisymm</a> (fun a' ha' => <a>by_contra</a> fun ha => _)\n    fun a' ha' => ha'.symm \u25b8 (p.mem_support_iff a).2 fun ha => <a>zero_ne_one</a> <| ha.symm.trans h,\n    fun h => <a>_root_.trans</a> (<a>symm</a> <| <a>tsum_eq_single</a> a\n      fun a' ha' => (p.apply_eq_zero_iff a').2 (h.symm \u25b8 ha')) p.tsum_coe\u27e9", [{"full_name": "Set.Subset.antisymm", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [370, 9], "def_end_pos": [370, 24]}, {"full_name": "by_contra", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [223, 7], "def_end_pos": [223, 16]}, {"full_name": "zero_ne_one", "def_path": "Mathlib/Algebra/NeZero.lean", "def_pos": [52, 15], "def_end_pos": [52, 26]}, {"full_name": "trans", "def_path": "Mathlib/Init/Algebra/Classes.lean", "def_pos": [308, 9], "def_end_pos": [308, 14]}, {"full_name": "symm", "def_path": "Mathlib/Init/Algebra/Classes.lean", "def_pos": [312, 9], "def_end_pos": [312, 13]}, {"full_name": "tsum_eq_single", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/Basic.lean", "def_pos": [520, 9], "def_end_pos": [520, 23]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\na : \u03b1\n\u22a2 \u2191p a = 1 \u2194 support p = {a}", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\na : \u03b1\nh : \u2191p a = 1\na' : \u03b1\nha' : a' \u2208 support p\nha : \u00aca' \u2208 {a}\n\u22a2 False"}, {"tactic": "suffices : 1 < \u2211' a, p a", "annotated_tactic": ["suffices : 1 < \u2211' a, p a", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\na : \u03b1\nh : \u2191p a = 1\na' : \u03b1\nha' : a' \u2208 support p\nha : \u00aca' \u2208 {a}\n\u22a2 False", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\na : \u03b1\nh : \u2191p a = 1\na' : \u03b1\nha' : a' \u2208 support p\nha : \u00aca' \u2208 {a}\nthis : 1 < \u2211' (a : \u03b1), \u2191p a\n\u22a2 False\n\ncase this\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\na : \u03b1\nh : \u2191p a = 1\na' : \u03b1\nha' : a' \u2208 support p\nha : \u00aca' \u2208 {a}\n\u22a2 1 < \u2211' (a : \u03b1), \u2191p a"}, {"tactic": "exact ne_of_lt this p.tsum_coe.symm", "annotated_tactic": ["exact <a>ne_of_lt</a> this p.tsum_coe.symm", [{"full_name": "ne_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [101, 9], "def_end_pos": [101, 17]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\na : \u03b1\nh : \u2191p a = 1\na' : \u03b1\nha' : a' \u2208 support p\nha : \u00aca' \u2208 {a}\nthis : 1 < \u2211' (a : \u03b1), \u2191p a\n\u22a2 False\n\ncase this\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\na : \u03b1\nh : \u2191p a = 1\na' : \u03b1\nha' : a' \u2208 support p\nha : \u00aca' \u2208 {a}\n\u22a2 1 < \u2211' (a : \u03b1), \u2191p a", "state_after": "case this\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\na : \u03b1\nh : \u2191p a = 1\na' : \u03b1\nha' : a' \u2208 support p\nha : \u00aca' \u2208 {a}\n\u22a2 1 < \u2211' (a : \u03b1), \u2191p a"}, {"tactic": "have : 0 < \u2211' b, ite (b = a) 0 (p b) := lt_of_le_of_ne' zero_le'\n  ((tsum_ne_zero_iff ENNReal.summable).2\n    \u27e8a', ite_ne_left_iff.2 \u27e8ha, Ne.symm <| (p.mem_support_iff a').2 ha'\u27e9\u27e9)", "annotated_tactic": ["have : 0 < \u2211' b, <a>ite</a> (b = a) 0 (p b) := <a>lt_of_le_of_ne'</a> <a>zero_le'</a>\n    ((<a>tsum_ne_zero_iff</a> <a>ENNReal.summable</a>).2\n      \u27e8a', <a>ite_ne_left_iff</a>.2 \u27e8ha, <a>Ne.symm</a> <| (p.mem_support_iff a').2 ha'\u27e9\u27e9)", [{"full_name": "ite", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [935, 21], "def_end_pos": [935, 24]}, {"full_name": "lt_of_le_of_ne'", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [103, 9], "def_end_pos": [103, 24]}, {"full_name": "zero_le'", "def_path": "Mathlib/Algebra/Order/WithZero.lean", "def_pos": [93, 9], "def_end_pos": [93, 17]}, {"full_name": "tsum_ne_zero_iff", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/Order.lean", "def_pos": [214, 9], "def_end_pos": [214, 25]}, {"full_name": "ENNReal.summable", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [777, 19], "def_end_pos": [777, 27]}, {"full_name": "ite_ne_left_iff", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [1174, 9], "def_end_pos": [1174, 24]}, {"full_name": "Ne.symm", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [575, 9], "def_end_pos": [575, 16]}]], "state_before": "case this\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\na : \u03b1\nh : \u2191p a = 1\na' : \u03b1\nha' : a' \u2208 support p\nha : \u00aca' \u2208 {a}\n\u22a2 1 < \u2211' (a : \u03b1), \u2191p a", "state_after": "case this\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\na : \u03b1\nh : \u2191p a = 1\na' : \u03b1\nha' : a' \u2208 support p\nha : \u00aca' \u2208 {a}\nthis : 0 < \u2211' (b : \u03b1), if b = a then 0 else \u2191p b\n\u22a2 1 < \u2211' (a : \u03b1), \u2191p a"}, {"tactic": "calc\n  1 = 1 + 0 := (add_zero 1).symm\n  _ < p a + \u2211' b, ite (b = a) 0 (p b) :=\n    (ENNReal.add_lt_add_of_le_of_lt ENNReal.one_ne_top (le_of_eq h.symm) this)\n  _ = ite (a = a) (p a) 0 + \u2211' b, ite (b = a) 0 (p b) := by rw [eq_self_iff_true, if_true]\n  _ = (\u2211' b, ite (b = a) (p b) 0) + \u2211' b, ite (b = a) 0 (p b) := by\n    congr\n    exact symm (tsum_eq_single a fun b hb => if_neg hb)\n  _ = \u2211' b, (ite (b = a) (p b) 0 + ite (b = a) 0 (p b)) := ENNReal.tsum_add.symm\n  _ = \u2211' b, p b := tsum_congr fun b => by split_ifs <;> simp only [zero_add, add_zero, le_rfl]", "annotated_tactic": ["calc\n    1 = 1 + 0 := (<a>add_zero</a> 1).<a>symm</a>\n    _ < p a + \u2211' b, <a>ite</a> (b = a) 0 (p b) :=\n      (<a>ENNReal.add_lt_add_of_le_of_lt</a> <a>ENNReal.one_ne_top</a> (<a>le_of_eq</a> h.symm) this)\n    _ = <a>ite</a> (a = a) (p a) 0 + \u2211' b, <a>ite</a> (b = a) 0 (p b) := by rw [<a>eq_self_iff_true</a>, <a>if_true</a>]\n    _ = (\u2211' b, <a>ite</a> (b = a) (p b) 0) + \u2211' b, <a>ite</a> (b = a) 0 (p b) := by\n      congr\n      exact <a>symm</a> (<a>tsum_eq_single</a> a fun b hb => <a>if_neg</a> hb)\n    _ = \u2211' b, (<a>ite</a> (b = a) (p b) 0 + <a>ite</a> (b = a) 0 (p b)) := ENNReal.tsum_add.symm\n    _ = \u2211' b, p b := <a>tsum_congr</a> fun b => by split_ifs <;> simp only [<a>zero_add</a>, <a>add_zero</a>, <a>le_rfl</a>]", [{"full_name": "add_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [469, 3], "def_end_pos": [469, 14]}, {"full_name": "Eq.symm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [310, 9], "def_end_pos": [310, 16]}, {"full_name": "ite", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [935, 21], "def_end_pos": [935, 24]}, {"full_name": "ENNReal.add_lt_add_of_le_of_lt", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [817, 19], "def_end_pos": [817, 41]}, {"full_name": "ENNReal.one_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [340, 17], "def_end_pos": [340, 27]}, {"full_name": "le_of_eq", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [72, 9], "def_end_pos": [72, 17]}, {"full_name": "ite", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [935, 21], "def_end_pos": [935, 24]}, {"full_name": "ite", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [935, 21], "def_end_pos": [935, 24]}, {"full_name": "eq_self_iff_true", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [86, 9], "def_end_pos": [86, 25]}, {"full_name": "if_true", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [727, 17], "def_end_pos": [727, 24]}, {"full_name": "ite", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [935, 21], "def_end_pos": [935, 24]}, {"full_name": "ite", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [935, 21], "def_end_pos": [935, 24]}, {"full_name": "symm", "def_path": "Mathlib/Init/Algebra/Classes.lean", "def_pos": [312, 9], "def_end_pos": [312, 13]}, {"full_name": "tsum_eq_single", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/Basic.lean", "def_pos": [520, 9], "def_end_pos": [520, 23]}, {"full_name": "if_neg", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [795, 9], "def_end_pos": [795, 15]}, {"full_name": "ite", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [935, 21], "def_end_pos": [935, 24]}, {"full_name": "ite", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [935, 21], "def_end_pos": [935, 24]}, {"full_name": "tsum_congr", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/Basic.lean", "def_pos": [498, 9], "def_end_pos": [498, 19]}, {"full_name": "zero_add", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [463, 3], "def_end_pos": [463, 14]}, {"full_name": "add_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [469, 3], "def_end_pos": [469, 14]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}]], "state_before": "case this\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\na : \u03b1\nh : \u2191p a = 1\na' : \u03b1\nha' : a' \u2208 support p\nha : \u00aca' \u2208 {a}\nthis : 0 < \u2211' (b : \u03b1), if b = a then 0 else \u2191p b\n\u22a2 1 < \u2211' (a : \u03b1), \u2191p a", "state_after": "no goals"}, {"tactic": "rw [eq_self_iff_true, if_true]", "annotated_tactic": ["rw [<a>eq_self_iff_true</a>, <a>if_true</a>]", [{"full_name": "eq_self_iff_true", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [86, 9], "def_end_pos": [86, 25]}, {"full_name": "if_true", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [727, 17], "def_end_pos": [727, 24]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\na : \u03b1\nh : \u2191p a = 1\na' : \u03b1\nha' : a' \u2208 support p\nha : \u00aca' \u2208 {a}\nthis : 0 < \u2211' (b : \u03b1), if b = a then 0 else \u2191p b\n\u22a2 (\u2191p a + \u2211' (b : \u03b1), if b = a then 0 else \u2191p b) = (if a = a then \u2191p a else 0) + \u2211' (b : \u03b1), if b = a then 0 else \u2191p b", "state_after": "no goals"}, {"tactic": "congr", "annotated_tactic": ["congr", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\na : \u03b1\nh : \u2191p a = 1\na' : \u03b1\nha' : a' \u2208 support p\nha : \u00aca' \u2208 {a}\nthis : 0 < \u2211' (b : \u03b1), if b = a then 0 else \u2191p b\n\u22a2 ((if a = a then \u2191p a else 0) + \u2211' (b : \u03b1), if b = a then 0 else \u2191p b) =\n    (\u2211' (b : \u03b1), if b = a then \u2191p b else 0) + \u2211' (b : \u03b1), if b = a then 0 else \u2191p b", "state_after": "case e_a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\na : \u03b1\nh : \u2191p a = 1\na' : \u03b1\nha' : a' \u2208 support p\nha : \u00aca' \u2208 {a}\nthis : 0 < \u2211' (b : \u03b1), if b = a then 0 else \u2191p b\n\u22a2 (if a = a then \u2191p a else 0) = \u2211' (b : \u03b1), if b = a then \u2191p b else 0"}, {"tactic": "exact symm (tsum_eq_single a fun b hb => if_neg hb)", "annotated_tactic": ["exact <a>symm</a> (<a>tsum_eq_single</a> a fun b hb => <a>if_neg</a> hb)", [{"full_name": "symm", "def_path": "Mathlib/Init/Algebra/Classes.lean", "def_pos": [312, 9], "def_end_pos": [312, 13]}, {"full_name": "tsum_eq_single", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/Basic.lean", "def_pos": [520, 9], "def_end_pos": [520, 23]}, {"full_name": "if_neg", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [795, 9], "def_end_pos": [795, 15]}]], "state_before": "case e_a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\na : \u03b1\nh : \u2191p a = 1\na' : \u03b1\nha' : a' \u2208 support p\nha : \u00aca' \u2208 {a}\nthis : 0 < \u2211' (b : \u03b1), if b = a then 0 else \u2191p b\n\u22a2 (if a = a then \u2191p a else 0) = \u2211' (b : \u03b1), if b = a then \u2191p b else 0", "state_after": "no goals"}, {"tactic": "split_ifs <;> simp only [zero_add, add_zero, le_rfl]", "annotated_tactic": ["split_ifs <;> simp only [<a>zero_add</a>, <a>add_zero</a>, <a>le_rfl</a>]", [{"full_name": "zero_add", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [463, 3], "def_end_pos": [463, 14]}, {"full_name": "add_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [469, 3], "def_end_pos": [469, 14]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\na : \u03b1\nh : \u2191p a = 1\na' : \u03b1\nha' : a' \u2208 support p\nha : \u00aca' \u2208 {a}\nthis : 0 < \u2211' (b : \u03b1), if b = a then 0 else \u2191p b\nb : \u03b1\n\u22a2 ((if b = a then \u2191p b else 0) + if b = a then 0 else \u2191p b) = \u2191p b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean", "full_name": "MeasureTheory.integral_comp_mul_deriv_Ioi", "start": [833, 1], "end": [840, 83], "traced_tactics": [{"tactic": "have hg2' : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) (Ici a) := by simpa [mul_comm] using hg2", "annotated_tactic": ["have hg2' : <a>IntegrableOn</a> (fun x => f' x \u2022 (g \u2218 f) x) (<a>Ici</a> a) := by simpa [<a>mul_comm</a>] using hg2", [{"full_name": "MeasureTheory.IntegrableOn", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [90, 5], "def_end_pos": [90, 17]}, {"full_name": "Set.Ici", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [74, 5], "def_end_pos": [74, 8]}, {"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [302, 9], "def_end_pos": [302, 17]}]], "state_before": "E : Type u_1\nf\u271d : \u211d \u2192 E\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf f' g : \u211d \u2192 \u211d\na : \u211d\nhf : ContinuousOn f (Ici a)\nhft : Tendsto f atTop atTop\nhff' : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 HasDerivWithinAt f (f' x) (Ioi x) x\nhg_cont : ContinuousOn g (f '' Ioi a)\nhg1 : IntegrableOn g (f '' Ici a)\nhg2 : IntegrableOn (fun x => (g \u2218 f) x * f' x) (Ici a)\n\u22a2 \u222b (x : \u211d) in Ioi a, (g \u2218 f) x * f' x = \u222b (u : \u211d) in Ioi (f a), g u", "state_after": "E : Type u_1\nf\u271d : \u211d \u2192 E\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf f' g : \u211d \u2192 \u211d\na : \u211d\nhf : ContinuousOn f (Ici a)\nhft : Tendsto f atTop atTop\nhff' : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 HasDerivWithinAt f (f' x) (Ioi x) x\nhg_cont : ContinuousOn g (f '' Ioi a)\nhg1 : IntegrableOn g (f '' Ici a)\nhg2 : IntegrableOn (fun x => (g \u2218 f) x * f' x) (Ici a)\nhg2' : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) (Ici a)\n\u22a2 \u222b (x : \u211d) in Ioi a, (g \u2218 f) x * f' x = \u222b (u : \u211d) in Ioi (f a), g u"}, {"tactic": "simpa [mul_comm] using integral_comp_smul_deriv_Ioi hf hft hff' hg_cont hg1 hg2'", "annotated_tactic": ["simpa [<a>mul_comm</a>] using <a>integral_comp_smul_deriv_Ioi</a> hf hft hff' hg_cont hg1 hg2'", [{"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [302, 9], "def_end_pos": [302, 17]}, {"full_name": "MeasureTheory.integral_comp_smul_deriv_Ioi", "def_path": "Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean", "def_pos": [805, 9], "def_end_pos": [805, 37]}]], "state_before": "E : Type u_1\nf\u271d : \u211d \u2192 E\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf f' g : \u211d \u2192 \u211d\na : \u211d\nhf : ContinuousOn f (Ici a)\nhft : Tendsto f atTop atTop\nhff' : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 HasDerivWithinAt f (f' x) (Ioi x) x\nhg_cont : ContinuousOn g (f '' Ioi a)\nhg1 : IntegrableOn g (f '' Ici a)\nhg2 : IntegrableOn (fun x => (g \u2218 f) x * f' x) (Ici a)\nhg2' : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) (Ici a)\n\u22a2 \u222b (x : \u211d) in Ioi a, (g \u2218 f) x * f' x = \u222b (u : \u211d) in Ioi (f a), g u", "state_after": "no goals"}, {"tactic": "simpa [mul_comm] using hg2", "annotated_tactic": ["simpa [<a>mul_comm</a>] using hg2", [{"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [302, 9], "def_end_pos": [302, 17]}]], "state_before": "E : Type u_1\nf\u271d : \u211d \u2192 E\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf f' g : \u211d \u2192 \u211d\na : \u211d\nhf : ContinuousOn f (Ici a)\nhft : Tendsto f atTop atTop\nhff' : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 HasDerivWithinAt f (f' x) (Ioi x) x\nhg_cont : ContinuousOn g (f '' Ioi a)\nhg1 : IntegrableOn g (f '' Ici a)\nhg2 : IntegrableOn (fun x => (g \u2218 f) x * f' x) (Ici a)\n\u22a2 IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) (Ici a)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "full_name": "Substring.ValidFor.foldl", "start": [938, 1], "end": [939, 78], "traced_tactics": [{"tactic": "simp [-List.append_assoc, Substring.foldl, foldlAux_of_valid]", "annotated_tactic": ["simp [-<a>List.append_assoc</a>, <a>Substring.foldl</a>, <a>foldlAux_of_valid</a>]", [{"full_name": "List.append_assoc", "def_path": "lake-packages/lean4/src/lean/Init/Data/List/Basic.lean", "def_pos": [103, 9], "def_end_pos": [103, 21]}, {"full_name": "Substring.foldl", "def_path": "lake-packages/lean4/src/lean/Init/Data/String/Basic.lean", "def_pos": [618, 15], "def_end_pos": [618, 20]}, {"full_name": "String.foldlAux_of_valid", "def_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "def_pos": [697, 9], "def_end_pos": [697, 26]}]], "state_before": "\u03b1 : Type u_1\nl m r : List Char\nf : \u03b1 \u2192 Char \u2192 \u03b1\ninit : \u03b1\n\u22a2 Substring.foldl f init\n      { str := { data := l ++ m ++ r }, startPos := { byteIdx := utf8Len l },\n        stopPos := { byteIdx := utf8Len l + utf8Len m } } =\n    List.foldl f init m", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Decomposition/SignedHahn.lean", "full_name": "MeasureTheory.SignedMeasure.exists_compl_positive_negative", "start": [370, 1], "end": [408, 57], "traced_tactics": [{"tactic": "obtain \u27e8f, _, hf\u2082, hf\u2081\u27e9 :=\n  exists_seq_tendsto_sInf \u27e80, @zero_mem_measureOfNegatives _ _ s\u27e9 bddBelow_measureOfNegatives", "annotated_tactic": ["obtain \u27e8f, _, hf\u2082, hf\u2081\u27e9 :=\n    <a>exists_seq_tendsto_sInf</a> \u27e80, @<a>zero_mem_measureOfNegatives</a> _ _ s\u27e9 <a>bddBelow_measureOfNegatives</a>", [{"full_name": "exists_seq_tendsto_sInf", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [2280, 9], "def_end_pos": [2280, 32]}, {"full_name": "MeasureTheory.SignedMeasure.zero_mem_measureOfNegatives", "def_path": "Mathlib/MeasureTheory/Decomposition/SignedHahn.lean", "def_pos": [339, 9], "def_end_pos": [339, 36]}, {"full_name": "MeasureTheory.SignedMeasure.bddBelow_measureOfNegatives", "def_path": "Mathlib/MeasureTheory/Decomposition/SignedHahn.lean", "def_pos": [343, 9], "def_end_pos": [343, 36]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : TopologicalSpace M\ninst\u271d : OrderedAddCommMonoid M\ns\u271d : SignedMeasure \u03b1\ni j : Set \u03b1\ns : SignedMeasure \u03b1\n\u22a2 \u2203 i, MeasurableSet i \u2227 restrict 0 i \u2264 restrict s i \u2227 restrict s i\u1d9c \u2264 restrict 0 i\u1d9c", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : TopologicalSpace M\ninst\u271d : OrderedAddCommMonoid M\ns\u271d : SignedMeasure \u03b1\ni j : Set \u03b1\ns : SignedMeasure \u03b1\nf : \u2115 \u2192 \u211d\nleft\u271d : Antitone f\nhf\u2082 : Tendsto f atTop (nhds (sInf (measureOfNegatives s)))\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measureOfNegatives s\n\u22a2 \u2203 i, MeasurableSet i \u2227 restrict 0 i \u2264 restrict s i \u2227 restrict s i\u1d9c \u2264 restrict 0 i\u1d9c"}, {"tactic": "choose B hB using hf\u2081", "annotated_tactic": ["choose B hB using hf\u2081", []], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : TopologicalSpace M\ninst\u271d : OrderedAddCommMonoid M\ns\u271d : SignedMeasure \u03b1\ni j : Set \u03b1\ns : SignedMeasure \u03b1\nf : \u2115 \u2192 \u211d\nleft\u271d : Antitone f\nhf\u2082 : Tendsto f atTop (nhds (sInf (measureOfNegatives s)))\nhf\u2081 : \u2200 (n : \u2115), f n \u2208 measureOfNegatives s\n\u22a2 \u2203 i, MeasurableSet i \u2227 restrict 0 i \u2264 restrict s i \u2227 restrict s i\u1d9c \u2264 restrict 0 i\u1d9c", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : TopologicalSpace M\ninst\u271d : OrderedAddCommMonoid M\ns\u271d : SignedMeasure \u03b1\ni j : Set \u03b1\ns : SignedMeasure \u03b1\nf : \u2115 \u2192 \u211d\nleft\u271d : Antitone f\nhf\u2082 : Tendsto f atTop (nhds (sInf (measureOfNegatives s)))\nB : \u2115 \u2192 Set \u03b1\nhB : \u2200 (n : \u2115), B n \u2208 {B | MeasurableSet B \u2227 restrict s B \u2264 restrict 0 B} \u2227 \u2191s (B n) = f n\n\u22a2 \u2203 i, MeasurableSet i \u2227 restrict 0 i \u2264 restrict s i \u2227 restrict s i\u1d9c \u2264 restrict 0 i\u1d9c"}, {"tactic": "have hB\u2081 : \u2200 n, MeasurableSet (B n) := fun n => (hB n).1.1", "annotated_tactic": ["have hB\u2081 : \u2200 n, <a>MeasurableSet</a> (B n) := fun n => (hB n).1.1", [{"full_name": "MeasurableSet", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [64, 5], "def_end_pos": [64, 18]}]], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : TopologicalSpace M\ninst\u271d : OrderedAddCommMonoid M\ns\u271d : SignedMeasure \u03b1\ni j : Set \u03b1\ns : SignedMeasure \u03b1\nf : \u2115 \u2192 \u211d\nleft\u271d : Antitone f\nhf\u2082 : Tendsto f atTop (nhds (sInf (measureOfNegatives s)))\nB : \u2115 \u2192 Set \u03b1\nhB : \u2200 (n : \u2115), B n \u2208 {B | MeasurableSet B \u2227 restrict s B \u2264 restrict 0 B} \u2227 \u2191s (B n) = f n\n\u22a2 \u2203 i, MeasurableSet i \u2227 restrict 0 i \u2264 restrict s i \u2227 restrict s i\u1d9c \u2264 restrict 0 i\u1d9c", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : TopologicalSpace M\ninst\u271d : OrderedAddCommMonoid M\ns\u271d : SignedMeasure \u03b1\ni j : Set \u03b1\ns : SignedMeasure \u03b1\nf : \u2115 \u2192 \u211d\nleft\u271d : Antitone f\nhf\u2082 : Tendsto f atTop (nhds (sInf (measureOfNegatives s)))\nB : \u2115 \u2192 Set \u03b1\nhB : \u2200 (n : \u2115), B n \u2208 {B | MeasurableSet B \u2227 restrict s B \u2264 restrict 0 B} \u2227 \u2191s (B n) = f n\nhB\u2081 : \u2200 (n : \u2115), MeasurableSet (B n)\n\u22a2 \u2203 i, MeasurableSet i \u2227 restrict 0 i \u2264 restrict s i \u2227 restrict s i\u1d9c \u2264 restrict 0 i\u1d9c"}, {"tactic": "have hB\u2082 : \u2200 n, s \u2264[B n] 0 := fun n => (hB n).1.2", "annotated_tactic": ["have hB\u2082 : \u2200 n, s \u2264[B n] 0 := fun n => (hB n).1.2", []], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : TopologicalSpace M\ninst\u271d : OrderedAddCommMonoid M\ns\u271d : SignedMeasure \u03b1\ni j : Set \u03b1\ns : SignedMeasure \u03b1\nf : \u2115 \u2192 \u211d\nleft\u271d : Antitone f\nhf\u2082 : Tendsto f atTop (nhds (sInf (measureOfNegatives s)))\nB : \u2115 \u2192 Set \u03b1\nhB : \u2200 (n : \u2115), B n \u2208 {B | MeasurableSet B \u2227 restrict s B \u2264 restrict 0 B} \u2227 \u2191s (B n) = f n\nhB\u2081 : \u2200 (n : \u2115), MeasurableSet (B n)\n\u22a2 \u2203 i, MeasurableSet i \u2227 restrict 0 i \u2264 restrict s i \u2227 restrict s i\u1d9c \u2264 restrict 0 i\u1d9c", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : TopologicalSpace M\ninst\u271d : OrderedAddCommMonoid M\ns\u271d : SignedMeasure \u03b1\ni j : Set \u03b1\ns : SignedMeasure \u03b1\nf : \u2115 \u2192 \u211d\nleft\u271d : Antitone f\nhf\u2082 : Tendsto f atTop (nhds (sInf (measureOfNegatives s)))\nB : \u2115 \u2192 Set \u03b1\nhB : \u2200 (n : \u2115), B n \u2208 {B | MeasurableSet B \u2227 restrict s B \u2264 restrict 0 B} \u2227 \u2191s (B n) = f n\nhB\u2081 : \u2200 (n : \u2115), MeasurableSet (B n)\nhB\u2082 : \u2200 (n : \u2115), restrict s (B n) \u2264 restrict 0 (B n)\n\u22a2 \u2203 i, MeasurableSet i \u2227 restrict 0 i \u2264 restrict s i \u2227 restrict s i\u1d9c \u2264 restrict 0 i\u1d9c"}, {"tactic": "set A := \u22c3 n, B n with hA", "annotated_tactic": ["set A := \u22c3 n, B n with hA", []], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : TopologicalSpace M\ninst\u271d : OrderedAddCommMonoid M\ns\u271d : SignedMeasure \u03b1\ni j : Set \u03b1\ns : SignedMeasure \u03b1\nf : \u2115 \u2192 \u211d\nleft\u271d : Antitone f\nhf\u2082 : Tendsto f atTop (nhds (sInf (measureOfNegatives s)))\nB : \u2115 \u2192 Set \u03b1\nhB : \u2200 (n : \u2115), B n \u2208 {B | MeasurableSet B \u2227 restrict s B \u2264 restrict 0 B} \u2227 \u2191s (B n) = f n\nhB\u2081 : \u2200 (n : \u2115), MeasurableSet (B n)\nhB\u2082 : \u2200 (n : \u2115), restrict s (B n) \u2264 restrict 0 (B n)\n\u22a2 \u2203 i, MeasurableSet i \u2227 restrict 0 i \u2264 restrict s i \u2227 restrict s i\u1d9c \u2264 restrict 0 i\u1d9c", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : TopologicalSpace M\ninst\u271d : OrderedAddCommMonoid M\ns\u271d : SignedMeasure \u03b1\ni j : Set \u03b1\ns : SignedMeasure \u03b1\nf : \u2115 \u2192 \u211d\nleft\u271d : Antitone f\nhf\u2082 : Tendsto f atTop (nhds (sInf (measureOfNegatives s)))\nB : \u2115 \u2192 Set \u03b1\nhB : \u2200 (n : \u2115), B n \u2208 {B | MeasurableSet B \u2227 restrict s B \u2264 restrict 0 B} \u2227 \u2191s (B n) = f n\nhB\u2081 : \u2200 (n : \u2115), MeasurableSet (B n)\nhB\u2082 : \u2200 (n : \u2115), restrict s (B n) \u2264 restrict 0 (B n)\nA : Set \u03b1 := \u22c3 n, B n\nhA : A = \u22c3 n, B n\n\u22a2 \u2203 i, MeasurableSet i \u2227 restrict 0 i \u2264 restrict s i \u2227 restrict s i\u1d9c \u2264 restrict 0 i\u1d9c"}, {"tactic": "have hA\u2081 : MeasurableSet A := MeasurableSet.iUnion hB\u2081", "annotated_tactic": ["have hA\u2081 : <a>MeasurableSet</a> A := <a>MeasurableSet.iUnion</a> hB\u2081", [{"full_name": "MeasurableSet", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [64, 5], "def_end_pos": [64, 18]}, {"full_name": "MeasurableSet.iUnion", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [115, 19], "def_end_pos": [115, 39]}]], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : TopologicalSpace M\ninst\u271d : OrderedAddCommMonoid M\ns\u271d : SignedMeasure \u03b1\ni j : Set \u03b1\ns : SignedMeasure \u03b1\nf : \u2115 \u2192 \u211d\nleft\u271d : Antitone f\nhf\u2082 : Tendsto f atTop (nhds (sInf (measureOfNegatives s)))\nB : \u2115 \u2192 Set \u03b1\nhB : \u2200 (n : \u2115), B n \u2208 {B | MeasurableSet B \u2227 restrict s B \u2264 restrict 0 B} \u2227 \u2191s (B n) = f n\nhB\u2081 : \u2200 (n : \u2115), MeasurableSet (B n)\nhB\u2082 : \u2200 (n : \u2115), restrict s (B n) \u2264 restrict 0 (B n)\nA : Set \u03b1 := \u22c3 n, B n\nhA : A = \u22c3 n, B n\n\u22a2 \u2203 i, MeasurableSet i \u2227 restrict 0 i \u2264 restrict s i \u2227 restrict s i\u1d9c \u2264 restrict 0 i\u1d9c", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : TopologicalSpace M\ninst\u271d : OrderedAddCommMonoid M\ns\u271d : SignedMeasure \u03b1\ni j : Set \u03b1\ns : SignedMeasure \u03b1\nf : \u2115 \u2192 \u211d\nleft\u271d : Antitone f\nhf\u2082 : Tendsto f atTop (nhds (sInf (measureOfNegatives s)))\nB : \u2115 \u2192 Set \u03b1\nhB : \u2200 (n : \u2115), B n \u2208 {B | MeasurableSet B \u2227 restrict s B \u2264 restrict 0 B} \u2227 \u2191s (B n) = f n\nhB\u2081 : \u2200 (n : \u2115), MeasurableSet (B n)\nhB\u2082 : \u2200 (n : \u2115), restrict s (B n) \u2264 restrict 0 (B n)\nA : Set \u03b1 := \u22c3 n, B n\nhA : A = \u22c3 n, B n\nhA\u2081 : MeasurableSet A\n\u22a2 \u2203 i, MeasurableSet i \u2227 restrict 0 i \u2264 restrict s i \u2227 restrict s i\u1d9c \u2264 restrict 0 i\u1d9c"}, {"tactic": "have hA\u2082 : s \u2264[A] 0 := restrict_le_restrict_iUnion _ _ hB\u2081 hB\u2082", "annotated_tactic": ["have hA\u2082 : s \u2264[A] 0 := <a>restrict_le_restrict_iUnion</a> _ _ hB\u2081 hB\u2082", [{"full_name": "MeasureTheory.VectorMeasure.restrict_le_restrict_iUnion", "def_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "def_pos": [935, 9], "def_end_pos": [935, 36]}]], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : TopologicalSpace M\ninst\u271d : OrderedAddCommMonoid M\ns\u271d : SignedMeasure \u03b1\ni j : Set \u03b1\ns : SignedMeasure \u03b1\nf : \u2115 \u2192 \u211d\nleft\u271d : Antitone f\nhf\u2082 : Tendsto f atTop (nhds (sInf (measureOfNegatives s)))\nB : \u2115 \u2192 Set \u03b1\nhB : \u2200 (n : \u2115), B n \u2208 {B | MeasurableSet B \u2227 restrict s B \u2264 restrict 0 B} \u2227 \u2191s (B n) = f n\nhB\u2081 : \u2200 (n : \u2115), MeasurableSet (B n)\nhB\u2082 : \u2200 (n : \u2115), restrict s (B n) \u2264 restrict 0 (B n)\nA : Set \u03b1 := \u22c3 n, B n\nhA : A = \u22c3 n, B n\nhA\u2081 : MeasurableSet A\n\u22a2 \u2203 i, MeasurableSet i \u2227 restrict 0 i \u2264 restrict s i \u2227 restrict s i\u1d9c \u2264 restrict 0 i\u1d9c", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : TopologicalSpace M\ninst\u271d : OrderedAddCommMonoid M\ns\u271d : SignedMeasure \u03b1\ni j : Set \u03b1\ns : SignedMeasure \u03b1\nf : \u2115 \u2192 \u211d\nleft\u271d : Antitone f\nhf\u2082 : Tendsto f atTop (nhds (sInf (measureOfNegatives s)))\nB : \u2115 \u2192 Set \u03b1\nhB : \u2200 (n : \u2115), B n \u2208 {B | MeasurableSet B \u2227 restrict s B \u2264 restrict 0 B} \u2227 \u2191s (B n) = f n\nhB\u2081 : \u2200 (n : \u2115), MeasurableSet (B n)\nhB\u2082 : \u2200 (n : \u2115), restrict s (B n) \u2264 restrict 0 (B n)\nA : Set \u03b1 := \u22c3 n, B n\nhA : A = \u22c3 n, B n\nhA\u2081 : MeasurableSet A\nhA\u2082 : restrict s A \u2264 restrict 0 A\n\u22a2 \u2203 i, MeasurableSet i \u2227 restrict 0 i \u2264 restrict s i \u2227 restrict s i\u1d9c \u2264 restrict 0 i\u1d9c"}, {"tactic": "refine' \u27e8A\u1d9c, hA\u2081.compl, _, (compl_compl A).symm \u25b8 hA\u2082\u27e9", "annotated_tactic": ["refine' \u27e8A\u1d9c, hA\u2081.compl, _, (<a>compl_compl</a> A).<a>symm</a> \u25b8 hA\u2082\u27e9", [{"full_name": "compl_compl", "def_path": "Mathlib/Order/BooleanAlgebra.lean", "def_pos": [634, 9], "def_end_pos": [634, 20]}, {"full_name": "Eq.symm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [310, 9], "def_end_pos": [310, 16]}]], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : TopologicalSpace M\ninst\u271d : OrderedAddCommMonoid M\ns\u271d : SignedMeasure \u03b1\ni j : Set \u03b1\ns : SignedMeasure \u03b1\nf : \u2115 \u2192 \u211d\nleft\u271d : Antitone f\nhf\u2082 : Tendsto f atTop (nhds (sInf (measureOfNegatives s)))\nB : \u2115 \u2192 Set \u03b1\nhB : \u2200 (n : \u2115), B n \u2208 {B | MeasurableSet B \u2227 restrict s B \u2264 restrict 0 B} \u2227 \u2191s (B n) = f n\nhB\u2081 : \u2200 (n : \u2115), MeasurableSet (B n)\nhB\u2082 : \u2200 (n : \u2115), restrict s (B n) \u2264 restrict 0 (B n)\nA : Set \u03b1 := \u22c3 n, B n\nhA : A = \u22c3 n, B n\nhA\u2081 : MeasurableSet A\nhA\u2082 : restrict s A \u2264 restrict 0 A\nhA\u2083 : \u2191s A = sInf (measureOfNegatives s)\n\u22a2 \u2203 i, MeasurableSet i \u2227 restrict 0 i \u2264 restrict s i \u2227 restrict s i\u1d9c \u2264 restrict 0 i\u1d9c", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : TopologicalSpace M\ninst\u271d : OrderedAddCommMonoid M\ns\u271d : SignedMeasure \u03b1\ni j : Set \u03b1\ns : SignedMeasure \u03b1\nf : \u2115 \u2192 \u211d\nleft\u271d : Antitone f\nhf\u2082 : Tendsto f atTop (nhds (sInf (measureOfNegatives s)))\nB : \u2115 \u2192 Set \u03b1\nhB : \u2200 (n : \u2115), B n \u2208 {B | MeasurableSet B \u2227 restrict s B \u2264 restrict 0 B} \u2227 \u2191s (B n) = f n\nhB\u2081 : \u2200 (n : \u2115), MeasurableSet (B n)\nhB\u2082 : \u2200 (n : \u2115), restrict s (B n) \u2264 restrict 0 (B n)\nA : Set \u03b1 := \u22c3 n, B n\nhA : A = \u22c3 n, B n\nhA\u2081 : MeasurableSet A\nhA\u2082 : restrict s A \u2264 restrict 0 A\nhA\u2083 : \u2191s A = sInf (measureOfNegatives s)\n\u22a2 restrict 0 A\u1d9c \u2264 restrict s A\u1d9c"}, {"tactic": "rw [restrict_le_restrict_iff _ _ hA\u2081.compl]", "annotated_tactic": ["rw [<a>restrict_le_restrict_iff</a> _ _ hA\u2081.compl]", [{"full_name": "MeasureTheory.VectorMeasure.restrict_le_restrict_iff", "def_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "def_pos": [861, 9], "def_end_pos": [861, 33]}]], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : TopologicalSpace M\ninst\u271d : OrderedAddCommMonoid M\ns\u271d : SignedMeasure \u03b1\ni j : Set \u03b1\ns : SignedMeasure \u03b1\nf : \u2115 \u2192 \u211d\nleft\u271d : Antitone f\nhf\u2082 : Tendsto f atTop (nhds (sInf (measureOfNegatives s)))\nB : \u2115 \u2192 Set \u03b1\nhB : \u2200 (n : \u2115), B n \u2208 {B | MeasurableSet B \u2227 restrict s B \u2264 restrict 0 B} \u2227 \u2191s (B n) = f n\nhB\u2081 : \u2200 (n : \u2115), MeasurableSet (B n)\nhB\u2082 : \u2200 (n : \u2115), restrict s (B n) \u2264 restrict 0 (B n)\nA : Set \u03b1 := \u22c3 n, B n\nhA : A = \u22c3 n, B n\nhA\u2081 : MeasurableSet A\nhA\u2082 : restrict s A \u2264 restrict 0 A\nhA\u2083 : \u2191s A = sInf (measureOfNegatives s)\n\u22a2 restrict 0 A\u1d9c \u2264 restrict s A\u1d9c", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : TopologicalSpace M\ninst\u271d : OrderedAddCommMonoid M\ns\u271d : SignedMeasure \u03b1\ni j : Set \u03b1\ns : SignedMeasure \u03b1\nf : \u2115 \u2192 \u211d\nleft\u271d : Antitone f\nhf\u2082 : Tendsto f atTop (nhds (sInf (measureOfNegatives s)))\nB : \u2115 \u2192 Set \u03b1\nhB : \u2200 (n : \u2115), B n \u2208 {B | MeasurableSet B \u2227 restrict s B \u2264 restrict 0 B} \u2227 \u2191s (B n) = f n\nhB\u2081 : \u2200 (n : \u2115), MeasurableSet (B n)\nhB\u2082 : \u2200 (n : \u2115), restrict s (B n) \u2264 restrict 0 (B n)\nA : Set \u03b1 := \u22c3 n, B n\nhA : A = \u22c3 n, B n\nhA\u2081 : MeasurableSet A\nhA\u2082 : restrict s A \u2264 restrict 0 A\nhA\u2083 : \u2191s A = sInf (measureOfNegatives s)\n\u22a2 \u2200 \u2983j : Set \u03b1\u2984, MeasurableSet j \u2192 j \u2286 A\u1d9c \u2192 \u21910 j \u2264 \u2191s j"}, {"tactic": "intro C _ hC\u2081", "annotated_tactic": ["intro C _ hC\u2081", []], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : TopologicalSpace M\ninst\u271d : OrderedAddCommMonoid M\ns\u271d : SignedMeasure \u03b1\ni j : Set \u03b1\ns : SignedMeasure \u03b1\nf : \u2115 \u2192 \u211d\nleft\u271d : Antitone f\nhf\u2082 : Tendsto f atTop (nhds (sInf (measureOfNegatives s)))\nB : \u2115 \u2192 Set \u03b1\nhB : \u2200 (n : \u2115), B n \u2208 {B | MeasurableSet B \u2227 restrict s B \u2264 restrict 0 B} \u2227 \u2191s (B n) = f n\nhB\u2081 : \u2200 (n : \u2115), MeasurableSet (B n)\nhB\u2082 : \u2200 (n : \u2115), restrict s (B n) \u2264 restrict 0 (B n)\nA : Set \u03b1 := \u22c3 n, B n\nhA : A = \u22c3 n, B n\nhA\u2081 : MeasurableSet A\nhA\u2082 : restrict s A \u2264 restrict 0 A\nhA\u2083 : \u2191s A = sInf (measureOfNegatives s)\n\u22a2 \u2200 \u2983j : Set \u03b1\u2984, MeasurableSet j \u2192 j \u2286 A\u1d9c \u2192 \u21910 j \u2264 \u2191s j", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : TopologicalSpace M\ninst\u271d : OrderedAddCommMonoid M\ns\u271d : SignedMeasure \u03b1\ni j : Set \u03b1\ns : SignedMeasure \u03b1\nf : \u2115 \u2192 \u211d\nleft\u271d : Antitone f\nhf\u2082 : Tendsto f atTop (nhds (sInf (measureOfNegatives s)))\nB : \u2115 \u2192 Set \u03b1\nhB : \u2200 (n : \u2115), B n \u2208 {B | MeasurableSet B \u2227 restrict s B \u2264 restrict 0 B} \u2227 \u2191s (B n) = f n\nhB\u2081 : \u2200 (n : \u2115), MeasurableSet (B n)\nhB\u2082 : \u2200 (n : \u2115), restrict s (B n) \u2264 restrict 0 (B n)\nA : Set \u03b1 := \u22c3 n, B n\nhA : A = \u22c3 n, B n\nhA\u2081 : MeasurableSet A\nhA\u2082 : restrict s A \u2264 restrict 0 A\nhA\u2083 : \u2191s A = sInf (measureOfNegatives s)\nC : Set \u03b1\na\u271d : MeasurableSet C\nhC\u2081 : C \u2286 A\u1d9c\n\u22a2 \u21910 C \u2264 \u2191s C"}, {"tactic": "by_contra' hC\u2082", "annotated_tactic": ["by_contra' hC\u2082", []], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : TopologicalSpace M\ninst\u271d : OrderedAddCommMonoid M\ns\u271d : SignedMeasure \u03b1\ni j : Set \u03b1\ns : SignedMeasure \u03b1\nf : \u2115 \u2192 \u211d\nleft\u271d : Antitone f\nhf\u2082 : Tendsto f atTop (nhds (sInf (measureOfNegatives s)))\nB : \u2115 \u2192 Set \u03b1\nhB : \u2200 (n : \u2115), B n \u2208 {B | MeasurableSet B \u2227 restrict s B \u2264 restrict 0 B} \u2227 \u2191s (B n) = f n\nhB\u2081 : \u2200 (n : \u2115), MeasurableSet (B n)\nhB\u2082 : \u2200 (n : \u2115), restrict s (B n) \u2264 restrict 0 (B n)\nA : Set \u03b1 := \u22c3 n, B n\nhA : A = \u22c3 n, B n\nhA\u2081 : MeasurableSet A\nhA\u2082 : restrict s A \u2264 restrict 0 A\nhA\u2083 : \u2191s A = sInf (measureOfNegatives s)\nC : Set \u03b1\na\u271d : MeasurableSet C\nhC\u2081 : C \u2286 A\u1d9c\n\u22a2 \u21910 C \u2264 \u2191s C", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : TopologicalSpace M\ninst\u271d : OrderedAddCommMonoid M\ns\u271d : SignedMeasure \u03b1\ni j : Set \u03b1\ns : SignedMeasure \u03b1\nf : \u2115 \u2192 \u211d\nleft\u271d : Antitone f\nhf\u2082 : Tendsto f atTop (nhds (sInf (measureOfNegatives s)))\nB : \u2115 \u2192 Set \u03b1\nhB : \u2200 (n : \u2115), B n \u2208 {B | MeasurableSet B \u2227 restrict s B \u2264 restrict 0 B} \u2227 \u2191s (B n) = f n\nhB\u2081 : \u2200 (n : \u2115), MeasurableSet (B n)\nhB\u2082 : \u2200 (n : \u2115), restrict s (B n) \u2264 restrict 0 (B n)\nA : Set \u03b1 := \u22c3 n, B n\nhA : A = \u22c3 n, B n\nhA\u2081 : MeasurableSet A\nhA\u2082 : restrict s A \u2264 restrict 0 A\nhA\u2083 : \u2191s A = sInf (measureOfNegatives s)\nC : Set \u03b1\na\u271d : MeasurableSet C\nhC\u2081 : C \u2286 A\u1d9c\nhC\u2082 : \u2191s C < \u21910 C\n\u22a2 False"}, {"tactic": "rcases exists_subset_restrict_nonpos hC\u2082 with \u27e8D, hD\u2081, hD, hD\u2082, hD\u2083\u27e9", "annotated_tactic": ["rcases <a>exists_subset_restrict_nonpos</a> hC\u2082 with \u27e8D, hD\u2081, hD, hD\u2082, hD\u2083\u27e9", [{"full_name": "MeasureTheory.SignedMeasure.exists_subset_restrict_nonpos", "def_path": "Mathlib/MeasureTheory/Decomposition/SignedHahn.lean", "def_pos": [266, 9], "def_end_pos": [266, 38]}]], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : TopologicalSpace M\ninst\u271d : OrderedAddCommMonoid M\ns\u271d : SignedMeasure \u03b1\ni j : Set \u03b1\ns : SignedMeasure \u03b1\nf : \u2115 \u2192 \u211d\nleft\u271d : Antitone f\nhf\u2082 : Tendsto f atTop (nhds (sInf (measureOfNegatives s)))\nB : \u2115 \u2192 Set \u03b1\nhB : \u2200 (n : \u2115), B n \u2208 {B | MeasurableSet B \u2227 restrict s B \u2264 restrict 0 B} \u2227 \u2191s (B n) = f n\nhB\u2081 : \u2200 (n : \u2115), MeasurableSet (B n)\nhB\u2082 : \u2200 (n : \u2115), restrict s (B n) \u2264 restrict 0 (B n)\nA : Set \u03b1 := \u22c3 n, B n\nhA : A = \u22c3 n, B n\nhA\u2081 : MeasurableSet A\nhA\u2082 : restrict s A \u2264 restrict 0 A\nhA\u2083 : \u2191s A = sInf (measureOfNegatives s)\nC : Set \u03b1\na\u271d : MeasurableSet C\nhC\u2081 : C \u2286 A\u1d9c\nhC\u2082 : \u2191s C < \u21910 C\n\u22a2 False", "state_after": "case intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : TopologicalSpace M\ninst\u271d : OrderedAddCommMonoid M\ns\u271d : SignedMeasure \u03b1\ni j : Set \u03b1\ns : SignedMeasure \u03b1\nf : \u2115 \u2192 \u211d\nleft\u271d : Antitone f\nhf\u2082 : Tendsto f atTop (nhds (sInf (measureOfNegatives s)))\nB : \u2115 \u2192 Set \u03b1\nhB : \u2200 (n : \u2115), B n \u2208 {B | MeasurableSet B \u2227 restrict s B \u2264 restrict 0 B} \u2227 \u2191s (B n) = f n\nhB\u2081 : \u2200 (n : \u2115), MeasurableSet (B n)\nhB\u2082 : \u2200 (n : \u2115), restrict s (B n) \u2264 restrict 0 (B n)\nA : Set \u03b1 := \u22c3 n, B n\nhA : A = \u22c3 n, B n\nhA\u2081 : MeasurableSet A\nhA\u2082 : restrict s A \u2264 restrict 0 A\nhA\u2083 : \u2191s A = sInf (measureOfNegatives s)\nC : Set \u03b1\na\u271d : MeasurableSet C\nhC\u2081 : C \u2286 A\u1d9c\nhC\u2082 : \u2191s C < \u21910 C\nD : Set \u03b1\nhD\u2081 : MeasurableSet D\nhD : D \u2286 C\nhD\u2082 : restrict s D \u2264 restrict 0 D\nhD\u2083 : \u2191s D < 0\n\u22a2 False"}, {"tactic": "have : s (A \u222a D) < sInf s.measureOfNegatives := by\n  rw [\u2190 hA\u2083,\n    of_union (Set.disjoint_of_subset_right (Set.Subset.trans hD hC\u2081) disjoint_compl_right) hA\u2081\n      hD\u2081]\n  linarith", "annotated_tactic": ["have : s (A \u222a D) < <a>sInf</a> s.measureOfNegatives := by\n    rw [\u2190 hA\u2083,\n      <a>of_union</a> (<a>Set.disjoint_of_subset_right</a> (<a>Set.Subset.trans</a> hD hC\u2081) <a>disjoint_compl_right</a>) hA\u2081\n        hD\u2081]\n    linarith", [{"full_name": "InfSet.sInf", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [62, 3], "def_end_pos": [62, 7]}, {"full_name": "MeasureTheory.VectorMeasure.of_union", "def_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "def_pos": [185, 9], "def_end_pos": [185, 17]}, {"full_name": "Set.disjoint_of_subset_right", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1576, 7], "def_end_pos": [1576, 31]}, {"full_name": "Set.Subset.trans", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [362, 9], "def_end_pos": [362, 21]}, {"full_name": "disjoint_compl_right", "def_path": "Mathlib/Order/Heyting/Basic.lean", "def_pos": [844, 9], "def_end_pos": [844, 29]}]], "state_before": "case intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : TopologicalSpace M\ninst\u271d : OrderedAddCommMonoid M\ns\u271d : SignedMeasure \u03b1\ni j : Set \u03b1\ns : SignedMeasure \u03b1\nf : \u2115 \u2192 \u211d\nleft\u271d : Antitone f\nhf\u2082 : Tendsto f atTop (nhds (sInf (measureOfNegatives s)))\nB : \u2115 \u2192 Set \u03b1\nhB : \u2200 (n : \u2115), B n \u2208 {B | MeasurableSet B \u2227 restrict s B \u2264 restrict 0 B} \u2227 \u2191s (B n) = f n\nhB\u2081 : \u2200 (n : \u2115), MeasurableSet (B n)\nhB\u2082 : \u2200 (n : \u2115), restrict s (B n) \u2264 restrict 0 (B n)\nA : Set \u03b1 := \u22c3 n, B n\nhA : A = \u22c3 n, B n\nhA\u2081 : MeasurableSet A\nhA\u2082 : restrict s A \u2264 restrict 0 A\nhA\u2083 : \u2191s A = sInf (measureOfNegatives s)\nC : Set \u03b1\na\u271d : MeasurableSet C\nhC\u2081 : C \u2286 A\u1d9c\nhC\u2082 : \u2191s C < \u21910 C\nD : Set \u03b1\nhD\u2081 : MeasurableSet D\nhD : D \u2286 C\nhD\u2082 : restrict s D \u2264 restrict 0 D\nhD\u2083 : \u2191s D < 0\n\u22a2 False", "state_after": "case intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : TopologicalSpace M\ninst\u271d : OrderedAddCommMonoid M\ns\u271d : SignedMeasure \u03b1\ni j : Set \u03b1\ns : SignedMeasure \u03b1\nf : \u2115 \u2192 \u211d\nleft\u271d : Antitone f\nhf\u2082 : Tendsto f atTop (nhds (sInf (measureOfNegatives s)))\nB : \u2115 \u2192 Set \u03b1\nhB : \u2200 (n : \u2115), B n \u2208 {B | MeasurableSet B \u2227 restrict s B \u2264 restrict 0 B} \u2227 \u2191s (B n) = f n\nhB\u2081 : \u2200 (n : \u2115), MeasurableSet (B n)\nhB\u2082 : \u2200 (n : \u2115), restrict s (B n) \u2264 restrict 0 (B n)\nA : Set \u03b1 := \u22c3 n, B n\nhA : A = \u22c3 n, B n\nhA\u2081 : MeasurableSet A\nhA\u2082 : restrict s A \u2264 restrict 0 A\nhA\u2083 : \u2191s A = sInf (measureOfNegatives s)\nC : Set \u03b1\na\u271d : MeasurableSet C\nhC\u2081 : C \u2286 A\u1d9c\nhC\u2082 : \u2191s C < \u21910 C\nD : Set \u03b1\nhD\u2081 : MeasurableSet D\nhD : D \u2286 C\nhD\u2082 : restrict s D \u2264 restrict 0 D\nhD\u2083 : \u2191s D < 0\nthis : \u2191s (A \u222a D) < sInf (measureOfNegatives s)\n\u22a2 False"}, {"tactic": "refine' not_le.2 this _", "annotated_tactic": ["refine' <a>not_le</a>.2 this _", [{"full_name": "not_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [373, 9], "def_end_pos": [373, 15]}]], "state_before": "case intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : TopologicalSpace M\ninst\u271d : OrderedAddCommMonoid M\ns\u271d : SignedMeasure \u03b1\ni j : Set \u03b1\ns : SignedMeasure \u03b1\nf : \u2115 \u2192 \u211d\nleft\u271d : Antitone f\nhf\u2082 : Tendsto f atTop (nhds (sInf (measureOfNegatives s)))\nB : \u2115 \u2192 Set \u03b1\nhB : \u2200 (n : \u2115), B n \u2208 {B | MeasurableSet B \u2227 restrict s B \u2264 restrict 0 B} \u2227 \u2191s (B n) = f n\nhB\u2081 : \u2200 (n : \u2115), MeasurableSet (B n)\nhB\u2082 : \u2200 (n : \u2115), restrict s (B n) \u2264 restrict 0 (B n)\nA : Set \u03b1 := \u22c3 n, B n\nhA : A = \u22c3 n, B n\nhA\u2081 : MeasurableSet A\nhA\u2082 : restrict s A \u2264 restrict 0 A\nhA\u2083 : \u2191s A = sInf (measureOfNegatives s)\nC : Set \u03b1\na\u271d : MeasurableSet C\nhC\u2081 : C \u2286 A\u1d9c\nhC\u2082 : \u2191s C < \u21910 C\nD : Set \u03b1\nhD\u2081 : MeasurableSet D\nhD : D \u2286 C\nhD\u2082 : restrict s D \u2264 restrict 0 D\nhD\u2083 : \u2191s D < 0\nthis : \u2191s (A \u222a D) < sInf (measureOfNegatives s)\n\u22a2 False", "state_after": "case intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : TopologicalSpace M\ninst\u271d : OrderedAddCommMonoid M\ns\u271d : SignedMeasure \u03b1\ni j : Set \u03b1\ns : SignedMeasure \u03b1\nf : \u2115 \u2192 \u211d\nleft\u271d : Antitone f\nhf\u2082 : Tendsto f atTop (nhds (sInf (measureOfNegatives s)))\nB : \u2115 \u2192 Set \u03b1\nhB : \u2200 (n : \u2115), B n \u2208 {B | MeasurableSet B \u2227 restrict s B \u2264 restrict 0 B} \u2227 \u2191s (B n) = f n\nhB\u2081 : \u2200 (n : \u2115), MeasurableSet (B n)\nhB\u2082 : \u2200 (n : \u2115), restrict s (B n) \u2264 restrict 0 (B n)\nA : Set \u03b1 := \u22c3 n, B n\nhA : A = \u22c3 n, B n\nhA\u2081 : MeasurableSet A\nhA\u2082 : restrict s A \u2264 restrict 0 A\nhA\u2083 : \u2191s A = sInf (measureOfNegatives s)\nC : Set \u03b1\na\u271d : MeasurableSet C\nhC\u2081 : C \u2286 A\u1d9c\nhC\u2082 : \u2191s C < \u21910 C\nD : Set \u03b1\nhD\u2081 : MeasurableSet D\nhD : D \u2286 C\nhD\u2082 : restrict s D \u2264 restrict 0 D\nhD\u2083 : \u2191s D < 0\nthis : \u2191s (A \u222a D) < sInf (measureOfNegatives s)\n\u22a2 sInf (measureOfNegatives s) \u2264 \u2191s (A \u222a D)"}, {"tactic": "refine' csInf_le bddBelow_measureOfNegatives \u27e8A \u222a D, \u27e8_, _\u27e9, rfl\u27e9", "annotated_tactic": ["refine' <a>csInf_le</a> <a>bddBelow_measureOfNegatives</a> \u27e8A \u222a D, \u27e8_, _\u27e9, <a>rfl</a>\u27e9", [{"full_name": "csInf_le", "def_path": "Mathlib/Order/ConditionallyCompleteLattice/Basic.lean", "def_pos": [465, 9], "def_end_pos": [465, 17]}, {"full_name": "MeasureTheory.SignedMeasure.bddBelow_measureOfNegatives", "def_path": "Mathlib/MeasureTheory/Decomposition/SignedHahn.lean", "def_pos": [343, 9], "def_end_pos": [343, 36]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "case intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : TopologicalSpace M\ninst\u271d : OrderedAddCommMonoid M\ns\u271d : SignedMeasure \u03b1\ni j : Set \u03b1\ns : SignedMeasure \u03b1\nf : \u2115 \u2192 \u211d\nleft\u271d : Antitone f\nhf\u2082 : Tendsto f atTop (nhds (sInf (measureOfNegatives s)))\nB : \u2115 \u2192 Set \u03b1\nhB : \u2200 (n : \u2115), B n \u2208 {B | MeasurableSet B \u2227 restrict s B \u2264 restrict 0 B} \u2227 \u2191s (B n) = f n\nhB\u2081 : \u2200 (n : \u2115), MeasurableSet (B n)\nhB\u2082 : \u2200 (n : \u2115), restrict s (B n) \u2264 restrict 0 (B n)\nA : Set \u03b1 := \u22c3 n, B n\nhA : A = \u22c3 n, B n\nhA\u2081 : MeasurableSet A\nhA\u2082 : restrict s A \u2264 restrict 0 A\nhA\u2083 : \u2191s A = sInf (measureOfNegatives s)\nC : Set \u03b1\na\u271d : MeasurableSet C\nhC\u2081 : C \u2286 A\u1d9c\nhC\u2082 : \u2191s C < \u21910 C\nD : Set \u03b1\nhD\u2081 : MeasurableSet D\nhD : D \u2286 C\nhD\u2082 : restrict s D \u2264 restrict 0 D\nhD\u2083 : \u2191s D < 0\nthis : \u2191s (A \u222a D) < sInf (measureOfNegatives s)\n\u22a2 sInf (measureOfNegatives s) \u2264 \u2191s (A \u222a D)", "state_after": "case intro.intro.intro.intro.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : TopologicalSpace M\ninst\u271d : OrderedAddCommMonoid M\ns\u271d : SignedMeasure \u03b1\ni j : Set \u03b1\ns : SignedMeasure \u03b1\nf : \u2115 \u2192 \u211d\nleft\u271d : Antitone f\nhf\u2082 : Tendsto f atTop (nhds (sInf (measureOfNegatives s)))\nB : \u2115 \u2192 Set \u03b1\nhB : \u2200 (n : \u2115), B n \u2208 {B | MeasurableSet B \u2227 restrict s B \u2264 restrict 0 B} \u2227 \u2191s (B n) = f n\nhB\u2081 : \u2200 (n : \u2115), MeasurableSet (B n)\nhB\u2082 : \u2200 (n : \u2115), restrict s (B n) \u2264 restrict 0 (B n)\nA : Set \u03b1 := \u22c3 n, B n\nhA : A = \u22c3 n, B n\nhA\u2081 : MeasurableSet A\nhA\u2082 : restrict s A \u2264 restrict 0 A\nhA\u2083 : \u2191s A = sInf (measureOfNegatives s)\nC : Set \u03b1\na\u271d : MeasurableSet C\nhC\u2081 : C \u2286 A\u1d9c\nhC\u2082 : \u2191s C < \u21910 C\nD : Set \u03b1\nhD\u2081 : MeasurableSet D\nhD : D \u2286 C\nhD\u2082 : restrict s D \u2264 restrict 0 D\nhD\u2083 : \u2191s D < 0\nthis : \u2191s (A \u222a D) < sInf (measureOfNegatives s)\n\u22a2 MeasurableSet (A \u222a D)\n\ncase intro.intro.intro.intro.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : TopologicalSpace M\ninst\u271d : OrderedAddCommMonoid M\ns\u271d : SignedMeasure \u03b1\ni j : Set \u03b1\ns : SignedMeasure \u03b1\nf : \u2115 \u2192 \u211d\nleft\u271d : Antitone f\nhf\u2082 : Tendsto f atTop (nhds (sInf (measureOfNegatives s)))\nB : \u2115 \u2192 Set \u03b1\nhB : \u2200 (n : \u2115), B n \u2208 {B | MeasurableSet B \u2227 restrict s B \u2264 restrict 0 B} \u2227 \u2191s (B n) = f n\nhB\u2081 : \u2200 (n : \u2115), MeasurableSet (B n)\nhB\u2082 : \u2200 (n : \u2115), restrict s (B n) \u2264 restrict 0 (B n)\nA : Set \u03b1 := \u22c3 n, B n\nhA : A = \u22c3 n, B n\nhA\u2081 : MeasurableSet A\nhA\u2082 : restrict s A \u2264 restrict 0 A\nhA\u2083 : \u2191s A = sInf (measureOfNegatives s)\nC : Set \u03b1\na\u271d : MeasurableSet C\nhC\u2081 : C \u2286 A\u1d9c\nhC\u2082 : \u2191s C < \u21910 C\nD : Set \u03b1\nhD\u2081 : MeasurableSet D\nhD : D \u2286 C\nhD\u2082 : restrict s D \u2264 restrict 0 D\nhD\u2083 : \u2191s D < 0\nthis : \u2191s (A \u222a D) < sInf (measureOfNegatives s)\n\u22a2 restrict s (A \u222a D) \u2264 restrict 0 (A \u222a D)"}, {"tactic": "apply le_antisymm", "annotated_tactic": ["apply <a>le_antisymm</a>", [{"full_name": "le_antisymm", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [188, 9], "def_end_pos": [188, 20]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : TopologicalSpace M\ninst\u271d : OrderedAddCommMonoid M\ns\u271d : SignedMeasure \u03b1\ni j : Set \u03b1\ns : SignedMeasure \u03b1\nf : \u2115 \u2192 \u211d\nleft\u271d : Antitone f\nhf\u2082 : Tendsto f atTop (nhds (sInf (measureOfNegatives s)))\nB : \u2115 \u2192 Set \u03b1\nhB : \u2200 (n : \u2115), B n \u2208 {B | MeasurableSet B \u2227 restrict s B \u2264 restrict 0 B} \u2227 \u2191s (B n) = f n\nhB\u2081 : \u2200 (n : \u2115), MeasurableSet (B n)\nhB\u2082 : \u2200 (n : \u2115), restrict s (B n) \u2264 restrict 0 (B n)\nA : Set \u03b1 := \u22c3 n, B n\nhA : A = \u22c3 n, B n\nhA\u2081 : MeasurableSet A\nhA\u2082 : restrict s A \u2264 restrict 0 A\n\u22a2 \u2191s A = sInf (measureOfNegatives s)", "state_after": "case a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : TopologicalSpace M\ninst\u271d : OrderedAddCommMonoid M\ns\u271d : SignedMeasure \u03b1\ni j : Set \u03b1\ns : SignedMeasure \u03b1\nf : \u2115 \u2192 \u211d\nleft\u271d : Antitone f\nhf\u2082 : Tendsto f atTop (nhds (sInf (measureOfNegatives s)))\nB : \u2115 \u2192 Set \u03b1\nhB : \u2200 (n : \u2115), B n \u2208 {B | MeasurableSet B \u2227 restrict s B \u2264 restrict 0 B} \u2227 \u2191s (B n) = f n\nhB\u2081 : \u2200 (n : \u2115), MeasurableSet (B n)\nhB\u2082 : \u2200 (n : \u2115), restrict s (B n) \u2264 restrict 0 (B n)\nA : Set \u03b1 := \u22c3 n, B n\nhA : A = \u22c3 n, B n\nhA\u2081 : MeasurableSet A\nhA\u2082 : restrict s A \u2264 restrict 0 A\n\u22a2 \u2191s A \u2264 sInf (measureOfNegatives s)\n\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : TopologicalSpace M\ninst\u271d : OrderedAddCommMonoid M\ns\u271d : SignedMeasure \u03b1\ni j : Set \u03b1\ns : SignedMeasure \u03b1\nf : \u2115 \u2192 \u211d\nleft\u271d : Antitone f\nhf\u2082 : Tendsto f atTop (nhds (sInf (measureOfNegatives s)))\nB : \u2115 \u2192 Set \u03b1\nhB : \u2200 (n : \u2115), B n \u2208 {B | MeasurableSet B \u2227 restrict s B \u2264 restrict 0 B} \u2227 \u2191s (B n) = f n\nhB\u2081 : \u2200 (n : \u2115), MeasurableSet (B n)\nhB\u2082 : \u2200 (n : \u2115), restrict s (B n) \u2264 restrict 0 (B n)\nA : Set \u03b1 := \u22c3 n, B n\nhA : A = \u22c3 n, B n\nhA\u2081 : MeasurableSet A\nhA\u2082 : restrict s A \u2264 restrict 0 A\n\u22a2 sInf (measureOfNegatives s) \u2264 \u2191s A"}, {"tactic": "refine' le_of_tendsto_of_tendsto tendsto_const_nhds hf\u2082 (eventually_of_forall fun n => _)", "annotated_tactic": ["refine' <a>le_of_tendsto_of_tendsto</a> <a>tendsto_const_nhds</a> hf\u2082 (<a>eventually_of_forall</a> fun n => _)", [{"full_name": "le_of_tendsto_of_tendsto", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [223, 9], "def_end_pos": [223, 33]}, {"full_name": "tendsto_const_nhds", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [1049, 9], "def_end_pos": [1049, 27]}, {"full_name": "Filter.eventually_of_forall", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1111, 9], "def_end_pos": [1111, 29]}]], "state_before": "case a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : TopologicalSpace M\ninst\u271d : OrderedAddCommMonoid M\ns\u271d : SignedMeasure \u03b1\ni j : Set \u03b1\ns : SignedMeasure \u03b1\nf : \u2115 \u2192 \u211d\nleft\u271d : Antitone f\nhf\u2082 : Tendsto f atTop (nhds (sInf (measureOfNegatives s)))\nB : \u2115 \u2192 Set \u03b1\nhB : \u2200 (n : \u2115), B n \u2208 {B | MeasurableSet B \u2227 restrict s B \u2264 restrict 0 B} \u2227 \u2191s (B n) = f n\nhB\u2081 : \u2200 (n : \u2115), MeasurableSet (B n)\nhB\u2082 : \u2200 (n : \u2115), restrict s (B n) \u2264 restrict 0 (B n)\nA : Set \u03b1 := \u22c3 n, B n\nhA : A = \u22c3 n, B n\nhA\u2081 : MeasurableSet A\nhA\u2082 : restrict s A \u2264 restrict 0 A\n\u22a2 \u2191s A \u2264 sInf (measureOfNegatives s)", "state_after": "case a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : TopologicalSpace M\ninst\u271d : OrderedAddCommMonoid M\ns\u271d : SignedMeasure \u03b1\ni j : Set \u03b1\ns : SignedMeasure \u03b1\nf : \u2115 \u2192 \u211d\nleft\u271d : Antitone f\nhf\u2082 : Tendsto f atTop (nhds (sInf (measureOfNegatives s)))\nB : \u2115 \u2192 Set \u03b1\nhB : \u2200 (n : \u2115), B n \u2208 {B | MeasurableSet B \u2227 restrict s B \u2264 restrict 0 B} \u2227 \u2191s (B n) = f n\nhB\u2081 : \u2200 (n : \u2115), MeasurableSet (B n)\nhB\u2082 : \u2200 (n : \u2115), restrict s (B n) \u2264 restrict 0 (B n)\nA : Set \u03b1 := \u22c3 n, B n\nhA : A = \u22c3 n, B n\nhA\u2081 : MeasurableSet A\nhA\u2082 : restrict s A \u2264 restrict 0 A\nn : \u2115\n\u22a2 (fun x => \u2191s A) n \u2264 f n"}, {"tactic": "rw [\u2190 (hB n).2, hA, \u2190 Set.diff_union_of_subset (Set.subset_iUnion _ n),\n  of_union Set.disjoint_sdiff_left _ (hB\u2081 n)]", "annotated_tactic": ["rw [\u2190 (hB n).2, hA, \u2190 <a>Set.diff_union_of_subset</a> (<a>Set.subset_iUnion</a> _ n),\n        <a>of_union</a> <a>Set.disjoint_sdiff_left</a> _ (hB\u2081 n)]", [{"full_name": "Set.diff_union_of_subset", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1956, 9], "def_end_pos": [1956, 29]}, {"full_name": "Set.subset_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [431, 9], "def_end_pos": [431, 22]}, {"full_name": "MeasureTheory.VectorMeasure.of_union", "def_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "def_pos": [185, 9], "def_end_pos": [185, 17]}, {"full_name": "Set.disjoint_sdiff_left", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1601, 7], "def_end_pos": [1601, 26]}]], "state_before": "case a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : TopologicalSpace M\ninst\u271d : OrderedAddCommMonoid M\ns\u271d : SignedMeasure \u03b1\ni j : Set \u03b1\ns : SignedMeasure \u03b1\nf : \u2115 \u2192 \u211d\nleft\u271d : Antitone f\nhf\u2082 : Tendsto f atTop (nhds (sInf (measureOfNegatives s)))\nB : \u2115 \u2192 Set \u03b1\nhB : \u2200 (n : \u2115), B n \u2208 {B | MeasurableSet B \u2227 restrict s B \u2264 restrict 0 B} \u2227 \u2191s (B n) = f n\nhB\u2081 : \u2200 (n : \u2115), MeasurableSet (B n)\nhB\u2082 : \u2200 (n : \u2115), restrict s (B n) \u2264 restrict 0 (B n)\nA : Set \u03b1 := \u22c3 n, B n\nhA : A = \u22c3 n, B n\nhA\u2081 : MeasurableSet A\nhA\u2082 : restrict s A \u2264 restrict 0 A\nn : \u2115\n\u22a2 (fun x => \u2191s A) n \u2264 f n", "state_after": "case a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : TopologicalSpace M\ninst\u271d : OrderedAddCommMonoid M\ns\u271d : SignedMeasure \u03b1\ni j : Set \u03b1\ns : SignedMeasure \u03b1\nf : \u2115 \u2192 \u211d\nleft\u271d : Antitone f\nhf\u2082 : Tendsto f atTop (nhds (sInf (measureOfNegatives s)))\nB : \u2115 \u2192 Set \u03b1\nhB : \u2200 (n : \u2115), B n \u2208 {B | MeasurableSet B \u2227 restrict s B \u2264 restrict 0 B} \u2227 \u2191s (B n) = f n\nhB\u2081 : \u2200 (n : \u2115), MeasurableSet (B n)\nhB\u2082 : \u2200 (n : \u2115), restrict s (B n) \u2264 restrict 0 (B n)\nA : Set \u03b1 := \u22c3 n, B n\nhA : A = \u22c3 n, B n\nhA\u2081 : MeasurableSet A\nhA\u2082 : restrict s A \u2264 restrict 0 A\nn : \u2115\n\u22a2 (fun x => \u2191s ((\u22c3 i, B i) \\ B n) + \u2191s (B n)) n \u2264 \u2191s (B n)\n\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : TopologicalSpace M\ninst\u271d : OrderedAddCommMonoid M\ns\u271d : SignedMeasure \u03b1\ni j : Set \u03b1\ns : SignedMeasure \u03b1\nf : \u2115 \u2192 \u211d\nleft\u271d : Antitone f\nhf\u2082 : Tendsto f atTop (nhds (sInf (measureOfNegatives s)))\nB : \u2115 \u2192 Set \u03b1\nhB : \u2200 (n : \u2115), B n \u2208 {B | MeasurableSet B \u2227 restrict s B \u2264 restrict 0 B} \u2227 \u2191s (B n) = f n\nhB\u2081 : \u2200 (n : \u2115), MeasurableSet (B n)\nhB\u2082 : \u2200 (n : \u2115), restrict s (B n) \u2264 restrict 0 (B n)\nA : Set \u03b1 := \u22c3 n, B n\nhA : A = \u22c3 n, B n\nhA\u2081 : MeasurableSet A\nhA\u2082 : restrict s A \u2264 restrict 0 A\nn : \u2115\n\u22a2 MeasurableSet ((\u22c3 i, B i) \\ B n)"}, {"tactic": "refine' add_le_of_nonpos_left _", "annotated_tactic": ["refine' <a>add_le_of_nonpos_left</a> _", [{"full_name": "add_le_of_nonpos_left", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [418, 15], "def_end_pos": [418, 36]}]], "state_before": "case a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : TopologicalSpace M\ninst\u271d : OrderedAddCommMonoid M\ns\u271d : SignedMeasure \u03b1\ni j : Set \u03b1\ns : SignedMeasure \u03b1\nf : \u2115 \u2192 \u211d\nleft\u271d : Antitone f\nhf\u2082 : Tendsto f atTop (nhds (sInf (measureOfNegatives s)))\nB : \u2115 \u2192 Set \u03b1\nhB : \u2200 (n : \u2115), B n \u2208 {B | MeasurableSet B \u2227 restrict s B \u2264 restrict 0 B} \u2227 \u2191s (B n) = f n\nhB\u2081 : \u2200 (n : \u2115), MeasurableSet (B n)\nhB\u2082 : \u2200 (n : \u2115), restrict s (B n) \u2264 restrict 0 (B n)\nA : Set \u03b1 := \u22c3 n, B n\nhA : A = \u22c3 n, B n\nhA\u2081 : MeasurableSet A\nhA\u2082 : restrict s A \u2264 restrict 0 A\nn : \u2115\n\u22a2 (fun x => \u2191s ((\u22c3 i, B i) \\ B n) + \u2191s (B n)) n \u2264 \u2191s (B n)", "state_after": "case a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : TopologicalSpace M\ninst\u271d : OrderedAddCommMonoid M\ns\u271d : SignedMeasure \u03b1\ni j : Set \u03b1\ns : SignedMeasure \u03b1\nf : \u2115 \u2192 \u211d\nleft\u271d : Antitone f\nhf\u2082 : Tendsto f atTop (nhds (sInf (measureOfNegatives s)))\nB : \u2115 \u2192 Set \u03b1\nhB : \u2200 (n : \u2115), B n \u2208 {B | MeasurableSet B \u2227 restrict s B \u2264 restrict 0 B} \u2227 \u2191s (B n) = f n\nhB\u2081 : \u2200 (n : \u2115), MeasurableSet (B n)\nhB\u2082 : \u2200 (n : \u2115), restrict s (B n) \u2264 restrict 0 (B n)\nA : Set \u03b1 := \u22c3 n, B n\nhA : A = \u22c3 n, B n\nhA\u2081 : MeasurableSet A\nhA\u2082 : restrict s A \u2264 restrict 0 A\nn : \u2115\n\u22a2 \u2191s ((\u22c3 i, B i) \\ B n) \u2264 0"}, {"tactic": "have : s \u2264[A] 0 :=\n  restrict_le_restrict_iUnion _ _ hB\u2081 fun m =>\n    let \u27e8_, h\u27e9 := (hB m).1\n    h", "annotated_tactic": ["have : s \u2264[A] 0 :=\n          <a>restrict_le_restrict_iUnion</a> _ _ hB\u2081 fun m =>\n            let \u27e8_, h\u27e9 := (hB m).1\n            h", [{"full_name": "MeasureTheory.VectorMeasure.restrict_le_restrict_iUnion", "def_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "def_pos": [935, 9], "def_end_pos": [935, 36]}]], "state_before": "case a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : TopologicalSpace M\ninst\u271d : OrderedAddCommMonoid M\ns\u271d : SignedMeasure \u03b1\ni j : Set \u03b1\ns : SignedMeasure \u03b1\nf : \u2115 \u2192 \u211d\nleft\u271d : Antitone f\nhf\u2082 : Tendsto f atTop (nhds (sInf (measureOfNegatives s)))\nB : \u2115 \u2192 Set \u03b1\nhB : \u2200 (n : \u2115), B n \u2208 {B | MeasurableSet B \u2227 restrict s B \u2264 restrict 0 B} \u2227 \u2191s (B n) = f n\nhB\u2081 : \u2200 (n : \u2115), MeasurableSet (B n)\nhB\u2082 : \u2200 (n : \u2115), restrict s (B n) \u2264 restrict 0 (B n)\nA : Set \u03b1 := \u22c3 n, B n\nhA : A = \u22c3 n, B n\nhA\u2081 : MeasurableSet A\nhA\u2082 : restrict s A \u2264 restrict 0 A\nn : \u2115\n\u22a2 \u2191s ((\u22c3 i, B i) \\ B n) \u2264 0", "state_after": "case a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : TopologicalSpace M\ninst\u271d : OrderedAddCommMonoid M\ns\u271d : SignedMeasure \u03b1\ni j : Set \u03b1\ns : SignedMeasure \u03b1\nf : \u2115 \u2192 \u211d\nleft\u271d : Antitone f\nhf\u2082 : Tendsto f atTop (nhds (sInf (measureOfNegatives s)))\nB : \u2115 \u2192 Set \u03b1\nhB : \u2200 (n : \u2115), B n \u2208 {B | MeasurableSet B \u2227 restrict s B \u2264 restrict 0 B} \u2227 \u2191s (B n) = f n\nhB\u2081 : \u2200 (n : \u2115), MeasurableSet (B n)\nhB\u2082 : \u2200 (n : \u2115), restrict s (B n) \u2264 restrict 0 (B n)\nA : Set \u03b1 := \u22c3 n, B n\nhA : A = \u22c3 n, B n\nhA\u2081 : MeasurableSet A\nhA\u2082 : restrict s A \u2264 restrict 0 A\nn : \u2115\nthis : restrict s A \u2264 restrict 0 A\n\u22a2 \u2191s ((\u22c3 i, B i) \\ B n) \u2264 0"}, {"tactic": "refine'\n  nonpos_of_restrict_le_zero _ (restrict_le_zero_subset _ _ (Set.diff_subset _ _) this)", "annotated_tactic": ["refine'\n          <a>nonpos_of_restrict_le_zero</a> _ (<a>restrict_le_zero_subset</a> _ _ (<a>Set.diff_subset</a> _ _) this)", [{"full_name": "MeasureTheory.VectorMeasure.nonpos_of_restrict_le_zero", "def_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "def_pos": [992, 9], "def_end_pos": [992, 35]}, {"full_name": "MeasureTheory.VectorMeasure.restrict_le_zero_subset", "def_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "def_pos": [1019, 9], "def_end_pos": [1019, 32]}, {"full_name": "Set.diff_subset", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1845, 9], "def_end_pos": [1845, 20]}]], "state_before": "case a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : TopologicalSpace M\ninst\u271d : OrderedAddCommMonoid M\ns\u271d : SignedMeasure \u03b1\ni j : Set \u03b1\ns : SignedMeasure \u03b1\nf : \u2115 \u2192 \u211d\nleft\u271d : Antitone f\nhf\u2082 : Tendsto f atTop (nhds (sInf (measureOfNegatives s)))\nB : \u2115 \u2192 Set \u03b1\nhB : \u2200 (n : \u2115), B n \u2208 {B | MeasurableSet B \u2227 restrict s B \u2264 restrict 0 B} \u2227 \u2191s (B n) = f n\nhB\u2081 : \u2200 (n : \u2115), MeasurableSet (B n)\nhB\u2082 : \u2200 (n : \u2115), restrict s (B n) \u2264 restrict 0 (B n)\nA : Set \u03b1 := \u22c3 n, B n\nhA : A = \u22c3 n, B n\nhA\u2081 : MeasurableSet A\nhA\u2082 : restrict s A \u2264 restrict 0 A\nn : \u2115\nthis : restrict s A \u2264 restrict 0 A\n\u22a2 \u2191s ((\u22c3 i, B i) \\ B n) \u2264 0", "state_after": "case a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : TopologicalSpace M\ninst\u271d : OrderedAddCommMonoid M\ns\u271d : SignedMeasure \u03b1\ni j : Set \u03b1\ns : SignedMeasure \u03b1\nf : \u2115 \u2192 \u211d\nleft\u271d : Antitone f\nhf\u2082 : Tendsto f atTop (nhds (sInf (measureOfNegatives s)))\nB : \u2115 \u2192 Set \u03b1\nhB : \u2200 (n : \u2115), B n \u2208 {B | MeasurableSet B \u2227 restrict s B \u2264 restrict 0 B} \u2227 \u2191s (B n) = f n\nhB\u2081 : \u2200 (n : \u2115), MeasurableSet (B n)\nhB\u2082 : \u2200 (n : \u2115), restrict s (B n) \u2264 restrict 0 (B n)\nA : Set \u03b1 := \u22c3 n, B n\nhA : A = \u22c3 n, B n\nhA\u2081 : MeasurableSet A\nhA\u2082 : restrict s A \u2264 restrict 0 A\nn : \u2115\nthis : restrict s A \u2264 restrict 0 A\n\u22a2 MeasurableSet (\u22c3 i, B i)"}, {"tactic": "exact MeasurableSet.iUnion hB\u2081", "annotated_tactic": ["exact <a>MeasurableSet.iUnion</a> hB\u2081", [{"full_name": "MeasurableSet.iUnion", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [115, 19], "def_end_pos": [115, 39]}]], "state_before": "case a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : TopologicalSpace M\ninst\u271d : OrderedAddCommMonoid M\ns\u271d : SignedMeasure \u03b1\ni j : Set \u03b1\ns : SignedMeasure \u03b1\nf : \u2115 \u2192 \u211d\nleft\u271d : Antitone f\nhf\u2082 : Tendsto f atTop (nhds (sInf (measureOfNegatives s)))\nB : \u2115 \u2192 Set \u03b1\nhB : \u2200 (n : \u2115), B n \u2208 {B | MeasurableSet B \u2227 restrict s B \u2264 restrict 0 B} \u2227 \u2191s (B n) = f n\nhB\u2081 : \u2200 (n : \u2115), MeasurableSet (B n)\nhB\u2082 : \u2200 (n : \u2115), restrict s (B n) \u2264 restrict 0 (B n)\nA : Set \u03b1 := \u22c3 n, B n\nhA : A = \u22c3 n, B n\nhA\u2081 : MeasurableSet A\nhA\u2082 : restrict s A \u2264 restrict 0 A\nn : \u2115\nthis : restrict s A \u2264 restrict 0 A\n\u22a2 MeasurableSet (\u22c3 i, B i)", "state_after": "no goals"}, {"tactic": "exact (MeasurableSet.iUnion hB\u2081).diff (hB\u2081 n)", "annotated_tactic": ["exact (<a>MeasurableSet.iUnion</a> hB\u2081).<a>diff</a> (hB\u2081 n)", [{"full_name": "MeasurableSet.iUnion", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [115, 19], "def_end_pos": [115, 39]}, {"full_name": "MeasurableSet.diff", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [205, 19], "def_end_pos": [205, 37]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : TopologicalSpace M\ninst\u271d : OrderedAddCommMonoid M\ns\u271d : SignedMeasure \u03b1\ni j : Set \u03b1\ns : SignedMeasure \u03b1\nf : \u2115 \u2192 \u211d\nleft\u271d : Antitone f\nhf\u2082 : Tendsto f atTop (nhds (sInf (measureOfNegatives s)))\nB : \u2115 \u2192 Set \u03b1\nhB : \u2200 (n : \u2115), B n \u2208 {B | MeasurableSet B \u2227 restrict s B \u2264 restrict 0 B} \u2227 \u2191s (B n) = f n\nhB\u2081 : \u2200 (n : \u2115), MeasurableSet (B n)\nhB\u2082 : \u2200 (n : \u2115), restrict s (B n) \u2264 restrict 0 (B n)\nA : Set \u03b1 := \u22c3 n, B n\nhA : A = \u22c3 n, B n\nhA\u2081 : MeasurableSet A\nhA\u2082 : restrict s A \u2264 restrict 0 A\nn : \u2115\n\u22a2 MeasurableSet ((\u22c3 i, B i) \\ B n)", "state_after": "no goals"}, {"tactic": "exact csInf_le bddBelow_measureOfNegatives \u27e8A, \u27e8hA\u2081, hA\u2082\u27e9, rfl\u27e9", "annotated_tactic": ["exact <a>csInf_le</a> <a>bddBelow_measureOfNegatives</a> \u27e8A, \u27e8hA\u2081, hA\u2082\u27e9, <a>rfl</a>\u27e9", [{"full_name": "csInf_le", "def_path": "Mathlib/Order/ConditionallyCompleteLattice/Basic.lean", "def_pos": [465, 9], "def_end_pos": [465, 17]}, {"full_name": "MeasureTheory.SignedMeasure.bddBelow_measureOfNegatives", "def_path": "Mathlib/MeasureTheory/Decomposition/SignedHahn.lean", "def_pos": [343, 9], "def_end_pos": [343, 36]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "case a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : TopologicalSpace M\ninst\u271d : OrderedAddCommMonoid M\ns\u271d : SignedMeasure \u03b1\ni j : Set \u03b1\ns : SignedMeasure \u03b1\nf : \u2115 \u2192 \u211d\nleft\u271d : Antitone f\nhf\u2082 : Tendsto f atTop (nhds (sInf (measureOfNegatives s)))\nB : \u2115 \u2192 Set \u03b1\nhB : \u2200 (n : \u2115), B n \u2208 {B | MeasurableSet B \u2227 restrict s B \u2264 restrict 0 B} \u2227 \u2191s (B n) = f n\nhB\u2081 : \u2200 (n : \u2115), MeasurableSet (B n)\nhB\u2082 : \u2200 (n : \u2115), restrict s (B n) \u2264 restrict 0 (B n)\nA : Set \u03b1 := \u22c3 n, B n\nhA : A = \u22c3 n, B n\nhA\u2081 : MeasurableSet A\nhA\u2082 : restrict s A \u2264 restrict 0 A\n\u22a2 sInf (measureOfNegatives s) \u2264 \u2191s A", "state_after": "no goals"}, {"tactic": "rw [\u2190 hA\u2083,\n  of_union (Set.disjoint_of_subset_right (Set.Subset.trans hD hC\u2081) disjoint_compl_right) hA\u2081\n    hD\u2081]", "annotated_tactic": ["rw [\u2190 hA\u2083,\n      <a>of_union</a> (<a>Set.disjoint_of_subset_right</a> (<a>Set.Subset.trans</a> hD hC\u2081) <a>disjoint_compl_right</a>) hA\u2081\n        hD\u2081]", [{"full_name": "MeasureTheory.VectorMeasure.of_union", "def_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "def_pos": [185, 9], "def_end_pos": [185, 17]}, {"full_name": "Set.disjoint_of_subset_right", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1576, 7], "def_end_pos": [1576, 31]}, {"full_name": "Set.Subset.trans", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [362, 9], "def_end_pos": [362, 21]}, {"full_name": "disjoint_compl_right", "def_path": "Mathlib/Order/Heyting/Basic.lean", "def_pos": [844, 9], "def_end_pos": [844, 29]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : TopologicalSpace M\ninst\u271d : OrderedAddCommMonoid M\ns\u271d : SignedMeasure \u03b1\ni j : Set \u03b1\ns : SignedMeasure \u03b1\nf : \u2115 \u2192 \u211d\nleft\u271d : Antitone f\nhf\u2082 : Tendsto f atTop (nhds (sInf (measureOfNegatives s)))\nB : \u2115 \u2192 Set \u03b1\nhB : \u2200 (n : \u2115), B n \u2208 {B | MeasurableSet B \u2227 restrict s B \u2264 restrict 0 B} \u2227 \u2191s (B n) = f n\nhB\u2081 : \u2200 (n : \u2115), MeasurableSet (B n)\nhB\u2082 : \u2200 (n : \u2115), restrict s (B n) \u2264 restrict 0 (B n)\nA : Set \u03b1 := \u22c3 n, B n\nhA : A = \u22c3 n, B n\nhA\u2081 : MeasurableSet A\nhA\u2082 : restrict s A \u2264 restrict 0 A\nhA\u2083 : \u2191s A = sInf (measureOfNegatives s)\nC : Set \u03b1\na\u271d : MeasurableSet C\nhC\u2081 : C \u2286 A\u1d9c\nhC\u2082 : \u2191s C < \u21910 C\nD : Set \u03b1\nhD\u2081 : MeasurableSet D\nhD : D \u2286 C\nhD\u2082 : restrict s D \u2264 restrict 0 D\nhD\u2083 : \u2191s D < 0\n\u22a2 \u2191s (A \u222a D) < sInf (measureOfNegatives s)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : TopologicalSpace M\ninst\u271d : OrderedAddCommMonoid M\ns\u271d : SignedMeasure \u03b1\ni j : Set \u03b1\ns : SignedMeasure \u03b1\nf : \u2115 \u2192 \u211d\nleft\u271d : Antitone f\nhf\u2082 : Tendsto f atTop (nhds (sInf (measureOfNegatives s)))\nB : \u2115 \u2192 Set \u03b1\nhB : \u2200 (n : \u2115), B n \u2208 {B | MeasurableSet B \u2227 restrict s B \u2264 restrict 0 B} \u2227 \u2191s (B n) = f n\nhB\u2081 : \u2200 (n : \u2115), MeasurableSet (B n)\nhB\u2082 : \u2200 (n : \u2115), restrict s (B n) \u2264 restrict 0 (B n)\nA : Set \u03b1 := \u22c3 n, B n\nhA : A = \u22c3 n, B n\nhA\u2081 : MeasurableSet A\nhA\u2082 : restrict s A \u2264 restrict 0 A\nhA\u2083 : \u2191s A = sInf (measureOfNegatives s)\nC : Set \u03b1\na\u271d : MeasurableSet C\nhC\u2081 : C \u2286 A\u1d9c\nhC\u2082 : \u2191s C < \u21910 C\nD : Set \u03b1\nhD\u2081 : MeasurableSet D\nhD : D \u2286 C\nhD\u2082 : restrict s D \u2264 restrict 0 D\nhD\u2083 : \u2191s D < 0\n\u22a2 \u2191s A + \u2191s D < \u2191s A"}, {"tactic": "linarith", "annotated_tactic": ["linarith", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : TopologicalSpace M\ninst\u271d : OrderedAddCommMonoid M\ns\u271d : SignedMeasure \u03b1\ni j : Set \u03b1\ns : SignedMeasure \u03b1\nf : \u2115 \u2192 \u211d\nleft\u271d : Antitone f\nhf\u2082 : Tendsto f atTop (nhds (sInf (measureOfNegatives s)))\nB : \u2115 \u2192 Set \u03b1\nhB : \u2200 (n : \u2115), B n \u2208 {B | MeasurableSet B \u2227 restrict s B \u2264 restrict 0 B} \u2227 \u2191s (B n) = f n\nhB\u2081 : \u2200 (n : \u2115), MeasurableSet (B n)\nhB\u2082 : \u2200 (n : \u2115), restrict s (B n) \u2264 restrict 0 (B n)\nA : Set \u03b1 := \u22c3 n, B n\nhA : A = \u22c3 n, B n\nhA\u2081 : MeasurableSet A\nhA\u2082 : restrict s A \u2264 restrict 0 A\nhA\u2083 : \u2191s A = sInf (measureOfNegatives s)\nC : Set \u03b1\na\u271d : MeasurableSet C\nhC\u2081 : C \u2286 A\u1d9c\nhC\u2082 : \u2191s C < \u21910 C\nD : Set \u03b1\nhD\u2081 : MeasurableSet D\nhD : D \u2286 C\nhD\u2082 : restrict s D \u2264 restrict 0 D\nhD\u2083 : \u2191s D < 0\n\u22a2 \u2191s A + \u2191s D < \u2191s A", "state_after": "no goals"}, {"tactic": "exact hA\u2081.union hD\u2081", "annotated_tactic": ["exact hA\u2081.union hD\u2081", []], "state_before": "case intro.intro.intro.intro.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : TopologicalSpace M\ninst\u271d : OrderedAddCommMonoid M\ns\u271d : SignedMeasure \u03b1\ni j : Set \u03b1\ns : SignedMeasure \u03b1\nf : \u2115 \u2192 \u211d\nleft\u271d : Antitone f\nhf\u2082 : Tendsto f atTop (nhds (sInf (measureOfNegatives s)))\nB : \u2115 \u2192 Set \u03b1\nhB : \u2200 (n : \u2115), B n \u2208 {B | MeasurableSet B \u2227 restrict s B \u2264 restrict 0 B} \u2227 \u2191s (B n) = f n\nhB\u2081 : \u2200 (n : \u2115), MeasurableSet (B n)\nhB\u2082 : \u2200 (n : \u2115), restrict s (B n) \u2264 restrict 0 (B n)\nA : Set \u03b1 := \u22c3 n, B n\nhA : A = \u22c3 n, B n\nhA\u2081 : MeasurableSet A\nhA\u2082 : restrict s A \u2264 restrict 0 A\nhA\u2083 : \u2191s A = sInf (measureOfNegatives s)\nC : Set \u03b1\na\u271d : MeasurableSet C\nhC\u2081 : C \u2286 A\u1d9c\nhC\u2082 : \u2191s C < \u21910 C\nD : Set \u03b1\nhD\u2081 : MeasurableSet D\nhD : D \u2286 C\nhD\u2082 : restrict s D \u2264 restrict 0 D\nhD\u2083 : \u2191s D < 0\nthis : \u2191s (A \u222a D) < sInf (measureOfNegatives s)\n\u22a2 MeasurableSet (A \u222a D)", "state_after": "no goals"}, {"tactic": "exact restrict_le_restrict_union _ _ hA\u2081 hA\u2082 hD\u2081 hD\u2082", "annotated_tactic": ["exact <a>restrict_le_restrict_union</a> _ _ hA\u2081 hA\u2082 hD\u2081 hD\u2082", [{"full_name": "MeasureTheory.VectorMeasure.restrict_le_restrict_union", "def_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "def_pos": [970, 9], "def_end_pos": [970, 35]}]], "state_before": "case intro.intro.intro.intro.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : TopologicalSpace M\ninst\u271d : OrderedAddCommMonoid M\ns\u271d : SignedMeasure \u03b1\ni j : Set \u03b1\ns : SignedMeasure \u03b1\nf : \u2115 \u2192 \u211d\nleft\u271d : Antitone f\nhf\u2082 : Tendsto f atTop (nhds (sInf (measureOfNegatives s)))\nB : \u2115 \u2192 Set \u03b1\nhB : \u2200 (n : \u2115), B n \u2208 {B | MeasurableSet B \u2227 restrict s B \u2264 restrict 0 B} \u2227 \u2191s (B n) = f n\nhB\u2081 : \u2200 (n : \u2115), MeasurableSet (B n)\nhB\u2082 : \u2200 (n : \u2115), restrict s (B n) \u2264 restrict 0 (B n)\nA : Set \u03b1 := \u22c3 n, B n\nhA : A = \u22c3 n, B n\nhA\u2081 : MeasurableSet A\nhA\u2082 : restrict s A \u2264 restrict 0 A\nhA\u2083 : \u2191s A = sInf (measureOfNegatives s)\nC : Set \u03b1\na\u271d : MeasurableSet C\nhC\u2081 : C \u2286 A\u1d9c\nhC\u2082 : \u2191s C < \u21910 C\nD : Set \u03b1\nhD\u2081 : MeasurableSet D\nhD : D \u2286 C\nhD\u2082 : restrict s D \u2264 restrict 0 D\nhD\u2083 : \u2191s D < 0\nthis : \u2191s (A \u222a D) < sInf (measureOfNegatives s)\n\u22a2 restrict s (A \u222a D) \u2264 restrict 0 (A \u222a D)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Functor.lean", "full_name": "Finset.id_traverse", "start": [200, 1], "end": [202, 23], "traced_tactics": [{"tactic": "rw [traverse, Multiset.id_traverse]", "annotated_tactic": ["rw [<a>traverse</a>, <a>Multiset.id_traverse</a>]", [{"full_name": "Finset.traverse", "def_path": "Mathlib/Data/Finset/Functor.lean", "def_pos": [195, 5], "def_end_pos": [195, 13]}, {"full_name": "Multiset.id_traverse", "def_path": "Mathlib/Data/Multiset/Functor.lean", "def_pos": [100, 9], "def_end_pos": [100, 20]}]], "state_before": "\u03b1 \u03b2 \u03b3 : Type u\nF G : Type u \u2192 Type u\ninst\u271d\u2074 : Applicative F\ninst\u271d\u00b3 : Applicative G\ninst\u271d\u00b2 : CommApplicative F\ninst\u271d\u00b9 : CommApplicative G\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\n\u22a2 traverse pure s = s", "state_after": "\u03b1 \u03b2 \u03b3 : Type u\nF G : Type u \u2192 Type u\ninst\u271d\u2074 : Applicative F\ninst\u271d\u00b3 : Applicative G\ninst\u271d\u00b2 : CommApplicative F\ninst\u271d\u00b9 : CommApplicative G\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\n\u22a2 Multiset.toFinset <$> s.val = s"}, {"tactic": "exact s.val_toFinset", "annotated_tactic": ["exact s.val_toFinset", []], "state_before": "\u03b1 \u03b2 \u03b3 : Type u\nF G : Type u \u2192 Type u\ninst\u271d\u2074 : Applicative F\ninst\u271d\u00b3 : Applicative G\ninst\u271d\u00b2 : CommApplicative F\ninst\u271d\u00b9 : CommApplicative G\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\n\u22a2 Multiset.toFinset <$> s.val = s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Kernel/CondCdf.lean", "full_name": "ProbabilityTheory.set_lintegral_iInf_gt_preCdf", "start": [329, 1], "end": [346, 64], "traced_tactics": [{"tactic": "refine' le_antisymm _ _", "annotated_tactic": ["refine' <a>le_antisymm</a> _ _", [{"full_name": "le_antisymm", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [188, 9], "def_end_pos": [188, 20]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nt : \u211a\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u222b\u207b (x : \u03b1) in s, \u2a05 r, preCdf \u03c1 (\u2191r) x \u2202Measure.fst \u03c1 = \u2191\u2191(Measure.IicSnd \u03c1 \u2191t) s", "state_after": "case refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nt : \u211a\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u222b\u207b (x : \u03b1) in s, \u2a05 r, preCdf \u03c1 (\u2191r) x \u2202Measure.fst \u03c1 \u2264 \u2191\u2191(Measure.IicSnd \u03c1 \u2191t) s\n\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nt : \u211a\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(Measure.IicSnd \u03c1 \u2191t) s \u2264 \u222b\u207b (x : \u03b1) in s, \u2a05 r, preCdf \u03c1 (\u2191r) x \u2202Measure.fst \u03c1"}, {"tactic": "calc\n  \u222b\u207b x in s, \u2a05 r : Ioi t, preCdf \u03c1 r x \u2202\u03c1.fst \u2264 \u2a05 q : Ioi t, \u03c1.IicSnd q s := le_iInf h\n  _ = \u03c1.IicSnd t s := Measure.iInf_IicSnd_gt \u03c1 t hs", "annotated_tactic": ["calc\n      \u222b\u207b x in s, \u2a05 r : <a>Ioi</a> t, <a>preCdf</a> \u03c1 r x \u2202\u03c1.fst \u2264 \u2a05 q : <a>Ioi</a> t, \u03c1.IicSnd q s := <a>le_iInf</a> h\n      _ = \u03c1.IicSnd t s := <a>Measure.iInf_IicSnd_gt</a> \u03c1 t hs", [{"full_name": "Set.Ioi", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [79, 5], "def_end_pos": [79, 8]}, {"full_name": "ProbabilityTheory.preCdf", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [294, 19], "def_end_pos": [294, 25]}, {"full_name": "Set.Ioi", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [79, 5], "def_end_pos": [79, 8]}, {"full_name": "le_iInf", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [879, 9], "def_end_pos": [879, 16]}, {"full_name": "MeasureTheory.Measure.iInf_IicSnd_gt", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [216, 9], "def_end_pos": [216, 23]}]], "state_before": "case refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nt : \u211a\ns : Set \u03b1\nhs : MeasurableSet s\nh : \u2200 (q : \u2191(Ioi t)), \u222b\u207b (x : \u03b1) in s, \u2a05 r, preCdf \u03c1 (\u2191r) x \u2202Measure.fst \u03c1 \u2264 \u2191\u2191(Measure.IicSnd \u03c1 \u2191\u2191q) s\n\u22a2 \u222b\u207b (x : \u03b1) in s, \u2a05 r, preCdf \u03c1 (\u2191r) x \u2202Measure.fst \u03c1 \u2264 \u2191\u2191(Measure.IicSnd \u03c1 \u2191t) s", "state_after": "no goals"}, {"tactic": "intro q", "annotated_tactic": ["intro q", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nt : \u211a\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2200 (q : \u2191(Ioi t)), \u222b\u207b (x : \u03b1) in s, \u2a05 r, preCdf \u03c1 (\u2191r) x \u2202Measure.fst \u03c1 \u2264 \u2191\u2191(Measure.IicSnd \u03c1 \u2191\u2191q) s", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nt : \u211a\ns : Set \u03b1\nhs : MeasurableSet s\nq : \u2191(Ioi t)\n\u22a2 \u222b\u207b (x : \u03b1) in s, \u2a05 r, preCdf \u03c1 (\u2191r) x \u2202Measure.fst \u03c1 \u2264 \u2191\u2191(Measure.IicSnd \u03c1 \u2191\u2191q) s"}, {"tactic": "rw [\u2190 set_lintegral_preCdf_fst \u03c1 _ hs]", "annotated_tactic": ["rw [\u2190 <a>set_lintegral_preCdf_fst</a> \u03c1 _ hs]", [{"full_name": "ProbabilityTheory.set_lintegral_preCdf_fst", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [307, 9], "def_end_pos": [307, 33]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nt : \u211a\ns : Set \u03b1\nhs : MeasurableSet s\nq : \u2191(Ioi t)\n\u22a2 \u222b\u207b (x : \u03b1) in s, \u2a05 r, preCdf \u03c1 (\u2191r) x \u2202Measure.fst \u03c1 \u2264 \u2191\u2191(Measure.IicSnd \u03c1 \u2191\u2191q) s", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nt : \u211a\ns : Set \u03b1\nhs : MeasurableSet s\nq : \u2191(Ioi t)\n\u22a2 \u222b\u207b (x : \u03b1) in s, \u2a05 r, preCdf \u03c1 (\u2191r) x \u2202Measure.fst \u03c1 \u2264 \u222b\u207b (x : \u03b1) in s, preCdf \u03c1 (\u2191q) x \u2202Measure.fst \u03c1"}, {"tactic": "refine' set_lintegral_mono_ae _ measurable_preCdf _", "annotated_tactic": ["refine' <a>set_lintegral_mono_ae</a> _ <a>measurable_preCdf</a> _", [{"full_name": "MeasureTheory.set_lintegral_mono_ae", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [282, 9], "def_end_pos": [282, 30]}, {"full_name": "ProbabilityTheory.measurable_preCdf", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [298, 9], "def_end_pos": [298, 26]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nt : \u211a\ns : Set \u03b1\nhs : MeasurableSet s\nq : \u2191(Ioi t)\n\u22a2 \u222b\u207b (x : \u03b1) in s, \u2a05 r, preCdf \u03c1 (\u2191r) x \u2202Measure.fst \u03c1 \u2264 \u222b\u207b (x : \u03b1) in s, preCdf \u03c1 (\u2191q) x \u2202Measure.fst \u03c1", "state_after": "case refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nt : \u211a\ns : Set \u03b1\nhs : MeasurableSet s\nq : \u2191(Ioi t)\n\u22a2 Measurable fun x => \u2a05 r, preCdf \u03c1 (\u2191r) x\n\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nt : \u211a\ns : Set \u03b1\nhs : MeasurableSet s\nq : \u2191(Ioi t)\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1, x \u2208 s \u2192 \u2a05 r, preCdf \u03c1 (\u2191r) x \u2264 preCdf \u03c1 (\u2191q) x"}, {"tactic": "exact measurable_iInf fun _ => measurable_preCdf", "annotated_tactic": ["exact <a>measurable_iInf</a> fun _ => <a>measurable_preCdf</a>", [{"full_name": "measurable_iInf", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [1385, 9], "def_end_pos": [1385, 24]}, {"full_name": "ProbabilityTheory.measurable_preCdf", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [298, 9], "def_end_pos": [298, 26]}]], "state_before": "case refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nt : \u211a\ns : Set \u03b1\nhs : MeasurableSet s\nq : \u2191(Ioi t)\n\u22a2 Measurable fun x => \u2a05 r, preCdf \u03c1 (\u2191r) x", "state_after": "no goals"}, {"tactic": "filter_upwards [monotone_preCdf _] with a _", "annotated_tactic": ["filter_upwards [<a>monotone_preCdf</a> _] with a _", [{"full_name": "ProbabilityTheory.monotone_preCdf", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [318, 9], "def_end_pos": [318, 24]}]], "state_before": "case refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nt : \u211a\ns : Set \u03b1\nhs : MeasurableSet s\nq : \u2191(Ioi t)\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1, x \u2208 s \u2192 \u2a05 r, preCdf \u03c1 (\u2191r) x \u2264 preCdf \u03c1 (\u2191q) x", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nt : \u211a\ns : Set \u03b1\nhs : MeasurableSet s\nq : \u2191(Ioi t)\na : \u03b1\na\u271d : Monotone fun r => preCdf \u03c1 r a\n\u22a2 a \u2208 s \u2192 \u2a05 r, preCdf \u03c1 (\u2191r) a \u2264 preCdf \u03c1 (\u2191q) a"}, {"tactic": "exact fun _ => iInf_le _ q", "annotated_tactic": ["exact fun _ => <a>iInf_le</a> _ q", [{"full_name": "iInf_le", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [814, 9], "def_end_pos": [814, 16]}]], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nt : \u211a\ns : Set \u03b1\nhs : MeasurableSet s\nq : \u2191(Ioi t)\na : \u03b1\na\u271d : Monotone fun r => preCdf \u03c1 r a\n\u22a2 a \u2208 s \u2192 \u2a05 r, preCdf \u03c1 (\u2191r) a \u2264 preCdf \u03c1 (\u2191q) a", "state_after": "no goals"}, {"tactic": "rw [(set_lintegral_preCdf_fst \u03c1 t hs).symm]", "annotated_tactic": ["rw [(<a>set_lintegral_preCdf_fst</a> \u03c1 t hs).<a>symm</a>]", [{"full_name": "ProbabilityTheory.set_lintegral_preCdf_fst", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [307, 9], "def_end_pos": [307, 33]}, {"full_name": "Eq.symm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [310, 9], "def_end_pos": [310, 16]}]], "state_before": "case refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nt : \u211a\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(Measure.IicSnd \u03c1 \u2191t) s \u2264 \u222b\u207b (x : \u03b1) in s, \u2a05 r, preCdf \u03c1 (\u2191r) x \u2202Measure.fst \u03c1", "state_after": "case refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nt : \u211a\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u222b\u207b (x : \u03b1) in s, preCdf \u03c1 t x \u2202Measure.fst \u03c1 \u2264 \u222b\u207b (x : \u03b1) in s, \u2a05 r, preCdf \u03c1 (\u2191r) x \u2202Measure.fst \u03c1"}, {"tactic": "refine' set_lintegral_mono_ae measurable_preCdf _ _", "annotated_tactic": ["refine' <a>set_lintegral_mono_ae</a> <a>measurable_preCdf</a> _ _", [{"full_name": "MeasureTheory.set_lintegral_mono_ae", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [282, 9], "def_end_pos": [282, 30]}, {"full_name": "ProbabilityTheory.measurable_preCdf", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [298, 9], "def_end_pos": [298, 26]}]], "state_before": "case refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nt : \u211a\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u222b\u207b (x : \u03b1) in s, preCdf \u03c1 t x \u2202Measure.fst \u03c1 \u2264 \u222b\u207b (x : \u03b1) in s, \u2a05 r, preCdf \u03c1 (\u2191r) x \u2202Measure.fst \u03c1", "state_after": "case refine'_2.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nt : \u211a\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 Measurable fun x => \u2a05 r, preCdf \u03c1 (\u2191r) x\n\ncase refine'_2.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nt : \u211a\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1, x \u2208 s \u2192 preCdf \u03c1 t x \u2264 \u2a05 r, preCdf \u03c1 (\u2191r) x"}, {"tactic": "exact measurable_iInf fun _ => measurable_preCdf", "annotated_tactic": ["exact <a>measurable_iInf</a> fun _ => <a>measurable_preCdf</a>", [{"full_name": "measurable_iInf", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [1385, 9], "def_end_pos": [1385, 24]}, {"full_name": "ProbabilityTheory.measurable_preCdf", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [298, 9], "def_end_pos": [298, 26]}]], "state_before": "case refine'_2.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nt : \u211a\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 Measurable fun x => \u2a05 r, preCdf \u03c1 (\u2191r) x", "state_after": "no goals"}, {"tactic": "filter_upwards [monotone_preCdf _] with a ha_mono", "annotated_tactic": ["filter_upwards [<a>monotone_preCdf</a> _] with a ha_mono", [{"full_name": "ProbabilityTheory.monotone_preCdf", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [318, 9], "def_end_pos": [318, 24]}]], "state_before": "case refine'_2.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nt : \u211a\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1, x \u2208 s \u2192 preCdf \u03c1 t x \u2264 \u2a05 r, preCdf \u03c1 (\u2191r) x", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nt : \u211a\ns : Set \u03b1\nhs : MeasurableSet s\na : \u03b1\nha_mono : Monotone fun r => preCdf \u03c1 r a\n\u22a2 a \u2208 s \u2192 preCdf \u03c1 t a \u2264 \u2a05 r, preCdf \u03c1 (\u2191r) a"}, {"tactic": "exact fun _ => le_iInf fun r => ha_mono (le_of_lt r.prop)", "annotated_tactic": ["exact fun _ => <a>le_iInf</a> fun r => ha_mono (<a>le_of_lt</a> r.prop)", [{"full_name": "le_iInf", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [879, 9], "def_end_pos": [879, 16]}, {"full_name": "le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [110, 9], "def_end_pos": [110, 17]}]], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nt : \u211a\ns : Set \u03b1\nhs : MeasurableSet s\na : \u03b1\nha_mono : Monotone fun r => preCdf \u03c1 r a\n\u22a2 a \u2208 s \u2192 preCdf \u03c1 t a \u2264 \u2a05 r, preCdf \u03c1 (\u2191r) a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "full_name": "MeasureTheory.extend_mono", "start": [1556, 1], "end": [1564, 38], "traced_tactics": [{"tactic": "refine' le_iInf _", "annotated_tactic": ["refine' <a>le_iInf</a> _", [{"full_name": "le_iInf", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [879, 9], "def_end_pos": [879, 16]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nm : (s : Set \u03b1) \u2192 MeasurableSet s \u2192 \u211d\u22650\u221e\nm0 : m \u2205 (_ : MeasurableSet \u2205) = 0\nmU :\n  \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), MeasurableSet (f i)),\n    Pairwise (Disjoint on f) \u2192\n      m (\u22c3 i, f i) (_ : MeasurableSet (\u22c3 b, f b)) = \u2211' (i : \u2115), m (f i) (_ : MeasurableSet (f i))\ns\u2081 s\u2082 : Set \u03b1\nh\u2081 : MeasurableSet s\u2081\nhs : s\u2081 \u2286 s\u2082\n\u22a2 extend m s\u2081 \u2264 extend m s\u2082", "state_after": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nm : (s : Set \u03b1) \u2192 MeasurableSet s \u2192 \u211d\u22650\u221e\nm0 : m \u2205 (_ : MeasurableSet \u2205) = 0\nmU :\n  \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), MeasurableSet (f i)),\n    Pairwise (Disjoint on f) \u2192\n      m (\u22c3 i, f i) (_ : MeasurableSet (\u22c3 b, f b)) = \u2211' (i : \u2115), m (f i) (_ : MeasurableSet (f i))\ns\u2081 s\u2082 : Set \u03b1\nh\u2081 : MeasurableSet s\u2081\nhs : s\u2081 \u2286 s\u2082\n\u22a2 \u2200 (i : (fun s => MeasurableSet s) s\u2082), extend m s\u2081 \u2264 m s\u2082 i"}, {"tactic": "intro h\u2082", "annotated_tactic": ["intro h\u2082", []], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nm : (s : Set \u03b1) \u2192 MeasurableSet s \u2192 \u211d\u22650\u221e\nm0 : m \u2205 (_ : MeasurableSet \u2205) = 0\nmU :\n  \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), MeasurableSet (f i)),\n    Pairwise (Disjoint on f) \u2192\n      m (\u22c3 i, f i) (_ : MeasurableSet (\u22c3 b, f b)) = \u2211' (i : \u2115), m (f i) (_ : MeasurableSet (f i))\ns\u2081 s\u2082 : Set \u03b1\nh\u2081 : MeasurableSet s\u2081\nhs : s\u2081 \u2286 s\u2082\n\u22a2 \u2200 (i : (fun s => MeasurableSet s) s\u2082), extend m s\u2081 \u2264 m s\u2082 i", "state_after": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nm : (s : Set \u03b1) \u2192 MeasurableSet s \u2192 \u211d\u22650\u221e\nm0 : m \u2205 (_ : MeasurableSet \u2205) = 0\nmU :\n  \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), MeasurableSet (f i)),\n    Pairwise (Disjoint on f) \u2192\n      m (\u22c3 i, f i) (_ : MeasurableSet (\u22c3 b, f b)) = \u2211' (i : \u2115), m (f i) (_ : MeasurableSet (f i))\ns\u2081 s\u2082 : Set \u03b1\nh\u2081 : MeasurableSet s\u2081\nhs : s\u2081 \u2286 s\u2082\nh\u2082 : (fun s => MeasurableSet s) s\u2082\n\u22a2 extend m s\u2081 \u2264 m s\u2082 h\u2082"}, {"tactic": "have :=\n  extend_union MeasurableSet.empty m0 MeasurableSet.iUnion mU disjoint_sdiff_self_right h\u2081\n    (h\u2082.diff h\u2081)", "annotated_tactic": ["have :=\n    <a>extend_union</a> <a>MeasurableSet.empty</a> m0 <a>MeasurableSet.iUnion</a> mU <a>disjoint_sdiff_self_right</a> h\u2081\n      (h\u2082.diff h\u2081)", [{"full_name": "MeasureTheory.extend_union", "def_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "def_pos": [1422, 9], "def_end_pos": [1422, 21]}, {"full_name": "MeasurableSet.empty", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [80, 9], "def_end_pos": [80, 28]}, {"full_name": "MeasurableSet.iUnion", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [115, 19], "def_end_pos": [115, 39]}, {"full_name": "disjoint_sdiff_self_right", "def_path": "Mathlib/Order/BooleanAlgebra.lean", "def_pos": [213, 9], "def_end_pos": [213, 34]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nm : (s : Set \u03b1) \u2192 MeasurableSet s \u2192 \u211d\u22650\u221e\nm0 : m \u2205 (_ : MeasurableSet \u2205) = 0\nmU :\n  \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), MeasurableSet (f i)),\n    Pairwise (Disjoint on f) \u2192\n      m (\u22c3 i, f i) (_ : MeasurableSet (\u22c3 b, f b)) = \u2211' (i : \u2115), m (f i) (_ : MeasurableSet (f i))\ns\u2081 s\u2082 : Set \u03b1\nh\u2081 : MeasurableSet s\u2081\nhs : s\u2081 \u2286 s\u2082\nh\u2082 : (fun s => MeasurableSet s) s\u2082\n\u22a2 extend m s\u2081 \u2264 m s\u2082 h\u2082", "state_after": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nm : (s : Set \u03b1) \u2192 MeasurableSet s \u2192 \u211d\u22650\u221e\nm0 : m \u2205 (_ : MeasurableSet \u2205) = 0\nmU :\n  \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), MeasurableSet (f i)),\n    Pairwise (Disjoint on f) \u2192\n      m (\u22c3 i, f i) (_ : MeasurableSet (\u22c3 b, f b)) = \u2211' (i : \u2115), m (f i) (_ : MeasurableSet (f i))\ns\u2081 s\u2082 : Set \u03b1\nh\u2081 : MeasurableSet s\u2081\nhs : s\u2081 \u2286 s\u2082\nh\u2082 : (fun s => MeasurableSet s) s\u2082\nthis : extend m (s\u2081 \u222a s\u2082 \\ s\u2081) = extend m s\u2081 + extend m (s\u2082 \\ s\u2081)\n\u22a2 extend m s\u2081 \u2264 m s\u2082 h\u2082"}, {"tactic": "rw [union_diff_cancel hs] at this", "annotated_tactic": ["rw [<a>union_diff_cancel</a> hs] at this", [{"full_name": "Set.union_diff_cancel", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1853, 9], "def_end_pos": [1853, 26]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nm : (s : Set \u03b1) \u2192 MeasurableSet s \u2192 \u211d\u22650\u221e\nm0 : m \u2205 (_ : MeasurableSet \u2205) = 0\nmU :\n  \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), MeasurableSet (f i)),\n    Pairwise (Disjoint on f) \u2192\n      m (\u22c3 i, f i) (_ : MeasurableSet (\u22c3 b, f b)) = \u2211' (i : \u2115), m (f i) (_ : MeasurableSet (f i))\ns\u2081 s\u2082 : Set \u03b1\nh\u2081 : MeasurableSet s\u2081\nhs : s\u2081 \u2286 s\u2082\nh\u2082 : (fun s => MeasurableSet s) s\u2082\nthis : extend m (s\u2081 \u222a s\u2082 \\ s\u2081) = extend m s\u2081 + extend m (s\u2082 \\ s\u2081)\n\u22a2 extend m s\u2081 \u2264 m s\u2082 h\u2082", "state_after": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nm : (s : Set \u03b1) \u2192 MeasurableSet s \u2192 \u211d\u22650\u221e\nm0 : m \u2205 (_ : MeasurableSet \u2205) = 0\nmU :\n  \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), MeasurableSet (f i)),\n    Pairwise (Disjoint on f) \u2192\n      m (\u22c3 i, f i) (_ : MeasurableSet (\u22c3 b, f b)) = \u2211' (i : \u2115), m (f i) (_ : MeasurableSet (f i))\ns\u2081 s\u2082 : Set \u03b1\nh\u2081 : MeasurableSet s\u2081\nhs : s\u2081 \u2286 s\u2082\nh\u2082 : (fun s => MeasurableSet s) s\u2082\nthis : extend m s\u2082 = extend m s\u2081 + extend m (s\u2082 \\ s\u2081)\n\u22a2 extend m s\u2081 \u2264 m s\u2082 h\u2082"}, {"tactic": "rw [\u2190 extend_eq m]", "annotated_tactic": ["rw [\u2190 <a>extend_eq</a> m]", [{"full_name": "MeasureTheory.extend_eq", "def_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "def_pos": [1316, 9], "def_end_pos": [1316, 18]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nm : (s : Set \u03b1) \u2192 MeasurableSet s \u2192 \u211d\u22650\u221e\nm0 : m \u2205 (_ : MeasurableSet \u2205) = 0\nmU :\n  \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), MeasurableSet (f i)),\n    Pairwise (Disjoint on f) \u2192\n      m (\u22c3 i, f i) (_ : MeasurableSet (\u22c3 b, f b)) = \u2211' (i : \u2115), m (f i) (_ : MeasurableSet (f i))\ns\u2081 s\u2082 : Set \u03b1\nh\u2081 : MeasurableSet s\u2081\nhs : s\u2081 \u2286 s\u2082\nh\u2082 : (fun s => MeasurableSet s) s\u2082\nthis : extend m s\u2082 = extend m s\u2081 + extend m (s\u2082 \\ s\u2081)\n\u22a2 extend m s\u2081 \u2264 m s\u2082 h\u2082", "state_after": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nm : (s : Set \u03b1) \u2192 MeasurableSet s \u2192 \u211d\u22650\u221e\nm0 : m \u2205 (_ : MeasurableSet \u2205) = 0\nmU :\n  \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), MeasurableSet (f i)),\n    Pairwise (Disjoint on f) \u2192\n      m (\u22c3 i, f i) (_ : MeasurableSet (\u22c3 b, f b)) = \u2211' (i : \u2115), m (f i) (_ : MeasurableSet (f i))\ns\u2081 s\u2082 : Set \u03b1\nh\u2081 : MeasurableSet s\u2081\nhs : s\u2081 \u2286 s\u2082\nh\u2082 : (fun s => MeasurableSet s) s\u2082\nthis : extend m s\u2082 = extend m s\u2081 + extend m (s\u2082 \\ s\u2081)\n\u22a2 extend m s\u2081 \u2264 extend m s\u2082"}, {"tactic": "exact le_iff_exists_add.2 \u27e8_, this\u27e9", "annotated_tactic": ["exact <a>le_iff_exists_add</a>.2 \u27e8_, this\u27e9", [{"full_name": "le_iff_exists_add", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [203, 3], "def_end_pos": [203, 14]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nm : (s : Set \u03b1) \u2192 MeasurableSet s \u2192 \u211d\u22650\u221e\nm0 : m \u2205 (_ : MeasurableSet \u2205) = 0\nmU :\n  \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), MeasurableSet (f i)),\n    Pairwise (Disjoint on f) \u2192\n      m (\u22c3 i, f i) (_ : MeasurableSet (\u22c3 b, f b)) = \u2211' (i : \u2115), m (f i) (_ : MeasurableSet (f i))\ns\u2081 s\u2082 : Set \u03b1\nh\u2081 : MeasurableSet s\u2081\nhs : s\u2081 \u2286 s\u2082\nh\u2082 : (fun s => MeasurableSet s) s\u2082\nthis : extend m s\u2082 = extend m s\u2081 + extend m (s\u2082 \\ s\u2081)\n\u22a2 extend m s\u2081 \u2264 extend m s\u2082", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/ProbabilityMassFunction/Integrals.lean", "full_name": "PMF.bernoulli_expectation", "start": [51, 1], "end": [52, 89], "traced_tactics": [{"tactic": "simp [integral_eq_sum]", "annotated_tactic": ["simp [<a>integral_eq_sum</a>]", [{"full_name": "PMF.integral_eq_sum", "def_path": "Mathlib/Probability/ProbabilityMassFunction/Integrals.lean", "def_pos": [43, 9], "def_end_pos": [43, 24]}]], "state_before": "p : \u211d\u22650\u221e\nh : p \u2264 1\n\u22a2 \u222b (b : Bool), bif b then 1 else 0 \u2202toMeasure (bernoulli p h) = ENNReal.toReal p", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Constructions/Pi.lean", "full_name": "generateFrom_eq_pi", "start": [133, 1], "end": [136, 43], "traced_tactics": [{"tactic": "rw [\u2190 funext hC, generateFrom_pi_eq h2C]", "annotated_tactic": ["rw [\u2190 <a>funext</a> hC, <a>generateFrom_pi_eq</a> h2C]", [{"full_name": "funext", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [1555, 9], "def_end_pos": [1555, 15]}, {"full_name": "generateFrom_pi_eq", "def_path": "Mathlib/MeasureTheory/Constructions/Pi.lean", "def_pos": [103, 9], "def_end_pos": [103, 27]}]], "state_before": "\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : \u03b9 \u2192 Type u_3\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : Finite \u03b9'\nh : (i : \u03b9) \u2192 MeasurableSpace (\u03b1 i)\nC : (i : \u03b9) \u2192 Set (Set (\u03b1 i))\nhC : \u2200 (i : \u03b9), generateFrom (C i) = h i\nh2C : \u2200 (i : \u03b9), IsCountablySpanning (C i)\n\u22a2 generateFrom (Set.pi univ '' Set.pi univ C) = MeasurableSpace.pi", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "full_name": "MeasureTheory.Lp.mem_Lp_of_nnnorm_ae_le_mul", "start": [393, 1], "end": [395, 85], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Fin/Lemmas.lean", "full_name": "Fin.pred_mk_succ", "start": [498, 1], "end": [501, 52], "traced_tactics": [{"tactic": "simp only [ext_iff, coe_pred, Nat.add_sub_cancel]", "annotated_tactic": ["simp only [<a>ext_iff</a>, <a>coe_pred</a>, <a>Nat.add_sub_cancel</a>]", [{"full_name": "Fin.ext_iff", "def_path": "lake-packages/std/Std/Data/Fin/Lemmas.lean", "def_pos": [38, 9], "def_end_pos": [38, 16]}, {"full_name": "Fin.coe_pred", "def_path": "lake-packages/std/Std/Data/Fin/Lemmas.lean", "def_pos": [483, 17], "def_end_pos": [483, 25]}, {"full_name": "Nat.add_sub_cancel", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [594, 19], "def_end_pos": [594, 33]}]], "state_before": "n i : Nat\nh : i < n + 1\n\u22a2 pred { val := i + 1, isLt := (_ : i + 1 < n + 1 + 1) } (_ : \u00ac{ val := i + 1, isLt := (_ : i + 1 < n + 1 + 1) } = 0) =\n    { val := i, isLt := h }", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "full_name": "MeasureTheory.OuterMeasure.smul_ofFunction", "start": [814, 1], "end": [820, 50], "traced_tactics": [{"tactic": "simp [m_empty]", "annotated_tactic": ["simp [m_empty]", []], "state_before": "\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\nc : \u211d\u22650\u221e\nhc : c \u2260 \u22a4\n\u22a2 (c \u2022 m) \u2205 = 0", "state_after": "no goals"}, {"tactic": "ext1 s", "annotated_tactic": ["ext1 s", []], "state_before": "\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\nc : \u211d\u22650\u221e\nhc : c \u2260 \u22a4\n\u22a2 c \u2022 OuterMeasure.ofFunction m m_empty = OuterMeasure.ofFunction (c \u2022 m) (_ : c \u2022 m \u2205 = 0)", "state_after": "case h\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\nc : \u211d\u22650\u221e\nhc : c \u2260 \u22a4\ns : Set \u03b1\n\u22a2 \u2191(c \u2022 OuterMeasure.ofFunction m m_empty) s = \u2191(OuterMeasure.ofFunction (c \u2022 m) (_ : c \u2022 m \u2205 = 0)) s"}, {"tactic": "haveI : Nonempty { t : \u2115 \u2192 Set \u03b1 // s \u2286 \u22c3 i, t i } := \u27e8\u27e8fun _ => s, subset_iUnion (fun _ => s) 0\u27e9\u27e9", "annotated_tactic": ["haveI : <a>Nonempty</a> { t : \u2115 \u2192 <a>Set</a> \u03b1 // s \u2286 \u22c3 i, t i } := \u27e8\u27e8fun _ => s, <a>subset_iUnion</a> (fun _ => s) 0\u27e9\u27e9", [{"full_name": "Nonempty", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [686, 17], "def_end_pos": [686, 25]}, {"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}, {"full_name": "Set.subset_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [431, 9], "def_end_pos": [431, 22]}]], "state_before": "case h\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\nc : \u211d\u22650\u221e\nhc : c \u2260 \u22a4\ns : Set \u03b1\n\u22a2 \u2191(c \u2022 OuterMeasure.ofFunction m m_empty) s = \u2191(OuterMeasure.ofFunction (c \u2022 m) (_ : c \u2022 m \u2205 = 0)) s", "state_after": "case h\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\nc : \u211d\u22650\u221e\nhc : c \u2260 \u22a4\ns : Set \u03b1\nthis : Nonempty { t // s \u2286 \u22c3 i, t i }\n\u22a2 \u2191(c \u2022 OuterMeasure.ofFunction m m_empty) s = \u2191(OuterMeasure.ofFunction (c \u2022 m) (_ : c \u2022 m \u2205 = 0)) s"}, {"tactic": "simp only [smul_apply, ofFunction_apply, ENNReal.tsum_mul_left, Pi.smul_apply, smul_eq_mul,\niInf_subtype']", "annotated_tactic": ["simp only [<a>smul_apply</a>, <a>ofFunction_apply</a>, <a>ENNReal.tsum_mul_left</a>, <a>Pi.smul_apply</a>, <a>smul_eq_mul</a>,\n  <a>iInf_subtype'</a>]", [{"full_name": "MeasureTheory.OuterMeasure.smul_apply", "def_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "def_pos": [310, 9], "def_end_pos": [310, 19]}, {"full_name": "MeasureTheory.OuterMeasure.ofFunction_apply", "def_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "def_pos": [691, 9], "def_end_pos": [691, 25]}, {"full_name": "ENNReal.tsum_mul_left", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [897, 19], "def_end_pos": [897, 32]}, {"full_name": "Pi.smul_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [116, 60], "def_end_pos": [116, 70]}, {"full_name": "smul_eq_mul", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [93, 9], "def_end_pos": [93, 20]}, {"full_name": "iInf_subtype'", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [1271, 9], "def_end_pos": [1271, 22]}]], "state_before": "case h\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\nc : \u211d\u22650\u221e\nhc : c \u2260 \u22a4\ns : Set \u03b1\nthis : Nonempty { t // s \u2286 \u22c3 i, t i }\n\u22a2 \u2191(c \u2022 OuterMeasure.ofFunction m m_empty) s = \u2191(OuterMeasure.ofFunction (c \u2022 m) (_ : c \u2022 m \u2205 = 0)) s", "state_after": "case h\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\nc : \u211d\u22650\u221e\nhc : c \u2260 \u22a4\ns : Set \u03b1\nthis : Nonempty { t // s \u2286 \u22c3 i, t i }\n\u22a2 c * \u2a05 x, \u2211' (n : \u2115), m (\u2191x n) = \u2a05 x, c * \u2211' (i : \u2115), m (\u2191x i)"}, {"tactic": "rw [ENNReal.iInf_mul_left fun h => (hc h).elim]", "annotated_tactic": ["rw [<a>ENNReal.iInf_mul_left</a> fun h => (hc h).<a>elim</a>]", [{"full_name": "ENNReal.iInf_mul_left", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [497, 9], "def_end_pos": [497, 22]}, {"full_name": "False.elim", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [223, 21], "def_end_pos": [223, 31]}]], "state_before": "case h\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\nc : \u211d\u22650\u221e\nhc : c \u2260 \u22a4\ns : Set \u03b1\nthis : Nonempty { t // s \u2286 \u22c3 i, t i }\n\u22a2 c * \u2a05 x, \u2211' (n : \u2115), m (\u2191x n) = \u2a05 x, c * \u2211' (i : \u2115), m (\u2191x i)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Halting.lean", "full_name": "ComputablePred.rice", "start": [197, 1], "end": [211, 18], "traced_tactics": [{"tactic": "cases' h with _ h", "annotated_tactic": ["cases' h with _ h", []], "state_before": "\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nC : Set (\u2115 \u2192. \u2115)\nh : ComputablePred fun c => eval c \u2208 C\nf g : \u2115 \u2192. \u2115\nhf : Nat.Partrec f\nhg : Nat.Partrec g\nfC : f \u2208 C\n\u22a2 g \u2208 C", "state_after": "case intro\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nC : Set (\u2115 \u2192. \u2115)\nf g : \u2115 \u2192. \u2115\nhf : Nat.Partrec f\nhg : Nat.Partrec g\nfC : f \u2208 C\nw\u271d : DecidablePred fun c => eval c \u2208 C\nh : Computable fun a => decide ((fun c => eval c \u2208 C) a)\n\u22a2 g \u2208 C"}, {"tactic": "obtain \u27e8c, e\u27e9 :=\n  fixed_point\u2082\n    (Partrec.cond (h.comp fst) ((Partrec.nat_iff.2 hg).comp snd).to\u2082\n        ((Partrec.nat_iff.2 hf).comp snd).to\u2082).to\u2082", "annotated_tactic": ["obtain \u27e8c, e\u27e9 :=\n    <a>fixed_point\u2082</a>\n      (<a>Partrec.cond</a> (h.comp <a>fst</a>) ((<a>Partrec.nat_iff</a>.2 hg).<a>comp</a> <a>snd</a>).<a>to\u2082</a>\n          ((<a>Partrec.nat_iff</a>.2 hf).<a>comp</a> <a>snd</a>).<a>to\u2082</a>).<a>to\u2082</a>", [{"full_name": "Nat.Partrec.Code.fixed_point\u2082", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [1188, 9], "def_end_pos": [1188, 21]}, {"full_name": "Partrec.cond", "def_path": "Mathlib/Computability/Halting.lean", "def_pos": [116, 9], "def_end_pos": [116, 13]}, {"full_name": "Computable.fst", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [302, 9], "def_end_pos": [302, 12]}, {"full_name": "Partrec.nat_iff", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [485, 9], "def_end_pos": [485, 16]}, {"full_name": "Partrec.comp", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [480, 16], "def_end_pos": [480, 20]}, {"full_name": "Computable.snd", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [306, 9], "def_end_pos": [306, 12]}, {"full_name": "Partrec.to\u2082", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [466, 9], "def_end_pos": [466, 12]}, {"full_name": "Partrec.nat_iff", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [485, 9], "def_end_pos": [485, 16]}, {"full_name": "Partrec.comp", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [480, 16], "def_end_pos": [480, 20]}, {"full_name": "Computable.snd", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [306, 9], "def_end_pos": [306, 12]}, {"full_name": "Partrec.to\u2082", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [466, 9], "def_end_pos": [466, 12]}, {"full_name": "Partrec.to\u2082", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [466, 9], "def_end_pos": [466, 12]}]], "state_before": "case intro\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nC : Set (\u2115 \u2192. \u2115)\nf g : \u2115 \u2192. \u2115\nhf : Nat.Partrec f\nhg : Nat.Partrec g\nfC : f \u2208 C\nw\u271d : DecidablePred fun c => eval c \u2208 C\nh : Computable fun a => decide ((fun c => eval c \u2208 C) a)\n\u22a2 g \u2208 C", "state_after": "case intro.intro\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nC : Set (\u2115 \u2192. \u2115)\nf g : \u2115 \u2192. \u2115\nhf : Nat.Partrec f\nhg : Nat.Partrec g\nfC : f \u2208 C\nw\u271d : DecidablePred fun c => eval c \u2208 C\nh : Computable fun a => decide ((fun c => eval c \u2208 C) a)\nc : Code\ne :\n  eval c = fun b =>\n    bif decide ((fun c => eval c \u2208 C) (c, b).1) then (fun a b => g (a, b).2) (c, b).1 (c, b).2\n    else (fun a b => f (a, b).2) (c, b).1 (c, b).2\n\u22a2 g \u2208 C"}, {"tactic": "simp at e", "annotated_tactic": ["simp at e", []], "state_before": "case intro.intro\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nC : Set (\u2115 \u2192. \u2115)\nf g : \u2115 \u2192. \u2115\nhf : Nat.Partrec f\nhg : Nat.Partrec g\nfC : f \u2208 C\nw\u271d : DecidablePred fun c => eval c \u2208 C\nh : Computable fun a => decide ((fun c => eval c \u2208 C) a)\nc : Code\ne :\n  eval c = fun b =>\n    bif decide ((fun c => eval c \u2208 C) (c, b).1) then (fun a b => g (a, b).2) (c, b).1 (c, b).2\n    else (fun a b => f (a, b).2) (c, b).1 (c, b).2\n\u22a2 g \u2208 C", "state_after": "case intro.intro\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nC : Set (\u2115 \u2192. \u2115)\nf g : \u2115 \u2192. \u2115\nhf : Nat.Partrec f\nhg : Nat.Partrec g\nfC : f \u2208 C\nw\u271d : DecidablePred fun c => eval c \u2208 C\nh : Computable fun a => decide ((fun c => eval c \u2208 C) a)\nc : Code\ne : eval c = fun b => if eval c \u2208 C then g b else f b\n\u22a2 g \u2208 C"}, {"tactic": "by_cases H : eval c \u2208 C", "annotated_tactic": ["by_cases H : <a>eval</a> c \u2208 C", [{"full_name": "Nat.Partrec.Code.eval", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [620, 5], "def_end_pos": [620, 9]}]], "state_before": "case intro.intro\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nC : Set (\u2115 \u2192. \u2115)\nf g : \u2115 \u2192. \u2115\nhf : Nat.Partrec f\nhg : Nat.Partrec g\nfC : f \u2208 C\nw\u271d : DecidablePred fun c => eval c \u2208 C\nh : Computable fun a => decide ((fun c => eval c \u2208 C) a)\nc : Code\ne : eval c = fun b => if eval c \u2208 C then g b else f b\n\u22a2 g \u2208 C", "state_after": "case pos\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nC : Set (\u2115 \u2192. \u2115)\nf g : \u2115 \u2192. \u2115\nhf : Nat.Partrec f\nhg : Nat.Partrec g\nfC : f \u2208 C\nw\u271d : DecidablePred fun c => eval c \u2208 C\nh : Computable fun a => decide ((fun c => eval c \u2208 C) a)\nc : Code\ne : eval c = fun b => if eval c \u2208 C then g b else f b\nH : eval c \u2208 C\n\u22a2 g \u2208 C\n\ncase neg\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nC : Set (\u2115 \u2192. \u2115)\nf g : \u2115 \u2192. \u2115\nhf : Nat.Partrec f\nhg : Nat.Partrec g\nfC : f \u2208 C\nw\u271d : DecidablePred fun c => eval c \u2208 C\nh : Computable fun a => decide ((fun c => eval c \u2208 C) a)\nc : Code\ne : eval c = fun b => if eval c \u2208 C then g b else f b\nH : \u00aceval c \u2208 C\n\u22a2 g \u2208 C"}, {"tactic": "simp only [H, if_true] at e", "annotated_tactic": ["simp only [H, <a>if_true</a>] at e", [{"full_name": "if_true", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [727, 17], "def_end_pos": [727, 24]}]], "state_before": "case pos\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nC : Set (\u2115 \u2192. \u2115)\nf g : \u2115 \u2192. \u2115\nhf : Nat.Partrec f\nhg : Nat.Partrec g\nfC : f \u2208 C\nw\u271d : DecidablePred fun c => eval c \u2208 C\nh : Computable fun a => decide ((fun c => eval c \u2208 C) a)\nc : Code\ne : eval c = fun b => if eval c \u2208 C then g b else f b\nH : eval c \u2208 C\n\u22a2 g \u2208 C", "state_after": "case pos\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nC : Set (\u2115 \u2192. \u2115)\nf g : \u2115 \u2192. \u2115\nhf : Nat.Partrec f\nhg : Nat.Partrec g\nfC : f \u2208 C\nw\u271d : DecidablePred fun c => eval c \u2208 C\nh : Computable fun a => decide ((fun c => eval c \u2208 C) a)\nc : Code\nH : eval c \u2208 C\ne : eval c = fun b => g b\n\u22a2 g \u2208 C"}, {"tactic": "change (fun b => g b) \u2208 C", "annotated_tactic": ["change (fun b => g b) \u2208 C", []], "state_before": "case pos\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nC : Set (\u2115 \u2192. \u2115)\nf g : \u2115 \u2192. \u2115\nhf : Nat.Partrec f\nhg : Nat.Partrec g\nfC : f \u2208 C\nw\u271d : DecidablePred fun c => eval c \u2208 C\nh : Computable fun a => decide ((fun c => eval c \u2208 C) a)\nc : Code\nH : eval c \u2208 C\ne : eval c = fun b => g b\n\u22a2 g \u2208 C", "state_after": "case pos\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nC : Set (\u2115 \u2192. \u2115)\nf g : \u2115 \u2192. \u2115\nhf : Nat.Partrec f\nhg : Nat.Partrec g\nfC : f \u2208 C\nw\u271d : DecidablePred fun c => eval c \u2208 C\nh : Computable fun a => decide ((fun c => eval c \u2208 C) a)\nc : Code\nH : eval c \u2208 C\ne : eval c = fun b => g b\n\u22a2 (fun b => g b) \u2208 C"}, {"tactic": "rwa [\u2190 e]", "annotated_tactic": ["rwa [\u2190 e]", []], "state_before": "case pos\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nC : Set (\u2115 \u2192. \u2115)\nf g : \u2115 \u2192. \u2115\nhf : Nat.Partrec f\nhg : Nat.Partrec g\nfC : f \u2208 C\nw\u271d : DecidablePred fun c => eval c \u2208 C\nh : Computable fun a => decide ((fun c => eval c \u2208 C) a)\nc : Code\nH : eval c \u2208 C\ne : eval c = fun b => g b\n\u22a2 (fun b => g b) \u2208 C", "state_after": "no goals"}, {"tactic": "simp only [H, if_false] at e", "annotated_tactic": ["simp only [H, <a>if_false</a>] at e", [{"full_name": "if_false", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [729, 17], "def_end_pos": [729, 25]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nC : Set (\u2115 \u2192. \u2115)\nf g : \u2115 \u2192. \u2115\nhf : Nat.Partrec f\nhg : Nat.Partrec g\nfC : f \u2208 C\nw\u271d : DecidablePred fun c => eval c \u2208 C\nh : Computable fun a => decide ((fun c => eval c \u2208 C) a)\nc : Code\ne : eval c = fun b => if eval c \u2208 C then g b else f b\nH : \u00aceval c \u2208 C\n\u22a2 g \u2208 C", "state_after": "case neg\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nC : Set (\u2115 \u2192. \u2115)\nf g : \u2115 \u2192. \u2115\nhf : Nat.Partrec f\nhg : Nat.Partrec g\nfC : f \u2208 C\nw\u271d : DecidablePred fun c => eval c \u2208 C\nh : Computable fun a => decide ((fun c => eval c \u2208 C) a)\nc : Code\nH : \u00aceval c \u2208 C\ne : eval c = fun b => f b\n\u22a2 g \u2208 C"}, {"tactic": "rw [e] at H", "annotated_tactic": ["rw [e] at H", []], "state_before": "case neg\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nC : Set (\u2115 \u2192. \u2115)\nf g : \u2115 \u2192. \u2115\nhf : Nat.Partrec f\nhg : Nat.Partrec g\nfC : f \u2208 C\nw\u271d : DecidablePred fun c => eval c \u2208 C\nh : Computable fun a => decide ((fun c => eval c \u2208 C) a)\nc : Code\nH : \u00aceval c \u2208 C\ne : eval c = fun b => f b\n\u22a2 g \u2208 C", "state_after": "case neg\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nC : Set (\u2115 \u2192. \u2115)\nf g : \u2115 \u2192. \u2115\nhf : Nat.Partrec f\nhg : Nat.Partrec g\nfC : f \u2208 C\nw\u271d : DecidablePred fun c => eval c \u2208 C\nh : Computable fun a => decide ((fun c => eval c \u2208 C) a)\nc : Code\nH : \u00ac(fun b => f b) \u2208 C\ne : eval c = fun b => f b\n\u22a2 g \u2208 C"}, {"tactic": "contradiction", "annotated_tactic": ["contradiction", []], "state_before": "case neg\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nC : Set (\u2115 \u2192. \u2115)\nf g : \u2115 \u2192. \u2115\nhf : Nat.Partrec f\nhg : Nat.Partrec g\nfC : f \u2208 C\nw\u271d : DecidablePred fun c => eval c \u2208 C\nh : Computable fun a => decide ((fun c => eval c \u2208 C) a)\nc : Code\nH : \u00ac(fun b => f b) \u2208 C\ne : eval c = fun b => f b\n\u22a2 g \u2208 C", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Haar/Basic.lean", "full_name": "MeasureTheory.Measure.haar.prehaar_empty", "start": [122, 1], "end": [123, 71], "traced_tactics": [{"tactic": "rw [prehaar, Compacts.coe_bot, index_empty, Nat.cast_zero, zero_div]", "annotated_tactic": ["rw [<a>prehaar</a>, <a>Compacts.coe_bot</a>, <a>index_empty</a>, <a>Nat.cast_zero</a>, <a>zero_div</a>]", [{"full_name": "MeasureTheory.Measure.haar.prehaar", "def_path": "Mathlib/MeasureTheory/Measure/Haar/Basic.lean", "def_pos": [116, 19], "def_end_pos": [116, 26]}, {"full_name": "TopologicalSpace.Compacts.coe_bot", "def_path": "Mathlib/Topology/Sets/Compacts.lean", "def_pos": [119, 9], "def_end_pos": [119, 16]}, {"full_name": "MeasureTheory.Measure.haar.index_empty", "def_path": "Mathlib/MeasureTheory/Measure/Haar/Basic.lean", "def_pos": [102, 9], "def_end_pos": [102, 20]}, {"full_name": "Nat.cast_zero", "def_path": "Mathlib/Data/Nat/Cast/Defs.lean", "def_pos": [114, 9], "def_end_pos": [114, 18]}, {"full_name": "zero_div", "def_path": "Mathlib/Algebra/GroupWithZero/Basic.lean", "def_pos": [291, 9], "def_end_pos": [291, 17]}]], "state_before": "G : Type u_1\ninst\u271d\u00b9 : Group G\ninst\u271d : TopologicalSpace G\nK\u2080 : PositiveCompacts G\nU : Set G\n\u22a2 prehaar (\u2191K\u2080) U \u22a5 = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean", "full_name": "MeasureTheory.tendsto_condexp_unique", "start": [391, 1], "end": [415, 91], "traced_tactics": [{"tactic": "by_cases hm : m \u2264 m0", "annotated_tactic": ["by_cases hm : m \u2264 m0", []], "state_before": "\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nfs gs : \u2115 \u2192 \u03b1 \u2192 F'\nf g : \u03b1 \u2192 F'\nhfs_int : \u2200 (n : \u2115), Integrable (fs n)\nhgs_int : \u2200 (n : \u2115), Integrable (gs n)\nhfs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n x) atTop (\ud835\udcdd (f x))\nhgs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => gs n x) atTop (\ud835\udcdd (g x))\nbound_fs : \u03b1 \u2192 \u211d\nh_int_bound_fs : Integrable bound_fs\nbound_gs : \u03b1 \u2192 \u211d\nh_int_bound_gs : Integrable bound_gs\nhfs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016fs n x\u2016 \u2264 bound_fs x\nhgs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016gs n x\u2016 \u2264 bound_gs x\nhfg : \u2200 (n : \u2115), \u03bc[fs n|m] =\u1d50[\u03bc] \u03bc[gs n|m]\n\u22a2 \u03bc[f|m] =\u1d50[\u03bc] \u03bc[g|m]", "state_after": "case pos\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nfs gs : \u2115 \u2192 \u03b1 \u2192 F'\nf g : \u03b1 \u2192 F'\nhfs_int : \u2200 (n : \u2115), Integrable (fs n)\nhgs_int : \u2200 (n : \u2115), Integrable (gs n)\nhfs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n x) atTop (\ud835\udcdd (f x))\nhgs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => gs n x) atTop (\ud835\udcdd (g x))\nbound_fs : \u03b1 \u2192 \u211d\nh_int_bound_fs : Integrable bound_fs\nbound_gs : \u03b1 \u2192 \u211d\nh_int_bound_gs : Integrable bound_gs\nhfs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016fs n x\u2016 \u2264 bound_fs x\nhgs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016gs n x\u2016 \u2264 bound_gs x\nhfg : \u2200 (n : \u2115), \u03bc[fs n|m] =\u1d50[\u03bc] \u03bc[gs n|m]\nhm : m \u2264 m0\n\u22a2 \u03bc[f|m] =\u1d50[\u03bc] \u03bc[g|m]\n\ncase neg\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nfs gs : \u2115 \u2192 \u03b1 \u2192 F'\nf g : \u03b1 \u2192 F'\nhfs_int : \u2200 (n : \u2115), Integrable (fs n)\nhgs_int : \u2200 (n : \u2115), Integrable (gs n)\nhfs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n x) atTop (\ud835\udcdd (f x))\nhgs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => gs n x) atTop (\ud835\udcdd (g x))\nbound_fs : \u03b1 \u2192 \u211d\nh_int_bound_fs : Integrable bound_fs\nbound_gs : \u03b1 \u2192 \u211d\nh_int_bound_gs : Integrable bound_gs\nhfs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016fs n x\u2016 \u2264 bound_fs x\nhgs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016gs n x\u2016 \u2264 bound_gs x\nhfg : \u2200 (n : \u2115), \u03bc[fs n|m] =\u1d50[\u03bc] \u03bc[gs n|m]\nhm : \u00acm \u2264 m0\n\u22a2 \u03bc[f|m] =\u1d50[\u03bc] \u03bc[g|m]"}, {"tactic": "swap", "annotated_tactic": ["swap", []], "state_before": "case pos\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nfs gs : \u2115 \u2192 \u03b1 \u2192 F'\nf g : \u03b1 \u2192 F'\nhfs_int : \u2200 (n : \u2115), Integrable (fs n)\nhgs_int : \u2200 (n : \u2115), Integrable (gs n)\nhfs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n x) atTop (\ud835\udcdd (f x))\nhgs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => gs n x) atTop (\ud835\udcdd (g x))\nbound_fs : \u03b1 \u2192 \u211d\nh_int_bound_fs : Integrable bound_fs\nbound_gs : \u03b1 \u2192 \u211d\nh_int_bound_gs : Integrable bound_gs\nhfs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016fs n x\u2016 \u2264 bound_fs x\nhgs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016gs n x\u2016 \u2264 bound_gs x\nhfg : \u2200 (n : \u2115), \u03bc[fs n|m] =\u1d50[\u03bc] \u03bc[gs n|m]\nhm : m \u2264 m0\n\u22a2 \u03bc[f|m] =\u1d50[\u03bc] \u03bc[g|m]\n\ncase neg\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nfs gs : \u2115 \u2192 \u03b1 \u2192 F'\nf g : \u03b1 \u2192 F'\nhfs_int : \u2200 (n : \u2115), Integrable (fs n)\nhgs_int : \u2200 (n : \u2115), Integrable (gs n)\nhfs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n x) atTop (\ud835\udcdd (f x))\nhgs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => gs n x) atTop (\ud835\udcdd (g x))\nbound_fs : \u03b1 \u2192 \u211d\nh_int_bound_fs : Integrable bound_fs\nbound_gs : \u03b1 \u2192 \u211d\nh_int_bound_gs : Integrable bound_gs\nhfs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016fs n x\u2016 \u2264 bound_fs x\nhgs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016gs n x\u2016 \u2264 bound_gs x\nhfg : \u2200 (n : \u2115), \u03bc[fs n|m] =\u1d50[\u03bc] \u03bc[gs n|m]\nhm : \u00acm \u2264 m0\n\u22a2 \u03bc[f|m] =\u1d50[\u03bc] \u03bc[g|m]", "state_after": "case neg\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nfs gs : \u2115 \u2192 \u03b1 \u2192 F'\nf g : \u03b1 \u2192 F'\nhfs_int : \u2200 (n : \u2115), Integrable (fs n)\nhgs_int : \u2200 (n : \u2115), Integrable (gs n)\nhfs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n x) atTop (\ud835\udcdd (f x))\nhgs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => gs n x) atTop (\ud835\udcdd (g x))\nbound_fs : \u03b1 \u2192 \u211d\nh_int_bound_fs : Integrable bound_fs\nbound_gs : \u03b1 \u2192 \u211d\nh_int_bound_gs : Integrable bound_gs\nhfs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016fs n x\u2016 \u2264 bound_fs x\nhgs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016gs n x\u2016 \u2264 bound_gs x\nhfg : \u2200 (n : \u2115), \u03bc[fs n|m] =\u1d50[\u03bc] \u03bc[gs n|m]\nhm : \u00acm \u2264 m0\n\u22a2 \u03bc[f|m] =\u1d50[\u03bc] \u03bc[g|m]\n\ncase pos\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nfs gs : \u2115 \u2192 \u03b1 \u2192 F'\nf g : \u03b1 \u2192 F'\nhfs_int : \u2200 (n : \u2115), Integrable (fs n)\nhgs_int : \u2200 (n : \u2115), Integrable (gs n)\nhfs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n x) atTop (\ud835\udcdd (f x))\nhgs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => gs n x) atTop (\ud835\udcdd (g x))\nbound_fs : \u03b1 \u2192 \u211d\nh_int_bound_fs : Integrable bound_fs\nbound_gs : \u03b1 \u2192 \u211d\nh_int_bound_gs : Integrable bound_gs\nhfs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016fs n x\u2016 \u2264 bound_fs x\nhgs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016gs n x\u2016 \u2264 bound_gs x\nhfg : \u2200 (n : \u2115), \u03bc[fs n|m] =\u1d50[\u03bc] \u03bc[gs n|m]\nhm : m \u2264 m0\n\u22a2 \u03bc[f|m] =\u1d50[\u03bc] \u03bc[g|m]"}, {"tactic": "by_cases h\u03bcm : SigmaFinite (\u03bc.trim hm)", "annotated_tactic": ["by_cases h\u03bcm : <a>SigmaFinite</a> (\u03bc.trim hm)", [{"full_name": "MeasureTheory.SigmaFinite", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3289, 7], "def_end_pos": [3289, 18]}]], "state_before": "case pos\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nfs gs : \u2115 \u2192 \u03b1 \u2192 F'\nf g : \u03b1 \u2192 F'\nhfs_int : \u2200 (n : \u2115), Integrable (fs n)\nhgs_int : \u2200 (n : \u2115), Integrable (gs n)\nhfs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n x) atTop (\ud835\udcdd (f x))\nhgs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => gs n x) atTop (\ud835\udcdd (g x))\nbound_fs : \u03b1 \u2192 \u211d\nh_int_bound_fs : Integrable bound_fs\nbound_gs : \u03b1 \u2192 \u211d\nh_int_bound_gs : Integrable bound_gs\nhfs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016fs n x\u2016 \u2264 bound_fs x\nhgs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016gs n x\u2016 \u2264 bound_gs x\nhfg : \u2200 (n : \u2115), \u03bc[fs n|m] =\u1d50[\u03bc] \u03bc[gs n|m]\nhm : m \u2264 m0\n\u22a2 \u03bc[f|m] =\u1d50[\u03bc] \u03bc[g|m]", "state_after": "case pos\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nfs gs : \u2115 \u2192 \u03b1 \u2192 F'\nf g : \u03b1 \u2192 F'\nhfs_int : \u2200 (n : \u2115), Integrable (fs n)\nhgs_int : \u2200 (n : \u2115), Integrable (gs n)\nhfs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n x) atTop (\ud835\udcdd (f x))\nhgs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => gs n x) atTop (\ud835\udcdd (g x))\nbound_fs : \u03b1 \u2192 \u211d\nh_int_bound_fs : Integrable bound_fs\nbound_gs : \u03b1 \u2192 \u211d\nh_int_bound_gs : Integrable bound_gs\nhfs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016fs n x\u2016 \u2264 bound_fs x\nhgs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016gs n x\u2016 \u2264 bound_gs x\nhfg : \u2200 (n : \u2115), \u03bc[fs n|m] =\u1d50[\u03bc] \u03bc[gs n|m]\nhm : m \u2264 m0\nh\u03bcm : SigmaFinite (Measure.trim \u03bc hm)\n\u22a2 \u03bc[f|m] =\u1d50[\u03bc] \u03bc[g|m]\n\ncase neg\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nfs gs : \u2115 \u2192 \u03b1 \u2192 F'\nf g : \u03b1 \u2192 F'\nhfs_int : \u2200 (n : \u2115), Integrable (fs n)\nhgs_int : \u2200 (n : \u2115), Integrable (gs n)\nhfs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n x) atTop (\ud835\udcdd (f x))\nhgs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => gs n x) atTop (\ud835\udcdd (g x))\nbound_fs : \u03b1 \u2192 \u211d\nh_int_bound_fs : Integrable bound_fs\nbound_gs : \u03b1 \u2192 \u211d\nh_int_bound_gs : Integrable bound_gs\nhfs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016fs n x\u2016 \u2264 bound_fs x\nhgs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016gs n x\u2016 \u2264 bound_gs x\nhfg : \u2200 (n : \u2115), \u03bc[fs n|m] =\u1d50[\u03bc] \u03bc[gs n|m]\nhm : m \u2264 m0\nh\u03bcm : \u00acSigmaFinite (Measure.trim \u03bc hm)\n\u22a2 \u03bc[f|m] =\u1d50[\u03bc] \u03bc[g|m]"}, {"tactic": "swap", "annotated_tactic": ["swap", []], "state_before": "case pos\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nfs gs : \u2115 \u2192 \u03b1 \u2192 F'\nf g : \u03b1 \u2192 F'\nhfs_int : \u2200 (n : \u2115), Integrable (fs n)\nhgs_int : \u2200 (n : \u2115), Integrable (gs n)\nhfs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n x) atTop (\ud835\udcdd (f x))\nhgs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => gs n x) atTop (\ud835\udcdd (g x))\nbound_fs : \u03b1 \u2192 \u211d\nh_int_bound_fs : Integrable bound_fs\nbound_gs : \u03b1 \u2192 \u211d\nh_int_bound_gs : Integrable bound_gs\nhfs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016fs n x\u2016 \u2264 bound_fs x\nhgs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016gs n x\u2016 \u2264 bound_gs x\nhfg : \u2200 (n : \u2115), \u03bc[fs n|m] =\u1d50[\u03bc] \u03bc[gs n|m]\nhm : m \u2264 m0\nh\u03bcm : SigmaFinite (Measure.trim \u03bc hm)\n\u22a2 \u03bc[f|m] =\u1d50[\u03bc] \u03bc[g|m]\n\ncase neg\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nfs gs : \u2115 \u2192 \u03b1 \u2192 F'\nf g : \u03b1 \u2192 F'\nhfs_int : \u2200 (n : \u2115), Integrable (fs n)\nhgs_int : \u2200 (n : \u2115), Integrable (gs n)\nhfs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n x) atTop (\ud835\udcdd (f x))\nhgs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => gs n x) atTop (\ud835\udcdd (g x))\nbound_fs : \u03b1 \u2192 \u211d\nh_int_bound_fs : Integrable bound_fs\nbound_gs : \u03b1 \u2192 \u211d\nh_int_bound_gs : Integrable bound_gs\nhfs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016fs n x\u2016 \u2264 bound_fs x\nhgs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016gs n x\u2016 \u2264 bound_gs x\nhfg : \u2200 (n : \u2115), \u03bc[fs n|m] =\u1d50[\u03bc] \u03bc[gs n|m]\nhm : m \u2264 m0\nh\u03bcm : \u00acSigmaFinite (Measure.trim \u03bc hm)\n\u22a2 \u03bc[f|m] =\u1d50[\u03bc] \u03bc[g|m]", "state_after": "case neg\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nfs gs : \u2115 \u2192 \u03b1 \u2192 F'\nf g : \u03b1 \u2192 F'\nhfs_int : \u2200 (n : \u2115), Integrable (fs n)\nhgs_int : \u2200 (n : \u2115), Integrable (gs n)\nhfs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n x) atTop (\ud835\udcdd (f x))\nhgs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => gs n x) atTop (\ud835\udcdd (g x))\nbound_fs : \u03b1 \u2192 \u211d\nh_int_bound_fs : Integrable bound_fs\nbound_gs : \u03b1 \u2192 \u211d\nh_int_bound_gs : Integrable bound_gs\nhfs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016fs n x\u2016 \u2264 bound_fs x\nhgs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016gs n x\u2016 \u2264 bound_gs x\nhfg : \u2200 (n : \u2115), \u03bc[fs n|m] =\u1d50[\u03bc] \u03bc[gs n|m]\nhm : m \u2264 m0\nh\u03bcm : \u00acSigmaFinite (Measure.trim \u03bc hm)\n\u22a2 \u03bc[f|m] =\u1d50[\u03bc] \u03bc[g|m]\n\ncase pos\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nfs gs : \u2115 \u2192 \u03b1 \u2192 F'\nf g : \u03b1 \u2192 F'\nhfs_int : \u2200 (n : \u2115), Integrable (fs n)\nhgs_int : \u2200 (n : \u2115), Integrable (gs n)\nhfs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n x) atTop (\ud835\udcdd (f x))\nhgs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => gs n x) atTop (\ud835\udcdd (g x))\nbound_fs : \u03b1 \u2192 \u211d\nh_int_bound_fs : Integrable bound_fs\nbound_gs : \u03b1 \u2192 \u211d\nh_int_bound_gs : Integrable bound_gs\nhfs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016fs n x\u2016 \u2264 bound_fs x\nhgs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016gs n x\u2016 \u2264 bound_gs x\nhfg : \u2200 (n : \u2115), \u03bc[fs n|m] =\u1d50[\u03bc] \u03bc[gs n|m]\nhm : m \u2264 m0\nh\u03bcm : SigmaFinite (Measure.trim \u03bc hm)\n\u22a2 \u03bc[f|m] =\u1d50[\u03bc] \u03bc[g|m]"}, {"tactic": "haveI : SigmaFinite (\u03bc.trim hm) := h\u03bcm", "annotated_tactic": ["haveI : <a>SigmaFinite</a> (\u03bc.trim hm) := h\u03bcm", [{"full_name": "MeasureTheory.SigmaFinite", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3289, 7], "def_end_pos": [3289, 18]}]], "state_before": "case pos\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nfs gs : \u2115 \u2192 \u03b1 \u2192 F'\nf g : \u03b1 \u2192 F'\nhfs_int : \u2200 (n : \u2115), Integrable (fs n)\nhgs_int : \u2200 (n : \u2115), Integrable (gs n)\nhfs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n x) atTop (\ud835\udcdd (f x))\nhgs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => gs n x) atTop (\ud835\udcdd (g x))\nbound_fs : \u03b1 \u2192 \u211d\nh_int_bound_fs : Integrable bound_fs\nbound_gs : \u03b1 \u2192 \u211d\nh_int_bound_gs : Integrable bound_gs\nhfs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016fs n x\u2016 \u2264 bound_fs x\nhgs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016gs n x\u2016 \u2264 bound_gs x\nhfg : \u2200 (n : \u2115), \u03bc[fs n|m] =\u1d50[\u03bc] \u03bc[gs n|m]\nhm : m \u2264 m0\nh\u03bcm : SigmaFinite (Measure.trim \u03bc hm)\n\u22a2 \u03bc[f|m] =\u1d50[\u03bc] \u03bc[g|m]", "state_after": "case pos\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nfs gs : \u2115 \u2192 \u03b1 \u2192 F'\nf g : \u03b1 \u2192 F'\nhfs_int : \u2200 (n : \u2115), Integrable (fs n)\nhgs_int : \u2200 (n : \u2115), Integrable (gs n)\nhfs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n x) atTop (\ud835\udcdd (f x))\nhgs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => gs n x) atTop (\ud835\udcdd (g x))\nbound_fs : \u03b1 \u2192 \u211d\nh_int_bound_fs : Integrable bound_fs\nbound_gs : \u03b1 \u2192 \u211d\nh_int_bound_gs : Integrable bound_gs\nhfs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016fs n x\u2016 \u2264 bound_fs x\nhgs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016gs n x\u2016 \u2264 bound_gs x\nhfg : \u2200 (n : \u2115), \u03bc[fs n|m] =\u1d50[\u03bc] \u03bc[gs n|m]\nhm : m \u2264 m0\nh\u03bcm this : SigmaFinite (Measure.trim \u03bc hm)\n\u22a2 \u03bc[f|m] =\u1d50[\u03bc] \u03bc[g|m]"}, {"tactic": "refine' (condexp_ae_eq_condexpL1 hm f).trans ((condexp_ae_eq_condexpL1 hm g).trans _).symm", "annotated_tactic": ["refine' (<a>condexp_ae_eq_condexpL1</a> hm f).<a>trans</a> ((<a>condexp_ae_eq_condexpL1</a> hm g).<a>trans</a> _).<a>symm</a>", [{"full_name": "MeasureTheory.condexp_ae_eq_condexpL1", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean", "def_pos": [136, 9], "def_end_pos": [136, 32]}, {"full_name": "Filter.EventuallyEq.trans", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1503, 9], "def_end_pos": [1503, 27]}, {"full_name": "MeasureTheory.condexp_ae_eq_condexpL1", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean", "def_pos": [136, 9], "def_end_pos": [136, 32]}, {"full_name": "Filter.EventuallyEq.trans", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1503, 9], "def_end_pos": [1503, 27]}, {"full_name": "Filter.EventuallyEq.symm", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1498, 9], "def_end_pos": [1498, 26]}]], "state_before": "case pos\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nfs gs : \u2115 \u2192 \u03b1 \u2192 F'\nf g : \u03b1 \u2192 F'\nhfs_int : \u2200 (n : \u2115), Integrable (fs n)\nhgs_int : \u2200 (n : \u2115), Integrable (gs n)\nhfs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n x) atTop (\ud835\udcdd (f x))\nhgs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => gs n x) atTop (\ud835\udcdd (g x))\nbound_fs : \u03b1 \u2192 \u211d\nh_int_bound_fs : Integrable bound_fs\nbound_gs : \u03b1 \u2192 \u211d\nh_int_bound_gs : Integrable bound_gs\nhfs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016fs n x\u2016 \u2264 bound_fs x\nhgs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016gs n x\u2016 \u2264 bound_gs x\nhfg : \u2200 (n : \u2115), \u03bc[fs n|m] =\u1d50[\u03bc] \u03bc[gs n|m]\nhm : m \u2264 m0\nh\u03bcm this : SigmaFinite (Measure.trim \u03bc hm)\n\u22a2 \u03bc[f|m] =\u1d50[\u03bc] \u03bc[g|m]", "state_after": "case pos\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nfs gs : \u2115 \u2192 \u03b1 \u2192 F'\nf g : \u03b1 \u2192 F'\nhfs_int : \u2200 (n : \u2115), Integrable (fs n)\nhgs_int : \u2200 (n : \u2115), Integrable (gs n)\nhfs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n x) atTop (\ud835\udcdd (f x))\nhgs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => gs n x) atTop (\ud835\udcdd (g x))\nbound_fs : \u03b1 \u2192 \u211d\nh_int_bound_fs : Integrable bound_fs\nbound_gs : \u03b1 \u2192 \u211d\nh_int_bound_gs : Integrable bound_gs\nhfs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016fs n x\u2016 \u2264 bound_fs x\nhgs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016gs n x\u2016 \u2264 bound_gs x\nhfg : \u2200 (n : \u2115), \u03bc[fs n|m] =\u1d50[\u03bc] \u03bc[gs n|m]\nhm : m \u2264 m0\nh\u03bcm this : SigmaFinite (Measure.trim \u03bc hm)\n\u22a2 \u2191\u2191(condexpL1 hm \u03bc g) =\u1d50[\u03bc] \u2191\u2191(condexpL1 hm \u03bc f)"}, {"tactic": "rw [\u2190 Lp.ext_iff]", "annotated_tactic": ["rw [\u2190 <a>Lp.ext_iff</a>]", [{"full_name": "MeasureTheory.Lp.ext_iff", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [167, 9], "def_end_pos": [167, 16]}]], "state_before": "case pos\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nfs gs : \u2115 \u2192 \u03b1 \u2192 F'\nf g : \u03b1 \u2192 F'\nhfs_int : \u2200 (n : \u2115), Integrable (fs n)\nhgs_int : \u2200 (n : \u2115), Integrable (gs n)\nhfs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n x) atTop (\ud835\udcdd (f x))\nhgs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => gs n x) atTop (\ud835\udcdd (g x))\nbound_fs : \u03b1 \u2192 \u211d\nh_int_bound_fs : Integrable bound_fs\nbound_gs : \u03b1 \u2192 \u211d\nh_int_bound_gs : Integrable bound_gs\nhfs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016fs n x\u2016 \u2264 bound_fs x\nhgs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016gs n x\u2016 \u2264 bound_gs x\nhfg : \u2200 (n : \u2115), \u03bc[fs n|m] =\u1d50[\u03bc] \u03bc[gs n|m]\nhm : m \u2264 m0\nh\u03bcm this : SigmaFinite (Measure.trim \u03bc hm)\n\u22a2 \u2191\u2191(condexpL1 hm \u03bc g) =\u1d50[\u03bc] \u2191\u2191(condexpL1 hm \u03bc f)", "state_after": "case pos\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nfs gs : \u2115 \u2192 \u03b1 \u2192 F'\nf g : \u03b1 \u2192 F'\nhfs_int : \u2200 (n : \u2115), Integrable (fs n)\nhgs_int : \u2200 (n : \u2115), Integrable (gs n)\nhfs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n x) atTop (\ud835\udcdd (f x))\nhgs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => gs n x) atTop (\ud835\udcdd (g x))\nbound_fs : \u03b1 \u2192 \u211d\nh_int_bound_fs : Integrable bound_fs\nbound_gs : \u03b1 \u2192 \u211d\nh_int_bound_gs : Integrable bound_gs\nhfs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016fs n x\u2016 \u2264 bound_fs x\nhgs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016gs n x\u2016 \u2264 bound_gs x\nhfg : \u2200 (n : \u2115), \u03bc[fs n|m] =\u1d50[\u03bc] \u03bc[gs n|m]\nhm : m \u2264 m0\nh\u03bcm this : SigmaFinite (Measure.trim \u03bc hm)\n\u22a2 condexpL1 hm \u03bc g = condexpL1 hm \u03bc f"}, {"tactic": "have hn_eq : \u2200 n, condexpL1 hm \u03bc (gs n) = condexpL1 hm \u03bc (fs n) := by\n  intro n\n  ext1\n  refine' (condexp_ae_eq_condexpL1 hm (gs n)).symm.trans ((hfg n).symm.trans _)\n  exact condexp_ae_eq_condexpL1 hm (fs n)", "annotated_tactic": ["have hn_eq : \u2200 n, <a>condexpL1</a> hm \u03bc (gs n) = <a>condexpL1</a> hm \u03bc (fs n) := by\n    intro n\n    ext1\n    refine' (<a>condexp_ae_eq_condexpL1</a> hm (gs n)).symm.trans ((hfg n).symm.trans _)\n    exact <a>condexp_ae_eq_condexpL1</a> hm (fs n)", [{"full_name": "MeasureTheory.condexpL1", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean", "def_pos": [514, 5], "def_end_pos": [514, 14]}, {"full_name": "MeasureTheory.condexpL1", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean", "def_pos": [514, 5], "def_end_pos": [514, 14]}, {"full_name": "MeasureTheory.condexp_ae_eq_condexpL1", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean", "def_pos": [136, 9], "def_end_pos": [136, 32]}, {"full_name": "MeasureTheory.condexp_ae_eq_condexpL1", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean", "def_pos": [136, 9], "def_end_pos": [136, 32]}]], "state_before": "case pos\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nfs gs : \u2115 \u2192 \u03b1 \u2192 F'\nf g : \u03b1 \u2192 F'\nhfs_int : \u2200 (n : \u2115), Integrable (fs n)\nhgs_int : \u2200 (n : \u2115), Integrable (gs n)\nhfs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n x) atTop (\ud835\udcdd (f x))\nhgs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => gs n x) atTop (\ud835\udcdd (g x))\nbound_fs : \u03b1 \u2192 \u211d\nh_int_bound_fs : Integrable bound_fs\nbound_gs : \u03b1 \u2192 \u211d\nh_int_bound_gs : Integrable bound_gs\nhfs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016fs n x\u2016 \u2264 bound_fs x\nhgs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016gs n x\u2016 \u2264 bound_gs x\nhfg : \u2200 (n : \u2115), \u03bc[fs n|m] =\u1d50[\u03bc] \u03bc[gs n|m]\nhm : m \u2264 m0\nh\u03bcm this : SigmaFinite (Measure.trim \u03bc hm)\n\u22a2 condexpL1 hm \u03bc g = condexpL1 hm \u03bc f", "state_after": "case pos\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nfs gs : \u2115 \u2192 \u03b1 \u2192 F'\nf g : \u03b1 \u2192 F'\nhfs_int : \u2200 (n : \u2115), Integrable (fs n)\nhgs_int : \u2200 (n : \u2115), Integrable (gs n)\nhfs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n x) atTop (\ud835\udcdd (f x))\nhgs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => gs n x) atTop (\ud835\udcdd (g x))\nbound_fs : \u03b1 \u2192 \u211d\nh_int_bound_fs : Integrable bound_fs\nbound_gs : \u03b1 \u2192 \u211d\nh_int_bound_gs : Integrable bound_gs\nhfs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016fs n x\u2016 \u2264 bound_fs x\nhgs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016gs n x\u2016 \u2264 bound_gs x\nhfg : \u2200 (n : \u2115), \u03bc[fs n|m] =\u1d50[\u03bc] \u03bc[gs n|m]\nhm : m \u2264 m0\nh\u03bcm this : SigmaFinite (Measure.trim \u03bc hm)\nhn_eq : \u2200 (n : \u2115), condexpL1 hm \u03bc (gs n) = condexpL1 hm \u03bc (fs n)\n\u22a2 condexpL1 hm \u03bc g = condexpL1 hm \u03bc f"}, {"tactic": "have hcond_fs : Tendsto (fun n => condexpL1 hm \u03bc (fs n)) atTop (\ud835\udcdd (condexpL1 hm \u03bc f)) :=\n  tendsto_condexpL1_of_dominated_convergence hm _ (fun n => (hfs_int n).1) h_int_bound_fs\n    hfs_bound hfs", "annotated_tactic": ["have hcond_fs : <a>Tendsto</a> (fun n => <a>condexpL1</a> hm \u03bc (fs n)) <a>atTop</a> (\ud835\udcdd (<a>condexpL1</a> hm \u03bc f)) :=\n    <a>tendsto_condexpL1_of_dominated_convergence</a> hm _ (fun n => (hfs_int n).1) h_int_bound_fs\n      hfs_bound hfs", [{"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2939, 5], "def_end_pos": [2939, 12]}, {"full_name": "MeasureTheory.condexpL1", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean", "def_pos": [514, 5], "def_end_pos": [514, 14]}, {"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}, {"full_name": "MeasureTheory.condexpL1", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean", "def_pos": [514, 5], "def_end_pos": [514, 14]}, {"full_name": "MeasureTheory.tendsto_condexpL1_of_dominated_convergence", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean", "def_pos": [378, 9], "def_end_pos": [378, 51]}]], "state_before": "case pos\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nfs gs : \u2115 \u2192 \u03b1 \u2192 F'\nf g : \u03b1 \u2192 F'\nhfs_int : \u2200 (n : \u2115), Integrable (fs n)\nhgs_int : \u2200 (n : \u2115), Integrable (gs n)\nhfs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n x) atTop (\ud835\udcdd (f x))\nhgs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => gs n x) atTop (\ud835\udcdd (g x))\nbound_fs : \u03b1 \u2192 \u211d\nh_int_bound_fs : Integrable bound_fs\nbound_gs : \u03b1 \u2192 \u211d\nh_int_bound_gs : Integrable bound_gs\nhfs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016fs n x\u2016 \u2264 bound_fs x\nhgs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016gs n x\u2016 \u2264 bound_gs x\nhfg : \u2200 (n : \u2115), \u03bc[fs n|m] =\u1d50[\u03bc] \u03bc[gs n|m]\nhm : m \u2264 m0\nh\u03bcm this : SigmaFinite (Measure.trim \u03bc hm)\nhn_eq : \u2200 (n : \u2115), condexpL1 hm \u03bc (gs n) = condexpL1 hm \u03bc (fs n)\n\u22a2 condexpL1 hm \u03bc g = condexpL1 hm \u03bc f", "state_after": "case pos\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nfs gs : \u2115 \u2192 \u03b1 \u2192 F'\nf g : \u03b1 \u2192 F'\nhfs_int : \u2200 (n : \u2115), Integrable (fs n)\nhgs_int : \u2200 (n : \u2115), Integrable (gs n)\nhfs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n x) atTop (\ud835\udcdd (f x))\nhgs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => gs n x) atTop (\ud835\udcdd (g x))\nbound_fs : \u03b1 \u2192 \u211d\nh_int_bound_fs : Integrable bound_fs\nbound_gs : \u03b1 \u2192 \u211d\nh_int_bound_gs : Integrable bound_gs\nhfs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016fs n x\u2016 \u2264 bound_fs x\nhgs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016gs n x\u2016 \u2264 bound_gs x\nhfg : \u2200 (n : \u2115), \u03bc[fs n|m] =\u1d50[\u03bc] \u03bc[gs n|m]\nhm : m \u2264 m0\nh\u03bcm this : SigmaFinite (Measure.trim \u03bc hm)\nhn_eq : \u2200 (n : \u2115), condexpL1 hm \u03bc (gs n) = condexpL1 hm \u03bc (fs n)\nhcond_fs : Tendsto (fun n => condexpL1 hm \u03bc (fs n)) atTop (\ud835\udcdd (condexpL1 hm \u03bc f))\n\u22a2 condexpL1 hm \u03bc g = condexpL1 hm \u03bc f"}, {"tactic": "have hcond_gs : Tendsto (fun n => condexpL1 hm \u03bc (gs n)) atTop (\ud835\udcdd (condexpL1 hm \u03bc g)) :=\n  tendsto_condexpL1_of_dominated_convergence hm _ (fun n => (hgs_int n).1) h_int_bound_gs\n    hgs_bound hgs", "annotated_tactic": ["have hcond_gs : <a>Tendsto</a> (fun n => <a>condexpL1</a> hm \u03bc (gs n)) <a>atTop</a> (\ud835\udcdd (<a>condexpL1</a> hm \u03bc g)) :=\n    <a>tendsto_condexpL1_of_dominated_convergence</a> hm _ (fun n => (hgs_int n).1) h_int_bound_gs\n      hgs_bound hgs", [{"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2939, 5], "def_end_pos": [2939, 12]}, {"full_name": "MeasureTheory.condexpL1", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean", "def_pos": [514, 5], "def_end_pos": [514, 14]}, {"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}, {"full_name": "MeasureTheory.condexpL1", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean", "def_pos": [514, 5], "def_end_pos": [514, 14]}, {"full_name": "MeasureTheory.tendsto_condexpL1_of_dominated_convergence", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean", "def_pos": [378, 9], "def_end_pos": [378, 51]}]], "state_before": "case pos\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nfs gs : \u2115 \u2192 \u03b1 \u2192 F'\nf g : \u03b1 \u2192 F'\nhfs_int : \u2200 (n : \u2115), Integrable (fs n)\nhgs_int : \u2200 (n : \u2115), Integrable (gs n)\nhfs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n x) atTop (\ud835\udcdd (f x))\nhgs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => gs n x) atTop (\ud835\udcdd (g x))\nbound_fs : \u03b1 \u2192 \u211d\nh_int_bound_fs : Integrable bound_fs\nbound_gs : \u03b1 \u2192 \u211d\nh_int_bound_gs : Integrable bound_gs\nhfs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016fs n x\u2016 \u2264 bound_fs x\nhgs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016gs n x\u2016 \u2264 bound_gs x\nhfg : \u2200 (n : \u2115), \u03bc[fs n|m] =\u1d50[\u03bc] \u03bc[gs n|m]\nhm : m \u2264 m0\nh\u03bcm this : SigmaFinite (Measure.trim \u03bc hm)\nhn_eq : \u2200 (n : \u2115), condexpL1 hm \u03bc (gs n) = condexpL1 hm \u03bc (fs n)\nhcond_fs : Tendsto (fun n => condexpL1 hm \u03bc (fs n)) atTop (\ud835\udcdd (condexpL1 hm \u03bc f))\n\u22a2 condexpL1 hm \u03bc g = condexpL1 hm \u03bc f", "state_after": "case pos\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nfs gs : \u2115 \u2192 \u03b1 \u2192 F'\nf g : \u03b1 \u2192 F'\nhfs_int : \u2200 (n : \u2115), Integrable (fs n)\nhgs_int : \u2200 (n : \u2115), Integrable (gs n)\nhfs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n x) atTop (\ud835\udcdd (f x))\nhgs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => gs n x) atTop (\ud835\udcdd (g x))\nbound_fs : \u03b1 \u2192 \u211d\nh_int_bound_fs : Integrable bound_fs\nbound_gs : \u03b1 \u2192 \u211d\nh_int_bound_gs : Integrable bound_gs\nhfs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016fs n x\u2016 \u2264 bound_fs x\nhgs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016gs n x\u2016 \u2264 bound_gs x\nhfg : \u2200 (n : \u2115), \u03bc[fs n|m] =\u1d50[\u03bc] \u03bc[gs n|m]\nhm : m \u2264 m0\nh\u03bcm this : SigmaFinite (Measure.trim \u03bc hm)\nhn_eq : \u2200 (n : \u2115), condexpL1 hm \u03bc (gs n) = condexpL1 hm \u03bc (fs n)\nhcond_fs : Tendsto (fun n => condexpL1 hm \u03bc (fs n)) atTop (\ud835\udcdd (condexpL1 hm \u03bc f))\nhcond_gs : Tendsto (fun n => condexpL1 hm \u03bc (gs n)) atTop (\ud835\udcdd (condexpL1 hm \u03bc g))\n\u22a2 condexpL1 hm \u03bc g = condexpL1 hm \u03bc f"}, {"tactic": "exact tendsto_nhds_unique_of_eventuallyEq hcond_gs hcond_fs (eventually_of_forall hn_eq)", "annotated_tactic": ["exact <a>tendsto_nhds_unique_of_eventuallyEq</a> hcond_gs hcond_fs (<a>eventually_of_forall</a> hn_eq)", [{"full_name": "tendsto_nhds_unique_of_eventuallyEq", "def_path": "Mathlib/Topology/Separation.lean", "def_pos": [1004, 9], "def_end_pos": [1004, 44]}, {"full_name": "Filter.eventually_of_forall", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1111, 9], "def_end_pos": [1111, 29]}]], "state_before": "case pos\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nfs gs : \u2115 \u2192 \u03b1 \u2192 F'\nf g : \u03b1 \u2192 F'\nhfs_int : \u2200 (n : \u2115), Integrable (fs n)\nhgs_int : \u2200 (n : \u2115), Integrable (gs n)\nhfs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n x) atTop (\ud835\udcdd (f x))\nhgs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => gs n x) atTop (\ud835\udcdd (g x))\nbound_fs : \u03b1 \u2192 \u211d\nh_int_bound_fs : Integrable bound_fs\nbound_gs : \u03b1 \u2192 \u211d\nh_int_bound_gs : Integrable bound_gs\nhfs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016fs n x\u2016 \u2264 bound_fs x\nhgs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016gs n x\u2016 \u2264 bound_gs x\nhfg : \u2200 (n : \u2115), \u03bc[fs n|m] =\u1d50[\u03bc] \u03bc[gs n|m]\nhm : m \u2264 m0\nh\u03bcm this : SigmaFinite (Measure.trim \u03bc hm)\nhn_eq : \u2200 (n : \u2115), condexpL1 hm \u03bc (gs n) = condexpL1 hm \u03bc (fs n)\nhcond_fs : Tendsto (fun n => condexpL1 hm \u03bc (fs n)) atTop (\ud835\udcdd (condexpL1 hm \u03bc f))\nhcond_gs : Tendsto (fun n => condexpL1 hm \u03bc (gs n)) atTop (\ud835\udcdd (condexpL1 hm \u03bc g))\n\u22a2 condexpL1 hm \u03bc g = condexpL1 hm \u03bc f", "state_after": "no goals"}, {"tactic": "simp_rw [condexp_of_not_le hm]", "annotated_tactic": ["simp_rw [<a>condexp_of_not_le</a> hm]", [{"full_name": "MeasureTheory.condexp_of_not_le", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean", "def_pos": [106, 9], "def_end_pos": [106, 26]}]], "state_before": "case neg\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nfs gs : \u2115 \u2192 \u03b1 \u2192 F'\nf g : \u03b1 \u2192 F'\nhfs_int : \u2200 (n : \u2115), Integrable (fs n)\nhgs_int : \u2200 (n : \u2115), Integrable (gs n)\nhfs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n x) atTop (\ud835\udcdd (f x))\nhgs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => gs n x) atTop (\ud835\udcdd (g x))\nbound_fs : \u03b1 \u2192 \u211d\nh_int_bound_fs : Integrable bound_fs\nbound_gs : \u03b1 \u2192 \u211d\nh_int_bound_gs : Integrable bound_gs\nhfs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016fs n x\u2016 \u2264 bound_fs x\nhgs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016gs n x\u2016 \u2264 bound_gs x\nhfg : \u2200 (n : \u2115), \u03bc[fs n|m] =\u1d50[\u03bc] \u03bc[gs n|m]\nhm : \u00acm \u2264 m0\n\u22a2 \u03bc[f|m] =\u1d50[\u03bc] \u03bc[g|m]", "state_after": "case neg\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nfs gs : \u2115 \u2192 \u03b1 \u2192 F'\nf g : \u03b1 \u2192 F'\nhfs_int : \u2200 (n : \u2115), Integrable (fs n)\nhgs_int : \u2200 (n : \u2115), Integrable (gs n)\nhfs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n x) atTop (\ud835\udcdd (f x))\nhgs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => gs n x) atTop (\ud835\udcdd (g x))\nbound_fs : \u03b1 \u2192 \u211d\nh_int_bound_fs : Integrable bound_fs\nbound_gs : \u03b1 \u2192 \u211d\nh_int_bound_gs : Integrable bound_gs\nhfs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016fs n x\u2016 \u2264 bound_fs x\nhgs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016gs n x\u2016 \u2264 bound_gs x\nhfg : \u2200 (n : \u2115), \u03bc[fs n|m] =\u1d50[\u03bc] \u03bc[gs n|m]\nhm : \u00acm \u2264 m0\n\u22a2 0 =\u1d50[\u03bc] 0"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case neg\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nfs gs : \u2115 \u2192 \u03b1 \u2192 F'\nf g : \u03b1 \u2192 F'\nhfs_int : \u2200 (n : \u2115), Integrable (fs n)\nhgs_int : \u2200 (n : \u2115), Integrable (gs n)\nhfs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n x) atTop (\ud835\udcdd (f x))\nhgs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => gs n x) atTop (\ud835\udcdd (g x))\nbound_fs : \u03b1 \u2192 \u211d\nh_int_bound_fs : Integrable bound_fs\nbound_gs : \u03b1 \u2192 \u211d\nh_int_bound_gs : Integrable bound_gs\nhfs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016fs n x\u2016 \u2264 bound_fs x\nhgs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016gs n x\u2016 \u2264 bound_gs x\nhfg : \u2200 (n : \u2115), \u03bc[fs n|m] =\u1d50[\u03bc] \u03bc[gs n|m]\nhm : \u00acm \u2264 m0\n\u22a2 0 =\u1d50[\u03bc] 0", "state_after": "no goals"}, {"tactic": "simp_rw [condexp_of_not_sigmaFinite hm h\u03bcm]", "annotated_tactic": ["simp_rw [<a>condexp_of_not_sigmaFinite</a> hm h\u03bcm]", [{"full_name": "MeasureTheory.condexp_of_not_sigmaFinite", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean", "def_pos": [109, 9], "def_end_pos": [109, 35]}]], "state_before": "case neg\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nfs gs : \u2115 \u2192 \u03b1 \u2192 F'\nf g : \u03b1 \u2192 F'\nhfs_int : \u2200 (n : \u2115), Integrable (fs n)\nhgs_int : \u2200 (n : \u2115), Integrable (gs n)\nhfs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n x) atTop (\ud835\udcdd (f x))\nhgs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => gs n x) atTop (\ud835\udcdd (g x))\nbound_fs : \u03b1 \u2192 \u211d\nh_int_bound_fs : Integrable bound_fs\nbound_gs : \u03b1 \u2192 \u211d\nh_int_bound_gs : Integrable bound_gs\nhfs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016fs n x\u2016 \u2264 bound_fs x\nhgs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016gs n x\u2016 \u2264 bound_gs x\nhfg : \u2200 (n : \u2115), \u03bc[fs n|m] =\u1d50[\u03bc] \u03bc[gs n|m]\nhm : m \u2264 m0\nh\u03bcm : \u00acSigmaFinite (Measure.trim \u03bc hm)\n\u22a2 \u03bc[f|m] =\u1d50[\u03bc] \u03bc[g|m]", "state_after": "case neg\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nfs gs : \u2115 \u2192 \u03b1 \u2192 F'\nf g : \u03b1 \u2192 F'\nhfs_int : \u2200 (n : \u2115), Integrable (fs n)\nhgs_int : \u2200 (n : \u2115), Integrable (gs n)\nhfs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n x) atTop (\ud835\udcdd (f x))\nhgs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => gs n x) atTop (\ud835\udcdd (g x))\nbound_fs : \u03b1 \u2192 \u211d\nh_int_bound_fs : Integrable bound_fs\nbound_gs : \u03b1 \u2192 \u211d\nh_int_bound_gs : Integrable bound_gs\nhfs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016fs n x\u2016 \u2264 bound_fs x\nhgs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016gs n x\u2016 \u2264 bound_gs x\nhfg : \u2200 (n : \u2115), \u03bc[fs n|m] =\u1d50[\u03bc] \u03bc[gs n|m]\nhm : m \u2264 m0\nh\u03bcm : \u00acSigmaFinite (Measure.trim \u03bc hm)\n\u22a2 0 =\u1d50[\u03bc] 0"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case neg\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nfs gs : \u2115 \u2192 \u03b1 \u2192 F'\nf g : \u03b1 \u2192 F'\nhfs_int : \u2200 (n : \u2115), Integrable (fs n)\nhgs_int : \u2200 (n : \u2115), Integrable (gs n)\nhfs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n x) atTop (\ud835\udcdd (f x))\nhgs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => gs n x) atTop (\ud835\udcdd (g x))\nbound_fs : \u03b1 \u2192 \u211d\nh_int_bound_fs : Integrable bound_fs\nbound_gs : \u03b1 \u2192 \u211d\nh_int_bound_gs : Integrable bound_gs\nhfs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016fs n x\u2016 \u2264 bound_fs x\nhgs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016gs n x\u2016 \u2264 bound_gs x\nhfg : \u2200 (n : \u2115), \u03bc[fs n|m] =\u1d50[\u03bc] \u03bc[gs n|m]\nhm : m \u2264 m0\nh\u03bcm : \u00acSigmaFinite (Measure.trim \u03bc hm)\n\u22a2 0 =\u1d50[\u03bc] 0", "state_after": "no goals"}, {"tactic": "intro n", "annotated_tactic": ["intro n", []], "state_before": "\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nfs gs : \u2115 \u2192 \u03b1 \u2192 F'\nf g : \u03b1 \u2192 F'\nhfs_int : \u2200 (n : \u2115), Integrable (fs n)\nhgs_int : \u2200 (n : \u2115), Integrable (gs n)\nhfs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n x) atTop (\ud835\udcdd (f x))\nhgs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => gs n x) atTop (\ud835\udcdd (g x))\nbound_fs : \u03b1 \u2192 \u211d\nh_int_bound_fs : Integrable bound_fs\nbound_gs : \u03b1 \u2192 \u211d\nh_int_bound_gs : Integrable bound_gs\nhfs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016fs n x\u2016 \u2264 bound_fs x\nhgs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016gs n x\u2016 \u2264 bound_gs x\nhfg : \u2200 (n : \u2115), \u03bc[fs n|m] =\u1d50[\u03bc] \u03bc[gs n|m]\nhm : m \u2264 m0\nh\u03bcm this : SigmaFinite (Measure.trim \u03bc hm)\n\u22a2 \u2200 (n : \u2115), condexpL1 hm \u03bc (gs n) = condexpL1 hm \u03bc (fs n)", "state_after": "\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nfs gs : \u2115 \u2192 \u03b1 \u2192 F'\nf g : \u03b1 \u2192 F'\nhfs_int : \u2200 (n : \u2115), Integrable (fs n)\nhgs_int : \u2200 (n : \u2115), Integrable (gs n)\nhfs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n x) atTop (\ud835\udcdd (f x))\nhgs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => gs n x) atTop (\ud835\udcdd (g x))\nbound_fs : \u03b1 \u2192 \u211d\nh_int_bound_fs : Integrable bound_fs\nbound_gs : \u03b1 \u2192 \u211d\nh_int_bound_gs : Integrable bound_gs\nhfs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016fs n x\u2016 \u2264 bound_fs x\nhgs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016gs n x\u2016 \u2264 bound_gs x\nhfg : \u2200 (n : \u2115), \u03bc[fs n|m] =\u1d50[\u03bc] \u03bc[gs n|m]\nhm : m \u2264 m0\nh\u03bcm this : SigmaFinite (Measure.trim \u03bc hm)\nn : \u2115\n\u22a2 condexpL1 hm \u03bc (gs n) = condexpL1 hm \u03bc (fs n)"}, {"tactic": "ext1", "annotated_tactic": ["ext1", []], "state_before": "\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nfs gs : \u2115 \u2192 \u03b1 \u2192 F'\nf g : \u03b1 \u2192 F'\nhfs_int : \u2200 (n : \u2115), Integrable (fs n)\nhgs_int : \u2200 (n : \u2115), Integrable (gs n)\nhfs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n x) atTop (\ud835\udcdd (f x))\nhgs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => gs n x) atTop (\ud835\udcdd (g x))\nbound_fs : \u03b1 \u2192 \u211d\nh_int_bound_fs : Integrable bound_fs\nbound_gs : \u03b1 \u2192 \u211d\nh_int_bound_gs : Integrable bound_gs\nhfs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016fs n x\u2016 \u2264 bound_fs x\nhgs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016gs n x\u2016 \u2264 bound_gs x\nhfg : \u2200 (n : \u2115), \u03bc[fs n|m] =\u1d50[\u03bc] \u03bc[gs n|m]\nhm : m \u2264 m0\nh\u03bcm this : SigmaFinite (Measure.trim \u03bc hm)\nn : \u2115\n\u22a2 condexpL1 hm \u03bc (gs n) = condexpL1 hm \u03bc (fs n)", "state_after": "case h\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nfs gs : \u2115 \u2192 \u03b1 \u2192 F'\nf g : \u03b1 \u2192 F'\nhfs_int : \u2200 (n : \u2115), Integrable (fs n)\nhgs_int : \u2200 (n : \u2115), Integrable (gs n)\nhfs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n x) atTop (\ud835\udcdd (f x))\nhgs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => gs n x) atTop (\ud835\udcdd (g x))\nbound_fs : \u03b1 \u2192 \u211d\nh_int_bound_fs : Integrable bound_fs\nbound_gs : \u03b1 \u2192 \u211d\nh_int_bound_gs : Integrable bound_gs\nhfs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016fs n x\u2016 \u2264 bound_fs x\nhgs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016gs n x\u2016 \u2264 bound_gs x\nhfg : \u2200 (n : \u2115), \u03bc[fs n|m] =\u1d50[\u03bc] \u03bc[gs n|m]\nhm : m \u2264 m0\nh\u03bcm this : SigmaFinite (Measure.trim \u03bc hm)\nn : \u2115\n\u22a2 \u2191\u2191(condexpL1 hm \u03bc (gs n)) =\u1d50[\u03bc] \u2191\u2191(condexpL1 hm \u03bc (fs n))"}, {"tactic": "refine' (condexp_ae_eq_condexpL1 hm (gs n)).symm.trans ((hfg n).symm.trans _)", "annotated_tactic": ["refine' (<a>condexp_ae_eq_condexpL1</a> hm (gs n)).symm.trans ((hfg n).symm.trans _)", [{"full_name": "MeasureTheory.condexp_ae_eq_condexpL1", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean", "def_pos": [136, 9], "def_end_pos": [136, 32]}]], "state_before": "case h\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nfs gs : \u2115 \u2192 \u03b1 \u2192 F'\nf g : \u03b1 \u2192 F'\nhfs_int : \u2200 (n : \u2115), Integrable (fs n)\nhgs_int : \u2200 (n : \u2115), Integrable (gs n)\nhfs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n x) atTop (\ud835\udcdd (f x))\nhgs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => gs n x) atTop (\ud835\udcdd (g x))\nbound_fs : \u03b1 \u2192 \u211d\nh_int_bound_fs : Integrable bound_fs\nbound_gs : \u03b1 \u2192 \u211d\nh_int_bound_gs : Integrable bound_gs\nhfs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016fs n x\u2016 \u2264 bound_fs x\nhgs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016gs n x\u2016 \u2264 bound_gs x\nhfg : \u2200 (n : \u2115), \u03bc[fs n|m] =\u1d50[\u03bc] \u03bc[gs n|m]\nhm : m \u2264 m0\nh\u03bcm this : SigmaFinite (Measure.trim \u03bc hm)\nn : \u2115\n\u22a2 \u2191\u2191(condexpL1 hm \u03bc (gs n)) =\u1d50[\u03bc] \u2191\u2191(condexpL1 hm \u03bc (fs n))", "state_after": "case h\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nfs gs : \u2115 \u2192 \u03b1 \u2192 F'\nf g : \u03b1 \u2192 F'\nhfs_int : \u2200 (n : \u2115), Integrable (fs n)\nhgs_int : \u2200 (n : \u2115), Integrable (gs n)\nhfs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n x) atTop (\ud835\udcdd (f x))\nhgs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => gs n x) atTop (\ud835\udcdd (g x))\nbound_fs : \u03b1 \u2192 \u211d\nh_int_bound_fs : Integrable bound_fs\nbound_gs : \u03b1 \u2192 \u211d\nh_int_bound_gs : Integrable bound_gs\nhfs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016fs n x\u2016 \u2264 bound_fs x\nhgs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016gs n x\u2016 \u2264 bound_gs x\nhfg : \u2200 (n : \u2115), \u03bc[fs n|m] =\u1d50[\u03bc] \u03bc[gs n|m]\nhm : m \u2264 m0\nh\u03bcm this : SigmaFinite (Measure.trim \u03bc hm)\nn : \u2115\n\u22a2 \u03bc[fs n|m] =\u1d50[\u03bc] \u2191\u2191(condexpL1 hm \u03bc (fs n))"}, {"tactic": "exact condexp_ae_eq_condexpL1 hm (fs n)", "annotated_tactic": ["exact <a>condexp_ae_eq_condexpL1</a> hm (fs n)", [{"full_name": "MeasureTheory.condexp_ae_eq_condexpL1", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean", "def_pos": [136, 9], "def_end_pos": [136, 32]}]], "state_before": "case h\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nfs gs : \u2115 \u2192 \u03b1 \u2192 F'\nf g : \u03b1 \u2192 F'\nhfs_int : \u2200 (n : \u2115), Integrable (fs n)\nhgs_int : \u2200 (n : \u2115), Integrable (gs n)\nhfs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n x) atTop (\ud835\udcdd (f x))\nhgs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => gs n x) atTop (\ud835\udcdd (g x))\nbound_fs : \u03b1 \u2192 \u211d\nh_int_bound_fs : Integrable bound_fs\nbound_gs : \u03b1 \u2192 \u211d\nh_int_bound_gs : Integrable bound_gs\nhfs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016fs n x\u2016 \u2264 bound_fs x\nhgs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016gs n x\u2016 \u2264 bound_gs x\nhfg : \u2200 (n : \u2115), \u03bc[fs n|m] =\u1d50[\u03bc] \u03bc[gs n|m]\nhm : m \u2264 m0\nh\u03bcm this : SigmaFinite (Measure.trim \u03bc hm)\nn : \u2115\n\u22a2 \u03bc[fs n|m] =\u1d50[\u03bc] \u2191\u2191(condexpL1 hm \u03bc (fs n))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Group/Measure.lean", "full_name": "MeasureTheory.measure_ne_zero_iff_nonempty_of_isMulLeftInvariant", "start": [605, 1], "end": [607, 90], "traced_tactics": [{"tactic": "simpa [null_iff_of_isMulLeftInvariant (\u03bc := \u03bc) hs, h\u03bc] using nonempty_iff_ne_empty.symm", "annotated_tactic": ["simpa [<a>null_iff_of_isMulLeftInvariant</a> (\u03bc := \u03bc) hs, h\u03bc] using nonempty_iff_ne_empty.symm", [{"full_name": "MeasureTheory.null_iff_of_isMulLeftInvariant", "def_path": "Mathlib/MeasureTheory/Group/Measure.lean", "def_pos": [596, 9], "def_end_pos": [596, 39]}]], "state_before": "\ud835\udd5c : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : MeasurableSpace H\ninst\u271d\u2075 : TopologicalSpace G\ninst\u271d\u2074 : BorelSpace G\n\u03bc : Measure G\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : TopologicalGroup G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : Regular \u03bc\nh\u03bc : \u03bc \u2260 0\ns : Set G\nhs : IsOpen s\n\u22a2 \u2191\u2191\u03bc s \u2260 0 \u2194 Set.Nonempty s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Kernel/Condexp.lean", "full_name": "MeasureTheory.AEStronglyMeasurable.integral_condexpKernel", "start": [100, 1], "end": [106, 43], "traced_tactics": [{"tactic": "simp_rw [condexpKernel_apply_eq_condDistrib]", "annotated_tactic": ["simp_rw [<a>condexpKernel_apply_eq_condDistrib</a>]", [{"full_name": "ProbabilityTheory.condexpKernel_apply_eq_condDistrib", "def_path": "Mathlib/Probability/Kernel/Condexp.lean", "def_pos": [78, 7], "def_end_pos": [78, 41]}]], "state_before": "\u03a9 : Type u_1\nF : Type u_2\ninst\u271d\u2077 : TopologicalSpace \u03a9\nm m\u03a9 : MeasurableSpace \u03a9\ninst\u271d\u2076 : PolishSpace \u03a9\ninst\u271d\u2075 : BorelSpace \u03a9\ninst\u271d\u2074 : Nonempty \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsFiniteMeasure \u03bc\ninst\u271d\u00b2 : NormedAddCommGroup F\nf : \u03a9 \u2192 F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nhf : AEStronglyMeasurable f \u03bc\n\u22a2 AEStronglyMeasurable (fun \u03c9 => \u222b (y : \u03a9), f y \u2202\u2191(condexpKernel \u03bc m) \u03c9) \u03bc", "state_after": "\u03a9 : Type u_1\nF : Type u_2\ninst\u271d\u2077 : TopologicalSpace \u03a9\nm m\u03a9 : MeasurableSpace \u03a9\ninst\u271d\u2076 : PolishSpace \u03a9\ninst\u271d\u2075 : BorelSpace \u03a9\ninst\u271d\u2074 : Nonempty \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsFiniteMeasure \u03bc\ninst\u271d\u00b2 : NormedAddCommGroup F\nf : \u03a9 \u2192 F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nhf : AEStronglyMeasurable f \u03bc\n\u22a2 AEStronglyMeasurable (fun \u03c9 => \u222b (y : \u03a9), f y \u2202\u2191(condDistrib id id \u03bc) (id \u03c9)) \u03bc"}, {"tactic": "exact AEStronglyMeasurable.integral_condDistrib\n  (aemeasurable_id'' \u03bc (inf_le_right : m \u2293 m\u03a9 \u2264 m\u03a9)) aemeasurable_id\n  (hf.comp_snd_map_prod_id inf_le_right)", "annotated_tactic": ["exact <a>AEStronglyMeasurable.integral_condDistrib</a>\n    (<a>aemeasurable_id''</a> \u03bc (<a>inf_le_right</a> : m \u2293 m\u03a9 \u2264 m\u03a9)) <a>aemeasurable_id</a>\n    (hf.comp_snd_map_prod_id <a>inf_le_right</a>)", [{"full_name": "MeasureTheory.AEStronglyMeasurable.integral_condDistrib", "def_path": "Mathlib/Probability/Kernel/CondDistrib.lean", "def_pos": [95, 9], "def_end_pos": [95, 71]}, {"full_name": "aemeasurable_id''", "def_path": "Mathlib/MeasureTheory/Measure/AEMeasurable.lean", "def_pos": [42, 9], "def_end_pos": [42, 26]}, {"full_name": "inf_le_right", "def_path": "Mathlib/Order/Lattice.lean", "def_pos": [399, 9], "def_end_pos": [399, 21]}, {"full_name": "aemeasurable_id", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [751, 9], "def_end_pos": [751, 24]}, {"full_name": "inf_le_right", "def_path": "Mathlib/Order/Lattice.lean", "def_pos": [399, 9], "def_end_pos": [399, 21]}]], "state_before": "\u03a9 : Type u_1\nF : Type u_2\ninst\u271d\u2077 : TopologicalSpace \u03a9\nm m\u03a9 : MeasurableSpace \u03a9\ninst\u271d\u2076 : PolishSpace \u03a9\ninst\u271d\u2075 : BorelSpace \u03a9\ninst\u271d\u2074 : Nonempty \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsFiniteMeasure \u03bc\ninst\u271d\u00b2 : NormedAddCommGroup F\nf : \u03a9 \u2192 F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nhf : AEStronglyMeasurable f \u03bc\n\u22a2 AEStronglyMeasurable (fun \u03c9 => \u222b (y : \u03a9), f y \u2202\u2191(condDistrib id id \u03bc) (id \u03c9)) \u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "full_name": "MeasureTheory.L1.ofReal_norm_eq_lintegral", "start": [1377, 1], "end": [1380, 45], "traced_tactics": [{"tactic": "rw [norm_def, ENNReal.ofReal_toReal]", "annotated_tactic": ["rw [<a>norm_def</a>, <a>ENNReal.ofReal_toReal</a>]", [{"full_name": "MeasureTheory.L1.norm_def", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [1361, 9], "def_end_pos": [1361, 17]}, {"full_name": "ENNReal.ofReal_toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [186, 9], "def_end_pos": [186, 22]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : NormedAddCommGroup \u03b3\nf : { x // x \u2208 Lp \u03b2 1 }\n\u22a2 ENNReal.ofReal \u2016f\u2016 = \u222b\u207b (x : \u03b1), \u2191\u2016\u2191\u2191f x\u2016\u208a \u2202\u03bc", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : NormedAddCommGroup \u03b3\nf : { x // x \u2208 Lp \u03b2 1 }\n\u22a2 \u222b\u207b (a : \u03b1), \u2191\u2016\u2191\u2191f a\u2016\u208a \u2202\u03bc \u2260 \u22a4"}, {"tactic": "exact ne_of_lt (hasFiniteIntegral_coeFn f)", "annotated_tactic": ["exact <a>ne_of_lt</a> (<a>hasFiniteIntegral_coeFn</a> f)", [{"full_name": "ne_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [101, 9], "def_end_pos": [101, 17]}, {"full_name": "MeasureTheory.L1.hasFiniteIntegral_coeFn", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [1329, 9], "def_end_pos": [1329, 32]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : NormedAddCommGroup \u03b3\nf : { x // x \u2208 Lp \u03b2 1 }\n\u22a2 \u222b\u207b (a : \u03b1), \u2191\u2016\u2191\u2191f a\u2016\u208a \u2202\u03bc \u2260 \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "Finset.insert_ne_empty", "start": [1174, 1], "end": [1175, 33], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Haar/Basic.lean", "full_name": "MeasureTheory.Measure.haar.chaar_nonneg", "start": [424, 1], "end": [425, 86], "traced_tactics": [{"tactic": "have := chaar_mem_haarProduct K\u2080 K (mem_univ _)", "annotated_tactic": ["have := <a>chaar_mem_haarProduct</a> K\u2080 K (<a>mem_univ</a> _)", [{"full_name": "MeasureTheory.Measure.haar.chaar_mem_haarProduct", "def_path": "Mathlib/MeasureTheory/Measure/Haar/Basic.lean", "def_pos": [410, 9], "def_end_pos": [410, 30]}, {"full_name": "Set.mem_univ", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [676, 9], "def_end_pos": [676, 17]}]], "state_before": "G : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK : Compacts G\n\u22a2 0 \u2264 chaar K\u2080 K", "state_after": "G : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK : Compacts G\nthis : chaar K\u2080 K \u2208 (fun K => Icc 0 \u2191(index \u2191K \u2191K\u2080)) K\n\u22a2 0 \u2264 chaar K\u2080 K"}, {"tactic": "rw [mem_Icc] at this", "annotated_tactic": ["rw [<a>mem_Icc</a>] at this", [{"full_name": "Set.mem_Icc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [131, 9], "def_end_pos": [131, 16]}]], "state_before": "G : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK : Compacts G\nthis : chaar K\u2080 K \u2208 (fun K => Icc 0 \u2191(index \u2191K \u2191K\u2080)) K\n\u22a2 0 \u2264 chaar K\u2080 K", "state_after": "G : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK : Compacts G\nthis : 0 \u2264 chaar K\u2080 K \u2227 chaar K\u2080 K \u2264 \u2191(index \u2191K \u2191K\u2080)\n\u22a2 0 \u2264 chaar K\u2080 K"}, {"tactic": "exact this.1", "annotated_tactic": ["exact this.1", []], "state_before": "G : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK : Compacts G\nthis : 0 \u2264 chaar K\u2080 K \u2227 chaar K\u2080 K \u2264 \u2191(index \u2191K \u2191K\u2080)\n\u22a2 0 \u2264 chaar K\u2080 K", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "full_name": "intervalIntegral.integral_non_aestronglyMeasurable_of_le", "start": [521, 1], "end": [523, 61], "traced_tactics": [{"tactic": "rwa [uIoc_of_le h]", "annotated_tactic": ["rwa [<a>uIoc_of_le</a> h]", [{"full_name": "Set.uIoc_of_le", "def_path": "Mathlib/Data/Set/Intervals/UnorderedInterval.lean", "def_pos": [288, 15], "def_end_pos": [288, 25]}]], "state_before": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\na b : \u211d\nf g : \u211d \u2192 E\n\u03bc : Measure \u211d\nh : a \u2264 b\nhf : \u00acAEStronglyMeasurable f (Measure.restrict \u03bc (Ioc a b))\n\u22a2 \u00acAEStronglyMeasurable (fun x => f x) (Measure.restrict \u03bc (\u0399 a b))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "full_name": "MeasureTheory.tendsto_lintegral_norm_of_dominated_convergence", "start": [327, 1], "end": [374, 12], "traced_tactics": [{"tactic": "have f_measurable : AEStronglyMeasurable f \u03bc :=\n  aestronglyMeasurable_of_tendsto_ae _ F_measurable h_lim", "annotated_tactic": ["have f_measurable : <a>AEStronglyMeasurable</a> f \u03bc :=\n    <a>aestronglyMeasurable_of_tendsto_ae</a> _ F_measurable h_lim", [{"full_name": "MeasureTheory.AEStronglyMeasurable", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [93, 5], "def_end_pos": [93, 25]}, {"full_name": "aestronglyMeasurable_of_tendsto_ae", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1664, 9], "def_end_pos": [1664, 50]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : NormedAddCommGroup \u03b3\nF\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\nf\u271d : \u03b1 \u2192 \u03b2\nbound\u271d : \u03b1 \u2192 \u211d\nF : \u2115 \u2192 \u03b1 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b2\nbound : \u03b1 \u2192 \u211d\nF_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (F n) \u03bc\nbound_hasFiniteIntegral : HasFiniteIntegral bound\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016F n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a))\n\u22a2 Tendsto (fun n => \u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016F n a - f a\u2016 \u2202\u03bc) atTop (\ud835\udcdd 0)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : NormedAddCommGroup \u03b3\nF\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\nf\u271d : \u03b1 \u2192 \u03b2\nbound\u271d : \u03b1 \u2192 \u211d\nF : \u2115 \u2192 \u03b1 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b2\nbound : \u03b1 \u2192 \u211d\nF_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (F n) \u03bc\nbound_hasFiniteIntegral : HasFiniteIntegral bound\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016F n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a))\nf_measurable : AEStronglyMeasurable f \u03bc\n\u22a2 Tendsto (fun n => \u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016F n a - f a\u2016 \u2202\u03bc) atTop (\ud835\udcdd 0)"}, {"tactic": "let b a := 2 * ENNReal.ofReal (bound a)", "annotated_tactic": ["let b a := 2 * <a>ENNReal.ofReal</a> (bound a)", [{"full_name": "ENNReal.ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [172, 29], "def_end_pos": [172, 35]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : NormedAddCommGroup \u03b3\nF\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\nf\u271d : \u03b1 \u2192 \u03b2\nbound\u271d : \u03b1 \u2192 \u211d\nF : \u2115 \u2192 \u03b1 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b2\nbound : \u03b1 \u2192 \u211d\nF_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (F n) \u03bc\nbound_hasFiniteIntegral : HasFiniteIntegral bound\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016F n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a))\nf_measurable : AEStronglyMeasurable f \u03bc\n\u22a2 Tendsto (fun n => \u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016F n a - f a\u2016 \u2202\u03bc) atTop (\ud835\udcdd 0)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : NormedAddCommGroup \u03b3\nF\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\nf\u271d : \u03b1 \u2192 \u03b2\nbound\u271d : \u03b1 \u2192 \u211d\nF : \u2115 \u2192 \u03b1 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b2\nbound : \u03b1 \u2192 \u211d\nF_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (F n) \u03bc\nbound_hasFiniteIntegral : HasFiniteIntegral bound\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016F n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a))\nf_measurable : AEStronglyMeasurable f \u03bc\nb : \u03b1 \u2192 \u211d\u22650\u221e := fun a => 2 * ENNReal.ofReal (bound a)\n\u22a2 Tendsto (fun n => \u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016F n a - f a\u2016 \u2202\u03bc) atTop (\ud835\udcdd 0)"}, {"tactic": "have h : \u2200\u1d50 a \u2202\u03bc, Tendsto (fun n => ENNReal.ofReal \u2016F n a - f a\u2016) atTop (\ud835\udcdd 0) := by\n  rw [\u2190 ENNReal.ofReal_zero]\n  refine' h_lim.mono fun a h => (continuous_ofReal.tendsto _).comp _\n  rwa [\u2190 tendsto_iff_norm_sub_tendsto_zero]", "annotated_tactic": ["have h : \u2200\u1d50 a \u2202\u03bc, <a>Tendsto</a> (fun n => <a>ENNReal.ofReal</a> \u2016F n a - f a\u2016) <a>atTop</a> (\ud835\udcdd 0) := by\n    rw [\u2190 <a>ENNReal.ofReal_zero</a>]\n    refine' h_lim.mono fun a h => (continuous_ofReal.tendsto _).<a>comp</a> _\n    rwa [\u2190 <a>tendsto_iff_norm_sub_tendsto_zero</a>]", [{"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2939, 5], "def_end_pos": [2939, 12]}, {"full_name": "ENNReal.ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [172, 29], "def_end_pos": [172, 35]}, {"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}, {"full_name": "ENNReal.ofReal_zero", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [245, 17], "def_end_pos": [245, 28]}, {"full_name": "Filter.Tendsto.comp", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [3032, 9], "def_end_pos": [3032, 21]}, {"full_name": "tendsto_iff_norm_sub_tendsto_zero", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [1079, 3], "def_end_pos": [1079, 14]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : NormedAddCommGroup \u03b3\nF\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\nf\u271d : \u03b1 \u2192 \u03b2\nbound\u271d : \u03b1 \u2192 \u211d\nF : \u2115 \u2192 \u03b1 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b2\nbound : \u03b1 \u2192 \u211d\nF_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (F n) \u03bc\nbound_hasFiniteIntegral : HasFiniteIntegral bound\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016F n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a))\nf_measurable : AEStronglyMeasurable f \u03bc\nb : \u03b1 \u2192 \u211d\u22650\u221e := fun a => 2 * ENNReal.ofReal (bound a)\nhb : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, ENNReal.ofReal \u2016F n a - f a\u2016 \u2264 b a\n\u22a2 Tendsto (fun n => \u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016F n a - f a\u2016 \u2202\u03bc) atTop (\ud835\udcdd 0)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : NormedAddCommGroup \u03b3\nF\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\nf\u271d : \u03b1 \u2192 \u03b2\nbound\u271d : \u03b1 \u2192 \u211d\nF : \u2115 \u2192 \u03b1 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b2\nbound : \u03b1 \u2192 \u211d\nF_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (F n) \u03bc\nbound_hasFiniteIntegral : HasFiniteIntegral bound\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016F n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a))\nf_measurable : AEStronglyMeasurable f \u03bc\nb : \u03b1 \u2192 \u211d\u22650\u221e := fun a => 2 * ENNReal.ofReal (bound a)\nhb : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, ENNReal.ofReal \u2016F n a - f a\u2016 \u2264 b a\nh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => ENNReal.ofReal \u2016F n a - f a\u2016) atTop (\ud835\udcdd 0)\n\u22a2 Tendsto (fun n => \u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016F n a - f a\u2016 \u2202\u03bc) atTop (\ud835\udcdd 0)"}, {"tactic": "suffices h : Tendsto (fun n => \u222b\u207b a, ENNReal.ofReal \u2016F n a - f a\u2016 \u2202\u03bc) atTop (\ud835\udcdd (\u222b\u207b _ : \u03b1, 0 \u2202\u03bc))", "annotated_tactic": ["suffices h : <a>Tendsto</a> (fun n => \u222b\u207b a, <a>ENNReal.ofReal</a> \u2016F n a - f a\u2016 \u2202\u03bc) <a>atTop</a> (\ud835\udcdd (\u222b\u207b _ : \u03b1, 0 \u2202\u03bc))", [{"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2939, 5], "def_end_pos": [2939, 12]}, {"full_name": "ENNReal.ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [172, 29], "def_end_pos": [172, 35]}, {"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : NormedAddCommGroup \u03b3\nF\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\nf\u271d : \u03b1 \u2192 \u03b2\nbound\u271d : \u03b1 \u2192 \u211d\nF : \u2115 \u2192 \u03b1 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b2\nbound : \u03b1 \u2192 \u211d\nF_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (F n) \u03bc\nbound_hasFiniteIntegral : HasFiniteIntegral bound\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016F n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a))\nf_measurable : AEStronglyMeasurable f \u03bc\nb : \u03b1 \u2192 \u211d\u22650\u221e := fun a => 2 * ENNReal.ofReal (bound a)\nhb : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, ENNReal.ofReal \u2016F n a - f a\u2016 \u2264 b a\nh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => ENNReal.ofReal \u2016F n a - f a\u2016) atTop (\ud835\udcdd 0)\n\u22a2 Tendsto (fun n => \u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016F n a - f a\u2016 \u2202\u03bc) atTop (\ud835\udcdd 0)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : NormedAddCommGroup \u03b3\nF\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\nf\u271d : \u03b1 \u2192 \u03b2\nbound\u271d : \u03b1 \u2192 \u211d\nF : \u2115 \u2192 \u03b1 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b2\nbound : \u03b1 \u2192 \u211d\nF_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (F n) \u03bc\nbound_hasFiniteIntegral : HasFiniteIntegral bound\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016F n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a))\nf_measurable : AEStronglyMeasurable f \u03bc\nb : \u03b1 \u2192 \u211d\u22650\u221e := fun a => 2 * ENNReal.ofReal (bound a)\nhb : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, ENNReal.ofReal \u2016F n a - f a\u2016 \u2264 b a\nh\u271d : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => ENNReal.ofReal \u2016F n a - f a\u2016) atTop (\ud835\udcdd 0)\nh : Tendsto (fun n => \u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016F n a - f a\u2016 \u2202\u03bc) atTop (\ud835\udcdd (\u222b\u207b (x : \u03b1), 0 \u2202\u03bc))\n\u22a2 Tendsto (fun n => \u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016F n a - f a\u2016 \u2202\u03bc) atTop (\ud835\udcdd 0)\n\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : NormedAddCommGroup \u03b3\nF\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\nf\u271d : \u03b1 \u2192 \u03b2\nbound\u271d : \u03b1 \u2192 \u211d\nF : \u2115 \u2192 \u03b1 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b2\nbound : \u03b1 \u2192 \u211d\nF_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (F n) \u03bc\nbound_hasFiniteIntegral : HasFiniteIntegral bound\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016F n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a))\nf_measurable : AEStronglyMeasurable f \u03bc\nb : \u03b1 \u2192 \u211d\u22650\u221e := fun a => 2 * ENNReal.ofReal (bound a)\nhb : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, ENNReal.ofReal \u2016F n a - f a\u2016 \u2264 b a\nh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => ENNReal.ofReal \u2016F n a - f a\u2016) atTop (\ud835\udcdd 0)\n\u22a2 Tendsto (fun n => \u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016F n a - f a\u2016 \u2202\u03bc) atTop (\ud835\udcdd (\u222b\u207b (x : \u03b1), 0 \u2202\u03bc))"}, {"tactic": "refine' tendsto_lintegral_of_dominated_convergence' _ _ hb _ _", "annotated_tactic": ["refine' <a>tendsto_lintegral_of_dominated_convergence'</a> _ _ hb _ _", [{"full_name": "MeasureTheory.tendsto_lintegral_of_dominated_convergence'", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [1077, 9], "def_end_pos": [1077, 52]}]], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : NormedAddCommGroup \u03b3\nF\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\nf\u271d : \u03b1 \u2192 \u03b2\nbound\u271d : \u03b1 \u2192 \u211d\nF : \u2115 \u2192 \u03b1 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b2\nbound : \u03b1 \u2192 \u211d\nF_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (F n) \u03bc\nbound_hasFiniteIntegral : HasFiniteIntegral bound\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016F n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a))\nf_measurable : AEStronglyMeasurable f \u03bc\nb : \u03b1 \u2192 \u211d\u22650\u221e := fun a => 2 * ENNReal.ofReal (bound a)\nhb : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, ENNReal.ofReal \u2016F n a - f a\u2016 \u2264 b a\nh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => ENNReal.ofReal \u2016F n a - f a\u2016) atTop (\ud835\udcdd 0)\n\u22a2 Tendsto (fun n => \u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016F n a - f a\u2016 \u2202\u03bc) atTop (\ud835\udcdd (\u222b\u207b (x : \u03b1), 0 \u2202\u03bc))", "state_after": "case h.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : NormedAddCommGroup \u03b3\nF\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\nf\u271d : \u03b1 \u2192 \u03b2\nbound\u271d : \u03b1 \u2192 \u211d\nF : \u2115 \u2192 \u03b1 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b2\nbound : \u03b1 \u2192 \u211d\nF_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (F n) \u03bc\nbound_hasFiniteIntegral : HasFiniteIntegral bound\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016F n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a))\nf_measurable : AEStronglyMeasurable f \u03bc\nb : \u03b1 \u2192 \u211d\u22650\u221e := fun a => 2 * ENNReal.ofReal (bound a)\nhb : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, ENNReal.ofReal \u2016F n a - f a\u2016 \u2264 b a\nh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => ENNReal.ofReal \u2016F n a - f a\u2016) atTop (\ud835\udcdd 0)\n\u22a2 \u2200 (n : \u2115), AEMeasurable fun a => ENNReal.ofReal \u2016F n a - f a\u2016\n\ncase h.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : NormedAddCommGroup \u03b3\nF\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\nf\u271d : \u03b1 \u2192 \u03b2\nbound\u271d : \u03b1 \u2192 \u211d\nF : \u2115 \u2192 \u03b1 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b2\nbound : \u03b1 \u2192 \u211d\nF_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (F n) \u03bc\nbound_hasFiniteIntegral : HasFiniteIntegral bound\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016F n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a))\nf_measurable : AEStronglyMeasurable f \u03bc\nb : \u03b1 \u2192 \u211d\u22650\u221e := fun a => 2 * ENNReal.ofReal (bound a)\nhb : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, ENNReal.ofReal \u2016F n a - f a\u2016 \u2264 b a\nh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => ENNReal.ofReal \u2016F n a - f a\u2016) atTop (\ud835\udcdd 0)\n\u22a2 \u222b\u207b (a : \u03b1), b a \u2202\u03bc \u2260 \u22a4\n\ncase h.refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : NormedAddCommGroup \u03b3\nF\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\nf\u271d : \u03b1 \u2192 \u03b2\nbound\u271d : \u03b1 \u2192 \u211d\nF : \u2115 \u2192 \u03b1 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b2\nbound : \u03b1 \u2192 \u211d\nF_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (F n) \u03bc\nbound_hasFiniteIntegral : HasFiniteIntegral bound\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016F n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a))\nf_measurable : AEStronglyMeasurable f \u03bc\nb : \u03b1 \u2192 \u211d\u22650\u221e := fun a => 2 * ENNReal.ofReal (bound a)\nhb : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, ENNReal.ofReal \u2016F n a - f a\u2016 \u2264 b a\nh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => ENNReal.ofReal \u2016F n a - f a\u2016) atTop (\ud835\udcdd 0)\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => ENNReal.ofReal \u2016F n a - f a\u2016) atTop (\ud835\udcdd 0)"}, {"tactic": "intro n", "annotated_tactic": ["intro n", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : NormedAddCommGroup \u03b3\nF\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\nf\u271d : \u03b1 \u2192 \u03b2\nbound\u271d : \u03b1 \u2192 \u211d\nF : \u2115 \u2192 \u03b1 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b2\nbound : \u03b1 \u2192 \u211d\nF_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (F n) \u03bc\nbound_hasFiniteIntegral : HasFiniteIntegral bound\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016F n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a))\nf_measurable : AEStronglyMeasurable f \u03bc\nb : \u03b1 \u2192 \u211d\u22650\u221e := fun a => 2 * ENNReal.ofReal (bound a)\n\u22a2 \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, ENNReal.ofReal \u2016F n a - f a\u2016 \u2264 b a", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : NormedAddCommGroup \u03b3\nF\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\nf\u271d : \u03b1 \u2192 \u03b2\nbound\u271d : \u03b1 \u2192 \u211d\nF : \u2115 \u2192 \u03b1 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b2\nbound : \u03b1 \u2192 \u211d\nF_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (F n) \u03bc\nbound_hasFiniteIntegral : HasFiniteIntegral bound\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016F n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a))\nf_measurable : AEStronglyMeasurable f \u03bc\nb : \u03b1 \u2192 \u211d\u22650\u221e := fun a => 2 * ENNReal.ofReal (bound a)\nn : \u2115\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, ENNReal.ofReal \u2016F n a - f a\u2016 \u2264 b a"}, {"tactic": "filter_upwards [all_ae_ofReal_F_le_bound h_bound n,\n  all_ae_ofReal_f_le_bound h_bound h_lim] with a h\u2081 h\u2082", "annotated_tactic": ["filter_upwards [<a>all_ae_ofReal_F_le_bound</a> h_bound n,\n      <a>all_ae_ofReal_f_le_bound</a> h_bound h_lim] with a h\u2081 h\u2082", [{"full_name": "MeasureTheory.all_ae_ofReal_F_le_bound", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [290, 9], "def_end_pos": [290, 33]}, {"full_name": "MeasureTheory.all_ae_ofReal_f_le_bound", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [301, 9], "def_end_pos": [301, 33]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : NormedAddCommGroup \u03b3\nF\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\nf\u271d : \u03b1 \u2192 \u03b2\nbound\u271d : \u03b1 \u2192 \u211d\nF : \u2115 \u2192 \u03b1 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b2\nbound : \u03b1 \u2192 \u211d\nF_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (F n) \u03bc\nbound_hasFiniteIntegral : HasFiniteIntegral bound\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016F n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a))\nf_measurable : AEStronglyMeasurable f \u03bc\nb : \u03b1 \u2192 \u211d\u22650\u221e := fun a => 2 * ENNReal.ofReal (bound a)\nn : \u2115\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, ENNReal.ofReal \u2016F n a - f a\u2016 \u2264 b a", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : NormedAddCommGroup \u03b3\nF\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\nf\u271d : \u03b1 \u2192 \u03b2\nbound\u271d : \u03b1 \u2192 \u211d\nF : \u2115 \u2192 \u03b1 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b2\nbound : \u03b1 \u2192 \u211d\nF_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (F n) \u03bc\nbound_hasFiniteIntegral : HasFiniteIntegral bound\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016F n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a))\nf_measurable : AEStronglyMeasurable f \u03bc\nb : \u03b1 \u2192 \u211d\u22650\u221e := fun a => 2 * ENNReal.ofReal (bound a)\nn : \u2115\na : \u03b1\nh\u2081 : ENNReal.ofReal \u2016F n a\u2016 \u2264 ENNReal.ofReal (bound a)\nh\u2082 : ENNReal.ofReal \u2016f a\u2016 \u2264 ENNReal.ofReal (bound a)\n\u22a2 ENNReal.ofReal \u2016F n a - f a\u2016 \u2264 b a"}, {"tactic": "rw [\u2190 ENNReal.ofReal_add]", "annotated_tactic": ["rw [\u2190 <a>ENNReal.ofReal_add</a>]", [{"full_name": "ENNReal.ofReal_add", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2025, 9], "def_end_pos": [2025, 19]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : NormedAddCommGroup \u03b3\nF\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\nf\u271d : \u03b1 \u2192 \u03b2\nbound\u271d : \u03b1 \u2192 \u211d\nF : \u2115 \u2192 \u03b1 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b2\nbound : \u03b1 \u2192 \u211d\nF_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (F n) \u03bc\nbound_hasFiniteIntegral : HasFiniteIntegral bound\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016F n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a))\nf_measurable : AEStronglyMeasurable f \u03bc\nb : \u03b1 \u2192 \u211d\u22650\u221e := fun a => 2 * ENNReal.ofReal (bound a)\nn : \u2115\na : \u03b1\nh\u2081 : ENNReal.ofReal \u2016F n a\u2016 \u2264 ENNReal.ofReal (bound a)\nh\u2082 : ENNReal.ofReal \u2016f a\u2016 \u2264 ENNReal.ofReal (bound a)\n\u22a2 ENNReal.ofReal \u2016F n a - f a\u2016 \u2264 ENNReal.ofReal \u2016F n a\u2016 + ENNReal.ofReal \u2016f a\u2016", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : NormedAddCommGroup \u03b3\nF\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\nf\u271d : \u03b1 \u2192 \u03b2\nbound\u271d : \u03b1 \u2192 \u211d\nF : \u2115 \u2192 \u03b1 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b2\nbound : \u03b1 \u2192 \u211d\nF_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (F n) \u03bc\nbound_hasFiniteIntegral : HasFiniteIntegral bound\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016F n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a))\nf_measurable : AEStronglyMeasurable f \u03bc\nb : \u03b1 \u2192 \u211d\u22650\u221e := fun a => 2 * ENNReal.ofReal (bound a)\nn : \u2115\na : \u03b1\nh\u2081 : ENNReal.ofReal \u2016F n a\u2016 \u2264 ENNReal.ofReal (bound a)\nh\u2082 : ENNReal.ofReal \u2016f a\u2016 \u2264 ENNReal.ofReal (bound a)\n\u22a2 ENNReal.ofReal \u2016F n a - f a\u2016 \u2264 ENNReal.ofReal (\u2016F n a\u2016 + \u2016f a\u2016)\n\ncase hp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : NormedAddCommGroup \u03b3\nF\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\nf\u271d : \u03b1 \u2192 \u03b2\nbound\u271d : \u03b1 \u2192 \u211d\nF : \u2115 \u2192 \u03b1 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b2\nbound : \u03b1 \u2192 \u211d\nF_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (F n) \u03bc\nbound_hasFiniteIntegral : HasFiniteIntegral bound\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016F n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a))\nf_measurable : AEStronglyMeasurable f \u03bc\nb : \u03b1 \u2192 \u211d\u22650\u221e := fun a => 2 * ENNReal.ofReal (bound a)\nn : \u2115\na : \u03b1\nh\u2081 : ENNReal.ofReal \u2016F n a\u2016 \u2264 ENNReal.ofReal (bound a)\nh\u2082 : ENNReal.ofReal \u2016f a\u2016 \u2264 ENNReal.ofReal (bound a)\n\u22a2 0 \u2264 \u2016F n a\u2016\n\ncase hq\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : NormedAddCommGroup \u03b3\nF\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\nf\u271d : \u03b1 \u2192 \u03b2\nbound\u271d : \u03b1 \u2192 \u211d\nF : \u2115 \u2192 \u03b1 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b2\nbound : \u03b1 \u2192 \u211d\nF_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (F n) \u03bc\nbound_hasFiniteIntegral : HasFiniteIntegral bound\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016F n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a))\nf_measurable : AEStronglyMeasurable f \u03bc\nb : \u03b1 \u2192 \u211d\u22650\u221e := fun a => 2 * ENNReal.ofReal (bound a)\nn : \u2115\na : \u03b1\nh\u2081 : ENNReal.ofReal \u2016F n a\u2016 \u2264 ENNReal.ofReal (bound a)\nh\u2082 : ENNReal.ofReal \u2016f a\u2016 \u2264 ENNReal.ofReal (bound a)\n\u22a2 0 \u2264 \u2016f a\u2016"}, {"tactic": "apply ofReal_le_ofReal", "annotated_tactic": ["apply <a>ofReal_le_ofReal</a>", [{"full_name": "ENNReal.ofReal_le_ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2135, 9], "def_end_pos": [2135, 25]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : NormedAddCommGroup \u03b3\nF\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\nf\u271d : \u03b1 \u2192 \u03b2\nbound\u271d : \u03b1 \u2192 \u211d\nF : \u2115 \u2192 \u03b1 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b2\nbound : \u03b1 \u2192 \u211d\nF_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (F n) \u03bc\nbound_hasFiniteIntegral : HasFiniteIntegral bound\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016F n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a))\nf_measurable : AEStronglyMeasurable f \u03bc\nb : \u03b1 \u2192 \u211d\u22650\u221e := fun a => 2 * ENNReal.ofReal (bound a)\nn : \u2115\na : \u03b1\nh\u2081 : ENNReal.ofReal \u2016F n a\u2016 \u2264 ENNReal.ofReal (bound a)\nh\u2082 : ENNReal.ofReal \u2016f a\u2016 \u2264 ENNReal.ofReal (bound a)\n\u22a2 ENNReal.ofReal \u2016F n a - f a\u2016 \u2264 ENNReal.ofReal (\u2016F n a\u2016 + \u2016f a\u2016)\n\ncase hp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : NormedAddCommGroup \u03b3\nF\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\nf\u271d : \u03b1 \u2192 \u03b2\nbound\u271d : \u03b1 \u2192 \u211d\nF : \u2115 \u2192 \u03b1 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b2\nbound : \u03b1 \u2192 \u211d\nF_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (F n) \u03bc\nbound_hasFiniteIntegral : HasFiniteIntegral bound\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016F n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a))\nf_measurable : AEStronglyMeasurable f \u03bc\nb : \u03b1 \u2192 \u211d\u22650\u221e := fun a => 2 * ENNReal.ofReal (bound a)\nn : \u2115\na : \u03b1\nh\u2081 : ENNReal.ofReal \u2016F n a\u2016 \u2264 ENNReal.ofReal (bound a)\nh\u2082 : ENNReal.ofReal \u2016f a\u2016 \u2264 ENNReal.ofReal (bound a)\n\u22a2 0 \u2264 \u2016F n a\u2016\n\ncase hq\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : NormedAddCommGroup \u03b3\nF\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\nf\u271d : \u03b1 \u2192 \u03b2\nbound\u271d : \u03b1 \u2192 \u211d\nF : \u2115 \u2192 \u03b1 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b2\nbound : \u03b1 \u2192 \u211d\nF_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (F n) \u03bc\nbound_hasFiniteIntegral : HasFiniteIntegral bound\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016F n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a))\nf_measurable : AEStronglyMeasurable f \u03bc\nb : \u03b1 \u2192 \u211d\u22650\u221e := fun a => 2 * ENNReal.ofReal (bound a)\nn : \u2115\na : \u03b1\nh\u2081 : ENNReal.ofReal \u2016F n a\u2016 \u2264 ENNReal.ofReal (bound a)\nh\u2082 : ENNReal.ofReal \u2016f a\u2016 \u2264 ENNReal.ofReal (bound a)\n\u22a2 0 \u2264 \u2016f a\u2016", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : NormedAddCommGroup \u03b3\nF\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\nf\u271d : \u03b1 \u2192 \u03b2\nbound\u271d : \u03b1 \u2192 \u211d\nF : \u2115 \u2192 \u03b1 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b2\nbound : \u03b1 \u2192 \u211d\nF_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (F n) \u03bc\nbound_hasFiniteIntegral : HasFiniteIntegral bound\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016F n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a))\nf_measurable : AEStronglyMeasurable f \u03bc\nb : \u03b1 \u2192 \u211d\u22650\u221e := fun a => 2 * ENNReal.ofReal (bound a)\nn : \u2115\na : \u03b1\nh\u2081 : ENNReal.ofReal \u2016F n a\u2016 \u2264 ENNReal.ofReal (bound a)\nh\u2082 : ENNReal.ofReal \u2016f a\u2016 \u2264 ENNReal.ofReal (bound a)\n\u22a2 \u2016F n a - f a\u2016 \u2264 \u2016F n a\u2016 + \u2016f a\u2016\n\ncase hp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : NormedAddCommGroup \u03b3\nF\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\nf\u271d : \u03b1 \u2192 \u03b2\nbound\u271d : \u03b1 \u2192 \u211d\nF : \u2115 \u2192 \u03b1 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b2\nbound : \u03b1 \u2192 \u211d\nF_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (F n) \u03bc\nbound_hasFiniteIntegral : HasFiniteIntegral bound\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016F n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a))\nf_measurable : AEStronglyMeasurable f \u03bc\nb : \u03b1 \u2192 \u211d\u22650\u221e := fun a => 2 * ENNReal.ofReal (bound a)\nn : \u2115\na : \u03b1\nh\u2081 : ENNReal.ofReal \u2016F n a\u2016 \u2264 ENNReal.ofReal (bound a)\nh\u2082 : ENNReal.ofReal \u2016f a\u2016 \u2264 ENNReal.ofReal (bound a)\n\u22a2 0 \u2264 \u2016F n a\u2016\n\ncase hq\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : NormedAddCommGroup \u03b3\nF\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\nf\u271d : \u03b1 \u2192 \u03b2\nbound\u271d : \u03b1 \u2192 \u211d\nF : \u2115 \u2192 \u03b1 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b2\nbound : \u03b1 \u2192 \u211d\nF_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (F n) \u03bc\nbound_hasFiniteIntegral : HasFiniteIntegral bound\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016F n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a))\nf_measurable : AEStronglyMeasurable f \u03bc\nb : \u03b1 \u2192 \u211d\u22650\u221e := fun a => 2 * ENNReal.ofReal (bound a)\nn : \u2115\na : \u03b1\nh\u2081 : ENNReal.ofReal \u2016F n a\u2016 \u2264 ENNReal.ofReal (bound a)\nh\u2082 : ENNReal.ofReal \u2016f a\u2016 \u2264 ENNReal.ofReal (bound a)\n\u22a2 0 \u2264 \u2016f a\u2016"}, {"tactic": "apply norm_sub_le", "annotated_tactic": ["apply <a>norm_sub_le</a>", [{"full_name": "norm_sub_le", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [553, 3], "def_end_pos": [553, 14]}]], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : NormedAddCommGroup \u03b3\nF\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\nf\u271d : \u03b1 \u2192 \u03b2\nbound\u271d : \u03b1 \u2192 \u211d\nF : \u2115 \u2192 \u03b1 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b2\nbound : \u03b1 \u2192 \u211d\nF_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (F n) \u03bc\nbound_hasFiniteIntegral : HasFiniteIntegral bound\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016F n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a))\nf_measurable : AEStronglyMeasurable f \u03bc\nb : \u03b1 \u2192 \u211d\u22650\u221e := fun a => 2 * ENNReal.ofReal (bound a)\nn : \u2115\na : \u03b1\nh\u2081 : ENNReal.ofReal \u2016F n a\u2016 \u2264 ENNReal.ofReal (bound a)\nh\u2082 : ENNReal.ofReal \u2016f a\u2016 \u2264 ENNReal.ofReal (bound a)\n\u22a2 \u2016F n a - f a\u2016 \u2264 \u2016F n a\u2016 + \u2016f a\u2016", "state_after": "no goals"}, {"tactic": "exact norm_nonneg _", "annotated_tactic": ["exact <a>norm_nonneg</a> _", [{"full_name": "norm_nonneg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [500, 30], "def_end_pos": [500, 41]}]], "state_before": "case hp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : NormedAddCommGroup \u03b3\nF\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\nf\u271d : \u03b1 \u2192 \u03b2\nbound\u271d : \u03b1 \u2192 \u211d\nF : \u2115 \u2192 \u03b1 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b2\nbound : \u03b1 \u2192 \u211d\nF_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (F n) \u03bc\nbound_hasFiniteIntegral : HasFiniteIntegral bound\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016F n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a))\nf_measurable : AEStronglyMeasurable f \u03bc\nb : \u03b1 \u2192 \u211d\u22650\u221e := fun a => 2 * ENNReal.ofReal (bound a)\nn : \u2115\na : \u03b1\nh\u2081 : ENNReal.ofReal \u2016F n a\u2016 \u2264 ENNReal.ofReal (bound a)\nh\u2082 : ENNReal.ofReal \u2016f a\u2016 \u2264 ENNReal.ofReal (bound a)\n\u22a2 0 \u2264 \u2016F n a\u2016", "state_after": "no goals"}, {"tactic": "exact norm_nonneg _", "annotated_tactic": ["exact <a>norm_nonneg</a> _", [{"full_name": "norm_nonneg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [500, 30], "def_end_pos": [500, 41]}]], "state_before": "case hq\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : NormedAddCommGroup \u03b3\nF\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\nf\u271d : \u03b1 \u2192 \u03b2\nbound\u271d : \u03b1 \u2192 \u211d\nF : \u2115 \u2192 \u03b1 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b2\nbound : \u03b1 \u2192 \u211d\nF_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (F n) \u03bc\nbound_hasFiniteIntegral : HasFiniteIntegral bound\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016F n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a))\nf_measurable : AEStronglyMeasurable f \u03bc\nb : \u03b1 \u2192 \u211d\u22650\u221e := fun a => 2 * ENNReal.ofReal (bound a)\nn : \u2115\na : \u03b1\nh\u2081 : ENNReal.ofReal \u2016F n a\u2016 \u2264 ENNReal.ofReal (bound a)\nh\u2082 : ENNReal.ofReal \u2016f a\u2016 \u2264 ENNReal.ofReal (bound a)\n\u22a2 0 \u2264 \u2016f a\u2016", "state_after": "no goals"}, {"tactic": "rw [\u2190 two_mul]", "annotated_tactic": ["rw [\u2190 <a>two_mul</a>]", [{"full_name": "two_mul", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [177, 9], "def_end_pos": [177, 16]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : NormedAddCommGroup \u03b3\nF\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\nf\u271d : \u03b1 \u2192 \u03b2\nbound\u271d : \u03b1 \u2192 \u211d\nF : \u2115 \u2192 \u03b1 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b2\nbound : \u03b1 \u2192 \u211d\nF_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (F n) \u03bc\nbound_hasFiniteIntegral : HasFiniteIntegral bound\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016F n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a))\nf_measurable : AEStronglyMeasurable f \u03bc\nb : \u03b1 \u2192 \u211d\u22650\u221e := fun a => 2 * ENNReal.ofReal (bound a)\nn : \u2115\na : \u03b1\nh\u2081 : ENNReal.ofReal \u2016F n a\u2016 \u2264 ENNReal.ofReal (bound a)\nh\u2082 : ENNReal.ofReal \u2016f a\u2016 \u2264 ENNReal.ofReal (bound a)\n\u22a2 ENNReal.ofReal (bound a) + ENNReal.ofReal (bound a) = b a", "state_after": "no goals"}, {"tactic": "rw [\u2190 ENNReal.ofReal_zero]", "annotated_tactic": ["rw [\u2190 <a>ENNReal.ofReal_zero</a>]", [{"full_name": "ENNReal.ofReal_zero", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [245, 17], "def_end_pos": [245, 28]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : NormedAddCommGroup \u03b3\nF\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\nf\u271d : \u03b1 \u2192 \u03b2\nbound\u271d : \u03b1 \u2192 \u211d\nF : \u2115 \u2192 \u03b1 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b2\nbound : \u03b1 \u2192 \u211d\nF_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (F n) \u03bc\nbound_hasFiniteIntegral : HasFiniteIntegral bound\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016F n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a))\nf_measurable : AEStronglyMeasurable f \u03bc\nb : \u03b1 \u2192 \u211d\u22650\u221e := fun a => 2 * ENNReal.ofReal (bound a)\nhb : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, ENNReal.ofReal \u2016F n a - f a\u2016 \u2264 b a\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => ENNReal.ofReal \u2016F n a - f a\u2016) atTop (\ud835\udcdd 0)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : NormedAddCommGroup \u03b3\nF\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\nf\u271d : \u03b1 \u2192 \u03b2\nbound\u271d : \u03b1 \u2192 \u211d\nF : \u2115 \u2192 \u03b1 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b2\nbound : \u03b1 \u2192 \u211d\nF_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (F n) \u03bc\nbound_hasFiniteIntegral : HasFiniteIntegral bound\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016F n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a))\nf_measurable : AEStronglyMeasurable f \u03bc\nb : \u03b1 \u2192 \u211d\u22650\u221e := fun a => 2 * ENNReal.ofReal (bound a)\nhb : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, ENNReal.ofReal \u2016F n a - f a\u2016 \u2264 b a\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => ENNReal.ofReal \u2016F n a - f a\u2016) atTop (\ud835\udcdd (ENNReal.ofReal 0))"}, {"tactic": "refine' h_lim.mono fun a h => (continuous_ofReal.tendsto _).comp _", "annotated_tactic": ["refine' h_lim.mono fun a h => (continuous_ofReal.tendsto _).<a>comp</a> _", [{"full_name": "Filter.Tendsto.comp", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [3032, 9], "def_end_pos": [3032, 21]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : NormedAddCommGroup \u03b3\nF\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\nf\u271d : \u03b1 \u2192 \u03b2\nbound\u271d : \u03b1 \u2192 \u211d\nF : \u2115 \u2192 \u03b1 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b2\nbound : \u03b1 \u2192 \u211d\nF_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (F n) \u03bc\nbound_hasFiniteIntegral : HasFiniteIntegral bound\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016F n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a))\nf_measurable : AEStronglyMeasurable f \u03bc\nb : \u03b1 \u2192 \u211d\u22650\u221e := fun a => 2 * ENNReal.ofReal (bound a)\nhb : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, ENNReal.ofReal \u2016F n a - f a\u2016 \u2264 b a\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => ENNReal.ofReal \u2016F n a - f a\u2016) atTop (\ud835\udcdd (ENNReal.ofReal 0))", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : NormedAddCommGroup \u03b3\nF\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\nf\u271d : \u03b1 \u2192 \u03b2\nbound\u271d : \u03b1 \u2192 \u211d\nF : \u2115 \u2192 \u03b1 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b2\nbound : \u03b1 \u2192 \u211d\nF_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (F n) \u03bc\nbound_hasFiniteIntegral : HasFiniteIntegral bound\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016F n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a))\nf_measurable : AEStronglyMeasurable f \u03bc\nb : \u03b1 \u2192 \u211d\u22650\u221e := fun a => 2 * ENNReal.ofReal (bound a)\nhb : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, ENNReal.ofReal \u2016F n a - f a\u2016 \u2264 b a\na : \u03b1\nh : Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a))\n\u22a2 Tendsto (fun n => \u2016F n a - f a\u2016) atTop (\ud835\udcdd 0)"}, {"tactic": "rwa [\u2190 tendsto_iff_norm_sub_tendsto_zero]", "annotated_tactic": ["rwa [\u2190 <a>tendsto_iff_norm_sub_tendsto_zero</a>]", [{"full_name": "tendsto_iff_norm_sub_tendsto_zero", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [1079, 3], "def_end_pos": [1079, 14]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : NormedAddCommGroup \u03b3\nF\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\nf\u271d : \u03b1 \u2192 \u03b2\nbound\u271d : \u03b1 \u2192 \u211d\nF : \u2115 \u2192 \u03b1 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b2\nbound : \u03b1 \u2192 \u211d\nF_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (F n) \u03bc\nbound_hasFiniteIntegral : HasFiniteIntegral bound\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016F n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a))\nf_measurable : AEStronglyMeasurable f \u03bc\nb : \u03b1 \u2192 \u211d\u22650\u221e := fun a => 2 * ENNReal.ofReal (bound a)\nhb : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, ENNReal.ofReal \u2016F n a - f a\u2016 \u2264 b a\na : \u03b1\nh : Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a))\n\u22a2 Tendsto (fun n => \u2016F n a - f a\u2016) atTop (\ud835\udcdd 0)", "state_after": "no goals"}, {"tactic": "rwa [lintegral_zero] at h", "annotated_tactic": ["rwa [<a>lintegral_zero</a>] at h", [{"full_name": "MeasureTheory.lintegral_zero", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [141, 9], "def_end_pos": [141, 23]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : NormedAddCommGroup \u03b3\nF\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\nf\u271d : \u03b1 \u2192 \u03b2\nbound\u271d : \u03b1 \u2192 \u211d\nF : \u2115 \u2192 \u03b1 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b2\nbound : \u03b1 \u2192 \u211d\nF_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (F n) \u03bc\nbound_hasFiniteIntegral : HasFiniteIntegral bound\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016F n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a))\nf_measurable : AEStronglyMeasurable f \u03bc\nb : \u03b1 \u2192 \u211d\u22650\u221e := fun a => 2 * ENNReal.ofReal (bound a)\nhb : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, ENNReal.ofReal \u2016F n a - f a\u2016 \u2264 b a\nh\u271d : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => ENNReal.ofReal \u2016F n a - f a\u2016) atTop (\ud835\udcdd 0)\nh : Tendsto (fun n => \u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016F n a - f a\u2016 \u2202\u03bc) atTop (\ud835\udcdd (\u222b\u207b (x : \u03b1), 0 \u2202\u03bc))\n\u22a2 Tendsto (fun n => \u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016F n a - f a\u2016 \u2202\u03bc) atTop (\ud835\udcdd 0)", "state_after": "no goals"}, {"tactic": "exact fun n =>\n  measurable_ofReal.comp_aemeasurable ((F_measurable n).sub f_measurable).norm.aemeasurable", "annotated_tactic": ["exact fun n =>\n      measurable_ofReal.comp_aemeasurable ((F_measurable n).<a>sub</a> f_measurable).norm.aemeasurable", [{"full_name": "MeasureTheory.AEStronglyMeasurable.sub", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1322, 3], "def_end_pos": [1322, 14]}]], "state_before": "case h.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : NormedAddCommGroup \u03b3\nF\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\nf\u271d : \u03b1 \u2192 \u03b2\nbound\u271d : \u03b1 \u2192 \u211d\nF : \u2115 \u2192 \u03b1 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b2\nbound : \u03b1 \u2192 \u211d\nF_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (F n) \u03bc\nbound_hasFiniteIntegral : HasFiniteIntegral bound\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016F n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a))\nf_measurable : AEStronglyMeasurable f \u03bc\nb : \u03b1 \u2192 \u211d\u22650\u221e := fun a => 2 * ENNReal.ofReal (bound a)\nhb : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, ENNReal.ofReal \u2016F n a - f a\u2016 \u2264 b a\nh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => ENNReal.ofReal \u2016F n a - f a\u2016) atTop (\ud835\udcdd 0)\n\u22a2 \u2200 (n : \u2115), AEMeasurable fun a => ENNReal.ofReal \u2016F n a - f a\u2016", "state_after": "no goals"}, {"tactic": "rw [hasFiniteIntegral_iff_ofReal] at bound_hasFiniteIntegral", "annotated_tactic": ["rw [<a>hasFiniteIntegral_iff_ofReal</a>] at bound_hasFiniteIntegral", [{"full_name": "MeasureTheory.hasFiniteIntegral_iff_ofReal", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [127, 9], "def_end_pos": [127, 37]}]], "state_before": "case h.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : NormedAddCommGroup \u03b3\nF\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\nf\u271d : \u03b1 \u2192 \u03b2\nbound\u271d : \u03b1 \u2192 \u211d\nF : \u2115 \u2192 \u03b1 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b2\nbound : \u03b1 \u2192 \u211d\nF_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (F n) \u03bc\nbound_hasFiniteIntegral : HasFiniteIntegral bound\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016F n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a))\nf_measurable : AEStronglyMeasurable f \u03bc\nb : \u03b1 \u2192 \u211d\u22650\u221e := fun a => 2 * ENNReal.ofReal (bound a)\nhb : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, ENNReal.ofReal \u2016F n a - f a\u2016 \u2264 b a\nh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => ENNReal.ofReal \u2016F n a - f a\u2016) atTop (\ud835\udcdd 0)\n\u22a2 \u222b\u207b (a : \u03b1), b a \u2202\u03bc \u2260 \u22a4", "state_after": "case h.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : NormedAddCommGroup \u03b3\nF\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\nf\u271d : \u03b1 \u2192 \u03b2\nbound\u271d : \u03b1 \u2192 \u211d\nF : \u2115 \u2192 \u03b1 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b2\nbound : \u03b1 \u2192 \u211d\nF_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (F n) \u03bc\nbound_hasFiniteIntegral : \u222b\u207b (a : \u03b1), ENNReal.ofReal (bound a) \u2202\u03bc < \u22a4\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016F n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a))\nf_measurable : AEStronglyMeasurable f \u03bc\nb : \u03b1 \u2192 \u211d\u22650\u221e := fun a => 2 * ENNReal.ofReal (bound a)\nhb : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, ENNReal.ofReal \u2016F n a - f a\u2016 \u2264 b a\nh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => ENNReal.ofReal \u2016F n a - f a\u2016) atTop (\ud835\udcdd 0)\n\u22a2 \u222b\u207b (a : \u03b1), b a \u2202\u03bc \u2260 \u22a4\n\ncase h.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : NormedAddCommGroup \u03b3\nF\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\nf\u271d : \u03b1 \u2192 \u03b2\nbound\u271d : \u03b1 \u2192 \u211d\nF : \u2115 \u2192 \u03b1 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b2\nbound : \u03b1 \u2192 \u211d\nF_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (F n) \u03bc\nbound_hasFiniteIntegral : HasFiniteIntegral bound\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016F n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a))\nf_measurable : AEStronglyMeasurable f \u03bc\nb : \u03b1 \u2192 \u211d\u22650\u221e := fun a => 2 * ENNReal.ofReal (bound a)\nhb : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, ENNReal.ofReal \u2016F n a - f a\u2016 \u2264 b a\nh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => ENNReal.ofReal \u2016F n a - f a\u2016) atTop (\ud835\udcdd 0)\n\u22a2 0 \u2264\u1d50[\u03bc] bound"}, {"tactic": "filter_upwards [h_bound 0] with _ h using le_trans (norm_nonneg _) h", "annotated_tactic": ["filter_upwards [h_bound 0] with _ h using <a>le_trans</a> (<a>norm_nonneg</a> _) h", [{"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}, {"full_name": "norm_nonneg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [500, 30], "def_end_pos": [500, 41]}]], "state_before": "case h.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : NormedAddCommGroup \u03b3\nF\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\nf\u271d : \u03b1 \u2192 \u03b2\nbound\u271d : \u03b1 \u2192 \u211d\nF : \u2115 \u2192 \u03b1 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b2\nbound : \u03b1 \u2192 \u211d\nF_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (F n) \u03bc\nbound_hasFiniteIntegral : HasFiniteIntegral bound\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016F n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a))\nf_measurable : AEStronglyMeasurable f \u03bc\nb : \u03b1 \u2192 \u211d\u22650\u221e := fun a => 2 * ENNReal.ofReal (bound a)\nhb : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, ENNReal.ofReal \u2016F n a - f a\u2016 \u2264 b a\nh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => ENNReal.ofReal \u2016F n a - f a\u2016) atTop (\ud835\udcdd 0)\n\u22a2 0 \u2264\u1d50[\u03bc] bound", "state_after": "no goals"}, {"tactic": "calc\n  \u222b\u207b a, b a \u2202\u03bc = 2 * \u222b\u207b a, ENNReal.ofReal (bound a) \u2202\u03bc := by\n    rw [lintegral_const_mul']\n    exact coe_ne_top\n  _ \u2260 \u221e := mul_ne_top coe_ne_top bound_hasFiniteIntegral.ne", "annotated_tactic": ["calc\n        \u222b\u207b a, b a \u2202\u03bc = 2 * \u222b\u207b a, <a>ENNReal.ofReal</a> (bound a) \u2202\u03bc := by\n          rw [<a>lintegral_const_mul'</a>]\n          exact <a>coe_ne_top</a>\n        _ \u2260 \u221e := <a>mul_ne_top</a> <a>coe_ne_top</a> bound_hasFiniteIntegral.ne", [{"full_name": "ENNReal.ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [172, 29], "def_end_pos": [172, 35]}, {"full_name": "MeasureTheory.lintegral_const_mul'", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [711, 9], "def_end_pos": [711, 29]}, {"full_name": "ENNReal.coe_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [302, 17], "def_end_pos": [302, 27]}, {"full_name": "ENNReal.mul_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [615, 9], "def_end_pos": [615, 19]}, {"full_name": "ENNReal.coe_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [302, 17], "def_end_pos": [302, 27]}]], "state_before": "case h.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : NormedAddCommGroup \u03b3\nF\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\nf\u271d : \u03b1 \u2192 \u03b2\nbound\u271d : \u03b1 \u2192 \u211d\nF : \u2115 \u2192 \u03b1 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b2\nbound : \u03b1 \u2192 \u211d\nF_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (F n) \u03bc\nbound_hasFiniteIntegral : \u222b\u207b (a : \u03b1), ENNReal.ofReal (bound a) \u2202\u03bc < \u22a4\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016F n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a))\nf_measurable : AEStronglyMeasurable f \u03bc\nb : \u03b1 \u2192 \u211d\u22650\u221e := fun a => 2 * ENNReal.ofReal (bound a)\nhb : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, ENNReal.ofReal \u2016F n a - f a\u2016 \u2264 b a\nh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => ENNReal.ofReal \u2016F n a - f a\u2016) atTop (\ud835\udcdd 0)\n\u22a2 \u222b\u207b (a : \u03b1), b a \u2202\u03bc \u2260 \u22a4", "state_after": "no goals"}, {"tactic": "rw [lintegral_const_mul']", "annotated_tactic": ["rw [<a>lintegral_const_mul'</a>]", [{"full_name": "MeasureTheory.lintegral_const_mul'", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [711, 9], "def_end_pos": [711, 29]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : NormedAddCommGroup \u03b3\nF\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\nf\u271d : \u03b1 \u2192 \u03b2\nbound\u271d : \u03b1 \u2192 \u211d\nF : \u2115 \u2192 \u03b1 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b2\nbound : \u03b1 \u2192 \u211d\nF_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (F n) \u03bc\nbound_hasFiniteIntegral : \u222b\u207b (a : \u03b1), ENNReal.ofReal (bound a) \u2202\u03bc < \u22a4\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016F n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a))\nf_measurable : AEStronglyMeasurable f \u03bc\nb : \u03b1 \u2192 \u211d\u22650\u221e := fun a => 2 * ENNReal.ofReal (bound a)\nhb : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, ENNReal.ofReal \u2016F n a - f a\u2016 \u2264 b a\nh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => ENNReal.ofReal \u2016F n a - f a\u2016) atTop (\ud835\udcdd 0)\n\u22a2 \u222b\u207b (a : \u03b1), b a \u2202\u03bc = 2 * \u222b\u207b (a : \u03b1), ENNReal.ofReal (bound a) \u2202\u03bc", "state_after": "case hr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : NormedAddCommGroup \u03b3\nF\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\nf\u271d : \u03b1 \u2192 \u03b2\nbound\u271d : \u03b1 \u2192 \u211d\nF : \u2115 \u2192 \u03b1 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b2\nbound : \u03b1 \u2192 \u211d\nF_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (F n) \u03bc\nbound_hasFiniteIntegral : \u222b\u207b (a : \u03b1), ENNReal.ofReal (bound a) \u2202\u03bc < \u22a4\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016F n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a))\nf_measurable : AEStronglyMeasurable f \u03bc\nb : \u03b1 \u2192 \u211d\u22650\u221e := fun a => 2 * ENNReal.ofReal (bound a)\nhb : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, ENNReal.ofReal \u2016F n a - f a\u2016 \u2264 b a\nh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => ENNReal.ofReal \u2016F n a - f a\u2016) atTop (\ud835\udcdd 0)\n\u22a2 2 \u2260 \u22a4"}, {"tactic": "exact coe_ne_top", "annotated_tactic": ["exact <a>coe_ne_top</a>", [{"full_name": "ENNReal.coe_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [302, 17], "def_end_pos": [302, 27]}]], "state_before": "case hr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : NormedAddCommGroup \u03b3\nF\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\nf\u271d : \u03b1 \u2192 \u03b2\nbound\u271d : \u03b1 \u2192 \u211d\nF : \u2115 \u2192 \u03b1 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b2\nbound : \u03b1 \u2192 \u211d\nF_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (F n) \u03bc\nbound_hasFiniteIntegral : \u222b\u207b (a : \u03b1), ENNReal.ofReal (bound a) \u2202\u03bc < \u22a4\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016F n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a))\nf_measurable : AEStronglyMeasurable f \u03bc\nb : \u03b1 \u2192 \u211d\u22650\u221e := fun a => 2 * ENNReal.ofReal (bound a)\nhb : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, ENNReal.ofReal \u2016F n a - f a\u2016 \u2264 b a\nh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => ENNReal.ofReal \u2016F n a - f a\u2016) atTop (\ud835\udcdd 0)\n\u22a2 2 \u2260 \u22a4", "state_after": "no goals"}, {"tactic": "exact h", "annotated_tactic": ["exact h", []], "state_before": "case h.refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : NormedAddCommGroup \u03b3\nF\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\nf\u271d : \u03b1 \u2192 \u03b2\nbound\u271d : \u03b1 \u2192 \u211d\nF : \u2115 \u2192 \u03b1 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b2\nbound : \u03b1 \u2192 \u211d\nF_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (F n) \u03bc\nbound_hasFiniteIntegral : HasFiniteIntegral bound\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016F n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a))\nf_measurable : AEStronglyMeasurable f \u03bc\nb : \u03b1 \u2192 \u211d\u22650\u221e := fun a => 2 * ENNReal.ofReal (bound a)\nhb : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, ENNReal.ofReal \u2016F n a - f a\u2016 \u2264 b a\nh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => ENNReal.ofReal \u2016F n a - f a\u2016) atTop (\ud835\udcdd 0)\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => ENNReal.ofReal \u2016F n a - f a\u2016) atTop (\ud835\udcdd 0)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/RBMap/Lemmas.lean", "full_name": "Std.RBSet.find?_insert_of_eq", "start": [722, 1], "end": [724, 46], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Basic.lean", "full_name": "Set.eq_of_inclusion_surjective", "start": [2840, 1], "end": [2844, 64], "traced_tactics": [{"tactic": "refine' Set.Subset.antisymm h (fun x hx => _)", "annotated_tactic": ["refine' <a>Set.Subset.antisymm</a> h (fun x hx => _)", [{"full_name": "Set.Subset.antisymm", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [370, 9], "def_end_pos": [370, 24]}]], "state_before": "\u03b1 : Type u_1\ns\u271d t\u271d u s t : Set \u03b1\nh : s \u2286 t\nh_surj : Surjective (inclusion h)\n\u22a2 s = t", "state_after": "\u03b1 : Type u_1\ns\u271d t\u271d u s t : Set \u03b1\nh : s \u2286 t\nh_surj : Surjective (inclusion h)\nx : \u03b1\nhx : x \u2208 t\n\u22a2 x \u2208 s"}, {"tactic": "obtain \u27e8y, hy\u27e9 := h_surj \u27e8x, hx\u27e9", "annotated_tactic": ["obtain \u27e8y, hy\u27e9 := h_surj \u27e8x, hx\u27e9", []], "state_before": "\u03b1 : Type u_1\ns\u271d t\u271d u s t : Set \u03b1\nh : s \u2286 t\nh_surj : Surjective (inclusion h)\nx : \u03b1\nhx : x \u2208 t\n\u22a2 x \u2208 s", "state_after": "case intro\n\u03b1 : Type u_1\ns\u271d t\u271d u s t : Set \u03b1\nh : s \u2286 t\nh_surj : Surjective (inclusion h)\nx : \u03b1\nhx : x \u2208 t\ny : \u2191s\nhy : inclusion h y = { val := x, property := hx }\n\u22a2 x \u2208 s"}, {"tactic": "exact mem_of_eq_of_mem (congr_arg Subtype.val hy).symm y.prop", "annotated_tactic": ["exact <a>mem_of_eq_of_mem</a> (<a>congr_arg</a> <a>Subtype.val</a> hy).<a>symm</a> y.prop", [{"full_name": "Set.mem_of_eq_of_mem", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [366, 9], "def_end_pos": [366, 25]}, {"full_name": "congr_arg", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [43, 7], "def_end_pos": [43, 16]}, {"full_name": "Subtype.val", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [564, 3], "def_end_pos": [564, 6]}, {"full_name": "Eq.symm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [310, 9], "def_end_pos": [310, 16]}]], "state_before": "case intro\n\u03b1 : Type u_1\ns\u271d t\u271d u s t : Set \u03b1\nh : s \u2286 t\nh_surj : Surjective (inclusion h)\nx : \u03b1\nhx : x \u2208 t\ny : \u2191s\nhy : inclusion h y = { val := x, property := hx }\n\u22a2 x \u2208 s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Kernel/Invariance.lean", "full_name": "ProbabilityTheory.kernel.Invariant.comp", "start": [87, 1], "end": [92, 34], "traced_tactics": [{"tactic": "cases' isEmpty_or_nonempty \u03b1 with _ h\u03b1", "annotated_tactic": ["cases' <a>isEmpty_or_nonempty</a> \u03b1 with _ h\u03b1", [{"full_name": "isEmpty_or_nonempty", "def_path": "Mathlib/Logic/IsEmpty.lean", "def_pos": [207, 9], "def_end_pos": [207, 28]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba \u03b7 : { x // x \u2208 kernel \u03b1 \u03b1 }\n\u03bc : Measure \u03b1\ninst\u271d : IsSFiniteKernel \u03ba\nh\u03ba : Invariant \u03ba \u03bc\nh\u03b7 : Invariant \u03b7 \u03bc\n\u22a2 Invariant (\u03ba \u2218\u2096 \u03b7) \u03bc", "state_after": "case inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba \u03b7 : { x // x \u2208 kernel \u03b1 \u03b1 }\n\u03bc : Measure \u03b1\ninst\u271d : IsSFiniteKernel \u03ba\nh\u03ba : Invariant \u03ba \u03bc\nh\u03b7 : Invariant \u03b7 \u03bc\nh\u271d : IsEmpty \u03b1\n\u22a2 Invariant (\u03ba \u2218\u2096 \u03b7) \u03bc\n\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba \u03b7 : { x // x \u2208 kernel \u03b1 \u03b1 }\n\u03bc : Measure \u03b1\ninst\u271d : IsSFiniteKernel \u03ba\nh\u03ba : Invariant \u03ba \u03bc\nh\u03b7 : Invariant \u03b7 \u03bc\nh\u03b1 : Nonempty \u03b1\n\u22a2 Invariant (\u03ba \u2218\u2096 \u03b7) \u03bc"}, {"tactic": "exact Subsingleton.elim _ _", "annotated_tactic": ["exact <a>Subsingleton.elim</a> _ _", [{"full_name": "Subsingleton.elim", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [873, 19], "def_end_pos": [873, 36]}]], "state_before": "case inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba \u03b7 : { x // x \u2208 kernel \u03b1 \u03b1 }\n\u03bc : Measure \u03b1\ninst\u271d : IsSFiniteKernel \u03ba\nh\u03ba : Invariant \u03ba \u03bc\nh\u03b7 : Invariant \u03b7 \u03bc\nh\u271d : IsEmpty \u03b1\n\u22a2 Invariant (\u03ba \u2218\u2096 \u03b7) \u03bc", "state_after": "no goals"}, {"tactic": "simp_rw [Invariant, \u2190 comp_const_apply_eq_bind (\u03ba \u2218\u2096 \u03b7) \u03bc h\u03b1.some, comp_assoc, h\u03b7.comp_const,\n  h\u03ba.comp_const, const_apply]", "annotated_tactic": ["simp_rw [<a>Invariant</a>, \u2190 <a>comp_const_apply_eq_bind</a> (\u03ba \u2218\u2096 \u03b7) \u03bc h\u03b1.some, <a>comp_assoc</a>, h\u03b7.comp_const,\n      h\u03ba.comp_const, <a>const_apply</a>]", [{"full_name": "ProbabilityTheory.kernel.Invariant", "def_path": "Mathlib/Probability/Kernel/Invariance.lean", "def_pos": [73, 5], "def_end_pos": [73, 14]}, {"full_name": "ProbabilityTheory.kernel.comp_const_apply_eq_bind", "def_path": "Mathlib/Probability/Kernel/Invariance.lean", "def_pos": [63, 9], "def_end_pos": [63, 33]}, {"full_name": "ProbabilityTheory.kernel.comp_assoc", "def_path": "Mathlib/Probability/Kernel/Composition.lean", "def_pos": [886, 9], "def_end_pos": [886, 19]}, {"full_name": "ProbabilityTheory.kernel.const_apply", "def_path": "Mathlib/Probability/Kernel/Basic.lean", "def_pos": [445, 9], "def_end_pos": [445, 20]}]], "state_before": "case inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba \u03b7 : { x // x \u2208 kernel \u03b1 \u03b1 }\n\u03bc : Measure \u03b1\ninst\u271d : IsSFiniteKernel \u03ba\nh\u03ba : Invariant \u03ba \u03bc\nh\u03b7 : Invariant \u03b7 \u03bc\nh\u03b1 : Nonempty \u03b1\n\u22a2 Invariant (\u03ba \u2218\u2096 \u03b7) \u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Finite.lean", "full_name": "Set.Finite.bddBelow_biUnion", "start": [1643, 1], "end": [1645, 45], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "full_name": "List.mem_inter_iff", "start": [1617, 9], "end": [1619, 49], "traced_tactics": [{"tactic": "cases l\u2081 <;> simp [List.inter_def, mem_filter]", "annotated_tactic": ["cases l\u2081 <;> simp [<a>List.inter_def</a>, <a>mem_filter</a>]", [{"full_name": "List.inter_def", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [1615, 9], "def_end_pos": [1615, 18]}, {"full_name": "List.mem_filter", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [1247, 9], "def_end_pos": [1247, 19]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nx : \u03b1\nl\u2081 l\u2082 : List \u03b1\n\u22a2 x \u2208 l\u2081 \u2229 l\u2082 \u2194 x \u2208 l\u2081 \u2227 x \u2208 l\u2082", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/List/Basic.lean", "full_name": "List.modifyNth_eq_modifyNthTR", "start": [557, 10], "end": [558, 56], "traced_tactics": [{"tactic": "funext \u03b1 f n l", "annotated_tactic": ["funext \u03b1 f n l", []], "state_before": "\u22a2 @modifyNth = @modifyNthTR", "state_after": "case h.h.h.h\n\u03b1 : Type u_1\nf : \u03b1 \u2192 \u03b1\nn : Nat\nl : List \u03b1\n\u22a2 modifyNth f n l = modifyNthTR f n l"}, {"tactic": "simp [modifyNthTR, modifyNthTR_go_eq]", "annotated_tactic": ["simp [<a>modifyNthTR</a>, <a>modifyNthTR_go_eq</a>]", [{"full_name": "List.modifyNthTR", "def_path": "lake-packages/std/Std/Data/List/Basic.lean", "def_pos": [545, 5], "def_end_pos": [545, 16]}, {"full_name": "List.modifyNthTR_go_eq", "def_path": "lake-packages/std/Std/Data/List/Basic.lean", "def_pos": [552, 9], "def_end_pos": [552, 26]}]], "state_before": "case h.h.h.h\n\u03b1 : Type u_1\nf : \u03b1 \u2192 \u03b1\nn : Nat\nl : List \u03b1\n\u22a2 modifyNth f n l = modifyNthTR f n l", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "Finset.nonempty_iff_eq_singleton_default", "start": [756, 1], "end": [757, 49], "traced_tactics": [{"tactic": "simp [eq_singleton_iff_nonempty_unique_mem]", "annotated_tactic": ["simp [<a>eq_singleton_iff_nonempty_unique_mem</a>]", [{"full_name": "Finset.eq_singleton_iff_nonempty_unique_mem", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [744, 9], "def_end_pos": [744, 45]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns\u271d : Finset \u03b1\na b : \u03b1\ninst\u271d : Unique \u03b1\ns : Finset \u03b1\n\u22a2 Finset.Nonempty s \u2194 s = {default}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "full_name": "Int.natAbs_dvd", "start": [652, 1], "end": [655, 41], "traced_tactics": [{"tactic": "rw [\u2190 e]", "annotated_tactic": ["rw [\u2190 e]", []], "state_before": "a b : Int\ne : a = \u2191(natAbs a)\n\u22a2 \u2191(natAbs a) \u2223 b \u2194 a \u2223 b", "state_after": "no goals"}, {"tactic": "rw [\u2190 Int.neg_dvd, \u2190 e]", "annotated_tactic": ["rw [\u2190 <a>Int.neg_dvd</a>, \u2190 e]", [{"full_name": "Int.neg_dvd", "def_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "def_pos": [604, 19], "def_end_pos": [604, 26]}]], "state_before": "a b : Int\ne : a = -\u2191(natAbs a)\n\u22a2 \u2191(natAbs a) \u2223 b \u2194 a \u2223 b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Kernel/CondCdf.lean", "full_name": "ProbabilityTheory.continuousWithinAt_condCdf'_Ici", "start": [753, 1], "end": [770, 22], "traced_tactics": [{"tactic": "rw [\u2190 continuousWithinAt_Ioi_iff_Ici]", "annotated_tactic": ["rw [\u2190 <a>continuousWithinAt_Ioi_iff_Ici</a>]", [{"full_name": "continuousWithinAt_Ioi_iff_Ici", "def_path": "Mathlib/Topology/Algebra/Order/LeftRight.lean", "def_pos": [34, 9], "def_end_pos": [34, 39]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nx : \u211d\n\u22a2 ContinuousWithinAt (condCdf' \u03c1 a) (Ici x) x", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nx : \u211d\n\u22a2 ContinuousWithinAt (condCdf' \u03c1 a) (Ioi x) x"}, {"tactic": "convert Monotone.tendsto_nhdsWithin_Ioi (monotone_condCdf' \u03c1 a) x", "annotated_tactic": ["convert <a>Monotone.tendsto_nhdsWithin_Ioi</a> (<a>monotone_condCdf'</a> \u03c1 a) x", [{"full_name": "Monotone.tendsto_nhdsWithin_Ioi", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [2974, 9], "def_end_pos": [2974, 40]}, {"full_name": "ProbabilityTheory.monotone_condCdf'", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [741, 9], "def_end_pos": [741, 26]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nx : \u211d\n\u22a2 ContinuousWithinAt (condCdf' \u03c1 a) (Ioi x) x", "state_after": "case a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nx : \u211d\n\u22a2 ContinuousWithinAt (condCdf' \u03c1 a) (Ioi x) x \u2194 Tendsto (condCdf' \u03c1 a) (\ud835\udcdd[Ioi x] x) (\ud835\udcdd (sInf (condCdf' \u03c1 a '' Ioi x)))"}, {"tactic": "rw [sInf_image']", "annotated_tactic": ["rw [<a>sInf_image'</a>]", [{"full_name": "sInf_image'", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [798, 9], "def_end_pos": [798, 20]}]], "state_before": "case a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nx : \u211d\n\u22a2 ContinuousWithinAt (condCdf' \u03c1 a) (Ioi x) x \u2194 Tendsto (condCdf' \u03c1 a) (\ud835\udcdd[Ioi x] x) (\ud835\udcdd (sInf (condCdf' \u03c1 a '' Ioi x)))", "state_after": "case a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nx : \u211d\n\u22a2 ContinuousWithinAt (condCdf' \u03c1 a) (Ioi x) x \u2194 Tendsto (condCdf' \u03c1 a) (\ud835\udcdd[Ioi x] x) (\ud835\udcdd (\u2a05 a_1, condCdf' \u03c1 a \u2191a_1))"}, {"tactic": "have h' : \u2a05 r : Ioi x, condCdf' \u03c1 a r = \u2a05 r : { r' : \u211a // x < r' }, condCdf' \u03c1 a r := by\n  refine' Real.iInf_Ioi_eq_iInf_rat_gt x _ (monotone_condCdf' \u03c1 a)\n  refine' \u27e80, fun z => _\u27e9\n  rintro \u27e8u, -, rfl\u27e9\n  exact condCdf'_nonneg \u03c1 a u", "annotated_tactic": ["have h' : \u2a05 r : <a>Ioi</a> x, <a>condCdf'</a> \u03c1 a r = \u2a05 r : { r' : \u211a // x < r' }, <a>condCdf'</a> \u03c1 a r := by\n    refine' <a>Real.iInf_Ioi_eq_iInf_rat_gt</a> x _ (<a>monotone_condCdf'</a> \u03c1 a)\n    refine' \u27e80, fun z => _\u27e9\n    rintro \u27e8u, -, rfl\u27e9\n    exact <a>condCdf'_nonneg</a> \u03c1 a u", [{"full_name": "Set.Ioi", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [79, 5], "def_end_pos": [79, 8]}, {"full_name": "ProbabilityTheory.condCdf'", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [707, 31], "def_end_pos": [707, 39]}, {"full_name": "ProbabilityTheory.condCdf'", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [707, 31], "def_end_pos": [707, 39]}, {"full_name": "Real.iInf_Ioi_eq_iInf_rat_gt", "def_path": "Mathlib/Data/Real/Basic.lean", "def_pos": [946, 9], "def_end_pos": [946, 32]}, {"full_name": "ProbabilityTheory.monotone_condCdf'", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [741, 9], "def_end_pos": [741, 26]}, {"full_name": "ProbabilityTheory.condCdf'_nonneg", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [728, 9], "def_end_pos": [728, 24]}]], "state_before": "case a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nx : \u211d\n\u22a2 ContinuousWithinAt (condCdf' \u03c1 a) (Ioi x) x \u2194 Tendsto (condCdf' \u03c1 a) (\ud835\udcdd[Ioi x] x) (\ud835\udcdd (\u2a05 a_1, condCdf' \u03c1 a \u2191a_1))", "state_after": "case a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nx : \u211d\nh' : \u2a05 r, condCdf' \u03c1 a \u2191r = \u2a05 r, condCdf' \u03c1 a \u2191\u2191r\n\u22a2 ContinuousWithinAt (condCdf' \u03c1 a) (Ioi x) x \u2194 Tendsto (condCdf' \u03c1 a) (\ud835\udcdd[Ioi x] x) (\ud835\udcdd (\u2a05 a_1, condCdf' \u03c1 a \u2191a_1))"}, {"tactic": "have h'' :\n  \u2a05 r : { r' : \u211a // x < r' }, condCdf' \u03c1 a r =\n    \u2a05 r : { r' : \u211a // x < r' }, condCdfRat \u03c1 a r := by\n  congr with r\n  exact condCdf'_eq_condCdfRat \u03c1 a r", "annotated_tactic": ["have h'' :\n    \u2a05 r : { r' : \u211a // x < r' }, <a>condCdf'</a> \u03c1 a r =\n      \u2a05 r : { r' : \u211a // x < r' }, <a>condCdfRat</a> \u03c1 a r := by\n    congr with r\n    exact <a>condCdf'_eq_condCdfRat</a> \u03c1 a r", [{"full_name": "ProbabilityTheory.condCdf'", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [707, 31], "def_end_pos": [707, 39]}, {"full_name": "ProbabilityTheory.condCdfRat", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [570, 19], "def_end_pos": [570, 29]}, {"full_name": "ProbabilityTheory.condCdf'_eq_condCdfRat", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [715, 9], "def_end_pos": [715, 31]}]], "state_before": "case a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nx : \u211d\nh' : \u2a05 r, condCdf' \u03c1 a \u2191r = \u2a05 r, condCdf' \u03c1 a \u2191\u2191r\n\u22a2 ContinuousWithinAt (condCdf' \u03c1 a) (Ioi x) x \u2194 Tendsto (condCdf' \u03c1 a) (\ud835\udcdd[Ioi x] x) (\ud835\udcdd (\u2a05 a_1, condCdf' \u03c1 a \u2191a_1))", "state_after": "case a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nx : \u211d\nh' : \u2a05 r, condCdf' \u03c1 a \u2191r = \u2a05 r, condCdf' \u03c1 a \u2191\u2191r\nh'' : \u2a05 r, condCdf' \u03c1 a \u2191\u2191r = \u2a05 r, condCdfRat \u03c1 a \u2191r\n\u22a2 ContinuousWithinAt (condCdf' \u03c1 a) (Ioi x) x \u2194 Tendsto (condCdf' \u03c1 a) (\ud835\udcdd[Ioi x] x) (\ud835\udcdd (\u2a05 a_1, condCdf' \u03c1 a \u2191a_1))"}, {"tactic": "rw [h', h'', ContinuousWithinAt]", "annotated_tactic": ["rw [h', h'', <a>ContinuousWithinAt</a>]", [{"full_name": "ContinuousWithinAt", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [518, 5], "def_end_pos": [518, 23]}]], "state_before": "case a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nx : \u211d\nh' : \u2a05 r, condCdf' \u03c1 a \u2191r = \u2a05 r, condCdf' \u03c1 a \u2191\u2191r\nh'' : \u2a05 r, condCdf' \u03c1 a \u2191\u2191r = \u2a05 r, condCdfRat \u03c1 a \u2191r\n\u22a2 ContinuousWithinAt (condCdf' \u03c1 a) (Ioi x) x \u2194 Tendsto (condCdf' \u03c1 a) (\ud835\udcdd[Ioi x] x) (\ud835\udcdd (\u2a05 a_1, condCdf' \u03c1 a \u2191a_1))", "state_after": "case a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nx : \u211d\nh' : \u2a05 r, condCdf' \u03c1 a \u2191r = \u2a05 r, condCdf' \u03c1 a \u2191\u2191r\nh'' : \u2a05 r, condCdf' \u03c1 a \u2191\u2191r = \u2a05 r, condCdfRat \u03c1 a \u2191r\n\u22a2 Tendsto (condCdf' \u03c1 a) (\ud835\udcdd[Ioi x] x) (\ud835\udcdd (condCdf' \u03c1 a x)) \u2194\n    Tendsto (condCdf' \u03c1 a) (\ud835\udcdd[Ioi x] x) (\ud835\udcdd (\u2a05 r, condCdfRat \u03c1 a \u2191r))"}, {"tactic": "congr!", "annotated_tactic": ["congr!", []], "state_before": "case a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nx : \u211d\nh' : \u2a05 r, condCdf' \u03c1 a \u2191r = \u2a05 r, condCdf' \u03c1 a \u2191\u2191r\nh'' : \u2a05 r, condCdf' \u03c1 a \u2191\u2191r = \u2a05 r, condCdfRat \u03c1 a \u2191r\n\u22a2 Tendsto (condCdf' \u03c1 a) (\ud835\udcdd[Ioi x] x) (\ud835\udcdd (condCdf' \u03c1 a x)) \u2194\n    Tendsto (condCdf' \u03c1 a) (\ud835\udcdd[Ioi x] x) (\ud835\udcdd (\u2a05 r, condCdfRat \u03c1 a \u2191r))", "state_after": "case a.a.h.e'_5.h.e'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nx : \u211d\nh' : \u2a05 r, condCdf' \u03c1 a \u2191r = \u2a05 r, condCdf' \u03c1 a \u2191\u2191r\nh'' : \u2a05 r, condCdf' \u03c1 a \u2191\u2191r = \u2a05 r, condCdfRat \u03c1 a \u2191r\n\u22a2 condCdf' \u03c1 a x = \u2a05 r, condCdfRat \u03c1 a \u2191r"}, {"tactic": "exact condCdf'_def'", "annotated_tactic": ["exact <a>condCdf'_def'</a>", [{"full_name": "ProbabilityTheory.condCdf'_def'", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [711, 9], "def_end_pos": [711, 22]}]], "state_before": "case a.a.h.e'_5.h.e'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nx : \u211d\nh' : \u2a05 r, condCdf' \u03c1 a \u2191r = \u2a05 r, condCdf' \u03c1 a \u2191\u2191r\nh'' : \u2a05 r, condCdf' \u03c1 a \u2191\u2191r = \u2a05 r, condCdfRat \u03c1 a \u2191r\n\u22a2 condCdf' \u03c1 a x = \u2a05 r, condCdfRat \u03c1 a \u2191r", "state_after": "no goals"}, {"tactic": "refine' Real.iInf_Ioi_eq_iInf_rat_gt x _ (monotone_condCdf' \u03c1 a)", "annotated_tactic": ["refine' <a>Real.iInf_Ioi_eq_iInf_rat_gt</a> x _ (<a>monotone_condCdf'</a> \u03c1 a)", [{"full_name": "Real.iInf_Ioi_eq_iInf_rat_gt", "def_path": "Mathlib/Data/Real/Basic.lean", "def_pos": [946, 9], "def_end_pos": [946, 32]}, {"full_name": "ProbabilityTheory.monotone_condCdf'", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [741, 9], "def_end_pos": [741, 26]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nx : \u211d\n\u22a2 \u2a05 r, condCdf' \u03c1 a \u2191r = \u2a05 r, condCdf' \u03c1 a \u2191\u2191r", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nx : \u211d\n\u22a2 BddBelow (condCdf' \u03c1 a '' Ioi x)"}, {"tactic": "refine' \u27e80, fun z => _\u27e9", "annotated_tactic": ["refine' \u27e80, fun z => _\u27e9", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nx : \u211d\n\u22a2 BddBelow (condCdf' \u03c1 a '' Ioi x)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nx z : \u211d\n\u22a2 z \u2208 condCdf' \u03c1 a '' Ioi x \u2192 0 \u2264 z"}, {"tactic": "rintro \u27e8u, -, rfl\u27e9", "annotated_tactic": ["rintro \u27e8u, -, rfl\u27e9", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nx z : \u211d\n\u22a2 z \u2208 condCdf' \u03c1 a '' Ioi x \u2192 0 \u2264 z", "state_after": "case intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nx u : \u211d\n\u22a2 0 \u2264 condCdf' \u03c1 a u"}, {"tactic": "exact condCdf'_nonneg \u03c1 a u", "annotated_tactic": ["exact <a>condCdf'_nonneg</a> \u03c1 a u", [{"full_name": "ProbabilityTheory.condCdf'_nonneg", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [728, 9], "def_end_pos": [728, 24]}]], "state_before": "case intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nx u : \u211d\n\u22a2 0 \u2264 condCdf' \u03c1 a u", "state_after": "no goals"}, {"tactic": "congr with r", "annotated_tactic": ["congr with r", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nx : \u211d\nh' : \u2a05 r, condCdf' \u03c1 a \u2191r = \u2a05 r, condCdf' \u03c1 a \u2191\u2191r\n\u22a2 \u2a05 r, condCdf' \u03c1 a \u2191\u2191r = \u2a05 r, condCdfRat \u03c1 a \u2191r", "state_after": "case e_s.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nx : \u211d\nh' : \u2a05 r, condCdf' \u03c1 a \u2191r = \u2a05 r, condCdf' \u03c1 a \u2191\u2191r\nr : { r' // x < \u2191r' }\n\u22a2 condCdf' \u03c1 a \u2191\u2191r = condCdfRat \u03c1 a \u2191r"}, {"tactic": "exact condCdf'_eq_condCdfRat \u03c1 a r", "annotated_tactic": ["exact <a>condCdf'_eq_condCdfRat</a> \u03c1 a r", [{"full_name": "ProbabilityTheory.condCdf'_eq_condCdfRat", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [715, 9], "def_end_pos": [715, 31]}]], "state_before": "case e_s.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nx : \u211d\nh' : \u2a05 r, condCdf' \u03c1 a \u2191r = \u2a05 r, condCdf' \u03c1 a \u2191\u2191r\nr : { r' // x < \u2191r' }\n\u22a2 condCdf' \u03c1 a \u2191\u2191r = condCdfRat \u03c1 a \u2191r", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Moments.lean", "full_name": "ProbabilityTheory.cgf_undef", "start": [178, 1], "end": [179, 42], "traced_tactics": [{"tactic": "simp only [cgf, mgf_undef hX, log_zero]", "annotated_tactic": ["simp only [<a>cgf</a>, <a>mgf_undef</a> hX, <a>log_zero</a>]", [{"full_name": "ProbabilityTheory.cgf", "def_path": "Mathlib/Probability/Moments.lean", "def_pos": [108, 5], "def_end_pos": [108, 8]}, {"full_name": "ProbabilityTheory.mgf_undef", "def_path": "Mathlib/Probability/Moments.lean", "def_pos": [174, 9], "def_end_pos": [174, 18]}, {"full_name": "Real.log_zero", "def_path": "Mathlib/Analysis/SpecialFunctions/Log/Basic.lean", "def_pos": [96, 9], "def_end_pos": [96, 17]}]], "state_before": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u211d\np : \u2115\n\u03bc : Measure \u03a9\nt : \u211d\nhX : \u00acIntegrable fun \u03c9 => rexp (t * X \u03c9)\n\u22a2 cgf X \u03bc t = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "full_name": "MeasurableSpace.generateFrom_singleton", "start": [1156, 9], "end": [1162, 94], "traced_tactics": [{"tactic": "classical\nletI : MeasurableSpace \u03b1 := generateFrom {s}\nrefine' le_antisymm (generateFrom_le fun t ht => \u27e8{True}, trivial, by simp [ht.symm]\u27e9) _\nrintro _ \u27e8u, -, rfl\u27e9\nexact (show MeasurableSet s from GenerateMeasurable.basic _ <| mem_singleton s).mem trivial", "annotated_tactic": ["classical\n  letI : <a>MeasurableSpace</a> \u03b1 := <a>generateFrom</a> {s}\n  refine' <a>le_antisymm</a> (<a>generateFrom_le</a> fun t ht => \u27e8{<a>True</a>}, <a>trivial</a>, by simp [ht.symm]\u27e9) _\n  rintro _ \u27e8u, -, rfl\u27e9\n  exact (show <a>MeasurableSet</a> s from <a>GenerateMeasurable.basic</a> _ <| <a>mem_singleton</a> s).<a>mem</a> <a>trivial</a>", [{"full_name": "MeasurableSpace", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [48, 20], "def_end_pos": [48, 35]}, {"full_name": "MeasurableSpace.generateFrom", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [363, 5], "def_end_pos": [363, 17]}, {"full_name": "le_antisymm", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [188, 9], "def_end_pos": [188, 20]}, {"full_name": "MeasurableSpace.generateFrom_le", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [384, 9], "def_end_pos": [384, 24]}, {"full_name": "True", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [176, 11], "def_end_pos": [176, 15]}, {"full_name": "trivial", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [514, 31], "def_end_pos": [514, 38]}, {"full_name": "MeasurableSet", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [64, 5], "def_end_pos": [64, 18]}, {"full_name": "MeasurableSpace.GenerateMeasurable.basic", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [355, 15], "def_end_pos": [355, 20]}, {"full_name": "Set.mem_singleton", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1289, 9], "def_end_pos": [1289, 22]}, {"full_name": "MeasurableSet.mem", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [1147, 11], "def_end_pos": [1147, 28]}, {"full_name": "trivial", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [514, 31], "def_end_pos": [514, 38]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u\u03b9\ns\u271d t u s : Set \u03b1\n\u22a2 generateFrom {s} = MeasurableSpace.comap (fun x => x \u2208 s) \u22a4", "state_after": "no goals"}, {"tactic": "letI : MeasurableSpace \u03b1 := generateFrom {s}", "annotated_tactic": ["letI : <a>MeasurableSpace</a> \u03b1 := <a>generateFrom</a> {s}", [{"full_name": "MeasurableSpace", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [48, 20], "def_end_pos": [48, 35]}, {"full_name": "MeasurableSpace.generateFrom", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [363, 5], "def_end_pos": [363, 17]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u\u03b9\ns\u271d t u s : Set \u03b1\n\u22a2 generateFrom {s} = MeasurableSpace.comap (fun x => x \u2208 s) \u22a4", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u\u03b9\ns\u271d t u s : Set \u03b1\nthis : MeasurableSpace \u03b1 := generateFrom {s}\n\u22a2 generateFrom {s} = MeasurableSpace.comap (fun x => x \u2208 s) \u22a4"}, {"tactic": "refine' le_antisymm (generateFrom_le fun t ht => \u27e8{True}, trivial, by simp [ht.symm]\u27e9) _", "annotated_tactic": ["refine' <a>le_antisymm</a> (<a>generateFrom_le</a> fun t ht => \u27e8{<a>True</a>}, <a>trivial</a>, by simp [ht.symm]\u27e9) _", [{"full_name": "le_antisymm", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [188, 9], "def_end_pos": [188, 20]}, {"full_name": "MeasurableSpace.generateFrom_le", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [384, 9], "def_end_pos": [384, 24]}, {"full_name": "True", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [176, 11], "def_end_pos": [176, 15]}, {"full_name": "trivial", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [514, 31], "def_end_pos": [514, 38]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u\u03b9\ns\u271d t u s : Set \u03b1\nthis : MeasurableSpace \u03b1 := generateFrom {s}\n\u22a2 generateFrom {s} = MeasurableSpace.comap (fun x => x \u2208 s) \u22a4", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u\u03b9\ns\u271d t u s : Set \u03b1\nthis : MeasurableSpace \u03b1 := generateFrom {s}\n\u22a2 MeasurableSpace.comap (fun x => x \u2208 s) \u22a4 \u2264 generateFrom {s}"}, {"tactic": "rintro _ \u27e8u, -, rfl\u27e9", "annotated_tactic": ["rintro _ \u27e8u, -, rfl\u27e9", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u\u03b9\ns\u271d t u s : Set \u03b1\nthis : MeasurableSpace \u03b1 := generateFrom {s}\n\u22a2 MeasurableSpace.comap (fun x => x \u2208 s) \u22a4 \u2264 generateFrom {s}", "state_after": "case intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u\u03b9\ns\u271d t u\u271d s : Set \u03b1\nthis : MeasurableSpace \u03b1 := generateFrom {s}\nu : Set Prop\n\u22a2 MeasurableSet ((fun x => x \u2208 s) \u207b\u00b9' u)"}, {"tactic": "exact (show MeasurableSet s from GenerateMeasurable.basic _ <| mem_singleton s).mem trivial", "annotated_tactic": ["exact (show <a>MeasurableSet</a> s from <a>GenerateMeasurable.basic</a> _ <| <a>mem_singleton</a> s).<a>mem</a> <a>trivial</a>", [{"full_name": "MeasurableSet", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [64, 5], "def_end_pos": [64, 18]}, {"full_name": "MeasurableSpace.GenerateMeasurable.basic", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [355, 15], "def_end_pos": [355, 20]}, {"full_name": "Set.mem_singleton", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1289, 9], "def_end_pos": [1289, 22]}, {"full_name": "MeasurableSet.mem", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [1147, 11], "def_end_pos": [1147, 28]}, {"full_name": "trivial", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [514, 31], "def_end_pos": [514, 38]}]], "state_before": "case intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u\u03b9\ns\u271d t u\u271d s : Set \u03b1\nthis : MeasurableSpace \u03b1 := generateFrom {s}\nu : Set Prop\n\u22a2 MeasurableSet ((fun x => x \u2208 s) \u207b\u00b9' u)", "state_after": "no goals"}, {"tactic": "simp [ht.symm]", "annotated_tactic": ["simp [ht.symm]", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u\u03b9\ns\u271d t\u271d u s : Set \u03b1\nthis : MeasurableSpace \u03b1 := generateFrom {s}\nt : Set \u03b1\nht : t \u2208 {s}\n\u22a2 (fun x => x \u2208 s) \u207b\u00b9' {True} = t", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Kernel/Disintegration.lean", "full_name": "ProbabilityTheory.kernel.const_unit_eq_compProd", "start": [360, 1], "end": [362, 86], "traced_tactics": [{"tactic": "simp_rw [condKernel_def]", "annotated_tactic": ["simp_rw [<a>condKernel_def</a>]", [{"full_name": "ProbabilityTheory.condKernel_def", "def_path": "Mathlib/Probability/Kernel/Disintegration.lean", "def_pos": [353, 9], "def_end_pos": [353, 23]}]], "state_before": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2075 : TopologicalSpace \u03a9\ninst\u271d\u2074 : PolishSpace \u03a9\ninst\u271d\u00b3 : MeasurableSpace \u03a9\ninst\u271d\u00b2 : BorelSpace \u03a9\ninst\u271d\u00b9 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d : IsFiniteMeasure \u03c1\n\u22a2 const Unit \u03c1 = const Unit (Measure.fst \u03c1) \u2297\u2096 prodMkLeft Unit (Measure.condKernel \u03c1)", "state_after": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2075 : TopologicalSpace \u03a9\ninst\u271d\u2074 : PolishSpace \u03a9\ninst\u271d\u00b3 : MeasurableSpace \u03a9\ninst\u271d\u00b2 : BorelSpace \u03a9\ninst\u271d\u00b9 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d : IsFiniteMeasure \u03c1\n\u22a2 const Unit \u03c1 =\n    const Unit (Measure.fst \u03c1) \u2297\u2096\n      prodMkLeft Unit (Exists.choose (_ : \u2203 \u03b7 _h, const Unit \u03c1 = const Unit (Measure.fst \u03c1) \u2297\u2096 prodMkLeft Unit \u03b7))"}, {"tactic": "exact (exists_cond_kernel \u03c1 Unit).choose_spec.choose_spec", "annotated_tactic": ["exact (<a>exists_cond_kernel</a> \u03c1 <a>Unit</a>).choose_spec.choose_spec", [{"full_name": "ProbabilityTheory.exists_cond_kernel", "def_path": "Mathlib/Probability/Kernel/Disintegration.lean", "def_pos": [265, 9], "def_end_pos": [265, 27]}, {"full_name": "Unit", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [129, 8], "def_end_pos": [129, 12]}]], "state_before": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2075 : TopologicalSpace \u03a9\ninst\u271d\u2074 : PolishSpace \u03a9\ninst\u271d\u00b3 : MeasurableSpace \u03a9\ninst\u271d\u00b2 : BorelSpace \u03a9\ninst\u271d\u00b9 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d : IsFiniteMeasure \u03c1\n\u22a2 const Unit \u03c1 =\n    const Unit (Measure.fst \u03c1) \u2297\u2096\n      prodMkLeft Unit (Exists.choose (_ : \u2203 \u03b7 _h, const Unit \u03c1 = const Unit (Measure.fst \u03c1) \u2297\u2096 prodMkLeft Unit \u03b7))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Intervals/WithBotTop.lean", "full_name": "WithBot.preimage_coe_bot", "start": [136, 1], "end": [137, 32], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Process/Stopping.lean", "full_name": "MeasureTheory.IsStoppingTime.measurableSet_inter_le", "start": [623, 1], "end": [652, 79], "traced_tactics": [{"tactic": "simp_rw [IsStoppingTime.measurableSet] at hs \u22a2", "annotated_tactic": ["simp_rw [<a>IsStoppingTime.measurableSet</a>] at hs \u22a2", [{"full_name": "MeasureTheory.IsStoppingTime.measurableSet", "def_path": "Mathlib/Probability/Process/Stopping.lean", "def_pos": [335, 19], "def_end_pos": [335, 32]}]], "state_before": "\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : LinearOrder \u03b9\nf : Filtration \u03b9 m\n\u03c4 \u03c0 : \u03a9 \u2192 \u03b9\ninst\u271d\u2074 : TopologicalSpace \u03b9\ninst\u271d\u00b3 : SecondCountableTopology \u03b9\ninst\u271d\u00b2 : OrderTopology \u03b9\ninst\u271d\u00b9 : MeasurableSpace \u03b9\ninst\u271d : BorelSpace \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\nh\u03c0 : IsStoppingTime f \u03c0\ns : Set \u03a9\nhs : MeasurableSet s\n\u22a2 MeasurableSet (s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c0 \u03c9})", "state_after": "\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : LinearOrder \u03b9\nf : Filtration \u03b9 m\n\u03c4 \u03c0 : \u03a9 \u2192 \u03b9\ninst\u271d\u2074 : TopologicalSpace \u03b9\ninst\u271d\u00b3 : SecondCountableTopology \u03b9\ninst\u271d\u00b2 : OrderTopology \u03b9\ninst\u271d\u00b9 : MeasurableSpace \u03b9\ninst\u271d : BorelSpace \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\nh\u03c0 : IsStoppingTime f \u03c0\ns : Set \u03a9\nhs : \u2200 (i : \u03b9), MeasurableSet (s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 i})\n\u22a2 \u2200 (i : \u03b9), MeasurableSet (s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c0 \u03c9} \u2229 {\u03c9 | min (\u03c4 \u03c9) (\u03c0 \u03c9) \u2264 i})"}, {"tactic": "intro i", "annotated_tactic": ["intro i", []], "state_before": "\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : LinearOrder \u03b9\nf : Filtration \u03b9 m\n\u03c4 \u03c0 : \u03a9 \u2192 \u03b9\ninst\u271d\u2074 : TopologicalSpace \u03b9\ninst\u271d\u00b3 : SecondCountableTopology \u03b9\ninst\u271d\u00b2 : OrderTopology \u03b9\ninst\u271d\u00b9 : MeasurableSpace \u03b9\ninst\u271d : BorelSpace \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\nh\u03c0 : IsStoppingTime f \u03c0\ns : Set \u03a9\nhs : \u2200 (i : \u03b9), MeasurableSet (s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 i})\n\u22a2 \u2200 (i : \u03b9), MeasurableSet (s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c0 \u03c9} \u2229 {\u03c9 | min (\u03c4 \u03c9) (\u03c0 \u03c9) \u2264 i})", "state_after": "\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : LinearOrder \u03b9\nf : Filtration \u03b9 m\n\u03c4 \u03c0 : \u03a9 \u2192 \u03b9\ninst\u271d\u2074 : TopologicalSpace \u03b9\ninst\u271d\u00b3 : SecondCountableTopology \u03b9\ninst\u271d\u00b2 : OrderTopology \u03b9\ninst\u271d\u00b9 : MeasurableSpace \u03b9\ninst\u271d : BorelSpace \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\nh\u03c0 : IsStoppingTime f \u03c0\ns : Set \u03a9\nhs : \u2200 (i : \u03b9), MeasurableSet (s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 i})\ni : \u03b9\n\u22a2 MeasurableSet (s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c0 \u03c9} \u2229 {\u03c9 | min (\u03c4 \u03c9) (\u03c0 \u03c9) \u2264 i})"}, {"tactic": "rw [this]", "annotated_tactic": ["rw [this]", []], "state_before": "\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : LinearOrder \u03b9\nf : Filtration \u03b9 m\n\u03c4 \u03c0 : \u03a9 \u2192 \u03b9\ninst\u271d\u2074 : TopologicalSpace \u03b9\ninst\u271d\u00b3 : SecondCountableTopology \u03b9\ninst\u271d\u00b2 : OrderTopology \u03b9\ninst\u271d\u00b9 : MeasurableSpace \u03b9\ninst\u271d : BorelSpace \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\nh\u03c0 : IsStoppingTime f \u03c0\ns : Set \u03a9\nhs : \u2200 (i : \u03b9), MeasurableSet (s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 i})\ni : \u03b9\nthis :\n  s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c0 \u03c9} \u2229 {\u03c9 | min (\u03c4 \u03c9) (\u03c0 \u03c9) \u2264 i} =\n    s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 i} \u2229 {\u03c9 | min (\u03c4 \u03c9) (\u03c0 \u03c9) \u2264 i} \u2229 {\u03c9 | min (\u03c4 \u03c9) i \u2264 min (min (\u03c4 \u03c9) (\u03c0 \u03c9)) i}\n\u22a2 MeasurableSet (s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c0 \u03c9} \u2229 {\u03c9 | min (\u03c4 \u03c9) (\u03c0 \u03c9) \u2264 i})", "state_after": "\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : LinearOrder \u03b9\nf : Filtration \u03b9 m\n\u03c4 \u03c0 : \u03a9 \u2192 \u03b9\ninst\u271d\u2074 : TopologicalSpace \u03b9\ninst\u271d\u00b3 : SecondCountableTopology \u03b9\ninst\u271d\u00b2 : OrderTopology \u03b9\ninst\u271d\u00b9 : MeasurableSpace \u03b9\ninst\u271d : BorelSpace \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\nh\u03c0 : IsStoppingTime f \u03c0\ns : Set \u03a9\nhs : \u2200 (i : \u03b9), MeasurableSet (s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 i})\ni : \u03b9\nthis :\n  s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c0 \u03c9} \u2229 {\u03c9 | min (\u03c4 \u03c9) (\u03c0 \u03c9) \u2264 i} =\n    s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 i} \u2229 {\u03c9 | min (\u03c4 \u03c9) (\u03c0 \u03c9) \u2264 i} \u2229 {\u03c9 | min (\u03c4 \u03c9) i \u2264 min (min (\u03c4 \u03c9) (\u03c0 \u03c9)) i}\n\u22a2 MeasurableSet (s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 i} \u2229 {\u03c9 | min (\u03c4 \u03c9) (\u03c0 \u03c9) \u2264 i} \u2229 {\u03c9 | min (\u03c4 \u03c9) i \u2264 min (min (\u03c4 \u03c9) (\u03c0 \u03c9)) i})"}, {"tactic": "refine' ((hs i).inter ((h\u03c4.min h\u03c0) i)).inter _", "annotated_tactic": ["refine' ((hs i).<a>inter</a> ((h\u03c4.min h\u03c0) i)).<a>inter</a> _", [{"full_name": "MeasurableSet.inter", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [198, 19], "def_end_pos": [198, 38]}, {"full_name": "MeasurableSet.inter", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [198, 19], "def_end_pos": [198, 38]}]], "state_before": "\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : LinearOrder \u03b9\nf : Filtration \u03b9 m\n\u03c4 \u03c0 : \u03a9 \u2192 \u03b9\ninst\u271d\u2074 : TopologicalSpace \u03b9\ninst\u271d\u00b3 : SecondCountableTopology \u03b9\ninst\u271d\u00b2 : OrderTopology \u03b9\ninst\u271d\u00b9 : MeasurableSpace \u03b9\ninst\u271d : BorelSpace \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\nh\u03c0 : IsStoppingTime f \u03c0\ns : Set \u03a9\nhs : \u2200 (i : \u03b9), MeasurableSet (s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 i})\ni : \u03b9\nthis :\n  s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c0 \u03c9} \u2229 {\u03c9 | min (\u03c4 \u03c9) (\u03c0 \u03c9) \u2264 i} =\n    s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 i} \u2229 {\u03c9 | min (\u03c4 \u03c9) (\u03c0 \u03c9) \u2264 i} \u2229 {\u03c9 | min (\u03c4 \u03c9) i \u2264 min (min (\u03c4 \u03c9) (\u03c0 \u03c9)) i}\n\u22a2 MeasurableSet (s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 i} \u2229 {\u03c9 | min (\u03c4 \u03c9) (\u03c0 \u03c9) \u2264 i} \u2229 {\u03c9 | min (\u03c4 \u03c9) i \u2264 min (min (\u03c4 \u03c9) (\u03c0 \u03c9)) i})", "state_after": "\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : LinearOrder \u03b9\nf : Filtration \u03b9 m\n\u03c4 \u03c0 : \u03a9 \u2192 \u03b9\ninst\u271d\u2074 : TopologicalSpace \u03b9\ninst\u271d\u00b3 : SecondCountableTopology \u03b9\ninst\u271d\u00b2 : OrderTopology \u03b9\ninst\u271d\u00b9 : MeasurableSpace \u03b9\ninst\u271d : BorelSpace \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\nh\u03c0 : IsStoppingTime f \u03c0\ns : Set \u03a9\nhs : \u2200 (i : \u03b9), MeasurableSet (s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 i})\ni : \u03b9\nthis :\n  s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c0 \u03c9} \u2229 {\u03c9 | min (\u03c4 \u03c9) (\u03c0 \u03c9) \u2264 i} =\n    s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 i} \u2229 {\u03c9 | min (\u03c4 \u03c9) (\u03c0 \u03c9) \u2264 i} \u2229 {\u03c9 | min (\u03c4 \u03c9) i \u2264 min (min (\u03c4 \u03c9) (\u03c0 \u03c9)) i}\n\u22a2 MeasurableSet {\u03c9 | min (\u03c4 \u03c9) i \u2264 min (min (\u03c4 \u03c9) (\u03c0 \u03c9)) i}"}, {"tactic": "apply @measurableSet_le _ _ _ _ _ (Filtration.seq f i) _ _ _ _ _ ?_ ?_", "annotated_tactic": ["apply @<a>measurableSet_le</a> _ _ _ _ _ (<a>Filtration.seq</a> f i) _ _ _ _ _ ?_ ?_", [{"full_name": "measurableSet_le", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [559, 9], "def_end_pos": [559, 25]}, {"full_name": "MeasureTheory.Filtration.seq", "def_path": "Mathlib/Probability/Process/Filtration.lean", "def_pos": [44, 3], "def_end_pos": [44, 6]}]], "state_before": "\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : LinearOrder \u03b9\nf : Filtration \u03b9 m\n\u03c4 \u03c0 : \u03a9 \u2192 \u03b9\ninst\u271d\u2074 : TopologicalSpace \u03b9\ninst\u271d\u00b3 : SecondCountableTopology \u03b9\ninst\u271d\u00b2 : OrderTopology \u03b9\ninst\u271d\u00b9 : MeasurableSpace \u03b9\ninst\u271d : BorelSpace \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\nh\u03c0 : IsStoppingTime f \u03c0\ns : Set \u03a9\nhs : \u2200 (i : \u03b9), MeasurableSet (s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 i})\ni : \u03b9\nthis :\n  s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c0 \u03c9} \u2229 {\u03c9 | min (\u03c4 \u03c9) (\u03c0 \u03c9) \u2264 i} =\n    s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 i} \u2229 {\u03c9 | min (\u03c4 \u03c9) (\u03c0 \u03c9) \u2264 i} \u2229 {\u03c9 | min (\u03c4 \u03c9) i \u2264 min (min (\u03c4 \u03c9) (\u03c0 \u03c9)) i}\n\u22a2 MeasurableSet {\u03c9 | min (\u03c4 \u03c9) i \u2264 min (min (\u03c4 \u03c9) (\u03c0 \u03c9)) i}", "state_after": "\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : LinearOrder \u03b9\nf : Filtration \u03b9 m\n\u03c4 \u03c0 : \u03a9 \u2192 \u03b9\ninst\u271d\u2074 : TopologicalSpace \u03b9\ninst\u271d\u00b3 : SecondCountableTopology \u03b9\ninst\u271d\u00b2 : OrderTopology \u03b9\ninst\u271d\u00b9 : MeasurableSpace \u03b9\ninst\u271d : BorelSpace \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\nh\u03c0 : IsStoppingTime f \u03c0\ns : Set \u03a9\nhs : \u2200 (i : \u03b9), MeasurableSet (s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 i})\ni : \u03b9\nthis :\n  s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c0 \u03c9} \u2229 {\u03c9 | min (\u03c4 \u03c9) (\u03c0 \u03c9) \u2264 i} =\n    s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 i} \u2229 {\u03c9 | min (\u03c4 \u03c9) (\u03c0 \u03c9) \u2264 i} \u2229 {\u03c9 | min (\u03c4 \u03c9) i \u2264 min (min (\u03c4 \u03c9) (\u03c0 \u03c9)) i}\n\u22a2 Measurable fun a => min (\u03c4 a) i\n\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : LinearOrder \u03b9\nf : Filtration \u03b9 m\n\u03c4 \u03c0 : \u03a9 \u2192 \u03b9\ninst\u271d\u2074 : TopologicalSpace \u03b9\ninst\u271d\u00b3 : SecondCountableTopology \u03b9\ninst\u271d\u00b2 : OrderTopology \u03b9\ninst\u271d\u00b9 : MeasurableSpace \u03b9\ninst\u271d : BorelSpace \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\nh\u03c0 : IsStoppingTime f \u03c0\ns : Set \u03a9\nhs : \u2200 (i : \u03b9), MeasurableSet (s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 i})\ni : \u03b9\nthis :\n  s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c0 \u03c9} \u2229 {\u03c9 | min (\u03c4 \u03c9) (\u03c0 \u03c9) \u2264 i} =\n    s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 i} \u2229 {\u03c9 | min (\u03c4 \u03c9) (\u03c0 \u03c9) \u2264 i} \u2229 {\u03c9 | min (\u03c4 \u03c9) i \u2264 min (min (\u03c4 \u03c9) (\u03c0 \u03c9)) i}\n\u22a2 Measurable fun a => min (min (\u03c4 a) (\u03c0 a)) i"}, {"tactic": "ext1 \u03c9", "annotated_tactic": ["ext1 \u03c9", []], "state_before": "\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : LinearOrder \u03b9\nf : Filtration \u03b9 m\n\u03c4 \u03c0 : \u03a9 \u2192 \u03b9\ninst\u271d\u2074 : TopologicalSpace \u03b9\ninst\u271d\u00b3 : SecondCountableTopology \u03b9\ninst\u271d\u00b2 : OrderTopology \u03b9\ninst\u271d\u00b9 : MeasurableSpace \u03b9\ninst\u271d : BorelSpace \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\nh\u03c0 : IsStoppingTime f \u03c0\ns : Set \u03a9\nhs : \u2200 (i : \u03b9), MeasurableSet (s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 i})\ni : \u03b9\n\u22a2 s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c0 \u03c9} \u2229 {\u03c9 | min (\u03c4 \u03c9) (\u03c0 \u03c9) \u2264 i} =\n    s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 i} \u2229 {\u03c9 | min (\u03c4 \u03c9) (\u03c0 \u03c9) \u2264 i} \u2229 {\u03c9 | min (\u03c4 \u03c9) i \u2264 min (min (\u03c4 \u03c9) (\u03c0 \u03c9)) i}", "state_after": "case h\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : LinearOrder \u03b9\nf : Filtration \u03b9 m\n\u03c4 \u03c0 : \u03a9 \u2192 \u03b9\ninst\u271d\u2074 : TopologicalSpace \u03b9\ninst\u271d\u00b3 : SecondCountableTopology \u03b9\ninst\u271d\u00b2 : OrderTopology \u03b9\ninst\u271d\u00b9 : MeasurableSpace \u03b9\ninst\u271d : BorelSpace \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\nh\u03c0 : IsStoppingTime f \u03c0\ns : Set \u03a9\nhs : \u2200 (i : \u03b9), MeasurableSet (s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 i})\ni : \u03b9\n\u03c9 : \u03a9\n\u22a2 \u03c9 \u2208 s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c0 \u03c9} \u2229 {\u03c9 | min (\u03c4 \u03c9) (\u03c0 \u03c9) \u2264 i} \u2194\n    \u03c9 \u2208 s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 i} \u2229 {\u03c9 | min (\u03c4 \u03c9) (\u03c0 \u03c9) \u2264 i} \u2229 {\u03c9 | min (\u03c4 \u03c9) i \u2264 min (min (\u03c4 \u03c9) (\u03c0 \u03c9)) i}"}, {"tactic": "simp only [min_le_iff, Set.mem_inter_iff, Set.mem_setOf_eq, le_min_iff, le_refl, true_and_iff,\n  and_true_iff, true_or_iff, or_true_iff]", "annotated_tactic": ["simp only [<a>min_le_iff</a>, <a>Set.mem_inter_iff</a>, <a>Set.mem_setOf_eq</a>, <a>le_min_iff</a>, <a>le_refl</a>, <a>true_and_iff</a>,\n      <a>and_true_iff</a>, <a>true_or_iff</a>, <a>or_true_iff</a>]", [{"full_name": "min_le_iff", "def_path": "Mathlib/Order/MinMax.lean", "def_pos": [43, 9], "def_end_pos": [43, 19]}, {"full_name": "Set.mem_inter_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [909, 9], "def_end_pos": [909, 22]}, {"full_name": "Set.mem_setOf_eq", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [256, 29], "def_end_pos": [256, 41]}, {"full_name": "le_min_iff", "def_path": "Mathlib/Order/MinMax.lean", "def_pos": [33, 9], "def_end_pos": [33, 19]}, {"full_name": "le_refl", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [50, 9], "def_end_pos": [50, 16]}, {"full_name": "true_and_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [147, 9], "def_end_pos": [147, 21]}, {"full_name": "and_true_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [145, 9], "def_end_pos": [145, 21]}, {"full_name": "true_or_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [182, 9], "def_end_pos": [182, 20]}, {"full_name": "or_true_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [184, 9], "def_end_pos": [184, 20]}]], "state_before": "case h\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : LinearOrder \u03b9\nf : Filtration \u03b9 m\n\u03c4 \u03c0 : \u03a9 \u2192 \u03b9\ninst\u271d\u2074 : TopologicalSpace \u03b9\ninst\u271d\u00b3 : SecondCountableTopology \u03b9\ninst\u271d\u00b2 : OrderTopology \u03b9\ninst\u271d\u00b9 : MeasurableSpace \u03b9\ninst\u271d : BorelSpace \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\nh\u03c0 : IsStoppingTime f \u03c0\ns : Set \u03a9\nhs : \u2200 (i : \u03b9), MeasurableSet (s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 i})\ni : \u03b9\n\u03c9 : \u03a9\n\u22a2 \u03c9 \u2208 s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c0 \u03c9} \u2229 {\u03c9 | min (\u03c4 \u03c9) (\u03c0 \u03c9) \u2264 i} \u2194\n    \u03c9 \u2208 s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 i} \u2229 {\u03c9 | min (\u03c4 \u03c9) (\u03c0 \u03c9) \u2264 i} \u2229 {\u03c9 | min (\u03c4 \u03c9) i \u2264 min (min (\u03c4 \u03c9) (\u03c0 \u03c9)) i}", "state_after": "case h\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : LinearOrder \u03b9\nf : Filtration \u03b9 m\n\u03c4 \u03c0 : \u03a9 \u2192 \u03b9\ninst\u271d\u2074 : TopologicalSpace \u03b9\ninst\u271d\u00b3 : SecondCountableTopology \u03b9\ninst\u271d\u00b2 : OrderTopology \u03b9\ninst\u271d\u00b9 : MeasurableSpace \u03b9\ninst\u271d : BorelSpace \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\nh\u03c0 : IsStoppingTime f \u03c0\ns : Set \u03a9\nhs : \u2200 (i : \u03b9), MeasurableSet (s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 i})\ni : \u03b9\n\u03c9 : \u03a9\n\u22a2 (\u03c9 \u2208 s \u2227 \u03c4 \u03c9 \u2264 \u03c0 \u03c9) \u2227 (\u03c4 \u03c9 \u2264 i \u2228 \u03c0 \u03c9 \u2264 i) \u2194 ((\u03c9 \u2208 s \u2227 \u03c4 \u03c9 \u2264 i) \u2227 (\u03c4 \u03c9 \u2264 i \u2228 \u03c0 \u03c9 \u2264 i)) \u2227 (\u03c4 \u03c9 \u2264 \u03c0 \u03c9 \u2228 i \u2264 \u03c0 \u03c9)"}, {"tactic": "by_cases h\u03c4i : \u03c4 \u03c9 \u2264 i", "annotated_tactic": ["by_cases h\u03c4i : \u03c4 \u03c9 \u2264 i", []], "state_before": "case h\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : LinearOrder \u03b9\nf : Filtration \u03b9 m\n\u03c4 \u03c0 : \u03a9 \u2192 \u03b9\ninst\u271d\u2074 : TopologicalSpace \u03b9\ninst\u271d\u00b3 : SecondCountableTopology \u03b9\ninst\u271d\u00b2 : OrderTopology \u03b9\ninst\u271d\u00b9 : MeasurableSpace \u03b9\ninst\u271d : BorelSpace \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\nh\u03c0 : IsStoppingTime f \u03c0\ns : Set \u03a9\nhs : \u2200 (i : \u03b9), MeasurableSet (s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 i})\ni : \u03b9\n\u03c9 : \u03a9\n\u22a2 (\u03c9 \u2208 s \u2227 \u03c4 \u03c9 \u2264 \u03c0 \u03c9) \u2227 (\u03c4 \u03c9 \u2264 i \u2228 \u03c0 \u03c9 \u2264 i) \u2194 ((\u03c9 \u2208 s \u2227 \u03c4 \u03c9 \u2264 i) \u2227 (\u03c4 \u03c9 \u2264 i \u2228 \u03c0 \u03c9 \u2264 i)) \u2227 (\u03c4 \u03c9 \u2264 \u03c0 \u03c9 \u2228 i \u2264 \u03c0 \u03c9)", "state_after": "case pos\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : LinearOrder \u03b9\nf : Filtration \u03b9 m\n\u03c4 \u03c0 : \u03a9 \u2192 \u03b9\ninst\u271d\u2074 : TopologicalSpace \u03b9\ninst\u271d\u00b3 : SecondCountableTopology \u03b9\ninst\u271d\u00b2 : OrderTopology \u03b9\ninst\u271d\u00b9 : MeasurableSpace \u03b9\ninst\u271d : BorelSpace \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\nh\u03c0 : IsStoppingTime f \u03c0\ns : Set \u03a9\nhs : \u2200 (i : \u03b9), MeasurableSet (s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 i})\ni : \u03b9\n\u03c9 : \u03a9\nh\u03c4i : \u03c4 \u03c9 \u2264 i\n\u22a2 (\u03c9 \u2208 s \u2227 \u03c4 \u03c9 \u2264 \u03c0 \u03c9) \u2227 (\u03c4 \u03c9 \u2264 i \u2228 \u03c0 \u03c9 \u2264 i) \u2194 ((\u03c9 \u2208 s \u2227 \u03c4 \u03c9 \u2264 i) \u2227 (\u03c4 \u03c9 \u2264 i \u2228 \u03c0 \u03c9 \u2264 i)) \u2227 (\u03c4 \u03c9 \u2264 \u03c0 \u03c9 \u2228 i \u2264 \u03c0 \u03c9)\n\ncase neg\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : LinearOrder \u03b9\nf : Filtration \u03b9 m\n\u03c4 \u03c0 : \u03a9 \u2192 \u03b9\ninst\u271d\u2074 : TopologicalSpace \u03b9\ninst\u271d\u00b3 : SecondCountableTopology \u03b9\ninst\u271d\u00b2 : OrderTopology \u03b9\ninst\u271d\u00b9 : MeasurableSpace \u03b9\ninst\u271d : BorelSpace \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\nh\u03c0 : IsStoppingTime f \u03c0\ns : Set \u03a9\nhs : \u2200 (i : \u03b9), MeasurableSet (s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 i})\ni : \u03b9\n\u03c9 : \u03a9\nh\u03c4i : \u00ac\u03c4 \u03c9 \u2264 i\n\u22a2 (\u03c9 \u2208 s \u2227 \u03c4 \u03c9 \u2264 \u03c0 \u03c9) \u2227 (\u03c4 \u03c9 \u2264 i \u2228 \u03c0 \u03c9 \u2264 i) \u2194 ((\u03c9 \u2208 s \u2227 \u03c4 \u03c9 \u2264 i) \u2227 (\u03c4 \u03c9 \u2264 i \u2228 \u03c0 \u03c9 \u2264 i)) \u2227 (\u03c4 \u03c9 \u2264 \u03c0 \u03c9 \u2228 i \u2264 \u03c0 \u03c9)"}, {"tactic": "simp only [h\u03c4i, false_or_iff, and_false_iff, false_and_iff, iff_false_iff, not_and, not_le,\n  and_imp]", "annotated_tactic": ["simp only [h\u03c4i, <a>false_or_iff</a>, <a>and_false_iff</a>, <a>false_and_iff</a>, <a>iff_false_iff</a>, <a>not_and</a>, <a>not_le</a>,\n      <a>and_imp</a>]", [{"full_name": "false_or_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [186, 9], "def_end_pos": [186, 21]}, {"full_name": "and_false_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [149, 9], "def_end_pos": [149, 22]}, {"full_name": "false_and_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [151, 9], "def_end_pos": [151, 22]}, {"full_name": "iff_false_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [201, 9], "def_end_pos": [201, 22]}, {"full_name": "not_and", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [316, 17], "def_end_pos": [316, 24]}, {"full_name": "not_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [373, 9], "def_end_pos": [373, 15]}, {"full_name": "and_imp", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [313, 17], "def_end_pos": [313, 24]}]], "state_before": "case neg\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : LinearOrder \u03b9\nf : Filtration \u03b9 m\n\u03c4 \u03c0 : \u03a9 \u2192 \u03b9\ninst\u271d\u2074 : TopologicalSpace \u03b9\ninst\u271d\u00b3 : SecondCountableTopology \u03b9\ninst\u271d\u00b2 : OrderTopology \u03b9\ninst\u271d\u00b9 : MeasurableSpace \u03b9\ninst\u271d : BorelSpace \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\nh\u03c0 : IsStoppingTime f \u03c0\ns : Set \u03a9\nhs : \u2200 (i : \u03b9), MeasurableSet (s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 i})\ni : \u03b9\n\u03c9 : \u03a9\nh\u03c4i : \u00ac\u03c4 \u03c9 \u2264 i\n\u22a2 (\u03c9 \u2208 s \u2227 \u03c4 \u03c9 \u2264 \u03c0 \u03c9) \u2227 (\u03c4 \u03c9 \u2264 i \u2228 \u03c0 \u03c9 \u2264 i) \u2194 ((\u03c9 \u2208 s \u2227 \u03c4 \u03c9 \u2264 i) \u2227 (\u03c4 \u03c9 \u2264 i \u2228 \u03c0 \u03c9 \u2264 i)) \u2227 (\u03c4 \u03c9 \u2264 \u03c0 \u03c9 \u2228 i \u2264 \u03c0 \u03c9)", "state_after": "case neg\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : LinearOrder \u03b9\nf : Filtration \u03b9 m\n\u03c4 \u03c0 : \u03a9 \u2192 \u03b9\ninst\u271d\u2074 : TopologicalSpace \u03b9\ninst\u271d\u00b3 : SecondCountableTopology \u03b9\ninst\u271d\u00b2 : OrderTopology \u03b9\ninst\u271d\u00b9 : MeasurableSpace \u03b9\ninst\u271d : BorelSpace \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\nh\u03c0 : IsStoppingTime f \u03c0\ns : Set \u03a9\nhs : \u2200 (i : \u03b9), MeasurableSet (s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 i})\ni : \u03b9\n\u03c9 : \u03a9\nh\u03c4i : \u00ac\u03c4 \u03c9 \u2264 i\n\u22a2 \u03c9 \u2208 s \u2192 \u03c4 \u03c9 \u2264 \u03c0 \u03c9 \u2192 i < \u03c0 \u03c9"}, {"tactic": "refine' fun _ h\u03c4_le_\u03c0 => lt_of_lt_of_le _ h\u03c4_le_\u03c0", "annotated_tactic": ["refine' fun _ h\u03c4_le_\u03c0 => <a>lt_of_lt_of_le</a> _ h\u03c4_le_\u03c0", [{"full_name": "lt_of_lt_of_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [115, 9], "def_end_pos": [115, 23]}]], "state_before": "case neg\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : LinearOrder \u03b9\nf : Filtration \u03b9 m\n\u03c4 \u03c0 : \u03a9 \u2192 \u03b9\ninst\u271d\u2074 : TopologicalSpace \u03b9\ninst\u271d\u00b3 : SecondCountableTopology \u03b9\ninst\u271d\u00b2 : OrderTopology \u03b9\ninst\u271d\u00b9 : MeasurableSpace \u03b9\ninst\u271d : BorelSpace \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\nh\u03c0 : IsStoppingTime f \u03c0\ns : Set \u03a9\nhs : \u2200 (i : \u03b9), MeasurableSet (s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 i})\ni : \u03b9\n\u03c9 : \u03a9\nh\u03c4i : \u00ac\u03c4 \u03c9 \u2264 i\n\u22a2 \u03c9 \u2208 s \u2192 \u03c4 \u03c9 \u2264 \u03c0 \u03c9 \u2192 i < \u03c0 \u03c9", "state_after": "case neg\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : LinearOrder \u03b9\nf : Filtration \u03b9 m\n\u03c4 \u03c0 : \u03a9 \u2192 \u03b9\ninst\u271d\u2074 : TopologicalSpace \u03b9\ninst\u271d\u00b3 : SecondCountableTopology \u03b9\ninst\u271d\u00b2 : OrderTopology \u03b9\ninst\u271d\u00b9 : MeasurableSpace \u03b9\ninst\u271d : BorelSpace \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\nh\u03c0 : IsStoppingTime f \u03c0\ns : Set \u03a9\nhs : \u2200 (i : \u03b9), MeasurableSet (s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 i})\ni : \u03b9\n\u03c9 : \u03a9\nh\u03c4i : \u00ac\u03c4 \u03c9 \u2264 i\nx\u271d : \u03c9 \u2208 s\nh\u03c4_le_\u03c0 : \u03c4 \u03c9 \u2264 \u03c0 \u03c9\n\u22a2 i < \u03c4 \u03c9"}, {"tactic": "rw [\u2190 not_le]", "annotated_tactic": ["rw [\u2190 <a>not_le</a>]", [{"full_name": "not_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [373, 9], "def_end_pos": [373, 15]}]], "state_before": "case neg\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : LinearOrder \u03b9\nf : Filtration \u03b9 m\n\u03c4 \u03c0 : \u03a9 \u2192 \u03b9\ninst\u271d\u2074 : TopologicalSpace \u03b9\ninst\u271d\u00b3 : SecondCountableTopology \u03b9\ninst\u271d\u00b2 : OrderTopology \u03b9\ninst\u271d\u00b9 : MeasurableSpace \u03b9\ninst\u271d : BorelSpace \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\nh\u03c0 : IsStoppingTime f \u03c0\ns : Set \u03a9\nhs : \u2200 (i : \u03b9), MeasurableSet (s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 i})\ni : \u03b9\n\u03c9 : \u03a9\nh\u03c4i : \u00ac\u03c4 \u03c9 \u2264 i\nx\u271d : \u03c9 \u2208 s\nh\u03c4_le_\u03c0 : \u03c4 \u03c9 \u2264 \u03c0 \u03c9\n\u22a2 i < \u03c4 \u03c9", "state_after": "case neg\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : LinearOrder \u03b9\nf : Filtration \u03b9 m\n\u03c4 \u03c0 : \u03a9 \u2192 \u03b9\ninst\u271d\u2074 : TopologicalSpace \u03b9\ninst\u271d\u00b3 : SecondCountableTopology \u03b9\ninst\u271d\u00b2 : OrderTopology \u03b9\ninst\u271d\u00b9 : MeasurableSpace \u03b9\ninst\u271d : BorelSpace \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\nh\u03c0 : IsStoppingTime f \u03c0\ns : Set \u03a9\nhs : \u2200 (i : \u03b9), MeasurableSet (s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 i})\ni : \u03b9\n\u03c9 : \u03a9\nh\u03c4i : \u00ac\u03c4 \u03c9 \u2264 i\nx\u271d : \u03c9 \u2208 s\nh\u03c4_le_\u03c0 : \u03c4 \u03c9 \u2264 \u03c0 \u03c9\n\u22a2 \u00ac\u03c4 \u03c9 \u2264 i"}, {"tactic": "exact h\u03c4i", "annotated_tactic": ["exact h\u03c4i", []], "state_before": "case neg\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : LinearOrder \u03b9\nf : Filtration \u03b9 m\n\u03c4 \u03c0 : \u03a9 \u2192 \u03b9\ninst\u271d\u2074 : TopologicalSpace \u03b9\ninst\u271d\u00b3 : SecondCountableTopology \u03b9\ninst\u271d\u00b2 : OrderTopology \u03b9\ninst\u271d\u00b9 : MeasurableSpace \u03b9\ninst\u271d : BorelSpace \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\nh\u03c0 : IsStoppingTime f \u03c0\ns : Set \u03a9\nhs : \u2200 (i : \u03b9), MeasurableSet (s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 i})\ni : \u03b9\n\u03c9 : \u03a9\nh\u03c4i : \u00ac\u03c4 \u03c9 \u2264 i\nx\u271d : \u03c9 \u2208 s\nh\u03c4_le_\u03c0 : \u03c4 \u03c9 \u2264 \u03c0 \u03c9\n\u22a2 \u00ac\u03c4 \u03c9 \u2264 i", "state_after": "no goals"}, {"tactic": "simp only [h\u03c4i, true_or_iff, and_true_iff, and_congr_right_iff]", "annotated_tactic": ["simp only [h\u03c4i, <a>true_or_iff</a>, <a>and_true_iff</a>, <a>and_congr_right_iff</a>]", [{"full_name": "true_or_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [182, 9], "def_end_pos": [182, 20]}, {"full_name": "and_true_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [145, 9], "def_end_pos": [145, 21]}, {"full_name": "and_congr_right_iff", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [219, 17], "def_end_pos": [219, 36]}]], "state_before": "case pos\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : LinearOrder \u03b9\nf : Filtration \u03b9 m\n\u03c4 \u03c0 : \u03a9 \u2192 \u03b9\ninst\u271d\u2074 : TopologicalSpace \u03b9\ninst\u271d\u00b3 : SecondCountableTopology \u03b9\ninst\u271d\u00b2 : OrderTopology \u03b9\ninst\u271d\u00b9 : MeasurableSpace \u03b9\ninst\u271d : BorelSpace \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\nh\u03c0 : IsStoppingTime f \u03c0\ns : Set \u03a9\nhs : \u2200 (i : \u03b9), MeasurableSet (s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 i})\ni : \u03b9\n\u03c9 : \u03a9\nh\u03c4i : \u03c4 \u03c9 \u2264 i\n\u22a2 (\u03c9 \u2208 s \u2227 \u03c4 \u03c9 \u2264 \u03c0 \u03c9) \u2227 (\u03c4 \u03c9 \u2264 i \u2228 \u03c0 \u03c9 \u2264 i) \u2194 ((\u03c9 \u2208 s \u2227 \u03c4 \u03c9 \u2264 i) \u2227 (\u03c4 \u03c9 \u2264 i \u2228 \u03c0 \u03c9 \u2264 i)) \u2227 (\u03c4 \u03c9 \u2264 \u03c0 \u03c9 \u2228 i \u2264 \u03c0 \u03c9)", "state_after": "case pos\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : LinearOrder \u03b9\nf : Filtration \u03b9 m\n\u03c4 \u03c0 : \u03a9 \u2192 \u03b9\ninst\u271d\u2074 : TopologicalSpace \u03b9\ninst\u271d\u00b3 : SecondCountableTopology \u03b9\ninst\u271d\u00b2 : OrderTopology \u03b9\ninst\u271d\u00b9 : MeasurableSpace \u03b9\ninst\u271d : BorelSpace \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\nh\u03c0 : IsStoppingTime f \u03c0\ns : Set \u03a9\nhs : \u2200 (i : \u03b9), MeasurableSet (s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 i})\ni : \u03b9\n\u03c9 : \u03a9\nh\u03c4i : \u03c4 \u03c9 \u2264 i\n\u22a2 \u03c9 \u2208 s \u2192 (\u03c4 \u03c9 \u2264 \u03c0 \u03c9 \u2194 \u03c4 \u03c9 \u2264 \u03c0 \u03c9 \u2228 i \u2264 \u03c0 \u03c9)"}, {"tactic": "intro", "annotated_tactic": ["intro", []], "state_before": "case pos\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : LinearOrder \u03b9\nf : Filtration \u03b9 m\n\u03c4 \u03c0 : \u03a9 \u2192 \u03b9\ninst\u271d\u2074 : TopologicalSpace \u03b9\ninst\u271d\u00b3 : SecondCountableTopology \u03b9\ninst\u271d\u00b2 : OrderTopology \u03b9\ninst\u271d\u00b9 : MeasurableSpace \u03b9\ninst\u271d : BorelSpace \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\nh\u03c0 : IsStoppingTime f \u03c0\ns : Set \u03a9\nhs : \u2200 (i : \u03b9), MeasurableSet (s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 i})\ni : \u03b9\n\u03c9 : \u03a9\nh\u03c4i : \u03c4 \u03c9 \u2264 i\n\u22a2 \u03c9 \u2208 s \u2192 (\u03c4 \u03c9 \u2264 \u03c0 \u03c9 \u2194 \u03c4 \u03c9 \u2264 \u03c0 \u03c9 \u2228 i \u2264 \u03c0 \u03c9)", "state_after": "case pos\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : LinearOrder \u03b9\nf : Filtration \u03b9 m\n\u03c4 \u03c0 : \u03a9 \u2192 \u03b9\ninst\u271d\u2074 : TopologicalSpace \u03b9\ninst\u271d\u00b3 : SecondCountableTopology \u03b9\ninst\u271d\u00b2 : OrderTopology \u03b9\ninst\u271d\u00b9 : MeasurableSpace \u03b9\ninst\u271d : BorelSpace \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\nh\u03c0 : IsStoppingTime f \u03c0\ns : Set \u03a9\nhs : \u2200 (i : \u03b9), MeasurableSet (s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 i})\ni : \u03b9\n\u03c9 : \u03a9\nh\u03c4i : \u03c4 \u03c9 \u2264 i\na\u271d : \u03c9 \u2208 s\n\u22a2 \u03c4 \u03c9 \u2264 \u03c0 \u03c9 \u2194 \u03c4 \u03c9 \u2264 \u03c0 \u03c9 \u2228 i \u2264 \u03c0 \u03c9"}, {"tactic": "constructor <;> intro h", "annotated_tactic": ["constructor <;> intro h", []], "state_before": "case pos\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : LinearOrder \u03b9\nf : Filtration \u03b9 m\n\u03c4 \u03c0 : \u03a9 \u2192 \u03b9\ninst\u271d\u2074 : TopologicalSpace \u03b9\ninst\u271d\u00b3 : SecondCountableTopology \u03b9\ninst\u271d\u00b2 : OrderTopology \u03b9\ninst\u271d\u00b9 : MeasurableSpace \u03b9\ninst\u271d : BorelSpace \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\nh\u03c0 : IsStoppingTime f \u03c0\ns : Set \u03a9\nhs : \u2200 (i : \u03b9), MeasurableSet (s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 i})\ni : \u03b9\n\u03c9 : \u03a9\nh\u03c4i : \u03c4 \u03c9 \u2264 i\na\u271d : \u03c9 \u2208 s\n\u22a2 \u03c4 \u03c9 \u2264 \u03c0 \u03c9 \u2194 \u03c4 \u03c9 \u2264 \u03c0 \u03c9 \u2228 i \u2264 \u03c0 \u03c9", "state_after": "case pos.mp\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : LinearOrder \u03b9\nf : Filtration \u03b9 m\n\u03c4 \u03c0 : \u03a9 \u2192 \u03b9\ninst\u271d\u2074 : TopologicalSpace \u03b9\ninst\u271d\u00b3 : SecondCountableTopology \u03b9\ninst\u271d\u00b2 : OrderTopology \u03b9\ninst\u271d\u00b9 : MeasurableSpace \u03b9\ninst\u271d : BorelSpace \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\nh\u03c0 : IsStoppingTime f \u03c0\ns : Set \u03a9\nhs : \u2200 (i : \u03b9), MeasurableSet (s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 i})\ni : \u03b9\n\u03c9 : \u03a9\nh\u03c4i : \u03c4 \u03c9 \u2264 i\na\u271d : \u03c9 \u2208 s\nh : \u03c4 \u03c9 \u2264 \u03c0 \u03c9\n\u22a2 \u03c4 \u03c9 \u2264 \u03c0 \u03c9 \u2228 i \u2264 \u03c0 \u03c9\n\ncase pos.mpr\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : LinearOrder \u03b9\nf : Filtration \u03b9 m\n\u03c4 \u03c0 : \u03a9 \u2192 \u03b9\ninst\u271d\u2074 : TopologicalSpace \u03b9\ninst\u271d\u00b3 : SecondCountableTopology \u03b9\ninst\u271d\u00b2 : OrderTopology \u03b9\ninst\u271d\u00b9 : MeasurableSpace \u03b9\ninst\u271d : BorelSpace \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\nh\u03c0 : IsStoppingTime f \u03c0\ns : Set \u03a9\nhs : \u2200 (i : \u03b9), MeasurableSet (s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 i})\ni : \u03b9\n\u03c9 : \u03a9\nh\u03c4i : \u03c4 \u03c9 \u2264 i\na\u271d : \u03c9 \u2208 s\nh : \u03c4 \u03c9 \u2264 \u03c0 \u03c9 \u2228 i \u2264 \u03c0 \u03c9\n\u22a2 \u03c4 \u03c9 \u2264 \u03c0 \u03c9"}, {"tactic": "exact Or.inl h", "annotated_tactic": ["exact <a>Or.inl</a> h", [{"full_name": "Or.inl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [517, 5], "def_end_pos": [517, 8]}]], "state_before": "case pos.mp\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : LinearOrder \u03b9\nf : Filtration \u03b9 m\n\u03c4 \u03c0 : \u03a9 \u2192 \u03b9\ninst\u271d\u2074 : TopologicalSpace \u03b9\ninst\u271d\u00b3 : SecondCountableTopology \u03b9\ninst\u271d\u00b2 : OrderTopology \u03b9\ninst\u271d\u00b9 : MeasurableSpace \u03b9\ninst\u271d : BorelSpace \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\nh\u03c0 : IsStoppingTime f \u03c0\ns : Set \u03a9\nhs : \u2200 (i : \u03b9), MeasurableSet (s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 i})\ni : \u03b9\n\u03c9 : \u03a9\nh\u03c4i : \u03c4 \u03c9 \u2264 i\na\u271d : \u03c9 \u2208 s\nh : \u03c4 \u03c9 \u2264 \u03c0 \u03c9\n\u22a2 \u03c4 \u03c9 \u2264 \u03c0 \u03c9 \u2228 i \u2264 \u03c0 \u03c9", "state_after": "no goals"}, {"tactic": "cases' h with h h", "annotated_tactic": ["cases' h with h h", []], "state_before": "case pos.mpr\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : LinearOrder \u03b9\nf : Filtration \u03b9 m\n\u03c4 \u03c0 : \u03a9 \u2192 \u03b9\ninst\u271d\u2074 : TopologicalSpace \u03b9\ninst\u271d\u00b3 : SecondCountableTopology \u03b9\ninst\u271d\u00b2 : OrderTopology \u03b9\ninst\u271d\u00b9 : MeasurableSpace \u03b9\ninst\u271d : BorelSpace \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\nh\u03c0 : IsStoppingTime f \u03c0\ns : Set \u03a9\nhs : \u2200 (i : \u03b9), MeasurableSet (s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 i})\ni : \u03b9\n\u03c9 : \u03a9\nh\u03c4i : \u03c4 \u03c9 \u2264 i\na\u271d : \u03c9 \u2208 s\nh : \u03c4 \u03c9 \u2264 \u03c0 \u03c9 \u2228 i \u2264 \u03c0 \u03c9\n\u22a2 \u03c4 \u03c9 \u2264 \u03c0 \u03c9", "state_after": "case pos.mpr.inl\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : LinearOrder \u03b9\nf : Filtration \u03b9 m\n\u03c4 \u03c0 : \u03a9 \u2192 \u03b9\ninst\u271d\u2074 : TopologicalSpace \u03b9\ninst\u271d\u00b3 : SecondCountableTopology \u03b9\ninst\u271d\u00b2 : OrderTopology \u03b9\ninst\u271d\u00b9 : MeasurableSpace \u03b9\ninst\u271d : BorelSpace \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\nh\u03c0 : IsStoppingTime f \u03c0\ns : Set \u03a9\nhs : \u2200 (i : \u03b9), MeasurableSet (s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 i})\ni : \u03b9\n\u03c9 : \u03a9\nh\u03c4i : \u03c4 \u03c9 \u2264 i\na\u271d : \u03c9 \u2208 s\nh : \u03c4 \u03c9 \u2264 \u03c0 \u03c9\n\u22a2 \u03c4 \u03c9 \u2264 \u03c0 \u03c9\n\ncase pos.mpr.inr\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : LinearOrder \u03b9\nf : Filtration \u03b9 m\n\u03c4 \u03c0 : \u03a9 \u2192 \u03b9\ninst\u271d\u2074 : TopologicalSpace \u03b9\ninst\u271d\u00b3 : SecondCountableTopology \u03b9\ninst\u271d\u00b2 : OrderTopology \u03b9\ninst\u271d\u00b9 : MeasurableSpace \u03b9\ninst\u271d : BorelSpace \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\nh\u03c0 : IsStoppingTime f \u03c0\ns : Set \u03a9\nhs : \u2200 (i : \u03b9), MeasurableSet (s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 i})\ni : \u03b9\n\u03c9 : \u03a9\nh\u03c4i : \u03c4 \u03c9 \u2264 i\na\u271d : \u03c9 \u2208 s\nh : i \u2264 \u03c0 \u03c9\n\u22a2 \u03c4 \u03c9 \u2264 \u03c0 \u03c9"}, {"tactic": "exact h", "annotated_tactic": ["exact h", []], "state_before": "case pos.mpr.inl\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : LinearOrder \u03b9\nf : Filtration \u03b9 m\n\u03c4 \u03c0 : \u03a9 \u2192 \u03b9\ninst\u271d\u2074 : TopologicalSpace \u03b9\ninst\u271d\u00b3 : SecondCountableTopology \u03b9\ninst\u271d\u00b2 : OrderTopology \u03b9\ninst\u271d\u00b9 : MeasurableSpace \u03b9\ninst\u271d : BorelSpace \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\nh\u03c0 : IsStoppingTime f \u03c0\ns : Set \u03a9\nhs : \u2200 (i : \u03b9), MeasurableSet (s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 i})\ni : \u03b9\n\u03c9 : \u03a9\nh\u03c4i : \u03c4 \u03c9 \u2264 i\na\u271d : \u03c9 \u2208 s\nh : \u03c4 \u03c9 \u2264 \u03c0 \u03c9\n\u22a2 \u03c4 \u03c9 \u2264 \u03c0 \u03c9", "state_after": "no goals"}, {"tactic": "exact h\u03c4i.trans h", "annotated_tactic": ["exact h\u03c4i.trans h", []], "state_before": "case pos.mpr.inr\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : LinearOrder \u03b9\nf : Filtration \u03b9 m\n\u03c4 \u03c0 : \u03a9 \u2192 \u03b9\ninst\u271d\u2074 : TopologicalSpace \u03b9\ninst\u271d\u00b3 : SecondCountableTopology \u03b9\ninst\u271d\u00b2 : OrderTopology \u03b9\ninst\u271d\u00b9 : MeasurableSpace \u03b9\ninst\u271d : BorelSpace \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\nh\u03c0 : IsStoppingTime f \u03c0\ns : Set \u03a9\nhs : \u2200 (i : \u03b9), MeasurableSet (s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 i})\ni : \u03b9\n\u03c9 : \u03a9\nh\u03c4i : \u03c4 \u03c9 \u2264 i\na\u271d : \u03c9 \u2208 s\nh : i \u2264 \u03c0 \u03c9\n\u22a2 \u03c4 \u03c9 \u2264 \u03c0 \u03c9", "state_after": "no goals"}, {"tactic": "exact (h\u03c4.min_const i).measurable_of_le fun _ => min_le_right _ _", "annotated_tactic": ["exact (h\u03c4.min_const i).<a>measurable_of_le</a> fun _ => <a>min_le_right</a> _ _", [{"full_name": "MeasureTheory.IsStoppingTime.measurable_of_le", "def_path": "Mathlib/Probability/Process/Stopping.lean", "def_pos": [585, 19], "def_end_pos": [585, 35]}, {"full_name": "min_le_right", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [40, 9], "def_end_pos": [40, 21]}]], "state_before": "\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : LinearOrder \u03b9\nf : Filtration \u03b9 m\n\u03c4 \u03c0 : \u03a9 \u2192 \u03b9\ninst\u271d\u2074 : TopologicalSpace \u03b9\ninst\u271d\u00b3 : SecondCountableTopology \u03b9\ninst\u271d\u00b2 : OrderTopology \u03b9\ninst\u271d\u00b9 : MeasurableSpace \u03b9\ninst\u271d : BorelSpace \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\nh\u03c0 : IsStoppingTime f \u03c0\ns : Set \u03a9\nhs : \u2200 (i : \u03b9), MeasurableSet (s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 i})\ni : \u03b9\nthis :\n  s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c0 \u03c9} \u2229 {\u03c9 | min (\u03c4 \u03c9) (\u03c0 \u03c9) \u2264 i} =\n    s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 i} \u2229 {\u03c9 | min (\u03c4 \u03c9) (\u03c0 \u03c9) \u2264 i} \u2229 {\u03c9 | min (\u03c4 \u03c9) i \u2264 min (min (\u03c4 \u03c9) (\u03c0 \u03c9)) i}\n\u22a2 Measurable fun a => min (\u03c4 a) i", "state_after": "no goals"}, {"tactic": "exact ((h\u03c4.min h\u03c0).min_const i).measurable_of_le fun _ => min_le_right _ _", "annotated_tactic": ["exact ((h\u03c4.min h\u03c0).<a>min_const</a> i).<a>measurable_of_le</a> fun _ => <a>min_le_right</a> _ _", [{"full_name": "MeasureTheory.IsStoppingTime.min_const", "def_path": "Mathlib/Probability/Process/Stopping.lean", "def_pos": [272, 19], "def_end_pos": [272, 28]}, {"full_name": "MeasureTheory.IsStoppingTime.measurable_of_le", "def_path": "Mathlib/Probability/Process/Stopping.lean", "def_pos": [585, 19], "def_end_pos": [585, 35]}, {"full_name": "min_le_right", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [40, 9], "def_end_pos": [40, 21]}]], "state_before": "\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : LinearOrder \u03b9\nf : Filtration \u03b9 m\n\u03c4 \u03c0 : \u03a9 \u2192 \u03b9\ninst\u271d\u2074 : TopologicalSpace \u03b9\ninst\u271d\u00b3 : SecondCountableTopology \u03b9\ninst\u271d\u00b2 : OrderTopology \u03b9\ninst\u271d\u00b9 : MeasurableSpace \u03b9\ninst\u271d : BorelSpace \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\nh\u03c0 : IsStoppingTime f \u03c0\ns : Set \u03a9\nhs : \u2200 (i : \u03b9), MeasurableSet (s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 i})\ni : \u03b9\nthis :\n  s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c0 \u03c9} \u2229 {\u03c9 | min (\u03c4 \u03c9) (\u03c0 \u03c9) \u2264 i} =\n    s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 i} \u2229 {\u03c9 | min (\u03c4 \u03c9) (\u03c0 \u03c9) \u2264 i} \u2229 {\u03c9 | min (\u03c4 \u03c9) i \u2264 min (min (\u03c4 \u03c9) (\u03c0 \u03c9)) i}\n\u22a2 Measurable fun a => min (min (\u03c4 a) (\u03c0 a)) i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "full_name": "MeasureTheory.SimpleFunc.integral_add_measure", "start": [448, 1], "end": [454, 50], "traced_tactics": [{"tactic": "simp_rw [integral_def]", "annotated_tactic": ["simp_rw [<a>integral_def</a>]", [{"full_name": "MeasureTheory.SimpleFunc.integral_def", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [313, 9], "def_end_pos": [313, 21]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\np : \u211d\u22650\u221e\nG : Type u_5\nF' : Type u_6\ninst\u271d\u2076 : NormedAddCommGroup G\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \u211d F'\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : SMulCommClass \u211d \ud835\udd5c E\n\u03bd : Measure \u03b1\nf : \u03b1 \u2192\u209b E\nhf : Integrable \u2191f\n\u22a2 integral (\u03bc + \u03bd) f = integral \u03bc f + integral \u03bd f", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\np : \u211d\u22650\u221e\nG : Type u_5\nF' : Type u_6\ninst\u271d\u2076 : NormedAddCommGroup G\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \u211d F'\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : SMulCommClass \u211d \ud835\udd5c E\n\u03bd : Measure \u03b1\nf : \u03b1 \u2192\u209b E\nhf : Integrable \u2191f\n\u22a2 setToSimpleFunc (weightedSMul (\u03bc + \u03bd)) f = setToSimpleFunc (weightedSMul \u03bc) f + setToSimpleFunc (weightedSMul \u03bd) f"}, {"tactic": "refine' setToSimpleFunc_add_left'\n  (weightedSMul \u03bc) (weightedSMul \u03bd) (weightedSMul (\u03bc + \u03bd)) (fun s _ h\u03bc\u03bds => _) hf", "annotated_tactic": ["refine' <a>setToSimpleFunc_add_left'</a>\n    (<a>weightedSMul</a> \u03bc) (<a>weightedSMul</a> \u03bd) (<a>weightedSMul</a> (\u03bc + \u03bd)) (fun s _ h\u03bc\u03bds => _) hf", [{"full_name": "MeasureTheory.SimpleFunc.setToSimpleFunc_add_left'", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [410, 9], "def_end_pos": [410, 34]}, {"full_name": "MeasureTheory.weightedSMul", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [166, 5], "def_end_pos": [166, 17]}, {"full_name": "MeasureTheory.weightedSMul", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [166, 5], "def_end_pos": [166, 17]}, {"full_name": "MeasureTheory.weightedSMul", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [166, 5], "def_end_pos": [166, 17]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\np : \u211d\u22650\u221e\nG : Type u_5\nF' : Type u_6\ninst\u271d\u2076 : NormedAddCommGroup G\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \u211d F'\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : SMulCommClass \u211d \ud835\udd5c E\n\u03bd : Measure \u03b1\nf : \u03b1 \u2192\u209b E\nhf : Integrable \u2191f\n\u22a2 setToSimpleFunc (weightedSMul (\u03bc + \u03bd)) f = setToSimpleFunc (weightedSMul \u03bc) f + setToSimpleFunc (weightedSMul \u03bd) f", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\np : \u211d\u22650\u221e\nG : Type u_5\nF' : Type u_6\ninst\u271d\u2076 : NormedAddCommGroup G\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \u211d F'\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : SMulCommClass \u211d \ud835\udd5c E\n\u03bd : Measure \u03b1\nf : \u03b1 \u2192\u209b E\nhf : Integrable \u2191f\ns : Set \u03b1\nx\u271d : MeasurableSet s\nh\u03bc\u03bds : \u2191\u2191(\u03bc + \u03bd) s < \u22a4\n\u22a2 weightedSMul (\u03bc + \u03bd) s = weightedSMul \u03bc s + weightedSMul \u03bd s"}, {"tactic": "rw [lt_top_iff_ne_top, Measure.coe_add, Pi.add_apply, ENNReal.add_ne_top] at h\u03bc\u03bds", "annotated_tactic": ["rw [<a>lt_top_iff_ne_top</a>, <a>Measure.coe_add</a>, <a>Pi.add_apply</a>, <a>ENNReal.add_ne_top</a>] at h\u03bc\u03bds", [{"full_name": "lt_top_iff_ne_top", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [173, 9], "def_end_pos": [173, 26]}, {"full_name": "MeasureTheory.Measure.coe_add", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [794, 9], "def_end_pos": [794, 16]}, {"full_name": "Pi.add_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [82, 3], "def_end_pos": [82, 14]}, {"full_name": "ENNReal.add_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [574, 9], "def_end_pos": [574, 19]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\np : \u211d\u22650\u221e\nG : Type u_5\nF' : Type u_6\ninst\u271d\u2076 : NormedAddCommGroup G\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \u211d F'\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : SMulCommClass \u211d \ud835\udd5c E\n\u03bd : Measure \u03b1\nf : \u03b1 \u2192\u209b E\nhf : Integrable \u2191f\ns : Set \u03b1\nx\u271d : MeasurableSet s\nh\u03bc\u03bds : \u2191\u2191(\u03bc + \u03bd) s < \u22a4\n\u22a2 weightedSMul (\u03bc + \u03bd) s = weightedSMul \u03bc s + weightedSMul \u03bd s", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\np : \u211d\u22650\u221e\nG : Type u_5\nF' : Type u_6\ninst\u271d\u2076 : NormedAddCommGroup G\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \u211d F'\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : SMulCommClass \u211d \ud835\udd5c E\n\u03bd : Measure \u03b1\nf : \u03b1 \u2192\u209b E\nhf : Integrable \u2191f\ns : Set \u03b1\nx\u271d : MeasurableSet s\nh\u03bc\u03bds : \u2191\u2191\u03bc s \u2260 \u22a4 \u2227 \u2191\u2191\u03bd s \u2260 \u22a4\n\u22a2 weightedSMul (\u03bc + \u03bd) s = weightedSMul \u03bc s + weightedSMul \u03bd s"}, {"tactic": "rw [weightedSMul_add_measure _ _ h\u03bc\u03bds.1 h\u03bc\u03bds.2]", "annotated_tactic": ["rw [<a>weightedSMul_add_measure</a> _ _ h\u03bc\u03bds.1 h\u03bc\u03bds.2]", [{"full_name": "MeasureTheory.weightedSMul_add_measure", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [184, 9], "def_end_pos": [184, 33]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\np : \u211d\u22650\u221e\nG : Type u_5\nF' : Type u_6\ninst\u271d\u2076 : NormedAddCommGroup G\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \u211d F'\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : SMulCommClass \u211d \ud835\udd5c E\n\u03bd : Measure \u03b1\nf : \u03b1 \u2192\u209b E\nhf : Integrable \u2191f\ns : Set \u03b1\nx\u271d : MeasurableSet s\nh\u03bc\u03bds : \u2191\u2191\u03bc s \u2260 \u22a4 \u2227 \u2191\u2191\u03bd s \u2260 \u22a4\n\u22a2 weightedSMul (\u03bc + \u03bd) s = weightedSMul \u03bc s + weightedSMul \u03bd s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Martingale/Convergence.lean", "full_name": "MeasureTheory.Submartingale.exists_ae_trim_tendsto_of_bdd", "start": [203, 1], "end": [211, 83], "traced_tactics": [{"tactic": "letI := (\u2a06 n, \u2131 n)", "annotated_tactic": ["letI := (\u2a06 n, \u2131 n)", []], "state_before": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200 (n : \u2115), snorm (f n) 1 \u03bc \u2264 \u2191R\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9) \u2202Measure.trim \u03bc (_ : sSup (Set.range fun n => \u2191\u2131 n) \u2264 m0), \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c)", "state_after": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200 (n : \u2115), snorm (f n) 1 \u03bc \u2264 \u2191R\nthis : MeasurableSpace \u03a9 := \u2a06 n, \u2191\u2131 n\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9) \u2202Measure.trim \u03bc (_ : sSup (Set.range fun n => \u2191\u2131 n) \u2264 m0), \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c)"}, {"tactic": "rw [ae_iff, trim_measurableSet_eq]", "annotated_tactic": ["rw [<a>ae_iff</a>, <a>trim_measurableSet_eq</a>]", [{"full_name": "MeasureTheory.ae_iff", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [388, 9], "def_end_pos": [388, 15]}, {"full_name": "MeasureTheory.trim_measurableSet_eq", "def_path": "Mathlib/MeasureTheory/Measure/Trim.lean", "def_pos": [53, 9], "def_end_pos": [53, 30]}]], "state_before": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200 (n : \u2115), snorm (f n) 1 \u03bc \u2264 \u2191R\nthis : MeasurableSpace \u03a9 := \u2a06 n, \u2191\u2131 n\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9) \u2202Measure.trim \u03bc (_ : sSup (Set.range fun n => \u2191\u2131 n) \u2264 m0), \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c)", "state_after": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200 (n : \u2115), snorm (f n) 1 \u03bc \u2264 \u2191R\nthis : MeasurableSpace \u03a9 := \u2a06 n, \u2191\u2131 n\n\u22a2 \u2191\u2191\u03bc {a | \u00ac\u2203 c, Tendsto (fun n => f n a) atTop (\ud835\udcdd c)} = 0\n\ncase hs\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200 (n : \u2115), snorm (f n) 1 \u03bc \u2264 \u2191R\nthis : MeasurableSpace \u03a9 := \u2a06 n, \u2191\u2131 n\n\u22a2 MeasurableSet {a | \u00ac\u2203 c, Tendsto (fun n => f n a) atTop (\ud835\udcdd c)}"}, {"tactic": "exact hf.exists_ae_tendsto_of_bdd hbdd", "annotated_tactic": ["exact hf.exists_ae_tendsto_of_bdd hbdd", []], "state_before": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200 (n : \u2115), snorm (f n) 1 \u03bc \u2264 \u2191R\nthis : MeasurableSpace \u03a9 := \u2a06 n, \u2191\u2131 n\n\u22a2 \u2191\u2191\u03bc {a | \u00ac\u2203 c, Tendsto (fun n => f n a) atTop (\ud835\udcdd c)} = 0", "state_after": "no goals"}, {"tactic": "exact MeasurableSet.compl $ measurableSet_exists_tendsto\n  fun n => (hf.stronglyMeasurable n).measurable.mono (le_sSup \u27e8n, rfl\u27e9) le_rfl", "annotated_tactic": ["exact <a>MeasurableSet.compl</a> $ <a>measurableSet_exists_tendsto</a>\n      fun n => (hf.stronglyMeasurable n).measurable.mono (<a>le_sSup</a> \u27e8n, <a>rfl</a>\u27e9) <a>le_rfl</a>", [{"full_name": "MeasurableSet.compl", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [87, 19], "def_end_pos": [87, 38]}, {"full_name": "MeasureTheory.measurableSet_exists_tendsto", "def_path": "Mathlib/MeasureTheory/Constructions/Polish.lean", "def_pos": [926, 9], "def_end_pos": [926, 37]}, {"full_name": "le_sSup", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [196, 9], "def_end_pos": [196, 16]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}]], "state_before": "case hs\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200 (n : \u2115), snorm (f n) 1 \u03bc \u2264 \u2191R\nthis : MeasurableSpace \u03a9 := \u2a06 n, \u2191\u2131 n\n\u22a2 MeasurableSet {a | \u00ac\u2203 c, Tendsto (fun n => f n a) atTop (\ud835\udcdd c)}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "Finset.inter_subset_ite", "start": [1848, 1], "end": [1849, 20], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "full_name": "MeasureTheory.L1.SimpleFunc.setToL1S_zero_left", "start": [695, 1], "end": [696, 36], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Function.lean", "full_name": "Set.eqOn_comp_right_iff", "start": [910, 1], "end": [911, 54], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "full_name": "measure_eq_measure_preimage_add_measure_tsum_Ico_zpow", "start": [1615, 1], "end": [1664, 26], "traced_tactics": [{"tactic": "rw [A, B, C, add_assoc]", "annotated_tactic": ["rw [A, B, C, <a>add_assoc</a>]", [{"full_name": "add_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [263, 3], "def_end_pos": [263, 14]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nA : \u2191\u2191\u03bc s = \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {0}) + \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioi 0)\nB : \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioi 0) = \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {\u22a4}) + \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioo 0 \u22a4)\nC : \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioo 0 \u22a4) = \u2211' (n : \u2124), \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ico (\u2191t ^ n) (\u2191t ^ (n + 1)))\n\u22a2 \u2191\u2191\u03bc s = \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {0}) + \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {\u22a4}) + \u2211' (n : \u2124), \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ico (\u2191t ^ n) (\u2191t ^ (n + 1)))", "state_after": "no goals"}, {"tactic": "rw [\u2190 measure_union]", "annotated_tactic": ["rw [\u2190 <a>measure_union</a>]", [{"full_name": "MeasureTheory.measure_union", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [124, 9], "def_end_pos": [124, 22]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\n\u22a2 \u2191\u2191\u03bc s = \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {0}) + \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioi 0)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\n\u22a2 \u2191\u2191\u03bc s = \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {0} \u222a s \u2229 f \u207b\u00b9' Ioi 0)\n\ncase hd\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\n\u22a2 Disjoint (s \u2229 f \u207b\u00b9' {0}) (s \u2229 f \u207b\u00b9' Ioi 0)\n\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\n\u22a2 MeasurableSet (s \u2229 f \u207b\u00b9' Ioi 0)"}, {"tactic": "congr 1", "annotated_tactic": ["congr 1", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\n\u22a2 \u2191\u2191\u03bc s = \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {0} \u222a s \u2229 f \u207b\u00b9' Ioi 0)", "state_after": "case e_a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\n\u22a2 s = s \u2229 f \u207b\u00b9' {0} \u222a s \u2229 f \u207b\u00b9' Ioi 0"}, {"tactic": "ext x", "annotated_tactic": ["ext x", []], "state_before": "case e_a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\n\u22a2 s = s \u2229 f \u207b\u00b9' {0} \u222a s \u2229 f \u207b\u00b9' Ioi 0", "state_after": "case e_a.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nx : \u03b1\n\u22a2 x \u2208 s \u2194 x \u2208 s \u2229 f \u207b\u00b9' {0} \u222a s \u2229 f \u207b\u00b9' Ioi 0"}, {"tactic": "have : 0 = f x \u2228 0 < f x := eq_or_lt_of_le bot_le", "annotated_tactic": ["have : 0 = f x \u2228 0 < f x := <a>eq_or_lt_of_le</a> <a>bot_le</a>", [{"full_name": "eq_or_lt_of_le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [414, 9], "def_end_pos": [414, 23]}, {"full_name": "bot_le", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [256, 9], "def_end_pos": [256, 15]}]], "state_before": "case e_a.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nx : \u03b1\n\u22a2 x \u2208 s \u2194 x \u2208 s \u2229 f \u207b\u00b9' {0} \u222a s \u2229 f \u207b\u00b9' Ioi 0", "state_after": "case e_a.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nx : \u03b1\nthis : 0 = f x \u2228 0 < f x\n\u22a2 x \u2208 s \u2194 x \u2208 s \u2229 f \u207b\u00b9' {0} \u222a s \u2229 f \u207b\u00b9' Ioi 0"}, {"tactic": "rw [eq_comm] at this", "annotated_tactic": ["rw [<a>eq_comm</a>] at this", [{"full_name": "eq_comm", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [104, 9], "def_end_pos": [104, 16]}]], "state_before": "case e_a.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nx : \u03b1\nthis : 0 = f x \u2228 0 < f x\n\u22a2 x \u2208 s \u2194 x \u2208 s \u2229 f \u207b\u00b9' {0} \u222a s \u2229 f \u207b\u00b9' Ioi 0", "state_after": "case e_a.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nx : \u03b1\nthis : f x = 0 \u2228 0 < f x\n\u22a2 x \u2208 s \u2194 x \u2208 s \u2229 f \u207b\u00b9' {0} \u222a s \u2229 f \u207b\u00b9' Ioi 0"}, {"tactic": "simp only [\u2190 and_or_left, this, mem_singleton_iff, mem_inter_iff, and_true_iff, mem_union,\n  mem_Ioi, mem_preimage]", "annotated_tactic": ["simp only [\u2190 <a>and_or_left</a>, this, <a>mem_singleton_iff</a>, <a>mem_inter_iff</a>, <a>and_true_iff</a>, <a>mem_union</a>,\n        <a>mem_Ioi</a>, <a>mem_preimage</a>]", [{"full_name": "and_or_left", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [321, 9], "def_end_pos": [321, 20]}, {"full_name": "Set.mem_singleton_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1273, 9], "def_end_pos": [1273, 26]}, {"full_name": "Set.mem_inter_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [909, 9], "def_end_pos": [909, 22]}, {"full_name": "and_true_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [145, 9], "def_end_pos": [145, 21]}, {"full_name": "Set.mem_union", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [767, 9], "def_end_pos": [767, 18]}, {"full_name": "Set.mem_Ioi", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [151, 9], "def_end_pos": [151, 16]}, {"full_name": "Set.mem_preimage", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [64, 9], "def_end_pos": [64, 21]}]], "state_before": "case e_a.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nx : \u03b1\nthis : f x = 0 \u2228 0 < f x\n\u22a2 x \u2208 s \u2194 x \u2208 s \u2229 f \u207b\u00b9' {0} \u222a s \u2229 f \u207b\u00b9' Ioi 0", "state_after": "no goals"}, {"tactic": "refine disjoint_left.2 fun x hx h'x => ?_", "annotated_tactic": ["refine <a>disjoint_left</a>.2 fun x hx h'x => ?_", [{"full_name": "Set.disjoint_left", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1546, 9], "def_end_pos": [1546, 22]}]], "state_before": "case hd\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\n\u22a2 Disjoint (s \u2229 f \u207b\u00b9' {0}) (s \u2229 f \u207b\u00b9' Ioi 0)", "state_after": "case hd\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nx : \u03b1\nhx : x \u2208 s \u2229 f \u207b\u00b9' {0}\nh'x : x \u2208 s \u2229 f \u207b\u00b9' Ioi 0\n\u22a2 False"}, {"tactic": "have : 0 < f x := h'x.2", "annotated_tactic": ["have : 0 < f x := h'x.2", []], "state_before": "case hd\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nx : \u03b1\nhx : x \u2208 s \u2229 f \u207b\u00b9' {0}\nh'x : x \u2208 s \u2229 f \u207b\u00b9' Ioi 0\n\u22a2 False", "state_after": "case hd\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nx : \u03b1\nhx : x \u2208 s \u2229 f \u207b\u00b9' {0}\nh'x : x \u2208 s \u2229 f \u207b\u00b9' Ioi 0\nthis : 0 < f x\n\u22a2 False"}, {"tactic": "exact lt_irrefl 0 (this.trans_le hx.2.le)", "annotated_tactic": ["exact <a>lt_irrefl</a> 0 (this.trans_le hx.2.<a>le</a>)", [{"full_name": "lt_irrefl", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [79, 9], "def_end_pos": [79, 18]}, {"full_name": "Eq.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [159, 7], "def_end_pos": [159, 12]}]], "state_before": "case hd\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nx : \u03b1\nhx : x \u2208 s \u2229 f \u207b\u00b9' {0}\nh'x : x \u2208 s \u2229 f \u207b\u00b9' Ioi 0\nthis : 0 < f x\n\u22a2 False", "state_after": "no goals"}, {"tactic": "exact hs.inter (hf measurableSet_Ioi)", "annotated_tactic": ["exact hs.inter (hf <a>measurableSet_Ioi</a>)", [{"full_name": "measurableSet_Ioi", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [579, 9], "def_end_pos": [579, 26]}]], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\n\u22a2 MeasurableSet (s \u2229 f \u207b\u00b9' Ioi 0)", "state_after": "no goals"}, {"tactic": "rw [\u2190 measure_union]", "annotated_tactic": ["rw [\u2190 <a>measure_union</a>]", [{"full_name": "MeasureTheory.measure_union", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [124, 9], "def_end_pos": [124, 22]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nA : \u2191\u2191\u03bc s = \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {0}) + \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioi 0)\n\u22a2 \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioi 0) = \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {\u22a4}) + \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioo 0 \u22a4)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nA : \u2191\u2191\u03bc s = \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {0}) + \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioi 0)\n\u22a2 \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioi 0) = \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {\u22a4} \u222a s \u2229 f \u207b\u00b9' Ioo 0 \u22a4)\n\ncase hd\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nA : \u2191\u2191\u03bc s = \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {0}) + \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioi 0)\n\u22a2 Disjoint (s \u2229 f \u207b\u00b9' {\u22a4}) (s \u2229 f \u207b\u00b9' Ioo 0 \u22a4)\n\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nA : \u2191\u2191\u03bc s = \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {0}) + \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioi 0)\n\u22a2 MeasurableSet (s \u2229 f \u207b\u00b9' Ioo 0 \u22a4)"}, {"tactic": "rw [\u2190 inter_union_distrib_left]", "annotated_tactic": ["rw [\u2190 <a>inter_union_distrib_left</a>]", [{"full_name": "Set.inter_union_distrib_left", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1059, 9], "def_end_pos": [1059, 33]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nA : \u2191\u2191\u03bc s = \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {0}) + \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioi 0)\n\u22a2 \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioi 0) = \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {\u22a4} \u222a s \u2229 f \u207b\u00b9' Ioo 0 \u22a4)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nA : \u2191\u2191\u03bc s = \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {0}) + \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioi 0)\n\u22a2 \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioi 0) = \u2191\u2191\u03bc (s \u2229 (f \u207b\u00b9' {\u22a4} \u222a f \u207b\u00b9' Ioo 0 \u22a4))"}, {"tactic": "congr", "annotated_tactic": ["congr", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nA : \u2191\u2191\u03bc s = \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {0}) + \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioi 0)\n\u22a2 \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioi 0) = \u2191\u2191\u03bc (s \u2229 (f \u207b\u00b9' {\u22a4} \u222a f \u207b\u00b9' Ioo 0 \u22a4))", "state_after": "case e_a.e_a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nA : \u2191\u2191\u03bc s = \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {0}) + \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioi 0)\n\u22a2 f \u207b\u00b9' Ioi 0 = f \u207b\u00b9' {\u22a4} \u222a f \u207b\u00b9' Ioo 0 \u22a4"}, {"tactic": "ext x", "annotated_tactic": ["ext x", []], "state_before": "case e_a.e_a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nA : \u2191\u2191\u03bc s = \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {0}) + \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioi 0)\n\u22a2 f \u207b\u00b9' Ioi 0 = f \u207b\u00b9' {\u22a4} \u222a f \u207b\u00b9' Ioo 0 \u22a4", "state_after": "case e_a.e_a.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nA : \u2191\u2191\u03bc s = \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {0}) + \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioi 0)\nx : \u03b1\n\u22a2 x \u2208 f \u207b\u00b9' Ioi 0 \u2194 x \u2208 f \u207b\u00b9' {\u22a4} \u222a f \u207b\u00b9' Ioo 0 \u22a4"}, {"tactic": "simp only [mem_singleton_iff, mem_union, mem_Ioo, mem_Ioi, mem_preimage]", "annotated_tactic": ["simp only [<a>mem_singleton_iff</a>, <a>mem_union</a>, <a>mem_Ioo</a>, <a>mem_Ioi</a>, <a>mem_preimage</a>]", [{"full_name": "Set.mem_singleton_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1273, 9], "def_end_pos": [1273, 26]}, {"full_name": "Set.mem_union", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [767, 9], "def_end_pos": [767, 18]}, {"full_name": "Set.mem_Ioo", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [116, 9], "def_end_pos": [116, 16]}, {"full_name": "Set.mem_Ioi", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [151, 9], "def_end_pos": [151, 16]}, {"full_name": "Set.mem_preimage", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [64, 9], "def_end_pos": [64, 21]}]], "state_before": "case e_a.e_a.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nA : \u2191\u2191\u03bc s = \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {0}) + \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioi 0)\nx : \u03b1\n\u22a2 x \u2208 f \u207b\u00b9' Ioi 0 \u2194 x \u2208 f \u207b\u00b9' {\u22a4} \u222a f \u207b\u00b9' Ioo 0 \u22a4", "state_after": "case e_a.e_a.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nA : \u2191\u2191\u03bc s = \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {0}) + \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioi 0)\nx : \u03b1\n\u22a2 0 < f x \u2194 f x = \u22a4 \u2228 0 < f x \u2227 f x < \u22a4"}, {"tactic": "have H : f x = \u221e \u2228 f x < \u221e := eq_or_lt_of_le le_top", "annotated_tactic": ["have H : f x = \u221e \u2228 f x < \u221e := <a>eq_or_lt_of_le</a> <a>le_top</a>", [{"full_name": "eq_or_lt_of_le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [414, 9], "def_end_pos": [414, 23]}, {"full_name": "le_top", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [98, 9], "def_end_pos": [98, 15]}]], "state_before": "case e_a.e_a.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nA : \u2191\u2191\u03bc s = \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {0}) + \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioi 0)\nx : \u03b1\n\u22a2 0 < f x \u2194 f x = \u22a4 \u2228 0 < f x \u2227 f x < \u22a4", "state_after": "case e_a.e_a.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nA : \u2191\u2191\u03bc s = \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {0}) + \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioi 0)\nx : \u03b1\nH : f x = \u22a4 \u2228 f x < \u22a4\n\u22a2 0 < f x \u2194 f x = \u22a4 \u2228 0 < f x \u2227 f x < \u22a4"}, {"tactic": "cases' H with H H", "annotated_tactic": ["cases' H with H H", []], "state_before": "case e_a.e_a.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nA : \u2191\u2191\u03bc s = \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {0}) + \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioi 0)\nx : \u03b1\nH : f x = \u22a4 \u2228 f x < \u22a4\n\u22a2 0 < f x \u2194 f x = \u22a4 \u2228 0 < f x \u2227 f x < \u22a4", "state_after": "case e_a.e_a.h.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nA : \u2191\u2191\u03bc s = \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {0}) + \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioi 0)\nx : \u03b1\nH : f x = \u22a4\n\u22a2 0 < f x \u2194 f x = \u22a4 \u2228 0 < f x \u2227 f x < \u22a4\n\ncase e_a.e_a.h.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nA : \u2191\u2191\u03bc s = \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {0}) + \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioi 0)\nx : \u03b1\nH : f x < \u22a4\n\u22a2 0 < f x \u2194 f x = \u22a4 \u2228 0 < f x \u2227 f x < \u22a4"}, {"tactic": "simp only [H, eq_self_iff_true, or_false_iff, WithTop.zero_lt_top, not_top_lt,\n  and_false_iff]", "annotated_tactic": ["simp only [H, <a>eq_self_iff_true</a>, <a>or_false_iff</a>, <a>WithTop.zero_lt_top</a>, <a>not_top_lt</a>,\n          <a>and_false_iff</a>]", [{"full_name": "eq_self_iff_true", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [86, 9], "def_end_pos": [86, 25]}, {"full_name": "or_false_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [188, 9], "def_end_pos": [188, 21]}, {"full_name": "WithTop.zero_lt_top", "def_path": "Mathlib/Algebra/Order/Monoid/WithTop.lean", "def_pos": [431, 9], "def_end_pos": [431, 20]}, {"full_name": "not_top_lt", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [118, 9], "def_end_pos": [118, 19]}, {"full_name": "and_false_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [149, 9], "def_end_pos": [149, 22]}]], "state_before": "case e_a.e_a.h.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nA : \u2191\u2191\u03bc s = \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {0}) + \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioi 0)\nx : \u03b1\nH : f x = \u22a4\n\u22a2 0 < f x \u2194 f x = \u22a4 \u2228 0 < f x \u2227 f x < \u22a4", "state_after": "no goals"}, {"tactic": "simp only [H, H.ne, and_true_iff, false_or_iff]", "annotated_tactic": ["simp only [H, H.ne, <a>and_true_iff</a>, <a>false_or_iff</a>]", [{"full_name": "and_true_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [145, 9], "def_end_pos": [145, 21]}, {"full_name": "false_or_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [186, 9], "def_end_pos": [186, 21]}]], "state_before": "case e_a.e_a.h.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nA : \u2191\u2191\u03bc s = \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {0}) + \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioi 0)\nx : \u03b1\nH : f x < \u22a4\n\u22a2 0 < f x \u2194 f x = \u22a4 \u2228 0 < f x \u2227 f x < \u22a4", "state_after": "no goals"}, {"tactic": "refine disjoint_left.2 fun x hx h'x => ?_", "annotated_tactic": ["refine <a>disjoint_left</a>.2 fun x hx h'x => ?_", [{"full_name": "Set.disjoint_left", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1546, 9], "def_end_pos": [1546, 22]}]], "state_before": "case hd\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nA : \u2191\u2191\u03bc s = \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {0}) + \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioi 0)\n\u22a2 Disjoint (s \u2229 f \u207b\u00b9' {\u22a4}) (s \u2229 f \u207b\u00b9' Ioo 0 \u22a4)", "state_after": "case hd\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nA : \u2191\u2191\u03bc s = \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {0}) + \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioi 0)\nx : \u03b1\nhx : x \u2208 s \u2229 f \u207b\u00b9' {\u22a4}\nh'x : x \u2208 s \u2229 f \u207b\u00b9' Ioo 0 \u22a4\n\u22a2 False"}, {"tactic": "have : f x < \u221e := h'x.2.2", "annotated_tactic": ["have : f x < \u221e := h'x.2.2", []], "state_before": "case hd\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nA : \u2191\u2191\u03bc s = \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {0}) + \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioi 0)\nx : \u03b1\nhx : x \u2208 s \u2229 f \u207b\u00b9' {\u22a4}\nh'x : x \u2208 s \u2229 f \u207b\u00b9' Ioo 0 \u22a4\n\u22a2 False", "state_after": "case hd\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nA : \u2191\u2191\u03bc s = \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {0}) + \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioi 0)\nx : \u03b1\nhx : x \u2208 s \u2229 f \u207b\u00b9' {\u22a4}\nh'x : x \u2208 s \u2229 f \u207b\u00b9' Ioo 0 \u22a4\nthis : f x < \u22a4\n\u22a2 False"}, {"tactic": "exact lt_irrefl _ (this.trans_le (le_of_eq hx.2.symm))", "annotated_tactic": ["exact <a>lt_irrefl</a> _ (this.trans_le (<a>le_of_eq</a> hx.2.<a>symm</a>))", [{"full_name": "lt_irrefl", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [79, 9], "def_end_pos": [79, 18]}, {"full_name": "le_of_eq", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [72, 9], "def_end_pos": [72, 17]}, {"full_name": "Eq.symm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [310, 9], "def_end_pos": [310, 16]}]], "state_before": "case hd\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nA : \u2191\u2191\u03bc s = \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {0}) + \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioi 0)\nx : \u03b1\nhx : x \u2208 s \u2229 f \u207b\u00b9' {\u22a4}\nh'x : x \u2208 s \u2229 f \u207b\u00b9' Ioo 0 \u22a4\nthis : f x < \u22a4\n\u22a2 False", "state_after": "no goals"}, {"tactic": "exact hs.inter (hf measurableSet_Ioo)", "annotated_tactic": ["exact hs.inter (hf <a>measurableSet_Ioo</a>)", [{"full_name": "measurableSet_Ioo", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [584, 9], "def_end_pos": [584, 26]}]], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nA : \u2191\u2191\u03bc s = \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {0}) + \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioi 0)\n\u22a2 MeasurableSet (s \u2229 f \u207b\u00b9' Ioo 0 \u22a4)", "state_after": "no goals"}, {"tactic": "rw [\u2190 measure_iUnion,\n  ENNReal.Ioo_zero_top_eq_iUnion_Ico_zpow (ENNReal.one_lt_coe_iff.2 ht) ENNReal.coe_ne_top,\n  preimage_iUnion, inter_iUnion]", "annotated_tactic": ["rw [\u2190 <a>measure_iUnion</a>,\n      <a>ENNReal.Ioo_zero_top_eq_iUnion_Ico_zpow</a> (<a>ENNReal.one_lt_coe_iff</a>.2 ht) <a>ENNReal.coe_ne_top</a>,\n      <a>preimage_iUnion</a>, <a>inter_iUnion</a>]", [{"full_name": "MeasureTheory.measure_iUnion", "def_path": "Mathlib/MeasureTheory/Measure/NullMeasurable.lean", "def_pos": [272, 9], "def_end_pos": [272, 23]}, {"full_name": "ENNReal.Ioo_zero_top_eq_iUnion_Ico_zpow", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1952, 9], "def_end_pos": [1952, 40]}, {"full_name": "ENNReal.one_lt_coe_iff", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [703, 9], "def_end_pos": [703, 23]}, {"full_name": "ENNReal.coe_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [302, 17], "def_end_pos": [302, 27]}, {"full_name": "Set.preimage_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [1854, 9], "def_end_pos": [1854, 24]}, {"full_name": "Set.inter_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [635, 9], "def_end_pos": [635, 21]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nA : \u2191\u2191\u03bc s = \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {0}) + \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioi 0)\nB : \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioi 0) = \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {\u22a4}) + \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioo 0 \u22a4)\n\u22a2 \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioo 0 \u22a4) = \u2211' (n : \u2124), \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ico (\u2191t ^ n) (\u2191t ^ (n + 1)))", "state_after": "case hn\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nA : \u2191\u2191\u03bc s = \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {0}) + \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioi 0)\nB : \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioi 0) = \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {\u22a4}) + \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioo 0 \u22a4)\n\u22a2 Pairwise (Disjoint on fun n => s \u2229 f \u207b\u00b9' Ico (\u2191t ^ n) (\u2191t ^ (n + 1)))\n\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nA : \u2191\u2191\u03bc s = \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {0}) + \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioi 0)\nB : \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioi 0) = \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {\u22a4}) + \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioo 0 \u22a4)\n\u22a2 \u2200 (i : \u2124), MeasurableSet (s \u2229 f \u207b\u00b9' Ico (\u2191t ^ i) (\u2191t ^ (i + 1)))"}, {"tactic": "intro i j", "annotated_tactic": ["intro i j", []], "state_before": "case hn\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nA : \u2191\u2191\u03bc s = \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {0}) + \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioi 0)\nB : \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioi 0) = \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {\u22a4}) + \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioo 0 \u22a4)\n\u22a2 Pairwise (Disjoint on fun n => s \u2229 f \u207b\u00b9' Ico (\u2191t ^ n) (\u2191t ^ (n + 1)))", "state_after": "case hn\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nA : \u2191\u2191\u03bc s = \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {0}) + \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioi 0)\nB : \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioi 0) = \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {\u22a4}) + \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioo 0 \u22a4)\ni j : \u2124\n\u22a2 i \u2260 j \u2192 (Disjoint on fun n => s \u2229 f \u207b\u00b9' Ico (\u2191t ^ n) (\u2191t ^ (n + 1))) i j"}, {"tactic": "simp only [Function.onFun]", "annotated_tactic": ["simp only [<a>Function.onFun</a>]", [{"full_name": "Function.onFun", "def_path": "Mathlib/Init/Function.lean", "def_pos": [49, 5], "def_end_pos": [49, 10]}]], "state_before": "case hn\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nA : \u2191\u2191\u03bc s = \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {0}) + \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioi 0)\nB : \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioi 0) = \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {\u22a4}) + \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioo 0 \u22a4)\ni j : \u2124\n\u22a2 i \u2260 j \u2192 (Disjoint on fun n => s \u2229 f \u207b\u00b9' Ico (\u2191t ^ n) (\u2191t ^ (n + 1))) i j", "state_after": "case hn\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nA : \u2191\u2191\u03bc s = \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {0}) + \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioi 0)\nB : \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioi 0) = \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {\u22a4}) + \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioo 0 \u22a4)\ni j : \u2124\n\u22a2 i \u2260 j \u2192 Disjoint (s \u2229 f \u207b\u00b9' Ico (\u2191t ^ i) (\u2191t ^ (i + 1))) (s \u2229 f \u207b\u00b9' Ico (\u2191t ^ j) (\u2191t ^ (j + 1)))"}, {"tactic": "intro hij", "annotated_tactic": ["intro hij", []], "state_before": "case hn\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nA : \u2191\u2191\u03bc s = \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {0}) + \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioi 0)\nB : \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioi 0) = \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {\u22a4}) + \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioo 0 \u22a4)\ni j : \u2124\n\u22a2 i \u2260 j \u2192 Disjoint (s \u2229 f \u207b\u00b9' Ico (\u2191t ^ i) (\u2191t ^ (i + 1))) (s \u2229 f \u207b\u00b9' Ico (\u2191t ^ j) (\u2191t ^ (j + 1)))", "state_after": "case hn\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nA : \u2191\u2191\u03bc s = \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {0}) + \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioi 0)\nB : \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioi 0) = \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {\u22a4}) + \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioo 0 \u22a4)\ni j : \u2124\nhij : i \u2260 j\n\u22a2 Disjoint (s \u2229 f \u207b\u00b9' Ico (\u2191t ^ i) (\u2191t ^ (i + 1))) (s \u2229 f \u207b\u00b9' Ico (\u2191t ^ j) (\u2191t ^ (j + 1)))"}, {"tactic": "wlog h : i < j generalizing i j", "annotated_tactic": ["wlog h : i < j generalizing i j", []], "state_before": "case hn\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nA : \u2191\u2191\u03bc s = \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {0}) + \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioi 0)\nB : \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioi 0) = \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {\u22a4}) + \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioo 0 \u22a4)\ni j : \u2124\nhij : i \u2260 j\n\u22a2 Disjoint (s \u2229 f \u207b\u00b9' Ico (\u2191t ^ i) (\u2191t ^ (i + 1))) (s \u2229 f \u207b\u00b9' Ico (\u2191t ^ j) (\u2191t ^ (j + 1)))", "state_after": "case hn.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nA : \u2191\u2191\u03bc s = \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {0}) + \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioi 0)\nB : \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioi 0) = \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {\u22a4}) + \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioo 0 \u22a4)\ni j : \u2124\nhij : i \u2260 j\nthis :\n  \u2200 \u2983i j : \u2124\u2984, i \u2260 j \u2192 i < j \u2192 Disjoint (s \u2229 f \u207b\u00b9' Ico (\u2191t ^ i) (\u2191t ^ (i + 1))) (s \u2229 f \u207b\u00b9' Ico (\u2191t ^ j) (\u2191t ^ (j + 1)))\nh : \u00aci < j\n\u22a2 Disjoint (s \u2229 f \u207b\u00b9' Ico (\u2191t ^ i) (\u2191t ^ (i + 1))) (s \u2229 f \u207b\u00b9' Ico (\u2191t ^ j) (\u2191t ^ (j + 1)))\n\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nA : \u2191\u2191\u03bc s = \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {0}) + \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioi 0)\nB : \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioi 0) = \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {\u22a4}) + \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioo 0 \u22a4)\ni j : \u2124\nhij : i \u2260 j\nh : i < j\n\u22a2 Disjoint (s \u2229 f \u207b\u00b9' Ico (\u2191t ^ i) (\u2191t ^ (i + 1))) (s \u2229 f \u207b\u00b9' Ico (\u2191t ^ j) (\u2191t ^ (j + 1)))"}, {"tactic": "refine disjoint_left.2 fun x hx h'x => lt_irrefl (f x) ?_", "annotated_tactic": ["refine <a>disjoint_left</a>.2 fun x hx h'x => <a>lt_irrefl</a> (f x) ?_", [{"full_name": "Set.disjoint_left", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1546, 9], "def_end_pos": [1546, 22]}, {"full_name": "lt_irrefl", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [79, 9], "def_end_pos": [79, 18]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nA : \u2191\u2191\u03bc s = \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {0}) + \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioi 0)\nB : \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioi 0) = \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {\u22a4}) + \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioo 0 \u22a4)\ni j : \u2124\nhij : i \u2260 j\nh : i < j\n\u22a2 Disjoint (s \u2229 f \u207b\u00b9' Ico (\u2191t ^ i) (\u2191t ^ (i + 1))) (s \u2229 f \u207b\u00b9' Ico (\u2191t ^ j) (\u2191t ^ (j + 1)))", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nA : \u2191\u2191\u03bc s = \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {0}) + \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioi 0)\nB : \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioi 0) = \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {\u22a4}) + \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioo 0 \u22a4)\ni j : \u2124\nhij : i \u2260 j\nh : i < j\nx : \u03b1\nhx : x \u2208 s \u2229 f \u207b\u00b9' Ico (\u2191t ^ i) (\u2191t ^ (i + 1))\nh'x : x \u2208 s \u2229 f \u207b\u00b9' Ico (\u2191t ^ j) (\u2191t ^ (j + 1))\n\u22a2 f x < f x"}, {"tactic": "calc\n  f x < (t : \u211d\u22650\u221e) ^ (i + 1) := hx.2.2\n  _ \u2264 (t : \u211d\u22650\u221e) ^ j := (ENNReal.zpow_le_of_le (ENNReal.one_le_coe_iff.2 ht.le) h)\n  _ \u2264 f x := h'x.2.1", "annotated_tactic": ["calc\n        f x < (t : \u211d\u22650\u221e) ^ (i + 1) := hx.2.2\n        _ \u2264 (t : \u211d\u22650\u221e) ^ j := (<a>ENNReal.zpow_le_of_le</a> (<a>ENNReal.one_le_coe_iff</a>.2 ht.le) h)\n        _ \u2264 f x := h'x.2.1", [{"full_name": "ENNReal.zpow_le_of_le", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1968, 9], "def_end_pos": [1968, 22]}, {"full_name": "ENNReal.one_le_coe_iff", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [691, 9], "def_end_pos": [691, 23]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nA : \u2191\u2191\u03bc s = \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {0}) + \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioi 0)\nB : \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioi 0) = \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {\u22a4}) + \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioo 0 \u22a4)\ni j : \u2124\nhij : i \u2260 j\nh : i < j\nx : \u03b1\nhx : x \u2208 s \u2229 f \u207b\u00b9' Ico (\u2191t ^ i) (\u2191t ^ (i + 1))\nh'x : x \u2208 s \u2229 f \u207b\u00b9' Ico (\u2191t ^ j) (\u2191t ^ (j + 1))\n\u22a2 f x < f x", "state_after": "no goals"}, {"tactic": "exact (this hij.symm (hij.lt_or_lt.resolve_left h)).symm", "annotated_tactic": ["exact (this hij.symm (hij.lt_or_lt.resolve_left h)).<a>symm</a>", [{"full_name": "Disjoint.symm", "def_path": "Mathlib/Order/Disjoint.lean", "def_pos": [50, 9], "def_end_pos": [50, 22]}]], "state_before": "case hn.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nA : \u2191\u2191\u03bc s = \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {0}) + \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioi 0)\nB : \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioi 0) = \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {\u22a4}) + \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioo 0 \u22a4)\ni j : \u2124\nhij : i \u2260 j\nthis :\n  \u2200 \u2983i j : \u2124\u2984, i \u2260 j \u2192 i < j \u2192 Disjoint (s \u2229 f \u207b\u00b9' Ico (\u2191t ^ i) (\u2191t ^ (i + 1))) (s \u2229 f \u207b\u00b9' Ico (\u2191t ^ j) (\u2191t ^ (j + 1)))\nh : \u00aci < j\n\u22a2 Disjoint (s \u2229 f \u207b\u00b9' Ico (\u2191t ^ i) (\u2191t ^ (i + 1))) (s \u2229 f \u207b\u00b9' Ico (\u2191t ^ j) (\u2191t ^ (j + 1)))", "state_after": "no goals"}, {"tactic": "intro n", "annotated_tactic": ["intro n", []], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nA : \u2191\u2191\u03bc s = \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {0}) + \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioi 0)\nB : \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioi 0) = \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {\u22a4}) + \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioo 0 \u22a4)\n\u22a2 \u2200 (i : \u2124), MeasurableSet (s \u2229 f \u207b\u00b9' Ico (\u2191t ^ i) (\u2191t ^ (i + 1)))", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nA : \u2191\u2191\u03bc s = \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {0}) + \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioi 0)\nB : \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioi 0) = \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {\u22a4}) + \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioo 0 \u22a4)\nn : \u2124\n\u22a2 MeasurableSet (s \u2229 f \u207b\u00b9' Ico (\u2191t ^ n) (\u2191t ^ (n + 1)))"}, {"tactic": "exact hs.inter (hf measurableSet_Ico)", "annotated_tactic": ["exact hs.inter (hf <a>measurableSet_Ico</a>)", [{"full_name": "measurableSet_Ico", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [594, 9], "def_end_pos": [594, 26]}]], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nA : \u2191\u2191\u03bc s = \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {0}) + \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioi 0)\nB : \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioi 0) = \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {\u22a4}) + \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioo 0 \u22a4)\nn : \u2124\n\u22a2 MeasurableSet (s \u2229 f \u207b\u00b9' Ico (\u2191t ^ n) (\u2191t ^ (n + 1)))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Tree.lean", "full_name": "Tree.numLeaves_pos", "start": [121, 1], "end": [123, 32], "traced_tactics": [{"tactic": "rw [numLeaves_eq_numNodes_succ]", "annotated_tactic": ["rw [<a>numLeaves_eq_numNodes_succ</a>]", [{"full_name": "Tree.numLeaves_eq_numNodes_succ", "def_path": "Mathlib/Data/Tree.lean", "def_pos": [117, 9], "def_end_pos": [117, 35]}]], "state_before": "\u03b1 : Type u\nx : Tree \u03b1\n\u22a2 0 < numLeaves x", "state_after": "\u03b1 : Type u\nx : Tree \u03b1\n\u22a2 0 < numNodes x + 1"}, {"tactic": "exact x.numNodes.zero_lt_succ", "annotated_tactic": ["exact x.numNodes.zero_lt_succ", []], "state_before": "\u03b1 : Type u\nx : Tree \u03b1\n\u22a2 0 < numNodes x + 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "full_name": "MeasureTheory.set_integral_eq_zero_of_forall_eq_zero", "start": [280, 1], "end": [282, 66], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/RBMap/Lemmas.lean", "full_name": "Std.RBSet.foldlM_eq_foldlM_toList", "start": [641, 1], "end": [642, 88], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "full_name": "Int.sub_ediv_of_dvd", "start": [806, 1], "end": [809, 39], "traced_tactics": [{"tactic": "rw [Int.sub_eq_add_neg, Int.sub_eq_add_neg, Int.add_ediv_of_dvd_right (Int.dvd_neg.2 hcb)]", "annotated_tactic": ["rw [<a>Int.sub_eq_add_neg</a>, <a>Int.sub_eq_add_neg</a>, <a>Int.add_ediv_of_dvd_right</a> (<a>Int.dvd_neg</a>.2 hcb)]", [{"full_name": "Int.sub_eq_add_neg", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [94, 19], "def_end_pos": [94, 33]}, {"full_name": "Int.sub_eq_add_neg", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [94, 19], "def_end_pos": [94, 33]}, {"full_name": "Int.add_ediv_of_dvd_right", "def_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "def_pos": [176, 9], "def_end_pos": [176, 30]}, {"full_name": "Int.dvd_neg", "def_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "def_pos": [608, 19], "def_end_pos": [608, 26]}]], "state_before": "a b c : Int\nhcb : c \u2223 b\n\u22a2 (a - b) / c = a / c - b / c", "state_after": "a b c : Int\nhcb : c \u2223 b\n\u22a2 a / c + -b / c = a / c + -(b / c)"}, {"tactic": "congr", "annotated_tactic": ["congr", []], "state_before": "a b c : Int\nhcb : c \u2223 b\n\u22a2 a / c + -b / c = a / c + -(b / c)", "state_after": "case e_a\na b c : Int\nhcb : c \u2223 b\n\u22a2 -b / c = -(b / c)"}, {"tactic": "exact Int.neg_ediv_of_dvd hcb", "annotated_tactic": ["exact <a>Int.neg_ediv_of_dvd</a> hcb", [{"full_name": "Int.neg_ediv_of_dvd", "def_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "def_pos": [802, 9], "def_end_pos": [802, 24]}]], "state_before": "case e_a\na b c : Int\nhcb : c \u2223 b\n\u22a2 -b / c = -(b / c)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Partrec.lean", "full_name": "Partrec.bind_decode\u2082_iff", "start": [595, 1], "end": [602, 95], "traced_tactics": [{"tactic": "simpa [encodek\u2082] using (nat_iff.2 h).comp (@Computable.encode \u03b1 _)", "annotated_tactic": ["simpa [<a>encodek\u2082</a>] using (<a>nat_iff</a>.2 h).<a>comp</a> (@<a>Computable.encode</a> \u03b1 _)", [{"full_name": "Encodable.encodek\u2082", "def_path": "Mathlib/Logic/Encodable/Basic.lean", "def_pos": [226, 9], "def_end_pos": [226, 17]}, {"full_name": "Partrec.nat_iff", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [485, 9], "def_end_pos": [485, 16]}, {"full_name": "Partrec.comp", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [480, 16], "def_end_pos": [480, 20]}, {"full_name": "Computable.encode", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [400, 19], "def_end_pos": [400, 25]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192. \u03c3\nh : Nat.Partrec fun n => Part.bind \u2191(decode\u2082 \u03b1 n) fun a => Part.map encode (f a)\n\u22a2 Partrec fun a => Part.map encode (f a)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/FundThmCalculus.lean", "full_name": "intervalIntegral.integral_eq_sub_of_hasDeriv_right_of_le", "start": [1171, 1], "end": [1178, 84], "traced_tactics": [{"tactic": "refine' (NormedSpace.eq_iff_forall_dual_eq \u211d).2 fun g => _", "annotated_tactic": ["refine' (<a>NormedSpace.eq_iff_forall_dual_eq</a> \u211d).2 fun g => _", [{"full_name": "NormedSpace.eq_iff_forall_dual_eq", "def_path": "Mathlib/Analysis/NormedSpace/Dual.lean", "def_pos": [137, 9], "def_end_pos": [137, 30]}]], "state_before": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf\u271d : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\nf f' : \u211d \u2192 E\na b : \u211d\nhab : a \u2264 b\nhcont : ContinuousOn f (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivWithinAt f (f' x) (Ioi x) x\nf'int : IntervalIntegrable f' volume a b\n\u22a2 \u222b (y : \u211d) in a..b, f' y = f b - f a", "state_after": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf\u271d : \u211d \u2192 E\ng' g\u271d \u03c6 : \u211d \u2192 \u211d\nf f' : \u211d \u2192 E\na b : \u211d\nhab : a \u2264 b\nhcont : ContinuousOn f (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivWithinAt f (f' x) (Ioi x) x\nf'int : IntervalIntegrable f' volume a b\ng : NormedSpace.Dual \u211d E\n\u22a2 \u2191g (\u222b (y : \u211d) in a..b, f' y) = \u2191g (f b - f a)"}, {"tactic": "rw [\u2190 g.intervalIntegral_comp_comm f'int, g.map_sub]", "annotated_tactic": ["rw [\u2190 g.intervalIntegral_comp_comm f'int, g.map_sub]", []], "state_before": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf\u271d : \u211d \u2192 E\ng' g\u271d \u03c6 : \u211d \u2192 \u211d\nf f' : \u211d \u2192 E\na b : \u211d\nhab : a \u2264 b\nhcont : ContinuousOn f (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivWithinAt f (f' x) (Ioi x) x\nf'int : IntervalIntegrable f' volume a b\ng : NormedSpace.Dual \u211d E\n\u22a2 \u2191g (\u222b (y : \u211d) in a..b, f' y) = \u2191g (f b - f a)", "state_after": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf\u271d : \u211d \u2192 E\ng' g\u271d \u03c6 : \u211d \u2192 \u211d\nf f' : \u211d \u2192 E\na b : \u211d\nhab : a \u2264 b\nhcont : ContinuousOn f (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivWithinAt f (f' x) (Ioi x) x\nf'int : IntervalIntegrable f' volume a b\ng : NormedSpace.Dual \u211d E\n\u22a2 \u222b (x : \u211d) in a..b, \u2191g (f' x) = \u2191g (f b) - \u2191g (f a)"}, {"tactic": "exact integral_eq_sub_of_hasDeriv_right_of_le_real hab (g.continuous.comp_continuousOn hcont)\n  (fun x hx => g.hasFDerivAt.comp_hasDerivWithinAt x (hderiv x hx))\n  (g.integrable_comp ((intervalIntegrable_iff_integrable_Icc_of_le hab).1 f'int))", "annotated_tactic": ["exact <a>integral_eq_sub_of_hasDeriv_right_of_le_real</a> hab (g.continuous.comp_continuousOn hcont)\n    (fun x hx => g.hasFDerivAt.comp_hasDerivWithinAt x (hderiv x hx))\n    (g.integrable_comp ((<a>intervalIntegrable_iff_integrable_Icc_of_le</a> hab).1 f'int))", [{"full_name": "intervalIntegral.integral_eq_sub_of_hasDeriv_right_of_le_real", "def_path": "Mathlib/MeasureTheory/Integral/FundThmCalculus.lean", "def_pos": [1159, 9], "def_end_pos": [1159, 53]}, {"full_name": "intervalIntegrable_iff_integrable_Icc_of_le", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [101, 9], "def_end_pos": [101, 52]}]], "state_before": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf\u271d : \u211d \u2192 E\ng' g\u271d \u03c6 : \u211d \u2192 \u211d\nf f' : \u211d \u2192 E\na b : \u211d\nhab : a \u2264 b\nhcont : ContinuousOn f (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivWithinAt f (f' x) (Ioi x) x\nf'int : IntervalIntegrable f' volume a b\ng : NormedSpace.Dual \u211d E\n\u22a2 \u222b (x : \u211d) in a..b, \u2191g (f' x) = \u2191g (f b) - \u2191g (f a)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/WithDensity.lean", "full_name": "MeasureTheory.withDensity_mul", "start": [411, 1], "end": [413, 51], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Holor.lean", "full_name": "Holor.cprankMax_nil", "start": [324, 1], "end": [326, 34], "traced_tactics": [{"tactic": "have h := CPRankMax.succ 0 x 0 (CPRankMax1.nil x) CPRankMax.zero", "annotated_tactic": ["have h := <a>CPRankMax.succ</a> 0 x 0 (<a>CPRankMax1.nil</a> x) <a>CPRankMax.zero</a>", [{"full_name": "Holor.CPRankMax.succ", "def_path": "Mathlib/Data/Holor.lean", "def_pos": [320, 5], "def_end_pos": [320, 9]}, {"full_name": "Holor.CPRankMax1.nil", "def_path": "Mathlib/Data/Holor.lean", "def_pos": [312, 5], "def_end_pos": [312, 8]}, {"full_name": "Holor.CPRankMax.zero", "def_path": "Mathlib/Data/Holor.lean", "def_pos": [319, 5], "def_end_pos": [319, 9]}]], "state_before": "\u03b1 : Type\nd : \u2115\nds ds\u2081 ds\u2082 ds\u2083 : List \u2115\ninst\u271d\u00b9 : Monoid \u03b1\ninst\u271d : AddMonoid \u03b1\nx : Holor \u03b1 []\n\u22a2 CPRankMax 1 x", "state_after": "\u03b1 : Type\nd : \u2115\nds ds\u2081 ds\u2082 ds\u2083 : List \u2115\ninst\u271d\u00b9 : Monoid \u03b1\ninst\u271d : AddMonoid \u03b1\nx : Holor \u03b1 []\nh : CPRankMax (0 + 1) (x + 0)\n\u22a2 CPRankMax 1 x"}, {"tactic": "rwa [add_zero x, zero_add] at h", "annotated_tactic": ["rwa [<a>add_zero</a> x, <a>zero_add</a>] at h", [{"full_name": "add_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [469, 3], "def_end_pos": [469, 14]}, {"full_name": "zero_add", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [463, 3], "def_end_pos": [463, 14]}]], "state_before": "\u03b1 : Type\nd : \u2115\nds ds\u2081 ds\u2082 ds\u2083 : List \u2115\ninst\u271d\u00b9 : Monoid \u03b1\ninst\u271d : AddMonoid \u03b1\nx : Holor \u03b1 []\nh : CPRankMax (0 + 1) (x + 0)\n\u22a2 CPRankMax 1 x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Card.lean", "full_name": "Set.one_lt_encard_iff", "start": [306, 1], "end": [307, 66], "traced_tactics": [{"tactic": "rw [\u2190not_iff_not, not_exists, not_lt, encard_le_one_iff]", "annotated_tactic": ["rw [\u2190<a>not_iff_not</a>, <a>not_exists</a>, <a>not_lt</a>, <a>encard_le_one_iff</a>]", [{"full_name": "not_iff_not", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [439, 9], "def_end_pos": [439, 20]}, {"full_name": "not_exists", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [422, 17], "def_end_pos": [422, 27]}, {"full_name": "not_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [368, 9], "def_end_pos": [368, 15]}, {"full_name": "Set.encard_le_one_iff", "def_path": "Mathlib/Data/Set/Card.lean", "def_pos": [299, 9], "def_end_pos": [299, 26]}]], "state_before": "\u03b1 : Type u_1\ns t : Set \u03b1\n\u22a2 1 < encard s \u2194 \u2203 a b, a \u2208 s \u2227 b \u2208 s \u2227 a \u2260 b", "state_after": "\u03b1 : Type u_1\ns t : Set \u03b1\n\u22a2 (\u2200 (a b : \u03b1), a \u2208 s \u2192 b \u2208 s \u2192 a = b) \u2194 \u2200 (x : \u03b1), \u00ac\u2203 b, x \u2208 s \u2227 b \u2208 s \u2227 x \u2260 b"}, {"tactic": "aesop", "annotated_tactic": ["aesop", []], "state_before": "\u03b1 : Type u_1\ns t : Set \u03b1\n\u22a2 (\u2200 (a b : \u03b1), a \u2208 s \u2192 b \u2208 s \u2192 a = b) \u2194 \u2200 (x : \u03b1), \u00ac\u2203 b, x \u2208 s \u2227 b \u2208 s \u2227 x \u2260 b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "full_name": "String.get_of_valid", "start": [223, 1], "end": [224, 43], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Kernel/IntegralCompProd.lean", "full_name": "ProbabilityTheory.hasFiniteIntegral_prod_mk_left", "start": [48, 1], "end": [61, 24], "traced_tactics": [{"tactic": "let t := toMeasurable ((\u03ba \u2297\u2096 \u03b7) a) s", "annotated_tactic": ["let t := <a>toMeasurable</a> ((\u03ba \u2297\u2096 \u03b7) a) s", [{"full_name": "MeasureTheory.toMeasurable", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [626, 17], "def_end_pos": [626, 29]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nE : Type u_4\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d\u00b2 : NormedAddCommGroup E\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d : IsSFiniteKernel \u03b7\na\u271d a : \u03b1\ns : Set (\u03b2 \u00d7 \u03b3)\nh2s : \u2191\u2191(\u2191(\u03ba \u2297\u2096 \u03b7) a) s \u2260 \u22a4\n\u22a2 HasFiniteIntegral fun b => ENNReal.toReal (\u2191\u2191(\u2191\u03b7 (a, b)) (Prod.mk b \u207b\u00b9' s))", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nE : Type u_4\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d\u00b2 : NormedAddCommGroup E\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d : IsSFiniteKernel \u03b7\na\u271d a : \u03b1\ns : Set (\u03b2 \u00d7 \u03b3)\nh2s : \u2191\u2191(\u2191(\u03ba \u2297\u2096 \u03b7) a) s \u2260 \u22a4\nt : Set (\u03b2 \u00d7 \u03b3) := toMeasurable (\u2191(\u03ba \u2297\u2096 \u03b7) a) s\n\u22a2 HasFiniteIntegral fun b => ENNReal.toReal (\u2191\u2191(\u2191\u03b7 (a, b)) (Prod.mk b \u207b\u00b9' s))"}, {"tactic": "simp_rw [HasFiniteIntegral, ennnorm_eq_ofReal toReal_nonneg]", "annotated_tactic": ["simp_rw [<a>HasFiniteIntegral</a>, <a>ennnorm_eq_ofReal</a> <a>toReal_nonneg</a>]", [{"full_name": "MeasureTheory.HasFiniteIntegral", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [106, 5], "def_end_pos": [106, 22]}, {"full_name": "Real.ennnorm_eq_ofReal", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [1808, 9], "def_end_pos": [1808, 26]}, {"full_name": "ENNReal.toReal_nonneg", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [221, 17], "def_end_pos": [221, 30]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nE : Type u_4\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d\u00b2 : NormedAddCommGroup E\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d : IsSFiniteKernel \u03b7\na\u271d a : \u03b1\ns : Set (\u03b2 \u00d7 \u03b3)\nh2s : \u2191\u2191(\u2191(\u03ba \u2297\u2096 \u03b7) a) s \u2260 \u22a4\nt : Set (\u03b2 \u00d7 \u03b3) := toMeasurable (\u2191(\u03ba \u2297\u2096 \u03b7) a) s\n\u22a2 HasFiniteIntegral fun b => ENNReal.toReal (\u2191\u2191(\u2191\u03b7 (a, b)) (Prod.mk b \u207b\u00b9' s))", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nE : Type u_4\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d\u00b2 : NormedAddCommGroup E\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d : IsSFiniteKernel \u03b7\na\u271d a : \u03b1\ns : Set (\u03b2 \u00d7 \u03b3)\nh2s : \u2191\u2191(\u2191(\u03ba \u2297\u2096 \u03b7) a) s \u2260 \u22a4\nt : Set (\u03b2 \u00d7 \u03b3) := toMeasurable (\u2191(\u03ba \u2297\u2096 \u03b7) a) s\n\u22a2 \u222b\u207b (a_1 : \u03b2), ENNReal.ofReal (ENNReal.toReal (\u2191\u2191(\u2191\u03b7 (a, a_1)) (Prod.mk a_1 \u207b\u00b9' s))) \u2202\u2191\u03ba a < \u22a4"}, {"tactic": "calc\n  \u222b\u207b b, ENNReal.ofReal (\u03b7 (a, b) (Prod.mk b \u207b\u00b9' s)).toReal \u2202\u03ba a\n  _ \u2264 \u222b\u207b b, \u03b7 (a, b) (Prod.mk b \u207b\u00b9' t) \u2202\u03ba a := by\n    refine' lintegral_mono_ae _\n    filter_upwards [ae_kernel_lt_top a h2s] with b hb\n    rw [ofReal_toReal hb.ne]\n    exact measure_mono (preimage_mono (subset_toMeasurable _ _))\n  _ \u2264 (\u03ba \u2297\u2096 \u03b7) a t := (le_compProd_apply _ _ _ _)\n  _ = (\u03ba \u2297\u2096 \u03b7) a s := (measure_toMeasurable s)\n  _ < \u22a4 := h2s.lt_top", "annotated_tactic": ["calc\n    \u222b\u207b b, <a>ENNReal.ofReal</a> (\u03b7 (a, b) (<a>Prod.mk</a> b \u207b\u00b9' s)).<a>toReal</a> \u2202\u03ba a\n    _ \u2264 \u222b\u207b b, \u03b7 (a, b) (<a>Prod.mk</a> b \u207b\u00b9' t) \u2202\u03ba a := by\n      refine' <a>lintegral_mono_ae</a> _\n      filter_upwards [<a>ae_kernel_lt_top</a> a h2s] with b hb\n      rw [<a>ofReal_toReal</a> hb.ne]\n      exact <a>measure_mono</a> (<a>preimage_mono</a> (<a>subset_toMeasurable</a> _ _))\n    _ \u2264 (\u03ba \u2297\u2096 \u03b7) a t := (<a>le_compProd_apply</a> _ _ _ _)\n    _ = (\u03ba \u2297\u2096 \u03b7) a s := (<a>measure_toMeasurable</a> s)\n    _ < \u22a4 := h2s.lt_top", [{"full_name": "ENNReal.ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [172, 29], "def_end_pos": [172, 35]}, {"full_name": "Prod.mk", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [466, 16], "def_end_pos": [466, 41]}, {"full_name": "ENNReal.toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [168, 15], "def_end_pos": [168, 21]}, {"full_name": "Prod.mk", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [466, 16], "def_end_pos": [466, 41]}, {"full_name": "MeasureTheory.lintegral_mono_ae", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [265, 9], "def_end_pos": [265, 26]}, {"full_name": "ProbabilityTheory.kernel.ae_kernel_lt_top", "def_path": "Mathlib/Probability/Kernel/Composition.lean", "def_pos": [285, 9], "def_end_pos": [285, 25]}, {"full_name": "ENNReal.ofReal_toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [186, 9], "def_end_pos": [186, 22]}, {"full_name": "MeasureTheory.measure_mono", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [193, 9], "def_end_pos": [193, 21]}, {"full_name": "Set.preimage_mono", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [74, 9], "def_end_pos": [74, 22]}, {"full_name": "MeasureTheory.subset_toMeasurable", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [633, 9], "def_end_pos": [633, 28]}, {"full_name": "ProbabilityTheory.kernel.le_compProd_apply", "def_path": "Mathlib/Probability/Kernel/Composition.lean", "def_pos": [248, 9], "def_end_pos": [248, 26]}, {"full_name": "MeasureTheory.measure_toMeasurable", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [653, 9], "def_end_pos": [653, 29]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nE : Type u_4\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d\u00b2 : NormedAddCommGroup E\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d : IsSFiniteKernel \u03b7\na\u271d a : \u03b1\ns : Set (\u03b2 \u00d7 \u03b3)\nh2s : \u2191\u2191(\u2191(\u03ba \u2297\u2096 \u03b7) a) s \u2260 \u22a4\nt : Set (\u03b2 \u00d7 \u03b3) := toMeasurable (\u2191(\u03ba \u2297\u2096 \u03b7) a) s\n\u22a2 \u222b\u207b (a_1 : \u03b2), ENNReal.ofReal (ENNReal.toReal (\u2191\u2191(\u2191\u03b7 (a, a_1)) (Prod.mk a_1 \u207b\u00b9' s))) \u2202\u2191\u03ba a < \u22a4", "state_after": "no goals"}, {"tactic": "refine' lintegral_mono_ae _", "annotated_tactic": ["refine' <a>lintegral_mono_ae</a> _", [{"full_name": "MeasureTheory.lintegral_mono_ae", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [265, 9], "def_end_pos": [265, 26]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nE : Type u_4\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d\u00b2 : NormedAddCommGroup E\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d : IsSFiniteKernel \u03b7\na\u271d a : \u03b1\ns : Set (\u03b2 \u00d7 \u03b3)\nh2s : \u2191\u2191(\u2191(\u03ba \u2297\u2096 \u03b7) a) s \u2260 \u22a4\nt : Set (\u03b2 \u00d7 \u03b3) := toMeasurable (\u2191(\u03ba \u2297\u2096 \u03b7) a) s\n\u22a2 \u222b\u207b (b : \u03b2), ENNReal.ofReal (ENNReal.toReal (\u2191\u2191(\u2191\u03b7 (a, b)) (Prod.mk b \u207b\u00b9' s))) \u2202\u2191\u03ba a \u2264\n    \u222b\u207b (b : \u03b2), \u2191\u2191(\u2191\u03b7 (a, b)) (Prod.mk b \u207b\u00b9' t) \u2202\u2191\u03ba a", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nE : Type u_4\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d\u00b2 : NormedAddCommGroup E\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d : IsSFiniteKernel \u03b7\na\u271d a : \u03b1\ns : Set (\u03b2 \u00d7 \u03b3)\nh2s : \u2191\u2191(\u2191(\u03ba \u2297\u2096 \u03b7) a) s \u2260 \u22a4\nt : Set (\u03b2 \u00d7 \u03b3) := toMeasurable (\u2191(\u03ba \u2297\u2096 \u03b7) a) s\n\u22a2 \u2200\u1d50 (a_1 : \u03b2) \u2202\u2191\u03ba a,\n    ENNReal.ofReal (ENNReal.toReal (\u2191\u2191(\u2191\u03b7 (a, a_1)) (Prod.mk a_1 \u207b\u00b9' s))) \u2264 \u2191\u2191(\u2191\u03b7 (a, a_1)) (Prod.mk a_1 \u207b\u00b9' t)"}, {"tactic": "filter_upwards [ae_kernel_lt_top a h2s] with b hb", "annotated_tactic": ["filter_upwards [<a>ae_kernel_lt_top</a> a h2s] with b hb", [{"full_name": "ProbabilityTheory.kernel.ae_kernel_lt_top", "def_path": "Mathlib/Probability/Kernel/Composition.lean", "def_pos": [285, 9], "def_end_pos": [285, 25]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nE : Type u_4\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d\u00b2 : NormedAddCommGroup E\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d : IsSFiniteKernel \u03b7\na\u271d a : \u03b1\ns : Set (\u03b2 \u00d7 \u03b3)\nh2s : \u2191\u2191(\u2191(\u03ba \u2297\u2096 \u03b7) a) s \u2260 \u22a4\nt : Set (\u03b2 \u00d7 \u03b3) := toMeasurable (\u2191(\u03ba \u2297\u2096 \u03b7) a) s\n\u22a2 \u2200\u1d50 (a_1 : \u03b2) \u2202\u2191\u03ba a,\n    ENNReal.ofReal (ENNReal.toReal (\u2191\u2191(\u2191\u03b7 (a, a_1)) (Prod.mk a_1 \u207b\u00b9' s))) \u2264 \u2191\u2191(\u2191\u03b7 (a, a_1)) (Prod.mk a_1 \u207b\u00b9' t)", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nE : Type u_4\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d\u00b2 : NormedAddCommGroup E\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d : IsSFiniteKernel \u03b7\na\u271d a : \u03b1\ns : Set (\u03b2 \u00d7 \u03b3)\nh2s : \u2191\u2191(\u2191(\u03ba \u2297\u2096 \u03b7) a) s \u2260 \u22a4\nt : Set (\u03b2 \u00d7 \u03b3) := toMeasurable (\u2191(\u03ba \u2297\u2096 \u03b7) a) s\nb : \u03b2\nhb : \u2191\u2191(\u2191\u03b7 (a, b)) (Prod.mk b \u207b\u00b9' s) < \u22a4\n\u22a2 ENNReal.ofReal (ENNReal.toReal (\u2191\u2191(\u2191\u03b7 (a, b)) (Prod.mk b \u207b\u00b9' s))) \u2264 \u2191\u2191(\u2191\u03b7 (a, b)) (Prod.mk b \u207b\u00b9' t)"}, {"tactic": "rw [ofReal_toReal hb.ne]", "annotated_tactic": ["rw [<a>ofReal_toReal</a> hb.ne]", [{"full_name": "ENNReal.ofReal_toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [186, 9], "def_end_pos": [186, 22]}]], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nE : Type u_4\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d\u00b2 : NormedAddCommGroup E\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d : IsSFiniteKernel \u03b7\na\u271d a : \u03b1\ns : Set (\u03b2 \u00d7 \u03b3)\nh2s : \u2191\u2191(\u2191(\u03ba \u2297\u2096 \u03b7) a) s \u2260 \u22a4\nt : Set (\u03b2 \u00d7 \u03b3) := toMeasurable (\u2191(\u03ba \u2297\u2096 \u03b7) a) s\nb : \u03b2\nhb : \u2191\u2191(\u2191\u03b7 (a, b)) (Prod.mk b \u207b\u00b9' s) < \u22a4\n\u22a2 ENNReal.ofReal (ENNReal.toReal (\u2191\u2191(\u2191\u03b7 (a, b)) (Prod.mk b \u207b\u00b9' s))) \u2264 \u2191\u2191(\u2191\u03b7 (a, b)) (Prod.mk b \u207b\u00b9' t)", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nE : Type u_4\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d\u00b2 : NormedAddCommGroup E\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d : IsSFiniteKernel \u03b7\na\u271d a : \u03b1\ns : Set (\u03b2 \u00d7 \u03b3)\nh2s : \u2191\u2191(\u2191(\u03ba \u2297\u2096 \u03b7) a) s \u2260 \u22a4\nt : Set (\u03b2 \u00d7 \u03b3) := toMeasurable (\u2191(\u03ba \u2297\u2096 \u03b7) a) s\nb : \u03b2\nhb : \u2191\u2191(\u2191\u03b7 (a, b)) (Prod.mk b \u207b\u00b9' s) < \u22a4\n\u22a2 \u2191\u2191(\u2191\u03b7 (a, b)) (Prod.mk b \u207b\u00b9' s) \u2264 \u2191\u2191(\u2191\u03b7 (a, b)) (Prod.mk b \u207b\u00b9' t)"}, {"tactic": "exact measure_mono (preimage_mono (subset_toMeasurable _ _))", "annotated_tactic": ["exact <a>measure_mono</a> (<a>preimage_mono</a> (<a>subset_toMeasurable</a> _ _))", [{"full_name": "MeasureTheory.measure_mono", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [193, 9], "def_end_pos": [193, 21]}, {"full_name": "Set.preimage_mono", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [74, 9], "def_end_pos": [74, 22]}, {"full_name": "MeasureTheory.subset_toMeasurable", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [633, 9], "def_end_pos": [633, 28]}]], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nE : Type u_4\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d\u00b2 : NormedAddCommGroup E\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d : IsSFiniteKernel \u03b7\na\u271d a : \u03b1\ns : Set (\u03b2 \u00d7 \u03b3)\nh2s : \u2191\u2191(\u2191(\u03ba \u2297\u2096 \u03b7) a) s \u2260 \u22a4\nt : Set (\u03b2 \u00d7 \u03b3) := toMeasurable (\u2191(\u03ba \u2297\u2096 \u03b7) a) s\nb : \u03b2\nhb : \u2191\u2191(\u2191\u03b7 (a, b)) (Prod.mk b \u207b\u00b9' s) < \u22a4\n\u22a2 \u2191\u2191(\u2191\u03b7 (a, b)) (Prod.mk b \u207b\u00b9' s) \u2264 \u2191\u2191(\u2191\u03b7 (a, b)) (Prod.mk b \u207b\u00b9' t)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean", "full_name": "MeasureTheory.aecover_Ioo_of_Ioo", "start": [195, 1], "end": [197, 87], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Constructions/Polish.lean", "full_name": "MeasurableSet.analyticSet", "start": [309, 1], "end": [319, 32], "traced_tactics": [{"tactic": "obtain \u27e8t', t't, t'_polish, s_closed, _\u27e9 :\n  \u2203 t' : TopologicalSpace \u03b1, t' \u2264 t \u2227 @PolishSpace \u03b1 t' \u2227 IsClosed[t'] s \u2227 IsOpen[t'] s :=\n  hs.isClopenable", "annotated_tactic": ["obtain \u27e8t', t't, t'_polish, s_closed, _\u27e9 :\n    \u2203 t' : <a>TopologicalSpace</a> \u03b1, t' \u2264 t \u2227 @<a>PolishSpace</a> \u03b1 t' \u2227 IsClosed[t'] s \u2227 IsOpen[t'] s :=\n    hs.isClopenable", [{"full_name": "TopologicalSpace", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [70, 7], "def_end_pos": [70, 23]}, {"full_name": "PolishSpace", "def_path": "Mathlib/Topology/MetricSpace/Polish.lean", "def_pos": [65, 7], "def_end_pos": [65, 18]}]], "state_before": "\u03b1\u271d : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\u271d\n\u03b1 : Type u_3\nt : TopologicalSpace \u03b1\ninst\u271d\u00b2 : PolishSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : BorelSpace \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 AnalyticSet s", "state_after": "case intro.intro.intro.intro\n\u03b1\u271d : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\u271d\n\u03b1 : Type u_3\nt : TopologicalSpace \u03b1\ninst\u271d\u00b2 : PolishSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : BorelSpace \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nt' : TopologicalSpace \u03b1\nt't : t' \u2264 t\nt'_polish : PolishSpace \u03b1\ns_closed : IsClosed s\nright\u271d : IsOpen s\n\u22a2 AnalyticSet s"}, {"tactic": "have A := @IsClosed.analyticSet \u03b1 t' t'_polish s s_closed", "annotated_tactic": ["have A := @<a>IsClosed.analyticSet</a> \u03b1 t' t'_polish s s_closed", [{"full_name": "IsClosed.analyticSet", "def_path": "Mathlib/MeasureTheory/Constructions/Polish.lean", "def_pos": [290, 9], "def_end_pos": [290, 36]}]], "state_before": "case intro.intro.intro.intro\n\u03b1\u271d : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\u271d\n\u03b1 : Type u_3\nt : TopologicalSpace \u03b1\ninst\u271d\u00b2 : PolishSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : BorelSpace \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nt' : TopologicalSpace \u03b1\nt't : t' \u2264 t\nt'_polish : PolishSpace \u03b1\ns_closed : IsClosed s\nright\u271d : IsOpen s\n\u22a2 AnalyticSet s", "state_after": "case intro.intro.intro.intro\n\u03b1\u271d : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\u271d\n\u03b1 : Type u_3\nt : TopologicalSpace \u03b1\ninst\u271d\u00b2 : PolishSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : BorelSpace \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nt' : TopologicalSpace \u03b1\nt't : t' \u2264 t\nt'_polish : PolishSpace \u03b1\ns_closed : IsClosed s\nright\u271d : IsOpen s\nA : AnalyticSet s\n\u22a2 AnalyticSet s"}, {"tactic": "convert @AnalyticSet.image_of_continuous \u03b1 t' \u03b1 t s A id (continuous_id_of_le t't)", "annotated_tactic": ["convert @<a>AnalyticSet.image_of_continuous</a> \u03b1 t' \u03b1 t s A <a>id</a> (<a>continuous_id_of_le</a> t't)", [{"full_name": "MeasureTheory.AnalyticSet.image_of_continuous", "def_path": "Mathlib/MeasureTheory/Constructions/Polish.lean", "def_pos": [225, 9], "def_end_pos": [225, 40]}, {"full_name": "id", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [33, 15], "def_end_pos": [33, 17]}, {"full_name": "continuous_id_of_le", "def_path": "Mathlib/Topology/Order.lean", "def_pos": [847, 9], "def_end_pos": [847, 28]}]], "state_before": "case intro.intro.intro.intro\n\u03b1\u271d : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\u271d\n\u03b1 : Type u_3\nt : TopologicalSpace \u03b1\ninst\u271d\u00b2 : PolishSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : BorelSpace \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nt' : TopologicalSpace \u03b1\nt't : t' \u2264 t\nt'_polish : PolishSpace \u03b1\ns_closed : IsClosed s\nright\u271d : IsOpen s\nA : AnalyticSet s\n\u22a2 AnalyticSet s", "state_after": "case h.e'_3\n\u03b1\u271d : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\u271d\n\u03b1 : Type u_3\nt : TopologicalSpace \u03b1\ninst\u271d\u00b2 : PolishSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : BorelSpace \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nt' : TopologicalSpace \u03b1\nt't : t' \u2264 t\nt'_polish : PolishSpace \u03b1\ns_closed : IsClosed s\nright\u271d : IsOpen s\nA : AnalyticSet s\n\u22a2 s = id '' s"}, {"tactic": "simp only [id.def, image_id']", "annotated_tactic": ["simp only [<a>id.def</a>, <a>image_id'</a>]", [{"full_name": "id.def", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [527, 9], "def_end_pos": [527, 15]}, {"full_name": "Set.image_id'", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [396, 9], "def_end_pos": [396, 18]}]], "state_before": "case h.e'_3\n\u03b1\u271d : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\u271d\n\u03b1 : Type u_3\nt : TopologicalSpace \u03b1\ninst\u271d\u00b2 : PolishSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : BorelSpace \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nt' : TopologicalSpace \u03b1\nt't : t' \u2264 t\nt'_polish : PolishSpace \u03b1\ns_closed : IsClosed s\nright\u271d : IsOpen s\nA : AnalyticSet s\n\u22a2 s = id '' s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Independence/Basic.lean", "full_name": "ProbabilityTheory.iIndepSet.indep_generateFrom_lt", "start": [426, 1], "end": [429, 50], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Part.lean", "full_name": "Part.inv_mem_inv", "start": [747, 1], "end": [748, 27], "traced_tactics": [{"tactic": "simp [inv_def]", "annotated_tactic": ["simp [<a>inv_def</a>]", [{"full_name": "Part.inv_def", "def_path": "Mathlib/Data/Part.lean", "def_pos": [703, 9], "def_end_pos": [703, 16]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : Inv \u03b1\na : Part \u03b1\nma : \u03b1\nha : ma \u2208 a\n\u22a2 ma\u207b\u00b9 \u2208 a\u207b\u00b9", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : Inv \u03b1\na : Part \u03b1\nma : \u03b1\nha : ma \u2208 a\n\u22a2 \u2203 a_1, a_1 \u2208 a \u2227 a_1\u207b\u00b9 = ma\u207b\u00b9"}, {"tactic": "aesop", "annotated_tactic": ["aesop", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : Inv \u03b1\na : Part \u03b1\nma : \u03b1\nha : ma \u2208 a\n\u22a2 \u2203 a_1, a_1 \u2208 a \u2227 a_1\u207b\u00b9 = ma\u207b\u00b9", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/NatAntidiagonal.lean", "full_name": "Finset.Nat.mem_antidiagonal", "start": [37, 1], "end": [38, 60], "traced_tactics": [{"tactic": "rw [antidiagonal, mem_def, Multiset.Nat.mem_antidiagonal]", "annotated_tactic": ["rw [<a>antidiagonal</a>, <a>mem_def</a>, <a>Multiset.Nat.mem_antidiagonal</a>]", [{"full_name": "Finset.Nat.antidiagonal", "def_path": "Mathlib/Data/Finset/NatAntidiagonal.lean", "def_pos": [31, 5], "def_end_pos": [31, 17]}, {"full_name": "Finset.mem_def", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [178, 9], "def_end_pos": [178, 16]}, {"full_name": "Multiset.Nat.mem_antidiagonal", "def_path": "Mathlib/Data/Multiset/NatAntidiagonal.lean", "def_pos": [36, 9], "def_end_pos": [36, 25]}]], "state_before": "n : \u2115\nx : \u2115 \u00d7 \u2115\n\u22a2 x \u2208 antidiagonal n \u2194 x.1 + x.2 = n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Card.lean", "full_name": "Set.ncard_le_ncard_insert", "start": [592, 1], "end": [596, 49], "traced_tactics": [{"tactic": "classical\nrefine'\n  s.finite_or_infinite.elim (fun h \u21a6 _) (fun h \u21a6 by (rw [h.ncard]; exact Nat.zero_le _))\nrw [ncard_insert_eq_ite h]; split_ifs <;> simp", "annotated_tactic": ["classical\n  refine'\n    s.finite_or_infinite.elim (fun h \u21a6 _) (fun h \u21a6 by (rw [h.ncard]; exact <a>Nat.zero_le</a> _))\n  rw [<a>ncard_insert_eq_ite</a> h]; split_ifs <;> simp", [{"full_name": "Nat.zero_le", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1578, 9], "def_end_pos": [1578, 20]}, {"full_name": "Set.ncard_insert_eq_ite", "def_path": "Mathlib/Data/Set/Card.lean", "def_pos": [585, 9], "def_end_pos": [585, 28]}]], "state_before": "\u03b1 : Type u_1\ns\u271d t : Set \u03b1\na : \u03b1\ns : Set \u03b1\n\u22a2 ncard s \u2264 ncard (insert a s)", "state_after": "no goals"}, {"tactic": "refine'\n  s.finite_or_infinite.elim (fun h \u21a6 _) (fun h \u21a6 by (rw [h.ncard]; exact Nat.zero_le _))", "annotated_tactic": ["refine'\n    s.finite_or_infinite.elim (fun h \u21a6 _) (fun h \u21a6 by (rw [h.ncard]; exact <a>Nat.zero_le</a> _))", [{"full_name": "Nat.zero_le", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1578, 9], "def_end_pos": [1578, 20]}]], "state_before": "\u03b1 : Type u_1\ns\u271d t : Set \u03b1\na : \u03b1\ns : Set \u03b1\n\u22a2 ncard s \u2264 ncard (insert a s)", "state_after": "\u03b1 : Type u_1\ns\u271d t : Set \u03b1\na : \u03b1\ns : Set \u03b1\nh : Set.Finite s\n\u22a2 ncard s \u2264 ncard (insert a s)"}, {"tactic": "rw [ncard_insert_eq_ite h]", "annotated_tactic": ["rw [<a>ncard_insert_eq_ite</a> h]", [{"full_name": "Set.ncard_insert_eq_ite", "def_path": "Mathlib/Data/Set/Card.lean", "def_pos": [585, 9], "def_end_pos": [585, 28]}]], "state_before": "\u03b1 : Type u_1\ns\u271d t : Set \u03b1\na : \u03b1\ns : Set \u03b1\nh : Set.Finite s\n\u22a2 ncard s \u2264 ncard (insert a s)", "state_after": "\u03b1 : Type u_1\ns\u271d t : Set \u03b1\na : \u03b1\ns : Set \u03b1\nh : Set.Finite s\n\u22a2 ncard s \u2264 if a \u2208 s then ncard s else ncard s + 1"}, {"tactic": "split_ifs <;> simp", "annotated_tactic": ["split_ifs <;> simp", []], "state_before": "\u03b1 : Type u_1\ns\u271d t : Set \u03b1\na : \u03b1\ns : Set \u03b1\nh : Set.Finite s\n\u22a2 ncard s \u2264 if a \u2208 s then ncard s else ncard s + 1", "state_after": "no goals"}, {"tactic": "(rw [h.ncard]; exact Nat.zero_le _)", "annotated_tactic": ["(rw [h.ncard]; exact <a>Nat.zero_le</a> _)", [{"full_name": "Nat.zero_le", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1578, 9], "def_end_pos": [1578, 20]}]], "state_before": "\u03b1 : Type u_1\ns\u271d t : Set \u03b1\na : \u03b1\ns : Set \u03b1\nh : Set.Infinite s\n\u22a2 ncard s \u2264 ncard (insert a s)", "state_after": "no goals"}, {"tactic": "rw [h.ncard]", "annotated_tactic": ["rw [h.ncard]", []], "state_before": "\u03b1 : Type u_1\ns\u271d t : Set \u03b1\na : \u03b1\ns : Set \u03b1\nh : Set.Infinite s\n\u22a2 ncard s \u2264 ncard (insert a s)", "state_after": "\u03b1 : Type u_1\ns\u271d t : Set \u03b1\na : \u03b1\ns : Set \u03b1\nh : Set.Infinite s\n\u22a2 0 \u2264 ncard (insert a s)"}, {"tactic": "exact Nat.zero_le _", "annotated_tactic": ["exact <a>Nat.zero_le</a> _", [{"full_name": "Nat.zero_le", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1578, 9], "def_end_pos": [1578, 20]}]], "state_before": "\u03b1 : Type u_1\ns\u271d t : Set \u03b1\na : \u03b1\ns : Set \u03b1\nh : Set.Infinite s\n\u22a2 0 \u2264 ncard (insert a s)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "full_name": "MeasureTheory.Measure.QuasiMeasurePreserving.exists_preimage_eq_of_preimage_ae", "start": [2336, 1], "end": [2343, 89], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Kernel/CondCdf.lean", "full_name": "MeasureTheory.Measure.tendsto_IicSnd_atBot", "start": [244, 1], "end": [266, 21], "traced_tactics": [{"tactic": "simp_rw [\u03c1.IicSnd_apply _ hs]", "annotated_tactic": ["simp_rw [\u03c1.IicSnd_apply _ hs]", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 Tendsto (fun r => \u2191\u2191(IicSnd \u03c1 \u2191r) s) atBot (\ud835\udcdd 0)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 Tendsto (fun r => \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191r)) atBot (\ud835\udcdd 0)"}, {"tactic": "have h_empty : \u03c1 (s \u00d7\u02e2 \u2205) = 0 := by simp only [prod_empty, measure_empty]", "annotated_tactic": ["have h_empty : \u03c1 (s \u00d7\u02e2 \u2205) = 0 := by simp only [<a>prod_empty</a>, <a>measure_empty</a>]", [{"full_name": "Set.prod_empty", "def_path": "Mathlib/Data/Set/Prod.lean", "def_pos": [113, 9], "def_end_pos": [113, 19]}, {"full_name": "MeasureTheory.measure_empty", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [185, 9], "def_end_pos": [185, 22]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 Tendsto (fun r => \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191r)) atBot (\ud835\udcdd 0)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nh_empty : \u2191\u2191\u03c1 (s \u00d7\u02e2 \u2205) = 0\n\u22a2 Tendsto (fun r => \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191r)) atBot (\ud835\udcdd 0)"}, {"tactic": "rw [\u2190 h_empty, \u2190 Real.iInter_Iic_rat, prod_iInter]", "annotated_tactic": ["rw [\u2190 h_empty, \u2190 <a>Real.iInter_Iic_rat</a>, <a>prod_iInter</a>]", [{"full_name": "Real.iInter_Iic_rat", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [84, 9], "def_end_pos": [84, 28]}, {"full_name": "prod_iInter", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [70, 9], "def_end_pos": [70, 20]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nh_empty : \u2191\u2191\u03c1 (s \u00d7\u02e2 \u2205) = 0\n\u22a2 Tendsto (fun r => \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191r)) atBot (\ud835\udcdd 0)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nh_empty : \u2191\u2191\u03c1 (s \u00d7\u02e2 \u2205) = 0\n\u22a2 Tendsto (fun r => \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191r)) atBot (\ud835\udcdd (\u2191\u2191\u03c1 (\u22c2 i, s \u00d7\u02e2 Iic \u2191i)))"}, {"tactic": "suffices h_neg :\n  Tendsto (fun r : \u211a => \u03c1 (s \u00d7\u02e2 Iic \u2191(-r))) atTop (\ud835\udcdd (\u03c1 (\u22c2 r : \u211a, s \u00d7\u02e2 Iic \u2191(-r))))", "annotated_tactic": ["suffices h_neg :\n    <a>Tendsto</a> (fun r : \u211a => \u03c1 (s \u00d7\u02e2 <a>Iic</a> \u2191(-r))) <a>atTop</a> (\ud835\udcdd (\u03c1 (\u22c2 r : \u211a, s \u00d7\u02e2 <a>Iic</a> \u2191(-r))))", [{"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2939, 5], "def_end_pos": [2939, 12]}, {"full_name": "Set.Iic", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [64, 5], "def_end_pos": [64, 8]}, {"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}, {"full_name": "Set.Iic", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [64, 5], "def_end_pos": [64, 8]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nh_empty : \u2191\u2191\u03c1 (s \u00d7\u02e2 \u2205) = 0\n\u22a2 Tendsto (fun r => \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191r)) atBot (\ud835\udcdd (\u2191\u2191\u03c1 (\u22c2 i, s \u00d7\u02e2 Iic \u2191i)))", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nh_empty : \u2191\u2191\u03c1 (s \u00d7\u02e2 \u2205) = 0\nh_neg : Tendsto (fun r => \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191(-r))) atTop (\ud835\udcdd (\u2191\u2191\u03c1 (\u22c2 r, s \u00d7\u02e2 Iic \u2191(-r))))\n\u22a2 Tendsto (fun r => \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191r)) atBot (\ud835\udcdd (\u2191\u2191\u03c1 (\u22c2 i, s \u00d7\u02e2 Iic \u2191i)))\n\ncase h_neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nh_empty : \u2191\u2191\u03c1 (s \u00d7\u02e2 \u2205) = 0\n\u22a2 Tendsto (fun r => \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191(-r))) atTop (\ud835\udcdd (\u2191\u2191\u03c1 (\u22c2 r, s \u00d7\u02e2 Iic \u2191(-r))))"}, {"tactic": "refine' tendsto_measure_iInter (fun q => hs.prod measurableSet_Iic) _ \u27e80, measure_ne_top \u03c1 _\u27e9", "annotated_tactic": ["refine' <a>tendsto_measure_iInter</a> (fun q => hs.prod <a>measurableSet_Iic</a>) _ \u27e80, <a>measure_ne_top</a> \u03c1 _\u27e9", [{"full_name": "MeasureTheory.tendsto_measure_iInter", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [538, 9], "def_end_pos": [538, 31]}, {"full_name": "measurableSet_Iic", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [515, 9], "def_end_pos": [515, 26]}, {"full_name": "MeasureTheory.measure_ne_top", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2875, 9], "def_end_pos": [2875, 23]}]], "state_before": "case h_neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nh_empty : \u2191\u2191\u03c1 (s \u00d7\u02e2 \u2205) = 0\n\u22a2 Tendsto (fun r => \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191(-r))) atTop (\ud835\udcdd (\u2191\u2191\u03c1 (\u22c2 r, s \u00d7\u02e2 Iic \u2191(-r))))", "state_after": "case h_neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nh_empty : \u2191\u2191\u03c1 (s \u00d7\u02e2 \u2205) = 0\n\u22a2 Antitone fun r => s \u00d7\u02e2 Iic \u2191(-r)"}, {"tactic": "refine' fun q r hqr => prod_subset_prod_iff.mpr (Or.inl \u27e8subset_rfl, fun x hx => _\u27e9)", "annotated_tactic": ["refine' fun q r hqr => prod_subset_prod_iff.mpr (<a>Or.inl</a> \u27e8<a>subset_rfl</a>, fun x hx => _\u27e9)", [{"full_name": "Or.inl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [517, 5], "def_end_pos": [517, 8]}, {"full_name": "subset_rfl", "def_path": "Mathlib/Order/RelClasses.lean", "def_pos": [627, 7], "def_end_pos": [627, 17]}]], "state_before": "case h_neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nh_empty : \u2191\u2191\u03c1 (s \u00d7\u02e2 \u2205) = 0\n\u22a2 Antitone fun r => s \u00d7\u02e2 Iic \u2191(-r)", "state_after": "case h_neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nh_empty : \u2191\u2191\u03c1 (s \u00d7\u02e2 \u2205) = 0\nq r : \u211a\nhqr : q \u2264 r\nx : \u211d\nhx : x \u2208 Iic \u2191(-r)\n\u22a2 x \u2208 Iic \u2191(-q)"}, {"tactic": "simp only [Rat.cast_neg, mem_Iic] at hx \u22a2", "annotated_tactic": ["simp only [<a>Rat.cast_neg</a>, <a>mem_Iic</a>] at hx \u22a2", [{"full_name": "Rat.cast_neg", "def_path": "Mathlib/Data/Rat/Cast/Defs.lean", "def_pos": [115, 9], "def_end_pos": [115, 17]}, {"full_name": "Set.mem_Iic", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [136, 9], "def_end_pos": [136, 16]}]], "state_before": "case h_neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nh_empty : \u2191\u2191\u03c1 (s \u00d7\u02e2 \u2205) = 0\nq r : \u211a\nhqr : q \u2264 r\nx : \u211d\nhx : x \u2208 Iic \u2191(-r)\n\u22a2 x \u2208 Iic \u2191(-q)", "state_after": "case h_neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nh_empty : \u2191\u2191\u03c1 (s \u00d7\u02e2 \u2205) = 0\nq r : \u211a\nhqr : q \u2264 r\nx : \u211d\nhx : x \u2264 -\u2191r\n\u22a2 x \u2264 -\u2191q"}, {"tactic": "refine' hx.trans (neg_le_neg _)", "annotated_tactic": ["refine' hx.trans (<a>neg_le_neg</a> _)", [{"full_name": "neg_le_neg", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [1238, 15], "def_end_pos": [1238, 25]}]], "state_before": "case h_neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nh_empty : \u2191\u2191\u03c1 (s \u00d7\u02e2 \u2205) = 0\nq r : \u211a\nhqr : q \u2264 r\nx : \u211d\nhx : x \u2264 -\u2191r\n\u22a2 x \u2264 -\u2191q", "state_after": "case h_neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nh_empty : \u2191\u2191\u03c1 (s \u00d7\u02e2 \u2205) = 0\nq r : \u211a\nhqr : q \u2264 r\nx : \u211d\nhx : x \u2264 -\u2191r\n\u22a2 \u2191q \u2264 \u2191r"}, {"tactic": "exact_mod_cast hqr", "annotated_tactic": ["exact_mod_cast hqr", []], "state_before": "case h_neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nh_empty : \u2191\u2191\u03c1 (s \u00d7\u02e2 \u2205) = 0\nq r : \u211a\nhqr : q \u2264 r\nx : \u211d\nhx : x \u2264 -\u2191r\n\u22a2 \u2191q \u2264 \u2191r", "state_after": "no goals"}, {"tactic": "simp only [prod_empty, measure_empty]", "annotated_tactic": ["simp only [<a>prod_empty</a>, <a>measure_empty</a>]", [{"full_name": "Set.prod_empty", "def_path": "Mathlib/Data/Set/Prod.lean", "def_pos": [113, 9], "def_end_pos": [113, 19]}, {"full_name": "MeasureTheory.measure_empty", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [185, 9], "def_end_pos": [185, 22]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191\u2191\u03c1 (s \u00d7\u02e2 \u2205) = 0", "state_after": "no goals"}, {"tactic": "rw [h_inter_eq] at h_neg", "annotated_tactic": ["rw [h_inter_eq] at h_neg", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nh_empty : \u2191\u2191\u03c1 (s \u00d7\u02e2 \u2205) = 0\nh_neg : Tendsto (fun r => \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191(-r))) atTop (\ud835\udcdd (\u2191\u2191\u03c1 (\u22c2 r, s \u00d7\u02e2 Iic \u2191(-r))))\nh_inter_eq : \u22c2 r, s \u00d7\u02e2 Iic \u2191(-r) = \u22c2 r, s \u00d7\u02e2 Iic \u2191r\n\u22a2 Tendsto (fun r => \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191r)) atBot (\ud835\udcdd (\u2191\u2191\u03c1 (\u22c2 i, s \u00d7\u02e2 Iic \u2191i)))", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nh_empty : \u2191\u2191\u03c1 (s \u00d7\u02e2 \u2205) = 0\nh_neg : Tendsto (fun r => \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191(-r))) atTop (\ud835\udcdd (\u2191\u2191\u03c1 (\u22c2 r, s \u00d7\u02e2 Iic \u2191r)))\nh_inter_eq : \u22c2 r, s \u00d7\u02e2 Iic \u2191(-r) = \u22c2 r, s \u00d7\u02e2 Iic \u2191r\n\u22a2 Tendsto (fun r => \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191r)) atBot (\ud835\udcdd (\u2191\u2191\u03c1 (\u22c2 i, s \u00d7\u02e2 Iic \u2191i)))"}, {"tactic": "have h_fun_eq : (fun r : \u211a => \u03c1 (s \u00d7\u02e2 Iic (r : \u211d))) = fun r : \u211a => \u03c1 (s \u00d7\u02e2 Iic \u2191(- -r)) := by\n  simp_rw [neg_neg]", "annotated_tactic": ["have h_fun_eq : (fun r : \u211a => \u03c1 (s \u00d7\u02e2 <a>Iic</a> (r : \u211d))) = fun r : \u211a => \u03c1 (s \u00d7\u02e2 <a>Iic</a> \u2191(- -r)) := by\n      simp_rw [<a>neg_neg</a>]", [{"full_name": "Set.Iic", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [64, 5], "def_end_pos": [64, 8]}, {"full_name": "Set.Iic", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [64, 5], "def_end_pos": [64, 8]}, {"full_name": "neg_neg", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [799, 3], "def_end_pos": [799, 14]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nh_empty : \u2191\u2191\u03c1 (s \u00d7\u02e2 \u2205) = 0\nh_neg : Tendsto (fun r => \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191(-r))) atTop (\ud835\udcdd (\u2191\u2191\u03c1 (\u22c2 r, s \u00d7\u02e2 Iic \u2191r)))\nh_inter_eq : \u22c2 r, s \u00d7\u02e2 Iic \u2191(-r) = \u22c2 r, s \u00d7\u02e2 Iic \u2191r\n\u22a2 Tendsto (fun r => \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191r)) atBot (\ud835\udcdd (\u2191\u2191\u03c1 (\u22c2 i, s \u00d7\u02e2 Iic \u2191i)))", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nh_empty : \u2191\u2191\u03c1 (s \u00d7\u02e2 \u2205) = 0\nh_neg : Tendsto (fun r => \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191(-r))) atTop (\ud835\udcdd (\u2191\u2191\u03c1 (\u22c2 r, s \u00d7\u02e2 Iic \u2191r)))\nh_inter_eq : \u22c2 r, s \u00d7\u02e2 Iic \u2191(-r) = \u22c2 r, s \u00d7\u02e2 Iic \u2191r\nh_fun_eq : (fun r => \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191r)) = fun r => \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191(- -r))\n\u22a2 Tendsto (fun r => \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191r)) atBot (\ud835\udcdd (\u2191\u2191\u03c1 (\u22c2 i, s \u00d7\u02e2 Iic \u2191i)))"}, {"tactic": "rw [h_fun_eq]", "annotated_tactic": ["rw [h_fun_eq]", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nh_empty : \u2191\u2191\u03c1 (s \u00d7\u02e2 \u2205) = 0\nh_neg : Tendsto (fun r => \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191(-r))) atTop (\ud835\udcdd (\u2191\u2191\u03c1 (\u22c2 r, s \u00d7\u02e2 Iic \u2191r)))\nh_inter_eq : \u22c2 r, s \u00d7\u02e2 Iic \u2191(-r) = \u22c2 r, s \u00d7\u02e2 Iic \u2191r\nh_fun_eq : (fun r => \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191r)) = fun r => \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191(- -r))\n\u22a2 Tendsto (fun r => \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191r)) atBot (\ud835\udcdd (\u2191\u2191\u03c1 (\u22c2 i, s \u00d7\u02e2 Iic \u2191i)))", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nh_empty : \u2191\u2191\u03c1 (s \u00d7\u02e2 \u2205) = 0\nh_neg : Tendsto (fun r => \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191(-r))) atTop (\ud835\udcdd (\u2191\u2191\u03c1 (\u22c2 r, s \u00d7\u02e2 Iic \u2191r)))\nh_inter_eq : \u22c2 r, s \u00d7\u02e2 Iic \u2191(-r) = \u22c2 r, s \u00d7\u02e2 Iic \u2191r\nh_fun_eq : (fun r => \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191r)) = fun r => \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191(- -r))\n\u22a2 Tendsto (fun r => \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191(- -r))) atBot (\ud835\udcdd (\u2191\u2191\u03c1 (\u22c2 i, s \u00d7\u02e2 Iic \u2191i)))"}, {"tactic": "exact h_neg.comp tendsto_neg_atBot_atTop", "annotated_tactic": ["exact h_neg.comp <a>tendsto_neg_atBot_atTop</a>", [{"full_name": "Filter.tendsto_neg_atBot_atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [897, 9], "def_end_pos": [897, 32]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nh_empty : \u2191\u2191\u03c1 (s \u00d7\u02e2 \u2205) = 0\nh_neg : Tendsto (fun r => \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191(-r))) atTop (\ud835\udcdd (\u2191\u2191\u03c1 (\u22c2 r, s \u00d7\u02e2 Iic \u2191r)))\nh_inter_eq : \u22c2 r, s \u00d7\u02e2 Iic \u2191(-r) = \u22c2 r, s \u00d7\u02e2 Iic \u2191r\nh_fun_eq : (fun r => \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191r)) = fun r => \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191(- -r))\n\u22a2 Tendsto (fun r => \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191(- -r))) atBot (\ud835\udcdd (\u2191\u2191\u03c1 (\u22c2 i, s \u00d7\u02e2 Iic \u2191i)))", "state_after": "no goals"}, {"tactic": "ext1 x", "annotated_tactic": ["ext1 x", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nh_empty : \u2191\u2191\u03c1 (s \u00d7\u02e2 \u2205) = 0\nh_neg : Tendsto (fun r => \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191(-r))) atTop (\ud835\udcdd (\u2191\u2191\u03c1 (\u22c2 r, s \u00d7\u02e2 Iic \u2191(-r))))\n\u22a2 \u22c2 r, s \u00d7\u02e2 Iic \u2191(-r) = \u22c2 r, s \u00d7\u02e2 Iic \u2191r", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nh_empty : \u2191\u2191\u03c1 (s \u00d7\u02e2 \u2205) = 0\nh_neg : Tendsto (fun r => \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191(-r))) atTop (\ud835\udcdd (\u2191\u2191\u03c1 (\u22c2 r, s \u00d7\u02e2 Iic \u2191(-r))))\nx : \u03b1 \u00d7 \u211d\n\u22a2 x \u2208 \u22c2 r, s \u00d7\u02e2 Iic \u2191(-r) \u2194 x \u2208 \u22c2 r, s \u00d7\u02e2 Iic \u2191r"}, {"tactic": "simp only [Rat.cast_eq_id, id.def, mem_iInter, mem_prod, mem_Iic]", "annotated_tactic": ["simp only [<a>Rat.cast_eq_id</a>, <a>id.def</a>, <a>mem_iInter</a>, <a>mem_prod</a>, <a>mem_Iic</a>]", [{"full_name": "Rat.cast_eq_id", "def_path": "Mathlib/Data/Rat/Cast/Defs.lean", "def_pos": [176, 9], "def_end_pos": [176, 19]}, {"full_name": "id.def", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [527, 9], "def_end_pos": [527, 15]}, {"full_name": "Set.mem_iInter", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [207, 9], "def_end_pos": [207, 19]}, {"full_name": "Set.mem_prod", "def_path": "Mathlib/Data/Set/Prod.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}, {"full_name": "Set.mem_Iic", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [136, 9], "def_end_pos": [136, 16]}]], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nh_empty : \u2191\u2191\u03c1 (s \u00d7\u02e2 \u2205) = 0\nh_neg : Tendsto (fun r => \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191(-r))) atTop (\ud835\udcdd (\u2191\u2191\u03c1 (\u22c2 r, s \u00d7\u02e2 Iic \u2191(-r))))\nx : \u03b1 \u00d7 \u211d\n\u22a2 x \u2208 \u22c2 r, s \u00d7\u02e2 Iic \u2191(-r) \u2194 x \u2208 \u22c2 r, s \u00d7\u02e2 Iic \u2191r", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nh_empty : \u2191\u2191\u03c1 (s \u00d7\u02e2 \u2205) = 0\nh_neg : Tendsto (fun r => \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191(-r))) atTop (\ud835\udcdd (\u2191\u2191\u03c1 (\u22c2 r, s \u00d7\u02e2 Iic \u2191(-r))))\nx : \u03b1 \u00d7 \u211d\n\u22a2 (\u2200 (i : \u211a), x.1 \u2208 s \u2227 x.2 \u2264 \u2191(-i)) \u2194 \u2200 (i : \u211a), x.1 \u2208 s \u2227 x.2 \u2264 \u2191i"}, {"tactic": "refine' \u27e8fun h i => \u27e8(h i).1, _\u27e9, fun h i => \u27e8(h i).1, _\u27e9\u27e9 <;> have h' := h (-i)", "annotated_tactic": ["refine' \u27e8fun h i => \u27e8(h i).1, _\u27e9, fun h i => \u27e8(h i).1, _\u27e9\u27e9 <;> have h' := h (-i)", []], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nh_empty : \u2191\u2191\u03c1 (s \u00d7\u02e2 \u2205) = 0\nh_neg : Tendsto (fun r => \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191(-r))) atTop (\ud835\udcdd (\u2191\u2191\u03c1 (\u22c2 r, s \u00d7\u02e2 Iic \u2191(-r))))\nx : \u03b1 \u00d7 \u211d\n\u22a2 (\u2200 (i : \u211a), x.1 \u2208 s \u2227 x.2 \u2264 \u2191(-i)) \u2194 \u2200 (i : \u211a), x.1 \u2208 s \u2227 x.2 \u2264 \u2191i", "state_after": "case h.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nh_empty : \u2191\u2191\u03c1 (s \u00d7\u02e2 \u2205) = 0\nh_neg : Tendsto (fun r => \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191(-r))) atTop (\ud835\udcdd (\u2191\u2191\u03c1 (\u22c2 r, s \u00d7\u02e2 Iic \u2191(-r))))\nx : \u03b1 \u00d7 \u211d\nh : \u2200 (i : \u211a), x.1 \u2208 s \u2227 x.2 \u2264 \u2191(-i)\ni : \u211a\nh' : x.1 \u2208 s \u2227 x.2 \u2264 \u2191(- -i)\n\u22a2 x.2 \u2264 \u2191i\n\ncase h.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nh_empty : \u2191\u2191\u03c1 (s \u00d7\u02e2 \u2205) = 0\nh_neg : Tendsto (fun r => \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191(-r))) atTop (\ud835\udcdd (\u2191\u2191\u03c1 (\u22c2 r, s \u00d7\u02e2 Iic \u2191(-r))))\nx : \u03b1 \u00d7 \u211d\nh : \u2200 (i : \u211a), x.1 \u2208 s \u2227 x.2 \u2264 \u2191i\ni : \u211a\nh' : x.1 \u2208 s \u2227 x.2 \u2264 \u2191(-i)\n\u22a2 x.2 \u2264 \u2191(-i)"}, {"tactic": "rw [neg_neg] at h'", "annotated_tactic": ["rw [<a>neg_neg</a>] at h'", [{"full_name": "neg_neg", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [799, 3], "def_end_pos": [799, 14]}]], "state_before": "case h.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nh_empty : \u2191\u2191\u03c1 (s \u00d7\u02e2 \u2205) = 0\nh_neg : Tendsto (fun r => \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191(-r))) atTop (\ud835\udcdd (\u2191\u2191\u03c1 (\u22c2 r, s \u00d7\u02e2 Iic \u2191(-r))))\nx : \u03b1 \u00d7 \u211d\nh : \u2200 (i : \u211a), x.1 \u2208 s \u2227 x.2 \u2264 \u2191(-i)\ni : \u211a\nh' : x.1 \u2208 s \u2227 x.2 \u2264 \u2191(- -i)\n\u22a2 x.2 \u2264 \u2191i", "state_after": "case h.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nh_empty : \u2191\u2191\u03c1 (s \u00d7\u02e2 \u2205) = 0\nh_neg : Tendsto (fun r => \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191(-r))) atTop (\ud835\udcdd (\u2191\u2191\u03c1 (\u22c2 r, s \u00d7\u02e2 Iic \u2191(-r))))\nx : \u03b1 \u00d7 \u211d\nh : \u2200 (i : \u211a), x.1 \u2208 s \u2227 x.2 \u2264 \u2191(-i)\ni : \u211a\nh' : x.1 \u2208 s \u2227 x.2 \u2264 \u2191i\n\u22a2 x.2 \u2264 \u2191i"}, {"tactic": "exact h'.2", "annotated_tactic": ["exact h'.2", []], "state_before": "case h.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nh_empty : \u2191\u2191\u03c1 (s \u00d7\u02e2 \u2205) = 0\nh_neg : Tendsto (fun r => \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191(-r))) atTop (\ud835\udcdd (\u2191\u2191\u03c1 (\u22c2 r, s \u00d7\u02e2 Iic \u2191(-r))))\nx : \u03b1 \u00d7 \u211d\nh : \u2200 (i : \u211a), x.1 \u2208 s \u2227 x.2 \u2264 \u2191(-i)\ni : \u211a\nh' : x.1 \u2208 s \u2227 x.2 \u2264 \u2191i\n\u22a2 x.2 \u2264 \u2191i", "state_after": "no goals"}, {"tactic": "exact h'.2", "annotated_tactic": ["exact h'.2", []], "state_before": "case h.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nh_empty : \u2191\u2191\u03c1 (s \u00d7\u02e2 \u2205) = 0\nh_neg : Tendsto (fun r => \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191(-r))) atTop (\ud835\udcdd (\u2191\u2191\u03c1 (\u22c2 r, s \u00d7\u02e2 Iic \u2191(-r))))\nx : \u03b1 \u00d7 \u211d\nh : \u2200 (i : \u211a), x.1 \u2208 s \u2227 x.2 \u2264 \u2191i\ni : \u211a\nh' : x.1 \u2208 s \u2227 x.2 \u2264 \u2191(-i)\n\u22a2 x.2 \u2264 \u2191(-i)", "state_after": "no goals"}, {"tactic": "simp_rw [neg_neg]", "annotated_tactic": ["simp_rw [<a>neg_neg</a>]", [{"full_name": "neg_neg", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [799, 3], "def_end_pos": [799, 14]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nh_empty : \u2191\u2191\u03c1 (s \u00d7\u02e2 \u2205) = 0\nh_neg : Tendsto (fun r => \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191(-r))) atTop (\ud835\udcdd (\u2191\u2191\u03c1 (\u22c2 r, s \u00d7\u02e2 Iic \u2191r)))\nh_inter_eq : \u22c2 r, s \u00d7\u02e2 Iic \u2191(-r) = \u22c2 r, s \u00d7\u02e2 Iic \u2191r\n\u22a2 (fun r => \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191r)) = fun r => \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191(- -r))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "full_name": "measurable_of_empty", "start": [276, 1], "end": [277, 26], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Partrec.lean", "full_name": "Computable.option_casesOn", "start": [682, 1], "end": [687, 35], "traced_tactics": [{"tactic": "cases o a <;> simp [encodek]", "annotated_tactic": ["cases o a <;> simp [<a>encodek</a>]", [{"full_name": "Encodable.encodek", "def_path": "Mathlib/Logic/Encodable/Basic.lean", "def_pos": [53, 3], "def_end_pos": [53, 10]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\no : \u03b1 \u2192 Option \u03b2\nf : \u03b1 \u2192 \u03c3\ng : \u03b1 \u2192 \u03b2 \u2192 \u03c3\nho : Computable o\nhf : Computable f\nhg : Computable\u2082 g\na : \u03b1\n\u22a2 (Nat.casesOn (encode (o a)) (Option.some (f a)) fun n => Option.map (g a) (decode n)) =\n    Option.some (Option.casesOn (o a) (f a) (g a))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/Variables.lean", "full_name": "MvPolynomial.exists_rename_eq_of_vars_subset_range", "start": [854, 1], "end": [865, 13], "traced_tactics": [{"tactic": "show (rename f).toRingHom.comp _ p = RingHom.id _ p", "annotated_tactic": ["show (<a>rename</a> f).toRingHom.comp _ p = <a>RingHom.id</a> _ p", [{"full_name": "MvPolynomial.rename", "def_path": "Mathlib/Data/MvPolynomial/Rename.lean", "def_pos": [56, 5], "def_end_pos": [56, 11]}, {"full_name": "RingHom.id", "def_path": "Mathlib/Algebra/Hom/Ring/Defs.lean", "def_pos": [632, 5], "def_end_pos": [632, 7]}]], "state_before": "R : Type u\nS : Type v\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nr : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u00b9 : CommSemiring R\np\u271d q : MvPolynomial \u03c3 R\ninst\u271d : CommSemiring S\np : MvPolynomial \u03c3 R\nf : \u03c4 \u2192 \u03c3\nhfi : Injective f\nhf : \u2191(vars p) \u2286 range f\n\u22a2 \u2191(rename f) (\u2191(aeval fun i => Option.elim' 0 X (partialInv f i)) p) = p", "state_after": "R : Type u\nS : Type v\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nr : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u00b9 : CommSemiring R\np\u271d q : MvPolynomial \u03c3 R\ninst\u271d : CommSemiring S\np : MvPolynomial \u03c3 R\nf : \u03c4 \u2192 \u03c3\nhfi : Injective f\nhf : \u2191(vars p) \u2286 range f\n\u22a2 \u2191(RingHom.comp \u2191(rename f) \u2191(aeval fun i => Option.elim' 0 X (partialInv f i))) p = \u2191(RingHom.id (MvPolynomial \u03c3 R)) p"}, {"tactic": "refine' hom_congr_vars _ _ _", "annotated_tactic": ["refine' <a>hom_congr_vars</a> _ _ _", [{"full_name": "MvPolynomial.hom_congr_vars", "def_path": "Mathlib/Data/MvPolynomial/Variables.lean", "def_pos": [845, 9], "def_end_pos": [845, 23]}]], "state_before": "R : Type u\nS : Type v\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nr : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u00b9 : CommSemiring R\np\u271d q : MvPolynomial \u03c3 R\ninst\u271d : CommSemiring S\np : MvPolynomial \u03c3 R\nf : \u03c4 \u2192 \u03c3\nhfi : Injective f\nhf : \u2191(vars p) \u2286 range f\n\u22a2 \u2191(RingHom.comp \u2191(rename f) \u2191(aeval fun i => Option.elim' 0 X (partialInv f i))) p = \u2191(RingHom.id (MvPolynomial \u03c3 R)) p", "state_after": "case refine'_1\nR : Type u\nS : Type v\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nr : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u00b9 : CommSemiring R\np\u271d q : MvPolynomial \u03c3 R\ninst\u271d : CommSemiring S\np : MvPolynomial \u03c3 R\nf : \u03c4 \u2192 \u03c3\nhfi : Injective f\nhf : \u2191(vars p) \u2286 range f\n\u22a2 RingHom.comp (RingHom.comp \u2191(rename f) \u2191(aeval fun i => Option.elim' 0 X (partialInv f i))) C =\n    RingHom.comp (RingHom.id (MvPolynomial \u03c3 R)) C\n\ncase refine'_2\nR : Type u\nS : Type v\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nr : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u00b9 : CommSemiring R\np\u271d q : MvPolynomial \u03c3 R\ninst\u271d : CommSemiring S\np : MvPolynomial \u03c3 R\nf : \u03c4 \u2192 \u03c3\nhfi : Injective f\nhf : \u2191(vars p) \u2286 range f\n\u22a2 \u2200 (i : \u03c3),\n    i \u2208 vars p \u2192\n      i \u2208 vars p \u2192\n        \u2191(RingHom.comp \u2191(rename f) \u2191(aeval fun i => Option.elim' 0 X (partialInv f i))) (X i) =\n          \u2191(RingHom.id (MvPolynomial \u03c3 R)) (X i)\n\ncase refine'_3\nR : Type u\nS : Type v\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nr : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u00b9 : CommSemiring R\np\u271d q : MvPolynomial \u03c3 R\ninst\u271d : CommSemiring S\np : MvPolynomial \u03c3 R\nf : \u03c4 \u2192 \u03c3\nhfi : Injective f\nhf : \u2191(vars p) \u2286 range f\n\u22a2 p = p"}, {"tactic": "ext1", "annotated_tactic": ["ext1", []], "state_before": "case refine'_1\nR : Type u\nS : Type v\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nr : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u00b9 : CommSemiring R\np\u271d q : MvPolynomial \u03c3 R\ninst\u271d : CommSemiring S\np : MvPolynomial \u03c3 R\nf : \u03c4 \u2192 \u03c3\nhfi : Injective f\nhf : \u2191(vars p) \u2286 range f\n\u22a2 RingHom.comp (RingHom.comp \u2191(rename f) \u2191(aeval fun i => Option.elim' 0 X (partialInv f i))) C =\n    RingHom.comp (RingHom.id (MvPolynomial \u03c3 R)) C", "state_after": "case refine'_1.a\nR : Type u\nS : Type v\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nr : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u00b9 : CommSemiring R\np\u271d q : MvPolynomial \u03c3 R\ninst\u271d : CommSemiring S\np : MvPolynomial \u03c3 R\nf : \u03c4 \u2192 \u03c3\nhfi : Injective f\nhf : \u2191(vars p) \u2286 range f\nx\u271d : R\n\u22a2 \u2191(RingHom.comp (RingHom.comp \u2191(rename f) \u2191(aeval fun i => Option.elim' 0 X (partialInv f i))) C) x\u271d =\n    \u2191(RingHom.comp (RingHom.id (MvPolynomial \u03c3 R)) C) x\u271d"}, {"tactic": "simp [algebraMap_eq]", "annotated_tactic": ["simp [<a>algebraMap_eq</a>]", [{"full_name": "MvPolynomial.algebraMap_eq", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [186, 9], "def_end_pos": [186, 22]}]], "state_before": "case refine'_1.a\nR : Type u\nS : Type v\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nr : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u00b9 : CommSemiring R\np\u271d q : MvPolynomial \u03c3 R\ninst\u271d : CommSemiring S\np : MvPolynomial \u03c3 R\nf : \u03c4 \u2192 \u03c3\nhfi : Injective f\nhf : \u2191(vars p) \u2286 range f\nx\u271d : R\n\u22a2 \u2191(RingHom.comp (RingHom.comp \u2191(rename f) \u2191(aeval fun i => Option.elim' 0 X (partialInv f i))) C) x\u271d =\n    \u2191(RingHom.comp (RingHom.id (MvPolynomial \u03c3 R)) C) x\u271d", "state_after": "no goals"}, {"tactic": "intro i hip _", "annotated_tactic": ["intro i hip _", []], "state_before": "case refine'_2\nR : Type u\nS : Type v\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nr : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u00b9 : CommSemiring R\np\u271d q : MvPolynomial \u03c3 R\ninst\u271d : CommSemiring S\np : MvPolynomial \u03c3 R\nf : \u03c4 \u2192 \u03c3\nhfi : Injective f\nhf : \u2191(vars p) \u2286 range f\n\u22a2 \u2200 (i : \u03c3),\n    i \u2208 vars p \u2192\n      i \u2208 vars p \u2192\n        \u2191(RingHom.comp \u2191(rename f) \u2191(aeval fun i => Option.elim' 0 X (partialInv f i))) (X i) =\n          \u2191(RingHom.id (MvPolynomial \u03c3 R)) (X i)", "state_after": "case refine'_2\nR : Type u\nS : Type v\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nr : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u00b9 : CommSemiring R\np\u271d q : MvPolynomial \u03c3 R\ninst\u271d : CommSemiring S\np : MvPolynomial \u03c3 R\nf : \u03c4 \u2192 \u03c3\nhfi : Injective f\nhf : \u2191(vars p) \u2286 range f\ni : \u03c3\nhip a\u271d : i \u2208 vars p\n\u22a2 \u2191(RingHom.comp \u2191(rename f) \u2191(aeval fun i => Option.elim' 0 X (partialInv f i))) (X i) =\n    \u2191(RingHom.id (MvPolynomial \u03c3 R)) (X i)"}, {"tactic": "rcases hf hip with \u27e8i, rfl\u27e9", "annotated_tactic": ["rcases hf hip with \u27e8i, rfl\u27e9", []], "state_before": "case refine'_2\nR : Type u\nS : Type v\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nr : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u00b9 : CommSemiring R\np\u271d q : MvPolynomial \u03c3 R\ninst\u271d : CommSemiring S\np : MvPolynomial \u03c3 R\nf : \u03c4 \u2192 \u03c3\nhfi : Injective f\nhf : \u2191(vars p) \u2286 range f\ni : \u03c3\nhip a\u271d : i \u2208 vars p\n\u22a2 \u2191(RingHom.comp \u2191(rename f) \u2191(aeval fun i => Option.elim' 0 X (partialInv f i))) (X i) =\n    \u2191(RingHom.id (MvPolynomial \u03c3 R)) (X i)", "state_after": "case refine'_2.intro\nR : Type u\nS : Type v\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nr : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u00b9 : CommSemiring R\np\u271d q : MvPolynomial \u03c3 R\ninst\u271d : CommSemiring S\np : MvPolynomial \u03c3 R\nf : \u03c4 \u2192 \u03c3\nhfi : Injective f\nhf : \u2191(vars p) \u2286 range f\ni : \u03c4\nhip a\u271d : f i \u2208 vars p\n\u22a2 \u2191(RingHom.comp \u2191(rename f) \u2191(aeval fun i => Option.elim' 0 X (partialInv f i))) (X (f i)) =\n    \u2191(RingHom.id (MvPolynomial \u03c3 R)) (X (f i))"}, {"tactic": "simp [partialInv_left hfi]", "annotated_tactic": ["simp [<a>partialInv_left</a> hfi]", [{"full_name": "Function.partialInv_left", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [419, 9], "def_end_pos": [419, 24]}]], "state_before": "case refine'_2.intro\nR : Type u\nS : Type v\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nr : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u00b9 : CommSemiring R\np\u271d q : MvPolynomial \u03c3 R\ninst\u271d : CommSemiring S\np : MvPolynomial \u03c3 R\nf : \u03c4 \u2192 \u03c3\nhfi : Injective f\nhf : \u2191(vars p) \u2286 range f\ni : \u03c4\nhip a\u271d : f i \u2208 vars p\n\u22a2 \u2191(RingHom.comp \u2191(rename f) \u2191(aeval fun i => Option.elim' 0 X (partialInv f i))) (X (f i)) =\n    \u2191(RingHom.id (MvPolynomial \u03c3 R)) (X (f i))", "state_after": "no goals"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case refine'_3\nR : Type u\nS : Type v\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nr : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u00b9 : CommSemiring R\np\u271d q : MvPolynomial \u03c3 R\ninst\u271d : CommSemiring S\np : MvPolynomial \u03c3 R\nf : \u03c4 \u2192 \u03c3\nhfi : Injective f\nhf : \u2191(vars p) \u2286 range f\n\u22a2 p = p", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finmap.lean", "full_name": "Finmap.mem_list_toFinmap", "start": [525, 1], "end": [535, 36], "traced_tactics": [{"tactic": "induction' xs with x xs", "annotated_tactic": ["induction' xs with x xs", []], "state_before": "\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nxs : List (Sigma \u03b2)\n\u22a2 a \u2208 List.toFinmap xs \u2194 \u2203 b, { fst := a, snd := b } \u2208 xs", "state_after": "case nil\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\na : \u03b1\n\u22a2 a \u2208 List.toFinmap [] \u2194 \u2203 b, { fst := a, snd := b } \u2208 []\n\ncase cons\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nx : Sigma \u03b2\nxs : List (Sigma \u03b2)\ntail_ih\u271d : a \u2208 List.toFinmap xs \u2194 \u2203 b, { fst := a, snd := b } \u2208 xs\n\u22a2 a \u2208 List.toFinmap (x :: xs) \u2194 \u2203 b, { fst := a, snd := b } \u2208 x :: xs"}, {"tactic": "cases' x with fst_i snd_i", "annotated_tactic": ["cases' x with fst_i snd_i", []], "state_before": "case cons\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nx : Sigma \u03b2\nxs : List (Sigma \u03b2)\ntail_ih\u271d : a \u2208 List.toFinmap xs \u2194 \u2203 b, { fst := a, snd := b } \u2208 xs\n\u22a2 a \u2208 List.toFinmap (x :: xs) \u2194 \u2203 b, { fst := a, snd := b } \u2208 x :: xs", "state_after": "case cons.mk\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nxs : List (Sigma \u03b2)\ntail_ih\u271d : a \u2208 List.toFinmap xs \u2194 \u2203 b, { fst := a, snd := b } \u2208 xs\nfst_i : \u03b1\nsnd_i : \u03b2 fst_i\n\u22a2 a \u2208 List.toFinmap ({ fst := fst_i, snd := snd_i } :: xs) \u2194\n    \u2203 b, { fst := a, snd := b } \u2208 { fst := fst_i, snd := snd_i } :: xs"}, {"tactic": "simp only [toFinmap_cons, *, exists_or, mem_cons, mem_insert, exists_and_left, Sigma.mk.inj_iff]", "annotated_tactic": ["simp only [<a>toFinmap_cons</a>, *, <a>exists_or</a>, <a>mem_cons</a>, <a>mem_insert</a>, <a>exists_and_left</a>, <a>Sigma.mk.inj_iff</a>]", [{"full_name": "Finmap.toFinmap_cons", "def_path": "Mathlib/Data/Finmap.lean", "def_pos": [520, 9], "def_end_pos": [520, 22]}, {"full_name": "exists_or", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [429, 9], "def_end_pos": [429, 18]}, {"full_name": "List.mem_cons", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [62, 17], "def_end_pos": [62, 25]}, {"full_name": "Finmap.mem_insert", "def_path": "Mathlib/Data/Finmap.lean", "def_pos": [493, 9], "def_end_pos": [493, 19]}, {"full_name": "exists_and_left", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [465, 17], "def_end_pos": [465, 32]}, {"full_name": "Sigma.mk.inj_iff", "def_path": "Mathlib/Data/Sigma/Basic.lean", "def_pos": [56, 9], "def_end_pos": [56, 19]}]], "state_before": "case cons.mk\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nxs : List (Sigma \u03b2)\ntail_ih\u271d : a \u2208 List.toFinmap xs \u2194 \u2203 b, { fst := a, snd := b } \u2208 xs\nfst_i : \u03b1\nsnd_i : \u03b2 fst_i\n\u22a2 a \u2208 List.toFinmap ({ fst := fst_i, snd := snd_i } :: xs) \u2194\n    \u2203 b, { fst := a, snd := b } \u2208 { fst := fst_i, snd := snd_i } :: xs", "state_after": "case cons.mk\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nxs : List (Sigma \u03b2)\ntail_ih\u271d : a \u2208 List.toFinmap xs \u2194 \u2203 b, { fst := a, snd := b } \u2208 xs\nfst_i : \u03b1\nsnd_i : \u03b2 fst_i\n\u22a2 (a = fst_i \u2228 \u2203 b, { fst := a, snd := b } \u2208 xs) \u2194 (a = fst_i \u2227 \u2203 x, HEq x snd_i) \u2228 \u2203 x, { fst := a, snd := x } \u2208 xs"}, {"tactic": "refine (or_congr_left <| and_iff_left_of_imp ?_).symm", "annotated_tactic": ["refine (<a>or_congr_left</a> <| <a>and_iff_left_of_imp</a> ?_).<a>symm</a>", [{"full_name": "or_congr_left", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [257, 9], "def_end_pos": [257, 22]}, {"full_name": "and_iff_left_of_imp", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [198, 9], "def_end_pos": [198, 28]}, {"full_name": "Iff.symm", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [671, 9], "def_end_pos": [671, 17]}]], "state_before": "case cons.mk\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nxs : List (Sigma \u03b2)\ntail_ih\u271d : a \u2208 List.toFinmap xs \u2194 \u2203 b, { fst := a, snd := b } \u2208 xs\nfst_i : \u03b1\nsnd_i : \u03b2 fst_i\n\u22a2 (a = fst_i \u2228 \u2203 b, { fst := a, snd := b } \u2208 xs) \u2194 (a = fst_i \u2227 \u2203 x, HEq x snd_i) \u2228 \u2203 x, { fst := a, snd := x } \u2208 xs", "state_after": "case cons.mk\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nxs : List (Sigma \u03b2)\ntail_ih\u271d : a \u2208 List.toFinmap xs \u2194 \u2203 b, { fst := a, snd := b } \u2208 xs\nfst_i : \u03b1\nsnd_i : \u03b2 fst_i\n\u22a2 a = fst_i \u2192 \u2203 x, HEq x snd_i"}, {"tactic": "rintro rfl", "annotated_tactic": ["rintro rfl", []], "state_before": "case cons.mk\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nxs : List (Sigma \u03b2)\ntail_ih\u271d : a \u2208 List.toFinmap xs \u2194 \u2203 b, { fst := a, snd := b } \u2208 xs\nfst_i : \u03b1\nsnd_i : \u03b2 fst_i\n\u22a2 a = fst_i \u2192 \u2203 x, HEq x snd_i", "state_after": "case cons.mk\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nxs : List (Sigma \u03b2)\ntail_ih\u271d : a \u2208 List.toFinmap xs \u2194 \u2203 b, { fst := a, snd := b } \u2208 xs\nsnd_i : \u03b2 a\n\u22a2 \u2203 x, HEq x snd_i"}, {"tactic": "simp only [exists_eq, heq_iff_eq]", "annotated_tactic": ["simp only [<a>exists_eq</a>, <a>heq_iff_eq</a>]", [{"full_name": "exists_eq", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [455, 17], "def_end_pos": [455, 26]}, {"full_name": "heq_iff_eq", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [703, 9], "def_end_pos": [703, 19]}]], "state_before": "case cons.mk\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nxs : List (Sigma \u03b2)\ntail_ih\u271d : a \u2208 List.toFinmap xs \u2194 \u2203 b, { fst := a, snd := b } \u2208 xs\nsnd_i : \u03b2 a\n\u22a2 \u2203 x, HEq x snd_i", "state_after": "no goals"}, {"tactic": "simp only [toFinmap_nil, not_mem_empty, find?, not_mem_nil, exists_false]", "annotated_tactic": ["simp only [<a>toFinmap_nil</a>, <a>not_mem_empty</a>, <a>find?</a>, <a>not_mem_nil</a>, <a>exists_false</a>]", [{"full_name": "Finmap.toFinmap_nil", "def_path": "Mathlib/Data/Finmap.lean", "def_pos": [221, 9], "def_end_pos": [221, 21]}, {"full_name": "Finmap.not_mem_empty", "def_path": "Mathlib/Data/Finmap.lean", "def_pos": [225, 9], "def_end_pos": [225, 22]}, {"full_name": "List.find?", "def_path": "lake-packages/lean4/src/lean/Init/Data/List/Basic.lean", "def_pos": [297, 5], "def_end_pos": [297, 10]}, {"full_name": "List.not_mem_nil", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [58, 17], "def_end_pos": [58, 28]}, {"full_name": "exists_false", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [433, 17], "def_end_pos": [433, 29]}]], "state_before": "case nil\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\na : \u03b1\n\u22a2 a \u2208 List.toFinmap [] \u2194 \u2203 b, { fst := a, snd := b } \u2208 []", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": 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\u03c3\ninst\u271d : Nontrivial R\n\u22a2 s \u2286 t \u2192 supported R s \u2264 supported R t", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/List.lean", "full_name": "Set.range_list_getD", "start": [54, 1], "end": [59, 88], "traced_tactics": [{"tactic": "simp only [range_list_get?, image_insert_eq, Option.getD, image_image, image_id']", "annotated_tactic": ["simp only [<a>range_list_get?</a>, <a>image_insert_eq</a>, <a>Option.getD</a>, <a>image_image</a>, <a>image_id'</a>]", [{"full_name": "Set.range_list_get?", "def_path": "Mathlib/Data/Set/List.lean", "def_pos": [46, 9], "def_end_pos": [46, 24]}, {"full_name": "Set.image_insert_eq", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [409, 9], "def_end_pos": [409, 24]}, {"full_name": "Option.getD", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2159, 21], "def_end_pos": [2159, 32]}, {"full_name": "Set.image_image", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [299, 9], "def_end_pos": [299, 20]}, {"full_name": "Set.image_id'", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [396, 9], "def_end_pos": [396, 18]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nl : List \u03b1\nd : \u03b1\n\u22a2 (fun o => Option.getD o d) '' range (get? l) = insert d {x | x \u2208 l}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Card.lean", "full_name": "Set.pred_ncard_le_ncard_diff_singleton", "start": [626, 1], "end": [633, 16], "traced_tactics": [{"tactic": "cases' s.finite_or_infinite with hs hs", "annotated_tactic": ["cases' s.finite_or_infinite with hs hs", []], "state_before": "\u03b1 : Type u_1\ns\u271d t s : Set \u03b1\na : \u03b1\n\u22a2 ncard s - 1 \u2264 ncard (s \\ {a})", "state_after": "case inl\n\u03b1 : Type u_1\ns\u271d t s : Set \u03b1\na : \u03b1\nhs : Set.Finite s\n\u22a2 ncard s - 1 \u2264 ncard (s \\ {a})\n\ncase inr\n\u03b1 : Type u_1\ns\u271d t s : Set \u03b1\na : \u03b1\nhs : Set.Infinite s\n\u22a2 ncard s - 1 \u2264 ncard (s \\ {a})"}, {"tactic": "convert Nat.zero_le _", "annotated_tactic": ["convert <a>Nat.zero_le</a> _", [{"full_name": "Nat.zero_le", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1578, 9], "def_end_pos": [1578, 20]}]], "state_before": "case inr\n\u03b1 : Type u_1\ns\u271d t s : Set \u03b1\na : \u03b1\nhs : Set.Infinite s\n\u22a2 ncard s - 1 \u2264 ncard (s \\ {a})", "state_after": "case h.e'_3\n\u03b1 : Type u_1\ns\u271d t s : Set \u03b1\na : \u03b1\nhs : Set.Infinite s\n\u22a2 ncard s - 1 = 0"}, {"tactic": "rw [hs.ncard]", "annotated_tactic": ["rw [hs.ncard]", []], "state_before": "case h.e'_3\n\u03b1 : Type u_1\ns\u271d t s : Set \u03b1\na : \u03b1\nhs : Set.Infinite s\n\u22a2 ncard s - 1 = 0", "state_after": "no goals"}, {"tactic": "by_cases h : a \u2208 s", "annotated_tactic": ["by_cases h : a \u2208 s", []], "state_before": "case inl\n\u03b1 : Type u_1\ns\u271d t s : Set \u03b1\na : \u03b1\nhs : Set.Finite s\n\u22a2 ncard s - 1 \u2264 ncard (s \\ {a})", "state_after": "case pos\n\u03b1 : Type u_1\ns\u271d t s : Set \u03b1\na : \u03b1\nhs : Set.Finite s\nh : a \u2208 s\n\u22a2 ncard s - 1 \u2264 ncard (s \\ {a})\n\ncase neg\n\u03b1 : Type u_1\ns\u271d t s : Set \u03b1\na : \u03b1\nhs : Set.Finite s\nh : \u00aca \u2208 s\n\u22a2 ncard s - 1 \u2264 ncard (s \\ {a})"}, {"tactic": "rw [diff_singleton_eq_self h]", "annotated_tactic": ["rw [<a>diff_singleton_eq_self</a> h]", [{"full_name": "Set.diff_singleton_eq_self", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [2068, 9], "def_end_pos": [2068, 31]}]], "state_before": "case neg\n\u03b1 : Type u_1\ns\u271d t s : Set \u03b1\na : \u03b1\nhs : Set.Finite s\nh : \u00aca \u2208 s\n\u22a2 ncard s - 1 \u2264 ncard (s \\ {a})", "state_after": "case neg\n\u03b1 : Type u_1\ns\u271d t s : Set \u03b1\na : \u03b1\nhs : Set.Finite s\nh : \u00aca \u2208 s\n\u22a2 ncard s - 1 \u2264 ncard s"}, {"tactic": "apply Nat.pred_le", "annotated_tactic": ["apply <a>Nat.pred_le</a>", [{"full_name": "Nat.pred_le", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [223, 9], "def_end_pos": [223, 16]}]], "state_before": "case neg\n\u03b1 : Type u_1\ns\u271d t s : Set \u03b1\na : \u03b1\nhs : Set.Finite s\nh : \u00aca \u2208 s\n\u22a2 ncard s - 1 \u2264 ncard s", "state_after": "no goals"}, {"tactic": "rw [ncard_diff_singleton_of_mem h hs]", "annotated_tactic": ["rw [<a>ncard_diff_singleton_of_mem</a> h hs]", [{"full_name": "Set.ncard_diff_singleton_of_mem", "def_path": "Mathlib/Data/Set/Card.lean", "def_pos": [609, 17], "def_end_pos": [609, 44]}]], "state_before": "case pos\n\u03b1 : Type u_1\ns\u271d t s : Set \u03b1\na : \u03b1\nhs : Set.Finite s\nh : a \u2208 s\n\u22a2 ncard s - 1 \u2264 ncard (s \\ {a})", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/List/Pairwise.lean", "full_name": "List.forall_mem_pwFilter", "start": [311, 1], "end": [328, 52], "traced_tactics": [{"tactic": "refine \u27e8?_, fun h b hb => h _ <| pwFilter_subset (R := R) _ hb\u27e9", "annotated_tactic": ["refine \u27e8?_, fun h b hb => h _ <| <a>pwFilter_subset</a> (R := R) _ hb\u27e9", [{"full_name": "List.pwFilter_subset", "def_path": "lake-packages/std/Std/Data/List/Pairwise.lean", "def_pos": [286, 9], "def_end_pos": [286, 24]}]], "state_before": "\u03b1 : Type u_1\nR : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel R\nneg_trans : \u2200 {x y z : \u03b1}, R x z \u2192 R x y \u2228 R y z\na : \u03b1\nl : List \u03b1\n\u22a2 (\u2200 (b : \u03b1), b \u2208 pwFilter R l \u2192 R a b) \u2194 \u2200 (b : \u03b1), b \u2208 l \u2192 R a b", "state_after": "\u03b1 : Type u_1\nR : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel R\nneg_trans : \u2200 {x y z : \u03b1}, R x z \u2192 R x y \u2228 R y z\na : \u03b1\nl : List \u03b1\n\u22a2 (\u2200 (b : \u03b1), b \u2208 pwFilter R l \u2192 R a b) \u2192 \u2200 (b : \u03b1), b \u2208 l \u2192 R a b"}, {"tactic": "induction l with\n| nil => exact fun _ _ h => (not_mem_nil _ h).elim\n| cons x l IH =>\n  simp only [forall_mem_cons]\n  if h : \u2200 y \u2208 pwFilter R l, R x y then\n    simpa [pwFilter_cons_of_pos h] using fun r H => \u27e8r, IH H\u27e9\n  else\n    refine pwFilter_cons_of_neg h \u25b8 fun H => \u27e8?_, IH H\u27e9\n    match e : find? (fun y => \u00acR x y) (pwFilter R l) with\n    | none => exact h.elim fun y hy => by simpa using find?_eq_none.1 e y hy\n    | some k =>\n      have := find?_some e\n      apply (neg_trans (H k (mem_of_find?_eq_some e))).resolve_right\n      rw [decide_eq_true_iff] at this; exact this", "annotated_tactic": ["induction l with\n  | <a>nil</a> => exact fun _ _ h => (<a>not_mem_nil</a> _ h).<a>elim</a>\n  | <a>cons</a> x l IH =>\n    simp only [<a>forall_mem_cons</a>]\n    if h : \u2200 y \u2208 <a>pwFilter</a> R l, R x y then\n      simpa [<a>pwFilter_cons_of_pos</a> h] using fun r H => \u27e8r, IH H\u27e9\n    else\n      refine <a>pwFilter_cons_of_neg</a> h \u25b8 fun H => \u27e8?_, IH H\u27e9\n      match e : <a>find?</a> (fun y => \u00acR x y) (<a>pwFilter</a> R l) with\n      | <a>none</a> => exact h.elim fun y hy => by simpa using <a>find?_eq_none</a>.1 e y hy\n      | <a>some</a> k =>\n        have := <a>find?_some</a> e\n        apply (neg_trans (H k (<a>mem_of_find?_eq_some</a> e))).<a>resolve_right</a>\n        rw [<a>decide_eq_true_iff</a>] at this; exact this", [{"full_name": "List.nil", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2184, 5], "def_end_pos": [2184, 8]}, {"full_name": "List.not_mem_nil", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [58, 17], "def_end_pos": [58, 28]}, {"full_name": "False.elim", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [223, 21], "def_end_pos": [223, 31]}, {"full_name": "List.cons", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2187, 5], "def_end_pos": [2187, 9]}, {"full_name": "List.forall_mem_cons", "def_path": "lake-packages/std/Std/Data/List/Init/Lemmas.lean", "def_pos": [120, 9], "def_end_pos": [120, 24]}, {"full_name": "List.pwFilter", "def_path": "lake-packages/std/Std/Data/List/Basic.lean", "def_pos": [1153, 5], "def_end_pos": [1153, 13]}, {"full_name": "List.pwFilter_cons_of_pos", "def_path": "lake-packages/std/Std/Data/List/Pairwise.lean", "def_pos": [258, 17], "def_end_pos": [258, 37]}, {"full_name": "List.pwFilter_cons_of_neg", "def_path": "lake-packages/std/Std/Data/List/Pairwise.lean", "def_pos": [261, 17], "def_end_pos": [261, 37]}, {"full_name": "List.find?", "def_path": "lake-packages/lean4/src/lean/Init/Data/List/Basic.lean", "def_pos": [297, 5], "def_end_pos": [297, 10]}, {"full_name": "List.pwFilter", "def_path": "lake-packages/std/Std/Data/List/Basic.lean", "def_pos": [1153, 5], "def_end_pos": [1153, 13]}, {"full_name": "Option.none", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2141, 5], "def_end_pos": [2141, 9]}, {"full_name": "List.find?_eq_none", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [1383, 9], "def_end_pos": [1383, 22]}, {"full_name": "Option.some", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2143, 5], "def_end_pos": [2143, 9]}, {"full_name": "List.find?_some", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [1386, 9], "def_end_pos": [1386, 19]}, {"full_name": "List.mem_of_find?_eq_some", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [1392, 17], "def_end_pos": [1392, 37]}, {"full_name": "Or.resolve_right", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [272, 9], "def_end_pos": [272, 25]}, {"full_name": "decide_eq_true_iff", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [537, 9], "def_end_pos": [537, 27]}]], "state_before": "\u03b1 : Type u_1\nR : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel R\nneg_trans : \u2200 {x y z : \u03b1}, R x z \u2192 R x y \u2228 R y z\na : \u03b1\nl : List \u03b1\n\u22a2 (\u2200 (b : \u03b1), b \u2208 pwFilter R l \u2192 R a b) \u2192 \u2200 (b : \u03b1), b \u2208 l \u2192 R a b", "state_after": "no goals"}, {"tactic": "exact fun _ _ h => (not_mem_nil _ h).elim", "annotated_tactic": ["exact fun _ _ h => (<a>not_mem_nil</a> _ h).<a>elim</a>", [{"full_name": "List.not_mem_nil", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [58, 17], "def_end_pos": [58, 28]}, {"full_name": "False.elim", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [223, 21], "def_end_pos": [223, 31]}]], "state_before": "case nil\n\u03b1 : Type u_1\nR : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel R\nneg_trans : \u2200 {x y z : \u03b1}, R x z \u2192 R x y \u2228 R y z\na : \u03b1\n\u22a2 (\u2200 (b : \u03b1), b \u2208 pwFilter R [] \u2192 R a b) \u2192 \u2200 (b : \u03b1), b \u2208 [] \u2192 R a b", "state_after": "no goals"}, {"tactic": "simp only [forall_mem_cons]", "annotated_tactic": ["simp only [<a>forall_mem_cons</a>]", [{"full_name": "List.forall_mem_cons", "def_path": "lake-packages/std/Std/Data/List/Init/Lemmas.lean", "def_pos": [120, 9], "def_end_pos": [120, 24]}]], "state_before": "case cons\n\u03b1 : Type u_1\nR : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel R\nneg_trans : \u2200 {x y z : \u03b1}, R x z \u2192 R x y \u2228 R y z\na x : \u03b1\nl : List \u03b1\nIH : (\u2200 (b : \u03b1), b \u2208 pwFilter R l \u2192 R a b) \u2192 \u2200 (b : \u03b1), b \u2208 l \u2192 R a b\n\u22a2 (\u2200 (b : \u03b1), b \u2208 pwFilter R (x :: l) \u2192 R a b) \u2192 \u2200 (b : \u03b1), b \u2208 x :: l \u2192 R a b", "state_after": "case cons\n\u03b1 : Type u_1\nR : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel R\nneg_trans : \u2200 {x y z : \u03b1}, R x z \u2192 R x y \u2228 R y z\na x : \u03b1\nl : List \u03b1\nIH : (\u2200 (b : \u03b1), b \u2208 pwFilter R l \u2192 R a b) \u2192 \u2200 (b : \u03b1), b \u2208 l \u2192 R a b\n\u22a2 (\u2200 (b : \u03b1), b \u2208 pwFilter R (x :: l) \u2192 R a b) \u2192 R a x \u2227 \u2200 (x : \u03b1), x \u2208 l \u2192 R a x"}, {"tactic": "if h : \u2200 y \u2208 pwFilter R l, R x y then\n  simpa [pwFilter_cons_of_pos h] using fun r H => \u27e8r, IH H\u27e9\nelse\n  refine pwFilter_cons_of_neg h \u25b8 fun H => \u27e8?_, IH H\u27e9\n  match e : find? (fun y => \u00acR x y) (pwFilter R l) with\n  | none => exact h.elim fun y hy => by simpa using find?_eq_none.1 e y hy\n  | some k =>\n    have := find?_some e\n    apply (neg_trans (H k (mem_of_find?_eq_some e))).resolve_right\n    rw [decide_eq_true_iff] at this; exact this", "annotated_tactic": ["if h : \u2200 y \u2208 <a>pwFilter</a> R l, R x y then\n      simpa [<a>pwFilter_cons_of_pos</a> h] using fun r H => \u27e8r, IH H\u27e9\n    else\n      refine <a>pwFilter_cons_of_neg</a> h \u25b8 fun H => \u27e8?_, IH H\u27e9\n      match e : <a>find?</a> (fun y => \u00acR x y) (<a>pwFilter</a> R l) with\n      | <a>none</a> => exact h.elim fun y hy => by simpa using <a>find?_eq_none</a>.1 e y hy\n      | <a>some</a> k =>\n        have := <a>find?_some</a> e\n        apply (neg_trans (H k (<a>mem_of_find?_eq_some</a> e))).<a>resolve_right</a>\n        rw [<a>decide_eq_true_iff</a>] at this; exact this", [{"full_name": "List.pwFilter", "def_path": "lake-packages/std/Std/Data/List/Basic.lean", "def_pos": [1153, 5], "def_end_pos": [1153, 13]}, {"full_name": "List.pwFilter_cons_of_pos", "def_path": "lake-packages/std/Std/Data/List/Pairwise.lean", "def_pos": [258, 17], "def_end_pos": [258, 37]}, {"full_name": "List.pwFilter_cons_of_neg", "def_path": "lake-packages/std/Std/Data/List/Pairwise.lean", "def_pos": [261, 17], "def_end_pos": [261, 37]}, {"full_name": "List.find?", "def_path": "lake-packages/lean4/src/lean/Init/Data/List/Basic.lean", "def_pos": [297, 5], "def_end_pos": [297, 10]}, {"full_name": "List.pwFilter", "def_path": "lake-packages/std/Std/Data/List/Basic.lean", "def_pos": [1153, 5], "def_end_pos": [1153, 13]}, {"full_name": "Option.none", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2141, 5], "def_end_pos": [2141, 9]}, {"full_name": "List.find?_eq_none", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [1383, 9], "def_end_pos": [1383, 22]}, {"full_name": "Option.some", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2143, 5], "def_end_pos": [2143, 9]}, {"full_name": "List.find?_some", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [1386, 9], "def_end_pos": [1386, 19]}, {"full_name": "List.mem_of_find?_eq_some", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [1392, 17], "def_end_pos": [1392, 37]}, {"full_name": "Or.resolve_right", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [272, 9], "def_end_pos": [272, 25]}, {"full_name": "decide_eq_true_iff", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [537, 9], "def_end_pos": [537, 27]}]], "state_before": "case cons\n\u03b1 : Type u_1\nR : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel R\nneg_trans : \u2200 {x y z : \u03b1}, R x z \u2192 R x y \u2228 R y z\na x : \u03b1\nl : List \u03b1\nIH : (\u2200 (b : \u03b1), b \u2208 pwFilter R l \u2192 R a b) \u2192 \u2200 (b : \u03b1), b \u2208 l \u2192 R a b\n\u22a2 (\u2200 (b : \u03b1), b \u2208 pwFilter R (x :: l) \u2192 R a b) \u2192 R a x \u2227 \u2200 (x : \u03b1), x \u2208 l \u2192 R a x", "state_after": "no goals"}, {"tactic": "simpa [pwFilter_cons_of_pos h] using fun r H => \u27e8r, IH H\u27e9", "annotated_tactic": ["simpa [<a>pwFilter_cons_of_pos</a> h] using fun r H => \u27e8r, IH H\u27e9", [{"full_name": "List.pwFilter_cons_of_pos", "def_path": "lake-packages/std/Std/Data/List/Pairwise.lean", "def_pos": [258, 17], "def_end_pos": [258, 37]}]], "state_before": "\u03b1 : Type u_1\nR : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel R\nneg_trans : \u2200 {x y z : \u03b1}, R x z \u2192 R x y \u2228 R y z\na x : \u03b1\nl : List \u03b1\nIH : (\u2200 (b : \u03b1), b \u2208 pwFilter R l \u2192 R a b) \u2192 \u2200 (b : \u03b1), b \u2208 l \u2192 R a b\nh : \u2200 (y : \u03b1), y \u2208 pwFilter R l \u2192 R x y\n\u22a2 (\u2200 (b : \u03b1), b \u2208 pwFilter R (x :: l) \u2192 R a b) \u2192 R a x \u2227 \u2200 (x : \u03b1), x \u2208 l \u2192 R a x", "state_after": "no goals"}, {"tactic": "refine pwFilter_cons_of_neg h \u25b8 fun H => \u27e8?_, IH H\u27e9", "annotated_tactic": ["refine <a>pwFilter_cons_of_neg</a> h \u25b8 fun H => \u27e8?_, IH H\u27e9", [{"full_name": "List.pwFilter_cons_of_neg", "def_path": "lake-packages/std/Std/Data/List/Pairwise.lean", "def_pos": [261, 17], "def_end_pos": [261, 37]}]], "state_before": "\u03b1 : Type u_1\nR : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel R\nneg_trans : \u2200 {x y z : \u03b1}, R x z \u2192 R x y \u2228 R y z\na x : \u03b1\nl : List \u03b1\nIH : (\u2200 (b : \u03b1), b \u2208 pwFilter R l \u2192 R a b) \u2192 \u2200 (b : \u03b1), b \u2208 l \u2192 R a b\nh : \u00ac\u2200 (y : \u03b1), y \u2208 pwFilter R l \u2192 R x y\n\u22a2 (\u2200 (b : \u03b1), b \u2208 pwFilter R (x :: l) \u2192 R a b) \u2192 R a x \u2227 \u2200 (x : \u03b1), x \u2208 l \u2192 R a x", "state_after": "\u03b1 : Type u_1\nR : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel R\nneg_trans : \u2200 {x y z : \u03b1}, R x z \u2192 R x y \u2228 R y z\na x : \u03b1\nl : List \u03b1\nIH : (\u2200 (b : \u03b1), b \u2208 pwFilter R l \u2192 R a b) \u2192 \u2200 (b : \u03b1), b \u2208 l \u2192 R a b\nh : \u00ac\u2200 (y : \u03b1), y \u2208 pwFilter R l \u2192 R x y\nH : \u2200 (b : \u03b1), b \u2208 pwFilter R l \u2192 R a b\n\u22a2 R a x"}, {"tactic": "match e : find? (fun y => \u00acR x y) (pwFilter R l) with\n| none => exact h.elim fun y hy => by simpa using find?_eq_none.1 e y hy\n| some k =>\n  have := find?_some e\n  apply (neg_trans (H k (mem_of_find?_eq_some e))).resolve_right\n  rw [decide_eq_true_iff] at this; exact this", "annotated_tactic": ["match e : <a>find?</a> (fun y => \u00acR x y) (<a>pwFilter</a> R l) with\n      | <a>none</a> => exact h.elim fun y hy => by simpa using <a>find?_eq_none</a>.1 e y hy\n      | <a>some</a> k =>\n        have := <a>find?_some</a> e\n        apply (neg_trans (H k (<a>mem_of_find?_eq_some</a> e))).<a>resolve_right</a>\n        rw [<a>decide_eq_true_iff</a>] at this; exact this", [{"full_name": "List.find?", "def_path": "lake-packages/lean4/src/lean/Init/Data/List/Basic.lean", "def_pos": [297, 5], "def_end_pos": [297, 10]}, {"full_name": "List.pwFilter", "def_path": "lake-packages/std/Std/Data/List/Basic.lean", "def_pos": [1153, 5], "def_end_pos": [1153, 13]}, {"full_name": "Option.none", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2141, 5], "def_end_pos": [2141, 9]}, {"full_name": "List.find?_eq_none", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [1383, 9], "def_end_pos": [1383, 22]}, {"full_name": "Option.some", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2143, 5], "def_end_pos": [2143, 9]}, {"full_name": "List.find?_some", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [1386, 9], "def_end_pos": [1386, 19]}, {"full_name": "List.mem_of_find?_eq_some", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [1392, 17], "def_end_pos": [1392, 37]}, {"full_name": "Or.resolve_right", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [272, 9], "def_end_pos": [272, 25]}, {"full_name": "decide_eq_true_iff", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [537, 9], "def_end_pos": [537, 27]}]], "state_before": "\u03b1 : Type u_1\nR : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel R\nneg_trans : \u2200 {x y z : \u03b1}, R x z \u2192 R x y \u2228 R y z\na x : \u03b1\nl : List \u03b1\nIH : (\u2200 (b : \u03b1), b \u2208 pwFilter R l \u2192 R a b) \u2192 \u2200 (b : \u03b1), b \u2208 l \u2192 R a b\nh : \u00ac\u2200 (y : \u03b1), y \u2208 pwFilter R l \u2192 R x y\nH : \u2200 (b : \u03b1), b \u2208 pwFilter R l \u2192 R a b\n\u22a2 R a x", "state_after": "no goals"}, {"tactic": "exact h.elim fun y hy => by simpa using find?_eq_none.1 e y hy", "annotated_tactic": ["exact h.elim fun y hy => by simpa using <a>find?_eq_none</a>.1 e y hy", [{"full_name": "List.find?_eq_none", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [1383, 9], "def_end_pos": [1383, 22]}]], "state_before": "\u03b1 : Type u_1\nR : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel R\nneg_trans : \u2200 {x y z : \u03b1}, R x z \u2192 R x y \u2228 R y z\na x : \u03b1\nl : List \u03b1\nIH : (\u2200 (b : \u03b1), b \u2208 pwFilter R l \u2192 R a b) \u2192 \u2200 (b : \u03b1), b \u2208 l \u2192 R a b\nh : \u00ac\u2200 (y : \u03b1), y \u2208 pwFilter R l \u2192 R x y\nH : \u2200 (b : \u03b1), b \u2208 pwFilter R l \u2192 R a b\ne : find? (fun y => decide \u00acR x y) (pwFilter R l) = none\n\u22a2 R a x", "state_after": "no goals"}, {"tactic": "simpa using find?_eq_none.1 e y hy", "annotated_tactic": ["simpa using <a>find?_eq_none</a>.1 e y hy", [{"full_name": "List.find?_eq_none", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [1383, 9], "def_end_pos": [1383, 22]}]], "state_before": "\u03b1 : Type u_1\nR : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel R\nneg_trans : \u2200 {x y z : \u03b1}, R x z \u2192 R x y \u2228 R y z\na x : \u03b1\nl : List \u03b1\nIH : (\u2200 (b : \u03b1), b \u2208 pwFilter R l \u2192 R a b) \u2192 \u2200 (b : \u03b1), b \u2208 l \u2192 R a b\nh : \u00ac\u2200 (y : \u03b1), y \u2208 pwFilter R l \u2192 R x y\nH : \u2200 (b : \u03b1), b \u2208 pwFilter R l \u2192 R a b\ne : find? (fun y => decide \u00acR x y) (pwFilter R l) = none\ny : \u03b1\nhy : y \u2208 pwFilter R l\n\u22a2 R x y", "state_after": "no goals"}, {"tactic": "have := find?_some e", "annotated_tactic": ["have := <a>find?_some</a> e", [{"full_name": "List.find?_some", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [1386, 9], "def_end_pos": [1386, 19]}]], "state_before": "\u03b1 : Type u_1\nR : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel R\nneg_trans : \u2200 {x y z : \u03b1}, R x z \u2192 R x y \u2228 R y z\na x : \u03b1\nl : List \u03b1\nIH : (\u2200 (b : \u03b1), b \u2208 pwFilter R l \u2192 R a b) \u2192 \u2200 (b : \u03b1), b \u2208 l \u2192 R a b\nh : \u00ac\u2200 (y : \u03b1), y \u2208 pwFilter R l \u2192 R x y\nH : \u2200 (b : \u03b1), b \u2208 pwFilter R l \u2192 R a b\nk : \u03b1\ne : find? (fun y => decide \u00acR x y) (pwFilter R l) = some k\n\u22a2 R a x", "state_after": "\u03b1 : Type u_1\nR : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel R\nneg_trans : \u2200 {x y z : \u03b1}, R x z \u2192 R x y \u2228 R y z\na x : \u03b1\nl : List \u03b1\nIH : (\u2200 (b : \u03b1), b \u2208 pwFilter R l \u2192 R a b) \u2192 \u2200 (b : \u03b1), b \u2208 l \u2192 R a b\nh : \u00ac\u2200 (y : \u03b1), y \u2208 pwFilter R l \u2192 R x y\nH : \u2200 (b : \u03b1), b \u2208 pwFilter R l \u2192 R a b\nk : \u03b1\ne : find? (fun y => decide \u00acR x y) (pwFilter R l) = some k\nthis : (decide \u00acR x k) = true\n\u22a2 R a x"}, {"tactic": "apply (neg_trans (H k (mem_of_find?_eq_some e))).resolve_right", "annotated_tactic": ["apply (neg_trans (H k (<a>mem_of_find?_eq_some</a> e))).<a>resolve_right</a>", [{"full_name": "List.mem_of_find?_eq_some", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [1392, 17], "def_end_pos": [1392, 37]}, {"full_name": "Or.resolve_right", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [272, 9], "def_end_pos": [272, 25]}]], "state_before": "\u03b1 : Type u_1\nR : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel R\nneg_trans : \u2200 {x y z : \u03b1}, R x z \u2192 R x y \u2228 R y z\na x : \u03b1\nl : List \u03b1\nIH : (\u2200 (b : \u03b1), b \u2208 pwFilter R l \u2192 R a b) \u2192 \u2200 (b : \u03b1), b \u2208 l \u2192 R a b\nh : \u00ac\u2200 (y : \u03b1), y \u2208 pwFilter R l \u2192 R x y\nH : \u2200 (b : \u03b1), b \u2208 pwFilter R l \u2192 R a b\nk : \u03b1\ne : find? (fun y => decide \u00acR x y) (pwFilter R l) = some k\nthis : (decide \u00acR x k) = true\n\u22a2 R a x", "state_after": "\u03b1 : Type u_1\nR : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel R\nneg_trans : \u2200 {x y z : \u03b1}, R x z \u2192 R x y \u2228 R y z\na x : \u03b1\nl : List \u03b1\nIH : (\u2200 (b : \u03b1), b \u2208 pwFilter R l \u2192 R a b) \u2192 \u2200 (b : \u03b1), b \u2208 l \u2192 R a b\nh : \u00ac\u2200 (y : \u03b1), y \u2208 pwFilter R l \u2192 R x y\nH : \u2200 (b : \u03b1), b \u2208 pwFilter R l \u2192 R a b\nk : \u03b1\ne : find? (fun y => decide \u00acR x y) (pwFilter R l) = some k\nthis : (decide \u00acR x k) = true\n\u22a2 \u00acR x k"}, {"tactic": "rw [decide_eq_true_iff] at this", "annotated_tactic": ["rw [<a>decide_eq_true_iff</a>] at this", [{"full_name": "decide_eq_true_iff", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [537, 9], "def_end_pos": [537, 27]}]], "state_before": "\u03b1 : Type u_1\nR : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel R\nneg_trans : \u2200 {x y z : \u03b1}, R x z \u2192 R x y \u2228 R y z\na x : \u03b1\nl : List \u03b1\nIH : (\u2200 (b : \u03b1), b \u2208 pwFilter R l \u2192 R a b) \u2192 \u2200 (b : \u03b1), b \u2208 l \u2192 R a b\nh : \u00ac\u2200 (y : \u03b1), y \u2208 pwFilter R l \u2192 R x y\nH : \u2200 (b : \u03b1), b \u2208 pwFilter R l \u2192 R a b\nk : \u03b1\ne : find? (fun y => decide \u00acR x y) (pwFilter R l) = some k\nthis : (decide \u00acR x k) = true\n\u22a2 \u00acR x k", "state_after": "\u03b1 : Type u_1\nR : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel R\nneg_trans : \u2200 {x y z : \u03b1}, R x z \u2192 R x y \u2228 R y z\na x : \u03b1\nl : List \u03b1\nIH : (\u2200 (b : \u03b1), b \u2208 pwFilter R l \u2192 R a b) \u2192 \u2200 (b : \u03b1), b \u2208 l \u2192 R a b\nh : \u00ac\u2200 (y : \u03b1), y \u2208 pwFilter R l \u2192 R x y\nH : \u2200 (b : \u03b1), b \u2208 pwFilter R l \u2192 R a b\nk : \u03b1\ne : find? (fun y => decide \u00acR x y) (pwFilter R l) = some k\nthis : \u00acR x k\n\u22a2 \u00acR x k"}, {"tactic": "exact this", "annotated_tactic": ["exact this", []], "state_before": "\u03b1 : Type u_1\nR : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel R\nneg_trans : \u2200 {x y z : \u03b1}, R x z \u2192 R x y \u2228 R y z\na x : \u03b1\nl : List \u03b1\nIH : (\u2200 (b : \u03b1), b \u2208 pwFilter R l \u2192 R a b) \u2192 \u2200 (b : \u03b1), b \u2208 l \u2192 R a b\nh : \u00ac\u2200 (y : \u03b1), y \u2208 pwFilter R l \u2192 R x y\nH : \u2200 (b : \u03b1), b \u2208 pwFilter R l \u2192 R a b\nk : \u03b1\ne : find? (fun y => decide \u00acR x y) (pwFilter R l) = some k\nthis : \u00acR x k\n\u22a2 \u00acR x k", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/FiniteMeasure.lean", "full_name": "MeasureTheory.FiniteMeasure.testAgainstNN_add", "start": [372, 1], "end": [376, 87], "traced_tactics": [{"tactic": "simp only [\u2190 ENNReal.coe_eq_coe, BoundedContinuousFunction.coe_add, ENNReal.coe_add, Pi.add_apply,\n  testAgainstNN_coe_eq]", "annotated_tactic": ["simp only [\u2190 <a>ENNReal.coe_eq_coe</a>, <a>BoundedContinuousFunction.coe_add</a>, <a>ENNReal.coe_add</a>, <a>Pi.add_apply</a>,\n    <a>testAgainstNN_coe_eq</a>]", [{"full_name": "ENNReal.coe_eq_coe", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [346, 28], "def_end_pos": [346, 38]}, {"full_name": "BoundedContinuousFunction.coe_add", "def_path": "Mathlib/Topology/ContinuousFunction/Bounded.lean", "def_pos": [683, 9], "def_end_pos": [683, 16]}, {"full_name": "ENNReal.coe_add", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [386, 28], "def_end_pos": [386, 35]}, {"full_name": "Pi.add_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [82, 3], "def_end_pos": [82, 14]}, {"full_name": "MeasureTheory.FiniteMeasure.testAgainstNN_coe_eq", "def_path": "Mathlib/MeasureTheory/Measure/FiniteMeasure.lean", "def_pos": [326, 9], "def_end_pos": [326, 29]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2075 : SMul R \u211d\u22650\ninst\u271d\u2074 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\nf\u2081 f\u2082 : \u03a9 \u2192\u1d47 \u211d\u22650\n\u22a2 testAgainstNN \u03bc (f\u2081 + f\u2082) = testAgainstNN \u03bc f\u2081 + testAgainstNN \u03bc f\u2082", "state_after": "\u03a9 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2075 : SMul R \u211d\u22650\ninst\u271d\u2074 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\nf\u2081 f\u2082 : \u03a9 \u2192\u1d47 \u211d\u22650\n\u22a2 \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f\u2081 \u03c9) + \u2191(\u2191f\u2082 \u03c9) \u2202\u2191\u03bc = \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f\u2081 \u03c9) \u2202\u2191\u03bc + \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f\u2082 \u03c9) \u2202\u2191\u03bc"}, {"tactic": "exact lintegral_add_left (BoundedContinuousFunction.measurable_coe_ennreal_comp _) _", "annotated_tactic": ["exact <a>lintegral_add_left</a> (<a>BoundedContinuousFunction.measurable_coe_ennreal_comp</a> _) _", [{"full_name": "MeasureTheory.lintegral_add_left", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [554, 9], "def_end_pos": [554, 27]}, {"full_name": "BoundedContinuousFunction.measurable_coe_ennreal_comp", "def_path": "Mathlib/MeasureTheory/Integral/BoundedContinuousFunction.lean", "def_pos": [27, 9], "def_end_pos": [27, 36]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2075 : SMul R \u211d\u22650\ninst\u271d\u2074 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\nf\u2081 f\u2082 : \u03a9 \u2192\u1d47 \u211d\u22650\n\u22a2 \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f\u2081 \u03c9) + \u2191(\u2191f\u2082 \u03c9) \u2202\u2191\u03bc = \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f\u2081 \u03c9) \u2202\u2191\u03bc + \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f\u2082 \u03c9) \u2202\u2191\u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Array/Lemmas.lean", "full_name": "Array.mapIdx_induction'", "start": [203, 1], "end": [207, 81], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/AEEqFun.lean", "full_name": "MeasureTheory.AEEqFun.toGerm_eq", "start": [447, 1], "end": [447, 92], "traced_tactics": [{"tactic": "rw [\u2190 mk_toGerm, mk_coeFn]", "annotated_tactic": ["rw [\u2190 <a>mk_toGerm</a>, <a>mk_coeFn</a>]", [{"full_name": "MeasureTheory.AEEqFun.mk_toGerm", "def_path": "Mathlib/MeasureTheory/Function/AEEqFun.lean", "def_pos": [443, 9], "def_end_pos": [443, 18]}, {"full_name": "MeasureTheory.AEEqFun.mk_coeFn", "def_path": "Mathlib/MeasureTheory/Function/AEEqFun.lean", "def_pos": [165, 9], "def_end_pos": [165, 17]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : TopologicalSpace \u03b4\nf : \u03b1 \u2192\u2098[\u03bc] \u03b2\n\u22a2 toGerm f = \u2191\u2191f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Partrec.lean", "full_name": "Partrec\u2082.unpaired'", "start": [505, 1], "end": [506, 38], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Powerset.lean", "full_name": "Finset.powerset_eq_singleton_empty", "start": [82, 1], "end": [83, 38], "traced_tactics": [{"tactic": "rw [\u2190 powerset_empty, powerset_inj]", "annotated_tactic": ["rw [\u2190 <a>powerset_empty</a>, <a>powerset_inj</a>]", [{"full_name": "Finset.powerset_empty", "def_path": "Mathlib/Data/Finset/Powerset.lean", "def_pos": [77, 9], "def_end_pos": [77, 23]}, {"full_name": "Finset.powerset_inj", "def_path": "Mathlib/Data/Finset/Powerset.lean", "def_pos": [72, 9], "def_end_pos": [72, 21]}]], "state_before": "\u03b1 : Type u_1\ns t : Finset \u03b1\n\u22a2 powerset s = {\u2205} \u2194 s = \u2205", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Prod.lean", "full_name": 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[{"tactic": "refine' (pi_eq fun s _ => _).symm", "annotated_tactic": ["refine' (<a>pi_eq</a> fun s _ => _).<a>symm</a>", [{"full_name": "MeasureTheory.Measure.pi_eq", "def_path": "Mathlib/MeasureTheory/Constructions/Pi.lean", "def_pos": [380, 9], "def_end_pos": [380, 14]}, {"full_name": "Eq.symm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [310, 9], "def_end_pos": [310, 16]}]], "state_before": "\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : \u03b9 \u2192 Type u_3\ninst\u271d\u00b2 : Fintype \u03b9\nm\u271d : (i : \u03b9) \u2192 OuterMeasure (\u03b1 i)\nm : (i : \u03b9) \u2192 MeasurableSpace (\u03b1 i)\n\u03bc : (i : \u03b9) \u2192 Measure (\u03b1 i)\ninst\u271d\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc i)\ninst\u271d : Fintype \u03b9'\nf : \u03b9' \u2243 \u03b9\n\u22a2 Measure.map (\u2191(MeasurableEquiv.piCongrLeft \u03b1 f)) (Measure.pi fun i' => \u03bc (\u2191f i')) = Measure.pi \u03bc", "state_after": "\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : \u03b9 \u2192 Type u_3\ninst\u271d\u00b2 : Fintype \u03b9\nm\u271d : (i : \u03b9) \u2192 OuterMeasure (\u03b1 i)\nm : (i : \u03b9) \u2192 MeasurableSpace (\u03b1 i)\n\u03bc : (i : \u03b9) \u2192 Measure (\u03b1 i)\ninst\u271d\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc i)\ninst\u271d : Fintype \u03b9'\nf : \u03b9' \u2243 \u03b9\ns : (i : \u03b9) \u2192 Set (\u03b1 i)\nx\u271d : \u2200 (i : \u03b9), MeasurableSet (s i)\n\u22a2 \u2191\u2191(Measure.map (\u2191(MeasurableEquiv.piCongrLeft \u03b1 f)) (Measure.pi fun i' => \u03bc (\u2191f i'))) (Set.pi univ s) =\n    \u220f i : \u03b9, \u2191\u2191(\u03bc i) (s i)"}, {"tactic": "rw [MeasurableEquiv.map_apply, MeasurableEquiv.coe_piCongrLeft f,\n  Equiv.piCongrLeft_preimage_univ_pi, pi_pi _ _, f.prod_comp (fun i => \u03bc i (s i))]", "annotated_tactic": ["rw [<a>MeasurableEquiv.map_apply</a>, <a>MeasurableEquiv.coe_piCongrLeft</a> f,\n      <a>Equiv.piCongrLeft_preimage_univ_pi</a>, <a>pi_pi</a> _ _, f.prod_comp (fun i => \u03bc i (s i))]", [{"full_name": "MeasurableEquiv.map_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [4218, 19], "def_end_pos": [4218, 28]}, {"full_name": "MeasurableEquiv.coe_piCongrLeft", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [1617, 9], "def_end_pos": [1617, 24]}, {"full_name": "Equiv.piCongrLeft_preimage_univ_pi", "def_path": "Mathlib/Data/Set/Prod.lean", "def_pos": [954, 9], "def_end_pos": [954, 37]}, {"full_name": "MeasureTheory.Measure.pi_pi", "def_path": "Mathlib/MeasureTheory/Constructions/Pi.lean", "def_pos": [394, 9], "def_end_pos": [394, 14]}]], "state_before": "\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : \u03b9 \u2192 Type u_3\ninst\u271d\u00b2 : Fintype \u03b9\nm\u271d : (i : \u03b9) \u2192 OuterMeasure (\u03b1 i)\nm : (i : \u03b9) \u2192 MeasurableSpace (\u03b1 i)\n\u03bc : (i : \u03b9) \u2192 Measure (\u03b1 i)\ninst\u271d\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc i)\ninst\u271d : Fintype \u03b9'\nf : \u03b9' \u2243 \u03b9\ns : (i : \u03b9) \u2192 Set (\u03b1 i)\nx\u271d : \u2200 (i : \u03b9), MeasurableSet (s i)\n\u22a2 \u2191\u2191(Measure.map (\u2191(MeasurableEquiv.piCongrLeft \u03b1 f)) (Measure.pi fun i' => \u03bc (\u2191f i'))) (Set.pi univ s) =\n    \u220f i : \u03b9, \u2191\u2191(\u03bc i) (s i)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/BinomialHeap/Basic.lean", "full_name": "Std.BinomialHeap.Imp.Heap.WF.singleton", "start": [349, 1], "end": [349, 79], "traced_tactics": [{"tactic": "decide", "annotated_tactic": ["decide", []], "state_before": "\u03b1\u271d : Type u_1\na : \u03b1\u271d\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\n\u22a2 0 \u2264 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "full_name": "MeasureTheory.integral_Ici_eq_integral_Ioi", "start": [699, 1], "end": [700, 55], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Function.lean", "full_name": "Set.restrict_piecewise_compl", "start": [112, 1], "end": [114, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "full_name": "MeasureTheory.FinStronglyMeasurable.neg", "start": [1080, 11], "end": [1085, 32], "traced_tactics": [{"tactic": "refine' \u27e8fun n => -hf.approx n, fun n => _, fun x => (hf.tendsto_approx x).neg\u27e9", "annotated_tactic": ["refine' \u27e8fun n => -hf.approx n, fun n => _, fun x => (hf.tendsto_approx x).<a>neg</a>\u27e9", [{"full_name": "Filter.Tendsto.neg", "def_path": "Mathlib/Topology/Algebra/Group/Basic.lean", "def_pos": [227, 3], "def_end_pos": [227, 14]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b3 : Countable \u03b9\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : AddGroup \u03b2\ninst\u271d : TopologicalAddGroup \u03b2\nhf : FinStronglyMeasurable f \u03bc\n\u22a2 FinStronglyMeasurable (-f) \u03bc", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b3 : Countable \u03b9\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : AddGroup \u03b2\ninst\u271d : TopologicalAddGroup \u03b2\nhf : FinStronglyMeasurable f \u03bc\nn : \u2115\n\u22a2 \u2191\u2191\u03bc (support \u2191((fun n => -FinStronglyMeasurable.approx hf n) n)) < \u22a4"}, {"tactic": "suffices \u03bc (Function.support fun x => -(hf.approx n) x) < \u221e by convert this", "annotated_tactic": ["suffices \u03bc (<a>Function.support</a> fun x => -(hf.approx n) x) < \u221e by convert this", [{"full_name": "Function.support", "def_path": "Mathlib/Algebra/Support.lean", "def_pos": [37, 5], "def_end_pos": [37, 12]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b3 : Countable \u03b9\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : AddGroup \u03b2\ninst\u271d : TopologicalAddGroup \u03b2\nhf : FinStronglyMeasurable f \u03bc\nn : \u2115\n\u22a2 \u2191\u2191\u03bc (support \u2191((fun n => -FinStronglyMeasurable.approx hf n) n)) < \u22a4", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b3 : Countable \u03b9\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : AddGroup \u03b2\ninst\u271d : TopologicalAddGroup \u03b2\nhf : FinStronglyMeasurable f \u03bc\nn : \u2115\n\u22a2 \u2191\u2191\u03bc (support fun x => -\u2191(FinStronglyMeasurable.approx hf n) x) < \u22a4"}, {"tactic": "rw [Function.support_neg (hf.approx n)]", "annotated_tactic": ["rw [<a>Function.support_neg</a> (hf.approx n)]", [{"full_name": "Function.support_neg", "def_path": "Mathlib/Algebra/Support.lean", "def_pos": [281, 3], "def_end_pos": [281, 14]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b3 : Countable \u03b9\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : AddGroup \u03b2\ninst\u271d : TopologicalAddGroup \u03b2\nhf : FinStronglyMeasurable f \u03bc\nn : \u2115\n\u22a2 \u2191\u2191\u03bc (support fun x => -\u2191(FinStronglyMeasurable.approx hf n) x) < \u22a4", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b3 : Countable \u03b9\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : AddGroup \u03b2\ninst\u271d : TopologicalAddGroup \u03b2\nhf : FinStronglyMeasurable f \u03bc\nn : \u2115\n\u22a2 \u2191\u2191\u03bc (support \u2191(FinStronglyMeasurable.approx hf n)) < \u22a4"}, {"tactic": "exact hf.fin_support_approx n", "annotated_tactic": ["exact hf.fin_support_approx n", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b3 : Countable \u03b9\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : AddGroup \u03b2\ninst\u271d : TopologicalAddGroup \u03b2\nhf : FinStronglyMeasurable f \u03bc\nn : \u2115\n\u22a2 \u2191\u2191\u03bc (support \u2191(FinStronglyMeasurable.approx hf n)) < \u22a4", "state_after": "no goals"}, {"tactic": "convert this", "annotated_tactic": ["convert this", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b3 : Countable \u03b9\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : AddGroup \u03b2\ninst\u271d : TopologicalAddGroup \u03b2\nhf : FinStronglyMeasurable f \u03bc\nn : \u2115\nthis : \u2191\u2191\u03bc (support fun x => -\u2191(FinStronglyMeasurable.approx hf n) x) < \u22a4\n\u22a2 \u2191\u2191\u03bc (support \u2191((fun n => -FinStronglyMeasurable.approx hf n) n)) < \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "full_name": "MeasureTheory.Lp.simpleFunc.toLp_toSimpleFunc", "start": [626, 1], "end": [628, 45], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Haar/OfBasis.lean", "full_name": "Basis.addHaar_eq_iff", "start": [236, 1], "end": [240, 49], "traced_tactics": [{"tactic": "rw [Basis.addHaar_def]", "annotated_tactic": ["rw [<a>Basis.addHaar_def</a>]", [{"full_name": "Basis.addHaar_def", "def_path": "Mathlib/MeasureTheory/Measure/Haar/OfBasis.lean", "def_pos": [220, 1], "def_end_pos": [223, 42]}]], "state_before": "\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u00b9\u2070 : Fintype \u03b9\ninst\u271d\u2079 : Fintype \u03b9'\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : MeasurableSpace E\ninst\u271d\u00b3 : BorelSpace E\ninst\u271d\u00b2 : SecondCountableTopology E\nb : Basis \u03b9 \u211d E\n\u03bc : Measure E\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : IsAddLeftInvariant \u03bc\n\u22a2 addHaar b = \u03bc \u2194 \u2191\u2191\u03bc \u2191(parallelepiped b) = 1", "state_after": "\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u00b9\u2070 : Fintype \u03b9\ninst\u271d\u2079 : Fintype \u03b9'\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : MeasurableSpace E\ninst\u271d\u00b3 : BorelSpace E\ninst\u271d\u00b2 : SecondCountableTopology E\nb : Basis \u03b9 \u211d E\n\u03bc : Measure E\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : IsAddLeftInvariant \u03bc\n\u22a2 addHaarMeasure (parallelepiped b) = \u03bc \u2194 \u2191\u2191\u03bc \u2191(parallelepiped b) = 1"}, {"tactic": "exact addHaarMeasure_eq_iff b.parallelepiped \u03bc", "annotated_tactic": ["exact <a>addHaarMeasure_eq_iff</a> b.parallelepiped \u03bc", [{"full_name": "MeasureTheory.Measure.addHaarMeasure_eq_iff", "def_path": "Mathlib/MeasureTheory/Measure/Haar/Basic.lean", "def_pos": [703, 3], "def_end_pos": [703, 14]}]], "state_before": "\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u00b9\u2070 : Fintype \u03b9\ninst\u271d\u2079 : Fintype \u03b9'\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : MeasurableSpace E\ninst\u271d\u00b3 : BorelSpace E\ninst\u271d\u00b2 : SecondCountableTopology E\nb : Basis \u03b9 \u211d E\n\u03bc : Measure E\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : IsAddLeftInvariant \u03bc\n\u22a2 addHaarMeasure (parallelepiped b) = \u03bc \u2194 \u2191\u2191\u03bc \u2191(parallelepiped b) = 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Regular.lean", "full_name": "MeasureTheory.Measure.InnerRegular.isCompact_isClosed", "start": [449, 1], "end": [460, 46], "traced_tactics": [{"tactic": "intro F hF r hr", "annotated_tactic": ["intro F hF r hr", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU s : Set \u03b1\n\u03b5 r : \u211d\u22650\u221e\nX : Type u_3\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : SigmaCompactSpace X\ninst\u271d : MeasurableSpace X\n\u03bc : Measure X\n\u22a2 InnerRegular \u03bc IsCompact IsClosed", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU s : Set \u03b1\n\u03b5 r\u271d : \u211d\u22650\u221e\nX : Type u_3\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : SigmaCompactSpace X\ninst\u271d : MeasurableSpace X\n\u03bc : Measure X\nF : Set X\nhF : IsClosed F\nr : \u211d\u22650\u221e\nhr : r < \u2191\u2191\u03bc F\n\u22a2 \u2203 K, K \u2286 F \u2227 IsCompact K \u2227 r < \u2191\u2191\u03bc K"}, {"tactic": "set B : \u2115 \u2192 Set X := compactCovering X", "annotated_tactic": ["set B : \u2115 \u2192 <a>Set</a> X := <a>compactCovering</a> X", [{"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}, {"full_name": "compactCovering", "def_path": "Mathlib/Topology/Compactness/SigmaCompact.lean", "def_pos": [197, 5], "def_end_pos": [197, 20]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU s : Set \u03b1\n\u03b5 r\u271d : \u211d\u22650\u221e\nX : Type u_3\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : SigmaCompactSpace X\ninst\u271d : MeasurableSpace X\n\u03bc : Measure X\nF : Set X\nhF : IsClosed F\nr : \u211d\u22650\u221e\nhr : r < \u2191\u2191\u03bc F\n\u22a2 \u2203 K, K \u2286 F \u2227 IsCompact K \u2227 r < \u2191\u2191\u03bc K", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU s : Set \u03b1\n\u03b5 r\u271d : \u211d\u22650\u221e\nX : Type u_3\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : SigmaCompactSpace X\ninst\u271d : MeasurableSpace X\n\u03bc : Measure X\nF : Set X\nhF : IsClosed F\nr : \u211d\u22650\u221e\nhr : r < \u2191\u2191\u03bc F\nB : \u2115 \u2192 Set X := compactCovering X\n\u22a2 \u2203 K, K \u2286 F \u2227 IsCompact K \u2227 r < \u2191\u2191\u03bc K"}, {"tactic": "have hBc : \u2200 n, IsCompact (F \u2229 B n) := fun n => (isCompact_compactCovering X n).inter_left hF", "annotated_tactic": ["have hBc : \u2200 n, <a>IsCompact</a> (F \u2229 B n) := fun n => (<a>isCompact_compactCovering</a> X n).<a>inter_left</a> hF", [{"full_name": "IsCompact", "def_path": "Mathlib/Topology/Compactness/Compact.lean", "def_pos": [40, 5], "def_end_pos": [40, 14]}, {"full_name": "isCompact_compactCovering", "def_path": "Mathlib/Topology/Compactness/SigmaCompact.lean", "def_pos": [201, 9], "def_end_pos": [201, 34]}, {"full_name": "IsCompact.inter_left", "def_path": "Mathlib/Topology/Compactness/Compact.lean", "def_pos": [91, 9], "def_end_pos": [91, 29]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU s : Set \u03b1\n\u03b5 r\u271d : \u211d\u22650\u221e\nX : Type u_3\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : SigmaCompactSpace X\ninst\u271d : MeasurableSpace X\n\u03bc : Measure X\nF : Set X\nhF : IsClosed F\nr : \u211d\u22650\u221e\nhr : r < \u2191\u2191\u03bc F\nB : \u2115 \u2192 Set X := compactCovering X\n\u22a2 \u2203 K, K \u2286 F \u2227 IsCompact K \u2227 r < \u2191\u2191\u03bc K", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU s : Set \u03b1\n\u03b5 r\u271d : \u211d\u22650\u221e\nX : Type u_3\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : SigmaCompactSpace X\ninst\u271d : MeasurableSpace X\n\u03bc : Measure X\nF : Set X\nhF : IsClosed F\nr : \u211d\u22650\u221e\nhr : r < \u2191\u2191\u03bc F\nB : \u2115 \u2192 Set X := compactCovering X\nhBc : \u2200 (n : \u2115), IsCompact (F \u2229 B n)\n\u22a2 \u2203 K, K \u2286 F \u2227 IsCompact K \u2227 r < \u2191\u2191\u03bc K"}, {"tactic": "have hBU : \u22c3 n, F \u2229 B n = F := by rw [\u2190 inter_iUnion, iUnion_compactCovering, Set.inter_univ]", "annotated_tactic": ["have hBU : \u22c3 n, F \u2229 B n = F := by rw [\u2190 <a>inter_iUnion</a>, <a>iUnion_compactCovering</a>, <a>Set.inter_univ</a>]", [{"full_name": "Set.inter_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [635, 9], "def_end_pos": [635, 21]}, {"full_name": "iUnion_compactCovering", "def_path": "Mathlib/Topology/Compactness/SigmaCompact.lean", "def_pos": [205, 9], "def_end_pos": [205, 31]}, {"full_name": "Set.inter_univ", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1012, 9], "def_end_pos": [1012, 19]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU s : Set \u03b1\n\u03b5 r\u271d : \u211d\u22650\u221e\nX : Type u_3\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : SigmaCompactSpace X\ninst\u271d : MeasurableSpace X\n\u03bc : Measure X\nF : Set X\nhF : IsClosed F\nr : \u211d\u22650\u221e\nhr : r < \u2191\u2191\u03bc F\nB : \u2115 \u2192 Set X := compactCovering X\nhBc : \u2200 (n : \u2115), IsCompact (F \u2229 B n)\n\u22a2 \u2203 K, K \u2286 F \u2227 IsCompact K \u2227 r < \u2191\u2191\u03bc K", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU s : Set \u03b1\n\u03b5 r\u271d : \u211d\u22650\u221e\nX : Type u_3\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : SigmaCompactSpace X\ninst\u271d : MeasurableSpace X\n\u03bc : Measure X\nF : Set X\nhF : IsClosed F\nr : \u211d\u22650\u221e\nhr : r < \u2191\u2191\u03bc F\nB : \u2115 \u2192 Set X := compactCovering X\nhBc : \u2200 (n : \u2115), IsCompact (F \u2229 B n)\nhBU : \u22c3 n, F \u2229 B n = F\n\u22a2 \u2203 K, K \u2286 F \u2227 IsCompact K \u2227 r < \u2191\u2191\u03bc K"}, {"tactic": "have : \u03bc F = \u2a06 n, \u03bc (F \u2229 B n) := by\n  rw [\u2190 measure_iUnion_eq_iSup, hBU]\n  exact Monotone.directed_le fun m n h => inter_subset_inter_right _ (compactCovering_subset _ h)", "annotated_tactic": ["have : \u03bc F = \u2a06 n, \u03bc (F \u2229 B n) := by\n    rw [\u2190 <a>measure_iUnion_eq_iSup</a>, hBU]\n    exact <a>Monotone.directed_le</a> fun m n h => <a>inter_subset_inter_right</a> _ (<a>compactCovering_subset</a> _ h)", [{"full_name": "MeasureTheory.measure_iUnion_eq_iSup", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [463, 9], "def_end_pos": [463, 31]}, {"full_name": "Monotone.directed_le", "def_path": "Mathlib/Order/Directed.lean", "def_pos": [112, 9], "def_end_pos": [112, 29]}, {"full_name": "Set.inter_subset_inter_right", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1032, 9], "def_end_pos": [1032, 33]}, {"full_name": "compactCovering_subset", "def_path": "Mathlib/Topology/Compactness/SigmaCompact.lean", "def_pos": [211, 9], "def_end_pos": [211, 31]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU s : Set \u03b1\n\u03b5 r\u271d : \u211d\u22650\u221e\nX : Type u_3\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : SigmaCompactSpace X\ninst\u271d : MeasurableSpace X\n\u03bc : Measure X\nF : Set X\nhF : IsClosed F\nr : \u211d\u22650\u221e\nhr : r < \u2191\u2191\u03bc F\nB : \u2115 \u2192 Set X := compactCovering X\nhBc : \u2200 (n : \u2115), IsCompact (F \u2229 B n)\nhBU : \u22c3 n, F \u2229 B n = F\n\u22a2 \u2203 K, K \u2286 F \u2227 IsCompact K \u2227 r < \u2191\u2191\u03bc K", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU s : Set \u03b1\n\u03b5 r\u271d : \u211d\u22650\u221e\nX : Type u_3\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : SigmaCompactSpace X\ninst\u271d : MeasurableSpace X\n\u03bc : Measure X\nF : Set X\nhF : IsClosed F\nr : \u211d\u22650\u221e\nhr : r < \u2191\u2191\u03bc F\nB : \u2115 \u2192 Set X := compactCovering X\nhBc : \u2200 (n : \u2115), IsCompact (F \u2229 B n)\nhBU : \u22c3 n, F \u2229 B n = F\nthis : \u2191\u2191\u03bc F = \u2a06 n, \u2191\u2191\u03bc (F \u2229 B n)\n\u22a2 \u2203 K, K \u2286 F \u2227 IsCompact K \u2227 r < \u2191\u2191\u03bc K"}, {"tactic": "rw [this] at hr", "annotated_tactic": ["rw [this] at hr", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU s : Set \u03b1\n\u03b5 r\u271d : \u211d\u22650\u221e\nX : Type u_3\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : SigmaCompactSpace X\ninst\u271d : MeasurableSpace X\n\u03bc : Measure X\nF : Set X\nhF : IsClosed F\nr : \u211d\u22650\u221e\nhr : r < \u2191\u2191\u03bc F\nB : \u2115 \u2192 Set X := compactCovering X\nhBc : \u2200 (n : \u2115), IsCompact (F \u2229 B n)\nhBU : \u22c3 n, F \u2229 B n = F\nthis : \u2191\u2191\u03bc F = \u2a06 n, \u2191\u2191\u03bc (F \u2229 B n)\n\u22a2 \u2203 K, K \u2286 F \u2227 IsCompact K \u2227 r < \u2191\u2191\u03bc K", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU s : Set \u03b1\n\u03b5 r\u271d : \u211d\u22650\u221e\nX : Type u_3\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : SigmaCompactSpace X\ninst\u271d : MeasurableSpace X\n\u03bc : Measure X\nF : Set X\nhF : IsClosed F\nr : \u211d\u22650\u221e\nB : \u2115 \u2192 Set X := compactCovering X\nhr : r < \u2a06 n, \u2191\u2191\u03bc (F \u2229 B n)\nhBc : \u2200 (n : \u2115), IsCompact (F \u2229 B n)\nhBU : \u22c3 n, F \u2229 B n = F\nthis : \u2191\u2191\u03bc F = \u2a06 n, \u2191\u2191\u03bc (F \u2229 B n)\n\u22a2 \u2203 K, K \u2286 F \u2227 IsCompact K \u2227 r < \u2191\u2191\u03bc K"}, {"tactic": "rcases lt_iSup_iff.1 hr with \u27e8n, hn\u27e9", "annotated_tactic": ["rcases <a>lt_iSup_iff</a>.1 hr with \u27e8n, hn\u27e9", [{"full_name": "lt_iSup_iff", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [668, 9], "def_end_pos": [668, 20]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU s : Set \u03b1\n\u03b5 r\u271d : \u211d\u22650\u221e\nX : Type u_3\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : SigmaCompactSpace X\ninst\u271d : MeasurableSpace X\n\u03bc : Measure X\nF : Set X\nhF : IsClosed F\nr : \u211d\u22650\u221e\nB : \u2115 \u2192 Set X := compactCovering X\nhr : r < \u2a06 n, \u2191\u2191\u03bc (F \u2229 B n)\nhBc : \u2200 (n : \u2115), IsCompact (F \u2229 B n)\nhBU : \u22c3 n, F \u2229 B n = F\nthis : \u2191\u2191\u03bc F = \u2a06 n, \u2191\u2191\u03bc (F \u2229 B n)\n\u22a2 \u2203 K, K \u2286 F \u2227 IsCompact K \u2227 r < \u2191\u2191\u03bc K", "state_after": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU s : Set \u03b1\n\u03b5 r\u271d : \u211d\u22650\u221e\nX : Type u_3\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : SigmaCompactSpace X\ninst\u271d : MeasurableSpace X\n\u03bc : Measure X\nF : Set X\nhF : IsClosed F\nr : \u211d\u22650\u221e\nB : \u2115 \u2192 Set X := compactCovering X\nhr : r < \u2a06 n, \u2191\u2191\u03bc (F \u2229 B n)\nhBc : \u2200 (n : \u2115), IsCompact (F \u2229 B n)\nhBU : \u22c3 n, F \u2229 B n = F\nthis : \u2191\u2191\u03bc F = \u2a06 n, \u2191\u2191\u03bc (F \u2229 B n)\nn : \u2115\nhn : r < \u2191\u2191\u03bc (F \u2229 B n)\n\u22a2 \u2203 K, K \u2286 F \u2227 IsCompact K \u2227 r < \u2191\u2191\u03bc K"}, {"tactic": "exact \u27e8_, inter_subset_left _ _, hBc n, hn\u27e9", "annotated_tactic": ["exact \u27e8_, <a>inter_subset_left</a> _ _, hBc n, hn\u27e9", [{"full_name": "Set.inter_subset_left", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [965, 9], "def_end_pos": [965, 26]}]], "state_before": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU s : Set \u03b1\n\u03b5 r\u271d : \u211d\u22650\u221e\nX : Type u_3\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : SigmaCompactSpace X\ninst\u271d : MeasurableSpace X\n\u03bc : Measure X\nF : Set X\nhF : IsClosed F\nr : \u211d\u22650\u221e\nB : \u2115 \u2192 Set X := compactCovering X\nhr : r < \u2a06 n, \u2191\u2191\u03bc (F \u2229 B n)\nhBc : \u2200 (n : \u2115), IsCompact (F \u2229 B n)\nhBU : \u22c3 n, F \u2229 B n = F\nthis : \u2191\u2191\u03bc F = \u2a06 n, \u2191\u2191\u03bc (F \u2229 B n)\nn : \u2115\nhn : r < \u2191\u2191\u03bc (F \u2229 B n)\n\u22a2 \u2203 K, K \u2286 F \u2227 IsCompact K \u2227 r < \u2191\u2191\u03bc K", "state_after": "no goals"}, {"tactic": "rw [\u2190 inter_iUnion, iUnion_compactCovering, Set.inter_univ]", "annotated_tactic": ["rw [\u2190 <a>inter_iUnion</a>, <a>iUnion_compactCovering</a>, <a>Set.inter_univ</a>]", [{"full_name": "Set.inter_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [635, 9], "def_end_pos": [635, 21]}, {"full_name": "iUnion_compactCovering", "def_path": "Mathlib/Topology/Compactness/SigmaCompact.lean", "def_pos": [205, 9], "def_end_pos": [205, 31]}, {"full_name": "Set.inter_univ", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1012, 9], "def_end_pos": [1012, 19]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU s : Set \u03b1\n\u03b5 r\u271d : \u211d\u22650\u221e\nX : Type u_3\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : SigmaCompactSpace X\ninst\u271d : MeasurableSpace X\n\u03bc : Measure X\nF : Set X\nhF : IsClosed F\nr : \u211d\u22650\u221e\nhr : r < \u2191\u2191\u03bc F\nB : \u2115 \u2192 Set X := compactCovering X\nhBc : \u2200 (n : \u2115), IsCompact (F \u2229 B n)\n\u22a2 \u22c3 n, F \u2229 B n = F", "state_after": "no goals"}, {"tactic": "rw [\u2190 measure_iUnion_eq_iSup, hBU]", "annotated_tactic": ["rw [\u2190 <a>measure_iUnion_eq_iSup</a>, hBU]", [{"full_name": "MeasureTheory.measure_iUnion_eq_iSup", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [463, 9], "def_end_pos": [463, 31]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU s : Set \u03b1\n\u03b5 r\u271d : \u211d\u22650\u221e\nX : Type u_3\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : SigmaCompactSpace X\ninst\u271d : MeasurableSpace X\n\u03bc : Measure X\nF : Set X\nhF : IsClosed F\nr : \u211d\u22650\u221e\nhr : r < \u2191\u2191\u03bc F\nB : \u2115 \u2192 Set X := compactCovering X\nhBc : \u2200 (n : \u2115), IsCompact (F \u2229 B n)\nhBU : \u22c3 n, F \u2229 B n = F\n\u22a2 \u2191\u2191\u03bc F = \u2a06 n, \u2191\u2191\u03bc (F \u2229 B n)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU s : Set \u03b1\n\u03b5 r\u271d : \u211d\u22650\u221e\nX : Type u_3\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : SigmaCompactSpace X\ninst\u271d : MeasurableSpace X\n\u03bc : Measure X\nF : Set X\nhF : IsClosed F\nr : \u211d\u22650\u221e\nhr : r < \u2191\u2191\u03bc F\nB : \u2115 \u2192 Set X := compactCovering X\nhBc : \u2200 (n : \u2115), IsCompact (F \u2229 B n)\nhBU : \u22c3 n, F \u2229 B n = F\n\u22a2 Directed (fun x x_1 => x \u2286 x_1) fun n => F \u2229 B n"}, {"tactic": "exact Monotone.directed_le fun m n h => inter_subset_inter_right _ (compactCovering_subset _ h)", "annotated_tactic": ["exact <a>Monotone.directed_le</a> fun m n h => <a>inter_subset_inter_right</a> _ (<a>compactCovering_subset</a> _ h)", [{"full_name": "Monotone.directed_le", "def_path": "Mathlib/Order/Directed.lean", "def_pos": [112, 9], "def_end_pos": [112, 29]}, {"full_name": "Set.inter_subset_inter_right", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1032, 9], "def_end_pos": [1032, 33]}, {"full_name": "compactCovering_subset", "def_path": "Mathlib/Topology/Compactness/SigmaCompact.lean", "def_pos": [211, 9], "def_end_pos": [211, 31]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU s : Set \u03b1\n\u03b5 r\u271d : \u211d\u22650\u221e\nX : Type u_3\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : SigmaCompactSpace X\ninst\u271d : MeasurableSpace X\n\u03bc : Measure X\nF : Set X\nhF : IsClosed F\nr : \u211d\u22650\u221e\nhr : r < \u2191\u2191\u03bc F\nB : \u2115 \u2192 Set X := compactCovering X\nhBc : \u2200 (n : \u2115), IsCompact (F \u2229 B n)\nhBU : \u22c3 n, F \u2229 B n = F\n\u22a2 Directed (fun x x_1 => x \u2286 x_1) fun n => F \u2229 B n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Opposite.lean", "full_name": "Opposite.equivToOpposite_coe", "start": [93, 1], "end": [94, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Image.lean", "full_name": "Sum.range_eq", "start": [1089, 1], "end": [1091, 26], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/ZMod/Basic.lean", "full_name": "ZMod.valMinAbs_natCast_eq_self", "start": [1137, 1], "end": [1140, 30], "traced_tactics": [{"tactic": "refine' \u27e8fun ha => _, valMinAbs_natCast_of_le_half\u27e9", "annotated_tactic": ["refine' \u27e8fun ha => _, <a>valMinAbs_natCast_of_le_half</a>\u27e9", [{"full_name": "ZMod.valMinAbs_natCast_of_le_half", "def_path": "Mathlib/Data/ZMod/Basic.lean", "def_pos": [1121, 9], "def_end_pos": [1121, 37]}]], "state_before": "n a : \u2115\ninst\u271d : NeZero n\n\u22a2 valMinAbs \u2191a = \u2191a \u2194 a \u2264 n / 2", "state_after": "n a : \u2115\ninst\u271d : NeZero n\nha : valMinAbs \u2191a = \u2191a\n\u22a2 a \u2264 n / 2"}, {"tactic": "rw [\u2190 Int.natAbs_ofNat a, \u2190 ha]", "annotated_tactic": ["rw [\u2190 <a>Int.natAbs_ofNat</a> a, \u2190 ha]", [{"full_name": "Int.natAbs_ofNat", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [138, 17], "def_end_pos": [138, 29]}]], "state_before": "n a : \u2115\ninst\u271d : NeZero n\nha : valMinAbs \u2191a = \u2191a\n\u22a2 a \u2264 n / 2", "state_after": "n a : \u2115\ninst\u271d : NeZero n\nha : valMinAbs \u2191a = \u2191a\n\u22a2 Int.natAbs (valMinAbs \u2191a) \u2264 n / 2"}, {"tactic": "exact natAbs_valMinAbs_le a", "annotated_tactic": ["exact <a>natAbs_valMinAbs_le</a> a", [{"full_name": "ZMod.natAbs_valMinAbs_le", "def_path": "Mathlib/Data/ZMod/Basic.lean", "def_pos": [1034, 9], "def_end_pos": [1034, 28]}]], "state_before": "n a : \u2115\ninst\u271d : NeZero n\nha : valMinAbs \u2191a = \u2191a\n\u22a2 Int.natAbs (valMinAbs \u2191a) \u2264 n / 2", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/TuringMachine.lean", "full_name": "Turing.TM2to1.tr_respects_aux\u2083", "start": [2667, 1], "end": [2675, 6], "traced_tactics": [{"tactic": "induction' n with n IH", "annotated_tactic": ["induction' n with n IH", []], "state_before": "K : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nq : Stmt\u2082\nv : \u03c3\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nn : \u2115\n\u22a2 Reaches\u2080 (TM1.step (tr M))\n    { l := some (ret q), var := v, Tape := (Tape.move Dir.right)^[n] (Tape.mk' \u2205 (addBottom L)) }\n    { l := some (ret q), var := v, Tape := Tape.mk' \u2205 (addBottom L) }", "state_after": "case zero\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nq : Stmt\u2082\nv : \u03c3\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\n\u22a2 Reaches\u2080 (TM1.step (tr M))\n    { l := some (ret q), var := v, Tape := (Tape.move Dir.right)^[Nat.zero] (Tape.mk' \u2205 (addBottom L)) }\n    { l := some (ret q), var := v, Tape := Tape.mk' \u2205 (addBottom L) }\n\ncase succ\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nq : Stmt\u2082\nv : \u03c3\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nn : \u2115\nIH :\n  Reaches\u2080 (TM1.step (tr M))\n    { l := some (ret q), var := v, Tape := (Tape.move Dir.right)^[n] (Tape.mk' \u2205 (addBottom L)) }\n    { l := some (ret q), var := v, Tape := Tape.mk' \u2205 (addBottom L) }\n\u22a2 Reaches\u2080 (TM1.step (tr M))\n    { l := some (ret q), var := v, Tape := (Tape.move Dir.right)^[Nat.succ n] (Tape.mk' \u2205 (addBottom L)) }\n    { l := some (ret q), var := v, Tape := Tape.mk' \u2205 (addBottom L) }"}, {"tactic": "refine' Reaches\u2080.head _ IH", "annotated_tactic": ["refine' <a>Reaches\u2080.head</a> _ IH", [{"full_name": "Turing.Reaches\u2080.head", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [792, 9], "def_end_pos": [792, 22]}]], "state_before": "case succ\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nq : Stmt\u2082\nv : \u03c3\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nn : \u2115\nIH :\n  Reaches\u2080 (TM1.step (tr M))\n    { l := some (ret q), var := v, Tape := (Tape.move Dir.right)^[n] (Tape.mk' \u2205 (addBottom L)) }\n    { l := some (ret q), var := v, Tape := Tape.mk' \u2205 (addBottom L) }\n\u22a2 Reaches\u2080 (TM1.step (tr M))\n    { l := some (ret q), var := v, Tape := (Tape.move Dir.right)^[Nat.succ n] (Tape.mk' \u2205 (addBottom L)) }\n    { l := some (ret q), var := v, Tape := Tape.mk' \u2205 (addBottom L) }", "state_after": "case succ\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nq : Stmt\u2082\nv : \u03c3\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nn : \u2115\nIH :\n  Reaches\u2080 (TM1.step (tr M))\n    { l := some (ret q), var := v, Tape := (Tape.move Dir.right)^[n] (Tape.mk' \u2205 (addBottom L)) }\n    { l := some (ret q), var := v, Tape := Tape.mk' \u2205 (addBottom L) }\n\u22a2 { l := some (ret q), var := v, Tape := (Tape.move Dir.right)^[n] (Tape.mk' \u2205 (addBottom L)) } \u2208\n    TM1.step (tr M)\n      { l := some (ret q), var := v, Tape := (Tape.move Dir.right)^[Nat.succ n] (Tape.mk' \u2205 (addBottom L)) }"}, {"tactic": "simp only [Option.mem_def, TM1.step]", "annotated_tactic": ["simp only [<a>Option.mem_def</a>, <a>TM1.step</a>]", [{"full_name": "Option.mem_def", "def_path": "lake-packages/std/Std/Data/Option/Basic.lean", "def_pos": [19, 17], "def_end_pos": [19, 24]}, {"full_name": "Turing.TM1.step", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1291, 5], "def_end_pos": [1291, 9]}]], "state_before": "case succ\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nq : Stmt\u2082\nv : \u03c3\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nn : \u2115\nIH :\n  Reaches\u2080 (TM1.step (tr M))\n    { l := some (ret q), var := v, Tape := (Tape.move Dir.right)^[n] (Tape.mk' \u2205 (addBottom L)) }\n    { l := some (ret q), var := v, Tape := Tape.mk' \u2205 (addBottom L) }\n\u22a2 { l := some (ret q), var := v, Tape := (Tape.move Dir.right)^[n] (Tape.mk' \u2205 (addBottom L)) } \u2208\n    TM1.step (tr M)\n      { l := some (ret q), var := v, Tape := (Tape.move Dir.right)^[Nat.succ n] (Tape.mk' \u2205 (addBottom L)) }", "state_after": "case succ\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nq : Stmt\u2082\nv : \u03c3\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nn : \u2115\nIH :\n  Reaches\u2080 (TM1.step (tr M))\n    { l := some (ret q), var := v, Tape := (Tape.move Dir.right)^[n] (Tape.mk' \u2205 (addBottom L)) }\n    { l := some (ret q), var := v, Tape := Tape.mk' \u2205 (addBottom L) }\n\u22a2 some (TM1.stepAux (tr M (ret q)) v ((Tape.move Dir.right)^[Nat.succ n] (Tape.mk' \u2205 (addBottom L)))) =\n    some { l := some (ret q), var := v, Tape := (Tape.move Dir.right)^[n] (Tape.mk' \u2205 (addBottom L)) }"}, {"tactic": "rw [Option.some_inj, tr, TM1.stepAux, Tape.move_right_n_head, Tape.mk'_nth_nat,\n  addBottom_nth_succ_fst, TM1.stepAux, iterate_succ', Function.comp_apply, Tape.move_right_left]", "annotated_tactic": ["rw [<a>Option.some_inj</a>, <a>tr</a>, <a>TM1.stepAux</a>, <a>Tape.move_right_n_head</a>, <a>Tape.mk'_nth_nat</a>,\n    <a>addBottom_nth_succ_fst</a>, <a>TM1.stepAux</a>, <a>iterate_succ'</a>, <a>Function.comp_apply</a>, <a>Tape.move_right_left</a>]", [{"full_name": "Option.some_inj", "def_path": "lake-packages/std/Std/Data/Option/Basic.lean", "def_pos": [27, 9], "def_end_pos": [27, 17]}, {"full_name": "Turing.TM2to1.tr", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [2637, 5], "def_end_pos": [2637, 7]}, {"full_name": "Turing.TM1.stepAux", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1281, 5], "def_end_pos": [1281, 12]}, {"full_name": "Turing.Tape.move_right_n_head", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [649, 9], "def_end_pos": [649, 31]}, {"full_name": "Turing.Tape.mk'_nth_nat", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [624, 9], "def_end_pos": [624, 25]}, {"full_name": "Turing.TM2to1.addBottom_nth_succ_fst", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [2391, 9], "def_end_pos": [2391, 31]}, {"full_name": "Turing.TM1.stepAux", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1281, 5], "def_end_pos": [1281, 12]}, {"full_name": "Function.iterate_succ'", "def_path": "Mathlib/Logic/Function/Iterate.lean", "def_pos": [186, 9], "def_end_pos": [186, 22]}, {"full_name": "Function.comp_apply", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [33, 17], "def_end_pos": [33, 36]}, {"full_name": "Turing.Tape.move_right_left", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [539, 9], "def_end_pos": [539, 29]}]], "state_before": "case succ\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nq : Stmt\u2082\nv : \u03c3\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nn : \u2115\nIH :\n  Reaches\u2080 (TM1.step (tr M))\n    { l := some (ret q), var := v, Tape := (Tape.move Dir.right)^[n] (Tape.mk' \u2205 (addBottom L)) }\n    { l := some (ret q), var := v, Tape := Tape.mk' \u2205 (addBottom L) }\n\u22a2 some (TM1.stepAux (tr M (ret q)) v ((Tape.move Dir.right)^[Nat.succ n] (Tape.mk' \u2205 (addBottom L)))) =\n    some { l := some (ret q), var := v, Tape := (Tape.move Dir.right)^[n] (Tape.mk' \u2205 (addBottom L)) }", "state_after": "case succ\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nq : Stmt\u2082\nv : \u03c3\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nn : \u2115\nIH :\n  Reaches\u2080 (TM1.step (tr M))\n    { l := some (ret q), var := v, Tape := (Tape.move Dir.right)^[n] (Tape.mk' \u2205 (addBottom L)) }\n    { l := some (ret q), var := v, Tape := Tape.mk' \u2205 (addBottom L) }\n\u22a2 (bif false then\n      TM1.stepAux (trNormal q) v (Tape.move Dir.right ((Tape.move Dir.right)^[n] (Tape.mk' \u2205 (addBottom L))))\n    else TM1.stepAux (goto fun x x => ret q) v ((Tape.move Dir.right)^[n] (Tape.mk' \u2205 (addBottom L)))) =\n    { l := some (ret q), var := v, Tape := (Tape.move Dir.right)^[n] (Tape.mk' \u2205 (addBottom L)) }"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case succ\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nq : Stmt\u2082\nv : \u03c3\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nn : \u2115\nIH :\n  Reaches\u2080 (TM1.step (tr M))\n    { l := some (ret q), var := v, Tape := (Tape.move Dir.right)^[n] (Tape.mk' \u2205 (addBottom L)) }\n    { l := some (ret q), var := v, Tape := Tape.mk' \u2205 (addBottom L) }\n\u22a2 (bif false then\n      TM1.stepAux (trNormal q) v (Tape.move Dir.right ((Tape.move Dir.right)^[n] (Tape.mk' \u2205 (addBottom L))))\n    else TM1.stepAux (goto fun x x => ret q) v ((Tape.move Dir.right)^[n] (Tape.mk' \u2205 (addBottom L)))) =\n    { l := some (ret q), var := v, Tape := (Tape.move Dir.right)^[n] (Tape.mk' \u2205 (addBottom L)) }", "state_after": "no goals"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case zero\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nq : Stmt\u2082\nv : \u03c3\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\n\u22a2 Reaches\u2080 (TM1.step (tr M))\n    { l := some (ret q), var := v, Tape := (Tape.move Dir.right)^[Nat.zero] (Tape.mk' \u2205 (addBottom L)) }\n    { l := some (ret q), var := v, Tape := Tape.mk' \u2205 (addBottom L) }", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Prod/TProd.lean", "full_name": "Set.elim_preimage_pi", "start": [173, 1], "end": [180, 13], "traced_tactics": [{"tactic": "have h2 : { i | i \u2208 l } = univ := by\n  ext i\n  simp [h]", "annotated_tactic": ["have h2 : { i | i \u2208 l } = <a>univ</a> := by\n    ext i\n    simp [h]", [{"full_name": "Set.univ", "def_path": "Mathlib/Init/Set.lean", "def_pos": [90, 5], "def_end_pos": [90, 9]}]], "state_before": "\u03b9 : Type u\n\u03b1 : \u03b9 \u2192 Type v\ni j : \u03b9\nl\u271d : List \u03b9\nf : (i : \u03b9) \u2192 \u03b1 i\ninst\u271d : DecidableEq \u03b9\nl : List \u03b9\nhnd : Nodup l\nh : \u2200 (i : \u03b9), i \u2208 l\nt : (i : \u03b9) \u2192 Set (\u03b1 i)\n\u22a2 TProd.elim' h \u207b\u00b9' pi univ t = Set.tprod l t", "state_after": "\u03b9 : Type u\n\u03b1 : \u03b9 \u2192 Type v\ni j : \u03b9\nl\u271d : List \u03b9\nf : (i : \u03b9) \u2192 \u03b1 i\ninst\u271d : DecidableEq \u03b9\nl : List \u03b9\nhnd : Nodup l\nh : \u2200 (i : \u03b9), i \u2208 l\nt : (i : \u03b9) \u2192 Set (\u03b1 i)\nh2 : {i | i \u2208 l} = univ\n\u22a2 TProd.elim' h \u207b\u00b9' pi univ t = Set.tprod l t"}, {"tactic": "rw [\u2190 h2, \u2190 mk_preimage_tprod, preimage_preimage]", "annotated_tactic": ["rw [\u2190 h2, \u2190 <a>mk_preimage_tprod</a>, <a>preimage_preimage</a>]", [{"full_name": "Set.mk_preimage_tprod", "def_path": "Mathlib/Data/Prod/TProd.lean", "def_pos": [157, 9], "def_end_pos": [157, 26]}, {"full_name": "Set.preimage_preimage", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [176, 9], "def_end_pos": [176, 26]}]], "state_before": "\u03b9 : Type u\n\u03b1 : \u03b9 \u2192 Type v\ni j : \u03b9\nl\u271d : List \u03b9\nf : (i : \u03b9) \u2192 \u03b1 i\ninst\u271d : DecidableEq \u03b9\nl : List \u03b9\nhnd : Nodup l\nh : \u2200 (i : \u03b9), i \u2208 l\nt : (i : \u03b9) \u2192 Set (\u03b1 i)\nh2 : {i | i \u2208 l} = univ\n\u22a2 TProd.elim' h \u207b\u00b9' pi univ t = Set.tprod l t", "state_after": "\u03b9 : Type u\n\u03b1 : \u03b9 \u2192 Type v\ni j : \u03b9\nl\u271d : List \u03b9\nf : (i : \u03b9) \u2192 \u03b1 i\ninst\u271d : DecidableEq \u03b9\nl : List \u03b9\nhnd : Nodup l\nh : \u2200 (i : \u03b9), i \u2208 l\nt : (i : \u03b9) \u2192 Set (\u03b1 i)\nh2 : {i | i \u2208 l} = univ\n\u22a2 (fun x => TProd.mk l (TProd.elim' h x)) \u207b\u00b9' Set.tprod l t = Set.tprod l t"}, {"tactic": "simp only [TProd.mk_elim hnd h]", "annotated_tactic": ["simp only [<a>TProd.mk_elim</a> hnd h]", [{"full_name": "List.TProd.mk_elim", "def_path": "Mathlib/Data/Prod/TProd.lean", "def_pos": [133, 9], "def_end_pos": [133, 16]}]], "state_before": "\u03b9 : Type u\n\u03b1 : \u03b9 \u2192 Type v\ni j : \u03b9\nl\u271d : List \u03b9\nf : (i : \u03b9) \u2192 \u03b1 i\ninst\u271d : DecidableEq \u03b9\nl : List \u03b9\nhnd : Nodup l\nh : \u2200 (i : \u03b9), i \u2208 l\nt : (i : \u03b9) \u2192 Set (\u03b1 i)\nh2 : {i | i \u2208 l} = univ\n\u22a2 (fun x => TProd.mk l (TProd.elim' h x)) \u207b\u00b9' Set.tprod l t = Set.tprod l t", "state_after": "\u03b9 : Type u\n\u03b1 : \u03b9 \u2192 Type v\ni j : \u03b9\nl\u271d : List \u03b9\nf : (i : \u03b9) \u2192 \u03b1 i\ninst\u271d : DecidableEq \u03b9\nl : List \u03b9\nhnd : Nodup l\nh : \u2200 (i : \u03b9), i \u2208 l\nt : (i : \u03b9) \u2192 Set (\u03b1 i)\nh2 : {i | i \u2208 l} = univ\n\u22a2 (fun x => x) \u207b\u00b9' Set.tprod l t = Set.tprod l t"}, {"tactic": "dsimp", "annotated_tactic": ["dsimp", []], "state_before": "\u03b9 : Type u\n\u03b1 : \u03b9 \u2192 Type v\ni j : \u03b9\nl\u271d : List \u03b9\nf : (i : \u03b9) \u2192 \u03b1 i\ninst\u271d : DecidableEq \u03b9\nl : List \u03b9\nhnd : Nodup l\nh : \u2200 (i : \u03b9), i \u2208 l\nt : (i : \u03b9) \u2192 Set (\u03b1 i)\nh2 : {i | i \u2208 l} = univ\n\u22a2 (fun x => x) \u207b\u00b9' Set.tprod l t = Set.tprod l t", "state_after": "\u03b9 : Type u\n\u03b1 : \u03b9 \u2192 Type v\ni j : \u03b9\nl\u271d : List \u03b9\nf : (i : \u03b9) \u2192 \u03b1 i\ninst\u271d : DecidableEq \u03b9\nl : List \u03b9\nhnd : Nodup l\nh : \u2200 (i : \u03b9), i \u2208 l\nt : (i : \u03b9) \u2192 Set (\u03b1 i)\nh2 : {i | i \u2208 l} = univ\n\u22a2 Set.tprod l t = Set.tprod l t"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\u03b9 : Type u\n\u03b1 : \u03b9 \u2192 Type v\ni j : \u03b9\nl\u271d : List \u03b9\nf : (i : \u03b9) \u2192 \u03b1 i\ninst\u271d : DecidableEq \u03b9\nl : List \u03b9\nhnd : Nodup l\nh : \u2200 (i : \u03b9), i \u2208 l\nt : (i : \u03b9) \u2192 Set (\u03b1 i)\nh2 : {i | i \u2208 l} = univ\n\u22a2 Set.tprod l t = Set.tprod l t", "state_after": "no goals"}, {"tactic": "ext i", "annotated_tactic": ["ext i", []], "state_before": "\u03b9 : Type u\n\u03b1 : \u03b9 \u2192 Type v\ni j : \u03b9\nl\u271d : List \u03b9\nf : (i : \u03b9) \u2192 \u03b1 i\ninst\u271d : DecidableEq \u03b9\nl : List \u03b9\nhnd : Nodup l\nh : \u2200 (i : \u03b9), i \u2208 l\nt : (i : \u03b9) \u2192 Set (\u03b1 i)\n\u22a2 {i | i \u2208 l} = univ", "state_after": "case h\n\u03b9 : Type u\n\u03b1 : \u03b9 \u2192 Type v\ni\u271d j : \u03b9\nl\u271d : List \u03b9\nf : (i : \u03b9) \u2192 \u03b1 i\ninst\u271d : DecidableEq \u03b9\nl : List \u03b9\nhnd : Nodup l\nh : \u2200 (i : \u03b9), i \u2208 l\nt : (i : \u03b9) \u2192 Set (\u03b1 i)\ni : \u03b9\n\u22a2 i \u2208 {i | i \u2208 l} \u2194 i \u2208 univ"}, {"tactic": "simp [h]", "annotated_tactic": ["simp [h]", []], "state_before": "case h\n\u03b9 : Type u\n\u03b1 : \u03b9 \u2192 Type v\ni\u271d j : \u03b9\nl\u271d : List \u03b9\nf : (i : \u03b9) \u2192 \u03b1 i\ninst\u271d : DecidableEq \u03b9\nl : List \u03b9\nhnd : Nodup l\nh : \u2200 (i : \u03b9), i \u2208 l\nt : (i : \u03b9) \u2192 Set (\u03b1 i)\ni : \u03b9\n\u22a2 i \u2208 {i | i \u2208 l} \u2194 i \u2208 univ", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Reduce.lean", "full_name": "Computable.equiv\u2082", "start": [225, 1], "end": [227, 51], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Lattice.lean", "full_name": "Finset.measurable_range_sup''", "start": [256, 1], "end": [260, 7], "traced_tactics": [{"tactic": "convert Finset.measurable_range_sup' hf using 1", "annotated_tactic": ["convert <a>Finset.measurable_range_sup'</a> hf using 1", [{"full_name": "Finset.measurable_range_sup'", "def_path": "Mathlib/MeasureTheory/Lattice.lean", "def_pos": [248, 9], "def_end_pos": [248, 37]}]], "state_before": "M : Type u_1\ninst\u271d\u00b3 : MeasurableSpace M\n\u03b1 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g : \u03b1 \u2192 M\n\u03b4 : Type u_3\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : SemilatticeSup \u03b1\ninst\u271d : MeasurableSup\u2082 \u03b1\nf : \u2115 \u2192 \u03b4 \u2192 \u03b1\nn : \u2115\nhf : \u2200 (k : \u2115), k \u2264 n \u2192 Measurable (f k)\n\u22a2 Measurable fun x => sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k x", "state_after": "case h.e'_5\nM : Type u_1\ninst\u271d\u00b3 : MeasurableSpace M\n\u03b1 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g : \u03b1 \u2192 M\n\u03b4 : Type u_3\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : SemilatticeSup \u03b1\ninst\u271d : MeasurableSup\u2082 \u03b1\nf : \u2115 \u2192 \u03b4 \u2192 \u03b1\nn : \u2115\nhf : \u2200 (k : \u2115), k \u2264 n \u2192 Measurable (f k)\n\u22a2 (fun x => sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k x) =\n    sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k"}, {"tactic": "ext x", "annotated_tactic": ["ext x", []], "state_before": "case h.e'_5\nM : Type u_1\ninst\u271d\u00b3 : MeasurableSpace M\n\u03b1 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g : \u03b1 \u2192 M\n\u03b4 : Type u_3\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : SemilatticeSup \u03b1\ninst\u271d : MeasurableSup\u2082 \u03b1\nf : \u2115 \u2192 \u03b4 \u2192 \u03b1\nn : \u2115\nhf : \u2200 (k : \u2115), k \u2264 n \u2192 Measurable (f k)\n\u22a2 (fun x => sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k x) =\n    sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k", "state_after": "case h.e'_5.h\nM : Type u_1\ninst\u271d\u00b3 : MeasurableSpace M\n\u03b1 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g : \u03b1 \u2192 M\n\u03b4 : Type u_3\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : SemilatticeSup \u03b1\ninst\u271d : MeasurableSup\u2082 \u03b1\nf : \u2115 \u2192 \u03b4 \u2192 \u03b1\nn : \u2115\nhf : \u2200 (k : \u2115), k \u2264 n \u2192 Measurable (f k)\nx : \u03b4\n\u22a2 (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k x) =\n    sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) (fun k => f k) x"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case h.e'_5.h\nM : Type u_1\ninst\u271d\u00b3 : MeasurableSpace M\n\u03b1 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g : \u03b1 \u2192 M\n\u03b4 : Type u_3\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : SemilatticeSup \u03b1\ninst\u271d : MeasurableSup\u2082 \u03b1\nf : \u2115 \u2192 \u03b4 \u2192 \u03b1\nn : \u2115\nhf : \u2200 (k : \u2115), k \u2264 n \u2192 Measurable (f k)\nx : \u03b4\n\u22a2 (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k x) =\n    sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) (fun k => f k) x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/LocallyFinite.lean", "full_name": "Finset.nonempty_uIcc", "start": [938, 1], "end": [939, 28], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "full_name": "String.Pos.zero_addString_byteIdx", "start": [143, 1], "end": [144, 60], "traced_tactics": [{"tactic": "simp only [addString_byteIdx, byteIdx_zero, Nat.zero_add]", "annotated_tactic": ["simp only [<a>addString_byteIdx</a>, <a>byteIdx_zero</a>, <a>Nat.zero_add</a>]", [{"full_name": "String.Pos.addString_byteIdx", "def_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "def_pos": [138, 17], "def_end_pos": [138, 34]}, {"full_name": "String.Pos.byteIdx_zero", "def_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "def_pos": [97, 17], "def_end_pos": [97, 29]}, {"full_name": "Nat.zero_add", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [114, 27], "def_end_pos": [114, 35]}]], "state_before": "s : String\n\u22a2 (0 + s).byteIdx = utf8ByteSize s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Kernel/WithDensity.lean", "full_name": "ProbabilityTheory.kernel.withDensity_kernel_sum", "start": [102, 1], "end": [111, 24], "traced_tactics": [{"tactic": "by_cases hf : Measurable (Function.uncurry f)", "annotated_tactic": ["by_cases hf : <a>Measurable</a> (<a>Function.uncurry</a> f)", [{"full_name": "Measurable", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [535, 5], "def_end_pos": [535, 15]}, {"full_name": "Function.uncurry", "def_path": "Mathlib/Init/Function.lean", "def_pos": [217, 5], "def_end_pos": [217, 12]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\ninst\u271d : Countable \u03b9\n\u03ba : \u03b9 \u2192 { x // x \u2208 kernel \u03b1 \u03b2 }\nh\u03ba : \u2200 (i : \u03b9), IsSFiniteKernel (\u03ba i)\nf : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\n\u22a2 withDensity (kernel.sum \u03ba) f = kernel.sum fun i => withDensity (\u03ba i) f", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\ninst\u271d : Countable \u03b9\n\u03ba : \u03b9 \u2192 { x // x \u2208 kernel \u03b1 \u03b2 }\nh\u03ba : \u2200 (i : \u03b9), IsSFiniteKernel (\u03ba i)\nf : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\nhf : Measurable (Function.uncurry f)\n\u22a2 withDensity (kernel.sum \u03ba) f = kernel.sum fun i => withDensity (\u03ba i) f\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\ninst\u271d : Countable \u03b9\n\u03ba : \u03b9 \u2192 { x // x \u2208 kernel \u03b1 \u03b2 }\nh\u03ba : \u2200 (i : \u03b9), IsSFiniteKernel (\u03ba i)\nf : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\nhf : \u00acMeasurable (Function.uncurry f)\n\u22a2 withDensity (kernel.sum \u03ba) f = kernel.sum fun i => withDensity (\u03ba i) f"}, {"tactic": "ext1 a", "annotated_tactic": ["ext1 a", []], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\ninst\u271d : Countable \u03b9\n\u03ba : \u03b9 \u2192 { x // x \u2208 kernel \u03b1 \u03b2 }\nh\u03ba : \u2200 (i : \u03b9), IsSFiniteKernel (\u03ba i)\nf : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\nhf : Measurable (Function.uncurry f)\n\u22a2 withDensity (kernel.sum \u03ba) f = kernel.sum fun i => withDensity (\u03ba i) f", "state_after": "case pos.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\ninst\u271d : Countable \u03b9\n\u03ba : \u03b9 \u2192 { x // x \u2208 kernel \u03b1 \u03b2 }\nh\u03ba : \u2200 (i : \u03b9), IsSFiniteKernel (\u03ba i)\nf : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\nhf : Measurable (Function.uncurry f)\na : \u03b1\n\u22a2 \u2191(withDensity (kernel.sum \u03ba) f) a = \u2191(kernel.sum fun i => withDensity (\u03ba i) f) a"}, {"tactic": "simp_rw [sum_apply, kernel.withDensity_apply _ hf, sum_apply,\n  withDensity_sum (fun n => \u03ba n a) (f a)]", "annotated_tactic": ["simp_rw [<a>sum_apply</a>, <a>kernel.withDensity_apply</a> _ hf, <a>sum_apply</a>,\n      <a>withDensity_sum</a> (fun n => \u03ba n a) (f a)]", [{"full_name": "ProbabilityTheory.kernel.sum_apply", "def_path": "Mathlib/Probability/Kernel/Basic.lean", "def_pos": [239, 9], "def_end_pos": [239, 18]}, {"full_name": "ProbabilityTheory.kernel.withDensity_apply", "def_path": "Mathlib/Probability/Kernel/WithDensity.lean", "def_pos": [61, 19], "def_end_pos": [61, 36]}, {"full_name": "ProbabilityTheory.kernel.sum_apply", "def_path": "Mathlib/Probability/Kernel/Basic.lean", "def_pos": [239, 9], "def_end_pos": [239, 18]}, {"full_name": "MeasureTheory.withDensity_sum", "def_path": "Mathlib/MeasureTheory/Measure/WithDensity.lean", "def_pos": [83, 9], "def_end_pos": [83, 24]}]], "state_before": "case pos.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\ninst\u271d : Countable \u03b9\n\u03ba : \u03b9 \u2192 { x // x \u2208 kernel \u03b1 \u03b2 }\nh\u03ba : \u2200 (i : \u03b9), IsSFiniteKernel (\u03ba i)\nf : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\nhf : Measurable (Function.uncurry f)\na : \u03b1\n\u22a2 \u2191(withDensity (kernel.sum \u03ba) f) a = \u2191(kernel.sum fun i => withDensity (\u03ba i) f) a", "state_after": "no goals"}, {"tactic": "simp_rw [withDensity_of_not_measurable _ hf]", "annotated_tactic": ["simp_rw [<a>withDensity_of_not_measurable</a> _ hf]", [{"full_name": "ProbabilityTheory.kernel.withDensity_of_not_measurable", "def_path": "Mathlib/Probability/Kernel/WithDensity.lean", "def_pos": [57, 9], "def_end_pos": [57, 38]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\ninst\u271d : Countable \u03b9\n\u03ba : \u03b9 \u2192 { x // x \u2208 kernel \u03b1 \u03b2 }\nh\u03ba : \u2200 (i : \u03b9), IsSFiniteKernel (\u03ba i)\nf : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\nhf : \u00acMeasurable (Function.uncurry f)\n\u22a2 withDensity (kernel.sum \u03ba) f = kernel.sum fun i => withDensity (\u03ba i) f", "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\ninst\u271d : Countable \u03b9\n\u03ba : \u03b9 \u2192 { x // x \u2208 kernel \u03b1 \u03b2 }\nh\u03ba : \u2200 (i : \u03b9), IsSFiniteKernel (\u03ba i)\nf : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\nhf : \u00acMeasurable (Function.uncurry f)\n\u22a2 0 = kernel.sum fun i => 0"}, {"tactic": "exact sum_zero.symm", "annotated_tactic": ["exact sum_zero.symm", []], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\ninst\u271d : Countable \u03b9\n\u03ba : \u03b9 \u2192 { x // x \u2208 kernel \u03b1 \u03b2 }\nh\u03ba : \u2200 (i : \u03b9), IsSFiniteKernel (\u03ba i)\nf : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\nhf : \u00acMeasurable (Function.uncurry f)\n\u22a2 0 = kernel.sum fun i => 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/NoncommProd.lean", "full_name": "Multiset.noncommProd_map", "start": [188, 1], "end": [191, 32], "traced_tactics": [{"tactic": "induction s using Quotient.inductionOn", "annotated_tactic": ["induction s using <a>Quotient.inductionOn</a>", [{"full_name": "Quotient.inductionOn", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [1367, 19], "def_end_pos": [1367, 30]}]], "state_before": "F : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b2 : Monoid \u03b1\ninst\u271d\u00b9 : Monoid \u03b2\ninst\u271d : MonoidHomClass F \u03b1 \u03b2\ns : Multiset \u03b1\ncomm : Set.Pairwise {x | x \u2208 s} Commute\nf : F\n\u22a2 \u2191f (noncommProd s comm) = noncommProd (map (\u2191f) s) (_ : Set.Pairwise {x | x \u2208 map (\u2191f) s} Commute)", "state_after": "case h\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b2 : Monoid \u03b1\ninst\u271d\u00b9 : Monoid \u03b2\ninst\u271d : MonoidHomClass F \u03b1 \u03b2\nf : F\na\u271d : List \u03b1\ncomm : Set.Pairwise {x | x \u2208 Quotient.mk (List.isSetoid \u03b1) a\u271d} Commute\n\u22a2 \u2191f (noncommProd (Quotient.mk (List.isSetoid \u03b1) a\u271d) comm) =\n    noncommProd (map (\u2191f) (Quotient.mk (List.isSetoid \u03b1) a\u271d))\n      (_ : Set.Pairwise {x | x \u2208 map (\u2191f) (Quotient.mk (List.isSetoid \u03b1) a\u271d)} Commute)"}, {"tactic": "simpa using map_list_prod f _", "annotated_tactic": ["simpa using <a>map_list_prod</a> f _", [{"full_name": "map_list_prod", "def_path": "Mathlib/Data/List/BigOperators/Basic.lean", "def_pos": [690, 9], "def_end_pos": [690, 22]}]], "state_before": "case h\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b2 : Monoid \u03b1\ninst\u271d\u00b9 : Monoid \u03b2\ninst\u271d : MonoidHomClass F \u03b1 \u03b2\nf : F\na\u271d : List \u03b1\ncomm : Set.Pairwise {x | x \u2208 Quotient.mk (List.isSetoid \u03b1) a\u271d} Commute\n\u22a2 \u2191f (noncommProd (Quotient.mk (List.isSetoid \u03b1) a\u271d) comm) =\n    noncommProd (map (\u2191f) (Quotient.mk (List.isSetoid \u03b1) a\u271d))\n      (_ : Set.Pairwise {x | x \u2208 map (\u2191f) (Quotient.mk (List.isSetoid \u03b1) a\u271d)} Commute)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Vector/Basic.lean", "full_name": "Vector.get_set_eq_if", "start": [636, 1], "end": [638, 52], "traced_tactics": [{"tactic": "split_ifs <;> (try simp [*])", "annotated_tactic": ["split_ifs <;> (try simp [*])", []], "state_before": "n : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nv : Vector \u03b1 n\ni j : Fin n\na : \u03b1\n\u22a2 get (set v i a) j = if i = j then a else get v j", "state_after": "case neg\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nv : Vector \u03b1 n\ni j : Fin n\na : \u03b1\nh\u271d : \u00aci = j\n\u22a2 get (set v i a) j = get v j"}, {"tactic": "rwa [get_set_of_ne]", "annotated_tactic": ["rwa [<a>get_set_of_ne</a>]", [{"full_name": "Vector.get_set_of_ne", "def_path": "Mathlib/Data/Vector/Basic.lean", "def_pos": [628, 9], "def_end_pos": [628, 22]}]], "state_before": "case neg\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nv : Vector \u03b1 n\ni j : Fin n\na : \u03b1\nh\u271d : \u00aci = j\n\u22a2 get (set v i a) j = get v j", "state_after": "no goals"}, {"tactic": "try simp [*]", "annotated_tactic": ["try simp [*]", []], "state_before": "case pos\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nv : Vector \u03b1 n\ni j : Fin n\na : \u03b1\nh\u271d : i = j\n\u22a2 get (set v i a) j = a", "state_after": "no goals"}, {"tactic": "simp [*]", "annotated_tactic": ["simp [*]", []], "state_before": "case pos\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nv : Vector \u03b1 n\ni j : Fin n\na : \u03b1\nh\u271d : i = j\n\u22a2 get (set v i a) j = a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "full_name": "MeasureTheory.VectorMeasure.AbsolutelyContinuous.refl", "start": [1089, 1], "end": [1090, 9], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/Basic.lean", "full_name": "MvPolynomial.constantCoeff_comp_algebraMap", "start": [928, 1], "end": [930, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "full_name": "Nat.lt_log2_self", "start": [858, 1], "end": [858, 89], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/NullMeasurable.lean", "full_name": "MeasureTheory.measure_union\u2080_aux", "start": [292, 1], "end": [295, 87], "traced_tactics": [{"tactic": "rw [union_eq_iUnion, measure_iUnion\u2080, tsum_fintype, Fintype.sum_bool, cond, cond]", "annotated_tactic": ["rw [<a>union_eq_iUnion</a>, <a>measure_iUnion\u2080</a>, <a>tsum_fintype</a>, <a>Fintype.sum_bool</a>, <a>cond</a>, <a>cond</a>]", [{"full_name": "Set.union_eq_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [1478, 9], "def_end_pos": [1478, 24]}, {"full_name": "MeasureTheory.measure_iUnion\u2080", "def_path": "Mathlib/MeasureTheory/Measure/NullMeasurable.lean", "def_pos": [282, 9], "def_end_pos": [282, 24]}, {"full_name": "tsum_fintype", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/Basic.lean", "def_pos": [503, 9], "def_end_pos": [503, 21]}, {"full_name": "Fintype.sum_bool", "def_path": "Mathlib/Data/Fintype/BigOperators.lean", "def_pos": [38, 3], "def_end_pos": [38, 14]}, {"full_name": "cond", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [971, 21], "def_end_pos": [971, 25]}, {"full_name": "cond", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [971, 21], "def_end_pos": [971, 25]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhs : NullMeasurableSet s\nht : NullMeasurableSet t\nhd : AEDisjoint \u03bc s t\n\u22a2 \u2191\u2191\u03bc (s \u222a t) = \u2191\u2191\u03bc s + \u2191\u2191\u03bc t", "state_after": "case hd\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhs : NullMeasurableSet s\nht : NullMeasurableSet t\nhd : AEDisjoint \u03bc s t\n\u22a2 Pairwise (AEDisjoint \u03bc on fun b => bif b then s else t)\n\ncase h\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhs : NullMeasurableSet s\nht : NullMeasurableSet t\nhd : AEDisjoint \u03bc s t\n\u22a2 \u2200 (i : Bool), NullMeasurableSet (bif i then s else t)"}, {"tactic": "exacts [(pairwise_on_bool AEDisjoint.symmetric).2 hd, fun b => Bool.casesOn b ht hs]", "annotated_tactic": ["exacts [(<a>pairwise_on_bool</a> <a>AEDisjoint.symmetric</a>).2 hd, fun b => <a>Bool.casesOn</a> b ht hs]", [{"full_name": "pairwise_on_bool", "def_path": "Mathlib/Data/Set/Pairwise/Basic.lean", "def_pos": [41, 9], "def_end_pos": [41, 25]}, {"full_name": "MeasureTheory.AEDisjoint.symmetric", "def_path": "Mathlib/MeasureTheory/Measure/AEDisjoint.lean", "def_pos": [58, 19], "def_end_pos": [58, 28]}, {"full_name": "Bool.casesOn", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [545, 11], "def_end_pos": [545, 15]}]], "state_before": "case hd\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhs : NullMeasurableSet s\nht : NullMeasurableSet t\nhd : AEDisjoint \u03bc s t\n\u22a2 Pairwise (AEDisjoint \u03bc on fun b => bif b then s else t)\n\ncase h\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhs : NullMeasurableSet s\nht : NullMeasurableSet t\nhd : AEDisjoint \u03bc s t\n\u22a2 \u2200 (i : Bool), NullMeasurableSet (bif i then s else t)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/AEMeasurable.lean", "full_name": "aemeasurable_restrict_of_measurable_subtype", "start": [301, 1], "end": [303, 65], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Kernel/Disintegration.lean", "full_name": "ProbabilityTheory.lintegral_condKernelReal_mem", "start": [136, 1], "end": [208, 64], "traced_tactics": [{"tactic": "apply MeasurableSpace.induction_on_inter generateFrom_prod.symm isPiSystem_prod _ _ _ _ hs", "annotated_tactic": ["apply <a>MeasurableSpace.induction_on_inter</a> generateFrom_prod.symm <a>isPiSystem_prod</a> _ _ _ _ hs", [{"full_name": "MeasurableSpace.induction_on_inter", "def_path": "Mathlib/MeasureTheory/PiSystem.lean", "def_pos": [745, 9], "def_end_pos": [745, 27]}, {"full_name": "isPiSystem_prod", "def_path": "Mathlib/MeasureTheory/Constructions/Prod/Basic.lean", "def_pos": [150, 9], "def_end_pos": [150, 24]}]], "state_before": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\n\u22a2 \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 s} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 s", "state_after": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\n\u22a2 \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 \u2205} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 \u2205\n\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\n\u22a2 \u2200 (t : Set (\u03b1 \u00d7 \u211d)),\n    t \u2208 image2 (fun x x_1 => x \u00d7\u02e2 x_1) {s | MeasurableSet s} {t | MeasurableSet t} \u2192\n      \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 t\n\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\n\u22a2 \u2200 (t : Set (\u03b1 \u00d7 \u211d)),\n    MeasurableSet t \u2192\n      \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 t \u2192\n        \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t\u1d9c} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 t\u1d9c\n\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\n\u22a2 \u2200 (f : \u2115 \u2192 Set (\u03b1 \u00d7 \u211d)),\n    Pairwise (Disjoint on f) \u2192\n      (\u2200 (i : \u2115), MeasurableSet (f i)) \u2192\n        (\u2200 (i : \u2115), \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 f i} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (f i)) \u2192\n          \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 \u22c3 i, f i} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (\u22c3 i, f i)"}, {"tactic": "simp only [mem_empty_iff_false, setOf_false, measure_empty, lintegral_const,\n  zero_mul]", "annotated_tactic": ["simp only [<a>mem_empty_iff_false</a>, <a>setOf_false</a>, <a>measure_empty</a>, <a>lintegral_const</a>,\n      <a>zero_mul</a>]", [{"full_name": "Set.mem_empty_iff_false", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [562, 9], "def_end_pos": [562, 28]}, {"full_name": "Set.setOf_false", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [567, 9], "def_end_pos": [567, 20]}, {"full_name": "MeasureTheory.measure_empty", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [185, 9], "def_end_pos": [185, 22]}, {"full_name": "MeasureTheory.lintegral_const", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [136, 9], "def_end_pos": [136, 24]}, {"full_name": "MulZeroClass.zero_mul", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [36, 3], "def_end_pos": [36, 11]}]], "state_before": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\n\u22a2 \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 \u2205} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 \u2205", "state_after": "no goals"}, {"tactic": "intro t ht", "annotated_tactic": ["intro t ht", []], "state_before": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\n\u22a2 \u2200 (t : Set (\u03b1 \u00d7 \u211d)),\n    t \u2208 image2 (fun x x_1 => x \u00d7\u02e2 x_1) {s | MeasurableSet s} {t | MeasurableSet t} \u2192\n      \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 t", "state_after": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nt : Set (\u03b1 \u00d7 \u211d)\nht : t \u2208 image2 (fun x x_1 => x \u00d7\u02e2 x_1) {s | MeasurableSet s} {t | MeasurableSet t}\n\u22a2 \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 t"}, {"tactic": "rw [mem_image2] at ht", "annotated_tactic": ["rw [<a>mem_image2</a>] at ht", [{"full_name": "Set.mem_image2", "def_path": "Mathlib/Data/Set/NAry.lean", "def_pos": [40, 9], "def_end_pos": [40, 19]}]], "state_before": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nt : Set (\u03b1 \u00d7 \u211d)\nht : t \u2208 image2 (fun x x_1 => x \u00d7\u02e2 x_1) {s | MeasurableSet s} {t | MeasurableSet t}\n\u22a2 \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 t", "state_after": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nt : Set (\u03b1 \u00d7 \u211d)\nht : \u2203 a b, a \u2208 {s | MeasurableSet s} \u2227 b \u2208 {t | MeasurableSet t} \u2227 a \u00d7\u02e2 b = t\n\u22a2 \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 t"}, {"tactic": "obtain \u27e8t\u2081, t\u2082, ht\u2081, ht\u2082, rfl\u27e9 := ht", "annotated_tactic": ["obtain \u27e8t\u2081, t\u2082, ht\u2081, ht\u2082, rfl\u27e9 := ht", []], "state_before": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nt : Set (\u03b1 \u00d7 \u211d)\nht : \u2203 a b, a \u2208 {s | MeasurableSet s} \u2227 b \u2208 {t | MeasurableSet t} \u2227 a \u00d7\u02e2 b = t\n\u22a2 \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 t", "state_after": "case intro.intro.intro.intro\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nt\u2081 : Set \u03b1\nt\u2082 : Set \u211d\nht\u2081 : t\u2081 \u2208 {s | MeasurableSet s}\nht\u2082 : t\u2082 \u2208 {t | MeasurableSet t}\n\u22a2 \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (t\u2081 \u00d7\u02e2 t\u2082)"}, {"tactic": "have h_prod_eq_snd : \u2200 a \u2208 t\u2081, {x : \u211d | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} = t\u2082 := by\n  intro a ha\n  simp only [ha, prod_mk_mem_set_prod_eq, true_and_iff, setOf_mem_eq]", "annotated_tactic": ["have h_prod_eq_snd : \u2200 a \u2208 t\u2081, {x : \u211d | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} = t\u2082 := by\n      intro a ha\n      simp only [ha, <a>prod_mk_mem_set_prod_eq</a>, <a>true_and_iff</a>, <a>setOf_mem_eq</a>]", [{"full_name": "Set.prod_mk_mem_set_prod_eq", "def_path": "Mathlib/Data/Set/Prod.lean", "def_pos": [62, 9], "def_end_pos": [62, 32]}, {"full_name": "true_and_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [147, 9], "def_end_pos": [147, 21]}, {"full_name": "Set.setOf_mem_eq", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [275, 9], "def_end_pos": [275, 21]}]], "state_before": "case intro.intro.intro.intro\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nt\u2081 : Set \u03b1\nt\u2082 : Set \u211d\nht\u2081 : t\u2081 \u2208 {s | MeasurableSet s}\nht\u2082 : t\u2082 \u2208 {t | MeasurableSet t}\n\u22a2 \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (t\u2081 \u00d7\u02e2 t\u2082)", "state_after": "case intro.intro.intro.intro\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nt\u2081 : Set \u03b1\nt\u2082 : Set \u211d\nht\u2081 : t\u2081 \u2208 {s | MeasurableSet s}\nht\u2082 : t\u2082 \u2208 {t | MeasurableSet t}\nh_prod_eq_snd : \u2200 (a : \u03b1), a \u2208 t\u2081 \u2192 {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} = t\u2082\n\u22a2 \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (t\u2081 \u00d7\u02e2 t\u2082)"}, {"tactic": "cases' eq_empty_or_nonempty t\u2082 with h h", "annotated_tactic": ["cases' <a>eq_empty_or_nonempty</a> t\u2082 with h h", [{"full_name": "Set.eq_empty_or_nonempty", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [635, 9], "def_end_pos": [635, 29]}]], "state_before": "case intro.intro.intro.intro\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nt\u2081 : Set \u03b1\nt\u2082 : Set \u211d\nht\u2081 : t\u2081 \u2208 {s | MeasurableSet s}\nht\u2082 : t\u2082 \u2208 {t | MeasurableSet t}\nh_prod_eq_snd : \u2200 (a : \u03b1), a \u2208 t\u2081 \u2192 {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} = t\u2082\n\u22a2 \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (t\u2081 \u00d7\u02e2 t\u2082)", "state_after": "case intro.intro.intro.intro.inl\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nt\u2081 : Set \u03b1\nt\u2082 : Set \u211d\nht\u2081 : t\u2081 \u2208 {s | MeasurableSet s}\nht\u2082 : t\u2082 \u2208 {t | MeasurableSet t}\nh_prod_eq_snd : \u2200 (a : \u03b1), a \u2208 t\u2081 \u2192 {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} = t\u2082\nh : t\u2082 = \u2205\n\u22a2 \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (t\u2081 \u00d7\u02e2 t\u2082)\n\ncase intro.intro.intro.intro.inr\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nt\u2081 : Set \u03b1\nt\u2082 : Set \u211d\nht\u2081 : t\u2081 \u2208 {s | MeasurableSet s}\nht\u2082 : t\u2082 \u2208 {t | MeasurableSet t}\nh_prod_eq_snd : \u2200 (a : \u03b1), a \u2208 t\u2081 \u2192 {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} = t\u2082\nh : Set.Nonempty t\u2082\n\u22a2 \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (t\u2081 \u00d7\u02e2 t\u2082)"}, {"tactic": "rw [\u2190 lintegral_add_compl _ ht\u2081]", "annotated_tactic": ["rw [\u2190 <a>lintegral_add_compl</a> _ ht\u2081]", [{"full_name": "MeasureTheory.lintegral_add_compl", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [1258, 9], "def_end_pos": [1258, 28]}]], "state_before": "case intro.intro.intro.intro.inr\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nt\u2081 : Set \u03b1\nt\u2082 : Set \u211d\nht\u2081 : t\u2081 \u2208 {s | MeasurableSet s}\nht\u2082 : t\u2082 \u2208 {t | MeasurableSet t}\nh_prod_eq_snd : \u2200 (a : \u03b1), a \u2208 t\u2081 \u2192 {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} = t\u2082\nh : Set.Nonempty t\u2082\n\u22a2 \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (t\u2081 \u00d7\u02e2 t\u2082)", "state_after": "case intro.intro.intro.intro.inr\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nt\u2081 : Set \u03b1\nt\u2082 : Set \u211d\nht\u2081 : t\u2081 \u2208 {s | MeasurableSet s}\nht\u2082 : t\u2082 \u2208 {t | MeasurableSet t}\nh_prod_eq_snd : \u2200 (a : \u03b1), a \u2208 t\u2081 \u2192 {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} = t\u2082\nh : Set.Nonempty t\u2082\n\u22a2 \u222b\u207b (x : \u03b1) in t\u2081, \u2191\u2191(\u2191(condKernelReal \u03c1) x) {x_1 | (x, x_1) \u2208 t\u2081 \u00d7\u02e2 t\u2082} \u2202Measure.fst \u03c1 +\n      \u222b\u207b (x : \u03b1) in t\u2081\u1d9c, \u2191\u2191(\u2191(condKernelReal \u03c1) x) {x_1 | (x, x_1) \u2208 t\u2081 \u00d7\u02e2 t\u2082} \u2202Measure.fst \u03c1 =\n    \u2191\u2191\u03c1 (t\u2081 \u00d7\u02e2 t\u2082)"}, {"tactic": "have h_eq1 : \u222b\u207b a in t\u2081, condKernelReal \u03c1 a {x : \u211d | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} \u2202\u03c1.fst =\n    \u222b\u207b a in t\u2081, condKernelReal \u03c1 a t\u2082 \u2202\u03c1.fst := by\n  refine' set_lintegral_congr_fun ht\u2081 (eventually_of_forall fun a ha => _)\n  rw [h_prod_eq_snd a ha]", "annotated_tactic": ["have h_eq1 : \u222b\u207b a in t\u2081, <a>condKernelReal</a> \u03c1 a {x : \u211d | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} \u2202\u03c1.fst =\n        \u222b\u207b a in t\u2081, <a>condKernelReal</a> \u03c1 a t\u2082 \u2202\u03c1.fst := by\n      refine' <a>set_lintegral_congr_fun</a> ht\u2081 (<a>eventually_of_forall</a> fun a ha => _)\n      rw [h_prod_eq_snd a ha]", [{"full_name": "ProbabilityTheory.condKernelReal", "def_path": "Mathlib/Probability/Kernel/Disintegration.lean", "def_pos": [65, 19], "def_end_pos": [65, 33]}, {"full_name": "ProbabilityTheory.condKernelReal", "def_path": "Mathlib/Probability/Kernel/Disintegration.lean", "def_pos": [65, 19], "def_end_pos": [65, 33]}, {"full_name": "MeasureTheory.set_lintegral_congr_fun", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [316, 9], "def_end_pos": [316, 32]}, {"full_name": "Filter.eventually_of_forall", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1111, 9], "def_end_pos": [1111, 29]}]], "state_before": "case intro.intro.intro.intro.inr\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nt\u2081 : Set \u03b1\nt\u2082 : Set \u211d\nht\u2081 : t\u2081 \u2208 {s | MeasurableSet s}\nht\u2082 : t\u2082 \u2208 {t | MeasurableSet t}\nh_prod_eq_snd : \u2200 (a : \u03b1), a \u2208 t\u2081 \u2192 {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} = t\u2082\nh : Set.Nonempty t\u2082\n\u22a2 \u222b\u207b (x : \u03b1) in t\u2081, \u2191\u2191(\u2191(condKernelReal \u03c1) x) {x_1 | (x, x_1) \u2208 t\u2081 \u00d7\u02e2 t\u2082} \u2202Measure.fst \u03c1 +\n      \u222b\u207b (x : \u03b1) in t\u2081\u1d9c, \u2191\u2191(\u2191(condKernelReal \u03c1) x) {x_1 | (x, x_1) \u2208 t\u2081 \u00d7\u02e2 t\u2082} \u2202Measure.fst \u03c1 =\n    \u2191\u2191\u03c1 (t\u2081 \u00d7\u02e2 t\u2082)", "state_after": "case intro.intro.intro.intro.inr\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nt\u2081 : Set \u03b1\nt\u2082 : Set \u211d\nht\u2081 : t\u2081 \u2208 {s | MeasurableSet s}\nht\u2082 : t\u2082 \u2208 {t | MeasurableSet t}\nh_prod_eq_snd : \u2200 (a : \u03b1), a \u2208 t\u2081 \u2192 {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} = t\u2082\nh : Set.Nonempty t\u2082\nh_eq1 :\n  \u222b\u207b (a : \u03b1) in t\u2081, \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} \u2202Measure.fst \u03c1 =\n    \u222b\u207b (a : \u03b1) in t\u2081, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t\u2082 \u2202Measure.fst \u03c1\n\u22a2 \u222b\u207b (x : \u03b1) in t\u2081, \u2191\u2191(\u2191(condKernelReal \u03c1) x) {x_1 | (x, x_1) \u2208 t\u2081 \u00d7\u02e2 t\u2082} \u2202Measure.fst \u03c1 +\n      \u222b\u207b (x : \u03b1) in t\u2081\u1d9c, \u2191\u2191(\u2191(condKernelReal \u03c1) x) {x_1 | (x, x_1) \u2208 t\u2081 \u00d7\u02e2 t\u2082} \u2202Measure.fst \u03c1 =\n    \u2191\u2191\u03c1 (t\u2081 \u00d7\u02e2 t\u2082)"}, {"tactic": "rw [h_eq1, h_eq2, add_zero]", "annotated_tactic": ["rw [h_eq1, h_eq2, <a>add_zero</a>]", [{"full_name": "add_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [469, 3], "def_end_pos": [469, 14]}]], "state_before": "case intro.intro.intro.intro.inr\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nt\u2081 : Set \u03b1\nt\u2082 : Set \u211d\nht\u2081 : t\u2081 \u2208 {s | MeasurableSet s}\nht\u2082 : t\u2082 \u2208 {t | MeasurableSet t}\nh_prod_eq_snd : \u2200 (a : \u03b1), a \u2208 t\u2081 \u2192 {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} = t\u2082\nh : Set.Nonempty t\u2082\nh_eq1 :\n  \u222b\u207b (a : \u03b1) in t\u2081, \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} \u2202Measure.fst \u03c1 =\n    \u222b\u207b (a : \u03b1) in t\u2081, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t\u2082 \u2202Measure.fst \u03c1\nh_eq2 : \u222b\u207b (a : \u03b1) in t\u2081\u1d9c, \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} \u2202Measure.fst \u03c1 = 0\n\u22a2 \u222b\u207b (x : \u03b1) in t\u2081, \u2191\u2191(\u2191(condKernelReal \u03c1) x) {x_1 | (x, x_1) \u2208 t\u2081 \u00d7\u02e2 t\u2082} \u2202Measure.fst \u03c1 +\n      \u222b\u207b (x : \u03b1) in t\u2081\u1d9c, \u2191\u2191(\u2191(condKernelReal \u03c1) x) {x_1 | (x, x_1) \u2208 t\u2081 \u00d7\u02e2 t\u2082} \u2202Measure.fst \u03c1 =\n    \u2191\u2191\u03c1 (t\u2081 \u00d7\u02e2 t\u2082)", "state_after": "case intro.intro.intro.intro.inr\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nt\u2081 : Set \u03b1\nt\u2082 : Set \u211d\nht\u2081 : t\u2081 \u2208 {s | MeasurableSet s}\nht\u2082 : t\u2082 \u2208 {t | MeasurableSet t}\nh_prod_eq_snd : \u2200 (a : \u03b1), a \u2208 t\u2081 \u2192 {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} = t\u2082\nh : Set.Nonempty t\u2082\nh_eq1 :\n  \u222b\u207b (a : \u03b1) in t\u2081, \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} \u2202Measure.fst \u03c1 =\n    \u222b\u207b (a : \u03b1) in t\u2081, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t\u2082 \u2202Measure.fst \u03c1\nh_eq2 : \u222b\u207b (a : \u03b1) in t\u2081\u1d9c, \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} \u2202Measure.fst \u03c1 = 0\n\u22a2 \u222b\u207b (a : \u03b1) in t\u2081, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t\u2082 \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (t\u2081 \u00d7\u02e2 t\u2082)"}, {"tactic": "exact set_lintegral_condKernelReal_prod \u03c1 ht\u2081 ht\u2082", "annotated_tactic": ["exact <a>set_lintegral_condKernelReal_prod</a> \u03c1 ht\u2081 ht\u2082", [{"full_name": "ProbabilityTheory.set_lintegral_condKernelReal_prod", "def_path": "Mathlib/Probability/Kernel/Disintegration.lean", "def_pos": [98, 9], "def_end_pos": [98, 42]}]], "state_before": "case intro.intro.intro.intro.inr\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nt\u2081 : Set \u03b1\nt\u2082 : Set \u211d\nht\u2081 : t\u2081 \u2208 {s | MeasurableSet s}\nht\u2082 : t\u2082 \u2208 {t | MeasurableSet t}\nh_prod_eq_snd : \u2200 (a : \u03b1), a \u2208 t\u2081 \u2192 {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} = t\u2082\nh : Set.Nonempty t\u2082\nh_eq1 :\n  \u222b\u207b (a : \u03b1) in t\u2081, \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} \u2202Measure.fst \u03c1 =\n    \u222b\u207b (a : \u03b1) in t\u2081, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t\u2082 \u2202Measure.fst \u03c1\nh_eq2 : \u222b\u207b (a : \u03b1) in t\u2081\u1d9c, \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} \u2202Measure.fst \u03c1 = 0\n\u22a2 \u222b\u207b (a : \u03b1) in t\u2081, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t\u2082 \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (t\u2081 \u00d7\u02e2 t\u2082)", "state_after": "no goals"}, {"tactic": "intro a ha", "annotated_tactic": ["intro a ha", []], "state_before": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nt\u2081 : Set \u03b1\nt\u2082 : Set \u211d\nht\u2081 : t\u2081 \u2208 {s | MeasurableSet s}\nht\u2082 : t\u2082 \u2208 {t | MeasurableSet t}\n\u22a2 \u2200 (a : \u03b1), a \u2208 t\u2081 \u2192 {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} = t\u2082", "state_after": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nt\u2081 : Set \u03b1\nt\u2082 : Set \u211d\nht\u2081 : t\u2081 \u2208 {s | MeasurableSet s}\nht\u2082 : t\u2082 \u2208 {t | MeasurableSet t}\na : \u03b1\nha : a \u2208 t\u2081\n\u22a2 {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} = t\u2082"}, {"tactic": "simp only [ha, prod_mk_mem_set_prod_eq, true_and_iff, setOf_mem_eq]", "annotated_tactic": ["simp only [ha, <a>prod_mk_mem_set_prod_eq</a>, <a>true_and_iff</a>, <a>setOf_mem_eq</a>]", [{"full_name": "Set.prod_mk_mem_set_prod_eq", "def_path": "Mathlib/Data/Set/Prod.lean", "def_pos": [62, 9], "def_end_pos": [62, 32]}, {"full_name": "true_and_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [147, 9], "def_end_pos": [147, 21]}, {"full_name": "Set.setOf_mem_eq", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [275, 9], "def_end_pos": [275, 21]}]], "state_before": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nt\u2081 : Set \u03b1\nt\u2082 : Set \u211d\nht\u2081 : t\u2081 \u2208 {s | MeasurableSet s}\nht\u2082 : t\u2082 \u2208 {t | MeasurableSet t}\na : \u03b1\nha : a \u2208 t\u2081\n\u22a2 {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} = t\u2082", "state_after": "no goals"}, {"tactic": "simp only [h, prod_empty, mem_empty_iff_false, setOf_false, measure_empty, lintegral_const,\n  zero_mul]", "annotated_tactic": ["simp only [h, <a>prod_empty</a>, <a>mem_empty_iff_false</a>, <a>setOf_false</a>, <a>measure_empty</a>, <a>lintegral_const</a>,\n        <a>zero_mul</a>]", [{"full_name": "Set.prod_empty", "def_path": "Mathlib/Data/Set/Prod.lean", "def_pos": [113, 9], "def_end_pos": [113, 19]}, {"full_name": "Set.mem_empty_iff_false", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [562, 9], "def_end_pos": [562, 28]}, {"full_name": "Set.setOf_false", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [567, 9], "def_end_pos": [567, 20]}, {"full_name": "MeasureTheory.measure_empty", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [185, 9], "def_end_pos": [185, 22]}, {"full_name": "MeasureTheory.lintegral_const", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [136, 9], "def_end_pos": [136, 24]}, {"full_name": "MulZeroClass.zero_mul", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [36, 3], "def_end_pos": [36, 11]}]], "state_before": "case intro.intro.intro.intro.inl\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nt\u2081 : Set \u03b1\nt\u2082 : Set \u211d\nht\u2081 : t\u2081 \u2208 {s | MeasurableSet s}\nht\u2082 : t\u2082 \u2208 {t | MeasurableSet t}\nh_prod_eq_snd : \u2200 (a : \u03b1), a \u2208 t\u2081 \u2192 {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} = t\u2082\nh : t\u2082 = \u2205\n\u22a2 \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (t\u2081 \u00d7\u02e2 t\u2082)", "state_after": "no goals"}, {"tactic": "refine' set_lintegral_congr_fun ht\u2081 (eventually_of_forall fun a ha => _)", "annotated_tactic": ["refine' <a>set_lintegral_congr_fun</a> ht\u2081 (<a>eventually_of_forall</a> fun a ha => _)", [{"full_name": "MeasureTheory.set_lintegral_congr_fun", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [316, 9], "def_end_pos": [316, 32]}, {"full_name": "Filter.eventually_of_forall", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1111, 9], "def_end_pos": [1111, 29]}]], "state_before": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nt\u2081 : Set \u03b1\nt\u2082 : Set \u211d\nht\u2081 : t\u2081 \u2208 {s | MeasurableSet s}\nht\u2082 : t\u2082 \u2208 {t | MeasurableSet t}\nh_prod_eq_snd : \u2200 (a : \u03b1), a \u2208 t\u2081 \u2192 {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} = t\u2082\nh : Set.Nonempty t\u2082\n\u22a2 \u222b\u207b (a : \u03b1) in t\u2081, \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} \u2202Measure.fst \u03c1 =\n    \u222b\u207b (a : \u03b1) in t\u2081, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t\u2082 \u2202Measure.fst \u03c1", "state_after": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nt\u2081 : Set \u03b1\nt\u2082 : Set \u211d\nht\u2081 : t\u2081 \u2208 {s | MeasurableSet s}\nht\u2082 : t\u2082 \u2208 {t | MeasurableSet t}\nh_prod_eq_snd : \u2200 (a : \u03b1), a \u2208 t\u2081 \u2192 {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} = t\u2082\nh : Set.Nonempty t\u2082\na : \u03b1\nha : a \u2208 t\u2081\n\u22a2 \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} = \u2191\u2191(\u2191(condKernelReal \u03c1) a) t\u2082"}, {"tactic": "rw [h_prod_eq_snd a ha]", "annotated_tactic": ["rw [h_prod_eq_snd a ha]", []], "state_before": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nt\u2081 : Set \u03b1\nt\u2082 : Set \u211d\nht\u2081 : t\u2081 \u2208 {s | MeasurableSet s}\nht\u2082 : t\u2082 \u2208 {t | MeasurableSet t}\nh_prod_eq_snd : \u2200 (a : \u03b1), a \u2208 t\u2081 \u2192 {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} = t\u2082\nh : Set.Nonempty t\u2082\na : \u03b1\nha : a \u2208 t\u2081\n\u22a2 \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} = \u2191\u2191(\u2191(condKernelReal \u03c1) a) t\u2082", "state_after": "no goals"}, {"tactic": "suffices h_eq_zero : \u2200 a \u2208 t\u2081\u1d9c, condKernelReal \u03c1 a {x : \u211d | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} = 0", "annotated_tactic": ["suffices h_eq_zero : \u2200 a \u2208 t\u2081\u1d9c, <a>condKernelReal</a> \u03c1 a {x : \u211d | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} = 0", [{"full_name": "ProbabilityTheory.condKernelReal", "def_path": "Mathlib/Probability/Kernel/Disintegration.lean", "def_pos": [65, 19], "def_end_pos": [65, 33]}]], "state_before": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nt\u2081 : Set \u03b1\nt\u2082 : Set \u211d\nht\u2081 : t\u2081 \u2208 {s | MeasurableSet s}\nht\u2082 : t\u2082 \u2208 {t | MeasurableSet t}\nh_prod_eq_snd : \u2200 (a : \u03b1), a \u2208 t\u2081 \u2192 {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} = t\u2082\nh : Set.Nonempty t\u2082\nh_eq1 :\n  \u222b\u207b (a : \u03b1) in t\u2081, \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} \u2202Measure.fst \u03c1 =\n    \u222b\u207b (a : \u03b1) in t\u2081, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t\u2082 \u2202Measure.fst \u03c1\n\u22a2 \u222b\u207b (a : \u03b1) in t\u2081\u1d9c, \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} \u2202Measure.fst \u03c1 = 0", "state_after": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nt\u2081 : Set \u03b1\nt\u2082 : Set \u211d\nht\u2081 : t\u2081 \u2208 {s | MeasurableSet s}\nht\u2082 : t\u2082 \u2208 {t | MeasurableSet t}\nh_prod_eq_snd : \u2200 (a : \u03b1), a \u2208 t\u2081 \u2192 {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} = t\u2082\nh : Set.Nonempty t\u2082\nh_eq1 :\n  \u222b\u207b (a : \u03b1) in t\u2081, \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} \u2202Measure.fst \u03c1 =\n    \u222b\u207b (a : \u03b1) in t\u2081, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t\u2082 \u2202Measure.fst \u03c1\nh_eq_zero : \u2200 (a : \u03b1), a \u2208 t\u2081\u1d9c \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} = 0\n\u22a2 \u222b\u207b (a : \u03b1) in t\u2081\u1d9c, \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} \u2202Measure.fst \u03c1 = 0\n\ncase h_eq_zero\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nt\u2081 : Set \u03b1\nt\u2082 : Set \u211d\nht\u2081 : t\u2081 \u2208 {s | MeasurableSet s}\nht\u2082 : t\u2082 \u2208 {t | MeasurableSet t}\nh_prod_eq_snd : \u2200 (a : \u03b1), a \u2208 t\u2081 \u2192 {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} = t\u2082\nh : Set.Nonempty t\u2082\nh_eq1 :\n  \u222b\u207b (a : \u03b1) in t\u2081, \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} \u2202Measure.fst \u03c1 =\n    \u222b\u207b (a : \u03b1) in t\u2081, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t\u2082 \u2202Measure.fst \u03c1\n\u22a2 \u2200 (a : \u03b1), a \u2208 t\u2081\u1d9c \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} = 0"}, {"tactic": "intro a hat\u2081", "annotated_tactic": ["intro a hat\u2081", []], "state_before": "case h_eq_zero\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nt\u2081 : Set \u03b1\nt\u2082 : Set \u211d\nht\u2081 : t\u2081 \u2208 {s | MeasurableSet s}\nht\u2082 : t\u2082 \u2208 {t | MeasurableSet t}\nh_prod_eq_snd : \u2200 (a : \u03b1), a \u2208 t\u2081 \u2192 {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} = t\u2082\nh : Set.Nonempty t\u2082\nh_eq1 :\n  \u222b\u207b (a : \u03b1) in t\u2081, \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} \u2202Measure.fst \u03c1 =\n    \u222b\u207b (a : \u03b1) in t\u2081, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t\u2082 \u2202Measure.fst \u03c1\n\u22a2 \u2200 (a : \u03b1), a \u2208 t\u2081\u1d9c \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} = 0", "state_after": "case h_eq_zero\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nt\u2081 : Set \u03b1\nt\u2082 : Set \u211d\nht\u2081 : t\u2081 \u2208 {s | MeasurableSet s}\nht\u2082 : t\u2082 \u2208 {t | MeasurableSet t}\nh_prod_eq_snd : \u2200 (a : \u03b1), a \u2208 t\u2081 \u2192 {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} = t\u2082\nh : Set.Nonempty t\u2082\nh_eq1 :\n  \u222b\u207b (a : \u03b1) in t\u2081, \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} \u2202Measure.fst \u03c1 =\n    \u222b\u207b (a : \u03b1) in t\u2081, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t\u2082 \u2202Measure.fst \u03c1\na : \u03b1\nhat\u2081 : a \u2208 t\u2081\u1d9c\n\u22a2 \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} = 0"}, {"tactic": "rw [mem_compl_iff] at hat\u2081", "annotated_tactic": ["rw [<a>mem_compl_iff</a>] at hat\u2081", [{"full_name": "Set.mem_compl_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1658, 9], "def_end_pos": [1658, 22]}]], "state_before": "case h_eq_zero\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nt\u2081 : Set \u03b1\nt\u2082 : Set \u211d\nht\u2081 : t\u2081 \u2208 {s | MeasurableSet s}\nht\u2082 : t\u2082 \u2208 {t | MeasurableSet t}\nh_prod_eq_snd : \u2200 (a : \u03b1), a \u2208 t\u2081 \u2192 {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} = t\u2082\nh : Set.Nonempty t\u2082\nh_eq1 :\n  \u222b\u207b (a : \u03b1) in t\u2081, \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} \u2202Measure.fst \u03c1 =\n    \u222b\u207b (a : \u03b1) in t\u2081, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t\u2082 \u2202Measure.fst \u03c1\na : \u03b1\nhat\u2081 : a \u2208 t\u2081\u1d9c\n\u22a2 \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} = 0", "state_after": "case h_eq_zero\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nt\u2081 : Set \u03b1\nt\u2082 : Set \u211d\nht\u2081 : t\u2081 \u2208 {s | MeasurableSet s}\nht\u2082 : t\u2082 \u2208 {t | MeasurableSet t}\nh_prod_eq_snd : \u2200 (a : \u03b1), a \u2208 t\u2081 \u2192 {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} = t\u2082\nh : Set.Nonempty t\u2082\nh_eq1 :\n  \u222b\u207b (a : \u03b1) in t\u2081, \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} \u2202Measure.fst \u03c1 =\n    \u222b\u207b (a : \u03b1) in t\u2081, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t\u2082 \u2202Measure.fst \u03c1\na : \u03b1\nhat\u2081 : \u00aca \u2208 t\u2081\n\u22a2 \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} = 0"}, {"tactic": "simp only [hat\u2081, prod_mk_mem_set_prod_eq, false_and_iff, setOf_false, measure_empty]", "annotated_tactic": ["simp only [hat\u2081, <a>prod_mk_mem_set_prod_eq</a>, <a>false_and_iff</a>, <a>setOf_false</a>, <a>measure_empty</a>]", [{"full_name": "Set.prod_mk_mem_set_prod_eq", "def_path": "Mathlib/Data/Set/Prod.lean", "def_pos": [62, 9], "def_end_pos": [62, 32]}, {"full_name": "false_and_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [151, 9], "def_end_pos": [151, 22]}, {"full_name": "Set.setOf_false", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [567, 9], "def_end_pos": [567, 20]}, {"full_name": "MeasureTheory.measure_empty", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [185, 9], "def_end_pos": [185, 22]}]], "state_before": "case h_eq_zero\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nt\u2081 : Set \u03b1\nt\u2082 : Set \u211d\nht\u2081 : t\u2081 \u2208 {s | MeasurableSet s}\nht\u2082 : t\u2082 \u2208 {t | MeasurableSet t}\nh_prod_eq_snd : \u2200 (a : \u03b1), a \u2208 t\u2081 \u2192 {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} = t\u2082\nh : Set.Nonempty t\u2082\nh_eq1 :\n  \u222b\u207b (a : \u03b1) in t\u2081, \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} \u2202Measure.fst \u03c1 =\n    \u222b\u207b (a : \u03b1) in t\u2081, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t\u2082 \u2202Measure.fst \u03c1\na : \u03b1\nhat\u2081 : \u00aca \u2208 t\u2081\n\u22a2 \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} = 0", "state_after": "no goals"}, {"tactic": "rw [set_lintegral_congr_fun ht\u2081.compl (eventually_of_forall h_eq_zero)]", "annotated_tactic": ["rw [<a>set_lintegral_congr_fun</a> ht\u2081.compl (<a>eventually_of_forall</a> h_eq_zero)]", [{"full_name": "MeasureTheory.set_lintegral_congr_fun", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [316, 9], "def_end_pos": [316, 32]}, {"full_name": "Filter.eventually_of_forall", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1111, 9], "def_end_pos": [1111, 29]}]], "state_before": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nt\u2081 : Set \u03b1\nt\u2082 : Set \u211d\nht\u2081 : t\u2081 \u2208 {s | MeasurableSet s}\nht\u2082 : t\u2082 \u2208 {t | MeasurableSet t}\nh_prod_eq_snd : \u2200 (a : \u03b1), a \u2208 t\u2081 \u2192 {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} = t\u2082\nh : Set.Nonempty t\u2082\nh_eq1 :\n  \u222b\u207b (a : \u03b1) in t\u2081, \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} \u2202Measure.fst \u03c1 =\n    \u222b\u207b (a : \u03b1) in t\u2081, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t\u2082 \u2202Measure.fst \u03c1\nh_eq_zero : \u2200 (a : \u03b1), a \u2208 t\u2081\u1d9c \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} = 0\n\u22a2 \u222b\u207b (a : \u03b1) in t\u2081\u1d9c, \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} \u2202Measure.fst \u03c1 = 0", "state_after": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nt\u2081 : Set \u03b1\nt\u2082 : Set \u211d\nht\u2081 : t\u2081 \u2208 {s | MeasurableSet s}\nht\u2082 : t\u2082 \u2208 {t | MeasurableSet t}\nh_prod_eq_snd : \u2200 (a : \u03b1), a \u2208 t\u2081 \u2192 {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} = t\u2082\nh : Set.Nonempty t\u2082\nh_eq1 :\n  \u222b\u207b (a : \u03b1) in t\u2081, \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} \u2202Measure.fst \u03c1 =\n    \u222b\u207b (a : \u03b1) in t\u2081, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t\u2082 \u2202Measure.fst \u03c1\nh_eq_zero : \u2200 (a : \u03b1), a \u2208 t\u2081\u1d9c \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} = 0\n\u22a2 \u222b\u207b (x : \u03b1) in t\u2081\u1d9c, 0 \u2202Measure.fst \u03c1 = 0"}, {"tactic": "simp only [lintegral_const, zero_mul]", "annotated_tactic": ["simp only [<a>lintegral_const</a>, <a>zero_mul</a>]", [{"full_name": "MeasureTheory.lintegral_const", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [136, 9], "def_end_pos": [136, 24]}, {"full_name": "MulZeroClass.zero_mul", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [36, 3], "def_end_pos": [36, 11]}]], "state_before": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nt\u2081 : Set \u03b1\nt\u2082 : Set \u211d\nht\u2081 : t\u2081 \u2208 {s | MeasurableSet s}\nht\u2082 : t\u2082 \u2208 {t | MeasurableSet t}\nh_prod_eq_snd : \u2200 (a : \u03b1), a \u2208 t\u2081 \u2192 {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} = t\u2082\nh : Set.Nonempty t\u2082\nh_eq1 :\n  \u222b\u207b (a : \u03b1) in t\u2081, \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} \u2202Measure.fst \u03c1 =\n    \u222b\u207b (a : \u03b1) in t\u2081, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t\u2082 \u2202Measure.fst \u03c1\nh_eq_zero : \u2200 (a : \u03b1), a \u2208 t\u2081\u1d9c \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} = 0\n\u22a2 \u222b\u207b (x : \u03b1) in t\u2081\u1d9c, 0 \u2202Measure.fst \u03c1 = 0", "state_after": "no goals"}, {"tactic": "intro t ht ht_eq", "annotated_tactic": ["intro t ht ht_eq", []], "state_before": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\n\u22a2 \u2200 (t : Set (\u03b1 \u00d7 \u211d)),\n    MeasurableSet t \u2192\n      \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 t \u2192\n        \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t\u1d9c} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 t\u1d9c", "state_after": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nt : Set (\u03b1 \u00d7 \u211d)\nht : MeasurableSet t\nht_eq : \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 t\n\u22a2 \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t\u1d9c} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 t\u1d9c"}, {"tactic": "congr with a : 1", "annotated_tactic": ["congr with a : 1", []], "state_before": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nt : Set (\u03b1 \u00d7 \u211d)\nht : MeasurableSet t\nht_eq : \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 t\n\u22a2 \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t}\u1d9c \u2202Measure.fst \u03c1 =\n    \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) univ - \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t} \u2202Measure.fst \u03c1", "state_after": "case e_f.h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nt : Set (\u03b1 \u00d7 \u211d)\nht : MeasurableSet t\nht_eq : \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 t\na : \u03b1\n\u22a2 \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t}\u1d9c =\n    \u2191\u2191(\u2191(condKernelReal \u03c1) a) univ - \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t}"}, {"tactic": "exact measure_compl (measurable_prod_mk_left ht) (measure_ne_top (condKernelReal \u03c1 a) _)", "annotated_tactic": ["exact <a>measure_compl</a> (<a>measurable_prod_mk_left</a> ht) (<a>measure_ne_top</a> (<a>condKernelReal</a> \u03c1 a) _)", [{"full_name": "MeasureTheory.measure_compl", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [303, 9], "def_end_pos": [303, 22]}, {"full_name": "measurable_prod_mk_left", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [736, 9], "def_end_pos": [736, 32]}, {"full_name": "MeasureTheory.measure_ne_top", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2875, 9], "def_end_pos": [2875, 23]}, {"full_name": "ProbabilityTheory.condKernelReal", "def_path": "Mathlib/Probability/Kernel/Disintegration.lean", "def_pos": [65, 19], "def_end_pos": [65, 33]}]], "state_before": "case e_f.h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nt : Set (\u03b1 \u00d7 \u211d)\nht : MeasurableSet t\nht_eq : \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 t\na : \u03b1\n\u22a2 \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t}\u1d9c =\n    \u2191\u2191(\u2191(condKernelReal \u03c1) a) univ - \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t}", "state_after": "no goals"}, {"tactic": "have h_le : (fun a => condKernelReal \u03c1 a {x : \u211d | (a, x) \u2208 t}) \u2264\u1d50[\u03c1.fst] fun a =>\n    condKernelReal \u03c1 a univ := eventually_of_forall fun a => measure_mono (subset_univ _)", "annotated_tactic": ["have h_le : (fun a => <a>condKernelReal</a> \u03c1 a {x : \u211d | (a, x) \u2208 t}) \u2264\u1d50[\u03c1.fst] fun a =>\n            <a>condKernelReal</a> \u03c1 a <a>univ</a> := <a>eventually_of_forall</a> fun a => <a>measure_mono</a> (<a>subset_univ</a> _)", [{"full_name": "ProbabilityTheory.condKernelReal", "def_path": "Mathlib/Probability/Kernel/Disintegration.lean", "def_pos": [65, 19], "def_end_pos": [65, 33]}, {"full_name": "ProbabilityTheory.condKernelReal", "def_path": "Mathlib/Probability/Kernel/Disintegration.lean", "def_pos": [65, 19], "def_end_pos": [65, 33]}, {"full_name": "Set.univ", "def_path": "Mathlib/Init/Set.lean", "def_pos": [90, 5], "def_end_pos": [90, 9]}, {"full_name": "Filter.eventually_of_forall", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1111, 9], "def_end_pos": [1111, 29]}, {"full_name": "MeasureTheory.measure_mono", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [193, 9], "def_end_pos": [193, 21]}, {"full_name": "Set.subset_univ", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [691, 9], "def_end_pos": [691, 20]}]], "state_before": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nt : Set (\u03b1 \u00d7 \u211d)\nht : MeasurableSet t\nht_eq : \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 t\n\u22a2 \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) univ - \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t} \u2202Measure.fst \u03c1 =\n    \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) univ \u2202Measure.fst \u03c1 -\n      \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t} \u2202Measure.fst \u03c1", "state_after": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nt : Set (\u03b1 \u00d7 \u211d)\nht : MeasurableSet t\nht_eq : \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 t\nh_le : (fun a => \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t}) \u2264\u1d50[Measure.fst \u03c1] fun a => \u2191\u2191(\u2191(condKernelReal \u03c1) a) univ\n\u22a2 \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) univ - \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t} \u2202Measure.fst \u03c1 =\n    \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) univ \u2202Measure.fst \u03c1 -\n      \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t} \u2202Measure.fst \u03c1"}, {"tactic": "rw [lintegral_sub _ _ h_le]", "annotated_tactic": ["rw [<a>lintegral_sub</a> _ _ h_le]", [{"full_name": "MeasureTheory.lintegral_sub", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [933, 9], "def_end_pos": [933, 22]}]], "state_before": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nt : Set (\u03b1 \u00d7 \u211d)\nht : MeasurableSet t\nht_eq : \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 t\nh_le : (fun a => \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t}) \u2264\u1d50[Measure.fst \u03c1] fun a => \u2191\u2191(\u2191(condKernelReal \u03c1) a) univ\n\u22a2 \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) univ - \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t} \u2202Measure.fst \u03c1 =\n    \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) univ \u2202Measure.fst \u03c1 -\n      \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t} \u2202Measure.fst \u03c1", "state_after": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nt : Set (\u03b1 \u00d7 \u211d)\nht : MeasurableSet t\nht_eq : \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 t\nh_le : (fun a => \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t}) \u2264\u1d50[Measure.fst \u03c1] fun a => \u2191\u2191(\u2191(condKernelReal \u03c1) a) univ\n\u22a2 Measurable fun a => \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t}\n\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nt : Set (\u03b1 \u00d7 \u211d)\nht : MeasurableSet t\nht_eq : \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 t\nh_le : (fun a => \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t}) \u2264\u1d50[Measure.fst \u03c1] fun a => \u2191\u2191(\u2191(condKernelReal \u03c1) a) univ\n\u22a2 \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t} \u2202Measure.fst \u03c1 \u2260 \u22a4"}, {"tactic": "refine' ((lintegral_mono_ae h_le).trans_lt _).ne", "annotated_tactic": ["refine' ((<a>lintegral_mono_ae</a> h_le).<a>trans_lt</a> _).<a>ne</a>", [{"full_name": "MeasureTheory.lintegral_mono_ae", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [265, 9], "def_end_pos": [265, 26]}, {"full_name": "LE.le.trans_lt", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [124, 7], "def_end_pos": [124, 21]}, {"full_name": "LT.lt.ne", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [152, 7], "def_end_pos": [152, 15]}]], "state_before": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nt : Set (\u03b1 \u00d7 \u211d)\nht : MeasurableSet t\nht_eq : \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 t\nh_le : (fun a => \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t}) \u2264\u1d50[Measure.fst \u03c1] fun a => \u2191\u2191(\u2191(condKernelReal \u03c1) a) univ\n\u22a2 \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t} \u2202Measure.fst \u03c1 \u2260 \u22a4", "state_after": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nt : Set (\u03b1 \u00d7 \u211d)\nht : MeasurableSet t\nht_eq : \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 t\nh_le : (fun a => \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t}) \u2264\u1d50[Measure.fst \u03c1] fun a => \u2191\u2191(\u2191(condKernelReal \u03c1) a) univ\n\u22a2 \u222b\u207b (a : \u03b1), (fun a => \u2191\u2191(\u2191(condKernelReal \u03c1) a) univ) a \u2202Measure.fst \u03c1 < \u22a4"}, {"tactic": "rw [lintegral_condKernelReal_univ]", "annotated_tactic": ["rw [<a>lintegral_condKernelReal_univ</a>]", [{"full_name": "ProbabilityTheory.lintegral_condKernelReal_univ", "def_path": "Mathlib/Probability/Kernel/Disintegration.lean", "def_pos": [90, 9], "def_end_pos": [90, 38]}]], "state_before": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nt : Set (\u03b1 \u00d7 \u211d)\nht : MeasurableSet t\nht_eq : \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 t\nh_le : (fun a => \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t}) \u2264\u1d50[Measure.fst \u03c1] fun a => \u2191\u2191(\u2191(condKernelReal \u03c1) a) univ\n\u22a2 \u222b\u207b (a : \u03b1), (fun a => \u2191\u2191(\u2191(condKernelReal \u03c1) a) univ) a \u2202Measure.fst \u03c1 < \u22a4", "state_after": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nt : Set (\u03b1 \u00d7 \u211d)\nht : MeasurableSet t\nht_eq : \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 t\nh_le : (fun a => \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t}) \u2264\u1d50[Measure.fst \u03c1] fun a => \u2191\u2191(\u2191(condKernelReal \u03c1) a) univ\n\u22a2 \u2191\u2191\u03c1 univ < \u22a4"}, {"tactic": "exact measure_lt_top \u03c1 univ", "annotated_tactic": ["exact <a>measure_lt_top</a> \u03c1 <a>univ</a>", [{"full_name": "MeasureTheory.measure_lt_top", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2866, 9], "def_end_pos": [2866, 23]}, {"full_name": "Set.univ", "def_path": "Mathlib/Init/Set.lean", "def_pos": [90, 5], "def_end_pos": [90, 9]}]], "state_before": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nt : Set (\u03b1 \u00d7 \u211d)\nht : MeasurableSet t\nht_eq : \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 t\nh_le : (fun a => \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t}) \u2264\u1d50[Measure.fst \u03c1] fun a => \u2191\u2191(\u2191(condKernelReal \u03c1) a) univ\n\u22a2 \u2191\u2191\u03c1 univ < \u22a4", "state_after": "no goals"}, {"tactic": "exact kernel.measurable_kernel_prod_mk_left ht", "annotated_tactic": ["exact <a>kernel.measurable_kernel_prod_mk_left</a> ht", [{"full_name": "ProbabilityTheory.kernel.measurable_kernel_prod_mk_left", "def_path": "Mathlib/Probability/Kernel/MeasurableIntegral.lean", "def_pos": [100, 9], "def_end_pos": [100, 39]}]], "state_before": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nt : Set (\u03b1 \u00d7 \u211d)\nht : MeasurableSet t\nht_eq : \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 t\nh_le : (fun a => \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t}) \u2264\u1d50[Measure.fst \u03c1] fun a => \u2191\u2191(\u2191(condKernelReal \u03c1) a) univ\n\u22a2 Measurable fun a => \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t}", "state_after": "no goals"}, {"tactic": "rw [ht_eq, lintegral_condKernelReal_univ]", "annotated_tactic": ["rw [ht_eq, <a>lintegral_condKernelReal_univ</a>]", [{"full_name": "ProbabilityTheory.lintegral_condKernelReal_univ", "def_path": "Mathlib/Probability/Kernel/Disintegration.lean", "def_pos": [90, 9], "def_end_pos": [90, 38]}]], "state_before": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nt : Set (\u03b1 \u00d7 \u211d)\nht : MeasurableSet t\nht_eq : \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 t\n\u22a2 \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) univ \u2202Measure.fst \u03c1 -\n      \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t} \u2202Measure.fst \u03c1 =\n    \u2191\u2191\u03c1 univ - \u2191\u2191\u03c1 t", "state_after": "no goals"}, {"tactic": "intro f hf_disj hf_meas hf_eq", "annotated_tactic": ["intro f hf_disj hf_meas hf_eq", []], "state_before": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\n\u22a2 \u2200 (f : \u2115 \u2192 Set (\u03b1 \u00d7 \u211d)),\n    Pairwise (Disjoint on f) \u2192\n      (\u2200 (i : \u2115), MeasurableSet (f i)) \u2192\n        (\u2200 (i : \u2115), \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 f i} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (f i)) \u2192\n          \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 \u22c3 i, f i} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (\u22c3 i, f i)", "state_after": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nf : \u2115 \u2192 Set (\u03b1 \u00d7 \u211d)\nhf_disj : Pairwise (Disjoint on f)\nhf_meas : \u2200 (i : \u2115), MeasurableSet (f i)\nhf_eq : \u2200 (i : \u2115), \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 f i} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (f i)\n\u22a2 \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 \u22c3 i, f i} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (\u22c3 i, f i)"}, {"tactic": "have h_eq : \u2200 a, {x | (a, x) \u2208 \u22c3 i, f i} = \u22c3 i, {x | (a, x) \u2208 f i} := by\n  intro a\n  ext1 x\n  simp only [mem_iUnion, mem_setOf_eq]", "annotated_tactic": ["have h_eq : \u2200 a, {x | (a, x) \u2208 \u22c3 i, f i} = \u22c3 i, {x | (a, x) \u2208 f i} := by\n      intro a\n      ext1 x\n      simp only [<a>mem_iUnion</a>, <a>mem_setOf_eq</a>]", [{"full_name": "Set.mem_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [201, 9], "def_end_pos": [201, 19]}, {"full_name": "Set.mem_setOf_eq", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [256, 29], "def_end_pos": [256, 41]}]], "state_before": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nf : \u2115 \u2192 Set (\u03b1 \u00d7 \u211d)\nhf_disj : Pairwise (Disjoint on f)\nhf_meas : \u2200 (i : \u2115), MeasurableSet (f i)\nhf_eq : \u2200 (i : \u2115), \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 f i} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (f i)\n\u22a2 \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 \u22c3 i, f i} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (\u22c3 i, f i)", "state_after": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nf : \u2115 \u2192 Set (\u03b1 \u00d7 \u211d)\nhf_disj : Pairwise (Disjoint on f)\nhf_meas : \u2200 (i : \u2115), MeasurableSet (f i)\nhf_eq : \u2200 (i : \u2115), \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 f i} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (f i)\nh_eq : \u2200 (a : \u03b1), {x | (a, x) \u2208 \u22c3 i, f i} = \u22c3 i, {x | (a, x) \u2208 f i}\n\u22a2 \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 \u22c3 i, f i} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (\u22c3 i, f i)"}, {"tactic": "simp_rw [h_eq]", "annotated_tactic": ["simp_rw [h_eq]", []], "state_before": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nf : \u2115 \u2192 Set (\u03b1 \u00d7 \u211d)\nhf_disj : Pairwise (Disjoint on f)\nhf_meas : \u2200 (i : \u2115), MeasurableSet (f i)\nhf_eq : \u2200 (i : \u2115), \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 f i} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (f i)\nh_eq : \u2200 (a : \u03b1), {x | (a, x) \u2208 \u22c3 i, f i} = \u22c3 i, {x | (a, x) \u2208 f i}\n\u22a2 \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 \u22c3 i, f i} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (\u22c3 i, f i)", "state_after": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nf : \u2115 \u2192 Set (\u03b1 \u00d7 \u211d)\nhf_disj : Pairwise (Disjoint on f)\nhf_meas : \u2200 (i : \u2115), MeasurableSet (f i)\nhf_eq : \u2200 (i : \u2115), \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 f i} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (f i)\nh_eq : \u2200 (a : \u03b1), {x | (a, x) \u2208 \u22c3 i, f i} = \u22c3 i, {x | (a, x) \u2208 f i}\n\u22a2 \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) (\u22c3 i, {x | (a, x) \u2208 f i}) \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (\u22c3 i, f i)"}, {"tactic": "have h_disj : \u2200 a, Pairwise (Disjoint on fun i => {x | (a, x) \u2208 f i}) := by\n  intro a i j hij\n  have h_disj := hf_disj hij\n  rw [Function.onFun, disjoint_iff_inter_eq_empty] at h_disj \u22a2\n  ext1 x\n  simp only [mem_inter_iff, mem_setOf_eq, mem_empty_iff_false, iff_false_iff]\n  intro h_mem_both\n  suffices (a, x) \u2208 \u2205 by rwa [mem_empty_iff_false] at this\n  rwa [\u2190 h_disj, mem_inter_iff]", "annotated_tactic": ["have h_disj : \u2200 a, <a>Pairwise</a> (<a>Disjoint</a> on fun i => {x | (a, x) \u2208 f i}) := by\n      intro a i j hij\n      have h_disj := hf_disj hij\n      rw [<a>Function.onFun</a>, <a>disjoint_iff_inter_eq_empty</a>] at h_disj \u22a2\n      ext1 x\n      simp only [<a>mem_inter_iff</a>, <a>mem_setOf_eq</a>, <a>mem_empty_iff_false</a>, <a>iff_false_iff</a>]\n      intro h_mem_both\n      suffices (a, x) \u2208 \u2205 by rwa [<a>mem_empty_iff_false</a>] at this\n      rwa [\u2190 h_disj, <a>mem_inter_iff</a>]", [{"full_name": "Pairwise", "def_path": "Mathlib/Logic/Pairwise.lean", "def_pos": [34, 5], "def_end_pos": [34, 13]}, {"full_name": "Disjoint", "def_path": "Mathlib/Order/Disjoint.lean", "def_pos": [41, 5], "def_end_pos": [41, 13]}, {"full_name": "Function.onFun", "def_path": "Mathlib/Init/Function.lean", "def_pos": [49, 5], "def_end_pos": [49, 10]}, {"full_name": "Set.disjoint_iff_inter_eq_empty", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1538, 9], "def_end_pos": [1538, 36]}, {"full_name": "Set.mem_inter_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [909, 9], "def_end_pos": [909, 22]}, {"full_name": "Set.mem_setOf_eq", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [256, 29], "def_end_pos": [256, 41]}, {"full_name": "Set.mem_empty_iff_false", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [562, 9], "def_end_pos": [562, 28]}, {"full_name": "iff_false_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [201, 9], "def_end_pos": [201, 22]}, {"full_name": "Set.mem_empty_iff_false", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [562, 9], "def_end_pos": [562, 28]}, {"full_name": "Set.mem_inter_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [909, 9], "def_end_pos": [909, 22]}]], "state_before": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nf : \u2115 \u2192 Set (\u03b1 \u00d7 \u211d)\nhf_disj : Pairwise (Disjoint on f)\nhf_meas : \u2200 (i : \u2115), MeasurableSet (f i)\nhf_eq : \u2200 (i : \u2115), \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 f i} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (f i)\nh_eq : \u2200 (a : \u03b1), {x | (a, x) \u2208 \u22c3 i, f i} = \u22c3 i, {x | (a, x) \u2208 f i}\n\u22a2 \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) (\u22c3 i, {x | (a, x) \u2208 f i}) \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (\u22c3 i, f i)", "state_after": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nf : \u2115 \u2192 Set (\u03b1 \u00d7 \u211d)\nhf_disj : Pairwise (Disjoint on f)\nhf_meas : \u2200 (i : \u2115), MeasurableSet (f i)\nhf_eq : \u2200 (i : \u2115), \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 f i} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (f i)\nh_eq : \u2200 (a : \u03b1), {x | (a, x) \u2208 \u22c3 i, f i} = \u22c3 i, {x | (a, x) \u2208 f i}\nh_disj : \u2200 (a : \u03b1), Pairwise (Disjoint on fun i => {x | (a, x) \u2208 f i})\n\u22a2 \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) (\u22c3 i, {x | (a, x) \u2208 f i}) \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (\u22c3 i, f i)"}, {"tactic": "calc\n  \u222b\u207b a, condKernelReal \u03c1 a (\u22c3 i, {x | (a, x) \u2208 f i}) \u2202\u03c1.fst =\n      \u222b\u207b a, \u2211' i, condKernelReal \u03c1 a {x | (a, x) \u2208 f i} \u2202\u03c1.fst := by\n    congr with a : 1\n    rw [measure_iUnion (h_disj a) fun i => measurable_prod_mk_left (hf_meas i)]\n  _ = \u2211' i, \u222b\u207b a, condKernelReal \u03c1 a {x | (a, x) \u2208 f i} \u2202\u03c1.fst :=\n    (lintegral_tsum fun i => (kernel.measurable_kernel_prod_mk_left (hf_meas i)).aemeasurable)\n  _ = \u2211' i, \u03c1 (f i) := by simp_rw [hf_eq]\n  _ = \u03c1 (iUnion f) := (measure_iUnion hf_disj hf_meas).symm", "annotated_tactic": ["calc\n      \u222b\u207b a, <a>condKernelReal</a> \u03c1 a (\u22c3 i, {x | (a, x) \u2208 f i}) \u2202\u03c1.fst =\n          \u222b\u207b a, \u2211' i, <a>condKernelReal</a> \u03c1 a {x | (a, x) \u2208 f i} \u2202\u03c1.fst := by\n        congr with a : 1\n        rw [<a>measure_iUnion</a> (h_disj a) fun i => <a>measurable_prod_mk_left</a> (hf_meas i)]\n      _ = \u2211' i, \u222b\u207b a, <a>condKernelReal</a> \u03c1 a {x | (a, x) \u2208 f i} \u2202\u03c1.fst :=\n        (<a>lintegral_tsum</a> fun i => (<a>kernel.measurable_kernel_prod_mk_left</a> (hf_meas i)).<a>aemeasurable</a>)\n      _ = \u2211' i, \u03c1 (f i) := by simp_rw [hf_eq]\n      _ = \u03c1 (<a>iUnion</a> f) := (<a>measure_iUnion</a> hf_disj hf_meas).<a>symm</a>", [{"full_name": "ProbabilityTheory.condKernelReal", "def_path": "Mathlib/Probability/Kernel/Disintegration.lean", "def_pos": [65, 19], "def_end_pos": [65, 33]}, {"full_name": "ProbabilityTheory.condKernelReal", "def_path": "Mathlib/Probability/Kernel/Disintegration.lean", "def_pos": [65, 19], "def_end_pos": [65, 33]}, {"full_name": "MeasureTheory.measure_iUnion", "def_path": "Mathlib/MeasureTheory/Measure/NullMeasurable.lean", "def_pos": [272, 9], "def_end_pos": [272, 23]}, {"full_name": "measurable_prod_mk_left", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [736, 9], "def_end_pos": [736, 32]}, {"full_name": "ProbabilityTheory.condKernelReal", "def_path": "Mathlib/Probability/Kernel/Disintegration.lean", "def_pos": [65, 19], "def_end_pos": [65, 33]}, {"full_name": "MeasureTheory.lintegral_tsum", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [1184, 9], "def_end_pos": [1184, 23]}, {"full_name": "ProbabilityTheory.kernel.measurable_kernel_prod_mk_left", "def_path": "Mathlib/Probability/Kernel/MeasurableIntegral.lean", "def_pos": [100, 9], "def_end_pos": [100, 39]}, {"full_name": "Measurable.aemeasurable", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [713, 9], "def_end_pos": [713, 32]}, {"full_name": "Set.iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [103, 5], "def_end_pos": [103, 11]}, {"full_name": "MeasureTheory.measure_iUnion", "def_path": "Mathlib/MeasureTheory/Measure/NullMeasurable.lean", "def_pos": [272, 9], "def_end_pos": [272, 23]}, {"full_name": "Eq.symm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [310, 9], "def_end_pos": [310, 16]}]], "state_before": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nf : \u2115 \u2192 Set (\u03b1 \u00d7 \u211d)\nhf_disj : Pairwise (Disjoint on f)\nhf_meas : \u2200 (i : \u2115), MeasurableSet (f i)\nhf_eq : \u2200 (i : \u2115), \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 f i} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (f i)\nh_eq : \u2200 (a : \u03b1), {x | (a, x) \u2208 \u22c3 i, f i} = \u22c3 i, {x | (a, x) \u2208 f i}\nh_disj : \u2200 (a : \u03b1), Pairwise (Disjoint on fun i => {x | (a, x) \u2208 f i})\n\u22a2 \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) (\u22c3 i, {x | (a, x) \u2208 f i}) \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (\u22c3 i, f i)", "state_after": "no goals"}, {"tactic": "intro a", "annotated_tactic": ["intro a", []], "state_before": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nf : \u2115 \u2192 Set (\u03b1 \u00d7 \u211d)\nhf_disj : Pairwise (Disjoint on f)\nhf_meas : \u2200 (i : \u2115), MeasurableSet (f i)\nhf_eq : \u2200 (i : \u2115), \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 f i} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (f i)\n\u22a2 \u2200 (a : \u03b1), {x | (a, x) \u2208 \u22c3 i, f i} = \u22c3 i, {x | (a, x) \u2208 f i}", "state_after": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nf : \u2115 \u2192 Set (\u03b1 \u00d7 \u211d)\nhf_disj : Pairwise (Disjoint on f)\nhf_meas : \u2200 (i : \u2115), MeasurableSet (f i)\nhf_eq : \u2200 (i : \u2115), \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 f i} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (f i)\na : \u03b1\n\u22a2 {x | (a, x) \u2208 \u22c3 i, f i} = \u22c3 i, {x | (a, x) \u2208 f i}"}, {"tactic": "ext1 x", "annotated_tactic": ["ext1 x", []], "state_before": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nf : \u2115 \u2192 Set (\u03b1 \u00d7 \u211d)\nhf_disj : Pairwise (Disjoint on f)\nhf_meas : \u2200 (i : \u2115), MeasurableSet (f i)\nhf_eq : \u2200 (i : \u2115), \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 f i} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (f i)\na : \u03b1\n\u22a2 {x | (a, x) \u2208 \u22c3 i, f i} = \u22c3 i, {x | (a, x) \u2208 f i}", "state_after": "case h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nf : \u2115 \u2192 Set (\u03b1 \u00d7 \u211d)\nhf_disj : Pairwise (Disjoint on f)\nhf_meas : \u2200 (i : \u2115), MeasurableSet (f i)\nhf_eq : \u2200 (i : \u2115), \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 f i} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (f i)\na : \u03b1\nx : \u211d\n\u22a2 x \u2208 {x | (a, x) \u2208 \u22c3 i, f i} \u2194 x \u2208 \u22c3 i, {x | (a, x) \u2208 f i}"}, {"tactic": "simp only [mem_iUnion, mem_setOf_eq]", "annotated_tactic": ["simp only [<a>mem_iUnion</a>, <a>mem_setOf_eq</a>]", [{"full_name": "Set.mem_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [201, 9], "def_end_pos": [201, 19]}, {"full_name": "Set.mem_setOf_eq", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [256, 29], "def_end_pos": [256, 41]}]], "state_before": "case h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nf : \u2115 \u2192 Set (\u03b1 \u00d7 \u211d)\nhf_disj : Pairwise (Disjoint on f)\nhf_meas : \u2200 (i : \u2115), MeasurableSet (f i)\nhf_eq : \u2200 (i : \u2115), \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 f i} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (f i)\na : \u03b1\nx : \u211d\n\u22a2 x \u2208 {x | (a, x) \u2208 \u22c3 i, f i} \u2194 x \u2208 \u22c3 i, {x | (a, x) \u2208 f i}", "state_after": "no goals"}, {"tactic": "intro a i j hij", "annotated_tactic": ["intro a i j hij", []], "state_before": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nf : \u2115 \u2192 Set (\u03b1 \u00d7 \u211d)\nhf_disj : Pairwise (Disjoint on f)\nhf_meas : \u2200 (i : \u2115), MeasurableSet (f i)\nhf_eq : \u2200 (i : \u2115), \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 f i} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (f i)\nh_eq : \u2200 (a : \u03b1), {x | (a, x) \u2208 \u22c3 i, f i} = \u22c3 i, {x | (a, x) \u2208 f i}\n\u22a2 \u2200 (a : \u03b1), Pairwise (Disjoint on fun i => {x | (a, x) \u2208 f i})", "state_after": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nf : \u2115 \u2192 Set (\u03b1 \u00d7 \u211d)\nhf_disj : Pairwise (Disjoint on f)\nhf_meas : \u2200 (i : \u2115), MeasurableSet (f i)\nhf_eq : \u2200 (i : \u2115), \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 f i} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (f i)\nh_eq : \u2200 (a : \u03b1), {x | (a, x) \u2208 \u22c3 i, f i} = \u22c3 i, {x | (a, x) \u2208 f i}\na : \u03b1\ni j : \u2115\nhij : i \u2260 j\n\u22a2 (Disjoint on fun i => {x | (a, x) \u2208 f i}) i j"}, {"tactic": "have h_disj := hf_disj hij", "annotated_tactic": ["have h_disj := hf_disj hij", []], "state_before": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nf : \u2115 \u2192 Set (\u03b1 \u00d7 \u211d)\nhf_disj : Pairwise (Disjoint on f)\nhf_meas : \u2200 (i : \u2115), MeasurableSet (f i)\nhf_eq : \u2200 (i : \u2115), \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 f i} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (f i)\nh_eq : \u2200 (a : \u03b1), {x | (a, x) \u2208 \u22c3 i, f i} = \u22c3 i, {x | (a, x) \u2208 f i}\na : \u03b1\ni j : \u2115\nhij : i \u2260 j\n\u22a2 (Disjoint on fun i => {x | (a, x) \u2208 f i}) i j", "state_after": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nf : \u2115 \u2192 Set (\u03b1 \u00d7 \u211d)\nhf_disj : Pairwise (Disjoint on f)\nhf_meas : \u2200 (i : \u2115), MeasurableSet (f i)\nhf_eq : \u2200 (i : \u2115), \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 f i} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (f i)\nh_eq : \u2200 (a : \u03b1), {x | (a, x) \u2208 \u22c3 i, f i} = \u22c3 i, {x | (a, x) \u2208 f i}\na : \u03b1\ni j : \u2115\nhij : i \u2260 j\nh_disj : (Disjoint on f) i j\n\u22a2 (Disjoint on fun i => {x | (a, x) \u2208 f i}) i j"}, {"tactic": "rw [Function.onFun, disjoint_iff_inter_eq_empty] at h_disj \u22a2", "annotated_tactic": ["rw [<a>Function.onFun</a>, <a>disjoint_iff_inter_eq_empty</a>] at h_disj \u22a2", [{"full_name": "Function.onFun", "def_path": "Mathlib/Init/Function.lean", "def_pos": [49, 5], "def_end_pos": [49, 10]}, {"full_name": "Set.disjoint_iff_inter_eq_empty", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1538, 9], "def_end_pos": [1538, 36]}]], "state_before": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nf : \u2115 \u2192 Set (\u03b1 \u00d7 \u211d)\nhf_disj : Pairwise (Disjoint on f)\nhf_meas : \u2200 (i : \u2115), MeasurableSet (f i)\nhf_eq : \u2200 (i : \u2115), \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 f i} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (f i)\nh_eq : \u2200 (a : \u03b1), {x | (a, x) \u2208 \u22c3 i, f i} = \u22c3 i, {x | (a, x) \u2208 f i}\na : \u03b1\ni j : \u2115\nhij : i \u2260 j\nh_disj : (Disjoint on f) i j\n\u22a2 (Disjoint on fun i => {x | (a, x) \u2208 f i}) i j", "state_after": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nf : \u2115 \u2192 Set (\u03b1 \u00d7 \u211d)\nhf_disj : Pairwise (Disjoint on f)\nhf_meas : \u2200 (i : \u2115), MeasurableSet (f i)\nhf_eq : \u2200 (i : \u2115), \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 f i} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (f i)\nh_eq : \u2200 (a : \u03b1), {x | (a, x) \u2208 \u22c3 i, f i} = \u22c3 i, {x | (a, x) \u2208 f i}\na : \u03b1\ni j : \u2115\nhij : i \u2260 j\nh_disj : f i \u2229 f j = \u2205\n\u22a2 {x | (a, x) \u2208 f i} \u2229 {x | (a, x) \u2208 f j} = \u2205"}, {"tactic": "ext1 x", "annotated_tactic": ["ext1 x", []], "state_before": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nf : \u2115 \u2192 Set (\u03b1 \u00d7 \u211d)\nhf_disj : Pairwise (Disjoint on f)\nhf_meas : \u2200 (i : \u2115), MeasurableSet (f i)\nhf_eq : \u2200 (i : \u2115), \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 f i} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (f i)\nh_eq : \u2200 (a : \u03b1), {x | (a, x) \u2208 \u22c3 i, f i} = \u22c3 i, {x | (a, x) \u2208 f i}\na : \u03b1\ni j : \u2115\nhij : i \u2260 j\nh_disj : f i \u2229 f j = \u2205\n\u22a2 {x | (a, x) \u2208 f i} \u2229 {x | (a, x) \u2208 f j} = \u2205", "state_after": "case h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nf : \u2115 \u2192 Set (\u03b1 \u00d7 \u211d)\nhf_disj : Pairwise (Disjoint on f)\nhf_meas : \u2200 (i : \u2115), MeasurableSet (f i)\nhf_eq : \u2200 (i : \u2115), \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 f i} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (f i)\nh_eq : \u2200 (a : \u03b1), {x | (a, x) \u2208 \u22c3 i, f i} = \u22c3 i, {x | (a, x) \u2208 f i}\na : \u03b1\ni j : \u2115\nhij : i \u2260 j\nh_disj : f i \u2229 f j = \u2205\nx : \u211d\n\u22a2 x \u2208 {x | (a, x) \u2208 f i} \u2229 {x | (a, x) \u2208 f j} \u2194 x \u2208 \u2205"}, {"tactic": "simp only [mem_inter_iff, mem_setOf_eq, mem_empty_iff_false, iff_false_iff]", "annotated_tactic": ["simp only [<a>mem_inter_iff</a>, <a>mem_setOf_eq</a>, <a>mem_empty_iff_false</a>, <a>iff_false_iff</a>]", [{"full_name": "Set.mem_inter_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [909, 9], "def_end_pos": [909, 22]}, {"full_name": "Set.mem_setOf_eq", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [256, 29], "def_end_pos": [256, 41]}, {"full_name": "Set.mem_empty_iff_false", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [562, 9], "def_end_pos": [562, 28]}, {"full_name": "iff_false_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [201, 9], "def_end_pos": [201, 22]}]], "state_before": "case h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nf : \u2115 \u2192 Set (\u03b1 \u00d7 \u211d)\nhf_disj : Pairwise (Disjoint on f)\nhf_meas : \u2200 (i : \u2115), MeasurableSet (f i)\nhf_eq : \u2200 (i : \u2115), \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 f i} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (f i)\nh_eq : \u2200 (a : \u03b1), {x | (a, x) \u2208 \u22c3 i, f i} = \u22c3 i, {x | (a, x) \u2208 f i}\na : \u03b1\ni j : \u2115\nhij : i \u2260 j\nh_disj : f i \u2229 f j = \u2205\nx : \u211d\n\u22a2 x \u2208 {x | (a, x) \u2208 f i} \u2229 {x | (a, x) \u2208 f j} \u2194 x \u2208 \u2205", "state_after": "case h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nf : \u2115 \u2192 Set (\u03b1 \u00d7 \u211d)\nhf_disj : Pairwise (Disjoint on f)\nhf_meas : \u2200 (i : \u2115), MeasurableSet (f i)\nhf_eq : \u2200 (i : \u2115), \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 f i} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (f i)\nh_eq : \u2200 (a : \u03b1), {x | (a, x) \u2208 \u22c3 i, f i} = \u22c3 i, {x | (a, x) \u2208 f i}\na : \u03b1\ni j : \u2115\nhij : i \u2260 j\nh_disj : f i \u2229 f j = \u2205\nx : \u211d\n\u22a2 \u00ac((a, x) \u2208 f i \u2227 (a, x) \u2208 f j)"}, {"tactic": "intro h_mem_both", "annotated_tactic": ["intro h_mem_both", []], "state_before": "case h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nf : \u2115 \u2192 Set (\u03b1 \u00d7 \u211d)\nhf_disj : Pairwise (Disjoint on f)\nhf_meas : \u2200 (i : \u2115), MeasurableSet (f i)\nhf_eq : \u2200 (i : \u2115), \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 f i} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (f i)\nh_eq : \u2200 (a : \u03b1), {x | (a, x) \u2208 \u22c3 i, f i} = \u22c3 i, {x | (a, x) \u2208 f i}\na : \u03b1\ni j : \u2115\nhij : i \u2260 j\nh_disj : f i \u2229 f j = \u2205\nx : \u211d\n\u22a2 \u00ac((a, x) \u2208 f i \u2227 (a, x) \u2208 f j)", "state_after": "case h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nf : \u2115 \u2192 Set (\u03b1 \u00d7 \u211d)\nhf_disj : Pairwise (Disjoint on f)\nhf_meas : \u2200 (i : \u2115), MeasurableSet (f i)\nhf_eq : \u2200 (i : \u2115), \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 f i} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (f i)\nh_eq : \u2200 (a : \u03b1), {x | (a, x) \u2208 \u22c3 i, f i} = \u22c3 i, {x | (a, x) \u2208 f i}\na : \u03b1\ni j : \u2115\nhij : i \u2260 j\nh_disj : f i \u2229 f j = \u2205\nx : \u211d\nh_mem_both : (a, x) \u2208 f i \u2227 (a, x) \u2208 f j\n\u22a2 False"}, {"tactic": "suffices (a, x) \u2208 \u2205 by rwa [mem_empty_iff_false] at this", "annotated_tactic": ["suffices (a, x) \u2208 \u2205 by rwa [<a>mem_empty_iff_false</a>] at this", [{"full_name": "Set.mem_empty_iff_false", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [562, 9], "def_end_pos": [562, 28]}]], "state_before": "case h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nf : \u2115 \u2192 Set (\u03b1 \u00d7 \u211d)\nhf_disj : Pairwise (Disjoint on f)\nhf_meas : \u2200 (i : \u2115), MeasurableSet (f i)\nhf_eq : \u2200 (i : \u2115), \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 f i} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (f i)\nh_eq : \u2200 (a : \u03b1), {x | (a, x) \u2208 \u22c3 i, f i} = \u22c3 i, {x | (a, x) \u2208 f i}\na : \u03b1\ni j : \u2115\nhij : i \u2260 j\nh_disj : f i \u2229 f j = \u2205\nx : \u211d\nh_mem_both : (a, x) \u2208 f i \u2227 (a, x) \u2208 f j\n\u22a2 False", "state_after": "case h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nf : \u2115 \u2192 Set (\u03b1 \u00d7 \u211d)\nhf_disj : Pairwise (Disjoint on f)\nhf_meas : \u2200 (i : \u2115), MeasurableSet (f i)\nhf_eq : \u2200 (i : \u2115), \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 f i} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (f i)\nh_eq : \u2200 (a : \u03b1), {x | (a, x) \u2208 \u22c3 i, f i} = \u22c3 i, {x | (a, x) \u2208 f i}\na : \u03b1\ni j : \u2115\nhij : i \u2260 j\nh_disj : f i \u2229 f j = \u2205\nx : \u211d\nh_mem_both : (a, x) \u2208 f i \u2227 (a, x) \u2208 f j\n\u22a2 (a, x) \u2208 \u2205"}, {"tactic": "rwa [\u2190 h_disj, mem_inter_iff]", "annotated_tactic": ["rwa [\u2190 h_disj, <a>mem_inter_iff</a>]", [{"full_name": "Set.mem_inter_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [909, 9], "def_end_pos": [909, 22]}]], "state_before": "case h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nf : \u2115 \u2192 Set (\u03b1 \u00d7 \u211d)\nhf_disj : Pairwise (Disjoint on f)\nhf_meas : \u2200 (i : \u2115), MeasurableSet (f i)\nhf_eq : \u2200 (i : \u2115), \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 f i} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (f i)\nh_eq : \u2200 (a : \u03b1), {x | (a, x) \u2208 \u22c3 i, f i} = \u22c3 i, {x | (a, x) \u2208 f i}\na : \u03b1\ni j : \u2115\nhij : i \u2260 j\nh_disj : f i \u2229 f j = \u2205\nx : \u211d\nh_mem_both : (a, x) \u2208 f i \u2227 (a, x) \u2208 f j\n\u22a2 (a, x) \u2208 \u2205", "state_after": "no goals"}, {"tactic": "rwa [mem_empty_iff_false] at this", "annotated_tactic": ["rwa [<a>mem_empty_iff_false</a>] at this", [{"full_name": "Set.mem_empty_iff_false", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [562, 9], "def_end_pos": [562, 28]}]], "state_before": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nf : \u2115 \u2192 Set (\u03b1 \u00d7 \u211d)\nhf_disj : Pairwise (Disjoint on f)\nhf_meas : \u2200 (i : \u2115), MeasurableSet (f i)\nhf_eq : \u2200 (i : \u2115), \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 f i} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (f i)\nh_eq : \u2200 (a : \u03b1), {x | (a, x) \u2208 \u22c3 i, f i} = \u22c3 i, {x | (a, x) \u2208 f i}\na : \u03b1\ni j : \u2115\nhij : i \u2260 j\nh_disj : f i \u2229 f j = \u2205\nx : \u211d\nh_mem_both : (a, x) \u2208 f i \u2227 (a, x) \u2208 f j\nthis : (a, x) \u2208 \u2205\n\u22a2 False", "state_after": "no goals"}, {"tactic": "congr with a : 1", "annotated_tactic": ["congr with a : 1", []], "state_before": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nf : \u2115 \u2192 Set (\u03b1 \u00d7 \u211d)\nhf_disj : Pairwise (Disjoint on f)\nhf_meas : \u2200 (i : \u2115), MeasurableSet (f i)\nhf_eq : \u2200 (i : \u2115), \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 f i} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (f i)\nh_eq : \u2200 (a : \u03b1), {x | (a, x) \u2208 \u22c3 i, f i} = \u22c3 i, {x | (a, x) \u2208 f i}\nh_disj : \u2200 (a : \u03b1), Pairwise (Disjoint on fun i => {x | (a, x) \u2208 f i})\n\u22a2 \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) (\u22c3 i, {x | (a, x) \u2208 f i}) \u2202Measure.fst \u03c1 =\n    \u222b\u207b (a : \u03b1), \u2211' (i : \u2115), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 f i} \u2202Measure.fst \u03c1", "state_after": "case e_f.h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nf : \u2115 \u2192 Set (\u03b1 \u00d7 \u211d)\nhf_disj : Pairwise (Disjoint on f)\nhf_meas : \u2200 (i : \u2115), MeasurableSet (f i)\nhf_eq : \u2200 (i : \u2115), \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 f i} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (f i)\nh_eq : \u2200 (a : \u03b1), {x | (a, x) \u2208 \u22c3 i, f i} = \u22c3 i, {x | (a, x) \u2208 f i}\nh_disj : \u2200 (a : \u03b1), Pairwise (Disjoint on fun i => {x | (a, x) \u2208 f i})\na : \u03b1\n\u22a2 \u2191\u2191(\u2191(condKernelReal \u03c1) a) (\u22c3 i, {x | (a, x) \u2208 f i}) = \u2211' (i : \u2115), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 f i}"}, {"tactic": "rw [measure_iUnion (h_disj a) fun i => measurable_prod_mk_left (hf_meas i)]", "annotated_tactic": ["rw [<a>measure_iUnion</a> (h_disj a) fun i => <a>measurable_prod_mk_left</a> (hf_meas i)]", [{"full_name": "MeasureTheory.measure_iUnion", "def_path": "Mathlib/MeasureTheory/Measure/NullMeasurable.lean", "def_pos": [272, 9], "def_end_pos": [272, 23]}, {"full_name": "measurable_prod_mk_left", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [736, 9], "def_end_pos": [736, 32]}]], "state_before": "case e_f.h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nf : \u2115 \u2192 Set (\u03b1 \u00d7 \u211d)\nhf_disj : Pairwise (Disjoint on f)\nhf_meas : \u2200 (i : \u2115), MeasurableSet (f i)\nhf_eq : \u2200 (i : \u2115), \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 f i} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (f i)\nh_eq : \u2200 (a : \u03b1), {x | (a, x) \u2208 \u22c3 i, f i} = \u22c3 i, {x | (a, x) \u2208 f i}\nh_disj : \u2200 (a : \u03b1), Pairwise (Disjoint on fun i => {x | (a, x) \u2208 f i})\na : \u03b1\n\u22a2 \u2191\u2191(\u2191(condKernelReal \u03c1) a) (\u22c3 i, {x | (a, x) \u2208 f i}) = \u2211' (i : \u2115), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 f i}", "state_after": "no goals"}, {"tactic": "simp_rw [hf_eq]", "annotated_tactic": ["simp_rw [hf_eq]", []], "state_before": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nf : \u2115 \u2192 Set (\u03b1 \u00d7 \u211d)\nhf_disj : Pairwise (Disjoint on f)\nhf_meas : \u2200 (i : \u2115), MeasurableSet (f i)\nhf_eq : \u2200 (i : \u2115), \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 f i} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (f i)\nh_eq : \u2200 (a : \u03b1), {x | (a, x) \u2208 \u22c3 i, f i} = \u22c3 i, {x | (a, x) \u2208 f i}\nh_disj : \u2200 (a : \u03b1), Pairwise (Disjoint on fun i => {x | (a, x) \u2208 f i})\n\u22a2 \u2211' (i : \u2115), \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 f i} \u2202Measure.fst \u03c1 = \u2211' (i : \u2115), \u2191\u2191\u03c1 (f i)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "full_name": "String.data_takeWhile", "start": [1120, 9], "end": [1121, 64], "traced_tactics": [{"tactic": "rw [takeWhile_eq]", "annotated_tactic": ["rw [<a>takeWhile_eq</a>]", [{"full_name": "String.takeWhile_eq", "def_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "def_pos": [1117, 9], "def_end_pos": [1117, 21]}]], "state_before": "p : Char \u2192 Bool\ns : String\n\u22a2 (takeWhile s p).data = List.takeWhile p s.data", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Process/Stopping.lean", "full_name": "MeasureTheory.stoppedProcess_eq_of_mem_finset", "start": [914, 1], "end": [935, 22], "traced_tactics": [{"tactic": "ext \u03c9", "annotated_tactic": ["ext \u03c9", []], "state_before": "\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nE : Type u_4\np : \u211d\u22650\u221e\nu : \u03b9 \u2192 \u03a9 \u2192 E\ninst\u271d\u00b9 : LinearOrder \u03b9\ninst\u271d : AddCommMonoid E\ns : Finset \u03b9\nn : \u03b9\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c4 \u03c9 < n \u2192 \u03c4 \u03c9 \u2208 s\n\u22a2 stoppedProcess u \u03c4 n =\n    Set.indicator {a | n \u2264 \u03c4 a} (u n) + \u2211 i in Finset.filter (fun x => x < n) s, Set.indicator {\u03c9 | \u03c4 \u03c9 = i} (u i)", "state_after": "case h\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nE : Type u_4\np : \u211d\u22650\u221e\nu : \u03b9 \u2192 \u03a9 \u2192 E\ninst\u271d\u00b9 : LinearOrder \u03b9\ninst\u271d : AddCommMonoid E\ns : Finset \u03b9\nn : \u03b9\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c4 \u03c9 < n \u2192 \u03c4 \u03c9 \u2208 s\n\u03c9 : \u03a9\n\u22a2 stoppedProcess u \u03c4 n \u03c9 =\n    (Set.indicator {a | n \u2264 \u03c4 a} (u n) + \u2211 i in Finset.filter (fun x => x < n) s, Set.indicator {\u03c9 | \u03c4 \u03c9 = i} (u i)) \u03c9"}, {"tactic": "rw [Pi.add_apply, Finset.sum_apply]", "annotated_tactic": ["rw [<a>Pi.add_apply</a>, <a>Finset.sum_apply</a>]", [{"full_name": "Pi.add_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [82, 3], "def_end_pos": [82, 14]}, {"full_name": "Finset.sum_apply", "def_path": "Mathlib/Algebra/BigOperators/Pi.lean", "def_pos": [41, 3], "def_end_pos": [41, 14]}]], "state_before": "case h\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nE : Type u_4\np : \u211d\u22650\u221e\nu : \u03b9 \u2192 \u03a9 \u2192 E\ninst\u271d\u00b9 : LinearOrder \u03b9\ninst\u271d : AddCommMonoid E\ns : Finset \u03b9\nn : \u03b9\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c4 \u03c9 < n \u2192 \u03c4 \u03c9 \u2208 s\n\u03c9 : \u03a9\n\u22a2 stoppedProcess u \u03c4 n \u03c9 =\n    (Set.indicator {a | n \u2264 \u03c4 a} (u n) + \u2211 i in Finset.filter (fun x => x < n) s, Set.indicator {\u03c9 | \u03c4 \u03c9 = i} (u i)) \u03c9", "state_after": "case h\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nE : Type u_4\np : \u211d\u22650\u221e\nu : \u03b9 \u2192 \u03a9 \u2192 E\ninst\u271d\u00b9 : LinearOrder \u03b9\ninst\u271d : AddCommMonoid E\ns : Finset \u03b9\nn : \u03b9\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c4 \u03c9 < n \u2192 \u03c4 \u03c9 \u2208 s\n\u03c9 : \u03a9\n\u22a2 stoppedProcess u \u03c4 n \u03c9 =\n    Set.indicator {a | n \u2264 \u03c4 a} (u n) \u03c9 + \u2211 c in Finset.filter (fun x => x < n) s, Set.indicator {\u03c9 | \u03c4 \u03c9 = c} (u c) \u03c9"}, {"tactic": "cases' le_or_lt n (\u03c4 \u03c9) with h h", "annotated_tactic": ["cases' <a>le_or_lt</a> n (\u03c4 \u03c9) with h h", [{"full_name": "le_or_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [340, 9], "def_end_pos": [340, 17]}]], "state_before": "case h\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nE : Type u_4\np : \u211d\u22650\u221e\nu : \u03b9 \u2192 \u03a9 \u2192 E\ninst\u271d\u00b9 : LinearOrder \u03b9\ninst\u271d : AddCommMonoid E\ns : Finset \u03b9\nn : \u03b9\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c4 \u03c9 < n \u2192 \u03c4 \u03c9 \u2208 s\n\u03c9 : \u03a9\n\u22a2 stoppedProcess u \u03c4 n \u03c9 =\n    Set.indicator {a | n \u2264 \u03c4 a} (u n) \u03c9 + \u2211 c in Finset.filter (fun x => x < n) s, Set.indicator {\u03c9 | \u03c4 \u03c9 = c} (u c) \u03c9", "state_after": "case h.inl\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nE : Type u_4\np : \u211d\u22650\u221e\nu : \u03b9 \u2192 \u03a9 \u2192 E\ninst\u271d\u00b9 : LinearOrder \u03b9\ninst\u271d : AddCommMonoid E\ns : Finset \u03b9\nn : \u03b9\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c4 \u03c9 < n \u2192 \u03c4 \u03c9 \u2208 s\n\u03c9 : \u03a9\nh : n \u2264 \u03c4 \u03c9\n\u22a2 stoppedProcess u \u03c4 n \u03c9 =\n    Set.indicator {a | n \u2264 \u03c4 a} (u n) \u03c9 + \u2211 c in Finset.filter (fun x => x < n) s, Set.indicator {\u03c9 | \u03c4 \u03c9 = c} (u c) \u03c9\n\ncase h.inr\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nE : Type u_4\np : \u211d\u22650\u221e\nu : \u03b9 \u2192 \u03a9 \u2192 E\ninst\u271d\u00b9 : LinearOrder \u03b9\ninst\u271d : AddCommMonoid E\ns : Finset \u03b9\nn : \u03b9\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c4 \u03c9 < n \u2192 \u03c4 \u03c9 \u2208 s\n\u03c9 : \u03a9\nh : \u03c4 \u03c9 < n\n\u22a2 stoppedProcess u \u03c4 n \u03c9 =\n    Set.indicator {a | n \u2264 \u03c4 a} (u n) \u03c9 + \u2211 c in Finset.filter (fun x => x < n) s, Set.indicator {\u03c9 | \u03c4 \u03c9 = c} (u c) \u03c9"}, {"tactic": "rw [stoppedProcess_eq_of_le h, Set.indicator_of_mem, Finset.sum_eq_zero, add_zero]", "annotated_tactic": ["rw [<a>stoppedProcess_eq_of_le</a> h, <a>Set.indicator_of_mem</a>, <a>Finset.sum_eq_zero</a>, <a>add_zero</a>]", [{"full_name": "MeasureTheory.stoppedProcess_eq_of_le", "def_path": "Mathlib/Probability/Process/Stopping.lean", "def_pos": [794, 9], "def_end_pos": [794, 32]}, {"full_name": "Set.indicator_of_mem", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [67, 3], "def_end_pos": [67, 14]}, {"full_name": "Finset.sum_eq_zero", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [728, 3], "def_end_pos": [728, 14]}, {"full_name": "add_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [469, 3], "def_end_pos": [469, 14]}]], "state_before": "case h.inl\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nE : Type u_4\np : \u211d\u22650\u221e\nu : \u03b9 \u2192 \u03a9 \u2192 E\ninst\u271d\u00b9 : LinearOrder \u03b9\ninst\u271d : AddCommMonoid E\ns : Finset \u03b9\nn : \u03b9\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c4 \u03c9 < n \u2192 \u03c4 \u03c9 \u2208 s\n\u03c9 : \u03a9\nh : n \u2264 \u03c4 \u03c9\n\u22a2 stoppedProcess u \u03c4 n \u03c9 =\n    Set.indicator {a | n \u2264 \u03c4 a} (u n) \u03c9 + \u2211 c in Finset.filter (fun x => x < n) s, Set.indicator {\u03c9 | \u03c4 \u03c9 = c} (u c) \u03c9", "state_after": "case h.inl\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nE : Type u_4\np : \u211d\u22650\u221e\nu : \u03b9 \u2192 \u03a9 \u2192 E\ninst\u271d\u00b9 : LinearOrder \u03b9\ninst\u271d : AddCommMonoid E\ns : Finset \u03b9\nn : \u03b9\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c4 \u03c9 < n \u2192 \u03c4 \u03c9 \u2208 s\n\u03c9 : \u03a9\nh : n \u2264 \u03c4 \u03c9\n\u22a2 \u2200 (x : \u03b9), x \u2208 Finset.filter (fun x => x < n) s \u2192 Set.indicator {\u03c9 | \u03c4 \u03c9 = x} (u x) \u03c9 = 0\n\ncase h.inl.h\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nE : Type u_4\np : \u211d\u22650\u221e\nu : \u03b9 \u2192 \u03a9 \u2192 E\ninst\u271d\u00b9 : LinearOrder \u03b9\ninst\u271d : AddCommMonoid E\ns : Finset \u03b9\nn : \u03b9\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c4 \u03c9 < n \u2192 \u03c4 \u03c9 \u2208 s\n\u03c9 : \u03a9\nh : n \u2264 \u03c4 \u03c9\n\u22a2 \u03c9 \u2208 {a | n \u2264 \u03c4 a}"}, {"tactic": "intro m hm", "annotated_tactic": ["intro m hm", []], "state_before": "case h.inl\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nE : Type u_4\np : \u211d\u22650\u221e\nu : \u03b9 \u2192 \u03a9 \u2192 E\ninst\u271d\u00b9 : LinearOrder \u03b9\ninst\u271d : AddCommMonoid E\ns : Finset \u03b9\nn : \u03b9\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c4 \u03c9 < n \u2192 \u03c4 \u03c9 \u2208 s\n\u03c9 : \u03a9\nh : n \u2264 \u03c4 \u03c9\n\u22a2 \u2200 (x : \u03b9), x \u2208 Finset.filter (fun x => x < n) s \u2192 Set.indicator {\u03c9 | \u03c4 \u03c9 = x} (u x) \u03c9 = 0", "state_after": "case h.inl\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nE : Type u_4\np : \u211d\u22650\u221e\nu : \u03b9 \u2192 \u03a9 \u2192 E\ninst\u271d\u00b9 : LinearOrder \u03b9\ninst\u271d : AddCommMonoid E\ns : Finset \u03b9\nn : \u03b9\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c4 \u03c9 < n \u2192 \u03c4 \u03c9 \u2208 s\n\u03c9 : \u03a9\nh : n \u2264 \u03c4 \u03c9\nm : \u03b9\nhm : m \u2208 Finset.filter (fun x => x < n) s\n\u22a2 Set.indicator {\u03c9 | \u03c4 \u03c9 = m} (u m) \u03c9 = 0"}, {"tactic": "refine' Set.indicator_of_not_mem _ _", "annotated_tactic": ["refine' <a>Set.indicator_of_not_mem</a> _ _", [{"full_name": "Set.indicator_of_not_mem", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [73, 3], "def_end_pos": [73, 14]}]], "state_before": "case h.inl\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nE : Type u_4\np : \u211d\u22650\u221e\nu : \u03b9 \u2192 \u03a9 \u2192 E\ninst\u271d\u00b9 : LinearOrder \u03b9\ninst\u271d : AddCommMonoid E\ns : Finset \u03b9\nn : \u03b9\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c4 \u03c9 < n \u2192 \u03c4 \u03c9 \u2208 s\n\u03c9 : \u03a9\nh : n \u2264 \u03c4 \u03c9\nm : \u03b9\nhm : m \u2208 Finset.filter (fun x => x < n) s\n\u22a2 Set.indicator {\u03c9 | \u03c4 \u03c9 = m} (u m) \u03c9 = 0", "state_after": "case h.inl\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nE : Type u_4\np : \u211d\u22650\u221e\nu : \u03b9 \u2192 \u03a9 \u2192 E\ninst\u271d\u00b9 : LinearOrder \u03b9\ninst\u271d : AddCommMonoid E\ns : Finset \u03b9\nn : \u03b9\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c4 \u03c9 < n \u2192 \u03c4 \u03c9 \u2208 s\n\u03c9 : \u03a9\nh : n \u2264 \u03c4 \u03c9\nm : \u03b9\nhm : m \u2208 Finset.filter (fun x => x < n) s\n\u22a2 \u00ac\u03c9 \u2208 {\u03c9 | \u03c4 \u03c9 = m}"}, {"tactic": "rw [Finset.mem_filter] at hm", "annotated_tactic": ["rw [<a>Finset.mem_filter</a>] at hm", [{"full_name": "Finset.mem_filter", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2708, 9], "def_end_pos": [2708, 19]}]], "state_before": "case h.inl\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nE : Type u_4\np : \u211d\u22650\u221e\nu : \u03b9 \u2192 \u03a9 \u2192 E\ninst\u271d\u00b9 : LinearOrder \u03b9\ninst\u271d : AddCommMonoid E\ns : Finset \u03b9\nn : \u03b9\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c4 \u03c9 < n \u2192 \u03c4 \u03c9 \u2208 s\n\u03c9 : \u03a9\nh : n \u2264 \u03c4 \u03c9\nm : \u03b9\nhm : m \u2208 Finset.filter (fun x => x < n) s\n\u22a2 \u00ac\u03c9 \u2208 {\u03c9 | \u03c4 \u03c9 = m}", "state_after": "case h.inl\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nE : Type u_4\np : \u211d\u22650\u221e\nu : \u03b9 \u2192 \u03a9 \u2192 E\ninst\u271d\u00b9 : LinearOrder \u03b9\ninst\u271d : AddCommMonoid E\ns : Finset \u03b9\nn : \u03b9\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c4 \u03c9 < n \u2192 \u03c4 \u03c9 \u2208 s\n\u03c9 : \u03a9\nh : n \u2264 \u03c4 \u03c9\nm : \u03b9\nhm : m \u2208 s \u2227 m < n\n\u22a2 \u00ac\u03c9 \u2208 {\u03c9 | \u03c4 \u03c9 = m}"}, {"tactic": "exact (hm.2.trans_le h).ne'", "annotated_tactic": ["exact (hm.2.<a>trans_le</a> h).<a>ne'</a>", [{"full_name": "LT.lt.trans_le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [148, 7], "def_end_pos": [148, 21]}, {"full_name": "LT.lt.ne'", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [328, 9], "def_end_pos": [328, 12]}]], "state_before": "case h.inl\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nE : Type u_4\np : \u211d\u22650\u221e\nu : \u03b9 \u2192 \u03a9 \u2192 E\ninst\u271d\u00b9 : LinearOrder \u03b9\ninst\u271d : AddCommMonoid E\ns : Finset \u03b9\nn : \u03b9\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c4 \u03c9 < n \u2192 \u03c4 \u03c9 \u2208 s\n\u03c9 : \u03a9\nh : n \u2264 \u03c4 \u03c9\nm : \u03b9\nhm : m \u2208 s \u2227 m < n\n\u22a2 \u00ac\u03c9 \u2208 {\u03c9 | \u03c4 \u03c9 = m}", "state_after": "no goals"}, {"tactic": "exact h", "annotated_tactic": ["exact h", []], "state_before": "case h.inl.h\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nE : Type u_4\np : \u211d\u22650\u221e\nu : \u03b9 \u2192 \u03a9 \u2192 E\ninst\u271d\u00b9 : LinearOrder \u03b9\ninst\u271d : AddCommMonoid E\ns : Finset \u03b9\nn : \u03b9\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c4 \u03c9 < n \u2192 \u03c4 \u03c9 \u2208 s\n\u03c9 : \u03a9\nh : n \u2264 \u03c4 \u03c9\n\u22a2 \u03c9 \u2208 {a | n \u2264 \u03c4 a}", "state_after": "no goals"}, {"tactic": "rw [stoppedProcess_eq_of_ge (le_of_lt h), Finset.sum_eq_single_of_mem (\u03c4 \u03c9)]", "annotated_tactic": ["rw [<a>stoppedProcess_eq_of_ge</a> (<a>le_of_lt</a> h), <a>Finset.sum_eq_single_of_mem</a> (\u03c4 \u03c9)]", [{"full_name": "MeasureTheory.stoppedProcess_eq_of_ge", "def_path": "Mathlib/Probability/Process/Stopping.lean", "def_pos": [798, 9], "def_end_pos": [798, 32]}, {"full_name": "le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [110, 9], "def_end_pos": [110, 17]}, {"full_name": "Finset.sum_eq_single_of_mem", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [784, 3], "def_end_pos": [784, 14]}]], "state_before": "case h.inr\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nE : Type u_4\np : \u211d\u22650\u221e\nu : \u03b9 \u2192 \u03a9 \u2192 E\ninst\u271d\u00b9 : LinearOrder \u03b9\ninst\u271d : AddCommMonoid E\ns : Finset \u03b9\nn : \u03b9\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c4 \u03c9 < n \u2192 \u03c4 \u03c9 \u2208 s\n\u03c9 : \u03a9\nh : \u03c4 \u03c9 < n\n\u22a2 stoppedProcess u \u03c4 n \u03c9 =\n    Set.indicator {a | n \u2264 \u03c4 a} (u n) \u03c9 + \u2211 c in Finset.filter (fun x => x < n) s, Set.indicator {\u03c9 | \u03c4 \u03c9 = c} (u c) \u03c9", "state_after": "case h.inr\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nE : Type u_4\np : \u211d\u22650\u221e\nu : \u03b9 \u2192 \u03a9 \u2192 E\ninst\u271d\u00b9 : LinearOrder \u03b9\ninst\u271d : AddCommMonoid E\ns : Finset \u03b9\nn : \u03b9\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c4 \u03c9 < n \u2192 \u03c4 \u03c9 \u2208 s\n\u03c9 : \u03a9\nh : \u03c4 \u03c9 < n\n\u22a2 u (\u03c4 \u03c9) \u03c9 = Set.indicator {a | n \u2264 \u03c4 a} (u n) \u03c9 + Set.indicator {\u03c9_1 | \u03c4 \u03c9_1 = \u03c4 \u03c9} (u (\u03c4 \u03c9)) \u03c9\n\ncase h.inr.h\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nE : Type u_4\np : \u211d\u22650\u221e\nu : \u03b9 \u2192 \u03a9 \u2192 E\ninst\u271d\u00b9 : LinearOrder \u03b9\ninst\u271d : AddCommMonoid E\ns : Finset \u03b9\nn : \u03b9\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c4 \u03c9 < n \u2192 \u03c4 \u03c9 \u2208 s\n\u03c9 : \u03a9\nh : \u03c4 \u03c9 < n\n\u22a2 \u03c4 \u03c9 \u2208 Finset.filter (fun x => x < n) s\n\ncase h.inr.h\u2080\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nE : Type u_4\np : \u211d\u22650\u221e\nu : \u03b9 \u2192 \u03a9 \u2192 E\ninst\u271d\u00b9 : LinearOrder \u03b9\ninst\u271d : AddCommMonoid E\ns : Finset \u03b9\nn : \u03b9\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c4 \u03c9 < n \u2192 \u03c4 \u03c9 \u2208 s\n\u03c9 : \u03a9\nh : \u03c4 \u03c9 < n\n\u22a2 \u2200 (b : \u03b9), b \u2208 Finset.filter (fun x => x < n) s \u2192 b \u2260 \u03c4 \u03c9 \u2192 Set.indicator {\u03c9 | \u03c4 \u03c9 = b} (u b) \u03c9 = 0"}, {"tactic": "rw [Set.indicator_of_not_mem, zero_add, Set.indicator_of_mem]", "annotated_tactic": ["rw [<a>Set.indicator_of_not_mem</a>, <a>zero_add</a>, <a>Set.indicator_of_mem</a>]", [{"full_name": "Set.indicator_of_not_mem", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [73, 3], "def_end_pos": [73, 14]}, {"full_name": "zero_add", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [463, 3], "def_end_pos": [463, 14]}, {"full_name": "Set.indicator_of_mem", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [67, 3], "def_end_pos": [67, 14]}]], "state_before": "case h.inr\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nE : Type u_4\np : \u211d\u22650\u221e\nu : \u03b9 \u2192 \u03a9 \u2192 E\ninst\u271d\u00b9 : LinearOrder \u03b9\ninst\u271d : AddCommMonoid E\ns : Finset \u03b9\nn : \u03b9\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c4 \u03c9 < n \u2192 \u03c4 \u03c9 \u2208 s\n\u03c9 : \u03a9\nh : \u03c4 \u03c9 < n\n\u22a2 u (\u03c4 \u03c9) \u03c9 = Set.indicator {a | n \u2264 \u03c4 a} (u n) \u03c9 + Set.indicator {\u03c9_1 | \u03c4 \u03c9_1 = \u03c4 \u03c9} (u (\u03c4 \u03c9)) \u03c9", "state_after": "case h.inr.h\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nE : Type u_4\np : \u211d\u22650\u221e\nu : \u03b9 \u2192 \u03a9 \u2192 E\ninst\u271d\u00b9 : LinearOrder \u03b9\ninst\u271d : AddCommMonoid E\ns : Finset \u03b9\nn : \u03b9\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c4 \u03c9 < n \u2192 \u03c4 \u03c9 \u2208 s\n\u03c9 : \u03a9\nh : \u03c4 \u03c9 < n\n\u22a2 \u03c9 \u2208 {\u03c9_1 | \u03c4 \u03c9_1 = \u03c4 \u03c9}\n\ncase h.inr.h\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nE : Type u_4\np : \u211d\u22650\u221e\nu : \u03b9 \u2192 \u03a9 \u2192 E\ninst\u271d\u00b9 : LinearOrder \u03b9\ninst\u271d : AddCommMonoid E\ns : Finset \u03b9\nn : \u03b9\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c4 \u03c9 < n \u2192 \u03c4 \u03c9 \u2208 s\n\u03c9 : \u03a9\nh : \u03c4 \u03c9 < n\n\u22a2 \u00ac\u03c9 \u2208 {a | n \u2264 \u03c4 a}"}, {"tactic": "exact rfl", "annotated_tactic": ["exact <a>rfl</a>", [{"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "case h.inr.h\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nE : Type u_4\np : \u211d\u22650\u221e\nu : \u03b9 \u2192 \u03a9 \u2192 E\ninst\u271d\u00b9 : LinearOrder \u03b9\ninst\u271d : AddCommMonoid E\ns : Finset \u03b9\nn : \u03b9\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c4 \u03c9 < n \u2192 \u03c4 \u03c9 \u2208 s\n\u03c9 : \u03a9\nh : \u03c4 \u03c9 < n\n\u22a2 \u03c9 \u2208 {\u03c9_1 | \u03c4 \u03c9_1 = \u03c4 \u03c9}", "state_after": "no goals"}, {"tactic": "exact not_le.2 h", "annotated_tactic": ["exact <a>not_le</a>.2 h", [{"full_name": "not_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [373, 9], "def_end_pos": [373, 15]}]], "state_before": "case h.inr.h\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nE : Type u_4\np : \u211d\u22650\u221e\nu : \u03b9 \u2192 \u03a9 \u2192 E\ninst\u271d\u00b9 : LinearOrder \u03b9\ninst\u271d : AddCommMonoid E\ns : Finset \u03b9\nn : \u03b9\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c4 \u03c9 < n \u2192 \u03c4 \u03c9 \u2208 s\n\u03c9 : \u03a9\nh : \u03c4 \u03c9 < n\n\u22a2 \u00ac\u03c9 \u2208 {a | n \u2264 \u03c4 a}", "state_after": "no goals"}, {"tactic": "rw [Finset.mem_filter]", "annotated_tactic": ["rw [<a>Finset.mem_filter</a>]", [{"full_name": "Finset.mem_filter", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2708, 9], "def_end_pos": [2708, 19]}]], "state_before": "case h.inr.h\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nE : Type u_4\np : \u211d\u22650\u221e\nu : \u03b9 \u2192 \u03a9 \u2192 E\ninst\u271d\u00b9 : LinearOrder \u03b9\ninst\u271d : AddCommMonoid E\ns : Finset \u03b9\nn : \u03b9\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c4 \u03c9 < n \u2192 \u03c4 \u03c9 \u2208 s\n\u03c9 : \u03a9\nh : \u03c4 \u03c9 < n\n\u22a2 \u03c4 \u03c9 \u2208 Finset.filter (fun x => x < n) s", "state_after": "case h.inr.h\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nE : Type u_4\np : \u211d\u22650\u221e\nu : \u03b9 \u2192 \u03a9 \u2192 E\ninst\u271d\u00b9 : LinearOrder \u03b9\ninst\u271d : AddCommMonoid E\ns : Finset \u03b9\nn : \u03b9\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c4 \u03c9 < n \u2192 \u03c4 \u03c9 \u2208 s\n\u03c9 : \u03a9\nh : \u03c4 \u03c9 < n\n\u22a2 \u03c4 \u03c9 \u2208 s \u2227 \u03c4 \u03c9 < n"}, {"tactic": "exact \u27e8hbdd \u03c9 h, h\u27e9", "annotated_tactic": ["exact \u27e8hbdd \u03c9 h, h\u27e9", []], "state_before": "case h.inr.h\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nE : Type u_4\np : \u211d\u22650\u221e\nu : \u03b9 \u2192 \u03a9 \u2192 E\ninst\u271d\u00b9 : LinearOrder \u03b9\ninst\u271d : AddCommMonoid E\ns : Finset \u03b9\nn : \u03b9\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c4 \u03c9 < n \u2192 \u03c4 \u03c9 \u2208 s\n\u03c9 : \u03a9\nh : \u03c4 \u03c9 < n\n\u22a2 \u03c4 \u03c9 \u2208 s \u2227 \u03c4 \u03c9 < n", "state_after": "no goals"}, {"tactic": "intro b _ hneq", "annotated_tactic": ["intro b _ hneq", []], "state_before": "case h.inr.h\u2080\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nE : Type u_4\np : \u211d\u22650\u221e\nu : \u03b9 \u2192 \u03a9 \u2192 E\ninst\u271d\u00b9 : LinearOrder \u03b9\ninst\u271d : AddCommMonoid E\ns : Finset \u03b9\nn : \u03b9\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c4 \u03c9 < n \u2192 \u03c4 \u03c9 \u2208 s\n\u03c9 : \u03a9\nh : \u03c4 \u03c9 < n\n\u22a2 \u2200 (b : \u03b9), b \u2208 Finset.filter (fun x => x < n) s \u2192 b \u2260 \u03c4 \u03c9 \u2192 Set.indicator {\u03c9 | \u03c4 \u03c9 = b} (u b) \u03c9 = 0", "state_after": "case h.inr.h\u2080\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nE : Type u_4\np : \u211d\u22650\u221e\nu : \u03b9 \u2192 \u03a9 \u2192 E\ninst\u271d\u00b9 : LinearOrder \u03b9\ninst\u271d : AddCommMonoid E\ns : Finset \u03b9\nn : \u03b9\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c4 \u03c9 < n \u2192 \u03c4 \u03c9 \u2208 s\n\u03c9 : \u03a9\nh : \u03c4 \u03c9 < n\nb : \u03b9\na\u271d : b \u2208 Finset.filter (fun x => x < n) s\nhneq : b \u2260 \u03c4 \u03c9\n\u22a2 Set.indicator {\u03c9 | \u03c4 \u03c9 = b} (u b) \u03c9 = 0"}, {"tactic": "rw [Set.indicator_of_not_mem]", "annotated_tactic": ["rw [<a>Set.indicator_of_not_mem</a>]", [{"full_name": "Set.indicator_of_not_mem", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [73, 3], "def_end_pos": [73, 14]}]], "state_before": "case h.inr.h\u2080\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nE : Type u_4\np : \u211d\u22650\u221e\nu : \u03b9 \u2192 \u03a9 \u2192 E\ninst\u271d\u00b9 : LinearOrder \u03b9\ninst\u271d : AddCommMonoid E\ns : Finset \u03b9\nn : \u03b9\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c4 \u03c9 < n \u2192 \u03c4 \u03c9 \u2208 s\n\u03c9 : \u03a9\nh : \u03c4 \u03c9 < n\nb : \u03b9\na\u271d : b \u2208 Finset.filter (fun x => x < n) s\nhneq : b \u2260 \u03c4 \u03c9\n\u22a2 Set.indicator {\u03c9 | \u03c4 \u03c9 = b} (u b) \u03c9 = 0", "state_after": "case h.inr.h\u2080.h\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nE : Type u_4\np : \u211d\u22650\u221e\nu : \u03b9 \u2192 \u03a9 \u2192 E\ninst\u271d\u00b9 : LinearOrder \u03b9\ninst\u271d : AddCommMonoid E\ns : Finset \u03b9\nn : \u03b9\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c4 \u03c9 < n \u2192 \u03c4 \u03c9 \u2208 s\n\u03c9 : \u03a9\nh : \u03c4 \u03c9 < n\nb : \u03b9\na\u271d : b \u2208 Finset.filter (fun x => x < n) s\nhneq : b \u2260 \u03c4 \u03c9\n\u22a2 \u00ac\u03c9 \u2208 {\u03c9 | \u03c4 \u03c9 = b}"}, {"tactic": "exact hneq.symm", "annotated_tactic": ["exact hneq.symm", []], "state_before": "case h.inr.h\u2080.h\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nE : Type u_4\np : \u211d\u22650\u221e\nu : \u03b9 \u2192 \u03a9 \u2192 E\ninst\u271d\u00b9 : LinearOrder \u03b9\ninst\u271d : AddCommMonoid E\ns : Finset \u03b9\nn : \u03b9\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c4 \u03c9 < n \u2192 \u03c4 \u03c9 \u2208 s\n\u03c9 : \u03a9\nh : \u03c4 \u03c9 < n\nb : \u03b9\na\u271d : b \u2208 Finset.filter (fun x => x < n) s\nhneq : b \u2260 \u03c4 \u03c9\n\u22a2 \u00ac\u03c9 \u2208 {\u03c9 | \u03c4 \u03c9 = b}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Group/Measure.lean", "full_name": "MeasureTheory.eventually_mul_right_iff", "start": [368, 1], "end": [370, 66], "traced_tactics": [{"tactic": "conv_rhs => rw [Filter.Eventually, \u2190 map_mul_right_ae \u03bc t]; rfl", "annotated_tactic": ["conv_rhs => rw [<a>Filter.Eventually</a>, \u2190 <a>map_mul_right_ae</a> \u03bc t]; rfl", [{"full_name": "Filter.Eventually", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1072, 15], "def_end_pos": [1072, 25]}, {"full_name": "MeasureTheory.map_mul_right_ae", "def_path": "Mathlib/MeasureTheory/Group/Measure.lean", "def_pos": [347, 9], "def_end_pos": [347, 25]}]], "state_before": "\ud835\udd5c : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : MeasurableSpace H\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : MeasurableMul G\n\u03bc : Measure G\ninst\u271d : IsMulRightInvariant \u03bc\nt : G\np : G \u2192 Prop\n\u22a2 (\u2200\u1d50 (x : G) \u2202\u03bc, p (x * t)) \u2194 \u2200\u1d50 (x : G) \u2202\u03bc, p x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Erased.lean", "full_name": "Erased.join_eq_out", "start": [118, 1], "end": [119, 18], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "full_name": "MeasureTheory.Measure.toOuterMeasure_top", "start": [1076, 1], "end": [1078, 85], "traced_tactics": [{"tactic": "rw [\u2190 OuterMeasure.toMeasure_top, toMeasure_toOuterMeasure, OuterMeasure.trim_top]", "annotated_tactic": ["rw [\u2190 <a>OuterMeasure.toMeasure_top</a>, <a>toMeasure_toOuterMeasure</a>, <a>OuterMeasure.trim_top</a>]", [{"full_name": "MeasureTheory.OuterMeasure.toMeasure_top", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1067, 9], "def_end_pos": [1067, 56]}, {"full_name": "MeasureTheory.toMeasure_toOuterMeasure", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [665, 9], "def_end_pos": [665, 33]}, {"full_name": "MeasureTheory.OuterMeasure.trim_top", "def_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "def_pos": [1664, 9], "def_end_pos": [1664, 17]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d : MeasurableSpace \u03b1\n\u22a2 \u2191\u22a4 = \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Image.lean", "full_name": "Set.preimage_inr_range_inl", "start": [948, 1], "end": [949, 44], "traced_tactics": [{"tactic": "rw [\u2190 image_univ, preimage_inr_image_inl]", "annotated_tactic": ["rw [\u2190 <a>image_univ</a>, <a>preimage_inr_image_inl</a>]", [{"full_name": "Set.image_univ", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [718, 9], "def_end_pos": [718, 19]}, {"full_name": "Set.preimage_inr_image_inl", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [937, 9], "def_end_pos": [937, 31]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\nf : \u03b9 \u2192 \u03b1\ns t : Set \u03b1\n\u22a2 Sum.inr \u207b\u00b9' range Sum.inl = \u2205", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "full_name": "MeasureTheory.setToFun_smul", "start": [1406, 1], "end": [1415, 66], "traced_tactics": [{"tactic": "by_cases hf : Integrable f \u03bc", "annotated_tactic": ["by_cases hf : <a>Integrable</a> f \u03bc", [{"full_name": "MeasureTheory.Integrable", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [442, 5], "def_end_pos": [442, 15]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : NormedAddCommGroup F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \ud835\udd5c F\nhT : DominatedFinMeasAdditive \u03bc T C\nh_smul : \u2200 (c : \ud835\udd5c) (s : Set \u03b1) (x : E), \u2191(T s) (c \u2022 x) = c \u2022 \u2191(T s) x\nc : \ud835\udd5c\nf : \u03b1 \u2192 E\n\u22a2 setToFun \u03bc T hT (c \u2022 f) = c \u2022 setToFun \u03bc T hT f", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : NormedAddCommGroup F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \ud835\udd5c F\nhT : DominatedFinMeasAdditive \u03bc T C\nh_smul : \u2200 (c : \ud835\udd5c) (s : Set \u03b1) (x : E), \u2191(T s) (c \u2022 x) = c \u2022 \u2191(T s) x\nc : \ud835\udd5c\nf : \u03b1 \u2192 E\nhf : Integrable f\n\u22a2 setToFun \u03bc T hT (c \u2022 f) = c \u2022 setToFun \u03bc T hT f\n\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : NormedAddCommGroup F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \ud835\udd5c F\nhT : DominatedFinMeasAdditive \u03bc T C\nh_smul : \u2200 (c : \ud835\udd5c) (s : Set \u03b1) (x : E), \u2191(T s) (c \u2022 x) = c \u2022 \u2191(T s) x\nc : \ud835\udd5c\nf : \u03b1 \u2192 E\nhf : \u00acIntegrable f\n\u22a2 setToFun \u03bc T hT (c \u2022 f) = c \u2022 setToFun \u03bc T hT f"}, {"tactic": "rw [setToFun_eq hT hf, setToFun_eq hT, Integrable.toL1_smul',\n  L1.setToL1_smul hT h_smul c _]", "annotated_tactic": ["rw [<a>setToFun_eq</a> hT hf, <a>setToFun_eq</a> hT, <a>Integrable.toL1_smul'</a>,\n      <a>L1.setToL1_smul</a> hT h_smul c _]", [{"full_name": "MeasureTheory.setToFun_eq", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [1276, 9], "def_end_pos": [1276, 20]}, {"full_name": "MeasureTheory.setToFun_eq", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [1276, 9], "def_end_pos": [1276, 20]}, {"full_name": "MeasureTheory.Integrable.toL1_smul'", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [1485, 9], "def_end_pos": [1485, 19]}, {"full_name": "MeasureTheory.L1.setToL1_smul", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [1129, 9], "def_end_pos": [1129, 21]}]], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : NormedAddCommGroup F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \ud835\udd5c F\nhT : DominatedFinMeasAdditive \u03bc T C\nh_smul : \u2200 (c : \ud835\udd5c) (s : Set \u03b1) (x : E), \u2191(T s) (c \u2022 x) = c \u2022 \u2191(T s) x\nc : \ud835\udd5c\nf : \u03b1 \u2192 E\nhf : Integrable f\n\u22a2 setToFun \u03bc T hT (c \u2022 f) = c \u2022 setToFun \u03bc T hT f", "state_after": "no goals"}, {"tactic": "by_cases hr : c = 0", "annotated_tactic": ["by_cases hr : c = 0", []], "state_before": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : NormedAddCommGroup F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \ud835\udd5c F\nhT : DominatedFinMeasAdditive \u03bc T C\nh_smul : \u2200 (c : \ud835\udd5c) (s : Set \u03b1) (x : E), \u2191(T s) (c \u2022 x) = c \u2022 \u2191(T s) x\nc : \ud835\udd5c\nf : \u03b1 \u2192 E\nhf : \u00acIntegrable f\n\u22a2 setToFun \u03bc T hT (c \u2022 f) = c \u2022 setToFun \u03bc T hT f", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : NormedAddCommGroup F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \ud835\udd5c F\nhT : DominatedFinMeasAdditive \u03bc T C\nh_smul : \u2200 (c : \ud835\udd5c) (s : Set \u03b1) (x : E), \u2191(T s) (c \u2022 x) = c \u2022 \u2191(T s) x\nc : \ud835\udd5c\nf : \u03b1 \u2192 E\nhf : \u00acIntegrable f\nhr : c = 0\n\u22a2 setToFun \u03bc T hT (c \u2022 f) = c \u2022 setToFun \u03bc T hT f\n\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : NormedAddCommGroup F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \ud835\udd5c F\nhT : DominatedFinMeasAdditive \u03bc T C\nh_smul : \u2200 (c : \ud835\udd5c) (s : Set \u03b1) (x : E), \u2191(T s) (c \u2022 x) = c \u2022 \u2191(T s) x\nc : \ud835\udd5c\nf : \u03b1 \u2192 E\nhf : \u00acIntegrable f\nhr : \u00acc = 0\n\u22a2 setToFun \u03bc T hT (c \u2022 f) = c \u2022 setToFun \u03bc T hT f"}, {"tactic": "rw [hr]", "annotated_tactic": ["rw [hr]", []], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : NormedAddCommGroup F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \ud835\udd5c F\nhT : DominatedFinMeasAdditive \u03bc T C\nh_smul : \u2200 (c : \ud835\udd5c) (s : Set \u03b1) (x : E), \u2191(T s) (c \u2022 x) = c \u2022 \u2191(T s) x\nc : \ud835\udd5c\nf : \u03b1 \u2192 E\nhf : \u00acIntegrable f\nhr : c = 0\n\u22a2 setToFun \u03bc T hT (c \u2022 f) = c \u2022 setToFun \u03bc T hT f", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : NormedAddCommGroup F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \ud835\udd5c F\nhT : DominatedFinMeasAdditive \u03bc T C\nh_smul : \u2200 (c : \ud835\udd5c) (s : Set \u03b1) (x : E), \u2191(T s) (c \u2022 x) = c \u2022 \u2191(T s) x\nc : \ud835\udd5c\nf : \u03b1 \u2192 E\nhf : \u00acIntegrable f\nhr : c = 0\n\u22a2 setToFun \u03bc T hT (0 \u2022 f) = 0 \u2022 setToFun \u03bc T hT f"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : NormedAddCommGroup F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \ud835\udd5c F\nhT : DominatedFinMeasAdditive \u03bc T C\nh_smul : \u2200 (c : \ud835\udd5c) (s : Set \u03b1) (x : E), \u2191(T s) (c \u2022 x) = c \u2022 \u2191(T s) x\nc : \ud835\udd5c\nf : \u03b1 \u2192 E\nhf : \u00acIntegrable f\nhr : c = 0\n\u22a2 setToFun \u03bc T hT (0 \u2022 f) = 0 \u2022 setToFun \u03bc T hT f", "state_after": "no goals"}, {"tactic": "have hf' : \u00acIntegrable (c \u2022 f) \u03bc := by rwa [integrable_smul_iff hr f]", "annotated_tactic": ["have hf' : \u00ac<a>Integrable</a> (c \u2022 f) \u03bc := by rwa [<a>integrable_smul_iff</a> hr f]", [{"full_name": "MeasureTheory.Integrable", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [442, 5], "def_end_pos": [442, 15]}, {"full_name": "MeasureTheory.integrable_smul_iff", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [1081, 9], "def_end_pos": [1081, 28]}]], "state_before": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : NormedAddCommGroup F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \ud835\udd5c F\nhT : DominatedFinMeasAdditive \u03bc T C\nh_smul : \u2200 (c : \ud835\udd5c) (s : Set \u03b1) (x : E), \u2191(T s) (c \u2022 x) = c \u2022 \u2191(T s) x\nc : \ud835\udd5c\nf : \u03b1 \u2192 E\nhf : \u00acIntegrable f\nhr : \u00acc = 0\n\u22a2 setToFun \u03bc T hT (c \u2022 f) = c \u2022 setToFun \u03bc T hT f", "state_after": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : NormedAddCommGroup F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \ud835\udd5c F\nhT : DominatedFinMeasAdditive \u03bc T C\nh_smul : \u2200 (c : \ud835\udd5c) (s : Set \u03b1) (x : E), \u2191(T s) (c \u2022 x) = c \u2022 \u2191(T s) x\nc : \ud835\udd5c\nf : \u03b1 \u2192 E\nhf : \u00acIntegrable f\nhr : \u00acc = 0\nhf' : \u00acIntegrable (c \u2022 f)\n\u22a2 setToFun \u03bc T hT (c \u2022 f) = c \u2022 setToFun \u03bc T hT f"}, {"tactic": "rw [setToFun_undef hT hf, setToFun_undef hT hf', smul_zero]", "annotated_tactic": ["rw [<a>setToFun_undef</a> hT hf, <a>setToFun_undef</a> hT hf', <a>smul_zero</a>]", [{"full_name": "MeasureTheory.setToFun_undef", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [1286, 9], "def_end_pos": [1286, 23]}, {"full_name": "MeasureTheory.setToFun_undef", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [1286, 9], "def_end_pos": [1286, 23]}, {"full_name": "smul_zero", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [732, 9], "def_end_pos": [732, 18]}]], "state_before": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : NormedAddCommGroup F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \ud835\udd5c F\nhT : DominatedFinMeasAdditive \u03bc T C\nh_smul : \u2200 (c : \ud835\udd5c) (s : Set \u03b1) (x : E), \u2191(T s) (c \u2022 x) = c \u2022 \u2191(T s) x\nc : \ud835\udd5c\nf : \u03b1 \u2192 E\nhf : \u00acIntegrable f\nhr : \u00acc = 0\nhf' : \u00acIntegrable (c \u2022 f)\n\u22a2 setToFun \u03bc T hT (c \u2022 f) = c \u2022 setToFun \u03bc T hT f", "state_after": "no goals"}, {"tactic": "rwa [integrable_smul_iff hr f]", "annotated_tactic": ["rwa [<a>integrable_smul_iff</a> hr f]", [{"full_name": "MeasureTheory.integrable_smul_iff", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [1081, 9], "def_end_pos": [1081, 28]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : NormedAddCommGroup F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \ud835\udd5c F\nhT : DominatedFinMeasAdditive \u03bc T C\nh_smul : \u2200 (c : \ud835\udd5c) (s : Set \u03b1) (x : E), \u2191(T s) (c \u2022 x) = c \u2022 \u2191(T s) x\nc : \ud835\udd5c\nf : \u03b1 \u2192 E\nhf : \u00acIntegrable f\nhr : \u00acc = 0\n\u22a2 \u00acIntegrable (c \u2022 f)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "full_name": "MeasureTheory.snorm'_add_le_of_le_one", "start": [783, 1], "end": [792, 63], "traced_tactics": [{"tactic": "refine' ENNReal.rpow_le_rpow _ (by simp [hq0] : 0 \u2264 1 / q)", "annotated_tactic": ["refine' <a>ENNReal.rpow_le_rpow</a> _ (by simp [hq0] : 0 \u2264 1 / q)", [{"full_name": "ENNReal.rpow_le_rpow", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [642, 9], "def_end_pos": [642, 21]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf g : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhq0 : 0 \u2264 q\nhq1 : q \u2264 1\n\u22a2 (\u222b\u207b (a : \u03b1), \u2191\u2016(f + g) a\u2016\u208a ^ q \u2202\u03bc) ^ (1 / q) \u2264\n    (\u222b\u207b (a : \u03b1), ((fun a => \u2191\u2016f a\u2016\u208a) + fun a => \u2191\u2016g a\u2016\u208a) a ^ q \u2202\u03bc) ^ (1 / q)", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf g : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhq0 : 0 \u2264 q\nhq1 : q \u2264 1\n\u22a2 \u222b\u207b (a : \u03b1), \u2191\u2016(f + g) a\u2016\u208a ^ q \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), ((fun a => \u2191\u2016f a\u2016\u208a) + fun a => \u2191\u2016g a\u2016\u208a) a ^ q \u2202\u03bc"}, {"tactic": "refine' lintegral_mono fun a => ENNReal.rpow_le_rpow _ hq0", "annotated_tactic": ["refine' <a>lintegral_mono</a> fun a => <a>ENNReal.rpow_le_rpow</a> _ hq0", [{"full_name": "MeasureTheory.lintegral_mono", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [99, 9], "def_end_pos": [99, 23]}, {"full_name": "ENNReal.rpow_le_rpow", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [642, 9], "def_end_pos": [642, 21]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf g : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhq0 : 0 \u2264 q\nhq1 : q \u2264 1\n\u22a2 \u222b\u207b (a : \u03b1), \u2191\u2016(f + g) a\u2016\u208a ^ q \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), ((fun a => \u2191\u2016f a\u2016\u208a) + fun a => \u2191\u2016g a\u2016\u208a) a ^ q \u2202\u03bc", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf g : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhq0 : 0 \u2264 q\nhq1 : q \u2264 1\na : \u03b1\n\u22a2 \u2191\u2016(f + g) a\u2016\u208a \u2264 ((fun a => \u2191\u2016f a\u2016\u208a) + fun a => \u2191\u2016g a\u2016\u208a) a"}, {"tactic": "simp only [Pi.add_apply, \u2190 ENNReal.coe_add, ENNReal.coe_le_coe, nnnorm_add_le]", "annotated_tactic": ["simp only [<a>Pi.add_apply</a>, \u2190 <a>ENNReal.coe_add</a>, <a>ENNReal.coe_le_coe</a>, <a>nnnorm_add_le</a>]", [{"full_name": "Pi.add_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [82, 3], "def_end_pos": [82, 14]}, {"full_name": "ENNReal.coe_add", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [386, 28], "def_end_pos": [386, 35]}, {"full_name": "ENNReal.coe_le_coe", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [349, 28], "def_end_pos": [349, 38]}, {"full_name": "nnnorm_add_le", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [946, 15], "def_end_pos": [946, 28]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf g : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhq0 : 0 \u2264 q\nhq1 : q \u2264 1\na : \u03b1\n\u22a2 \u2191\u2016(f + g) a\u2016\u208a \u2264 ((fun a => \u2191\u2016f a\u2016\u208a) + fun a => \u2191\u2016g a\u2016\u208a) a", "state_after": "no goals"}, {"tactic": "simp [hq0]", "annotated_tactic": ["simp [hq0]", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf g : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhq0 : 0 \u2264 q\nhq1 : q \u2264 1\n\u22a2 0 \u2264 1 / q", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Pairwise/Basic.lean", "full_name": "Set.pairwise_insert", "start": [151, 1], "end": [154, 15], "traced_tactics": [{"tactic": "simp only [insert_eq, pairwise_union, pairwise_singleton, true_and_iff, mem_singleton_iff,\n  forall_eq]", "annotated_tactic": ["simp only [<a>insert_eq</a>, <a>pairwise_union</a>, <a>pairwise_singleton</a>, <a>true_and_iff</a>, <a>mem_singleton_iff</a>,\n    <a>forall_eq</a>]", [{"full_name": "Set.insert_eq", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1310, 9], "def_end_pos": [1310, 18]}, {"full_name": "Set.pairwise_union", "def_path": "Mathlib/Data/Set/Pairwise/Basic.lean", "def_pos": [137, 9], "def_end_pos": [137, 23]}, {"full_name": "Set.pairwise_singleton", "def_path": "Mathlib/Data/Set/Pairwise/Basic.lean", "def_pos": [89, 9], "def_end_pos": [89, 27]}, {"full_name": "true_and_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [147, 9], "def_end_pos": [147, 21]}, {"full_name": "Set.mem_singleton_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1273, 9], "def_end_pos": [1273, 26]}, {"full_name": "forall_eq", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [450, 17], "def_end_pos": [450, 26]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\nf g : \u03b9 \u2192 \u03b1\ns t u : Set \u03b1\na b : \u03b1\n\u22a2 Set.Pairwise (insert a s) r \u2194 Set.Pairwise s r \u2227 \u2200 (b : \u03b1), b \u2208 s \u2192 a \u2260 b \u2192 r a b \u2227 r b a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Holor.lean", "full_name": "Holor.mul_assoc0", "start": [194, 1], "end": [201, 22], "traced_tactics": [{"tactic": "rw [assocLeft]", "annotated_tactic": ["rw [<a>assocLeft</a>]", [{"full_name": "Holor.assocLeft", "def_path": "Mathlib/Data/Holor.lean", "def_pos": [190, 5], "def_end_pos": [190, 14]}]], "state_before": "\u03b1 : Type\nd : \u2115\nds ds\u2081 ds\u2082 ds\u2083 : List \u2115\ninst\u271d : Semigroup \u03b1\nx : Holor \u03b1 ds\u2081\ny : Holor \u03b1 ds\u2082\nz : Holor \u03b1 ds\u2083\nt : HolorIndex (ds\u2081 ++ ds\u2082 ++ ds\u2083)\n\u22a2 (x \u2297 y \u2297 z) t = assocLeft (x \u2297 (y \u2297 z)) t", "state_after": "\u03b1 : Type\nd : \u2115\nds ds\u2081 ds\u2082 ds\u2083 : List \u2115\ninst\u271d : Semigroup \u03b1\nx : Holor \u03b1 ds\u2081\ny : Holor \u03b1 ds\u2082\nz : Holor \u03b1 ds\u2083\nt : HolorIndex (ds\u2081 ++ ds\u2082 ++ ds\u2083)\n\u22a2 (x \u2297 y \u2297 z) t = cast (_ : Holor \u03b1 (ds\u2081 ++ (ds\u2082 ++ ds\u2083)) = Holor \u03b1 (ds\u2081 ++ ds\u2082 ++ ds\u2083)) (x \u2297 (y \u2297 z)) t"}, {"tactic": "unfold mul", "annotated_tactic": ["unfold <a>mul</a>", [{"full_name": "Holor.mul", "def_path": "Mathlib/Data/Holor.lean", "def_pos": [175, 5], "def_end_pos": [175, 8]}]], "state_before": "\u03b1 : Type\nd : \u2115\nds ds\u2081 ds\u2082 ds\u2083 : List \u2115\ninst\u271d : Semigroup \u03b1\nx : Holor \u03b1 ds\u2081\ny : Holor \u03b1 ds\u2082\nz : Holor \u03b1 ds\u2083\nt : HolorIndex (ds\u2081 ++ ds\u2082 ++ ds\u2083)\n\u22a2 (x \u2297 y \u2297 z) t = cast (_ : Holor \u03b1 (ds\u2081 ++ (ds\u2082 ++ ds\u2083)) = Holor \u03b1 (ds\u2081 ++ ds\u2082 ++ ds\u2083)) (x \u2297 (y \u2297 z)) t", "state_after": "\u03b1 : Type\nd : \u2115\nds ds\u2081 ds\u2082 ds\u2083 : List \u2115\ninst\u271d : Semigroup \u03b1\nx : Holor \u03b1 ds\u2081\ny : Holor \u03b1 ds\u2082\nz : Holor \u03b1 ds\u2083\nt : HolorIndex (ds\u2081 ++ ds\u2082 ++ ds\u2083)\n\u22a2 x (HolorIndex.take (HolorIndex.take t)) * y (HolorIndex.drop (HolorIndex.take t)) * z (HolorIndex.drop t) =\n    cast (_ : Holor \u03b1 (ds\u2081 ++ (ds\u2082 ++ ds\u2083)) = Holor \u03b1 (ds\u2081 ++ ds\u2082 ++ ds\u2083))\n      (fun t =>\n        x (HolorIndex.take t) * (y (HolorIndex.take (HolorIndex.drop t)) * z (HolorIndex.drop (HolorIndex.drop t))))\n      t"}, {"tactic": "rw [mul_assoc, \u2190HolorIndex.take_take, \u2190HolorIndex.drop_take, \u2190HolorIndex.drop_drop, cast_type]", "annotated_tactic": ["rw [<a>mul_assoc</a>, \u2190<a>HolorIndex.take_take</a>, \u2190<a>HolorIndex.drop_take</a>, \u2190<a>HolorIndex.drop_drop</a>, <a>cast_type</a>]", [{"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [264, 9], "def_end_pos": [264, 18]}, {"full_name": "HolorIndex.take_take", "def_path": "Mathlib/Data/Holor.lean", "def_pos": [71, 9], "def_end_pos": [71, 18]}, {"full_name": "HolorIndex.drop_take", "def_path": "Mathlib/Data/Holor.lean", "def_pos": [77, 9], "def_end_pos": [77, 18]}, {"full_name": "HolorIndex.drop_drop", "def_path": "Mathlib/Data/Holor.lean", "def_pos": [81, 9], "def_end_pos": [81, 18]}, {"full_name": "Holor.cast_type", "def_path": "Mathlib/Data/Holor.lean", "def_pos": [181, 9], "def_end_pos": [181, 18]}]], "state_before": "\u03b1 : Type\nd : \u2115\nds ds\u2081 ds\u2082 ds\u2083 : List \u2115\ninst\u271d : Semigroup \u03b1\nx : Holor \u03b1 ds\u2081\ny : Holor \u03b1 ds\u2082\nz : Holor \u03b1 ds\u2083\nt : HolorIndex (ds\u2081 ++ ds\u2082 ++ ds\u2083)\n\u22a2 x (HolorIndex.take (HolorIndex.take t)) * y (HolorIndex.drop (HolorIndex.take t)) * z (HolorIndex.drop t) =\n    cast (_ : Holor \u03b1 (ds\u2081 ++ (ds\u2082 ++ ds\u2083)) = Holor \u03b1 (ds\u2081 ++ ds\u2082 ++ ds\u2083))\n      (fun t =>\n        x (HolorIndex.take t) * (y (HolorIndex.take (HolorIndex.drop t)) * z (HolorIndex.drop (HolorIndex.drop t))))\n      t", "state_after": "\u03b1 : Type\nd : \u2115\nds ds\u2081 ds\u2082 ds\u2083 : List \u2115\ninst\u271d : Semigroup \u03b1\nx : Holor \u03b1 ds\u2081\ny : Holor \u03b1 ds\u2082\nz : Holor \u03b1 ds\u2083\nt : HolorIndex (ds\u2081 ++ ds\u2082 ++ ds\u2083)\n\u22a2 x (HolorIndex.take (HolorIndex.assocRight t)) *\n      (y (HolorIndex.take (HolorIndex.drop (HolorIndex.assocRight t))) *\n        z (HolorIndex.drop (HolorIndex.drop (HolorIndex.assocRight t)))) =\n    (fun t =>\n        x (HolorIndex.take (cast (_ : HolorIndex (ds\u2081 ++ ds\u2082 ++ ds\u2083) = HolorIndex (ds\u2081 ++ (ds\u2082 ++ ds\u2083))) t)) *\n          (y\n              (HolorIndex.take\n                (HolorIndex.drop (cast (_ : HolorIndex (ds\u2081 ++ ds\u2082 ++ ds\u2083) = HolorIndex (ds\u2081 ++ (ds\u2082 ++ ds\u2083))) t))) *\n            z\n              (HolorIndex.drop\n                (HolorIndex.drop (cast (_ : HolorIndex (ds\u2081 ++ ds\u2082 ++ ds\u2083) = HolorIndex (ds\u2081 ++ (ds\u2082 ++ ds\u2083))) t)))))\n      t\n\ncase eq\n\u03b1 : Type\nd : \u2115\nds ds\u2081 ds\u2082 ds\u2083 : List \u2115\ninst\u271d : Semigroup \u03b1\nx : Holor \u03b1 ds\u2081\ny : Holor \u03b1 ds\u2082\nz : Holor \u03b1 ds\u2083\nt : HolorIndex (ds\u2081 ++ ds\u2082 ++ ds\u2083)\n\u22a2 ds\u2081 ++ (ds\u2082 ++ ds\u2083) = ds\u2081 ++ ds\u2082 ++ ds\u2083"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\u03b1 : Type\nd : \u2115\nds ds\u2081 ds\u2082 ds\u2083 : List \u2115\ninst\u271d : Semigroup \u03b1\nx : Holor \u03b1 ds\u2081\ny : Holor \u03b1 ds\u2082\nz : Holor \u03b1 ds\u2083\nt : HolorIndex (ds\u2081 ++ ds\u2082 ++ ds\u2083)\n\u22a2 x (HolorIndex.take (HolorIndex.assocRight t)) *\n      (y (HolorIndex.take (HolorIndex.drop (HolorIndex.assocRight t))) *\n        z (HolorIndex.drop (HolorIndex.drop (HolorIndex.assocRight t)))) =\n    (fun t =>\n        x (HolorIndex.take (cast (_ : HolorIndex (ds\u2081 ++ ds\u2082 ++ ds\u2083) = HolorIndex (ds\u2081 ++ (ds\u2082 ++ ds\u2083))) t)) *\n          (y\n              (HolorIndex.take\n                (HolorIndex.drop (cast (_ : HolorIndex (ds\u2081 ++ ds\u2082 ++ ds\u2083) = HolorIndex (ds\u2081 ++ (ds\u2082 ++ ds\u2083))) t))) *\n            z\n              (HolorIndex.drop\n                (HolorIndex.drop (cast (_ : HolorIndex (ds\u2081 ++ ds\u2082 ++ ds\u2083) = HolorIndex (ds\u2081 ++ (ds\u2082 ++ ds\u2083))) t)))))\n      t\n\ncase eq\n\u03b1 : Type\nd : \u2115\nds ds\u2081 ds\u2082 ds\u2083 : List \u2115\ninst\u271d : Semigroup \u03b1\nx : Holor \u03b1 ds\u2081\ny : Holor \u03b1 ds\u2082\nz : Holor \u03b1 ds\u2083\nt : HolorIndex (ds\u2081 ++ ds\u2082 ++ ds\u2083)\n\u22a2 ds\u2081 ++ (ds\u2082 ++ ds\u2083) = ds\u2081 ++ ds\u2082 ++ ds\u2083", "state_after": "case eq\n\u03b1 : Type\nd : \u2115\nds ds\u2081 ds\u2082 ds\u2083 : List \u2115\ninst\u271d : Semigroup \u03b1\nx : Holor \u03b1 ds\u2081\ny : Holor \u03b1 ds\u2082\nz : Holor \u03b1 ds\u2083\nt : HolorIndex (ds\u2081 ++ ds\u2082 ++ ds\u2083)\n\u22a2 ds\u2081 ++ (ds\u2082 ++ ds\u2083) = ds\u2081 ++ ds\u2082 ++ ds\u2083"}, {"tactic": "rw [append_assoc]", "annotated_tactic": ["rw [<a>append_assoc</a>]", [{"full_name": "List.append_assoc", "def_path": "lake-packages/lean4/src/lean/Init/Data/List/Basic.lean", "def_pos": [103, 9], "def_end_pos": [103, 21]}]], "state_before": "case eq\n\u03b1 : Type\nd : \u2115\nds ds\u2081 ds\u2082 ds\u2083 : List \u2115\ninst\u271d : Semigroup \u03b1\nx : Holor \u03b1 ds\u2081\ny : Holor \u03b1 ds\u2082\nz : Holor \u03b1 ds\u2083\nt : HolorIndex (ds\u2081 ++ ds\u2082 ++ ds\u2083)\n\u22a2 ds\u2081 ++ (ds\u2082 ++ ds\u2083) = ds\u2081 ++ ds\u2082 ++ ds\u2083", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "full_name": "MeasureTheory.integrable_add_measure", "start": [561, 1], "end": [563, 91], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/StrongLaw.lean", "full_name": "MeasureTheory.AEStronglyMeasurable.mem\u2112p_truncation", "start": [132, 1], "end": [134, 97], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Independence/Basic.lean", "full_name": "ProbabilityTheory.iIndepFun.indepFun", "start": [342, 1], "end": [346, 41], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/PeakFunction.lean", "full_name": "tendsto_set_integral_pow_smul_of_unique_maximum_of_isCompact_of_integrableOn", "start": [294, 1], "end": [309, 79], "traced_tactics": [{"tactic": "have : x\u2080 \u2208 s := by rw [\u2190 hs.isClosed.closure_eq]; exact closure_mono interior_subset h\u2080", "annotated_tactic": ["have : x\u2080 \u2208 s := by rw [\u2190 hs.isClosed.closure_eq]; exact <a>closure_mono</a> <a>interior_subset</a> h\u2080", [{"full_name": "closure_mono", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [475, 9], "def_end_pos": [475, 21]}, {"full_name": "interior_subset", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [302, 9], "def_end_pos": [302, 24]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2077 : TopologicalSpace \u03b1\ninst\u271d\u2076 : BorelSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : MetrizableSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsOpenPosMeasure \u03bc\nhs : IsCompact s\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 closure (interior s)\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u22a2 Tendsto (fun n => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 \u2022 \u222b (x : \u03b1) in s, c x ^ n \u2022 g x \u2202\u03bc) atTop (\ud835\udcdd (g x\u2080))", "state_after": "\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2077 : TopologicalSpace \u03b1\ninst\u271d\u2076 : BorelSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : MetrizableSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsOpenPosMeasure \u03bc\nhs : IsCompact s\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 closure (interior s)\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nthis : x\u2080 \u2208 s\n\u22a2 Tendsto (fun n => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 \u2022 \u222b (x : \u03b1) in s, c x ^ n \u2022 g x \u2202\u03bc) atTop (\ud835\udcdd (g x\u2080))"}, {"tactic": "apply\n  tendsto_set_integral_pow_smul_of_unique_maximum_of_isCompact_of_measure_nhdsWithin_pos hs _ hc\n    h'c hnc hnc\u2080 this hmg hcg", "annotated_tactic": ["apply\n    <a>tendsto_set_integral_pow_smul_of_unique_maximum_of_isCompact_of_measure_nhdsWithin_pos</a> hs _ hc\n      h'c hnc hnc\u2080 this hmg hcg", [{"full_name": "tendsto_set_integral_pow_smul_of_unique_maximum_of_isCompact_of_measure_nhdsWithin_pos", "def_path": "Mathlib/MeasureTheory/Integral/PeakFunction.lean", "def_pos": [195, 9], "def_end_pos": [195, 95]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2077 : TopologicalSpace \u03b1\ninst\u271d\u2076 : BorelSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : MetrizableSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsOpenPosMeasure \u03bc\nhs : IsCompact s\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 closure (interior s)\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nthis : x\u2080 \u2208 s\n\u22a2 Tendsto (fun n => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 \u2022 \u222b (x : \u03b1) in s, c x ^ n \u2022 g x \u2202\u03bc) atTop (\ud835\udcdd (g x\u2080))", "state_after": "\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2077 : TopologicalSpace \u03b1\ninst\u271d\u2076 : BorelSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : MetrizableSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsOpenPosMeasure \u03bc\nhs : IsCompact s\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 closure (interior s)\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nthis : x\u2080 \u2208 s\n\u22a2 \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)"}, {"tactic": "intro u u_open x\u2080_u", "annotated_tactic": ["intro u u_open x\u2080_u", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2077 : TopologicalSpace \u03b1\ninst\u271d\u2076 : BorelSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : MetrizableSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsOpenPosMeasure \u03bc\nhs : IsCompact s\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 closure (interior s)\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nthis : x\u2080 \u2208 s\n\u22a2 \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)", "state_after": "\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2077 : TopologicalSpace \u03b1\ninst\u271d\u2076 : BorelSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : MetrizableSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsOpenPosMeasure \u03bc\nhs : IsCompact s\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 closure (interior s)\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nthis : x\u2080 \u2208 s\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080_u : x\u2080 \u2208 u\n\u22a2 0 < \u2191\u2191\u03bc (u \u2229 s)"}, {"tactic": "calc\n  0 < \u03bc (u \u2229 interior s) :=\n    (u_open.inter isOpen_interior).measure_pos \u03bc (_root_.mem_closure_iff.1 h\u2080 u u_open x\u2080_u)\n  _ \u2264 \u03bc (u \u2229 s) := measure_mono (inter_subset_inter_right _ interior_subset)", "annotated_tactic": ["calc\n    0 < \u03bc (u \u2229 <a>interior</a> s) :=\n      (u_open.inter <a>isOpen_interior</a>).<a>measure_pos</a> \u03bc (<a>_root_.mem_closure_iff</a>.1 h\u2080 u u_open x\u2080_u)\n    _ \u2264 \u03bc (u \u2229 s) := <a>measure_mono</a> (<a>inter_subset_inter_right</a> _ <a>interior_subset</a>)", [{"full_name": "interior", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [288, 5], "def_end_pos": [288, 13]}, {"full_name": "isOpen_interior", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [298, 9], "def_end_pos": [298, 24]}, {"full_name": "IsOpen.measure_pos", "def_path": "Mathlib/MeasureTheory/Measure/OpenPos.lean", "def_pos": [46, 9], "def_end_pos": [46, 34]}, {"full_name": "mem_closure_iff", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [576, 9], "def_end_pos": [576, 24]}, {"full_name": "MeasureTheory.measure_mono", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [193, 9], "def_end_pos": [193, 21]}, {"full_name": "Set.inter_subset_inter_right", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1032, 9], "def_end_pos": [1032, 33]}, {"full_name": "interior_subset", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [302, 9], "def_end_pos": [302, 24]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2077 : TopologicalSpace \u03b1\ninst\u271d\u2076 : BorelSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : MetrizableSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsOpenPosMeasure \u03bc\nhs : IsCompact s\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 closure (interior s)\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nthis : x\u2080 \u2208 s\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080_u : x\u2080 \u2208 u\n\u22a2 0 < \u2191\u2191\u03bc (u \u2229 s)", "state_after": "no goals"}, {"tactic": "rw [\u2190 hs.isClosed.closure_eq]", "annotated_tactic": ["rw [\u2190 hs.isClosed.closure_eq]", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2077 : TopologicalSpace \u03b1\ninst\u271d\u2076 : BorelSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : MetrizableSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsOpenPosMeasure \u03bc\nhs : IsCompact s\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 closure (interior s)\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u22a2 x\u2080 \u2208 s", "state_after": "\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2077 : TopologicalSpace \u03b1\ninst\u271d\u2076 : BorelSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : MetrizableSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsOpenPosMeasure \u03bc\nhs : IsCompact s\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 closure (interior s)\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u22a2 x\u2080 \u2208 closure s"}, {"tactic": "exact closure_mono interior_subset h\u2080", "annotated_tactic": ["exact <a>closure_mono</a> <a>interior_subset</a> h\u2080", [{"full_name": "closure_mono", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [475, 9], "def_end_pos": [475, 21]}, {"full_name": "interior_subset", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [302, 9], "def_end_pos": [302, 24]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2077 : TopologicalSpace \u03b1\ninst\u271d\u2076 : BorelSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : MetrizableSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsOpenPosMeasure \u03bc\nhs : IsCompact s\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 closure (interior s)\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u22a2 x\u2080 \u2208 closure s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Process/Stopping.lean", "full_name": "MeasureTheory.IsStoppingTime.measurableSpace_le", "start": [379, 1], "end": [388, 31], "traced_tactics": [{"tactic": "cases isEmpty_or_nonempty \u03b9", "annotated_tactic": ["cases <a>isEmpty_or_nonempty</a> \u03b9", [{"full_name": "isEmpty_or_nonempty", "def_path": "Mathlib/Logic/IsEmpty.lean", "def_pos": [207, 9], "def_end_pos": [207, 28]}]], "state_before": "\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u00b2 : Preorder \u03b9\u271d\nf\u271d : Filtration \u03b9\u271d m\n\u03c4\u271d \u03c0 : \u03a9 \u2192 \u03b9\u271d\n\u03b9 : Type u_4\ninst\u271d\u00b9 : SemilatticeSup \u03b9\nf : Filtration \u03b9 m\n\u03c4 : \u03a9 \u2192 \u03b9\ninst\u271d : IsCountablyGenerated atTop\nh\u03c4 : IsStoppingTime f \u03c4\n\u22a2 IsStoppingTime.measurableSpace h\u03c4 \u2264 m", "state_after": "case inl\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u00b2 : Preorder \u03b9\u271d\nf\u271d : Filtration \u03b9\u271d m\n\u03c4\u271d \u03c0 : \u03a9 \u2192 \u03b9\u271d\n\u03b9 : Type u_4\ninst\u271d\u00b9 : SemilatticeSup \u03b9\nf : Filtration \u03b9 m\n\u03c4 : \u03a9 \u2192 \u03b9\ninst\u271d : IsCountablyGenerated atTop\nh\u03c4 : IsStoppingTime f \u03c4\nh\u271d : IsEmpty \u03b9\n\u22a2 IsStoppingTime.measurableSpace h\u03c4 \u2264 m\n\ncase inr\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u00b2 : Preorder \u03b9\u271d\nf\u271d : Filtration \u03b9\u271d m\n\u03c4\u271d \u03c0 : \u03a9 \u2192 \u03b9\u271d\n\u03b9 : Type u_4\ninst\u271d\u00b9 : SemilatticeSup \u03b9\nf : Filtration \u03b9 m\n\u03c4 : \u03a9 \u2192 \u03b9\ninst\u271d : IsCountablyGenerated atTop\nh\u03c4 : IsStoppingTime f \u03c4\nh\u271d : Nonempty \u03b9\n\u22a2 IsStoppingTime.measurableSpace h\u03c4 \u2264 m"}, {"tactic": "exact measurableSpace_le' h\u03c4", "annotated_tactic": ["exact <a>measurableSpace_le'</a> h\u03c4", [{"full_name": "MeasureTheory.IsStoppingTime.measurableSpace_le'", "def_path": "Mathlib/Probability/Process/Stopping.lean", "def_pos": [363, 9], "def_end_pos": [363, 28]}]], "state_before": "case inr\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u00b2 : Preorder \u03b9\u271d\nf\u271d : Filtration \u03b9\u271d m\n\u03c4\u271d \u03c0 : \u03a9 \u2192 \u03b9\u271d\n\u03b9 : Type u_4\ninst\u271d\u00b9 : SemilatticeSup \u03b9\nf : Filtration \u03b9 m\n\u03c4 : \u03a9 \u2192 \u03b9\ninst\u271d : IsCountablyGenerated atTop\nh\u03c4 : IsStoppingTime f \u03c4\nh\u271d : Nonempty \u03b9\n\u22a2 IsStoppingTime.measurableSpace h\u03c4 \u2264 m", "state_after": "no goals"}, {"tactic": "haveI : IsEmpty \u03a9 := \u27e8fun \u03c9 => IsEmpty.false (\u03c4 \u03c9)\u27e9", "annotated_tactic": ["haveI : <a>IsEmpty</a> \u03a9 := \u27e8fun \u03c9 => <a>IsEmpty.false</a> (\u03c4 \u03c9)\u27e9", [{"full_name": "IsEmpty", "def_path": "Mathlib/Logic/IsEmpty.lean", "def_pos": [26, 7], "def_end_pos": [26, 14]}, {"full_name": "IsEmpty.false", "def_path": "Mathlib/Logic/IsEmpty.lean", "def_pos": [27, 13], "def_end_pos": [27, 18]}]], "state_before": "case inl\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u00b2 : Preorder \u03b9\u271d\nf\u271d : Filtration \u03b9\u271d m\n\u03c4\u271d \u03c0 : \u03a9 \u2192 \u03b9\u271d\n\u03b9 : Type u_4\ninst\u271d\u00b9 : SemilatticeSup \u03b9\nf : Filtration \u03b9 m\n\u03c4 : \u03a9 \u2192 \u03b9\ninst\u271d : IsCountablyGenerated atTop\nh\u03c4 : IsStoppingTime f \u03c4\nh\u271d : IsEmpty \u03b9\n\u22a2 IsStoppingTime.measurableSpace h\u03c4 \u2264 m", "state_after": "case inl\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u00b2 : Preorder \u03b9\u271d\nf\u271d : Filtration \u03b9\u271d m\n\u03c4\u271d \u03c0 : \u03a9 \u2192 \u03b9\u271d\n\u03b9 : Type u_4\ninst\u271d\u00b9 : SemilatticeSup \u03b9\nf : Filtration \u03b9 m\n\u03c4 : \u03a9 \u2192 \u03b9\ninst\u271d : IsCountablyGenerated atTop\nh\u03c4 : IsStoppingTime f \u03c4\nh\u271d : IsEmpty \u03b9\nthis : IsEmpty \u03a9\n\u22a2 IsStoppingTime.measurableSpace h\u03c4 \u2264 m"}, {"tactic": "intro s _", "annotated_tactic": ["intro s _", []], "state_before": "case inl\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u00b2 : Preorder \u03b9\u271d\nf\u271d : Filtration \u03b9\u271d m\n\u03c4\u271d \u03c0 : \u03a9 \u2192 \u03b9\u271d\n\u03b9 : Type u_4\ninst\u271d\u00b9 : SemilatticeSup \u03b9\nf : Filtration \u03b9 m\n\u03c4 : \u03a9 \u2192 \u03b9\ninst\u271d : IsCountablyGenerated atTop\nh\u03c4 : IsStoppingTime f \u03c4\nh\u271d : IsEmpty \u03b9\nthis : IsEmpty \u03a9\n\u22a2 IsStoppingTime.measurableSpace h\u03c4 \u2264 m", "state_after": "case inl\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u00b2 : Preorder \u03b9\u271d\nf\u271d : Filtration \u03b9\u271d m\n\u03c4\u271d \u03c0 : \u03a9 \u2192 \u03b9\u271d\n\u03b9 : Type u_4\ninst\u271d\u00b9 : SemilatticeSup \u03b9\nf : Filtration \u03b9 m\n\u03c4 : \u03a9 \u2192 \u03b9\ninst\u271d : IsCountablyGenerated atTop\nh\u03c4 : IsStoppingTime f \u03c4\nh\u271d : IsEmpty \u03b9\nthis : IsEmpty \u03a9\ns : Set \u03a9\na\u271d : MeasurableSet s\n\u22a2 MeasurableSet s"}, {"tactic": "suffices hs : s = \u2205", "annotated_tactic": ["suffices hs : s = \u2205", []], "state_before": "case inl\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u00b2 : Preorder \u03b9\u271d\nf\u271d : Filtration \u03b9\u271d m\n\u03c4\u271d \u03c0 : \u03a9 \u2192 \u03b9\u271d\n\u03b9 : Type u_4\ninst\u271d\u00b9 : SemilatticeSup \u03b9\nf : Filtration \u03b9 m\n\u03c4 : \u03a9 \u2192 \u03b9\ninst\u271d : IsCountablyGenerated atTop\nh\u03c4 : IsStoppingTime f \u03c4\nh\u271d : IsEmpty \u03b9\nthis : IsEmpty \u03a9\ns : Set \u03a9\na\u271d : MeasurableSet s\n\u22a2 MeasurableSet s", "state_after": "case inl\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u00b2 : Preorder \u03b9\u271d\nf\u271d : Filtration \u03b9\u271d m\n\u03c4\u271d \u03c0 : \u03a9 \u2192 \u03b9\u271d\n\u03b9 : Type u_4\ninst\u271d\u00b9 : SemilatticeSup \u03b9\nf : Filtration \u03b9 m\n\u03c4 : \u03a9 \u2192 \u03b9\ninst\u271d : IsCountablyGenerated atTop\nh\u03c4 : IsStoppingTime f \u03c4\nh\u271d : IsEmpty \u03b9\nthis : IsEmpty \u03a9\ns : Set \u03a9\na\u271d : MeasurableSet s\nhs : s = \u2205\n\u22a2 MeasurableSet s\n\ncase hs\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u00b2 : Preorder \u03b9\u271d\nf\u271d : Filtration \u03b9\u271d m\n\u03c4\u271d \u03c0 : \u03a9 \u2192 \u03b9\u271d\n\u03b9 : Type u_4\ninst\u271d\u00b9 : SemilatticeSup \u03b9\nf : Filtration \u03b9 m\n\u03c4 : \u03a9 \u2192 \u03b9\ninst\u271d : IsCountablyGenerated atTop\nh\u03c4 : IsStoppingTime f \u03c4\nh\u271d : IsEmpty \u03b9\nthis : IsEmpty \u03a9\ns : Set \u03a9\na\u271d : MeasurableSet s\n\u22a2 s = \u2205"}, {"tactic": "haveI : Unique (Set \u03a9) := Set.uniqueEmpty", "annotated_tactic": ["haveI : <a>Unique</a> (<a>Set</a> \u03a9) := <a>Set.uniqueEmpty</a>", [{"full_name": "Unique", "def_path": "Mathlib/Logic/Unique.lean", "def_pos": [55, 11], "def_end_pos": [55, 17]}, {"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}, {"full_name": "Set.uniqueEmpty", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [599, 10], "def_end_pos": [599, 21]}]], "state_before": "case hs\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u00b2 : Preorder \u03b9\u271d\nf\u271d : Filtration \u03b9\u271d m\n\u03c4\u271d \u03c0 : \u03a9 \u2192 \u03b9\u271d\n\u03b9 : Type u_4\ninst\u271d\u00b9 : SemilatticeSup \u03b9\nf : Filtration \u03b9 m\n\u03c4 : \u03a9 \u2192 \u03b9\ninst\u271d : IsCountablyGenerated atTop\nh\u03c4 : IsStoppingTime f \u03c4\nh\u271d : IsEmpty \u03b9\nthis : IsEmpty \u03a9\ns : Set \u03a9\na\u271d : MeasurableSet s\n\u22a2 s = \u2205", "state_after": "case hs\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u00b2 : Preorder \u03b9\u271d\nf\u271d : Filtration \u03b9\u271d m\n\u03c4\u271d \u03c0 : \u03a9 \u2192 \u03b9\u271d\n\u03b9 : Type u_4\ninst\u271d\u00b9 : SemilatticeSup \u03b9\nf : Filtration \u03b9 m\n\u03c4 : \u03a9 \u2192 \u03b9\ninst\u271d : IsCountablyGenerated atTop\nh\u03c4 : IsStoppingTime f \u03c4\nh\u271d : IsEmpty \u03b9\nthis\u271d : IsEmpty \u03a9\ns : Set \u03a9\na\u271d : MeasurableSet s\nthis : Unique (Set \u03a9)\n\u22a2 s = \u2205"}, {"tactic": "rw [Unique.eq_default s, Unique.eq_default \u2205]", "annotated_tactic": ["rw [<a>Unique.eq_default</a> s, <a>Unique.eq_default</a> \u2205]", [{"full_name": "Unique.eq_default", "def_path": "Mathlib/Logic/Unique.lean", "def_pos": [143, 9], "def_end_pos": [143, 19]}, {"full_name": "Unique.eq_default", "def_path": "Mathlib/Logic/Unique.lean", "def_pos": [143, 9], "def_end_pos": [143, 19]}]], "state_before": "case hs\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u00b2 : Preorder \u03b9\u271d\nf\u271d : Filtration \u03b9\u271d m\n\u03c4\u271d \u03c0 : \u03a9 \u2192 \u03b9\u271d\n\u03b9 : Type u_4\ninst\u271d\u00b9 : SemilatticeSup \u03b9\nf : Filtration \u03b9 m\n\u03c4 : \u03a9 \u2192 \u03b9\ninst\u271d : IsCountablyGenerated atTop\nh\u03c4 : IsStoppingTime f \u03c4\nh\u271d : IsEmpty \u03b9\nthis\u271d : IsEmpty \u03a9\ns : Set \u03a9\na\u271d : MeasurableSet s\nthis : Unique (Set \u03a9)\n\u22a2 s = \u2205", "state_after": "no goals"}, {"tactic": "rw [hs]", "annotated_tactic": ["rw [hs]", []], "state_before": "case inl\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u00b2 : Preorder \u03b9\u271d\nf\u271d : Filtration \u03b9\u271d m\n\u03c4\u271d \u03c0 : \u03a9 \u2192 \u03b9\u271d\n\u03b9 : Type u_4\ninst\u271d\u00b9 : SemilatticeSup \u03b9\nf : Filtration \u03b9 m\n\u03c4 : \u03a9 \u2192 \u03b9\ninst\u271d : IsCountablyGenerated atTop\nh\u03c4 : IsStoppingTime f \u03c4\nh\u271d : IsEmpty \u03b9\nthis : IsEmpty \u03a9\ns : Set \u03a9\na\u271d : MeasurableSet s\nhs : s = \u2205\n\u22a2 MeasurableSet s", "state_after": "case inl\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u00b2 : Preorder \u03b9\u271d\nf\u271d : Filtration \u03b9\u271d m\n\u03c4\u271d \u03c0 : \u03a9 \u2192 \u03b9\u271d\n\u03b9 : Type u_4\ninst\u271d\u00b9 : SemilatticeSup \u03b9\nf : Filtration \u03b9 m\n\u03c4 : \u03a9 \u2192 \u03b9\ninst\u271d : IsCountablyGenerated atTop\nh\u03c4 : IsStoppingTime f \u03c4\nh\u271d : IsEmpty \u03b9\nthis : IsEmpty \u03a9\ns : Set \u03a9\na\u271d : MeasurableSet s\nhs : s = \u2205\n\u22a2 MeasurableSet \u2205"}, {"tactic": "exact MeasurableSet.empty", "annotated_tactic": ["exact <a>MeasurableSet.empty</a>", [{"full_name": "MeasurableSet.empty", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [80, 9], "def_end_pos": [80, 28]}]], "state_before": "case inl\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u00b2 : Preorder \u03b9\u271d\nf\u271d : Filtration \u03b9\u271d m\n\u03c4\u271d \u03c0 : \u03a9 \u2192 \u03b9\u271d\n\u03b9 : Type u_4\ninst\u271d\u00b9 : SemilatticeSup \u03b9\nf : Filtration \u03b9 m\n\u03c4 : \u03a9 \u2192 \u03b9\ninst\u271d : IsCountablyGenerated atTop\nh\u03c4 : IsStoppingTime f \u03c4\nh\u271d : IsEmpty \u03b9\nthis : IsEmpty \u03a9\ns : Set \u03a9\na\u271d : MeasurableSet s\nhs : s = \u2205\n\u22a2 MeasurableSet \u2205", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean", "full_name": "MeasureTheory.AECover.integral_tendsto_of_countably_generated", "start": [483, 1], "end": [491, 32], "traced_tactics": [{"tactic": "convert h using 2", "annotated_tactic": ["convert h using 2", []], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\nE : Type u_3\ninst\u271d\u2074 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nl : Filter \u03b9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : IsCountablyGenerated l\n\u03c6 : \u03b9 \u2192 Set \u03b1\nh\u03c6 : AECover \u03bc l \u03c6\nf : \u03b1 \u2192 E\nhfi : Integrable f\nh : Tendsto (fun i => \u222b (x : \u03b1), indicator (\u03c6 i) f x \u2202\u03bc) l (\ud835\udcdd (\u222b (x : \u03b1), f x \u2202\u03bc))\n\u22a2 Tendsto (fun i => \u222b (x : \u03b1) in \u03c6 i, f x \u2202\u03bc) l (\ud835\udcdd (\u222b (x : \u03b1), f x \u2202\u03bc))", "state_after": "case h.e'_3.h\n\u03b1 : Type u_1\n\u03b9 : Type u_2\nE : Type u_3\ninst\u271d\u2074 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nl : Filter \u03b9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : IsCountablyGenerated l\n\u03c6 : \u03b9 \u2192 Set \u03b1\nh\u03c6 : AECover \u03bc l \u03c6\nf : \u03b1 \u2192 E\nhfi : Integrable f\nh : Tendsto (fun i => \u222b (x : \u03b1), indicator (\u03c6 i) f x \u2202\u03bc) l (\ud835\udcdd (\u222b (x : \u03b1), f x \u2202\u03bc))\nx\u271d : \u03b9\n\u22a2 \u222b (x : \u03b1) in \u03c6 x\u271d, f x \u2202\u03bc = \u222b (x : \u03b1), indicator (\u03c6 x\u271d) f x \u2202\u03bc"}, {"tactic": "rw [integral_indicator (h\u03c6.measurableSet _)]", "annotated_tactic": ["rw [<a>integral_indicator</a> (h\u03c6.measurableSet _)]", [{"full_name": "MeasureTheory.integral_indicator", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [169, 9], "def_end_pos": [169, 27]}]], "state_before": "case h.e'_3.h\n\u03b1 : Type u_1\n\u03b9 : Type u_2\nE : Type u_3\ninst\u271d\u2074 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nl : Filter \u03b9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : IsCountablyGenerated l\n\u03c6 : \u03b9 \u2192 Set \u03b1\nh\u03c6 : AECover \u03bc l \u03c6\nf : \u03b1 \u2192 E\nhfi : Integrable f\nh : Tendsto (fun i => \u222b (x : \u03b1), indicator (\u03c6 i) f x \u2202\u03bc) l (\ud835\udcdd (\u222b (x : \u03b1), f x \u2202\u03bc))\nx\u271d : \u03b9\n\u22a2 \u222b (x : \u03b1) in \u03c6 x\u271d, f x \u2202\u03bc = \u222b (x : \u03b1), indicator (\u03c6 x\u271d) f x \u2202\u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean", "full_name": "MeasureTheory.integral_comp_rpow_Ioi", "start": [844, 1], "end": [866, 70], "traced_tactics": [{"tactic": "let S := Ioi (0 : \u211d)", "annotated_tactic": ["let S := <a>Ioi</a> (0 : \u211d)", [{"full_name": "Set.Ioi", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [79, 5], "def_end_pos": [79, 8]}]], "state_before": "E : Type u_1\nf : \u211d \u2192 E\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\ng : \u211d \u2192 E\np : \u211d\nhp : p \u2260 0\n\u22a2 \u222b (x : \u211d) in Ioi 0, (|p| * x ^ (p - 1)) \u2022 g (x ^ p) = \u222b (y : \u211d) in Ioi 0, g y", "state_after": "E : Type u_1\nf : \u211d \u2192 E\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\ng : \u211d \u2192 E\np : \u211d\nhp : p \u2260 0\nS : Set \u211d := Ioi 0\n\u22a2 \u222b (x : \u211d) in Ioi 0, (|p| * x ^ (p - 1)) \u2022 g (x ^ p) = \u222b (y : \u211d) in Ioi 0, g y"}, {"tactic": "have a1 : \u2200 x : \u211d, x \u2208 S \u2192 HasDerivWithinAt (fun t : \u211d => t ^ p) (p * x ^ (p - 1)) S x :=\n  fun x hx => (hasDerivAt_rpow_const (Or.inl (mem_Ioi.mp hx).ne')).hasDerivWithinAt", "annotated_tactic": ["have a1 : \u2200 x : \u211d, x \u2208 S \u2192 <a>HasDerivWithinAt</a> (fun t : \u211d => t ^ p) (p * x ^ (p - 1)) S x :=\n    fun x hx => (<a>hasDerivAt_rpow_const</a> (<a>Or.inl</a> (mem_Ioi.mp hx).<a>ne'</a>)).<a>hasDerivWithinAt</a>", [{"full_name": "HasDerivWithinAt", "def_path": "Mathlib/Analysis/Calculus/Deriv/Basic.lean", "def_pos": [115, 5], "def_end_pos": [115, 21]}, {"full_name": "Real.hasDerivAt_rpow_const", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Deriv.lean", "def_pos": [350, 9], "def_end_pos": [350, 30]}, {"full_name": "Or.inl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [517, 5], "def_end_pos": [517, 8]}, {"full_name": "LT.lt.ne'", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [328, 9], "def_end_pos": [328, 12]}, {"full_name": "HasDerivAt.hasDerivWithinAt", "def_path": "Mathlib/Analysis/Calculus/Deriv/Basic.lean", "def_pos": [388, 9], "def_end_pos": [388, 36]}]], "state_before": "E : Type u_1\nf : \u211d \u2192 E\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\ng : \u211d \u2192 E\np : \u211d\nhp : p \u2260 0\nS : Set \u211d := Ioi 0\n\u22a2 \u222b (x : \u211d) in Ioi 0, (|p| * x ^ (p - 1)) \u2022 g (x ^ p) = \u222b (y : \u211d) in Ioi 0, g y", "state_after": "E : Type u_1\nf : \u211d \u2192 E\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\ng : \u211d \u2192 E\np : \u211d\nhp : p \u2260 0\nS : Set \u211d := Ioi 0\na1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x\n\u22a2 \u222b (x : \u211d) in Ioi 0, (|p| * x ^ (p - 1)) \u2022 g (x ^ p) = \u222b (y : \u211d) in Ioi 0, g y"}, {"tactic": "have := integral_image_eq_integral_abs_deriv_smul measurableSet_Ioi a1 a2 g", "annotated_tactic": ["have := <a>integral_image_eq_integral_abs_deriv_smul</a> <a>measurableSet_Ioi</a> a1 a2 g", [{"full_name": "MeasureTheory.integral_image_eq_integral_abs_deriv_smul", "def_path": "Mathlib/MeasureTheory/Function/Jacobian.lean", "def_pos": [1248, 9], "def_end_pos": [1248, 50]}, {"full_name": "measurableSet_Ioi", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [579, 9], "def_end_pos": [579, 26]}]], "state_before": "E : Type u_1\nf : \u211d \u2192 E\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\ng : \u211d \u2192 E\np : \u211d\nhp : p \u2260 0\nS : Set \u211d := Ioi 0\na1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x\na2 : InjOn (fun x => x ^ p) S\na3 : (fun t => t ^ p) '' S = S\n\u22a2 \u222b (x : \u211d) in Ioi 0, (|p| * x ^ (p - 1)) \u2022 g (x ^ p) = \u222b (y : \u211d) in Ioi 0, g y", "state_after": "E : Type u_1\nf : \u211d \u2192 E\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\ng : \u211d \u2192 E\np : \u211d\nhp : p \u2260 0\nS : Set \u211d := Ioi 0\na1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x\na2 : InjOn (fun x => x ^ p) S\na3 : (fun t => t ^ p) '' S = S\nthis : \u222b (x : \u211d) in (fun t => t ^ p) '' Ioi 0, g x = \u222b (x : \u211d) in Ioi 0, |p * x ^ (p - 1)| \u2022 g (x ^ p)\n\u22a2 \u222b (x : \u211d) in Ioi 0, (|p| * x ^ (p - 1)) \u2022 g (x ^ p) = \u222b (y : \u211d) in Ioi 0, g y"}, {"tactic": "rw [a3] at this", "annotated_tactic": ["rw [a3] at this", []], "state_before": "E : Type u_1\nf : \u211d \u2192 E\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\ng : \u211d \u2192 E\np : \u211d\nhp : p \u2260 0\nS : Set \u211d := Ioi 0\na1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x\na2 : InjOn (fun x => x ^ p) S\na3 : (fun t => t ^ p) '' S = S\nthis : \u222b (x : \u211d) in (fun t => t ^ p) '' Ioi 0, g x = \u222b (x : \u211d) in Ioi 0, |p * x ^ (p - 1)| \u2022 g (x ^ p)\n\u22a2 \u222b (x : \u211d) in Ioi 0, (|p| * x ^ (p - 1)) \u2022 g (x ^ p) = \u222b (y : \u211d) in Ioi 0, g y", "state_after": "E : Type u_1\nf : \u211d \u2192 E\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\ng : \u211d \u2192 E\np : \u211d\nhp : p \u2260 0\nS : Set \u211d := Ioi 0\na1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x\na2 : InjOn (fun x => x ^ p) S\na3 : (fun t => t ^ p) '' S = S\nthis : \u222b (x : \u211d) in S, g x = \u222b (x : \u211d) in Ioi 0, |p * x ^ (p - 1)| \u2022 g (x ^ p)\n\u22a2 \u222b (x : \u211d) in Ioi 0, (|p| * x ^ (p - 1)) \u2022 g (x ^ p) = \u222b (y : \u211d) in Ioi 0, g y"}, {"tactic": "rw [this]", "annotated_tactic": ["rw [this]", []], "state_before": "E : Type u_1\nf : \u211d \u2192 E\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\ng : \u211d \u2192 E\np : \u211d\nhp : p \u2260 0\nS : Set \u211d := Ioi 0\na1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x\na2 : InjOn (fun x => x ^ p) S\na3 : (fun t => t ^ p) '' S = S\nthis : \u222b (x : \u211d) in S, g x = \u222b (x : \u211d) in Ioi 0, |p * x ^ (p - 1)| \u2022 g (x ^ p)\n\u22a2 \u222b (x : \u211d) in Ioi 0, (|p| * x ^ (p - 1)) \u2022 g (x ^ p) = \u222b (y : \u211d) in Ioi 0, g y", "state_after": "E : Type u_1\nf : \u211d \u2192 E\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\ng : \u211d \u2192 E\np : \u211d\nhp : p \u2260 0\nS : Set \u211d := Ioi 0\na1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x\na2 : InjOn (fun x => x ^ p) S\na3 : (fun t => t ^ p) '' S = S\nthis : \u222b (x : \u211d) in S, g x = \u222b (x : \u211d) in Ioi 0, |p * x ^ (p - 1)| \u2022 g (x ^ p)\n\u22a2 \u222b (x : \u211d) in Ioi 0, (|p| * x ^ (p - 1)) \u2022 g (x ^ p) = \u222b (x : \u211d) in Ioi 0, |p * x ^ (p - 1)| \u2022 g (x ^ p)"}, {"tactic": "refine' set_integral_congr measurableSet_Ioi _", "annotated_tactic": ["refine' <a>set_integral_congr</a> <a>measurableSet_Ioi</a> _", [{"full_name": "MeasureTheory.set_integral_congr", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [87, 9], "def_end_pos": [87, 27]}, {"full_name": "measurableSet_Ioi", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [579, 9], "def_end_pos": [579, 26]}]], "state_before": "E : Type u_1\nf : \u211d \u2192 E\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\ng : \u211d \u2192 E\np : \u211d\nhp : p \u2260 0\nS : Set \u211d := Ioi 0\na1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x\na2 : InjOn (fun x => x ^ p) S\na3 : (fun t => t ^ p) '' S = S\nthis : \u222b (x : \u211d) in S, g x = \u222b (x : \u211d) in Ioi 0, |p * x ^ (p - 1)| \u2022 g (x ^ p)\n\u22a2 \u222b (x : \u211d) in Ioi 0, (|p| * x ^ (p - 1)) \u2022 g (x ^ p) = \u222b (x : \u211d) in Ioi 0, |p * x ^ (p - 1)| \u2022 g (x ^ p)", "state_after": "E : Type u_1\nf : \u211d \u2192 E\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\ng : \u211d \u2192 E\np : \u211d\nhp : p \u2260 0\nS : Set \u211d := Ioi 0\na1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x\na2 : InjOn (fun x => x ^ p) S\na3 : (fun t => t ^ p) '' S = S\nthis : \u222b (x : \u211d) in S, g x = \u222b (x : \u211d) in Ioi 0, |p * x ^ (p - 1)| \u2022 g (x ^ p)\n\u22a2 EqOn (fun x => (|p| * x ^ (p - 1)) \u2022 g (x ^ p)) (fun x => |p * x ^ (p - 1)| \u2022 g (x ^ p)) (Ioi 0)"}, {"tactic": "intro x hx", "annotated_tactic": ["intro x hx", []], "state_before": "E : Type u_1\nf : \u211d \u2192 E\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\ng : \u211d \u2192 E\np : \u211d\nhp : p \u2260 0\nS : Set \u211d := Ioi 0\na1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x\na2 : InjOn (fun x => x ^ p) S\na3 : (fun t => t ^ p) '' S = S\nthis : \u222b (x : \u211d) in S, g x = \u222b (x : \u211d) in Ioi 0, |p * x ^ (p - 1)| \u2022 g (x ^ p)\n\u22a2 EqOn (fun x => (|p| * x ^ (p - 1)) \u2022 g (x ^ p)) (fun x => |p * x ^ (p - 1)| \u2022 g (x ^ p)) (Ioi 0)", "state_after": "E : Type u_1\nf : \u211d \u2192 E\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\ng : \u211d \u2192 E\np : \u211d\nhp : p \u2260 0\nS : Set \u211d := Ioi 0\na1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x\na2 : InjOn (fun x => x ^ p) S\na3 : (fun t => t ^ p) '' S = S\nthis : \u222b (x : \u211d) in S, g x = \u222b (x : \u211d) in Ioi 0, |p * x ^ (p - 1)| \u2022 g (x ^ p)\nx : \u211d\nhx : x \u2208 Ioi 0\n\u22a2 (fun x => (|p| * x ^ (p - 1)) \u2022 g (x ^ p)) x = (fun x => |p * x ^ (p - 1)| \u2022 g (x ^ p)) x"}, {"tactic": "dsimp only", "annotated_tactic": ["dsimp only", []], "state_before": "E : Type u_1\nf : \u211d \u2192 E\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\ng : \u211d \u2192 E\np : \u211d\nhp : p \u2260 0\nS : Set \u211d := Ioi 0\na1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x\na2 : InjOn (fun x => x ^ p) S\na3 : (fun t => t ^ p) '' S = S\nthis : \u222b (x : \u211d) in S, g x = \u222b (x : \u211d) in Ioi 0, |p * x ^ (p - 1)| \u2022 g (x ^ p)\nx : \u211d\nhx : x \u2208 Ioi 0\n\u22a2 (fun x => (|p| * x ^ (p - 1)) \u2022 g (x ^ p)) x = (fun x => |p * x ^ (p - 1)| \u2022 g (x ^ p)) x", "state_after": "E : Type u_1\nf : \u211d \u2192 E\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\ng : \u211d \u2192 E\np : \u211d\nhp : p \u2260 0\nS : Set \u211d := Ioi 0\na1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x\na2 : InjOn (fun x => x ^ p) S\na3 : (fun t => t ^ p) '' S = S\nthis : \u222b (x : \u211d) in S, g x = \u222b (x : \u211d) in Ioi 0, |p * x ^ (p - 1)| \u2022 g (x ^ p)\nx : \u211d\nhx : x \u2208 Ioi 0\n\u22a2 (|p| * x ^ (p - 1)) \u2022 g (x ^ p) = |p * x ^ (p - 1)| \u2022 g (x ^ p)"}, {"tactic": "rw [abs_mul, abs_of_nonneg (rpow_nonneg_of_nonneg (le_of_lt hx) _)]", "annotated_tactic": ["rw [<a>abs_mul</a>, <a>abs_of_nonneg</a> (<a>rpow_nonneg_of_nonneg</a> (<a>le_of_lt</a> hx) _)]", [{"full_name": "abs_mul", "def_path": "Mathlib/Algebra/Order/Ring/Abs.lean", "def_pos": [33, 9], "def_end_pos": [33, 16]}, {"full_name": "abs_of_nonneg", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [107, 9], "def_end_pos": [107, 22]}, {"full_name": "Real.rpow_nonneg_of_nonneg", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [141, 9], "def_end_pos": [141, 30]}, {"full_name": "le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [110, 9], "def_end_pos": [110, 17]}]], "state_before": "E : Type u_1\nf : \u211d \u2192 E\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\ng : \u211d \u2192 E\np : \u211d\nhp : p \u2260 0\nS : Set \u211d := Ioi 0\na1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x\na2 : InjOn (fun x => x ^ p) S\na3 : (fun t => t ^ p) '' S = S\nthis : \u222b (x : \u211d) in S, g x = \u222b (x : \u211d) in Ioi 0, |p * x ^ (p - 1)| \u2022 g (x ^ p)\nx : \u211d\nhx : x \u2208 Ioi 0\n\u22a2 (|p| * x ^ (p - 1)) \u2022 g (x ^ p) = |p * x ^ (p - 1)| \u2022 g (x ^ p)", "state_after": "no goals"}, {"tactic": "rcases lt_or_gt_of_ne hp with (h | h)", "annotated_tactic": ["rcases <a>lt_or_gt_of_ne</a> hp with (h | h)", [{"full_name": "lt_or_gt_of_ne", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [352, 9], "def_end_pos": [352, 23]}]], "state_before": "E : Type u_1\nf : \u211d \u2192 E\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\ng : \u211d \u2192 E\np : \u211d\nhp : p \u2260 0\nS : Set \u211d := Ioi 0\na1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x\n\u22a2 InjOn (fun x => x ^ p) S", "state_after": "case inl\nE : Type u_1\nf : \u211d \u2192 E\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\ng : \u211d \u2192 E\np : \u211d\nhp : p \u2260 0\nS : Set \u211d := Ioi 0\na1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x\nh : p < 0\n\u22a2 InjOn (fun x => x ^ p) S\n\ncase inr\nE : Type u_1\nf : \u211d \u2192 E\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\ng : \u211d \u2192 E\np : \u211d\nhp : p \u2260 0\nS : Set \u211d := Ioi 0\na1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x\nh : p > 0\n\u22a2 InjOn (fun x => x ^ p) S"}, {"tactic": "exact StrictMonoOn.injOn fun x hx y _ hxy => rpow_lt_rpow (mem_Ioi.mp hx).le hxy h", "annotated_tactic": ["exact <a>StrictMonoOn.injOn</a> fun x hx y _ hxy => <a>rpow_lt_rpow</a> (mem_Ioi.mp hx).<a>le</a> hxy h", [{"full_name": "StrictMonoOn.injOn", "def_path": "Mathlib/Data/Set/Function.lean", "def_pos": [1579, 9], "def_end_pos": [1579, 27]}, {"full_name": "Real.rpow_lt_rpow", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [420, 9], "def_end_pos": [420, 21]}, {"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [142, 7], "def_end_pos": [142, 15]}]], "state_before": "case inr\nE : Type u_1\nf : \u211d \u2192 E\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\ng : \u211d \u2192 E\np : \u211d\nhp : p \u2260 0\nS : Set \u211d := Ioi 0\na1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x\nh : p > 0\n\u22a2 InjOn (fun x => x ^ p) S", "state_after": "no goals"}, {"tactic": "apply StrictAntiOn.injOn", "annotated_tactic": ["apply <a>StrictAntiOn.injOn</a>", [{"full_name": "StrictAntiOn.injOn", "def_path": "Mathlib/Data/Set/Function.lean", "def_pos": [1584, 9], "def_end_pos": [1584, 27]}]], "state_before": "case inl\nE : Type u_1\nf : \u211d \u2192 E\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\ng : \u211d \u2192 E\np : \u211d\nhp : p \u2260 0\nS : Set \u211d := Ioi 0\na1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x\nh : p < 0\n\u22a2 InjOn (fun x => x ^ p) S", "state_after": "case inl.H\nE : Type u_1\nf : \u211d \u2192 E\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\ng : \u211d \u2192 E\np : \u211d\nhp : p \u2260 0\nS : Set \u211d := Ioi 0\na1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x\nh : p < 0\n\u22a2 StrictAntiOn (fun x => x ^ p) S"}, {"tactic": "intro x hx y hy hxy", "annotated_tactic": ["intro x hx y hy hxy", []], "state_before": "case inl.H\nE : Type u_1\nf : \u211d \u2192 E\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\ng : \u211d \u2192 E\np : \u211d\nhp : p \u2260 0\nS : Set \u211d := Ioi 0\na1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x\nh : p < 0\n\u22a2 StrictAntiOn (fun x => x ^ p) S", "state_after": "case inl.H\nE : Type u_1\nf : \u211d \u2192 E\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\ng : \u211d \u2192 E\np : \u211d\nhp : p \u2260 0\nS : Set \u211d := Ioi 0\na1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x\nh : p < 0\nx : \u211d\nhx : x \u2208 S\ny : \u211d\nhy : y \u2208 S\nhxy : x < y\n\u22a2 (fun x => x ^ p) y < (fun x => x ^ p) x"}, {"tactic": "rw [\u2190 inv_lt_inv (rpow_pos_of_pos hx p) (rpow_pos_of_pos hy p), \u2190 rpow_neg (le_of_lt hx),\n  \u2190 rpow_neg (le_of_lt hy)]", "annotated_tactic": ["rw [\u2190 <a>inv_lt_inv</a> (<a>rpow_pos_of_pos</a> hx p) (<a>rpow_pos_of_pos</a> hy p), \u2190 <a>rpow_neg</a> (<a>le_of_lt</a> hx),\n        \u2190 <a>rpow_neg</a> (<a>le_of_lt</a> hy)]", [{"full_name": "inv_lt_inv", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [275, 9], "def_end_pos": [275, 19]}, {"full_name": "Real.rpow_pos_of_pos", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [92, 9], "def_end_pos": [92, 24]}, {"full_name": "Real.rpow_pos_of_pos", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [92, 9], "def_end_pos": [92, 24]}, {"full_name": "Real.rpow_neg", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [221, 9], "def_end_pos": [221, 17]}, {"full_name": "le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [110, 9], "def_end_pos": [110, 17]}, {"full_name": "Real.rpow_neg", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [221, 9], "def_end_pos": [221, 17]}, {"full_name": "le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [110, 9], "def_end_pos": [110, 17]}]], "state_before": "case inl.H\nE : Type u_1\nf : \u211d \u2192 E\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\ng : \u211d \u2192 E\np : \u211d\nhp : p \u2260 0\nS : Set \u211d := Ioi 0\na1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x\nh : p < 0\nx : \u211d\nhx : x \u2208 S\ny : \u211d\nhy : y \u2208 S\nhxy : x < y\n\u22a2 (fun x => x ^ p) y < (fun x => x ^ p) x", "state_after": "case inl.H\nE : Type u_1\nf : \u211d \u2192 E\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\ng : \u211d \u2192 E\np : \u211d\nhp : p \u2260 0\nS : Set \u211d := Ioi 0\na1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x\nh : p < 0\nx : \u211d\nhx : x \u2208 S\ny : \u211d\nhy : y \u2208 S\nhxy : x < y\n\u22a2 x ^ (-p) < y ^ (-p)"}, {"tactic": "exact rpow_lt_rpow (le_of_lt hx) hxy (neg_pos.mpr h)", "annotated_tactic": ["exact <a>rpow_lt_rpow</a> (<a>le_of_lt</a> hx) hxy (neg_pos.mpr h)", [{"full_name": "Real.rpow_lt_rpow", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [420, 9], "def_end_pos": [420, 21]}, {"full_name": "le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [110, 9], "def_end_pos": [110, 17]}]], "state_before": "case inl.H\nE : Type u_1\nf : \u211d \u2192 E\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\ng : \u211d \u2192 E\np : \u211d\nhp : p \u2260 0\nS : Set \u211d := Ioi 0\na1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x\nh : p < 0\nx : \u211d\nhx : x \u2208 S\ny : \u211d\nhy : y \u2208 S\nhxy : x < y\n\u22a2 x ^ (-p) < y ^ (-p)", "state_after": "no goals"}, {"tactic": "ext1 x", "annotated_tactic": ["ext1 x", []], "state_before": "E : Type u_1\nf : \u211d \u2192 E\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\ng : \u211d \u2192 E\np : \u211d\nhp : p \u2260 0\nS : Set \u211d := Ioi 0\na1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x\na2 : InjOn (fun x => x ^ p) S\n\u22a2 (fun t => t ^ p) '' S = S", "state_after": "case h\nE : Type u_1\nf : \u211d \u2192 E\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\ng : \u211d \u2192 E\np : \u211d\nhp : p \u2260 0\nS : Set \u211d := Ioi 0\na1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x\na2 : InjOn (fun x => x ^ p) S\nx : \u211d\n\u22a2 x \u2208 (fun t => t ^ p) '' S \u2194 x \u2208 S"}, {"tactic": "rw [mem_image]", "annotated_tactic": ["rw [<a>mem_image</a>]", [{"full_name": "Set.mem_image", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [231, 9], "def_end_pos": [231, 18]}]], "state_before": "case h\nE : Type u_1\nf : \u211d \u2192 E\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\ng : \u211d \u2192 E\np : \u211d\nhp : p \u2260 0\nS : Set \u211d := Ioi 0\na1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x\na2 : InjOn (fun x => x ^ p) S\nx : \u211d\n\u22a2 x \u2208 (fun t => t ^ p) '' S \u2194 x \u2208 S", "state_after": "case h\nE : Type u_1\nf : \u211d \u2192 E\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\ng : \u211d \u2192 E\np : \u211d\nhp : p \u2260 0\nS : Set \u211d := Ioi 0\na1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x\na2 : InjOn (fun x => x ^ p) S\nx : \u211d\n\u22a2 (\u2203 x_1, x_1 \u2208 S \u2227 x_1 ^ p = x) \u2194 x \u2208 S"}, {"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "case h\nE : Type u_1\nf : \u211d \u2192 E\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\ng : \u211d \u2192 E\np : \u211d\nhp : p \u2260 0\nS : Set \u211d := Ioi 0\na1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x\na2 : InjOn (fun x => x ^ p) S\nx : \u211d\n\u22a2 (\u2203 x_1, x_1 \u2208 S \u2227 x_1 ^ p = x) \u2194 x \u2208 S", "state_after": "case h.mp\nE : Type u_1\nf : \u211d \u2192 E\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\ng : \u211d \u2192 E\np : \u211d\nhp : p \u2260 0\nS : Set \u211d := Ioi 0\na1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x\na2 : InjOn (fun x => x ^ p) S\nx : \u211d\n\u22a2 (\u2203 x_1, x_1 \u2208 S \u2227 x_1 ^ p = x) \u2192 x \u2208 S\n\ncase h.mpr\nE : Type u_1\nf : \u211d \u2192 E\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\ng : \u211d \u2192 E\np : \u211d\nhp : p \u2260 0\nS : Set \u211d := Ioi 0\na1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x\na2 : InjOn (fun x => x ^ p) S\nx : \u211d\n\u22a2 x \u2208 S \u2192 \u2203 x_1, x_1 \u2208 S \u2227 x_1 ^ p = x"}, {"tactic": "rintro \u27e8y, hy, rfl\u27e9", "annotated_tactic": ["rintro \u27e8y, hy, rfl\u27e9", []], "state_before": "case h.mp\nE : Type u_1\nf : \u211d \u2192 E\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\ng : \u211d \u2192 E\np : \u211d\nhp : p \u2260 0\nS : Set \u211d := Ioi 0\na1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x\na2 : InjOn (fun x => x ^ p) S\nx : \u211d\n\u22a2 (\u2203 x_1, x_1 \u2208 S \u2227 x_1 ^ p = x) \u2192 x \u2208 S", "state_after": "case h.mp.intro.intro\nE : Type u_1\nf : \u211d \u2192 E\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\ng : \u211d \u2192 E\np : \u211d\nhp : p \u2260 0\nS : Set \u211d := Ioi 0\na1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x\na2 : InjOn (fun x => x ^ p) S\ny : \u211d\nhy : y \u2208 S\n\u22a2 y ^ p \u2208 S"}, {"tactic": "exact rpow_pos_of_pos hy p", "annotated_tactic": ["exact <a>rpow_pos_of_pos</a> hy p", [{"full_name": "Real.rpow_pos_of_pos", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [92, 9], "def_end_pos": [92, 24]}]], "state_before": "case h.mp.intro.intro\nE : Type u_1\nf : \u211d \u2192 E\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\ng : \u211d \u2192 E\np : \u211d\nhp : p \u2260 0\nS : Set \u211d := Ioi 0\na1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x\na2 : InjOn (fun x => x ^ p) S\ny : \u211d\nhy : y \u2208 S\n\u22a2 y ^ p \u2208 S", "state_after": "no goals"}, {"tactic": "intro hx", "annotated_tactic": ["intro hx", []], "state_before": "case h.mpr\nE : Type u_1\nf : \u211d \u2192 E\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\ng : \u211d \u2192 E\np : \u211d\nhp : p \u2260 0\nS : Set \u211d := Ioi 0\na1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x\na2 : InjOn (fun x => x ^ p) S\nx : \u211d\n\u22a2 x \u2208 S \u2192 \u2203 x_1, x_1 \u2208 S \u2227 x_1 ^ p = x", "state_after": "case h.mpr\nE : Type u_1\nf : \u211d \u2192 E\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\ng : \u211d \u2192 E\np : \u211d\nhp : p \u2260 0\nS : Set \u211d := Ioi 0\na1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x\na2 : InjOn (fun x => x ^ p) S\nx : \u211d\nhx : x \u2208 S\n\u22a2 \u2203 x_1, x_1 \u2208 S \u2227 x_1 ^ p = x"}, {"tactic": "refine' \u27e8x ^ (1 / p), rpow_pos_of_pos hx _, _\u27e9", "annotated_tactic": ["refine' \u27e8x ^ (1 / p), <a>rpow_pos_of_pos</a> hx _, _\u27e9", [{"full_name": "Real.rpow_pos_of_pos", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [92, 9], "def_end_pos": [92, 24]}]], "state_before": "case h.mpr\nE : Type u_1\nf : \u211d \u2192 E\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\ng : \u211d \u2192 E\np : \u211d\nhp : p \u2260 0\nS : Set \u211d := Ioi 0\na1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x\na2 : InjOn (fun x => x ^ p) S\nx : \u211d\nhx : x \u2208 S\n\u22a2 \u2203 x_1, x_1 \u2208 S \u2227 x_1 ^ p = x", "state_after": "case h.mpr\nE : Type u_1\nf : \u211d \u2192 E\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\ng : \u211d \u2192 E\np : \u211d\nhp : p \u2260 0\nS : Set \u211d := Ioi 0\na1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x\na2 : InjOn (fun x => x ^ p) S\nx : \u211d\nhx : x \u2208 S\n\u22a2 (x ^ (1 / p)) ^ p = x"}, {"tactic": "rw [\u2190 rpow_mul (le_of_lt hx), one_div_mul_cancel hp, rpow_one]", "annotated_tactic": ["rw [\u2190 <a>rpow_mul</a> (<a>le_of_lt</a> hx), <a>one_div_mul_cancel</a> hp, <a>rpow_one</a>]", [{"full_name": "Real.rpow_mul", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [317, 9], "def_end_pos": [317, 17]}, {"full_name": "le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [110, 9], "def_end_pos": [110, 17]}, {"full_name": "one_div_mul_cancel", "def_path": "Mathlib/Algebra/GroupWithZero/Units/Lemmas.lean", "def_pos": [79, 9], "def_end_pos": [79, 27]}, {"full_name": "Real.rpow_one", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [126, 9], "def_end_pos": [126, 17]}]], "state_before": "case h.mpr\nE : Type u_1\nf : \u211d \u2192 E\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\ng : \u211d \u2192 E\np : \u211d\nhp : p \u2260 0\nS : Set \u211d := Ioi 0\na1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x\na2 : InjOn (fun x => x ^ p) S\nx : \u211d\nhx : x \u2208 S\n\u22a2 (x ^ (1 / p)) ^ p = x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/TuringMachine.lean", "full_name": "Turing.Tape.mk'_nth_nat", "start": [624, 1], "end": [626, 42], "traced_tactics": [{"tactic": "rw [\u2190 Tape.right\u2080_nth, Tape.mk'_right\u2080]", "annotated_tactic": ["rw [\u2190 <a>Tape.right\u2080_nth</a>, <a>Tape.mk'_right\u2080</a>]", [{"full_name": "Turing.Tape.right\u2080_nth", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [618, 9], "def_end_pos": [618, 24]}, {"full_name": "Turing.Tape.mk'_right\u2080", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [565, 9], "def_end_pos": [565, 24]}]], "state_before": "\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\nL R : ListBlank \u0393\nn : \u2115\n\u22a2 nth (mk' L R) \u2191n = ListBlank.nth R n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "full_name": "Substring.ValidFor.isEmpty", "start": [831, 1], "end": [832, 49], "traced_tactics": [{"tactic": "simp [Substring.isEmpty, h.bsize]", "annotated_tactic": ["simp [<a>Substring.isEmpty</a>, h.bsize]", [{"full_name": "Substring.isEmpty", "def_path": "lake-packages/lean4/src/lean/Init/Data/String/Basic.lean", "def_pos": [522, 15], "def_end_pos": [522, 22]}]], "state_before": "l m r : List Char\nx\u271d : Substring\nh : ValidFor l m r x\u271d\n\u22a2 Substring.isEmpty x\u271d = true \u2194 m = []", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "full_name": "List.exists_of_set", "start": [905, 1], "end": [907, 55], "traced_tactics": [{"tactic": "rw [set_eq_modifyNth]", "annotated_tactic": ["rw [<a>set_eq_modifyNth</a>]", [{"full_name": "List.set_eq_modifyNth", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [889, 9], "def_end_pos": [889, 25]}]], "state_before": "\u03b1 : Type u_1\nn : Nat\na' : \u03b1\nl : List \u03b1\nh : n < length l\n\u22a2 \u2203 l\u2081 a l\u2082, l = l\u2081 ++ a :: l\u2082 \u2227 length l\u2081 = n \u2227 set l n a' = l\u2081 ++ a' :: l\u2082", "state_after": "\u03b1 : Type u_1\nn : Nat\na' : \u03b1\nl : List \u03b1\nh : n < length l\n\u22a2 \u2203 l\u2081 a l\u2082, l = l\u2081 ++ a :: l\u2082 \u2227 length l\u2081 = n \u2227 modifyNth (fun x => a') n l = l\u2081 ++ a' :: l\u2082"}, {"tactic": "exact exists_of_modifyNth _ h", "annotated_tactic": ["exact <a>exists_of_modifyNth</a> _ h", [{"full_name": "List.exists_of_modifyNth", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [881, 9], "def_end_pos": [881, 28]}]], "state_before": "\u03b1 : Type u_1\nn : Nat\na' : \u03b1\nl : List \u03b1\nh : n < length l\n\u22a2 \u2203 l\u2081 a l\u2082, l = l\u2081 ++ a :: l\u2082 \u2227 length l\u2081 = n \u2227 modifyNth (fun x => a') n l = l\u2081 ++ a' :: l\u2082", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Array/Init/Lemmas.lean", "full_name": "Array.toListAppend_eq", "start": [92, 9], "end": [93, 43], "traced_tactics": [{"tactic": "simp [toListAppend, foldr_eq_foldr_data]", "annotated_tactic": ["simp [<a>toListAppend</a>, <a>foldr_eq_foldr_data</a>]", [{"full_name": "Array.toListAppend", "def_path": "lake-packages/std/Std/Data/Array/Init/Basic.lean", "def_pos": [18, 15], "def_end_pos": [18, 27]}, {"full_name": "Array.foldr_eq_foldr_data", "def_path": "lake-packages/std/Std/Data/Array/Init/Lemmas.lean", "def_pos": [71, 9], "def_end_pos": [71, 28]}]], "state_before": "\u03b1 : Type u_1\narr : Array \u03b1\nl : List \u03b1\n\u22a2 toListAppend arr l = arr.data ++ l", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "full_name": "MeasureTheory.integral_finset_sum", "start": [881, 1], "end": [886, 24], "traced_tactics": [{"tactic": "by_cases hG : CompleteSpace G", "annotated_tactic": ["by_cases hG : <a>CompleteSpace</a> G", [{"full_name": "CompleteSpace", "def_path": "Mathlib/Topology/UniformSpace/Cauchy.lean", "def_pos": [397, 7], "def_end_pos": [397, 20]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b9 : Type u_6\ns : Finset \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 G\nhf : \u2200 (i : \u03b9), i \u2208 s \u2192 Integrable (f i)\n\u22a2 \u222b (a : \u03b1), \u2211 i in s, f i a \u2202\u03bc = \u2211 i in s, \u222b (a : \u03b1), f i a \u2202\u03bc", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b9 : Type u_6\ns : Finset \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 G\nhf : \u2200 (i : \u03b9), i \u2208 s \u2192 Integrable (f i)\nhG : CompleteSpace G\n\u22a2 \u222b (a : \u03b1), \u2211 i in s, f i a \u2202\u03bc = \u2211 i in s, \u222b (a : \u03b1), f i a \u2202\u03bc\n\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b9 : Type u_6\ns : Finset \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 G\nhf : \u2200 (i : \u03b9), i \u2208 s \u2192 Integrable (f i)\nhG : \u00acCompleteSpace G\n\u22a2 \u222b (a : \u03b1), \u2211 i in s, f i a \u2202\u03bc = \u2211 i in s, \u222b (a : \u03b1), f i a \u2202\u03bc"}, {"tactic": "simp only [integral, hG, L1.integral]", "annotated_tactic": ["simp only [<a>integral</a>, hG, <a>L1.integral</a>]", [{"full_name": "MeasureTheory.integral", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [791, 17], "def_end_pos": [791, 25]}, {"full_name": "MeasureTheory.L1.integral", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [666, 17], "def_end_pos": [666, 25]}]], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b9 : Type u_6\ns : Finset \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 G\nhf : \u2200 (i : \u03b9), i \u2208 s \u2192 Integrable (f i)\nhG : CompleteSpace G\n\u22a2 \u222b (a : \u03b1), \u2211 i in s, f i a \u2202\u03bc = \u2211 i in s, \u222b (a : \u03b1), f i a \u2202\u03bc", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b9 : Type u_6\ns : Finset \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 G\nhf : \u2200 (i : \u03b9), i \u2208 s \u2192 Integrable (f i)\nhG : CompleteSpace G\n\u22a2 (if h : True then\n      if hf : Integrable fun a => \u2211 i in s, f i a then \u2191L1.integralCLM (Integrable.toL1 (fun a => \u2211 i in s, f i a) hf)\n      else 0\n    else 0) =\n    \u2211 x in s,\n      if h : True then\n        if hf : Integrable fun a => f x a then \u2191L1.integralCLM (Integrable.toL1 (fun a => f x a) hf) else 0\n      else 0"}, {"tactic": "exact setToFun_finset_sum (dominatedFinMeasAdditive_weightedSMul _) s hf", "annotated_tactic": ["exact <a>setToFun_finset_sum</a> (<a>dominatedFinMeasAdditive_weightedSMul</a> _) s hf", [{"full_name": "MeasureTheory.setToFun_finset_sum", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [1386, 9], "def_end_pos": [1386, 28]}, {"full_name": "MeasureTheory.dominatedFinMeasAdditive_weightedSMul", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [243, 9], "def_end_pos": [243, 46]}]], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b9 : Type u_6\ns : Finset \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 G\nhf : \u2200 (i : \u03b9), i \u2208 s \u2192 Integrable (f i)\nhG : CompleteSpace G\n\u22a2 (if h : True then\n      if hf : Integrable fun a => \u2211 i in s, f i a then \u2191L1.integralCLM (Integrable.toL1 (fun a => \u2211 i in s, f i a) hf)\n      else 0\n    else 0) =\n    \u2211 x in s,\n      if h : True then\n        if hf : Integrable fun a => f x a then \u2191L1.integralCLM (Integrable.toL1 (fun a => f x a) hf) else 0\n      else 0", "state_after": "no goals"}, {"tactic": "simp [integral, hG]", "annotated_tactic": ["simp [<a>integral</a>, hG]", [{"full_name": "MeasureTheory.integral", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [791, 17], "def_end_pos": [791, 25]}]], "state_before": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b9 : Type u_6\ns : Finset \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 G\nhf : \u2200 (i : \u03b9), i \u2208 s \u2192 Integrable (f i)\nhG : \u00acCompleteSpace G\n\u22a2 \u222b (a : \u03b1), \u2211 i in s, f i a \u2202\u03bc = \u2211 i in s, \u222b (a : \u03b1), f i a \u2202\u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Int/ModEq.lean", "full_name": "Int.ModEq.cancel_left_div_gcd", "start": [237, 1], "end": [238, 57], "traced_tactics": [{"tactic": "simpa [mul_comm] using h", "annotated_tactic": ["simpa [<a>mul_comm</a>] using h", [{"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [302, 9], "def_end_pos": [302, 17]}]], "state_before": "m n a b c d : \u2124\nhm : 0 < m\nh : c * a \u2261 c * b [ZMOD m]\n\u22a2 a * c \u2261 b * c [ZMOD m]", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Moments.lean", "full_name": "ProbabilityTheory.aestronglyMeasurable_exp_mul_sum", "start": [263, 1], "end": [274, 81], "traced_tactics": [{"tactic": "induction' s using Finset.induction_on with i s hi_notin_s h_rec h_int", "annotated_tactic": ["induction' s using <a>Finset.induction_on</a> with i s hi_notin_s h_rec h_int", [{"full_name": "Finset.induction_on", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1251, 19], "def_end_pos": [1251, 31]}]], "state_before": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\np : \u2115\n\u03bc : Measure \u03a9\nt : \u211d\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\ns : Finset \u03b9\nh_int : \u2200 (i : \u03b9), i \u2208 s \u2192 AEStronglyMeasurable (fun \u03c9 => rexp (t * X i \u03c9)) \u03bc\n\u22a2 AEStronglyMeasurable (fun \u03c9 => rexp (t * Finset.sum s (fun i => X i) \u03c9)) \u03bc", "state_after": "case empty\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\np : \u2115\n\u03bc : Measure \u03a9\nt : \u211d\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\ns : Finset \u03b9\nh_int\u271d : \u2200 (i : \u03b9), i \u2208 s \u2192 AEStronglyMeasurable (fun \u03c9 => rexp (t * X i \u03c9)) \u03bc\nh_int : \u2200 (i : \u03b9), i \u2208 \u2205 \u2192 AEStronglyMeasurable (fun \u03c9 => rexp (t * X i \u03c9)) \u03bc\n\u22a2 AEStronglyMeasurable (fun \u03c9 => rexp (t * Finset.sum \u2205 (fun i => X i) \u03c9)) \u03bc\n\ncase insert\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\np : \u2115\n\u03bc : Measure \u03a9\nt : \u211d\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\ns\u271d : Finset \u03b9\nh_int\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 AEStronglyMeasurable (fun \u03c9 => rexp (t * X i \u03c9)) \u03bc\ni : \u03b9\ns : Finset \u03b9\nhi_notin_s : \u00aci \u2208 s\nh_rec :\n  (\u2200 (i : \u03b9), i \u2208 s \u2192 AEStronglyMeasurable (fun \u03c9 => rexp (t * X i \u03c9)) \u03bc) \u2192\n    AEStronglyMeasurable (fun \u03c9 => rexp (t * Finset.sum s (fun i => X i) \u03c9)) \u03bc\nh_int : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i s \u2192 AEStronglyMeasurable (fun \u03c9 => rexp (t * X i_1 \u03c9)) \u03bc\n\u22a2 AEStronglyMeasurable (fun \u03c9 => rexp (t * Finset.sum (insert i s) (fun i => X i) \u03c9)) \u03bc"}, {"tactic": "simp only [Pi.zero_apply, sum_apply, sum_empty, mul_zero, exp_zero]", "annotated_tactic": ["simp only [<a>Pi.zero_apply</a>, <a>sum_apply</a>, <a>sum_empty</a>, <a>mul_zero</a>, <a>exp_zero</a>]", [{"full_name": "Pi.zero_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [46, 3], "def_end_pos": [46, 14]}, {"full_name": "Finset.sum_apply", "def_path": "Mathlib/Algebra/BigOperators/Pi.lean", "def_pos": [41, 3], "def_end_pos": [41, 14]}, {"full_name": "Finset.sum_empty", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [298, 3], "def_end_pos": [298, 14]}, {"full_name": "MulZeroClass.mul_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [38, 3], "def_end_pos": [38, 11]}, {"full_name": "Real.exp_zero", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [1135, 9], "def_end_pos": [1135, 17]}]], "state_before": "case empty\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\np : \u2115\n\u03bc : Measure \u03a9\nt : \u211d\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\ns : Finset \u03b9\nh_int\u271d : \u2200 (i : \u03b9), i \u2208 s \u2192 AEStronglyMeasurable (fun \u03c9 => rexp (t * X i \u03c9)) \u03bc\nh_int : \u2200 (i : \u03b9), i \u2208 \u2205 \u2192 AEStronglyMeasurable (fun \u03c9 => rexp (t * X i \u03c9)) \u03bc\n\u22a2 AEStronglyMeasurable (fun \u03c9 => rexp (t * Finset.sum \u2205 (fun i => X i) \u03c9)) \u03bc", "state_after": "case empty\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\np : \u2115\n\u03bc : Measure \u03a9\nt : \u211d\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\ns : Finset \u03b9\nh_int\u271d : \u2200 (i : \u03b9), i \u2208 s \u2192 AEStronglyMeasurable (fun \u03c9 => rexp (t * X i \u03c9)) \u03bc\nh_int : \u2200 (i : \u03b9), i \u2208 \u2205 \u2192 AEStronglyMeasurable (fun \u03c9 => rexp (t * X i \u03c9)) \u03bc\n\u22a2 AEStronglyMeasurable (fun \u03c9 => 1) \u03bc"}, {"tactic": "exact aestronglyMeasurable_const", "annotated_tactic": ["exact <a>aestronglyMeasurable_const</a>", [{"full_name": "MeasureTheory.aestronglyMeasurable_const", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1155, 9], "def_end_pos": [1155, 35]}]], "state_before": "case empty\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\np : \u2115\n\u03bc : Measure \u03a9\nt : \u211d\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\ns : Finset \u03b9\nh_int\u271d : \u2200 (i : \u03b9), i \u2208 s \u2192 AEStronglyMeasurable (fun \u03c9 => rexp (t * X i \u03c9)) \u03bc\nh_int : \u2200 (i : \u03b9), i \u2208 \u2205 \u2192 AEStronglyMeasurable (fun \u03c9 => rexp (t * X i \u03c9)) \u03bc\n\u22a2 AEStronglyMeasurable (fun \u03c9 => 1) \u03bc", "state_after": "no goals"}, {"tactic": "have : \u2200 i : \u03b9, i \u2208 s \u2192 AEStronglyMeasurable (fun \u03c9 : \u03a9 => exp (t * X i \u03c9)) \u03bc := fun i hi =>\n  h_int i (mem_insert_of_mem hi)", "annotated_tactic": ["have : \u2200 i : \u03b9, i \u2208 s \u2192 <a>AEStronglyMeasurable</a> (fun \u03c9 : \u03a9 => <a>exp</a> (t * X i \u03c9)) \u03bc := fun i hi =>\n      h_int i (<a>mem_insert_of_mem</a> hi)", [{"full_name": "MeasureTheory.AEStronglyMeasurable", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [93, 5], "def_end_pos": [93, 25]}, {"full_name": "Real.exp", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [434, 12], "def_end_pos": [434, 15]}, {"full_name": "Finset.mem_insert_of_mem", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1095, 9], "def_end_pos": [1095, 26]}]], "state_before": "case insert\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\np : \u2115\n\u03bc : Measure \u03a9\nt : \u211d\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\ns\u271d : Finset \u03b9\nh_int\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 AEStronglyMeasurable (fun \u03c9 => rexp (t * X i \u03c9)) \u03bc\ni : \u03b9\ns : Finset \u03b9\nhi_notin_s : \u00aci \u2208 s\nh_rec :\n  (\u2200 (i : \u03b9), i \u2208 s \u2192 AEStronglyMeasurable (fun \u03c9 => rexp (t * X i \u03c9)) \u03bc) \u2192\n    AEStronglyMeasurable (fun \u03c9 => rexp (t * Finset.sum s (fun i => X i) \u03c9)) \u03bc\nh_int : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i s \u2192 AEStronglyMeasurable (fun \u03c9 => rexp (t * X i_1 \u03c9)) \u03bc\n\u22a2 AEStronglyMeasurable (fun \u03c9 => rexp (t * Finset.sum (insert i s) (fun i => X i) \u03c9)) \u03bc", "state_after": "case insert\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\np : \u2115\n\u03bc : Measure \u03a9\nt : \u211d\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\ns\u271d : Finset \u03b9\nh_int\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 AEStronglyMeasurable (fun \u03c9 => rexp (t * X i \u03c9)) \u03bc\ni : \u03b9\ns : Finset \u03b9\nhi_notin_s : \u00aci \u2208 s\nh_rec :\n  (\u2200 (i : \u03b9), i \u2208 s \u2192 AEStronglyMeasurable (fun \u03c9 => rexp (t * X i \u03c9)) \u03bc) \u2192\n    AEStronglyMeasurable (fun \u03c9 => rexp (t * Finset.sum s (fun i => X i) \u03c9)) \u03bc\nh_int : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i s \u2192 AEStronglyMeasurable (fun \u03c9 => rexp (t * X i_1 \u03c9)) \u03bc\nthis : \u2200 (i : \u03b9), i \u2208 s \u2192 AEStronglyMeasurable (fun \u03c9 => rexp (t * X i \u03c9)) \u03bc\n\u22a2 AEStronglyMeasurable (fun \u03c9 => rexp (t * Finset.sum (insert i s) (fun i => X i) \u03c9)) \u03bc"}, {"tactic": "specialize h_rec this", "annotated_tactic": ["specialize h_rec this", []], "state_before": "case insert\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\np : \u2115\n\u03bc : Measure \u03a9\nt : \u211d\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\ns\u271d : Finset \u03b9\nh_int\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 AEStronglyMeasurable (fun \u03c9 => rexp (t * X i \u03c9)) \u03bc\ni : \u03b9\ns : Finset \u03b9\nhi_notin_s : \u00aci \u2208 s\nh_rec :\n  (\u2200 (i : \u03b9), i \u2208 s \u2192 AEStronglyMeasurable (fun \u03c9 => rexp (t * X i \u03c9)) \u03bc) \u2192\n    AEStronglyMeasurable (fun \u03c9 => rexp (t * Finset.sum s (fun i => X i) \u03c9)) \u03bc\nh_int : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i s \u2192 AEStronglyMeasurable (fun \u03c9 => rexp (t * X i_1 \u03c9)) \u03bc\nthis : \u2200 (i : \u03b9), i \u2208 s \u2192 AEStronglyMeasurable (fun \u03c9 => rexp (t * X i \u03c9)) \u03bc\n\u22a2 AEStronglyMeasurable (fun \u03c9 => rexp (t * Finset.sum (insert i s) (fun i => X i) \u03c9)) \u03bc", "state_after": "case insert\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\np : \u2115\n\u03bc : Measure \u03a9\nt : \u211d\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\ns\u271d : Finset \u03b9\nh_int\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 AEStronglyMeasurable (fun \u03c9 => rexp (t * X i \u03c9)) \u03bc\ni : \u03b9\ns : Finset \u03b9\nhi_notin_s : \u00aci \u2208 s\nh_int : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i s \u2192 AEStronglyMeasurable (fun \u03c9 => rexp (t * X i_1 \u03c9)) \u03bc\nthis : \u2200 (i : \u03b9), i \u2208 s \u2192 AEStronglyMeasurable (fun \u03c9 => rexp (t * X i \u03c9)) \u03bc\nh_rec : AEStronglyMeasurable (fun \u03c9 => rexp (t * Finset.sum s (fun i => X i) \u03c9)) \u03bc\n\u22a2 AEStronglyMeasurable (fun \u03c9 => rexp (t * Finset.sum (insert i s) (fun i => X i) \u03c9)) \u03bc"}, {"tactic": "rw [sum_insert hi_notin_s]", "annotated_tactic": ["rw [<a>sum_insert</a> hi_notin_s]", [{"full_name": "Finset.sum_insert", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [316, 3], "def_end_pos": [316, 14]}]], "state_before": "case insert\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\np : \u2115\n\u03bc : Measure \u03a9\nt : \u211d\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\ns\u271d : Finset \u03b9\nh_int\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 AEStronglyMeasurable (fun \u03c9 => rexp (t * X i \u03c9)) \u03bc\ni : \u03b9\ns : Finset \u03b9\nhi_notin_s : \u00aci \u2208 s\nh_int : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i s \u2192 AEStronglyMeasurable (fun \u03c9 => rexp (t * X i_1 \u03c9)) \u03bc\nthis : \u2200 (i : \u03b9), i \u2208 s \u2192 AEStronglyMeasurable (fun \u03c9 => rexp (t * X i \u03c9)) \u03bc\nh_rec : AEStronglyMeasurable (fun \u03c9 => rexp (t * Finset.sum s (fun i => X i) \u03c9)) \u03bc\n\u22a2 AEStronglyMeasurable (fun \u03c9 => rexp (t * Finset.sum (insert i s) (fun i => X i) \u03c9)) \u03bc", "state_after": "case insert\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\np : \u2115\n\u03bc : Measure \u03a9\nt : \u211d\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\ns\u271d : Finset \u03b9\nh_int\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 AEStronglyMeasurable (fun \u03c9 => rexp (t * X i \u03c9)) \u03bc\ni : \u03b9\ns : Finset \u03b9\nhi_notin_s : \u00aci \u2208 s\nh_int : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i s \u2192 AEStronglyMeasurable (fun \u03c9 => rexp (t * X i_1 \u03c9)) \u03bc\nthis : \u2200 (i : \u03b9), i \u2208 s \u2192 AEStronglyMeasurable (fun \u03c9 => rexp (t * X i \u03c9)) \u03bc\nh_rec : AEStronglyMeasurable (fun \u03c9 => rexp (t * Finset.sum s (fun i => X i) \u03c9)) \u03bc\n\u22a2 AEStronglyMeasurable (fun \u03c9 => rexp (t * (X i + \u2211 x in s, X x) \u03c9)) \u03bc"}, {"tactic": "apply aestronglyMeasurable_exp_mul_add (h_int i (mem_insert_self _ _)) h_rec", "annotated_tactic": ["apply <a>aestronglyMeasurable_exp_mul_add</a> (h_int i (<a>mem_insert_self</a> _ _)) h_rec", [{"full_name": "ProbabilityTheory.aestronglyMeasurable_exp_mul_add", "def_path": "Mathlib/Probability/Moments.lean", "def_pos": [255, 9], "def_end_pos": [255, 41]}, {"full_name": "Finset.mem_insert_self", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1091, 9], "def_end_pos": [1091, 24]}]], "state_before": "case insert\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\np : \u2115\n\u03bc : Measure \u03a9\nt : \u211d\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\ns\u271d : Finset \u03b9\nh_int\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 AEStronglyMeasurable (fun \u03c9 => rexp (t * X i \u03c9)) \u03bc\ni : \u03b9\ns : Finset \u03b9\nhi_notin_s : \u00aci \u2208 s\nh_int : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i s \u2192 AEStronglyMeasurable (fun \u03c9 => rexp (t * X i_1 \u03c9)) \u03bc\nthis : \u2200 (i : \u03b9), i \u2208 s \u2192 AEStronglyMeasurable (fun \u03c9 => rexp (t * X i \u03c9)) \u03bc\nh_rec : AEStronglyMeasurable (fun \u03c9 => rexp (t * Finset.sum s (fun i => X i) \u03c9)) \u03bc\n\u22a2 AEStronglyMeasurable (fun \u03c9 => rexp (t * (X i + \u2211 x in s, X x) \u03c9)) \u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Card.lean", "full_name": "Set.ncard_image_iff", "start": [661, 1], "end": [663, 63], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Intervals/OrdConnectedComponent.lean", "full_name": "Set.disjoint_left_ordSeparatingSet", "start": [176, 1], "end": [179, 70], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/RBMap/Lemmas.lean", "full_name": "Std.RBSet.mem_toList_insert", "start": [706, 1], "end": [709, 53], "traced_tactics": [{"tactic": "let \u27e8ht\u2081, _, _, ht\u2082\u27e9 := t.2.out", "annotated_tactic": ["let \u27e8ht\u2081, _, _, ht\u2082\u27e9 := t.2.<a>out</a>", [{"full_name": "Std.RBNode.WF.out", "def_path": "lake-packages/std/Std/Data/RBMap/WF.lean", "def_pos": [466, 9], "def_end_pos": [466, 15]}]], "state_before": "\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nv' v : \u03b1\ninst\u271d : TransCmp cmp\nt : RBSet \u03b1 cmp\n\u22a2 v' \u2208 toList (insert t v) \u2194 v' \u2208 toList t \u2227 find? t v \u2260 some v' \u2228 v' = v", "state_after": "\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nv' v : \u03b1\ninst\u271d : TransCmp cmp\nt : RBSet \u03b1 cmp\nht\u2081 : RBNode.Ordered cmp t.val\nw\u271d\u00b9 : RBColor\nw\u271d : Nat\nht\u2082 : RBNode.Balanced t.val w\u271d\u00b9 w\u271d\n\u22a2 v' \u2208 toList (insert t v) \u2194 v' \u2208 toList t \u2227 find? t v \u2260 some v' \u2228 v' = v"}, {"tactic": "simpa [mem_toList] using RBNode.mem_insert ht\u2082 ht\u2081", "annotated_tactic": ["simpa [<a>mem_toList</a>] using <a>RBNode.mem_insert</a> ht\u2082 ht\u2081", [{"full_name": "Std.RBSet.mem_toList", "def_path": "lake-packages/std/Std/Data/RBMap/Lemmas.lean", "def_pos": [624, 9], "def_end_pos": [624, 19]}, {"full_name": "Std.RBNode.mem_insert", "def_path": "lake-packages/std/Std/Data/RBMap/Lemmas.lean", "def_pos": [590, 9], "def_end_pos": [590, 19]}]], "state_before": "\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nv' v : \u03b1\ninst\u271d : TransCmp cmp\nt : RBSet \u03b1 cmp\nht\u2081 : RBNode.Ordered cmp t.val\nw\u271d\u00b9 : RBColor\nw\u271d : Nat\nht\u2082 : RBNode.Balanced t.val w\u271d\u00b9 w\u271d\n\u22a2 v' \u2208 toList (insert t v) \u2194 v' \u2208 toList t \u2227 find? t v \u2260 some v' \u2228 v' = v", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean", "full_name": "measurableSet_graph", "start": [527, 1], "end": [528, 74], "traced_tactics": [{"tactic": "simpa using measurableSet_region_between_cc hf hf MeasurableSet.univ", "annotated_tactic": ["simpa using <a>measurableSet_region_between_cc</a> hf hf <a>MeasurableSet.univ</a>", [{"full_name": "measurableSet_region_between_cc", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean", "def_pos": [515, 9], "def_end_pos": [515, 40]}, {"full_name": "MeasurableSet.univ", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [101, 19], "def_end_pos": [101, 37]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\nhf : Measurable f\n\u22a2 MeasurableSet {p | p.2 = f p.1}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "full_name": "intervalIntegral.integral_div", "start": [627, 1], "end": [629, 61], "traced_tactics": [{"tactic": "simpa only [div_eq_mul_inv] using integral_mul_const r\u207b\u00b9 f", "annotated_tactic": ["simpa only [<a>div_eq_mul_inv</a>] using <a>integral_mul_const</a> r\u207b\u00b9 f", [{"full_name": "div_eq_mul_inv", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [977, 9], "def_end_pos": [977, 23]}, {"full_name": "intervalIntegral.integral_mul_const", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [621, 9], "def_end_pos": [621, 27]}]], "state_before": "\u03b9 : Type u_1\n\ud835\udd5c\u271d : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : NormedSpace \u211d E\na b : \u211d\nf\u271d g : \u211d \u2192 E\n\u03bc : Measure \u211d\n\ud835\udd5c : Type u_6\ninst\u271d : IsROrC \ud835\udd5c\nr : \ud835\udd5c\nf : \u211d \u2192 \ud835\udd5c\n\u22a2 \u222b (x : \u211d) in a..b, f x / r \u2202\u03bc = (\u222b (x : \u211d) in a..b, f x \u2202\u03bc) / r", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Num/Lemmas.lean", "full_name": "Num.ofNat'_succ", "start": [252, 1], "end": [261, 44], "traced_tactics": [{"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03b1 : Type u_1\n\u22a2 ofNat' (0 + 1) = ofNat' 0 + 1", "state_after": "no goals"}, {"tactic": "cases b", "annotated_tactic": ["cases b", []], "state_before": "\u03b1 : Type u_1\nb : Bool\nn : \u2115\nih : ofNat' (n + 1) = ofNat' n + 1\n\u22a2 ofNat' (Nat.bit b n + 1) = ofNat' (Nat.bit b n) + 1", "state_after": "case false\n\u03b1 : Type u_1\nn : \u2115\nih : ofNat' (n + 1) = ofNat' n + 1\n\u22a2 ofNat' (Nat.bit false n + 1) = ofNat' (Nat.bit false n) + 1\n\ncase true\n\u03b1 : Type u_1\nn : \u2115\nih : ofNat' (n + 1) = ofNat' n + 1\n\u22a2 ofNat' (Nat.bit true n + 1) = ofNat' (Nat.bit true n) + 1"}, {"tactic": "erw [ofNat'_bit true n, ofNat'_bit]", "annotated_tactic": ["erw [<a>ofNat'_bit</a> <a>true</a> n, <a>ofNat'_bit</a>]", [{"full_name": "Num.ofNat'_bit", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [239, 9], "def_end_pos": [239, 19]}, {"full_name": "Bool.true", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [549, 5], "def_end_pos": [549, 9]}, {"full_name": "Num.ofNat'_bit", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [239, 9], "def_end_pos": [239, 19]}]], "state_before": "case false\n\u03b1 : Type u_1\nn : \u2115\nih : ofNat' (n + 1) = ofNat' n + 1\n\u22a2 ofNat' (Nat.bit false n + 1) = ofNat' (Nat.bit false n) + 1", "state_after": "case false\n\u03b1 : Type u_1\nn : \u2115\nih : ofNat' (n + 1) = ofNat' n + 1\n\u22a2 cond true Num.bit1 Num.bit0 (ofNat' n) = cond false Num.bit1 Num.bit0 (ofNat' n) + 1"}, {"tactic": "simp only [\u2190 bit1_of_bit1, \u2190 bit0_of_bit0, cond, _root_.bit1]", "annotated_tactic": ["simp only [\u2190 <a>bit1_of_bit1</a>, \u2190 <a>bit0_of_bit0</a>, <a>cond</a>, <a>_root_.bit1</a>]", [{"full_name": "Num.bit1_of_bit1", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [230, 9], "def_end_pos": [230, 21]}, {"full_name": "Num.bit0_of_bit0", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [225, 9], "def_end_pos": [225, 21]}, {"full_name": "cond", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [971, 21], "def_end_pos": [971, 25]}, {"full_name": "bit1", "def_path": "Mathlib/Init/ZeroOne.lean", "def_pos": [39, 34], "def_end_pos": [39, 38]}]], "state_before": "case false\n\u03b1 : Type u_1\nn : \u2115\nih : ofNat' (n + 1) = ofNat' n + 1\n\u22a2 cond true Num.bit1 Num.bit0 (ofNat' n) = cond false Num.bit1 Num.bit0 (ofNat' n) + 1", "state_after": "no goals"}, {"tactic": "erw [show n.bit true + 1 = (n + 1).bit false by\n  simp [Nat.bit, _root_.bit1, _root_.bit0]; exact Nat.add_left_comm n 1 1,\n  ofNat'_bit, ofNat'_bit, ih]", "annotated_tactic": ["erw [show n.bit <a>true</a> + 1 = (n + 1).<a>bit</a> <a>false</a> by\n        simp [<a>Nat.bit</a>, <a>_root_.bit1</a>, <a>_root_.bit0</a>]; exact <a>Nat.add_left_comm</a> n 1 1,\n        <a>ofNat'_bit</a>, <a>ofNat'_bit</a>, ih]", [{"full_name": "Bool.true", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [549, 5], "def_end_pos": [549, 9]}, {"full_name": "Nat.bit", "def_path": "Mathlib/Init/Data/Nat/Bitwise.lean", "def_pos": [148, 5], "def_end_pos": [148, 8]}, {"full_name": "Bool.false", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [547, 5], "def_end_pos": [547, 10]}, {"full_name": "Nat.bit", "def_path": "Mathlib/Init/Data/Nat/Bitwise.lean", "def_pos": [148, 5], "def_end_pos": [148, 8]}, {"full_name": "bit1", "def_path": "Mathlib/Init/ZeroOne.lean", "def_pos": [39, 34], "def_end_pos": [39, 38]}, {"full_name": "bit0", "def_path": "Mathlib/Init/ZeroOne.lean", "def_pos": [36, 34], "def_end_pos": [36, 38]}, {"full_name": "Nat.add_left_comm", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [142, 19], "def_end_pos": [142, 32]}, {"full_name": "Num.ofNat'_bit", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [239, 9], "def_end_pos": [239, 19]}, {"full_name": "Num.ofNat'_bit", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [239, 9], "def_end_pos": [239, 19]}]], "state_before": "case true\n\u03b1 : Type u_1\nn : \u2115\nih : ofNat' (n + 1) = ofNat' n + 1\n\u22a2 ofNat' (Nat.bit true n + 1) = ofNat' (Nat.bit true n) + 1", "state_after": "case true\n\u03b1 : Type u_1\nn : \u2115\nih : ofNat' (n + 1) = ofNat' n + 1\n\u22a2 cond false Num.bit1 Num.bit0 (ofNat' n + 1) = cond true Num.bit1 Num.bit0 (ofNat' n) + 1"}, {"tactic": "simp only [cond, add_one, bit1_succ]", "annotated_tactic": ["simp only [<a>cond</a>, <a>add_one</a>, <a>bit1_succ</a>]", [{"full_name": "cond", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [971, 21], "def_end_pos": [971, 25]}, {"full_name": "Num.add_one", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [214, 9], "def_end_pos": [214, 16]}, {"full_name": "Num.bit1_succ", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [247, 9], "def_end_pos": [247, 18]}]], "state_before": "case true\n\u03b1 : Type u_1\nn : \u2115\nih : ofNat' (n + 1) = ofNat' n + 1\n\u22a2 cond false Num.bit1 Num.bit0 (ofNat' n + 1) = cond true Num.bit1 Num.bit0 (ofNat' n) + 1", "state_after": "no goals"}, {"tactic": "simp [Nat.bit, _root_.bit1, _root_.bit0]", "annotated_tactic": ["simp [<a>Nat.bit</a>, <a>_root_.bit1</a>, <a>_root_.bit0</a>]", [{"full_name": "Nat.bit", "def_path": "Mathlib/Init/Data/Nat/Bitwise.lean", "def_pos": [148, 5], "def_end_pos": [148, 8]}, {"full_name": "bit1", "def_path": "Mathlib/Init/ZeroOne.lean", "def_pos": [39, 34], "def_end_pos": [39, 38]}, {"full_name": "bit0", "def_path": "Mathlib/Init/ZeroOne.lean", "def_pos": [36, 34], "def_end_pos": [36, 38]}]], "state_before": "\u03b1 : Type u_1\nn : \u2115\nih : ofNat' (n + 1) = ofNat' n + 1\n\u22a2 Nat.bit true n + 1 = Nat.bit false (n + 1)", "state_after": "\u03b1 : Type u_1\nn : \u2115\nih : ofNat' (n + 1) = ofNat' n + 1\n\u22a2 n + (1 + 1) = 1 + (n + 1)"}, {"tactic": "exact Nat.add_left_comm n 1 1", "annotated_tactic": ["exact <a>Nat.add_left_comm</a> n 1 1", [{"full_name": "Nat.add_left_comm", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [142, 19], "def_end_pos": [142, 32]}]], "state_before": "\u03b1 : Type u_1\nn : \u2115\nih : ofNat' (n + 1) = ofNat' n + 1\n\u22a2 n + (1 + 1) = 1 + (n + 1)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "full_name": "MeasureTheory.continuousOn_of_dominated", "start": [1103, 1], "end": [1112, 44], "traced_tactics": [{"tactic": "by_cases hG : CompleteSpace G", "annotated_tactic": ["by_cases hG : <a>CompleteSpace</a> G", [{"full_name": "CompleteSpace", "def_path": "Mathlib/Topology/UniformSpace/Cauchy.lean", "def_pos": [397, 7], "def_end_pos": [397, 20]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF\u271d : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\u271d\ninst\u271d\u2075 : NormedSpace \u211d F\u271d\ninst\u271d\u2074 : CompleteSpace F\u271d\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nF : X \u2192 \u03b1 \u2192 G\nbound : \u03b1 \u2192 \u211d\ns : Set X\nhF_meas : \u2200 (x : X), x \u2208 s \u2192 AEStronglyMeasurable (F x) \u03bc\nh_bound : \u2200 (x : X), x \u2208 s \u2192 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016F x a\u2016 \u2264 bound a\nbound_integrable : Integrable bound\nh_cont : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, ContinuousOn (fun x => F x a) s\n\u22a2 ContinuousOn (fun x => \u222b (a : \u03b1), F x a \u2202\u03bc) s", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF\u271d : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\u271d\ninst\u271d\u2075 : NormedSpace \u211d F\u271d\ninst\u271d\u2074 : CompleteSpace F\u271d\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nF : X \u2192 \u03b1 \u2192 G\nbound : \u03b1 \u2192 \u211d\ns : Set X\nhF_meas : \u2200 (x : X), x \u2208 s \u2192 AEStronglyMeasurable (F x) \u03bc\nh_bound : \u2200 (x : X), x \u2208 s \u2192 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016F x a\u2016 \u2264 bound a\nbound_integrable : Integrable bound\nh_cont : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, ContinuousOn (fun x => F x a) s\nhG : CompleteSpace G\n\u22a2 ContinuousOn (fun x => \u222b (a : \u03b1), F x a \u2202\u03bc) s\n\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF\u271d : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\u271d\ninst\u271d\u2075 : NormedSpace \u211d F\u271d\ninst\u271d\u2074 : CompleteSpace F\u271d\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nF : X \u2192 \u03b1 \u2192 G\nbound : \u03b1 \u2192 \u211d\ns : Set X\nhF_meas : \u2200 (x : X), x \u2208 s \u2192 AEStronglyMeasurable (F x) \u03bc\nh_bound : \u2200 (x : X), x \u2208 s \u2192 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016F x a\u2016 \u2264 bound a\nbound_integrable : Integrable bound\nh_cont : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, ContinuousOn (fun x => F x a) s\nhG : \u00acCompleteSpace G\n\u22a2 ContinuousOn (fun x => \u222b (a : \u03b1), F x a \u2202\u03bc) s"}, {"tactic": "simp only [integral, hG, L1.integral]", "annotated_tactic": ["simp only [<a>integral</a>, hG, <a>L1.integral</a>]", [{"full_name": "MeasureTheory.integral", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [791, 17], "def_end_pos": [791, 25]}, {"full_name": "MeasureTheory.L1.integral", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [666, 17], "def_end_pos": [666, 25]}]], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF\u271d : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\u271d\ninst\u271d\u2075 : NormedSpace \u211d F\u271d\ninst\u271d\u2074 : CompleteSpace F\u271d\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nF : X \u2192 \u03b1 \u2192 G\nbound : \u03b1 \u2192 \u211d\ns : Set X\nhF_meas : \u2200 (x : X), x \u2208 s \u2192 AEStronglyMeasurable (F x) \u03bc\nh_bound : \u2200 (x : X), x \u2208 s \u2192 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016F x a\u2016 \u2264 bound a\nbound_integrable : Integrable bound\nh_cont : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, ContinuousOn (fun x => F x a) s\nhG : CompleteSpace G\n\u22a2 ContinuousOn (fun x => \u222b (a : \u03b1), F x a \u2202\u03bc) s", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF\u271d : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\u271d\ninst\u271d\u2075 : NormedSpace \u211d F\u271d\ninst\u271d\u2074 : CompleteSpace F\u271d\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nF : X \u2192 \u03b1 \u2192 G\nbound : \u03b1 \u2192 \u211d\ns : Set X\nhF_meas : \u2200 (x : X), x \u2208 s \u2192 AEStronglyMeasurable (F x) \u03bc\nh_bound : \u2200 (x : X), x \u2208 s \u2192 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016F x a\u2016 \u2264 bound a\nbound_integrable : Integrable bound\nh_cont : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, ContinuousOn (fun x => F x a) s\nhG : CompleteSpace G\n\u22a2 ContinuousOn\n    (fun x =>\n      if h : True then\n        if hf : Integrable fun a => F x a then \u2191L1.integralCLM (Integrable.toL1 (fun a => F x a) hf) else 0\n      else 0)\n    s"}, {"tactic": "exact continuousOn_setToFun_of_dominated (dominatedFinMeasAdditive_weightedSMul \u03bc)\n  hF_meas h_bound bound_integrable h_cont", "annotated_tactic": ["exact <a>continuousOn_setToFun_of_dominated</a> (<a>dominatedFinMeasAdditive_weightedSMul</a> \u03bc)\n      hF_meas h_bound bound_integrable h_cont", [{"full_name": "MeasureTheory.continuousOn_setToFun_of_dominated", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [1800, 9], "def_end_pos": [1800, 43]}, {"full_name": "MeasureTheory.dominatedFinMeasAdditive_weightedSMul", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [243, 9], "def_end_pos": [243, 46]}]], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF\u271d : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\u271d\ninst\u271d\u2075 : NormedSpace \u211d F\u271d\ninst\u271d\u2074 : CompleteSpace F\u271d\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nF : X \u2192 \u03b1 \u2192 G\nbound : \u03b1 \u2192 \u211d\ns : Set X\nhF_meas : \u2200 (x : X), x \u2208 s \u2192 AEStronglyMeasurable (F x) \u03bc\nh_bound : \u2200 (x : X), x \u2208 s \u2192 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016F x a\u2016 \u2264 bound a\nbound_integrable : Integrable bound\nh_cont : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, ContinuousOn (fun x => F x a) s\nhG : CompleteSpace G\n\u22a2 ContinuousOn\n    (fun x =>\n      if h : True then\n        if hf : Integrable fun a => F x a then \u2191L1.integralCLM (Integrable.toL1 (fun a => F x a) hf) else 0\n      else 0)\n    s", "state_after": "no goals"}, {"tactic": "simp [integral, hG, continuousOn_const]", "annotated_tactic": ["simp [<a>integral</a>, hG, <a>continuousOn_const</a>]", [{"full_name": "MeasureTheory.integral", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [791, 17], "def_end_pos": [791, 25]}, {"full_name": "continuousOn_const", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [1025, 9], "def_end_pos": [1025, 27]}]], "state_before": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF\u271d : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\u271d\ninst\u271d\u2075 : NormedSpace \u211d F\u271d\ninst\u271d\u2074 : CompleteSpace F\u271d\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nF : X \u2192 \u03b1 \u2192 G\nbound : \u03b1 \u2192 \u211d\ns : Set X\nhF_meas : \u2200 (x : X), x \u2208 s \u2192 AEStronglyMeasurable (F x) \u03bc\nh_bound : \u2200 (x : X), x \u2208 s \u2192 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016F x a\u2016 \u2264 bound a\nbound_integrable : Integrable bound\nh_cont : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, ContinuousOn (fun x => F x a) s\nhG : \u00acCompleteSpace G\n\u22a2 ContinuousOn (fun x => \u222b (a : \u03b1), F x a \u2202\u03bc) s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "full_name": "MeasureTheory.SimpleFunc.measure_lt_top_of_mem\u2112p_indicator", "start": [417, 1], "end": [422, 55], "traced_tactics": [{"tactic": "have : Function.support (const \u03b1 c) = Set.univ := Function.support_const hc", "annotated_tactic": ["have : <a>Function.support</a> (<a>const</a> \u03b1 c) = <a>Set.univ</a> := <a>Function.support_const</a> hc", [{"full_name": "Function.support", "def_path": "Mathlib/Algebra/Support.lean", "def_pos": [37, 5], "def_end_pos": [37, 12]}, {"full_name": "MeasureTheory.SimpleFunc.const", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [145, 5], "def_end_pos": [145, 10]}, {"full_name": "Set.univ", "def_path": "Mathlib/Init/Set.lean", "def_pos": [90, 5], "def_end_pos": [90, 9]}, {"full_name": "Function.support_const", "def_path": "Mathlib/Algebra/Support.lean", "def_pos": [139, 3], "def_end_pos": [139, 14]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\nhp_pos : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nc : E\nhc : c \u2260 0\ns : Set \u03b1\nhs : MeasurableSet s\nhcs : Mem\u2112p (\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0))) p\n\u22a2 \u2191\u2191\u03bc s < \u22a4", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\nhp_pos : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nc : E\nhc : c \u2260 0\ns : Set \u03b1\nhs : MeasurableSet s\nhcs : Mem\u2112p (\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0))) p\nthis : support \u2191(const \u03b1 c) = Set.univ\n\u22a2 \u2191\u2191\u03bc s < \u22a4"}, {"tactic": "simpa only [mem\u2112p_iff_finMeasSupp hp_pos hp_ne_top, finMeasSupp_iff_support,\n  support_indicator, Set.inter_univ, this] using hcs", "annotated_tactic": ["simpa only [<a>mem\u2112p_iff_finMeasSupp</a> hp_pos hp_ne_top, <a>finMeasSupp_iff_support</a>,\n    <a>support_indicator</a>, <a>Set.inter_univ</a>, this] using hcs", [{"full_name": "MeasureTheory.SimpleFunc.mem\u2112p_iff_finMeasSupp", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "def_pos": [366, 9], "def_end_pos": [366, 30]}, {"full_name": "MeasureTheory.SimpleFunc.finMeasSupp_iff_support", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [1170, 9], "def_end_pos": [1170, 32]}, {"full_name": "MeasureTheory.SimpleFunc.support_indicator", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [256, 9], "def_end_pos": [256, 26]}, {"full_name": "Set.inter_univ", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1012, 9], "def_end_pos": [1012, 19]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\nhp_pos : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nc : E\nhc : c \u2260 0\ns : Set \u03b1\nhs : MeasurableSet s\nhcs : Mem\u2112p (\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0))) p\nthis : support \u2191(const \u03b1 c) = Set.univ\n\u22a2 \u2191\u2191\u03bc s < \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/LocallyFinite.lean", "full_name": "Finset.uIcc_eq_union", "start": [1041, 1], "end": [1044, 28], "traced_tactics": [{"tactic": "push_cast", "annotated_tactic": ["push_cast", []], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\n\u22a2 \u2191[[a, b]] = \u2191(Icc a b \u222a Icc b a)", "state_after": "\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\n\u22a2 Set.uIcc a b = Set.Icc a b \u222a Set.Icc b a"}, {"tactic": "exact Set.uIcc_eq_union", "annotated_tactic": ["exact <a>Set.uIcc_eq_union</a>", [{"full_name": "Set.uIcc_eq_union", "def_path": "Mathlib/Data/Set/Intervals/UnorderedInterval.lean", "def_pos": [229, 7], "def_end_pos": [229, 20]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\n\u22a2 Set.uIcc a b = Set.Icc a b \u222a Set.Icc b a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Primrec.lean", "full_name": "Primrec.nat_sqrt", "start": [1567, 1], "end": [1568, 45], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "full_name": "intervalIntegral.integral_interval_sub_interval_comm'", "start": [951, 1], "end": [956, 36], "traced_tactics": [{"tactic": "rw [integral_interval_sub_interval_comm hab hcd hac, integral_symm b d, integral_symm a c,\n  sub_neg_eq_add, sub_eq_neg_add]", "annotated_tactic": ["rw [<a>integral_interval_sub_interval_comm</a> hab hcd hac, <a>integral_symm</a> b d, <a>integral_symm</a> a c,\n    <a>sub_neg_eq_add</a>, <a>sub_eq_neg_add</a>]", [{"full_name": "intervalIntegral.integral_interval_sub_interval_comm", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [943, 9], "def_end_pos": [943, 44]}, {"full_name": "intervalIntegral.integral_symm", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [474, 9], "def_end_pos": [474, 22]}, {"full_name": "intervalIntegral.integral_symm", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [474, 9], "def_end_pos": [474, 22]}, {"full_name": "sub_neg_eq_add", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [453, 3], "def_end_pos": [453, 14]}, {"full_name": "sub_eq_neg_add", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [491, 3], "def_end_pos": [491, 14]}]], "state_before": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\na b c d : \u211d\nf g : \u211d \u2192 E\n\u03bc : Measure \u211d\nhab : IntervalIntegrable f \u03bc a b\nhcd : IntervalIntegrable f \u03bc c d\nhac : IntervalIntegrable f \u03bc a c\n\u22a2 \u222b (x : \u211d) in a..b, f x \u2202\u03bc - \u222b (x : \u211d) in c..d, f x \u2202\u03bc = \u222b (x : \u211d) in d..b, f x \u2202\u03bc - \u222b (x : \u211d) in c..a, f x \u2202\u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "full_name": "String.valid_next", "start": [277, 1], "end": [282, 45], "traced_tactics": [{"tactic": "match s, p, h with\n| \u27e8_\u27e9, \u27e8_\u27e9, \u27e8cs, [], rfl, rfl\u27e9 => simp at h\u2082\n| \u27e8_\u27e9, \u27e8_\u27e9, \u27e8cs, c::cs', rfl, rfl\u27e9 =>\n  rw [utf8ByteSize.go_eq, next_of_valid]\n  simpa using Pos.Valid.mk (cs ++ [c]) cs'", "annotated_tactic": ["match s, p, h with\n  | \u27e8_\u27e9, \u27e8_\u27e9, \u27e8cs, [], <a>rfl</a>, <a>rfl</a>\u27e9 => simp at h\u2082\n  | \u27e8_\u27e9, \u27e8_\u27e9, \u27e8cs, c::cs', <a>rfl</a>, <a>rfl</a>\u27e9 =>\n    rw [<a>utf8ByteSize.go_eq</a>, <a>next_of_valid</a>]\n    simpa using <a>Pos.Valid.mk</a> (cs ++ [c]) cs'", [{"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}, {"full_name": "Nat", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1038, 11], "def_end_pos": [1038, 14]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}, {"full_name": "Nat", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1038, 11], "def_end_pos": [1038, 14]}, {"full_name": "String.utf8ByteSize.go_eq", "def_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "def_pos": [56, 17], "def_end_pos": [56, 35]}, {"full_name": "String.next_of_valid", "def_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "def_pos": [269, 9], "def_end_pos": [269, 22]}, {"full_name": "String.Pos.Valid.mk", "def_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "def_pos": [168, 9], "def_end_pos": [168, 17]}]], "state_before": "s : String\np : Pos\nh : Pos.Valid s p\nh\u2082 : p < endPos s\n\u22a2 Pos.Valid s (next s p)", "state_after": "no goals"}, {"tactic": "simp at h\u2082", "annotated_tactic": ["simp at h\u2082", []], "state_before": "s : String\np : Pos\nh : Pos.Valid s p\ncs : List Char\nh\u2082 : { byteIdx := utf8ByteSize.go cs } < endPos { data := cs ++ [] }\n\u22a2 Pos.Valid { data := cs ++ [] } (next { data := cs ++ [] } { byteIdx := utf8ByteSize.go cs })", "state_after": "no goals"}, {"tactic": "rw [utf8ByteSize.go_eq, next_of_valid]", "annotated_tactic": ["rw [<a>utf8ByteSize.go_eq</a>, <a>next_of_valid</a>]", [{"full_name": "String.utf8ByteSize.go_eq", "def_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "def_pos": [56, 17], "def_end_pos": [56, 35]}, {"full_name": "String.next_of_valid", "def_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "def_pos": [269, 9], "def_end_pos": [269, 22]}]], "state_before": "s : String\np : Pos\nh : Pos.Valid s p\ncs : List Char\nc : Char\ncs' : List Char\nh\u2082 : { byteIdx := utf8ByteSize.go cs } < endPos { data := cs ++ c :: cs' }\n\u22a2 Pos.Valid { data := cs ++ c :: cs' } (next { data := cs ++ c :: cs' } { byteIdx := utf8ByteSize.go cs })", "state_after": "s : String\np : Pos\nh : Pos.Valid s p\ncs : List Char\nc : Char\ncs' : List Char\nh\u2082 : { byteIdx := utf8ByteSize.go cs } < endPos { data := cs ++ c :: cs' }\n\u22a2 Pos.Valid { data := cs ++ c :: cs' } { byteIdx := utf8Len cs + csize c }"}, {"tactic": "simpa using Pos.Valid.mk (cs ++ [c]) cs'", "annotated_tactic": ["simpa using <a>Pos.Valid.mk</a> (cs ++ [c]) cs'", [{"full_name": "String.Pos.Valid.mk", "def_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "def_pos": [168, 9], "def_end_pos": [168, 17]}]], "state_before": "s : String\np : Pos\nh : Pos.Valid s p\ncs : List Char\nc : Char\ncs' : List Char\nh\u2082 : { byteIdx := utf8ByteSize.go cs } < endPos { data := cs ++ c :: cs' }\n\u22a2 Pos.Valid { data := cs ++ c :: cs' } { byteIdx := utf8Len cs + csize c }", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/CircleTransform.lean", "full_name": "Complex.abs_circleTransformBoundingFunction_le", "start": [128, 1], "end": [139, 55], "traced_tactics": [{"tactic": "have cts := continuousOn_abs_circleTransformBoundingFunction hr z", "annotated_tactic": ["have cts := <a>continuousOn_abs_circleTransformBoundingFunction</a> hr z", [{"full_name": "Complex.continuousOn_abs_circleTransformBoundingFunction", "def_path": "Mathlib/MeasureTheory/Integral/CircleTransform.lean", "def_pos": [113, 9], "def_end_pos": [113, 57]}]], "state_before": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR r : \u211d\nhr : r < R\nhr' : 0 \u2264 r\nz : \u2102\n\u22a2 \u2203 x,\n    \u2200 (y : \u2191(closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])),\n      \u2191abs (circleTransformBoundingFunction R z \u2191y) \u2264 \u2191abs (circleTransformBoundingFunction R z \u2191x)", "state_after": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR r : \u211d\nhr : r < R\nhr' : 0 \u2264 r\nz : \u2102\ncts : ContinuousOn (\u2191abs \u2218 fun t => circleTransformBoundingFunction R z t) (closedBall z r \u00d7\u02e2 univ)\n\u22a2 \u2203 x,\n    \u2200 (y : \u2191(closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])),\n      \u2191abs (circleTransformBoundingFunction R z \u2191y) \u2264 \u2191abs (circleTransformBoundingFunction R z \u2191x)"}, {"tactic": "have comp : IsCompact (closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]]) := by\n  apply_rules [IsCompact.prod, ProperSpace.isCompact_closedBall z r, isCompact_uIcc]", "annotated_tactic": ["have comp : <a>IsCompact</a> (<a>closedBall</a> z r \u00d7\u02e2 [[0, 2 * \u03c0]]) := by\n    apply_rules [<a>IsCompact.prod</a>, <a>ProperSpace.isCompact_closedBall</a> z r, <a>isCompact_uIcc</a>]", [{"full_name": "IsCompact", "def_path": "Mathlib/Topology/Compactness/Compact.lean", "def_pos": [40, 5], "def_end_pos": [40, 14]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "IsCompact.prod", "def_path": "Mathlib/Topology/Compactness/Compact.lean", "def_pos": [911, 9], "def_end_pos": [911, 23]}, {"full_name": "ProperSpace.isCompact_closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [2199, 3], "def_end_pos": [2199, 23]}, {"full_name": "isCompact_uIcc", "def_path": "Mathlib/Topology/Algebra/Order/Compact.lean", "def_pos": [133, 9], "def_end_pos": [133, 23]}]], "state_before": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR r : \u211d\nhr : r < R\nhr' : 0 \u2264 r\nz : \u2102\ncts : ContinuousOn (\u2191abs \u2218 fun t => circleTransformBoundingFunction R z t) (closedBall z r \u00d7\u02e2 univ)\n\u22a2 \u2203 x,\n    \u2200 (y : \u2191(closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])),\n      \u2191abs (circleTransformBoundingFunction R z \u2191y) \u2264 \u2191abs (circleTransformBoundingFunction R z \u2191x)", "state_after": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR r : \u211d\nhr : r < R\nhr' : 0 \u2264 r\nz : \u2102\ncts : ContinuousOn (\u2191abs \u2218 fun t => circleTransformBoundingFunction R z t) (closedBall z r \u00d7\u02e2 univ)\ncomp : IsCompact (closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])\n\u22a2 \u2203 x,\n    \u2200 (y : \u2191(closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])),\n      \u2191abs (circleTransformBoundingFunction R z \u2191y) \u2264 \u2191abs (circleTransformBoundingFunction R z \u2191x)"}, {"tactic": "have none : (closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]]).Nonempty :=\n  (nonempty_closedBall.2 hr').prod nonempty_uIcc", "annotated_tactic": ["have none : (<a>closedBall</a> z r \u00d7\u02e2 [[0, 2 * \u03c0]]).<a>Nonempty</a> :=\n    (<a>nonempty_closedBall</a>.2 hr').<a>prod</a> <a>nonempty_uIcc</a>", [{"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "Set.Nonempty", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [439, 15], "def_end_pos": [439, 23]}, {"full_name": "Metric.nonempty_closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [520, 9], "def_end_pos": [520, 28]}, {"full_name": "Set.Nonempty.prod", "def_path": "Mathlib/Data/Set/Prod.lean", "def_pos": [335, 9], "def_end_pos": [335, 22]}, {"full_name": "Set.nonempty_uIcc", "def_path": "Mathlib/Data/Set/Intervals/UnorderedInterval.lean", "def_pos": [89, 15], "def_end_pos": [89, 28]}]], "state_before": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR r : \u211d\nhr : r < R\nhr' : 0 \u2264 r\nz : \u2102\ncts : ContinuousOn (\u2191abs \u2218 fun t => circleTransformBoundingFunction R z t) (closedBall z r \u00d7\u02e2 univ)\ncomp : IsCompact (closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])\n\u22a2 \u2203 x,\n    \u2200 (y : \u2191(closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])),\n      \u2191abs (circleTransformBoundingFunction R z \u2191y) \u2264 \u2191abs (circleTransformBoundingFunction R z \u2191x)", "state_after": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR r : \u211d\nhr : r < R\nhr' : 0 \u2264 r\nz : \u2102\ncts : ContinuousOn (\u2191abs \u2218 fun t => circleTransformBoundingFunction R z t) (closedBall z r \u00d7\u02e2 univ)\ncomp : IsCompact (closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])\nnone : Set.Nonempty (closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])\n\u22a2 \u2203 x,\n    \u2200 (y : \u2191(closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])),\n      \u2191abs (circleTransformBoundingFunction R z \u2191y) \u2264 \u2191abs (circleTransformBoundingFunction R z \u2191x)"}, {"tactic": "have := IsCompact.exists_isMaxOn comp none (cts.mono\n  (by intro z; simp only [mem_prod, mem_closedBall, mem_univ, and_true_iff, and_imp]; tauto))", "annotated_tactic": ["have := <a>IsCompact.exists_isMaxOn</a> comp none (cts.mono\n    (by intro z; simp only [<a>mem_prod</a>, <a>mem_closedBall</a>, <a>mem_univ</a>, <a>and_true_iff</a>, <a>and_imp</a>]; tauto))", [{"full_name": "IsCompact.exists_isMaxOn", "def_path": "Mathlib/Topology/Algebra/Order/Compact.lean", "def_pos": [212, 9], "def_end_pos": [212, 33]}, {"full_name": "Set.mem_prod", "def_path": "Mathlib/Data/Set/Prod.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}, {"full_name": "Metric.mem_closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [478, 17], "def_end_pos": [478, 31]}, {"full_name": "Set.mem_univ", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [676, 9], "def_end_pos": [676, 17]}, {"full_name": "and_true_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [145, 9], "def_end_pos": [145, 21]}, {"full_name": "and_imp", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [313, 17], "def_end_pos": [313, 24]}]], "state_before": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR r : \u211d\nhr : r < R\nhr' : 0 \u2264 r\nz : \u2102\ncts : ContinuousOn (\u2191abs \u2218 fun t => circleTransformBoundingFunction R z t) (closedBall z r \u00d7\u02e2 univ)\ncomp : IsCompact (closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])\nnone : Set.Nonempty (closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])\n\u22a2 \u2203 x,\n    \u2200 (y : \u2191(closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])),\n      \u2191abs (circleTransformBoundingFunction R z \u2191y) \u2264 \u2191abs (circleTransformBoundingFunction R z \u2191x)", "state_after": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR r : \u211d\nhr : r < R\nhr' : 0 \u2264 r\nz : \u2102\ncts : ContinuousOn (\u2191abs \u2218 fun t => circleTransformBoundingFunction R z t) (closedBall z r \u00d7\u02e2 univ)\ncomp : IsCompact (closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])\nnone : Set.Nonempty (closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])\nthis :\n  \u2203 x,\n    x \u2208 closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]] \u2227\n      IsMaxOn (\u2191abs \u2218 fun t => circleTransformBoundingFunction R z t) (closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]]) x\n\u22a2 \u2203 x,\n    \u2200 (y : \u2191(closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])),\n      \u2191abs (circleTransformBoundingFunction R z \u2191y) \u2264 \u2191abs (circleTransformBoundingFunction R z \u2191x)"}, {"tactic": "simp only [IsMaxOn, IsMaxFilter] at this", "annotated_tactic": ["simp only [<a>IsMaxOn</a>, <a>IsMaxFilter</a>] at this", [{"full_name": "IsMaxOn", "def_path": "Mathlib/Order/Filter/Extr.lean", "def_pos": [116, 5], "def_end_pos": [116, 12]}, {"full_name": "IsMaxFilter", "def_path": "Mathlib/Order/Filter/Extr.lean", "def_pos": [101, 5], "def_end_pos": [101, 16]}]], "state_before": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR r : \u211d\nhr : r < R\nhr' : 0 \u2264 r\nz : \u2102\ncts : ContinuousOn (\u2191abs \u2218 fun t => circleTransformBoundingFunction R z t) (closedBall z r \u00d7\u02e2 univ)\ncomp : IsCompact (closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])\nnone : Set.Nonempty (closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])\nthis :\n  \u2203 x,\n    x \u2208 closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]] \u2227\n      IsMaxOn (\u2191abs \u2218 fun t => circleTransformBoundingFunction R z t) (closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]]) x\n\u22a2 \u2203 x,\n    \u2200 (y : \u2191(closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])),\n      \u2191abs (circleTransformBoundingFunction R z \u2191y) \u2264 \u2191abs (circleTransformBoundingFunction R z \u2191x)", "state_after": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR r : \u211d\nhr : r < R\nhr' : 0 \u2264 r\nz : \u2102\ncts : ContinuousOn (\u2191abs \u2218 fun t => circleTransformBoundingFunction R z t) (closedBall z r \u00d7\u02e2 univ)\ncomp : IsCompact (closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])\nnone : Set.Nonempty (closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])\nthis :\n  \u2203 x,\n    x \u2208 closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]] \u2227\n      \u2200\u1da0 (x_1 : \u2102 \u00d7 \u211d) in \ud835\udcdf (closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]]),\n        (\u2191abs \u2218 fun t => circleTransformBoundingFunction R z t) x_1 \u2264\n          (\u2191abs \u2218 fun t => circleTransformBoundingFunction R z t) x\n\u22a2 \u2203 x,\n    \u2200 (y : \u2191(closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])),\n      \u2191abs (circleTransformBoundingFunction R z \u2191y) \u2264 \u2191abs (circleTransformBoundingFunction R z \u2191x)"}, {"tactic": "simpa [SetCoe.forall, Subtype.coe_mk, SetCoe.exists]", "annotated_tactic": ["simpa [<a>SetCoe.forall</a>, <a>Subtype.coe_mk</a>, <a>SetCoe.exists</a>]", [{"full_name": "SetCoe.forall", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [185, 9], "def_end_pos": [185, 22]}, {"full_name": "Subtype.coe_mk", "def_path": "Mathlib/Data/Subtype.lean", "def_pos": [99, 9], "def_end_pos": [99, 15]}, {"full_name": "SetCoe.exists", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [190, 9], "def_end_pos": [190, 22]}]], "state_before": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR r : \u211d\nhr : r < R\nhr' : 0 \u2264 r\nz : \u2102\ncts : ContinuousOn (\u2191abs \u2218 fun t => circleTransformBoundingFunction R z t) (closedBall z r \u00d7\u02e2 univ)\ncomp : IsCompact (closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])\nnone : Set.Nonempty (closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])\nthis :\n  \u2203 x,\n    x \u2208 closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]] \u2227\n      \u2200\u1da0 (x_1 : \u2102 \u00d7 \u211d) in \ud835\udcdf (closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]]),\n        (\u2191abs \u2218 fun t => circleTransformBoundingFunction R z t) x_1 \u2264\n          (\u2191abs \u2218 fun t => circleTransformBoundingFunction R z t) x\n\u22a2 \u2203 x,\n    \u2200 (y : \u2191(closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])),\n      \u2191abs (circleTransformBoundingFunction R z \u2191y) \u2264 \u2191abs (circleTransformBoundingFunction R z \u2191x)", "state_after": "no goals"}, {"tactic": "apply_rules [IsCompact.prod, ProperSpace.isCompact_closedBall z r, isCompact_uIcc]", "annotated_tactic": ["apply_rules [<a>IsCompact.prod</a>, <a>ProperSpace.isCompact_closedBall</a> z r, <a>isCompact_uIcc</a>]", [{"full_name": "IsCompact.prod", "def_path": "Mathlib/Topology/Compactness/Compact.lean", "def_pos": [911, 9], "def_end_pos": [911, 23]}, {"full_name": "ProperSpace.isCompact_closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [2199, 3], "def_end_pos": [2199, 23]}, {"full_name": "isCompact_uIcc", "def_path": "Mathlib/Topology/Algebra/Order/Compact.lean", "def_pos": [133, 9], "def_end_pos": [133, 23]}]], "state_before": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR r : \u211d\nhr : r < R\nhr' : 0 \u2264 r\nz : \u2102\ncts : ContinuousOn (\u2191abs \u2218 fun t => circleTransformBoundingFunction R z t) (closedBall z r \u00d7\u02e2 univ)\n\u22a2 IsCompact (closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])", "state_after": "no goals"}, {"tactic": "intro z", "annotated_tactic": ["intro z", []], "state_before": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR r : \u211d\nhr : r < R\nhr' : 0 \u2264 r\nz : \u2102\ncts : ContinuousOn (\u2191abs \u2218 fun t => circleTransformBoundingFunction R z t) (closedBall z r \u00d7\u02e2 univ)\ncomp : IsCompact (closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])\nnone : Set.Nonempty (closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])\n\u22a2 closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]] \u2286 closedBall z r \u00d7\u02e2 univ", "state_after": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d\u00b9 w : \u2102\nR r : \u211d\nhr : r < R\nhr' : 0 \u2264 r\nz\u271d : \u2102\ncts : ContinuousOn (\u2191abs \u2218 fun t => circleTransformBoundingFunction R z\u271d t) (closedBall z\u271d r \u00d7\u02e2 univ)\ncomp : IsCompact (closedBall z\u271d r \u00d7\u02e2 [[0, 2 * \u03c0]])\nnone : Set.Nonempty (closedBall z\u271d r \u00d7\u02e2 [[0, 2 * \u03c0]])\nz : \u2102 \u00d7 \u211d\n\u22a2 z \u2208 closedBall z\u271d r \u00d7\u02e2 [[0, 2 * \u03c0]] \u2192 z \u2208 closedBall z\u271d r \u00d7\u02e2 univ"}, {"tactic": "simp only [mem_prod, mem_closedBall, mem_univ, and_true_iff, and_imp]", "annotated_tactic": ["simp only [<a>mem_prod</a>, <a>mem_closedBall</a>, <a>mem_univ</a>, <a>and_true_iff</a>, <a>and_imp</a>]", [{"full_name": "Set.mem_prod", "def_path": "Mathlib/Data/Set/Prod.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}, {"full_name": "Metric.mem_closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [478, 17], "def_end_pos": [478, 31]}, {"full_name": "Set.mem_univ", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [676, 9], "def_end_pos": [676, 17]}, {"full_name": "and_true_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [145, 9], "def_end_pos": [145, 21]}, {"full_name": "and_imp", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [313, 17], "def_end_pos": [313, 24]}]], "state_before": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d\u00b9 w : \u2102\nR r : \u211d\nhr : r < R\nhr' : 0 \u2264 r\nz\u271d : \u2102\ncts : ContinuousOn (\u2191abs \u2218 fun t => circleTransformBoundingFunction R z\u271d t) (closedBall z\u271d r \u00d7\u02e2 univ)\ncomp : IsCompact (closedBall z\u271d r \u00d7\u02e2 [[0, 2 * \u03c0]])\nnone : Set.Nonempty (closedBall z\u271d r \u00d7\u02e2 [[0, 2 * \u03c0]])\nz : \u2102 \u00d7 \u211d\n\u22a2 z \u2208 closedBall z\u271d r \u00d7\u02e2 [[0, 2 * \u03c0]] \u2192 z \u2208 closedBall z\u271d r \u00d7\u02e2 univ", "state_after": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d\u00b9 w : \u2102\nR r : \u211d\nhr : r < R\nhr' : 0 \u2264 r\nz\u271d : \u2102\ncts : ContinuousOn (\u2191abs \u2218 fun t => circleTransformBoundingFunction R z\u271d t) (closedBall z\u271d r \u00d7\u02e2 univ)\ncomp : IsCompact (closedBall z\u271d r \u00d7\u02e2 [[0, 2 * \u03c0]])\nnone : Set.Nonempty (closedBall z\u271d r \u00d7\u02e2 [[0, 2 * \u03c0]])\nz : \u2102 \u00d7 \u211d\n\u22a2 dist z.1 z\u271d \u2264 r \u2192 z.2 \u2208 [[0, 2 * \u03c0]] \u2192 dist z.1 z\u271d \u2264 r"}, {"tactic": "tauto", "annotated_tactic": ["tauto", []], "state_before": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d\u00b9 w : \u2102\nR r : \u211d\nhr : r < R\nhr' : 0 \u2264 r\nz\u271d : \u2102\ncts : ContinuousOn (\u2191abs \u2218 fun t => circleTransformBoundingFunction R z\u271d t) (closedBall z\u271d r \u00d7\u02e2 univ)\ncomp : IsCompact (closedBall z\u271d r \u00d7\u02e2 [[0, 2 * \u03c0]])\nnone : Set.Nonempty (closedBall z\u271d r \u00d7\u02e2 [[0, 2 * \u03c0]])\nz : \u2102 \u00d7 \u211d\n\u22a2 dist z.1 z\u271d \u2264 r \u2192 z.2 \u2208 [[0, 2 * \u03c0]] \u2192 dist z.1 z\u271d \u2264 r", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Intervals/Group.lean", "full_name": "Set.pairwise_disjoint_Ioc_add_int_cast", "start": [233, 1], "end": [236, 46], "traced_tactics": [{"tactic": "simpa only [zsmul_one, Int.cast_add, Int.cast_one, \u2190 add_assoc] using\n  pairwise_disjoint_Ioc_add_zsmul a (1 : \u03b1)", "annotated_tactic": ["simpa only [<a>zsmul_one</a>, <a>Int.cast_add</a>, <a>Int.cast_one</a>, \u2190 <a>add_assoc</a>] using\n    <a>pairwise_disjoint_Ioc_add_zsmul</a> a (1 : \u03b1)", [{"full_name": "zsmul_one", "def_path": "Mathlib/Algebra/GroupPower/Lemmas.lean", "def_pos": [130, 9], "def_end_pos": [130, 18]}, {"full_name": "Int.cast_add", "def_path": "Mathlib/Data/Int/Cast/Basic.lean", "def_pos": [108, 9], "def_end_pos": [108, 17]}, {"full_name": "Int.cast_one", "def_path": "Mathlib/Data/Int/Cast/Basic.lean", "def_pos": [77, 9], "def_end_pos": [77, 17]}, {"full_name": "add_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [263, 3], "def_end_pos": [263, 14]}, {"full_name": "Set.pairwise_disjoint_Ioc_add_zsmul", "def_path": "Mathlib/Data/Set/Intervals/Group.lean", "def_pos": [167, 3], "def_end_pos": [167, 14]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : OrderedRing \u03b1\na : \u03b1\n\u22a2 Pairwise (Disjoint on fun n => Ioc (a + \u2191n) (a + \u2191n + 1))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Fin/Lemmas.lean", "full_name": "Fin.castLE_comp_castLE", "start": [292, 9], "end": [294, 31], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Halting.lean", "full_name": "Nat.Partrec.merge'", "start": [27, 1], "end": [57, 39], "traced_tactics": [{"tactic": "obtain \u27e8cf, rfl\u27e9 := Code.exists_code.1 hf", "annotated_tactic": ["obtain \u27e8cf, rfl\u27e9 := <a>Code.exists_code</a>.1 hf", [{"full_name": "Nat.Partrec.Code.exists_code", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [699, 9], "def_end_pos": [699, 20]}]], "state_before": "f g : \u2115 \u2192. \u2115\nhf : Partrec f\nhg : Partrec g\n\u22a2 \u2203 h, Partrec h \u2227 \u2200 (a : \u2115), (\u2200 (x : \u2115), x \u2208 h a \u2192 x \u2208 f a \u2228 x \u2208 g a) \u2227 ((h a).Dom \u2194 (f a).Dom \u2228 (g a).Dom)", "state_after": "case intro\ng : \u2115 \u2192. \u2115\nhg : Partrec g\ncf : Code\nhf : Partrec (Code.eval cf)\n\u22a2 \u2203 h,\n    Partrec h \u2227\n      \u2200 (a : \u2115), (\u2200 (x : \u2115), x \u2208 h a \u2192 x \u2208 Code.eval cf a \u2228 x \u2208 g a) \u2227 ((h a).Dom \u2194 (Code.eval cf a).Dom \u2228 (g a).Dom)"}, {"tactic": "obtain \u27e8cg, rfl\u27e9 := Code.exists_code.1 hg", "annotated_tactic": ["obtain \u27e8cg, rfl\u27e9 := <a>Code.exists_code</a>.1 hg", [{"full_name": "Nat.Partrec.Code.exists_code", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [699, 9], "def_end_pos": [699, 20]}]], "state_before": "case intro\ng : \u2115 \u2192. \u2115\nhg : Partrec g\ncf : Code\nhf : Partrec (Code.eval cf)\n\u22a2 \u2203 h,\n    Partrec h \u2227\n      \u2200 (a : \u2115), (\u2200 (x : \u2115), x \u2208 h a \u2192 x \u2208 Code.eval cf a \u2228 x \u2208 g a) \u2227 ((h a).Dom \u2194 (Code.eval cf a).Dom \u2228 (g a).Dom)", "state_after": "case intro.intro\ncf : Code\nhf : Partrec (Code.eval cf)\ncg : Code\nhg : Partrec (Code.eval cg)\n\u22a2 \u2203 h,\n    Partrec h \u2227\n      \u2200 (a : \u2115),\n        (\u2200 (x : \u2115), x \u2208 h a \u2192 x \u2208 Code.eval cf a \u2228 x \u2208 Code.eval cg a) \u2227\n          ((h a).Dom \u2194 (Code.eval cf a).Dom \u2228 (Code.eval cg a).Dom)"}, {"tactic": "have : Nat.Partrec fun n => Nat.rfindOpt fun k => cf.evaln k n <|> cg.evaln k n :=\n  Partrec.nat_iff.1\n    (Partrec.rfindOpt <|\n      Primrec.option_orElse.to_comp.comp\n        (Code.evaln_prim.to_comp.comp <| (snd.pair (const cf)).pair fst)\n        (Code.evaln_prim.to_comp.comp <| (snd.pair (const cg)).pair fst))", "annotated_tactic": ["have : <a>Nat.Partrec</a> fun n => <a>Nat.rfindOpt</a> fun k => cf.evaln k n <|> cg.evaln k n :=\n    <a>Partrec.nat_iff</a>.1\n      (<a>Partrec.rfindOpt</a> <|\n        Primrec.option_orElse.to_comp.comp\n          (Code.evaln_prim.to_comp.comp <| (snd.pair (<a>const</a> cf)).<a>pair</a> <a>fst</a>)\n          (Code.evaln_prim.to_comp.comp <| (snd.pair (<a>const</a> cg)).<a>pair</a> <a>fst</a>))", [{"full_name": "Nat.Partrec", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [155, 11], "def_end_pos": [155, 18]}, {"full_name": "Nat.rfindOpt", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [124, 5], "def_end_pos": [124, 13]}, {"full_name": "Partrec.nat_iff", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [485, 9], "def_end_pos": [485, 16]}, {"full_name": "Partrec.rfindOpt", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [576, 9], "def_end_pos": [576, 17]}, {"full_name": "Computable.const", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [284, 9], "def_end_pos": [284, 14]}, {"full_name": "Computable.pair", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [310, 16], "def_end_pos": [310, 20]}, {"full_name": "Computable.fst", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [302, 9], "def_end_pos": [302, 12]}, {"full_name": "Computable.const", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [284, 9], "def_end_pos": [284, 14]}, {"full_name": "Computable.pair", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [310, 16], "def_end_pos": [310, 20]}, {"full_name": "Computable.fst", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [302, 9], "def_end_pos": [302, 12]}]], "state_before": "case intro.intro\ncf : Code\nhf : Partrec (Code.eval cf)\ncg : Code\nhg : Partrec (Code.eval cg)\n\u22a2 \u2203 h,\n    Partrec h \u2227\n      \u2200 (a : \u2115),\n        (\u2200 (x : \u2115), x \u2208 h a \u2192 x \u2208 Code.eval cf a \u2228 x \u2208 Code.eval cg a) \u2227\n          ((h a).Dom \u2194 (Code.eval cf a).Dom \u2228 (Code.eval cg a).Dom)", "state_after": "case intro.intro\ncf : Code\nhf : Partrec (Code.eval cf)\ncg : Code\nhg : Partrec (Code.eval cg)\nthis : Partrec fun n => rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n\n\u22a2 \u2203 h,\n    Partrec h \u2227\n      \u2200 (a : \u2115),\n        (\u2200 (x : \u2115), x \u2208 h a \u2192 x \u2208 Code.eval cf a \u2228 x \u2208 Code.eval cg a) \u2227\n          ((h a).Dom \u2194 (Code.eval cf a).Dom \u2228 (Code.eval cg a).Dom)"}, {"tactic": "refine' \u27e8_, this, fun n => _\u27e9", "annotated_tactic": ["refine' \u27e8_, this, fun n => _\u27e9", []], "state_before": "case intro.intro\ncf : Code\nhf : Partrec (Code.eval cf)\ncg : Code\nhg : Partrec (Code.eval cg)\nthis : Partrec fun n => rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n\n\u22a2 \u2203 h,\n    Partrec h \u2227\n      \u2200 (a : \u2115),\n        (\u2200 (x : \u2115), x \u2208 h a \u2192 x \u2208 Code.eval cf a \u2228 x \u2208 Code.eval cg a) \u2227\n          ((h a).Dom \u2194 (Code.eval cf a).Dom \u2228 (Code.eval cg a).Dom)", "state_after": "case intro.intro\ncf : Code\nhf : Partrec (Code.eval cf)\ncg : Code\nhg : Partrec (Code.eval cg)\nthis : Partrec fun n => rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n\nn : \u2115\n\u22a2 (\u2200 (x : \u2115),\n      (x \u2208 rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n) \u2192\n        x \u2208 Code.eval cf n \u2228 x \u2208 Code.eval cg n) \u2227\n    ((rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n).Dom \u2194\n      (Code.eval cf n).Dom \u2228 (Code.eval cg n).Dom)"}, {"tactic": "suffices", "annotated_tactic": ["suffices", []], "state_before": "case intro.intro\ncf : Code\nhf : Partrec (Code.eval cf)\ncg : Code\nhg : Partrec (Code.eval cg)\nthis : Partrec fun n => rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n\nn : \u2115\n\u22a2 (\u2200 (x : \u2115),\n      (x \u2208 rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n) \u2192\n        x \u2208 Code.eval cf n \u2228 x \u2208 Code.eval cg n) \u2227\n    ((rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n).Dom \u2194\n      (Code.eval cf n).Dom \u2228 (Code.eval cg n).Dom)", "state_after": "case intro.intro\ncf : Code\nhf : Partrec (Code.eval cf)\ncg : Code\nhg : Partrec (Code.eval cg)\nthis\u271d : Partrec fun n => rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n\nn : \u2115\nthis : ?m.104784\n\u22a2 (\u2200 (x : \u2115),\n      (x \u2208 rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n) \u2192\n        x \u2208 Code.eval cf n \u2228 x \u2208 Code.eval cg n) \u2227\n    ((rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n).Dom \u2194\n      (Code.eval cf n).Dom \u2228 (Code.eval cg n).Dom)\n\ncase this\ncf : Code\nhf : Partrec (Code.eval cf)\ncg : Code\nhg : Partrec (Code.eval cg)\nthis : Partrec fun n => rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n\nn : \u2115\n\u22a2 ?m.104784"}, {"tactic": "refine' \u27e8this, \u27e8fun h => (this _ \u27e8h, rfl\u27e9).imp Exists.fst Exists.fst, _\u27e9\u27e9", "annotated_tactic": ["refine' \u27e8this, \u27e8fun h => (this _ \u27e8h, <a>rfl</a>\u27e9).<a>imp</a> <a>Exists.fst</a> <a>Exists.fst</a>, _\u27e9\u27e9", [{"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}, {"full_name": "Or.imp", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [249, 9], "def_end_pos": [249, 15]}, {"full_name": "Exists.fst", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [864, 9], "def_end_pos": [864, 19]}, {"full_name": "Exists.fst", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [864, 9], "def_end_pos": [864, 19]}]], "state_before": "case intro.intro\ncf : Code\nhf : Partrec (Code.eval cf)\ncg : Code\nhg : Partrec (Code.eval cg)\nthis\u271d : Partrec fun n => rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n\nn : \u2115\nthis : ?m.104784\n\u22a2 (\u2200 (x : \u2115),\n      (x \u2208 rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n) \u2192\n        x \u2208 Code.eval cf n \u2228 x \u2208 Code.eval cg n) \u2227\n    ((rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n).Dom \u2194\n      (Code.eval cf n).Dom \u2228 (Code.eval cg n).Dom)\n\ncase this\ncf : Code\nhf : Partrec (Code.eval cf)\ncg : Code\nhg : Partrec (Code.eval cg)\nthis : Partrec fun n => rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n\nn : \u2115\n\u22a2 ?m.104784", "state_after": "case intro.intro\ncf : Code\nhf : Partrec (Code.eval cf)\ncg : Code\nhg : Partrec (Code.eval cg)\nthis\u271d : Partrec fun n => rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n\nn : \u2115\nthis :\n  \u2200 (x : \u2115),\n    (x \u2208 rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n) \u2192\n      x \u2208 Code.eval cf n \u2228 x \u2208 Code.eval cg n\n\u22a2 (Code.eval cf n).Dom \u2228 (Code.eval cg n).Dom \u2192\n    (rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n).Dom\n\ncase this\ncf : Code\nhf : Partrec (Code.eval cf)\ncg : Code\nhg : Partrec (Code.eval cg)\nthis : Partrec fun n => rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n\nn : \u2115\n\u22a2 \u2200 (x : \u2115),\n    (x \u2208 rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n) \u2192\n      x \u2208 Code.eval cf n \u2228 x \u2208 Code.eval cg n"}, {"tactic": "intro x h", "annotated_tactic": ["intro x h", []], "state_before": "case this\ncf : Code\nhf : Partrec (Code.eval cf)\ncg : Code\nhg : Partrec (Code.eval cg)\nthis : Partrec fun n => rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n\nn : \u2115\n\u22a2 \u2200 (x : \u2115),\n    (x \u2208 rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n) \u2192\n      x \u2208 Code.eval cf n \u2228 x \u2208 Code.eval cg n", "state_after": "case this\ncf : Code\nhf : Partrec (Code.eval cf)\ncg : Code\nhg : Partrec (Code.eval cg)\nthis : Partrec fun n => rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n\nn x : \u2115\nh : x \u2208 rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n\n\u22a2 x \u2208 Code.eval cf n \u2228 x \u2208 Code.eval cg n"}, {"tactic": "obtain \u27e8k, e\u27e9 := Nat.rfindOpt_spec h", "annotated_tactic": ["obtain \u27e8k, e\u27e9 := <a>Nat.rfindOpt_spec</a> h", [{"full_name": "Nat.rfindOpt_spec", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [128, 9], "def_end_pos": [128, 22]}]], "state_before": "case this\ncf : Code\nhf : Partrec (Code.eval cf)\ncg : Code\nhg : Partrec (Code.eval cg)\nthis : Partrec fun n => rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n\nn x : \u2115\nh : x \u2208 rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n\n\u22a2 x \u2208 Code.eval cf n \u2228 x \u2208 Code.eval cg n", "state_after": "case this.intro\ncf : Code\nhf : Partrec (Code.eval cf)\ncg : Code\nhg : Partrec (Code.eval cg)\nthis : Partrec fun n => rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n\nn x : \u2115\nh : x \u2208 rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n\nk : \u2115\ne : x \u2208 HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n\n\u22a2 x \u2208 Code.eval cf n \u2228 x \u2208 Code.eval cg n"}, {"tactic": "revert e", "annotated_tactic": ["revert e", []], "state_before": "case this.intro\ncf : Code\nhf : Partrec (Code.eval cf)\ncg : Code\nhg : Partrec (Code.eval cg)\nthis : Partrec fun n => rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n\nn x : \u2115\nh : x \u2208 rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n\nk : \u2115\ne : x \u2208 HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n\n\u22a2 x \u2208 Code.eval cf n \u2228 x \u2208 Code.eval cg n", "state_after": "case this.intro\ncf : Code\nhf : Partrec (Code.eval cf)\ncg : Code\nhg : Partrec (Code.eval cg)\nthis : Partrec fun n => rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n\nn x : \u2115\nh : x \u2208 rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n\nk : \u2115\n\u22a2 (x \u2208 HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n) \u2192 x \u2208 Code.eval cf n \u2228 x \u2208 Code.eval cg n"}, {"tactic": "simp only [Option.mem_def]", "annotated_tactic": ["simp only [<a>Option.mem_def</a>]", [{"full_name": "Option.mem_def", "def_path": "lake-packages/std/Std/Data/Option/Basic.lean", "def_pos": [19, 17], "def_end_pos": [19, 24]}]], "state_before": "case this.intro\ncf : Code\nhf : Partrec (Code.eval cf)\ncg : Code\nhg : Partrec (Code.eval cg)\nthis : Partrec fun n => rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n\nn x : \u2115\nh : x \u2208 rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n\nk : \u2115\n\u22a2 (x \u2208 HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n) \u2192 x \u2208 Code.eval cf n \u2228 x \u2208 Code.eval cg n", "state_after": "case this.intro\ncf : Code\nhf : Partrec (Code.eval cf)\ncg : Code\nhg : Partrec (Code.eval cg)\nthis : Partrec fun n => rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n\nn x : \u2115\nh : x \u2208 rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n\nk : \u2115\n\u22a2 (HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n) = Option.some x \u2192\n    x \u2208 Code.eval cf n \u2228 x \u2208 Code.eval cg n"}, {"tactic": "cases' e' : cf.evaln k n with y <;> simp <;> intro e", "annotated_tactic": ["cases' e' : cf.evaln k n with y <;> simp <;> intro e", []], "state_before": "case this.intro\ncf : Code\nhf : Partrec (Code.eval cf)\ncg : Code\nhg : Partrec (Code.eval cg)\nthis : Partrec fun n => rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n\nn x : \u2115\nh : x \u2208 rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n\nk : \u2115\n\u22a2 (HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n) = Option.some x \u2192\n    x \u2208 Code.eval cf n \u2228 x \u2208 Code.eval cg n", "state_after": "case this.intro.none\ncf : Code\nhf : Partrec (Code.eval cf)\ncg : Code\nhg : Partrec (Code.eval cg)\nthis : Partrec fun n => rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n\nn x : \u2115\nh : x \u2208 rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n\nk : \u2115\ne' : Code.evaln k cf n = Option.none\ne : Code.evaln k cg n = Option.some x\n\u22a2 x \u2208 Code.eval cf n \u2228 x \u2208 Code.eval cg n\n\ncase this.intro.some\ncf : Code\nhf : Partrec (Code.eval cf)\ncg : Code\nhg : Partrec (Code.eval cg)\nthis : Partrec fun n => rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n\nn x : \u2115\nh : x \u2208 rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n\nk y : \u2115\ne' : Code.evaln k cf n = Option.some y\ne : y = x\n\u22a2 x \u2208 Code.eval cf n \u2228 x \u2208 Code.eval cg n"}, {"tactic": "intro h", "annotated_tactic": ["intro h", []], "state_before": "case intro.intro\ncf : Code\nhf : Partrec (Code.eval cf)\ncg : Code\nhg : Partrec (Code.eval cg)\nthis\u271d : Partrec fun n => rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n\nn : \u2115\nthis :\n  \u2200 (x : \u2115),\n    (x \u2208 rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n) \u2192\n      x \u2208 Code.eval cf n \u2228 x \u2208 Code.eval cg n\n\u22a2 (Code.eval cf n).Dom \u2228 (Code.eval cg n).Dom \u2192\n    (rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n).Dom", "state_after": "case intro.intro\ncf : Code\nhf : Partrec (Code.eval cf)\ncg : Code\nhg : Partrec (Code.eval cg)\nthis\u271d : Partrec fun n => rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n\nn : \u2115\nthis :\n  \u2200 (x : \u2115),\n    (x \u2208 rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n) \u2192\n      x \u2208 Code.eval cf n \u2228 x \u2208 Code.eval cg n\nh : (Code.eval cf n).Dom \u2228 (Code.eval cg n).Dom\n\u22a2 (rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n).Dom"}, {"tactic": "rw [Nat.rfindOpt_dom]", "annotated_tactic": ["rw [<a>Nat.rfindOpt_dom</a>]", [{"full_name": "Nat.rfindOpt_dom", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [133, 9], "def_end_pos": [133, 21]}]], "state_before": "case intro.intro\ncf : Code\nhf : Partrec (Code.eval cf)\ncg : Code\nhg : Partrec (Code.eval cg)\nthis\u271d : Partrec fun n => rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n\nn : \u2115\nthis :\n  \u2200 (x : \u2115),\n    (x \u2208 rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n) \u2192\n      x \u2208 Code.eval cf n \u2228 x \u2208 Code.eval cg n\nh : (Code.eval cf n).Dom \u2228 (Code.eval cg n).Dom\n\u22a2 (rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n).Dom", "state_after": "case intro.intro\ncf : Code\nhf : Partrec (Code.eval cf)\ncg : Code\nhg : Partrec (Code.eval cg)\nthis\u271d : Partrec fun n => rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n\nn : \u2115\nthis :\n  \u2200 (x : \u2115),\n    (x \u2208 rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n) \u2192\n      x \u2208 Code.eval cf n \u2228 x \u2208 Code.eval cg n\nh : (Code.eval cf n).Dom \u2228 (Code.eval cg n).Dom\n\u22a2 \u2203 n_1 a, a \u2208 HOrElse.hOrElse (Code.evaln n_1 cf n) fun x => Code.evaln n_1 cg n"}, {"tactic": "simp only [dom_iff_mem, Code.evaln_complete, Option.mem_def] at h", "annotated_tactic": ["simp only [<a>dom_iff_mem</a>, <a>Code.evaln_complete</a>, <a>Option.mem_def</a>] at h", [{"full_name": "Part.dom_iff_mem", "def_path": "Mathlib/Data/Part.lean", "def_pos": [101, 9], "def_end_pos": [101, 20]}, {"full_name": "Nat.Partrec.Code.evaln_complete", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [858, 9], "def_end_pos": [858, 23]}, {"full_name": "Option.mem_def", "def_path": "lake-packages/std/Std/Data/Option/Basic.lean", "def_pos": [19, 17], "def_end_pos": [19, 24]}]], "state_before": "case intro.intro\ncf : Code\nhf : Partrec (Code.eval cf)\ncg : Code\nhg : Partrec (Code.eval cg)\nthis\u271d : Partrec fun n => rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n\nn : \u2115\nthis :\n  \u2200 (x : \u2115),\n    (x \u2208 rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n) \u2192\n      x \u2208 Code.eval cf n \u2228 x \u2208 Code.eval cg n\nh : (Code.eval cf n).Dom \u2228 (Code.eval cg n).Dom\n\u22a2 \u2203 n_1 a, a \u2208 HOrElse.hOrElse (Code.evaln n_1 cf n) fun x => Code.evaln n_1 cg n", "state_after": "case intro.intro\ncf : Code\nhf : Partrec (Code.eval cf)\ncg : Code\nhg : Partrec (Code.eval cg)\nthis\u271d : Partrec fun n => rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n\nn : \u2115\nthis :\n  \u2200 (x : \u2115),\n    (x \u2208 rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n) \u2192\n      x \u2208 Code.eval cf n \u2228 x \u2208 Code.eval cg n\nh : (\u2203 y k, Code.evaln k cf n = Option.some y) \u2228 \u2203 y k, Code.evaln k cg n = Option.some y\n\u22a2 \u2203 n_1 a, a \u2208 HOrElse.hOrElse (Code.evaln n_1 cf n) fun x => Code.evaln n_1 cg n"}, {"tactic": "obtain \u27e8x, k, e\u27e9 | \u27e8x, k, e\u27e9 := h", "annotated_tactic": ["obtain \u27e8x, k, e\u27e9 | \u27e8x, k, e\u27e9 := h", []], "state_before": "case intro.intro\ncf : Code\nhf : Partrec (Code.eval cf)\ncg : Code\nhg : Partrec (Code.eval cg)\nthis\u271d : Partrec fun n => rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n\nn : \u2115\nthis :\n  \u2200 (x : \u2115),\n    (x \u2208 rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n) \u2192\n      x \u2208 Code.eval cf n \u2228 x \u2208 Code.eval cg n\nh : (\u2203 y k, Code.evaln k cf n = Option.some y) \u2228 \u2203 y k, Code.evaln k cg n = Option.some y\n\u22a2 \u2203 n_1 a, a \u2208 HOrElse.hOrElse (Code.evaln n_1 cf n) fun x => Code.evaln n_1 cg n", "state_after": "case intro.intro.inl.intro.intro\ncf : Code\nhf : Partrec (Code.eval cf)\ncg : Code\nhg : Partrec (Code.eval cg)\nthis\u271d : Partrec fun n => rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n\nn : \u2115\nthis :\n  \u2200 (x : \u2115),\n    (x \u2208 rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n) \u2192\n      x \u2208 Code.eval cf n \u2228 x \u2208 Code.eval cg n\nx k : \u2115\ne : Code.evaln k cf n = Option.some x\n\u22a2 \u2203 n_1 a, a \u2208 HOrElse.hOrElse (Code.evaln n_1 cf n) fun x => Code.evaln n_1 cg n\n\ncase intro.intro.inr.intro.intro\ncf : Code\nhf : Partrec (Code.eval cf)\ncg : Code\nhg : Partrec (Code.eval cg)\nthis\u271d : Partrec fun n => rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n\nn : \u2115\nthis :\n  \u2200 (x : \u2115),\n    (x \u2208 rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n) \u2192\n      x \u2208 Code.eval cf n \u2228 x \u2208 Code.eval cg n\nx k : \u2115\ne : Code.evaln k cg n = Option.some x\n\u22a2 \u2203 n_1 a, a \u2208 HOrElse.hOrElse (Code.evaln n_1 cf n) fun x => Code.evaln n_1 cg n"}, {"tactic": "refine' \u27e8k, x, _\u27e9", "annotated_tactic": ["refine' \u27e8k, x, _\u27e9", []], "state_before": "case intro.intro.inl.intro.intro\ncf : Code\nhf : Partrec (Code.eval cf)\ncg : Code\nhg : Partrec (Code.eval cg)\nthis\u271d : Partrec fun n => rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n\nn : \u2115\nthis :\n  \u2200 (x : \u2115),\n    (x \u2208 rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n) \u2192\n      x \u2208 Code.eval cf n \u2228 x \u2208 Code.eval cg n\nx k : \u2115\ne : Code.evaln k cf n = Option.some x\n\u22a2 \u2203 n_1 a, a \u2208 HOrElse.hOrElse (Code.evaln n_1 cf n) fun x => Code.evaln n_1 cg n", "state_after": "case intro.intro.inl.intro.intro\ncf : Code\nhf : Partrec (Code.eval cf)\ncg : Code\nhg : Partrec (Code.eval cg)\nthis\u271d : Partrec fun n => rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n\nn : \u2115\nthis :\n  \u2200 (x : \u2115),\n    (x \u2208 rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n) \u2192\n      x \u2208 Code.eval cf n \u2228 x \u2208 Code.eval cg n\nx k : \u2115\ne : Code.evaln k cf n = Option.some x\n\u22a2 x \u2208 HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n"}, {"tactic": "simp only [e, Option.some_orElse, Option.mem_def]", "annotated_tactic": ["simp only [e, <a>Option.some_orElse</a>, <a>Option.mem_def</a>]", [{"full_name": "Option.some_orElse", "def_path": "lake-packages/std/Std/Data/Option/Lemmas.lean", "def_pos": [169, 17], "def_end_pos": [169, 28]}, {"full_name": "Option.mem_def", "def_path": "lake-packages/std/Std/Data/Option/Basic.lean", "def_pos": [19, 17], "def_end_pos": [19, 24]}]], "state_before": "case intro.intro.inl.intro.intro\ncf : Code\nhf : Partrec (Code.eval cf)\ncg : Code\nhg : Partrec (Code.eval cg)\nthis\u271d : Partrec fun n => rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n\nn : \u2115\nthis :\n  \u2200 (x : \u2115),\n    (x \u2208 rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n) \u2192\n      x \u2208 Code.eval cf n \u2228 x \u2208 Code.eval cg n\nx k : \u2115\ne : Code.evaln k cf n = Option.some x\n\u22a2 x \u2208 HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n", "state_after": "no goals"}, {"tactic": "refine' \u27e8k, _\u27e9", "annotated_tactic": ["refine' \u27e8k, _\u27e9", []], "state_before": "case intro.intro.inr.intro.intro\ncf : Code\nhf : Partrec (Code.eval cf)\ncg : Code\nhg : Partrec (Code.eval cg)\nthis\u271d : Partrec fun n => rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n\nn : \u2115\nthis :\n  \u2200 (x : \u2115),\n    (x \u2208 rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n) \u2192\n      x \u2208 Code.eval cf n \u2228 x \u2208 Code.eval cg n\nx k : \u2115\ne : Code.evaln k cg n = Option.some x\n\u22a2 \u2203 n_1 a, a \u2208 HOrElse.hOrElse (Code.evaln n_1 cf n) fun x => Code.evaln n_1 cg n", "state_after": "case intro.intro.inr.intro.intro\ncf : Code\nhf : Partrec (Code.eval cf)\ncg : Code\nhg : Partrec (Code.eval cg)\nthis\u271d : Partrec fun n => rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n\nn : \u2115\nthis :\n  \u2200 (x : \u2115),\n    (x \u2208 rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n) \u2192\n      x \u2208 Code.eval cf n \u2228 x \u2208 Code.eval cg n\nx k : \u2115\ne : Code.evaln k cg n = Option.some x\n\u22a2 \u2203 a, a \u2208 HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n"}, {"tactic": "cases' cf.evaln k n with y", "annotated_tactic": ["cases' cf.evaln k n with y", []], "state_before": "case intro.intro.inr.intro.intro\ncf : Code\nhf : Partrec (Code.eval cf)\ncg : Code\nhg : Partrec (Code.eval cg)\nthis\u271d : Partrec fun n => rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n\nn : \u2115\nthis :\n  \u2200 (x : \u2115),\n    (x \u2208 rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n) \u2192\n      x \u2208 Code.eval cf n \u2228 x \u2208 Code.eval cg n\nx k : \u2115\ne : Code.evaln k cg n = Option.some x\n\u22a2 \u2203 a, a \u2208 HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n", "state_after": "case intro.intro.inr.intro.intro.none\ncf : Code\nhf : Partrec (Code.eval cf)\ncg : Code\nhg : Partrec (Code.eval cg)\nthis\u271d : Partrec fun n => rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n\nn : \u2115\nthis :\n  \u2200 (x : \u2115),\n    (x \u2208 rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n) \u2192\n      x \u2208 Code.eval cf n \u2228 x \u2208 Code.eval cg n\nx k : \u2115\ne : Code.evaln k cg n = Option.some x\n\u22a2 \u2203 a, a \u2208 HOrElse.hOrElse Option.none fun x => Code.evaln k cg n\n\ncase intro.intro.inr.intro.intro.some\ncf : Code\nhf : Partrec (Code.eval cf)\ncg : Code\nhg : Partrec (Code.eval cg)\nthis\u271d : Partrec fun n => rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n\nn : \u2115\nthis :\n  \u2200 (x : \u2115),\n    (x \u2208 rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n) \u2192\n      x \u2208 Code.eval cf n \u2228 x \u2208 Code.eval cg n\nx k : \u2115\ne : Code.evaln k cg n = Option.some x\ny : \u2115\n\u22a2 \u2203 a, a \u2208 HOrElse.hOrElse (Option.some y) fun x => Code.evaln k cg n"}, {"tactic": "exact \u27e8x, by simp only [e, Option.mem_def, Option.none_orElse]\u27e9", "annotated_tactic": ["exact \u27e8x, by simp only [e, <a>Option.mem_def</a>, <a>Option.none_orElse</a>]\u27e9", [{"full_name": "Option.mem_def", "def_path": "lake-packages/std/Std/Data/Option/Basic.lean", "def_pos": [19, 17], "def_end_pos": [19, 24]}, {"full_name": "Option.none_orElse", "def_path": "lake-packages/std/Std/Data/Option/Lemmas.lean", "def_pos": [171, 17], "def_end_pos": [171, 28]}]], "state_before": "case intro.intro.inr.intro.intro.none\ncf : Code\nhf : Partrec (Code.eval cf)\ncg : Code\nhg : Partrec (Code.eval cg)\nthis\u271d : Partrec fun n => rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n\nn : \u2115\nthis :\n  \u2200 (x : \u2115),\n    (x \u2208 rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n) \u2192\n      x \u2208 Code.eval cf n \u2228 x \u2208 Code.eval cg n\nx k : \u2115\ne : Code.evaln k cg n = Option.some x\n\u22a2 \u2203 a, a \u2208 HOrElse.hOrElse Option.none fun x => Code.evaln k cg n", "state_after": "no goals"}, {"tactic": "simp only [e, Option.mem_def, Option.none_orElse]", "annotated_tactic": ["simp only [e, <a>Option.mem_def</a>, <a>Option.none_orElse</a>]", [{"full_name": "Option.mem_def", "def_path": "lake-packages/std/Std/Data/Option/Basic.lean", "def_pos": [19, 17], "def_end_pos": [19, 24]}, {"full_name": "Option.none_orElse", "def_path": "lake-packages/std/Std/Data/Option/Lemmas.lean", "def_pos": [171, 17], "def_end_pos": [171, 28]}]], "state_before": "cf : Code\nhf : Partrec (Code.eval cf)\ncg : Code\nhg : Partrec (Code.eval cg)\nthis\u271d : Partrec fun n => rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n\nn : \u2115\nthis :\n  \u2200 (x : \u2115),\n    (x \u2208 rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n) \u2192\n      x \u2208 Code.eval cf n \u2228 x \u2208 Code.eval cg n\nx k : \u2115\ne : Code.evaln k cg n = Option.some x\n\u22a2 x \u2208 HOrElse.hOrElse Option.none fun x => Code.evaln k cg n", "state_after": "no goals"}, {"tactic": "exact \u27e8y, by simp only [Option.some_orElse, Option.mem_def]\u27e9", "annotated_tactic": ["exact \u27e8y, by simp only [<a>Option.some_orElse</a>, <a>Option.mem_def</a>]\u27e9", [{"full_name": "Option.some_orElse", "def_path": "lake-packages/std/Std/Data/Option/Lemmas.lean", "def_pos": [169, 17], "def_end_pos": [169, 28]}, {"full_name": "Option.mem_def", "def_path": "lake-packages/std/Std/Data/Option/Basic.lean", "def_pos": [19, 17], "def_end_pos": [19, 24]}]], "state_before": "case intro.intro.inr.intro.intro.some\ncf : Code\nhf : Partrec (Code.eval cf)\ncg : Code\nhg : Partrec (Code.eval cg)\nthis\u271d : Partrec fun n => rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n\nn : \u2115\nthis :\n  \u2200 (x : \u2115),\n    (x \u2208 rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n) \u2192\n      x \u2208 Code.eval cf n \u2228 x \u2208 Code.eval cg n\nx k : \u2115\ne : Code.evaln k cg n = Option.some x\ny : \u2115\n\u22a2 \u2203 a, a \u2208 HOrElse.hOrElse (Option.some y) fun x => Code.evaln k cg n", "state_after": "no goals"}, {"tactic": "simp only [Option.some_orElse, Option.mem_def]", "annotated_tactic": ["simp only [<a>Option.some_orElse</a>, <a>Option.mem_def</a>]", [{"full_name": "Option.some_orElse", "def_path": "lake-packages/std/Std/Data/Option/Lemmas.lean", "def_pos": [169, 17], "def_end_pos": [169, 28]}, {"full_name": "Option.mem_def", "def_path": "lake-packages/std/Std/Data/Option/Basic.lean", "def_pos": [19, 17], "def_end_pos": [19, 24]}]], "state_before": "cf : Code\nhf : Partrec (Code.eval cf)\ncg : Code\nhg : Partrec (Code.eval cg)\nthis\u271d : Partrec fun n => rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n\nn : \u2115\nthis :\n  \u2200 (x : \u2115),\n    (x \u2208 rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n) \u2192\n      x \u2208 Code.eval cf n \u2228 x \u2208 Code.eval cg n\nx k : \u2115\ne : Code.evaln k cg n = Option.some x\ny : \u2115\n\u22a2 y \u2208 HOrElse.hOrElse (Option.some y) fun x => Code.evaln k cg n", "state_after": "no goals"}, {"tactic": "exact Or.inr (Code.evaln_sound e)", "annotated_tactic": ["exact <a>Or.inr</a> (<a>Code.evaln_sound</a> e)", [{"full_name": "Or.inr", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [519, 5], "def_end_pos": [519, 8]}, {"full_name": "Nat.Partrec.Code.evaln_sound", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [818, 9], "def_end_pos": [818, 20]}]], "state_before": "case this.intro.none\ncf : Code\nhf : Partrec (Code.eval cf)\ncg : Code\nhg : Partrec (Code.eval cg)\nthis : Partrec fun n => rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n\nn x : \u2115\nh : x \u2208 rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n\nk : \u2115\ne' : Code.evaln k cf n = Option.none\ne : Code.evaln k cg n = Option.some x\n\u22a2 x \u2208 Code.eval cf n \u2228 x \u2208 Code.eval cg n", "state_after": "no goals"}, {"tactic": "subst y", "annotated_tactic": ["subst y", []], "state_before": "case this.intro.some\ncf : Code\nhf : Partrec (Code.eval cf)\ncg : Code\nhg : Partrec (Code.eval cg)\nthis : Partrec fun n => rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n\nn x : \u2115\nh : x \u2208 rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n\nk y : \u2115\ne' : Code.evaln k cf n = Option.some y\ne : y = x\n\u22a2 x \u2208 Code.eval cf n \u2228 x \u2208 Code.eval cg n", "state_after": "case this.intro.some\ncf : Code\nhf : Partrec (Code.eval cf)\ncg : Code\nhg : Partrec (Code.eval cg)\nthis : Partrec fun n => rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n\nn x : \u2115\nh : x \u2208 rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n\nk : \u2115\ne' : Code.evaln k cf n = Option.some x\n\u22a2 x \u2208 Code.eval cf n \u2228 x \u2208 Code.eval cg n"}, {"tactic": "exact Or.inl (Code.evaln_sound e')", "annotated_tactic": ["exact <a>Or.inl</a> (<a>Code.evaln_sound</a> e')", [{"full_name": "Or.inl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [517, 5], "def_end_pos": [517, 8]}, {"full_name": "Nat.Partrec.Code.evaln_sound", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [818, 9], "def_end_pos": [818, 20]}]], "state_before": "case this.intro.some\ncf : Code\nhf : Partrec (Code.eval cf)\ncg : Code\nhg : Partrec (Code.eval cg)\nthis : Partrec fun n => rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n\nn x : \u2115\nh : x \u2208 rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n\nk : \u2115\ne' : Code.evaln k cf n = Option.some x\n\u22a2 x \u2208 Code.eval cf n \u2228 x \u2208 Code.eval cg n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "Finset.piecewise_singleton", "start": [2552, 1], "end": [2553, 61], "traced_tactics": [{"tactic": "rw [\u2190 insert_emptyc_eq, piecewise_insert, piecewise_empty]", "annotated_tactic": ["rw [\u2190 <a>insert_emptyc_eq</a>, <a>piecewise_insert</a>, <a>piecewise_empty</a>]", [{"full_name": "IsLawfulSingleton.insert_emptyc_eq", "def_path": "lake-packages/std/Std/Classes/SetNotation.lean", "def_pos": [116, 3], "def_end_pos": [116, 19]}, {"full_name": "Finset.piecewise_insert", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2532, 9], "def_end_pos": [2532, 25]}, {"full_name": "Finset.piecewise_empty", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2497, 9], "def_end_pos": [2497, 24]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : \u03b1 \u2192 Sort u_4\ns : Finset \u03b1\nf g : (i : \u03b1) \u2192 \u03b4 i\ninst\u271d\u00b9 : (j : \u03b1) \u2192 Decidable (j \u2208 s)\ninst\u271d : DecidableEq \u03b1\ni : \u03b1\n\u22a2 piecewise {i} f g = update g i (f i)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Hausdorff.lean", "full_name": "LipschitzWith.hausdorffMeasure_image_le", "start": [794, 1], "end": [796, 53], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Variance.lean", "full_name": "ProbabilityTheory.IndepFun.variance_sum", "start": [320, 1], "end": [379, 67], "traced_tactics": [{"tactic": "induction' s using Finset.induction_on with k s ks IH", "annotated_tactic": ["induction' s using <a>Finset.induction_on</a> with k s ks IH", [{"full_name": "Finset.induction_on", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1251, 19], "def_end_pos": [1251, 31]}]], "state_before": "\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\n\u03b9 : Type u_2\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\ns : Finset \u03b9\nhs : \u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2\nh : Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)\n\u22a2 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119", "state_after": "case empty\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\n\u03b9 : Type u_2\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\ns : Finset \u03b9\nhs\u271d : \u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2\nh\u271d : Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)\nhs : \u2200 (i : \u03b9), i \u2208 \u2205 \u2192 Mem\u2112p (X i) 2\nh : Set.Pairwise \u2191\u2205 fun i j => IndepFun (X i) (X j)\n\u22a2 variance (\u2211 i in \u2205, X i) \u2119 = \u2211 i in \u2205, variance (X i) \u2119\n\ncase insert\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\n\u03b9 : Type u_2\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\ns\u271d : Finset \u03b9\nhs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2\nh\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j)\nk : \u03b9\ns : Finset \u03b9\nks : \u00ack \u2208 s\nIH :\n  (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192\n    (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119\nhs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2\nh : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j)\n\u22a2 variance (\u2211 i in insert k s, X i) \u2119 = \u2211 i in insert k s, variance (X i) \u2119"}, {"tactic": "rw [variance_def' (mem\u2112p_finset_sum' _ hs), sum_insert ks, sum_insert ks]", "annotated_tactic": ["rw [<a>variance_def'</a> (<a>mem\u2112p_finset_sum'</a> _ hs), <a>sum_insert</a> ks, <a>sum_insert</a> ks]", [{"full_name": "ProbabilityTheory.variance_def'", "def_path": "Mathlib/Probability/Variance.lean", "def_pos": [210, 9], "def_end_pos": [210, 22]}, {"full_name": "MeasureTheory.mem\u2112p_finset_sum'", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [1254, 9], "def_end_pos": [1254, 26]}, {"full_name": "Finset.sum_insert", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [316, 3], "def_end_pos": [316, 14]}, {"full_name": "Finset.sum_insert", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [316, 3], "def_end_pos": [316, 14]}]], "state_before": "case insert\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\n\u03b9 : Type u_2\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\ns\u271d : Finset \u03b9\nhs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2\nh\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j)\nk : \u03b9\ns : Finset \u03b9\nks : \u00ack \u2208 s\nIH :\n  (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192\n    (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119\nhs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2\nh : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j)\n\u22a2 variance (\u2211 i in insert k s, X i) \u2119 = \u2211 i in insert k s, variance (X i) \u2119", "state_after": "case insert\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\n\u03b9 : Type u_2\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\ns\u271d : Finset \u03b9\nhs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2\nh\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j)\nk : \u03b9\ns : Finset \u03b9\nks : \u00ack \u2208 s\nIH :\n  (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192\n    (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119\nhs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2\nh : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j)\n\u22a2 (\u222b (a : \u03a9), ((X k + \u2211 x in s, X x) ^ 2) a) - (\u222b (a : \u03a9), (X k + \u2211 x in s, X x) a) ^ 2 =\n    variance (X k) \u2119 + \u2211 x in s, variance (X x) \u2119"}, {"tactic": "simp only [add_sq']", "annotated_tactic": ["simp only [<a>add_sq'</a>]", [{"full_name": "add_sq'", "def_path": "Mathlib/Algebra/GroupPower/Ring.lean", "def_pos": [174, 9], "def_end_pos": [174, 16]}]], "state_before": "case insert\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\n\u03b9 : Type u_2\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\ns\u271d : Finset \u03b9\nhs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2\nh\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j)\nk : \u03b9\ns : Finset \u03b9\nks : \u00ack \u2208 s\nIH :\n  (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192\n    (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119\nhs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2\nh : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j)\n\u22a2 (\u222b (a : \u03a9), ((X k + \u2211 x in s, X x) ^ 2) a) - (\u222b (a : \u03a9), (X k + \u2211 x in s, X x) a) ^ 2 =\n    variance (X k) \u2119 + \u2211 x in s, variance (X x) \u2119", "state_after": "case insert\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\n\u03b9 : Type u_2\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\ns\u271d : Finset \u03b9\nhs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2\nh\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j)\nk : \u03b9\ns : Finset \u03b9\nks : \u00ack \u2208 s\nIH :\n  (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192\n    (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119\nhs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2\nh : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j)\n\u22a2 (\u222b (a : \u03a9), (X k ^ 2 + (\u2211 x in s, X x) ^ 2 + 2 * X k * \u2211 x in s, X x) a) - (\u222b (a : \u03a9), (X k + \u2211 x in s, X x) a) ^ 2 =\n    variance (X k) \u2119 + \u2211 x in s, variance (X x) \u2119"}, {"tactic": "simp only [Finset.sum_empty, variance_zero]", "annotated_tactic": ["simp only [<a>Finset.sum_empty</a>, <a>variance_zero</a>]", [{"full_name": "Finset.sum_empty", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [298, 3], "def_end_pos": [298, 14]}, {"full_name": "ProbabilityTheory.variance_zero", "def_path": "Mathlib/Probability/Variance.lean", "def_pos": [180, 9], "def_end_pos": [180, 22]}]], "state_before": "case empty\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\n\u03b9 : Type u_2\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\ns : Finset \u03b9\nhs\u271d : \u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2\nh\u271d : Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)\nhs : \u2200 (i : \u03b9), i \u2208 \u2205 \u2192 Mem\u2112p (X i) 2\nh : Set.Pairwise \u2191\u2205 fun i j => IndepFun (X i) (X j)\n\u22a2 variance (\u2211 i in \u2205, X i) \u2119 = \u2211 i in \u2205, variance (X i) \u2119", "state_after": "no goals"}, {"tactic": "rw [integral_add', integral_add', integral_add']", "annotated_tactic": ["rw [<a>integral_add'</a>, <a>integral_add'</a>, <a>integral_add'</a>]", [{"full_name": "MeasureTheory.integral_add'", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [876, 9], "def_end_pos": [876, 22]}, {"full_name": "MeasureTheory.integral_add'", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [876, 9], "def_end_pos": [876, 22]}, {"full_name": "MeasureTheory.integral_add'", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [876, 9], "def_end_pos": [876, 22]}]], "state_before": "\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\n\u03b9 : Type u_2\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\ns\u271d : Finset \u03b9\nhs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2\nh\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j)\nk : \u03b9\ns : Finset \u03b9\nks : \u00ack \u2208 s\nIH :\n  (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192\n    (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119\nhs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2\nh : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j)\n\u22a2 (\u222b (a : \u03a9), (X k ^ 2 + (\u2211 i in s, X i) ^ 2 + 2 * X k * \u2211 i in s, X i) a) - (\u222b (a : \u03a9), (X k + \u2211 i in s, X i) a) ^ 2 =\n    (((\u222b (a : \u03a9), (X k ^ 2) a) + \u222b (a : \u03a9), ((\u2211 i in s, X i) ^ 2) a) + \u222b (a : \u03a9), (2 * X k * \u2211 i in s, X i) a) -\n      ((\u222b (a : \u03a9), X k a) + \u222b (a : \u03a9), Finset.sum s (fun i => X i) a) ^ 2", "state_after": "case hf\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\n\u03b9 : Type u_2\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\ns\u271d : Finset \u03b9\nhs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2\nh\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j)\nk : \u03b9\ns : Finset \u03b9\nks : \u00ack \u2208 s\nIH :\n  (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192\n    (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119\nhs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2\nh : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j)\n\u22a2 Integrable (X k)\n\ncase hg\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\n\u03b9 : Type u_2\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\ns\u271d : Finset \u03b9\nhs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2\nh\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j)\nk : \u03b9\ns : Finset \u03b9\nks : \u00ack \u2208 s\nIH :\n  (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192\n    (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119\nhs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2\nh : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j)\n\u22a2 Integrable (\u2211 i in s, X i)\n\ncase hf\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\n\u03b9 : Type u_2\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\ns\u271d : Finset \u03b9\nhs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2\nh\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j)\nk : \u03b9\ns : Finset \u03b9\nks : \u00ack \u2208 s\nIH :\n  (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192\n    (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119\nhs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2\nh : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j)\n\u22a2 Integrable (X k ^ 2)\n\ncase hg\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\n\u03b9 : Type u_2\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\ns\u271d : Finset \u03b9\nhs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2\nh\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j)\nk : \u03b9\ns : Finset \u03b9\nks : \u00ack \u2208 s\nIH :\n  (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192\n    (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119\nhs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2\nh : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j)\n\u22a2 Integrable ((\u2211 i in s, X i) ^ 2)\n\ncase hf\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\n\u03b9 : Type u_2\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\ns\u271d : Finset \u03b9\nhs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2\nh\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j)\nk : \u03b9\ns : Finset \u03b9\nks : \u00ack \u2208 s\nIH :\n  (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192\n    (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119\nhs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2\nh : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j)\n\u22a2 Integrable (X k ^ 2 + (\u2211 i in s, X i) ^ 2)\n\ncase hg\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\n\u03b9 : Type u_2\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\ns\u271d : Finset \u03b9\nhs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2\nh\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j)\nk : \u03b9\ns : Finset \u03b9\nks : \u00ack \u2208 s\nIH :\n  (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192\n    (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119\nhs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2\nh : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j)\n\u22a2 Integrable (2 * X k * \u2211 i in s, X i)"}, {"tactic": "exact Mem\u2112p.integrable one_le_two (hs _ (mem_insert_self _ _))", "annotated_tactic": ["exact <a>Mem\u2112p.integrable</a> <a>one_le_two</a> (hs _ (<a>mem_insert_self</a> _ _))", [{"full_name": "MeasureTheory.Mem\u2112p.integrable", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [807, 9], "def_end_pos": [807, 25]}, {"full_name": "one_le_two", "def_path": "Mathlib/Algebra/Order/Monoid/NatCast.lean", "def_pos": [50, 7], "def_end_pos": [50, 17]}, {"full_name": "Finset.mem_insert_self", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1091, 9], "def_end_pos": [1091, 24]}]], "state_before": "case hf\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\n\u03b9 : Type u_2\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\ns\u271d : Finset \u03b9\nhs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2\nh\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j)\nk : \u03b9\ns : Finset \u03b9\nks : \u00ack \u2208 s\nIH :\n  (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192\n    (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119\nhs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2\nh : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j)\n\u22a2 Integrable (X k)", "state_after": "no goals"}, {"tactic": "apply integrable_finset_sum' _ fun i hi => ?_", "annotated_tactic": ["apply <a>integrable_finset_sum'</a> _ fun i hi => ?_", [{"full_name": "MeasureTheory.integrable_finset_sum'", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [682, 9], "def_end_pos": [682, 31]}]], "state_before": "case hg\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\n\u03b9 : Type u_2\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\ns\u271d : Finset \u03b9\nhs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2\nh\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j)\nk : \u03b9\ns : Finset \u03b9\nks : \u00ack \u2208 s\nIH :\n  (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192\n    (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119\nhs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2\nh : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j)\n\u22a2 Integrable (\u2211 i in s, X i)", "state_after": "\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\n\u03b9 : Type u_2\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\ns\u271d : Finset \u03b9\nhs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2\nh\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j)\nk : \u03b9\ns : Finset \u03b9\nks : \u00ack \u2208 s\nIH :\n  (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192\n    (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119\nhs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2\nh : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j)\ni : \u03b9\nhi : i \u2208 s\n\u22a2 Integrable (X i)"}, {"tactic": "exact Mem\u2112p.integrable one_le_two (hs _ (mem_insert_of_mem hi))", "annotated_tactic": ["exact <a>Mem\u2112p.integrable</a> <a>one_le_two</a> (hs _ (<a>mem_insert_of_mem</a> hi))", [{"full_name": "MeasureTheory.Mem\u2112p.integrable", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [807, 9], "def_end_pos": [807, 25]}, {"full_name": "one_le_two", "def_path": "Mathlib/Algebra/Order/Monoid/NatCast.lean", "def_pos": [50, 7], "def_end_pos": [50, 17]}, {"full_name": "Finset.mem_insert_of_mem", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1095, 9], "def_end_pos": [1095, 26]}]], "state_before": "\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\n\u03b9 : Type u_2\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\ns\u271d : Finset \u03b9\nhs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2\nh\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j)\nk : \u03b9\ns : Finset \u03b9\nks : \u00ack \u2208 s\nIH :\n  (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192\n    (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119\nhs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2\nh : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j)\ni : \u03b9\nhi : i \u2208 s\n\u22a2 Integrable (X i)", "state_after": "no goals"}, {"tactic": "exact Mem\u2112p.integrable_sq (hs _ (mem_insert_self _ _))", "annotated_tactic": ["exact <a>Mem\u2112p.integrable_sq</a> (hs _ (<a>mem_insert_self</a> _ _))", [{"full_name": "MeasureTheory.Mem\u2112p.integrable_sq", "def_path": "Mathlib/MeasureTheory/Function/L2Space.lean", "def_pos": [44, 9], "def_end_pos": [44, 28]}, {"full_name": "Finset.mem_insert_self", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1091, 9], "def_end_pos": [1091, 24]}]], "state_before": "case hf\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\n\u03b9 : Type u_2\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\ns\u271d : Finset \u03b9\nhs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2\nh\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j)\nk : \u03b9\ns : Finset \u03b9\nks : \u00ack \u2208 s\nIH :\n  (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192\n    (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119\nhs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2\nh : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j)\n\u22a2 Integrable (X k ^ 2)", "state_after": "no goals"}, {"tactic": "apply Mem\u2112p.integrable_sq", "annotated_tactic": ["apply <a>Mem\u2112p.integrable_sq</a>", [{"full_name": "MeasureTheory.Mem\u2112p.integrable_sq", "def_path": "Mathlib/MeasureTheory/Function/L2Space.lean", "def_pos": [44, 9], "def_end_pos": [44, 28]}]], "state_before": "case hg\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\n\u03b9 : Type u_2\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\ns\u271d : Finset \u03b9\nhs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2\nh\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j)\nk : \u03b9\ns : Finset \u03b9\nks : \u00ack \u2208 s\nIH :\n  (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192\n    (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119\nhs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2\nh : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j)\n\u22a2 Integrable ((\u2211 i in s, X i) ^ 2)", "state_after": "case hg.h\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\n\u03b9 : Type u_2\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\ns\u271d : Finset \u03b9\nhs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2\nh\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j)\nk : \u03b9\ns : Finset \u03b9\nks : \u00ack \u2208 s\nIH :\n  (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192\n    (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119\nhs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2\nh : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j)\n\u22a2 Mem\u2112p (fun x => Finset.sum s (fun i => X i) x) 2"}, {"tactic": "exact mem\u2112p_finset_sum' _ fun i hi => hs _ (mem_insert_of_mem hi)", "annotated_tactic": ["exact <a>mem\u2112p_finset_sum'</a> _ fun i hi => hs _ (<a>mem_insert_of_mem</a> hi)", [{"full_name": "MeasureTheory.mem\u2112p_finset_sum'", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [1254, 9], "def_end_pos": [1254, 26]}, {"full_name": "Finset.mem_insert_of_mem", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1095, 9], "def_end_pos": [1095, 26]}]], "state_before": "case hg.h\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\n\u03b9 : Type u_2\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\ns\u271d : Finset \u03b9\nhs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2\nh\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j)\nk : \u03b9\ns : Finset \u03b9\nks : \u00ack \u2208 s\nIH :\n  (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192\n    (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119\nhs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2\nh : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j)\n\u22a2 Mem\u2112p (fun x => Finset.sum s (fun i => X i) x) 2", "state_after": "no goals"}, {"tactic": "apply Integrable.add", "annotated_tactic": ["apply <a>Integrable.add</a>", [{"full_name": "MeasureTheory.Integrable.add", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [677, 9], "def_end_pos": [677, 23]}]], "state_before": "case hf\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\n\u03b9 : Type u_2\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\ns\u271d : Finset \u03b9\nhs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2\nh\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j)\nk : \u03b9\ns : Finset \u03b9\nks : \u00ack \u2208 s\nIH :\n  (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192\n    (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119\nhs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2\nh : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j)\n\u22a2 Integrable (X k ^ 2 + (\u2211 i in s, X i) ^ 2)", "state_after": "case hf.hf\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\n\u03b9 : Type u_2\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\ns\u271d : Finset \u03b9\nhs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2\nh\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j)\nk : \u03b9\ns : Finset \u03b9\nks : \u00ack \u2208 s\nIH :\n  (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192\n    (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119\nhs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2\nh : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j)\n\u22a2 Integrable (X k ^ 2)\n\ncase hf.hg\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\n\u03b9 : Type u_2\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\ns\u271d : Finset \u03b9\nhs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2\nh\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j)\nk : \u03b9\ns : Finset \u03b9\nks : \u00ack \u2208 s\nIH :\n  (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192\n    (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119\nhs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2\nh : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j)\n\u22a2 Integrable ((\u2211 i in s, X i) ^ 2)"}, {"tactic": "exact Mem\u2112p.integrable_sq (hs _ (mem_insert_self _ _))", "annotated_tactic": ["exact <a>Mem\u2112p.integrable_sq</a> (hs _ (<a>mem_insert_self</a> _ _))", [{"full_name": "MeasureTheory.Mem\u2112p.integrable_sq", "def_path": "Mathlib/MeasureTheory/Function/L2Space.lean", "def_pos": [44, 9], "def_end_pos": [44, 28]}, {"full_name": "Finset.mem_insert_self", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1091, 9], "def_end_pos": [1091, 24]}]], "state_before": "case hf.hf\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\n\u03b9 : Type u_2\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\ns\u271d : Finset \u03b9\nhs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2\nh\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j)\nk : \u03b9\ns : Finset \u03b9\nks : \u00ack \u2208 s\nIH :\n  (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192\n    (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119\nhs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2\nh : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j)\n\u22a2 Integrable (X k ^ 2)", "state_after": "no goals"}, {"tactic": "apply Mem\u2112p.integrable_sq", "annotated_tactic": ["apply <a>Mem\u2112p.integrable_sq</a>", [{"full_name": "MeasureTheory.Mem\u2112p.integrable_sq", "def_path": "Mathlib/MeasureTheory/Function/L2Space.lean", "def_pos": [44, 9], "def_end_pos": [44, 28]}]], "state_before": "case hf.hg\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\n\u03b9 : Type u_2\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\ns\u271d : Finset \u03b9\nhs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2\nh\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j)\nk : \u03b9\ns : Finset \u03b9\nks : \u00ack \u2208 s\nIH :\n  (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192\n    (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119\nhs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2\nh : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j)\n\u22a2 Integrable ((\u2211 i in s, X i) ^ 2)", "state_after": "case hf.hg.h\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\n\u03b9 : Type u_2\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\ns\u271d : Finset \u03b9\nhs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2\nh\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j)\nk : \u03b9\ns : Finset \u03b9\nks : \u00ack \u2208 s\nIH :\n  (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192\n    (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119\nhs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2\nh : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j)\n\u22a2 Mem\u2112p (fun x => Finset.sum s (fun i => X i) x) 2"}, {"tactic": "exact mem\u2112p_finset_sum' _ fun i hi => hs _ (mem_insert_of_mem hi)", "annotated_tactic": ["exact <a>mem\u2112p_finset_sum'</a> _ fun i hi => hs _ (<a>mem_insert_of_mem</a> hi)", [{"full_name": "MeasureTheory.mem\u2112p_finset_sum'", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [1254, 9], "def_end_pos": [1254, 26]}, {"full_name": "Finset.mem_insert_of_mem", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1095, 9], "def_end_pos": [1095, 26]}]], "state_before": "case hf.hg.h\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\n\u03b9 : Type u_2\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\ns\u271d : Finset \u03b9\nhs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2\nh\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j)\nk : \u03b9\ns : Finset \u03b9\nks : \u00ack \u2208 s\nIH :\n  (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192\n    (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119\nhs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2\nh : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j)\n\u22a2 Mem\u2112p (fun x => Finset.sum s (fun i => X i) x) 2", "state_after": "no goals"}, {"tactic": "rw [mul_assoc]", "annotated_tactic": ["rw [<a>mul_assoc</a>]", [{"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [264, 9], "def_end_pos": [264, 18]}]], "state_before": "case hg\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\n\u03b9 : Type u_2\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\ns\u271d : Finset \u03b9\nhs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2\nh\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j)\nk : \u03b9\ns : Finset \u03b9\nks : \u00ack \u2208 s\nIH :\n  (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192\n    (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119\nhs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2\nh : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j)\n\u22a2 Integrable (2 * X k * \u2211 i in s, X i)", "state_after": "case hg\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\n\u03b9 : Type u_2\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\ns\u271d : Finset \u03b9\nhs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2\nh\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j)\nk : \u03b9\ns : Finset \u03b9\nks : \u00ack \u2208 s\nIH :\n  (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192\n    (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119\nhs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2\nh : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j)\n\u22a2 Integrable (2 * (X k * \u2211 i in s, X i))"}, {"tactic": "apply Integrable.const_mul _ (2 : \u211d)", "annotated_tactic": ["apply <a>Integrable.const_mul</a> _ (2 : \u211d)", [{"full_name": "MeasureTheory.Integrable.const_mul", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [1128, 9], "def_end_pos": [1128, 29]}]], "state_before": "case hg\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\n\u03b9 : Type u_2\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\ns\u271d : Finset \u03b9\nhs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2\nh\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j)\nk : \u03b9\ns : Finset \u03b9\nks : \u00ack \u2208 s\nIH :\n  (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192\n    (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119\nhs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2\nh : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j)\n\u22a2 Integrable (2 * (X k * \u2211 i in s, X i))", "state_after": "\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\n\u03b9 : Type u_2\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\ns\u271d : Finset \u03b9\nhs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2\nh\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j)\nk : \u03b9\ns : Finset \u03b9\nks : \u00ack \u2208 s\nIH :\n  (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192\n    (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119\nhs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2\nh : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j)\n\u22a2 Integrable fun x => (X k * \u2211 i in s, X i) x"}, {"tactic": "simp only [mul_sum, sum_apply, Pi.mul_apply]", "annotated_tactic": ["simp only [<a>mul_sum</a>, <a>sum_apply</a>, <a>Pi.mul_apply</a>]", [{"full_name": "Finset.mul_sum", "def_path": "Mathlib/Algebra/BigOperators/Ring.lean", "def_pos": [55, 9], "def_end_pos": [55, 16]}, {"full_name": "Finset.sum_apply", "def_path": "Mathlib/Algebra/BigOperators/Pi.lean", "def_pos": [41, 3], "def_end_pos": [41, 14]}, {"full_name": "Pi.mul_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [83, 9], "def_end_pos": [83, 18]}]], "state_before": "\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\n\u03b9 : Type u_2\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\ns\u271d : Finset \u03b9\nhs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2\nh\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j)\nk : \u03b9\ns : Finset \u03b9\nks : \u00ack \u2208 s\nIH :\n  (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192\n    (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119\nhs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2\nh : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j)\n\u22a2 Integrable fun x => (X k * \u2211 i in s, X i) x", "state_after": "\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\n\u03b9 : Type u_2\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\ns\u271d : Finset \u03b9\nhs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2\nh\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j)\nk : \u03b9\ns : Finset \u03b9\nks : \u00ack \u2208 s\nIH :\n  (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192\n    (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119\nhs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2\nh : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j)\n\u22a2 Integrable fun x => \u2211 x_1 in s, X k x * X x_1 x"}, {"tactic": "apply integrable_finset_sum _ fun i hi => ?_", "annotated_tactic": ["apply <a>integrable_finset_sum</a> _ fun i hi => ?_", [{"full_name": "MeasureTheory.integrable_finset_sum", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [688, 9], "def_end_pos": [688, 30]}]], "state_before": "\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\n\u03b9 : Type u_2\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\ns\u271d : Finset \u03b9\nhs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2\nh\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j)\nk : \u03b9\ns : Finset \u03b9\nks : \u00ack \u2208 s\nIH :\n  (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192\n    (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119\nhs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2\nh : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j)\n\u22a2 Integrable fun x => \u2211 x_1 in s, X k x * X x_1 x", "state_after": "\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\n\u03b9 : Type u_2\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\ns\u271d : Finset \u03b9\nhs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2\nh\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j)\nk : \u03b9\ns : Finset \u03b9\nks : \u00ack \u2208 s\nIH :\n  (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192\n    (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119\nhs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2\nh : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j)\ni : \u03b9\nhi : i \u2208 s\n\u22a2 Integrable fun a => X k a * X i a"}, {"tactic": "apply IndepFun.integrable_mul _ (Mem\u2112p.integrable one_le_two (hs _ (mem_insert_self _ _)))\n  (Mem\u2112p.integrable one_le_two (hs _ (mem_insert_of_mem hi)))", "annotated_tactic": ["apply <a>IndepFun.integrable_mul</a> _ (<a>Mem\u2112p.integrable</a> <a>one_le_two</a> (hs _ (<a>mem_insert_self</a> _ _)))\n          (<a>Mem\u2112p.integrable</a> <a>one_le_two</a> (hs _ (<a>mem_insert_of_mem</a> hi)))", [{"full_name": "ProbabilityTheory.IndepFun.integrable_mul", "def_path": "Mathlib/Probability/Integration.lean", "def_pos": [138, 9], "def_end_pos": [138, 32]}, {"full_name": "MeasureTheory.Mem\u2112p.integrable", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [807, 9], "def_end_pos": [807, 25]}, {"full_name": "one_le_two", "def_path": "Mathlib/Algebra/Order/Monoid/NatCast.lean", "def_pos": [50, 7], "def_end_pos": [50, 17]}, {"full_name": "Finset.mem_insert_self", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1091, 9], "def_end_pos": [1091, 24]}, {"full_name": "MeasureTheory.Mem\u2112p.integrable", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [807, 9], "def_end_pos": [807, 25]}, {"full_name": "one_le_two", "def_path": "Mathlib/Algebra/Order/Monoid/NatCast.lean", "def_pos": [50, 7], "def_end_pos": [50, 17]}, {"full_name": "Finset.mem_insert_of_mem", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1095, 9], "def_end_pos": [1095, 26]}]], "state_before": "\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\n\u03b9 : Type u_2\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\ns\u271d : Finset \u03b9\nhs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2\nh\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j)\nk : \u03b9\ns : Finset \u03b9\nks : \u00ack \u2208 s\nIH :\n  (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192\n    (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119\nhs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2\nh : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j)\ni : \u03b9\nhi : i \u2208 s\n\u22a2 Integrable fun a => X k a * X i a", "state_after": "\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\n\u03b9 : Type u_2\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\ns\u271d : Finset \u03b9\nhs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2\nh\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j)\nk : \u03b9\ns : Finset \u03b9\nks : \u00ack \u2208 s\nIH :\n  (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192\n    (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119\nhs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2\nh : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j)\ni : \u03b9\nhi : i \u2208 s\n\u22a2 IndepFun (X k) (X i)"}, {"tactic": "apply h (mem_insert_self _ _) (mem_insert_of_mem hi)", "annotated_tactic": ["apply h (<a>mem_insert_self</a> _ _) (<a>mem_insert_of_mem</a> hi)", [{"full_name": "Finset.mem_insert_self", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1091, 9], "def_end_pos": [1091, 24]}, {"full_name": "Finset.mem_insert_of_mem", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1095, 9], "def_end_pos": [1095, 26]}]], "state_before": "\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\n\u03b9 : Type u_2\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\ns\u271d : Finset \u03b9\nhs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2\nh\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j)\nk : \u03b9\ns : Finset \u03b9\nks : \u00ack \u2208 s\nIH :\n  (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192\n    (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119\nhs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2\nh : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j)\ni : \u03b9\nhi : i \u2208 s\n\u22a2 IndepFun (X k) (X i)", "state_after": "\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\n\u03b9 : Type u_2\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\ns\u271d : Finset \u03b9\nhs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2\nh\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j)\nk : \u03b9\ns : Finset \u03b9\nks : \u00ack \u2208 s\nIH :\n  (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192\n    (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119\nhs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2\nh : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j)\ni : \u03b9\nhi : i \u2208 s\n\u22a2 k \u2260 i"}, {"tactic": "exact fun hki => ks (hki.symm \u25b8 hi)", "annotated_tactic": ["exact fun hki => ks (hki.symm \u25b8 hi)", []], "state_before": "\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\n\u03b9 : Type u_2\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\ns\u271d : Finset \u03b9\nhs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2\nh\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j)\nk : \u03b9\ns : Finset \u03b9\nks : \u00ack \u2208 s\nIH :\n  (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192\n    (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119\nhs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2\nh : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j)\ni : \u03b9\nhi : i \u2208 s\n\u22a2 k \u2260 i", "state_after": "no goals"}, {"tactic": "rw [variance_def' (hs _ (mem_insert_self _ _)),\n  variance_def' (mem\u2112p_finset_sum' _ fun i hi => hs _ (mem_insert_of_mem hi))]", "annotated_tactic": ["rw [<a>variance_def'</a> (hs _ (<a>mem_insert_self</a> _ _)),\n        <a>variance_def'</a> (<a>mem\u2112p_finset_sum'</a> _ fun i hi => hs _ (<a>mem_insert_of_mem</a> hi))]", [{"full_name": "ProbabilityTheory.variance_def'", "def_path": "Mathlib/Probability/Variance.lean", "def_pos": [210, 9], "def_end_pos": [210, 22]}, {"full_name": "Finset.mem_insert_self", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1091, 9], "def_end_pos": [1091, 24]}, {"full_name": "ProbabilityTheory.variance_def'", "def_path": "Mathlib/Probability/Variance.lean", "def_pos": [210, 9], "def_end_pos": [210, 22]}, {"full_name": "MeasureTheory.mem\u2112p_finset_sum'", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [1254, 9], "def_end_pos": [1254, 26]}, {"full_name": "Finset.mem_insert_of_mem", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1095, 9], "def_end_pos": [1095, 26]}]], "state_before": "\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\n\u03b9 : Type u_2\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\ns\u271d : Finset \u03b9\nhs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2\nh\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j)\nk : \u03b9\ns : Finset \u03b9\nks : \u00ack \u2208 s\nIH :\n  (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192\n    (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119\nhs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2\nh : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j)\n\u22a2 (((\u222b (a : \u03a9), (X k ^ 2) a) + \u222b (a : \u03a9), ((\u2211 i in s, X i) ^ 2) a) + \u222b (a : \u03a9), (2 * X k * \u2211 i in s, X i) a) -\n      ((\u222b (a : \u03a9), X k a) + \u222b (a : \u03a9), Finset.sum s (fun i => X i) a) ^ 2 =\n    variance (X k) \u2119 + variance (\u2211 i in s, X i) \u2119 +\n      ((\u222b (a : \u03a9), (2 * X k * \u2211 i in s, X i) a) - (2 * \u222b (a : \u03a9), X k a) * \u222b (a : \u03a9), Finset.sum s (fun i => X i) a)", "state_after": "\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\n\u03b9 : Type u_2\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\ns\u271d : Finset \u03b9\nhs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2\nh\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j)\nk : \u03b9\ns : Finset \u03b9\nks : \u00ack \u2208 s\nIH :\n  (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192\n    (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119\nhs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2\nh : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j)\n\u22a2 (((\u222b (a : \u03a9), (X k ^ 2) a) + \u222b (a : \u03a9), ((\u2211 i in s, X i) ^ 2) a) + \u222b (a : \u03a9), (2 * X k * \u2211 i in s, X i) a) -\n      ((\u222b (a : \u03a9), X k a) + \u222b (a : \u03a9), Finset.sum s (fun i => X i) a) ^ 2 =\n    (\u222b (a : \u03a9), (X k ^ 2) a) - (\u222b (a : \u03a9), X k a) ^ 2 +\n        ((\u222b (a : \u03a9), ((\u2211 i in s, X i) ^ 2) a) - (\u222b (a : \u03a9), Finset.sum s (fun i => X i) a) ^ 2) +\n      ((\u222b (a : \u03a9), (2 * X k * \u2211 i in s, X i) a) - (2 * \u222b (a : \u03a9), X k a) * \u222b (a : \u03a9), Finset.sum s (fun i => X i) a)"}, {"tactic": "ring", "annotated_tactic": ["ring", []], "state_before": "\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\n\u03b9 : Type u_2\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\ns\u271d : Finset \u03b9\nhs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2\nh\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j)\nk : \u03b9\ns : Finset \u03b9\nks : \u00ack \u2208 s\nIH :\n  (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192\n    (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119\nhs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2\nh : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j)\n\u22a2 (((\u222b (a : \u03a9), (X k ^ 2) a) + \u222b (a : \u03a9), ((\u2211 i in s, X i) ^ 2) a) + \u222b (a : \u03a9), (2 * X k * \u2211 i in s, X i) a) -\n      ((\u222b (a : \u03a9), X k a) + \u222b (a : \u03a9), Finset.sum s (fun i => X i) a) ^ 2 =\n    (\u222b (a : \u03a9), (X k ^ 2) a) - (\u222b (a : \u03a9), X k a) ^ 2 +\n        ((\u222b (a : \u03a9), ((\u2211 i in s, X i) ^ 2) a) - (\u222b (a : \u03a9), Finset.sum s (fun i => X i) a) ^ 2) +\n      ((\u222b (a : \u03a9), (2 * X k * \u2211 i in s, X i) a) - (2 * \u222b (a : \u03a9), X k a) * \u222b (a : \u03a9), Finset.sum s (fun i => X i) a)", "state_after": "no goals"}, {"tactic": "simp only [mul_assoc, integral_mul_left, Pi.mul_apply, Pi.one_apply, sum_apply,\n  add_right_eq_self, mul_sum]", "annotated_tactic": ["simp only [<a>mul_assoc</a>, <a>integral_mul_left</a>, <a>Pi.mul_apply</a>, <a>Pi.one_apply</a>, <a>sum_apply</a>,\n        <a>add_right_eq_self</a>, <a>mul_sum</a>]", [{"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [264, 9], "def_end_pos": [264, 18]}, {"full_name": "MeasureTheory.integral_mul_left", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [923, 9], "def_end_pos": [923, 26]}, {"full_name": "Pi.mul_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [83, 9], "def_end_pos": [83, 18]}, {"full_name": "Pi.one_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [47, 9], "def_end_pos": [47, 18]}, {"full_name": "Finset.sum_apply", "def_path": "Mathlib/Algebra/BigOperators/Pi.lean", "def_pos": [41, 3], "def_end_pos": [41, 14]}, {"full_name": "add_right_eq_self", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [180, 3], "def_end_pos": [180, 14]}, {"full_name": "Finset.mul_sum", "def_path": "Mathlib/Algebra/BigOperators/Ring.lean", "def_pos": [55, 9], "def_end_pos": [55, 16]}]], "state_before": "\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\n\u03b9 : Type u_2\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\ns\u271d : Finset \u03b9\nhs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2\nh\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j)\nk : \u03b9\ns : Finset \u03b9\nks : \u00ack \u2208 s\nIH :\n  (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192\n    (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119\nhs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2\nh : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j)\n\u22a2 variance (X k) \u2119 + variance (\u2211 i in s, X i) \u2119 +\n      ((\u222b (a : \u03a9), (2 * X k * \u2211 i in s, X i) a) - (2 * \u222b (a : \u03a9), X k a) * \u222b (a : \u03a9), Finset.sum s (fun i => X i) a) =\n    variance (X k) \u2119 + variance (\u2211 i in s, X i) \u2119", "state_after": "\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\n\u03b9 : Type u_2\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\ns\u271d : Finset \u03b9\nhs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2\nh\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j)\nk : \u03b9\ns : Finset \u03b9\nks : \u00ack \u2208 s\nIH :\n  (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192\n    (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119\nhs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2\nh : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j)\n\u22a2 (\u222b (a : \u03a9), \u2211 x in s, OfNat.ofNat 2 a * (X k a * X x a)) - 2 * ((\u222b (a : \u03a9), X k a) * \u222b (a : \u03a9), \u2211 c in s, X c a) = 0"}, {"tactic": "rw [integral_finset_sum s fun i hi => ?_]", "annotated_tactic": ["rw [<a>integral_finset_sum</a> s fun i hi => ?_]", [{"full_name": "MeasureTheory.integral_finset_sum", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [881, 9], "def_end_pos": [881, 28]}]], "state_before": "\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\n\u03b9 : Type u_2\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\ns\u271d : Finset \u03b9\nhs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2\nh\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j)\nk : \u03b9\ns : Finset \u03b9\nks : \u00ack \u2208 s\nIH :\n  (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192\n    (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119\nhs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2\nh : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j)\n\u22a2 (\u222b (a : \u03a9), \u2211 x in s, OfNat.ofNat 2 a * (X k a * X x a)) - 2 * ((\u222b (a : \u03a9), X k a) * \u222b (a : \u03a9), \u2211 c in s, X c a) = 0", "state_after": "\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\n\u03b9 : Type u_2\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\ns\u271d : Finset \u03b9\nhs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2\nh\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j)\nk : \u03b9\ns : Finset \u03b9\nks : \u00ack \u2208 s\nIH :\n  (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192\n    (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119\nhs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2\nh : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j)\n\u22a2 (\u2211 i in s, \u222b (a : \u03a9), OfNat.ofNat 2 a * (X k a * X i a)) - 2 * ((\u222b (a : \u03a9), X k a) * \u222b (a : \u03a9), \u2211 c in s, X c a) = 0\n\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\n\u03b9 : Type u_2\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\ns\u271d : Finset \u03b9\nhs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2\nh\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j)\nk : \u03b9\ns : Finset \u03b9\nks : \u00ack \u2208 s\nIH :\n  (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192\n    (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119\nhs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2\nh : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j)\ni : \u03b9\nhi : i \u2208 s\n\u22a2 Integrable fun a => OfNat.ofNat 2 a * (X k a * X i a)"}, {"tactic": "swap", "annotated_tactic": ["swap", []], "state_before": "\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\n\u03b9 : Type u_2\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\ns\u271d : Finset \u03b9\nhs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2\nh\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j)\nk : \u03b9\ns : Finset \u03b9\nks : \u00ack \u2208 s\nIH :\n  (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192\n    (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119\nhs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2\nh : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j)\n\u22a2 (\u2211 i in s, \u222b (a : \u03a9), OfNat.ofNat 2 a * (X k a * X i a)) - 2 * ((\u222b (a : \u03a9), X k a) * \u222b (a : \u03a9), \u2211 c in s, X c a) = 0\n\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\n\u03b9 : Type u_2\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\ns\u271d : Finset \u03b9\nhs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2\nh\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j)\nk : \u03b9\ns : Finset \u03b9\nks : \u00ack \u2208 s\nIH :\n  (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192\n    (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119\nhs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2\nh : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j)\ni : \u03b9\nhi : i \u2208 s\n\u22a2 Integrable fun a => OfNat.ofNat 2 a * (X k a * X i a)", "state_after": "\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\n\u03b9 : Type u_2\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\ns\u271d : Finset \u03b9\nhs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2\nh\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j)\nk : \u03b9\ns : Finset \u03b9\nks : \u00ack \u2208 s\nIH :\n  (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192\n    (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119\nhs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2\nh : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j)\ni : \u03b9\nhi : i \u2208 s\n\u22a2 Integrable fun a => OfNat.ofNat 2 a * (X k a * X i a)\n\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\n\u03b9 : Type u_2\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\ns\u271d : Finset \u03b9\nhs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2\nh\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j)\nk : \u03b9\ns : Finset \u03b9\nks : \u00ack \u2208 s\nIH :\n  (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192\n    (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119\nhs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2\nh : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j)\n\u22a2 (\u2211 i in s, \u222b (a : \u03a9), OfNat.ofNat 2 a * (X k a * X i a)) - 2 * ((\u222b (a : \u03a9), X k a) * \u222b (a : \u03a9), \u2211 c in s, X c a) = 0"}, {"tactic": "rw [integral_finset_sum s fun i hi =>\n    Mem\u2112p.integrable one_le_two (hs _ (mem_insert_of_mem hi)),\n  mul_sum, mul_sum, \u2190 sum_sub_distrib]", "annotated_tactic": ["rw [<a>integral_finset_sum</a> s fun i hi =>\n          <a>Mem\u2112p.integrable</a> <a>one_le_two</a> (hs _ (<a>mem_insert_of_mem</a> hi)),\n        <a>mul_sum</a>, <a>mul_sum</a>, \u2190 <a>sum_sub_distrib</a>]", [{"full_name": "MeasureTheory.integral_finset_sum", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [881, 9], "def_end_pos": [881, 28]}, {"full_name": "MeasureTheory.Mem\u2112p.integrable", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [807, 9], "def_end_pos": [807, 25]}, {"full_name": "one_le_two", "def_path": "Mathlib/Algebra/Order/Monoid/NatCast.lean", "def_pos": [50, 7], "def_end_pos": [50, 17]}, {"full_name": "Finset.mem_insert_of_mem", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1095, 9], "def_end_pos": [1095, 26]}, {"full_name": "Finset.mul_sum", "def_path": "Mathlib/Algebra/BigOperators/Ring.lean", "def_pos": [55, 9], "def_end_pos": [55, 16]}, {"full_name": "Finset.mul_sum", "def_path": "Mathlib/Algebra/BigOperators/Ring.lean", "def_pos": [55, 9], "def_end_pos": [55, 16]}, {"full_name": "Finset.sum_sub_distrib", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [1820, 3], "def_end_pos": [1820, 14]}]], "state_before": "\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\n\u03b9 : Type u_2\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\ns\u271d : Finset \u03b9\nhs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2\nh\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j)\nk : \u03b9\ns : Finset \u03b9\nks : \u00ack \u2208 s\nIH :\n  (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192\n    (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119\nhs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2\nh : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j)\n\u22a2 (\u2211 i in s, \u222b (a : \u03a9), OfNat.ofNat 2 a * (X k a * X i a)) - 2 * ((\u222b (a : \u03a9), X k a) * \u222b (a : \u03a9), \u2211 c in s, X c a) = 0", "state_after": "\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\n\u03b9 : Type u_2\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\ns\u271d : Finset \u03b9\nhs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2\nh\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j)\nk : \u03b9\ns : Finset \u03b9\nks : \u00ack \u2208 s\nIH :\n  (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192\n    (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119\nhs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2\nh : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j)\n\u22a2 \u2211 x in s, ((\u222b (a : \u03a9), OfNat.ofNat 2 a * (X k a * X x a)) - 2 * ((\u222b (a : \u03a9), X k a) * \u222b (a : \u03a9), X x a)) = 0"}, {"tactic": "apply Finset.sum_eq_zero fun i hi => ?_", "annotated_tactic": ["apply <a>Finset.sum_eq_zero</a> fun i hi => ?_", [{"full_name": "Finset.sum_eq_zero", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [728, 3], "def_end_pos": [728, 14]}]], "state_before": "\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\n\u03b9 : Type u_2\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\ns\u271d : Finset \u03b9\nhs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2\nh\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j)\nk : \u03b9\ns : Finset \u03b9\nks : \u00ack \u2208 s\nIH :\n  (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192\n    (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119\nhs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2\nh : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j)\n\u22a2 \u2211 x in s, ((\u222b (a : \u03a9), OfNat.ofNat 2 a * (X k a * X x a)) - 2 * ((\u222b (a : \u03a9), X k a) * \u222b (a : \u03a9), X x a)) = 0", "state_after": "\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\n\u03b9 : Type u_2\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\ns\u271d : Finset \u03b9\nhs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2\nh\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j)\nk : \u03b9\ns : Finset \u03b9\nks : \u00ack \u2208 s\nIH :\n  (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192\n    (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119\nhs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2\nh : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j)\ni : \u03b9\nhi : i \u2208 s\n\u22a2 (\u222b (a : \u03a9), OfNat.ofNat 2 a * (X k a * X i a)) - 2 * ((\u222b (a : \u03a9), X k a) * \u222b (a : \u03a9), X i a) = 0"}, {"tactic": "have : \u2200 (a : \u03a9), @OfNat.ofNat (\u03a9 \u2192 \u211d) 2 instOfNat a = (2 : \u211d) := fun a => rfl", "annotated_tactic": ["have : \u2200 (a : \u03a9), @<a>OfNat.ofNat</a> (\u03a9 \u2192 \u211d) 2 <a>instOfNat</a> a = (2 : \u211d) := fun a => <a>rfl</a>", [{"full_name": "OfNat.ofNat", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1066, 3], "def_end_pos": [1066, 8]}, {"full_name": "instOfNat", "def_path": "Mathlib/Data/Nat/Cast/Defs.lean", "def_pos": [55, 10], "def_end_pos": [55, 19]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\n\u03b9 : Type u_2\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\ns\u271d : Finset \u03b9\nhs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2\nh\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j)\nk : \u03b9\ns : Finset \u03b9\nks : \u00ack \u2208 s\nIH :\n  (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192\n    (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119\nhs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2\nh : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j)\ni : \u03b9\nhi : i \u2208 s\n\u22a2 (\u222b (a : \u03a9), OfNat.ofNat 2 a * (X k a * X i a)) - 2 * ((\u222b (a : \u03a9), X k a) * \u222b (a : \u03a9), X i a) = 0", "state_after": "\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\n\u03b9 : Type u_2\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\ns\u271d : Finset \u03b9\nhs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2\nh\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j)\nk : \u03b9\ns : Finset \u03b9\nks : \u00ack \u2208 s\nIH :\n  (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192\n    (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119\nhs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2\nh : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j)\ni : \u03b9\nhi : i \u2208 s\nthis : \u2200 (a : \u03a9), OfNat.ofNat 2 a = 2\n\u22a2 (\u222b (a : \u03a9), OfNat.ofNat 2 a * (X k a * X i a)) - 2 * ((\u222b (a : \u03a9), X k a) * \u222b (a : \u03a9), X i a) = 0"}, {"tactic": "conv_lhs => enter [1, 2, a]; rw [this]", "annotated_tactic": ["conv_lhs => enter [1, 2, a]; rw [this]", []], "state_before": "\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\n\u03b9 : Type u_2\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\ns\u271d : Finset \u03b9\nhs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2\nh\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j)\nk : \u03b9\ns : Finset \u03b9\nks : \u00ack \u2208 s\nIH :\n  (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192\n    (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119\nhs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2\nh : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j)\ni : \u03b9\nhi : i \u2208 s\nthis : \u2200 (a : \u03a9), OfNat.ofNat 2 a = 2\n\u22a2 (\u222b (a : \u03a9), OfNat.ofNat 2 a * (X k a * X i a)) - 2 * ((\u222b (a : \u03a9), X k a) * \u222b (a : \u03a9), X i a) = 0", "state_after": "\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\n\u03b9 : Type u_2\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\ns\u271d : Finset \u03b9\nhs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2\nh\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j)\nk : \u03b9\ns : Finset \u03b9\nks : \u00ack \u2208 s\nIH :\n  (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192\n    (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119\nhs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2\nh : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j)\ni : \u03b9\nhi : i \u2208 s\nthis : \u2200 (a : \u03a9), OfNat.ofNat 2 a = 2\n\u22a2 (\u222b (a : \u03a9), 2 * (X k a * X i a)) - 2 * ((\u222b (a : \u03a9), X k a) * \u222b (a : \u03a9), X i a) = 0"}, {"tactic": "rw [integral_mul_left, IndepFun.integral_mul', sub_self]", "annotated_tactic": ["rw [<a>integral_mul_left</a>, <a>IndepFun.integral_mul'</a>, <a>sub_self</a>]", [{"full_name": "MeasureTheory.integral_mul_left", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [923, 9], "def_end_pos": [923, 26]}, {"full_name": "ProbabilityTheory.IndepFun.integral_mul'", "def_path": "Mathlib/Probability/Integration.lean", "def_pos": [296, 9], "def_end_pos": [296, 31]}, {"full_name": "sub_self", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [734, 30], "def_end_pos": [734, 38]}]], "state_before": "\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\n\u03b9 : Type u_2\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\ns\u271d : Finset \u03b9\nhs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2\nh\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j)\nk : \u03b9\ns : Finset \u03b9\nks : \u00ack \u2208 s\nIH :\n  (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192\n    (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119\nhs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2\nh : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j)\ni : \u03b9\nhi : i \u2208 s\nthis : \u2200 (a : \u03a9), OfNat.ofNat 2 a = 2\n\u22a2 (\u222b (a : \u03a9), 2 * (X k a * X i a)) - 2 * ((\u222b (a : \u03a9), X k a) * \u222b (a : \u03a9), X i a) = 0", "state_after": "case hXY\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\n\u03b9 : Type u_2\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\ns\u271d : Finset \u03b9\nhs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2\nh\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j)\nk : \u03b9\ns : Finset \u03b9\nks : \u00ack \u2208 s\nIH :\n  (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192\n    (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119\nhs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2\nh : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j)\ni : \u03b9\nhi : i \u2208 s\nthis : \u2200 (a : \u03a9), OfNat.ofNat 2 a = 2\n\u22a2 IndepFun (fun a => X k a) fun a => X i a\n\ncase hX\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\n\u03b9 : Type u_2\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\ns\u271d : Finset \u03b9\nhs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2\nh\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j)\nk : \u03b9\ns : Finset \u03b9\nks : \u00ack \u2208 s\nIH :\n  (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192\n    (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119\nhs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2\nh : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j)\ni : \u03b9\nhi : i \u2208 s\nthis : \u2200 (a : \u03a9), OfNat.ofNat 2 a = 2\n\u22a2 AEStronglyMeasurable (fun a => X k a) \u2119\n\ncase hY\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\n\u03b9 : Type u_2\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\ns\u271d : Finset \u03b9\nhs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2\nh\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j)\nk : \u03b9\ns : Finset \u03b9\nks : \u00ack \u2208 s\nIH :\n  (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192\n    (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119\nhs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2\nh : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j)\ni : \u03b9\nhi : i \u2208 s\nthis : \u2200 (a : \u03a9), OfNat.ofNat 2 a = 2\n\u22a2 AEStronglyMeasurable (fun a => X i a) \u2119"}, {"tactic": "apply Integrable.const_mul _ (2 : \u211d)", "annotated_tactic": ["apply <a>Integrable.const_mul</a> _ (2 : \u211d)", [{"full_name": "MeasureTheory.Integrable.const_mul", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [1128, 9], "def_end_pos": [1128, 29]}]], "state_before": "\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\n\u03b9 : Type u_2\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\ns\u271d : Finset \u03b9\nhs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2\nh\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j)\nk : \u03b9\ns : Finset \u03b9\nks : \u00ack \u2208 s\nIH :\n  (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192\n    (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119\nhs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2\nh : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j)\ni : \u03b9\nhi : i \u2208 s\n\u22a2 Integrable fun a => OfNat.ofNat 2 a * (X k a * X i a)", "state_after": "\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\n\u03b9 : Type u_2\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\ns\u271d : Finset \u03b9\nhs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2\nh\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j)\nk : \u03b9\ns : Finset \u03b9\nks : \u00ack \u2208 s\nIH :\n  (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192\n    (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119\nhs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2\nh : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j)\ni : \u03b9\nhi : i \u2208 s\n\u22a2 Integrable fun x => X k x * X i x"}, {"tactic": "apply IndepFun.integrable_mul _ (Mem\u2112p.integrable one_le_two (hs _ (mem_insert_self _ _)))\n  (Mem\u2112p.integrable one_le_two (hs _ (mem_insert_of_mem hi)))", "annotated_tactic": ["apply <a>IndepFun.integrable_mul</a> _ (<a>Mem\u2112p.integrable</a> <a>one_le_two</a> (hs _ (<a>mem_insert_self</a> _ _)))\n          (<a>Mem\u2112p.integrable</a> <a>one_le_two</a> (hs _ (<a>mem_insert_of_mem</a> hi)))", [{"full_name": "ProbabilityTheory.IndepFun.integrable_mul", "def_path": "Mathlib/Probability/Integration.lean", "def_pos": [138, 9], "def_end_pos": [138, 32]}, {"full_name": "MeasureTheory.Mem\u2112p.integrable", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [807, 9], "def_end_pos": [807, 25]}, {"full_name": "one_le_two", "def_path": "Mathlib/Algebra/Order/Monoid/NatCast.lean", "def_pos": [50, 7], "def_end_pos": [50, 17]}, {"full_name": "Finset.mem_insert_self", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1091, 9], "def_end_pos": [1091, 24]}, {"full_name": "MeasureTheory.Mem\u2112p.integrable", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [807, 9], "def_end_pos": [807, 25]}, {"full_name": "one_le_two", "def_path": "Mathlib/Algebra/Order/Monoid/NatCast.lean", "def_pos": [50, 7], "def_end_pos": [50, 17]}, {"full_name": "Finset.mem_insert_of_mem", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1095, 9], "def_end_pos": [1095, 26]}]], "state_before": "\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\n\u03b9 : Type u_2\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\ns\u271d : Finset \u03b9\nhs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2\nh\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j)\nk : \u03b9\ns : Finset \u03b9\nks : \u00ack \u2208 s\nIH :\n  (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192\n    (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119\nhs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2\nh : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j)\ni : \u03b9\nhi : i \u2208 s\n\u22a2 Integrable fun x => X k x * X i x", "state_after": "\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\n\u03b9 : Type u_2\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\ns\u271d : Finset \u03b9\nhs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2\nh\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j)\nk : \u03b9\ns : Finset \u03b9\nks : \u00ack \u2208 s\nIH :\n  (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192\n    (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119\nhs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2\nh : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j)\ni : \u03b9\nhi : i \u2208 s\n\u22a2 IndepFun (X k) (X i)"}, {"tactic": "apply h (mem_insert_self _ _) (mem_insert_of_mem hi)", "annotated_tactic": ["apply h (<a>mem_insert_self</a> _ _) (<a>mem_insert_of_mem</a> hi)", [{"full_name": "Finset.mem_insert_self", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1091, 9], "def_end_pos": [1091, 24]}, {"full_name": "Finset.mem_insert_of_mem", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1095, 9], "def_end_pos": [1095, 26]}]], "state_before": "case hXY\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\n\u03b9 : Type u_2\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\ns\u271d : Finset \u03b9\nhs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2\nh\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j)\nk : \u03b9\ns : Finset \u03b9\nks : \u00ack \u2208 s\nIH :\n  (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192\n    (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119\nhs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2\nh : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j)\ni : \u03b9\nhi : i \u2208 s\nthis : \u2200 (a : \u03a9), OfNat.ofNat 2 a = 2\n\u22a2 IndepFun (fun a => X k a) fun a => X i a", "state_after": "case hXY\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\n\u03b9 : Type u_2\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\ns\u271d : Finset \u03b9\nhs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2\nh\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j)\nk : \u03b9\ns : Finset \u03b9\nks : \u00ack \u2208 s\nIH :\n  (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192\n    (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119\nhs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2\nh : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j)\ni : \u03b9\nhi : i \u2208 s\nthis : \u2200 (a : \u03a9), OfNat.ofNat 2 a = 2\n\u22a2 k \u2260 i"}, {"tactic": "exact fun hki => ks (hki.symm \u25b8 hi)", "annotated_tactic": ["exact fun hki => ks (hki.symm \u25b8 hi)", []], "state_before": "case hXY\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\n\u03b9 : Type u_2\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\ns\u271d : Finset \u03b9\nhs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2\nh\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j)\nk : \u03b9\ns : Finset \u03b9\nks : \u00ack \u2208 s\nIH :\n  (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192\n    (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119\nhs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2\nh : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j)\ni : \u03b9\nhi : i \u2208 s\nthis : \u2200 (a : \u03a9), OfNat.ofNat 2 a = 2\n\u22a2 k \u2260 i", "state_after": "no goals"}, {"tactic": "exact Mem\u2112p.aestronglyMeasurable (hs _ (mem_insert_self _ _))", "annotated_tactic": ["exact <a>Mem\u2112p.aestronglyMeasurable</a> (hs _ (<a>mem_insert_self</a> _ _))", [{"full_name": "MeasureTheory.Mem\u2112p.aestronglyMeasurable", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [120, 9], "def_end_pos": [120, 35]}, {"full_name": "Finset.mem_insert_self", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1091, 9], "def_end_pos": [1091, 24]}]], "state_before": "case hX\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\n\u03b9 : Type u_2\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\ns\u271d : Finset \u03b9\nhs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2\nh\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j)\nk : \u03b9\ns : Finset \u03b9\nks : \u00ack \u2208 s\nIH :\n  (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192\n    (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119\nhs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2\nh : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j)\ni : \u03b9\nhi : i \u2208 s\nthis : \u2200 (a : \u03a9), OfNat.ofNat 2 a = 2\n\u22a2 AEStronglyMeasurable (fun a => X k a) \u2119", "state_after": "no goals"}, {"tactic": "exact Mem\u2112p.aestronglyMeasurable (hs _ (mem_insert_of_mem hi))", "annotated_tactic": ["exact <a>Mem\u2112p.aestronglyMeasurable</a> (hs _ (<a>mem_insert_of_mem</a> hi))", [{"full_name": "MeasureTheory.Mem\u2112p.aestronglyMeasurable", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [120, 9], "def_end_pos": [120, 35]}, {"full_name": "Finset.mem_insert_of_mem", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1095, 9], "def_end_pos": [1095, 26]}]], "state_before": "case hY\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\n\u03b9 : Type u_2\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\ns\u271d : Finset \u03b9\nhs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2\nh\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j)\nk : \u03b9\ns : Finset \u03b9\nks : \u00ack \u2208 s\nIH :\n  (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192\n    (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119\nhs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2\nh : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j)\ni : \u03b9\nhi : i \u2208 s\nthis : \u2200 (a : \u03a9), OfNat.ofNat 2 a = 2\n\u22a2 AEStronglyMeasurable (fun a => X i a) \u2119", "state_after": "no goals"}, {"tactic": "rw [IH (fun i hi => hs i (mem_insert_of_mem hi))\n    (h.mono (by simp only [coe_insert, Set.subset_insert]))]", "annotated_tactic": ["rw [IH (fun i hi => hs i (<a>mem_insert_of_mem</a> hi))\n          (h.mono (by simp only [<a>coe_insert</a>, <a>Set.subset_insert</a>]))]", [{"full_name": "Finset.mem_insert_of_mem", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1095, 9], "def_end_pos": [1095, 26]}, {"full_name": "Finset.coe_insert", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1113, 9], "def_end_pos": [1113, 19]}, {"full_name": "Set.subset_insert", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1123, 9], "def_end_pos": [1123, 22]}]], "state_before": "\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\n\u03b9 : Type u_2\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\ns\u271d : Finset \u03b9\nhs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2\nh\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j)\nk : \u03b9\ns : Finset \u03b9\nks : \u00ack \u2208 s\nIH :\n  (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192\n    (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119\nhs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2\nh : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j)\n\u22a2 variance (X k) \u2119 + variance (\u2211 i in s, X i) \u2119 = variance (X k) \u2119 + \u2211 i in s, variance (X i) \u2119", "state_after": "no goals"}, {"tactic": "simp only [coe_insert, Set.subset_insert]", "annotated_tactic": ["simp only [<a>coe_insert</a>, <a>Set.subset_insert</a>]", [{"full_name": "Finset.coe_insert", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1113, 9], "def_end_pos": [1113, 19]}, {"full_name": "Set.subset_insert", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1123, 9], "def_end_pos": [1123, 22]}]], "state_before": "\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\n\u03b9 : Type u_2\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\ns\u271d : Finset \u03b9\nhs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2\nh\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j)\nk : \u03b9\ns : Finset \u03b9\nks : \u00ack \u2208 s\nIH :\n  (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192\n    (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119\nhs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2\nh : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j)\n\u22a2 \u2191s \u2286 \u2191(insert k s)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/ZMod/Quotient.lean", "full_name": "MulAction.zpowersQuotientStabilizerEquiv_symm_apply", "start": [132, 1], "end": [134, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Lp.lean", "full_name": "MeasureTheory.Integrable.aefinStronglyMeasurable", "start": [67, 1], "end": [68, 91], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Density.lean", "full_name": "MeasureTheory.pdf.ae_lt_top", "start": [139, 8], "end": [146, 21], "traced_tactics": [{"tactic": "by_cases hpdf : HasPDF X \u2119 \u03bc", "annotated_tactic": ["by_cases hpdf : <a>HasPDF</a> X \u2119 \u03bc", [{"full_name": "MeasureTheory.HasPDF", "def_path": "Mathlib/Probability/Density.lean", "def_pos": [67, 7], "def_end_pos": [67, 13]}]], "state_before": "\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b9 : MeasurableSpace E\nm : MeasurableSpace \u03a9\n\u2119 : Measure \u03a9\n\u03bc\u271d : Measure E\ninst\u271d : IsFiniteMeasure \u2119\n\u03bc : Measure E\nX : \u03a9 \u2192 E\n\u22a2 \u2200\u1d50 (x : E) \u2202\u03bc, pdf X \u2119 x < \u22a4", "state_after": "case pos\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b9 : MeasurableSpace E\nm : MeasurableSpace \u03a9\n\u2119 : Measure \u03a9\n\u03bc\u271d : Measure E\ninst\u271d : IsFiniteMeasure \u2119\n\u03bc : Measure E\nX : \u03a9 \u2192 E\nhpdf : HasPDF X \u2119\n\u22a2 \u2200\u1d50 (x : E) \u2202\u03bc, pdf X \u2119 x < \u22a4\n\ncase neg\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b9 : MeasurableSpace E\nm : MeasurableSpace \u03a9\n\u2119 : Measure \u03a9\n\u03bc\u271d : Measure E\ninst\u271d : IsFiniteMeasure \u2119\n\u03bc : Measure E\nX : \u03a9 \u2192 E\nhpdf : \u00acHasPDF X \u2119\n\u22a2 \u2200\u1d50 (x : E) \u2202\u03bc, pdf X \u2119 x < \u22a4"}, {"tactic": "haveI := hpdf", "annotated_tactic": ["haveI := hpdf", []], "state_before": "case pos\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b9 : MeasurableSpace E\nm : MeasurableSpace \u03a9\n\u2119 : Measure \u03a9\n\u03bc\u271d : Measure E\ninst\u271d : IsFiniteMeasure \u2119\n\u03bc : Measure E\nX : \u03a9 \u2192 E\nhpdf : HasPDF X \u2119\n\u22a2 \u2200\u1d50 (x : E) \u2202\u03bc, pdf X \u2119 x < \u22a4", "state_after": "case pos\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b9 : MeasurableSpace E\nm : MeasurableSpace \u03a9\n\u2119 : Measure \u03a9\n\u03bc\u271d : Measure E\ninst\u271d : IsFiniteMeasure \u2119\n\u03bc : Measure E\nX : \u03a9 \u2192 E\nhpdf this : HasPDF X \u2119\n\u22a2 \u2200\u1d50 (x : E) \u2202\u03bc, pdf X \u2119 x < \u22a4"}, {"tactic": "refine' ae_lt_top (measurable_pdf X \u2119 \u03bc) _", "annotated_tactic": ["refine' <a>ae_lt_top</a> (<a>measurable_pdf</a> X \u2119 \u03bc) _", [{"full_name": "MeasureTheory.ae_lt_top", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [1522, 9], "def_end_pos": [1522, 18]}, {"full_name": "MeasureTheory.measurable_pdf", "def_path": "Mathlib/Probability/Density.lean", "def_pos": [109, 9], "def_end_pos": [109, 23]}]], "state_before": "case pos\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b9 : MeasurableSpace E\nm : MeasurableSpace \u03a9\n\u2119 : Measure \u03a9\n\u03bc\u271d : Measure E\ninst\u271d : IsFiniteMeasure \u2119\n\u03bc : Measure E\nX : \u03a9 \u2192 E\nhpdf this : HasPDF X \u2119\n\u22a2 \u2200\u1d50 (x : E) \u2202\u03bc, pdf X \u2119 x < \u22a4", "state_after": "case pos\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b9 : MeasurableSpace E\nm : MeasurableSpace \u03a9\n\u2119 : Measure \u03a9\n\u03bc\u271d : Measure E\ninst\u271d : IsFiniteMeasure \u2119\n\u03bc : Measure E\nX : \u03a9 \u2192 E\nhpdf this : HasPDF X \u2119\n\u22a2 \u222b\u207b (x : E), pdf X \u2119 x \u2202\u03bc \u2260 \u22a4"}, {"tactic": "rw [lintegral_eq_measure_univ]", "annotated_tactic": ["rw [<a>lintegral_eq_measure_univ</a>]", [{"full_name": "MeasureTheory.pdf.lintegral_eq_measure_univ", "def_path": "Mathlib/Probability/Density.lean", "def_pos": [133, 9], "def_end_pos": [133, 34]}]], "state_before": "case pos\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b9 : MeasurableSpace E\nm : MeasurableSpace \u03a9\n\u2119 : Measure \u03a9\n\u03bc\u271d : Measure E\ninst\u271d : IsFiniteMeasure \u2119\n\u03bc : Measure E\nX : \u03a9 \u2192 E\nhpdf this : HasPDF X \u2119\n\u22a2 \u222b\u207b (x : E), pdf X \u2119 x \u2202\u03bc \u2260 \u22a4", "state_after": "case pos\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b9 : MeasurableSpace E\nm : MeasurableSpace \u03a9\n\u2119 : Measure \u03a9\n\u03bc\u271d : Measure E\ninst\u271d : IsFiniteMeasure \u2119\n\u03bc : Measure E\nX : \u03a9 \u2192 E\nhpdf this : HasPDF X \u2119\n\u22a2 \u2191\u2191\u2119 Set.univ \u2260 \u22a4"}, {"tactic": "exact (measure_lt_top _ _).ne", "annotated_tactic": ["exact (<a>measure_lt_top</a> _ _).<a>ne</a>", [{"full_name": "MeasureTheory.measure_lt_top", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2866, 9], "def_end_pos": [2866, 23]}, {"full_name": "LT.lt.ne", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [152, 7], "def_end_pos": [152, 15]}]], "state_before": "case pos\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b9 : MeasurableSpace E\nm : MeasurableSpace \u03a9\n\u2119 : Measure \u03a9\n\u03bc\u271d : Measure E\ninst\u271d : IsFiniteMeasure \u2119\n\u03bc : Measure E\nX : \u03a9 \u2192 E\nhpdf this : HasPDF X \u2119\n\u22a2 \u2191\u2191\u2119 Set.univ \u2260 \u22a4", "state_after": "no goals"}, {"tactic": "simp [pdf, hpdf]", "annotated_tactic": ["simp [<a>pdf</a>, hpdf]", [{"full_name": "MeasureTheory.pdf", "def_path": "Mathlib/Probability/Density.lean", "def_pos": [82, 5], "def_end_pos": [82, 8]}]], "state_before": "case neg\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b9 : MeasurableSpace E\nm : MeasurableSpace \u03a9\n\u2119 : Measure \u03a9\n\u03bc\u271d : Measure E\ninst\u271d : IsFiniteMeasure \u2119\n\u03bc : Measure E\nX : \u03a9 \u2192 E\nhpdf : \u00acHasPDF X \u2119\n\u22a2 \u2200\u1d50 (x : E) \u2202\u03bc, pdf X \u2119 x < \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/FinEnum.lean", "full_name": "FinEnum.mem_toList", "start": [73, 1], "end": [74, 38], "traced_tactics": [{"tactic": "simp [toList]", "annotated_tactic": ["simp [<a>toList</a>]", [{"full_name": "FinEnum.toList", "def_path": "Mathlib/Data/FinEnum.lean", "def_pos": [66, 5], "def_end_pos": [66, 11]}]], "state_before": "\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : FinEnum \u03b1\nx : \u03b1\n\u22a2 x \u2208 toList \u03b1", "state_after": "\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : FinEnum \u03b1\nx : \u03b1\n\u22a2 \u2203 a, \u2191equiv.symm a = x"}, {"tactic": "exists equiv x", "annotated_tactic": ["exists <a>equiv</a> x", [{"full_name": "FinEnum.equiv", "def_path": "Mathlib/Data/FinEnum.lean", "def_pos": [32, 3], "def_end_pos": [32, 8]}]], "state_before": "\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : FinEnum \u03b1\nx : \u03b1\n\u22a2 \u2203 a, \u2191equiv.symm a = x", "state_after": "\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : FinEnum \u03b1\nx : \u03b1\n\u22a2 \u2191equiv.symm (\u2191equiv x) = x"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : FinEnum \u03b1\nx : \u03b1\n\u22a2 \u2191equiv.symm (\u2191equiv x) = x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "full_name": "MeasureTheory.hasSum_integral_measure", "start": [1510, 1], "end": [1529, 33], "traced_tactics": [{"tactic": "have hfi : \u2200 i, Integrable f (\u03bc i) := fun i => hf.mono_measure (Measure.le_sum _ _)", "annotated_tactic": ["have hfi : \u2200 i, <a>Integrable</a> f (\u03bc i) := fun i => hf.mono_measure (<a>Measure.le_sum</a> _ _)", [{"full_name": "MeasureTheory.Integrable", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [442, 5], "def_end_pos": [442, 15]}, {"full_name": "MeasureTheory.Measure.le_sum", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1993, 9], "def_end_pos": [1993, 15]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b9 : Type u_7\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 G\n\u03bc : \u03b9 \u2192 Measure \u03b1\nhf : Integrable f\n\u22a2 HasSum (fun i => \u222b (a : \u03b1), f a \u2202\u03bc i) (\u222b (a : \u03b1), f a \u2202Measure.sum \u03bc)", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b9 : Type u_7\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 G\n\u03bc : \u03b9 \u2192 Measure \u03b1\nhf : Integrable f\nhfi : \u2200 (i : \u03b9), Integrable f\n\u22a2 HasSum (fun i => \u222b (a : \u03b1), f a \u2202\u03bc i) (\u222b (a : \u03b1), f a \u2202Measure.sum \u03bc)"}, {"tactic": "simp only [HasSum, \u2190 integral_finset_sum_measure fun i _ => hfi i]", "annotated_tactic": ["simp only [<a>HasSum</a>, \u2190 <a>integral_finset_sum_measure</a> fun i _ => hfi i]", [{"full_name": "HasSum", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/Basic.lean", "def_pos": [57, 5], "def_end_pos": [57, 11]}, {"full_name": "MeasureTheory.integral_finset_sum_measure", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1492, 9], "def_end_pos": [1492, 36]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b9 : Type u_7\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 G\n\u03bc : \u03b9 \u2192 Measure \u03b1\nhf : Integrable f\nhfi : \u2200 (i : \u03b9), Integrable f\n\u22a2 HasSum (fun i => \u222b (a : \u03b1), f a \u2202\u03bc i) (\u222b (a : \u03b1), f a \u2202Measure.sum \u03bc)", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b9 : Type u_7\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 G\n\u03bc : \u03b9 \u2192 Measure \u03b1\nhf : Integrable f\nhfi : \u2200 (i : \u03b9), Integrable f\n\u22a2 Tendsto (fun s => \u222b (a : \u03b1), f a \u2202\u2211 i in s, \u03bc i) atTop (\ud835\udcdd (\u222b (a : \u03b1), f a \u2202Measure.sum \u03bc))"}, {"tactic": "refine' Metric.nhds_basis_ball.tendsto_right_iff.mpr fun \u03b5 \u03b50 => _", "annotated_tactic": ["refine' Metric.nhds_basis_ball.tendsto_right_iff.mpr fun \u03b5 \u03b50 => _", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b9 : Type u_7\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 G\n\u03bc : \u03b9 \u2192 Measure \u03b1\nhf : Integrable f\nhfi : \u2200 (i : \u03b9), Integrable f\n\u22a2 Tendsto (fun s => \u222b (a : \u03b1), f a \u2202\u2211 i in s, \u03bc i) atTop (\ud835\udcdd (\u222b (a : \u03b1), f a \u2202Measure.sum \u03bc))", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b9 : Type u_7\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 G\n\u03bc : \u03b9 \u2192 Measure \u03b1\nhf : Integrable f\nhfi : \u2200 (i : \u03b9), Integrable f\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\n\u22a2 \u2200\u1da0 (x : Finset \u03b9) in atTop, \u222b (a : \u03b1), f a \u2202\u2211 i in x, \u03bc i \u2208 Metric.ball (\u222b (a : \u03b1), f a \u2202Measure.sum \u03bc) \u03b5"}, {"tactic": "lift \u03b5 to \u211d\u22650 using \u03b50.le", "annotated_tactic": ["lift \u03b5 to \u211d\u22650 using \u03b50.le", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b9 : Type u_7\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 G\n\u03bc : \u03b9 \u2192 Measure \u03b1\nhf : Integrable f\nhfi : \u2200 (i : \u03b9), Integrable f\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\n\u22a2 \u2200\u1da0 (x : Finset \u03b9) in atTop, \u222b (a : \u03b1), f a \u2202\u2211 i in x, \u03bc i \u2208 Metric.ball (\u222b (a : \u03b1), f a \u2202Measure.sum \u03bc) \u03b5", "state_after": "case intro\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b9 : Type u_7\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 G\n\u03bc : \u03b9 \u2192 Measure \u03b1\nhf : Integrable f\nhfi : \u2200 (i : \u03b9), Integrable f\n\u03b5 : \u211d\u22650\n\u03b50 : 0 < \u2191\u03b5\n\u22a2 \u2200\u1da0 (x : Finset \u03b9) in atTop, \u222b (a : \u03b1), f a \u2202\u2211 i in x, \u03bc i \u2208 Metric.ball (\u222b (a : \u03b1), f a \u2202Measure.sum \u03bc) \u2191\u03b5"}, {"tactic": "have hf_lt : (\u222b\u207b x, \u2016f x\u2016\u208a \u2202Measure.sum \u03bc) < \u221e := hf.2", "annotated_tactic": ["have hf_lt : (\u222b\u207b x, \u2016f x\u2016\u208a \u2202<a>Measure.sum</a> \u03bc) < \u221e := hf.2", [{"full_name": "MeasureTheory.Measure.sum", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1978, 5], "def_end_pos": [1978, 8]}]], "state_before": "case intro\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b9 : Type u_7\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 G\n\u03bc : \u03b9 \u2192 Measure \u03b1\nhf : Integrable f\nhfi : \u2200 (i : \u03b9), Integrable f\n\u03b5 : \u211d\u22650\n\u03b50 : 0 < \u2191\u03b5\n\u22a2 \u2200\u1da0 (x : Finset \u03b9) in atTop, \u222b (a : \u03b1), f a \u2202\u2211 i in x, \u03bc i \u2208 Metric.ball (\u222b (a : \u03b1), f a \u2202Measure.sum \u03bc) \u2191\u03b5", "state_after": "case intro\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b9 : Type u_7\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 G\n\u03bc : \u03b9 \u2192 Measure \u03b1\nhf : Integrable f\nhfi : \u2200 (i : \u03b9), Integrable f\n\u03b5 : \u211d\u22650\n\u03b50 : 0 < \u2191\u03b5\nhf_lt : \u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc < \u22a4\n\u22a2 \u2200\u1da0 (x : Finset \u03b9) in atTop, \u222b (a : \u03b1), f a \u2202\u2211 i in x, \u03bc i \u2208 Metric.ball (\u222b (a : \u03b1), f a \u2202Measure.sum \u03bc) \u2191\u03b5"}, {"tactic": "have hmem : \u2200\u1da0 y in \ud835\udcdd (\u222b\u207b x, \u2016f x\u2016\u208a \u2202Measure.sum \u03bc), (\u222b\u207b x, \u2016f x\u2016\u208a \u2202Measure.sum \u03bc) < y + \u03b5 := by\n  refine' tendsto_id.add tendsto_const_nhds (lt_mem_nhds (\u03b1 := \u211d\u22650\u221e) <| ENNReal.lt_add_right _ _)\n  exacts [hf_lt.ne, ENNReal.coe_ne_zero.2 (NNReal.coe_ne_zero.1 \u03b50.ne')]", "annotated_tactic": ["have hmem : \u2200\u1da0 y in \ud835\udcdd (\u222b\u207b x, \u2016f x\u2016\u208a \u2202<a>Measure.sum</a> \u03bc), (\u222b\u207b x, \u2016f x\u2016\u208a \u2202<a>Measure.sum</a> \u03bc) < y + \u03b5 := by\n    refine' tendsto_id.add <a>tendsto_const_nhds</a> (<a>lt_mem_nhds</a> (\u03b1 := \u211d\u22650\u221e) <| <a>ENNReal.lt_add_right</a> _ _)\n    exacts [hf_lt.ne, <a>ENNReal.coe_ne_zero</a>.2 (<a>NNReal.coe_ne_zero</a>.1 \u03b50.ne')]", [{"full_name": "MeasureTheory.Measure.sum", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1978, 5], "def_end_pos": [1978, 8]}, {"full_name": "MeasureTheory.Measure.sum", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1978, 5], "def_end_pos": [1978, 8]}, {"full_name": "tendsto_const_nhds", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [1049, 9], "def_end_pos": [1049, 27]}, {"full_name": "lt_mem_nhds", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [897, 9], "def_end_pos": [897, 20]}, {"full_name": "ENNReal.lt_add_right", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [829, 9], "def_end_pos": [829, 21]}, {"full_name": "ENNReal.coe_ne_zero", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [383, 9], "def_end_pos": [383, 20]}, {"full_name": "NNReal.coe_ne_zero", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [222, 9], "def_end_pos": [222, 20]}]], "state_before": "case intro\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b9 : Type u_7\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 G\n\u03bc : \u03b9 \u2192 Measure \u03b1\nhf : Integrable f\nhfi : \u2200 (i : \u03b9), Integrable f\n\u03b5 : \u211d\u22650\n\u03b50 : 0 < \u2191\u03b5\nhf_lt : \u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc < \u22a4\n\u22a2 \u2200\u1da0 (x : Finset \u03b9) in atTop, \u222b (a : \u03b1), f a \u2202\u2211 i in x, \u03bc i \u2208 Metric.ball (\u222b (a : \u03b1), f a \u2202Measure.sum \u03bc) \u2191\u03b5", "state_after": "case intro\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b9 : Type u_7\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 G\n\u03bc : \u03b9 \u2192 Measure \u03b1\nhf : Integrable f\nhfi : \u2200 (i : \u03b9), Integrable f\n\u03b5 : \u211d\u22650\n\u03b50 : 0 < \u2191\u03b5\nhf_lt : \u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc < \u22a4\nhmem : \u2200\u1da0 (y : \u211d\u22650\u221e) in \ud835\udcdd (\u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc), \u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc < y + \u2191\u03b5\n\u22a2 \u2200\u1da0 (x : Finset \u03b9) in atTop, \u222b (a : \u03b1), f a \u2202\u2211 i in x, \u03bc i \u2208 Metric.ball (\u222b (a : \u03b1), f a \u2202Measure.sum \u03bc) \u2191\u03b5"}, {"tactic": "refine' ((hasSum_lintegral_measure (fun x => \u2016f x\u2016\u208a) \u03bc).eventually hmem).mono fun s hs => _", "annotated_tactic": ["refine' ((<a>hasSum_lintegral_measure</a> (fun x => \u2016f x\u2016\u208a) \u03bc).<a>eventually</a> hmem).<a>mono</a> fun s hs => _", [{"full_name": "MeasureTheory.hasSum_lintegral_measure", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [613, 9], "def_end_pos": [613, 33]}, {"full_name": "Filter.Tendsto.eventually", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2953, 9], "def_end_pos": [2953, 27]}, {"full_name": "Filter.Eventually.mono", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1140, 9], "def_end_pos": [1140, 24]}]], "state_before": "case intro\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b9 : Type u_7\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 G\n\u03bc : \u03b9 \u2192 Measure \u03b1\nhf : Integrable f\nhfi : \u2200 (i : \u03b9), Integrable f\n\u03b5 : \u211d\u22650\n\u03b50 : 0 < \u2191\u03b5\nhf_lt : \u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc < \u22a4\nhmem : \u2200\u1da0 (y : \u211d\u22650\u221e) in \ud835\udcdd (\u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc), \u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc < y + \u2191\u03b5\n\u22a2 \u2200\u1da0 (x : Finset \u03b9) in atTop, \u222b (a : \u03b1), f a \u2202\u2211 i in x, \u03bc i \u2208 Metric.ball (\u222b (a : \u03b1), f a \u2202Measure.sum \u03bc) \u2191\u03b5", "state_after": "case intro\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b9 : Type u_7\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 G\n\u03bc : \u03b9 \u2192 Measure \u03b1\nhf : Integrable f\nhfi : \u2200 (i : \u03b9), Integrable f\n\u03b5 : \u211d\u22650\n\u03b50 : 0 < \u2191\u03b5\nhf_lt : \u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc < \u22a4\nhmem : \u2200\u1da0 (y : \u211d\u22650\u221e) in \ud835\udcdd (\u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc), \u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc < y + \u2191\u03b5\ns : Finset \u03b9\nhs : \u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc < \u2211 b in s, (fun i => \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a \u2202\u03bc i) b + \u2191\u03b5\n\u22a2 \u222b (a : \u03b1), f a \u2202\u2211 i in s, \u03bc i \u2208 Metric.ball (\u222b (a : \u03b1), f a \u2202Measure.sum \u03bc) \u2191\u03b5"}, {"tactic": "obtain \u27e8\u03bd, h\u03bd\u27e9 : \u2203 \u03bd, (\u2211 i in s, \u03bc i) + \u03bd = Measure.sum \u03bc := by\n  refine' \u27e8Measure.sum fun i : \u21a5(s\u1d9c : Set \u03b9) => \u03bc i, _\u27e9\n  simpa only [\u2190 Measure.sum_coe_finset] using Measure.sum_add_sum_compl (s : Set \u03b9) \u03bc", "annotated_tactic": ["obtain \u27e8\u03bd, h\u03bd\u27e9 : \u2203 \u03bd, (\u2211 i in s, \u03bc i) + \u03bd = <a>Measure.sum</a> \u03bc := by\n    refine' \u27e8<a>Measure.sum</a> fun i : \u21a5(s\u1d9c : <a>Set</a> \u03b9) => \u03bc i, _\u27e9\n    simpa only [\u2190 <a>Measure.sum_coe_finset</a>] using <a>Measure.sum_add_sum_compl</a> (s : <a>Set</a> \u03b9) \u03bc", [{"full_name": "MeasureTheory.Measure.sum", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1978, 5], "def_end_pos": [1978, 8]}, {"full_name": "MeasureTheory.Measure.sum", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1978, 5], "def_end_pos": [1978, 8]}, {"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}, {"full_name": "MeasureTheory.Measure.sum_coe_finset", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2040, 9], "def_end_pos": [2040, 23]}, {"full_name": "MeasureTheory.Measure.sum_add_sum_compl", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2070, 9], "def_end_pos": [2070, 26]}, {"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}]], "state_before": "case intro\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b9 : Type u_7\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 G\n\u03bc : \u03b9 \u2192 Measure \u03b1\nhf : Integrable f\nhfi : \u2200 (i : \u03b9), Integrable f\n\u03b5 : \u211d\u22650\n\u03b50 : 0 < \u2191\u03b5\nhf_lt : \u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc < \u22a4\nhmem : \u2200\u1da0 (y : \u211d\u22650\u221e) in \ud835\udcdd (\u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc), \u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc < y + \u2191\u03b5\ns : Finset \u03b9\nhs : \u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc < \u2211 b in s, (fun i => \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a \u2202\u03bc i) b + \u2191\u03b5\n\u22a2 \u222b (a : \u03b1), f a \u2202\u2211 i in s, \u03bc i \u2208 Metric.ball (\u222b (a : \u03b1), f a \u2202Measure.sum \u03bc) \u2191\u03b5", "state_after": "case intro.intro\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\n\u03bd\u271d : Measure \u03b1\n\u03b9 : Type u_7\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 G\n\u03bc : \u03b9 \u2192 Measure \u03b1\nhf : Integrable f\nhfi : \u2200 (i : \u03b9), Integrable f\n\u03b5 : \u211d\u22650\n\u03b50 : 0 < \u2191\u03b5\nhf_lt : \u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc < \u22a4\nhmem : \u2200\u1da0 (y : \u211d\u22650\u221e) in \ud835\udcdd (\u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc), \u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc < y + \u2191\u03b5\ns : Finset \u03b9\nhs : \u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc < \u2211 b in s, (fun i => \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a \u2202\u03bc i) b + \u2191\u03b5\n\u03bd : Measure \u03b1\nh\u03bd : \u2211 i in s, \u03bc i + \u03bd = Measure.sum \u03bc\n\u22a2 \u222b (a : \u03b1), f a \u2202\u2211 i in s, \u03bc i \u2208 Metric.ball (\u222b (a : \u03b1), f a \u2202Measure.sum \u03bc) \u2191\u03b5"}, {"tactic": "rw [Metric.mem_ball, \u2190 coe_nndist, NNReal.coe_lt_coe, \u2190 ENNReal.coe_lt_coe, \u2190 h\u03bd]", "annotated_tactic": ["rw [<a>Metric.mem_ball</a>, \u2190 <a>coe_nndist</a>, <a>NNReal.coe_lt_coe</a>, \u2190 <a>ENNReal.coe_lt_coe</a>, \u2190 h\u03bd]", [{"full_name": "Metric.mem_ball", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [414, 9], "def_end_pos": [414, 17]}, {"full_name": "coe_nndist", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [310, 9], "def_end_pos": [310, 19]}, {"full_name": "NNReal.coe_lt_coe", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [371, 19], "def_end_pos": [371, 29]}, {"full_name": "ENNReal.coe_lt_coe", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [352, 28], "def_end_pos": [352, 38]}]], "state_before": "case intro.intro\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\n\u03bd\u271d : Measure \u03b1\n\u03b9 : Type u_7\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 G\n\u03bc : \u03b9 \u2192 Measure \u03b1\nhf : Integrable f\nhfi : \u2200 (i : \u03b9), Integrable f\n\u03b5 : \u211d\u22650\n\u03b50 : 0 < \u2191\u03b5\nhf_lt : \u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc < \u22a4\nhmem : \u2200\u1da0 (y : \u211d\u22650\u221e) in \ud835\udcdd (\u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc), \u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc < y + \u2191\u03b5\ns : Finset \u03b9\nhs : \u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc < \u2211 b in s, (fun i => \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a \u2202\u03bc i) b + \u2191\u03b5\n\u03bd : Measure \u03b1\nh\u03bd : \u2211 i in s, \u03bc i + \u03bd = Measure.sum \u03bc\n\u22a2 \u222b (a : \u03b1), f a \u2202\u2211 i in s, \u03bc i \u2208 Metric.ball (\u222b (a : \u03b1), f a \u2202Measure.sum \u03bc) \u2191\u03b5", "state_after": "case intro.intro\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\n\u03bd\u271d : Measure \u03b1\n\u03b9 : Type u_7\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 G\n\u03bc : \u03b9 \u2192 Measure \u03b1\nhf : Integrable f\nhfi : \u2200 (i : \u03b9), Integrable f\n\u03b5 : \u211d\u22650\n\u03b50 : 0 < \u2191\u03b5\nhf_lt : \u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc < \u22a4\nhmem : \u2200\u1da0 (y : \u211d\u22650\u221e) in \ud835\udcdd (\u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc), \u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc < y + \u2191\u03b5\ns : Finset \u03b9\nhs : \u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc < \u2211 b in s, (fun i => \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a \u2202\u03bc i) b + \u2191\u03b5\n\u03bd : Measure \u03b1\nh\u03bd : \u2211 i in s, \u03bc i + \u03bd = Measure.sum \u03bc\n\u22a2 \u2191(nndist (\u222b (a : \u03b1), f a \u2202\u2211 i in s, \u03bc i) (\u222b (a : \u03b1), f a \u2202(\u2211 i in s, \u03bc i + \u03bd))) < \u2191\u03b5"}, {"tactic": "rw [\u2190 h\u03bd, integrable_add_measure] at hf", "annotated_tactic": ["rw [\u2190 h\u03bd, <a>integrable_add_measure</a>] at hf", [{"full_name": "MeasureTheory.integrable_add_measure", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [561, 9], "def_end_pos": [561, 31]}]], "state_before": "case intro.intro\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\n\u03bd\u271d : Measure \u03b1\n\u03b9 : Type u_7\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 G\n\u03bc : \u03b9 \u2192 Measure \u03b1\nhf : Integrable f\nhfi : \u2200 (i : \u03b9), Integrable f\n\u03b5 : \u211d\u22650\n\u03b50 : 0 < \u2191\u03b5\nhf_lt : \u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc < \u22a4\nhmem : \u2200\u1da0 (y : \u211d\u22650\u221e) in \ud835\udcdd (\u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc), \u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc < y + \u2191\u03b5\ns : Finset \u03b9\nhs : \u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc < \u2211 b in s, (fun i => \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a \u2202\u03bc i) b + \u2191\u03b5\n\u03bd : Measure \u03b1\nh\u03bd : \u2211 i in s, \u03bc i + \u03bd = Measure.sum \u03bc\n\u22a2 \u2191(nndist (\u222b (a : \u03b1), f a \u2202\u2211 i in s, \u03bc i) (\u222b (a : \u03b1), f a \u2202(\u2211 i in s, \u03bc i + \u03bd))) < \u2191\u03b5", "state_after": "case intro.intro\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\n\u03bd\u271d : Measure \u03b1\n\u03b9 : Type u_7\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 G\n\u03bc : \u03b9 \u2192 Measure \u03b1\nhfi : \u2200 (i : \u03b9), Integrable f\n\u03b5 : \u211d\u22650\n\u03b50 : 0 < \u2191\u03b5\nhf_lt : \u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc < \u22a4\nhmem : \u2200\u1da0 (y : \u211d\u22650\u221e) in \ud835\udcdd (\u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc), \u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc < y + \u2191\u03b5\ns : Finset \u03b9\nhs : \u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc < \u2211 b in s, (fun i => \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a \u2202\u03bc i) b + \u2191\u03b5\n\u03bd : Measure \u03b1\nhf : Integrable f \u2227 Integrable f\nh\u03bd : \u2211 i in s, \u03bc i + \u03bd = Measure.sum \u03bc\n\u22a2 \u2191(nndist (\u222b (a : \u03b1), f a \u2202\u2211 i in s, \u03bc i) (\u222b (a : \u03b1), f a \u2202(\u2211 i in s, \u03bc i + \u03bd))) < \u2191\u03b5"}, {"tactic": "refine' (nndist_integral_add_measure_le_lintegral hf.1 hf.2).trans_lt _", "annotated_tactic": ["refine' (<a>nndist_integral_add_measure_le_lintegral</a> hf.1 hf.2).<a>trans_lt</a> _", [{"full_name": "MeasureTheory.nndist_integral_add_measure_le_lintegral", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1503, 9], "def_end_pos": [1503, 49]}, {"full_name": "LE.le.trans_lt", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [124, 7], "def_end_pos": [124, 21]}]], "state_before": "case intro.intro\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\n\u03bd\u271d : Measure \u03b1\n\u03b9 : Type u_7\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 G\n\u03bc : \u03b9 \u2192 Measure \u03b1\nhfi : \u2200 (i : \u03b9), Integrable f\n\u03b5 : \u211d\u22650\n\u03b50 : 0 < \u2191\u03b5\nhf_lt : \u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc < \u22a4\nhmem : \u2200\u1da0 (y : \u211d\u22650\u221e) in \ud835\udcdd (\u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc), \u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc < y + \u2191\u03b5\ns : Finset \u03b9\nhs : \u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc < \u2211 b in s, (fun i => \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a \u2202\u03bc i) b + \u2191\u03b5\n\u03bd : Measure \u03b1\nhf : Integrable f \u2227 Integrable f\nh\u03bd : \u2211 i in s, \u03bc i + \u03bd = Measure.sum \u03bc\n\u22a2 \u2191(nndist (\u222b (a : \u03b1), f a \u2202\u2211 i in s, \u03bc i) (\u222b (a : \u03b1), f a \u2202(\u2211 i in s, \u03bc i + \u03bd))) < \u2191\u03b5", "state_after": "case intro.intro\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\n\u03bd\u271d : Measure \u03b1\n\u03b9 : Type u_7\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 G\n\u03bc : \u03b9 \u2192 Measure \u03b1\nhfi : \u2200 (i : \u03b9), Integrable f\n\u03b5 : \u211d\u22650\n\u03b50 : 0 < \u2191\u03b5\nhf_lt : \u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc < \u22a4\nhmem : \u2200\u1da0 (y : \u211d\u22650\u221e) in \ud835\udcdd (\u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc), \u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc < y + \u2191\u03b5\ns : Finset \u03b9\nhs : \u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc < \u2211 b in s, (fun i => \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a \u2202\u03bc i) b + \u2191\u03b5\n\u03bd : Measure \u03b1\nhf : Integrable f \u2227 Integrable f\nh\u03bd : \u2211 i in s, \u03bc i + \u03bd = Measure.sum \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202\u03bd < \u2191\u03b5"}, {"tactic": "rw [\u2190 h\u03bd, lintegral_add_measure, lintegral_finset_sum_measure] at hs", "annotated_tactic": ["rw [\u2190 h\u03bd, <a>lintegral_add_measure</a>, <a>lintegral_finset_sum_measure</a>] at hs", [{"full_name": "MeasureTheory.lintegral_add_measure", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [619, 9], "def_end_pos": [619, 30]}, {"full_name": "MeasureTheory.lintegral_finset_sum_measure", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [625, 9], "def_end_pos": [625, 37]}]], "state_before": "case intro.intro\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\n\u03bd\u271d : Measure \u03b1\n\u03b9 : Type u_7\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 G\n\u03bc : \u03b9 \u2192 Measure \u03b1\nhfi : \u2200 (i : \u03b9), Integrable f\n\u03b5 : \u211d\u22650\n\u03b50 : 0 < \u2191\u03b5\nhf_lt : \u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc < \u22a4\nhmem : \u2200\u1da0 (y : \u211d\u22650\u221e) in \ud835\udcdd (\u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc), \u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc < y + \u2191\u03b5\ns : Finset \u03b9\nhs : \u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc < \u2211 b in s, (fun i => \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a \u2202\u03bc i) b + \u2191\u03b5\n\u03bd : Measure \u03b1\nhf : Integrable f \u2227 Integrable f\nh\u03bd : \u2211 i in s, \u03bc i + \u03bd = Measure.sum \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202\u03bd < \u2191\u03b5", "state_after": "case intro.intro\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\n\u03bd\u271d : Measure \u03b1\n\u03b9 : Type u_7\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 G\n\u03bc : \u03b9 \u2192 Measure \u03b1\nhfi : \u2200 (i : \u03b9), Integrable f\n\u03b5 : \u211d\u22650\n\u03b50 : 0 < \u2191\u03b5\nhf_lt : \u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc < \u22a4\nhmem : \u2200\u1da0 (y : \u211d\u22650\u221e) in \ud835\udcdd (\u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc), \u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc < y + \u2191\u03b5\ns : Finset \u03b9\n\u03bd : Measure \u03b1\nhs : \u2211 i in s, \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a \u2202\u03bc i + \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a \u2202\u03bd < \u2211 b in s, (fun i => \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a \u2202\u03bc i) b + \u2191\u03b5\nhf : Integrable f \u2227 Integrable f\nh\u03bd : \u2211 i in s, \u03bc i + \u03bd = Measure.sum \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202\u03bd < \u2191\u03b5"}, {"tactic": "exact lt_of_add_lt_add_left hs", "annotated_tactic": ["exact <a>lt_of_add_lt_add_left</a> hs", [{"full_name": "lt_of_add_lt_add_left", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [127, 15], "def_end_pos": [127, 36]}]], "state_before": "case intro.intro\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\n\u03bd\u271d : Measure \u03b1\n\u03b9 : Type u_7\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 G\n\u03bc : \u03b9 \u2192 Measure \u03b1\nhfi : \u2200 (i : \u03b9), Integrable f\n\u03b5 : \u211d\u22650\n\u03b50 : 0 < \u2191\u03b5\nhf_lt : \u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc < \u22a4\nhmem : \u2200\u1da0 (y : \u211d\u22650\u221e) in \ud835\udcdd (\u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc), \u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc < y + \u2191\u03b5\ns : Finset \u03b9\n\u03bd : Measure \u03b1\nhs : \u2211 i in s, \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a \u2202\u03bc i + \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a \u2202\u03bd < \u2211 b in s, (fun i => \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a \u2202\u03bc i) b + \u2191\u03b5\nhf : Integrable f \u2227 Integrable f\nh\u03bd : \u2211 i in s, \u03bc i + \u03bd = Measure.sum \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202\u03bd < \u2191\u03b5", "state_after": "no goals"}, {"tactic": "refine' tendsto_id.add tendsto_const_nhds (lt_mem_nhds (\u03b1 := \u211d\u22650\u221e) <| ENNReal.lt_add_right _ _)", "annotated_tactic": ["refine' tendsto_id.add <a>tendsto_const_nhds</a> (<a>lt_mem_nhds</a> (\u03b1 := \u211d\u22650\u221e) <| <a>ENNReal.lt_add_right</a> _ _)", [{"full_name": "tendsto_const_nhds", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [1049, 9], "def_end_pos": [1049, 27]}, {"full_name": "lt_mem_nhds", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [897, 9], "def_end_pos": [897, 20]}, {"full_name": "ENNReal.lt_add_right", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [829, 9], "def_end_pos": [829, 21]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b9 : Type u_7\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 G\n\u03bc : \u03b9 \u2192 Measure \u03b1\nhf : Integrable f\nhfi : \u2200 (i : \u03b9), Integrable f\n\u03b5 : \u211d\u22650\n\u03b50 : 0 < \u2191\u03b5\nhf_lt : \u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc < \u22a4\n\u22a2 \u2200\u1da0 (y : \u211d\u22650\u221e) in \ud835\udcdd (\u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc), \u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc < y + \u2191\u03b5", "state_after": "case refine'_1\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b9 : Type u_7\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 G\n\u03bc : \u03b9 \u2192 Measure \u03b1\nhf : Integrable f\nhfi : \u2200 (i : \u03b9), Integrable f\n\u03b5 : \u211d\u22650\n\u03b50 : 0 < \u2191\u03b5\nhf_lt : \u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc < \u22a4\n\u22a2 \u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc \u2260 \u22a4\n\ncase refine'_2\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b9 : Type u_7\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 G\n\u03bc : \u03b9 \u2192 Measure \u03b1\nhf : Integrable f\nhfi : \u2200 (i : \u03b9), Integrable f\n\u03b5 : \u211d\u22650\n\u03b50 : 0 < \u2191\u03b5\nhf_lt : \u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc < \u22a4\n\u22a2 \u2191\u03b5 \u2260 0"}, {"tactic": "exacts [hf_lt.ne, ENNReal.coe_ne_zero.2 (NNReal.coe_ne_zero.1 \u03b50.ne')]", "annotated_tactic": ["exacts [hf_lt.ne, <a>ENNReal.coe_ne_zero</a>.2 (<a>NNReal.coe_ne_zero</a>.1 \u03b50.ne')]", [{"full_name": "ENNReal.coe_ne_zero", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [383, 9], "def_end_pos": [383, 20]}, {"full_name": "NNReal.coe_ne_zero", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [222, 9], "def_end_pos": [222, 20]}]], "state_before": "case refine'_1\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b9 : Type u_7\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 G\n\u03bc : \u03b9 \u2192 Measure \u03b1\nhf : Integrable f\nhfi : \u2200 (i : \u03b9), Integrable f\n\u03b5 : \u211d\u22650\n\u03b50 : 0 < \u2191\u03b5\nhf_lt : \u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc < \u22a4\n\u22a2 \u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc \u2260 \u22a4\n\ncase refine'_2\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b9 : Type u_7\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 G\n\u03bc : \u03b9 \u2192 Measure \u03b1\nhf : Integrable f\nhfi : \u2200 (i : \u03b9), Integrable f\n\u03b5 : \u211d\u22650\n\u03b50 : 0 < \u2191\u03b5\nhf_lt : \u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc < \u22a4\n\u22a2 \u2191\u03b5 \u2260 0", "state_after": "no goals"}, {"tactic": "refine' \u27e8Measure.sum fun i : \u21a5(s\u1d9c : Set \u03b9) => \u03bc i, _\u27e9", "annotated_tactic": ["refine' \u27e8<a>Measure.sum</a> fun i : \u21a5(s\u1d9c : <a>Set</a> \u03b9) => \u03bc i, _\u27e9", [{"full_name": "MeasureTheory.Measure.sum", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1978, 5], "def_end_pos": [1978, 8]}, {"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b9 : Type u_7\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 G\n\u03bc : \u03b9 \u2192 Measure \u03b1\nhf : Integrable f\nhfi : \u2200 (i : \u03b9), Integrable f\n\u03b5 : \u211d\u22650\n\u03b50 : 0 < \u2191\u03b5\nhf_lt : \u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc < \u22a4\nhmem : \u2200\u1da0 (y : \u211d\u22650\u221e) in \ud835\udcdd (\u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc), \u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc < y + \u2191\u03b5\ns : Finset \u03b9\nhs : \u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc < \u2211 b in s, (fun i => \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a \u2202\u03bc i) b + \u2191\u03b5\n\u22a2 \u2203 \u03bd, \u2211 i in s, \u03bc i + \u03bd = Measure.sum \u03bc", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b9 : Type u_7\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 G\n\u03bc : \u03b9 \u2192 Measure \u03b1\nhf : Integrable f\nhfi : \u2200 (i : \u03b9), Integrable f\n\u03b5 : \u211d\u22650\n\u03b50 : 0 < \u2191\u03b5\nhf_lt : \u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc < \u22a4\nhmem : \u2200\u1da0 (y : \u211d\u22650\u221e) in \ud835\udcdd (\u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc), \u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc < y + \u2191\u03b5\ns : Finset \u03b9\nhs : \u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc < \u2211 b in s, (fun i => \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a \u2202\u03bc i) b + \u2191\u03b5\n\u22a2 (\u2211 i in s, \u03bc i + Measure.sum fun i => \u03bc \u2191i) = Measure.sum \u03bc"}, {"tactic": "simpa only [\u2190 Measure.sum_coe_finset] using Measure.sum_add_sum_compl (s : Set \u03b9) \u03bc", "annotated_tactic": ["simpa only [\u2190 <a>Measure.sum_coe_finset</a>] using <a>Measure.sum_add_sum_compl</a> (s : <a>Set</a> \u03b9) \u03bc", [{"full_name": "MeasureTheory.Measure.sum_coe_finset", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2040, 9], "def_end_pos": [2040, 23]}, {"full_name": "MeasureTheory.Measure.sum_add_sum_compl", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2070, 9], "def_end_pos": [2070, 26]}, {"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\n\u03bd : Measure \u03b1\n\u03b9 : Type u_7\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 G\n\u03bc : \u03b9 \u2192 Measure \u03b1\nhf : Integrable f\nhfi : \u2200 (i : \u03b9), Integrable f\n\u03b5 : \u211d\u22650\n\u03b50 : 0 < \u2191\u03b5\nhf_lt : \u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc < \u22a4\nhmem : \u2200\u1da0 (y : \u211d\u22650\u221e) in \ud835\udcdd (\u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc), \u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc < y + \u2191\u03b5\ns : Finset \u03b9\nhs : \u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc < \u2211 b in s, (fun i => \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a \u2202\u03bc i) b + \u2191\u03b5\n\u22a2 (\u2211 i in s, \u03bc i + Measure.sum fun i => \u03bc \u2191i) = Measure.sum \u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "full_name": "MeasureTheory.OuterMeasure.ext", "start": [238, 1], "end": [239, 31], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "full_name": "MeasureTheory.Measure.QuasiMeasurePreserving.image_zpow_ae_eq", "start": [2294, 1], "end": [2308, 46], "traced_tactics": [{"tactic": "rw [Equiv.image_eq_preimage]", "annotated_tactic": ["rw [<a>Equiv.image_eq_preimage</a>]", [{"full_name": "Equiv.image_eq_preimage", "def_path": "Mathlib/Logic/Equiv/Set.lean", "def_pos": [40, 19], "def_end_pos": [40, 36]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\n\u03bca \u03bca' : Measure \u03b1\n\u03bcb \u03bcb' : Measure \u03b2\n\u03bcc : Measure \u03b3\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\ne : \u03b1 \u2243 \u03b1\nhe : QuasiMeasurePreserving \u2191e\nhe' : QuasiMeasurePreserving \u2191e.symm\nk : \u2124\nhs : \u2191e '' s =\u1d50[\u03bc] s\n\u22a2 \u2191(e ^ k) '' s =\u1d50[\u03bc] s", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\n\u03bca \u03bca' : Measure \u03b1\n\u03bcb \u03bcb' : Measure \u03b2\n\u03bcc : Measure \u03b3\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\ne : \u03b1 \u2243 \u03b1\nhe : QuasiMeasurePreserving \u2191e\nhe' : QuasiMeasurePreserving \u2191e.symm\nk : \u2124\nhs : \u2191e '' s =\u1d50[\u03bc] s\n\u22a2 \u2191(e ^ k).symm \u207b\u00b9' s =\u1d50[\u03bc] s"}, {"tactic": "obtain \u27e8k, rfl | rfl\u27e9 := k.eq_nat_or_neg", "annotated_tactic": ["obtain \u27e8k, rfl | rfl\u27e9 := k.eq_nat_or_neg", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\n\u03bca \u03bca' : Measure \u03b1\n\u03bcb \u03bcb' : Measure \u03b2\n\u03bcc : Measure \u03b3\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\ne : \u03b1 \u2243 \u03b1\nhe : QuasiMeasurePreserving \u2191e\nhe' : QuasiMeasurePreserving \u2191e.symm\nk : \u2124\nhs : \u2191e '' s =\u1d50[\u03bc] s\n\u22a2 \u2191(e ^ k).symm \u207b\u00b9' s =\u1d50[\u03bc] s", "state_after": "case intro.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\n\u03bca \u03bca' : Measure \u03b1\n\u03bcb \u03bcb' : Measure \u03b2\n\u03bcc : Measure \u03b3\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\ne : \u03b1 \u2243 \u03b1\nhe : QuasiMeasurePreserving \u2191e\nhe' : QuasiMeasurePreserving \u2191e.symm\nhs : \u2191e '' s =\u1d50[\u03bc] s\nk : \u2115\n\u22a2 \u2191(e ^ \u2191k).symm \u207b\u00b9' s =\u1d50[\u03bc] s\n\ncase intro.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\n\u03bca \u03bca' : Measure \u03b1\n\u03bcb \u03bcb' : Measure \u03b2\n\u03bcc : Measure \u03b3\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\ne : \u03b1 \u2243 \u03b1\nhe : QuasiMeasurePreserving \u2191e\nhe' : QuasiMeasurePreserving \u2191e.symm\nhs : \u2191e '' s =\u1d50[\u03bc] s\nk : \u2115\n\u22a2 \u2191(e ^ (-\u2191k)).symm \u207b\u00b9' s =\u1d50[\u03bc] s"}, {"tactic": "replace hs : (\u21d1e\u207b\u00b9) \u207b\u00b9' s =\u1d50[\u03bc] s", "annotated_tactic": ["replace hs : (\u21d1e\u207b\u00b9) \u207b\u00b9' s =\u1d50[\u03bc] s", []], "state_before": "case intro.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\n\u03bca \u03bca' : Measure \u03b1\n\u03bcb \u03bcb' : Measure \u03b2\n\u03bcc : Measure \u03b3\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\ne : \u03b1 \u2243 \u03b1\nhe : QuasiMeasurePreserving \u2191e\nhe' : QuasiMeasurePreserving \u2191e.symm\nhs : \u2191e '' s =\u1d50[\u03bc] s\nk : \u2115\n\u22a2 \u2191(e ^ \u2191k).symm \u207b\u00b9' s =\u1d50[\u03bc] s", "state_after": "case hs\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\n\u03bca \u03bca' : Measure \u03b1\n\u03bcb \u03bcb' : Measure \u03b2\n\u03bcc : Measure \u03b3\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\ne : \u03b1 \u2243 \u03b1\nhe : QuasiMeasurePreserving \u2191e\nhe' : QuasiMeasurePreserving \u2191e.symm\nhs : \u2191e '' s =\u1d50[\u03bc] s\nk : \u2115\n\u22a2 \u2191e\u207b\u00b9 \u207b\u00b9' s =\u1d50[\u03bc] s\n\ncase intro.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\n\u03bca \u03bca' : Measure \u03b1\n\u03bcb \u03bcb' : Measure \u03b2\n\u03bcc : Measure \u03b3\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\ne : \u03b1 \u2243 \u03b1\nhe : QuasiMeasurePreserving \u2191e\nhe' : QuasiMeasurePreserving \u2191e.symm\nk : \u2115\nhs : \u2191e\u207b\u00b9 \u207b\u00b9' s =\u1d50[\u03bc] s\n\u22a2 \u2191(e ^ \u2191k).symm \u207b\u00b9' s =\u1d50[\u03bc] s"}, {"tactic": "replace he' : (\u21d1e\u207b\u00b9)^[k] \u207b\u00b9' s =\u1d50[\u03bc] s := he'.preimage_iterate_ae_eq k hs", "annotated_tactic": ["replace he' : (\u21d1e\u207b\u00b9)^[k] \u207b\u00b9' s =\u1d50[\u03bc] s := he'.preimage_iterate_ae_eq k hs", []], "state_before": "case intro.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\n\u03bca \u03bca' : Measure \u03b1\n\u03bcb \u03bcb' : Measure \u03b2\n\u03bcc : Measure \u03b3\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\ne : \u03b1 \u2243 \u03b1\nhe : QuasiMeasurePreserving \u2191e\nhe' : QuasiMeasurePreserving \u2191e.symm\nk : \u2115\nhs : \u2191e\u207b\u00b9 \u207b\u00b9' s =\u1d50[\u03bc] s\n\u22a2 \u2191(e ^ \u2191k).symm \u207b\u00b9' s =\u1d50[\u03bc] s", "state_after": "case intro.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\n\u03bca \u03bca' : Measure \u03b1\n\u03bcb \u03bcb' : Measure \u03b2\n\u03bcc : Measure \u03b3\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\ne : \u03b1 \u2243 \u03b1\nhe : QuasiMeasurePreserving \u2191e\nk : \u2115\nhs : \u2191e\u207b\u00b9 \u207b\u00b9' s =\u1d50[\u03bc] s\nhe' : (\u2191e\u207b\u00b9)^[k] \u207b\u00b9' s =\u1d50[\u03bc] s\n\u22a2 \u2191(e ^ \u2191k).symm \u207b\u00b9' s =\u1d50[\u03bc] s"}, {"tactic": "rwa [Equiv.Perm.iterate_eq_pow e\u207b\u00b9 k, inv_pow e k] at he'", "annotated_tactic": ["rwa [<a>Equiv.Perm.iterate_eq_pow</a> e\u207b\u00b9 k, <a>inv_pow</a> e k] at he'", [{"full_name": "Equiv.Perm.iterate_eq_pow", "def_path": "Mathlib/GroupTheory/Perm/Basic.lean", "def_pos": [107, 15], "def_end_pos": [107, 29]}, {"full_name": "inv_pow", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [317, 9], "def_end_pos": [317, 16]}]], "state_before": "case intro.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\n\u03bca \u03bca' : Measure \u03b1\n\u03bcb \u03bcb' : Measure \u03b2\n\u03bcc : Measure \u03b3\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\ne : \u03b1 \u2243 \u03b1\nhe : QuasiMeasurePreserving \u2191e\nk : \u2115\nhs : \u2191e\u207b\u00b9 \u207b\u00b9' s =\u1d50[\u03bc] s\nhe' : (\u2191e\u207b\u00b9)^[k] \u207b\u00b9' s =\u1d50[\u03bc] s\n\u22a2 \u2191(e ^ \u2191k).symm \u207b\u00b9' s =\u1d50[\u03bc] s", "state_after": "no goals"}, {"tactic": "rwa [Equiv.image_eq_preimage] at hs", "annotated_tactic": ["rwa [<a>Equiv.image_eq_preimage</a>] at hs", [{"full_name": "Equiv.image_eq_preimage", "def_path": "Mathlib/Logic/Equiv/Set.lean", "def_pos": [40, 19], "def_end_pos": [40, 36]}]], "state_before": "case hs\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\n\u03bca \u03bca' : Measure \u03b1\n\u03bcb \u03bcb' : Measure \u03b2\n\u03bcc : Measure \u03b3\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\ne : \u03b1 \u2243 \u03b1\nhe : QuasiMeasurePreserving \u2191e\nhe' : QuasiMeasurePreserving \u2191e.symm\nhs : \u2191e '' s =\u1d50[\u03bc] s\nk : \u2115\n\u22a2 \u2191e\u207b\u00b9 \u207b\u00b9' s =\u1d50[\u03bc] s", "state_after": "no goals"}, {"tactic": "rw [zpow_neg, zpow_ofNat]", "annotated_tactic": ["rw [<a>zpow_neg</a>, <a>zpow_ofNat</a>]", [{"full_name": "zpow_neg", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [332, 9], "def_end_pos": [332, 17]}, {"full_name": "zpow_ofNat", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [948, 9], "def_end_pos": [948, 19]}]], "state_before": "case intro.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\n\u03bca \u03bca' : Measure \u03b1\n\u03bcb \u03bcb' : Measure \u03b2\n\u03bcc : Measure \u03b3\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\ne : \u03b1 \u2243 \u03b1\nhe : QuasiMeasurePreserving \u2191e\nhe' : QuasiMeasurePreserving \u2191e.symm\nhs : \u2191e '' s =\u1d50[\u03bc] s\nk : \u2115\n\u22a2 \u2191(e ^ (-\u2191k)).symm \u207b\u00b9' s =\u1d50[\u03bc] s", "state_after": "case intro.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\n\u03bca \u03bca' : Measure \u03b1\n\u03bcb \u03bcb' : Measure \u03b2\n\u03bcc : Measure \u03b3\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\ne : \u03b1 \u2243 \u03b1\nhe : QuasiMeasurePreserving \u2191e\nhe' : QuasiMeasurePreserving \u2191e.symm\nhs : \u2191e '' s =\u1d50[\u03bc] s\nk : \u2115\n\u22a2 \u2191(e ^ k)\u207b\u00b9.symm \u207b\u00b9' s =\u1d50[\u03bc] s"}, {"tactic": "replace hs : e \u207b\u00b9' s =\u1d50[\u03bc] s", "annotated_tactic": ["replace hs : e \u207b\u00b9' s =\u1d50[\u03bc] s", []], "state_before": "case intro.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\n\u03bca \u03bca' : Measure \u03b1\n\u03bcb \u03bcb' : Measure \u03b2\n\u03bcc : Measure \u03b3\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\ne : \u03b1 \u2243 \u03b1\nhe : QuasiMeasurePreserving \u2191e\nhe' : QuasiMeasurePreserving \u2191e.symm\nhs : \u2191e '' s =\u1d50[\u03bc] s\nk : \u2115\n\u22a2 \u2191(e ^ k)\u207b\u00b9.symm \u207b\u00b9' s =\u1d50[\u03bc] s", "state_after": "case hs\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\n\u03bca \u03bca' : Measure \u03b1\n\u03bcb \u03bcb' : Measure \u03b2\n\u03bcc : Measure \u03b3\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\ne : \u03b1 \u2243 \u03b1\nhe : QuasiMeasurePreserving \u2191e\nhe' : QuasiMeasurePreserving \u2191e.symm\nhs : \u2191e '' s =\u1d50[\u03bc] s\nk : \u2115\n\u22a2 \u2191e \u207b\u00b9' s =\u1d50[\u03bc] s\n\ncase intro.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\n\u03bca \u03bca' : Measure \u03b1\n\u03bcb \u03bcb' : Measure \u03b2\n\u03bcc : Measure \u03b3\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\ne : \u03b1 \u2243 \u03b1\nhe : QuasiMeasurePreserving \u2191e\nhe' : QuasiMeasurePreserving \u2191e.symm\nk : \u2115\nhs : \u2191e \u207b\u00b9' s =\u1d50[\u03bc] s\n\u22a2 \u2191(e ^ k)\u207b\u00b9.symm \u207b\u00b9' s =\u1d50[\u03bc] s"}, {"tactic": "replace he : (\u21d1e)^[k] \u207b\u00b9' s =\u1d50[\u03bc] s := he.preimage_iterate_ae_eq k hs", "annotated_tactic": ["replace he : (\u21d1e)^[k] \u207b\u00b9' s =\u1d50[\u03bc] s := he.preimage_iterate_ae_eq k hs", []], "state_before": "case intro.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\n\u03bca \u03bca' : Measure \u03b1\n\u03bcb \u03bcb' : Measure \u03b2\n\u03bcc : Measure \u03b3\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\ne : \u03b1 \u2243 \u03b1\nhe : QuasiMeasurePreserving \u2191e\nhe' : QuasiMeasurePreserving \u2191e.symm\nk : \u2115\nhs : \u2191e \u207b\u00b9' s =\u1d50[\u03bc] s\n\u22a2 \u2191(e ^ k)\u207b\u00b9.symm \u207b\u00b9' s =\u1d50[\u03bc] s", "state_after": "case intro.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\n\u03bca \u03bca' : Measure \u03b1\n\u03bcb \u03bcb' : Measure \u03b2\n\u03bcc : Measure \u03b3\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\ne : \u03b1 \u2243 \u03b1\nhe' : QuasiMeasurePreserving \u2191e.symm\nk : \u2115\nhs : \u2191e \u207b\u00b9' s =\u1d50[\u03bc] s\nhe : (\u2191e)^[k] \u207b\u00b9' s =\u1d50[\u03bc] s\n\u22a2 \u2191(e ^ k)\u207b\u00b9.symm \u207b\u00b9' s =\u1d50[\u03bc] s"}, {"tactic": "rwa [Equiv.Perm.iterate_eq_pow e k] at he", "annotated_tactic": ["rwa [<a>Equiv.Perm.iterate_eq_pow</a> e k] at he", [{"full_name": "Equiv.Perm.iterate_eq_pow", "def_path": "Mathlib/GroupTheory/Perm/Basic.lean", "def_pos": [107, 15], "def_end_pos": [107, 29]}]], "state_before": "case intro.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\n\u03bca \u03bca' : Measure \u03b1\n\u03bcb \u03bcb' : Measure \u03b2\n\u03bcc : Measure \u03b3\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\ne : \u03b1 \u2243 \u03b1\nhe' : QuasiMeasurePreserving \u2191e.symm\nk : \u2115\nhs : \u2191e \u207b\u00b9' s =\u1d50[\u03bc] s\nhe : (\u2191e)^[k] \u207b\u00b9' s =\u1d50[\u03bc] s\n\u22a2 \u2191(e ^ k)\u207b\u00b9.symm \u207b\u00b9' s =\u1d50[\u03bc] s", "state_after": "no goals"}, {"tactic": "convert he.preimage_ae_eq hs.symm", "annotated_tactic": ["convert he.preimage_ae_eq hs.symm", []], "state_before": "case hs\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\n\u03bca \u03bca' : Measure \u03b1\n\u03bcb \u03bcb' : Measure \u03b2\n\u03bcc : Measure \u03b3\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\ne : \u03b1 \u2243 \u03b1\nhe : QuasiMeasurePreserving \u2191e\nhe' : QuasiMeasurePreserving \u2191e.symm\nhs : \u2191e '' s =\u1d50[\u03bc] s\nk : \u2115\n\u22a2 \u2191e \u207b\u00b9' s =\u1d50[\u03bc] s", "state_after": "case h.e'_5\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\n\u03bca \u03bca' : Measure \u03b1\n\u03bcb \u03bcb' : Measure \u03b2\n\u03bcc : Measure \u03b3\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\ne : \u03b1 \u2243 \u03b1\nhe : QuasiMeasurePreserving \u2191e\nhe' : QuasiMeasurePreserving \u2191e.symm\nhs : \u2191e '' s =\u1d50[\u03bc] s\nk : \u2115\n\u22a2 s = \u2191e \u207b\u00b9' (\u2191e '' s)"}, {"tactic": "rw [Equiv.preimage_image]", "annotated_tactic": ["rw [<a>Equiv.preimage_image</a>]", [{"full_name": "Equiv.preimage_image", "def_path": "Mathlib/Logic/Equiv/Set.lean", "def_pos": [98, 9], "def_end_pos": [98, 23]}]], "state_before": "case h.e'_5\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\n\u03bca \u03bca' : Measure \u03b1\n\u03bcb \u03bcb' : Measure \u03b2\n\u03bcc : Measure \u03b3\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\ne : \u03b1 \u2243 \u03b1\nhe : QuasiMeasurePreserving \u2191e\nhe' : QuasiMeasurePreserving \u2191e.symm\nhs : \u2191e '' s =\u1d50[\u03bc] s\nk : \u2115\n\u22a2 s = \u2191e \u207b\u00b9' (\u2191e '' s)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/ProbabilityMeasure.lean", "full_name": "MeasureTheory.ProbabilityMeasure.coeFn_univ_ne_zero", "start": [149, 1], "end": [150, 61], "traced_tactics": [{"tactic": "simp only [coeFn_univ, Ne.def, one_ne_zero, not_false_iff]", "annotated_tactic": ["simp only [<a>coeFn_univ</a>, <a>Ne.def</a>, <a>one_ne_zero</a>, <a>not_false_iff</a>]", [{"full_name": "MeasureTheory.ProbabilityMeasure.coeFn_univ", "def_path": "Mathlib/MeasureTheory/Measure/ProbabilityMeasure.lean", "def_pos": [145, 9], "def_end_pos": [145, 19]}, {"full_name": "Ne.def", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [59, 9], "def_end_pos": [59, 15]}, {"full_name": "one_ne_zero", "def_path": "Mathlib/Algebra/NeZero.lean", "def_pos": [55, 15], "def_end_pos": [55, 26]}, {"full_name": "not_false_iff", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [82, 9], "def_end_pos": [82, 22]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d : MeasurableSpace \u03a9\n\u03bd : ProbabilityMeasure \u03a9\n\u22a2 (fun s => ENNReal.toNNReal (\u2191\u2191\u2191\u03bd s)) univ \u2260 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Decomposition/SignedHahn.lean", "full_name": "MeasureTheory.SignedMeasure.someExistsOneDivLT_spec", "start": [126, 9], "end": [131, 59], "traced_tactics": [{"tactic": "rw [someExistsOneDivLT, dif_pos hi]", "annotated_tactic": ["rw [<a>someExistsOneDivLT</a>, <a>dif_pos</a> hi]", [{"full_name": "_private.Mathlib.MeasureTheory.Decomposition.SignedHahn.0.MeasureTheory.SignedMeasure.someExistsOneDivLT", "def_path": "Mathlib/MeasureTheory/Decomposition/SignedHahn.lean", "def_pos": [123, 13], "def_end_pos": [123, 31]}, {"full_name": "dif_pos", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [807, 9], "def_end_pos": [807, 16]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : TopologicalSpace M\ninst\u271d : OrderedAddCommMonoid M\ns : SignedMeasure \u03b1\ni j : Set \u03b1\nhi : \u00acrestrict s i \u2264 restrict 0 i\n\u22a2 MeasureTheory.SignedMeasure.someExistsOneDivLT s i \u2286 i \u2227\n    MeasurableSet (MeasureTheory.SignedMeasure.someExistsOneDivLT s i) \u2227\n      1 / (\u2191(MeasureTheory.SignedMeasure.findExistsOneDivLT s i) + 1) <\n        \u2191s (MeasureTheory.SignedMeasure.someExistsOneDivLT s i)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : TopologicalSpace M\ninst\u271d : OrderedAddCommMonoid M\ns : SignedMeasure \u03b1\ni j : Set \u03b1\nhi : \u00acrestrict s i \u2264 restrict 0 i\n\u22a2 Classical.choose\n        (_ : MeasureTheory.SignedMeasure.ExistsOneDivLT s i (MeasureTheory.SignedMeasure.findExistsOneDivLT s i)) \u2286\n      i \u2227\n    MeasurableSet\n        (Classical.choose\n          (_ : MeasureTheory.SignedMeasure.ExistsOneDivLT s i (MeasureTheory.SignedMeasure.findExistsOneDivLT s i))) \u2227\n      1 / (\u2191(MeasureTheory.SignedMeasure.findExistsOneDivLT s i) + 1) <\n        \u2191s\n          (Classical.choose\n            (_ : MeasureTheory.SignedMeasure.ExistsOneDivLT s i (MeasureTheory.SignedMeasure.findExistsOneDivLT s i)))"}, {"tactic": "exact Classical.choose_spec (findExistsOneDivLT_spec hi)", "annotated_tactic": ["exact <a>Classical.choose_spec</a> (<a>findExistsOneDivLT_spec</a> hi)", [{"full_name": "Classical.choose_spec", "def_path": "lake-packages/lean4/src/lean/Init/Classical.lean", "def_pos": [22, 9], "def_end_pos": [22, 20]}, {"full_name": "_private.Mathlib.MeasureTheory.Decomposition.SignedHahn.0.MeasureTheory.SignedMeasure.findExistsOneDivLT_spec", "def_path": "Mathlib/MeasureTheory/Decomposition/SignedHahn.lean", "def_pos": [110, 17], "def_end_pos": [110, 40]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : TopologicalSpace M\ninst\u271d : OrderedAddCommMonoid M\ns : SignedMeasure \u03b1\ni j : Set \u03b1\nhi : \u00acrestrict s i \u2264 restrict 0 i\n\u22a2 Classical.choose\n        (_ : MeasureTheory.SignedMeasure.ExistsOneDivLT s i (MeasureTheory.SignedMeasure.findExistsOneDivLT s i)) \u2286\n      i \u2227\n    MeasurableSet\n        (Classical.choose\n          (_ : MeasureTheory.SignedMeasure.ExistsOneDivLT s i (MeasureTheory.SignedMeasure.findExistsOneDivLT s i))) \u2227\n      1 / (\u2191(MeasureTheory.SignedMeasure.findExistsOneDivLT s i) + 1) <\n        \u2191s\n          (Classical.choose\n            (_ : MeasureTheory.SignedMeasure.ExistsOneDivLT s i (MeasureTheory.SignedMeasure.findExistsOneDivLT s i)))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finmap.lean", "full_name": "Finmap.lookup_erase", "start": [446, 1], "end": [447, 41], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Decomposition/SignedHahn.lean", "full_name": "MeasureTheory.SignedMeasure.existsNatOneDivLTMeasure_of_not_negative", "start": [99, 9], "end": [103, 23], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Lattice.lean", "full_name": "Finset.inf'_biUnion", "start": [987, 1], "end": [990, 34], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/EssSup.lean", "full_name": "essInf_antitone_measure", "start": [203, 1], "end": [205, 29], "traced_tactics": [{"tactic": "refine' liminf_le_liminf_of_le (Measure.ae_le_iff_absolutelyContinuous.mpr h\u03bc\u03bd) _ _", "annotated_tactic": ["refine' <a>liminf_le_liminf_of_le</a> (Measure.ae_le_iff_absolutelyContinuous.mpr h\u03bc\u03bd) _ _", [{"full_name": "Filter.liminf_le_liminf_of_le", "def_path": "Mathlib/Order/LiminfLimsup.lean", "def_pos": [618, 9], "def_end_pos": [618, 31]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : CompleteLattice \u03b2\nf : \u03b1 \u2192 \u03b2\nh\u03bc\u03bd : \u03bc \u226a \u03bd\n\u22a2 essInf f \u03bd \u2264 essInf f \u03bc", "state_after": "case refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : CompleteLattice \u03b2\nf : \u03b1 \u2192 \u03b2\nh\u03bc\u03bd : \u03bc \u226a \u03bd\n\u22a2 IsBoundedUnder (fun x x_1 => x \u2265 x_1) (Measure.ae \u03bd) f\n\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : CompleteLattice \u03b2\nf : \u03b1 \u2192 \u03b2\nh\u03bc\u03bd : \u03bc \u226a \u03bd\n\u22a2 IsCoboundedUnder (fun x x_1 => x \u2265 x_1) (Measure.ae \u03bc) f"}, {"tactic": "all_goals isBoundedDefault", "annotated_tactic": ["all_goals isBoundedDefault", []], "state_before": "case refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : CompleteLattice \u03b2\nf : \u03b1 \u2192 \u03b2\nh\u03bc\u03bd : \u03bc \u226a \u03bd\n\u22a2 IsBoundedUnder (fun x x_1 => x \u2265 x_1) (Measure.ae \u03bd) f\n\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : CompleteLattice \u03b2\nf : \u03b1 \u2192 \u03b2\nh\u03bc\u03bd : \u03bc \u226a \u03bd\n\u22a2 IsCoboundedUnder (fun x x_1 => x \u2265 x_1) (Measure.ae \u03bc) f", "state_after": "no goals"}, {"tactic": "isBoundedDefault", "annotated_tactic": ["isBoundedDefault", []], "state_before": "case refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : CompleteLattice \u03b2\nf : \u03b1 \u2192 \u03b2\nh\u03bc\u03bd : \u03bc \u226a \u03bd\n\u22a2 IsCoboundedUnder (fun x x_1 => x \u2265 x_1) (Measure.ae \u03bc) f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Card.lean", "full_name": "Set.Infinite.encard_eq", "start": [87, 1], "end": [90, 70], "traced_tactics": [{"tactic": "have := h.to_subtype", "annotated_tactic": ["have := h.to_subtype", []], "state_before": "\u03b1 : Type u_1\ns\u271d t s : Set \u03b1\nh : Set.Infinite s\n\u22a2 encard s = \u22a4", "state_after": "\u03b1 : Type u_1\ns\u271d t s : Set \u03b1\nh : Set.Infinite s\nthis : Infinite \u2191s\n\u22a2 encard s = \u22a4"}, {"tactic": "rw [encard, \u2190PartENat.withTopEquiv.symm.injective.eq_iff, Equiv.symm_apply_apply,\n  PartENat.withTopEquiv_symm_top, PartENat.card_eq_top_of_infinite]", "annotated_tactic": ["rw [<a>encard</a>, \u2190PartENat.withTopEquiv.symm.injective.eq_iff, <a>Equiv.symm_apply_apply</a>,\n    <a>PartENat.withTopEquiv_symm_top</a>, <a>PartENat.card_eq_top_of_infinite</a>]", [{"full_name": "Set.encard", "def_path": "Mathlib/Data/Set/Card.lean", "def_pos": [66, 19], "def_end_pos": [66, 25]}, {"full_name": "Equiv.symm_apply_apply", "def_path": "Mathlib/Logic/Equiv/Defs.lean", "def_pos": [283, 17], "def_end_pos": [283, 33]}, {"full_name": "PartENat.withTopEquiv_symm_top", "def_path": "Mathlib/Data/Nat/PartENat.lean", "def_pos": [718, 9], "def_end_pos": [718, 30]}, {"full_name": "PartENat.card_eq_top_of_infinite", "def_path": "Mathlib/SetTheory/Cardinal/Finite.lean", "def_pos": [153, 9], "def_end_pos": [153, 32]}]], "state_before": "\u03b1 : Type u_1\ns\u271d t s : Set \u03b1\nh : Set.Infinite s\nthis : Infinite \u2191s\n\u22a2 encard s = \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/AEEqFun.lean", "full_name": "MeasureTheory.AEEqFun.coeFn_comp\u2082", "start": [385, 1], "end": [388, 17], "traced_tactics": [{"tactic": "rw [comp\u2082_eq_mk]", "annotated_tactic": ["rw [<a>comp\u2082_eq_mk</a>]", [{"full_name": "MeasureTheory.AEEqFun.comp\u2082_eq_mk", "def_path": "Mathlib/MeasureTheory/Function/AEEqFun.lean", "def_pos": [377, 9], "def_end_pos": [377, 20]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : TopologicalSpace \u03b4\ng : \u03b2 \u2192 \u03b3 \u2192 \u03b4\nhg : Continuous (uncurry g)\nf\u2081 : \u03b1 \u2192\u2098[\u03bc] \u03b2\nf\u2082 : \u03b1 \u2192\u2098[\u03bc] \u03b3\n\u22a2 \u2191(comp\u2082 g hg f\u2081 f\u2082) =\u1d50[\u03bc] fun a => g (\u2191f\u2081 a) (\u2191f\u2082 a)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : TopologicalSpace \u03b4\ng : \u03b2 \u2192 \u03b3 \u2192 \u03b4\nhg : Continuous (uncurry g)\nf\u2081 : \u03b1 \u2192\u2098[\u03bc] \u03b2\nf\u2082 : \u03b1 \u2192\u2098[\u03bc] \u03b3\n\u22a2 \u2191(mk (fun a => g (\u2191f\u2081 a) (\u2191f\u2082 a)) (_ : AEStronglyMeasurable (fun x => uncurry g (\u2191f\u2081 x, \u2191f\u2082 x)) \u03bc)) =\u1d50[\u03bc] fun a =>\n    g (\u2191f\u2081 a) (\u2191f\u2082 a)"}, {"tactic": "apply coeFn_mk", "annotated_tactic": ["apply <a>coeFn_mk</a>", [{"full_name": "MeasureTheory.AEEqFun.coeFn_mk", "def_path": "Mathlib/MeasureTheory/Function/AEEqFun.lean", "def_pos": [182, 9], "def_end_pos": [182, 17]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : TopologicalSpace \u03b4\ng : \u03b2 \u2192 \u03b3 \u2192 \u03b4\nhg : Continuous (uncurry g)\nf\u2081 : \u03b1 \u2192\u2098[\u03bc] \u03b2\nf\u2082 : \u03b1 \u2192\u2098[\u03bc] \u03b3\n\u22a2 \u2191(mk (fun a => g (\u2191f\u2081 a) (\u2191f\u2082 a)) (_ : AEStronglyMeasurable (fun x => uncurry g (\u2191f\u2081 x, \u2191f\u2082 x)) \u03bc)) =\u1d50[\u03bc] fun a =>\n    g (\u2191f\u2081 a) (\u2191f\u2082 a)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "full_name": "Int.subNatNat_add_negSucc", "start": [245, 1], "end": [260, 61], "traced_tactics": [{"tactic": "have h := Nat.lt_or_ge m n", "annotated_tactic": ["have h := <a>Nat.lt_or_ge</a> m n", [{"full_name": "Nat.lt_or_ge", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1644, 19], "def_end_pos": [1644, 31]}]], "state_before": "m n k : Nat\n\u22a2 subNatNat m n + -[k+1] = subNatNat m (n + succ k)", "state_after": "m n k : Nat\nh : m < n \u2228 m \u2265 n\n\u22a2 subNatNat m n + -[k+1] = subNatNat m (n + succ k)"}, {"tactic": "cases h with\n| inr h' =>\n  rw [subNatNat_of_le h']\n  simp\n  rw [subNatNat_sub h', Nat.add_comm]\n| inl h' =>\n  have h\u2082 : m < n + succ k := Nat.lt_of_lt_of_le h' (le_add_right _ _)\n  have h\u2083 : m \u2264 n + k := le_of_succ_le_succ h\u2082\n  rw [subNatNat_of_lt h', subNatNat_of_lt h\u2082]\n  simp [Nat.add_comm]\n  rw [\u2190 add_succ, succ_pred_eq_of_pos (Nat.sub_pos_of_lt h'), add_succ, succ_sub h\u2083,\n    Nat.pred_succ]\n  rw [Nat.add_comm n, Nat.add_sub_assoc (Nat.le_of_lt h')]", "annotated_tactic": ["cases h with\n  | <a>inr</a> h' =>\n    rw [<a>subNatNat_of_le</a> h']\n    simp\n    rw [<a>subNatNat_sub</a> h', <a>Nat.add_comm</a>]\n  | <a>inl</a> h' =>\n    have h\u2082 : m < n + <a>succ</a> k := <a>Nat.lt_of_lt_of_le</a> h' (<a>le_add_right</a> _ _)\n    have h\u2083 : m \u2264 n + k := <a>le_of_succ_le_succ</a> h\u2082\n    rw [<a>subNatNat_of_lt</a> h', <a>subNatNat_of_lt</a> h\u2082]\n    simp [<a>Nat.add_comm</a>]\n    rw [\u2190 <a>add_succ</a>, <a>succ_pred_eq_of_pos</a> (<a>Nat.sub_pos_of_lt</a> h'), <a>add_succ</a>, <a>succ_sub</a> h\u2083,\n      <a>Nat.pred_succ</a>]\n    rw [<a>Nat.add_comm</a> n, <a>Nat.add_sub_assoc</a> (<a>Nat.le_of_lt</a> h')]", [{"full_name": "Or.inr", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [519, 5], "def_end_pos": [519, 8]}, {"full_name": "Int.subNatNat_of_le", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [130, 9], "def_end_pos": [130, 24]}, {"full_name": "Int.subNatNat_sub", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [233, 9], "def_end_pos": [233, 22]}, {"full_name": "Nat.add_comm", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [131, 19], "def_end_pos": [131, 27]}, {"full_name": "Or.inl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [517, 5], "def_end_pos": [517, 8]}, {"full_name": "Nat.succ", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1044, 5], "def_end_pos": [1044, 9]}, {"full_name": "Nat.lt_of_lt_of_le", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [259, 19], "def_end_pos": [259, 33]}, {"full_name": "Nat.le_add_right", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [340, 9], "def_end_pos": [340, 21]}, {"full_name": "Nat.le_of_succ_le_succ", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1628, 9], "def_end_pos": [1628, 31]}, {"full_name": "Int.subNatNat_of_lt", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [133, 9], "def_end_pos": [133, 24]}, {"full_name": "Int.subNatNat_of_lt", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [133, 9], "def_end_pos": [133, 24]}, {"full_name": "Nat.add_comm", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [131, 19], "def_end_pos": [131, 27]}, {"full_name": "Nat.add_succ", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [122, 9], "def_end_pos": [122, 17]}, {"full_name": "Nat.succ_pred_eq_of_pos", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [232, 9], "def_end_pos": [232, 28]}, {"full_name": "Nat.sub_pos_of_lt", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [404, 19], "def_end_pos": [404, 32]}, {"full_name": "Nat.add_succ", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [122, 9], "def_end_pos": [122, 17]}, {"full_name": "Nat.succ_sub", "def_path": "lake-packages/std/Std/Data/Nat/Init/Lemmas.lean", "def_pos": [10, 9], "def_end_pos": [10, 17]}, {"full_name": "Nat.pred_succ", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [686, 27], "def_end_pos": [686, 36]}, {"full_name": "Nat.add_comm", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [131, 19], "def_end_pos": [131, 27]}, {"full_name": "Nat.add_sub_assoc", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [602, 19], "def_end_pos": [602, 32]}, {"full_name": "Nat.le_of_lt", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [283, 19], "def_end_pos": [283, 27]}]], "state_before": "m n k : Nat\nh : m < n \u2228 m \u2265 n\n\u22a2 subNatNat m n + -[k+1] = subNatNat m (n + succ k)", "state_after": "no goals"}, {"tactic": "rw [subNatNat_of_le h']", "annotated_tactic": ["rw [<a>subNatNat_of_le</a> h']", [{"full_name": "Int.subNatNat_of_le", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [130, 9], "def_end_pos": [130, 24]}]], "state_before": "case inr\nm n k : Nat\nh' : m \u2265 n\n\u22a2 subNatNat m n + -[k+1] = subNatNat m (n + succ k)", "state_after": "case inr\nm n k : Nat\nh' : m \u2265 n\n\u22a2 \u2191(m - n) + -[k+1] = subNatNat m (n + succ k)"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case inr\nm n k : Nat\nh' : m \u2265 n\n\u22a2 \u2191(m - n) + -[k+1] = subNatNat m (n + succ k)", "state_after": "case inr\nm n k : Nat\nh' : m \u2265 n\n\u22a2 subNatNat (m - n) (succ k) = subNatNat m (n + succ k)"}, {"tactic": "rw [subNatNat_sub h', Nat.add_comm]", "annotated_tactic": ["rw [<a>subNatNat_sub</a> h', <a>Nat.add_comm</a>]", [{"full_name": "Int.subNatNat_sub", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [233, 9], "def_end_pos": [233, 22]}, {"full_name": "Nat.add_comm", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [131, 19], "def_end_pos": [131, 27]}]], "state_before": "case inr\nm n k : Nat\nh' : m \u2265 n\n\u22a2 subNatNat (m - n) (succ k) = subNatNat m (n + succ k)", "state_after": "no goals"}, {"tactic": "have h\u2082 : m < n + succ k := Nat.lt_of_lt_of_le h' (le_add_right _ _)", "annotated_tactic": ["have h\u2082 : m < n + <a>succ</a> k := <a>Nat.lt_of_lt_of_le</a> h' (<a>le_add_right</a> _ _)", [{"full_name": "Nat.succ", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1044, 5], "def_end_pos": [1044, 9]}, {"full_name": "Nat.lt_of_lt_of_le", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [259, 19], "def_end_pos": [259, 33]}, {"full_name": "Nat.le_add_right", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [340, 9], "def_end_pos": [340, 21]}]], "state_before": "case inl\nm n k : Nat\nh' : m < n\n\u22a2 subNatNat m n + -[k+1] = subNatNat m (n + succ k)", "state_after": "case inl\nm n k : Nat\nh' : m < n\nh\u2082 : m < n + succ k\n\u22a2 subNatNat m n + -[k+1] = subNatNat m (n + succ k)"}, {"tactic": "have h\u2083 : m \u2264 n + k := le_of_succ_le_succ h\u2082", "annotated_tactic": ["have h\u2083 : m \u2264 n + k := <a>le_of_succ_le_succ</a> h\u2082", [{"full_name": "Nat.le_of_succ_le_succ", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1628, 9], "def_end_pos": [1628, 31]}]], "state_before": "case inl\nm n k : Nat\nh' : m < n\nh\u2082 : m < n + succ k\n\u22a2 subNatNat m n + -[k+1] = subNatNat m (n + succ k)", "state_after": "case inl\nm n k : Nat\nh' : m < n\nh\u2082 : m < n + succ k\nh\u2083 : m \u2264 n + k\n\u22a2 subNatNat m n + -[k+1] = subNatNat m (n + succ k)"}, {"tactic": "rw [subNatNat_of_lt h', subNatNat_of_lt h\u2082]", "annotated_tactic": ["rw [<a>subNatNat_of_lt</a> h', <a>subNatNat_of_lt</a> h\u2082]", [{"full_name": "Int.subNatNat_of_lt", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [133, 9], "def_end_pos": [133, 24]}, {"full_name": "Int.subNatNat_of_lt", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [133, 9], "def_end_pos": [133, 24]}]], "state_before": "case inl\nm n k : Nat\nh' : m < n\nh\u2082 : m < n + succ k\nh\u2083 : m \u2264 n + k\n\u22a2 subNatNat m n + -[k+1] = subNatNat m (n + succ k)", "state_after": "case inl\nm n k : Nat\nh' : m < n\nh\u2082 : m < n + succ k\nh\u2083 : m \u2264 n + k\n\u22a2 -[pred (n - m)+1] + -[k+1] = -[pred (n + succ k - m)+1]"}, {"tactic": "simp [Nat.add_comm]", "annotated_tactic": ["simp [<a>Nat.add_comm</a>]", [{"full_name": "Nat.add_comm", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [131, 19], "def_end_pos": [131, 27]}]], "state_before": "case inl\nm n k : Nat\nh' : m < n\nh\u2082 : m < n + succ k\nh\u2083 : m \u2264 n + k\n\u22a2 -[pred (n - m)+1] + -[k+1] = -[pred (n + succ k - m)+1]", "state_after": "case inl\nm n k : Nat\nh' : m < n\nh\u2082 : m < n + succ k\nh\u2083 : m \u2264 n + k\n\u22a2 succ (k + pred (n - m)) = pred (n + succ k - m)"}, {"tactic": "rw [\u2190 add_succ, succ_pred_eq_of_pos (Nat.sub_pos_of_lt h'), add_succ, succ_sub h\u2083,\n  Nat.pred_succ]", "annotated_tactic": ["rw [\u2190 <a>add_succ</a>, <a>succ_pred_eq_of_pos</a> (<a>Nat.sub_pos_of_lt</a> h'), <a>add_succ</a>, <a>succ_sub</a> h\u2083,\n      <a>Nat.pred_succ</a>]", [{"full_name": "Nat.add_succ", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [122, 9], "def_end_pos": [122, 17]}, {"full_name": "Nat.succ_pred_eq_of_pos", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [232, 9], "def_end_pos": [232, 28]}, {"full_name": "Nat.sub_pos_of_lt", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [404, 19], "def_end_pos": [404, 32]}, {"full_name": "Nat.add_succ", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [122, 9], "def_end_pos": [122, 17]}, {"full_name": "Nat.succ_sub", "def_path": "lake-packages/std/Std/Data/Nat/Init/Lemmas.lean", "def_pos": [10, 9], "def_end_pos": [10, 17]}, {"full_name": "Nat.pred_succ", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [686, 27], "def_end_pos": [686, 36]}]], "state_before": "case inl\nm n k : Nat\nh' : m < n\nh\u2082 : m < n + succ k\nh\u2083 : m \u2264 n + k\n\u22a2 succ (k + pred (n - m)) = pred (n + succ k - m)", "state_after": "case inl\nm n k : Nat\nh' : m < n\nh\u2082 : m < n + succ k\nh\u2083 : m \u2264 n + k\n\u22a2 k + (n - m) = n + k - m"}, {"tactic": "rw [Nat.add_comm n, Nat.add_sub_assoc (Nat.le_of_lt h')]", "annotated_tactic": ["rw [<a>Nat.add_comm</a> n, <a>Nat.add_sub_assoc</a> (<a>Nat.le_of_lt</a> h')]", [{"full_name": "Nat.add_comm", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [131, 19], "def_end_pos": [131, 27]}, {"full_name": "Nat.add_sub_assoc", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [602, 19], "def_end_pos": [602, 32]}, {"full_name": "Nat.le_of_lt", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [283, 19], "def_end_pos": [283, 27]}]], "state_before": "case inl\nm n k : Nat\nh' : m < n\nh\u2082 : m < n + succ k\nh\u2083 : m \u2264 n + k\n\u22a2 k + (n - m) = n + k - m", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Finite.lean", "full_name": "Set.iInter_iUnion_of_monotone", "start": [1541, 1], "end": [1544, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Function.lean", "full_name": "Set.RightInvOn.mapsTo", "start": [1145, 1], "end": [1146, 24], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/Egorov.lean", "full_name": "MeasureTheory.tendstoUniformlyOn_of_ae_tendsto'", "start": [213, 1], "end": [220, 31], "traced_tactics": [{"tactic": "have \u27e8t, _, ht, htendsto\u27e9 := tendstoUniformlyOn_of_ae_tendsto hf hg MeasurableSet.univ\n  (measure_ne_top \u03bc Set.univ) (by filter_upwards [hfg] with _ htendsto _ using htendsto) h\u03b5", "annotated_tactic": ["have \u27e8t, _, ht, htendsto\u27e9 := <a>tendstoUniformlyOn_of_ae_tendsto</a> hf hg <a>MeasurableSet.univ</a>\n    (<a>measure_ne_top</a> \u03bc <a>Set.univ</a>) (by filter_upwards [hfg] with _ htendsto _ using htendsto) h\u03b5", [{"full_name": "MeasureTheory.tendstoUniformlyOn_of_ae_tendsto", "def_path": "Mathlib/MeasureTheory/Function/Egorov.lean", "def_pos": [200, 9], "def_end_pos": [200, 41]}, {"full_name": "MeasurableSet.univ", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [101, 19], "def_end_pos": [101, 37]}, {"full_name": "MeasureTheory.measure_ne_top", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2875, 9], "def_end_pos": [2875, 23]}, {"full_name": "Set.univ", "def_path": "Mathlib/Init/Set.lean", "def_pos": [90, 5], "def_end_pos": [90, 9]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u2075 : MetricSpace \u03b2\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : SemilatticeSup \u03b9\ninst\u271d\u00b3 : Nonempty \u03b9\ninst\u271d\u00b2 : Countable \u03b9\n\u03b3 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace \u03b3\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\ns : Set \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nhf : \u2200 (n : \u03b9), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u22a2 \u2203 t, MeasurableSet t \u2227 \u2191\u2191\u03bc t \u2264 ENNReal.ofReal \u03b5 \u2227 TendstoUniformlyOn f g atTop t\u1d9c", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u2075 : MetricSpace \u03b2\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : SemilatticeSup \u03b9\ninst\u271d\u00b3 : Nonempty \u03b9\ninst\u271d\u00b2 : Countable \u03b9\n\u03b3 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace \u03b3\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\ns : Set \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nhf : \u2200 (n : \u03b9), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nt : Set \u03b1\nw\u271d : t \u2286 univ\nht : MeasurableSet t\nhtendsto : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal \u03b5 \u2227 TendstoUniformlyOn (fun n => f n) g atTop (univ \\ t)\n\u22a2 \u2203 t, MeasurableSet t \u2227 \u2191\u2191\u03bc t \u2264 ENNReal.ofReal \u03b5 \u2227 TendstoUniformlyOn f g atTop t\u1d9c"}, {"tactic": "refine' \u27e8_, ht, _\u27e9", "annotated_tactic": ["refine' \u27e8_, ht, _\u27e9", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u2075 : MetricSpace \u03b2\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : SemilatticeSup \u03b9\ninst\u271d\u00b3 : Nonempty \u03b9\ninst\u271d\u00b2 : Countable \u03b9\n\u03b3 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace \u03b3\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\ns : Set \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nhf : \u2200 (n : \u03b9), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nt : Set \u03b1\nw\u271d : t \u2286 univ\nht : MeasurableSet t\nhtendsto : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal \u03b5 \u2227 TendstoUniformlyOn (fun n => f n) g atTop (univ \\ t)\n\u22a2 \u2203 t, MeasurableSet t \u2227 \u2191\u2191\u03bc t \u2264 ENNReal.ofReal \u03b5 \u2227 TendstoUniformlyOn f g atTop t\u1d9c", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u2075 : MetricSpace \u03b2\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : SemilatticeSup \u03b9\ninst\u271d\u00b3 : Nonempty \u03b9\ninst\u271d\u00b2 : Countable \u03b9\n\u03b3 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace \u03b3\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\ns : Set \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nhf : \u2200 (n : \u03b9), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nt : Set \u03b1\nw\u271d : t \u2286 univ\nht : MeasurableSet t\nhtendsto : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal \u03b5 \u2227 TendstoUniformlyOn (fun n => f n) g atTop (univ \\ t)\n\u22a2 \u2191\u2191\u03bc t \u2264 ENNReal.ofReal \u03b5 \u2227 TendstoUniformlyOn f g atTop t\u1d9c"}, {"tactic": "rwa [Set.compl_eq_univ_diff]", "annotated_tactic": ["rwa [<a>Set.compl_eq_univ_diff</a>]", [{"full_name": "Set.compl_eq_univ_diff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1916, 9], "def_end_pos": [1916, 27]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u2075 : MetricSpace \u03b2\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : SemilatticeSup \u03b9\ninst\u271d\u00b3 : Nonempty \u03b9\ninst\u271d\u00b2 : Countable \u03b9\n\u03b3 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace \u03b3\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\ns : Set \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nhf : \u2200 (n : \u03b9), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nt : Set \u03b1\nw\u271d : t \u2286 univ\nht : MeasurableSet t\nhtendsto : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal \u03b5 \u2227 TendstoUniformlyOn (fun n => f n) g atTop (univ \\ t)\n\u22a2 \u2191\u2191\u03bc t \u2264 ENNReal.ofReal \u03b5 \u2227 TendstoUniformlyOn f g atTop t\u1d9c", "state_after": "no goals"}, {"tactic": "filter_upwards [hfg] with _ htendsto _ using htendsto", "annotated_tactic": ["filter_upwards [hfg] with _ htendsto _ using htendsto", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u2075 : MetricSpace \u03b2\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : SemilatticeSup \u03b9\ninst\u271d\u00b3 : Nonempty \u03b9\ninst\u271d\u00b2 : Countable \u03b9\n\u03b3 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace \u03b3\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\ns : Set \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nhf : \u2200 (n : \u03b9), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 univ \u2192 Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/NoncommProd.lean", "full_name": "Multiset.noncommFoldr_coe", "start": [52, 1], "end": [56, 41], "traced_tactics": [{"tactic": "simp only [noncommFoldr, coe_foldr, coe_attach, List.attach, Function.comp]", "annotated_tactic": ["simp only [<a>noncommFoldr</a>, <a>coe_foldr</a>, <a>coe_attach</a>, <a>List.attach</a>, <a>Function.comp</a>]", [{"full_name": "Multiset.noncommFoldr", "def_path": "Mathlib/Data/Finset/NoncommProd.lean", "def_pos": [42, 5], "def_end_pos": [42, 17]}, {"full_name": "Multiset.coe_foldr", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [1408, 9], "def_end_pos": [1408, 18]}, {"full_name": "Multiset.coe_attach", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [1506, 9], "def_end_pos": [1506, 19]}, {"full_name": "List.attach", "def_path": "lake-packages/std/Std/Data/List/Basic.lean", "def_pos": [767, 34], "def_end_pos": [767, 40]}, {"full_name": "Function.comp", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [52, 15], "def_end_pos": [52, 28]}]], "state_before": "F : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\nl : List \u03b1\ncomm : Set.Pairwise {x | x \u2208 \u2191l} fun x y => \u2200 (b : \u03b2), f x (f y b) = f y (f x b)\nb : \u03b2\n\u22a2 noncommFoldr f (\u2191l) comm b = List.foldr f b l", "state_after": "F : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\nl : List \u03b1\ncomm : Set.Pairwise {x | x \u2208 \u2191l} fun x y => \u2200 (b : \u03b2), f x (f y b) = f y (f x b)\nb : \u03b2\n\u22a2 List.foldr (fun x => f \u2191x) b (List.pmap Subtype.mk l (_ : \u2200 (x : \u03b1), x \u2208 l \u2192 x \u2208 l)) = List.foldr f b l"}, {"tactic": "rw [\u2190 List.foldr_map]", "annotated_tactic": ["rw [\u2190 <a>List.foldr_map</a>]", [{"full_name": "List.foldr_map", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [1583, 9], "def_end_pos": [1583, 18]}]], "state_before": "F : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\nl : List \u03b1\ncomm : Set.Pairwise {x | x \u2208 \u2191l} fun x y => \u2200 (b : \u03b2), f x (f y b) = f y (f x b)\nb : \u03b2\n\u22a2 List.foldr (fun x => f \u2191x) b (List.pmap Subtype.mk l (_ : \u2200 (x : \u03b1), x \u2208 l \u2192 x \u2208 l)) = List.foldr f b l", "state_after": "F : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\nl : List \u03b1\ncomm : Set.Pairwise {x | x \u2208 \u2191l} fun x y => \u2200 (b : \u03b2), f x (f y b) = f y (f x b)\nb : \u03b2\n\u22a2 List.foldr f b (List.map (fun x => \u2191x) (List.pmap Subtype.mk l (_ : \u2200 (x : \u03b1), x \u2208 l \u2192 x \u2208 l))) = List.foldr f b l"}, {"tactic": "simp [List.map_pmap, List.pmap_eq_map]", "annotated_tactic": ["simp [<a>List.map_pmap</a>, <a>List.pmap_eq_map</a>]", [{"full_name": "List.map_pmap", "def_path": "Mathlib/Data/List/Basic.lean", "def_pos": [3013, 9], "def_end_pos": [3013, 17]}, {"full_name": "List.pmap_eq_map", "def_path": "Mathlib/Data/List/Basic.lean", "def_pos": [3001, 9], "def_end_pos": [3001, 20]}]], "state_before": "F : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\nl : List \u03b1\ncomm : Set.Pairwise {x | x \u2208 \u2191l} fun x y => \u2200 (b : \u03b2), f x (f y b) = f y (f x b)\nb : \u03b2\n\u22a2 List.foldr f b (List.map (fun x => \u2191x) (List.pmap Subtype.mk l (_ : \u2200 (x : \u03b1), x \u2208 l \u2192 x \u2208 l))) = List.foldr f b l", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Card.lean", "full_name": "Set.exists_ne_of_one_lt_encard", "start": [309, 1], "end": [313, 26], "traced_tactics": [{"tactic": "by_contra' h'", "annotated_tactic": ["by_contra' h'", []], "state_before": "\u03b1 : Type u_1\ns t : Set \u03b1\nh : 1 < encard s\na : \u03b1\n\u22a2 \u2203 b, b \u2208 s \u2227 b \u2260 a", "state_after": "\u03b1 : Type u_1\ns t : Set \u03b1\nh : 1 < encard s\na : \u03b1\nh' : \u2200 (b : \u03b1), b \u2208 s \u2192 b = a\n\u22a2 False"}, {"tactic": "obtain \u27e8b, b', hb, hb', hne\u27e9 := one_lt_encard_iff.1 h", "annotated_tactic": ["obtain \u27e8b, b', hb, hb', hne\u27e9 := <a>one_lt_encard_iff</a>.1 h", [{"full_name": "Set.one_lt_encard_iff", "def_path": "Mathlib/Data/Set/Card.lean", "def_pos": [306, 9], "def_end_pos": [306, 26]}]], "state_before": "\u03b1 : Type u_1\ns t : Set \u03b1\nh : 1 < encard s\na : \u03b1\nh' : \u2200 (b : \u03b1), b \u2208 s \u2192 b = a\n\u22a2 False", "state_after": "case intro.intro.intro.intro\n\u03b1 : Type u_1\ns t : Set \u03b1\nh : 1 < encard s\na : \u03b1\nh' : \u2200 (b : \u03b1), b \u2208 s \u2192 b = a\nb b' : \u03b1\nhb : b \u2208 s\nhb' : b' \u2208 s\nhne : b \u2260 b'\n\u22a2 False"}, {"tactic": "apply hne", "annotated_tactic": ["apply hne", []], "state_before": "case intro.intro.intro.intro\n\u03b1 : Type u_1\ns t : Set \u03b1\nh : 1 < encard s\na : \u03b1\nh' : \u2200 (b : \u03b1), b \u2208 s \u2192 b = a\nb b' : \u03b1\nhb : b \u2208 s\nhb' : b' \u2208 s\nhne : b \u2260 b'\n\u22a2 False", "state_after": "case intro.intro.intro.intro\n\u03b1 : Type u_1\ns t : Set \u03b1\nh : 1 < encard s\na : \u03b1\nh' : \u2200 (b : \u03b1), b \u2208 s \u2192 b = a\nb b' : \u03b1\nhb : b \u2208 s\nhb' : b' \u2208 s\nhne : b \u2260 b'\n\u22a2 b = b'"}, {"tactic": "rw [h' b hb, h' b' hb']", "annotated_tactic": ["rw [h' b hb, h' b' hb']", []], "state_before": "case intro.intro.intro.intro\n\u03b1 : Type u_1\ns t : Set \u03b1\nh : 1 < encard s\na : \u03b1\nh' : \u2200 (b : \u03b1), b \u2208 s \u2192 b = a\nb b' : \u03b1\nhb : b \u2208 s\nhb' : b' \u2208 s\nhne : b \u2260 b'\n\u22a2 b = b'", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/LocallyFinite.lean", "full_name": "Finset.Ico_union_Ico_eq_Ico", "start": [794, 1], "end": [796, 89], "traced_tactics": [{"tactic": "rw [\u2190 coe_inj, coe_union, coe_Ico, coe_Ico, coe_Ico, Set.Ico_union_Ico_eq_Ico hab hbc]", "annotated_tactic": ["rw [\u2190 <a>coe_inj</a>, <a>coe_union</a>, <a>coe_Ico</a>, <a>coe_Ico</a>, <a>coe_Ico</a>, <a>Set.Ico_union_Ico_eq_Ico</a> hab hbc]", [{"full_name": "Finset.coe_inj", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [244, 9], "def_end_pos": [244, 16]}, {"full_name": "Finset.coe_union", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1399, 9], "def_end_pos": [1399, 18]}, {"full_name": "Finset.coe_Ico", "def_path": "Mathlib/Order/LocallyFinite.lean", "def_pos": [351, 9], "def_end_pos": [351, 16]}, {"full_name": "Finset.coe_Ico", "def_path": "Mathlib/Order/LocallyFinite.lean", "def_pos": [351, 9], "def_end_pos": [351, 16]}, {"full_name": "Finset.coe_Ico", "def_path": "Mathlib/Order/LocallyFinite.lean", "def_pos": [351, 9], "def_end_pos": [351, 16]}, {"full_name": "Set.Ico_union_Ico_eq_Ico", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [1529, 9], "def_end_pos": [1529, 29]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na\u271d b\u271d a b c : \u03b1\nhab : a \u2264 b\nhbc : b \u2264 c\n\u22a2 Ico a b \u222a Ico b c = Ico a c", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/EssSup.lean", "full_name": "ENNReal.coe_essSup", "start": [332, 1], "end": [336, 82], "traced_tactics": [{"tactic": "simp [essSup, limsup, limsSup, eventually_map, ENNReal.forall_ennreal]", "annotated_tactic": ["simp [<a>essSup</a>, <a>limsup</a>, <a>limsSup</a>, <a>eventually_map</a>, <a>ENNReal.forall_ennreal</a>]", [{"full_name": "essSup", "def_path": "Mathlib/MeasureTheory/Function/EssSup.lean", "def_pos": [44, 5], "def_end_pos": [44, 11]}, {"full_name": "Filter.limsup", "def_path": "Mathlib/Order/LiminfLimsup.lean", "def_pos": [420, 5], "def_end_pos": [420, 11]}, {"full_name": "Filter.limsSup", "def_path": "Mathlib/Order/LiminfLimsup.lean", "def_pos": [405, 5], "def_end_pos": [405, 12]}, {"full_name": "Filter.eventually_map", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1845, 9], "def_end_pos": [1845, 23]}, {"full_name": "ENNReal.forall_ennreal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [255, 9], "def_end_pos": [255, 23]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf\u271d : \u03b1 \u2192 \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\nhf : IsBoundedUnder (fun x x_1 => x \u2264 x_1) (Measure.ae \u03bc) f\nr : \u211d\u22650\u221e\n\u22a2 r \u2264 \u2a05 a \u2208 fun x => sets (map f (Measure.ae \u03bc)) {x_1 | (fun x_2 => (fun x x_3 => x \u2264 x_3) x_2 x) x_1}, \u2191a \u2194\n    r \u2264 essSup (fun x => \u2191(f x)) \u03bc", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf\u271d : \u03b1 \u2192 \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\nhf : IsBoundedUnder (fun x x_1 => x \u2264 x_1) (Measure.ae \u03bc) f\nr : \u211d\u22650\u221e\n\u22a2 (\u2200 (i : \u211d\u22650), (i \u2208 fun x => sets (map f (Measure.ae \u03bc)) {x_1 | x_1 \u2264 x}) \u2192 r \u2264 \u2191i) \u2194\n    \u2200 (r_1 : \u211d\u22650), (\u2200\u1d50 (a : \u03b1) \u2202\u03bc, f a \u2264 r_1) \u2192 r \u2264 \u2191r_1"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf\u271d : \u03b1 \u2192 \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\nhf : IsBoundedUnder (fun x x_1 => x \u2264 x_1) (Measure.ae \u03bc) f\nr : \u211d\u22650\u221e\n\u22a2 (\u2200 (i : \u211d\u22650), (i \u2208 fun x => sets (map f (Measure.ae \u03bc)) {x_1 | x_1 \u2264 x}) \u2192 r \u2264 \u2191i) \u2194\n    \u2200 (r_1 : \u211d\u22650), (\u2200\u1d50 (a : \u03b1) \u2202\u03bc, f a \u2264 r_1) \u2192 r \u2264 \u2191r_1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Martingale/Basic.lean", "full_name": "MeasureTheory.Supermartingale.smul_nonneg", "start": [350, 1], "end": [356, 38], "traced_tactics": [{"tactic": "refine' \u27e8hf.1.smul c, fun i j hij => _, fun i => (hf.2.2 i).smul c\u27e9", "annotated_tactic": ["refine' \u27e8hf.1.<a>smul</a> c, fun i j hij => _, fun i => (hf.2.2 i).<a>smul</a> c\u27e9", [{"full_name": "MeasureTheory.Adapted.smul", "def_path": "Mathlib/Probability/Process/Adapted.lean", "def_pos": [73, 19], "def_end_pos": [73, 23]}, {"full_name": "MeasureTheory.Integrable.smul", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [1071, 9], "def_end_pos": [1071, 24]}]], "state_before": "\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2077 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\nF : Type u_4\ninst\u271d\u00b3 : NormedLatticeAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : OrderedSMul \u211d F\nf : \u03b9 \u2192 \u03a9 \u2192 F\nc : \u211d\nhc : 0 \u2264 c\nhf : Supermartingale f \u2131 \u03bc\n\u22a2 Supermartingale (c \u2022 f) \u2131 \u03bc", "state_after": "\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2077 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\nF : Type u_4\ninst\u271d\u00b3 : NormedLatticeAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : OrderedSMul \u211d F\nf : \u03b9 \u2192 \u03a9 \u2192 F\nc : \u211d\nhc : 0 \u2264 c\nhf : Supermartingale f \u2131 \u03bc\ni j : \u03b9\nhij : i \u2264 j\n\u22a2 \u03bc[(c \u2022 f) j|\u2191\u2131 i] \u2264\u1d50[\u03bc] (c \u2022 f) i"}, {"tactic": "refine' (condexp_smul c (f j)).le.trans _", "annotated_tactic": ["refine' (<a>condexp_smul</a> c (f j)).le.trans _", [{"full_name": "MeasureTheory.condexp_smul", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean", "def_pos": [303, 9], "def_end_pos": [303, 21]}]], "state_before": "\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2077 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\nF : Type u_4\ninst\u271d\u00b3 : NormedLatticeAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : OrderedSMul \u211d F\nf : \u03b9 \u2192 \u03a9 \u2192 F\nc : \u211d\nhc : 0 \u2264 c\nhf : Supermartingale f \u2131 \u03bc\ni j : \u03b9\nhij : i \u2264 j\n\u22a2 \u03bc[(c \u2022 f) j|\u2191\u2131 i] \u2264\u1d50[\u03bc] (c \u2022 f) i", "state_after": "\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2077 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\nF : Type u_4\ninst\u271d\u00b3 : NormedLatticeAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : OrderedSMul \u211d F\nf : \u03b9 \u2192 \u03a9 \u2192 F\nc : \u211d\nhc : 0 \u2264 c\nhf : Supermartingale f \u2131 \u03bc\ni j : \u03b9\nhij : i \u2264 j\n\u22a2 c \u2022 \u03bc[f j|\u2191\u2131 i] \u2264\u1d50[\u03bc] (c \u2022 f) i"}, {"tactic": "filter_upwards [hf.2.1 i j hij] with _ hle", "annotated_tactic": ["filter_upwards [hf.2.1 i j hij] with _ hle", []], "state_before": "\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2077 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\nF : Type u_4\ninst\u271d\u00b3 : NormedLatticeAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : OrderedSMul \u211d F\nf : \u03b9 \u2192 \u03a9 \u2192 F\nc : \u211d\nhc : 0 \u2264 c\nhf : Supermartingale f \u2131 \u03bc\ni j : \u03b9\nhij : i \u2264 j\n\u22a2 c \u2022 \u03bc[f j|\u2191\u2131 i] \u2264\u1d50[\u03bc] (c \u2022 f) i", "state_after": "case h\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2077 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\nF : Type u_4\ninst\u271d\u00b3 : NormedLatticeAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : OrderedSMul \u211d F\nf : \u03b9 \u2192 \u03a9 \u2192 F\nc : \u211d\nhc : 0 \u2264 c\nhf : Supermartingale f \u2131 \u03bc\ni j : \u03b9\nhij : i \u2264 j\na\u271d : \u03a9\nhle : (\u03bc[f j|\u2191\u2131 i]) a\u271d \u2264 f i a\u271d\n\u22a2 (c \u2022 \u03bc[f j|\u2191\u2131 i]) a\u271d \u2264 (c \u2022 f) i a\u271d"}, {"tactic": "simp_rw [Pi.smul_apply]", "annotated_tactic": ["simp_rw [<a>Pi.smul_apply</a>]", [{"full_name": "Pi.smul_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [116, 60], "def_end_pos": [116, 70]}]], "state_before": "case h\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2077 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\nF : Type u_4\ninst\u271d\u00b3 : NormedLatticeAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : OrderedSMul \u211d F\nf : \u03b9 \u2192 \u03a9 \u2192 F\nc : \u211d\nhc : 0 \u2264 c\nhf : Supermartingale f \u2131 \u03bc\ni j : \u03b9\nhij : i \u2264 j\na\u271d : \u03a9\nhle : (\u03bc[f j|\u2191\u2131 i]) a\u271d \u2264 f i a\u271d\n\u22a2 (c \u2022 \u03bc[f j|\u2191\u2131 i]) a\u271d \u2264 (c \u2022 f) i a\u271d", "state_after": "case h\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2077 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\nF : Type u_4\ninst\u271d\u00b3 : NormedLatticeAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : OrderedSMul \u211d F\nf : \u03b9 \u2192 \u03a9 \u2192 F\nc : \u211d\nhc : 0 \u2264 c\nhf : Supermartingale f \u2131 \u03bc\ni j : \u03b9\nhij : i \u2264 j\na\u271d : \u03a9\nhle : (\u03bc[f j|\u2191\u2131 i]) a\u271d \u2264 f i a\u271d\n\u22a2 c \u2022 (\u03bc[f j|\u2191\u2131 i]) a\u271d \u2264 c \u2022 f i a\u271d"}, {"tactic": "exact smul_le_smul_of_nonneg hle hc", "annotated_tactic": ["exact <a>smul_le_smul_of_nonneg</a> hle hc", [{"full_name": "smul_le_smul_of_nonneg", "def_path": "Mathlib/Algebra/Order/SMul.lean", "def_pos": [94, 19], "def_end_pos": [94, 41]}]], "state_before": "case h\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2077 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\nF : Type u_4\ninst\u271d\u00b3 : NormedLatticeAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : OrderedSMul \u211d F\nf : \u03b9 \u2192 \u03a9 \u2192 F\nc : \u211d\nhc : 0 \u2264 c\nhf : Supermartingale f \u2131 \u03bc\ni j : \u03b9\nhij : i \u2264 j\na\u271d : \u03a9\nhle : (\u03bc[f j|\u2191\u2131 i]) a\u271d \u2264 f i a\u271d\n\u22a2 c \u2022 (\u03bc[f j|\u2191\u2131 i]) a\u271d \u2264 c \u2022 f i a\u271d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Bundle.lean", "full_name": "Bundle.TotalSpace.mk_inj", "start": [74, 1], "end": [75, 28], "traced_tactics": [{"tactic": "simp [TotalSpace.ext_iff]", "annotated_tactic": ["simp [<a>TotalSpace.ext_iff</a>]", [{"full_name": "Bundle.TotalSpace.ext_iff", "def_path": "Mathlib/Data/Bundle.lean", "def_pos": [51, 3], "def_end_pos": [51, 6]}]], "state_before": "B : Type u_1\nF : Type u_2\nE : B \u2192 Type u_3\nb : B\ny y' : E b\n\u22a2 mk' F b y = mk' F b y' \u2194 y = y'", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/TuringMachine.lean", "full_name": "Turing.Tape.map_mk\u2081", "start": [729, 1], "end": [731, 21], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "full_name": "MeasureTheory.Measure.QuasiMeasurePreserving.mono", "start": [2231, 1], "end": [2233, 33], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "full_name": "MeasureTheory.OuterMeasure.boundedBy_eq_self", "start": [862, 1], "end": [863, 77], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/PImage.lean", "full_name": "Part.coe_toFinset", "start": [49, 1], "end": [50, 32], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/CircleIntegral.lean", "full_name": "circleIntegral.norm_integral_lt_of_norm_le_const_of_lt", "start": [421, 1], "end": [438, 51], "traced_tactics": [{"tactic": "rw [\u2190 _root_.abs_of_pos hR, \u2190 image_circleMap_Ioc] at hlt", "annotated_tactic": ["rw [\u2190 <a>_root_.abs_of_pos</a> hR, \u2190 <a>image_circleMap_Ioc</a>] at hlt", [{"full_name": "abs_of_pos", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [111, 9], "def_end_pos": [111, 19]}, {"full_name": "image_circleMap_Ioc", "def_path": "Mathlib/MeasureTheory/Integral/CircleIntegral.lean", "def_pos": [154, 9], "def_end_pos": [154, 28]}]], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf : \u2102 \u2192 E\nc : \u2102\nR C : \u211d\nhR : 0 < R\nhc : ContinuousOn f (sphere c R)\nhf : \u2200 (z : \u2102), z \u2208 sphere c R \u2192 \u2016f z\u2016 \u2264 C\nhlt : \u2203 z, z \u2208 sphere c R \u2227 \u2016f z\u2016 < C\n\u22a2 \u2016\u222e (z : \u2102) in C(c, R), f z\u2016 < 2 * \u03c0 * R * C", "state_after": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf : \u2102 \u2192 E\nc : \u2102\nR C : \u211d\nhR : 0 < R\nhc : ContinuousOn f (sphere c R)\nhf : \u2200 (z : \u2102), z \u2208 sphere c R \u2192 \u2016f z\u2016 \u2264 C\nhlt : \u2203 z, z \u2208 circleMap c R '' Ioc 0 (2 * \u03c0) \u2227 \u2016f z\u2016 < C\n\u22a2 \u2016\u222e (z : \u2102) in C(c, R), f z\u2016 < 2 * \u03c0 * R * C"}, {"tactic": "rcases hlt with \u27e8_, \u27e8\u03b8\u2080, hmem, rfl\u27e9, hlt\u27e9", "annotated_tactic": ["rcases hlt with \u27e8_, \u27e8\u03b8\u2080, hmem, rfl\u27e9, hlt\u27e9", []], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf : \u2102 \u2192 E\nc : \u2102\nR C : \u211d\nhR : 0 < R\nhc : ContinuousOn f (sphere c R)\nhf : \u2200 (z : \u2102), z \u2208 sphere c R \u2192 \u2016f z\u2016 \u2264 C\nhlt : \u2203 z, z \u2208 circleMap c R '' Ioc 0 (2 * \u03c0) \u2227 \u2016f z\u2016 < C\n\u22a2 \u2016\u222e (z : \u2102) in C(c, R), f z\u2016 < 2 * \u03c0 * R * C", "state_after": "case intro.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf : \u2102 \u2192 E\nc : \u2102\nR C : \u211d\nhR : 0 < R\nhc : ContinuousOn f (sphere c R)\nhf : \u2200 (z : \u2102), z \u2208 sphere c R \u2192 \u2016f z\u2016 \u2264 C\n\u03b8\u2080 : \u211d\nhmem : \u03b8\u2080 \u2208 Ioc 0 (2 * \u03c0)\nhlt : \u2016f (circleMap c R \u03b8\u2080)\u2016 < C\n\u22a2 \u2016\u222e (z : \u2102) in C(c, R), f z\u2016 < 2 * \u03c0 * R * C"}, {"tactic": "simp only [norm_smul, deriv_circleMap, norm_eq_abs, map_mul, abs_I, mul_one,\n  abs_circleMap_zero, abs_of_pos hR]", "annotated_tactic": ["simp only [<a>norm_smul</a>, <a>deriv_circleMap</a>, <a>norm_eq_abs</a>, <a>map_mul</a>, <a>abs_I</a>, <a>mul_one</a>,\n        <a>abs_circleMap_zero</a>, <a>abs_of_pos</a> hR]", [{"full_name": "norm_smul", "def_path": "Mathlib/Analysis/Normed/MulAction.lean", "def_pos": [89, 9], "def_end_pos": [89, 18]}, {"full_name": "deriv_circleMap", "def_path": "Mathlib/MeasureTheory/Integral/CircleIntegral.lean", "def_pos": [195, 9], "def_end_pos": [195, 24]}, {"full_name": "Complex.norm_eq_abs", "def_path": "Mathlib/Analysis/Complex/Basic.lean", "def_pos": [51, 9], "def_end_pos": [51, 20]}, {"full_name": "map_mul", "def_path": "Mathlib/Algebra/Hom/Group/Defs.lean", "def_pos": [299, 9], "def_end_pos": [299, 16]}, {"full_name": "Complex.abs_I", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [1018, 9], "def_end_pos": [1018, 14]}, {"full_name": "mul_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [470, 9], "def_end_pos": [470, 16]}, {"full_name": "abs_circleMap_zero", "def_path": "Mathlib/MeasureTheory/Integral/CircleIntegral.lean", "def_pos": [116, 9], "def_end_pos": [116, 27]}, {"full_name": "abs_of_pos", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [111, 9], "def_end_pos": [111, 19]}]], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf : \u2102 \u2192 E\nc : \u2102\nR C : \u211d\nhR : 0 < R\nhc : ContinuousOn f (sphere c R)\nhf : \u2200 (z : \u2102), z \u2208 sphere c R \u2192 \u2016f z\u2016 \u2264 C\n\u03b8\u2080 : \u211d\nhmem : \u03b8\u2080 \u2208 Ioc 0 (2 * \u03c0)\nhlt : \u2016f (circleMap c R \u03b8\u2080)\u2016 < C\n\u22a2 \u222b (\u03b8 : \u211d) in 0 ..2 * \u03c0, \u2016deriv (circleMap c R) \u03b8 \u2022 f (circleMap c R \u03b8)\u2016 < \u222b (x : \u211d) in 0 ..2 * \u03c0, R * C", "state_after": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf : \u2102 \u2192 E\nc : \u2102\nR C : \u211d\nhR : 0 < R\nhc : ContinuousOn f (sphere c R)\nhf : \u2200 (z : \u2102), z \u2208 sphere c R \u2192 \u2016f z\u2016 \u2264 C\n\u03b8\u2080 : \u211d\nhmem : \u03b8\u2080 \u2208 Ioc 0 (2 * \u03c0)\nhlt : \u2016f (circleMap c R \u03b8\u2080)\u2016 < C\n\u22a2 \u222b (\u03b8 : \u211d) in 0 ..2 * \u03c0, R * \u2016f (circleMap c R \u03b8)\u2016 < \u222b (x : \u211d) in 0 ..2 * \u03c0, R * C"}, {"tactic": "refine' intervalIntegral.integral_lt_integral_of_continuousOn_of_le_of_exists_lt\n    Real.two_pi_pos _ continuousOn_const (fun \u03b8 _ => _) \u27e8\u03b8\u2080, Ioc_subset_Icc_self hmem, _\u27e9", "annotated_tactic": ["refine' <a>intervalIntegral.integral_lt_integral_of_continuousOn_of_le_of_exists_lt</a>\n          <a>Real.two_pi_pos</a> _ <a>continuousOn_const</a> (fun \u03b8 _ => _) \u27e8\u03b8\u2080, <a>Ioc_subset_Icc_self</a> hmem, _\u27e9", [{"full_name": "intervalIntegral.integral_lt_integral_of_continuousOn_of_le_of_exists_lt", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [1337, 9], "def_end_pos": [1337, 64]}, {"full_name": "Real.two_pi_pos", "def_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean", "def_pos": [186, 9], "def_end_pos": [186, 19]}, {"full_name": "continuousOn_const", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [1025, 9], "def_end_pos": [1025, 27]}, {"full_name": "Set.Ioc_subset_Icc_self", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [513, 9], "def_end_pos": [513, 28]}]], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf : \u2102 \u2192 E\nc : \u2102\nR C : \u211d\nhR : 0 < R\nhc : ContinuousOn f (sphere c R)\nhf : \u2200 (z : \u2102), z \u2208 sphere c R \u2192 \u2016f z\u2016 \u2264 C\n\u03b8\u2080 : \u211d\nhmem : \u03b8\u2080 \u2208 Ioc 0 (2 * \u03c0)\nhlt : \u2016f (circleMap c R \u03b8\u2080)\u2016 < C\n\u22a2 \u222b (\u03b8 : \u211d) in 0 ..2 * \u03c0, R * \u2016f (circleMap c R \u03b8)\u2016 < \u222b (x : \u211d) in 0 ..2 * \u03c0, R * C", "state_after": "case refine'_1\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf : \u2102 \u2192 E\nc : \u2102\nR C : \u211d\nhR : 0 < R\nhc : ContinuousOn f (sphere c R)\nhf : \u2200 (z : \u2102), z \u2208 sphere c R \u2192 \u2016f z\u2016 \u2264 C\n\u03b8\u2080 : \u211d\nhmem : \u03b8\u2080 \u2208 Ioc 0 (2 * \u03c0)\nhlt : \u2016f (circleMap c R \u03b8\u2080)\u2016 < C\n\u22a2 ContinuousOn (fun \u03b8 => R * \u2016f (circleMap c R \u03b8)\u2016) (Icc 0 (2 * \u03c0))\n\ncase refine'_2\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf : \u2102 \u2192 E\nc : \u2102\nR C : \u211d\nhR : 0 < R\nhc : ContinuousOn f (sphere c R)\nhf : \u2200 (z : \u2102), z \u2208 sphere c R \u2192 \u2016f z\u2016 \u2264 C\n\u03b8\u2080 : \u211d\nhmem : \u03b8\u2080 \u2208 Ioc 0 (2 * \u03c0)\nhlt : \u2016f (circleMap c R \u03b8\u2080)\u2016 < C\n\u03b8 : \u211d\nx\u271d : \u03b8 \u2208 Ioc 0 (2 * \u03c0)\n\u22a2 R * \u2016f (circleMap c R \u03b8)\u2016 \u2264 R * C\n\ncase refine'_3\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf : \u2102 \u2192 E\nc : \u2102\nR C : \u211d\nhR : 0 < R\nhc : ContinuousOn f (sphere c R)\nhf : \u2200 (z : \u2102), z \u2208 sphere c R \u2192 \u2016f z\u2016 \u2264 C\n\u03b8\u2080 : \u211d\nhmem : \u03b8\u2080 \u2208 Ioc 0 (2 * \u03c0)\nhlt : \u2016f (circleMap c R \u03b8\u2080)\u2016 < C\n\u22a2 R * \u2016f (circleMap c R \u03b8\u2080)\u2016 < R * C"}, {"tactic": "exact continuousOn_const.mul (hc.comp (continuous_circleMap _ _).continuousOn fun \u03b8 _ =>\n  circleMap_mem_sphere _ hR.le _).norm", "annotated_tactic": ["exact continuousOn_const.mul (hc.comp (<a>continuous_circleMap</a> _ _).<a>continuousOn</a> fun \u03b8 _ =>\n          <a>circleMap_mem_sphere</a> _ hR.le _).<a>norm</a>", [{"full_name": "continuous_circleMap", "def_path": "Mathlib/MeasureTheory/Integral/CircleIntegral.lean", "def_pos": [185, 9], "def_end_pos": [185, 29]}, {"full_name": "Continuous.continuousOn", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [947, 9], "def_end_pos": [947, 32]}, {"full_name": "circleMap_mem_sphere", "def_path": "Mathlib/MeasureTheory/Integral/CircleIntegral.lean", "def_pos": [122, 9], "def_end_pos": [122, 29]}, {"full_name": "ContinuousOn.norm", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [1296, 15], "def_end_pos": [1296, 32]}]], "state_before": "case refine'_1\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf : \u2102 \u2192 E\nc : \u2102\nR C : \u211d\nhR : 0 < R\nhc : ContinuousOn f (sphere c R)\nhf : \u2200 (z : \u2102), z \u2208 sphere c R \u2192 \u2016f z\u2016 \u2264 C\n\u03b8\u2080 : \u211d\nhmem : \u03b8\u2080 \u2208 Ioc 0 (2 * \u03c0)\nhlt : \u2016f (circleMap c R \u03b8\u2080)\u2016 < C\n\u22a2 ContinuousOn (fun \u03b8 => R * \u2016f (circleMap c R \u03b8)\u2016) (Icc 0 (2 * \u03c0))", "state_after": "no goals"}, {"tactic": "exact mul_le_mul_of_nonneg_left (hf _ <| circleMap_mem_sphere _ hR.le _) hR.le", "annotated_tactic": ["exact <a>mul_le_mul_of_nonneg_left</a> (hf _ <| <a>circleMap_mem_sphere</a> _ hR.le _) hR.le", [{"full_name": "mul_le_mul_of_nonneg_left", "def_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "def_pos": [152, 9], "def_end_pos": [152, 34]}, {"full_name": "circleMap_mem_sphere", "def_path": "Mathlib/MeasureTheory/Integral/CircleIntegral.lean", "def_pos": [122, 9], "def_end_pos": [122, 29]}]], "state_before": "case refine'_2\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf : \u2102 \u2192 E\nc : \u2102\nR C : \u211d\nhR : 0 < R\nhc : ContinuousOn f (sphere c R)\nhf : \u2200 (z : \u2102), z \u2208 sphere c R \u2192 \u2016f z\u2016 \u2264 C\n\u03b8\u2080 : \u211d\nhmem : \u03b8\u2080 \u2208 Ioc 0 (2 * \u03c0)\nhlt : \u2016f (circleMap c R \u03b8\u2080)\u2016 < C\n\u03b8 : \u211d\nx\u271d : \u03b8 \u2208 Ioc 0 (2 * \u03c0)\n\u22a2 R * \u2016f (circleMap c R \u03b8)\u2016 \u2264 R * C", "state_after": "no goals"}, {"tactic": "exact (mul_lt_mul_left hR).2 hlt", "annotated_tactic": ["exact (<a>mul_lt_mul_left</a> hR).2 hlt", [{"full_name": "mul_lt_mul_left", "def_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "def_pos": [197, 9], "def_end_pos": [197, 24]}]], "state_before": "case refine'_3\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf : \u2102 \u2192 E\nc : \u2102\nR C : \u211d\nhR : 0 < R\nhc : ContinuousOn f (sphere c R)\nhf : \u2200 (z : \u2102), z \u2208 sphere c R \u2192 \u2016f z\u2016 \u2264 C\n\u03b8\u2080 : \u211d\nhmem : \u03b8\u2080 \u2208 Ioc 0 (2 * \u03c0)\nhlt : \u2016f (circleMap c R \u03b8\u2080)\u2016 < C\n\u22a2 R * \u2016f (circleMap c R \u03b8\u2080)\u2016 < R * C", "state_after": "no goals"}, {"tactic": "simp [mul_assoc]", "annotated_tactic": ["simp [<a>mul_assoc</a>]", [{"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [264, 9], "def_end_pos": [264, 18]}]], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf : \u2102 \u2192 E\nc : \u2102\nR C : \u211d\nhR : 0 < R\nhc : ContinuousOn f (sphere c R)\nhf : \u2200 (z : \u2102), z \u2208 sphere c R \u2192 \u2016f z\u2016 \u2264 C\n\u03b8\u2080 : \u211d\nhmem : \u03b8\u2080 \u2208 Ioc 0 (2 * \u03c0)\nhlt : \u2016f (circleMap c R \u03b8\u2080)\u2016 < C\n\u22a2 \u222b (x : \u211d) in 0 ..2 * \u03c0, R * C = 2 * \u03c0 * R * C", "state_after": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf : \u2102 \u2192 E\nc : \u2102\nR C : \u211d\nhR : 0 < R\nhc : ContinuousOn f (sphere c R)\nhf : \u2200 (z : \u2102), z \u2208 sphere c R \u2192 \u2016f z\u2016 \u2264 C\n\u03b8\u2080 : \u211d\nhmem : \u03b8\u2080 \u2208 Ioc 0 (2 * \u03c0)\nhlt : \u2016f (circleMap c R \u03b8\u2080)\u2016 < C\n\u22a2 R * (2 * (\u03c0 * C)) = 2 * (\u03c0 * (R * C))"}, {"tactic": "ring", "annotated_tactic": ["ring", []], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf : \u2102 \u2192 E\nc : \u2102\nR C : \u211d\nhR : 0 < R\nhc : ContinuousOn f (sphere c R)\nhf : \u2200 (z : \u2102), z \u2208 sphere c R \u2192 \u2016f z\u2016 \u2264 C\n\u03b8\u2080 : \u211d\nhmem : \u03b8\u2080 \u2208 Ioc 0 (2 * \u03c0)\nhlt : \u2016f (circleMap c R \u03b8\u2080)\u2016 < C\n\u22a2 R * (2 * (\u03c0 * C)) = 2 * (\u03c0 * (R * C))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "full_name": "MeasureTheory.StronglyMeasurable.ite", "start": [768, 11], "end": [771, 40], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Array/Lemmas.lean", "full_name": "Array.mapM_eq_mapM_data", "start": [163, 1], "end": [169, 60], "traced_tactics": [{"tactic": "rw [mapM_eq_foldlM, foldlM_eq_foldlM_data, \u2190 List.foldrM_reverse]", "annotated_tactic": ["rw [<a>mapM_eq_foldlM</a>, <a>foldlM_eq_foldlM_data</a>, \u2190 <a>List.foldrM_reverse</a>]", [{"full_name": "Array.mapM_eq_foldlM", "def_path": "lake-packages/std/Std/Data/Array/Init/Lemmas.lean", "def_pos": [143, 9], "def_end_pos": [143, 23]}, {"full_name": "Array.foldlM_eq_foldlM_data", "def_path": "lake-packages/std/Std/Data/Array/Init/Lemmas.lean", "def_pos": [42, 9], "def_end_pos": [42, 30]}, {"full_name": "List.foldrM_reverse", "def_path": "lake-packages/std/Std/Data/List/Init/Lemmas.lean", "def_pos": [201, 17], "def_end_pos": [201, 31]}]], "state_before": "m : Type u_1 \u2192 Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_1\ninst\u271d\u00b9 : Monad m\ninst\u271d : LawfulMonad m\nf : \u03b1 \u2192 m \u03b2\narr : Array \u03b1\n\u22a2 mapM f arr = do\n    let __do_lift \u2190 List.mapM f arr.data\n    pure { data := __do_lift }", "state_after": "m : Type u_1 \u2192 Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_1\ninst\u271d\u00b9 : Monad m\ninst\u271d : LawfulMonad m\nf : \u03b1 \u2192 m \u03b2\narr : Array \u03b1\n\u22a2 List.foldrM (fun a bs => push bs <$> f a) #[] (List.reverse arr.data) = do\n    let __do_lift \u2190 List.mapM f arr.data\n    pure { data := __do_lift }"}, {"tactic": "conv => rhs; rw [\u2190 List.reverse_reverse arr.data]", "annotated_tactic": ["conv => rhs; rw [\u2190 <a>List.reverse_reverse</a> arr.data]", [{"full_name": "List.reverse_reverse", "def_path": "lake-packages/lean4/src/lean/Init/Data/List/Basic.lean", "def_pos": [67, 17], "def_end_pos": [67, 32]}]], "state_before": "m : Type u_1 \u2192 Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_1\ninst\u271d\u00b9 : Monad m\ninst\u271d : LawfulMonad m\nf : \u03b1 \u2192 m \u03b2\narr : Array \u03b1\n\u22a2 List.foldrM (fun a bs => push bs <$> f a) #[] (List.reverse arr.data) = do\n    let __do_lift \u2190 List.mapM f arr.data\n    pure { data := __do_lift }", "state_after": "m : Type u_1 \u2192 Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_1\ninst\u271d\u00b9 : Monad m\ninst\u271d : LawfulMonad m\nf : \u03b1 \u2192 m \u03b2\narr : Array \u03b1\n\u22a2 List.foldrM (fun a bs => push bs <$> f a) #[] (List.reverse arr.data) = do\n    let __do_lift \u2190 List.mapM f (List.reverse (List.reverse arr.data))\n    pure { data := __do_lift }"}, {"tactic": "induction arr.data.reverse with\n| nil => simp; rfl\n| cons a l ih => simp [ih]; simp [map_eq_pure_bind, push]", "annotated_tactic": ["induction arr.data.reverse with\n  | <a>nil</a> => simp; rfl\n  | <a>cons</a> a l ih => simp [ih]; simp [<a>map_eq_pure_bind</a>, <a>push</a>]", [{"full_name": "List.nil", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2184, 5], "def_end_pos": [2184, 8]}, {"full_name": "List.cons", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2187, 5], "def_end_pos": [2187, 9]}, {"full_name": "map_eq_pure_bind", "def_path": "lake-packages/lean4/src/lean/Init/Control/Lawful.lean", "def_pos": [62, 9], "def_end_pos": [62, 25]}, {"full_name": "Array.push", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2541, 5], "def_end_pos": [2541, 15]}]], "state_before": "m : Type u_1 \u2192 Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_1\ninst\u271d\u00b9 : Monad m\ninst\u271d : LawfulMonad m\nf : \u03b1 \u2192 m \u03b2\narr : Array \u03b1\n\u22a2 List.foldrM (fun a bs => push bs <$> f a) #[] (List.reverse arr.data) = do\n    let __do_lift \u2190 List.mapM f (List.reverse (List.reverse arr.data))\n    pure { data := __do_lift }", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case nil\nm : Type u_1 \u2192 Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_1\ninst\u271d\u00b9 : Monad m\ninst\u271d : LawfulMonad m\nf : \u03b1 \u2192 m \u03b2\narr : Array \u03b1\n\u22a2 List.foldrM (fun a bs => push bs <$> f a) #[] [] = do\n    let __do_lift \u2190 List.mapM f (List.reverse [])\n    pure { data := __do_lift }", "state_after": "case nil\nm : Type u_1 \u2192 Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_1\ninst\u271d\u00b9 : Monad m\ninst\u271d : LawfulMonad m\nf : \u03b1 \u2192 m \u03b2\narr : Array \u03b1\n\u22a2 pure #[] = pure { data := [] }"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case nil\nm : Type u_1 \u2192 Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_1\ninst\u271d\u00b9 : Monad m\ninst\u271d : LawfulMonad m\nf : \u03b1 \u2192 m \u03b2\narr : Array \u03b1\n\u22a2 pure #[] = pure { data := [] }", "state_after": "no goals"}, {"tactic": "simp [ih]", "annotated_tactic": ["simp [ih]", []], "state_before": "case cons\nm : Type u_1 \u2192 Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_1\ninst\u271d\u00b9 : Monad m\ninst\u271d : LawfulMonad m\nf : \u03b1 \u2192 m \u03b2\narr : Array \u03b1\na : \u03b1\nl : List \u03b1\nih :\n  List.foldrM (fun a bs => push bs <$> f a) #[] l = do\n    let __do_lift \u2190 List.mapM f (List.reverse l)\n    pure { data := __do_lift }\n\u22a2 List.foldrM (fun a bs => push bs <$> f a) #[] (a :: l) = do\n    let __do_lift \u2190 List.mapM f (List.reverse (a :: l))\n    pure { data := __do_lift }", "state_after": "case cons\nm : Type u_1 \u2192 Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_1\ninst\u271d\u00b9 : Monad m\ninst\u271d : LawfulMonad m\nf : \u03b1 \u2192 m \u03b2\narr : Array \u03b1\na : \u03b1\nl : List \u03b1\nih :\n  List.foldrM (fun a bs => push bs <$> f a) #[] l = do\n    let __do_lift \u2190 List.mapM f (List.reverse l)\n    pure { data := __do_lift }\n\u22a2 (do\n      let x \u2190 List.mapM f (List.reverse l)\n      push { data := x } <$> f a) =\n    do\n    let x \u2190 List.mapM f (List.reverse l)\n    let x_1 \u2190 f a\n    pure { data := x ++ [x_1] }"}, {"tactic": "simp [map_eq_pure_bind, push]", "annotated_tactic": ["simp [<a>map_eq_pure_bind</a>, <a>push</a>]", [{"full_name": "map_eq_pure_bind", "def_path": "lake-packages/lean4/src/lean/Init/Control/Lawful.lean", "def_pos": [62, 9], "def_end_pos": [62, 25]}, {"full_name": "Array.push", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2541, 5], "def_end_pos": [2541, 15]}]], "state_before": "case cons\nm : Type u_1 \u2192 Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_1\ninst\u271d\u00b9 : Monad m\ninst\u271d : LawfulMonad m\nf : \u03b1 \u2192 m \u03b2\narr : Array \u03b1\na : \u03b1\nl : List \u03b1\nih :\n  List.foldrM (fun a bs => push bs <$> f a) #[] l = do\n    let __do_lift \u2190 List.mapM f (List.reverse l)\n    pure { data := __do_lift }\n\u22a2 (do\n      let x \u2190 List.mapM f (List.reverse l)\n      push { data := x } <$> f a) =\n    do\n    let x \u2190 List.mapM f (List.reverse l)\n    let x_1 \u2190 f a\n    pure { data := x ++ [x_1] }", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Primrec.lean", "full_name": "Primrec.dom_fintype", "start": [782, 1], "end": [787, 66], "traced_tactics": [{"tactic": "haveI := decidableEqOfEncodable \u03b1", "annotated_tactic": ["haveI := <a>decidableEqOfEncodable</a> \u03b1", [{"full_name": "Encodable.decidableEqOfEncodable", "def_path": "Mathlib/Logic/Encodable/Basic.lean", "def_pos": [85, 5], "def_end_pos": [85, 27]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03c3 : Type u_5\ninst\u271d\u2075 : Primcodable \u03b1\ninst\u271d\u2074 : Primcodable \u03b2\ninst\u271d\u00b3 : Primcodable \u03b3\ninst\u271d\u00b2 : Primcodable \u03b4\ninst\u271d\u00b9 : Primcodable \u03c3\ninst\u271d : Fintype \u03b1\nf : \u03b1 \u2192 \u03c3\nl : List \u03b1\nleft\u271d : List.Nodup l\nm : \u2200 (x : \u03b1), x \u2208 l\n\u22a2 Primrec fun a => some (f a)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03c3 : Type u_5\ninst\u271d\u2075 : Primcodable \u03b1\ninst\u271d\u2074 : Primcodable \u03b2\ninst\u271d\u00b3 : Primcodable \u03b3\ninst\u271d\u00b2 : Primcodable \u03b4\ninst\u271d\u00b9 : Primcodable \u03c3\ninst\u271d : Fintype \u03b1\nf : \u03b1 \u2192 \u03c3\nl : List \u03b1\nleft\u271d : List.Nodup l\nm : \u2200 (x : \u03b1), x \u2208 l\nthis : DecidableEq \u03b1\n\u22a2 Primrec fun a => some (f a)"}, {"tactic": "refine ((list_get?\u2081 (l.map f)).comp (list_indexOf\u2081 l)).of_eq fun a => ?_", "annotated_tactic": ["refine ((<a>list_get?\u2081</a> (l.map f)).<a>comp</a> (<a>list_indexOf\u2081</a> l)).<a>of_eq</a> fun a => ?_", [{"full_name": "Primrec.list_get?\u2081", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [375, 9], "def_end_pos": [375, 19]}, {"full_name": "Primrec.comp", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [259, 9], "def_end_pos": [259, 13]}, {"full_name": "Primrec.list_indexOf\u2081", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [778, 9], "def_end_pos": [778, 22]}, {"full_name": "Primrec.of_eq", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [246, 9], "def_end_pos": [246, 14]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03c3 : Type u_5\ninst\u271d\u2075 : Primcodable \u03b1\ninst\u271d\u2074 : Primcodable \u03b2\ninst\u271d\u00b3 : Primcodable \u03b3\ninst\u271d\u00b2 : Primcodable \u03b4\ninst\u271d\u00b9 : Primcodable \u03c3\ninst\u271d : Fintype \u03b1\nf : \u03b1 \u2192 \u03c3\nl : List \u03b1\nleft\u271d : List.Nodup l\nm : \u2200 (x : \u03b1), x \u2208 l\nthis : DecidableEq \u03b1\n\u22a2 Primrec fun a => some (f a)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03c3 : Type u_5\ninst\u271d\u2075 : Primcodable \u03b1\ninst\u271d\u2074 : Primcodable \u03b2\ninst\u271d\u00b3 : Primcodable \u03b3\ninst\u271d\u00b2 : Primcodable \u03b4\ninst\u271d\u00b9 : Primcodable \u03c3\ninst\u271d : Fintype \u03b1\nf : \u03b1 \u2192 \u03c3\nl : List \u03b1\nleft\u271d : List.Nodup l\nm : \u2200 (x : \u03b1), x \u2208 l\nthis : DecidableEq \u03b1\na : \u03b1\n\u22a2 List.get? (List.map f l) (List.indexOf a l) = some (f a)"}, {"tactic": "rw [List.get?_map, List.indexOf_get? (m a), Option.map_some']", "annotated_tactic": ["rw [<a>List.get?_map</a>, <a>List.indexOf_get?</a> (m a), <a>Option.map_some'</a>]", [{"full_name": "List.get?_map", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [655, 17], "def_end_pos": [655, 25]}, {"full_name": "List.indexOf_get?", "def_path": "Mathlib/Data/List/Basic.lean", "def_pos": [1357, 9], "def_end_pos": [1357, 21]}, {"full_name": "Option.map_some'", "def_path": "lake-packages/std/Std/Data/Option/Init/Lemmas.lean", "def_pos": [20, 17], "def_end_pos": [20, 26]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03c3 : Type u_5\ninst\u271d\u2075 : Primcodable \u03b1\ninst\u271d\u2074 : Primcodable \u03b2\ninst\u271d\u00b3 : Primcodable \u03b3\ninst\u271d\u00b2 : Primcodable \u03b4\ninst\u271d\u00b9 : Primcodable \u03c3\ninst\u271d : Fintype \u03b1\nf : \u03b1 \u2192 \u03c3\nl : List \u03b1\nleft\u271d : List.Nodup l\nm : \u2200 (x : \u03b1), x \u2208 l\nthis : DecidableEq \u03b1\na : \u03b1\n\u22a2 List.get? (List.map f l) (List.indexOf a l) = some (f a)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Primrec.lean", "full_name": "list_casesOn'", "start": [879, 9], "end": [890, 85], "traced_tactics": [{"tactic": "cases' f a with b l <;> simp [encodek]", "annotated_tactic": ["cases' f a with b l <;> simp [<a>encodek</a>]", [{"full_name": "Encodable.encodek", "def_path": "Mathlib/Logic/Encodable/Basic.lean", "def_pos": [53, 3], "def_end_pos": [53, 10]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03c3 : Type u_3\ninst\u271d\u00b2 : Primcodable \u03b1\ninst\u271d\u00b9 : Primcodable \u03b2\ninst\u271d : Primcodable \u03c3\nH : Nat.Primrec fun n => encode (decode n)\nf : \u03b1 \u2192 List \u03b2\ng : \u03b1 \u2192 \u03c3\nh : \u03b1 \u2192 \u03b2 \u00d7 List \u03b2 \u2192 \u03c3\nhf : Primrec f\nhg : Primrec g\nhh : Primrec\u2082 h\nthis\u271d : Primcodable (List \u03b2) := prim H\nthis : Primrec fun a => Option.map (fun o => Option.casesOn o (g a) (h a)) (decode (encode (f a)))\na : \u03b1\n\u22a2 Option.map (fun o => Option.casesOn o (g a) (h a)) (decode (encode (f a))) =\n    some (List.casesOn (f a) (g a) fun b l => h a (b, l))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Primrec.lean", "full_name": "Primrec.beq", "start": [744, 11], "end": [748, 39], "traced_tactics": [{"tactic": "simp [le_antisymm_iff]", "annotated_tactic": ["simp [<a>le_antisymm_iff</a>]", [{"full_name": "le_antisymm_iff", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [192, 9], "def_end_pos": [192, 24]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03c3 : Type u_5\ninst\u271d\u2075 : Primcodable \u03b1\ninst\u271d\u2074 : Primcodable \u03b2\ninst\u271d\u00b3 : Primcodable \u03b3\ninst\u271d\u00b2 : Primcodable \u03b4\ninst\u271d\u00b9 : Primcodable \u03c3\ninst\u271d : DecidableEq \u03b1\na : \u2115 \u00d7 \u2115\n\u22a2 (fun x x_1 => x \u2264 x_1) a.1 a.2 \u2227 (fun x x_1 => x \u2264 x_1) a.2 a.1 \u2194 (fun a b => a = b) a.1 a.2", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/PartrecCode.lean", "full_name": "Nat.Partrec.Code.eval_prec_zero", "start": [639, 1], "end": [642, 20], "traced_tactics": [{"tactic": "rw [eval, Nat.unpaired, Nat.unpair_pair]", "annotated_tactic": ["rw [<a>eval</a>, <a>Nat.unpaired</a>, <a>Nat.unpair_pair</a>]", [{"full_name": "Nat.Partrec.Code.eval", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [620, 5], "def_end_pos": [620, 9]}, {"full_name": "Nat.unpaired", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [70, 5], "def_end_pos": [70, 13]}, {"full_name": "Nat.unpair_pair", "def_path": "Mathlib/Data/Nat/Pairing.lean", "def_pos": [65, 9], "def_end_pos": [65, 20]}]], "state_before": "cf cg : Code\na : \u2115\n\u22a2 eval (prec cf cg) (Nat.pair a 0) = eval cf a", "state_after": "cf cg : Code\na : \u2115\n\u22a2 Nat.rec (eval cf (a, 0).1)\n      (fun y IH => do\n        let i \u2190 IH\n        eval cg (Nat.pair (a, 0).1 (Nat.pair y i)))\n      (a, 0).2 =\n    eval cf a"}, {"tactic": "simp (config := { Lean.Meta.Simp.neutralConfig with proj := true }) only []", "annotated_tactic": ["simp (config := { <a>Lean.Meta.Simp.neutralConfig</a> with proj := <a>true</a> }) only []", [{"full_name": "Lean.Meta.Simp.neutralConfig", "def_path": "lake-packages/lean4/src/lean/Init/Meta.lean", "def_pos": [1270, 5], "def_end_pos": [1270, 18]}, {"full_name": "Bool.true", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [549, 5], "def_end_pos": [549, 9]}]], "state_before": "cf cg : Code\na : \u2115\n\u22a2 Nat.rec (eval cf (a, 0).1)\n      (fun y IH => do\n        let i \u2190 IH\n        eval cg (Nat.pair (a, 0).1 (Nat.pair y i)))\n      (a, 0).2 =\n    eval cf a", "state_after": "cf cg : Code\na : \u2115\n\u22a2 Nat.rec (eval cf a)\n      (fun y IH => do\n        let i \u2190 IH\n        eval cg (Nat.pair a (Nat.pair y i)))\n      0 =\n    eval cf a"}, {"tactic": "rw [Nat.rec_zero]", "annotated_tactic": ["rw [<a>Nat.rec_zero</a>]", [{"full_name": "Nat.rec_zero", "def_path": "Mathlib/Data/Nat/Basic.lean", "def_pos": [404, 9], "def_end_pos": [404, 17]}]], "state_before": "cf cg : Code\na : \u2115\n\u22a2 Nat.rec (eval cf a)\n      (fun y IH => do\n        let i \u2190 IH\n        eval cg (Nat.pair a (Nat.pair y i)))\n      0 =\n    eval cf a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Metrizable.lean", "full_name": "measurable_of_tendsto_metrizable'", "start": [66, 1], "end": [82, 41], "traced_tactics": [{"tactic": "letI : PseudoMetricSpace \u03b2 := pseudoMetrizableSpacePseudoMetric \u03b2", "annotated_tactic": ["letI : <a>PseudoMetricSpace</a> \u03b2 := <a>pseudoMetrizableSpacePseudoMetric</a> \u03b2", [{"full_name": "PseudoMetricSpace", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [123, 7], "def_end_pos": [123, 24]}, {"full_name": "TopologicalSpace.pseudoMetrizableSpacePseudoMetric", "def_path": "Mathlib/Topology/MetricSpace/Metrizable.lean", "def_pos": [48, 19], "def_end_pos": [48, 52]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : PseudoMetrizableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : BorelSpace \u03b2\n\u03b9 : Type u_3\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nu : Filter \u03b9\ninst\u271d\u00b9 : NeBot u\ninst\u271d : IsCountablyGenerated u\nhf : \u2200 (i : \u03b9), Measurable (f i)\nlim : Tendsto f u (\ud835\udcdd g)\n\u22a2 Measurable g", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : PseudoMetrizableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : BorelSpace \u03b2\n\u03b9 : Type u_3\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nu : Filter \u03b9\ninst\u271d\u00b9 : NeBot u\ninst\u271d : IsCountablyGenerated u\nhf : \u2200 (i : \u03b9), Measurable (f i)\nlim : Tendsto f u (\ud835\udcdd g)\nthis : PseudoMetricSpace \u03b2 := pseudoMetrizableSpacePseudoMetric \u03b2\n\u22a2 Measurable g"}, {"tactic": "apply measurable_of_is_closed'", "annotated_tactic": ["apply <a>measurable_of_is_closed'</a>", [{"full_name": "measurable_of_is_closed'", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [365, 9], "def_end_pos": [365, 33]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : PseudoMetrizableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : BorelSpace \u03b2\n\u03b9 : Type u_3\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nu : Filter \u03b9\ninst\u271d\u00b9 : NeBot u\ninst\u271d : IsCountablyGenerated u\nhf : \u2200 (i : \u03b9), Measurable (f i)\nlim : Tendsto f u (\ud835\udcdd g)\nthis : PseudoMetricSpace \u03b2 := pseudoMetrizableSpacePseudoMetric \u03b2\n\u22a2 Measurable g", "state_after": "case hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : PseudoMetrizableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : BorelSpace \u03b2\n\u03b9 : Type u_3\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nu : Filter \u03b9\ninst\u271d\u00b9 : NeBot u\ninst\u271d : IsCountablyGenerated u\nhf : \u2200 (i : \u03b9), Measurable (f i)\nlim : Tendsto f u (\ud835\udcdd g)\nthis : PseudoMetricSpace \u03b2 := pseudoMetrizableSpacePseudoMetric \u03b2\n\u22a2 \u2200 (s : Set \u03b2), IsClosed s \u2192 Set.Nonempty s \u2192 s \u2260 Set.univ \u2192 MeasurableSet (g \u207b\u00b9' s)"}, {"tactic": "intro s h1s h2s h3s", "annotated_tactic": ["intro s h1s h2s h3s", []], "state_before": "case hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : PseudoMetrizableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : BorelSpace \u03b2\n\u03b9 : Type u_3\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nu : Filter \u03b9\ninst\u271d\u00b9 : NeBot u\ninst\u271d : IsCountablyGenerated u\nhf : \u2200 (i : \u03b9), Measurable (f i)\nlim : Tendsto f u (\ud835\udcdd g)\nthis : PseudoMetricSpace \u03b2 := pseudoMetrizableSpacePseudoMetric \u03b2\n\u22a2 \u2200 (s : Set \u03b2), IsClosed s \u2192 Set.Nonempty s \u2192 s \u2260 Set.univ \u2192 MeasurableSet (g \u207b\u00b9' s)", "state_after": "case hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : PseudoMetrizableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : BorelSpace \u03b2\n\u03b9 : Type u_3\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nu : Filter \u03b9\ninst\u271d\u00b9 : NeBot u\ninst\u271d : IsCountablyGenerated u\nhf : \u2200 (i : \u03b9), Measurable (f i)\nlim : Tendsto f u (\ud835\udcdd g)\nthis : PseudoMetricSpace \u03b2 := pseudoMetrizableSpacePseudoMetric \u03b2\ns : Set \u03b2\nh1s : IsClosed s\nh2s : Set.Nonempty s\nh3s : s \u2260 Set.univ\n\u22a2 MeasurableSet (g \u207b\u00b9' s)"}, {"tactic": "have : Measurable fun x => infNndist (g x) s := by\n  suffices : Tendsto (fun i x => infNndist (f i x) s) u (\ud835\udcdd fun x => infNndist (g x) s)\n  exact measurable_of_tendsto_nnreal' u (fun i => (hf i).infNndist) this\n  rw [tendsto_pi_nhds] at lim \u22a2\n  intro x\n  exact ((continuous_infNndist_pt s).tendsto (g x)).comp (lim x)", "annotated_tactic": ["have : <a>Measurable</a> fun x => <a>infNndist</a> (g x) s := by\n    suffices : <a>Tendsto</a> (fun i x => <a>infNndist</a> (f i x) s) u (\ud835\udcdd fun x => <a>infNndist</a> (g x) s)\n    exact <a>measurable_of_tendsto_nnreal'</a> u (fun i => (hf i).<a>infNndist</a>) this\n    rw [<a>tendsto_pi_nhds</a>] at lim \u22a2\n    intro x\n    exact ((<a>continuous_infNndist_pt</a> s).<a>tendsto</a> (g x)).<a>comp</a> (lim x)", [{"full_name": "Measurable", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [535, 5], "def_end_pos": [535, 15]}, {"full_name": "Metric.infNndist", "def_path": "Mathlib/Topology/MetricSpace/HausdorffDistance.lean", "def_pos": [655, 5], "def_end_pos": [655, 14]}, {"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2939, 5], "def_end_pos": [2939, 12]}, {"full_name": "Metric.infNndist", "def_path": "Mathlib/Topology/MetricSpace/HausdorffDistance.lean", "def_pos": [655, 5], "def_end_pos": [655, 14]}, {"full_name": "Metric.infNndist", "def_path": "Mathlib/Topology/MetricSpace/HausdorffDistance.lean", "def_pos": [655, 5], "def_end_pos": [655, 14]}, {"full_name": "measurable_of_tendsto_nnreal'", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Metrizable.lean", "def_pos": [49, 9], "def_end_pos": [49, 38]}, {"full_name": "Measurable.infNndist", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [1702, 9], "def_end_pos": [1702, 29]}, {"full_name": "tendsto_pi_nhds", "def_path": "Mathlib/Topology/Constructions.lean", "def_pos": [1231, 9], "def_end_pos": [1231, 24]}, {"full_name": "Metric.continuous_infNndist_pt", "def_path": "Mathlib/Topology/MetricSpace/HausdorffDistance.lean", "def_pos": [675, 9], "def_end_pos": [675, 32]}, {"full_name": "Continuous.tendsto", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [1697, 9], "def_end_pos": [1697, 27]}, {"full_name": "Filter.Tendsto.comp", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [3032, 9], "def_end_pos": [3032, 21]}]], "state_before": "case hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : PseudoMetrizableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : BorelSpace \u03b2\n\u03b9 : Type u_3\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nu : Filter \u03b9\ninst\u271d\u00b9 : NeBot u\ninst\u271d : IsCountablyGenerated u\nhf : \u2200 (i : \u03b9), Measurable (f i)\nlim : Tendsto f u (\ud835\udcdd g)\nthis : PseudoMetricSpace \u03b2 := pseudoMetrizableSpacePseudoMetric \u03b2\ns : Set \u03b2\nh1s : IsClosed s\nh2s : Set.Nonempty s\nh3s : s \u2260 Set.univ\n\u22a2 MeasurableSet (g \u207b\u00b9' s)", "state_after": "case hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : PseudoMetrizableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : BorelSpace \u03b2\n\u03b9 : Type u_3\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nu : Filter \u03b9\ninst\u271d\u00b9 : NeBot u\ninst\u271d : IsCountablyGenerated u\nhf : \u2200 (i : \u03b9), Measurable (f i)\nlim : Tendsto f u (\ud835\udcdd g)\nthis\u271d : PseudoMetricSpace \u03b2 := pseudoMetrizableSpacePseudoMetric \u03b2\ns : Set \u03b2\nh1s : IsClosed s\nh2s : Set.Nonempty s\nh3s : s \u2260 Set.univ\nthis : Measurable fun x => infNndist (g x) s\n\u22a2 MeasurableSet (g \u207b\u00b9' s)"}, {"tactic": "have h4s : g \u207b\u00b9' s = (fun x => infNndist (g x) s) \u207b\u00b9' {0} := by\n  ext x\n  simp [h1s, \u2190 h1s.mem_iff_infDist_zero h2s, \u2190 NNReal.coe_eq_zero]", "annotated_tactic": ["have h4s : g \u207b\u00b9' s = (fun x => <a>infNndist</a> (g x) s) \u207b\u00b9' {0} := by\n    ext x\n    simp [h1s, \u2190 h1s.mem_iff_infDist_zero h2s, \u2190 <a>NNReal.coe_eq_zero</a>]", [{"full_name": "Metric.infNndist", "def_path": "Mathlib/Topology/MetricSpace/HausdorffDistance.lean", "def_pos": [655, 5], "def_end_pos": [655, 14]}, {"full_name": "NNReal.coe_eq_zero", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [212, 19], "def_end_pos": [212, 30]}]], "state_before": "case hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : PseudoMetrizableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : BorelSpace \u03b2\n\u03b9 : Type u_3\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nu : Filter \u03b9\ninst\u271d\u00b9 : NeBot u\ninst\u271d : IsCountablyGenerated u\nhf : \u2200 (i : \u03b9), Measurable (f i)\nlim : Tendsto f u (\ud835\udcdd g)\nthis\u271d : PseudoMetricSpace \u03b2 := pseudoMetrizableSpacePseudoMetric \u03b2\ns : Set \u03b2\nh1s : IsClosed s\nh2s : Set.Nonempty s\nh3s : s \u2260 Set.univ\nthis : Measurable fun x => infNndist (g x) s\n\u22a2 MeasurableSet (g \u207b\u00b9' s)", "state_after": "case hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : PseudoMetrizableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : BorelSpace \u03b2\n\u03b9 : Type u_3\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nu : Filter \u03b9\ninst\u271d\u00b9 : NeBot u\ninst\u271d : IsCountablyGenerated u\nhf : \u2200 (i : \u03b9), Measurable (f i)\nlim : Tendsto f u (\ud835\udcdd g)\nthis\u271d : PseudoMetricSpace \u03b2 := pseudoMetrizableSpacePseudoMetric \u03b2\ns : Set \u03b2\nh1s : IsClosed s\nh2s : Set.Nonempty s\nh3s : s \u2260 Set.univ\nthis : Measurable fun x => infNndist (g x) s\nh4s : g \u207b\u00b9' s = (fun x => infNndist (g x) s) \u207b\u00b9' {0}\n\u22a2 MeasurableSet (g \u207b\u00b9' s)"}, {"tactic": "rw [h4s]", "annotated_tactic": ["rw [h4s]", []], "state_before": "case hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : PseudoMetrizableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : BorelSpace \u03b2\n\u03b9 : Type u_3\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nu : Filter \u03b9\ninst\u271d\u00b9 : NeBot u\ninst\u271d : IsCountablyGenerated u\nhf : \u2200 (i : \u03b9), Measurable (f i)\nlim : Tendsto f u (\ud835\udcdd g)\nthis\u271d : PseudoMetricSpace \u03b2 := pseudoMetrizableSpacePseudoMetric \u03b2\ns : Set \u03b2\nh1s : IsClosed s\nh2s : Set.Nonempty s\nh3s : s \u2260 Set.univ\nthis : Measurable fun x => infNndist (g x) s\nh4s : g \u207b\u00b9' s = (fun x => infNndist (g x) s) \u207b\u00b9' {0}\n\u22a2 MeasurableSet (g \u207b\u00b9' s)", "state_after": "case hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : PseudoMetrizableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : BorelSpace \u03b2\n\u03b9 : Type u_3\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nu : Filter \u03b9\ninst\u271d\u00b9 : NeBot u\ninst\u271d : IsCountablyGenerated u\nhf : \u2200 (i : \u03b9), Measurable (f i)\nlim : Tendsto f u (\ud835\udcdd g)\nthis\u271d : PseudoMetricSpace \u03b2 := pseudoMetrizableSpacePseudoMetric \u03b2\ns : Set \u03b2\nh1s : IsClosed s\nh2s : Set.Nonempty s\nh3s : s \u2260 Set.univ\nthis : Measurable fun x => infNndist (g x) s\nh4s : g \u207b\u00b9' s = (fun x => infNndist (g x) s) \u207b\u00b9' {0}\n\u22a2 MeasurableSet ((fun x => infNndist (g x) s) \u207b\u00b9' {0})"}, {"tactic": "exact this (measurableSet_singleton 0)", "annotated_tactic": ["exact this (<a>measurableSet_singleton</a> 0)", [{"full_name": "MeasurableSingletonClass.measurableSet_singleton", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [269, 3], "def_end_pos": [269, 26]}]], "state_before": "case hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : PseudoMetrizableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : BorelSpace \u03b2\n\u03b9 : Type u_3\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nu : Filter \u03b9\ninst\u271d\u00b9 : NeBot u\ninst\u271d : IsCountablyGenerated u\nhf : \u2200 (i : \u03b9), Measurable (f i)\nlim : Tendsto f u (\ud835\udcdd g)\nthis\u271d : PseudoMetricSpace \u03b2 := pseudoMetrizableSpacePseudoMetric \u03b2\ns : Set \u03b2\nh1s : IsClosed s\nh2s : Set.Nonempty s\nh3s : s \u2260 Set.univ\nthis : Measurable fun x => infNndist (g x) s\nh4s : g \u207b\u00b9' s = (fun x => infNndist (g x) s) \u207b\u00b9' {0}\n\u22a2 MeasurableSet ((fun x => infNndist (g x) s) \u207b\u00b9' {0})", "state_after": "no goals"}, {"tactic": "suffices : Tendsto (fun i x => infNndist (f i x) s) u (\ud835\udcdd fun x => infNndist (g x) s)", "annotated_tactic": ["suffices : <a>Tendsto</a> (fun i x => <a>infNndist</a> (f i x) s) u (\ud835\udcdd fun x => <a>infNndist</a> (g x) s)", [{"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2939, 5], "def_end_pos": [2939, 12]}, {"full_name": "Metric.infNndist", "def_path": "Mathlib/Topology/MetricSpace/HausdorffDistance.lean", "def_pos": [655, 5], "def_end_pos": [655, 14]}, {"full_name": "Metric.infNndist", "def_path": "Mathlib/Topology/MetricSpace/HausdorffDistance.lean", "def_pos": [655, 5], "def_end_pos": [655, 14]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : PseudoMetrizableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : BorelSpace \u03b2\n\u03b9 : Type u_3\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nu : Filter \u03b9\ninst\u271d\u00b9 : NeBot u\ninst\u271d : IsCountablyGenerated u\nhf : \u2200 (i : \u03b9), Measurable (f i)\nlim : Tendsto f u (\ud835\udcdd g)\nthis : PseudoMetricSpace \u03b2 := pseudoMetrizableSpacePseudoMetric \u03b2\ns : Set \u03b2\nh1s : IsClosed s\nh2s : Set.Nonempty s\nh3s : s \u2260 Set.univ\n\u22a2 Measurable fun x => infNndist (g x) s", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : PseudoMetrizableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : BorelSpace \u03b2\n\u03b9 : Type u_3\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nu : Filter \u03b9\ninst\u271d\u00b9 : NeBot u\ninst\u271d : IsCountablyGenerated u\nhf : \u2200 (i : \u03b9), Measurable (f i)\nlim : Tendsto f u (\ud835\udcdd g)\nthis\u271d : PseudoMetricSpace \u03b2 := pseudoMetrizableSpacePseudoMetric \u03b2\ns : Set \u03b2\nh1s : IsClosed s\nh2s : Set.Nonempty s\nh3s : s \u2260 Set.univ\nthis : Tendsto (fun i x => infNndist (f i x) s) u (\ud835\udcdd fun x => infNndist (g x) s)\n\u22a2 Measurable fun x => infNndist (g x) s\n\ncase this\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : PseudoMetrizableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : BorelSpace \u03b2\n\u03b9 : Type u_3\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nu : Filter \u03b9\ninst\u271d\u00b9 : NeBot u\ninst\u271d : IsCountablyGenerated u\nhf : \u2200 (i : \u03b9), Measurable (f i)\nlim : Tendsto f u (\ud835\udcdd g)\nthis : PseudoMetricSpace \u03b2 := pseudoMetrizableSpacePseudoMetric \u03b2\ns : Set \u03b2\nh1s : IsClosed s\nh2s : Set.Nonempty s\nh3s : s \u2260 Set.univ\n\u22a2 Tendsto (fun i x => infNndist (f i x) s) u (\ud835\udcdd fun x => infNndist (g x) s)"}, {"tactic": "exact measurable_of_tendsto_nnreal' u (fun i => (hf i).infNndist) this", "annotated_tactic": ["exact <a>measurable_of_tendsto_nnreal'</a> u (fun i => (hf i).<a>infNndist</a>) this", [{"full_name": "measurable_of_tendsto_nnreal'", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Metrizable.lean", "def_pos": [49, 9], "def_end_pos": [49, 38]}, {"full_name": "Measurable.infNndist", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [1702, 9], "def_end_pos": [1702, 29]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : PseudoMetrizableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : BorelSpace \u03b2\n\u03b9 : Type u_3\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nu : Filter \u03b9\ninst\u271d\u00b9 : NeBot u\ninst\u271d : IsCountablyGenerated u\nhf : \u2200 (i : \u03b9), Measurable (f i)\nlim : Tendsto f u (\ud835\udcdd g)\nthis\u271d : PseudoMetricSpace \u03b2 := pseudoMetrizableSpacePseudoMetric \u03b2\ns : Set \u03b2\nh1s : IsClosed s\nh2s : Set.Nonempty s\nh3s : s \u2260 Set.univ\nthis : Tendsto (fun i x => infNndist (f i x) s) u (\ud835\udcdd fun x => infNndist (g x) s)\n\u22a2 Measurable fun x => infNndist (g x) s\n\ncase this\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : PseudoMetrizableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : BorelSpace \u03b2\n\u03b9 : Type u_3\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nu : Filter \u03b9\ninst\u271d\u00b9 : NeBot u\ninst\u271d : IsCountablyGenerated u\nhf : \u2200 (i : \u03b9), Measurable (f i)\nlim : Tendsto f u (\ud835\udcdd g)\nthis : PseudoMetricSpace \u03b2 := pseudoMetrizableSpacePseudoMetric \u03b2\ns : Set \u03b2\nh1s : IsClosed s\nh2s : Set.Nonempty s\nh3s : s \u2260 Set.univ\n\u22a2 Tendsto (fun i x => infNndist (f i x) s) u (\ud835\udcdd fun x => infNndist (g x) s)", "state_after": "case this\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : PseudoMetrizableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : BorelSpace \u03b2\n\u03b9 : Type u_3\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nu : Filter \u03b9\ninst\u271d\u00b9 : NeBot u\ninst\u271d : IsCountablyGenerated u\nhf : \u2200 (i : \u03b9), Measurable (f i)\nlim : Tendsto f u (\ud835\udcdd g)\nthis : PseudoMetricSpace \u03b2 := pseudoMetrizableSpacePseudoMetric \u03b2\ns : Set \u03b2\nh1s : IsClosed s\nh2s : Set.Nonempty s\nh3s : s \u2260 Set.univ\n\u22a2 Tendsto (fun i x => infNndist (f i x) s) u (\ud835\udcdd fun x => infNndist (g x) s)"}, {"tactic": "rw [tendsto_pi_nhds] at lim \u22a2", "annotated_tactic": ["rw [<a>tendsto_pi_nhds</a>] at lim \u22a2", [{"full_name": "tendsto_pi_nhds", "def_path": "Mathlib/Topology/Constructions.lean", "def_pos": [1231, 9], "def_end_pos": [1231, 24]}]], "state_before": "case this\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : PseudoMetrizableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : BorelSpace \u03b2\n\u03b9 : Type u_3\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nu : Filter \u03b9\ninst\u271d\u00b9 : NeBot u\ninst\u271d : IsCountablyGenerated u\nhf : \u2200 (i : \u03b9), Measurable (f i)\nlim : Tendsto f u (\ud835\udcdd g)\nthis : PseudoMetricSpace \u03b2 := pseudoMetrizableSpacePseudoMetric \u03b2\ns : Set \u03b2\nh1s : IsClosed s\nh2s : Set.Nonempty s\nh3s : s \u2260 Set.univ\n\u22a2 Tendsto (fun i x => infNndist (f i x) s) u (\ud835\udcdd fun x => infNndist (g x) s)", "state_after": "case this\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : PseudoMetrizableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : BorelSpace \u03b2\n\u03b9 : Type u_3\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nu : Filter \u03b9\ninst\u271d\u00b9 : NeBot u\ninst\u271d : IsCountablyGenerated u\nhf : \u2200 (i : \u03b9), Measurable (f i)\nlim : \u2200 (x : \u03b1), Tendsto (fun i => f i x) u (\ud835\udcdd (g x))\nthis : PseudoMetricSpace \u03b2 := pseudoMetrizableSpacePseudoMetric \u03b2\ns : Set \u03b2\nh1s : IsClosed s\nh2s : Set.Nonempty s\nh3s : s \u2260 Set.univ\n\u22a2 \u2200 (x : \u03b1), Tendsto (fun i => infNndist (f i x) s) u (\ud835\udcdd (infNndist (g x) s))"}, {"tactic": "intro x", "annotated_tactic": ["intro x", []], "state_before": "case this\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : PseudoMetrizableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : BorelSpace \u03b2\n\u03b9 : Type u_3\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nu : Filter \u03b9\ninst\u271d\u00b9 : NeBot u\ninst\u271d : IsCountablyGenerated u\nhf : \u2200 (i : \u03b9), Measurable (f i)\nlim : \u2200 (x : \u03b1), Tendsto (fun i => f i x) u (\ud835\udcdd (g x))\nthis : PseudoMetricSpace \u03b2 := pseudoMetrizableSpacePseudoMetric \u03b2\ns : Set \u03b2\nh1s : IsClosed s\nh2s : Set.Nonempty s\nh3s : s \u2260 Set.univ\n\u22a2 \u2200 (x : \u03b1), Tendsto (fun i => infNndist (f i x) s) u (\ud835\udcdd (infNndist (g x) s))", "state_after": "case this\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : PseudoMetrizableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : BorelSpace \u03b2\n\u03b9 : Type u_3\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nu : Filter \u03b9\ninst\u271d\u00b9 : NeBot u\ninst\u271d : IsCountablyGenerated u\nhf : \u2200 (i : \u03b9), Measurable (f i)\nlim : \u2200 (x : \u03b1), Tendsto (fun i => f i x) u (\ud835\udcdd (g x))\nthis : PseudoMetricSpace \u03b2 := pseudoMetrizableSpacePseudoMetric \u03b2\ns : Set \u03b2\nh1s : IsClosed s\nh2s : Set.Nonempty s\nh3s : s \u2260 Set.univ\nx : \u03b1\n\u22a2 Tendsto (fun i => infNndist (f i x) s) u (\ud835\udcdd (infNndist (g x) s))"}, {"tactic": "exact ((continuous_infNndist_pt s).tendsto (g x)).comp (lim x)", "annotated_tactic": ["exact ((<a>continuous_infNndist_pt</a> s).<a>tendsto</a> (g x)).<a>comp</a> (lim x)", [{"full_name": "Metric.continuous_infNndist_pt", "def_path": "Mathlib/Topology/MetricSpace/HausdorffDistance.lean", "def_pos": [675, 9], "def_end_pos": [675, 32]}, {"full_name": "Continuous.tendsto", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [1697, 9], "def_end_pos": [1697, 27]}, {"full_name": "Filter.Tendsto.comp", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [3032, 9], "def_end_pos": [3032, 21]}]], "state_before": "case this\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : PseudoMetrizableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : BorelSpace \u03b2\n\u03b9 : Type u_3\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nu : Filter \u03b9\ninst\u271d\u00b9 : NeBot u\ninst\u271d : IsCountablyGenerated u\nhf : \u2200 (i : \u03b9), Measurable (f i)\nlim : \u2200 (x : \u03b1), Tendsto (fun i => f i x) u (\ud835\udcdd (g x))\nthis : PseudoMetricSpace \u03b2 := pseudoMetrizableSpacePseudoMetric \u03b2\ns : Set \u03b2\nh1s : IsClosed s\nh2s : Set.Nonempty s\nh3s : s \u2260 Set.univ\nx : \u03b1\n\u22a2 Tendsto (fun i => infNndist (f i x) s) u (\ud835\udcdd (infNndist (g x) s))", "state_after": "no goals"}, {"tactic": "ext x", "annotated_tactic": ["ext x", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : PseudoMetrizableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : BorelSpace \u03b2\n\u03b9 : Type u_3\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nu : Filter \u03b9\ninst\u271d\u00b9 : NeBot u\ninst\u271d : IsCountablyGenerated u\nhf : \u2200 (i : \u03b9), Measurable (f i)\nlim : Tendsto f u (\ud835\udcdd g)\nthis\u271d : PseudoMetricSpace \u03b2 := pseudoMetrizableSpacePseudoMetric \u03b2\ns : Set \u03b2\nh1s : IsClosed s\nh2s : Set.Nonempty s\nh3s : s \u2260 Set.univ\nthis : Measurable fun x => infNndist (g x) s\n\u22a2 g \u207b\u00b9' s = (fun x => infNndist (g x) s) \u207b\u00b9' {0}", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : PseudoMetrizableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : BorelSpace \u03b2\n\u03b9 : Type u_3\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nu : Filter \u03b9\ninst\u271d\u00b9 : NeBot u\ninst\u271d : IsCountablyGenerated u\nhf : \u2200 (i : \u03b9), Measurable (f i)\nlim : Tendsto f u (\ud835\udcdd g)\nthis\u271d : PseudoMetricSpace \u03b2 := pseudoMetrizableSpacePseudoMetric \u03b2\ns : Set \u03b2\nh1s : IsClosed s\nh2s : Set.Nonempty s\nh3s : s \u2260 Set.univ\nthis : Measurable fun x => infNndist (g x) s\nx : \u03b1\n\u22a2 x \u2208 g \u207b\u00b9' s \u2194 x \u2208 (fun x => infNndist (g x) s) \u207b\u00b9' {0}"}, {"tactic": "simp [h1s, \u2190 h1s.mem_iff_infDist_zero h2s, \u2190 NNReal.coe_eq_zero]", "annotated_tactic": ["simp [h1s, \u2190 h1s.mem_iff_infDist_zero h2s, \u2190 <a>NNReal.coe_eq_zero</a>]", [{"full_name": "NNReal.coe_eq_zero", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [212, 19], "def_end_pos": [212, 30]}]], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : PseudoMetrizableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : BorelSpace \u03b2\n\u03b9 : Type u_3\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nu : Filter \u03b9\ninst\u271d\u00b9 : NeBot u\ninst\u271d : IsCountablyGenerated u\nhf : \u2200 (i : \u03b9), Measurable (f i)\nlim : Tendsto f u (\ud835\udcdd g)\nthis\u271d : PseudoMetricSpace \u03b2 := pseudoMetrizableSpacePseudoMetric \u03b2\ns : Set \u03b2\nh1s : IsClosed s\nh2s : Set.Nonempty s\nh3s : s \u2260 Set.univ\nthis : Measurable fun x => infNndist (g x) s\nx : \u03b1\n\u22a2 x \u2208 g \u207b\u00b9' s \u2194 x \u2208 (fun x => infNndist (g x) s) \u207b\u00b9' {0}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean", "full_name": "MeasureTheory.condexp_mono", "start": [347, 1], "end": [356, 78], "traced_tactics": [{"tactic": "by_cases hm : m \u2264 m0", "annotated_tactic": ["by_cases hm : m \u2264 m0", []], "state_before": "\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\ninst\u271d\u2077 : NormedAddCommGroup F'\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedLatticeAddCommGroup E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : OrderedSMul \u211d E\nf g : \u03b1 \u2192 E\nhf : Integrable f\nhg : Integrable g\nhfg : f \u2264\u1d50[\u03bc] g\n\u22a2 \u03bc[f|m] \u2264\u1d50[\u03bc] \u03bc[g|m]", "state_after": "case pos\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\ninst\u271d\u2077 : NormedAddCommGroup F'\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedLatticeAddCommGroup E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : OrderedSMul \u211d E\nf g : \u03b1 \u2192 E\nhf : Integrable f\nhg : Integrable g\nhfg : f \u2264\u1d50[\u03bc] g\nhm : m \u2264 m0\n\u22a2 \u03bc[f|m] \u2264\u1d50[\u03bc] \u03bc[g|m]\n\ncase neg\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\ninst\u271d\u2077 : NormedAddCommGroup F'\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedLatticeAddCommGroup E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : OrderedSMul \u211d E\nf g : \u03b1 \u2192 E\nhf : Integrable f\nhg : Integrable g\nhfg : f \u2264\u1d50[\u03bc] g\nhm : \u00acm \u2264 m0\n\u22a2 \u03bc[f|m] \u2264\u1d50[\u03bc] \u03bc[g|m]"}, {"tactic": "swap", "annotated_tactic": ["swap", []], "state_before": "case pos\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\ninst\u271d\u2077 : NormedAddCommGroup F'\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedLatticeAddCommGroup E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : OrderedSMul \u211d E\nf g : \u03b1 \u2192 E\nhf : Integrable f\nhg : Integrable g\nhfg : f \u2264\u1d50[\u03bc] g\nhm : m \u2264 m0\n\u22a2 \u03bc[f|m] \u2264\u1d50[\u03bc] \u03bc[g|m]\n\ncase neg\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\ninst\u271d\u2077 : NormedAddCommGroup F'\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedLatticeAddCommGroup E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : OrderedSMul \u211d E\nf g : \u03b1 \u2192 E\nhf : Integrable f\nhg : Integrable g\nhfg : f \u2264\u1d50[\u03bc] g\nhm : \u00acm \u2264 m0\n\u22a2 \u03bc[f|m] \u2264\u1d50[\u03bc] \u03bc[g|m]", "state_after": "case neg\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\ninst\u271d\u2077 : NormedAddCommGroup F'\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedLatticeAddCommGroup E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : OrderedSMul \u211d E\nf g : \u03b1 \u2192 E\nhf : Integrable f\nhg : Integrable g\nhfg : f \u2264\u1d50[\u03bc] g\nhm : \u00acm \u2264 m0\n\u22a2 \u03bc[f|m] \u2264\u1d50[\u03bc] \u03bc[g|m]\n\ncase pos\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\ninst\u271d\u2077 : NormedAddCommGroup F'\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedLatticeAddCommGroup E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : OrderedSMul \u211d E\nf g : \u03b1 \u2192 E\nhf : Integrable f\nhg : Integrable g\nhfg : f \u2264\u1d50[\u03bc] g\nhm : m \u2264 m0\n\u22a2 \u03bc[f|m] \u2264\u1d50[\u03bc] \u03bc[g|m]"}, {"tactic": "by_cases h\u03bcm : SigmaFinite (\u03bc.trim hm)", "annotated_tactic": ["by_cases h\u03bcm : <a>SigmaFinite</a> (\u03bc.trim hm)", [{"full_name": "MeasureTheory.SigmaFinite", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3289, 7], "def_end_pos": [3289, 18]}]], "state_before": "case pos\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\ninst\u271d\u2077 : NormedAddCommGroup F'\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedLatticeAddCommGroup E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : OrderedSMul \u211d E\nf g : \u03b1 \u2192 E\nhf : Integrable f\nhg : Integrable g\nhfg : f \u2264\u1d50[\u03bc] g\nhm : m \u2264 m0\n\u22a2 \u03bc[f|m] \u2264\u1d50[\u03bc] \u03bc[g|m]", "state_after": "case pos\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\ninst\u271d\u2077 : NormedAddCommGroup F'\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedLatticeAddCommGroup E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : OrderedSMul \u211d E\nf g : \u03b1 \u2192 E\nhf : Integrable f\nhg : Integrable g\nhfg : f \u2264\u1d50[\u03bc] g\nhm : m \u2264 m0\nh\u03bcm : SigmaFinite (Measure.trim \u03bc hm)\n\u22a2 \u03bc[f|m] \u2264\u1d50[\u03bc] \u03bc[g|m]\n\ncase neg\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\ninst\u271d\u2077 : NormedAddCommGroup F'\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedLatticeAddCommGroup E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : OrderedSMul \u211d E\nf g : \u03b1 \u2192 E\nhf : Integrable f\nhg : Integrable g\nhfg : f \u2264\u1d50[\u03bc] g\nhm : m \u2264 m0\nh\u03bcm : \u00acSigmaFinite (Measure.trim \u03bc hm)\n\u22a2 \u03bc[f|m] \u2264\u1d50[\u03bc] \u03bc[g|m]"}, {"tactic": "swap", "annotated_tactic": ["swap", []], "state_before": "case pos\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\ninst\u271d\u2077 : NormedAddCommGroup F'\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedLatticeAddCommGroup E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : OrderedSMul \u211d E\nf g : \u03b1 \u2192 E\nhf : Integrable f\nhg : Integrable g\nhfg : f \u2264\u1d50[\u03bc] g\nhm : m \u2264 m0\nh\u03bcm : SigmaFinite (Measure.trim \u03bc hm)\n\u22a2 \u03bc[f|m] \u2264\u1d50[\u03bc] \u03bc[g|m]\n\ncase neg\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\ninst\u271d\u2077 : NormedAddCommGroup F'\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedLatticeAddCommGroup E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : OrderedSMul \u211d E\nf g : \u03b1 \u2192 E\nhf : Integrable f\nhg : Integrable g\nhfg : f \u2264\u1d50[\u03bc] g\nhm : m \u2264 m0\nh\u03bcm : \u00acSigmaFinite (Measure.trim \u03bc hm)\n\u22a2 \u03bc[f|m] \u2264\u1d50[\u03bc] \u03bc[g|m]", "state_after": "case neg\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\ninst\u271d\u2077 : NormedAddCommGroup F'\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedLatticeAddCommGroup E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : OrderedSMul \u211d E\nf g : \u03b1 \u2192 E\nhf : Integrable f\nhg : Integrable g\nhfg : f \u2264\u1d50[\u03bc] g\nhm : m \u2264 m0\nh\u03bcm : \u00acSigmaFinite (Measure.trim \u03bc hm)\n\u22a2 \u03bc[f|m] \u2264\u1d50[\u03bc] \u03bc[g|m]\n\ncase pos\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\ninst\u271d\u2077 : NormedAddCommGroup F'\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedLatticeAddCommGroup E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : OrderedSMul \u211d E\nf g : \u03b1 \u2192 E\nhf : Integrable f\nhg : Integrable g\nhfg : f \u2264\u1d50[\u03bc] g\nhm : m \u2264 m0\nh\u03bcm : SigmaFinite (Measure.trim \u03bc hm)\n\u22a2 \u03bc[f|m] \u2264\u1d50[\u03bc] \u03bc[g|m]"}, {"tactic": "haveI : SigmaFinite (\u03bc.trim hm) := h\u03bcm", "annotated_tactic": ["haveI : <a>SigmaFinite</a> (\u03bc.trim hm) := h\u03bcm", [{"full_name": "MeasureTheory.SigmaFinite", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3289, 7], "def_end_pos": [3289, 18]}]], "state_before": "case pos\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\ninst\u271d\u2077 : NormedAddCommGroup F'\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedLatticeAddCommGroup E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : OrderedSMul \u211d E\nf g : \u03b1 \u2192 E\nhf : Integrable f\nhg : Integrable g\nhfg : f \u2264\u1d50[\u03bc] g\nhm : m \u2264 m0\nh\u03bcm : SigmaFinite (Measure.trim \u03bc hm)\n\u22a2 \u03bc[f|m] \u2264\u1d50[\u03bc] \u03bc[g|m]", "state_after": "case pos\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\ninst\u271d\u2077 : NormedAddCommGroup F'\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedLatticeAddCommGroup E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : OrderedSMul \u211d E\nf g : \u03b1 \u2192 E\nhf : Integrable f\nhg : Integrable g\nhfg : f \u2264\u1d50[\u03bc] g\nhm : m \u2264 m0\nh\u03bcm this : SigmaFinite (Measure.trim \u03bc hm)\n\u22a2 \u03bc[f|m] \u2264\u1d50[\u03bc] \u03bc[g|m]"}, {"tactic": "exact (condexp_ae_eq_condexpL1 hm _).trans_le\n  ((condexpL1_mono hf hg hfg).trans_eq (condexp_ae_eq_condexpL1 hm _).symm)", "annotated_tactic": ["exact (<a>condexp_ae_eq_condexpL1</a> hm _).<a>trans_le</a>\n    ((<a>condexpL1_mono</a> hf hg hfg).<a>trans_eq</a> (<a>condexp_ae_eq_condexpL1</a> hm _).<a>symm</a>)", [{"full_name": "MeasureTheory.condexp_ae_eq_condexpL1", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean", "def_pos": [136, 9], "def_end_pos": [136, 32]}, {"full_name": "Filter.EventuallyEq.trans_le", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1684, 9], "def_end_pos": [1684, 30]}, {"full_name": "MeasureTheory.condexpL1_mono", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean", "def_pos": [592, 9], "def_end_pos": [592, 23]}, {"full_name": "Filter.EventuallyLE.trans_eq", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1693, 9], "def_end_pos": [1693, 30]}, {"full_name": "MeasureTheory.condexp_ae_eq_condexpL1", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean", "def_pos": [136, 9], "def_end_pos": [136, 32]}, {"full_name": "Filter.EventuallyEq.symm", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1498, 9], "def_end_pos": [1498, 26]}]], "state_before": "case pos\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\ninst\u271d\u2077 : NormedAddCommGroup F'\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedLatticeAddCommGroup E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : OrderedSMul \u211d E\nf g : \u03b1 \u2192 E\nhf : Integrable f\nhg : Integrable g\nhfg : f \u2264\u1d50[\u03bc] g\nhm : m \u2264 m0\nh\u03bcm this : SigmaFinite (Measure.trim \u03bc hm)\n\u22a2 \u03bc[f|m] \u2264\u1d50[\u03bc] \u03bc[g|m]", "state_after": "no goals"}, {"tactic": "simp_rw [condexp_of_not_le hm]", "annotated_tactic": ["simp_rw [<a>condexp_of_not_le</a> hm]", [{"full_name": "MeasureTheory.condexp_of_not_le", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean", "def_pos": [106, 9], "def_end_pos": [106, 26]}]], "state_before": "case neg\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\ninst\u271d\u2077 : NormedAddCommGroup F'\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedLatticeAddCommGroup E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : OrderedSMul \u211d E\nf g : \u03b1 \u2192 E\nhf : Integrable f\nhg : Integrable g\nhfg : f \u2264\u1d50[\u03bc] g\nhm : \u00acm \u2264 m0\n\u22a2 \u03bc[f|m] \u2264\u1d50[\u03bc] \u03bc[g|m]", "state_after": "case neg\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\ninst\u271d\u2077 : NormedAddCommGroup F'\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedLatticeAddCommGroup E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : OrderedSMul \u211d E\nf g : \u03b1 \u2192 E\nhf : Integrable f\nhg : Integrable g\nhfg : f \u2264\u1d50[\u03bc] g\nhm : \u00acm \u2264 m0\n\u22a2 0 \u2264\u1d50[\u03bc] 0"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case neg\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\ninst\u271d\u2077 : NormedAddCommGroup F'\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedLatticeAddCommGroup E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : OrderedSMul \u211d E\nf g : \u03b1 \u2192 E\nhf : Integrable f\nhg : Integrable g\nhfg : f \u2264\u1d50[\u03bc] g\nhm : \u00acm \u2264 m0\n\u22a2 0 \u2264\u1d50[\u03bc] 0", "state_after": "no goals"}, {"tactic": "simp_rw [condexp_of_not_sigmaFinite hm h\u03bcm]", "annotated_tactic": ["simp_rw [<a>condexp_of_not_sigmaFinite</a> hm h\u03bcm]", [{"full_name": "MeasureTheory.condexp_of_not_sigmaFinite", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean", "def_pos": [109, 9], "def_end_pos": [109, 35]}]], "state_before": "case neg\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\ninst\u271d\u2077 : NormedAddCommGroup F'\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedLatticeAddCommGroup E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : OrderedSMul \u211d E\nf g : \u03b1 \u2192 E\nhf : Integrable f\nhg : Integrable g\nhfg : f \u2264\u1d50[\u03bc] g\nhm : m \u2264 m0\nh\u03bcm : \u00acSigmaFinite (Measure.trim \u03bc hm)\n\u22a2 \u03bc[f|m] \u2264\u1d50[\u03bc] \u03bc[g|m]", "state_after": "case neg\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\ninst\u271d\u2077 : NormedAddCommGroup F'\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedLatticeAddCommGroup E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : OrderedSMul \u211d E\nf g : \u03b1 \u2192 E\nhf : Integrable f\nhg : Integrable g\nhfg : f \u2264\u1d50[\u03bc] g\nhm : m \u2264 m0\nh\u03bcm : \u00acSigmaFinite (Measure.trim \u03bc hm)\n\u22a2 0 \u2264\u1d50[\u03bc] 0"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case neg\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\ninst\u271d\u2077 : NormedAddCommGroup F'\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedLatticeAddCommGroup E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : OrderedSMul \u211d E\nf g : \u03b1 \u2192 E\nhf : Integrable f\nhg : Integrable g\nhfg : f \u2264\u1d50[\u03bc] g\nhm : m \u2264 m0\nh\u03bcm : \u00acSigmaFinite (Measure.trim \u03bc hm)\n\u22a2 0 \u2264\u1d50[\u03bc] 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Constructions/Prod/Basic.lean", "full_name": "MeasureTheory.MeasurePreserving.skew_product", "start": [718, 1], "end": [744, 40], "traced_tactics": [{"tactic": "have : Measurable fun p : \u03b1 \u00d7 \u03b3 => (f p.1, g p.1 p.2) := (hf.1.comp measurable_fst).prod_mk hgm", "annotated_tactic": ["have : <a>Measurable</a> fun p : \u03b1 \u00d7 \u03b3 => (f p.1, g p.1 p.2) := (hf.1.<a>comp</a> <a>measurable_fst</a>).<a>prod_mk</a> hgm", [{"full_name": "Measurable", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [535, 5], "def_end_pos": [535, 15]}, {"full_name": "Measurable.comp", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [557, 19], "def_end_pos": [557, 34]}, {"full_name": "measurable_fst", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [692, 9], "def_end_pos": [692, 23]}, {"full_name": "Measurable.prod_mk", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [726, 9], "def_end_pos": [726, 27]}]], "state_before": "\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03b4 : Type u_7\ninst\u271d\u00b2 : MeasurableSpace \u03b4\n\u03bca : Measure \u03b1\n\u03bcb : Measure \u03b2\n\u03bcc : Measure \u03b3\n\u03bcd : Measure \u03b4\ninst\u271d\u00b9 : SigmaFinite \u03bcb\ninst\u271d : SigmaFinite \u03bcd\nf : \u03b1 \u2192 \u03b2\nhf : MeasurePreserving f\ng : \u03b1 \u2192 \u03b3 \u2192 \u03b4\nhgm : Measurable (uncurry g)\nhg : \u2200\u1d50 (x : \u03b1) \u2202\u03bca, map (g x) \u03bcc = \u03bcd\n\u22a2 MeasurePreserving fun p => (f p.1, g p.1 p.2)", "state_after": "\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03b4 : Type u_7\ninst\u271d\u00b2 : MeasurableSpace \u03b4\n\u03bca : Measure \u03b1\n\u03bcb : Measure \u03b2\n\u03bcc : Measure \u03b3\n\u03bcd : Measure \u03b4\ninst\u271d\u00b9 : SigmaFinite \u03bcb\ninst\u271d : SigmaFinite \u03bcd\nf : \u03b1 \u2192 \u03b2\nhf : MeasurePreserving f\ng : \u03b1 \u2192 \u03b3 \u2192 \u03b4\nhgm : Measurable (uncurry g)\nhg : \u2200\u1d50 (x : \u03b1) \u2202\u03bca, map (g x) \u03bcc = \u03bcd\nthis : Measurable fun p => (f p.1, g p.1 p.2)\n\u22a2 MeasurePreserving fun p => (f p.1, g p.1 p.2)"}, {"tactic": "rcases eq_or_ne \u03bca 0 with (rfl | ha)", "annotated_tactic": ["rcases <a>eq_or_ne</a> \u03bca 0 with (rfl | ha)", [{"full_name": "eq_or_ne", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [209, 9], "def_end_pos": [209, 17]}]], "state_before": "\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03b4 : Type u_7\ninst\u271d\u00b2 : MeasurableSpace \u03b4\n\u03bca : Measure \u03b1\n\u03bcb : Measure \u03b2\n\u03bcc : Measure \u03b3\n\u03bcd : Measure \u03b4\ninst\u271d\u00b9 : SigmaFinite \u03bcb\ninst\u271d : SigmaFinite \u03bcd\nf : \u03b1 \u2192 \u03b2\nhf : MeasurePreserving f\ng : \u03b1 \u2192 \u03b3 \u2192 \u03b4\nhgm : Measurable (uncurry g)\nhg : \u2200\u1d50 (x : \u03b1) \u2202\u03bca, map (g x) \u03bcc = \u03bcd\nthis : Measurable fun p => (f p.1, g p.1 p.2)\n\u22a2 MeasurePreserving fun p => (f p.1, g p.1 p.2)", "state_after": "case inl\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03b4 : Type u_7\ninst\u271d\u00b2 : MeasurableSpace \u03b4\n\u03bcb : Measure \u03b2\n\u03bcc : Measure \u03b3\n\u03bcd : Measure \u03b4\ninst\u271d\u00b9 : SigmaFinite \u03bcb\ninst\u271d : SigmaFinite \u03bcd\nf : \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b3 \u2192 \u03b4\nhgm : Measurable (uncurry g)\nthis : Measurable fun p => (f p.1, g p.1 p.2)\nhf : MeasurePreserving f\nhg : \u2200\u1d50 (x : \u03b1) \u22020, map (g x) \u03bcc = \u03bcd\n\u22a2 MeasurePreserving fun p => (f p.1, g p.1 p.2)\n\ncase inr\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03b4 : Type u_7\ninst\u271d\u00b2 : MeasurableSpace \u03b4\n\u03bca : Measure \u03b1\n\u03bcb : Measure \u03b2\n\u03bcc : Measure \u03b3\n\u03bcd : Measure \u03b4\ninst\u271d\u00b9 : SigmaFinite \u03bcb\ninst\u271d : SigmaFinite \u03bcd\nf : \u03b1 \u2192 \u03b2\nhf : MeasurePreserving f\ng : \u03b1 \u2192 \u03b3 \u2192 \u03b4\nhgm : Measurable (uncurry g)\nhg : \u2200\u1d50 (x : \u03b1) \u2202\u03bca, map (g x) \u03bcc = \u03bcd\nthis : Measurable fun p => (f p.1, g p.1 p.2)\nha : \u03bca \u2260 0\n\u22a2 MeasurePreserving fun p => (f p.1, g p.1 p.2)"}, {"tactic": "have sf : SigmaFinite \u03bcc := by\n  rcases (ae_neBot.2 ha).nonempty_of_mem hg with \u27e8x, hx : map (g x) \u03bcc = \u03bcd\u27e9\n  exact SigmaFinite.of_map _ hgm.of_uncurry_left.aemeasurable (by rwa [hx])", "annotated_tactic": ["have sf : <a>SigmaFinite</a> \u03bcc := by\n      rcases (<a>ae_neBot</a>.2 ha).<a>nonempty_of_mem</a> hg with \u27e8x, hx : <a>map</a> (g x) \u03bcc = \u03bcd\u27e9\n      exact <a>SigmaFinite.of_map</a> _ hgm.of_uncurry_left.aemeasurable (by rwa [hx])", [{"full_name": "MeasureTheory.SigmaFinite", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3289, 7], "def_end_pos": [3289, 18]}, {"full_name": "MeasureTheory.ae_neBot", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2444, 9], "def_end_pos": [2444, 17]}, {"full_name": "Filter.NeBot.nonempty_of_mem", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [701, 9], "def_end_pos": [701, 30]}, {"full_name": "MeasureTheory.Measure.map", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1163, 17], "def_end_pos": [1163, 20]}, {"full_name": "MeasureTheory.SigmaFinite.of_map", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3767, 9], "def_end_pos": [3767, 27]}]], "state_before": "case inr\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03b4 : Type u_7\ninst\u271d\u00b2 : MeasurableSpace \u03b4\n\u03bca : Measure \u03b1\n\u03bcb : Measure \u03b2\n\u03bcc : Measure \u03b3\n\u03bcd : Measure \u03b4\ninst\u271d\u00b9 : SigmaFinite \u03bcb\ninst\u271d : SigmaFinite \u03bcd\nf : \u03b1 \u2192 \u03b2\nhf : MeasurePreserving f\ng : \u03b1 \u2192 \u03b3 \u2192 \u03b4\nhgm : Measurable (uncurry g)\nhg : \u2200\u1d50 (x : \u03b1) \u2202\u03bca, map (g x) \u03bcc = \u03bcd\nthis : Measurable fun p => (f p.1, g p.1 p.2)\nha : \u03bca \u2260 0\n\u22a2 MeasurePreserving fun p => (f p.1, g p.1 p.2)", "state_after": "case inr\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03b4 : Type u_7\ninst\u271d\u00b2 : MeasurableSpace \u03b4\n\u03bca : Measure \u03b1\n\u03bcb : Measure \u03b2\n\u03bcc : Measure \u03b3\n\u03bcd : Measure \u03b4\ninst\u271d\u00b9 : SigmaFinite \u03bcb\ninst\u271d : SigmaFinite \u03bcd\nf : \u03b1 \u2192 \u03b2\nhf : MeasurePreserving f\ng : \u03b1 \u2192 \u03b3 \u2192 \u03b4\nhgm : Measurable (uncurry g)\nhg : \u2200\u1d50 (x : \u03b1) \u2202\u03bca, map (g x) \u03bcc = \u03bcd\nthis : Measurable fun p => (f p.1, g p.1 p.2)\nha : \u03bca \u2260 0\nsf : SigmaFinite \u03bcc\n\u22a2 MeasurePreserving fun p => (f p.1, g p.1 p.2)"}, {"tactic": "refine' \u27e8this, (prod_eq fun s t hs ht => _).symm\u27e9", "annotated_tactic": ["refine' \u27e8this, (<a>prod_eq</a> fun s t hs ht => _).<a>symm</a>\u27e9", [{"full_name": "MeasureTheory.Measure.prod_eq", "def_path": "Mathlib/MeasureTheory/Constructions/Prod/Basic.lean", "def_pos": [565, 9], "def_end_pos": [565, 16]}, {"full_name": "Eq.symm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [310, 9], "def_end_pos": [310, 16]}]], "state_before": "case inr\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03b4 : Type u_7\ninst\u271d\u00b2 : MeasurableSpace \u03b4\n\u03bca : Measure \u03b1\n\u03bcb : Measure \u03b2\n\u03bcc : Measure \u03b3\n\u03bcd : Measure \u03b4\ninst\u271d\u00b9 : SigmaFinite \u03bcb\ninst\u271d : SigmaFinite \u03bcd\nf : \u03b1 \u2192 \u03b2\nhf : MeasurePreserving f\ng : \u03b1 \u2192 \u03b3 \u2192 \u03b4\nhgm : Measurable (uncurry g)\nhg : \u2200\u1d50 (x : \u03b1) \u2202\u03bca, map (g x) \u03bcc = \u03bcd\nthis : Measurable fun p => (f p.1, g p.1 p.2)\nha : \u03bca \u2260 0\nsf : SigmaFinite \u03bcc\n\u22a2 MeasurePreserving fun p => (f p.1, g p.1 p.2)", "state_after": "case inr\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03b4 : Type u_7\ninst\u271d\u00b2 : MeasurableSpace \u03b4\n\u03bca : Measure \u03b1\n\u03bcb : Measure \u03b2\n\u03bcc : Measure \u03b3\n\u03bcd : Measure \u03b4\ninst\u271d\u00b9 : SigmaFinite \u03bcb\ninst\u271d : SigmaFinite \u03bcd\nf : \u03b1 \u2192 \u03b2\nhf : MeasurePreserving f\ng : \u03b1 \u2192 \u03b3 \u2192 \u03b4\nhgm : Measurable (uncurry g)\nhg : \u2200\u1d50 (x : \u03b1) \u2202\u03bca, map (g x) \u03bcc = \u03bcd\nthis : Measurable fun p => (f p.1, g p.1 p.2)\nha : \u03bca \u2260 0\nsf : SigmaFinite \u03bcc\ns : Set \u03b2\nt : Set \u03b4\nhs : MeasurableSet s\nht : MeasurableSet t\n\u22a2 \u2191\u2191(map (fun p => (f p.1, g p.1 p.2)) (Measure.prod \u03bca \u03bcc)) (s \u00d7\u02e2 t) = \u2191\u2191\u03bcb s * \u2191\u2191\u03bcd t"}, {"tactic": "rw [map_apply this (hs.prod ht)]", "annotated_tactic": ["rw [<a>map_apply</a> this (hs.prod ht)]", [{"full_name": "MeasureTheory.Measure.map_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1236, 9], "def_end_pos": [1236, 18]}]], "state_before": "case inr\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03b4 : Type u_7\ninst\u271d\u00b2 : MeasurableSpace \u03b4\n\u03bca : Measure \u03b1\n\u03bcb : Measure \u03b2\n\u03bcc : Measure \u03b3\n\u03bcd : Measure \u03b4\ninst\u271d\u00b9 : SigmaFinite \u03bcb\ninst\u271d : SigmaFinite \u03bcd\nf : \u03b1 \u2192 \u03b2\nhf : MeasurePreserving f\ng : \u03b1 \u2192 \u03b3 \u2192 \u03b4\nhgm : Measurable (uncurry g)\nhg : \u2200\u1d50 (x : \u03b1) \u2202\u03bca, map (g x) \u03bcc = \u03bcd\nthis : Measurable fun p => (f p.1, g p.1 p.2)\nha : \u03bca \u2260 0\nsf : SigmaFinite \u03bcc\ns : Set \u03b2\nt : Set \u03b4\nhs : MeasurableSet s\nht : MeasurableSet t\n\u22a2 \u2191\u2191(map (fun p => (f p.1, g p.1 p.2)) (Measure.prod \u03bca \u03bcc)) (s \u00d7\u02e2 t) = \u2191\u2191\u03bcb s * \u2191\u2191\u03bcd t", "state_after": "case inr\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03b4 : Type u_7\ninst\u271d\u00b2 : MeasurableSpace \u03b4\n\u03bca : Measure \u03b1\n\u03bcb : Measure \u03b2\n\u03bcc : Measure \u03b3\n\u03bcd : Measure \u03b4\ninst\u271d\u00b9 : SigmaFinite \u03bcb\ninst\u271d : SigmaFinite \u03bcd\nf : \u03b1 \u2192 \u03b2\nhf : MeasurePreserving f\ng : \u03b1 \u2192 \u03b3 \u2192 \u03b4\nhgm : Measurable (uncurry g)\nhg : \u2200\u1d50 (x : \u03b1) \u2202\u03bca, map (g x) \u03bcc = \u03bcd\nthis : Measurable fun p => (f p.1, g p.1 p.2)\nha : \u03bca \u2260 0\nsf : SigmaFinite \u03bcc\ns : Set \u03b2\nt : Set \u03b4\nhs : MeasurableSet s\nht : MeasurableSet t\n\u22a2 \u2191\u2191(Measure.prod \u03bca \u03bcc) ((fun p => (f p.1, g p.1 p.2)) \u207b\u00b9' s \u00d7\u02e2 t) = \u2191\u2191\u03bcb s * \u2191\u2191\u03bcd t"}, {"tactic": "refine' (prod_apply (this <| hs.prod ht)).trans _", "annotated_tactic": ["refine' (<a>prod_apply</a> (this <| hs.prod ht)).<a>trans</a> _", [{"full_name": "MeasureTheory.Measure.prod_apply", "def_path": "Mathlib/MeasureTheory/Constructions/Prod/Basic.lean", "def_pos": [307, 9], "def_end_pos": [307, 19]}, {"full_name": "Eq.trans", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [322, 9], "def_end_pos": [322, 17]}]], "state_before": "case inr\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03b4 : Type u_7\ninst\u271d\u00b2 : MeasurableSpace \u03b4\n\u03bca : Measure \u03b1\n\u03bcb : Measure \u03b2\n\u03bcc : Measure \u03b3\n\u03bcd : Measure \u03b4\ninst\u271d\u00b9 : SigmaFinite \u03bcb\ninst\u271d : SigmaFinite \u03bcd\nf : \u03b1 \u2192 \u03b2\nhf : MeasurePreserving f\ng : \u03b1 \u2192 \u03b3 \u2192 \u03b4\nhgm : Measurable (uncurry g)\nhg : \u2200\u1d50 (x : \u03b1) \u2202\u03bca, map (g x) \u03bcc = \u03bcd\nthis : Measurable fun p => (f p.1, g p.1 p.2)\nha : \u03bca \u2260 0\nsf : SigmaFinite \u03bcc\ns : Set \u03b2\nt : Set \u03b4\nhs : MeasurableSet s\nht : MeasurableSet t\n\u22a2 \u2191\u2191(Measure.prod \u03bca \u03bcc) ((fun p => (f p.1, g p.1 p.2)) \u207b\u00b9' s \u00d7\u02e2 t) = \u2191\u2191\u03bcb s * \u2191\u2191\u03bcd t", "state_after": "case inr\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03b4 : Type u_7\ninst\u271d\u00b2 : MeasurableSpace \u03b4\n\u03bca : Measure \u03b1\n\u03bcb : Measure \u03b2\n\u03bcc : Measure \u03b3\n\u03bcd : Measure \u03b4\ninst\u271d\u00b9 : SigmaFinite \u03bcb\ninst\u271d : SigmaFinite \u03bcd\nf : \u03b1 \u2192 \u03b2\nhf : MeasurePreserving f\ng : \u03b1 \u2192 \u03b3 \u2192 \u03b4\nhgm : Measurable (uncurry g)\nhg : \u2200\u1d50 (x : \u03b1) \u2202\u03bca, map (g x) \u03bcc = \u03bcd\nthis : Measurable fun p => (f p.1, g p.1 p.2)\nha : \u03bca \u2260 0\nsf : SigmaFinite \u03bcc\ns : Set \u03b2\nt : Set \u03b4\nhs : MeasurableSet s\nht : MeasurableSet t\n\u22a2 \u222b\u207b (x : \u03b1), \u2191\u2191\u03bcc (Prod.mk x \u207b\u00b9' ((fun p => (f p.1, g p.1 p.2)) \u207b\u00b9' s \u00d7\u02e2 t)) \u2202\u03bca = \u2191\u2191\u03bcb s * \u2191\u2191\u03bcd t"}, {"tactic": "have : \u2200\u1d50 x \u2202\u03bca,\n    \u03bcc ((fun y => (f x, g x y)) \u207b\u00b9' s \u00d7\u02e2 t) = indicator (f \u207b\u00b9' s) (fun _ => \u03bcd t) x := by\n  refine' hg.mono fun x hx => _\n  subst hx\n  simp only [mk_preimage_prod_right_fn_eq_if, indicator_apply, mem_preimage]\n  split_ifs\n  exacts [(map_apply hgm.of_uncurry_left ht).symm, measure_empty]", "annotated_tactic": ["have : \u2200\u1d50 x \u2202\u03bca,\n        \u03bcc ((fun y => (f x, g x y)) \u207b\u00b9' s \u00d7\u02e2 t) = <a>indicator</a> (f \u207b\u00b9' s) (fun _ => \u03bcd t) x := by\n      refine' hg.mono fun x hx => _\n      subst hx\n      simp only [<a>mk_preimage_prod_right_fn_eq_if</a>, <a>indicator_apply</a>, <a>mem_preimage</a>]\n      split_ifs\n      exacts [(<a>map_apply</a> hgm.of_uncurry_left ht).<a>symm</a>, <a>measure_empty</a>]", [{"full_name": "Set.indicator", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [46, 3], "def_end_pos": [46, 14]}, {"full_name": "Set.mk_preimage_prod_right_fn_eq_if", "def_path": "Mathlib/Data/Set/Prod.lean", "def_pos": [286, 9], "def_end_pos": [286, 40]}, {"full_name": "Set.indicator_apply", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [59, 3], "def_end_pos": [59, 14]}, {"full_name": "Set.mem_preimage", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [64, 9], "def_end_pos": [64, 21]}, {"full_name": "MeasureTheory.Measure.map_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1236, 9], "def_end_pos": [1236, 18]}, {"full_name": "Eq.symm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [310, 9], "def_end_pos": [310, 16]}, {"full_name": "MeasureTheory.measure_empty", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [185, 9], "def_end_pos": [185, 22]}]], "state_before": "case inr\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03b4 : Type u_7\ninst\u271d\u00b2 : MeasurableSpace \u03b4\n\u03bca : Measure \u03b1\n\u03bcb : Measure \u03b2\n\u03bcc : Measure \u03b3\n\u03bcd : Measure \u03b4\ninst\u271d\u00b9 : SigmaFinite \u03bcb\ninst\u271d : SigmaFinite \u03bcd\nf : \u03b1 \u2192 \u03b2\nhf : MeasurePreserving f\ng : \u03b1 \u2192 \u03b3 \u2192 \u03b4\nhgm : Measurable (uncurry g)\nhg : \u2200\u1d50 (x : \u03b1) \u2202\u03bca, map (g x) \u03bcc = \u03bcd\nthis : Measurable fun p => (f p.1, g p.1 p.2)\nha : \u03bca \u2260 0\nsf : SigmaFinite \u03bcc\ns : Set \u03b2\nt : Set \u03b4\nhs : MeasurableSet s\nht : MeasurableSet t\n\u22a2 \u222b\u207b (x : \u03b1), \u2191\u2191\u03bcc (Prod.mk x \u207b\u00b9' ((fun p => (f p.1, g p.1 p.2)) \u207b\u00b9' s \u00d7\u02e2 t)) \u2202\u03bca = \u2191\u2191\u03bcb s * \u2191\u2191\u03bcd t", "state_after": "case inr\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03b4 : Type u_7\ninst\u271d\u00b2 : MeasurableSpace \u03b4\n\u03bca : Measure \u03b1\n\u03bcb : Measure \u03b2\n\u03bcc : Measure \u03b3\n\u03bcd : Measure \u03b4\ninst\u271d\u00b9 : SigmaFinite \u03bcb\ninst\u271d : SigmaFinite \u03bcd\nf : \u03b1 \u2192 \u03b2\nhf : MeasurePreserving f\ng : \u03b1 \u2192 \u03b3 \u2192 \u03b4\nhgm : Measurable (uncurry g)\nhg : \u2200\u1d50 (x : \u03b1) \u2202\u03bca, map (g x) \u03bcc = \u03bcd\nthis\u271d : Measurable fun p => (f p.1, g p.1 p.2)\nha : \u03bca \u2260 0\nsf : SigmaFinite \u03bcc\ns : Set \u03b2\nt : Set \u03b4\nhs : MeasurableSet s\nht : MeasurableSet t\nthis : \u2200\u1d50 (x : \u03b1) \u2202\u03bca, \u2191\u2191\u03bcc ((fun y => (f x, g x y)) \u207b\u00b9' s \u00d7\u02e2 t) = indicator (f \u207b\u00b9' s) (fun x => \u2191\u2191\u03bcd t) x\n\u22a2 \u222b\u207b (x : \u03b1), \u2191\u2191\u03bcc (Prod.mk x \u207b\u00b9' ((fun p => (f p.1, g p.1 p.2)) \u207b\u00b9' s \u00d7\u02e2 t)) \u2202\u03bca = \u2191\u2191\u03bcb s * \u2191\u2191\u03bcd t"}, {"tactic": "simp only [preimage_preimage]", "annotated_tactic": ["simp only [<a>preimage_preimage</a>]", [{"full_name": "Set.preimage_preimage", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [176, 9], "def_end_pos": [176, 26]}]], "state_before": "case inr\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03b4 : Type u_7\ninst\u271d\u00b2 : MeasurableSpace \u03b4\n\u03bca : Measure \u03b1\n\u03bcb : Measure \u03b2\n\u03bcc : Measure \u03b3\n\u03bcd : Measure \u03b4\ninst\u271d\u00b9 : SigmaFinite \u03bcb\ninst\u271d : SigmaFinite \u03bcd\nf : \u03b1 \u2192 \u03b2\nhf : MeasurePreserving f\ng : \u03b1 \u2192 \u03b3 \u2192 \u03b4\nhgm : Measurable (uncurry g)\nhg : \u2200\u1d50 (x : \u03b1) \u2202\u03bca, map (g x) \u03bcc = \u03bcd\nthis\u271d : Measurable fun p => (f p.1, g p.1 p.2)\nha : \u03bca \u2260 0\nsf : SigmaFinite \u03bcc\ns : Set \u03b2\nt : Set \u03b4\nhs : MeasurableSet s\nht : MeasurableSet t\nthis : \u2200\u1d50 (x : \u03b1) \u2202\u03bca, \u2191\u2191\u03bcc ((fun y => (f x, g x y)) \u207b\u00b9' s \u00d7\u02e2 t) = indicator (f \u207b\u00b9' s) (fun x => \u2191\u2191\u03bcd t) x\n\u22a2 \u222b\u207b (x : \u03b1), \u2191\u2191\u03bcc (Prod.mk x \u207b\u00b9' ((fun p => (f p.1, g p.1 p.2)) \u207b\u00b9' s \u00d7\u02e2 t)) \u2202\u03bca = \u2191\u2191\u03bcb s * \u2191\u2191\u03bcd t", "state_after": "case inr\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03b4 : Type u_7\ninst\u271d\u00b2 : MeasurableSpace \u03b4\n\u03bca : Measure \u03b1\n\u03bcb : Measure \u03b2\n\u03bcc : Measure \u03b3\n\u03bcd : Measure \u03b4\ninst\u271d\u00b9 : SigmaFinite \u03bcb\ninst\u271d : SigmaFinite \u03bcd\nf : \u03b1 \u2192 \u03b2\nhf : MeasurePreserving f\ng : \u03b1 \u2192 \u03b3 \u2192 \u03b4\nhgm : Measurable (uncurry g)\nhg : \u2200\u1d50 (x : \u03b1) \u2202\u03bca, map (g x) \u03bcc = \u03bcd\nthis\u271d : Measurable fun p => (f p.1, g p.1 p.2)\nha : \u03bca \u2260 0\nsf : SigmaFinite \u03bcc\ns : Set \u03b2\nt : Set \u03b4\nhs : MeasurableSet s\nht : MeasurableSet t\nthis : \u2200\u1d50 (x : \u03b1) \u2202\u03bca, \u2191\u2191\u03bcc ((fun y => (f x, g x y)) \u207b\u00b9' s \u00d7\u02e2 t) = indicator (f \u207b\u00b9' s) (fun x => \u2191\u2191\u03bcd t) x\n\u22a2 \u222b\u207b (x : \u03b1), \u2191\u2191\u03bcc ((fun x_1 => (f x, g x x_1)) \u207b\u00b9' s \u00d7\u02e2 t) \u2202\u03bca = \u2191\u2191\u03bcb s * \u2191\u2191\u03bcd t"}, {"tactic": "rw [lintegral_congr_ae this, lintegral_indicator _ (hf.1 hs), set_lintegral_const,\n  hf.measure_preimage hs, mul_comm]", "annotated_tactic": ["rw [<a>lintegral_congr_ae</a> this, <a>lintegral_indicator</a> _ (hf.1 hs), <a>set_lintegral_const</a>,\n      hf.measure_preimage hs, <a>mul_comm</a>]", [{"full_name": "MeasureTheory.lintegral_congr_ae", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [304, 9], "def_end_pos": [304, 27]}, {"full_name": "MeasureTheory.lintegral_indicator", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [762, 9], "def_end_pos": [762, 28]}, {"full_name": "MeasureTheory.set_lintegral_const", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [152, 9], "def_end_pos": [152, 28]}, {"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [302, 9], "def_end_pos": [302, 17]}]], "state_before": "case inr\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03b4 : Type u_7\ninst\u271d\u00b2 : MeasurableSpace \u03b4\n\u03bca : Measure \u03b1\n\u03bcb : Measure \u03b2\n\u03bcc : Measure \u03b3\n\u03bcd : Measure \u03b4\ninst\u271d\u00b9 : SigmaFinite \u03bcb\ninst\u271d : SigmaFinite \u03bcd\nf : \u03b1 \u2192 \u03b2\nhf : MeasurePreserving f\ng : \u03b1 \u2192 \u03b3 \u2192 \u03b4\nhgm : Measurable (uncurry g)\nhg : \u2200\u1d50 (x : \u03b1) \u2202\u03bca, map (g x) \u03bcc = \u03bcd\nthis\u271d : Measurable fun p => (f p.1, g p.1 p.2)\nha : \u03bca \u2260 0\nsf : SigmaFinite \u03bcc\ns : Set \u03b2\nt : Set \u03b4\nhs : MeasurableSet s\nht : MeasurableSet t\nthis : \u2200\u1d50 (x : \u03b1) \u2202\u03bca, \u2191\u2191\u03bcc ((fun y => (f x, g x y)) \u207b\u00b9' s \u00d7\u02e2 t) = indicator (f \u207b\u00b9' s) (fun x => \u2191\u2191\u03bcd t) x\n\u22a2 \u222b\u207b (x : \u03b1), \u2191\u2191\u03bcc ((fun x_1 => (f x, g x x_1)) \u207b\u00b9' s \u00d7\u02e2 t) \u2202\u03bca = \u2191\u2191\u03bcb s * \u2191\u2191\u03bcd t", "state_after": "no goals"}, {"tactic": "exact \u27e8this, by simp only [Measure.map_zero]\u27e9", "annotated_tactic": ["exact \u27e8this, by simp only [<a>Measure.map_zero</a>]\u27e9", [{"full_name": "MeasureTheory.Measure.map_zero", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1183, 9], "def_end_pos": [1183, 17]}]], "state_before": "case inl\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03b4 : Type u_7\ninst\u271d\u00b2 : MeasurableSpace \u03b4\n\u03bcb : Measure \u03b2\n\u03bcc : Measure \u03b3\n\u03bcd : Measure \u03b4\ninst\u271d\u00b9 : SigmaFinite \u03bcb\ninst\u271d : SigmaFinite \u03bcd\nf : \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b3 \u2192 \u03b4\nhgm : Measurable (uncurry g)\nthis : Measurable fun p => (f p.1, g p.1 p.2)\nhf : MeasurePreserving f\nhg : \u2200\u1d50 (x : \u03b1) \u22020, map (g x) \u03bcc = \u03bcd\n\u22a2 MeasurePreserving fun p => (f p.1, g p.1 p.2)", "state_after": "no goals"}, {"tactic": "simp only [Measure.map_zero]", "annotated_tactic": ["simp only [<a>Measure.map_zero</a>]", [{"full_name": "MeasureTheory.Measure.map_zero", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1183, 9], "def_end_pos": [1183, 17]}]], "state_before": "\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03b4 : Type u_7\ninst\u271d\u00b2 : MeasurableSpace \u03b4\n\u03bcb : Measure \u03b2\n\u03bcc : Measure \u03b3\n\u03bcd : Measure \u03b4\ninst\u271d\u00b9 : SigmaFinite \u03bcb\ninst\u271d : SigmaFinite \u03bcd\nf : \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b3 \u2192 \u03b4\nhgm : Measurable (uncurry g)\nthis : Measurable fun p => (f p.1, g p.1 p.2)\nhf : MeasurePreserving f\nhg : \u2200\u1d50 (x : \u03b1) \u22020, map (g x) \u03bcc = \u03bcd\n\u22a2 map (fun p => (f p.1, g p.1 p.2)) 0 = 0", "state_after": "no goals"}, {"tactic": "rcases (ae_neBot.2 ha).nonempty_of_mem hg with \u27e8x, hx : map (g x) \u03bcc = \u03bcd\u27e9", "annotated_tactic": ["rcases (<a>ae_neBot</a>.2 ha).<a>nonempty_of_mem</a> hg with \u27e8x, hx : <a>map</a> (g x) \u03bcc = \u03bcd\u27e9", [{"full_name": "MeasureTheory.ae_neBot", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2444, 9], "def_end_pos": [2444, 17]}, {"full_name": "Filter.NeBot.nonempty_of_mem", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [701, 9], "def_end_pos": [701, 30]}, {"full_name": "MeasureTheory.Measure.map", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1163, 17], "def_end_pos": [1163, 20]}]], "state_before": "\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03b4 : Type u_7\ninst\u271d\u00b2 : MeasurableSpace \u03b4\n\u03bca : Measure \u03b1\n\u03bcb : Measure \u03b2\n\u03bcc : Measure \u03b3\n\u03bcd : Measure \u03b4\ninst\u271d\u00b9 : SigmaFinite \u03bcb\ninst\u271d : SigmaFinite \u03bcd\nf : \u03b1 \u2192 \u03b2\nhf : MeasurePreserving f\ng : \u03b1 \u2192 \u03b3 \u2192 \u03b4\nhgm : Measurable (uncurry g)\nhg : \u2200\u1d50 (x : \u03b1) \u2202\u03bca, map (g x) \u03bcc = \u03bcd\nthis : Measurable fun p => (f p.1, g p.1 p.2)\nha : \u03bca \u2260 0\n\u22a2 SigmaFinite \u03bcc", "state_after": "case intro\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03b4 : Type u_7\ninst\u271d\u00b2 : MeasurableSpace \u03b4\n\u03bca : Measure \u03b1\n\u03bcb : Measure \u03b2\n\u03bcc : Measure \u03b3\n\u03bcd : Measure \u03b4\ninst\u271d\u00b9 : SigmaFinite \u03bcb\ninst\u271d : SigmaFinite \u03bcd\nf : \u03b1 \u2192 \u03b2\nhf : MeasurePreserving f\ng : \u03b1 \u2192 \u03b3 \u2192 \u03b4\nhgm : Measurable (uncurry g)\nhg : \u2200\u1d50 (x : \u03b1) \u2202\u03bca, map (g x) \u03bcc = \u03bcd\nthis : Measurable fun p => (f p.1, g p.1 p.2)\nha : \u03bca \u2260 0\nx : \u03b1\nhx : map (g x) \u03bcc = \u03bcd\n\u22a2 SigmaFinite \u03bcc"}, {"tactic": "exact SigmaFinite.of_map _ hgm.of_uncurry_left.aemeasurable (by rwa [hx])", "annotated_tactic": ["exact <a>SigmaFinite.of_map</a> _ hgm.of_uncurry_left.aemeasurable (by rwa [hx])", [{"full_name": "MeasureTheory.SigmaFinite.of_map", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3767, 9], "def_end_pos": [3767, 27]}]], "state_before": "case intro\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03b4 : Type u_7\ninst\u271d\u00b2 : MeasurableSpace \u03b4\n\u03bca : Measure \u03b1\n\u03bcb : Measure \u03b2\n\u03bcc : Measure \u03b3\n\u03bcd : Measure \u03b4\ninst\u271d\u00b9 : SigmaFinite \u03bcb\ninst\u271d : SigmaFinite \u03bcd\nf : \u03b1 \u2192 \u03b2\nhf : MeasurePreserving f\ng : \u03b1 \u2192 \u03b3 \u2192 \u03b4\nhgm : Measurable (uncurry g)\nhg : \u2200\u1d50 (x : \u03b1) \u2202\u03bca, map (g x) \u03bcc = \u03bcd\nthis : Measurable fun p => (f p.1, g p.1 p.2)\nha : \u03bca \u2260 0\nx : \u03b1\nhx : map (g x) \u03bcc = \u03bcd\n\u22a2 SigmaFinite \u03bcc", "state_after": "no goals"}, {"tactic": "rwa [hx]", "annotated_tactic": ["rwa [hx]", []], "state_before": "\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03b4 : Type u_7\ninst\u271d\u00b2 : MeasurableSpace \u03b4\n\u03bca : Measure \u03b1\n\u03bcb : Measure \u03b2\n\u03bcc : Measure \u03b3\n\u03bcd : Measure \u03b4\ninst\u271d\u00b9 : SigmaFinite \u03bcb\ninst\u271d : SigmaFinite \u03bcd\nf : \u03b1 \u2192 \u03b2\nhf : MeasurePreserving f\ng : \u03b1 \u2192 \u03b3 \u2192 \u03b4\nhgm : Measurable (uncurry g)\nhg : \u2200\u1d50 (x : \u03b1) \u2202\u03bca, map (g x) \u03bcc = \u03bcd\nthis : Measurable fun p => (f p.1, g p.1 p.2)\nha : \u03bca \u2260 0\nx : \u03b1\nhx : map (g x) \u03bcc = \u03bcd\n\u22a2 SigmaFinite (map (g ?m.119880) \u03bcc)", "state_after": "no goals"}, {"tactic": "refine' hg.mono fun x hx => _", "annotated_tactic": ["refine' hg.mono fun x hx => _", []], "state_before": "\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03b4 : Type u_7\ninst\u271d\u00b2 : MeasurableSpace \u03b4\n\u03bca : Measure \u03b1\n\u03bcb : Measure \u03b2\n\u03bcc : Measure \u03b3\n\u03bcd : Measure \u03b4\ninst\u271d\u00b9 : SigmaFinite \u03bcb\ninst\u271d : SigmaFinite \u03bcd\nf : \u03b1 \u2192 \u03b2\nhf : MeasurePreserving f\ng : \u03b1 \u2192 \u03b3 \u2192 \u03b4\nhgm : Measurable (uncurry g)\nhg : \u2200\u1d50 (x : \u03b1) \u2202\u03bca, map (g x) \u03bcc = \u03bcd\nthis : Measurable fun p => (f p.1, g p.1 p.2)\nha : \u03bca \u2260 0\nsf : SigmaFinite \u03bcc\ns : Set \u03b2\nt : Set \u03b4\nhs : MeasurableSet s\nht : MeasurableSet t\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bca, \u2191\u2191\u03bcc ((fun y => (f x, g x y)) \u207b\u00b9' s \u00d7\u02e2 t) = indicator (f \u207b\u00b9' s) (fun x => \u2191\u2191\u03bcd t) x", "state_after": "\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03b4 : Type u_7\ninst\u271d\u00b2 : MeasurableSpace \u03b4\n\u03bca : Measure \u03b1\n\u03bcb : Measure \u03b2\n\u03bcc : Measure \u03b3\n\u03bcd : Measure \u03b4\ninst\u271d\u00b9 : SigmaFinite \u03bcb\ninst\u271d : SigmaFinite \u03bcd\nf : \u03b1 \u2192 \u03b2\nhf : MeasurePreserving f\ng : \u03b1 \u2192 \u03b3 \u2192 \u03b4\nhgm : Measurable (uncurry g)\nhg : \u2200\u1d50 (x : \u03b1) \u2202\u03bca, map (g x) \u03bcc = \u03bcd\nthis : Measurable fun p => (f p.1, g p.1 p.2)\nha : \u03bca \u2260 0\nsf : SigmaFinite \u03bcc\ns : Set \u03b2\nt : Set \u03b4\nhs : MeasurableSet s\nht : MeasurableSet t\nx : \u03b1\nhx : map (g x) \u03bcc = \u03bcd\n\u22a2 \u2191\u2191\u03bcc ((fun y => (f x, g x y)) \u207b\u00b9' s \u00d7\u02e2 t) = indicator (f \u207b\u00b9' s) (fun x => \u2191\u2191\u03bcd t) x"}, {"tactic": "subst hx", "annotated_tactic": ["subst hx", []], "state_before": "\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03b4 : Type u_7\ninst\u271d\u00b2 : MeasurableSpace \u03b4\n\u03bca : Measure \u03b1\n\u03bcb : Measure \u03b2\n\u03bcc : Measure \u03b3\n\u03bcd : Measure \u03b4\ninst\u271d\u00b9 : SigmaFinite \u03bcb\ninst\u271d : SigmaFinite \u03bcd\nf : \u03b1 \u2192 \u03b2\nhf : MeasurePreserving f\ng : \u03b1 \u2192 \u03b3 \u2192 \u03b4\nhgm : Measurable (uncurry g)\nhg : \u2200\u1d50 (x : \u03b1) \u2202\u03bca, map (g x) \u03bcc = \u03bcd\nthis : Measurable fun p => (f p.1, g p.1 p.2)\nha : \u03bca \u2260 0\nsf : SigmaFinite \u03bcc\ns : Set \u03b2\nt : Set \u03b4\nhs : MeasurableSet s\nht : MeasurableSet t\nx : \u03b1\nhx : map (g x) \u03bcc = \u03bcd\n\u22a2 \u2191\u2191\u03bcc ((fun y => (f x, g x y)) \u207b\u00b9' s \u00d7\u02e2 t) = indicator (f \u207b\u00b9' s) (fun x => \u2191\u2191\u03bcd t) x", "state_after": "\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03b4 : Type u_7\ninst\u271d\u00b2 : MeasurableSpace \u03b4\n\u03bca : Measure \u03b1\n\u03bcb : Measure \u03b2\n\u03bcc : Measure \u03b3\ninst\u271d\u00b9 : SigmaFinite \u03bcb\nf : \u03b1 \u2192 \u03b2\nhf : MeasurePreserving f\ng : \u03b1 \u2192 \u03b3 \u2192 \u03b4\nhgm : Measurable (uncurry g)\nthis : Measurable fun p => (f p.1, g p.1 p.2)\nha : \u03bca \u2260 0\nsf : SigmaFinite \u03bcc\ns : Set \u03b2\nt : Set \u03b4\nhs : MeasurableSet s\nht : MeasurableSet t\nx : \u03b1\ninst\u271d : SigmaFinite (map (g x) \u03bcc)\nhg : \u2200\u1d50 (x_1 : \u03b1) \u2202\u03bca, map (g x_1) \u03bcc = map (g x) \u03bcc\n\u22a2 \u2191\u2191\u03bcc ((fun y => (f x, g x y)) \u207b\u00b9' s \u00d7\u02e2 t) = indicator (f \u207b\u00b9' s) (fun x_1 => \u2191\u2191(map (g x) \u03bcc) t) x"}, {"tactic": "simp only [mk_preimage_prod_right_fn_eq_if, indicator_apply, mem_preimage]", "annotated_tactic": ["simp only [<a>mk_preimage_prod_right_fn_eq_if</a>, <a>indicator_apply</a>, <a>mem_preimage</a>]", [{"full_name": "Set.mk_preimage_prod_right_fn_eq_if", "def_path": "Mathlib/Data/Set/Prod.lean", "def_pos": [286, 9], "def_end_pos": [286, 40]}, {"full_name": "Set.indicator_apply", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [59, 3], "def_end_pos": [59, 14]}, {"full_name": "Set.mem_preimage", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [64, 9], "def_end_pos": [64, 21]}]], "state_before": "\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03b4 : Type u_7\ninst\u271d\u00b2 : MeasurableSpace \u03b4\n\u03bca : Measure \u03b1\n\u03bcb : Measure \u03b2\n\u03bcc : Measure \u03b3\ninst\u271d\u00b9 : SigmaFinite \u03bcb\nf : \u03b1 \u2192 \u03b2\nhf : MeasurePreserving f\ng : \u03b1 \u2192 \u03b3 \u2192 \u03b4\nhgm : Measurable (uncurry g)\nthis : Measurable fun p => (f p.1, g p.1 p.2)\nha : \u03bca \u2260 0\nsf : SigmaFinite \u03bcc\ns : Set \u03b2\nt : Set \u03b4\nhs : MeasurableSet s\nht : MeasurableSet t\nx : \u03b1\ninst\u271d : SigmaFinite (map (g x) \u03bcc)\nhg : \u2200\u1d50 (x_1 : \u03b1) \u2202\u03bca, map (g x_1) \u03bcc = map (g x) \u03bcc\n\u22a2 \u2191\u2191\u03bcc ((fun y => (f x, g x y)) \u207b\u00b9' s \u00d7\u02e2 t) = indicator (f \u207b\u00b9' s) (fun x_1 => \u2191\u2191(map (g x) \u03bcc) t) x", "state_after": "\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03b4 : Type u_7\ninst\u271d\u00b2 : MeasurableSpace \u03b4\n\u03bca : Measure \u03b1\n\u03bcb : Measure \u03b2\n\u03bcc : Measure \u03b3\ninst\u271d\u00b9 : SigmaFinite \u03bcb\nf : \u03b1 \u2192 \u03b2\nhf : MeasurePreserving f\ng : \u03b1 \u2192 \u03b3 \u2192 \u03b4\nhgm : Measurable (uncurry g)\nthis : Measurable fun p => (f p.1, g p.1 p.2)\nha : \u03bca \u2260 0\nsf : SigmaFinite \u03bcc\ns : Set \u03b2\nt : Set \u03b4\nhs : MeasurableSet s\nht : MeasurableSet t\nx : \u03b1\ninst\u271d : SigmaFinite (map (g x) \u03bcc)\nhg : \u2200\u1d50 (x_1 : \u03b1) \u2202\u03bca, map (g x_1) \u03bcc = map (g x) \u03bcc\n\u22a2 \u2191\u2191\u03bcc (if f x \u2208 s then (fun b => g x b) \u207b\u00b9' t else \u2205) = if f x \u2208 s then \u2191\u2191(map (g x) \u03bcc) t else 0"}, {"tactic": "split_ifs", "annotated_tactic": ["split_ifs", []], "state_before": "\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03b4 : Type u_7\ninst\u271d\u00b2 : MeasurableSpace \u03b4\n\u03bca : Measure \u03b1\n\u03bcb : Measure \u03b2\n\u03bcc : Measure \u03b3\ninst\u271d\u00b9 : SigmaFinite \u03bcb\nf : \u03b1 \u2192 \u03b2\nhf : MeasurePreserving f\ng : \u03b1 \u2192 \u03b3 \u2192 \u03b4\nhgm : Measurable (uncurry g)\nthis : Measurable fun p => (f p.1, g p.1 p.2)\nha : \u03bca \u2260 0\nsf : SigmaFinite \u03bcc\ns : Set \u03b2\nt : Set \u03b4\nhs : MeasurableSet s\nht : MeasurableSet t\nx : \u03b1\ninst\u271d : SigmaFinite (map (g x) \u03bcc)\nhg : \u2200\u1d50 (x_1 : \u03b1) \u2202\u03bca, map (g x_1) \u03bcc = map (g x) \u03bcc\n\u22a2 \u2191\u2191\u03bcc (if f x \u2208 s then (fun b => g x b) \u207b\u00b9' t else \u2205) = if f x \u2208 s then \u2191\u2191(map (g x) \u03bcc) t else 0", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03b4 : Type u_7\ninst\u271d\u00b2 : MeasurableSpace \u03b4\n\u03bca : Measure \u03b1\n\u03bcb : Measure \u03b2\n\u03bcc : Measure \u03b3\ninst\u271d\u00b9 : SigmaFinite \u03bcb\nf : \u03b1 \u2192 \u03b2\nhf : MeasurePreserving f\ng : \u03b1 \u2192 \u03b3 \u2192 \u03b4\nhgm : Measurable (uncurry g)\nthis : Measurable fun p => (f p.1, g p.1 p.2)\nha : \u03bca \u2260 0\nsf : SigmaFinite \u03bcc\ns : Set \u03b2\nt : Set \u03b4\nhs : MeasurableSet s\nht : MeasurableSet t\nx : \u03b1\ninst\u271d : SigmaFinite (map (g x) \u03bcc)\nhg : \u2200\u1d50 (x_1 : \u03b1) \u2202\u03bca, map (g x_1) \u03bcc = map (g x) \u03bcc\nh\u271d : f x \u2208 s\n\u22a2 \u2191\u2191\u03bcc ((fun b => g x b) \u207b\u00b9' t) = \u2191\u2191(map (g x) \u03bcc) t\n\ncase neg\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03b4 : Type u_7\ninst\u271d\u00b2 : MeasurableSpace \u03b4\n\u03bca : Measure \u03b1\n\u03bcb : Measure \u03b2\n\u03bcc : Measure \u03b3\ninst\u271d\u00b9 : SigmaFinite \u03bcb\nf : \u03b1 \u2192 \u03b2\nhf : MeasurePreserving f\ng : \u03b1 \u2192 \u03b3 \u2192 \u03b4\nhgm : Measurable (uncurry g)\nthis : Measurable fun p => (f p.1, g p.1 p.2)\nha : \u03bca \u2260 0\nsf : SigmaFinite \u03bcc\ns : Set \u03b2\nt : Set \u03b4\nhs : MeasurableSet s\nht : MeasurableSet t\nx : \u03b1\ninst\u271d : SigmaFinite (map (g x) \u03bcc)\nhg : \u2200\u1d50 (x_1 : \u03b1) \u2202\u03bca, map (g x_1) \u03bcc = map (g x) \u03bcc\nh\u271d : \u00acf x \u2208 s\n\u22a2 \u2191\u2191\u03bcc \u2205 = 0"}, {"tactic": "exacts [(map_apply hgm.of_uncurry_left ht).symm, measure_empty]", "annotated_tactic": ["exacts [(<a>map_apply</a> hgm.of_uncurry_left ht).<a>symm</a>, <a>measure_empty</a>]", [{"full_name": "MeasureTheory.Measure.map_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1236, 9], "def_end_pos": [1236, 18]}, {"full_name": "Eq.symm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [310, 9], "def_end_pos": [310, 16]}, {"full_name": "MeasureTheory.measure_empty", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [185, 9], "def_end_pos": [185, 22]}]], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03b4 : Type u_7\ninst\u271d\u00b2 : MeasurableSpace \u03b4\n\u03bca : Measure \u03b1\n\u03bcb : Measure \u03b2\n\u03bcc : Measure \u03b3\ninst\u271d\u00b9 : SigmaFinite \u03bcb\nf : \u03b1 \u2192 \u03b2\nhf : MeasurePreserving f\ng : \u03b1 \u2192 \u03b3 \u2192 \u03b4\nhgm : Measurable (uncurry g)\nthis : Measurable fun p => (f p.1, g p.1 p.2)\nha : \u03bca \u2260 0\nsf : SigmaFinite \u03bcc\ns : Set \u03b2\nt : Set \u03b4\nhs : MeasurableSet s\nht : MeasurableSet t\nx : \u03b1\ninst\u271d : SigmaFinite (map (g x) \u03bcc)\nhg : \u2200\u1d50 (x_1 : \u03b1) \u2202\u03bca, map (g x_1) \u03bcc = map (g x) \u03bcc\nh\u271d : f x \u2208 s\n\u22a2 \u2191\u2191\u03bcc ((fun b => g x b) \u207b\u00b9' t) = \u2191\u2191(map (g x) \u03bcc) t\n\ncase neg\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03b4 : Type u_7\ninst\u271d\u00b2 : MeasurableSpace \u03b4\n\u03bca : Measure \u03b1\n\u03bcb : Measure \u03b2\n\u03bcc : Measure \u03b3\ninst\u271d\u00b9 : SigmaFinite \u03bcb\nf : \u03b1 \u2192 \u03b2\nhf : MeasurePreserving f\ng : \u03b1 \u2192 \u03b3 \u2192 \u03b4\nhgm : Measurable (uncurry g)\nthis : Measurable fun p => (f p.1, g p.1 p.2)\nha : \u03bca \u2260 0\nsf : SigmaFinite \u03bcc\ns : Set \u03b2\nt : Set \u03b4\nhs : MeasurableSet s\nht : MeasurableSet t\nx : \u03b1\ninst\u271d : SigmaFinite (map (g x) \u03bcc)\nhg : \u2200\u1d50 (x_1 : \u03b1) \u2202\u03bca, map (g x_1) \u03bcc = map (g x) \u03bcc\nh\u271d : \u00acf x \u2208 s\n\u22a2 \u2191\u2191\u03bcc \u2205 = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Kernel/IntegralCompProd.lean", "full_name": "ProbabilityTheory.kernel.integral_integral_sub'", "start": [209, 1], "end": [213, 37], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Countable.lean", "full_name": "Set.countable_univ_pi", "start": [267, 1], "end": [270, 58], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/Average.lean", "full_name": "MeasureTheory.measure_smul_average", "start": [290, 1], "end": [296, 39], "traced_tactics": [{"tactic": "cases' eq_or_ne \u03bc 0 with h\u03bc h\u03bc", "annotated_tactic": ["cases' <a>eq_or_ne</a> \u03bc 0 with h\u03bc h\u03bc", [{"full_name": "eq_or_ne", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [209, 9], "def_end_pos": [209, 17]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\nf\u271d g : \u03b1 \u2192 E\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 E\n\u22a2 ENNReal.toReal (\u2191\u2191\u03bc univ) \u2022 \u2a0d (x : \u03b1), f x \u2202\u03bc = \u222b (x : \u03b1), f x \u2202\u03bc", "state_after": "case inl\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\nf\u271d g : \u03b1 \u2192 E\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 E\nh\u03bc : \u03bc = 0\n\u22a2 ENNReal.toReal (\u2191\u2191\u03bc univ) \u2022 \u2a0d (x : \u03b1), f x \u2202\u03bc = \u222b (x : \u03b1), f x \u2202\u03bc\n\ncase inr\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\nf\u271d g : \u03b1 \u2192 E\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 E\nh\u03bc : \u03bc \u2260 0\n\u22a2 ENNReal.toReal (\u2191\u2191\u03bc univ) \u2022 \u2a0d (x : \u03b1), f x \u2202\u03bc = \u222b (x : \u03b1), f x \u2202\u03bc"}, {"tactic": "rw [h\u03bc, integral_zero_measure, average_zero_measure, smul_zero]", "annotated_tactic": ["rw [h\u03bc, <a>integral_zero_measure</a>, <a>average_zero_measure</a>, <a>smul_zero</a>]", [{"full_name": "MeasureTheory.integral_zero_measure", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1484, 9], "def_end_pos": [1484, 30]}, {"full_name": "MeasureTheory.average_zero_measure", "def_path": "Mathlib/MeasureTheory/Integral/Average.lean", "def_pos": [268, 9], "def_end_pos": [268, 29]}, {"full_name": "smul_zero", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [732, 9], "def_end_pos": [732, 18]}]], "state_before": "case inl\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\nf\u271d g : \u03b1 \u2192 E\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 E\nh\u03bc : \u03bc = 0\n\u22a2 ENNReal.toReal (\u2191\u2191\u03bc univ) \u2022 \u2a0d (x : \u03b1), f x \u2202\u03bc = \u222b (x : \u03b1), f x \u2202\u03bc", "state_after": "no goals"}, {"tactic": "rw [average_eq, smul_inv_smul\u2080]", "annotated_tactic": ["rw [<a>average_eq</a>, <a>smul_inv_smul\u2080</a>]", [{"full_name": "MeasureTheory.average_eq", "def_path": "Mathlib/MeasureTheory/Integral/Average.lean", "def_pos": [281, 9], "def_end_pos": [281, 19]}, {"full_name": "smul_inv_smul\u2080", "def_path": "Mathlib/GroupTheory/GroupAction/Group.lean", "def_pos": [197, 9], "def_end_pos": [197, 23]}]], "state_before": "case inr\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\nf\u271d g : \u03b1 \u2192 E\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 E\nh\u03bc : \u03bc \u2260 0\n\u22a2 ENNReal.toReal (\u2191\u2191\u03bc univ) \u2022 \u2a0d (x : \u03b1), f x \u2202\u03bc = \u222b (x : \u03b1), f x \u2202\u03bc", "state_after": "case inr.hc\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\nf\u271d g : \u03b1 \u2192 E\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 E\nh\u03bc : \u03bc \u2260 0\n\u22a2 ENNReal.toReal (\u2191\u2191\u03bc univ) \u2260 0"}, {"tactic": "refine' (ENNReal.toReal_pos _ <| measure_ne_top _ _).ne'", "annotated_tactic": ["refine' (<a>ENNReal.toReal_pos</a> _ <| <a>measure_ne_top</a> _ _).<a>ne'</a>", [{"full_name": "ENNReal.toReal_pos", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2131, 9], "def_end_pos": [2131, 19]}, {"full_name": "MeasureTheory.measure_ne_top", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2875, 9], "def_end_pos": [2875, 23]}, {"full_name": "LT.lt.ne'", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [328, 9], "def_end_pos": [328, 12]}]], "state_before": "case inr.hc\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\nf\u271d g : \u03b1 \u2192 E\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 E\nh\u03bc : \u03bc \u2260 0\n\u22a2 ENNReal.toReal (\u2191\u2191\u03bc univ) \u2260 0", "state_after": "case inr.hc\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\nf\u271d g : \u03b1 \u2192 E\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 E\nh\u03bc : \u03bc \u2260 0\n\u22a2 \u2191\u2191\u03bc univ \u2260 0"}, {"tactic": "rwa [Ne.def, measure_univ_eq_zero]", "annotated_tactic": ["rwa [<a>Ne.def</a>, <a>measure_univ_eq_zero</a>]", [{"full_name": "Ne.def", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [59, 9], "def_end_pos": [59, 15]}, {"full_name": "MeasureTheory.Measure.measure_univ_eq_zero", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1100, 9], "def_end_pos": [1100, 29]}]], "state_before": "case inr.hc\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\nf\u271d g : \u03b1 \u2192 E\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 E\nh\u03bc : \u03bc \u2260 0\n\u22a2 \u2191\u2191\u03bc univ \u2260 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "full_name": "BoundedContinuousFunction.mem_Lp", "start": [1729, 1], "end": [1732, 39], "traced_tactics": [{"tactic": "refine' Lp.mem_Lp_of_ae_bound \u2016f\u2016 _", "annotated_tactic": ["refine' <a>Lp.mem_Lp_of_ae_bound</a> \u2016f\u2016 _", [{"full_name": "MeasureTheory.Lp.mem_Lp_of_ae_bound", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [418, 9], "def_end_pos": [418, 27]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopologyEither \u03b1 E\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192\u1d47 E\n\u22a2 ContinuousMap.toAEEqFun \u03bc f.toContinuousMap \u2208 Lp E p", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopologyEither \u03b1 E\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192\u1d47 E\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016\u2191(ContinuousMap.toAEEqFun \u03bc f.toContinuousMap) x\u2016 \u2264 \u2016f\u2016"}, {"tactic": "filter_upwards [f.toContinuousMap.coeFn_toAEEqFun \u03bc] with x _", "annotated_tactic": ["filter_upwards [f.toContinuousMap.coeFn_toAEEqFun \u03bc] with x _", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopologyEither \u03b1 E\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192\u1d47 E\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016\u2191(ContinuousMap.toAEEqFun \u03bc f.toContinuousMap) x\u2016 \u2264 \u2016f\u2016", "state_after": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopologyEither \u03b1 E\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192\u1d47 E\nx : \u03b1\na\u271d : \u2191(ContinuousMap.toAEEqFun \u03bc f.toContinuousMap) x = \u2191f.toContinuousMap x\n\u22a2 \u2016\u2191(ContinuousMap.toAEEqFun \u03bc f.toContinuousMap) x\u2016 \u2264 \u2016f\u2016"}, {"tactic": "convert f.norm_coe_le_norm x using 2", "annotated_tactic": ["convert f.norm_coe_le_norm x using 2", []], "state_before": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopologyEither \u03b1 E\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192\u1d47 E\nx : \u03b1\na\u271d : \u2191(ContinuousMap.toAEEqFun \u03bc f.toContinuousMap) x = \u2191f.toContinuousMap x\n\u22a2 \u2016\u2191(ContinuousMap.toAEEqFun \u03bc f.toContinuousMap) x\u2016 \u2264 \u2016f\u2016", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Haar/InnerProductSpace.lean", "full_name": "EuclideanSpace.volume_preserving_measurableEquiv", "start": [84, 1], "end": [90, 61], "traced_tactics": [{"tactic": "suffices volume = map (EuclideanSpace.measurableEquiv \u03b9).symm volume by\n  convert ((EuclideanSpace.measurableEquiv \u03b9).symm.measurable.measurePreserving _).symm", "annotated_tactic": ["suffices <a>volume</a> = <a>map</a> (<a>EuclideanSpace.measurableEquiv</a> \u03b9).<a>symm</a> <a>volume</a> by\n    convert ((<a>EuclideanSpace.measurableEquiv</a> \u03b9).symm.measurable.measurePreserving _).<a>symm</a>", [{"full_name": "MeasureTheory.MeasureSpace.volume", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [663, 3], "def_end_pos": [663, 9]}, {"full_name": "MeasureTheory.Measure.map", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1163, 17], "def_end_pos": [1163, 20]}, {"full_name": "EuclideanSpace.measurableEquiv", "def_path": "Mathlib/MeasureTheory/Measure/Haar/OfBasis.lean", "def_pos": [288, 15], "def_end_pos": [288, 30]}, {"full_name": "MeasurableEquiv.symm", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [1325, 5], "def_end_pos": [1325, 9]}, {"full_name": "MeasureTheory.MeasureSpace.volume", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [663, 3], "def_end_pos": [663, 9]}, {"full_name": "EuclideanSpace.measurableEquiv", "def_path": "Mathlib/MeasureTheory/Measure/Haar/OfBasis.lean", "def_pos": [288, 15], "def_end_pos": [288, 30]}, {"full_name": "MeasureTheory.MeasurePreserving.symm", "def_path": "Mathlib/Dynamics/Ergodic/MeasurePreserving.lean", "def_pos": [70, 9], "def_end_pos": [70, 13]}]], "state_before": "\u03b9\u271d : Type u_1\nF : Type u_2\ninst\u271d\u2076 : Fintype \u03b9\u271d\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\ninst\u271d\u00b3 : FiniteDimensional \u211d F\ninst\u271d\u00b2 : MeasurableSpace F\ninst\u271d\u00b9 : BorelSpace F\n\u03b9 : Type u_3\ninst\u271d : Fintype \u03b9\n\u22a2 MeasurePreserving \u2191(EuclideanSpace.measurableEquiv \u03b9)", "state_after": "\u03b9\u271d : Type u_1\nF : Type u_2\ninst\u271d\u2076 : Fintype \u03b9\u271d\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\ninst\u271d\u00b3 : FiniteDimensional \u211d F\ninst\u271d\u00b2 : MeasurableSpace F\ninst\u271d\u00b9 : BorelSpace F\n\u03b9 : Type u_3\ninst\u271d : Fintype \u03b9\n\u22a2 volume = map (\u2191(MeasurableEquiv.symm (EuclideanSpace.measurableEquiv \u03b9))) volume"}, {"tactic": "rw [\u2190 addHaarMeasure_eq_volume_pi, \u2190 Basis.parallelepiped_basisFun, \u2190 Basis.addHaar_def,\n  coe_measurableEquiv_symm, \u2190 PiLp.continuousLinearEquiv_symm_apply 2 \u211d, Basis.map_addHaar]", "annotated_tactic": ["rw [\u2190 <a>addHaarMeasure_eq_volume_pi</a>, \u2190 <a>Basis.parallelepiped_basisFun</a>, \u2190 <a>Basis.addHaar_def</a>,\n    <a>coe_measurableEquiv_symm</a>, \u2190 <a>PiLp.continuousLinearEquiv_symm_apply</a> 2 \u211d, <a>Basis.map_addHaar</a>]", [{"full_name": "MeasureTheory.addHaarMeasure_eq_volume_pi", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean", "def_pos": [120, 9], "def_end_pos": [120, 36]}, {"full_name": "Basis.parallelepiped_basisFun", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean", "def_pos": [74, 9], "def_end_pos": [74, 38]}, {"full_name": "Basis.addHaar_def", "def_path": "Mathlib/MeasureTheory/Measure/Haar/OfBasis.lean", "def_pos": [220, 1], "def_end_pos": [223, 42]}, {"full_name": "EuclideanSpace.coe_measurableEquiv_symm", "def_path": "Mathlib/MeasureTheory/Measure/Haar/OfBasis.lean", "def_pos": [297, 9], "def_end_pos": [297, 33]}, {"full_name": "PiLp.continuousLinearEquiv_symm_apply", "def_path": "Mathlib/Analysis/NormedSpace/PiLp.lean", "def_pos": [871, 54], "def_end_pos": [871, 64]}, {"full_name": "Basis.map_addHaar", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean", "def_pos": [95, 9], "def_end_pos": [95, 26]}]], "state_before": "\u03b9\u271d : Type u_1\nF : Type u_2\ninst\u271d\u2076 : Fintype \u03b9\u271d\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\ninst\u271d\u00b3 : FiniteDimensional \u211d F\ninst\u271d\u00b2 : MeasurableSpace F\ninst\u271d\u00b9 : BorelSpace F\n\u03b9 : Type u_3\ninst\u271d : Fintype \u03b9\n\u22a2 volume = map (\u2191(MeasurableEquiv.symm (EuclideanSpace.measurableEquiv \u03b9))) volume", "state_after": "\u03b9\u271d : Type u_1\nF : Type u_2\ninst\u271d\u2076 : Fintype \u03b9\u271d\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\ninst\u271d\u00b3 : FiniteDimensional \u211d F\ninst\u271d\u00b2 : MeasurableSpace F\ninst\u271d\u00b9 : BorelSpace F\n\u03b9 : Type u_3\ninst\u271d : Fintype \u03b9\n\u22a2 volume =\n    Basis.addHaar\n      (Basis.map (Pi.basisFun \u211d \u03b9)\n        (ContinuousLinearEquiv.symm (PiLp.continuousLinearEquiv 2 \u211d fun a => \u211d)).toLinearEquiv)"}, {"tactic": "exact (EuclideanSpace.basisFun _ _).addHaar_eq_volume.symm", "annotated_tactic": ["exact (<a>EuclideanSpace.basisFun</a> _ _).addHaar_eq_volume.symm", [{"full_name": "EuclideanSpace.basisFun", "def_path": "Mathlib/Analysis/InnerProductSpace/PiL2.lean", "def_pos": [617, 19], "def_end_pos": [617, 27]}]], "state_before": "\u03b9\u271d : Type u_1\nF : Type u_2\ninst\u271d\u2076 : Fintype \u03b9\u271d\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\ninst\u271d\u00b3 : FiniteDimensional \u211d F\ninst\u271d\u00b2 : MeasurableSpace F\ninst\u271d\u00b9 : BorelSpace F\n\u03b9 : Type u_3\ninst\u271d : Fintype \u03b9\n\u22a2 volume =\n    Basis.addHaar\n      (Basis.map (Pi.basisFun \u211d \u03b9)\n        (ContinuousLinearEquiv.symm (PiLp.continuousLinearEquiv 2 \u211d fun a => \u211d)).toLinearEquiv)", "state_after": "no goals"}, {"tactic": "convert ((EuclideanSpace.measurableEquiv \u03b9).symm.measurable.measurePreserving _).symm", "annotated_tactic": ["convert ((<a>EuclideanSpace.measurableEquiv</a> \u03b9).symm.measurable.measurePreserving _).<a>symm</a>", [{"full_name": "EuclideanSpace.measurableEquiv", "def_path": "Mathlib/MeasureTheory/Measure/Haar/OfBasis.lean", "def_pos": [288, 15], "def_end_pos": [288, 30]}, {"full_name": "MeasureTheory.MeasurePreserving.symm", "def_path": "Mathlib/Dynamics/Ergodic/MeasurePreserving.lean", "def_pos": [70, 9], "def_end_pos": [70, 13]}]], "state_before": "\u03b9\u271d : Type u_1\nF : Type u_2\ninst\u271d\u2076 : Fintype \u03b9\u271d\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\ninst\u271d\u00b3 : FiniteDimensional \u211d F\ninst\u271d\u00b2 : MeasurableSpace F\ninst\u271d\u00b9 : BorelSpace F\n\u03b9 : Type u_3\ninst\u271d : Fintype \u03b9\nthis : volume = map (\u2191(MeasurableEquiv.symm (EuclideanSpace.measurableEquiv \u03b9))) volume\n\u22a2 MeasurePreserving \u2191(EuclideanSpace.measurableEquiv \u03b9)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Intervals/WithBotTop.lean", "full_name": "WithBot.image_coe_Icc", "start": [215, 1], "end": [218, 78], "traced_tactics": [{"tactic": "rw [\u2190 preimage_coe_Icc, image_preimage_eq_inter_range, range_coe,\n  inter_eq_self_of_subset_left\n    (Subset.trans Icc_subset_Ici_self <| Ici_subset_Ioi.2 <| bot_lt_coe a)]", "annotated_tactic": ["rw [\u2190 <a>preimage_coe_Icc</a>, <a>image_preimage_eq_inter_range</a>, <a>range_coe</a>,\n    <a>inter_eq_self_of_subset_left</a>\n      (<a>Subset.trans</a> <a>Icc_subset_Ici_self</a> <| <a>Ici_subset_Ioi</a>.2 <| <a>bot_lt_coe</a> a)]", [{"full_name": "WithBot.preimage_coe_Icc", "def_path": "Mathlib/Data/Set/Intervals/WithBotTop.lean", "def_pos": [167, 9], "def_end_pos": [167, 25]}, {"full_name": "Set.image_preimage_eq_inter_range", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [796, 9], "def_end_pos": [796, 38]}, {"full_name": "WithBot.range_coe", "def_path": "Mathlib/Data/Set/Intervals/WithBotTop.lean", "def_pos": [142, 9], "def_end_pos": [142, 18]}, {"full_name": "Set.inter_eq_self_of_subset_left", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [987, 9], "def_end_pos": [987, 37]}, {"full_name": "Set.Subset.trans", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [362, 9], "def_end_pos": [362, 21]}, {"full_name": "Set.Icc_subset_Ici_self", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [471, 9], "def_end_pos": [471, 28]}, {"full_name": "Set.Ici_subset_Ioi", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [423, 9], "def_end_pos": [423, 23]}, {"full_name": "WithBot.bot_lt_coe", "def_path": "Mathlib/Order/WithBot.lean", "def_pos": [289, 9], "def_end_pos": [289, 19]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : PartialOrder \u03b1\na b : \u03b1\n\u22a2 some '' Icc a b = Icc \u2191a \u2191b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "full_name": "MeasureTheory.VectorMeasure.ext_iff", "start": [127, 1], "end": [135, 36], "traced_tactics": [{"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : TopologicalSpace M\nv w : VectorMeasure \u03b1 M\n\u22a2 v = w \u2194 \u2200 (i : Set \u03b1), MeasurableSet i \u2192 \u2191v i = \u2191w i", "state_after": "case mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : TopologicalSpace M\nv w : VectorMeasure \u03b1 M\n\u22a2 v = w \u2192 \u2200 (i : Set \u03b1), MeasurableSet i \u2192 \u2191v i = \u2191w i\n\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : TopologicalSpace M\nv w : VectorMeasure \u03b1 M\n\u22a2 (\u2200 (i : Set \u03b1), MeasurableSet i \u2192 \u2191v i = \u2191w i) \u2192 v = w"}, {"tactic": "rintro rfl _ _", "annotated_tactic": ["rintro rfl _ _", []], "state_before": "case mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : TopologicalSpace M\nv w : VectorMeasure \u03b1 M\n\u22a2 v = w \u2192 \u2200 (i : Set \u03b1), MeasurableSet i \u2192 \u2191v i = \u2191w i", "state_after": "case mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : TopologicalSpace M\nv : VectorMeasure \u03b1 M\ni\u271d : Set \u03b1\na\u271d : MeasurableSet i\u271d\n\u22a2 \u2191v i\u271d = \u2191v i\u271d"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : TopologicalSpace M\nv : VectorMeasure \u03b1 M\ni\u271d : Set \u03b1\na\u271d : MeasurableSet i\u271d\n\u22a2 \u2191v i\u271d = \u2191v i\u271d", "state_after": "no goals"}, {"tactic": "rw [ext_iff']", "annotated_tactic": ["rw [<a>ext_iff'</a>]", [{"full_name": "MeasureTheory.VectorMeasure.ext_iff'", "def_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "def_pos": [123, 9], "def_end_pos": [123, 17]}]], "state_before": "case mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : TopologicalSpace M\nv w : VectorMeasure \u03b1 M\n\u22a2 (\u2200 (i : Set \u03b1), MeasurableSet i \u2192 \u2191v i = \u2191w i) \u2192 v = w", "state_after": "case mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : TopologicalSpace M\nv w : VectorMeasure \u03b1 M\n\u22a2 (\u2200 (i : Set \u03b1), MeasurableSet i \u2192 \u2191v i = \u2191w i) \u2192 \u2200 (i : Set \u03b1), \u2191v i = \u2191w i"}, {"tactic": "intro h i", "annotated_tactic": ["intro h i", []], "state_before": "case mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : TopologicalSpace M\nv w : VectorMeasure \u03b1 M\n\u22a2 (\u2200 (i : Set \u03b1), MeasurableSet i \u2192 \u2191v i = \u2191w i) \u2192 \u2200 (i : Set \u03b1), \u2191v i = \u2191w i", "state_after": "case mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : TopologicalSpace M\nv w : VectorMeasure \u03b1 M\nh : \u2200 (i : Set \u03b1), MeasurableSet i \u2192 \u2191v i = \u2191w i\ni : Set \u03b1\n\u22a2 \u2191v i = \u2191w i"}, {"tactic": "by_cases hi : MeasurableSet i", "annotated_tactic": ["by_cases hi : <a>MeasurableSet</a> i", [{"full_name": "MeasurableSet", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [64, 5], "def_end_pos": [64, 18]}]], "state_before": "case mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : TopologicalSpace M\nv w : VectorMeasure \u03b1 M\nh : \u2200 (i : Set \u03b1), MeasurableSet i \u2192 \u2191v i = \u2191w i\ni : Set \u03b1\n\u22a2 \u2191v i = \u2191w i", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : TopologicalSpace M\nv w : VectorMeasure \u03b1 M\nh : \u2200 (i : Set \u03b1), MeasurableSet i \u2192 \u2191v i = \u2191w i\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 \u2191v i = \u2191w i\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : TopologicalSpace M\nv w : VectorMeasure \u03b1 M\nh : \u2200 (i : Set \u03b1), MeasurableSet i \u2192 \u2191v i = \u2191w i\ni : Set \u03b1\nhi : \u00acMeasurableSet i\n\u22a2 \u2191v i = \u2191w i"}, {"tactic": "exact h i hi", "annotated_tactic": ["exact h i hi", []], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : TopologicalSpace M\nv w : VectorMeasure \u03b1 M\nh : \u2200 (i : Set \u03b1), MeasurableSet i \u2192 \u2191v i = \u2191w i\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 \u2191v i = \u2191w i", "state_after": "no goals"}, {"tactic": "simp_rw [not_measurable _ hi]", "annotated_tactic": ["simp_rw [<a>not_measurable</a> _ hi]", [{"full_name": "MeasureTheory.VectorMeasure.not_measurable", "def_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "def_pos": [102, 9], "def_end_pos": [102, 23]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : TopologicalSpace M\nv w : VectorMeasure \u03b1 M\nh : \u2200 (i : Set \u03b1), MeasurableSet i \u2192 \u2191v i = \u2191w i\ni : Set \u03b1\nhi : \u00acMeasurableSet i\n\u22a2 \u2191v i = \u2191w i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/TMToPartrec.lean", "full_name": "Turing.PartrecToTM2.trNum_natEnd", "start": [1461, 1], "end": [1462, 45], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "Finset.pair_comm", "start": [1159, 1], "end": [1160, 20], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Lattice.lean", "full_name": "Finset.min_eq_top", "start": [1363, 1], "end": [1368, 33], "traced_tactics": [{"tactic": "let \u27e8a, ha\u27e9 := min_of_nonempty H", "annotated_tactic": ["let \u27e8a, ha\u27e9 := <a>min_of_nonempty</a> H", [{"full_name": "Finset.min_of_nonempty", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [1358, 9], "def_end_pos": [1358, 24]}]], "state_before": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Type u_5\n\u03ba : Type u_6\ninst\u271d : LinearOrder \u03b1\ns : Finset \u03b1\nh : Finset.min s = \u22a4\nH : Finset.Nonempty s\n\u22a2 s = \u2205", "state_after": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Type u_5\n\u03ba : Type u_6\ninst\u271d : LinearOrder \u03b1\ns : Finset \u03b1\nh : Finset.min s = \u22a4\nH : Finset.Nonempty s\na : \u03b1\nha : Finset.min s = \u2191a\n\u22a2 s = \u2205"}, {"tactic": "rw [h] at ha", "annotated_tactic": ["rw [h] at ha", []], "state_before": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Type u_5\n\u03ba : Type u_6\ninst\u271d : LinearOrder \u03b1\ns : Finset \u03b1\nh : Finset.min s = \u22a4\nH : Finset.Nonempty s\na : \u03b1\nha : Finset.min s = \u2191a\n\u22a2 s = \u2205", "state_after": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Type u_5\n\u03ba : Type u_6\ninst\u271d : LinearOrder \u03b1\ns : Finset \u03b1\nh : Finset.min s = \u22a4\nH : Finset.Nonempty s\na : \u03b1\nha : \u22a4 = \u2191a\n\u22a2 s = \u2205"}, {"tactic": "cases ha", "annotated_tactic": ["cases ha", []], "state_before": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Type u_5\n\u03ba : Type u_6\ninst\u271d : LinearOrder \u03b1\ns : Finset \u03b1\nh : Finset.min s = \u22a4\nH : Finset.Nonempty s\na : \u03b1\nha : \u22a4 = \u2191a\n\u22a2 s = \u2205", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean", "full_name": "MeasureTheory.Measure.map_linearMap_addHaar_pi_eq_smul_addHaar", "start": [227, 1], "end": [235, 66], "traced_tactics": [{"tactic": "cases nonempty_fintype \u03b9", "annotated_tactic": ["cases <a>nonempty_fintype</a> \u03b9", [{"full_name": "nonempty_fintype", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [442, 9], "def_end_pos": [442, 25]}]], "state_before": "\u03b9 : Type u_1\ninst\u271d\u00b9 : Finite \u03b9\nf : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d\nhf : \u2191LinearMap.det f \u2260 0\n\u03bc : Measure (\u03b9 \u2192 \u211d)\ninst\u271d : IsAddHaarMeasure \u03bc\n\u22a2 map (\u2191f) \u03bc = ENNReal.ofReal |(\u2191LinearMap.det f)\u207b\u00b9| \u2022 \u03bc", "state_after": "case intro\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Finite \u03b9\nf : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d\nhf : \u2191LinearMap.det f \u2260 0\n\u03bc : Measure (\u03b9 \u2192 \u211d)\ninst\u271d : IsAddHaarMeasure \u03bc\nval\u271d : Fintype \u03b9\n\u22a2 map (\u2191f) \u03bc = ENNReal.ofReal |(\u2191LinearMap.det f)\u207b\u00b9| \u2022 \u03bc"}, {"tactic": "have := addHaarMeasure_unique \u03bc (piIcc01 \u03b9)", "annotated_tactic": ["have := <a>addHaarMeasure_unique</a> \u03bc (<a>piIcc01</a> \u03b9)", [{"full_name": "MeasureTheory.Measure.addHaarMeasure_unique", "def_path": "Mathlib/MeasureTheory/Measure/Haar/Basic.lean", "def_pos": [688, 3], "def_end_pos": [688, 14]}, {"full_name": "TopologicalSpace.PositiveCompacts.piIcc01", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean", "def_pos": [64, 5], "def_end_pos": [64, 46]}]], "state_before": "case intro\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Finite \u03b9\nf : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d\nhf : \u2191LinearMap.det f \u2260 0\n\u03bc : Measure (\u03b9 \u2192 \u211d)\ninst\u271d : IsAddHaarMeasure \u03bc\nval\u271d : Fintype \u03b9\n\u22a2 map (\u2191f) \u03bc = ENNReal.ofReal |(\u2191LinearMap.det f)\u207b\u00b9| \u2022 \u03bc", "state_after": "case intro\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Finite \u03b9\nf : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d\nhf : \u2191LinearMap.det f \u2260 0\n\u03bc : Measure (\u03b9 \u2192 \u211d)\ninst\u271d : IsAddHaarMeasure \u03bc\nval\u271d : Fintype \u03b9\nthis : \u03bc = \u2191\u2191\u03bc \u2191(piIcc01 \u03b9) \u2022 addHaarMeasure (piIcc01 \u03b9)\n\u22a2 map (\u2191f) \u03bc = ENNReal.ofReal |(\u2191LinearMap.det f)\u207b\u00b9| \u2022 \u03bc"}, {"tactic": "rw [this, addHaarMeasure_eq_volume_pi, Measure.map_smul,\n  Real.map_linearMap_volume_pi_eq_smul_volume_pi hf, smul_comm]", "annotated_tactic": ["rw [this, <a>addHaarMeasure_eq_volume_pi</a>, <a>Measure.map_smul</a>,\n    <a>Real.map_linearMap_volume_pi_eq_smul_volume_pi</a> hf, <a>smul_comm</a>]", [{"full_name": "MeasureTheory.addHaarMeasure_eq_volume_pi", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean", "def_pos": [120, 9], "def_end_pos": [120, 36]}, {"full_name": "MeasureTheory.Measure.map_smul", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1202, 19], "def_end_pos": [1202, 27]}, {"full_name": "Real.map_linearMap_volume_pi_eq_smul_volume_pi", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean", "def_pos": [446, 9], "def_end_pos": [446, 50]}, {"full_name": "SMulCommClass.smul_comm", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [187, 3], "def_end_pos": [187, 12]}]], "state_before": "case intro\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Finite \u03b9\nf : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d\nhf : \u2191LinearMap.det f \u2260 0\n\u03bc : Measure (\u03b9 \u2192 \u211d)\ninst\u271d : IsAddHaarMeasure \u03bc\nval\u271d : Fintype \u03b9\nthis : \u03bc = \u2191\u2191\u03bc \u2191(piIcc01 \u03b9) \u2022 addHaarMeasure (piIcc01 \u03b9)\n\u22a2 map (\u2191f) \u03bc = ENNReal.ofReal |(\u2191LinearMap.det f)\u207b\u00b9| \u2022 \u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Group/FundamentalDomain.lean", "full_name": "MeasureTheory.IsFundamentalDomain.measure_fundamentalFrontier", "start": [673, 1], "end": [676, 21], "traced_tactics": [{"tactic": "simpa only [fundamentalFrontier, iUnion\u2082_inter, measure_iUnion_null_iff', one_smul,\n  measure_iUnion_null_iff, inter_comm s, Function.onFun] using fun g (hg : g \u2260 1) =>\n  hs.aedisjoint hg", "annotated_tactic": ["simpa only [<a>fundamentalFrontier</a>, <a>iUnion\u2082_inter</a>, <a>measure_iUnion_null_iff'</a>, <a>one_smul</a>,\n    <a>measure_iUnion_null_iff</a>, <a>inter_comm</a> s, <a>Function.onFun</a>] using fun g (hg : g \u2260 1) =>\n    hs.aedisjoint hg", [{"full_name": "MeasureTheory.fundamentalFrontier", "def_path": "Mathlib/MeasureTheory/Group/FundamentalDomain.lean", "def_pos": [544, 5], "def_end_pos": [544, 24]}, {"full_name": "Set.iUnion\u2082_inter", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [1127, 9], "def_end_pos": [1127, 22]}, {"full_name": "MeasureTheory.measure_iUnion_null_iff'", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [284, 9], "def_end_pos": [284, 33]}, {"full_name": "one_smul", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [492, 9], "def_end_pos": [492, 17]}, {"full_name": "MeasureTheory.measure_iUnion_null_iff", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [275, 9], "def_end_pos": [275, 32]}, {"full_name": "Set.inter_comm", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [940, 9], "def_end_pos": [940, 19]}, {"full_name": "Function.onFun", "def_path": "Mathlib/Init/Function.lean", "def_pos": [49, 5], "def_end_pos": [49, 10]}]], "state_before": "G : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b3 : Countable G\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : MulAction G \u03b1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nhs : IsFundamentalDomain G s\n\u22a2 \u2191\u2191\u03bc (fundamentalFrontier G s) = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "full_name": "MeasureTheory.lintegral_strict_mono_of_ae_le_of_ae_lt_on", "start": [961, 1], "end": [965, 86], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Primrec.lean", "full_name": "list_foldl'", "start": [892, 9], "end": [916, 34], "traced_tactics": [{"tactic": "letI := prim H", "annotated_tactic": ["letI := <a>prim</a> H", [{"full_name": "_private.Mathlib.Computability.Primrec.0.prim", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [877, 13], "def_end_pos": [877, 17]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03c3 : Type u_3\ninst\u271d\u00b2 : Primcodable \u03b1\ninst\u271d\u00b9 : Primcodable \u03b2\ninst\u271d : Primcodable \u03c3\nH : Nat.Primrec fun n => encode (decode n)\nf : \u03b1 \u2192 List \u03b2\ng : \u03b1 \u2192 \u03c3\nh : \u03b1 \u2192 \u03c3 \u00d7 \u03b2 \u2192 \u03c3\nhf : Primrec f\nhg : Primrec g\nhh : Primrec\u2082 h\n\u22a2 Primrec fun a => List.foldl (fun s b => h a (s, b)) (g a) (f a)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03c3 : Type u_3\ninst\u271d\u00b2 : Primcodable \u03b1\ninst\u271d\u00b9 : Primcodable \u03b2\ninst\u271d : Primcodable \u03c3\nH : Nat.Primrec fun n => encode (decode n)\nf : \u03b1 \u2192 List \u03b2\ng : \u03b1 \u2192 \u03c3\nh : \u03b1 \u2192 \u03c3 \u00d7 \u03b2 \u2192 \u03c3\nhf : Primrec f\nhg : Primrec g\nhh : Primrec\u2082 h\nthis : Primcodable (List \u03b2) := prim H\n\u22a2 Primrec fun a => List.foldl (fun s b => h a (s, b)) (g a) (f a)"}, {"tactic": "let G (a : \u03b1) (IH : \u03c3 \u00d7 List \u03b2) : \u03c3 \u00d7 List \u03b2 := List.casesOn IH.2 IH fun b l => (h a (IH.1, b), l)", "annotated_tactic": ["let G (a : \u03b1) (IH : \u03c3 \u00d7 <a>List</a> \u03b2) : \u03c3 \u00d7 <a>List</a> \u03b2 := <a>List.casesOn</a> IH.2 IH fun b l => (h a (IH.1, b), l)", [{"full_name": "List", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2182, 11], "def_end_pos": [2182, 15]}, {"full_name": "List", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2182, 11], "def_end_pos": [2182, 15]}, {"full_name": "List.casesOn", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2182, 11], "def_end_pos": [2182, 15]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03c3 : Type u_3\ninst\u271d\u00b2 : Primcodable \u03b1\ninst\u271d\u00b9 : Primcodable \u03b2\ninst\u271d : Primcodable \u03c3\nH : Nat.Primrec fun n => encode (decode n)\nf : \u03b1 \u2192 List \u03b2\ng : \u03b1 \u2192 \u03c3\nh : \u03b1 \u2192 \u03c3 \u00d7 \u03b2 \u2192 \u03c3\nhf : Primrec f\nhg : Primrec g\nhh : Primrec\u2082 h\nthis : Primcodable (List \u03b2) := prim H\n\u22a2 Primrec fun a => List.foldl (fun s b => h a (s, b)) (g a) (f a)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03c3 : Type u_3\ninst\u271d\u00b2 : Primcodable \u03b1\ninst\u271d\u00b9 : Primcodable \u03b2\ninst\u271d : Primcodable \u03c3\nH : Nat.Primrec fun n => encode (decode n)\nf : \u03b1 \u2192 List \u03b2\ng : \u03b1 \u2192 \u03c3\nh : \u03b1 \u2192 \u03c3 \u00d7 \u03b2 \u2192 \u03c3\nhf : Primrec f\nhg : Primrec g\nhh : Primrec\u2082 h\nthis : Primcodable (List \u03b2) := prim H\nG : \u03b1 \u2192 \u03c3 \u00d7 List \u03b2 \u2192 \u03c3 \u00d7 List \u03b2 := fun a IH => List.casesOn IH.2 IH fun b l => (h a (IH.1, b), l)\n\u22a2 Primrec fun a => List.foldl (fun s b => h a (s, b)) (g a) (f a)"}, {"tactic": "have hG : Primrec\u2082 G := list_casesOn' H (snd.comp snd) snd <|\n  to\u2082 <|\n  pair (hh.comp (fst.comp fst) <| pair ((fst.comp snd).comp fst) (fst.comp snd))\n    (snd.comp snd)", "annotated_tactic": ["have hG : <a>Primrec\u2082</a> G := <a>list_casesOn'</a> H (snd.comp <a>snd</a>) <a>snd</a> <|\n    <a>to\u2082</a> <|\n    <a>pair</a> (hh.comp (fst.comp <a>fst</a>) <| <a>pair</a> ((fst.comp <a>snd</a>).<a>comp</a> <a>fst</a>) (fst.comp <a>snd</a>))\n      (snd.comp <a>snd</a>)", [{"full_name": "Primrec\u2082", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [389, 5], "def_end_pos": [389, 13]}, {"full_name": "_private.Mathlib.Computability.Primrec.0.list_casesOn'", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [879, 17], "def_end_pos": [879, 30]}, {"full_name": "Primrec.snd", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [351, 9], "def_end_pos": [351, 12]}, {"full_name": "Primrec.snd", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [351, 9], "def_end_pos": [351, 12]}, {"full_name": "Primrec.to\u2082", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [555, 9], "def_end_pos": [555, 12]}, {"full_name": "Primrec.pair", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [362, 9], "def_end_pos": [362, 13]}, {"full_name": "Primrec.fst", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [340, 9], "def_end_pos": [340, 12]}, {"full_name": "Primrec.pair", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [362, 9], "def_end_pos": [362, 13]}, {"full_name": "Primrec.snd", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [351, 9], "def_end_pos": [351, 12]}, {"full_name": "Primrec.comp", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [259, 9], "def_end_pos": [259, 13]}, {"full_name": "Primrec.fst", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [340, 9], "def_end_pos": [340, 12]}, {"full_name": "Primrec.snd", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [351, 9], "def_end_pos": [351, 12]}, {"full_name": "Primrec.snd", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [351, 9], "def_end_pos": [351, 12]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03c3 : Type u_3\ninst\u271d\u00b2 : Primcodable \u03b1\ninst\u271d\u00b9 : Primcodable \u03b2\ninst\u271d : Primcodable \u03c3\nH : Nat.Primrec fun n => encode (decode n)\nf : \u03b1 \u2192 List \u03b2\ng : \u03b1 \u2192 \u03c3\nh : \u03b1 \u2192 \u03c3 \u00d7 \u03b2 \u2192 \u03c3\nhf : Primrec f\nhg : Primrec g\nhh : Primrec\u2082 h\nthis : Primcodable (List \u03b2) := prim H\nG : \u03b1 \u2192 \u03c3 \u00d7 List \u03b2 \u2192 \u03c3 \u00d7 List \u03b2 := fun a IH => List.casesOn IH.2 IH fun b l => (h a (IH.1, b), l)\n\u22a2 Primrec fun a => List.foldl (fun s b => h a (s, b)) (g a) (f a)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03c3 : Type u_3\ninst\u271d\u00b2 : Primcodable \u03b1\ninst\u271d\u00b9 : Primcodable \u03b2\ninst\u271d : Primcodable \u03c3\nH : Nat.Primrec fun n => encode (decode n)\nf : \u03b1 \u2192 List \u03b2\ng : \u03b1 \u2192 \u03c3\nh : \u03b1 \u2192 \u03c3 \u00d7 \u03b2 \u2192 \u03c3\nhf : Primrec f\nhg : Primrec g\nhh : Primrec\u2082 h\nthis : Primcodable (List \u03b2) := prim H\nG : \u03b1 \u2192 \u03c3 \u00d7 List \u03b2 \u2192 \u03c3 \u00d7 List \u03b2 := fun a IH => List.casesOn IH.2 IH fun b l => (h a (IH.1, b), l)\nhG : Primrec\u2082 G\n\u22a2 Primrec fun a => List.foldl (fun s b => h a (s, b)) (g a) (f a)"}, {"tactic": "let F := fun (a : \u03b1) (n : \u2115) => (G a)^[n] (g a, f a)", "annotated_tactic": ["let F := fun (a : \u03b1) (n : \u2115) => (G a)^[n] (g a, f a)", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03c3 : Type u_3\ninst\u271d\u00b2 : Primcodable \u03b1\ninst\u271d\u00b9 : Primcodable \u03b2\ninst\u271d : Primcodable \u03c3\nH : Nat.Primrec fun n => encode (decode n)\nf : \u03b1 \u2192 List \u03b2\ng : \u03b1 \u2192 \u03c3\nh : \u03b1 \u2192 \u03c3 \u00d7 \u03b2 \u2192 \u03c3\nhf : Primrec f\nhg : Primrec g\nhh : Primrec\u2082 h\nthis : Primcodable (List \u03b2) := prim H\nG : \u03b1 \u2192 \u03c3 \u00d7 List \u03b2 \u2192 \u03c3 \u00d7 List \u03b2 := fun a IH => List.casesOn IH.2 IH fun b l => (h a (IH.1, b), l)\nhG : Primrec\u2082 G\n\u22a2 Primrec fun a => List.foldl (fun s b => h a (s, b)) (g a) (f a)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03c3 : Type u_3\ninst\u271d\u00b2 : Primcodable \u03b1\ninst\u271d\u00b9 : Primcodable \u03b2\ninst\u271d : Primcodable \u03c3\nH : Nat.Primrec fun n => encode (decode n)\nf : \u03b1 \u2192 List \u03b2\ng : \u03b1 \u2192 \u03c3\nh : \u03b1 \u2192 \u03c3 \u00d7 \u03b2 \u2192 \u03c3\nhf : Primrec f\nhg : Primrec g\nhh : Primrec\u2082 h\nthis : Primcodable (List \u03b2) := prim H\nG : \u03b1 \u2192 \u03c3 \u00d7 List \u03b2 \u2192 \u03c3 \u00d7 List \u03b2 := fun a IH => List.casesOn IH.2 IH fun b l => (h a (IH.1, b), l)\nhG : Primrec\u2082 G\nF : \u03b1 \u2192 \u2115 \u2192 \u03c3 \u00d7 List \u03b2 := fun a n => (G a)^[n] (g a, f a)\n\u22a2 Primrec fun a => List.foldl (fun s b => h a (s, b)) (g a) (f a)"}, {"tactic": "have hF : Primrec fun a => (F a (encode (f a))).1 :=\n  (fst.comp <|\n    nat_iterate (encode_iff.2 hf) (pair hg hf) <|\n    hG)", "annotated_tactic": ["have hF : <a>Primrec</a> fun a => (F a (<a>encode</a> (f a))).1 :=\n    (fst.comp <|\n      <a>nat_iterate</a> (<a>encode_iff</a>.2 hf) (<a>pair</a> hg hf) <|\n      hG)", [{"full_name": "Primrec", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [207, 5], "def_end_pos": [207, 12]}, {"full_name": "Encodable.encode", "def_path": "Mathlib/Logic/Encodable/Basic.lean", "def_pos": [47, 3], "def_end_pos": [47, 9]}, {"full_name": "Primrec.nat_iterate", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [604, 9], "def_end_pos": [604, 20]}, {"full_name": "Primrec.encode_iff", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [272, 9], "def_end_pos": [272, 19]}, {"full_name": "Primrec.pair", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [362, 9], "def_end_pos": [362, 13]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03c3 : Type u_3\ninst\u271d\u00b2 : Primcodable \u03b1\ninst\u271d\u00b9 : Primcodable \u03b2\ninst\u271d : Primcodable \u03c3\nH : Nat.Primrec fun n => encode (decode n)\nf : \u03b1 \u2192 List \u03b2\ng : \u03b1 \u2192 \u03c3\nh : \u03b1 \u2192 \u03c3 \u00d7 \u03b2 \u2192 \u03c3\nhf : Primrec f\nhg : Primrec g\nhh : Primrec\u2082 h\nthis : Primcodable (List \u03b2) := prim H\nG : \u03b1 \u2192 \u03c3 \u00d7 List \u03b2 \u2192 \u03c3 \u00d7 List \u03b2 := fun a IH => List.casesOn IH.2 IH fun b l => (h a (IH.1, b), l)\nhG : Primrec\u2082 G\nF : \u03b1 \u2192 \u2115 \u2192 \u03c3 \u00d7 List \u03b2 := fun a n => (G a)^[n] (g a, f a)\n\u22a2 Primrec fun a => List.foldl (fun s b => h a (s, b)) (g a) (f a)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03c3 : Type u_3\ninst\u271d\u00b2 : Primcodable \u03b1\ninst\u271d\u00b9 : Primcodable \u03b2\ninst\u271d : Primcodable \u03c3\nH : Nat.Primrec fun n => encode (decode n)\nf : \u03b1 \u2192 List \u03b2\ng : \u03b1 \u2192 \u03c3\nh : \u03b1 \u2192 \u03c3 \u00d7 \u03b2 \u2192 \u03c3\nhf : Primrec f\nhg : Primrec g\nhh : Primrec\u2082 h\nthis : Primcodable (List \u03b2) := prim H\nG : \u03b1 \u2192 \u03c3 \u00d7 List \u03b2 \u2192 \u03c3 \u00d7 List \u03b2 := fun a IH => List.casesOn IH.2 IH fun b l => (h a (IH.1, b), l)\nhG : Primrec\u2082 G\nF : \u03b1 \u2192 \u2115 \u2192 \u03c3 \u00d7 List \u03b2 := fun a n => (G a)^[n] (g a, f a)\nhF : Primrec fun a => (F a (encode (f a))).1\n\u22a2 Primrec fun a => List.foldl (fun s b => h a (s, b)) (g a) (f a)"}, {"tactic": "suffices \u2200 a n, F a n = (((f a).take n).foldl (fun s b => h a (s, b)) (g a), (f a).drop n) by\n  refine hF.of_eq fun a => ?_\n  rw [this, List.take_all_of_le (length_le_encode _)]", "annotated_tactic": ["suffices \u2200 a n, F a n = (((f a).<a>take</a> n).<a>foldl</a> (fun s b => h a (s, b)) (g a), (f a).<a>drop</a> n) by\n    refine hF.of_eq fun a => ?_\n    rw [this, <a>List.take_all_of_le</a> (<a>length_le_encode</a> _)]", [{"full_name": "List.take", "def_path": "lake-packages/lean4/src/lean/Init/Data/List/Basic.lean", "def_pos": [494, 5], "def_end_pos": [494, 9]}, {"full_name": "List.foldl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2212, 5], "def_end_pos": [2212, 15]}, {"full_name": "List.drop", "def_path": "lake-packages/lean4/src/lean/Init/Data/List/Basic.lean", "def_pos": [475, 5], "def_end_pos": [475, 9]}, {"full_name": "List.take_all_of_le", "def_path": "Mathlib/Data/List/Basic.lean", "def_pos": [1867, 9], "def_end_pos": [1867, 23]}, {"full_name": "Encodable.length_le_encode", "def_path": "Mathlib/Logic/Equiv/List.lean", "def_pos": [83, 9], "def_end_pos": [83, 25]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03c3 : Type u_3\ninst\u271d\u00b2 : Primcodable \u03b1\ninst\u271d\u00b9 : Primcodable \u03b2\ninst\u271d : Primcodable \u03c3\nH : Nat.Primrec fun n => encode (decode n)\nf : \u03b1 \u2192 List \u03b2\ng : \u03b1 \u2192 \u03c3\nh : \u03b1 \u2192 \u03c3 \u00d7 \u03b2 \u2192 \u03c3\nhf : Primrec f\nhg : Primrec g\nhh : Primrec\u2082 h\nthis : Primcodable (List \u03b2) := prim H\nG : \u03b1 \u2192 \u03c3 \u00d7 List \u03b2 \u2192 \u03c3 \u00d7 List \u03b2 := fun a IH => List.casesOn IH.2 IH fun b l => (h a (IH.1, b), l)\nhG : Primrec\u2082 G\nF : \u03b1 \u2192 \u2115 \u2192 \u03c3 \u00d7 List \u03b2 := fun a n => (G a)^[n] (g a, f a)\nhF : Primrec fun a => (F a (encode (f a))).1\n\u22a2 Primrec fun a => List.foldl (fun s b => h a (s, b)) (g a) (f a)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03c3 : Type u_3\ninst\u271d\u00b2 : Primcodable \u03b1\ninst\u271d\u00b9 : Primcodable \u03b2\ninst\u271d : Primcodable \u03c3\nH : Nat.Primrec fun n => encode (decode n)\nf : \u03b1 \u2192 List \u03b2\ng : \u03b1 \u2192 \u03c3\nh : \u03b1 \u2192 \u03c3 \u00d7 \u03b2 \u2192 \u03c3\nhf : Primrec f\nhg : Primrec g\nhh : Primrec\u2082 h\nthis : Primcodable (List \u03b2) := prim H\nG : \u03b1 \u2192 \u03c3 \u00d7 List \u03b2 \u2192 \u03c3 \u00d7 List \u03b2 := fun a IH => List.casesOn IH.2 IH fun b l => (h a (IH.1, b), l)\nhG : Primrec\u2082 G\nF : \u03b1 \u2192 \u2115 \u2192 \u03c3 \u00d7 List \u03b2 := fun a n => (G a)^[n] (g a, f a)\nhF : Primrec fun a => (F a (encode (f a))).1\n\u22a2 \u2200 (a : \u03b1) (n : \u2115), F a n = (List.foldl (fun s b => h a (s, b)) (g a) (List.take n (f a)), List.drop n (f a))"}, {"tactic": "introv", "annotated_tactic": ["introv", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03c3 : Type u_3\ninst\u271d\u00b2 : Primcodable \u03b1\ninst\u271d\u00b9 : Primcodable \u03b2\ninst\u271d : Primcodable \u03c3\nH : Nat.Primrec fun n => encode (decode n)\nf : \u03b1 \u2192 List \u03b2\ng : \u03b1 \u2192 \u03c3\nh : \u03b1 \u2192 \u03c3 \u00d7 \u03b2 \u2192 \u03c3\nhf : Primrec f\nhg : Primrec g\nhh : Primrec\u2082 h\nthis : Primcodable (List \u03b2) := prim H\nG : \u03b1 \u2192 \u03c3 \u00d7 List \u03b2 \u2192 \u03c3 \u00d7 List \u03b2 := fun a IH => List.casesOn IH.2 IH fun b l => (h a (IH.1, b), l)\nhG : Primrec\u2082 G\nF : \u03b1 \u2192 \u2115 \u2192 \u03c3 \u00d7 List \u03b2 := fun a n => (G a)^[n] (g a, f a)\nhF : Primrec fun a => (F a (encode (f a))).1\n\u22a2 \u2200 (a : \u03b1) (n : \u2115), F a n = (List.foldl (fun s b => h a (s, b)) (g a) (List.take n (f a)), List.drop n (f a))", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03c3 : Type u_3\ninst\u271d\u00b2 : Primcodable \u03b1\ninst\u271d\u00b9 : Primcodable \u03b2\ninst\u271d : Primcodable \u03c3\nH : Nat.Primrec fun n => encode (decode n)\nf : \u03b1 \u2192 List \u03b2\ng : \u03b1 \u2192 \u03c3\nh : \u03b1 \u2192 \u03c3 \u00d7 \u03b2 \u2192 \u03c3\nhf : Primrec f\nhg : Primrec g\nhh : Primrec\u2082 h\nthis : Primcodable (List \u03b2) := prim H\nG : \u03b1 \u2192 \u03c3 \u00d7 List \u03b2 \u2192 \u03c3 \u00d7 List \u03b2 := fun a IH => List.casesOn IH.2 IH fun b l => (h a (IH.1, b), l)\nhG : Primrec\u2082 G\nF : \u03b1 \u2192 \u2115 \u2192 \u03c3 \u00d7 List \u03b2 := fun a n => (G a)^[n] (g a, f a)\nhF : Primrec fun a => (F a (encode (f a))).1\na : \u03b1\nn : \u2115\n\u22a2 F a n = (List.foldl (fun s b => h a (s, b)) (g a) (List.take n (f a)), List.drop n (f a))"}, {"tactic": "dsimp only", "annotated_tactic": ["dsimp only", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03c3 : Type u_3\ninst\u271d\u00b2 : Primcodable \u03b1\ninst\u271d\u00b9 : Primcodable \u03b2\ninst\u271d : Primcodable \u03c3\nH : Nat.Primrec fun n => encode (decode n)\nf : \u03b1 \u2192 List \u03b2\ng : \u03b1 \u2192 \u03c3\nh : \u03b1 \u2192 \u03c3 \u00d7 \u03b2 \u2192 \u03c3\nhf : Primrec f\nhg : Primrec g\nhh : Primrec\u2082 h\nthis : Primcodable (List \u03b2) := prim H\nG : \u03b1 \u2192 \u03c3 \u00d7 List \u03b2 \u2192 \u03c3 \u00d7 List \u03b2 := fun a IH => List.casesOn IH.2 IH fun b l => (h a (IH.1, b), l)\nhG : Primrec\u2082 G\nF : \u03b1 \u2192 \u2115 \u2192 \u03c3 \u00d7 List \u03b2 := fun a n => (G a)^[n] (g a, f a)\nhF : Primrec fun a => (F a (encode (f a))).1\na : \u03b1\nn : \u2115\n\u22a2 F a n = (List.foldl (fun s b => h a (s, b)) (g a) (List.take n (f a)), List.drop n (f a))", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03c3 : Type u_3\ninst\u271d\u00b2 : Primcodable \u03b1\ninst\u271d\u00b9 : Primcodable \u03b2\ninst\u271d : Primcodable \u03c3\nH : Nat.Primrec fun n => encode (decode n)\nf : \u03b1 \u2192 List \u03b2\ng : \u03b1 \u2192 \u03c3\nh : \u03b1 \u2192 \u03c3 \u00d7 \u03b2 \u2192 \u03c3\nhf : Primrec f\nhg : Primrec g\nhh : Primrec\u2082 h\nthis : Primcodable (List \u03b2) := prim H\nG : \u03b1 \u2192 \u03c3 \u00d7 List \u03b2 \u2192 \u03c3 \u00d7 List \u03b2 := fun a IH => List.casesOn IH.2 IH fun b l => (h a (IH.1, b), l)\nhG : Primrec\u2082 G\nF : \u03b1 \u2192 \u2115 \u2192 \u03c3 \u00d7 List \u03b2 := fun a n => (G a)^[n] (g a, f a)\nhF : Primrec fun a => (F a (encode (f a))).1\na : \u03b1\nn : \u2115\n\u22a2 (fun IH => List.rec IH (fun head tail tail_ih => (h a (IH.1, head), tail)) IH.2)^[n] (g a, f a) =\n    (List.foldl (fun s b => h a (s, b)) (g a) (List.take n (f a)), List.drop n (f a))"}, {"tactic": "generalize f a = l", "annotated_tactic": ["generalize f a = l", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03c3 : Type u_3\ninst\u271d\u00b2 : Primcodable \u03b1\ninst\u271d\u00b9 : Primcodable \u03b2\ninst\u271d : Primcodable \u03c3\nH : Nat.Primrec fun n => encode (decode n)\nf : \u03b1 \u2192 List \u03b2\ng : \u03b1 \u2192 \u03c3\nh : \u03b1 \u2192 \u03c3 \u00d7 \u03b2 \u2192 \u03c3\nhf : Primrec f\nhg : Primrec g\nhh : Primrec\u2082 h\nthis : Primcodable (List \u03b2) := prim H\nG : \u03b1 \u2192 \u03c3 \u00d7 List \u03b2 \u2192 \u03c3 \u00d7 List \u03b2 := fun a IH => List.casesOn IH.2 IH fun b l => (h a (IH.1, b), l)\nhG : Primrec\u2082 G\nF : \u03b1 \u2192 \u2115 \u2192 \u03c3 \u00d7 List \u03b2 := fun a n => (G a)^[n] (g a, f a)\nhF : Primrec fun a => (F a (encode (f a))).1\na : \u03b1\nn : \u2115\n\u22a2 (fun IH => List.rec IH (fun head tail tail_ih => (h a (IH.1, head), tail)) IH.2)^[n] (g a, f a) =\n    (List.foldl (fun s b => h a (s, b)) (g a) (List.take n (f a)), List.drop n (f a))", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03c3 : Type u_3\ninst\u271d\u00b2 : Primcodable \u03b1\ninst\u271d\u00b9 : Primcodable \u03b2\ninst\u271d : Primcodable \u03c3\nH : Nat.Primrec fun n => encode (decode n)\nf : \u03b1 \u2192 List \u03b2\ng : \u03b1 \u2192 \u03c3\nh : \u03b1 \u2192 \u03c3 \u00d7 \u03b2 \u2192 \u03c3\nhf : Primrec f\nhg : Primrec g\nhh : Primrec\u2082 h\nthis : Primcodable (List \u03b2) := prim H\nG : \u03b1 \u2192 \u03c3 \u00d7 List \u03b2 \u2192 \u03c3 \u00d7 List \u03b2 := fun a IH => List.casesOn IH.2 IH fun b l => (h a (IH.1, b), l)\nhG : Primrec\u2082 G\nF : \u03b1 \u2192 \u2115 \u2192 \u03c3 \u00d7 List \u03b2 := fun a n => (G a)^[n] (g a, f a)\nhF : Primrec fun a => (F a (encode (f a))).1\na : \u03b1\nn : \u2115\nl : List \u03b2\n\u22a2 (fun IH => List.rec IH (fun head tail tail_ih => (h a (IH.1, head), tail)) IH.2)^[n] (g a, l) =\n    (List.foldl (fun s b => h a (s, b)) (g a) (List.take n l), List.drop n l)"}, {"tactic": "generalize g a = x", "annotated_tactic": ["generalize g a = x", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03c3 : Type u_3\ninst\u271d\u00b2 : Primcodable \u03b1\ninst\u271d\u00b9 : Primcodable \u03b2\ninst\u271d : Primcodable \u03c3\nH : Nat.Primrec fun n => encode (decode n)\nf : \u03b1 \u2192 List \u03b2\ng : \u03b1 \u2192 \u03c3\nh : \u03b1 \u2192 \u03c3 \u00d7 \u03b2 \u2192 \u03c3\nhf : Primrec f\nhg : Primrec g\nhh : Primrec\u2082 h\nthis : Primcodable (List \u03b2) := prim H\nG : \u03b1 \u2192 \u03c3 \u00d7 List \u03b2 \u2192 \u03c3 \u00d7 List \u03b2 := fun a IH => List.casesOn IH.2 IH fun b l => (h a (IH.1, b), l)\nhG : Primrec\u2082 G\nF : \u03b1 \u2192 \u2115 \u2192 \u03c3 \u00d7 List \u03b2 := fun a n => (G a)^[n] (g a, f a)\nhF : Primrec fun a => (F a (encode (f a))).1\na : \u03b1\nn : \u2115\nl : List \u03b2\n\u22a2 (fun IH => List.rec IH (fun head tail tail_ih => (h a (IH.1, head), tail)) IH.2)^[n] (g a, l) =\n    (List.foldl (fun s b => h a (s, b)) (g a) (List.take n l), List.drop n l)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03c3 : Type u_3\ninst\u271d\u00b2 : Primcodable \u03b1\ninst\u271d\u00b9 : Primcodable \u03b2\ninst\u271d : Primcodable \u03c3\nH : Nat.Primrec fun n => encode (decode n)\nf : \u03b1 \u2192 List \u03b2\ng : \u03b1 \u2192 \u03c3\nh : \u03b1 \u2192 \u03c3 \u00d7 \u03b2 \u2192 \u03c3\nhf : Primrec f\nhg : Primrec g\nhh : Primrec\u2082 h\nthis : Primcodable (List \u03b2) := prim H\nG : \u03b1 \u2192 \u03c3 \u00d7 List \u03b2 \u2192 \u03c3 \u00d7 List \u03b2 := fun a IH => List.casesOn IH.2 IH fun b l => (h a (IH.1, b), l)\nhG : Primrec\u2082 G\nF : \u03b1 \u2192 \u2115 \u2192 \u03c3 \u00d7 List \u03b2 := fun a n => (G a)^[n] (g a, f a)\nhF : Primrec fun a => (F a (encode (f a))).1\na : \u03b1\nn : \u2115\nl : List \u03b2\nx : \u03c3\n\u22a2 (fun IH => List.rec IH (fun head tail tail_ih => (h a (IH.1, head), tail)) IH.2)^[n] (x, l) =\n    (List.foldl (fun s b => h a (s, b)) x (List.take n l), List.drop n l)"}, {"tactic": "induction' n with n IH generalizing l x", "annotated_tactic": ["induction' n with n IH generalizing l x", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03c3 : Type u_3\ninst\u271d\u00b2 : Primcodable \u03b1\ninst\u271d\u00b9 : Primcodable \u03b2\ninst\u271d : Primcodable \u03c3\nH : Nat.Primrec fun n => encode (decode n)\nf : \u03b1 \u2192 List \u03b2\ng : \u03b1 \u2192 \u03c3\nh : \u03b1 \u2192 \u03c3 \u00d7 \u03b2 \u2192 \u03c3\nhf : Primrec f\nhg : Primrec g\nhh : Primrec\u2082 h\nthis : Primcodable (List \u03b2) := prim H\nG : \u03b1 \u2192 \u03c3 \u00d7 List \u03b2 \u2192 \u03c3 \u00d7 List \u03b2 := fun a IH => List.casesOn IH.2 IH fun b l => (h a (IH.1, b), l)\nhG : Primrec\u2082 G\nF : \u03b1 \u2192 \u2115 \u2192 \u03c3 \u00d7 List \u03b2 := fun a n => (G a)^[n] (g a, f a)\nhF : Primrec fun a => (F a (encode (f a))).1\na : \u03b1\nn : \u2115\nl : List \u03b2\nx : \u03c3\n\u22a2 (fun IH => List.rec IH (fun head tail tail_ih => (h a (IH.1, head), tail)) IH.2)^[n] (x, l) =\n    (List.foldl (fun s b => h a (s, b)) x (List.take n l), List.drop n l)", "state_after": "case zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03c3 : Type u_3\ninst\u271d\u00b2 : Primcodable \u03b1\ninst\u271d\u00b9 : Primcodable \u03b2\ninst\u271d : Primcodable \u03c3\nH : Nat.Primrec fun n => encode (decode n)\nf : \u03b1 \u2192 List \u03b2\ng : \u03b1 \u2192 \u03c3\nh : \u03b1 \u2192 \u03c3 \u00d7 \u03b2 \u2192 \u03c3\nhf : Primrec f\nhg : Primrec g\nhh : Primrec\u2082 h\nthis : Primcodable (List \u03b2) := prim H\nG : \u03b1 \u2192 \u03c3 \u00d7 List \u03b2 \u2192 \u03c3 \u00d7 List \u03b2 := fun a IH => List.casesOn IH.2 IH fun b l => (h a (IH.1, b), l)\nhG : Primrec\u2082 G\nF : \u03b1 \u2192 \u2115 \u2192 \u03c3 \u00d7 List \u03b2 := fun a n => (G a)^[n] (g a, f a)\nhF : Primrec fun a => (F a (encode (f a))).1\na : \u03b1\nl\u271d : List \u03b2\nx\u271d : \u03c3\nl : List \u03b2\nx : \u03c3\n\u22a2 (fun IH => List.rec IH (fun head tail tail_ih => (h a (IH.1, head), tail)) IH.2)^[Nat.zero] (x, l) =\n    (List.foldl (fun s b => h a (s, b)) x (List.take Nat.zero l), List.drop Nat.zero l)\n\ncase succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03c3 : Type u_3\ninst\u271d\u00b2 : Primcodable \u03b1\ninst\u271d\u00b9 : Primcodable \u03b2\ninst\u271d : Primcodable \u03c3\nH : Nat.Primrec fun n => encode (decode n)\nf : \u03b1 \u2192 List \u03b2\ng : \u03b1 \u2192 \u03c3\nh : \u03b1 \u2192 \u03c3 \u00d7 \u03b2 \u2192 \u03c3\nhf : Primrec f\nhg : Primrec g\nhh : Primrec\u2082 h\nthis : Primcodable (List \u03b2) := prim H\nG : \u03b1 \u2192 \u03c3 \u00d7 List \u03b2 \u2192 \u03c3 \u00d7 List \u03b2 := fun a IH => List.casesOn IH.2 IH fun b l => (h a (IH.1, b), l)\nhG : Primrec\u2082 G\nF : \u03b1 \u2192 \u2115 \u2192 \u03c3 \u00d7 List \u03b2 := fun a n => (G a)^[n] (g a, f a)\nhF : Primrec fun a => (F a (encode (f a))).1\na : \u03b1\nl\u271d : List \u03b2\nx\u271d : \u03c3\nn : \u2115\nIH :\n  \u2200 (l : List \u03b2) (x : \u03c3),\n    (fun IH => List.rec IH (fun head tail tail_ih => (h a (IH.1, head), tail)) IH.2)^[n] (x, l) =\n      (List.foldl (fun s b => h a (s, b)) x (List.take n l), List.drop n l)\nl : List \u03b2\nx : \u03c3\n\u22a2 (fun IH => List.rec IH (fun head tail tail_ih => (h a (IH.1, head), tail)) IH.2)^[Nat.succ n] (x, l) =\n    (List.foldl (fun s b => h a (s, b)) x (List.take (Nat.succ n) l), List.drop (Nat.succ n) l)"}, {"tactic": "simp only [iterate_succ, comp_apply]", "annotated_tactic": ["simp only [<a>iterate_succ</a>, <a>comp_apply</a>]", [{"full_name": "Function.iterate_succ", "def_path": "Mathlib/Logic/Function/Iterate.lean", "def_pos": [62, 9], "def_end_pos": [62, 21]}, {"full_name": "Function.comp_apply", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [33, 17], "def_end_pos": [33, 36]}]], "state_before": "case succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03c3 : Type u_3\ninst\u271d\u00b2 : Primcodable \u03b1\ninst\u271d\u00b9 : Primcodable \u03b2\ninst\u271d : Primcodable \u03c3\nH : Nat.Primrec fun n => encode (decode n)\nf : \u03b1 \u2192 List \u03b2\ng : \u03b1 \u2192 \u03c3\nh : \u03b1 \u2192 \u03c3 \u00d7 \u03b2 \u2192 \u03c3\nhf : Primrec f\nhg : Primrec g\nhh : Primrec\u2082 h\nthis : Primcodable (List \u03b2) := prim H\nG : \u03b1 \u2192 \u03c3 \u00d7 List \u03b2 \u2192 \u03c3 \u00d7 List \u03b2 := fun a IH => List.casesOn IH.2 IH fun b l => (h a (IH.1, b), l)\nhG : Primrec\u2082 G\nF : \u03b1 \u2192 \u2115 \u2192 \u03c3 \u00d7 List \u03b2 := fun a n => (G a)^[n] (g a, f a)\nhF : Primrec fun a => (F a (encode (f a))).1\na : \u03b1\nl\u271d : List \u03b2\nx\u271d : \u03c3\nn : \u2115\nIH :\n  \u2200 (l : List \u03b2) (x : \u03c3),\n    (fun IH => List.rec IH (fun head tail tail_ih => (h a (IH.1, head), tail)) IH.2)^[n] (x, l) =\n      (List.foldl (fun s b => h a (s, b)) x (List.take n l), List.drop n l)\nl : List \u03b2\nx : \u03c3\n\u22a2 (fun IH => List.rec IH (fun head tail tail_ih => (h a (IH.1, head), tail)) IH.2)^[Nat.succ n] (x, l) =\n    (List.foldl (fun s b => h a (s, b)) x (List.take (Nat.succ n) l), List.drop (Nat.succ n) l)", "state_after": "case succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03c3 : Type u_3\ninst\u271d\u00b2 : Primcodable \u03b1\ninst\u271d\u00b9 : Primcodable \u03b2\ninst\u271d : Primcodable \u03c3\nH : Nat.Primrec fun n => encode (decode n)\nf : \u03b1 \u2192 List \u03b2\ng : \u03b1 \u2192 \u03c3\nh : \u03b1 \u2192 \u03c3 \u00d7 \u03b2 \u2192 \u03c3\nhf : Primrec f\nhg : Primrec g\nhh : Primrec\u2082 h\nthis : Primcodable (List \u03b2) := prim H\nG : \u03b1 \u2192 \u03c3 \u00d7 List \u03b2 \u2192 \u03c3 \u00d7 List \u03b2 := fun a IH => List.casesOn IH.2 IH fun b l => (h a (IH.1, b), l)\nhG : Primrec\u2082 G\nF : \u03b1 \u2192 \u2115 \u2192 \u03c3 \u00d7 List \u03b2 := fun a n => (G a)^[n] (g a, f a)\nhF : Primrec fun a => (F a (encode (f a))).1\na : \u03b1\nl\u271d : List \u03b2\nx\u271d : \u03c3\nn : \u2115\nIH :\n  \u2200 (l : List \u03b2) (x : \u03c3),\n    (fun IH => List.rec IH (fun head tail tail_ih => (h a (IH.1, head), tail)) IH.2)^[n] (x, l) =\n      (List.foldl (fun s b => h a (s, b)) x (List.take n l), List.drop n l)\nl : List \u03b2\nx : \u03c3\n\u22a2 (fun IH => List.rec IH (fun head tail tail_ih => (h a (IH.1, head), tail)) IH.2)^[n]\n      (List.rec (x, l) (fun head tail tail_ih => (h a (x, head), tail)) l) =\n    (List.foldl (fun s b => h a (s, b)) x (List.take (Nat.succ n) l), List.drop (Nat.succ n) l)"}, {"tactic": "cases' l with b l <;> simp [IH]", "annotated_tactic": ["cases' l with b l <;> simp [IH]", []], "state_before": "case succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03c3 : Type u_3\ninst\u271d\u00b2 : Primcodable \u03b1\ninst\u271d\u00b9 : Primcodable \u03b2\ninst\u271d : Primcodable \u03c3\nH : Nat.Primrec fun n => encode (decode n)\nf : \u03b1 \u2192 List \u03b2\ng : \u03b1 \u2192 \u03c3\nh : \u03b1 \u2192 \u03c3 \u00d7 \u03b2 \u2192 \u03c3\nhf : Primrec f\nhg : Primrec g\nhh : Primrec\u2082 h\nthis : Primcodable (List \u03b2) := prim H\nG : \u03b1 \u2192 \u03c3 \u00d7 List \u03b2 \u2192 \u03c3 \u00d7 List \u03b2 := fun a IH => List.casesOn IH.2 IH fun b l => (h a (IH.1, b), l)\nhG : Primrec\u2082 G\nF : \u03b1 \u2192 \u2115 \u2192 \u03c3 \u00d7 List \u03b2 := fun a n => (G a)^[n] (g a, f a)\nhF : Primrec fun a => (F a (encode (f a))).1\na : \u03b1\nl\u271d : List \u03b2\nx\u271d : \u03c3\nn : \u2115\nIH :\n  \u2200 (l : List \u03b2) (x : \u03c3),\n    (fun IH => List.rec IH (fun head tail tail_ih => (h a (IH.1, head), tail)) IH.2)^[n] (x, l) =\n      (List.foldl (fun s b => h a (s, b)) x (List.take n l), List.drop n l)\nl : List \u03b2\nx : \u03c3\n\u22a2 (fun IH => List.rec IH (fun head tail tail_ih => (h a (IH.1, head), tail)) IH.2)^[n]\n      (List.rec (x, l) (fun head tail tail_ih => (h a (x, head), tail)) l) =\n    (List.foldl (fun s b => h a (s, b)) x (List.take (Nat.succ n) l), List.drop (Nat.succ n) l)", "state_after": "no goals"}, {"tactic": "refine hF.of_eq fun a => ?_", "annotated_tactic": ["refine hF.of_eq fun a => ?_", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03c3 : Type u_3\ninst\u271d\u00b2 : Primcodable \u03b1\ninst\u271d\u00b9 : Primcodable \u03b2\ninst\u271d : Primcodable \u03c3\nH : Nat.Primrec fun n => encode (decode n)\nf : \u03b1 \u2192 List \u03b2\ng : \u03b1 \u2192 \u03c3\nh : \u03b1 \u2192 \u03c3 \u00d7 \u03b2 \u2192 \u03c3\nhf : Primrec f\nhg : Primrec g\nhh : Primrec\u2082 h\nthis\u271d : Primcodable (List \u03b2) := prim H\nG : \u03b1 \u2192 \u03c3 \u00d7 List \u03b2 \u2192 \u03c3 \u00d7 List \u03b2 := fun a IH => List.casesOn IH.2 IH fun b l => (h a (IH.1, b), l)\nhG : Primrec\u2082 G\nF : \u03b1 \u2192 \u2115 \u2192 \u03c3 \u00d7 List \u03b2 := fun a n => (G a)^[n] (g a, f a)\nhF : Primrec fun a => (F a (encode (f a))).1\nthis : \u2200 (a : \u03b1) (n : \u2115), F a n = (List.foldl (fun s b => h a (s, b)) (g a) (List.take n (f a)), List.drop n (f a))\n\u22a2 Primrec fun a => List.foldl (fun s b => h a (s, b)) (g a) (f a)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03c3 : Type u_3\ninst\u271d\u00b2 : Primcodable \u03b1\ninst\u271d\u00b9 : Primcodable \u03b2\ninst\u271d : Primcodable \u03c3\nH : Nat.Primrec fun n => encode (decode n)\nf : \u03b1 \u2192 List \u03b2\ng : \u03b1 \u2192 \u03c3\nh : \u03b1 \u2192 \u03c3 \u00d7 \u03b2 \u2192 \u03c3\nhf : Primrec f\nhg : Primrec g\nhh : Primrec\u2082 h\nthis\u271d : Primcodable (List \u03b2) := prim H\nG : \u03b1 \u2192 \u03c3 \u00d7 List \u03b2 \u2192 \u03c3 \u00d7 List \u03b2 := fun a IH => List.casesOn IH.2 IH fun b l => (h a (IH.1, b), l)\nhG : Primrec\u2082 G\nF : \u03b1 \u2192 \u2115 \u2192 \u03c3 \u00d7 List \u03b2 := fun a n => (G a)^[n] (g a, f a)\nhF : Primrec fun a => (F a (encode (f a))).1\nthis : \u2200 (a : \u03b1) (n : \u2115), F a n = (List.foldl (fun s b => h a (s, b)) (g a) (List.take n (f a)), List.drop n (f a))\na : \u03b1\n\u22a2 (F a (encode (f a))).1 = List.foldl (fun s b => h a (s, b)) (g a) (f a)"}, {"tactic": "rw [this, List.take_all_of_le (length_le_encode _)]", "annotated_tactic": ["rw [this, <a>List.take_all_of_le</a> (<a>length_le_encode</a> _)]", [{"full_name": "List.take_all_of_le", "def_path": "Mathlib/Data/List/Basic.lean", "def_pos": [1867, 9], "def_end_pos": [1867, 23]}, {"full_name": "Encodable.length_le_encode", "def_path": "Mathlib/Logic/Equiv/List.lean", "def_pos": [83, 9], "def_end_pos": [83, 25]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03c3 : Type u_3\ninst\u271d\u00b2 : Primcodable \u03b1\ninst\u271d\u00b9 : Primcodable \u03b2\ninst\u271d : Primcodable \u03c3\nH : Nat.Primrec fun n => encode (decode n)\nf : \u03b1 \u2192 List \u03b2\ng : \u03b1 \u2192 \u03c3\nh : \u03b1 \u2192 \u03c3 \u00d7 \u03b2 \u2192 \u03c3\nhf : Primrec f\nhg : Primrec g\nhh : Primrec\u2082 h\nthis\u271d : Primcodable (List \u03b2) := prim H\nG : \u03b1 \u2192 \u03c3 \u00d7 List \u03b2 \u2192 \u03c3 \u00d7 List \u03b2 := fun a IH => List.casesOn IH.2 IH fun b l => (h a (IH.1, b), l)\nhG : Primrec\u2082 G\nF : \u03b1 \u2192 \u2115 \u2192 \u03c3 \u00d7 List \u03b2 := fun a n => (G a)^[n] (g a, f a)\nhF : Primrec fun a => (F a (encode (f a))).1\nthis : \u2200 (a : \u03b1) (n : \u2115), F a n = (List.foldl (fun s b => h a (s, b)) (g a) (List.take n (f a)), List.drop n (f a))\na : \u03b1\n\u22a2 (F a (encode (f a))).1 = List.foldl (fun s b => h a (s, b)) (g a) (f a)", "state_after": "no goals"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03c3 : Type u_3\ninst\u271d\u00b2 : Primcodable \u03b1\ninst\u271d\u00b9 : Primcodable \u03b2\ninst\u271d : Primcodable \u03c3\nH : Nat.Primrec fun n => encode (decode n)\nf : \u03b1 \u2192 List \u03b2\ng : \u03b1 \u2192 \u03c3\nh : \u03b1 \u2192 \u03c3 \u00d7 \u03b2 \u2192 \u03c3\nhf : Primrec f\nhg : Primrec g\nhh : Primrec\u2082 h\nthis : Primcodable (List \u03b2) := prim H\nG : \u03b1 \u2192 \u03c3 \u00d7 List \u03b2 \u2192 \u03c3 \u00d7 List \u03b2 := fun a IH => List.casesOn IH.2 IH fun b l => (h a (IH.1, b), l)\nhG : Primrec\u2082 G\nF : \u03b1 \u2192 \u2115 \u2192 \u03c3 \u00d7 List \u03b2 := fun a n => (G a)^[n] (g a, f a)\nhF : Primrec fun a => (F a (encode (f a))).1\na : \u03b1\nl\u271d : List \u03b2\nx\u271d : \u03c3\nl : List \u03b2\nx : \u03c3\n\u22a2 (fun IH => List.rec IH (fun head tail tail_ih => (h a (IH.1, head), tail)) IH.2)^[Nat.zero] (x, l) =\n    (List.foldl (fun s b => h a (s, b)) x (List.take Nat.zero l), List.drop Nat.zero l)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "full_name": "MeasureTheory.SimpleFunc.mem\u2112p_of_finite_measure_preimage", "start": [336, 1], "end": [348, 86], "traced_tactics": [{"tactic": "by_cases hp0 : p = 0", "annotated_tactic": ["by_cases hp0 : p = 0", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np\u271d p : \u211d\u22650\u221e\nf : \u03b1 \u2192\u209b E\nhf : \u2200 (y : E), y \u2260 0 \u2192 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4\n\u22a2 Mem\u2112p (\u2191f) p", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np\u271d p : \u211d\u22650\u221e\nf : \u03b1 \u2192\u209b E\nhf : \u2200 (y : E), y \u2260 0 \u2192 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4\nhp0 : p = 0\n\u22a2 Mem\u2112p (\u2191f) p\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np\u271d p : \u211d\u22650\u221e\nf : \u03b1 \u2192\u209b E\nhf : \u2200 (y : E), y \u2260 0 \u2192 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4\nhp0 : \u00acp = 0\n\u22a2 Mem\u2112p (\u2191f) p"}, {"tactic": "by_cases hp_top : p = \u221e", "annotated_tactic": ["by_cases hp_top : p = \u221e", []], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np\u271d p : \u211d\u22650\u221e\nf : \u03b1 \u2192\u209b E\nhf : \u2200 (y : E), y \u2260 0 \u2192 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4\nhp0 : \u00acp = 0\n\u22a2 Mem\u2112p (\u2191f) p", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np\u271d p : \u211d\u22650\u221e\nf : \u03b1 \u2192\u209b E\nhf : \u2200 (y : E), y \u2260 0 \u2192 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4\nhp0 : \u00acp = 0\nhp_top : p = \u22a4\n\u22a2 Mem\u2112p (\u2191f) p\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np\u271d p : \u211d\u22650\u221e\nf : \u03b1 \u2192\u209b E\nhf : \u2200 (y : E), y \u2260 0 \u2192 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4\nhp0 : \u00acp = 0\nhp_top : \u00acp = \u22a4\n\u22a2 Mem\u2112p (\u2191f) p"}, {"tactic": "refine' \u27e8f.aestronglyMeasurable, _\u27e9", "annotated_tactic": ["refine' \u27e8f.aestronglyMeasurable, _\u27e9", []], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np\u271d p : \u211d\u22650\u221e\nf : \u03b1 \u2192\u209b E\nhf : \u2200 (y : E), y \u2260 0 \u2192 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4\nhp0 : \u00acp = 0\nhp_top : \u00acp = \u22a4\n\u22a2 Mem\u2112p (\u2191f) p", "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np\u271d p : \u211d\u22650\u221e\nf : \u03b1 \u2192\u209b E\nhf : \u2200 (y : E), y \u2260 0 \u2192 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4\nhp0 : \u00acp = 0\nhp_top : \u00acp = \u22a4\n\u22a2 snorm (\u2191f) p \u03bc < \u22a4"}, {"tactic": "rw [snorm_eq_snorm' hp0 hp_top, f.snorm'_eq]", "annotated_tactic": ["rw [<a>snorm_eq_snorm'</a> hp0 hp_top, f.snorm'_eq]", [{"full_name": "MeasureTheory.snorm_eq_snorm'", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [88, 9], "def_end_pos": [88, 24]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np\u271d p : \u211d\u22650\u221e\nf : \u03b1 \u2192\u209b E\nhf : \u2200 (y : E), y \u2260 0 \u2192 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4\nhp0 : \u00acp = 0\nhp_top : \u00acp = \u22a4\n\u22a2 snorm (\u2191f) p \u03bc < \u22a4", "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np\u271d p : \u211d\u22650\u221e\nf : \u03b1 \u2192\u209b E\nhf : \u2200 (y : E), y \u2260 0 \u2192 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4\nhp0 : \u00acp = 0\nhp_top : \u00acp = \u22a4\n\u22a2 (\u2211 y in SimpleFunc.range f, \u2191\u2016y\u2016\u208a ^ ENNReal.toReal p * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y})) ^ (1 / ENNReal.toReal p) < \u22a4"}, {"tactic": "refine' ENNReal.rpow_lt_top_of_nonneg (by simp) (ENNReal.sum_lt_top_iff.mpr fun y _ => _).ne", "annotated_tactic": ["refine' <a>ENNReal.rpow_lt_top_of_nonneg</a> (by simp) (ENNReal.sum_lt_top_iff.mpr fun y _ => _).<a>ne</a>", [{"full_name": "ENNReal.rpow_lt_top_of_nonneg", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [503, 9], "def_end_pos": [503, 30]}, {"full_name": "LT.lt.ne", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [152, 7], "def_end_pos": [152, 15]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np\u271d p : \u211d\u22650\u221e\nf : \u03b1 \u2192\u209b E\nhf : \u2200 (y : E), y \u2260 0 \u2192 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4\nhp0 : \u00acp = 0\nhp_top : \u00acp = \u22a4\n\u22a2 (\u2211 y in SimpleFunc.range f, \u2191\u2016y\u2016\u208a ^ ENNReal.toReal p * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y})) ^ (1 / ENNReal.toReal p) < \u22a4", "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np\u271d p : \u211d\u22650\u221e\nf : \u03b1 \u2192\u209b E\nhf : \u2200 (y : E), y \u2260 0 \u2192 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4\nhp0 : \u00acp = 0\nhp_top : \u00acp = \u22a4\ny : E\nx\u271d : y \u2208 SimpleFunc.range f\n\u22a2 \u2191\u2016y\u2016\u208a ^ ENNReal.toReal p * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4"}, {"tactic": "by_cases hy0 : y = 0", "annotated_tactic": ["by_cases hy0 : y = 0", []], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np\u271d p : \u211d\u22650\u221e\nf : \u03b1 \u2192\u209b E\nhf : \u2200 (y : E), y \u2260 0 \u2192 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4\nhp0 : \u00acp = 0\nhp_top : \u00acp = \u22a4\ny : E\nx\u271d : y \u2208 SimpleFunc.range f\n\u22a2 \u2191\u2016y\u2016\u208a ^ ENNReal.toReal p * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np\u271d p : \u211d\u22650\u221e\nf : \u03b1 \u2192\u209b E\nhf : \u2200 (y : E), y \u2260 0 \u2192 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4\nhp0 : \u00acp = 0\nhp_top : \u00acp = \u22a4\ny : E\nx\u271d : y \u2208 SimpleFunc.range f\nhy0 : y = 0\n\u22a2 \u2191\u2016y\u2016\u208a ^ ENNReal.toReal p * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np\u271d p : \u211d\u22650\u221e\nf : \u03b1 \u2192\u209b E\nhf : \u2200 (y : E), y \u2260 0 \u2192 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4\nhp0 : \u00acp = 0\nhp_top : \u00acp = \u22a4\ny : E\nx\u271d : y \u2208 SimpleFunc.range f\nhy0 : \u00acy = 0\n\u22a2 \u2191\u2016y\u2016\u208a ^ ENNReal.toReal p * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4"}, {"tactic": "rw [hp0, mem\u2112p_zero_iff_aestronglyMeasurable]", "annotated_tactic": ["rw [hp0, <a>mem\u2112p_zero_iff_aestronglyMeasurable</a>]", [{"full_name": "MeasureTheory.mem\u2112p_zero_iff_aestronglyMeasurable", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [179, 9], "def_end_pos": [179, 44]}]], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np\u271d p : \u211d\u22650\u221e\nf : \u03b1 \u2192\u209b E\nhf : \u2200 (y : E), y \u2260 0 \u2192 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4\nhp0 : p = 0\n\u22a2 Mem\u2112p (\u2191f) p", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np\u271d p : \u211d\u22650\u221e\nf : \u03b1 \u2192\u209b E\nhf : \u2200 (y : E), y \u2260 0 \u2192 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4\nhp0 : p = 0\n\u22a2 AEStronglyMeasurable (\u2191f) \u03bc"}, {"tactic": "exact f.aestronglyMeasurable", "annotated_tactic": ["exact f.aestronglyMeasurable", []], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np\u271d p : \u211d\u22650\u221e\nf : \u03b1 \u2192\u209b E\nhf : \u2200 (y : E), y \u2260 0 \u2192 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4\nhp0 : p = 0\n\u22a2 AEStronglyMeasurable (\u2191f) \u03bc", "state_after": "no goals"}, {"tactic": "rw [hp_top]", "annotated_tactic": ["rw [hp_top]", []], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np\u271d p : \u211d\u22650\u221e\nf : \u03b1 \u2192\u209b E\nhf : \u2200 (y : E), y \u2260 0 \u2192 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4\nhp0 : \u00acp = 0\nhp_top : p = \u22a4\n\u22a2 Mem\u2112p (\u2191f) p", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np\u271d p : \u211d\u22650\u221e\nf : \u03b1 \u2192\u209b E\nhf : \u2200 (y : E), y \u2260 0 \u2192 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4\nhp0 : \u00acp = 0\nhp_top : p = \u22a4\n\u22a2 Mem\u2112p \u2191f \u22a4"}, {"tactic": "exact mem\u2112p_top f \u03bc", "annotated_tactic": ["exact <a>mem\u2112p_top</a> f \u03bc", [{"full_name": "MeasureTheory.SimpleFunc.mem\u2112p_top", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "def_pos": [295, 9], "def_end_pos": [295, 18]}]], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np\u271d p : \u211d\u22650\u221e\nf : \u03b1 \u2192\u209b E\nhf : \u2200 (y : E), y \u2260 0 \u2192 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4\nhp0 : \u00acp = 0\nhp_top : p = \u22a4\n\u22a2 Mem\u2112p \u2191f \u22a4", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np\u271d p : \u211d\u22650\u221e\nf : \u03b1 \u2192\u209b E\nhf : \u2200 (y : E), y \u2260 0 \u2192 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4\nhp0 : \u00acp = 0\nhp_top : \u00acp = \u22a4\n\u22a2 0 \u2264 1 / ENNReal.toReal p", "state_after": "no goals"}, {"tactic": "simp [hy0, ENNReal.toReal_pos hp0 hp_top]", "annotated_tactic": ["simp [hy0, <a>ENNReal.toReal_pos</a> hp0 hp_top]", [{"full_name": "ENNReal.toReal_pos", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2131, 9], "def_end_pos": [2131, 19]}]], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np\u271d p : \u211d\u22650\u221e\nf : \u03b1 \u2192\u209b E\nhf : \u2200 (y : E), y \u2260 0 \u2192 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4\nhp0 : \u00acp = 0\nhp_top : \u00acp = \u22a4\ny : E\nx\u271d : y \u2208 SimpleFunc.range f\nhy0 : y = 0\n\u22a2 \u2191\u2016y\u2016\u208a ^ ENNReal.toReal p * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4", "state_after": "no goals"}, {"tactic": "refine' ENNReal.mul_lt_top _ (hf y hy0).ne", "annotated_tactic": ["refine' <a>ENNReal.mul_lt_top</a> _ (hf y hy0).<a>ne</a>", [{"full_name": "ENNReal.mul_lt_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [612, 9], "def_end_pos": [612, 19]}, {"full_name": "LT.lt.ne", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [152, 7], "def_end_pos": [152, 15]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np\u271d p : \u211d\u22650\u221e\nf : \u03b1 \u2192\u209b E\nhf : \u2200 (y : E), y \u2260 0 \u2192 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4\nhp0 : \u00acp = 0\nhp_top : \u00acp = \u22a4\ny : E\nx\u271d : y \u2208 SimpleFunc.range f\nhy0 : \u00acy = 0\n\u22a2 \u2191\u2016y\u2016\u208a ^ ENNReal.toReal p * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4", "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np\u271d p : \u211d\u22650\u221e\nf : \u03b1 \u2192\u209b E\nhf : \u2200 (y : E), y \u2260 0 \u2192 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4\nhp0 : \u00acp = 0\nhp_top : \u00acp = \u22a4\ny : E\nx\u271d : y \u2208 SimpleFunc.range f\nhy0 : \u00acy = 0\n\u22a2 \u2191\u2016y\u2016\u208a ^ ENNReal.toReal p \u2260 \u22a4"}, {"tactic": "exact (ENNReal.rpow_lt_top_of_nonneg ENNReal.toReal_nonneg ENNReal.coe_ne_top).ne", "annotated_tactic": ["exact (<a>ENNReal.rpow_lt_top_of_nonneg</a> <a>ENNReal.toReal_nonneg</a> <a>ENNReal.coe_ne_top</a>).<a>ne</a>", [{"full_name": "ENNReal.rpow_lt_top_of_nonneg", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [503, 9], "def_end_pos": [503, 30]}, {"full_name": "ENNReal.toReal_nonneg", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [221, 17], "def_end_pos": [221, 30]}, {"full_name": "ENNReal.coe_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [302, 17], "def_end_pos": [302, 27]}, {"full_name": "LT.lt.ne", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [152, 7], "def_end_pos": [152, 15]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np\u271d p : \u211d\u22650\u221e\nf : \u03b1 \u2192\u209b E\nhf : \u2200 (y : E), y \u2260 0 \u2192 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4\nhp0 : \u00acp = 0\nhp_top : \u00acp = \u22a4\ny : E\nx\u271d : y \u2208 SimpleFunc.range f\nhy0 : \u00acy = 0\n\u22a2 \u2191\u2016y\u2016\u208a ^ ENNReal.toReal p \u2260 \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/AEEqOfIntegral.lean", "full_name": "MeasureTheory.ae_eq_zero_restrict_of_forall_set_integral_eq_zero", "start": [379, 1], "end": [391, 41], "traced_tactics": [{"tactic": "rcases (hf_int_finite t ht h\u03bct.lt_top).aestronglyMeasurable.isSeparable_ae_range with\n  \u27e8u, u_sep, hu\u27e9", "annotated_tactic": ["rcases (hf_int_finite t ht h\u03bct.lt_top).aestronglyMeasurable.isSeparable_ae_range with\n    \u27e8u, u_sep, hu\u27e9", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = 0\nt : Set \u03b1\nht : MeasurableSet t\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\n\u22a2 f =\u1d50[Measure.restrict \u03bc t] 0", "state_after": "case intro.intro\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = 0\nt : Set \u03b1\nht : MeasurableSet t\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\nu : Set E\nu_sep : IsSeparable u\nhu : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc t, f x \u2208 u\n\u22a2 f =\u1d50[Measure.restrict \u03bc t] 0"}, {"tactic": "refine' ae_eq_zero_of_forall_dual_of_isSeparable \u211d u_sep (fun c => _) hu", "annotated_tactic": ["refine' <a>ae_eq_zero_of_forall_dual_of_isSeparable</a> \u211d u_sep (fun c => _) hu", [{"full_name": "MeasureTheory.ae_eq_zero_of_forall_dual_of_isSeparable", "def_path": "Mathlib/MeasureTheory/Function/AEEqOfIntegral.lean", "def_pos": [75, 9], "def_end_pos": [75, 49]}]], "state_before": "case intro.intro\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = 0\nt : Set \u03b1\nht : MeasurableSet t\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\nu : Set E\nu_sep : IsSeparable u\nhu : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc t, f x \u2208 u\n\u22a2 f =\u1d50[Measure.restrict \u03bc t] 0", "state_after": "case intro.intro\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = 0\nt : Set \u03b1\nht : MeasurableSet t\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\nu : Set E\nu_sep : IsSeparable u\nhu : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc t, f x \u2208 u\nc : Dual \u211d E\n\u22a2 (fun x => \u2191c (f x)) =\u1d50[Measure.restrict \u03bc t] 0"}, {"tactic": "refine' ae_eq_zero_restrict_of_forall_set_integral_eq_zero_real _ _ ht h\u03bct", "annotated_tactic": ["refine' <a>ae_eq_zero_restrict_of_forall_set_integral_eq_zero_real</a> _ _ ht h\u03bct", [{"full_name": "MeasureTheory.ae_eq_zero_restrict_of_forall_set_integral_eq_zero_real", "def_path": "Mathlib/MeasureTheory/Function/AEEqOfIntegral.lean", "def_pos": [355, 9], "def_end_pos": [355, 64]}]], "state_before": "case intro.intro\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = 0\nt : Set \u03b1\nht : MeasurableSet t\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\nu : Set E\nu_sep : IsSeparable u\nhu : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc t, f x \u2208 u\nc : Dual \u211d E\n\u22a2 (fun x => \u2191c (f x)) =\u1d50[Measure.restrict \u03bc t] 0", "state_after": "case intro.intro.refine'_1\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = 0\nt : Set \u03b1\nht : MeasurableSet t\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\nu : Set E\nu_sep : IsSeparable u\nhu : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc t, f x \u2208 u\nc : Dual \u211d E\n\u22a2 \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn (fun x => \u2191c (f x)) s\n\ncase intro.intro.refine'_2\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = 0\nt : Set \u03b1\nht : MeasurableSet t\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\nu : Set E\nu_sep : IsSeparable u\nhu : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc t, f x \u2208 u\nc : Dual \u211d E\n\u22a2 \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191c (f x) \u2202\u03bc = 0"}, {"tactic": "intro s hs h\u03bcs", "annotated_tactic": ["intro s hs h\u03bcs", []], "state_before": "case intro.intro.refine'_1\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = 0\nt : Set \u03b1\nht : MeasurableSet t\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\nu : Set E\nu_sep : IsSeparable u\nhu : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc t, f x \u2208 u\nc : Dual \u211d E\n\u22a2 \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn (fun x => \u2191c (f x)) s", "state_after": "case intro.intro.refine'_1\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t\u271d : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = 0\nt : Set \u03b1\nht : MeasurableSet t\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\nu : Set E\nu_sep : IsSeparable u\nhu : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc t, f x \u2208 u\nc : Dual \u211d E\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 IntegrableOn (fun x => \u2191c (f x)) s"}, {"tactic": "exact ContinuousLinearMap.integrable_comp c (hf_int_finite s hs h\u03bcs)", "annotated_tactic": ["exact <a>ContinuousLinearMap.integrable_comp</a> c (hf_int_finite s hs h\u03bcs)", [{"full_name": "ContinuousLinearMap.integrable_comp", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [1517, 9], "def_end_pos": [1517, 44]}]], "state_before": "case intro.intro.refine'_1\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t\u271d : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = 0\nt : Set \u03b1\nht : MeasurableSet t\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\nu : Set E\nu_sep : IsSeparable u\nhu : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc t, f x \u2208 u\nc : Dual \u211d E\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 IntegrableOn (fun x => \u2191c (f x)) s", "state_after": "no goals"}, {"tactic": "intro s hs h\u03bcs", "annotated_tactic": ["intro s hs h\u03bcs", []], "state_before": "case intro.intro.refine'_2\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = 0\nt : Set \u03b1\nht : MeasurableSet t\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\nu : Set E\nu_sep : IsSeparable u\nhu : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc t, f x \u2208 u\nc : Dual \u211d E\n\u22a2 \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191c (f x) \u2202\u03bc = 0", "state_after": "case intro.intro.refine'_2\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t\u271d : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = 0\nt : Set \u03b1\nht : MeasurableSet t\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\nu : Set E\nu_sep : IsSeparable u\nhu : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc t, f x \u2208 u\nc : Dual \u211d E\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u222b (x : \u03b1) in s, \u2191c (f x) \u2202\u03bc = 0"}, {"tactic": "rw [ContinuousLinearMap.integral_comp_comm c (hf_int_finite s hs h\u03bcs), hf_zero s hs h\u03bcs]", "annotated_tactic": ["rw [<a>ContinuousLinearMap.integral_comp_comm</a> c (hf_int_finite s hs h\u03bcs), hf_zero s hs h\u03bcs]", [{"full_name": "ContinuousLinearMap.integral_comp_comm", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [1120, 9], "def_end_pos": [1120, 27]}]], "state_before": "case intro.intro.refine'_2\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t\u271d : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = 0\nt : Set \u03b1\nht : MeasurableSet t\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\nu : Set E\nu_sep : IsSeparable u\nhu : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc t, f x \u2208 u\nc : Dual \u211d E\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u222b (x : \u03b1) in s, \u2191c (f x) \u2202\u03bc = 0", "state_after": "case intro.intro.refine'_2\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t\u271d : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = 0\nt : Set \u03b1\nht : MeasurableSet t\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\nu : Set E\nu_sep : IsSeparable u\nhu : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc t, f x \u2208 u\nc : Dual \u211d E\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u2191c 0 = 0"}, {"tactic": "exact ContinuousLinearMap.map_zero _", "annotated_tactic": ["exact <a>ContinuousLinearMap.map_zero</a> _", [{"full_name": "ContinuousLinearMap.map_zero", "def_path": "Mathlib/Topology/Algebra/Module/Basic.lean", "def_pos": [506, 19], "def_end_pos": [506, 27]}]], "state_before": "case intro.intro.refine'_2\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t\u271d : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = 0\nt : Set \u03b1\nht : MeasurableSet t\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\nu : Set E\nu_sep : IsSeparable u\nhu : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc t, f x \u2208 u\nc : Dual \u211d E\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u2191c 0 = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Fin/Lemmas.lean", "full_name": "Fin.lastCases_castSucc", "start": [684, 9], "end": [686, 31], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "full_name": "Int.max_le", "start": [719, 11], "end": [721, 62], "traced_tactics": [{"tactic": "rw [Int.max_def]", "annotated_tactic": ["rw [<a>Int.max_def</a>]", [{"full_name": "Int.max_def", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [688, 19], "def_end_pos": [688, 26]}]], "state_before": "a b c : Int\nx\u271d : a \u2264 c \u2227 b \u2264 c\nh\u2081 : a \u2264 c\nh\u2082 : b \u2264 c\n\u22a2 max a b \u2264 c", "state_after": "a b c : Int\nx\u271d : a \u2264 c \u2227 b \u2264 c\nh\u2081 : a \u2264 c\nh\u2082 : b \u2264 c\n\u22a2 (if a \u2264 b then b else a) \u2264 c"}, {"tactic": "split <;> assumption", "annotated_tactic": ["split <;> assumption", []], "state_before": "a b c : Int\nx\u271d : a \u2264 c \u2227 b \u2264 c\nh\u2081 : a \u2264 c\nh\u2082 : b \u2264 c\n\u22a2 (if a \u2264 b then b else a) \u2264 c", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Vector/Basic.lean", "full_name": "Vector.removeNth_insertNth'", "start": [574, 1], "end": [592, 33], "traced_tactics": [{"tactic": "dsimp [insertNth, removeNth, Fin.succAbove, Fin.predAbove]", "annotated_tactic": ["dsimp [<a>insertNth</a>, <a>removeNth</a>, <a>Fin.succAbove</a>, <a>Fin.predAbove</a>]", [{"full_name": "Vector.insertNth", "def_path": "Mathlib/Data/Vector/Basic.lean", "def_pos": [552, 5], "def_end_pos": [552, 14]}, {"full_name": "Vector.removeNth", "def_path": "Mathlib/Data/Vector.lean", "def_pos": [165, 5], "def_end_pos": [165, 14]}, {"full_name": "Fin.succAbove", "def_path": "Mathlib/Data/Fin/Basic.lean", "def_pos": [1298, 5], "def_end_pos": [1298, 14]}, {"full_name": "Fin.predAbove", "def_path": "Mathlib/Data/Fin/Basic.lean", "def_pos": [1547, 5], "def_end_pos": [1547, 14]}]], "state_before": "n : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na : \u03b1\nv : Vector \u03b1 (n + 1)\ni : \u2115\nhi : i < n + 1\nj : \u2115\nhj : j < n + 2\n\u22a2 removeNth (Fin.succAbove { val := j, isLt := hj } { val := i, isLt := hi }) (insertNth a { val := j, isLt := hj } v) =\n    insertNth a (Fin.predAbove { val := i, isLt := hi } { val := j, isLt := hj }) (removeNth { val := i, isLt := hi } v)", "state_after": "n : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na : \u03b1\nv : Vector \u03b1 (n + 1)\ni : \u2115\nhi : i < n + 1\nj : \u2115\nhj : j < n + 2\n\u22a2 {\n      val :=\n        List.removeNth (List.insertNth j a \u2191v)\n          \u2191(if i < j then { val := i, isLt := (_ : i < Nat.succ (n + 1)) }\n            else { val := i + 1, isLt := (_ : Nat.succ i < Nat.succ (n + 1)) }),\n      property :=\n        (_ :\n          List.length\n              (List.removeNth (List.insertNth j a \u2191v)\n                \u2191(if i < j then { val := i, isLt := (_ : i < Nat.succ (n + 1)) }\n                  else { val := i + 1, isLt := (_ : Nat.succ i < Nat.succ (n + 1)) })) =\n            n + 1 + 1 - 1) } =\n    {\n      val :=\n        List.insertNth\n          (\u2191(if h : { val := i, isLt := (_ : i < Nat.succ (n + 1)) } < { val := j, isLt := hj } then\n              Fin.pred { val := j, isLt := hj } (_ : { val := j, isLt := hj } \u2260 0)\n            else { val := j, isLt := (_ : \u2191{ val := j, isLt := hj } < n + 1) }))\n          a\n          \u2191(match v with\n            | { val := l, property := p } =>\n              { val := List.removeNth l i,\n                property := (_ : List.length (List.removeNth l \u2191{ val := i, isLt := hi }) = n + 1 - 1) }),\n      property :=\n        (_ :\n          List.length\n              (List.insertNth\n                (\u2191(if h : { val := i, isLt := (_ : i < Nat.succ (n + 1)) } < { val := j, isLt := hj } then\n                    Fin.pred { val := j, isLt := hj } (_ : { val := j, isLt := hj } \u2260 0)\n                  else { val := j, isLt := (_ : \u2191{ val := j, isLt := hj } < n + 1) }))\n                a\n                \u2191(match v with\n                  | { val := l, property := p } =>\n                    { val := List.removeNth l i,\n                      property := (_ : List.length (List.removeNth l \u2191{ val := i, isLt := hi }) = n + 1 - 1) })) =\n            n + 1) }"}, {"tactic": "rw [Subtype.mk_eq_mk]", "annotated_tactic": ["rw [<a>Subtype.mk_eq_mk</a>]", [{"full_name": "Subtype.mk_eq_mk", "def_path": "Mathlib/Data/Subtype.lean", "def_pos": [107, 9], "def_end_pos": [107, 17]}]], "state_before": "n : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na : \u03b1\nv : Vector \u03b1 (n + 1)\ni : \u2115\nhi : i < n + 1\nj : \u2115\nhj : j < n + 2\n\u22a2 {\n      val :=\n        List.removeNth (List.insertNth j a \u2191v)\n          \u2191(if i < j then { val := i, isLt := (_ : i < Nat.succ (n + 1)) }\n            else { val := i + 1, isLt := (_ : Nat.succ i < Nat.succ (n + 1)) }),\n      property :=\n        (_ :\n          List.length\n              (List.removeNth (List.insertNth j a \u2191v)\n                \u2191(if i < j then { val := i, isLt := (_ : i < Nat.succ (n + 1)) }\n                  else { val := i + 1, isLt := (_ : Nat.succ i < Nat.succ (n + 1)) })) =\n            n + 1 + 1 - 1) } =\n    {\n      val :=\n        List.insertNth\n          (\u2191(if h : { val := i, isLt := (_ : i < Nat.succ (n + 1)) } < { val := j, isLt := hj } then\n              Fin.pred { val := j, isLt := hj } (_ : { val := j, isLt := hj } \u2260 0)\n            else { val := j, isLt := (_ : \u2191{ val := j, isLt := hj } < n + 1) }))\n          a\n          \u2191(match v with\n            | { val := l, property := p } =>\n              { val := List.removeNth l i,\n                property := (_ : List.length (List.removeNth l \u2191{ val := i, isLt := hi }) = n + 1 - 1) }),\n      property :=\n        (_ :\n          List.length\n              (List.insertNth\n                (\u2191(if h : { val := i, isLt := (_ : i < Nat.succ (n + 1)) } < { val := j, isLt := hj } then\n                    Fin.pred { val := j, isLt := hj } (_ : { val := j, isLt := hj } \u2260 0)\n                  else { val := j, isLt := (_ : \u2191{ val := j, isLt := hj } < n + 1) }))\n                a\n                \u2191(match v with\n                  | { val := l, property := p } =>\n                    { val := List.removeNth l i,\n                      property := (_ : List.length (List.removeNth l \u2191{ val := i, isLt := hi }) = n + 1 - 1) })) =\n            n + 1) }", "state_after": "n : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na : \u03b1\nv : Vector \u03b1 (n + 1)\ni : \u2115\nhi : i < n + 1\nj : \u2115\nhj : j < n + 2\n\u22a2 List.removeNth (List.insertNth j a \u2191v)\n      \u2191(if i < j then { val := i, isLt := (_ : i < Nat.succ (n + 1)) }\n        else { val := i + 1, isLt := (_ : Nat.succ i < Nat.succ (n + 1)) }) =\n    List.insertNth\n      (\u2191(if h : { val := i, isLt := (_ : i < Nat.succ (n + 1)) } < { val := j, isLt := hj } then\n          Fin.pred { val := j, isLt := hj } (_ : { val := j, isLt := hj } \u2260 0)\n        else { val := j, isLt := (_ : \u2191{ val := j, isLt := hj } < n + 1) }))\n      a\n      \u2191(match v with\n        | { val := l, property := p } =>\n          { val := List.removeNth l i,\n            property := (_ : List.length (List.removeNth l \u2191{ val := i, isLt := hi }) = n + 1 - 1) })"}, {"tactic": "simp only [Fin.lt_iff_val_lt_val]", "annotated_tactic": ["simp only [<a>Fin.lt_iff_val_lt_val</a>]", [{"full_name": "Fin.lt_iff_val_lt_val", "def_path": "Mathlib/Data/Fin/Basic.lean", "def_pos": [215, 9], "def_end_pos": [215, 26]}]], "state_before": "n : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na : \u03b1\nv : Vector \u03b1 (n + 1)\ni : \u2115\nhi : i < n + 1\nj : \u2115\nhj : j < n + 2\n\u22a2 List.removeNth (List.insertNth j a \u2191v)\n      \u2191(if i < j then { val := i, isLt := (_ : i < Nat.succ (n + 1)) }\n        else { val := i + 1, isLt := (_ : Nat.succ i < Nat.succ (n + 1)) }) =\n    List.insertNth\n      (\u2191(if h : { val := i, isLt := (_ : i < Nat.succ (n + 1)) } < { val := j, isLt := hj } then\n          Fin.pred { val := j, isLt := hj } (_ : { val := j, isLt := hj } \u2260 0)\n        else { val := j, isLt := (_ : \u2191{ val := j, isLt := hj } < n + 1) }))\n      a\n      \u2191(match v with\n        | { val := l, property := p } =>\n          { val := List.removeNth l i,\n            property := (_ : List.length (List.removeNth l \u2191{ val := i, isLt := hi }) = n + 1 - 1) })", "state_after": "n : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na : \u03b1\nv : Vector \u03b1 (n + 1)\ni : \u2115\nhi : i < n + 1\nj : \u2115\nhj : j < n + 2\n\u22a2 List.removeNth (List.insertNth j a \u2191v)\n      \u2191(if i < j then { val := i, isLt := (_ : i < Nat.succ (n + 1)) }\n        else { val := i + 1, isLt := (_ : Nat.succ i < Nat.succ (n + 1)) }) =\n    List.insertNth\n      (\u2191(if h : i < j then Fin.pred { val := j, isLt := hj } (_ : { val := j, isLt := hj } \u2260 0)\n        else { val := j, isLt := (_ : \u2191{ val := j, isLt := hj } < n + 1) }))\n      a\n      \u2191(match v with\n        | { val := l, property := p } =>\n          { val := List.removeNth l i,\n            property := (_ : List.length (List.removeNth l \u2191{ val := i, isLt := hi }) = n + 1 - 1) })"}, {"tactic": "split_ifs with hij", "annotated_tactic": ["split_ifs with hij", []], "state_before": "n : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na : \u03b1\nv : Vector \u03b1 (n + 1)\ni : \u2115\nhi : i < n + 1\nj : \u2115\nhj : j < n + 2\n\u22a2 List.removeNth (List.insertNth j a \u2191v)\n      \u2191(if i < j then { val := i, isLt := (_ : i < Nat.succ (n + 1)) }\n        else { val := i + 1, isLt := (_ : Nat.succ i < Nat.succ (n + 1)) }) =\n    List.insertNth\n      (\u2191(if h : i < j then Fin.pred { val := j, isLt := hj } (_ : { val := j, isLt := hj } \u2260 0)\n        else { val := j, isLt := (_ : \u2191{ val := j, isLt := hj } < n + 1) }))\n      a\n      \u2191(match v with\n        | { val := l, property := p } =>\n          { val := List.removeNth l i,\n            property := (_ : List.length (List.removeNth l \u2191{ val := i, isLt := hi }) = n + 1 - 1) })", "state_after": "case pos\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na : \u03b1\nv : Vector \u03b1 (n + 1)\ni : \u2115\nhi : i < n + 1\nj : \u2115\nhj : j < n + 2\nhij : i < j\n\u22a2 List.removeNth (List.insertNth j a \u2191v) \u2191{ val := i, isLt := (_ : i < Nat.succ (n + 1)) } =\n    List.insertNth (\u2191(Fin.pred { val := j, isLt := hj } (_ : { val := j, isLt := hj } \u2260 0))) a\n      \u2191(match v with\n        | { val := l, property := p } =>\n          { val := List.removeNth l i,\n            property := (_ : List.length (List.removeNth l \u2191{ val := i, isLt := hi }) = n + 1 - 1) })\n\ncase neg\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na : \u03b1\nv : Vector \u03b1 (n + 1)\ni : \u2115\nhi : i < n + 1\nj : \u2115\nhj : j < n + 2\nhij : \u00aci < j\n\u22a2 List.removeNth (List.insertNth j a \u2191v) \u2191{ val := i + 1, isLt := (_ : Nat.succ i < Nat.succ (n + 1)) } =\n    List.insertNth (\u2191{ val := j, isLt := (_ : \u2191{ val := j, isLt := hj } < n + 1) }) a\n      \u2191(match v with\n        | { val := l, property := p } =>\n          { val := List.removeNth l i,\n            property := (_ : List.length (List.removeNth l \u2191{ val := i, isLt := hi }) = n + 1 - 1) })"}, {"tactic": "rcases Nat.exists_eq_succ_of_ne_zero\n  (Nat.pos_iff_ne_zero.1 (lt_of_le_of_lt (Nat.zero_le _) hij)) with \u27e8j, rfl\u27e9", "annotated_tactic": ["rcases <a>Nat.exists_eq_succ_of_ne_zero</a>\n        (<a>Nat.pos_iff_ne_zero</a>.1 (<a>lt_of_le_of_lt</a> (<a>Nat.zero_le</a> _) hij)) with \u27e8j, rfl\u27e9", [{"full_name": "Nat.exists_eq_succ_of_ne_zero", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [238, 9], "def_end_pos": [238, 34]}, {"full_name": "Nat.pos_iff_ne_zero", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [204, 19], "def_end_pos": [204, 34]}, {"full_name": "lt_of_le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [122, 9], "def_end_pos": [122, 23]}, {"full_name": "Nat.zero_le", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1578, 9], "def_end_pos": [1578, 20]}]], "state_before": "case pos\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na : \u03b1\nv : Vector \u03b1 (n + 1)\ni : \u2115\nhi : i < n + 1\nj : \u2115\nhj : j < n + 2\nhij : i < j\n\u22a2 List.removeNth (List.insertNth j a \u2191v) \u2191{ val := i, isLt := (_ : i < Nat.succ (n + 1)) } =\n    List.insertNth (\u2191(Fin.pred { val := j, isLt := hj } (_ : { val := j, isLt := hj } \u2260 0))) a\n      \u2191(match v with\n        | { val := l, property := p } =>\n          { val := List.removeNth l i,\n            property := (_ : List.length (List.removeNth l \u2191{ val := i, isLt := hi }) = n + 1 - 1) })", "state_after": "case pos.intro\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na : \u03b1\nv : Vector \u03b1 (n + 1)\ni : \u2115\nhi : i < n + 1\nj : \u2115\nhj : Nat.succ j < n + 2\nhij : i < Nat.succ j\n\u22a2 List.removeNth (List.insertNth (Nat.succ j) a \u2191v) \u2191{ val := i, isLt := (_ : i < Nat.succ (n + 1)) } =\n    List.insertNth (\u2191(Fin.pred { val := Nat.succ j, isLt := hj } (_ : { val := Nat.succ j, isLt := hj } \u2260 0))) a\n      \u2191(match v with\n        | { val := l, property := p } =>\n          { val := List.removeNth l i,\n            property := (_ : List.length (List.removeNth l \u2191{ val := i, isLt := hi }) = n + 1 - 1) })"}, {"tactic": "rw [\u2190 List.insertNth_removeNth_of_ge]", "annotated_tactic": ["rw [\u2190 <a>List.insertNth_removeNth_of_ge</a>]", [{"full_name": "List.insertNth_removeNth_of_ge", "def_path": "Mathlib/Data/List/Basic.lean", "def_pos": [1563, 9], "def_end_pos": [1563, 34]}]], "state_before": "case pos.intro\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na : \u03b1\nv : Vector \u03b1 (n + 1)\ni : \u2115\nhi : i < n + 1\nj : \u2115\nhj : Nat.succ j < n + 2\nhij : i < Nat.succ j\n\u22a2 List.removeNth (List.insertNth (Nat.succ j) a \u2191v) \u2191{ val := i, isLt := (_ : i < Nat.succ (n + 1)) } =\n    List.insertNth (\u2191(Fin.pred { val := Nat.succ j, isLt := hj } (_ : { val := Nat.succ j, isLt := hj } \u2260 0))) a\n      \u2191(match v with\n        | { val := l, property := p } =>\n          { val := List.removeNth l i,\n            property := (_ : List.length (List.removeNth l \u2191{ val := i, isLt := hi }) = n + 1 - 1) })", "state_after": "case pos.intro\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na : \u03b1\nv : Vector \u03b1 (n + 1)\ni : \u2115\nhi : i < n + 1\nj : \u2115\nhj : Nat.succ j < n + 2\nhij : i < Nat.succ j\n\u22a2 List.insertNth j a (List.removeNth \u2191v \u2191{ val := i, isLt := (_ : i < Nat.succ (n + 1)) }) =\n    List.insertNth (\u2191(Fin.pred { val := Nat.succ j, isLt := hj } (_ : { val := Nat.succ j, isLt := hj } \u2260 0))) a\n      \u2191(match v with\n        | { val := l, property := p } =>\n          { val := List.removeNth l i,\n            property := (_ : List.length (List.removeNth l \u2191{ val := i, isLt := hi }) = n + 1 - 1) })\n\ncase pos.intro.a\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na : \u03b1\nv : Vector \u03b1 (n + 1)\ni : \u2115\nhi : i < n + 1\nj : \u2115\nhj : Nat.succ j < n + 2\nhij : i < Nat.succ j\n\u22a2 \u2191{ val := i, isLt := (_ : i < Nat.succ (n + 1)) } < List.length \u2191v\n\ncase pos.intro.a\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na : \u03b1\nv : Vector \u03b1 (n + 1)\ni : \u2115\nhi : i < n + 1\nj : \u2115\nhj : Nat.succ j < n + 2\nhij : i < Nat.succ j\n\u22a2 \u2191{ val := i, isLt := (_ : i < Nat.succ (n + 1)) } \u2264 j"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case pos.intro\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na : \u03b1\nv : Vector \u03b1 (n + 1)\ni : \u2115\nhi : i < n + 1\nj : \u2115\nhj : Nat.succ j < n + 2\nhij : i < Nat.succ j\n\u22a2 List.insertNth j a (List.removeNth \u2191v \u2191{ val := i, isLt := (_ : i < Nat.succ (n + 1)) }) =\n    List.insertNth (\u2191(Fin.pred { val := Nat.succ j, isLt := hj } (_ : { val := Nat.succ j, isLt := hj } \u2260 0))) a\n      \u2191(match v with\n        | { val := l, property := p } =>\n          { val := List.removeNth l i,\n            property := (_ : List.length (List.removeNth l \u2191{ val := i, isLt := hi }) = n + 1 - 1) })", "state_after": "case pos.intro\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na : \u03b1\nv : Vector \u03b1 (n + 1)\ni : \u2115\nhi : i < n + 1\nj : \u2115\nhj : Nat.succ j < n + 2\nhij : i < Nat.succ j\n\u22a2 List.insertNth j a (List.removeNth (\u2191v) i) =\n    List.insertNth j a\n      \u2191(match v with\n        | { val := l, property := p } =>\n          { val := List.removeNth l i,\n            property := (_ : List.length (List.removeNth l \u2191{ val := i, isLt := hi }) = n + 1 - 1) })"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case pos.intro\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na : \u03b1\nv : Vector \u03b1 (n + 1)\ni : \u2115\nhi : i < n + 1\nj : \u2115\nhj : Nat.succ j < n + 2\nhij : i < Nat.succ j\n\u22a2 List.insertNth j a (List.removeNth (\u2191v) i) =\n    List.insertNth j a\n      \u2191(match v with\n        | { val := l, property := p } =>\n          { val := List.removeNth l i,\n            property := (_ : List.length (List.removeNth l \u2191{ val := i, isLt := hi }) = n + 1 - 1) })", "state_after": "no goals"}, {"tactic": "simpa", "annotated_tactic": ["simpa", []], "state_before": "case pos.intro.a\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na : \u03b1\nv : Vector \u03b1 (n + 1)\ni : \u2115\nhi : i < n + 1\nj : \u2115\nhj : Nat.succ j < n + 2\nhij : i < Nat.succ j\n\u22a2 \u2191{ val := i, isLt := (_ : i < Nat.succ (n + 1)) } < List.length \u2191v", "state_after": "no goals"}, {"tactic": "simpa [Nat.lt_succ_iff] using hij", "annotated_tactic": ["simpa [<a>Nat.lt_succ_iff</a>] using hij", [{"full_name": "Nat.lt_succ_iff", "def_path": "Mathlib/Data/Nat/Basic.lean", "def_pos": [207, 9], "def_end_pos": [207, 20]}]], "state_before": "case pos.intro.a\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na : \u03b1\nv : Vector \u03b1 (n + 1)\ni : \u2115\nhi : i < n + 1\nj : \u2115\nhj : Nat.succ j < n + 2\nhij : i < Nat.succ j\n\u22a2 \u2191{ val := i, isLt := (_ : i < Nat.succ (n + 1)) } \u2264 j", "state_after": "no goals"}, {"tactic": "dsimp", "annotated_tactic": ["dsimp", []], "state_before": "case neg\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na : \u03b1\nv : Vector \u03b1 (n + 1)\ni : \u2115\nhi : i < n + 1\nj : \u2115\nhj : j < n + 2\nhij : \u00aci < j\n\u22a2 List.removeNth (List.insertNth j a \u2191v) \u2191{ val := i + 1, isLt := (_ : Nat.succ i < Nat.succ (n + 1)) } =\n    List.insertNth (\u2191{ val := j, isLt := (_ : \u2191{ val := j, isLt := hj } < n + 1) }) a\n      \u2191(match v with\n        | { val := l, property := p } =>\n          { val := List.removeNth l i,\n            property := (_ : List.length (List.removeNth l \u2191{ val := i, isLt := hi }) = n + 1 - 1) })", "state_after": "case neg\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na : \u03b1\nv : Vector \u03b1 (n + 1)\ni : \u2115\nhi : i < n + 1\nj : \u2115\nhj : j < n + 2\nhij : \u00aci < j\n\u22a2 List.removeNth (List.insertNth j a \u2191v) (i + 1) =\n    List.insertNth j a\n      \u2191(match v with\n        | { val := l, property := p } =>\n          { val := List.removeNth l i,\n            property := (_ : List.length (List.removeNth l \u2191{ val := i, isLt := hi }) = n + 1 - 1) })"}, {"tactic": "rw [\u2190 List.insertNth_removeNth_of_le i j _ _ _]", "annotated_tactic": ["rw [\u2190 <a>List.insertNth_removeNth_of_le</a> i j _ _ _]", [{"full_name": "List.insertNth_removeNth_of_le", "def_path": "Mathlib/Data/List/Basic.lean", "def_pos": [1574, 9], "def_end_pos": [1574, 34]}]], "state_before": "case neg\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na : \u03b1\nv : Vector \u03b1 (n + 1)\ni : \u2115\nhi : i < n + 1\nj : \u2115\nhj : j < n + 2\nhij : \u00aci < j\n\u22a2 List.removeNth (List.insertNth j a \u2191v) (i + 1) =\n    List.insertNth j a\n      \u2191(match v with\n        | { val := l, property := p } =>\n          { val := List.removeNth l i,\n            property := (_ : List.length (List.removeNth l \u2191{ val := i, isLt := hi }) = n + 1 - 1) })", "state_after": "case neg\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na : \u03b1\nv : Vector \u03b1 (n + 1)\ni : \u2115\nhi : i < n + 1\nj : \u2115\nhj : j < n + 2\nhij : \u00aci < j\n\u22a2 List.insertNth j a (List.removeNth (\u2191v) i) =\n    List.insertNth j a\n      \u2191(match v with\n        | { val := l, property := p } =>\n          { val := List.removeNth l i,\n            property := (_ : List.length (List.removeNth l \u2191{ val := i, isLt := hi }) = n + 1 - 1) })\n\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na : \u03b1\nv : Vector \u03b1 (n + 1)\ni : \u2115\nhi : i < n + 1\nj : \u2115\nhj : j < n + 2\nhij : \u00aci < j\n\u22a2 i < List.length \u2191v\n\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na : \u03b1\nv : Vector \u03b1 (n + 1)\ni : \u2115\nhi : i < n + 1\nj : \u2115\nhj : j < n + 2\nhij : \u00aci < j\n\u22a2 j \u2264 i"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case neg\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na : \u03b1\nv : Vector \u03b1 (n + 1)\ni : \u2115\nhi : i < n + 1\nj : \u2115\nhj : j < n + 2\nhij : \u00aci < j\n\u22a2 List.insertNth j a (List.removeNth (\u2191v) i) =\n    List.insertNth j a\n      \u2191(match v with\n        | { val := l, property := p } =>\n          { val := List.removeNth l i,\n            property := (_ : List.length (List.removeNth l \u2191{ val := i, isLt := hi }) = n + 1 - 1) })", "state_after": "no goals"}, {"tactic": "simpa", "annotated_tactic": ["simpa", []], "state_before": "n : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na : \u03b1\nv : Vector \u03b1 (n + 1)\ni : \u2115\nhi : i < n + 1\nj : \u2115\nhj : j < n + 2\nhij : \u00aci < j\n\u22a2 i < List.length \u2191v", "state_after": "no goals"}, {"tactic": "simpa [not_lt] using hij", "annotated_tactic": ["simpa [<a>not_lt</a>] using hij", [{"full_name": "not_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [368, 9], "def_end_pos": [368, 15]}]], "state_before": "n : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na : \u03b1\nv : Vector \u03b1 (n + 1)\ni : \u2115\nhi : i < n + 1\nj : \u2115\nhj : j < n + 2\nhij : \u00aci < j\n\u22a2 j \u2264 i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Int/ModEq.lean", "full_name": "Int.neg_modEq_neg", "start": [113, 1], "end": [115, 67], "traced_tactics": [{"tactic": "simp [-sub_neg_eq_add, neg_sub_neg, modEq_iff_dvd, dvd_sub_comm]", "annotated_tactic": ["simp [-<a>sub_neg_eq_add</a>, <a>neg_sub_neg</a>, <a>modEq_iff_dvd</a>, <a>dvd_sub_comm</a>]", [{"full_name": "sub_neg_eq_add", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [453, 3], "def_end_pos": [453, 14]}, {"full_name": "neg_sub_neg", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [506, 3], "def_end_pos": [506, 14]}, {"full_name": "Int.modEq_iff_dvd", "def_path": "Mathlib/Data/Int/ModEq.lean", "def_pos": [94, 9], "def_end_pos": [94, 22]}, {"full_name": "dvd_sub_comm", "def_path": "Mathlib/Algebra/Ring/Divisibility/Basic.lean", "def_pos": [123, 9], "def_end_pos": [123, 21]}]], "state_before": "m n a b c d : \u2124\n\u22a2 -a \u2261 -b [ZMOD n] \u2194 a \u2261 b [ZMOD n]", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "full_name": "MeasureTheory.L1.integral_eq_norm_posPart_sub", "start": [753, 1], "end": [766, 52], "traced_tactics": [{"tactic": "refine' @isClosed_property _ _ _ ((\u2191) : (\u03b1 \u2192\u2081\u209b[\u03bc] \u211d) \u2192 \u03b1 \u2192\u2081[\u03bc] \u211d)\n    (fun f : \u03b1 \u2192\u2081[\u03bc] \u211d => integral f = \u2016Lp.posPart f\u2016 - \u2016Lp.negPart f\u2016)\n    (simpleFunc.denseRange one_ne_top) (isClosed_eq _ _) _ f", "annotated_tactic": ["refine' @<a>isClosed_property</a> _ _ _ ((\u2191) : (\u03b1 \u2192\u2081\u209b[\u03bc] \u211d) \u2192 \u03b1 \u2192\u2081[\u03bc] \u211d)\n      (fun f : \u03b1 \u2192\u2081[\u03bc] \u211d => <a>integral</a> f = \u2016<a>Lp.posPart</a> f\u2016 - \u2016<a>Lp.negPart</a> f\u2016)\n      (<a>simpleFunc.denseRange</a> <a>one_ne_top</a>) (<a>isClosed_eq</a> _ _) _ f", [{"full_name": "isClosed_property", "def_path": "Mathlib/Topology/DenseEmbedding.lean", "def_pos": [308, 9], "def_end_pos": [308, 26]}, {"full_name": "MeasureTheory.L1.integral", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [666, 17], "def_end_pos": [666, 25]}, {"full_name": "MeasureTheory.Lp.posPart", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [1240, 5], "def_end_pos": [1240, 12]}, {"full_name": "MeasureTheory.Lp.negPart", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [1245, 5], "def_end_pos": [1245, 12]}, {"full_name": "MeasureTheory.Lp.simpleFunc.denseRange", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "def_pos": [786, 19], "def_end_pos": [786, 29]}, {"full_name": "ENNReal.one_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [340, 17], "def_end_pos": [340, 27]}, {"full_name": "isClosed_eq", "def_path": "Mathlib/Topology/Separation.lean", "def_pos": [1217, 9], "def_end_pos": [1217, 20]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedAddCommGroup F\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace E\nf : { x // x \u2208 Lp \u211d 1 }\n\u22a2 integral f = \u2016Lp.posPart f\u2016 - \u2016Lp.negPart f\u2016", "state_after": "case refine'_1\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedAddCommGroup F\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace E\nf : { x // x \u2208 Lp \u211d 1 }\n\u22a2 Continuous fun x => integral x\n\ncase refine'_2\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedAddCommGroup F\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace E\nf : { x // x \u2208 Lp \u211d 1 }\n\u22a2 Continuous fun x => \u2016Lp.posPart x\u2016 - \u2016Lp.negPart x\u2016\n\ncase refine'_3\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedAddCommGroup F\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace E\nf : { x // x \u2208 Lp \u211d 1 }\n\u22a2 \u2200 (a : { x // x \u2208 \u2191(simpleFunc \u211d 1 \u03bc) }), (fun f => integral f = \u2016Lp.posPart f\u2016 - \u2016Lp.negPart f\u2016) \u2191a"}, {"tactic": "simp only [integral]", "annotated_tactic": ["simp only [<a>integral</a>]", [{"full_name": "MeasureTheory.L1.integral", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [666, 17], "def_end_pos": [666, 25]}]], "state_before": "case refine'_1\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedAddCommGroup F\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace E\nf : { x // x \u2208 Lp \u211d 1 }\n\u22a2 Continuous fun x => integral x", "state_after": "case refine'_1\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedAddCommGroup F\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace E\nf : { x // x \u2208 Lp \u211d 1 }\n\u22a2 Continuous fun x => \u2191integralCLM x"}, {"tactic": "exact cont _", "annotated_tactic": ["exact <a>cont</a> _", [{"full_name": "ContinuousLinearMap.cont", "def_path": "Mathlib/Topology/Algebra/Module/Basic.lean", "def_pos": [252, 3], "def_end_pos": [252, 7]}]], "state_before": "case refine'_1\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedAddCommGroup F\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace E\nf : { x // x \u2208 Lp \u211d 1 }\n\u22a2 Continuous fun x => \u2191integralCLM x", "state_after": "no goals"}, {"tactic": "refine' Continuous.sub (continuous_norm.comp Lp.continuous_posPart)\n  (continuous_norm.comp Lp.continuous_negPart)", "annotated_tactic": ["refine' <a>Continuous.sub</a> (continuous_norm.comp <a>Lp.continuous_posPart</a>)\n      (continuous_norm.comp <a>Lp.continuous_negPart</a>)", [{"full_name": "Continuous.sub", "def_path": "Mathlib/Topology/Algebra/Group/Basic.lean", "def_pos": [1104, 36], "def_end_pos": [1104, 39]}, {"full_name": "MeasureTheory.Lp.continuous_posPart", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [1268, 9], "def_end_pos": [1268, 27]}, {"full_name": "MeasureTheory.Lp.continuous_negPart", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [1272, 9], "def_end_pos": [1272, 27]}]], "state_before": "case refine'_2\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedAddCommGroup F\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace E\nf : { x // x \u2208 Lp \u211d 1 }\n\u22a2 Continuous fun x => \u2016Lp.posPart x\u2016 - \u2016Lp.negPart x\u2016", "state_after": "no goals"}, {"tactic": "intro s", "annotated_tactic": ["intro s", []], "state_before": "case refine'_3\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedAddCommGroup F\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace E\nf : { x // x \u2208 Lp \u211d 1 }\n\u22a2 \u2200 (a : { x // x \u2208 \u2191(simpleFunc \u211d 1 \u03bc) }), (fun f => integral f = \u2016Lp.posPart f\u2016 - \u2016Lp.negPart f\u2016) \u2191a", "state_after": "case refine'_3\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedAddCommGroup F\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace E\nf : { x // x \u2208 Lp \u211d 1 }\ns : { x // x \u2208 \u2191(simpleFunc \u211d 1 \u03bc) }\n\u22a2 integral \u2191s = \u2016Lp.posPart \u2191s\u2016 - \u2016Lp.negPart \u2191s\u2016"}, {"tactic": "norm_cast", "annotated_tactic": ["norm_cast", []], "state_before": "case refine'_3\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedAddCommGroup F\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace E\nf : { x // x \u2208 Lp \u211d 1 }\ns : { x // x \u2208 \u2191(simpleFunc \u211d 1 \u03bc) }\n\u22a2 integral \u2191s = \u2016Lp.posPart \u2191s\u2016 - \u2016Lp.negPart \u2191s\u2016", "state_after": "case refine'_3\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedAddCommGroup F\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace E\nf : { x // x \u2208 Lp \u211d 1 }\ns : { x // x \u2208 \u2191(simpleFunc \u211d 1 \u03bc) }\n\u22a2 SimpleFunc.integral s = \u2016SimpleFunc.posPart s\u2016 - \u2016SimpleFunc.negPart s\u2016"}, {"tactic": "exact SimpleFunc.integral_eq_norm_posPart_sub _", "annotated_tactic": ["exact <a>SimpleFunc.integral_eq_norm_posPart_sub</a> _", [{"full_name": "MeasureTheory.L1.SimpleFunc.integral_eq_norm_posPart_sub", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [606, 9], "def_end_pos": [606, 37]}]], "state_before": "case refine'_3\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedAddCommGroup F\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace E\nf : { x // x \u2208 Lp \u211d 1 }\ns : { x // x \u2208 \u2191(simpleFunc \u211d 1 \u03bc) }\n\u22a2 SimpleFunc.integral s = \u2016SimpleFunc.posPart s\u2016 - \u2016SimpleFunc.negPart s\u2016", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "full_name": "MeasureTheory.exists_nonempty_inter_of_measure_univ_lt_tsum_measure", "start": [415, 1], "end": [422, 38], "traced_tactics": [{"tactic": "contrapose! H", "annotated_tactic": ["contrapose! H", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : \u03b9 \u2192 Set \u03b1\nhs : \u2200 (i : \u03b9), MeasurableSet (s i)\nH : \u2191\u2191\u03bc univ < \u2211' (i : \u03b9), \u2191\u2191\u03bc (s i)\n\u22a2 \u2203 i j _h, Set.Nonempty (s i \u2229 s j)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : \u03b9 \u2192 Set \u03b1\nhs : \u2200 (i : \u03b9), MeasurableSet (s i)\nH : \u2200 (i j : \u03b9), i \u2260 j \u2192 \u00acSet.Nonempty (s i \u2229 s j)\n\u22a2 \u2211' (i : \u03b9), \u2191\u2191\u03bc (s i) \u2264 \u2191\u2191\u03bc univ"}, {"tactic": "apply tsum_measure_le_measure_univ hs", "annotated_tactic": ["apply <a>tsum_measure_le_measure_univ</a> hs", [{"full_name": "MeasureTheory.tsum_measure_le_measure_univ", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [406, 9], "def_end_pos": [406, 37]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : \u03b9 \u2192 Set \u03b1\nhs : \u2200 (i : \u03b9), MeasurableSet (s i)\nH : \u2200 (i j : \u03b9), i \u2260 j \u2192 \u00acSet.Nonempty (s i \u2229 s j)\n\u22a2 \u2211' (i : \u03b9), \u2191\u2191\u03bc (s i) \u2264 \u2191\u2191\u03bc univ", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : \u03b9 \u2192 Set \u03b1\nhs : \u2200 (i : \u03b9), MeasurableSet (s i)\nH : \u2200 (i j : \u03b9), i \u2260 j \u2192 \u00acSet.Nonempty (s i \u2229 s j)\n\u22a2 Pairwise (Disjoint on fun i => s i)"}, {"tactic": "intro i j hij", "annotated_tactic": ["intro i j hij", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : \u03b9 \u2192 Set \u03b1\nhs : \u2200 (i : \u03b9), MeasurableSet (s i)\nH : \u2200 (i j : \u03b9), i \u2260 j \u2192 \u00acSet.Nonempty (s i \u2229 s j)\n\u22a2 Pairwise (Disjoint on fun i => s i)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : \u03b9 \u2192 Set \u03b1\nhs : \u2200 (i : \u03b9), MeasurableSet (s i)\nH : \u2200 (i j : \u03b9), i \u2260 j \u2192 \u00acSet.Nonempty (s i \u2229 s j)\ni j : \u03b9\nhij : i \u2260 j\n\u22a2 (Disjoint on fun i => s i) i j"}, {"tactic": "rw [Function.onFun, disjoint_iff_inf_le]", "annotated_tactic": ["rw [<a>Function.onFun</a>, <a>disjoint_iff_inf_le</a>]", [{"full_name": "Function.onFun", "def_path": "Mathlib/Init/Function.lean", "def_pos": [49, 5], "def_end_pos": [49, 10]}, {"full_name": "disjoint_iff_inf_le", "def_path": "Mathlib/Order/Disjoint.lean", "def_pos": [122, 9], "def_end_pos": [122, 28]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : \u03b9 \u2192 Set \u03b1\nhs : \u2200 (i : \u03b9), MeasurableSet (s i)\nH : \u2200 (i j : \u03b9), i \u2260 j \u2192 \u00acSet.Nonempty (s i \u2229 s j)\ni j : \u03b9\nhij : i \u2260 j\n\u22a2 (Disjoint on fun i => s i) i j", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : \u03b9 \u2192 Set \u03b1\nhs : \u2200 (i : \u03b9), MeasurableSet (s i)\nH : \u2200 (i j : \u03b9), i \u2260 j \u2192 \u00acSet.Nonempty (s i \u2229 s j)\ni j : \u03b9\nhij : i \u2260 j\n\u22a2 s i \u2293 s j \u2264 \u22a5"}, {"tactic": "exact fun x hx => H i j hij \u27e8x, hx\u27e9", "annotated_tactic": ["exact fun x hx => H i j hij \u27e8x, hx\u27e9", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : \u03b9 \u2192 Set \u03b1\nhs : \u2200 (i : \u03b9), MeasurableSet (s i)\nH : \u2200 (i j : \u03b9), i \u2260 j \u2192 \u00acSet.Nonempty (s i \u2229 s j)\ni j : \u03b9\nhij : i \u2260 j\n\u22a2 s i \u2293 s j \u2264 \u22a5", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "full_name": "MeasureTheory.integral_norm_eq_pos_sub_neg", "start": [449, 1], "end": [471, 80], "traced_tactics": [{"tactic": "rw [\u2190 integral_add_compl\u2080 h_meas hfi.norm]", "annotated_tactic": ["rw [\u2190 <a>integral_add_compl\u2080</a> h_meas hfi.norm]", [{"full_name": "MeasureTheory.integral_add_compl\u2080", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [155, 9], "def_end_pos": [155, 28]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\nf\u271d g : \u03b1 \u2192 E\ns t : Set \u03b1\n\u03bc \u03bd : Measure \u03b1\nl l' : Filter \u03b1\ninst\u271d : NormedSpace \u211d E\nf : \u03b1 \u2192 \u211d\nhfi : Integrable f\nh_meas : NullMeasurableSet {x | 0 \u2264 f x}\n\u22a2 \u222b (x : \u03b1), \u2016f x\u2016 \u2202\u03bc = \u222b (x : \u03b1) in {x | 0 \u2264 f x}, \u2016f x\u2016 \u2202\u03bc + \u222b (x : \u03b1) in {x | 0 \u2264 f x}\u1d9c, \u2016f x\u2016 \u2202\u03bc", "state_after": "no goals"}, {"tactic": "congr 1", "annotated_tactic": ["congr 1", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\nf\u271d g : \u03b1 \u2192 E\ns t : Set \u03b1\n\u03bc \u03bd : Measure \u03b1\nl l' : Filter \u03b1\ninst\u271d : NormedSpace \u211d E\nf : \u03b1 \u2192 \u211d\nhfi : Integrable f\nh_meas : NullMeasurableSet {x | 0 \u2264 f x}\n\u22a2 \u222b (x : \u03b1) in {x | 0 \u2264 f x}, \u2016f x\u2016 \u2202\u03bc + \u222b (x : \u03b1) in {x | 0 \u2264 f x}\u1d9c, \u2016f x\u2016 \u2202\u03bc =\n    \u222b (x : \u03b1) in {x | 0 \u2264 f x}, f x \u2202\u03bc + \u222b (x : \u03b1) in {x | 0 \u2264 f x}\u1d9c, \u2016f x\u2016 \u2202\u03bc", "state_after": "case e_a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\nf\u271d g : \u03b1 \u2192 E\ns t : Set \u03b1\n\u03bc \u03bd : Measure \u03b1\nl l' : Filter \u03b1\ninst\u271d : NormedSpace \u211d E\nf : \u03b1 \u2192 \u211d\nhfi : Integrable f\nh_meas : NullMeasurableSet {x | 0 \u2264 f x}\n\u22a2 \u222b (x : \u03b1) in {x | 0 \u2264 f x}, \u2016f x\u2016 \u2202\u03bc = \u222b (x : \u03b1) in {x | 0 \u2264 f x}, f x \u2202\u03bc"}, {"tactic": "refine' set_integral_congr\u2080 h_meas fun x hx => _", "annotated_tactic": ["refine' <a>set_integral_congr\u2080</a> h_meas fun x hx => _", [{"full_name": "MeasureTheory.set_integral_congr\u2080", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [82, 9], "def_end_pos": [82, 28]}]], "state_before": "case e_a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\nf\u271d g : \u03b1 \u2192 E\ns t : Set \u03b1\n\u03bc \u03bd : Measure \u03b1\nl l' : Filter \u03b1\ninst\u271d : NormedSpace \u211d E\nf : \u03b1 \u2192 \u211d\nhfi : Integrable f\nh_meas : NullMeasurableSet {x | 0 \u2264 f x}\n\u22a2 \u222b (x : \u03b1) in {x | 0 \u2264 f x}, \u2016f x\u2016 \u2202\u03bc = \u222b (x : \u03b1) in {x | 0 \u2264 f x}, f x \u2202\u03bc", "state_after": "case e_a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\nf\u271d g : \u03b1 \u2192 E\ns t : Set \u03b1\n\u03bc \u03bd : Measure \u03b1\nl l' : Filter \u03b1\ninst\u271d : NormedSpace \u211d E\nf : \u03b1 \u2192 \u211d\nhfi : Integrable f\nh_meas : NullMeasurableSet {x | 0 \u2264 f x}\nx : \u03b1\nhx : x \u2208 {x | 0 \u2264 f x}\n\u22a2 \u2016f x\u2016 = f x"}, {"tactic": "rw [Real.norm_eq_abs, abs_eq_self.mpr _]", "annotated_tactic": ["rw [<a>Real.norm_eq_abs</a>, abs_eq_self.mpr _]", [{"full_name": "Real.norm_eq_abs", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [1761, 9], "def_end_pos": [1761, 20]}]], "state_before": "case e_a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\nf\u271d g : \u03b1 \u2192 E\ns t : Set \u03b1\n\u03bc \u03bd : Measure \u03b1\nl l' : Filter \u03b1\ninst\u271d : NormedSpace \u211d E\nf : \u03b1 \u2192 \u211d\nhfi : Integrable f\nh_meas : NullMeasurableSet {x | 0 \u2264 f x}\nx : \u03b1\nhx : x \u2208 {x | 0 \u2264 f x}\n\u22a2 \u2016f x\u2016 = f x", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\nf\u271d g : \u03b1 \u2192 E\ns t : Set \u03b1\n\u03bc \u03bd : Measure \u03b1\nl l' : Filter \u03b1\ninst\u271d : NormedSpace \u211d E\nf : \u03b1 \u2192 \u211d\nhfi : Integrable f\nh_meas : NullMeasurableSet {x | 0 \u2264 f x}\nx : \u03b1\nhx : x \u2208 {x | 0 \u2264 f x}\n\u22a2 0 \u2264 f x"}, {"tactic": "exact hx", "annotated_tactic": ["exact hx", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\nf\u271d g : \u03b1 \u2192 E\ns t : Set \u03b1\n\u03bc \u03bd : Measure \u03b1\nl l' : Filter \u03b1\ninst\u271d : NormedSpace \u211d E\nf : \u03b1 \u2192 \u211d\nhfi : Integrable f\nh_meas : NullMeasurableSet {x | 0 \u2264 f x}\nx : \u03b1\nhx : x \u2208 {x | 0 \u2264 f x}\n\u22a2 0 \u2264 f x", "state_after": "no goals"}, {"tactic": "congr 1", "annotated_tactic": ["congr 1", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\nf\u271d g : \u03b1 \u2192 E\ns t : Set \u03b1\n\u03bc \u03bd : Measure \u03b1\nl l' : Filter \u03b1\ninst\u271d : NormedSpace \u211d E\nf : \u03b1 \u2192 \u211d\nhfi : Integrable f\nh_meas : NullMeasurableSet {x | 0 \u2264 f x}\n\u22a2 \u222b (x : \u03b1) in {x | 0 \u2264 f x}, f x \u2202\u03bc + \u222b (x : \u03b1) in {x | 0 \u2264 f x}\u1d9c, \u2016f x\u2016 \u2202\u03bc =\n    \u222b (x : \u03b1) in {x | 0 \u2264 f x}, f x \u2202\u03bc - \u222b (x : \u03b1) in {x | 0 \u2264 f x}\u1d9c, f x \u2202\u03bc", "state_after": "case e_a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\nf\u271d g : \u03b1 \u2192 E\ns t : Set \u03b1\n\u03bc \u03bd : Measure \u03b1\nl l' : Filter \u03b1\ninst\u271d : NormedSpace \u211d E\nf : \u03b1 \u2192 \u211d\nhfi : Integrable f\nh_meas : NullMeasurableSet {x | 0 \u2264 f x}\n\u22a2 \u222b (x : \u03b1) in {x | 0 \u2264 f x}\u1d9c, \u2016f x\u2016 \u2202\u03bc = -\u222b (x : \u03b1) in {x | 0 \u2264 f x}\u1d9c, f x \u2202\u03bc"}, {"tactic": "rw [\u2190 integral_neg]", "annotated_tactic": ["rw [\u2190 <a>integral_neg</a>]", [{"full_name": "MeasureTheory.integral_neg", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [890, 9], "def_end_pos": [890, 21]}]], "state_before": "case e_a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\nf\u271d g : \u03b1 \u2192 E\ns t : Set \u03b1\n\u03bc \u03bd : Measure \u03b1\nl l' : Filter \u03b1\ninst\u271d : NormedSpace \u211d E\nf : \u03b1 \u2192 \u211d\nhfi : Integrable f\nh_meas : NullMeasurableSet {x | 0 \u2264 f x}\n\u22a2 \u222b (x : \u03b1) in {x | 0 \u2264 f x}\u1d9c, \u2016f x\u2016 \u2202\u03bc = -\u222b (x : \u03b1) in {x | 0 \u2264 f x}\u1d9c, f x \u2202\u03bc", "state_after": "case e_a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\nf\u271d g : \u03b1 \u2192 E\ns t : Set \u03b1\n\u03bc \u03bd : Measure \u03b1\nl l' : Filter \u03b1\ninst\u271d : NormedSpace \u211d E\nf : \u03b1 \u2192 \u211d\nhfi : Integrable f\nh_meas : NullMeasurableSet {x | 0 \u2264 f x}\n\u22a2 \u222b (x : \u03b1) in {x | 0 \u2264 f x}\u1d9c, \u2016f x\u2016 \u2202\u03bc = \u222b (a : \u03b1) in {x | 0 \u2264 f x}\u1d9c, -f a \u2202\u03bc"}, {"tactic": "refine' set_integral_congr\u2080 h_meas.compl fun x hx => _", "annotated_tactic": ["refine' <a>set_integral_congr\u2080</a> h_meas.compl fun x hx => _", [{"full_name": "MeasureTheory.set_integral_congr\u2080", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [82, 9], "def_end_pos": [82, 28]}]], "state_before": "case e_a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\nf\u271d g : \u03b1 \u2192 E\ns t : Set \u03b1\n\u03bc \u03bd : Measure \u03b1\nl l' : Filter \u03b1\ninst\u271d : NormedSpace \u211d E\nf : \u03b1 \u2192 \u211d\nhfi : Integrable f\nh_meas : NullMeasurableSet {x | 0 \u2264 f x}\n\u22a2 \u222b (x : \u03b1) in {x | 0 \u2264 f x}\u1d9c, \u2016f x\u2016 \u2202\u03bc = \u222b (a : \u03b1) in {x | 0 \u2264 f x}\u1d9c, -f a \u2202\u03bc", "state_after": "case e_a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\nf\u271d g : \u03b1 \u2192 E\ns t : Set \u03b1\n\u03bc \u03bd : Measure \u03b1\nl l' : Filter \u03b1\ninst\u271d : NormedSpace \u211d E\nf : \u03b1 \u2192 \u211d\nhfi : Integrable f\nh_meas : NullMeasurableSet {x | 0 \u2264 f x}\nx : \u03b1\nhx : x \u2208 {x | 0 \u2264 f x}\u1d9c\n\u22a2 \u2016f x\u2016 = -f x"}, {"tactic": "rw [Real.norm_eq_abs, abs_eq_neg_self.mpr _]", "annotated_tactic": ["rw [<a>Real.norm_eq_abs</a>, abs_eq_neg_self.mpr _]", [{"full_name": "Real.norm_eq_abs", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [1761, 9], "def_end_pos": [1761, 20]}]], "state_before": "case e_a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\nf\u271d g : \u03b1 \u2192 E\ns t : Set \u03b1\n\u03bc \u03bd : Measure \u03b1\nl l' : Filter \u03b1\ninst\u271d : NormedSpace \u211d E\nf : \u03b1 \u2192 \u211d\nhfi : Integrable f\nh_meas : NullMeasurableSet {x | 0 \u2264 f x}\nx : \u03b1\nhx : x \u2208 {x | 0 \u2264 f x}\u1d9c\n\u22a2 \u2016f x\u2016 = -f x", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\nf\u271d g : \u03b1 \u2192 E\ns t : Set \u03b1\n\u03bc \u03bd : Measure \u03b1\nl l' : Filter \u03b1\ninst\u271d : NormedSpace \u211d E\nf : \u03b1 \u2192 \u211d\nhfi : Integrable f\nh_meas : NullMeasurableSet {x | 0 \u2264 f x}\nx : \u03b1\nhx : x \u2208 {x | 0 \u2264 f x}\u1d9c\n\u22a2 f x \u2264 0"}, {"tactic": "rw [Set.mem_compl_iff, Set.nmem_setOf_iff] at hx", "annotated_tactic": ["rw [<a>Set.mem_compl_iff</a>, <a>Set.nmem_setOf_iff</a>] at hx", [{"full_name": "Set.mem_compl_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1658, 9], "def_end_pos": [1658, 22]}, {"full_name": "Set.nmem_setOf_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [270, 9], "def_end_pos": [270, 23]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\nf\u271d g : \u03b1 \u2192 E\ns t : Set \u03b1\n\u03bc \u03bd : Measure \u03b1\nl l' : Filter \u03b1\ninst\u271d : NormedSpace \u211d E\nf : \u03b1 \u2192 \u211d\nhfi : Integrable f\nh_meas : NullMeasurableSet {x | 0 \u2264 f x}\nx : \u03b1\nhx : x \u2208 {x | 0 \u2264 f x}\u1d9c\n\u22a2 f x \u2264 0", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\nf\u271d g : \u03b1 \u2192 E\ns t : Set \u03b1\n\u03bc \u03bd : Measure \u03b1\nl l' : Filter \u03b1\ninst\u271d : NormedSpace \u211d E\nf : \u03b1 \u2192 \u211d\nhfi : Integrable f\nh_meas : NullMeasurableSet {x | 0 \u2264 f x}\nx : \u03b1\nhx : \u00ac0 \u2264 f x\n\u22a2 f x \u2264 0"}, {"tactic": "linarith", "annotated_tactic": ["linarith", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\nf\u271d g : \u03b1 \u2192 E\ns t : Set \u03b1\n\u03bc \u03bd : Measure \u03b1\nl l' : Filter \u03b1\ninst\u271d : NormedSpace \u211d E\nf : \u03b1 \u2192 \u211d\nhfi : Integrable f\nh_meas : NullMeasurableSet {x | 0 \u2264 f x}\nx : \u03b1\nhx : \u00ac0 \u2264 f x\n\u22a2 f x \u2264 0", "state_after": "no goals"}, {"tactic": "rw [\u2190 set_integral_neg_eq_set_integral_nonpos hfi.1]", "annotated_tactic": ["rw [\u2190 <a>set_integral_neg_eq_set_integral_nonpos</a> hfi.1]", [{"full_name": "MeasureTheory.set_integral_neg_eq_set_integral_nonpos", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [436, 9], "def_end_pos": [436, 48]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\nf\u271d g : \u03b1 \u2192 E\ns t : Set \u03b1\n\u03bc \u03bd : Measure \u03b1\nl l' : Filter \u03b1\ninst\u271d : NormedSpace \u211d E\nf : \u03b1 \u2192 \u211d\nhfi : Integrable f\nh_meas : NullMeasurableSet {x | 0 \u2264 f x}\n\u22a2 \u222b (x : \u03b1) in {x | 0 \u2264 f x}, f x \u2202\u03bc - \u222b (x : \u03b1) in {x | 0 \u2264 f x}\u1d9c, f x \u2202\u03bc =\n    \u222b (x : \u03b1) in {x | 0 \u2264 f x}, f x \u2202\u03bc - \u222b (x : \u03b1) in {x | f x \u2264 0}, f x \u2202\u03bc", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\nf\u271d g : \u03b1 \u2192 E\ns t : Set \u03b1\n\u03bc \u03bd : Measure \u03b1\nl l' : Filter \u03b1\ninst\u271d : NormedSpace \u211d E\nf : \u03b1 \u2192 \u211d\nhfi : Integrable f\nh_meas : NullMeasurableSet {x | 0 \u2264 f x}\n\u22a2 \u222b (x : \u03b1) in {x | 0 \u2264 f x}, f x \u2202\u03bc - \u222b (x : \u03b1) in {x | 0 \u2264 f x}\u1d9c, f x \u2202\u03bc =\n    \u222b (x : \u03b1) in {x | 0 \u2264 f x}, f x \u2202\u03bc - \u222b (x : \u03b1) in {x | f x < 0}, f x \u2202\u03bc"}, {"tactic": "congr", "annotated_tactic": ["congr", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\nf\u271d g : \u03b1 \u2192 E\ns t : Set \u03b1\n\u03bc \u03bd : Measure \u03b1\nl l' : Filter \u03b1\ninst\u271d : NormedSpace \u211d E\nf : \u03b1 \u2192 \u211d\nhfi : Integrable f\nh_meas : NullMeasurableSet {x | 0 \u2264 f x}\n\u22a2 \u222b (x : \u03b1) in {x | 0 \u2264 f x}, f x \u2202\u03bc - \u222b (x : \u03b1) in {x | 0 \u2264 f x}\u1d9c, f x \u2202\u03bc =\n    \u222b (x : \u03b1) in {x | 0 \u2264 f x}, f x \u2202\u03bc - \u222b (x : \u03b1) in {x | f x < 0}, f x \u2202\u03bc", "state_after": "case e_a.e_\u03bc.e_s\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\nf\u271d g : \u03b1 \u2192 E\ns t : Set \u03b1\n\u03bc \u03bd : Measure \u03b1\nl l' : Filter \u03b1\ninst\u271d : NormedSpace \u211d E\nf : \u03b1 \u2192 \u211d\nhfi : Integrable f\nh_meas : NullMeasurableSet {x | 0 \u2264 f x}\n\u22a2 {x | 0 \u2264 f x}\u1d9c = {x | f x < 0}"}, {"tactic": "ext1 x", "annotated_tactic": ["ext1 x", []], "state_before": "case e_a.e_\u03bc.e_s\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\nf\u271d g : \u03b1 \u2192 E\ns t : Set \u03b1\n\u03bc \u03bd : Measure \u03b1\nl l' : Filter \u03b1\ninst\u271d : NormedSpace \u211d E\nf : \u03b1 \u2192 \u211d\nhfi : Integrable f\nh_meas : NullMeasurableSet {x | 0 \u2264 f x}\n\u22a2 {x | 0 \u2264 f x}\u1d9c = {x | f x < 0}", "state_after": "case e_a.e_\u03bc.e_s.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\nf\u271d g : \u03b1 \u2192 E\ns t : Set \u03b1\n\u03bc \u03bd : Measure \u03b1\nl l' : Filter \u03b1\ninst\u271d : NormedSpace \u211d E\nf : \u03b1 \u2192 \u211d\nhfi : Integrable f\nh_meas : NullMeasurableSet {x | 0 \u2264 f x}\nx : \u03b1\n\u22a2 x \u2208 {x | 0 \u2264 f x}\u1d9c \u2194 x \u2208 {x | f x < 0}"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case e_a.e_\u03bc.e_s.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\nf\u271d g : \u03b1 \u2192 E\ns t : Set \u03b1\n\u03bc \u03bd : Measure \u03b1\nl l' : Filter \u03b1\ninst\u271d : NormedSpace \u211d E\nf : \u03b1 \u2192 \u211d\nhfi : Integrable f\nh_meas : NullMeasurableSet {x | 0 \u2264 f x}\nx : \u03b1\n\u22a2 x \u2208 {x | 0 \u2264 f x}\u1d9c \u2194 x \u2208 {x | f x < 0}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/LpSpace/DomAct/Basic.lean", "full_name": "DomMulAct.dist_smul_Lp", "start": [99, 1], "end": [100, 48], "traced_tactics": [{"tactic": "simp only [dist, \u2190 smul_Lp_sub, norm_smul_Lp]", "annotated_tactic": ["simp only [<a>dist</a>, \u2190 <a>smul_Lp_sub</a>, <a>norm_smul_Lp</a>]", [{"full_name": "Dist.dist", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [94, 3], "def_end_pos": [94, 7]}, {"full_name": "DomMulAct.smul_Lp_sub", "def_path": "Mathlib/MeasureTheory/Function/LpSpace/DomAct/Basic.lean", "def_pos": [82, 9], "def_end_pos": [82, 20]}, {"full_name": "DomMulAct.norm_smul_Lp", "def_path": "Mathlib/MeasureTheory/Function/LpSpace/DomAct/Basic.lean", "def_pos": [91, 9], "def_end_pos": [91, 21]}]], "state_before": "M : Type u_1\nN : Type u_2\n\u03b1 : Type u_3\nE : Type u_4\ninst\u271d\u2076 : MeasurableSpace M\ninst\u271d\u2075 : MeasurableSpace N\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b2 : SMul M \u03b1\ninst\u271d\u00b9 : SMulInvariantMeasure M \u03b1 \u03bc\ninst\u271d : MeasurableSMul M \u03b1\nc : M\u1d48\u1d50\u1d43\nf g : { x // x \u2208 Lp E p }\n\u22a2 dist (c \u2022 f) (c \u2022 g) = dist f g", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "full_name": "MeasureTheory.SimpleFunc.FinMeasSupp.pair", "start": [1205, 11], "end": [1210, 35], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "full_name": "Int.sign_ofNat_of_nonzero", "start": [207, 1], "end": [209, 38], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Portmanteau.lean", "full_name": "MeasureTheory.tendsto_measure_of_null_frontier", "start": [247, 1], "end": [254, 74], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Regular.lean", "full_name": "MeasureTheory.Measure.Regular.smul", "start": [541, 11], "end": [544, 38], "traced_tactics": [{"tactic": "haveI := OuterRegular.smul \u03bc hx", "annotated_tactic": ["haveI := <a>OuterRegular.smul</a> \u03bc hx", [{"full_name": "MeasureTheory.Measure.OuterRegular.smul", "def_path": "Mathlib/MeasureTheory/Measure/Regular.lean", "def_pos": [291, 19], "def_end_pos": [291, 23]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : Regular \u03bc\nx : \u211d\u22650\u221e\nhx : x \u2260 \u22a4\n\u22a2 Regular (x \u2022 \u03bc)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : Regular \u03bc\nx : \u211d\u22650\u221e\nhx : x \u2260 \u22a4\nthis : OuterRegular (x \u2022 \u03bc)\n\u22a2 Regular (x \u2022 \u03bc)"}, {"tactic": "haveI := IsFiniteMeasureOnCompacts.smul \u03bc hx", "annotated_tactic": ["haveI := <a>IsFiniteMeasureOnCompacts.smul</a> \u03bc hx", [{"full_name": "MeasureTheory.IsFiniteMeasureOnCompacts.smul", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3883, 19], "def_end_pos": [3883, 49]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : Regular \u03bc\nx : \u211d\u22650\u221e\nhx : x \u2260 \u22a4\nthis : OuterRegular (x \u2022 \u03bc)\n\u22a2 Regular (x \u2022 \u03bc)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : Regular \u03bc\nx : \u211d\u22650\u221e\nhx : x \u2260 \u22a4\nthis\u271d : OuterRegular (x \u2022 \u03bc)\nthis : IsFiniteMeasureOnCompacts (x \u2022 \u03bc)\n\u22a2 Regular (x \u2022 \u03bc)"}, {"tactic": "exact \u27e8Regular.innerRegular.smul x\u27e9", "annotated_tactic": ["exact \u27e8Regular.innerRegular.smul x\u27e9", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : Regular \u03bc\nx : \u211d\u22650\u221e\nhx : x \u2260 \u22a4\nthis\u271d : OuterRegular (x \u2022 \u03bc)\nthis : IsFiniteMeasureOnCompacts (x \u2022 \u03bc)\n\u22a2 Regular (x \u2022 \u03bc)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Martingale/Upcrossing.lean", "full_name": "MeasureTheory.sub_eq_zero_of_upcrossingsBefore_lt", "start": [588, 1], "end": [596, 96], "traced_tactics": [{"tactic": "have : N \u2264 upperCrossingTime a b f N n \u03c9 := by\n  rw [upcrossingsBefore] at hn\n  rw [\u2190 not_lt]\n  exact fun h => not_le.2 hn (le_csSup (upperCrossingTime_lt_bddAbove hab) h)", "annotated_tactic": ["have : N \u2264 <a>upperCrossingTime</a> a b f N n \u03c9 := by\n    rw [<a>upcrossingsBefore</a>] at hn\n    rw [\u2190 <a>not_lt</a>]\n    exact fun h => <a>not_le</a>.2 hn (<a>le_csSup</a> (<a>upperCrossingTime_lt_bddAbove</a> hab) h)", [{"full_name": "MeasureTheory.upperCrossingTime", "def_path": "Mathlib/Probability/Martingale/Upcrossing.lean", "def_pos": [139, 19], "def_end_pos": [139, 36]}, {"full_name": "MeasureTheory.upcrossingsBefore", "def_path": "Mathlib/Probability/Martingale/Upcrossing.lean", "def_pos": [450, 19], "def_end_pos": [450, 36]}, {"full_name": "not_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [368, 9], "def_end_pos": [368, 15]}, {"full_name": "not_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [373, 9], "def_end_pos": [373, 15]}, {"full_name": "le_csSup", "def_path": "Mathlib/Order/ConditionallyCompleteLattice/Basic.lean", "def_pos": [457, 9], "def_end_pos": [457, 17]}, {"full_name": "MeasureTheory.upperCrossingTime_lt_bddAbove", "def_path": "Mathlib/Probability/Martingale/Upcrossing.lean", "def_pos": [302, 9], "def_end_pos": [302, 38]}]], "state_before": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\nhab : a < b\nhn : upcrossingsBefore a b f N \u03c9 < n\n\u22a2 stoppedValue f (upperCrossingTime a b f N (n + 1)) \u03c9 - stoppedValue f (lowerCrossingTime a b f N n) \u03c9 = 0", "state_after": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\nhab : a < b\nhn : upcrossingsBefore a b f N \u03c9 < n\nthis : N \u2264 upperCrossingTime a b f N n \u03c9\n\u22a2 stoppedValue f (upperCrossingTime a b f N (n + 1)) \u03c9 - stoppedValue f (lowerCrossingTime a b f N n) \u03c9 = 0"}, {"tactic": "simp [stoppedValue, upperCrossingTime_stabilize' (Nat.le_succ n) this,\n  lowerCrossingTime_stabilize' le_rfl (le_trans this upperCrossingTime_le_lowerCrossingTime)]", "annotated_tactic": ["simp [<a>stoppedValue</a>, <a>upperCrossingTime_stabilize'</a> (<a>Nat.le_succ</a> n) this,\n    <a>lowerCrossingTime_stabilize'</a> <a>le_rfl</a> (<a>le_trans</a> this <a>upperCrossingTime_le_lowerCrossingTime</a>)]", [{"full_name": "MeasureTheory.stoppedValue", "def_path": "Mathlib/Probability/Process/Stopping.lean", "def_pos": [768, 5], "def_end_pos": [768, 17]}, {"full_name": "MeasureTheory.upperCrossingTime_stabilize'", "def_path": "Mathlib/Probability/Martingale/Upcrossing.lean", "def_pos": [284, 9], "def_end_pos": [284, 37]}, {"full_name": "Nat.le_succ", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1599, 9], "def_end_pos": [1599, 20]}, {"full_name": "MeasureTheory.lowerCrossingTime_stabilize'", "def_path": "Mathlib/Probability/Martingale/Upcrossing.lean", "def_pos": [279, 9], "def_end_pos": [279, 37]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}, {"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}, {"full_name": "MeasureTheory.upperCrossingTime_le_lowerCrossingTime", "def_path": "Mathlib/Probability/Martingale/Upcrossing.lean", "def_pos": [202, 9], "def_end_pos": [202, 47]}]], "state_before": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\nhab : a < b\nhn : upcrossingsBefore a b f N \u03c9 < n\nthis : N \u2264 upperCrossingTime a b f N n \u03c9\n\u22a2 stoppedValue f (upperCrossingTime a b f N (n + 1)) \u03c9 - stoppedValue f (lowerCrossingTime a b f N n) \u03c9 = 0", "state_after": "no goals"}, {"tactic": "rw [upcrossingsBefore] at hn", "annotated_tactic": ["rw [<a>upcrossingsBefore</a>] at hn", [{"full_name": "MeasureTheory.upcrossingsBefore", "def_path": "Mathlib/Probability/Martingale/Upcrossing.lean", "def_pos": [450, 19], "def_end_pos": [450, 36]}]], "state_before": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\nhab : a < b\nhn : upcrossingsBefore a b f N \u03c9 < n\n\u22a2 N \u2264 upperCrossingTime a b f N n \u03c9", "state_after": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\nhab : a < b\nhn : sSup {n | upperCrossingTime a b f N n \u03c9 < N} < n\n\u22a2 N \u2264 upperCrossingTime a b f N n \u03c9"}, {"tactic": "rw [\u2190 not_lt]", "annotated_tactic": ["rw [\u2190 <a>not_lt</a>]", [{"full_name": "not_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [368, 9], "def_end_pos": [368, 15]}]], "state_before": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\nhab : a < b\nhn : sSup {n | upperCrossingTime a b f N n \u03c9 < N} < n\n\u22a2 N \u2264 upperCrossingTime a b f N n \u03c9", "state_after": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\nhab : a < b\nhn : sSup {n | upperCrossingTime a b f N n \u03c9 < N} < n\n\u22a2 \u00acupperCrossingTime a b f N n \u03c9 < N"}, {"tactic": "exact fun h => not_le.2 hn (le_csSup (upperCrossingTime_lt_bddAbove hab) h)", "annotated_tactic": ["exact fun h => <a>not_le</a>.2 hn (<a>le_csSup</a> (<a>upperCrossingTime_lt_bddAbove</a> hab) h)", [{"full_name": "not_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [373, 9], "def_end_pos": [373, 15]}, {"full_name": "le_csSup", "def_path": "Mathlib/Order/ConditionallyCompleteLattice/Basic.lean", "def_pos": [457, 9], "def_end_pos": [457, 17]}, {"full_name": "MeasureTheory.upperCrossingTime_lt_bddAbove", "def_path": "Mathlib/Probability/Martingale/Upcrossing.lean", "def_pos": [302, 9], "def_end_pos": [302, 38]}]], "state_before": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9 : \u03a9\n\u2131 : Filtration \u2115 m0\nhab : a < b\nhn : sSup {n | upperCrossingTime a b f N n \u03c9 < N} < n\n\u22a2 \u00acupperCrossingTime a b f N n \u03c9 < N", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "full_name": "MeasurableSpace.map_const", "start": [190, 9], "end": [191, 100], "traced_tactics": [{"tactic": "by_cases b \u2208 s <;> simp [*, map_def] <;> rw [Set.preimage_id'] <;> simp", "annotated_tactic": ["by_cases b \u2208 s <;> simp [*, <a>map_def</a>] <;> rw [<a>Set.preimage_id'</a>] <;> simp", [{"full_name": "MeasurableSpace.map_def", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [88, 7], "def_end_pos": [88, 14]}, {"full_name": "Set.preimage_id'", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [133, 9], "def_end_pos": [133, 21]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u\u03b9\ns\u271d t u : Set \u03b1\nm : MeasurableSpace \u03b1\nb : \u03b2\ns : Set \u03b2\nx\u271d : MeasurableSet s\n\u22a2 MeasurableSet s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Pointwise.lean", "full_name": "Finset.pairwiseDisjoint_smul_iff", "start": [1877, 1], "end": [1879, 83], "traced_tactics": [{"tactic": "simp_rw [\u2190 pairwiseDisjoint_coe, coe_smul_finset, Set.pairwiseDisjoint_smul_iff]", "annotated_tactic": ["simp_rw [\u2190 <a>pairwiseDisjoint_coe</a>, <a>coe_smul_finset</a>, <a>Set.pairwiseDisjoint_smul_iff</a>]", [{"full_name": "Finset.pairwiseDisjoint_coe", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1009, 9], "def_end_pos": [1009, 29]}, {"full_name": "Finset.coe_smul_finset", "def_path": "Mathlib/Data/Finset/Pointwise.lean", "def_pos": [1599, 9], "def_end_pos": [1599, 24]}, {"full_name": "Set.pairwiseDisjoint_smul_iff", "def_path": "Mathlib/Data/Set/Pointwise/SMul.lean", "def_pos": [868, 9], "def_end_pos": [868, 34]}]], "state_before": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b9 : LeftCancelSemigroup \u03b1\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d : Finset \u03b1\na : \u03b1\ns : Set \u03b1\nt : Finset \u03b1\n\u22a2 (Set.PairwiseDisjoint s fun x => x \u2022 t) \u2194 Set.InjOn (fun p => p.1 * p.2) (s \u00d7\u02e2 \u2191t)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "full_name": "MeasureTheory.OuterMeasure.toMeasure_zero", "start": [4269, 1], "end": [4272, 42], "traced_tactics": [{"tactic": "rw [\u2190 Measure.measure_univ_eq_zero, toMeasure_apply _ _ MeasurableSet.univ,\n  OuterMeasure.coe_zero, Pi.zero_apply]", "annotated_tactic": ["rw [\u2190 <a>Measure.measure_univ_eq_zero</a>, <a>toMeasure_apply</a> _ _ <a>MeasurableSet.univ</a>,\n    <a>OuterMeasure.coe_zero</a>, <a>Pi.zero_apply</a>]", [{"full_name": "MeasureTheory.Measure.measure_univ_eq_zero", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1100, 9], "def_end_pos": [1100, 29]}, {"full_name": "MeasureTheory.toMeasure_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [674, 9], "def_end_pos": [674, 24]}, {"full_name": "MeasurableSet.univ", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [101, 19], "def_end_pos": [101, 37]}, {"full_name": "MeasureTheory.OuterMeasure.coe_zero", "def_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "def_pos": [257, 9], "def_end_pos": [257, 17]}, {"full_name": "Pi.zero_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [46, 3], "def_end_pos": [46, 14]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d : MeasurableSpace \u03b1\n\u22a2 toMeasure 0 (_ : inst\u271d \u2264 OuterMeasure.caratheodory 0) = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/NoncommProd.lean", "full_name": "Multiset.noncommProd_eq_pow_card", "start": [196, 1], "end": [200, 36], "traced_tactics": [{"tactic": "induction s using Quotient.inductionOn", "annotated_tactic": ["induction s using <a>Quotient.inductionOn</a>", [{"full_name": "Quotient.inductionOn", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [1367, 19], "def_end_pos": [1367, 30]}]], "state_before": "F : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b1\ninst\u271d : Monoid \u03b2\ns : Multiset \u03b1\ncomm : Set.Pairwise {x | x \u2208 s} Commute\nm : \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 x = m\n\u22a2 noncommProd s comm = m ^ \u2191card s", "state_after": "case h\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b1\ninst\u271d : Monoid \u03b2\nm : \u03b1\na\u271d : List \u03b1\ncomm : Set.Pairwise {x | x \u2208 Quotient.mk (List.isSetoid \u03b1) a\u271d} Commute\nh : \u2200 (x : \u03b1), x \u2208 Quotient.mk (List.isSetoid \u03b1) a\u271d \u2192 x = m\n\u22a2 noncommProd (Quotient.mk (List.isSetoid \u03b1) a\u271d) comm = m ^ \u2191card (Quotient.mk (List.isSetoid \u03b1) a\u271d)"}, {"tactic": "simp only [quot_mk_to_coe, noncommProd_coe, coe_card, mem_coe] at *", "annotated_tactic": ["simp only [<a>quot_mk_to_coe</a>, <a>noncommProd_coe</a>, <a>coe_card</a>, <a>mem_coe</a>] at *", [{"full_name": "Multiset.quot_mk_to_coe", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [43, 9], "def_end_pos": [43, 23]}, {"full_name": "Multiset.noncommProd_coe", "def_path": "Mathlib/Data/Finset/NoncommProd.lean", "def_pos": [124, 9], "def_end_pos": [124, 24]}, {"full_name": "Multiset.coe_card", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [720, 9], "def_end_pos": [720, 17]}, {"full_name": "Multiset.mem_coe", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [226, 9], "def_end_pos": [226, 16]}]], "state_before": "case h\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b1\ninst\u271d : Monoid \u03b2\nm : \u03b1\na\u271d : List \u03b1\ncomm : Set.Pairwise {x | x \u2208 Quotient.mk (List.isSetoid \u03b1) a\u271d} Commute\nh : \u2200 (x : \u03b1), x \u2208 Quotient.mk (List.isSetoid \u03b1) a\u271d \u2192 x = m\n\u22a2 noncommProd (Quotient.mk (List.isSetoid \u03b1) a\u271d) comm = m ^ \u2191card (Quotient.mk (List.isSetoid \u03b1) a\u271d)", "state_after": "case h\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b1\ninst\u271d : Monoid \u03b2\nm : \u03b1\na\u271d : List \u03b1\ncomm : Set.Pairwise {x | x \u2208 Quotient.mk (List.isSetoid \u03b1) a\u271d} Commute\nh : \u2200 (x : \u03b1), x \u2208 a\u271d \u2192 x = m\n\u22a2 List.prod a\u271d = m ^ List.length a\u271d"}, {"tactic": "exact List.prod_eq_pow_card _ m h", "annotated_tactic": ["exact <a>List.prod_eq_pow_card</a> _ m h", [{"full_name": "List.prod_eq_pow_card", "def_path": "Mathlib/Data/List/BigOperators/Basic.lean", "def_pos": [86, 9], "def_end_pos": [86, 25]}]], "state_before": "case h\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b1\ninst\u271d : Monoid \u03b2\nm : \u03b1\na\u271d : List \u03b1\ncomm : Set.Pairwise {x | x \u2208 Quotient.mk (List.isSetoid \u03b1) a\u271d} Commute\nh : \u2200 (x : \u03b1), x \u2208 a\u271d \u2192 x = m\n\u22a2 List.prod a\u271d = m ^ List.length a\u271d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/FiniteMeasure.lean", "full_name": "MeasureTheory.FiniteMeasure.tendsto_testAgainstNN_of_le_const", "start": [627, 1], "end": [633, 34], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "full_name": "MeasureTheory.SimpleFunc.integrable_pair", "start": [379, 1], "end": [381, 65], "traced_tactics": [{"tactic": "simpa only [integrable_iff_finMeasSupp] using FinMeasSupp.pair", "annotated_tactic": ["simpa only [<a>integrable_iff_finMeasSupp</a>] using <a>FinMeasSupp.pair</a>", [{"full_name": "MeasureTheory.SimpleFunc.integrable_iff_finMeasSupp", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "def_pos": [371, 9], "def_end_pos": [371, 35]}, {"full_name": "MeasureTheory.SimpleFunc.FinMeasSupp.pair", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [1205, 19], "def_end_pos": [1205, 23]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\nf : \u03b1 \u2192\u209b E\ng : \u03b1 \u2192\u209b F\n\u22a2 Integrable \u2191f \u2192 Integrable \u2191g \u2192 Integrable \u2191(pair f g)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "full_name": "MeasureTheory.L1.SimpleFunc.norm_eq_sum_mul", "start": [658, 1], "end": [673, 96], "traced_tactics": [{"tactic": "rw [norm_toSimpleFunc, snorm_one_eq_lintegral_nnnorm]", "annotated_tactic": ["rw [<a>norm_toSimpleFunc</a>, <a>snorm_one_eq_lintegral_nnnorm</a>]", [{"full_name": "MeasureTheory.Lp.simpleFunc.norm_toSimpleFunc", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "def_pos": [678, 9], "def_end_pos": [678, 26]}, {"full_name": "MeasureTheory.snorm_one_eq_lintegral_nnnorm", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [97, 9], "def_end_pos": [97, 38]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : { x // x \u2208 simpleFunc G 1 \u03bc }\n\u22a2 \u2016f\u2016 = \u2211 x in SimpleFunc.range (toSimpleFunc f), ENNReal.toReal (\u2191\u2191\u03bc (\u2191(toSimpleFunc f) \u207b\u00b9' {x})) * \u2016x\u2016", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : { x // x \u2208 simpleFunc G 1 \u03bc }\n\u22a2 ENNReal.toReal (\u222b\u207b (x : \u03b1), \u2191\u2016\u2191(toSimpleFunc f) x\u2016\u208a \u2202\u03bc) =\n    \u2211 x in SimpleFunc.range (toSimpleFunc f), ENNReal.toReal (\u2191\u2191\u03bc (\u2191(toSimpleFunc f) \u207b\u00b9' {x})) * \u2016x\u2016"}, {"tactic": "have h_eq := SimpleFunc.map_apply (fun x => (\u2016x\u2016\u208a : \u211d\u22650\u221e)) (toSimpleFunc f)", "annotated_tactic": ["have h_eq := <a>SimpleFunc.map_apply</a> (fun x => (\u2016x\u2016\u208a : \u211d\u22650\u221e)) (<a>toSimpleFunc</a> f)", [{"full_name": "MeasureTheory.SimpleFunc.map_apply", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [294, 9], "def_end_pos": [294, 18]}, {"full_name": "MeasureTheory.Lp.simpleFunc.toSimpleFunc", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "def_pos": [587, 5], "def_end_pos": [587, 17]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : { x // x \u2208 simpleFunc G 1 \u03bc }\n\u22a2 ENNReal.toReal (\u222b\u207b (x : \u03b1), \u2191\u2016\u2191(toSimpleFunc f) x\u2016\u208a \u2202\u03bc) =\n    \u2211 x in SimpleFunc.range (toSimpleFunc f), ENNReal.toReal (\u2191\u2191\u03bc (\u2191(toSimpleFunc f) \u207b\u00b9' {x})) * \u2016x\u2016", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : { x // x \u2208 simpleFunc G 1 \u03bc }\nh_eq : \u2200 (a : \u03b1), \u2191(SimpleFunc.map (fun x => \u2191\u2016x\u2016\u208a) (toSimpleFunc f)) a = \u2191\u2016\u2191(toSimpleFunc f) a\u2016\u208a\n\u22a2 ENNReal.toReal (\u222b\u207b (x : \u03b1), \u2191\u2016\u2191(toSimpleFunc f) x\u2016\u208a \u2202\u03bc) =\n    \u2211 x in SimpleFunc.range (toSimpleFunc f), ENNReal.toReal (\u2191\u2191\u03bc (\u2191(toSimpleFunc f) \u207b\u00b9' {x})) * \u2016x\u2016"}, {"tactic": "simp_rw [\u2190 h_eq]", "annotated_tactic": ["simp_rw [\u2190 h_eq]", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : { x // x \u2208 simpleFunc G 1 \u03bc }\nh_eq : \u2200 (a : \u03b1), \u2191(SimpleFunc.map (fun x => \u2191\u2016x\u2016\u208a) (toSimpleFunc f)) a = \u2191\u2016\u2191(toSimpleFunc f) a\u2016\u208a\n\u22a2 ENNReal.toReal (\u222b\u207b (x : \u03b1), \u2191\u2016\u2191(toSimpleFunc f) x\u2016\u208a \u2202\u03bc) =\n    \u2211 x in SimpleFunc.range (toSimpleFunc f), ENNReal.toReal (\u2191\u2191\u03bc (\u2191(toSimpleFunc f) \u207b\u00b9' {x})) * \u2016x\u2016", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : { x // x \u2208 simpleFunc G 1 \u03bc }\nh_eq : \u2200 (a : \u03b1), \u2191(SimpleFunc.map (fun x => \u2191\u2016x\u2016\u208a) (toSimpleFunc f)) a = \u2191\u2016\u2191(toSimpleFunc f) a\u2016\u208a\n\u22a2 ENNReal.toReal (\u222b\u207b (x : \u03b1), \u2191(SimpleFunc.map (fun x => \u2191\u2016x\u2016\u208a) (toSimpleFunc f)) x \u2202\u03bc) =\n    \u2211 x in SimpleFunc.range (toSimpleFunc f), ENNReal.toReal (\u2191\u2191\u03bc (\u2191(toSimpleFunc f) \u207b\u00b9' {x})) * \u2016x\u2016"}, {"tactic": "rw [SimpleFunc.lintegral_eq_lintegral, SimpleFunc.map_lintegral, ENNReal.toReal_sum]", "annotated_tactic": ["rw [<a>SimpleFunc.lintegral_eq_lintegral</a>, <a>SimpleFunc.map_lintegral</a>, <a>ENNReal.toReal_sum</a>]", [{"full_name": "MeasureTheory.SimpleFunc.lintegral_eq_lintegral", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [80, 9], "def_end_pos": [80, 42]}, {"full_name": "MeasureTheory.SimpleFunc.map_lintegral", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [976, 9], "def_end_pos": [976, 22]}, {"full_name": "ENNReal.toReal_sum", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1288, 9], "def_end_pos": [1288, 19]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : { x // x \u2208 simpleFunc G 1 \u03bc }\nh_eq : \u2200 (a : \u03b1), \u2191(SimpleFunc.map (fun x => \u2191\u2016x\u2016\u208a) (toSimpleFunc f)) a = \u2191\u2016\u2191(toSimpleFunc f) a\u2016\u208a\n\u22a2 ENNReal.toReal (\u222b\u207b (x : \u03b1), \u2191(SimpleFunc.map (fun x => \u2191\u2016x\u2016\u208a) (toSimpleFunc f)) x \u2202\u03bc) =\n    \u2211 x in SimpleFunc.range (toSimpleFunc f), ENNReal.toReal (\u2191\u2191\u03bc (\u2191(toSimpleFunc f) \u207b\u00b9' {x})) * \u2016x\u2016", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : { x // x \u2208 simpleFunc G 1 \u03bc }\nh_eq : \u2200 (a : \u03b1), \u2191(SimpleFunc.map (fun x => \u2191\u2016x\u2016\u208a) (toSimpleFunc f)) a = \u2191\u2016\u2191(toSimpleFunc f) a\u2016\u208a\n\u22a2 \u2211 a in SimpleFunc.range (toSimpleFunc f), ENNReal.toReal (\u2191\u2016a\u2016\u208a * \u2191\u2191\u03bc (\u2191(toSimpleFunc f) \u207b\u00b9' {a})) =\n    \u2211 x in SimpleFunc.range (toSimpleFunc f), ENNReal.toReal (\u2191\u2191\u03bc (\u2191(toSimpleFunc f) \u207b\u00b9' {x})) * \u2016x\u2016\n\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : { x // x \u2208 simpleFunc G 1 \u03bc }\nh_eq : \u2200 (a : \u03b1), \u2191(SimpleFunc.map (fun x => \u2191\u2016x\u2016\u208a) (toSimpleFunc f)) a = \u2191\u2016\u2191(toSimpleFunc f) a\u2016\u208a\n\u22a2 \u2200 (a : G), a \u2208 SimpleFunc.range (toSimpleFunc f) \u2192 \u2191\u2016a\u2016\u208a * \u2191\u2191\u03bc (\u2191(toSimpleFunc f) \u207b\u00b9' {a}) \u2260 \u22a4"}, {"tactic": "congr", "annotated_tactic": ["congr", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : { x // x \u2208 simpleFunc G 1 \u03bc }\nh_eq : \u2200 (a : \u03b1), \u2191(SimpleFunc.map (fun x => \u2191\u2016x\u2016\u208a) (toSimpleFunc f)) a = \u2191\u2016\u2191(toSimpleFunc f) a\u2016\u208a\n\u22a2 \u2211 a in SimpleFunc.range (toSimpleFunc f), ENNReal.toReal (\u2191\u2016a\u2016\u208a * \u2191\u2191\u03bc (\u2191(toSimpleFunc f) \u207b\u00b9' {a})) =\n    \u2211 x in SimpleFunc.range (toSimpleFunc f), ENNReal.toReal (\u2191\u2191\u03bc (\u2191(toSimpleFunc f) \u207b\u00b9' {x})) * \u2016x\u2016", "state_after": "case e_f\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : { x // x \u2208 simpleFunc G 1 \u03bc }\nh_eq : \u2200 (a : \u03b1), \u2191(SimpleFunc.map (fun x => \u2191\u2016x\u2016\u208a) (toSimpleFunc f)) a = \u2191\u2016\u2191(toSimpleFunc f) a\u2016\u208a\n\u22a2 (fun a => ENNReal.toReal (\u2191\u2016a\u2016\u208a * \u2191\u2191\u03bc (\u2191(toSimpleFunc f) \u207b\u00b9' {a}))) = fun x =>\n    ENNReal.toReal (\u2191\u2191\u03bc (\u2191(toSimpleFunc f) \u207b\u00b9' {x})) * \u2016x\u2016"}, {"tactic": "ext1 x", "annotated_tactic": ["ext1 x", []], "state_before": "case e_f\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : { x // x \u2208 simpleFunc G 1 \u03bc }\nh_eq : \u2200 (a : \u03b1), \u2191(SimpleFunc.map (fun x => \u2191\u2016x\u2016\u208a) (toSimpleFunc f)) a = \u2191\u2016\u2191(toSimpleFunc f) a\u2016\u208a\n\u22a2 (fun a => ENNReal.toReal (\u2191\u2016a\u2016\u208a * \u2191\u2191\u03bc (\u2191(toSimpleFunc f) \u207b\u00b9' {a}))) = fun x =>\n    ENNReal.toReal (\u2191\u2191\u03bc (\u2191(toSimpleFunc f) \u207b\u00b9' {x})) * \u2016x\u2016", "state_after": "case e_f.h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : { x // x \u2208 simpleFunc G 1 \u03bc }\nh_eq : \u2200 (a : \u03b1), \u2191(SimpleFunc.map (fun x => \u2191\u2016x\u2016\u208a) (toSimpleFunc f)) a = \u2191\u2016\u2191(toSimpleFunc f) a\u2016\u208a\nx : G\n\u22a2 ENNReal.toReal (\u2191\u2016x\u2016\u208a * \u2191\u2191\u03bc (\u2191(toSimpleFunc f) \u207b\u00b9' {x})) = ENNReal.toReal (\u2191\u2191\u03bc (\u2191(toSimpleFunc f) \u207b\u00b9' {x})) * \u2016x\u2016"}, {"tactic": "rw [ENNReal.toReal_mul, mul_comm, \u2190 ofReal_norm_eq_coe_nnnorm,\n  ENNReal.toReal_ofReal (norm_nonneg _)]", "annotated_tactic": ["rw [<a>ENNReal.toReal_mul</a>, <a>mul_comm</a>, \u2190 <a>ofReal_norm_eq_coe_nnnorm</a>,\n      <a>ENNReal.toReal_ofReal</a> (<a>norm_nonneg</a> _)]", [{"full_name": "ENNReal.toReal_mul", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2296, 9], "def_end_pos": [2296, 19]}, {"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [302, 9], "def_end_pos": [302, 17]}, {"full_name": "ofReal_norm_eq_coe_nnnorm", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [999, 15], "def_end_pos": [999, 40]}, {"full_name": "ENNReal.toReal_ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [191, 9], "def_end_pos": [191, 22]}, {"full_name": "norm_nonneg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [500, 30], "def_end_pos": [500, 41]}]], "state_before": "case e_f.h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : { x // x \u2208 simpleFunc G 1 \u03bc }\nh_eq : \u2200 (a : \u03b1), \u2191(SimpleFunc.map (fun x => \u2191\u2016x\u2016\u208a) (toSimpleFunc f)) a = \u2191\u2016\u2191(toSimpleFunc f) a\u2016\u208a\nx : G\n\u22a2 ENNReal.toReal (\u2191\u2016x\u2016\u208a * \u2191\u2191\u03bc (\u2191(toSimpleFunc f) \u207b\u00b9' {x})) = ENNReal.toReal (\u2191\u2191\u03bc (\u2191(toSimpleFunc f) \u207b\u00b9' {x})) * \u2016x\u2016", "state_after": "no goals"}, {"tactic": "intro x _", "annotated_tactic": ["intro x _", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : { x // x \u2208 simpleFunc G 1 \u03bc }\nh_eq : \u2200 (a : \u03b1), \u2191(SimpleFunc.map (fun x => \u2191\u2016x\u2016\u208a) (toSimpleFunc f)) a = \u2191\u2016\u2191(toSimpleFunc f) a\u2016\u208a\n\u22a2 \u2200 (a : G), a \u2208 SimpleFunc.range (toSimpleFunc f) \u2192 \u2191\u2016a\u2016\u208a * \u2191\u2191\u03bc (\u2191(toSimpleFunc f) \u207b\u00b9' {a}) \u2260 \u22a4", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : { x // x \u2208 simpleFunc G 1 \u03bc }\nh_eq : \u2200 (a : \u03b1), \u2191(SimpleFunc.map (fun x => \u2191\u2016x\u2016\u208a) (toSimpleFunc f)) a = \u2191\u2016\u2191(toSimpleFunc f) a\u2016\u208a\nx : G\na\u271d : x \u2208 SimpleFunc.range (toSimpleFunc f)\n\u22a2 \u2191\u2016x\u2016\u208a * \u2191\u2191\u03bc (\u2191(toSimpleFunc f) \u207b\u00b9' {x}) \u2260 \u22a4"}, {"tactic": "by_cases hx0 : x = 0", "annotated_tactic": ["by_cases hx0 : x = 0", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : { x // x \u2208 simpleFunc G 1 \u03bc }\nh_eq : \u2200 (a : \u03b1), \u2191(SimpleFunc.map (fun x => \u2191\u2016x\u2016\u208a) (toSimpleFunc f)) a = \u2191\u2016\u2191(toSimpleFunc f) a\u2016\u208a\nx : G\na\u271d : x \u2208 SimpleFunc.range (toSimpleFunc f)\n\u22a2 \u2191\u2016x\u2016\u208a * \u2191\u2191\u03bc (\u2191(toSimpleFunc f) \u207b\u00b9' {x}) \u2260 \u22a4", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : { x // x \u2208 simpleFunc G 1 \u03bc }\nh_eq : \u2200 (a : \u03b1), \u2191(SimpleFunc.map (fun x => \u2191\u2016x\u2016\u208a) (toSimpleFunc f)) a = \u2191\u2016\u2191(toSimpleFunc f) a\u2016\u208a\nx : G\na\u271d : x \u2208 SimpleFunc.range (toSimpleFunc f)\nhx0 : x = 0\n\u22a2 \u2191\u2016x\u2016\u208a * \u2191\u2191\u03bc (\u2191(toSimpleFunc f) \u207b\u00b9' {x}) \u2260 \u22a4\n\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : { x // x \u2208 simpleFunc G 1 \u03bc }\nh_eq : \u2200 (a : \u03b1), \u2191(SimpleFunc.map (fun x => \u2191\u2016x\u2016\u208a) (toSimpleFunc f)) a = \u2191\u2016\u2191(toSimpleFunc f) a\u2016\u208a\nx : G\na\u271d : x \u2208 SimpleFunc.range (toSimpleFunc f)\nhx0 : \u00acx = 0\n\u22a2 \u2191\u2016x\u2016\u208a * \u2191\u2191\u03bc (\u2191(toSimpleFunc f) \u207b\u00b9' {x}) \u2260 \u22a4"}, {"tactic": "rw [hx0]", "annotated_tactic": ["rw [hx0]", []], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : { x // x \u2208 simpleFunc G 1 \u03bc }\nh_eq : \u2200 (a : \u03b1), \u2191(SimpleFunc.map (fun x => \u2191\u2016x\u2016\u208a) (toSimpleFunc f)) a = \u2191\u2016\u2191(toSimpleFunc f) a\u2016\u208a\nx : G\na\u271d : x \u2208 SimpleFunc.range (toSimpleFunc f)\nhx0 : x = 0\n\u22a2 \u2191\u2016x\u2016\u208a * \u2191\u2191\u03bc (\u2191(toSimpleFunc f) \u207b\u00b9' {x}) \u2260 \u22a4", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : { x // x \u2208 simpleFunc G 1 \u03bc }\nh_eq : \u2200 (a : \u03b1), \u2191(SimpleFunc.map (fun x => \u2191\u2016x\u2016\u208a) (toSimpleFunc f)) a = \u2191\u2016\u2191(toSimpleFunc f) a\u2016\u208a\nx : G\na\u271d : x \u2208 SimpleFunc.range (toSimpleFunc f)\nhx0 : x = 0\n\u22a2 \u2191\u20160\u2016\u208a * \u2191\u2191\u03bc (\u2191(toSimpleFunc f) \u207b\u00b9' {0}) \u2260 \u22a4"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : { x // x \u2208 simpleFunc G 1 \u03bc }\nh_eq : \u2200 (a : \u03b1), \u2191(SimpleFunc.map (fun x => \u2191\u2016x\u2016\u208a) (toSimpleFunc f)) a = \u2191\u2016\u2191(toSimpleFunc f) a\u2016\u208a\nx : G\na\u271d : x \u2208 SimpleFunc.range (toSimpleFunc f)\nhx0 : x = 0\n\u22a2 \u2191\u20160\u2016\u208a * \u2191\u2191\u03bc (\u2191(toSimpleFunc f) \u207b\u00b9' {0}) \u2260 \u22a4", "state_after": "no goals"}, {"tactic": "exact\n  ENNReal.mul_ne_top ENNReal.coe_ne_top\n    (SimpleFunc.measure_preimage_lt_top_of_integrable _ (SimpleFunc.integrable f) hx0).ne", "annotated_tactic": ["exact\n        <a>ENNReal.mul_ne_top</a> <a>ENNReal.coe_ne_top</a>\n          (<a>SimpleFunc.measure_preimage_lt_top_of_integrable</a> _ (<a>SimpleFunc.integrable</a> f) hx0).<a>ne</a>", [{"full_name": "ENNReal.mul_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [615, 9], "def_end_pos": [615, 19]}, {"full_name": "ENNReal.coe_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [302, 17], "def_end_pos": [302, 27]}, {"full_name": "MeasureTheory.SimpleFunc.measure_preimage_lt_top_of_integrable", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "def_pos": [394, 9], "def_end_pos": [394, 46]}, {"full_name": "MeasureTheory.L1.SimpleFunc.integrable", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "def_pos": [1040, 19], "def_end_pos": [1040, 43]}, {"full_name": "LT.lt.ne", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [152, 7], "def_end_pos": [152, 15]}]], "state_before": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : { x // x \u2208 simpleFunc G 1 \u03bc }\nh_eq : \u2200 (a : \u03b1), \u2191(SimpleFunc.map (fun x => \u2191\u2016x\u2016\u208a) (toSimpleFunc f)) a = \u2191\u2016\u2191(toSimpleFunc f) a\u2016\u208a\nx : G\na\u271d : x \u2208 SimpleFunc.range (toSimpleFunc f)\nhx0 : \u00acx = 0\n\u22a2 \u2191\u2016x\u2016\u208a * \u2191\u2191\u03bc (\u2191(toSimpleFunc f) \u207b\u00b9' {x}) \u2260 \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Group/Prod.lean", "full_name": "MeasureTheory.quasiMeasurePreserving_mul_left", "start": [498, 1], "end": [508, 13], "traced_tactics": [{"tactic": "have :=\n  (quasiMeasurePreserving_mul_right \u03bc.inv g\u207b\u00b9).mono (inv_absolutelyContinuous \u03bc.inv)\n    (absolutelyContinuous_inv \u03bc.inv)", "annotated_tactic": ["have :=\n    (<a>quasiMeasurePreserving_mul_right</a> \u03bc.inv g\u207b\u00b9).<a>mono</a> (<a>inv_absolutelyContinuous</a> \u03bc.inv)\n      (<a>absolutelyContinuous_inv</a> \u03bc.inv)", [{"full_name": "MeasureTheory.quasiMeasurePreserving_mul_right", "def_path": "Mathlib/MeasureTheory/Group/Prod.lean", "def_pos": [486, 9], "def_end_pos": [486, 41]}, {"full_name": "MeasureTheory.Measure.QuasiMeasurePreserving.mono", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2231, 9], "def_end_pos": [2231, 13]}, {"full_name": "MeasureTheory.inv_absolutelyContinuous", "def_path": "Mathlib/MeasureTheory/Group/Prod.lean", "def_pos": [182, 9], "def_end_pos": [182, 33]}, {"full_name": "MeasureTheory.absolutelyContinuous_inv", "def_path": "Mathlib/MeasureTheory/Group/Prod.lean", "def_pos": [188, 9], "def_end_pos": [188, 33]}]], "state_before": "G : Type u_1\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b3 : SigmaFinite \u03bd\ninst\u271d\u00b2 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b9 : MeasurableInv G\ninst\u271d : IsMulRightInvariant \u03bc\ng : G\n\u22a2 QuasiMeasurePreserving fun h => g * h", "state_after": "G : Type u_1\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b3 : SigmaFinite \u03bd\ninst\u271d\u00b2 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b9 : MeasurableInv G\ninst\u271d : IsMulRightInvariant \u03bc\ng : G\nthis : QuasiMeasurePreserving fun h => h * g\u207b\u00b9\n\u22a2 QuasiMeasurePreserving fun h => g * h"}, {"tactic": "have :=\n  (quasiMeasurePreserving_inv_of_right_invariant \u03bc).comp\n    (this.comp (quasiMeasurePreserving_inv_of_right_invariant \u03bc))", "annotated_tactic": ["have :=\n    (<a>quasiMeasurePreserving_inv_of_right_invariant</a> \u03bc).<a>comp</a>\n      (this.comp (<a>quasiMeasurePreserving_inv_of_right_invariant</a> \u03bc))", [{"full_name": "MeasureTheory.quasiMeasurePreserving_inv_of_right_invariant", "def_path": "Mathlib/MeasureTheory/Group/Prod.lean", "def_pos": [437, 9], "def_end_pos": [437, 54]}, {"full_name": "MeasureTheory.Measure.QuasiMeasurePreserving.comp", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2236, 19], "def_end_pos": [2236, 23]}, {"full_name": "MeasureTheory.quasiMeasurePreserving_inv_of_right_invariant", "def_path": "Mathlib/MeasureTheory/Group/Prod.lean", "def_pos": [437, 9], "def_end_pos": [437, 54]}]], "state_before": "G : Type u_1\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b3 : SigmaFinite \u03bd\ninst\u271d\u00b2 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b9 : MeasurableInv G\ninst\u271d : IsMulRightInvariant \u03bc\ng : G\nthis : QuasiMeasurePreserving fun h => h * g\u207b\u00b9\n\u22a2 QuasiMeasurePreserving fun h => g * h", "state_after": "G : Type u_1\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b3 : SigmaFinite \u03bd\ninst\u271d\u00b2 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b9 : MeasurableInv G\ninst\u271d : IsMulRightInvariant \u03bc\ng : G\nthis\u271d : QuasiMeasurePreserving fun h => h * g\u207b\u00b9\nthis : QuasiMeasurePreserving (Inv.inv \u2218 (fun h => h * g\u207b\u00b9) \u2218 Inv.inv)\n\u22a2 QuasiMeasurePreserving fun h => g * h"}, {"tactic": "simp_rw [Function.comp, mul_inv_rev, inv_inv] at this", "annotated_tactic": ["simp_rw [<a>Function.comp</a>, <a>mul_inv_rev</a>, <a>inv_inv</a>] at this", [{"full_name": "Function.comp", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [52, 15], "def_end_pos": [52, 28]}, {"full_name": "mul_inv_rev", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [1050, 9], "def_end_pos": [1050, 20]}, {"full_name": "inv_inv", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [800, 9], "def_end_pos": [800, 16]}]], "state_before": "G : Type u_1\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b3 : SigmaFinite \u03bd\ninst\u271d\u00b2 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b9 : MeasurableInv G\ninst\u271d : IsMulRightInvariant \u03bc\ng : G\nthis\u271d : QuasiMeasurePreserving fun h => h * g\u207b\u00b9\nthis : QuasiMeasurePreserving (Inv.inv \u2218 (fun h => h * g\u207b\u00b9) \u2218 Inv.inv)\n\u22a2 QuasiMeasurePreserving fun h => g * h", "state_after": "G : Type u_1\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b3 : SigmaFinite \u03bd\ninst\u271d\u00b2 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b9 : MeasurableInv G\ninst\u271d : IsMulRightInvariant \u03bc\ng : G\nthis\u271d : QuasiMeasurePreserving fun h => h * g\u207b\u00b9\nthis : QuasiMeasurePreserving fun x => g * x\n\u22a2 QuasiMeasurePreserving fun h => g * h"}, {"tactic": "exact this", "annotated_tactic": ["exact this", []], "state_before": "G : Type u_1\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b3 : SigmaFinite \u03bd\ninst\u271d\u00b2 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b9 : MeasurableInv G\ninst\u271d : IsMulRightInvariant \u03bc\ng : G\nthis\u271d : QuasiMeasurePreserving fun h => h * g\u207b\u00b9\nthis : QuasiMeasurePreserving fun x => g * x\n\u22a2 QuasiMeasurePreserving fun h => g * h", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/UniformIntegrable.lean", "full_name": "MeasureTheory.uniformIntegrable_finite", "start": [761, 1], "end": [776, 61], "traced_tactics": [{"tactic": "cases nonempty_fintype \u03b9", "annotated_tactic": ["cases <a>nonempty_fintype</a> \u03b9", [{"full_name": "nonempty_fintype", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [442, 9], "def_end_pos": [442, 25]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ninst\u271d : Finite \u03b9\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\nhf : \u2200 (i : \u03b9), Mem\u2112p (f i) p\n\u22a2 UniformIntegrable f p \u03bc", "state_after": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ninst\u271d : Finite \u03b9\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\nhf : \u2200 (i : \u03b9), Mem\u2112p (f i) p\nval\u271d : Fintype \u03b9\n\u22a2 UniformIntegrable f p \u03bc"}, {"tactic": "refine' \u27e8fun n => (hf n).1, unifIntegrable_finite \u03bc hp_one hp_top hf, _\u27e9", "annotated_tactic": ["refine' \u27e8fun n => (hf n).1, <a>unifIntegrable_finite</a> \u03bc hp_one hp_top hf, _\u27e9", [{"full_name": "MeasureTheory.unifIntegrable_finite", "def_path": "Mathlib/MeasureTheory/Function/UniformIntegrable.lean", "def_pos": [450, 9], "def_end_pos": [450, 30]}]], "state_before": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ninst\u271d : Finite \u03b9\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\nhf : \u2200 (i : \u03b9), Mem\u2112p (f i) p\nval\u271d : Fintype \u03b9\n\u22a2 UniformIntegrable f p \u03bc", "state_after": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ninst\u271d : Finite \u03b9\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\nhf : \u2200 (i : \u03b9), Mem\u2112p (f i) p\nval\u271d : Fintype \u03b9\n\u22a2 \u2203 C, \u2200 (i : \u03b9), snorm (f i) p \u03bc \u2264 \u2191C"}, {"tactic": "by_cases h\u03b9 : Nonempty \u03b9", "annotated_tactic": ["by_cases h\u03b9 : <a>Nonempty</a> \u03b9", [{"full_name": "Nonempty", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [686, 17], "def_end_pos": [686, 25]}]], "state_before": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ninst\u271d : Finite \u03b9\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\nhf : \u2200 (i : \u03b9), Mem\u2112p (f i) p\nval\u271d : Fintype \u03b9\n\u22a2 \u2203 C, \u2200 (i : \u03b9), snorm (f i) p \u03bc \u2264 \u2191C", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ninst\u271d : Finite \u03b9\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\nhf : \u2200 (i : \u03b9), Mem\u2112p (f i) p\nval\u271d : Fintype \u03b9\nh\u03b9 : Nonempty \u03b9\n\u22a2 \u2203 C, \u2200 (i : \u03b9), snorm (f i) p \u03bc \u2264 \u2191C\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ninst\u271d : Finite \u03b9\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\nhf : \u2200 (i : \u03b9), Mem\u2112p (f i) p\nval\u271d : Fintype \u03b9\nh\u03b9 : \u00acNonempty \u03b9\n\u22a2 \u2203 C, \u2200 (i : \u03b9), snorm (f i) p \u03bc \u2264 \u2191C"}, {"tactic": "choose _ hf using hf", "annotated_tactic": ["choose _ hf using hf", []], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ninst\u271d : Finite \u03b9\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\nhf : \u2200 (i : \u03b9), Mem\u2112p (f i) p\nval\u271d : Fintype \u03b9\nh\u03b9 : Nonempty \u03b9\n\u22a2 \u2203 C, \u2200 (i : \u03b9), snorm (f i) p \u03bc \u2264 \u2191C", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ninst\u271d : Finite \u03b9\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\nval\u271d : Fintype \u03b9\nh\u03b9 : Nonempty \u03b9\nh\u271d : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\nhf : \u2200 (i : \u03b9), snorm (f i) p \u03bc < \u22a4\n\u22a2 \u2203 C, \u2200 (i : \u03b9), snorm (f i) p \u03bc \u2264 \u2191C"}, {"tactic": "set C := (Finset.univ.image fun i : \u03b9 => snorm (f i) p \u03bc).max'\n  \u27e8snorm (f h\u03b9.some) p \u03bc, Finset.mem_image.2 \u27e8h\u03b9.some, Finset.mem_univ _, rfl\u27e9\u27e9", "annotated_tactic": ["set C := (Finset.univ.image fun i : \u03b9 => <a>snorm</a> (f i) p \u03bc).<a>max'</a>\n      \u27e8<a>snorm</a> (f h\u03b9.some) p \u03bc, <a>Finset.mem_image</a>.2 \u27e8h\u03b9.some, <a>Finset.mem_univ</a> _, <a>rfl</a>\u27e9\u27e9", [{"full_name": "MeasureTheory.snorm", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [84, 5], "def_end_pos": [84, 10]}, {"full_name": "Finset.max'", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [1409, 5], "def_end_pos": [1409, 9]}, {"full_name": "MeasureTheory.snorm", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [84, 5], "def_end_pos": [84, 10]}, {"full_name": "Finset.mem_image", "def_path": "Mathlib/Data/Finset/Image.lean", "def_pos": [330, 9], "def_end_pos": [330, 18]}, {"full_name": "Finset.mem_univ", "def_path": "Mathlib/Data/Fintype/Basic.lean", "def_pos": [72, 9], "def_end_pos": [72, 17]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ninst\u271d : Finite \u03b9\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\nval\u271d : Fintype \u03b9\nh\u03b9 : Nonempty \u03b9\nh\u271d : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\nhf : \u2200 (i : \u03b9), snorm (f i) p \u03bc < \u22a4\n\u22a2 \u2203 C, \u2200 (i : \u03b9), snorm (f i) p \u03bc \u2264 \u2191C", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ninst\u271d : Finite \u03b9\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\nval\u271d : Fintype \u03b9\nh\u03b9 : Nonempty \u03b9\nh\u271d : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\nhf : \u2200 (i : \u03b9), snorm (f i) p \u03bc < \u22a4\nC : \u211d\u22650\u221e :=\n  Finset.max' (Finset.image (fun i => snorm (f i) p \u03bc) Finset.univ)\n    (_ : \u2203 x, x \u2208 Finset.image (fun i => snorm (f i) p \u03bc) Finset.univ)\n\u22a2 \u2203 C, \u2200 (i : \u03b9), snorm (f i) p \u03bc \u2264 \u2191C"}, {"tactic": "refine' \u27e8C.toNNReal, fun i => _\u27e9", "annotated_tactic": ["refine' \u27e8C.toNNReal, fun i => _\u27e9", []], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ninst\u271d : Finite \u03b9\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\nval\u271d : Fintype \u03b9\nh\u03b9 : Nonempty \u03b9\nh\u271d : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\nhf : \u2200 (i : \u03b9), snorm (f i) p \u03bc < \u22a4\nC : \u211d\u22650\u221e :=\n  Finset.max' (Finset.image (fun i => snorm (f i) p \u03bc) Finset.univ)\n    (_ : \u2203 x, x \u2208 Finset.image (fun i => snorm (f i) p \u03bc) Finset.univ)\n\u22a2 \u2203 C, \u2200 (i : \u03b9), snorm (f i) p \u03bc \u2264 \u2191C", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ninst\u271d : Finite \u03b9\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\nval\u271d : Fintype \u03b9\nh\u03b9 : Nonempty \u03b9\nh\u271d : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\nhf : \u2200 (i : \u03b9), snorm (f i) p \u03bc < \u22a4\nC : \u211d\u22650\u221e :=\n  Finset.max' (Finset.image (fun i => snorm (f i) p \u03bc) Finset.univ)\n    (_ : \u2203 x, x \u2208 Finset.image (fun i => snorm (f i) p \u03bc) Finset.univ)\ni : \u03b9\n\u22a2 snorm (f i) p \u03bc \u2264 \u2191(ENNReal.toNNReal C)"}, {"tactic": "rw [ENNReal.coe_toNNReal]", "annotated_tactic": ["rw [<a>ENNReal.coe_toNNReal</a>]", [{"full_name": "ENNReal.coe_toNNReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [180, 9], "def_end_pos": [180, 21]}]], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ninst\u271d : Finite \u03b9\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\nval\u271d : Fintype \u03b9\nh\u03b9 : Nonempty \u03b9\nh\u271d : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\nhf : \u2200 (i : \u03b9), snorm (f i) p \u03bc < \u22a4\nC : \u211d\u22650\u221e :=\n  Finset.max' (Finset.image (fun i => snorm (f i) p \u03bc) Finset.univ)\n    (_ : \u2203 x, x \u2208 Finset.image (fun i => snorm (f i) p \u03bc) Finset.univ)\ni : \u03b9\n\u22a2 snorm (f i) p \u03bc \u2264 \u2191(ENNReal.toNNReal C)", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ninst\u271d : Finite \u03b9\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\nval\u271d : Fintype \u03b9\nh\u03b9 : Nonempty \u03b9\nh\u271d : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\nhf : \u2200 (i : \u03b9), snorm (f i) p \u03bc < \u22a4\nC : \u211d\u22650\u221e :=\n  Finset.max' (Finset.image (fun i => snorm (f i) p \u03bc) Finset.univ)\n    (_ : \u2203 x, x \u2208 Finset.image (fun i => snorm (f i) p \u03bc) Finset.univ)\ni : \u03b9\n\u22a2 snorm (f i) p \u03bc \u2264 C\n\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ninst\u271d : Finite \u03b9\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\nval\u271d : Fintype \u03b9\nh\u03b9 : Nonempty \u03b9\nh\u271d : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\nhf : \u2200 (i : \u03b9), snorm (f i) p \u03bc < \u22a4\nC : \u211d\u22650\u221e :=\n  Finset.max' (Finset.image (fun i => snorm (f i) p \u03bc) Finset.univ)\n    (_ : \u2203 x, x \u2208 Finset.image (fun i => snorm (f i) p \u03bc) Finset.univ)\ni : \u03b9\n\u22a2 C \u2260 \u22a4"}, {"tactic": "exact Finset.le_max' (\u03b1 := \u211d\u22650\u221e) _ _ (Finset.mem_image.2 \u27e8i, Finset.mem_univ _, rfl\u27e9)", "annotated_tactic": ["exact <a>Finset.le_max'</a> (\u03b1 := \u211d\u22650\u221e) _ _ (<a>Finset.mem_image</a>.2 \u27e8i, <a>Finset.mem_univ</a> _, <a>rfl</a>\u27e9)", [{"full_name": "Finset.le_max'", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [1445, 9], "def_end_pos": [1445, 16]}, {"full_name": "Finset.mem_image", "def_path": "Mathlib/Data/Finset/Image.lean", "def_pos": [330, 9], "def_end_pos": [330, 18]}, {"full_name": "Finset.mem_univ", "def_path": "Mathlib/Data/Fintype/Basic.lean", "def_pos": [72, 9], "def_end_pos": [72, 17]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ninst\u271d : Finite \u03b9\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\nval\u271d : Fintype \u03b9\nh\u03b9 : Nonempty \u03b9\nh\u271d : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\nhf : \u2200 (i : \u03b9), snorm (f i) p \u03bc < \u22a4\nC : \u211d\u22650\u221e :=\n  Finset.max' (Finset.image (fun i => snorm (f i) p \u03bc) Finset.univ)\n    (_ : \u2203 x, x \u2208 Finset.image (fun i => snorm (f i) p \u03bc) Finset.univ)\ni : \u03b9\n\u22a2 snorm (f i) p \u03bc \u2264 C", "state_after": "no goals"}, {"tactic": "refine' ne_of_lt ((Finset.max'_lt_iff _ _).2 fun y hy => _)", "annotated_tactic": ["refine' <a>ne_of_lt</a> ((<a>Finset.max'_lt_iff</a> _ _).2 fun y hy => _)", [{"full_name": "ne_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [101, 9], "def_end_pos": [101, 17]}, {"full_name": "Finset.max'_lt_iff", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [1463, 9], "def_end_pos": [1463, 20]}]], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ninst\u271d : Finite \u03b9\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\nval\u271d : Fintype \u03b9\nh\u03b9 : Nonempty \u03b9\nh\u271d : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\nhf : \u2200 (i : \u03b9), snorm (f i) p \u03bc < \u22a4\nC : \u211d\u22650\u221e :=\n  Finset.max' (Finset.image (fun i => snorm (f i) p \u03bc) Finset.univ)\n    (_ : \u2203 x, x \u2208 Finset.image (fun i => snorm (f i) p \u03bc) Finset.univ)\ni : \u03b9\n\u22a2 C \u2260 \u22a4", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ninst\u271d : Finite \u03b9\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\nval\u271d : Fintype \u03b9\nh\u03b9 : Nonempty \u03b9\nh\u271d : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\nhf : \u2200 (i : \u03b9), snorm (f i) p \u03bc < \u22a4\nC : \u211d\u22650\u221e :=\n  Finset.max' (Finset.image (fun i => snorm (f i) p \u03bc) Finset.univ)\n    (_ : \u2203 x, x \u2208 Finset.image (fun i => snorm (f i) p \u03bc) Finset.univ)\ni : \u03b9\ny : \u211d\u22650\u221e\nhy : y \u2208 Finset.image (fun i => snorm (f i) p \u03bc) Finset.univ\n\u22a2 y < \u22a4"}, {"tactic": "rw [Finset.mem_image] at hy", "annotated_tactic": ["rw [<a>Finset.mem_image</a>] at hy", [{"full_name": "Finset.mem_image", "def_path": "Mathlib/Data/Finset/Image.lean", "def_pos": [330, 9], "def_end_pos": [330, 18]}]], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ninst\u271d : Finite \u03b9\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\nval\u271d : Fintype \u03b9\nh\u03b9 : Nonempty \u03b9\nh\u271d : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\nhf : \u2200 (i : \u03b9), snorm (f i) p \u03bc < \u22a4\nC : \u211d\u22650\u221e :=\n  Finset.max' (Finset.image (fun i => snorm (f i) p \u03bc) Finset.univ)\n    (_ : \u2203 x, x \u2208 Finset.image (fun i => snorm (f i) p \u03bc) Finset.univ)\ni : \u03b9\ny : \u211d\u22650\u221e\nhy : y \u2208 Finset.image (fun i => snorm (f i) p \u03bc) Finset.univ\n\u22a2 y < \u22a4", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ninst\u271d : Finite \u03b9\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\nval\u271d : Fintype \u03b9\nh\u03b9 : Nonempty \u03b9\nh\u271d : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\nhf : \u2200 (i : \u03b9), snorm (f i) p \u03bc < \u22a4\nC : \u211d\u22650\u221e :=\n  Finset.max' (Finset.image (fun i => snorm (f i) p \u03bc) Finset.univ)\n    (_ : \u2203 x, x \u2208 Finset.image (fun i => snorm (f i) p \u03bc) Finset.univ)\ni : \u03b9\ny : \u211d\u22650\u221e\nhy : \u2203 a, a \u2208 Finset.univ \u2227 snorm (f a) p \u03bc = y\n\u22a2 y < \u22a4"}, {"tactic": "obtain \u27e8i, -, rfl\u27e9 := hy", "annotated_tactic": ["obtain \u27e8i, -, rfl\u27e9 := hy", []], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ninst\u271d : Finite \u03b9\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\nval\u271d : Fintype \u03b9\nh\u03b9 : Nonempty \u03b9\nh\u271d : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\nhf : \u2200 (i : \u03b9), snorm (f i) p \u03bc < \u22a4\nC : \u211d\u22650\u221e :=\n  Finset.max' (Finset.image (fun i => snorm (f i) p \u03bc) Finset.univ)\n    (_ : \u2203 x, x \u2208 Finset.image (fun i => snorm (f i) p \u03bc) Finset.univ)\ni : \u03b9\ny : \u211d\u22650\u221e\nhy : \u2203 a, a \u2208 Finset.univ \u2227 snorm (f a) p \u03bc = y\n\u22a2 y < \u22a4", "state_after": "case pos.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ninst\u271d : Finite \u03b9\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\nval\u271d : Fintype \u03b9\nh\u03b9 : Nonempty \u03b9\nh\u271d : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\nhf : \u2200 (i : \u03b9), snorm (f i) p \u03bc < \u22a4\nC : \u211d\u22650\u221e :=\n  Finset.max' (Finset.image (fun i => snorm (f i) p \u03bc) Finset.univ)\n    (_ : \u2203 x, x \u2208 Finset.image (fun i => snorm (f i) p \u03bc) Finset.univ)\ni\u271d i : \u03b9\n\u22a2 snorm (f i) p \u03bc < \u22a4"}, {"tactic": "exact hf i", "annotated_tactic": ["exact hf i", []], "state_before": "case pos.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ninst\u271d : Finite \u03b9\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\nval\u271d : Fintype \u03b9\nh\u03b9 : Nonempty \u03b9\nh\u271d : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc\nhf : \u2200 (i : \u03b9), snorm (f i) p \u03bc < \u22a4\nC : \u211d\u22650\u221e :=\n  Finset.max' (Finset.image (fun i => snorm (f i) p \u03bc) Finset.univ)\n    (_ : \u2203 x, x \u2208 Finset.image (fun i => snorm (f i) p \u03bc) Finset.univ)\ni\u271d i : \u03b9\n\u22a2 snorm (f i) p \u03bc < \u22a4", "state_after": "no goals"}, {"tactic": "exact \u27e80, fun i => False.elim <| h\u03b9 <| Nonempty.intro i\u27e9", "annotated_tactic": ["exact \u27e80, fun i => <a>False.elim</a> <| h\u03b9 <| <a>Nonempty.intro</a> i\u27e9", [{"full_name": "False.elim", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [223, 21], "def_end_pos": [223, 31]}, {"full_name": "Nonempty.intro", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [688, 5], "def_end_pos": [688, 10]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ninst\u271d : Finite \u03b9\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\nhf : \u2200 (i : \u03b9), Mem\u2112p (f i) p\nval\u271d : Fintype \u03b9\nh\u03b9 : \u00acNonempty \u03b9\n\u22a2 \u2203 C, \u2200 (i : \u03b9), snorm (f i) p \u03bc \u2264 \u2191C", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean", "full_name": "MeasureTheory.integrableOn_Ioi_of_intervalIntegral_norm_tendsto", "start": [583, 1], "end": [587, 68], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/Equiv.lean", "full_name": "MvPolynomial.eval_eq_eval_mv_eval'", "start": [389, 1], "end": [410, 32], "traced_tactics": [{"tactic": "let \u03c6 : (MvPolynomial (Fin n) R)[X] \u2192\u2090[R] R[X] :=\n  { Polynomial.mapRingHom (eval s) with\n    commutes' := fun r => by\n      convert Polynomial.map_C (eval s)\n      exact (eval_C _).symm }", "annotated_tactic": ["let \u03c6 : (<a>MvPolynomial</a> (<a>Fin</a> n) R)[X] \u2192\u2090[R] R[X] :=\n    { <a>Polynomial.mapRingHom</a> (<a>eval</a> s) with\n      commutes' := fun r => by\n        convert <a>Polynomial.map_C</a> (<a>eval</a> s)\n        exact (<a>eval_C</a> _).<a>symm</a> }", [{"full_name": "MvPolynomial", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [89, 5], "def_end_pos": [89, 17]}, {"full_name": "Fin", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1745, 11], "def_end_pos": [1745, 14]}, {"full_name": "Polynomial.mapRingHom", "def_path": "Mathlib/Data/Polynomial/Eval.lean", "def_pos": [766, 5], "def_end_pos": [766, 15]}, {"full_name": "MvPolynomial.eval", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [1145, 5], "def_end_pos": [1145, 9]}, {"full_name": "Polynomial.map_C", "def_path": "Mathlib/Data/Polynomial/Eval.lean", "def_pos": [718, 9], "def_end_pos": [718, 14]}, {"full_name": "MvPolynomial.eval", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [1145, 5], "def_end_pos": [1145, 9]}, {"full_name": "MvPolynomial.eval_C", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [1164, 9], "def_end_pos": [1164, 15]}, {"full_name": "Eq.symm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [310, 9], "def_end_pos": [310, 16]}]], "state_before": "R : Type u\nS\u2081 : Type v\nS\u2082 : Type w\nS\u2083 : Type x\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\ns\u271d : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\nn : \u2115\ns : Fin n \u2192 R\ny : R\nf : MvPolynomial (Fin (n + 1)) R\n\u22a2 \u2191(eval (Fin.cons y s)) f = Polynomial.eval y (Polynomial.map (eval s) (\u2191(finSuccEquiv R n) f))", "state_after": "R : Type u\nS\u2081 : Type v\nS\u2082 : Type w\nS\u2083 : Type x\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\ns\u271d : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\nn : \u2115\ns : Fin n \u2192 R\ny : R\nf : MvPolynomial (Fin (n + 1)) R\n\u03c6 : (MvPolynomial (Fin n) R)[X] \u2192\u2090[R] R[X] :=\n  let src := mapRingHom (eval s);\n  {\n    toRingHom :=\n      { toMonoidHom := \u2191src, map_zero' := (_ : OneHom.toFun (\u2191\u2191src) 0 = 0),\n        map_add' :=\n          (_ :\n            \u2200 (x y : (MvPolynomial (Fin n) R)[X]),\n              OneHom.toFun (\u2191\u2191src) (x + y) = OneHom.toFun (\u2191\u2191src) x + OneHom.toFun (\u2191\u2191src) y) },\n    commutes' :=\n      (_ :\n        \u2200 (r : R),\n          OneHom.toFun\n              (\u2191\u2191{ toMonoidHom := \u2191src, map_zero' := (_ : OneHom.toFun (\u2191\u2191src) 0 = 0),\n                    map_add' :=\n                      (_ :\n                        \u2200 (x y : (MvPolynomial (Fin n) R)[X]),\n                          OneHom.toFun (\u2191\u2191src) (x + y) = OneHom.toFun (\u2191\u2191src) x + OneHom.toFun (\u2191\u2191src) y) })\n              (\u2191(algebraMap R (MvPolynomial (Fin n) R)[X]) r) =\n            \u2191(algebraMap R R[X]) r) }\n\u22a2 \u2191(eval (Fin.cons y s)) f = Polynomial.eval y (Polynomial.map (eval s) (\u2191(finSuccEquiv R n) f))"}, {"tactic": "show\n  aeval (Fin.cons y s : Fin (n + 1) \u2192 R) f =\n    (Polynomial.aeval y).comp (\u03c6.comp (finSuccEquiv R n).toAlgHom) f", "annotated_tactic": ["show\n    <a>aeval</a> (<a>Fin.cons</a> y s : <a>Fin</a> (n + 1) \u2192 R) f =\n      (<a>Polynomial.aeval</a> y).<a>comp</a> (\u03c6.comp (<a>finSuccEquiv</a> R n).<a>toAlgHom</a>) f", [{"full_name": "MvPolynomial.aeval", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [1461, 5], "def_end_pos": [1461, 10]}, {"full_name": "Fin.cons", "def_path": "Mathlib/Data/Fin/Tuple/Basic.lean", "def_pos": [66, 5], "def_end_pos": [66, 9]}, {"full_name": "Fin", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1745, 11], "def_end_pos": [1745, 14]}, {"full_name": "Polynomial.aeval", "def_path": "Mathlib/Data/Polynomial/AlgebraMap.lean", "def_pos": [166, 5], "def_end_pos": [166, 10]}, {"full_name": "AlgHom.comp", "def_path": "Mathlib/Algebra/Algebra/Hom.lean", "def_pos": [327, 5], "def_end_pos": [327, 9]}, {"full_name": "MvPolynomial.finSuccEquiv", "def_path": "Mathlib/Data/MvPolynomial/Equiv.lean", "def_pos": [316, 5], "def_end_pos": [316, 17]}, {"full_name": "AlgEquiv.toAlgHom", "def_path": "Mathlib/Algebra/Algebra/Equiv.lean", "def_pos": [255, 5], "def_end_pos": [255, 13]}]], "state_before": "R : Type u\nS\u2081 : Type v\nS\u2082 : Type w\nS\u2083 : Type x\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\ns\u271d : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\nn : \u2115\ns : Fin n \u2192 R\ny : R\nf : MvPolynomial (Fin (n + 1)) R\n\u03c6 : (MvPolynomial (Fin n) R)[X] \u2192\u2090[R] R[X] :=\n  let src := mapRingHom (eval s);\n  {\n    toRingHom :=\n      { toMonoidHom := \u2191src, map_zero' := (_ : OneHom.toFun (\u2191\u2191src) 0 = 0),\n        map_add' :=\n          (_ :\n            \u2200 (x y : (MvPolynomial (Fin n) R)[X]),\n              OneHom.toFun (\u2191\u2191src) (x + y) = OneHom.toFun (\u2191\u2191src) x + OneHom.toFun (\u2191\u2191src) y) },\n    commutes' :=\n      (_ :\n        \u2200 (r : R),\n          OneHom.toFun\n              (\u2191\u2191{ toMonoidHom := \u2191src, map_zero' := (_ : OneHom.toFun (\u2191\u2191src) 0 = 0),\n                    map_add' :=\n                      (_ :\n                        \u2200 (x y : (MvPolynomial (Fin n) R)[X]),\n                          OneHom.toFun (\u2191\u2191src) (x + y) = OneHom.toFun (\u2191\u2191src) x + OneHom.toFun (\u2191\u2191src) y) })\n              (\u2191(algebraMap R (MvPolynomial (Fin n) R)[X]) r) =\n            \u2191(algebraMap R R[X]) r) }\n\u22a2 \u2191(eval (Fin.cons y s)) f = Polynomial.eval y (Polynomial.map (eval s) (\u2191(finSuccEquiv R n) f))", "state_after": "R : Type u\nS\u2081 : Type v\nS\u2082 : Type w\nS\u2083 : Type x\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\ns\u271d : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\nn : \u2115\ns : Fin n \u2192 R\ny : R\nf : MvPolynomial (Fin (n + 1)) R\n\u03c6 : (MvPolynomial (Fin n) R)[X] \u2192\u2090[R] R[X] :=\n  let src := mapRingHom (eval s);\n  {\n    toRingHom :=\n      { toMonoidHom := \u2191src, map_zero' := (_ : OneHom.toFun (\u2191\u2191src) 0 = 0),\n        map_add' :=\n          (_ :\n            \u2200 (x y : (MvPolynomial (Fin n) R)[X]),\n              OneHom.toFun (\u2191\u2191src) (x + y) = OneHom.toFun (\u2191\u2191src) x + OneHom.toFun (\u2191\u2191src) y) },\n    commutes' :=\n      (_ :\n        \u2200 (r : R),\n          OneHom.toFun\n              (\u2191\u2191{ toMonoidHom := \u2191src, map_zero' := (_ : OneHom.toFun (\u2191\u2191src) 0 = 0),\n                    map_add' :=\n                      (_ :\n                        \u2200 (x y : (MvPolynomial (Fin n) R)[X]),\n                          OneHom.toFun (\u2191\u2191src) (x + y) = OneHom.toFun (\u2191\u2191src) x + OneHom.toFun (\u2191\u2191src) y) })\n              (\u2191(algebraMap R (MvPolynomial (Fin n) R)[X]) r) =\n            \u2191(algebraMap R R[X]) r) }\n\u22a2 \u2191(aeval (Fin.cons y s)) f = \u2191(AlgHom.comp (Polynomial.aeval y) (AlgHom.comp \u03c6 \u2191(finSuccEquiv R n))) f"}, {"tactic": "congr 2", "annotated_tactic": ["congr 2", []], "state_before": "R : Type u\nS\u2081 : Type v\nS\u2082 : Type w\nS\u2083 : Type x\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\ns\u271d : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\nn : \u2115\ns : Fin n \u2192 R\ny : R\nf : MvPolynomial (Fin (n + 1)) R\n\u03c6 : (MvPolynomial (Fin n) R)[X] \u2192\u2090[R] R[X] :=\n  let src := mapRingHom (eval s);\n  {\n    toRingHom :=\n      { toMonoidHom := \u2191src, map_zero' := (_ : OneHom.toFun (\u2191\u2191src) 0 = 0),\n        map_add' :=\n          (_ :\n            \u2200 (x y : (MvPolynomial (Fin n) R)[X]),\n              OneHom.toFun (\u2191\u2191src) (x + y) = OneHom.toFun (\u2191\u2191src) x + OneHom.toFun (\u2191\u2191src) y) },\n    commutes' :=\n      (_ :\n        \u2200 (r : R),\n          OneHom.toFun\n              (\u2191\u2191{ toMonoidHom := \u2191src, map_zero' := (_ : OneHom.toFun (\u2191\u2191src) 0 = 0),\n                    map_add' :=\n                      (_ :\n                        \u2200 (x y : (MvPolynomial (Fin n) R)[X]),\n                          OneHom.toFun (\u2191\u2191src) (x + y) = OneHom.toFun (\u2191\u2191src) x + OneHom.toFun (\u2191\u2191src) y) })\n              (\u2191(algebraMap R (MvPolynomial (Fin n) R)[X]) r) =\n            \u2191(algebraMap R R[X]) r) }\n\u22a2 \u2191(aeval (Fin.cons y s)) f = \u2191(AlgHom.comp (Polynomial.aeval y) (AlgHom.comp \u03c6 \u2191(finSuccEquiv R n))) f", "state_after": "case e_a\nR : Type u\nS\u2081 : Type v\nS\u2082 : Type w\nS\u2083 : Type x\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\ns\u271d : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\nn : \u2115\ns : Fin n \u2192 R\ny : R\nf : MvPolynomial (Fin (n + 1)) R\n\u03c6 : (MvPolynomial (Fin n) R)[X] \u2192\u2090[R] R[X] :=\n  let src := mapRingHom (eval s);\n  {\n    toRingHom :=\n      { toMonoidHom := \u2191src, map_zero' := (_ : OneHom.toFun (\u2191\u2191src) 0 = 0),\n        map_add' :=\n          (_ :\n            \u2200 (x y : (MvPolynomial (Fin n) R)[X]),\n              OneHom.toFun (\u2191\u2191src) (x + y) = OneHom.toFun (\u2191\u2191src) x + OneHom.toFun (\u2191\u2191src) y) },\n    commutes' :=\n      (_ :\n        \u2200 (r : R),\n          OneHom.toFun\n              (\u2191\u2191{ toMonoidHom := \u2191src, map_zero' := (_ : OneHom.toFun (\u2191\u2191src) 0 = 0),\n                    map_add' :=\n                      (_ :\n                        \u2200 (x y : (MvPolynomial (Fin n) R)[X]),\n                          OneHom.toFun (\u2191\u2191src) (x + y) = OneHom.toFun (\u2191\u2191src) x + OneHom.toFun (\u2191\u2191src) y) })\n              (\u2191(algebraMap R (MvPolynomial (Fin n) R)[X]) r) =\n            \u2191(algebraMap R R[X]) r) }\n\u22a2 aeval (Fin.cons y s) = AlgHom.comp (Polynomial.aeval y) (AlgHom.comp \u03c6 \u2191(finSuccEquiv R n))"}, {"tactic": "apply MvPolynomial.algHom_ext", "annotated_tactic": ["apply <a>MvPolynomial.algHom_ext</a>", [{"full_name": "MvPolynomial.algHom_ext", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [497, 9], "def_end_pos": [497, 19]}]], "state_before": "case e_a\nR : Type u\nS\u2081 : Type v\nS\u2082 : Type w\nS\u2083 : Type x\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\ns\u271d : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\nn : \u2115\ns : Fin n \u2192 R\ny : R\nf : MvPolynomial (Fin (n + 1)) R\n\u03c6 : (MvPolynomial (Fin n) R)[X] \u2192\u2090[R] R[X] :=\n  let src := mapRingHom (eval s);\n  {\n    toRingHom :=\n      { toMonoidHom := \u2191src, map_zero' := (_ : OneHom.toFun (\u2191\u2191src) 0 = 0),\n        map_add' :=\n          (_ :\n            \u2200 (x y : (MvPolynomial (Fin n) R)[X]),\n              OneHom.toFun (\u2191\u2191src) (x + y) = OneHom.toFun (\u2191\u2191src) x + OneHom.toFun (\u2191\u2191src) y) },\n    commutes' :=\n      (_ :\n        \u2200 (r : R),\n          OneHom.toFun\n              (\u2191\u2191{ toMonoidHom := \u2191src, map_zero' := (_ : OneHom.toFun (\u2191\u2191src) 0 = 0),\n                    map_add' :=\n                      (_ :\n                        \u2200 (x y : (MvPolynomial (Fin n) R)[X]),\n                          OneHom.toFun (\u2191\u2191src) (x + y) = OneHom.toFun (\u2191\u2191src) x + OneHom.toFun (\u2191\u2191src) y) })\n              (\u2191(algebraMap R (MvPolynomial (Fin n) R)[X]) r) =\n            \u2191(algebraMap R R[X]) r) }\n\u22a2 aeval (Fin.cons y s) = AlgHom.comp (Polynomial.aeval y) (AlgHom.comp \u03c6 \u2191(finSuccEquiv R n))", "state_after": "case e_a.hf\nR : Type u\nS\u2081 : Type v\nS\u2082 : Type w\nS\u2083 : Type x\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\ns\u271d : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\nn : \u2115\ns : Fin n \u2192 R\ny : R\nf : MvPolynomial (Fin (n + 1)) R\n\u03c6 : (MvPolynomial (Fin n) R)[X] \u2192\u2090[R] R[X] :=\n  let src := mapRingHom (eval s);\n  {\n    toRingHom :=\n      { toMonoidHom := \u2191src, map_zero' := (_ : OneHom.toFun (\u2191\u2191src) 0 = 0),\n        map_add' :=\n          (_ :\n            \u2200 (x y : (MvPolynomial (Fin n) R)[X]),\n              OneHom.toFun (\u2191\u2191src) (x + y) = OneHom.toFun (\u2191\u2191src) x + OneHom.toFun (\u2191\u2191src) y) },\n    commutes' :=\n      (_ :\n        \u2200 (r : R),\n          OneHom.toFun\n              (\u2191\u2191{ toMonoidHom := \u2191src, map_zero' := (_ : OneHom.toFun (\u2191\u2191src) 0 = 0),\n                    map_add' :=\n                      (_ :\n                        \u2200 (x y : (MvPolynomial (Fin n) R)[X]),\n                          OneHom.toFun (\u2191\u2191src) (x + y) = OneHom.toFun (\u2191\u2191src) x + OneHom.toFun (\u2191\u2191src) y) })\n              (\u2191(algebraMap R (MvPolynomial (Fin n) R)[X]) r) =\n            \u2191(algebraMap R R[X]) r) }\n\u22a2 \u2200 (i : Fin (n + 1)),\n    \u2191(aeval (Fin.cons y s)) (X i) = \u2191(AlgHom.comp (Polynomial.aeval y) (AlgHom.comp \u03c6 \u2191(finSuccEquiv R n))) (X i)"}, {"tactic": "rw [Fin.forall_fin_succ]", "annotated_tactic": ["rw [<a>Fin.forall_fin_succ</a>]", [{"full_name": "Fin.forall_fin_succ", "def_path": "lake-packages/std/Std/Data/Fin/Lemmas.lean", "def_pos": [627, 9], "def_end_pos": [627, 24]}]], "state_before": "case e_a.hf\nR : Type u\nS\u2081 : Type v\nS\u2082 : Type w\nS\u2083 : Type x\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\ns\u271d : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\nn : \u2115\ns : Fin n \u2192 R\ny : R\nf : MvPolynomial (Fin (n + 1)) R\n\u03c6 : (MvPolynomial (Fin n) R)[X] \u2192\u2090[R] R[X] :=\n  let src := mapRingHom (eval s);\n  {\n    toRingHom :=\n      { toMonoidHom := \u2191src, map_zero' := (_ : OneHom.toFun (\u2191\u2191src) 0 = 0),\n        map_add' :=\n          (_ :\n            \u2200 (x y : (MvPolynomial (Fin n) R)[X]),\n              OneHom.toFun (\u2191\u2191src) (x + y) = OneHom.toFun (\u2191\u2191src) x + OneHom.toFun (\u2191\u2191src) y) },\n    commutes' :=\n      (_ :\n        \u2200 (r : R),\n          OneHom.toFun\n              (\u2191\u2191{ toMonoidHom := \u2191src, map_zero' := (_ : OneHom.toFun (\u2191\u2191src) 0 = 0),\n                    map_add' :=\n                      (_ :\n                        \u2200 (x y : (MvPolynomial (Fin n) R)[X]),\n                          OneHom.toFun (\u2191\u2191src) (x + y) = OneHom.toFun (\u2191\u2191src) x + OneHom.toFun (\u2191\u2191src) y) })\n              (\u2191(algebraMap R (MvPolynomial (Fin n) R)[X]) r) =\n            \u2191(algebraMap R R[X]) r) }\n\u22a2 \u2200 (i : Fin (n + 1)),\n    \u2191(aeval (Fin.cons y s)) (X i) = \u2191(AlgHom.comp (Polynomial.aeval y) (AlgHom.comp \u03c6 \u2191(finSuccEquiv R n))) (X i)", "state_after": "case e_a.hf\nR : Type u\nS\u2081 : Type v\nS\u2082 : Type w\nS\u2083 : Type x\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\ns\u271d : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\nn : \u2115\ns : Fin n \u2192 R\ny : R\nf : MvPolynomial (Fin (n + 1)) R\n\u03c6 : (MvPolynomial (Fin n) R)[X] \u2192\u2090[R] R[X] :=\n  let src := mapRingHom (eval s);\n  {\n    toRingHom :=\n      { toMonoidHom := \u2191src, map_zero' := (_ : OneHom.toFun (\u2191\u2191src) 0 = 0),\n        map_add' :=\n          (_ :\n            \u2200 (x y : (MvPolynomial (Fin n) R)[X]),\n              OneHom.toFun (\u2191\u2191src) (x + y) = OneHom.toFun (\u2191\u2191src) x + OneHom.toFun (\u2191\u2191src) y) },\n    commutes' :=\n      (_ :\n        \u2200 (r : R),\n          OneHom.toFun\n              (\u2191\u2191{ toMonoidHom := \u2191src, map_zero' := (_ : OneHom.toFun (\u2191\u2191src) 0 = 0),\n                    map_add' :=\n                      (_ :\n                        \u2200 (x y : (MvPolynomial (Fin n) R)[X]),\n                          OneHom.toFun (\u2191\u2191src) (x + y) = OneHom.toFun (\u2191\u2191src) x + OneHom.toFun (\u2191\u2191src) y) })\n              (\u2191(algebraMap R (MvPolynomial (Fin n) R)[X]) r) =\n            \u2191(algebraMap R R[X]) r) }\n\u22a2 \u2191(aeval (Fin.cons y s)) (X 0) = \u2191(AlgHom.comp (Polynomial.aeval y) (AlgHom.comp \u03c6 \u2191(finSuccEquiv R n))) (X 0) \u2227\n    \u2200 (i : Fin n),\n      \u2191(aeval (Fin.cons y s)) (X (Fin.succ i)) =\n        \u2191(AlgHom.comp (Polynomial.aeval y) (AlgHom.comp \u03c6 \u2191(finSuccEquiv R n))) (X (Fin.succ i))"}, {"tactic": "simp only [aeval_X, Fin.cons_zero, AlgEquiv.toAlgHom_eq_coe, AlgHom.coe_comp,\n  Polynomial.coe_aeval_eq_eval, Polynomial.map_C, AlgHom.coe_mk, RingHom.toFun_eq_coe,\n  Polynomial.coe_mapRingHom, comp_apply, finSuccEquiv_apply, eval\u2082Hom_X',\n  Fin.cases_zero, Polynomial.map_X, Polynomial.eval_X, Fin.cons_succ,\n  Fin.cases_succ, eval_X, Polynomial.eval_C,\n  RingHom.coe_mk, MonoidHom.coe_coe, AlgHom.coe_coe, implies_true, and_self,\n  RingHom.toMonoidHom_eq_coe]", "annotated_tactic": ["simp only [<a>aeval_X</a>, <a>Fin.cons_zero</a>, <a>AlgEquiv.toAlgHom_eq_coe</a>, <a>AlgHom.coe_comp</a>,\n    <a>Polynomial.coe_aeval_eq_eval</a>, <a>Polynomial.map_C</a>, <a>AlgHom.coe_mk</a>, <a>RingHom.toFun_eq_coe</a>,\n    <a>Polynomial.coe_mapRingHom</a>, <a>comp_apply</a>, <a>finSuccEquiv_apply</a>, <a>eval\u2082Hom_X'</a>,\n    <a>Fin.cases_zero</a>, <a>Polynomial.map_X</a>, <a>Polynomial.eval_X</a>, <a>Fin.cons_succ</a>,\n    <a>Fin.cases_succ</a>, <a>eval_X</a>, <a>Polynomial.eval_C</a>,\n    <a>RingHom.coe_mk</a>, <a>MonoidHom.coe_coe</a>, <a>AlgHom.coe_coe</a>, <a>implies_true</a>, <a>and_self</a>,\n    <a>RingHom.toMonoidHom_eq_coe</a>]", [{"full_name": "MvPolynomial.aeval_X", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [1474, 9], "def_end_pos": [1474, 16]}, {"full_name": "Fin.cons_zero", "def_path": "Mathlib/Data/Fin/Tuple/Basic.lean", "def_pos": [78, 9], "def_end_pos": [78, 18]}, {"full_name": "AlgEquiv.toAlgHom_eq_coe", "def_path": "Mathlib/Algebra/Algebra/Equiv.lean", "def_pos": [262, 9], "def_end_pos": [262, 24]}, {"full_name": "AlgHom.coe_comp", "def_path": "Mathlib/Algebra/Algebra/Hom.lean", "def_pos": [333, 9], "def_end_pos": [333, 17]}, {"full_name": "Polynomial.coe_aeval_eq_eval", "def_path": "Mathlib/Data/Polynomial/AlgebraMap.lean", "def_pos": [296, 9], "def_end_pos": [296, 26]}, {"full_name": "Polynomial.map_C", "def_path": "Mathlib/Data/Polynomial/Eval.lean", "def_pos": [718, 9], "def_end_pos": [718, 14]}, {"full_name": "AlgHom.coe_mk", "def_path": "Mathlib/Algebra/Algebra/Hom.lean", "def_pos": [157, 9], "def_end_pos": [157, 15]}, {"full_name": "RingHom.toFun_eq_coe", "def_path": "Mathlib/Algebra/Hom/Ring/Defs.lean", "def_pos": [444, 9], "def_end_pos": [444, 21]}, {"full_name": "Polynomial.coe_mapRingHom", "def_path": "Mathlib/Data/Polynomial/Eval.lean", "def_pos": [775, 9], "def_end_pos": [775, 23]}, {"full_name": "Function.comp_apply", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [33, 17], "def_end_pos": [33, 36]}, {"full_name": "MvPolynomial.finSuccEquiv_apply", "def_path": "Mathlib/Data/MvPolynomial/Equiv.lean", "def_pos": [332, 9], "def_end_pos": [332, 27]}, {"full_name": "MvPolynomial.eval\u2082Hom_X'", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [1068, 9], "def_end_pos": [1068, 20]}, {"full_name": "Fin.cases_zero", "def_path": "lake-packages/std/Std/Data/Fin/Lemmas.lean", "def_pos": [617, 17], "def_end_pos": [617, 27]}, {"full_name": "Polynomial.map_X", "def_path": "Mathlib/Data/Polynomial/Eval.lean", "def_pos": [723, 9], "def_end_pos": [723, 14]}, {"full_name": "Polynomial.eval_X", "def_path": "Mathlib/Data/Polynomial/Eval.lean", "def_pos": [373, 9], "def_end_pos": [373, 15]}, {"full_name": "Fin.cons_succ", "def_path": "Mathlib/Data/Fin/Tuple/Basic.lean", "def_pos": [74, 9], "def_end_pos": [74, 18]}, {"full_name": "Fin.cases_succ", "def_path": "lake-packages/std/Std/Data/Fin/Lemmas.lean", "def_pos": [620, 17], "def_end_pos": [620, 27]}, {"full_name": "MvPolynomial.eval_X", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [1169, 9], "def_end_pos": [1169, 15]}, {"full_name": "Polynomial.eval_C", "def_path": "Mathlib/Data/Polynomial/Eval.lean", "def_pos": [359, 9], "def_end_pos": [359, 15]}, {"full_name": "RingHom.coe_mk", "def_path": "Mathlib/Algebra/Hom/Ring/Defs.lean", "def_pos": [449, 9], "def_end_pos": [449, 15]}, {"full_name": "MonoidHom.coe_coe", "def_path": "Mathlib/Algebra/Hom/Group/Defs.lean", "def_pos": [395, 9], "def_end_pos": [395, 26]}, {"full_name": "AlgHom.coe_coe", "def_path": "Mathlib/Algebra/Algebra/Hom.lean", "def_pos": [123, 19], "def_end_pos": [123, 26]}, {"full_name": "implies_true", "def_path": "lake-packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [98, 17], "def_end_pos": [98, 29]}, {"full_name": "and_self", "def_path": "lake-packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [82, 17], "def_end_pos": [82, 25]}, {"full_name": "RingHom.toMonoidHom_eq_coe", "def_path": "Mathlib/Algebra/Hom/Ring/Defs.lean", "def_pos": [468, 9], "def_end_pos": [468, 27]}]], "state_before": "case e_a.hf\nR : Type u\nS\u2081 : Type v\nS\u2082 : Type w\nS\u2083 : Type x\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\ns\u271d : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\nn : \u2115\ns : Fin n \u2192 R\ny : R\nf : MvPolynomial (Fin (n + 1)) R\n\u03c6 : (MvPolynomial (Fin n) R)[X] \u2192\u2090[R] R[X] :=\n  let src := mapRingHom (eval s);\n  {\n    toRingHom :=\n      { toMonoidHom := \u2191src, map_zero' := (_ : OneHom.toFun (\u2191\u2191src) 0 = 0),\n        map_add' :=\n          (_ :\n            \u2200 (x y : (MvPolynomial (Fin n) R)[X]),\n              OneHom.toFun (\u2191\u2191src) (x + y) = OneHom.toFun (\u2191\u2191src) x + OneHom.toFun (\u2191\u2191src) y) },\n    commutes' :=\n      (_ :\n        \u2200 (r : R),\n          OneHom.toFun\n              (\u2191\u2191{ toMonoidHom := \u2191src, map_zero' := (_ : OneHom.toFun (\u2191\u2191src) 0 = 0),\n                    map_add' :=\n                      (_ :\n                        \u2200 (x y : (MvPolynomial (Fin n) R)[X]),\n                          OneHom.toFun (\u2191\u2191src) (x + y) = OneHom.toFun (\u2191\u2191src) x + OneHom.toFun (\u2191\u2191src) y) })\n              (\u2191(algebraMap R (MvPolynomial (Fin n) R)[X]) r) =\n            \u2191(algebraMap R R[X]) r) }\n\u22a2 \u2191(aeval (Fin.cons y s)) (X 0) = \u2191(AlgHom.comp (Polynomial.aeval y) (AlgHom.comp \u03c6 \u2191(finSuccEquiv R n))) (X 0) \u2227\n    \u2200 (i : Fin n),\n      \u2191(aeval (Fin.cons y s)) (X (Fin.succ i)) =\n        \u2191(AlgHom.comp (Polynomial.aeval y) (AlgHom.comp \u03c6 \u2191(finSuccEquiv R n))) (X (Fin.succ i))", "state_after": "no goals"}, {"tactic": "convert Polynomial.map_C (eval s)", "annotated_tactic": ["convert <a>Polynomial.map_C</a> (<a>eval</a> s)", [{"full_name": "Polynomial.map_C", "def_path": "Mathlib/Data/Polynomial/Eval.lean", "def_pos": [718, 9], "def_end_pos": [718, 14]}, {"full_name": "MvPolynomial.eval", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [1145, 5], "def_end_pos": [1145, 9]}]], "state_before": "R : Type u\nS\u2081 : Type v\nS\u2082 : Type w\nS\u2083 : Type x\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\ns\u271d : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\nn : \u2115\ns : Fin n \u2192 R\ny : R\nf : MvPolynomial (Fin (n + 1)) R\nsrc\u271d : (MvPolynomial (Fin n) R)[X] \u2192+* R[X] := mapRingHom (eval s)\nr : R\n\u22a2 OneHom.toFun\n      (\u2191\u2191{ toMonoidHom := \u2191src\u271d, map_zero' := (_ : OneHom.toFun (\u2191\u2191src\u271d) 0 = 0),\n            map_add' :=\n              (_ :\n                \u2200 (x y : (MvPolynomial (Fin n) R)[X]),\n                  OneHom.toFun (\u2191\u2191src\u271d) (x + y) = OneHom.toFun (\u2191\u2191src\u271d) x + OneHom.toFun (\u2191\u2191src\u271d) y) })\n      (\u2191(algebraMap R (MvPolynomial (Fin n) R)[X]) r) =\n    \u2191(algebraMap R R[X]) r", "state_after": "case h.e'_3.h.e'_6\nR : Type u\nS\u2081 : Type v\nS\u2082 : Type w\nS\u2083 : Type x\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\ns\u271d : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\nn : \u2115\ns : Fin n \u2192 R\ny : R\nf : MvPolynomial (Fin (n + 1)) R\nsrc\u271d : (MvPolynomial (Fin n) R)[X] \u2192+* R[X] := mapRingHom (eval s)\nr : R\n\u22a2 r = \u2191(eval s) (\u2191(algebraMap R (MvPolynomial (Fin n) R)) r)"}, {"tactic": "exact (eval_C _).symm", "annotated_tactic": ["exact (<a>eval_C</a> _).<a>symm</a>", [{"full_name": "MvPolynomial.eval_C", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [1164, 9], "def_end_pos": [1164, 15]}, {"full_name": "Eq.symm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [310, 9], "def_end_pos": [310, 16]}]], "state_before": "case h.e'_3.h.e'_6\nR : Type u\nS\u2081 : Type v\nS\u2082 : Type w\nS\u2083 : Type x\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\ns\u271d : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\nn : \u2115\ns : Fin n \u2192 R\ny : R\nf : MvPolynomial (Fin (n + 1)) R\nsrc\u271d : (MvPolynomial (Fin n) R)[X] \u2192+* R[X] := mapRingHom (eval s)\nr : R\n\u22a2 r = \u2191(eval s) (\u2191(algebraMap R (MvPolynomial (Fin n) R)) r)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Group/Prod.lean", "full_name": "MeasureTheory.ae_measure_preimage_mul_right_lt_top", "start": [276, 1], "end": [288, 38], "traced_tactics": [{"tactic": "refine' ae_of_forall_measure_lt_top_ae_restrict' \u03bd.inv _ _", "annotated_tactic": ["refine' <a>ae_of_forall_measure_lt_top_ae_restrict'</a> \u03bd.inv _ _", [{"full_name": "MeasureTheory.ae_of_forall_measure_lt_top_ae_restrict'", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3783, 9], "def_end_pos": [3783, 49]}]], "state_before": "G : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nsm : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\n\u22a2 \u2200\u1d50 (x : G) \u2202\u03bc, \u2191\u2191\u03bd ((fun y => y * x) \u207b\u00b9' s) < \u22a4", "state_after": "G : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nsm : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\n\u22a2 \u2200 (s_1 : Set G),\n    MeasurableSet s_1 \u2192\n      \u2191\u2191\u03bc s_1 < \u22a4 \u2192 \u2191\u2191(Measure.inv \u03bd) s_1 < \u22a4 \u2192 \u2200\u1d50 (x : G) \u2202Measure.restrict \u03bc s_1, \u2191\u2191\u03bd ((fun y => y * x) \u207b\u00b9' s) < \u22a4"}, {"tactic": "intro A hA _ h3A", "annotated_tactic": ["intro A hA _ h3A", []], "state_before": "G : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nsm : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\n\u22a2 \u2200 (s_1 : Set G),\n    MeasurableSet s_1 \u2192\n      \u2191\u2191\u03bc s_1 < \u22a4 \u2192 \u2191\u2191(Measure.inv \u03bd) s_1 < \u22a4 \u2192 \u2200\u1d50 (x : G) \u2202Measure.restrict \u03bc s_1, \u2191\u2191\u03bd ((fun y => y * x) \u207b\u00b9' s) < \u22a4", "state_after": "G : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nsm : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nA : Set G\nhA : MeasurableSet A\na\u271d : \u2191\u2191\u03bc A < \u22a4\nh3A : \u2191\u2191(Measure.inv \u03bd) A < \u22a4\n\u22a2 \u2200\u1d50 (x : G) \u2202Measure.restrict \u03bc A, \u2191\u2191\u03bd ((fun y => y * x) \u207b\u00b9' s) < \u22a4"}, {"tactic": "simp only [\u03bd.inv_apply] at h3A", "annotated_tactic": ["simp only [\u03bd.inv_apply] at h3A", []], "state_before": "G : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nsm : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nA : Set G\nhA : MeasurableSet A\na\u271d : \u2191\u2191\u03bc A < \u22a4\nh3A : \u2191\u2191(Measure.inv \u03bd) A < \u22a4\n\u22a2 \u2200\u1d50 (x : G) \u2202Measure.restrict \u03bc A, \u2191\u2191\u03bd ((fun y => y * x) \u207b\u00b9' s) < \u22a4", "state_after": "G : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nsm : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nA : Set G\nhA : MeasurableSet A\na\u271d : \u2191\u2191\u03bc A < \u22a4\nh3A : \u2191\u2191\u03bd A\u207b\u00b9 < \u22a4\n\u22a2 \u2200\u1d50 (x : G) \u2202Measure.restrict \u03bc A, \u2191\u2191\u03bd ((fun y => y * x) \u207b\u00b9' s) < \u22a4"}, {"tactic": "apply ae_lt_top (measurable_measure_mul_right \u03bd sm)", "annotated_tactic": ["apply <a>ae_lt_top</a> (<a>measurable_measure_mul_right</a> \u03bd sm)", [{"full_name": "MeasureTheory.ae_lt_top", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [1522, 9], "def_end_pos": [1522, 18]}, {"full_name": "MeasureTheory.measurable_measure_mul_right", "def_path": "Mathlib/MeasureTheory/Group/Prod.lean", "def_pos": [105, 9], "def_end_pos": [105, 37]}]], "state_before": "G : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nsm : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nA : Set G\nhA : MeasurableSet A\na\u271d : \u2191\u2191\u03bc A < \u22a4\nh3A : \u2191\u2191\u03bd A\u207b\u00b9 < \u22a4\n\u22a2 \u2200\u1d50 (x : G) \u2202Measure.restrict \u03bc A, \u2191\u2191\u03bd ((fun y => y * x) \u207b\u00b9' s) < \u22a4", "state_after": "G : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nsm : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nA : Set G\nhA : MeasurableSet A\na\u271d : \u2191\u2191\u03bc A < \u22a4\nh3A : \u2191\u2191\u03bd A\u207b\u00b9 < \u22a4\n\u22a2 \u222b\u207b (x : G) in A, \u2191\u2191\u03bd ((fun y => y * x) \u207b\u00b9' s) \u2202\u03bc \u2260 \u22a4"}, {"tactic": "have h1 := measure_mul_lintegral_eq \u03bc \u03bd sm (A\u207b\u00b9.indicator 1) (measurable_one.indicator hA.inv)", "annotated_tactic": ["have h1 := <a>measure_mul_lintegral_eq</a> \u03bc \u03bd sm (A\u207b\u00b9.<a>indicator</a> 1) (measurable_one.indicator hA.inv)", [{"full_name": "MeasureTheory.measure_mul_lintegral_eq", "def_path": "Mathlib/MeasureTheory/Group/Prod.lean", "def_pos": [244, 9], "def_end_pos": [244, 33]}, {"full_name": "Set.indicator", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [46, 3], "def_end_pos": [46, 14]}]], "state_before": "G : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nsm : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nA : Set G\nhA : MeasurableSet A\na\u271d : \u2191\u2191\u03bc A < \u22a4\nh3A : \u2191\u2191\u03bd A\u207b\u00b9 < \u22a4\n\u22a2 \u222b\u207b (x : G) in A, \u2191\u2191\u03bd ((fun y => y * x) \u207b\u00b9' s) \u2202\u03bc \u2260 \u22a4", "state_after": "G : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nsm : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nA : Set G\nhA : MeasurableSet A\na\u271d : \u2191\u2191\u03bc A < \u22a4\nh3A : \u2191\u2191\u03bd A\u207b\u00b9 < \u22a4\nh1 : \u2191\u2191\u03bc s * \u222b\u207b (y : G), indicator A\u207b\u00b9 1 y \u2202\u03bd = \u222b\u207b (x : G), \u2191\u2191\u03bd ((fun z => z * x) \u207b\u00b9' s) * indicator A\u207b\u00b9 1 x\u207b\u00b9 \u2202\u03bc\n\u22a2 \u222b\u207b (x : G) in A, \u2191\u2191\u03bd ((fun y => y * x) \u207b\u00b9' s) \u2202\u03bc \u2260 \u22a4"}, {"tactic": "rw [lintegral_indicator _ hA.inv] at h1", "annotated_tactic": ["rw [<a>lintegral_indicator</a> _ hA.inv] at h1", [{"full_name": "MeasureTheory.lintegral_indicator", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [762, 9], "def_end_pos": [762, 28]}]], "state_before": "G : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nsm : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nA : Set G\nhA : MeasurableSet A\na\u271d : \u2191\u2191\u03bc A < \u22a4\nh3A : \u2191\u2191\u03bd A\u207b\u00b9 < \u22a4\nh1 : \u2191\u2191\u03bc s * \u222b\u207b (y : G), indicator A\u207b\u00b9 1 y \u2202\u03bd = \u222b\u207b (x : G), \u2191\u2191\u03bd ((fun z => z * x) \u207b\u00b9' s) * indicator A\u207b\u00b9 1 x\u207b\u00b9 \u2202\u03bc\n\u22a2 \u222b\u207b (x : G) in A, \u2191\u2191\u03bd ((fun y => y * x) \u207b\u00b9' s) \u2202\u03bc \u2260 \u22a4", "state_after": "G : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nsm : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nA : Set G\nhA : MeasurableSet A\na\u271d : \u2191\u2191\u03bc A < \u22a4\nh3A : \u2191\u2191\u03bd A\u207b\u00b9 < \u22a4\nh1 : \u2191\u2191\u03bc s * \u222b\u207b (a : G) in A\u207b\u00b9, OfNat.ofNat 1 a \u2202\u03bd = \u222b\u207b (x : G), \u2191\u2191\u03bd ((fun z => z * x) \u207b\u00b9' s) * indicator A\u207b\u00b9 1 x\u207b\u00b9 \u2202\u03bc\n\u22a2 \u222b\u207b (x : G) in A, \u2191\u2191\u03bd ((fun y => y * x) \u207b\u00b9' s) \u2202\u03bc \u2260 \u22a4"}, {"tactic": "simp_rw [Pi.one_apply, set_lintegral_one, \u2190 image_inv, indicator_image inv_injective, image_inv, \u2190\n  indicator_mul_right _ fun x => \u03bd ((fun y => y * x) \u207b\u00b9' s), Function.comp, Pi.one_apply,\n  mul_one] at h1", "annotated_tactic": ["simp_rw [<a>Pi.one_apply</a>, <a>set_lintegral_one</a>, \u2190 <a>image_inv</a>, <a>indicator_image</a> <a>inv_injective</a>, <a>image_inv</a>, \u2190\n    <a>indicator_mul_right</a> _ fun x => \u03bd ((fun y => y * x) \u207b\u00b9' s), <a>Function.comp</a>, <a>Pi.one_apply</a>,\n    <a>mul_one</a>] at h1", [{"full_name": "Pi.one_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [47, 9], "def_end_pos": [47, 18]}, {"full_name": "MeasureTheory.set_lintegral_one", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [156, 9], "def_end_pos": [156, 26]}, {"full_name": "Set.image_inv", "def_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "def_pos": [257, 9], "def_end_pos": [257, 18]}, {"full_name": "Set.indicator_image", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [253, 3], "def_end_pos": [253, 14]}, {"full_name": "inv_injective", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [251, 9], "def_end_pos": [251, 22]}, {"full_name": "Set.image_inv", "def_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "def_pos": [257, 9], "def_end_pos": [257, 18]}, {"full_name": "Set.indicator_mul_right", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [722, 9], "def_end_pos": [722, 28]}, {"full_name": "Function.comp", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [52, 15], "def_end_pos": [52, 28]}, {"full_name": "Pi.one_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [47, 9], "def_end_pos": [47, 18]}, {"full_name": "mul_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [470, 9], "def_end_pos": [470, 16]}]], "state_before": "G : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nsm : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nA : Set G\nhA : MeasurableSet A\na\u271d : \u2191\u2191\u03bc A < \u22a4\nh3A : \u2191\u2191\u03bd A\u207b\u00b9 < \u22a4\nh1 : \u2191\u2191\u03bc s * \u222b\u207b (a : G) in A\u207b\u00b9, OfNat.ofNat 1 a \u2202\u03bd = \u222b\u207b (x : G), \u2191\u2191\u03bd ((fun z => z * x) \u207b\u00b9' s) * indicator A\u207b\u00b9 1 x\u207b\u00b9 \u2202\u03bc\n\u22a2 \u222b\u207b (x : G) in A, \u2191\u2191\u03bd ((fun y => y * x) \u207b\u00b9' s) \u2202\u03bc \u2260 \u22a4", "state_after": "G : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nsm : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nA : Set G\nhA : MeasurableSet A\na\u271d : \u2191\u2191\u03bc A < \u22a4\nh3A : \u2191\u2191\u03bd A\u207b\u00b9 < \u22a4\nh1 : \u2191\u2191\u03bc s * \u2191\u2191\u03bd A\u207b\u00b9 = \u222b\u207b (x : G), indicator A (fun a => \u2191\u2191\u03bd ((fun y => y * a) \u207b\u00b9' s)) x \u2202\u03bc\n\u22a2 \u222b\u207b (x : G) in A, \u2191\u2191\u03bd ((fun y => y * x) \u207b\u00b9' s) \u2202\u03bc \u2260 \u22a4"}, {"tactic": "rw [\u2190 lintegral_indicator _ hA, \u2190 h1]", "annotated_tactic": ["rw [\u2190 <a>lintegral_indicator</a> _ hA, \u2190 h1]", [{"full_name": "MeasureTheory.lintegral_indicator", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [762, 9], "def_end_pos": [762, 28]}]], "state_before": "G : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nsm : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nA : Set G\nhA : MeasurableSet A\na\u271d : \u2191\u2191\u03bc A < \u22a4\nh3A : \u2191\u2191\u03bd A\u207b\u00b9 < \u22a4\nh1 : \u2191\u2191\u03bc s * \u2191\u2191\u03bd A\u207b\u00b9 = \u222b\u207b (x : G), indicator A (fun a => \u2191\u2191\u03bd ((fun y => y * a) \u207b\u00b9' s)) x \u2202\u03bc\n\u22a2 \u222b\u207b (x : G) in A, \u2191\u2191\u03bd ((fun y => y * x) \u207b\u00b9' s) \u2202\u03bc \u2260 \u22a4", "state_after": "G : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nsm : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nA : Set G\nhA : MeasurableSet A\na\u271d : \u2191\u2191\u03bc A < \u22a4\nh3A : \u2191\u2191\u03bd A\u207b\u00b9 < \u22a4\nh1 : \u2191\u2191\u03bc s * \u2191\u2191\u03bd A\u207b\u00b9 = \u222b\u207b (x : G), indicator A (fun a => \u2191\u2191\u03bd ((fun y => y * a) \u207b\u00b9' s)) x \u2202\u03bc\n\u22a2 \u2191\u2191\u03bc s * \u2191\u2191\u03bd A\u207b\u00b9 \u2260 \u22a4"}, {"tactic": "exact ENNReal.mul_ne_top h\u03bcs h3A.ne", "annotated_tactic": ["exact <a>ENNReal.mul_ne_top</a> h\u03bcs h3A.ne", [{"full_name": "ENNReal.mul_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [615, 9], "def_end_pos": [615, 19]}]], "state_before": "G : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nsm : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nA : Set G\nhA : MeasurableSet A\na\u271d : \u2191\u2191\u03bc A < \u22a4\nh3A : \u2191\u2191\u03bd A\u207b\u00b9 < \u22a4\nh1 : \u2191\u2191\u03bc s * \u2191\u2191\u03bd A\u207b\u00b9 = \u222b\u207b (x : G), indicator A (fun a => \u2191\u2191\u03bd ((fun y => y * a) \u207b\u00b9' s)) x \u2202\u03bc\n\u22a2 \u2191\u2191\u03bc s * \u2191\u2191\u03bd A\u207b\u00b9 \u2260 \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/CommRing.lean", "full_name": "MvPolynomial.support_sub", "start": [88, 1], "end": [90, 22], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean", "full_name": "MeasureTheory.Measure.addHaar_preimage_linearEquiv", "start": [295, 1], "end": [299, 39], "traced_tactics": [{"tactic": "have A : LinearMap.det (f : E \u2192\u2097[\u211d] E) \u2260 0 := (LinearEquiv.isUnit_det' f).ne_zero", "annotated_tactic": ["have A : <a>LinearMap.det</a> (f : E \u2192\u2097[\u211d] E) \u2260 0 := (<a>LinearEquiv.isUnit_det'</a> f).<a>ne_zero</a>", [{"full_name": "LinearMap.det", "def_path": "Mathlib/LinearAlgebra/Determinant.lean", "def_pos": [178, 27], "def_end_pos": [178, 30]}, {"full_name": "LinearEquiv.isUnit_det'", "def_path": "Mathlib/LinearAlgebra/Determinant.lean", "def_pos": [434, 9], "def_end_pos": [434, 32]}, {"full_name": "IsUnit.ne_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Units/Basic.lean", "def_pos": [52, 9], "def_end_pos": [52, 16]}]], "state_before": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf : E \u2243\u2097[\u211d] E\ns : Set E\n\u22a2 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' s) = ENNReal.ofReal |\u2191LinearMap.det \u2191(LinearEquiv.symm f)| * \u2191\u2191\u03bc s", "state_after": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf : E \u2243\u2097[\u211d] E\ns : Set E\nA : \u2191LinearMap.det \u2191f \u2260 0\n\u22a2 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' s) = ENNReal.ofReal |\u2191LinearMap.det \u2191(LinearEquiv.symm f)| * \u2191\u2191\u03bc s"}, {"tactic": "convert addHaar_preimage_linearMap \u03bc A s", "annotated_tactic": ["convert <a>addHaar_preimage_linearMap</a> \u03bc A s", [{"full_name": "MeasureTheory.Measure.addHaar_preimage_linearMap", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean", "def_pos": [273, 9], "def_end_pos": [273, 35]}]], "state_before": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf : E \u2243\u2097[\u211d] E\ns : Set E\nA : \u2191LinearMap.det \u2191f \u2260 0\n\u22a2 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' s) = ENNReal.ofReal |\u2191LinearMap.det \u2191(LinearEquiv.symm f)| * \u2191\u2191\u03bc s", "state_after": "case h.e'_3.h.e'_5.h.e'_1.h.e'_3\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf : E \u2243\u2097[\u211d] E\ns : Set E\nA : \u2191LinearMap.det \u2191f \u2260 0\n\u22a2 \u2191LinearMap.det \u2191(LinearEquiv.symm f) = (\u2191LinearMap.det \u2191f)\u207b\u00b9"}, {"tactic": "simp only [LinearEquiv.det_coe_symm]", "annotated_tactic": ["simp only [<a>LinearEquiv.det_coe_symm</a>]", [{"full_name": "LinearEquiv.det_coe_symm", "def_path": "Mathlib/LinearAlgebra/Determinant.lean", "def_pos": [440, 9], "def_end_pos": [440, 33]}]], "state_before": "case h.e'_3.h.e'_5.h.e'_1.h.e'_3\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf : E \u2243\u2097[\u211d] E\ns : Set E\nA : \u2191LinearMap.det \u2191f \u2260 0\n\u22a2 \u2191LinearMap.det \u2191(LinearEquiv.symm f) = (\u2191LinearMap.det \u2191f)\u207b\u00b9", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Stieltjes.lean", "full_name": "StieltjesFunction.measure_Ioo", "start": [396, 1], "end": [409, 52], "traced_tactics": [{"tactic": "rcases le_or_lt b a with (hab | hab)", "annotated_tactic": ["rcases <a>le_or_lt</a> b a with (hab | hab)", [{"full_name": "le_or_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [340, 9], "def_end_pos": [340, 17]}]], "state_before": "f : StieltjesFunction\na b : \u211d\n\u22a2 \u2191\u2191(StieltjesFunction.measure f) (Ioo a b) = ofReal (leftLim (\u2191f) b - \u2191f a)", "state_after": "case inl\nf : StieltjesFunction\na b : \u211d\nhab : b \u2264 a\n\u22a2 \u2191\u2191(StieltjesFunction.measure f) (Ioo a b) = ofReal (leftLim (\u2191f) b - \u2191f a)\n\ncase inr\nf : StieltjesFunction\na b : \u211d\nhab : a < b\n\u22a2 \u2191\u2191(StieltjesFunction.measure f) (Ioo a b) = ofReal (leftLim (\u2191f) b - \u2191f a)"}, {"tactic": "simp only [hab, measure_empty, Ioo_eq_empty, not_lt]", "annotated_tactic": ["simp only [hab, <a>measure_empty</a>, <a>Ioo_eq_empty</a>, <a>not_lt</a>]", [{"full_name": "MeasureTheory.measure_empty", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [185, 9], "def_end_pos": [185, 22]}, {"full_name": "Set.Ioo_eq_empty", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [373, 9], "def_end_pos": [373, 21]}, {"full_name": "not_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [368, 9], "def_end_pos": [368, 15]}]], "state_before": "case inl\nf : StieltjesFunction\na b : \u211d\nhab : b \u2264 a\n\u22a2 \u2191\u2191(StieltjesFunction.measure f) (Ioo a b) = ofReal (leftLim (\u2191f) b - \u2191f a)", "state_after": "case inl\nf : StieltjesFunction\na b : \u211d\nhab : b \u2264 a\n\u22a2 0 = ofReal (leftLim (\u2191f) b - \u2191f a)"}, {"tactic": "symm", "annotated_tactic": ["symm", []], "state_before": "case inl\nf : StieltjesFunction\na b : \u211d\nhab : b \u2264 a\n\u22a2 0 = ofReal (leftLim (\u2191f) b - \u2191f a)", "state_after": "case inl\nf : StieltjesFunction\na b : \u211d\nhab : b \u2264 a\n\u22a2 ofReal (leftLim (\u2191f) b - \u2191f a) = 0"}, {"tactic": "simp [ENNReal.ofReal_eq_zero, f.mono.leftLim_le hab]", "annotated_tactic": ["simp [<a>ENNReal.ofReal_eq_zero</a>, f.mono.leftLim_le hab]", [{"full_name": "ENNReal.ofReal_eq_zero", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2170, 9], "def_end_pos": [2170, 23]}]], "state_before": "case inl\nf : StieltjesFunction\na b : \u211d\nhab : b \u2264 a\n\u22a2 ofReal (leftLim (\u2191f) b - \u2191f a) = 0", "state_after": "no goals"}, {"tactic": "have A : Disjoint (Ioo a b) {b} := by simp", "annotated_tactic": ["have A : <a>Disjoint</a> (<a>Ioo</a> a b) {b} := by simp", [{"full_name": "Disjoint", "def_path": "Mathlib/Order/Disjoint.lean", "def_pos": [41, 5], "def_end_pos": [41, 13]}, {"full_name": "Set.Ioo", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [44, 5], "def_end_pos": [44, 8]}]], "state_before": "case inr\nf : StieltjesFunction\na b : \u211d\nhab : a < b\n\u22a2 \u2191\u2191(StieltjesFunction.measure f) (Ioo a b) = ofReal (leftLim (\u2191f) b - \u2191f a)", "state_after": "case inr\nf : StieltjesFunction\na b : \u211d\nhab : a < b\nA : Disjoint (Ioo a b) {b}\n\u22a2 \u2191\u2191(StieltjesFunction.measure f) (Ioo a b) = ofReal (leftLim (\u2191f) b - \u2191f a)"}, {"tactic": "have D : f b - f a = f b - leftLim f b + (leftLim f b - f a) := by abel", "annotated_tactic": ["have D : f b - f a = f b - <a>leftLim</a> f b + (<a>leftLim</a> f b - f a) := by abel", [{"full_name": "Function.leftLim", "def_path": "Mathlib/Topology/Algebra/Order/LeftRightLim.lean", "def_pos": [48, 19], "def_end_pos": [48, 35]}, {"full_name": "Function.leftLim", "def_path": "Mathlib/Topology/Algebra/Order/LeftRightLim.lean", "def_pos": [48, 19], "def_end_pos": [48, 35]}]], "state_before": "case inr\nf : StieltjesFunction\na b : \u211d\nhab : a < b\nA : Disjoint (Ioo a b) {b}\n\u22a2 \u2191\u2191(StieltjesFunction.measure f) (Ioo a b) = ofReal (leftLim (\u2191f) b - \u2191f a)", "state_after": "case inr\nf : StieltjesFunction\na b : \u211d\nhab : a < b\nA : Disjoint (Ioo a b) {b}\nD : \u2191f b - \u2191f a = \u2191f b - leftLim (\u2191f) b + (leftLim (\u2191f) b - \u2191f a)\n\u22a2 \u2191\u2191(StieltjesFunction.measure f) (Ioo a b) = ofReal (leftLim (\u2191f) b - \u2191f a)"}, {"tactic": "have := f.measure_Ioc a b", "annotated_tactic": ["have := f.measure_Ioc a b", []], "state_before": "case inr\nf : StieltjesFunction\na b : \u211d\nhab : a < b\nA : Disjoint (Ioo a b) {b}\nD : \u2191f b - \u2191f a = \u2191f b - leftLim (\u2191f) b + (leftLim (\u2191f) b - \u2191f a)\n\u22a2 \u2191\u2191(StieltjesFunction.measure f) (Ioo a b) = ofReal (leftLim (\u2191f) b - \u2191f a)", "state_after": "case inr\nf : StieltjesFunction\na b : \u211d\nhab : a < b\nA : Disjoint (Ioo a b) {b}\nD : \u2191f b - \u2191f a = \u2191f b - leftLim (\u2191f) b + (leftLim (\u2191f) b - \u2191f a)\nthis : \u2191\u2191(StieltjesFunction.measure f) (Ioc a b) = ofReal (\u2191f b - \u2191f a)\n\u22a2 \u2191\u2191(StieltjesFunction.measure f) (Ioo a b) = ofReal (leftLim (\u2191f) b - \u2191f a)"}, {"tactic": "simp only [\u2190 Ioo_union_Icc_eq_Ioc hab le_rfl, measure_singleton,\n  measure_union A (measurableSet_singleton b), Icc_self] at this", "annotated_tactic": ["simp only [\u2190 <a>Ioo_union_Icc_eq_Ioc</a> hab <a>le_rfl</a>, <a>measure_singleton</a>,\n      <a>measure_union</a> A (<a>measurableSet_singleton</a> b), <a>Icc_self</a>] at this", [{"full_name": "Set.Ioo_union_Icc_eq_Ioc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [1570, 9], "def_end_pos": [1570, 29]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}, {"full_name": "StieltjesFunction.measure_singleton", "def_path": "Mathlib/MeasureTheory/Measure/Stieltjes.lean", "def_pos": [358, 9], "def_end_pos": [358, 26]}, {"full_name": "MeasureTheory.measure_union", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [124, 9], "def_end_pos": [124, 22]}, {"full_name": "MeasurableSingletonClass.measurableSet_singleton", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [269, 3], "def_end_pos": [269, 26]}, {"full_name": "Set.Icc_self", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [755, 9], "def_end_pos": [755, 17]}]], "state_before": "case inr\nf : StieltjesFunction\na b : \u211d\nhab : a < b\nA : Disjoint (Ioo a b) {b}\nD : \u2191f b - \u2191f a = \u2191f b - leftLim (\u2191f) b + (leftLim (\u2191f) b - \u2191f a)\nthis : \u2191\u2191(StieltjesFunction.measure f) (Ioc a b) = ofReal (\u2191f b - \u2191f a)\n\u22a2 \u2191\u2191(StieltjesFunction.measure f) (Ioo a b) = ofReal (leftLim (\u2191f) b - \u2191f a)", "state_after": "case inr\nf : StieltjesFunction\na b : \u211d\nhab : a < b\nA : Disjoint (Ioo a b) {b}\nD : \u2191f b - \u2191f a = \u2191f b - leftLim (\u2191f) b + (leftLim (\u2191f) b - \u2191f a)\nthis : \u2191\u2191(StieltjesFunction.measure f) (Ioo a b) + ofReal (\u2191f b - leftLim (\u2191f) b) = ofReal (\u2191f b - \u2191f a)\n\u22a2 \u2191\u2191(StieltjesFunction.measure f) (Ioo a b) = ofReal (leftLim (\u2191f) b - \u2191f a)"}, {"tactic": "rw [D, ENNReal.ofReal_add, add_comm] at this", "annotated_tactic": ["rw [D, <a>ENNReal.ofReal_add</a>, <a>add_comm</a>] at this", [{"full_name": "ENNReal.ofReal_add", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2025, 9], "def_end_pos": [2025, 19]}, {"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [301, 3], "def_end_pos": [301, 14]}]], "state_before": "case inr\nf : StieltjesFunction\na b : \u211d\nhab : a < b\nA : Disjoint (Ioo a b) {b}\nD : \u2191f b - \u2191f a = \u2191f b - leftLim (\u2191f) b + (leftLim (\u2191f) b - \u2191f a)\nthis : \u2191\u2191(StieltjesFunction.measure f) (Ioo a b) + ofReal (\u2191f b - leftLim (\u2191f) b) = ofReal (\u2191f b - \u2191f a)\n\u22a2 \u2191\u2191(StieltjesFunction.measure f) (Ioo a b) = ofReal (leftLim (\u2191f) b - \u2191f a)", "state_after": "case inr\nf : StieltjesFunction\na b : \u211d\nhab : a < b\nA : Disjoint (Ioo a b) {b}\nD : \u2191f b - \u2191f a = \u2191f b - leftLim (\u2191f) b + (leftLim (\u2191f) b - \u2191f a)\nthis :\n  ofReal (\u2191f b - leftLim (\u2191f) b) + \u2191\u2191(StieltjesFunction.measure f) (Ioo a b) =\n    ofReal (\u2191f b - leftLim (\u2191f) b) + ofReal (leftLim (\u2191f) b - \u2191f a)\n\u22a2 \u2191\u2191(StieltjesFunction.measure f) (Ioo a b) = ofReal (leftLim (\u2191f) b - \u2191f a)\n\ncase inr.hp\nf : StieltjesFunction\na b : \u211d\nhab : a < b\nA : Disjoint (Ioo a b) {b}\nD : \u2191f b - \u2191f a = \u2191f b - leftLim (\u2191f) b + (leftLim (\u2191f) b - \u2191f a)\nthis :\n  \u2191\u2191(StieltjesFunction.measure f) (Ioo a b) + ofReal (\u2191f b - leftLim (\u2191f) b) =\n    ofReal (\u2191f b - leftLim (\u2191f) b + (leftLim (\u2191f) b - \u2191f a))\n\u22a2 0 \u2264 \u2191f b - leftLim (\u2191f) b\n\ncase inr.hq\nf : StieltjesFunction\na b : \u211d\nhab : a < b\nA : Disjoint (Ioo a b) {b}\nD : \u2191f b - \u2191f a = \u2191f b - leftLim (\u2191f) b + (leftLim (\u2191f) b - \u2191f a)\nthis :\n  \u2191\u2191(StieltjesFunction.measure f) (Ioo a b) + ofReal (\u2191f b - leftLim (\u2191f) b) =\n    ofReal (\u2191f b - leftLim (\u2191f) b + (leftLim (\u2191f) b - \u2191f a))\n\u22a2 0 \u2264 leftLim (\u2191f) b - \u2191f a"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "f : StieltjesFunction\na b : \u211d\nhab : a < b\n\u22a2 Disjoint (Ioo a b) {b}", "state_after": "no goals"}, {"tactic": "abel", "annotated_tactic": ["abel", []], "state_before": "f : StieltjesFunction\na b : \u211d\nhab : a < b\nA : Disjoint (Ioo a b) {b}\n\u22a2 \u2191f b - \u2191f a = \u2191f b - leftLim (\u2191f) b + (leftLim (\u2191f) b - \u2191f a)", "state_after": "no goals"}, {"tactic": "simpa only [ENNReal.add_right_inj ENNReal.ofReal_ne_top]", "annotated_tactic": ["simpa only [<a>ENNReal.add_right_inj</a> <a>ENNReal.ofReal_ne_top</a>]", [{"full_name": "ENNReal.add_right_inj", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1120, 9], "def_end_pos": [1120, 22]}, {"full_name": "ENNReal.ofReal_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [311, 17], "def_end_pos": [311, 30]}]], "state_before": "case inr\nf : StieltjesFunction\na b : \u211d\nhab : a < b\nA : Disjoint (Ioo a b) {b}\nD : \u2191f b - \u2191f a = \u2191f b - leftLim (\u2191f) b + (leftLim (\u2191f) b - \u2191f a)\nthis :\n  ofReal (\u2191f b - leftLim (\u2191f) b) + \u2191\u2191(StieltjesFunction.measure f) (Ioo a b) =\n    ofReal (\u2191f b - leftLim (\u2191f) b) + ofReal (leftLim (\u2191f) b - \u2191f a)\n\u22a2 \u2191\u2191(StieltjesFunction.measure f) (Ioo a b) = ofReal (leftLim (\u2191f) b - \u2191f a)", "state_after": "no goals"}, {"tactic": "simp only [f.mono.leftLim_le le_rfl, sub_nonneg]", "annotated_tactic": ["simp only [f.mono.leftLim_le <a>le_rfl</a>, <a>sub_nonneg</a>]", [{"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}, {"full_name": "sub_nonneg", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [720, 30], "def_end_pos": [720, 40]}]], "state_before": "case inr.hp\nf : StieltjesFunction\na b : \u211d\nhab : a < b\nA : Disjoint (Ioo a b) {b}\nD : \u2191f b - \u2191f a = \u2191f b - leftLim (\u2191f) b + (leftLim (\u2191f) b - \u2191f a)\nthis :\n  \u2191\u2191(StieltjesFunction.measure f) (Ioo a b) + ofReal (\u2191f b - leftLim (\u2191f) b) =\n    ofReal (\u2191f b - leftLim (\u2191f) b + (leftLim (\u2191f) b - \u2191f a))\n\u22a2 0 \u2264 \u2191f b - leftLim (\u2191f) b", "state_after": "no goals"}, {"tactic": "simp only [f.mono.le_leftLim hab, sub_nonneg]", "annotated_tactic": ["simp only [f.mono.le_leftLim hab, <a>sub_nonneg</a>]", [{"full_name": "sub_nonneg", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [720, 30], "def_end_pos": [720, 40]}]], "state_before": "case inr.hq\nf : StieltjesFunction\na b : \u211d\nhab : a < b\nA : Disjoint (Ioo a b) {b}\nD : \u2191f b - \u2191f a = \u2191f b - leftLim (\u2191f) b + (leftLim (\u2191f) b - \u2191f a)\nthis :\n  \u2191\u2191(StieltjesFunction.measure f) (Ioo a b) + ofReal (\u2191f b - leftLim (\u2191f) b) =\n    ofReal (\u2191f b - leftLim (\u2191f) b + (leftLim (\u2191f) b - \u2191f a))\n\u22a2 0 \u2264 leftLim (\u2191f) b - \u2191f a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/RegularExpressions.lean", "full_name": "RegularExpression.zero_def", "start": [87, 1], "end": [88, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Group/Prod.lean", "full_name": "MeasureTheory.ae_measure_preimage_mul_right_lt_top_of_ne_zero", "start": [293, 1], "end": [300, 43], "traced_tactics": [{"tactic": "refine' (ae_measure_preimage_mul_right_lt_top \u03bd \u03bd sm h3s).filter_mono _", "annotated_tactic": ["refine' (<a>ae_measure_preimage_mul_right_lt_top</a> \u03bd \u03bd sm h3s).<a>filter_mono</a> _", [{"full_name": "MeasureTheory.ae_measure_preimage_mul_right_lt_top", "def_path": "Mathlib/MeasureTheory/Group/Prod.lean", "def_pos": [276, 9], "def_end_pos": [276, 45]}, {"full_name": "Filter.Eventually.filter_mono", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1093, 9], "def_end_pos": [1093, 31]}]], "state_before": "G : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nsm : MeasurableSet s\nh2s : \u2191\u2191\u03bd s \u2260 0\nh3s : \u2191\u2191\u03bd s \u2260 \u22a4\n\u22a2 \u2200\u1d50 (x : G) \u2202\u03bc, \u2191\u2191\u03bd ((fun y => y * x) \u207b\u00b9' s) < \u22a4", "state_after": "G : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nsm : MeasurableSet s\nh2s : \u2191\u2191\u03bd s \u2260 0\nh3s : \u2191\u2191\u03bd s \u2260 \u22a4\n\u22a2 ae \u03bc \u2264 ae \u03bd"}, {"tactic": "refine' (absolutelyContinuous_of_isMulLeftInvariant \u03bc \u03bd _).ae_le", "annotated_tactic": ["refine' (<a>absolutelyContinuous_of_isMulLeftInvariant</a> \u03bc \u03bd _).<a>ae_le</a>", [{"full_name": "MeasureTheory.absolutelyContinuous_of_isMulLeftInvariant", "def_path": "Mathlib/MeasureTheory/Group/Prod.lean", "def_pos": [266, 9], "def_end_pos": [266, 51]}, {"full_name": "MeasureTheory.Measure.AbsolutelyContinuous.ae_le", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2180, 49], "def_end_pos": [2180, 75]}]], "state_before": "G : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nsm : MeasurableSet s\nh2s : \u2191\u2191\u03bd s \u2260 0\nh3s : \u2191\u2191\u03bd s \u2260 \u22a4\n\u22a2 ae \u03bc \u2264 ae \u03bd", "state_after": "G : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nsm : MeasurableSet s\nh2s : \u2191\u2191\u03bd s \u2260 0\nh3s : \u2191\u2191\u03bd s \u2260 \u22a4\n\u22a2 \u03bd \u2260 0"}, {"tactic": "refine' mt _ h2s", "annotated_tactic": ["refine' <a>mt</a> _ h2s", [{"full_name": "mt", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [516, 9], "def_end_pos": [516, 11]}]], "state_before": "G : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nsm : MeasurableSet s\nh2s : \u2191\u2191\u03bd s \u2260 0\nh3s : \u2191\u2191\u03bd s \u2260 \u22a4\n\u22a2 \u03bd \u2260 0", "state_after": "G : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nsm : MeasurableSet s\nh2s : \u2191\u2191\u03bd s \u2260 0\nh3s : \u2191\u2191\u03bd s \u2260 \u22a4\n\u22a2 \u03bd = 0 \u2192 \u2191\u2191\u03bd s = 0"}, {"tactic": "intro h\u03bd", "annotated_tactic": ["intro h\u03bd", []], "state_before": "G : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nsm : MeasurableSet s\nh2s : \u2191\u2191\u03bd s \u2260 0\nh3s : \u2191\u2191\u03bd s \u2260 \u22a4\n\u22a2 \u03bd = 0 \u2192 \u2191\u2191\u03bd s = 0", "state_after": "G : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nsm : MeasurableSet s\nh2s : \u2191\u2191\u03bd s \u2260 0\nh3s : \u2191\u2191\u03bd s \u2260 \u22a4\nh\u03bd : \u03bd = 0\n\u22a2 \u2191\u2191\u03bd s = 0"}, {"tactic": "rw [h\u03bd, Measure.coe_zero, Pi.zero_apply]", "annotated_tactic": ["rw [h\u03bd, <a>Measure.coe_zero</a>, <a>Pi.zero_apply</a>]", [{"full_name": "MeasureTheory.Measure.coe_zero", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [760, 9], "def_end_pos": [760, 17]}, {"full_name": "Pi.zero_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [46, 3], "def_end_pos": [46, 14]}]], "state_before": "G : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nsm : MeasurableSet s\nh2s : \u2191\u2191\u03bd s \u2260 0\nh3s : \u2191\u2191\u03bd s \u2260 \u22a4\nh\u03bd : \u03bd = 0\n\u22a2 \u2191\u2191\u03bd s = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/RegularExpressions.lean", "full_name": "RegularExpression.add_rmatch_iff", "start": [238, 1], "end": [244, 17], "traced_tactics": [{"tactic": "induction' x with _ _ ih generalizing P Q", "annotated_tactic": ["induction' x with _ _ ih generalizing P Q", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\nP Q : RegularExpression \u03b1\nx : List \u03b1\n\u22a2 rmatch (P + Q) x = true \u2194 rmatch P x = true \u2228 rmatch Q x = true", "state_after": "case nil\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\nP\u271d Q\u271d P Q : RegularExpression \u03b1\n\u22a2 rmatch (P + Q) [] = true \u2194 rmatch P [] = true \u2228 rmatch Q [] = true\n\ncase cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\nP\u271d Q\u271d : RegularExpression \u03b1\nhead\u271d : \u03b1\ntail\u271d : List \u03b1\nih : \u2200 (P Q : RegularExpression \u03b1), rmatch (P + Q) tail\u271d = true \u2194 rmatch P tail\u271d = true \u2228 rmatch Q tail\u271d = true\nP Q : RegularExpression \u03b1\n\u22a2 rmatch (P + Q) (head\u271d :: tail\u271d) = true \u2194 rmatch P (head\u271d :: tail\u271d) = true \u2228 rmatch Q (head\u271d :: tail\u271d) = true"}, {"tactic": "simp only [rmatch, matchEpsilon, Bool.or_coe_iff]", "annotated_tactic": ["simp only [<a>rmatch</a>, <a>matchEpsilon</a>, <a>Bool.or_coe_iff</a>]", [{"full_name": "RegularExpression.rmatch", "def_path": "Mathlib/Computability/RegularExpressions.lean", "def_pos": [210, 5], "def_end_pos": [210, 11]}, {"full_name": "RegularExpression.matchEpsilon", "def_path": "Mathlib/Computability/RegularExpressions.lean", "def_pos": [157, 5], "def_end_pos": [157, 17]}, {"full_name": "Bool.or_coe_iff", "def_path": "Mathlib/Init/Data/Bool/Lemmas.lean", "def_pos": [164, 9], "def_end_pos": [164, 19]}]], "state_before": "case nil\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\nP\u271d Q\u271d P Q : RegularExpression \u03b1\n\u22a2 rmatch (P + Q) [] = true \u2194 rmatch P [] = true \u2228 rmatch Q [] = true", "state_after": "no goals"}, {"tactic": "repeat' rw [rmatch]", "annotated_tactic": ["repeat' rw [rmatch]", []], "state_before": "case cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\nP\u271d Q\u271d : RegularExpression \u03b1\nhead\u271d : \u03b1\ntail\u271d : List \u03b1\nih : \u2200 (P Q : RegularExpression \u03b1), rmatch (P + Q) tail\u271d = true \u2194 rmatch P tail\u271d = true \u2228 rmatch Q tail\u271d = true\nP Q : RegularExpression \u03b1\n\u22a2 rmatch (P + Q) (head\u271d :: tail\u271d) = true \u2194 rmatch P (head\u271d :: tail\u271d) = true \u2228 rmatch Q (head\u271d :: tail\u271d) = true", "state_after": "case cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\nP\u271d Q\u271d : RegularExpression \u03b1\nhead\u271d : \u03b1\ntail\u271d : List \u03b1\nih : \u2200 (P Q : RegularExpression \u03b1), rmatch (P + Q) tail\u271d = true \u2194 rmatch P tail\u271d = true \u2228 rmatch Q tail\u271d = true\nP Q : RegularExpression \u03b1\n\u22a2 rmatch (deriv (P + Q) head\u271d) tail\u271d = true \u2194 rmatch (deriv P head\u271d) tail\u271d = true \u2228 rmatch (deriv Q head\u271d) tail\u271d = true"}, {"tactic": "rw [deriv_add]", "annotated_tactic": ["rw [<a>deriv_add</a>]", [{"full_name": "RegularExpression.deriv_add", "def_path": "Mathlib/Computability/RegularExpressions.lean", "def_pos": [199, 9], "def_end_pos": [199, 18]}]], "state_before": "case cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\nP\u271d Q\u271d : RegularExpression \u03b1\nhead\u271d : \u03b1\ntail\u271d : List \u03b1\nih : \u2200 (P Q : RegularExpression \u03b1), rmatch (P + Q) tail\u271d = true \u2194 rmatch P tail\u271d = true \u2228 rmatch Q tail\u271d = true\nP Q : RegularExpression \u03b1\n\u22a2 rmatch (deriv (P + Q) head\u271d) tail\u271d = true \u2194 rmatch (deriv P head\u271d) tail\u271d = true \u2228 rmatch (deriv Q head\u271d) tail\u271d = true", "state_after": "case cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\nP\u271d Q\u271d : RegularExpression \u03b1\nhead\u271d : \u03b1\ntail\u271d : List \u03b1\nih : \u2200 (P Q : RegularExpression \u03b1), rmatch (P + Q) tail\u271d = true \u2194 rmatch P tail\u271d = true \u2228 rmatch Q tail\u271d = true\nP Q : RegularExpression \u03b1\n\u22a2 rmatch (deriv P head\u271d + deriv Q head\u271d) tail\u271d = true \u2194\n    rmatch (deriv P head\u271d) tail\u271d = true \u2228 rmatch (deriv Q head\u271d) tail\u271d = true"}, {"tactic": "exact ih _ _", "annotated_tactic": ["exact ih _ _", []], "state_before": "case cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\nP\u271d Q\u271d : RegularExpression \u03b1\nhead\u271d : \u03b1\ntail\u271d : List \u03b1\nih : \u2200 (P Q : RegularExpression \u03b1), rmatch (P + Q) tail\u271d = true \u2194 rmatch P tail\u271d = true \u2228 rmatch Q tail\u271d = true\nP Q : RegularExpression \u03b1\n\u22a2 rmatch (deriv P head\u271d + deriv Q head\u271d) tail\u271d = true \u2194\n    rmatch (deriv P head\u271d) tail\u271d = true \u2228 rmatch (deriv Q head\u271d) tail\u271d = true", "state_after": "no goals"}, {"tactic": "rw [rmatch]", "annotated_tactic": ["rw [rmatch]", []], "state_before": "case cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\nP\u271d Q\u271d : RegularExpression \u03b1\nhead\u271d : \u03b1\ntail\u271d : List \u03b1\nih : \u2200 (P Q : RegularExpression \u03b1), rmatch (P + Q) tail\u271d = true \u2194 rmatch P tail\u271d = true \u2228 rmatch Q tail\u271d = true\nP Q : RegularExpression \u03b1\n\u22a2 rmatch (deriv (P + Q) head\u271d) tail\u271d = true \u2194 rmatch (deriv P head\u271d) tail\u271d = true \u2228 rmatch Q (head\u271d :: tail\u271d) = true", "state_after": "case cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\nP\u271d Q\u271d : RegularExpression \u03b1\nhead\u271d : \u03b1\ntail\u271d : List \u03b1\nih : \u2200 (P Q : RegularExpression \u03b1), rmatch (P + Q) tail\u271d = true \u2194 rmatch P tail\u271d = true \u2228 rmatch Q tail\u271d = true\nP Q : RegularExpression \u03b1\n\u22a2 rmatch (deriv (P + Q) head\u271d) tail\u271d = true \u2194 rmatch (deriv P head\u271d) tail\u271d = true \u2228 rmatch (deriv Q head\u271d) tail\u271d = true"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/Equiv.lean", "full_name": "MvPolynomial.degree_finSuccEquiv", "start": [492, 1], "end": [502, 31], "traced_tactics": [{"tactic": "have h\u2080 : \u2200 {\u03b1 \u03b2 : Type _} (f : \u03b1 \u2192 \u03b2), (fun x => x) \u2218 f = f := fun f => rfl", "annotated_tactic": ["have h\u2080 : \u2200 {\u03b1 \u03b2 : Type _} (f : \u03b1 \u2192 \u03b2), (fun x => x) \u2218 f = f := fun f => <a>rfl</a>", [{"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "R : Type u\nS\u2081 : Type v\nS\u2082 : Type w\nS\u2083 : Type x\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\nn : \u2115\nf : MvPolynomial (Fin (n + 1)) R\nh : f \u2260 0\n\u22a2 degree (\u2191(finSuccEquiv R n) f) = \u2191(degreeOf 0 f)", "state_after": "R : Type u\nS\u2081 : Type v\nS\u2082 : Type w\nS\u2083 : Type x\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\nn : \u2115\nf : MvPolynomial (Fin (n + 1)) R\nh : f \u2260 0\nh\u2080 : \u2200 {\u03b1 : Type ?u.1271073} {\u03b2 : Type ?u.1271076} (f : \u03b1 \u2192 \u03b2), (fun x => x) \u2218 f = f\n\u22a2 degree (\u2191(finSuccEquiv R n) f) = \u2191(degreeOf 0 f)"}, {"tactic": "have h\u2081 : \u2200 {\u03b1 \u03b2 : Type _} (f : \u03b1 \u2192 \u03b2), f \u2218 (fun x => x) = f := fun f => rfl", "annotated_tactic": ["have h\u2081 : \u2200 {\u03b1 \u03b2 : Type _} (f : \u03b1 \u2192 \u03b2), f \u2218 (fun x => x) = f := fun f => <a>rfl</a>", [{"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "R : Type u\nS\u2081 : Type v\nS\u2082 : Type w\nS\u2083 : Type x\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\nn : \u2115\nf : MvPolynomial (Fin (n + 1)) R\nh : f \u2260 0\nh\u2080 : \u2200 {\u03b1 : Type ?u.1271073} {\u03b2 : Type ?u.1271076} (f : \u03b1 \u2192 \u03b2), (fun x => x) \u2218 f = f\n\u22a2 degree (\u2191(finSuccEquiv R n) f) = \u2191(degreeOf 0 f)", "state_after": "R : Type u\nS\u2081 : Type v\nS\u2082 : Type w\nS\u2083 : Type x\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\nn : \u2115\nf : MvPolynomial (Fin (n + 1)) R\nh : f \u2260 0\nh\u2080 : \u2200 {\u03b1 : Type ?u.1271073} {\u03b2 : Type ?u.1271076} (f : \u03b1 \u2192 \u03b2), (fun x => x) \u2218 f = f\nh\u2081 : \u2200 {\u03b1 : Type ?u.1271138} {\u03b2 : Type ?u.1271141} (f : \u03b1 \u2192 \u03b2), (f \u2218 fun x => x) = f\n\u22a2 degree (\u2191(finSuccEquiv R n) f) = \u2191(degreeOf 0 f)"}, {"tactic": "have h\u2082 : WithBot.some = Nat.cast := rfl", "annotated_tactic": ["have h\u2082 : <a>WithBot.some</a> = <a>Nat.cast</a> := <a>rfl</a>", [{"full_name": "WithBot.some", "def_path": "Mathlib/Order/WithBot.lean", "def_pos": [42, 27], "def_end_pos": [42, 31]}, {"full_name": "Nat.cast", "def_path": "lake-packages/std/Std/Classes/Cast.lean", "def_pos": [19, 48], "def_end_pos": [19, 56]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "R : Type u\nS\u2081 : Type v\nS\u2082 : Type w\nS\u2083 : Type x\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\nn : \u2115\nf : MvPolynomial (Fin (n + 1)) R\nh : f \u2260 0\nh\u2080 : \u2200 {\u03b1 : Type ?u.1271073} {\u03b2 : Type ?u.1271076} (f : \u03b1 \u2192 \u03b2), (fun x => x) \u2218 f = f\nh\u2081 : \u2200 {\u03b1 : Type ?u.1271138} {\u03b2 : Type ?u.1271141} (f : \u03b1 \u2192 \u03b2), (f \u2218 fun x => x) = f\n\u22a2 degree (\u2191(finSuccEquiv R n) f) = \u2191(degreeOf 0 f)", "state_after": "R : Type u\nS\u2081 : Type v\nS\u2082 : Type w\nS\u2083 : Type x\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\nn : \u2115\nf : MvPolynomial (Fin (n + 1)) R\nh : f \u2260 0\nh\u2080 : \u2200 {\u03b1 : Type ?u.1271073} {\u03b2 : Type ?u.1271076} (f : \u03b1 \u2192 \u03b2), (fun x => x) \u2218 f = f\nh\u2081 : \u2200 {\u03b1 : Type ?u.1271138} {\u03b2 : Type ?u.1271141} (f : \u03b1 \u2192 \u03b2), (f \u2218 fun x => x) = f\nh\u2082 : WithBot.some = Nat.cast\n\u22a2 degree (\u2191(finSuccEquiv R n) f) = \u2191(degreeOf 0 f)"}, {"tactic": "have h' : ((finSuccEquiv R n f).support.sup fun x => x) = degreeOf 0 f := by\n  rw [degreeOf_eq_sup, finSuccEquiv_support f, Finset.sup_image, h\u2080]", "annotated_tactic": ["have h' : ((<a>finSuccEquiv</a> R n f).support.sup fun x => x) = <a>degreeOf</a> 0 f := by\n    rw [<a>degreeOf_eq_sup</a>, <a>finSuccEquiv_support</a> f, <a>Finset.sup_image</a>, h\u2080]", [{"full_name": "MvPolynomial.finSuccEquiv", "def_path": "Mathlib/Data/MvPolynomial/Equiv.lean", "def_pos": [316, 5], "def_end_pos": [316, 17]}, {"full_name": "MvPolynomial.degreeOf", "def_path": "Mathlib/Data/MvPolynomial/Variables.lean", "def_pos": [485, 5], "def_end_pos": [485, 13]}, {"full_name": "MvPolynomial.degreeOf_eq_sup", "def_path": "Mathlib/Data/MvPolynomial/Variables.lean", "def_pos": [494, 9], "def_end_pos": [494, 24]}, {"full_name": "MvPolynomial.finSuccEquiv_support", "def_path": "Mathlib/Data/MvPolynomial/Equiv.lean", "def_pos": [447, 9], "def_end_pos": [447, 29]}, {"full_name": "Finset.sup_image", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [63, 9], "def_end_pos": [63, 18]}]], "state_before": "R : Type u\nS\u2081 : Type v\nS\u2082 : Type w\nS\u2083 : Type x\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\nn : \u2115\nf : MvPolynomial (Fin (n + 1)) R\nh : f \u2260 0\nh\u2080 : \u2200 {\u03b1 : Type ?u.1271073} {\u03b2 : Type ?u.1271076} (f : \u03b1 \u2192 \u03b2), (fun x => x) \u2218 f = f\nh\u2081 : \u2200 {\u03b1 : Type ?u.1271138} {\u03b2 : Type ?u.1271141} (f : \u03b1 \u2192 \u03b2), (f \u2218 fun x => x) = f\nh\u2082 : WithBot.some = Nat.cast\n\u22a2 degree (\u2191(finSuccEquiv R n) f) = \u2191(degreeOf 0 f)", "state_after": "R : Type u\nS\u2081 : Type v\nS\u2082 : Type w\nS\u2083 : Type x\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\nn : \u2115\nf : MvPolynomial (Fin (n + 1)) R\nh : f \u2260 0\nh\u2080 : \u2200 {\u03b1 \u03b2 : Type} (f : \u03b1 \u2192 \u03b2), (fun x => x) \u2218 f = f\nh\u2081 : \u2200 {\u03b1 : Type ?u.1271138} {\u03b2 : Type ?u.1271141} (f : \u03b1 \u2192 \u03b2), (f \u2218 fun x => x) = f\nh\u2082 : WithBot.some = Nat.cast\nh' : (Finset.sup (Polynomial.support (\u2191(finSuccEquiv R n) f)) fun x => x) = degreeOf 0 f\n\u22a2 degree (\u2191(finSuccEquiv R n) f) = \u2191(degreeOf 0 f)"}, {"tactic": "rw [Polynomial.degree, \u2190 h', \u2190 h\u2082, Finset.coe_sup_of_nonempty (support_finSuccEquiv_nonempty h),\n  Finset.max_eq_sup_coe, h\u2081]", "annotated_tactic": ["rw [<a>Polynomial.degree</a>, \u2190 h', \u2190 h\u2082, <a>Finset.coe_sup_of_nonempty</a> (<a>support_finSuccEquiv_nonempty</a> h),\n    <a>Finset.max_eq_sup_coe</a>, h\u2081]", [{"full_name": "Polynomial.degree", "def_path": "Mathlib/Data/Polynomial/Degree/Definitions.lean", "def_pos": [53, 5], "def_end_pos": [53, 11]}, {"full_name": "Finset.coe_sup_of_nonempty", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [1062, 9], "def_end_pos": [1062, 28]}, {"full_name": "MvPolynomial.support_finSuccEquiv_nonempty", "def_path": "Mathlib/Data/MvPolynomial/Equiv.lean", "def_pos": [480, 9], "def_end_pos": [480, 38]}, {"full_name": "Finset.max_eq_sup_coe", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [1247, 9], "def_end_pos": [1247, 23]}]], "state_before": "R : Type u\nS\u2081 : Type v\nS\u2082 : Type w\nS\u2083 : Type x\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\nn : \u2115\nf : MvPolynomial (Fin (n + 1)) R\nh : f \u2260 0\nh\u2080 : \u2200 {\u03b1 \u03b2 : Type} (f : \u03b1 \u2192 \u03b2), (fun x => x) \u2218 f = f\nh\u2081 : \u2200 {\u03b1 : Type ?u.1271138} {\u03b2 : Type ?u.1271141} (f : \u03b1 \u2192 \u03b2), (f \u2218 fun x => x) = f\nh\u2082 : WithBot.some = Nat.cast\nh' : (Finset.sup (Polynomial.support (\u2191(finSuccEquiv R n) f)) fun x => x) = degreeOf 0 f\n\u22a2 degree (\u2191(finSuccEquiv R n) f) = \u2191(degreeOf 0 f)", "state_after": "no goals"}, {"tactic": "rw [degreeOf_eq_sup, finSuccEquiv_support f, Finset.sup_image, h\u2080]", "annotated_tactic": ["rw [<a>degreeOf_eq_sup</a>, <a>finSuccEquiv_support</a> f, <a>Finset.sup_image</a>, h\u2080]", [{"full_name": "MvPolynomial.degreeOf_eq_sup", "def_path": "Mathlib/Data/MvPolynomial/Variables.lean", "def_pos": [494, 9], "def_end_pos": [494, 24]}, {"full_name": "MvPolynomial.finSuccEquiv_support", "def_path": "Mathlib/Data/MvPolynomial/Equiv.lean", "def_pos": [447, 9], "def_end_pos": [447, 29]}, {"full_name": "Finset.sup_image", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [63, 9], "def_end_pos": [63, 18]}]], "state_before": "R : Type u\nS\u2081 : Type v\nS\u2082 : Type w\nS\u2083 : Type x\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\nn : \u2115\nf : MvPolynomial (Fin (n + 1)) R\nh : f \u2260 0\nh\u2080 : \u2200 {\u03b1 : Type ?u.1271073} {\u03b2 : Type ?u.1271076} (f : \u03b1 \u2192 \u03b2), (fun x => x) \u2218 f = f\nh\u2081 : \u2200 {\u03b1 : Type ?u.1271138} {\u03b2 : Type ?u.1271141} (f : \u03b1 \u2192 \u03b2), (f \u2218 fun x => x) = f\nh\u2082 : WithBot.some = Nat.cast\n\u22a2 (Finset.sup (Polynomial.support (\u2191(finSuccEquiv R n) f)) fun x => x) = degreeOf 0 f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Prod.lean", "full_name": "Set.univ_pi_update", "start": [841, 1], "end": [844, 62], "traced_tactics": [{"tactic": "rw [compl_eq_univ_diff, \u2190 pi_update_of_mem (mem_univ _)]", "annotated_tactic": ["rw [<a>compl_eq_univ_diff</a>, \u2190 <a>pi_update_of_mem</a> (<a>mem_univ</a> _)]", [{"full_name": "Set.compl_eq_univ_diff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1916, 9], "def_end_pos": [1916, 27]}, {"full_name": "Set.pi_update_of_mem", "def_path": "Mathlib/Data/Set/Prod.lean", "def_pos": [831, 9], "def_end_pos": [831, 25]}, {"full_name": "Set.mem_univ", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [676, 9], "def_end_pos": [676, 17]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2\u271d : \u03b9 \u2192 Type u_3\ns s\u2081 s\u2082 : Set \u03b9\nt\u271d t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\ni\u271d : \u03b9\ninst\u271d : DecidableEq \u03b9\n\u03b2 : \u03b9 \u2192 Type u_4\ni : \u03b9\nf : (j : \u03b9) \u2192 \u03b1 j\na : \u03b1 i\nt : (j : \u03b9) \u2192 \u03b1 j \u2192 Set (\u03b2 j)\n\u22a2 (pi univ fun j => t j (update f i a j)) = {x | x i \u2208 t i a} \u2229 pi {i}\u1d9c fun j => t j (f j)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "full_name": "Substring.ValidFor.dropWhile", "start": [960, 1], "end": [965, 90], "traced_tactics": [{"tactic": "simp only [Substring.dropWhile, takeWhileAux_of_valid]", "annotated_tactic": ["simp only [<a>Substring.dropWhile</a>, <a>takeWhileAux_of_valid</a>]", [{"full_name": "Substring.dropWhile", "def_path": "lake-packages/lean4/src/lean/Init/Data/String/Basic.lean", "def_pos": [650, 15], "def_end_pos": [650, 24]}, {"full_name": "String.takeWhileAux_of_valid", "def_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "def_pos": [764, 9], "def_end_pos": [764, 30]}]], "state_before": "l m r : List Char\np : Char \u2192 Bool\n\u22a2 ValidFor (l ++ List.takeWhile p m) (List.dropWhile p m) r\n    (Substring.dropWhile\n      { str := { data := l ++ m ++ r }, startPos := { byteIdx := utf8Len l },\n        stopPos := { byteIdx := utf8Len l + utf8Len m } }\n      p)", "state_after": "l m r : List Char\np : Char \u2192 Bool\n\u22a2 ValidFor (l ++ List.takeWhile p m) (List.dropWhile p m) r\n    { str := { data := l ++ m ++ r }, startPos := { byteIdx := utf8Len l + utf8Len (List.takeWhile p m) },\n      stopPos := { byteIdx := utf8Len l + utf8Len m } }"}, {"tactic": "refine' .of_eq .. <;> simp", "annotated_tactic": ["refine' .of_eq .. <;> simp", []], "state_before": "l m r : List Char\np : Char \u2192 Bool\n\u22a2 ValidFor (l ++ List.takeWhile p m) (List.dropWhile p m) r\n    { str := { data := l ++ m ++ r }, startPos := { byteIdx := utf8Len l + utf8Len (List.takeWhile p m) },\n      stopPos := { byteIdx := utf8Len l + utf8Len m } }", "state_after": "case refine'_3\nl m r : List Char\np : Char \u2192 Bool\n\u22a2 utf8Len l + utf8Len m = utf8Len l + utf8Len (List.takeWhile p m) + utf8Len (List.dropWhile p m)"}, {"tactic": "rw [Nat.add_assoc, \u2190 utf8Len_append (m.takeWhile p), List.takeWhile_append_dropWhile]", "annotated_tactic": ["rw [<a>Nat.add_assoc</a>, \u2190 <a>utf8Len_append</a> (m.takeWhile p), <a>List.takeWhile_append_dropWhile</a>]", [{"full_name": "Nat.add_assoc", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [138, 19], "def_end_pos": [138, 28]}, {"full_name": "String.utf8Len_append", "def_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "def_pos": [64, 17], "def_end_pos": [64, 31]}, {"full_name": "List.takeWhile_append_dropWhile", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [1939, 17], "def_end_pos": [1939, 43]}]], "state_before": "case refine'_3\nl m r : List Char\np : Char \u2192 Bool\n\u22a2 utf8Len l + utf8Len m = utf8Len l + utf8Len (List.takeWhile p m) + utf8Len (List.dropWhile p m)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "full_name": "isFiniteMeasure_iff_isFiniteMeasureOnCompacts_of_compactSpace", "start": [4326, 1], "end": [4331, 39], "traced_tactics": [{"tactic": "constructor <;> intros", "annotated_tactic": ["constructor <;> intros", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompactSpace \u03b1\n\u22a2 IsFiniteMeasure \u03bc \u2194 IsFiniteMeasureOnCompacts \u03bc", "state_after": "case mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompactSpace \u03b1\na\u271d : IsFiniteMeasure \u03bc\n\u22a2 IsFiniteMeasureOnCompacts \u03bc\n\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompactSpace \u03b1\na\u271d : IsFiniteMeasureOnCompacts \u03bc\n\u22a2 IsFiniteMeasure \u03bc"}, {"tactic": "infer_instance", "annotated_tactic": ["infer_instance", []], "state_before": "case mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompactSpace \u03b1\na\u271d : IsFiniteMeasure \u03bc\n\u22a2 IsFiniteMeasureOnCompacts \u03bc", "state_after": "no goals"}, {"tactic": "exact CompactSpace.isFiniteMeasure", "annotated_tactic": ["exact <a>CompactSpace.isFiniteMeasure</a>", [{"full_name": "MeasureTheory.CompactSpace.isFiniteMeasure", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3890, 9], "def_end_pos": [3890, 37]}]], "state_before": "case mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompactSpace \u03b1\na\u271d : IsFiniteMeasureOnCompacts \u03bc\n\u22a2 IsFiniteMeasure \u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Num/Lemmas.lean", "full_name": "PosNum.natSize_to_nat", "start": [579, 1], "end": [579, 94], "traced_tactics": [{"tactic": "rw [\u2190 size_eq_natSize, size_to_nat]", "annotated_tactic": ["rw [\u2190 <a>size_eq_natSize</a>, <a>size_to_nat</a>]", [{"full_name": "PosNum.size_eq_natSize", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [573, 9], "def_end_pos": [573, 24]}, {"full_name": "PosNum.size_to_nat", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [566, 9], "def_end_pos": [566, 20]}]], "state_before": "\u03b1 : Type u_1\nn : PosNum\n\u22a2 natSize n = Nat.size \u2191n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Moments.lean", "full_name": "ProbabilityTheory.iIndepFun.cgf_sum", "start": [317, 1], "end": [324, 37], "traced_tactics": [{"tactic": "simp_rw [cgf]", "annotated_tactic": ["simp_rw [<a>cgf</a>]", [{"full_name": "ProbabilityTheory.cgf", "def_path": "Mathlib/Probability/Moments.lean", "def_pos": [108, 5], "def_end_pos": [108, 8]}]], "state_before": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\np : \u2115\n\u03bc : Measure \u03a9\nt : \u211d\ninst\u271d : IsProbabilityMeasure \u03bc\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\nh_indep : iIndepFun (fun i => inferInstance) X\nh_meas : \u2200 (i : \u03b9), Measurable (X i)\ns : Finset \u03b9\nh_int : \u2200 (i : \u03b9), i \u2208 s \u2192 Integrable fun \u03c9 => rexp (t * X i \u03c9)\n\u22a2 cgf (\u2211 i in s, X i) \u03bc t = \u2211 i in s, cgf (X i) \u03bc t", "state_after": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\np : \u2115\n\u03bc : Measure \u03a9\nt : \u211d\ninst\u271d : IsProbabilityMeasure \u03bc\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\nh_indep : iIndepFun (fun i => inferInstance) X\nh_meas : \u2200 (i : \u03b9), Measurable (X i)\ns : Finset \u03b9\nh_int : \u2200 (i : \u03b9), i \u2208 s \u2192 Integrable fun \u03c9 => rexp (t * X i \u03c9)\n\u22a2 log (mgf (\u2211 i in s, X i) \u03bc t) = \u2211 x in s, log (mgf (X x) \u03bc t)"}, {"tactic": "rw [\u2190 log_prod _ _ fun j hj => ?_]", "annotated_tactic": ["rw [\u2190 <a>log_prod</a> _ _ fun j hj => ?_]", [{"full_name": "Real.log_prod", "def_path": "Mathlib/Analysis/SpecialFunctions/Log/Basic.lean", "def_pos": [380, 9], "def_end_pos": [380, 17]}]], "state_before": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\np : \u2115\n\u03bc : Measure \u03a9\nt : \u211d\ninst\u271d : IsProbabilityMeasure \u03bc\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\nh_indep : iIndepFun (fun i => inferInstance) X\nh_meas : \u2200 (i : \u03b9), Measurable (X i)\ns : Finset \u03b9\nh_int : \u2200 (i : \u03b9), i \u2208 s \u2192 Integrable fun \u03c9 => rexp (t * X i \u03c9)\n\u22a2 log (mgf (\u2211 i in s, X i) \u03bc t) = \u2211 x in s, log (mgf (X x) \u03bc t)", "state_after": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\np : \u2115\n\u03bc : Measure \u03a9\nt : \u211d\ninst\u271d : IsProbabilityMeasure \u03bc\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\nh_indep : iIndepFun (fun i => inferInstance) X\nh_meas : \u2200 (i : \u03b9), Measurable (X i)\ns : Finset \u03b9\nh_int : \u2200 (i : \u03b9), i \u2208 s \u2192 Integrable fun \u03c9 => rexp (t * X i \u03c9)\n\u22a2 log (mgf (\u2211 i in s, X i) \u03bc t) = log (\u220f i in s, mgf (X i) \u03bc t)\n\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\np : \u2115\n\u03bc : Measure \u03a9\nt : \u211d\ninst\u271d : IsProbabilityMeasure \u03bc\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\nh_indep : iIndepFun (fun i => inferInstance) X\nh_meas : \u2200 (i : \u03b9), Measurable (X i)\ns : Finset \u03b9\nh_int : \u2200 (i : \u03b9), i \u2208 s \u2192 Integrable fun \u03c9 => rexp (t * X i \u03c9)\nj : \u03b9\nhj : j \u2208 s\n\u22a2 mgf (X j) \u03bc t \u2260 0"}, {"tactic": "rw [h_indep.mgf_sum h_meas]", "annotated_tactic": ["rw [h_indep.mgf_sum h_meas]", []], "state_before": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\np : \u2115\n\u03bc : Measure \u03a9\nt : \u211d\ninst\u271d : IsProbabilityMeasure \u03bc\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\nh_indep : iIndepFun (fun i => inferInstance) X\nh_meas : \u2200 (i : \u03b9), Measurable (X i)\ns : Finset \u03b9\nh_int : \u2200 (i : \u03b9), i \u2208 s \u2192 Integrable fun \u03c9 => rexp (t * X i \u03c9)\n\u22a2 log (mgf (\u2211 i in s, X i) \u03bc t) = log (\u220f i in s, mgf (X i) \u03bc t)", "state_after": "no goals"}, {"tactic": "exact (mgf_pos (h_int j hj)).ne'", "annotated_tactic": ["exact (<a>mgf_pos</a> (h_int j hj)).<a>ne'</a>", [{"full_name": "ProbabilityTheory.mgf_pos", "def_path": "Mathlib/Probability/Moments.lean", "def_pos": [208, 9], "def_end_pos": [208, 16]}, {"full_name": "LT.lt.ne'", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [328, 9], "def_end_pos": [328, 12]}]], "state_before": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\np : \u2115\n\u03bc : Measure \u03a9\nt : \u211d\ninst\u271d : IsProbabilityMeasure \u03bc\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\nh_indep : iIndepFun (fun i => inferInstance) X\nh_meas : \u2200 (i : \u03b9), Measurable (X i)\ns : Finset \u03b9\nh_int : \u2200 (i : \u03b9), i \u2208 s \u2192 Integrable fun \u03c9 => rexp (t * X i \u03c9)\nj : \u03b9\nhj : j \u2208 s\n\u22a2 mgf (X j) \u03bc t \u2260 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "full_name": "MeasureTheory.limsup_lintegral_le", "start": [1038, 1], "end": [1054, 92], "traced_tactics": [{"tactic": "refine' (lintegral_iInf _ _ _).symm", "annotated_tactic": ["refine' (<a>lintegral_iInf</a> _ _ _).<a>symm</a>", [{"full_name": "MeasureTheory.lintegral_iInf", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [1014, 9], "def_end_pos": [1014, 23]}, {"full_name": "Eq.symm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [310, 9], "def_end_pos": [310, 16]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\ng : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : \u2200 (n : \u2115), Measurable (f n)\nh_bound : \u2200 (n : \u2115), f n \u2264\u1d50[\u03bc] g\nh_fin : \u222b\u207b (a : \u03b1), g a \u2202\u03bc \u2260 \u22a4\n\u22a2 \u2a05 n, \u222b\u207b (a : \u03b1), \u2a06 i, \u2a06 (_ : i \u2265 n), f i a \u2202\u03bc = \u222b\u207b (a : \u03b1), \u2a05 n, \u2a06 i, \u2a06 (_ : i \u2265 n), f i a \u2202\u03bc", "state_after": "case refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\ng : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : \u2200 (n : \u2115), Measurable (f n)\nh_bound : \u2200 (n : \u2115), f n \u2264\u1d50[\u03bc] g\nh_fin : \u222b\u207b (a : \u03b1), g a \u2202\u03bc \u2260 \u22a4\n\u22a2 \u2200 (n : \u2115), Measurable fun a => \u2a06 i, \u2a06 (_ : i \u2265 n), f i a\n\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\ng : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : \u2200 (n : \u2115), Measurable (f n)\nh_bound : \u2200 (n : \u2115), f n \u2264\u1d50[\u03bc] g\nh_fin : \u222b\u207b (a : \u03b1), g a \u2202\u03bc \u2260 \u22a4\n\u22a2 Antitone fun n a => \u2a06 i, \u2a06 (_ : i \u2265 n), f i a\n\ncase refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\ng : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : \u2200 (n : \u2115), Measurable (f n)\nh_bound : \u2200 (n : \u2115), f n \u2264\u1d50[\u03bc] g\nh_fin : \u222b\u207b (a : \u03b1), g a \u2202\u03bc \u2260 \u22a4\n\u22a2 \u222b\u207b (a : \u03b1), \u2a06 i, \u2a06 (_ : i \u2265 0), f i a \u2202\u03bc \u2260 \u22a4"}, {"tactic": "intro n", "annotated_tactic": ["intro n", []], "state_before": "case refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\ng : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : \u2200 (n : \u2115), Measurable (f n)\nh_bound : \u2200 (n : \u2115), f n \u2264\u1d50[\u03bc] g\nh_fin : \u222b\u207b (a : \u03b1), g a \u2202\u03bc \u2260 \u22a4\n\u22a2 \u2200 (n : \u2115), Measurable fun a => \u2a06 i, \u2a06 (_ : i \u2265 n), f i a", "state_after": "case refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\ng : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : \u2200 (n : \u2115), Measurable (f n)\nh_bound : \u2200 (n : \u2115), f n \u2264\u1d50[\u03bc] g\nh_fin : \u222b\u207b (a : \u03b1), g a \u2202\u03bc \u2260 \u22a4\nn : \u2115\n\u22a2 Measurable fun a => \u2a06 i, \u2a06 (_ : i \u2265 n), f i a"}, {"tactic": "exact measurable_biSup _ (to_countable _) (fun i _ \u21a6 hf_meas i)", "annotated_tactic": ["exact <a>measurable_biSup</a> _ (<a>to_countable</a> _) (fun i _ \u21a6 hf_meas i)", [{"full_name": "measurable_biSup", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [1412, 9], "def_end_pos": [1412, 25]}, {"full_name": "Set.to_countable", "def_path": "Mathlib/Data/Set/Countable.lean", "def_pos": [41, 9], "def_end_pos": [41, 21]}]], "state_before": "case refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\ng : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : \u2200 (n : \u2115), Measurable (f n)\nh_bound : \u2200 (n : \u2115), f n \u2264\u1d50[\u03bc] g\nh_fin : \u222b\u207b (a : \u03b1), g a \u2202\u03bc \u2260 \u22a4\nn : \u2115\n\u22a2 Measurable fun a => \u2a06 i, \u2a06 (_ : i \u2265 n), f i a", "state_after": "no goals"}, {"tactic": "intro n m hnm a", "annotated_tactic": ["intro n m hnm a", []], "state_before": "case refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\ng : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : \u2200 (n : \u2115), Measurable (f n)\nh_bound : \u2200 (n : \u2115), f n \u2264\u1d50[\u03bc] g\nh_fin : \u222b\u207b (a : \u03b1), g a \u2202\u03bc \u2260 \u22a4\n\u22a2 Antitone fun n a => \u2a06 i, \u2a06 (_ : i \u2265 n), f i a", "state_after": "case refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\ng : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : \u2200 (n : \u2115), Measurable (f n)\nh_bound : \u2200 (n : \u2115), f n \u2264\u1d50[\u03bc] g\nh_fin : \u222b\u207b (a : \u03b1), g a \u2202\u03bc \u2260 \u22a4\nn m : \u2115\nhnm : n \u2264 m\na : \u03b1\n\u22a2 (fun n a => \u2a06 i, \u2a06 (_ : i \u2265 n), f i a) m a \u2264 (fun n a => \u2a06 i, \u2a06 (_ : i \u2265 n), f i a) n a"}, {"tactic": "exact iSup_le_iSup_of_subset fun i hi => le_trans hnm hi", "annotated_tactic": ["exact <a>iSup_le_iSup_of_subset</a> fun i hi => <a>le_trans</a> hnm hi", [{"full_name": "iSup_le_iSup_of_subset", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [1502, 9], "def_end_pos": [1502, 31]}, {"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}]], "state_before": "case refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\ng : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : \u2200 (n : \u2115), Measurable (f n)\nh_bound : \u2200 (n : \u2115), f n \u2264\u1d50[\u03bc] g\nh_fin : \u222b\u207b (a : \u03b1), g a \u2202\u03bc \u2260 \u22a4\nn m : \u2115\nhnm : n \u2264 m\na : \u03b1\n\u22a2 (fun n a => \u2a06 i, \u2a06 (_ : i \u2265 n), f i a) m a \u2264 (fun n a => \u2a06 i, \u2a06 (_ : i \u2265 n), f i a) n a", "state_after": "no goals"}, {"tactic": "refine' ne_top_of_le_ne_top h_fin (lintegral_mono_ae _)", "annotated_tactic": ["refine' <a>ne_top_of_le_ne_top</a> h_fin (<a>lintegral_mono_ae</a> _)", [{"full_name": "ne_top_of_le_ne_top", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [194, 9], "def_end_pos": [194, 28]}, {"full_name": "MeasureTheory.lintegral_mono_ae", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [265, 9], "def_end_pos": [265, 26]}]], "state_before": "case refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\ng : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : \u2200 (n : \u2115), Measurable (f n)\nh_bound : \u2200 (n : \u2115), f n \u2264\u1d50[\u03bc] g\nh_fin : \u222b\u207b (a : \u03b1), g a \u2202\u03bc \u2260 \u22a4\n\u22a2 \u222b\u207b (a : \u03b1), \u2a06 i, \u2a06 (_ : i \u2265 0), f i a \u2202\u03bc \u2260 \u22a4", "state_after": "case refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\ng : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : \u2200 (n : \u2115), Measurable (f n)\nh_bound : \u2200 (n : \u2115), f n \u2264\u1d50[\u03bc] g\nh_fin : \u222b\u207b (a : \u03b1), g a \u2202\u03bc \u2260 \u22a4\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2a06 i, \u2a06 (_ : i \u2265 0), f i a \u2264 g a"}, {"tactic": "refine' (ae_all_iff.2 h_bound).mono fun n hn => _", "annotated_tactic": ["refine' (<a>ae_all_iff</a>.2 h_bound).<a>mono</a> fun n hn => _", [{"full_name": "MeasureTheory.ae_all_iff", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [422, 9], "def_end_pos": [422, 19]}, {"full_name": "Filter.Eventually.mono", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1140, 9], "def_end_pos": [1140, 24]}]], "state_before": "case refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\ng : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : \u2200 (n : \u2115), Measurable (f n)\nh_bound : \u2200 (n : \u2115), f n \u2264\u1d50[\u03bc] g\nh_fin : \u222b\u207b (a : \u03b1), g a \u2202\u03bc \u2260 \u22a4\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2a06 i, \u2a06 (_ : i \u2265 0), f i a \u2264 g a", "state_after": "case refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\ng : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : \u2200 (n : \u2115), Measurable (f n)\nh_bound : \u2200 (n : \u2115), f n \u2264\u1d50[\u03bc] g\nh_fin : \u222b\u207b (a : \u03b1), g a \u2202\u03bc \u2260 \u22a4\nn : \u03b1\nhn : \u2200 (i : \u2115), f i n \u2264 g n\n\u22a2 \u2a06 i, \u2a06 (_ : i \u2265 0), f i n \u2264 g n"}, {"tactic": "exact iSup_le fun i => iSup_le fun _ => hn i", "annotated_tactic": ["exact <a>iSup_le</a> fun i => <a>iSup_le</a> fun _ => hn i", [{"full_name": "iSup_le", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [875, 9], "def_end_pos": [875, 16]}, {"full_name": "iSup_le", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [875, 9], "def_end_pos": [875, 16]}]], "state_before": "case refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\ng : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : \u2200 (n : \u2115), Measurable (f n)\nh_bound : \u2200 (n : \u2115), f n \u2264\u1d50[\u03bc] g\nh_fin : \u222b\u207b (a : \u03b1), g a \u2202\u03bc \u2260 \u22a4\nn : \u03b1\nhn : \u2200 (i : \u2115), f i n \u2264 g n\n\u22a2 \u2a06 i, \u2a06 (_ : i \u2265 0), f i n \u2264 g n", "state_after": "no goals"}, {"tactic": "simp only [limsup_eq_iInf_iSup_of_nat]", "annotated_tactic": ["simp only [<a>limsup_eq_iInf_iSup_of_nat</a>]", [{"full_name": "Filter.limsup_eq_iInf_iSup_of_nat", "def_path": "Mathlib/Order/LiminfLimsup.lean", "def_pos": [796, 9], "def_end_pos": [796, 35]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\ng : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : \u2200 (n : \u2115), Measurable (f n)\nh_bound : \u2200 (n : \u2115), f n \u2264\u1d50[\u03bc] g\nh_fin : \u222b\u207b (a : \u03b1), g a \u2202\u03bc \u2260 \u22a4\n\u22a2 \u222b\u207b (a : \u03b1), \u2a05 n, \u2a06 i, \u2a06 (_ : i \u2265 n), f i a \u2202\u03bc = \u222b\u207b (a : \u03b1), limsup (fun n => f n a) atTop \u2202\u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/ZMod/Basic.lean", "full_name": "ZMod.ne_neg_self", "start": [847, 1], "end": [848, 68], "traced_tactics": [{"tactic": "rwa [Ne, eq_neg_iff_add_eq_zero, add_self_eq_zero_iff_eq_zero hn]", "annotated_tactic": ["rwa [<a>Ne</a>, <a>eq_neg_iff_add_eq_zero</a>, <a>add_self_eq_zero_iff_eq_zero</a> hn]", [{"full_name": "Ne", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [560, 18], "def_end_pos": [560, 20]}, {"full_name": "eq_neg_iff_add_eq_zero", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [666, 3], "def_end_pos": [666, 14]}, {"full_name": "ZMod.add_self_eq_zero_iff_eq_zero", "def_path": "Mathlib/Data/ZMod/Basic.lean", "def_pos": [842, 9], "def_end_pos": [842, 37]}]], "state_before": "n : \u2115\nhn : Odd n\na : ZMod n\nha : a \u2260 0\n\u22a2 a \u2260 -a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Part.lean", "full_name": "Part.getOrElse_none", "start": [280, 1], "end": [281, 43], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/WithDensity.lean", "full_name": "MeasureTheory.ae_withDensity_iff", "start": [223, 1], "end": [228, 81], "traced_tactics": [{"tactic": "rw [ae_iff, ae_iff, withDensity_apply_eq_zero hf, iff_iff_eq]", "annotated_tactic": ["rw [<a>ae_iff</a>, <a>ae_iff</a>, <a>withDensity_apply_eq_zero</a> hf, <a>iff_iff_eq</a>]", [{"full_name": "MeasureTheory.ae_iff", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [388, 9], "def_end_pos": [388, 15]}, {"full_name": "MeasureTheory.ae_iff", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [388, 9], "def_end_pos": [388, 15]}, {"full_name": "MeasureTheory.withDensity_apply_eq_zero", "def_path": "Mathlib/MeasureTheory/Measure/WithDensity.lean", "def_pos": [189, 9], "def_end_pos": [189, 34]}, {"full_name": "iff_iff_eq", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [51, 9], "def_end_pos": [51, 19]}]], "state_before": "\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u03b1 \u2192 Prop\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u22a2 (\u2200\u1d50 (x : \u03b1) \u2202withDensity \u03bc f, p x) \u2194 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2260 0 \u2192 p x", "state_after": "\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u03b1 \u2192 Prop\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u22a2 (\u2191\u2191\u03bc ({x | f x \u2260 0} \u2229 {a | \u00acp a}) = 0) = (\u2191\u2191\u03bc {a | \u00ac(f a \u2260 0 \u2192 p a)} = 0)"}, {"tactic": "congr", "annotated_tactic": ["congr", []], "state_before": "\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u03b1 \u2192 Prop\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u22a2 (\u2191\u2191\u03bc ({x | f x \u2260 0} \u2229 {a | \u00acp a}) = 0) = (\u2191\u2191\u03bc {a | \u00ac(f a \u2260 0 \u2192 p a)} = 0)", "state_after": "case e_a.e_a\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u03b1 \u2192 Prop\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u22a2 {x | f x \u2260 0} \u2229 {a | \u00acp a} = {a | \u00ac(f a \u2260 0 \u2192 p a)}"}, {"tactic": "ext x", "annotated_tactic": ["ext x", []], "state_before": "case e_a.e_a\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u03b1 \u2192 Prop\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u22a2 {x | f x \u2260 0} \u2229 {a | \u00acp a} = {a | \u00ac(f a \u2260 0 \u2192 p a)}", "state_after": "case e_a.e_a.h\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u03b1 \u2192 Prop\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nx : \u03b1\n\u22a2 x \u2208 {x | f x \u2260 0} \u2229 {a | \u00acp a} \u2194 x \u2208 {a | \u00ac(f a \u2260 0 \u2192 p a)}"}, {"tactic": "simp only [exists_prop, mem_inter_iff, iff_self_iff, mem_setOf_eq, not_forall]", "annotated_tactic": ["simp only [<a>exists_prop</a>, <a>mem_inter_iff</a>, <a>iff_self_iff</a>, <a>mem_setOf_eq</a>, <a>not_forall</a>]", [{"full_name": "exists_prop", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [485, 17], "def_end_pos": [485, 28]}, {"full_name": "Set.mem_inter_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [909, 9], "def_end_pos": [909, 22]}, {"full_name": "iff_self_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [207, 9], "def_end_pos": [207, 21]}, {"full_name": "Set.mem_setOf_eq", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [256, 29], "def_end_pos": [256, 41]}, {"full_name": "Classical.not_forall", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [686, 9], "def_end_pos": [686, 19]}]], "state_before": "case e_a.e_a.h\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u03b1 \u2192 Prop\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nx : \u03b1\n\u22a2 x \u2208 {x | f x \u2260 0} \u2229 {a | \u00acp a} \u2194 x \u2208 {a | \u00ac(f a \u2260 0 \u2192 p a)}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "full_name": "Int.add_lt_add", "start": [803, 11], "end": [804, 70], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "full_name": "MeasureTheory.L1.SimpleFunc.setToL1S_zero_left'", "start": [699, 1], "end": [701, 70], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "full_name": "Real.borel_eq_generateFrom_Ici_rat", "start": [1889, 1], "end": [1895, 76], "traced_tactics": [{"tactic": "rw [borel_eq_generateFrom_Iio_rat, iUnion_singleton_eq_range, iUnion_singleton_eq_range]", "annotated_tactic": ["rw [<a>borel_eq_generateFrom_Iio_rat</a>, <a>iUnion_singleton_eq_range</a>, <a>iUnion_singleton_eq_range</a>]", [{"full_name": "Real.borel_eq_generateFrom_Iio_rat", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [1857, 9], "def_end_pos": [1857, 38]}, {"full_name": "Set.iUnion_singleton_eq_range", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [1441, 9], "def_end_pos": [1441, 34]}, {"full_name": "Set.iUnion_singleton_eq_range", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [1441, 9], "def_end_pos": [1441, 34]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns t u : Set \u03b1\n\u22a2 borel \u211d = MeasurableSpace.generateFrom (\u22c3 a, {Ici \u2191a})", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns t u : Set \u03b1\n\u22a2 MeasurableSpace.generateFrom (range fun a => Iio \u2191a) = MeasurableSpace.generateFrom (range fun a => Ici \u2191a)"}, {"tactic": "refine le_antisymm (generateFrom_le ?_) (generateFrom_le ?_) <;>\nrintro _ \u27e8q, rfl\u27e9 <;>\ndsimp only <;>\n[rw [\u2190 compl_Ici]; rw [\u2190 compl_Iio]] <;>\nexact MeasurableSet.compl (GenerateMeasurable.basic _ (mem_range_self q))", "annotated_tactic": ["refine <a>le_antisymm</a> (<a>generateFrom_le</a> ?_) (<a>generateFrom_le</a> ?_) <;>\n  rintro _ \u27e8q, rfl\u27e9 <;>\n  dsimp only <;>\n  [rw [\u2190 <a>compl_Ici</a>]; rw [\u2190 <a>compl_Iio</a>]] <;>\n  exact <a>MeasurableSet.compl</a> (<a>GenerateMeasurable.basic</a> _ (<a>mem_range_self</a> q))", [{"full_name": "le_antisymm", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [188, 9], "def_end_pos": [188, 20]}, {"full_name": "MeasurableSpace.generateFrom_le", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [384, 9], "def_end_pos": [384, 24]}, {"full_name": "MeasurableSpace.generateFrom_le", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [384, 9], "def_end_pos": [384, 24]}, {"full_name": "Set.compl_Ici", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [1079, 9], "def_end_pos": [1079, 18]}, {"full_name": "Set.compl_Iio", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [1084, 9], "def_end_pos": [1084, 18]}, {"full_name": "MeasurableSet.compl", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [87, 19], "def_end_pos": [87, 38]}, {"full_name": "MeasurableSpace.GenerateMeasurable.basic", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [355, 15], "def_end_pos": [355, 20]}, {"full_name": "Set.mem_range_self", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [680, 9], "def_end_pos": [680, 23]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns t u : Set \u03b1\n\u22a2 MeasurableSpace.generateFrom (range fun a => Iio \u2191a) = MeasurableSpace.generateFrom (range fun a => Ici \u2191a)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Lattice.lean", "full_name": "Finset.ofDual_min'", "start": [1522, 1], "end": [1527, 6], "traced_tactics": [{"tactic": "rw [\u2190 WithBot.coe_eq_coe]", "annotated_tactic": ["rw [\u2190 <a>WithBot.coe_eq_coe</a>]", [{"full_name": "WithBot.coe_eq_coe", "def_path": "Mathlib/Order/WithBot.lean", "def_pos": [130, 9], "def_end_pos": [130, 19]}]], "state_before": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Type u_5\n\u03ba : Type u_6\ninst\u271d : LinearOrder \u03b1\ns\u271d : Finset \u03b1\nH : Finset.Nonempty s\u271d\nx : \u03b1\ns : Finset \u03b1\u1d52\u1d48\nhs : Finset.Nonempty s\n\u22a2 \u2191ofDual (min' s hs) = max' (image (\u2191ofDual) s) (_ : Finset.Nonempty (image (\u2191ofDual) s))", "state_after": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Type u_5\n\u03ba : Type u_6\ninst\u271d : LinearOrder \u03b1\ns\u271d : Finset \u03b1\nH : Finset.Nonempty s\u271d\nx : \u03b1\ns : Finset \u03b1\u1d52\u1d48\nhs : Finset.Nonempty s\n\u22a2 \u2191(\u2191ofDual (min' s hs)) = \u2191(max' (image (\u2191ofDual) s) (_ : Finset.Nonempty (image (\u2191ofDual) s)))"}, {"tactic": "simp only [min'_eq_inf', id_eq, ofDual_inf', Function.comp_apply, coe_sup', max'_eq_sup',\n  sup_image]", "annotated_tactic": ["simp only [<a>min'_eq_inf'</a>, <a>id_eq</a>, <a>ofDual_inf'</a>, <a>Function.comp_apply</a>, <a>coe_sup'</a>, <a>max'_eq_sup'</a>,\n    <a>sup_image</a>]", [{"full_name": "Finset.min'_eq_inf'", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [1476, 9], "def_end_pos": [1476, 21]}, {"full_name": "id_eq", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [284, 17], "def_end_pos": [284, 22]}, {"full_name": "Finset.ofDual_inf'", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [1130, 9], "def_end_pos": [1130, 20]}, {"full_name": "Function.comp_apply", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [33, 17], "def_end_pos": [33, 36]}, {"full_name": "Finset.coe_sup'", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [772, 9], "def_end_pos": [772, 17]}, {"full_name": "Finset.max'_eq_sup'", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [1472, 9], "def_end_pos": [1472, 21]}, {"full_name": "Finset.sup_image", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [63, 9], "def_end_pos": [63, 18]}]], "state_before": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Type u_5\n\u03ba : Type u_6\ninst\u271d : LinearOrder \u03b1\ns\u271d : Finset \u03b1\nH : Finset.Nonempty s\u271d\nx : \u03b1\ns : Finset \u03b1\u1d52\u1d48\nhs : Finset.Nonempty s\n\u22a2 \u2191(\u2191ofDual (min' s hs)) = \u2191(max' (image (\u2191ofDual) s) (_ : Finset.Nonempty (image (\u2191ofDual) s)))", "state_after": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Type u_5\n\u03ba : Type u_6\ninst\u271d : LinearOrder \u03b1\ns\u271d : Finset \u03b1\nH : Finset.Nonempty s\u271d\nx : \u03b1\ns : Finset \u03b1\u1d52\u1d48\nhs : Finset.Nonempty s\n\u22a2 sup s (WithBot.some \u2218 fun x => \u2191ofDual x) = sup s ((WithBot.some \u2218 fun x => x) \u2218 \u2191ofDual)"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Type u_5\n\u03ba : Type u_6\ninst\u271d : LinearOrder \u03b1\ns\u271d : Finset \u03b1\nH : Finset.Nonempty s\u271d\nx : \u03b1\ns : Finset \u03b1\u1d52\u1d48\nhs : Finset.Nonempty s\n\u22a2 sup s (WithBot.some \u2218 fun x => \u2191ofDual x) = sup s ((WithBot.some \u2218 fun x => x) \u2218 \u2191ofDual)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/Layercake.lean", "full_name": "MeasureTheory.Integrable.integral_eq_integral_meas_lt", "start": [577, 1], "end": [596, 94], "traced_tactics": [{"tactic": "have key := lintegral_eq_lintegral_meas_lt \u03bc f_nn f_intble.aemeasurable", "annotated_tactic": ["have key := <a>lintegral_eq_lintegral_meas_lt</a> \u03bc f_nn f_intble.aemeasurable", [{"full_name": "MeasureTheory.lintegral_eq_lintegral_meas_lt", "def_path": "Mathlib/MeasureTheory/Integral/Layercake.lean", "def_pos": [536, 9], "def_end_pos": [536, 39]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nf_intble : Integrable f\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\n\u22a2 \u222b (\u03c9 : \u03b1), f \u03c9 \u2202\u03bc = \u222b (t : \u211d) in Ioi 0, ENNReal.toReal (\u2191\u2191\u03bc {a | t < f a})", "state_after": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nf_intble : Integrable f\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nkey : \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9) \u2202\u03bc = \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a}\n\u22a2 \u222b (\u03c9 : \u03b1), f \u03c9 \u2202\u03bc = \u222b (t : \u211d) in Ioi 0, ENNReal.toReal (\u2191\u2191\u03bc {a | t < f a})"}, {"tactic": "have lhs_finite : \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9) \u2202\u03bc < \u221e := Integrable.lintegral_lt_top f_intble", "annotated_tactic": ["have lhs_finite : \u222b\u207b (\u03c9 : \u03b1), <a>ENNReal.ofReal</a> (f \u03c9) \u2202\u03bc < \u221e := <a>Integrable.lintegral_lt_top</a> f_intble", [{"full_name": "ENNReal.ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [172, 29], "def_end_pos": [172, 35]}, {"full_name": "MeasureTheory.Integrable.lintegral_lt_top", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [373, 9], "def_end_pos": [373, 36]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nf_intble : Integrable f\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nkey : \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9) \u2202\u03bc = \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a}\n\u22a2 \u222b (\u03c9 : \u03b1), f \u03c9 \u2202\u03bc = \u222b (t : \u211d) in Ioi 0, ENNReal.toReal (\u2191\u2191\u03bc {a | t < f a})", "state_after": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nf_intble : Integrable f\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nkey : \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9) \u2202\u03bc = \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a}\nlhs_finite : \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9) \u2202\u03bc < \u22a4\n\u22a2 \u222b (\u03c9 : \u03b1), f \u03c9 \u2202\u03bc = \u222b (t : \u211d) in Ioi 0, ENNReal.toReal (\u2191\u2191\u03bc {a | t < f a})"}, {"tactic": "have rhs_finite : \u222b\u207b (t : \u211d) in Set.Ioi 0, \u03bc {a | t < f a} < \u221e := by simp only [\u2190 key, lhs_finite]", "annotated_tactic": ["have rhs_finite : \u222b\u207b (t : \u211d) in <a>Set.Ioi</a> 0, \u03bc {a | t < f a} < \u221e := by simp only [\u2190 key, lhs_finite]", [{"full_name": "Set.Ioi", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [79, 5], "def_end_pos": [79, 8]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nf_intble : Integrable f\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nkey : \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9) \u2202\u03bc = \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a}\nlhs_finite : \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9) \u2202\u03bc < \u22a4\n\u22a2 \u222b (\u03c9 : \u03b1), f \u03c9 \u2202\u03bc = \u222b (t : \u211d) in Ioi 0, ENNReal.toReal (\u2191\u2191\u03bc {a | t < f a})", "state_after": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nf_intble : Integrable f\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nkey : \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9) \u2202\u03bc = \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a}\nlhs_finite : \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9) \u2202\u03bc < \u22a4\nrhs_finite : \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a} < \u22a4\n\u22a2 \u222b (\u03c9 : \u03b1), f \u03c9 \u2202\u03bc = \u222b (t : \u211d) in Ioi 0, ENNReal.toReal (\u2191\u2191\u03bc {a | t < f a})"}, {"tactic": "have rhs_integrand_finite : \u2200 (t : \u211d), t > 0 \u2192 \u03bc {a | t < f a} < \u221e :=\n  fun t ht \u21a6 measure_gt_lt_top f_intble ht", "annotated_tactic": ["have rhs_integrand_finite : \u2200 (t : \u211d), t > 0 \u2192 \u03bc {a | t < f a} < \u221e :=\n    fun t ht \u21a6 <a>measure_gt_lt_top</a> f_intble ht", [{"full_name": "MeasureTheory.Integrable.measure_gt_lt_top", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [853, 7], "def_end_pos": [853, 35]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nf_intble : Integrable f\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nkey : \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9) \u2202\u03bc = \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a}\nlhs_finite : \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9) \u2202\u03bc < \u22a4\nrhs_finite : \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a} < \u22a4\n\u22a2 \u222b (\u03c9 : \u03b1), f \u03c9 \u2202\u03bc = \u222b (t : \u211d) in Ioi 0, ENNReal.toReal (\u2191\u2191\u03bc {a | t < f a})", "state_after": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nf_intble : Integrable f\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nkey : \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9) \u2202\u03bc = \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a}\nlhs_finite : \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9) \u2202\u03bc < \u22a4\nrhs_finite : \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a} < \u22a4\nrhs_integrand_finite : \u2200 (t : \u211d), t > 0 \u2192 \u2191\u2191\u03bc {a | t < f a} < \u22a4\n\u22a2 \u222b (\u03c9 : \u03b1), f \u03c9 \u2202\u03bc = \u222b (t : \u211d) in Ioi 0, ENNReal.toReal (\u2191\u2191\u03bc {a | t < f a})"}, {"tactic": "convert (ENNReal.toReal_eq_toReal lhs_finite.ne rhs_finite.ne).mpr key", "annotated_tactic": ["convert (<a>ENNReal.toReal_eq_toReal</a> lhs_finite.ne rhs_finite.ne).<a>mpr</a> key", [{"full_name": "ENNReal.toReal_eq_toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2327, 9], "def_end_pos": [2327, 25]}, {"full_name": "Iff.mpr", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [92, 3], "def_end_pos": [92, 6]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nf_intble : Integrable f\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nkey : \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9) \u2202\u03bc = \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a}\nlhs_finite : \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9) \u2202\u03bc < \u22a4\nrhs_finite : \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a} < \u22a4\nrhs_integrand_finite : \u2200 (t : \u211d), t > 0 \u2192 \u2191\u2191\u03bc {a | t < f a} < \u22a4\n\u22a2 \u222b (\u03c9 : \u03b1), f \u03c9 \u2202\u03bc = \u222b (t : \u211d) in Ioi 0, ENNReal.toReal (\u2191\u2191\u03bc {a | t < f a})", "state_after": "case h.e'_2\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nf_intble : Integrable f\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nkey : \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9) \u2202\u03bc = \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a}\nlhs_finite : \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9) \u2202\u03bc < \u22a4\nrhs_finite : \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a} < \u22a4\nrhs_integrand_finite : \u2200 (t : \u211d), t > 0 \u2192 \u2191\u2191\u03bc {a | t < f a} < \u22a4\n\u22a2 \u222b (\u03c9 : \u03b1), f \u03c9 \u2202\u03bc = ENNReal.toReal (\u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9) \u2202\u03bc)\n\ncase h.e'_3\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nf_intble : Integrable f\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nkey : \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9) \u2202\u03bc = \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a}\nlhs_finite : \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9) \u2202\u03bc < \u22a4\nrhs_finite : \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a} < \u22a4\nrhs_integrand_finite : \u2200 (t : \u211d), t > 0 \u2192 \u2191\u2191\u03bc {a | t < f a} < \u22a4\n\u22a2 \u222b (t : \u211d) in Ioi 0, ENNReal.toReal (\u2191\u2191\u03bc {a | t < f a}) = ENNReal.toReal (\u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a})"}, {"tactic": "simp only [\u2190 key, lhs_finite]", "annotated_tactic": ["simp only [\u2190 key, lhs_finite]", []], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nf_intble : Integrable f\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nkey : \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9) \u2202\u03bc = \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a}\nlhs_finite : \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9) \u2202\u03bc < \u22a4\n\u22a2 \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a} < \u22a4", "state_after": "no goals"}, {"tactic": "exact integral_eq_lintegral_of_nonneg_ae f_nn f_intble.aestronglyMeasurable", "annotated_tactic": ["exact <a>integral_eq_lintegral_of_nonneg_ae</a> f_nn f_intble.aestronglyMeasurable", [{"full_name": "MeasureTheory.integral_eq_lintegral_of_nonneg_ae", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1158, 9], "def_end_pos": [1158, 43]}]], "state_before": "case h.e'_2\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nf_intble : Integrable f\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nkey : \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9) \u2202\u03bc = \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a}\nlhs_finite : \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9) \u2202\u03bc < \u22a4\nrhs_finite : \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a} < \u22a4\nrhs_integrand_finite : \u2200 (t : \u211d), t > 0 \u2192 \u2191\u2191\u03bc {a | t < f a} < \u22a4\n\u22a2 \u222b (\u03c9 : \u03b1), f \u03c9 \u2202\u03bc = ENNReal.toReal (\u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9) \u2202\u03bc)", "state_after": "no goals"}, {"tactic": "have aux := @integral_eq_lintegral_of_nonneg_ae _ _ ((volume : Measure \u211d).restrict (Set.Ioi 0))\n  (fun t \u21a6 ENNReal.toReal (\u03bc {a : \u03b1 | t < f a})) ?_ ?_", "annotated_tactic": ["have aux := @<a>integral_eq_lintegral_of_nonneg_ae</a> _ _ ((<a>volume</a> : <a>Measure</a> \u211d).<a>restrict</a> (<a>Set.Ioi</a> 0))\n      (fun t \u21a6 <a>ENNReal.toReal</a> (\u03bc {a : \u03b1 | t < f a})) ?_ ?_", [{"full_name": "MeasureTheory.integral_eq_lintegral_of_nonneg_ae", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1158, 9], "def_end_pos": [1158, 43]}, {"full_name": "MeasureTheory.MeasureSpace.volume", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [663, 3], "def_end_pos": [663, 9]}, {"full_name": "MeasureTheory.Measure", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [74, 11], "def_end_pos": [74, 18]}, {"full_name": "MeasureTheory.Measure.restrict", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1503, 5], "def_end_pos": [1503, 13]}, {"full_name": "Set.Ioi", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [79, 5], "def_end_pos": [79, 8]}, {"full_name": "ENNReal.toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [168, 15], "def_end_pos": [168, 21]}]], "state_before": "case h.e'_3\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nf_intble : Integrable f\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nkey : \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9) \u2202\u03bc = \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a}\nlhs_finite : \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9) \u2202\u03bc < \u22a4\nrhs_finite : \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a} < \u22a4\nrhs_integrand_finite : \u2200 (t : \u211d), t > 0 \u2192 \u2191\u2191\u03bc {a | t < f a} < \u22a4\n\u22a2 \u222b (t : \u211d) in Ioi 0, ENNReal.toReal (\u2191\u2191\u03bc {a | t < f a}) = ENNReal.toReal (\u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a})", "state_after": "case h.e'_3.refine_3\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nf_intble : Integrable f\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nkey : \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9) \u2202\u03bc = \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a}\nlhs_finite : \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9) \u2202\u03bc < \u22a4\nrhs_finite : \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a} < \u22a4\nrhs_integrand_finite : \u2200 (t : \u211d), t > 0 \u2192 \u2191\u2191\u03bc {a | t < f a} < \u22a4\naux :\n  \u222b (a : \u211d) in Ioi 0, ENNReal.toReal (\u2191\u2191\u03bc {a_1 | a < f a_1}) =\n    ENNReal.toReal (\u222b\u207b (a : \u211d) in Ioi 0, ENNReal.ofReal (ENNReal.toReal (\u2191\u2191\u03bc {a_1 | a < f a_1})))\n\u22a2 \u222b (t : \u211d) in Ioi 0, ENNReal.toReal (\u2191\u2191\u03bc {a | t < f a}) = ENNReal.toReal (\u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a})\n\ncase h.e'_3.refine_1\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nf_intble : Integrable f\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nkey : \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9) \u2202\u03bc = \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a}\nlhs_finite : \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9) \u2202\u03bc < \u22a4\nrhs_finite : \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a} < \u22a4\nrhs_integrand_finite : \u2200 (t : \u211d), t > 0 \u2192 \u2191\u2191\u03bc {a | t < f a} < \u22a4\n\u22a2 0 \u2264\u1da0[ae (Measure.restrict volume (Ioi 0))] fun t => ENNReal.toReal (\u2191\u2191\u03bc {a | t < f a})\n\ncase h.e'_3.refine_2\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nf_intble : Integrable f\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nkey : \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9) \u2202\u03bc = \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a}\nlhs_finite : \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9) \u2202\u03bc < \u22a4\nrhs_finite : \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a} < \u22a4\nrhs_integrand_finite : \u2200 (t : \u211d), t > 0 \u2192 \u2191\u2191\u03bc {a | t < f a} < \u22a4\n\u22a2 AEStronglyMeasurable (fun t => ENNReal.toReal (\u2191\u2191\u03bc {a | t < f a})) (Measure.restrict volume (Ioi 0))"}, {"tactic": "rw [aux]", "annotated_tactic": ["rw [aux]", []], "state_before": "case h.e'_3.refine_3\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nf_intble : Integrable f\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nkey : \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9) \u2202\u03bc = \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a}\nlhs_finite : \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9) \u2202\u03bc < \u22a4\nrhs_finite : \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a} < \u22a4\nrhs_integrand_finite : \u2200 (t : \u211d), t > 0 \u2192 \u2191\u2191\u03bc {a | t < f a} < \u22a4\naux :\n  \u222b (a : \u211d) in Ioi 0, ENNReal.toReal (\u2191\u2191\u03bc {a_1 | a < f a_1}) =\n    ENNReal.toReal (\u222b\u207b (a : \u211d) in Ioi 0, ENNReal.ofReal (ENNReal.toReal (\u2191\u2191\u03bc {a_1 | a < f a_1})))\n\u22a2 \u222b (t : \u211d) in Ioi 0, ENNReal.toReal (\u2191\u2191\u03bc {a | t < f a}) = ENNReal.toReal (\u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a})", "state_after": "case h.e'_3.refine_3\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nf_intble : Integrable f\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nkey : \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9) \u2202\u03bc = \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a}\nlhs_finite : \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9) \u2202\u03bc < \u22a4\nrhs_finite : \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a} < \u22a4\nrhs_integrand_finite : \u2200 (t : \u211d), t > 0 \u2192 \u2191\u2191\u03bc {a | t < f a} < \u22a4\naux :\n  \u222b (a : \u211d) in Ioi 0, ENNReal.toReal (\u2191\u2191\u03bc {a_1 | a < f a_1}) =\n    ENNReal.toReal (\u222b\u207b (a : \u211d) in Ioi 0, ENNReal.ofReal (ENNReal.toReal (\u2191\u2191\u03bc {a_1 | a < f a_1})))\n\u22a2 ENNReal.toReal (\u222b\u207b (a : \u211d) in Ioi 0, ENNReal.ofReal (ENNReal.toReal (\u2191\u2191\u03bc {a_1 | a < f a_1}))) =\n    ENNReal.toReal (\u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a})"}, {"tactic": "congr 1", "annotated_tactic": ["congr 1", []], "state_before": "case h.e'_3.refine_3\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nf_intble : Integrable f\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nkey : \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9) \u2202\u03bc = \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a}\nlhs_finite : \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9) \u2202\u03bc < \u22a4\nrhs_finite : \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a} < \u22a4\nrhs_integrand_finite : \u2200 (t : \u211d), t > 0 \u2192 \u2191\u2191\u03bc {a | t < f a} < \u22a4\naux :\n  \u222b (a : \u211d) in Ioi 0, ENNReal.toReal (\u2191\u2191\u03bc {a_1 | a < f a_1}) =\n    ENNReal.toReal (\u222b\u207b (a : \u211d) in Ioi 0, ENNReal.ofReal (ENNReal.toReal (\u2191\u2191\u03bc {a_1 | a < f a_1})))\n\u22a2 ENNReal.toReal (\u222b\u207b (a : \u211d) in Ioi 0, ENNReal.ofReal (ENNReal.toReal (\u2191\u2191\u03bc {a_1 | a < f a_1}))) =\n    ENNReal.toReal (\u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a})", "state_after": "case h.e'_3.refine_3.e_a\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nf_intble : Integrable f\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nkey : \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9) \u2202\u03bc = \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a}\nlhs_finite : \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9) \u2202\u03bc < \u22a4\nrhs_finite : \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a} < \u22a4\nrhs_integrand_finite : \u2200 (t : \u211d), t > 0 \u2192 \u2191\u2191\u03bc {a | t < f a} < \u22a4\naux :\n  \u222b (a : \u211d) in Ioi 0, ENNReal.toReal (\u2191\u2191\u03bc {a_1 | a < f a_1}) =\n    ENNReal.toReal (\u222b\u207b (a : \u211d) in Ioi 0, ENNReal.ofReal (ENNReal.toReal (\u2191\u2191\u03bc {a_1 | a < f a_1})))\n\u22a2 \u222b\u207b (a : \u211d) in Ioi 0, ENNReal.ofReal (ENNReal.toReal (\u2191\u2191\u03bc {a_1 | a < f a_1})) = \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a}"}, {"tactic": "apply set_lintegral_congr_fun measurableSet_Ioi (eventually_of_forall _)", "annotated_tactic": ["apply <a>set_lintegral_congr_fun</a> <a>measurableSet_Ioi</a> (<a>eventually_of_forall</a> _)", [{"full_name": "MeasureTheory.set_lintegral_congr_fun", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [316, 9], "def_end_pos": [316, 32]}, {"full_name": "measurableSet_Ioi", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [579, 9], "def_end_pos": [579, 26]}, {"full_name": "Filter.eventually_of_forall", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1111, 9], "def_end_pos": [1111, 29]}]], "state_before": "case h.e'_3.refine_3.e_a\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nf_intble : Integrable f\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nkey : \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9) \u2202\u03bc = \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a}\nlhs_finite : \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9) \u2202\u03bc < \u22a4\nrhs_finite : \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a} < \u22a4\nrhs_integrand_finite : \u2200 (t : \u211d), t > 0 \u2192 \u2191\u2191\u03bc {a | t < f a} < \u22a4\naux :\n  \u222b (a : \u211d) in Ioi 0, ENNReal.toReal (\u2191\u2191\u03bc {a_1 | a < f a_1}) =\n    ENNReal.toReal (\u222b\u207b (a : \u211d) in Ioi 0, ENNReal.ofReal (ENNReal.toReal (\u2191\u2191\u03bc {a_1 | a < f a_1})))\n\u22a2 \u222b\u207b (a : \u211d) in Ioi 0, ENNReal.ofReal (ENNReal.toReal (\u2191\u2191\u03bc {a_1 | a < f a_1})) = \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a}", "state_after": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nf_intble : Integrable f\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nkey : \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9) \u2202\u03bc = \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a}\nlhs_finite : \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9) \u2202\u03bc < \u22a4\nrhs_finite : \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a} < \u22a4\nrhs_integrand_finite : \u2200 (t : \u211d), t > 0 \u2192 \u2191\u2191\u03bc {a | t < f a} < \u22a4\naux :\n  \u222b (a : \u211d) in Ioi 0, ENNReal.toReal (\u2191\u2191\u03bc {a_1 | a < f a_1}) =\n    ENNReal.toReal (\u222b\u207b (a : \u211d) in Ioi 0, ENNReal.ofReal (ENNReal.toReal (\u2191\u2191\u03bc {a_1 | a < f a_1})))\n\u22a2 \u2200 (x : \u211d), x \u2208 Ioi 0 \u2192 ENNReal.ofReal (ENNReal.toReal (\u2191\u2191\u03bc {a | x < f a})) = \u2191\u2191\u03bc {a | x < f a}"}, {"tactic": "exact fun t t_pos \u21a6 ENNReal.ofReal_toReal (rhs_integrand_finite t t_pos).ne", "annotated_tactic": ["exact fun t t_pos \u21a6 <a>ENNReal.ofReal_toReal</a> (rhs_integrand_finite t t_pos).<a>ne</a>", [{"full_name": "ENNReal.ofReal_toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [186, 9], "def_end_pos": [186, 22]}, {"full_name": "LT.lt.ne", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [152, 7], "def_end_pos": [152, 15]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nf_intble : Integrable f\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nkey : \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9) \u2202\u03bc = \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a}\nlhs_finite : \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9) \u2202\u03bc < \u22a4\nrhs_finite : \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a} < \u22a4\nrhs_integrand_finite : \u2200 (t : \u211d), t > 0 \u2192 \u2191\u2191\u03bc {a | t < f a} < \u22a4\naux :\n  \u222b (a : \u211d) in Ioi 0, ENNReal.toReal (\u2191\u2191\u03bc {a_1 | a < f a_1}) =\n    ENNReal.toReal (\u222b\u207b (a : \u211d) in Ioi 0, ENNReal.ofReal (ENNReal.toReal (\u2191\u2191\u03bc {a_1 | a < f a_1})))\n\u22a2 \u2200 (x : \u211d), x \u2208 Ioi 0 \u2192 ENNReal.ofReal (ENNReal.toReal (\u2191\u2191\u03bc {a | x < f a})) = \u2191\u2191\u03bc {a | x < f a}", "state_after": "no goals"}, {"tactic": "exact eventually_of_forall (fun x \u21a6 by simp only [Pi.zero_apply, ENNReal.toReal_nonneg])", "annotated_tactic": ["exact <a>eventually_of_forall</a> (fun x \u21a6 by simp only [<a>Pi.zero_apply</a>, <a>ENNReal.toReal_nonneg</a>])", [{"full_name": "Filter.eventually_of_forall", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1111, 9], "def_end_pos": [1111, 29]}, {"full_name": "Pi.zero_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [46, 3], "def_end_pos": [46, 14]}, {"full_name": "ENNReal.toReal_nonneg", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [221, 17], "def_end_pos": [221, 30]}]], "state_before": "case h.e'_3.refine_1\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nf_intble : Integrable f\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nkey : \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9) \u2202\u03bc = \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a}\nlhs_finite : \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9) \u2202\u03bc < \u22a4\nrhs_finite : \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a} < \u22a4\nrhs_integrand_finite : \u2200 (t : \u211d), t > 0 \u2192 \u2191\u2191\u03bc {a | t < f a} < \u22a4\n\u22a2 0 \u2264\u1da0[ae (Measure.restrict volume (Ioi 0))] fun t => ENNReal.toReal (\u2191\u2191\u03bc {a | t < f a})", "state_after": "no goals"}, {"tactic": "simp only [Pi.zero_apply, ENNReal.toReal_nonneg]", "annotated_tactic": ["simp only [<a>Pi.zero_apply</a>, <a>ENNReal.toReal_nonneg</a>]", [{"full_name": "Pi.zero_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [46, 3], "def_end_pos": [46, 14]}, {"full_name": "ENNReal.toReal_nonneg", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [221, 17], "def_end_pos": [221, 30]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nf_intble : Integrable f\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nkey : \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9) \u2202\u03bc = \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a}\nlhs_finite : \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9) \u2202\u03bc < \u22a4\nrhs_finite : \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a} < \u22a4\nrhs_integrand_finite : \u2200 (t : \u211d), t > 0 \u2192 \u2191\u2191\u03bc {a | t < f a} < \u22a4\nx : \u211d\n\u22a2 OfNat.ofNat 0 x \u2264 (fun t => ENNReal.toReal (\u2191\u2191\u03bc {a | t < f a})) x", "state_after": "no goals"}, {"tactic": "apply Measurable.aestronglyMeasurable", "annotated_tactic": ["apply <a>Measurable.aestronglyMeasurable</a>", [{"full_name": "Measurable.aestronglyMeasurable", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1285, 9], "def_end_pos": [1285, 47]}]], "state_before": "case h.e'_3.refine_2\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nf_intble : Integrable f\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nkey : \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9) \u2202\u03bc = \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a}\nlhs_finite : \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9) \u2202\u03bc < \u22a4\nrhs_finite : \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a} < \u22a4\nrhs_integrand_finite : \u2200 (t : \u211d), t > 0 \u2192 \u2191\u2191\u03bc {a | t < f a} < \u22a4\n\u22a2 AEStronglyMeasurable (fun t => ENNReal.toReal (\u2191\u2191\u03bc {a | t < f a})) (Measure.restrict volume (Ioi 0))", "state_after": "case h.e'_3.refine_2.hf\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nf_intble : Integrable f\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nkey : \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9) \u2202\u03bc = \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a}\nlhs_finite : \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9) \u2202\u03bc < \u22a4\nrhs_finite : \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a} < \u22a4\nrhs_integrand_finite : \u2200 (t : \u211d), t > 0 \u2192 \u2191\u2191\u03bc {a | t < f a} < \u22a4\n\u22a2 Measurable fun t => ENNReal.toReal (\u2191\u2191\u03bc {a | t < f a})"}, {"tactic": "refine Measurable.ennreal_toReal ?_", "annotated_tactic": ["refine <a>Measurable.ennreal_toReal</a> ?_", [{"full_name": "Measurable.ennreal_toReal", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [2123, 9], "def_end_pos": [2123, 34]}]], "state_before": "case h.e'_3.refine_2.hf\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nf_intble : Integrable f\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nkey : \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9) \u2202\u03bc = \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a}\nlhs_finite : \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9) \u2202\u03bc < \u22a4\nrhs_finite : \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a} < \u22a4\nrhs_integrand_finite : \u2200 (t : \u211d), t > 0 \u2192 \u2191\u2191\u03bc {a | t < f a} < \u22a4\n\u22a2 Measurable fun t => ENNReal.toReal (\u2191\u2191\u03bc {a | t < f a})", "state_after": "case h.e'_3.refine_2.hf\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nf_intble : Integrable f\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nkey : \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9) \u2202\u03bc = \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a}\nlhs_finite : \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9) \u2202\u03bc < \u22a4\nrhs_finite : \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a} < \u22a4\nrhs_integrand_finite : \u2200 (t : \u211d), t > 0 \u2192 \u2191\u2191\u03bc {a | t < f a} < \u22a4\n\u22a2 Measurable fun t => \u2191\u2191\u03bc {a | t < f a}"}, {"tactic": "exact Antitone.measurable (fun _ _ hst \u21a6 measure_mono (fun _ h \u21a6 lt_of_le_of_lt hst h))", "annotated_tactic": ["exact <a>Antitone.measurable</a> (fun _ _ hst \u21a6 <a>measure_mono</a> (fun _ h \u21a6 <a>lt_of_le_of_lt</a> hst h))", [{"full_name": "Antitone.measurable", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [1255, 19], "def_end_pos": [1255, 38]}, {"full_name": "MeasureTheory.measure_mono", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [193, 9], "def_end_pos": [193, 21]}, {"full_name": "lt_of_le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [122, 9], "def_end_pos": [122, 23]}]], "state_before": "case h.e'_3.refine_2.hf\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nf_intble : Integrable f\nf_nn : 0 \u2264\u1da0[ae \u03bc] f\nkey : \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9) \u2202\u03bc = \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a}\nlhs_finite : \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9) \u2202\u03bc < \u22a4\nrhs_finite : \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a} < \u22a4\nrhs_integrand_finite : \u2200 (t : \u211d), t > 0 \u2192 \u2191\u2191\u03bc {a | t < f a} < \u22a4\n\u22a2 Measurable fun t => \u2191\u2191\u03bc {a | t < f a}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/Division.lean", "full_name": "MvPolynomial.modMonomial_X", "start": [195, 1], "end": [196, 25], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "full_name": "MeasureTheory.Integrable.add'", "start": [668, 1], "end": [674, 71], "traced_tactics": [{"tactic": "exact_mod_cast nnnorm_add_le _ _", "annotated_tactic": ["exact_mod_cast <a>nnnorm_add_le</a> _ _", [{"full_name": "nnnorm_add_le", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [946, 15], "def_end_pos": [946, 28]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : NormedAddCommGroup \u03b3\nf g : \u03b1 \u2192 \u03b2\nhf : Integrable f\nhg : Integrable g\na : \u03b1\n\u22a2 \u2191\u2016f a + g a\u2016\u208a \u2264 \u2191\u2016f a\u2016\u208a + \u2191\u2016g a\u2016\u208a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Num/Prime.lean", "full_name": "Num.minFac_to_nat", "start": [119, 1], "end": [121, 36], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Variance.lean", "full_name": "ProbabilityTheory.variance_mul", "start": [188, 1], "end": [191, 6], "traced_tactics": [{"tactic": "rw [variance, evariance_mul, ENNReal.toReal_mul, ENNReal.toReal_ofReal (sq_nonneg _)]", "annotated_tactic": ["rw [<a>variance</a>, <a>evariance_mul</a>, <a>ENNReal.toReal_mul</a>, <a>ENNReal.toReal_ofReal</a> (<a>sq_nonneg</a> _)]", [{"full_name": "ProbabilityTheory.variance", "def_path": "Mathlib/Probability/Variance.lean", "def_pos": [61, 5], "def_end_pos": [61, 13]}, {"full_name": "ProbabilityTheory.evariance_mul", "def_path": "Mathlib/Probability/Variance.lean", "def_pos": [162, 9], "def_end_pos": [162, 22]}, {"full_name": "ENNReal.toReal_mul", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2296, 9], "def_end_pos": [2296, 19]}, {"full_name": "ENNReal.toReal_ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [191, 9], "def_end_pos": [191, 22]}, {"full_name": "sq_nonneg", "def_path": "Mathlib/Algebra/GroupPower/Order.lean", "def_pos": [645, 9], "def_end_pos": [645, 18]}]], "state_before": "\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc\u271d : Measure \u03a9\nc : \u211d\nX : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\n\u22a2 variance (fun \u03c9 => c * X \u03c9) \u03bc = c ^ 2 * variance X \u03bc", "state_after": "\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc\u271d : Measure \u03a9\nc : \u211d\nX : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\n\u22a2 c ^ 2 * ENNReal.toReal (evariance (fun \u03c9 => X \u03c9) \u03bc) = c ^ 2 * variance X \u03bc"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc\u271d : Measure \u03a9\nc : \u211d\nX : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\n\u22a2 c ^ 2 * ENNReal.toReal (evariance (fun \u03c9 => X \u03c9) \u03bc) = c ^ 2 * variance X \u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/Jacobian.lean", "full_name": "MeasureTheory.addHaar_image_le_mul_of_det_lt", "start": [285, 1], "end": [387, 26], "traced_tactics": [{"tactic": "apply nhdsWithin_le_nhds", "annotated_tactic": ["apply <a>nhdsWithin_le_nhds</a>", [{"full_name": "nhdsWithin_le_nhds", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [204, 9], "def_end_pos": [204, 27]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\n\u22a2 \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd[Ioi 0] 0, \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u2191\u2191\u03bc (f '' s) \u2264 \u2191m * \u2191\u2191\u03bc s", "state_after": "case a\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\n\u22a2 {x | (fun \u03b4 => \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u2191\u2191\u03bc (f '' s) \u2264 \u2191m * \u2191\u2191\u03bc s) x} \u2208 \ud835\udcdd 0"}, {"tactic": "let d := ENNReal.ofReal |A.det|", "annotated_tactic": ["let d := <a>ENNReal.ofReal</a> |A.det|", [{"full_name": "ENNReal.ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [172, 29], "def_end_pos": [172, 35]}]], "state_before": "case a\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\n\u22a2 {x | (fun \u03b4 => \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u2191\u2191\u03bc (f '' s) \u2264 \u2191m * \u2191\u2191\u03bc s) x} \u2208 \ud835\udcdd 0", "state_after": "case a\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u22a2 {x | (fun \u03b4 => \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u2191\u2191\u03bc (f '' s) \u2264 \u2191m * \u2191\u2191\u03bc s) x} \u2208 \ud835\udcdd 0"}, {"tactic": "obtain \u27e8\u03b5, h\u03b5, \u03b5pos\u27e9 :\n  \u2203 \u03b5 : \u211d, \u03bc (closedBall 0 \u03b5 + A '' closedBall 0 1) < m * \u03bc (closedBall 0 1) \u2227 0 < \u03b5 := by\n  have HC : IsCompact (A '' closedBall 0 1) :=\n    (ProperSpace.isCompact_closedBall _ _).image A.continuous\n  have L0 :\n    Tendsto (fun \u03b5 => \u03bc (cthickening \u03b5 (A '' closedBall 0 1))) (\ud835\udcdd[>] 0)\n      (\ud835\udcdd (\u03bc (A '' closedBall 0 1))) := by\n    apply Tendsto.mono_left _ nhdsWithin_le_nhds\n    exact tendsto_measure_cthickening_of_isCompact HC\n  have L1 :\n    Tendsto (fun \u03b5 => \u03bc (closedBall 0 \u03b5 + A '' closedBall 0 1)) (\ud835\udcdd[>] 0)\n      (\ud835\udcdd (\u03bc (A '' closedBall 0 1))) := by\n    apply L0.congr' _\n    filter_upwards [self_mem_nhdsWithin] with r hr\n    rw [\u2190 HC.add_closedBall_zero (le_of_lt hr), add_comm]\n  have L2 :\n    Tendsto (fun \u03b5 => \u03bc (closedBall 0 \u03b5 + A '' closedBall 0 1)) (\ud835\udcdd[>] 0)\n      (\ud835\udcdd (d * \u03bc (closedBall 0 1))) := by\n    convert L1\n    exact (addHaar_image_continuousLinearMap _ _ _).symm\n  have I : d * \u03bc (closedBall 0 1) < m * \u03bc (closedBall 0 1) :=\n    (ENNReal.mul_lt_mul_right (measure_closedBall_pos \u03bc _ zero_lt_one).ne'\n          measure_closedBall_lt_top.ne).2\n      hm\n  have H :\n    \u2200\u1da0 b : \u211d in \ud835\udcdd[>] 0, \u03bc (closedBall 0 b + A '' closedBall 0 1) < m * \u03bc (closedBall 0 1) :=\n    (tendsto_order.1 L2).2 _ I\n  exact (H.and self_mem_nhdsWithin).exists", "annotated_tactic": ["obtain \u27e8\u03b5, h\u03b5, \u03b5pos\u27e9 :\n    \u2203 \u03b5 : \u211d, \u03bc (<a>closedBall</a> 0 \u03b5 + A '' <a>closedBall</a> 0 1) < m * \u03bc (<a>closedBall</a> 0 1) \u2227 0 < \u03b5 := by\n    have HC : <a>IsCompact</a> (A '' <a>closedBall</a> 0 1) :=\n      (<a>ProperSpace.isCompact_closedBall</a> _ _).<a>image</a> A.continuous\n    have L0 :\n      <a>Tendsto</a> (fun \u03b5 => \u03bc (<a>cthickening</a> \u03b5 (A '' <a>closedBall</a> 0 1))) (\ud835\udcdd[>] 0)\n        (\ud835\udcdd (\u03bc (A '' <a>closedBall</a> 0 1))) := by\n      apply <a>Tendsto.mono_left</a> _ <a>nhdsWithin_le_nhds</a>\n      exact <a>tendsto_measure_cthickening_of_isCompact</a> HC\n    have L1 :\n      <a>Tendsto</a> (fun \u03b5 => \u03bc (<a>closedBall</a> 0 \u03b5 + A '' <a>closedBall</a> 0 1)) (\ud835\udcdd[>] 0)\n        (\ud835\udcdd (\u03bc (A '' <a>closedBall</a> 0 1))) := by\n      apply L0.congr' _\n      filter_upwards [<a>self_mem_nhdsWithin</a>] with r hr\n      rw [\u2190 HC.add_closedBall_zero (<a>le_of_lt</a> hr), <a>add_comm</a>]\n    have L2 :\n      <a>Tendsto</a> (fun \u03b5 => \u03bc (<a>closedBall</a> 0 \u03b5 + A '' <a>closedBall</a> 0 1)) (\ud835\udcdd[>] 0)\n        (\ud835\udcdd (d * \u03bc (<a>closedBall</a> 0 1))) := by\n      convert L1\n      exact (<a>addHaar_image_continuousLinearMap</a> _ _ _).<a>symm</a>\n    have I : d * \u03bc (<a>closedBall</a> 0 1) < m * \u03bc (<a>closedBall</a> 0 1) :=\n      (<a>ENNReal.mul_lt_mul_right</a> (<a>measure_closedBall_pos</a> \u03bc _ <a>zero_lt_one</a>).<a>ne'</a>\n            measure_closedBall_lt_top.ne).2\n        hm\n    have H :\n      \u2200\u1da0 b : \u211d in \ud835\udcdd[>] 0, \u03bc (<a>closedBall</a> 0 b + A '' <a>closedBall</a> 0 1) < m * \u03bc (<a>closedBall</a> 0 1) :=\n      (<a>tendsto_order</a>.1 L2).2 _ I\n    exact (H.and <a>self_mem_nhdsWithin</a>).<a>exists</a>", [{"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "IsCompact", "def_path": "Mathlib/Topology/Compactness/Compact.lean", "def_pos": [40, 5], "def_end_pos": [40, 14]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "ProperSpace.isCompact_closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [2199, 3], "def_end_pos": [2199, 23]}, {"full_name": "IsCompact.image", "def_path": "Mathlib/Topology/Compactness/Compact.lean", "def_pos": [121, 9], "def_end_pos": [121, 24]}, {"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2939, 5], "def_end_pos": [2939, 12]}, {"full_name": "Metric.cthickening", "def_path": "Mathlib/Topology/MetricSpace/HausdorffDistance.lean", "def_pos": [1027, 5], "def_end_pos": [1027, 16]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "Filter.Tendsto.mono_left", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [3036, 9], "def_end_pos": [3036, 26]}, {"full_name": "nhdsWithin_le_nhds", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [204, 9], "def_end_pos": [204, 27]}, {"full_name": "tendsto_measure_cthickening_of_isCompact", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [1840, 9], "def_end_pos": [1840, 49]}, {"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2939, 5], "def_end_pos": [2939, 12]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "self_mem_nhdsWithin", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [151, 9], "def_end_pos": [151, 28]}, {"full_name": "le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [110, 9], "def_end_pos": [110, 17]}, {"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [301, 3], "def_end_pos": [301, 14]}, {"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2939, 5], "def_end_pos": [2939, 12]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "MeasureTheory.Measure.addHaar_image_continuousLinearMap", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean", "def_pos": [329, 9], "def_end_pos": [329, 42]}, {"full_name": "Eq.symm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [310, 9], "def_end_pos": [310, 16]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "ENNReal.mul_lt_mul_right", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1080, 9], "def_end_pos": [1080, 25]}, {"full_name": "Metric.measure_closedBall_pos", "def_path": "Mathlib/MeasureTheory/Measure/OpenPos.lean", "def_pos": [227, 9], "def_end_pos": [227, 31]}, {"full_name": "zero_lt_one", "def_path": "Mathlib/Algebra/Order/ZeroLEOne.lean", "def_pos": [39, 15], "def_end_pos": [39, 26]}, {"full_name": "LT.lt.ne'", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [328, 9], "def_end_pos": [328, 12]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "tendsto_order", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [919, 9], "def_end_pos": [919, 22]}, {"full_name": "self_mem_nhdsWithin", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [151, 9], "def_end_pos": [151, 28]}, {"full_name": "Filter.Eventually.exists", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1308, 9], "def_end_pos": [1308, 26]}]], "state_before": "case a\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u22a2 {x | (fun \u03b4 => \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u2191\u2191\u03bc (f '' s) \u2264 \u2191m * \u2191\u2191\u03bc s) x} \u2208 \ud835\udcdd 0", "state_after": "case a.intro.intro\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\n\u22a2 {x | (fun \u03b4 => \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u2191\u2191\u03bc (f '' s) \u2264 \u2191m * \u2191\u2191\u03bc s) x} \u2208 \ud835\udcdd 0"}, {"tactic": "have : Iio (\u27e8\u03b5, \u03b5pos.le\u27e9 : \u211d\u22650) \u2208 \ud835\udcdd (0 : \u211d\u22650) := by apply Iio_mem_nhds; exact \u03b5pos", "annotated_tactic": ["have : <a>Iio</a> (\u27e8\u03b5, \u03b5pos.le\u27e9 : \u211d\u22650) \u2208 \ud835\udcdd (0 : \u211d\u22650) := by apply <a>Iio_mem_nhds</a>; exact \u03b5pos", [{"full_name": "Set.Iio", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [54, 5], "def_end_pos": [54, 8]}, {"full_name": "Iio_mem_nhds", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [359, 9], "def_end_pos": [359, 21]}]], "state_before": "case a.intro.intro\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\n\u22a2 {x | (fun \u03b4 => \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u2191\u2191\u03bc (f '' s) \u2264 \u2191m * \u2191\u2191\u03bc s) x} \u2208 \ud835\udcdd 0", "state_after": "case a.intro.intro\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0\n\u22a2 {x | (fun \u03b4 => \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u2191\u2191\u03bc (f '' s) \u2264 \u2191m * \u2191\u2191\u03bc s) x} \u2208 \ud835\udcdd 0"}, {"tactic": "filter_upwards [this]", "annotated_tactic": ["filter_upwards [this]", []], "state_before": "case a.intro.intro\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0\n\u22a2 {x | (fun \u03b4 => \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u2191\u2191\u03bc (f '' s) \u2264 \u2191m * \u2191\u2191\u03bc s) x} \u2208 \ud835\udcdd 0", "state_after": "case h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0\n\u22a2 \u2200 (a : \u211d\u22650),\n    a \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2192\n      \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s a \u2192 \u2191\u2191\u03bc (f '' s) \u2264 \u2191m * \u2191\u2191\u03bc s"}, {"tactic": "intro \u03b4 h\u03b4 s f hf", "annotated_tactic": ["intro \u03b4 h\u03b4 s f hf", []], "state_before": "case h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0\n\u22a2 \u2200 (a : \u211d\u22650),\n    a \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2192\n      \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s a \u2192 \u2191\u2191\u03bc (f '' s) \u2264 \u2191m * \u2191\u2191\u03bc s", "state_after": "case h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\nh\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\n\u22a2 \u2191\u2191\u03bc (f '' s) \u2264 \u2191m * \u2191\u2191\u03bc s"}, {"tactic": "have J : \u2200\u1da0 a in \ud835\udcdd[>] (0 : \u211d\u22650\u221e), \u03bc (f '' s) \u2264 m * (\u03bc s + a) := by\n  filter_upwards [self_mem_nhdsWithin] with a ha\n  change 0 < a at ha\n  obtain \u27e8t, r, t_count, ts, rpos, st, \u03bct\u27e9 :\n    \u2203 (t : Set E) (r : E \u2192 \u211d),\n      t.Countable \u2227\n        t \u2286 s \u2227\n          (\u2200 x : E, x \u2208 t \u2192 0 < r x) \u2227\n            (s \u2286 \u22c3 x \u2208 t, closedBall x (r x)) \u2227\n              (\u2211' x : \u21a5t, \u03bc (closedBall (\u2191x) (r \u2191x))) \u2264 \u03bc s + a :=\n    Besicovitch.exists_closedBall_covering_tsum_measure_le \u03bc ha.ne' (fun _ => Ioi 0) s\n      fun x _ \u03b4 \u03b4pos => \u27e8\u03b4 / 2, by simp [half_pos \u03b4pos, \u03b4pos]\u27e9\n  haveI : Encodable t := t_count.toEncodable\n  calc\n    \u03bc (f '' s) \u2264 \u03bc (\u22c3 x : t, f '' (s \u2229 closedBall x (r x))) := by\n      rw [biUnion_eq_iUnion] at st\n      apply measure_mono\n      rw [\u2190 image_iUnion, \u2190 inter_iUnion]\n      exact image_subset _ (subset_inter (Subset.refl _) st)\n    _ \u2264 \u2211' x : t, \u03bc (f '' (s \u2229 closedBall x (r x))) := (measure_iUnion_le _)\n    _ \u2264 \u2211' x : t, m * \u03bc (closedBall x (r x)) :=\n      (ENNReal.tsum_le_tsum fun x => I x (r x) (ts x.2) (rpos x x.2).le)\n    _ \u2264 m * (\u03bc s + a) := by rw [ENNReal.tsum_mul_left]; exact mul_le_mul_left' \u03bct _", "annotated_tactic": ["have J : \u2200\u1da0 a in \ud835\udcdd[>] (0 : \u211d\u22650\u221e), \u03bc (f '' s) \u2264 m * (\u03bc s + a) := by\n    filter_upwards [<a>self_mem_nhdsWithin</a>] with a ha\n    change 0 < a at ha\n    obtain \u27e8t, r, t_count, ts, rpos, st, \u03bct\u27e9 :\n      \u2203 (t : <a>Set</a> E) (r : E \u2192 \u211d),\n        t.Countable \u2227\n          t \u2286 s \u2227\n            (\u2200 x : E, x \u2208 t \u2192 0 < r x) \u2227\n              (s \u2286 \u22c3 x \u2208 t, <a>closedBall</a> x (r x)) \u2227\n                (\u2211' x : \u21a5t, \u03bc (<a>closedBall</a> (\u2191x) (r \u2191x))) \u2264 \u03bc s + a :=\n      <a>Besicovitch.exists_closedBall_covering_tsum_measure_le</a> \u03bc ha.ne' (fun _ => <a>Ioi</a> 0) s\n        fun x _ \u03b4 \u03b4pos => \u27e8\u03b4 / 2, by simp [<a>half_pos</a> \u03b4pos, \u03b4pos]\u27e9\n    haveI : <a>Encodable</a> t := t_count.toEncodable\n    calc\n      \u03bc (f '' s) \u2264 \u03bc (\u22c3 x : t, f '' (s \u2229 <a>closedBall</a> x (r x))) := by\n        rw [<a>biUnion_eq_iUnion</a>] at st\n        apply <a>measure_mono</a>\n        rw [\u2190 <a>image_iUnion</a>, \u2190 <a>inter_iUnion</a>]\n        exact <a>image_subset</a> _ (<a>subset_inter</a> (<a>Subset.refl</a> _) st)\n      _ \u2264 \u2211' x : t, \u03bc (f '' (s \u2229 <a>closedBall</a> x (r x))) := (<a>measure_iUnion_le</a> _)\n      _ \u2264 \u2211' x : t, m * \u03bc (<a>closedBall</a> x (r x)) :=\n        (<a>ENNReal.tsum_le_tsum</a> fun x => I x (r x) (ts x.2) (rpos x x.2).<a>le</a>)\n      _ \u2264 m * (\u03bc s + a) := by rw [<a>ENNReal.tsum_mul_left</a>]; exact <a>mul_le_mul_left'</a> \u03bct _", [{"full_name": "self_mem_nhdsWithin", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [151, 9], "def_end_pos": [151, 28]}, {"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "Besicovitch.exists_closedBall_covering_tsum_measure_le", "def_path": "Mathlib/MeasureTheory/Covering/Besicovitch.lean", "def_pos": [916, 9], "def_end_pos": [916, 51]}, {"full_name": "Set.Ioi", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [79, 5], "def_end_pos": [79, 8]}, {"full_name": "half_pos", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [504, 9], "def_end_pos": [504, 17]}, {"full_name": "Encodable", "def_path": "Mathlib/Logic/Encodable/Basic.lean", "def_pos": [45, 7], "def_end_pos": [45, 16]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "Set.biUnion_eq_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [1010, 9], "def_end_pos": [1010, 26]}, {"full_name": "MeasureTheory.measure_mono", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [193, 9], "def_end_pos": [193, 21]}, {"full_name": "Set.image_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [1791, 9], "def_end_pos": [1791, 21]}, {"full_name": "Set.inter_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [635, 9], "def_end_pos": [635, 21]}, {"full_name": "Set.image_subset", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [321, 9], "def_end_pos": [321, 21]}, {"full_name": "Set.subset_inter", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [972, 9], "def_end_pos": [972, 21]}, {"full_name": "Set.Subset.refl", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [354, 9], "def_end_pos": [354, 20]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "MeasureTheory.measure_iUnion_le", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [240, 9], "def_end_pos": [240, 26]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "ENNReal.tsum_le_tsum", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [827, 19], "def_end_pos": [827, 31]}, {"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [142, 7], "def_end_pos": [142, 15]}, {"full_name": "ENNReal.tsum_mul_left", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [897, 19], "def_end_pos": [897, 32]}, {"full_name": "mul_le_mul_left'", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [50, 9], "def_end_pos": [50, 25]}]], "state_before": "case h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\nh\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nI : \u2200 (x : E) (r : \u211d), x \u2208 s \u2192 0 \u2264 r \u2192 \u2191\u2191\u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u2191\u2191\u03bc (closedBall x r)\n\u22a2 \u2191\u2191\u03bc (f '' s) \u2264 \u2191m * \u2191\u2191\u03bc s", "state_after": "case h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\nh\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nI : \u2200 (x : E) (r : \u211d), x \u2208 s \u2192 0 \u2264 r \u2192 \u2191\u2191\u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u2191\u2191\u03bc (closedBall x r)\nJ : \u2200\u1da0 (a : \u211d\u22650\u221e) in \ud835\udcdd[Ioi 0] 0, \u2191\u2191\u03bc (f '' s) \u2264 \u2191m * (\u2191\u2191\u03bc s + a)\n\u22a2 \u2191\u2191\u03bc (f '' s) \u2264 \u2191m * \u2191\u2191\u03bc s"}, {"tactic": "have L : Tendsto (fun a => (m : \u211d\u22650\u221e) * (\u03bc s + a)) (\ud835\udcdd[>] 0) (\ud835\udcdd (m * (\u03bc s + 0))) := by\n  apply Tendsto.mono_left _ nhdsWithin_le_nhds\n  apply ENNReal.Tendsto.const_mul (tendsto_const_nhds.add tendsto_id)\n  simp only [ENNReal.coe_ne_top, Ne.def, or_true_iff, not_false_iff]", "annotated_tactic": ["have L : <a>Tendsto</a> (fun a => (m : \u211d\u22650\u221e) * (\u03bc s + a)) (\ud835\udcdd[>] 0) (\ud835\udcdd (m * (\u03bc s + 0))) := by\n    apply <a>Tendsto.mono_left</a> _ <a>nhdsWithin_le_nhds</a>\n    apply <a>ENNReal.Tendsto.const_mul</a> (tendsto_const_nhds.add <a>tendsto_id</a>)\n    simp only [<a>ENNReal.coe_ne_top</a>, <a>Ne.def</a>, <a>or_true_iff</a>, <a>not_false_iff</a>]", [{"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2939, 5], "def_end_pos": [2939, 12]}, {"full_name": "Filter.Tendsto.mono_left", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [3036, 9], "def_end_pos": [3036, 26]}, {"full_name": "nhdsWithin_le_nhds", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [204, 9], "def_end_pos": [204, 27]}, {"full_name": "ENNReal.Tendsto.const_mul", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [373, 19], "def_end_pos": [373, 36]}, {"full_name": "Filter.tendsto_id", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [3028, 9], "def_end_pos": [3028, 19]}, {"full_name": "ENNReal.coe_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [302, 17], "def_end_pos": [302, 27]}, {"full_name": "Ne.def", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [59, 9], "def_end_pos": [59, 15]}, {"full_name": "or_true_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [184, 9], "def_end_pos": [184, 20]}, {"full_name": "not_false_iff", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [82, 9], "def_end_pos": [82, 22]}]], "state_before": "case h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\nh\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nI : \u2200 (x : E) (r : \u211d), x \u2208 s \u2192 0 \u2264 r \u2192 \u2191\u2191\u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u2191\u2191\u03bc (closedBall x r)\nJ : \u2200\u1da0 (a : \u211d\u22650\u221e) in \ud835\udcdd[Ioi 0] 0, \u2191\u2191\u03bc (f '' s) \u2264 \u2191m * (\u2191\u2191\u03bc s + a)\n\u22a2 \u2191\u2191\u03bc (f '' s) \u2264 \u2191m * \u2191\u2191\u03bc s", "state_after": "case h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\nh\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nI : \u2200 (x : E) (r : \u211d), x \u2208 s \u2192 0 \u2264 r \u2192 \u2191\u2191\u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u2191\u2191\u03bc (closedBall x r)\nJ : \u2200\u1da0 (a : \u211d\u22650\u221e) in \ud835\udcdd[Ioi 0] 0, \u2191\u2191\u03bc (f '' s) \u2264 \u2191m * (\u2191\u2191\u03bc s + a)\nL : Tendsto (fun a => \u2191m * (\u2191\u2191\u03bc s + a)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191m * (\u2191\u2191\u03bc s + 0)))\n\u22a2 \u2191\u2191\u03bc (f '' s) \u2264 \u2191m * \u2191\u2191\u03bc s"}, {"tactic": "rw [add_zero] at L", "annotated_tactic": ["rw [<a>add_zero</a>] at L", [{"full_name": "add_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [469, 3], "def_end_pos": [469, 14]}]], "state_before": "case h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\nh\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nI : \u2200 (x : E) (r : \u211d), x \u2208 s \u2192 0 \u2264 r \u2192 \u2191\u2191\u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u2191\u2191\u03bc (closedBall x r)\nJ : \u2200\u1da0 (a : \u211d\u22650\u221e) in \ud835\udcdd[Ioi 0] 0, \u2191\u2191\u03bc (f '' s) \u2264 \u2191m * (\u2191\u2191\u03bc s + a)\nL : Tendsto (fun a => \u2191m * (\u2191\u2191\u03bc s + a)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191m * (\u2191\u2191\u03bc s + 0)))\n\u22a2 \u2191\u2191\u03bc (f '' s) \u2264 \u2191m * \u2191\u2191\u03bc s", "state_after": "case h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\nh\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nI : \u2200 (x : E) (r : \u211d), x \u2208 s \u2192 0 \u2264 r \u2192 \u2191\u2191\u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u2191\u2191\u03bc (closedBall x r)\nJ : \u2200\u1da0 (a : \u211d\u22650\u221e) in \ud835\udcdd[Ioi 0] 0, \u2191\u2191\u03bc (f '' s) \u2264 \u2191m * (\u2191\u2191\u03bc s + a)\nL : Tendsto (fun a => \u2191m * (\u2191\u2191\u03bc s + a)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191m * \u2191\u2191\u03bc s))\n\u22a2 \u2191\u2191\u03bc (f '' s) \u2264 \u2191m * \u2191\u2191\u03bc s"}, {"tactic": "exact ge_of_tendsto L J", "annotated_tactic": ["exact <a>ge_of_tendsto</a> L J", [{"full_name": "ge_of_tendsto", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [168, 9], "def_end_pos": [168, 22]}]], "state_before": "case h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\nh\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nI : \u2200 (x : E) (r : \u211d), x \u2208 s \u2192 0 \u2264 r \u2192 \u2191\u2191\u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u2191\u2191\u03bc (closedBall x r)\nJ : \u2200\u1da0 (a : \u211d\u22650\u221e) in \ud835\udcdd[Ioi 0] 0, \u2191\u2191\u03bc (f '' s) \u2264 \u2191m * (\u2191\u2191\u03bc s + a)\nL : Tendsto (fun a => \u2191m * (\u2191\u2191\u03bc s + a)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191m * \u2191\u2191\u03bc s))\n\u22a2 \u2191\u2191\u03bc (f '' s) \u2264 \u2191m * \u2191\u2191\u03bc s", "state_after": "no goals"}, {"tactic": "have HC : IsCompact (A '' closedBall 0 1) :=\n  (ProperSpace.isCompact_closedBall _ _).image A.continuous", "annotated_tactic": ["have HC : <a>IsCompact</a> (A '' <a>closedBall</a> 0 1) :=\n      (<a>ProperSpace.isCompact_closedBall</a> _ _).<a>image</a> A.continuous", [{"full_name": "IsCompact", "def_path": "Mathlib/Topology/Compactness/Compact.lean", "def_pos": [40, 5], "def_end_pos": [40, 14]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "ProperSpace.isCompact_closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [2199, 3], "def_end_pos": [2199, 23]}, {"full_name": "IsCompact.image", "def_path": "Mathlib/Topology/Compactness/Compact.lean", "def_pos": [121, 9], "def_end_pos": [121, 24]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u22a2 \u2203 \u03b5, \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1) \u2227 0 < \u03b5", "state_after": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\nHC : IsCompact (\u2191A '' closedBall 0 1)\n\u22a2 \u2203 \u03b5, \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1) \u2227 0 < \u03b5"}, {"tactic": "have L0 :\n  Tendsto (fun \u03b5 => \u03bc (cthickening \u03b5 (A '' closedBall 0 1))) (\ud835\udcdd[>] 0)\n    (\ud835\udcdd (\u03bc (A '' closedBall 0 1))) := by\n  apply Tendsto.mono_left _ nhdsWithin_le_nhds\n  exact tendsto_measure_cthickening_of_isCompact HC", "annotated_tactic": ["have L0 :\n      <a>Tendsto</a> (fun \u03b5 => \u03bc (<a>cthickening</a> \u03b5 (A '' <a>closedBall</a> 0 1))) (\ud835\udcdd[>] 0)\n        (\ud835\udcdd (\u03bc (A '' <a>closedBall</a> 0 1))) := by\n      apply <a>Tendsto.mono_left</a> _ <a>nhdsWithin_le_nhds</a>\n      exact <a>tendsto_measure_cthickening_of_isCompact</a> HC", [{"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2939, 5], "def_end_pos": [2939, 12]}, {"full_name": "Metric.cthickening", "def_path": "Mathlib/Topology/MetricSpace/HausdorffDistance.lean", "def_pos": [1027, 5], "def_end_pos": [1027, 16]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "Filter.Tendsto.mono_left", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [3036, 9], "def_end_pos": [3036, 26]}, {"full_name": "nhdsWithin_le_nhds", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [204, 9], "def_end_pos": [204, 27]}, {"full_name": "tendsto_measure_cthickening_of_isCompact", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [1840, 9], "def_end_pos": [1840, 49]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\nHC : IsCompact (\u2191A '' closedBall 0 1)\n\u22a2 \u2203 \u03b5, \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1) \u2227 0 < \u03b5", "state_after": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\nHC : IsCompact (\u2191A '' closedBall 0 1)\nL0 : Tendsto (fun \u03b5 => \u2191\u2191\u03bc (cthickening \u03b5 (\u2191A '' closedBall 0 1))) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (\u2191A '' closedBall 0 1)))\n\u22a2 \u2203 \u03b5, \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1) \u2227 0 < \u03b5"}, {"tactic": "have L1 :\n  Tendsto (fun \u03b5 => \u03bc (closedBall 0 \u03b5 + A '' closedBall 0 1)) (\ud835\udcdd[>] 0)\n    (\ud835\udcdd (\u03bc (A '' closedBall 0 1))) := by\n  apply L0.congr' _\n  filter_upwards [self_mem_nhdsWithin] with r hr\n  rw [\u2190 HC.add_closedBall_zero (le_of_lt hr), add_comm]", "annotated_tactic": ["have L1 :\n      <a>Tendsto</a> (fun \u03b5 => \u03bc (<a>closedBall</a> 0 \u03b5 + A '' <a>closedBall</a> 0 1)) (\ud835\udcdd[>] 0)\n        (\ud835\udcdd (\u03bc (A '' <a>closedBall</a> 0 1))) := by\n      apply L0.congr' _\n      filter_upwards [<a>self_mem_nhdsWithin</a>] with r hr\n      rw [\u2190 HC.add_closedBall_zero (<a>le_of_lt</a> hr), <a>add_comm</a>]", [{"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2939, 5], "def_end_pos": [2939, 12]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "self_mem_nhdsWithin", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [151, 9], "def_end_pos": [151, 28]}, {"full_name": "le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [110, 9], "def_end_pos": [110, 17]}, {"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [301, 3], "def_end_pos": [301, 14]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\nHC : IsCompact (\u2191A '' closedBall 0 1)\nL0 : Tendsto (fun \u03b5 => \u2191\u2191\u03bc (cthickening \u03b5 (\u2191A '' closedBall 0 1))) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (\u2191A '' closedBall 0 1)))\n\u22a2 \u2203 \u03b5, \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1) \u2227 0 < \u03b5", "state_after": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\nHC : IsCompact (\u2191A '' closedBall 0 1)\nL0 : Tendsto (fun \u03b5 => \u2191\u2191\u03bc (cthickening \u03b5 (\u2191A '' closedBall 0 1))) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (\u2191A '' closedBall 0 1)))\nL1 : Tendsto (fun \u03b5 => \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (\u2191A '' closedBall 0 1)))\n\u22a2 \u2203 \u03b5, \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1) \u2227 0 < \u03b5"}, {"tactic": "have L2 :\n  Tendsto (fun \u03b5 => \u03bc (closedBall 0 \u03b5 + A '' closedBall 0 1)) (\ud835\udcdd[>] 0)\n    (\ud835\udcdd (d * \u03bc (closedBall 0 1))) := by\n  convert L1\n  exact (addHaar_image_continuousLinearMap _ _ _).symm", "annotated_tactic": ["have L2 :\n      <a>Tendsto</a> (fun \u03b5 => \u03bc (<a>closedBall</a> 0 \u03b5 + A '' <a>closedBall</a> 0 1)) (\ud835\udcdd[>] 0)\n        (\ud835\udcdd (d * \u03bc (<a>closedBall</a> 0 1))) := by\n      convert L1\n      exact (<a>addHaar_image_continuousLinearMap</a> _ _ _).<a>symm</a>", [{"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2939, 5], "def_end_pos": [2939, 12]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "MeasureTheory.Measure.addHaar_image_continuousLinearMap", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean", "def_pos": [329, 9], "def_end_pos": [329, 42]}, {"full_name": "Eq.symm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [310, 9], "def_end_pos": [310, 16]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\nHC : IsCompact (\u2191A '' closedBall 0 1)\nL0 : Tendsto (fun \u03b5 => \u2191\u2191\u03bc (cthickening \u03b5 (\u2191A '' closedBall 0 1))) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (\u2191A '' closedBall 0 1)))\nL1 : Tendsto (fun \u03b5 => \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (\u2191A '' closedBall 0 1)))\n\u22a2 \u2203 \u03b5, \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1) \u2227 0 < \u03b5", "state_after": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\nHC : IsCompact (\u2191A '' closedBall 0 1)\nL0 : Tendsto (fun \u03b5 => \u2191\u2191\u03bc (cthickening \u03b5 (\u2191A '' closedBall 0 1))) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (\u2191A '' closedBall 0 1)))\nL1 : Tendsto (fun \u03b5 => \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (\u2191A '' closedBall 0 1)))\nL2 : Tendsto (fun \u03b5 => \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (d * \u2191\u2191\u03bc (closedBall 0 1)))\n\u22a2 \u2203 \u03b5, \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1) \u2227 0 < \u03b5"}, {"tactic": "have I : d * \u03bc (closedBall 0 1) < m * \u03bc (closedBall 0 1) :=\n  (ENNReal.mul_lt_mul_right (measure_closedBall_pos \u03bc _ zero_lt_one).ne'\n        measure_closedBall_lt_top.ne).2\n    hm", "annotated_tactic": ["have I : d * \u03bc (<a>closedBall</a> 0 1) < m * \u03bc (<a>closedBall</a> 0 1) :=\n      (<a>ENNReal.mul_lt_mul_right</a> (<a>measure_closedBall_pos</a> \u03bc _ <a>zero_lt_one</a>).<a>ne'</a>\n            measure_closedBall_lt_top.ne).2\n        hm", [{"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "ENNReal.mul_lt_mul_right", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1080, 9], "def_end_pos": [1080, 25]}, {"full_name": "Metric.measure_closedBall_pos", "def_path": "Mathlib/MeasureTheory/Measure/OpenPos.lean", "def_pos": [227, 9], "def_end_pos": [227, 31]}, {"full_name": "zero_lt_one", "def_path": "Mathlib/Algebra/Order/ZeroLEOne.lean", "def_pos": [39, 15], "def_end_pos": [39, 26]}, {"full_name": "LT.lt.ne'", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [328, 9], "def_end_pos": [328, 12]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\nHC : IsCompact (\u2191A '' closedBall 0 1)\nL0 : Tendsto (fun \u03b5 => \u2191\u2191\u03bc (cthickening \u03b5 (\u2191A '' closedBall 0 1))) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (\u2191A '' closedBall 0 1)))\nL1 : Tendsto (fun \u03b5 => \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (\u2191A '' closedBall 0 1)))\nL2 : Tendsto (fun \u03b5 => \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (d * \u2191\u2191\u03bc (closedBall 0 1)))\n\u22a2 \u2203 \u03b5, \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1) \u2227 0 < \u03b5", "state_after": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\nHC : IsCompact (\u2191A '' closedBall 0 1)\nL0 : Tendsto (fun \u03b5 => \u2191\u2191\u03bc (cthickening \u03b5 (\u2191A '' closedBall 0 1))) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (\u2191A '' closedBall 0 1)))\nL1 : Tendsto (fun \u03b5 => \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (\u2191A '' closedBall 0 1)))\nL2 : Tendsto (fun \u03b5 => \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (d * \u2191\u2191\u03bc (closedBall 0 1)))\nI : d * \u2191\u2191\u03bc (closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u22a2 \u2203 \u03b5, \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1) \u2227 0 < \u03b5"}, {"tactic": "have H :\n  \u2200\u1da0 b : \u211d in \ud835\udcdd[>] 0, \u03bc (closedBall 0 b + A '' closedBall 0 1) < m * \u03bc (closedBall 0 1) :=\n  (tendsto_order.1 L2).2 _ I", "annotated_tactic": ["have H :\n      \u2200\u1da0 b : \u211d in \ud835\udcdd[>] 0, \u03bc (<a>closedBall</a> 0 b + A '' <a>closedBall</a> 0 1) < m * \u03bc (<a>closedBall</a> 0 1) :=\n      (<a>tendsto_order</a>.1 L2).2 _ I", [{"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "tendsto_order", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [919, 9], "def_end_pos": [919, 22]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\nHC : IsCompact (\u2191A '' closedBall 0 1)\nL0 : Tendsto (fun \u03b5 => \u2191\u2191\u03bc (cthickening \u03b5 (\u2191A '' closedBall 0 1))) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (\u2191A '' closedBall 0 1)))\nL1 : Tendsto (fun \u03b5 => \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (\u2191A '' closedBall 0 1)))\nL2 : Tendsto (fun \u03b5 => \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (d * \u2191\u2191\u03bc (closedBall 0 1)))\nI : d * \u2191\u2191\u03bc (closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u22a2 \u2203 \u03b5, \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1) \u2227 0 < \u03b5", "state_after": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\nHC : IsCompact (\u2191A '' closedBall 0 1)\nL0 : Tendsto (fun \u03b5 => \u2191\u2191\u03bc (cthickening \u03b5 (\u2191A '' closedBall 0 1))) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (\u2191A '' closedBall 0 1)))\nL1 : Tendsto (fun \u03b5 => \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (\u2191A '' closedBall 0 1)))\nL2 : Tendsto (fun \u03b5 => \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (d * \u2191\u2191\u03bc (closedBall 0 1)))\nI : d * \u2191\u2191\u03bc (closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\nH : \u2200\u1da0 (b : \u211d) in \ud835\udcdd[Ioi 0] 0, \u2191\u2191\u03bc (closedBall 0 b + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u22a2 \u2203 \u03b5, \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1) \u2227 0 < \u03b5"}, {"tactic": "exact (H.and self_mem_nhdsWithin).exists", "annotated_tactic": ["exact (H.and <a>self_mem_nhdsWithin</a>).<a>exists</a>", [{"full_name": "self_mem_nhdsWithin", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [151, 9], "def_end_pos": [151, 28]}, {"full_name": "Filter.Eventually.exists", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1308, 9], "def_end_pos": [1308, 26]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\nHC : IsCompact (\u2191A '' closedBall 0 1)\nL0 : Tendsto (fun \u03b5 => \u2191\u2191\u03bc (cthickening \u03b5 (\u2191A '' closedBall 0 1))) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (\u2191A '' closedBall 0 1)))\nL1 : Tendsto (fun \u03b5 => \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (\u2191A '' closedBall 0 1)))\nL2 : Tendsto (fun \u03b5 => \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (d * \u2191\u2191\u03bc (closedBall 0 1)))\nI : d * \u2191\u2191\u03bc (closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\nH : \u2200\u1da0 (b : \u211d) in \ud835\udcdd[Ioi 0] 0, \u2191\u2191\u03bc (closedBall 0 b + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u22a2 \u2203 \u03b5, \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1) \u2227 0 < \u03b5", "state_after": "no goals"}, {"tactic": "apply Tendsto.mono_left _ nhdsWithin_le_nhds", "annotated_tactic": ["apply <a>Tendsto.mono_left</a> _ <a>nhdsWithin_le_nhds</a>", [{"full_name": "Filter.Tendsto.mono_left", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [3036, 9], "def_end_pos": [3036, 26]}, {"full_name": "nhdsWithin_le_nhds", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [204, 9], "def_end_pos": [204, 27]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\nHC : IsCompact (\u2191A '' closedBall 0 1)\n\u22a2 Tendsto (fun \u03b5 => \u2191\u2191\u03bc (cthickening \u03b5 (\u2191A '' closedBall 0 1))) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (\u2191A '' closedBall 0 1)))", "state_after": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\nHC : IsCompact (\u2191A '' closedBall 0 1)\n\u22a2 Tendsto (fun \u03b5 => \u2191\u2191\u03bc (cthickening \u03b5 (\u2191A '' closedBall 0 1))) (\ud835\udcdd 0) (\ud835\udcdd (\u2191\u2191\u03bc (\u2191A '' closedBall 0 1)))"}, {"tactic": "exact tendsto_measure_cthickening_of_isCompact HC", "annotated_tactic": ["exact <a>tendsto_measure_cthickening_of_isCompact</a> HC", [{"full_name": "tendsto_measure_cthickening_of_isCompact", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [1840, 9], "def_end_pos": [1840, 49]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\nHC : IsCompact (\u2191A '' closedBall 0 1)\n\u22a2 Tendsto (fun \u03b5 => \u2191\u2191\u03bc (cthickening \u03b5 (\u2191A '' closedBall 0 1))) (\ud835\udcdd 0) (\ud835\udcdd (\u2191\u2191\u03bc (\u2191A '' closedBall 0 1)))", "state_after": "no goals"}, {"tactic": "apply L0.congr' _", "annotated_tactic": ["apply L0.congr' _", []], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\nHC : IsCompact (\u2191A '' closedBall 0 1)\nL0 : Tendsto (fun \u03b5 => \u2191\u2191\u03bc (cthickening \u03b5 (\u2191A '' closedBall 0 1))) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (\u2191A '' closedBall 0 1)))\n\u22a2 Tendsto (fun \u03b5 => \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (\u2191A '' closedBall 0 1)))", "state_after": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\nHC : IsCompact (\u2191A '' closedBall 0 1)\nL0 : Tendsto (fun \u03b5 => \u2191\u2191\u03bc (cthickening \u03b5 (\u2191A '' closedBall 0 1))) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (\u2191A '' closedBall 0 1)))\n\u22a2 (fun \u03b5 => \u2191\u2191\u03bc (cthickening \u03b5 (\u2191A '' closedBall 0 1))) =\u1da0[\ud835\udcdd[Ioi 0] 0] fun \u03b5 =>\n    \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1)"}, {"tactic": "filter_upwards [self_mem_nhdsWithin] with r hr", "annotated_tactic": ["filter_upwards [<a>self_mem_nhdsWithin</a>] with r hr", [{"full_name": "self_mem_nhdsWithin", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [151, 9], "def_end_pos": [151, 28]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\nHC : IsCompact (\u2191A '' closedBall 0 1)\nL0 : Tendsto (fun \u03b5 => \u2191\u2191\u03bc (cthickening \u03b5 (\u2191A '' closedBall 0 1))) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (\u2191A '' closedBall 0 1)))\n\u22a2 (fun \u03b5 => \u2191\u2191\u03bc (cthickening \u03b5 (\u2191A '' closedBall 0 1))) =\u1da0[\ud835\udcdd[Ioi 0] 0] fun \u03b5 =>\n    \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1)", "state_after": "case h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\nHC : IsCompact (\u2191A '' closedBall 0 1)\nL0 : Tendsto (fun \u03b5 => \u2191\u2191\u03bc (cthickening \u03b5 (\u2191A '' closedBall 0 1))) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (\u2191A '' closedBall 0 1)))\nr : \u211d\nhr : r \u2208 Ioi 0\n\u22a2 \u2191\u2191\u03bc (cthickening r (\u2191A '' closedBall 0 1)) = \u2191\u2191\u03bc (closedBall 0 r + \u2191A '' closedBall 0 1)"}, {"tactic": "rw [\u2190 HC.add_closedBall_zero (le_of_lt hr), add_comm]", "annotated_tactic": ["rw [\u2190 HC.add_closedBall_zero (<a>le_of_lt</a> hr), <a>add_comm</a>]", [{"full_name": "le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [110, 9], "def_end_pos": [110, 17]}, {"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [301, 3], "def_end_pos": [301, 14]}]], "state_before": "case h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\nHC : IsCompact (\u2191A '' closedBall 0 1)\nL0 : Tendsto (fun \u03b5 => \u2191\u2191\u03bc (cthickening \u03b5 (\u2191A '' closedBall 0 1))) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (\u2191A '' closedBall 0 1)))\nr : \u211d\nhr : r \u2208 Ioi 0\n\u22a2 \u2191\u2191\u03bc (cthickening r (\u2191A '' closedBall 0 1)) = \u2191\u2191\u03bc (closedBall 0 r + \u2191A '' closedBall 0 1)", "state_after": "no goals"}, {"tactic": "convert L1", "annotated_tactic": ["convert L1", []], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\nHC : IsCompact (\u2191A '' closedBall 0 1)\nL0 : Tendsto (fun \u03b5 => \u2191\u2191\u03bc (cthickening \u03b5 (\u2191A '' closedBall 0 1))) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (\u2191A '' closedBall 0 1)))\nL1 : Tendsto (fun \u03b5 => \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (\u2191A '' closedBall 0 1)))\n\u22a2 Tendsto (fun \u03b5 => \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (d * \u2191\u2191\u03bc (closedBall 0 1)))", "state_after": "case h.e'_5.h.e'_3\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\nHC : IsCompact (\u2191A '' closedBall 0 1)\nL0 : Tendsto (fun \u03b5 => \u2191\u2191\u03bc (cthickening \u03b5 (\u2191A '' closedBall 0 1))) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (\u2191A '' closedBall 0 1)))\nL1 : Tendsto (fun \u03b5 => \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (\u2191A '' closedBall 0 1)))\n\u22a2 d * \u2191\u2191\u03bc (closedBall 0 1) = \u2191\u2191\u03bc (\u2191A '' closedBall 0 1)"}, {"tactic": "exact (addHaar_image_continuousLinearMap _ _ _).symm", "annotated_tactic": ["exact (<a>addHaar_image_continuousLinearMap</a> _ _ _).<a>symm</a>", [{"full_name": "MeasureTheory.Measure.addHaar_image_continuousLinearMap", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean", "def_pos": [329, 9], "def_end_pos": [329, 42]}, {"full_name": "Eq.symm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [310, 9], "def_end_pos": [310, 16]}]], "state_before": "case h.e'_5.h.e'_3\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\nHC : IsCompact (\u2191A '' closedBall 0 1)\nL0 : Tendsto (fun \u03b5 => \u2191\u2191\u03bc (cthickening \u03b5 (\u2191A '' closedBall 0 1))) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (\u2191A '' closedBall 0 1)))\nL1 : Tendsto (fun \u03b5 => \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (\u2191A '' closedBall 0 1)))\n\u22a2 d * \u2191\u2191\u03bc (closedBall 0 1) = \u2191\u2191\u03bc (\u2191A '' closedBall 0 1)", "state_after": "no goals"}, {"tactic": "apply Iio_mem_nhds", "annotated_tactic": ["apply <a>Iio_mem_nhds</a>", [{"full_name": "Iio_mem_nhds", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [359, 9], "def_end_pos": [359, 21]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\n\u22a2 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0", "state_after": "case h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\n\u22a2 0 < { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }"}, {"tactic": "exact \u03b5pos", "annotated_tactic": ["exact \u03b5pos", []], "state_before": "case h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\n\u22a2 0 < { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }", "state_after": "no goals"}, {"tactic": "intro x r xs r0", "annotated_tactic": ["intro x r xs r0", []], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\nh\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\n\u22a2 \u2200 (x : E) (r : \u211d), x \u2208 s \u2192 0 \u2264 r \u2192 \u2191\u2191\u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u2191\u2191\u03bc (closedBall x r)", "state_after": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\nh\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nx : E\nr : \u211d\nxs : x \u2208 s\nr0 : 0 \u2264 r\n\u22a2 \u2191\u2191\u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u2191\u2191\u03bc (closedBall x r)"}, {"tactic": "have :\n  A '' closedBall 0 r + closedBall (f x) (\u03b5 * r) =\n    {f x} + r \u2022 (A '' closedBall 0 1 + closedBall 0 \u03b5) := by\n  rw [smul_add, \u2190 add_assoc, add_comm {f x}, add_assoc, smul_closedBall _ _ \u03b5pos.le, smul_zero,\n    singleton_add_closedBall_zero, \u2190 image_smul_set \u211d E E A, smul_closedBall _ _ zero_le_one,\n    smul_zero, Real.norm_eq_abs, abs_of_nonneg r0, mul_one, mul_comm]", "annotated_tactic": ["have :\n      A '' <a>closedBall</a> 0 r + <a>closedBall</a> (f x) (\u03b5 * r) =\n        {f x} + r \u2022 (A '' <a>closedBall</a> 0 1 + <a>closedBall</a> 0 \u03b5) := by\n      rw [<a>smul_add</a>, \u2190 <a>add_assoc</a>, <a>add_comm</a> {f x}, <a>add_assoc</a>, <a>smul_closedBall</a> _ _ \u03b5pos.le, <a>smul_zero</a>,\n        <a>singleton_add_closedBall_zero</a>, \u2190 <a>image_smul_set</a> \u211d E E A, <a>smul_closedBall</a> _ _ <a>zero_le_one</a>,\n        <a>smul_zero</a>, <a>Real.norm_eq_abs</a>, <a>abs_of_nonneg</a> r0, <a>mul_one</a>, <a>mul_comm</a>]", [{"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "smul_add", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [807, 9], "def_end_pos": [807, 17]}, {"full_name": "add_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [263, 3], "def_end_pos": [263, 14]}, {"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [301, 3], "def_end_pos": [301, 14]}, {"full_name": "add_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [263, 3], "def_end_pos": [263, 14]}, {"full_name": "smul_closedBall", "def_path": "Mathlib/Analysis/NormedSpace/Pointwise.lean", "def_pos": [388, 9], "def_end_pos": [388, 24]}, {"full_name": "smul_zero", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [732, 9], "def_end_pos": [732, 18]}, {"full_name": "singleton_add_closedBall_zero", "def_path": "Mathlib/Analysis/Normed/Group/Pointwise.lean", "def_pos": [177, 3], "def_end_pos": [177, 14]}, {"full_name": "image_smul_set", "def_path": "Mathlib/Algebra/Module/LinearMap.lean", "def_pos": [394, 9], "def_end_pos": [394, 30]}, {"full_name": "smul_closedBall", "def_path": "Mathlib/Analysis/NormedSpace/Pointwise.lean", "def_pos": [388, 9], "def_end_pos": [388, 24]}, {"full_name": "zero_le_one", "def_path": "Mathlib/Algebra/Order/ZeroLEOne.lean", "def_pos": [26, 15], "def_end_pos": [26, 26]}, {"full_name": "smul_zero", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [732, 9], "def_end_pos": [732, 18]}, {"full_name": "Real.norm_eq_abs", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [1761, 9], "def_end_pos": [1761, 20]}, {"full_name": "abs_of_nonneg", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [107, 9], "def_end_pos": [107, 22]}, {"full_name": "mul_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [470, 9], "def_end_pos": [470, 16]}, {"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [302, 9], "def_end_pos": [302, 17]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\nh\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nx : E\nr : \u211d\nxs : x \u2208 s\nr0 : 0 \u2264 r\nK : f '' (s \u2229 closedBall x r) \u2286 \u2191A '' closedBall 0 r + closedBall (f x) (\u03b5 * r)\n\u22a2 \u2191\u2191\u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u2191\u2191\u03bc (closedBall x r)", "state_after": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis\u271d : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\nh\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nx : E\nr : \u211d\nxs : x \u2208 s\nr0 : 0 \u2264 r\nK : f '' (s \u2229 closedBall x r) \u2286 \u2191A '' closedBall 0 r + closedBall (f x) (\u03b5 * r)\nthis : \u2191A '' closedBall 0 r + closedBall (f x) (\u03b5 * r) = {f x} + r \u2022 (\u2191A '' closedBall 0 1 + closedBall 0 \u03b5)\n\u22a2 \u2191\u2191\u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u2191\u2191\u03bc (closedBall x r)"}, {"tactic": "rw [this] at K", "annotated_tactic": ["rw [this] at K", []], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis\u271d : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\nh\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nx : E\nr : \u211d\nxs : x \u2208 s\nr0 : 0 \u2264 r\nK : f '' (s \u2229 closedBall x r) \u2286 \u2191A '' closedBall 0 r + closedBall (f x) (\u03b5 * r)\nthis : \u2191A '' closedBall 0 r + closedBall (f x) (\u03b5 * r) = {f x} + r \u2022 (\u2191A '' closedBall 0 1 + closedBall 0 \u03b5)\n\u22a2 \u2191\u2191\u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u2191\u2191\u03bc (closedBall x r)", "state_after": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis\u271d : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\nh\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nx : E\nr : \u211d\nxs : x \u2208 s\nr0 : 0 \u2264 r\nK : f '' (s \u2229 closedBall x r) \u2286 {f x} + r \u2022 (\u2191A '' closedBall 0 1 + closedBall 0 \u03b5)\nthis : \u2191A '' closedBall 0 r + closedBall (f x) (\u03b5 * r) = {f x} + r \u2022 (\u2191A '' closedBall 0 1 + closedBall 0 \u03b5)\n\u22a2 \u2191\u2191\u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u2191\u2191\u03bc (closedBall x r)"}, {"tactic": "calc\n  \u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u03bc ({f x} + r \u2022 (A '' closedBall 0 1 + closedBall 0 \u03b5)) :=\n    measure_mono K\n  _ = ENNReal.ofReal (r ^ finrank \u211d E) * \u03bc (A '' closedBall 0 1 + closedBall 0 \u03b5) := by\n    simp only [abs_of_nonneg r0, addHaar_smul, image_add_left, abs_pow, singleton_add,\n      measure_preimage_add]\n  _ \u2264 ENNReal.ofReal (r ^ finrank \u211d E) * (m * \u03bc (closedBall 0 1)) := by\n    rw [add_comm]; exact mul_le_mul_left' h\u03b5.le _\n  _ = m * \u03bc (closedBall x r) := by simp only [addHaar_closedBall' \u03bc _ r0]; ring", "annotated_tactic": ["calc\n      \u03bc (f '' (s \u2229 <a>closedBall</a> x r)) \u2264 \u03bc ({f x} + r \u2022 (A '' <a>closedBall</a> 0 1 + <a>closedBall</a> 0 \u03b5)) :=\n        <a>measure_mono</a> K\n      _ = <a>ENNReal.ofReal</a> (r ^ <a>finrank</a> \u211d E) * \u03bc (A '' <a>closedBall</a> 0 1 + <a>closedBall</a> 0 \u03b5) := by\n        simp only [<a>abs_of_nonneg</a> r0, <a>addHaar_smul</a>, <a>image_add_left</a>, <a>abs_pow</a>, <a>singleton_add</a>,\n          <a>measure_preimage_add</a>]\n      _ \u2264 <a>ENNReal.ofReal</a> (r ^ <a>finrank</a> \u211d E) * (m * \u03bc (<a>closedBall</a> 0 1)) := by\n        rw [<a>add_comm</a>]; exact <a>mul_le_mul_left'</a> h\u03b5.le _\n      _ = m * \u03bc (<a>closedBall</a> x r) := by simp only [<a>addHaar_closedBall'</a> \u03bc _ r0]; ring", [{"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "MeasureTheory.measure_mono", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [193, 9], "def_end_pos": [193, 21]}, {"full_name": "ENNReal.ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [172, 29], "def_end_pos": [172, 35]}, {"full_name": "FiniteDimensional.finrank", "def_path": "Mathlib/LinearAlgebra/Finrank.lean", "def_pos": [58, 19], "def_end_pos": [58, 26]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "abs_of_nonneg", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [107, 9], "def_end_pos": [107, 22]}, {"full_name": "MeasureTheory.Measure.addHaar_smul", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean", "def_pos": [371, 9], "def_end_pos": [371, 21]}, {"full_name": "Set.image_add_left", "def_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "def_pos": [1198, 3], "def_end_pos": [1198, 14]}, {"full_name": "abs_pow", "def_path": "Mathlib/Algebra/GroupPower/Lemmas.lean", "def_pos": [705, 9], "def_end_pos": [705, 16]}, {"full_name": "Set.singleton_add", "def_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "def_pos": [402, 3], "def_end_pos": [402, 14]}, {"full_name": "MeasureTheory.measure_preimage_add", "def_path": "Mathlib/MeasureTheory/Group/Measure.lean", "def_pos": [317, 3], "def_end_pos": [317, 14]}, {"full_name": "ENNReal.ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [172, 29], "def_end_pos": [172, 35]}, {"full_name": "FiniteDimensional.finrank", "def_path": "Mathlib/LinearAlgebra/Finrank.lean", "def_pos": [58, 19], "def_end_pos": [58, 26]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [301, 3], "def_end_pos": [301, 14]}, {"full_name": "mul_le_mul_left'", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [50, 9], "def_end_pos": [50, 25]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "MeasureTheory.Measure.addHaar_closedBall'", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean", "def_pos": [481, 9], "def_end_pos": [481, 28]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis\u271d : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\nh\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nx : E\nr : \u211d\nxs : x \u2208 s\nr0 : 0 \u2264 r\nK : f '' (s \u2229 closedBall x r) \u2286 {f x} + r \u2022 (\u2191A '' closedBall 0 1 + closedBall 0 \u03b5)\nthis : \u2191A '' closedBall 0 r + closedBall (f x) (\u03b5 * r) = {f x} + r \u2022 (\u2191A '' closedBall 0 1 + closedBall 0 \u03b5)\n\u22a2 \u2191\u2191\u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u2191\u2191\u03bc (closedBall x r)", "state_after": "no goals"}, {"tactic": "rintro y \u27e8z, \u27e8zs, zr\u27e9, rfl\u27e9", "annotated_tactic": ["rintro y \u27e8z, \u27e8zs, zr\u27e9, rfl\u27e9", []], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\nh\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nx : E\nr : \u211d\nxs : x \u2208 s\nr0 : 0 \u2264 r\n\u22a2 f '' (s \u2229 closedBall x r) \u2286 \u2191A '' closedBall 0 r + closedBall (f x) (\u03b5 * r)", "state_after": "case intro.intro.intro\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\nh\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nx : E\nr : \u211d\nxs : x \u2208 s\nr0 : 0 \u2264 r\nz : E\nzs : z \u2208 s\nzr : z \u2208 closedBall x r\n\u22a2 f z \u2208 \u2191A '' closedBall 0 r + closedBall (f x) (\u03b5 * r)"}, {"tactic": "apply Set.mem_add.2 \u27e8A (z - x), f z - f x - A (z - x) + f x, _, _, _\u27e9", "annotated_tactic": ["apply <a>Set.mem_add</a>.2 \u27e8A (z - x), f z - f x - A (z - x) + f x, _, _, _\u27e9", [{"full_name": "Set.mem_add", "def_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "def_pos": [335, 3], "def_end_pos": [335, 14]}]], "state_before": "case intro.intro.intro\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\nh\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nx : E\nr : \u211d\nxs : x \u2208 s\nr0 : 0 \u2264 r\nz : E\nzs : z \u2208 s\nzr : z \u2208 closedBall x r\n\u22a2 f z \u2208 \u2191A '' closedBall 0 r + closedBall (f x) (\u03b5 * r)", "state_after": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\nh\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nx : E\nr : \u211d\nxs : x \u2208 s\nr0 : 0 \u2264 r\nz : E\nzs : z \u2208 s\nzr : z \u2208 closedBall x r\n\u22a2 \u2191A (z - x) \u2208 \u2191A '' closedBall 0 r\n\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\nh\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nx : E\nr : \u211d\nxs : x \u2208 s\nr0 : 0 \u2264 r\nz : E\nzs : z \u2208 s\nzr : z \u2208 closedBall x r\n\u22a2 f z - f x - \u2191A (z - x) + f x \u2208 closedBall (f x) (\u03b5 * r)\n\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\nh\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nx : E\nr : \u211d\nxs : x \u2208 s\nr0 : 0 \u2264 r\nz : E\nzs : z \u2208 s\nzr : z \u2208 closedBall x r\n\u22a2 \u2191A (z - x) + (f z - f x - \u2191A (z - x) + f x) = f z"}, {"tactic": "apply mem_image_of_mem", "annotated_tactic": ["apply <a>mem_image_of_mem</a>", [{"full_name": "Set.mem_image_of_mem", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [240, 9], "def_end_pos": [240, 25]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\nh\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nx : E\nr : \u211d\nxs : x \u2208 s\nr0 : 0 \u2264 r\nz : E\nzs : z \u2208 s\nzr : z \u2208 closedBall x r\n\u22a2 \u2191A (z - x) \u2208 \u2191A '' closedBall 0 r", "state_after": "case h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\nh\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nx : E\nr : \u211d\nxs : x \u2208 s\nr0 : 0 \u2264 r\nz : E\nzs : z \u2208 s\nzr : z \u2208 closedBall x r\n\u22a2 z - x \u2208 closedBall 0 r"}, {"tactic": "simpa only [dist_eq_norm, mem_closedBall, mem_closedBall_zero_iff, sub_zero] using zr", "annotated_tactic": ["simpa only [<a>dist_eq_norm</a>, <a>mem_closedBall</a>, <a>mem_closedBall_zero_iff</a>, <a>sub_zero</a>] using zr", [{"full_name": "dist_eq_norm", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [383, 7], "def_end_pos": [383, 19]}, {"full_name": "Metric.mem_closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [478, 17], "def_end_pos": [478, 31]}, {"full_name": "mem_closedBall_zero_iff", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [652, 3], "def_end_pos": [652, 14]}, {"full_name": "sub_zero", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [339, 3], "def_end_pos": [339, 14]}]], "state_before": "case h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\nh\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nx : E\nr : \u211d\nxs : x \u2208 s\nr0 : 0 \u2264 r\nz : E\nzs : z \u2208 s\nzr : z \u2208 closedBall x r\n\u22a2 z - x \u2208 closedBall 0 r", "state_after": "no goals"}, {"tactic": "rw [mem_closedBall_iff_norm, add_sub_cancel]", "annotated_tactic": ["rw [<a>mem_closedBall_iff_norm</a>, <a>add_sub_cancel</a>]", [{"full_name": "mem_closedBall_iff_norm", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [646, 15], "def_end_pos": [646, 38]}, {"full_name": "add_sub_cancel", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [739, 30], "def_end_pos": [739, 44]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\nh\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nx : E\nr : \u211d\nxs : x \u2208 s\nr0 : 0 \u2264 r\nz : E\nzs : z \u2208 s\nzr : z \u2208 closedBall x r\n\u22a2 f z - f x - \u2191A (z - x) + f x \u2208 closedBall (f x) (\u03b5 * r)", "state_after": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\nh\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nx : E\nr : \u211d\nxs : x \u2208 s\nr0 : 0 \u2264 r\nz : E\nzs : z \u2208 s\nzr : z \u2208 closedBall x r\n\u22a2 \u2016f z - f x - \u2191A (z - x)\u2016 \u2264 \u03b5 * r"}, {"tactic": "calc\n  \u2016f z - f x - A (z - x)\u2016 \u2264 \u03b4 * \u2016z - x\u2016 := hf _ zs _ xs\n  _ \u2264 \u03b5 * r :=\n    mul_le_mul (le_of_lt h\u03b4) (mem_closedBall_iff_norm.1 zr) (norm_nonneg _) \u03b5pos.le", "annotated_tactic": ["calc\n          \u2016f z - f x - A (z - x)\u2016 \u2264 \u03b4 * \u2016z - x\u2016 := hf _ zs _ xs\n          _ \u2264 \u03b5 * r :=\n            <a>mul_le_mul</a> (<a>le_of_lt</a> h\u03b4) (<a>mem_closedBall_iff_norm</a>.1 zr) (<a>norm_nonneg</a> _) \u03b5pos.le", [{"full_name": "mul_le_mul", "def_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "def_pos": [414, 9], "def_end_pos": [414, 19]}, {"full_name": "le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [110, 9], "def_end_pos": [110, 17]}, {"full_name": "mem_closedBall_iff_norm", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [646, 15], "def_end_pos": [646, 38]}, {"full_name": "norm_nonneg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [500, 30], "def_end_pos": [500, 41]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\nh\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nx : E\nr : \u211d\nxs : x \u2208 s\nr0 : 0 \u2264 r\nz : E\nzs : z \u2208 s\nzr : z \u2208 closedBall x r\n\u22a2 \u2016f z - f x - \u2191A (z - x)\u2016 \u2264 \u03b5 * r", "state_after": "no goals"}, {"tactic": "simp only [map_sub, Pi.sub_apply]", "annotated_tactic": ["simp only [<a>map_sub</a>, <a>Pi.sub_apply</a>]", [{"full_name": "map_sub", "def_path": "Mathlib/Algebra/Hom/Group/Defs.lean", "def_pos": [428, 3], "def_end_pos": [428, 14]}, {"full_name": "Pi.sub_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [200, 3], "def_end_pos": [200, 14]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\nh\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nx : E\nr : \u211d\nxs : x \u2208 s\nr0 : 0 \u2264 r\nz : E\nzs : z \u2208 s\nzr : z \u2208 closedBall x r\n\u22a2 \u2191A (z - x) + (f z - f x - \u2191A (z - x) + f x) = f z", "state_after": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\nh\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nx : E\nr : \u211d\nxs : x \u2208 s\nr0 : 0 \u2264 r\nz : E\nzs : z \u2208 s\nzr : z \u2208 closedBall x r\n\u22a2 \u2191A z - \u2191A x + (f z - f x - (\u2191A z - \u2191A x) + f x) = f z"}, {"tactic": "abel", "annotated_tactic": ["abel", []], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\nh\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nx : E\nr : \u211d\nxs : x \u2208 s\nr0 : 0 \u2264 r\nz : E\nzs : z \u2208 s\nzr : z \u2208 closedBall x r\n\u22a2 \u2191A z - \u2191A x + (f z - f x - (\u2191A z - \u2191A x) + f x) = f z", "state_after": "no goals"}, {"tactic": "rw [smul_add, \u2190 add_assoc, add_comm {f x}, add_assoc, smul_closedBall _ _ \u03b5pos.le, smul_zero,\n  singleton_add_closedBall_zero, \u2190 image_smul_set \u211d E E A, smul_closedBall _ _ zero_le_one,\n  smul_zero, Real.norm_eq_abs, abs_of_nonneg r0, mul_one, mul_comm]", "annotated_tactic": ["rw [<a>smul_add</a>, \u2190 <a>add_assoc</a>, <a>add_comm</a> {f x}, <a>add_assoc</a>, <a>smul_closedBall</a> _ _ \u03b5pos.le, <a>smul_zero</a>,\n        <a>singleton_add_closedBall_zero</a>, \u2190 <a>image_smul_set</a> \u211d E E A, <a>smul_closedBall</a> _ _ <a>zero_le_one</a>,\n        <a>smul_zero</a>, <a>Real.norm_eq_abs</a>, <a>abs_of_nonneg</a> r0, <a>mul_one</a>, <a>mul_comm</a>]", [{"full_name": "smul_add", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [807, 9], "def_end_pos": [807, 17]}, {"full_name": "add_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [263, 3], "def_end_pos": [263, 14]}, {"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [301, 3], "def_end_pos": [301, 14]}, {"full_name": "add_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [263, 3], "def_end_pos": [263, 14]}, {"full_name": "smul_closedBall", "def_path": "Mathlib/Analysis/NormedSpace/Pointwise.lean", "def_pos": [388, 9], "def_end_pos": [388, 24]}, {"full_name": "smul_zero", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [732, 9], "def_end_pos": [732, 18]}, {"full_name": "singleton_add_closedBall_zero", "def_path": "Mathlib/Analysis/Normed/Group/Pointwise.lean", "def_pos": [177, 3], "def_end_pos": [177, 14]}, {"full_name": "image_smul_set", "def_path": "Mathlib/Algebra/Module/LinearMap.lean", "def_pos": [394, 9], "def_end_pos": [394, 30]}, {"full_name": "smul_closedBall", "def_path": "Mathlib/Analysis/NormedSpace/Pointwise.lean", "def_pos": [388, 9], "def_end_pos": [388, 24]}, {"full_name": "zero_le_one", "def_path": "Mathlib/Algebra/Order/ZeroLEOne.lean", "def_pos": [26, 15], "def_end_pos": [26, 26]}, {"full_name": "smul_zero", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [732, 9], "def_end_pos": [732, 18]}, {"full_name": "Real.norm_eq_abs", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [1761, 9], "def_end_pos": [1761, 20]}, {"full_name": "abs_of_nonneg", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [107, 9], "def_end_pos": [107, 22]}, {"full_name": "mul_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [470, 9], "def_end_pos": [470, 16]}, {"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [302, 9], "def_end_pos": [302, 17]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\nh\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nx : E\nr : \u211d\nxs : x \u2208 s\nr0 : 0 \u2264 r\nK : f '' (s \u2229 closedBall x r) \u2286 \u2191A '' closedBall 0 r + closedBall (f x) (\u03b5 * r)\n\u22a2 \u2191A '' closedBall 0 r + closedBall (f x) (\u03b5 * r) = {f x} + r \u2022 (\u2191A '' closedBall 0 1 + closedBall 0 \u03b5)", "state_after": "no goals"}, {"tactic": "simp only [abs_of_nonneg r0, addHaar_smul, image_add_left, abs_pow, singleton_add,\n  measure_preimage_add]", "annotated_tactic": ["simp only [<a>abs_of_nonneg</a> r0, <a>addHaar_smul</a>, <a>image_add_left</a>, <a>abs_pow</a>, <a>singleton_add</a>,\n          <a>measure_preimage_add</a>]", [{"full_name": "abs_of_nonneg", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [107, 9], "def_end_pos": [107, 22]}, {"full_name": "MeasureTheory.Measure.addHaar_smul", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean", "def_pos": [371, 9], "def_end_pos": [371, 21]}, {"full_name": "Set.image_add_left", "def_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "def_pos": [1198, 3], "def_end_pos": [1198, 14]}, {"full_name": "abs_pow", "def_path": "Mathlib/Algebra/GroupPower/Lemmas.lean", "def_pos": [705, 9], "def_end_pos": [705, 16]}, {"full_name": "Set.singleton_add", "def_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "def_pos": [402, 3], "def_end_pos": [402, 14]}, {"full_name": "MeasureTheory.measure_preimage_add", "def_path": "Mathlib/MeasureTheory/Group/Measure.lean", "def_pos": [317, 3], "def_end_pos": [317, 14]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis\u271d : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\nh\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nx : E\nr : \u211d\nxs : x \u2208 s\nr0 : 0 \u2264 r\nK : f '' (s \u2229 closedBall x r) \u2286 {f x} + r \u2022 (\u2191A '' closedBall 0 1 + closedBall 0 \u03b5)\nthis : \u2191A '' closedBall 0 r + closedBall (f x) (\u03b5 * r) = {f x} + r \u2022 (\u2191A '' closedBall 0 1 + closedBall 0 \u03b5)\n\u22a2 \u2191\u2191\u03bc ({f x} + r \u2022 (\u2191A '' closedBall 0 1 + closedBall 0 \u03b5)) =\n    ENNReal.ofReal (r ^ finrank \u211d E) * \u2191\u2191\u03bc (\u2191A '' closedBall 0 1 + closedBall 0 \u03b5)", "state_after": "no goals"}, {"tactic": "rw [add_comm]", "annotated_tactic": ["rw [<a>add_comm</a>]", [{"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [301, 3], "def_end_pos": [301, 14]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis\u271d : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\nh\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nx : E\nr : \u211d\nxs : x \u2208 s\nr0 : 0 \u2264 r\nK : f '' (s \u2229 closedBall x r) \u2286 {f x} + r \u2022 (\u2191A '' closedBall 0 1 + closedBall 0 \u03b5)\nthis : \u2191A '' closedBall 0 r + closedBall (f x) (\u03b5 * r) = {f x} + r \u2022 (\u2191A '' closedBall 0 1 + closedBall 0 \u03b5)\n\u22a2 ENNReal.ofReal (r ^ finrank \u211d E) * \u2191\u2191\u03bc (\u2191A '' closedBall 0 1 + closedBall 0 \u03b5) \u2264\n    ENNReal.ofReal (r ^ finrank \u211d E) * (\u2191m * \u2191\u2191\u03bc (closedBall 0 1))", "state_after": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis\u271d : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\nh\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nx : E\nr : \u211d\nxs : x \u2208 s\nr0 : 0 \u2264 r\nK : f '' (s \u2229 closedBall x r) \u2286 {f x} + r \u2022 (\u2191A '' closedBall 0 1 + closedBall 0 \u03b5)\nthis : \u2191A '' closedBall 0 r + closedBall (f x) (\u03b5 * r) = {f x} + r \u2022 (\u2191A '' closedBall 0 1 + closedBall 0 \u03b5)\n\u22a2 ENNReal.ofReal (r ^ finrank \u211d E) * \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) \u2264\n    ENNReal.ofReal (r ^ finrank \u211d E) * (\u2191m * \u2191\u2191\u03bc (closedBall 0 1))"}, {"tactic": "exact mul_le_mul_left' h\u03b5.le _", "annotated_tactic": ["exact <a>mul_le_mul_left'</a> h\u03b5.le _", [{"full_name": "mul_le_mul_left'", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [50, 9], "def_end_pos": [50, 25]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis\u271d : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\nh\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nx : E\nr : \u211d\nxs : x \u2208 s\nr0 : 0 \u2264 r\nK : f '' (s \u2229 closedBall x r) \u2286 {f x} + r \u2022 (\u2191A '' closedBall 0 1 + closedBall 0 \u03b5)\nthis : \u2191A '' closedBall 0 r + closedBall (f x) (\u03b5 * r) = {f x} + r \u2022 (\u2191A '' closedBall 0 1 + closedBall 0 \u03b5)\n\u22a2 ENNReal.ofReal (r ^ finrank \u211d E) * \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) \u2264\n    ENNReal.ofReal (r ^ finrank \u211d E) * (\u2191m * \u2191\u2191\u03bc (closedBall 0 1))", "state_after": "no goals"}, {"tactic": "simp only [addHaar_closedBall' \u03bc _ r0]", "annotated_tactic": ["simp only [<a>addHaar_closedBall'</a> \u03bc _ r0]", [{"full_name": "MeasureTheory.Measure.addHaar_closedBall'", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean", "def_pos": [481, 9], "def_end_pos": [481, 28]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis\u271d : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\nh\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nx : E\nr : \u211d\nxs : x \u2208 s\nr0 : 0 \u2264 r\nK : f '' (s \u2229 closedBall x r) \u2286 {f x} + r \u2022 (\u2191A '' closedBall 0 1 + closedBall 0 \u03b5)\nthis : \u2191A '' closedBall 0 r + closedBall (f x) (\u03b5 * r) = {f x} + r \u2022 (\u2191A '' closedBall 0 1 + closedBall 0 \u03b5)\n\u22a2 ENNReal.ofReal (r ^ finrank \u211d E) * (\u2191m * \u2191\u2191\u03bc (closedBall 0 1)) = \u2191m * \u2191\u2191\u03bc (closedBall x r)", "state_after": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis\u271d : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\nh\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nx : E\nr : \u211d\nxs : x \u2208 s\nr0 : 0 \u2264 r\nK : f '' (s \u2229 closedBall x r) \u2286 {f x} + r \u2022 (\u2191A '' closedBall 0 1 + closedBall 0 \u03b5)\nthis : \u2191A '' closedBall 0 r + closedBall (f x) (\u03b5 * r) = {f x} + r \u2022 (\u2191A '' closedBall 0 1 + closedBall 0 \u03b5)\n\u22a2 ENNReal.ofReal (r ^ finrank \u211d E) * (\u2191m * \u2191\u2191\u03bc (closedBall 0 1)) =\n    \u2191m * (ENNReal.ofReal (r ^ finrank \u211d E) * \u2191\u2191\u03bc (closedBall 0 1))"}, {"tactic": "ring", "annotated_tactic": ["ring", []], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis\u271d : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\nh\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nx : E\nr : \u211d\nxs : x \u2208 s\nr0 : 0 \u2264 r\nK : f '' (s \u2229 closedBall x r) \u2286 {f x} + r \u2022 (\u2191A '' closedBall 0 1 + closedBall 0 \u03b5)\nthis : \u2191A '' closedBall 0 r + closedBall (f x) (\u03b5 * r) = {f x} + r \u2022 (\u2191A '' closedBall 0 1 + closedBall 0 \u03b5)\n\u22a2 ENNReal.ofReal (r ^ finrank \u211d E) * (\u2191m * \u2191\u2191\u03bc (closedBall 0 1)) =\n    \u2191m * (ENNReal.ofReal (r ^ finrank \u211d E) * \u2191\u2191\u03bc (closedBall 0 1))", "state_after": "no goals"}, {"tactic": "filter_upwards [self_mem_nhdsWithin] with a ha", "annotated_tactic": ["filter_upwards [<a>self_mem_nhdsWithin</a>] with a ha", [{"full_name": "self_mem_nhdsWithin", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [151, 9], "def_end_pos": [151, 28]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\nh\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nI : \u2200 (x : E) (r : \u211d), x \u2208 s \u2192 0 \u2264 r \u2192 \u2191\u2191\u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u2191\u2191\u03bc (closedBall x r)\n\u22a2 \u2200\u1da0 (a : \u211d\u22650\u221e) in \ud835\udcdd[Ioi 0] 0, \u2191\u2191\u03bc (f '' s) \u2264 \u2191m * (\u2191\u2191\u03bc s + a)", "state_after": "case h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\nh\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nI : \u2200 (x : E) (r : \u211d), x \u2208 s \u2192 0 \u2264 r \u2192 \u2191\u2191\u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u2191\u2191\u03bc (closedBall x r)\na : \u211d\u22650\u221e\nha : a \u2208 Ioi 0\n\u22a2 \u2191\u2191\u03bc (f '' s) \u2264 \u2191m * (\u2191\u2191\u03bc s + a)"}, {"tactic": "change 0 < a at ha", "annotated_tactic": ["change 0 < a at ha", []], "state_before": "case h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\nh\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nI : \u2200 (x : E) (r : \u211d), x \u2208 s \u2192 0 \u2264 r \u2192 \u2191\u2191\u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u2191\u2191\u03bc (closedBall x r)\na : \u211d\u22650\u221e\nha : a \u2208 Ioi 0\n\u22a2 \u2191\u2191\u03bc (f '' s) \u2264 \u2191m * (\u2191\u2191\u03bc s + a)", "state_after": "case h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\nh\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nI : \u2200 (x : E) (r : \u211d), x \u2208 s \u2192 0 \u2264 r \u2192 \u2191\u2191\u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u2191\u2191\u03bc (closedBall x r)\na : \u211d\u22650\u221e\nha : 0 < a\n\u22a2 \u2191\u2191\u03bc (f '' s) \u2264 \u2191m * (\u2191\u2191\u03bc s + a)"}, {"tactic": "obtain \u27e8t, r, t_count, ts, rpos, st, \u03bct\u27e9 :\n  \u2203 (t : Set E) (r : E \u2192 \u211d),\n    t.Countable \u2227\n      t \u2286 s \u2227\n        (\u2200 x : E, x \u2208 t \u2192 0 < r x) \u2227\n          (s \u2286 \u22c3 x \u2208 t, closedBall x (r x)) \u2227\n            (\u2211' x : \u21a5t, \u03bc (closedBall (\u2191x) (r \u2191x))) \u2264 \u03bc s + a :=\n  Besicovitch.exists_closedBall_covering_tsum_measure_le \u03bc ha.ne' (fun _ => Ioi 0) s\n    fun x _ \u03b4 \u03b4pos => \u27e8\u03b4 / 2, by simp [half_pos \u03b4pos, \u03b4pos]\u27e9", "annotated_tactic": ["obtain \u27e8t, r, t_count, ts, rpos, st, \u03bct\u27e9 :\n      \u2203 (t : <a>Set</a> E) (r : E \u2192 \u211d),\n        t.Countable \u2227\n          t \u2286 s \u2227\n            (\u2200 x : E, x \u2208 t \u2192 0 < r x) \u2227\n              (s \u2286 \u22c3 x \u2208 t, <a>closedBall</a> x (r x)) \u2227\n                (\u2211' x : \u21a5t, \u03bc (<a>closedBall</a> (\u2191x) (r \u2191x))) \u2264 \u03bc s + a :=\n      <a>Besicovitch.exists_closedBall_covering_tsum_measure_le</a> \u03bc ha.ne' (fun _ => <a>Ioi</a> 0) s\n        fun x _ \u03b4 \u03b4pos => \u27e8\u03b4 / 2, by simp [<a>half_pos</a> \u03b4pos, \u03b4pos]\u27e9", [{"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "Besicovitch.exists_closedBall_covering_tsum_measure_le", "def_path": "Mathlib/MeasureTheory/Covering/Besicovitch.lean", "def_pos": [916, 9], "def_end_pos": [916, 51]}, {"full_name": "Set.Ioi", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [79, 5], "def_end_pos": [79, 8]}, {"full_name": "half_pos", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [504, 9], "def_end_pos": [504, 17]}]], "state_before": "case h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\nh\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nI : \u2200 (x : E) (r : \u211d), x \u2208 s \u2192 0 \u2264 r \u2192 \u2191\u2191\u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u2191\u2191\u03bc (closedBall x r)\na : \u211d\u22650\u221e\nha : 0 < a\n\u22a2 \u2191\u2191\u03bc (f '' s) \u2264 \u2191m * (\u2191\u2191\u03bc s + a)", "state_after": "case h.intro.intro.intro.intro.intro.intro\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\nh\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nI : \u2200 (x : E) (r : \u211d), x \u2208 s \u2192 0 \u2264 r \u2192 \u2191\u2191\u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u2191\u2191\u03bc (closedBall x r)\na : \u211d\u22650\u221e\nha : 0 < a\nt : Set E\nr : E \u2192 \u211d\nt_count : Set.Countable t\nts : t \u2286 s\nrpos : \u2200 (x : E), x \u2208 t \u2192 0 < r x\nst : s \u2286 \u22c3 x \u2208 t, closedBall x (r x)\n\u03bct : \u2211' (x : \u2191t), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + a\n\u22a2 \u2191\u2191\u03bc (f '' s) \u2264 \u2191m * (\u2191\u2191\u03bc s + a)"}, {"tactic": "haveI : Encodable t := t_count.toEncodable", "annotated_tactic": ["haveI : <a>Encodable</a> t := t_count.toEncodable", [{"full_name": "Encodable", "def_path": "Mathlib/Logic/Encodable/Basic.lean", "def_pos": [45, 7], "def_end_pos": [45, 16]}]], "state_before": "case h.intro.intro.intro.intro.intro.intro\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\nh\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nI : \u2200 (x : E) (r : \u211d), x \u2208 s \u2192 0 \u2264 r \u2192 \u2191\u2191\u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u2191\u2191\u03bc (closedBall x r)\na : \u211d\u22650\u221e\nha : 0 < a\nt : Set E\nr : E \u2192 \u211d\nt_count : Set.Countable t\nts : t \u2286 s\nrpos : \u2200 (x : E), x \u2208 t \u2192 0 < r x\nst : s \u2286 \u22c3 x \u2208 t, closedBall x (r x)\n\u03bct : \u2211' (x : \u2191t), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + a\n\u22a2 \u2191\u2191\u03bc (f '' s) \u2264 \u2191m * (\u2191\u2191\u03bc s + a)", "state_after": "case h.intro.intro.intro.intro.intro.intro\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis\u271d : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\nh\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nI : \u2200 (x : E) (r : \u211d), x \u2208 s \u2192 0 \u2264 r \u2192 \u2191\u2191\u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u2191\u2191\u03bc (closedBall x r)\na : \u211d\u22650\u221e\nha : 0 < a\nt : Set E\nr : E \u2192 \u211d\nt_count : Set.Countable t\nts : t \u2286 s\nrpos : \u2200 (x : E), x \u2208 t \u2192 0 < r x\nst : s \u2286 \u22c3 x \u2208 t, closedBall x (r x)\n\u03bct : \u2211' (x : \u2191t), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + a\nthis : Encodable \u2191t\n\u22a2 \u2191\u2191\u03bc (f '' s) \u2264 \u2191m * (\u2191\u2191\u03bc s + a)"}, {"tactic": "calc\n  \u03bc (f '' s) \u2264 \u03bc (\u22c3 x : t, f '' (s \u2229 closedBall x (r x))) := by\n    rw [biUnion_eq_iUnion] at st\n    apply measure_mono\n    rw [\u2190 image_iUnion, \u2190 inter_iUnion]\n    exact image_subset _ (subset_inter (Subset.refl _) st)\n  _ \u2264 \u2211' x : t, \u03bc (f '' (s \u2229 closedBall x (r x))) := (measure_iUnion_le _)\n  _ \u2264 \u2211' x : t, m * \u03bc (closedBall x (r x)) :=\n    (ENNReal.tsum_le_tsum fun x => I x (r x) (ts x.2) (rpos x x.2).le)\n  _ \u2264 m * (\u03bc s + a) := by rw [ENNReal.tsum_mul_left]; exact mul_le_mul_left' \u03bct _", "annotated_tactic": ["calc\n      \u03bc (f '' s) \u2264 \u03bc (\u22c3 x : t, f '' (s \u2229 <a>closedBall</a> x (r x))) := by\n        rw [<a>biUnion_eq_iUnion</a>] at st\n        apply <a>measure_mono</a>\n        rw [\u2190 <a>image_iUnion</a>, \u2190 <a>inter_iUnion</a>]\n        exact <a>image_subset</a> _ (<a>subset_inter</a> (<a>Subset.refl</a> _) st)\n      _ \u2264 \u2211' x : t, \u03bc (f '' (s \u2229 <a>closedBall</a> x (r x))) := (<a>measure_iUnion_le</a> _)\n      _ \u2264 \u2211' x : t, m * \u03bc (<a>closedBall</a> x (r x)) :=\n        (<a>ENNReal.tsum_le_tsum</a> fun x => I x (r x) (ts x.2) (rpos x x.2).<a>le</a>)\n      _ \u2264 m * (\u03bc s + a) := by rw [<a>ENNReal.tsum_mul_left</a>]; exact <a>mul_le_mul_left'</a> \u03bct _", [{"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "Set.biUnion_eq_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [1010, 9], "def_end_pos": [1010, 26]}, {"full_name": "MeasureTheory.measure_mono", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [193, 9], "def_end_pos": [193, 21]}, {"full_name": "Set.image_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [1791, 9], "def_end_pos": [1791, 21]}, {"full_name": "Set.inter_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [635, 9], "def_end_pos": [635, 21]}, {"full_name": "Set.image_subset", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [321, 9], "def_end_pos": [321, 21]}, {"full_name": "Set.subset_inter", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [972, 9], "def_end_pos": [972, 21]}, {"full_name": "Set.Subset.refl", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [354, 9], "def_end_pos": [354, 20]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "MeasureTheory.measure_iUnion_le", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [240, 9], "def_end_pos": [240, 26]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "ENNReal.tsum_le_tsum", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [827, 19], "def_end_pos": [827, 31]}, {"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [142, 7], "def_end_pos": [142, 15]}, {"full_name": "ENNReal.tsum_mul_left", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [897, 19], "def_end_pos": [897, 32]}, {"full_name": "mul_le_mul_left'", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [50, 9], "def_end_pos": [50, 25]}]], "state_before": "case h.intro.intro.intro.intro.intro.intro\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis\u271d : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\nh\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nI : \u2200 (x : E) (r : \u211d), x \u2208 s \u2192 0 \u2264 r \u2192 \u2191\u2191\u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u2191\u2191\u03bc (closedBall x r)\na : \u211d\u22650\u221e\nha : 0 < a\nt : Set E\nr : E \u2192 \u211d\nt_count : Set.Countable t\nts : t \u2286 s\nrpos : \u2200 (x : E), x \u2208 t \u2192 0 < r x\nst : s \u2286 \u22c3 x \u2208 t, closedBall x (r x)\n\u03bct : \u2211' (x : \u2191t), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + a\nthis : Encodable \u2191t\n\u22a2 \u2191\u2191\u03bc (f '' s) \u2264 \u2191m * (\u2191\u2191\u03bc s + a)", "state_after": "no goals"}, {"tactic": "simp [half_pos \u03b4pos, \u03b4pos]", "annotated_tactic": ["simp [<a>half_pos</a> \u03b4pos, \u03b4pos]", [{"full_name": "half_pos", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [504, 9], "def_end_pos": [504, 17]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0\n\u03b4\u271d : \u211d\u22650\nh\u03b4 : \u03b4\u271d \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\u271d\nI : \u2200 (x : E) (r : \u211d), x \u2208 s \u2192 0 \u2264 r \u2192 \u2191\u2191\u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u2191\u2191\u03bc (closedBall x r)\na : \u211d\u22650\u221e\nha : 0 < a\nx : E\nx\u271d : x \u2208 s\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\n\u22a2 \u03b4 / 2 \u2208 (fun x => Ioi 0) x \u2229 Ioo 0 \u03b4", "state_after": "no goals"}, {"tactic": "rw [biUnion_eq_iUnion] at st", "annotated_tactic": ["rw [<a>biUnion_eq_iUnion</a>] at st", [{"full_name": "Set.biUnion_eq_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [1010, 9], "def_end_pos": [1010, 26]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis\u271d : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\nh\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nI : \u2200 (x : E) (r : \u211d), x \u2208 s \u2192 0 \u2264 r \u2192 \u2191\u2191\u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u2191\u2191\u03bc (closedBall x r)\na : \u211d\u22650\u221e\nha : 0 < a\nt : Set E\nr : E \u2192 \u211d\nt_count : Set.Countable t\nts : t \u2286 s\nrpos : \u2200 (x : E), x \u2208 t \u2192 0 < r x\nst : s \u2286 \u22c3 x \u2208 t, closedBall x (r x)\n\u03bct : \u2211' (x : \u2191t), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + a\nthis : Encodable \u2191t\n\u22a2 \u2191\u2191\u03bc (f '' s) \u2264 \u2191\u2191\u03bc (\u22c3 x, f '' (s \u2229 closedBall (\u2191x) (r \u2191x)))", "state_after": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis\u271d : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\nh\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nI : \u2200 (x : E) (r : \u211d), x \u2208 s \u2192 0 \u2264 r \u2192 \u2191\u2191\u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u2191\u2191\u03bc (closedBall x r)\na : \u211d\u22650\u221e\nha : 0 < a\nt : Set E\nr : E \u2192 \u211d\nt_count : Set.Countable t\nts : t \u2286 s\nrpos : \u2200 (x : E), x \u2208 t \u2192 0 < r x\nst : s \u2286 \u22c3 x, closedBall (\u2191x) (r \u2191x)\n\u03bct : \u2211' (x : \u2191t), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + a\nthis : Encodable \u2191t\n\u22a2 \u2191\u2191\u03bc (f '' s) \u2264 \u2191\u2191\u03bc (\u22c3 x, f '' (s \u2229 closedBall (\u2191x) (r \u2191x)))"}, {"tactic": "apply measure_mono", "annotated_tactic": ["apply <a>measure_mono</a>", [{"full_name": "MeasureTheory.measure_mono", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [193, 9], "def_end_pos": [193, 21]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis\u271d : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\nh\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nI : \u2200 (x : E) (r : \u211d), x \u2208 s \u2192 0 \u2264 r \u2192 \u2191\u2191\u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u2191\u2191\u03bc (closedBall x r)\na : \u211d\u22650\u221e\nha : 0 < a\nt : Set E\nr : E \u2192 \u211d\nt_count : Set.Countable t\nts : t \u2286 s\nrpos : \u2200 (x : E), x \u2208 t \u2192 0 < r x\nst : s \u2286 \u22c3 x, closedBall (\u2191x) (r \u2191x)\n\u03bct : \u2211' (x : \u2191t), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + a\nthis : Encodable \u2191t\n\u22a2 \u2191\u2191\u03bc (f '' s) \u2264 \u2191\u2191\u03bc (\u22c3 x, f '' (s \u2229 closedBall (\u2191x) (r \u2191x)))", "state_after": "case h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis\u271d : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\nh\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nI : \u2200 (x : E) (r : \u211d), x \u2208 s \u2192 0 \u2264 r \u2192 \u2191\u2191\u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u2191\u2191\u03bc (closedBall x r)\na : \u211d\u22650\u221e\nha : 0 < a\nt : Set E\nr : E \u2192 \u211d\nt_count : Set.Countable t\nts : t \u2286 s\nrpos : \u2200 (x : E), x \u2208 t \u2192 0 < r x\nst : s \u2286 \u22c3 x, closedBall (\u2191x) (r \u2191x)\n\u03bct : \u2211' (x : \u2191t), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + a\nthis : Encodable \u2191t\n\u22a2 f '' s \u2286 \u22c3 x, f '' (s \u2229 closedBall (\u2191x) (r \u2191x))"}, {"tactic": "rw [\u2190 image_iUnion, \u2190 inter_iUnion]", "annotated_tactic": ["rw [\u2190 <a>image_iUnion</a>, \u2190 <a>inter_iUnion</a>]", [{"full_name": "Set.image_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [1791, 9], "def_end_pos": [1791, 21]}, {"full_name": "Set.inter_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [635, 9], "def_end_pos": [635, 21]}]], "state_before": "case h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis\u271d : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\nh\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nI : \u2200 (x : E) (r : \u211d), x \u2208 s \u2192 0 \u2264 r \u2192 \u2191\u2191\u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u2191\u2191\u03bc (closedBall x r)\na : \u211d\u22650\u221e\nha : 0 < a\nt : Set E\nr : E \u2192 \u211d\nt_count : Set.Countable t\nts : t \u2286 s\nrpos : \u2200 (x : E), x \u2208 t \u2192 0 < r x\nst : s \u2286 \u22c3 x, closedBall (\u2191x) (r \u2191x)\n\u03bct : \u2211' (x : \u2191t), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + a\nthis : Encodable \u2191t\n\u22a2 f '' s \u2286 \u22c3 x, f '' (s \u2229 closedBall (\u2191x) (r \u2191x))", "state_after": "case h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis\u271d : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\nh\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nI : \u2200 (x : E) (r : \u211d), x \u2208 s \u2192 0 \u2264 r \u2192 \u2191\u2191\u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u2191\u2191\u03bc (closedBall x r)\na : \u211d\u22650\u221e\nha : 0 < a\nt : Set E\nr : E \u2192 \u211d\nt_count : Set.Countable t\nts : t \u2286 s\nrpos : \u2200 (x : E), x \u2208 t \u2192 0 < r x\nst : s \u2286 \u22c3 x, closedBall (\u2191x) (r \u2191x)\n\u03bct : \u2211' (x : \u2191t), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + a\nthis : Encodable \u2191t\n\u22a2 f '' s \u2286 f '' (s \u2229 \u22c3 i, closedBall (\u2191i) (r \u2191i))"}, {"tactic": "exact image_subset _ (subset_inter (Subset.refl _) st)", "annotated_tactic": ["exact <a>image_subset</a> _ (<a>subset_inter</a> (<a>Subset.refl</a> _) st)", [{"full_name": "Set.image_subset", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [321, 9], "def_end_pos": [321, 21]}, {"full_name": "Set.subset_inter", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [972, 9], "def_end_pos": [972, 21]}, {"full_name": "Set.Subset.refl", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [354, 9], "def_end_pos": [354, 20]}]], "state_before": "case h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis\u271d : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\nh\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nI : \u2200 (x : E) (r : \u211d), x \u2208 s \u2192 0 \u2264 r \u2192 \u2191\u2191\u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u2191\u2191\u03bc (closedBall x r)\na : \u211d\u22650\u221e\nha : 0 < a\nt : Set E\nr : E \u2192 \u211d\nt_count : Set.Countable t\nts : t \u2286 s\nrpos : \u2200 (x : E), x \u2208 t \u2192 0 < r x\nst : s \u2286 \u22c3 x, closedBall (\u2191x) (r \u2191x)\n\u03bct : \u2211' (x : \u2191t), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + a\nthis : Encodable \u2191t\n\u22a2 f '' s \u2286 f '' (s \u2229 \u22c3 i, closedBall (\u2191i) (r \u2191i))", "state_after": "no goals"}, {"tactic": "rw [ENNReal.tsum_mul_left]", "annotated_tactic": ["rw [<a>ENNReal.tsum_mul_left</a>]", [{"full_name": "ENNReal.tsum_mul_left", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [897, 19], "def_end_pos": [897, 32]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis\u271d : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\nh\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nI : \u2200 (x : E) (r : \u211d), x \u2208 s \u2192 0 \u2264 r \u2192 \u2191\u2191\u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u2191\u2191\u03bc (closedBall x r)\na : \u211d\u22650\u221e\nha : 0 < a\nt : Set E\nr : E \u2192 \u211d\nt_count : Set.Countable t\nts : t \u2286 s\nrpos : \u2200 (x : E), x \u2208 t \u2192 0 < r x\nst : s \u2286 \u22c3 x \u2208 t, closedBall x (r x)\n\u03bct : \u2211' (x : \u2191t), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + a\nthis : Encodable \u2191t\n\u22a2 \u2211' (x : \u2191t), \u2191m * \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191m * (\u2191\u2191\u03bc s + a)", "state_after": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis\u271d : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\nh\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nI : \u2200 (x : E) (r : \u211d), x \u2208 s \u2192 0 \u2264 r \u2192 \u2191\u2191\u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u2191\u2191\u03bc (closedBall x r)\na : \u211d\u22650\u221e\nha : 0 < a\nt : Set E\nr : E \u2192 \u211d\nt_count : Set.Countable t\nts : t \u2286 s\nrpos : \u2200 (x : E), x \u2208 t \u2192 0 < r x\nst : s \u2286 \u22c3 x \u2208 t, closedBall x (r x)\n\u03bct : \u2211' (x : \u2191t), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + a\nthis : Encodable \u2191t\n\u22a2 \u2191m * \u2211' (i : \u2191t), \u2191\u2191\u03bc (closedBall (\u2191i) (r \u2191i)) \u2264 \u2191m * (\u2191\u2191\u03bc s + a)"}, {"tactic": "exact mul_le_mul_left' \u03bct _", "annotated_tactic": ["exact <a>mul_le_mul_left'</a> \u03bct _", [{"full_name": "mul_le_mul_left'", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [50, 9], "def_end_pos": [50, 25]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis\u271d : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\nh\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nI : \u2200 (x : E) (r : \u211d), x \u2208 s \u2192 0 \u2264 r \u2192 \u2191\u2191\u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u2191\u2191\u03bc (closedBall x r)\na : \u211d\u22650\u221e\nha : 0 < a\nt : Set E\nr : E \u2192 \u211d\nt_count : Set.Countable t\nts : t \u2286 s\nrpos : \u2200 (x : E), x \u2208 t \u2192 0 < r x\nst : s \u2286 \u22c3 x \u2208 t, closedBall x (r x)\n\u03bct : \u2211' (x : \u2191t), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + a\nthis : Encodable \u2191t\n\u22a2 \u2191m * \u2211' (i : \u2191t), \u2191\u2191\u03bc (closedBall (\u2191i) (r \u2191i)) \u2264 \u2191m * (\u2191\u2191\u03bc s + a)", "state_after": "no goals"}, {"tactic": "apply Tendsto.mono_left _ nhdsWithin_le_nhds", "annotated_tactic": ["apply <a>Tendsto.mono_left</a> _ <a>nhdsWithin_le_nhds</a>", [{"full_name": "Filter.Tendsto.mono_left", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [3036, 9], "def_end_pos": [3036, 26]}, {"full_name": "nhdsWithin_le_nhds", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [204, 9], "def_end_pos": [204, 27]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\nh\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nI : \u2200 (x : E) (r : \u211d), x \u2208 s \u2192 0 \u2264 r \u2192 \u2191\u2191\u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u2191\u2191\u03bc (closedBall x r)\nJ : \u2200\u1da0 (a : \u211d\u22650\u221e) in \ud835\udcdd[Ioi 0] 0, \u2191\u2191\u03bc (f '' s) \u2264 \u2191m * (\u2191\u2191\u03bc s + a)\n\u22a2 Tendsto (fun a => \u2191m * (\u2191\u2191\u03bc s + a)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191m * (\u2191\u2191\u03bc s + 0)))", "state_after": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\nh\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nI : \u2200 (x : E) (r : \u211d), x \u2208 s \u2192 0 \u2264 r \u2192 \u2191\u2191\u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u2191\u2191\u03bc (closedBall x r)\nJ : \u2200\u1da0 (a : \u211d\u22650\u221e) in \ud835\udcdd[Ioi 0] 0, \u2191\u2191\u03bc (f '' s) \u2264 \u2191m * (\u2191\u2191\u03bc s + a)\n\u22a2 Tendsto (fun a => \u2191m * (\u2191\u2191\u03bc s + a)) (\ud835\udcdd 0) (\ud835\udcdd (\u2191m * (\u2191\u2191\u03bc s + 0)))"}, {"tactic": "apply ENNReal.Tendsto.const_mul (tendsto_const_nhds.add tendsto_id)", "annotated_tactic": ["apply <a>ENNReal.Tendsto.const_mul</a> (tendsto_const_nhds.add <a>tendsto_id</a>)", [{"full_name": "ENNReal.Tendsto.const_mul", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [373, 19], "def_end_pos": [373, 36]}, {"full_name": "Filter.tendsto_id", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [3028, 9], "def_end_pos": [3028, 19]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\nh\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nI : \u2200 (x : E) (r : \u211d), x \u2208 s \u2192 0 \u2264 r \u2192 \u2191\u2191\u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u2191\u2191\u03bc (closedBall x r)\nJ : \u2200\u1da0 (a : \u211d\u22650\u221e) in \ud835\udcdd[Ioi 0] 0, \u2191\u2191\u03bc (f '' s) \u2264 \u2191m * (\u2191\u2191\u03bc s + a)\n\u22a2 Tendsto (fun a => \u2191m * (\u2191\u2191\u03bc s + a)) (\ud835\udcdd 0) (\ud835\udcdd (\u2191m * (\u2191\u2191\u03bc s + 0)))", "state_after": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\nh\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nI : \u2200 (x : E) (r : \u211d), x \u2208 s \u2192 0 \u2264 r \u2192 \u2191\u2191\u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u2191\u2191\u03bc (closedBall x r)\nJ : \u2200\u1da0 (a : \u211d\u22650\u221e) in \ud835\udcdd[Ioi 0] 0, \u2191\u2191\u03bc (f '' s) \u2264 \u2191m * (\u2191\u2191\u03bc s + a)\n\u22a2 \u2191\u2191\u03bc s + 0 \u2260 0 \u2228 \u2191m \u2260 \u22a4"}, {"tactic": "simp only [ENNReal.coe_ne_top, Ne.def, or_true_iff, not_false_iff]", "annotated_tactic": ["simp only [<a>ENNReal.coe_ne_top</a>, <a>Ne.def</a>, <a>or_true_iff</a>, <a>not_false_iff</a>]", [{"full_name": "ENNReal.coe_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [302, 17], "def_end_pos": [302, 27]}, {"full_name": "Ne.def", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [59, 9], "def_end_pos": [59, 15]}, {"full_name": "or_true_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [184, 9], "def_end_pos": [184, 20]}, {"full_name": "not_false_iff", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [82, 9], "def_end_pos": [82, 22]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\nh\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nI : \u2200 (x : E) (r : \u211d), x \u2208 s \u2192 0 \u2264 r \u2192 \u2191\u2191\u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u2191\u2191\u03bc (closedBall x r)\nJ : \u2200\u1da0 (a : \u211d\u22650\u221e) in \ud835\udcdd[Ioi 0] 0, \u2191\u2191\u03bc (f '' s) \u2264 \u2191m * (\u2191\u2191\u03bc s + a)\n\u22a2 \u2191\u2191\u03bc s + 0 \u2260 0 \u2228 \u2191m \u2260 \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Hausdorff.lean", "full_name": "MeasureTheory.OuterMeasure.mkMetric_nnreal_smul", "start": [393, 1], "end": [396, 69], "traced_tactics": [{"tactic": "rw [ENNReal.smul_def, ENNReal.smul_def,\n  mkMetric_smul m ENNReal.coe_ne_top (ENNReal.coe_ne_zero.mpr hc)]", "annotated_tactic": ["rw [<a>ENNReal.smul_def</a>, <a>ENNReal.smul_def</a>,\n    <a>mkMetric_smul</a> m <a>ENNReal.coe_ne_top</a> (ENNReal.coe_ne_zero.mpr hc)]", [{"full_name": "ENNReal.smul_def", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [503, 9], "def_end_pos": [503, 17]}, {"full_name": "ENNReal.smul_def", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [503, 9], "def_end_pos": [503, 17]}, {"full_name": "MeasureTheory.OuterMeasure.mkMetric_smul", "def_path": "Mathlib/MeasureTheory/Measure/Hausdorff.lean", "def_pos": [387, 9], "def_end_pos": [387, 22]}, {"full_name": "ENNReal.coe_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [302, 17], "def_end_pos": [302, 27]}]], "state_before": "\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nc : \u211d\u22650\nhc : c \u2260 0\n\u22a2 mkMetric (c \u2022 m) = c \u2022 mkMetric m", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/ProbabilityMassFunction/Constructions.lean", "full_name": "PMF.mem_support_filter_iff", "start": [282, 1], "end": [283, 70], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "full_name": "Substring.ValidFor.takeWhile", "start": [953, 1], "end": [958, 62], "traced_tactics": [{"tactic": "simp only [Substring.takeWhile, takeWhileAux_of_valid]", "annotated_tactic": ["simp only [<a>Substring.takeWhile</a>, <a>takeWhileAux_of_valid</a>]", [{"full_name": "Substring.takeWhile", "def_path": "lake-packages/lean4/src/lean/Init/Data/String/Basic.lean", "def_pos": [645, 15], "def_end_pos": [645, 24]}, {"full_name": "String.takeWhileAux_of_valid", "def_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "def_pos": [764, 9], "def_end_pos": [764, 30]}]], "state_before": "l m r : List Char\np : Char \u2192 Bool\n\u22a2 ValidFor l (List.takeWhile p m) (List.dropWhile p m ++ r)\n    (Substring.takeWhile\n      { str := { data := l ++ m ++ r }, startPos := { byteIdx := utf8Len l },\n        stopPos := { byteIdx := utf8Len l + utf8Len m } }\n      p)", "state_after": "l m r : List Char\np : Char \u2192 Bool\n\u22a2 ValidFor l (List.takeWhile p m) (List.dropWhile p m ++ r)\n    { str := { data := l ++ m ++ r }, startPos := { byteIdx := utf8Len l },\n      stopPos := { byteIdx := utf8Len l + utf8Len (List.takeWhile p m) } }"}, {"tactic": "refine' .of_eq .. <;> simp", "annotated_tactic": ["refine' .of_eq .. <;> simp", []], "state_before": "l m r : List Char\np : Char \u2192 Bool\n\u22a2 ValidFor l (List.takeWhile p m) (List.dropWhile p m ++ r)\n    { str := { data := l ++ m ++ r }, startPos := { byteIdx := utf8Len l },\n      stopPos := { byteIdx := utf8Len l + utf8Len (List.takeWhile p m) } }", "state_after": "case refine'_1\nl m r : List Char\np : Char \u2192 Bool\n\u22a2 m ++ r = List.takeWhile p m ++ (List.dropWhile p m ++ r)"}, {"tactic": "rw [\u2190 List.append_assoc, List.takeWhile_append_dropWhile]", "annotated_tactic": ["rw [\u2190 <a>List.append_assoc</a>, <a>List.takeWhile_append_dropWhile</a>]", [{"full_name": "List.append_assoc", "def_path": "lake-packages/lean4/src/lean/Init/Data/List/Basic.lean", "def_pos": [103, 9], "def_end_pos": [103, 21]}, {"full_name": "List.takeWhile_append_dropWhile", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [1939, 17], "def_end_pos": [1939, 43]}]], "state_before": "case refine'_1\nl m r : List Char\np : Char \u2192 Bool\n\u22a2 m ++ r = List.takeWhile p m ++ (List.dropWhile p m ++ r)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Real.lean", "full_name": "MeasureTheory.set_integral_abs_condexp_le", "start": [117, 1], "end": [141, 65], "traced_tactics": [{"tactic": "by_cases hnm : m \u2264 m0", "annotated_tactic": ["by_cases hnm : m \u2264 m0", []], "state_before": "\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nf : \u03b1 \u2192 \u211d\n\u22a2 \u222b (x : \u03b1) in s, |(\u03bc[f|m]) x| \u2202\u03bc \u2264 \u222b (x : \u03b1) in s, |f x| \u2202\u03bc", "state_after": "case pos\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nf : \u03b1 \u2192 \u211d\nhnm : m \u2264 m0\n\u22a2 \u222b (x : \u03b1) in s, |(\u03bc[f|m]) x| \u2202\u03bc \u2264 \u222b (x : \u03b1) in s, |f x| \u2202\u03bc\n\ncase neg\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nf : \u03b1 \u2192 \u211d\nhnm : \u00acm \u2264 m0\n\u22a2 \u222b (x : \u03b1) in s, |(\u03bc[f|m]) x| \u2202\u03bc \u2264 \u222b (x : \u03b1) in s, |f x| \u2202\u03bc"}, {"tactic": "swap", "annotated_tactic": ["swap", []], "state_before": "case pos\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nf : \u03b1 \u2192 \u211d\nhnm : m \u2264 m0\n\u22a2 \u222b (x : \u03b1) in s, |(\u03bc[f|m]) x| \u2202\u03bc \u2264 \u222b (x : \u03b1) in s, |f x| \u2202\u03bc\n\ncase neg\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nf : \u03b1 \u2192 \u211d\nhnm : \u00acm \u2264 m0\n\u22a2 \u222b (x : \u03b1) in s, |(\u03bc[f|m]) x| \u2202\u03bc \u2264 \u222b (x : \u03b1) in s, |f x| \u2202\u03bc", "state_after": "case neg\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nf : \u03b1 \u2192 \u211d\nhnm : \u00acm \u2264 m0\n\u22a2 \u222b (x : \u03b1) in s, |(\u03bc[f|m]) x| \u2202\u03bc \u2264 \u222b (x : \u03b1) in s, |f x| \u2202\u03bc\n\ncase pos\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nf : \u03b1 \u2192 \u211d\nhnm : m \u2264 m0\n\u22a2 \u222b (x : \u03b1) in s, |(\u03bc[f|m]) x| \u2202\u03bc \u2264 \u222b (x : \u03b1) in s, |f x| \u2202\u03bc"}, {"tactic": "by_cases hfint : Integrable f \u03bc", "annotated_tactic": ["by_cases hfint : <a>Integrable</a> f \u03bc", [{"full_name": "MeasureTheory.Integrable", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [442, 5], "def_end_pos": [442, 15]}]], "state_before": "case pos\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nf : \u03b1 \u2192 \u211d\nhnm : m \u2264 m0\n\u22a2 \u222b (x : \u03b1) in s, |(\u03bc[f|m]) x| \u2202\u03bc \u2264 \u222b (x : \u03b1) in s, |f x| \u2202\u03bc", "state_after": "case pos\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nf : \u03b1 \u2192 \u211d\nhnm : m \u2264 m0\nhfint : Integrable f\n\u22a2 \u222b (x : \u03b1) in s, |(\u03bc[f|m]) x| \u2202\u03bc \u2264 \u222b (x : \u03b1) in s, |f x| \u2202\u03bc\n\ncase neg\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nf : \u03b1 \u2192 \u211d\nhnm : m \u2264 m0\nhfint : \u00acIntegrable f\n\u22a2 \u222b (x : \u03b1) in s, |(\u03bc[f|m]) x| \u2202\u03bc \u2264 \u222b (x : \u03b1) in s, |f x| \u2202\u03bc"}, {"tactic": "swap", "annotated_tactic": ["swap", []], "state_before": "case pos\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nf : \u03b1 \u2192 \u211d\nhnm : m \u2264 m0\nhfint : Integrable f\n\u22a2 \u222b (x : \u03b1) in s, |(\u03bc[f|m]) x| \u2202\u03bc \u2264 \u222b (x : \u03b1) in s, |f x| \u2202\u03bc\n\ncase neg\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nf : \u03b1 \u2192 \u211d\nhnm : m \u2264 m0\nhfint : \u00acIntegrable f\n\u22a2 \u222b (x : \u03b1) in s, |(\u03bc[f|m]) x| \u2202\u03bc \u2264 \u222b (x : \u03b1) in s, |f x| \u2202\u03bc", "state_after": "case neg\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nf : \u03b1 \u2192 \u211d\nhnm : m \u2264 m0\nhfint : \u00acIntegrable f\n\u22a2 \u222b (x : \u03b1) in s, |(\u03bc[f|m]) x| \u2202\u03bc \u2264 \u222b (x : \u03b1) in s, |f x| \u2202\u03bc\n\ncase pos\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nf : \u03b1 \u2192 \u211d\nhnm : m \u2264 m0\nhfint : Integrable f\n\u22a2 \u222b (x : \u03b1) in s, |(\u03bc[f|m]) x| \u2202\u03bc \u2264 \u222b (x : \u03b1) in s, |f x| \u2202\u03bc"}, {"tactic": "rw [this, \u2190 integral_indicator]", "annotated_tactic": ["rw [this, \u2190 <a>integral_indicator</a>]", [{"full_name": "MeasureTheory.integral_indicator", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [169, 9], "def_end_pos": [169, 27]}]], "state_before": "case pos\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nf : \u03b1 \u2192 \u211d\nhnm : m \u2264 m0\nhfint : Integrable f\nthis : \u222b (x : \u03b1) in s, |(\u03bc[f|m]) x| \u2202\u03bc = \u222b (x : \u03b1), |(\u03bc[Set.indicator s f|m]) x| \u2202\u03bc\n\u22a2 \u222b (x : \u03b1) in s, |(\u03bc[f|m]) x| \u2202\u03bc \u2264 \u222b (x : \u03b1) in s, |f x| \u2202\u03bc", "state_after": "case pos\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nf : \u03b1 \u2192 \u211d\nhnm : m \u2264 m0\nhfint : Integrable f\nthis : \u222b (x : \u03b1) in s, |(\u03bc[f|m]) x| \u2202\u03bc = \u222b (x : \u03b1), |(\u03bc[Set.indicator s f|m]) x| \u2202\u03bc\n\u22a2 \u222b (x : \u03b1), |(\u03bc[Set.indicator s f|m]) x| \u2202\u03bc \u2264 \u222b (x : \u03b1), Set.indicator s (fun x => |f x|) x \u2202\u03bc\n\ncase pos\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nf : \u03b1 \u2192 \u211d\nhnm : m \u2264 m0\nhfint : Integrable f\nthis : \u222b (x : \u03b1) in s, |(\u03bc[f|m]) x| \u2202\u03bc = \u222b (x : \u03b1), |(\u03bc[Set.indicator s f|m]) x| \u2202\u03bc\n\u22a2 MeasurableSet s"}, {"tactic": "swap", "annotated_tactic": ["swap", []], "state_before": "case pos\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nf : \u03b1 \u2192 \u211d\nhnm : m \u2264 m0\nhfint : Integrable f\nthis : \u222b (x : \u03b1) in s, |(\u03bc[f|m]) x| \u2202\u03bc = \u222b (x : \u03b1), |(\u03bc[Set.indicator s f|m]) x| \u2202\u03bc\n\u22a2 \u222b (x : \u03b1), |(\u03bc[Set.indicator s f|m]) x| \u2202\u03bc \u2264 \u222b (x : \u03b1), Set.indicator s (fun x => |f x|) x \u2202\u03bc\n\ncase pos\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nf : \u03b1 \u2192 \u211d\nhnm : m \u2264 m0\nhfint : Integrable f\nthis : \u222b (x : \u03b1) in s, |(\u03bc[f|m]) x| \u2202\u03bc = \u222b (x : \u03b1), |(\u03bc[Set.indicator s f|m]) x| \u2202\u03bc\n\u22a2 MeasurableSet s", "state_after": "case pos\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nf : \u03b1 \u2192 \u211d\nhnm : m \u2264 m0\nhfint : Integrable f\nthis : \u222b (x : \u03b1) in s, |(\u03bc[f|m]) x| \u2202\u03bc = \u222b (x : \u03b1), |(\u03bc[Set.indicator s f|m]) x| \u2202\u03bc\n\u22a2 MeasurableSet s\n\ncase pos\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nf : \u03b1 \u2192 \u211d\nhnm : m \u2264 m0\nhfint : Integrable f\nthis : \u222b (x : \u03b1) in s, |(\u03bc[f|m]) x| \u2202\u03bc = \u222b (x : \u03b1), |(\u03bc[Set.indicator s f|m]) x| \u2202\u03bc\n\u22a2 \u222b (x : \u03b1), |(\u03bc[Set.indicator s f|m]) x| \u2202\u03bc \u2264 \u222b (x : \u03b1), Set.indicator s (fun x => |f x|) x \u2202\u03bc"}, {"tactic": "refine' (integral_abs_condexp_le _).trans\n  (le_of_eq <| integral_congr_ae <| eventually_of_forall fun x => _)", "annotated_tactic": ["refine' (<a>integral_abs_condexp_le</a> _).<a>trans</a>\n    (<a>le_of_eq</a> <| <a>integral_congr_ae</a> <| <a>eventually_of_forall</a> fun x => _)", [{"full_name": "MeasureTheory.integral_abs_condexp_le", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Real.lean", "def_pos": [93, 9], "def_end_pos": [93, 32]}, {"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [120, 7], "def_end_pos": [120, 18]}, {"full_name": "le_of_eq", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [72, 9], "def_end_pos": [72, 17]}, {"full_name": "MeasureTheory.integral_congr_ae", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [938, 9], "def_end_pos": [938, 26]}, {"full_name": "Filter.eventually_of_forall", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1111, 9], "def_end_pos": [1111, 29]}]], "state_before": "case pos\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nf : \u03b1 \u2192 \u211d\nhnm : m \u2264 m0\nhfint : Integrable f\nthis : \u222b (x : \u03b1) in s, |(\u03bc[f|m]) x| \u2202\u03bc = \u222b (x : \u03b1), |(\u03bc[Set.indicator s f|m]) x| \u2202\u03bc\n\u22a2 \u222b (x : \u03b1), |(\u03bc[Set.indicator s f|m]) x| \u2202\u03bc \u2264 \u222b (x : \u03b1), Set.indicator s (fun x => |f x|) x \u2202\u03bc", "state_after": "case pos\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nf : \u03b1 \u2192 \u211d\nhnm : m \u2264 m0\nhfint : Integrable f\nthis : \u222b (x : \u03b1) in s, |(\u03bc[f|m]) x| \u2202\u03bc = \u222b (x : \u03b1), |(\u03bc[Set.indicator s f|m]) x| \u2202\u03bc\nx : \u03b1\n\u22a2 (fun x => |Set.indicator s f x|) x = (fun x => Set.indicator s (fun x => |f x|) x) x"}, {"tactic": "simp_rw [\u2190 Real.norm_eq_abs, norm_indicator_eq_indicator_norm]", "annotated_tactic": ["simp_rw [\u2190 <a>Real.norm_eq_abs</a>, <a>norm_indicator_eq_indicator_norm</a>]", [{"full_name": "Real.norm_eq_abs", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [1761, 9], "def_end_pos": [1761, 20]}, {"full_name": "norm_indicator_eq_indicator_norm", "def_path": "Mathlib/Analysis/NormedSpace/IndicatorFunction.lean", "def_pos": [25, 9], "def_end_pos": [25, 41]}]], "state_before": "case pos\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nf : \u03b1 \u2192 \u211d\nhnm : m \u2264 m0\nhfint : Integrable f\nthis : \u222b (x : \u03b1) in s, |(\u03bc[f|m]) x| \u2202\u03bc = \u222b (x : \u03b1), |(\u03bc[Set.indicator s f|m]) x| \u2202\u03bc\nx : \u03b1\n\u22a2 (fun x => |Set.indicator s f x|) x = (fun x => Set.indicator s (fun x => |f x|) x) x", "state_after": "no goals"}, {"tactic": "simp_rw [condexp_of_not_le hnm, Pi.zero_apply, abs_zero, integral_zero]", "annotated_tactic": ["simp_rw [<a>condexp_of_not_le</a> hnm, <a>Pi.zero_apply</a>, <a>abs_zero</a>, <a>integral_zero</a>]", [{"full_name": "MeasureTheory.condexp_of_not_le", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean", "def_pos": [106, 9], "def_end_pos": [106, 26]}, {"full_name": "Pi.zero_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [46, 3], "def_end_pos": [46, 14]}, {"full_name": "abs_zero", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [128, 9], "def_end_pos": [128, 17]}, {"full_name": "MeasureTheory.integral_zero", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [850, 9], "def_end_pos": [850, 22]}]], "state_before": "case neg\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nf : \u03b1 \u2192 \u211d\nhnm : \u00acm \u2264 m0\n\u22a2 \u222b (x : \u03b1) in s, |(\u03bc[f|m]) x| \u2202\u03bc \u2264 \u222b (x : \u03b1) in s, |f x| \u2202\u03bc", "state_after": "case neg\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nf : \u03b1 \u2192 \u211d\nhnm : \u00acm \u2264 m0\n\u22a2 0 \u2264 \u222b (x : \u03b1) in s, |f x| \u2202\u03bc"}, {"tactic": "exact integral_nonneg fun x => abs_nonneg _", "annotated_tactic": ["exact <a>integral_nonneg</a> fun x => <a>abs_nonneg</a> _", [{"full_name": "MeasureTheory.integral_nonneg", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1251, 9], "def_end_pos": [1251, 24]}, {"full_name": "abs_nonneg", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [169, 9], "def_end_pos": [169, 19]}]], "state_before": "case neg\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nf : \u03b1 \u2192 \u211d\nhnm : \u00acm \u2264 m0\n\u22a2 0 \u2264 \u222b (x : \u03b1) in s, |f x| \u2202\u03bc", "state_after": "no goals"}, {"tactic": "simp only [condexp_undef hfint, Pi.zero_apply, abs_zero, integral_const, Algebra.id.smul_eq_mul,\n  mul_zero]", "annotated_tactic": ["simp only [<a>condexp_undef</a> hfint, <a>Pi.zero_apply</a>, <a>abs_zero</a>, <a>integral_const</a>, <a>Algebra.id.smul_eq_mul</a>,\n      <a>mul_zero</a>]", [{"full_name": "MeasureTheory.condexp_undef", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean", "def_pos": [159, 9], "def_end_pos": [159, 22]}, {"full_name": "Pi.zero_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [46, 3], "def_end_pos": [46, 14]}, {"full_name": "abs_zero", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [128, 9], "def_end_pos": [128, 17]}, {"full_name": "MeasureTheory.integral_const", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1409, 9], "def_end_pos": [1409, 23]}, {"full_name": "Algebra.id.smul_eq_mul", "def_path": "Mathlib/Algebra/Algebra/Basic.lean", "def_pos": [453, 9], "def_end_pos": [453, 20]}, {"full_name": "MulZeroClass.mul_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [38, 3], "def_end_pos": [38, 11]}]], "state_before": "case neg\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nf : \u03b1 \u2192 \u211d\nhnm : m \u2264 m0\nhfint : \u00acIntegrable f\n\u22a2 \u222b (x : \u03b1) in s, |(\u03bc[f|m]) x| \u2202\u03bc \u2264 \u222b (x : \u03b1) in s, |f x| \u2202\u03bc", "state_after": "case neg\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nf : \u03b1 \u2192 \u211d\nhnm : m \u2264 m0\nhfint : \u00acIntegrable f\n\u22a2 0 \u2264 \u222b (x : \u03b1) in s, |f x| \u2202\u03bc"}, {"tactic": "exact integral_nonneg fun x => abs_nonneg _", "annotated_tactic": ["exact <a>integral_nonneg</a> fun x => <a>abs_nonneg</a> _", [{"full_name": "MeasureTheory.integral_nonneg", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1251, 9], "def_end_pos": [1251, 24]}, {"full_name": "abs_nonneg", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [169, 9], "def_end_pos": [169, 19]}]], "state_before": "case neg\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nf : \u03b1 \u2192 \u211d\nhnm : m \u2264 m0\nhfint : \u00acIntegrable f\n\u22a2 0 \u2264 \u222b (x : \u03b1) in s, |f x| \u2202\u03bc", "state_after": "no goals"}, {"tactic": "rw [\u2190 integral_indicator]", "annotated_tactic": ["rw [\u2190 <a>integral_indicator</a>]", [{"full_name": "MeasureTheory.integral_indicator", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [169, 9], "def_end_pos": [169, 27]}]], "state_before": "\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nf : \u03b1 \u2192 \u211d\nhnm : m \u2264 m0\nhfint : Integrable f\n\u22a2 \u222b (x : \u03b1) in s, |(\u03bc[f|m]) x| \u2202\u03bc = \u222b (x : \u03b1), |(\u03bc[Set.indicator s f|m]) x| \u2202\u03bc", "state_after": "\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nf : \u03b1 \u2192 \u211d\nhnm : m \u2264 m0\nhfint : Integrable f\n\u22a2 \u222b (x : \u03b1), Set.indicator s (fun x => |(\u03bc[f|m]) x|) x \u2202\u03bc = \u222b (x : \u03b1), |(\u03bc[Set.indicator s f|m]) x| \u2202\u03bc\n\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nf : \u03b1 \u2192 \u211d\nhnm : m \u2264 m0\nhfint : Integrable f\n\u22a2 MeasurableSet s"}, {"tactic": "swap", "annotated_tactic": ["swap", []], "state_before": "\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nf : \u03b1 \u2192 \u211d\nhnm : m \u2264 m0\nhfint : Integrable f\n\u22a2 \u222b (x : \u03b1), Set.indicator s (fun x => |(\u03bc[f|m]) x|) x \u2202\u03bc = \u222b (x : \u03b1), |(\u03bc[Set.indicator s f|m]) x| \u2202\u03bc\n\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nf : \u03b1 \u2192 \u211d\nhnm : m \u2264 m0\nhfint : Integrable f\n\u22a2 MeasurableSet s", "state_after": "\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nf : \u03b1 \u2192 \u211d\nhnm : m \u2264 m0\nhfint : Integrable f\n\u22a2 MeasurableSet s\n\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nf : \u03b1 \u2192 \u211d\nhnm : m \u2264 m0\nhfint : Integrable f\n\u22a2 \u222b (x : \u03b1), Set.indicator s (fun x => |(\u03bc[f|m]) x|) x \u2202\u03bc = \u222b (x : \u03b1), |(\u03bc[Set.indicator s f|m]) x| \u2202\u03bc"}, {"tactic": "refine' integral_congr_ae _", "annotated_tactic": ["refine' <a>integral_congr_ae</a> _", [{"full_name": "MeasureTheory.integral_congr_ae", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [938, 9], "def_end_pos": [938, 26]}]], "state_before": "\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nf : \u03b1 \u2192 \u211d\nhnm : m \u2264 m0\nhfint : Integrable f\n\u22a2 \u222b (x : \u03b1), Set.indicator s (fun x => |(\u03bc[f|m]) x|) x \u2202\u03bc = \u222b (x : \u03b1), |(\u03bc[Set.indicator s f|m]) x| \u2202\u03bc", "state_after": "\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nf : \u03b1 \u2192 \u211d\nhnm : m \u2264 m0\nhfint : Integrable f\n\u22a2 (fun x => Set.indicator s (fun x => |(\u03bc[f|m]) x|) x) =\u1d50[\u03bc] fun x => |(\u03bc[Set.indicator s f|m]) x|"}, {"tactic": "have : (fun x => |(\u03bc[s.indicator f|m]) x|) =\u1d50[\u03bc] fun x => |s.indicator (\u03bc[f|m]) x| :=\n  EventuallyEq.fun_comp (condexp_indicator hfint hs) _", "annotated_tactic": ["have : (fun x => |(\u03bc[s.indicator f|m]) x|) =\u1d50[\u03bc] fun x => |s.indicator (\u03bc[f|m]) x| :=\n      <a>EventuallyEq.fun_comp</a> (<a>condexp_indicator</a> hfint hs) _", [{"full_name": "Filter.EventuallyEq.fun_comp", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1520, 9], "def_end_pos": [1520, 30]}, {"full_name": "MeasureTheory.condexp_indicator", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Indicator.lean", "def_pos": [75, 9], "def_end_pos": [75, 26]}]], "state_before": "\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nf : \u03b1 \u2192 \u211d\nhnm : m \u2264 m0\nhfint : Integrable f\n\u22a2 (fun x => Set.indicator s (fun x => |(\u03bc[f|m]) x|) x) =\u1d50[\u03bc] fun x => |(\u03bc[Set.indicator s f|m]) x|", "state_after": "\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nf : \u03b1 \u2192 \u211d\nhnm : m \u2264 m0\nhfint : Integrable f\nthis : (fun x => |(\u03bc[Set.indicator s f|m]) x|) =\u1d50[\u03bc] fun x => |Set.indicator s (\u03bc[f|m]) x|\n\u22a2 (fun x => Set.indicator s (fun x => |(\u03bc[f|m]) x|) x) =\u1d50[\u03bc] fun x => |(\u03bc[Set.indicator s f|m]) x|"}, {"tactic": "refine' EventuallyEq.trans (eventually_of_forall fun x => _) this.symm", "annotated_tactic": ["refine' <a>EventuallyEq.trans</a> (<a>eventually_of_forall</a> fun x => _) this.symm", [{"full_name": "Filter.EventuallyEq.trans", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1503, 9], "def_end_pos": [1503, 27]}, {"full_name": "Filter.eventually_of_forall", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1111, 9], "def_end_pos": [1111, 29]}]], "state_before": "\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nf : \u03b1 \u2192 \u211d\nhnm : m \u2264 m0\nhfint : Integrable f\nthis : (fun x => |(\u03bc[Set.indicator s f|m]) x|) =\u1d50[\u03bc] fun x => |Set.indicator s (\u03bc[f|m]) x|\n\u22a2 (fun x => Set.indicator s (fun x => |(\u03bc[f|m]) x|) x) =\u1d50[\u03bc] fun x => |(\u03bc[Set.indicator s f|m]) x|", "state_after": "\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nf : \u03b1 \u2192 \u211d\nhnm : m \u2264 m0\nhfint : Integrable f\nthis : (fun x => |(\u03bc[Set.indicator s f|m]) x|) =\u1d50[\u03bc] fun x => |Set.indicator s (\u03bc[f|m]) x|\nx : \u03b1\n\u22a2 (fun x => Set.indicator s (fun x => |(\u03bc[f|m]) x|) x) x = |Set.indicator s (\u03bc[f|m]) x|"}, {"tactic": "rw [\u2190 Real.norm_eq_abs, norm_indicator_eq_indicator_norm]", "annotated_tactic": ["rw [\u2190 <a>Real.norm_eq_abs</a>, <a>norm_indicator_eq_indicator_norm</a>]", [{"full_name": "Real.norm_eq_abs", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [1761, 9], "def_end_pos": [1761, 20]}, {"full_name": "norm_indicator_eq_indicator_norm", "def_path": "Mathlib/Analysis/NormedSpace/IndicatorFunction.lean", "def_pos": [25, 9], "def_end_pos": [25, 41]}]], "state_before": "\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nf : \u03b1 \u2192 \u211d\nhnm : m \u2264 m0\nhfint : Integrable f\nthis : (fun x => |(\u03bc[Set.indicator s f|m]) x|) =\u1d50[\u03bc] fun x => |Set.indicator s (\u03bc[f|m]) x|\nx : \u03b1\n\u22a2 (fun x => Set.indicator s (fun x => |(\u03bc[f|m]) x|) x) x = |Set.indicator s (\u03bc[f|m]) x|", "state_after": "\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nf : \u03b1 \u2192 \u211d\nhnm : m \u2264 m0\nhfint : Integrable f\nthis : (fun x => |(\u03bc[Set.indicator s f|m]) x|) =\u1d50[\u03bc] fun x => |Set.indicator s (\u03bc[f|m]) x|\nx : \u03b1\n\u22a2 (fun x => Set.indicator s (fun x => |(\u03bc[f|m]) x|) x) x = Set.indicator s (fun a => \u2016(\u03bc[f|m]) a\u2016) x"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nf : \u03b1 \u2192 \u211d\nhnm : m \u2264 m0\nhfint : Integrable f\nthis : (fun x => |(\u03bc[Set.indicator s f|m]) x|) =\u1d50[\u03bc] fun x => |Set.indicator s (\u03bc[f|m]) x|\nx : \u03b1\n\u22a2 (fun x => Set.indicator s (fun x => |(\u03bc[f|m]) x|) x) x = Set.indicator s (fun a => \u2016(\u03bc[f|m]) a\u2016) x", "state_after": "no goals"}, {"tactic": "exact hnm _ hs", "annotated_tactic": ["exact hnm _ hs", []], "state_before": "\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nf : \u03b1 \u2192 \u211d\nhnm : m \u2264 m0\nhfint : Integrable f\n\u22a2 MeasurableSet s", "state_after": "no goals"}, {"tactic": "exact hnm _ hs", "annotated_tactic": ["exact hnm _ hs", []], "state_before": "case pos\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nf : \u03b1 \u2192 \u211d\nhnm : m \u2264 m0\nhfint : Integrable f\nthis : \u222b (x : \u03b1) in s, |(\u03bc[f|m]) x| \u2202\u03bc = \u222b (x : \u03b1), |(\u03bc[Set.indicator s f|m]) x| \u2202\u03bc\n\u22a2 MeasurableSet s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean", "full_name": "ae_restrict_of_ae_restrict_inter_Ioo", "start": [644, 1], "end": [673, 28], "traced_tactics": [{"tactic": "let T : s \u00d7 s \u2192 Set \u211d := fun p => Ioo p.1 p.2", "annotated_tactic": ["let T : s \u00d7 s \u2192 <a>Set</a> \u211d := fun p => <a>Ioo</a> p.1 p.2", [{"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}, {"full_name": "Set.Ioo", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [44, 5], "def_end_pos": [44, 8]}]], "state_before": "\u03bc : Measure \u211d\ninst\u271d : NoAtoms \u03bc\ns : Set \u211d\np : \u211d \u2192 Prop\nh : \u2200 (a b : \u211d), a \u2208 s \u2192 b \u2208 s \u2192 a < b \u2192 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \u2229 Ioo a b), p x\n\u22a2 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc s, p x", "state_after": "\u03bc : Measure \u211d\ninst\u271d : NoAtoms \u03bc\ns : Set \u211d\np : \u211d \u2192 Prop\nh : \u2200 (a b : \u211d), a \u2208 s \u2192 b \u2208 s \u2192 a < b \u2192 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \u2229 Ioo a b), p x\nT : \u2191s \u00d7 \u2191s \u2192 Set \u211d := fun p => Ioo \u2191p.1 \u2191p.2\n\u22a2 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc s, p x"}, {"tactic": "let u := \u22c3 i : \u21a5s \u00d7 \u21a5s, T i", "annotated_tactic": ["let u := \u22c3 i : \u21a5s \u00d7 \u21a5s, T i", []], "state_before": "\u03bc : Measure \u211d\ninst\u271d : NoAtoms \u03bc\ns : Set \u211d\np : \u211d \u2192 Prop\nh : \u2200 (a b : \u211d), a \u2208 s \u2192 b \u2208 s \u2192 a < b \u2192 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \u2229 Ioo a b), p x\nT : \u2191s \u00d7 \u2191s \u2192 Set \u211d := fun p => Ioo \u2191p.1 \u2191p.2\n\u22a2 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc s, p x", "state_after": "\u03bc : Measure \u211d\ninst\u271d : NoAtoms \u03bc\ns : Set \u211d\np : \u211d \u2192 Prop\nh : \u2200 (a b : \u211d), a \u2208 s \u2192 b \u2208 s \u2192 a < b \u2192 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \u2229 Ioo a b), p x\nT : \u2191s \u00d7 \u2191s \u2192 Set \u211d := fun p => Ioo \u2191p.1 \u2191p.2\nu : Set \u211d := \u22c3 i, T i\n\u22a2 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc s, p x"}, {"tactic": "have hfinite : (s \\ u).Finite := s.finite_diff_iUnion_Ioo'", "annotated_tactic": ["have hfinite : (s \\ u).<a>Finite</a> := s.finite_diff_iUnion_Ioo'", [{"full_name": "Set.Finite", "def_path": "Mathlib/Data/Set/Finite.lean", "def_pos": [61, 11], "def_end_pos": [61, 17]}]], "state_before": "\u03bc : Measure \u211d\ninst\u271d : NoAtoms \u03bc\ns : Set \u211d\np : \u211d \u2192 Prop\nh : \u2200 (a b : \u211d), a \u2208 s \u2192 b \u2208 s \u2192 a < b \u2192 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \u2229 Ioo a b), p x\nT : \u2191s \u00d7 \u2191s \u2192 Set \u211d := fun p => Ioo \u2191p.1 \u2191p.2\nu : Set \u211d := \u22c3 i, T i\n\u22a2 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc s, p x", "state_after": "\u03bc : Measure \u211d\ninst\u271d : NoAtoms \u03bc\ns : Set \u211d\np : \u211d \u2192 Prop\nh : \u2200 (a b : \u211d), a \u2208 s \u2192 b \u2208 s \u2192 a < b \u2192 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \u2229 Ioo a b), p x\nT : \u2191s \u00d7 \u2191s \u2192 Set \u211d := fun p => Ioo \u2191p.1 \u2191p.2\nu : Set \u211d := \u22c3 i, T i\nhfinite : Set.Finite (s \\ u)\n\u22a2 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc s, p x"}, {"tactic": "obtain \u27e8A, A_count, hA\u27e9 :\n  \u2203 A : Set (\u21a5s \u00d7 \u21a5s), A.Countable \u2227 \u22c3 i \u2208 A, T i = \u22c3 i : \u21a5s \u00d7 \u21a5s, T i :=\n  isOpen_iUnion_countable _ fun p => isOpen_Ioo", "annotated_tactic": ["obtain \u27e8A, A_count, hA\u27e9 :\n    \u2203 A : <a>Set</a> (\u21a5s \u00d7 \u21a5s), A.Countable \u2227 \u22c3 i \u2208 A, T i = \u22c3 i : \u21a5s \u00d7 \u21a5s, T i :=\n    <a>isOpen_iUnion_countable</a> _ fun p => <a>isOpen_Ioo</a>", [{"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}, {"full_name": "TopologicalSpace.isOpen_iUnion_countable", "def_path": "Mathlib/Topology/Bases.lean", "def_pos": [794, 9], "def_end_pos": [794, 32]}, {"full_name": "isOpen_Ioo", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [336, 9], "def_end_pos": [336, 19]}]], "state_before": "\u03bc : Measure \u211d\ninst\u271d : NoAtoms \u03bc\ns : Set \u211d\np : \u211d \u2192 Prop\nh : \u2200 (a b : \u211d), a \u2208 s \u2192 b \u2208 s \u2192 a < b \u2192 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \u2229 Ioo a b), p x\nT : \u2191s \u00d7 \u2191s \u2192 Set \u211d := fun p => Ioo \u2191p.1 \u2191p.2\nu : Set \u211d := \u22c3 i, T i\nhfinite : Set.Finite (s \\ u)\n\u22a2 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc s, p x", "state_after": "case intro.intro\n\u03bc : Measure \u211d\ninst\u271d : NoAtoms \u03bc\ns : Set \u211d\np : \u211d \u2192 Prop\nh : \u2200 (a b : \u211d), a \u2208 s \u2192 b \u2208 s \u2192 a < b \u2192 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \u2229 Ioo a b), p x\nT : \u2191s \u00d7 \u2191s \u2192 Set \u211d := fun p => Ioo \u2191p.1 \u2191p.2\nu : Set \u211d := \u22c3 i, T i\nhfinite : Set.Finite (s \\ u)\nA : Set (\u2191s \u00d7 \u2191s)\nA_count : Set.Countable A\nhA : \u22c3 i \u2208 A, T i = \u22c3 i, T i\n\u22a2 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc s, p x"}, {"tactic": "apply ae_restrict_of_ae_restrict_of_subset this", "annotated_tactic": ["apply <a>ae_restrict_of_ae_restrict_of_subset</a> this", [{"full_name": "MeasureTheory.ae_restrict_of_ae_restrict_of_subset", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2606, 9], "def_end_pos": [2606, 45]}]], "state_before": "case intro.intro\n\u03bc : Measure \u211d\ninst\u271d : NoAtoms \u03bc\ns : Set \u211d\np : \u211d \u2192 Prop\nh : \u2200 (a b : \u211d), a \u2208 s \u2192 b \u2208 s \u2192 a < b \u2192 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \u2229 Ioo a b), p x\nT : \u2191s \u00d7 \u2191s \u2192 Set \u211d := fun p => Ioo \u2191p.1 \u2191p.2\nu : Set \u211d := \u22c3 i, T i\nhfinite : Set.Finite (s \\ u)\nA : Set (\u2191s \u00d7 \u2191s)\nA_count : Set.Countable A\nhA : \u22c3 i \u2208 A, T i = \u22c3 i, T i\nthis : s \u2286 s \\ u \u222a \u22c3 p \u2208 A, s \u2229 T p\n\u22a2 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc s, p x", "state_after": "case intro.intro\n\u03bc : Measure \u211d\ninst\u271d : NoAtoms \u03bc\ns : Set \u211d\np : \u211d \u2192 Prop\nh : \u2200 (a b : \u211d), a \u2208 s \u2192 b \u2208 s \u2192 a < b \u2192 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \u2229 Ioo a b), p x\nT : \u2191s \u00d7 \u2191s \u2192 Set \u211d := fun p => Ioo \u2191p.1 \u2191p.2\nu : Set \u211d := \u22c3 i, T i\nhfinite : Set.Finite (s \\ u)\nA : Set (\u2191s \u00d7 \u2191s)\nA_count : Set.Countable A\nhA : \u22c3 i \u2208 A, T i = \u22c3 i, T i\nthis : s \u2286 s \\ u \u222a \u22c3 p \u2208 A, s \u2229 T p\n\u22a2 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \\ u \u222a \u22c3 p \u2208 A, s \u2229 T p), p x"}, {"tactic": "rw [ae_restrict_union_iff, ae_restrict_biUnion_iff _ A_count]", "annotated_tactic": ["rw [<a>ae_restrict_union_iff</a>, <a>ae_restrict_biUnion_iff</a> _ A_count]", [{"full_name": "MeasureTheory.ae_restrict_union_iff", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2518, 9], "def_end_pos": [2518, 30]}, {"full_name": "MeasureTheory.ae_restrict_biUnion_iff", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2522, 9], "def_end_pos": [2522, 32]}]], "state_before": "case intro.intro\n\u03bc : Measure \u211d\ninst\u271d : NoAtoms \u03bc\ns : Set \u211d\np : \u211d \u2192 Prop\nh : \u2200 (a b : \u211d), a \u2208 s \u2192 b \u2208 s \u2192 a < b \u2192 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \u2229 Ioo a b), p x\nT : \u2191s \u00d7 \u2191s \u2192 Set \u211d := fun p => Ioo \u2191p.1 \u2191p.2\nu : Set \u211d := \u22c3 i, T i\nhfinite : Set.Finite (s \\ u)\nA : Set (\u2191s \u00d7 \u2191s)\nA_count : Set.Countable A\nhA : \u22c3 i \u2208 A, T i = \u22c3 i, T i\nthis : s \u2286 s \\ u \u222a \u22c3 p \u2208 A, s \u2229 T p\n\u22a2 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \\ u \u222a \u22c3 p \u2208 A, s \u2229 T p), p x", "state_after": "case intro.intro\n\u03bc : Measure \u211d\ninst\u271d : NoAtoms \u03bc\ns : Set \u211d\np : \u211d \u2192 Prop\nh : \u2200 (a b : \u211d), a \u2208 s \u2192 b \u2208 s \u2192 a < b \u2192 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \u2229 Ioo a b), p x\nT : \u2191s \u00d7 \u2191s \u2192 Set \u211d := fun p => Ioo \u2191p.1 \u2191p.2\nu : Set \u211d := \u22c3 i, T i\nhfinite : Set.Finite (s \\ u)\nA : Set (\u2191s \u00d7 \u2191s)\nA_count : Set.Countable A\nhA : \u22c3 i \u2208 A, T i = \u22c3 i, T i\nthis : s \u2286 s \\ u \u222a \u22c3 p \u2208 A, s \u2229 T p\n\u22a2 (\u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \\ u), p x) \u2227 \u2200 (i : \u2191s \u00d7 \u2191s), i \u2208 A \u2192 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \u2229 T i), p x"}, {"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "case intro.intro\n\u03bc : Measure \u211d\ninst\u271d : NoAtoms \u03bc\ns : Set \u211d\np : \u211d \u2192 Prop\nh : \u2200 (a b : \u211d), a \u2208 s \u2192 b \u2208 s \u2192 a < b \u2192 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \u2229 Ioo a b), p x\nT : \u2191s \u00d7 \u2191s \u2192 Set \u211d := fun p => Ioo \u2191p.1 \u2191p.2\nu : Set \u211d := \u22c3 i, T i\nhfinite : Set.Finite (s \\ u)\nA : Set (\u2191s \u00d7 \u2191s)\nA_count : Set.Countable A\nhA : \u22c3 i \u2208 A, T i = \u22c3 i, T i\nthis : s \u2286 s \\ u \u222a \u22c3 p \u2208 A, s \u2229 T p\n\u22a2 (\u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \\ u), p x) \u2227 \u2200 (i : \u2191s \u00d7 \u2191s), i \u2208 A \u2192 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \u2229 T i), p x", "state_after": "case intro.intro.left\n\u03bc : Measure \u211d\ninst\u271d : NoAtoms \u03bc\ns : Set \u211d\np : \u211d \u2192 Prop\nh : \u2200 (a b : \u211d), a \u2208 s \u2192 b \u2208 s \u2192 a < b \u2192 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \u2229 Ioo a b), p x\nT : \u2191s \u00d7 \u2191s \u2192 Set \u211d := fun p => Ioo \u2191p.1 \u2191p.2\nu : Set \u211d := \u22c3 i, T i\nhfinite : Set.Finite (s \\ u)\nA : Set (\u2191s \u00d7 \u2191s)\nA_count : Set.Countable A\nhA : \u22c3 i \u2208 A, T i = \u22c3 i, T i\nthis : s \u2286 s \\ u \u222a \u22c3 p \u2208 A, s \u2229 T p\n\u22a2 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \\ u), p x\n\ncase intro.intro.right\n\u03bc : Measure \u211d\ninst\u271d : NoAtoms \u03bc\ns : Set \u211d\np : \u211d \u2192 Prop\nh : \u2200 (a b : \u211d), a \u2208 s \u2192 b \u2208 s \u2192 a < b \u2192 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \u2229 Ioo a b), p x\nT : \u2191s \u00d7 \u2191s \u2192 Set \u211d := fun p => Ioo \u2191p.1 \u2191p.2\nu : Set \u211d := \u22c3 i, T i\nhfinite : Set.Finite (s \\ u)\nA : Set (\u2191s \u00d7 \u2191s)\nA_count : Set.Countable A\nhA : \u22c3 i \u2208 A, T i = \u22c3 i, T i\nthis : s \u2286 s \\ u \u222a \u22c3 p \u2208 A, s \u2229 T p\n\u22a2 \u2200 (i : \u2191s \u00d7 \u2191s), i \u2208 A \u2192 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \u2229 T i), p x"}, {"tactic": "intro x hx", "annotated_tactic": ["intro x hx", []], "state_before": "\u03bc : Measure \u211d\ninst\u271d : NoAtoms \u03bc\ns : Set \u211d\np : \u211d \u2192 Prop\nh : \u2200 (a b : \u211d), a \u2208 s \u2192 b \u2208 s \u2192 a < b \u2192 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \u2229 Ioo a b), p x\nT : \u2191s \u00d7 \u2191s \u2192 Set \u211d := fun p => Ioo \u2191p.1 \u2191p.2\nu : Set \u211d := \u22c3 i, T i\nhfinite : Set.Finite (s \\ u)\nA : Set (\u2191s \u00d7 \u2191s)\nA_count : Set.Countable A\nhA : \u22c3 i \u2208 A, T i = \u22c3 i, T i\n\u22a2 s \u2286 s \\ u \u222a \u22c3 p \u2208 A, s \u2229 T p", "state_after": "\u03bc : Measure \u211d\ninst\u271d : NoAtoms \u03bc\ns : Set \u211d\np : \u211d \u2192 Prop\nh : \u2200 (a b : \u211d), a \u2208 s \u2192 b \u2208 s \u2192 a < b \u2192 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \u2229 Ioo a b), p x\nT : \u2191s \u00d7 \u2191s \u2192 Set \u211d := fun p => Ioo \u2191p.1 \u2191p.2\nu : Set \u211d := \u22c3 i, T i\nhfinite : Set.Finite (s \\ u)\nA : Set (\u2191s \u00d7 \u2191s)\nA_count : Set.Countable A\nhA : \u22c3 i \u2208 A, T i = \u22c3 i, T i\nx : \u211d\nhx : x \u2208 s\n\u22a2 x \u2208 s \\ u \u222a \u22c3 p \u2208 A, s \u2229 T p"}, {"tactic": "by_cases h'x : x \u2208 \u22c3 i : \u21a5s \u00d7 \u21a5s, T i", "annotated_tactic": ["by_cases h'x : x \u2208 \u22c3 i : \u21a5s \u00d7 \u21a5s, T i", []], "state_before": "\u03bc : Measure \u211d\ninst\u271d : NoAtoms \u03bc\ns : Set \u211d\np : \u211d \u2192 Prop\nh : \u2200 (a b : \u211d), a \u2208 s \u2192 b \u2208 s \u2192 a < b \u2192 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \u2229 Ioo a b), p x\nT : \u2191s \u00d7 \u2191s \u2192 Set \u211d := fun p => Ioo \u2191p.1 \u2191p.2\nu : Set \u211d := \u22c3 i, T i\nhfinite : Set.Finite (s \\ u)\nA : Set (\u2191s \u00d7 \u2191s)\nA_count : Set.Countable A\nhA : \u22c3 i \u2208 A, T i = \u22c3 i, T i\nx : \u211d\nhx : x \u2208 s\n\u22a2 x \u2208 s \\ u \u222a \u22c3 p \u2208 A, s \u2229 T p", "state_after": "case pos\n\u03bc : Measure \u211d\ninst\u271d : NoAtoms \u03bc\ns : Set \u211d\np : \u211d \u2192 Prop\nh : \u2200 (a b : \u211d), a \u2208 s \u2192 b \u2208 s \u2192 a < b \u2192 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \u2229 Ioo a b), p x\nT : \u2191s \u00d7 \u2191s \u2192 Set \u211d := fun p => Ioo \u2191p.1 \u2191p.2\nu : Set \u211d := \u22c3 i, T i\nhfinite : Set.Finite (s \\ u)\nA : Set (\u2191s \u00d7 \u2191s)\nA_count : Set.Countable A\nhA : \u22c3 i \u2208 A, T i = \u22c3 i, T i\nx : \u211d\nhx : x \u2208 s\nh'x : x \u2208 \u22c3 i, T i\n\u22a2 x \u2208 s \\ u \u222a \u22c3 p \u2208 A, s \u2229 T p\n\ncase neg\n\u03bc : Measure \u211d\ninst\u271d : NoAtoms \u03bc\ns : Set \u211d\np : \u211d \u2192 Prop\nh : \u2200 (a b : \u211d), a \u2208 s \u2192 b \u2208 s \u2192 a < b \u2192 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \u2229 Ioo a b), p x\nT : \u2191s \u00d7 \u2191s \u2192 Set \u211d := fun p => Ioo \u2191p.1 \u2191p.2\nu : Set \u211d := \u22c3 i, T i\nhfinite : Set.Finite (s \\ u)\nA : Set (\u2191s \u00d7 \u2191s)\nA_count : Set.Countable A\nhA : \u22c3 i \u2208 A, T i = \u22c3 i, T i\nx : \u211d\nhx : x \u2208 s\nh'x : \u00acx \u2208 \u22c3 i, T i\n\u22a2 x \u2208 s \\ u \u222a \u22c3 p \u2208 A, s \u2229 T p"}, {"tactic": "rw [\u2190 hA] at h'x", "annotated_tactic": ["rw [\u2190 hA] at h'x", []], "state_before": "case pos\n\u03bc : Measure \u211d\ninst\u271d : NoAtoms \u03bc\ns : Set \u211d\np : \u211d \u2192 Prop\nh : \u2200 (a b : \u211d), a \u2208 s \u2192 b \u2208 s \u2192 a < b \u2192 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \u2229 Ioo a b), p x\nT : \u2191s \u00d7 \u2191s \u2192 Set \u211d := fun p => Ioo \u2191p.1 \u2191p.2\nu : Set \u211d := \u22c3 i, T i\nhfinite : Set.Finite (s \\ u)\nA : Set (\u2191s \u00d7 \u2191s)\nA_count : Set.Countable A\nhA : \u22c3 i \u2208 A, T i = \u22c3 i, T i\nx : \u211d\nhx : x \u2208 s\nh'x : x \u2208 \u22c3 i, T i\n\u22a2 x \u2208 s \\ u \u222a \u22c3 p \u2208 A, s \u2229 T p", "state_after": "case pos\n\u03bc : Measure \u211d\ninst\u271d : NoAtoms \u03bc\ns : Set \u211d\np : \u211d \u2192 Prop\nh : \u2200 (a b : \u211d), a \u2208 s \u2192 b \u2208 s \u2192 a < b \u2192 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \u2229 Ioo a b), p x\nT : \u2191s \u00d7 \u2191s \u2192 Set \u211d := fun p => Ioo \u2191p.1 \u2191p.2\nu : Set \u211d := \u22c3 i, T i\nhfinite : Set.Finite (s \\ u)\nA : Set (\u2191s \u00d7 \u2191s)\nA_count : Set.Countable A\nhA : \u22c3 i \u2208 A, T i = \u22c3 i, T i\nx : \u211d\nhx : x \u2208 s\nh'x : x \u2208 \u22c3 i \u2208 A, T i\n\u22a2 x \u2208 s \\ u \u222a \u22c3 p \u2208 A, s \u2229 T p"}, {"tactic": "obtain \u27e8p, pA, xp\u27e9 : \u2203 p : \u21a5s \u00d7 \u21a5s, p \u2208 A \u2227 x \u2208 T p := by\n  simpa only [mem_iUnion, exists_prop, SetCoe.exists, exists_and_right] using h'x", "annotated_tactic": ["obtain \u27e8p, pA, xp\u27e9 : \u2203 p : \u21a5s \u00d7 \u21a5s, p \u2208 A \u2227 x \u2208 T p := by\n        simpa only [<a>mem_iUnion</a>, <a>exists_prop</a>, <a>SetCoe.exists</a>, <a>exists_and_right</a>] using h'x", [{"full_name": "Set.mem_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [201, 9], "def_end_pos": [201, 19]}, {"full_name": "exists_prop", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [485, 17], "def_end_pos": [485, 28]}, {"full_name": "SetCoe.exists", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [190, 9], "def_end_pos": [190, 22]}, {"full_name": "exists_and_right", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [468, 17], "def_end_pos": [468, 33]}]], "state_before": "case pos\n\u03bc : Measure \u211d\ninst\u271d : NoAtoms \u03bc\ns : Set \u211d\np : \u211d \u2192 Prop\nh : \u2200 (a b : \u211d), a \u2208 s \u2192 b \u2208 s \u2192 a < b \u2192 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \u2229 Ioo a b), p x\nT : \u2191s \u00d7 \u2191s \u2192 Set \u211d := fun p => Ioo \u2191p.1 \u2191p.2\nu : Set \u211d := \u22c3 i, T i\nhfinite : Set.Finite (s \\ u)\nA : Set (\u2191s \u00d7 \u2191s)\nA_count : Set.Countable A\nhA : \u22c3 i \u2208 A, T i = \u22c3 i, T i\nx : \u211d\nhx : x \u2208 s\nh'x : x \u2208 \u22c3 i \u2208 A, T i\n\u22a2 x \u2208 s \\ u \u222a \u22c3 p \u2208 A, s \u2229 T p", "state_after": "case pos.intro.intro\n\u03bc : Measure \u211d\ninst\u271d : NoAtoms \u03bc\ns : Set \u211d\np\u271d : \u211d \u2192 Prop\nh : \u2200 (a b : \u211d), a \u2208 s \u2192 b \u2208 s \u2192 a < b \u2192 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \u2229 Ioo a b), p\u271d x\nT : \u2191s \u00d7 \u2191s \u2192 Set \u211d := fun p => Ioo \u2191p.1 \u2191p.2\nu : Set \u211d := \u22c3 i, T i\nhfinite : Set.Finite (s \\ u)\nA : Set (\u2191s \u00d7 \u2191s)\nA_count : Set.Countable A\nhA : \u22c3 i \u2208 A, T i = \u22c3 i, T i\nx : \u211d\nhx : x \u2208 s\nh'x : x \u2208 \u22c3 i \u2208 A, T i\np : \u2191s \u00d7 \u2191s\npA : p \u2208 A\nxp : x \u2208 T p\n\u22a2 x \u2208 s \\ u \u222a \u22c3 p \u2208 A, s \u2229 T p"}, {"tactic": "right", "annotated_tactic": ["right", []], "state_before": "case pos.intro.intro\n\u03bc : Measure \u211d\ninst\u271d : NoAtoms \u03bc\ns : Set \u211d\np\u271d : \u211d \u2192 Prop\nh : \u2200 (a b : \u211d), a \u2208 s \u2192 b \u2208 s \u2192 a < b \u2192 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \u2229 Ioo a b), p\u271d x\nT : \u2191s \u00d7 \u2191s \u2192 Set \u211d := fun p => Ioo \u2191p.1 \u2191p.2\nu : Set \u211d := \u22c3 i, T i\nhfinite : Set.Finite (s \\ u)\nA : Set (\u2191s \u00d7 \u2191s)\nA_count : Set.Countable A\nhA : \u22c3 i \u2208 A, T i = \u22c3 i, T i\nx : \u211d\nhx : x \u2208 s\nh'x : x \u2208 \u22c3 i \u2208 A, T i\np : \u2191s \u00d7 \u2191s\npA : p \u2208 A\nxp : x \u2208 T p\n\u22a2 x \u2208 s \\ u \u222a \u22c3 p \u2208 A, s \u2229 T p", "state_after": "case pos.intro.intro.h\n\u03bc : Measure \u211d\ninst\u271d : NoAtoms \u03bc\ns : Set \u211d\np\u271d : \u211d \u2192 Prop\nh : \u2200 (a b : \u211d), a \u2208 s \u2192 b \u2208 s \u2192 a < b \u2192 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \u2229 Ioo a b), p\u271d x\nT : \u2191s \u00d7 \u2191s \u2192 Set \u211d := fun p => Ioo \u2191p.1 \u2191p.2\nu : Set \u211d := \u22c3 i, T i\nhfinite : Set.Finite (s \\ u)\nA : Set (\u2191s \u00d7 \u2191s)\nA_count : Set.Countable A\nhA : \u22c3 i \u2208 A, T i = \u22c3 i, T i\nx : \u211d\nhx : x \u2208 s\nh'x : x \u2208 \u22c3 i \u2208 A, T i\np : \u2191s \u00d7 \u2191s\npA : p \u2208 A\nxp : x \u2208 T p\n\u22a2 x \u2208 \u22c3 p \u2208 A, s \u2229 T p"}, {"tactic": "exact mem_biUnion pA \u27e8hx, xp\u27e9", "annotated_tactic": ["exact <a>mem_biUnion</a> pA \u27e8hx, xp\u27e9", [{"full_name": "Set.mem_biUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [966, 9], "def_end_pos": [966, 20]}]], "state_before": "case pos.intro.intro.h\n\u03bc : Measure \u211d\ninst\u271d : NoAtoms \u03bc\ns : Set \u211d\np\u271d : \u211d \u2192 Prop\nh : \u2200 (a b : \u211d), a \u2208 s \u2192 b \u2208 s \u2192 a < b \u2192 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \u2229 Ioo a b), p\u271d x\nT : \u2191s \u00d7 \u2191s \u2192 Set \u211d := fun p => Ioo \u2191p.1 \u2191p.2\nu : Set \u211d := \u22c3 i, T i\nhfinite : Set.Finite (s \\ u)\nA : Set (\u2191s \u00d7 \u2191s)\nA_count : Set.Countable A\nhA : \u22c3 i \u2208 A, T i = \u22c3 i, T i\nx : \u211d\nhx : x \u2208 s\nh'x : x \u2208 \u22c3 i \u2208 A, T i\np : \u2191s \u00d7 \u2191s\npA : p \u2208 A\nxp : x \u2208 T p\n\u22a2 x \u2208 \u22c3 p \u2208 A, s \u2229 T p", "state_after": "no goals"}, {"tactic": "simpa only [mem_iUnion, exists_prop, SetCoe.exists, exists_and_right] using h'x", "annotated_tactic": ["simpa only [<a>mem_iUnion</a>, <a>exists_prop</a>, <a>SetCoe.exists</a>, <a>exists_and_right</a>] using h'x", [{"full_name": "Set.mem_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [201, 9], "def_end_pos": [201, 19]}, {"full_name": "exists_prop", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [485, 17], "def_end_pos": [485, 28]}, {"full_name": "SetCoe.exists", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [190, 9], "def_end_pos": [190, 22]}, {"full_name": "exists_and_right", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [468, 17], "def_end_pos": [468, 33]}]], "state_before": "\u03bc : Measure \u211d\ninst\u271d : NoAtoms \u03bc\ns : Set \u211d\np : \u211d \u2192 Prop\nh : \u2200 (a b : \u211d), a \u2208 s \u2192 b \u2208 s \u2192 a < b \u2192 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \u2229 Ioo a b), p x\nT : \u2191s \u00d7 \u2191s \u2192 Set \u211d := fun p => Ioo \u2191p.1 \u2191p.2\nu : Set \u211d := \u22c3 i, T i\nhfinite : Set.Finite (s \\ u)\nA : Set (\u2191s \u00d7 \u2191s)\nA_count : Set.Countable A\nhA : \u22c3 i \u2208 A, T i = \u22c3 i, T i\nx : \u211d\nhx : x \u2208 s\nh'x : x \u2208 \u22c3 i \u2208 A, T i\n\u22a2 \u2203 p, p \u2208 A \u2227 x \u2208 T p", "state_after": "no goals"}, {"tactic": "exact Or.inl \u27e8hx, h'x\u27e9", "annotated_tactic": ["exact <a>Or.inl</a> \u27e8hx, h'x\u27e9", [{"full_name": "Or.inl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [517, 5], "def_end_pos": [517, 8]}]], "state_before": "case neg\n\u03bc : Measure \u211d\ninst\u271d : NoAtoms \u03bc\ns : Set \u211d\np : \u211d \u2192 Prop\nh : \u2200 (a b : \u211d), a \u2208 s \u2192 b \u2208 s \u2192 a < b \u2192 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \u2229 Ioo a b), p x\nT : \u2191s \u00d7 \u2191s \u2192 Set \u211d := fun p => Ioo \u2191p.1 \u2191p.2\nu : Set \u211d := \u22c3 i, T i\nhfinite : Set.Finite (s \\ u)\nA : Set (\u2191s \u00d7 \u2191s)\nA_count : Set.Countable A\nhA : \u22c3 i \u2208 A, T i = \u22c3 i, T i\nx : \u211d\nhx : x \u2208 s\nh'x : \u00acx \u2208 \u22c3 i, T i\n\u22a2 x \u2208 s \\ u \u222a \u22c3 p \u2208 A, s \u2229 T p", "state_after": "no goals"}, {"tactic": "have : \u03bc.restrict (s \\ u) = 0 := by simp only [restrict_eq_zero, hfinite.measure_zero]", "annotated_tactic": ["have : \u03bc.restrict (s \\ u) = 0 := by simp only [<a>restrict_eq_zero</a>, hfinite.measure_zero]", [{"full_name": "MeasureTheory.Measure.restrict_eq_zero", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1683, 9], "def_end_pos": [1683, 25]}]], "state_before": "case intro.intro.left\n\u03bc : Measure \u211d\ninst\u271d : NoAtoms \u03bc\ns : Set \u211d\np : \u211d \u2192 Prop\nh : \u2200 (a b : \u211d), a \u2208 s \u2192 b \u2208 s \u2192 a < b \u2192 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \u2229 Ioo a b), p x\nT : \u2191s \u00d7 \u2191s \u2192 Set \u211d := fun p => Ioo \u2191p.1 \u2191p.2\nu : Set \u211d := \u22c3 i, T i\nhfinite : Set.Finite (s \\ u)\nA : Set (\u2191s \u00d7 \u2191s)\nA_count : Set.Countable A\nhA : \u22c3 i \u2208 A, T i = \u22c3 i, T i\nthis : s \u2286 s \\ u \u222a \u22c3 p \u2208 A, s \u2229 T p\n\u22a2 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \\ u), p x", "state_after": "case intro.intro.left\n\u03bc : Measure \u211d\ninst\u271d : NoAtoms \u03bc\ns : Set \u211d\np : \u211d \u2192 Prop\nh : \u2200 (a b : \u211d), a \u2208 s \u2192 b \u2208 s \u2192 a < b \u2192 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \u2229 Ioo a b), p x\nT : \u2191s \u00d7 \u2191s \u2192 Set \u211d := fun p => Ioo \u2191p.1 \u2191p.2\nu : Set \u211d := \u22c3 i, T i\nhfinite : Set.Finite (s \\ u)\nA : Set (\u2191s \u00d7 \u2191s)\nA_count : Set.Countable A\nhA : \u22c3 i \u2208 A, T i = \u22c3 i, T i\nthis\u271d : s \u2286 s \\ u \u222a \u22c3 p \u2208 A, s \u2229 T p\nthis : Measure.restrict \u03bc (s \\ u) = 0\n\u22a2 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \\ u), p x"}, {"tactic": "simp only [this, ae_zero, eventually_bot]", "annotated_tactic": ["simp only [this, <a>ae_zero</a>, <a>eventually_bot</a>]", [{"full_name": "MeasureTheory.ae_zero", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2451, 9], "def_end_pos": [2451, 16]}, {"full_name": "Filter.eventually_bot", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1213, 9], "def_end_pos": [1213, 23]}]], "state_before": "case intro.intro.left\n\u03bc : Measure \u211d\ninst\u271d : NoAtoms \u03bc\ns : Set \u211d\np : \u211d \u2192 Prop\nh : \u2200 (a b : \u211d), a \u2208 s \u2192 b \u2208 s \u2192 a < b \u2192 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \u2229 Ioo a b), p x\nT : \u2191s \u00d7 \u2191s \u2192 Set \u211d := fun p => Ioo \u2191p.1 \u2191p.2\nu : Set \u211d := \u22c3 i, T i\nhfinite : Set.Finite (s \\ u)\nA : Set (\u2191s \u00d7 \u2191s)\nA_count : Set.Countable A\nhA : \u22c3 i \u2208 A, T i = \u22c3 i, T i\nthis\u271d : s \u2286 s \\ u \u222a \u22c3 p \u2208 A, s \u2229 T p\nthis : Measure.restrict \u03bc (s \\ u) = 0\n\u22a2 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \\ u), p x", "state_after": "no goals"}, {"tactic": "simp only [restrict_eq_zero, hfinite.measure_zero]", "annotated_tactic": ["simp only [<a>restrict_eq_zero</a>, hfinite.measure_zero]", [{"full_name": "MeasureTheory.Measure.restrict_eq_zero", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1683, 9], "def_end_pos": [1683, 25]}]], "state_before": "\u03bc : Measure \u211d\ninst\u271d : NoAtoms \u03bc\ns : Set \u211d\np : \u211d \u2192 Prop\nh : \u2200 (a b : \u211d), a \u2208 s \u2192 b \u2208 s \u2192 a < b \u2192 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \u2229 Ioo a b), p x\nT : \u2191s \u00d7 \u2191s \u2192 Set \u211d := fun p => Ioo \u2191p.1 \u2191p.2\nu : Set \u211d := \u22c3 i, T i\nhfinite : Set.Finite (s \\ u)\nA : Set (\u2191s \u00d7 \u2191s)\nA_count : Set.Countable A\nhA : \u22c3 i \u2208 A, T i = \u22c3 i, T i\nthis : s \u2286 s \\ u \u222a \u22c3 p \u2208 A, s \u2229 T p\n\u22a2 Measure.restrict \u03bc (s \\ u) = 0", "state_after": "no goals"}, {"tactic": "rintro \u27e8\u27e8a, as\u27e9, \u27e8b, bs\u27e9\u27e9 -", "annotated_tactic": ["rintro \u27e8\u27e8a, as\u27e9, \u27e8b, bs\u27e9\u27e9 -", []], "state_before": "case intro.intro.right\n\u03bc : Measure \u211d\ninst\u271d : NoAtoms \u03bc\ns : Set \u211d\np : \u211d \u2192 Prop\nh : \u2200 (a b : \u211d), a \u2208 s \u2192 b \u2208 s \u2192 a < b \u2192 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \u2229 Ioo a b), p x\nT : \u2191s \u00d7 \u2191s \u2192 Set \u211d := fun p => Ioo \u2191p.1 \u2191p.2\nu : Set \u211d := \u22c3 i, T i\nhfinite : Set.Finite (s \\ u)\nA : Set (\u2191s \u00d7 \u2191s)\nA_count : Set.Countable A\nhA : \u22c3 i \u2208 A, T i = \u22c3 i, T i\nthis : s \u2286 s \\ u \u222a \u22c3 p \u2208 A, s \u2229 T p\n\u22a2 \u2200 (i : \u2191s \u00d7 \u2191s), i \u2208 A \u2192 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \u2229 T i), p x", "state_after": "case intro.intro.right.mk.mk.mk\n\u03bc : Measure \u211d\ninst\u271d : NoAtoms \u03bc\ns : Set \u211d\np : \u211d \u2192 Prop\nh : \u2200 (a b : \u211d), a \u2208 s \u2192 b \u2208 s \u2192 a < b \u2192 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \u2229 Ioo a b), p x\nT : \u2191s \u00d7 \u2191s \u2192 Set \u211d := fun p => Ioo \u2191p.1 \u2191p.2\nu : Set \u211d := \u22c3 i, T i\nhfinite : Set.Finite (s \\ u)\nA : Set (\u2191s \u00d7 \u2191s)\nA_count : Set.Countable A\nhA : \u22c3 i \u2208 A, T i = \u22c3 i, T i\nthis : s \u2286 s \\ u \u222a \u22c3 p \u2208 A, s \u2229 T p\na : \u211d\nas : a \u2208 s\nb : \u211d\nbs : b \u2208 s\n\u22a2 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \u2229 T ({ val := a, property := as }, { val := b, property := bs })), p x"}, {"tactic": "dsimp", "annotated_tactic": ["dsimp", []], "state_before": "case intro.intro.right.mk.mk.mk\n\u03bc : Measure \u211d\ninst\u271d : NoAtoms \u03bc\ns : Set \u211d\np : \u211d \u2192 Prop\nh : \u2200 (a b : \u211d), a \u2208 s \u2192 b \u2208 s \u2192 a < b \u2192 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \u2229 Ioo a b), p x\nT : \u2191s \u00d7 \u2191s \u2192 Set \u211d := fun p => Ioo \u2191p.1 \u2191p.2\nu : Set \u211d := \u22c3 i, T i\nhfinite : Set.Finite (s \\ u)\nA : Set (\u2191s \u00d7 \u2191s)\nA_count : Set.Countable A\nhA : \u22c3 i \u2208 A, T i = \u22c3 i, T i\nthis : s \u2286 s \\ u \u222a \u22c3 p \u2208 A, s \u2229 T p\na : \u211d\nas : a \u2208 s\nb : \u211d\nbs : b \u2208 s\n\u22a2 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \u2229 T ({ val := a, property := as }, { val := b, property := bs })), p x", "state_after": "case intro.intro.right.mk.mk.mk\n\u03bc : Measure \u211d\ninst\u271d : NoAtoms \u03bc\ns : Set \u211d\np : \u211d \u2192 Prop\nh : \u2200 (a b : \u211d), a \u2208 s \u2192 b \u2208 s \u2192 a < b \u2192 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \u2229 Ioo a b), p x\nT : \u2191s \u00d7 \u2191s \u2192 Set \u211d := fun p => Ioo \u2191p.1 \u2191p.2\nu : Set \u211d := \u22c3 i, T i\nhfinite : Set.Finite (s \\ u)\nA : Set (\u2191s \u00d7 \u2191s)\nA_count : Set.Countable A\nhA : \u22c3 i \u2208 A, T i = \u22c3 i, T i\nthis : s \u2286 s \\ u \u222a \u22c3 p \u2208 A, s \u2229 T p\na : \u211d\nas : a \u2208 s\nb : \u211d\nbs : b \u2208 s\n\u22a2 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \u2229 Ioo a b), p x"}, {"tactic": "rcases le_or_lt b a with (hba | hab)", "annotated_tactic": ["rcases <a>le_or_lt</a> b a with (hba | hab)", [{"full_name": "le_or_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [340, 9], "def_end_pos": [340, 17]}]], "state_before": "case intro.intro.right.mk.mk.mk\n\u03bc : Measure \u211d\ninst\u271d : NoAtoms \u03bc\ns : Set \u211d\np : \u211d \u2192 Prop\nh : \u2200 (a b : \u211d), a \u2208 s \u2192 b \u2208 s \u2192 a < b \u2192 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \u2229 Ioo a b), p x\nT : \u2191s \u00d7 \u2191s \u2192 Set \u211d := fun p => Ioo \u2191p.1 \u2191p.2\nu : Set \u211d := \u22c3 i, T i\nhfinite : Set.Finite (s \\ u)\nA : Set (\u2191s \u00d7 \u2191s)\nA_count : Set.Countable A\nhA : \u22c3 i \u2208 A, T i = \u22c3 i, T i\nthis : s \u2286 s \\ u \u222a \u22c3 p \u2208 A, s \u2229 T p\na : \u211d\nas : a \u2208 s\nb : \u211d\nbs : b \u2208 s\n\u22a2 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \u2229 Ioo a b), p x", "state_after": "case intro.intro.right.mk.mk.mk.inl\n\u03bc : Measure \u211d\ninst\u271d : NoAtoms \u03bc\ns : Set \u211d\np : \u211d \u2192 Prop\nh : \u2200 (a b : \u211d), a \u2208 s \u2192 b \u2208 s \u2192 a < b \u2192 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \u2229 Ioo a b), p x\nT : \u2191s \u00d7 \u2191s \u2192 Set \u211d := fun p => Ioo \u2191p.1 \u2191p.2\nu : Set \u211d := \u22c3 i, T i\nhfinite : Set.Finite (s \\ u)\nA : Set (\u2191s \u00d7 \u2191s)\nA_count : Set.Countable A\nhA : \u22c3 i \u2208 A, T i = \u22c3 i, T i\nthis : s \u2286 s \\ u \u222a \u22c3 p \u2208 A, s \u2229 T p\na : \u211d\nas : a \u2208 s\nb : \u211d\nbs : b \u2208 s\nhba : b \u2264 a\n\u22a2 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \u2229 Ioo a b), p x\n\ncase intro.intro.right.mk.mk.mk.inr\n\u03bc : Measure \u211d\ninst\u271d : NoAtoms \u03bc\ns : Set \u211d\np : \u211d \u2192 Prop\nh : \u2200 (a b : \u211d), a \u2208 s \u2192 b \u2208 s \u2192 a < b \u2192 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \u2229 Ioo a b), p x\nT : \u2191s \u00d7 \u2191s \u2192 Set \u211d := fun p => Ioo \u2191p.1 \u2191p.2\nu : Set \u211d := \u22c3 i, T i\nhfinite : Set.Finite (s \\ u)\nA : Set (\u2191s \u00d7 \u2191s)\nA_count : Set.Countable A\nhA : \u22c3 i \u2208 A, T i = \u22c3 i, T i\nthis : s \u2286 s \\ u \u222a \u22c3 p \u2208 A, s \u2229 T p\na : \u211d\nas : a \u2208 s\nb : \u211d\nbs : b \u2208 s\nhab : a < b\n\u22a2 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \u2229 Ioo a b), p x"}, {"tactic": "simp only [Ioo_eq_empty_of_le hba, inter_empty, restrict_empty, ae_zero, eventually_bot]", "annotated_tactic": ["simp only [<a>Ioo_eq_empty_of_le</a> hba, <a>inter_empty</a>, <a>restrict_empty</a>, <a>ae_zero</a>, <a>eventually_bot</a>]", [{"full_name": "Set.Ioo_eq_empty_of_le", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [393, 9], "def_end_pos": [393, 27]}, {"full_name": "Set.inter_empty", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [931, 9], "def_end_pos": [931, 20]}, {"full_name": "MeasureTheory.Measure.restrict_empty", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1696, 9], "def_end_pos": [1696, 23]}, {"full_name": "MeasureTheory.ae_zero", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2451, 9], "def_end_pos": [2451, 16]}, {"full_name": "Filter.eventually_bot", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1213, 9], "def_end_pos": [1213, 23]}]], "state_before": "case intro.intro.right.mk.mk.mk.inl\n\u03bc : Measure \u211d\ninst\u271d : NoAtoms \u03bc\ns : Set \u211d\np : \u211d \u2192 Prop\nh : \u2200 (a b : \u211d), a \u2208 s \u2192 b \u2208 s \u2192 a < b \u2192 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \u2229 Ioo a b), p x\nT : \u2191s \u00d7 \u2191s \u2192 Set \u211d := fun p => Ioo \u2191p.1 \u2191p.2\nu : Set \u211d := \u22c3 i, T i\nhfinite : Set.Finite (s \\ u)\nA : Set (\u2191s \u00d7 \u2191s)\nA_count : Set.Countable A\nhA : \u22c3 i \u2208 A, T i = \u22c3 i, T i\nthis : s \u2286 s \\ u \u222a \u22c3 p \u2208 A, s \u2229 T p\na : \u211d\nas : a \u2208 s\nb : \u211d\nbs : b \u2208 s\nhba : b \u2264 a\n\u22a2 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \u2229 Ioo a b), p x", "state_after": "no goals"}, {"tactic": "exact h a b as bs hab", "annotated_tactic": ["exact h a b as bs hab", []], "state_before": "case intro.intro.right.mk.mk.mk.inr\n\u03bc : Measure \u211d\ninst\u271d : NoAtoms \u03bc\ns : Set \u211d\np : \u211d \u2192 Prop\nh : \u2200 (a b : \u211d), a \u2208 s \u2192 b \u2208 s \u2192 a < b \u2192 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \u2229 Ioo a b), p x\nT : \u2191s \u00d7 \u2191s \u2192 Set \u211d := fun p => Ioo \u2191p.1 \u2191p.2\nu : Set \u211d := \u22c3 i, T i\nhfinite : Set.Finite (s \\ u)\nA : Set (\u2191s \u00d7 \u2191s)\nA_count : Set.Countable A\nhA : \u22c3 i \u2208 A, T i = \u22c3 i, T i\nthis : s \u2286 s \\ u \u222a \u22c3 p \u2208 A, s \u2229 T p\na : \u211d\nas : a \u2208 s\nb : \u211d\nbs : b \u2208 s\nhab : a < b\n\u22a2 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \u2229 Ioo a b), p x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Finite.lean", "full_name": "Finite.Set.subset", "start": [613, 11], "end": [615, 17], "traced_tactics": [{"tactic": "rw [\u2190 sep_eq_of_subset h]", "annotated_tactic": ["rw [\u2190 <a>sep_eq_of_subset</a> h]", [{"full_name": "Set.sep_eq_of_subset", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1441, 9], "def_end_pos": [1441, 25]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\n\u03b3 : Type x\ns t : Set \u03b1\ninst\u271d : Finite \u2191s\nh : t \u2286 s\n\u22a2 Finite \u2191t", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\n\u03b3 : Type x\ns t : Set \u03b1\ninst\u271d : Finite \u2191s\nh : t \u2286 s\n\u22a2 Finite \u2191{x | x \u2208 s \u2227 x \u2208 t}"}, {"tactic": "infer_instance", "annotated_tactic": ["infer_instance", []], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\n\u03b3 : Type x\ns t : Set \u03b1\ninst\u271d : Finite \u2191s\nh : t \u2286 s\n\u22a2 Finite \u2191{x | x \u2208 s \u2227 x \u2208 t}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "full_name": "Nat.recDiagOn_succ_zero", "start": [101, 1], "end": [106, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "full_name": "MeasureTheory.Integrable.induction", "start": [1058, 1], "end": [1067, 70], "traced_tactics": [{"tactic": "simp only [\u2190 mem\u2112p_one_iff_integrable] at *", "annotated_tactic": ["simp only [\u2190 <a>mem\u2112p_one_iff_integrable</a>] at *", [{"full_name": "MeasureTheory.mem\u2112p_one_iff_integrable", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [453, 9], "def_end_pos": [453, 33]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nP : (\u03b1 \u2192 E) \u2192 Prop\nh_ind : \u2200 (c : E) \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 P (indicator s fun x => c)\nh_add : \u2200 \u2983f g : \u03b1 \u2192 E\u2984, Disjoint (support f) (support g) \u2192 Integrable f \u2192 Integrable g \u2192 P f \u2192 P g \u2192 P (f + g)\nh_closed : IsClosed {f | P \u2191\u2191f}\nh_ae : \u2200 \u2983f g : \u03b1 \u2192 E\u2984, f =\u1d50[\u03bc] g \u2192 Integrable f \u2192 P f \u2192 P g\n\u22a2 \u2200 \u2983f : \u03b1 \u2192 E\u2984, Integrable f \u2192 P f", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nP : (\u03b1 \u2192 E) \u2192 Prop\nh_ind : \u2200 (c : E) \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 P (indicator s fun x => c)\nh_closed : IsClosed {f | P \u2191\u2191f}\nh_add : \u2200 \u2983f g : \u03b1 \u2192 E\u2984, Disjoint (support f) (support g) \u2192 Mem\u2112p f 1 \u2192 Mem\u2112p g 1 \u2192 P f \u2192 P g \u2192 P (f + g)\nh_ae : \u2200 \u2983f g : \u03b1 \u2192 E\u2984, f =\u1d50[\u03bc] g \u2192 Mem\u2112p f 1 \u2192 P f \u2192 P g\n\u22a2 \u2200 \u2983f : \u03b1 \u2192 E\u2984, Mem\u2112p f 1 \u2192 P f"}, {"tactic": "exact Mem\u2112p.induction one_ne_top (P := P) h_ind h_add h_closed h_ae", "annotated_tactic": ["exact <a>Mem\u2112p.induction</a> <a>one_ne_top</a> (P := P) h_ind h_add h_closed h_ae", [{"full_name": "MeasureTheory.Mem\u2112p.induction", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "def_pos": [950, 9], "def_end_pos": [950, 24]}, {"full_name": "ENNReal.one_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [340, 17], "def_end_pos": [340, 27]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nP : (\u03b1 \u2192 E) \u2192 Prop\nh_ind : \u2200 (c : E) \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 P (indicator s fun x => c)\nh_closed : IsClosed {f | P \u2191\u2191f}\nh_add : \u2200 \u2983f g : \u03b1 \u2192 E\u2984, Disjoint (support f) (support g) \u2192 Mem\u2112p f 1 \u2192 Mem\u2112p g 1 \u2192 P f \u2192 P g \u2192 P (f + g)\nh_ae : \u2200 \u2983f g : \u03b1 \u2192 E\u2984, f =\u1d50[\u03bc] g \u2192 Mem\u2112p f 1 \u2192 P f \u2192 P g\n\u22a2 \u2200 \u2983f : \u03b1 \u2192 E\u2984, Mem\u2112p f 1 \u2192 P f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Decomposition/SignedHahn.lean", "full_name": "MeasureTheory.SignedMeasure.zero_mem_measureOfNegatives", "start": [339, 1], "end": [340, 61], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "full_name": "ContinuousMap.toLp_norm_eq_toLp_norm_coe", "start": [1895, 1], "end": [1898, 59], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Card.lean", "full_name": "Set.le_ncard_diff", "start": [879, 1], "end": [881, 86], "traced_tactics": [{"tactic": "rw [add_comm]", "annotated_tactic": ["rw [<a>add_comm</a>]", [{"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [301, 3], "def_end_pos": [301, 14]}]], "state_before": "\u03b1 : Type u_1\ns\u271d t\u271d s t : Set \u03b1\nhs : autoParam (Set.Finite s) _auto\u271d\n\u22a2 ncard t \u2264 ncard s + ncard (t \\ s)", "state_after": "\u03b1 : Type u_1\ns\u271d t\u271d s t : Set \u03b1\nhs : autoParam (Set.Finite s) _auto\u271d\n\u22a2 ncard t \u2264 ncard (t \\ s) + ncard s"}, {"tactic": "apply ncard_le_ncard_diff_add_ncard _ _ hs", "annotated_tactic": ["apply <a>ncard_le_ncard_diff_add_ncard</a> _ _ hs", [{"full_name": "Set.ncard_le_ncard_diff_add_ncard", "def_path": "Mathlib/Data/Set/Card.lean", "def_pos": [869, 9], "def_end_pos": [869, 38]}]], "state_before": "\u03b1 : Type u_1\ns\u271d t\u271d s t : Set \u03b1\nhs : autoParam (Set.Finite s) _auto\u271d\n\u22a2 ncard t \u2264 ncard (t \\ s) + ncard s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/MulAntidiagonal.lean", "full_name": "Finset.isPwo_support_mulAntidiagonal", "start": [109, 1], "end": [110, 54], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "full_name": "MeasureTheory.snorm_eq_zero_iff", "start": [764, 1], "end": [769, 60], "traced_tactics": [{"tactic": "by_cases h_top : p = \u221e", "annotated_tactic": ["by_cases h_top : p = \u221e", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nh0 : p \u2260 0\n\u22a2 snorm f p \u03bc = 0 \u2194 f =\u1d50[\u03bc] 0", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nh0 : p \u2260 0\nh_top : p = \u22a4\n\u22a2 snorm f p \u03bc = 0 \u2194 f =\u1d50[\u03bc] 0\n\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nh0 : p \u2260 0\nh_top : \u00acp = \u22a4\n\u22a2 snorm f p \u03bc = 0 \u2194 f =\u1d50[\u03bc] 0"}, {"tactic": "rw [snorm_eq_snorm' h0 h_top]", "annotated_tactic": ["rw [<a>snorm_eq_snorm'</a> h0 h_top]", [{"full_name": "MeasureTheory.snorm_eq_snorm'", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [88, 9], "def_end_pos": [88, 24]}]], "state_before": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nh0 : p \u2260 0\nh_top : \u00acp = \u22a4\n\u22a2 snorm f p \u03bc = 0 \u2194 f =\u1d50[\u03bc] 0", "state_after": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nh0 : p \u2260 0\nh_top : \u00acp = \u22a4\n\u22a2 snorm' f (ENNReal.toReal p) \u03bc = 0 \u2194 f =\u1d50[\u03bc] 0"}, {"tactic": "exact snorm'_eq_zero_iff (ENNReal.toReal_pos h0 h_top) hf", "annotated_tactic": ["exact <a>snorm'_eq_zero_iff</a> (<a>ENNReal.toReal_pos</a> h0 h_top) hf", [{"full_name": "MeasureTheory.snorm'_eq_zero_iff", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [749, 9], "def_end_pos": [749, 27]}, {"full_name": "ENNReal.toReal_pos", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2131, 9], "def_end_pos": [2131, 19]}]], "state_before": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nh0 : p \u2260 0\nh_top : \u00acp = \u22a4\n\u22a2 snorm' f (ENNReal.toReal p) \u03bc = 0 \u2194 f =\u1d50[\u03bc] 0", "state_after": "no goals"}, {"tactic": "rw [h_top, snorm_exponent_top, snormEssSup_eq_zero_iff]", "annotated_tactic": ["rw [h_top, <a>snorm_exponent_top</a>, <a>snormEssSup_eq_zero_iff</a>]", [{"full_name": "MeasureTheory.snorm_exponent_top", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [103, 9], "def_end_pos": [103, 27]}, {"full_name": "MeasureTheory.snormEssSup_eq_zero_iff", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [760, 9], "def_end_pos": [760, 32]}]], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nh0 : p \u2260 0\nh_top : p = \u22a4\n\u22a2 snorm f p \u03bc = 0 \u2194 f =\u1d50[\u03bc] 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Function.lean", "full_name": "Set.image_eq_iff_surjOn_mapsTo", "start": [883, 1], "end": [886, 45], "traced_tactics": [{"tactic": "refine' \u27e8_, fun h => h.1.image_eq_of_mapsTo h.2\u27e9", "annotated_tactic": ["refine' \u27e8_, fun h => h.1.<a>image_eq_of_mapsTo</a> h.2\u27e9", [{"full_name": "Set.SurjOn.image_eq_of_mapsTo", "def_path": "Mathlib/Data/Set/Function.lean", "def_pos": [879, 9], "def_end_pos": [879, 34]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03c0 : \u03b1 \u2192 Type u_5\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\np : Set \u03b3\nf f\u2081 f\u2082 f\u2083 : \u03b1 \u2192 \u03b2\ng g\u2081 g\u2082 : \u03b2 \u2192 \u03b3\nf' f\u2081' f\u2082' : \u03b2 \u2192 \u03b1\ng' : \u03b3 \u2192 \u03b2\na : \u03b1\nb : \u03b2\n\u22a2 f '' s = t \u2194 SurjOn f s t \u2227 MapsTo f s t", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03c0 : \u03b1 \u2192 Type u_5\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\np : Set \u03b3\nf f\u2081 f\u2082 f\u2083 : \u03b1 \u2192 \u03b2\ng g\u2081 g\u2082 : \u03b2 \u2192 \u03b3\nf' f\u2081' f\u2082' : \u03b2 \u2192 \u03b1\ng' : \u03b3 \u2192 \u03b2\na : \u03b1\nb : \u03b2\n\u22a2 f '' s = t \u2192 SurjOn f s t \u2227 MapsTo f s t"}, {"tactic": "rintro rfl", "annotated_tactic": ["rintro rfl", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03c0 : \u03b1 \u2192 Type u_5\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\np : Set \u03b3\nf f\u2081 f\u2082 f\u2083 : \u03b1 \u2192 \u03b2\ng g\u2081 g\u2082 : \u03b2 \u2192 \u03b3\nf' f\u2081' f\u2082' : \u03b2 \u2192 \u03b1\ng' : \u03b3 \u2192 \u03b2\na : \u03b1\nb : \u03b2\n\u22a2 f '' s = t \u2192 SurjOn f s t \u2227 MapsTo f s t", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03c0 : \u03b1 \u2192 Type u_5\ns s\u2081 s\u2082 : Set \u03b1\nt\u2081 t\u2082 : Set \u03b2\np : Set \u03b3\nf f\u2081 f\u2082 f\u2083 : \u03b1 \u2192 \u03b2\ng g\u2081 g\u2082 : \u03b2 \u2192 \u03b3\nf' f\u2081' f\u2082' : \u03b2 \u2192 \u03b1\ng' : \u03b3 \u2192 \u03b2\na : \u03b1\nb : \u03b2\n\u22a2 SurjOn f s (f '' s) \u2227 MapsTo f s (f '' s)"}, {"tactic": "exact \u27e8s.surjOn_image f, s.mapsTo_image f\u27e9", "annotated_tactic": ["exact \u27e8s.surjOn_image f, s.mapsTo_image f\u27e9", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03c0 : \u03b1 \u2192 Type u_5\ns s\u2081 s\u2082 : Set \u03b1\nt\u2081 t\u2082 : Set \u03b2\np : Set \u03b3\nf f\u2081 f\u2082 f\u2083 : \u03b1 \u2192 \u03b2\ng g\u2081 g\u2082 : \u03b2 \u2192 \u03b3\nf' f\u2081' f\u2082' : \u03b2 \u2192 \u03b1\ng' : \u03b3 \u2192 \u03b2\na : \u03b1\nb : \u03b2\n\u22a2 SurjOn f s (f '' s) \u2227 MapsTo f s (f '' s)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/RBMap/Lemmas.lean", "full_name": "Std.RBSet.find?_insert_of_ne", "start": [726, 1], "end": [732, 30], "traced_tactics": [{"tactic": "refine Option.ext fun u =>\n  find?_some.trans <| .trans (and_congr_left fun h' => ?_) find?_some.symm", "annotated_tactic": ["refine <a>Option.ext</a> fun u =>\n    find?_some.trans <| .trans (<a>and_congr_left</a> fun h' => ?_) find?_some.symm", [{"full_name": "Option.ext", "def_path": "lake-packages/std/Std/Data/Option/Lemmas.lean", "def_pos": [43, 16], "def_end_pos": [43, 19]}, {"full_name": "and_congr_left", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [164, 9], "def_end_pos": [164, 23]}]], "state_before": "\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nv' v : \u03b1\ninst\u271d : TransCmp cmp\nt : RBSet \u03b1 cmp\nh : cmp v' v \u2260 Ordering.eq\n\u22a2 find? (insert t v) v' = find? t v'", "state_after": "\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nv' v : \u03b1\ninst\u271d : TransCmp cmp\nt : RBSet \u03b1 cmp\nh : cmp v' v \u2260 Ordering.eq\nu : \u03b1\nh' : cmp v' u = Ordering.eq\n\u22a2 u \u2208 toList (insert t v) \u2194 u \u2208 toList t"}, {"tactic": "rw [mem_toList_insert, or_iff_left, and_iff_left]", "annotated_tactic": ["rw [<a>mem_toList_insert</a>, <a>or_iff_left</a>, <a>and_iff_left</a>]", [{"full_name": "Std.RBSet.mem_toList_insert", "def_path": "lake-packages/std/Std/Data/RBMap/Lemmas.lean", "def_pos": [706, 9], "def_end_pos": [706, 26]}, {"full_name": "or_iff_left", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [301, 9], "def_end_pos": [301, 20]}, {"full_name": "and_iff_left", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [204, 9], "def_end_pos": [204, 21]}]], "state_before": "\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nv' v : \u03b1\ninst\u271d : TransCmp cmp\nt : RBSet \u03b1 cmp\nh : cmp v' v \u2260 Ordering.eq\nu : \u03b1\nh' : cmp v' u = Ordering.eq\n\u22a2 u \u2208 toList (insert t v) \u2194 u \u2208 toList t", "state_after": "\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nv' v : \u03b1\ninst\u271d : TransCmp cmp\nt : RBSet \u03b1 cmp\nh : cmp v' v \u2260 Ordering.eq\nu : \u03b1\nh' : cmp v' u = Ordering.eq\n\u22a2 find? t v \u2260 some u\n\n\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nv' v : \u03b1\ninst\u271d : TransCmp cmp\nt : RBSet \u03b1 cmp\nh : cmp v' v \u2260 Ordering.eq\nu : \u03b1\nh' : cmp v' u = Ordering.eq\n\u22a2 \u00acu = v"}, {"tactic": "exact mt (fun h => by rwa [TransCmp.cmp_congr_right (find?_some_eq_eq h)]) h", "annotated_tactic": ["exact <a>mt</a> (fun h => by rwa [<a>TransCmp.cmp_congr_right</a> (<a>find?_some_eq_eq</a> h)]) h", [{"full_name": "mt", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [516, 9], "def_end_pos": [516, 11]}, {"full_name": "Std.TransCmp.cmp_congr_right", "def_path": "lake-packages/std/Std/Classes/Order.lean", "def_pos": [88, 9], "def_end_pos": [88, 24]}, {"full_name": "Std.RBSet.find?_some_eq_eq", "def_path": "lake-packages/std/Std/Data/RBMap/Lemmas.lean", "def_pos": [655, 9], "def_end_pos": [655, 25]}]], "state_before": "\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nv' v : \u03b1\ninst\u271d : TransCmp cmp\nt : RBSet \u03b1 cmp\nh : cmp v' v \u2260 Ordering.eq\nu : \u03b1\nh' : cmp v' u = Ordering.eq\n\u22a2 find? t v \u2260 some u", "state_after": "no goals"}, {"tactic": "rwa [TransCmp.cmp_congr_right (find?_some_eq_eq h)]", "annotated_tactic": ["rwa [<a>TransCmp.cmp_congr_right</a> (<a>find?_some_eq_eq</a> h)]", [{"full_name": "Std.TransCmp.cmp_congr_right", "def_path": "lake-packages/std/Std/Classes/Order.lean", "def_pos": [88, 9], "def_end_pos": [88, 24]}, {"full_name": "Std.RBSet.find?_some_eq_eq", "def_path": "lake-packages/std/Std/Data/RBMap/Lemmas.lean", "def_pos": [655, 9], "def_end_pos": [655, 25]}]], "state_before": "\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nv' v : \u03b1\ninst\u271d : TransCmp cmp\nt : RBSet \u03b1 cmp\nh\u271d : cmp v' v \u2260 Ordering.eq\nu : \u03b1\nh' : cmp v' u = Ordering.eq\nh : find? t v = some u\n\u22a2 cmp v' v = Ordering.eq", "state_after": "no goals"}, {"tactic": "rintro rfl", "annotated_tactic": ["rintro rfl", []], "state_before": "\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nv' v : \u03b1\ninst\u271d : TransCmp cmp\nt : RBSet \u03b1 cmp\nh : cmp v' v \u2260 Ordering.eq\nu : \u03b1\nh' : cmp v' u = Ordering.eq\n\u22a2 \u00acu = v", "state_after": "\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nv' : \u03b1\ninst\u271d : TransCmp cmp\nt : RBSet \u03b1 cmp\nu : \u03b1\nh' : cmp v' u = Ordering.eq\nh : cmp v' u \u2260 Ordering.eq\n\u22a2 False"}, {"tactic": "contradiction", "annotated_tactic": ["contradiction", []], "state_before": "\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nv' : \u03b1\ninst\u271d : TransCmp cmp\nt : RBSet \u03b1 cmp\nu : \u03b1\nh' : cmp v' u = Ordering.eq\nh : cmp v' u \u2260 Ordering.eq\n\u22a2 False", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "full_name": "MeasureTheory.mem\u2112p_congr_ae", "start": [553, 1], "end": [554, 72], "traced_tactics": [{"tactic": "simp only [Mem\u2112p, snorm_congr_ae hfg, aestronglyMeasurable_congr hfg]", "annotated_tactic": ["simp only [<a>Mem\u2112p</a>, <a>snorm_congr_ae</a> hfg, <a>aestronglyMeasurable_congr</a> hfg]", [{"full_name": "MeasureTheory.Mem\u2112p", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [108, 5], "def_end_pos": [108, 10]}, {"full_name": "MeasureTheory.snorm_congr_ae", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [549, 9], "def_end_pos": [549, 23]}, {"full_name": "aestronglyMeasurable_congr", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1232, 9], "def_end_pos": [1232, 42]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf g : \u03b1 \u2192 E\nhfg : f =\u1d50[\u03bc] g\n\u22a2 Mem\u2112p f p \u2194 Mem\u2112p g p", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/PiSystem.lean", "full_name": "subset_piiUnionInter", "start": [494, 1], "end": [502, 30], "traced_tactics": [{"tactic": "have h_ss : {i} \u2286 S := by\n  intro j hj\n  rw [mem_singleton_iff] at hj\n  rwa [hj]", "annotated_tactic": ["have h_ss : {i} \u2286 S := by\n    intro j hj\n    rw [<a>mem_singleton_iff</a>] at hj\n    rwa [hj]", [{"full_name": "Set.mem_singleton_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1273, 9], "def_end_pos": [1273, 26]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03c0 : \u03b9 \u2192 Set (Set \u03b1)\nS : Set \u03b9\ni : \u03b9\nhis : i \u2208 S\n\u22a2 \u03c0 i \u2286 piiUnionInter \u03c0 S", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03c0 : \u03b9 \u2192 Set (Set \u03b1)\nS : Set \u03b9\ni : \u03b9\nhis : i \u2208 S\nh_ss : {i} \u2286 S\n\u22a2 \u03c0 i \u2286 piiUnionInter \u03c0 S"}, {"tactic": "refine' Subset.trans _ (piiUnionInter_mono_right h_ss)", "annotated_tactic": ["refine' <a>Subset.trans</a> _ (<a>piiUnionInter_mono_right</a> h_ss)", [{"full_name": "Set.Subset.trans", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [362, 9], "def_end_pos": [362, 21]}, {"full_name": "piiUnionInter_mono_right", "def_path": "Mathlib/MeasureTheory/PiSystem.lean", "def_pos": [481, 9], "def_end_pos": [481, 33]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03c0 : \u03b9 \u2192 Set (Set \u03b1)\nS : Set \u03b9\ni : \u03b9\nhis : i \u2208 S\nh_ss : {i} \u2286 S\n\u22a2 \u03c0 i \u2286 piiUnionInter \u03c0 S", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03c0 : \u03b9 \u2192 Set (Set \u03b1)\nS : Set \u03b9\ni : \u03b9\nhis : i \u2208 S\nh_ss : {i} \u2286 S\n\u22a2 \u03c0 i \u2286 piiUnionInter \u03c0 {i}"}, {"tactic": "rw [piiUnionInter_singleton]", "annotated_tactic": ["rw [<a>piiUnionInter_singleton</a>]", [{"full_name": "piiUnionInter_singleton", "def_path": "Mathlib/MeasureTheory/PiSystem.lean", "def_pos": [376, 9], "def_end_pos": [376, 32]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03c0 : \u03b9 \u2192 Set (Set \u03b1)\nS : Set \u03b9\ni : \u03b9\nhis : i \u2208 S\nh_ss : {i} \u2286 S\n\u22a2 \u03c0 i \u2286 piiUnionInter \u03c0 {i}", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03c0 : \u03b9 \u2192 Set (Set \u03b1)\nS : Set \u03b9\ni : \u03b9\nhis : i \u2208 S\nh_ss : {i} \u2286 S\n\u22a2 \u03c0 i \u2286 \u03c0 i \u222a {univ}"}, {"tactic": "exact subset_union_left _ _", "annotated_tactic": ["exact <a>subset_union_left</a> _ _", [{"full_name": "Set.subset_union_left", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [829, 9], "def_end_pos": [829, 26]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03c0 : \u03b9 \u2192 Set (Set \u03b1)\nS : Set \u03b9\ni : \u03b9\nhis : i \u2208 S\nh_ss : {i} \u2286 S\n\u22a2 \u03c0 i \u2286 \u03c0 i \u222a {univ}", "state_after": "no goals"}, {"tactic": "intro j hj", "annotated_tactic": ["intro j hj", []], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03c0 : \u03b9 \u2192 Set (Set \u03b1)\nS : Set \u03b9\ni : \u03b9\nhis : i \u2208 S\n\u22a2 {i} \u2286 S", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03c0 : \u03b9 \u2192 Set (Set \u03b1)\nS : Set \u03b9\ni : \u03b9\nhis : i \u2208 S\nj : \u03b9\nhj : j \u2208 {i}\n\u22a2 j \u2208 S"}, {"tactic": "rw [mem_singleton_iff] at hj", "annotated_tactic": ["rw [<a>mem_singleton_iff</a>] at hj", [{"full_name": "Set.mem_singleton_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1273, 9], "def_end_pos": [1273, 26]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03c0 : \u03b9 \u2192 Set (Set \u03b1)\nS : Set \u03b9\ni : \u03b9\nhis : i \u2208 S\nj : \u03b9\nhj : j \u2208 {i}\n\u22a2 j \u2208 S", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03c0 : \u03b9 \u2192 Set (Set \u03b1)\nS : Set \u03b9\ni : \u03b9\nhis : i \u2208 S\nj : \u03b9\nhj : j = i\n\u22a2 j \u2208 S"}, {"tactic": "rwa [hj]", "annotated_tactic": ["rwa [hj]", []], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03c0 : \u03b9 \u2192 Set (Set \u03b1)\nS : Set \u03b9\ni : \u03b9\nhis : i \u2208 S\nj : \u03b9\nhj : j = i\n\u22a2 j \u2208 S", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Stieltjes.lean", "full_name": "StieltjesFunction.outer_Ioc", "start": [224, 1], "end": [284, 80], "traced_tactics": [{"tactic": "refine'\n  le_antisymm\n    (by\n      rw [\u2190 f.length_Ioc]\n      apply outer_le_length)\n    (le_iInf\u2082 fun s hs => ENNReal.le_of_forall_pos_le_add fun \u03b5 \u03b5pos h => _)", "annotated_tactic": ["refine'\n    <a>le_antisymm</a>\n      (by\n        rw [\u2190 f.length_Ioc]\n        apply <a>outer_le_length</a>)\n      (<a>le_iInf\u2082</a> fun s hs => <a>ENNReal.le_of_forall_pos_le_add</a> fun \u03b5 \u03b5pos h => _)", [{"full_name": "le_antisymm", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [188, 9], "def_end_pos": [188, 20]}, {"full_name": "StieltjesFunction.outer_le_length", "def_path": "Mathlib/MeasureTheory/Measure/Stieltjes.lean", "def_pos": [181, 9], "def_end_pos": [181, 24]}, {"full_name": "le_iInf\u2082", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [887, 9], "def_end_pos": [887, 17]}, {"full_name": "ENNReal.le_of_forall_pos_le_add", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [867, 9], "def_end_pos": [867, 32]}]], "state_before": "f : StieltjesFunction\na b : \u211d\n\u22a2 \u2191(StieltjesFunction.outer f) (Ioc a b) = ofReal (\u2191f b - \u2191f a)", "state_after": "f : StieltjesFunction\na b : \u211d\ns : \u2115 \u2192 Set \u211d\nhs : Ioc a b \u2286 \u22c3 i, s i\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh : \u2211' (i : \u2115), length f (s i) < \u22a4\n\u22a2 ofReal (\u2191f b - \u2191f a) \u2264 \u2211' (i : \u2115), length f (s i) + \u2191\u03b5"}, {"tactic": "let \u03b4 := \u03b5 / 2", "annotated_tactic": ["let \u03b4 := \u03b5 / 2", []], "state_before": "f : StieltjesFunction\na b : \u211d\ns : \u2115 \u2192 Set \u211d\nhs : Ioc a b \u2286 \u22c3 i, s i\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh : \u2211' (i : \u2115), length f (s i) < \u22a4\n\u22a2 ofReal (\u2191f b - \u2191f a) \u2264 \u2211' (i : \u2115), length f (s i) + \u2191\u03b5", "state_after": "f : StieltjesFunction\na b : \u211d\ns : \u2115 \u2192 Set \u211d\nhs : Ioc a b \u2286 \u22c3 i, s i\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh : \u2211' (i : \u2115), length f (s i) < \u22a4\n\u03b4 : \u211d\u22650 := \u03b5 / 2\n\u22a2 ofReal (\u2191f b - \u2191f a) \u2264 \u2211' (i : \u2115), length f (s i) + \u2191\u03b5"}, {"tactic": "have \u03b4pos : 0 < (\u03b4 : \u211d\u22650\u221e) := by simpa using \u03b5pos.ne'", "annotated_tactic": ["have \u03b4pos : 0 < (\u03b4 : \u211d\u22650\u221e) := by simpa using \u03b5pos.ne'", []], "state_before": "f : StieltjesFunction\na b : \u211d\ns : \u2115 \u2192 Set \u211d\nhs : Ioc a b \u2286 \u22c3 i, s i\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh : \u2211' (i : \u2115), length f (s i) < \u22a4\n\u03b4 : \u211d\u22650 := \u03b5 / 2\n\u22a2 ofReal (\u2191f b - \u2191f a) \u2264 \u2211' (i : \u2115), length f (s i) + \u2191\u03b5", "state_after": "f : StieltjesFunction\na b : \u211d\ns : \u2115 \u2192 Set \u211d\nhs : Ioc a b \u2286 \u22c3 i, s i\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh : \u2211' (i : \u2115), length f (s i) < \u22a4\n\u03b4 : \u211d\u22650 := \u03b5 / 2\n\u03b4pos : 0 < \u2191\u03b4\n\u22a2 ofReal (\u2191f b - \u2191f a) \u2264 \u2211' (i : \u2115), length f (s i) + \u2191\u03b5"}, {"tactic": "rcases ENNReal.exists_pos_sum_of_countable \u03b4pos.ne' \u2115 with \u27e8\u03b5', \u03b5'0, h\u03b5\u27e9", "annotated_tactic": ["rcases <a>ENNReal.exists_pos_sum_of_countable</a> \u03b4pos.ne' \u2115 with \u27e8\u03b5', \u03b5'0, h\u03b5\u27e9", [{"full_name": "ENNReal.exists_pos_sum_of_countable", "def_path": "Mathlib/Analysis/SpecificLimits/Basic.lean", "def_pos": [528, 9], "def_end_pos": [528, 36]}]], "state_before": "f : StieltjesFunction\na b : \u211d\ns : \u2115 \u2192 Set \u211d\nhs : Ioc a b \u2286 \u22c3 i, s i\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh : \u2211' (i : \u2115), length f (s i) < \u22a4\n\u03b4 : \u211d\u22650 := \u03b5 / 2\n\u03b4pos : 0 < \u2191\u03b4\n\u22a2 ofReal (\u2191f b - \u2191f a) \u2264 \u2211' (i : \u2115), length f (s i) + \u2191\u03b5", "state_after": "case intro.intro\nf : StieltjesFunction\na b : \u211d\ns : \u2115 \u2192 Set \u211d\nhs : Ioc a b \u2286 \u22c3 i, s i\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh : \u2211' (i : \u2115), length f (s i) < \u22a4\n\u03b4 : \u211d\u22650 := \u03b5 / 2\n\u03b4pos : 0 < \u2191\u03b4\n\u03b5' : \u2115 \u2192 \u211d\u22650\n\u03b5'0 : \u2200 (i : \u2115), 0 < \u03b5' i\nh\u03b5 : \u2211' (i : \u2115), \u2191(\u03b5' i) < \u2191\u03b4\n\u22a2 ofReal (\u2191f b - \u2191f a) \u2264 \u2211' (i : \u2115), length f (s i) + \u2191\u03b5"}, {"tactic": "obtain \u27e8a', ha', aa'\u27e9 : \u2203 a', f a' - f a < \u03b4 \u2227 a < a' := by\n  have A : ContinuousWithinAt (fun r => f r - f a) (Ioi a) a := by\n    refine' ContinuousWithinAt.sub _ continuousWithinAt_const\n    exact (f.right_continuous a).mono Ioi_subset_Ici_self\n  have B : f a - f a < \u03b4 := by rwa [sub_self, NNReal.coe_pos, \u2190 ENNReal.coe_pos]\n  exact (((tendsto_order.1 A).2 _ B).and self_mem_nhdsWithin).exists", "annotated_tactic": ["obtain \u27e8a', ha', aa'\u27e9 : \u2203 a', f a' - f a < \u03b4 \u2227 a < a' := by\n    have A : <a>ContinuousWithinAt</a> (fun r => f r - f a) (<a>Ioi</a> a) a := by\n      refine' <a>ContinuousWithinAt.sub</a> _ <a>continuousWithinAt_const</a>\n      exact (f.right_continuous a).<a>mono</a> <a>Ioi_subset_Ici_self</a>\n    have B : f a - f a < \u03b4 := by rwa [<a>sub_self</a>, <a>NNReal.coe_pos</a>, \u2190 <a>ENNReal.coe_pos</a>]\n    exact (((<a>tendsto_order</a>.1 A).2 _ B).<a>and</a> <a>self_mem_nhdsWithin</a>).<a>exists</a>", [{"full_name": "ContinuousWithinAt", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [518, 5], "def_end_pos": [518, 23]}, {"full_name": "Set.Ioi", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [79, 5], "def_end_pos": [79, 8]}, {"full_name": "ContinuousWithinAt.sub", "def_path": "Mathlib/Topology/Algebra/Group/Basic.lean", "def_pos": [1129, 15], "def_end_pos": [1129, 18]}, {"full_name": "continuousWithinAt_const", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [1029, 9], "def_end_pos": [1029, 33]}, {"full_name": "ContinuousWithinAt.mono", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [733, 9], "def_end_pos": [733, 32]}, {"full_name": "Set.Ioi_subset_Ici_self", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [532, 9], "def_end_pos": [532, 28]}, {"full_name": "sub_self", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [734, 30], "def_end_pos": [734, 38]}, {"full_name": "NNReal.coe_pos", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [376, 19], "def_end_pos": [376, 26]}, {"full_name": "ENNReal.coe_pos", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [380, 28], "def_end_pos": [380, 35]}, {"full_name": "tendsto_order", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [919, 9], "def_end_pos": [919, 22]}, {"full_name": "Filter.Eventually.and", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1103, 19], "def_end_pos": [1103, 33]}, {"full_name": "self_mem_nhdsWithin", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [151, 9], "def_end_pos": [151, 28]}, {"full_name": "Filter.Eventually.exists", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1308, 9], "def_end_pos": [1308, 26]}]], "state_before": "case intro.intro\nf : StieltjesFunction\na b : \u211d\ns : \u2115 \u2192 Set \u211d\nhs : Ioc a b \u2286 \u22c3 i, s i\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh : \u2211' (i : \u2115), length f (s i) < \u22a4\n\u03b4 : \u211d\u22650 := \u03b5 / 2\n\u03b4pos : 0 < \u2191\u03b4\n\u03b5' : \u2115 \u2192 \u211d\u22650\n\u03b5'0 : \u2200 (i : \u2115), 0 < \u03b5' i\nh\u03b5 : \u2211' (i : \u2115), \u2191(\u03b5' i) < \u2191\u03b4\n\u22a2 ofReal (\u2191f b - \u2191f a) \u2264 \u2211' (i : \u2115), length f (s i) + \u2191\u03b5", "state_after": "case intro.intro.intro.intro\nf : StieltjesFunction\na b : \u211d\ns : \u2115 \u2192 Set \u211d\nhs : Ioc a b \u2286 \u22c3 i, s i\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh : \u2211' (i : \u2115), length f (s i) < \u22a4\n\u03b4 : \u211d\u22650 := \u03b5 / 2\n\u03b4pos : 0 < \u2191\u03b4\n\u03b5' : \u2115 \u2192 \u211d\u22650\n\u03b5'0 : \u2200 (i : \u2115), 0 < \u03b5' i\nh\u03b5 : \u2211' (i : \u2115), \u2191(\u03b5' i) < \u2191\u03b4\na' : \u211d\nha' : \u2191f a' - \u2191f a < \u2191\u03b4\naa' : a < a'\n\u22a2 ofReal (\u2191f b - \u2191f a) \u2264 \u2211' (i : \u2115), length f (s i) + \u2191\u03b5"}, {"tactic": "have : \u2200 i, \u2203 p : \u211d \u00d7 \u211d, s i \u2286 Ioo p.1 p.2 \u2227\n    (ofReal (f p.2 - f p.1) : \u211d\u22650\u221e) < f.length (s i) + \u03b5' i := by\n  intro i\n  have hl :=\n    ENNReal.lt_add_right ((ENNReal.le_tsum i).trans_lt h).ne (ENNReal.coe_ne_zero.2 (\u03b5'0 i).ne')\n  conv at hl =>\n    lhs\n    rw [length]\n  simp only [iInf_lt_iff, exists_prop] at hl\n  rcases hl with \u27e8p, q', spq, hq'\u27e9\n  have : ContinuousWithinAt (fun r => ofReal (f r - f p)) (Ioi q') q' := by\n    apply ENNReal.continuous_ofReal.continuousAt.comp_continuousWithinAt\n    refine' ContinuousWithinAt.sub _ continuousWithinAt_const\n    exact (f.right_continuous q').mono Ioi_subset_Ici_self\n  rcases (((tendsto_order.1 this).2 _ hq').and self_mem_nhdsWithin).exists with \u27e8q, hq, q'q\u27e9\n  exact \u27e8\u27e8p, q\u27e9, spq.trans (Ioc_subset_Ioo_right q'q), hq\u27e9", "annotated_tactic": ["have : \u2200 i, \u2203 p : \u211d \u00d7 \u211d, s i \u2286 <a>Ioo</a> p.1 p.2 \u2227\n      (<a>ofReal</a> (f p.2 - f p.1) : \u211d\u22650\u221e) < f.length (s i) + \u03b5' i := by\n    intro i\n    have hl :=\n      <a>ENNReal.lt_add_right</a> ((<a>ENNReal.le_tsum</a> i).<a>trans_lt</a> h).<a>ne</a> (<a>ENNReal.coe_ne_zero</a>.2 (\u03b5'0 i).<a>ne'</a>)\n    conv at hl =>\n      lhs\n      rw [<a>length</a>]\n    simp only [<a>iInf_lt_iff</a>, <a>exists_prop</a>] at hl\n    rcases hl with \u27e8p, q', spq, hq'\u27e9\n    have : <a>ContinuousWithinAt</a> (fun r => <a>ofReal</a> (f r - f p)) (<a>Ioi</a> q') q' := by\n      apply ENNReal.continuous_ofReal.continuousAt.comp_continuousWithinAt\n      refine' <a>ContinuousWithinAt.sub</a> _ <a>continuousWithinAt_const</a>\n      exact (f.right_continuous q').<a>mono</a> <a>Ioi_subset_Ici_self</a>\n    rcases (((<a>tendsto_order</a>.1 this).2 _ hq').<a>and</a> <a>self_mem_nhdsWithin</a>).<a>exists</a> with \u27e8q, hq, q'q\u27e9\n    exact \u27e8\u27e8p, q\u27e9, spq.trans (<a>Ioc_subset_Ioo_right</a> q'q), hq\u27e9", [{"full_name": "Set.Ioo", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [44, 5], "def_end_pos": [44, 8]}, {"full_name": "ENNReal.ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [172, 29], "def_end_pos": [172, 35]}, {"full_name": "ENNReal.lt_add_right", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [829, 9], "def_end_pos": [829, 21]}, {"full_name": "ENNReal.le_tsum", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [857, 19], "def_end_pos": [857, 26]}, {"full_name": "LE.le.trans_lt", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [124, 7], "def_end_pos": [124, 21]}, {"full_name": "LT.lt.ne", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [152, 7], "def_end_pos": [152, 15]}, {"full_name": "ENNReal.coe_ne_zero", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [383, 9], "def_end_pos": [383, 20]}, {"full_name": "LT.lt.ne'", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [328, 9], "def_end_pos": [328, 12]}, {"full_name": "StieltjesFunction.length", "def_path": "Mathlib/MeasureTheory/Measure/Stieltjes.lean", "def_pos": [149, 5], "def_end_pos": [149, 11]}, {"full_name": "iInf_lt_iff", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [672, 9], "def_end_pos": [672, 20]}, {"full_name": "exists_prop", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [485, 17], "def_end_pos": [485, 28]}, {"full_name": "ContinuousWithinAt", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [518, 5], "def_end_pos": [518, 23]}, {"full_name": "ENNReal.ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [172, 29], "def_end_pos": [172, 35]}, {"full_name": "Set.Ioi", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [79, 5], "def_end_pos": [79, 8]}, {"full_name": "ContinuousWithinAt.sub", "def_path": "Mathlib/Topology/Algebra/Group/Basic.lean", "def_pos": [1129, 15], "def_end_pos": [1129, 18]}, {"full_name": "continuousWithinAt_const", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [1029, 9], "def_end_pos": [1029, 33]}, {"full_name": "ContinuousWithinAt.mono", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [733, 9], "def_end_pos": [733, 32]}, {"full_name": "Set.Ioi_subset_Ici_self", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [532, 9], "def_end_pos": [532, 28]}, {"full_name": "tendsto_order", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [919, 9], "def_end_pos": [919, 22]}, {"full_name": "Filter.Eventually.and", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1103, 19], "def_end_pos": [1103, 33]}, {"full_name": "self_mem_nhdsWithin", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [151, 9], "def_end_pos": [151, 28]}, {"full_name": "Filter.Eventually.exists", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1308, 9], "def_end_pos": [1308, 26]}, {"full_name": "Set.Ioc_subset_Ioo_right", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [496, 9], "def_end_pos": [496, 29]}]], "state_before": "case intro.intro.intro.intro\nf : StieltjesFunction\na b : \u211d\ns : \u2115 \u2192 Set \u211d\nhs : Ioc a b \u2286 \u22c3 i, s i\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh : \u2211' (i : \u2115), length f (s i) < \u22a4\n\u03b4 : \u211d\u22650 := \u03b5 / 2\n\u03b4pos : 0 < \u2191\u03b4\n\u03b5' : \u2115 \u2192 \u211d\u22650\n\u03b5'0 : \u2200 (i : \u2115), 0 < \u03b5' i\nh\u03b5 : \u2211' (i : \u2115), \u2191(\u03b5' i) < \u2191\u03b4\na' : \u211d\nha' : \u2191f a' - \u2191f a < \u2191\u03b4\naa' : a < a'\n\u22a2 ofReal (\u2191f b - \u2191f a) \u2264 \u2211' (i : \u2115), length f (s i) + \u2191\u03b5", "state_after": "case intro.intro.intro.intro\nf : StieltjesFunction\na b : \u211d\ns : \u2115 \u2192 Set \u211d\nhs : Ioc a b \u2286 \u22c3 i, s i\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh : \u2211' (i : \u2115), length f (s i) < \u22a4\n\u03b4 : \u211d\u22650 := \u03b5 / 2\n\u03b4pos : 0 < \u2191\u03b4\n\u03b5' : \u2115 \u2192 \u211d\u22650\n\u03b5'0 : \u2200 (i : \u2115), 0 < \u03b5' i\nh\u03b5 : \u2211' (i : \u2115), \u2191(\u03b5' i) < \u2191\u03b4\na' : \u211d\nha' : \u2191f a' - \u2191f a < \u2191\u03b4\naa' : a < a'\nthis : \u2200 (i : \u2115), \u2203 p, s i \u2286 Ioo p.1 p.2 \u2227 ofReal (\u2191f p.2 - \u2191f p.1) < length f (s i) + \u2191(\u03b5' i)\n\u22a2 ofReal (\u2191f b - \u2191f a) \u2264 \u2211' (i : \u2115), length f (s i) + \u2191\u03b5"}, {"tactic": "choose g hg using this", "annotated_tactic": ["choose g hg using this", []], "state_before": "case intro.intro.intro.intro\nf : StieltjesFunction\na b : \u211d\ns : \u2115 \u2192 Set \u211d\nhs : Ioc a b \u2286 \u22c3 i, s i\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh : \u2211' (i : \u2115), length f (s i) < \u22a4\n\u03b4 : \u211d\u22650 := \u03b5 / 2\n\u03b4pos : 0 < \u2191\u03b4\n\u03b5' : \u2115 \u2192 \u211d\u22650\n\u03b5'0 : \u2200 (i : \u2115), 0 < \u03b5' i\nh\u03b5 : \u2211' (i : \u2115), \u2191(\u03b5' i) < \u2191\u03b4\na' : \u211d\nha' : \u2191f a' - \u2191f a < \u2191\u03b4\naa' : a < a'\nthis : \u2200 (i : \u2115), \u2203 p, s i \u2286 Ioo p.1 p.2 \u2227 ofReal (\u2191f p.2 - \u2191f p.1) < length f (s i) + \u2191(\u03b5' i)\n\u22a2 ofReal (\u2191f b - \u2191f a) \u2264 \u2211' (i : \u2115), length f (s i) + \u2191\u03b5", "state_after": "case intro.intro.intro.intro\nf : StieltjesFunction\na b : \u211d\ns : \u2115 \u2192 Set \u211d\nhs : Ioc a b \u2286 \u22c3 i, s i\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh : \u2211' (i : \u2115), length f (s i) < \u22a4\n\u03b4 : \u211d\u22650 := \u03b5 / 2\n\u03b4pos : 0 < \u2191\u03b4\n\u03b5' : \u2115 \u2192 \u211d\u22650\n\u03b5'0 : \u2200 (i : \u2115), 0 < \u03b5' i\nh\u03b5 : \u2211' (i : \u2115), \u2191(\u03b5' i) < \u2191\u03b4\na' : \u211d\nha' : \u2191f a' - \u2191f a < \u2191\u03b4\naa' : a < a'\ng : \u2115 \u2192 \u211d \u00d7 \u211d\nhg : \u2200 (i : \u2115), s i \u2286 Ioo (g i).1 (g i).2 \u2227 ofReal (\u2191f (g i).2 - \u2191f (g i).1) < length f (s i) + \u2191(\u03b5' i)\n\u22a2 ofReal (\u2191f b - \u2191f a) \u2264 \u2211' (i : \u2115), length f (s i) + \u2191\u03b5"}, {"tactic": "have I_subset : Icc a' b \u2286 \u22c3 i, Ioo (g i).1 (g i).2 :=\n  calc\n    Icc a' b \u2286 Ioc a b := fun x hx => \u27e8aa'.trans_le hx.1, hx.2\u27e9\n    _ \u2286 \u22c3 i, s i := hs\n    _ \u2286 \u22c3 i, Ioo (g i).1 (g i).2 := iUnion_mono fun i => (hg i).1", "annotated_tactic": ["have I_subset : <a>Icc</a> a' b \u2286 \u22c3 i, <a>Ioo</a> (g i).1 (g i).2 :=\n    calc\n      <a>Icc</a> a' b \u2286 <a>Ioc</a> a b := fun x hx => \u27e8aa'.trans_le hx.1, hx.2\u27e9\n      _ \u2286 \u22c3 i, s i := hs\n      _ \u2286 \u22c3 i, <a>Ioo</a> (g i).1 (g i).2 := <a>iUnion_mono</a> fun i => (hg i).1", [{"full_name": "Set.Icc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [59, 5], "def_end_pos": [59, 8]}, {"full_name": "Set.Ioo", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [44, 5], "def_end_pos": [44, 8]}, {"full_name": "Set.Icc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [59, 5], "def_end_pos": [59, 8]}, {"full_name": "Set.Ioc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [69, 5], "def_end_pos": [69, 8]}, {"full_name": "Set.Ioo", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [44, 5], "def_end_pos": [44, 8]}, {"full_name": "Set.iUnion_mono", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [478, 9], "def_end_pos": [478, 20]}]], "state_before": "case intro.intro.intro.intro\nf : StieltjesFunction\na b : \u211d\ns : \u2115 \u2192 Set \u211d\nhs : Ioc a b \u2286 \u22c3 i, s i\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh : \u2211' (i : \u2115), length f (s i) < \u22a4\n\u03b4 : \u211d\u22650 := \u03b5 / 2\n\u03b4pos : 0 < \u2191\u03b4\n\u03b5' : \u2115 \u2192 \u211d\u22650\n\u03b5'0 : \u2200 (i : \u2115), 0 < \u03b5' i\nh\u03b5 : \u2211' (i : \u2115), \u2191(\u03b5' i) < \u2191\u03b4\na' : \u211d\nha' : \u2191f a' - \u2191f a < \u2191\u03b4\naa' : a < a'\ng : \u2115 \u2192 \u211d \u00d7 \u211d\nhg : \u2200 (i : \u2115), s i \u2286 Ioo (g i).1 (g i).2 \u2227 ofReal (\u2191f (g i).2 - \u2191f (g i).1) < length f (s i) + \u2191(\u03b5' i)\n\u22a2 ofReal (\u2191f b - \u2191f a) \u2264 \u2211' (i : \u2115), length f (s i) + \u2191\u03b5", "state_after": "case intro.intro.intro.intro\nf : StieltjesFunction\na b : \u211d\ns : \u2115 \u2192 Set \u211d\nhs : Ioc a b \u2286 \u22c3 i, s i\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh : \u2211' (i : \u2115), length f (s i) < \u22a4\n\u03b4 : \u211d\u22650 := \u03b5 / 2\n\u03b4pos : 0 < \u2191\u03b4\n\u03b5' : \u2115 \u2192 \u211d\u22650\n\u03b5'0 : \u2200 (i : \u2115), 0 < \u03b5' i\nh\u03b5 : \u2211' (i : \u2115), \u2191(\u03b5' i) < \u2191\u03b4\na' : \u211d\nha' : \u2191f a' - \u2191f a < \u2191\u03b4\naa' : a < a'\ng : \u2115 \u2192 \u211d \u00d7 \u211d\nhg : \u2200 (i : \u2115), s i \u2286 Ioo (g i).1 (g i).2 \u2227 ofReal (\u2191f (g i).2 - \u2191f (g i).1) < length f (s i) + \u2191(\u03b5' i)\nI_subset : Icc a' b \u2286 \u22c3 i, Ioo (g i).1 (g i).2\n\u22a2 ofReal (\u2191f b - \u2191f a) \u2264 \u2211' (i : \u2115), length f (s i) + \u2191\u03b5"}, {"tactic": "calc\n  ofReal (f b - f a) = ofReal (f b - f a' + (f a' - f a)) := by rw [sub_add_sub_cancel]\n  _ \u2264 ofReal (f b - f a') + ofReal (f a' - f a) := ENNReal.ofReal_add_le\n  _ \u2264 \u2211' i, ofReal (f (g i).2 - f (g i).1) + ofReal \u03b4 :=\n    (add_le_add (f.length_subadditive_Icc_Ioo I_subset) (ENNReal.ofReal_le_ofReal ha'.le))\n  _ \u2264 \u2211' i, (f.length (s i) + \u03b5' i) + \u03b4 :=\n    (add_le_add (ENNReal.tsum_le_tsum fun i => (hg i).2.le)\n      (by simp only [ENNReal.ofReal_coe_nnreal, le_rfl]))\n  _ = \u2211' i, f.length (s i) + \u2211' i, (\u03b5' i : \u211d\u22650\u221e) + \u03b4 := by rw [ENNReal.tsum_add]\n  _ \u2264 \u2211' i, f.length (s i) + \u03b4 + \u03b4 := (add_le_add (add_le_add le_rfl h\u03b5.le) le_rfl)\n  _ = \u2211' i : \u2115, f.length (s i) + \u03b5 := by simp [add_assoc, ENNReal.add_halves]", "annotated_tactic": ["calc\n    <a>ofReal</a> (f b - f a) = <a>ofReal</a> (f b - f a' + (f a' - f a)) := by rw [<a>sub_add_sub_cancel</a>]\n    _ \u2264 <a>ofReal</a> (f b - f a') + <a>ofReal</a> (f a' - f a) := <a>ENNReal.ofReal_add_le</a>\n    _ \u2264 \u2211' i, <a>ofReal</a> (f (g i).2 - f (g i).1) + <a>ofReal</a> \u03b4 :=\n      (<a>add_le_add</a> (f.length_subadditive_Icc_Ioo I_subset) (<a>ENNReal.ofReal_le_ofReal</a> ha'.le))\n    _ \u2264 \u2211' i, (f.length (s i) + \u03b5' i) + \u03b4 :=\n      (<a>add_le_add</a> (<a>ENNReal.tsum_le_tsum</a> fun i => (hg i).2.<a>le</a>)\n        (by simp only [<a>ENNReal.ofReal_coe_nnreal</a>, <a>le_rfl</a>]))\n    _ = \u2211' i, f.length (s i) + \u2211' i, (\u03b5' i : \u211d\u22650\u221e) + \u03b4 := by rw [<a>ENNReal.tsum_add</a>]\n    _ \u2264 \u2211' i, f.length (s i) + \u03b4 + \u03b4 := (<a>add_le_add</a> (<a>add_le_add</a> <a>le_rfl</a> h\u03b5.le) <a>le_rfl</a>)\n    _ = \u2211' i : \u2115, f.length (s i) + \u03b5 := by simp [<a>add_assoc</a>, <a>ENNReal.add_halves</a>]", [{"full_name": "ENNReal.ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [172, 29], "def_end_pos": [172, 35]}, {"full_name": "ENNReal.ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [172, 29], "def_end_pos": [172, 35]}, {"full_name": "sub_add_sub_cancel", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [789, 30], "def_end_pos": [789, 48]}, {"full_name": "ENNReal.ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [172, 29], "def_end_pos": [172, 35]}, {"full_name": "ENNReal.ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [172, 29], "def_end_pos": [172, 35]}, {"full_name": "ENNReal.ofReal_add_le", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2031, 9], "def_end_pos": [2031, 22]}, {"full_name": "ENNReal.ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [172, 29], "def_end_pos": [172, 35]}, {"full_name": "ENNReal.ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [172, 29], "def_end_pos": [172, 35]}, {"full_name": "add_le_add", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [205, 15], "def_end_pos": [205, 25]}, {"full_name": "ENNReal.ofReal_le_ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2135, 9], "def_end_pos": [2135, 25]}, {"full_name": "add_le_add", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [205, 15], "def_end_pos": [205, 25]}, {"full_name": "ENNReal.tsum_le_tsum", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [827, 19], "def_end_pos": [827, 31]}, {"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [142, 7], "def_end_pos": [142, 15]}, {"full_name": "ENNReal.ofReal_coe_nnreal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [212, 17], "def_end_pos": [212, 34]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}, {"full_name": "ENNReal.tsum_add", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [823, 19], "def_end_pos": [823, 27]}, {"full_name": "add_le_add", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [205, 15], "def_end_pos": [205, 25]}, {"full_name": "add_le_add", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [205, 15], "def_end_pos": [205, 25]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}, {"full_name": "add_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [263, 3], "def_end_pos": [263, 14]}, {"full_name": "ENNReal.add_halves", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1781, 19], "def_end_pos": [1781, 29]}]], "state_before": "case intro.intro.intro.intro\nf : StieltjesFunction\na b : \u211d\ns : \u2115 \u2192 Set \u211d\nhs : Ioc a b \u2286 \u22c3 i, s i\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh : \u2211' (i : \u2115), length f (s i) < \u22a4\n\u03b4 : \u211d\u22650 := \u03b5 / 2\n\u03b4pos : 0 < \u2191\u03b4\n\u03b5' : \u2115 \u2192 \u211d\u22650\n\u03b5'0 : \u2200 (i : \u2115), 0 < \u03b5' i\nh\u03b5 : \u2211' (i : \u2115), \u2191(\u03b5' i) < \u2191\u03b4\na' : \u211d\nha' : \u2191f a' - \u2191f a < \u2191\u03b4\naa' : a < a'\ng : \u2115 \u2192 \u211d \u00d7 \u211d\nhg : \u2200 (i : \u2115), s i \u2286 Ioo (g i).1 (g i).2 \u2227 ofReal (\u2191f (g i).2 - \u2191f (g i).1) < length f (s i) + \u2191(\u03b5' i)\nI_subset : Icc a' b \u2286 \u22c3 i, Ioo (g i).1 (g i).2\n\u22a2 ofReal (\u2191f b - \u2191f a) \u2264 \u2211' (i : \u2115), length f (s i) + \u2191\u03b5", "state_after": "no goals"}, {"tactic": "rw [\u2190 f.length_Ioc]", "annotated_tactic": ["rw [\u2190 f.length_Ioc]", []], "state_before": "f : StieltjesFunction\na b : \u211d\n\u22a2 \u2191(StieltjesFunction.outer f) (Ioc a b) \u2264 ofReal (\u2191f b - \u2191f a)", "state_after": "f : StieltjesFunction\na b : \u211d\n\u22a2 \u2191(StieltjesFunction.outer f) (Ioc a b) \u2264 length f (Ioc a b)"}, {"tactic": "apply outer_le_length", "annotated_tactic": ["apply <a>outer_le_length</a>", [{"full_name": "StieltjesFunction.outer_le_length", "def_path": "Mathlib/MeasureTheory/Measure/Stieltjes.lean", "def_pos": [181, 9], "def_end_pos": [181, 24]}]], "state_before": "f : StieltjesFunction\na b : \u211d\n\u22a2 \u2191(StieltjesFunction.outer f) (Ioc a b) \u2264 length f (Ioc a b)", "state_after": "no goals"}, {"tactic": "simpa using \u03b5pos.ne'", "annotated_tactic": ["simpa using \u03b5pos.ne'", []], "state_before": "f : StieltjesFunction\na b : \u211d\ns : \u2115 \u2192 Set \u211d\nhs : Ioc a b \u2286 \u22c3 i, s i\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh : \u2211' (i : \u2115), length f (s i) < \u22a4\n\u03b4 : \u211d\u22650 := \u03b5 / 2\n\u22a2 0 < \u2191\u03b4", "state_after": "no goals"}, {"tactic": "have A : ContinuousWithinAt (fun r => f r - f a) (Ioi a) a := by\n  refine' ContinuousWithinAt.sub _ continuousWithinAt_const\n  exact (f.right_continuous a).mono Ioi_subset_Ici_self", "annotated_tactic": ["have A : <a>ContinuousWithinAt</a> (fun r => f r - f a) (<a>Ioi</a> a) a := by\n      refine' <a>ContinuousWithinAt.sub</a> _ <a>continuousWithinAt_const</a>\n      exact (f.right_continuous a).<a>mono</a> <a>Ioi_subset_Ici_self</a>", [{"full_name": "ContinuousWithinAt", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [518, 5], "def_end_pos": [518, 23]}, {"full_name": "Set.Ioi", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [79, 5], "def_end_pos": [79, 8]}, {"full_name": "ContinuousWithinAt.sub", "def_path": "Mathlib/Topology/Algebra/Group/Basic.lean", "def_pos": [1129, 15], "def_end_pos": [1129, 18]}, {"full_name": "continuousWithinAt_const", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [1029, 9], "def_end_pos": [1029, 33]}, {"full_name": "ContinuousWithinAt.mono", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [733, 9], "def_end_pos": [733, 32]}, {"full_name": "Set.Ioi_subset_Ici_self", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [532, 9], "def_end_pos": [532, 28]}]], "state_before": "f : StieltjesFunction\na b : \u211d\ns : \u2115 \u2192 Set \u211d\nhs : Ioc a b \u2286 \u22c3 i, s i\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh : \u2211' (i : \u2115), length f (s i) < \u22a4\n\u03b4 : \u211d\u22650 := \u03b5 / 2\n\u03b4pos : 0 < \u2191\u03b4\n\u03b5' : \u2115 \u2192 \u211d\u22650\n\u03b5'0 : \u2200 (i : \u2115), 0 < \u03b5' i\nh\u03b5 : \u2211' (i : \u2115), \u2191(\u03b5' i) < \u2191\u03b4\n\u22a2 \u2203 a', \u2191f a' - \u2191f a < \u2191\u03b4 \u2227 a < a'", "state_after": "f : StieltjesFunction\na b : \u211d\ns : \u2115 \u2192 Set \u211d\nhs : Ioc a b \u2286 \u22c3 i, s i\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh : \u2211' (i : \u2115), length f (s i) < \u22a4\n\u03b4 : \u211d\u22650 := \u03b5 / 2\n\u03b4pos : 0 < \u2191\u03b4\n\u03b5' : \u2115 \u2192 \u211d\u22650\n\u03b5'0 : \u2200 (i : \u2115), 0 < \u03b5' i\nh\u03b5 : \u2211' (i : \u2115), \u2191(\u03b5' i) < \u2191\u03b4\nA : ContinuousWithinAt (fun r => \u2191f r - \u2191f a) (Ioi a) a\n\u22a2 \u2203 a', \u2191f a' - \u2191f a < \u2191\u03b4 \u2227 a < a'"}, {"tactic": "have B : f a - f a < \u03b4 := by rwa [sub_self, NNReal.coe_pos, \u2190 ENNReal.coe_pos]", "annotated_tactic": ["have B : f a - f a < \u03b4 := by rwa [<a>sub_self</a>, <a>NNReal.coe_pos</a>, \u2190 <a>ENNReal.coe_pos</a>]", [{"full_name": "sub_self", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [734, 30], "def_end_pos": [734, 38]}, {"full_name": "NNReal.coe_pos", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [376, 19], "def_end_pos": [376, 26]}, {"full_name": "ENNReal.coe_pos", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [380, 28], "def_end_pos": [380, 35]}]], "state_before": "f : StieltjesFunction\na b : \u211d\ns : \u2115 \u2192 Set \u211d\nhs : Ioc a b \u2286 \u22c3 i, s i\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh : \u2211' (i : \u2115), length f (s i) < \u22a4\n\u03b4 : \u211d\u22650 := \u03b5 / 2\n\u03b4pos : 0 < \u2191\u03b4\n\u03b5' : \u2115 \u2192 \u211d\u22650\n\u03b5'0 : \u2200 (i : \u2115), 0 < \u03b5' i\nh\u03b5 : \u2211' (i : \u2115), \u2191(\u03b5' i) < \u2191\u03b4\nA : ContinuousWithinAt (fun r => \u2191f r - \u2191f a) (Ioi a) a\n\u22a2 \u2203 a', \u2191f a' - \u2191f a < \u2191\u03b4 \u2227 a < a'", "state_after": "f : StieltjesFunction\na b : \u211d\ns : \u2115 \u2192 Set \u211d\nhs : Ioc a b \u2286 \u22c3 i, s i\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh : \u2211' (i : \u2115), length f (s i) < \u22a4\n\u03b4 : \u211d\u22650 := \u03b5 / 2\n\u03b4pos : 0 < \u2191\u03b4\n\u03b5' : \u2115 \u2192 \u211d\u22650\n\u03b5'0 : \u2200 (i : \u2115), 0 < \u03b5' i\nh\u03b5 : \u2211' (i : \u2115), \u2191(\u03b5' i) < \u2191\u03b4\nA : ContinuousWithinAt (fun r => \u2191f r - \u2191f a) (Ioi a) a\nB : \u2191f a - \u2191f a < \u2191\u03b4\n\u22a2 \u2203 a', \u2191f a' - \u2191f a < \u2191\u03b4 \u2227 a < a'"}, {"tactic": "exact (((tendsto_order.1 A).2 _ B).and self_mem_nhdsWithin).exists", "annotated_tactic": ["exact (((<a>tendsto_order</a>.1 A).2 _ B).<a>and</a> <a>self_mem_nhdsWithin</a>).<a>exists</a>", [{"full_name": "tendsto_order", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [919, 9], "def_end_pos": [919, 22]}, {"full_name": "Filter.Eventually.and", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1103, 19], "def_end_pos": [1103, 33]}, {"full_name": "self_mem_nhdsWithin", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [151, 9], "def_end_pos": [151, 28]}, {"full_name": "Filter.Eventually.exists", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1308, 9], "def_end_pos": [1308, 26]}]], "state_before": "f : StieltjesFunction\na b : \u211d\ns : \u2115 \u2192 Set \u211d\nhs : Ioc a b \u2286 \u22c3 i, s i\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh : \u2211' (i : \u2115), length f (s i) < \u22a4\n\u03b4 : \u211d\u22650 := \u03b5 / 2\n\u03b4pos : 0 < \u2191\u03b4\n\u03b5' : \u2115 \u2192 \u211d\u22650\n\u03b5'0 : \u2200 (i : \u2115), 0 < \u03b5' i\nh\u03b5 : \u2211' (i : \u2115), \u2191(\u03b5' i) < \u2191\u03b4\nA : ContinuousWithinAt (fun r => \u2191f r - \u2191f a) (Ioi a) a\nB : \u2191f a - \u2191f a < \u2191\u03b4\n\u22a2 \u2203 a', \u2191f a' - \u2191f a < \u2191\u03b4 \u2227 a < a'", "state_after": "no goals"}, {"tactic": "refine' ContinuousWithinAt.sub _ continuousWithinAt_const", "annotated_tactic": ["refine' <a>ContinuousWithinAt.sub</a> _ <a>continuousWithinAt_const</a>", [{"full_name": "ContinuousWithinAt.sub", "def_path": "Mathlib/Topology/Algebra/Group/Basic.lean", "def_pos": [1129, 15], "def_end_pos": [1129, 18]}, {"full_name": "continuousWithinAt_const", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [1029, 9], "def_end_pos": [1029, 33]}]], "state_before": "f : StieltjesFunction\na b : \u211d\ns : \u2115 \u2192 Set \u211d\nhs : Ioc a b \u2286 \u22c3 i, s i\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh : \u2211' (i : \u2115), length f (s i) < \u22a4\n\u03b4 : \u211d\u22650 := \u03b5 / 2\n\u03b4pos : 0 < \u2191\u03b4\n\u03b5' : \u2115 \u2192 \u211d\u22650\n\u03b5'0 : \u2200 (i : \u2115), 0 < \u03b5' i\nh\u03b5 : \u2211' (i : \u2115), \u2191(\u03b5' i) < \u2191\u03b4\n\u22a2 ContinuousWithinAt (fun r => \u2191f r - \u2191f a) (Ioi a) a", "state_after": "f : StieltjesFunction\na b : \u211d\ns : \u2115 \u2192 Set \u211d\nhs : Ioc a b \u2286 \u22c3 i, s i\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh : \u2211' (i : \u2115), length f (s i) < \u22a4\n\u03b4 : \u211d\u22650 := \u03b5 / 2\n\u03b4pos : 0 < \u2191\u03b4\n\u03b5' : \u2115 \u2192 \u211d\u22650\n\u03b5'0 : \u2200 (i : \u2115), 0 < \u03b5' i\nh\u03b5 : \u2211' (i : \u2115), \u2191(\u03b5' i) < \u2191\u03b4\n\u22a2 ContinuousWithinAt (fun r => \u2191f r) (Ioi a) a"}, {"tactic": "exact (f.right_continuous a).mono Ioi_subset_Ici_self", "annotated_tactic": ["exact (f.right_continuous a).<a>mono</a> <a>Ioi_subset_Ici_self</a>", [{"full_name": "ContinuousWithinAt.mono", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [733, 9], "def_end_pos": [733, 32]}, {"full_name": "Set.Ioi_subset_Ici_self", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [532, 9], "def_end_pos": [532, 28]}]], "state_before": "f : StieltjesFunction\na b : \u211d\ns : \u2115 \u2192 Set \u211d\nhs : Ioc a b \u2286 \u22c3 i, s i\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh : \u2211' (i : \u2115), length f (s i) < \u22a4\n\u03b4 : \u211d\u22650 := \u03b5 / 2\n\u03b4pos : 0 < \u2191\u03b4\n\u03b5' : \u2115 \u2192 \u211d\u22650\n\u03b5'0 : \u2200 (i : \u2115), 0 < \u03b5' i\nh\u03b5 : \u2211' (i : \u2115), \u2191(\u03b5' i) < \u2191\u03b4\n\u22a2 ContinuousWithinAt (fun r => \u2191f r) (Ioi a) a", "state_after": "no goals"}, {"tactic": "rwa [sub_self, NNReal.coe_pos, \u2190 ENNReal.coe_pos]", "annotated_tactic": ["rwa [<a>sub_self</a>, <a>NNReal.coe_pos</a>, \u2190 <a>ENNReal.coe_pos</a>]", [{"full_name": "sub_self", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [734, 30], "def_end_pos": [734, 38]}, {"full_name": "NNReal.coe_pos", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [376, 19], "def_end_pos": [376, 26]}, {"full_name": "ENNReal.coe_pos", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [380, 28], "def_end_pos": [380, 35]}]], "state_before": "f : StieltjesFunction\na b : \u211d\ns : \u2115 \u2192 Set \u211d\nhs : Ioc a b \u2286 \u22c3 i, s i\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh : \u2211' (i : \u2115), length f (s i) < \u22a4\n\u03b4 : \u211d\u22650 := \u03b5 / 2\n\u03b4pos : 0 < \u2191\u03b4\n\u03b5' : \u2115 \u2192 \u211d\u22650\n\u03b5'0 : \u2200 (i : \u2115), 0 < \u03b5' i\nh\u03b5 : \u2211' (i : \u2115), \u2191(\u03b5' i) < \u2191\u03b4\nA : ContinuousWithinAt (fun r => \u2191f r - \u2191f a) (Ioi a) a\n\u22a2 \u2191f a - \u2191f a < \u2191\u03b4", "state_after": "no goals"}, {"tactic": "intro i", "annotated_tactic": ["intro i", []], "state_before": "f : StieltjesFunction\na b : \u211d\ns : \u2115 \u2192 Set \u211d\nhs : Ioc a b \u2286 \u22c3 i, s i\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh : \u2211' (i : \u2115), length f (s i) < \u22a4\n\u03b4 : \u211d\u22650 := \u03b5 / 2\n\u03b4pos : 0 < \u2191\u03b4\n\u03b5' : \u2115 \u2192 \u211d\u22650\n\u03b5'0 : \u2200 (i : \u2115), 0 < \u03b5' i\nh\u03b5 : \u2211' (i : \u2115), \u2191(\u03b5' i) < \u2191\u03b4\na' : \u211d\nha' : \u2191f a' - \u2191f a < \u2191\u03b4\naa' : a < a'\n\u22a2 \u2200 (i : \u2115), \u2203 p, s i \u2286 Ioo p.1 p.2 \u2227 ofReal (\u2191f p.2 - \u2191f p.1) < length f (s i) + \u2191(\u03b5' i)", "state_after": "f : StieltjesFunction\na b : \u211d\ns : \u2115 \u2192 Set \u211d\nhs : Ioc a b \u2286 \u22c3 i, s i\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh : \u2211' (i : \u2115), length f (s i) < \u22a4\n\u03b4 : \u211d\u22650 := \u03b5 / 2\n\u03b4pos : 0 < \u2191\u03b4\n\u03b5' : \u2115 \u2192 \u211d\u22650\n\u03b5'0 : \u2200 (i : \u2115), 0 < \u03b5' i\nh\u03b5 : \u2211' (i : \u2115), \u2191(\u03b5' i) < \u2191\u03b4\na' : \u211d\nha' : \u2191f a' - \u2191f a < \u2191\u03b4\naa' : a < a'\ni : \u2115\n\u22a2 \u2203 p, s i \u2286 Ioo p.1 p.2 \u2227 ofReal (\u2191f p.2 - \u2191f p.1) < length f (s i) + \u2191(\u03b5' i)"}, {"tactic": "have hl :=\n  ENNReal.lt_add_right ((ENNReal.le_tsum i).trans_lt h).ne (ENNReal.coe_ne_zero.2 (\u03b5'0 i).ne')", "annotated_tactic": ["have hl :=\n      <a>ENNReal.lt_add_right</a> ((<a>ENNReal.le_tsum</a> i).<a>trans_lt</a> h).<a>ne</a> (<a>ENNReal.coe_ne_zero</a>.2 (\u03b5'0 i).<a>ne'</a>)", [{"full_name": "ENNReal.lt_add_right", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [829, 9], "def_end_pos": [829, 21]}, {"full_name": "ENNReal.le_tsum", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [857, 19], "def_end_pos": [857, 26]}, {"full_name": "LE.le.trans_lt", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [124, 7], "def_end_pos": [124, 21]}, {"full_name": "LT.lt.ne", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [152, 7], "def_end_pos": [152, 15]}, {"full_name": "ENNReal.coe_ne_zero", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [383, 9], "def_end_pos": [383, 20]}, {"full_name": "LT.lt.ne'", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [328, 9], "def_end_pos": [328, 12]}]], "state_before": "f : StieltjesFunction\na b : \u211d\ns : \u2115 \u2192 Set \u211d\nhs : Ioc a b \u2286 \u22c3 i, s i\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh : \u2211' (i : \u2115), length f (s i) < \u22a4\n\u03b4 : \u211d\u22650 := \u03b5 / 2\n\u03b4pos : 0 < \u2191\u03b4\n\u03b5' : \u2115 \u2192 \u211d\u22650\n\u03b5'0 : \u2200 (i : \u2115), 0 < \u03b5' i\nh\u03b5 : \u2211' (i : \u2115), \u2191(\u03b5' i) < \u2191\u03b4\na' : \u211d\nha' : \u2191f a' - \u2191f a < \u2191\u03b4\naa' : a < a'\ni : \u2115\n\u22a2 \u2203 p, s i \u2286 Ioo p.1 p.2 \u2227 ofReal (\u2191f p.2 - \u2191f p.1) < length f (s i) + \u2191(\u03b5' i)", "state_after": "f : StieltjesFunction\na b : \u211d\ns : \u2115 \u2192 Set \u211d\nhs : Ioc a b \u2286 \u22c3 i, s i\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh : \u2211' (i : \u2115), length f (s i) < \u22a4\n\u03b4 : \u211d\u22650 := \u03b5 / 2\n\u03b4pos : 0 < \u2191\u03b4\n\u03b5' : \u2115 \u2192 \u211d\u22650\n\u03b5'0 : \u2200 (i : \u2115), 0 < \u03b5' i\nh\u03b5 : \u2211' (i : \u2115), \u2191(\u03b5' i) < \u2191\u03b4\na' : \u211d\nha' : \u2191f a' - \u2191f a < \u2191\u03b4\naa' : a < a'\ni : \u2115\nhl : length f (s i) < length f (s i) + \u2191(\u03b5' i)\n\u22a2 \u2203 p, s i \u2286 Ioo p.1 p.2 \u2227 ofReal (\u2191f p.2 - \u2191f p.1) < length f (s i) + \u2191(\u03b5' i)"}, {"tactic": "conv at hl =>\n  lhs\n  rw [length]", "annotated_tactic": ["conv at hl =>\n      lhs\n      rw [<a>length</a>]", [{"full_name": "StieltjesFunction.length", "def_path": "Mathlib/MeasureTheory/Measure/Stieltjes.lean", "def_pos": [149, 5], "def_end_pos": [149, 11]}]], "state_before": "f : StieltjesFunction\na b : \u211d\ns : \u2115 \u2192 Set \u211d\nhs : Ioc a b \u2286 \u22c3 i, s i\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh : \u2211' (i : \u2115), length f (s i) < \u22a4\n\u03b4 : \u211d\u22650 := \u03b5 / 2\n\u03b4pos : 0 < \u2191\u03b4\n\u03b5' : \u2115 \u2192 \u211d\u22650\n\u03b5'0 : \u2200 (i : \u2115), 0 < \u03b5' i\nh\u03b5 : \u2211' (i : \u2115), \u2191(\u03b5' i) < \u2191\u03b4\na' : \u211d\nha' : \u2191f a' - \u2191f a < \u2191\u03b4\naa' : a < a'\ni : \u2115\nhl : length f (s i) < length f (s i) + \u2191(\u03b5' i)\n\u22a2 \u2203 p, s i \u2286 Ioo p.1 p.2 \u2227 ofReal (\u2191f p.2 - \u2191f p.1) < length f (s i) + \u2191(\u03b5' i)", "state_after": "f : StieltjesFunction\na b : \u211d\ns : \u2115 \u2192 Set \u211d\nhs : Ioc a b \u2286 \u22c3 i, s i\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh : \u2211' (i : \u2115), length f (s i) < \u22a4\n\u03b4 : \u211d\u22650 := \u03b5 / 2\n\u03b4pos : 0 < \u2191\u03b4\n\u03b5' : \u2115 \u2192 \u211d\u22650\n\u03b5'0 : \u2200 (i : \u2115), 0 < \u03b5' i\nh\u03b5 : \u2211' (i : \u2115), \u2191(\u03b5' i) < \u2191\u03b4\na' : \u211d\nha' : \u2191f a' - \u2191f a < \u2191\u03b4\naa' : a < a'\ni : \u2115\nhl : \u2a05 a, \u2a05 b, \u2a05 (_ : s i \u2286 Ioc a b), ofReal (\u2191f b - \u2191f a) < length f (s i) + \u2191(\u03b5' i)\n\u22a2 \u2203 p, s i \u2286 Ioo p.1 p.2 \u2227 ofReal (\u2191f p.2 - \u2191f p.1) < length f (s i) + \u2191(\u03b5' i)"}, {"tactic": "simp only [iInf_lt_iff, exists_prop] at hl", "annotated_tactic": ["simp only [<a>iInf_lt_iff</a>, <a>exists_prop</a>] at hl", [{"full_name": "iInf_lt_iff", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [672, 9], "def_end_pos": [672, 20]}, {"full_name": "exists_prop", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [485, 17], "def_end_pos": [485, 28]}]], "state_before": "f : StieltjesFunction\na b : \u211d\ns : \u2115 \u2192 Set \u211d\nhs : Ioc a b \u2286 \u22c3 i, s i\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh : \u2211' (i : \u2115), length f (s i) < \u22a4\n\u03b4 : \u211d\u22650 := \u03b5 / 2\n\u03b4pos : 0 < \u2191\u03b4\n\u03b5' : \u2115 \u2192 \u211d\u22650\n\u03b5'0 : \u2200 (i : \u2115), 0 < \u03b5' i\nh\u03b5 : \u2211' (i : \u2115), \u2191(\u03b5' i) < \u2191\u03b4\na' : \u211d\nha' : \u2191f a' - \u2191f a < \u2191\u03b4\naa' : a < a'\ni : \u2115\nhl : \u2a05 a, \u2a05 b, \u2a05 (_ : s i \u2286 Ioc a b), ofReal (\u2191f b - \u2191f a) < length f (s i) + \u2191(\u03b5' i)\n\u22a2 \u2203 p, s i \u2286 Ioo p.1 p.2 \u2227 ofReal (\u2191f p.2 - \u2191f p.1) < length f (s i) + \u2191(\u03b5' i)", "state_after": "f : StieltjesFunction\na b : \u211d\ns : \u2115 \u2192 Set \u211d\nhs : Ioc a b \u2286 \u22c3 i, s i\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh : \u2211' (i : \u2115), length f (s i) < \u22a4\n\u03b4 : \u211d\u22650 := \u03b5 / 2\n\u03b4pos : 0 < \u2191\u03b4\n\u03b5' : \u2115 \u2192 \u211d\u22650\n\u03b5'0 : \u2200 (i : \u2115), 0 < \u03b5' i\nh\u03b5 : \u2211' (i : \u2115), \u2191(\u03b5' i) < \u2191\u03b4\na' : \u211d\nha' : \u2191f a' - \u2191f a < \u2191\u03b4\naa' : a < a'\ni : \u2115\nhl : \u2203 i_1 i_2, s i \u2286 Ioc i_1 i_2 \u2227 ofReal (\u2191f i_2 - \u2191f i_1) < length f (s i) + \u2191(\u03b5' i)\n\u22a2 \u2203 p, s i \u2286 Ioo p.1 p.2 \u2227 ofReal (\u2191f p.2 - \u2191f p.1) < length f (s i) + \u2191(\u03b5' i)"}, {"tactic": "rcases hl with \u27e8p, q', spq, hq'\u27e9", "annotated_tactic": ["rcases hl with \u27e8p, q', spq, hq'\u27e9", []], "state_before": "f : StieltjesFunction\na b : \u211d\ns : \u2115 \u2192 Set \u211d\nhs : Ioc a b \u2286 \u22c3 i, s i\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh : \u2211' (i : \u2115), length f (s i) < \u22a4\n\u03b4 : \u211d\u22650 := \u03b5 / 2\n\u03b4pos : 0 < \u2191\u03b4\n\u03b5' : \u2115 \u2192 \u211d\u22650\n\u03b5'0 : \u2200 (i : \u2115), 0 < \u03b5' i\nh\u03b5 : \u2211' (i : \u2115), \u2191(\u03b5' i) < \u2191\u03b4\na' : \u211d\nha' : \u2191f a' - \u2191f a < \u2191\u03b4\naa' : a < a'\ni : \u2115\nhl : \u2203 i_1 i_2, s i \u2286 Ioc i_1 i_2 \u2227 ofReal (\u2191f i_2 - \u2191f i_1) < length f (s i) + \u2191(\u03b5' i)\n\u22a2 \u2203 p, s i \u2286 Ioo p.1 p.2 \u2227 ofReal (\u2191f p.2 - \u2191f p.1) < length f (s i) + \u2191(\u03b5' i)", "state_after": "case intro.intro.intro\nf : StieltjesFunction\na b : \u211d\ns : \u2115 \u2192 Set \u211d\nhs : Ioc a b \u2286 \u22c3 i, s i\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh : \u2211' (i : \u2115), length f (s i) < \u22a4\n\u03b4 : \u211d\u22650 := \u03b5 / 2\n\u03b4pos : 0 < \u2191\u03b4\n\u03b5' : \u2115 \u2192 \u211d\u22650\n\u03b5'0 : \u2200 (i : \u2115), 0 < \u03b5' i\nh\u03b5 : \u2211' (i : \u2115), \u2191(\u03b5' i) < \u2191\u03b4\na' : \u211d\nha' : \u2191f a' - \u2191f a < \u2191\u03b4\naa' : a < a'\ni : \u2115\np q' : \u211d\nspq : s i \u2286 Ioc p q'\nhq' : ofReal (\u2191f q' - \u2191f p) < length f (s i) + \u2191(\u03b5' i)\n\u22a2 \u2203 p, s i \u2286 Ioo p.1 p.2 \u2227 ofReal (\u2191f p.2 - \u2191f p.1) < length f (s i) + \u2191(\u03b5' i)"}, {"tactic": "have : ContinuousWithinAt (fun r => ofReal (f r - f p)) (Ioi q') q' := by\n  apply ENNReal.continuous_ofReal.continuousAt.comp_continuousWithinAt\n  refine' ContinuousWithinAt.sub _ continuousWithinAt_const\n  exact (f.right_continuous q').mono Ioi_subset_Ici_self", "annotated_tactic": ["have : <a>ContinuousWithinAt</a> (fun r => <a>ofReal</a> (f r - f p)) (<a>Ioi</a> q') q' := by\n      apply ENNReal.continuous_ofReal.continuousAt.comp_continuousWithinAt\n      refine' <a>ContinuousWithinAt.sub</a> _ <a>continuousWithinAt_const</a>\n      exact (f.right_continuous q').<a>mono</a> <a>Ioi_subset_Ici_self</a>", [{"full_name": "ContinuousWithinAt", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [518, 5], "def_end_pos": [518, 23]}, {"full_name": "ENNReal.ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [172, 29], "def_end_pos": [172, 35]}, {"full_name": "Set.Ioi", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [79, 5], "def_end_pos": [79, 8]}, {"full_name": "ContinuousWithinAt.sub", "def_path": "Mathlib/Topology/Algebra/Group/Basic.lean", "def_pos": [1129, 15], "def_end_pos": [1129, 18]}, {"full_name": "continuousWithinAt_const", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [1029, 9], "def_end_pos": [1029, 33]}, {"full_name": "ContinuousWithinAt.mono", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [733, 9], "def_end_pos": [733, 32]}, {"full_name": "Set.Ioi_subset_Ici_self", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [532, 9], "def_end_pos": [532, 28]}]], "state_before": "case intro.intro.intro\nf : StieltjesFunction\na b : \u211d\ns : \u2115 \u2192 Set \u211d\nhs : Ioc a b \u2286 \u22c3 i, s i\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh : \u2211' (i : \u2115), length f (s i) < \u22a4\n\u03b4 : \u211d\u22650 := \u03b5 / 2\n\u03b4pos : 0 < \u2191\u03b4\n\u03b5' : \u2115 \u2192 \u211d\u22650\n\u03b5'0 : \u2200 (i : \u2115), 0 < \u03b5' i\nh\u03b5 : \u2211' (i : \u2115), \u2191(\u03b5' i) < \u2191\u03b4\na' : \u211d\nha' : \u2191f a' - \u2191f a < \u2191\u03b4\naa' : a < a'\ni : \u2115\np q' : \u211d\nspq : s i \u2286 Ioc p q'\nhq' : ofReal (\u2191f q' - \u2191f p) < length f (s i) + \u2191(\u03b5' i)\n\u22a2 \u2203 p, s i \u2286 Ioo p.1 p.2 \u2227 ofReal (\u2191f p.2 - \u2191f p.1) < length f (s i) + \u2191(\u03b5' i)", "state_after": "case intro.intro.intro\nf : StieltjesFunction\na b : \u211d\ns : \u2115 \u2192 Set \u211d\nhs : Ioc a b \u2286 \u22c3 i, s i\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh : \u2211' (i : \u2115), length f (s i) < \u22a4\n\u03b4 : \u211d\u22650 := \u03b5 / 2\n\u03b4pos : 0 < \u2191\u03b4\n\u03b5' : \u2115 \u2192 \u211d\u22650\n\u03b5'0 : \u2200 (i : \u2115), 0 < \u03b5' i\nh\u03b5 : \u2211' (i : \u2115), \u2191(\u03b5' i) < \u2191\u03b4\na' : \u211d\nha' : \u2191f a' - \u2191f a < \u2191\u03b4\naa' : a < a'\ni : \u2115\np q' : \u211d\nspq : s i \u2286 Ioc p q'\nhq' : ofReal (\u2191f q' - \u2191f p) < length f (s i) + \u2191(\u03b5' i)\nthis : ContinuousWithinAt (fun r => ofReal (\u2191f r - \u2191f p)) (Ioi q') q'\n\u22a2 \u2203 p, s i \u2286 Ioo p.1 p.2 \u2227 ofReal (\u2191f p.2 - \u2191f p.1) < length f (s i) + \u2191(\u03b5' i)"}, {"tactic": "rcases (((tendsto_order.1 this).2 _ hq').and self_mem_nhdsWithin).exists with \u27e8q, hq, q'q\u27e9", "annotated_tactic": ["rcases (((<a>tendsto_order</a>.1 this).2 _ hq').<a>and</a> <a>self_mem_nhdsWithin</a>).<a>exists</a> with \u27e8q, hq, q'q\u27e9", [{"full_name": "tendsto_order", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [919, 9], "def_end_pos": [919, 22]}, {"full_name": "Filter.Eventually.and", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1103, 19], "def_end_pos": [1103, 33]}, {"full_name": "self_mem_nhdsWithin", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [151, 9], "def_end_pos": [151, 28]}, {"full_name": "Filter.Eventually.exists", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1308, 9], "def_end_pos": [1308, 26]}]], "state_before": "case intro.intro.intro\nf : StieltjesFunction\na b : \u211d\ns : \u2115 \u2192 Set \u211d\nhs : Ioc a b \u2286 \u22c3 i, s i\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh : \u2211' (i : \u2115), length f (s i) < \u22a4\n\u03b4 : \u211d\u22650 := \u03b5 / 2\n\u03b4pos : 0 < \u2191\u03b4\n\u03b5' : \u2115 \u2192 \u211d\u22650\n\u03b5'0 : \u2200 (i : \u2115), 0 < \u03b5' i\nh\u03b5 : \u2211' (i : \u2115), \u2191(\u03b5' i) < \u2191\u03b4\na' : \u211d\nha' : \u2191f a' - \u2191f a < \u2191\u03b4\naa' : a < a'\ni : \u2115\np q' : \u211d\nspq : s i \u2286 Ioc p q'\nhq' : ofReal (\u2191f q' - \u2191f p) < length f (s i) + \u2191(\u03b5' i)\nthis : ContinuousWithinAt (fun r => ofReal (\u2191f r - \u2191f p)) (Ioi q') q'\n\u22a2 \u2203 p, s i \u2286 Ioo p.1 p.2 \u2227 ofReal (\u2191f p.2 - \u2191f p.1) < length f (s i) + \u2191(\u03b5' i)", "state_after": "case intro.intro.intro.intro.intro\nf : StieltjesFunction\na b : \u211d\ns : \u2115 \u2192 Set \u211d\nhs : Ioc a b \u2286 \u22c3 i, s i\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh : \u2211' (i : \u2115), length f (s i) < \u22a4\n\u03b4 : \u211d\u22650 := \u03b5 / 2\n\u03b4pos : 0 < \u2191\u03b4\n\u03b5' : \u2115 \u2192 \u211d\u22650\n\u03b5'0 : \u2200 (i : \u2115), 0 < \u03b5' i\nh\u03b5 : \u2211' (i : \u2115), \u2191(\u03b5' i) < \u2191\u03b4\na' : \u211d\nha' : \u2191f a' - \u2191f a < \u2191\u03b4\naa' : a < a'\ni : \u2115\np q' : \u211d\nspq : s i \u2286 Ioc p q'\nhq' : ofReal (\u2191f q' - \u2191f p) < length f (s i) + \u2191(\u03b5' i)\nthis : ContinuousWithinAt (fun r => ofReal (\u2191f r - \u2191f p)) (Ioi q') q'\nq : \u211d\nhq : ofReal (\u2191f q - \u2191f p) < length f (s i) + \u2191(\u03b5' i)\nq'q : q' < q\n\u22a2 \u2203 p, s i \u2286 Ioo p.1 p.2 \u2227 ofReal (\u2191f p.2 - \u2191f p.1) < length f (s i) + \u2191(\u03b5' i)"}, {"tactic": "exact \u27e8\u27e8p, q\u27e9, spq.trans (Ioc_subset_Ioo_right q'q), hq\u27e9", "annotated_tactic": ["exact \u27e8\u27e8p, q\u27e9, spq.trans (<a>Ioc_subset_Ioo_right</a> q'q), hq\u27e9", [{"full_name": "Set.Ioc_subset_Ioo_right", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [496, 9], "def_end_pos": [496, 29]}]], "state_before": "case intro.intro.intro.intro.intro\nf : StieltjesFunction\na b : \u211d\ns : \u2115 \u2192 Set \u211d\nhs : Ioc a b \u2286 \u22c3 i, s i\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh : \u2211' (i : \u2115), length f (s i) < \u22a4\n\u03b4 : \u211d\u22650 := \u03b5 / 2\n\u03b4pos : 0 < \u2191\u03b4\n\u03b5' : \u2115 \u2192 \u211d\u22650\n\u03b5'0 : \u2200 (i : \u2115), 0 < \u03b5' i\nh\u03b5 : \u2211' (i : \u2115), \u2191(\u03b5' i) < \u2191\u03b4\na' : \u211d\nha' : \u2191f a' - \u2191f a < \u2191\u03b4\naa' : a < a'\ni : \u2115\np q' : \u211d\nspq : s i \u2286 Ioc p q'\nhq' : ofReal (\u2191f q' - \u2191f p) < length f (s i) + \u2191(\u03b5' i)\nthis : ContinuousWithinAt (fun r => ofReal (\u2191f r - \u2191f p)) (Ioi q') q'\nq : \u211d\nhq : ofReal (\u2191f q - \u2191f p) < length f (s i) + \u2191(\u03b5' i)\nq'q : q' < q\n\u22a2 \u2203 p, s i \u2286 Ioo p.1 p.2 \u2227 ofReal (\u2191f p.2 - \u2191f p.1) < length f (s i) + \u2191(\u03b5' i)", "state_after": "no goals"}, {"tactic": "apply ENNReal.continuous_ofReal.continuousAt.comp_continuousWithinAt", "annotated_tactic": ["apply ENNReal.continuous_ofReal.continuousAt.comp_continuousWithinAt", []], "state_before": "f : StieltjesFunction\na b : \u211d\ns : \u2115 \u2192 Set \u211d\nhs : Ioc a b \u2286 \u22c3 i, s i\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh : \u2211' (i : \u2115), length f (s i) < \u22a4\n\u03b4 : \u211d\u22650 := \u03b5 / 2\n\u03b4pos : 0 < \u2191\u03b4\n\u03b5' : \u2115 \u2192 \u211d\u22650\n\u03b5'0 : \u2200 (i : \u2115), 0 < \u03b5' i\nh\u03b5 : \u2211' (i : \u2115), \u2191(\u03b5' i) < \u2191\u03b4\na' : \u211d\nha' : \u2191f a' - \u2191f a < \u2191\u03b4\naa' : a < a'\ni : \u2115\np q' : \u211d\nspq : s i \u2286 Ioc p q'\nhq' : ofReal (\u2191f q' - \u2191f p) < length f (s i) + \u2191(\u03b5' i)\n\u22a2 ContinuousWithinAt (fun r => ofReal (\u2191f r - \u2191f p)) (Ioi q') q'", "state_after": "f : StieltjesFunction\na b : \u211d\ns : \u2115 \u2192 Set \u211d\nhs : Ioc a b \u2286 \u22c3 i, s i\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh : \u2211' (i : \u2115), length f (s i) < \u22a4\n\u03b4 : \u211d\u22650 := \u03b5 / 2\n\u03b4pos : 0 < \u2191\u03b4\n\u03b5' : \u2115 \u2192 \u211d\u22650\n\u03b5'0 : \u2200 (i : \u2115), 0 < \u03b5' i\nh\u03b5 : \u2211' (i : \u2115), \u2191(\u03b5' i) < \u2191\u03b4\na' : \u211d\nha' : \u2191f a' - \u2191f a < \u2191\u03b4\naa' : a < a'\ni : \u2115\np q' : \u211d\nspq : s i \u2286 Ioc p q'\nhq' : ofReal (\u2191f q' - \u2191f p) < length f (s i) + \u2191(\u03b5' i)\n\u22a2 ContinuousWithinAt (fun r => \u2191f r - \u2191f p) (Ioi q') q'"}, {"tactic": "refine' ContinuousWithinAt.sub _ continuousWithinAt_const", "annotated_tactic": ["refine' <a>ContinuousWithinAt.sub</a> _ <a>continuousWithinAt_const</a>", [{"full_name": "ContinuousWithinAt.sub", "def_path": "Mathlib/Topology/Algebra/Group/Basic.lean", "def_pos": [1129, 15], "def_end_pos": [1129, 18]}, {"full_name": "continuousWithinAt_const", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [1029, 9], "def_end_pos": [1029, 33]}]], "state_before": "f : StieltjesFunction\na b : \u211d\ns : \u2115 \u2192 Set \u211d\nhs : Ioc a b \u2286 \u22c3 i, s i\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh : \u2211' (i : \u2115), length f (s i) < \u22a4\n\u03b4 : \u211d\u22650 := \u03b5 / 2\n\u03b4pos : 0 < \u2191\u03b4\n\u03b5' : \u2115 \u2192 \u211d\u22650\n\u03b5'0 : \u2200 (i : \u2115), 0 < \u03b5' i\nh\u03b5 : \u2211' (i : \u2115), \u2191(\u03b5' i) < \u2191\u03b4\na' : \u211d\nha' : \u2191f a' - \u2191f a < \u2191\u03b4\naa' : a < a'\ni : \u2115\np q' : \u211d\nspq : s i \u2286 Ioc p q'\nhq' : ofReal (\u2191f q' - \u2191f p) < length f (s i) + \u2191(\u03b5' i)\n\u22a2 ContinuousWithinAt (fun r => \u2191f r - \u2191f p) (Ioi q') q'", "state_after": "f : StieltjesFunction\na b : \u211d\ns : \u2115 \u2192 Set \u211d\nhs : Ioc a b \u2286 \u22c3 i, s i\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh : \u2211' (i : \u2115), length f (s i) < \u22a4\n\u03b4 : \u211d\u22650 := \u03b5 / 2\n\u03b4pos : 0 < \u2191\u03b4\n\u03b5' : \u2115 \u2192 \u211d\u22650\n\u03b5'0 : \u2200 (i : \u2115), 0 < \u03b5' i\nh\u03b5 : \u2211' (i : \u2115), \u2191(\u03b5' i) < \u2191\u03b4\na' : \u211d\nha' : \u2191f a' - \u2191f a < \u2191\u03b4\naa' : a < a'\ni : \u2115\np q' : \u211d\nspq : s i \u2286 Ioc p q'\nhq' : ofReal (\u2191f q' - \u2191f p) < length f (s i) + \u2191(\u03b5' i)\n\u22a2 ContinuousWithinAt (fun r => \u2191f r) (Ioi q') q'"}, {"tactic": "exact (f.right_continuous q').mono Ioi_subset_Ici_self", "annotated_tactic": ["exact (f.right_continuous q').<a>mono</a> <a>Ioi_subset_Ici_self</a>", [{"full_name": "ContinuousWithinAt.mono", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [733, 9], "def_end_pos": [733, 32]}, {"full_name": "Set.Ioi_subset_Ici_self", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [532, 9], "def_end_pos": [532, 28]}]], "state_before": "f : StieltjesFunction\na b : \u211d\ns : \u2115 \u2192 Set \u211d\nhs : Ioc a b \u2286 \u22c3 i, s i\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh : \u2211' (i : \u2115), length f (s i) < \u22a4\n\u03b4 : \u211d\u22650 := \u03b5 / 2\n\u03b4pos : 0 < \u2191\u03b4\n\u03b5' : \u2115 \u2192 \u211d\u22650\n\u03b5'0 : \u2200 (i : \u2115), 0 < \u03b5' i\nh\u03b5 : \u2211' (i : \u2115), \u2191(\u03b5' i) < \u2191\u03b4\na' : \u211d\nha' : \u2191f a' - \u2191f a < \u2191\u03b4\naa' : a < a'\ni : \u2115\np q' : \u211d\nspq : s i \u2286 Ioc p q'\nhq' : ofReal (\u2191f q' - \u2191f p) < length f (s i) + \u2191(\u03b5' i)\n\u22a2 ContinuousWithinAt (fun r => \u2191f r) (Ioi q') q'", "state_after": "no goals"}, {"tactic": "rw [sub_add_sub_cancel]", "annotated_tactic": ["rw [<a>sub_add_sub_cancel</a>]", [{"full_name": "sub_add_sub_cancel", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [789, 30], "def_end_pos": [789, 48]}]], "state_before": "f : StieltjesFunction\na b : \u211d\ns : \u2115 \u2192 Set \u211d\nhs : Ioc a b \u2286 \u22c3 i, s i\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh : \u2211' (i : \u2115), length f (s i) < \u22a4\n\u03b4 : \u211d\u22650 := \u03b5 / 2\n\u03b4pos : 0 < \u2191\u03b4\n\u03b5' : \u2115 \u2192 \u211d\u22650\n\u03b5'0 : \u2200 (i : \u2115), 0 < \u03b5' i\nh\u03b5 : \u2211' (i : \u2115), \u2191(\u03b5' i) < \u2191\u03b4\na' : \u211d\nha' : \u2191f a' - \u2191f a < \u2191\u03b4\naa' : a < a'\ng : \u2115 \u2192 \u211d \u00d7 \u211d\nhg : \u2200 (i : \u2115), s i \u2286 Ioo (g i).1 (g i).2 \u2227 ofReal (\u2191f (g i).2 - \u2191f (g i).1) < length f (s i) + \u2191(\u03b5' i)\nI_subset : Icc a' b \u2286 \u22c3 i, Ioo (g i).1 (g i).2\n\u22a2 ofReal (\u2191f b - \u2191f a) = ofReal (\u2191f b - \u2191f a' + (\u2191f a' - \u2191f a))", "state_after": "no goals"}, {"tactic": "simp only [ENNReal.ofReal_coe_nnreal, le_rfl]", "annotated_tactic": ["simp only [<a>ENNReal.ofReal_coe_nnreal</a>, <a>le_rfl</a>]", [{"full_name": "ENNReal.ofReal_coe_nnreal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [212, 17], "def_end_pos": [212, 34]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}]], "state_before": "f : StieltjesFunction\na b : \u211d\ns : \u2115 \u2192 Set \u211d\nhs : Ioc a b \u2286 \u22c3 i, s i\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh : \u2211' (i : \u2115), length f (s i) < \u22a4\n\u03b4 : \u211d\u22650 := \u03b5 / 2\n\u03b4pos : 0 < \u2191\u03b4\n\u03b5' : \u2115 \u2192 \u211d\u22650\n\u03b5'0 : \u2200 (i : \u2115), 0 < \u03b5' i\nh\u03b5 : \u2211' (i : \u2115), \u2191(\u03b5' i) < \u2191\u03b4\na' : \u211d\nha' : \u2191f a' - \u2191f a < \u2191\u03b4\naa' : a < a'\ng : \u2115 \u2192 \u211d \u00d7 \u211d\nhg : \u2200 (i : \u2115), s i \u2286 Ioo (g i).1 (g i).2 \u2227 ofReal (\u2191f (g i).2 - \u2191f (g i).1) < length f (s i) + \u2191(\u03b5' i)\nI_subset : Icc a' b \u2286 \u22c3 i, Ioo (g i).1 (g i).2\n\u22a2 ofReal \u2191\u03b4 \u2264 \u2191\u03b4", "state_after": "no goals"}, {"tactic": "rw [ENNReal.tsum_add]", "annotated_tactic": ["rw [<a>ENNReal.tsum_add</a>]", [{"full_name": "ENNReal.tsum_add", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [823, 19], "def_end_pos": [823, 27]}]], "state_before": "f : StieltjesFunction\na b : \u211d\ns : \u2115 \u2192 Set \u211d\nhs : Ioc a b \u2286 \u22c3 i, s i\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh : \u2211' (i : \u2115), length f (s i) < \u22a4\n\u03b4 : \u211d\u22650 := \u03b5 / 2\n\u03b4pos : 0 < \u2191\u03b4\n\u03b5' : \u2115 \u2192 \u211d\u22650\n\u03b5'0 : \u2200 (i : \u2115), 0 < \u03b5' i\nh\u03b5 : \u2211' (i : \u2115), \u2191(\u03b5' i) < \u2191\u03b4\na' : \u211d\nha' : \u2191f a' - \u2191f a < \u2191\u03b4\naa' : a < a'\ng : \u2115 \u2192 \u211d \u00d7 \u211d\nhg : \u2200 (i : \u2115), s i \u2286 Ioo (g i).1 (g i).2 \u2227 ofReal (\u2191f (g i).2 - \u2191f (g i).1) < length f (s i) + \u2191(\u03b5' i)\nI_subset : Icc a' b \u2286 \u22c3 i, Ioo (g i).1 (g i).2\n\u22a2 \u2211' (i : \u2115), (length f (s i) + \u2191(\u03b5' i)) + \u2191\u03b4 = \u2211' (i : \u2115), length f (s i) + \u2211' (i : \u2115), \u2191(\u03b5' i) + \u2191\u03b4", "state_after": "no goals"}, {"tactic": "simp [add_assoc, ENNReal.add_halves]", "annotated_tactic": ["simp [<a>add_assoc</a>, <a>ENNReal.add_halves</a>]", [{"full_name": "add_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [263, 3], "def_end_pos": [263, 14]}, {"full_name": "ENNReal.add_halves", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1781, 19], "def_end_pos": [1781, 29]}]], "state_before": "f : StieltjesFunction\na b : \u211d\ns : \u2115 \u2192 Set \u211d\nhs : Ioc a b \u2286 \u22c3 i, s i\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh : \u2211' (i : \u2115), length f (s i) < \u22a4\n\u03b4 : \u211d\u22650 := \u03b5 / 2\n\u03b4pos : 0 < \u2191\u03b4\n\u03b5' : \u2115 \u2192 \u211d\u22650\n\u03b5'0 : \u2200 (i : \u2115), 0 < \u03b5' i\nh\u03b5 : \u2211' (i : \u2115), \u2191(\u03b5' i) < \u2191\u03b4\na' : \u211d\nha' : \u2191f a' - \u2191f a < \u2191\u03b4\naa' : a < a'\ng : \u2115 \u2192 \u211d \u00d7 \u211d\nhg : \u2200 (i : \u2115), s i \u2286 Ioo (g i).1 (g i).2 \u2227 ofReal (\u2191f (g i).2 - \u2191f (g i).1) < length f (s i) + \u2191(\u03b5' i)\nI_subset : Icc a' b \u2286 \u22c3 i, Ioo (g i).1 (g i).2\n\u22a2 \u2211' (i : \u2115), length f (s i) + \u2191\u03b4 + \u2191\u03b4 = \u2211' (i : \u2115), length f (s i) + \u2191\u03b5", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/Monad.lean", "full_name": "MvPolynomial.bind\u2082_bind\u2082", "start": [239, 1], "end": [241, 45], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/Rename.lean", "full_name": "MvPolynomial.killCompl_rename_app", "start": [143, 1], "end": [144, 48], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/PairingHeap.lean", "full_name": "Std.PairingHeapImp.Heap.size_deleteMin", "start": [139, 1], "end": [141, 68], "traced_tactics": [{"tactic": "cases h with cases eq | node a c => rw [size_combine, size, size]", "annotated_tactic": ["cases h with cases eq | <a>node</a> a c => rw [<a>size_combine</a>, <a>size</a>, <a>size</a>]", [{"full_name": "Std.PairingHeapImp.Heap.NoSibling.node", "def_path": "lake-packages/std/Std/Data/PairingHeap.lean", "def_pos": [83, 5], "def_end_pos": [83, 9]}, {"full_name": "Std.PairingHeapImp.Heap.size_combine", "def_path": "lake-packages/std/Std/Data/PairingHeap.lean", "def_pos": [130, 9], "def_end_pos": [130, 26]}, {"full_name": "Std.PairingHeapImp.Heap.size", "def_path": "lake-packages/std/Std/Data/PairingHeap.lean", "def_pos": [30, 5], "def_end_pos": [30, 14]}, {"full_name": "Std.PairingHeapImp.Heap.size", "def_path": "lake-packages/std/Std/Data/PairingHeap.lean", "def_pos": [30, 5], "def_end_pos": [30, 14]}]], "state_before": "\u03b1 : Type u_1\nle : \u03b1 \u2192 \u03b1 \u2192 Bool\na : \u03b1\ns' s : Heap \u03b1\nh : NoSibling s\neq : deleteMin le s = some (a, s')\n\u22a2 size s = size s' + 1", "state_after": "no goals"}, {"tactic": "rw [size_combine, size, size]", "annotated_tactic": ["rw [<a>size_combine</a>, <a>size</a>, <a>size</a>]", [{"full_name": "Std.PairingHeapImp.Heap.size_combine", "def_path": "lake-packages/std/Std/Data/PairingHeap.lean", "def_pos": [130, 9], "def_end_pos": [130, 26]}, {"full_name": "Std.PairingHeapImp.Heap.size", "def_path": "lake-packages/std/Std/Data/PairingHeap.lean", "def_pos": [30, 5], "def_end_pos": [30, 14]}, {"full_name": "Std.PairingHeapImp.Heap.size", "def_path": "lake-packages/std/Std/Data/PairingHeap.lean", "def_pos": [30, 5], "def_end_pos": [30, 14]}]], "state_before": "case node.refl\n\u03b1 : Type u_1\nle : \u03b1 \u2192 \u03b1 \u2192 Bool\na : \u03b1\nc : Heap \u03b1\n\u22a2 size (node a c nil) = size (combine le c) + 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "full_name": "MeasureTheory.integral_Icc_eq_integral_Ico", "start": [679, 1], "end": [680, 55], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean", "full_name": "MeasureTheory.aestronglyMeasurable'_condexpL1Clm", "start": [465, 1], "end": [480, 44], "traced_tactics": [{"tactic": "refine' @Lp.induction _ _ _ _ _ _ _ ENNReal.one_ne_top\n  (fun f : \u03b1 \u2192\u2081[\u03bc] F' => AEStronglyMeasurable' m (condexpL1Clm F' hm \u03bc f) \u03bc) _ _ _ f", "annotated_tactic": ["refine' @<a>Lp.induction</a> _ _ _ _ _ _ _ <a>ENNReal.one_ne_top</a>\n    (fun f : \u03b1 \u2192\u2081[\u03bc] F' => <a>AEStronglyMeasurable'</a> m (<a>condexpL1Clm</a> F' hm \u03bc f) \u03bc) _ _ _ f", [{"full_name": "MeasureTheory.Lp.induction", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "def_pos": [924, 9], "def_end_pos": [924, 21]}, {"full_name": "ENNReal.one_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [340, 17], "def_end_pos": [340, 27]}, {"full_name": "MeasureTheory.AEStronglyMeasurable'", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean", "def_pos": [49, 5], "def_end_pos": [49, 26]}, {"full_name": "MeasureTheory.condexpL1Clm", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean", "def_pos": [380, 5], "def_end_pos": [380, 17]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup F'\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2076 : NormedSpace \u211d F'\ninst\u271d\u2075 : CompleteSpace F'\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedAddCommGroup G'\ninst\u271d\u00b2 : NormedSpace \u211d G'\ninst\u271d\u00b9 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nf\u271d g : \u03b1 \u2192 F'\ns : Set \u03b1\nf : { x // x \u2208 Lp F' 1 }\n\u22a2 AEStronglyMeasurable' m (\u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) f)) \u03bc", "state_after": "case refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup F'\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2076 : NormedSpace \u211d F'\ninst\u271d\u2075 : CompleteSpace F'\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedAddCommGroup G'\ninst\u271d\u00b2 : NormedSpace \u211d G'\ninst\u271d\u00b9 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nf\u271d g : \u03b1 \u2192 F'\ns : Set \u03b1\nf : { x // x \u2208 Lp F' 1 }\n\u22a2 \u2200 (c : F') {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n    (fun f => AEStronglyMeasurable' m (\u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) f)) \u03bc)\n      \u2191(simpleFunc.indicatorConst 1 hs (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\n\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup F'\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2076 : NormedSpace \u211d F'\ninst\u271d\u2075 : CompleteSpace F'\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedAddCommGroup G'\ninst\u271d\u00b2 : NormedSpace \u211d G'\ninst\u271d\u00b9 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nf\u271d g : \u03b1 \u2192 F'\ns : Set \u03b1\nf : { x // x \u2208 Lp F' 1 }\n\u22a2 \u2200 \u2983f g : \u03b1 \u2192 F'\u2984 (hf : Mem\u2112p f 1) (hg : Mem\u2112p g 1),\n    Disjoint (Function.support f) (Function.support g) \u2192\n      (fun f => AEStronglyMeasurable' m (\u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) f)) \u03bc) (Mem\u2112p.toLp f hf) \u2192\n        (fun f => AEStronglyMeasurable' m (\u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) f)) \u03bc) (Mem\u2112p.toLp g hg) \u2192\n          (fun f => AEStronglyMeasurable' m (\u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) f)) \u03bc) (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\n\ncase refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup F'\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2076 : NormedSpace \u211d F'\ninst\u271d\u2075 : CompleteSpace F'\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedAddCommGroup G'\ninst\u271d\u00b2 : NormedSpace \u211d G'\ninst\u271d\u00b9 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nf\u271d g : \u03b1 \u2192 F'\ns : Set \u03b1\nf : { x // x \u2208 Lp F' 1 }\n\u22a2 IsClosed {f | (fun f => AEStronglyMeasurable' m (\u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) f)) \u03bc) f}"}, {"tactic": "intro c s hs h\u03bcs", "annotated_tactic": ["intro c s hs h\u03bcs", []], "state_before": "case refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup F'\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2076 : NormedSpace \u211d F'\ninst\u271d\u2075 : CompleteSpace F'\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedAddCommGroup G'\ninst\u271d\u00b2 : NormedSpace \u211d G'\ninst\u271d\u00b9 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nf\u271d g : \u03b1 \u2192 F'\ns : Set \u03b1\nf : { x // x \u2208 Lp F' 1 }\n\u22a2 \u2200 (c : F') {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n    (fun f => AEStronglyMeasurable' m (\u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) f)) \u03bc)\n      \u2191(simpleFunc.indicatorConst 1 hs (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)", "state_after": "case refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup F'\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2076 : NormedSpace \u211d F'\ninst\u271d\u2075 : CompleteSpace F'\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedAddCommGroup G'\ninst\u271d\u00b2 : NormedSpace \u211d G'\ninst\u271d\u00b9 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nf\u271d g : \u03b1 \u2192 F'\ns\u271d : Set \u03b1\nf : { x // x \u2208 Lp F' 1 }\nc : F'\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 AEStronglyMeasurable' m (\u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) \u2191(simpleFunc.indicatorConst 1 hs (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c))) \u03bc"}, {"tactic": "rw [condexpL1Clm_indicatorConst hs h\u03bcs.ne c]", "annotated_tactic": ["rw [<a>condexpL1Clm_indicatorConst</a> hs h\u03bcs.ne c]", [{"full_name": "MeasureTheory.condexpL1Clm_indicatorConst", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean", "def_pos": [398, 9], "def_end_pos": [398, 36]}]], "state_before": "case refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup F'\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2076 : NormedSpace \u211d F'\ninst\u271d\u2075 : CompleteSpace F'\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedAddCommGroup G'\ninst\u271d\u00b2 : NormedSpace \u211d G'\ninst\u271d\u00b9 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nf\u271d g : \u03b1 \u2192 F'\ns\u271d : Set \u03b1\nf : { x // x \u2208 Lp F' 1 }\nc : F'\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 AEStronglyMeasurable' m (\u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) \u2191(simpleFunc.indicatorConst 1 hs (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c))) \u03bc", "state_after": "case refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup F'\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2076 : NormedSpace \u211d F'\ninst\u271d\u2075 : CompleteSpace F'\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedAddCommGroup G'\ninst\u271d\u00b2 : NormedSpace \u211d G'\ninst\u271d\u00b9 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nf\u271d g : \u03b1 \u2192 F'\ns\u271d : Set \u03b1\nf : { x // x \u2208 Lp F' 1 }\nc : F'\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 AEStronglyMeasurable' m (\u2191\u2191(\u2191(condexpInd F' hm \u03bc s) c)) \u03bc"}, {"tactic": "exact aestronglyMeasurable'_condexpInd hs h\u03bcs.ne c", "annotated_tactic": ["exact <a>aestronglyMeasurable'_condexpInd</a> hs h\u03bcs.ne c", [{"full_name": "MeasureTheory.aestronglyMeasurable'_condexpInd", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean", "def_pos": [285, 9], "def_end_pos": [285, 41]}]], "state_before": "case refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup F'\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2076 : NormedSpace \u211d F'\ninst\u271d\u2075 : CompleteSpace F'\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedAddCommGroup G'\ninst\u271d\u00b2 : NormedSpace \u211d G'\ninst\u271d\u00b9 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nf\u271d g : \u03b1 \u2192 F'\ns\u271d : Set \u03b1\nf : { x // x \u2208 Lp F' 1 }\nc : F'\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 AEStronglyMeasurable' m (\u2191\u2191(\u2191(condexpInd F' hm \u03bc s) c)) \u03bc", "state_after": "no goals"}, {"tactic": "intro f g hf hg _ hfm hgm", "annotated_tactic": ["intro f g hf hg _ hfm hgm", []], "state_before": "case refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup F'\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2076 : NormedSpace \u211d F'\ninst\u271d\u2075 : CompleteSpace F'\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedAddCommGroup G'\ninst\u271d\u00b2 : NormedSpace \u211d G'\ninst\u271d\u00b9 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nf\u271d g : \u03b1 \u2192 F'\ns : Set \u03b1\nf : { x // x \u2208 Lp F' 1 }\n\u22a2 \u2200 \u2983f g : \u03b1 \u2192 F'\u2984 (hf : Mem\u2112p f 1) (hg : Mem\u2112p g 1),\n    Disjoint (Function.support f) (Function.support g) \u2192\n      (fun f => AEStronglyMeasurable' m (\u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) f)) \u03bc) (Mem\u2112p.toLp f hf) \u2192\n        (fun f => AEStronglyMeasurable' m (\u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) f)) \u03bc) (Mem\u2112p.toLp g hg) \u2192\n          (fun f => AEStronglyMeasurable' m (\u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) f)) \u03bc) (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)", "state_after": "case refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup F'\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2076 : NormedSpace \u211d F'\ninst\u271d\u2075 : CompleteSpace F'\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedAddCommGroup G'\ninst\u271d\u00b2 : NormedSpace \u211d G'\ninst\u271d\u00b9 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nf\u271d\u00b9 g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nf\u271d : { x // x \u2208 Lp F' 1 }\nf g : \u03b1 \u2192 F'\nhf : Mem\u2112p f 1\nhg : Mem\u2112p g 1\na\u271d : Disjoint (Function.support f) (Function.support g)\nhfm : AEStronglyMeasurable' m (\u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) (Mem\u2112p.toLp f hf))) \u03bc\nhgm : AEStronglyMeasurable' m (\u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) (Mem\u2112p.toLp g hg))) \u03bc\n\u22a2 AEStronglyMeasurable' m (\u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg))) \u03bc"}, {"tactic": "rw [(condexpL1Clm F' hm \u03bc).map_add]", "annotated_tactic": ["rw [(<a>condexpL1Clm</a> F' hm \u03bc).<a>map_add</a>]", [{"full_name": "MeasureTheory.condexpL1Clm", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean", "def_pos": [380, 5], "def_end_pos": [380, 17]}, {"full_name": "ContinuousLinearMap.map_add", "def_path": "Mathlib/Topology/Algebra/Module/Basic.lean", "def_pos": [510, 19], "def_end_pos": [510, 26]}]], "state_before": "case refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup F'\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2076 : NormedSpace \u211d F'\ninst\u271d\u2075 : CompleteSpace F'\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedAddCommGroup G'\ninst\u271d\u00b2 : NormedSpace \u211d G'\ninst\u271d\u00b9 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nf\u271d\u00b9 g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nf\u271d : { x // x \u2208 Lp F' 1 }\nf g : \u03b1 \u2192 F'\nhf : Mem\u2112p f 1\nhg : Mem\u2112p g 1\na\u271d : Disjoint (Function.support f) (Function.support g)\nhfm : AEStronglyMeasurable' m (\u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) (Mem\u2112p.toLp f hf))) \u03bc\nhgm : AEStronglyMeasurable' m (\u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) (Mem\u2112p.toLp g hg))) \u03bc\n\u22a2 AEStronglyMeasurable' m (\u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg))) \u03bc", "state_after": "case refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup F'\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2076 : NormedSpace \u211d F'\ninst\u271d\u2075 : CompleteSpace F'\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedAddCommGroup G'\ninst\u271d\u00b2 : NormedSpace \u211d G'\ninst\u271d\u00b9 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nf\u271d\u00b9 g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nf\u271d : { x // x \u2208 Lp F' 1 }\nf g : \u03b1 \u2192 F'\nhf : Mem\u2112p f 1\nhg : Mem\u2112p g 1\na\u271d : Disjoint (Function.support f) (Function.support g)\nhfm : AEStronglyMeasurable' m (\u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) (Mem\u2112p.toLp f hf))) \u03bc\nhgm : AEStronglyMeasurable' m (\u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) (Mem\u2112p.toLp g hg))) \u03bc\n\u22a2 AEStronglyMeasurable' m (\u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) (Mem\u2112p.toLp f hf) + \u2191(condexpL1Clm F' hm \u03bc) (Mem\u2112p.toLp g hg))) \u03bc"}, {"tactic": "refine' AEStronglyMeasurable'.congr _ (coeFn_add _ _).symm", "annotated_tactic": ["refine' <a>AEStronglyMeasurable'.congr</a> _ (<a>coeFn_add</a> _ _).<a>symm</a>", [{"full_name": "MeasureTheory.AEStronglyMeasurable'.congr", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean", "def_pos": [59, 9], "def_end_pos": [59, 14]}, {"full_name": "MeasureTheory.Lp.coeFn_add", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [232, 9], "def_end_pos": [232, 18]}, {"full_name": "Filter.EventuallyEq.symm", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1498, 9], "def_end_pos": [1498, 26]}]], "state_before": "case refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup F'\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2076 : NormedSpace \u211d F'\ninst\u271d\u2075 : CompleteSpace F'\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedAddCommGroup G'\ninst\u271d\u00b2 : NormedSpace \u211d G'\ninst\u271d\u00b9 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nf\u271d\u00b9 g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nf\u271d : { x // x \u2208 Lp F' 1 }\nf g : \u03b1 \u2192 F'\nhf : Mem\u2112p f 1\nhg : Mem\u2112p g 1\na\u271d : Disjoint (Function.support f) (Function.support g)\nhfm : AEStronglyMeasurable' m (\u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) (Mem\u2112p.toLp f hf))) \u03bc\nhgm : AEStronglyMeasurable' m (\u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) (Mem\u2112p.toLp g hg))) \u03bc\n\u22a2 AEStronglyMeasurable' m (\u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) (Mem\u2112p.toLp f hf) + \u2191(condexpL1Clm F' hm \u03bc) (Mem\u2112p.toLp g hg))) \u03bc", "state_after": "case refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup F'\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2076 : NormedSpace \u211d F'\ninst\u271d\u2075 : CompleteSpace F'\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedAddCommGroup G'\ninst\u271d\u00b2 : NormedSpace \u211d G'\ninst\u271d\u00b9 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nf\u271d\u00b9 g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nf\u271d : { x // x \u2208 Lp F' 1 }\nf g : \u03b1 \u2192 F'\nhf : Mem\u2112p f 1\nhg : Mem\u2112p g 1\na\u271d : Disjoint (Function.support f) (Function.support g)\nhfm : AEStronglyMeasurable' m (\u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) (Mem\u2112p.toLp f hf))) \u03bc\nhgm : AEStronglyMeasurable' m (\u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) (Mem\u2112p.toLp g hg))) \u03bc\n\u22a2 AEStronglyMeasurable' m\n    (\u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) (Mem\u2112p.toLp f hf)) + \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) (Mem\u2112p.toLp g hg))) \u03bc"}, {"tactic": "exact AEStronglyMeasurable'.add hfm hgm", "annotated_tactic": ["exact <a>AEStronglyMeasurable'.add</a> hfm hgm", [{"full_name": "MeasureTheory.AEStronglyMeasurable'.add", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean", "def_pos": [67, 9], "def_end_pos": [67, 12]}]], "state_before": "case refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup F'\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2076 : NormedSpace \u211d F'\ninst\u271d\u2075 : CompleteSpace F'\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedAddCommGroup G'\ninst\u271d\u00b2 : NormedSpace \u211d G'\ninst\u271d\u00b9 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nf\u271d\u00b9 g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nf\u271d : { x // x \u2208 Lp F' 1 }\nf g : \u03b1 \u2192 F'\nhf : Mem\u2112p f 1\nhg : Mem\u2112p g 1\na\u271d : Disjoint (Function.support f) (Function.support g)\nhfm : AEStronglyMeasurable' m (\u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) (Mem\u2112p.toLp f hf))) \u03bc\nhgm : AEStronglyMeasurable' m (\u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) (Mem\u2112p.toLp g hg))) \u03bc\n\u22a2 AEStronglyMeasurable' m\n    (\u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) (Mem\u2112p.toLp f hf)) + \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) (Mem\u2112p.toLp g hg))) \u03bc", "state_after": "no goals"}, {"tactic": "have : {f : Lp F' 1 \u03bc | AEStronglyMeasurable' m (condexpL1Clm F' hm \u03bc f) \u03bc} =\n    condexpL1Clm F' hm \u03bc \u207b\u00b9' {f | AEStronglyMeasurable' m f \u03bc} := rfl", "annotated_tactic": ["have : {f : <a>Lp</a> F' 1 \u03bc | <a>AEStronglyMeasurable'</a> m (<a>condexpL1Clm</a> F' hm \u03bc f) \u03bc} =\n        <a>condexpL1Clm</a> F' hm \u03bc \u207b\u00b9' {f | <a>AEStronglyMeasurable'</a> m f \u03bc} := <a>rfl</a>", [{"full_name": "MeasureTheory.Lp", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [98, 5], "def_end_pos": [98, 7]}, {"full_name": "MeasureTheory.AEStronglyMeasurable'", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean", "def_pos": [49, 5], "def_end_pos": [49, 26]}, {"full_name": "MeasureTheory.condexpL1Clm", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean", "def_pos": [380, 5], "def_end_pos": [380, 17]}, {"full_name": "MeasureTheory.condexpL1Clm", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean", "def_pos": [380, 5], "def_end_pos": [380, 17]}, {"full_name": "MeasureTheory.AEStronglyMeasurable'", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean", "def_pos": [49, 5], "def_end_pos": [49, 26]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "case refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup F'\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2076 : NormedSpace \u211d F'\ninst\u271d\u2075 : CompleteSpace F'\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedAddCommGroup G'\ninst\u271d\u00b2 : NormedSpace \u211d G'\ninst\u271d\u00b9 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nf\u271d g : \u03b1 \u2192 F'\ns : Set \u03b1\nf : { x // x \u2208 Lp F' 1 }\n\u22a2 IsClosed {f | (fun f => AEStronglyMeasurable' m (\u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) f)) \u03bc) f}", "state_after": "case refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup F'\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2076 : NormedSpace \u211d F'\ninst\u271d\u2075 : CompleteSpace F'\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedAddCommGroup G'\ninst\u271d\u00b2 : NormedSpace \u211d G'\ninst\u271d\u00b9 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nf\u271d g : \u03b1 \u2192 F'\ns : Set \u03b1\nf : { x // x \u2208 Lp F' 1 }\nthis :\n  {f | AEStronglyMeasurable' m (\u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) f)) \u03bc} =\n    \u2191(condexpL1Clm F' hm \u03bc) \u207b\u00b9' {f | AEStronglyMeasurable' m (\u2191\u2191f) \u03bc}\n\u22a2 IsClosed {f | (fun f => AEStronglyMeasurable' m (\u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) f)) \u03bc) f}"}, {"tactic": "rw [this]", "annotated_tactic": ["rw [this]", []], "state_before": "case refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup F'\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2076 : NormedSpace \u211d F'\ninst\u271d\u2075 : CompleteSpace F'\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedAddCommGroup G'\ninst\u271d\u00b2 : NormedSpace \u211d G'\ninst\u271d\u00b9 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nf\u271d g : \u03b1 \u2192 F'\ns : Set \u03b1\nf : { x // x \u2208 Lp F' 1 }\nthis :\n  {f | AEStronglyMeasurable' m (\u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) f)) \u03bc} =\n    \u2191(condexpL1Clm F' hm \u03bc) \u207b\u00b9' {f | AEStronglyMeasurable' m (\u2191\u2191f) \u03bc}\n\u22a2 IsClosed {f | (fun f => AEStronglyMeasurable' m (\u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) f)) \u03bc) f}", "state_after": "case refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup F'\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2076 : NormedSpace \u211d F'\ninst\u271d\u2075 : CompleteSpace F'\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedAddCommGroup G'\ninst\u271d\u00b2 : NormedSpace \u211d G'\ninst\u271d\u00b9 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nf\u271d g : \u03b1 \u2192 F'\ns : Set \u03b1\nf : { x // x \u2208 Lp F' 1 }\nthis :\n  {f | AEStronglyMeasurable' m (\u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) f)) \u03bc} =\n    \u2191(condexpL1Clm F' hm \u03bc) \u207b\u00b9' {f | AEStronglyMeasurable' m (\u2191\u2191f) \u03bc}\n\u22a2 IsClosed (\u2191(condexpL1Clm F' hm \u03bc) \u207b\u00b9' {f | AEStronglyMeasurable' m (\u2191\u2191f) \u03bc})"}, {"tactic": "refine' IsClosed.preimage (condexpL1Clm F' hm \u03bc).continuous _", "annotated_tactic": ["refine' <a>IsClosed.preimage</a> (<a>condexpL1Clm</a> F' hm \u03bc).<a>continuous</a> _", [{"full_name": "IsClosed.preimage", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [1752, 9], "def_end_pos": [1752, 26]}, {"full_name": "MeasureTheory.condexpL1Clm", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean", "def_pos": [380, 5], "def_end_pos": [380, 17]}, {"full_name": "ContinuousLinearMap.continuous", "def_path": "Mathlib/Topology/Algebra/Module/Basic.lean", "def_pos": [448, 19], "def_end_pos": [448, 29]}]], "state_before": "case refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup F'\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2076 : NormedSpace \u211d F'\ninst\u271d\u2075 : CompleteSpace F'\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedAddCommGroup G'\ninst\u271d\u00b2 : NormedSpace \u211d G'\ninst\u271d\u00b9 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nf\u271d g : \u03b1 \u2192 F'\ns : Set \u03b1\nf : { x // x \u2208 Lp F' 1 }\nthis :\n  {f | AEStronglyMeasurable' m (\u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) f)) \u03bc} =\n    \u2191(condexpL1Clm F' hm \u03bc) \u207b\u00b9' {f | AEStronglyMeasurable' m (\u2191\u2191f) \u03bc}\n\u22a2 IsClosed (\u2191(condexpL1Clm F' hm \u03bc) \u207b\u00b9' {f | AEStronglyMeasurable' m (\u2191\u2191f) \u03bc})", "state_after": "case refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup F'\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2076 : NormedSpace \u211d F'\ninst\u271d\u2075 : CompleteSpace F'\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedAddCommGroup G'\ninst\u271d\u00b2 : NormedSpace \u211d G'\ninst\u271d\u00b9 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nf\u271d g : \u03b1 \u2192 F'\ns : Set \u03b1\nf : { x // x \u2208 Lp F' 1 }\nthis :\n  {f | AEStronglyMeasurable' m (\u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) f)) \u03bc} =\n    \u2191(condexpL1Clm F' hm \u03bc) \u207b\u00b9' {f | AEStronglyMeasurable' m (\u2191\u2191f) \u03bc}\n\u22a2 IsClosed {f | AEStronglyMeasurable' m (\u2191\u2191f) \u03bc}"}, {"tactic": "exact isClosed_aeStronglyMeasurable' hm", "annotated_tactic": ["exact <a>isClosed_aeStronglyMeasurable'</a> hm", [{"full_name": "MeasureTheory.isClosed_aeStronglyMeasurable'", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean", "def_pos": [510, 9], "def_end_pos": [510, 39]}]], "state_before": "case refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup F'\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2076 : NormedSpace \u211d F'\ninst\u271d\u2075 : CompleteSpace F'\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedAddCommGroup G'\ninst\u271d\u00b2 : NormedSpace \u211d G'\ninst\u271d\u00b9 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nf\u271d g : \u03b1 \u2192 F'\ns : Set \u03b1\nf : { x // x \u2208 Lp F' 1 }\nthis :\n  {f | AEStronglyMeasurable' m (\u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) f)) \u03bc} =\n    \u2191(condexpL1Clm F' hm \u03bc) \u207b\u00b9' {f | AEStronglyMeasurable' m (\u2191\u2191f) \u03bc}\n\u22a2 IsClosed {f | AEStronglyMeasurable' m (\u2191\u2191f) \u03bc}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Card.lean", "full_name": "Finset.exists_subset_or_subset_of_two_mul_lt_card", "start": [508, 1], "end": [516, 55], "traced_tactics": [{"tactic": "have h\u2081 : (X \u2229 (Y \\ X)).card = 0 := Finset.card_eq_zero.mpr (Finset.inter_sdiff_self X Y)", "annotated_tactic": ["have h\u2081 : (X \u2229 (Y \\ X)).<a>card</a> = 0 := Finset.card_eq_zero.mpr (<a>Finset.inter_sdiff_self</a> X Y)", [{"full_name": "Finset.card", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [43, 5], "def_end_pos": [43, 9]}, {"full_name": "Finset.inter_sdiff_self", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2082, 9], "def_end_pos": [2082, 25]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn\u271d : \u2115\ninst\u271d : DecidableEq \u03b1\nX Y : Finset \u03b1\nn : \u2115\nhXY : 2 * n < card (X \u222a Y)\n\u22a2 \u2203 C, n < card C \u2227 (C \u2286 X \u2228 C \u2286 Y)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn\u271d : \u2115\ninst\u271d : DecidableEq \u03b1\nX Y : Finset \u03b1\nn : \u2115\nhXY : 2 * n < card (X \u222a Y)\nh\u2081 : card (X \u2229 (Y \\ X)) = 0\n\u22a2 \u2203 C, n < card C \u2227 (C \u2286 X \u2228 C \u2286 Y)"}, {"tactic": "have h\u2082 : (X \u222a Y).card = X.card + (Y \\ X).card := by\n  rw [\u2190 card_union_add_card_inter X (Y \\ X), Finset.union_sdiff_self_eq_union, h\u2081, add_zero]", "annotated_tactic": ["have h\u2082 : (X \u222a Y).<a>card</a> = X.card + (Y \\ X).<a>card</a> := by\n    rw [\u2190 <a>card_union_add_card_inter</a> X (Y \\ X), <a>Finset.union_sdiff_self_eq_union</a>, h\u2081, <a>add_zero</a>]", [{"full_name": "Finset.card", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [43, 5], "def_end_pos": [43, 9]}, {"full_name": "Finset.card", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [43, 5], "def_end_pos": [43, 9]}, {"full_name": "Finset.card_union_add_card_inter", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [417, 9], "def_end_pos": [417, 34]}, {"full_name": "Finset.union_sdiff_self_eq_union", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2155, 9], "def_end_pos": [2155, 34]}, {"full_name": "add_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [469, 3], "def_end_pos": [469, 14]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn\u271d : \u2115\ninst\u271d : DecidableEq \u03b1\nX Y : Finset \u03b1\nn : \u2115\nhXY : 2 * n < card (X \u222a Y)\nh\u2081 : card (X \u2229 (Y \\ X)) = 0\n\u22a2 \u2203 C, n < card C \u2227 (C \u2286 X \u2228 C \u2286 Y)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn\u271d : \u2115\ninst\u271d : DecidableEq \u03b1\nX Y : Finset \u03b1\nn : \u2115\nhXY : 2 * n < card (X \u222a Y)\nh\u2081 : card (X \u2229 (Y \\ X)) = 0\nh\u2082 : card (X \u222a Y) = card X + card (Y \\ X)\n\u22a2 \u2203 C, n < card C \u2227 (C \u2286 X \u2228 C \u2286 Y)"}, {"tactic": "rw [h\u2082, two_mul] at hXY", "annotated_tactic": ["rw [h\u2082, <a>two_mul</a>] at hXY", [{"full_name": "two_mul", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [177, 9], "def_end_pos": [177, 16]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn\u271d : \u2115\ninst\u271d : DecidableEq \u03b1\nX Y : Finset \u03b1\nn : \u2115\nhXY : 2 * n < card (X \u222a Y)\nh\u2081 : card (X \u2229 (Y \\ X)) = 0\nh\u2082 : card (X \u222a Y) = card X + card (Y \\ X)\n\u22a2 \u2203 C, n < card C \u2227 (C \u2286 X \u2228 C \u2286 Y)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn\u271d : \u2115\ninst\u271d : DecidableEq \u03b1\nX Y : Finset \u03b1\nn : \u2115\nhXY : n + n < card X + card (Y \\ X)\nh\u2081 : card (X \u2229 (Y \\ X)) = 0\nh\u2082 : card (X \u222a Y) = card X + card (Y \\ X)\n\u22a2 \u2203 C, n < card C \u2227 (C \u2286 X \u2228 C \u2286 Y)"}, {"tactic": "rcases lt_or_lt_of_add_lt_add hXY with (h | h)", "annotated_tactic": ["rcases <a>lt_or_lt_of_add_lt_add</a> hXY with (h | h)", [{"full_name": "lt_or_lt_of_add_lt_add", "def_path": "Mathlib/Algebra/Order/Monoid/MinMax.lean", "def_pos": [80, 3], "def_end_pos": [80, 14]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn\u271d : \u2115\ninst\u271d : DecidableEq \u03b1\nX Y : Finset \u03b1\nn : \u2115\nhXY : n + n < card X + card (Y \\ X)\nh\u2081 : card (X \u2229 (Y \\ X)) = 0\nh\u2082 : card (X \u222a Y) = card X + card (Y \\ X)\n\u22a2 \u2203 C, n < card C \u2227 (C \u2286 X \u2228 C \u2286 Y)", "state_after": "case inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn\u271d : \u2115\ninst\u271d : DecidableEq \u03b1\nX Y : Finset \u03b1\nn : \u2115\nhXY : n + n < card X + card (Y \\ X)\nh\u2081 : card (X \u2229 (Y \\ X)) = 0\nh\u2082 : card (X \u222a Y) = card X + card (Y \\ X)\nh : n < card X\n\u22a2 \u2203 C, n < card C \u2227 (C \u2286 X \u2228 C \u2286 Y)\n\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn\u271d : \u2115\ninst\u271d : DecidableEq \u03b1\nX Y : Finset \u03b1\nn : \u2115\nhXY : n + n < card X + card (Y \\ X)\nh\u2081 : card (X \u2229 (Y \\ X)) = 0\nh\u2082 : card (X \u222a Y) = card X + card (Y \\ X)\nh : n < card (Y \\ X)\n\u22a2 \u2203 C, n < card C \u2227 (C \u2286 X \u2228 C \u2286 Y)"}, {"tactic": "rw [\u2190 card_union_add_card_inter X (Y \\ X), Finset.union_sdiff_self_eq_union, h\u2081, add_zero]", "annotated_tactic": ["rw [\u2190 <a>card_union_add_card_inter</a> X (Y \\ X), <a>Finset.union_sdiff_self_eq_union</a>, h\u2081, <a>add_zero</a>]", [{"full_name": "Finset.card_union_add_card_inter", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [417, 9], "def_end_pos": [417, 34]}, {"full_name": "Finset.union_sdiff_self_eq_union", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2155, 9], "def_end_pos": [2155, 34]}, {"full_name": "add_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [469, 3], "def_end_pos": [469, 14]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn\u271d : \u2115\ninst\u271d : DecidableEq \u03b1\nX Y : Finset \u03b1\nn : \u2115\nhXY : 2 * n < card (X \u222a Y)\nh\u2081 : card (X \u2229 (Y \\ X)) = 0\n\u22a2 card (X \u222a Y) = card X + card (Y \\ X)", "state_after": "no goals"}, {"tactic": "exact \u27e8X, h, Or.inl (Finset.Subset.refl X)\u27e9", "annotated_tactic": ["exact \u27e8X, h, <a>Or.inl</a> (<a>Finset.Subset.refl</a> X)\u27e9", [{"full_name": "Or.inl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [517, 5], "def_end_pos": [517, 8]}, {"full_name": "Finset.Subset.refl", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [339, 9], "def_end_pos": [339, 20]}]], "state_before": "case inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn\u271d : \u2115\ninst\u271d : DecidableEq \u03b1\nX Y : Finset \u03b1\nn : \u2115\nhXY : n + n < card X + card (Y \\ X)\nh\u2081 : card (X \u2229 (Y \\ X)) = 0\nh\u2082 : card (X \u222a Y) = card X + card (Y \\ X)\nh : n < card X\n\u22a2 \u2203 C, n < card C \u2227 (C \u2286 X \u2228 C \u2286 Y)", "state_after": "no goals"}, {"tactic": "exact \u27e8Y \\ X, h, Or.inr (Finset.sdiff_subset Y X)\u27e9", "annotated_tactic": ["exact \u27e8Y \\ X, h, <a>Or.inr</a> (<a>Finset.sdiff_subset</a> Y X)\u27e9", [{"full_name": "Or.inr", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [519, 5], "def_end_pos": [519, 8]}, {"full_name": "Finset.sdiff_subset", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2236, 9], "def_end_pos": [2236, 21]}]], "state_before": "case inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn\u271d : \u2115\ninst\u271d : DecidableEq \u03b1\nX Y : Finset \u03b1\nn : \u2115\nhXY : n + n < card X + card (Y \\ X)\nh\u2081 : card (X \u2229 (Y \\ X)) = 0\nh\u2082 : card (X \u222a Y) = card X + card (Y \\ X)\nh : n < card (Y \\ X)\n\u22a2 \u2203 C, n < card C \u2227 (C \u2286 X \u2228 C \u2286 Y)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Quot.lean", "full_name": "Quot.factor_mk_eq", "start": [90, 1], "end": [92, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Martingale/Convergence.lean", "full_name": "MeasureTheory.not_frequently_of_upcrossings_lt_top", "start": [111, 1], "end": [128, 91], "traced_tactics": [{"tactic": "rw [\u2190 lt_top_iff_ne_top, upcrossings_lt_top_iff] at h\u03c9", "annotated_tactic": ["rw [\u2190 <a>lt_top_iff_ne_top</a>, <a>upcrossings_lt_top_iff</a>] at h\u03c9", [{"full_name": "lt_top_iff_ne_top", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [173, 9], "def_end_pos": [173, 26]}, {"full_name": "MeasureTheory.upcrossings_lt_top_iff", "def_path": "Mathlib/Probability/Martingale/Upcrossing.lean", "def_pos": [833, 9], "def_end_pos": [833, 31]}]], "state_before": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nR : \u211d\u22650\nhab : a < b\nh\u03c9 : upcrossings a b f \u03c9 \u2260 \u22a4\n\u22a2 \u00ac((\u2203\u1da0 (n : \u2115) in atTop, f n \u03c9 < a) \u2227 \u2203\u1da0 (n : \u2115) in atTop, b < f n \u03c9)", "state_after": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nR : \u211d\u22650\nhab : a < b\nh\u03c9 : \u2203 k, \u2200 (N : \u2115), upcrossingsBefore a b f N \u03c9 \u2264 k\n\u22a2 \u00ac((\u2203\u1da0 (n : \u2115) in atTop, f n \u03c9 < a) \u2227 \u2203\u1da0 (n : \u2115) in atTop, b < f n \u03c9)"}, {"tactic": "replace h\u03c9 : \u2203 k, \u2200 N, upcrossingsBefore a b f N \u03c9 < k", "annotated_tactic": ["replace h\u03c9 : \u2203 k, \u2200 N, <a>upcrossingsBefore</a> a b f N \u03c9 < k", [{"full_name": "MeasureTheory.upcrossingsBefore", "def_path": "Mathlib/Probability/Martingale/Upcrossing.lean", "def_pos": [450, 19], "def_end_pos": [450, 36]}]], "state_before": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nR : \u211d\u22650\nhab : a < b\nh\u03c9 : \u2203 k, \u2200 (N : \u2115), upcrossingsBefore a b f N \u03c9 \u2264 k\n\u22a2 \u00ac((\u2203\u1da0 (n : \u2115) in atTop, f n \u03c9 < a) \u2227 \u2203\u1da0 (n : \u2115) in atTop, b < f n \u03c9)", "state_after": "case h\u03c9\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nR : \u211d\u22650\nhab : a < b\nh\u03c9 : \u2203 k, \u2200 (N : \u2115), upcrossingsBefore a b f N \u03c9 \u2264 k\n\u22a2 \u2203 k, \u2200 (N : \u2115), upcrossingsBefore a b f N \u03c9 < k\n\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nR : \u211d\u22650\nhab : a < b\nh\u03c9 : \u2203 k, \u2200 (N : \u2115), upcrossingsBefore a b f N \u03c9 < k\n\u22a2 \u00ac((\u2203\u1da0 (n : \u2115) in atTop, f n \u03c9 < a) \u2227 \u2203\u1da0 (n : \u2115) in atTop, b < f n \u03c9)"}, {"tactic": "rintro \u27e8h\u2081, h\u2082\u27e9", "annotated_tactic": ["rintro \u27e8h\u2081, h\u2082\u27e9", []], "state_before": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nR : \u211d\u22650\nhab : a < b\nh\u03c9 : \u2203 k, \u2200 (N : \u2115), upcrossingsBefore a b f N \u03c9 < k\n\u22a2 \u00ac((\u2203\u1da0 (n : \u2115) in atTop, f n \u03c9 < a) \u2227 \u2203\u1da0 (n : \u2115) in atTop, b < f n \u03c9)", "state_after": "case intro\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nR : \u211d\u22650\nhab : a < b\nh\u03c9 : \u2203 k, \u2200 (N : \u2115), upcrossingsBefore a b f N \u03c9 < k\nh\u2081 : \u2203\u1da0 (n : \u2115) in atTop, f n \u03c9 < a\nh\u2082 : \u2203\u1da0 (n : \u2115) in atTop, b < f n \u03c9\n\u22a2 False"}, {"tactic": "rw [frequently_atTop] at h\u2081 h\u2082", "annotated_tactic": ["rw [<a>frequently_atTop</a>] at h\u2081 h\u2082", [{"full_name": "Filter.frequently_atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [349, 9], "def_end_pos": [349, 25]}]], "state_before": "case intro\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nR : \u211d\u22650\nhab : a < b\nh\u03c9 : \u2203 k, \u2200 (N : \u2115), upcrossingsBefore a b f N \u03c9 < k\nh\u2081 : \u2203\u1da0 (n : \u2115) in atTop, f n \u03c9 < a\nh\u2082 : \u2203\u1da0 (n : \u2115) in atTop, b < f n \u03c9\n\u22a2 False", "state_after": "case intro\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nR : \u211d\u22650\nhab : a < b\nh\u03c9 : \u2203 k, \u2200 (N : \u2115), upcrossingsBefore a b f N \u03c9 < k\nh\u2081 : \u2200 (a_1 : \u2115), \u2203 b, b \u2265 a_1 \u2227 f b \u03c9 < a\nh\u2082 : \u2200 (a : \u2115), \u2203 b_1, b_1 \u2265 a \u2227 b < f b_1 \u03c9\n\u22a2 False"}, {"tactic": "refine' Classical.not_not.2 h\u03c9 _", "annotated_tactic": ["refine' <a>Classical.not_not</a>.2 h\u03c9 _", [{"full_name": "Classical.not_not", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [683, 24], "def_end_pos": [683, 31]}]], "state_before": "case intro\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nR : \u211d\u22650\nhab : a < b\nh\u03c9 : \u2203 k, \u2200 (N : \u2115), upcrossingsBefore a b f N \u03c9 < k\nh\u2081 : \u2200 (a_1 : \u2115), \u2203 b, b \u2265 a_1 \u2227 f b \u03c9 < a\nh\u2082 : \u2200 (a : \u2115), \u2203 b_1, b_1 \u2265 a \u2227 b < f b_1 \u03c9\n\u22a2 False", "state_after": "case intro\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nR : \u211d\u22650\nhab : a < b\nh\u03c9 : \u2203 k, \u2200 (N : \u2115), upcrossingsBefore a b f N \u03c9 < k\nh\u2081 : \u2200 (a_1 : \u2115), \u2203 b, b \u2265 a_1 \u2227 f b \u03c9 < a\nh\u2082 : \u2200 (a : \u2115), \u2203 b_1, b_1 \u2265 a \u2227 b < f b_1 \u03c9\n\u22a2 \u00ac\u2203 k, \u2200 (N : \u2115), upcrossingsBefore a b f N \u03c9 < k"}, {"tactic": "push_neg", "annotated_tactic": ["push_neg", []], "state_before": "case intro\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nR : \u211d\u22650\nhab : a < b\nh\u03c9 : \u2203 k, \u2200 (N : \u2115), upcrossingsBefore a b f N \u03c9 < k\nh\u2081 : \u2200 (a_1 : \u2115), \u2203 b, b \u2265 a_1 \u2227 f b \u03c9 < a\nh\u2082 : \u2200 (a : \u2115), \u2203 b_1, b_1 \u2265 a \u2227 b < f b_1 \u03c9\n\u22a2 \u00ac\u2203 k, \u2200 (N : \u2115), upcrossingsBefore a b f N \u03c9 < k", "state_after": "case intro\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nR : \u211d\u22650\nhab : a < b\nh\u03c9 : \u2203 k, \u2200 (N : \u2115), upcrossingsBefore a b f N \u03c9 < k\nh\u2081 : \u2200 (a_1 : \u2115), \u2203 b, b \u2265 a_1 \u2227 f b \u03c9 < a\nh\u2082 : \u2200 (a : \u2115), \u2203 b_1, b_1 \u2265 a \u2227 b < f b_1 \u03c9\n\u22a2 \u2200 (k : \u2115), \u2203 N, k \u2264 upcrossingsBefore a b f N \u03c9"}, {"tactic": "intro k", "annotated_tactic": ["intro k", []], "state_before": "case intro\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nR : \u211d\u22650\nhab : a < b\nh\u03c9 : \u2203 k, \u2200 (N : \u2115), upcrossingsBefore a b f N \u03c9 < k\nh\u2081 : \u2200 (a_1 : \u2115), \u2203 b, b \u2265 a_1 \u2227 f b \u03c9 < a\nh\u2082 : \u2200 (a : \u2115), \u2203 b_1, b_1 \u2265 a \u2227 b < f b_1 \u03c9\n\u22a2 \u2200 (k : \u2115), \u2203 N, k \u2264 upcrossingsBefore a b f N \u03c9", "state_after": "case intro\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nR : \u211d\u22650\nhab : a < b\nh\u03c9 : \u2203 k, \u2200 (N : \u2115), upcrossingsBefore a b f N \u03c9 < k\nh\u2081 : \u2200 (a_1 : \u2115), \u2203 b, b \u2265 a_1 \u2227 f b \u03c9 < a\nh\u2082 : \u2200 (a : \u2115), \u2203 b_1, b_1 \u2265 a \u2227 b < f b_1 \u03c9\nk : \u2115\n\u22a2 \u2203 N, k \u2264 upcrossingsBefore a b f N \u03c9"}, {"tactic": "induction' k with k ih", "annotated_tactic": ["induction' k with k ih", []], "state_before": "case intro\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nR : \u211d\u22650\nhab : a < b\nh\u03c9 : \u2203 k, \u2200 (N : \u2115), upcrossingsBefore a b f N \u03c9 < k\nh\u2081 : \u2200 (a_1 : \u2115), \u2203 b, b \u2265 a_1 \u2227 f b \u03c9 < a\nh\u2082 : \u2200 (a : \u2115), \u2203 b_1, b_1 \u2265 a \u2227 b < f b_1 \u03c9\nk : \u2115\n\u22a2 \u2203 N, k \u2264 upcrossingsBefore a b f N \u03c9", "state_after": "case intro.zero\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nR : \u211d\u22650\nhab : a < b\nh\u03c9 : \u2203 k, \u2200 (N : \u2115), upcrossingsBefore a b f N \u03c9 < k\nh\u2081 : \u2200 (a_1 : \u2115), \u2203 b, b \u2265 a_1 \u2227 f b \u03c9 < a\nh\u2082 : \u2200 (a : \u2115), \u2203 b_1, b_1 \u2265 a \u2227 b < f b_1 \u03c9\n\u22a2 \u2203 N, Nat.zero \u2264 upcrossingsBefore a b f N \u03c9\n\ncase intro.succ\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nR : \u211d\u22650\nhab : a < b\nh\u03c9 : \u2203 k, \u2200 (N : \u2115), upcrossingsBefore a b f N \u03c9 < k\nh\u2081 : \u2200 (a_1 : \u2115), \u2203 b, b \u2265 a_1 \u2227 f b \u03c9 < a\nh\u2082 : \u2200 (a : \u2115), \u2203 b_1, b_1 \u2265 a \u2227 b < f b_1 \u03c9\nk : \u2115\nih : \u2203 N, k \u2264 upcrossingsBefore a b f N \u03c9\n\u22a2 \u2203 N, Nat.succ k \u2264 upcrossingsBefore a b f N \u03c9"}, {"tactic": "obtain \u27e8k, hk\u27e9 := h\u03c9", "annotated_tactic": ["obtain \u27e8k, hk\u27e9 := h\u03c9", []], "state_before": "case h\u03c9\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nR : \u211d\u22650\nhab : a < b\nh\u03c9 : \u2203 k, \u2200 (N : \u2115), upcrossingsBefore a b f N \u03c9 \u2264 k\n\u22a2 \u2203 k, \u2200 (N : \u2115), upcrossingsBefore a b f N \u03c9 < k", "state_after": "case h\u03c9.intro\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nR : \u211d\u22650\nhab : a < b\nk : \u2115\nhk : \u2200 (N : \u2115), upcrossingsBefore a b f N \u03c9 \u2264 k\n\u22a2 \u2203 k, \u2200 (N : \u2115), upcrossingsBefore a b f N \u03c9 < k"}, {"tactic": "exact \u27e8k + 1, fun N => lt_of_le_of_lt (hk N) k.lt_succ_self\u27e9", "annotated_tactic": ["exact \u27e8k + 1, fun N => <a>lt_of_le_of_lt</a> (hk N) k.lt_succ_self\u27e9", [{"full_name": "lt_of_le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [122, 9], "def_end_pos": [122, 23]}]], "state_before": "case h\u03c9.intro\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nR : \u211d\u22650\nhab : a < b\nk : \u2115\nhk : \u2200 (N : \u2115), upcrossingsBefore a b f N \u03c9 \u2264 k\n\u22a2 \u2203 k, \u2200 (N : \u2115), upcrossingsBefore a b f N \u03c9 < k", "state_after": "no goals"}, {"tactic": "simp only [Nat.zero_eq, zero_le, exists_const]", "annotated_tactic": ["simp only [<a>Nat.zero_eq</a>, <a>zero_le</a>, <a>exists_const</a>]", [{"full_name": "Nat.zero_eq", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [83, 17], "def_end_pos": [83, 24]}, {"full_name": "zero_le", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [217, 30], "def_end_pos": [217, 37]}, {"full_name": "exists_const", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [369, 17], "def_end_pos": [369, 29]}]], "state_before": "case intro.zero\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nR : \u211d\u22650\nhab : a < b\nh\u03c9 : \u2203 k, \u2200 (N : \u2115), upcrossingsBefore a b f N \u03c9 < k\nh\u2081 : \u2200 (a_1 : \u2115), \u2203 b, b \u2265 a_1 \u2227 f b \u03c9 < a\nh\u2082 : \u2200 (a : \u2115), \u2203 b_1, b_1 \u2265 a \u2227 b < f b_1 \u03c9\n\u22a2 \u2203 N, Nat.zero \u2264 upcrossingsBefore a b f N \u03c9", "state_after": "no goals"}, {"tactic": "obtain \u27e8N, hN\u27e9 := ih", "annotated_tactic": ["obtain \u27e8N, hN\u27e9 := ih", []], "state_before": "case intro.succ\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nR : \u211d\u22650\nhab : a < b\nh\u03c9 : \u2203 k, \u2200 (N : \u2115), upcrossingsBefore a b f N \u03c9 < k\nh\u2081 : \u2200 (a_1 : \u2115), \u2203 b, b \u2265 a_1 \u2227 f b \u03c9 < a\nh\u2082 : \u2200 (a : \u2115), \u2203 b_1, b_1 \u2265 a \u2227 b < f b_1 \u03c9\nk : \u2115\nih : \u2203 N, k \u2264 upcrossingsBefore a b f N \u03c9\n\u22a2 \u2203 N, Nat.succ k \u2264 upcrossingsBefore a b f N \u03c9", "state_after": "case intro.succ.intro\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nR : \u211d\u22650\nhab : a < b\nh\u03c9 : \u2203 k, \u2200 (N : \u2115), upcrossingsBefore a b f N \u03c9 < k\nh\u2081 : \u2200 (a_1 : \u2115), \u2203 b, b \u2265 a_1 \u2227 f b \u03c9 < a\nh\u2082 : \u2200 (a : \u2115), \u2203 b_1, b_1 \u2265 a \u2227 b < f b_1 \u03c9\nk N : \u2115\nhN : k \u2264 upcrossingsBefore a b f N \u03c9\n\u22a2 \u2203 N, Nat.succ k \u2264 upcrossingsBefore a b f N \u03c9"}, {"tactic": "obtain \u27e8N\u2081, hN\u2081, hN\u2081'\u27e9 := h\u2081 N", "annotated_tactic": ["obtain \u27e8N\u2081, hN\u2081, hN\u2081'\u27e9 := h\u2081 N", []], "state_before": "case intro.succ.intro\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nR : \u211d\u22650\nhab : a < b\nh\u03c9 : \u2203 k, \u2200 (N : \u2115), upcrossingsBefore a b f N \u03c9 < k\nh\u2081 : \u2200 (a_1 : \u2115), \u2203 b, b \u2265 a_1 \u2227 f b \u03c9 < a\nh\u2082 : \u2200 (a : \u2115), \u2203 b_1, b_1 \u2265 a \u2227 b < f b_1 \u03c9\nk N : \u2115\nhN : k \u2264 upcrossingsBefore a b f N \u03c9\n\u22a2 \u2203 N, Nat.succ k \u2264 upcrossingsBefore a b f N \u03c9", "state_after": "case intro.succ.intro.intro.intro\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nR : \u211d\u22650\nhab : a < b\nh\u03c9 : \u2203 k, \u2200 (N : \u2115), upcrossingsBefore a b f N \u03c9 < k\nh\u2081 : \u2200 (a_1 : \u2115), \u2203 b, b \u2265 a_1 \u2227 f b \u03c9 < a\nh\u2082 : \u2200 (a : \u2115), \u2203 b_1, b_1 \u2265 a \u2227 b < f b_1 \u03c9\nk N : \u2115\nhN : k \u2264 upcrossingsBefore a b f N \u03c9\nN\u2081 : \u2115\nhN\u2081 : N\u2081 \u2265 N\nhN\u2081' : f N\u2081 \u03c9 < a\n\u22a2 \u2203 N, Nat.succ k \u2264 upcrossingsBefore a b f N \u03c9"}, {"tactic": "obtain \u27e8N\u2082, hN\u2082, hN\u2082'\u27e9 := h\u2082 N\u2081", "annotated_tactic": ["obtain \u27e8N\u2082, hN\u2082, hN\u2082'\u27e9 := h\u2082 N\u2081", []], "state_before": "case intro.succ.intro.intro.intro\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nR : \u211d\u22650\nhab : a < b\nh\u03c9 : \u2203 k, \u2200 (N : \u2115), upcrossingsBefore a b f N \u03c9 < k\nh\u2081 : \u2200 (a_1 : \u2115), \u2203 b, b \u2265 a_1 \u2227 f b \u03c9 < a\nh\u2082 : \u2200 (a : \u2115), \u2203 b_1, b_1 \u2265 a \u2227 b < f b_1 \u03c9\nk N : \u2115\nhN : k \u2264 upcrossingsBefore a b f N \u03c9\nN\u2081 : \u2115\nhN\u2081 : N\u2081 \u2265 N\nhN\u2081' : f N\u2081 \u03c9 < a\n\u22a2 \u2203 N, Nat.succ k \u2264 upcrossingsBefore a b f N \u03c9", "state_after": "case intro.succ.intro.intro.intro.intro.intro\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nR : \u211d\u22650\nhab : a < b\nh\u03c9 : \u2203 k, \u2200 (N : \u2115), upcrossingsBefore a b f N \u03c9 < k\nh\u2081 : \u2200 (a_1 : \u2115), \u2203 b, b \u2265 a_1 \u2227 f b \u03c9 < a\nh\u2082 : \u2200 (a : \u2115), \u2203 b_1, b_1 \u2265 a \u2227 b < f b_1 \u03c9\nk N : \u2115\nhN : k \u2264 upcrossingsBefore a b f N \u03c9\nN\u2081 : \u2115\nhN\u2081 : N\u2081 \u2265 N\nhN\u2081' : f N\u2081 \u03c9 < a\nN\u2082 : \u2115\nhN\u2082 : N\u2082 \u2265 N\u2081\nhN\u2082' : b < f N\u2082 \u03c9\n\u22a2 \u2203 N, Nat.succ k \u2264 upcrossingsBefore a b f N \u03c9"}, {"tactic": "exact \u27e8N\u2082 + 1, Nat.succ_le_of_lt <|\n  lt_of_le_of_lt hN (upcrossingsBefore_lt_of_exists_upcrossing hab hN\u2081 hN\u2081' hN\u2082 hN\u2082')\u27e9", "annotated_tactic": ["exact \u27e8N\u2082 + 1, <a>Nat.succ_le_of_lt</a> <|\n      <a>lt_of_le_of_lt</a> hN (<a>upcrossingsBefore_lt_of_exists_upcrossing</a> hab hN\u2081 hN\u2081' hN\u2082 hN\u2082')\u27e9", [{"full_name": "Nat.succ_le_of_lt", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [313, 9], "def_end_pos": [313, 22]}, {"full_name": "lt_of_le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [122, 9], "def_end_pos": [122, 23]}, {"full_name": "MeasureTheory.upcrossingsBefore_lt_of_exists_upcrossing", "def_path": "Mathlib/Probability/Martingale/Upcrossing.lean", "def_pos": [554, 9], "def_end_pos": [554, 50]}]], "state_before": "case intro.succ.intro.intro.intro.intro.intro\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nR : \u211d\u22650\nhab : a < b\nh\u03c9 : \u2203 k, \u2200 (N : \u2115), upcrossingsBefore a b f N \u03c9 < k\nh\u2081 : \u2200 (a_1 : \u2115), \u2203 b, b \u2265 a_1 \u2227 f b \u03c9 < a\nh\u2082 : \u2200 (a : \u2115), \u2203 b_1, b_1 \u2265 a \u2227 b < f b_1 \u03c9\nk N : \u2115\nhN : k \u2264 upcrossingsBefore a b f N \u03c9\nN\u2081 : \u2115\nhN\u2081 : N\u2081 \u2265 N\nhN\u2081' : f N\u2081 \u03c9 < a\nN\u2082 : \u2115\nhN\u2082 : N\u2082 \u2265 N\u2081\nhN\u2082' : b < f N\u2082 \u03c9\n\u22a2 \u2203 N, Nat.succ k \u2264 upcrossingsBefore a b f N \u03c9", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "full_name": "MeasureTheory.Integrable.edist_toL1_zero", "start": [1471, 1], "end": [1475, 29], "traced_tactics": [{"tactic": "simp only [toL1, Lp.edist_toLp_zero, snorm, one_ne_zero, snorm', one_toReal, ENNReal.rpow_one,\n  ne_eq, not_false_eq_true, div_self, ite_false]", "annotated_tactic": ["simp only [<a>toL1</a>, <a>Lp.edist_toLp_zero</a>, <a>snorm</a>, <a>one_ne_zero</a>, <a>snorm'</a>, <a>one_toReal</a>, <a>ENNReal.rpow_one</a>,\n    <a>ne_eq</a>, <a>not_false_eq_true</a>, <a>div_self</a>, <a>ite_false</a>]", [{"full_name": "MeasureTheory.Integrable.toL1", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [1402, 5], "def_end_pos": [1402, 9]}, {"full_name": "MeasureTheory.Lp.edist_toLp_zero", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [315, 9], "def_end_pos": [315, 24]}, {"full_name": "MeasureTheory.snorm", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [84, 5], "def_end_pos": [84, 10]}, {"full_name": "one_ne_zero", "def_path": "Mathlib/Algebra/NeZero.lean", "def_pos": [55, 15], "def_end_pos": [55, 26]}, {"full_name": "MeasureTheory.snorm'", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [73, 5], "def_end_pos": [73, 11]}, {"full_name": "ENNReal.one_toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [230, 17], "def_end_pos": [230, 27]}, {"full_name": "ENNReal.rpow_one", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [450, 9], "def_end_pos": [450, 17]}, {"full_name": "ne_eq", "def_path": "lake-packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [76, 17], "def_end_pos": [76, 22]}, {"full_name": "not_false_eq_true", "def_path": "lake-packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [100, 17], "def_end_pos": [100, 34]}, {"full_name": "div_self", "def_path": "Mathlib/Algebra/GroupWithZero/Units/Lemmas.lean", "def_pos": [29, 9], "def_end_pos": [29, 17]}, {"full_name": "ite_false", "def_path": "lake-packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [78, 17], "def_end_pos": [78, 26]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : NormedAddCommGroup \u03b3\nf : \u03b1 \u2192 \u03b2\nhf : Integrable f\n\u22a2 edist (toL1 f hf) 0 = \u222b\u207b (a : \u03b1), edist (f a) 0 \u2202\u03bc", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : NormedAddCommGroup \u03b3\nf : \u03b1 \u2192 \u03b2\nhf : Integrable f\n\u22a2 \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a \u2202\u03bc = \u222b\u207b (a : \u03b1), edist (f a) 0 \u2202\u03bc"}, {"tactic": "simp [edist_eq_coe_nnnorm]", "annotated_tactic": ["simp [<a>edist_eq_coe_nnnorm</a>]", [{"full_name": "edist_eq_coe_nnnorm", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [1011, 15], "def_end_pos": [1011, 34]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : NormedAddCommGroup \u03b3\nf : \u03b1 \u2192 \u03b2\nhf : Integrable f\n\u22a2 \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a \u2202\u03bc = \u222b\u207b (a : \u03b1), edist (f a) 0 \u2202\u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean", "full_name": "MeasureTheory.set_integral_condexp", "start": [221, 1], "end": [224, 37], "traced_tactics": [{"tactic": "rw [set_integral_congr_ae (hm s hs) ((condexp_ae_eq_condexpL1 hm f).mono fun x hx _ => hx)]", "annotated_tactic": ["rw [<a>set_integral_congr_ae</a> (hm s hs) ((<a>condexp_ae_eq_condexpL1</a> hm f).<a>mono</a> fun x hx _ => hx)]", [{"full_name": "MeasureTheory.set_integral_congr_ae", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [77, 9], "def_end_pos": [77, 30]}, {"full_name": "MeasureTheory.condexp_ae_eq_condexpL1", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean", "def_pos": [136, 9], "def_end_pos": [136, 32]}, {"full_name": "Filter.Eventually.mono", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1140, 9], "def_end_pos": [1140, 24]}]], "state_before": "\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nhf : Integrable f\nhs : MeasurableSet s\n\u22a2 \u222b (x : \u03b1) in s, (\u03bc[f|m]) x \u2202\u03bc = \u222b (x : \u03b1) in s, f x \u2202\u03bc", "state_after": "\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nhf : Integrable f\nhs : MeasurableSet s\n\u22a2 \u222b (x : \u03b1) in s, \u2191\u2191(condexpL1 hm \u03bc f) x \u2202\u03bc = \u222b (x : \u03b1) in s, f x \u2202\u03bc"}, {"tactic": "exact set_integral_condexpL1 hf hs", "annotated_tactic": ["exact <a>set_integral_condexpL1</a> hf hs", [{"full_name": "MeasureTheory.set_integral_condexpL1", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean", "def_pos": [558, 9], "def_end_pos": [558, 31]}]], "state_before": "\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nhf : Integrable f\nhs : MeasurableSet s\n\u22a2 \u222b (x : \u03b1) in s, \u2191\u2191(condexpL1 hm \u03bc f) x \u2202\u03bc = \u222b (x : \u03b1) in s, f x \u2202\u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/BitVec/Basic.lean", "full_name": "Std.BitVec.ofNat_eq_ofNat", "start": [100, 9], "end": [100, 87], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Kernel/Basic.lean", "full_name": "ProbabilityTheory.kernel.set_integral_deterministic'", "start": [419, 1], "end": [423, 63], "traced_tactics": [{"tactic": "rw [kernel.deterministic_apply, set_integral_dirac' hf _ hs]", "annotated_tactic": ["rw [<a>kernel.deterministic_apply</a>, <a>set_integral_dirac'</a> hf _ hs]", [{"full_name": "ProbabilityTheory.kernel.deterministic_apply", "def_path": "Mathlib/Probability/Kernel/Basic.lean", "def_pos": [365, 9], "def_end_pos": [365, 28]}, {"full_name": "MeasureTheory.set_integral_dirac'", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1676, 9], "def_end_pos": [1676, 28]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\nE : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf : \u03b2 \u2192 E\ng : \u03b1 \u2192 \u03b2\na : \u03b1\nhg : Measurable g\nhf : StronglyMeasurable f\ns : Set \u03b2\nhs : MeasurableSet s\ninst\u271d : Decidable (g a \u2208 s)\n\u22a2 \u222b (x : \u03b2) in s, f x \u2202\u2191(deterministic g hg) a = if g a \u2208 s then f (g a) else 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Vector3.lean", "full_name": "Vector3.consElim_cons", "start": [125, 1], "end": [126, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/NAry.lean", "full_name": "Finset.image\u2082_inter_left", "start": [195, 1], "end": [199, 31], "traced_tactics": [{"tactic": "push_cast", "annotated_tactic": ["push_cast", []], "state_before": "\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\ninst\u271d\u2078 : DecidableEq \u03b1'\ninst\u271d\u2077 : DecidableEq \u03b2'\ninst\u271d\u2076 : DecidableEq \u03b3\ninst\u271d\u2075 : DecidableEq \u03b3'\ninst\u271d\u2074 : DecidableEq \u03b4\ninst\u271d\u00b3 : DecidableEq \u03b4'\ninst\u271d\u00b2 : DecidableEq \u03b5\ninst\u271d\u00b9 : DecidableEq \u03b5'\nf f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Finset \u03b1\nt t' : Finset \u03b2\nu u' : Finset \u03b3\na a' : \u03b1\nb b' : \u03b2\nc : \u03b3\ninst\u271d : DecidableEq \u03b1\nhf : Injective2 f\n\u22a2 \u2191(image\u2082 f (s \u2229 s') t) = \u2191(image\u2082 f s t \u2229 image\u2082 f s' t)", "state_after": "\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\ninst\u271d\u2078 : DecidableEq \u03b1'\ninst\u271d\u2077 : DecidableEq \u03b2'\ninst\u271d\u2076 : DecidableEq \u03b3\ninst\u271d\u2075 : DecidableEq \u03b3'\ninst\u271d\u2074 : DecidableEq \u03b4\ninst\u271d\u00b3 : DecidableEq \u03b4'\ninst\u271d\u00b2 : DecidableEq \u03b5\ninst\u271d\u00b9 : DecidableEq \u03b5'\nf f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Finset \u03b1\nt t' : Finset \u03b2\nu u' : Finset \u03b3\na a' : \u03b1\nb b' : \u03b2\nc : \u03b3\ninst\u271d : DecidableEq \u03b1\nhf : Injective2 f\n\u22a2 image2 f (\u2191s \u2229 \u2191s') \u2191t = image2 f \u2191s \u2191t \u2229 image2 f \u2191s' \u2191t"}, {"tactic": "exact image2_inter_left hf", "annotated_tactic": ["exact <a>image2_inter_left</a> hf", [{"full_name": "Set.image2_inter_left", "def_path": "Mathlib/Data/Set/NAry.lean", "def_pos": [136, 7], "def_end_pos": [136, 24]}]], "state_before": "\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\ninst\u271d\u2078 : DecidableEq \u03b1'\ninst\u271d\u2077 : DecidableEq \u03b2'\ninst\u271d\u2076 : DecidableEq \u03b3\ninst\u271d\u2075 : DecidableEq \u03b3'\ninst\u271d\u2074 : DecidableEq \u03b4\ninst\u271d\u00b3 : DecidableEq \u03b4'\ninst\u271d\u00b2 : DecidableEq \u03b5\ninst\u271d\u00b9 : DecidableEq \u03b5'\nf f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Finset \u03b1\nt t' : Finset \u03b2\nu u' : Finset \u03b3\na a' : \u03b1\nb b' : \u03b2\nc : \u03b3\ninst\u271d : DecidableEq \u03b1\nhf : Injective2 f\n\u22a2 image2 f (\u2191s \u2229 \u2191s') \u2191t = image2 f \u2191s \u2191t \u2229 image2 f \u2191s' \u2191t", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/RieszMarkovKakutani.lean", "full_name": "rieszContentAux_sup_le", "start": [95, 1], "end": [113, 71], "traced_tactics": [{"tactic": "apply NNReal.le_of_forall_pos_le_add", "annotated_tactic": ["apply <a>NNReal.le_of_forall_pos_le_add</a>", [{"full_name": "NNReal.le_of_forall_pos_le_add", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [551, 16], "def_end_pos": [551, 39]}]], "state_before": "X : Type u_1\ninst\u271d : TopologicalSpace X\n\u039b : (X \u2192\u1d47 \u211d\u22650) \u2192\u2097[\u211d\u22650] \u211d\u22650\nK1 K2 : Compacts X\n\u22a2 rieszContentAux \u039b (K1 \u2294 K2) \u2264 rieszContentAux \u039b K1 + rieszContentAux \u039b K2", "state_after": "case h\nX : Type u_1\ninst\u271d : TopologicalSpace X\n\u039b : (X \u2192\u1d47 \u211d\u22650) \u2192\u2097[\u211d\u22650] \u211d\u22650\nK1 K2 : Compacts X\n\u22a2 \u2200 (\u03b5 : \u211d\u22650), 0 < \u03b5 \u2192 rieszContentAux \u039b (K1 \u2294 K2) \u2264 rieszContentAux \u039b K1 + rieszContentAux \u039b K2 + \u03b5"}, {"tactic": "intro \u03b5 \u03b5pos", "annotated_tactic": ["intro \u03b5 \u03b5pos", []], "state_before": "case h\nX : Type u_1\ninst\u271d : TopologicalSpace X\n\u039b : (X \u2192\u1d47 \u211d\u22650) \u2192\u2097[\u211d\u22650] \u211d\u22650\nK1 K2 : Compacts X\n\u22a2 \u2200 (\u03b5 : \u211d\u22650), 0 < \u03b5 \u2192 rieszContentAux \u039b (K1 \u2294 K2) \u2264 rieszContentAux \u039b K1 + rieszContentAux \u039b K2 + \u03b5", "state_after": "case h\nX : Type u_1\ninst\u271d : TopologicalSpace X\n\u039b : (X \u2192\u1d47 \u211d\u22650) \u2192\u2097[\u211d\u22650] \u211d\u22650\nK1 K2 : Compacts X\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\n\u22a2 rieszContentAux \u039b (K1 \u2294 K2) \u2264 rieszContentAux \u039b K1 + rieszContentAux \u039b K2 + \u03b5"}, {"tactic": "obtain \u27e8f1, f_test_function_K1\u27e9 := exists_lt_rieszContentAux_add_pos \u039b K1 (half_pos \u03b5pos)", "annotated_tactic": ["obtain \u27e8f1, f_test_function_K1\u27e9 := <a>exists_lt_rieszContentAux_add_pos</a> \u039b K1 (<a>half_pos</a> \u03b5pos)", [{"full_name": "exists_lt_rieszContentAux_add_pos", "def_path": "Mathlib/MeasureTheory/Integral/RieszMarkovKakutani.lean", "def_pos": [82, 9], "def_end_pos": [82, 42]}, {"full_name": "half_pos", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [504, 9], "def_end_pos": [504, 17]}]], "state_before": "case h\nX : Type u_1\ninst\u271d : TopologicalSpace X\n\u039b : (X \u2192\u1d47 \u211d\u22650) \u2192\u2097[\u211d\u22650] \u211d\u22650\nK1 K2 : Compacts X\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\n\u22a2 rieszContentAux \u039b (K1 \u2294 K2) \u2264 rieszContentAux \u039b K1 + rieszContentAux \u039b K2 + \u03b5", "state_after": "case h.intro\nX : Type u_1\ninst\u271d : TopologicalSpace X\n\u039b : (X \u2192\u1d47 \u211d\u22650) \u2192\u2097[\u211d\u22650] \u211d\u22650\nK1 K2 : Compacts X\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nf1 : X \u2192\u1d47 \u211d\u22650\nf_test_function_K1 : (\u2200 (x : X), x \u2208 K1 \u2192 1 \u2264 \u2191f1 x) \u2227 \u2191\u039b f1 < rieszContentAux \u039b K1 + \u03b5 / 2\n\u22a2 rieszContentAux \u039b (K1 \u2294 K2) \u2264 rieszContentAux \u039b K1 + rieszContentAux \u039b K2 + \u03b5"}, {"tactic": "obtain \u27e8f2, f_test_function_K2\u27e9 := exists_lt_rieszContentAux_add_pos \u039b K2 (half_pos \u03b5pos)", "annotated_tactic": ["obtain \u27e8f2, f_test_function_K2\u27e9 := <a>exists_lt_rieszContentAux_add_pos</a> \u039b K2 (<a>half_pos</a> \u03b5pos)", [{"full_name": "exists_lt_rieszContentAux_add_pos", "def_path": "Mathlib/MeasureTheory/Integral/RieszMarkovKakutani.lean", "def_pos": [82, 9], "def_end_pos": [82, 42]}, {"full_name": "half_pos", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [504, 9], "def_end_pos": [504, 17]}]], "state_before": "case h.intro\nX : Type u_1\ninst\u271d : TopologicalSpace X\n\u039b : (X \u2192\u1d47 \u211d\u22650) \u2192\u2097[\u211d\u22650] \u211d\u22650\nK1 K2 : Compacts X\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nf1 : X \u2192\u1d47 \u211d\u22650\nf_test_function_K1 : (\u2200 (x : X), x \u2208 K1 \u2192 1 \u2264 \u2191f1 x) \u2227 \u2191\u039b f1 < rieszContentAux \u039b K1 + \u03b5 / 2\n\u22a2 rieszContentAux \u039b (K1 \u2294 K2) \u2264 rieszContentAux \u039b K1 + rieszContentAux \u039b K2 + \u03b5", "state_after": "case h.intro.intro\nX : Type u_1\ninst\u271d : TopologicalSpace X\n\u039b : (X \u2192\u1d47 \u211d\u22650) \u2192\u2097[\u211d\u22650] \u211d\u22650\nK1 K2 : Compacts X\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nf1 : X \u2192\u1d47 \u211d\u22650\nf_test_function_K1 : (\u2200 (x : X), x \u2208 K1 \u2192 1 \u2264 \u2191f1 x) \u2227 \u2191\u039b f1 < rieszContentAux \u039b K1 + \u03b5 / 2\nf2 : X \u2192\u1d47 \u211d\u22650\nf_test_function_K2 : (\u2200 (x : X), x \u2208 K2 \u2192 1 \u2264 \u2191f2 x) \u2227 \u2191\u039b f2 < rieszContentAux \u039b K2 + \u03b5 / 2\n\u22a2 rieszContentAux \u039b (K1 \u2294 K2) \u2264 rieszContentAux \u039b K1 + rieszContentAux \u039b K2 + \u03b5"}, {"tactic": "apply (rieszContentAux_le \u039b f_test_function_union).trans (le_of_lt _)", "annotated_tactic": ["apply (<a>rieszContentAux_le</a> \u039b f_test_function_union).<a>trans</a> (<a>le_of_lt</a> _)", [{"full_name": "rieszContentAux_le", "def_path": "Mathlib/MeasureTheory/Integral/RieszMarkovKakutani.lean", "def_pos": [74, 9], "def_end_pos": [74, 27]}, {"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [120, 7], "def_end_pos": [120, 18]}, {"full_name": "le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [110, 9], "def_end_pos": [110, 17]}]], "state_before": "case h.intro.intro\nX : Type u_1\ninst\u271d : TopologicalSpace X\n\u039b : (X \u2192\u1d47 \u211d\u22650) \u2192\u2097[\u211d\u22650] \u211d\u22650\nK1 K2 : Compacts X\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nf1 : X \u2192\u1d47 \u211d\u22650\nf_test_function_K1 : (\u2200 (x : X), x \u2208 K1 \u2192 1 \u2264 \u2191f1 x) \u2227 \u2191\u039b f1 < rieszContentAux \u039b K1 + \u03b5 / 2\nf2 : X \u2192\u1d47 \u211d\u22650\nf_test_function_K2 : (\u2200 (x : X), x \u2208 K2 \u2192 1 \u2264 \u2191f2 x) \u2227 \u2191\u039b f2 < rieszContentAux \u039b K2 + \u03b5 / 2\nf_test_function_union : \u2200 (x : X), x \u2208 K1 \u2294 K2 \u2192 1 \u2264 \u2191(f1 + f2) x\n\u22a2 rieszContentAux \u039b (K1 \u2294 K2) \u2264 rieszContentAux \u039b K1 + rieszContentAux \u039b K2 + \u03b5", "state_after": "X : Type u_1\ninst\u271d : TopologicalSpace X\n\u039b : (X \u2192\u1d47 \u211d\u22650) \u2192\u2097[\u211d\u22650] \u211d\u22650\nK1 K2 : Compacts X\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nf1 : X \u2192\u1d47 \u211d\u22650\nf_test_function_K1 : (\u2200 (x : X), x \u2208 K1 \u2192 1 \u2264 \u2191f1 x) \u2227 \u2191\u039b f1 < rieszContentAux \u039b K1 + \u03b5 / 2\nf2 : X \u2192\u1d47 \u211d\u22650\nf_test_function_K2 : (\u2200 (x : X), x \u2208 K2 \u2192 1 \u2264 \u2191f2 x) \u2227 \u2191\u039b f2 < rieszContentAux \u039b K2 + \u03b5 / 2\nf_test_function_union : \u2200 (x : X), x \u2208 K1 \u2294 K2 \u2192 1 \u2264 \u2191(f1 + f2) x\n\u22a2 \u2191\u039b (f1 + f2) < rieszContentAux \u039b K1 + rieszContentAux \u039b K2 + \u03b5"}, {"tactic": "rw [map_add]", "annotated_tactic": ["rw [<a>map_add</a>]", [{"full_name": "map_add", "def_path": "Mathlib/Algebra/Hom/Group/Defs.lean", "def_pos": [298, 3], "def_end_pos": [298, 14]}]], "state_before": "X : Type u_1\ninst\u271d : TopologicalSpace X\n\u039b : (X \u2192\u1d47 \u211d\u22650) \u2192\u2097[\u211d\u22650] \u211d\u22650\nK1 K2 : Compacts X\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nf1 : X \u2192\u1d47 \u211d\u22650\nf_test_function_K1 : (\u2200 (x : X), x \u2208 K1 \u2192 1 \u2264 \u2191f1 x) \u2227 \u2191\u039b f1 < rieszContentAux \u039b K1 + \u03b5 / 2\nf2 : X \u2192\u1d47 \u211d\u22650\nf_test_function_K2 : (\u2200 (x : X), x \u2208 K2 \u2192 1 \u2264 \u2191f2 x) \u2227 \u2191\u039b f2 < rieszContentAux \u039b K2 + \u03b5 / 2\nf_test_function_union : \u2200 (x : X), x \u2208 K1 \u2294 K2 \u2192 1 \u2264 \u2191(f1 + f2) x\n\u22a2 \u2191\u039b (f1 + f2) < rieszContentAux \u039b K1 + rieszContentAux \u039b K2 + \u03b5", "state_after": "X : Type u_1\ninst\u271d : TopologicalSpace X\n\u039b : (X \u2192\u1d47 \u211d\u22650) \u2192\u2097[\u211d\u22650] \u211d\u22650\nK1 K2 : Compacts X\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nf1 : X \u2192\u1d47 \u211d\u22650\nf_test_function_K1 : (\u2200 (x : X), x \u2208 K1 \u2192 1 \u2264 \u2191f1 x) \u2227 \u2191\u039b f1 < rieszContentAux \u039b K1 + \u03b5 / 2\nf2 : X \u2192\u1d47 \u211d\u22650\nf_test_function_K2 : (\u2200 (x : X), x \u2208 K2 \u2192 1 \u2264 \u2191f2 x) \u2227 \u2191\u039b f2 < rieszContentAux \u039b K2 + \u03b5 / 2\nf_test_function_union : \u2200 (x : X), x \u2208 K1 \u2294 K2 \u2192 1 \u2264 \u2191(f1 + f2) x\n\u22a2 \u2191\u039b f1 + \u2191\u039b f2 < rieszContentAux \u039b K1 + rieszContentAux \u039b K2 + \u03b5"}, {"tactic": "apply lt_of_lt_of_le (_root_.add_lt_add f_test_function_K1.right f_test_function_K2.right)\n  (le_of_eq _)", "annotated_tactic": ["apply <a>lt_of_lt_of_le</a> (<a>_root_.add_lt_add</a> f_test_function_K1.right f_test_function_K2.right)\n    (<a>le_of_eq</a> _)", [{"full_name": "lt_of_lt_of_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [115, 9], "def_end_pos": [115, 23]}, {"full_name": "add_lt_add", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [167, 7], "def_end_pos": [167, 17]}, {"full_name": "le_of_eq", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [72, 9], "def_end_pos": [72, 17]}]], "state_before": "X : Type u_1\ninst\u271d : TopologicalSpace X\n\u039b : (X \u2192\u1d47 \u211d\u22650) \u2192\u2097[\u211d\u22650] \u211d\u22650\nK1 K2 : Compacts X\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nf1 : X \u2192\u1d47 \u211d\u22650\nf_test_function_K1 : (\u2200 (x : X), x \u2208 K1 \u2192 1 \u2264 \u2191f1 x) \u2227 \u2191\u039b f1 < rieszContentAux \u039b K1 + \u03b5 / 2\nf2 : X \u2192\u1d47 \u211d\u22650\nf_test_function_K2 : (\u2200 (x : X), x \u2208 K2 \u2192 1 \u2264 \u2191f2 x) \u2227 \u2191\u039b f2 < rieszContentAux \u039b K2 + \u03b5 / 2\nf_test_function_union : \u2200 (x : X), x \u2208 K1 \u2294 K2 \u2192 1 \u2264 \u2191(f1 + f2) x\n\u22a2 \u2191\u039b f1 + \u2191\u039b f2 < rieszContentAux \u039b K1 + rieszContentAux \u039b K2 + \u03b5", "state_after": "X : Type u_1\ninst\u271d : TopologicalSpace X\n\u039b : (X \u2192\u1d47 \u211d\u22650) \u2192\u2097[\u211d\u22650] \u211d\u22650\nK1 K2 : Compacts X\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nf1 : X \u2192\u1d47 \u211d\u22650\nf_test_function_K1 : (\u2200 (x : X), x \u2208 K1 \u2192 1 \u2264 \u2191f1 x) \u2227 \u2191\u039b f1 < rieszContentAux \u039b K1 + \u03b5 / 2\nf2 : X \u2192\u1d47 \u211d\u22650\nf_test_function_K2 : (\u2200 (x : X), x \u2208 K2 \u2192 1 \u2264 \u2191f2 x) \u2227 \u2191\u039b f2 < rieszContentAux \u039b K2 + \u03b5 / 2\nf_test_function_union : \u2200 (x : X), x \u2208 K1 \u2294 K2 \u2192 1 \u2264 \u2191(f1 + f2) x\n\u22a2 rieszContentAux \u039b K1 + \u03b5 / 2 + (rieszContentAux \u039b K2 + \u03b5 / 2) = rieszContentAux \u039b K1 + rieszContentAux \u039b K2 + \u03b5"}, {"tactic": "rw [add_assoc, add_comm (\u03b5 / 2), add_assoc, add_halves \u03b5, add_assoc]", "annotated_tactic": ["rw [<a>add_assoc</a>, <a>add_comm</a> (\u03b5 / 2), <a>add_assoc</a>, <a>add_halves</a> \u03b5, <a>add_assoc</a>]", [{"full_name": "add_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [263, 3], "def_end_pos": [263, 14]}, {"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [301, 3], "def_end_pos": [301, 14]}, {"full_name": "add_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [263, 3], "def_end_pos": [263, 14]}, {"full_name": "add_halves", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [495, 9], "def_end_pos": [495, 19]}, {"full_name": "add_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [263, 3], "def_end_pos": [263, 14]}]], "state_before": "X : Type u_1\ninst\u271d : TopologicalSpace X\n\u039b : (X \u2192\u1d47 \u211d\u22650) \u2192\u2097[\u211d\u22650] \u211d\u22650\nK1 K2 : Compacts X\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nf1 : X \u2192\u1d47 \u211d\u22650\nf_test_function_K1 : (\u2200 (x : X), x \u2208 K1 \u2192 1 \u2264 \u2191f1 x) \u2227 \u2191\u039b f1 < rieszContentAux \u039b K1 + \u03b5 / 2\nf2 : X \u2192\u1d47 \u211d\u22650\nf_test_function_K2 : (\u2200 (x : X), x \u2208 K2 \u2192 1 \u2264 \u2191f2 x) \u2227 \u2191\u039b f2 < rieszContentAux \u039b K2 + \u03b5 / 2\nf_test_function_union : \u2200 (x : X), x \u2208 K1 \u2294 K2 \u2192 1 \u2264 \u2191(f1 + f2) x\n\u22a2 rieszContentAux \u039b K1 + \u03b5 / 2 + (rieszContentAux \u039b K2 + \u03b5 / 2) = rieszContentAux \u039b K1 + rieszContentAux \u039b K2 + \u03b5", "state_after": "no goals"}, {"tactic": "rintro x (x_in_K1 | x_in_K2)", "annotated_tactic": ["rintro x (x_in_K1 | x_in_K2)", []], "state_before": "X : Type u_1\ninst\u271d : TopologicalSpace X\n\u039b : (X \u2192\u1d47 \u211d\u22650) \u2192\u2097[\u211d\u22650] \u211d\u22650\nK1 K2 : Compacts X\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nf1 : X \u2192\u1d47 \u211d\u22650\nf_test_function_K1 : (\u2200 (x : X), x \u2208 K1 \u2192 1 \u2264 \u2191f1 x) \u2227 \u2191\u039b f1 < rieszContentAux \u039b K1 + \u03b5 / 2\nf2 : X \u2192\u1d47 \u211d\u22650\nf_test_function_K2 : (\u2200 (x : X), x \u2208 K2 \u2192 1 \u2264 \u2191f2 x) \u2227 \u2191\u039b f2 < rieszContentAux \u039b K2 + \u03b5 / 2\n\u22a2 \u2200 (x : X), x \u2208 K1 \u2294 K2 \u2192 1 \u2264 \u2191(f1 + f2) x", "state_after": "case inl\nX : Type u_1\ninst\u271d : TopologicalSpace X\n\u039b : (X \u2192\u1d47 \u211d\u22650) \u2192\u2097[\u211d\u22650] \u211d\u22650\nK1 K2 : Compacts X\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nf1 : X \u2192\u1d47 \u211d\u22650\nf_test_function_K1 : (\u2200 (x : X), x \u2208 K1 \u2192 1 \u2264 \u2191f1 x) \u2227 \u2191\u039b f1 < rieszContentAux \u039b K1 + \u03b5 / 2\nf2 : X \u2192\u1d47 \u211d\u22650\nf_test_function_K2 : (\u2200 (x : X), x \u2208 K2 \u2192 1 \u2264 \u2191f2 x) \u2227 \u2191\u039b f2 < rieszContentAux \u039b K2 + \u03b5 / 2\nx : X\nx_in_K1 : x \u2208 \u2191K1\n\u22a2 1 \u2264 \u2191(f1 + f2) x\n\ncase inr\nX : Type u_1\ninst\u271d : TopologicalSpace X\n\u039b : (X \u2192\u1d47 \u211d\u22650) \u2192\u2097[\u211d\u22650] \u211d\u22650\nK1 K2 : Compacts X\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nf1 : X \u2192\u1d47 \u211d\u22650\nf_test_function_K1 : (\u2200 (x : X), x \u2208 K1 \u2192 1 \u2264 \u2191f1 x) \u2227 \u2191\u039b f1 < rieszContentAux \u039b K1 + \u03b5 / 2\nf2 : X \u2192\u1d47 \u211d\u22650\nf_test_function_K2 : (\u2200 (x : X), x \u2208 K2 \u2192 1 \u2264 \u2191f2 x) \u2227 \u2191\u039b f2 < rieszContentAux \u039b K2 + \u03b5 / 2\nx : X\nx_in_K2 : x \u2208 \u2191K2\n\u22a2 1 \u2264 \u2191(f1 + f2) x"}, {"tactic": "exact le_add_right (f_test_function_K1.left x x_in_K1)", "annotated_tactic": ["exact <a>le_add_right</a> (f_test_function_K1.left x x_in_K1)", [{"full_name": "le_add_right", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [278, 3], "def_end_pos": [278, 14]}]], "state_before": "case inl\nX : Type u_1\ninst\u271d : TopologicalSpace X\n\u039b : (X \u2192\u1d47 \u211d\u22650) \u2192\u2097[\u211d\u22650] \u211d\u22650\nK1 K2 : Compacts X\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nf1 : X \u2192\u1d47 \u211d\u22650\nf_test_function_K1 : (\u2200 (x : X), x \u2208 K1 \u2192 1 \u2264 \u2191f1 x) \u2227 \u2191\u039b f1 < rieszContentAux \u039b K1 + \u03b5 / 2\nf2 : X \u2192\u1d47 \u211d\u22650\nf_test_function_K2 : (\u2200 (x : X), x \u2208 K2 \u2192 1 \u2264 \u2191f2 x) \u2227 \u2191\u039b f2 < rieszContentAux \u039b K2 + \u03b5 / 2\nx : X\nx_in_K1 : x \u2208 \u2191K1\n\u22a2 1 \u2264 \u2191(f1 + f2) x", "state_after": "no goals"}, {"tactic": "exact le_add_left (f_test_function_K2.left x x_in_K2)", "annotated_tactic": ["exact <a>le_add_left</a> (f_test_function_K2.left x x_in_K2)", [{"full_name": "le_add_left", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [270, 3], "def_end_pos": [270, 14]}]], "state_before": "case inr\nX : Type u_1\ninst\u271d : TopologicalSpace X\n\u039b : (X \u2192\u1d47 \u211d\u22650) \u2192\u2097[\u211d\u22650] \u211d\u22650\nK1 K2 : Compacts X\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nf1 : X \u2192\u1d47 \u211d\u22650\nf_test_function_K1 : (\u2200 (x : X), x \u2208 K1 \u2192 1 \u2264 \u2191f1 x) \u2227 \u2191\u039b f1 < rieszContentAux \u039b K1 + \u03b5 / 2\nf2 : X \u2192\u1d47 \u211d\u22650\nf_test_function_K2 : (\u2200 (x : X), x \u2208 K2 \u2192 1 \u2264 \u2191f2 x) \u2227 \u2191\u039b f2 < rieszContentAux \u039b K2 + \u03b5 / 2\nx : X\nx_in_K2 : x \u2208 \u2191K2\n\u22a2 1 \u2264 \u2191(f1 + f2) x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Decomposition/Lebesgue.lean", "full_name": "MeasureTheory.Measure.LebesgueDecomposition.sup_mem_measurableLE", "start": [466, 1], "end": [475, 64], "traced_tactics": [{"tactic": "simp_rw [ENNReal.sup_eq_max]", "annotated_tactic": ["simp_rw [<a>ENNReal.sup_eq_max</a>]", [{"full_name": "ENNReal.sup_eq_max", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [771, 17], "def_end_pos": [771, 27]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : f \u2208 measurableLE \u03bc \u03bd\nhg : g \u2208 measurableLE \u03bc \u03bd\n\u22a2 (fun a => f a \u2294 g a) \u2208 measurableLE \u03bc \u03bd", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : f \u2208 measurableLE \u03bc \u03bd\nhg : g \u2208 measurableLE \u03bc \u03bd\n\u22a2 (fun a => max (f a) (g a)) \u2208 measurableLE \u03bc \u03bd"}, {"tactic": "refine' \u27e8Measurable.max hf.1 hg.1, fun A hA => _\u27e9", "annotated_tactic": ["refine' \u27e8<a>Measurable.max</a> hf.1 hg.1, fun A hA => _\u27e9", [{"full_name": "Measurable.max", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [862, 9], "def_end_pos": [862, 23]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : f \u2208 measurableLE \u03bc \u03bd\nhg : g \u2208 measurableLE \u03bc \u03bd\n\u22a2 (fun a => max (f a) (g a)) \u2208 measurableLE \u03bc \u03bd", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : f \u2208 measurableLE \u03bc \u03bd\nhg : g \u2208 measurableLE \u03bc \u03bd\nA : Set \u03b1\nhA : MeasurableSet A\n\u22a2 \u222b\u207b (x : \u03b1) in A, (fun a => max (f a) (g a)) x \u2202\u03bc \u2264 \u2191\u2191\u03bd A"}, {"tactic": "have h\u2081 := hA.inter (measurableSet_le hf.1 hg.1)", "annotated_tactic": ["have h\u2081 := hA.inter (<a>measurableSet_le</a> hf.1 hg.1)", [{"full_name": "measurableSet_le", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [559, 9], "def_end_pos": [559, 25]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : f \u2208 measurableLE \u03bc \u03bd\nhg : g \u2208 measurableLE \u03bc \u03bd\nA : Set \u03b1\nhA : MeasurableSet A\n\u22a2 \u222b\u207b (x : \u03b1) in A, (fun a => max (f a) (g a)) x \u2202\u03bc \u2264 \u2191\u2191\u03bd A", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : f \u2208 measurableLE \u03bc \u03bd\nhg : g \u2208 measurableLE \u03bc \u03bd\nA : Set \u03b1\nhA : MeasurableSet A\nh\u2081 : MeasurableSet (A \u2229 {a | f a \u2264 g a})\n\u22a2 \u222b\u207b (x : \u03b1) in A, (fun a => max (f a) (g a)) x \u2202\u03bc \u2264 \u2191\u2191\u03bd A"}, {"tactic": "have h\u2082 := hA.inter (measurableSet_lt hg.1 hf.1)", "annotated_tactic": ["have h\u2082 := hA.inter (<a>measurableSet_lt</a> hg.1 hf.1)", [{"full_name": "measurableSet_lt", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [616, 9], "def_end_pos": [616, 25]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : f \u2208 measurableLE \u03bc \u03bd\nhg : g \u2208 measurableLE \u03bc \u03bd\nA : Set \u03b1\nhA : MeasurableSet A\nh\u2081 : MeasurableSet (A \u2229 {a | f a \u2264 g a})\n\u22a2 \u222b\u207b (x : \u03b1) in A, (fun a => max (f a) (g a)) x \u2202\u03bc \u2264 \u2191\u2191\u03bd A", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : f \u2208 measurableLE \u03bc \u03bd\nhg : g \u2208 measurableLE \u03bc \u03bd\nA : Set \u03b1\nhA : MeasurableSet A\nh\u2081 : MeasurableSet (A \u2229 {a | f a \u2264 g a})\nh\u2082 : MeasurableSet (A \u2229 {a | g a < f a})\n\u22a2 \u222b\u207b (x : \u03b1) in A, (fun a => max (f a) (g a)) x \u2202\u03bc \u2264 \u2191\u2191\u03bd A"}, {"tactic": "rw [set_lintegral_max hf.1 hg.1]", "annotated_tactic": ["rw [<a>set_lintegral_max</a> hf.1 hg.1]", [{"full_name": "MeasureTheory.set_lintegral_max", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [1274, 9], "def_end_pos": [1274, 26]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : f \u2208 measurableLE \u03bc \u03bd\nhg : g \u2208 measurableLE \u03bc \u03bd\nA : Set \u03b1\nhA : MeasurableSet A\nh\u2081 : MeasurableSet (A \u2229 {a | f a \u2264 g a})\nh\u2082 : MeasurableSet (A \u2229 {a | g a < f a})\n\u22a2 \u222b\u207b (x : \u03b1) in A, (fun a => max (f a) (g a)) x \u2202\u03bc \u2264 \u2191\u2191\u03bd A", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : f \u2208 measurableLE \u03bc \u03bd\nhg : g \u2208 measurableLE \u03bc \u03bd\nA : Set \u03b1\nhA : MeasurableSet A\nh\u2081 : MeasurableSet (A \u2229 {a | f a \u2264 g a})\nh\u2082 : MeasurableSet (A \u2229 {a | g a < f a})\n\u22a2 \u222b\u207b (x : \u03b1) in A \u2229 {x | f x \u2264 g x}, g x \u2202\u03bc + \u222b\u207b (x : \u03b1) in A \u2229 {x | g x < f x}, f x \u2202\u03bc \u2264 \u2191\u2191\u03bd A"}, {"tactic": "refine' (add_le_add (hg.2 _ h\u2081) (hf.2 _ h\u2082)).trans_eq _", "annotated_tactic": ["refine' (<a>add_le_add</a> (hg.2 _ h\u2081) (hf.2 _ h\u2082)).<a>trans_eq</a> _", [{"full_name": "add_le_add", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [205, 15], "def_end_pos": [205, 25]}, {"full_name": "LE.le.trans_eq", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [211, 7], "def_end_pos": [211, 21]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : f \u2208 measurableLE \u03bc \u03bd\nhg : g \u2208 measurableLE \u03bc \u03bd\nA : Set \u03b1\nhA : MeasurableSet A\nh\u2081 : MeasurableSet (A \u2229 {a | f a \u2264 g a})\nh\u2082 : MeasurableSet (A \u2229 {a | g a < f a})\n\u22a2 \u222b\u207b (x : \u03b1) in A \u2229 {x | f x \u2264 g x}, g x \u2202\u03bc + \u222b\u207b (x : \u03b1) in A \u2229 {x | g x < f x}, f x \u2202\u03bc \u2264 \u2191\u2191\u03bd A", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : f \u2208 measurableLE \u03bc \u03bd\nhg : g \u2208 measurableLE \u03bc \u03bd\nA : Set \u03b1\nhA : MeasurableSet A\nh\u2081 : MeasurableSet (A \u2229 {a | f a \u2264 g a})\nh\u2082 : MeasurableSet (A \u2229 {a | g a < f a})\n\u22a2 \u2191\u2191\u03bd (A \u2229 {a | f a \u2264 g a}) + \u2191\u2191\u03bd (A \u2229 {a | g a < f a}) = \u2191\u2191\u03bd A"}, {"tactic": "simp only [\u2190 not_le, \u2190 compl_setOf, \u2190 diff_eq]", "annotated_tactic": ["simp only [\u2190 <a>not_le</a>, \u2190 <a>compl_setOf</a>, \u2190 <a>diff_eq</a>]", [{"full_name": "not_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [373, 9], "def_end_pos": [373, 15]}, {"full_name": "Set.compl_setOf", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1649, 9], "def_end_pos": [1649, 20]}, {"full_name": "Set.diff_eq", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1814, 9], "def_end_pos": [1814, 16]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : f \u2208 measurableLE \u03bc \u03bd\nhg : g \u2208 measurableLE \u03bc \u03bd\nA : Set \u03b1\nhA : MeasurableSet A\nh\u2081 : MeasurableSet (A \u2229 {a | f a \u2264 g a})\nh\u2082 : MeasurableSet (A \u2229 {a | g a < f a})\n\u22a2 \u2191\u2191\u03bd (A \u2229 {a | f a \u2264 g a}) + \u2191\u2191\u03bd (A \u2229 {a | g a < f a}) = \u2191\u2191\u03bd A", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : f \u2208 measurableLE \u03bc \u03bd\nhg : g \u2208 measurableLE \u03bc \u03bd\nA : Set \u03b1\nhA : MeasurableSet A\nh\u2081 : MeasurableSet (A \u2229 {a | f a \u2264 g a})\nh\u2082 : MeasurableSet (A \u2229 {a | g a < f a})\n\u22a2 \u2191\u2191\u03bd (A \u2229 {a | f a \u2264 g a}) + \u2191\u2191\u03bd (A \\ {a | f a \u2264 g a}) = \u2191\u2191\u03bd A"}, {"tactic": "exact measure_inter_add_diff _ (measurableSet_le hf.1 hg.1)", "annotated_tactic": ["exact <a>measure_inter_add_diff</a> _ (<a>measurableSet_le</a> hf.1 hg.1)", [{"full_name": "MeasureTheory.measure_inter_add_diff", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [132, 9], "def_end_pos": [132, 31]}, {"full_name": "measurableSet_le", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [559, 9], "def_end_pos": [559, 25]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : f \u2208 measurableLE \u03bc \u03bd\nhg : g \u2208 measurableLE \u03bc \u03bd\nA : Set \u03b1\nhA : MeasurableSet A\nh\u2081 : MeasurableSet (A \u2229 {a | f a \u2264 g a})\nh\u2082 : MeasurableSet (A \u2229 {a | g a < f a})\n\u22a2 \u2191\u2191\u03bd (A \u2229 {a | f a \u2264 g a}) + \u2191\u2191\u03bd (A \\ {a | f a \u2264 g a}) = \u2191\u2191\u03bd A", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "full_name": "MeasureTheory.SimpleFunc.FinMeasSupp.map\u2082", "start": [1213, 11], "end": [1215, 21], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/QPF/Multivariate/Constructions/Cofix.lean", "full_name": "MvQPF.Cofix.bisim_rel", "start": [258, 1], "end": [280, 19], "traced_tactics": [{"tactic": "let r' (x y) := x = y \u2228 r x y", "annotated_tactic": ["let r' (x y) := x = y \u2228 r x y", []], "state_before": "n : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\n\u22a2 \u2200 (x y : Cofix F \u03b1), r x y \u2192 x = y", "state_after": "n : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nr' : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => x = y \u2228 r x y\n\u22a2 \u2200 (x y : Cofix F \u03b1), r x y \u2192 x = y"}, {"tactic": "intro x y rxy", "annotated_tactic": ["intro x y rxy", []], "state_before": "n : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nr' : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => x = y \u2228 r x y\n\u22a2 \u2200 (x y : Cofix F \u03b1), r x y \u2192 x = y", "state_after": "n : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nr' : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => x = y \u2228 r x y\nx y : Cofix F \u03b1\nrxy : r x y\n\u22a2 x = y"}, {"tactic": "apply Cofix.bisim_aux r'", "annotated_tactic": ["apply <a>Cofix.bisim_aux</a> r'", [{"full_name": "_private.Mathlib.Data.QPF.Multivariate.Constructions.Cofix.0.MvQPF.Cofix.bisim_aux", "def_path": "Mathlib/Data/QPF/Multivariate/Constructions/Cofix.lean", "def_pos": [214, 17], "def_end_pos": [214, 32]}]], "state_before": "n : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nr' : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => x = y \u2228 r x y\nx y : Cofix F \u03b1\nrxy : r x y\n\u22a2 x = y", "state_after": "case h'\nn : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nr' : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => x = y \u2228 r x y\nx y : Cofix F \u03b1\nrxy : r x y\n\u22a2 \u2200 (x : Cofix F \u03b1), r' x x\n\ncase h\nn : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nr' : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => x = y \u2228 r x y\nx y : Cofix F \u03b1\nrxy : r x y\n\u22a2 \u2200 (x y : Cofix F \u03b1), r' x y \u2192 (TypeVec.id ::: Quot.mk r') <$$> dest x = (TypeVec.id ::: Quot.mk r') <$$> dest y\n\ncase a\nn : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nr' : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => x = y \u2228 r x y\nx y : Cofix F \u03b1\nrxy : r x y\n\u22a2 r' x y"}, {"tactic": "right", "annotated_tactic": ["right", []], "state_before": "case a\nn : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nr' : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => x = y \u2228 r x y\nx y : Cofix F \u03b1\nrxy : r x y\n\u22a2 r' x y", "state_after": "case a.h\nn : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nr' : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => x = y \u2228 r x y\nx y : Cofix F \u03b1\nrxy : r x y\n\u22a2 r x y"}, {"tactic": "exact rxy", "annotated_tactic": ["exact rxy", []], "state_before": "case a.h\nn : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nr' : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => x = y \u2228 r x y\nx y : Cofix F \u03b1\nrxy : r x y\n\u22a2 r x y", "state_after": "no goals"}, {"tactic": "intro x", "annotated_tactic": ["intro x", []], "state_before": "case h'\nn : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nr' : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => x = y \u2228 r x y\nx y : Cofix F \u03b1\nrxy : r x y\n\u22a2 \u2200 (x : Cofix F \u03b1), r' x x", "state_after": "case h'\nn : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nr' : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => x = y \u2228 r x y\nx\u271d y : Cofix F \u03b1\nrxy : r x\u271d y\nx : Cofix F \u03b1\n\u22a2 r' x x"}, {"tactic": "left", "annotated_tactic": ["left", []], "state_before": "case h'\nn : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nr' : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => x = y \u2228 r x y\nx\u271d y : Cofix F \u03b1\nrxy : r x\u271d y\nx : Cofix F \u03b1\n\u22a2 r' x x", "state_after": "case h'.h\nn : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nr' : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => x = y \u2228 r x y\nx\u271d y : Cofix F \u03b1\nrxy : r x\u271d y\nx : Cofix F \u03b1\n\u22a2 x = x"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case h'.h\nn : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nr' : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => x = y \u2228 r x y\nx\u271d y : Cofix F \u03b1\nrxy : r x\u271d y\nx : Cofix F \u03b1\n\u22a2 x = x", "state_after": "no goals"}, {"tactic": "intro x y r'xy", "annotated_tactic": ["intro x y r'xy", []], "state_before": "case h\nn : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nr' : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => x = y \u2228 r x y\nx y : Cofix F \u03b1\nrxy : r x y\n\u22a2 \u2200 (x y : Cofix F \u03b1), r' x y \u2192 (TypeVec.id ::: Quot.mk r') <$$> dest x = (TypeVec.id ::: Quot.mk r') <$$> dest y", "state_after": "case h\nn : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nr' : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => x = y \u2228 r x y\nx\u271d y\u271d : Cofix F \u03b1\nrxy : r x\u271d y\u271d\nx y : Cofix F \u03b1\nr'xy : r' x y\n\u22a2 (TypeVec.id ::: Quot.mk r') <$$> dest x = (TypeVec.id ::: Quot.mk r') <$$> dest y"}, {"tactic": "cases r'xy", "annotated_tactic": ["cases r'xy", []], "state_before": "case h\nn : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nr' : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => x = y \u2228 r x y\nx\u271d y\u271d : Cofix F \u03b1\nrxy : r x\u271d y\u271d\nx y : Cofix F \u03b1\nr'xy : r' x y\n\u22a2 (TypeVec.id ::: Quot.mk r') <$$> dest x = (TypeVec.id ::: Quot.mk r') <$$> dest y", "state_after": "case h.inl\nn : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nr' : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => x = y \u2228 r x y\nx\u271d y\u271d : Cofix F \u03b1\nrxy : r x\u271d y\u271d\nx y : Cofix F \u03b1\nh\u271d : x = y\n\u22a2 (TypeVec.id ::: Quot.mk r') <$$> dest x = (TypeVec.id ::: Quot.mk r') <$$> dest y\n\ncase h.inr\nn : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nr' : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => x = y \u2228 r x y\nx\u271d y\u271d : Cofix F \u03b1\nrxy : r x\u271d y\u271d\nx y : Cofix F \u03b1\nh\u271d : r x y\n\u22a2 (TypeVec.id ::: Quot.mk r') <$$> dest x = (TypeVec.id ::: Quot.mk r') <$$> dest y"}, {"tactic": "case inl h =>\n  rw [h]", "annotated_tactic": ["case inl h =>\n      rw [h]", []], "state_before": "case h.inl\nn : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nr' : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => x = y \u2228 r x y\nx\u271d y\u271d : Cofix F \u03b1\nrxy : r x\u271d y\u271d\nx y : Cofix F \u03b1\nh\u271d : x = y\n\u22a2 (TypeVec.id ::: Quot.mk r') <$$> dest x = (TypeVec.id ::: Quot.mk r') <$$> dest y\n\ncase h.inr\nn : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nr' : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => x = y \u2228 r x y\nx\u271d y\u271d : Cofix F \u03b1\nrxy : r x\u271d y\u271d\nx y : Cofix F \u03b1\nh\u271d : r x y\n\u22a2 (TypeVec.id ::: Quot.mk r') <$$> dest x = (TypeVec.id ::: Quot.mk r') <$$> dest y", "state_after": "case h.inr\nn : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nr' : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => x = y \u2228 r x y\nx\u271d y\u271d : Cofix F \u03b1\nrxy : r x\u271d y\u271d\nx y : Cofix F \u03b1\nh\u271d : r x y\n\u22a2 (TypeVec.id ::: Quot.mk r') <$$> dest x = (TypeVec.id ::: Quot.mk r') <$$> dest y"}, {"tactic": "case inr r'xy =>\n  have : \u2200 x y, r x y \u2192 r' x y := fun x y h => Or.inr h\n  rw [\u2190 Quot.factor_mk_eq _ _ this]\n  dsimp\n  rw [appendFun_comp_id]\n  rw [@comp_map _ _ _ q _ _ _ (appendFun id (Quot.mk r)),\n    @comp_map _ _ _ q _ _ _ (appendFun id (Quot.mk r))]\n  rw [h _ _ r'xy]", "annotated_tactic": ["case inr r'xy =>\n      have : \u2200 x y, r x y \u2192 r' x y := fun x y h => <a>Or.inr</a> h\n      rw [\u2190 <a>Quot.factor_mk_eq</a> _ _ this]\n      dsimp\n      rw [<a>appendFun_comp_id</a>]\n      rw [@<a>comp_map</a> _ _ _ q _ _ _ (<a>appendFun</a> <a>id</a> (<a>Quot.mk</a> r)),\n        @<a>comp_map</a> _ _ _ q _ _ _ (<a>appendFun</a> <a>id</a> (<a>Quot.mk</a> r))]\n      rw [h _ _ r'xy]", [{"full_name": "Or.inr", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [519, 5], "def_end_pos": [519, 8]}, {"full_name": "Quot.factor_mk_eq", "def_path": "Mathlib/Data/Quot.lean", "def_pos": [90, 9], "def_end_pos": [90, 21]}, {"full_name": "TypeVec.appendFun_comp_id", "def_path": "Mathlib/Data/TypeVec.lean", "def_pos": [270, 9], "def_end_pos": [270, 26]}, {"full_name": "MvQPF.comp_map", "def_path": "Mathlib/Data/QPF/Multivariate/Basic.lean", "def_pos": [112, 9], "def_end_pos": [112, 17]}, {"full_name": "TypeVec.appendFun", "def_path": "Mathlib/Data/TypeVec.lean", "def_pos": [150, 5], "def_end_pos": [150, 14]}, {"full_name": "TypeVec.id", "def_path": "Mathlib/Data/TypeVec.lean", "def_pos": [69, 5], "def_end_pos": [69, 7]}, {"full_name": "Quot.mk", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [406, 14], "def_end_pos": [406, 21]}, {"full_name": "MvQPF.comp_map", "def_path": "Mathlib/Data/QPF/Multivariate/Basic.lean", "def_pos": [112, 9], "def_end_pos": [112, 17]}, {"full_name": "TypeVec.appendFun", "def_path": "Mathlib/Data/TypeVec.lean", "def_pos": [150, 5], "def_end_pos": [150, 14]}, {"full_name": "TypeVec.id", "def_path": "Mathlib/Data/TypeVec.lean", "def_pos": [69, 5], "def_end_pos": [69, 7]}, {"full_name": "Quot.mk", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [406, 14], "def_end_pos": [406, 21]}]], "state_before": "case h.inr\nn : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nr' : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => x = y \u2228 r x y\nx\u271d y\u271d : Cofix F \u03b1\nrxy : r x\u271d y\u271d\nx y : Cofix F \u03b1\nh\u271d : r x y\n\u22a2 (TypeVec.id ::: Quot.mk r') <$$> dest x = (TypeVec.id ::: Quot.mk r') <$$> dest y", "state_after": "no goals"}, {"tactic": "rw [h]", "annotated_tactic": ["rw [h]", []], "state_before": "n : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh\u271d : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nr' : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => x = y \u2228 r x y\nx\u271d y\u271d : Cofix F \u03b1\nrxy : r x\u271d y\u271d\nx y : Cofix F \u03b1\nh : x = y\n\u22a2 (TypeVec.id ::: Quot.mk r') <$$> dest x = (TypeVec.id ::: Quot.mk r') <$$> dest y", "state_after": "no goals"}, {"tactic": "have : \u2200 x y, r x y \u2192 r' x y := fun x y h => Or.inr h", "annotated_tactic": ["have : \u2200 x y, r x y \u2192 r' x y := fun x y h => <a>Or.inr</a> h", [{"full_name": "Or.inr", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [519, 5], "def_end_pos": [519, 8]}]], "state_before": "n : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nr' : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => x = y \u2228 r x y\nx\u271d y\u271d : Cofix F \u03b1\nrxy : r x\u271d y\u271d\nx y : Cofix F \u03b1\nr'xy : r x y\n\u22a2 (TypeVec.id ::: Quot.mk r') <$$> dest x = (TypeVec.id ::: Quot.mk r') <$$> dest y", "state_after": "n : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nr' : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => x = y \u2228 r x y\nx\u271d y\u271d : Cofix F \u03b1\nrxy : r x\u271d y\u271d\nx y : Cofix F \u03b1\nr'xy : r x y\nthis : \u2200 (x y : Cofix F \u03b1), r x y \u2192 r' x y\n\u22a2 (TypeVec.id ::: Quot.mk r') <$$> dest x = (TypeVec.id ::: Quot.mk r') <$$> dest y"}, {"tactic": "rw [\u2190 Quot.factor_mk_eq _ _ this]", "annotated_tactic": ["rw [\u2190 <a>Quot.factor_mk_eq</a> _ _ this]", [{"full_name": "Quot.factor_mk_eq", "def_path": "Mathlib/Data/Quot.lean", "def_pos": [90, 9], "def_end_pos": [90, 21]}]], "state_before": "n : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nr' : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => x = y \u2228 r x y\nx\u271d y\u271d : Cofix F \u03b1\nrxy : r x\u271d y\u271d\nx y : Cofix F \u03b1\nr'xy : r x y\nthis : \u2200 (x y : Cofix F \u03b1), r x y \u2192 r' x y\n\u22a2 (TypeVec.id ::: Quot.mk r') <$$> dest x = (TypeVec.id ::: Quot.mk r') <$$> dest y", "state_after": "n : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nr' : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => x = y \u2228 r x y\nx\u271d y\u271d : Cofix F \u03b1\nrxy : r x\u271d y\u271d\nx y : Cofix F \u03b1\nr'xy : r x y\nthis : \u2200 (x y : Cofix F \u03b1), r x y \u2192 r' x y\n\u22a2 (TypeVec.id ::: Quot.factor (fun x y => r x y) (fun x y => r' x y) this \u2218 Quot.mk fun x y => r x y) <$$> dest x =\n    (TypeVec.id ::: Quot.factor (fun x y => r x y) (fun x y => r' x y) this \u2218 Quot.mk fun x y => r x y) <$$> dest y"}, {"tactic": "dsimp", "annotated_tactic": ["dsimp", []], "state_before": "n : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nr' : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => x = y \u2228 r x y\nx\u271d y\u271d : Cofix F \u03b1\nrxy : r x\u271d y\u271d\nx y : Cofix F \u03b1\nr'xy : r x y\nthis : \u2200 (x y : Cofix F \u03b1), r x y \u2192 r' x y\n\u22a2 (TypeVec.id ::: Quot.factor (fun x y => r x y) (fun x y => r' x y) this \u2218 Quot.mk fun x y => r x y) <$$> dest x =\n    (TypeVec.id ::: Quot.factor (fun x y => r x y) (fun x y => r' x y) this \u2218 Quot.mk fun x y => r x y) <$$> dest y", "state_after": "n : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nr' : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => x = y \u2228 r x y\nx\u271d y\u271d : Cofix F \u03b1\nrxy : r x\u271d y\u271d\nx y : Cofix F \u03b1\nr'xy : r x y\nthis : \u2200 (x y : Cofix F \u03b1), r x y \u2192 r' x y\n\u22a2 (TypeVec.id ::: Quot.factor (fun x y => r x y) (fun x y => x = y \u2228 r x y) this \u2218 Quot.mk fun x y => r x y) <$$>\n      dest x =\n    (TypeVec.id ::: Quot.factor (fun x y => r x y) (fun x y => x = y \u2228 r x y) this \u2218 Quot.mk fun x y => r x y) <$$>\n      dest y"}, {"tactic": "rw [appendFun_comp_id]", "annotated_tactic": ["rw [<a>appendFun_comp_id</a>]", [{"full_name": "TypeVec.appendFun_comp_id", "def_path": "Mathlib/Data/TypeVec.lean", "def_pos": [270, 9], "def_end_pos": [270, 26]}]], "state_before": "n : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nr' : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => x = y \u2228 r x y\nx\u271d y\u271d : Cofix F \u03b1\nrxy : r x\u271d y\u271d\nx y : Cofix F \u03b1\nr'xy : r x y\nthis : \u2200 (x y : Cofix F \u03b1), r x y \u2192 r' x y\n\u22a2 (TypeVec.id ::: Quot.factor (fun x y => r x y) (fun x y => x = y \u2228 r x y) this \u2218 Quot.mk fun x y => r x y) <$$>\n      dest x =\n    (TypeVec.id ::: Quot.factor (fun x y => r x y) (fun x y => x = y \u2228 r x y) this \u2218 Quot.mk fun x y => r x y) <$$>\n      dest y", "state_after": "n : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nr' : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => x = y \u2228 r x y\nx\u271d y\u271d : Cofix F \u03b1\nrxy : r x\u271d y\u271d\nx y : Cofix F \u03b1\nr'xy : r x y\nthis : \u2200 (x y : Cofix F \u03b1), r x y \u2192 r' x y\n\u22a2 ((TypeVec.id ::: Quot.factor (fun x y => r x y) (fun x y => x = y \u2228 r x y) this) \u229a\n        (TypeVec.id ::: Quot.mk fun x y => r x y)) <$$>\n      dest x =\n    ((TypeVec.id ::: Quot.factor (fun x y => r x y) (fun x y => x = y \u2228 r x y) this) \u229a\n        (TypeVec.id ::: Quot.mk fun x y => r x y)) <$$>\n      dest y"}, {"tactic": "rw [@comp_map _ _ _ q _ _ _ (appendFun id (Quot.mk r)),\n  @comp_map _ _ _ q _ _ _ (appendFun id (Quot.mk r))]", "annotated_tactic": ["rw [@<a>comp_map</a> _ _ _ q _ _ _ (<a>appendFun</a> <a>id</a> (<a>Quot.mk</a> r)),\n        @<a>comp_map</a> _ _ _ q _ _ _ (<a>appendFun</a> <a>id</a> (<a>Quot.mk</a> r))]", [{"full_name": "MvQPF.comp_map", "def_path": "Mathlib/Data/QPF/Multivariate/Basic.lean", "def_pos": [112, 9], "def_end_pos": [112, 17]}, {"full_name": "TypeVec.appendFun", "def_path": "Mathlib/Data/TypeVec.lean", "def_pos": [150, 5], "def_end_pos": [150, 14]}, {"full_name": "TypeVec.id", "def_path": "Mathlib/Data/TypeVec.lean", "def_pos": [69, 5], "def_end_pos": [69, 7]}, {"full_name": "Quot.mk", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [406, 14], "def_end_pos": [406, 21]}, {"full_name": "MvQPF.comp_map", "def_path": "Mathlib/Data/QPF/Multivariate/Basic.lean", "def_pos": [112, 9], "def_end_pos": [112, 17]}, {"full_name": "TypeVec.appendFun", "def_path": "Mathlib/Data/TypeVec.lean", "def_pos": [150, 5], "def_end_pos": [150, 14]}, {"full_name": "TypeVec.id", "def_path": "Mathlib/Data/TypeVec.lean", "def_pos": [69, 5], "def_end_pos": [69, 7]}, {"full_name": "Quot.mk", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [406, 14], "def_end_pos": [406, 21]}]], "state_before": "n : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nr' : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => x = y \u2228 r x y\nx\u271d y\u271d : Cofix F \u03b1\nrxy : r x\u271d y\u271d\nx y : Cofix F \u03b1\nr'xy : r x y\nthis : \u2200 (x y : Cofix F \u03b1), r x y \u2192 r' x y\n\u22a2 ((TypeVec.id ::: Quot.factor (fun x y => r x y) (fun x y => x = y \u2228 r x y) this) \u229a\n        (TypeVec.id ::: Quot.mk fun x y => r x y)) <$$>\n      dest x =\n    ((TypeVec.id ::: Quot.factor (fun x y => r x y) (fun x y => x = y \u2228 r x y) this) \u229a\n        (TypeVec.id ::: Quot.mk fun x y => r x y)) <$$>\n      dest y", "state_after": "n : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nr' : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => x = y \u2228 r x y\nx\u271d y\u271d : Cofix F \u03b1\nrxy : r x\u271d y\u271d\nx y : Cofix F \u03b1\nr'xy : r x y\nthis : \u2200 (x y : Cofix F \u03b1), r x y \u2192 r' x y\n\u22a2 (TypeVec.id ::: Quot.factor (fun x y => r x y) (fun x y => x = y \u2228 r x y) this) <$$>\n      (TypeVec.id ::: Quot.mk r) <$$> dest x =\n    (TypeVec.id ::: Quot.factor (fun x y => r x y) (fun x y => x = y \u2228 r x y) this) <$$>\n      (TypeVec.id ::: Quot.mk r) <$$> dest y"}, {"tactic": "rw [h _ _ r'xy]", "annotated_tactic": ["rw [h _ _ r'xy]", []], "state_before": "n : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nr' : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => x = y \u2228 r x y\nx\u271d y\u271d : Cofix F \u03b1\nrxy : r x\u271d y\u271d\nx y : Cofix F \u03b1\nr'xy : r x y\nthis : \u2200 (x y : Cofix F \u03b1), r x y \u2192 r' x y\n\u22a2 (TypeVec.id ::: Quot.factor (fun x y => r x y) (fun x y => x = y \u2228 r x y) this) <$$>\n      (TypeVec.id ::: Quot.mk r) <$$> dest x =\n    (TypeVec.id ::: Quot.factor (fun x y => r x y) (fun x y => x = y \u2228 r x y) this) <$$>\n      (TypeVec.id ::: Quot.mk r) <$$> dest y", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/ProbabilityMassFunction/Monad.lean", "full_name": "PMF.toMeasure_bind_apply", "start": [187, 1], "end": [192, 97], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Rat/Lemmas.lean", "full_name": "Rat.divInt_mul_right", "start": [152, 1], "end": [153, 53], "traced_tactics": [{"tactic": "simp [\u2190 divInt_mul_left (d := d) a0, Int.mul_comm]", "annotated_tactic": ["simp [\u2190 <a>divInt_mul_left</a> (d := d) a0, <a>Int.mul_comm</a>]", [{"full_name": "Rat.divInt_mul_left", "def_path": "lake-packages/std/Std/Data/Rat/Lemmas.lean", "def_pos": [148, 9], "def_end_pos": [148, 24]}, {"full_name": "Int.mul_comm", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [380, 19], "def_end_pos": [380, 27]}]], "state_before": "n d a : Int\na0 : a \u2260 0\n\u22a2 n * a /. (d * a) = n /. d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "full_name": "tendsto_measure_cthickening_of_isClosed", "start": [1793, 1], "end": [1797, 28], "traced_tactics": [{"tactic": "convert tendsto_measure_cthickening hs", "annotated_tactic": ["convert <a>tendsto_measure_cthickening</a> hs", [{"full_name": "tendsto_measure_cthickening", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [1775, 9], "def_end_pos": [1775, 36]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t u : Set \u03b1\ninst\u271d\u00b3 : PseudoEMetricSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\nx : \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\ns : Set \u03b1\nhs : \u2203 R, R > 0 \u2227 \u2191\u2191\u03bc (cthickening R s) \u2260 \u22a4\nh's : IsClosed s\n\u22a2 Tendsto (fun r => \u2191\u2191\u03bc (cthickening r s)) (\ud835\udcdd 0) (\ud835\udcdd (\u2191\u2191\u03bc s))", "state_after": "case h.e'_5.h.e'_3.h.e'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t u : Set \u03b1\ninst\u271d\u00b3 : PseudoEMetricSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\nx : \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\ns : Set \u03b1\nhs : \u2203 R, R > 0 \u2227 \u2191\u2191\u03bc (cthickening R s) \u2260 \u22a4\nh's : IsClosed s\n\u22a2 s = closure s"}, {"tactic": "exact h's.closure_eq.symm", "annotated_tactic": ["exact h's.closure_eq.symm", []], "state_before": "case h.e'_5.h.e'_3.h.e'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9 : Sort y\ns\u271d t u : Set \u03b1\ninst\u271d\u00b3 : PseudoEMetricSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\nx : \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\ns : Set \u03b1\nhs : \u2203 R, R > 0 \u2227 \u2191\u2191\u03bc (cthickening R s) \u2260 \u22a4\nh's : IsClosed s\n\u22a2 s = closure s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "full_name": "MeasureTheory.Lp.norm_const_le", "start": [870, 1], "end": [872, 33], "traced_tactics": [{"tactic": "rw [\u2190 indicatorConstLp_univ]", "annotated_tactic": ["rw [\u2190 <a>indicatorConstLp_univ</a>]", [{"full_name": "MeasureTheory.indicatorConstLp_univ", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [854, 7], "def_end_pos": [854, 28]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : IsFiniteMeasure \u03bc\nc : E\n\u22a2 \u2016\u2191(Lp.const p \u03bc) c\u2016 \u2264 \u2016c\u2016 * ENNReal.toReal (\u2191\u2191\u03bc Set.univ) ^ (1 / ENNReal.toReal p)", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : IsFiniteMeasure \u03bc\nc : E\n\u22a2 \u2016indicatorConstLp p (_ : MeasurableSet Set.univ) (_ : \u2191\u2191\u03bc Set.univ \u2260 \u22a4) c\u2016 \u2264\n    \u2016c\u2016 * ENNReal.toReal (\u2191\u2191\u03bc Set.univ) ^ (1 / ENNReal.toReal p)"}, {"tactic": "exact norm_indicatorConstLp_le", "annotated_tactic": ["exact <a>norm_indicatorConstLp_le</a>", [{"full_name": "MeasureTheory.norm_indicatorConstLp_le", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [782, 9], "def_end_pos": [782, 33]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : IsFiniteMeasure \u03bc\nc : E\n\u22a2 \u2016indicatorConstLp p (_ : MeasurableSet Set.univ) (_ : \u2191\u2191\u03bc Set.univ \u2260 \u22a4) c\u2016 \u2264\n    \u2016c\u2016 * ENNReal.toReal (\u2191\u2191\u03bc Set.univ) ^ (1 / ENNReal.toReal p)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Pointwise/Finite.lean", "full_name": "Group.card_pow_eq_card_pow_card_univ", "start": [175, 1], "end": [200, 57], "traced_tactics": [{"tactic": "have hG : 0 < Fintype.card G := Fintype.card_pos_iff.mpr \u27e81\u27e9", "annotated_tactic": ["have hG : 0 < <a>Fintype.card</a> G := Fintype.card_pos_iff.mpr \u27e81\u27e9", [{"full_name": "Fintype.card", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [61, 5], "def_end_pos": [61, 9]}]], "state_before": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nG : Type u_5\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Fintype G\nS : Set G\ninst\u271d : (k : \u2115) \u2192 DecidablePred fun x => x \u2208 S ^ k\n\u22a2 \u2200 (k : \u2115), Fintype.card G \u2264 k \u2192 Fintype.card \u2191(S ^ k) = Fintype.card \u2191(S ^ Fintype.card G)", "state_after": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nG : Type u_5\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Fintype G\nS : Set G\ninst\u271d : (k : \u2115) \u2192 DecidablePred fun x => x \u2208 S ^ k\nhG : 0 < Fintype.card G\n\u22a2 \u2200 (k : \u2115), Fintype.card G \u2264 k \u2192 Fintype.card \u2191(S ^ k) = Fintype.card \u2191(S ^ Fintype.card G)"}, {"tactic": "by_cases hS : S = \u2205", "annotated_tactic": ["by_cases hS : S = \u2205", []], "state_before": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nG : Type u_5\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Fintype G\nS : Set G\ninst\u271d : (k : \u2115) \u2192 DecidablePred fun x => x \u2208 S ^ k\nhG : 0 < Fintype.card G\n\u22a2 \u2200 (k : \u2115), Fintype.card G \u2264 k \u2192 Fintype.card \u2191(S ^ k) = Fintype.card \u2191(S ^ Fintype.card G)", "state_after": "case pos\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nG : Type u_5\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Fintype G\nS : Set G\ninst\u271d : (k : \u2115) \u2192 DecidablePred fun x => x \u2208 S ^ k\nhG : 0 < Fintype.card G\nhS : S = \u2205\n\u22a2 \u2200 (k : \u2115), Fintype.card G \u2264 k \u2192 Fintype.card \u2191(S ^ k) = Fintype.card \u2191(S ^ Fintype.card G)\n\ncase neg\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nG : Type u_5\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Fintype G\nS : Set G\ninst\u271d : (k : \u2115) \u2192 DecidablePred fun x => x \u2208 S ^ k\nhG : 0 < Fintype.card G\nhS : \u00acS = \u2205\n\u22a2 \u2200 (k : \u2115), Fintype.card G \u2264 k \u2192 Fintype.card \u2191(S ^ k) = Fintype.card \u2191(S ^ Fintype.card G)"}, {"tactic": "obtain \u27e8a, ha\u27e9 := Set.nonempty_iff_ne_empty.2 hS", "annotated_tactic": ["obtain \u27e8a, ha\u27e9 := <a>Set.nonempty_iff_ne_empty</a>.2 hS", [{"full_name": "Set.nonempty_iff_ne_empty", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [610, 9], "def_end_pos": [610, 30]}]], "state_before": "case neg\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nG : Type u_5\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Fintype G\nS : Set G\ninst\u271d : (k : \u2115) \u2192 DecidablePred fun x => x \u2208 S ^ k\nhG : 0 < Fintype.card G\nhS : \u00acS = \u2205\n\u22a2 \u2200 (k : \u2115), Fintype.card G \u2264 k \u2192 Fintype.card \u2191(S ^ k) = Fintype.card \u2191(S ^ Fintype.card G)", "state_after": "case neg.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nG : Type u_5\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Fintype G\nS : Set G\ninst\u271d : (k : \u2115) \u2192 DecidablePred fun x => x \u2208 S ^ k\nhG : 0 < Fintype.card G\nhS : \u00acS = \u2205\na : G\nha : a \u2208 S\n\u22a2 \u2200 (k : \u2115), Fintype.card G \u2264 k \u2192 Fintype.card \u2191(S ^ k) = Fintype.card \u2191(S ^ Fintype.card G)"}, {"tactic": "have key : \u2200 (a) (s t : Set G) [Fintype s] [Fintype t],\n    (\u2200 b : G, b \u2208 s \u2192 a * b \u2208 t) \u2192 Fintype.card s \u2264 Fintype.card t := by\n  refine' fun a s t _ _ h \u21a6 Fintype.card_le_of_injective (fun \u27e8b, hb\u27e9 \u21a6 \u27e8a * b, h b hb\u27e9) _\n  rintro \u27e8b, hb\u27e9 \u27e8c, hc\u27e9 hbc\n  exact Subtype.ext (mul_left_cancel (Subtype.ext_iff.mp hbc))", "annotated_tactic": ["have key : \u2200 (a) (s t : <a>Set</a> G) [<a>Fintype</a> s] [<a>Fintype</a> t],\n      (\u2200 b : G, b \u2208 s \u2192 a * b \u2208 t) \u2192 <a>Fintype.card</a> s \u2264 <a>Fintype.card</a> t := by\n    refine' fun a s t _ _ h \u21a6 <a>Fintype.card_le_of_injective</a> (fun \u27e8b, hb\u27e9 \u21a6 \u27e8a * b, h b hb\u27e9) _\n    rintro \u27e8b, hb\u27e9 \u27e8c, hc\u27e9 hbc\n    exact <a>Subtype.ext</a> (<a>mul_left_cancel</a> (Subtype.ext_iff.mp hbc))", [{"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}, {"full_name": "Fintype", "def_path": "Mathlib/Data/Fintype/Basic.lean", "def_pos": [54, 7], "def_end_pos": [54, 14]}, {"full_name": "Fintype", "def_path": "Mathlib/Data/Fintype/Basic.lean", "def_pos": [54, 7], "def_end_pos": [54, 14]}, {"full_name": "Fintype.card", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [61, 5], "def_end_pos": [61, 9]}, {"full_name": "Fintype.card", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [61, 5], "def_end_pos": [61, 9]}, {"full_name": "Fintype.card_le_of_injective", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [479, 9], "def_end_pos": [479, 29]}, {"full_name": "Subtype.ext", "def_path": "Mathlib/Data/Subtype.lean", "def_pos": [65, 19], "def_end_pos": [65, 22]}, {"full_name": "mul_left_cancel", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [172, 9], "def_end_pos": [172, 24]}]], "state_before": "case neg.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nG : Type u_5\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Fintype G\nS : Set G\ninst\u271d : (k : \u2115) \u2192 DecidablePred fun x => x \u2208 S ^ k\nhG : 0 < Fintype.card G\nhS : \u00acS = \u2205\na : G\nha : a \u2208 S\n\u22a2 \u2200 (k : \u2115), Fintype.card G \u2264 k \u2192 Fintype.card \u2191(S ^ k) = Fintype.card \u2191(S ^ Fintype.card G)", "state_after": "case neg.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nG : Type u_5\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Fintype G\nS : Set G\ninst\u271d : (k : \u2115) \u2192 DecidablePred fun x => x \u2208 S ^ k\nhG : 0 < Fintype.card G\nhS : \u00acS = \u2205\na : G\nha : a \u2208 S\nkey :\n  \u2200 (a : G) (s t : Set G) [inst : Fintype \u2191s] [inst_1 : Fintype \u2191t],\n    (\u2200 (b : G), b \u2208 s \u2192 a * b \u2208 t) \u2192 Fintype.card \u2191s \u2264 Fintype.card \u2191t\n\u22a2 \u2200 (k : \u2115), Fintype.card G \u2264 k \u2192 Fintype.card \u2191(S ^ k) = Fintype.card \u2191(S ^ Fintype.card G)"}, {"tactic": "have mono : Monotone (fun n \u21a6 Fintype.card (\u21a5(S ^ n)) : \u2115 \u2192 \u2115) :=\n  monotone_nat_of_le_succ fun n \u21a6 key a _ _ fun b hb \u21a6 Set.mul_mem_mul ha hb", "annotated_tactic": ["have mono : <a>Monotone</a> (fun n \u21a6 <a>Fintype.card</a> (\u21a5(S ^ n)) : \u2115 \u2192 \u2115) :=\n    <a>monotone_nat_of_le_succ</a> fun n \u21a6 key a _ _ fun b hb \u21a6 <a>Set.mul_mem_mul</a> ha hb", [{"full_name": "Monotone", "def_path": "Mathlib/Order/Monotone/Basic.lean", "def_pos": [77, 5], "def_end_pos": [77, 13]}, {"full_name": "Fintype.card", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [61, 5], "def_end_pos": [61, 9]}, {"full_name": "monotone_nat_of_le_succ", "def_path": "Mathlib/Order/Monotone/Basic.lean", "def_pos": [1025, 9], "def_end_pos": [1025, 32]}, {"full_name": "Set.mul_mem_mul", "def_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "def_pos": [342, 9], "def_end_pos": [342, 20]}]], "state_before": "case neg.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nG : Type u_5\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Fintype G\nS : Set G\ninst\u271d : (k : \u2115) \u2192 DecidablePred fun x => x \u2208 S ^ k\nhG : 0 < Fintype.card G\nhS : \u00acS = \u2205\na : G\nha : a \u2208 S\nkey :\n  \u2200 (a : G) (s t : Set G) [inst : Fintype \u2191s] [inst_1 : Fintype \u2191t],\n    (\u2200 (b : G), b \u2208 s \u2192 a * b \u2208 t) \u2192 Fintype.card \u2191s \u2264 Fintype.card \u2191t\n\u22a2 \u2200 (k : \u2115), Fintype.card G \u2264 k \u2192 Fintype.card \u2191(S ^ k) = Fintype.card \u2191(S ^ Fintype.card G)", "state_after": "case neg.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nG : Type u_5\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Fintype G\nS : Set G\ninst\u271d : (k : \u2115) \u2192 DecidablePred fun x => x \u2208 S ^ k\nhG : 0 < Fintype.card G\nhS : \u00acS = \u2205\na : G\nha : a \u2208 S\nkey :\n  \u2200 (a : G) (s t : Set G) [inst : Fintype \u2191s] [inst_1 : Fintype \u2191t],\n    (\u2200 (b : G), b \u2208 s \u2192 a * b \u2208 t) \u2192 Fintype.card \u2191s \u2264 Fintype.card \u2191t\nmono : Monotone fun n => Fintype.card \u2191(S ^ n)\n\u22a2 \u2200 (k : \u2115), Fintype.card G \u2264 k \u2192 Fintype.card \u2191(S ^ k) = Fintype.card \u2191(S ^ Fintype.card G)"}, {"tactic": "refine' card_pow_eq_card_pow_card_univ_aux mono (fun n \u21a6 set_fintype_card_le_univ (S ^ n))\n  fun n h \u21a6 le_antisymm (mono (n + 1).le_succ) (key a\u207b\u00b9 (S ^ (n + 2)) (S ^ (n + 1)) _)", "annotated_tactic": ["refine' <a>card_pow_eq_card_pow_card_univ_aux</a> mono (fun n \u21a6 <a>set_fintype_card_le_univ</a> (S ^ n))\n    fun n h \u21a6 <a>le_antisymm</a> (mono (n + 1).<a>le_succ</a>) (key a\u207b\u00b9 (S ^ (n + 2)) (S ^ (n + 1)) _)", [{"full_name": "Group.card_pow_eq_card_pow_card_univ_aux", "def_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "def_pos": [1350, 9], "def_end_pos": [1350, 43]}, {"full_name": "set_fintype_card_le_univ", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [775, 9], "def_end_pos": [775, 33]}, {"full_name": "le_antisymm", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [188, 9], "def_end_pos": [188, 20]}, {"full_name": "Nat.le_succ", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1599, 9], "def_end_pos": [1599, 20]}]], "state_before": "case neg.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nG : Type u_5\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Fintype G\nS : Set G\ninst\u271d : (k : \u2115) \u2192 DecidablePred fun x => x \u2208 S ^ k\nhG : 0 < Fintype.card G\nhS : \u00acS = \u2205\na : G\nha : a \u2208 S\nkey :\n  \u2200 (a : G) (s t : Set G) [inst : Fintype \u2191s] [inst_1 : Fintype \u2191t],\n    (\u2200 (b : G), b \u2208 s \u2192 a * b \u2208 t) \u2192 Fintype.card \u2191s \u2264 Fintype.card \u2191t\nmono : Monotone fun n => Fintype.card \u2191(S ^ n)\n\u22a2 \u2200 (k : \u2115), Fintype.card G \u2264 k \u2192 Fintype.card \u2191(S ^ k) = Fintype.card \u2191(S ^ Fintype.card G)", "state_after": "case neg.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nG : Type u_5\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Fintype G\nS : Set G\ninst\u271d : (k : \u2115) \u2192 DecidablePred fun x => x \u2208 S ^ k\nhG : 0 < Fintype.card G\nhS : \u00acS = \u2205\na : G\nha : a \u2208 S\nkey :\n  \u2200 (a : G) (s t : Set G) [inst : Fintype \u2191s] [inst_1 : Fintype \u2191t],\n    (\u2200 (b : G), b \u2208 s \u2192 a * b \u2208 t) \u2192 Fintype.card \u2191s \u2264 Fintype.card \u2191t\nmono : Monotone fun n => Fintype.card \u2191(S ^ n)\nn : \u2115\nh : Fintype.card \u2191(S ^ n) = Fintype.card \u2191(S ^ (n + 1))\n\u22a2 \u2200 (b : G), b \u2208 S ^ (n + 2) \u2192 a\u207b\u00b9 * b \u2208 S ^ (n + 1)"}, {"tactic": "replace h\u2082 : {a} * S ^ n = S ^ (n + 1)", "annotated_tactic": ["replace h\u2082 : {a} * S ^ n = S ^ (n + 1)", []], "state_before": "case neg.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nG : Type u_5\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Fintype G\nS : Set G\ninst\u271d : (k : \u2115) \u2192 DecidablePred fun x => x \u2208 S ^ k\nhG : 0 < Fintype.card G\nhS : \u00acS = \u2205\na : G\nha : a \u2208 S\nkey :\n  \u2200 (a : G) (s t : Set G) [inst : Fintype \u2191s] [inst_1 : Fintype \u2191t],\n    (\u2200 (b : G), b \u2208 s \u2192 a * b \u2208 t) \u2192 Fintype.card \u2191s \u2264 Fintype.card \u2191t\nmono : Monotone fun n => Fintype.card \u2191(S ^ n)\nn : \u2115\nh : Fintype.card \u2191(S ^ n) = Fintype.card \u2191(S ^ (n + 1))\n\u22a2 \u2200 (b : G), b \u2208 S ^ (n + 2) \u2192 a\u207b\u00b9 * b \u2208 S ^ (n + 1)", "state_after": "case h\u2082\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nG : Type u_5\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Fintype G\nS : Set G\ninst\u271d : (k : \u2115) \u2192 DecidablePred fun x => x \u2208 S ^ k\nhG : 0 < Fintype.card G\nhS : \u00acS = \u2205\na : G\nha : a \u2208 S\nkey :\n  \u2200 (a : G) (s t : Set G) [inst : Fintype \u2191s] [inst_1 : Fintype \u2191t],\n    (\u2200 (b : G), b \u2208 s \u2192 a * b \u2208 t) \u2192 Fintype.card \u2191s \u2264 Fintype.card \u2191t\nmono : Monotone fun n => Fintype.card \u2191(S ^ n)\nn : \u2115\nh : Fintype.card \u2191(S ^ n) = Fintype.card \u2191(S ^ (n + 1))\n\u22a2 {a} * S ^ n = S ^ (n + 1)\n\ncase neg.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nG : Type u_5\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Fintype G\nS : Set G\ninst\u271d : (k : \u2115) \u2192 DecidablePred fun x => x \u2208 S ^ k\nhG : 0 < Fintype.card G\nhS : \u00acS = \u2205\na : G\nha : a \u2208 S\nkey :\n  \u2200 (a : G) (s t : Set G) [inst : Fintype \u2191s] [inst_1 : Fintype \u2191t],\n    (\u2200 (b : G), b \u2208 s \u2192 a * b \u2208 t) \u2192 Fintype.card \u2191s \u2264 Fintype.card \u2191t\nmono : Monotone fun n => Fintype.card \u2191(S ^ n)\nn : \u2115\nh : Fintype.card \u2191(S ^ n) = Fintype.card \u2191(S ^ (n + 1))\nh\u2082 : {a} * S ^ n = S ^ (n + 1)\n\u22a2 \u2200 (b : G), b \u2208 S ^ (n + 2) \u2192 a\u207b\u00b9 * b \u2208 S ^ (n + 1)"}, {"tactic": "rw [pow_succ', \u2190 h\u2082, mul_assoc, \u2190 pow_succ', h\u2082]", "annotated_tactic": ["rw [<a>pow_succ'</a>, \u2190 h\u2082, <a>mul_assoc</a>, \u2190 <a>pow_succ'</a>, h\u2082]", [{"full_name": "pow_succ'", "def_path": "Mathlib/Algebra/Group/Commute/Defs.lean", "def_pos": [213, 9], "def_end_pos": [213, 25]}, {"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [264, 9], "def_end_pos": [264, 18]}, {"full_name": "pow_succ'", "def_path": "Mathlib/Algebra/Group/Commute/Defs.lean", "def_pos": [213, 9], "def_end_pos": [213, 25]}]], "state_before": "case neg.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nG : Type u_5\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Fintype G\nS : Set G\ninst\u271d : (k : \u2115) \u2192 DecidablePred fun x => x \u2208 S ^ k\nhG : 0 < Fintype.card G\nhS : \u00acS = \u2205\na : G\nha : a \u2208 S\nkey :\n  \u2200 (a : G) (s t : Set G) [inst : Fintype \u2191s] [inst_1 : Fintype \u2191t],\n    (\u2200 (b : G), b \u2208 s \u2192 a * b \u2208 t) \u2192 Fintype.card \u2191s \u2264 Fintype.card \u2191t\nmono : Monotone fun n => Fintype.card \u2191(S ^ n)\nn : \u2115\nh : Fintype.card \u2191(S ^ n) = Fintype.card \u2191(S ^ (n + 1))\nh\u2082 : {a} * S ^ n = S ^ (n + 1)\n\u22a2 \u2200 (b : G), b \u2208 S ^ (n + 2) \u2192 a\u207b\u00b9 * b \u2208 S ^ (n + 1)", "state_after": "case neg.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nG : Type u_5\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Fintype G\nS : Set G\ninst\u271d : (k : \u2115) \u2192 DecidablePred fun x => x \u2208 S ^ k\nhG : 0 < Fintype.card G\nhS : \u00acS = \u2205\na : G\nha : a \u2208 S\nkey :\n  \u2200 (a : G) (s t : Set G) [inst : Fintype \u2191s] [inst_1 : Fintype \u2191t],\n    (\u2200 (b : G), b \u2208 s \u2192 a * b \u2208 t) \u2192 Fintype.card \u2191s \u2264 Fintype.card \u2191t\nmono : Monotone fun n => Fintype.card \u2191(S ^ n)\nn : \u2115\nh : Fintype.card \u2191(S ^ n) = Fintype.card \u2191(S ^ (n + 1))\nh\u2082 : {a} * S ^ n = S ^ (n + 1)\n\u22a2 \u2200 (b : G), b \u2208 {a} * S ^ (n + 1) \u2192 a\u207b\u00b9 * b \u2208 S ^ (n + 1)"}, {"tactic": "rintro _ \u27e8b, c, hb, hc, rfl\u27e9", "annotated_tactic": ["rintro _ \u27e8b, c, hb, hc, rfl\u27e9", []], "state_before": "case neg.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nG : Type u_5\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Fintype G\nS : Set G\ninst\u271d : (k : \u2115) \u2192 DecidablePred fun x => x \u2208 S ^ k\nhG : 0 < Fintype.card G\nhS : \u00acS = \u2205\na : G\nha : a \u2208 S\nkey :\n  \u2200 (a : G) (s t : Set G) [inst : Fintype \u2191s] [inst_1 : Fintype \u2191t],\n    (\u2200 (b : G), b \u2208 s \u2192 a * b \u2208 t) \u2192 Fintype.card \u2191s \u2264 Fintype.card \u2191t\nmono : Monotone fun n => Fintype.card \u2191(S ^ n)\nn : \u2115\nh : Fintype.card \u2191(S ^ n) = Fintype.card \u2191(S ^ (n + 1))\nh\u2082 : {a} * S ^ n = S ^ (n + 1)\n\u22a2 \u2200 (b : G), b \u2208 {a} * S ^ (n + 1) \u2192 a\u207b\u00b9 * b \u2208 S ^ (n + 1)", "state_after": "case neg.intro.intro.intro.intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nG : Type u_5\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Fintype G\nS : Set G\ninst\u271d : (k : \u2115) \u2192 DecidablePred fun x => x \u2208 S ^ k\nhG : 0 < Fintype.card G\nhS : \u00acS = \u2205\na : G\nha : a \u2208 S\nkey :\n  \u2200 (a : G) (s t : Set G) [inst : Fintype \u2191s] [inst_1 : Fintype \u2191t],\n    (\u2200 (b : G), b \u2208 s \u2192 a * b \u2208 t) \u2192 Fintype.card \u2191s \u2264 Fintype.card \u2191t\nmono : Monotone fun n => Fintype.card \u2191(S ^ n)\nn : \u2115\nh : Fintype.card \u2191(S ^ n) = Fintype.card \u2191(S ^ (n + 1))\nh\u2082 : {a} * S ^ n = S ^ (n + 1)\nb c : G\nhb : b \u2208 {a}\nhc : c \u2208 S ^ (n + 1)\n\u22a2 a\u207b\u00b9 * (fun x x_1 => x * x_1) b c \u2208 S ^ (n + 1)"}, {"tactic": "rwa [Set.mem_singleton_iff.mp hb, inv_mul_cancel_left]", "annotated_tactic": ["rwa [Set.mem_singleton_iff.mp hb, <a>inv_mul_cancel_left</a>]", [{"full_name": "inv_mul_cancel_left", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [1147, 9], "def_end_pos": [1147, 28]}]], "state_before": "case neg.intro.intro.intro.intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nG : Type u_5\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Fintype G\nS : Set G\ninst\u271d : (k : \u2115) \u2192 DecidablePred fun x => x \u2208 S ^ k\nhG : 0 < Fintype.card G\nhS : \u00acS = \u2205\na : G\nha : a \u2208 S\nkey :\n  \u2200 (a : G) (s t : Set G) [inst : Fintype \u2191s] [inst_1 : Fintype \u2191t],\n    (\u2200 (b : G), b \u2208 s \u2192 a * b \u2208 t) \u2192 Fintype.card \u2191s \u2264 Fintype.card \u2191t\nmono : Monotone fun n => Fintype.card \u2191(S ^ n)\nn : \u2115\nh : Fintype.card \u2191(S ^ n) = Fintype.card \u2191(S ^ (n + 1))\nh\u2082 : {a} * S ^ n = S ^ (n + 1)\nb c : G\nhb : b \u2208 {a}\nhc : c \u2208 S ^ (n + 1)\n\u22a2 a\u207b\u00b9 * (fun x x_1 => x * x_1) b c \u2208 S ^ (n + 1)", "state_after": "no goals"}, {"tactic": "refine' fun k hk \u21a6 Fintype.card_congr _", "annotated_tactic": ["refine' fun k hk \u21a6 <a>Fintype.card_congr</a> _", [{"full_name": "Fintype.card_congr", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [149, 9], "def_end_pos": [149, 19]}]], "state_before": "case pos\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nG : Type u_5\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Fintype G\nS : Set G\ninst\u271d : (k : \u2115) \u2192 DecidablePred fun x => x \u2208 S ^ k\nhG : 0 < Fintype.card G\nhS : S = \u2205\n\u22a2 \u2200 (k : \u2115), Fintype.card G \u2264 k \u2192 Fintype.card \u2191(S ^ k) = Fintype.card \u2191(S ^ Fintype.card G)", "state_after": "case pos\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nG : Type u_5\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Fintype G\nS : Set G\ninst\u271d : (k : \u2115) \u2192 DecidablePred fun x => x \u2208 S ^ k\nhG : 0 < Fintype.card G\nhS : S = \u2205\nk : \u2115\nhk : Fintype.card G \u2264 k\n\u22a2 \u2191(S ^ k) \u2243 \u2191(S ^ Fintype.card G)"}, {"tactic": "rw [hS, empty_pow (ne_of_gt (lt_of_lt_of_le hG hk)), empty_pow (ne_of_gt hG)]", "annotated_tactic": ["rw [hS, <a>empty_pow</a> (<a>ne_of_gt</a> (<a>lt_of_lt_of_le</a> hG hk)), <a>empty_pow</a> (<a>ne_of_gt</a> hG)]", [{"full_name": "Set.empty_pow", "def_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "def_pos": [977, 9], "def_end_pos": [977, 18]}, {"full_name": "ne_of_gt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [104, 9], "def_end_pos": [104, 17]}, {"full_name": "lt_of_lt_of_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [115, 9], "def_end_pos": [115, 23]}, {"full_name": "Set.empty_pow", "def_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "def_pos": [977, 9], "def_end_pos": [977, 18]}, {"full_name": "ne_of_gt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [104, 9], "def_end_pos": [104, 17]}]], "state_before": "case pos\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nG : Type u_5\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Fintype G\nS : Set G\ninst\u271d : (k : \u2115) \u2192 DecidablePred fun x => x \u2208 S ^ k\nhG : 0 < Fintype.card G\nhS : S = \u2205\nk : \u2115\nhk : Fintype.card G \u2264 k\n\u22a2 \u2191(S ^ k) \u2243 \u2191(S ^ Fintype.card G)", "state_after": "no goals"}, {"tactic": "refine' fun a s t _ _ h \u21a6 Fintype.card_le_of_injective (fun \u27e8b, hb\u27e9 \u21a6 \u27e8a * b, h b hb\u27e9) _", "annotated_tactic": ["refine' fun a s t _ _ h \u21a6 <a>Fintype.card_le_of_injective</a> (fun \u27e8b, hb\u27e9 \u21a6 \u27e8a * b, h b hb\u27e9) _", [{"full_name": "Fintype.card_le_of_injective", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [479, 9], "def_end_pos": [479, 29]}]], "state_before": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nG : Type u_5\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Fintype G\nS : Set G\ninst\u271d : (k : \u2115) \u2192 DecidablePred fun x => x \u2208 S ^ k\nhG : 0 < Fintype.card G\nhS : \u00acS = \u2205\na : G\nha : a \u2208 S\n\u22a2 \u2200 (a : G) (s t : Set G) [inst : Fintype \u2191s] [inst_1 : Fintype \u2191t],\n    (\u2200 (b : G), b \u2208 s \u2192 a * b \u2208 t) \u2192 Fintype.card \u2191s \u2264 Fintype.card \u2191t", "state_after": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nG : Type u_5\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Fintype G\nS : Set G\ninst\u271d : (k : \u2115) \u2192 DecidablePred fun x => x \u2208 S ^ k\nhG : 0 < Fintype.card G\nhS : \u00acS = \u2205\na\u271d : G\nha : a\u271d \u2208 S\na : G\ns t : Set G\nx\u271d\u00b9 : Fintype \u2191s\nx\u271d : Fintype \u2191t\nh : \u2200 (b : G), b \u2208 s \u2192 a * b \u2208 t\n\u22a2 Function.Injective fun x =>\n    match x with\n    | { val := b, property := hb } => { val := a * b, property := (_ : a * b \u2208 t) }"}, {"tactic": "rintro \u27e8b, hb\u27e9 \u27e8c, hc\u27e9 hbc", "annotated_tactic": ["rintro \u27e8b, hb\u27e9 \u27e8c, hc\u27e9 hbc", []], "state_before": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nG : Type u_5\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Fintype G\nS : Set G\ninst\u271d : (k : \u2115) \u2192 DecidablePred fun x => x \u2208 S ^ k\nhG : 0 < Fintype.card G\nhS : \u00acS = \u2205\na\u271d : G\nha : a\u271d \u2208 S\na : G\ns t : Set G\nx\u271d\u00b9 : Fintype \u2191s\nx\u271d : Fintype \u2191t\nh : \u2200 (b : G), b \u2208 s \u2192 a * b \u2208 t\n\u22a2 Function.Injective fun x =>\n    match x with\n    | { val := b, property := hb } => { val := a * b, property := (_ : a * b \u2208 t) }", "state_after": "case mk.mk\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nG : Type u_5\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Fintype G\nS : Set G\ninst\u271d : (k : \u2115) \u2192 DecidablePred fun x => x \u2208 S ^ k\nhG : 0 < Fintype.card G\nhS : \u00acS = \u2205\na\u271d : G\nha : a\u271d \u2208 S\na : G\ns t : Set G\nx\u271d\u00b9 : Fintype \u2191s\nx\u271d : Fintype \u2191t\nh : \u2200 (b : G), b \u2208 s \u2192 a * b \u2208 t\nb : G\nhb : b \u2208 s\nc : G\nhc : c \u2208 s\nhbc :\n  (fun x =>\n        match x with\n        | { val := b, property := hb } => { val := a * b, property := (_ : a * b \u2208 t) })\n      { val := b, property := hb } =\n    (fun x =>\n        match x with\n        | { val := b, property := hb } => { val := a * b, property := (_ : a * b \u2208 t) })\n      { val := c, property := hc }\n\u22a2 { val := b, property := hb } = { val := c, property := hc }"}, {"tactic": "exact Subtype.ext (mul_left_cancel (Subtype.ext_iff.mp hbc))", "annotated_tactic": ["exact <a>Subtype.ext</a> (<a>mul_left_cancel</a> (Subtype.ext_iff.mp hbc))", [{"full_name": "Subtype.ext", "def_path": "Mathlib/Data/Subtype.lean", "def_pos": [65, 19], "def_end_pos": [65, 22]}, {"full_name": "mul_left_cancel", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [172, 9], "def_end_pos": [172, 24]}]], "state_before": "case mk.mk\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nG : Type u_5\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Fintype G\nS : Set G\ninst\u271d : (k : \u2115) \u2192 DecidablePred fun x => x \u2208 S ^ k\nhG : 0 < Fintype.card G\nhS : \u00acS = \u2205\na\u271d : G\nha : a\u271d \u2208 S\na : G\ns t : Set G\nx\u271d\u00b9 : Fintype \u2191s\nx\u271d : Fintype \u2191t\nh : \u2200 (b : G), b \u2208 s \u2192 a * b \u2208 t\nb : G\nhb : b \u2208 s\nc : G\nhc : c \u2208 s\nhbc :\n  (fun x =>\n        match x with\n        | { val := b, property := hb } => { val := a * b, property := (_ : a * b \u2208 t) })\n      { val := b, property := hb } =\n    (fun x =>\n        match x with\n        | { val := b, property := hb } => { val := a * b, property := (_ : a * b \u2208 t) })\n      { val := c, property := hc }\n\u22a2 { val := b, property := hb } = { val := c, property := hc }", "state_after": "no goals"}, {"tactic": "have : Fintype (Set.singleton a * S ^ n) := by\n  classical!\n  apply fintypeMul", "annotated_tactic": ["have : <a>Fintype</a> (<a>Set.singleton</a> a * S ^ n) := by\n      classical!\n      apply <a>fintypeMul</a>", [{"full_name": "Fintype", "def_path": "Mathlib/Data/Fintype/Basic.lean", "def_pos": [54, 7], "def_end_pos": [54, 14]}, {"full_name": "Set.singleton", "def_path": "Mathlib/Init/Set.lean", "def_pos": [97, 15], "def_end_pos": [97, 24]}, {"full_name": "Set.fintypeMul", "def_path": "Mathlib/Data/Set/Pointwise/Finite.lean", "def_pos": [56, 5], "def_end_pos": [56, 15]}]], "state_before": "case h\u2082\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nG : Type u_5\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Fintype G\nS : Set G\ninst\u271d : (k : \u2115) \u2192 DecidablePred fun x => x \u2208 S ^ k\nhG : 0 < Fintype.card G\nhS : \u00acS = \u2205\na : G\nha : a \u2208 S\nkey :\n  \u2200 (a : G) (s t : Set G) [inst : Fintype \u2191s] [inst_1 : Fintype \u2191t],\n    (\u2200 (b : G), b \u2208 s \u2192 a * b \u2208 t) \u2192 Fintype.card \u2191s \u2264 Fintype.card \u2191t\nmono : Monotone fun n => Fintype.card \u2191(S ^ n)\nn : \u2115\nh : Fintype.card \u2191(S ^ n) = Fintype.card \u2191(S ^ (n + 1))\n\u22a2 {a} * S ^ n = S ^ (n + 1)", "state_after": "case h\u2082\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nG : Type u_5\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Fintype G\nS : Set G\ninst\u271d : (k : \u2115) \u2192 DecidablePred fun x => x \u2208 S ^ k\nhG : 0 < Fintype.card G\nhS : \u00acS = \u2205\na : G\nha : a \u2208 S\nkey :\n  \u2200 (a : G) (s t : Set G) [inst : Fintype \u2191s] [inst_1 : Fintype \u2191t],\n    (\u2200 (b : G), b \u2208 s \u2192 a * b \u2208 t) \u2192 Fintype.card \u2191s \u2264 Fintype.card \u2191t\nmono : Monotone fun n => Fintype.card \u2191(S ^ n)\nn : \u2115\nh : Fintype.card \u2191(S ^ n) = Fintype.card \u2191(S ^ (n + 1))\nthis : Fintype \u2191(Set.singleton a * S ^ n)\n\u22a2 {a} * S ^ n = S ^ (n + 1)"}, {"tactic": "refine' Set.eq_of_subset_of_card_le _ (le_trans (ge_of_eq h) _)", "annotated_tactic": ["refine' <a>Set.eq_of_subset_of_card_le</a> _ (<a>le_trans</a> (<a>ge_of_eq</a> h) _)", [{"full_name": "Set.eq_of_subset_of_card_le", "def_path": "Mathlib/Data/Set/Finite.lean", "def_pos": [1255, 9], "def_end_pos": [1255, 32]}, {"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}, {"full_name": "ge_of_eq", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [352, 9], "def_end_pos": [352, 17]}]], "state_before": "case h\u2082\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nG : Type u_5\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Fintype G\nS : Set G\ninst\u271d : (k : \u2115) \u2192 DecidablePred fun x => x \u2208 S ^ k\nhG : 0 < Fintype.card G\nhS : \u00acS = \u2205\na : G\nha : a \u2208 S\nkey :\n  \u2200 (a : G) (s t : Set G) [inst : Fintype \u2191s] [inst_1 : Fintype \u2191t],\n    (\u2200 (b : G), b \u2208 s \u2192 a * b \u2208 t) \u2192 Fintype.card \u2191s \u2264 Fintype.card \u2191t\nmono : Monotone fun n => Fintype.card \u2191(S ^ n)\nn : \u2115\nh : Fintype.card \u2191(S ^ n) = Fintype.card \u2191(S ^ (n + 1))\nthis : Fintype \u2191(Set.singleton a * S ^ n)\n\u22a2 {a} * S ^ n = S ^ (n + 1)", "state_after": "case h\u2082.refine'_1\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nG : Type u_5\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Fintype G\nS : Set G\ninst\u271d : (k : \u2115) \u2192 DecidablePred fun x => x \u2208 S ^ k\nhG : 0 < Fintype.card G\nhS : \u00acS = \u2205\na : G\nha : a \u2208 S\nkey :\n  \u2200 (a : G) (s t : Set G) [inst : Fintype \u2191s] [inst_1 : Fintype \u2191t],\n    (\u2200 (b : G), b \u2208 s \u2192 a * b \u2208 t) \u2192 Fintype.card \u2191s \u2264 Fintype.card \u2191t\nmono : Monotone fun n => Fintype.card \u2191(S ^ n)\nn : \u2115\nh : Fintype.card \u2191(S ^ n) = Fintype.card \u2191(S ^ (n + 1))\nthis : Fintype \u2191(Set.singleton a * S ^ n)\n\u22a2 {a} * S ^ n \u2286 S ^ (n + 1)\n\ncase h\u2082.refine'_2\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nG : Type u_5\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Fintype G\nS : Set G\ninst\u271d : (k : \u2115) \u2192 DecidablePred fun x => x \u2208 S ^ k\nhG : 0 < Fintype.card G\nhS : \u00acS = \u2205\na : G\nha : a \u2208 S\nkey :\n  \u2200 (a : G) (s t : Set G) [inst : Fintype \u2191s] [inst_1 : Fintype \u2191t],\n    (\u2200 (b : G), b \u2208 s \u2192 a * b \u2208 t) \u2192 Fintype.card \u2191s \u2264 Fintype.card \u2191t\nmono : Monotone fun n => Fintype.card \u2191(S ^ n)\nn : \u2115\nh : Fintype.card \u2191(S ^ n) = Fintype.card \u2191(S ^ (n + 1))\nthis : Fintype \u2191(Set.singleton a * S ^ n)\n\u22a2 Fintype.card \u2191(S ^ n) \u2264 Fintype.card \u2191({a} * S ^ n)"}, {"tactic": "classical!", "annotated_tactic": ["classical!", []], "state_before": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nG : Type u_5\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Fintype G\nS : Set G\ninst\u271d : (k : \u2115) \u2192 DecidablePred fun x => x \u2208 S ^ k\nhG : 0 < Fintype.card G\nhS : \u00acS = \u2205\na : G\nha : a \u2208 S\nkey :\n  \u2200 (a : G) (s t : Set G) [inst : Fintype \u2191s] [inst_1 : Fintype \u2191t],\n    (\u2200 (b : G), b \u2208 s \u2192 a * b \u2208 t) \u2192 Fintype.card \u2191s \u2264 Fintype.card \u2191t\nmono : Monotone fun n => Fintype.card \u2191(S ^ n)\nn : \u2115\nh : Fintype.card \u2191(S ^ n) = Fintype.card \u2191(S ^ (n + 1))\n\u22a2 Fintype \u2191(Set.singleton a * S ^ n)", "state_after": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nG : Type u_5\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Fintype G\nS : Set G\ninst\u271d : (k : \u2115) \u2192 DecidablePred fun x => x \u2208 S ^ k\nhG : 0 < Fintype.card G\nhS : \u00acS = \u2205\na : G\nha : a \u2208 S\nkey :\n  \u2200 (a : G) (s t : Set G) [inst : Fintype \u2191s] [inst_1 : Fintype \u2191t],\n    (\u2200 (b : G), b \u2208 s \u2192 a * b \u2208 t) \u2192 Fintype.card \u2191s \u2264 Fintype.card \u2191t\nmono : Monotone fun n => Fintype.card \u2191(S ^ n)\nn : \u2115\nh : Fintype.card \u2191(S ^ n) = Fintype.card \u2191(S ^ (n + 1))\nem\u271d : (a : Prop) \u2192 Decidable a\n\u22a2 Fintype \u2191(Set.singleton a * S ^ n)"}, {"tactic": "apply fintypeMul", "annotated_tactic": ["apply <a>fintypeMul</a>", [{"full_name": "Set.fintypeMul", "def_path": "Mathlib/Data/Set/Pointwise/Finite.lean", "def_pos": [56, 5], "def_end_pos": [56, 15]}]], "state_before": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nG : Type u_5\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Fintype G\nS : Set G\ninst\u271d : (k : \u2115) \u2192 DecidablePred fun x => x \u2208 S ^ k\nhG : 0 < Fintype.card G\nhS : \u00acS = \u2205\na : G\nha : a \u2208 S\nkey :\n  \u2200 (a : G) (s t : Set G) [inst : Fintype \u2191s] [inst_1 : Fintype \u2191t],\n    (\u2200 (b : G), b \u2208 s \u2192 a * b \u2208 t) \u2192 Fintype.card \u2191s \u2264 Fintype.card \u2191t\nmono : Monotone fun n => Fintype.card \u2191(S ^ n)\nn : \u2115\nh : Fintype.card \u2191(S ^ n) = Fintype.card \u2191(S ^ (n + 1))\nem\u271d : (a : Prop) \u2192 Decidable a\n\u22a2 Fintype \u2191(Set.singleton a * S ^ n)", "state_after": "no goals"}, {"tactic": "exact mul_subset_mul (Set.singleton_subset_iff.mpr ha) Set.Subset.rfl", "annotated_tactic": ["exact <a>mul_subset_mul</a> (Set.singleton_subset_iff.mpr ha) <a>Set.Subset.rfl</a>", [{"full_name": "Set.mul_subset_mul", "def_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "def_pos": [416, 9], "def_end_pos": [416, 23]}, {"full_name": "Set.Subset.rfl", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [357, 9], "def_end_pos": [357, 19]}]], "state_before": "case h\u2082.refine'_1\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nG : Type u_5\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Fintype G\nS : Set G\ninst\u271d : (k : \u2115) \u2192 DecidablePred fun x => x \u2208 S ^ k\nhG : 0 < Fintype.card G\nhS : \u00acS = \u2205\na : G\nha : a \u2208 S\nkey :\n  \u2200 (a : G) (s t : Set G) [inst : Fintype \u2191s] [inst_1 : Fintype \u2191t],\n    (\u2200 (b : G), b \u2208 s \u2192 a * b \u2208 t) \u2192 Fintype.card \u2191s \u2264 Fintype.card \u2191t\nmono : Monotone fun n => Fintype.card \u2191(S ^ n)\nn : \u2115\nh : Fintype.card \u2191(S ^ n) = Fintype.card \u2191(S ^ (n + 1))\nthis : Fintype \u2191(Set.singleton a * S ^ n)\n\u22a2 {a} * S ^ n \u2286 S ^ (n + 1)", "state_after": "no goals"}, {"tactic": "convert key a (S ^ n) ({a} * S ^ n) fun b hb \u21a6 Set.mul_mem_mul (Set.mem_singleton a) hb", "annotated_tactic": ["convert key a (S ^ n) ({a} * S ^ n) fun b hb \u21a6 <a>Set.mul_mem_mul</a> (<a>Set.mem_singleton</a> a) hb", [{"full_name": "Set.mul_mem_mul", "def_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "def_pos": [342, 9], "def_end_pos": [342, 20]}, {"full_name": "Set.mem_singleton", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1289, 9], "def_end_pos": [1289, 22]}]], "state_before": "case h\u2082.refine'_2\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nG : Type u_5\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Fintype G\nS : Set G\ninst\u271d : (k : \u2115) \u2192 DecidablePred fun x => x \u2208 S ^ k\nhG : 0 < Fintype.card G\nhS : \u00acS = \u2205\na : G\nha : a \u2208 S\nkey :\n  \u2200 (a : G) (s t : Set G) [inst : Fintype \u2191s] [inst_1 : Fintype \u2191t],\n    (\u2200 (b : G), b \u2208 s \u2192 a * b \u2208 t) \u2192 Fintype.card \u2191s \u2264 Fintype.card \u2191t\nmono : Monotone fun n => Fintype.card \u2191(S ^ n)\nn : \u2115\nh : Fintype.card \u2191(S ^ n) = Fintype.card \u2191(S ^ (n + 1))\nthis : Fintype \u2191(Set.singleton a * S ^ n)\n\u22a2 Fintype.card \u2191(S ^ n) \u2264 Fintype.card \u2191({a} * S ^ n)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/StrongLaw.lean", "full_name": "ProbabilityTheory.tendsto_integral_truncation", "start": [203, 1], "end": [215, 39], "traced_tactics": [{"tactic": "refine' tendsto_integral_filter_of_dominated_convergence (fun x => abs (f x)) _ _ _ _", "annotated_tactic": ["refine' <a>tendsto_integral_filter_of_dominated_convergence</a> (fun x => <a>abs</a> (f x)) _ _ _ _", [{"full_name": "MeasureTheory.tendsto_integral_filter_of_dominated_convergence", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1036, 9], "def_end_pos": [1036, 57]}, {"full_name": "Abs.abs", "def_path": "Mathlib/Algebra/Abs.lean", "def_pos": [41, 3], "def_end_pos": [41, 6]}]], "state_before": "\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d f : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u22a2 Tendsto (fun A => \u222b (x : \u03b1), truncation f A x \u2202\u03bc) atTop (\ud835\udcdd (\u222b (x : \u03b1), f x \u2202\u03bc))", "state_after": "case refine'_1\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d f : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u22a2 \u2200\u1da0 (n : \u211d) in atTop, AEStronglyMeasurable (fun x => truncation f n x) \u03bc\n\ncase refine'_2\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d f : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u22a2 \u2200\u1da0 (n : \u211d) in atTop, \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016truncation f n a\u2016 \u2264 (fun x => |f x|) a\n\ncase refine'_3\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d f : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u22a2 Integrable fun x => |f x|\n\ncase refine'_4\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d f : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => truncation f n a) atTop (\ud835\udcdd (f a))"}, {"tactic": "exact eventually_of_forall fun A => hf.aestronglyMeasurable.truncation", "annotated_tactic": ["exact <a>eventually_of_forall</a> fun A => hf.aestronglyMeasurable.truncation", [{"full_name": "Filter.eventually_of_forall", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1111, 9], "def_end_pos": [1111, 29]}]], "state_before": "case refine'_1\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d f : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u22a2 \u2200\u1da0 (n : \u211d) in atTop, AEStronglyMeasurable (fun x => truncation f n x) \u03bc", "state_after": "no goals"}, {"tactic": "apply eventually_of_forall fun A => ?_", "annotated_tactic": ["apply <a>eventually_of_forall</a> fun A => ?_", [{"full_name": "Filter.eventually_of_forall", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1111, 9], "def_end_pos": [1111, 29]}]], "state_before": "case refine'_2\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d f : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u22a2 \u2200\u1da0 (n : \u211d) in atTop, \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016truncation f n a\u2016 \u2264 (fun x => |f x|) a", "state_after": "\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d f : \u03b1 \u2192 \u211d\nhf : Integrable f\nA : \u211d\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016truncation f A a\u2016 \u2264 (fun x => |f x|) a"}, {"tactic": "apply eventually_of_forall fun x => ?_", "annotated_tactic": ["apply <a>eventually_of_forall</a> fun x => ?_", [{"full_name": "Filter.eventually_of_forall", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1111, 9], "def_end_pos": [1111, 29]}]], "state_before": "\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d f : \u03b1 \u2192 \u211d\nhf : Integrable f\nA : \u211d\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016truncation f A a\u2016 \u2264 (fun x => |f x|) a", "state_after": "\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d f : \u03b1 \u2192 \u211d\nhf : Integrable f\nA : \u211d\nx : \u03b1\n\u22a2 \u2016truncation f A x\u2016 \u2264 (fun x => |f x|) x"}, {"tactic": "rw [Real.norm_eq_abs]", "annotated_tactic": ["rw [<a>Real.norm_eq_abs</a>]", [{"full_name": "Real.norm_eq_abs", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [1761, 9], "def_end_pos": [1761, 20]}]], "state_before": "\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d f : \u03b1 \u2192 \u211d\nhf : Integrable f\nA : \u211d\nx : \u03b1\n\u22a2 \u2016truncation f A x\u2016 \u2264 (fun x => |f x|) x", "state_after": "\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d f : \u03b1 \u2192 \u211d\nhf : Integrable f\nA : \u211d\nx : \u03b1\n\u22a2 |truncation f A x| \u2264 (fun x => |f x|) x"}, {"tactic": "exact abs_truncation_le_abs_self _ _ _", "annotated_tactic": ["exact <a>abs_truncation_le_abs_self</a> _ _ _", [{"full_name": "ProbabilityTheory.abs_truncation_le_abs_self", "def_path": "Mathlib/Probability/StrongLaw.lean", "def_pos": [101, 9], "def_end_pos": [101, 35]}]], "state_before": "\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d f : \u03b1 \u2192 \u211d\nhf : Integrable f\nA : \u211d\nx : \u03b1\n\u22a2 |truncation f A x| \u2264 (fun x => |f x|) x", "state_after": "no goals"}, {"tactic": "apply hf.abs", "annotated_tactic": ["apply hf.abs", []], "state_before": "case refine'_3\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d f : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u22a2 Integrable fun x => |f x|", "state_after": "no goals"}, {"tactic": "apply eventually_of_forall fun x => ?_", "annotated_tactic": ["apply <a>eventually_of_forall</a> fun x => ?_", [{"full_name": "Filter.eventually_of_forall", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1111, 9], "def_end_pos": [1111, 29]}]], "state_before": "case refine'_4\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d f : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => truncation f n a) atTop (\ud835\udcdd (f a))", "state_after": "\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d f : \u03b1 \u2192 \u211d\nhf : Integrable f\nx : \u03b1\n\u22a2 Tendsto (fun n => truncation f n x) atTop (\ud835\udcdd (f x))"}, {"tactic": "apply tendsto_const_nhds.congr' _", "annotated_tactic": ["apply tendsto_const_nhds.congr' _", []], "state_before": "\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d f : \u03b1 \u2192 \u211d\nhf : Integrable f\nx : \u03b1\n\u22a2 Tendsto (fun n => truncation f n x) atTop (\ud835\udcdd (f x))", "state_after": "\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d f : \u03b1 \u2192 \u211d\nhf : Integrable f\nx : \u03b1\n\u22a2 (fun x_1 => f x) =\u1da0[atTop] fun n => truncation f n x"}, {"tactic": "filter_upwards [Ioi_mem_atTop (abs (f x))] with A hA", "annotated_tactic": ["filter_upwards [<a>Ioi_mem_atTop</a> (<a>abs</a> (f x))] with A hA", [{"full_name": "Filter.Ioi_mem_atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [61, 9], "def_end_pos": [61, 22]}, {"full_name": "Abs.abs", "def_path": "Mathlib/Algebra/Abs.lean", "def_pos": [41, 3], "def_end_pos": [41, 6]}]], "state_before": "\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d f : \u03b1 \u2192 \u211d\nhf : Integrable f\nx : \u03b1\n\u22a2 (fun x_1 => f x) =\u1da0[atTop] fun n => truncation f n x", "state_after": "case h\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d f : \u03b1 \u2192 \u211d\nhf : Integrable f\nx : \u03b1\nA : \u211d\nhA : A \u2208 Set.Ioi |f x|\n\u22a2 f x = truncation f A x"}, {"tactic": "exact (truncation_eq_self hA).symm", "annotated_tactic": ["exact (<a>truncation_eq_self</a> hA).<a>symm</a>", [{"full_name": "ProbabilityTheory.truncation_eq_self", "def_path": "Mathlib/Probability/StrongLaw.lean", "def_pos": [108, 9], "def_end_pos": [108, 27]}, {"full_name": "Eq.symm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [310, 9], "def_end_pos": [310, 16]}]], "state_before": "case h\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d f : \u03b1 \u2192 \u211d\nhf : Integrable f\nx : \u03b1\nA : \u211d\nhA : A \u2208 Set.Ioi |f x|\n\u22a2 f x = truncation f A x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Martingale/Upcrossing.lean", "full_name": "MeasureTheory.upperCrossingTime_eq_of_bound_le", "start": [329, 1], "end": [332, 90], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Nat/Gcd.lean", "full_name": "Nat.gcd_gcd_self_left_right", "start": [168, 9], "end": [169, 42], "traced_tactics": [{"tactic": "rw [gcd_comm, gcd_gcd_self_right_right]", "annotated_tactic": ["rw [<a>gcd_comm</a>, <a>gcd_gcd_self_right_right</a>]", [{"full_name": "Nat.gcd_comm", "def_path": "lake-packages/std/Std/Data/Nat/Gcd.lean", "def_pos": [59, 9], "def_end_pos": [59, 17]}, {"full_name": "Nat.gcd_gcd_self_right_right", "def_path": "lake-packages/std/Std/Data/Nat/Gcd.lean", "def_pos": [165, 17], "def_end_pos": [165, 41]}]], "state_before": "m n : Nat\n\u22a2 gcd (gcd n m) m = gcd n m", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "full_name": "MeasureTheory.snorm_map_measure", "start": [920, 1], "end": [929, 6], "traced_tactics": [{"tactic": "by_cases hp_zero : p = 0", "annotated_tactic": ["by_cases hp_zero : p = 0", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\n\u03b2 : Type u_5\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 E\nhg : AEStronglyMeasurable g (Measure.map f \u03bc)\nhf : AEMeasurable f\n\u22a2 snorm g p (Measure.map f \u03bc) = snorm (g \u2218 f) p \u03bc", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\n\u03b2 : Type u_5\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 E\nhg : AEStronglyMeasurable g (Measure.map f \u03bc)\nhf : AEMeasurable f\nhp_zero : p = 0\n\u22a2 snorm g p (Measure.map f \u03bc) = snorm (g \u2218 f) p \u03bc\n\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\n\u03b2 : Type u_5\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 E\nhg : AEStronglyMeasurable g (Measure.map f \u03bc)\nhf : AEMeasurable f\nhp_zero : \u00acp = 0\n\u22a2 snorm g p (Measure.map f \u03bc) = snorm (g \u2218 f) p \u03bc"}, {"tactic": "by_cases hp_top : p = \u221e", "annotated_tactic": ["by_cases hp_top : p = \u221e", []], "state_before": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\n\u03b2 : Type u_5\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 E\nhg : AEStronglyMeasurable g (Measure.map f \u03bc)\nhf : AEMeasurable f\nhp_zero : \u00acp = 0\n\u22a2 snorm g p (Measure.map f \u03bc) = snorm (g \u2218 f) p \u03bc", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\n\u03b2 : Type u_5\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 E\nhg : AEStronglyMeasurable g (Measure.map f \u03bc)\nhf : AEMeasurable f\nhp_zero : \u00acp = 0\nhp_top : p = \u22a4\n\u22a2 snorm g p (Measure.map f \u03bc) = snorm (g \u2218 f) p \u03bc\n\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\n\u03b2 : Type u_5\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 E\nhg : AEStronglyMeasurable g (Measure.map f \u03bc)\nhf : AEMeasurable f\nhp_zero : \u00acp = 0\nhp_top : \u00acp = \u22a4\n\u22a2 snorm g p (Measure.map f \u03bc) = snorm (g \u2218 f) p \u03bc"}, {"tactic": "simp_rw [snorm_eq_lintegral_rpow_nnnorm hp_zero hp_top]", "annotated_tactic": ["simp_rw [<a>snorm_eq_lintegral_rpow_nnnorm</a> hp_zero hp_top]", [{"full_name": "MeasureTheory.snorm_eq_lintegral_rpow_nnnorm", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [92, 9], "def_end_pos": [92, 39]}]], "state_before": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\n\u03b2 : Type u_5\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 E\nhg : AEStronglyMeasurable g (Measure.map f \u03bc)\nhf : AEMeasurable f\nhp_zero : \u00acp = 0\nhp_top : \u00acp = \u22a4\n\u22a2 snorm g p (Measure.map f \u03bc) = snorm (g \u2218 f) p \u03bc", "state_after": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\n\u03b2 : Type u_5\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 E\nhg : AEStronglyMeasurable g (Measure.map f \u03bc)\nhf : AEMeasurable f\nhp_zero : \u00acp = 0\nhp_top : \u00acp = \u22a4\n\u22a2 (\u222b\u207b (x : \u03b2), \u2191\u2016g x\u2016\u208a ^ ENNReal.toReal p \u2202Measure.map f \u03bc) ^ (1 / ENNReal.toReal p) =\n    (\u222b\u207b (x : \u03b1), \u2191\u2016(g \u2218 f) x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) ^ (1 / ENNReal.toReal p)"}, {"tactic": "rw [lintegral_map' (hg.ennnorm.pow_const p.toReal) hf]", "annotated_tactic": ["rw [<a>lintegral_map'</a> (hg.ennnorm.pow_const p.toReal) hf]", [{"full_name": "MeasureTheory.lintegral_map'", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [1289, 9], "def_end_pos": [1289, 23]}]], "state_before": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\n\u03b2 : Type u_5\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 E\nhg : AEStronglyMeasurable g (Measure.map f \u03bc)\nhf : AEMeasurable f\nhp_zero : \u00acp = 0\nhp_top : \u00acp = \u22a4\n\u22a2 (\u222b\u207b (x : \u03b2), \u2191\u2016g x\u2016\u208a ^ ENNReal.toReal p \u2202Measure.map f \u03bc) ^ (1 / ENNReal.toReal p) =\n    (\u222b\u207b (x : \u03b1), \u2191\u2016(g \u2218 f) x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) ^ (1 / ENNReal.toReal p)", "state_after": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\n\u03b2 : Type u_5\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 E\nhg : AEStronglyMeasurable g (Measure.map f \u03bc)\nhf : AEMeasurable f\nhp_zero : \u00acp = 0\nhp_top : \u00acp = \u22a4\n\u22a2 (\u222b\u207b (a : \u03b1), \u2191\u2016g (f a)\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) ^ (1 / ENNReal.toReal p) =\n    (\u222b\u207b (x : \u03b1), \u2191\u2016(g \u2218 f) x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) ^ (1 / ENNReal.toReal p)"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\n\u03b2 : Type u_5\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 E\nhg : AEStronglyMeasurable g (Measure.map f \u03bc)\nhf : AEMeasurable f\nhp_zero : \u00acp = 0\nhp_top : \u00acp = \u22a4\n\u22a2 (\u222b\u207b (a : \u03b1), \u2191\u2016g (f a)\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) ^ (1 / ENNReal.toReal p) =\n    (\u222b\u207b (x : \u03b1), \u2191\u2016(g \u2218 f) x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) ^ (1 / ENNReal.toReal p)", "state_after": "no goals"}, {"tactic": "simp only [hp_zero, snorm_exponent_zero]", "annotated_tactic": ["simp only [hp_zero, <a>snorm_exponent_zero</a>]", [{"full_name": "MeasureTheory.snorm_exponent_zero", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [176, 9], "def_end_pos": [176, 28]}]], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\n\u03b2 : Type u_5\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 E\nhg : AEStronglyMeasurable g (Measure.map f \u03bc)\nhf : AEMeasurable f\nhp_zero : p = 0\n\u22a2 snorm g p (Measure.map f \u03bc) = snorm (g \u2218 f) p \u03bc", "state_after": "no goals"}, {"tactic": "simp_rw [hp_top, snorm_exponent_top]", "annotated_tactic": ["simp_rw [hp_top, <a>snorm_exponent_top</a>]", [{"full_name": "MeasureTheory.snorm_exponent_top", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [103, 9], "def_end_pos": [103, 27]}]], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\n\u03b2 : Type u_5\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 E\nhg : AEStronglyMeasurable g (Measure.map f \u03bc)\nhf : AEMeasurable f\nhp_zero : \u00acp = 0\nhp_top : p = \u22a4\n\u22a2 snorm g p (Measure.map f \u03bc) = snorm (g \u2218 f) p \u03bc", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\n\u03b2 : Type u_5\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 E\nhg : AEStronglyMeasurable g (Measure.map f \u03bc)\nhf : AEMeasurable f\nhp_zero : \u00acp = 0\nhp_top : p = \u22a4\n\u22a2 snormEssSup g (Measure.map f \u03bc) = snormEssSup (g \u2218 f) \u03bc"}, {"tactic": "exact snormEssSup_map_measure hg hf", "annotated_tactic": ["exact <a>snormEssSup_map_measure</a> hg hf", [{"full_name": "MeasureTheory.snormEssSup_map_measure", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [915, 9], "def_end_pos": [915, 32]}]], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\n\u03b2 : Type u_5\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 E\nhg : AEStronglyMeasurable g (Measure.map f \u03bc)\nhf : AEMeasurable f\nhp_zero : \u00acp = 0\nhp_top : p = \u22a4\n\u22a2 snormEssSup g (Measure.map f \u03bc) = snormEssSup (g \u2218 f) \u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/QPF/Multivariate/Basic.lean", "full_name": "MvQPF.suppPreservation_iff_liftpPreservation", "start": [267, 1], "end": [279, 38], "traced_tactics": [{"tactic": "constructor <;> intro h", "annotated_tactic": ["constructor <;> intro h", []], "state_before": "n : \u2115\nF : TypeVec.{u} n \u2192 Type u_1\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u22a2 SuppPreservation \u2194 LiftPPreservation", "state_after": "case mp\nn : \u2115\nF : TypeVec.{u} n \u2192 Type u_1\ninst\u271d : MvFunctor F\nq : MvQPF F\nh : SuppPreservation\n\u22a2 LiftPPreservation\n\ncase mpr\nn : \u2115\nF : TypeVec.{u} n \u2192 Type u_1\ninst\u271d : MvFunctor F\nq : MvQPF F\nh : LiftPPreservation\n\u22a2 SuppPreservation"}, {"tactic": "rintro \u03b1 p \u27e8a, f\u27e9", "annotated_tactic": ["rintro \u03b1 p \u27e8a, f\u27e9", []], "state_before": "case mp\nn : \u2115\nF : TypeVec.{u} n \u2192 Type u_1\ninst\u271d : MvFunctor F\nq : MvQPF F\nh : SuppPreservation\n\u22a2 LiftPPreservation", "state_after": "case mp.mk\nn : \u2115\nF : TypeVec.{u} n \u2192 Type u_1\ninst\u271d : MvFunctor F\nq : MvQPF F\nh : SuppPreservation\n\u03b1 : TypeVec.{u} n\np : \u2983i : Fin2 n\u2984 \u2192 \u03b1 i \u2192 Prop\na : (P F).A\nf : MvPFunctor.B (P F) a \u27f9 \u03b1\n\u22a2 LiftP p (abs { fst := a, snd := f }) \u2194 LiftP p { fst := a, snd := f }"}, {"tactic": "have h' := h", "annotated_tactic": ["have h' := h", []], "state_before": "case mp.mk\nn : \u2115\nF : TypeVec.{u} n \u2192 Type u_1\ninst\u271d : MvFunctor F\nq : MvQPF F\nh : SuppPreservation\n\u03b1 : TypeVec.{u} n\np : \u2983i : Fin2 n\u2984 \u2192 \u03b1 i \u2192 Prop\na : (P F).A\nf : MvPFunctor.B (P F) a \u27f9 \u03b1\n\u22a2 LiftP p (abs { fst := a, snd := f }) \u2194 LiftP p { fst := a, snd := f }", "state_after": "case mp.mk\nn : \u2115\nF : TypeVec.{u} n \u2192 Type u_1\ninst\u271d : MvFunctor F\nq : MvQPF F\nh : SuppPreservation\n\u03b1 : TypeVec.{u} n\np : \u2983i : Fin2 n\u2984 \u2192 \u03b1 i \u2192 Prop\na : (P F).A\nf : MvPFunctor.B (P F) a \u27f9 \u03b1\nh' : SuppPreservation\n\u22a2 LiftP p (abs { fst := a, snd := f }) \u2194 LiftP p { fst := a, snd := f }"}, {"tactic": "rw [suppPreservation_iff_isUniform] at h'", "annotated_tactic": ["rw [<a>suppPreservation_iff_isUniform</a>] at h'", [{"full_name": "MvQPF.suppPreservation_iff_isUniform", "def_path": "Mathlib/Data/QPF/Multivariate/Basic.lean", "def_pos": [258, 9], "def_end_pos": [258, 39]}]], "state_before": "case mp.mk\nn : \u2115\nF : TypeVec.{u} n \u2192 Type u_1\ninst\u271d : MvFunctor F\nq : MvQPF F\nh : SuppPreservation\n\u03b1 : TypeVec.{u} n\np : \u2983i : Fin2 n\u2984 \u2192 \u03b1 i \u2192 Prop\na : (P F).A\nf : MvPFunctor.B (P F) a \u27f9 \u03b1\nh' : SuppPreservation\n\u22a2 LiftP p (abs { fst := a, snd := f }) \u2194 LiftP p { fst := a, snd := f }", "state_after": "case mp.mk\nn : \u2115\nF : TypeVec.{u} n \u2192 Type u_1\ninst\u271d : MvFunctor F\nq : MvQPF F\nh : SuppPreservation\n\u03b1 : TypeVec.{u} n\np : \u2983i : Fin2 n\u2984 \u2192 \u03b1 i \u2192 Prop\na : (P F).A\nf : MvPFunctor.B (P F) a \u27f9 \u03b1\nh' : IsUniform\n\u22a2 LiftP p (abs { fst := a, snd := f }) \u2194 LiftP p { fst := a, snd := f }"}, {"tactic": "dsimp only [SuppPreservation, supp] at h", "annotated_tactic": ["dsimp only [<a>SuppPreservation</a>, <a>supp</a>] at h", [{"full_name": "MvQPF.SuppPreservation", "def_path": "Mathlib/Data/QPF/Multivariate/Basic.lean", "def_pos": [223, 5], "def_end_pos": [223, 21]}, {"full_name": "MvFunctor.supp", "def_path": "Mathlib/Control/Functor/Multivariate.lean", "def_pos": [57, 5], "def_end_pos": [57, 9]}]], "state_before": "case mp.mk\nn : \u2115\nF : TypeVec.{u} n \u2192 Type u_1\ninst\u271d : MvFunctor F\nq : MvQPF F\nh : SuppPreservation\n\u03b1 : TypeVec.{u} n\np : \u2983i : Fin2 n\u2984 \u2192 \u03b1 i \u2192 Prop\na : (P F).A\nf : MvPFunctor.B (P F) a \u27f9 \u03b1\nh' : IsUniform\n\u22a2 LiftP p (abs { fst := a, snd := f }) \u2194 LiftP p { fst := a, snd := f }", "state_after": "case mp.mk\nn : \u2115\nF : TypeVec.{u} n \u2192 Type u_1\ninst\u271d : MvFunctor F\nq : MvQPF F\nh :\n  \u2200 \u2983\u03b1 : TypeVec.{u} n\u2984 (x : \u2191(P F) \u03b1),\n    (fun i => {y | \u2200 \u2983P : (i : Fin2 n) \u2192 \u03b1 i \u2192 Prop\u2984, LiftP P (abs x) \u2192 P i y}) = fun i =>\n      {y | \u2200 \u2983P : (i : Fin2 n) \u2192 \u03b1 i \u2192 Prop\u2984, LiftP P x \u2192 P i y}\n\u03b1 : TypeVec.{u} n\np : \u2983i : Fin2 n\u2984 \u2192 \u03b1 i \u2192 Prop\na : (P F).A\nf : MvPFunctor.B (P F) a \u27f9 \u03b1\nh' : IsUniform\n\u22a2 LiftP p (abs { fst := a, snd := f }) \u2194 LiftP p { fst := a, snd := f }"}, {"tactic": "simp only [liftP_iff_of_isUniform, supp_eq_of_isUniform, MvPFunctor.liftP_iff', h',\n  image_univ, mem_range, exists_imp]", "annotated_tactic": ["simp only [<a>liftP_iff_of_isUniform</a>, <a>supp_eq_of_isUniform</a>, <a>MvPFunctor.liftP_iff'</a>, h',\n      <a>image_univ</a>, <a>mem_range</a>, <a>exists_imp</a>]", [{"full_name": "MvQPF.liftP_iff_of_isUniform", "def_path": "Mathlib/Data/QPF/Multivariate/Basic.lean", "def_pos": [236, 9], "def_end_pos": [236, 31]}, {"full_name": "MvQPF.supp_eq_of_isUniform", "def_path": "Mathlib/Data/QPF/Multivariate/Basic.lean", "def_pos": [227, 9], "def_end_pos": [227, 29]}, {"full_name": "MvPFunctor.liftP_iff'", "def_path": "Mathlib/Data/PFunctor/Multivariate/Basic.lean", "def_pos": [173, 9], "def_end_pos": [173, 19]}, {"full_name": "Set.image_univ", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [718, 9], "def_end_pos": [718, 19]}, {"full_name": "Set.mem_range", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [673, 9], "def_end_pos": [673, 18]}, {"full_name": "exists_imp", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [367, 9], "def_end_pos": [367, 19]}]], "state_before": "case mp.mk\nn : \u2115\nF : TypeVec.{u} n \u2192 Type u_1\ninst\u271d : MvFunctor F\nq : MvQPF F\nh :\n  \u2200 \u2983\u03b1 : TypeVec.{u} n\u2984 (x : \u2191(P F) \u03b1),\n    (fun i => {y | \u2200 \u2983P : (i : Fin2 n) \u2192 \u03b1 i \u2192 Prop\u2984, LiftP P (abs x) \u2192 P i y}) = fun i =>\n      {y | \u2200 \u2983P : (i : Fin2 n) \u2192 \u03b1 i \u2192 Prop\u2984, LiftP P x \u2192 P i y}\n\u03b1 : TypeVec.{u} n\np : \u2983i : Fin2 n\u2984 \u2192 \u03b1 i \u2192 Prop\na : (P F).A\nf : MvPFunctor.B (P F) a \u27f9 \u03b1\nh' : IsUniform\n\u22a2 LiftP p (abs { fst := a, snd := f }) \u2194 LiftP p { fst := a, snd := f }", "state_after": "case mp.mk\nn : \u2115\nF : TypeVec.{u} n \u2192 Type u_1\ninst\u271d : MvFunctor F\nq : MvQPF F\nh :\n  \u2200 \u2983\u03b1 : TypeVec.{u} n\u2984 (x : \u2191(P F) \u03b1),\n    (fun i => {y | \u2200 \u2983P : (i : Fin2 n) \u2192 \u03b1 i \u2192 Prop\u2984, LiftP P (abs x) \u2192 P i y}) = fun i =>\n      {y | \u2200 \u2983P : (i : Fin2 n) \u2192 \u03b1 i \u2192 Prop\u2984, LiftP P x \u2192 P i y}\n\u03b1 : TypeVec.{u} n\np : \u2983i : Fin2 n\u2984 \u2192 \u03b1 i \u2192 Prop\na : (P F).A\nf : MvPFunctor.B (P F) a \u27f9 \u03b1\nh' : IsUniform\n\u22a2 (\u2200 (i : Fin2 n) (u : \u03b1 i) (x : MvPFunctor.B (P F) a i), f i x = u \u2192 p u) \u2194\n    \u2200 (i : Fin2 n) (x : MvPFunctor.B (P F) a i), p (f i x)"}, {"tactic": "constructor <;> intros <;> subst_vars <;> solve_by_elim", "annotated_tactic": ["constructor <;> intros <;> subst_vars <;> solve_by_elim", []], "state_before": "case mp.mk\nn : \u2115\nF : TypeVec.{u} n \u2192 Type u_1\ninst\u271d : MvFunctor F\nq : MvQPF F\nh :\n  \u2200 \u2983\u03b1 : TypeVec.{u} n\u2984 (x : \u2191(P F) \u03b1),\n    (fun i => {y | \u2200 \u2983P : (i : Fin2 n) \u2192 \u03b1 i \u2192 Prop\u2984, LiftP P (abs x) \u2192 P i y}) = fun i =>\n      {y | \u2200 \u2983P : (i : Fin2 n) \u2192 \u03b1 i \u2192 Prop\u2984, LiftP P x \u2192 P i y}\n\u03b1 : TypeVec.{u} n\np : \u2983i : Fin2 n\u2984 \u2192 \u03b1 i \u2192 Prop\na : (P F).A\nf : MvPFunctor.B (P F) a \u27f9 \u03b1\nh' : IsUniform\n\u22a2 (\u2200 (i : Fin2 n) (u : \u03b1 i) (x : MvPFunctor.B (P F) a i), f i x = u \u2192 p u) \u2194\n    \u2200 (i : Fin2 n) (x : MvPFunctor.B (P F) a i), p (f i x)", "state_after": "no goals"}, {"tactic": "rintro \u03b1 \u27e8a, f\u27e9", "annotated_tactic": ["rintro \u03b1 \u27e8a, f\u27e9", []], "state_before": "case mpr\nn : \u2115\nF : TypeVec.{u} n \u2192 Type u_1\ninst\u271d : MvFunctor F\nq : MvQPF F\nh : LiftPPreservation\n\u22a2 SuppPreservation", "state_after": "case mpr.mk\nn : \u2115\nF : TypeVec.{u} n \u2192 Type u_1\ninst\u271d : MvFunctor F\nq : MvQPF F\nh : LiftPPreservation\n\u03b1 : TypeVec.{u} n\na : (P F).A\nf : MvPFunctor.B (P F) a \u27f9 \u03b1\n\u22a2 supp (abs { fst := a, snd := f }) = supp { fst := a, snd := f }"}, {"tactic": "simp only [LiftPPreservation] at h", "annotated_tactic": ["simp only [<a>LiftPPreservation</a>] at h", [{"full_name": "MvQPF.LiftPPreservation", "def_path": "Mathlib/Data/QPF/Multivariate/Basic.lean", "def_pos": [218, 5], "def_end_pos": [218, 22]}]], "state_before": "case mpr.mk\nn : \u2115\nF : TypeVec.{u} n \u2192 Type u_1\ninst\u271d : MvFunctor F\nq : MvQPF F\nh : LiftPPreservation\n\u03b1 : TypeVec.{u} n\na : (P F).A\nf : MvPFunctor.B (P F) a \u27f9 \u03b1\n\u22a2 supp (abs { fst := a, snd := f }) = supp { fst := a, snd := f }", "state_after": "case mpr.mk\nn : \u2115\nF : TypeVec.{u} n \u2192 Type u_1\ninst\u271d : MvFunctor F\nq : MvQPF F\nh : \u2200 \u2983\u03b1 : TypeVec.{u} n\u2984 (p : \u2983i : Fin2 n\u2984 \u2192 \u03b1 i \u2192 Prop) (x : \u2191(P F) \u03b1), LiftP p (abs x) \u2194 LiftP p x\n\u03b1 : TypeVec.{u} n\na : (P F).A\nf : MvPFunctor.B (P F) a \u27f9 \u03b1\n\u22a2 supp (abs { fst := a, snd := f }) = supp { fst := a, snd := f }"}, {"tactic": "ext", "annotated_tactic": ["ext", []], "state_before": "case mpr.mk\nn : \u2115\nF : TypeVec.{u} n \u2192 Type u_1\ninst\u271d : MvFunctor F\nq : MvQPF F\nh : \u2200 \u2983\u03b1 : TypeVec.{u} n\u2984 (p : \u2983i : Fin2 n\u2984 \u2192 \u03b1 i \u2192 Prop) (x : \u2191(P F) \u03b1), LiftP p (abs x) \u2194 LiftP p x\n\u03b1 : TypeVec.{u} n\na : (P F).A\nf : MvPFunctor.B (P F) a \u27f9 \u03b1\n\u22a2 supp (abs { fst := a, snd := f }) = supp { fst := a, snd := f }", "state_after": "case mpr.mk.h.h\nn : \u2115\nF : TypeVec.{u} n \u2192 Type u_1\ninst\u271d : MvFunctor F\nq : MvQPF F\nh : \u2200 \u2983\u03b1 : TypeVec.{u} n\u2984 (p : \u2983i : Fin2 n\u2984 \u2192 \u03b1 i \u2192 Prop) (x : \u2191(P F) \u03b1), LiftP p (abs x) \u2194 LiftP p x\n\u03b1 : TypeVec.{u} n\na : (P F).A\nf : MvPFunctor.B (P F) a \u27f9 \u03b1\nx\u271d\u00b9 : Fin2 n\nx\u271d : \u03b1 x\u271d\u00b9\n\u22a2 x\u271d \u2208 supp (abs { fst := a, snd := f }) x\u271d\u00b9 \u2194 x\u271d \u2208 supp { fst := a, snd := f } x\u271d\u00b9"}, {"tactic": "simp only [supp, h, mem_setOf_eq]", "annotated_tactic": ["simp only [<a>supp</a>, h, <a>mem_setOf_eq</a>]", [{"full_name": "MvFunctor.supp", "def_path": "Mathlib/Control/Functor/Multivariate.lean", "def_pos": [57, 5], "def_end_pos": [57, 9]}, {"full_name": "Set.mem_setOf_eq", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [256, 29], "def_end_pos": [256, 41]}]], "state_before": "case mpr.mk.h.h\nn : \u2115\nF : TypeVec.{u} n \u2192 Type u_1\ninst\u271d : MvFunctor F\nq : MvQPF F\nh : \u2200 \u2983\u03b1 : TypeVec.{u} n\u2984 (p : \u2983i : Fin2 n\u2984 \u2192 \u03b1 i \u2192 Prop) (x : \u2191(P F) \u03b1), LiftP p (abs x) \u2194 LiftP p x\n\u03b1 : TypeVec.{u} n\na : (P F).A\nf : MvPFunctor.B (P F) a \u27f9 \u03b1\nx\u271d\u00b9 : Fin2 n\nx\u271d : \u03b1 x\u271d\u00b9\n\u22a2 x\u271d \u2208 supp (abs { fst := a, snd := f }) x\u271d\u00b9 \u2194 x\u271d \u2208 supp { fst := a, snd := f } x\u271d\u00b9", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "full_name": "String.Iterator.Valid.toEnd", "start": [668, 1], "end": [668, 83], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Setoid/Partition.lean", "full_name": "IndexedPartition.proj_eq_iff", "start": [415, 1], "end": [416, 18], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/List/Basic.lean", "full_name": "List.filterMap_eq_filterMapTR", "start": [103, 10], "end": [108, 24], "traced_tactics": [{"tactic": "funext \u03b1 \u03b2 f l", "annotated_tactic": ["funext \u03b1 \u03b2 f l", []], "state_before": "\u22a2 @filterMap = @filterMapTR", "state_after": "case h.h.h.h\n\u03b1 : Type u_2\n\u03b2 : Type u_1\nf : \u03b1 \u2192 Option \u03b2\nl : List \u03b1\n\u22a2 filterMap f l = filterMapTR f l"}, {"tactic": "exact (go l #[]).symm", "annotated_tactic": ["exact (go l #[]).<a>symm</a>", [{"full_name": "Eq.symm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [310, 9], "def_end_pos": [310, 16]}]], "state_before": "case h.h.h.h\n\u03b1 : Type u_2\n\u03b2 : Type u_1\nf : \u03b1 \u2192 Option \u03b2\nl : List \u03b1\n\u22a2 filterMap f l = filterMapTR f l", "state_after": "no goals"}, {"tactic": "simp [filterMapTR.go, filterMap]", "annotated_tactic": ["simp [<a>filterMapTR.go</a>, <a>filterMap</a>]", [{"full_name": "List.filterMapTR.go", "def_path": "lake-packages/std/Std/Data/List/Basic.lean", "def_pos": [97, 17], "def_end_pos": [97, 19]}, {"full_name": "List.filterMap", "def_path": "lake-packages/lean4/src/lean/Init/Data/List/Basic.lean", "def_pos": [214, 19], "def_end_pos": [214, 28]}]], "state_before": "\u03b1 : Type u_2\n\u03b2 : Type u_1\nf : \u03b1 \u2192 Option \u03b2\nl : List \u03b1\nacc : Array \u03b2\n\u22a2 filterMapTR.go f [] acc = acc.data ++ filterMap f []", "state_after": "no goals"}, {"tactic": "simp [filterMapTR.go, filterMap, go as]", "annotated_tactic": ["simp [<a>filterMapTR.go</a>, <a>filterMap</a>, go as]", [{"full_name": "List.filterMapTR.go", "def_path": "lake-packages/std/Std/Data/List/Basic.lean", "def_pos": [97, 17], "def_end_pos": [97, 19]}, {"full_name": "List.filterMap", "def_path": "lake-packages/lean4/src/lean/Init/Data/List/Basic.lean", "def_pos": [214, 19], "def_end_pos": [214, 28]}]], "state_before": "\u03b1 : Type u_2\n\u03b2 : Type u_1\nf : \u03b1 \u2192 Option \u03b2\nl : List \u03b1\na : \u03b1\nas : List \u03b1\nacc : Array \u03b2\n\u22a2 filterMapTR.go f (a :: as) acc = acc.data ++ filterMap f (a :: as)", "state_after": "\u03b1 : Type u_2\n\u03b2 : Type u_1\nf : \u03b1 \u2192 Option \u03b2\nl : List \u03b1\na : \u03b1\nas : List \u03b1\nacc : Array \u03b2\n\u22a2 (match f a with\n    | none => acc.data ++ filterMap f as\n    | some b => acc.data ++ b :: filterMap f as) =\n    acc.data ++\n      match f a with\n      | none => filterMap f as\n      | some b => b :: filterMap f as"}, {"tactic": "split <;> simp [*]", "annotated_tactic": ["split <;> simp [*]", []], "state_before": "\u03b1 : Type u_2\n\u03b2 : Type u_1\nf : \u03b1 \u2192 Option \u03b2\nl : List \u03b1\na : \u03b1\nas : List \u03b1\nacc : Array \u03b2\n\u22a2 (match f a with\n    | none => acc.data ++ filterMap f as\n    | some b => acc.data ++ b :: filterMap f as) =\n    acc.data ++\n      match f a with\n      | none => filterMap f as\n      | some b => b :: filterMap f as", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/MutuallySingular.lean", "full_name": "MeasureTheory.Measure.MutuallySingular.add_right", "start": [114, 1], "end": [115, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/RBMap/Alter.lean", "full_name": "Std.RBNode.Path.zoom_fill", "start": [54, 1], "end": [55, 35], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Independence/Basic.lean", "full_name": "ProbabilityTheory.IndepFun.ae_eq", "start": [567, 1], "end": [571, 18], "traced_tactics": [{"tactic": "refine kernel.IndepFun.ae_eq hfg ?_ ?_ <;>\n  simp only [ae_dirac_eq, Filter.eventually_pure, kernel.const_apply]", "annotated_tactic": ["refine <a>kernel.IndepFun.ae_eq</a> hfg ?_ ?_ <;>\n    simp only [<a>ae_dirac_eq</a>, <a>Filter.eventually_pure</a>, <a>kernel.const_apply</a>]", [{"full_name": "ProbabilityTheory.kernel.IndepFun.ae_eq", "def_path": "Mathlib/Probability/Independence/Kernel.lean", "def_pos": [715, 9], "def_end_pos": [715, 23]}, {"full_name": "MeasureTheory.ae_dirac_eq", "def_path": "Mathlib/MeasureTheory/Measure/Dirac.lean", "def_pos": [117, 9], "def_end_pos": [117, 20]}, {"full_name": "Filter.eventually_pure", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2055, 9], "def_end_pos": [2055, 24]}, {"full_name": "ProbabilityTheory.kernel.const_apply", "def_path": "Mathlib/Probability/Kernel/Basic.lean", "def_pos": [445, 9], "def_end_pos": [445, 20]}]], "state_before": "\u03a9 : Type u_1\n\u03b9 : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\nm\u03a9 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nf\u271d : \u03a9 \u2192 \u03b2\ng\u271d : \u03a9 \u2192 \u03b2'\nm\u03b2 : MeasurableSpace \u03b2\nf g f' g' : \u03a9 \u2192 \u03b2\nhfg : IndepFun f g\nhf : f =\u1d50[\u03bc] f'\nhg : g =\u1d50[\u03bc] g'\n\u22a2 IndepFun f' g'", "state_after": "case refine_1\n\u03a9 : Type u_1\n\u03b9 : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\nm\u03a9 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nf\u271d : \u03a9 \u2192 \u03b2\ng\u271d : \u03a9 \u2192 \u03b2'\nm\u03b2 : MeasurableSpace \u03b2\nf g f' g' : \u03a9 \u2192 \u03b2\nhfg : IndepFun f g\nhf : f =\u1d50[\u03bc] f'\nhg : g =\u1d50[\u03bc] g'\n\u22a2 f =\u1d50[\u03bc] f'\n\ncase refine_2\n\u03a9 : Type u_1\n\u03b9 : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\nm\u03a9 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nf\u271d : \u03a9 \u2192 \u03b2\ng\u271d : \u03a9 \u2192 \u03b2'\nm\u03b2 : MeasurableSpace \u03b2\nf g f' g' : \u03a9 \u2192 \u03b2\nhfg : IndepFun f g\nhf : f =\u1d50[\u03bc] f'\nhg : g =\u1d50[\u03bc] g'\n\u22a2 g =\u1d50[\u03bc] g'"}, {"tactic": "exacts [hf, hg]", "annotated_tactic": ["exacts [hf, hg]", []], "state_before": "case refine_1\n\u03a9 : Type u_1\n\u03b9 : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\nm\u03a9 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nf\u271d : \u03a9 \u2192 \u03b2\ng\u271d : \u03a9 \u2192 \u03b2'\nm\u03b2 : MeasurableSpace \u03b2\nf g f' g' : \u03a9 \u2192 \u03b2\nhfg : IndepFun f g\nhf : f =\u1d50[\u03bc] f'\nhg : g =\u1d50[\u03bc] g'\n\u22a2 f =\u1d50[\u03bc] f'\n\ncase refine_2\n\u03a9 : Type u_1\n\u03b9 : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\nm\u03a9 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nf\u271d : \u03a9 \u2192 \u03b2\ng\u271d : \u03a9 \u2192 \u03b2'\nm\u03b2 : MeasurableSpace \u03b2\nf g f' g' : \u03a9 \u2192 \u03b2\nhfg : IndepFun f g\nhf : f =\u1d50[\u03bc] f'\nhg : g =\u1d50[\u03bc] g'\n\u22a2 g =\u1d50[\u03bc] g'", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Num/Lemmas.lean", "full_name": "PosNum.cast_to_znum", "start": [1169, 1], "end": [1172, 74], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/LocallyIntegrable.lean", "full_name": "MeasureTheory.locallyIntegrableOn_const", "start": [274, 1], "end": [276, 52], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean", "full_name": "MeasureTheory.Measure.addHaar_smul", "start": [371, 1], "end": [384, 36], "traced_tactics": [{"tactic": "rcases ne_or_eq r 0 with (h | rfl)", "annotated_tactic": ["rcases <a>ne_or_eq</a> r 0 with (h | rfl)", [{"full_name": "ne_or_eq", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [212, 9], "def_end_pos": [212, 17]}]], "state_before": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nr : \u211d\ns : Set E\n\u22a2 \u2191\u2191\u03bc (r \u2022 s) = ENNReal.ofReal |r ^ finrank \u211d E| * \u2191\u2191\u03bc s", "state_after": "case inl\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nr : \u211d\ns : Set E\nh : r \u2260 0\n\u22a2 \u2191\u2191\u03bc (r \u2022 s) = ENNReal.ofReal |r ^ finrank \u211d E| * \u2191\u2191\u03bc s\n\ncase inr\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns : Set E\n\u22a2 \u2191\u2191\u03bc (0 \u2022 s) = ENNReal.ofReal |0 ^ finrank \u211d E| * \u2191\u2191\u03bc s"}, {"tactic": "rcases eq_empty_or_nonempty s with (rfl | hs)", "annotated_tactic": ["rcases <a>eq_empty_or_nonempty</a> s with (rfl | hs)", [{"full_name": "Set.eq_empty_or_nonempty", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [635, 9], "def_end_pos": [635, 29]}]], "state_before": "case inr\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns : Set E\n\u22a2 \u2191\u2191\u03bc (0 \u2022 s) = ENNReal.ofReal |0 ^ finrank \u211d E| * \u2191\u2191\u03bc s", "state_after": "case inr.inl\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u22a2 \u2191\u2191\u03bc (0 \u2022 \u2205) = ENNReal.ofReal |0 ^ finrank \u211d E| * \u2191\u2191\u03bc \u2205\n\ncase inr.inr\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns : Set E\nhs : Set.Nonempty s\n\u22a2 \u2191\u2191\u03bc (0 \u2022 s) = ENNReal.ofReal |0 ^ finrank \u211d E| * \u2191\u2191\u03bc s"}, {"tactic": "rw [zero_smul_set hs, \u2190 singleton_zero]", "annotated_tactic": ["rw [<a>zero_smul_set</a> hs, \u2190 <a>singleton_zero</a>]", [{"full_name": "Set.zero_smul_set", "def_path": "Mathlib/Data/Set/Pointwise/SMul.lean", "def_pos": [812, 9], "def_end_pos": [812, 22]}, {"full_name": "Set.singleton_zero", "def_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "def_pos": [93, 3], "def_end_pos": [93, 14]}]], "state_before": "case inr.inr\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns : Set E\nhs : Set.Nonempty s\n\u22a2 \u2191\u2191\u03bc (0 \u2022 s) = ENNReal.ofReal |0 ^ finrank \u211d E| * \u2191\u2191\u03bc s", "state_after": "case inr.inr\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns : Set E\nhs : Set.Nonempty s\n\u22a2 \u2191\u2191\u03bc {0} = ENNReal.ofReal |0 ^ finrank \u211d E| * \u2191\u2191\u03bc s"}, {"tactic": "by_cases h : finrank \u211d E = 0", "annotated_tactic": ["by_cases h : <a>finrank</a> \u211d E = 0", [{"full_name": "FiniteDimensional.finrank", "def_path": "Mathlib/LinearAlgebra/Finrank.lean", "def_pos": [58, 19], "def_end_pos": [58, 26]}]], "state_before": "case inr.inr\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns : Set E\nhs : Set.Nonempty s\n\u22a2 \u2191\u2191\u03bc {0} = ENNReal.ofReal |0 ^ finrank \u211d E| * \u2191\u2191\u03bc s", "state_after": "case pos\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns : Set E\nhs : Set.Nonempty s\nh : finrank \u211d E = 0\n\u22a2 \u2191\u2191\u03bc {0} = ENNReal.ofReal |0 ^ finrank \u211d E| * \u2191\u2191\u03bc s\n\ncase neg\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns : Set E\nhs : Set.Nonempty s\nh : \u00acfinrank \u211d E = 0\n\u22a2 \u2191\u2191\u03bc {0} = ENNReal.ofReal |0 ^ finrank \u211d E| * \u2191\u2191\u03bc s"}, {"tactic": "rw [\u2190 preimage_smul_inv\u2080 h, addHaar_preimage_smul \u03bc (inv_ne_zero h), inv_pow, inv_inv]", "annotated_tactic": ["rw [\u2190 <a>preimage_smul_inv\u2080</a> h, <a>addHaar_preimage_smul</a> \u03bc (<a>inv_ne_zero</a> h), <a>inv_pow</a>, <a>inv_inv</a>]", [{"full_name": "Set.preimage_smul_inv\u2080", "def_path": "Mathlib/Data/Set/Pointwise/SMul.lean", "def_pos": [1031, 9], "def_end_pos": [1031, 27]}, {"full_name": "MeasureTheory.Measure.addHaar_preimage_smul", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean", "def_pos": [359, 9], "def_end_pos": [359, 30]}, {"full_name": "inv_ne_zero", "def_path": "Mathlib/Algebra/GroupWithZero/NeZero.lean", "def_pos": [49, 9], "def_end_pos": [49, 20]}, {"full_name": "inv_pow", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [317, 9], "def_end_pos": [317, 16]}, {"full_name": "inv_inv", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [800, 9], "def_end_pos": [800, 16]}]], "state_before": "case inl\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nr : \u211d\ns : Set E\nh : r \u2260 0\n\u22a2 \u2191\u2191\u03bc (r \u2022 s) = ENNReal.ofReal |r ^ finrank \u211d E| * \u2191\u2191\u03bc s", "state_after": "no goals"}, {"tactic": "simp only [measure_empty, mul_zero, smul_set_empty]", "annotated_tactic": ["simp only [<a>measure_empty</a>, <a>mul_zero</a>, <a>smul_set_empty</a>]", [{"full_name": "MeasureTheory.measure_empty", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [185, 9], "def_end_pos": [185, 22]}, {"full_name": "MulZeroClass.mul_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [38, 3], "def_end_pos": [38, 11]}, {"full_name": "Set.smul_set_empty", "def_path": "Mathlib/Data/Set/Pointwise/SMul.lean", "def_pos": [330, 9], "def_end_pos": [330, 23]}]], "state_before": "case inr.inl\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u22a2 \u2191\u2191\u03bc (0 \u2022 \u2205) = ENNReal.ofReal |0 ^ finrank \u211d E| * \u2191\u2191\u03bc \u2205", "state_after": "no goals"}, {"tactic": "haveI : Subsingleton E := finrank_zero_iff.1 h", "annotated_tactic": ["haveI : <a>Subsingleton</a> E := <a>finrank_zero_iff</a>.1 h", [{"full_name": "Subsingleton", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [867, 7], "def_end_pos": [867, 19]}, {"full_name": "FiniteDimensional.finrank_zero_iff", "def_path": "Mathlib/LinearAlgebra/FiniteDimensional.lean", "def_pos": [340, 9], "def_end_pos": [340, 25]}]], "state_before": "case pos\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns : Set E\nhs : Set.Nonempty s\nh : finrank \u211d E = 0\n\u22a2 \u2191\u2191\u03bc {0} = ENNReal.ofReal |0 ^ finrank \u211d E| * \u2191\u2191\u03bc s", "state_after": "case pos\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns : Set E\nhs : Set.Nonempty s\nh : finrank \u211d E = 0\nthis : Subsingleton E\n\u22a2 \u2191\u2191\u03bc {0} = ENNReal.ofReal |0 ^ finrank \u211d E| * \u2191\u2191\u03bc s"}, {"tactic": "simp only [h, one_mul, ENNReal.ofReal_one, abs_one, Subsingleton.eq_univ_of_nonempty hs,\n  pow_zero, Subsingleton.eq_univ_of_nonempty (singleton_nonempty (0 : E))]", "annotated_tactic": ["simp only [h, <a>one_mul</a>, <a>ENNReal.ofReal_one</a>, <a>abs_one</a>, <a>Subsingleton.eq_univ_of_nonempty</a> hs,\n      <a>pow_zero</a>, <a>Subsingleton.eq_univ_of_nonempty</a> (<a>singleton_nonempty</a> (0 : E))]", [{"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [464, 9], "def_end_pos": [464, 16]}, {"full_name": "ENNReal.ofReal_one", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [248, 17], "def_end_pos": [248, 27]}, {"full_name": "abs_one", "def_path": "Mathlib/Algebra/Order/Ring/Abs.lean", "def_pos": [24, 9], "def_end_pos": [24, 16]}, {"full_name": "Subsingleton.eq_univ_of_nonempty", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [2863, 9], "def_end_pos": [2863, 28]}, {"full_name": "pow_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [639, 9], "def_end_pos": [639, 17]}, {"full_name": "Subsingleton.eq_univ_of_nonempty", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [2863, 9], "def_end_pos": [2863, 28]}, {"full_name": "Set.singleton_nonempty", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1315, 9], "def_end_pos": [1315, 27]}]], "state_before": "case pos\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns : Set E\nhs : Set.Nonempty s\nh : finrank \u211d E = 0\nthis : Subsingleton E\n\u22a2 \u2191\u2191\u03bc {0} = ENNReal.ofReal |0 ^ finrank \u211d E| * \u2191\u2191\u03bc s", "state_after": "no goals"}, {"tactic": "haveI : Nontrivial E := nontrivial_of_finrank_pos (bot_lt_iff_ne_bot.2 h)", "annotated_tactic": ["haveI : <a>Nontrivial</a> E := <a>nontrivial_of_finrank_pos</a> (<a>bot_lt_iff_ne_bot</a>.2 h)", [{"full_name": "Nontrivial", "def_path": "Mathlib/Logic/Nontrivial/Defs.lean", "def_pos": [29, 7], "def_end_pos": [29, 17]}, {"full_name": "FiniteDimensional.nontrivial_of_finrank_pos", "def_path": "Mathlib/LinearAlgebra/Finrank.lean", "def_pos": [110, 9], "def_end_pos": [110, 34]}, {"full_name": "bot_lt_iff_ne_bot", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [371, 9], "def_end_pos": [371, 26]}]], "state_before": "case neg\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns : Set E\nhs : Set.Nonempty s\nh : \u00acfinrank \u211d E = 0\n\u22a2 \u2191\u2191\u03bc {0} = ENNReal.ofReal |0 ^ finrank \u211d E| * \u2191\u2191\u03bc s", "state_after": "case neg\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns : Set E\nhs : Set.Nonempty s\nh : \u00acfinrank \u211d E = 0\nthis : Nontrivial E\n\u22a2 \u2191\u2191\u03bc {0} = ENNReal.ofReal |0 ^ finrank \u211d E| * \u2191\u2191\u03bc s"}, {"tactic": "simp only [h, zero_mul, ENNReal.ofReal_zero, abs_zero, Ne.def, not_false_iff,\n  zero_pow', measure_singleton]", "annotated_tactic": ["simp only [h, <a>zero_mul</a>, <a>ENNReal.ofReal_zero</a>, <a>abs_zero</a>, <a>Ne.def</a>, <a>not_false_iff</a>,\n      <a>zero_pow'</a>, <a>measure_singleton</a>]", [{"full_name": "MulZeroClass.zero_mul", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [36, 3], "def_end_pos": [36, 11]}, {"full_name": "ENNReal.ofReal_zero", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [245, 17], "def_end_pos": [245, 28]}, {"full_name": "abs_zero", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [128, 9], "def_end_pos": [128, 17]}, {"full_name": "Ne.def", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [59, 9], "def_end_pos": [59, 15]}, {"full_name": "not_false_iff", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [82, 9], "def_end_pos": [82, 22]}, {"full_name": "zero_pow'", "def_path": "Mathlib/Algebra/GroupPower/Ring.lean", "def_pos": [37, 9], "def_end_pos": [37, 18]}, {"full_name": "MeasureTheory.NoAtoms.measure_singleton", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3103, 3], "def_end_pos": [3103, 20]}]], "state_before": "case neg\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns : Set E\nhs : Set.Nonempty s\nh : \u00acfinrank \u211d E = 0\nthis : Nontrivial E\n\u22a2 \u2191\u2191\u03bc {0} = ENNReal.ofReal |0 ^ finrank \u211d E| * \u2191\u2191\u03bc s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Logic.lean", "full_name": "exists\u2084_congr", "start": [405, 1], "end": [407, 45], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Martingale/Basic.lean", "full_name": "MeasureTheory.martingale_of_condexp_sub_eq_zero_nat", "start": [464, 1], "end": [472, 56], "traced_tactics": [{"tactic": "refine' martingale_iff.2 \u27e8supermartingale_of_condexp_sub_nonneg_nat hadp hint fun i => _,\n  submartingale_of_condexp_sub_nonneg_nat hadp hint fun i => (hf i).symm.le\u27e9", "annotated_tactic": ["refine' <a>martingale_iff</a>.2 \u27e8<a>supermartingale_of_condexp_sub_nonneg_nat</a> hadp hint fun i => _,\n    <a>submartingale_of_condexp_sub_nonneg_nat</a> hadp hint fun i => (hf i).symm.le\u27e9", [{"full_name": "MeasureTheory.martingale_iff", "def_path": "Mathlib/Probability/Martingale/Basic.lean", "def_pos": [145, 9], "def_end_pos": [145, 23]}, {"full_name": "MeasureTheory.supermartingale_of_condexp_sub_nonneg_nat", "def_path": "Mathlib/Probability/Martingale/Basic.lean", "def_pos": [456, 9], "def_end_pos": [456, 50]}, {"full_name": "MeasureTheory.submartingale_of_condexp_sub_nonneg_nat", "def_path": "Mathlib/Probability/Martingale/Basic.lean", "def_pos": [448, 9], "def_end_pos": [448, 48]}]], "state_before": "\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2074 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\n\ud835\udca2 : Filtration \u2115 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nhadp : Adapted \ud835\udca2 f\nhint : \u2200 (i : \u2115), Integrable (f i)\nhf : \u2200 (i : \u2115), \u03bc[f (i + 1) - f i|\u2191\ud835\udca2 i] =\u1d50[\u03bc] 0\n\u22a2 Martingale f \ud835\udca2 \u03bc", "state_after": "\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2074 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\n\ud835\udca2 : Filtration \u2115 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nhadp : Adapted \ud835\udca2 f\nhint : \u2200 (i : \u2115), Integrable (f i)\nhf : \u2200 (i : \u2115), \u03bc[f (i + 1) - f i|\u2191\ud835\udca2 i] =\u1d50[\u03bc] 0\ni : \u2115\n\u22a2 0 \u2264\u1d50[\u03bc] \u03bc[f i - f (i + 1)|\u2191\ud835\udca2 i]"}, {"tactic": "rw [\u2190 neg_sub]", "annotated_tactic": ["rw [\u2190 <a>neg_sub</a>]", [{"full_name": "neg_sub", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [400, 3], "def_end_pos": [400, 14]}]], "state_before": "\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2074 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\n\ud835\udca2 : Filtration \u2115 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nhadp : Adapted \ud835\udca2 f\nhint : \u2200 (i : \u2115), Integrable (f i)\nhf : \u2200 (i : \u2115), \u03bc[f (i + 1) - f i|\u2191\ud835\udca2 i] =\u1d50[\u03bc] 0\ni : \u2115\n\u22a2 0 \u2264\u1d50[\u03bc] \u03bc[f i - f (i + 1)|\u2191\ud835\udca2 i]", "state_after": "\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2074 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\n\ud835\udca2 : Filtration \u2115 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nhadp : Adapted \ud835\udca2 f\nhint : \u2200 (i : \u2115), Integrable (f i)\nhf : \u2200 (i : \u2115), \u03bc[f (i + 1) - f i|\u2191\ud835\udca2 i] =\u1d50[\u03bc] 0\ni : \u2115\n\u22a2 0 \u2264\u1d50[\u03bc] \u03bc[-(f (i + 1) - f i)|\u2191\ud835\udca2 i]"}, {"tactic": "refine' (EventuallyEq.trans _ (condexp_neg _).symm).le", "annotated_tactic": ["refine' (<a>EventuallyEq.trans</a> _ (<a>condexp_neg</a> _).<a>symm</a>).<a>le</a>", [{"full_name": "Filter.EventuallyEq.trans", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1503, 9], "def_end_pos": [1503, 27]}, {"full_name": "MeasureTheory.condexp_neg", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean", "def_pos": [316, 9], "def_end_pos": [316, 20]}, {"full_name": "Filter.EventuallyEq.symm", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1498, 9], "def_end_pos": [1498, 26]}, {"full_name": "Filter.EventuallyEq.le", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1661, 9], "def_end_pos": [1661, 24]}]], "state_before": "\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2074 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\n\ud835\udca2 : Filtration \u2115 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nhadp : Adapted \ud835\udca2 f\nhint : \u2200 (i : \u2115), Integrable (f i)\nhf : \u2200 (i : \u2115), \u03bc[f (i + 1) - f i|\u2191\ud835\udca2 i] =\u1d50[\u03bc] 0\ni : \u2115\n\u22a2 0 \u2264\u1d50[\u03bc] \u03bc[-(f (i + 1) - f i)|\u2191\ud835\udca2 i]", "state_after": "\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2074 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\n\ud835\udca2 : Filtration \u2115 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nhadp : Adapted \ud835\udca2 f\nhint : \u2200 (i : \u2115), Integrable (f i)\nhf : \u2200 (i : \u2115), \u03bc[f (i + 1) - f i|\u2191\ud835\udca2 i] =\u1d50[\u03bc] 0\ni : \u2115\n\u22a2 0 =\u1d50[\u03bc] -\u03bc[f (i + 1) - f i|\u2191\ud835\udca2 i]"}, {"tactic": "filter_upwards [hf i] with x hx", "annotated_tactic": ["filter_upwards [hf i] with x hx", []], "state_before": "\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2074 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\n\ud835\udca2 : Filtration \u2115 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nhadp : Adapted \ud835\udca2 f\nhint : \u2200 (i : \u2115), Integrable (f i)\nhf : \u2200 (i : \u2115), \u03bc[f (i + 1) - f i|\u2191\ud835\udca2 i] =\u1d50[\u03bc] 0\ni : \u2115\n\u22a2 0 =\u1d50[\u03bc] -\u03bc[f (i + 1) - f i|\u2191\ud835\udca2 i]", "state_after": "case h\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2074 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\n\ud835\udca2 : Filtration \u2115 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nhadp : Adapted \ud835\udca2 f\nhint : \u2200 (i : \u2115), Integrable (f i)\nhf : \u2200 (i : \u2115), \u03bc[f (i + 1) - f i|\u2191\ud835\udca2 i] =\u1d50[\u03bc] 0\ni : \u2115\nx : \u03a9\nhx : (\u03bc[f (i + 1) - f i|\u2191\ud835\udca2 i]) x = OfNat.ofNat 0 x\n\u22a2 OfNat.ofNat 0 x = (-\u03bc[f (i + 1) - f i|\u2191\ud835\udca2 i]) x"}, {"tactic": "simpa only [Pi.zero_apply, Pi.neg_apply, zero_eq_neg]", "annotated_tactic": ["simpa only [<a>Pi.zero_apply</a>, <a>Pi.neg_apply</a>, <a>zero_eq_neg</a>]", [{"full_name": "Pi.zero_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [46, 3], "def_end_pos": [46, 14]}, {"full_name": "Pi.neg_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [170, 3], "def_end_pos": [170, 14]}, {"full_name": "zero_eq_neg", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [428, 3], "def_end_pos": [428, 14]}]], "state_before": "case h\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2074 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\n\ud835\udca2 : Filtration \u2115 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nhadp : Adapted \ud835\udca2 f\nhint : \u2200 (i : \u2115), Integrable (f i)\nhf : \u2200 (i : \u2115), \u03bc[f (i + 1) - f i|\u2191\ud835\udca2 i] =\u1d50[\u03bc] 0\ni : \u2115\nx : \u03a9\nhx : (\u03bc[f (i + 1) - f i|\u2191\ud835\udca2 i]) x = OfNat.ofNat 0 x\n\u22a2 OfNat.ofNat 0 x = (-\u03bc[f (i + 1) - f i|\u2191\ud835\udca2 i]) x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Martingale/OptionalStopping.lean", "full_name": "MeasureTheory.maximal_ineq", "start": [141, 1], "end": [218, 90], "traced_tactics": [{"tactic": "rwa [hadd, ENNReal.add_le_add_iff_right ENNReal.ofReal_ne_top] at this", "annotated_tactic": ["rwa [hadd, <a>ENNReal.add_le_add_iff_right</a> <a>ENNReal.ofReal_ne_top</a>] at this", [{"full_name": "ENNReal.add_le_add_iff_right", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [805, 19], "def_end_pos": [805, 39]}, {"full_name": "ENNReal.ofReal_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [311, 17], "def_end_pos": [311, 30]}]], "state_before": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\nhnonneg : 0 \u2264 f\n\u03b5 : \u211d\u22650\nn : \u2115\nthis :\n  \u03b5 \u2022 \u2191\u2191\u03bc {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9} +\n      ENNReal.ofReal\n        (\u222b (\u03c9 : \u03a9) in {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5},\n          f n \u03c9 \u2202\u03bc) \u2264\n    ENNReal.ofReal (\u222b (x : \u03a9), f n x \u2202\u03bc)\nhadd :\n  ENNReal.ofReal (\u222b (\u03c9 : \u03a9), f n \u03c9 \u2202\u03bc) =\n    ENNReal.ofReal\n        (\u222b (\u03c9 : \u03a9) in {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9}, f n \u03c9 \u2202\u03bc) +\n      ENNReal.ofReal\n        (\u222b (\u03c9 : \u03a9) in {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5}, f n \u03c9 \u2202\u03bc)\n\u22a2 \u03b5 \u2022 \u2191\u2191\u03bc {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9} \u2264\n    ENNReal.ofReal\n      (\u222b (\u03c9 : \u03a9) in {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9}, f n \u03c9 \u2202\u03bc)", "state_after": "no goals"}, {"tactic": "rw [\u2190 ENNReal.ofReal_add, \u2190 integral_union]", "annotated_tactic": ["rw [\u2190 <a>ENNReal.ofReal_add</a>, \u2190 <a>integral_union</a>]", [{"full_name": "ENNReal.ofReal_add", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2025, 9], "def_end_pos": [2025, 19]}, {"full_name": "MeasureTheory.integral_union", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [102, 9], "def_end_pos": [102, 23]}]], "state_before": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\nhnonneg : 0 \u2264 f\n\u03b5 : \u211d\u22650\nn : \u2115\nthis :\n  \u03b5 \u2022 \u2191\u2191\u03bc {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9} +\n      ENNReal.ofReal\n        (\u222b (\u03c9 : \u03a9) in {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5},\n          f n \u03c9 \u2202\u03bc) \u2264\n    ENNReal.ofReal (\u222b (x : \u03a9), f n x \u2202\u03bc)\n\u22a2 ENNReal.ofReal (\u222b (\u03c9 : \u03a9), f n \u03c9 \u2202\u03bc) =\n    ENNReal.ofReal\n        (\u222b (\u03c9 : \u03a9) in {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9}, f n \u03c9 \u2202\u03bc) +\n      ENNReal.ofReal\n        (\u222b (\u03c9 : \u03a9) in {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5}, f n \u03c9 \u2202\u03bc)", "state_after": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\nhnonneg : 0 \u2264 f\n\u03b5 : \u211d\u22650\nn : \u2115\nthis :\n  \u03b5 \u2022 \u2191\u2191\u03bc {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9} +\n      ENNReal.ofReal\n        (\u222b (\u03c9 : \u03a9) in {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5},\n          f n \u03c9 \u2202\u03bc) \u2264\n    ENNReal.ofReal (\u222b (x : \u03a9), f n x \u2202\u03bc)\n\u22a2 ENNReal.ofReal (\u222b (\u03c9 : \u03a9), f n \u03c9 \u2202\u03bc) =\n    ENNReal.ofReal\n      (\u222b (x : \u03a9) in\n        {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9} \u222a\n          {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5},\n        f n x \u2202\u03bc)\n\ncase hst\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\nhnonneg : 0 \u2264 f\n\u03b5 : \u211d\u22650\nn : \u2115\nthis :\n  \u03b5 \u2022 \u2191\u2191\u03bc {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9} +\n      ENNReal.ofReal\n        (\u222b (\u03c9 : \u03a9) in {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5},\n          f n \u03c9 \u2202\u03bc) \u2264\n    ENNReal.ofReal (\u222b (x : \u03a9), f n x \u2202\u03bc)\n\u22a2 Disjoint {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9}\n    {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5}\n\ncase ht\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\nhnonneg : 0 \u2264 f\n\u03b5 : \u211d\u22650\nn : \u2115\nthis :\n  \u03b5 \u2022 \u2191\u2191\u03bc {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9} +\n      ENNReal.ofReal\n        (\u222b (\u03c9 : \u03a9) in {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5},\n          f n \u03c9 \u2202\u03bc) \u2264\n    ENNReal.ofReal (\u222b (x : \u03a9), f n x \u2202\u03bc)\n\u22a2 MeasurableSet {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5}\n\ncase hfs\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\nhnonneg : 0 \u2264 f\n\u03b5 : \u211d\u22650\nn : \u2115\nthis :\n  \u03b5 \u2022 \u2191\u2191\u03bc {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9} +\n      ENNReal.ofReal\n        (\u222b (\u03c9 : \u03a9) in {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5},\n          f n \u03c9 \u2202\u03bc) \u2264\n    ENNReal.ofReal (\u222b (x : \u03a9), f n x \u2202\u03bc)\n\u22a2 IntegrableOn (fun \u03c9 => f n \u03c9) {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9}\n\ncase hft\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\nhnonneg : 0 \u2264 f\n\u03b5 : \u211d\u22650\nn : \u2115\nthis :\n  \u03b5 \u2022 \u2191\u2191\u03bc {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9} +\n      ENNReal.ofReal\n        (\u222b (\u03c9 : \u03a9) in {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5},\n          f n \u03c9 \u2202\u03bc) \u2264\n    ENNReal.ofReal (\u222b (x : \u03a9), f n x \u2202\u03bc)\n\u22a2 IntegrableOn (fun \u03c9 => f n \u03c9) {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5}\n\ncase hp\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\nhnonneg : 0 \u2264 f\n\u03b5 : \u211d\u22650\nn : \u2115\nthis :\n  \u03b5 \u2022 \u2191\u2191\u03bc {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9} +\n      ENNReal.ofReal\n        (\u222b (\u03c9 : \u03a9) in {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5},\n          f n \u03c9 \u2202\u03bc) \u2264\n    ENNReal.ofReal (\u222b (x : \u03a9), f n x \u2202\u03bc)\n\u22a2 0 \u2264 \u222b (\u03c9 : \u03a9) in {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9}, f n \u03c9 \u2202\u03bc\n\ncase hq\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\nhnonneg : 0 \u2264 f\n\u03b5 : \u211d\u22650\nn : \u2115\nthis :\n  \u03b5 \u2022 \u2191\u2191\u03bc {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9} +\n      ENNReal.ofReal\n        (\u222b (\u03c9 : \u03a9) in {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5},\n          f n \u03c9 \u2202\u03bc) \u2264\n    ENNReal.ofReal (\u222b (x : \u03a9), f n x \u2202\u03bc)\n\u22a2 0 \u2264 \u222b (\u03c9 : \u03a9) in {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5}, f n \u03c9 \u2202\u03bc"}, {"tactic": "exacts [(hsub.integrable _).integrableOn, (hsub.integrable _).integrableOn,\n  integral_nonneg (hnonneg _), integral_nonneg (hnonneg _)]", "annotated_tactic": ["exacts [(hsub.integrable _).<a>integrableOn</a>, (hsub.integrable _).<a>integrableOn</a>,\n        <a>integral_nonneg</a> (hnonneg _), <a>integral_nonneg</a> (hnonneg _)]", [{"full_name": "MeasureTheory.Integrable.integrableOn", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [163, 9], "def_end_pos": [163, 32]}, {"full_name": "MeasureTheory.Integrable.integrableOn", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [163, 9], "def_end_pos": [163, 32]}, {"full_name": "MeasureTheory.integral_nonneg", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1251, 9], "def_end_pos": [1251, 24]}, {"full_name": "MeasureTheory.integral_nonneg", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1251, 9], "def_end_pos": [1251, 24]}]], "state_before": "case hfs\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\nhnonneg : 0 \u2264 f\n\u03b5 : \u211d\u22650\nn : \u2115\nthis :\n  \u03b5 \u2022 \u2191\u2191\u03bc {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9} +\n      ENNReal.ofReal\n        (\u222b (\u03c9 : \u03a9) in {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5},\n          f n \u03c9 \u2202\u03bc) \u2264\n    ENNReal.ofReal (\u222b (x : \u03a9), f n x \u2202\u03bc)\n\u22a2 IntegrableOn (fun \u03c9 => f n \u03c9) {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9}\n\ncase hft\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\nhnonneg : 0 \u2264 f\n\u03b5 : \u211d\u22650\nn : \u2115\nthis :\n  \u03b5 \u2022 \u2191\u2191\u03bc {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9} +\n      ENNReal.ofReal\n        (\u222b (\u03c9 : \u03a9) in {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5},\n          f n \u03c9 \u2202\u03bc) \u2264\n    ENNReal.ofReal (\u222b (x : \u03a9), f n x \u2202\u03bc)\n\u22a2 IntegrableOn (fun \u03c9 => f n \u03c9) {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5}\n\ncase hp\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\nhnonneg : 0 \u2264 f\n\u03b5 : \u211d\u22650\nn : \u2115\nthis :\n  \u03b5 \u2022 \u2191\u2191\u03bc {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9} +\n      ENNReal.ofReal\n        (\u222b (\u03c9 : \u03a9) in {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5},\n          f n \u03c9 \u2202\u03bc) \u2264\n    ENNReal.ofReal (\u222b (x : \u03a9), f n x \u2202\u03bc)\n\u22a2 0 \u2264 \u222b (\u03c9 : \u03a9) in {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9}, f n \u03c9 \u2202\u03bc\n\ncase hq\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\nhnonneg : 0 \u2264 f\n\u03b5 : \u211d\u22650\nn : \u2115\nthis :\n  \u03b5 \u2022 \u2191\u2191\u03bc {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9} +\n      ENNReal.ofReal\n        (\u222b (\u03c9 : \u03a9) in {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5},\n          f n \u03c9 \u2202\u03bc) \u2264\n    ENNReal.ofReal (\u222b (x : \u03a9), f n x \u2202\u03bc)\n\u22a2 0 \u2264 \u222b (\u03c9 : \u03a9) in {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5}, f n \u03c9 \u2202\u03bc", "state_after": "no goals"}, {"tactic": "rw [\u2190 integral_univ]", "annotated_tactic": ["rw [\u2190 <a>integral_univ</a>]", [{"full_name": "MeasureTheory.integral_univ", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [152, 9], "def_end_pos": [152, 22]}]], "state_before": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\nhnonneg : 0 \u2264 f\n\u03b5 : \u211d\u22650\nn : \u2115\nthis :\n  \u03b5 \u2022 \u2191\u2191\u03bc {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9} +\n      ENNReal.ofReal\n        (\u222b (\u03c9 : \u03a9) in {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5},\n          f n \u03c9 \u2202\u03bc) \u2264\n    ENNReal.ofReal (\u222b (x : \u03a9), f n x \u2202\u03bc)\n\u22a2 ENNReal.ofReal (\u222b (\u03c9 : \u03a9), f n \u03c9 \u2202\u03bc) =\n    ENNReal.ofReal\n      (\u222b (x : \u03a9) in\n        {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9} \u222a\n          {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5},\n        f n x \u2202\u03bc)", "state_after": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\nhnonneg : 0 \u2264 f\n\u03b5 : \u211d\u22650\nn : \u2115\nthis :\n  \u03b5 \u2022 \u2191\u2191\u03bc {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9} +\n      ENNReal.ofReal\n        (\u222b (\u03c9 : \u03a9) in {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5},\n          f n \u03c9 \u2202\u03bc) \u2264\n    ENNReal.ofReal (\u222b (x : \u03a9), f n x \u2202\u03bc)\n\u22a2 ENNReal.ofReal (\u222b (x : \u03a9) in Set.univ, f n x \u2202\u03bc) =\n    ENNReal.ofReal\n      (\u222b (x : \u03a9) in\n        {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9} \u222a\n          {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5},\n        f n x \u2202\u03bc)"}, {"tactic": "convert rfl", "annotated_tactic": ["convert <a>rfl</a>", [{"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\nhnonneg : 0 \u2264 f\n\u03b5 : \u211d\u22650\nn : \u2115\nthis :\n  \u03b5 \u2022 \u2191\u2191\u03bc {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9} +\n      ENNReal.ofReal\n        (\u222b (\u03c9 : \u03a9) in {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5},\n          f n \u03c9 \u2202\u03bc) \u2264\n    ENNReal.ofReal (\u222b (x : \u03a9), f n x \u2202\u03bc)\n\u22a2 ENNReal.ofReal (\u222b (x : \u03a9) in Set.univ, f n x \u2202\u03bc) =\n    ENNReal.ofReal\n      (\u222b (x : \u03a9) in\n        {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9} \u222a\n          {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5},\n        f n x \u2202\u03bc)", "state_after": "case h.e'_3.h.e'_1.h.e'_6.h.e'_4\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\nhnonneg : 0 \u2264 f\n\u03b5 : \u211d\u22650\nn : \u2115\nthis :\n  \u03b5 \u2022 \u2191\u2191\u03bc {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9} +\n      ENNReal.ofReal\n        (\u222b (\u03c9 : \u03a9) in {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5},\n          f n \u03c9 \u2202\u03bc) \u2264\n    ENNReal.ofReal (\u222b (x : \u03a9), f n x \u2202\u03bc)\n\u22a2 {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9} \u222a\n      {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5} =\n    Set.univ"}, {"tactic": "ext \u03c9", "annotated_tactic": ["ext \u03c9", []], "state_before": "case h.e'_3.h.e'_1.h.e'_6.h.e'_4\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\nhnonneg : 0 \u2264 f\n\u03b5 : \u211d\u22650\nn : \u2115\nthis :\n  \u03b5 \u2022 \u2191\u2191\u03bc {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9} +\n      ENNReal.ofReal\n        (\u222b (\u03c9 : \u03a9) in {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5},\n          f n \u03c9 \u2202\u03bc) \u2264\n    ENNReal.ofReal (\u222b (x : \u03a9), f n x \u2202\u03bc)\n\u22a2 {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9} \u222a\n      {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5} =\n    Set.univ", "state_after": "case h.e'_3.h.e'_1.h.e'_6.h.e'_4.h\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\nhnonneg : 0 \u2264 f\n\u03b5 : \u211d\u22650\nn : \u2115\nthis :\n  \u03b5 \u2022 \u2191\u2191\u03bc {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9} +\n      ENNReal.ofReal\n        (\u222b (\u03c9 : \u03a9) in {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5},\n          f n \u03c9 \u2202\u03bc) \u2264\n    ENNReal.ofReal (\u222b (x : \u03a9), f n x \u2202\u03bc)\n\u03c9 : \u03a9\n\u22a2 \u03c9 \u2208\n      {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9} \u222a\n        {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5} \u2194\n    \u03c9 \u2208 Set.univ"}, {"tactic": "change (\u03b5 : \u211d) \u2264 _ \u2228 _ < (\u03b5 : \u211d) \u2194 _", "annotated_tactic": ["change (\u03b5 : \u211d) \u2264 _ \u2228 _ < (\u03b5 : \u211d) \u2194 _", []], "state_before": "case h.e'_3.h.e'_1.h.e'_6.h.e'_4.h\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\nhnonneg : 0 \u2264 f\n\u03b5 : \u211d\u22650\nn : \u2115\nthis :\n  \u03b5 \u2022 \u2191\u2191\u03bc {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9} +\n      ENNReal.ofReal\n        (\u222b (\u03c9 : \u03a9) in {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5},\n          f n \u03c9 \u2202\u03bc) \u2264\n    ENNReal.ofReal (\u222b (x : \u03a9), f n x \u2202\u03bc)\n\u03c9 : \u03a9\n\u22a2 \u03c9 \u2208\n      {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9} \u222a\n        {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5} \u2194\n    \u03c9 \u2208 Set.univ", "state_after": "case h.e'_3.h.e'_1.h.e'_6.h.e'_4.h\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\nhnonneg : 0 \u2264 f\n\u03b5 : \u211d\u22650\nn : \u2115\nthis :\n  \u03b5 \u2022 \u2191\u2191\u03bc {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9} +\n      ENNReal.ofReal\n        (\u222b (\u03c9 : \u03a9) in {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5},\n          f n \u03c9 \u2202\u03bc) \u2264\n    ENNReal.ofReal (\u222b (x : \u03a9), f n x \u2202\u03bc)\n\u03c9 : \u03a9\n\u22a2 (\u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) \u2228\n      (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5 \u2194\n    \u03c9 \u2208 Set.univ"}, {"tactic": "simp only [le_or_lt, Set.mem_univ]", "annotated_tactic": ["simp only [<a>le_or_lt</a>, <a>Set.mem_univ</a>]", [{"full_name": "le_or_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [340, 9], "def_end_pos": [340, 17]}, {"full_name": "Set.mem_univ", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [676, 9], "def_end_pos": [676, 17]}]], "state_before": "case h.e'_3.h.e'_1.h.e'_6.h.e'_4.h\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\nhnonneg : 0 \u2264 f\n\u03b5 : \u211d\u22650\nn : \u2115\nthis :\n  \u03b5 \u2022 \u2191\u2191\u03bc {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9} +\n      ENNReal.ofReal\n        (\u222b (\u03c9 : \u03a9) in {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5},\n          f n \u03c9 \u2202\u03bc) \u2264\n    ENNReal.ofReal (\u222b (x : \u03a9), f n x \u2202\u03bc)\n\u03c9 : \u03a9\n\u22a2 (\u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) \u2228\n      (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5 \u2194\n    \u03c9 \u2208 Set.univ", "state_after": "no goals"}, {"tactic": "rw [disjoint_iff_inf_le]", "annotated_tactic": ["rw [<a>disjoint_iff_inf_le</a>]", [{"full_name": "disjoint_iff_inf_le", "def_path": "Mathlib/Order/Disjoint.lean", "def_pos": [122, 9], "def_end_pos": [122, 28]}]], "state_before": "case hst\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\nhnonneg : 0 \u2264 f\n\u03b5 : \u211d\u22650\nn : \u2115\nthis :\n  \u03b5 \u2022 \u2191\u2191\u03bc {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9} +\n      ENNReal.ofReal\n        (\u222b (\u03c9 : \u03a9) in {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5},\n          f n \u03c9 \u2202\u03bc) \u2264\n    ENNReal.ofReal (\u222b (x : \u03a9), f n x \u2202\u03bc)\n\u22a2 Disjoint {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9}\n    {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5}", "state_after": "case hst\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\nhnonneg : 0 \u2264 f\n\u03b5 : \u211d\u22650\nn : \u2115\nthis :\n  \u03b5 \u2022 \u2191\u2191\u03bc {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9} +\n      ENNReal.ofReal\n        (\u222b (\u03c9 : \u03a9) in {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5},\n          f n \u03c9 \u2202\u03bc) \u2264\n    ENNReal.ofReal (\u222b (x : \u03a9), f n x \u2202\u03bc)\n\u22a2 {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9} \u2293\n      {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5} \u2264\n    \u22a5"}, {"tactic": "rintro \u03c9 \u27e8h\u03c9\u2081, h\u03c9\u2082\u27e9", "annotated_tactic": ["rintro \u03c9 \u27e8h\u03c9\u2081, h\u03c9\u2082\u27e9", []], "state_before": "case hst\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\nhnonneg : 0 \u2264 f\n\u03b5 : \u211d\u22650\nn : \u2115\nthis :\n  \u03b5 \u2022 \u2191\u2191\u03bc {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9} +\n      ENNReal.ofReal\n        (\u222b (\u03c9 : \u03a9) in {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5},\n          f n \u03c9 \u2202\u03bc) \u2264\n    ENNReal.ofReal (\u222b (x : \u03a9), f n x \u2202\u03bc)\n\u22a2 {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9} \u2293\n      {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5} \u2264\n    \u22a5", "state_after": "case hst.intro\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\nhnonneg : 0 \u2264 f\n\u03b5 : \u211d\u22650\nn : \u2115\nthis :\n  \u03b5 \u2022 \u2191\u2191\u03bc {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9} +\n      ENNReal.ofReal\n        (\u222b (\u03c9 : \u03a9) in {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5},\n          f n \u03c9 \u2202\u03bc) \u2264\n    ENNReal.ofReal (\u222b (x : \u03a9), f n x \u2202\u03bc)\n\u03c9 : \u03a9\nh\u03c9\u2081 : \u03c9 \u2208 {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9}\nh\u03c9\u2082 : \u03c9 \u2208 {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5}\n\u22a2 \u03c9 \u2208 \u22a5"}, {"tactic": "change (\u03b5 : \u211d) \u2264 _ at h\u03c9\u2081", "annotated_tactic": ["change (\u03b5 : \u211d) \u2264 _ at h\u03c9\u2081", []], "state_before": "case hst.intro\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\nhnonneg : 0 \u2264 f\n\u03b5 : \u211d\u22650\nn : \u2115\nthis :\n  \u03b5 \u2022 \u2191\u2191\u03bc {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9} +\n      ENNReal.ofReal\n        (\u222b (\u03c9 : \u03a9) in {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5},\n          f n \u03c9 \u2202\u03bc) \u2264\n    ENNReal.ofReal (\u222b (x : \u03a9), f n x \u2202\u03bc)\n\u03c9 : \u03a9\nh\u03c9\u2081 : \u03c9 \u2208 {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9}\nh\u03c9\u2082 : \u03c9 \u2208 {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5}\n\u22a2 \u03c9 \u2208 \u22a5", "state_after": "case hst.intro\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\nhnonneg : 0 \u2264 f\n\u03b5 : \u211d\u22650\nn : \u2115\nthis :\n  \u03b5 \u2022 \u2191\u2191\u03bc {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9} +\n      ENNReal.ofReal\n        (\u222b (\u03c9 : \u03a9) in {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5},\n          f n \u03c9 \u2202\u03bc) \u2264\n    ENNReal.ofReal (\u222b (x : \u03a9), f n x \u2202\u03bc)\n\u03c9 : \u03a9\nh\u03c9\u2082 : \u03c9 \u2208 {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5}\nh\u03c9\u2081 : \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9\n\u22a2 \u03c9 \u2208 \u22a5"}, {"tactic": "change _ < (\u03b5 : \u211d) at h\u03c9\u2082", "annotated_tactic": ["change _ < (\u03b5 : \u211d) at h\u03c9\u2082", []], "state_before": "case hst.intro\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\nhnonneg : 0 \u2264 f\n\u03b5 : \u211d\u22650\nn : \u2115\nthis :\n  \u03b5 \u2022 \u2191\u2191\u03bc {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9} +\n      ENNReal.ofReal\n        (\u222b (\u03c9 : \u03a9) in {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5},\n          f n \u03c9 \u2202\u03bc) \u2264\n    ENNReal.ofReal (\u222b (x : \u03a9), f n x \u2202\u03bc)\n\u03c9 : \u03a9\nh\u03c9\u2082 : \u03c9 \u2208 {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5}\nh\u03c9\u2081 : \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9\n\u22a2 \u03c9 \u2208 \u22a5", "state_after": "case hst.intro\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\nhnonneg : 0 \u2264 f\n\u03b5 : \u211d\u22650\nn : \u2115\nthis :\n  \u03b5 \u2022 \u2191\u2191\u03bc {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9} +\n      ENNReal.ofReal\n        (\u222b (\u03c9 : \u03a9) in {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5},\n          f n \u03c9 \u2202\u03bc) \u2264\n    ENNReal.ofReal (\u222b (x : \u03a9), f n x \u2202\u03bc)\n\u03c9 : \u03a9\nh\u03c9\u2081 : \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9\nh\u03c9\u2082 : (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5\n\u22a2 \u03c9 \u2208 \u22a5"}, {"tactic": "exact (not_le.2 h\u03c9\u2082) h\u03c9\u2081", "annotated_tactic": ["exact (<a>not_le</a>.2 h\u03c9\u2082) h\u03c9\u2081", [{"full_name": "not_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [373, 9], "def_end_pos": [373, 15]}]], "state_before": "case hst.intro\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\nhnonneg : 0 \u2264 f\n\u03b5 : \u211d\u22650\nn : \u2115\nthis :\n  \u03b5 \u2022 \u2191\u2191\u03bc {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9} +\n      ENNReal.ofReal\n        (\u222b (\u03c9 : \u03a9) in {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5},\n          f n \u03c9 \u2202\u03bc) \u2264\n    ENNReal.ofReal (\u222b (x : \u03a9), f n x \u2202\u03bc)\n\u03c9 : \u03a9\nh\u03c9\u2081 : \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9\nh\u03c9\u2082 : (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5\n\u22a2 \u03c9 \u2208 \u22a5", "state_after": "no goals"}, {"tactic": "exact measurableSet_lt (Finset.measurable_range_sup'' fun n _ =>\n  (hsub.stronglyMeasurable n).measurable.le (\ud835\udca2.le n)) measurable_const", "annotated_tactic": ["exact <a>measurableSet_lt</a> (<a>Finset.measurable_range_sup''</a> fun n _ =>\n          (hsub.stronglyMeasurable n).measurable.le (\ud835\udca2.le n)) <a>measurable_const</a>", [{"full_name": "measurableSet_lt", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [616, 9], "def_end_pos": [616, 25]}, {"full_name": "Finset.measurable_range_sup''", "def_path": "Mathlib/MeasureTheory/Lattice.lean", "def_pos": [256, 9], "def_end_pos": [256, 38]}, {"full_name": "measurable_const", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [570, 9], "def_end_pos": [570, 25]}]], "state_before": "case ht\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\nhnonneg : 0 \u2264 f\n\u03b5 : \u211d\u22650\nn : \u2115\nthis :\n  \u03b5 \u2022 \u2191\u2191\u03bc {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9} +\n      ENNReal.ofReal\n        (\u222b (\u03c9 : \u03a9) in {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5},\n          f n \u03c9 \u2202\u03bc) \u2264\n    ENNReal.ofReal (\u222b (x : \u03a9), f n x \u2202\u03bc)\n\u22a2 MeasurableSet {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5}", "state_after": "no goals"}, {"tactic": "refine' add_le_add (smul_le_stoppedValue_hitting hsub _)\n  (ENNReal.ofReal_le_ofReal (set_integral_mono_on (hsub.integrable n).integrableOn\n    (Integrable.integrableOn (hsub.integrable_stoppedValue\n      (hitting_isStoppingTime hsub.adapted measurableSet_Ici) hitting_le))\n        (measurableSet_lt (Finset.measurable_range_sup'' fun n _ =>\n          (hsub.stronglyMeasurable n).measurable.le (\ud835\udca2.le n)) measurable_const) _))", "annotated_tactic": ["refine' <a>add_le_add</a> (<a>smul_le_stoppedValue_hitting</a> hsub _)\n        (<a>ENNReal.ofReal_le_ofReal</a> (<a>set_integral_mono_on</a> (hsub.integrable n).<a>integrableOn</a>\n          (<a>Integrable.integrableOn</a> (hsub.integrable_stoppedValue\n            (<a>hitting_isStoppingTime</a> hsub.adapted <a>measurableSet_Ici</a>) <a>hitting_le</a>))\n              (<a>measurableSet_lt</a> (<a>Finset.measurable_range_sup''</a> fun n _ =>\n                (hsub.stronglyMeasurable n).measurable.le (\ud835\udca2.le n)) <a>measurable_const</a>) _))", [{"full_name": "add_le_add", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [205, 15], "def_end_pos": [205, 25]}, {"full_name": "MeasureTheory.smul_le_stoppedValue_hitting", "def_path": "Mathlib/Probability/Martingale/OptionalStopping.lean", "def_pos": [112, 9], "def_end_pos": [112, 37]}, {"full_name": "ENNReal.ofReal_le_ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2135, 9], "def_end_pos": [2135, 25]}, {"full_name": "MeasureTheory.set_integral_mono_on", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [721, 9], "def_end_pos": [721, 29]}, {"full_name": "MeasureTheory.Integrable.integrableOn", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [163, 9], "def_end_pos": [163, 32]}, {"full_name": "MeasureTheory.Integrable.integrableOn", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [163, 9], "def_end_pos": [163, 32]}, {"full_name": "MeasureTheory.hitting_isStoppingTime", "def_path": "Mathlib/Probability/Process/HittingTime.lean", "def_pos": [227, 9], "def_end_pos": [227, 31]}, {"full_name": "measurableSet_Ici", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [510, 9], "def_end_pos": [510, 26]}, {"full_name": "MeasureTheory.hitting_le", "def_path": "Mathlib/Probability/Process/HittingTime.lean", "def_pos": [71, 9], "def_end_pos": [71, 19]}, {"full_name": "measurableSet_lt", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [616, 9], "def_end_pos": [616, 25]}, {"full_name": "Finset.measurable_range_sup''", "def_path": "Mathlib/MeasureTheory/Lattice.lean", "def_pos": [256, 9], "def_end_pos": [256, 38]}, {"full_name": "measurable_const", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [570, 9], "def_end_pos": [570, 25]}]], "state_before": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\nhnonneg : 0 \u2264 f\n\u03b5 : \u211d\u22650\nn : \u2115\n\u22a2 \u03b5 \u2022 \u2191\u2191\u03bc {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9} +\n      ENNReal.ofReal\n        (\u222b (\u03c9 : \u03a9) in {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5},\n          f n \u03c9 \u2202\u03bc) \u2264\n    ENNReal.ofReal\n        (\u222b (\u03c9 : \u03a9) in {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9},\n          stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) \u03c9 \u2202\u03bc) +\n      ENNReal.ofReal\n        (\u222b (\u03c9 : \u03a9) in {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5},\n          stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) \u03c9 \u2202\u03bc)", "state_after": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\nhnonneg : 0 \u2264 f\n\u03b5 : \u211d\u22650\nn : \u2115\n\u22a2 \u2200 (x : \u03a9),\n    x \u2208 {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5} \u2192\n      f n x \u2264 stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) x"}, {"tactic": "intro \u03c9 h\u03c9", "annotated_tactic": ["intro \u03c9 h\u03c9", []], "state_before": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\nhnonneg : 0 \u2264 f\n\u03b5 : \u211d\u22650\nn : \u2115\n\u22a2 \u2200 (x : \u03a9),\n    x \u2208 {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5} \u2192\n      f n x \u2264 stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) x", "state_after": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\nhnonneg : 0 \u2264 f\n\u03b5 : \u211d\u22650\nn : \u2115\n\u03c9 : \u03a9\nh\u03c9 : \u03c9 \u2208 {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5}\n\u22a2 f n \u03c9 \u2264 stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) \u03c9"}, {"tactic": "rw [Set.mem_setOf_eq] at h\u03c9", "annotated_tactic": ["rw [<a>Set.mem_setOf_eq</a>] at h\u03c9", [{"full_name": "Set.mem_setOf_eq", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [256, 29], "def_end_pos": [256, 41]}]], "state_before": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\nhnonneg : 0 \u2264 f\n\u03b5 : \u211d\u22650\nn : \u2115\n\u03c9 : \u03a9\nh\u03c9 : \u03c9 \u2208 {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5}\n\u22a2 f n \u03c9 \u2264 stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) \u03c9", "state_after": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\nhnonneg : 0 \u2264 f\n\u03b5 : \u211d\u22650\nn : \u2115\n\u03c9 : \u03a9\nh\u03c9 : (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5\n\u22a2 f n \u03c9 \u2264 stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) \u03c9"}, {"tactic": "have : hitting f {y : \u211d | \u2191\u03b5 \u2264 y} 0 n \u03c9 = n := by\n  classical simp only [hitting, Set.mem_setOf_eq, exists_prop, Pi.coe_nat, Nat.cast_id,\n    ite_eq_right_iff, forall_exists_index, and_imp]\n  intro m hm h\u03b5m\n  exact False.elim\n    ((not_le.2 h\u03c9) ((le_sup'_iff _).2 \u27e8m, mem_range.2 (Nat.lt_succ_of_le hm.2), h\u03b5m\u27e9))", "annotated_tactic": ["have : <a>hitting</a> f {y : \u211d | \u2191\u03b5 \u2264 y} 0 n \u03c9 = n := by\n        classical simp only [<a>hitting</a>, <a>Set.mem_setOf_eq</a>, <a>exists_prop</a>, <a>Pi.coe_nat</a>, <a>Nat.cast_id</a>,\n          <a>ite_eq_right_iff</a>, <a>forall_exists_index</a>, <a>and_imp</a>]\n        intro m hm h\u03b5m\n        exact <a>False.elim</a>\n          ((<a>not_le</a>.2 h\u03c9) ((<a>le_sup'_iff</a> _).2 \u27e8m, <a>mem_range</a>.2 (<a>Nat.lt_succ_of_le</a> hm.2), h\u03b5m\u27e9))", [{"full_name": "MeasureTheory.hitting", "def_path": "Mathlib/Probability/Process/HittingTime.lean", "def_pos": [51, 19], "def_end_pos": [51, 26]}, {"full_name": "MeasureTheory.hitting", "def_path": "Mathlib/Probability/Process/HittingTime.lean", "def_pos": [51, 19], "def_end_pos": [51, 26]}, {"full_name": "Set.mem_setOf_eq", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [256, 29], "def_end_pos": [256, 41]}, {"full_name": "exists_prop", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [485, 17], "def_end_pos": [485, 28]}, {"full_name": "Pi.coe_nat", "def_path": "Mathlib/Data/Nat/Cast/Basic.lean", "def_pos": [195, 9], "def_end_pos": [195, 16]}, {"full_name": "Nat.cast_id", "def_path": "Mathlib/Data/Nat/Cast/Basic.lean", "def_pos": [167, 9], "def_end_pos": [167, 20]}, {"full_name": "ite_eq_right_iff", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [1162, 17], "def_end_pos": [1162, 33]}, {"full_name": "forall_exists_index", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [356, 17], "def_end_pos": [356, 36]}, {"full_name": "and_imp", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [313, 17], "def_end_pos": [313, 24]}, {"full_name": "False.elim", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [223, 21], "def_end_pos": [223, 31]}, {"full_name": "not_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [373, 9], "def_end_pos": [373, 15]}, {"full_name": "Finset.le_sup'_iff", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [1175, 9], "def_end_pos": [1175, 20]}, {"full_name": "Finset.mem_range", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3037, 9], "def_end_pos": [3037, 18]}, {"full_name": "Nat.lt_succ_of_le", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [212, 9], "def_end_pos": [212, 22]}]], "state_before": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\nhnonneg : 0 \u2264 f\n\u03b5 : \u211d\u22650\nn : \u2115\n\u03c9 : \u03a9\nh\u03c9 : (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5\n\u22a2 f n \u03c9 \u2264 stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) \u03c9", "state_after": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\nhnonneg : 0 \u2264 f\n\u03b5 : \u211d\u22650\nn : \u2115\n\u03c9 : \u03a9\nh\u03c9 : (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5\nthis : hitting f {y | \u2191\u03b5 \u2264 y} 0 n \u03c9 = n\n\u22a2 f n \u03c9 \u2264 stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) \u03c9"}, {"tactic": "simp_rw [stoppedValue, this, le_rfl]", "annotated_tactic": ["simp_rw [<a>stoppedValue</a>, this, <a>le_rfl</a>]", [{"full_name": "MeasureTheory.stoppedValue", "def_path": "Mathlib/Probability/Process/Stopping.lean", "def_pos": [768, 5], "def_end_pos": [768, 17]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}]], "state_before": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\nhnonneg : 0 \u2264 f\n\u03b5 : \u211d\u22650\nn : \u2115\n\u03c9 : \u03a9\nh\u03c9 : (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5\nthis : hitting f {y | \u2191\u03b5 \u2264 y} 0 n \u03c9 = n\n\u22a2 f n \u03c9 \u2264 stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) \u03c9", "state_after": "no goals"}, {"tactic": "classical simp only [hitting, Set.mem_setOf_eq, exists_prop, Pi.coe_nat, Nat.cast_id,\n  ite_eq_right_iff, forall_exists_index, and_imp]", "annotated_tactic": ["classical simp only [<a>hitting</a>, <a>Set.mem_setOf_eq</a>, <a>exists_prop</a>, <a>Pi.coe_nat</a>, <a>Nat.cast_id</a>,\n          <a>ite_eq_right_iff</a>, <a>forall_exists_index</a>, <a>and_imp</a>]", [{"full_name": "MeasureTheory.hitting", "def_path": "Mathlib/Probability/Process/HittingTime.lean", "def_pos": [51, 19], "def_end_pos": [51, 26]}, {"full_name": "Set.mem_setOf_eq", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [256, 29], "def_end_pos": [256, 41]}, {"full_name": "exists_prop", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [485, 17], "def_end_pos": [485, 28]}, {"full_name": "Pi.coe_nat", "def_path": "Mathlib/Data/Nat/Cast/Basic.lean", "def_pos": [195, 9], "def_end_pos": [195, 16]}, {"full_name": "Nat.cast_id", "def_path": "Mathlib/Data/Nat/Cast/Basic.lean", "def_pos": [167, 9], "def_end_pos": [167, 20]}, {"full_name": "ite_eq_right_iff", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [1162, 17], "def_end_pos": [1162, 33]}, {"full_name": "forall_exists_index", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [356, 17], "def_end_pos": [356, 36]}, {"full_name": "and_imp", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [313, 17], "def_end_pos": [313, 24]}]], "state_before": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\nhnonneg : 0 \u2264 f\n\u03b5 : \u211d\u22650\nn : \u2115\n\u03c9 : \u03a9\nh\u03c9 : (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5\n\u22a2 hitting f {y | \u2191\u03b5 \u2264 y} 0 n \u03c9 = n", "state_after": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\nhnonneg : 0 \u2264 f\n\u03b5 : \u211d\u22650\nn : \u2115\n\u03c9 : \u03a9\nh\u03c9 : (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5\n\u22a2 \u2200 (x : \u2115), x \u2208 Set.Icc 0 n \u2192 \u2191\u03b5 \u2264 f x \u03c9 \u2192 sInf (Set.Icc 0 n \u2229 {i | \u2191\u03b5 \u2264 f i \u03c9}) = n"}, {"tactic": "intro m hm h\u03b5m", "annotated_tactic": ["intro m hm h\u03b5m", []], "state_before": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\nhnonneg : 0 \u2264 f\n\u03b5 : \u211d\u22650\nn : \u2115\n\u03c9 : \u03a9\nh\u03c9 : (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5\n\u22a2 \u2200 (x : \u2115), x \u2208 Set.Icc 0 n \u2192 \u2191\u03b5 \u2264 f x \u03c9 \u2192 sInf (Set.Icc 0 n \u2229 {i | \u2191\u03b5 \u2264 f i \u03c9}) = n", "state_after": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\nhnonneg : 0 \u2264 f\n\u03b5 : \u211d\u22650\nn : \u2115\n\u03c9 : \u03a9\nh\u03c9 : (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5\nm : \u2115\nhm : m \u2208 Set.Icc 0 n\nh\u03b5m : \u2191\u03b5 \u2264 f m \u03c9\n\u22a2 sInf (Set.Icc 0 n \u2229 {i | \u2191\u03b5 \u2264 f i \u03c9}) = n"}, {"tactic": "exact False.elim\n  ((not_le.2 h\u03c9) ((le_sup'_iff _).2 \u27e8m, mem_range.2 (Nat.lt_succ_of_le hm.2), h\u03b5m\u27e9))", "annotated_tactic": ["exact <a>False.elim</a>\n          ((<a>not_le</a>.2 h\u03c9) ((<a>le_sup'_iff</a> _).2 \u27e8m, <a>mem_range</a>.2 (<a>Nat.lt_succ_of_le</a> hm.2), h\u03b5m\u27e9))", [{"full_name": "False.elim", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [223, 21], "def_end_pos": [223, 31]}, {"full_name": "not_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [373, 9], "def_end_pos": [373, 15]}, {"full_name": "Finset.le_sup'_iff", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [1175, 9], "def_end_pos": [1175, 20]}, {"full_name": "Finset.mem_range", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3037, 9], "def_end_pos": [3037, 18]}, {"full_name": "Nat.lt_succ_of_le", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [212, 9], "def_end_pos": [212, 22]}]], "state_before": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\nhnonneg : 0 \u2264 f\n\u03b5 : \u211d\u22650\nn : \u2115\n\u03c9 : \u03a9\nh\u03c9 : (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5\nm : \u2115\nhm : m \u2208 Set.Icc 0 n\nh\u03b5m : \u2191\u03b5 \u2264 f m \u03c9\n\u22a2 sInf (Set.Icc 0 n \u2229 {i | \u2191\u03b5 \u2264 f i \u03c9}) = n", "state_after": "no goals"}, {"tactic": "simp only [hitting, Set.mem_setOf_eq, exists_prop, Pi.coe_nat, Nat.cast_id,\nite_eq_right_iff, forall_exists_index, and_imp]", "annotated_tactic": ["simp only [<a>hitting</a>, <a>Set.mem_setOf_eq</a>, <a>exists_prop</a>, <a>Pi.coe_nat</a>, <a>Nat.cast_id</a>,\n          <a>ite_eq_right_iff</a>, <a>forall_exists_index</a>, <a>and_imp</a>]", [{"full_name": "MeasureTheory.hitting", "def_path": "Mathlib/Probability/Process/HittingTime.lean", "def_pos": [51, 19], "def_end_pos": [51, 26]}, {"full_name": "Set.mem_setOf_eq", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [256, 29], "def_end_pos": [256, 41]}, {"full_name": "exists_prop", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [485, 17], "def_end_pos": [485, 28]}, {"full_name": "Pi.coe_nat", "def_path": "Mathlib/Data/Nat/Cast/Basic.lean", "def_pos": [195, 9], "def_end_pos": [195, 16]}, {"full_name": "Nat.cast_id", "def_path": "Mathlib/Data/Nat/Cast/Basic.lean", "def_pos": [167, 9], "def_end_pos": [167, 20]}, {"full_name": "ite_eq_right_iff", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [1162, 17], "def_end_pos": [1162, 33]}, {"full_name": "forall_exists_index", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [356, 17], "def_end_pos": [356, 36]}, {"full_name": "and_imp", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [313, 17], "def_end_pos": [313, 24]}]], "state_before": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\nhnonneg : 0 \u2264 f\n\u03b5 : \u211d\u22650\nn : \u2115\n\u03c9 : \u03a9\nh\u03c9 : (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5\n\u22a2 hitting f {y | \u2191\u03b5 \u2264 y} 0 n \u03c9 = n", "state_after": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\nhnonneg : 0 \u2264 f\n\u03b5 : \u211d\u22650\nn : \u2115\n\u03c9 : \u03a9\nh\u03c9 : (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5\n\u22a2 \u2200 (x : \u2115), x \u2208 Set.Icc 0 n \u2192 \u2191\u03b5 \u2264 f x \u03c9 \u2192 sInf (Set.Icc 0 n \u2229 {i | \u2191\u03b5 \u2264 f i \u03c9}) = n"}, {"tactic": "rw [\u2190 ENNReal.ofReal_add, \u2190 integral_union]", "annotated_tactic": ["rw [\u2190 <a>ENNReal.ofReal_add</a>, \u2190 <a>integral_union</a>]", [{"full_name": "ENNReal.ofReal_add", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2025, 9], "def_end_pos": [2025, 19]}, {"full_name": "MeasureTheory.integral_union", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [102, 9], "def_end_pos": [102, 23]}]], "state_before": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\nhnonneg : 0 \u2264 f\n\u03b5 : \u211d\u22650\nn : \u2115\n\u22a2 ENNReal.ofReal\n        (\u222b (\u03c9 : \u03a9) in {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9},\n          stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) \u03c9 \u2202\u03bc) +\n      ENNReal.ofReal\n        (\u222b (\u03c9 : \u03a9) in {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5},\n          stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) \u03c9 \u2202\u03bc) =\n    ENNReal.ofReal (\u222b (\u03c9 : \u03a9), stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) \u03c9 \u2202\u03bc)", "state_after": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\nhnonneg : 0 \u2264 f\n\u03b5 : \u211d\u22650\nn : \u2115\n\u22a2 ENNReal.ofReal\n      (\u222b (x : \u03a9) in\n        {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9} \u222a\n          {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5},\n        stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) x \u2202\u03bc) =\n    ENNReal.ofReal (\u222b (\u03c9 : \u03a9), stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) \u03c9 \u2202\u03bc)\n\ncase hst\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\nhnonneg : 0 \u2264 f\n\u03b5 : \u211d\u22650\nn : \u2115\n\u22a2 Disjoint {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9}\n    {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5}\n\ncase ht\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\nhnonneg : 0 \u2264 f\n\u03b5 : \u211d\u22650\nn : \u2115\n\u22a2 MeasurableSet {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5}\n\ncase hfs\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\nhnonneg : 0 \u2264 f\n\u03b5 : \u211d\u22650\nn : \u2115\n\u22a2 IntegrableOn (fun \u03c9 => stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) \u03c9)\n    {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9}\n\ncase hft\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\nhnonneg : 0 \u2264 f\n\u03b5 : \u211d\u22650\nn : \u2115\n\u22a2 IntegrableOn (fun \u03c9 => stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) \u03c9)\n    {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5}\n\ncase hp\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\nhnonneg : 0 \u2264 f\n\u03b5 : \u211d\u22650\nn : \u2115\n\u22a2 0 \u2264\n    \u222b (\u03c9 : \u03a9) in {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9},\n      stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) \u03c9 \u2202\u03bc\n\ncase hq\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\nhnonneg : 0 \u2264 f\n\u03b5 : \u211d\u22650\nn : \u2115\n\u22a2 0 \u2264\n    \u222b (\u03c9 : \u03a9) in {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5},\n      stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) \u03c9 \u2202\u03bc"}, {"tactic": "exacts [integral_nonneg fun x => hnonneg _ _, integral_nonneg fun x => hnonneg _ _]", "annotated_tactic": ["exacts [<a>integral_nonneg</a> fun x => hnonneg _ _, <a>integral_nonneg</a> fun x => hnonneg _ _]", [{"full_name": "MeasureTheory.integral_nonneg", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1251, 9], "def_end_pos": [1251, 24]}, {"full_name": "MeasureTheory.integral_nonneg", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1251, 9], "def_end_pos": [1251, 24]}]], "state_before": "case hp\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\nhnonneg : 0 \u2264 f\n\u03b5 : \u211d\u22650\nn : \u2115\n\u22a2 0 \u2264\n    \u222b (\u03c9 : \u03a9) in {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9},\n      stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) \u03c9 \u2202\u03bc\n\ncase hq\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\nhnonneg : 0 \u2264 f\n\u03b5 : \u211d\u22650\nn : \u2115\n\u22a2 0 \u2264\n    \u222b (\u03c9 : \u03a9) in {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5},\n      stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) \u03c9 \u2202\u03bc", "state_after": "no goals"}, {"tactic": "rw [\u2190 integral_univ (\u03bc := \u03bc)]", "annotated_tactic": ["rw [\u2190 <a>integral_univ</a> (\u03bc := \u03bc)]", [{"full_name": "MeasureTheory.integral_univ", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [152, 9], "def_end_pos": [152, 22]}]], "state_before": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\nhnonneg : 0 \u2264 f\n\u03b5 : \u211d\u22650\nn : \u2115\n\u22a2 ENNReal.ofReal\n      (\u222b (x : \u03a9) in\n        {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9} \u222a\n          {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5},\n        stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) x \u2202\u03bc) =\n    ENNReal.ofReal (\u222b (\u03c9 : \u03a9), stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) \u03c9 \u2202\u03bc)", "state_after": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\nhnonneg : 0 \u2264 f\n\u03b5 : \u211d\u22650\nn : \u2115\n\u22a2 ENNReal.ofReal\n      (\u222b (x : \u03a9) in\n        {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9} \u222a\n          {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5},\n        stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) x \u2202\u03bc) =\n    ENNReal.ofReal (\u222b (x : \u03a9) in Set.univ, stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) x \u2202\u03bc)"}, {"tactic": "convert rfl", "annotated_tactic": ["convert <a>rfl</a>", [{"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\nhnonneg : 0 \u2264 f\n\u03b5 : \u211d\u22650\nn : \u2115\n\u22a2 ENNReal.ofReal\n      (\u222b (x : \u03a9) in\n        {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9} \u222a\n          {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5},\n        stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) x \u2202\u03bc) =\n    ENNReal.ofReal (\u222b (x : \u03a9) in Set.univ, stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) x \u2202\u03bc)", "state_after": "case h.e'_3.h.e'_1.h.e'_6.h.e'_4\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\nhnonneg : 0 \u2264 f\n\u03b5 : \u211d\u22650\nn : \u2115\n\u22a2 Set.univ =\n    {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9} \u222a\n      {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5}"}, {"tactic": "ext \u03c9", "annotated_tactic": ["ext \u03c9", []], "state_before": "case h.e'_3.h.e'_1.h.e'_6.h.e'_4\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\nhnonneg : 0 \u2264 f\n\u03b5 : \u211d\u22650\nn : \u2115\n\u22a2 Set.univ =\n    {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9} \u222a\n      {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5}", "state_after": "case h.e'_3.h.e'_1.h.e'_6.h.e'_4.h\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\nhnonneg : 0 \u2264 f\n\u03b5 : \u211d\u22650\nn : \u2115\n\u03c9 : \u03a9\n\u22a2 \u03c9 \u2208 Set.univ \u2194\n    \u03c9 \u2208\n      {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9} \u222a\n        {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5}"}, {"tactic": "change _ \u2194 (\u03b5 : \u211d) \u2264 _ \u2228 _ < (\u03b5 : \u211d)", "annotated_tactic": ["change _ \u2194 (\u03b5 : \u211d) \u2264 _ \u2228 _ < (\u03b5 : \u211d)", []], "state_before": "case h.e'_3.h.e'_1.h.e'_6.h.e'_4.h\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\nhnonneg : 0 \u2264 f\n\u03b5 : \u211d\u22650\nn : \u2115\n\u03c9 : \u03a9\n\u22a2 \u03c9 \u2208 Set.univ \u2194\n    \u03c9 \u2208\n      {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9} \u222a\n        {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5}", "state_after": "case h.e'_3.h.e'_1.h.e'_6.h.e'_4.h\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\nhnonneg : 0 \u2264 f\n\u03b5 : \u211d\u22650\nn : \u2115\n\u03c9 : \u03a9\n\u22a2 \u03c9 \u2208 Set.univ \u2194\n    (\u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) \u2228\n      (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5"}, {"tactic": "simp only [le_or_lt, Set.mem_univ]", "annotated_tactic": ["simp only [<a>le_or_lt</a>, <a>Set.mem_univ</a>]", [{"full_name": "le_or_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [340, 9], "def_end_pos": [340, 17]}, {"full_name": "Set.mem_univ", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [676, 9], "def_end_pos": [676, 17]}]], "state_before": "case h.e'_3.h.e'_1.h.e'_6.h.e'_4.h\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\nhnonneg : 0 \u2264 f\n\u03b5 : \u211d\u22650\nn : \u2115\n\u03c9 : \u03a9\n\u22a2 \u03c9 \u2208 Set.univ \u2194\n    (\u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) \u2228\n      (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5", "state_after": "no goals"}, {"tactic": "rw [disjoint_iff_inf_le]", "annotated_tactic": ["rw [<a>disjoint_iff_inf_le</a>]", [{"full_name": "disjoint_iff_inf_le", "def_path": "Mathlib/Order/Disjoint.lean", "def_pos": [122, 9], "def_end_pos": [122, 28]}]], "state_before": "case hst\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\nhnonneg : 0 \u2264 f\n\u03b5 : \u211d\u22650\nn : \u2115\n\u22a2 Disjoint {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9}\n    {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5}", "state_after": "case hst\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\nhnonneg : 0 \u2264 f\n\u03b5 : \u211d\u22650\nn : \u2115\n\u22a2 {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9} \u2293\n      {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5} \u2264\n    \u22a5"}, {"tactic": "rintro \u03c9 \u27e8h\u03c9\u2081, h\u03c9\u2082\u27e9", "annotated_tactic": ["rintro \u03c9 \u27e8h\u03c9\u2081, h\u03c9\u2082\u27e9", []], "state_before": "case hst\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\nhnonneg : 0 \u2264 f\n\u03b5 : \u211d\u22650\nn : \u2115\n\u22a2 {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9} \u2293\n      {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5} \u2264\n    \u22a5", "state_after": "case hst.intro\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\nhnonneg : 0 \u2264 f\n\u03b5 : \u211d\u22650\nn : \u2115\n\u03c9 : \u03a9\nh\u03c9\u2081 : \u03c9 \u2208 {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9}\nh\u03c9\u2082 : \u03c9 \u2208 {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5}\n\u22a2 \u03c9 \u2208 \u22a5"}, {"tactic": "change (\u03b5 : \u211d) \u2264 _ at h\u03c9\u2081", "annotated_tactic": ["change (\u03b5 : \u211d) \u2264 _ at h\u03c9\u2081", []], "state_before": "case hst.intro\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\nhnonneg : 0 \u2264 f\n\u03b5 : \u211d\u22650\nn : \u2115\n\u03c9 : \u03a9\nh\u03c9\u2081 : \u03c9 \u2208 {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9}\nh\u03c9\u2082 : \u03c9 \u2208 {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5}\n\u22a2 \u03c9 \u2208 \u22a5", "state_after": "case hst.intro\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\nhnonneg : 0 \u2264 f\n\u03b5 : \u211d\u22650\nn : \u2115\n\u03c9 : \u03a9\nh\u03c9\u2082 : \u03c9 \u2208 {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5}\nh\u03c9\u2081 : \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9\n\u22a2 \u03c9 \u2208 \u22a5"}, {"tactic": "change _ < (\u03b5 : \u211d) at h\u03c9\u2082", "annotated_tactic": ["change _ < (\u03b5 : \u211d) at h\u03c9\u2082", []], "state_before": "case hst.intro\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\nhnonneg : 0 \u2264 f\n\u03b5 : \u211d\u22650\nn : \u2115\n\u03c9 : \u03a9\nh\u03c9\u2082 : \u03c9 \u2208 {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5}\nh\u03c9\u2081 : \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9\n\u22a2 \u03c9 \u2208 \u22a5", "state_after": "case hst.intro\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\nhnonneg : 0 \u2264 f\n\u03b5 : \u211d\u22650\nn : \u2115\n\u03c9 : \u03a9\nh\u03c9\u2081 : \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9\nh\u03c9\u2082 : (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5\n\u22a2 \u03c9 \u2208 \u22a5"}, {"tactic": "exact (not_le.2 h\u03c9\u2082) h\u03c9\u2081", "annotated_tactic": ["exact (<a>not_le</a>.2 h\u03c9\u2082) h\u03c9\u2081", [{"full_name": "not_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [373, 9], "def_end_pos": [373, 15]}]], "state_before": "case hst.intro\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\nhnonneg : 0 \u2264 f\n\u03b5 : \u211d\u22650\nn : \u2115\n\u03c9 : \u03a9\nh\u03c9\u2081 : \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9\nh\u03c9\u2082 : (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5\n\u22a2 \u03c9 \u2208 \u22a5", "state_after": "no goals"}, {"tactic": "exact measurableSet_lt (Finset.measurable_range_sup'' fun n _ =>\n  (hsub.stronglyMeasurable n).measurable.le (\ud835\udca2.le n)) measurable_const", "annotated_tactic": ["exact <a>measurableSet_lt</a> (<a>Finset.measurable_range_sup''</a> fun n _ =>\n          (hsub.stronglyMeasurable n).measurable.le (\ud835\udca2.le n)) <a>measurable_const</a>", [{"full_name": "measurableSet_lt", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [616, 9], "def_end_pos": [616, 25]}, {"full_name": "Finset.measurable_range_sup''", "def_path": "Mathlib/MeasureTheory/Lattice.lean", "def_pos": [256, 9], "def_end_pos": [256, 38]}, {"full_name": "measurable_const", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [570, 9], "def_end_pos": [570, 25]}]], "state_before": "case ht\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\nhnonneg : 0 \u2264 f\n\u03b5 : \u211d\u22650\nn : \u2115\n\u22a2 MeasurableSet {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5}", "state_after": "no goals"}, {"tactic": "exact Integrable.integrableOn (hsub.integrable_stoppedValue\n  (hitting_isStoppingTime hsub.adapted measurableSet_Ici) hitting_le)", "annotated_tactic": ["exact <a>Integrable.integrableOn</a> (hsub.integrable_stoppedValue\n          (<a>hitting_isStoppingTime</a> hsub.adapted <a>measurableSet_Ici</a>) <a>hitting_le</a>)", [{"full_name": "MeasureTheory.Integrable.integrableOn", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [163, 9], "def_end_pos": [163, 32]}, {"full_name": "MeasureTheory.hitting_isStoppingTime", "def_path": "Mathlib/Probability/Process/HittingTime.lean", "def_pos": [227, 9], "def_end_pos": [227, 31]}, {"full_name": "measurableSet_Ici", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [510, 9], "def_end_pos": [510, 26]}, {"full_name": "MeasureTheory.hitting_le", "def_path": "Mathlib/Probability/Process/HittingTime.lean", "def_pos": [71, 9], "def_end_pos": [71, 19]}]], "state_before": "case hfs\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\nhnonneg : 0 \u2264 f\n\u03b5 : \u211d\u22650\nn : \u2115\n\u22a2 IntegrableOn (fun \u03c9 => stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) \u03c9)\n    {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9}", "state_after": "no goals"}, {"tactic": "exact Integrable.integrableOn (hsub.integrable_stoppedValue\n  (hitting_isStoppingTime hsub.adapted measurableSet_Ici) hitting_le)", "annotated_tactic": ["exact <a>Integrable.integrableOn</a> (hsub.integrable_stoppedValue\n          (<a>hitting_isStoppingTime</a> hsub.adapted <a>measurableSet_Ici</a>) <a>hitting_le</a>)", [{"full_name": "MeasureTheory.Integrable.integrableOn", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [163, 9], "def_end_pos": [163, 32]}, {"full_name": "MeasureTheory.hitting_isStoppingTime", "def_path": "Mathlib/Probability/Process/HittingTime.lean", "def_pos": [227, 9], "def_end_pos": [227, 31]}, {"full_name": "measurableSet_Ici", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [510, 9], "def_end_pos": [510, 26]}, {"full_name": "MeasureTheory.hitting_le", "def_path": "Mathlib/Probability/Process/HittingTime.lean", "def_pos": [71, 9], "def_end_pos": [71, 19]}]], "state_before": "case hft\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\nhnonneg : 0 \u2264 f\n\u03b5 : \u211d\u22650\nn : \u2115\n\u22a2 IntegrableOn (fun \u03c9 => stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) \u03c9)\n    {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5}", "state_after": "no goals"}, {"tactic": "refine' ENNReal.ofReal_le_ofReal _", "annotated_tactic": ["refine' <a>ENNReal.ofReal_le_ofReal</a> _", [{"full_name": "ENNReal.ofReal_le_ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2135, 9], "def_end_pos": [2135, 25]}]], "state_before": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\nhnonneg : 0 \u2264 f\n\u03b5 : \u211d\u22650\nn : \u2115\n\u22a2 ENNReal.ofReal (\u222b (\u03c9 : \u03a9), stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) \u03c9 \u2202\u03bc) \u2264 ENNReal.ofReal (\u222b (x : \u03a9), f n x \u2202\u03bc)", "state_after": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\nhnonneg : 0 \u2264 f\n\u03b5 : \u211d\u22650\nn : \u2115\n\u22a2 \u222b (\u03c9 : \u03a9), stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) \u03c9 \u2202\u03bc \u2264 \u222b (x : \u03a9), f n x \u2202\u03bc"}, {"tactic": "rw [\u2190 stoppedValue_const f n]", "annotated_tactic": ["rw [\u2190 <a>stoppedValue_const</a> f n]", [{"full_name": "MeasureTheory.stoppedValue_const", "def_path": "Mathlib/Probability/Process/Stopping.lean", "def_pos": [771, 9], "def_end_pos": [771, 27]}]], "state_before": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\nhnonneg : 0 \u2264 f\n\u03b5 : \u211d\u22650\nn : \u2115\n\u22a2 \u222b (\u03c9 : \u03a9), stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) \u03c9 \u2202\u03bc \u2264 \u222b (x : \u03a9), f n x \u2202\u03bc", "state_after": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\nhnonneg : 0 \u2264 f\n\u03b5 : \u211d\u22650\nn : \u2115\n\u22a2 \u222b (\u03c9 : \u03a9), stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) \u03c9 \u2202\u03bc \u2264 \u222b (x : \u03a9), stoppedValue f (fun x => n) x \u2202\u03bc"}, {"tactic": "exact hsub.expected_stoppedValue_mono (hitting_isStoppingTime hsub.adapted measurableSet_Ici)\n  (isStoppingTime_const _ _) (fun \u03c9 => hitting_le \u03c9) (fun _ => le_rfl : \u2200 _, n \u2264 n)", "annotated_tactic": ["exact hsub.expected_stoppedValue_mono (<a>hitting_isStoppingTime</a> hsub.adapted <a>measurableSet_Ici</a>)\n        (<a>isStoppingTime_const</a> _ _) (fun \u03c9 => <a>hitting_le</a> \u03c9) (fun _ => <a>le_rfl</a> : \u2200 _, n \u2264 n)", [{"full_name": "MeasureTheory.hitting_isStoppingTime", "def_path": "Mathlib/Probability/Process/HittingTime.lean", "def_pos": [227, 9], "def_end_pos": [227, 31]}, {"full_name": "measurableSet_Ici", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [510, 9], "def_end_pos": [510, 26]}, {"full_name": "MeasureTheory.isStoppingTime_const", "def_path": "Mathlib/Probability/Process/Stopping.lean", "def_pos": [57, 9], "def_end_pos": [57, 29]}, {"full_name": "MeasureTheory.hitting_le", "def_path": "Mathlib/Probability/Process/HittingTime.lean", "def_pos": [71, 9], "def_end_pos": [71, 19]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}]], "state_before": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\nhnonneg : 0 \u2264 f\n\u03b5 : \u211d\u22650\nn : \u2115\n\u22a2 \u222b (\u03c9 : \u03a9), stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) \u03c9 \u2202\u03bc \u2264 \u222b (x : \u03a9), stoppedValue f (fun x => n) x \u2202\u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Logic.lean", "full_name": "exists\u2085_congr", "start": [415, 1], "end": [418, 45], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Prod.lean", "full_name": "Set.disjoint_pi", "start": [761, 1], "end": [762, 80], "traced_tactics": [{"tactic": "simp only [disjoint_iff_inter_eq_empty, \u2190 pi_inter_distrib, pi_eq_empty_iff']", "annotated_tactic": ["simp only [<a>disjoint_iff_inter_eq_empty</a>, \u2190 <a>pi_inter_distrib</a>, <a>pi_eq_empty_iff'</a>]", [{"full_name": "Set.disjoint_iff_inter_eq_empty", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1538, 9], "def_end_pos": [1538, 36]}, {"full_name": "Set.pi_inter_distrib", "def_path": "Mathlib/Data/Set/Prod.lean", "def_pos": [701, 9], "def_end_pos": [701, 25]}, {"full_name": "Set.pi_eq_empty_iff'", "def_path": "Mathlib/Data/Set/Prod.lean", "def_pos": [757, 9], "def_end_pos": [757, 25]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\ns s\u2081 s\u2082 : Set \u03b9\nt t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\ni : \u03b9\ninst\u271d : \u2200 (i : \u03b9), Nonempty (\u03b1 i)\n\u22a2 Disjoint (pi s t\u2081) (pi s t\u2082) \u2194 \u2203 i, i \u2208 s \u2227 Disjoint (t\u2081 i) (t\u2082 i)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Pointwise/Interval.lean", "full_name": "Set.image_mul_left_Icc", "start": [744, 1], "end": [746, 74], "traced_tactics": [{"tactic": "convert image_mul_right_Icc hbc ha using 1 <;> simp only [mul_comm _ a]", "annotated_tactic": ["convert <a>image_mul_right_Icc</a> hbc ha using 1 <;> simp only [<a>mul_comm</a> _ a]", [{"full_name": "Set.image_mul_right_Icc", "def_path": "Mathlib/Data/Set/Pointwise/Interval.lean", "def_pos": [731, 9], "def_end_pos": [731, 28]}, {"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [302, 9], "def_end_pos": [302, 17]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na\u271d a b c : \u03b1\nha : 0 \u2264 a\nhbc : b \u2264 c\n\u22a2 (fun x x_1 => x * x_1) a '' Icc b c = Icc (a * b) (a * c)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Sign.lean", "full_name": "exists_signed_sum_aux", "start": [492, 9], "end": [503, 68], "traced_tactics": [{"tactic": "refine'\n  \u27e8(\u03a3 _ : { x // x \u2208 s }, \u2115), Finset.univ.sigma fun a => range (f a).natAbs,\n    fun a => sign (f a.1), fun a => a.1, fun a => a.1.2, _, _\u27e9", "annotated_tactic": ["refine'\n    \u27e8(\u03a3 _ : { x // x \u2208 s }, \u2115), Finset.univ.sigma fun a => <a>range</a> (f a).<a>natAbs</a>,\n      fun a => <a>sign</a> (f a.1), fun a => a.1, fun a => a.1.2, _, _\u27e9", [{"full_name": "Finset.range", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3027, 5], "def_end_pos": [3027, 10]}, {"full_name": "Int.natAbs", "def_path": "lake-packages/lean4/src/lean/Init/Data/Int/Basic.lean", "def_pos": [242, 5], "def_end_pos": [242, 11]}, {"full_name": "SignType.sign", "def_path": "Mathlib/Data/Sign.lean", "def_pos": [301, 5], "def_end_pos": [301, 18]}]], "state_before": "\u03b1\u271d \u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\nf : \u03b1 \u2192 \u2124\n\u22a2 \u2203 \u03b2 t sgn g,\n    (\u2200 (b : \u03b2), g b \u2208 s) \u2227\n      card t = \u2211 a in s, Int.natAbs (f a) \u2227 \u2200 (a : \u03b1), a \u2208 s \u2192 (\u2211 b in t, if g b = a then \u2191(sgn b) else 0) = f a", "state_after": "case refine'_1\n\u03b1\u271d \u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\nf : \u03b1 \u2192 \u2124\n\u22a2 card (Finset.sigma univ fun a => range (Int.natAbs (f \u2191a))) = \u2211 a in s, Int.natAbs (f a)\n\ncase refine'_2\n\u03b1\u271d \u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\nf : \u03b1 \u2192 \u2124\n\u22a2 \u2200 (a : \u03b1),\n    a \u2208 s \u2192\n      (\u2211 b in Finset.sigma univ fun a => range (Int.natAbs (f \u2191a)),\n          if (fun a => \u2191a.fst) b = a then \u2191((fun a => \u2191sign (f \u2191a.fst)) b) else 0) =\n        f a"}, {"tactic": "simp [sum_attach (f := fun a => (f a).natAbs)]", "annotated_tactic": ["simp [<a>sum_attach</a> (f := fun a => (f a).<a>natAbs</a>)]", [{"full_name": "Finset.sum_attach", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [851, 3], "def_end_pos": [851, 14]}, {"full_name": "Int.natAbs", "def_path": "lake-packages/lean4/src/lean/Init/Data/Int/Basic.lean", "def_pos": [242, 5], "def_end_pos": [242, 11]}]], "state_before": "case refine'_1\n\u03b1\u271d \u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\nf : \u03b1 \u2192 \u2124\n\u22a2 card (Finset.sigma univ fun a => range (Int.natAbs (f \u2191a))) = \u2211 a in s, Int.natAbs (f a)", "state_after": "no goals"}, {"tactic": "intro x hx", "annotated_tactic": ["intro x hx", []], "state_before": "case refine'_2\n\u03b1\u271d \u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\nf : \u03b1 \u2192 \u2124\n\u22a2 \u2200 (a : \u03b1),\n    a \u2208 s \u2192\n      (\u2211 b in Finset.sigma univ fun a => range (Int.natAbs (f \u2191a)),\n          if (fun a => \u2191a.fst) b = a then \u2191((fun a => \u2191sign (f \u2191a.fst)) b) else 0) =\n        f a", "state_after": "case refine'_2\n\u03b1\u271d \u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\nf : \u03b1 \u2192 \u2124\nx : \u03b1\nhx : x \u2208 s\n\u22a2 (\u2211 b in Finset.sigma univ fun a => range (Int.natAbs (f \u2191a)),\n      if (fun a => \u2191a.fst) b = x then \u2191((fun a => \u2191sign (f \u2191a.fst)) b) else 0) =\n    f x"}, {"tactic": "simp [sum_sigma, hx, \u2190 Int.sign_eq_sign, Int.sign_mul_abs, mul_comm |f _|,\n  sum_attach (s := s) (f := fun y => if y = x then f y else 0)]", "annotated_tactic": ["simp [<a>sum_sigma</a>, hx, \u2190 <a>Int.sign_eq_sign</a>, <a>Int.sign_mul_abs</a>, <a>mul_comm</a> |f _|,\n      <a>sum_attach</a> (s := s) (f := fun y => if y = x then f y else 0)]", [{"full_name": "Finset.sum_sigma", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [523, 3], "def_end_pos": [523, 14]}, {"full_name": "Int.sign_eq_sign", "def_path": "Mathlib/Data/Sign.lean", "def_pos": [480, 9], "def_end_pos": [480, 21]}, {"full_name": "Int.sign_mul_abs", "def_path": "Mathlib/Data/Int/Order/Basic.lean", "def_pos": [71, 9], "def_end_pos": [71, 21]}, {"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [302, 9], "def_end_pos": [302, 17]}, {"full_name": "Finset.sum_attach", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [851, 3], "def_end_pos": [851, 14]}]], "state_before": "case refine'_2\n\u03b1\u271d \u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\nf : \u03b1 \u2192 \u2124\nx : \u03b1\nhx : x \u2208 s\n\u22a2 (\u2211 b in Finset.sigma univ fun a => range (Int.natAbs (f \u2191a)),\n      if (fun a => \u2191a.fst) b = x then \u2191((fun a => \u2191sign (f \u2191a.fst)) b) else 0) =\n    f x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "full_name": "MeasureTheory.Measure.ae_sum_iff'", "start": [2026, 1], "end": [2028, 29], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/RBMap/Lemmas.lean", "full_name": "Std.RBNode.size_lt_depth", "start": [30, 1], "end": [37, 73], "traced_tactics": [{"tactic": "decide", "annotated_tactic": ["decide", []], "state_before": "\u03b1 : Type u_1\n\u22a2 0 < 1", "state_after": "no goals"}, {"tactic": "rw [size, depth, Nat.add_right_comm, Nat.pow_succ, Nat.mul_two]", "annotated_tactic": ["rw [<a>size</a>, <a>depth</a>, <a>Nat.add_right_comm</a>, <a>Nat.pow_succ</a>, <a>Nat.mul_two</a>]", [{"full_name": "Std.RBNode.size", "def_path": "lake-packages/std/Std/Data/RBMap/Basic.lean", "def_pos": [346, 13], "def_end_pos": [346, 17]}, {"full_name": "Std.RBNode.depth", "def_path": "lake-packages/std/Std/Data/RBMap/Lemmas.lean", "def_pos": [26, 5], "def_end_pos": [26, 10]}, {"full_name": "Nat.add_right_comm", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [145, 19], "def_end_pos": [145, 33]}, {"full_name": "Nat.pow_succ", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [473, 9], "def_end_pos": [473, 17]}, {"full_name": "Nat.mul_two", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [506, 19], "def_end_pos": [506, 26]}]], "state_before": "\u03b1 : Type u_1\nc\u271d : RBColor\na : RBNode \u03b1\nv\u271d : \u03b1\nb : RBNode \u03b1\n\u22a2 size (node c\u271d a v\u271d b) < 2 ^ depth (node c\u271d a v\u271d b)", "state_after": "\u03b1 : Type u_1\nc\u271d : RBColor\na : RBNode \u03b1\nv\u271d : \u03b1\nb : RBNode \u03b1\n\u22a2 size a + 1 + size b < 2 ^ max (depth a) (depth b) + 2 ^ max (depth a) (depth b)"}, {"tactic": "refine Nat.add_le_add\n  (Nat.lt_of_lt_of_le a.size_lt_depth ?_) (Nat.lt_of_lt_of_le b.size_lt_depth ?_)", "annotated_tactic": ["refine <a>Nat.add_le_add</a>\n      (<a>Nat.lt_of_lt_of_le</a> a.size_lt_depth ?_) (<a>Nat.lt_of_lt_of_le</a> b.size_lt_depth ?_)", [{"full_name": "Nat.add_le_add", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [404, 9], "def_end_pos": [404, 19]}, {"full_name": "Nat.lt_of_lt_of_le", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [259, 19], "def_end_pos": [259, 33]}, {"full_name": "Nat.lt_of_lt_of_le", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [259, 19], "def_end_pos": [259, 33]}]], "state_before": "\u03b1 : Type u_1\nc\u271d : RBColor\na : RBNode \u03b1\nv\u271d : \u03b1\nb : RBNode \u03b1\n\u22a2 size a + 1 + size b < 2 ^ max (depth a) (depth b) + 2 ^ max (depth a) (depth b)", "state_after": "case refine_1\n\u03b1 : Type u_1\nc\u271d : RBColor\na : RBNode \u03b1\nv\u271d : \u03b1\nb : RBNode \u03b1\n\u22a2 2 ^ depth a \u2264 2 ^ max (depth a) (depth b)\n\ncase refine_2\n\u03b1 : Type u_1\nc\u271d : RBColor\na : RBNode \u03b1\nv\u271d : \u03b1\nb : RBNode \u03b1\n\u22a2 2 ^ depth b \u2264 2 ^ max (depth a) (depth b)"}, {"tactic": "exact Nat.pow_le_pow_of_le_right (by decide) (Nat.le_max_left ..)", "annotated_tactic": ["exact <a>Nat.pow_le_pow_of_le_right</a> (by decide) (<a>Nat.le_max_left</a> ..)", [{"full_name": "Nat.pow_le_pow_of_le_right", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [482, 9], "def_end_pos": [482, 31]}, {"full_name": "Nat.le_max_left", "def_path": "lake-packages/std/Std/Data/Nat/Init/Lemmas.lean", "def_pos": [46, 19], "def_end_pos": [46, 30]}]], "state_before": "case refine_1\n\u03b1 : Type u_1\nc\u271d : RBColor\na : RBNode \u03b1\nv\u271d : \u03b1\nb : RBNode \u03b1\n\u22a2 2 ^ depth a \u2264 2 ^ max (depth a) (depth b)", "state_after": "no goals"}, {"tactic": "decide", "annotated_tactic": ["decide", []], "state_before": "\u03b1 : Type u_1\nc\u271d : RBColor\na : RBNode \u03b1\nv\u271d : \u03b1\nb : RBNode \u03b1\n\u22a2 2 > 0", "state_after": "no goals"}, {"tactic": "exact Nat.pow_le_pow_of_le_right (by decide) (Nat.le_max_right ..)", "annotated_tactic": ["exact <a>Nat.pow_le_pow_of_le_right</a> (by decide) (<a>Nat.le_max_right</a> ..)", [{"full_name": "Nat.pow_le_pow_of_le_right", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [482, 9], "def_end_pos": [482, 31]}, {"full_name": "Nat.le_max_right", "def_path": "lake-packages/std/Std/Data/Nat/Init/Lemmas.lean", "def_pos": [48, 19], "def_end_pos": [48, 31]}]], "state_before": "case refine_2\n\u03b1 : Type u_1\nc\u271d : RBColor\na : RBNode \u03b1\nv\u271d : \u03b1\nb : RBNode \u03b1\n\u22a2 2 ^ depth b \u2264 2 ^ max (depth a) (depth b)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "full_name": "MeasureTheory.measure_liminf_eq_zero", "start": [610, 1], "end": [614, 65], "traced_tactics": [{"tactic": "rw [\u2190 le_zero_iff]", "annotated_tactic": ["rw [\u2190 <a>le_zero_iff</a>]", [{"full_name": "le_zero_iff", "def_path": "Mathlib/Algebra/Order/WithZero.lean", "def_pos": [102, 9], "def_end_pos": [102, 20]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\ns : \u2115 \u2192 Set \u03b1\nh : \u2211' (i : \u2115), \u2191\u2191\u03bc (s i) \u2260 \u22a4\n\u22a2 \u2191\u2191\u03bc (liminf s atTop) = 0", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\ns : \u2115 \u2192 Set \u03b1\nh : \u2211' (i : \u2115), \u2191\u2191\u03bc (s i) \u2260 \u22a4\n\u22a2 \u2191\u2191\u03bc (liminf s atTop) \u2264 0"}, {"tactic": "have : liminf s atTop \u2264 limsup s atTop := liminf_le_limsup", "annotated_tactic": ["have : <a>liminf</a> s <a>atTop</a> \u2264 <a>limsup</a> s <a>atTop</a> := <a>liminf_le_limsup</a>", [{"full_name": "Filter.liminf", "def_path": "Mathlib/Order/LiminfLimsup.lean", "def_pos": [426, 5], "def_end_pos": [426, 11]}, {"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}, {"full_name": "Filter.limsup", "def_path": "Mathlib/Order/LiminfLimsup.lean", "def_pos": [420, 5], "def_end_pos": [420, 11]}, {"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}, {"full_name": "Filter.isBounded_ge_of_bot", "def_path": "Mathlib/Order/LiminfLimsup.lean", "def_pos": [310, 9], "def_end_pos": [310, 28]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\ns : \u2115 \u2192 Set \u03b1\nh : \u2211' (i : \u2115), \u2191\u2191\u03bc (s i) \u2260 \u22a4\n\u22a2 \u2191\u2191\u03bc (liminf s atTop) \u2264 0", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\ns : \u2115 \u2192 Set \u03b1\nh : \u2211' (i : \u2115), \u2191\u2191\u03bc (s i) \u2260 \u22a4\nthis : liminf s atTop \u2264 limsup s atTop\n\u22a2 \u2191\u2191\u03bc (liminf s atTop) \u2264 0"}, {"tactic": "exact (\u03bc.mono this).trans (by simp [measure_limsup_eq_zero h])", "annotated_tactic": ["exact (\u03bc.mono this).<a>trans</a> (by simp [<a>measure_limsup_eq_zero</a> h])", [{"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [120, 7], "def_end_pos": [120, 18]}, {"full_name": "MeasureTheory.measure_limsup_eq_zero", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [585, 9], "def_end_pos": [585, 31]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\ns : \u2115 \u2192 Set \u03b1\nh : \u2211' (i : \u2115), \u2191\u2191\u03bc (s i) \u2260 \u22a4\nthis : liminf s atTop \u2264 limsup s atTop\n\u22a2 \u2191\u2191\u03bc (liminf s atTop) \u2264 0", "state_after": "no goals"}, {"tactic": "simp [measure_limsup_eq_zero h]", "annotated_tactic": ["simp [<a>measure_limsup_eq_zero</a> h]", [{"full_name": "MeasureTheory.measure_limsup_eq_zero", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [585, 9], "def_end_pos": [585, 31]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\ns : \u2115 \u2192 Set \u03b1\nh : \u2211' (i : \u2115), \u2191\u2191\u03bc (s i) \u2260 \u22a4\nthis : liminf s atTop \u2264 limsup s atTop\n\u22a2 \u2191\u2191\u03bc (limsup s atTop) \u2264 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/Jacobian.lean", "full_name": "MeasureTheory.addHaar_image_le_lintegral_abs_det_fderiv_aux2", "start": [885, 1], "end": [900, 69], "traced_tactics": [{"tactic": "have :\n  Tendsto (fun \u03b5 : \u211d\u22650 => (\u222b\u207b x in s, ENNReal.ofReal |(f' x).det| \u2202\u03bc) + 2 * \u03b5 * \u03bc s) (\ud835\udcdd[>] 0)\n    (\ud835\udcdd ((\u222b\u207b x in s, ENNReal.ofReal |(f' x).det| \u2202\u03bc) + 2 * (0 : \u211d\u22650) * \u03bc s)) := by\n  apply Tendsto.mono_left _ nhdsWithin_le_nhds\n  refine' tendsto_const_nhds.add _\n  refine' ENNReal.Tendsto.mul_const _ (Or.inr h's)\n  exact ENNReal.Tendsto.const_mul (ENNReal.tendsto_coe.2 tendsto_id) (Or.inr ENNReal.coe_ne_top)", "annotated_tactic": ["have :\n    <a>Tendsto</a> (fun \u03b5 : \u211d\u22650 => (\u222b\u207b x in s, <a>ENNReal.ofReal</a> |(f' x).<a>det</a>| \u2202\u03bc) + 2 * \u03b5 * \u03bc s) (\ud835\udcdd[>] 0)\n      (\ud835\udcdd ((\u222b\u207b x in s, <a>ENNReal.ofReal</a> |(f' x).<a>det</a>| \u2202\u03bc) + 2 * (0 : \u211d\u22650) * \u03bc s)) := by\n    apply <a>Tendsto.mono_left</a> _ <a>nhdsWithin_le_nhds</a>\n    refine' tendsto_const_nhds.add _\n    refine' <a>ENNReal.Tendsto.mul_const</a> _ (<a>Or.inr</a> h's)\n    exact <a>ENNReal.Tendsto.const_mul</a> (<a>ENNReal.tendsto_coe</a>.2 <a>tendsto_id</a>) (<a>Or.inr</a> <a>ENNReal.coe_ne_top</a>)", [{"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2939, 5], "def_end_pos": [2939, 12]}, {"full_name": "ENNReal.ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [172, 29], "def_end_pos": [172, 35]}, {"full_name": "ContinuousLinearMap.det", "def_path": "Mathlib/Topology/Algebra/Module/Determinant.lean", "def_pos": [22, 19], "def_end_pos": [22, 22]}, {"full_name": "ENNReal.ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [172, 29], "def_end_pos": [172, 35]}, {"full_name": "ContinuousLinearMap.det", "def_path": "Mathlib/Topology/Algebra/Module/Determinant.lean", "def_pos": [22, 19], "def_end_pos": [22, 22]}, {"full_name": "Filter.Tendsto.mono_left", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [3036, 9], "def_end_pos": [3036, 26]}, {"full_name": "nhdsWithin_le_nhds", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [204, 9], "def_end_pos": [204, 27]}, {"full_name": "ENNReal.Tendsto.mul_const", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [379, 19], "def_end_pos": [379, 36]}, {"full_name": "Or.inr", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [519, 5], "def_end_pos": [519, 8]}, {"full_name": "ENNReal.Tendsto.const_mul", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [373, 19], "def_end_pos": [373, 36]}, {"full_name": "ENNReal.tendsto_coe", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [71, 9], "def_end_pos": [71, 20]}, {"full_name": "Filter.tendsto_id", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [3028, 9], "def_end_pos": [3028, 19]}, {"full_name": "Or.inr", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [519, 5], "def_end_pos": [519, 8]}, {"full_name": "ENNReal.coe_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [302, 17], "def_end_pos": [302, 27]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nh's : \u2191\u2191\u03bc s \u2260 \u22a4\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u22a2 \u2191\u2191\u03bc (f '' s) \u2264 \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc", "state_after": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nh's : \u2191\u2191\u03bc s \u2260 \u22a4\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nthis :\n  Tendsto (fun \u03b5 => \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc + 2 * \u2191\u03b5 * \u2191\u2191\u03bc s) (\ud835\udcdd[Ioi 0] 0)\n    (\ud835\udcdd (\u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc + 2 * \u21910 * \u2191\u2191\u03bc s))\n\u22a2 \u2191\u2191\u03bc (f '' s) \u2264 \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc"}, {"tactic": "simp only [add_zero, zero_mul, mul_zero, ENNReal.coe_zero] at this", "annotated_tactic": ["simp only [<a>add_zero</a>, <a>zero_mul</a>, <a>mul_zero</a>, <a>ENNReal.coe_zero</a>] at this", [{"full_name": "add_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [469, 3], "def_end_pos": [469, 14]}, {"full_name": "MulZeroClass.zero_mul", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [36, 3], "def_end_pos": [36, 11]}, {"full_name": "MulZeroClass.mul_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [38, 3], "def_end_pos": [38, 11]}, {"full_name": "ENNReal.coe_zero", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [215, 28], "def_end_pos": [215, 36]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nh's : \u2191\u2191\u03bc s \u2260 \u22a4\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nthis :\n  Tendsto (fun \u03b5 => \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc + 2 * \u2191\u03b5 * \u2191\u2191\u03bc s) (\ud835\udcdd[Ioi 0] 0)\n    (\ud835\udcdd (\u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc + 2 * \u21910 * \u2191\u2191\u03bc s))\n\u22a2 \u2191\u2191\u03bc (f '' s) \u2264 \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc", "state_after": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nh's : \u2191\u2191\u03bc s \u2260 \u22a4\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nthis :\n  Tendsto (fun \u03b5 => \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc + 2 * \u2191\u03b5 * \u2191\u2191\u03bc s) (\ud835\udcdd[Ioi 0] 0)\n    (\ud835\udcdd (\u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc))\n\u22a2 \u2191\u2191\u03bc (f '' s) \u2264 \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc"}, {"tactic": "apply ge_of_tendsto this", "annotated_tactic": ["apply <a>ge_of_tendsto</a> this", [{"full_name": "ge_of_tendsto", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [168, 9], "def_end_pos": [168, 22]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nh's : \u2191\u2191\u03bc s \u2260 \u22a4\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nthis :\n  Tendsto (fun \u03b5 => \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc + 2 * \u2191\u03b5 * \u2191\u2191\u03bc s) (\ud835\udcdd[Ioi 0] 0)\n    (\ud835\udcdd (\u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc))\n\u22a2 \u2191\u2191\u03bc (f '' s) \u2264 \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc", "state_after": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nh's : \u2191\u2191\u03bc s \u2260 \u22a4\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nthis :\n  Tendsto (fun \u03b5 => \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc + 2 * \u2191\u03b5 * \u2191\u2191\u03bc s) (\ud835\udcdd[Ioi 0] 0)\n    (\ud835\udcdd (\u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc))\n\u22a2 \u2200\u1da0 (c : \u211d\u22650) in \ud835\udcdd[Ioi 0] 0,\n    \u2191\u2191\u03bc (f '' s) \u2264 \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc + 2 * \u2191c * \u2191\u2191\u03bc s"}, {"tactic": "filter_upwards [self_mem_nhdsWithin]", "annotated_tactic": ["filter_upwards [<a>self_mem_nhdsWithin</a>]", [{"full_name": "self_mem_nhdsWithin", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [151, 9], "def_end_pos": [151, 28]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nh's : \u2191\u2191\u03bc s \u2260 \u22a4\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nthis :\n  Tendsto (fun \u03b5 => \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc + 2 * \u2191\u03b5 * \u2191\u2191\u03bc s) (\ud835\udcdd[Ioi 0] 0)\n    (\ud835\udcdd (\u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc))\n\u22a2 \u2200\u1da0 (c : \u211d\u22650) in \ud835\udcdd[Ioi 0] 0,\n    \u2191\u2191\u03bc (f '' s) \u2264 \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc + 2 * \u2191c * \u2191\u2191\u03bc s", "state_after": "case h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nh's : \u2191\u2191\u03bc s \u2260 \u22a4\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nthis :\n  Tendsto (fun \u03b5 => \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc + 2 * \u2191\u03b5 * \u2191\u2191\u03bc s) (\ud835\udcdd[Ioi 0] 0)\n    (\ud835\udcdd (\u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc))\n\u22a2 \u2200 (a : \u211d\u22650),\n    a \u2208 Ioi 0 \u2192 \u2191\u2191\u03bc (f '' s) \u2264 \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc + 2 * \u2191a * \u2191\u2191\u03bc s"}, {"tactic": "rintro \u03b5 (\u03b5pos : 0 < \u03b5)", "annotated_tactic": ["rintro \u03b5 (\u03b5pos : 0 < \u03b5)", []], "state_before": "case h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nh's : \u2191\u2191\u03bc s \u2260 \u22a4\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nthis :\n  Tendsto (fun \u03b5 => \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc + 2 * \u2191\u03b5 * \u2191\u2191\u03bc s) (\ud835\udcdd[Ioi 0] 0)\n    (\ud835\udcdd (\u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc))\n\u22a2 \u2200 (a : \u211d\u22650),\n    a \u2208 Ioi 0 \u2192 \u2191\u2191\u03bc (f '' s) \u2264 \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc + 2 * \u2191a * \u2191\u2191\u03bc s", "state_after": "case h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nh's : \u2191\u2191\u03bc s \u2260 \u22a4\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nthis :\n  Tendsto (fun \u03b5 => \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc + 2 * \u2191\u03b5 * \u2191\u2191\u03bc s) (\ud835\udcdd[Ioi 0] 0)\n    (\ud835\udcdd (\u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc))\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\n\u22a2 \u2191\u2191\u03bc (f '' s) \u2264 \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc + 2 * \u2191\u03b5 * \u2191\u2191\u03bc s"}, {"tactic": "exact addHaar_image_le_lintegral_abs_det_fderiv_aux1 \u03bc hs hf' \u03b5pos", "annotated_tactic": ["exact <a>addHaar_image_le_lintegral_abs_det_fderiv_aux1</a> \u03bc hs hf' \u03b5pos", [{"full_name": "MeasureTheory.addHaar_image_le_lintegral_abs_det_fderiv_aux1", "def_path": "Mathlib/MeasureTheory/Function/Jacobian.lean", "def_pos": [802, 9], "def_end_pos": [802, 55]}]], "state_before": "case h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nh's : \u2191\u2191\u03bc s \u2260 \u22a4\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nthis :\n  Tendsto (fun \u03b5 => \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc + 2 * \u2191\u03b5 * \u2191\u2191\u03bc s) (\ud835\udcdd[Ioi 0] 0)\n    (\ud835\udcdd (\u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc))\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\n\u22a2 \u2191\u2191\u03bc (f '' s) \u2264 \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc + 2 * \u2191\u03b5 * \u2191\u2191\u03bc s", "state_after": "no goals"}, {"tactic": "apply Tendsto.mono_left _ nhdsWithin_le_nhds", "annotated_tactic": ["apply <a>Tendsto.mono_left</a> _ <a>nhdsWithin_le_nhds</a>", [{"full_name": "Filter.Tendsto.mono_left", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [3036, 9], "def_end_pos": [3036, 26]}, {"full_name": "nhdsWithin_le_nhds", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [204, 9], "def_end_pos": [204, 27]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nh's : \u2191\u2191\u03bc s \u2260 \u22a4\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u22a2 Tendsto (fun \u03b5 => \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc + 2 * \u2191\u03b5 * \u2191\u2191\u03bc s) (\ud835\udcdd[Ioi 0] 0)\n    (\ud835\udcdd (\u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc + 2 * \u21910 * \u2191\u2191\u03bc s))", "state_after": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nh's : \u2191\u2191\u03bc s \u2260 \u22a4\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u22a2 Tendsto (fun \u03b5 => \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc + 2 * \u2191\u03b5 * \u2191\u2191\u03bc s) (\ud835\udcdd 0)\n    (\ud835\udcdd (\u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc + 2 * \u21910 * \u2191\u2191\u03bc s))"}, {"tactic": "refine' tendsto_const_nhds.add _", "annotated_tactic": ["refine' tendsto_const_nhds.add _", []], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nh's : \u2191\u2191\u03bc s \u2260 \u22a4\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u22a2 Tendsto (fun \u03b5 => \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc + 2 * \u2191\u03b5 * \u2191\u2191\u03bc s) (\ud835\udcdd 0)\n    (\ud835\udcdd (\u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc + 2 * \u21910 * \u2191\u2191\u03bc s))", "state_after": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nh's : \u2191\u2191\u03bc s \u2260 \u22a4\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u22a2 Tendsto (fun \u03b5 => 2 * \u2191\u03b5 * \u2191\u2191\u03bc s) (\ud835\udcdd 0) (\ud835\udcdd (2 * \u21910 * \u2191\u2191\u03bc s))"}, {"tactic": "refine' ENNReal.Tendsto.mul_const _ (Or.inr h's)", "annotated_tactic": ["refine' <a>ENNReal.Tendsto.mul_const</a> _ (<a>Or.inr</a> h's)", [{"full_name": "ENNReal.Tendsto.mul_const", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [379, 19], "def_end_pos": [379, 36]}, {"full_name": "Or.inr", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [519, 5], "def_end_pos": [519, 8]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nh's : \u2191\u2191\u03bc s \u2260 \u22a4\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u22a2 Tendsto (fun \u03b5 => 2 * \u2191\u03b5 * \u2191\u2191\u03bc s) (\ud835\udcdd 0) (\ud835\udcdd (2 * \u21910 * \u2191\u2191\u03bc s))", "state_after": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nh's : \u2191\u2191\u03bc s \u2260 \u22a4\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u22a2 Tendsto (fun \u03b5 => 2 * \u2191\u03b5) (\ud835\udcdd 0) (\ud835\udcdd (2 * \u21910))"}, {"tactic": "exact ENNReal.Tendsto.const_mul (ENNReal.tendsto_coe.2 tendsto_id) (Or.inr ENNReal.coe_ne_top)", "annotated_tactic": ["exact <a>ENNReal.Tendsto.const_mul</a> (<a>ENNReal.tendsto_coe</a>.2 <a>tendsto_id</a>) (<a>Or.inr</a> <a>ENNReal.coe_ne_top</a>)", [{"full_name": "ENNReal.Tendsto.const_mul", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [373, 19], "def_end_pos": [373, 36]}, {"full_name": "ENNReal.tendsto_coe", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [71, 9], "def_end_pos": [71, 20]}, {"full_name": "Filter.tendsto_id", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [3028, 9], "def_end_pos": [3028, 19]}, {"full_name": "Or.inr", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [519, 5], "def_end_pos": [519, 8]}, {"full_name": "ENNReal.coe_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [302, 17], "def_end_pos": [302, 27]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nh's : \u2191\u2191\u03bc s \u2260 \u22a4\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u22a2 Tendsto (fun \u03b5 => 2 * \u2191\u03b5) (\ud835\udcdd 0) (\ud835\udcdd (2 * \u21910))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Group/Integral.lean", "full_name": "MeasureTheory.Integrable.comp_div_left", "start": [136, 1], "end": [138, 84], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "full_name": "MeasureTheory.setToFun_congr_left", "start": [1296, 1], "end": [1301, 34], "traced_tactics": [{"tactic": "by_cases hf : Integrable f \u03bc", "annotated_tactic": ["by_cases hf : <a>Integrable</a> f \u03bc", [{"full_name": "MeasureTheory.Integrable", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [442, 5], "def_end_pos": [442, 15]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc T' C'\nh : T = T'\nf : \u03b1 \u2192 E\n\u22a2 setToFun \u03bc T hT f = setToFun \u03bc T' hT' f", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc T' C'\nh : T = T'\nf : \u03b1 \u2192 E\nhf : Integrable f\n\u22a2 setToFun \u03bc T hT f = setToFun \u03bc T' hT' f\n\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc T' C'\nh : T = T'\nf : \u03b1 \u2192 E\nhf : \u00acIntegrable f\n\u22a2 setToFun \u03bc T hT f = setToFun \u03bc T' hT' f"}, {"tactic": "simp_rw [setToFun_eq _ hf, L1.setToL1_congr_left T T' hT hT' h]", "annotated_tactic": ["simp_rw [<a>setToFun_eq</a> _ hf, <a>L1.setToL1_congr_left</a> T T' hT hT' h]", [{"full_name": "MeasureTheory.setToFun_eq", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [1276, 9], "def_end_pos": [1276, 20]}, {"full_name": "MeasureTheory.L1.setToL1_congr_left", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [1054, 9], "def_end_pos": [1054, 27]}]], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc T' C'\nh : T = T'\nf : \u03b1 \u2192 E\nhf : Integrable f\n\u22a2 setToFun \u03bc T hT f = setToFun \u03bc T' hT' f", "state_after": "no goals"}, {"tactic": "simp_rw [setToFun_undef _ hf]", "annotated_tactic": ["simp_rw [<a>setToFun_undef</a> _ hf]", [{"full_name": "MeasureTheory.setToFun_undef", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [1286, 9], "def_end_pos": [1286, 23]}]], "state_before": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc T' C'\nh : T = T'\nf : \u03b1 \u2192 E\nhf : \u00acIntegrable f\n\u22a2 setToFun \u03bc T hT f = setToFun \u03bc T' hT' f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Intervals/Monoid.lean", "full_name": "Set.image_const_add_Ici", "start": [114, 1], "end": [115, 46], "traced_tactics": [{"tactic": "simp only [add_comm a, image_add_const_Ici]", "annotated_tactic": ["simp only [<a>add_comm</a> a, <a>image_add_const_Ici</a>]", [{"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [301, 3], "def_end_pos": [301, 14]}, {"full_name": "Set.image_add_const_Ici", "def_path": "Mathlib/Data/Set/Intervals/Monoid.lean", "def_pos": [79, 9], "def_end_pos": [79, 28]}]], "state_before": "M : Type u_1\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid M\ninst\u271d : ExistsAddOfLE M\na b c d : M\n\u22a2 (fun x => a + x) '' Ici b = Ici (a + b)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Group/Prod.lean", "full_name": "MeasureTheory.measure_eq_div_smul", "start": [352, 1], "end": [356, 94], "traced_tactics": [{"tactic": "ext1 t ht", "annotated_tactic": ["ext1 t ht", []], "state_before": "G : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nhs : MeasurableSet s\nh2s : \u2191\u2191\u03bd s \u2260 0\nh3s : \u2191\u2191\u03bd s \u2260 \u22a4\n\u22a2 \u03bc = (\u2191\u2191\u03bc s / \u2191\u2191\u03bd s) \u2022 \u03bd", "state_after": "case h\nG : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nhs : MeasurableSet s\nh2s : \u2191\u2191\u03bd s \u2260 0\nh3s : \u2191\u2191\u03bd s \u2260 \u22a4\nt : Set G\nht : MeasurableSet t\n\u22a2 \u2191\u2191\u03bc t = \u2191\u2191((\u2191\u2191\u03bc s / \u2191\u2191\u03bd s) \u2022 \u03bd) t"}, {"tactic": "rw [smul_apply, smul_eq_mul, mul_comm, \u2190 mul_div_assoc, mul_comm,\n  measure_mul_measure_eq \u03bc \u03bd hs ht h2s h3s, mul_div_assoc, ENNReal.mul_div_cancel' h2s h3s]", "annotated_tactic": ["rw [<a>smul_apply</a>, <a>smul_eq_mul</a>, <a>mul_comm</a>, \u2190 <a>mul_div_assoc</a>, <a>mul_comm</a>,\n    <a>measure_mul_measure_eq</a> \u03bc \u03bd hs ht h2s h3s, <a>mul_div_assoc</a>, <a>ENNReal.mul_div_cancel'</a> h2s h3s]", [{"full_name": "MeasureTheory.Measure.smul_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [841, 9], "def_end_pos": [841, 19]}, {"full_name": "smul_eq_mul", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [93, 9], "def_end_pos": [93, 20]}, {"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [302, 9], "def_end_pos": [302, 17]}, {"full_name": "mul_div_assoc", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [306, 9], "def_end_pos": [306, 22]}, {"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [302, 9], "def_end_pos": [302, 17]}, {"full_name": "MeasureTheory.measure_mul_measure_eq", "def_path": "Mathlib/MeasureTheory/Group/Prod.lean", "def_pos": [336, 9], "def_end_pos": [336, 31]}, {"full_name": "mul_div_assoc", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [306, 9], "def_end_pos": [306, 22]}, {"full_name": "ENNReal.mul_div_cancel'", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1432, 19], "def_end_pos": [1432, 34]}]], "state_before": "case h\nG : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nhs : MeasurableSet s\nh2s : \u2191\u2191\u03bd s \u2260 0\nh3s : \u2191\u2191\u03bd s \u2260 \u22a4\nt : Set G\nht : MeasurableSet t\n\u22a2 \u2191\u2191\u03bc t = \u2191\u2191((\u2191\u2191\u03bc s / \u2191\u2191\u03bd s) \u2022 \u03bd) t", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/LocallyFinite.lean", "full_name": "Finset.map_add_left_Icc", "start": [1079, 1], "end": [1082, 38], "traced_tactics": [{"tactic": "rw [\u2190 coe_inj, coe_map, coe_Icc, coe_Icc]", "annotated_tactic": ["rw [\u2190 <a>coe_inj</a>, <a>coe_map</a>, <a>coe_Icc</a>, <a>coe_Icc</a>]", [{"full_name": "Finset.coe_inj", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [244, 9], "def_end_pos": [244, 16]}, {"full_name": "Finset.coe_map", "def_path": "Mathlib/Data/Finset/Image.lean", "def_pos": [107, 9], "def_end_pos": [107, 16]}, {"full_name": "Finset.coe_Icc", "def_path": "Mathlib/Order/LocallyFinite.lean", "def_pos": [346, 9], "def_end_pos": [346, 16]}, {"full_name": "Finset.coe_Icc", "def_path": "Mathlib/Order/LocallyFinite.lean", "def_pos": [346, 9], "def_end_pos": [346, 16]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : OrderedCancelAddCommMonoid \u03b1\ninst\u271d\u00b9 : ExistsAddOfLE \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na b c : \u03b1\n\u22a2 map (addLeftEmbedding c) (Icc a b) = Icc (c + a) (c + b)", "state_after": "\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : OrderedCancelAddCommMonoid \u03b1\ninst\u271d\u00b9 : ExistsAddOfLE \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na b c : \u03b1\n\u22a2 \u2191(addLeftEmbedding c) '' Set.Icc a b = Set.Icc (c + a) (c + b)"}, {"tactic": "exact Set.image_const_add_Icc _ _ _", "annotated_tactic": ["exact <a>Set.image_const_add_Icc</a> _ _ _", [{"full_name": "Set.image_const_add_Icc", "def_path": "Mathlib/Data/Set/Intervals/Monoid.lean", "def_pos": [124, 9], "def_end_pos": [124, 28]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : OrderedCancelAddCommMonoid \u03b1\ninst\u271d\u00b9 : ExistsAddOfLE \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na b c : \u03b1\n\u22a2 \u2191(addLeftEmbedding c) '' Set.Icc a b = Set.Icc (c + a) (c + b)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "full_name": "MeasureTheory.lintegral_map", "start": [1281, 1], "end": [1286, 51], "traced_tactics": [{"tactic": "erw [lintegral_eq_iSup_eapprox_lintegral hf, lintegral_eq_iSup_eapprox_lintegral (hf.comp hg)]", "annotated_tactic": ["erw [<a>lintegral_eq_iSup_eapprox_lintegral</a> hf, <a>lintegral_eq_iSup_eapprox_lintegral</a> (hf.comp hg)]", [{"full_name": "MeasureTheory.lintegral_eq_iSup_eapprox_lintegral", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [451, 9], "def_end_pos": [451, 44]}, {"full_name": "MeasureTheory.lintegral_eq_iSup_eapprox_lintegral", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [451, 9], "def_end_pos": [451, 44]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b2 \u2192 \u211d\u22650\u221e\ng : \u03b1 \u2192 \u03b2\nhf : Measurable f\nhg : Measurable g\n\u22a2 \u222b\u207b (a : \u03b2), f a \u2202Measure.map g \u03bc = \u222b\u207b (a : \u03b1), f (g a) \u2202\u03bc", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b2 \u2192 \u211d\u22650\u221e\ng : \u03b1 \u2192 \u03b2\nhf : Measurable f\nhg : Measurable g\n\u22a2 \u2a06 n, SimpleFunc.lintegral (eapprox f n) (Measure.map g \u03bc) = \u2a06 n, SimpleFunc.lintegral (eapprox (f \u2218 g) n) \u03bc"}, {"tactic": "congr with n : 1", "annotated_tactic": ["congr with n : 1", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b2 \u2192 \u211d\u22650\u221e\ng : \u03b1 \u2192 \u03b2\nhf : Measurable f\nhg : Measurable g\n\u22a2 \u2a06 n, SimpleFunc.lintegral (eapprox f n) (Measure.map g \u03bc) = \u2a06 n, SimpleFunc.lintegral (eapprox (f \u2218 g) n) \u03bc", "state_after": "case e_s.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b2 \u2192 \u211d\u22650\u221e\ng : \u03b1 \u2192 \u03b2\nhf : Measurable f\nhg : Measurable g\nn : \u2115\n\u22a2 SimpleFunc.lintegral (eapprox f n) (Measure.map g \u03bc) = SimpleFunc.lintegral (eapprox (f \u2218 g) n) \u03bc"}, {"tactic": "convert SimpleFunc.lintegral_map _ hg", "annotated_tactic": ["convert <a>SimpleFunc.lintegral_map</a> _ hg", [{"full_name": "MeasureTheory.SimpleFunc.lintegral_map", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [1139, 9], "def_end_pos": [1139, 22]}]], "state_before": "case e_s.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b2 \u2192 \u211d\u22650\u221e\ng : \u03b1 \u2192 \u03b2\nhf : Measurable f\nhg : Measurable g\nn : \u2115\n\u22a2 SimpleFunc.lintegral (eapprox f n) (Measure.map g \u03bc) = SimpleFunc.lintegral (eapprox (f \u2218 g) n) \u03bc", "state_after": "case h.e'_3.h.e'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b2 \u2192 \u211d\u22650\u221e\ng : \u03b1 \u2192 \u03b2\nhf : Measurable f\nhg : Measurable g\nn : \u2115\n\u22a2 eapprox (f \u2218 g) n = comp (eapprox f n) g hg"}, {"tactic": "ext1 x", "annotated_tactic": ["ext1 x", []], "state_before": "case h.e'_3.h.e'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b2 \u2192 \u211d\u22650\u221e\ng : \u03b1 \u2192 \u03b2\nhf : Measurable f\nhg : Measurable g\nn : \u2115\n\u22a2 eapprox (f \u2218 g) n = comp (eapprox f n) g hg", "state_after": "case h.e'_3.h.e'_3.H\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b2 \u2192 \u211d\u22650\u221e\ng : \u03b1 \u2192 \u03b2\nhf : Measurable f\nhg : Measurable g\nn : \u2115\nx : \u03b1\n\u22a2 \u2191(eapprox (f \u2218 g) n) x = \u2191(comp (eapprox f n) g hg) x"}, {"tactic": "simp only [eapprox_comp hf hg, coe_comp]", "annotated_tactic": ["simp only [<a>eapprox_comp</a> hf hg, <a>coe_comp</a>]", [{"full_name": "MeasureTheory.SimpleFunc.eapprox_comp", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [910, 9], "def_end_pos": [910, 21]}, {"full_name": "MeasureTheory.SimpleFunc.coe_comp", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [337, 9], "def_end_pos": [337, 17]}]], "state_before": "case h.e'_3.h.e'_3.H\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b2 \u2192 \u211d\u22650\u221e\ng : \u03b1 \u2192 \u03b2\nhf : Measurable f\nhg : Measurable g\nn : \u2115\nx : \u03b1\n\u22a2 \u2191(eapprox (f \u2218 g) n) x = \u2191(comp (eapprox f n) g hg) x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "full_name": "MeasureTheory.measure_union_add_inter'", "start": [147, 1], "end": [149, 70], "traced_tactics": [{"tactic": "rw [union_comm, inter_comm, measure_union_add_inter t hs, add_comm]", "annotated_tactic": ["rw [<a>union_comm</a>, <a>inter_comm</a>, <a>measure_union_add_inter</a> t hs, <a>add_comm</a>]", [{"full_name": "Set.union_comm", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [786, 9], "def_end_pos": [786, 19]}, {"full_name": "Set.inter_comm", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [940, 9], "def_end_pos": [940, 19]}, {"full_name": "MeasureTheory.measure_union_add_inter", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [140, 9], "def_end_pos": [140, 32]}, {"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [301, 3], "def_end_pos": [301, 14]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns s\u2081 s\u2082 t\u271d : Set \u03b1\nhs : MeasurableSet s\nt : Set \u03b1\n\u22a2 \u2191\u2191\u03bc (s \u222a t) + \u2191\u2191\u03bc (s \u2229 t) = \u2191\u2191\u03bc s + \u2191\u2191\u03bc t", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Fin/Lemmas.lean", "full_name": "Fin.val_ne_iff", "start": [40, 1], "end": [40, 74], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/StrongLaw.lean", "full_name": "ProbabilityTheory.strong_law_aux1", "start": [416, 1], "end": [522, 36], "traced_tactics": [{"tactic": "have c_pos : 0 < c := zero_lt_one.trans c_one", "annotated_tactic": ["have c_pos : 0 < c := zero_lt_one.trans c_one", []], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9),\n    \u2200\u1da0 (n : \u2115) in atTop,\n      |\u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 -\n            \u222b (a : \u03a9), Finset.sum (range \u230ac ^ n\u230b\u208a) (fun i => truncation (X i) \u2191i) a| <\n        \u03b5 * \u2191\u230ac ^ n\u230b\u208a", "state_after": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9),\n    \u2200\u1da0 (n : \u2115) in atTop,\n      |\u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 -\n            \u222b (a : \u03a9), Finset.sum (range \u230ac ^ n\u230b\u208a) (fun i => truncation (X i) \u2191i) a| <\n        \u03b5 * \u2191\u230ac ^ n\u230b\u208a"}, {"tactic": "have hX : \u2200 i, AEStronglyMeasurable (X i) \u2119 := fun i =>\n  (hident i).symm.aestronglyMeasurable_snd hint.1", "annotated_tactic": ["have hX : \u2200 i, <a>AEStronglyMeasurable</a> (X i) \u2119 := fun i =>\n    (hident i).symm.aestronglyMeasurable_snd hint.1", [{"full_name": "MeasureTheory.AEStronglyMeasurable", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [93, 5], "def_end_pos": [93, 25]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9),\n    \u2200\u1da0 (n : \u2115) in atTop,\n      |\u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 -\n            \u222b (a : \u03a9), Finset.sum (range \u230ac ^ n\u230b\u208a) (fun i => truncation (X i) \u2191i) a| <\n        \u03b5 * \u2191\u230ac ^ n\u230b\u208a", "state_after": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9),\n    \u2200\u1da0 (n : \u2115) in atTop,\n      |\u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 -\n            \u222b (a : \u03a9), Finset.sum (range \u230ac ^ n\u230b\u208a) (fun i => truncation (X i) \u2191i) a| <\n        \u03b5 * \u2191\u230ac ^ n\u230b\u208a"}, {"tactic": "have A : \u2200 i, StronglyMeasurable (indicator (Set.Ioc (-i : \u211d) i) id) := fun i =>\n  stronglyMeasurable_id.indicator measurableSet_Ioc", "annotated_tactic": ["have A : \u2200 i, <a>StronglyMeasurable</a> (<a>indicator</a> (<a>Set.Ioc</a> (-i : \u211d) i) <a>id</a>) := fun i =>\n    stronglyMeasurable_id.indicator <a>measurableSet_Ioc</a>", [{"full_name": "MeasureTheory.StronglyMeasurable", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [78, 5], "def_end_pos": [78, 23]}, {"full_name": "Set.indicator", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [46, 3], "def_end_pos": [46, 14]}, {"full_name": "Set.Ioc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [69, 5], "def_end_pos": [69, 8]}, {"full_name": "id", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [33, 15], "def_end_pos": [33, 17]}, {"full_name": "measurableSet_Ioc", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [589, 9], "def_end_pos": [589, 26]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9),\n    \u2200\u1da0 (n : \u2115) in atTop,\n      |\u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 -\n            \u222b (a : \u03a9), Finset.sum (range \u230ac ^ n\u230b\u208a) (fun i => truncation (X i) \u2191i) a| <\n        \u03b5 * \u2191\u230ac ^ n\u230b\u208a", "state_after": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9),\n    \u2200\u1da0 (n : \u2115) in atTop,\n      |\u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 -\n            \u222b (a : \u03a9), Finset.sum (range \u230ac ^ n\u230b\u208a) (fun i => truncation (X i) \u2191i) a| <\n        \u03b5 * \u2191\u230ac ^ n\u230b\u208a"}, {"tactic": "set Y := fun n : \u2115 => truncation (X n) n", "annotated_tactic": ["set Y := fun n : \u2115 => <a>truncation</a> (X n) n", [{"full_name": "ProbabilityTheory.truncation", "def_path": "Mathlib/Probability/StrongLaw.lean", "def_pos": [78, 5], "def_end_pos": [78, 15]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9),\n    \u2200\u1da0 (n : \u2115) in atTop,\n      |\u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 -\n            \u222b (a : \u03a9), Finset.sum (range \u230ac ^ n\u230b\u208a) (fun i => truncation (X i) \u2191i) a| <\n        \u03b5 * \u2191\u230ac ^ n\u230b\u208a", "state_after": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9),\n    \u2200\u1da0 (n : \u2115) in atTop,\n      |\u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 - \u222b (a : \u03a9), Finset.sum (range \u230ac ^ n\u230b\u208a) Y a| < \u03b5 * \u2191\u230ac ^ n\u230b\u208a"}, {"tactic": "set S := fun n => \u2211 i in range n, Y i with hS", "annotated_tactic": ["set S := fun n => \u2211 i in <a>range</a> n, Y i with hS", [{"full_name": "Finset.range", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3027, 5], "def_end_pos": [3027, 10]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9),\n    \u2200\u1da0 (n : \u2115) in atTop,\n      |\u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 - \u222b (a : \u03a9), Finset.sum (range \u230ac ^ n\u230b\u208a) Y a| < \u03b5 * \u2191\u230ac ^ n\u230b\u208a", "state_after": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9),\n    \u2200\u1da0 (n : \u2115) in atTop,\n      |\u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 - \u222b (a : \u03a9), Finset.sum (range \u230ac ^ n\u230b\u208a) Y a| < \u03b5 * \u2191\u230ac ^ n\u230b\u208a"}, {"tactic": "let u : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a", "annotated_tactic": ["let u : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a", []], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9),\n    \u2200\u1da0 (n : \u2115) in atTop,\n      |\u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 - \u222b (a : \u03a9), Finset.sum (range \u230ac ^ n\u230b\u208a) Y a| < \u03b5 * \u2191\u230ac ^ n\u230b\u208a", "state_after": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9),\n    \u2200\u1da0 (n : \u2115) in atTop,\n      |\u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 - \u222b (a : \u03a9), Finset.sum (range \u230ac ^ n\u230b\u208a) Y a| < \u03b5 * \u2191\u230ac ^ n\u230b\u208a"}, {"tactic": "have u_mono : Monotone u := fun i j hij => Nat.floor_mono (pow_le_pow c_one.le hij)", "annotated_tactic": ["have u_mono : <a>Monotone</a> u := fun i j hij => <a>Nat.floor_mono</a> (<a>pow_le_pow</a> c_one.le hij)", [{"full_name": "Monotone", "def_path": "Mathlib/Order/Monotone/Basic.lean", "def_pos": [77, 5], "def_end_pos": [77, 13]}, {"full_name": "Nat.floor_mono", "def_path": "Mathlib/Algebra/Order/Floor.lean", "def_pos": [185, 9], "def_end_pos": [185, 19]}, {"full_name": "pow_le_pow", "def_path": "Mathlib/Algebra/GroupPower/Order.lean", "def_pos": [437, 9], "def_end_pos": [437, 19]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9),\n    \u2200\u1da0 (n : \u2115) in atTop,\n      |\u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 - \u222b (a : \u03a9), Finset.sum (range \u230ac ^ n\u230b\u208a) Y a| < \u03b5 * \u2191\u230ac ^ n\u230b\u208a", "state_after": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9),\n    \u2200\u1da0 (n : \u2115) in atTop,\n      |\u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 - \u222b (a : \u03a9), Finset.sum (range \u230ac ^ n\u230b\u208a) Y a| < \u03b5 * \u2191\u230ac ^ n\u230b\u208a"}, {"tactic": "have I1 : \u2200 K, \u2211 j in range K, ((j : \u211d) ^ 2)\u207b\u00b9 * Var[Y j] \u2264 2 * \ud835\udd3c[X 0] := by\n  intro K\n  calc\n    \u2211 j in range K, ((j : \u211d) ^ 2)\u207b\u00b9 * Var[Y j] \u2264\n        \u2211 j in range K, ((j : \u211d) ^ 2)\u207b\u00b9 * \ud835\udd3c[truncation (X 0) j ^ 2] := by\n      apply sum_le_sum fun j _ => ?_\n      refine' mul_le_mul_of_nonneg_left _ (inv_nonneg.2 (sq_nonneg _))\n      rw [(hident j).truncation.variance_eq]\n      exact variance_le_expectation_sq (hX 0).truncation\n    _ \u2264 2 * \ud835\udd3c[X 0] := sum_variance_truncation_le hint (hnonneg 0) K", "annotated_tactic": ["have I1 : \u2200 K, \u2211 j in <a>range</a> K, ((j : \u211d) ^ 2)\u207b\u00b9 * Var[Y j] \u2264 2 * \ud835\udd3c[X 0] := by\n    intro K\n    calc\n      \u2211 j in <a>range</a> K, ((j : \u211d) ^ 2)\u207b\u00b9 * Var[Y j] \u2264\n          \u2211 j in <a>range</a> K, ((j : \u211d) ^ 2)\u207b\u00b9 * \ud835\udd3c[<a>truncation</a> (X 0) j ^ 2] := by\n        apply <a>sum_le_sum</a> fun j _ => ?_\n        refine' <a>mul_le_mul_of_nonneg_left</a> _ (<a>inv_nonneg</a>.2 (<a>sq_nonneg</a> _))\n        rw [(hident j).truncation.variance_eq]\n        exact <a>variance_le_expectation_sq</a> (hX 0).<a>truncation</a>\n      _ \u2264 2 * \ud835\udd3c[X 0] := <a>sum_variance_truncation_le</a> hint (hnonneg 0) K", [{"full_name": "Finset.range", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3027, 5], "def_end_pos": [3027, 10]}, {"full_name": "Finset.range", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3027, 5], "def_end_pos": [3027, 10]}, {"full_name": "Finset.range", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3027, 5], "def_end_pos": [3027, 10]}, {"full_name": "ProbabilityTheory.truncation", "def_path": "Mathlib/Probability/StrongLaw.lean", "def_pos": [78, 5], "def_end_pos": [78, 15]}, {"full_name": "Finset.sum_le_sum", "def_path": "Mathlib/Algebra/BigOperators/Order.lean", "def_pos": [111, 15], "def_end_pos": [111, 25]}, {"full_name": "mul_le_mul_of_nonneg_left", "def_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "def_pos": [152, 9], "def_end_pos": [152, 34]}, {"full_name": "inv_nonneg", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [58, 9], "def_end_pos": [58, 19]}, {"full_name": "sq_nonneg", "def_path": "Mathlib/Algebra/GroupPower/Order.lean", "def_pos": [645, 9], "def_end_pos": [645, 18]}, {"full_name": "ProbabilityTheory.variance_le_expectation_sq", "def_path": "Mathlib/Probability/Variance.lean", "def_pos": [223, 9], "def_end_pos": [223, 35]}, {"full_name": "MeasureTheory.AEStronglyMeasurable.truncation", "def_path": "Mathlib/Probability/StrongLaw.lean", "def_pos": [84, 9], "def_end_pos": [84, 61]}, {"full_name": "ProbabilityTheory.sum_variance_truncation_le", "def_path": "Mathlib/Probability/StrongLaw.lean", "def_pos": [339, 9], "def_end_pos": [339, 35]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9),\n    \u2200\u1da0 (n : \u2115) in atTop,\n      |\u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 - \u222b (a : \u03a9), Finset.sum (range \u230ac ^ n\u230b\u208a) Y a| < \u03b5 * \u2191\u230ac ^ n\u230b\u208a", "state_after": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9),\n    \u2200\u1da0 (n : \u2115) in atTop,\n      |\u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 - \u222b (a : \u03a9), Finset.sum (range \u230ac ^ n\u230b\u208a) Y a| < \u03b5 * \u2191\u230ac ^ n\u230b\u208a"}, {"tactic": "let C := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \ud835\udd3c[X 0])", "annotated_tactic": ["let C := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \ud835\udd3c[X 0])", []], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9),\n    \u2200\u1da0 (n : \u2115) in atTop,\n      |\u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 - \u222b (a : \u03a9), Finset.sum (range \u230ac ^ n\u230b\u208a) Y a| < \u03b5 * \u2191\u230ac ^ n\u230b\u208a", "state_after": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9),\n    \u2200\u1da0 (n : \u2115) in atTop,\n      |\u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 - \u222b (a : \u03a9), Finset.sum (range \u230ac ^ n\u230b\u208a) Y a| < \u03b5 * \u2191\u230ac ^ n\u230b\u208a"}, {"tactic": "have I4 : (\u2211' i, \u2119 {\u03c9 | (u i * \u03b5 : \u211d) \u2264 |S (u i) \u03c9 - \ud835\udd3c[S (u i)]|}) < \u221e :=\n  (le_of_tendsto_of_tendsto' (ENNReal.tendsto_nat_tsum _) tendsto_const_nhds I3).trans_lt\n    ENNReal.ofReal_lt_top", "annotated_tactic": ["have I4 : (\u2211' i, \u2119 {\u03c9 | (u i * \u03b5 : \u211d) \u2264 |S (u i) \u03c9 - \ud835\udd3c[S (u i)]|}) < \u221e :=\n    (<a>le_of_tendsto_of_tendsto'</a> (<a>ENNReal.tendsto_nat_tsum</a> _) <a>tendsto_const_nhds</a> I3).<a>trans_lt</a>\n      <a>ENNReal.ofReal_lt_top</a>", [{"full_name": "le_of_tendsto_of_tendsto'", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [232, 9], "def_end_pos": [232, 34]}, {"full_name": "ENNReal.tendsto_nat_tsum", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [929, 9], "def_end_pos": [929, 25]}, {"full_name": "tendsto_const_nhds", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [1049, 9], "def_end_pos": [1049, 27]}, {"full_name": "LE.le.trans_lt", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [124, 7], "def_end_pos": [124, 21]}, {"full_name": "ENNReal.ofReal_lt_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [314, 17], "def_end_pos": [314, 30]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nI2 : \u2200 (N : \u2115), \u2211 i in range N, (\u2191(u i) ^ 2)\u207b\u00b9 * variance (S (u i)) \u2119 \u2264 C\nI3 : \u2200 (N : \u2115), \u2211 i in range N, \u2191\u2191\u2119 {\u03c9 | \u2191(u i) * \u03b5 \u2264 |S (u i) \u03c9 - \u222b (a : \u03a9), S (u i) a|} \u2264 ENNReal.ofReal (\u03b5\u207b\u00b9 ^ 2 * C)\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9),\n    \u2200\u1da0 (n : \u2115) in atTop,\n      |\u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 - \u222b (a : \u03a9), Finset.sum (range \u230ac ^ n\u230b\u208a) Y a| < \u03b5 * \u2191\u230ac ^ n\u230b\u208a", "state_after": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nI2 : \u2200 (N : \u2115), \u2211 i in range N, (\u2191(u i) ^ 2)\u207b\u00b9 * variance (S (u i)) \u2119 \u2264 C\nI3 : \u2200 (N : \u2115), \u2211 i in range N, \u2191\u2191\u2119 {\u03c9 | \u2191(u i) * \u03b5 \u2264 |S (u i) \u03c9 - \u222b (a : \u03a9), S (u i) a|} \u2264 ENNReal.ofReal (\u03b5\u207b\u00b9 ^ 2 * C)\nI4 : \u2211' (i : \u2115), \u2191\u2191\u2119 {\u03c9 | \u2191(u i) * \u03b5 \u2264 |S (u i) \u03c9 - \u222b (a : \u03a9), S (u i) a|} < \u22a4\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9),\n    \u2200\u1da0 (n : \u2115) in atTop,\n      |\u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 - \u222b (a : \u03a9), Finset.sum (range \u230ac ^ n\u230b\u208a) Y a| < \u03b5 * \u2191\u230ac ^ n\u230b\u208a"}, {"tactic": "filter_upwards [ae_eventually_not_mem I4.ne] with \u03c9 h\u03c9", "annotated_tactic": ["filter_upwards [<a>ae_eventually_not_mem</a> I4.ne] with \u03c9 h\u03c9", [{"full_name": "MeasureTheory.ae_eventually_not_mem", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2710, 9], "def_end_pos": [2710, 30]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nI2 : \u2200 (N : \u2115), \u2211 i in range N, (\u2191(u i) ^ 2)\u207b\u00b9 * variance (S (u i)) \u2119 \u2264 C\nI3 : \u2200 (N : \u2115), \u2211 i in range N, \u2191\u2191\u2119 {\u03c9 | \u2191(u i) * \u03b5 \u2264 |S (u i) \u03c9 - \u222b (a : \u03a9), S (u i) a|} \u2264 ENNReal.ofReal (\u03b5\u207b\u00b9 ^ 2 * C)\nI4 : \u2211' (i : \u2115), \u2191\u2191\u2119 {\u03c9 | \u2191(u i) * \u03b5 \u2264 |S (u i) \u03c9 - \u222b (a : \u03a9), S (u i) a|} < \u22a4\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9),\n    \u2200\u1da0 (n : \u2115) in atTop,\n      |\u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 - \u222b (a : \u03a9), Finset.sum (range \u230ac ^ n\u230b\u208a) Y a| < \u03b5 * \u2191\u230ac ^ n\u230b\u208a", "state_after": "case h\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nI2 : \u2200 (N : \u2115), \u2211 i in range N, (\u2191(u i) ^ 2)\u207b\u00b9 * variance (S (u i)) \u2119 \u2264 C\nI3 : \u2200 (N : \u2115), \u2211 i in range N, \u2191\u2191\u2119 {\u03c9 | \u2191(u i) * \u03b5 \u2264 |S (u i) \u03c9 - \u222b (a : \u03a9), S (u i) a|} \u2264 ENNReal.ofReal (\u03b5\u207b\u00b9 ^ 2 * C)\nI4 : \u2211' (i : \u2115), \u2191\u2191\u2119 {\u03c9 | \u2191(u i) * \u03b5 \u2264 |S (u i) \u03c9 - \u222b (a : \u03a9), S (u i) a|} < \u22a4\n\u03c9 : \u03a9\nh\u03c9 : \u2200\u1da0 (n : \u2115) in atTop, \u00ac\u2191(u n) * \u03b5 \u2264 |S (u n) \u03c9 - \u222b (a : \u03a9), S (u n) a|\n\u22a2 \u2200\u1da0 (n : \u2115) in atTop,\n    |\u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 - \u222b (a : \u03a9), Finset.sum (range \u230ac ^ n\u230b\u208a) Y a| < \u03b5 * \u2191\u230ac ^ n\u230b\u208a"}, {"tactic": "simp_rw [not_le, mul_comm, sum_apply] at h\u03c9", "annotated_tactic": ["simp_rw [<a>not_le</a>, <a>mul_comm</a>, <a>sum_apply</a>] at h\u03c9", [{"full_name": "not_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [373, 9], "def_end_pos": [373, 15]}, {"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [302, 9], "def_end_pos": [302, 17]}, {"full_name": "Finset.sum_apply", "def_path": "Mathlib/Algebra/BigOperators/Pi.lean", "def_pos": [41, 3], "def_end_pos": [41, 14]}]], "state_before": "case h\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nI2 : \u2200 (N : \u2115), \u2211 i in range N, (\u2191(u i) ^ 2)\u207b\u00b9 * variance (S (u i)) \u2119 \u2264 C\nI3 : \u2200 (N : \u2115), \u2211 i in range N, \u2191\u2191\u2119 {\u03c9 | \u2191(u i) * \u03b5 \u2264 |S (u i) \u03c9 - \u222b (a : \u03a9), S (u i) a|} \u2264 ENNReal.ofReal (\u03b5\u207b\u00b9 ^ 2 * C)\nI4 : \u2211' (i : \u2115), \u2191\u2191\u2119 {\u03c9 | \u2191(u i) * \u03b5 \u2264 |S (u i) \u03c9 - \u222b (a : \u03a9), S (u i) a|} < \u22a4\n\u03c9 : \u03a9\nh\u03c9 : \u2200\u1da0 (n : \u2115) in atTop, \u00ac\u2191(u n) * \u03b5 \u2264 |S (u n) \u03c9 - \u222b (a : \u03a9), S (u n) a|\n\u22a2 \u2200\u1da0 (n : \u2115) in atTop,\n    |\u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 - \u222b (a : \u03a9), Finset.sum (range \u230ac ^ n\u230b\u208a) Y a| < \u03b5 * \u2191\u230ac ^ n\u230b\u208a", "state_after": "case h\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nI2 : \u2200 (N : \u2115), \u2211 i in range N, (\u2191(u i) ^ 2)\u207b\u00b9 * variance (S (u i)) \u2119 \u2264 C\nI3 : \u2200 (N : \u2115), \u2211 i in range N, \u2191\u2191\u2119 {\u03c9 | \u2191(u i) * \u03b5 \u2264 |S (u i) \u03c9 - \u222b (a : \u03a9), S (u i) a|} \u2264 ENNReal.ofReal (\u03b5\u207b\u00b9 ^ 2 * C)\nI4 : \u2211' (i : \u2115), \u2191\u2191\u2119 {\u03c9 | \u2191(u i) * \u03b5 \u2264 |S (u i) \u03c9 - \u222b (a : \u03a9), S (u i) a|} < \u22a4\n\u03c9 : \u03a9\nh\u03c9 :\n  \u2200\u1da0 (n : \u2115) in atTop,\n    |\u2211 c in range \u230ac ^ n\u230b\u208a, truncation (X c) (\u2191c) \u03c9 - \u222b (a : \u03a9), \u2211 c in range \u230ac ^ n\u230b\u208a, truncation (X c) (\u2191c) a| <\n      \u03b5 * \u2191\u230ac ^ n\u230b\u208a\n\u22a2 \u2200\u1da0 (n : \u2115) in atTop,\n    |\u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 - \u222b (a : \u03a9), Finset.sum (range \u230ac ^ n\u230b\u208a) Y a| < \u03b5 * \u2191\u230ac ^ n\u230b\u208a"}, {"tactic": "convert h\u03c9", "annotated_tactic": ["convert h\u03c9", []], "state_before": "case h\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nI2 : \u2200 (N : \u2115), \u2211 i in range N, (\u2191(u i) ^ 2)\u207b\u00b9 * variance (S (u i)) \u2119 \u2264 C\nI3 : \u2200 (N : \u2115), \u2211 i in range N, \u2191\u2191\u2119 {\u03c9 | \u2191(u i) * \u03b5 \u2264 |S (u i) \u03c9 - \u222b (a : \u03a9), S (u i) a|} \u2264 ENNReal.ofReal (\u03b5\u207b\u00b9 ^ 2 * C)\nI4 : \u2211' (i : \u2115), \u2191\u2191\u2119 {\u03c9 | \u2191(u i) * \u03b5 \u2264 |S (u i) \u03c9 - \u222b (a : \u03a9), S (u i) a|} < \u22a4\n\u03c9 : \u03a9\nh\u03c9 :\n  \u2200\u1da0 (n : \u2115) in atTop,\n    |\u2211 c in range \u230ac ^ n\u230b\u208a, truncation (X c) (\u2191c) \u03c9 - \u222b (a : \u03a9), \u2211 c in range \u230ac ^ n\u230b\u208a, truncation (X c) (\u2191c) a| <\n      \u03b5 * \u2191\u230ac ^ n\u230b\u208a\n\u22a2 \u2200\u1da0 (n : \u2115) in atTop,\n    |\u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 - \u222b (a : \u03a9), Finset.sum (range \u230ac ^ n\u230b\u208a) Y a| < \u03b5 * \u2191\u230ac ^ n\u230b\u208a", "state_after": "case h.e'_2.h.h.e'_3.h.e'_3.h.e'_6.h.e'_7.h\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nI2 : \u2200 (N : \u2115), \u2211 i in range N, (\u2191(u i) ^ 2)\u207b\u00b9 * variance (S (u i)) \u2119 \u2264 C\nI3 : \u2200 (N : \u2115), \u2211 i in range N, \u2191\u2191\u2119 {\u03c9 | \u2191(u i) * \u03b5 \u2264 |S (u i) \u03c9 - \u222b (a : \u03a9), S (u i) a|} \u2264 ENNReal.ofReal (\u03b5\u207b\u00b9 ^ 2 * C)\nI4 : \u2211' (i : \u2115), \u2191\u2191\u2119 {\u03c9 | \u2191(u i) * \u03b5 \u2264 |S (u i) \u03c9 - \u222b (a : \u03a9), S (u i) a|} < \u22a4\n\u03c9 : \u03a9\nh\u03c9 :\n  \u2200\u1da0 (n : \u2115) in atTop,\n    |\u2211 c in range \u230ac ^ n\u230b\u208a, truncation (X c) (\u2191c) \u03c9 - \u222b (a : \u03a9), \u2211 c in range \u230ac ^ n\u230b\u208a, truncation (X c) (\u2191c) a| <\n      \u03b5 * \u2191\u230ac ^ n\u230b\u208a\nx\u271d\u00b9 : \u2115\nx\u271d : \u03a9\n\u22a2 Finset.sum (range \u230ac ^ x\u271d\u00b9\u230b\u208a) Y x\u271d = \u2211 c in range \u230ac ^ x\u271d\u00b9\u230b\u208a, truncation (X c) (\u2191c) x\u271d"}, {"tactic": "simp only [sum_apply]", "annotated_tactic": ["simp only [<a>sum_apply</a>]", [{"full_name": "Finset.sum_apply", "def_path": "Mathlib/Algebra/BigOperators/Pi.lean", "def_pos": [41, 3], "def_end_pos": [41, 14]}]], "state_before": "case h.e'_2.h.h.e'_3.h.e'_3.h.e'_6.h.e'_7.h\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nI2 : \u2200 (N : \u2115), \u2211 i in range N, (\u2191(u i) ^ 2)\u207b\u00b9 * variance (S (u i)) \u2119 \u2264 C\nI3 : \u2200 (N : \u2115), \u2211 i in range N, \u2191\u2191\u2119 {\u03c9 | \u2191(u i) * \u03b5 \u2264 |S (u i) \u03c9 - \u222b (a : \u03a9), S (u i) a|} \u2264 ENNReal.ofReal (\u03b5\u207b\u00b9 ^ 2 * C)\nI4 : \u2211' (i : \u2115), \u2191\u2191\u2119 {\u03c9 | \u2191(u i) * \u03b5 \u2264 |S (u i) \u03c9 - \u222b (a : \u03a9), S (u i) a|} < \u22a4\n\u03c9 : \u03a9\nh\u03c9 :\n  \u2200\u1da0 (n : \u2115) in atTop,\n    |\u2211 c in range \u230ac ^ n\u230b\u208a, truncation (X c) (\u2191c) \u03c9 - \u222b (a : \u03a9), \u2211 c in range \u230ac ^ n\u230b\u208a, truncation (X c) (\u2191c) a| <\n      \u03b5 * \u2191\u230ac ^ n\u230b\u208a\nx\u271d\u00b9 : \u2115\nx\u271d : \u03a9\n\u22a2 Finset.sum (range \u230ac ^ x\u271d\u00b9\u230b\u208a) Y x\u271d = \u2211 c in range \u230ac ^ x\u271d\u00b9\u230b\u208a, truncation (X c) (\u2191c) x\u271d", "state_after": "no goals"}, {"tactic": "intro K", "annotated_tactic": ["intro K", []], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\n\u22a2 \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a", "state_after": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nK : \u2115\n\u22a2 \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a"}, {"tactic": "calc\n  \u2211 j in range K, ((j : \u211d) ^ 2)\u207b\u00b9 * Var[Y j] \u2264\n      \u2211 j in range K, ((j : \u211d) ^ 2)\u207b\u00b9 * \ud835\udd3c[truncation (X 0) j ^ 2] := by\n    apply sum_le_sum fun j _ => ?_\n    refine' mul_le_mul_of_nonneg_left _ (inv_nonneg.2 (sq_nonneg _))\n    rw [(hident j).truncation.variance_eq]\n    exact variance_le_expectation_sq (hX 0).truncation\n  _ \u2264 2 * \ud835\udd3c[X 0] := sum_variance_truncation_le hint (hnonneg 0) K", "annotated_tactic": ["calc\n      \u2211 j in <a>range</a> K, ((j : \u211d) ^ 2)\u207b\u00b9 * Var[Y j] \u2264\n          \u2211 j in <a>range</a> K, ((j : \u211d) ^ 2)\u207b\u00b9 * \ud835\udd3c[<a>truncation</a> (X 0) j ^ 2] := by\n        apply <a>sum_le_sum</a> fun j _ => ?_\n        refine' <a>mul_le_mul_of_nonneg_left</a> _ (<a>inv_nonneg</a>.2 (<a>sq_nonneg</a> _))\n        rw [(hident j).truncation.variance_eq]\n        exact <a>variance_le_expectation_sq</a> (hX 0).<a>truncation</a>\n      _ \u2264 2 * \ud835\udd3c[X 0] := <a>sum_variance_truncation_le</a> hint (hnonneg 0) K", [{"full_name": "Finset.range", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3027, 5], "def_end_pos": [3027, 10]}, {"full_name": "Finset.range", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3027, 5], "def_end_pos": [3027, 10]}, {"full_name": "ProbabilityTheory.truncation", "def_path": "Mathlib/Probability/StrongLaw.lean", "def_pos": [78, 5], "def_end_pos": [78, 15]}, {"full_name": "Finset.sum_le_sum", "def_path": "Mathlib/Algebra/BigOperators/Order.lean", "def_pos": [111, 15], "def_end_pos": [111, 25]}, {"full_name": "mul_le_mul_of_nonneg_left", "def_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "def_pos": [152, 9], "def_end_pos": [152, 34]}, {"full_name": "inv_nonneg", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [58, 9], "def_end_pos": [58, 19]}, {"full_name": "sq_nonneg", "def_path": "Mathlib/Algebra/GroupPower/Order.lean", "def_pos": [645, 9], "def_end_pos": [645, 18]}, {"full_name": "ProbabilityTheory.variance_le_expectation_sq", "def_path": "Mathlib/Probability/Variance.lean", "def_pos": [223, 9], "def_end_pos": [223, 35]}, {"full_name": "MeasureTheory.AEStronglyMeasurable.truncation", "def_path": "Mathlib/Probability/StrongLaw.lean", "def_pos": [84, 9], "def_end_pos": [84, 61]}, {"full_name": "ProbabilityTheory.sum_variance_truncation_le", "def_path": "Mathlib/Probability/StrongLaw.lean", "def_pos": [339, 9], "def_end_pos": [339, 35]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nK : \u2115\n\u22a2 \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a", "state_after": "no goals"}, {"tactic": "apply sum_le_sum fun j _ => ?_", "annotated_tactic": ["apply <a>sum_le_sum</a> fun j _ => ?_", [{"full_name": "Finset.sum_le_sum", "def_path": "Mathlib/Algebra/BigOperators/Order.lean", "def_pos": [111, 15], "def_end_pos": [111, 25]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nK : \u2115\n\u22a2 \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * \u222b (a : \u03a9), (truncation (X 0) \u2191j ^ 2) a", "state_after": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nK j : \u2115\nx\u271d : j \u2208 range K\n\u22a2 (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 (\u2191j ^ 2)\u207b\u00b9 * \u222b (a : \u03a9), (truncation (X 0) \u2191j ^ 2) a"}, {"tactic": "refine' mul_le_mul_of_nonneg_left _ (inv_nonneg.2 (sq_nonneg _))", "annotated_tactic": ["refine' <a>mul_le_mul_of_nonneg_left</a> _ (<a>inv_nonneg</a>.2 (<a>sq_nonneg</a> _))", [{"full_name": "mul_le_mul_of_nonneg_left", "def_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "def_pos": [152, 9], "def_end_pos": [152, 34]}, {"full_name": "inv_nonneg", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [58, 9], "def_end_pos": [58, 19]}, {"full_name": "sq_nonneg", "def_path": "Mathlib/Algebra/GroupPower/Order.lean", "def_pos": [645, 9], "def_end_pos": [645, 18]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nK j : \u2115\nx\u271d : j \u2208 range K\n\u22a2 (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 (\u2191j ^ 2)\u207b\u00b9 * \u222b (a : \u03a9), (truncation (X 0) \u2191j ^ 2) a", "state_after": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nK j : \u2115\nx\u271d : j \u2208 range K\n\u22a2 variance (Y j) \u2119 \u2264 \u222b (a : \u03a9), (truncation (X 0) \u2191j ^ 2) a"}, {"tactic": "rw [(hident j).truncation.variance_eq]", "annotated_tactic": ["rw [(hident j).truncation.variance_eq]", []], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nK j : \u2115\nx\u271d : j \u2208 range K\n\u22a2 variance (Y j) \u2119 \u2264 \u222b (a : \u03a9), (truncation (X 0) \u2191j ^ 2) a", "state_after": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nK j : \u2115\nx\u271d : j \u2208 range K\n\u22a2 variance (truncation (X 0) \u2191j) \u2119 \u2264 \u222b (a : \u03a9), (truncation (X 0) \u2191j ^ 2) a"}, {"tactic": "exact variance_le_expectation_sq (hX 0).truncation", "annotated_tactic": ["exact <a>variance_le_expectation_sq</a> (hX 0).<a>truncation</a>", [{"full_name": "ProbabilityTheory.variance_le_expectation_sq", "def_path": "Mathlib/Probability/Variance.lean", "def_pos": [223, 9], "def_end_pos": [223, 35]}, {"full_name": "MeasureTheory.AEStronglyMeasurable.truncation", "def_path": "Mathlib/Probability/StrongLaw.lean", "def_pos": [84, 9], "def_end_pos": [84, 61]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nK j : \u2115\nx\u271d : j \u2208 range K\n\u22a2 variance (truncation (X 0) \u2191j) \u2119 \u2264 \u222b (a : \u03a9), (truncation (X 0) \u2191j ^ 2) a", "state_after": "no goals"}, {"tactic": "intro N", "annotated_tactic": ["intro N", []], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\n\u22a2 \u2200 (N : \u2115), \u2211 i in range N, (\u2191(u i) ^ 2)\u207b\u00b9 * variance (S (u i)) \u2119 \u2264 C", "state_after": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nN : \u2115\n\u22a2 \u2211 i in range N, (\u2191(u i) ^ 2)\u207b\u00b9 * variance (S (u i)) \u2119 \u2264 C"}, {"tactic": "congr 1 with i", "annotated_tactic": ["congr 1 with i", []], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nN : \u2115\n\u22a2 \u2211 i in range N, (\u2191(u i) ^ 2)\u207b\u00b9 * variance (S (u i)) \u2119 =\n    \u2211 i in range N, (\u2191(u i) ^ 2)\u207b\u00b9 * \u2211 j in range (u i), variance (Y j) \u2119", "state_after": "case e_f.h\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nN i : \u2115\n\u22a2 (\u2191(u i) ^ 2)\u207b\u00b9 * variance (S (u i)) \u2119 = (\u2191(u i) ^ 2)\u207b\u00b9 * \u2211 j in range (u i), variance (Y j) \u2119"}, {"tactic": "congr 1", "annotated_tactic": ["congr 1", []], "state_before": "case e_f.h\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nN i : \u2115\n\u22a2 (\u2191(u i) ^ 2)\u207b\u00b9 * variance (S (u i)) \u2119 = (\u2191(u i) ^ 2)\u207b\u00b9 * \u2211 j in range (u i), variance (Y j) \u2119", "state_after": "case e_f.h.e_a\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nN i : \u2115\n\u22a2 variance (S (u i)) \u2119 = \u2211 j in range (u i), variance (Y j) \u2119"}, {"tactic": "rw [hS, IndepFun.variance_sum]", "annotated_tactic": ["rw [hS, <a>IndepFun.variance_sum</a>]", [{"full_name": "ProbabilityTheory.IndepFun.variance_sum", "def_path": "Mathlib/Probability/Variance.lean", "def_pos": [320, 9], "def_end_pos": [320, 30]}]], "state_before": "case e_f.h.e_a\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nN i : \u2115\n\u22a2 variance (S (u i)) \u2119 = \u2211 j in range (u i), variance (Y j) \u2119", "state_after": "case e_f.h.e_a.hs\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nN i : \u2115\n\u22a2 \u2200 (i_1 : \u2115), i_1 \u2208 range (u i) \u2192 Mem\u2112p (Y i_1) 2\n\ncase e_f.h.e_a.h\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nN i : \u2115\n\u22a2 Set.Pairwise \u2191(range (u i)) fun i j => IndepFun (Y i) (Y j)"}, {"tactic": "intro j _", "annotated_tactic": ["intro j _", []], "state_before": "case e_f.h.e_a.hs\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nN i : \u2115\n\u22a2 \u2200 (i_1 : \u2115), i_1 \u2208 range (u i) \u2192 Mem\u2112p (Y i_1) 2", "state_after": "case e_f.h.e_a.hs\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nN i j : \u2115\na\u271d : j \u2208 range (u i)\n\u22a2 Mem\u2112p (Y j) 2"}, {"tactic": "exact (hident j).aestronglyMeasurable_fst.mem\u2112p_truncation", "annotated_tactic": ["exact (hident j).aestronglyMeasurable_fst.mem\u2112p_truncation", []], "state_before": "case e_f.h.e_a.hs\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nN i j : \u2115\na\u271d : j \u2208 range (u i)\n\u22a2 Mem\u2112p (Y j) 2", "state_after": "no goals"}, {"tactic": "intro k _ l _ hkl", "annotated_tactic": ["intro k _ l _ hkl", []], "state_before": "case e_f.h.e_a.h\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nN i : \u2115\n\u22a2 Set.Pairwise \u2191(range (u i)) fun i j => IndepFun (Y i) (Y j)", "state_after": "case e_f.h.e_a.h\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nN i k : \u2115\na\u271d\u00b9 : k \u2208 \u2191(range (u i))\nl : \u2115\na\u271d : l \u2208 \u2191(range (u i))\nhkl : k \u2260 l\n\u22a2 IndepFun (Y k) (Y l)"}, {"tactic": "exact (hindep hkl).comp (A k).measurable (A l).measurable", "annotated_tactic": ["exact (hindep hkl).<a>comp</a> (A k).<a>measurable</a> (A l).<a>measurable</a>", [{"full_name": "ProbabilityTheory.IndepFun.comp", "def_path": "Mathlib/Probability/Independence/Basic.lean", "def_pos": [574, 9], "def_end_pos": [574, 22]}, {"full_name": "MeasureTheory.StronglyMeasurable.measurable", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [341, 19], "def_end_pos": [341, 29]}, {"full_name": "MeasureTheory.StronglyMeasurable.measurable", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [341, 19], "def_end_pos": [341, 29]}]], "state_before": "case e_f.h.e_a.h\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nN i k : \u2115\na\u271d\u00b9 : k \u2208 \u2191(range (u i))\nl : \u2115\na\u271d : l \u2208 \u2191(range (u i))\nhkl : k \u2260 l\n\u22a2 IndepFun (Y k) (Y l)", "state_after": "no goals"}, {"tactic": "simp_rw [mul_sum, sum_mul, sum_sigma']", "annotated_tactic": ["simp_rw [<a>mul_sum</a>, <a>sum_mul</a>, <a>sum_sigma'</a>]", [{"full_name": "Finset.mul_sum", "def_path": "Mathlib/Algebra/BigOperators/Ring.lean", "def_pos": [55, 9], "def_end_pos": [55, 16]}, {"full_name": "Finset.sum_mul", "def_path": "Mathlib/Algebra/BigOperators/Ring.lean", "def_pos": [51, 9], "def_end_pos": [51, 16]}, {"full_name": "Finset.sum_sigma'", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [531, 3], "def_end_pos": [531, 14]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nN : \u2115\n\u22a2 \u2211 i in range N, (\u2191(u i) ^ 2)\u207b\u00b9 * \u2211 j in range (u i), variance (Y j) \u2119 =\n    \u2211 j in range (u (N - 1)), (\u2211 i in filter (fun i => j < u i) (range N), (\u2191(u i) ^ 2)\u207b\u00b9) * variance (Y j) \u2119", "state_after": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nN : \u2115\n\u22a2 \u2211 x in Finset.sigma (range N) fun a => range \u230ac ^ a\u230b\u208a,\n      (\u2191\u230ac ^ x.fst\u230b\u208a ^ 2)\u207b\u00b9 * variance (truncation (X x.snd) \u2191x.snd) \u2119 =\n    \u2211 x in Finset.sigma (range \u230ac ^ (N - 1)\u230b\u208a) fun a => filter (fun i => a < \u230ac ^ i\u230b\u208a) (range N),\n      (\u2191\u230ac ^ x.snd\u230b\u208a ^ 2)\u207b\u00b9 * variance (truncation (X x.fst) \u2191x.fst) \u2119"}, {"tactic": "refine' sum_bij' (fun (p : \u03a3 _ : \u2115, \u2115) _ => (\u27e8p.2, p.1\u27e9 : \u03a3 _ : \u2115, \u2115)) _ (fun a _ => rfl)\n  (fun (p : \u03a3 _ : \u2115, \u2115) _ => (\u27e8p.2, p.1\u27e9 : \u03a3 _ : \u2115, \u2115)) _ _ _", "annotated_tactic": ["refine' <a>sum_bij'</a> (fun (p : \u03a3 _ : \u2115, \u2115) _ => (\u27e8p.2, p.1\u27e9 : \u03a3 _ : \u2115, \u2115)) _ (fun a _ => <a>rfl</a>)\n          (fun (p : \u03a3 _ : \u2115, \u2115) _ => (\u27e8p.2, p.1\u27e9 : \u03a3 _ : \u2115, \u2115)) _ _ _", [{"full_name": "Finset.sum_bij'", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [560, 3], "def_end_pos": [560, 14]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nN : \u2115\n\u22a2 \u2211 x in Finset.sigma (range N) fun a => range \u230ac ^ a\u230b\u208a,\n      (\u2191\u230ac ^ x.fst\u230b\u208a ^ 2)\u207b\u00b9 * variance (truncation (X x.snd) \u2191x.snd) \u2119 =\n    \u2211 x in Finset.sigma (range \u230ac ^ (N - 1)\u230b\u208a) fun a => filter (fun i => a < \u230ac ^ i\u230b\u208a) (range N),\n      (\u2191\u230ac ^ x.snd\u230b\u208a ^ 2)\u207b\u00b9 * variance (truncation (X x.fst) \u2191x.fst) \u2119", "state_after": "case refine'_1\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nN : \u2115\n\u22a2 \u2200 (a : (_ : \u2115) \u00d7 \u2115) (ha : a \u2208 Finset.sigma (range N) fun a => range \u230ac ^ a\u230b\u208a),\n    (fun p x => { fst := p.snd, snd := p.fst }) a ha \u2208\n      Finset.sigma (range \u230ac ^ (N - 1)\u230b\u208a) fun a => filter (fun i => a < \u230ac ^ i\u230b\u208a) (range N)\n\ncase refine'_2\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nN : \u2115\n\u22a2 \u2200 (a : (_ : \u2115) \u00d7 \u2115) (ha : a \u2208 Finset.sigma (range \u230ac ^ (N - 1)\u230b\u208a) fun a => filter (fun i => a < \u230ac ^ i\u230b\u208a) (range N)),\n    (fun p x => { fst := p.snd, snd := p.fst }) a ha \u2208 Finset.sigma (range N) fun a => range \u230ac ^ a\u230b\u208a\n\ncase refine'_3\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nN : \u2115\n\u22a2 \u2200 (a : (_ : \u2115) \u00d7 \u2115) (ha : a \u2208 Finset.sigma (range N) fun a => range \u230ac ^ a\u230b\u208a),\n    (fun p x => { fst := p.snd, snd := p.fst }) ((fun p x => { fst := p.snd, snd := p.fst }) a ha)\n        (_ : (fun p x => { fst := p.snd, snd := p.fst }) a ha \u2208 ?m.218496) =\n      a\n\ncase refine'_4\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nN : \u2115\n\u22a2 \u2200 (a : (_ : \u2115) \u00d7 \u2115) (ha : a \u2208 Finset.sigma (range \u230ac ^ (N - 1)\u230b\u208a) fun a => filter (fun i => a < \u230ac ^ i\u230b\u208a) (range N)),\n    (fun p x => { fst := p.snd, snd := p.fst }) ((fun p x => { fst := p.snd, snd := p.fst }) a ha)\n        (_ : (fun p x => { fst := p.snd, snd := p.fst }) a ha \u2208 ?m.218495) =\n      a"}, {"tactic": "rintro \u27e8i, j\u27e9 hij", "annotated_tactic": ["rintro \u27e8i, j\u27e9 hij", []], "state_before": "case refine'_1\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nN : \u2115\n\u22a2 \u2200 (a : (_ : \u2115) \u00d7 \u2115) (ha : a \u2208 Finset.sigma (range N) fun a => range \u230ac ^ a\u230b\u208a),\n    (fun p x => { fst := p.snd, snd := p.fst }) a ha \u2208\n      Finset.sigma (range \u230ac ^ (N - 1)\u230b\u208a) fun a => filter (fun i => a < \u230ac ^ i\u230b\u208a) (range N)", "state_after": "case refine'_1.mk\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nN i j : \u2115\nhij : { fst := i, snd := j } \u2208 Finset.sigma (range N) fun a => range \u230ac ^ a\u230b\u208a\n\u22a2 (fun p x => { fst := p.snd, snd := p.fst }) { fst := i, snd := j } hij \u2208\n    Finset.sigma (range \u230ac ^ (N - 1)\u230b\u208a) fun a => filter (fun i => a < \u230ac ^ i\u230b\u208a) (range N)"}, {"tactic": "simp only [mem_sigma, mem_range] at hij", "annotated_tactic": ["simp only [<a>mem_sigma</a>, <a>mem_range</a>] at hij", [{"full_name": "Finset.mem_sigma", "def_path": "Mathlib/Data/Finset/Sigma.lean", "def_pos": [49, 9], "def_end_pos": [49, 18]}, {"full_name": "Finset.mem_range", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3037, 9], "def_end_pos": [3037, 18]}]], "state_before": "case refine'_1.mk\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nN i j : \u2115\nhij : { fst := i, snd := j } \u2208 Finset.sigma (range N) fun a => range \u230ac ^ a\u230b\u208a\n\u22a2 (fun p x => { fst := p.snd, snd := p.fst }) { fst := i, snd := j } hij \u2208\n    Finset.sigma (range \u230ac ^ (N - 1)\u230b\u208a) fun a => filter (fun i => a < \u230ac ^ i\u230b\u208a) (range N)", "state_after": "case refine'_1.mk\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nN i j : \u2115\nhij\u271d : { fst := i, snd := j } \u2208 Finset.sigma (range N) fun a => range \u230ac ^ a\u230b\u208a\nhij : i < N \u2227 j < \u230ac ^ i\u230b\u208a\n\u22a2 (fun p x => { fst := p.snd, snd := p.fst }) { fst := i, snd := j } hij\u271d \u2208\n    Finset.sigma (range \u230ac ^ (N - 1)\u230b\u208a) fun a => filter (fun i => a < \u230ac ^ i\u230b\u208a) (range N)"}, {"tactic": "simp only [hij.1, hij.2, mem_sigma, mem_range, mem_filter, and_true_iff]", "annotated_tactic": ["simp only [hij.1, hij.2, <a>mem_sigma</a>, <a>mem_range</a>, <a>mem_filter</a>, <a>and_true_iff</a>]", [{"full_name": "Finset.mem_sigma", "def_path": "Mathlib/Data/Finset/Sigma.lean", "def_pos": [49, 9], "def_end_pos": [49, 18]}, {"full_name": "Finset.mem_range", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3037, 9], "def_end_pos": [3037, 18]}, {"full_name": "Finset.mem_filter", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2708, 9], "def_end_pos": [2708, 19]}, {"full_name": "and_true_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [145, 9], "def_end_pos": [145, 21]}]], "state_before": "case refine'_1.mk\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nN i j : \u2115\nhij\u271d : { fst := i, snd := j } \u2208 Finset.sigma (range N) fun a => range \u230ac ^ a\u230b\u208a\nhij : i < N \u2227 j < \u230ac ^ i\u230b\u208a\n\u22a2 (fun p x => { fst := p.snd, snd := p.fst }) { fst := i, snd := j } hij\u271d \u2208\n    Finset.sigma (range \u230ac ^ (N - 1)\u230b\u208a) fun a => filter (fun i => a < \u230ac ^ i\u230b\u208a) (range N)", "state_after": "case refine'_1.mk\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nN i j : \u2115\nhij\u271d : { fst := i, snd := j } \u2208 Finset.sigma (range N) fun a => range \u230ac ^ a\u230b\u208a\nhij : i < N \u2227 j < \u230ac ^ i\u230b\u208a\n\u22a2 j < \u230ac ^ (N - 1)\u230b\u208a"}, {"tactic": "exact hij.2.trans_le (u_mono (Nat.le_pred_of_lt hij.1))", "annotated_tactic": ["exact hij.2.<a>trans_le</a> (u_mono (<a>Nat.le_pred_of_lt</a> hij.1))", [{"full_name": "LT.lt.trans_le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [148, 7], "def_end_pos": [148, 21]}, {"full_name": "Nat.le_pred_of_lt", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [268, 9], "def_end_pos": [268, 22]}]], "state_before": "case refine'_1.mk\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nN i j : \u2115\nhij\u271d : { fst := i, snd := j } \u2208 Finset.sigma (range N) fun a => range \u230ac ^ a\u230b\u208a\nhij : i < N \u2227 j < \u230ac ^ i\u230b\u208a\n\u22a2 j < \u230ac ^ (N - 1)\u230b\u208a", "state_after": "no goals"}, {"tactic": "rintro \u27e8i, j\u27e9 hij", "annotated_tactic": ["rintro \u27e8i, j\u27e9 hij", []], "state_before": "case refine'_2\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nN : \u2115\n\u22a2 \u2200 (a : (_ : \u2115) \u00d7 \u2115) (ha : a \u2208 Finset.sigma (range \u230ac ^ (N - 1)\u230b\u208a) fun a => filter (fun i => a < \u230ac ^ i\u230b\u208a) (range N)),\n    (fun p x => { fst := p.snd, snd := p.fst }) a ha \u2208 Finset.sigma (range N) fun a => range \u230ac ^ a\u230b\u208a", "state_after": "case refine'_2.mk\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nN i j : \u2115\nhij : { fst := i, snd := j } \u2208 Finset.sigma (range \u230ac ^ (N - 1)\u230b\u208a) fun a => filter (fun i => a < \u230ac ^ i\u230b\u208a) (range N)\n\u22a2 (fun p x => { fst := p.snd, snd := p.fst }) { fst := i, snd := j } hij \u2208\n    Finset.sigma (range N) fun a => range \u230ac ^ a\u230b\u208a"}, {"tactic": "simp only [mem_sigma, mem_range, mem_filter] at hij", "annotated_tactic": ["simp only [<a>mem_sigma</a>, <a>mem_range</a>, <a>mem_filter</a>] at hij", [{"full_name": "Finset.mem_sigma", "def_path": "Mathlib/Data/Finset/Sigma.lean", "def_pos": [49, 9], "def_end_pos": [49, 18]}, {"full_name": "Finset.mem_range", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3037, 9], "def_end_pos": [3037, 18]}, {"full_name": "Finset.mem_filter", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2708, 9], "def_end_pos": [2708, 19]}]], "state_before": "case refine'_2.mk\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nN i j : \u2115\nhij : { fst := i, snd := j } \u2208 Finset.sigma (range \u230ac ^ (N - 1)\u230b\u208a) fun a => filter (fun i => a < \u230ac ^ i\u230b\u208a) (range N)\n\u22a2 (fun p x => { fst := p.snd, snd := p.fst }) { fst := i, snd := j } hij \u2208\n    Finset.sigma (range N) fun a => range \u230ac ^ a\u230b\u208a", "state_after": "case refine'_2.mk\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nN i j : \u2115\nhij\u271d : { fst := i, snd := j } \u2208 Finset.sigma (range \u230ac ^ (N - 1)\u230b\u208a) fun a => filter (fun i => a < \u230ac ^ i\u230b\u208a) (range N)\nhij : i < \u230ac ^ (N - 1)\u230b\u208a \u2227 j < N \u2227 i < \u230ac ^ j\u230b\u208a\n\u22a2 (fun p x => { fst := p.snd, snd := p.fst }) { fst := i, snd := j } hij\u271d \u2208\n    Finset.sigma (range N) fun a => range \u230ac ^ a\u230b\u208a"}, {"tactic": "simp only [hij.2.1, hij.2.2, mem_sigma, mem_range, and_self_iff]", "annotated_tactic": ["simp only [hij.2.1, hij.2.2, <a>mem_sigma</a>, <a>mem_range</a>, <a>and_self_iff</a>]", [{"full_name": "Finset.mem_sigma", "def_path": "Mathlib/Data/Finset/Sigma.lean", "def_pos": [49, 9], "def_end_pos": [49, 18]}, {"full_name": "Finset.mem_range", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3037, 9], "def_end_pos": [3037, 18]}, {"full_name": "and_self_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [155, 9], "def_end_pos": [155, 21]}]], "state_before": "case refine'_2.mk\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nN i j : \u2115\nhij\u271d : { fst := i, snd := j } \u2208 Finset.sigma (range \u230ac ^ (N - 1)\u230b\u208a) fun a => filter (fun i => a < \u230ac ^ i\u230b\u208a) (range N)\nhij : i < \u230ac ^ (N - 1)\u230b\u208a \u2227 j < N \u2227 i < \u230ac ^ j\u230b\u208a\n\u22a2 (fun p x => { fst := p.snd, snd := p.fst }) { fst := i, snd := j } hij\u271d \u2208\n    Finset.sigma (range N) fun a => range \u230ac ^ a\u230b\u208a", "state_after": "no goals"}, {"tactic": "rintro \u27e8i, j\u27e9 hij", "annotated_tactic": ["rintro \u27e8i, j\u27e9 hij", []], "state_before": "case refine'_3\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nN : \u2115\n\u22a2 \u2200 (a : (_ : \u2115) \u00d7 \u2115) (ha : a \u2208 Finset.sigma (range N) fun a => range \u230ac ^ a\u230b\u208a),\n    (fun p x => { fst := p.snd, snd := p.fst }) ((fun p x => { fst := p.snd, snd := p.fst }) a ha)\n        (_ :\n          (fun p x => { fst := p.snd, snd := p.fst }) a ha \u2208\n            Finset.sigma (range \u230ac ^ (N - 1)\u230b\u208a) fun a => filter (fun i => a < \u230ac ^ i\u230b\u208a) (range N)) =\n      a", "state_after": "case refine'_3.mk\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nN i j : \u2115\nhij : { fst := i, snd := j } \u2208 Finset.sigma (range N) fun a => range \u230ac ^ a\u230b\u208a\n\u22a2 (fun p x => { fst := p.snd, snd := p.fst }) ((fun p x => { fst := p.snd, snd := p.fst }) { fst := i, snd := j } hij)\n      (_ :\n        (fun p x => { fst := p.snd, snd := p.fst }) { fst := i, snd := j } hij \u2208\n          Finset.sigma (range \u230ac ^ (N - 1)\u230b\u208a) fun a => filter (fun i => a < \u230ac ^ i\u230b\u208a) (range N)) =\n    { fst := i, snd := j }"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case refine'_3.mk\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nN i j : \u2115\nhij : { fst := i, snd := j } \u2208 Finset.sigma (range N) fun a => range \u230ac ^ a\u230b\u208a\n\u22a2 (fun p x => { fst := p.snd, snd := p.fst }) ((fun p x => { fst := p.snd, snd := p.fst }) { fst := i, snd := j } hij)\n      (_ :\n        (fun p x => { fst := p.snd, snd := p.fst }) { fst := i, snd := j } hij \u2208\n          Finset.sigma (range \u230ac ^ (N - 1)\u230b\u208a) fun a => filter (fun i => a < \u230ac ^ i\u230b\u208a) (range N)) =\n    { fst := i, snd := j }", "state_after": "no goals"}, {"tactic": "rintro \u27e8i, j\u27e9 hij", "annotated_tactic": ["rintro \u27e8i, j\u27e9 hij", []], "state_before": "case refine'_4\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nN : \u2115\n\u22a2 \u2200 (a : (_ : \u2115) \u00d7 \u2115) (ha : a \u2208 Finset.sigma (range \u230ac ^ (N - 1)\u230b\u208a) fun a => filter (fun i => a < \u230ac ^ i\u230b\u208a) (range N)),\n    (fun p x => { fst := p.snd, snd := p.fst }) ((fun p x => { fst := p.snd, snd := p.fst }) a ha)\n        (_ : (fun p x => { fst := p.snd, snd := p.fst }) a ha \u2208 Finset.sigma (range N) fun a => range \u230ac ^ a\u230b\u208a) =\n      a", "state_after": "case refine'_4.mk\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nN i j : \u2115\nhij : { fst := i, snd := j } \u2208 Finset.sigma (range \u230ac ^ (N - 1)\u230b\u208a) fun a => filter (fun i => a < \u230ac ^ i\u230b\u208a) (range N)\n\u22a2 (fun p x => { fst := p.snd, snd := p.fst }) ((fun p x => { fst := p.snd, snd := p.fst }) { fst := i, snd := j } hij)\n      (_ :\n        (fun p x => { fst := p.snd, snd := p.fst }) { fst := i, snd := j } hij \u2208\n          Finset.sigma (range N) fun a => range \u230ac ^ a\u230b\u208a) =\n    { fst := i, snd := j }"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case refine'_4.mk\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nN i j : \u2115\nhij : { fst := i, snd := j } \u2208 Finset.sigma (range \u230ac ^ (N - 1)\u230b\u208a) fun a => filter (fun i => a < \u230ac ^ i\u230b\u208a) (range N)\n\u22a2 (fun p x => { fst := p.snd, snd := p.fst }) ((fun p x => { fst := p.snd, snd := p.fst }) { fst := i, snd := j } hij)\n      (_ :\n        (fun p x => { fst := p.snd, snd := p.fst }) { fst := i, snd := j } hij \u2208\n          Finset.sigma (range N) fun a => range \u230ac ^ a\u230b\u208a) =\n    { fst := i, snd := j }", "state_after": "no goals"}, {"tactic": "apply sum_le_sum fun j hj => ?_", "annotated_tactic": ["apply <a>sum_le_sum</a> fun j hj => ?_", [{"full_name": "Finset.sum_le_sum", "def_path": "Mathlib/Algebra/BigOperators/Order.lean", "def_pos": [111, 15], "def_end_pos": [111, 25]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nN : \u2115\n\u22a2 \u2211 j in range (u (N - 1)), (\u2211 i in filter (fun i => j < u i) (range N), (\u2191(u i) ^ 2)\u207b\u00b9) * variance (Y j) \u2119 \u2264\n    \u2211 j in range (u (N - 1)), c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 / \u2191j ^ 2 * variance (Y j) \u2119", "state_after": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nN j : \u2115\nhj : j \u2208 range (u (N - 1))\n\u22a2 (\u2211 i in filter (fun i => j < u i) (range N), (\u2191(u i) ^ 2)\u207b\u00b9) * variance (Y j) \u2119 \u2264\n    c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 / \u2191j ^ 2 * variance (Y j) \u2119"}, {"tactic": "rcases @eq_zero_or_pos _ _ j with (rfl | hj)", "annotated_tactic": ["rcases @<a>eq_zero_or_pos</a> _ _ j with (rfl | hj)", [{"full_name": "eq_zero_or_pos", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [249, 3], "def_end_pos": [249, 14]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nN j : \u2115\nhj : j \u2208 range (u (N - 1))\n\u22a2 (\u2211 i in filter (fun i => j < u i) (range N), (\u2191(u i) ^ 2)\u207b\u00b9) * variance (Y j) \u2119 \u2264\n    c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 / \u2191j ^ 2 * variance (Y j) \u2119", "state_after": "case inl\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nN : \u2115\nhj : 0 \u2208 range (u (N - 1))\n\u22a2 (\u2211 i in filter (fun i => 0 < u i) (range N), (\u2191(u i) ^ 2)\u207b\u00b9) * variance (Y 0) \u2119 \u2264\n    c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 / \u21910 ^ 2 * variance (Y 0) \u2119\n\ncase inr\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nN j : \u2115\nhj\u271d : j \u2208 range (u (N - 1))\nhj : 0 < j\n\u22a2 (\u2211 i in filter (fun i => j < u i) (range N), (\u2191(u i) ^ 2)\u207b\u00b9) * variance (Y j) \u2119 \u2264\n    c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 / \u2191j ^ 2 * variance (Y j) \u2119"}, {"tactic": "apply mul_le_mul_of_nonneg_right _ (variance_nonneg _ _)", "annotated_tactic": ["apply <a>mul_le_mul_of_nonneg_right</a> _ (<a>variance_nonneg</a> _ _)", [{"full_name": "mul_le_mul_of_nonneg_right", "def_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "def_pos": [156, 9], "def_end_pos": [156, 35]}, {"full_name": "ProbabilityTheory.variance_nonneg", "def_path": "Mathlib/Probability/Variance.lean", "def_pos": [184, 9], "def_end_pos": [184, 24]}]], "state_before": "case inr\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nN j : \u2115\nhj\u271d : j \u2208 range (u (N - 1))\nhj : 0 < j\n\u22a2 (\u2211 i in filter (fun i => j < u i) (range N), (\u2191(u i) ^ 2)\u207b\u00b9) * variance (Y j) \u2119 \u2264\n    c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 / \u2191j ^ 2 * variance (Y j) \u2119", "state_after": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nN j : \u2115\nhj\u271d : j \u2208 range (u (N - 1))\nhj : 0 < j\n\u22a2 \u2211 i in filter (fun i => j < u i) (range N), (\u2191(u i) ^ 2)\u207b\u00b9 \u2264 c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 / \u2191j ^ 2"}, {"tactic": "convert sum_div_nat_floor_pow_sq_le_div_sq N (Nat.cast_pos.2 hj) c_one using 2", "annotated_tactic": ["convert <a>sum_div_nat_floor_pow_sq_le_div_sq</a> N (<a>Nat.cast_pos</a>.2 hj) c_one using 2", [{"full_name": "sum_div_nat_floor_pow_sq_le_div_sq", "def_path": "Mathlib/Analysis/SpecificLimits/FloorPow.lean", "def_pos": [319, 9], "def_end_pos": [319, 43]}, {"full_name": "Nat.cast_pos", "def_path": "Mathlib/Data/Nat/Cast/Order.lean", "def_pos": [72, 9], "def_end_pos": [72, 17]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nN j : \u2115\nhj\u271d : j \u2208 range (u (N - 1))\nhj : 0 < j\n\u22a2 \u2211 i in filter (fun i => j < u i) (range N), (\u2191(u i) ^ 2)\u207b\u00b9 \u2264 c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 / \u2191j ^ 2", "state_after": "case h.e'_3.h\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nN j : \u2115\nhj\u271d : j \u2208 range (u (N - 1))\nhj : 0 < j\n\u22a2 filter (fun i => j < u i) (range N) = filter (fun x => \u2191j < \u2191\u230ac ^ x\u230b\u208a) (range N)\n\ncase h.e'_3.a\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nN j : \u2115\nhj\u271d : j \u2208 range (u (N - 1))\nhj : 0 < j\nx\u271d : \u2115\na\u271d : x\u271d \u2208 filter (fun x => \u2191j < \u2191\u230ac ^ x\u230b\u208a) (range N)\n\u22a2 (\u2191(u x\u271d) ^ 2)\u207b\u00b9 = 1 / \u2191\u230ac ^ x\u271d\u230b\u208a ^ 2"}, {"tactic": "simp only [Nat.cast_zero, zero_pow', Ne.def, bit0_eq_zero, Nat.one_ne_zero,\n  not_false_iff, div_zero, zero_mul]", "annotated_tactic": ["simp only [<a>Nat.cast_zero</a>, <a>zero_pow'</a>, <a>Ne.def</a>, <a>bit0_eq_zero</a>, <a>Nat.one_ne_zero</a>,\n            <a>not_false_iff</a>, <a>div_zero</a>, <a>zero_mul</a>]", [{"full_name": "Nat.cast_zero", "def_path": "Mathlib/Data/Nat/Cast/Defs.lean", "def_pos": [114, 9], "def_end_pos": [114, 18]}, {"full_name": "zero_pow'", "def_path": "Mathlib/Algebra/GroupPower/Ring.lean", "def_pos": [37, 9], "def_end_pos": [37, 18]}, {"full_name": "Ne.def", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [59, 9], "def_end_pos": [59, 15]}, {"full_name": "bit0_eq_zero", "def_path": "Mathlib/Algebra/CharZero/Lemmas.lean", "def_pos": [77, 9], "def_end_pos": [77, 21]}, {"full_name": "Nat.one_ne_zero", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [426, 19], "def_end_pos": [426, 30]}, {"full_name": "not_false_iff", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [82, 9], "def_end_pos": [82, 22]}, {"full_name": "div_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Basic.lean", "def_pos": [295, 9], "def_end_pos": [295, 17]}, {"full_name": "MulZeroClass.zero_mul", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [36, 3], "def_end_pos": [36, 11]}]], "state_before": "case inl\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nN : \u2115\nhj : 0 \u2208 range (u (N - 1))\n\u22a2 (\u2211 i in filter (fun i => 0 < u i) (range N), (\u2191(u i) ^ 2)\u207b\u00b9) * variance (Y 0) \u2119 \u2264\n    c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 / \u21910 ^ 2 * variance (Y 0) \u2119", "state_after": "case inl\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nN : \u2115\nhj : 0 \u2208 range (u (N - 1))\n\u22a2 (\u2211 x in filter (fun i => 0 < \u230ac ^ i\u230b\u208a) (range N), (\u2191\u230ac ^ x\u230b\u208a ^ 2)\u207b\u00b9) * variance (truncation (X 0) 0) \u2119 \u2264 0"}, {"tactic": "simp only [Nat.cast_zero, truncation_zero, variance_zero, mul_zero, le_rfl]", "annotated_tactic": ["simp only [<a>Nat.cast_zero</a>, <a>truncation_zero</a>, <a>variance_zero</a>, <a>mul_zero</a>, <a>le_rfl</a>]", [{"full_name": "Nat.cast_zero", "def_path": "Mathlib/Data/Nat/Cast/Defs.lean", "def_pos": [114, 9], "def_end_pos": [114, 18]}, {"full_name": "ProbabilityTheory.truncation_zero", "def_path": "Mathlib/Probability/StrongLaw.lean", "def_pos": [98, 9], "def_end_pos": [98, 24]}, {"full_name": "ProbabilityTheory.variance_zero", "def_path": "Mathlib/Probability/Variance.lean", "def_pos": [180, 9], "def_end_pos": [180, 22]}, {"full_name": "MulZeroClass.mul_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [38, 3], "def_end_pos": [38, 11]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}]], "state_before": "case inl\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nN : \u2115\nhj : 0 \u2208 range (u (N - 1))\n\u22a2 (\u2211 x in filter (fun i => 0 < \u230ac ^ i\u230b\u208a) (range N), (\u2191\u230ac ^ x\u230b\u208a ^ 2)\u207b\u00b9) * variance (truncation (X 0) 0) \u2119 \u2264 0", "state_after": "no goals"}, {"tactic": "simp only [Nat.cast_lt]", "annotated_tactic": ["simp only [<a>Nat.cast_lt</a>]", [{"full_name": "Nat.cast_lt", "def_path": "Mathlib/Data/Nat/Cast/Order.lean", "def_pos": [96, 9], "def_end_pos": [96, 16]}]], "state_before": "case h.e'_3.h\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nN j : \u2115\nhj\u271d : j \u2208 range (u (N - 1))\nhj : 0 < j\n\u22a2 filter (fun i => j < u i) (range N) = filter (fun x => \u2191j < \u2191\u230ac ^ x\u230b\u208a) (range N)", "state_after": "no goals"}, {"tactic": "simp only [one_div]", "annotated_tactic": ["simp only [<a>one_div</a>]", [{"full_name": "one_div", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [318, 9], "def_end_pos": [318, 16]}]], "state_before": "case h.e'_3.a\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nN j : \u2115\nhj\u271d : j \u2208 range (u (N - 1))\nhj : 0 < j\nx\u271d : \u2115\na\u271d : x\u271d \u2208 filter (fun x => \u2191j < \u2191\u230ac ^ x\u230b\u208a) (range N)\n\u22a2 (\u2191(u x\u271d) ^ 2)\u207b\u00b9 = 1 / \u2191\u230ac ^ x\u271d\u230b\u208a ^ 2", "state_after": "no goals"}, {"tactic": "simp_rw [mul_sum, div_eq_mul_inv, mul_assoc]", "annotated_tactic": ["simp_rw [<a>mul_sum</a>, <a>div_eq_mul_inv</a>, <a>mul_assoc</a>]", [{"full_name": "Finset.mul_sum", "def_path": "Mathlib/Algebra/BigOperators/Ring.lean", "def_pos": [55, 9], "def_end_pos": [55, 16]}, {"full_name": "div_eq_mul_inv", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [977, 9], "def_end_pos": [977, 23]}, {"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [264, 9], "def_end_pos": [264, 18]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nN : \u2115\n\u22a2 \u2211 j in range (u (N - 1)), c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 / \u2191j ^ 2 * variance (Y j) \u2119 =\n    c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * \u2211 j in range (u (N - 1)), (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119", "state_after": "no goals"}, {"tactic": "apply mul_le_mul_of_nonneg_left (I1 _)", "annotated_tactic": ["apply <a>mul_le_mul_of_nonneg_left</a> (I1 _)", [{"full_name": "mul_le_mul_of_nonneg_left", "def_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "def_pos": [152, 9], "def_end_pos": [152, 34]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nN : \u2115\n\u22a2 c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * \u2211 j in range (u (N - 1)), (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264\n    c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)", "state_after": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nN : \u2115\n\u22a2 0 \u2264 c ^ 5 * (c - 1)\u207b\u00b9 ^ 3"}, {"tactic": "apply mul_nonneg (pow_nonneg c_pos.le _)", "annotated_tactic": ["apply <a>mul_nonneg</a> (<a>pow_nonneg</a> c_pos.le _)", [{"full_name": "mul_nonneg", "def_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "def_pos": [380, 7], "def_end_pos": [380, 17]}, {"full_name": "pow_nonneg", "def_path": "Mathlib/Algebra/Order/Ring/Defs.lean", "def_pos": [244, 9], "def_end_pos": [244, 19]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nN : \u2115\n\u22a2 0 \u2264 c ^ 5 * (c - 1)\u207b\u00b9 ^ 3", "state_after": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nN : \u2115\n\u22a2 0 \u2264 (c - 1)\u207b\u00b9 ^ 3"}, {"tactic": "exact pow_nonneg (inv_nonneg.2 (sub_nonneg.2 c_one.le)) _", "annotated_tactic": ["exact <a>pow_nonneg</a> (<a>inv_nonneg</a>.2 (<a>sub_nonneg</a>.2 c_one.le)) _", [{"full_name": "pow_nonneg", "def_path": "Mathlib/Algebra/Order/Ring/Defs.lean", "def_pos": [244, 9], "def_end_pos": [244, 19]}, {"full_name": "inv_nonneg", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [58, 9], "def_end_pos": [58, 19]}, {"full_name": "sub_nonneg", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [720, 30], "def_end_pos": [720, 40]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nN : \u2115\n\u22a2 0 \u2264 (c - 1)\u207b\u00b9 ^ 3", "state_after": "no goals"}, {"tactic": "intro N", "annotated_tactic": ["intro N", []], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nI2 : \u2200 (N : \u2115), \u2211 i in range N, (\u2191(u i) ^ 2)\u207b\u00b9 * variance (S (u i)) \u2119 \u2264 C\n\u22a2 \u2200 (N : \u2115), \u2211 i in range N, \u2191\u2191\u2119 {\u03c9 | \u2191(u i) * \u03b5 \u2264 |S (u i) \u03c9 - \u222b (a : \u03a9), S (u i) a|} \u2264 ENNReal.ofReal (\u03b5\u207b\u00b9 ^ 2 * C)", "state_after": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nI2 : \u2200 (N : \u2115), \u2211 i in range N, (\u2191(u i) ^ 2)\u207b\u00b9 * variance (S (u i)) \u2119 \u2264 C\nN : \u2115\n\u22a2 \u2211 i in range N, \u2191\u2191\u2119 {\u03c9 | \u2191(u i) * \u03b5 \u2264 |S (u i) \u03c9 - \u222b (a : \u03a9), S (u i) a|} \u2264 ENNReal.ofReal (\u03b5\u207b\u00b9 ^ 2 * C)"}, {"tactic": "refine' sum_le_sum fun i _ => _", "annotated_tactic": ["refine' <a>sum_le_sum</a> fun i _ => _", [{"full_name": "Finset.sum_le_sum", "def_path": "Mathlib/Algebra/BigOperators/Order.lean", "def_pos": [111, 15], "def_end_pos": [111, 25]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nI2 : \u2200 (N : \u2115), \u2211 i in range N, (\u2191(u i) ^ 2)\u207b\u00b9 * variance (S (u i)) \u2119 \u2264 C\nN : \u2115\n\u22a2 \u2211 i in range N, \u2191\u2191\u2119 {\u03c9 | \u2191(u i) * \u03b5 \u2264 |S (u i) \u03c9 - \u222b (a : \u03a9), S (u i) a|} \u2264\n    \u2211 i in range N, ENNReal.ofReal (variance (S (u i)) \u2119 / (\u2191(u i) * \u03b5) ^ 2)", "state_after": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nI2 : \u2200 (N : \u2115), \u2211 i in range N, (\u2191(u i) ^ 2)\u207b\u00b9 * variance (S (u i)) \u2119 \u2264 C\nN i : \u2115\nx\u271d : i \u2208 range N\n\u22a2 \u2191\u2191\u2119 {\u03c9 | \u2191(u i) * \u03b5 \u2264 |S (u i) \u03c9 - \u222b (a : \u03a9), S (u i) a|} \u2264 ENNReal.ofReal (variance (S (u i)) \u2119 / (\u2191(u i) * \u03b5) ^ 2)"}, {"tactic": "apply meas_ge_le_variance_div_sq", "annotated_tactic": ["apply <a>meas_ge_le_variance_div_sq</a>", [{"full_name": "ProbabilityTheory.meas_ge_le_variance_div_sq", "def_path": "Mathlib/Probability/Variance.lean", "def_pos": [285, 9], "def_end_pos": [285, 35]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nI2 : \u2200 (N : \u2115), \u2211 i in range N, (\u2191(u i) ^ 2)\u207b\u00b9 * variance (S (u i)) \u2119 \u2264 C\nN i : \u2115\nx\u271d : i \u2208 range N\n\u22a2 \u2191\u2191\u2119 {\u03c9 | \u2191(u i) * \u03b5 \u2264 |S (u i) \u03c9 - \u222b (a : \u03a9), S (u i) a|} \u2264 ENNReal.ofReal (variance (S (u i)) \u2119 / (\u2191(u i) * \u03b5) ^ 2)", "state_after": "case hX\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nI2 : \u2200 (N : \u2115), \u2211 i in range N, (\u2191(u i) ^ 2)\u207b\u00b9 * variance (S (u i)) \u2119 \u2264 C\nN i : \u2115\nx\u271d : i \u2208 range N\n\u22a2 Mem\u2112p (fun \u03c9 => S (u i) \u03c9) 2\n\ncase hc\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nI2 : \u2200 (N : \u2115), \u2211 i in range N, (\u2191(u i) ^ 2)\u207b\u00b9 * variance (S (u i)) \u2119 \u2264 C\nN i : \u2115\nx\u271d : i \u2208 range N\n\u22a2 0 < \u2191(u i) * \u03b5"}, {"tactic": "exact mem\u2112p_finset_sum' _ fun j _ => (hident j).aestronglyMeasurable_fst.mem\u2112p_truncation", "annotated_tactic": ["exact <a>mem\u2112p_finset_sum'</a> _ fun j _ => (hident j).aestronglyMeasurable_fst.mem\u2112p_truncation", [{"full_name": "MeasureTheory.mem\u2112p_finset_sum'", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [1254, 9], "def_end_pos": [1254, 26]}]], "state_before": "case hX\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nI2 : \u2200 (N : \u2115), \u2211 i in range N, (\u2191(u i) ^ 2)\u207b\u00b9 * variance (S (u i)) \u2119 \u2264 C\nN i : \u2115\nx\u271d : i \u2208 range N\n\u22a2 Mem\u2112p (fun \u03c9 => S (u i) \u03c9) 2", "state_after": "no goals"}, {"tactic": "apply mul_pos (Nat.cast_pos.2 _) \u03b5pos", "annotated_tactic": ["apply <a>mul_pos</a> (<a>Nat.cast_pos</a>.2 _) \u03b5pos", [{"full_name": "mul_pos", "def_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "def_pos": [345, 7], "def_end_pos": [345, 14]}, {"full_name": "Nat.cast_pos", "def_path": "Mathlib/Data/Nat/Cast/Order.lean", "def_pos": [72, 9], "def_end_pos": [72, 17]}]], "state_before": "case hc\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nI2 : \u2200 (N : \u2115), \u2211 i in range N, (\u2191(u i) ^ 2)\u207b\u00b9 * variance (S (u i)) \u2119 \u2264 C\nN i : \u2115\nx\u271d : i \u2208 range N\n\u22a2 0 < \u2191(u i) * \u03b5", "state_after": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nI2 : \u2200 (N : \u2115), \u2211 i in range N, (\u2191(u i) ^ 2)\u207b\u00b9 * variance (S (u i)) \u2119 \u2264 C\nN i : \u2115\nx\u271d : i \u2208 range N\n\u22a2 0 < u i"}, {"tactic": "refine' zero_lt_one.trans_le _", "annotated_tactic": ["refine' zero_lt_one.trans_le _", []], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nI2 : \u2200 (N : \u2115), \u2211 i in range N, (\u2191(u i) ^ 2)\u207b\u00b9 * variance (S (u i)) \u2119 \u2264 C\nN i : \u2115\nx\u271d : i \u2208 range N\n\u22a2 0 < u i", "state_after": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nI2 : \u2200 (N : \u2115), \u2211 i in range N, (\u2191(u i) ^ 2)\u207b\u00b9 * variance (S (u i)) \u2119 \u2264 C\nN i : \u2115\nx\u271d : i \u2208 range N\n\u22a2 1 \u2264 u i"}, {"tactic": "apply Nat.le_floor", "annotated_tactic": ["apply <a>Nat.le_floor</a>", [{"full_name": "Nat.le_floor", "def_path": "Mathlib/Algebra/Order/Floor.lean", "def_pos": [134, 9], "def_end_pos": [134, 17]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nI2 : \u2200 (N : \u2115), \u2211 i in range N, (\u2191(u i) ^ 2)\u207b\u00b9 * variance (S (u i)) \u2119 \u2264 C\nN i : \u2115\nx\u271d : i \u2208 range N\n\u22a2 1 \u2264 u i", "state_after": "case h\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nI2 : \u2200 (N : \u2115), \u2211 i in range N, (\u2191(u i) ^ 2)\u207b\u00b9 * variance (S (u i)) \u2119 \u2264 C\nN i : \u2115\nx\u271d : i \u2208 range N\n\u22a2 \u21911 \u2264 c ^ i"}, {"tactic": "rw [Nat.cast_one]", "annotated_tactic": ["rw [<a>Nat.cast_one</a>]", [{"full_name": "Nat.cast_one", "def_path": "Mathlib/Data/Nat/Cast/Defs.lean", "def_pos": [141, 9], "def_end_pos": [141, 17]}]], "state_before": "case h\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nI2 : \u2200 (N : \u2115), \u2211 i in range N, (\u2191(u i) ^ 2)\u207b\u00b9 * variance (S (u i)) \u2119 \u2264 C\nN i : \u2115\nx\u271d : i \u2208 range N\n\u22a2 \u21911 \u2264 c ^ i", "state_after": "case h\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nI2 : \u2200 (N : \u2115), \u2211 i in range N, (\u2191(u i) ^ 2)\u207b\u00b9 * variance (S (u i)) \u2119 \u2264 C\nN i : \u2115\nx\u271d : i \u2208 range N\n\u22a2 1 \u2264 c ^ i"}, {"tactic": "apply one_le_pow_of_one_le c_one.le", "annotated_tactic": ["apply <a>one_le_pow_of_one_le</a> c_one.le", [{"full_name": "one_le_pow_of_one_le", "def_path": "Mathlib/Algebra/GroupPower/Order.lean", "def_pos": [423, 9], "def_end_pos": [423, 29]}]], "state_before": "case h\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nI2 : \u2200 (N : \u2115), \u2211 i in range N, (\u2191(u i) ^ 2)\u207b\u00b9 * variance (S (u i)) \u2119 \u2264 C\nN i : \u2115\nx\u271d : i \u2208 range N\n\u22a2 1 \u2264 c ^ i", "state_after": "no goals"}, {"tactic": "rw [ENNReal.ofReal_sum_of_nonneg fun i _ => ?_]", "annotated_tactic": ["rw [<a>ENNReal.ofReal_sum_of_nonneg</a> fun i _ => ?_]", [{"full_name": "ENNReal.ofReal_sum_of_nonneg", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1294, 9], "def_end_pos": [1294, 29]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nI2 : \u2200 (N : \u2115), \u2211 i in range N, (\u2191(u i) ^ 2)\u207b\u00b9 * variance (S (u i)) \u2119 \u2264 C\nN : \u2115\n\u22a2 \u2211 i in range N, ENNReal.ofReal (variance (S (u i)) \u2119 / (\u2191(u i) * \u03b5) ^ 2) =\n    ENNReal.ofReal (\u2211 i in range N, variance (S (u i)) \u2119 / (\u2191(u i) * \u03b5) ^ 2)", "state_after": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nI2 : \u2200 (N : \u2115), \u2211 i in range N, (\u2191(u i) ^ 2)\u207b\u00b9 * variance (S (u i)) \u2119 \u2264 C\nN i : \u2115\nx\u271d : i \u2208 range N\n\u22a2 0 \u2264 variance (S (u i)) \u2119 / (\u2191(u i) * \u03b5) ^ 2"}, {"tactic": "exact div_nonneg (variance_nonneg _ _) (sq_nonneg _)", "annotated_tactic": ["exact <a>div_nonneg</a> (<a>variance_nonneg</a> _ _) (<a>sq_nonneg</a> _)", [{"full_name": "div_nonneg", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [94, 9], "def_end_pos": [94, 19]}, {"full_name": "ProbabilityTheory.variance_nonneg", "def_path": "Mathlib/Probability/Variance.lean", "def_pos": [184, 9], "def_end_pos": [184, 24]}, {"full_name": "sq_nonneg", "def_path": "Mathlib/Algebra/GroupPower/Order.lean", "def_pos": [645, 9], "def_end_pos": [645, 18]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nI2 : \u2200 (N : \u2115), \u2211 i in range N, (\u2191(u i) ^ 2)\u207b\u00b9 * variance (S (u i)) \u2119 \u2264 C\nN i : \u2115\nx\u271d : i \u2208 range N\n\u22a2 0 \u2264 variance (S (u i)) \u2119 / (\u2191(u i) * \u03b5) ^ 2", "state_after": "no goals"}, {"tactic": "apply ENNReal.ofReal_le_ofReal", "annotated_tactic": ["apply <a>ENNReal.ofReal_le_ofReal</a>", [{"full_name": "ENNReal.ofReal_le_ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2135, 9], "def_end_pos": [2135, 25]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nI2 : \u2200 (N : \u2115), \u2211 i in range N, (\u2191(u i) ^ 2)\u207b\u00b9 * variance (S (u i)) \u2119 \u2264 C\nN : \u2115\n\u22a2 ENNReal.ofReal (\u2211 i in range N, variance (S (u i)) \u2119 / (\u2191(u i) * \u03b5) ^ 2) \u2264 ENNReal.ofReal (\u03b5\u207b\u00b9 ^ 2 * C)", "state_after": "case h\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nI2 : \u2200 (N : \u2115), \u2211 i in range N, (\u2191(u i) ^ 2)\u207b\u00b9 * variance (S (u i)) \u2119 \u2264 C\nN : \u2115\n\u22a2 \u2211 i in range N, variance (S (u i)) \u2119 / (\u2191(u i) * \u03b5) ^ 2 \u2264 \u03b5\u207b\u00b9 ^ 2 * C"}, {"tactic": "conv_lhs =>\n  enter [2, i]\n  rw [div_eq_inv_mul, \u2190 inv_pow, mul_inv, mul_comm _ \u03b5\u207b\u00b9, mul_pow, mul_assoc]", "annotated_tactic": ["conv_lhs =>\n          enter [2, i]\n          rw [<a>div_eq_inv_mul</a>, \u2190 <a>inv_pow</a>, <a>mul_inv</a>, <a>mul_comm</a> _ \u03b5\u207b\u00b9, <a>mul_pow</a>, <a>mul_assoc</a>]", [{"full_name": "div_eq_inv_mul", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [492, 9], "def_end_pos": [492, 23]}, {"full_name": "inv_pow", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [317, 9], "def_end_pos": [317, 16]}, {"full_name": "mul_inv", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [482, 9], "def_end_pos": [482, 16]}, {"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [302, 9], "def_end_pos": [302, 17]}, {"full_name": "mul_pow", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [257, 9], "def_end_pos": [257, 16]}, {"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [264, 9], "def_end_pos": [264, 18]}]], "state_before": "case h\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nI2 : \u2200 (N : \u2115), \u2211 i in range N, (\u2191(u i) ^ 2)\u207b\u00b9 * variance (S (u i)) \u2119 \u2264 C\nN : \u2115\n\u22a2 \u2211 i in range N, variance (S (u i)) \u2119 / (\u2191(u i) * \u03b5) ^ 2 \u2264 \u03b5\u207b\u00b9 ^ 2 * C", "state_after": "case h\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nI2 : \u2200 (N : \u2115), \u2211 i in range N, (\u2191(u i) ^ 2)\u207b\u00b9 * variance (S (u i)) \u2119 \u2264 C\nN : \u2115\n\u22a2 \u2211 i in range N, \u03b5\u207b\u00b9 ^ 2 * ((\u2191(u i))\u207b\u00b9 ^ 2 * variance (S (u i)) \u2119) \u2264 \u03b5\u207b\u00b9 ^ 2 * C"}, {"tactic": "rw [\u2190 mul_sum]", "annotated_tactic": ["rw [\u2190 <a>mul_sum</a>]", [{"full_name": "Finset.mul_sum", "def_path": "Mathlib/Algebra/BigOperators/Ring.lean", "def_pos": [55, 9], "def_end_pos": [55, 16]}]], "state_before": "case h\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nI2 : \u2200 (N : \u2115), \u2211 i in range N, (\u2191(u i) ^ 2)\u207b\u00b9 * variance (S (u i)) \u2119 \u2264 C\nN : \u2115\n\u22a2 \u2211 i in range N, \u03b5\u207b\u00b9 ^ 2 * ((\u2191(u i))\u207b\u00b9 ^ 2 * variance (S (u i)) \u2119) \u2264 \u03b5\u207b\u00b9 ^ 2 * C", "state_after": "case h\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nI2 : \u2200 (N : \u2115), \u2211 i in range N, (\u2191(u i) ^ 2)\u207b\u00b9 * variance (S (u i)) \u2119 \u2264 C\nN : \u2115\n\u22a2 \u03b5\u207b\u00b9 ^ 2 * \u2211 x in range N, (\u2191(u x))\u207b\u00b9 ^ 2 * variance (S (u x)) \u2119 \u2264 \u03b5\u207b\u00b9 ^ 2 * C"}, {"tactic": "refine' mul_le_mul_of_nonneg_left _ (sq_nonneg _)", "annotated_tactic": ["refine' <a>mul_le_mul_of_nonneg_left</a> _ (<a>sq_nonneg</a> _)", [{"full_name": "mul_le_mul_of_nonneg_left", "def_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "def_pos": [152, 9], "def_end_pos": [152, 34]}, {"full_name": "sq_nonneg", "def_path": "Mathlib/Algebra/GroupPower/Order.lean", "def_pos": [645, 9], "def_end_pos": [645, 18]}]], "state_before": "case h\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nI2 : \u2200 (N : \u2115), \u2211 i in range N, (\u2191(u i) ^ 2)\u207b\u00b9 * variance (S (u i)) \u2119 \u2264 C\nN : \u2115\n\u22a2 \u03b5\u207b\u00b9 ^ 2 * \u2211 x in range N, (\u2191(u x))\u207b\u00b9 ^ 2 * variance (S (u x)) \u2119 \u2264 \u03b5\u207b\u00b9 ^ 2 * C", "state_after": "case h\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nI2 : \u2200 (N : \u2115), \u2211 i in range N, (\u2191(u i) ^ 2)\u207b\u00b9 * variance (S (u i)) \u2119 \u2264 C\nN : \u2115\n\u22a2 \u2211 x in range N, (\u2191(u x))\u207b\u00b9 ^ 2 * variance (S (u x)) \u2119 \u2264 C"}, {"tactic": "conv_lhs => enter [2, i]; rw [inv_pow]", "annotated_tactic": ["conv_lhs => enter [2, i]; rw [<a>inv_pow</a>]", [{"full_name": "inv_pow", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [317, 9], "def_end_pos": [317, 16]}]], "state_before": "case h\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nI2 : \u2200 (N : \u2115), \u2211 i in range N, (\u2191(u i) ^ 2)\u207b\u00b9 * variance (S (u i)) \u2119 \u2264 C\nN : \u2115\n\u22a2 \u2211 x in range N, (\u2191(u x))\u207b\u00b9 ^ 2 * variance (S (u x)) \u2119 \u2264 C", "state_after": "case h\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nI2 : \u2200 (N : \u2115), \u2211 i in range N, (\u2191(u i) ^ 2)\u207b\u00b9 * variance (S (u i)) \u2119 \u2264 C\nN : \u2115\n\u22a2 \u2211 i in range N, (\u2191(u i) ^ 2)\u207b\u00b9 * variance (S (u i)) \u2119 \u2264 C"}, {"tactic": "exact I2 N", "annotated_tactic": ["exact I2 N", []], "state_before": "case h\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nI2 : \u2200 (N : \u2115), \u2211 i in range N, (\u2191(u i) ^ 2)\u207b\u00b9 * variance (S (u i)) \u2119 \u2264 C\nN : \u2115\n\u22a2 \u2211 i in range N, (\u2191(u i) ^ 2)\u207b\u00b9 * variance (S (u i)) \u2119 \u2264 C", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Group/Arithmetic.lean", "full_name": "aemeasurable_const_smul_iff\u2080", "start": [761, 1], "end": [763, 48], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/RBMap/Lemmas.lean", "full_name": "Std.RBSet.mem_insert_of_mem_toList", "start": [696, 1], "end": [700, 100], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/PImage.lean", "full_name": "Finset.pimage_mono", "start": [116, 1], "end": [117, 62], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Constructions/Prod/Basic.lean", "full_name": "IsCountablySpanning.prod", "start": [80, 1], "end": [84, 52], "traced_tactics": [{"tactic": "rcases hC, hD with \u27e8\u27e8s, h1s, h2s\u27e9, t, h1t, h2t\u27e9", "annotated_tactic": ["rcases hC, hD with \u27e8\u27e8s, h1s, h2s\u27e9, t, h1t, h2t\u27e9", []], "state_before": "\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\nC : Set (Set \u03b1)\nD : Set (Set \u03b2)\nhC : IsCountablySpanning C\nhD : IsCountablySpanning D\n\u22a2 IsCountablySpanning (image2 (fun x x_1 => x \u00d7\u02e2 x_1) C D)", "state_after": "case intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\nC : Set (Set \u03b1)\nD : Set (Set \u03b2)\ns : \u2115 \u2192 Set \u03b1\nh1s : \u2200 (n : \u2115), s n \u2208 C\nh2s : \u22c3 n, s n = univ\nt : \u2115 \u2192 Set \u03b2\nh1t : \u2200 (n : \u2115), t n \u2208 D\nh2t : \u22c3 n, t n = univ\n\u22a2 IsCountablySpanning (image2 (fun x x_1 => x \u00d7\u02e2 x_1) C D)"}, {"tactic": "refine' \u27e8fun n => s n.unpair.1 \u00d7\u02e2 t n.unpair.2, fun n => mem_image2_of_mem (h1s _) (h1t _), _\u27e9", "annotated_tactic": ["refine' \u27e8fun n => s n.unpair.1 \u00d7\u02e2 t n.unpair.2, fun n => <a>mem_image2_of_mem</a> (h1s _) (h1t _), _\u27e9", [{"full_name": "Set.mem_image2_of_mem", "def_path": "Mathlib/Data/Set/NAry.lean", "def_pos": [44, 9], "def_end_pos": [44, 26]}]], "state_before": "case intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\nC : Set (Set \u03b1)\nD : Set (Set \u03b2)\ns : \u2115 \u2192 Set \u03b1\nh1s : \u2200 (n : \u2115), s n \u2208 C\nh2s : \u22c3 n, s n = univ\nt : \u2115 \u2192 Set \u03b2\nh1t : \u2200 (n : \u2115), t n \u2208 D\nh2t : \u22c3 n, t n = univ\n\u22a2 IsCountablySpanning (image2 (fun x x_1 => x \u00d7\u02e2 x_1) C D)", "state_after": "case intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\nC : Set (Set \u03b1)\nD : Set (Set \u03b2)\ns : \u2115 \u2192 Set \u03b1\nh1s : \u2200 (n : \u2115), s n \u2208 C\nh2s : \u22c3 n, s n = univ\nt : \u2115 \u2192 Set \u03b2\nh1t : \u2200 (n : \u2115), t n \u2208 D\nh2t : \u22c3 n, t n = univ\n\u22a2 \u22c3 n, (fun n => s (Nat.unpair n).1 \u00d7\u02e2 t (Nat.unpair n).2) n = univ"}, {"tactic": "rw [iUnion_unpair_prod, h2s, h2t, univ_prod_univ]", "annotated_tactic": ["rw [<a>iUnion_unpair_prod</a>, h2s, h2t, <a>univ_prod_univ</a>]", [{"full_name": "Set.iUnion_unpair_prod", "def_path": "Mathlib/Data/Nat/Pairing.lean", "def_pos": [203, 9], "def_end_pos": [203, 27]}, {"full_name": "Set.univ_prod_univ", "def_path": "Mathlib/Data/Set/Prod.lean", "def_pos": [125, 9], "def_end_pos": [125, 23]}]], "state_before": "case intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\nC : Set (Set \u03b1)\nD : Set (Set \u03b2)\ns : \u2115 \u2192 Set \u03b1\nh1s : \u2200 (n : \u2115), s n \u2208 C\nh2s : \u22c3 n, s n = univ\nt : \u2115 \u2192 Set \u03b2\nh1t : \u2200 (n : \u2115), t n \u2208 D\nh2t : \u22c3 n, t n = univ\n\u22a2 \u22c3 n, (fun n => s (Nat.unpair n).1 \u00d7\u02e2 t (Nat.unpair n).2) n = univ", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Group/Measure.lean", "full_name": "MeasureTheory.measure_pos_iff_nonempty_of_isMulLeftInvariant", "start": [612, 1], "end": [614, 85], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Setoid/Partition.lean", "full_name": "IndexedPartition.equivQuotient_index_apply", "start": [431, 1], "end": [432, 39], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/Division.lean", "full_name": "MvPolynomial.X_mul_modMonomial", "start": [181, 1], "end": [183, 31], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Array/Lemmas.lean", "full_name": "Array.get_set_ne", "start": [91, 9], "end": [93, 58], "traced_tactics": [{"tactic": "simp [*]", "annotated_tactic": ["simp [*]", []], "state_before": "\u03b1 : Type ?u.13187\na : Array \u03b1\ni : Fin (size a)\nj : Nat\nv : \u03b1\nhj : j < size a\nh : i.val \u2260 j\n\u22a2 j < size (set a i v)", "state_after": "no goals"}, {"tactic": "simp only [set, getElem_eq_data_get, List.get_set_ne h]", "annotated_tactic": ["simp only [<a>set</a>, <a>getElem_eq_data_get</a>, <a>List.get_set_ne</a> h]", [{"full_name": "Array.set", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2587, 5], "def_end_pos": [2587, 14]}, {"full_name": "Array.getElem_eq_data_get", "def_path": "lake-packages/std/Std/Data/Array/Init/Lemmas.lean", "def_pos": [28, 9], "def_end_pos": [28, 28]}, {"full_name": "List.get_set_ne", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [956, 17], "def_end_pos": [956, 27]}]], "state_before": "\u03b1 : Type u_1\na : Array \u03b1\ni : Fin (size a)\nj : Nat\nv : \u03b1\nhj : j < size a\nh : i.val \u2260 j\n\u22a2 (set a i v)[j] = a[j]", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Group/Action.lean", "full_name": "MeasureTheory.measure_isOpen_pos_of_smulInvariant_of_ne_zero", "start": [271, 1], "end": [274, 73], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "full_name": "integral_withDensity_eq_integral_smul", "start": [1269, 1], "end": [1314, 57], "traced_tactics": [{"tactic": "by_cases hE : CompleteSpace E", "annotated_tactic": ["by_cases hE : <a>CompleteSpace</a> E", [{"full_name": "CompleteSpace", "def_path": "Mathlib/Topology/UniformSpace/Cauchy.lean", "def_pos": [397, 7], "def_end_pos": [397, 20]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\n\u22a2 (\u222b (a : \u03b1), g a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 g a \u2202\u03bc", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\n\u22a2 (\u222b (a : \u03b1), g a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 g a \u2202\u03bc\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : \u00acCompleteSpace E\n\u22a2 (\u222b (a : \u03b1), g a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 g a \u2202\u03bc"}, {"tactic": "swap", "annotated_tactic": ["swap", []], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\n\u22a2 (\u222b (a : \u03b1), g a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 g a \u2202\u03bc\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : \u00acCompleteSpace E\n\u22a2 (\u222b (a : \u03b1), g a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 g a \u2202\u03bc", "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : \u00acCompleteSpace E\n\u22a2 (\u222b (a : \u03b1), g a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 g a \u2202\u03bc\n\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\n\u22a2 (\u222b (a : \u03b1), g a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 g a \u2202\u03bc"}, {"tactic": "by_cases hg : Integrable g (\u03bc.withDensity fun x => f x)", "annotated_tactic": ["by_cases hg : <a>Integrable</a> g (\u03bc.withDensity fun x => f x)", [{"full_name": "MeasureTheory.Integrable", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [442, 5], "def_end_pos": [442, 15]}]], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\n\u22a2 (\u222b (a : \u03b1), g a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 g a \u2202\u03bc", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\n\u22a2 (\u222b (a : \u03b1), g a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 g a \u2202\u03bc\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : \u00acIntegrable g\n\u22a2 (\u222b (a : \u03b1), g a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 g a \u2202\u03bc"}, {"tactic": "swap", "annotated_tactic": ["swap", []], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\n\u22a2 (\u222b (a : \u03b1), g a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 g a \u2202\u03bc\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : \u00acIntegrable g\n\u22a2 (\u222b (a : \u03b1), g a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 g a \u2202\u03bc", "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : \u00acIntegrable g\n\u22a2 (\u222b (a : \u03b1), g a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 g a \u2202\u03bc\n\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\n\u22a2 (\u222b (a : \u03b1), g a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 g a \u2202\u03bc"}, {"tactic": "refine' Integrable.induction\n  (P := fun g => \u222b a, g a \u2202\u03bc.withDensity (fun x => f x) = \u222b a, f a \u2022 g a \u2202\u03bc) _ _ _ _ hg", "annotated_tactic": ["refine' <a>Integrable.induction</a>\n    (P := fun g => \u222b a, g a \u2202\u03bc.withDensity (fun x => f x) = \u222b a, f a \u2022 g a \u2202\u03bc) _ _ _ _ hg", [{"full_name": "MeasureTheory.Integrable.induction", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "def_pos": [1058, 9], "def_end_pos": [1058, 29]}]], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\n\u22a2 (\u222b (a : \u03b1), g a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 g a \u2202\u03bc", "state_after": "case pos.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\n\u22a2 \u2200 (c : E) \u2983s : Set \u03b1\u2984,\n    MeasurableSet s \u2192\n      \u2191\u2191(Measure.withDensity \u03bc fun x => \u2191(f x)) s < \u22a4 \u2192\n        (fun g => (\u222b (a : \u03b1), g a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 g a \u2202\u03bc)\n          (indicator s fun x => c)\n\ncase pos.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\n\u22a2 \u2200 \u2983f_1 g : \u03b1 \u2192 E\u2984,\n    Disjoint (support f_1) (support g) \u2192\n      Integrable f_1 \u2192\n        Integrable g \u2192\n          (fun g => (\u222b (a : \u03b1), g a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 g a \u2202\u03bc) f_1 \u2192\n            (fun g => (\u222b (a : \u03b1), g a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 g a \u2202\u03bc) g \u2192\n              (fun g => (\u222b (a : \u03b1), g a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 g a \u2202\u03bc) (f_1 + g)\n\ncase pos.refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\n\u22a2 IsClosed {f_1 | (fun g => (\u222b (a : \u03b1), g a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 g a \u2202\u03bc) \u2191\u2191f_1}\n\ncase pos.refine'_4\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\n\u22a2 \u2200 \u2983f_1 g : \u03b1 \u2192 E\u2984,\n    f_1 =\u1d50[Measure.withDensity \u03bc fun x => \u2191(f x)] g \u2192\n      Integrable f_1 \u2192\n        (fun g => (\u222b (a : \u03b1), g a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 g a \u2202\u03bc) f_1 \u2192\n          (fun g => (\u222b (a : \u03b1), g a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 g a \u2202\u03bc) g"}, {"tactic": "simp [integral, hE]", "annotated_tactic": ["simp [<a>integral</a>, hE]", [{"full_name": "MeasureTheory.integral", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [791, 17], "def_end_pos": [791, 25]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : \u00acCompleteSpace E\n\u22a2 (\u222b (a : \u03b1), g a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 g a \u2202\u03bc", "state_after": "no goals"}, {"tactic": "rw [integral_undef hg, integral_undef]", "annotated_tactic": ["rw [<a>integral_undef</a> hg, <a>integral_undef</a>]", [{"full_name": "MeasureTheory.integral_undef", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [836, 9], "def_end_pos": [836, 23]}, {"full_name": "MeasureTheory.integral_undef", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [836, 9], "def_end_pos": [836, 23]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : \u00acIntegrable g\n\u22a2 (\u222b (a : \u03b1), g a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 g a \u2202\u03bc", "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : \u00acIntegrable g\n\u22a2 \u00acIntegrable fun a => f a \u2022 g a"}, {"tactic": "rwa [\u2190 integrable_withDensity_iff_integrable_smul f_meas]", "annotated_tactic": ["rwa [\u2190 <a>integrable_withDensity_iff_integrable_smul</a> f_meas]", [{"full_name": "MeasureTheory.integrable_withDensity_iff_integrable_smul", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [918, 9], "def_end_pos": [918, 51]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : \u00acIntegrable g\n\u22a2 \u00acIntegrable fun a => f a \u2022 g a", "state_after": "no goals"}, {"tactic": "intro c s s_meas hs", "annotated_tactic": ["intro c s s_meas hs", []], "state_before": "case pos.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\n\u22a2 \u2200 (c : E) \u2983s : Set \u03b1\u2984,\n    MeasurableSet s \u2192\n      \u2191\u2191(Measure.withDensity \u03bc fun x => \u2191(f x)) s < \u22a4 \u2192\n        (fun g => (\u222b (a : \u03b1), g a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 g a \u2202\u03bc)\n          (indicator s fun x => c)", "state_after": "case pos.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nc : E\ns : Set \u03b1\ns_meas : MeasurableSet s\nhs : \u2191\u2191(Measure.withDensity \u03bc fun x => \u2191(f x)) s < \u22a4\n\u22a2 (\u222b (a : \u03b1), indicator s (fun x => c) a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) =\n    \u222b (a : \u03b1), f a \u2022 indicator s (fun x => c) a \u2202\u03bc"}, {"tactic": "rw [integral_indicator s_meas]", "annotated_tactic": ["rw [<a>integral_indicator</a> s_meas]", [{"full_name": "MeasureTheory.integral_indicator", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [169, 9], "def_end_pos": [169, 27]}]], "state_before": "case pos.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nc : E\ns : Set \u03b1\ns_meas : MeasurableSet s\nhs : \u2191\u2191(Measure.withDensity \u03bc fun x => \u2191(f x)) s < \u22a4\n\u22a2 (\u222b (a : \u03b1), indicator s (fun x => c) a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) =\n    \u222b (a : \u03b1), f a \u2022 indicator s (fun x => c) a \u2202\u03bc", "state_after": "case pos.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nc : E\ns : Set \u03b1\ns_meas : MeasurableSet s\nhs : \u2191\u2191(Measure.withDensity \u03bc fun x => \u2191(f x)) s < \u22a4\n\u22a2 (\u222b (x : \u03b1) in s, c \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 indicator s (fun x => c) a \u2202\u03bc"}, {"tactic": "simp_rw [\u2190 indicator_smul_apply, integral_indicator s_meas]", "annotated_tactic": ["simp_rw [\u2190 <a>indicator_smul_apply</a>, <a>integral_indicator</a> s_meas]", [{"full_name": "Set.indicator_smul_apply", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [481, 9], "def_end_pos": [481, 29]}, {"full_name": "MeasureTheory.integral_indicator", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [169, 9], "def_end_pos": [169, 27]}]], "state_before": "case pos.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nc : E\ns : Set \u03b1\ns_meas : MeasurableSet s\nhs : \u2191\u2191(Measure.withDensity \u03bc fun x => \u2191(f x)) s < \u22a4\n\u22a2 (\u222b (x : \u03b1) in s, c \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 indicator s (fun x => c) a \u2202\u03bc", "state_after": "case pos.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nc : E\ns : Set \u03b1\ns_meas : MeasurableSet s\nhs : \u2191\u2191(Measure.withDensity \u03bc fun x => \u2191(f x)) s < \u22a4\n\u22a2 (\u222b (x : \u03b1) in s, c \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (x : \u03b1) in s, f x \u2022 c \u2202\u03bc"}, {"tactic": "simp only [s_meas, integral_const, Measure.restrict_apply', univ_inter, withDensity_apply]", "annotated_tactic": ["simp only [s_meas, <a>integral_const</a>, <a>Measure.restrict_apply'</a>, <a>univ_inter</a>, <a>withDensity_apply</a>]", [{"full_name": "MeasureTheory.integral_const", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1409, 9], "def_end_pos": [1409, 23]}, {"full_name": "MeasureTheory.Measure.restrict_apply'", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1567, 9], "def_end_pos": [1567, 24]}, {"full_name": "Set.univ_inter", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1017, 9], "def_end_pos": [1017, 19]}, {"full_name": "MeasureTheory.withDensity_apply", "def_path": "Mathlib/MeasureTheory/Measure/WithDensity.lean", "def_pos": [39, 9], "def_end_pos": [39, 26]}]], "state_before": "case pos.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nc : E\ns : Set \u03b1\ns_meas : MeasurableSet s\nhs : \u2191\u2191(Measure.withDensity \u03bc fun x => \u2191(f x)) s < \u22a4\n\u22a2 (\u222b (x : \u03b1) in s, c \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (x : \u03b1) in s, f x \u2022 c \u2202\u03bc", "state_after": "case pos.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nc : E\ns : Set \u03b1\ns_meas : MeasurableSet s\nhs : \u2191\u2191(Measure.withDensity \u03bc fun x => \u2191(f x)) s < \u22a4\n\u22a2 ENNReal.toReal (\u222b\u207b (x : \u03b1) in s, \u2191(f x) \u2202\u03bc) \u2022 c = \u222b (x : \u03b1) in s, f x \u2022 c \u2202\u03bc"}, {"tactic": "rw [lintegral_coe_eq_integral, ENNReal.toReal_ofReal, \u2190 integral_smul_const]", "annotated_tactic": ["rw [<a>lintegral_coe_eq_integral</a>, <a>ENNReal.toReal_ofReal</a>, \u2190 <a>integral_smul_const</a>]", [{"full_name": "MeasureTheory.lintegral_coe_eq_integral", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1208, 9], "def_end_pos": [1208, 34]}, {"full_name": "ENNReal.toReal_ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [191, 9], "def_end_pos": [191, 22]}, {"full_name": "integral_smul_const", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [1257, 9], "def_end_pos": [1257, 28]}]], "state_before": "case pos.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nc : E\ns : Set \u03b1\ns_meas : MeasurableSet s\nhs : \u2191\u2191(Measure.withDensity \u03bc fun x => \u2191(f x)) s < \u22a4\n\u22a2 ENNReal.toReal (\u222b\u207b (x : \u03b1) in s, \u2191(f x) \u2202\u03bc) \u2022 c = \u222b (x : \u03b1) in s, f x \u2022 c \u2202\u03bc", "state_after": "case pos.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nc : E\ns : Set \u03b1\ns_meas : MeasurableSet s\nhs : \u2191\u2191(Measure.withDensity \u03bc fun x => \u2191(f x)) s < \u22a4\n\u22a2 \u222b (x : \u03b1) in s, \u2191(f x) \u2022 c \u2202\u03bc = \u222b (x : \u03b1) in s, f x \u2022 c \u2202\u03bc\n\ncase pos.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nc : E\ns : Set \u03b1\ns_meas : MeasurableSet s\nhs : \u2191\u2191(Measure.withDensity \u03bc fun x => \u2191(f x)) s < \u22a4\n\u22a2 0 \u2264 \u222b (a : \u03b1) in s, \u2191(f a) \u2202\u03bc\n\ncase pos.refine'_1.hfi\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nc : E\ns : Set \u03b1\ns_meas : MeasurableSet s\nhs : \u2191\u2191(Measure.withDensity \u03bc fun x => \u2191(f x)) s < \u22a4\n\u22a2 Integrable fun x => \u2191(f x)"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case pos.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nc : E\ns : Set \u03b1\ns_meas : MeasurableSet s\nhs : \u2191\u2191(Measure.withDensity \u03bc fun x => \u2191(f x)) s < \u22a4\n\u22a2 \u222b (x : \u03b1) in s, \u2191(f x) \u2022 c \u2202\u03bc = \u222b (x : \u03b1) in s, f x \u2022 c \u2202\u03bc", "state_after": "no goals"}, {"tactic": "exact integral_nonneg fun x => NNReal.coe_nonneg _", "annotated_tactic": ["exact <a>integral_nonneg</a> fun x => <a>NNReal.coe_nonneg</a> _", [{"full_name": "MeasureTheory.integral_nonneg", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1251, 9], "def_end_pos": [1251, 24]}, {"full_name": "NNReal.coe_nonneg", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [134, 9], "def_end_pos": [134, 19]}]], "state_before": "case pos.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nc : E\ns : Set \u03b1\ns_meas : MeasurableSet s\nhs : \u2191\u2191(Measure.withDensity \u03bc fun x => \u2191(f x)) s < \u22a4\n\u22a2 0 \u2264 \u222b (a : \u03b1) in s, \u2191(f a) \u2202\u03bc", "state_after": "no goals"}, {"tactic": "refine' \u27e8f_meas.coe_nnreal_real.aemeasurable.aestronglyMeasurable, _\u27e9", "annotated_tactic": ["refine' \u27e8f_meas.coe_nnreal_real.aemeasurable.aestronglyMeasurable, _\u27e9", []], "state_before": "case pos.refine'_1.hfi\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nc : E\ns : Set \u03b1\ns_meas : MeasurableSet s\nhs : \u2191\u2191(Measure.withDensity \u03bc fun x => \u2191(f x)) s < \u22a4\n\u22a2 Integrable fun x => \u2191(f x)", "state_after": "case pos.refine'_1.hfi\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nc : E\ns : Set \u03b1\ns_meas : MeasurableSet s\nhs : \u2191\u2191(Measure.withDensity \u03bc fun x => \u2191(f x)) s < \u22a4\n\u22a2 HasFiniteIntegral fun x => \u2191(f x)"}, {"tactic": "rw [withDensity_apply _ s_meas] at hs", "annotated_tactic": ["rw [<a>withDensity_apply</a> _ s_meas] at hs", [{"full_name": "MeasureTheory.withDensity_apply", "def_path": "Mathlib/MeasureTheory/Measure/WithDensity.lean", "def_pos": [39, 9], "def_end_pos": [39, 26]}]], "state_before": "case pos.refine'_1.hfi\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nc : E\ns : Set \u03b1\ns_meas : MeasurableSet s\nhs : \u2191\u2191(Measure.withDensity \u03bc fun x => \u2191(f x)) s < \u22a4\n\u22a2 HasFiniteIntegral fun x => \u2191(f x)", "state_after": "case pos.refine'_1.hfi\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nc : E\ns : Set \u03b1\ns_meas : MeasurableSet s\nhs : \u222b\u207b (a : \u03b1) in s, \u2191(f a) \u2202\u03bc < \u22a4\n\u22a2 HasFiniteIntegral fun x => \u2191(f x)"}, {"tactic": "rw [HasFiniteIntegral]", "annotated_tactic": ["rw [<a>HasFiniteIntegral</a>]", [{"full_name": "MeasureTheory.HasFiniteIntegral", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [106, 5], "def_end_pos": [106, 22]}]], "state_before": "case pos.refine'_1.hfi\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nc : E\ns : Set \u03b1\ns_meas : MeasurableSet s\nhs : \u222b\u207b (a : \u03b1) in s, \u2191(f a) \u2202\u03bc < \u22a4\n\u22a2 HasFiniteIntegral fun x => \u2191(f x)", "state_after": "case pos.refine'_1.hfi\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nc : E\ns : Set \u03b1\ns_meas : MeasurableSet s\nhs : \u222b\u207b (a : \u03b1) in s, \u2191(f a) \u2202\u03bc < \u22a4\n\u22a2 \u222b\u207b (a : \u03b1) in s, \u2191\u2016\u2191(f a)\u2016\u208a \u2202\u03bc < \u22a4"}, {"tactic": "convert hs with x", "annotated_tactic": ["convert hs with x", []], "state_before": "case pos.refine'_1.hfi\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nc : E\ns : Set \u03b1\ns_meas : MeasurableSet s\nhs : \u222b\u207b (a : \u03b1) in s, \u2191(f a) \u2202\u03bc < \u22a4\n\u22a2 \u222b\u207b (a : \u03b1) in s, \u2191\u2016\u2191(f a)\u2016\u208a \u2202\u03bc < \u22a4", "state_after": "case h.e'_3.h.e'_4.h.h.e'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nc : E\ns : Set \u03b1\ns_meas : MeasurableSet s\nhs : \u222b\u207b (a : \u03b1) in s, \u2191(f a) \u2202\u03bc < \u22a4\nx : \u03b1\n\u22a2 \u2016\u2191(f x)\u2016\u208a = f x"}, {"tactic": "simp only [NNReal.nnnorm_eq]", "annotated_tactic": ["simp only [<a>NNReal.nnnorm_eq</a>]", [{"full_name": "NNReal.nnnorm_eq", "def_path": "Mathlib/Analysis/Normed/Field/Basic.lean", "def_pos": [830, 9], "def_end_pos": [830, 18]}]], "state_before": "case h.e'_3.h.e'_4.h.h.e'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nc : E\ns : Set \u03b1\ns_meas : MeasurableSet s\nhs : \u222b\u207b (a : \u03b1) in s, \u2191(f a) \u2202\u03bc < \u22a4\nx : \u03b1\n\u22a2 \u2016\u2191(f x)\u2016\u208a = f x", "state_after": "no goals"}, {"tactic": "intro u u' _ u_int u'_int h h'", "annotated_tactic": ["intro u u' _ u_int u'_int h h'", []], "state_before": "case pos.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\n\u22a2 \u2200 \u2983f_1 g : \u03b1 \u2192 E\u2984,\n    Disjoint (support f_1) (support g) \u2192\n      Integrable f_1 \u2192\n        Integrable g \u2192\n          (fun g => (\u222b (a : \u03b1), g a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 g a \u2202\u03bc) f_1 \u2192\n            (fun g => (\u222b (a : \u03b1), g a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 g a \u2202\u03bc) g \u2192\n              (fun g => (\u222b (a : \u03b1), g a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 g a \u2202\u03bc) (f_1 + g)", "state_after": "case pos.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nu u' : \u03b1 \u2192 E\na\u271d : Disjoint (support u) (support u')\nu_int : Integrable u\nu'_int : Integrable u'\nh : (\u222b (a : \u03b1), u a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 u a \u2202\u03bc\nh' : (\u222b (a : \u03b1), u' a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 u' a \u2202\u03bc\n\u22a2 (\u222b (a : \u03b1), (u + u') a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 (u + u') a \u2202\u03bc"}, {"tactic": "change\n  (\u222b a : \u03b1, u a + u' a \u2202\u03bc.withDensity fun x : \u03b1 => \u2191(f x)) = \u222b a : \u03b1, f a \u2022 (u a + u' a) \u2202\u03bc", "annotated_tactic": ["change\n      (\u222b a : \u03b1, u a + u' a \u2202\u03bc.withDensity fun x : \u03b1 => \u2191(f x)) = \u222b a : \u03b1, f a \u2022 (u a + u' a) \u2202\u03bc", []], "state_before": "case pos.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nu u' : \u03b1 \u2192 E\na\u271d : Disjoint (support u) (support u')\nu_int : Integrable u\nu'_int : Integrable u'\nh : (\u222b (a : \u03b1), u a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 u a \u2202\u03bc\nh' : (\u222b (a : \u03b1), u' a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 u' a \u2202\u03bc\n\u22a2 (\u222b (a : \u03b1), (u + u') a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 (u + u') a \u2202\u03bc", "state_after": "case pos.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nu u' : \u03b1 \u2192 E\na\u271d : Disjoint (support u) (support u')\nu_int : Integrable u\nu'_int : Integrable u'\nh : (\u222b (a : \u03b1), u a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 u a \u2202\u03bc\nh' : (\u222b (a : \u03b1), u' a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 u' a \u2202\u03bc\n\u22a2 (\u222b (a : \u03b1), u a + u' a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 (u a + u' a) \u2202\u03bc"}, {"tactic": "simp_rw [smul_add]", "annotated_tactic": ["simp_rw [<a>smul_add</a>]", [{"full_name": "smul_add", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [807, 9], "def_end_pos": [807, 17]}]], "state_before": "case pos.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nu u' : \u03b1 \u2192 E\na\u271d : Disjoint (support u) (support u')\nu_int : Integrable u\nu'_int : Integrable u'\nh : (\u222b (a : \u03b1), u a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 u a \u2202\u03bc\nh' : (\u222b (a : \u03b1), u' a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 u' a \u2202\u03bc\n\u22a2 (\u222b (a : \u03b1), u a + u' a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 (u a + u' a) \u2202\u03bc", "state_after": "case pos.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nu u' : \u03b1 \u2192 E\na\u271d : Disjoint (support u) (support u')\nu_int : Integrable u\nu'_int : Integrable u'\nh : (\u222b (a : \u03b1), u a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 u a \u2202\u03bc\nh' : (\u222b (a : \u03b1), u' a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 u' a \u2202\u03bc\n\u22a2 (\u222b (a : \u03b1), u a + u' a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 u a + f a \u2022 u' a \u2202\u03bc"}, {"tactic": "rw [integral_add u_int u'_int, h, h', integral_add]", "annotated_tactic": ["rw [<a>integral_add</a> u_int u'_int, h, h', <a>integral_add</a>]", [{"full_name": "MeasureTheory.integral_add", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [868, 9], "def_end_pos": [868, 21]}, {"full_name": "MeasureTheory.integral_add", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [868, 9], "def_end_pos": [868, 21]}]], "state_before": "case pos.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nu u' : \u03b1 \u2192 E\na\u271d : Disjoint (support u) (support u')\nu_int : Integrable u\nu'_int : Integrable u'\nh : (\u222b (a : \u03b1), u a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 u a \u2202\u03bc\nh' : (\u222b (a : \u03b1), u' a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 u' a \u2202\u03bc\n\u22a2 (\u222b (a : \u03b1), u a + u' a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 u a + f a \u2022 u' a \u2202\u03bc", "state_after": "case pos.refine'_2.hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nu u' : \u03b1 \u2192 E\na\u271d : Disjoint (support u) (support u')\nu_int : Integrable u\nu'_int : Integrable u'\nh : (\u222b (a : \u03b1), u a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 u a \u2202\u03bc\nh' : (\u222b (a : \u03b1), u' a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 u' a \u2202\u03bc\n\u22a2 Integrable fun a => f a \u2022 u a\n\ncase pos.refine'_2.hg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nu u' : \u03b1 \u2192 E\na\u271d : Disjoint (support u) (support u')\nu_int : Integrable u\nu'_int : Integrable u'\nh : (\u222b (a : \u03b1), u a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 u a \u2202\u03bc\nh' : (\u222b (a : \u03b1), u' a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 u' a \u2202\u03bc\n\u22a2 Integrable fun a => f a \u2022 u' a"}, {"tactic": "exact (integrable_withDensity_iff_integrable_smul f_meas).1 u_int", "annotated_tactic": ["exact (<a>integrable_withDensity_iff_integrable_smul</a> f_meas).1 u_int", [{"full_name": "MeasureTheory.integrable_withDensity_iff_integrable_smul", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [918, 9], "def_end_pos": [918, 51]}]], "state_before": "case pos.refine'_2.hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nu u' : \u03b1 \u2192 E\na\u271d : Disjoint (support u) (support u')\nu_int : Integrable u\nu'_int : Integrable u'\nh : (\u222b (a : \u03b1), u a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 u a \u2202\u03bc\nh' : (\u222b (a : \u03b1), u' a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 u' a \u2202\u03bc\n\u22a2 Integrable fun a => f a \u2022 u a", "state_after": "no goals"}, {"tactic": "exact (integrable_withDensity_iff_integrable_smul f_meas).1 u'_int", "annotated_tactic": ["exact (<a>integrable_withDensity_iff_integrable_smul</a> f_meas).1 u'_int", [{"full_name": "MeasureTheory.integrable_withDensity_iff_integrable_smul", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [918, 9], "def_end_pos": [918, 51]}]], "state_before": "case pos.refine'_2.hg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nu u' : \u03b1 \u2192 E\na\u271d : Disjoint (support u) (support u')\nu_int : Integrable u\nu'_int : Integrable u'\nh : (\u222b (a : \u03b1), u a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 u a \u2202\u03bc\nh' : (\u222b (a : \u03b1), u' a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 u' a \u2202\u03bc\n\u22a2 Integrable fun a => f a \u2022 u' a", "state_after": "no goals"}, {"tactic": "have C1 :\n  Continuous fun u : Lp E 1 (\u03bc.withDensity fun x => f x) =>\n    \u222b x, u x \u2202\u03bc.withDensity fun x => f x :=\n  continuous_integral", "annotated_tactic": ["have C1 :\n      <a>Continuous</a> fun u : <a>Lp</a> E 1 (\u03bc.withDensity fun x => f x) =>\n        \u222b x, u x \u2202\u03bc.withDensity fun x => f x :=\n      <a>continuous_integral</a>", [{"full_name": "Continuous", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [1591, 11], "def_end_pos": [1591, 21]}, {"full_name": "MeasureTheory.Lp", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [98, 5], "def_end_pos": [98, 7]}, {"full_name": "MeasureTheory.continuous_integral", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [957, 9], "def_end_pos": [957, 28]}]], "state_before": "case pos.refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\n\u22a2 IsClosed {f_1 | (fun g => (\u222b (a : \u03b1), g a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 g a \u2202\u03bc) \u2191\u2191f_1}", "state_after": "case pos.refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nC1 : Continuous fun u => \u222b (x : \u03b1), \u2191\u2191u x \u2202Measure.withDensity \u03bc fun x => \u2191(f x)\n\u22a2 IsClosed {f_1 | (fun g => (\u222b (a : \u03b1), g a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 g a \u2202\u03bc) \u2191\u2191f_1}"}, {"tactic": "have C2 : Continuous fun u : Lp E 1 (\u03bc.withDensity fun x => f x) => \u222b x, f x \u2022 u x \u2202\u03bc := by\n  have : Continuous ((fun u : Lp E 1 \u03bc => \u222b x, u x \u2202\u03bc) \u2218 withDensitySMulLI (E := E) \u03bc f_meas) :=\n    continuous_integral.comp (withDensitySMulLI (E := E) \u03bc f_meas).continuous\n  convert this with u\n  simp only [Function.comp_apply, withDensitySMulLI_apply]\n  exact integral_congr_ae (mem\u21121_smul_of_L1_withDensity f_meas u).coeFn_toLp.symm", "annotated_tactic": ["have C2 : <a>Continuous</a> fun u : <a>Lp</a> E 1 (\u03bc.withDensity fun x => f x) => \u222b x, f x \u2022 u x \u2202\u03bc := by\n      have : <a>Continuous</a> ((fun u : <a>Lp</a> E 1 \u03bc => \u222b x, u x \u2202\u03bc) \u2218 <a>withDensitySMulLI</a> (E := E) \u03bc f_meas) :=\n        continuous_integral.comp (<a>withDensitySMulLI</a> (E := E) \u03bc f_meas).<a>continuous</a>\n      convert this with u\n      simp only [<a>Function.comp_apply</a>, <a>withDensitySMulLI_apply</a>]\n      exact <a>integral_congr_ae</a> (<a>mem\u21121_smul_of_L1_withDensity</a> f_meas u).coeFn_toLp.symm", [{"full_name": "Continuous", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [1591, 11], "def_end_pos": [1591, 21]}, {"full_name": "MeasureTheory.Lp", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [98, 5], "def_end_pos": [98, 7]}, {"full_name": "Continuous", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [1591, 11], "def_end_pos": [1591, 21]}, {"full_name": "MeasureTheory.Lp", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [98, 5], "def_end_pos": [98, 7]}, {"full_name": "MeasureTheory.withDensitySMulLI", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [978, 19], "def_end_pos": [978, 36]}, {"full_name": "MeasureTheory.withDensitySMulLI", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [978, 19], "def_end_pos": [978, 36]}, {"full_name": "LinearIsometry.continuous", "def_path": "Mathlib/Analysis/NormedSpace/LinearIsometry.lean", "def_pos": [301, 19], "def_end_pos": [301, 29]}, {"full_name": "Function.comp_apply", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [33, 17], "def_end_pos": [33, 36]}, {"full_name": "MeasureTheory.withDensitySMulLI_apply", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [1027, 9], "def_end_pos": [1027, 32]}, {"full_name": "MeasureTheory.integral_congr_ae", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [938, 9], "def_end_pos": [938, 26]}, {"full_name": "MeasureTheory.mem\u21121_smul_of_L1_withDensity", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [968, 9], "def_end_pos": [968, 37]}]], "state_before": "case pos.refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nC1 : Continuous fun u => \u222b (x : \u03b1), \u2191\u2191u x \u2202Measure.withDensity \u03bc fun x => \u2191(f x)\n\u22a2 IsClosed {f_1 | (fun g => (\u222b (a : \u03b1), g a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 g a \u2202\u03bc) \u2191\u2191f_1}", "state_after": "case pos.refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nC1 : Continuous fun u => \u222b (x : \u03b1), \u2191\u2191u x \u2202Measure.withDensity \u03bc fun x => \u2191(f x)\nC2 : Continuous fun u => \u222b (x : \u03b1), f x \u2022 \u2191\u2191u x \u2202\u03bc\n\u22a2 IsClosed {f_1 | (fun g => (\u222b (a : \u03b1), g a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 g a \u2202\u03bc) \u2191\u2191f_1}"}, {"tactic": "exact isClosed_eq C1 C2", "annotated_tactic": ["exact <a>isClosed_eq</a> C1 C2", [{"full_name": "isClosed_eq", "def_path": "Mathlib/Topology/Separation.lean", "def_pos": [1217, 9], "def_end_pos": [1217, 20]}]], "state_before": "case pos.refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nC1 : Continuous fun u => \u222b (x : \u03b1), \u2191\u2191u x \u2202Measure.withDensity \u03bc fun x => \u2191(f x)\nC2 : Continuous fun u => \u222b (x : \u03b1), f x \u2022 \u2191\u2191u x \u2202\u03bc\n\u22a2 IsClosed {f_1 | (fun g => (\u222b (a : \u03b1), g a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 g a \u2202\u03bc) \u2191\u2191f_1}", "state_after": "no goals"}, {"tactic": "have : Continuous ((fun u : Lp E 1 \u03bc => \u222b x, u x \u2202\u03bc) \u2218 withDensitySMulLI (E := E) \u03bc f_meas) :=\n  continuous_integral.comp (withDensitySMulLI (E := E) \u03bc f_meas).continuous", "annotated_tactic": ["have : <a>Continuous</a> ((fun u : <a>Lp</a> E 1 \u03bc => \u222b x, u x \u2202\u03bc) \u2218 <a>withDensitySMulLI</a> (E := E) \u03bc f_meas) :=\n        continuous_integral.comp (<a>withDensitySMulLI</a> (E := E) \u03bc f_meas).<a>continuous</a>", [{"full_name": "Continuous", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [1591, 11], "def_end_pos": [1591, 21]}, {"full_name": "MeasureTheory.Lp", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [98, 5], "def_end_pos": [98, 7]}, {"full_name": "MeasureTheory.withDensitySMulLI", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [978, 19], "def_end_pos": [978, 36]}, {"full_name": "MeasureTheory.withDensitySMulLI", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [978, 19], "def_end_pos": [978, 36]}, {"full_name": "LinearIsometry.continuous", "def_path": "Mathlib/Analysis/NormedSpace/LinearIsometry.lean", "def_pos": [301, 19], "def_end_pos": [301, 29]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nC1 : Continuous fun u => \u222b (x : \u03b1), \u2191\u2191u x \u2202Measure.withDensity \u03bc fun x => \u2191(f x)\n\u22a2 Continuous fun u => \u222b (x : \u03b1), f x \u2022 \u2191\u2191u x \u2202\u03bc", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nC1 : Continuous fun u => \u222b (x : \u03b1), \u2191\u2191u x \u2202Measure.withDensity \u03bc fun x => \u2191(f x)\nthis : Continuous ((fun u => \u222b (x : \u03b1), \u2191\u2191u x \u2202\u03bc) \u2218 \u2191(withDensitySMulLI \u03bc f_meas))\n\u22a2 Continuous fun u => \u222b (x : \u03b1), f x \u2022 \u2191\u2191u x \u2202\u03bc"}, {"tactic": "convert this with u", "annotated_tactic": ["convert this with u", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nC1 : Continuous fun u => \u222b (x : \u03b1), \u2191\u2191u x \u2202Measure.withDensity \u03bc fun x => \u2191(f x)\nthis : Continuous ((fun u => \u222b (x : \u03b1), \u2191\u2191u x \u2202\u03bc) \u2218 \u2191(withDensitySMulLI \u03bc f_meas))\n\u22a2 Continuous fun u => \u222b (x : \u03b1), f x \u2022 \u2191\u2191u x \u2202\u03bc", "state_after": "case h.e'_5.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nC1 : Continuous fun u => \u222b (x : \u03b1), \u2191\u2191u x \u2202Measure.withDensity \u03bc fun x => \u2191(f x)\nthis : Continuous ((fun u => \u222b (x : \u03b1), \u2191\u2191u x \u2202\u03bc) \u2218 \u2191(withDensitySMulLI \u03bc f_meas))\nu : { x // x \u2208 Lp E 1 }\n\u22a2 \u222b (x : \u03b1), f x \u2022 \u2191\u2191u x \u2202\u03bc = ((fun u => \u222b (x : \u03b1), \u2191\u2191u x \u2202\u03bc) \u2218 \u2191(withDensitySMulLI \u03bc f_meas)) u"}, {"tactic": "simp only [Function.comp_apply, withDensitySMulLI_apply]", "annotated_tactic": ["simp only [<a>Function.comp_apply</a>, <a>withDensitySMulLI_apply</a>]", [{"full_name": "Function.comp_apply", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [33, 17], "def_end_pos": [33, 36]}, {"full_name": "MeasureTheory.withDensitySMulLI_apply", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [1027, 9], "def_end_pos": [1027, 32]}]], "state_before": "case h.e'_5.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nC1 : Continuous fun u => \u222b (x : \u03b1), \u2191\u2191u x \u2202Measure.withDensity \u03bc fun x => \u2191(f x)\nthis : Continuous ((fun u => \u222b (x : \u03b1), \u2191\u2191u x \u2202\u03bc) \u2218 \u2191(withDensitySMulLI \u03bc f_meas))\nu : { x // x \u2208 Lp E 1 }\n\u22a2 \u222b (x : \u03b1), f x \u2022 \u2191\u2191u x \u2202\u03bc = ((fun u => \u222b (x : \u03b1), \u2191\u2191u x \u2202\u03bc) \u2218 \u2191(withDensitySMulLI \u03bc f_meas)) u", "state_after": "case h.e'_5.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nC1 : Continuous fun u => \u222b (x : \u03b1), \u2191\u2191u x \u2202Measure.withDensity \u03bc fun x => \u2191(f x)\nthis : Continuous ((fun u => \u222b (x : \u03b1), \u2191\u2191u x \u2202\u03bc) \u2218 \u2191(withDensitySMulLI \u03bc f_meas))\nu : { x // x \u2208 Lp E 1 }\n\u22a2 \u222b (x : \u03b1), f x \u2022 \u2191\u2191u x \u2202\u03bc = \u222b (x : \u03b1), \u2191\u2191(Mem\u2112p.toLp (fun x => f x \u2022 \u2191\u2191u x) (_ : Mem\u2112p (fun x => f x \u2022 \u2191\u2191u x) 1)) x \u2202\u03bc"}, {"tactic": "exact integral_congr_ae (mem\u21121_smul_of_L1_withDensity f_meas u).coeFn_toLp.symm", "annotated_tactic": ["exact <a>integral_congr_ae</a> (<a>mem\u21121_smul_of_L1_withDensity</a> f_meas u).coeFn_toLp.symm", [{"full_name": "MeasureTheory.integral_congr_ae", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [938, 9], "def_end_pos": [938, 26]}, {"full_name": "MeasureTheory.mem\u21121_smul_of_L1_withDensity", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [968, 9], "def_end_pos": [968, 37]}]], "state_before": "case h.e'_5.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nC1 : Continuous fun u => \u222b (x : \u03b1), \u2191\u2191u x \u2202Measure.withDensity \u03bc fun x => \u2191(f x)\nthis : Continuous ((fun u => \u222b (x : \u03b1), \u2191\u2191u x \u2202\u03bc) \u2218 \u2191(withDensitySMulLI \u03bc f_meas))\nu : { x // x \u2208 Lp E 1 }\n\u22a2 \u222b (x : \u03b1), f x \u2022 \u2191\u2191u x \u2202\u03bc = \u222b (x : \u03b1), \u2191\u2191(Mem\u2112p.toLp (fun x => f x \u2022 \u2191\u2191u x) (_ : Mem\u2112p (fun x => f x \u2022 \u2191\u2191u x) 1)) x \u2202\u03bc", "state_after": "no goals"}, {"tactic": "intro u v huv _ hu", "annotated_tactic": ["intro u v huv _ hu", []], "state_before": "case pos.refine'_4\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\n\u22a2 \u2200 \u2983f_1 g : \u03b1 \u2192 E\u2984,\n    f_1 =\u1d50[Measure.withDensity \u03bc fun x => \u2191(f x)] g \u2192\n      Integrable f_1 \u2192\n        (fun g => (\u222b (a : \u03b1), g a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 g a \u2202\u03bc) f_1 \u2192\n          (fun g => (\u222b (a : \u03b1), g a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 g a \u2202\u03bc) g", "state_after": "case pos.refine'_4\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nu v : \u03b1 \u2192 E\nhuv : u =\u1d50[Measure.withDensity \u03bc fun x => \u2191(f x)] v\na\u271d : Integrable u\nhu : (\u222b (a : \u03b1), u a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 u a \u2202\u03bc\n\u22a2 (\u222b (a : \u03b1), v a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 v a \u2202\u03bc"}, {"tactic": "rw [\u2190 integral_congr_ae huv, hu]", "annotated_tactic": ["rw [\u2190 <a>integral_congr_ae</a> huv, hu]", [{"full_name": "MeasureTheory.integral_congr_ae", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [938, 9], "def_end_pos": [938, 26]}]], "state_before": "case pos.refine'_4\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nu v : \u03b1 \u2192 E\nhuv : u =\u1d50[Measure.withDensity \u03bc fun x => \u2191(f x)] v\na\u271d : Integrable u\nhu : (\u222b (a : \u03b1), u a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 u a \u2202\u03bc\n\u22a2 (\u222b (a : \u03b1), v a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 v a \u2202\u03bc", "state_after": "case pos.refine'_4\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nu v : \u03b1 \u2192 E\nhuv : u =\u1d50[Measure.withDensity \u03bc fun x => \u2191(f x)] v\na\u271d : Integrable u\nhu : (\u222b (a : \u03b1), u a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 u a \u2202\u03bc\n\u22a2 \u222b (a : \u03b1), f a \u2022 u a \u2202\u03bc = \u222b (a : \u03b1), f a \u2022 v a \u2202\u03bc"}, {"tactic": "apply integral_congr_ae", "annotated_tactic": ["apply <a>integral_congr_ae</a>", [{"full_name": "MeasureTheory.integral_congr_ae", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [938, 9], "def_end_pos": [938, 26]}]], "state_before": "case pos.refine'_4\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nu v : \u03b1 \u2192 E\nhuv : u =\u1d50[Measure.withDensity \u03bc fun x => \u2191(f x)] v\na\u271d : Integrable u\nhu : (\u222b (a : \u03b1), u a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 u a \u2202\u03bc\n\u22a2 \u222b (a : \u03b1), f a \u2022 u a \u2202\u03bc = \u222b (a : \u03b1), f a \u2022 v a \u2202\u03bc", "state_after": "case pos.refine'_4.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nu v : \u03b1 \u2192 E\nhuv : u =\u1d50[Measure.withDensity \u03bc fun x => \u2191(f x)] v\na\u271d : Integrable u\nhu : (\u222b (a : \u03b1), u a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 u a \u2202\u03bc\n\u22a2 (fun a => f a \u2022 u a) =\u1d50[\u03bc] fun a => f a \u2022 v a"}, {"tactic": "filter_upwards [(ae_withDensity_iff f_meas.coe_nnreal_ennreal).1 huv] with x hx", "annotated_tactic": ["filter_upwards [(<a>ae_withDensity_iff</a> f_meas.coe_nnreal_ennreal).1 huv] with x hx", [{"full_name": "MeasureTheory.ae_withDensity_iff", "def_path": "Mathlib/MeasureTheory/Measure/WithDensity.lean", "def_pos": [223, 9], "def_end_pos": [223, 27]}]], "state_before": "case pos.refine'_4.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nu v : \u03b1 \u2192 E\nhuv : u =\u1d50[Measure.withDensity \u03bc fun x => \u2191(f x)] v\na\u271d : Integrable u\nhu : (\u222b (a : \u03b1), u a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 u a \u2202\u03bc\n\u22a2 (fun a => f a \u2022 u a) =\u1d50[\u03bc] fun a => f a \u2022 v a", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nu v : \u03b1 \u2192 E\nhuv : u =\u1d50[Measure.withDensity \u03bc fun x => \u2191(f x)] v\na\u271d : Integrable u\nhu : (\u222b (a : \u03b1), u a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 u a \u2202\u03bc\nx : \u03b1\nhx : \u2191(f x) \u2260 0 \u2192 u x = v x\n\u22a2 f x \u2022 u x = f x \u2022 v x"}, {"tactic": "rcases eq_or_ne (f x) 0 with (h'x | h'x)", "annotated_tactic": ["rcases <a>eq_or_ne</a> (f x) 0 with (h'x | h'x)", [{"full_name": "eq_or_ne", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [209, 9], "def_end_pos": [209, 17]}]], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nu v : \u03b1 \u2192 E\nhuv : u =\u1d50[Measure.withDensity \u03bc fun x => \u2191(f x)] v\na\u271d : Integrable u\nhu : (\u222b (a : \u03b1), u a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 u a \u2202\u03bc\nx : \u03b1\nhx : \u2191(f x) \u2260 0 \u2192 u x = v x\n\u22a2 f x \u2022 u x = f x \u2022 v x", "state_after": "case h.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nu v : \u03b1 \u2192 E\nhuv : u =\u1d50[Measure.withDensity \u03bc fun x => \u2191(f x)] v\na\u271d : Integrable u\nhu : (\u222b (a : \u03b1), u a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 u a \u2202\u03bc\nx : \u03b1\nhx : \u2191(f x) \u2260 0 \u2192 u x = v x\nh'x : f x = 0\n\u22a2 f x \u2022 u x = f x \u2022 v x\n\ncase h.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nu v : \u03b1 \u2192 E\nhuv : u =\u1d50[Measure.withDensity \u03bc fun x => \u2191(f x)] v\na\u271d : Integrable u\nhu : (\u222b (a : \u03b1), u a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 u a \u2202\u03bc\nx : \u03b1\nhx : \u2191(f x) \u2260 0 \u2192 u x = v x\nh'x : f x \u2260 0\n\u22a2 f x \u2022 u x = f x \u2022 v x"}, {"tactic": "simp only [h'x, zero_smul]", "annotated_tactic": ["simp only [h'x, <a>zero_smul</a>]", [{"full_name": "zero_smul", "def_path": "Mathlib/Algebra/SMulWithZero.lean", "def_pos": [70, 9], "def_end_pos": [70, 18]}]], "state_before": "case h.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nu v : \u03b1 \u2192 E\nhuv : u =\u1d50[Measure.withDensity \u03bc fun x => \u2191(f x)] v\na\u271d : Integrable u\nhu : (\u222b (a : \u03b1), u a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 u a \u2202\u03bc\nx : \u03b1\nhx : \u2191(f x) \u2260 0 \u2192 u x = v x\nh'x : f x = 0\n\u22a2 f x \u2022 u x = f x \u2022 v x", "state_after": "no goals"}, {"tactic": "rw [hx _]", "annotated_tactic": ["rw [hx _]", []], "state_before": "case h.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nu v : \u03b1 \u2192 E\nhuv : u =\u1d50[Measure.withDensity \u03bc fun x => \u2191(f x)] v\na\u271d : Integrable u\nhu : (\u222b (a : \u03b1), u a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 u a \u2202\u03bc\nx : \u03b1\nhx : \u2191(f x) \u2260 0 \u2192 u x = v x\nh'x : f x \u2260 0\n\u22a2 f x \u2022 u x = f x \u2022 v x", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nu v : \u03b1 \u2192 E\nhuv : u =\u1d50[Measure.withDensity \u03bc fun x => \u2191(f x)] v\na\u271d : Integrable u\nhu : (\u222b (a : \u03b1), u a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 u a \u2202\u03bc\nx : \u03b1\nhx : \u2191(f x) \u2260 0 \u2192 u x = v x\nh'x : f x \u2260 0\n\u22a2 \u2191(f x) \u2260 0"}, {"tactic": "simpa only [Ne.def, ENNReal.coe_eq_zero] using h'x", "annotated_tactic": ["simpa only [<a>Ne.def</a>, <a>ENNReal.coe_eq_zero</a>] using h'x", [{"full_name": "Ne.def", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [59, 9], "def_end_pos": [59, 15]}, {"full_name": "ENNReal.coe_eq_zero", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [368, 28], "def_end_pos": [368, 39]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nu v : \u03b1 \u2192 E\nhuv : u =\u1d50[Measure.withDensity \u03bc fun x => \u2191(f x)] v\na\u271d : Integrable u\nhu : (\u222b (a : \u03b1), u a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 u a \u2202\u03bc\nx : \u03b1\nhx : \u2191(f x) \u2260 0 \u2192 u x = v x\nh'x : f x \u2260 0\n\u22a2 \u2191(f x) \u2260 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finmap.lean", "full_name": "Finmap.insert_insert_of_ne", "start": [514, 1], "end": [517, 82], "traced_tactics": [{"tactic": "simp only [insert_toFinmap, AList.toFinmap_eq, AList.insert_insert_of_ne _ h]", "annotated_tactic": ["simp only [<a>insert_toFinmap</a>, <a>AList.toFinmap_eq</a>, <a>AList.insert_insert_of_ne</a> _ h]", [{"full_name": "Finmap.insert_toFinmap", "def_path": "Mathlib/Data/Finmap.lean", "def_pos": [480, 9], "def_end_pos": [480, 24]}, {"full_name": "AList.toFinmap_eq", "def_path": "Mathlib/Data/Finmap.lean", "def_pos": [75, 9], "def_end_pos": [75, 26]}, {"full_name": "AList.insert_insert_of_ne", "def_path": "Mathlib/Data/List/AList.lean", "def_pos": [331, 9], "def_end_pos": [331, 28]}]], "state_before": "\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\na a' : \u03b1\nb : \u03b2 a\nb' : \u03b2 a'\ns\u271d : Finmap \u03b2\nh : a \u2260 a'\ns : AList \u03b2\n\u22a2 insert a' b' (insert a b \u27e6s\u27e7) = insert a b (insert a' b' \u27e6s\u27e7)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Partrec.lean", "full_name": "Vector.mOfFn_part_some", "start": [620, 1], "end": [622, 20], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Haar/Basic.lean", "full_name": "MeasureTheory.Measure.haar.chaar_sup_le", "start": [469, 1], "end": [479, 62], "traced_tactics": [{"tactic": "let eval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082)", "annotated_tactic": ["let eval : (<a>Compacts</a> G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082)", [{"full_name": "TopologicalSpace.Compacts", "def_path": "Mathlib/Topology/Sets/Compacts.lean", "def_pos": [36, 11], "def_end_pos": [36, 19]}]], "state_before": "G : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\n\u22a2 chaar K\u2080 (K\u2081 \u2294 K\u2082) \u2264 chaar K\u2080 K\u2081 + chaar K\u2080 K\u2082", "state_after": "G : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082)\n\u22a2 chaar K\u2080 (K\u2081 \u2294 K\u2082) \u2264 chaar K\u2080 K\u2081 + chaar K\u2080 K\u2082"}, {"tactic": "have : Continuous eval := by\n  exact ((continuous_apply K\u2081).add (continuous_apply K\u2082)).sub (continuous_apply (K\u2081 \u2294 K\u2082))", "annotated_tactic": ["have : <a>Continuous</a> eval := by\n    exact ((<a>continuous_apply</a> K\u2081).<a>add</a> (<a>continuous_apply</a> K\u2082)).<a>sub</a> (<a>continuous_apply</a> (K\u2081 \u2294 K\u2082))", [{"full_name": "Continuous", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [1591, 11], "def_end_pos": [1591, 21]}, {"full_name": "continuous_apply", "def_path": "Mathlib/Topology/Constructions.lean", "def_pos": [1208, 9], "def_end_pos": [1208, 25]}, {"full_name": "Continuous.add", "def_path": "Mathlib/Topology/Algebra/Monoid.lean", "def_pos": [86, 3], "def_end_pos": [86, 14]}, {"full_name": "continuous_apply", "def_path": "Mathlib/Topology/Constructions.lean", "def_pos": [1208, 9], "def_end_pos": [1208, 25]}, {"full_name": "Continuous.sub", "def_path": "Mathlib/Topology/Algebra/Group/Basic.lean", "def_pos": [1104, 36], "def_end_pos": [1104, 39]}, {"full_name": "continuous_apply", "def_path": "Mathlib/Topology/Constructions.lean", "def_pos": [1208, 9], "def_end_pos": [1208, 25]}]], "state_before": "G : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082)\n\u22a2 chaar K\u2080 (K\u2081 \u2294 K\u2082) \u2264 chaar K\u2080 K\u2081 + chaar K\u2080 K\u2082", "state_after": "G : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082)\nthis : Continuous eval\n\u22a2 chaar K\u2080 (K\u2081 \u2294 K\u2082) \u2264 chaar K\u2080 K\u2081 + chaar K\u2080 K\u2082"}, {"tactic": "rw [\u2190 sub_nonneg]", "annotated_tactic": ["rw [\u2190 <a>sub_nonneg</a>]", [{"full_name": "sub_nonneg", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [720, 30], "def_end_pos": [720, 40]}]], "state_before": "G : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082)\nthis : Continuous eval\n\u22a2 chaar K\u2080 (K\u2081 \u2294 K\u2082) \u2264 chaar K\u2080 K\u2081 + chaar K\u2080 K\u2082", "state_after": "G : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082)\nthis : Continuous eval\n\u22a2 0 \u2264 chaar K\u2080 K\u2081 + chaar K\u2080 K\u2082 - chaar K\u2080 (K\u2081 \u2294 K\u2082)"}, {"tactic": "show chaar K\u2080 \u2208 eval \u207b\u00b9' Ici (0 : \u211d)", "annotated_tactic": ["show <a>chaar</a> K\u2080 \u2208 eval \u207b\u00b9' <a>Ici</a> (0 : \u211d)", [{"full_name": "MeasureTheory.Measure.haar.chaar", "def_path": "Mathlib/MeasureTheory/Measure/Haar/Basic.lean", "def_pos": [404, 19], "def_end_pos": [404, 24]}, {"full_name": "Set.Ici", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [74, 5], "def_end_pos": [74, 8]}]], "state_before": "G : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082)\nthis : Continuous eval\n\u22a2 0 \u2264 chaar K\u2080 K\u2081 + chaar K\u2080 K\u2082 - chaar K\u2080 (K\u2081 \u2294 K\u2082)", "state_after": "G : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082)\nthis : Continuous eval\n\u22a2 chaar K\u2080 \u2208 eval \u207b\u00b9' Ici 0"}, {"tactic": "apply mem_of_subset_of_mem _ (chaar_mem_clPrehaar K\u2080 \u22a4)", "annotated_tactic": ["apply <a>mem_of_subset_of_mem</a> _ (<a>chaar_mem_clPrehaar</a> K\u2080 \u22a4)", [{"full_name": "Set.mem_of_subset_of_mem", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [383, 9], "def_end_pos": [383, 29]}, {"full_name": "MeasureTheory.Measure.haar.chaar_mem_clPrehaar", "def_path": "Mathlib/MeasureTheory/Measure/Haar/Basic.lean", "def_pos": [416, 9], "def_end_pos": [416, 28]}]], "state_before": "G : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082)\nthis : Continuous eval\n\u22a2 chaar K\u2080 \u2208 eval \u207b\u00b9' Ici 0", "state_after": "G : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082)\nthis : Continuous eval\n\u22a2 clPrehaar \u2191K\u2080 \u22a4 \u2286 eval \u207b\u00b9' Ici 0"}, {"tactic": "unfold clPrehaar", "annotated_tactic": ["unfold <a>clPrehaar</a>", [{"full_name": "MeasureTheory.Measure.haar.clPrehaar", "def_path": "Mathlib/MeasureTheory/Measure/Haar/Basic.lean", "def_pos": [154, 5], "def_end_pos": [154, 14]}]], "state_before": "G : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082)\nthis : Continuous eval\n\u22a2 clPrehaar \u2191K\u2080 \u22a4 \u2286 eval \u207b\u00b9' Ici 0", "state_after": "G : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082)\nthis : Continuous eval\n\u22a2 closure (prehaar \u2191K\u2080 '' {U | U \u2286 \u2191\u22a4.toOpens \u2227 IsOpen U \u2227 1 \u2208 U}) \u2286 eval \u207b\u00b9' Ici 0"}, {"tactic": "rw [IsClosed.closure_subset_iff]", "annotated_tactic": ["rw [<a>IsClosed.closure_subset_iff</a>]", [{"full_name": "IsClosed.closure_subset_iff", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [465, 9], "def_end_pos": [465, 36]}]], "state_before": "G : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082)\nthis : Continuous eval\n\u22a2 closure (prehaar \u2191K\u2080 '' {U | U \u2286 \u2191\u22a4.toOpens \u2227 IsOpen U \u2227 1 \u2208 U}) \u2286 eval \u207b\u00b9' Ici 0", "state_after": "G : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082)\nthis : Continuous eval\n\u22a2 prehaar \u2191K\u2080 '' {U | U \u2286 \u2191\u22a4.toOpens \u2227 IsOpen U \u2227 1 \u2208 U} \u2286 eval \u207b\u00b9' Ici 0\n\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082)\nthis : Continuous eval\n\u22a2 IsClosed (eval \u207b\u00b9' Ici 0)"}, {"tactic": "exact ((continuous_apply K\u2081).add (continuous_apply K\u2082)).sub (continuous_apply (K\u2081 \u2294 K\u2082))", "annotated_tactic": ["exact ((<a>continuous_apply</a> K\u2081).<a>add</a> (<a>continuous_apply</a> K\u2082)).<a>sub</a> (<a>continuous_apply</a> (K\u2081 \u2294 K\u2082))", [{"full_name": "continuous_apply", "def_path": "Mathlib/Topology/Constructions.lean", "def_pos": [1208, 9], "def_end_pos": [1208, 25]}, {"full_name": "Continuous.add", "def_path": "Mathlib/Topology/Algebra/Monoid.lean", "def_pos": [86, 3], "def_end_pos": [86, 14]}, {"full_name": "continuous_apply", "def_path": "Mathlib/Topology/Constructions.lean", "def_pos": [1208, 9], "def_end_pos": [1208, 25]}, {"full_name": "Continuous.sub", "def_path": "Mathlib/Topology/Algebra/Group/Basic.lean", "def_pos": [1104, 36], "def_end_pos": [1104, 39]}, {"full_name": "continuous_apply", "def_path": "Mathlib/Topology/Constructions.lean", "def_pos": [1208, 9], "def_end_pos": [1208, 25]}]], "state_before": "G : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082)\n\u22a2 Continuous eval", "state_after": "no goals"}, {"tactic": "rintro _ \u27e8U, \u27e8_, h2U, h3U\u27e9, rfl\u27e9", "annotated_tactic": ["rintro _ \u27e8U, \u27e8_, h2U, h3U\u27e9, rfl\u27e9", []], "state_before": "G : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082)\nthis : Continuous eval\n\u22a2 prehaar \u2191K\u2080 '' {U | U \u2286 \u2191\u22a4.toOpens \u2227 IsOpen U \u2227 1 \u2208 U} \u2286 eval \u207b\u00b9' Ici 0", "state_after": "case intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082)\nthis : Continuous eval\nU : Set G\nleft\u271d : U \u2286 \u2191\u22a4.toOpens\nh2U : IsOpen U\nh3U : 1 \u2208 U\n\u22a2 prehaar (\u2191K\u2080) U \u2208 eval \u207b\u00b9' Ici 0"}, {"tactic": "simp only [mem_preimage, mem_Ici, sub_nonneg]", "annotated_tactic": ["simp only [<a>mem_preimage</a>, <a>mem_Ici</a>, <a>sub_nonneg</a>]", [{"full_name": "Set.mem_preimage", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [64, 9], "def_end_pos": [64, 21]}, {"full_name": "Set.mem_Ici", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [146, 9], "def_end_pos": [146, 16]}, {"full_name": "sub_nonneg", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [720, 30], "def_end_pos": [720, 40]}]], "state_before": "case intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082)\nthis : Continuous eval\nU : Set G\nleft\u271d : U \u2286 \u2191\u22a4.toOpens\nh2U : IsOpen U\nh3U : 1 \u2208 U\n\u22a2 prehaar (\u2191K\u2080) U \u2208 eval \u207b\u00b9' Ici 0", "state_after": "case intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082)\nthis : Continuous eval\nU : Set G\nleft\u271d : U \u2286 \u2191\u22a4.toOpens\nh2U : IsOpen U\nh3U : 1 \u2208 U\n\u22a2 prehaar (\u2191K\u2080) U (K\u2081 \u2294 K\u2082) \u2264 prehaar (\u2191K\u2080) U K\u2081 + prehaar (\u2191K\u2080) U K\u2082"}, {"tactic": "apply prehaar_sup_le", "annotated_tactic": ["apply <a>prehaar_sup_le</a>", [{"full_name": "MeasureTheory.Measure.haar.prehaar_sup_le", "def_path": "Mathlib/MeasureTheory/Measure/Haar/Basic.lean", "def_pos": [332, 9], "def_end_pos": [332, 23]}]], "state_before": "case intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082)\nthis : Continuous eval\nU : Set G\nleft\u271d : U \u2286 \u2191\u22a4.toOpens\nh2U : IsOpen U\nh3U : 1 \u2208 U\n\u22a2 prehaar (\u2191K\u2080) U (K\u2081 \u2294 K\u2082) \u2264 prehaar (\u2191K\u2080) U K\u2081 + prehaar (\u2191K\u2080) U K\u2082", "state_after": "case intro.intro.intro.intro.hU\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082)\nthis : Continuous eval\nU : Set G\nleft\u271d : U \u2286 \u2191\u22a4.toOpens\nh2U : IsOpen U\nh3U : 1 \u2208 U\n\u22a2 Set.Nonempty (interior U)"}, {"tactic": "rw [h2U.interior_eq]", "annotated_tactic": ["rw [h2U.interior_eq]", []], "state_before": "case intro.intro.intro.intro.hU\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082)\nthis : Continuous eval\nU : Set G\nleft\u271d : U \u2286 \u2191\u22a4.toOpens\nh2U : IsOpen U\nh3U : 1 \u2208 U\n\u22a2 Set.Nonempty (interior U)", "state_after": "case intro.intro.intro.intro.hU\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082)\nthis : Continuous eval\nU : Set G\nleft\u271d : U \u2286 \u2191\u22a4.toOpens\nh2U : IsOpen U\nh3U : 1 \u2208 U\n\u22a2 Set.Nonempty U"}, {"tactic": "exact \u27e81, h3U\u27e9", "annotated_tactic": ["exact \u27e81, h3U\u27e9", []], "state_before": "case intro.intro.intro.intro.hU\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082)\nthis : Continuous eval\nU : Set G\nleft\u271d : U \u2286 \u2191\u22a4.toOpens\nh2U : IsOpen U\nh3U : 1 \u2208 U\n\u22a2 Set.Nonempty U", "state_after": "no goals"}, {"tactic": "apply continuous_iff_isClosed.mp this", "annotated_tactic": ["apply continuous_iff_isClosed.mp this", []], "state_before": "G : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082)\nthis : Continuous eval\n\u22a2 IsClosed (eval \u207b\u00b9' Ici 0)", "state_after": "case a\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082)\nthis : Continuous eval\n\u22a2 IsClosed (Ici 0)"}, {"tactic": "exact isClosed_Ici", "annotated_tactic": ["exact <a>isClosed_Ici</a>", [{"full_name": "isClosed_Ici", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [159, 9], "def_end_pos": [159, 21]}]], "state_before": "case a\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082)\nthis : Continuous eval\n\u22a2 IsClosed (Ici 0)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Fin/Lemmas.lean", "full_name": "USize.modn_toNat", "start": [750, 9], "end": [751, 18], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Covering/Differentiation.lean", "full_name": "VitaliFamily.ae_tendsto_lintegral_nnnorm_sub_div'_of_integrable", "start": [801, 1], "end": [874, 57], "traced_tactics": [{"tactic": "let A := MeasureTheory.Measure.finiteSpanningSetsInOpen' \u03bc", "annotated_tactic": ["let A := <a>MeasureTheory.Measure.finiteSpanningSetsInOpen'</a> \u03bc", [{"full_name": "MeasureTheory.Measure.finiteSpanningSetsInOpen'", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [4364, 1], "def_end_pos": [4403, 31]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f\nh'f : StronglyMeasurable f\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - f x\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd 0)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f\nh'f : StronglyMeasurable f\nA : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - f x\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd 0)"}, {"tactic": "rcases h'f.isSeparable_range with \u27e8t, t_count, ht\u27e9", "annotated_tactic": ["rcases h'f.isSeparable_range with \u27e8t, t_count, ht\u27e9", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f\nh'f : StronglyMeasurable f\nA : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - f x\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd 0)", "state_after": "case intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f\nh'f : StronglyMeasurable f\nA : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc\nt : Set E\nt_count : Set.Countable t\nht : range f \u2286 closure t\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - f x\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd 0)"}, {"tactic": "filter_upwards [main, v.ae_eventually_measure_pos] with x hx h'x", "annotated_tactic": ["filter_upwards [main, v.ae_eventually_measure_pos] with x hx h'x", []], "state_before": "case intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f\nh'f : StronglyMeasurable f\nA : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc\nt : Set E\nt_count : Set.Countable t\nht : range f \u2286 closure t\nmain :\n  \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    \u2200 (n : \u2115) (c : E),\n      c \u2208 t \u2192\n        Tendsto\n          (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)\n          (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a)\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - f x\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd 0)", "state_after": "case h\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f\nh'f : StronglyMeasurable f\nA : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc\nt : Set E\nt_count : Set.Countable t\nht : range f \u2286 closure t\nmain :\n  \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    \u2200 (n : \u2115) (c : E),\n      c \u2208 t \u2192\n        Tendsto\n          (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)\n          (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a)\nx : \u03b1\nhx :\n  \u2200 (n : \u2115) (c : E),\n    c \u2208 t \u2192\n      Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)\n        (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a)\nh'x : \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a\n\u22a2 Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - f x\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd 0)"}, {"tactic": "have M :\n  \u2200 c \u2208 t, Tendsto (fun a => (\u222b\u207b y in a, \u2016f y - c\u2016\u208a \u2202\u03bc) / \u03bc a)\n    (v.filterAt x) (\ud835\udcdd \u2016f x - c\u2016\u208a) := by\n  intro c hc\n  obtain \u27e8n, xn\u27e9 : \u2203 n, x \u2208 A.set n := by simpa [\u2190 A.spanning] using mem_univ x\n  specialize hx n c hc\n  simp only [xn, indicator_of_mem] at hx\n  apply hx.congr' _\n  filter_upwards [v.eventually_filterAt_subset_of_nhds (IsOpen.mem_nhds (A.set_mem n) xn),\n    v.eventually_filterAt_measurableSet x] with a ha h'a\n  congr 1\n  apply set_lintegral_congr_fun h'a\n  apply eventually_of_forall fun y => ?_\n  intro hy\n  simp only [ha hy, indicator_of_mem]", "annotated_tactic": ["have M :\n    \u2200 c \u2208 t, <a>Tendsto</a> (fun a => (\u222b\u207b y in a, \u2016f y - c\u2016\u208a \u2202\u03bc) / \u03bc a)\n      (v.filterAt x) (\ud835\udcdd \u2016f x - c\u2016\u208a) := by\n    intro c hc\n    obtain \u27e8n, xn\u27e9 : \u2203 n, x \u2208 A.set n := by simpa [\u2190 A.spanning] using <a>mem_univ</a> x\n    specialize hx n c hc\n    simp only [xn, <a>indicator_of_mem</a>] at hx\n    apply hx.congr' _\n    filter_upwards [v.eventually_filterAt_subset_of_nhds (<a>IsOpen.mem_nhds</a> (A.set_mem n) xn),\n      v.eventually_filterAt_measurableSet x] with a ha h'a\n    congr 1\n    apply <a>set_lintegral_congr_fun</a> h'a\n    apply <a>eventually_of_forall</a> fun y => ?_\n    intro hy\n    simp only [ha hy, <a>indicator_of_mem</a>]", [{"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2939, 5], "def_end_pos": [2939, 12]}, {"full_name": "Set.mem_univ", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [676, 9], "def_end_pos": [676, 17]}, {"full_name": "Set.indicator_of_mem", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [67, 3], "def_end_pos": [67, 14]}, {"full_name": "IsOpen.mem_nhds", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [928, 9], "def_end_pos": [928, 24]}, {"full_name": "MeasureTheory.set_lintegral_congr_fun", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [316, 9], "def_end_pos": [316, 32]}, {"full_name": "Filter.eventually_of_forall", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1111, 9], "def_end_pos": [1111, 29]}, {"full_name": "Set.indicator_of_mem", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [67, 3], "def_end_pos": [67, 14]}]], "state_before": "case h\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f\nh'f : StronglyMeasurable f\nA : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc\nt : Set E\nt_count : Set.Countable t\nht : range f \u2286 closure t\nmain :\n  \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    \u2200 (n : \u2115) (c : E),\n      c \u2208 t \u2192\n        Tendsto\n          (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)\n          (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a)\nx : \u03b1\nhx :\n  \u2200 (n : \u2115) (c : E),\n    c \u2208 t \u2192\n      Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)\n        (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a)\nh'x : \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a\n\u22a2 Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - f x\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd 0)", "state_after": "case h\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f\nh'f : StronglyMeasurable f\nA : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc\nt : Set E\nt_count : Set.Countable t\nht : range f \u2286 closure t\nmain :\n  \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    \u2200 (n : \u2115) (c : E),\n      c \u2208 t \u2192\n        Tendsto\n          (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)\n          (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a)\nx : \u03b1\nhx :\n  \u2200 (n : \u2115) (c : E),\n    c \u2208 t \u2192\n      Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)\n        (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a)\nh'x : \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a\nM : \u2200 (c : E), c \u2208 t \u2192 Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a)\n\u22a2 Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - f x\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd 0)"}, {"tactic": "apply ENNReal.tendsto_nhds_zero.2 fun \u03b5 \u03b5pos => ?_", "annotated_tactic": ["apply <a>ENNReal.tendsto_nhds_zero</a>.2 fun \u03b5 \u03b5pos => ?_", [{"full_name": "ENNReal.tendsto_nhds_zero", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [279, 19], "def_end_pos": [279, 36]}]], "state_before": "case h\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f\nh'f : StronglyMeasurable f\nA : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc\nt : Set E\nt_count : Set.Countable t\nht : range f \u2286 closure t\nmain :\n  \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    \u2200 (n : \u2115) (c : E),\n      c \u2208 t \u2192\n        Tendsto\n          (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)\n          (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a)\nx : \u03b1\nhx :\n  \u2200 (n : \u2115) (c : E),\n    c \u2208 t \u2192\n      Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)\n        (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a)\nh'x : \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a\nM : \u2200 (c : E), c \u2208 t \u2192 Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a)\n\u22a2 Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - f x\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd 0)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f\nh'f : StronglyMeasurable f\nA : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc\nt : Set E\nt_count : Set.Countable t\nht : range f \u2286 closure t\nmain :\n  \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    \u2200 (n : \u2115) (c : E),\n      c \u2208 t \u2192\n        Tendsto\n          (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)\n          (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a)\nx : \u03b1\nhx :\n  \u2200 (n : \u2115) (c : E),\n    c \u2208 t \u2192\n      Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)\n        (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a)\nh'x : \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a\nM : \u2200 (c : E), c \u2208 t \u2192 Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a)\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\n\u22a2 \u2200\u1da0 (x_1 : Set \u03b1) in filterAt v x, (\u222b\u207b (y : \u03b1) in x_1, \u2191\u2016f y - f x\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc x_1 \u2264 \u03b5"}, {"tactic": "obtain \u27e8c, ct, xc\u27e9 : \u2203 c \u2208 t, (\u2016f x - c\u2016\u208a : \u211d\u22650\u221e) < \u03b5 / 2 := by\n  simp_rw [\u2190 edist_eq_coe_nnnorm_sub]\n  have : f x \u2208 closure t := ht (mem_range_self _)\n  exact EMetric.mem_closure_iff.1 this (\u03b5 / 2) (ENNReal.half_pos (ne_of_gt \u03b5pos))", "annotated_tactic": ["obtain \u27e8c, ct, xc\u27e9 : \u2203 c \u2208 t, (\u2016f x - c\u2016\u208a : \u211d\u22650\u221e) < \u03b5 / 2 := by\n    simp_rw [\u2190 <a>edist_eq_coe_nnnorm_sub</a>]\n    have : f x \u2208 <a>closure</a> t := ht (<a>mem_range_self</a> _)\n    exact <a>EMetric.mem_closure_iff</a>.1 this (\u03b5 / 2) (<a>ENNReal.half_pos</a> (<a>ne_of_gt</a> \u03b5pos))", [{"full_name": "edist_eq_coe_nnnorm_sub", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [1005, 3], "def_end_pos": [1005, 14]}, {"full_name": "closure", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [422, 5], "def_end_pos": [422, 12]}, {"full_name": "Set.mem_range_self", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [680, 9], "def_end_pos": [680, 23]}, {"full_name": "EMetric.mem_closure_iff", "def_path": "Mathlib/Topology/EMetricSpace/Basic.lean", "def_pos": [720, 9], "def_end_pos": [720, 24]}, {"full_name": "ENNReal.half_pos", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1796, 19], "def_end_pos": [1796, 27]}, {"full_name": "ne_of_gt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [104, 9], "def_end_pos": [104, 17]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f\nh'f : StronglyMeasurable f\nA : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc\nt : Set E\nt_count : Set.Countable t\nht : range f \u2286 closure t\nmain :\n  \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    \u2200 (n : \u2115) (c : E),\n      c \u2208 t \u2192\n        Tendsto\n          (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)\n          (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a)\nx : \u03b1\nhx :\n  \u2200 (n : \u2115) (c : E),\n    c \u2208 t \u2192\n      Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)\n        (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a)\nh'x : \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a\nM : \u2200 (c : E), c \u2208 t \u2192 Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a)\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\n\u22a2 \u2200\u1da0 (x_1 : Set \u03b1) in filterAt v x, (\u222b\u207b (y : \u03b1) in x_1, \u2191\u2016f y - f x\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc x_1 \u2264 \u03b5", "state_after": "case intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f\nh'f : StronglyMeasurable f\nA : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc\nt : Set E\nt_count : Set.Countable t\nht : range f \u2286 closure t\nmain :\n  \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    \u2200 (n : \u2115) (c : E),\n      c \u2208 t \u2192\n        Tendsto\n          (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)\n          (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a)\nx : \u03b1\nhx :\n  \u2200 (n : \u2115) (c : E),\n    c \u2208 t \u2192\n      Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)\n        (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a)\nh'x : \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a\nM : \u2200 (c : E), c \u2208 t \u2192 Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a)\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\nc : E\nct : c \u2208 t\nxc : \u2191\u2016f x - c\u2016\u208a < \u03b5 / 2\n\u22a2 \u2200\u1da0 (x_1 : Set \u03b1) in filterAt v x, (\u222b\u207b (y : \u03b1) in x_1, \u2191\u2016f y - f x\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc x_1 \u2264 \u03b5"}, {"tactic": "filter_upwards [(tendsto_order.1 (M c ct)).2 (\u03b5 / 2) xc, h'x, v.eventually_measure_lt_top x] with\n  a ha h'a h''a", "annotated_tactic": ["filter_upwards [(<a>tendsto_order</a>.1 (M c ct)).2 (\u03b5 / 2) xc, h'x, v.eventually_measure_lt_top x] with\n    a ha h'a h''a", [{"full_name": "tendsto_order", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [919, 9], "def_end_pos": [919, 22]}]], "state_before": "case intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f\nh'f : StronglyMeasurable f\nA : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc\nt : Set E\nt_count : Set.Countable t\nht : range f \u2286 closure t\nmain :\n  \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    \u2200 (n : \u2115) (c : E),\n      c \u2208 t \u2192\n        Tendsto\n          (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)\n          (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a)\nx : \u03b1\nhx :\n  \u2200 (n : \u2115) (c : E),\n    c \u2208 t \u2192\n      Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)\n        (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a)\nh'x : \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a\nM : \u2200 (c : E), c \u2208 t \u2192 Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a)\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\nc : E\nct : c \u2208 t\nxc : \u2191\u2016f x - c\u2016\u208a < \u03b5 / 2\n\u22a2 \u2200\u1da0 (x_1 : Set \u03b1) in filterAt v x, (\u222b\u207b (y : \u03b1) in x_1, \u2191\u2016f y - f x\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc x_1 \u2264 \u03b5", "state_after": "case h\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f\nh'f : StronglyMeasurable f\nA : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc\nt : Set E\nt_count : Set.Countable t\nht : range f \u2286 closure t\nmain :\n  \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    \u2200 (n : \u2115) (c : E),\n      c \u2208 t \u2192\n        Tendsto\n          (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)\n          (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a)\nx : \u03b1\nhx :\n  \u2200 (n : \u2115) (c : E),\n    c \u2208 t \u2192\n      Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)\n        (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a)\nh'x : \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a\nM : \u2200 (c : E), c \u2208 t \u2192 Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a)\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\nc : E\nct : c \u2208 t\nxc : \u2191\u2016f x - c\u2016\u208a < \u03b5 / 2\na : Set \u03b1\nha : (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a < \u03b5 / 2\nh'a : 0 < \u2191\u2191\u03bc a\nh''a : \u2191\u2191\u03bc a < \u22a4\n\u22a2 (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - f x\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a \u2264 \u03b5"}, {"tactic": "apply ENNReal.div_le_of_le_mul", "annotated_tactic": ["apply <a>ENNReal.div_le_of_le_mul</a>", [{"full_name": "ENNReal.div_le_of_le_mul", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1635, 9], "def_end_pos": [1635, 25]}]], "state_before": "case h\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f\nh'f : StronglyMeasurable f\nA : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc\nt : Set E\nt_count : Set.Countable t\nht : range f \u2286 closure t\nmain :\n  \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    \u2200 (n : \u2115) (c : E),\n      c \u2208 t \u2192\n        Tendsto\n          (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)\n          (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a)\nx : \u03b1\nhx :\n  \u2200 (n : \u2115) (c : E),\n    c \u2208 t \u2192\n      Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)\n        (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a)\nh'x : \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a\nM : \u2200 (c : E), c \u2208 t \u2192 Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a)\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\nc : E\nct : c \u2208 t\nxc : \u2191\u2016f x - c\u2016\u208a < \u03b5 / 2\na : Set \u03b1\nha : (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a < \u03b5 / 2\nh'a : 0 < \u2191\u2191\u03bc a\nh''a : \u2191\u2191\u03bc a < \u22a4\n\u22a2 (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - f x\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a \u2264 \u03b5", "state_after": "case h.h\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f\nh'f : StronglyMeasurable f\nA : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc\nt : Set E\nt_count : Set.Countable t\nht : range f \u2286 closure t\nmain :\n  \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    \u2200 (n : \u2115) (c : E),\n      c \u2208 t \u2192\n        Tendsto\n          (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)\n          (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a)\nx : \u03b1\nhx :\n  \u2200 (n : \u2115) (c : E),\n    c \u2208 t \u2192\n      Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)\n        (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a)\nh'x : \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a\nM : \u2200 (c : E), c \u2208 t \u2192 Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a)\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\nc : E\nct : c \u2208 t\nxc : \u2191\u2016f x - c\u2016\u208a < \u03b5 / 2\na : Set \u03b1\nha : (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a < \u03b5 / 2\nh'a : 0 < \u2191\u2191\u03bc a\nh''a : \u2191\u2191\u03bc a < \u22a4\n\u22a2 \u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - f x\u2016\u208a \u2202\u03bc \u2264 \u03b5 * \u2191\u2191\u03bc a"}, {"tactic": "simp_rw [ae_all_iff, ae_ball_iff t_count]", "annotated_tactic": ["simp_rw [<a>ae_all_iff</a>, <a>ae_ball_iff</a> t_count]", [{"full_name": "MeasureTheory.ae_all_iff", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [422, 9], "def_end_pos": [422, 19]}, {"full_name": "MeasureTheory.ae_ball_iff", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [431, 9], "def_end_pos": [431, 20]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f\nh'f : StronglyMeasurable f\nA : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc\nt : Set E\nt_count : Set.Countable t\nht : range f \u2286 closure t\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    \u2200 (n : \u2115) (c : E),\n      c \u2208 t \u2192\n        Tendsto\n          (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)\n          (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f\nh'f : StronglyMeasurable f\nA : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc\nt : Set E\nt_count : Set.Countable t\nht : range f \u2286 closure t\n\u22a2 \u2200 (i : \u2115) (i_1 : E),\n    i_1 \u2208 t \u2192\n      \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n        Tendsto\n          (fun a =>\n            (\u222b\u207b (y : \u03b1) in a,\n                \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set (finiteSpanningSetsInOpen' \u03bc) i) (fun x => i_1) y\u2016\u208a \u2202\u03bc) /\n              \u2191\u2191\u03bc a)\n          (filterAt v x)\n          (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set (finiteSpanningSetsInOpen' \u03bc) i) (fun x => i_1) x\u2016\u208a)"}, {"tactic": "intro n c _", "annotated_tactic": ["intro n c _", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f\nh'f : StronglyMeasurable f\nA : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc\nt : Set E\nt_count : Set.Countable t\nht : range f \u2286 closure t\n\u22a2 \u2200 (i : \u2115) (i_1 : E),\n    i_1 \u2208 t \u2192\n      \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n        Tendsto\n          (fun a =>\n            (\u222b\u207b (y : \u03b1) in a,\n                \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set (finiteSpanningSetsInOpen' \u03bc) i) (fun x => i_1) y\u2016\u208a \u2202\u03bc) /\n              \u2191\u2191\u03bc a)\n          (filterAt v x)\n          (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set (finiteSpanningSetsInOpen' \u03bc) i) (fun x => i_1) x\u2016\u208a)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f\nh'f : StronglyMeasurable f\nA : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc\nt : Set E\nt_count : Set.Countable t\nht : range f \u2286 closure t\nn : \u2115\nc : E\nhi\u271d : c \u2208 t\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    Tendsto\n      (fun a =>\n        (\u222b\u207b (y : \u03b1) in a,\n            \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set (finiteSpanningSetsInOpen' \u03bc) n) (fun x => c) y\u2016\u208a \u2202\u03bc) /\n          \u2191\u2191\u03bc a)\n      (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set (finiteSpanningSetsInOpen' \u03bc) n) (fun x => c) x\u2016\u208a)"}, {"tactic": "apply ae_tendsto_lintegral_div'", "annotated_tactic": ["apply <a>ae_tendsto_lintegral_div'</a>", [{"full_name": "VitaliFamily.ae_tendsto_lintegral_div'", "def_path": "Mathlib/MeasureTheory/Covering/Differentiation.lean", "def_pos": [775, 9], "def_end_pos": [775, 34]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f\nh'f : StronglyMeasurable f\nA : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc\nt : Set E\nt_count : Set.Countable t\nht : range f \u2286 closure t\nn : \u2115\nc : E\nhi\u271d : c \u2208 t\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    Tendsto\n      (fun a =>\n        (\u222b\u207b (y : \u03b1) in a,\n            \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set (finiteSpanningSetsInOpen' \u03bc) n) (fun x => c) y\u2016\u208a \u2202\u03bc) /\n          \u2191\u2191\u03bc a)\n      (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set (finiteSpanningSetsInOpen' \u03bc) n) (fun x => c) x\u2016\u208a)", "state_after": "case hf\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f\nh'f : StronglyMeasurable f\nA : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc\nt : Set E\nt_count : Set.Countable t\nht : range f \u2286 closure t\nn : \u2115\nc : E\nhi\u271d : c \u2208 t\n\u22a2 Measurable fun y => \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set (finiteSpanningSetsInOpen' \u03bc) n) (fun x => c) y\u2016\u208a\n\ncase h'f\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f\nh'f : StronglyMeasurable f\nA : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc\nt : Set E\nt_count : Set.Countable t\nht : range f \u2286 closure t\nn : \u2115\nc : E\nhi\u271d : c \u2208 t\n\u22a2 \u222b\u207b (y : \u03b1), \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set (finiteSpanningSetsInOpen' \u03bc) n) (fun x => c) y\u2016\u208a \u2202\u03bc \u2260 \u22a4"}, {"tactic": "refine' (h'f.sub _).ennnorm", "annotated_tactic": ["refine' (h'f.sub _).<a>ennnorm</a>", [{"full_name": "MeasureTheory.StronglyMeasurable.ennnorm", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [851, 19], "def_end_pos": [851, 26]}]], "state_before": "case hf\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f\nh'f : StronglyMeasurable f\nA : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc\nt : Set E\nt_count : Set.Countable t\nht : range f \u2286 closure t\nn : \u2115\nc : E\nhi\u271d : c \u2208 t\n\u22a2 Measurable fun y => \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set (finiteSpanningSetsInOpen' \u03bc) n) (fun x => c) y\u2016\u208a", "state_after": "case hf\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f\nh'f : StronglyMeasurable f\nA : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc\nt : Set E\nt_count : Set.Countable t\nht : range f \u2286 closure t\nn : \u2115\nc : E\nhi\u271d : c \u2208 t\n\u22a2 StronglyMeasurable fun y => indicator (FiniteSpanningSetsIn.set (finiteSpanningSetsInOpen' \u03bc) n) (fun x => c) y"}, {"tactic": "exact stronglyMeasurable_const.indicator (IsOpen.measurableSet (A.set_mem n))", "annotated_tactic": ["exact stronglyMeasurable_const.indicator (<a>IsOpen.measurableSet</a> (A.set_mem n))", [{"full_name": "IsOpen.measurableSet", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [316, 9], "def_end_pos": [316, 29]}]], "state_before": "case hf\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f\nh'f : StronglyMeasurable f\nA : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc\nt : Set E\nt_count : Set.Countable t\nht : range f \u2286 closure t\nn : \u2115\nc : E\nhi\u271d : c \u2208 t\n\u22a2 StronglyMeasurable fun y => indicator (FiniteSpanningSetsIn.set (finiteSpanningSetsInOpen' \u03bc) n) (fun x => c) y", "state_after": "no goals"}, {"tactic": "apply ne_of_lt", "annotated_tactic": ["apply <a>ne_of_lt</a>", [{"full_name": "ne_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [101, 9], "def_end_pos": [101, 17]}]], "state_before": "case h'f\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f\nh'f : StronglyMeasurable f\nA : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc\nt : Set E\nt_count : Set.Countable t\nht : range f \u2286 closure t\nn : \u2115\nc : E\nhi\u271d : c \u2208 t\n\u22a2 \u222b\u207b (y : \u03b1), \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set (finiteSpanningSetsInOpen' \u03bc) n) (fun x => c) y\u2016\u208a \u2202\u03bc \u2260 \u22a4", "state_after": "case h'f.h\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f\nh'f : StronglyMeasurable f\nA : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc\nt : Set E\nt_count : Set.Countable t\nht : range f \u2286 closure t\nn : \u2115\nc : E\nhi\u271d : c \u2208 t\n\u22a2 \u222b\u207b (y : \u03b1), \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set (finiteSpanningSetsInOpen' \u03bc) n) (fun x => c) y\u2016\u208a \u2202\u03bc < \u22a4"}, {"tactic": "calc\n  (\u222b\u207b y, \u2191\u2016f y - (A.set n).indicator (fun _ : \u03b1 => c) y\u2016\u208a \u2202\u03bc) \u2264\n      \u222b\u207b y, \u2016f y\u2016\u208a + \u2016(A.set n).indicator (fun _ : \u03b1 => c) y\u2016\u208a \u2202\u03bc := by\n    apply lintegral_mono\n    intro x\n    dsimp\n    rw [\u2190 ENNReal.coe_add]\n    exact ENNReal.coe_le_coe.2 (nnnorm_sub_le _ _)\n  _ = (\u222b\u207b y, \u2016f y\u2016\u208a \u2202\u03bc) + \u222b\u207b y, \u2016(A.set n).indicator (fun _ : \u03b1 => c) y\u2016\u208a \u2202\u03bc :=\n    (lintegral_add_left h'f.ennnorm _)\n  _ < \u221e + \u221e :=\n    haveI I : Integrable ((A.set n).indicator fun _ : \u03b1 => c) \u03bc := by\n      simp only [integrable_indicator_iff (IsOpen.measurableSet (A.set_mem n)),\n        integrableOn_const, A.finite n, or_true_iff]\n    ENNReal.add_lt_add hf.2 I.2", "annotated_tactic": ["calc\n        (\u222b\u207b y, \u2191\u2016f y - (A.set n).<a>indicator</a> (fun _ : \u03b1 => c) y\u2016\u208a \u2202\u03bc) \u2264\n            \u222b\u207b y, \u2016f y\u2016\u208a + \u2016(A.set n).<a>indicator</a> (fun _ : \u03b1 => c) y\u2016\u208a \u2202\u03bc := by\n          apply <a>lintegral_mono</a>\n          intro x\n          dsimp\n          rw [\u2190 <a>ENNReal.coe_add</a>]\n          exact <a>ENNReal.coe_le_coe</a>.2 (<a>nnnorm_sub_le</a> _ _)\n        _ = (\u222b\u207b y, \u2016f y\u2016\u208a \u2202\u03bc) + \u222b\u207b y, \u2016(A.set n).<a>indicator</a> (fun _ : \u03b1 => c) y\u2016\u208a \u2202\u03bc :=\n          (<a>lintegral_add_left</a> h'f.ennnorm _)\n        _ < \u221e + \u221e :=\n          haveI I : <a>Integrable</a> ((A.set n).<a>indicator</a> fun _ : \u03b1 => c) \u03bc := by\n            simp only [<a>integrable_indicator_iff</a> (<a>IsOpen.measurableSet</a> (A.set_mem n)),\n              <a>integrableOn_const</a>, A.finite n, <a>or_true_iff</a>]\n          <a>ENNReal.add_lt_add</a> hf.2 I.2", [{"full_name": "Set.indicator", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [46, 3], "def_end_pos": [46, 14]}, {"full_name": "Set.indicator", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [46, 3], "def_end_pos": [46, 14]}, {"full_name": "MeasureTheory.lintegral_mono", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [99, 9], "def_end_pos": [99, 23]}, {"full_name": "ENNReal.coe_add", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [386, 28], "def_end_pos": [386, 35]}, {"full_name": "ENNReal.coe_le_coe", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [349, 28], "def_end_pos": [349, 38]}, {"full_name": "nnnorm_sub_le", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [963, 3], "def_end_pos": [963, 14]}, {"full_name": "Set.indicator", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [46, 3], "def_end_pos": [46, 14]}, {"full_name": "MeasureTheory.lintegral_add_left", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [554, 9], "def_end_pos": [554, 27]}, {"full_name": "MeasureTheory.Integrable", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [442, 5], "def_end_pos": [442, 15]}, {"full_name": "Set.indicator", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [46, 3], "def_end_pos": [46, 14]}, {"full_name": "MeasureTheory.integrable_indicator_iff", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [257, 9], "def_end_pos": [257, 33]}, {"full_name": "IsOpen.measurableSet", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [316, 9], "def_end_pos": [316, 29]}, {"full_name": "MeasureTheory.integrableOn_const", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [119, 9], "def_end_pos": [119, 27]}, {"full_name": "or_true_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [184, 9], "def_end_pos": [184, 20]}, {"full_name": "ENNReal.add_lt_add", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [931, 19], "def_end_pos": [931, 29]}]], "state_before": "case h'f.h\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f\nh'f : StronglyMeasurable f\nA : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc\nt : Set E\nt_count : Set.Countable t\nht : range f \u2286 closure t\nn : \u2115\nc : E\nhi\u271d : c \u2208 t\n\u22a2 \u222b\u207b (y : \u03b1), \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set (finiteSpanningSetsInOpen' \u03bc) n) (fun x => c) y\u2016\u208a \u2202\u03bc < \u22a4", "state_after": "no goals"}, {"tactic": "apply lintegral_mono", "annotated_tactic": ["apply <a>lintegral_mono</a>", [{"full_name": "MeasureTheory.lintegral_mono", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [99, 9], "def_end_pos": [99, 23]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f\nh'f : StronglyMeasurable f\nA : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc\nt : Set E\nt_count : Set.Countable t\nht : range f \u2286 closure t\nn : \u2115\nc : E\nhi\u271d : c \u2208 t\n\u22a2 \u222b\u207b (y : \u03b1), \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc \u2264\n    \u222b\u207b (y : \u03b1), \u2191\u2016f y\u2016\u208a + \u2191\u2016indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc", "state_after": "case hfg\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f\nh'f : StronglyMeasurable f\nA : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc\nt : Set E\nt_count : Set.Countable t\nht : range f \u2286 closure t\nn : \u2115\nc : E\nhi\u271d : c \u2208 t\n\u22a2 (fun a => \u2191\u2016f a - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) a\u2016\u208a) \u2264 fun a =>\n    \u2191\u2016f a\u2016\u208a + \u2191\u2016indicator (FiniteSpanningSetsIn.set A n) (fun x => c) a\u2016\u208a"}, {"tactic": "intro x", "annotated_tactic": ["intro x", []], "state_before": "case hfg\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f\nh'f : StronglyMeasurable f\nA : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc\nt : Set E\nt_count : Set.Countable t\nht : range f \u2286 closure t\nn : \u2115\nc : E\nhi\u271d : c \u2208 t\n\u22a2 (fun a => \u2191\u2016f a - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) a\u2016\u208a) \u2264 fun a =>\n    \u2191\u2016f a\u2016\u208a + \u2191\u2016indicator (FiniteSpanningSetsIn.set A n) (fun x => c) a\u2016\u208a", "state_after": "case hfg\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f\nh'f : StronglyMeasurable f\nA : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc\nt : Set E\nt_count : Set.Countable t\nht : range f \u2286 closure t\nn : \u2115\nc : E\nhi\u271d : c \u2208 t\nx : \u03b1\n\u22a2 (fun a => \u2191\u2016f a - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) a\u2016\u208a) x \u2264\n    (fun a => \u2191\u2016f a\u2016\u208a + \u2191\u2016indicator (FiniteSpanningSetsIn.set A n) (fun x => c) a\u2016\u208a) x"}, {"tactic": "dsimp", "annotated_tactic": ["dsimp", []], "state_before": "case hfg\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f\nh'f : StronglyMeasurable f\nA : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc\nt : Set E\nt_count : Set.Countable t\nht : range f \u2286 closure t\nn : \u2115\nc : E\nhi\u271d : c \u2208 t\nx : \u03b1\n\u22a2 (fun a => \u2191\u2016f a - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) a\u2016\u208a) x \u2264\n    (fun a => \u2191\u2016f a\u2016\u208a + \u2191\u2016indicator (FiniteSpanningSetsIn.set A n) (fun x => c) a\u2016\u208a) x", "state_after": "case hfg\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f\nh'f : StronglyMeasurable f\nA : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc\nt : Set E\nt_count : Set.Countable t\nht : range f \u2286 closure t\nn : \u2115\nc : E\nhi\u271d : c \u2208 t\nx : \u03b1\n\u22a2 \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set (finiteSpanningSetsInOpen' \u03bc) n) (fun x => c) x\u2016\u208a \u2264\n    \u2191\u2016f x\u2016\u208a + \u2191\u2016indicator (FiniteSpanningSetsIn.set (finiteSpanningSetsInOpen' \u03bc) n) (fun x => c) x\u2016\u208a"}, {"tactic": "rw [\u2190 ENNReal.coe_add]", "annotated_tactic": ["rw [\u2190 <a>ENNReal.coe_add</a>]", [{"full_name": "ENNReal.coe_add", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [386, 28], "def_end_pos": [386, 35]}]], "state_before": "case hfg\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f\nh'f : StronglyMeasurable f\nA : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc\nt : Set E\nt_count : Set.Countable t\nht : range f \u2286 closure t\nn : \u2115\nc : E\nhi\u271d : c \u2208 t\nx : \u03b1\n\u22a2 \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set (finiteSpanningSetsInOpen' \u03bc) n) (fun x => c) x\u2016\u208a \u2264\n    \u2191\u2016f x\u2016\u208a + \u2191\u2016indicator (FiniteSpanningSetsIn.set (finiteSpanningSetsInOpen' \u03bc) n) (fun x => c) x\u2016\u208a", "state_after": "case hfg\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f\nh'f : StronglyMeasurable f\nA : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc\nt : Set E\nt_count : Set.Countable t\nht : range f \u2286 closure t\nn : \u2115\nc : E\nhi\u271d : c \u2208 t\nx : \u03b1\n\u22a2 \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set (finiteSpanningSetsInOpen' \u03bc) n) (fun x => c) x\u2016\u208a \u2264\n    \u2191(\u2016f x\u2016\u208a + \u2016indicator (FiniteSpanningSetsIn.set (finiteSpanningSetsInOpen' \u03bc) n) (fun x => c) x\u2016\u208a)"}, {"tactic": "exact ENNReal.coe_le_coe.2 (nnnorm_sub_le _ _)", "annotated_tactic": ["exact <a>ENNReal.coe_le_coe</a>.2 (<a>nnnorm_sub_le</a> _ _)", [{"full_name": "ENNReal.coe_le_coe", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [349, 28], "def_end_pos": [349, 38]}, {"full_name": "nnnorm_sub_le", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [963, 3], "def_end_pos": [963, 14]}]], "state_before": "case hfg\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f\nh'f : StronglyMeasurable f\nA : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc\nt : Set E\nt_count : Set.Countable t\nht : range f \u2286 closure t\nn : \u2115\nc : E\nhi\u271d : c \u2208 t\nx : \u03b1\n\u22a2 \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set (finiteSpanningSetsInOpen' \u03bc) n) (fun x => c) x\u2016\u208a \u2264\n    \u2191(\u2016f x\u2016\u208a + \u2016indicator (FiniteSpanningSetsIn.set (finiteSpanningSetsInOpen' \u03bc) n) (fun x => c) x\u2016\u208a)", "state_after": "no goals"}, {"tactic": "simp only [integrable_indicator_iff (IsOpen.measurableSet (A.set_mem n)),\n  integrableOn_const, A.finite n, or_true_iff]", "annotated_tactic": ["simp only [<a>integrable_indicator_iff</a> (<a>IsOpen.measurableSet</a> (A.set_mem n)),\n              <a>integrableOn_const</a>, A.finite n, <a>or_true_iff</a>]", [{"full_name": "MeasureTheory.integrable_indicator_iff", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [257, 9], "def_end_pos": [257, 33]}, {"full_name": "IsOpen.measurableSet", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [316, 9], "def_end_pos": [316, 29]}, {"full_name": "MeasureTheory.integrableOn_const", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [119, 9], "def_end_pos": [119, 27]}, {"full_name": "or_true_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [184, 9], "def_end_pos": [184, 20]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f\nh'f : StronglyMeasurable f\nA : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc\nt : Set E\nt_count : Set.Countable t\nht : range f \u2286 closure t\nn : \u2115\nc : E\nhi\u271d : c \u2208 t\n\u22a2 Integrable (indicator (FiniteSpanningSetsIn.set A n) fun x => c)", "state_after": "no goals"}, {"tactic": "intro c hc", "annotated_tactic": ["intro c hc", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f\nh'f : StronglyMeasurable f\nA : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc\nt : Set E\nt_count : Set.Countable t\nht : range f \u2286 closure t\nmain :\n  \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    \u2200 (n : \u2115) (c : E),\n      c \u2208 t \u2192\n        Tendsto\n          (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)\n          (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a)\nx : \u03b1\nhx :\n  \u2200 (n : \u2115) (c : E),\n    c \u2208 t \u2192\n      Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)\n        (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a)\nh'x : \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a\n\u22a2 \u2200 (c : E), c \u2208 t \u2192 Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f\nh'f : StronglyMeasurable f\nA : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc\nt : Set E\nt_count : Set.Countable t\nht : range f \u2286 closure t\nmain :\n  \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    \u2200 (n : \u2115) (c : E),\n      c \u2208 t \u2192\n        Tendsto\n          (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)\n          (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a)\nx : \u03b1\nhx :\n  \u2200 (n : \u2115) (c : E),\n    c \u2208 t \u2192\n      Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)\n        (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a)\nh'x : \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a\nc : E\nhc : c \u2208 t\n\u22a2 Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a)"}, {"tactic": "obtain \u27e8n, xn\u27e9 : \u2203 n, x \u2208 A.set n := by simpa [\u2190 A.spanning] using mem_univ x", "annotated_tactic": ["obtain \u27e8n, xn\u27e9 : \u2203 n, x \u2208 A.set n := by simpa [\u2190 A.spanning] using <a>mem_univ</a> x", [{"full_name": "Set.mem_univ", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [676, 9], "def_end_pos": [676, 17]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f\nh'f : StronglyMeasurable f\nA : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc\nt : Set E\nt_count : Set.Countable t\nht : range f \u2286 closure t\nmain :\n  \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    \u2200 (n : \u2115) (c : E),\n      c \u2208 t \u2192\n        Tendsto\n          (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)\n          (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a)\nx : \u03b1\nhx :\n  \u2200 (n : \u2115) (c : E),\n    c \u2208 t \u2192\n      Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)\n        (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a)\nh'x : \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a\nc : E\nhc : c \u2208 t\n\u22a2 Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a)", "state_after": "case intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f\nh'f : StronglyMeasurable f\nA : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc\nt : Set E\nt_count : Set.Countable t\nht : range f \u2286 closure t\nmain :\n  \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    \u2200 (n : \u2115) (c : E),\n      c \u2208 t \u2192\n        Tendsto\n          (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)\n          (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a)\nx : \u03b1\nhx :\n  \u2200 (n : \u2115) (c : E),\n    c \u2208 t \u2192\n      Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)\n        (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a)\nh'x : \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a\nc : E\nhc : c \u2208 t\nn : \u2115\nxn : x \u2208 FiniteSpanningSetsIn.set A n\n\u22a2 Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a)"}, {"tactic": "specialize hx n c hc", "annotated_tactic": ["specialize hx n c hc", []], "state_before": "case intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f\nh'f : StronglyMeasurable f\nA : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc\nt : Set E\nt_count : Set.Countable t\nht : range f \u2286 closure t\nmain :\n  \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    \u2200 (n : \u2115) (c : E),\n      c \u2208 t \u2192\n        Tendsto\n          (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)\n          (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a)\nx : \u03b1\nhx :\n  \u2200 (n : \u2115) (c : E),\n    c \u2208 t \u2192\n      Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)\n        (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a)\nh'x : \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a\nc : E\nhc : c \u2208 t\nn : \u2115\nxn : x \u2208 FiniteSpanningSetsIn.set A n\n\u22a2 Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a)", "state_after": "case intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f\nh'f : StronglyMeasurable f\nA : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc\nt : Set E\nt_count : Set.Countable t\nht : range f \u2286 closure t\nmain :\n  \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    \u2200 (n : \u2115) (c : E),\n      c \u2208 t \u2192\n        Tendsto\n          (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)\n          (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a)\nx : \u03b1\nh'x : \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a\nc : E\nhc : c \u2208 t\nn : \u2115\nxn : x \u2208 FiniteSpanningSetsIn.set A n\nhx :\n  Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)\n    (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a)\n\u22a2 Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a)"}, {"tactic": "simp only [xn, indicator_of_mem] at hx", "annotated_tactic": ["simp only [xn, <a>indicator_of_mem</a>] at hx", [{"full_name": "Set.indicator_of_mem", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [67, 3], "def_end_pos": [67, 14]}]], "state_before": "case intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f\nh'f : StronglyMeasurable f\nA : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc\nt : Set E\nt_count : Set.Countable t\nht : range f \u2286 closure t\nmain :\n  \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    \u2200 (n : \u2115) (c : E),\n      c \u2208 t \u2192\n        Tendsto\n          (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)\n          (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a)\nx : \u03b1\nh'x : \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a\nc : E\nhc : c \u2208 t\nn : \u2115\nxn : x \u2208 FiniteSpanningSetsIn.set A n\nhx :\n  Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)\n    (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a)\n\u22a2 Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a)", "state_after": "case intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f\nh'f : StronglyMeasurable f\nA : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc\nt : Set E\nt_count : Set.Countable t\nht : range f \u2286 closure t\nmain :\n  \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    \u2200 (n : \u2115) (c : E),\n      c \u2208 t \u2192\n        Tendsto\n          (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)\n          (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a)\nx : \u03b1\nh'x : \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a\nc : E\nhc : c \u2208 t\nn : \u2115\nxn : x \u2208 FiniteSpanningSetsIn.set A n\nhx :\n  Tendsto\n    (fun a =>\n      (\u222b\u207b (y : \u03b1) in a,\n          \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set (finiteSpanningSetsInOpen' \u03bc) n) (fun x => c) y\u2016\u208a \u2202\u03bc) /\n        \u2191\u2191\u03bc a)\n    (filterAt v x) (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a)\n\u22a2 Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a)"}, {"tactic": "apply hx.congr' _", "annotated_tactic": ["apply hx.congr' _", []], "state_before": "case intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f\nh'f : StronglyMeasurable f\nA : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc\nt : Set E\nt_count : Set.Countable t\nht : range f \u2286 closure t\nmain :\n  \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    \u2200 (n : \u2115) (c : E),\n      c \u2208 t \u2192\n        Tendsto\n          (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)\n          (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a)\nx : \u03b1\nh'x : \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a\nc : E\nhc : c \u2208 t\nn : \u2115\nxn : x \u2208 FiniteSpanningSetsIn.set A n\nhx :\n  Tendsto\n    (fun a =>\n      (\u222b\u207b (y : \u03b1) in a,\n          \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set (finiteSpanningSetsInOpen' \u03bc) n) (fun x => c) y\u2016\u208a \u2202\u03bc) /\n        \u2191\u2191\u03bc a)\n    (filterAt v x) (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a)\n\u22a2 Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f\nh'f : StronglyMeasurable f\nA : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc\nt : Set E\nt_count : Set.Countable t\nht : range f \u2286 closure t\nmain :\n  \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    \u2200 (n : \u2115) (c : E),\n      c \u2208 t \u2192\n        Tendsto\n          (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)\n          (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a)\nx : \u03b1\nh'x : \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a\nc : E\nhc : c \u2208 t\nn : \u2115\nxn : x \u2208 FiniteSpanningSetsIn.set A n\nhx :\n  Tendsto\n    (fun a =>\n      (\u222b\u207b (y : \u03b1) in a,\n          \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set (finiteSpanningSetsInOpen' \u03bc) n) (fun x => c) y\u2016\u208a \u2202\u03bc) /\n        \u2191\u2191\u03bc a)\n    (filterAt v x) (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a)\n\u22a2 (fun a =>\n      (\u222b\u207b (y : \u03b1) in a,\n          \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set (finiteSpanningSetsInOpen' \u03bc) n) (fun x => c) y\u2016\u208a \u2202\u03bc) /\n        \u2191\u2191\u03bc a) =\u1da0[filterAt v x]\n    fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a"}, {"tactic": "filter_upwards [v.eventually_filterAt_subset_of_nhds (IsOpen.mem_nhds (A.set_mem n) xn),\n  v.eventually_filterAt_measurableSet x] with a ha h'a", "annotated_tactic": ["filter_upwards [v.eventually_filterAt_subset_of_nhds (<a>IsOpen.mem_nhds</a> (A.set_mem n) xn),\n      v.eventually_filterAt_measurableSet x] with a ha h'a", [{"full_name": "IsOpen.mem_nhds", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [928, 9], "def_end_pos": [928, 24]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f\nh'f : StronglyMeasurable f\nA : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc\nt : Set E\nt_count : Set.Countable t\nht : range f \u2286 closure t\nmain :\n  \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    \u2200 (n : \u2115) (c : E),\n      c \u2208 t \u2192\n        Tendsto\n          (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)\n          (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a)\nx : \u03b1\nh'x : \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a\nc : E\nhc : c \u2208 t\nn : \u2115\nxn : x \u2208 FiniteSpanningSetsIn.set A n\nhx :\n  Tendsto\n    (fun a =>\n      (\u222b\u207b (y : \u03b1) in a,\n          \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set (finiteSpanningSetsInOpen' \u03bc) n) (fun x => c) y\u2016\u208a \u2202\u03bc) /\n        \u2191\u2191\u03bc a)\n    (filterAt v x) (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a)\n\u22a2 (fun a =>\n      (\u222b\u207b (y : \u03b1) in a,\n          \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set (finiteSpanningSetsInOpen' \u03bc) n) (fun x => c) y\u2016\u208a \u2202\u03bc) /\n        \u2191\u2191\u03bc a) =\u1da0[filterAt v x]\n    fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a", "state_after": "case h\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f\nh'f : StronglyMeasurable f\nA : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc\nt : Set E\nt_count : Set.Countable t\nht : range f \u2286 closure t\nmain :\n  \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    \u2200 (n : \u2115) (c : E),\n      c \u2208 t \u2192\n        Tendsto\n          (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)\n          (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a)\nx : \u03b1\nh'x : \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a\nc : E\nhc : c \u2208 t\nn : \u2115\nxn : x \u2208 FiniteSpanningSetsIn.set A n\nhx :\n  Tendsto\n    (fun a =>\n      (\u222b\u207b (y : \u03b1) in a,\n          \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set (finiteSpanningSetsInOpen' \u03bc) n) (fun x => c) y\u2016\u208a \u2202\u03bc) /\n        \u2191\u2191\u03bc a)\n    (filterAt v x) (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a)\na : Set \u03b1\nha : a \u2286 FiniteSpanningSetsIn.set A n\nh'a : MeasurableSet a\n\u22a2 (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set (finiteSpanningSetsInOpen' \u03bc) n) (fun x => c) y\u2016\u208a \u2202\u03bc) /\n      \u2191\u2191\u03bc a =\n    (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a"}, {"tactic": "congr 1", "annotated_tactic": ["congr 1", []], "state_before": "case h\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f\nh'f : StronglyMeasurable f\nA : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc\nt : Set E\nt_count : Set.Countable t\nht : range f \u2286 closure t\nmain :\n  \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    \u2200 (n : \u2115) (c : E),\n      c \u2208 t \u2192\n        Tendsto\n          (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)\n          (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a)\nx : \u03b1\nh'x : \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a\nc : E\nhc : c \u2208 t\nn : \u2115\nxn : x \u2208 FiniteSpanningSetsIn.set A n\nhx :\n  Tendsto\n    (fun a =>\n      (\u222b\u207b (y : \u03b1) in a,\n          \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set (finiteSpanningSetsInOpen' \u03bc) n) (fun x => c) y\u2016\u208a \u2202\u03bc) /\n        \u2191\u2191\u03bc a)\n    (filterAt v x) (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a)\na : Set \u03b1\nha : a \u2286 FiniteSpanningSetsIn.set A n\nh'a : MeasurableSet a\n\u22a2 (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set (finiteSpanningSetsInOpen' \u03bc) n) (fun x => c) y\u2016\u208a \u2202\u03bc) /\n      \u2191\u2191\u03bc a =\n    (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a", "state_after": "case h.e_a\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f\nh'f : StronglyMeasurable f\nA : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc\nt : Set E\nt_count : Set.Countable t\nht : range f \u2286 closure t\nmain :\n  \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    \u2200 (n : \u2115) (c : E),\n      c \u2208 t \u2192\n        Tendsto\n          (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)\n          (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a)\nx : \u03b1\nh'x : \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a\nc : E\nhc : c \u2208 t\nn : \u2115\nxn : x \u2208 FiniteSpanningSetsIn.set A n\nhx :\n  Tendsto\n    (fun a =>\n      (\u222b\u207b (y : \u03b1) in a,\n          \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set (finiteSpanningSetsInOpen' \u03bc) n) (fun x => c) y\u2016\u208a \u2202\u03bc) /\n        \u2191\u2191\u03bc a)\n    (filterAt v x) (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a)\na : Set \u03b1\nha : a \u2286 FiniteSpanningSetsIn.set A n\nh'a : MeasurableSet a\n\u22a2 \u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set (finiteSpanningSetsInOpen' \u03bc) n) (fun x => c) y\u2016\u208a \u2202\u03bc =\n    \u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc"}, {"tactic": "apply set_lintegral_congr_fun h'a", "annotated_tactic": ["apply <a>set_lintegral_congr_fun</a> h'a", [{"full_name": "MeasureTheory.set_lintegral_congr_fun", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [316, 9], "def_end_pos": [316, 32]}]], "state_before": "case h.e_a\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f\nh'f : StronglyMeasurable f\nA : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc\nt : Set E\nt_count : Set.Countable t\nht : range f \u2286 closure t\nmain :\n  \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    \u2200 (n : \u2115) (c : E),\n      c \u2208 t \u2192\n        Tendsto\n          (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)\n          (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a)\nx : \u03b1\nh'x : \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a\nc : E\nhc : c \u2208 t\nn : \u2115\nxn : x \u2208 FiniteSpanningSetsIn.set A n\nhx :\n  Tendsto\n    (fun a =>\n      (\u222b\u207b (y : \u03b1) in a,\n          \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set (finiteSpanningSetsInOpen' \u03bc) n) (fun x => c) y\u2016\u208a \u2202\u03bc) /\n        \u2191\u2191\u03bc a)\n    (filterAt v x) (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a)\na : Set \u03b1\nha : a \u2286 FiniteSpanningSetsIn.set A n\nh'a : MeasurableSet a\n\u22a2 \u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set (finiteSpanningSetsInOpen' \u03bc) n) (fun x => c) y\u2016\u208a \u2202\u03bc =\n    \u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc", "state_after": "case h.e_a\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f\nh'f : StronglyMeasurable f\nA : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc\nt : Set E\nt_count : Set.Countable t\nht : range f \u2286 closure t\nmain :\n  \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    \u2200 (n : \u2115) (c : E),\n      c \u2208 t \u2192\n        Tendsto\n          (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)\n          (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a)\nx : \u03b1\nh'x : \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a\nc : E\nhc : c \u2208 t\nn : \u2115\nxn : x \u2208 FiniteSpanningSetsIn.set A n\nhx :\n  Tendsto\n    (fun a =>\n      (\u222b\u207b (y : \u03b1) in a,\n          \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set (finiteSpanningSetsInOpen' \u03bc) n) (fun x => c) y\u2016\u208a \u2202\u03bc) /\n        \u2191\u2191\u03bc a)\n    (filterAt v x) (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a)\na : Set \u03b1\nha : a \u2286 FiniteSpanningSetsIn.set A n\nh'a : MeasurableSet a\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    x \u2208 a \u2192 \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set (finiteSpanningSetsInOpen' \u03bc) n) (fun x => c) x\u2016\u208a = \u2191\u2016f x - c\u2016\u208a"}, {"tactic": "apply eventually_of_forall fun y => ?_", "annotated_tactic": ["apply <a>eventually_of_forall</a> fun y => ?_", [{"full_name": "Filter.eventually_of_forall", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1111, 9], "def_end_pos": [1111, 29]}]], "state_before": "case h.e_a\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f\nh'f : StronglyMeasurable f\nA : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc\nt : Set E\nt_count : Set.Countable t\nht : range f \u2286 closure t\nmain :\n  \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    \u2200 (n : \u2115) (c : E),\n      c \u2208 t \u2192\n        Tendsto\n          (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)\n          (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a)\nx : \u03b1\nh'x : \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a\nc : E\nhc : c \u2208 t\nn : \u2115\nxn : x \u2208 FiniteSpanningSetsIn.set A n\nhx :\n  Tendsto\n    (fun a =>\n      (\u222b\u207b (y : \u03b1) in a,\n          \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set (finiteSpanningSetsInOpen' \u03bc) n) (fun x => c) y\u2016\u208a \u2202\u03bc) /\n        \u2191\u2191\u03bc a)\n    (filterAt v x) (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a)\na : Set \u03b1\nha : a \u2286 FiniteSpanningSetsIn.set A n\nh'a : MeasurableSet a\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    x \u2208 a \u2192 \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set (finiteSpanningSetsInOpen' \u03bc) n) (fun x => c) x\u2016\u208a = \u2191\u2016f x - c\u2016\u208a", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f\nh'f : StronglyMeasurable f\nA : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc\nt : Set E\nt_count : Set.Countable t\nht : range f \u2286 closure t\nmain :\n  \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    \u2200 (n : \u2115) (c : E),\n      c \u2208 t \u2192\n        Tendsto\n          (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)\n          (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a)\nx : \u03b1\nh'x : \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a\nc : E\nhc : c \u2208 t\nn : \u2115\nxn : x \u2208 FiniteSpanningSetsIn.set A n\nhx :\n  Tendsto\n    (fun a =>\n      (\u222b\u207b (y : \u03b1) in a,\n          \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set (finiteSpanningSetsInOpen' \u03bc) n) (fun x => c) y\u2016\u208a \u2202\u03bc) /\n        \u2191\u2191\u03bc a)\n    (filterAt v x) (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a)\na : Set \u03b1\nha : a \u2286 FiniteSpanningSetsIn.set A n\nh'a : MeasurableSet a\ny : \u03b1\n\u22a2 y \u2208 a \u2192 \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set (finiteSpanningSetsInOpen' \u03bc) n) (fun x => c) y\u2016\u208a = \u2191\u2016f y - c\u2016\u208a"}, {"tactic": "intro hy", "annotated_tactic": ["intro hy", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f\nh'f : StronglyMeasurable f\nA : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc\nt : Set E\nt_count : Set.Countable t\nht : range f \u2286 closure t\nmain :\n  \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    \u2200 (n : \u2115) (c : E),\n      c \u2208 t \u2192\n        Tendsto\n          (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)\n          (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a)\nx : \u03b1\nh'x : \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a\nc : E\nhc : c \u2208 t\nn : \u2115\nxn : x \u2208 FiniteSpanningSetsIn.set A n\nhx :\n  Tendsto\n    (fun a =>\n      (\u222b\u207b (y : \u03b1) in a,\n          \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set (finiteSpanningSetsInOpen' \u03bc) n) (fun x => c) y\u2016\u208a \u2202\u03bc) /\n        \u2191\u2191\u03bc a)\n    (filterAt v x) (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a)\na : Set \u03b1\nha : a \u2286 FiniteSpanningSetsIn.set A n\nh'a : MeasurableSet a\ny : \u03b1\n\u22a2 y \u2208 a \u2192 \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set (finiteSpanningSetsInOpen' \u03bc) n) (fun x => c) y\u2016\u208a = \u2191\u2016f y - c\u2016\u208a", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f\nh'f : StronglyMeasurable f\nA : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc\nt : Set E\nt_count : Set.Countable t\nht : range f \u2286 closure t\nmain :\n  \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    \u2200 (n : \u2115) (c : E),\n      c \u2208 t \u2192\n        Tendsto\n          (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)\n          (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a)\nx : \u03b1\nh'x : \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a\nc : E\nhc : c \u2208 t\nn : \u2115\nxn : x \u2208 FiniteSpanningSetsIn.set A n\nhx :\n  Tendsto\n    (fun a =>\n      (\u222b\u207b (y : \u03b1) in a,\n          \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set (finiteSpanningSetsInOpen' \u03bc) n) (fun x => c) y\u2016\u208a \u2202\u03bc) /\n        \u2191\u2191\u03bc a)\n    (filterAt v x) (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a)\na : Set \u03b1\nha : a \u2286 FiniteSpanningSetsIn.set A n\nh'a : MeasurableSet a\ny : \u03b1\nhy : y \u2208 a\n\u22a2 \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set (finiteSpanningSetsInOpen' \u03bc) n) (fun x => c) y\u2016\u208a = \u2191\u2016f y - c\u2016\u208a"}, {"tactic": "simp only [ha hy, indicator_of_mem]", "annotated_tactic": ["simp only [ha hy, <a>indicator_of_mem</a>]", [{"full_name": "Set.indicator_of_mem", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [67, 3], "def_end_pos": [67, 14]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f\nh'f : StronglyMeasurable f\nA : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc\nt : Set E\nt_count : Set.Countable t\nht : range f \u2286 closure t\nmain :\n  \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    \u2200 (n : \u2115) (c : E),\n      c \u2208 t \u2192\n        Tendsto\n          (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)\n          (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a)\nx : \u03b1\nh'x : \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a\nc : E\nhc : c \u2208 t\nn : \u2115\nxn : x \u2208 FiniteSpanningSetsIn.set A n\nhx :\n  Tendsto\n    (fun a =>\n      (\u222b\u207b (y : \u03b1) in a,\n          \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set (finiteSpanningSetsInOpen' \u03bc) n) (fun x => c) y\u2016\u208a \u2202\u03bc) /\n        \u2191\u2191\u03bc a)\n    (filterAt v x) (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a)\na : Set \u03b1\nha : a \u2286 FiniteSpanningSetsIn.set A n\nh'a : MeasurableSet a\ny : \u03b1\nhy : y \u2208 a\n\u22a2 \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set (finiteSpanningSetsInOpen' \u03bc) n) (fun x => c) y\u2016\u208a = \u2191\u2016f y - c\u2016\u208a", "state_after": "no goals"}, {"tactic": "simpa [\u2190 A.spanning] using mem_univ x", "annotated_tactic": ["simpa [\u2190 A.spanning] using <a>mem_univ</a> x", [{"full_name": "Set.mem_univ", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [676, 9], "def_end_pos": [676, 17]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f\nh'f : StronglyMeasurable f\nA : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc\nt : Set E\nt_count : Set.Countable t\nht : range f \u2286 closure t\nmain :\n  \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    \u2200 (n : \u2115) (c : E),\n      c \u2208 t \u2192\n        Tendsto\n          (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)\n          (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a)\nx : \u03b1\nhx :\n  \u2200 (n : \u2115) (c : E),\n    c \u2208 t \u2192\n      Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)\n        (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a)\nh'x : \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a\nc : E\nhc : c \u2208 t\n\u22a2 \u2203 n, x \u2208 FiniteSpanningSetsIn.set A n", "state_after": "no goals"}, {"tactic": "simp_rw [\u2190 edist_eq_coe_nnnorm_sub]", "annotated_tactic": ["simp_rw [\u2190 <a>edist_eq_coe_nnnorm_sub</a>]", [{"full_name": "edist_eq_coe_nnnorm_sub", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [1005, 3], "def_end_pos": [1005, 14]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f\nh'f : StronglyMeasurable f\nA : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc\nt : Set E\nt_count : Set.Countable t\nht : range f \u2286 closure t\nmain :\n  \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    \u2200 (n : \u2115) (c : E),\n      c \u2208 t \u2192\n        Tendsto\n          (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)\n          (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a)\nx : \u03b1\nhx :\n  \u2200 (n : \u2115) (c : E),\n    c \u2208 t \u2192\n      Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)\n        (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a)\nh'x : \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a\nM : \u2200 (c : E), c \u2208 t \u2192 Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a)\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\n\u22a2 \u2203 c, c \u2208 t \u2227 \u2191\u2016f x - c\u2016\u208a < \u03b5 / 2", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f\nh'f : StronglyMeasurable f\nA : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc\nt : Set E\nt_count : Set.Countable t\nht : range f \u2286 closure t\nmain :\n  \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    \u2200 (n : \u2115) (c : E),\n      c \u2208 t \u2192\n        Tendsto\n          (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)\n          (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a)\nx : \u03b1\nhx :\n  \u2200 (n : \u2115) (c : E),\n    c \u2208 t \u2192\n      Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)\n        (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a)\nh'x : \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a\nM : \u2200 (c : E), c \u2208 t \u2192 Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a)\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\n\u22a2 \u2203 c, c \u2208 t \u2227 edist (f x) c < \u03b5 / 2"}, {"tactic": "have : f x \u2208 closure t := ht (mem_range_self _)", "annotated_tactic": ["have : f x \u2208 <a>closure</a> t := ht (<a>mem_range_self</a> _)", [{"full_name": "closure", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [422, 5], "def_end_pos": [422, 12]}, {"full_name": "Set.mem_range_self", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [680, 9], "def_end_pos": [680, 23]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f\nh'f : StronglyMeasurable f\nA : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc\nt : Set E\nt_count : Set.Countable t\nht : range f \u2286 closure t\nmain :\n  \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    \u2200 (n : \u2115) (c : E),\n      c \u2208 t \u2192\n        Tendsto\n          (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)\n          (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a)\nx : \u03b1\nhx :\n  \u2200 (n : \u2115) (c : E),\n    c \u2208 t \u2192\n      Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)\n        (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a)\nh'x : \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a\nM : \u2200 (c : E), c \u2208 t \u2192 Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a)\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\n\u22a2 \u2203 c, c \u2208 t \u2227 edist (f x) c < \u03b5 / 2", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f\nh'f : StronglyMeasurable f\nA : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc\nt : Set E\nt_count : Set.Countable t\nht : range f \u2286 closure t\nmain :\n  \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    \u2200 (n : \u2115) (c : E),\n      c \u2208 t \u2192\n        Tendsto\n          (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)\n          (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a)\nx : \u03b1\nhx :\n  \u2200 (n : \u2115) (c : E),\n    c \u2208 t \u2192\n      Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)\n        (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a)\nh'x : \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a\nM : \u2200 (c : E), c \u2208 t \u2192 Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a)\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\nthis : f x \u2208 closure t\n\u22a2 \u2203 c, c \u2208 t \u2227 edist (f x) c < \u03b5 / 2"}, {"tactic": "exact EMetric.mem_closure_iff.1 this (\u03b5 / 2) (ENNReal.half_pos (ne_of_gt \u03b5pos))", "annotated_tactic": ["exact <a>EMetric.mem_closure_iff</a>.1 this (\u03b5 / 2) (<a>ENNReal.half_pos</a> (<a>ne_of_gt</a> \u03b5pos))", [{"full_name": "EMetric.mem_closure_iff", "def_path": "Mathlib/Topology/EMetricSpace/Basic.lean", "def_pos": [720, 9], "def_end_pos": [720, 24]}, {"full_name": "ENNReal.half_pos", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1796, 19], "def_end_pos": [1796, 27]}, {"full_name": "ne_of_gt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [104, 9], "def_end_pos": [104, 17]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f\nh'f : StronglyMeasurable f\nA : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc\nt : Set E\nt_count : Set.Countable t\nht : range f \u2286 closure t\nmain :\n  \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    \u2200 (n : \u2115) (c : E),\n      c \u2208 t \u2192\n        Tendsto\n          (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)\n          (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a)\nx : \u03b1\nhx :\n  \u2200 (n : \u2115) (c : E),\n    c \u2208 t \u2192\n      Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)\n        (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a)\nh'x : \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a\nM : \u2200 (c : E), c \u2208 t \u2192 Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a)\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\nthis : f x \u2208 closure t\n\u22a2 \u2203 c, c \u2208 t \u2227 edist (f x) c < \u03b5 / 2", "state_after": "no goals"}, {"tactic": "apply lintegral_mono fun x => ?_", "annotated_tactic": ["apply <a>lintegral_mono</a> fun x => ?_", [{"full_name": "MeasureTheory.lintegral_mono", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [99, 9], "def_end_pos": [99, 23]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f\nh'f : StronglyMeasurable f\nA : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc\nt : Set E\nt_count : Set.Countable t\nht : range f \u2286 closure t\nmain :\n  \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    \u2200 (n : \u2115) (c : E),\n      c \u2208 t \u2192\n        Tendsto\n          (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)\n          (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a)\nx : \u03b1\nhx :\n  \u2200 (n : \u2115) (c : E),\n    c \u2208 t \u2192\n      Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)\n        (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a)\nh'x : \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a\nM : \u2200 (c : E), c \u2208 t \u2192 Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a)\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\nc : E\nct : c \u2208 t\nxc : \u2191\u2016f x - c\u2016\u208a < \u03b5 / 2\na : Set \u03b1\nha : (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a < \u03b5 / 2\nh'a : 0 < \u2191\u2191\u03bc a\nh''a : \u2191\u2191\u03bc a < \u22a4\n\u22a2 \u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - f x\u2016\u208a \u2202\u03bc \u2264 \u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a + \u2191\u2016f x - c\u2016\u208a \u2202\u03bc", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f\nh'f : StronglyMeasurable f\nA : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc\nt : Set E\nt_count : Set.Countable t\nht : range f \u2286 closure t\nmain :\n  \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    \u2200 (n : \u2115) (c : E),\n      c \u2208 t \u2192\n        Tendsto\n          (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)\n          (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a)\nx\u271d : \u03b1\nhx :\n  \u2200 (n : \u2115) (c : E),\n    c \u2208 t \u2192\n      Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)\n        (filterAt v x\u271d) (\ud835\udcdd \u2191\u2016f x\u271d - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u271d\u2016\u208a)\nh'x : \u2200\u1da0 (a : Set \u03b1) in filterAt v x\u271d, 0 < \u2191\u2191\u03bc a\nM : \u2200 (c : E), c \u2208 t \u2192 Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x\u271d) (\ud835\udcdd \u2191\u2016f x\u271d - c\u2016\u208a)\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\nc : E\nct : c \u2208 t\nxc : \u2191\u2016f x\u271d - c\u2016\u208a < \u03b5 / 2\na : Set \u03b1\nha : (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a < \u03b5 / 2\nh'a : 0 < \u2191\u2191\u03bc a\nh''a : \u2191\u2191\u03bc a < \u22a4\nx : \u03b1\n\u22a2 \u2191\u2016f x - f x\u271d\u2016\u208a \u2264 \u2191\u2016f x - c\u2016\u208a + \u2191\u2016f x\u271d - c\u2016\u208a"}, {"tactic": "simpa only [\u2190 edist_eq_coe_nnnorm_sub] using edist_triangle_right _ _ _", "annotated_tactic": ["simpa only [\u2190 <a>edist_eq_coe_nnnorm_sub</a>] using <a>edist_triangle_right</a> _ _ _", [{"full_name": "edist_eq_coe_nnnorm_sub", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [1005, 3], "def_end_pos": [1005, 14]}, {"full_name": "edist_triangle_right", "def_path": "Mathlib/Topology/EMetricSpace/Basic.lean", "def_pos": [100, 9], "def_end_pos": [100, 29]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f\nh'f : StronglyMeasurable f\nA : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc\nt : Set E\nt_count : Set.Countable t\nht : range f \u2286 closure t\nmain :\n  \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    \u2200 (n : \u2115) (c : E),\n      c \u2208 t \u2192\n        Tendsto\n          (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)\n          (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a)\nx\u271d : \u03b1\nhx :\n  \u2200 (n : \u2115) (c : E),\n    c \u2208 t \u2192\n      Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)\n        (filterAt v x\u271d) (\ud835\udcdd \u2191\u2016f x\u271d - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u271d\u2016\u208a)\nh'x : \u2200\u1da0 (a : Set \u03b1) in filterAt v x\u271d, 0 < \u2191\u2191\u03bc a\nM : \u2200 (c : E), c \u2208 t \u2192 Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x\u271d) (\ud835\udcdd \u2191\u2016f x\u271d - c\u2016\u208a)\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\nc : E\nct : c \u2208 t\nxc : \u2191\u2016f x\u271d - c\u2016\u208a < \u03b5 / 2\na : Set \u03b1\nha : (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a < \u03b5 / 2\nh'a : 0 < \u2191\u2191\u03bc a\nh''a : \u2191\u2191\u03bc a < \u22a4\nx : \u03b1\n\u22a2 \u2191\u2016f x - f x\u271d\u2016\u208a \u2264 \u2191\u2016f x - c\u2016\u208a + \u2191\u2016f x\u271d - c\u2016\u208a", "state_after": "no goals"}, {"tactic": "refine' add_le_add _ _", "annotated_tactic": ["refine' <a>add_le_add</a> _ _", [{"full_name": "add_le_add", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [205, 15], "def_end_pos": [205, 25]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f\nh'f : StronglyMeasurable f\nA : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc\nt : Set E\nt_count : Set.Countable t\nht : range f \u2286 closure t\nmain :\n  \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    \u2200 (n : \u2115) (c : E),\n      c \u2208 t \u2192\n        Tendsto\n          (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)\n          (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a)\nx : \u03b1\nhx :\n  \u2200 (n : \u2115) (c : E),\n    c \u2208 t \u2192\n      Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)\n        (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a)\nh'x : \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a\nM : \u2200 (c : E), c \u2208 t \u2192 Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a)\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\nc : E\nct : c \u2208 t\nxc : \u2191\u2016f x - c\u2016\u208a < \u03b5 / 2\na : Set \u03b1\nha : (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a < \u03b5 / 2\nh'a : 0 < \u2191\u2191\u03bc a\nh''a : \u2191\u2191\u03bc a < \u22a4\n\u22a2 \u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc + \u222b\u207b (x_1 : \u03b1) in a, \u2191\u2016f x - c\u2016\u208a \u2202\u03bc \u2264 \u03b5 / 2 * \u2191\u2191\u03bc a + \u03b5 / 2 * \u2191\u2191\u03bc a", "state_after": "case refine'_1\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f\nh'f : StronglyMeasurable f\nA : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc\nt : Set E\nt_count : Set.Countable t\nht : range f \u2286 closure t\nmain :\n  \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    \u2200 (n : \u2115) (c : E),\n      c \u2208 t \u2192\n        Tendsto\n          (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)\n          (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a)\nx : \u03b1\nhx :\n  \u2200 (n : \u2115) (c : E),\n    c \u2208 t \u2192\n      Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)\n        (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a)\nh'x : \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a\nM : \u2200 (c : E), c \u2208 t \u2192 Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a)\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\nc : E\nct : c \u2208 t\nxc : \u2191\u2016f x - c\u2016\u208a < \u03b5 / 2\na : Set \u03b1\nha : (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a < \u03b5 / 2\nh'a : 0 < \u2191\u2191\u03bc a\nh''a : \u2191\u2191\u03bc a < \u22a4\n\u22a2 \u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc \u2264 \u03b5 / 2 * \u2191\u2191\u03bc a\n\ncase refine'_2\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f\nh'f : StronglyMeasurable f\nA : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc\nt : Set E\nt_count : Set.Countable t\nht : range f \u2286 closure t\nmain :\n  \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    \u2200 (n : \u2115) (c : E),\n      c \u2208 t \u2192\n        Tendsto\n          (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)\n          (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a)\nx : \u03b1\nhx :\n  \u2200 (n : \u2115) (c : E),\n    c \u2208 t \u2192\n      Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)\n        (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a)\nh'x : \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a\nM : \u2200 (c : E), c \u2208 t \u2192 Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a)\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\nc : E\nct : c \u2208 t\nxc : \u2191\u2016f x - c\u2016\u208a < \u03b5 / 2\na : Set \u03b1\nha : (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a < \u03b5 / 2\nh'a : 0 < \u2191\u2191\u03bc a\nh''a : \u2191\u2191\u03bc a < \u22a4\n\u22a2 \u222b\u207b (x_1 : \u03b1) in a, \u2191\u2016f x - c\u2016\u208a \u2202\u03bc \u2264 \u03b5 / 2 * \u2191\u2191\u03bc a"}, {"tactic": "rw [ENNReal.div_lt_iff (Or.inl h'a.ne') (Or.inl h''a.ne)] at ha", "annotated_tactic": ["rw [<a>ENNReal.div_lt_iff</a> (<a>Or.inl</a> h'a.ne') (<a>Or.inl</a> h''a.ne)] at ha", [{"full_name": "ENNReal.div_lt_iff", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1658, 19], "def_end_pos": [1658, 29]}, {"full_name": "Or.inl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [517, 5], "def_end_pos": [517, 8]}, {"full_name": "Or.inl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [517, 5], "def_end_pos": [517, 8]}]], "state_before": "case refine'_1\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f\nh'f : StronglyMeasurable f\nA : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc\nt : Set E\nt_count : Set.Countable t\nht : range f \u2286 closure t\nmain :\n  \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    \u2200 (n : \u2115) (c : E),\n      c \u2208 t \u2192\n        Tendsto\n          (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)\n          (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a)\nx : \u03b1\nhx :\n  \u2200 (n : \u2115) (c : E),\n    c \u2208 t \u2192\n      Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)\n        (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a)\nh'x : \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a\nM : \u2200 (c : E), c \u2208 t \u2192 Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a)\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\nc : E\nct : c \u2208 t\nxc : \u2191\u2016f x - c\u2016\u208a < \u03b5 / 2\na : Set \u03b1\nha : (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a < \u03b5 / 2\nh'a : 0 < \u2191\u2191\u03bc a\nh''a : \u2191\u2191\u03bc a < \u22a4\n\u22a2 \u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc \u2264 \u03b5 / 2 * \u2191\u2191\u03bc a", "state_after": "case refine'_1\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f\nh'f : StronglyMeasurable f\nA : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc\nt : Set E\nt_count : Set.Countable t\nht : range f \u2286 closure t\nmain :\n  \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    \u2200 (n : \u2115) (c : E),\n      c \u2208 t \u2192\n        Tendsto\n          (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)\n          (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a)\nx : \u03b1\nhx :\n  \u2200 (n : \u2115) (c : E),\n    c \u2208 t \u2192\n      Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)\n        (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a)\nh'x : \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a\nM : \u2200 (c : E), c \u2208 t \u2192 Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a)\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\nc : E\nct : c \u2208 t\nxc : \u2191\u2016f x - c\u2016\u208a < \u03b5 / 2\na : Set \u03b1\nha : \u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc < \u03b5 / 2 * \u2191\u2191\u03bc a\nh'a : 0 < \u2191\u2191\u03bc a\nh''a : \u2191\u2191\u03bc a < \u22a4\n\u22a2 \u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc \u2264 \u03b5 / 2 * \u2191\u2191\u03bc a"}, {"tactic": "exact ha.le", "annotated_tactic": ["exact ha.le", []], "state_before": "case refine'_1\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f\nh'f : StronglyMeasurable f\nA : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc\nt : Set E\nt_count : Set.Countable t\nht : range f \u2286 closure t\nmain :\n  \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    \u2200 (n : \u2115) (c : E),\n      c \u2208 t \u2192\n        Tendsto\n          (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)\n          (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a)\nx : \u03b1\nhx :\n  \u2200 (n : \u2115) (c : E),\n    c \u2208 t \u2192\n      Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)\n        (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a)\nh'x : \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a\nM : \u2200 (c : E), c \u2208 t \u2192 Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a)\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\nc : E\nct : c \u2208 t\nxc : \u2191\u2016f x - c\u2016\u208a < \u03b5 / 2\na : Set \u03b1\nha : \u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc < \u03b5 / 2 * \u2191\u2191\u03bc a\nh'a : 0 < \u2191\u2191\u03bc a\nh''a : \u2191\u2191\u03bc a < \u22a4\n\u22a2 \u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc \u2264 \u03b5 / 2 * \u2191\u2191\u03bc a", "state_after": "no goals"}, {"tactic": "simp only [lintegral_const, Measure.restrict_apply, MeasurableSet.univ, univ_inter]", "annotated_tactic": ["simp only [<a>lintegral_const</a>, <a>Measure.restrict_apply</a>, <a>MeasurableSet.univ</a>, <a>univ_inter</a>]", [{"full_name": "MeasureTheory.lintegral_const", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [136, 9], "def_end_pos": [136, 24]}, {"full_name": "MeasureTheory.Measure.restrict_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1533, 9], "def_end_pos": [1533, 23]}, {"full_name": "MeasurableSet.univ", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [101, 19], "def_end_pos": [101, 37]}, {"full_name": "Set.univ_inter", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1017, 9], "def_end_pos": [1017, 19]}]], "state_before": "case refine'_2\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f\nh'f : StronglyMeasurable f\nA : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc\nt : Set E\nt_count : Set.Countable t\nht : range f \u2286 closure t\nmain :\n  \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    \u2200 (n : \u2115) (c : E),\n      c \u2208 t \u2192\n        Tendsto\n          (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)\n          (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a)\nx : \u03b1\nhx :\n  \u2200 (n : \u2115) (c : E),\n    c \u2208 t \u2192\n      Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)\n        (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a)\nh'x : \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a\nM : \u2200 (c : E), c \u2208 t \u2192 Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a)\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\nc : E\nct : c \u2208 t\nxc : \u2191\u2016f x - c\u2016\u208a < \u03b5 / 2\na : Set \u03b1\nha : (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a < \u03b5 / 2\nh'a : 0 < \u2191\u2191\u03bc a\nh''a : \u2191\u2191\u03bc a < \u22a4\n\u22a2 \u222b\u207b (x_1 : \u03b1) in a, \u2191\u2016f x - c\u2016\u208a \u2202\u03bc \u2264 \u03b5 / 2 * \u2191\u2191\u03bc a", "state_after": "case refine'_2\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f\nh'f : StronglyMeasurable f\nA : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc\nt : Set E\nt_count : Set.Countable t\nht : range f \u2286 closure t\nmain :\n  \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    \u2200 (n : \u2115) (c : E),\n      c \u2208 t \u2192\n        Tendsto\n          (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)\n          (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a)\nx : \u03b1\nhx :\n  \u2200 (n : \u2115) (c : E),\n    c \u2208 t \u2192\n      Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)\n        (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a)\nh'x : \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a\nM : \u2200 (c : E), c \u2208 t \u2192 Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a)\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\nc : E\nct : c \u2208 t\nxc : \u2191\u2016f x - c\u2016\u208a < \u03b5 / 2\na : Set \u03b1\nha : (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a < \u03b5 / 2\nh'a : 0 < \u2191\u2191\u03bc a\nh''a : \u2191\u2191\u03bc a < \u22a4\n\u22a2 \u2191\u2016f x - c\u2016\u208a * \u2191\u2191\u03bc a \u2264 \u03b5 / 2 * \u2191\u2191\u03bc a"}, {"tactic": "exact mul_le_mul_right' xc.le _", "annotated_tactic": ["exact <a>mul_le_mul_right'</a> xc.le _", [{"full_name": "mul_le_mul_right'", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [67, 9], "def_end_pos": [67, 26]}]], "state_before": "case refine'_2\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f\nh'f : StronglyMeasurable f\nA : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc\nt : Set E\nt_count : Set.Countable t\nht : range f \u2286 closure t\nmain :\n  \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    \u2200 (n : \u2115) (c : E),\n      c \u2208 t \u2192\n        Tendsto\n          (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)\n          (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a)\nx : \u03b1\nhx :\n  \u2200 (n : \u2115) (c : E),\n    c \u2208 t \u2192\n      Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)\n        (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a)\nh'x : \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a\nM : \u2200 (c : E), c \u2208 t \u2192 Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a)\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\nc : E\nct : c \u2208 t\nxc : \u2191\u2016f x - c\u2016\u208a < \u03b5 / 2\na : Set \u03b1\nha : (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a < \u03b5 / 2\nh'a : 0 < \u2191\u2191\u03bc a\nh''a : \u2191\u2191\u03bc a < \u22a4\n\u22a2 \u2191\u2016f x - c\u2016\u208a * \u2191\u2191\u03bc a \u2264 \u03b5 / 2 * \u2191\u2191\u03bc a", "state_after": "no goals"}, {"tactic": "rw [\u2190 add_mul, ENNReal.add_halves]", "annotated_tactic": ["rw [\u2190 <a>add_mul</a>, <a>ENNReal.add_halves</a>]", [{"full_name": "add_mul", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [91, 7], "def_end_pos": [91, 14]}, {"full_name": "ENNReal.add_halves", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1781, 19], "def_end_pos": [1781, 29]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f\nh'f : StronglyMeasurable f\nA : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc\nt : Set E\nt_count : Set.Countable t\nht : range f \u2286 closure t\nmain :\n  \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    \u2200 (n : \u2115) (c : E),\n      c \u2208 t \u2192\n        Tendsto\n          (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)\n          (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a)\nx : \u03b1\nhx :\n  \u2200 (n : \u2115) (c : E),\n    c \u2208 t \u2192\n      Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)\n        (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a)\nh'x : \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a\nM : \u2200 (c : E), c \u2208 t \u2192 Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a)\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\nc : E\nct : c \u2208 t\nxc : \u2191\u2016f x - c\u2016\u208a < \u03b5 / 2\na : Set \u03b1\nha : (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a < \u03b5 / 2\nh'a : 0 < \u2191\u2191\u03bc a\nh''a : \u2191\u2191\u03bc a < \u22a4\n\u22a2 \u03b5 / 2 * \u2191\u2191\u03bc a + \u03b5 / 2 * \u2191\u2191\u03bc a = \u03b5 * \u2191\u2191\u03bc a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Intervals/Group.lean", "full_name": "Set.pairwise_disjoint_Ico_zpow", "start": [214, 1], "end": [216, 64], "traced_tactics": [{"tactic": "simpa only [one_mul] using pairwise_disjoint_Ico_mul_zpow 1 b", "annotated_tactic": ["simpa only [<a>one_mul</a>] using <a>pairwise_disjoint_Ico_mul_zpow</a> 1 b", [{"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [464, 9], "def_end_pos": [464, 16]}, {"full_name": "Set.pairwise_disjoint_Ico_mul_zpow", "def_path": "Mathlib/Data/Set/Intervals/Group.lean", "def_pos": [184, 9], "def_end_pos": [184, 39]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : OrderedCommGroup \u03b1\na b : \u03b1\n\u22a2 Pairwise (Disjoint on fun n => Ico (b ^ n) (b ^ (n + 1)))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Nat/Gcd.lean", "full_name": "Nat.Coprime.gcd_mul", "start": [419, 1], "end": [424, 39], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "full_name": "Set.empty_pow", "start": [977, 1], "end": [978, 97], "traced_tactics": [{"tactic": "rw [\u2190 tsub_add_cancel_of_le (Nat.succ_le_of_lt <| Nat.pos_of_ne_zero hn), pow_succ, empty_mul]", "annotated_tactic": ["rw [\u2190 <a>tsub_add_cancel_of_le</a> (<a>Nat.succ_le_of_lt</a> <| <a>Nat.pos_of_ne_zero</a> hn), <a>pow_succ</a>, <a>empty_mul</a>]", [{"full_name": "tsub_add_cancel_of_le", "def_path": "Mathlib/Algebra/Order/Sub/Canonical.lean", "def_pos": [30, 9], "def_end_pos": [30, 30]}, {"full_name": "Nat.succ_le_of_lt", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [313, 9], "def_end_pos": [313, 22]}, {"full_name": "Nat.pos_of_ne_zero", "def_path": "lake-packages/std/Std/Data/Nat/Init/Lemmas.lean", "def_pos": [25, 19], "def_end_pos": [25, 33]}, {"full_name": "pow_succ", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [645, 9], "def_end_pos": [645, 17]}, {"full_name": "Set.empty_mul", "def_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "def_pos": [355, 9], "def_end_pos": [355, 18]}]], "state_before": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : Monoid \u03b1\ns t : Set \u03b1\na : \u03b1\nm n\u271d n : \u2115\nhn : n \u2260 0\n\u22a2 \u2205 ^ n = \u2205", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Primrec.lean", "full_name": "Primrec\u2082.unpaired", "start": [437, 1], "end": [438, 75], "traced_tactics": [{"tactic": "simpa using h.comp natPair", "annotated_tactic": ["simpa using h.comp <a>natPair</a>", [{"full_name": "Primrec\u2082.natPair", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [434, 9], "def_end_pos": [434, 16]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03c3 : Type u_3\ninst\u271d\u00b2 : Primcodable \u03b1\ninst\u271d\u00b9 : Primcodable \u03b2\ninst\u271d : Primcodable \u03c3\nf : \u2115 \u2192 \u2115 \u2192 \u03b1\nh : Primrec (Nat.unpaired f)\n\u22a2 Primrec\u2082 f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Num/Lemmas.lean", "full_name": "Num.gcd_to_nat_aux", "start": [1662, 1], "end": [1684, 25], "traced_tactics": [{"tactic": "simp only [gcdAux, cast_pos]", "annotated_tactic": ["simp only [<a>gcdAux</a>, <a>cast_pos</a>]", [{"full_name": "Num.gcdAux", "def_path": "Mathlib/Data/Num/Basic.lean", "def_pos": [626, 5], "def_end_pos": [626, 11]}, {"full_name": "Num.cast_pos", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [287, 9], "def_end_pos": [287, 17]}]], "state_before": "n : \u2115\na : PosNum\nb : Num\nab : pos a \u2264 b\nh : natSize (pos a * b) \u2264 Nat.succ n\n\u22a2 \u2191(gcdAux (Nat.succ n) (pos a) b) = Nat.gcd \u2191(pos a) \u2191b", "state_after": "n : \u2115\na : PosNum\nb : Num\nab : pos a \u2264 b\nh : natSize (pos a * b) \u2264 Nat.succ n\n\u22a2 \u2191(gcdAux n (b % pos a) (pos a)) = Nat.gcd \u2191a \u2191b"}, {"tactic": "rw [Nat.gcd_rec, gcd_to_nat_aux, mod_to_nat]", "annotated_tactic": ["rw [<a>Nat.gcd_rec</a>, gcd_to_nat_aux, <a>mod_to_nat</a>]", [{"full_name": "Nat.gcd_rec", "def_path": "lake-packages/std/Std/Data/Nat/Gcd.lean", "def_pos": [16, 9], "def_end_pos": [16, 16]}, {"full_name": "Num.mod_to_nat", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [1655, 9], "def_end_pos": [1655, 19]}]], "state_before": "n : \u2115\na : PosNum\nb : Num\nab : pos a \u2264 b\nh : natSize (pos a * b) \u2264 Nat.succ n\n\u22a2 \u2191(gcdAux n (b % pos a) (pos a)) = Nat.gcd \u2191a \u2191b", "state_after": "n : \u2115\na : PosNum\nb : Num\nab : pos a \u2264 b\nh : natSize (pos a * b) \u2264 Nat.succ n\n\u22a2 Nat.gcd (\u2191b % \u2191(pos a)) \u2191(pos a) = Nat.gcd (\u2191b % \u2191a) \u2191a\n\ncase a\nn : \u2115\na : PosNum\nb : Num\nab : pos a \u2264 b\nh : natSize (pos a * b) \u2264 Nat.succ n\n\u22a2 b % pos a \u2264 pos a\n\ncase a\nn : \u2115\na : PosNum\nb : Num\nab : pos a \u2264 b\nh : natSize (pos a * b) \u2264 Nat.succ n\n\u22a2 natSize (b % pos a * pos a) \u2264 n"}, {"tactic": "rw [natSize_to_nat, mul_to_nat, Nat.size_le] at h \u22a2", "annotated_tactic": ["rw [<a>natSize_to_nat</a>, <a>mul_to_nat</a>, <a>Nat.size_le</a>] at h \u22a2", [{"full_name": "Num.natSize_to_nat", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [760, 9], "def_end_pos": [760, 23]}, {"full_name": "Num.mul_to_nat", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [315, 9], "def_end_pos": [315, 19]}, {"full_name": "Nat.size_le", "def_path": "Mathlib/Data/Nat/Size.lean", "def_pos": [124, 9], "def_end_pos": [124, 16]}]], "state_before": "case a\nn : \u2115\na : PosNum\nb : Num\nab : pos a \u2264 b\nh : natSize (pos a * b) \u2264 Nat.succ n\n\u22a2 natSize (b % pos a * pos a) \u2264 n", "state_after": "case a\nn : \u2115\na : PosNum\nb : Num\nab : pos a \u2264 b\nh : \u2191(pos a) * \u2191b < 2 ^ Nat.succ n\n\u22a2 \u2191(b % pos a) * \u2191(pos a) < 2 ^ n"}, {"tactic": "rw [mod_to_nat, mul_comm]", "annotated_tactic": ["rw [<a>mod_to_nat</a>, <a>mul_comm</a>]", [{"full_name": "Num.mod_to_nat", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [1655, 9], "def_end_pos": [1655, 19]}, {"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [302, 9], "def_end_pos": [302, 17]}]], "state_before": "case a\nn : \u2115\na : PosNum\nb : Num\nab : pos a \u2264 b\nh : \u2191(pos a) * \u2191b < 2 ^ Nat.succ n\n\u22a2 \u2191(b % pos a) * \u2191(pos a) < 2 ^ n", "state_after": "case a\nn : \u2115\na : PosNum\nb : Num\nab : pos a \u2264 b\nh : \u2191(pos a) * \u2191b < 2 ^ Nat.succ n\n\u22a2 \u2191(pos a) * (\u2191b % \u2191(pos a)) < 2 ^ n"}, {"tactic": "rw [pow_succ', \u2190 Nat.mod_add_div b (pos a)] at h", "annotated_tactic": ["rw [<a>pow_succ'</a>, \u2190 <a>Nat.mod_add_div</a> b (<a>pos</a> a)] at h", [{"full_name": "pow_succ'", "def_path": "Mathlib/Algebra/Group/Commute/Defs.lean", "def_pos": [213, 9], "def_end_pos": [213, 25]}, {"full_name": "Nat.mod_add_div", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [550, 9], "def_end_pos": [550, 20]}, {"full_name": "Num.pos", "def_path": "Mathlib/Data/Num/Basic.lean", "def_pos": [45, 5], "def_end_pos": [45, 8]}]], "state_before": "case a\nn : \u2115\na : PosNum\nb : Num\nab : pos a \u2264 b\nh : \u2191(pos a) * \u2191b < 2 ^ Nat.succ n\n\u22a2 \u2191(pos a) * (\u2191b % \u2191(pos a)) < 2 ^ n", "state_after": "case a\nn : \u2115\na : PosNum\nb : Num\nab : pos a \u2264 b\nh : \u2191(pos a) * (\u2191b % \u2191(pos a) + \u2191(pos a) * (\u2191b / \u2191(pos a))) < 2 ^ n * 2\n\u22a2 \u2191(pos a) * (\u2191b % \u2191(pos a)) < 2 ^ n"}, {"tactic": "refine' lt_of_mul_lt_mul_right (lt_of_le_of_lt _ h) (Nat.zero_le 2)", "annotated_tactic": ["refine' <a>lt_of_mul_lt_mul_right</a> (<a>lt_of_le_of_lt</a> _ h) (<a>Nat.zero_le</a> 2)", [{"full_name": "lt_of_mul_lt_mul_right", "def_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "def_pos": [172, 9], "def_end_pos": [172, 31]}, {"full_name": "lt_of_le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [122, 9], "def_end_pos": [122, 23]}, {"full_name": "Nat.zero_le", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1578, 9], "def_end_pos": [1578, 20]}]], "state_before": "case a\nn : \u2115\na : PosNum\nb : Num\nab : pos a \u2264 b\nh : \u2191(pos a) * (\u2191b % \u2191(pos a) + \u2191(pos a) * (\u2191b / \u2191(pos a))) < 2 ^ n * 2\n\u22a2 \u2191(pos a) * (\u2191b % \u2191(pos a)) < 2 ^ n", "state_after": "case a\nn : \u2115\na : PosNum\nb : Num\nab : pos a \u2264 b\nh : \u2191(pos a) * (\u2191b % \u2191(pos a) + \u2191(pos a) * (\u2191b / \u2191(pos a))) < 2 ^ n * 2\n\u22a2 \u2191(pos a) * (\u2191b % \u2191(pos a)) * 2 \u2264 \u2191(pos a) * (\u2191b % \u2191(pos a) + \u2191(pos a) * (\u2191b / \u2191(pos a)))"}, {"tactic": "rw [mul_two, mul_add]", "annotated_tactic": ["rw [<a>mul_two</a>, <a>mul_add</a>]", [{"full_name": "mul_two", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [188, 9], "def_end_pos": [188, 16]}, {"full_name": "mul_add", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [83, 7], "def_end_pos": [83, 14]}]], "state_before": "case a\nn : \u2115\na : PosNum\nb : Num\nab : pos a \u2264 b\nh : \u2191(pos a) * (\u2191b % \u2191(pos a) + \u2191(pos a) * (\u2191b / \u2191(pos a))) < 2 ^ n * 2\n\u22a2 \u2191(pos a) * (\u2191b % \u2191(pos a)) * 2 \u2264 \u2191(pos a) * (\u2191b % \u2191(pos a) + \u2191(pos a) * (\u2191b / \u2191(pos a)))", "state_after": "case a\nn : \u2115\na : PosNum\nb : Num\nab : pos a \u2264 b\nh : \u2191(pos a) * (\u2191b % \u2191(pos a) + \u2191(pos a) * (\u2191b / \u2191(pos a))) < 2 ^ n * 2\n\u22a2 \u2191(pos a) * (\u2191b % \u2191(pos a)) + \u2191(pos a) * (\u2191b % \u2191(pos a)) \u2264\n    \u2191(pos a) * (\u2191b % \u2191(pos a)) + \u2191(pos a) * (\u2191(pos a) * (\u2191b / \u2191(pos a)))"}, {"tactic": "refine'\n  add_le_add_left\n    (Nat.mul_le_mul_left _ (le_trans (le_of_lt (Nat.mod_lt _ (PosNum.cast_pos _))) _)) _", "annotated_tactic": ["refine'\n      <a>add_le_add_left</a>\n        (<a>Nat.mul_le_mul_left</a> _ (<a>le_trans</a> (<a>le_of_lt</a> (<a>Nat.mod_lt</a> _ (<a>PosNum.cast_pos</a> _))) _)) _", [{"full_name": "add_le_add_left", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [49, 15], "def_end_pos": [49, 30]}, {"full_name": "Nat.mul_le_mul_left", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [437, 9], "def_end_pos": [437, 24]}, {"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}, {"full_name": "le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [110, 9], "def_end_pos": [110, 17]}, {"full_name": "Nat.mod_lt", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Div.lean", "def_pos": [122, 9], "def_end_pos": [122, 15]}, {"full_name": "PosNum.cast_pos", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [681, 9], "def_end_pos": [681, 17]}]], "state_before": "case a\nn : \u2115\na : PosNum\nb : Num\nab : pos a \u2264 b\nh : \u2191(pos a) * (\u2191b % \u2191(pos a) + \u2191(pos a) * (\u2191b / \u2191(pos a))) < 2 ^ n * 2\n\u22a2 \u2191(pos a) * (\u2191b % \u2191(pos a)) + \u2191(pos a) * (\u2191b % \u2191(pos a)) \u2264\n    \u2191(pos a) * (\u2191b % \u2191(pos a)) + \u2191(pos a) * (\u2191(pos a) * (\u2191b / \u2191(pos a)))", "state_after": "case a\nn : \u2115\na : PosNum\nb : Num\nab : pos a \u2264 b\nh : \u2191(pos a) * (\u2191b % \u2191(pos a) + \u2191(pos a) * (\u2191b / \u2191(pos a))) < 2 ^ n * 2\n\u22a2 \u2191a \u2264 \u2191(pos a) * (\u2191b / \u2191(pos a))"}, {"tactic": "suffices 1 \u2264 _ by simpa using Nat.mul_le_mul_left (pos a) this", "annotated_tactic": ["suffices 1 \u2264 _ by simpa using <a>Nat.mul_le_mul_left</a> (<a>pos</a> a) this", [{"full_name": "Nat.mul_le_mul_left", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [437, 9], "def_end_pos": [437, 24]}, {"full_name": "Num.pos", "def_path": "Mathlib/Data/Num/Basic.lean", "def_pos": [45, 5], "def_end_pos": [45, 8]}]], "state_before": "case a\nn : \u2115\na : PosNum\nb : Num\nab : pos a \u2264 b\nh : \u2191(pos a) * (\u2191b % \u2191(pos a) + \u2191(pos a) * (\u2191b / \u2191(pos a))) < 2 ^ n * 2\n\u22a2 \u2191a \u2264 \u2191(pos a) * (\u2191b / \u2191(pos a))", "state_after": "case a\nn : \u2115\na : PosNum\nb : Num\nab : pos a \u2264 b\nh : \u2191(pos a) * (\u2191b % \u2191(pos a) + \u2191(pos a) * (\u2191b / \u2191(pos a))) < 2 ^ n * 2\n\u22a2 1 \u2264 \u2191b / \u2191a"}, {"tactic": "rw [Nat.le_div_iff_mul_le a.cast_pos, one_mul]", "annotated_tactic": ["rw [<a>Nat.le_div_iff_mul_le</a> a.cast_pos, <a>one_mul</a>]", [{"full_name": "Nat.le_div_iff_mul_le", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [565, 9], "def_end_pos": [565, 26]}, {"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [464, 9], "def_end_pos": [464, 16]}]], "state_before": "case a\nn : \u2115\na : PosNum\nb : Num\nab : pos a \u2264 b\nh : \u2191(pos a) * (\u2191b % \u2191(pos a) + \u2191(pos a) * (\u2191b / \u2191(pos a))) < 2 ^ n * 2\n\u22a2 1 \u2264 \u2191b / \u2191a", "state_after": "case a\nn : \u2115\na : PosNum\nb : Num\nab : pos a \u2264 b\nh : \u2191(pos a) * (\u2191b % \u2191(pos a) + \u2191(pos a) * (\u2191b / \u2191(pos a))) < 2 ^ n * 2\n\u22a2 \u2191a \u2264 \u2191b"}, {"tactic": "exact le_to_nat.2 ab", "annotated_tactic": ["exact <a>le_to_nat</a>.2 ab", [{"full_name": "Num.le_to_nat", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [341, 9], "def_end_pos": [341, 18]}]], "state_before": "case a\nn : \u2115\na : PosNum\nb : Num\nab : pos a \u2264 b\nh : \u2191(pos a) * (\u2191b % \u2191(pos a) + \u2191(pos a) * (\u2191b / \u2191(pos a))) < 2 ^ n * 2\n\u22a2 \u2191a \u2264 \u2191b", "state_after": "no goals"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "n : \u2115\na : PosNum\nb : Num\nab : pos a \u2264 b\nh : natSize (pos a * b) \u2264 Nat.succ n\n\u22a2 Nat.gcd (\u2191b % \u2191(pos a)) \u2191(pos a) = Nat.gcd (\u2191b % \u2191a) \u2191a", "state_after": "no goals"}, {"tactic": "rw [\u2190 le_to_nat, mod_to_nat]", "annotated_tactic": ["rw [\u2190 <a>le_to_nat</a>, <a>mod_to_nat</a>]", [{"full_name": "Num.le_to_nat", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [341, 9], "def_end_pos": [341, 18]}, {"full_name": "Num.mod_to_nat", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [1655, 9], "def_end_pos": [1655, 19]}]], "state_before": "case a\nn : \u2115\na : PosNum\nb : Num\nab : pos a \u2264 b\nh : natSize (pos a * b) \u2264 Nat.succ n\n\u22a2 b % pos a \u2264 pos a", "state_after": "case a\nn : \u2115\na : PosNum\nb : Num\nab : pos a \u2264 b\nh : natSize (pos a * b) \u2264 Nat.succ n\n\u22a2 \u2191b % \u2191(pos a) \u2264 \u2191(pos a)"}, {"tactic": "exact le_of_lt (Nat.mod_lt _ (PosNum.cast_pos _))", "annotated_tactic": ["exact <a>le_of_lt</a> (<a>Nat.mod_lt</a> _ (<a>PosNum.cast_pos</a> _))", [{"full_name": "le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [110, 9], "def_end_pos": [110, 17]}, {"full_name": "Nat.mod_lt", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Div.lean", "def_pos": [122, 9], "def_end_pos": [122, 15]}, {"full_name": "PosNum.cast_pos", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [681, 9], "def_end_pos": [681, 17]}]], "state_before": "case a\nn : \u2115\na : PosNum\nb : Num\nab : pos a \u2264 b\nh : natSize (pos a * b) \u2264 Nat.succ n\n\u22a2 \u2191b % \u2191(pos a) \u2264 \u2191(pos a)", "state_after": "no goals"}, {"tactic": "simpa using Nat.mul_le_mul_left (pos a) this", "annotated_tactic": ["simpa using <a>Nat.mul_le_mul_left</a> (<a>pos</a> a) this", [{"full_name": "Nat.mul_le_mul_left", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [437, 9], "def_end_pos": [437, 24]}, {"full_name": "Num.pos", "def_path": "Mathlib/Data/Num/Basic.lean", "def_pos": [45, 5], "def_end_pos": [45, 8]}]], "state_before": "n : \u2115\na : PosNum\nb : Num\nab : pos a \u2264 b\nh : \u2191(pos a) * (\u2191b % \u2191(pos a) + \u2191(pos a) * (\u2191b / \u2191(pos a))) < 2 ^ n * 2\nthis : 1 \u2264 ?m.1181137\n\u22a2 \u2191a \u2264 \u2191(pos a) * (\u2191b / \u2191(pos a))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Rel.lean", "full_name": "Rel.image_univ", "start": [168, 1], "end": [170, 26], "traced_tactics": [{"tactic": "ext y", "annotated_tactic": ["ext y", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : Rel \u03b1 \u03b2\n\u22a2 image r Set.univ = codom r", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : Rel \u03b1 \u03b2\ny : \u03b2\n\u22a2 y \u2208 image r Set.univ \u2194 y \u2208 codom r"}, {"tactic": "simp [mem_image, codom]", "annotated_tactic": ["simp [<a>mem_image</a>, <a>codom</a>]", [{"full_name": "Rel.mem_image", "def_path": "Mathlib/Data/Rel.lean", "def_pos": [133, 9], "def_end_pos": [133, 18]}, {"full_name": "Rel.codom", "def_path": "Mathlib/Data/Rel.lean", "def_pos": [77, 5], "def_end_pos": [77, 10]}]], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : Rel \u03b1 \u03b2\ny : \u03b2\n\u22a2 y \u2208 image r Set.univ \u2194 y \u2208 codom r", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "full_name": "MeasureTheory.ae_restrict_biUnion_finset_iff", "start": [2528, 1], "end": [2530, 73], "traced_tactics": [{"tactic": "simp_rw [Filter.Eventually, ae_restrict_biUnion_finset_eq s, mem_iSup]", "annotated_tactic": ["simp_rw [<a>Filter.Eventually</a>, <a>ae_restrict_biUnion_finset_eq</a> s, <a>mem_iSup</a>]", [{"full_name": "Filter.Eventually", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1072, 15], "def_end_pos": [1072, 25]}, {"full_name": "MeasureTheory.ae_restrict_biUnion_finset_eq", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2509, 9], "def_end_pos": [2509, 38]}, {"full_name": "Filter.mem_iSup", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [582, 9], "def_end_pos": [582, 17]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\ns : \u03b9 \u2192 Set \u03b1\nt : Finset \u03b9\np : \u03b1 \u2192 Prop\n\u22a2 (\u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc (\u22c3 i \u2208 t, s i), p x) \u2194 \u2200 (i : \u03b9), i \u2208 t \u2192 \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc (s i), p x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Pointwise/BigOperators.lean", "full_name": "Set.image_finset_prod", "start": [52, 1], "end": [54, 87], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finmap.lean", "full_name": "Finmap.disjoint_union_left", "start": [670, 1], "end": [671, 59], "traced_tactics": [{"tactic": "simp [Disjoint, Finmap.mem_union, or_imp, forall_and]", "annotated_tactic": ["simp [<a>Disjoint</a>, <a>Finmap.mem_union</a>, <a>or_imp</a>, <a>forall_and</a>]", [{"full_name": "Finmap.Disjoint", "def_path": "Mathlib/Data/Finmap.lean", "def_pos": [648, 5], "def_end_pos": [648, 13]}, {"full_name": "Finmap.mem_union", "def_path": "Mathlib/Data/Finmap.lean", "def_pos": [569, 9], "def_end_pos": [569, 18]}, {"full_name": "or_imp", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [337, 9], "def_end_pos": [337, 15]}, {"full_name": "forall_and", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [426, 9], "def_end_pos": [426, 19]}]], "state_before": "\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\nx y z : Finmap \u03b2\n\u22a2 Disjoint (x \u222a y) z \u2194 Disjoint x z \u2227 Disjoint y z", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Kernel/Condexp.lean", "full_name": "ProbabilityTheory.stronglyMeasurable_condexpKernel", "start": [96, 1], "end": [98, 62], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean", "full_name": "MeasureTheory.condexpIndL1Fin_add", "start": [92, 1], "end": [102, 6], "traced_tactics": [{"tactic": "ext1", "annotated_tactic": ["ext1", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : CompleteSpace F'\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \u211d G'\ninst\u271d\u00b2 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx y : G\n\u22a2 condexpIndL1Fin hm hs h\u03bcs (x + y) = condexpIndL1Fin hm hs h\u03bcs x + condexpIndL1Fin hm hs h\u03bcs y", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : CompleteSpace F'\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \u211d G'\ninst\u271d\u00b2 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx y : G\n\u22a2 \u2191\u2191(condexpIndL1Fin hm hs h\u03bcs (x + y)) =\u1d50[\u03bc] \u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x + condexpIndL1Fin hm hs h\u03bcs y)"}, {"tactic": "refine' (Mem\u2112p.coeFn_toLp q).trans _", "annotated_tactic": ["refine' (<a>Mem\u2112p.coeFn_toLp</a> <a>q</a>).<a>trans</a> _", [{"full_name": "MeasureTheory.Mem\u2112p.coeFn_toLp", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [119, 9], "def_end_pos": [119, 19]}, {"full_name": "_private.Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1.0.MeasureTheory.q", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean", "def_pos": [88, 17], "def_end_pos": [88, 18]}, {"full_name": "Filter.EventuallyEq.trans", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1503, 9], "def_end_pos": [1503, 27]}]], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : CompleteSpace F'\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \u211d G'\ninst\u271d\u00b2 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx y : G\n\u22a2 \u2191\u2191(condexpIndL1Fin hm hs h\u03bcs (x + y)) =\u1d50[\u03bc] \u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x + condexpIndL1Fin hm hs h\u03bcs y)", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : CompleteSpace F'\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \u211d G'\ninst\u271d\u00b2 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx y : G\n\u22a2 \u2191\u2191(condexpIndSMul hm hs h\u03bcs (x + y)) =\u1d50[\u03bc] \u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x + condexpIndL1Fin hm hs h\u03bcs y)"}, {"tactic": "refine' EventuallyEq.trans _ (Lp.coeFn_add _ _).symm", "annotated_tactic": ["refine' <a>EventuallyEq.trans</a> _ (<a>Lp.coeFn_add</a> _ _).<a>symm</a>", [{"full_name": "Filter.EventuallyEq.trans", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1503, 9], "def_end_pos": [1503, 27]}, {"full_name": "MeasureTheory.Lp.coeFn_add", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [232, 9], "def_end_pos": [232, 18]}, {"full_name": "Filter.EventuallyEq.symm", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1498, 9], "def_end_pos": [1498, 26]}]], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : CompleteSpace F'\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \u211d G'\ninst\u271d\u00b2 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx y : G\n\u22a2 \u2191\u2191(condexpIndSMul hm hs h\u03bcs (x + y)) =\u1d50[\u03bc] \u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x + condexpIndL1Fin hm hs h\u03bcs y)", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : CompleteSpace F'\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \u211d G'\ninst\u271d\u00b2 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx y : G\n\u22a2 \u2191\u2191(condexpIndSMul hm hs h\u03bcs (x + y)) =\u1d50[\u03bc] \u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) + \u2191\u2191(condexpIndL1Fin hm hs h\u03bcs y)"}, {"tactic": "refine' EventuallyEq.trans _\n  (EventuallyEq.add (Mem\u2112p.coeFn_toLp q).symm (Mem\u2112p.coeFn_toLp q).symm)", "annotated_tactic": ["refine' <a>EventuallyEq.trans</a> _\n    (<a>EventuallyEq.add</a> (<a>Mem\u2112p.coeFn_toLp</a> <a>q</a>).<a>symm</a> (<a>Mem\u2112p.coeFn_toLp</a> <a>q</a>).<a>symm</a>)", [{"full_name": "Filter.EventuallyEq.trans", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1503, 9], "def_end_pos": [1503, 27]}, {"full_name": "Filter.EventuallyEq.add", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1530, 3], "def_end_pos": [1530, 14]}, {"full_name": "MeasureTheory.Mem\u2112p.coeFn_toLp", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [119, 9], "def_end_pos": [119, 19]}, {"full_name": "_private.Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1.0.MeasureTheory.q", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean", "def_pos": [88, 17], "def_end_pos": [88, 18]}, {"full_name": "Filter.EventuallyEq.symm", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1498, 9], "def_end_pos": [1498, 26]}, {"full_name": "MeasureTheory.Mem\u2112p.coeFn_toLp", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [119, 9], "def_end_pos": [119, 19]}, {"full_name": "_private.Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1.0.MeasureTheory.q", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean", "def_pos": [88, 17], "def_end_pos": [88, 18]}, {"full_name": "Filter.EventuallyEq.symm", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1498, 9], "def_end_pos": [1498, 26]}]], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : CompleteSpace F'\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \u211d G'\ninst\u271d\u00b2 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx y : G\n\u22a2 \u2191\u2191(condexpIndSMul hm hs h\u03bcs (x + y)) =\u1d50[\u03bc] \u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) + \u2191\u2191(condexpIndL1Fin hm hs h\u03bcs y)", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : CompleteSpace F'\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \u211d G'\ninst\u271d\u00b2 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx y : G\n\u22a2 \u2191\u2191(condexpIndSMul hm hs h\u03bcs (x + y)) =\u1d50[\u03bc] fun x_1 =>\n    \u2191\u2191(condexpIndSMul hm hs h\u03bcs x) x_1 + \u2191\u2191(condexpIndSMul hm hs h\u03bcs y) x_1"}, {"tactic": "rw [condexpIndSMul_add]", "annotated_tactic": ["rw [<a>condexpIndSMul_add</a>]", [{"full_name": "MeasureTheory.condexpIndSMul_add", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL2.lean", "def_pos": [402, 9], "def_end_pos": [402, 27]}]], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : CompleteSpace F'\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \u211d G'\ninst\u271d\u00b2 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx y : G\n\u22a2 \u2191\u2191(condexpIndSMul hm hs h\u03bcs (x + y)) =\u1d50[\u03bc] fun x_1 =>\n    \u2191\u2191(condexpIndSMul hm hs h\u03bcs x) x_1 + \u2191\u2191(condexpIndSMul hm hs h\u03bcs y) x_1", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : CompleteSpace F'\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \u211d G'\ninst\u271d\u00b2 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx y : G\n\u22a2 \u2191\u2191(condexpIndSMul hm hs h\u03bcs x + condexpIndSMul hm hs h\u03bcs y) =\u1d50[\u03bc] fun x_1 =>\n    \u2191\u2191(condexpIndSMul hm hs h\u03bcs x) x_1 + \u2191\u2191(condexpIndSMul hm hs h\u03bcs y) x_1"}, {"tactic": "refine' (Lp.coeFn_add _ _).trans (eventually_of_forall fun a => _)", "annotated_tactic": ["refine' (<a>Lp.coeFn_add</a> _ _).<a>trans</a> (<a>eventually_of_forall</a> fun a => _)", [{"full_name": "MeasureTheory.Lp.coeFn_add", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [232, 9], "def_end_pos": [232, 18]}, {"full_name": "Filter.EventuallyEq.trans", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1503, 9], "def_end_pos": [1503, 27]}, {"full_name": "Filter.eventually_of_forall", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1111, 9], "def_end_pos": [1111, 29]}]], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : CompleteSpace F'\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \u211d G'\ninst\u271d\u00b2 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx y : G\n\u22a2 \u2191\u2191(condexpIndSMul hm hs h\u03bcs x + condexpIndSMul hm hs h\u03bcs y) =\u1d50[\u03bc] fun x_1 =>\n    \u2191\u2191(condexpIndSMul hm hs h\u03bcs x) x_1 + \u2191\u2191(condexpIndSMul hm hs h\u03bcs y) x_1", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : CompleteSpace F'\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \u211d G'\ninst\u271d\u00b2 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx y : G\na : \u03b1\n\u22a2 (\u2191\u2191(condexpIndSMul hm hs h\u03bcs x) + \u2191\u2191(condexpIndSMul hm hs h\u03bcs y)) a =\n    (fun x_1 => \u2191\u2191(condexpIndSMul hm hs h\u03bcs x) x_1 + \u2191\u2191(condexpIndSMul hm hs h\u03bcs y) x_1) a"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : CompleteSpace F'\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \u211d G'\ninst\u271d\u00b2 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx y : G\na : \u03b1\n\u22a2 (\u2191\u2191(condexpIndSMul hm hs h\u03bcs x) + \u2191\u2191(condexpIndSMul hm hs h\u03bcs y)) a =\n    (fun x_1 => \u2191\u2191(condexpIndSMul hm hs h\u03bcs x) x_1 + \u2191\u2191(condexpIndSMul hm hs h\u03bcs y) x_1) a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/QPF/Multivariate/Constructions/Cofix.lean", "full_name": "MvQPF.liftR_map_last", "start": [390, 1], "end": [419, 67], "traced_tactics": [{"tactic": "dsimp", "annotated_tactic": ["dsimp", []], "state_before": "n : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\nlawful : LawfulMvFunctor F\n\u03b1 : TypeVec.{u} n\n\u03b9 \u03b9' : Type u\nR : \u03b9' \u2192 \u03b9' \u2192 Prop\nx : F (\u03b1 ::: \u03b9)\nf g : \u03b9 \u2192 \u03b9'\nhh : \u2200 (x : \u03b9), R (f x) (g x)\nh : \u03b9 \u2192 { x // uncurry R x } := fun x => { val := (f x, g x), property := (_ : R (f x) (g x)) }\nb : \u03b1 ::: \u03b9 \u27f9 Subtype_ (repeatEq \u03b1) ::: { x // uncurry R x } := diagSub ::: h\nc : Subtype_ (repeatEq \u03b1) ::: { x // uncurry R x } \u27f9\n  (fun i => { x // ofRepeat (RelLast' \u03b1 R (Fin2.fs i) x) }) ::: Subtype (uncurry R) :=\n  ofSubtype (repeatEq \u03b1) ::: _root_.id\n\u22a2 subtypeVal (RelLast' \u03b1 R) \u229a toSubtype (RelLast' \u03b1 R) \u229a fromAppend1DropLast \u229a c \u229a b =\n    ((TypeVec.id ::: f) \u2297' (TypeVec.id ::: g)) \u229a prod.diag", "state_after": "n : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\nlawful : LawfulMvFunctor F\n\u03b1 : TypeVec.{u} n\n\u03b9 \u03b9' : Type u\nR : \u03b9' \u2192 \u03b9' \u2192 Prop\nx : F (\u03b1 ::: \u03b9)\nf g : \u03b9 \u2192 \u03b9'\nhh : \u2200 (x : \u03b9), R (f x) (g x)\nh : \u03b9 \u2192 { x // uncurry R x } := fun x => { val := (f x, g x), property := (_ : R (f x) (g x)) }\nb : \u03b1 ::: \u03b9 \u27f9 Subtype_ (repeatEq \u03b1) ::: { x // uncurry R x } := diagSub ::: h\nc : Subtype_ (repeatEq \u03b1) ::: { x // uncurry R x } \u27f9\n  (fun i => { x // ofRepeat (RelLast' \u03b1 R (Fin2.fs i) x) }) ::: Subtype (uncurry R) :=\n  ofSubtype (repeatEq \u03b1) ::: _root_.id\n\u22a2 subtypeVal (RelLast' \u03b1 R) \u229a\n      toSubtype (RelLast' \u03b1 R) \u229a\n        fromAppend1DropLast \u229a\n          (ofSubtype (repeatEq \u03b1) ::: _root_.id) \u229a\n            (diagSub ::: fun x => { val := (f x, g x), property := (_ : R (f x) (g x)) }) =\n    ((TypeVec.id ::: f) \u2297' (TypeVec.id ::: g)) \u229a prod.diag"}, {"tactic": "apply eq_of_drop_last_eq", "annotated_tactic": ["apply <a>eq_of_drop_last_eq</a>", [{"full_name": "TypeVec.eq_of_drop_last_eq", "def_path": "Mathlib/Data/TypeVec.lean", "def_pos": [171, 9], "def_end_pos": [171, 27]}]], "state_before": "n : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\nlawful : LawfulMvFunctor F\n\u03b1 : TypeVec.{u} n\n\u03b9 \u03b9' : Type u\nR : \u03b9' \u2192 \u03b9' \u2192 Prop\nx : F (\u03b1 ::: \u03b9)\nf g : \u03b9 \u2192 \u03b9'\nhh : \u2200 (x : \u03b9), R (f x) (g x)\nh : \u03b9 \u2192 { x // uncurry R x } := fun x => { val := (f x, g x), property := (_ : R (f x) (g x)) }\nb : \u03b1 ::: \u03b9 \u27f9 Subtype_ (repeatEq \u03b1) ::: { x // uncurry R x } := diagSub ::: h\nc : Subtype_ (repeatEq \u03b1) ::: { x // uncurry R x } \u27f9\n  (fun i => { x // ofRepeat (RelLast' \u03b1 R (Fin2.fs i) x) }) ::: Subtype (uncurry R) :=\n  ofSubtype (repeatEq \u03b1) ::: _root_.id\n\u22a2 subtypeVal (RelLast' \u03b1 R) \u229a\n      toSubtype (RelLast' \u03b1 R) \u229a\n        fromAppend1DropLast \u229a\n          (ofSubtype (repeatEq \u03b1) ::: _root_.id) \u229a\n            (diagSub ::: fun x => { val := (f x, g x), property := (_ : R (f x) (g x)) }) =\n    ((TypeVec.id ::: f) \u2297' (TypeVec.id ::: g)) \u229a prod.diag", "state_after": "case h\u2080\nn : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\nlawful : LawfulMvFunctor F\n\u03b1 : TypeVec.{u} n\n\u03b9 \u03b9' : Type u\nR : \u03b9' \u2192 \u03b9' \u2192 Prop\nx : F (\u03b1 ::: \u03b9)\nf g : \u03b9 \u2192 \u03b9'\nhh : \u2200 (x : \u03b9), R (f x) (g x)\nh : \u03b9 \u2192 { x // uncurry R x } := fun x => { val := (f x, g x), property := (_ : R (f x) (g x)) }\nb : \u03b1 ::: \u03b9 \u27f9 Subtype_ (repeatEq \u03b1) ::: { x // uncurry R x } := diagSub ::: h\nc : Subtype_ (repeatEq \u03b1) ::: { x // uncurry R x } \u27f9\n  (fun i => { x // ofRepeat (RelLast' \u03b1 R (Fin2.fs i) x) }) ::: Subtype (uncurry R) :=\n  ofSubtype (repeatEq \u03b1) ::: _root_.id\n\u22a2 dropFun\n      (subtypeVal (RelLast' \u03b1 R) \u229a\n        toSubtype (RelLast' \u03b1 R) \u229a\n          fromAppend1DropLast \u229a\n            (ofSubtype (repeatEq \u03b1) ::: _root_.id) \u229a\n              (diagSub ::: fun x => { val := (f x, g x), property := (_ : R (f x) (g x)) })) =\n    dropFun (((TypeVec.id ::: f) \u2297' (TypeVec.id ::: g)) \u229a prod.diag)\n\ncase h\u2081\nn : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\nlawful : LawfulMvFunctor F\n\u03b1 : TypeVec.{u} n\n\u03b9 \u03b9' : Type u\nR : \u03b9' \u2192 \u03b9' \u2192 Prop\nx : F (\u03b1 ::: \u03b9)\nf g : \u03b9 \u2192 \u03b9'\nhh : \u2200 (x : \u03b9), R (f x) (g x)\nh : \u03b9 \u2192 { x // uncurry R x } := fun x => { val := (f x, g x), property := (_ : R (f x) (g x)) }\nb : \u03b1 ::: \u03b9 \u27f9 Subtype_ (repeatEq \u03b1) ::: { x // uncurry R x } := diagSub ::: h\nc : Subtype_ (repeatEq \u03b1) ::: { x // uncurry R x } \u27f9\n  (fun i => { x // ofRepeat (RelLast' \u03b1 R (Fin2.fs i) x) }) ::: Subtype (uncurry R) :=\n  ofSubtype (repeatEq \u03b1) ::: _root_.id\n\u22a2 lastFun\n      (subtypeVal (RelLast' \u03b1 R) \u229a\n        toSubtype (RelLast' \u03b1 R) \u229a\n          fromAppend1DropLast \u229a\n            (ofSubtype (repeatEq \u03b1) ::: _root_.id) \u229a\n              (diagSub ::: fun x => { val := (f x, g x), property := (_ : R (f x) (g x)) })) =\n    lastFun (((TypeVec.id ::: f) \u2297' (TypeVec.id ::: g)) \u229a prod.diag)"}, {"tactic": "simp only [lastFun_from_append1_drop_last, lastFun_toSubtype, lastFun_appendFun,\n  lastFun_subtypeVal, comp.left_id, lastFun_comp, lastFun_prod]", "annotated_tactic": ["simp only [<a>lastFun_from_append1_drop_last</a>, <a>lastFun_toSubtype</a>, <a>lastFun_appendFun</a>,\n      <a>lastFun_subtypeVal</a>, <a>comp.left_id</a>, <a>lastFun_comp</a>, <a>lastFun_prod</a>]", [{"full_name": "TypeVec.lastFun_from_append1_drop_last", "def_path": "Mathlib/Data/TypeVec.lean", "def_pos": [751, 9], "def_end_pos": [751, 39]}, {"full_name": "TypeVec.lastFun_toSubtype", "def_path": "Mathlib/Data/TypeVec.lean", "def_pos": [700, 9], "def_end_pos": [700, 26]}, {"full_name": "TypeVec.lastFun_appendFun", "def_path": "Mathlib/Data/TypeVec.lean", "def_pos": [219, 9], "def_end_pos": [219, 26]}, {"full_name": "TypeVec.lastFun_subtypeVal", "def_path": "Mathlib/Data/TypeVec.lean", "def_pos": [687, 9], "def_end_pos": [687, 27]}, {"full_name": "Function.comp.left_id", "def_path": "Mathlib/Init/Function.lean", "def_pos": [95, 9], "def_end_pos": [95, 21]}, {"full_name": "TypeVec.lastFun_comp", "def_path": "Mathlib/Data/TypeVec.lean", "def_pos": [282, 9], "def_end_pos": [282, 21]}, {"full_name": "TypeVec.lastFun_prod", "def_path": "Mathlib/Data/TypeVec.lean", "def_pos": [738, 9], "def_end_pos": [738, 21]}]], "state_before": "case h\u2081\nn : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\nlawful : LawfulMvFunctor F\n\u03b1 : TypeVec.{u} n\n\u03b9 \u03b9' : Type u\nR : \u03b9' \u2192 \u03b9' \u2192 Prop\nx : F (\u03b1 ::: \u03b9)\nf g : \u03b9 \u2192 \u03b9'\nhh : \u2200 (x : \u03b9), R (f x) (g x)\nh : \u03b9 \u2192 { x // uncurry R x } := fun x => { val := (f x, g x), property := (_ : R (f x) (g x)) }\nb : \u03b1 ::: \u03b9 \u27f9 Subtype_ (repeatEq \u03b1) ::: { x // uncurry R x } := diagSub ::: h\nc : Subtype_ (repeatEq \u03b1) ::: { x // uncurry R x } \u27f9\n  (fun i => { x // ofRepeat (RelLast' \u03b1 R (Fin2.fs i) x) }) ::: Subtype (uncurry R) :=\n  ofSubtype (repeatEq \u03b1) ::: _root_.id\n\u22a2 lastFun\n      (subtypeVal (RelLast' \u03b1 R) \u229a\n        toSubtype (RelLast' \u03b1 R) \u229a\n          fromAppend1DropLast \u229a\n            (ofSubtype (repeatEq \u03b1) ::: _root_.id) \u229a\n              (diagSub ::: fun x => { val := (f x, g x), property := (_ : R (f x) (g x)) })) =\n    lastFun (((TypeVec.id ::: f) \u2297' (TypeVec.id ::: g)) \u229a prod.diag)", "state_after": "case h\u2081\nn : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\nlawful : LawfulMvFunctor F\n\u03b1 : TypeVec.{u} n\n\u03b9 \u03b9' : Type u\nR : \u03b9' \u2192 \u03b9' \u2192 Prop\nx : F (\u03b1 ::: \u03b9)\nf g : \u03b9 \u2192 \u03b9'\nhh : \u2200 (x : \u03b9), R (f x) (g x)\nh : \u03b9 \u2192 { x // uncurry R x } := fun x => { val := (f x, g x), property := (_ : R (f x) (g x)) }\nb : \u03b1 ::: \u03b9 \u27f9 Subtype_ (repeatEq \u03b1) ::: { x // uncurry R x } := diagSub ::: h\nc : Subtype_ (repeatEq \u03b1) ::: { x // uncurry R x } \u27f9\n  (fun i => { x // ofRepeat (RelLast' \u03b1 R (Fin2.fs i) x) }) ::: Subtype (uncurry R) :=\n  ofSubtype (repeatEq \u03b1) ::: _root_.id\n\u22a2 (Subtype.val \u2218 fun x => { val := (f x, g x), property := (_ : R (f x) (g x)) }) = Prod.map f g \u2218 lastFun prod.diag"}, {"tactic": "ext1", "annotated_tactic": ["ext1", []], "state_before": "case h\u2081\nn : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\nlawful : LawfulMvFunctor F\n\u03b1 : TypeVec.{u} n\n\u03b9 \u03b9' : Type u\nR : \u03b9' \u2192 \u03b9' \u2192 Prop\nx : F (\u03b1 ::: \u03b9)\nf g : \u03b9 \u2192 \u03b9'\nhh : \u2200 (x : \u03b9), R (f x) (g x)\nh : \u03b9 \u2192 { x // uncurry R x } := fun x => { val := (f x, g x), property := (_ : R (f x) (g x)) }\nb : \u03b1 ::: \u03b9 \u27f9 Subtype_ (repeatEq \u03b1) ::: { x // uncurry R x } := diagSub ::: h\nc : Subtype_ (repeatEq \u03b1) ::: { x // uncurry R x } \u27f9\n  (fun i => { x // ofRepeat (RelLast' \u03b1 R (Fin2.fs i) x) }) ::: Subtype (uncurry R) :=\n  ofSubtype (repeatEq \u03b1) ::: _root_.id\n\u22a2 (Subtype.val \u2218 fun x => { val := (f x, g x), property := (_ : R (f x) (g x)) }) = Prod.map f g \u2218 lastFun prod.diag", "state_after": "case h\u2081.h\nn : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\nlawful : LawfulMvFunctor F\n\u03b1 : TypeVec.{u} n\n\u03b9 \u03b9' : Type u\nR : \u03b9' \u2192 \u03b9' \u2192 Prop\nx : F (\u03b1 ::: \u03b9)\nf g : \u03b9 \u2192 \u03b9'\nhh : \u2200 (x : \u03b9), R (f x) (g x)\nh : \u03b9 \u2192 { x // uncurry R x } := fun x => { val := (f x, g x), property := (_ : R (f x) (g x)) }\nb : \u03b1 ::: \u03b9 \u27f9 Subtype_ (repeatEq \u03b1) ::: { x // uncurry R x } := diagSub ::: h\nc : Subtype_ (repeatEq \u03b1) ::: { x // uncurry R x } \u27f9\n  (fun i => { x // ofRepeat (RelLast' \u03b1 R (Fin2.fs i) x) }) ::: Subtype (uncurry R) :=\n  ofSubtype (repeatEq \u03b1) ::: _root_.id\nx\u271d : TypeVec.last (\u03b1 ::: \u03b9)\n\u22a2 (Subtype.val \u2218 fun x => { val := (f x, g x), property := (_ : R (f x) (g x)) }) x\u271d =\n    (Prod.map f g \u2218 lastFun prod.diag) x\u271d"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case h\u2081.h\nn : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\nlawful : LawfulMvFunctor F\n\u03b1 : TypeVec.{u} n\n\u03b9 \u03b9' : Type u\nR : \u03b9' \u2192 \u03b9' \u2192 Prop\nx : F (\u03b1 ::: \u03b9)\nf g : \u03b9 \u2192 \u03b9'\nhh : \u2200 (x : \u03b9), R (f x) (g x)\nh : \u03b9 \u2192 { x // uncurry R x } := fun x => { val := (f x, g x), property := (_ : R (f x) (g x)) }\nb : \u03b1 ::: \u03b9 \u27f9 Subtype_ (repeatEq \u03b1) ::: { x // uncurry R x } := diagSub ::: h\nc : Subtype_ (repeatEq \u03b1) ::: { x // uncurry R x } \u27f9\n  (fun i => { x // ofRepeat (RelLast' \u03b1 R (Fin2.fs i) x) }) ::: Subtype (uncurry R) :=\n  ofSubtype (repeatEq \u03b1) ::: _root_.id\nx\u271d : TypeVec.last (\u03b1 ::: \u03b9)\n\u22a2 (Subtype.val \u2218 fun x => { val := (f x, g x), property := (_ : R (f x) (g x)) }) x\u271d =\n    (Prod.map f g \u2218 lastFun prod.diag) x\u271d", "state_after": "no goals"}, {"tactic": "dsimp", "annotated_tactic": ["dsimp", []], "state_before": "case h\u2080\nn : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\nlawful : LawfulMvFunctor F\n\u03b1 : TypeVec.{u} n\n\u03b9 \u03b9' : Type u\nR : \u03b9' \u2192 \u03b9' \u2192 Prop\nx : F (\u03b1 ::: \u03b9)\nf g : \u03b9 \u2192 \u03b9'\nhh : \u2200 (x : \u03b9), R (f x) (g x)\nh : \u03b9 \u2192 { x // uncurry R x } := fun x => { val := (f x, g x), property := (_ : R (f x) (g x)) }\nb : \u03b1 ::: \u03b9 \u27f9 Subtype_ (repeatEq \u03b1) ::: { x // uncurry R x } := diagSub ::: h\nc : Subtype_ (repeatEq \u03b1) ::: { x // uncurry R x } \u27f9\n  (fun i => { x // ofRepeat (RelLast' \u03b1 R (Fin2.fs i) x) }) ::: Subtype (uncurry R) :=\n  ofSubtype (repeatEq \u03b1) ::: _root_.id\n\u22a2 dropFun\n      (subtypeVal (RelLast' \u03b1 R) \u229a\n        toSubtype (RelLast' \u03b1 R) \u229a\n          fromAppend1DropLast \u229a\n            (ofSubtype (repeatEq \u03b1) ::: _root_.id) \u229a\n              (diagSub ::: fun x => { val := (f x, g x), property := (_ : R (f x) (g x)) })) =\n    dropFun (((TypeVec.id ::: f) \u2297' (TypeVec.id ::: g)) \u229a prod.diag)", "state_after": "case h\u2080\nn : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\nlawful : LawfulMvFunctor F\n\u03b1 : TypeVec.{u} n\n\u03b9 \u03b9' : Type u\nR : \u03b9' \u2192 \u03b9' \u2192 Prop\nx : F (\u03b1 ::: \u03b9)\nf g : \u03b9 \u2192 \u03b9'\nhh : \u2200 (x : \u03b9), R (f x) (g x)\nh : \u03b9 \u2192 { x // uncurry R x } := fun x => { val := (f x, g x), property := (_ : R (f x) (g x)) }\nb : \u03b1 ::: \u03b9 \u27f9 Subtype_ (repeatEq \u03b1) ::: { x // uncurry R x } := diagSub ::: h\nc : Subtype_ (repeatEq \u03b1) ::: { x // uncurry R x } \u27f9\n  (fun i => { x // ofRepeat (RelLast' \u03b1 R (Fin2.fs i) x) }) ::: Subtype (uncurry R) :=\n  ofSubtype (repeatEq \u03b1) ::: _root_.id\n\u22a2 subtypeVal (repeatEq \u03b1) \u229a dropFun (toSubtype (RelLast' \u03b1 R)) \u229a ofSubtype (repeatEq \u03b1) \u229a diagSub =\n    dropFun ((TypeVec.id ::: f) \u2297' (TypeVec.id ::: g)) \u229a dropFun prod.diag"}, {"tactic": "simp only [prod_map_id, dropFun_prod, dropFun_appendFun, dropFun_diag, id_comp,\n  dropFun_toSubtype]", "annotated_tactic": ["simp only [<a>prod_map_id</a>, <a>dropFun_prod</a>, <a>dropFun_appendFun</a>, <a>dropFun_diag</a>, <a>id_comp</a>,\n        <a>dropFun_toSubtype</a>]", [{"full_name": "TypeVec.prod_map_id", "def_path": "Mathlib/Data/TypeVec.lean", "def_pos": [762, 9], "def_end_pos": [762, 20]}, {"full_name": "TypeVec.dropFun_prod", "def_path": "Mathlib/Data/TypeVec.lean", "def_pos": [731, 9], "def_end_pos": [731, 21]}, {"full_name": "TypeVec.dropFun_appendFun", "def_path": "Mathlib/Data/TypeVec.lean", "def_pos": [213, 9], "def_end_pos": [213, 26]}, {"full_name": "TypeVec.dropFun_diag", "def_path": "Mathlib/Data/TypeVec.lean", "def_pos": [675, 9], "def_end_pos": [675, 21]}, {"full_name": "TypeVec.id_comp", "def_path": "Mathlib/Data/TypeVec.lean", "def_pos": [79, 9], "def_end_pos": [79, 16]}, {"full_name": "TypeVec.dropFun_toSubtype", "def_path": "Mathlib/Data/TypeVec.lean", "def_pos": [693, 9], "def_end_pos": [693, 26]}]], "state_before": "case h\u2080\nn : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\nlawful : LawfulMvFunctor F\n\u03b1 : TypeVec.{u} n\n\u03b9 \u03b9' : Type u\nR : \u03b9' \u2192 \u03b9' \u2192 Prop\nx : F (\u03b1 ::: \u03b9)\nf g : \u03b9 \u2192 \u03b9'\nhh : \u2200 (x : \u03b9), R (f x) (g x)\nh : \u03b9 \u2192 { x // uncurry R x } := fun x => { val := (f x, g x), property := (_ : R (f x) (g x)) }\nb : \u03b1 ::: \u03b9 \u27f9 Subtype_ (repeatEq \u03b1) ::: { x // uncurry R x } := diagSub ::: h\nc : Subtype_ (repeatEq \u03b1) ::: { x // uncurry R x } \u27f9\n  (fun i => { x // ofRepeat (RelLast' \u03b1 R (Fin2.fs i) x) }) ::: Subtype (uncurry R) :=\n  ofSubtype (repeatEq \u03b1) ::: _root_.id\n\u22a2 subtypeVal (repeatEq \u03b1) \u229a dropFun (toSubtype (RelLast' \u03b1 R)) \u229a ofSubtype (repeatEq \u03b1) \u229a diagSub =\n    dropFun ((TypeVec.id ::: f) \u2297' (TypeVec.id ::: g)) \u229a dropFun prod.diag", "state_after": "case h\u2080\nn : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\nlawful : LawfulMvFunctor F\n\u03b1 : TypeVec.{u} n\n\u03b9 \u03b9' : Type u\nR : \u03b9' \u2192 \u03b9' \u2192 Prop\nx : F (\u03b1 ::: \u03b9)\nf g : \u03b9 \u2192 \u03b9'\nhh : \u2200 (x : \u03b9), R (f x) (g x)\nh : \u03b9 \u2192 { x // uncurry R x } := fun x => { val := (f x, g x), property := (_ : R (f x) (g x)) }\nb : \u03b1 ::: \u03b9 \u27f9 Subtype_ (repeatEq \u03b1) ::: { x // uncurry R x } := diagSub ::: h\nc : Subtype_ (repeatEq \u03b1) ::: { x // uncurry R x } \u27f9\n  (fun i => { x // ofRepeat (RelLast' \u03b1 R (Fin2.fs i) x) }) ::: Subtype (uncurry R) :=\n  ofSubtype (repeatEq \u03b1) ::: _root_.id\n\u22a2 subtypeVal (repeatEq \u03b1) \u229a (toSubtype fun i x => RelLast' \u03b1 R (Fin2.fs i) x) \u229a ofSubtype (repeatEq \u03b1) \u229a diagSub =\n    prod.diag"}, {"tactic": "erw [toSubtype_of_subtype_assoc, id_comp]", "annotated_tactic": ["erw [<a>toSubtype_of_subtype_assoc</a>, <a>id_comp</a>]", [{"full_name": "TypeVec.toSubtype_of_subtype_assoc", "def_path": "Mathlib/Data/TypeVec.lean", "def_pos": [797, 9], "def_end_pos": [797, 35]}, {"full_name": "TypeVec.id_comp", "def_path": "Mathlib/Data/TypeVec.lean", "def_pos": [79, 9], "def_end_pos": [79, 16]}]], "state_before": "case h\u2080\nn : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\nlawful : LawfulMvFunctor F\n\u03b1 : TypeVec.{u} n\n\u03b9 \u03b9' : Type u\nR : \u03b9' \u2192 \u03b9' \u2192 Prop\nx : F (\u03b1 ::: \u03b9)\nf g : \u03b9 \u2192 \u03b9'\nhh : \u2200 (x : \u03b9), R (f x) (g x)\nh : \u03b9 \u2192 { x // uncurry R x } := fun x => { val := (f x, g x), property := (_ : R (f x) (g x)) }\nb : \u03b1 ::: \u03b9 \u27f9 Subtype_ (repeatEq \u03b1) ::: { x // uncurry R x } := diagSub ::: h\nc : Subtype_ (repeatEq \u03b1) ::: { x // uncurry R x } \u27f9\n  (fun i => { x // ofRepeat (RelLast' \u03b1 R (Fin2.fs i) x) }) ::: Subtype (uncurry R) :=\n  ofSubtype (repeatEq \u03b1) ::: _root_.id\n\u22a2 subtypeVal (repeatEq \u03b1) \u229a (toSubtype fun i x => RelLast' \u03b1 R (Fin2.fs i) x) \u229a ofSubtype (repeatEq \u03b1) \u229a diagSub =\n    prod.diag", "state_after": "case h\u2080\nn : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\nlawful : LawfulMvFunctor F\n\u03b1 : TypeVec.{u} n\n\u03b9 \u03b9' : Type u\nR : \u03b9' \u2192 \u03b9' \u2192 Prop\nx : F (\u03b1 ::: \u03b9)\nf g : \u03b9 \u2192 \u03b9'\nhh : \u2200 (x : \u03b9), R (f x) (g x)\nh : \u03b9 \u2192 { x // uncurry R x } := fun x => { val := (f x, g x), property := (_ : R (f x) (g x)) }\nb : \u03b1 ::: \u03b9 \u27f9 Subtype_ (repeatEq \u03b1) ::: { x // uncurry R x } := diagSub ::: h\nc : Subtype_ (repeatEq \u03b1) ::: { x // uncurry R x } \u27f9\n  (fun i => { x // ofRepeat (RelLast' \u03b1 R (Fin2.fs i) x) }) ::: Subtype (uncurry R) :=\n  ofSubtype (repeatEq \u03b1) ::: _root_.id\n\u22a2 (fun i x => subtypeVal (repeatEq \u03b1) i (diagSub i x)) = prod.diag"}, {"tactic": "clear liftR_map_last q mvf lawful F x R f g hh h b c", "annotated_tactic": ["clear liftR_map_last q mvf lawful F x R f g hh h b c", []], "state_before": "case h\u2080\nn : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\nlawful : LawfulMvFunctor F\n\u03b1 : TypeVec.{u} n\n\u03b9 \u03b9' : Type u\nR : \u03b9' \u2192 \u03b9' \u2192 Prop\nx : F (\u03b1 ::: \u03b9)\nf g : \u03b9 \u2192 \u03b9'\nhh : \u2200 (x : \u03b9), R (f x) (g x)\nh : \u03b9 \u2192 { x // uncurry R x } := fun x => { val := (f x, g x), property := (_ : R (f x) (g x)) }\nb : \u03b1 ::: \u03b9 \u27f9 Subtype_ (repeatEq \u03b1) ::: { x // uncurry R x } := diagSub ::: h\nc : Subtype_ (repeatEq \u03b1) ::: { x // uncurry R x } \u27f9\n  (fun i => { x // ofRepeat (RelLast' \u03b1 R (Fin2.fs i) x) }) ::: Subtype (uncurry R) :=\n  ofSubtype (repeatEq \u03b1) ::: _root_.id\n\u22a2 (fun i x => subtypeVal (repeatEq \u03b1) i (diagSub i x)) = prod.diag", "state_after": "case h\u2080\nn : \u2115\n\u03b1 : TypeVec.{u} n\n\u03b9 \u03b9' : Type u\n\u22a2 (fun i x => subtypeVal (repeatEq \u03b1) i (diagSub i x)) = prod.diag"}, {"tactic": "ext (i x) : 2", "annotated_tactic": ["ext (i x) : 2", []], "state_before": "case h\u2080\nn : \u2115\n\u03b1 : TypeVec.{u} n\n\u03b9 \u03b9' : Type u\n\u22a2 (fun i x => subtypeVal (repeatEq \u03b1) i (diagSub i x)) = prod.diag", "state_after": "case h\u2080.a.h\nn : \u2115\n\u03b1 : TypeVec.{u} n\n\u03b9 \u03b9' : Type u\ni : Fin2 n\nx : TypeVec.drop (\u03b1 ::: \u03b9) i\n\u22a2 subtypeVal (repeatEq \u03b1) i (diagSub i x) = prod.diag i x"}, {"tactic": "induction i with\n| fz => rfl\n| fs _ ih =>\n  apply ih", "annotated_tactic": ["induction i with\n      | <a>fz</a> => rfl\n      | <a>fs</a> _ ih =>\n        apply ih", [{"full_name": "Fin2.fz", "def_path": "Mathlib/Data/Fin/Fin2.lean", "def_pos": [40, 5], "def_end_pos": [40, 7]}, {"full_name": "Fin2.fs", "def_path": "Mathlib/Data/Fin/Fin2.lean", "def_pos": [42, 5], "def_end_pos": [42, 7]}]], "state_before": "case h\u2080.a.h\nn : \u2115\n\u03b1 : TypeVec.{u} n\n\u03b9 \u03b9' : Type u\ni : Fin2 n\nx : TypeVec.drop (\u03b1 ::: \u03b9) i\n\u22a2 subtypeVal (repeatEq \u03b1) i (diagSub i x) = prod.diag i x", "state_after": "no goals"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case h\u2080.a.h.fz\nn : \u2115\n\u03b9 \u03b9' : Type u\nn\u271d : \u2115\n\u03b1 : TypeVec.{u} (Nat.succ n\u271d)\nx : TypeVec.drop (\u03b1 ::: \u03b9) Fin2.fz\n\u22a2 subtypeVal (repeatEq \u03b1) Fin2.fz (diagSub Fin2.fz x) = prod.diag Fin2.fz x", "state_after": "no goals"}, {"tactic": "apply ih", "annotated_tactic": ["apply ih", []], "state_before": "case h\u2080.a.h.fs\nn : \u2115\n\u03b9 \u03b9' : Type u\nn\u271d : \u2115\na\u271d : Fin2 n\u271d\nih : \u2200 {\u03b1 : TypeVec.{u} n\u271d} (x : TypeVec.drop (\u03b1 ::: \u03b9) a\u271d), subtypeVal (repeatEq \u03b1) a\u271d (diagSub a\u271d x) = prod.diag a\u271d x\n\u03b1 : TypeVec.{u} (Nat.succ n\u271d)\nx : TypeVec.drop (\u03b1 ::: \u03b9) (Fin2.fs a\u271d)\n\u22a2 subtypeVal (repeatEq \u03b1) (Fin2.fs a\u271d) (diagSub (Fin2.fs a\u271d) x) = prod.diag (Fin2.fs a\u271d) x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Intervals/WithBotTop.lean", "full_name": "WithBot.image_coe_Ico", "start": [226, 1], "end": [229, 78], "traced_tactics": [{"tactic": "rw [\u2190 preimage_coe_Ico, image_preimage_eq_inter_range, range_coe,\n  inter_eq_self_of_subset_left\n    (Subset.trans Ico_subset_Ici_self <| Ici_subset_Ioi.2 <| bot_lt_coe a)]", "annotated_tactic": ["rw [\u2190 <a>preimage_coe_Ico</a>, <a>image_preimage_eq_inter_range</a>, <a>range_coe</a>,\n    <a>inter_eq_self_of_subset_left</a>\n      (<a>Subset.trans</a> <a>Ico_subset_Ici_self</a> <| <a>Ici_subset_Ioi</a>.2 <| <a>bot_lt_coe</a> a)]", [{"full_name": "WithBot.preimage_coe_Ico", "def_path": "Mathlib/Data/Set/Intervals/WithBotTop.lean", "def_pos": [171, 9], "def_end_pos": [171, 25]}, {"full_name": "Set.image_preimage_eq_inter_range", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [796, 9], "def_end_pos": [796, 38]}, {"full_name": "WithBot.range_coe", "def_path": "Mathlib/Data/Set/Intervals/WithBotTop.lean", "def_pos": [142, 9], "def_end_pos": [142, 18]}, {"full_name": "Set.inter_eq_self_of_subset_left", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [987, 9], "def_end_pos": [987, 37]}, {"full_name": "Set.Subset.trans", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [362, 9], "def_end_pos": [362, 21]}, {"full_name": "Set.Ico_subset_Ici_self", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [538, 9], "def_end_pos": [538, 28]}, {"full_name": "Set.Ici_subset_Ioi", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [423, 9], "def_end_pos": [423, 23]}, {"full_name": "WithBot.bot_lt_coe", "def_path": "Mathlib/Order/WithBot.lean", "def_pos": [289, 9], "def_end_pos": [289, 19]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : PartialOrder \u03b1\na b : \u03b1\n\u22a2 some '' Ico a b = Ico \u2191a \u2191b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/ZMod/Quotient.lean", "full_name": "order_eq_card_zpowers'", "start": [208, 1], "end": [210, 67], "traced_tactics": [{"tactic": "have := Nat.card_congr (MulAction.orbitZpowersEquiv a (1 : \u03b1))", "annotated_tactic": ["have := <a>Nat.card_congr</a> (<a>MulAction.orbitZpowersEquiv</a> a (1 : \u03b1))", [{"full_name": "Nat.card_congr", "def_path": "Mathlib/SetTheory/Cardinal/Finite.lean", "def_pos": [55, 9], "def_end_pos": [55, 19]}, {"full_name": "MulAction.orbitZpowersEquiv", "def_path": "Mathlib/Data/ZMod/Quotient.lean", "def_pos": [138, 19], "def_end_pos": [138, 36]}]], "state_before": "n : \u2115\nA : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : AddGroup A\ninst\u271d\u00b9 : Ring R\n\u03b1 : Type u_3\ninst\u271d : Group \u03b1\na : \u03b1\n\u22a2 orderOf a = Nat.card { x // x \u2208 zpowers a }", "state_after": "n : \u2115\nA : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : AddGroup A\ninst\u271d\u00b9 : Ring R\n\u03b1 : Type u_3\ninst\u271d : Group \u03b1\na : \u03b1\nthis :\n  Nat.card \u2191(MulAction.orbit { x // x \u2208 zpowers a } 1) =\n    Nat.card (ZMod (Function.minimalPeriod ((fun x x_1 => x \u2022 x_1) a) 1))\n\u22a2 orderOf a = Nat.card { x // x \u2208 zpowers a }"}, {"tactic": "rwa [Nat.card_zmod, orbit_subgroup_one_eq_self, eq_comm] at this", "annotated_tactic": ["rwa [<a>Nat.card_zmod</a>, <a>orbit_subgroup_one_eq_self</a>, <a>eq_comm</a>] at this", [{"full_name": "Nat.card_zmod", "def_path": "Mathlib/SetTheory/Cardinal/Finite.lean", "def_pos": [131, 9], "def_end_pos": [131, 18]}, {"full_name": "orbit_subgroup_one_eq_self", "def_path": "Mathlib/GroupTheory/Coset.lean", "def_pos": [244, 9], "def_end_pos": [244, 35]}, {"full_name": "eq_comm", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [104, 9], "def_end_pos": [104, 16]}]], "state_before": "n : \u2115\nA : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : AddGroup A\ninst\u271d\u00b9 : Ring R\n\u03b1 : Type u_3\ninst\u271d : Group \u03b1\na : \u03b1\nthis :\n  Nat.card \u2191(MulAction.orbit { x // x \u2208 zpowers a } 1) =\n    Nat.card (ZMod (Function.minimalPeriod ((fun x x_1 => x \u2022 x_1) a) 1))\n\u22a2 orderOf a = Nat.card { x // x \u2208 zpowers a }", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Int/Basic.lean", "full_name": "Int.pred_neg_pred", "start": [177, 1], "end": [177, 83], "traced_tactics": [{"tactic": "rw [neg_pred, pred_succ]", "annotated_tactic": ["rw [<a>neg_pred</a>, <a>pred_succ</a>]", [{"full_name": "Int.neg_pred", "def_path": "Mathlib/Data/Int/Basic.lean", "def_pos": [173, 9], "def_end_pos": [173, 17]}, {"full_name": "Int.pred_succ", "def_path": "Mathlib/Data/Int/Basic.lean", "def_pos": [161, 9], "def_end_pos": [161, 18]}]], "state_before": "a : \u2124\n\u22a2 pred (-pred a) = -a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Card.lean", "full_name": "Set.ncard_inter_add_ncard_union", "start": [836, 1], "end": [838, 55], "traced_tactics": [{"tactic": "rw [add_comm, ncard_union_add_ncard_inter _ _ hs ht]", "annotated_tactic": ["rw [<a>add_comm</a>, <a>ncard_union_add_ncard_inter</a> _ _ hs ht]", [{"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [301, 3], "def_end_pos": [301, 14]}, {"full_name": "Set.ncard_union_add_ncard_inter", "def_path": "Mathlib/Data/Set/Card.lean", "def_pos": [830, 9], "def_end_pos": [830, 36]}]], "state_before": "\u03b1 : Type u_1\ns\u271d t\u271d s t : Set \u03b1\nhs : autoParam (Set.Finite s) _auto\u271d\nht : autoParam (Set.Finite t) _auto\u271d\n\u22a2 ncard (s \u2229 t) + ncard (s \u222a t) = ncard s + ncard t", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Intervals/Pi.lean", "full_name": "Set.image_mulSingle_Ioc_left", "start": [237, 1], "end": [239, 30], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Reduce.lean", "full_name": "ComputablePred.computable_of_oneOneReducible", "start": [142, 1], "end": [144, 47], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Analysis/Topology.lean", "full_name": "Ctop.Realizer.ext", "start": [168, 1], "end": [170, 79], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Primrec.lean", "full_name": "Primrec.nat_sub", "start": [670, 1], "end": [671, 39], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/MulAntidiagonal.lean", "full_name": "Set.MulAntidiagonal.eq_of_fst_le_fst_of_snd_le_snd", "start": [107, 1], "end": [112, 83], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "full_name": "measurable_of_restrict_of_restrict_compl", "start": [654, 1], "end": [656, 86], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Intervals/Pi.lean", "full_name": "Set.piecewise_mem_Icc", "start": [47, 1], "end": [51, 66], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Independence/ZeroOne.lean", "full_name": "ProbabilityTheory.indep_iSup_limsup", "start": [97, 1], "end": [107, 21], "traced_tactics": [{"tactic": "suffices (\u2a06 a, \u2a06 n \u2208 ns a, s n) = \u2a06 n, s n by\n  rw [\u2190 this]\n  exact indep_iSup_directed_limsup h_le h_indep hf hns hnsp", "annotated_tactic": ["suffices (\u2a06 a, \u2a06 n \u2208 ns a, s n) = \u2a06 n, s n by\n    rw [\u2190 this]\n    exact <a>indep_iSup_directed_limsup</a> h_le h_indep hf hns hnsp", [{"full_name": "ProbabilityTheory.indep_iSup_directed_limsup", "def_path": "Mathlib/Probability/Independence/ZeroOne.lean", "def_pos": [83, 9], "def_end_pos": [83, 35]}]], "state_before": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm m0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d : IsProbabilityMeasure \u03bc\ns : \u03b9 \u2192 MeasurableSpace \u03a9\n\u03b1 : Type u_3\np : Set \u03b9 \u2192 Prop\nf : Filter \u03b9\nns : \u03b1 \u2192 Set \u03b9\nh_le : \u2200 (n : \u03b9), s n \u2264 m0\nh_indep : iIndep s\nhf : \u2200 (t : Set \u03b9), p t \u2192 t\u1d9c \u2208 f\nhns : Directed (fun x x_1 => x \u2264 x_1) ns\nhnsp : \u2200 (a : \u03b1), p (ns a)\nhns_univ : \u2200 (n : \u03b9), \u2203 a, n \u2208 ns a\n\u22a2 Indep (\u2a06 n, s n) (limsup s f)", "state_after": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm m0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d : IsProbabilityMeasure \u03bc\ns : \u03b9 \u2192 MeasurableSpace \u03a9\n\u03b1 : Type u_3\np : Set \u03b9 \u2192 Prop\nf : Filter \u03b9\nns : \u03b1 \u2192 Set \u03b9\nh_le : \u2200 (n : \u03b9), s n \u2264 m0\nh_indep : iIndep s\nhf : \u2200 (t : Set \u03b9), p t \u2192 t\u1d9c \u2208 f\nhns : Directed (fun x x_1 => x \u2264 x_1) ns\nhnsp : \u2200 (a : \u03b1), p (ns a)\nhns_univ : \u2200 (n : \u03b9), \u2203 a, n \u2208 ns a\n\u22a2 \u2a06 a, \u2a06 n \u2208 ns a, s n = \u2a06 n, s n"}, {"tactic": "rw [iSup_comm]", "annotated_tactic": ["rw [<a>iSup_comm</a>]", [{"full_name": "iSup_comm", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [1196, 9], "def_end_pos": [1196, 18]}]], "state_before": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm m0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d : IsProbabilityMeasure \u03bc\ns : \u03b9 \u2192 MeasurableSpace \u03a9\n\u03b1 : Type u_3\np : Set \u03b9 \u2192 Prop\nf : Filter \u03b9\nns : \u03b1 \u2192 Set \u03b9\nh_le : \u2200 (n : \u03b9), s n \u2264 m0\nh_indep : iIndep s\nhf : \u2200 (t : Set \u03b9), p t \u2192 t\u1d9c \u2208 f\nhns : Directed (fun x x_1 => x \u2264 x_1) ns\nhnsp : \u2200 (a : \u03b1), p (ns a)\nhns_univ : \u2200 (n : \u03b9), \u2203 a, n \u2208 ns a\n\u22a2 \u2a06 a, \u2a06 n \u2208 ns a, s n = \u2a06 n, s n", "state_after": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm m0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d : IsProbabilityMeasure \u03bc\ns : \u03b9 \u2192 MeasurableSpace \u03a9\n\u03b1 : Type u_3\np : Set \u03b9 \u2192 Prop\nf : Filter \u03b9\nns : \u03b1 \u2192 Set \u03b9\nh_le : \u2200 (n : \u03b9), s n \u2264 m0\nh_indep : iIndep s\nhf : \u2200 (t : Set \u03b9), p t \u2192 t\u1d9c \u2208 f\nhns : Directed (fun x x_1 => x \u2264 x_1) ns\nhnsp : \u2200 (a : \u03b1), p (ns a)\nhns_univ : \u2200 (n : \u03b9), \u2203 a, n \u2208 ns a\n\u22a2 \u2a06 j, \u2a06 i, \u2a06 (_ : j \u2208 ns i), s j = \u2a06 n, s n"}, {"tactic": "refine' iSup_congr fun n => _", "annotated_tactic": ["refine' <a>iSup_congr</a> fun n => _", [{"full_name": "iSup_congr", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [692, 9], "def_end_pos": [692, 19]}]], "state_before": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm m0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d : IsProbabilityMeasure \u03bc\ns : \u03b9 \u2192 MeasurableSpace \u03a9\n\u03b1 : Type u_3\np : Set \u03b9 \u2192 Prop\nf : Filter \u03b9\nns : \u03b1 \u2192 Set \u03b9\nh_le : \u2200 (n : \u03b9), s n \u2264 m0\nh_indep : iIndep s\nhf : \u2200 (t : Set \u03b9), p t \u2192 t\u1d9c \u2208 f\nhns : Directed (fun x x_1 => x \u2264 x_1) ns\nhnsp : \u2200 (a : \u03b1), p (ns a)\nhns_univ : \u2200 (n : \u03b9), \u2203 a, n \u2208 ns a\n\u22a2 \u2a06 j, \u2a06 i, \u2a06 (_ : j \u2208 ns i), s j = \u2a06 n, s n", "state_after": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm m0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d : IsProbabilityMeasure \u03bc\ns : \u03b9 \u2192 MeasurableSpace \u03a9\n\u03b1 : Type u_3\np : Set \u03b9 \u2192 Prop\nf : Filter \u03b9\nns : \u03b1 \u2192 Set \u03b9\nh_le : \u2200 (n : \u03b9), s n \u2264 m0\nh_indep : iIndep s\nhf : \u2200 (t : Set \u03b9), p t \u2192 t\u1d9c \u2208 f\nhns : Directed (fun x x_1 => x \u2264 x_1) ns\nhnsp : \u2200 (a : \u03b1), p (ns a)\nhns_univ : \u2200 (n : \u03b9), \u2203 a, n \u2208 ns a\nn : \u03b9\n\u22a2 \u2a06 i, \u2a06 (_ : n \u2208 ns i), s n = s n"}, {"tactic": "have h : \u2a06 (i : \u03b1) (_ : n \u2208 ns i), s n = \u2a06 _ : \u2203 i, n \u2208 ns i, s n := by rw [iSup_exists]", "annotated_tactic": ["have h : \u2a06 (i : \u03b1) (_ : n \u2208 ns i), s n = \u2a06 _ : \u2203 i, n \u2208 ns i, s n := by rw [<a>iSup_exists</a>]", [{"full_name": "iSup_exists", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [1369, 9], "def_end_pos": [1369, 20]}]], "state_before": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm m0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d : IsProbabilityMeasure \u03bc\ns : \u03b9 \u2192 MeasurableSpace \u03a9\n\u03b1 : Type u_3\np : Set \u03b9 \u2192 Prop\nf : Filter \u03b9\nns : \u03b1 \u2192 Set \u03b9\nh_le : \u2200 (n : \u03b9), s n \u2264 m0\nh_indep : iIndep s\nhf : \u2200 (t : Set \u03b9), p t \u2192 t\u1d9c \u2208 f\nhns : Directed (fun x x_1 => x \u2264 x_1) ns\nhnsp : \u2200 (a : \u03b1), p (ns a)\nhns_univ : \u2200 (n : \u03b9), \u2203 a, n \u2208 ns a\nn : \u03b9\n\u22a2 \u2a06 i, \u2a06 (_ : n \u2208 ns i), s n = s n", "state_after": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm m0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d : IsProbabilityMeasure \u03bc\ns : \u03b9 \u2192 MeasurableSpace \u03a9\n\u03b1 : Type u_3\np : Set \u03b9 \u2192 Prop\nf : Filter \u03b9\nns : \u03b1 \u2192 Set \u03b9\nh_le : \u2200 (n : \u03b9), s n \u2264 m0\nh_indep : iIndep s\nhf : \u2200 (t : Set \u03b9), p t \u2192 t\u1d9c \u2208 f\nhns : Directed (fun x x_1 => x \u2264 x_1) ns\nhnsp : \u2200 (a : \u03b1), p (ns a)\nhns_univ : \u2200 (n : \u03b9), \u2203 a, n \u2208 ns a\nn : \u03b9\nh : \u2a06 i, \u2a06 (_ : n \u2208 ns i), s n = \u2a06 (_ : \u2203 i, n \u2208 ns i), s n\n\u22a2 \u2a06 i, \u2a06 (_ : n \u2208 ns i), s n = s n"}, {"tactic": "haveI : Nonempty (\u2203 i : \u03b1, n \u2208 ns i) := \u27e8hns_univ n\u27e9", "annotated_tactic": ["haveI : <a>Nonempty</a> (\u2203 i : \u03b1, n \u2208 ns i) := \u27e8hns_univ n\u27e9", [{"full_name": "Nonempty", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [686, 17], "def_end_pos": [686, 25]}]], "state_before": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm m0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d : IsProbabilityMeasure \u03bc\ns : \u03b9 \u2192 MeasurableSpace \u03a9\n\u03b1 : Type u_3\np : Set \u03b9 \u2192 Prop\nf : Filter \u03b9\nns : \u03b1 \u2192 Set \u03b9\nh_le : \u2200 (n : \u03b9), s n \u2264 m0\nh_indep : iIndep s\nhf : \u2200 (t : Set \u03b9), p t \u2192 t\u1d9c \u2208 f\nhns : Directed (fun x x_1 => x \u2264 x_1) ns\nhnsp : \u2200 (a : \u03b1), p (ns a)\nhns_univ : \u2200 (n : \u03b9), \u2203 a, n \u2208 ns a\nn : \u03b9\nh : \u2a06 i, \u2a06 (_ : n \u2208 ns i), s n = \u2a06 (_ : \u2203 i, n \u2208 ns i), s n\n\u22a2 \u2a06 i, \u2a06 (_ : n \u2208 ns i), s n = s n", "state_after": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm m0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d : IsProbabilityMeasure \u03bc\ns : \u03b9 \u2192 MeasurableSpace \u03a9\n\u03b1 : Type u_3\np : Set \u03b9 \u2192 Prop\nf : Filter \u03b9\nns : \u03b1 \u2192 Set \u03b9\nh_le : \u2200 (n : \u03b9), s n \u2264 m0\nh_indep : iIndep s\nhf : \u2200 (t : Set \u03b9), p t \u2192 t\u1d9c \u2208 f\nhns : Directed (fun x x_1 => x \u2264 x_1) ns\nhnsp : \u2200 (a : \u03b1), p (ns a)\nhns_univ : \u2200 (n : \u03b9), \u2203 a, n \u2208 ns a\nn : \u03b9\nh : \u2a06 i, \u2a06 (_ : n \u2208 ns i), s n = \u2a06 (_ : \u2203 i, n \u2208 ns i), s n\nthis : Nonempty (\u2203 i, n \u2208 ns i)\n\u22a2 \u2a06 i, \u2a06 (_ : n \u2208 ns i), s n = s n"}, {"tactic": "rw [h, iSup_const]", "annotated_tactic": ["rw [h, <a>iSup_const</a>]", [{"full_name": "iSup_const", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [1105, 9], "def_end_pos": [1105, 19]}]], "state_before": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm m0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d : IsProbabilityMeasure \u03bc\ns : \u03b9 \u2192 MeasurableSpace \u03a9\n\u03b1 : Type u_3\np : Set \u03b9 \u2192 Prop\nf : Filter \u03b9\nns : \u03b1 \u2192 Set \u03b9\nh_le : \u2200 (n : \u03b9), s n \u2264 m0\nh_indep : iIndep s\nhf : \u2200 (t : Set \u03b9), p t \u2192 t\u1d9c \u2208 f\nhns : Directed (fun x x_1 => x \u2264 x_1) ns\nhnsp : \u2200 (a : \u03b1), p (ns a)\nhns_univ : \u2200 (n : \u03b9), \u2203 a, n \u2208 ns a\nn : \u03b9\nh : \u2a06 i, \u2a06 (_ : n \u2208 ns i), s n = \u2a06 (_ : \u2203 i, n \u2208 ns i), s n\nthis : Nonempty (\u2203 i, n \u2208 ns i)\n\u22a2 \u2a06 i, \u2a06 (_ : n \u2208 ns i), s n = s n", "state_after": "no goals"}, {"tactic": "rw [\u2190 this]", "annotated_tactic": ["rw [\u2190 this]", []], "state_before": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm m0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d : IsProbabilityMeasure \u03bc\ns : \u03b9 \u2192 MeasurableSpace \u03a9\n\u03b1 : Type u_3\np : Set \u03b9 \u2192 Prop\nf : Filter \u03b9\nns : \u03b1 \u2192 Set \u03b9\nh_le : \u2200 (n : \u03b9), s n \u2264 m0\nh_indep : iIndep s\nhf : \u2200 (t : Set \u03b9), p t \u2192 t\u1d9c \u2208 f\nhns : Directed (fun x x_1 => x \u2264 x_1) ns\nhnsp : \u2200 (a : \u03b1), p (ns a)\nhns_univ : \u2200 (n : \u03b9), \u2203 a, n \u2208 ns a\nthis : \u2a06 a, \u2a06 n \u2208 ns a, s n = \u2a06 n, s n\n\u22a2 Indep (\u2a06 n, s n) (limsup s f)", "state_after": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm m0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d : IsProbabilityMeasure \u03bc\ns : \u03b9 \u2192 MeasurableSpace \u03a9\n\u03b1 : Type u_3\np : Set \u03b9 \u2192 Prop\nf : Filter \u03b9\nns : \u03b1 \u2192 Set \u03b9\nh_le : \u2200 (n : \u03b9), s n \u2264 m0\nh_indep : iIndep s\nhf : \u2200 (t : Set \u03b9), p t \u2192 t\u1d9c \u2208 f\nhns : Directed (fun x x_1 => x \u2264 x_1) ns\nhnsp : \u2200 (a : \u03b1), p (ns a)\nhns_univ : \u2200 (n : \u03b9), \u2203 a, n \u2208 ns a\nthis : \u2a06 a, \u2a06 n \u2208 ns a, s n = \u2a06 n, s n\n\u22a2 Indep (\u2a06 a, \u2a06 n \u2208 ns a, s n) (limsup s f)"}, {"tactic": "exact indep_iSup_directed_limsup h_le h_indep hf hns hnsp", "annotated_tactic": ["exact <a>indep_iSup_directed_limsup</a> h_le h_indep hf hns hnsp", [{"full_name": "ProbabilityTheory.indep_iSup_directed_limsup", "def_path": "Mathlib/Probability/Independence/ZeroOne.lean", "def_pos": [83, 9], "def_end_pos": [83, 35]}]], "state_before": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm m0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d : IsProbabilityMeasure \u03bc\ns : \u03b9 \u2192 MeasurableSpace \u03a9\n\u03b1 : Type u_3\np : Set \u03b9 \u2192 Prop\nf : Filter \u03b9\nns : \u03b1 \u2192 Set \u03b9\nh_le : \u2200 (n : \u03b9), s n \u2264 m0\nh_indep : iIndep s\nhf : \u2200 (t : Set \u03b9), p t \u2192 t\u1d9c \u2208 f\nhns : Directed (fun x x_1 => x \u2264 x_1) ns\nhnsp : \u2200 (a : \u03b1), p (ns a)\nhns_univ : \u2200 (n : \u03b9), \u2203 a, n \u2208 ns a\nthis : \u2a06 a, \u2a06 n \u2208 ns a, s n = \u2a06 n, s n\n\u22a2 Indep (\u2a06 a, \u2a06 n \u2208 ns a, s n) (limsup s f)", "state_after": "no goals"}, {"tactic": "rw [iSup_exists]", "annotated_tactic": ["rw [<a>iSup_exists</a>]", [{"full_name": "iSup_exists", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [1369, 9], "def_end_pos": [1369, 20]}]], "state_before": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm m0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d : IsProbabilityMeasure \u03bc\ns : \u03b9 \u2192 MeasurableSpace \u03a9\n\u03b1 : Type u_3\np : Set \u03b9 \u2192 Prop\nf : Filter \u03b9\nns : \u03b1 \u2192 Set \u03b9\nh_le : \u2200 (n : \u03b9), s n \u2264 m0\nh_indep : iIndep s\nhf : \u2200 (t : Set \u03b9), p t \u2192 t\u1d9c \u2208 f\nhns : Directed (fun x x_1 => x \u2264 x_1) ns\nhnsp : \u2200 (a : \u03b1), p (ns a)\nhns_univ : \u2200 (n : \u03b9), \u2203 a, n \u2208 ns a\nn : \u03b9\n\u22a2 \u2a06 i, \u2a06 (_ : n \u2208 ns i), s n = \u2a06 (_ : \u2203 i, n \u2208 ns i), s n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Function.lean", "full_name": "Set.piecewise_range_comp", "start": [1474, 1], "end": [1476, 30], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Card.lean", "full_name": "Set.exists_superset_subset_encard_eq", "start": [363, 1], "end": [374, 75], "traced_tactics": [{"tactic": "obtain (hs | hs) := eq_or_ne s.encard \u22a4", "annotated_tactic": ["obtain (hs | hs) := <a>eq_or_ne</a> s.encard \u22a4", [{"full_name": "eq_or_ne", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [209, 9], "def_end_pos": [209, 17]}]], "state_before": "\u03b1 : Type u_1\ns t : Set \u03b1\nk : (fun x => \u2115\u221e) (PartENat.card \u2191s)\nhst : s \u2286 t\nhsk : encard s \u2264 k\nhkt : k \u2264 encard t\n\u22a2 \u2203 r, s \u2286 r \u2227 r \u2286 t \u2227 encard r = k", "state_after": "case inl\n\u03b1 : Type u_1\ns t : Set \u03b1\nk : (fun x => \u2115\u221e) (PartENat.card \u2191s)\nhst : s \u2286 t\nhsk : encard s \u2264 k\nhkt : k \u2264 encard t\nhs : encard s = \u22a4\n\u22a2 \u2203 r, s \u2286 r \u2227 r \u2286 t \u2227 encard r = k\n\ncase inr\n\u03b1 : Type u_1\ns t : Set \u03b1\nk : (fun x => \u2115\u221e) (PartENat.card \u2191s)\nhst : s \u2286 t\nhsk : encard s \u2264 k\nhkt : k \u2264 encard t\nhs : encard s \u2260 \u22a4\n\u22a2 \u2203 r, s \u2286 r \u2227 r \u2286 t \u2227 encard r = k"}, {"tactic": "obtain \u27e8k, rfl\u27e9 := exists_add_of_le hsk", "annotated_tactic": ["obtain \u27e8k, rfl\u27e9 := <a>exists_add_of_le</a> hsk", [{"full_name": "ExistsAddOfLE.exists_add_of_le", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [37, 3], "def_end_pos": [37, 19]}]], "state_before": "case inr\n\u03b1 : Type u_1\ns t : Set \u03b1\nk : (fun x => \u2115\u221e) (PartENat.card \u2191s)\nhst : s \u2286 t\nhsk : encard s \u2264 k\nhkt : k \u2264 encard t\nhs : encard s \u2260 \u22a4\n\u22a2 \u2203 r, s \u2286 r \u2227 r \u2286 t \u2227 encard r = k", "state_after": "case inr.intro\n\u03b1 : Type u_1\ns t : Set \u03b1\nhst : s \u2286 t\nhs : encard s \u2260 \u22a4\nk : \u2115\u221e\nhsk : encard s \u2264 encard s + k\nhkt : encard s + k \u2264 encard t\n\u22a2 \u2203 r, s \u2286 r \u2227 r \u2286 t \u2227 encard r = encard s + k"}, {"tactic": "obtain \u27e8k', hk'\u27e9 := exists_add_of_le hkt", "annotated_tactic": ["obtain \u27e8k', hk'\u27e9 := <a>exists_add_of_le</a> hkt", [{"full_name": "ExistsAddOfLE.exists_add_of_le", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [37, 3], "def_end_pos": [37, 19]}]], "state_before": "case inr.intro\n\u03b1 : Type u_1\ns t : Set \u03b1\nhst : s \u2286 t\nhs : encard s \u2260 \u22a4\nk : \u2115\u221e\nhsk : encard s \u2264 encard s + k\nhkt : encard s + k \u2264 encard t\n\u22a2 \u2203 r, s \u2286 r \u2227 r \u2286 t \u2227 encard r = encard s + k", "state_after": "case inr.intro.intro\n\u03b1 : Type u_1\ns t : Set \u03b1\nhst : s \u2286 t\nhs : encard s \u2260 \u22a4\nk : \u2115\u221e\nhsk : encard s \u2264 encard s + k\nhkt : encard s + k \u2264 encard t\nk' : \u2115\u221e\nhk' : encard t = encard s + k + k'\n\u22a2 \u2203 r, s \u2286 r \u2227 r \u2286 t \u2227 encard r = encard s + k"}, {"tactic": "have hk : k \u2264 encard (t \\ s)", "annotated_tactic": ["have hk : k \u2264 <a>encard</a> (t \\ s)", [{"full_name": "Set.encard", "def_path": "Mathlib/Data/Set/Card.lean", "def_pos": [66, 19], "def_end_pos": [66, 25]}]], "state_before": "case inr.intro.intro\n\u03b1 : Type u_1\ns t : Set \u03b1\nhst : s \u2286 t\nhs : encard s \u2260 \u22a4\nk : \u2115\u221e\nhsk : encard s \u2264 encard s + k\nhkt : encard s + k \u2264 encard t\nk' : \u2115\u221e\nhk' : encard t = encard s + k + k'\n\u22a2 \u2203 r, s \u2286 r \u2227 r \u2286 t \u2227 encard r = encard s + k", "state_after": "case hk\n\u03b1 : Type u_1\ns t : Set \u03b1\nhst : s \u2286 t\nhs : encard s \u2260 \u22a4\nk : \u2115\u221e\nhsk : encard s \u2264 encard s + k\nhkt : encard s + k \u2264 encard t\nk' : \u2115\u221e\nhk' : encard t = encard s + k + k'\n\u22a2 k \u2264 encard (t \\ s)\n\ncase inr.intro.intro\n\u03b1 : Type u_1\ns t : Set \u03b1\nhst : s \u2286 t\nhs : encard s \u2260 \u22a4\nk : \u2115\u221e\nhsk : encard s \u2264 encard s + k\nhkt : encard s + k \u2264 encard t\nk' : \u2115\u221e\nhk' : encard t = encard s + k + k'\nhk : k \u2264 encard (t \\ s)\n\u22a2 \u2203 r, s \u2286 r \u2227 r \u2286 t \u2227 encard r = encard s + k"}, {"tactic": "obtain \u27e8r', hr', rfl\u27e9 := exists_subset_encard_eq hk", "annotated_tactic": ["obtain \u27e8r', hr', rfl\u27e9 := <a>exists_subset_encard_eq</a> hk", [{"full_name": "Set.exists_subset_encard_eq", "def_path": "Mathlib/Data/Set/Card.lean", "def_pos": [351, 9], "def_end_pos": [351, 32]}]], "state_before": "case inr.intro.intro\n\u03b1 : Type u_1\ns t : Set \u03b1\nhst : s \u2286 t\nhs : encard s \u2260 \u22a4\nk : \u2115\u221e\nhsk : encard s \u2264 encard s + k\nhkt : encard s + k \u2264 encard t\nk' : \u2115\u221e\nhk' : encard t = encard s + k + k'\nhk : k \u2264 encard (t \\ s)\n\u22a2 \u2203 r, s \u2286 r \u2227 r \u2286 t \u2227 encard r = encard s + k", "state_after": "case inr.intro.intro.intro.intro\n\u03b1 : Type u_1\ns t : Set \u03b1\nhst : s \u2286 t\nhs : encard s \u2260 \u22a4\nk' : \u2115\u221e\nr' : Set \u03b1\nhr' : r' \u2286 t \\ s\nhsk : encard s \u2264 encard s + encard r'\nhkt : encard s + encard r' \u2264 encard t\nhk' : encard t = encard s + encard r' + k'\nhk : encard r' \u2264 encard (t \\ s)\n\u22a2 \u2203 r, s \u2286 r \u2227 r \u2286 t \u2227 encard r = encard s + encard r'"}, {"tactic": "refine' \u27e8s \u222a r', subset_union_left _ _, union_subset hst (hr'.trans (diff_subset _ _)), _\u27e9", "annotated_tactic": ["refine' \u27e8s \u222a r', <a>subset_union_left</a> _ _, <a>union_subset</a> hst (hr'.trans (<a>diff_subset</a> _ _)), _\u27e9", [{"full_name": "Set.subset_union_left", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [829, 9], "def_end_pos": [829, 26]}, {"full_name": "Set.union_subset", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [836, 9], "def_end_pos": [836, 21]}, {"full_name": "Set.diff_subset", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1845, 9], "def_end_pos": [1845, 20]}]], "state_before": "case inr.intro.intro.intro.intro\n\u03b1 : Type u_1\ns t : Set \u03b1\nhst : s \u2286 t\nhs : encard s \u2260 \u22a4\nk' : \u2115\u221e\nr' : Set \u03b1\nhr' : r' \u2286 t \\ s\nhsk : encard s \u2264 encard s + encard r'\nhkt : encard s + encard r' \u2264 encard t\nhk' : encard t = encard s + encard r' + k'\nhk : encard r' \u2264 encard (t \\ s)\n\u22a2 \u2203 r, s \u2286 r \u2227 r \u2286 t \u2227 encard r = encard s + encard r'", "state_after": "case inr.intro.intro.intro.intro\n\u03b1 : Type u_1\ns t : Set \u03b1\nhst : s \u2286 t\nhs : encard s \u2260 \u22a4\nk' : \u2115\u221e\nr' : Set \u03b1\nhr' : r' \u2286 t \\ s\nhsk : encard s \u2264 encard s + encard r'\nhkt : encard s + encard r' \u2264 encard t\nhk' : encard t = encard s + encard r' + k'\nhk : encard r' \u2264 encard (t \\ s)\n\u22a2 encard (s \u222a r') = encard s + encard r'"}, {"tactic": "rw [encard_union_eq (disjoint_of_subset_right hr' disjoint_sdiff_right)]", "annotated_tactic": ["rw [<a>encard_union_eq</a> (<a>disjoint_of_subset_right</a> hr' <a>disjoint_sdiff_right</a>)]", [{"full_name": "Set.encard_union_eq", "def_path": "Mathlib/Data/Set/Card.lean", "def_pos": [113, 9], "def_end_pos": [113, 24]}, {"full_name": "Set.disjoint_of_subset_right", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1576, 7], "def_end_pos": [1576, 31]}, {"full_name": "Set.disjoint_sdiff_right", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1604, 7], "def_end_pos": [1604, 27]}]], "state_before": "case inr.intro.intro.intro.intro\n\u03b1 : Type u_1\ns t : Set \u03b1\nhst : s \u2286 t\nhs : encard s \u2260 \u22a4\nk' : \u2115\u221e\nr' : Set \u03b1\nhr' : r' \u2286 t \\ s\nhsk : encard s \u2264 encard s + encard r'\nhkt : encard s + encard r' \u2264 encard t\nhk' : encard t = encard s + encard r' + k'\nhk : encard r' \u2264 encard (t \\ s)\n\u22a2 encard (s \u222a r') = encard s + encard r'", "state_after": "no goals"}, {"tactic": "rw [hs, top_le_iff] at hsk", "annotated_tactic": ["rw [hs, <a>top_le_iff</a>] at hsk", [{"full_name": "top_le_iff", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [157, 9], "def_end_pos": [157, 19]}]], "state_before": "case inl\n\u03b1 : Type u_1\ns t : Set \u03b1\nk : (fun x => \u2115\u221e) (PartENat.card \u2191s)\nhst : s \u2286 t\nhsk : encard s \u2264 k\nhkt : k \u2264 encard t\nhs : encard s = \u22a4\n\u22a2 \u2203 r, s \u2286 r \u2227 r \u2286 t \u2227 encard r = k", "state_after": "case inl\n\u03b1 : Type u_1\ns t : Set \u03b1\nk : (fun x => \u2115\u221e) (PartENat.card \u2191s)\nhst : s \u2286 t\nhsk : k = \u22a4\nhkt : k \u2264 encard t\nhs : encard s = \u22a4\n\u22a2 \u2203 r, s \u2286 r \u2227 r \u2286 t \u2227 encard r = k"}, {"tactic": "subst hsk", "annotated_tactic": ["subst hsk", []], "state_before": "case inl\n\u03b1 : Type u_1\ns t : Set \u03b1\nk : (fun x => \u2115\u221e) (PartENat.card \u2191s)\nhst : s \u2286 t\nhsk : k = \u22a4\nhkt : k \u2264 encard t\nhs : encard s = \u22a4\n\u22a2 \u2203 r, s \u2286 r \u2227 r \u2286 t \u2227 encard r = k", "state_after": "case inl\n\u03b1 : Type u_1\ns t : Set \u03b1\nhst : s \u2286 t\nhs : encard s = \u22a4\nhkt : \u22a4 \u2264 encard t\n\u22a2 \u2203 r, s \u2286 r \u2227 r \u2286 t \u2227 encard r = \u22a4"}, {"tactic": "exact \u27e8s, Subset.rfl, hst, hs\u27e9", "annotated_tactic": ["exact \u27e8s, <a>Subset.rfl</a>, hst, hs\u27e9", [{"full_name": "Set.Subset.rfl", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [357, 9], "def_end_pos": [357, 19]}]], "state_before": "case inl\n\u03b1 : Type u_1\ns t : Set \u03b1\nhst : s \u2286 t\nhs : encard s = \u22a4\nhkt : \u22a4 \u2264 encard t\n\u22a2 \u2203 r, s \u2286 r \u2227 r \u2286 t \u2227 encard r = \u22a4", "state_after": "no goals"}, {"tactic": "rw [\u2190encard_diff_add_encard_of_subset hst, add_comm] at hkt", "annotated_tactic": ["rw [\u2190<a>encard_diff_add_encard_of_subset</a> hst, <a>add_comm</a>] at hkt", [{"full_name": "Set.encard_diff_add_encard_of_subset", "def_path": "Mathlib/Data/Set/Card.lean", "def_pos": [160, 9], "def_end_pos": [160, 41]}, {"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [301, 3], "def_end_pos": [301, 14]}]], "state_before": "case hk\n\u03b1 : Type u_1\ns t : Set \u03b1\nhst : s \u2286 t\nhs : encard s \u2260 \u22a4\nk : \u2115\u221e\nhsk : encard s \u2264 encard s + k\nhkt : encard s + k \u2264 encard t\nk' : \u2115\u221e\nhk' : encard t = encard s + k + k'\n\u22a2 k \u2264 encard (t \\ s)", "state_after": "case hk\n\u03b1 : Type u_1\ns t : Set \u03b1\nhst : s \u2286 t\nhs : encard s \u2260 \u22a4\nk : \u2115\u221e\nhsk : encard s \u2264 encard s + k\nhkt : k + encard s \u2264 encard (t \\ s) + encard s\nk' : \u2115\u221e\nhk' : encard t = encard s + k + k'\n\u22a2 k \u2264 encard (t \\ s)"}, {"tactic": "exact WithTop.le_of_add_le_add_right hs hkt", "annotated_tactic": ["exact <a>WithTop.le_of_add_le_add_right</a> hs hkt", [{"full_name": "WithTop.le_of_add_le_add_right", "def_path": "Mathlib/Algebra/Order/Monoid/WithTop.lean", "def_pos": [251, 19], "def_end_pos": [251, 41]}]], "state_before": "case hk\n\u03b1 : Type u_1\ns t : Set \u03b1\nhst : s \u2286 t\nhs : encard s \u2260 \u22a4\nk : \u2115\u221e\nhsk : encard s \u2264 encard s + k\nhkt : k + encard s \u2264 encard (t \\ s) + encard s\nk' : \u2115\u221e\nhk' : encard t = encard s + k + k'\n\u22a2 k \u2264 encard (t \\ s)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finmap.lean", "full_name": "Finmap.insert_insert", "start": [509, 1], "end": [511, 78], "traced_tactics": [{"tactic": "simp only [insert_toFinmap, AList.insert_insert]", "annotated_tactic": ["simp only [<a>insert_toFinmap</a>, <a>AList.insert_insert</a>]", [{"full_name": "Finmap.insert_toFinmap", "def_path": "Mathlib/Data/Finmap.lean", "def_pos": [480, 9], "def_end_pos": [480, 24]}, {"full_name": "AList.insert_insert", "def_path": "Mathlib/Data/List/AList.lean", "def_pos": [326, 9], "def_end_pos": [326, 22]}]], "state_before": "\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nb b' : \u03b2 a\ns\u271d : Finmap \u03b2\ns : AList \u03b2\n\u22a2 insert a b' (insert a b \u27e6s\u27e7) = insert a b' \u27e6s\u27e7", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/TuringMachine.lean", "full_name": "Turing.TM2to1.tr_respects_aux", "start": [2678, 1], "end": [2696, 11], "traced_tactics": [{"tactic": "simp only [trNormal_run, step_run]", "annotated_tactic": ["simp only [<a>trNormal_run</a>, <a>step_run</a>]", [{"full_name": "Turing.TM2to1.trNormal_run", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [2526, 9], "def_end_pos": [2526, 21]}, {"full_name": "Turing.TM2to1.step_run", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [2506, 9], "def_end_pos": [2506, 17]}]], "state_before": "K : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nq : TM2.Stmt (fun k => \u0393 k) \u039b \u03c3\nv : \u03c3\nT : ListBlank ((i : K) \u2192 Option (\u0393 i))\nk : K\nS : (k : K) \u2192 List (\u0393 k)\nhT : \u2200 (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.reverse (List.map some (S k)))\no : StAct k\nIH :\n  \u2200 {v : \u03c3} {S : (k : K) \u2192 List (\u0393 k)} {T : ListBlank ((k : K) \u2192 Option (\u0393 k))},\n    (\u2200 (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.reverse (List.map some (S k)))) \u2192\n      \u2203 b,\n        TrCfg (TM2.stepAux q v S) b \u2227\n          Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q) v (Tape.mk' \u2205 (addBottom T))) b\n\u22a2 \u2203 b,\n    TrCfg (TM2.stepAux (stRun o q) v S) b \u2227\n      Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal (stRun o q)) v (Tape.mk' \u2205 (addBottom T))) b", "state_after": "K : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nq : TM2.Stmt (fun k => \u0393 k) \u039b \u03c3\nv : \u03c3\nT : ListBlank ((i : K) \u2192 Option (\u0393 i))\nk : K\nS : (k : K) \u2192 List (\u0393 k)\nhT : \u2200 (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.reverse (List.map some (S k)))\no : StAct k\nIH :\n  \u2200 {v : \u03c3} {S : (k : K) \u2192 List (\u0393 k)} {T : ListBlank ((k : K) \u2192 Option (\u0393 k))},\n    (\u2200 (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.reverse (List.map some (S k)))) \u2192\n      \u2203 b,\n        TrCfg (TM2.stepAux q v S) b \u2227\n          Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q) v (Tape.mk' \u2205 (addBottom T))) b\n\u22a2 \u2203 b,\n    TrCfg (TM2.stepAux q (stVar v (S k) o) (update S k (stWrite v (S k) o))) b \u2227\n      Reaches (TM1.step (tr M)) (TM1.stepAux (goto fun x x => go k o q) v (Tape.mk' \u2205 (addBottom T))) b"}, {"tactic": "have hgo := tr_respects_aux\u2081 M o q v (hT k) _ le_rfl", "annotated_tactic": ["have hgo := <a>tr_respects_aux\u2081</a> M o q v (hT k) _ <a>le_rfl</a>", [{"full_name": "Turing.TM2to1.tr_respects_aux\u2081", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [2655, 9], "def_end_pos": [2655, 25]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}]], "state_before": "K : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nq : TM2.Stmt (fun k => \u0393 k) \u039b \u03c3\nv : \u03c3\nT : ListBlank ((i : K) \u2192 Option (\u0393 i))\nk : K\nS : (k : K) \u2192 List (\u0393 k)\nhT : \u2200 (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.reverse (List.map some (S k)))\no : StAct k\nIH :\n  \u2200 {v : \u03c3} {S : (k : K) \u2192 List (\u0393 k)} {T : ListBlank ((k : K) \u2192 Option (\u0393 k))},\n    (\u2200 (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.reverse (List.map some (S k)))) \u2192\n      \u2203 b,\n        TrCfg (TM2.stepAux q v S) b \u2227\n          Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q) v (Tape.mk' \u2205 (addBottom T))) b\n\u22a2 \u2203 b,\n    TrCfg (TM2.stepAux q (stVar v (S k) o) (update S k (stWrite v (S k) o))) b \u2227\n      Reaches (TM1.step (tr M)) (TM1.stepAux (goto fun x x => go k o q) v (Tape.mk' \u2205 (addBottom T))) b", "state_after": "K : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nq : TM2.Stmt (fun k => \u0393 k) \u039b \u03c3\nv : \u03c3\nT : ListBlank ((i : K) \u2192 Option (\u0393 i))\nk : K\nS : (k : K) \u2192 List (\u0393 k)\nhT : \u2200 (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.reverse (List.map some (S k)))\no : StAct k\nIH :\n  \u2200 {v : \u03c3} {S : (k : K) \u2192 List (\u0393 k)} {T : ListBlank ((k : K) \u2192 Option (\u0393 k))},\n    (\u2200 (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.reverse (List.map some (S k)))) \u2192\n      \u2203 b,\n        TrCfg (TM2.stepAux q v S) b \u2227\n          Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q) v (Tape.mk' \u2205 (addBottom T))) b\nhgo :\n  Reaches\u2080 (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' \u2205 (addBottom T) }\n    { l := some (go k o q), var := v, Tape := (Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom T)) }\n\u22a2 \u2203 b,\n    TrCfg (TM2.stepAux q (stVar v (S k) o) (update S k (stWrite v (S k) o))) b \u2227\n      Reaches (TM1.step (tr M)) (TM1.stepAux (goto fun x x => go k o q) v (Tape.mk' \u2205 (addBottom T))) b"}, {"tactic": "obtain \u27e8T', hT', hrun\u27e9 := tr_respects_aux\u2082 hT o", "annotated_tactic": ["obtain \u27e8T', hT', hrun\u27e9 := <a>tr_respects_aux\u2082</a> hT o", [{"full_name": "Turing.TM2to1.tr_respects_aux\u2082", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [2548, 9], "def_end_pos": [2548, 25]}]], "state_before": "K : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nq : TM2.Stmt (fun k => \u0393 k) \u039b \u03c3\nv : \u03c3\nT : ListBlank ((i : K) \u2192 Option (\u0393 i))\nk : K\nS : (k : K) \u2192 List (\u0393 k)\nhT : \u2200 (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.reverse (List.map some (S k)))\no : StAct k\nIH :\n  \u2200 {v : \u03c3} {S : (k : K) \u2192 List (\u0393 k)} {T : ListBlank ((k : K) \u2192 Option (\u0393 k))},\n    (\u2200 (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.reverse (List.map some (S k)))) \u2192\n      \u2203 b,\n        TrCfg (TM2.stepAux q v S) b \u2227\n          Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q) v (Tape.mk' \u2205 (addBottom T))) b\nhgo :\n  Reaches\u2080 (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' \u2205 (addBottom T) }\n    { l := some (go k o q), var := v, Tape := (Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom T)) }\n\u22a2 \u2203 b,\n    TrCfg (TM2.stepAux q (stVar v (S k) o) (update S k (stWrite v (S k) o))) b \u2227\n      Reaches (TM1.step (tr M)) (TM1.stepAux (goto fun x x => go k o q) v (Tape.mk' \u2205 (addBottom T))) b", "state_after": "case intro.intro\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nq : TM2.Stmt (fun k => \u0393 k) \u039b \u03c3\nv : \u03c3\nT : ListBlank ((i : K) \u2192 Option (\u0393 i))\nk : K\nS : (k : K) \u2192 List (\u0393 k)\nhT : \u2200 (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.reverse (List.map some (S k)))\no : StAct k\nIH :\n  \u2200 {v : \u03c3} {S : (k : K) \u2192 List (\u0393 k)} {T : ListBlank ((k : K) \u2192 Option (\u0393 k))},\n    (\u2200 (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.reverse (List.map some (S k)))) \u2192\n      \u2203 b,\n        TrCfg (TM2.stepAux q v S) b \u2227\n          Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q) v (Tape.mk' \u2205 (addBottom T))) b\nhgo :\n  Reaches\u2080 (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' \u2205 (addBottom T) }\n    { l := some (go k o q), var := v, Tape := (Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom T)) }\nT' : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhT' :\n  \u2200 (k_1 : K),\n    ListBlank.map (proj k_1) T' =\n      ListBlank.mk (List.reverse (List.map some (update (fun k => S k) k (stWrite ?m.723096 (S k) o) k_1)))\nhrun :\n  TM1.stepAux (trStAct ?m.723095 o) ?m.723096 ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom T))) =\n    TM1.stepAux ?m.723095 (stVar ?m.723096 (S k) o)\n      ((Tape.move Dir.right)^[List.length (update (fun k => S k) k (stWrite ?m.723096 (S k) o) k)]\n        (Tape.mk' \u2205 (addBottom T')))\n\u22a2 \u2203 b,\n    TrCfg (TM2.stepAux q (stVar v (S k) o) (update S k (stWrite v (S k) o))) b \u2227\n      Reaches (TM1.step (tr M)) (TM1.stepAux (goto fun x x => go k o q) v (Tape.mk' \u2205 (addBottom T))) b"}, {"tactic": "have := hgo.tail' rfl", "annotated_tactic": ["have := hgo.tail' <a>rfl</a>", [{"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "case intro.intro\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nq : TM2.Stmt (fun k => \u0393 k) \u039b \u03c3\nv : \u03c3\nT : ListBlank ((i : K) \u2192 Option (\u0393 i))\nk : K\nS : (k : K) \u2192 List (\u0393 k)\nhT : \u2200 (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.reverse (List.map some (S k)))\no : StAct k\nIH :\n  \u2200 {v : \u03c3} {S : (k : K) \u2192 List (\u0393 k)} {T : ListBlank ((k : K) \u2192 Option (\u0393 k))},\n    (\u2200 (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.reverse (List.map some (S k)))) \u2192\n      \u2203 b,\n        TrCfg (TM2.stepAux q v S) b \u2227\n          Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q) v (Tape.mk' \u2205 (addBottom T))) b\nhgo :\n  Reaches\u2080 (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' \u2205 (addBottom T) }\n    { l := some (go k o q), var := v, Tape := (Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom T)) }\nT' : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhT' :\n  \u2200 (k_1 : K),\n    ListBlank.map (proj k_1) T' =\n      ListBlank.mk (List.reverse (List.map some (update (fun k => S k) k (stWrite ?m.723096 (S k) o) k_1)))\nhrun :\n  TM1.stepAux (trStAct ?m.723095 o) ?m.723096 ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom T))) =\n    TM1.stepAux ?m.723095 (stVar ?m.723096 (S k) o)\n      ((Tape.move Dir.right)^[List.length (update (fun k => S k) k (stWrite ?m.723096 (S k) o) k)]\n        (Tape.mk' \u2205 (addBottom T')))\n\u22a2 \u2203 b,\n    TrCfg (TM2.stepAux q (stVar v (S k) o) (update S k (stWrite v (S k) o))) b \u2227\n      Reaches (TM1.step (tr M)) (TM1.stepAux (goto fun x x => go k o q) v (Tape.mk' \u2205 (addBottom T))) b", "state_after": "case intro.intro\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nq : TM2.Stmt (fun k => \u0393 k) \u039b \u03c3\nv : \u03c3\nT : ListBlank ((i : K) \u2192 Option (\u0393 i))\nk : K\nS : (k : K) \u2192 List (\u0393 k)\nhT : \u2200 (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.reverse (List.map some (S k)))\no : StAct k\nIH :\n  \u2200 {v : \u03c3} {S : (k : K) \u2192 List (\u0393 k)} {T : ListBlank ((k : K) \u2192 Option (\u0393 k))},\n    (\u2200 (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.reverse (List.map some (S k)))) \u2192\n      \u2203 b,\n        TrCfg (TM2.stepAux q v S) b \u2227\n          Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q) v (Tape.mk' \u2205 (addBottom T))) b\nhgo :\n  Reaches\u2080 (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' \u2205 (addBottom T) }\n    { l := some (go k o q), var := v, Tape := (Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom T)) }\nT' : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhT' :\n  \u2200 (k_1 : K),\n    ListBlank.map (proj k_1) T' =\n      ListBlank.mk (List.reverse (List.map some (update (fun k => S k) k (stWrite ?m.723096 (S k) o) k_1)))\nhrun :\n  TM1.stepAux (trStAct ?m.723095 o) ?m.723096 ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom T))) =\n    TM1.stepAux ?m.723095 (stVar ?m.723096 (S k) o)\n      ((Tape.move Dir.right)^[List.length (update (fun k => S k) k (stWrite ?m.723096 (S k) o) k)]\n        (Tape.mk' \u2205 (addBottom T')))\nthis :\n  Reaches\u2081 (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' \u2205 (addBottom T) }\n    (TM1.stepAux (tr M (go k o q)) v ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom T))))\n\u22a2 \u2203 b,\n    TrCfg (TM2.stepAux q (stVar v (S k) o) (update S k (stWrite v (S k) o))) b \u2227\n      Reaches (TM1.step (tr M)) (TM1.stepAux (goto fun x x => go k o q) v (Tape.mk' \u2205 (addBottom T))) b"}, {"tactic": "rw [tr, TM1.stepAux, Tape.move_right_n_head, Tape.mk'_nth_nat, addBottom_nth_snd,\n  stk_nth_val _ (hT k), List.get?_len_le (le_of_eq (List.length_reverse _)), Option.isNone, cond,\n  hrun, TM1.stepAux] at this", "annotated_tactic": ["rw [<a>tr</a>, <a>TM1.stepAux</a>, <a>Tape.move_right_n_head</a>, <a>Tape.mk'_nth_nat</a>, <a>addBottom_nth_snd</a>,\n    <a>stk_nth_val</a> _ (hT k), <a>List.get?_len_le</a> (<a>le_of_eq</a> (<a>List.length_reverse</a> _)), <a>Option.isNone</a>, <a>cond</a>,\n    hrun, <a>TM1.stepAux</a>] at this", [{"full_name": "Turing.TM2to1.tr", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [2637, 5], "def_end_pos": [2637, 7]}, {"full_name": "Turing.TM1.stepAux", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1281, 5], "def_end_pos": [1281, 12]}, {"full_name": "Turing.Tape.move_right_n_head", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [649, 9], "def_end_pos": [649, 31]}, {"full_name": "Turing.Tape.mk'_nth_nat", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [624, 9], "def_end_pos": [624, 25]}, {"full_name": "Turing.TM2to1.addBottom_nth_snd", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [2386, 9], "def_end_pos": [2386, 26]}, {"full_name": "Turing.TM2to1.stk_nth_val", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [2325, 9], "def_end_pos": [2325, 20]}, {"full_name": "List.get?_len_le", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [592, 9], "def_end_pos": [592, 20]}, {"full_name": "le_of_eq", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [72, 9], "def_end_pos": [72, 17]}, {"full_name": "List.length_reverse", "def_path": "lake-packages/lean4/src/lean/Init/Data/List/Basic.lean", "def_pos": [800, 17], "def_end_pos": [800, 31]}, {"full_name": "Option.isNone", "def_path": "lake-packages/lean4/src/lean/Init/Data/Option/Basic.lean", "def_pos": [25, 15], "def_end_pos": [25, 21]}, {"full_name": "cond", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [971, 21], "def_end_pos": [971, 25]}, {"full_name": "Turing.TM1.stepAux", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1281, 5], "def_end_pos": [1281, 12]}]], "state_before": "case intro.intro\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nq : TM2.Stmt (fun k => \u0393 k) \u039b \u03c3\nv : \u03c3\nT : ListBlank ((i : K) \u2192 Option (\u0393 i))\nk : K\nS : (k : K) \u2192 List (\u0393 k)\nhT : \u2200 (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.reverse (List.map some (S k)))\no : StAct k\nIH :\n  \u2200 {v : \u03c3} {S : (k : K) \u2192 List (\u0393 k)} {T : ListBlank ((k : K) \u2192 Option (\u0393 k))},\n    (\u2200 (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.reverse (List.map some (S k)))) \u2192\n      \u2203 b,\n        TrCfg (TM2.stepAux q v S) b \u2227\n          Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q) v (Tape.mk' \u2205 (addBottom T))) b\nhgo :\n  Reaches\u2080 (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' \u2205 (addBottom T) }\n    { l := some (go k o q), var := v, Tape := (Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom T)) }\nT' : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhT' :\n  \u2200 (k_1 : K),\n    ListBlank.map (proj k_1) T' =\n      ListBlank.mk (List.reverse (List.map some (update (fun k => S k) k (stWrite ?m.723096 (S k) o) k_1)))\nhrun :\n  TM1.stepAux (trStAct ?m.723095 o) ?m.723096 ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom T))) =\n    TM1.stepAux ?m.723095 (stVar ?m.723096 (S k) o)\n      ((Tape.move Dir.right)^[List.length (update (fun k => S k) k (stWrite ?m.723096 (S k) o) k)]\n        (Tape.mk' \u2205 (addBottom T')))\nthis :\n  Reaches\u2081 (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' \u2205 (addBottom T) }\n    (TM1.stepAux (tr M (go k o q)) v ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom T))))\n\u22a2 \u2203 b,\n    TrCfg (TM2.stepAux q (stVar v (S k) o) (update S k (stWrite v (S k) o))) b \u2227\n      Reaches (TM1.step (tr M)) (TM1.stepAux (goto fun x x => go k o q) v (Tape.mk' \u2205 (addBottom T))) b", "state_after": "case intro.intro\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nq : TM2.Stmt (fun k => \u0393 k) \u039b \u03c3\nv : \u03c3\nT : ListBlank ((i : K) \u2192 Option (\u0393 i))\nk : K\nS : (k : K) \u2192 List (\u0393 k)\nhT : \u2200 (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.reverse (List.map some (S k)))\no : StAct k\nIH :\n  \u2200 {v : \u03c3} {S : (k : K) \u2192 List (\u0393 k)} {T : ListBlank ((k : K) \u2192 Option (\u0393 k))},\n    (\u2200 (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.reverse (List.map some (S k)))) \u2192\n      \u2203 b,\n        TrCfg (TM2.stepAux q v S) b \u2227\n          Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q) v (Tape.mk' \u2205 (addBottom T))) b\nhgo :\n  Reaches\u2080 (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' \u2205 (addBottom T) }\n    { l := some (go k o q), var := v, Tape := (Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom T)) }\nT' : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhT' :\n  \u2200 (k_1 : K),\n    ListBlank.map (proj k_1) T' =\n      ListBlank.mk (List.reverse (List.map some (update (fun k => S k) k (stWrite v (S k) o) k_1)))\nhrun :\n  TM1.stepAux (trStAct (goto fun x x => ret q) o) v\n      ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom T))) =\n    TM1.stepAux (goto fun x x => ret q) (stVar v (S k) o)\n      ((Tape.move Dir.right)^[List.length (update (fun k => S k) k (stWrite v (S k) o) k)] (Tape.mk' \u2205 (addBottom T')))\nthis :\n  Reaches\u2081 (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' \u2205 (addBottom T) }\n    (match\n      match none with\n      | some val => false\n      | none => true with\n    | true =>\n      TM1.stepAux (goto fun x x => ret q) (stVar v (S k) o)\n        ((Tape.move Dir.right)^[List.length (update (fun k => S k) k (stWrite v (S k) o) k)]\n          (Tape.mk' \u2205 (addBottom T')))\n    | false =>\n      TM1.stepAux (goto fun x x => go k o q) v\n        (Tape.move Dir.right ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom T)))))\n\u22a2 \u2203 b,\n    TrCfg (TM2.stepAux q (stVar v (S k) o) (update S k (stWrite v (S k) o))) b \u2227\n      Reaches (TM1.step (tr M)) (TM1.stepAux (goto fun x x => go k o q) v (Tape.mk' \u2205 (addBottom T))) b"}, {"tactic": "obtain \u27e8c, gc, rc\u27e9 := IH hT'", "annotated_tactic": ["obtain \u27e8c, gc, rc\u27e9 := IH hT'", []], "state_before": "case intro.intro\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nq : TM2.Stmt (fun k => \u0393 k) \u039b \u03c3\nv : \u03c3\nT : ListBlank ((i : K) \u2192 Option (\u0393 i))\nk : K\nS : (k : K) \u2192 List (\u0393 k)\nhT : \u2200 (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.reverse (List.map some (S k)))\no : StAct k\nIH :\n  \u2200 {v : \u03c3} {S : (k : K) \u2192 List (\u0393 k)} {T : ListBlank ((k : K) \u2192 Option (\u0393 k))},\n    (\u2200 (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.reverse (List.map some (S k)))) \u2192\n      \u2203 b,\n        TrCfg (TM2.stepAux q v S) b \u2227\n          Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q) v (Tape.mk' \u2205 (addBottom T))) b\nhgo :\n  Reaches\u2080 (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' \u2205 (addBottom T) }\n    { l := some (go k o q), var := v, Tape := (Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom T)) }\nT' : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhT' :\n  \u2200 (k_1 : K),\n    ListBlank.map (proj k_1) T' =\n      ListBlank.mk (List.reverse (List.map some (update (fun k => S k) k (stWrite v (S k) o) k_1)))\nhrun :\n  TM1.stepAux (trStAct (goto fun x x => ret q) o) v\n      ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom T))) =\n    TM1.stepAux (goto fun x x => ret q) (stVar v (S k) o)\n      ((Tape.move Dir.right)^[List.length (update (fun k => S k) k (stWrite v (S k) o) k)] (Tape.mk' \u2205 (addBottom T')))\nthis :\n  Reaches\u2081 (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' \u2205 (addBottom T) }\n    (match\n      match none with\n      | some val => false\n      | none => true with\n    | true =>\n      TM1.stepAux (goto fun x x => ret q) (stVar v (S k) o)\n        ((Tape.move Dir.right)^[List.length (update (fun k => S k) k (stWrite v (S k) o) k)]\n          (Tape.mk' \u2205 (addBottom T')))\n    | false =>\n      TM1.stepAux (goto fun x x => go k o q) v\n        (Tape.move Dir.right ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom T)))))\n\u22a2 \u2203 b,\n    TrCfg (TM2.stepAux q (stVar v (S k) o) (update S k (stWrite v (S k) o))) b \u2227\n      Reaches (TM1.step (tr M)) (TM1.stepAux (goto fun x x => go k o q) v (Tape.mk' \u2205 (addBottom T))) b", "state_after": "case intro.intro.intro.intro\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nq : TM2.Stmt (fun k => \u0393 k) \u039b \u03c3\nv : \u03c3\nT : ListBlank ((i : K) \u2192 Option (\u0393 i))\nk : K\nS : (k : K) \u2192 List (\u0393 k)\nhT : \u2200 (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.reverse (List.map some (S k)))\no : StAct k\nIH :\n  \u2200 {v : \u03c3} {S : (k : K) \u2192 List (\u0393 k)} {T : ListBlank ((k : K) \u2192 Option (\u0393 k))},\n    (\u2200 (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.reverse (List.map some (S k)))) \u2192\n      \u2203 b,\n        TrCfg (TM2.stepAux q v S) b \u2227\n          Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q) v (Tape.mk' \u2205 (addBottom T))) b\nhgo :\n  Reaches\u2080 (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' \u2205 (addBottom T) }\n    { l := some (go k o q), var := v, Tape := (Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom T)) }\nT' : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhT' :\n  \u2200 (k_1 : K),\n    ListBlank.map (proj k_1) T' =\n      ListBlank.mk (List.reverse (List.map some (update (fun k => S k) k (stWrite v (S k) o) k_1)))\nhrun :\n  TM1.stepAux (trStAct (goto fun x x => ret q) o) v\n      ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom T))) =\n    TM1.stepAux (goto fun x x => ret q) (stVar v (S k) o)\n      ((Tape.move Dir.right)^[List.length (update (fun k => S k) k (stWrite v (S k) o) k)] (Tape.mk' \u2205 (addBottom T')))\nthis :\n  Reaches\u2081 (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' \u2205 (addBottom T) }\n    (match\n      match none with\n      | some val => false\n      | none => true with\n    | true =>\n      TM1.stepAux (goto fun x x => ret q) (stVar v (S k) o)\n        ((Tape.move Dir.right)^[List.length (update (fun k => S k) k (stWrite v (S k) o) k)]\n          (Tape.mk' \u2205 (addBottom T')))\n    | false =>\n      TM1.stepAux (goto fun x x => go k o q) v\n        (Tape.move Dir.right ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom T)))))\nc : TM1.Cfg \u0393' \u039b' \u03c3\ngc : TrCfg (TM2.stepAux q ?m.726996 fun k_1 => update (fun k => S k) k (stWrite v (S k) o) k_1) c\nrc : Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q) ?m.726996 (Tape.mk' \u2205 (addBottom T'))) c\n\u22a2 \u2203 b,\n    TrCfg (TM2.stepAux q (stVar v (S k) o) (update S k (stWrite v (S k) o))) b \u2227\n      Reaches (TM1.step (tr M)) (TM1.stepAux (goto fun x x => go k o q) v (Tape.mk' \u2205 (addBottom T))) b"}, {"tactic": "refine' \u27e8c, gc, (this.to\u2080.trans (tr_respects_aux\u2083 M _) c (TransGen.head' rfl _)).to_reflTransGen\u27e9", "annotated_tactic": ["refine' \u27e8c, gc, (this.to\u2080.trans (<a>tr_respects_aux\u2083</a> M _) c (<a>TransGen.head'</a> <a>rfl</a> _)).<a>to_reflTransGen</a>\u27e9", [{"full_name": "Turing.TM2to1.tr_respects_aux\u2083", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [2667, 9], "def_end_pos": [2667, 25]}, {"full_name": "Relation.TransGen.head'", "def_path": "Mathlib/Logic/Relation.lean", "def_pos": [375, 9], "def_end_pos": [375, 14]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}, {"full_name": "Relation.TransGen.to_reflTransGen", "def_path": "Mathlib/Logic/Relation.lean", "def_pos": [351, 9], "def_end_pos": [351, 24]}]], "state_before": "case intro.intro.intro.intro\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nq : TM2.Stmt (fun k => \u0393 k) \u039b \u03c3\nv : \u03c3\nT : ListBlank ((i : K) \u2192 Option (\u0393 i))\nk : K\nS : (k : K) \u2192 List (\u0393 k)\nhT : \u2200 (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.reverse (List.map some (S k)))\no : StAct k\nIH :\n  \u2200 {v : \u03c3} {S : (k : K) \u2192 List (\u0393 k)} {T : ListBlank ((k : K) \u2192 Option (\u0393 k))},\n    (\u2200 (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.reverse (List.map some (S k)))) \u2192\n      \u2203 b,\n        TrCfg (TM2.stepAux q v S) b \u2227\n          Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q) v (Tape.mk' \u2205 (addBottom T))) b\nhgo :\n  Reaches\u2080 (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' \u2205 (addBottom T) }\n    { l := some (go k o q), var := v, Tape := (Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom T)) }\nT' : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhT' :\n  \u2200 (k_1 : K),\n    ListBlank.map (proj k_1) T' =\n      ListBlank.mk (List.reverse (List.map some (update (fun k => S k) k (stWrite v (S k) o) k_1)))\nhrun :\n  TM1.stepAux (trStAct (goto fun x x => ret q) o) v\n      ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom T))) =\n    TM1.stepAux (goto fun x x => ret q) (stVar v (S k) o)\n      ((Tape.move Dir.right)^[List.length (update (fun k => S k) k (stWrite v (S k) o) k)] (Tape.mk' \u2205 (addBottom T')))\nthis :\n  Reaches\u2081 (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' \u2205 (addBottom T) }\n    (match\n      match none with\n      | some val => false\n      | none => true with\n    | true =>\n      TM1.stepAux (goto fun x x => ret q) (stVar v (S k) o)\n        ((Tape.move Dir.right)^[List.length (update (fun k => S k) k (stWrite v (S k) o) k)]\n          (Tape.mk' \u2205 (addBottom T')))\n    | false =>\n      TM1.stepAux (goto fun x x => go k o q) v\n        (Tape.move Dir.right ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom T)))))\nc : TM1.Cfg \u0393' \u039b' \u03c3\ngc : TrCfg (TM2.stepAux q ?m.726996 fun k_1 => update (fun k => S k) k (stWrite v (S k) o) k_1) c\nrc : Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q) ?m.726996 (Tape.mk' \u2205 (addBottom T'))) c\n\u22a2 \u2203 b,\n    TrCfg (TM2.stepAux q (stVar v (S k) o) (update S k (stWrite v (S k) o))) b \u2227\n      Reaches (TM1.step (tr M)) (TM1.stepAux (goto fun x x => go k o q) v (Tape.mk' \u2205 (addBottom T))) b", "state_after": "case intro.intro.intro.intro\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nq : TM2.Stmt (fun k => \u0393 k) \u039b \u03c3\nv : \u03c3\nT : ListBlank ((i : K) \u2192 Option (\u0393 i))\nk : K\nS : (k : K) \u2192 List (\u0393 k)\nhT : \u2200 (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.reverse (List.map some (S k)))\no : StAct k\nIH :\n  \u2200 {v : \u03c3} {S : (k : K) \u2192 List (\u0393 k)} {T : ListBlank ((k : K) \u2192 Option (\u0393 k))},\n    (\u2200 (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.reverse (List.map some (S k)))) \u2192\n      \u2203 b,\n        TrCfg (TM2.stepAux q v S) b \u2227\n          Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q) v (Tape.mk' \u2205 (addBottom T))) b\nhgo :\n  Reaches\u2080 (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' \u2205 (addBottom T) }\n    { l := some (go k o q), var := v, Tape := (Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom T)) }\nT' : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhT' :\n  \u2200 (k_1 : K),\n    ListBlank.map (proj k_1) T' =\n      ListBlank.mk (List.reverse (List.map some (update (fun k => S k) k (stWrite v (S k) o) k_1)))\nhrun :\n  TM1.stepAux (trStAct (goto fun x x => ret q) o) v\n      ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom T))) =\n    TM1.stepAux (goto fun x x => ret q) (stVar v (S k) o)\n      ((Tape.move Dir.right)^[List.length (update (fun k => S k) k (stWrite v (S k) o) k)] (Tape.mk' \u2205 (addBottom T')))\nthis :\n  Reaches\u2081 (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' \u2205 (addBottom T) }\n    (match\n      match none with\n      | some val => false\n      | none => true with\n    | true =>\n      TM1.stepAux (goto fun x x => ret q) (stVar v (S k) o)\n        ((Tape.move Dir.right)^[List.length (update (fun k => S k) k (stWrite v (S k) o) k)]\n          (Tape.mk' \u2205 (addBottom T')))\n    | false =>\n      TM1.stepAux (goto fun x x => go k o q) v\n        (Tape.move Dir.right ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom T)))))\nc : TM1.Cfg \u0393' \u039b' \u03c3\ngc : TrCfg (TM2.stepAux q (stVar v (S k) o) fun k_1 => update (fun k => S k) k (stWrite v (S k) o) k_1) c\nrc : Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q) (stVar v (S k) o) (Tape.mk' \u2205 (addBottom T'))) c\n\u22a2 ReflTransGen (fun a b => b \u2208 TM1.step (tr M) a)\n    (TM1.stepAux (tr M (ret q)) (stVar v (S k) o) (Tape.mk' \u2205 (addBottom T'))) c"}, {"tactic": "rw [tr, TM1.stepAux, Tape.mk'_head, addBottom_head_fst]", "annotated_tactic": ["rw [<a>tr</a>, <a>TM1.stepAux</a>, <a>Tape.mk'_head</a>, <a>addBottom_head_fst</a>]", [{"full_name": "Turing.TM2to1.tr", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [2637, 5], "def_end_pos": [2637, 7]}, {"full_name": "Turing.TM1.stepAux", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1281, 5], "def_end_pos": [1281, 12]}, {"full_name": "Turing.Tape.mk'_head", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [555, 9], "def_end_pos": [555, 22]}, {"full_name": "Turing.TM2to1.addBottom_head_fst", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [2396, 9], "def_end_pos": [2396, 27]}]], "state_before": "case intro.intro.intro.intro\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nq : TM2.Stmt (fun k => \u0393 k) \u039b \u03c3\nv : \u03c3\nT : ListBlank ((i : K) \u2192 Option (\u0393 i))\nk : K\nS : (k : K) \u2192 List (\u0393 k)\nhT : \u2200 (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.reverse (List.map some (S k)))\no : StAct k\nIH :\n  \u2200 {v : \u03c3} {S : (k : K) \u2192 List (\u0393 k)} {T : ListBlank ((k : K) \u2192 Option (\u0393 k))},\n    (\u2200 (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.reverse (List.map some (S k)))) \u2192\n      \u2203 b,\n        TrCfg (TM2.stepAux q v S) b \u2227\n          Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q) v (Tape.mk' \u2205 (addBottom T))) b\nhgo :\n  Reaches\u2080 (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' \u2205 (addBottom T) }\n    { l := some (go k o q), var := v, Tape := (Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom T)) }\nT' : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhT' :\n  \u2200 (k_1 : K),\n    ListBlank.map (proj k_1) T' =\n      ListBlank.mk (List.reverse (List.map some (update (fun k => S k) k (stWrite v (S k) o) k_1)))\nhrun :\n  TM1.stepAux (trStAct (goto fun x x => ret q) o) v\n      ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom T))) =\n    TM1.stepAux (goto fun x x => ret q) (stVar v (S k) o)\n      ((Tape.move Dir.right)^[List.length (update (fun k => S k) k (stWrite v (S k) o) k)] (Tape.mk' \u2205 (addBottom T')))\nthis :\n  Reaches\u2081 (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' \u2205 (addBottom T) }\n    (match\n      match none with\n      | some val => false\n      | none => true with\n    | true =>\n      TM1.stepAux (goto fun x x => ret q) (stVar v (S k) o)\n        ((Tape.move Dir.right)^[List.length (update (fun k => S k) k (stWrite v (S k) o) k)]\n          (Tape.mk' \u2205 (addBottom T')))\n    | false =>\n      TM1.stepAux (goto fun x x => go k o q) v\n        (Tape.move Dir.right ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom T)))))\nc : TM1.Cfg \u0393' \u039b' \u03c3\ngc : TrCfg (TM2.stepAux q (stVar v (S k) o) fun k_1 => update (fun k => S k) k (stWrite v (S k) o) k_1) c\nrc : Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q) (stVar v (S k) o) (Tape.mk' \u2205 (addBottom T'))) c\n\u22a2 ReflTransGen (fun a b => b \u2208 TM1.step (tr M) a)\n    (TM1.stepAux (tr M (ret q)) (stVar v (S k) o) (Tape.mk' \u2205 (addBottom T'))) c", "state_after": "case intro.intro.intro.intro\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nq : TM2.Stmt (fun k => \u0393 k) \u039b \u03c3\nv : \u03c3\nT : ListBlank ((i : K) \u2192 Option (\u0393 i))\nk : K\nS : (k : K) \u2192 List (\u0393 k)\nhT : \u2200 (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.reverse (List.map some (S k)))\no : StAct k\nIH :\n  \u2200 {v : \u03c3} {S : (k : K) \u2192 List (\u0393 k)} {T : ListBlank ((k : K) \u2192 Option (\u0393 k))},\n    (\u2200 (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.reverse (List.map some (S k)))) \u2192\n      \u2203 b,\n        TrCfg (TM2.stepAux q v S) b \u2227\n          Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q) v (Tape.mk' \u2205 (addBottom T))) b\nhgo :\n  Reaches\u2080 (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' \u2205 (addBottom T) }\n    { l := some (go k o q), var := v, Tape := (Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom T)) }\nT' : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhT' :\n  \u2200 (k_1 : K),\n    ListBlank.map (proj k_1) T' =\n      ListBlank.mk (List.reverse (List.map some (update (fun k => S k) k (stWrite v (S k) o) k_1)))\nhrun :\n  TM1.stepAux (trStAct (goto fun x x => ret q) o) v\n      ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom T))) =\n    TM1.stepAux (goto fun x x => ret q) (stVar v (S k) o)\n      ((Tape.move Dir.right)^[List.length (update (fun k => S k) k (stWrite v (S k) o) k)] (Tape.mk' \u2205 (addBottom T')))\nthis :\n  Reaches\u2081 (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' \u2205 (addBottom T) }\n    (match\n      match none with\n      | some val => false\n      | none => true with\n    | true =>\n      TM1.stepAux (goto fun x x => ret q) (stVar v (S k) o)\n        ((Tape.move Dir.right)^[List.length (update (fun k => S k) k (stWrite v (S k) o) k)]\n          (Tape.mk' \u2205 (addBottom T')))\n    | false =>\n      TM1.stepAux (goto fun x x => go k o q) v\n        (Tape.move Dir.right ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom T)))))\nc : TM1.Cfg \u0393' \u039b' \u03c3\ngc : TrCfg (TM2.stepAux q (stVar v (S k) o) fun k_1 => update (fun k => S k) k (stWrite v (S k) o) k_1) c\nrc : Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q) (stVar v (S k) o) (Tape.mk' \u2205 (addBottom T'))) c\n\u22a2 ReflTransGen (fun a b => b \u2208 TM1.step (tr M) a)\n    (bif true then TM1.stepAux (trNormal q) (stVar v (S k) o) (Tape.mk' \u2205 (addBottom T'))\n    else TM1.stepAux (move Dir.left (goto fun x x => ret q)) (stVar v (S k) o) (Tape.mk' \u2205 (addBottom T')))\n    c"}, {"tactic": "exact rc", "annotated_tactic": ["exact rc", []], "state_before": "case intro.intro.intro.intro\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nq : TM2.Stmt (fun k => \u0393 k) \u039b \u03c3\nv : \u03c3\nT : ListBlank ((i : K) \u2192 Option (\u0393 i))\nk : K\nS : (k : K) \u2192 List (\u0393 k)\nhT : \u2200 (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.reverse (List.map some (S k)))\no : StAct k\nIH :\n  \u2200 {v : \u03c3} {S : (k : K) \u2192 List (\u0393 k)} {T : ListBlank ((k : K) \u2192 Option (\u0393 k))},\n    (\u2200 (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.reverse (List.map some (S k)))) \u2192\n      \u2203 b,\n        TrCfg (TM2.stepAux q v S) b \u2227\n          Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q) v (Tape.mk' \u2205 (addBottom T))) b\nhgo :\n  Reaches\u2080 (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' \u2205 (addBottom T) }\n    { l := some (go k o q), var := v, Tape := (Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom T)) }\nT' : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhT' :\n  \u2200 (k_1 : K),\n    ListBlank.map (proj k_1) T' =\n      ListBlank.mk (List.reverse (List.map some (update (fun k => S k) k (stWrite v (S k) o) k_1)))\nhrun :\n  TM1.stepAux (trStAct (goto fun x x => ret q) o) v\n      ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom T))) =\n    TM1.stepAux (goto fun x x => ret q) (stVar v (S k) o)\n      ((Tape.move Dir.right)^[List.length (update (fun k => S k) k (stWrite v (S k) o) k)] (Tape.mk' \u2205 (addBottom T')))\nthis :\n  Reaches\u2081 (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' \u2205 (addBottom T) }\n    (match\n      match none with\n      | some val => false\n      | none => true with\n    | true =>\n      TM1.stepAux (goto fun x x => ret q) (stVar v (S k) o)\n        ((Tape.move Dir.right)^[List.length (update (fun k => S k) k (stWrite v (S k) o) k)]\n          (Tape.mk' \u2205 (addBottom T')))\n    | false =>\n      TM1.stepAux (goto fun x x => go k o q) v\n        (Tape.move Dir.right ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom T)))))\nc : TM1.Cfg \u0393' \u039b' \u03c3\ngc : TrCfg (TM2.stepAux q (stVar v (S k) o) fun k_1 => update (fun k => S k) k (stWrite v (S k) o) k_1) c\nrc : Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q) (stVar v (S k) o) (Tape.mk' \u2205 (addBottom T'))) c\n\u22a2 ReflTransGen (fun a b => b \u2208 TM1.step (tr M) a)\n    (bif true then TM1.stepAux (trNormal q) (stVar v (S k) o) (Tape.mk' \u2205 (addBottom T'))\n    else TM1.stepAux (move Dir.left (goto fun x x => ret q)) (stVar v (S k) o) (Tape.mk' \u2205 (addBottom T')))\n    c", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Independence/Kernel.lean", "full_name": "ProbabilityTheory.kernel.IndepSets.indep", "start": [353, 1], "end": [389, 56], "traced_tactics": [{"tactic": "intros t1 t2 ht1 ht2", "annotated_tactic": ["intros t1 t2 ht1 ht2", []], "state_before": "\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\nm1 m2 m : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\ninst\u271d : IsMarkovKernel \u03ba\np1 p2 : Set (Set \u03a9)\nh1 : m1 \u2264 m\nh2 : m2 \u2264 m\nhp1 : IsPiSystem p1\nhp2 : IsPiSystem p2\nhpm1 : m1 = generateFrom p1\nhpm2 : m2 = generateFrom p2\nhyp : IndepSets p1 p2 \u03ba\n\u22a2 Indep m1 m2 \u03ba", "state_after": "\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\nm1 m2 m : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\ninst\u271d : IsMarkovKernel \u03ba\np1 p2 : Set (Set \u03a9)\nh1 : m1 \u2264 m\nh2 : m2 \u2264 m\nhp1 : IsPiSystem p1\nhp2 : IsPiSystem p2\nhpm1 : m1 = generateFrom p1\nhpm2 : m2 = generateFrom p2\nhyp : IndepSets p1 p2 \u03ba\nt1 t2 : Set \u03a9\nht1 : t1 \u2208 {s | MeasurableSet s}\nht2 : t2 \u2208 {s | MeasurableSet s}\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (t1 \u2229 t2) = \u2191\u2191(\u2191\u03ba a) t1 * \u2191\u2191(\u2191\u03ba a) t2"}, {"tactic": "refine @induction_on_inter _ (fun t \u21a6 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u03ba a (t \u2229 t2) = \u03ba a t * \u03ba a t2) _ m1 hpm1 hp1\n  ?_ ?_ ?_ ?_ _ ht1", "annotated_tactic": ["refine @<a>induction_on_inter</a> _ (fun t \u21a6 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u03ba a (t \u2229 t2) = \u03ba a t * \u03ba a t2) _ m1 hpm1 hp1\n    ?_ ?_ ?_ ?_ _ ht1", [{"full_name": "MeasurableSpace.induction_on_inter", "def_path": "Mathlib/MeasureTheory/PiSystem.lean", "def_pos": [745, 9], "def_end_pos": [745, 27]}]], "state_before": "\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\nm1 m2 m : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\ninst\u271d : IsMarkovKernel \u03ba\np1 p2 : Set (Set \u03a9)\nh1 : m1 \u2264 m\nh2 : m2 \u2264 m\nhp1 : IsPiSystem p1\nhp2 : IsPiSystem p2\nhpm1 : m1 = generateFrom p1\nhpm2 : m2 = generateFrom p2\nhyp : IndepSets p1 p2 \u03ba\nt1 t2 : Set \u03a9\nht1 : t1 \u2208 {s | MeasurableSet s}\nht2 : t2 \u2208 {s | MeasurableSet s}\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (t1 \u2229 t2) = \u2191\u2191(\u2191\u03ba a) t1 * \u2191\u2191(\u2191\u03ba a) t2", "state_after": "case refine_1\n\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\nm1 m2 m : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\ninst\u271d : IsMarkovKernel \u03ba\np1 p2 : Set (Set \u03a9)\nh1 : m1 \u2264 m\nh2 : m2 \u2264 m\nhp1 : IsPiSystem p1\nhp2 : IsPiSystem p2\nhpm1 : m1 = generateFrom p1\nhpm2 : m2 = generateFrom p2\nhyp : IndepSets p1 p2 \u03ba\nt1 t2 : Set \u03a9\nht1 : t1 \u2208 {s | MeasurableSet s}\nht2 : t2 \u2208 {s | MeasurableSet s}\n\u22a2 (fun t => \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (t \u2229 t2) = \u2191\u2191(\u2191\u03ba a) t * \u2191\u2191(\u2191\u03ba a) t2) \u2205\n\ncase refine_2\n\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\nm1 m2 m : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\ninst\u271d : IsMarkovKernel \u03ba\np1 p2 : Set (Set \u03a9)\nh1 : m1 \u2264 m\nh2 : m2 \u2264 m\nhp1 : IsPiSystem p1\nhp2 : IsPiSystem p2\nhpm1 : m1 = generateFrom p1\nhpm2 : m2 = generateFrom p2\nhyp : IndepSets p1 p2 \u03ba\nt1 t2 : Set \u03a9\nht1 : t1 \u2208 {s | MeasurableSet s}\nht2 : t2 \u2208 {s | MeasurableSet s}\n\u22a2 \u2200 (t : Set \u03a9), t \u2208 p1 \u2192 (fun t => \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (t \u2229 t2) = \u2191\u2191(\u2191\u03ba a) t * \u2191\u2191(\u2191\u03ba a) t2) t\n\ncase refine_3\n\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\nm1 m2 m : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\ninst\u271d : IsMarkovKernel \u03ba\np1 p2 : Set (Set \u03a9)\nh1 : m1 \u2264 m\nh2 : m2 \u2264 m\nhp1 : IsPiSystem p1\nhp2 : IsPiSystem p2\nhpm1 : m1 = generateFrom p1\nhpm2 : m2 = generateFrom p2\nhyp : IndepSets p1 p2 \u03ba\nt1 t2 : Set \u03a9\nht1 : t1 \u2208 {s | MeasurableSet s}\nht2 : t2 \u2208 {s | MeasurableSet s}\n\u22a2 \u2200 (t : Set \u03a9),\n    MeasurableSet t \u2192\n      (fun t => \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (t \u2229 t2) = \u2191\u2191(\u2191\u03ba a) t * \u2191\u2191(\u2191\u03ba a) t2) t \u2192\n        (fun t => \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (t \u2229 t2) = \u2191\u2191(\u2191\u03ba a) t * \u2191\u2191(\u2191\u03ba a) t2) t\u1d9c\n\ncase refine_4\n\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\nm1 m2 m : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\ninst\u271d : IsMarkovKernel \u03ba\np1 p2 : Set (Set \u03a9)\nh1 : m1 \u2264 m\nh2 : m2 \u2264 m\nhp1 : IsPiSystem p1\nhp2 : IsPiSystem p2\nhpm1 : m1 = generateFrom p1\nhpm2 : m2 = generateFrom p2\nhyp : IndepSets p1 p2 \u03ba\nt1 t2 : Set \u03a9\nht1 : t1 \u2208 {s | MeasurableSet s}\nht2 : t2 \u2208 {s | MeasurableSet s}\n\u22a2 \u2200 (f : \u2115 \u2192 Set \u03a9),\n    Pairwise (Disjoint on f) \u2192\n      (\u2200 (i : \u2115), MeasurableSet (f i)) \u2192\n        (\u2200 (i : \u2115), (fun t => \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (t \u2229 t2) = \u2191\u2191(\u2191\u03ba a) t * \u2191\u2191(\u2191\u03ba a) t2) (f i)) \u2192\n          (fun t => \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (t \u2229 t2) = \u2191\u2191(\u2191\u03ba a) t * \u2191\u2191(\u2191\u03ba a) t2) (\u22c3 i, f i)"}, {"tactic": "simp only [Set.empty_inter, measure_empty, zero_mul, eq_self_iff_true,\n  Filter.eventually_true]", "annotated_tactic": ["simp only [<a>Set.empty_inter</a>, <a>measure_empty</a>, <a>zero_mul</a>, <a>eq_self_iff_true</a>,\n      <a>Filter.eventually_true</a>]", [{"full_name": "Set.empty_inter", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [936, 9], "def_end_pos": [936, 20]}, {"full_name": "MeasureTheory.measure_empty", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [185, 9], "def_end_pos": [185, 22]}, {"full_name": "MulZeroClass.zero_mul", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [36, 3], "def_end_pos": [36, 11]}, {"full_name": "eq_self_iff_true", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [86, 9], "def_end_pos": [86, 25]}, {"full_name": "Filter.eventually_true", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1108, 17], "def_end_pos": [1108, 32]}]], "state_before": "case refine_1\n\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\nm1 m2 m : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\ninst\u271d : IsMarkovKernel \u03ba\np1 p2 : Set (Set \u03a9)\nh1 : m1 \u2264 m\nh2 : m2 \u2264 m\nhp1 : IsPiSystem p1\nhp2 : IsPiSystem p2\nhpm1 : m1 = generateFrom p1\nhpm2 : m2 = generateFrom p2\nhyp : IndepSets p1 p2 \u03ba\nt1 t2 : Set \u03a9\nht1 : t1 \u2208 {s | MeasurableSet s}\nht2 : t2 \u2208 {s | MeasurableSet s}\n\u22a2 (fun t => \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (t \u2229 t2) = \u2191\u2191(\u2191\u03ba a) t * \u2191\u2191(\u2191\u03ba a) t2) \u2205", "state_after": "no goals"}, {"tactic": "intros t ht_mem_p1", "annotated_tactic": ["intros t ht_mem_p1", []], "state_before": "case refine_2\n\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\nm1 m2 m : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\ninst\u271d : IsMarkovKernel \u03ba\np1 p2 : Set (Set \u03a9)\nh1 : m1 \u2264 m\nh2 : m2 \u2264 m\nhp1 : IsPiSystem p1\nhp2 : IsPiSystem p2\nhpm1 : m1 = generateFrom p1\nhpm2 : m2 = generateFrom p2\nhyp : IndepSets p1 p2 \u03ba\nt1 t2 : Set \u03a9\nht1 : t1 \u2208 {s | MeasurableSet s}\nht2 : t2 \u2208 {s | MeasurableSet s}\n\u22a2 \u2200 (t : Set \u03a9), t \u2208 p1 \u2192 (fun t => \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (t \u2229 t2) = \u2191\u2191(\u2191\u03ba a) t * \u2191\u2191(\u2191\u03ba a) t2) t", "state_after": "case refine_2\n\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\nm1 m2 m : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\ninst\u271d : IsMarkovKernel \u03ba\np1 p2 : Set (Set \u03a9)\nh1 : m1 \u2264 m\nh2 : m2 \u2264 m\nhp1 : IsPiSystem p1\nhp2 : IsPiSystem p2\nhpm1 : m1 = generateFrom p1\nhpm2 : m2 = generateFrom p2\nhyp : IndepSets p1 p2 \u03ba\nt1 t2 : Set \u03a9\nht1 : t1 \u2208 {s | MeasurableSet s}\nht2 : t2 \u2208 {s | MeasurableSet s}\nt : Set \u03a9\nht_mem_p1 : t \u2208 p1\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (t \u2229 t2) = \u2191\u2191(\u2191\u03ba a) t * \u2191\u2191(\u2191\u03ba a) t2"}, {"tactic": "have ht1 : MeasurableSet[m] t := by\n  refine h1 _ ?_\n  rw [hpm1]\n  exact measurableSet_generateFrom ht_mem_p1", "annotated_tactic": ["have ht1 : MeasurableSet[m] t := by\n      refine h1 _ ?_\n      rw [hpm1]\n      exact <a>measurableSet_generateFrom</a> ht_mem_p1", [{"full_name": "MeasurableSpace.measurableSet_generateFrom", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [370, 9], "def_end_pos": [370, 35]}]], "state_before": "case refine_2\n\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\nm1 m2 m : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\ninst\u271d : IsMarkovKernel \u03ba\np1 p2 : Set (Set \u03a9)\nh1 : m1 \u2264 m\nh2 : m2 \u2264 m\nhp1 : IsPiSystem p1\nhp2 : IsPiSystem p2\nhpm1 : m1 = generateFrom p1\nhpm2 : m2 = generateFrom p2\nhyp : IndepSets p1 p2 \u03ba\nt1 t2 : Set \u03a9\nht1 : t1 \u2208 {s | MeasurableSet s}\nht2 : t2 \u2208 {s | MeasurableSet s}\nt : Set \u03a9\nht_mem_p1 : t \u2208 p1\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (t \u2229 t2) = \u2191\u2191(\u2191\u03ba a) t * \u2191\u2191(\u2191\u03ba a) t2", "state_after": "case refine_2\n\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\nm1 m2 m : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\ninst\u271d : IsMarkovKernel \u03ba\np1 p2 : Set (Set \u03a9)\nh1 : m1 \u2264 m\nh2 : m2 \u2264 m\nhp1 : IsPiSystem p1\nhp2 : IsPiSystem p2\nhpm1 : m1 = generateFrom p1\nhpm2 : m2 = generateFrom p2\nhyp : IndepSets p1 p2 \u03ba\nt1 t2 : Set \u03a9\nht1\u271d : t1 \u2208 {s | MeasurableSet s}\nht2 : t2 \u2208 {s | MeasurableSet s}\nt : Set \u03a9\nht_mem_p1 : t \u2208 p1\nht1 : MeasurableSet t\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (t \u2229 t2) = \u2191\u2191(\u2191\u03ba a) t * \u2191\u2191(\u2191\u03ba a) t2"}, {"tactic": "exact IndepSets.indep_aux h2 hp2 hpm2 hyp ht_mem_p1 ht1 ht2", "annotated_tactic": ["exact <a>IndepSets.indep_aux</a> h2 hp2 hpm2 hyp ht_mem_p1 ht1 ht2", [{"full_name": "ProbabilityTheory.kernel.IndepSets.indep_aux", "def_path": "Mathlib/Probability/Independence/Kernel.lean", "def_pos": [321, 9], "def_end_pos": [321, 28]}]], "state_before": "case refine_2\n\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\nm1 m2 m : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\ninst\u271d : IsMarkovKernel \u03ba\np1 p2 : Set (Set \u03a9)\nh1 : m1 \u2264 m\nh2 : m2 \u2264 m\nhp1 : IsPiSystem p1\nhp2 : IsPiSystem p2\nhpm1 : m1 = generateFrom p1\nhpm2 : m2 = generateFrom p2\nhyp : IndepSets p1 p2 \u03ba\nt1 t2 : Set \u03a9\nht1\u271d : t1 \u2208 {s | MeasurableSet s}\nht2 : t2 \u2208 {s | MeasurableSet s}\nt : Set \u03a9\nht_mem_p1 : t \u2208 p1\nht1 : MeasurableSet t\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (t \u2229 t2) = \u2191\u2191(\u2191\u03ba a) t * \u2191\u2191(\u2191\u03ba a) t2", "state_after": "no goals"}, {"tactic": "exact measurableSet_generateFrom ht_mem_p1", "annotated_tactic": ["exact <a>measurableSet_generateFrom</a> ht_mem_p1", [{"full_name": "MeasurableSpace.measurableSet_generateFrom", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [370, 9], "def_end_pos": [370, 35]}]], "state_before": "\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\nm1 m2 m : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\ninst\u271d : IsMarkovKernel \u03ba\np1 p2 : Set (Set \u03a9)\nh1 : m1 \u2264 m\nh2 : m2 \u2264 m\nhp1 : IsPiSystem p1\nhp2 : IsPiSystem p2\nhpm1 : m1 = generateFrom p1\nhpm2 : m2 = generateFrom p2\nhyp : IndepSets p1 p2 \u03ba\nt1 t2 : Set \u03a9\nht1 : t1 \u2208 {s | MeasurableSet s}\nht2 : t2 \u2208 {s | MeasurableSet s}\nt : Set \u03a9\nht_mem_p1 : t \u2208 p1\n\u22a2 MeasurableSet t", "state_after": "no goals"}, {"tactic": "intros t ht h", "annotated_tactic": ["intros t ht h", []], "state_before": "case refine_3\n\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\nm1 m2 m : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\ninst\u271d : IsMarkovKernel \u03ba\np1 p2 : Set (Set \u03a9)\nh1 : m1 \u2264 m\nh2 : m2 \u2264 m\nhp1 : IsPiSystem p1\nhp2 : IsPiSystem p2\nhpm1 : m1 = generateFrom p1\nhpm2 : m2 = generateFrom p2\nhyp : IndepSets p1 p2 \u03ba\nt1 t2 : Set \u03a9\nht1 : t1 \u2208 {s | MeasurableSet s}\nht2 : t2 \u2208 {s | MeasurableSet s}\n\u22a2 \u2200 (t : Set \u03a9),\n    MeasurableSet t \u2192\n      (fun t => \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (t \u2229 t2) = \u2191\u2191(\u2191\u03ba a) t * \u2191\u2191(\u2191\u03ba a) t2) t \u2192\n        (fun t => \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (t \u2229 t2) = \u2191\u2191(\u2191\u03ba a) t * \u2191\u2191(\u2191\u03ba a) t2) t\u1d9c", "state_after": "case refine_3\n\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\nm1 m2 m : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\ninst\u271d : IsMarkovKernel \u03ba\np1 p2 : Set (Set \u03a9)\nh1 : m1 \u2264 m\nh2 : m2 \u2264 m\nhp1 : IsPiSystem p1\nhp2 : IsPiSystem p2\nhpm1 : m1 = generateFrom p1\nhpm2 : m2 = generateFrom p2\nhyp : IndepSets p1 p2 \u03ba\nt1 t2 : Set \u03a9\nht1 : t1 \u2208 {s | MeasurableSet s}\nht2 : t2 \u2208 {s | MeasurableSet s}\nt : Set \u03a9\nht : MeasurableSet t\nh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (t \u2229 t2) = \u2191\u2191(\u2191\u03ba a) t * \u2191\u2191(\u2191\u03ba a) t2\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (t\u1d9c \u2229 t2) = \u2191\u2191(\u2191\u03ba a) t\u1d9c * \u2191\u2191(\u2191\u03ba a) t2"}, {"tactic": "filter_upwards [h] with a ha", "annotated_tactic": ["filter_upwards [h] with a ha", []], "state_before": "case refine_3\n\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\nm1 m2 m : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\ninst\u271d : IsMarkovKernel \u03ba\np1 p2 : Set (Set \u03a9)\nh1 : m1 \u2264 m\nh2 : m2 \u2264 m\nhp1 : IsPiSystem p1\nhp2 : IsPiSystem p2\nhpm1 : m1 = generateFrom p1\nhpm2 : m2 = generateFrom p2\nhyp : IndepSets p1 p2 \u03ba\nt1 t2 : Set \u03a9\nht1 : t1 \u2208 {s | MeasurableSet s}\nht2 : t2 \u2208 {s | MeasurableSet s}\nt : Set \u03a9\nht : MeasurableSet t\nh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (t \u2229 t2) = \u2191\u2191(\u2191\u03ba a) t * \u2191\u2191(\u2191\u03ba a) t2\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (t\u1d9c \u2229 t2) = \u2191\u2191(\u2191\u03ba a) t\u1d9c * \u2191\u2191(\u2191\u03ba a) t2", "state_after": "case h\n\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\nm1 m2 m : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\ninst\u271d : IsMarkovKernel \u03ba\np1 p2 : Set (Set \u03a9)\nh1 : m1 \u2264 m\nh2 : m2 \u2264 m\nhp1 : IsPiSystem p1\nhp2 : IsPiSystem p2\nhpm1 : m1 = generateFrom p1\nhpm2 : m2 = generateFrom p2\nhyp : IndepSets p1 p2 \u03ba\nt1 t2 : Set \u03a9\nht1 : t1 \u2208 {s | MeasurableSet s}\nht2 : t2 \u2208 {s | MeasurableSet s}\nt : Set \u03a9\nht : MeasurableSet t\nh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (t \u2229 t2) = \u2191\u2191(\u2191\u03ba a) t * \u2191\u2191(\u2191\u03ba a) t2\na : \u03b1\nha : \u2191\u2191(\u2191\u03ba a) (t \u2229 t2) = \u2191\u2191(\u2191\u03ba a) t * \u2191\u2191(\u2191\u03ba a) t2\n\u22a2 \u2191\u2191(\u2191\u03ba a) (t\u1d9c \u2229 t2) = \u2191\u2191(\u2191\u03ba a) t\u1d9c * \u2191\u2191(\u2191\u03ba a) t2"}, {"tactic": "have : t\u1d9c \u2229 t2 = t2 \\ (t \u2229 t2) := by\n  rw [Set.inter_comm t, Set.diff_self_inter, Set.diff_eq_compl_inter]", "annotated_tactic": ["have : t\u1d9c \u2229 t2 = t2 \\ (t \u2229 t2) := by\n      rw [<a>Set.inter_comm</a> t, <a>Set.diff_self_inter</a>, <a>Set.diff_eq_compl_inter</a>]", [{"full_name": "Set.inter_comm", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [940, 9], "def_end_pos": [940, 19]}, {"full_name": "Set.diff_self_inter", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [2063, 9], "def_end_pos": [2063, 24]}, {"full_name": "Set.diff_eq_compl_inter", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1838, 9], "def_end_pos": [1838, 28]}]], "state_before": "case h\n\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\nm1 m2 m : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\ninst\u271d : IsMarkovKernel \u03ba\np1 p2 : Set (Set \u03a9)\nh1 : m1 \u2264 m\nh2 : m2 \u2264 m\nhp1 : IsPiSystem p1\nhp2 : IsPiSystem p2\nhpm1 : m1 = generateFrom p1\nhpm2 : m2 = generateFrom p2\nhyp : IndepSets p1 p2 \u03ba\nt1 t2 : Set \u03a9\nht1 : t1 \u2208 {s | MeasurableSet s}\nht2 : t2 \u2208 {s | MeasurableSet s}\nt : Set \u03a9\nht : MeasurableSet t\nh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (t \u2229 t2) = \u2191\u2191(\u2191\u03ba a) t * \u2191\u2191(\u2191\u03ba a) t2\na : \u03b1\nha : \u2191\u2191(\u2191\u03ba a) (t \u2229 t2) = \u2191\u2191(\u2191\u03ba a) t * \u2191\u2191(\u2191\u03ba a) t2\n\u22a2 \u2191\u2191(\u2191\u03ba a) (t\u1d9c \u2229 t2) = \u2191\u2191(\u2191\u03ba a) t\u1d9c * \u2191\u2191(\u2191\u03ba a) t2", "state_after": "case h\n\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\nm1 m2 m : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\ninst\u271d : IsMarkovKernel \u03ba\np1 p2 : Set (Set \u03a9)\nh1 : m1 \u2264 m\nh2 : m2 \u2264 m\nhp1 : IsPiSystem p1\nhp2 : IsPiSystem p2\nhpm1 : m1 = generateFrom p1\nhpm2 : m2 = generateFrom p2\nhyp : IndepSets p1 p2 \u03ba\nt1 t2 : Set \u03a9\nht1 : t1 \u2208 {s | MeasurableSet s}\nht2 : t2 \u2208 {s | MeasurableSet s}\nt : Set \u03a9\nht : MeasurableSet t\nh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (t \u2229 t2) = \u2191\u2191(\u2191\u03ba a) t * \u2191\u2191(\u2191\u03ba a) t2\na : \u03b1\nha : \u2191\u2191(\u2191\u03ba a) (t \u2229 t2) = \u2191\u2191(\u2191\u03ba a) t * \u2191\u2191(\u2191\u03ba a) t2\nthis : t\u1d9c \u2229 t2 = t2 \\ (t \u2229 t2)\n\u22a2 \u2191\u2191(\u2191\u03ba a) (t\u1d9c \u2229 t2) = \u2191\u2191(\u2191\u03ba a) t\u1d9c * \u2191\u2191(\u2191\u03ba a) t2"}, {"tactic": "rw [this, Set.inter_comm t t2,\n  measure_diff (Set.inter_subset_left _ _) ((h2 _ ht2).inter (h1 _ ht))\n    (measure_ne_top (\u03ba a) _),\n  Set.inter_comm, ha, measure_compl (h1 _ ht) (measure_ne_top (\u03ba a) t), measure_univ,\n  mul_comm (1 - \u03ba a t), ENNReal.mul_sub (fun _ _ \u21a6 measure_ne_top (\u03ba a) _), mul_one, mul_comm]", "annotated_tactic": ["rw [this, <a>Set.inter_comm</a> t t2,\n      <a>measure_diff</a> (<a>Set.inter_subset_left</a> _ _) ((h2 _ ht2).<a>inter</a> (h1 _ ht))\n        (<a>measure_ne_top</a> (\u03ba a) _),\n      <a>Set.inter_comm</a>, ha, <a>measure_compl</a> (h1 _ ht) (<a>measure_ne_top</a> (\u03ba a) t), <a>measure_univ</a>,\n      <a>mul_comm</a> (1 - \u03ba a t), <a>ENNReal.mul_sub</a> (fun _ _ \u21a6 <a>measure_ne_top</a> (\u03ba a) _), <a>mul_one</a>, <a>mul_comm</a>]", [{"full_name": "Set.inter_comm", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [940, 9], "def_end_pos": [940, 19]}, {"full_name": "MeasureTheory.measure_diff", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [252, 9], "def_end_pos": [252, 21]}, {"full_name": "Set.inter_subset_left", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [965, 9], "def_end_pos": [965, 26]}, {"full_name": "MeasurableSet.inter", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [198, 19], "def_end_pos": [198, 38]}, {"full_name": "MeasureTheory.measure_ne_top", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2875, 9], "def_end_pos": [2875, 23]}, {"full_name": "Set.inter_comm", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [940, 9], "def_end_pos": [940, 19]}, {"full_name": "MeasureTheory.measure_compl", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [303, 9], "def_end_pos": [303, 22]}, {"full_name": "MeasureTheory.measure_ne_top", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2875, 9], "def_end_pos": [2875, 23]}, {"full_name": "MeasureTheory.IsProbabilityMeasure.measure_univ", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3027, 3], "def_end_pos": [3027, 15]}, {"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [302, 9], "def_end_pos": [302, 17]}, {"full_name": "ENNReal.mul_sub", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1241, 9], "def_end_pos": [1241, 16]}, {"full_name": "MeasureTheory.measure_ne_top", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2875, 9], "def_end_pos": [2875, 23]}, {"full_name": "mul_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [470, 9], "def_end_pos": [470, 16]}, {"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [302, 9], "def_end_pos": [302, 17]}]], "state_before": "case h\n\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\nm1 m2 m : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\ninst\u271d : IsMarkovKernel \u03ba\np1 p2 : Set (Set \u03a9)\nh1 : m1 \u2264 m\nh2 : m2 \u2264 m\nhp1 : IsPiSystem p1\nhp2 : IsPiSystem p2\nhpm1 : m1 = generateFrom p1\nhpm2 : m2 = generateFrom p2\nhyp : IndepSets p1 p2 \u03ba\nt1 t2 : Set \u03a9\nht1 : t1 \u2208 {s | MeasurableSet s}\nht2 : t2 \u2208 {s | MeasurableSet s}\nt : Set \u03a9\nht : MeasurableSet t\nh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (t \u2229 t2) = \u2191\u2191(\u2191\u03ba a) t * \u2191\u2191(\u2191\u03ba a) t2\na : \u03b1\nha : \u2191\u2191(\u2191\u03ba a) (t \u2229 t2) = \u2191\u2191(\u2191\u03ba a) t * \u2191\u2191(\u2191\u03ba a) t2\nthis : t\u1d9c \u2229 t2 = t2 \\ (t \u2229 t2)\n\u22a2 \u2191\u2191(\u2191\u03ba a) (t\u1d9c \u2229 t2) = \u2191\u2191(\u2191\u03ba a) t\u1d9c * \u2191\u2191(\u2191\u03ba a) t2", "state_after": "no goals"}, {"tactic": "rw [Set.inter_comm t, Set.diff_self_inter, Set.diff_eq_compl_inter]", "annotated_tactic": ["rw [<a>Set.inter_comm</a> t, <a>Set.diff_self_inter</a>, <a>Set.diff_eq_compl_inter</a>]", [{"full_name": "Set.inter_comm", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [940, 9], "def_end_pos": [940, 19]}, {"full_name": "Set.diff_self_inter", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [2063, 9], "def_end_pos": [2063, 24]}, {"full_name": "Set.diff_eq_compl_inter", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1838, 9], "def_end_pos": [1838, 28]}]], "state_before": "\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\nm1 m2 m : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\ninst\u271d : IsMarkovKernel \u03ba\np1 p2 : Set (Set \u03a9)\nh1 : m1 \u2264 m\nh2 : m2 \u2264 m\nhp1 : IsPiSystem p1\nhp2 : IsPiSystem p2\nhpm1 : m1 = generateFrom p1\nhpm2 : m2 = generateFrom p2\nhyp : IndepSets p1 p2 \u03ba\nt1 t2 : Set \u03a9\nht1 : t1 \u2208 {s | MeasurableSet s}\nht2 : t2 \u2208 {s | MeasurableSet s}\nt : Set \u03a9\nht : MeasurableSet t\nh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (t \u2229 t2) = \u2191\u2191(\u2191\u03ba a) t * \u2191\u2191(\u2191\u03ba a) t2\na : \u03b1\nha : \u2191\u2191(\u2191\u03ba a) (t \u2229 t2) = \u2191\u2191(\u2191\u03ba a) t * \u2191\u2191(\u2191\u03ba a) t2\n\u22a2 t\u1d9c \u2229 t2 = t2 \\ (t \u2229 t2)", "state_after": "no goals"}, {"tactic": "intros f hf_disj hf_meas h", "annotated_tactic": ["intros f hf_disj hf_meas h", []], "state_before": "case refine_4\n\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\nm1 m2 m : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\ninst\u271d : IsMarkovKernel \u03ba\np1 p2 : Set (Set \u03a9)\nh1 : m1 \u2264 m\nh2 : m2 \u2264 m\nhp1 : IsPiSystem p1\nhp2 : IsPiSystem p2\nhpm1 : m1 = generateFrom p1\nhpm2 : m2 = generateFrom p2\nhyp : IndepSets p1 p2 \u03ba\nt1 t2 : Set \u03a9\nht1 : t1 \u2208 {s | MeasurableSet s}\nht2 : t2 \u2208 {s | MeasurableSet s}\n\u22a2 \u2200 (f : \u2115 \u2192 Set \u03a9),\n    Pairwise (Disjoint on f) \u2192\n      (\u2200 (i : \u2115), MeasurableSet (f i)) \u2192\n        (\u2200 (i : \u2115), (fun t => \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (t \u2229 t2) = \u2191\u2191(\u2191\u03ba a) t * \u2191\u2191(\u2191\u03ba a) t2) (f i)) \u2192\n          (fun t => \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (t \u2229 t2) = \u2191\u2191(\u2191\u03ba a) t * \u2191\u2191(\u2191\u03ba a) t2) (\u22c3 i, f i)", "state_after": "case refine_4\n\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\nm1 m2 m : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\ninst\u271d : IsMarkovKernel \u03ba\np1 p2 : Set (Set \u03a9)\nh1 : m1 \u2264 m\nh2 : m2 \u2264 m\nhp1 : IsPiSystem p1\nhp2 : IsPiSystem p2\nhpm1 : m1 = generateFrom p1\nhpm2 : m2 = generateFrom p2\nhyp : IndepSets p1 p2 \u03ba\nt1 t2 : Set \u03a9\nht1 : t1 \u2208 {s | MeasurableSet s}\nht2 : t2 \u2208 {s | MeasurableSet s}\nf : \u2115 \u2192 Set \u03a9\nhf_disj : Pairwise (Disjoint on f)\nhf_meas : \u2200 (i : \u2115), MeasurableSet (f i)\nh : \u2200 (i : \u2115), (fun t => \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (t \u2229 t2) = \u2191\u2191(\u2191\u03ba a) t * \u2191\u2191(\u2191\u03ba a) t2) (f i)\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) ((\u22c3 i, f i) \u2229 t2) = \u2191\u2191(\u2191\u03ba a) (\u22c3 i, f i) * \u2191\u2191(\u2191\u03ba a) t2"}, {"tactic": "rw [\u2190 ae_all_iff] at h", "annotated_tactic": ["rw [\u2190 <a>ae_all_iff</a>] at h", [{"full_name": "MeasureTheory.ae_all_iff", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [422, 9], "def_end_pos": [422, 19]}]], "state_before": "case refine_4\n\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\nm1 m2 m : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\ninst\u271d : IsMarkovKernel \u03ba\np1 p2 : Set (Set \u03a9)\nh1 : m1 \u2264 m\nh2 : m2 \u2264 m\nhp1 : IsPiSystem p1\nhp2 : IsPiSystem p2\nhpm1 : m1 = generateFrom p1\nhpm2 : m2 = generateFrom p2\nhyp : IndepSets p1 p2 \u03ba\nt1 t2 : Set \u03a9\nht1 : t1 \u2208 {s | MeasurableSet s}\nht2 : t2 \u2208 {s | MeasurableSet s}\nf : \u2115 \u2192 Set \u03a9\nhf_disj : Pairwise (Disjoint on f)\nhf_meas : \u2200 (i : \u2115), MeasurableSet (f i)\nh : \u2200 (i : \u2115), (fun t => \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (t \u2229 t2) = \u2191\u2191(\u2191\u03ba a) t * \u2191\u2191(\u2191\u03ba a) t2) (f i)\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) ((\u22c3 i, f i) \u2229 t2) = \u2191\u2191(\u2191\u03ba a) (\u22c3 i, f i) * \u2191\u2191(\u2191\u03ba a) t2", "state_after": "case refine_4\n\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\nm1 m2 m : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\ninst\u271d : IsMarkovKernel \u03ba\np1 p2 : Set (Set \u03a9)\nh1 : m1 \u2264 m\nh2 : m2 \u2264 m\nhp1 : IsPiSystem p1\nhp2 : IsPiSystem p2\nhpm1 : m1 = generateFrom p1\nhpm2 : m2 = generateFrom p2\nhyp : IndepSets p1 p2 \u03ba\nt1 t2 : Set \u03a9\nht1 : t1 \u2208 {s | MeasurableSet s}\nht2 : t2 \u2208 {s | MeasurableSet s}\nf : \u2115 \u2192 Set \u03a9\nhf_disj : Pairwise (Disjoint on f)\nhf_meas : \u2200 (i : \u2115), MeasurableSet (f i)\nh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (i : \u2115), \u2191\u2191(\u2191\u03ba a) (f i \u2229 t2) = \u2191\u2191(\u2191\u03ba a) (f i) * \u2191\u2191(\u2191\u03ba a) t2\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) ((\u22c3 i, f i) \u2229 t2) = \u2191\u2191(\u2191\u03ba a) (\u22c3 i, f i) * \u2191\u2191(\u2191\u03ba a) t2"}, {"tactic": "filter_upwards [h] with a ha", "annotated_tactic": ["filter_upwards [h] with a ha", []], "state_before": "case refine_4\n\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\nm1 m2 m : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\ninst\u271d : IsMarkovKernel \u03ba\np1 p2 : Set (Set \u03a9)\nh1 : m1 \u2264 m\nh2 : m2 \u2264 m\nhp1 : IsPiSystem p1\nhp2 : IsPiSystem p2\nhpm1 : m1 = generateFrom p1\nhpm2 : m2 = generateFrom p2\nhyp : IndepSets p1 p2 \u03ba\nt1 t2 : Set \u03a9\nht1 : t1 \u2208 {s | MeasurableSet s}\nht2 : t2 \u2208 {s | MeasurableSet s}\nf : \u2115 \u2192 Set \u03a9\nhf_disj : Pairwise (Disjoint on f)\nhf_meas : \u2200 (i : \u2115), MeasurableSet (f i)\nh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (i : \u2115), \u2191\u2191(\u2191\u03ba a) (f i \u2229 t2) = \u2191\u2191(\u2191\u03ba a) (f i) * \u2191\u2191(\u2191\u03ba a) t2\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) ((\u22c3 i, f i) \u2229 t2) = \u2191\u2191(\u2191\u03ba a) (\u22c3 i, f i) * \u2191\u2191(\u2191\u03ba a) t2", "state_after": "case h\n\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\nm1 m2 m : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\ninst\u271d : IsMarkovKernel \u03ba\np1 p2 : Set (Set \u03a9)\nh1 : m1 \u2264 m\nh2 : m2 \u2264 m\nhp1 : IsPiSystem p1\nhp2 : IsPiSystem p2\nhpm1 : m1 = generateFrom p1\nhpm2 : m2 = generateFrom p2\nhyp : IndepSets p1 p2 \u03ba\nt1 t2 : Set \u03a9\nht1 : t1 \u2208 {s | MeasurableSet s}\nht2 : t2 \u2208 {s | MeasurableSet s}\nf : \u2115 \u2192 Set \u03a9\nhf_disj : Pairwise (Disjoint on f)\nhf_meas : \u2200 (i : \u2115), MeasurableSet (f i)\nh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (i : \u2115), \u2191\u2191(\u2191\u03ba a) (f i \u2229 t2) = \u2191\u2191(\u2191\u03ba a) (f i) * \u2191\u2191(\u2191\u03ba a) t2\na : \u03b1\nha : \u2200 (i : \u2115), \u2191\u2191(\u2191\u03ba a) (f i \u2229 t2) = \u2191\u2191(\u2191\u03ba a) (f i) * \u2191\u2191(\u2191\u03ba a) t2\n\u22a2 \u2191\u2191(\u2191\u03ba a) ((\u22c3 i, f i) \u2229 t2) = \u2191\u2191(\u2191\u03ba a) (\u22c3 i, f i) * \u2191\u2191(\u2191\u03ba a) t2"}, {"tactic": "rw [Set.inter_comm, Set.inter_iUnion, measure_iUnion]", "annotated_tactic": ["rw [<a>Set.inter_comm</a>, <a>Set.inter_iUnion</a>, <a>measure_iUnion</a>]", [{"full_name": "Set.inter_comm", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [940, 9], "def_end_pos": [940, 19]}, {"full_name": "Set.inter_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [635, 9], "def_end_pos": [635, 21]}, {"full_name": "MeasureTheory.measure_iUnion", "def_path": "Mathlib/MeasureTheory/Measure/NullMeasurable.lean", "def_pos": [272, 9], "def_end_pos": [272, 23]}]], "state_before": "case h\n\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\nm1 m2 m : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\ninst\u271d : IsMarkovKernel \u03ba\np1 p2 : Set (Set \u03a9)\nh1 : m1 \u2264 m\nh2 : m2 \u2264 m\nhp1 : IsPiSystem p1\nhp2 : IsPiSystem p2\nhpm1 : m1 = generateFrom p1\nhpm2 : m2 = generateFrom p2\nhyp : IndepSets p1 p2 \u03ba\nt1 t2 : Set \u03a9\nht1 : t1 \u2208 {s | MeasurableSet s}\nht2 : t2 \u2208 {s | MeasurableSet s}\nf : \u2115 \u2192 Set \u03a9\nhf_disj : Pairwise (Disjoint on f)\nhf_meas : \u2200 (i : \u2115), MeasurableSet (f i)\nh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (i : \u2115), \u2191\u2191(\u2191\u03ba a) (f i \u2229 t2) = \u2191\u2191(\u2191\u03ba a) (f i) * \u2191\u2191(\u2191\u03ba a) t2\na : \u03b1\nha : \u2200 (i : \u2115), \u2191\u2191(\u2191\u03ba a) (f i \u2229 t2) = \u2191\u2191(\u2191\u03ba a) (f i) * \u2191\u2191(\u2191\u03ba a) t2\n\u22a2 \u2191\u2191(\u2191\u03ba a) ((\u22c3 i, f i) \u2229 t2) = \u2191\u2191(\u2191\u03ba a) (\u22c3 i, f i) * \u2191\u2191(\u2191\u03ba a) t2", "state_after": "case h\n\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\nm1 m2 m : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\ninst\u271d : IsMarkovKernel \u03ba\np1 p2 : Set (Set \u03a9)\nh1 : m1 \u2264 m\nh2 : m2 \u2264 m\nhp1 : IsPiSystem p1\nhp2 : IsPiSystem p2\nhpm1 : m1 = generateFrom p1\nhpm2 : m2 = generateFrom p2\nhyp : IndepSets p1 p2 \u03ba\nt1 t2 : Set \u03a9\nht1 : t1 \u2208 {s | MeasurableSet s}\nht2 : t2 \u2208 {s | MeasurableSet s}\nf : \u2115 \u2192 Set \u03a9\nhf_disj : Pairwise (Disjoint on f)\nhf_meas : \u2200 (i : \u2115), MeasurableSet (f i)\nh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (i : \u2115), \u2191\u2191(\u2191\u03ba a) (f i \u2229 t2) = \u2191\u2191(\u2191\u03ba a) (f i) * \u2191\u2191(\u2191\u03ba a) t2\na : \u03b1\nha : \u2200 (i : \u2115), \u2191\u2191(\u2191\u03ba a) (f i \u2229 t2) = \u2191\u2191(\u2191\u03ba a) (f i) * \u2191\u2191(\u2191\u03ba a) t2\n\u22a2 \u2211' (i : \u2115), \u2191\u2191(\u2191\u03ba a) (t2 \u2229 f i) = \u2191\u2191(\u2191\u03ba a) (\u22c3 i, f i) * \u2191\u2191(\u2191\u03ba a) t2\n\ncase h.hn\n\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\nm1 m2 m : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\ninst\u271d : IsMarkovKernel \u03ba\np1 p2 : Set (Set \u03a9)\nh1 : m1 \u2264 m\nh2 : m2 \u2264 m\nhp1 : IsPiSystem p1\nhp2 : IsPiSystem p2\nhpm1 : m1 = generateFrom p1\nhpm2 : m2 = generateFrom p2\nhyp : IndepSets p1 p2 \u03ba\nt1 t2 : Set \u03a9\nht1 : t1 \u2208 {s | MeasurableSet s}\nht2 : t2 \u2208 {s | MeasurableSet s}\nf : \u2115 \u2192 Set \u03a9\nhf_disj : Pairwise (Disjoint on f)\nhf_meas : \u2200 (i : \u2115), MeasurableSet (f i)\nh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (i : \u2115), \u2191\u2191(\u2191\u03ba a) (f i \u2229 t2) = \u2191\u2191(\u2191\u03ba a) (f i) * \u2191\u2191(\u2191\u03ba a) t2\na : \u03b1\nha : \u2200 (i : \u2115), \u2191\u2191(\u2191\u03ba a) (f i \u2229 t2) = \u2191\u2191(\u2191\u03ba a) (f i) * \u2191\u2191(\u2191\u03ba a) t2\n\u22a2 Pairwise (Disjoint on fun i => t2 \u2229 f i)\n\ncase h.h\n\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\nm1 m2 m : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\ninst\u271d : IsMarkovKernel \u03ba\np1 p2 : Set (Set \u03a9)\nh1 : m1 \u2264 m\nh2 : m2 \u2264 m\nhp1 : IsPiSystem p1\nhp2 : IsPiSystem p2\nhpm1 : m1 = generateFrom p1\nhpm2 : m2 = generateFrom p2\nhyp : IndepSets p1 p2 \u03ba\nt1 t2 : Set \u03a9\nht1 : t1 \u2208 {s | MeasurableSet s}\nht2 : t2 \u2208 {s | MeasurableSet s}\nf : \u2115 \u2192 Set \u03a9\nhf_disj : Pairwise (Disjoint on f)\nhf_meas : \u2200 (i : \u2115), MeasurableSet (f i)\nh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (i : \u2115), \u2191\u2191(\u2191\u03ba a) (f i \u2229 t2) = \u2191\u2191(\u2191\u03ba a) (f i) * \u2191\u2191(\u2191\u03ba a) t2\na : \u03b1\nha : \u2200 (i : \u2115), \u2191\u2191(\u2191\u03ba a) (f i \u2229 t2) = \u2191\u2191(\u2191\u03ba a) (f i) * \u2191\u2191(\u2191\u03ba a) t2\n\u22a2 \u2200 (i : \u2115), MeasurableSet (t2 \u2229 f i)"}, {"tactic": "rw [measure_iUnion hf_disj (fun i \u21a6 h1 _ (hf_meas i))]", "annotated_tactic": ["rw [<a>measure_iUnion</a> hf_disj (fun i \u21a6 h1 _ (hf_meas i))]", [{"full_name": "MeasureTheory.measure_iUnion", "def_path": "Mathlib/MeasureTheory/Measure/NullMeasurable.lean", "def_pos": [272, 9], "def_end_pos": [272, 23]}]], "state_before": "case h\n\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\nm1 m2 m : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\ninst\u271d : IsMarkovKernel \u03ba\np1 p2 : Set (Set \u03a9)\nh1 : m1 \u2264 m\nh2 : m2 \u2264 m\nhp1 : IsPiSystem p1\nhp2 : IsPiSystem p2\nhpm1 : m1 = generateFrom p1\nhpm2 : m2 = generateFrom p2\nhyp : IndepSets p1 p2 \u03ba\nt1 t2 : Set \u03a9\nht1 : t1 \u2208 {s | MeasurableSet s}\nht2 : t2 \u2208 {s | MeasurableSet s}\nf : \u2115 \u2192 Set \u03a9\nhf_disj : Pairwise (Disjoint on f)\nhf_meas : \u2200 (i : \u2115), MeasurableSet (f i)\nh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (i : \u2115), \u2191\u2191(\u2191\u03ba a) (f i \u2229 t2) = \u2191\u2191(\u2191\u03ba a) (f i) * \u2191\u2191(\u2191\u03ba a) t2\na : \u03b1\nha : \u2200 (i : \u2115), \u2191\u2191(\u2191\u03ba a) (f i \u2229 t2) = \u2191\u2191(\u2191\u03ba a) (f i) * \u2191\u2191(\u2191\u03ba a) t2\n\u22a2 \u2211' (i : \u2115), \u2191\u2191(\u2191\u03ba a) (t2 \u2229 f i) = \u2191\u2191(\u2191\u03ba a) (\u22c3 i, f i) * \u2191\u2191(\u2191\u03ba a) t2", "state_after": "case h\n\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\nm1 m2 m : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\ninst\u271d : IsMarkovKernel \u03ba\np1 p2 : Set (Set \u03a9)\nh1 : m1 \u2264 m\nh2 : m2 \u2264 m\nhp1 : IsPiSystem p1\nhp2 : IsPiSystem p2\nhpm1 : m1 = generateFrom p1\nhpm2 : m2 = generateFrom p2\nhyp : IndepSets p1 p2 \u03ba\nt1 t2 : Set \u03a9\nht1 : t1 \u2208 {s | MeasurableSet s}\nht2 : t2 \u2208 {s | MeasurableSet s}\nf : \u2115 \u2192 Set \u03a9\nhf_disj : Pairwise (Disjoint on f)\nhf_meas : \u2200 (i : \u2115), MeasurableSet (f i)\nh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (i : \u2115), \u2191\u2191(\u2191\u03ba a) (f i \u2229 t2) = \u2191\u2191(\u2191\u03ba a) (f i) * \u2191\u2191(\u2191\u03ba a) t2\na : \u03b1\nha : \u2200 (i : \u2115), \u2191\u2191(\u2191\u03ba a) (f i \u2229 t2) = \u2191\u2191(\u2191\u03ba a) (f i) * \u2191\u2191(\u2191\u03ba a) t2\n\u22a2 \u2211' (i : \u2115), \u2191\u2191(\u2191\u03ba a) (t2 \u2229 f i) = (\u2211' (i : \u2115), \u2191\u2191(\u2191\u03ba a) (f i)) * \u2191\u2191(\u2191\u03ba a) t2"}, {"tactic": "rw [\u2190 ENNReal.tsum_mul_right]", "annotated_tactic": ["rw [\u2190 <a>ENNReal.tsum_mul_right</a>]", [{"full_name": "ENNReal.tsum_mul_right", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [907, 19], "def_end_pos": [907, 33]}]], "state_before": "case h\n\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\nm1 m2 m : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\ninst\u271d : IsMarkovKernel \u03ba\np1 p2 : Set (Set \u03a9)\nh1 : m1 \u2264 m\nh2 : m2 \u2264 m\nhp1 : IsPiSystem p1\nhp2 : IsPiSystem p2\nhpm1 : m1 = generateFrom p1\nhpm2 : m2 = generateFrom p2\nhyp : IndepSets p1 p2 \u03ba\nt1 t2 : Set \u03a9\nht1 : t1 \u2208 {s | MeasurableSet s}\nht2 : t2 \u2208 {s | MeasurableSet s}\nf : \u2115 \u2192 Set \u03a9\nhf_disj : Pairwise (Disjoint on f)\nhf_meas : \u2200 (i : \u2115), MeasurableSet (f i)\nh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (i : \u2115), \u2191\u2191(\u2191\u03ba a) (f i \u2229 t2) = \u2191\u2191(\u2191\u03ba a) (f i) * \u2191\u2191(\u2191\u03ba a) t2\na : \u03b1\nha : \u2200 (i : \u2115), \u2191\u2191(\u2191\u03ba a) (f i \u2229 t2) = \u2191\u2191(\u2191\u03ba a) (f i) * \u2191\u2191(\u2191\u03ba a) t2\n\u22a2 \u2211' (i : \u2115), \u2191\u2191(\u2191\u03ba a) (t2 \u2229 f i) = (\u2211' (i : \u2115), \u2191\u2191(\u2191\u03ba a) (f i)) * \u2191\u2191(\u2191\u03ba a) t2", "state_after": "case h\n\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\nm1 m2 m : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\ninst\u271d : IsMarkovKernel \u03ba\np1 p2 : Set (Set \u03a9)\nh1 : m1 \u2264 m\nh2 : m2 \u2264 m\nhp1 : IsPiSystem p1\nhp2 : IsPiSystem p2\nhpm1 : m1 = generateFrom p1\nhpm2 : m2 = generateFrom p2\nhyp : IndepSets p1 p2 \u03ba\nt1 t2 : Set \u03a9\nht1 : t1 \u2208 {s | MeasurableSet s}\nht2 : t2 \u2208 {s | MeasurableSet s}\nf : \u2115 \u2192 Set \u03a9\nhf_disj : Pairwise (Disjoint on f)\nhf_meas : \u2200 (i : \u2115), MeasurableSet (f i)\nh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (i : \u2115), \u2191\u2191(\u2191\u03ba a) (f i \u2229 t2) = \u2191\u2191(\u2191\u03ba a) (f i) * \u2191\u2191(\u2191\u03ba a) t2\na : \u03b1\nha : \u2200 (i : \u2115), \u2191\u2191(\u2191\u03ba a) (f i \u2229 t2) = \u2191\u2191(\u2191\u03ba a) (f i) * \u2191\u2191(\u2191\u03ba a) t2\n\u22a2 \u2211' (i : \u2115), \u2191\u2191(\u2191\u03ba a) (t2 \u2229 f i) = \u2211' (i : \u2115), \u2191\u2191(\u2191\u03ba a) (f i) * \u2191\u2191(\u2191\u03ba a) t2"}, {"tactic": "congr 1 with i", "annotated_tactic": ["congr 1 with i", []], "state_before": "case h\n\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\nm1 m2 m : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\ninst\u271d : IsMarkovKernel \u03ba\np1 p2 : Set (Set \u03a9)\nh1 : m1 \u2264 m\nh2 : m2 \u2264 m\nhp1 : IsPiSystem p1\nhp2 : IsPiSystem p2\nhpm1 : m1 = generateFrom p1\nhpm2 : m2 = generateFrom p2\nhyp : IndepSets p1 p2 \u03ba\nt1 t2 : Set \u03a9\nht1 : t1 \u2208 {s | MeasurableSet s}\nht2 : t2 \u2208 {s | MeasurableSet s}\nf : \u2115 \u2192 Set \u03a9\nhf_disj : Pairwise (Disjoint on f)\nhf_meas : \u2200 (i : \u2115), MeasurableSet (f i)\nh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (i : \u2115), \u2191\u2191(\u2191\u03ba a) (f i \u2229 t2) = \u2191\u2191(\u2191\u03ba a) (f i) * \u2191\u2191(\u2191\u03ba a) t2\na : \u03b1\nha : \u2200 (i : \u2115), \u2191\u2191(\u2191\u03ba a) (f i \u2229 t2) = \u2191\u2191(\u2191\u03ba a) (f i) * \u2191\u2191(\u2191\u03ba a) t2\n\u22a2 \u2211' (i : \u2115), \u2191\u2191(\u2191\u03ba a) (t2 \u2229 f i) = \u2211' (i : \u2115), \u2191\u2191(\u2191\u03ba a) (f i) * \u2191\u2191(\u2191\u03ba a) t2", "state_after": "case h.e_f.h\n\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\nm1 m2 m : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\ninst\u271d : IsMarkovKernel \u03ba\np1 p2 : Set (Set \u03a9)\nh1 : m1 \u2264 m\nh2 : m2 \u2264 m\nhp1 : IsPiSystem p1\nhp2 : IsPiSystem p2\nhpm1 : m1 = generateFrom p1\nhpm2 : m2 = generateFrom p2\nhyp : IndepSets p1 p2 \u03ba\nt1 t2 : Set \u03a9\nht1 : t1 \u2208 {s | MeasurableSet s}\nht2 : t2 \u2208 {s | MeasurableSet s}\nf : \u2115 \u2192 Set \u03a9\nhf_disj : Pairwise (Disjoint on f)\nhf_meas : \u2200 (i : \u2115), MeasurableSet (f i)\nh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (i : \u2115), \u2191\u2191(\u2191\u03ba a) (f i \u2229 t2) = \u2191\u2191(\u2191\u03ba a) (f i) * \u2191\u2191(\u2191\u03ba a) t2\na : \u03b1\nha : \u2200 (i : \u2115), \u2191\u2191(\u2191\u03ba a) (f i \u2229 t2) = \u2191\u2191(\u2191\u03ba a) (f i) * \u2191\u2191(\u2191\u03ba a) t2\ni : \u2115\n\u22a2 \u2191\u2191(\u2191\u03ba a) (t2 \u2229 f i) = \u2191\u2191(\u2191\u03ba a) (f i) * \u2191\u2191(\u2191\u03ba a) t2"}, {"tactic": "rw [Set.inter_comm t2, ha i]", "annotated_tactic": ["rw [<a>Set.inter_comm</a> t2, ha i]", [{"full_name": "Set.inter_comm", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [940, 9], "def_end_pos": [940, 19]}]], "state_before": "case h.e_f.h\n\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\nm1 m2 m : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\ninst\u271d : IsMarkovKernel \u03ba\np1 p2 : Set (Set \u03a9)\nh1 : m1 \u2264 m\nh2 : m2 \u2264 m\nhp1 : IsPiSystem p1\nhp2 : IsPiSystem p2\nhpm1 : m1 = generateFrom p1\nhpm2 : m2 = generateFrom p2\nhyp : IndepSets p1 p2 \u03ba\nt1 t2 : Set \u03a9\nht1 : t1 \u2208 {s | MeasurableSet s}\nht2 : t2 \u2208 {s | MeasurableSet s}\nf : \u2115 \u2192 Set \u03a9\nhf_disj : Pairwise (Disjoint on f)\nhf_meas : \u2200 (i : \u2115), MeasurableSet (f i)\nh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (i : \u2115), \u2191\u2191(\u2191\u03ba a) (f i \u2229 t2) = \u2191\u2191(\u2191\u03ba a) (f i) * \u2191\u2191(\u2191\u03ba a) t2\na : \u03b1\nha : \u2200 (i : \u2115), \u2191\u2191(\u2191\u03ba a) (f i \u2229 t2) = \u2191\u2191(\u2191\u03ba a) (f i) * \u2191\u2191(\u2191\u03ba a) t2\ni : \u2115\n\u22a2 \u2191\u2191(\u2191\u03ba a) (t2 \u2229 f i) = \u2191\u2191(\u2191\u03ba a) (f i) * \u2191\u2191(\u2191\u03ba a) t2", "state_after": "no goals"}, {"tactic": "intros i j hij", "annotated_tactic": ["intros i j hij", []], "state_before": "case h.hn\n\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\nm1 m2 m : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\ninst\u271d : IsMarkovKernel \u03ba\np1 p2 : Set (Set \u03a9)\nh1 : m1 \u2264 m\nh2 : m2 \u2264 m\nhp1 : IsPiSystem p1\nhp2 : IsPiSystem p2\nhpm1 : m1 = generateFrom p1\nhpm2 : m2 = generateFrom p2\nhyp : IndepSets p1 p2 \u03ba\nt1 t2 : Set \u03a9\nht1 : t1 \u2208 {s | MeasurableSet s}\nht2 : t2 \u2208 {s | MeasurableSet s}\nf : \u2115 \u2192 Set \u03a9\nhf_disj : Pairwise (Disjoint on f)\nhf_meas : \u2200 (i : \u2115), MeasurableSet (f i)\nh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (i : \u2115), \u2191\u2191(\u2191\u03ba a) (f i \u2229 t2) = \u2191\u2191(\u2191\u03ba a) (f i) * \u2191\u2191(\u2191\u03ba a) t2\na : \u03b1\nha : \u2200 (i : \u2115), \u2191\u2191(\u2191\u03ba a) (f i \u2229 t2) = \u2191\u2191(\u2191\u03ba a) (f i) * \u2191\u2191(\u2191\u03ba a) t2\n\u22a2 Pairwise (Disjoint on fun i => t2 \u2229 f i)", "state_after": "case h.hn\n\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\nm1 m2 m : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\ninst\u271d : IsMarkovKernel \u03ba\np1 p2 : Set (Set \u03a9)\nh1 : m1 \u2264 m\nh2 : m2 \u2264 m\nhp1 : IsPiSystem p1\nhp2 : IsPiSystem p2\nhpm1 : m1 = generateFrom p1\nhpm2 : m2 = generateFrom p2\nhyp : IndepSets p1 p2 \u03ba\nt1 t2 : Set \u03a9\nht1 : t1 \u2208 {s | MeasurableSet s}\nht2 : t2 \u2208 {s | MeasurableSet s}\nf : \u2115 \u2192 Set \u03a9\nhf_disj : Pairwise (Disjoint on f)\nhf_meas : \u2200 (i : \u2115), MeasurableSet (f i)\nh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (i : \u2115), \u2191\u2191(\u2191\u03ba a) (f i \u2229 t2) = \u2191\u2191(\u2191\u03ba a) (f i) * \u2191\u2191(\u2191\u03ba a) t2\na : \u03b1\nha : \u2200 (i : \u2115), \u2191\u2191(\u2191\u03ba a) (f i \u2229 t2) = \u2191\u2191(\u2191\u03ba a) (f i) * \u2191\u2191(\u2191\u03ba a) t2\ni j : \u2115\nhij : i \u2260 j\n\u22a2 (Disjoint on fun i => t2 \u2229 f i) i j"}, {"tactic": "rw [Function.onFun, Set.inter_comm t2, Set.inter_comm t2]", "annotated_tactic": ["rw [<a>Function.onFun</a>, <a>Set.inter_comm</a> t2, <a>Set.inter_comm</a> t2]", [{"full_name": "Function.onFun", "def_path": "Mathlib/Init/Function.lean", "def_pos": [49, 5], "def_end_pos": [49, 10]}, {"full_name": "Set.inter_comm", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [940, 9], "def_end_pos": [940, 19]}, {"full_name": "Set.inter_comm", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [940, 9], "def_end_pos": [940, 19]}]], "state_before": "case h.hn\n\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\nm1 m2 m : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\ninst\u271d : IsMarkovKernel \u03ba\np1 p2 : Set (Set \u03a9)\nh1 : m1 \u2264 m\nh2 : m2 \u2264 m\nhp1 : IsPiSystem p1\nhp2 : IsPiSystem p2\nhpm1 : m1 = generateFrom p1\nhpm2 : m2 = generateFrom p2\nhyp : IndepSets p1 p2 \u03ba\nt1 t2 : Set \u03a9\nht1 : t1 \u2208 {s | MeasurableSet s}\nht2 : t2 \u2208 {s | MeasurableSet s}\nf : \u2115 \u2192 Set \u03a9\nhf_disj : Pairwise (Disjoint on f)\nhf_meas : \u2200 (i : \u2115), MeasurableSet (f i)\nh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (i : \u2115), \u2191\u2191(\u2191\u03ba a) (f i \u2229 t2) = \u2191\u2191(\u2191\u03ba a) (f i) * \u2191\u2191(\u2191\u03ba a) t2\na : \u03b1\nha : \u2200 (i : \u2115), \u2191\u2191(\u2191\u03ba a) (f i \u2229 t2) = \u2191\u2191(\u2191\u03ba a) (f i) * \u2191\u2191(\u2191\u03ba a) t2\ni j : \u2115\nhij : i \u2260 j\n\u22a2 (Disjoint on fun i => t2 \u2229 f i) i j", "state_after": "case h.hn\n\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\nm1 m2 m : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\ninst\u271d : IsMarkovKernel \u03ba\np1 p2 : Set (Set \u03a9)\nh1 : m1 \u2264 m\nh2 : m2 \u2264 m\nhp1 : IsPiSystem p1\nhp2 : IsPiSystem p2\nhpm1 : m1 = generateFrom p1\nhpm2 : m2 = generateFrom p2\nhyp : IndepSets p1 p2 \u03ba\nt1 t2 : Set \u03a9\nht1 : t1 \u2208 {s | MeasurableSet s}\nht2 : t2 \u2208 {s | MeasurableSet s}\nf : \u2115 \u2192 Set \u03a9\nhf_disj : Pairwise (Disjoint on f)\nhf_meas : \u2200 (i : \u2115), MeasurableSet (f i)\nh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (i : \u2115), \u2191\u2191(\u2191\u03ba a) (f i \u2229 t2) = \u2191\u2191(\u2191\u03ba a) (f i) * \u2191\u2191(\u2191\u03ba a) t2\na : \u03b1\nha : \u2200 (i : \u2115), \u2191\u2191(\u2191\u03ba a) (f i \u2229 t2) = \u2191\u2191(\u2191\u03ba a) (f i) * \u2191\u2191(\u2191\u03ba a) t2\ni j : \u2115\nhij : i \u2260 j\n\u22a2 Disjoint (f i \u2229 t2) (f j \u2229 t2)"}, {"tactic": "exact Disjoint.inter_left _ (Disjoint.inter_right _ (hf_disj hij))", "annotated_tactic": ["exact <a>Disjoint.inter_left</a> _ (<a>Disjoint.inter_right</a> _ (hf_disj hij))", [{"full_name": "Disjoint.inter_left", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [2996, 9], "def_end_pos": [2996, 19]}, {"full_name": "Disjoint.inter_right", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [3004, 9], "def_end_pos": [3004, 20]}]], "state_before": "case h.hn\n\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\nm1 m2 m : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\ninst\u271d : IsMarkovKernel \u03ba\np1 p2 : Set (Set \u03a9)\nh1 : m1 \u2264 m\nh2 : m2 \u2264 m\nhp1 : IsPiSystem p1\nhp2 : IsPiSystem p2\nhpm1 : m1 = generateFrom p1\nhpm2 : m2 = generateFrom p2\nhyp : IndepSets p1 p2 \u03ba\nt1 t2 : Set \u03a9\nht1 : t1 \u2208 {s | MeasurableSet s}\nht2 : t2 \u2208 {s | MeasurableSet s}\nf : \u2115 \u2192 Set \u03a9\nhf_disj : Pairwise (Disjoint on f)\nhf_meas : \u2200 (i : \u2115), MeasurableSet (f i)\nh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (i : \u2115), \u2191\u2191(\u2191\u03ba a) (f i \u2229 t2) = \u2191\u2191(\u2191\u03ba a) (f i) * \u2191\u2191(\u2191\u03ba a) t2\na : \u03b1\nha : \u2200 (i : \u2115), \u2191\u2191(\u2191\u03ba a) (f i \u2229 t2) = \u2191\u2191(\u2191\u03ba a) (f i) * \u2191\u2191(\u2191\u03ba a) t2\ni j : \u2115\nhij : i \u2260 j\n\u22a2 Disjoint (f i \u2229 t2) (f j \u2229 t2)", "state_after": "no goals"}, {"tactic": "exact fun i \u21a6 (h2 _ ht2).inter (h1 _ (hf_meas i))", "annotated_tactic": ["exact fun i \u21a6 (h2 _ ht2).<a>inter</a> (h1 _ (hf_meas i))", [{"full_name": "MeasurableSet.inter", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [198, 19], "def_end_pos": [198, 38]}]], "state_before": "case h.h\n\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\nm1 m2 m : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\ninst\u271d : IsMarkovKernel \u03ba\np1 p2 : Set (Set \u03a9)\nh1 : m1 \u2264 m\nh2 : m2 \u2264 m\nhp1 : IsPiSystem p1\nhp2 : IsPiSystem p2\nhpm1 : m1 = generateFrom p1\nhpm2 : m2 = generateFrom p2\nhyp : IndepSets p1 p2 \u03ba\nt1 t2 : Set \u03a9\nht1 : t1 \u2208 {s | MeasurableSet s}\nht2 : t2 \u2208 {s | MeasurableSet s}\nf : \u2115 \u2192 Set \u03a9\nhf_disj : Pairwise (Disjoint on f)\nhf_meas : \u2200 (i : \u2115), MeasurableSet (f i)\nh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (i : \u2115), \u2191\u2191(\u2191\u03ba a) (f i \u2229 t2) = \u2191\u2191(\u2191\u03ba a) (f i) * \u2191\u2191(\u2191\u03ba a) t2\na : \u03b1\nha : \u2200 (i : \u2115), \u2191\u2191(\u2191\u03ba a) (f i \u2229 t2) = \u2191\u2191(\u2191\u03ba a) (f i) * \u2191\u2191(\u2191\u03ba a) t2\n\u22a2 \u2200 (i : \u2115), MeasurableSet (t2 \u2229 f i)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Partrec.lean", "full_name": "Partrec.sum_casesOn_right", "start": [787, 1], "end": [801, 65], "traced_tactics": [{"tactic": "cases f a <;> simp", "annotated_tactic": ["cases f a <;> simp", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 \u03b2 \u2295 \u03b3\ng : \u03b1 \u2192 \u03b2 \u2192 \u03c3\nh : \u03b1 \u2192 \u03b3 \u2192. \u03c3\nhf : Computable f\nhg : Computable\u2082 g\nhh : Partrec\u2082 h\nthis :\n  Partrec fun a =>\n    Option.casesOn (Sum.casesOn (f a) (fun x => Option.none) Option.some)\n      (Part.some (Sum.casesOn (f a) (fun b => Option.some (g a b)) fun x => Option.none)) fun c =>\n      Part.map Option.some (h a c)\na : \u03b1\n\u22a2 (Option.casesOn (Sum.casesOn (f a) (fun x => Option.none) Option.some)\n      (Part.some (Sum.casesOn (f a) (fun b => Option.some (g a b)) fun x => Option.none)) fun c =>\n      Part.map Option.some (h a c)) =\n    Part.map Option.some (Sum.casesOn (f a) (fun b => Part.some (g a b)) (h a))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "full_name": "MeasureTheory.VectorMeasure.of_disjoint_iUnion", "start": [180, 1], "end": [182, 51], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "full_name": "MeasureTheory.lintegral_iSup", "start": [345, 1], "end": [411, 34], "traced_tactics": [{"tactic": "set c : \u211d\u22650 \u2192 \u211d\u22650\u221e := (\u2191)", "annotated_tactic": ["set c : \u211d\u22650 \u2192 \u211d\u22650\u221e := (\u2191)", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\n\u22a2 \u222b\u207b (a : \u03b1), \u2a06 n, f n a \u2202\u03bc = \u2a06 n, \u222b\u207b (a : \u03b1), f n a \u2202\u03bc", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\n\u22a2 \u222b\u207b (a : \u03b1), \u2a06 n, f n a \u2202\u03bc = \u2a06 n, \u222b\u207b (a : \u03b1), f n a \u2202\u03bc"}, {"tactic": "set F := fun a : \u03b1 => \u2a06 n, f n a", "annotated_tactic": ["set F := fun a : \u03b1 => \u2a06 n, f n a", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\n\u22a2 \u222b\u207b (a : \u03b1), \u2a06 n, f n a \u2202\u03bc = \u2a06 n, \u222b\u207b (a : \u03b1), f n a \u2202\u03bc", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\n\u22a2 lintegral \u03bc F = \u2a06 n, \u222b\u207b (a : \u03b1), f n a \u2202\u03bc"}, {"tactic": "have _ : Measurable F := measurable_iSup hf", "annotated_tactic": ["have _ : <a>Measurable</a> F := <a>measurable_iSup</a> hf", [{"full_name": "Measurable", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [535, 5], "def_end_pos": [535, 15]}, {"full_name": "measurable_iSup", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [1360, 9], "def_end_pos": [1360, 24]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\n\u22a2 lintegral \u03bc F = \u2a06 n, \u222b\u207b (a : \u03b1), f n a \u2202\u03bc", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\n\u22a2 lintegral \u03bc F = \u2a06 n, \u222b\u207b (a : \u03b1), f n a \u2202\u03bc"}, {"tactic": "refine' le_antisymm _ (iSup_lintegral_le _)", "annotated_tactic": ["refine' <a>le_antisymm</a> _ (<a>iSup_lintegral_le</a> _)", [{"full_name": "le_antisymm", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [188, 9], "def_end_pos": [188, 20]}, {"full_name": "MeasureTheory.iSup_lintegral_le", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [241, 9], "def_end_pos": [241, 26]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\n\u22a2 lintegral \u03bc F = \u2a06 n, \u222b\u207b (a : \u03b1), f n a \u2202\u03bc", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\n\u22a2 lintegral \u03bc F \u2264 \u2a06 n, \u222b\u207b (a : \u03b1), f n a \u2202\u03bc"}, {"tactic": "rw [lintegral_eq_nnreal]", "annotated_tactic": ["rw [<a>lintegral_eq_nnreal</a>]", [{"full_name": "MeasureTheory.lintegral_eq_nnreal", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [197, 9], "def_end_pos": [197, 28]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\n\u22a2 lintegral \u03bc F \u2264 \u2a06 n, \u222b\u207b (a : \u03b1), f n a \u2202\u03bc", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\n\u22a2 \u2a06 \u03c6, \u2a06 (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 \u2a06 n, f n x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc \u2264\n    \u2a06 n, \u222b\u207b (a : \u03b1), f n a \u2202\u03bc"}, {"tactic": "refine' iSup_le fun s => iSup_le fun hsf => _", "annotated_tactic": ["refine' <a>iSup_le</a> fun s => <a>iSup_le</a> fun hsf => _", [{"full_name": "iSup_le", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [875, 9], "def_end_pos": [875, 16]}, {"full_name": "iSup_le", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [875, 9], "def_end_pos": [875, 16]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\n\u22a2 \u2a06 \u03c6, \u2a06 (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 \u2a06 n, f n x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc \u2264\n    \u2a06 n, \u222b\u207b (a : \u03b1), f n a \u2202\u03bc", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\n\u22a2 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some s) \u03bc \u2264 \u2a06 n, \u222b\u207b (a : \u03b1), f n a \u2202\u03bc"}, {"tactic": "refine' ENNReal.le_of_forall_lt_one_mul_le fun a ha => _", "annotated_tactic": ["refine' <a>ENNReal.le_of_forall_lt_one_mul_le</a> fun a ha => _", [{"full_name": "ENNReal.le_of_forall_lt_one_mul_le", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [476, 9], "def_end_pos": [476, 35]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\n\u22a2 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some s) \u03bc \u2264 \u2a06 n, \u222b\u207b (a : \u03b1), f n a \u2202\u03bc", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\na : \u211d\u22650\u221e\nha : a < 1\n\u22a2 a * SimpleFunc.lintegral (SimpleFunc.map ENNReal.some s) \u03bc \u2264 \u2a06 n, \u222b\u207b (a : \u03b1), f n a \u2202\u03bc"}, {"tactic": "rcases ENNReal.lt_iff_exists_coe.1 ha with \u27e8r, rfl, _\u27e9", "annotated_tactic": ["rcases <a>ENNReal.lt_iff_exists_coe</a>.1 ha with \u27e8r, rfl, _\u27e9", [{"full_name": "ENNReal.lt_iff_exists_coe", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [746, 9], "def_end_pos": [746, 26]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\na : \u211d\u22650\u221e\nha : a < 1\n\u22a2 a * SimpleFunc.lintegral (SimpleFunc.map ENNReal.some s) \u03bc \u2264 \u2a06 n, \u222b\u207b (a : \u03b1), f n a \u2202\u03bc", "state_after": "case intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha : \u2191r < 1\n\u22a2 \u2191r * SimpleFunc.lintegral (SimpleFunc.map ENNReal.some s) \u03bc \u2264 \u2a06 n, \u222b\u207b (a : \u03b1), f n a \u2202\u03bc"}, {"tactic": "have ha : r < 1 := ENNReal.coe_lt_coe.1 ha", "annotated_tactic": ["have ha : r < 1 := <a>ENNReal.coe_lt_coe</a>.1 ha", [{"full_name": "ENNReal.coe_lt_coe", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [352, 28], "def_end_pos": [352, 38]}]], "state_before": "case intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha : \u2191r < 1\n\u22a2 \u2191r * SimpleFunc.lintegral (SimpleFunc.map ENNReal.some s) \u03bc \u2264 \u2a06 n, \u222b\u207b (a : \u03b1), f n a \u2202\u03bc", "state_after": "case intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\n\u22a2 \u2191r * SimpleFunc.lintegral (SimpleFunc.map ENNReal.some s) \u03bc \u2264 \u2a06 n, \u222b\u207b (a : \u03b1), f n a \u2202\u03bc"}, {"tactic": "let rs := s.map fun a => r * a", "annotated_tactic": ["let rs := s.map fun a => r * a", []], "state_before": "case intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\n\u22a2 \u2191r * SimpleFunc.lintegral (SimpleFunc.map ENNReal.some s) \u03bc \u2264 \u2a06 n, \u222b\u207b (a : \u03b1), f n a \u2202\u03bc", "state_after": "case intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\n\u22a2 \u2191r * SimpleFunc.lintegral (SimpleFunc.map ENNReal.some s) \u03bc \u2264 \u2a06 n, \u222b\u207b (a : \u03b1), f n a \u2202\u03bc"}, {"tactic": "have eq_rs : (const \u03b1 r : \u03b1 \u2192\u209b \u211d\u22650\u221e) * map c s = rs.map c := by\n  ext1 a\n  exact ENNReal.coe_mul.symm", "annotated_tactic": ["have eq_rs : (<a>const</a> \u03b1 r : \u03b1 \u2192\u209b \u211d\u22650\u221e) * <a>map</a> c s = rs.map c := by\n    ext1 a\n    exact ENNReal.coe_mul.symm", [{"full_name": "MeasureTheory.SimpleFunc.const", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [145, 5], "def_end_pos": [145, 10]}, {"full_name": "MeasureTheory.SimpleFunc.map", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [290, 5], "def_end_pos": [290, 8]}]], "state_before": "case intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\n\u22a2 \u2191r * SimpleFunc.lintegral (SimpleFunc.map ENNReal.some s) \u03bc \u2264 \u2a06 n, \u222b\u207b (a : \u03b1), f n a \u2202\u03bc", "state_after": "case intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\n\u22a2 \u2191r * SimpleFunc.lintegral (SimpleFunc.map ENNReal.some s) \u03bc \u2264 \u2a06 n, \u222b\u207b (a : \u03b1), f n a \u2202\u03bc"}, {"tactic": "have mono : \u2200 r : \u211d\u22650\u221e, Monotone fun n => rs.map c \u207b\u00b9' {r} \u2229 { a | r \u2264 f n a } := by\n  intro r i j h\n  refine' inter_subset_inter (Subset.refl _) _\n  intro x (hx : r \u2264 f i x)\n  exact le_trans hx (h_mono h x)", "annotated_tactic": ["have mono : \u2200 r : \u211d\u22650\u221e, <a>Monotone</a> fun n => rs.map c \u207b\u00b9' {r} \u2229 { a | r \u2264 f n a } := by\n    intro r i j h\n    refine' <a>inter_subset_inter</a> (<a>Subset.refl</a> _) _\n    intro x (hx : r \u2264 f i x)\n    exact <a>le_trans</a> hx (h_mono h x)", [{"full_name": "Monotone", "def_path": "Mathlib/Order/Monotone/Basic.lean", "def_pos": [77, 5], "def_end_pos": [77, 13]}, {"full_name": "Set.inter_subset_inter", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1022, 9], "def_end_pos": [1022, 27]}, {"full_name": "Set.Subset.refl", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [354, 9], "def_end_pos": [354, 20]}, {"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}]], "state_before": "case intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\neq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a}\n\u22a2 \u2191r * SimpleFunc.lintegral (SimpleFunc.map ENNReal.some s) \u03bc \u2264 \u2a06 n, \u222b\u207b (a : \u03b1), f n a \u2202\u03bc", "state_after": "case intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\neq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a}\nmono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}\n\u22a2 \u2191r * SimpleFunc.lintegral (SimpleFunc.map ENNReal.some s) \u03bc \u2264 \u2a06 n, \u222b\u207b (a : \u03b1), f n a \u2202\u03bc"}, {"tactic": "have h_meas : \u2200 n, MeasurableSet { a : \u03b1 | (\u21d1(map c rs)) a \u2264 f n a } := fun n =>\n  measurableSet_le (SimpleFunc.measurable _) (hf n)", "annotated_tactic": ["have h_meas : \u2200 n, <a>MeasurableSet</a> { a : \u03b1 | (\u21d1(<a>map</a> c rs)) a \u2264 f n a } := fun n =>\n    <a>measurableSet_le</a> (<a>SimpleFunc.measurable</a> _) (hf n)", [{"full_name": "MeasurableSet", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [64, 5], "def_end_pos": [64, 18]}, {"full_name": "MeasureTheory.SimpleFunc.map", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [290, 5], "def_end_pos": [290, 8]}, {"full_name": "measurableSet_le", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [559, 9], "def_end_pos": [559, 25]}, {"full_name": "MeasureTheory.SimpleFunc.measurable", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [201, 19], "def_end_pos": [201, 29]}]], "state_before": "case intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\neq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a}\nmono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}\n\u22a2 \u2191r * SimpleFunc.lintegral (SimpleFunc.map ENNReal.some s) \u03bc \u2264 \u2a06 n, \u222b\u207b (a : \u03b1), f n a \u2202\u03bc", "state_after": "case intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\neq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a}\nmono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}\nh_meas : \u2200 (n : \u2115), MeasurableSet {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}\n\u22a2 \u2191r * SimpleFunc.lintegral (SimpleFunc.map ENNReal.some s) \u03bc \u2264 \u2a06 n, \u222b\u207b (a : \u03b1), f n a \u2202\u03bc"}, {"tactic": "ext1 a", "annotated_tactic": ["ext1 a", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\n\u22a2 const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs", "state_after": "case H\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\na : \u03b1\n\u22a2 \u2191(const \u03b1 \u2191r * SimpleFunc.map c s) a = \u2191(SimpleFunc.map c rs) a"}, {"tactic": "exact ENNReal.coe_mul.symm", "annotated_tactic": ["exact ENNReal.coe_mul.symm", []], "state_before": "case H\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\na : \u03b1\n\u22a2 \u2191(const \u03b1 \u2191r * SimpleFunc.map c s) a = \u2191(SimpleFunc.map c rs) a", "state_after": "no goals"}, {"tactic": "intro p", "annotated_tactic": ["intro p", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\n\u22a2 \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a}", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\np : \u211d\u22650\u221e\n\u22a2 \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a}"}, {"tactic": "rw [\u2190 inter_iUnion]", "annotated_tactic": ["rw [\u2190 <a>inter_iUnion</a>]", [{"full_name": "Set.inter_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [635, 9], "def_end_pos": [635, 21]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\np : \u211d\u22650\u221e\n\u22a2 \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a}", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\np : \u211d\u22650\u221e\n\u22a2 \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 \u22c3 i, {a | p \u2264 f i a}"}, {"tactic": "nth_rw 1 [\u2190 inter_univ (map c rs \u207b\u00b9' {p})]", "annotated_tactic": ["nth_rw 1 [\u2190 <a>inter_univ</a> (<a>map</a> c rs \u207b\u00b9' {p})]", [{"full_name": "Set.inter_univ", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1012, 9], "def_end_pos": [1012, 19]}, {"full_name": "MeasureTheory.SimpleFunc.map", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [290, 5], "def_end_pos": [290, 8]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\np : \u211d\u22650\u221e\n\u22a2 \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 \u22c3 i, {a | p \u2264 f i a}", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\np : \u211d\u22650\u221e\n\u22a2 \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 univ = \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 \u22c3 i, {a | p \u2264 f i a}"}, {"tactic": "refine' Set.ext fun x => and_congr_right fun hx => true_iff_iff.2 _", "annotated_tactic": ["refine' <a>Set.ext</a> fun x => <a>and_congr_right</a> fun hx => <a>true_iff_iff</a>.2 _", [{"full_name": "Set.ext", "def_path": "Mathlib/Init/Set.lean", "def_pos": [54, 9], "def_end_pos": [54, 12]}, {"full_name": "and_congr_right", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [161, 9], "def_end_pos": [161, 24]}, {"full_name": "true_iff_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [198, 9], "def_end_pos": [198, 21]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\np : \u211d\u22650\u221e\n\u22a2 \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 univ = \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 \u22c3 i, {a | p \u2264 f i a}", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\np : \u211d\u22650\u221e\nx : \u03b1\nhx : x \u2208 \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p}\n\u22a2 x \u2208 \u22c3 i, {a | p \u2264 f i a}"}, {"tactic": "by_cases p_eq : p = 0", "annotated_tactic": ["by_cases p_eq : p = 0", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\np : \u211d\u22650\u221e\nx : \u03b1\nhx : x \u2208 \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p}\n\u22a2 x \u2208 \u22c3 i, {a | p \u2264 f i a}", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\np : \u211d\u22650\u221e\nx : \u03b1\nhx : x \u2208 \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p}\np_eq : p = 0\n\u22a2 x \u2208 \u22c3 i, {a | p \u2264 f i a}\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\np : \u211d\u22650\u221e\nx : \u03b1\nhx : x \u2208 \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p}\np_eq : \u00acp = 0\n\u22a2 x \u2208 \u22c3 i, {a | p \u2264 f i a}"}, {"tactic": "simp only [coe_map, mem_preimage, Function.comp_apply, mem_singleton_iff] at hx", "annotated_tactic": ["simp only [<a>coe_map</a>, <a>mem_preimage</a>, <a>Function.comp_apply</a>, <a>mem_singleton_iff</a>] at hx", [{"full_name": "MeasureTheory.SimpleFunc.coe_map", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [303, 9], "def_end_pos": [303, 16]}, {"full_name": "Set.mem_preimage", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [64, 9], "def_end_pos": [64, 21]}, {"full_name": "Function.comp_apply", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [33, 17], "def_end_pos": [33, 36]}, {"full_name": "Set.mem_singleton_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1273, 9], "def_end_pos": [1273, 26]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\np : \u211d\u22650\u221e\nx : \u03b1\nhx : x \u2208 \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p}\np_eq : \u00acp = 0\n\u22a2 x \u2208 \u22c3 i, {a | p \u2264 f i a}", "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\np : \u211d\u22650\u221e\nx : \u03b1\np_eq : \u00acp = 0\nhx : \u2191(r * \u2191s x) = p\n\u22a2 x \u2208 \u22c3 i, {a | p \u2264 f i a}"}, {"tactic": "subst hx", "annotated_tactic": ["subst hx", []], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\np : \u211d\u22650\u221e\nx : \u03b1\np_eq : \u00acp = 0\nhx : \u2191(r * \u2191s x) = p\n\u22a2 x \u2208 \u22c3 i, {a | p \u2264 f i a}", "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\nx : \u03b1\np_eq : \u00ac\u2191(r * \u2191s x) = 0\n\u22a2 x \u2208 \u22c3 i, {a | \u2191(r * \u2191s x) \u2264 f i a}"}, {"tactic": "have : r * s x \u2260 0 := by rwa [Ne, \u2190 ENNReal.coe_eq_zero]", "annotated_tactic": ["have : r * s x \u2260 0 := by rwa [<a>Ne</a>, \u2190 <a>ENNReal.coe_eq_zero</a>]", [{"full_name": "Ne", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [560, 18], "def_end_pos": [560, 20]}, {"full_name": "ENNReal.coe_eq_zero", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [368, 28], "def_end_pos": [368, 39]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\nx : \u03b1\np_eq : \u00ac\u2191(r * \u2191s x) = 0\n\u22a2 x \u2208 \u22c3 i, {a | \u2191(r * \u2191s x) \u2264 f i a}", "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\nx : \u03b1\np_eq : \u00ac\u2191(r * \u2191s x) = 0\nthis : r * \u2191s x \u2260 0\n\u22a2 x \u2208 \u22c3 i, {a | \u2191(r * \u2191s x) \u2264 f i a}"}, {"tactic": "have : s x \u2260 0 := by\n  refine' mt _ this\n  intro h\n  rw [h, mul_zero]", "annotated_tactic": ["have : s x \u2260 0 := by\n      refine' <a>mt</a> _ this\n      intro h\n      rw [h, <a>mul_zero</a>]", [{"full_name": "mt", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [516, 9], "def_end_pos": [516, 11]}, {"full_name": "MulZeroClass.mul_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [38, 3], "def_end_pos": [38, 11]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\nx : \u03b1\np_eq : \u00ac\u2191(r * \u2191s x) = 0\nthis : r * \u2191s x \u2260 0\n\u22a2 x \u2208 \u22c3 i, {a | \u2191(r * \u2191s x) \u2264 f i a}", "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\nx : \u03b1\np_eq : \u00ac\u2191(r * \u2191s x) = 0\nthis\u271d : r * \u2191s x \u2260 0\nthis : \u2191s x \u2260 0\n\u22a2 x \u2208 \u22c3 i, {a | \u2191(r * \u2191s x) \u2264 f i a}"}, {"tactic": "have : (rs.map c) x < \u2a06 n : \u2115, f n x := by\n  refine' lt_of_lt_of_le (ENNReal.coe_lt_coe.2 _) (hsf x)\n  suffices r * s x < 1 * s x by simpa\n  exact mul_lt_mul_of_pos_right ha (pos_iff_ne_zero.2 this)", "annotated_tactic": ["have : (rs.map c) x < \u2a06 n : \u2115, f n x := by\n      refine' <a>lt_of_lt_of_le</a> (<a>ENNReal.coe_lt_coe</a>.2 _) (hsf x)\n      suffices r * s x < 1 * s x by simpa\n      exact <a>mul_lt_mul_of_pos_right</a> ha (<a>pos_iff_ne_zero</a>.2 this)", [{"full_name": "lt_of_lt_of_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [115, 9], "def_end_pos": [115, 23]}, {"full_name": "ENNReal.coe_lt_coe", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [352, 28], "def_end_pos": [352, 38]}, {"full_name": "mul_lt_mul_of_pos_right", "def_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "def_pos": [164, 9], "def_end_pos": [164, 32]}, {"full_name": "pos_iff_ne_zero", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [243, 3], "def_end_pos": [243, 14]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\nx : \u03b1\np_eq : \u00ac\u2191(r * \u2191s x) = 0\nthis\u271d : r * \u2191s x \u2260 0\nthis : \u2191s x \u2260 0\n\u22a2 x \u2208 \u22c3 i, {a | \u2191(r * \u2191s x) \u2264 f i a}", "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\nx : \u03b1\np_eq : \u00ac\u2191(r * \u2191s x) = 0\nthis\u271d\u00b9 : r * \u2191s x \u2260 0\nthis\u271d : \u2191s x \u2260 0\nthis : \u2191(SimpleFunc.map c rs) x < \u2a06 n, f n x\n\u22a2 x \u2208 \u22c3 i, {a | \u2191(r * \u2191s x) \u2264 f i a}"}, {"tactic": "rcases lt_iSup_iff.1 this with \u27e8i, hi\u27e9", "annotated_tactic": ["rcases <a>lt_iSup_iff</a>.1 this with \u27e8i, hi\u27e9", [{"full_name": "lt_iSup_iff", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [668, 9], "def_end_pos": [668, 20]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\nx : \u03b1\np_eq : \u00ac\u2191(r * \u2191s x) = 0\nthis\u271d\u00b9 : r * \u2191s x \u2260 0\nthis\u271d : \u2191s x \u2260 0\nthis : \u2191(SimpleFunc.map c rs) x < \u2a06 n, f n x\n\u22a2 x \u2208 \u22c3 i, {a | \u2191(r * \u2191s x) \u2264 f i a}", "state_after": "case neg.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\nx : \u03b1\np_eq : \u00ac\u2191(r * \u2191s x) = 0\nthis\u271d\u00b9 : r * \u2191s x \u2260 0\nthis\u271d : \u2191s x \u2260 0\nthis : \u2191(SimpleFunc.map c rs) x < \u2a06 n, f n x\ni : \u2115\nhi : \u2191(SimpleFunc.map c rs) x < f i x\n\u22a2 x \u2208 \u22c3 i, {a | \u2191(r * \u2191s x) \u2264 f i a}"}, {"tactic": "exact mem_iUnion.2 \u27e8i, le_of_lt hi\u27e9", "annotated_tactic": ["exact <a>mem_iUnion</a>.2 \u27e8i, <a>le_of_lt</a> hi\u27e9", [{"full_name": "Set.mem_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [201, 9], "def_end_pos": [201, 19]}, {"full_name": "le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [110, 9], "def_end_pos": [110, 17]}]], "state_before": "case neg.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\nx : \u03b1\np_eq : \u00ac\u2191(r * \u2191s x) = 0\nthis\u271d\u00b9 : r * \u2191s x \u2260 0\nthis\u271d : \u2191s x \u2260 0\nthis : \u2191(SimpleFunc.map c rs) x < \u2a06 n, f n x\ni : \u2115\nhi : \u2191(SimpleFunc.map c rs) x < f i x\n\u22a2 x \u2208 \u22c3 i, {a | \u2191(r * \u2191s x) \u2264 f i a}", "state_after": "no goals"}, {"tactic": "simp [p_eq]", "annotated_tactic": ["simp [p_eq]", []], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\np : \u211d\u22650\u221e\nx : \u03b1\nhx : x \u2208 \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p}\np_eq : p = 0\n\u22a2 x \u2208 \u22c3 i, {a | p \u2264 f i a}", "state_after": "no goals"}, {"tactic": "rwa [Ne, \u2190 ENNReal.coe_eq_zero]", "annotated_tactic": ["rwa [<a>Ne</a>, \u2190 <a>ENNReal.coe_eq_zero</a>]", [{"full_name": "Ne", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [560, 18], "def_end_pos": [560, 20]}, {"full_name": "ENNReal.coe_eq_zero", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [368, 28], "def_end_pos": [368, 39]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\nx : \u03b1\np_eq : \u00ac\u2191(r * \u2191s x) = 0\n\u22a2 r * \u2191s x \u2260 0", "state_after": "no goals"}, {"tactic": "refine' mt _ this", "annotated_tactic": ["refine' <a>mt</a> _ this", [{"full_name": "mt", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [516, 9], "def_end_pos": [516, 11]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\nx : \u03b1\np_eq : \u00ac\u2191(r * \u2191s x) = 0\nthis : r * \u2191s x \u2260 0\n\u22a2 \u2191s x \u2260 0", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\nx : \u03b1\np_eq : \u00ac\u2191(r * \u2191s x) = 0\nthis : r * \u2191s x \u2260 0\n\u22a2 \u2191s x = 0 \u2192 r * \u2191s x = 0"}, {"tactic": "intro h", "annotated_tactic": ["intro h", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\nx : \u03b1\np_eq : \u00ac\u2191(r * \u2191s x) = 0\nthis : r * \u2191s x \u2260 0\n\u22a2 \u2191s x = 0 \u2192 r * \u2191s x = 0", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\nx : \u03b1\np_eq : \u00ac\u2191(r * \u2191s x) = 0\nthis : r * \u2191s x \u2260 0\nh : \u2191s x = 0\n\u22a2 r * \u2191s x = 0"}, {"tactic": "rw [h, mul_zero]", "annotated_tactic": ["rw [h, <a>mul_zero</a>]", [{"full_name": "MulZeroClass.mul_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [38, 3], "def_end_pos": [38, 11]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\nx : \u03b1\np_eq : \u00ac\u2191(r * \u2191s x) = 0\nthis : r * \u2191s x \u2260 0\nh : \u2191s x = 0\n\u22a2 r * \u2191s x = 0", "state_after": "no goals"}, {"tactic": "refine' lt_of_lt_of_le (ENNReal.coe_lt_coe.2 _) (hsf x)", "annotated_tactic": ["refine' <a>lt_of_lt_of_le</a> (<a>ENNReal.coe_lt_coe</a>.2 _) (hsf x)", [{"full_name": "lt_of_lt_of_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [115, 9], "def_end_pos": [115, 23]}, {"full_name": "ENNReal.coe_lt_coe", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [352, 28], "def_end_pos": [352, 38]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\nx : \u03b1\np_eq : \u00ac\u2191(r * \u2191s x) = 0\nthis\u271d : r * \u2191s x \u2260 0\nthis : \u2191s x \u2260 0\n\u22a2 \u2191(SimpleFunc.map c rs) x < \u2a06 n, f n x", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\nx : \u03b1\np_eq : \u00ac\u2191(r * \u2191s x) = 0\nthis\u271d : r * \u2191s x \u2260 0\nthis : \u2191s x \u2260 0\n\u22a2 \u2191rs x < \u2191s x"}, {"tactic": "suffices r * s x < 1 * s x by simpa", "annotated_tactic": ["suffices r * s x < 1 * s x by simpa", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\nx : \u03b1\np_eq : \u00ac\u2191(r * \u2191s x) = 0\nthis\u271d : r * \u2191s x \u2260 0\nthis : \u2191s x \u2260 0\n\u22a2 \u2191rs x < \u2191s x", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\nx : \u03b1\np_eq : \u00ac\u2191(r * \u2191s x) = 0\nthis\u271d : r * \u2191s x \u2260 0\nthis : \u2191s x \u2260 0\n\u22a2 r * \u2191s x < 1 * \u2191s x"}, {"tactic": "exact mul_lt_mul_of_pos_right ha (pos_iff_ne_zero.2 this)", "annotated_tactic": ["exact <a>mul_lt_mul_of_pos_right</a> ha (<a>pos_iff_ne_zero</a>.2 this)", [{"full_name": "mul_lt_mul_of_pos_right", "def_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "def_pos": [164, 9], "def_end_pos": [164, 32]}, {"full_name": "pos_iff_ne_zero", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [243, 3], "def_end_pos": [243, 14]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\nx : \u03b1\np_eq : \u00ac\u2191(r * \u2191s x) = 0\nthis\u271d : r * \u2191s x \u2260 0\nthis : \u2191s x \u2260 0\n\u22a2 r * \u2191s x < 1 * \u2191s x", "state_after": "no goals"}, {"tactic": "simpa", "annotated_tactic": ["simpa", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\nx : \u03b1\np_eq : \u00ac\u2191(r * \u2191s x) = 0\nthis\u271d\u00b9 : r * \u2191s x \u2260 0\nthis\u271d : \u2191s x \u2260 0\nthis : r * \u2191s x < 1 * \u2191s x\n\u22a2 \u2191rs x < \u2191s x", "state_after": "no goals"}, {"tactic": "intro r i j h", "annotated_tactic": ["intro r i j h", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\neq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a}\n\u22a2 \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr\u271d : \u211d\u22650\nright\u271d ha\u271d : \u2191r\u271d < 1\nha : r\u271d < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r\u271d * a) s\neq_rs : const \u03b1 \u2191r\u271d * SimpleFunc.map c s = SimpleFunc.map c rs\neq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a}\nr : \u211d\u22650\u221e\ni j : \u2115\nh : i \u2264 j\n\u22a2 (fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}) i \u2264\n    (fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}) j"}, {"tactic": "refine' inter_subset_inter (Subset.refl _) _", "annotated_tactic": ["refine' <a>inter_subset_inter</a> (<a>Subset.refl</a> _) _", [{"full_name": "Set.inter_subset_inter", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1022, 9], "def_end_pos": [1022, 27]}, {"full_name": "Set.Subset.refl", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [354, 9], "def_end_pos": [354, 20]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr\u271d : \u211d\u22650\nright\u271d ha\u271d : \u2191r\u271d < 1\nha : r\u271d < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r\u271d * a) s\neq_rs : const \u03b1 \u2191r\u271d * SimpleFunc.map c s = SimpleFunc.map c rs\neq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a}\nr : \u211d\u22650\u221e\ni j : \u2115\nh : i \u2264 j\n\u22a2 (fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}) i \u2264\n    (fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}) j", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr\u271d : \u211d\u22650\nright\u271d ha\u271d : \u2191r\u271d < 1\nha : r\u271d < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r\u271d * a) s\neq_rs : const \u03b1 \u2191r\u271d * SimpleFunc.map c s = SimpleFunc.map c rs\neq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a}\nr : \u211d\u22650\u221e\ni j : \u2115\nh : i \u2264 j\n\u22a2 {a | r \u2264 f i a} \u2286 {a | r \u2264 f j a}"}, {"tactic": "intro x (hx : r \u2264 f i x)", "annotated_tactic": ["intro x (hx : r \u2264 f i x)", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr\u271d : \u211d\u22650\nright\u271d ha\u271d : \u2191r\u271d < 1\nha : r\u271d < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r\u271d * a) s\neq_rs : const \u03b1 \u2191r\u271d * SimpleFunc.map c s = SimpleFunc.map c rs\neq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a}\nr : \u211d\u22650\u221e\ni j : \u2115\nh : i \u2264 j\n\u22a2 {a | r \u2264 f i a} \u2286 {a | r \u2264 f j a}", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr\u271d : \u211d\u22650\nright\u271d ha\u271d : \u2191r\u271d < 1\nha : r\u271d < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r\u271d * a) s\neq_rs : const \u03b1 \u2191r\u271d * SimpleFunc.map c s = SimpleFunc.map c rs\neq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a}\nr : \u211d\u22650\u221e\ni j : \u2115\nh : i \u2264 j\nx : \u03b1\nhx : r \u2264 f i x\n\u22a2 x \u2208 {a | r \u2264 f j a}"}, {"tactic": "exact le_trans hx (h_mono h x)", "annotated_tactic": ["exact <a>le_trans</a> hx (h_mono h x)", [{"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr\u271d : \u211d\u22650\nright\u271d ha\u271d : \u2191r\u271d < 1\nha : r\u271d < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r\u271d * a) s\neq_rs : const \u03b1 \u2191r\u271d * SimpleFunc.map c s = SimpleFunc.map c rs\neq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a}\nr : \u211d\u22650\u221e\ni j : \u2115\nh : i \u2264 j\nx : \u03b1\nhx : r \u2264 f i x\n\u22a2 x \u2208 {a | r \u2264 f j a}", "state_after": "no goals"}, {"tactic": "rw [\u2190 const_mul_lintegral, eq_rs, SimpleFunc.lintegral]", "annotated_tactic": ["rw [\u2190 <a>const_mul_lintegral</a>, eq_rs, <a>SimpleFunc.lintegral</a>]", [{"full_name": "MeasureTheory.SimpleFunc.const_mul_lintegral", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [1003, 9], "def_end_pos": [1003, 28]}, {"full_name": "MeasureTheory.SimpleFunc.lintegral", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [950, 5], "def_end_pos": [950, 14]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\neq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a}\nmono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}\nh_meas : \u2200 (n : \u2115), MeasurableSet {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}\n\u22a2 \u2191r * SimpleFunc.lintegral (SimpleFunc.map c s) \u03bc =\n    \u2211 r in SimpleFunc.range (SimpleFunc.map c rs), r * \u2191\u2191\u03bc (\u2191(SimpleFunc.map c rs) \u207b\u00b9' {r})", "state_after": "no goals"}, {"tactic": "simp only [(eq _).symm]", "annotated_tactic": ["simp only [(eq _).<a>symm</a>]", [{"full_name": "Eq.symm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [310, 9], "def_end_pos": [310, 16]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\neq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a}\nmono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}\nh_meas : \u2200 (n : \u2115), MeasurableSet {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}\n\u22a2 \u2211 r in SimpleFunc.range (SimpleFunc.map c rs), r * \u2191\u2191\u03bc (\u2191(SimpleFunc.map c rs) \u207b\u00b9' {r}) =\n    \u2211 r in SimpleFunc.range (SimpleFunc.map c rs), r * \u2191\u2191\u03bc (\u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a})", "state_after": "no goals"}, {"tactic": "rw [measure_iUnion_eq_iSup (directed_of_sup <| mono x), ENNReal.mul_iSup]", "annotated_tactic": ["rw [<a>measure_iUnion_eq_iSup</a> (<a>directed_of_sup</a> <| mono x), <a>ENNReal.mul_iSup</a>]", [{"full_name": "MeasureTheory.measure_iUnion_eq_iSup", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [463, 9], "def_end_pos": [463, 31]}, {"full_name": "directed_of_sup", "def_path": "Mathlib/Order/Directed.lean", "def_pos": [107, 9], "def_end_pos": [107, 24]}, {"full_name": "ENNReal.mul_iSup", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [638, 9], "def_end_pos": [638, 17]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d\u00b9 : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\neq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a}\nmono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}\nh_meas : \u2200 (n : \u2115), MeasurableSet {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}\nx : \u211d\u22650\u221e\nx\u271d : x \u2208 SimpleFunc.range (SimpleFunc.map c rs)\n\u22a2 x * \u2191\u2191\u03bc (\u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {x} \u2229 {a | x \u2264 f n a}) =\n    \u2a06 n, x * \u2191\u2191\u03bc (\u2191(SimpleFunc.map c rs) \u207b\u00b9' {x} \u2229 {a | x \u2264 f n a})", "state_after": "no goals"}, {"tactic": "rw [ENNReal.finset_sum_iSup_nat]", "annotated_tactic": ["rw [<a>ENNReal.finset_sum_iSup_nat</a>]", [{"full_name": "ENNReal.finset_sum_iSup_nat", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [627, 9], "def_end_pos": [627, 28]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\neq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a}\nmono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}\nh_meas : \u2200 (n : \u2115), MeasurableSet {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}\n\u22a2 \u2211 r in SimpleFunc.range (SimpleFunc.map c rs), \u2a06 n, r * \u2191\u2191\u03bc (\u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}) =\n    \u2a06 n, \u2211 r in SimpleFunc.range (SimpleFunc.map c rs), r * \u2191\u2191\u03bc (\u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a})", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\neq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a}\nmono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}\nh_meas : \u2200 (n : \u2115), MeasurableSet {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}\n\u22a2 \u2200 (a : \u211d\u22650\u221e), Monotone fun n => a * \u2191\u2191\u03bc (\u2191(SimpleFunc.map c rs) \u207b\u00b9' {a} \u2229 {a_1 | a \u2264 f n a_1})"}, {"tactic": "intro p i j h", "annotated_tactic": ["intro p i j h", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\neq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a}\nmono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}\nh_meas : \u2200 (n : \u2115), MeasurableSet {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}\n\u22a2 \u2200 (a : \u211d\u22650\u221e), Monotone fun n => a * \u2191\u2191\u03bc (\u2191(SimpleFunc.map c rs) \u207b\u00b9' {a} \u2229 {a_1 | a \u2264 f n a_1})", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\neq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a}\nmono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}\nh_meas : \u2200 (n : \u2115), MeasurableSet {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}\np : \u211d\u22650\u221e\ni j : \u2115\nh : i \u2264 j\n\u22a2 (fun n => p * \u2191\u2191\u03bc (\u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a})) i \u2264\n    (fun n => p * \u2191\u2191\u03bc (\u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a})) j"}, {"tactic": "exact mul_le_mul_left' (measure_mono <| mono p h) _", "annotated_tactic": ["exact <a>mul_le_mul_left'</a> (<a>measure_mono</a> <| mono p h) _", [{"full_name": "mul_le_mul_left'", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [50, 9], "def_end_pos": [50, 25]}, {"full_name": "MeasureTheory.measure_mono", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [193, 9], "def_end_pos": [193, 21]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\neq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a}\nmono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}\nh_meas : \u2200 (n : \u2115), MeasurableSet {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}\np : \u211d\u22650\u221e\ni j : \u2115\nh : i \u2264 j\n\u22a2 (fun n => p * \u2191\u2191\u03bc (\u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a})) i \u2264\n    (fun n => p * \u2191\u2191\u03bc (\u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a})) j", "state_after": "no goals"}, {"tactic": "refine' iSup_mono fun n => _", "annotated_tactic": ["refine' <a>iSup_mono</a> fun n => _", [{"full_name": "iSup_mono", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [900, 9], "def_end_pos": [900, 18]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\neq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a}\nmono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}\nh_meas : \u2200 (n : \u2115), MeasurableSet {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}\n\u22a2 \u2a06 n, \u2211 r in SimpleFunc.range (SimpleFunc.map c rs), r * \u2191\u2191\u03bc (\u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}) \u2264\n    \u2a06 n, SimpleFunc.lintegral (restrict (SimpleFunc.map c rs) {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}) \u03bc", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\neq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a}\nmono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}\nh_meas : \u2200 (n : \u2115), MeasurableSet {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}\nn : \u2115\n\u22a2 \u2211 r in SimpleFunc.range (SimpleFunc.map c rs), r * \u2191\u2191\u03bc (\u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}) \u2264\n    SimpleFunc.lintegral (restrict (SimpleFunc.map c rs) {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}) \u03bc"}, {"tactic": "rw [restrict_lintegral _ (h_meas n)]", "annotated_tactic": ["rw [<a>restrict_lintegral</a> _ (h_meas n)]", [{"full_name": "MeasureTheory.SimpleFunc.restrict_lintegral", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [1048, 9], "def_end_pos": [1048, 27]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\neq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a}\nmono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}\nh_meas : \u2200 (n : \u2115), MeasurableSet {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}\nn : \u2115\n\u22a2 \u2211 r in SimpleFunc.range (SimpleFunc.map c rs), r * \u2191\u2191\u03bc (\u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}) \u2264\n    SimpleFunc.lintegral (restrict (SimpleFunc.map c rs) {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}) \u03bc", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\neq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a}\nmono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}\nh_meas : \u2200 (n : \u2115), MeasurableSet {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}\nn : \u2115\n\u22a2 \u2211 r in SimpleFunc.range (SimpleFunc.map c rs), r * \u2191\u2191\u03bc (\u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}) \u2264\n    \u2211 r in SimpleFunc.range (SimpleFunc.map c rs),\n      r * \u2191\u2191\u03bc (\u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a})"}, {"tactic": "refine' le_of_eq (Finset.sum_congr rfl fun r _ => _)", "annotated_tactic": ["refine' <a>le_of_eq</a> (<a>Finset.sum_congr</a> <a>rfl</a> fun r _ => _)", [{"full_name": "le_of_eq", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [72, 9], "def_end_pos": [72, 17]}, {"full_name": "Finset.sum_congr", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [376, 3], "def_end_pos": [376, 14]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\neq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a}\nmono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}\nh_meas : \u2200 (n : \u2115), MeasurableSet {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}\nn : \u2115\n\u22a2 \u2211 r in SimpleFunc.range (SimpleFunc.map c rs), r * \u2191\u2191\u03bc (\u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}) \u2264\n    \u2211 r in SimpleFunc.range (SimpleFunc.map c rs),\n      r * \u2191\u2191\u03bc (\u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a})", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d\u00b9 : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr\u271d : \u211d\u22650\nright\u271d ha\u271d : \u2191r\u271d < 1\nha : r\u271d < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r\u271d * a) s\neq_rs : const \u03b1 \u2191r\u271d * SimpleFunc.map c s = SimpleFunc.map c rs\neq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a}\nmono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}\nh_meas : \u2200 (n : \u2115), MeasurableSet {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}\nn : \u2115\nr : \u211d\u22650\u221e\nx\u271d : r \u2208 SimpleFunc.range (SimpleFunc.map c rs)\n\u22a2 r * \u2191\u2191\u03bc (\u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}) =\n    r * \u2191\u2191\u03bc (\u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a})"}, {"tactic": "congr 2 with a", "annotated_tactic": ["congr 2 with a", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d\u00b9 : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr\u271d : \u211d\u22650\nright\u271d ha\u271d : \u2191r\u271d < 1\nha : r\u271d < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r\u271d * a) s\neq_rs : const \u03b1 \u2191r\u271d * SimpleFunc.map c s = SimpleFunc.map c rs\neq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a}\nmono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}\nh_meas : \u2200 (n : \u2115), MeasurableSet {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}\nn : \u2115\nr : \u211d\u22650\u221e\nx\u271d : r \u2208 SimpleFunc.range (SimpleFunc.map c rs)\n\u22a2 r * \u2191\u2191\u03bc (\u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}) =\n    r * \u2191\u2191\u03bc (\u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a})", "state_after": "case e_a.e_a.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d\u00b9 : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr\u271d : \u211d\u22650\nright\u271d ha\u271d : \u2191r\u271d < 1\nha : r\u271d < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r\u271d * a) s\neq_rs : const \u03b1 \u2191r\u271d * SimpleFunc.map c s = SimpleFunc.map c rs\neq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a}\nmono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}\nh_meas : \u2200 (n : \u2115), MeasurableSet {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}\nn : \u2115\nr : \u211d\u22650\u221e\nx\u271d : r \u2208 SimpleFunc.range (SimpleFunc.map c rs)\na : \u03b1\n\u22a2 a \u2208 \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a} \u2194\n    a \u2208 \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}"}, {"tactic": "refine' and_congr_right _", "annotated_tactic": ["refine' <a>and_congr_right</a> _", [{"full_name": "and_congr_right", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [161, 9], "def_end_pos": [161, 24]}]], "state_before": "case e_a.e_a.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d\u00b9 : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr\u271d : \u211d\u22650\nright\u271d ha\u271d : \u2191r\u271d < 1\nha : r\u271d < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r\u271d * a) s\neq_rs : const \u03b1 \u2191r\u271d * SimpleFunc.map c s = SimpleFunc.map c rs\neq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a}\nmono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}\nh_meas : \u2200 (n : \u2115), MeasurableSet {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}\nn : \u2115\nr : \u211d\u22650\u221e\nx\u271d : r \u2208 SimpleFunc.range (SimpleFunc.map c rs)\na : \u03b1\n\u22a2 a \u2208 \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a} \u2194\n    a \u2208 \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}", "state_after": "case e_a.e_a.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d\u00b9 : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr\u271d : \u211d\u22650\nright\u271d ha\u271d : \u2191r\u271d < 1\nha : r\u271d < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r\u271d * a) s\neq_rs : const \u03b1 \u2191r\u271d * SimpleFunc.map c s = SimpleFunc.map c rs\neq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a}\nmono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}\nh_meas : \u2200 (n : \u2115), MeasurableSet {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}\nn : \u2115\nr : \u211d\u22650\u221e\nx\u271d : r \u2208 SimpleFunc.range (SimpleFunc.map c rs)\na : \u03b1\n\u22a2 a \u2208 \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2192 (a \u2208 {a | r \u2264 f n a} \u2194 a \u2208 {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a})"}, {"tactic": "simp (config := { contextual := true })", "annotated_tactic": ["simp (config := { contextual := <a>true</a> })", [{"full_name": "Bool.true", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [549, 5], "def_end_pos": [549, 9]}]], "state_before": "case e_a.e_a.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d\u00b9 : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr\u271d : \u211d\u22650\nright\u271d ha\u271d : \u2191r\u271d < 1\nha : r\u271d < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r\u271d * a) s\neq_rs : const \u03b1 \u2191r\u271d * SimpleFunc.map c s = SimpleFunc.map c rs\neq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a}\nmono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}\nh_meas : \u2200 (n : \u2115), MeasurableSet {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}\nn : \u2115\nr : \u211d\u22650\u221e\nx\u271d : r \u2208 SimpleFunc.range (SimpleFunc.map c rs)\na : \u03b1\n\u22a2 a \u2208 \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2192 (a \u2208 {a | r \u2264 f n a} \u2194 a \u2208 {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a})", "state_after": "no goals"}, {"tactic": "refine' iSup_mono fun n => _", "annotated_tactic": ["refine' <a>iSup_mono</a> fun n => _", [{"full_name": "iSup_mono", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [900, 9], "def_end_pos": [900, 18]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\neq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a}\nmono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}\nh_meas : \u2200 (n : \u2115), MeasurableSet {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}\n\u22a2 \u2a06 n, SimpleFunc.lintegral (restrict (SimpleFunc.map c rs) {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}) \u03bc \u2264\n    \u2a06 n, \u222b\u207b (a : \u03b1), f n a \u2202\u03bc", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\neq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a}\nmono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}\nh_meas : \u2200 (n : \u2115), MeasurableSet {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}\nn : \u2115\n\u22a2 SimpleFunc.lintegral (restrict (SimpleFunc.map c rs) {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}) \u03bc \u2264 \u222b\u207b (a : \u03b1), f n a \u2202\u03bc"}, {"tactic": "rw [\u2190 SimpleFunc.lintegral_eq_lintegral]", "annotated_tactic": ["rw [\u2190 <a>SimpleFunc.lintegral_eq_lintegral</a>]", [{"full_name": "MeasureTheory.SimpleFunc.lintegral_eq_lintegral", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [80, 9], "def_end_pos": [80, 42]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\neq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a}\nmono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}\nh_meas : \u2200 (n : \u2115), MeasurableSet {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}\nn : \u2115\n\u22a2 SimpleFunc.lintegral (restrict (SimpleFunc.map c rs) {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}) \u03bc \u2264 \u222b\u207b (a : \u03b1), f n a \u2202\u03bc", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\neq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a}\nmono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}\nh_meas : \u2200 (n : \u2115), MeasurableSet {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}\nn : \u2115\n\u22a2 \u222b\u207b (a : \u03b1), \u2191(restrict (SimpleFunc.map c rs) {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}) a \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), f n a \u2202\u03bc"}, {"tactic": "refine' lintegral_mono fun a => _", "annotated_tactic": ["refine' <a>lintegral_mono</a> fun a => _", [{"full_name": "MeasureTheory.lintegral_mono", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [99, 9], "def_end_pos": [99, 23]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\neq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a}\nmono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}\nh_meas : \u2200 (n : \u2115), MeasurableSet {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}\nn : \u2115\n\u22a2 \u222b\u207b (a : \u03b1), \u2191(restrict (SimpleFunc.map c rs) {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}) a \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), f n a \u2202\u03bc", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\neq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a}\nmono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}\nh_meas : \u2200 (n : \u2115), MeasurableSet {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}\nn : \u2115\na : \u03b1\n\u22a2 \u2191(restrict (SimpleFunc.map c rs) {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}) a \u2264 f n a"}, {"tactic": "simp only [map_apply] at h_meas", "annotated_tactic": ["simp only [<a>map_apply</a>] at h_meas", [{"full_name": "MeasureTheory.SimpleFunc.map_apply", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [294, 9], "def_end_pos": [294, 18]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\neq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a}\nmono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}\nh_meas : \u2200 (n : \u2115), MeasurableSet {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}\nn : \u2115\na : \u03b1\n\u22a2 \u2191(restrict (SimpleFunc.map c rs) {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}) a \u2264 f n a", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\neq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a}\nmono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}\nh_meas : \u2200 (n : \u2115), MeasurableSet {a | \u2191(r * \u2191s a) \u2264 f n a}\nn : \u2115\na : \u03b1\n\u22a2 \u2191(restrict (SimpleFunc.map c rs) {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}) a \u2264 f n a"}, {"tactic": "exact indicator_apply_le id", "annotated_tactic": ["exact <a>indicator_apply_le</a> <a>id</a>", [{"full_name": "Set.indicator_apply_le", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [906, 3], "def_end_pos": [906, 14]}, {"full_name": "id", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [33, 15], "def_end_pos": [33, 17]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\neq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a}\nmono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}\nh_meas : \u2200 (n : \u2115), MeasurableSet {a | \u2191(r * \u2191s a) \u2264 f n a}\nn : \u2115\na : \u03b1\n\u22a2 indicator {a | \u2191(r * \u2191s a) \u2264 f n a} (fun x => \u2191(r * \u2191s x)) a \u2264 f n a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Pointwise/Interval.lean", "full_name": "Set.image_mul_left_Ioo", "start": [754, 1], "end": [756, 73], "traced_tactics": [{"tactic": "convert image_mul_right_Ioo b c h using 1 <;> simp only [mul_comm _ a]", "annotated_tactic": ["convert <a>image_mul_right_Ioo</a> b c h using 1 <;> simp only [<a>mul_comm</a> _ a]", [{"full_name": "Set.image_mul_right_Ioo", "def_path": "Mathlib/Data/Set/Pointwise/Interval.lean", "def_pos": [749, 9], "def_end_pos": [749, 28]}, {"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [302, 9], "def_end_pos": [302, 17]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na\u271d a : \u03b1\nh : 0 < a\nb c : \u03b1\n\u22a2 (fun x x_1 => x * x_1) a '' Ioo b c = Ioo (a * b) (a * c)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Image.lean", "full_name": "Set.disjoint_image_iff", "start": [1619, 1], "end": [1620, 54], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/CondCount.lean", "full_name": "ProbabilityTheory.condCount_inter_self", "start": [100, 1], "end": [101, 53], "traced_tactics": [{"tactic": "rw [condCount, cond_inter_self _ hs.measurableSet]", "annotated_tactic": ["rw [<a>condCount</a>, <a>cond_inter_self</a> _ hs.measurableSet]", [{"full_name": "ProbabilityTheory.condCount", "def_path": "Mathlib/Probability/CondCount.lean", "def_pos": [54, 5], "def_end_pos": [54, 14]}, {"full_name": "ProbabilityTheory.cond_inter_self", "def_path": "Mathlib/Probability/ConditionalProbability.lean", "def_pos": [106, 9], "def_end_pos": [106, 24]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nhs : Set.Finite s\n\u22a2 \u2191\u2191(condCount s) (s \u2229 t) = \u2191\u2191(condCount s) t", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Function.lean", "full_name": "Set.injOn_iff_invFunOn_image_image_eq_self", "start": [1278, 1], "end": [1281, 77], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Intervals/OrdConnectedComponent.lean", "full_name": "Set.dual_ordConnectedComponent", "start": [36, 1], "end": [40, 8], "traced_tactics": [{"tactic": "rw [mem_ordConnectedComponent, dual_uIcc]", "annotated_tactic": ["rw [<a>mem_ordConnectedComponent</a>, <a>dual_uIcc</a>]", [{"full_name": "Set.mem_ordConnectedComponent", "def_path": "Mathlib/Data/Set/Intervals/OrdConnectedComponent.lean", "def_pos": [32, 9], "def_end_pos": [32, 34]}, {"full_name": "Set.dual_uIcc", "def_path": "Mathlib/Data/Set/Intervals/UnorderedInterval.lean", "def_pos": [65, 15], "def_end_pos": [65, 24]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\ns t : Set \u03b1\nx\u271d y z x : \u03b1\n\u22a2 \u2191toDual x \u2208 ordConnectedComponent (\u2191ofDual \u207b\u00b9' s) (\u2191toDual x\u271d) \u2194 \u2191toDual x \u2208 \u2191ofDual \u207b\u00b9' ordConnectedComponent s x\u271d", "state_after": "\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\ns t : Set \u03b1\nx\u271d y z x : \u03b1\n\u22a2 \u2191ofDual \u207b\u00b9' [[x\u271d, x]] \u2286 \u2191ofDual \u207b\u00b9' s \u2194 \u2191toDual x \u2208 \u2191ofDual \u207b\u00b9' ordConnectedComponent s x\u271d"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\ns t : Set \u03b1\nx\u271d y z x : \u03b1\n\u22a2 \u2191ofDual \u207b\u00b9' [[x\u271d, x]] \u2286 \u2191ofDual \u207b\u00b9' s \u2194 \u2191toDual x \u2208 \u2191ofDual \u207b\u00b9' ordConnectedComponent s x\u271d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Portmanteau.lean", "full_name": "MeasureTheory.ProbabilityMeasure.tendsto_measure_of_null_frontier_of_tendsto", "start": [424, 1], "end": [431, 68], "traced_tactics": [{"tactic": "have E_nullbdry' : (\u03bc : Measure \u03a9) (frontier E) = 0 := by\n  rw [\u2190 ProbabilityMeasure.ennreal_coeFn_eq_coeFn_toMeasure, E_nullbdry, ENNReal.coe_zero]", "annotated_tactic": ["have E_nullbdry' : (\u03bc : <a>Measure</a> \u03a9) (<a>frontier</a> E) = 0 := by\n    rw [\u2190 <a>ProbabilityMeasure.ennreal_coeFn_eq_coeFn_toMeasure</a>, E_nullbdry, <a>ENNReal.coe_zero</a>]", [{"full_name": "MeasureTheory.Measure", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [74, 11], "def_end_pos": [74, 18]}, {"full_name": "frontier", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [703, 5], "def_end_pos": [703, 13]}, {"full_name": "MeasureTheory.ProbabilityMeasure.ennreal_coeFn_eq_coeFn_toMeasure", "def_path": "Mathlib/MeasureTheory/Measure/ProbabilityMeasure.lean", "def_pos": [171, 9], "def_end_pos": [171, 41]}, {"full_name": "ENNReal.coe_zero", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [215, 28], "def_end_pos": [215, 36]}]], "state_before": "\u03a9\u271d : Type u_1\ninst\u271d\u00b3 : MeasurableSpace \u03a9\u271d\n\u03a9 : Type u_2\n\u03b9 : Type u_3\nL : Filter \u03b9\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : PseudoEMetricSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03bc : ProbabilityMeasure \u03a9\n\u03bcs : \u03b9 \u2192 ProbabilityMeasure \u03a9\n\u03bcs_lim : Tendsto \u03bcs L (\ud835\udcdd \u03bc)\nE : Set \u03a9\nE_nullbdry : (fun s => ENNReal.toNNReal (\u2191\u2191\u2191\u03bc s)) (frontier E) = 0\n\u22a2 Tendsto (fun i => (fun s => ENNReal.toNNReal (\u2191\u2191\u2191(\u03bcs i) s)) E) L (\ud835\udcdd ((fun s => ENNReal.toNNReal (\u2191\u2191\u2191\u03bc s)) E))", "state_after": "\u03a9\u271d : Type u_1\ninst\u271d\u00b3 : MeasurableSpace \u03a9\u271d\n\u03a9 : Type u_2\n\u03b9 : Type u_3\nL : Filter \u03b9\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : PseudoEMetricSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03bc : ProbabilityMeasure \u03a9\n\u03bcs : \u03b9 \u2192 ProbabilityMeasure \u03a9\n\u03bcs_lim : Tendsto \u03bcs L (\ud835\udcdd \u03bc)\nE : Set \u03a9\nE_nullbdry : (fun s => ENNReal.toNNReal (\u2191\u2191\u2191\u03bc s)) (frontier E) = 0\nE_nullbdry' : \u2191\u2191\u2191\u03bc (frontier E) = 0\n\u22a2 Tendsto (fun i => (fun s => ENNReal.toNNReal (\u2191\u2191\u2191(\u03bcs i) s)) E) L (\ud835\udcdd ((fun s => ENNReal.toNNReal (\u2191\u2191\u2191\u03bc s)) E))"}, {"tactic": "have key := ProbabilityMeasure.tendsto_measure_of_null_frontier_of_tendsto' \u03bcs_lim E_nullbdry'", "annotated_tactic": ["have key := <a>ProbabilityMeasure.tendsto_measure_of_null_frontier_of_tendsto'</a> \u03bcs_lim E_nullbdry'", [{"full_name": "MeasureTheory.ProbabilityMeasure.tendsto_measure_of_null_frontier_of_tendsto'", "def_path": "Mathlib/MeasureTheory/Measure/Portmanteau.lean", "def_pos": [406, 9], "def_end_pos": [406, 72]}]], "state_before": "\u03a9\u271d : Type u_1\ninst\u271d\u00b3 : MeasurableSpace \u03a9\u271d\n\u03a9 : Type u_2\n\u03b9 : Type u_3\nL : Filter \u03b9\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : PseudoEMetricSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03bc : ProbabilityMeasure \u03a9\n\u03bcs : \u03b9 \u2192 ProbabilityMeasure \u03a9\n\u03bcs_lim : Tendsto \u03bcs L (\ud835\udcdd \u03bc)\nE : Set \u03a9\nE_nullbdry : (fun s => ENNReal.toNNReal (\u2191\u2191\u2191\u03bc s)) (frontier E) = 0\nE_nullbdry' : \u2191\u2191\u2191\u03bc (frontier E) = 0\n\u22a2 Tendsto (fun i => (fun s => ENNReal.toNNReal (\u2191\u2191\u2191(\u03bcs i) s)) E) L (\ud835\udcdd ((fun s => ENNReal.toNNReal (\u2191\u2191\u2191\u03bc s)) E))", "state_after": "\u03a9\u271d : Type u_1\ninst\u271d\u00b3 : MeasurableSpace \u03a9\u271d\n\u03a9 : Type u_2\n\u03b9 : Type u_3\nL : Filter \u03b9\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : PseudoEMetricSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03bc : ProbabilityMeasure \u03a9\n\u03bcs : \u03b9 \u2192 ProbabilityMeasure \u03a9\n\u03bcs_lim : Tendsto \u03bcs L (\ud835\udcdd \u03bc)\nE : Set \u03a9\nE_nullbdry : (fun s => ENNReal.toNNReal (\u2191\u2191\u2191\u03bc s)) (frontier E) = 0\nE_nullbdry' : \u2191\u2191\u2191\u03bc (frontier E) = 0\nkey : Tendsto (fun i => \u2191\u2191\u2191(\u03bcs i) E) L (\ud835\udcdd (\u2191\u2191\u2191\u03bc E))\n\u22a2 Tendsto (fun i => (fun s => ENNReal.toNNReal (\u2191\u2191\u2191(\u03bcs i) s)) E) L (\ud835\udcdd ((fun s => ENNReal.toNNReal (\u2191\u2191\u2191\u03bc s)) E))"}, {"tactic": "exact (ENNReal.tendsto_toNNReal (measure_ne_top (\u2191\u03bc) E)).comp key", "annotated_tactic": ["exact (<a>ENNReal.tendsto_toNNReal</a> (<a>measure_ne_top</a> (\u2191\u03bc) E)).<a>comp</a> key", [{"full_name": "ENNReal.tendsto_toNNReal", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [113, 9], "def_end_pos": [113, 25]}, {"full_name": "MeasureTheory.measure_ne_top", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2875, 9], "def_end_pos": [2875, 23]}, {"full_name": "Filter.Tendsto.comp", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [3032, 9], "def_end_pos": [3032, 21]}]], "state_before": "\u03a9\u271d : Type u_1\ninst\u271d\u00b3 : MeasurableSpace \u03a9\u271d\n\u03a9 : Type u_2\n\u03b9 : Type u_3\nL : Filter \u03b9\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : PseudoEMetricSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03bc : ProbabilityMeasure \u03a9\n\u03bcs : \u03b9 \u2192 ProbabilityMeasure \u03a9\n\u03bcs_lim : Tendsto \u03bcs L (\ud835\udcdd \u03bc)\nE : Set \u03a9\nE_nullbdry : (fun s => ENNReal.toNNReal (\u2191\u2191\u2191\u03bc s)) (frontier E) = 0\nE_nullbdry' : \u2191\u2191\u2191\u03bc (frontier E) = 0\nkey : Tendsto (fun i => \u2191\u2191\u2191(\u03bcs i) E) L (\ud835\udcdd (\u2191\u2191\u2191\u03bc E))\n\u22a2 Tendsto (fun i => (fun s => ENNReal.toNNReal (\u2191\u2191\u2191(\u03bcs i) s)) E) L (\ud835\udcdd ((fun s => ENNReal.toNNReal (\u2191\u2191\u2191\u03bc s)) E))", "state_after": "no goals"}, {"tactic": "rw [\u2190 ProbabilityMeasure.ennreal_coeFn_eq_coeFn_toMeasure, E_nullbdry, ENNReal.coe_zero]", "annotated_tactic": ["rw [\u2190 <a>ProbabilityMeasure.ennreal_coeFn_eq_coeFn_toMeasure</a>, E_nullbdry, <a>ENNReal.coe_zero</a>]", [{"full_name": "MeasureTheory.ProbabilityMeasure.ennreal_coeFn_eq_coeFn_toMeasure", "def_path": "Mathlib/MeasureTheory/Measure/ProbabilityMeasure.lean", "def_pos": [171, 9], "def_end_pos": [171, 41]}, {"full_name": "ENNReal.coe_zero", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [215, 28], "def_end_pos": [215, 36]}]], "state_before": "\u03a9\u271d : Type u_1\ninst\u271d\u00b3 : MeasurableSpace \u03a9\u271d\n\u03a9 : Type u_2\n\u03b9 : Type u_3\nL : Filter \u03b9\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : PseudoEMetricSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03bc : ProbabilityMeasure \u03a9\n\u03bcs : \u03b9 \u2192 ProbabilityMeasure \u03a9\n\u03bcs_lim : Tendsto \u03bcs L (\ud835\udcdd \u03bc)\nE : Set \u03a9\nE_nullbdry : (fun s => ENNReal.toNNReal (\u2191\u2191\u2191\u03bc s)) (frontier E) = 0\n\u22a2 \u2191\u2191\u2191\u03bc (frontier E) = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/Monad.lean", "full_name": "MvPolynomial.map_bind\u2081", "start": [290, 1], "end": [293, 6], "traced_tactics": [{"tactic": "rw [hom_bind\u2081, map_comp_C, \u2190 eval\u2082Hom_map_hom]", "annotated_tactic": ["rw [<a>hom_bind\u2081</a>, <a>map_comp_C</a>, \u2190 <a>eval\u2082Hom_map_hom</a>]", [{"full_name": "MvPolynomial.hom_bind\u2081", "def_path": "Mathlib/Data/MvPolynomial/Monad.lean", "def_pos": [285, 9], "def_end_pos": [285, 18]}, {"full_name": "MvPolynomial.map_comp_C", "def_path": "Mathlib/Data/MvPolynomial/Monad.lean", "def_pos": [278, 9], "def_end_pos": [278, 19]}, {"full_name": "MvPolynomial.eval\u2082Hom_map_hom", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [1363, 9], "def_end_pos": [1363, 25]}]], "state_before": "\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : CommSemiring T\nf\u271d : \u03c3 \u2192 MvPolynomial \u03c4 R\nf : R \u2192+* S\ng : \u03c3 \u2192 MvPolynomial \u03c4 R\n\u03c6 : MvPolynomial \u03c3 R\n\u22a2 \u2191(map f) (\u2191(bind\u2081 g) \u03c6) = \u2191(bind\u2081 fun i => \u2191(map f) (g i)) (\u2191(map f) \u03c6)", "state_after": "\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : CommSemiring T\nf\u271d : \u03c3 \u2192 MvPolynomial \u03c4 R\nf : R \u2192+* S\ng : \u03c3 \u2192 MvPolynomial \u03c4 R\n\u03c6 : MvPolynomial \u03c3 R\n\u22a2 \u2191(eval\u2082Hom C fun i => \u2191(map f) (g i)) (\u2191(map f) \u03c6) = \u2191(bind\u2081 fun i => \u2191(map f) (g i)) (\u2191(map f) \u03c6)"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : CommSemiring T\nf\u271d : \u03c3 \u2192 MvPolynomial \u03c4 R\nf : R \u2192+* S\ng : \u03c3 \u2192 MvPolynomial \u03c4 R\n\u03c6 : MvPolynomial \u03c3 R\n\u22a2 \u2191(eval\u2082Hom C fun i => \u2191(map f) (g i)) (\u2191(map f) \u03c6) = \u2191(bind\u2081 fun i => \u2191(map f) (g i)) (\u2191(map f) \u03c6)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "full_name": "MeasureTheory.Measure.comap\u2097_eq_comap", "start": [1341, 1], "end": [1344, 70], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "full_name": "List.isSublist_iff_sublist", "start": [492, 1], "end": [508, 24], "traced_tactics": [{"tactic": "cases l\u2081 <;> cases l\u2082 <;> simp [isSublist]", "annotated_tactic": ["cases l\u2081 <;> cases l\u2082 <;> simp [<a>isSublist</a>]", [{"full_name": "List.isSublist", "def_path": "lake-packages/std/Std/Data/List/Basic.lean", "def_pos": [449, 5], "def_end_pos": [449, 14]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082 : List \u03b1\n\u22a2 isSublist l\u2081 l\u2082 = true \u2194 l\u2081 <+ l\u2082", "state_after": "case cons.cons\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nhead\u271d\u00b9 : \u03b1\ntail\u271d\u00b9 : List \u03b1\nhead\u271d : \u03b1\ntail\u271d : List \u03b1\n\u22a2 (if head\u271d\u00b9 = head\u271d then isSublist tail\u271d\u00b9 tail\u271d else isSublist (head\u271d\u00b9 :: tail\u271d\u00b9) tail\u271d) = true \u2194\n    head\u271d\u00b9 :: tail\u271d\u00b9 <+ head\u271d :: tail\u271d"}, {"tactic": "simp [h_eq, cons_sublist_cons, isSublist_iff_sublist]", "annotated_tactic": ["simp [h_eq, <a>cons_sublist_cons</a>, isSublist_iff_sublist]", [{"full_name": "List.cons_sublist_cons", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [405, 9], "def_end_pos": [405, 26]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nhd\u2081 : \u03b1\ntl\u2081 : List \u03b1\nhd\u2082 : \u03b1\ntl\u2082 : List \u03b1\nh_eq : hd\u2081 = hd\u2082\n\u22a2 (if hd\u2081 = hd\u2082 then isSublist tl\u2081 tl\u2082 else isSublist (hd\u2081 :: tl\u2081) tl\u2082) = true \u2194 hd\u2081 :: tl\u2081 <+ hd\u2082 :: tl\u2082", "state_after": "no goals"}, {"tactic": "simp only [h_eq]", "annotated_tactic": ["simp only [h_eq]", []], "state_before": "\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nhd\u2081 : \u03b1\ntl\u2081 : List \u03b1\nhd\u2082 : \u03b1\ntl\u2082 : List \u03b1\nh_eq : \u00achd\u2081 = hd\u2082\n\u22a2 (if hd\u2081 = hd\u2082 then isSublist tl\u2081 tl\u2082 else isSublist (hd\u2081 :: tl\u2081) tl\u2082) = true \u2194 hd\u2081 :: tl\u2081 <+ hd\u2082 :: tl\u2082", "state_after": "\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nhd\u2081 : \u03b1\ntl\u2081 : List \u03b1\nhd\u2082 : \u03b1\ntl\u2082 : List \u03b1\nh_eq : \u00achd\u2081 = hd\u2082\n\u22a2 (if False then isSublist tl\u2081 tl\u2082 else isSublist (hd\u2081 :: tl\u2081) tl\u2082) = true \u2194 hd\u2081 :: tl\u2081 <+ hd\u2082 :: tl\u2082"}, {"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nhd\u2081 : \u03b1\ntl\u2081 : List \u03b1\nhd\u2082 : \u03b1\ntl\u2082 : List \u03b1\nh_eq : \u00achd\u2081 = hd\u2082\n\u22a2 (if False then isSublist tl\u2081 tl\u2082 else isSublist (hd\u2081 :: tl\u2081) tl\u2082) = true \u2194 hd\u2081 :: tl\u2081 <+ hd\u2082 :: tl\u2082", "state_after": "case mp\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nhd\u2081 : \u03b1\ntl\u2081 : List \u03b1\nhd\u2082 : \u03b1\ntl\u2082 : List \u03b1\nh_eq : \u00achd\u2081 = hd\u2082\n\u22a2 (if False then isSublist tl\u2081 tl\u2082 else isSublist (hd\u2081 :: tl\u2081) tl\u2082) = true \u2192 hd\u2081 :: tl\u2081 <+ hd\u2082 :: tl\u2082\n\ncase mpr\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nhd\u2081 : \u03b1\ntl\u2081 : List \u03b1\nhd\u2082 : \u03b1\ntl\u2082 : List \u03b1\nh_eq : \u00achd\u2081 = hd\u2082\n\u22a2 hd\u2081 :: tl\u2081 <+ hd\u2082 :: tl\u2082 \u2192 (if False then isSublist tl\u2081 tl\u2082 else isSublist (hd\u2081 :: tl\u2081) tl\u2082) = true"}, {"tactic": "intro h_sub", "annotated_tactic": ["intro h_sub", []], "state_before": "case mp\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nhd\u2081 : \u03b1\ntl\u2081 : List \u03b1\nhd\u2082 : \u03b1\ntl\u2082 : List \u03b1\nh_eq : \u00achd\u2081 = hd\u2082\n\u22a2 (if False then isSublist tl\u2081 tl\u2082 else isSublist (hd\u2081 :: tl\u2081) tl\u2082) = true \u2192 hd\u2081 :: tl\u2081 <+ hd\u2082 :: tl\u2082", "state_after": "case mp\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nhd\u2081 : \u03b1\ntl\u2081 : List \u03b1\nhd\u2082 : \u03b1\ntl\u2082 : List \u03b1\nh_eq : \u00achd\u2081 = hd\u2082\nh_sub : (if False then isSublist tl\u2081 tl\u2082 else isSublist (hd\u2081 :: tl\u2081) tl\u2082) = true\n\u22a2 hd\u2081 :: tl\u2081 <+ hd\u2082 :: tl\u2082"}, {"tactic": "apply Sublist.cons", "annotated_tactic": ["apply <a>Sublist.cons</a>", [{"full_name": "List.Sublist.cons", "def_path": "lake-packages/std/Std/Data/List/Basic.lean", "def_pos": [442, 5], "def_end_pos": [442, 9]}]], "state_before": "case mp\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nhd\u2081 : \u03b1\ntl\u2081 : List \u03b1\nhd\u2082 : \u03b1\ntl\u2082 : List \u03b1\nh_eq : \u00achd\u2081 = hd\u2082\nh_sub : (if False then isSublist tl\u2081 tl\u2082 else isSublist (hd\u2081 :: tl\u2081) tl\u2082) = true\n\u22a2 hd\u2081 :: tl\u2081 <+ hd\u2082 :: tl\u2082", "state_after": "case mp.a\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nhd\u2081 : \u03b1\ntl\u2081 : List \u03b1\nhd\u2082 : \u03b1\ntl\u2082 : List \u03b1\nh_eq : \u00achd\u2081 = hd\u2082\nh_sub : (if False then isSublist tl\u2081 tl\u2082 else isSublist (hd\u2081 :: tl\u2081) tl\u2082) = true\n\u22a2 hd\u2081 :: tl\u2081 <+ tl\u2082"}, {"tactic": "exact isSublist_iff_sublist.mp h_sub", "annotated_tactic": ["exact isSublist_iff_sublist.mp h_sub", []], "state_before": "case mp.a\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nhd\u2081 : \u03b1\ntl\u2081 : List \u03b1\nhd\u2082 : \u03b1\ntl\u2082 : List \u03b1\nh_eq : \u00achd\u2081 = hd\u2082\nh_sub : (if False then isSublist tl\u2081 tl\u2082 else isSublist (hd\u2081 :: tl\u2081) tl\u2082) = true\n\u22a2 hd\u2081 :: tl\u2081 <+ tl\u2082", "state_after": "no goals"}, {"tactic": "intro h_sub", "annotated_tactic": ["intro h_sub", []], "state_before": "case mpr\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nhd\u2081 : \u03b1\ntl\u2081 : List \u03b1\nhd\u2082 : \u03b1\ntl\u2082 : List \u03b1\nh_eq : \u00achd\u2081 = hd\u2082\n\u22a2 hd\u2081 :: tl\u2081 <+ hd\u2082 :: tl\u2082 \u2192 (if False then isSublist tl\u2081 tl\u2082 else isSublist (hd\u2081 :: tl\u2081) tl\u2082) = true", "state_after": "case mpr\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nhd\u2081 : \u03b1\ntl\u2081 : List \u03b1\nhd\u2082 : \u03b1\ntl\u2082 : List \u03b1\nh_eq : \u00achd\u2081 = hd\u2082\nh_sub : hd\u2081 :: tl\u2081 <+ hd\u2082 :: tl\u2082\n\u22a2 (if False then isSublist tl\u2081 tl\u2082 else isSublist (hd\u2081 :: tl\u2081) tl\u2082) = true"}, {"tactic": "cases h_sub", "annotated_tactic": ["cases h_sub", []], "state_before": "case mpr\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nhd\u2081 : \u03b1\ntl\u2081 : List \u03b1\nhd\u2082 : \u03b1\ntl\u2082 : List \u03b1\nh_eq : \u00achd\u2081 = hd\u2082\nh_sub : hd\u2081 :: tl\u2081 <+ hd\u2082 :: tl\u2082\n\u22a2 (if False then isSublist tl\u2081 tl\u2082 else isSublist (hd\u2081 :: tl\u2081) tl\u2082) = true", "state_after": "case mpr.cons\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nhd\u2081 : \u03b1\ntl\u2081 : List \u03b1\nhd\u2082 : \u03b1\ntl\u2082 : List \u03b1\nh_eq : \u00achd\u2081 = hd\u2082\na\u271d : hd\u2081 :: tl\u2081 <+ tl\u2082\n\u22a2 (if False then isSublist tl\u2081 tl\u2082 else isSublist (hd\u2081 :: tl\u2081) tl\u2082) = true\n\ncase mpr.cons\u2082\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nhd\u2081 : \u03b1\ntl\u2081 tl\u2082 : List \u03b1\nh_eq : \u00achd\u2081 = hd\u2081\na\u271d : tl\u2081 <+ tl\u2082\n\u22a2 (if False then isSublist tl\u2081 tl\u2082 else isSublist (hd\u2081 :: tl\u2081) tl\u2082) = true"}, {"tactic": "case cons h_sub =>\n  exact isSublist_iff_sublist.mpr h_sub", "annotated_tactic": ["case cons h_sub =>\n          exact isSublist_iff_sublist.mpr h_sub", []], "state_before": "case mpr.cons\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nhd\u2081 : \u03b1\ntl\u2081 : List \u03b1\nhd\u2082 : \u03b1\ntl\u2082 : List \u03b1\nh_eq : \u00achd\u2081 = hd\u2082\na\u271d : hd\u2081 :: tl\u2081 <+ tl\u2082\n\u22a2 (if False then isSublist tl\u2081 tl\u2082 else isSublist (hd\u2081 :: tl\u2081) tl\u2082) = true\n\ncase mpr.cons\u2082\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nhd\u2081 : \u03b1\ntl\u2081 tl\u2082 : List \u03b1\nh_eq : \u00achd\u2081 = hd\u2081\na\u271d : tl\u2081 <+ tl\u2082\n\u22a2 (if False then isSublist tl\u2081 tl\u2082 else isSublist (hd\u2081 :: tl\u2081) tl\u2082) = true", "state_after": "case mpr.cons\u2082\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nhd\u2081 : \u03b1\ntl\u2081 tl\u2082 : List \u03b1\nh_eq : \u00achd\u2081 = hd\u2081\na\u271d : tl\u2081 <+ tl\u2082\n\u22a2 (if False then isSublist tl\u2081 tl\u2082 else isSublist (hd\u2081 :: tl\u2081) tl\u2082) = true"}, {"tactic": "case cons\u2082 =>\n  contradiction", "annotated_tactic": ["case cons\u2082 =>\n          contradiction", []], "state_before": "case mpr.cons\u2082\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nhd\u2081 : \u03b1\ntl\u2081 tl\u2082 : List \u03b1\nh_eq : \u00achd\u2081 = hd\u2081\na\u271d : tl\u2081 <+ tl\u2082\n\u22a2 (if False then isSublist tl\u2081 tl\u2082 else isSublist (hd\u2081 :: tl\u2081) tl\u2082) = true", "state_after": "no goals"}, {"tactic": "exact isSublist_iff_sublist.mpr h_sub", "annotated_tactic": ["exact isSublist_iff_sublist.mpr h_sub", []], "state_before": "\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nhd\u2081 : \u03b1\ntl\u2081 : List \u03b1\nhd\u2082 : \u03b1\ntl\u2082 : List \u03b1\nh_eq : \u00achd\u2081 = hd\u2082\nh_sub : hd\u2081 :: tl\u2081 <+ tl\u2082\n\u22a2 (if False then isSublist tl\u2081 tl\u2082 else isSublist (hd\u2081 :: tl\u2081) tl\u2082) = true", "state_after": "no goals"}, {"tactic": "contradiction", "annotated_tactic": ["contradiction", []], "state_before": "\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nhd\u2081 : \u03b1\ntl\u2081 tl\u2082 : List \u03b1\nh_eq : \u00achd\u2081 = hd\u2081\na\u271d : tl\u2081 <+ tl\u2082\n\u22a2 (if False then isSublist tl\u2081 tl\u2082 else isSublist (hd\u2081 :: tl\u2081) tl\u2082) = true", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "full_name": "BoundedContinuousFunction.Lp_norm_le", "start": [1749, 1], "end": [1752, 17], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/FiniteMeasure.lean", "full_name": "MeasureTheory.FiniteMeasure.continuous_testAgainstNN_eval", "start": [460, 1], "end": [464, 82], "traced_tactics": [{"tactic": "show Continuous ((fun \u03c6 : WeakDual \u211d\u22650 (\u03a9 \u2192\u1d47 \u211d\u22650) => \u03c6 f) \u2218 toWeakDualBCNN)", "annotated_tactic": ["show <a>Continuous</a> ((fun \u03c6 : <a>WeakDual</a> \u211d\u22650 (\u03a9 \u2192\u1d47 \u211d\u22650) => \u03c6 f) \u2218 <a>toWeakDualBCNN</a>)", [{"full_name": "Continuous", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [1591, 11], "def_end_pos": [1591, 21]}, {"full_name": "WeakDual", "def_path": "Mathlib/Topology/Algebra/Module/WeakDual.lean", "def_pos": [213, 5], "def_end_pos": [213, 13]}, {"full_name": "MeasureTheory.FiniteMeasure.toWeakDualBCNN", "def_path": "Mathlib/MeasureTheory/Measure/FiniteMeasure.lean", "def_pos": [430, 5], "def_end_pos": [430, 19]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2075 : SMul R \u211d\u22650\ninst\u271d\u2074 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\nf : \u03a9 \u2192\u1d47 \u211d\u22650\n\u22a2 Continuous fun \u03bc => testAgainstNN \u03bc f", "state_after": "\u03a9 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2075 : SMul R \u211d\u22650\ninst\u271d\u2074 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\nf : \u03a9 \u2192\u1d47 \u211d\u22650\n\u22a2 Continuous ((fun \u03c6 => \u2191\u03c6 f) \u2218 toWeakDualBCNN)"}, {"tactic": "refine Continuous.comp ?_ (toWeakDualBCNN_continuous (\u03a9 := \u03a9))", "annotated_tactic": ["refine <a>Continuous.comp</a> ?_ (<a>toWeakDualBCNN_continuous</a> (\u03a9 := \u03a9))", [{"full_name": "Continuous.comp", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [1673, 9], "def_end_pos": [1673, 24]}, {"full_name": "MeasureTheory.FiniteMeasure.toWeakDualBCNN_continuous", "def_path": "Mathlib/MeasureTheory/Measure/FiniteMeasure.lean", "def_pos": [454, 9], "def_end_pos": [454, 34]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2075 : SMul R \u211d\u22650\ninst\u271d\u2074 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\nf : \u03a9 \u2192\u1d47 \u211d\u22650\n\u22a2 Continuous ((fun \u03c6 => \u2191\u03c6 f) \u2218 toWeakDualBCNN)", "state_after": "\u03a9 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2075 : SMul R \u211d\u22650\ninst\u271d\u2074 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\nf : \u03a9 \u2192\u1d47 \u211d\u22650\n\u22a2 Continuous fun \u03c6 => \u2191\u03c6 f"}, {"tactic": "exact @WeakBilin.eval_continuous _ _ _ _ _ _ ContinuousLinearMap.module _ _ _ _", "annotated_tactic": ["exact @<a>WeakBilin.eval_continuous</a> _ _ _ _ _ _ <a>ContinuousLinearMap.module</a> _ _ _ _", [{"full_name": "WeakBilin.eval_continuous", "def_path": "Mathlib/Topology/Algebra/Module/WeakDual.lean", "def_pos": [121, 9], "def_end_pos": [121, 24]}, {"full_name": "ContinuousLinearMap.module", "def_path": "Mathlib/Topology/Algebra/Module/Basic.lean", "def_pos": [1647, 10], "def_end_pos": [1647, 16]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2075 : SMul R \u211d\u22650\ninst\u271d\u2074 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\nf : \u03a9 \u2192\u1d47 \u211d\u22650\n\u22a2 Continuous fun \u03c6 => \u2191\u03c6 f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Kernel/Basic.lean", "full_name": "ProbabilityTheory.kernel.set_lintegral_deterministic'", "start": [393, 1], "end": [396, 62], "traced_tactics": [{"tactic": "rw [kernel.deterministic_apply, set_lintegral_dirac' hf hs]", "annotated_tactic": ["rw [<a>kernel.deterministic_apply</a>, <a>set_lintegral_dirac'</a> hf hs]", [{"full_name": "ProbabilityTheory.kernel.deterministic_apply", "def_path": "Mathlib/Probability/Kernel/Basic.lean", "def_pos": [365, 9], "def_end_pos": [365, 28]}, {"full_name": "MeasureTheory.set_lintegral_dirac'", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [1404, 9], "def_end_pos": [1404, 29]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\nf : \u03b2 \u2192 \u211d\u22650\u221e\ng : \u03b1 \u2192 \u03b2\na : \u03b1\nhg : Measurable g\nhf : Measurable f\ns : Set \u03b2\nhs : MeasurableSet s\ninst\u271d : Decidable (g a \u2208 s)\n\u22a2 \u222b\u207b (x : \u03b2) in s, f x \u2202\u2191(deterministic g hg) a = if g a \u2208 s then f (g a) else 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/NoncommProd.lean", "full_name": "MonoidHom.pi_ext", "start": [441, 1], "end": [446, 34], "traced_tactics": [{"tactic": "cases nonempty_fintype \u03b9", "annotated_tactic": ["cases <a>nonempty_fintype</a> \u03b9", [{"full_name": "nonempty_fintype", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [442, 9], "def_end_pos": [442, 25]}]], "state_before": "F : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u2074 : Monoid \u03b2\ninst\u271d\u00b3 : Monoid \u03b3\nM : \u03b9 \u2192 Type u_6\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Monoid (M i)\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : DecidableEq \u03b9\nf g : ((i : \u03b9) \u2192 M i) \u2192* \u03b3\nh : \u2200 (i : \u03b9) (x : M i), \u2191f (Pi.mulSingle i x) = \u2191g (Pi.mulSingle i x)\n\u22a2 f = g", "state_after": "case intro\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u2074 : Monoid \u03b2\ninst\u271d\u00b3 : Monoid \u03b3\nM : \u03b9 \u2192 Type u_6\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Monoid (M i)\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : DecidableEq \u03b9\nf g : ((i : \u03b9) \u2192 M i) \u2192* \u03b3\nh : \u2200 (i : \u03b9) (x : M i), \u2191f (Pi.mulSingle i x) = \u2191g (Pi.mulSingle i x)\nval\u271d : Fintype \u03b9\n\u22a2 f = g"}, {"tactic": "ext x", "annotated_tactic": ["ext x", []], "state_before": "case intro\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u2074 : Monoid \u03b2\ninst\u271d\u00b3 : Monoid \u03b3\nM : \u03b9 \u2192 Type u_6\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Monoid (M i)\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : DecidableEq \u03b9\nf g : ((i : \u03b9) \u2192 M i) \u2192* \u03b3\nh : \u2200 (i : \u03b9) (x : M i), \u2191f (Pi.mulSingle i x) = \u2191g (Pi.mulSingle i x)\nval\u271d : Fintype \u03b9\n\u22a2 f = g", "state_after": "case intro.h\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u2074 : Monoid \u03b2\ninst\u271d\u00b3 : Monoid \u03b3\nM : \u03b9 \u2192 Type u_6\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Monoid (M i)\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : DecidableEq \u03b9\nf g : ((i : \u03b9) \u2192 M i) \u2192* \u03b3\nh : \u2200 (i : \u03b9) (x : M i), \u2191f (Pi.mulSingle i x) = \u2191g (Pi.mulSingle i x)\nval\u271d : Fintype \u03b9\nx : (i : \u03b9) \u2192 M i\n\u22a2 \u2191f x = \u2191g x"}, {"tactic": "rw [\u2190 noncommProd_mul_single x, univ.noncommProd_map, univ.noncommProd_map]", "annotated_tactic": ["rw [\u2190 <a>noncommProd_mul_single</a> x, univ.noncommProd_map, univ.noncommProd_map]", [{"full_name": "Finset.noncommProd_mul_single", "def_path": "Mathlib/Data/Finset/NoncommProd.lean", "def_pos": [416, 9], "def_end_pos": [416, 31]}]], "state_before": "case intro.h\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u2074 : Monoid \u03b2\ninst\u271d\u00b3 : Monoid \u03b3\nM : \u03b9 \u2192 Type u_6\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Monoid (M i)\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : DecidableEq \u03b9\nf g : ((i : \u03b9) \u2192 M i) \u2192* \u03b3\nh : \u2200 (i : \u03b9) (x : M i), \u2191f (Pi.mulSingle i x) = \u2191g (Pi.mulSingle i x)\nval\u271d : Fintype \u03b9\nx : (i : \u03b9) \u2192 M i\n\u22a2 \u2191f x = \u2191g x", "state_after": "case intro.h\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u2074 : Monoid \u03b2\ninst\u271d\u00b3 : Monoid \u03b3\nM : \u03b9 \u2192 Type u_6\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Monoid (M i)\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : DecidableEq \u03b9\nf g : ((i : \u03b9) \u2192 M i) \u2192* \u03b3\nh : \u2200 (i : \u03b9) (x : M i), \u2191f (Pi.mulSingle i x) = \u2191g (Pi.mulSingle i x)\nval\u271d : Fintype \u03b9\nx : (i : \u03b9) \u2192 M i\n\u22a2 noncommProd univ (fun i => \u2191f (Pi.mulSingle i (x i)))\n      (_ :\n        \u2200 (x_1 : \u03b9),\n          x_1 \u2208 \u2191univ \u2192\n            \u2200 (y : \u03b9), y \u2208 \u2191univ \u2192 x_1 \u2260 y \u2192 Commute (\u2191f (Pi.mulSingle x_1 (x x_1))) (\u2191f (Pi.mulSingle y (x y)))) =\n    noncommProd univ (fun i => \u2191g (Pi.mulSingle i (x i)))\n      (_ :\n        \u2200 (x_1 : \u03b9),\n          x_1 \u2208 \u2191univ \u2192\n            \u2200 (y : \u03b9), y \u2208 \u2191univ \u2192 x_1 \u2260 y \u2192 Commute (\u2191g (Pi.mulSingle x_1 (x x_1))) (\u2191g (Pi.mulSingle y (x y))))"}, {"tactic": "congr 1 with i", "annotated_tactic": ["congr 1 with i", []], "state_before": "case intro.h\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u2074 : Monoid \u03b2\ninst\u271d\u00b3 : Monoid \u03b3\nM : \u03b9 \u2192 Type u_6\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Monoid (M i)\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : DecidableEq \u03b9\nf g : ((i : \u03b9) \u2192 M i) \u2192* \u03b3\nh : \u2200 (i : \u03b9) (x : M i), \u2191f (Pi.mulSingle i x) = \u2191g (Pi.mulSingle i x)\nval\u271d : Fintype \u03b9\nx : (i : \u03b9) \u2192 M i\n\u22a2 noncommProd univ (fun i => \u2191f (Pi.mulSingle i (x i)))\n      (_ :\n        \u2200 (x_1 : \u03b9),\n          x_1 \u2208 \u2191univ \u2192\n            \u2200 (y : \u03b9), y \u2208 \u2191univ \u2192 x_1 \u2260 y \u2192 Commute (\u2191f (Pi.mulSingle x_1 (x x_1))) (\u2191f (Pi.mulSingle y (x y)))) =\n    noncommProd univ (fun i => \u2191g (Pi.mulSingle i (x i)))\n      (_ :\n        \u2200 (x_1 : \u03b9),\n          x_1 \u2208 \u2191univ \u2192\n            \u2200 (y : \u03b9), y \u2208 \u2191univ \u2192 x_1 \u2260 y \u2192 Commute (\u2191g (Pi.mulSingle x_1 (x x_1))) (\u2191g (Pi.mulSingle y (x y))))", "state_after": "case intro.h.e_f.h\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u2074 : Monoid \u03b2\ninst\u271d\u00b3 : Monoid \u03b3\nM : \u03b9 \u2192 Type u_6\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Monoid (M i)\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : DecidableEq \u03b9\nf g : ((i : \u03b9) \u2192 M i) \u2192* \u03b3\nh : \u2200 (i : \u03b9) (x : M i), \u2191f (Pi.mulSingle i x) = \u2191g (Pi.mulSingle i x)\nval\u271d : Fintype \u03b9\nx : (i : \u03b9) \u2192 M i\ni : \u03b9\n\u22a2 \u2191f (Pi.mulSingle i (x i)) = \u2191g (Pi.mulSingle i (x i))"}, {"tactic": "exact h i (x i)", "annotated_tactic": ["exact h i (x i)", []], "state_before": "case intro.h.e_f.h\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u2074 : Monoid \u03b2\ninst\u271d\u00b3 : Monoid \u03b3\nM : \u03b9 \u2192 Type u_6\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Monoid (M i)\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : DecidableEq \u03b9\nf g : ((i : \u03b9) \u2192 M i) \u2192* \u03b3\nh : \u2200 (i : \u03b9) (x : M i), \u2191f (Pi.mulSingle i x) = \u2191g (Pi.mulSingle i x)\nval\u271d : Fintype \u03b9\nx : (i : \u03b9) \u2192 M i\ni : \u03b9\n\u22a2 \u2191f (Pi.mulSingle i (x i)) = \u2191g (Pi.mulSingle i (x i))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "full_name": "MeasureTheory.setToFun_finset_sum", "start": [1386, 1], "end": [1389, 52], "traced_tactics": [{"tactic": "convert setToFun_finset_sum' hT s hf with a", "annotated_tactic": ["convert <a>setToFun_finset_sum'</a> hT s hf with a", [{"full_name": "MeasureTheory.setToFun_finset_sum'", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [1371, 9], "def_end_pos": [1371, 29]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\n\u03b9 : Type u_7\ns : Finset \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 E\nhf : \u2200 (i : \u03b9), i \u2208 s \u2192 Integrable (f i)\n\u22a2 (setToFun \u03bc T hT fun a => \u2211 i in s, f i a) = \u2211 i in s, setToFun \u03bc T hT (f i)", "state_after": "case h.e'_2.h.e'_14.h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\n\u03b9 : Type u_7\ns : Finset \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 E\nhf : \u2200 (i : \u03b9), i \u2208 s \u2192 Integrable (f i)\na : \u03b1\n\u22a2 \u2211 i in s, f i a = Finset.sum s (fun i => f i) a"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case h.e'_2.h.e'_14.h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\n\u03b9 : Type u_7\ns : Finset \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 E\nhf : \u2200 (i : \u03b9), i \u2208 s \u2192 Integrable (f i)\na : \u03b1\n\u22a2 \u2211 i in s, f i a = Finset.sum s (fun i => f i) a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Rat/Lemmas.lean", "full_name": "Rat.divInt_zero", "start": [131, 9], "end": [131, 61], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "full_name": "MeasureTheory.Measure.isFiniteMeasure_map", "start": [2947, 1], "end": [2954, 44], "traced_tactics": [{"tactic": "by_cases hf : AEMeasurable f \u03bc", "annotated_tactic": ["by_cases hf : <a>AEMeasurable</a> f \u03bc", [{"full_name": "AEMeasurable", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [708, 5], "def_end_pos": [708, 17]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 \u03b2\n\u22a2 IsFiniteMeasure (map f \u03bc)", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 \u03b2\nhf : AEMeasurable f\n\u22a2 IsFiniteMeasure (map f \u03bc)\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 \u03b2\nhf : \u00acAEMeasurable f\n\u22a2 IsFiniteMeasure (map f \u03bc)"}, {"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 \u03b2\nhf : AEMeasurable f\n\u22a2 IsFiniteMeasure (map f \u03bc)", "state_after": "case pos.measure_univ_lt_top\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 \u03b2\nhf : AEMeasurable f\n\u22a2 \u2191\u2191(map f \u03bc) univ < \u22a4"}, {"tactic": "rw [map_apply_of_aemeasurable hf MeasurableSet.univ]", "annotated_tactic": ["rw [<a>map_apply_of_aemeasurable</a> hf <a>MeasurableSet.univ</a>]", [{"full_name": "MeasureTheory.Measure.map_apply_of_aemeasurable", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1228, 9], "def_end_pos": [1228, 34]}, {"full_name": "MeasurableSet.univ", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [101, 19], "def_end_pos": [101, 37]}]], "state_before": "case pos.measure_univ_lt_top\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 \u03b2\nhf : AEMeasurable f\n\u22a2 \u2191\u2191(map f \u03bc) univ < \u22a4", "state_after": "case pos.measure_univ_lt_top\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 \u03b2\nhf : AEMeasurable f\n\u22a2 \u2191\u2191\u03bc (f \u207b\u00b9' univ) < \u22a4"}, {"tactic": "exact measure_lt_top \u03bc _", "annotated_tactic": ["exact <a>measure_lt_top</a> \u03bc _", [{"full_name": "MeasureTheory.measure_lt_top", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2866, 9], "def_end_pos": [2866, 23]}]], "state_before": "case pos.measure_univ_lt_top\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 \u03b2\nhf : AEMeasurable f\n\u22a2 \u2191\u2191\u03bc (f \u207b\u00b9' univ) < \u22a4", "state_after": "no goals"}, {"tactic": "rw [map_of_not_aemeasurable hf]", "annotated_tactic": ["rw [<a>map_of_not_aemeasurable</a> hf]", [{"full_name": "MeasureTheory.Measure.map_of_not_aemeasurable", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1187, 9], "def_end_pos": [1187, 32]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 \u03b2\nhf : \u00acAEMeasurable f\n\u22a2 IsFiniteMeasure (map f \u03bc)", "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 \u03b2\nhf : \u00acAEMeasurable f\n\u22a2 IsFiniteMeasure 0"}, {"tactic": "exact MeasureTheory.isFiniteMeasureZero", "annotated_tactic": ["exact <a>MeasureTheory.isFiniteMeasureZero</a>", [{"full_name": "MeasureTheory.isFiniteMeasureZero", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2907, 10], "def_end_pos": [2907, 29]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 \u03b2\nhf : \u00acAEMeasurable f\n\u22a2 IsFiniteMeasure 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Basic.lean", "full_name": "Set.compl_subset_compl", "start": [1756, 1], "end": [1757, 39], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/BinomialHeap/Basic.lean", "full_name": "Std.BinomialHeap.Imp.Heap.WF.deleteMin", "start": [450, 1], "end": [460, 59], "traced_tactics": [{"tactic": "cases s with cases eq | cons r a c s => ?_", "annotated_tactic": ["cases s with cases eq | <a>cons</a> r a c s => ?_", [{"full_name": "Std.BinomialHeap.Imp.Heap.cons", "def_path": "lake-packages/std/Std/Data/BinomialHeap/Basic.lean", "def_pos": [77, 5], "def_end_pos": [77, 9]}]], "state_before": "\u03b1 : Type u_1\nle : \u03b1 \u2192 \u03b1 \u2192 Bool\nn : Nat\na : \u03b1\ns' s : Heap \u03b1\nh : WF le n s\neq : Heap.deleteMin le s = some (a, s')\n\u22a2 WF le 0 s'", "state_after": "case cons.refl\n\u03b1 : Type u_1\nle : \u03b1 \u2192 \u03b1 \u2192 Bool\nn r : Nat\na : \u03b1\nc : HeapNode \u03b1\ns : Heap \u03b1\nh : WF le n (cons r a c s)\n\u22a2 WF le 0\n    (Heap.merge le\n      (HeapNode.toHeap (Heap.findMin le (cons r a c) s { before := id, val := a, node := c, next := s }).node)\n      (FindMin.before (Heap.findMin le (cons r a c) s { before := id, val := a, node := c, next := s })\n        (Heap.findMin le (cons r a c) s { before := id, val := a, node := c, next := s }).next))"}, {"tactic": "have : (s.findMin le (cons r a c) \u27e8id, a, c, s\u27e9).WF le :=\n  let \u27e8_, h\u2082, h\u2083\u27e9 := h\n  h\u2083.findMin \u27e8_, fun h => h.of_le (Nat.zero_le _), h\u2082, h\u2083\u27e9\n    fun h => \u27e8Nat.zero_le _, h\u2082, h\u27e9", "annotated_tactic": ["have : (s.findMin le (<a>cons</a> r a c) \u27e8<a>id</a>, a, c, s\u27e9).<a>WF</a> le :=\n    let \u27e8_, h\u2082, h\u2083\u27e9 := h\n    h\u2083.findMin \u27e8_, fun h => h.of_le (<a>Nat.zero_le</a> _), h\u2082, h\u2083\u27e9\n      fun h => \u27e8<a>Nat.zero_le</a> _, h\u2082, h\u27e9", [{"full_name": "Std.BinomialHeap.Imp.Heap.cons", "def_path": "lake-packages/std/Std/Data/BinomialHeap/Basic.lean", "def_pos": [77, 5], "def_end_pos": [77, 9]}, {"full_name": "id", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [33, 15], "def_end_pos": [33, 17]}, {"full_name": "Std.BinomialHeap.Imp.FindMin.WF", "def_path": "lake-packages/std/Std/Data/BinomialHeap/Basic.lean", "def_pos": [427, 11], "def_end_pos": [427, 21]}, {"full_name": "Nat.zero_le", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1578, 9], "def_end_pos": [1578, 20]}, {"full_name": "Nat.zero_le", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1578, 9], "def_end_pos": [1578, 20]}]], "state_before": "case cons.refl\n\u03b1 : Type u_1\nle : \u03b1 \u2192 \u03b1 \u2192 Bool\nn r : Nat\na : \u03b1\nc : HeapNode \u03b1\ns : Heap \u03b1\nh : WF le n (cons r a c s)\n\u22a2 WF le 0\n    (Heap.merge le\n      (HeapNode.toHeap (Heap.findMin le (cons r a c) s { before := id, val := a, node := c, next := s }).node)\n      (FindMin.before (Heap.findMin le (cons r a c) s { before := id, val := a, node := c, next := s })\n        (Heap.findMin le (cons r a c) s { before := id, val := a, node := c, next := s }).next))", "state_after": "case cons.refl\n\u03b1 : Type u_1\nle : \u03b1 \u2192 \u03b1 \u2192 Bool\nn r : Nat\na : \u03b1\nc : HeapNode \u03b1\ns : Heap \u03b1\nh : WF le n (cons r a c s)\nthis : FindMin.WF le (Heap.findMin le (cons r a c) s { before := id, val := a, node := c, next := s })\n\u22a2 WF le 0\n    (Heap.merge le\n      (HeapNode.toHeap (Heap.findMin le (cons r a c) s { before := id, val := a, node := c, next := s }).node)\n      (FindMin.before (Heap.findMin le (cons r a c) s { before := id, val := a, node := c, next := s })\n        (Heap.findMin le (cons r a c) s { before := id, val := a, node := c, next := s }).next))"}, {"tactic": "revert this", "annotated_tactic": ["revert this", []], "state_before": "case cons.refl\n\u03b1 : Type u_1\nle : \u03b1 \u2192 \u03b1 \u2192 Bool\nn r : Nat\na : \u03b1\nc : HeapNode \u03b1\ns : Heap \u03b1\nh : WF le n (cons r a c s)\nthis : FindMin.WF le (Heap.findMin le (cons r a c) s { before := id, val := a, node := c, next := s })\n\u22a2 WF le 0\n    (Heap.merge le\n      (HeapNode.toHeap (Heap.findMin le (cons r a c) s { before := id, val := a, node := c, next := s }).node)\n      (FindMin.before (Heap.findMin le (cons r a c) s { before := id, val := a, node := c, next := s })\n        (Heap.findMin le (cons r a c) s { before := id, val := a, node := c, next := s }).next))", "state_after": "case cons.refl\n\u03b1 : Type u_1\nle : \u03b1 \u2192 \u03b1 \u2192 Bool\nn r : Nat\na : \u03b1\nc : HeapNode \u03b1\ns : Heap \u03b1\nh : WF le n (cons r a c s)\n\u22a2 FindMin.WF le (Heap.findMin le (cons r a c) s { before := id, val := a, node := c, next := s }) \u2192\n    WF le 0\n      (Heap.merge le\n        (HeapNode.toHeap (Heap.findMin le (cons r a c) s { before := id, val := a, node := c, next := s }).node)\n        (FindMin.before (Heap.findMin le (cons r a c) s { before := id, val := a, node := c, next := s })\n          (Heap.findMin le (cons r a c) s { before := id, val := a, node := c, next := s }).next))"}, {"tactic": "let { before, val, node, next } := s.findMin le (cons r a c) \u27e8id, a, c, s\u27e9", "annotated_tactic": ["let { before, val, node, next } := s.findMin le (<a>cons</a> r a c) \u27e8<a>id</a>, a, c, s\u27e9", [{"full_name": "Std.BinomialHeap.Imp.Heap.cons", "def_path": "lake-packages/std/Std/Data/BinomialHeap/Basic.lean", "def_pos": [77, 5], "def_end_pos": [77, 9]}, {"full_name": "id", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [33, 15], "def_end_pos": [33, 17]}]], "state_before": "case cons.refl\n\u03b1 : Type u_1\nle : \u03b1 \u2192 \u03b1 \u2192 Bool\nn r : Nat\na : \u03b1\nc : HeapNode \u03b1\ns : Heap \u03b1\nh : WF le n (cons r a c s)\n\u22a2 FindMin.WF le (Heap.findMin le (cons r a c) s { before := id, val := a, node := c, next := s }) \u2192\n    WF le 0\n      (Heap.merge le\n        (HeapNode.toHeap (Heap.findMin le (cons r a c) s { before := id, val := a, node := c, next := s }).node)\n        (FindMin.before (Heap.findMin le (cons r a c) s { before := id, val := a, node := c, next := s })\n          (Heap.findMin le (cons r a c) s { before := id, val := a, node := c, next := s }).next))", "state_after": "case cons.refl\n\u03b1 : Type u_1\nle : \u03b1 \u2192 \u03b1 \u2192 Bool\nn r : Nat\na : \u03b1\nc : HeapNode \u03b1\ns : Heap \u03b1\nh : WF le n (cons r a c s)\nbefore : Heap \u03b1 \u2192 Heap \u03b1\nval : \u03b1\nnode : HeapNode \u03b1\nnext : Heap \u03b1\n\u22a2 FindMin.WF le { before := before, val := val, node := node, next := next } \u2192\n    WF le 0\n      (Heap.merge le (HeapNode.toHeap { before := before, val := val, node := node, next := next }.node)\n        (FindMin.before { before := before, val := val, node := node, next := next }\n          { before := before, val := val, node := node, next := next }.next))"}, {"tactic": "intro \u27e8_, hk, ih\u2081, ih\u2082\u27e9", "annotated_tactic": ["intro \u27e8_, hk, ih\u2081, ih\u2082\u27e9", []], "state_before": "case cons.refl\n\u03b1 : Type u_1\nle : \u03b1 \u2192 \u03b1 \u2192 Bool\nn r : Nat\na : \u03b1\nc : HeapNode \u03b1\ns : Heap \u03b1\nh : WF le n (cons r a c s)\nbefore : Heap \u03b1 \u2192 Heap \u03b1\nval : \u03b1\nnode : HeapNode \u03b1\nnext : Heap \u03b1\n\u22a2 FindMin.WF le { before := before, val := val, node := node, next := next } \u2192\n    WF le 0\n      (Heap.merge le (HeapNode.toHeap { before := before, val := val, node := node, next := next }.node)\n        (FindMin.before { before := before, val := val, node := node, next := next }\n          { before := before, val := val, node := node, next := next }.next))", "state_after": "case cons.refl\n\u03b1 : Type u_1\nle : \u03b1 \u2192 \u03b1 \u2192 Bool\nn r : Nat\na : \u03b1\nc : HeapNode \u03b1\ns : Heap \u03b1\nh : WF le n (cons r a c s)\nbefore : Heap \u03b1 \u2192 Heap \u03b1\nval : \u03b1\nnode : HeapNode \u03b1\nnext : Heap \u03b1\nrank\u271d : Nat\nhk :\n  \u2200 {s : Heap \u03b1},\n    WF le rank\u271d s \u2192 WF le 0 (FindMin.before { before := before, val := val, node := node, next := next } s)\nih\u2081 :\n  HeapNode.WF le { before := before, val := val, node := node, next := next }.val\n    { before := before, val := val, node := node, next := next }.node rank\u271d\nih\u2082 : WF le (rank\u271d + 1) { before := before, val := val, node := node, next := next }.next\n\u22a2 WF le 0\n    (Heap.merge le (HeapNode.toHeap { before := before, val := val, node := node, next := next }.node)\n      (FindMin.before { before := before, val := val, node := node, next := next }\n        { before := before, val := val, node := node, next := next }.next))"}, {"tactic": "exact ih\u2081.toHeap.merge <| hk (ih\u2082.of_le (Nat.le_succ _))", "annotated_tactic": ["exact ih\u2081.toHeap.merge <| hk (ih\u2082.of_le (<a>Nat.le_succ</a> _))", [{"full_name": "Nat.le_succ", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1599, 9], "def_end_pos": [1599, 20]}]], "state_before": "case cons.refl\n\u03b1 : Type u_1\nle : \u03b1 \u2192 \u03b1 \u2192 Bool\nn r : Nat\na : \u03b1\nc : HeapNode \u03b1\ns : Heap \u03b1\nh : WF le n (cons r a c s)\nbefore : Heap \u03b1 \u2192 Heap \u03b1\nval : \u03b1\nnode : HeapNode \u03b1\nnext : Heap \u03b1\nrank\u271d : Nat\nhk :\n  \u2200 {s : Heap \u03b1},\n    WF le rank\u271d s \u2192 WF le 0 (FindMin.before { before := before, val := val, node := node, next := next } s)\nih\u2081 :\n  HeapNode.WF le { before := before, val := val, node := node, next := next }.val\n    { before := before, val := val, node := node, next := next }.node rank\u271d\nih\u2082 : WF le (rank\u271d + 1) { before := before, val := val, node := node, next := next }.next\n\u22a2 WF le 0\n    (Heap.merge le (HeapNode.toHeap { before := before, val := val, node := node, next := next }.node)\n      (FindMin.before { before := before, val := val, node := node, next := next }\n        { before := before, val := val, node := node, next := next }.next))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Kernel/WithDensity.lean", "full_name": "ProbabilityTheory.kernel.withDensity_tsum", "start": [114, 1], "end": [134, 40], "traced_tactics": [{"tactic": "have h_sum_a : \u2200 a, Summable fun n => f n a := fun a => Pi.summable.mpr fun b => ENNReal.summable", "annotated_tactic": ["have h_sum_a : \u2200 a, <a>Summable</a> fun n => f n a := fun a => Pi.summable.mpr fun b => <a>ENNReal.summable</a>", [{"full_name": "Summable", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/Basic.lean", "def_pos": [62, 5], "def_end_pos": [62, 13]}, {"full_name": "ENNReal.summable", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [777, 19], "def_end_pos": [777, 27]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\ninst\u271d\u00b9 : Countable \u03b9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsSFiniteKernel \u03ba\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\nhf : \u2200 (i : \u03b9), Measurable (Function.uncurry (f i))\n\u22a2 withDensity \u03ba (\u2211' (n : \u03b9), f n) = kernel.sum fun n => withDensity \u03ba (f n)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\ninst\u271d\u00b9 : Countable \u03b9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsSFiniteKernel \u03ba\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\nhf : \u2200 (i : \u03b9), Measurable (Function.uncurry (f i))\nh_sum_a : \u2200 (a : \u03b1), Summable fun n => f n a\n\u22a2 withDensity \u03ba (\u2211' (n : \u03b9), f n) = kernel.sum fun n => withDensity \u03ba (f n)"}, {"tactic": "have h_sum : Summable fun n => f n := Pi.summable.mpr h_sum_a", "annotated_tactic": ["have h_sum : <a>Summable</a> fun n => f n := Pi.summable.mpr h_sum_a", [{"full_name": "Summable", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/Basic.lean", "def_pos": [62, 5], "def_end_pos": [62, 13]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\ninst\u271d\u00b9 : Countable \u03b9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsSFiniteKernel \u03ba\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\nhf : \u2200 (i : \u03b9), Measurable (Function.uncurry (f i))\nh_sum_a : \u2200 (a : \u03b1), Summable fun n => f n a\n\u22a2 withDensity \u03ba (\u2211' (n : \u03b9), f n) = kernel.sum fun n => withDensity \u03ba (f n)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\ninst\u271d\u00b9 : Countable \u03b9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsSFiniteKernel \u03ba\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\nhf : \u2200 (i : \u03b9), Measurable (Function.uncurry (f i))\nh_sum_a : \u2200 (a : \u03b1), Summable fun n => f n a\nh_sum : Summable fun n => f n\n\u22a2 withDensity \u03ba (\u2211' (n : \u03b9), f n) = kernel.sum fun n => withDensity \u03ba (f n)"}, {"tactic": "ext a s hs", "annotated_tactic": ["ext a s hs", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\ninst\u271d\u00b9 : Countable \u03b9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsSFiniteKernel \u03ba\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\nhf : \u2200 (i : \u03b9), Measurable (Function.uncurry (f i))\nh_sum_a : \u2200 (a : \u03b1), Summable fun n => f n a\nh_sum : Summable fun n => f n\n\u22a2 withDensity \u03ba (\u2211' (n : \u03b9), f n) = kernel.sum fun n => withDensity \u03ba (f n)", "state_after": "case h.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\ninst\u271d\u00b9 : Countable \u03b9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsSFiniteKernel \u03ba\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\nhf : \u2200 (i : \u03b9), Measurable (Function.uncurry (f i))\nh_sum_a : \u2200 (a : \u03b1), Summable fun n => f n a\nh_sum : Summable fun n => f n\na : \u03b1\ns : Set \u03b2\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(\u2191(withDensity \u03ba (\u2211' (n : \u03b9), f n)) a) s = \u2191\u2191(\u2191(kernel.sum fun n => withDensity \u03ba (f n)) a) s"}, {"tactic": "rw [sum_apply' _ a hs, withDensity_apply' \u03ba _ a hs]", "annotated_tactic": ["rw [<a>sum_apply'</a> _ a hs, <a>withDensity_apply'</a> \u03ba _ a hs]", [{"full_name": "ProbabilityTheory.kernel.sum_apply'", "def_path": "Mathlib/Probability/Kernel/Basic.lean", "def_pos": [244, 9], "def_end_pos": [244, 19]}, {"full_name": "ProbabilityTheory.kernel.withDensity_apply'", "def_path": "Mathlib/Probability/Kernel/WithDensity.lean", "def_pos": [69, 9], "def_end_pos": [69, 27]}]], "state_before": "case h.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\ninst\u271d\u00b9 : Countable \u03b9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsSFiniteKernel \u03ba\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\nhf : \u2200 (i : \u03b9), Measurable (Function.uncurry (f i))\nh_sum_a : \u2200 (a : \u03b1), Summable fun n => f n a\nh_sum : Summable fun n => f n\na : \u03b1\ns : Set \u03b2\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(\u2191(withDensity \u03ba (\u2211' (n : \u03b9), f n)) a) s = \u2191\u2191(\u2191(kernel.sum fun n => withDensity \u03ba (f n)) a) s", "state_after": "case h.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\ninst\u271d\u00b9 : Countable \u03b9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsSFiniteKernel \u03ba\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\nhf : \u2200 (i : \u03b9), Measurable (Function.uncurry (f i))\nh_sum_a : \u2200 (a : \u03b1), Summable fun n => f n a\nh_sum : Summable fun n => f n\na : \u03b1\ns : Set \u03b2\nhs : MeasurableSet s\n\u22a2 \u222b\u207b (b : \u03b2) in s, tsum (fun n => f n) a b \u2202\u2191\u03ba a = \u2211' (n : \u03b9), \u2191\u2191(\u2191(withDensity \u03ba (f n)) a) s\n\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\ninst\u271d\u00b9 : Countable \u03b9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsSFiniteKernel \u03ba\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\nhf : \u2200 (i : \u03b9), Measurable (Function.uncurry (f i))\nh_sum_a : \u2200 (a : \u03b1), Summable fun n => f n a\nh_sum : Summable fun n => f n\na : \u03b1\ns : Set \u03b2\nhs : MeasurableSet s\n\u22a2 Measurable (Function.uncurry (\u2211' (n : \u03b9), f n))"}, {"tactic": "swap", "annotated_tactic": ["swap", []], "state_before": "case h.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\ninst\u271d\u00b9 : Countable \u03b9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsSFiniteKernel \u03ba\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\nhf : \u2200 (i : \u03b9), Measurable (Function.uncurry (f i))\nh_sum_a : \u2200 (a : \u03b1), Summable fun n => f n a\nh_sum : Summable fun n => f n\na : \u03b1\ns : Set \u03b2\nhs : MeasurableSet s\n\u22a2 \u222b\u207b (b : \u03b2) in s, tsum (fun n => f n) a b \u2202\u2191\u03ba a = \u2211' (n : \u03b9), \u2191\u2191(\u2191(withDensity \u03ba (f n)) a) s\n\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\ninst\u271d\u00b9 : Countable \u03b9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsSFiniteKernel \u03ba\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\nhf : \u2200 (i : \u03b9), Measurable (Function.uncurry (f i))\nh_sum_a : \u2200 (a : \u03b1), Summable fun n => f n a\nh_sum : Summable fun n => f n\na : \u03b1\ns : Set \u03b2\nhs : MeasurableSet s\n\u22a2 Measurable (Function.uncurry (\u2211' (n : \u03b9), f n))", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\ninst\u271d\u00b9 : Countable \u03b9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsSFiniteKernel \u03ba\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\nhf : \u2200 (i : \u03b9), Measurable (Function.uncurry (f i))\nh_sum_a : \u2200 (a : \u03b1), Summable fun n => f n a\nh_sum : Summable fun n => f n\na : \u03b1\ns : Set \u03b2\nhs : MeasurableSet s\n\u22a2 Measurable (Function.uncurry (\u2211' (n : \u03b9), f n))\n\ncase h.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\ninst\u271d\u00b9 : Countable \u03b9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsSFiniteKernel \u03ba\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\nhf : \u2200 (i : \u03b9), Measurable (Function.uncurry (f i))\nh_sum_a : \u2200 (a : \u03b1), Summable fun n => f n a\nh_sum : Summable fun n => f n\na : \u03b1\ns : Set \u03b2\nhs : MeasurableSet s\n\u22a2 \u222b\u207b (b : \u03b2) in s, tsum (fun n => f n) a b \u2202\u2191\u03ba a = \u2211' (n : \u03b9), \u2191\u2191(\u2191(withDensity \u03ba (f n)) a) s"}, {"tactic": "have : \u222b\u207b b in s, (\u2211' n, f n) a b \u2202\u03ba a = \u222b\u207b b in s, \u2211' n, (fun b => f n a b) b \u2202\u03ba a := by\n  congr with b\n  rw [tsum_apply h_sum, tsum_apply (h_sum_a a)]", "annotated_tactic": ["have : \u222b\u207b b in s, (\u2211' n, f n) a b \u2202\u03ba a = \u222b\u207b b in s, \u2211' n, (fun b => f n a b) b \u2202\u03ba a := by\n    congr with b\n    rw [<a>tsum_apply</a> h_sum, <a>tsum_apply</a> (h_sum_a a)]", [{"full_name": "tsum_apply", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/Basic.lean", "def_pos": [1353, 9], "def_end_pos": [1353, 19]}, {"full_name": "tsum_apply", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/Basic.lean", "def_pos": [1353, 9], "def_end_pos": [1353, 19]}]], "state_before": "case h.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\ninst\u271d\u00b9 : Countable \u03b9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsSFiniteKernel \u03ba\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\nhf : \u2200 (i : \u03b9), Measurable (Function.uncurry (f i))\nh_sum_a : \u2200 (a : \u03b1), Summable fun n => f n a\nh_sum : Summable fun n => f n\na : \u03b1\ns : Set \u03b2\nhs : MeasurableSet s\n\u22a2 \u222b\u207b (b : \u03b2) in s, tsum (fun n => f n) a b \u2202\u2191\u03ba a = \u2211' (n : \u03b9), \u2191\u2191(\u2191(withDensity \u03ba (f n)) a) s", "state_after": "case h.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\ninst\u271d\u00b9 : Countable \u03b9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsSFiniteKernel \u03ba\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\nhf : \u2200 (i : \u03b9), Measurable (Function.uncurry (f i))\nh_sum_a : \u2200 (a : \u03b1), Summable fun n => f n a\nh_sum : Summable fun n => f n\na : \u03b1\ns : Set \u03b2\nhs : MeasurableSet s\nthis : \u222b\u207b (b : \u03b2) in s, tsum (fun n => f n) a b \u2202\u2191\u03ba a = \u222b\u207b (b : \u03b2) in s, \u2211' (n : \u03b9), (fun b => f n a b) b \u2202\u2191\u03ba a\n\u22a2 \u222b\u207b (b : \u03b2) in s, tsum (fun n => f n) a b \u2202\u2191\u03ba a = \u2211' (n : \u03b9), \u2191\u2191(\u2191(withDensity \u03ba (f n)) a) s"}, {"tactic": "rw [this, lintegral_tsum fun n => (Measurable.of_uncurry_left (hf n)).aemeasurable]", "annotated_tactic": ["rw [this, <a>lintegral_tsum</a> fun n => (<a>Measurable.of_uncurry_left</a> (hf n)).<a>aemeasurable</a>]", [{"full_name": "MeasureTheory.lintegral_tsum", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [1184, 9], "def_end_pos": [1184, 23]}, {"full_name": "Measurable.of_uncurry_left", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [744, 9], "def_end_pos": [744, 35]}, {"full_name": "Measurable.aemeasurable", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [713, 9], "def_end_pos": [713, 32]}]], "state_before": "case h.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\ninst\u271d\u00b9 : Countable \u03b9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsSFiniteKernel \u03ba\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\nhf : \u2200 (i : \u03b9), Measurable (Function.uncurry (f i))\nh_sum_a : \u2200 (a : \u03b1), Summable fun n => f n a\nh_sum : Summable fun n => f n\na : \u03b1\ns : Set \u03b2\nhs : MeasurableSet s\nthis : \u222b\u207b (b : \u03b2) in s, tsum (fun n => f n) a b \u2202\u2191\u03ba a = \u222b\u207b (b : \u03b2) in s, \u2211' (n : \u03b9), (fun b => f n a b) b \u2202\u2191\u03ba a\n\u22a2 \u222b\u207b (b : \u03b2) in s, tsum (fun n => f n) a b \u2202\u2191\u03ba a = \u2211' (n : \u03b9), \u2191\u2191(\u2191(withDensity \u03ba (f n)) a) s", "state_after": "case h.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\ninst\u271d\u00b9 : Countable \u03b9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsSFiniteKernel \u03ba\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\nhf : \u2200 (i : \u03b9), Measurable (Function.uncurry (f i))\nh_sum_a : \u2200 (a : \u03b1), Summable fun n => f n a\nh_sum : Summable fun n => f n\na : \u03b1\ns : Set \u03b2\nhs : MeasurableSet s\nthis : \u222b\u207b (b : \u03b2) in s, tsum (fun n => f n) a b \u2202\u2191\u03ba a = \u222b\u207b (b : \u03b2) in s, \u2211' (n : \u03b9), (fun b => f n a b) b \u2202\u2191\u03ba a\n\u22a2 \u2211' (i : \u03b9), \u222b\u207b (a_1 : \u03b2) in s, f i a a_1 \u2202\u2191\u03ba a = \u2211' (n : \u03b9), \u2191\u2191(\u2191(withDensity \u03ba (f n)) a) s"}, {"tactic": "congr with n", "annotated_tactic": ["congr with n", []], "state_before": "case h.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\ninst\u271d\u00b9 : Countable \u03b9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsSFiniteKernel \u03ba\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\nhf : \u2200 (i : \u03b9), Measurable (Function.uncurry (f i))\nh_sum_a : \u2200 (a : \u03b1), Summable fun n => f n a\nh_sum : Summable fun n => f n\na : \u03b1\ns : Set \u03b2\nhs : MeasurableSet s\nthis : \u222b\u207b (b : \u03b2) in s, tsum (fun n => f n) a b \u2202\u2191\u03ba a = \u222b\u207b (b : \u03b2) in s, \u2211' (n : \u03b9), (fun b => f n a b) b \u2202\u2191\u03ba a\n\u22a2 \u2211' (i : \u03b9), \u222b\u207b (a_1 : \u03b2) in s, f i a a_1 \u2202\u2191\u03ba a = \u2211' (n : \u03b9), \u2191\u2191(\u2191(withDensity \u03ba (f n)) a) s", "state_after": "case h.h.e_f.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\ninst\u271d\u00b9 : Countable \u03b9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsSFiniteKernel \u03ba\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\nhf : \u2200 (i : \u03b9), Measurable (Function.uncurry (f i))\nh_sum_a : \u2200 (a : \u03b1), Summable fun n => f n a\nh_sum : Summable fun n => f n\na : \u03b1\ns : Set \u03b2\nhs : MeasurableSet s\nthis : \u222b\u207b (b : \u03b2) in s, tsum (fun n => f n) a b \u2202\u2191\u03ba a = \u222b\u207b (b : \u03b2) in s, \u2211' (n : \u03b9), (fun b => f n a b) b \u2202\u2191\u03ba a\nn : \u03b9\n\u22a2 \u222b\u207b (a_1 : \u03b2) in s, f n a a_1 \u2202\u2191\u03ba a = \u2191\u2191(\u2191(withDensity \u03ba (f n)) a) s"}, {"tactic": "rw [withDensity_apply' _ (hf n) a hs]", "annotated_tactic": ["rw [<a>withDensity_apply'</a> _ (hf n) a hs]", [{"full_name": "ProbabilityTheory.kernel.withDensity_apply'", "def_path": "Mathlib/Probability/Kernel/WithDensity.lean", "def_pos": [69, 9], "def_end_pos": [69, 27]}]], "state_before": "case h.h.e_f.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\ninst\u271d\u00b9 : Countable \u03b9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsSFiniteKernel \u03ba\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\nhf : \u2200 (i : \u03b9), Measurable (Function.uncurry (f i))\nh_sum_a : \u2200 (a : \u03b1), Summable fun n => f n a\nh_sum : Summable fun n => f n\na : \u03b1\ns : Set \u03b2\nhs : MeasurableSet s\nthis : \u222b\u207b (b : \u03b2) in s, tsum (fun n => f n) a b \u2202\u2191\u03ba a = \u222b\u207b (b : \u03b2) in s, \u2211' (n : \u03b9), (fun b => f n a b) b \u2202\u2191\u03ba a\nn : \u03b9\n\u22a2 \u222b\u207b (a_1 : \u03b2) in s, f n a a_1 \u2202\u2191\u03ba a = \u2191\u2191(\u2191(withDensity \u03ba (f n)) a) s", "state_after": "no goals"}, {"tactic": "have : Function.uncurry (\u2211' n, f n) = \u2211' n, Function.uncurry (f n) := by\n  ext1 p\n  simp only [Function.uncurry_def]\n  rw [tsum_apply h_sum, tsum_apply (h_sum_a _), tsum_apply]\n  exact Pi.summable.mpr fun p => ENNReal.summable", "annotated_tactic": ["have : <a>Function.uncurry</a> (\u2211' n, f n) = \u2211' n, <a>Function.uncurry</a> (f n) := by\n      ext1 p\n      simp only [<a>Function.uncurry_def</a>]\n      rw [<a>tsum_apply</a> h_sum, <a>tsum_apply</a> (h_sum_a _), <a>tsum_apply</a>]\n      exact Pi.summable.mpr fun p => <a>ENNReal.summable</a>", [{"full_name": "Function.uncurry", "def_path": "Mathlib/Init/Function.lean", "def_pos": [217, 5], "def_end_pos": [217, 12]}, {"full_name": "Function.uncurry", "def_path": "Mathlib/Init/Function.lean", "def_pos": [217, 5], "def_end_pos": [217, 12]}, {"full_name": "Function.uncurry_def", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [812, 9], "def_end_pos": [812, 20]}, {"full_name": "tsum_apply", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/Basic.lean", "def_pos": [1353, 9], "def_end_pos": [1353, 19]}, {"full_name": "tsum_apply", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/Basic.lean", "def_pos": [1353, 9], "def_end_pos": [1353, 19]}, {"full_name": "tsum_apply", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/Basic.lean", "def_pos": [1353, 9], "def_end_pos": [1353, 19]}, {"full_name": "ENNReal.summable", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [777, 19], "def_end_pos": [777, 27]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\ninst\u271d\u00b9 : Countable \u03b9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsSFiniteKernel \u03ba\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\nhf : \u2200 (i : \u03b9), Measurable (Function.uncurry (f i))\nh_sum_a : \u2200 (a : \u03b1), Summable fun n => f n a\nh_sum : Summable fun n => f n\na : \u03b1\ns : Set \u03b2\nhs : MeasurableSet s\n\u22a2 Measurable (Function.uncurry (\u2211' (n : \u03b9), f n))", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\ninst\u271d\u00b9 : Countable \u03b9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsSFiniteKernel \u03ba\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\nhf : \u2200 (i : \u03b9), Measurable (Function.uncurry (f i))\nh_sum_a : \u2200 (a : \u03b1), Summable fun n => f n a\nh_sum : Summable fun n => f n\na : \u03b1\ns : Set \u03b2\nhs : MeasurableSet s\nthis : Function.uncurry (\u2211' (n : \u03b9), f n) = \u2211' (n : \u03b9), Function.uncurry (f n)\n\u22a2 Measurable (Function.uncurry (\u2211' (n : \u03b9), f n))"}, {"tactic": "rw [this]", "annotated_tactic": ["rw [this]", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\ninst\u271d\u00b9 : Countable \u03b9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsSFiniteKernel \u03ba\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\nhf : \u2200 (i : \u03b9), Measurable (Function.uncurry (f i))\nh_sum_a : \u2200 (a : \u03b1), Summable fun n => f n a\nh_sum : Summable fun n => f n\na : \u03b1\ns : Set \u03b2\nhs : MeasurableSet s\nthis : Function.uncurry (\u2211' (n : \u03b9), f n) = \u2211' (n : \u03b9), Function.uncurry (f n)\n\u22a2 Measurable (Function.uncurry (\u2211' (n : \u03b9), f n))", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\ninst\u271d\u00b9 : Countable \u03b9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsSFiniteKernel \u03ba\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\nhf : \u2200 (i : \u03b9), Measurable (Function.uncurry (f i))\nh_sum_a : \u2200 (a : \u03b1), Summable fun n => f n a\nh_sum : Summable fun n => f n\na : \u03b1\ns : Set \u03b2\nhs : MeasurableSet s\nthis : Function.uncurry (\u2211' (n : \u03b9), f n) = \u2211' (n : \u03b9), Function.uncurry (f n)\n\u22a2 Measurable (\u2211' (n : \u03b9), Function.uncurry (f n))"}, {"tactic": "exact Measurable.ennreal_tsum' hf", "annotated_tactic": ["exact <a>Measurable.ennreal_tsum'</a> hf", [{"full_name": "Measurable.ennreal_tsum'", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [2144, 9], "def_end_pos": [2144, 33]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\ninst\u271d\u00b9 : Countable \u03b9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsSFiniteKernel \u03ba\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\nhf : \u2200 (i : \u03b9), Measurable (Function.uncurry (f i))\nh_sum_a : \u2200 (a : \u03b1), Summable fun n => f n a\nh_sum : Summable fun n => f n\na : \u03b1\ns : Set \u03b2\nhs : MeasurableSet s\nthis : Function.uncurry (\u2211' (n : \u03b9), f n) = \u2211' (n : \u03b9), Function.uncurry (f n)\n\u22a2 Measurable (\u2211' (n : \u03b9), Function.uncurry (f n))", "state_after": "no goals"}, {"tactic": "ext1 p", "annotated_tactic": ["ext1 p", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\ninst\u271d\u00b9 : Countable \u03b9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsSFiniteKernel \u03ba\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\nhf : \u2200 (i : \u03b9), Measurable (Function.uncurry (f i))\nh_sum_a : \u2200 (a : \u03b1), Summable fun n => f n a\nh_sum : Summable fun n => f n\na : \u03b1\ns : Set \u03b2\nhs : MeasurableSet s\n\u22a2 Function.uncurry (\u2211' (n : \u03b9), f n) = \u2211' (n : \u03b9), Function.uncurry (f n)", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\ninst\u271d\u00b9 : Countable \u03b9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsSFiniteKernel \u03ba\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\nhf : \u2200 (i : \u03b9), Measurable (Function.uncurry (f i))\nh_sum_a : \u2200 (a : \u03b1), Summable fun n => f n a\nh_sum : Summable fun n => f n\na : \u03b1\ns : Set \u03b2\nhs : MeasurableSet s\np : \u03b1 \u00d7 \u03b2\n\u22a2 Function.uncurry (\u2211' (n : \u03b9), f n) p = tsum (fun n => Function.uncurry (f n)) p"}, {"tactic": "simp only [Function.uncurry_def]", "annotated_tactic": ["simp only [<a>Function.uncurry_def</a>]", [{"full_name": "Function.uncurry_def", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [812, 9], "def_end_pos": [812, 20]}]], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\ninst\u271d\u00b9 : Countable \u03b9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsSFiniteKernel \u03ba\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\nhf : \u2200 (i : \u03b9), Measurable (Function.uncurry (f i))\nh_sum_a : \u2200 (a : \u03b1), Summable fun n => f n a\nh_sum : Summable fun n => f n\na : \u03b1\ns : Set \u03b2\nhs : MeasurableSet s\np : \u03b1 \u00d7 \u03b2\n\u22a2 Function.uncurry (\u2211' (n : \u03b9), f n) p = tsum (fun n => Function.uncurry (f n)) p", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\ninst\u271d\u00b9 : Countable \u03b9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsSFiniteKernel \u03ba\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\nhf : \u2200 (i : \u03b9), Measurable (Function.uncurry (f i))\nh_sum_a : \u2200 (a : \u03b1), Summable fun n => f n a\nh_sum : Summable fun n => f n\na : \u03b1\ns : Set \u03b2\nhs : MeasurableSet s\np : \u03b1 \u00d7 \u03b2\n\u22a2 tsum (fun n => f n) p.1 p.2 = tsum (fun n p => f n p.1 p.2) p"}, {"tactic": "rw [tsum_apply h_sum, tsum_apply (h_sum_a _), tsum_apply]", "annotated_tactic": ["rw [<a>tsum_apply</a> h_sum, <a>tsum_apply</a> (h_sum_a _), <a>tsum_apply</a>]", [{"full_name": "tsum_apply", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/Basic.lean", "def_pos": [1353, 9], "def_end_pos": [1353, 19]}, {"full_name": "tsum_apply", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/Basic.lean", "def_pos": [1353, 9], "def_end_pos": [1353, 19]}, {"full_name": "tsum_apply", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/Basic.lean", "def_pos": [1353, 9], "def_end_pos": [1353, 19]}]], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\ninst\u271d\u00b9 : Countable \u03b9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsSFiniteKernel \u03ba\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\nhf : \u2200 (i : \u03b9), Measurable (Function.uncurry (f i))\nh_sum_a : \u2200 (a : \u03b1), Summable fun n => f n a\nh_sum : Summable fun n => f n\na : \u03b1\ns : Set \u03b2\nhs : MeasurableSet s\np : \u03b1 \u00d7 \u03b2\n\u22a2 tsum (fun n => f n) p.1 p.2 = tsum (fun n p => f n p.1 p.2) p", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\ninst\u271d\u00b9 : Countable \u03b9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsSFiniteKernel \u03ba\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\nhf : \u2200 (i : \u03b9), Measurable (Function.uncurry (f i))\nh_sum_a : \u2200 (a : \u03b1), Summable fun n => f n a\nh_sum : Summable fun n => f n\na : \u03b1\ns : Set \u03b2\nhs : MeasurableSet s\np : \u03b1 \u00d7 \u03b2\n\u22a2 Summable fun n p => f n p.1 p.2"}, {"tactic": "exact Pi.summable.mpr fun p => ENNReal.summable", "annotated_tactic": ["exact Pi.summable.mpr fun p => <a>ENNReal.summable</a>", [{"full_name": "ENNReal.summable", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [777, 19], "def_end_pos": [777, 27]}]], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\ninst\u271d\u00b9 : Countable \u03b9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsSFiniteKernel \u03ba\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\nhf : \u2200 (i : \u03b9), Measurable (Function.uncurry (f i))\nh_sum_a : \u2200 (a : \u03b1), Summable fun n => f n a\nh_sum : Summable fun n => f n\na : \u03b1\ns : Set \u03b2\nhs : MeasurableSet s\np : \u03b1 \u00d7 \u03b2\n\u22a2 Summable fun n p => f n p.1 p.2", "state_after": "no goals"}, {"tactic": "congr with b", "annotated_tactic": ["congr with b", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\ninst\u271d\u00b9 : Countable \u03b9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsSFiniteKernel \u03ba\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\nhf : \u2200 (i : \u03b9), Measurable (Function.uncurry (f i))\nh_sum_a : \u2200 (a : \u03b1), Summable fun n => f n a\nh_sum : Summable fun n => f n\na : \u03b1\ns : Set \u03b2\nhs : MeasurableSet s\n\u22a2 \u222b\u207b (b : \u03b2) in s, tsum (fun n => f n) a b \u2202\u2191\u03ba a = \u222b\u207b (b : \u03b2) in s, \u2211' (n : \u03b9), (fun b => f n a b) b \u2202\u2191\u03ba a", "state_after": "case e_f.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\ninst\u271d\u00b9 : Countable \u03b9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsSFiniteKernel \u03ba\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\nhf : \u2200 (i : \u03b9), Measurable (Function.uncurry (f i))\nh_sum_a : \u2200 (a : \u03b1), Summable fun n => f n a\nh_sum : Summable fun n => f n\na : \u03b1\ns : Set \u03b2\nhs : MeasurableSet s\nb : \u03b2\n\u22a2 tsum (fun n => f n) a b = \u2211' (n : \u03b9), (fun b => f n a b) b"}, {"tactic": "rw [tsum_apply h_sum, tsum_apply (h_sum_a a)]", "annotated_tactic": ["rw [<a>tsum_apply</a> h_sum, <a>tsum_apply</a> (h_sum_a a)]", [{"full_name": "tsum_apply", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/Basic.lean", "def_pos": [1353, 9], "def_end_pos": [1353, 19]}, {"full_name": "tsum_apply", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/Basic.lean", "def_pos": [1353, 9], "def_end_pos": [1353, 19]}]], "state_before": "case e_f.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\ninst\u271d\u00b9 : Countable \u03b9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsSFiniteKernel \u03ba\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\nhf : \u2200 (i : \u03b9), Measurable (Function.uncurry (f i))\nh_sum_a : \u2200 (a : \u03b1), Summable fun n => f n a\nh_sum : Summable fun n => f n\na : \u03b1\ns : Set \u03b2\nhs : MeasurableSet s\nb : \u03b2\n\u22a2 tsum (fun n => f n) a b = \u2211' (n : \u03b9), (fun b => f n a b) b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/FiniteMeasure.lean", "full_name": "MeasureTheory.FiniteMeasure.restrict_mass", "start": [304, 1], "end": [305, 70], "traced_tactics": [{"tactic": "simp only [mass, restrict_apply \u03bc A MeasurableSet.univ, univ_inter]", "annotated_tactic": ["simp only [<a>mass</a>, <a>restrict_apply</a> \u03bc A <a>MeasurableSet.univ</a>, <a>univ_inter</a>]", [{"full_name": "MeasureTheory.FiniteMeasure.mass", "def_path": "Mathlib/MeasureTheory/Measure/FiniteMeasure.lean", "def_pos": [163, 5], "def_end_pos": [163, 9]}, {"full_name": "MeasureTheory.FiniteMeasure.restrict_apply", "def_path": "Mathlib/MeasureTheory/Measure/FiniteMeasure.lean", "def_pos": [298, 9], "def_end_pos": [298, 23]}, {"full_name": "MeasurableSet.univ", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [101, 19], "def_end_pos": [101, 37]}, {"full_name": "Set.univ_inter", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1017, 9], "def_end_pos": [1017, 19]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u2074 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u00b3 : SMul R \u211d\u22650\ninst\u271d\u00b2 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b9 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\n\u03bc : FiniteMeasure \u03a9\nA : Set \u03a9\n\u22a2 mass (restrict \u03bc A) = (fun s => ENNReal.toNNReal (\u2191\u2191\u2191\u03bc s)) A", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Group/Prod.lean", "full_name": "MeasureTheory.quasiMeasurePreserving_div", "start": [474, 1], "end": [477, 94], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Group/Prod.lean", "full_name": "MeasureTheory.measure_lintegral_div_measure", "start": [322, 1], "end": [331, 78], "traced_tactics": [{"tactic": "set g := fun y => f y\u207b\u00b9 / \u03bd ((fun x => x * y\u207b\u00b9) \u207b\u00b9' s)", "annotated_tactic": ["set g := fun y => f y\u207b\u00b9 / \u03bd ((fun x => x * y\u207b\u00b9) \u207b\u00b9' s)", []], "state_before": "G : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nsm : MeasurableSet s\nh2s : \u2191\u2191\u03bd s \u2260 0\nh3s : \u2191\u2191\u03bd s \u2260 \u22a4\nf : G \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u22a2 \u2191\u2191\u03bc s * \u222b\u207b (y : G), f y\u207b\u00b9 / \u2191\u2191\u03bd ((fun x => x * y\u207b\u00b9) \u207b\u00b9' s) \u2202\u03bd = \u222b\u207b (x : G), f x \u2202\u03bc", "state_after": "G : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nsm : MeasurableSet s\nh2s : \u2191\u2191\u03bd s \u2260 0\nh3s : \u2191\u2191\u03bd s \u2260 \u22a4\nf : G \u2192 \u211d\u22650\u221e\nhf : Measurable f\ng : G \u2192 \u211d\u22650\u221e := fun y => f y\u207b\u00b9 / \u2191\u2191\u03bd ((fun x => x * y\u207b\u00b9) \u207b\u00b9' s)\n\u22a2 \u2191\u2191\u03bc s * lintegral \u03bd g = \u222b\u207b (x : G), f x \u2202\u03bc"}, {"tactic": "have hg : Measurable g :=\n  (hf.comp measurable_inv).div ((measurable_measure_mul_right \u03bd sm).comp measurable_inv)", "annotated_tactic": ["have hg : <a>Measurable</a> g :=\n    (hf.comp <a>measurable_inv</a>).<a>div</a> ((<a>measurable_measure_mul_right</a> \u03bd sm).<a>comp</a> <a>measurable_inv</a>)", [{"full_name": "Measurable", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [535, 5], "def_end_pos": [535, 15]}, {"full_name": "MeasurableInv.measurable_inv", "def_path": "Mathlib/MeasureTheory/Group/Arithmetic.lean", "def_pos": [427, 3], "def_end_pos": [427, 17]}, {"full_name": "Measurable.div", "def_path": "Mathlib/MeasureTheory/Group/Arithmetic.lean", "def_pos": [334, 9], "def_end_pos": [334, 23]}, {"full_name": "MeasureTheory.measurable_measure_mul_right", "def_path": "Mathlib/MeasureTheory/Group/Prod.lean", "def_pos": [105, 9], "def_end_pos": [105, 37]}, {"full_name": "Measurable.comp", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [557, 19], "def_end_pos": [557, 34]}, {"full_name": "MeasurableInv.measurable_inv", "def_path": "Mathlib/MeasureTheory/Group/Arithmetic.lean", "def_pos": [427, 3], "def_end_pos": [427, 17]}]], "state_before": "G : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nsm : MeasurableSet s\nh2s : \u2191\u2191\u03bd s \u2260 0\nh3s : \u2191\u2191\u03bd s \u2260 \u22a4\nf : G \u2192 \u211d\u22650\u221e\nhf : Measurable f\ng : G \u2192 \u211d\u22650\u221e := fun y => f y\u207b\u00b9 / \u2191\u2191\u03bd ((fun x => x * y\u207b\u00b9) \u207b\u00b9' s)\n\u22a2 \u2191\u2191\u03bc s * lintegral \u03bd g = \u222b\u207b (x : G), f x \u2202\u03bc", "state_after": "G : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nsm : MeasurableSet s\nh2s : \u2191\u2191\u03bd s \u2260 0\nh3s : \u2191\u2191\u03bd s \u2260 \u22a4\nf : G \u2192 \u211d\u22650\u221e\nhf : Measurable f\ng : G \u2192 \u211d\u22650\u221e := fun y => f y\u207b\u00b9 / \u2191\u2191\u03bd ((fun x => x * y\u207b\u00b9) \u207b\u00b9' s)\nhg : Measurable g\n\u22a2 \u2191\u2191\u03bc s * lintegral \u03bd g = \u222b\u207b (x : G), f x \u2202\u03bc"}, {"tactic": "simp_rw [measure_mul_lintegral_eq \u03bc \u03bd sm g hg, inv_inv]", "annotated_tactic": ["simp_rw [<a>measure_mul_lintegral_eq</a> \u03bc \u03bd sm g hg, <a>inv_inv</a>]", [{"full_name": "MeasureTheory.measure_mul_lintegral_eq", "def_path": "Mathlib/MeasureTheory/Group/Prod.lean", "def_pos": [244, 9], "def_end_pos": [244, 33]}, {"full_name": "inv_inv", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [800, 9], "def_end_pos": [800, 16]}]], "state_before": "G : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nsm : MeasurableSet s\nh2s : \u2191\u2191\u03bd s \u2260 0\nh3s : \u2191\u2191\u03bd s \u2260 \u22a4\nf : G \u2192 \u211d\u22650\u221e\nhf : Measurable f\ng : G \u2192 \u211d\u22650\u221e := fun y => f y\u207b\u00b9 / \u2191\u2191\u03bd ((fun x => x * y\u207b\u00b9) \u207b\u00b9' s)\nhg : Measurable g\n\u22a2 \u2191\u2191\u03bc s * lintegral \u03bd g = \u222b\u207b (x : G), f x \u2202\u03bc", "state_after": "G : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nsm : MeasurableSet s\nh2s : \u2191\u2191\u03bd s \u2260 0\nh3s : \u2191\u2191\u03bd s \u2260 \u22a4\nf : G \u2192 \u211d\u22650\u221e\nhf : Measurable f\ng : G \u2192 \u211d\u22650\u221e := fun y => f y\u207b\u00b9 / \u2191\u2191\u03bd ((fun x => x * y\u207b\u00b9) \u207b\u00b9' s)\nhg : Measurable g\n\u22a2 \u222b\u207b (x : G), \u2191\u2191\u03bd ((fun z => z * x) \u207b\u00b9' s) * (f x / \u2191\u2191\u03bd ((fun z => z * x) \u207b\u00b9' s)) \u2202\u03bc = \u222b\u207b (x : G), f x \u2202\u03bc"}, {"tactic": "refine' lintegral_congr_ae _", "annotated_tactic": ["refine' <a>lintegral_congr_ae</a> _", [{"full_name": "MeasureTheory.lintegral_congr_ae", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [304, 9], "def_end_pos": [304, 27]}]], "state_before": "G : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nsm : MeasurableSet s\nh2s : \u2191\u2191\u03bd s \u2260 0\nh3s : \u2191\u2191\u03bd s \u2260 \u22a4\nf : G \u2192 \u211d\u22650\u221e\nhf : Measurable f\ng : G \u2192 \u211d\u22650\u221e := fun y => f y\u207b\u00b9 / \u2191\u2191\u03bd ((fun x => x * y\u207b\u00b9) \u207b\u00b9' s)\nhg : Measurable g\n\u22a2 \u222b\u207b (x : G), \u2191\u2191\u03bd ((fun z => z * x) \u207b\u00b9' s) * (f x / \u2191\u2191\u03bd ((fun z => z * x) \u207b\u00b9' s)) \u2202\u03bc = \u222b\u207b (x : G), f x \u2202\u03bc", "state_after": "G : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nsm : MeasurableSet s\nh2s : \u2191\u2191\u03bd s \u2260 0\nh3s : \u2191\u2191\u03bd s \u2260 \u22a4\nf : G \u2192 \u211d\u22650\u221e\nhf : Measurable f\ng : G \u2192 \u211d\u22650\u221e := fun y => f y\u207b\u00b9 / \u2191\u2191\u03bd ((fun x => x * y\u207b\u00b9) \u207b\u00b9' s)\nhg : Measurable g\n\u22a2 (fun x => \u2191\u2191\u03bd ((fun z => z * x) \u207b\u00b9' s) * (f x / \u2191\u2191\u03bd ((fun z => z * x) \u207b\u00b9' s))) =\u1da0[ae \u03bc] fun x => f x"}, {"tactic": "refine' (ae_measure_preimage_mul_right_lt_top_of_ne_zero \u03bc \u03bd sm h2s h3s).mono fun x hx => _", "annotated_tactic": ["refine' (<a>ae_measure_preimage_mul_right_lt_top_of_ne_zero</a> \u03bc \u03bd sm h2s h3s).<a>mono</a> fun x hx => _", [{"full_name": "MeasureTheory.ae_measure_preimage_mul_right_lt_top_of_ne_zero", "def_path": "Mathlib/MeasureTheory/Group/Prod.lean", "def_pos": [293, 9], "def_end_pos": [293, 56]}, {"full_name": "Filter.Eventually.mono", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1140, 9], "def_end_pos": [1140, 24]}]], "state_before": "G : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nsm : MeasurableSet s\nh2s : \u2191\u2191\u03bd s \u2260 0\nh3s : \u2191\u2191\u03bd s \u2260 \u22a4\nf : G \u2192 \u211d\u22650\u221e\nhf : Measurable f\ng : G \u2192 \u211d\u22650\u221e := fun y => f y\u207b\u00b9 / \u2191\u2191\u03bd ((fun x => x * y\u207b\u00b9) \u207b\u00b9' s)\nhg : Measurable g\n\u22a2 (fun x => \u2191\u2191\u03bd ((fun z => z * x) \u207b\u00b9' s) * (f x / \u2191\u2191\u03bd ((fun z => z * x) \u207b\u00b9' s))) =\u1da0[ae \u03bc] fun x => f x", "state_after": "G : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nsm : MeasurableSet s\nh2s : \u2191\u2191\u03bd s \u2260 0\nh3s : \u2191\u2191\u03bd s \u2260 \u22a4\nf : G \u2192 \u211d\u22650\u221e\nhf : Measurable f\ng : G \u2192 \u211d\u22650\u221e := fun y => f y\u207b\u00b9 / \u2191\u2191\u03bd ((fun x => x * y\u207b\u00b9) \u207b\u00b9' s)\nhg : Measurable g\nx : G\nhx : \u2191\u2191\u03bd ((fun y => y * x) \u207b\u00b9' s) < \u22a4\n\u22a2 (fun x => \u2191\u2191\u03bd ((fun z => z * x) \u207b\u00b9' s) * (f x / \u2191\u2191\u03bd ((fun z => z * x) \u207b\u00b9' s))) x = (fun x => f x) x"}, {"tactic": "simp_rw [ENNReal.mul_div_cancel' (measure_mul_right_ne_zero \u03bd h2s _) hx.ne]", "annotated_tactic": ["simp_rw [<a>ENNReal.mul_div_cancel'</a> (<a>measure_mul_right_ne_zero</a> \u03bd h2s _) hx.ne]", [{"full_name": "ENNReal.mul_div_cancel'", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1432, 19], "def_end_pos": [1432, 34]}, {"full_name": "MeasureTheory.measure_mul_right_ne_zero", "def_path": "Mathlib/MeasureTheory/Group/Prod.lean", "def_pos": [221, 9], "def_end_pos": [221, 34]}]], "state_before": "G : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nsm : MeasurableSet s\nh2s : \u2191\u2191\u03bd s \u2260 0\nh3s : \u2191\u2191\u03bd s \u2260 \u22a4\nf : G \u2192 \u211d\u22650\u221e\nhf : Measurable f\ng : G \u2192 \u211d\u22650\u221e := fun y => f y\u207b\u00b9 / \u2191\u2191\u03bd ((fun x => x * y\u207b\u00b9) \u207b\u00b9' s)\nhg : Measurable g\nx : G\nhx : \u2191\u2191\u03bd ((fun y => y * x) \u207b\u00b9' s) < \u22a4\n\u22a2 (fun x => \u2191\u2191\u03bd ((fun z => z * x) \u207b\u00b9' s) * (f x / \u2191\u2191\u03bd ((fun z => z * x) \u207b\u00b9' s))) x = (fun x => f x) x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Moments.lean", "full_name": "ProbabilityTheory.iIndepFun.mgf_sum", "start": [302, 1], "end": [313, 37], "traced_tactics": [{"tactic": "induction' s using Finset.induction_on with i s hi_notin_s h_rec h_int", "annotated_tactic": ["induction' s using <a>Finset.induction_on</a> with i s hi_notin_s h_rec h_int", [{"full_name": "Finset.induction_on", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1251, 19], "def_end_pos": [1251, 31]}]], "state_before": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\np : \u2115\n\u03bc : Measure \u03a9\nt : \u211d\ninst\u271d : IsProbabilityMeasure \u03bc\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\nh_indep : iIndepFun (fun i => inferInstance) X\nh_meas : \u2200 (i : \u03b9), Measurable (X i)\ns : Finset \u03b9\n\u22a2 mgf (\u2211 i in s, X i) \u03bc t = \u220f i in s, mgf (X i) \u03bc t", "state_after": "case empty\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\np : \u2115\n\u03bc : Measure \u03a9\nt : \u211d\ninst\u271d : IsProbabilityMeasure \u03bc\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\nh_indep : iIndepFun (fun i => inferInstance) X\nh_meas : \u2200 (i : \u03b9), Measurable (X i)\n\u22a2 mgf (\u2211 i in \u2205, X i) \u03bc t = \u220f i in \u2205, mgf (X i) \u03bc t\n\ncase insert\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\np : \u2115\n\u03bc : Measure \u03a9\nt : \u211d\ninst\u271d : IsProbabilityMeasure \u03bc\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\nh_indep : iIndepFun (fun i => inferInstance) X\nh_meas : \u2200 (i : \u03b9), Measurable (X i)\ni : \u03b9\ns : Finset \u03b9\nhi_notin_s : \u00aci \u2208 s\nh_rec : mgf (\u2211 i in s, X i) \u03bc t = \u220f i in s, mgf (X i) \u03bc t\n\u22a2 mgf (\u2211 i in insert i s, X i) \u03bc t = \u220f i in insert i s, mgf (X i) \u03bc t"}, {"tactic": "simp only [sum_empty, mgf_zero_fun, measure_univ, ENNReal.one_toReal, prod_empty]", "annotated_tactic": ["simp only [<a>sum_empty</a>, <a>mgf_zero_fun</a>, <a>measure_univ</a>, <a>ENNReal.one_toReal</a>, <a>prod_empty</a>]", [{"full_name": "Finset.sum_empty", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [298, 3], "def_end_pos": [298, 14]}, {"full_name": "ProbabilityTheory.mgf_zero_fun", "def_path": "Mathlib/Probability/Moments.lean", "def_pos": [113, 9], "def_end_pos": [113, 21]}, {"full_name": "MeasureTheory.IsProbabilityMeasure.measure_univ", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3027, 3], "def_end_pos": [3027, 15]}, {"full_name": "ENNReal.one_toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [230, 17], "def_end_pos": [230, 27]}, {"full_name": "Finset.prod_empty", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [299, 9], "def_end_pos": [299, 19]}]], "state_before": "case empty\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\np : \u2115\n\u03bc : Measure \u03a9\nt : \u211d\ninst\u271d : IsProbabilityMeasure \u03bc\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\nh_indep : iIndepFun (fun i => inferInstance) X\nh_meas : \u2200 (i : \u03b9), Measurable (X i)\n\u22a2 mgf (\u2211 i in \u2205, X i) \u03bc t = \u220f i in \u2205, mgf (X i) \u03bc t", "state_after": "no goals"}, {"tactic": "have h_int' : \u2200 i : \u03b9, AEStronglyMeasurable (fun \u03c9 : \u03a9 => exp (t * X i \u03c9)) \u03bc := fun i =>\n  ((h_meas i).const_mul t).exp.aestronglyMeasurable", "annotated_tactic": ["have h_int' : \u2200 i : \u03b9, <a>AEStronglyMeasurable</a> (fun \u03c9 : \u03a9 => <a>exp</a> (t * X i \u03c9)) \u03bc := fun i =>\n      ((h_meas i).<a>const_mul</a> t).exp.aestronglyMeasurable", [{"full_name": "MeasureTheory.AEStronglyMeasurable", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [93, 5], "def_end_pos": [93, 25]}, {"full_name": "Real.exp", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [434, 12], "def_end_pos": [434, 15]}, {"full_name": "Measurable.const_mul", "def_path": "Mathlib/MeasureTheory/Group/Arithmetic.lean", "def_pos": [106, 9], "def_end_pos": [106, 29]}]], "state_before": "case insert\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\np : \u2115\n\u03bc : Measure \u03a9\nt : \u211d\ninst\u271d : IsProbabilityMeasure \u03bc\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\nh_indep : iIndepFun (fun i => inferInstance) X\nh_meas : \u2200 (i : \u03b9), Measurable (X i)\ni : \u03b9\ns : Finset \u03b9\nhi_notin_s : \u00aci \u2208 s\nh_rec : mgf (\u2211 i in s, X i) \u03bc t = \u220f i in s, mgf (X i) \u03bc t\n\u22a2 mgf (\u2211 i in insert i s, X i) \u03bc t = \u220f i in insert i s, mgf (X i) \u03bc t", "state_after": "case insert\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\np : \u2115\n\u03bc : Measure \u03a9\nt : \u211d\ninst\u271d : IsProbabilityMeasure \u03bc\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\nh_indep : iIndepFun (fun i => inferInstance) X\nh_meas : \u2200 (i : \u03b9), Measurable (X i)\ni : \u03b9\ns : Finset \u03b9\nhi_notin_s : \u00aci \u2208 s\nh_rec : mgf (\u2211 i in s, X i) \u03bc t = \u220f i in s, mgf (X i) \u03bc t\nh_int' : \u2200 (i : \u03b9), AEStronglyMeasurable (fun \u03c9 => rexp (t * X i \u03c9)) \u03bc\n\u22a2 mgf (\u2211 i in insert i s, X i) \u03bc t = \u220f i in insert i s, mgf (X i) \u03bc t"}, {"tactic": "rw [sum_insert hi_notin_s,\n  IndepFun.mgf_add (h_indep.indepFun_finset_sum_of_not_mem h_meas hi_notin_s).symm (h_int' i)\n    (aestronglyMeasurable_exp_mul_sum fun i _ => h_int' i),\n  h_rec, prod_insert hi_notin_s]", "annotated_tactic": ["rw [<a>sum_insert</a> hi_notin_s,\n      <a>IndepFun.mgf_add</a> (h_indep.indepFun_finset_sum_of_not_mem h_meas hi_notin_s).<a>symm</a> (h_int' i)\n        (<a>aestronglyMeasurable_exp_mul_sum</a> fun i _ => h_int' i),\n      h_rec, <a>prod_insert</a> hi_notin_s]", [{"full_name": "Finset.sum_insert", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [316, 3], "def_end_pos": [316, 14]}, {"full_name": "ProbabilityTheory.IndepFun.mgf_add", "def_path": "Mathlib/Probability/Moments.lean", "def_pos": [227, 9], "def_end_pos": [227, 25]}, {"full_name": "ProbabilityTheory.IndepFun.symm", "def_path": "Mathlib/Probability/Independence/Basic.lean", "def_pos": [563, 16], "def_end_pos": [563, 29]}, {"full_name": "ProbabilityTheory.aestronglyMeasurable_exp_mul_sum", "def_path": "Mathlib/Probability/Moments.lean", "def_pos": [263, 9], "def_end_pos": [263, 41]}, {"full_name": "Finset.prod_insert", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [317, 9], "def_end_pos": [317, 20]}]], "state_before": "case insert\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\np : \u2115\n\u03bc : Measure \u03a9\nt : \u211d\ninst\u271d : IsProbabilityMeasure \u03bc\nX : \u03b9 \u2192 \u03a9 \u2192 \u211d\nh_indep : iIndepFun (fun i => inferInstance) X\nh_meas : \u2200 (i : \u03b9), Measurable (X i)\ni : \u03b9\ns : Finset \u03b9\nhi_notin_s : \u00aci \u2208 s\nh_rec : mgf (\u2211 i in s, X i) \u03bc t = \u220f i in s, mgf (X i) \u03bc t\nh_int' : \u2200 (i : \u03b9), AEStronglyMeasurable (fun \u03c9 => rexp (t * X i \u03c9)) \u03bc\n\u22a2 mgf (\u2211 i in insert i s, X i) \u03bc t = \u220f i in insert i s, mgf (X i) \u03bc t", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "full_name": "MeasureTheory.OuterMeasure.map_ofFunction", "start": [792, 1], "end": [802, 29], "traced_tactics": [{"tactic": "refine' (map_ofFunction_le _).antisymm fun s => _", "annotated_tactic": ["refine' (<a>map_ofFunction_le</a> _).<a>antisymm</a> fun s => _", [{"full_name": "MeasureTheory.OuterMeasure.map_ofFunction_le", "def_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "def_pos": [784, 9], "def_end_pos": [784, 26]}, {"full_name": "LE.le.antisymm", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [128, 7], "def_end_pos": [128, 21]}]], "state_before": "\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b2\nhf : Injective f\n\u22a2 \u2191(map f) (OuterMeasure.ofFunction m m_empty) = OuterMeasure.ofFunction (fun s => m (f \u207b\u00b9' s)) m_empty", "state_after": "\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b2\nhf : Injective f\ns : Set \u03b2\n\u22a2 \u2191(OuterMeasure.ofFunction (fun s => m (f \u207b\u00b9' s)) m_empty) s \u2264 \u2191(\u2191(map f) (OuterMeasure.ofFunction m m_empty)) s"}, {"tactic": "simp only [ofFunction_apply, map_apply, le_iInf_iff]", "annotated_tactic": ["simp only [<a>ofFunction_apply</a>, <a>map_apply</a>, <a>le_iInf_iff</a>]", [{"full_name": "MeasureTheory.OuterMeasure.ofFunction_apply", "def_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "def_pos": [691, 9], "def_end_pos": [691, 25]}, {"full_name": "MeasureTheory.OuterMeasure.map_apply", "def_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "def_pos": [451, 9], "def_end_pos": [451, 18]}, {"full_name": "le_iInf_iff", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [969, 9], "def_end_pos": [969, 20]}]], "state_before": "\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b2\nhf : Injective f\ns : Set \u03b2\n\u22a2 \u2191(OuterMeasure.ofFunction (fun s => m (f \u207b\u00b9' s)) m_empty) s \u2264 \u2191(\u2191(map f) (OuterMeasure.ofFunction m m_empty)) s", "state_after": "\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b2\nhf : Injective f\ns : Set \u03b2\n\u22a2 \u2200 (i : \u2115 \u2192 Set \u03b1), f \u207b\u00b9' s \u2286 iUnion i \u2192 \u2a05 t, \u2a05 (_ : s \u2286 iUnion t), \u2211' (n : \u2115), m (f \u207b\u00b9' t n) \u2264 \u2211' (n : \u2115), m (i n)"}, {"tactic": "intro t ht", "annotated_tactic": ["intro t ht", []], "state_before": "\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b2\nhf : Injective f\ns : Set \u03b2\n\u22a2 \u2200 (i : \u2115 \u2192 Set \u03b1), f \u207b\u00b9' s \u2286 iUnion i \u2192 \u2a05 t, \u2a05 (_ : s \u2286 iUnion t), \u2211' (n : \u2115), m (f \u207b\u00b9' t n) \u2264 \u2211' (n : \u2115), m (i n)", "state_after": "\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b2\nhf : Injective f\ns : Set \u03b2\nt : \u2115 \u2192 Set \u03b1\nht : f \u207b\u00b9' s \u2286 iUnion t\n\u22a2 \u2a05 t, \u2a05 (_ : s \u2286 iUnion t), \u2211' (n : \u2115), m (f \u207b\u00b9' t n) \u2264 \u2211' (n : \u2115), m (t n)"}, {"tactic": "refine' iInf_le_of_le (fun n => (range f)\u1d9c \u222a f '' t n) (iInf_le_of_le _ _)", "annotated_tactic": ["refine' <a>iInf_le_of_le</a> (fun n => (<a>range</a> f)\u1d9c \u222a f '' t n) (<a>iInf_le_of_le</a> _ _)", [{"full_name": "iInf_le_of_le", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [853, 9], "def_end_pos": [853, 22]}, {"full_name": "Set.range", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [668, 5], "def_end_pos": [668, 10]}, {"full_name": "iInf_le_of_le", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [853, 9], "def_end_pos": [853, 22]}]], "state_before": "\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b2\nhf : Injective f\ns : Set \u03b2\nt : \u2115 \u2192 Set \u03b1\nht : f \u207b\u00b9' s \u2286 iUnion t\n\u22a2 \u2a05 t, \u2a05 (_ : s \u2286 iUnion t), \u2211' (n : \u2115), m (f \u207b\u00b9' t n) \u2264 \u2211' (n : \u2115), m (t n)", "state_after": "case refine'_1\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b2\nhf : Injective f\ns : Set \u03b2\nt : \u2115 \u2192 Set \u03b1\nht : f \u207b\u00b9' s \u2286 iUnion t\n\u22a2 s \u2286 \u22c3 n, (range f)\u1d9c \u222a f '' t n\n\ncase refine'_2\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b2\nhf : Injective f\ns : Set \u03b2\nt : \u2115 \u2192 Set \u03b1\nht : f \u207b\u00b9' s \u2286 iUnion t\n\u22a2 \u2211' (n : \u2115), m (f \u207b\u00b9' (fun n => (range f)\u1d9c \u222a f '' t n) n) \u2264 \u2211' (n : \u2115), m (t n)"}, {"tactic": "rw [\u2190 union_iUnion, \u2190 inter_subset, \u2190 image_preimage_eq_inter_range, \u2190 image_iUnion]", "annotated_tactic": ["rw [\u2190 <a>union_iUnion</a>, \u2190 <a>inter_subset</a>, \u2190 <a>image_preimage_eq_inter_range</a>, \u2190 <a>image_iUnion</a>]", [{"full_name": "Set.union_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [653, 9], "def_end_pos": [653, 21]}, {"full_name": "Set.inter_subset", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1803, 9], "def_end_pos": [1803, 21]}, {"full_name": "Set.image_preimage_eq_inter_range", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [796, 9], "def_end_pos": [796, 38]}, {"full_name": "Set.image_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [1791, 9], "def_end_pos": [1791, 21]}]], "state_before": "case refine'_1\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b2\nhf : Injective f\ns : Set \u03b2\nt : \u2115 \u2192 Set \u03b1\nht : f \u207b\u00b9' s \u2286 iUnion t\n\u22a2 s \u2286 \u22c3 n, (range f)\u1d9c \u222a f '' t n", "state_after": "case refine'_1\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b2\nhf : Injective f\ns : Set \u03b2\nt : \u2115 \u2192 Set \u03b1\nht : f \u207b\u00b9' s \u2286 iUnion t\n\u22a2 f '' (f \u207b\u00b9' s) \u2286 f '' \u22c3 i, t i"}, {"tactic": "exact image_subset _ ht", "annotated_tactic": ["exact <a>image_subset</a> _ ht", [{"full_name": "Set.image_subset", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [321, 9], "def_end_pos": [321, 21]}]], "state_before": "case refine'_1\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b2\nhf : Injective f\ns : Set \u03b2\nt : \u2115 \u2192 Set \u03b1\nht : f \u207b\u00b9' s \u2286 iUnion t\n\u22a2 f '' (f \u207b\u00b9' s) \u2286 f '' \u22c3 i, t i", "state_after": "no goals"}, {"tactic": "refine' ENNReal.tsum_le_tsum fun n => le_of_eq _", "annotated_tactic": ["refine' <a>ENNReal.tsum_le_tsum</a> fun n => <a>le_of_eq</a> _", [{"full_name": "ENNReal.tsum_le_tsum", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [827, 19], "def_end_pos": [827, 31]}, {"full_name": "le_of_eq", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [72, 9], "def_end_pos": [72, 17]}]], "state_before": "case refine'_2\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b2\nhf : Injective f\ns : Set \u03b2\nt : \u2115 \u2192 Set \u03b1\nht : f \u207b\u00b9' s \u2286 iUnion t\n\u22a2 \u2211' (n : \u2115), m (f \u207b\u00b9' (fun n => (range f)\u1d9c \u222a f '' t n) n) \u2264 \u2211' (n : \u2115), m (t n)", "state_after": "case refine'_2\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b2\nhf : Injective f\ns : Set \u03b2\nt : \u2115 \u2192 Set \u03b1\nht : f \u207b\u00b9' s \u2286 iUnion t\nn : \u2115\n\u22a2 m (f \u207b\u00b9' (fun n => (range f)\u1d9c \u222a f '' t n) n) = m (t n)"}, {"tactic": "simp [hf.preimage_image]", "annotated_tactic": ["simp [hf.preimage_image]", []], "state_before": "case refine'_2\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b2\nhf : Injective f\ns : Set \u03b2\nt : \u2115 \u2192 Set \u03b1\nht : f \u207b\u00b9' s \u2286 iUnion t\nn : \u2115\n\u22a2 m (f \u207b\u00b9' (fun n => (range f)\u1d9c \u222a f '' t n) n) = m (t n)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Count.lean", "full_name": "MeasureTheory.Measure.count_singleton", "start": [161, 1], "end": [162, 47], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "full_name": "MeasureTheory.Integrable.bdd_mul", "start": [728, 1], "end": [744, 66], "traced_tactics": [{"tactic": "cases' isEmpty_or_nonempty \u03b1 with h\u03b1 h\u03b1", "annotated_tactic": ["cases' <a>isEmpty_or_nonempty</a> \u03b1 with h\u03b1 h\u03b1", [{"full_name": "isEmpty_or_nonempty", "def_path": "Mathlib/Logic/IsEmpty.lean", "def_pos": [207, 9], "def_end_pos": [207, 28]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b4\ninst\u271d\u00b2 : NormedAddCommGroup \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b3\nF : Type u_5\ninst\u271d : NormedDivisionRing F\nf g : \u03b1 \u2192 F\nhint : Integrable g\nhm : AEStronglyMeasurable f \u03bc\nhfbdd : \u2203 C, \u2200 (x : \u03b1), \u2016f x\u2016 \u2264 C\n\u22a2 Integrable fun x => f x * g x", "state_after": "case inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b4\ninst\u271d\u00b2 : NormedAddCommGroup \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b3\nF : Type u_5\ninst\u271d : NormedDivisionRing F\nf g : \u03b1 \u2192 F\nhint : Integrable g\nhm : AEStronglyMeasurable f \u03bc\nhfbdd : \u2203 C, \u2200 (x : \u03b1), \u2016f x\u2016 \u2264 C\nh\u03b1 : IsEmpty \u03b1\n\u22a2 Integrable fun x => f x * g x\n\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b4\ninst\u271d\u00b2 : NormedAddCommGroup \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b3\nF : Type u_5\ninst\u271d : NormedDivisionRing F\nf g : \u03b1 \u2192 F\nhint : Integrable g\nhm : AEStronglyMeasurable f \u03bc\nhfbdd : \u2203 C, \u2200 (x : \u03b1), \u2016f x\u2016 \u2264 C\nh\u03b1 : Nonempty \u03b1\n\u22a2 Integrable fun x => f x * g x"}, {"tactic": "exact integrable_zero_measure", "annotated_tactic": ["exact <a>integrable_zero_measure</a>", [{"full_name": "MeasureTheory.integrable_zero_measure", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [567, 9], "def_end_pos": [567, 32]}]], "state_before": "case inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b4\ninst\u271d\u00b2 : NormedAddCommGroup \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b3\nF : Type u_5\ninst\u271d : NormedDivisionRing F\nf g : \u03b1 \u2192 F\nhint : Integrable g\nhm : AEStronglyMeasurable f \u03bc\nhfbdd : \u2203 C, \u2200 (x : \u03b1), \u2016f x\u2016 \u2264 C\nh\u03b1 : IsEmpty \u03b1\n\u22a2 Integrable fun x => f x * g x", "state_after": "no goals"}, {"tactic": "refine' \u27e8hm.mul hint.1, _\u27e9", "annotated_tactic": ["refine' \u27e8hm.mul hint.1, _\u27e9", []], "state_before": "case inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b4\ninst\u271d\u00b2 : NormedAddCommGroup \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b3\nF : Type u_5\ninst\u271d : NormedDivisionRing F\nf g : \u03b1 \u2192 F\nhint : Integrable g\nhm : AEStronglyMeasurable f \u03bc\nhfbdd : \u2203 C, \u2200 (x : \u03b1), \u2016f x\u2016 \u2264 C\nh\u03b1 : Nonempty \u03b1\n\u22a2 Integrable fun x => f x * g x", "state_after": "case inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b4\ninst\u271d\u00b2 : NormedAddCommGroup \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b3\nF : Type u_5\ninst\u271d : NormedDivisionRing F\nf g : \u03b1 \u2192 F\nhint : Integrable g\nhm : AEStronglyMeasurable f \u03bc\nhfbdd : \u2203 C, \u2200 (x : \u03b1), \u2016f x\u2016 \u2264 C\nh\u03b1 : Nonempty \u03b1\n\u22a2 HasFiniteIntegral fun x => f x * g x"}, {"tactic": "obtain \u27e8C, hC\u27e9 := hfbdd", "annotated_tactic": ["obtain \u27e8C, hC\u27e9 := hfbdd", []], "state_before": "case inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b4\ninst\u271d\u00b2 : NormedAddCommGroup \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b3\nF : Type u_5\ninst\u271d : NormedDivisionRing F\nf g : \u03b1 \u2192 F\nhint : Integrable g\nhm : AEStronglyMeasurable f \u03bc\nhfbdd : \u2203 C, \u2200 (x : \u03b1), \u2016f x\u2016 \u2264 C\nh\u03b1 : Nonempty \u03b1\n\u22a2 HasFiniteIntegral fun x => f x * g x", "state_after": "case inr.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b4\ninst\u271d\u00b2 : NormedAddCommGroup \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b3\nF : Type u_5\ninst\u271d : NormedDivisionRing F\nf g : \u03b1 \u2192 F\nhint : Integrable g\nhm : AEStronglyMeasurable f \u03bc\nh\u03b1 : Nonempty \u03b1\nC : \u211d\nhC : \u2200 (x : \u03b1), \u2016f x\u2016 \u2264 C\n\u22a2 HasFiniteIntegral fun x => f x * g x"}, {"tactic": "have hCnonneg : 0 \u2264 C := le_trans (norm_nonneg _) (hC h\u03b1.some)", "annotated_tactic": ["have hCnonneg : 0 \u2264 C := <a>le_trans</a> (<a>norm_nonneg</a> _) (hC h\u03b1.some)", [{"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}, {"full_name": "norm_nonneg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [500, 30], "def_end_pos": [500, 41]}]], "state_before": "case inr.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b4\ninst\u271d\u00b2 : NormedAddCommGroup \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b3\nF : Type u_5\ninst\u271d : NormedDivisionRing F\nf g : \u03b1 \u2192 F\nhint : Integrable g\nhm : AEStronglyMeasurable f \u03bc\nh\u03b1 : Nonempty \u03b1\nC : \u211d\nhC : \u2200 (x : \u03b1), \u2016f x\u2016 \u2264 C\n\u22a2 HasFiniteIntegral fun x => f x * g x", "state_after": "case inr.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b4\ninst\u271d\u00b2 : NormedAddCommGroup \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b3\nF : Type u_5\ninst\u271d : NormedDivisionRing F\nf g : \u03b1 \u2192 F\nhint : Integrable g\nhm : AEStronglyMeasurable f \u03bc\nh\u03b1 : Nonempty \u03b1\nC : \u211d\nhC : \u2200 (x : \u03b1), \u2016f x\u2016 \u2264 C\nhCnonneg : 0 \u2264 C\n\u22a2 HasFiniteIntegral fun x => f x * g x"}, {"tactic": "have : (fun x => \u2016f x * g x\u2016\u208a) \u2264 fun x => \u27e8C, hCnonneg\u27e9 * \u2016g x\u2016\u208a := by\n  intro x\n  simp only [nnnorm_mul]\n  exact mul_le_mul_of_nonneg_right (hC x) (zero_le _)", "annotated_tactic": ["have : (fun x => \u2016f x * g x\u2016\u208a) \u2264 fun x => \u27e8C, hCnonneg\u27e9 * \u2016g x\u2016\u208a := by\n      intro x\n      simp only [<a>nnnorm_mul</a>]\n      exact <a>mul_le_mul_of_nonneg_right</a> (hC x) (<a>zero_le</a> _)", [{"full_name": "nnnorm_mul", "def_path": "Mathlib/Analysis/Normed/Field/Basic.lean", "def_pos": [521, 9], "def_end_pos": [521, 19]}, {"full_name": "mul_le_mul_of_nonneg_right", "def_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "def_pos": [156, 9], "def_end_pos": [156, 35]}, {"full_name": "zero_le", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [217, 30], "def_end_pos": [217, 37]}]], "state_before": "case inr.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b4\ninst\u271d\u00b2 : NormedAddCommGroup \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b3\nF : Type u_5\ninst\u271d : NormedDivisionRing F\nf g : \u03b1 \u2192 F\nhint : Integrable g\nhm : AEStronglyMeasurable f \u03bc\nh\u03b1 : Nonempty \u03b1\nC : \u211d\nhC : \u2200 (x : \u03b1), \u2016f x\u2016 \u2264 C\nhCnonneg : 0 \u2264 C\n\u22a2 HasFiniteIntegral fun x => f x * g x", "state_after": "case inr.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b4\ninst\u271d\u00b2 : NormedAddCommGroup \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b3\nF : Type u_5\ninst\u271d : NormedDivisionRing F\nf g : \u03b1 \u2192 F\nhint : Integrable g\nhm : AEStronglyMeasurable f \u03bc\nh\u03b1 : Nonempty \u03b1\nC : \u211d\nhC : \u2200 (x : \u03b1), \u2016f x\u2016 \u2264 C\nhCnonneg : 0 \u2264 C\nthis : (fun x => \u2016f x * g x\u2016\u208a) \u2264 fun x => { val := C, property := hCnonneg } * \u2016g x\u2016\u208a\n\u22a2 HasFiniteIntegral fun x => f x * g x"}, {"tactic": "refine' lt_of_le_of_lt (lintegral_mono_nnreal this) _", "annotated_tactic": ["refine' <a>lt_of_le_of_lt</a> (<a>lintegral_mono_nnreal</a> this) _", [{"full_name": "lt_of_le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [122, 9], "def_end_pos": [122, 23]}, {"full_name": "MeasureTheory.lintegral_mono_nnreal", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [108, 9], "def_end_pos": [108, 30]}]], "state_before": "case inr.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b4\ninst\u271d\u00b2 : NormedAddCommGroup \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b3\nF : Type u_5\ninst\u271d : NormedDivisionRing F\nf g : \u03b1 \u2192 F\nhint : Integrable g\nhm : AEStronglyMeasurable f \u03bc\nh\u03b1 : Nonempty \u03b1\nC : \u211d\nhC : \u2200 (x : \u03b1), \u2016f x\u2016 \u2264 C\nhCnonneg : 0 \u2264 C\nthis : (fun x => \u2016f x * g x\u2016\u208a) \u2264 fun x => { val := C, property := hCnonneg } * \u2016g x\u2016\u208a\n\u22a2 HasFiniteIntegral fun x => f x * g x", "state_after": "case inr.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b4\ninst\u271d\u00b2 : NormedAddCommGroup \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b3\nF : Type u_5\ninst\u271d : NormedDivisionRing F\nf g : \u03b1 \u2192 F\nhint : Integrable g\nhm : AEStronglyMeasurable f \u03bc\nh\u03b1 : Nonempty \u03b1\nC : \u211d\nhC : \u2200 (x : \u03b1), \u2016f x\u2016 \u2264 C\nhCnonneg : 0 \u2264 C\nthis : (fun x => \u2016f x * g x\u2016\u208a) \u2264 fun x => { val := C, property := hCnonneg } * \u2016g x\u2016\u208a\n\u22a2 \u222b\u207b (a : \u03b1), \u2191({ val := C, property := hCnonneg } * \u2016g a\u2016\u208a) \u2202\u03bc < \u22a4"}, {"tactic": "simp only [ENNReal.coe_mul]", "annotated_tactic": ["simp only [<a>ENNReal.coe_mul</a>]", [{"full_name": "ENNReal.coe_mul", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [390, 9], "def_end_pos": [390, 16]}]], "state_before": "case inr.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b4\ninst\u271d\u00b2 : NormedAddCommGroup \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b3\nF : Type u_5\ninst\u271d : NormedDivisionRing F\nf g : \u03b1 \u2192 F\nhint : Integrable g\nhm : AEStronglyMeasurable f \u03bc\nh\u03b1 : Nonempty \u03b1\nC : \u211d\nhC : \u2200 (x : \u03b1), \u2016f x\u2016 \u2264 C\nhCnonneg : 0 \u2264 C\nthis : (fun x => \u2016f x * g x\u2016\u208a) \u2264 fun x => { val := C, property := hCnonneg } * \u2016g x\u2016\u208a\n\u22a2 \u222b\u207b (a : \u03b1), \u2191({ val := C, property := hCnonneg } * \u2016g a\u2016\u208a) \u2202\u03bc < \u22a4", "state_after": "case inr.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b4\ninst\u271d\u00b2 : NormedAddCommGroup \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b3\nF : Type u_5\ninst\u271d : NormedDivisionRing F\nf g : \u03b1 \u2192 F\nhint : Integrable g\nhm : AEStronglyMeasurable f \u03bc\nh\u03b1 : Nonempty \u03b1\nC : \u211d\nhC : \u2200 (x : \u03b1), \u2016f x\u2016 \u2264 C\nhCnonneg : 0 \u2264 C\nthis : (fun x => \u2016f x * g x\u2016\u208a) \u2264 fun x => { val := C, property := hCnonneg } * \u2016g x\u2016\u208a\n\u22a2 \u222b\u207b (a : \u03b1), \u2191{ val := C, property := hCnonneg } * \u2191\u2016g a\u2016\u208a \u2202\u03bc < \u22a4"}, {"tactic": "rw [lintegral_const_mul' _ _ ENNReal.coe_ne_top]", "annotated_tactic": ["rw [<a>lintegral_const_mul'</a> _ _ <a>ENNReal.coe_ne_top</a>]", [{"full_name": "MeasureTheory.lintegral_const_mul'", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [711, 9], "def_end_pos": [711, 29]}, {"full_name": "ENNReal.coe_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [302, 17], "def_end_pos": [302, 27]}]], "state_before": "case inr.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b4\ninst\u271d\u00b2 : NormedAddCommGroup \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b3\nF : Type u_5\ninst\u271d : NormedDivisionRing F\nf g : \u03b1 \u2192 F\nhint : Integrable g\nhm : AEStronglyMeasurable f \u03bc\nh\u03b1 : Nonempty \u03b1\nC : \u211d\nhC : \u2200 (x : \u03b1), \u2016f x\u2016 \u2264 C\nhCnonneg : 0 \u2264 C\nthis : (fun x => \u2016f x * g x\u2016\u208a) \u2264 fun x => { val := C, property := hCnonneg } * \u2016g x\u2016\u208a\n\u22a2 \u222b\u207b (a : \u03b1), \u2191{ val := C, property := hCnonneg } * \u2191\u2016g a\u2016\u208a \u2202\u03bc < \u22a4", "state_after": "case inr.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b4\ninst\u271d\u00b2 : NormedAddCommGroup \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b3\nF : Type u_5\ninst\u271d : NormedDivisionRing F\nf g : \u03b1 \u2192 F\nhint : Integrable g\nhm : AEStronglyMeasurable f \u03bc\nh\u03b1 : Nonempty \u03b1\nC : \u211d\nhC : \u2200 (x : \u03b1), \u2016f x\u2016 \u2264 C\nhCnonneg : 0 \u2264 C\nthis : (fun x => \u2016f x * g x\u2016\u208a) \u2264 fun x => { val := C, property := hCnonneg } * \u2016g x\u2016\u208a\n\u22a2 \u2191{ val := C, property := hCnonneg } * \u222b\u207b (a : \u03b1), \u2191\u2016g a\u2016\u208a \u2202\u03bc < \u22a4"}, {"tactic": "exact ENNReal.mul_lt_top ENNReal.coe_ne_top (ne_of_lt hint.2)", "annotated_tactic": ["exact <a>ENNReal.mul_lt_top</a> <a>ENNReal.coe_ne_top</a> (<a>ne_of_lt</a> hint.2)", [{"full_name": "ENNReal.mul_lt_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [612, 9], "def_end_pos": [612, 19]}, {"full_name": "ENNReal.coe_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [302, 17], "def_end_pos": [302, 27]}, {"full_name": "ne_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [101, 9], "def_end_pos": [101, 17]}]], "state_before": "case inr.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b4\ninst\u271d\u00b2 : NormedAddCommGroup \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b3\nF : Type u_5\ninst\u271d : NormedDivisionRing F\nf g : \u03b1 \u2192 F\nhint : Integrable g\nhm : AEStronglyMeasurable f \u03bc\nh\u03b1 : Nonempty \u03b1\nC : \u211d\nhC : \u2200 (x : \u03b1), \u2016f x\u2016 \u2264 C\nhCnonneg : 0 \u2264 C\nthis : (fun x => \u2016f x * g x\u2016\u208a) \u2264 fun x => { val := C, property := hCnonneg } * \u2016g x\u2016\u208a\n\u22a2 \u2191{ val := C, property := hCnonneg } * \u222b\u207b (a : \u03b1), \u2191\u2016g a\u2016\u208a \u2202\u03bc < \u22a4", "state_after": "no goals"}, {"tactic": "intro x", "annotated_tactic": ["intro x", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b4\ninst\u271d\u00b2 : NormedAddCommGroup \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b3\nF : Type u_5\ninst\u271d : NormedDivisionRing F\nf g : \u03b1 \u2192 F\nhint : Integrable g\nhm : AEStronglyMeasurable f \u03bc\nh\u03b1 : Nonempty \u03b1\nC : \u211d\nhC : \u2200 (x : \u03b1), \u2016f x\u2016 \u2264 C\nhCnonneg : 0 \u2264 C\n\u22a2 (fun x => \u2016f x * g x\u2016\u208a) \u2264 fun x => { val := C, property := hCnonneg } * \u2016g x\u2016\u208a", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b4\ninst\u271d\u00b2 : NormedAddCommGroup \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b3\nF : Type u_5\ninst\u271d : NormedDivisionRing F\nf g : \u03b1 \u2192 F\nhint : Integrable g\nhm : AEStronglyMeasurable f \u03bc\nh\u03b1 : Nonempty \u03b1\nC : \u211d\nhC : \u2200 (x : \u03b1), \u2016f x\u2016 \u2264 C\nhCnonneg : 0 \u2264 C\nx : \u03b1\n\u22a2 (fun x => \u2016f x * g x\u2016\u208a) x \u2264 (fun x => { val := C, property := hCnonneg } * \u2016g x\u2016\u208a) x"}, {"tactic": "simp only [nnnorm_mul]", "annotated_tactic": ["simp only [<a>nnnorm_mul</a>]", [{"full_name": "nnnorm_mul", "def_path": "Mathlib/Analysis/Normed/Field/Basic.lean", "def_pos": [521, 9], "def_end_pos": [521, 19]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b4\ninst\u271d\u00b2 : NormedAddCommGroup \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b3\nF : Type u_5\ninst\u271d : NormedDivisionRing F\nf g : \u03b1 \u2192 F\nhint : Integrable g\nhm : AEStronglyMeasurable f \u03bc\nh\u03b1 : Nonempty \u03b1\nC : \u211d\nhC : \u2200 (x : \u03b1), \u2016f x\u2016 \u2264 C\nhCnonneg : 0 \u2264 C\nx : \u03b1\n\u22a2 (fun x => \u2016f x * g x\u2016\u208a) x \u2264 (fun x => { val := C, property := hCnonneg } * \u2016g x\u2016\u208a) x", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b4\ninst\u271d\u00b2 : NormedAddCommGroup \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b3\nF : Type u_5\ninst\u271d : NormedDivisionRing F\nf g : \u03b1 \u2192 F\nhint : Integrable g\nhm : AEStronglyMeasurable f \u03bc\nh\u03b1 : Nonempty \u03b1\nC : \u211d\nhC : \u2200 (x : \u03b1), \u2016f x\u2016 \u2264 C\nhCnonneg : 0 \u2264 C\nx : \u03b1\n\u22a2 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a \u2264 { val := C, property := hCnonneg } * \u2016g x\u2016\u208a"}, {"tactic": "exact mul_le_mul_of_nonneg_right (hC x) (zero_le _)", "annotated_tactic": ["exact <a>mul_le_mul_of_nonneg_right</a> (hC x) (<a>zero_le</a> _)", [{"full_name": "mul_le_mul_of_nonneg_right", "def_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "def_pos": [156, 9], "def_end_pos": [156, 35]}, {"full_name": "zero_le", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [217, 30], "def_end_pos": [217, 37]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b4\ninst\u271d\u00b2 : NormedAddCommGroup \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b3\nF : Type u_5\ninst\u271d : NormedDivisionRing F\nf g : \u03b1 \u2192 F\nhint : Integrable g\nhm : AEStronglyMeasurable f \u03bc\nh\u03b1 : Nonempty \u03b1\nC : \u211d\nhC : \u2200 (x : \u03b1), \u2016f x\u2016 \u2264 C\nhCnonneg : 0 \u2264 C\nx : \u03b1\n\u22a2 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a \u2264 { val := C, property := hCnonneg } * \u2016g x\u2016\u208a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Array/Lemmas.lean", "full_name": "Array.reverse_data", "start": [230, 9], "end": [265, 26], "traced_tactics": [{"tactic": "simp only [reverse]", "annotated_tactic": ["simp only [<a>reverse</a>]", [{"full_name": "Array.reverse", "def_path": "lake-packages/lean4/src/lean/Init/Data/Array/Basic.lean", "def_pos": [645, 5], "def_end_pos": [645, 12]}]], "state_before": "\u03b1 : Type u_1\na : Array \u03b1\n\u22a2 (reverse a).data = List.reverse a.data", "state_after": "\u03b1 : Type u_1\na : Array \u03b1\n\u22a2 (if h : size a \u2264 1 then a\n      else\n        reverse.loop a 0 { val := size a - 1, isLt := (_ : Nat.pred (Nat.sub (size a) 0) < Nat.sub (size a) 0) }).data =\n    List.reverse a.data"}, {"tactic": "split", "annotated_tactic": ["split", []], "state_before": "\u03b1 : Type u_1\na : Array \u03b1\n\u22a2 (if h : size a \u2264 1 then a\n      else\n        reverse.loop a 0 { val := size a - 1, isLt := (_ : Nat.pred (Nat.sub (size a) 0) < Nat.sub (size a) 0) }).data =\n    List.reverse a.data", "state_after": "case inl\n\u03b1 : Type u_1\na : Array \u03b1\nh\u271d : size a \u2264 1\n\u22a2 a.data = List.reverse a.data\n\ncase inr\n\u03b1 : Type u_1\na : Array \u03b1\nh\u271d : \u00acsize a \u2264 1\n\u22a2 (reverse.loop a 0 { val := size a - 1, isLt := (_ : Nat.pred (Nat.sub (size a) 0) < Nat.sub (size a) 0) }).data =\n    List.reverse a.data"}, {"tactic": "rw [reverse.loop]", "annotated_tactic": ["rw [<a>reverse.loop</a>]", [{"full_name": "Array.reverse.loop", "def_path": "lake-packages/lean4/src/lean/Init/Data/Array/Basic.lean", "def_pos": [651, 3], "def_end_pos": [651, 7]}]], "state_before": "\u03b1 : Type u_1\na as : Array \u03b1\ni j : Nat\nhj : j < size as\nh : i + j + 1 = size a\nh\u2082 : size as = size a\nH : \u2200 (k : Nat), List.get? as.data k = if i \u2264 k \u2227 k \u2264 j then List.get? a.data k else List.get? (List.reverse a.data) k\nk : Nat\n\u22a2 List.get? (reverse.loop as i { val := j, isLt := hj }).data k = List.get? (List.reverse a.data) k", "state_after": "\u03b1 : Type u_1\na as : Array \u03b1\ni j : Nat\nhj : j < size as\nh : i + j + 1 = size a\nh\u2082 : size as = size a\nH : \u2200 (k : Nat), List.get? as.data k = if i \u2264 k \u2227 k \u2264 j then List.get? a.data k else List.get? (List.reverse a.data) k\nk : Nat\n\u22a2 List.get?\n      (if h : i < { val := j, isLt := hj }.val then\n          let_fun this := (_ : { val := j, isLt := hj }.val - 1 - (i + 1) < { val := j, isLt := hj }.val - i);\n          let as_1 := swap as { val := i, isLt := (_ : i < size as) } { val := j, isLt := hj };\n          let_fun this := (_ : { val := j, isLt := hj }.val - 1 < size as_1);\n          reverse.loop as_1 (i + 1) { val := { val := j, isLt := hj }.val - 1, isLt := this }\n        else as).data\n      k =\n    List.get? (List.reverse a.data) k"}, {"tactic": "dsimp", "annotated_tactic": ["dsimp", []], "state_before": "\u03b1 : Type u_1\na as : Array \u03b1\ni j : Nat\nhj : j < size as\nh : i + j + 1 = size a\nh\u2082 : size as = size a\nH : \u2200 (k : Nat), List.get? as.data k = if i \u2264 k \u2227 k \u2264 j then List.get? a.data k else List.get? (List.reverse a.data) k\nk : Nat\n\u22a2 List.get?\n      (if h : i < { val := j, isLt := hj }.val then\n          let_fun this := (_ : { val := j, isLt := hj }.val - 1 - (i + 1) < { val := j, isLt := hj }.val - i);\n          let as_1 := swap as { val := i, isLt := (_ : i < size as) } { val := j, isLt := hj };\n          let_fun this := (_ : { val := j, isLt := hj }.val - 1 < size as_1);\n          reverse.loop as_1 (i + 1) { val := { val := j, isLt := hj }.val - 1, isLt := this }\n        else as).data\n      k =\n    List.get? (List.reverse a.data) k", "state_after": "\u03b1 : Type u_1\na as : Array \u03b1\ni j : Nat\nhj : j < size as\nh : i + j + 1 = size a\nh\u2082 : size as = size a\nH : \u2200 (k : Nat), List.get? as.data k = if i \u2264 k \u2227 k \u2264 j then List.get? a.data k else List.get? (List.reverse a.data) k\nk : Nat\n\u22a2 List.get?\n      (if h : i < j then\n          reverse.loop (swap as { val := i, isLt := (_ : i < size as) } { val := j, isLt := hj }) (i + 1)\n            { val := j - 1,\n              isLt := (_ : j - 1 < size (swap as { val := i, isLt := (_ : i < size as) } { val := j, isLt := hj })) }\n        else as).data\n      k =\n    List.get? (List.reverse a.data) k"}, {"tactic": "split <;> rename_i h\u2081", "annotated_tactic": ["split <;> rename_i h\u2081", []], "state_before": "\u03b1 : Type u_1\na as : Array \u03b1\ni j : Nat\nhj : j < size as\nh : i + j + 1 = size a\nh\u2082 : size as = size a\nH : \u2200 (k : Nat), List.get? as.data k = if i \u2264 k \u2227 k \u2264 j then List.get? a.data k else List.get? (List.reverse a.data) k\nk : Nat\n\u22a2 List.get?\n      (if h : i < j then\n          reverse.loop (swap as { val := i, isLt := (_ : i < size as) } { val := j, isLt := hj }) (i + 1)\n            { val := j - 1,\n              isLt := (_ : j - 1 < size (swap as { val := i, isLt := (_ : i < size as) } { val := j, isLt := hj })) }\n        else as).data\n      k =\n    List.get? (List.reverse a.data) k", "state_after": "case inl\n\u03b1 : Type u_1\na as : Array \u03b1\ni j : Nat\nhj : j < size as\nh : i + j + 1 = size a\nh\u2082 : size as = size a\nH : \u2200 (k : Nat), List.get? as.data k = if i \u2264 k \u2227 k \u2264 j then List.get? a.data k else List.get? (List.reverse a.data) k\nk : Nat\nh\u2081 : i < j\n\u22a2 List.get?\n      (reverse.loop (swap as { val := i, isLt := (_ : i < size as) } { val := j, isLt := hj }) (i + 1)\n          { val := j - 1,\n            isLt :=\n              (_ : j - 1 < size (swap as { val := i, isLt := (_ : i < size as) } { val := j, isLt := hj })) }).data\n      k =\n    List.get? (List.reverse a.data) k\n\ncase inr\n\u03b1 : Type u_1\na as : Array \u03b1\ni j : Nat\nhj : j < size as\nh : i + j + 1 = size a\nh\u2082 : size as = size a\nH : \u2200 (k : Nat), List.get? as.data k = if i \u2264 k \u2227 k \u2264 j then List.get? a.data k else List.get? (List.reverse a.data) k\nk : Nat\nh\u2081 : \u00aci < j\n\u22a2 List.get? as.data k = List.get? (List.reverse a.data) k"}, {"tactic": "have := reverse.termination h\u2081", "annotated_tactic": ["have := <a>reverse.termination</a> h\u2081", [{"full_name": "Array.reverse.termination", "def_path": "lake-packages/lean4/src/lean/Init/Data/Array/Basic.lean", "def_pos": [641, 9], "def_end_pos": [641, 28]}]], "state_before": "case inl\n\u03b1 : Type u_1\na as : Array \u03b1\ni j : Nat\nhj : j < size as\nh : i + j + 1 = size a\nh\u2082 : size as = size a\nH : \u2200 (k : Nat), List.get? as.data k = if i \u2264 k \u2227 k \u2264 j then List.get? a.data k else List.get? (List.reverse a.data) k\nk : Nat\nh\u2081 : i < j\n\u22a2 List.get?\n      (reverse.loop (swap as { val := i, isLt := (_ : i < size as) } { val := j, isLt := hj }) (i + 1)\n          { val := j - 1,\n            isLt :=\n              (_ : j - 1 < size (swap as { val := i, isLt := (_ : i < size as) } { val := j, isLt := hj })) }).data\n      k =\n    List.get? (List.reverse a.data) k", "state_after": "case inl\n\u03b1 : Type u_1\na as : Array \u03b1\ni j : Nat\nhj : j < size as\nh : i + j + 1 = size a\nh\u2082 : size as = size a\nH : \u2200 (k : Nat), List.get? as.data k = if i \u2264 k \u2227 k \u2264 j then List.get? a.data k else List.get? (List.reverse a.data) k\nk : Nat\nh\u2081 : i < j\nthis : j - 1 - (i + 1) < j - i\n\u22a2 List.get?\n      (reverse.loop (swap as { val := i, isLt := (_ : i < size as) } { val := j, isLt := hj }) (i + 1)\n          { val := j - 1,\n            isLt :=\n              (_ : j - 1 < size (swap as { val := i, isLt := (_ : i < size as) } { val := j, isLt := hj })) }).data\n      k =\n    List.get? (List.reverse a.data) k"}, {"tactic": "match j with | j+1 => ?_", "annotated_tactic": ["match j with | j+1 => ?_", []], "state_before": "case inl\n\u03b1 : Type u_1\na as : Array \u03b1\ni j : Nat\nhj : j < size as\nh : i + j + 1 = size a\nh\u2082 : size as = size a\nH : \u2200 (k : Nat), List.get? as.data k = if i \u2264 k \u2227 k \u2264 j then List.get? a.data k else List.get? (List.reverse a.data) k\nk : Nat\nh\u2081 : i < j\nthis : j - 1 - (i + 1) < j - i\n\u22a2 List.get?\n      (reverse.loop (swap as { val := i, isLt := (_ : i < size as) } { val := j, isLt := hj }) (i + 1)\n          { val := j - 1,\n            isLt :=\n              (_ : j - 1 < size (swap as { val := i, isLt := (_ : i < size as) } { val := j, isLt := hj })) }).data\n      k =\n    List.get? (List.reverse a.data) k", "state_after": "case inl\n\u03b1 : Type u_1\na as : Array \u03b1\ni j\u271d : Nat\nh\u2082 : size as = size a\nk j : Nat\nhj : j + 1 < size as\nh : i + (j + 1) + 1 = size a\nH :\n  \u2200 (k : Nat), List.get? as.data k = if i \u2264 k \u2227 k \u2264 j + 1 then List.get? a.data k else List.get? (List.reverse a.data) k\nh\u2081 : i < j + 1\nthis : j + 1 - 1 - (i + 1) < j + 1 - i\n\u22a2 List.get?\n      (reverse.loop (swap as { val := i, isLt := (_ : i < size as) } { val := j + 1, isLt := hj }) (i + 1)\n          { val := j + 1 - 1,\n            isLt :=\n              (_ :\n                j + 1 - 1 < size (swap as { val := i, isLt := (_ : i < size as) } { val := j + 1, isLt := hj })) }).data\n      k =\n    List.get? (List.reverse a.data) k"}, {"tactic": "simp at *", "annotated_tactic": ["simp at *", []], "state_before": "case inl\n\u03b1 : Type u_1\na as : Array \u03b1\ni j\u271d : Nat\nh\u2082 : size as = size a\nk j : Nat\nhj : j + 1 < size as\nh : i + (j + 1) + 1 = size a\nH :\n  \u2200 (k : Nat), List.get? as.data k = if i \u2264 k \u2227 k \u2264 j + 1 then List.get? a.data k else List.get? (List.reverse a.data) k\nh\u2081 : i < j + 1\nthis : j + 1 - 1 - (i + 1) < j + 1 - i\n\u22a2 List.get?\n      (reverse.loop (swap as { val := i, isLt := (_ : i < size as) } { val := j + 1, isLt := hj }) (i + 1)\n          { val := j + 1 - 1,\n            isLt :=\n              (_ :\n                j + 1 - 1 < size (swap as { val := i, isLt := (_ : i < size as) } { val := j + 1, isLt := hj })) }).data\n      k =\n    List.get? (List.reverse a.data) k", "state_after": "case inl\n\u03b1 : Type u_1\na as : Array \u03b1\ni j\u271d : Nat\nh\u2082 : size as = size a\nk j : Nat\nhj : j + 1 < size as\nh : i + (j + 1) + 1 = size a\nH :\n  \u2200 (k : Nat), List.get? as.data k = if i \u2264 k \u2227 k \u2264 j + 1 then List.get? a.data k else List.get? (List.reverse a.data) k\nh\u2081 : i < j + 1\nthis : j - (i + 1) < j + 1 - i\n\u22a2 List.get?\n      (reverse.loop (swap as { val := i, isLt := (_ : i < size as) } { val := j + 1, isLt := hj }) (i + 1)\n          { val := j,\n            isLt :=\n              (_ : j < size (swap as { val := i, isLt := (_ : i < size as) } { val := j + 1, isLt := hj })) }).data\n      k =\n    List.get? (List.reverse a.data) k"}, {"tactic": "rwa [Nat.add_right_comm i]", "annotated_tactic": ["rwa [<a>Nat.add_right_comm</a> i]", [{"full_name": "Nat.add_right_comm", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [145, 19], "def_end_pos": [145, 33]}]], "state_before": "case inl.h\n\u03b1 : Type u_1\na as : Array \u03b1\ni j\u271d : Nat\nh\u2082 : size as = size a\nk j : Nat\nhj : j + 1 < size as\nh : i + (j + 1) + 1 = size a\nH :\n  \u2200 (k : Nat), List.get? as.data k = if i \u2264 k \u2227 k \u2264 j + 1 then List.get? a.data k else List.get? (List.reverse a.data) k\nh\u2081 : i < j + 1\nthis : j - (i + 1) < j + 1 - i\n\u22a2 i + 1 + j + 1 = size a", "state_after": "no goals"}, {"tactic": "simp [size_swap, h\u2082]", "annotated_tactic": ["simp [<a>size_swap</a>, h\u2082]", [{"full_name": "Array.size_swap", "def_path": "lake-packages/lean4/src/lean/Init/Data/Array/Basic.lean", "def_pos": [632, 17], "def_end_pos": [632, 26]}]], "state_before": "case inl.h\u2082\n\u03b1 : Type u_1\na as : Array \u03b1\ni j\u271d : Nat\nh\u2082 : size as = size a\nk j : Nat\nhj : j + 1 < size as\nh : i + (j + 1) + 1 = size a\nH :\n  \u2200 (k : Nat), List.get? as.data k = if i \u2264 k \u2227 k \u2264 j + 1 then List.get? a.data k else List.get? (List.reverse a.data) k\nh\u2081 : i < j + 1\nthis : j - (i + 1) < j + 1 - i\n\u22a2 size (swap as { val := i, isLt := (_ : i < size as) } { val := j + 1, isLt := hj }) = size a", "state_after": "no goals"}, {"tactic": "intro k", "annotated_tactic": ["intro k", []], "state_before": "case inl.H\n\u03b1 : Type u_1\na as : Array \u03b1\ni j\u271d : Nat\nh\u2082 : size as = size a\nk j : Nat\nhj : j + 1 < size as\nh : i + (j + 1) + 1 = size a\nH :\n  \u2200 (k : Nat), List.get? as.data k = if i \u2264 k \u2227 k \u2264 j + 1 then List.get? a.data k else List.get? (List.reverse a.data) k\nh\u2081 : i < j + 1\nthis : j - (i + 1) < j + 1 - i\n\u22a2 \u2200 (k : Nat),\n    List.get? (swap as { val := i, isLt := (_ : i < size as) } { val := j + 1, isLt := hj }).data k =\n      if i + 1 \u2264 k \u2227 k \u2264 j then List.get? a.data k else List.get? (List.reverse a.data) k", "state_after": "case inl.H\n\u03b1 : Type u_1\na as : Array \u03b1\ni j\u271d : Nat\nh\u2082 : size as = size a\nk\u271d j : Nat\nhj : j + 1 < size as\nh : i + (j + 1) + 1 = size a\nH :\n  \u2200 (k : Nat), List.get? as.data k = if i \u2264 k \u2227 k \u2264 j + 1 then List.get? a.data k else List.get? (List.reverse a.data) k\nh\u2081 : i < j + 1\nthis : j - (i + 1) < j + 1 - i\nk : Nat\n\u22a2 List.get? (swap as { val := i, isLt := (_ : i < size as) } { val := j + 1, isLt := hj }).data k =\n    if i + 1 \u2264 k \u2227 k \u2264 j then List.get? a.data k else List.get? (List.reverse a.data) k"}, {"tactic": "rw [\u2190 getElem?_eq_data_get?, get?_swap]", "annotated_tactic": ["rw [\u2190 <a>getElem?_eq_data_get?</a>, <a>get?_swap</a>]", [{"full_name": "Array.getElem?_eq_data_get?", "def_path": "lake-packages/std/Std/Data/Array/Lemmas.lean", "def_pos": [56, 9], "def_end_pos": [56, 30]}, {"full_name": "Array.get?_swap", "def_path": "lake-packages/std/Std/Data/Array/Lemmas.lean", "def_pos": [122, 9], "def_end_pos": [122, 18]}]], "state_before": "case inl.H\n\u03b1 : Type u_1\na as : Array \u03b1\ni j\u271d : Nat\nh\u2082 : size as = size a\nk\u271d j : Nat\nhj : j + 1 < size as\nh : i + (j + 1) + 1 = size a\nH :\n  \u2200 (k : Nat), List.get? as.data k = if i \u2264 k \u2227 k \u2264 j + 1 then List.get? a.data k else List.get? (List.reverse a.data) k\nh\u2081 : i < j + 1\nthis : j - (i + 1) < j + 1 - i\nk : Nat\n\u22a2 List.get? (swap as { val := i, isLt := (_ : i < size as) } { val := j + 1, isLt := hj }).data k =\n    if i + 1 \u2264 k \u2227 k \u2264 j then List.get? a.data k else List.get? (List.reverse a.data) k", "state_after": "case inl.H\n\u03b1 : Type u_1\na as : Array \u03b1\ni j\u271d : Nat\nh\u2082 : size as = size a\nk\u271d j : Nat\nhj : j + 1 < size as\nh : i + (j + 1) + 1 = size a\nH :\n  \u2200 (k : Nat), List.get? as.data k = if i \u2264 k \u2227 k \u2264 j + 1 then List.get? a.data k else List.get? (List.reverse a.data) k\nh\u2081 : i < j + 1\nthis : j - (i + 1) < j + 1 - i\nk : Nat\n\u22a2 (if { val := j + 1, isLt := hj }.val = k then some as[{ val := i, isLt := (_ : i < size as) }.val]\n    else\n      if { val := i, isLt := (_ : i < size as) }.val = k then some as[{ val := j + 1, isLt := hj }.val] else as[k]?) =\n    if i + 1 \u2264 k \u2227 k \u2264 j then List.get? a.data k else List.get? (List.reverse a.data) k"}, {"tactic": "simp [getElem?_eq_data_get?, getElem_eq_data_get, \u2190 List.get?_eq_get, H, Nat.le_of_lt h\u2081]", "annotated_tactic": ["simp [<a>getElem?_eq_data_get?</a>, <a>getElem_eq_data_get</a>, \u2190 <a>List.get?_eq_get</a>, H, <a>Nat.le_of_lt</a> h\u2081]", [{"full_name": "Array.getElem?_eq_data_get?", "def_path": "lake-packages/std/Std/Data/Array/Lemmas.lean", "def_pos": [56, 9], "def_end_pos": [56, 30]}, {"full_name": "Array.getElem_eq_data_get", "def_path": "lake-packages/std/Std/Data/Array/Init/Lemmas.lean", "def_pos": [28, 9], "def_end_pos": [28, 28]}, {"full_name": "List.get?_eq_get", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [581, 9], "def_end_pos": [581, 20]}, {"full_name": "Nat.le_of_lt", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [283, 19], "def_end_pos": [283, 27]}]], "state_before": "case inl.H\n\u03b1 : Type u_1\na as : Array \u03b1\ni j\u271d : Nat\nh\u2082 : size as = size a\nk\u271d j : Nat\nhj : j + 1 < size as\nh : i + (j + 1) + 1 = size a\nH :\n  \u2200 (k : Nat), List.get? as.data k = if i \u2264 k \u2227 k \u2264 j + 1 then List.get? a.data k else List.get? (List.reverse a.data) k\nh\u2081 : i < j + 1\nthis : j - (i + 1) < j + 1 - i\nk : Nat\n\u22a2 (if { val := j + 1, isLt := hj }.val = k then some as[{ val := i, isLt := (_ : i < size as) }.val]\n    else\n      if { val := i, isLt := (_ : i < size as) }.val = k then some as[{ val := j + 1, isLt := hj }.val] else as[k]?) =\n    if i + 1 \u2264 k \u2227 k \u2264 j then List.get? a.data k else List.get? (List.reverse a.data) k", "state_after": "case inl.H\n\u03b1 : Type u_1\na as : Array \u03b1\ni j\u271d : Nat\nh\u2082 : size as = size a\nk\u271d j : Nat\nhj : j + 1 < size as\nh : i + (j + 1) + 1 = size a\nH :\n  \u2200 (k : Nat), List.get? as.data k = if i \u2264 k \u2227 k \u2264 j + 1 then List.get? a.data k else List.get? (List.reverse a.data) k\nh\u2081 : i < j + 1\nthis : j - (i + 1) < j + 1 - i\nk : Nat\n\u22a2 (if j + 1 = k then List.get? a.data i\n    else\n      if i = k then List.get? a.data (j + 1)\n      else if i \u2264 k \u2227 k \u2264 j + 1 then List.get? a.data k else List.get? (List.reverse a.data) k) =\n    if i + 1 \u2264 k \u2227 k \u2264 j then List.get? a.data k else List.get? (List.reverse a.data) k"}, {"tactic": "split <;> rename_i h\u2082", "annotated_tactic": ["split <;> rename_i h\u2082", []], "state_before": "case inl.H\n\u03b1 : Type u_1\na as : Array \u03b1\ni j\u271d : Nat\nh\u2082 : size as = size a\nk\u271d j : Nat\nhj : j + 1 < size as\nh : i + (j + 1) + 1 = size a\nH :\n  \u2200 (k : Nat), List.get? as.data k = if i \u2264 k \u2227 k \u2264 j + 1 then List.get? a.data k else List.get? (List.reverse a.data) k\nh\u2081 : i < j + 1\nthis : j - (i + 1) < j + 1 - i\nk : Nat\n\u22a2 (if j + 1 = k then List.get? a.data i\n    else\n      if i = k then List.get? a.data (j + 1)\n      else if i \u2264 k \u2227 k \u2264 j + 1 then List.get? a.data k else List.get? (List.reverse a.data) k) =\n    if i + 1 \u2264 k \u2227 k \u2264 j then List.get? a.data k else List.get? (List.reverse a.data) k", "state_after": "case inl.H.inl\n\u03b1 : Type u_1\na as : Array \u03b1\ni j\u271d : Nat\nh\u2082\u271d : size as = size a\nk\u271d j : Nat\nhj : j + 1 < size as\nh : i + (j + 1) + 1 = size a\nH :\n  \u2200 (k : Nat), List.get? as.data k = if i \u2264 k \u2227 k \u2264 j + 1 then List.get? a.data k else List.get? (List.reverse a.data) k\nh\u2081 : i < j + 1\nthis : j - (i + 1) < j + 1 - i\nk : Nat\nh\u2082 : j + 1 = k\n\u22a2 List.get? a.data i = if i + 1 \u2264 k \u2227 k \u2264 j then List.get? a.data k else List.get? (List.reverse a.data) k\n\ncase inl.H.inr\n\u03b1 : Type u_1\na as : Array \u03b1\ni j\u271d : Nat\nh\u2082\u271d : size as = size a\nk\u271d j : Nat\nhj : j + 1 < size as\nh : i + (j + 1) + 1 = size a\nH :\n  \u2200 (k : Nat), List.get? as.data k = if i \u2264 k \u2227 k \u2264 j + 1 then List.get? a.data k else List.get? (List.reverse a.data) k\nh\u2081 : i < j + 1\nthis : j - (i + 1) < j + 1 - i\nk : Nat\nh\u2082 : \u00acj + 1 = k\n\u22a2 (if i = k then List.get? a.data (j + 1)\n    else if i \u2264 k \u2227 k \u2264 j + 1 then List.get? a.data k else List.get? (List.reverse a.data) k) =\n    if i + 1 \u2264 k \u2227 k \u2264 j then List.get? a.data k else List.get? (List.reverse a.data) k"}, {"tactic": "split <;> rename_i h\u2083", "annotated_tactic": ["split <;> rename_i h\u2083", []], "state_before": "case inl.H.inr\n\u03b1 : Type u_1\na as : Array \u03b1\ni j\u271d : Nat\nh\u2082\u271d : size as = size a\nk\u271d j : Nat\nhj : j + 1 < size as\nh : i + (j + 1) + 1 = size a\nH :\n  \u2200 (k : Nat), List.get? as.data k = if i \u2264 k \u2227 k \u2264 j + 1 then List.get? a.data k else List.get? (List.reverse a.data) k\nh\u2081 : i < j + 1\nthis : j - (i + 1) < j + 1 - i\nk : Nat\nh\u2082 : \u00acj + 1 = k\n\u22a2 (if i = k then List.get? a.data (j + 1)\n    else if i \u2264 k \u2227 k \u2264 j + 1 then List.get? a.data k else List.get? (List.reverse a.data) k) =\n    if i + 1 \u2264 k \u2227 k \u2264 j then List.get? a.data k else List.get? (List.reverse a.data) k", "state_after": "case inl.H.inr.inl\n\u03b1 : Type u_1\na as : Array \u03b1\ni j\u271d : Nat\nh\u2082\u271d : size as = size a\nk\u271d j : Nat\nhj : j + 1 < size as\nh : i + (j + 1) + 1 = size a\nH :\n  \u2200 (k : Nat), List.get? as.data k = if i \u2264 k \u2227 k \u2264 j + 1 then List.get? a.data k else List.get? (List.reverse a.data) k\nh\u2081 : i < j + 1\nthis : j - (i + 1) < j + 1 - i\nk : Nat\nh\u2082 : \u00acj + 1 = k\nh\u2083 : i = k\n\u22a2 List.get? a.data (j + 1) = if i + 1 \u2264 k \u2227 k \u2264 j then List.get? a.data k else List.get? (List.reverse a.data) k\n\ncase inl.H.inr.inr\n\u03b1 : Type u_1\na as : Array \u03b1\ni j\u271d : Nat\nh\u2082\u271d : size as = size a\nk\u271d j : Nat\nhj : j + 1 < size as\nh : i + (j + 1) + 1 = size a\nH :\n  \u2200 (k : Nat), List.get? as.data k = if i \u2264 k \u2227 k \u2264 j + 1 then List.get? a.data k else List.get? (List.reverse a.data) k\nh\u2081 : i < j + 1\nthis : j - (i + 1) < j + 1 - i\nk : Nat\nh\u2082 : \u00acj + 1 = k\nh\u2083 : \u00aci = k\n\u22a2 (if i \u2264 k \u2227 k \u2264 j + 1 then List.get? a.data k else List.get? (List.reverse a.data) k) =\n    if i + 1 \u2264 k \u2227 k \u2264 j then List.get? a.data k else List.get? (List.reverse a.data) k"}, {"tactic": "simp only [Nat.succ_le, Nat.lt_iff_le_and_ne.trans (and_iff_left h\u2083),\n  Nat.lt_succ.symm.trans (Nat.lt_iff_le_and_ne.trans (and_iff_left (Ne.symm h\u2082)))]", "annotated_tactic": ["simp only [<a>Nat.succ_le</a>, Nat.lt_iff_le_and_ne.trans (<a>and_iff_left</a> h\u2083),\n          Nat.lt_succ.symm.trans (Nat.lt_iff_le_and_ne.trans (<a>and_iff_left</a> (<a>Ne.symm</a> h\u2082)))]", [{"full_name": "Nat.succ_le", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [221, 9], "def_end_pos": [221, 16]}, {"full_name": "and_iff_left", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [204, 9], "def_end_pos": [204, 21]}, {"full_name": "and_iff_left", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [204, 9], "def_end_pos": [204, 21]}, {"full_name": "Ne.symm", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [575, 9], "def_end_pos": [575, 16]}]], "state_before": "case inl.H.inr.inr\n\u03b1 : Type u_1\na as : Array \u03b1\ni j\u271d : Nat\nh\u2082\u271d : size as = size a\nk\u271d j : Nat\nhj : j + 1 < size as\nh : i + (j + 1) + 1 = size a\nH :\n  \u2200 (k : Nat), List.get? as.data k = if i \u2264 k \u2227 k \u2264 j + 1 then List.get? a.data k else List.get? (List.reverse a.data) k\nh\u2081 : i < j + 1\nthis : j - (i + 1) < j + 1 - i\nk : Nat\nh\u2082 : \u00acj + 1 = k\nh\u2083 : \u00aci = k\n\u22a2 (if i \u2264 k \u2227 k \u2264 j + 1 then List.get? a.data k else List.get? (List.reverse a.data) k) =\n    if i + 1 \u2264 k \u2227 k \u2264 j then List.get? a.data k else List.get? (List.reverse a.data) k", "state_after": "no goals"}, {"tactic": "simp [\u2190 h\u2082, Nat.not_le.2 (Nat.lt_succ_self _)]", "annotated_tactic": ["simp [\u2190 h\u2082, <a>Nat.not_le</a>.2 (<a>Nat.lt_succ_self</a> _)]", [{"full_name": "Nat.not_le", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [147, 27], "def_end_pos": [147, 33]}, {"full_name": "Nat.lt_succ_self", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [294, 9], "def_end_pos": [294, 21]}]], "state_before": "case inl.H.inl\n\u03b1 : Type u_1\na as : Array \u03b1\ni j\u271d : Nat\nh\u2082\u271d : size as = size a\nk\u271d j : Nat\nhj : j + 1 < size as\nh : i + (j + 1) + 1 = size a\nH :\n  \u2200 (k : Nat), List.get? as.data k = if i \u2264 k \u2227 k \u2264 j + 1 then List.get? a.data k else List.get? (List.reverse a.data) k\nh\u2081 : i < j + 1\nthis : j - (i + 1) < j + 1 - i\nk : Nat\nh\u2082 : j + 1 = k\n\u22a2 List.get? a.data i = if i + 1 \u2264 k \u2227 k \u2264 j then List.get? a.data k else List.get? (List.reverse a.data) k", "state_after": "case inl.H.inl\n\u03b1 : Type u_1\na as : Array \u03b1\ni j\u271d : Nat\nh\u2082\u271d : size as = size a\nk\u271d j : Nat\nhj : j + 1 < size as\nh : i + (j + 1) + 1 = size a\nH :\n  \u2200 (k : Nat), List.get? as.data k = if i \u2264 k \u2227 k \u2264 j + 1 then List.get? a.data k else List.get? (List.reverse a.data) k\nh\u2081 : i < j + 1\nthis : j - (i + 1) < j + 1 - i\nk : Nat\nh\u2082 : j + 1 = k\n\u22a2 List.get? a.data i = List.get? (List.reverse a.data) (j + 1)"}, {"tactic": "exact (List.get?_reverse' _ _ (Eq.trans (by simp_arith) h)).symm", "annotated_tactic": ["exact (<a>List.get?_reverse'</a> _ _ (<a>Eq.trans</a> (by simp_arith) h)).<a>symm</a>", [{"full_name": "List.get?_reverse'", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [753, 9], "def_end_pos": [753, 22]}, {"full_name": "Eq.trans", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [322, 9], "def_end_pos": [322, 17]}, {"full_name": "Eq.symm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [310, 9], "def_end_pos": [310, 16]}]], "state_before": "case inl.H.inl\n\u03b1 : Type u_1\na as : Array \u03b1\ni j\u271d : Nat\nh\u2082\u271d : size as = size a\nk\u271d j : Nat\nhj : j + 1 < size as\nh : i + (j + 1) + 1 = size a\nH :\n  \u2200 (k : Nat), List.get? as.data k = if i \u2264 k \u2227 k \u2264 j + 1 then List.get? a.data k else List.get? (List.reverse a.data) k\nh\u2081 : i < j + 1\nthis : j - (i + 1) < j + 1 - i\nk : Nat\nh\u2082 : j + 1 = k\n\u22a2 List.get? a.data i = List.get? (List.reverse a.data) (j + 1)", "state_after": "no goals"}, {"tactic": "simp_arith", "annotated_tactic": ["simp_arith", []], "state_before": "\u03b1 : Type u_1\na as : Array \u03b1\ni j\u271d : Nat\nh\u2082\u271d : size as = size a\nk\u271d j : Nat\nhj : j + 1 < size as\nh : i + (j + 1) + 1 = size a\nH :\n  \u2200 (k : Nat), List.get? as.data k = if i \u2264 k \u2227 k \u2264 j + 1 then List.get? a.data k else List.get? (List.reverse a.data) k\nh\u2081 : i < j + 1\nthis : j - (i + 1) < j + 1 - i\nk : Nat\nh\u2082 : j + 1 = k\n\u22a2 j + 1 + i + 1 = i + (j + 1) + 1", "state_after": "no goals"}, {"tactic": "simp [\u2190 h\u2083, Nat.not_le.2 (Nat.lt_succ_self _)]", "annotated_tactic": ["simp [\u2190 h\u2083, <a>Nat.not_le</a>.2 (<a>Nat.lt_succ_self</a> _)]", [{"full_name": "Nat.not_le", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [147, 27], "def_end_pos": [147, 33]}, {"full_name": "Nat.lt_succ_self", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [294, 9], "def_end_pos": [294, 21]}]], "state_before": "case inl.H.inr.inl\n\u03b1 : Type u_1\na as : Array \u03b1\ni j\u271d : Nat\nh\u2082\u271d : size as = size a\nk\u271d j : Nat\nhj : j + 1 < size as\nh : i + (j + 1) + 1 = size a\nH :\n  \u2200 (k : Nat), List.get? as.data k = if i \u2264 k \u2227 k \u2264 j + 1 then List.get? a.data k else List.get? (List.reverse a.data) k\nh\u2081 : i < j + 1\nthis : j - (i + 1) < j + 1 - i\nk : Nat\nh\u2082 : \u00acj + 1 = k\nh\u2083 : i = k\n\u22a2 List.get? a.data (j + 1) = if i + 1 \u2264 k \u2227 k \u2264 j then List.get? a.data k else List.get? (List.reverse a.data) k", "state_after": "case inl.H.inr.inl\n\u03b1 : Type u_1\na as : Array \u03b1\ni j\u271d : Nat\nh\u2082\u271d : size as = size a\nk\u271d j : Nat\nhj : j + 1 < size as\nh : i + (j + 1) + 1 = size a\nH :\n  \u2200 (k : Nat), List.get? as.data k = if i \u2264 k \u2227 k \u2264 j + 1 then List.get? a.data k else List.get? (List.reverse a.data) k\nh\u2081 : i < j + 1\nthis : j - (i + 1) < j + 1 - i\nk : Nat\nh\u2082 : \u00acj + 1 = k\nh\u2083 : i = k\n\u22a2 List.get? a.data (j + 1) = List.get? (List.reverse a.data) i"}, {"tactic": "exact (List.get?_reverse' _ _ (Eq.trans (by simp_arith) h)).symm", "annotated_tactic": ["exact (<a>List.get?_reverse'</a> _ _ (<a>Eq.trans</a> (by simp_arith) h)).<a>symm</a>", [{"full_name": "List.get?_reverse'", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [753, 9], "def_end_pos": [753, 22]}, {"full_name": "Eq.trans", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [322, 9], "def_end_pos": [322, 17]}, {"full_name": "Eq.symm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [310, 9], "def_end_pos": [310, 16]}]], "state_before": "case inl.H.inr.inl\n\u03b1 : Type u_1\na as : Array \u03b1\ni j\u271d : Nat\nh\u2082\u271d : size as = size a\nk\u271d j : Nat\nhj : j + 1 < size as\nh : i + (j + 1) + 1 = size a\nH :\n  \u2200 (k : Nat), List.get? as.data k = if i \u2264 k \u2227 k \u2264 j + 1 then List.get? a.data k else List.get? (List.reverse a.data) k\nh\u2081 : i < j + 1\nthis : j - (i + 1) < j + 1 - i\nk : Nat\nh\u2082 : \u00acj + 1 = k\nh\u2083 : i = k\n\u22a2 List.get? a.data (j + 1) = List.get? (List.reverse a.data) i", "state_after": "no goals"}, {"tactic": "simp_arith", "annotated_tactic": ["simp_arith", []], "state_before": "\u03b1 : Type u_1\na as : Array \u03b1\ni j\u271d : Nat\nh\u2082\u271d : size as = size a\nk\u271d j : Nat\nhj : j + 1 < size as\nh : i + (j + 1) + 1 = size a\nH :\n  \u2200 (k : Nat), List.get? as.data k = if i \u2264 k \u2227 k \u2264 j + 1 then List.get? a.data k else List.get? (List.reverse a.data) k\nh\u2081 : i < j + 1\nthis : j - (i + 1) < j + 1 - i\nk : Nat\nh\u2082 : \u00acj + 1 = k\nh\u2083 : i = k\n\u22a2 i + (j + 1) + 1 = i + (j + 1) + 1", "state_after": "no goals"}, {"tactic": "rw [H]", "annotated_tactic": ["rw [H]", []], "state_before": "case inr\n\u03b1 : Type u_1\na as : Array \u03b1\ni j : Nat\nhj : j < size as\nh : i + j + 1 = size a\nh\u2082 : size as = size a\nH : \u2200 (k : Nat), List.get? as.data k = if i \u2264 k \u2227 k \u2264 j then List.get? a.data k else List.get? (List.reverse a.data) k\nk : Nat\nh\u2081 : \u00aci < j\n\u22a2 List.get? as.data k = List.get? (List.reverse a.data) k", "state_after": "case inr\n\u03b1 : Type u_1\na as : Array \u03b1\ni j : Nat\nhj : j < size as\nh : i + j + 1 = size a\nh\u2082 : size as = size a\nH : \u2200 (k : Nat), List.get? as.data k = if i \u2264 k \u2227 k \u2264 j then List.get? a.data k else List.get? (List.reverse a.data) k\nk : Nat\nh\u2081 : \u00aci < j\n\u22a2 (if i \u2264 k \u2227 k \u2264 j then List.get? a.data k else List.get? (List.reverse a.data) k) = List.get? (List.reverse a.data) k"}, {"tactic": "split <;> rename_i h\u2082", "annotated_tactic": ["split <;> rename_i h\u2082", []], "state_before": "case inr\n\u03b1 : Type u_1\na as : Array \u03b1\ni j : Nat\nhj : j < size as\nh : i + j + 1 = size a\nh\u2082 : size as = size a\nH : \u2200 (k : Nat), List.get? as.data k = if i \u2264 k \u2227 k \u2264 j then List.get? a.data k else List.get? (List.reverse a.data) k\nk : Nat\nh\u2081 : \u00aci < j\n\u22a2 (if i \u2264 k \u2227 k \u2264 j then List.get? a.data k else List.get? (List.reverse a.data) k) = List.get? (List.reverse a.data) k", "state_after": "case inr.inl\n\u03b1 : Type u_1\na as : Array \u03b1\ni j : Nat\nhj : j < size as\nh : i + j + 1 = size a\nh\u2082\u271d : size as = size a\nH : \u2200 (k : Nat), List.get? as.data k = if i \u2264 k \u2227 k \u2264 j then List.get? a.data k else List.get? (List.reverse a.data) k\nk : Nat\nh\u2081 : \u00aci < j\nh\u2082 : i \u2264 k \u2227 k \u2264 j\n\u22a2 List.get? a.data k = List.get? (List.reverse a.data) k\n\ncase inr.inr\n\u03b1 : Type u_1\na as : Array \u03b1\ni j : Nat\nhj : j < size as\nh : i + j + 1 = size a\nh\u2082\u271d : size as = size a\nH : \u2200 (k : Nat), List.get? as.data k = if i \u2264 k \u2227 k \u2264 j then List.get? a.data k else List.get? (List.reverse a.data) k\nk : Nat\nh\u2081 : \u00aci < j\nh\u2082 : \u00ac(i \u2264 k \u2227 k \u2264 j)\n\u22a2 List.get? (List.reverse a.data) k = List.get? (List.reverse a.data) k"}, {"tactic": "cases Nat.le_antisymm (Nat.not_lt.1 h\u2081) (Nat.le_trans h\u2082.1 h\u2082.2)", "annotated_tactic": ["cases <a>Nat.le_antisymm</a> (<a>Nat.not_lt</a>.1 h\u2081) (<a>Nat.le_trans</a> h\u2082.1 h\u2082.2)", [{"full_name": "Nat.le_antisymm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1665, 19], "def_end_pos": [1665, 34]}, {"full_name": "Nat.not_lt", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [150, 27], "def_end_pos": [150, 33]}, {"full_name": "Nat.le_trans", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1592, 19], "def_end_pos": [1592, 31]}]], "state_before": "case inr.inl\n\u03b1 : Type u_1\na as : Array \u03b1\ni j : Nat\nhj : j < size as\nh : i + j + 1 = size a\nh\u2082\u271d : size as = size a\nH : \u2200 (k : Nat), List.get? as.data k = if i \u2264 k \u2227 k \u2264 j then List.get? a.data k else List.get? (List.reverse a.data) k\nk : Nat\nh\u2081 : \u00aci < j\nh\u2082 : i \u2264 k \u2227 k \u2264 j\n\u22a2 List.get? a.data k = List.get? (List.reverse a.data) k", "state_after": "case inr.inl.refl\n\u03b1 : Type u_1\na as : Array \u03b1\ni : Nat\nh\u2082\u271d : size as = size a\nk : Nat\nhj : i < size as\nh : i + i + 1 = size a\nH : \u2200 (k : Nat), List.get? as.data k = if i \u2264 k \u2227 k \u2264 i then List.get? a.data k else List.get? (List.reverse a.data) k\nh\u2081 : \u00aci < i\nh\u2082 : i \u2264 k \u2227 k \u2264 i\n\u22a2 List.get? a.data k = List.get? (List.reverse a.data) k"}, {"tactic": "cases Nat.le_antisymm h\u2082.1 h\u2082.2", "annotated_tactic": ["cases <a>Nat.le_antisymm</a> h\u2082.1 h\u2082.2", [{"full_name": "Nat.le_antisymm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1665, 19], "def_end_pos": [1665, 34]}]], "state_before": "case inr.inl.refl\n\u03b1 : Type u_1\na as : Array \u03b1\ni : Nat\nh\u2082\u271d : size as = size a\nk : Nat\nhj : i < size as\nh : i + i + 1 = size a\nH : \u2200 (k : Nat), List.get? as.data k = if i \u2264 k \u2227 k \u2264 i then List.get? a.data k else List.get? (List.reverse a.data) k\nh\u2081 : \u00aci < i\nh\u2082 : i \u2264 k \u2227 k \u2264 i\n\u22a2 List.get? a.data k = List.get? (List.reverse a.data) k", "state_after": "case inr.inl.refl.refl\n\u03b1 : Type u_1\na as : Array \u03b1\ni : Nat\nh\u2082\u271d : size as = size a\nhj : i < size as\nh : i + i + 1 = size a\nH : \u2200 (k : Nat), List.get? as.data k = if i \u2264 k \u2227 k \u2264 i then List.get? a.data k else List.get? (List.reverse a.data) k\nh\u2081 : \u00aci < i\nh\u2082 : i \u2264 i \u2227 i \u2264 i\n\u22a2 List.get? a.data i = List.get? (List.reverse a.data) i"}, {"tactic": "exact (List.get?_reverse' _ _ h).symm", "annotated_tactic": ["exact (<a>List.get?_reverse'</a> _ _ h).<a>symm</a>", [{"full_name": "List.get?_reverse'", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [753, 9], "def_end_pos": [753, 22]}, {"full_name": "Eq.symm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [310, 9], "def_end_pos": [310, 16]}]], "state_before": "case inr.inl.refl.refl\n\u03b1 : Type u_1\na as : Array \u03b1\ni : Nat\nh\u2082\u271d : size as = size a\nhj : i < size as\nh : i + i + 1 = size a\nH : \u2200 (k : Nat), List.get? as.data k = if i \u2264 k \u2227 k \u2264 i then List.get? a.data k else List.get? (List.reverse a.data) k\nh\u2081 : \u00aci < i\nh\u2082 : i \u2264 i \u2227 i \u2264 i\n\u22a2 List.get? a.data i = List.get? (List.reverse a.data) i", "state_after": "no goals"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case inr.inr\n\u03b1 : Type u_1\na as : Array \u03b1\ni j : Nat\nhj : j < size as\nh : i + j + 1 = size a\nh\u2082\u271d : size as = size a\nH : \u2200 (k : Nat), List.get? as.data k = if i \u2264 k \u2227 k \u2264 j then List.get? a.data k else List.get? (List.reverse a.data) k\nk : Nat\nh\u2081 : \u00aci < j\nh\u2082 : \u00ac(i \u2264 k \u2227 k \u2264 j)\n\u22a2 List.get? (List.reverse a.data) k = List.get? (List.reverse a.data) k", "state_after": "no goals"}, {"tactic": "match a with | \u27e8[]\u27e9 | \u27e8[_]\u27e9 => rfl", "annotated_tactic": ["match a with | \u27e8[]\u27e9 | \u27e8[_]\u27e9 => rfl", []], "state_before": "case inl\n\u03b1 : Type u_1\na : Array \u03b1\nh\u271d : size a \u2264 1\n\u22a2 a.data = List.reverse a.data", "state_after": "no goals"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\u03b1 : Type u_1\na : Array \u03b1\nhead\u271d : \u03b1\nh\u271d : size { data := [head\u271d] } \u2264 1\n\u22a2 { data := [head\u271d] }.data = List.reverse { data := [head\u271d] }.data", "state_after": "no goals"}, {"tactic": "have := Nat.sub_add_cancel (Nat.le_of_not_le \u2039_\u203a)", "annotated_tactic": ["have := <a>Nat.sub_add_cancel</a> (<a>Nat.le_of_not_le</a> \u2039_\u203a)", [{"full_name": "Nat.sub_add_cancel", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [583, 19], "def_end_pos": [583, 33]}, {"full_name": "Nat.le_of_not_le", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [138, 19], "def_end_pos": [138, 31]}]], "state_before": "case inr\n\u03b1 : Type u_1\na : Array \u03b1\nh\u271d : \u00acsize a \u2264 1\n\u22a2 (reverse.loop a 0 { val := size a - 1, isLt := (_ : Nat.pred (Nat.sub (size a) 0) < Nat.sub (size a) 0) }).data =\n    List.reverse a.data", "state_after": "case inr\n\u03b1 : Type u_1\na : Array \u03b1\nh\u271d : \u00acsize a \u2264 1\nthis : size a - 1 + 1 = size a\n\u22a2 (reverse.loop a 0 { val := size a - 1, isLt := (_ : Nat.pred (Nat.sub (size a) 0) < Nat.sub (size a) 0) }).data =\n    List.reverse a.data"}, {"tactic": "refine List.ext <| go _ _ _ _ (by simp [this]) rfl fun k => ?_", "annotated_tactic": ["refine <a>List.ext</a> <| go _ _ _ _ (by simp [this]) <a>rfl</a> fun k => ?_", [{"full_name": "List.ext", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [736, 16], "def_end_pos": [736, 19]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "case inr\n\u03b1 : Type u_1\na : Array \u03b1\nh\u271d : \u00acsize a \u2264 1\nthis : size a - 1 + 1 = size a\n\u22a2 (reverse.loop a 0 { val := size a - 1, isLt := (_ : Nat.pred (Nat.sub (size a) 0) < Nat.sub (size a) 0) }).data =\n    List.reverse a.data", "state_after": "case inr\n\u03b1 : Type u_1\na : Array \u03b1\nh\u271d : \u00acsize a \u2264 1\nthis : size a - 1 + 1 = size a\nk : Nat\n\u22a2 List.get? a.data k = if 0 \u2264 k \u2227 k \u2264 size a - 1 then List.get? a.data k else List.get? (List.reverse a.data) k"}, {"tactic": "split", "annotated_tactic": ["split", []], "state_before": "case inr\n\u03b1 : Type u_1\na : Array \u03b1\nh\u271d : \u00acsize a \u2264 1\nthis : size a - 1 + 1 = size a\nk : Nat\n\u22a2 List.get? a.data k = if 0 \u2264 k \u2227 k \u2264 size a - 1 then List.get? a.data k else List.get? (List.reverse a.data) k", "state_after": "case inr.inl\n\u03b1 : Type u_1\na : Array \u03b1\nh\u271d\u00b9 : \u00acsize a \u2264 1\nthis : size a - 1 + 1 = size a\nk : Nat\nh\u271d : 0 \u2264 k \u2227 k \u2264 size a - 1\n\u22a2 List.get? a.data k = List.get? a.data k\n\ncase inr.inr\n\u03b1 : Type u_1\na : Array \u03b1\nh\u271d\u00b9 : \u00acsize a \u2264 1\nthis : size a - 1 + 1 = size a\nk : Nat\nh\u271d : \u00ac(0 \u2264 k \u2227 k \u2264 size a - 1)\n\u22a2 List.get? a.data k = List.get? (List.reverse a.data) k"}, {"tactic": "{rfl}", "annotated_tactic": ["{rfl}", []], "state_before": "case inr.inl\n\u03b1 : Type u_1\na : Array \u03b1\nh\u271d\u00b9 : \u00acsize a \u2264 1\nthis : size a - 1 + 1 = size a\nk : Nat\nh\u271d : 0 \u2264 k \u2227 k \u2264 size a - 1\n\u22a2 List.get? a.data k = List.get? a.data k\n\ncase inr.inr\n\u03b1 : Type u_1\na : Array \u03b1\nh\u271d\u00b9 : \u00acsize a \u2264 1\nthis : size a - 1 + 1 = size a\nk : Nat\nh\u271d : \u00ac(0 \u2264 k \u2227 k \u2264 size a - 1)\n\u22a2 List.get? a.data k = List.get? (List.reverse a.data) k", "state_after": "case inr.inr\n\u03b1 : Type u_1\na : Array \u03b1\nh\u271d\u00b9 : \u00acsize a \u2264 1\nthis : size a - 1 + 1 = size a\nk : Nat\nh\u271d : \u00ac(0 \u2264 k \u2227 k \u2264 size a - 1)\n\u22a2 List.get? a.data k = List.get? (List.reverse a.data) k"}, {"tactic": "rename_i h", "annotated_tactic": ["rename_i h", []], "state_before": "case inr.inr\n\u03b1 : Type u_1\na : Array \u03b1\nh\u271d\u00b9 : \u00acsize a \u2264 1\nthis : size a - 1 + 1 = size a\nk : Nat\nh\u271d : \u00ac(0 \u2264 k \u2227 k \u2264 size a - 1)\n\u22a2 List.get? a.data k = List.get? (List.reverse a.data) k", "state_after": "case inr.inr\n\u03b1 : Type u_1\na : Array \u03b1\nh\u271d : \u00acsize a \u2264 1\nthis : size a - 1 + 1 = size a\nk : Nat\nh : \u00ac(0 \u2264 k \u2227 k \u2264 size a - 1)\n\u22a2 List.get? a.data k = List.get? (List.reverse a.data) k"}, {"tactic": "simp [\u2190 show k < _ + 1 \u2194 _ from Nat.lt_succ (n := a.size - 1), this] at h", "annotated_tactic": ["simp [\u2190 show k < _ + 1 \u2194 _ from <a>Nat.lt_succ</a> (n := a.size - 1), this] at h", [{"full_name": "Nat.lt_succ", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [223, 9], "def_end_pos": [223, 16]}]], "state_before": "case inr.inr\n\u03b1 : Type u_1\na : Array \u03b1\nh\u271d : \u00acsize a \u2264 1\nthis : size a - 1 + 1 = size a\nk : Nat\nh : \u00ac(0 \u2264 k \u2227 k \u2264 size a - 1)\n\u22a2 List.get? a.data k = List.get? (List.reverse a.data) k", "state_after": "case inr.inr\n\u03b1 : Type u_1\na : Array \u03b1\nh\u271d : \u00acsize a \u2264 1\nthis : size a - 1 + 1 = size a\nk : Nat\nh : size a \u2264 k\n\u22a2 List.get? a.data k = List.get? (List.reverse a.data) k"}, {"tactic": "rw [List.get?_eq_none.2 \u2039_\u203a, List.get?_eq_none.2 (a.data.length_reverse \u25b8 \u2039_\u203a)]", "annotated_tactic": ["rw [<a>List.get?_eq_none</a>.2 \u2039_\u203a, <a>List.get?_eq_none</a>.2 (a.data.length_reverse \u25b8 \u2039_\u203a)]", [{"full_name": "List.get?_eq_none", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [606, 17], "def_end_pos": [606, 29]}, {"full_name": "List.get?_eq_none", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [606, 17], "def_end_pos": [606, 29]}]], "state_before": "case inr.inr\n\u03b1 : Type u_1\na : Array \u03b1\nh\u271d : \u00acsize a \u2264 1\nthis : size a - 1 + 1 = size a\nk : Nat\nh : size a \u2264 k\n\u22a2 List.get? a.data k = List.get? (List.reverse a.data) k", "state_after": "no goals"}, {"tactic": "simp [this]", "annotated_tactic": ["simp [this]", []], "state_before": "\u03b1 : Type u_1\na : Array \u03b1\nh\u271d : \u00acsize a \u2264 1\nthis : size a - 1 + 1 = size a\n\u22a2 0 + (size a - 1) + 1 = size a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "full_name": "MeasureTheory.L1.SimpleFunc.integral_eq_norm_posPart_sub", "start": [606, 1], "end": [627, 58], "traced_tactics": [{"tactic": "have ae_eq\u2081 : (toSimpleFunc f).posPart =\u1d50[\u03bc] (toSimpleFunc (posPart f)).map norm := by\n  filter_upwards [posPart_toSimpleFunc f] with _ h\n  rw [SimpleFunc.map_apply, h]\n  conv_lhs => rw [\u2190 SimpleFunc.posPart_map_norm, SimpleFunc.map_apply]", "annotated_tactic": ["have ae_eq\u2081 : (<a>toSimpleFunc</a> f).<a>posPart</a> =\u1d50[\u03bc] (<a>toSimpleFunc</a> (<a>posPart</a> f)).<a>map</a> <a>norm</a> := by\n    filter_upwards [<a>posPart_toSimpleFunc</a> f] with _ h\n    rw [<a>SimpleFunc.map_apply</a>, h]\n    conv_lhs => rw [\u2190 <a>SimpleFunc.posPart_map_norm</a>, <a>SimpleFunc.map_apply</a>]", [{"full_name": "MeasureTheory.Lp.simpleFunc.toSimpleFunc", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "def_pos": [587, 5], "def_end_pos": [587, 17]}, {"full_name": "MeasureTheory.SimpleFunc.posPart", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [265, 5], "def_end_pos": [265, 12]}, {"full_name": "MeasureTheory.Lp.simpleFunc.toSimpleFunc", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "def_pos": [587, 5], "def_end_pos": [587, 17]}, {"full_name": "MeasureTheory.L1.SimpleFunc.posPart", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [478, 12], "def_end_pos": [478, 19]}, {"full_name": "MeasureTheory.SimpleFunc.map", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [290, 5], "def_end_pos": [290, 8]}, {"full_name": "Norm.norm", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [59, 3], "def_end_pos": [59, 7]}, {"full_name": "MeasureTheory.L1.SimpleFunc.posPart_toSimpleFunc", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [582, 9], "def_end_pos": [582, 29]}, {"full_name": "MeasureTheory.SimpleFunc.map_apply", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [294, 9], "def_end_pos": [294, 18]}, {"full_name": "MeasureTheory.SimpleFunc.posPart_map_norm", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [274, 9], "def_end_pos": [274, 25]}, {"full_name": "MeasureTheory.SimpleFunc.map_apply", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [294, 9], "def_end_pos": [294, 18]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedAddCommGroup F\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2078 : NormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\nF' : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\nE' : Type u_6\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : NormedSpace \u211d E'\ninst\u271d : NormedSpace \ud835\udd5c E'\nf : { x // x \u2208 simpleFunc \u211d 1 \u03bc }\n\u22a2 integral f = \u2016posPart f\u2016 - \u2016negPart f\u2016", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedAddCommGroup F\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2078 : NormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\nF' : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\nE' : Type u_6\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : NormedSpace \u211d E'\ninst\u271d : NormedSpace \ud835\udd5c E'\nf : { x // x \u2208 simpleFunc \u211d 1 \u03bc }\nae_eq\u2081 : \u2191(MeasureTheory.SimpleFunc.posPart (toSimpleFunc f)) =\u1d50[\u03bc] \u2191(SimpleFunc.map norm (toSimpleFunc (posPart f)))\n\u22a2 integral f = \u2016posPart f\u2016 - \u2016negPart f\u2016"}, {"tactic": "have ae_eq\u2082 : (toSimpleFunc f).negPart =\u1d50[\u03bc] (toSimpleFunc (negPart f)).map norm := by\n  filter_upwards [negPart_toSimpleFunc f] with _ h\n  rw [SimpleFunc.map_apply, h]\n  conv_lhs => rw [\u2190 SimpleFunc.negPart_map_norm, SimpleFunc.map_apply]", "annotated_tactic": ["have ae_eq\u2082 : (<a>toSimpleFunc</a> f).<a>negPart</a> =\u1d50[\u03bc] (<a>toSimpleFunc</a> (<a>negPart</a> f)).<a>map</a> <a>norm</a> := by\n    filter_upwards [<a>negPart_toSimpleFunc</a> f] with _ h\n    rw [<a>SimpleFunc.map_apply</a>, h]\n    conv_lhs => rw [\u2190 <a>SimpleFunc.negPart_map_norm</a>, <a>SimpleFunc.map_apply</a>]", [{"full_name": "MeasureTheory.Lp.simpleFunc.toSimpleFunc", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "def_pos": [587, 5], "def_end_pos": [587, 17]}, {"full_name": "MeasureTheory.SimpleFunc.negPart", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [270, 5], "def_end_pos": [270, 12]}, {"full_name": "MeasureTheory.Lp.simpleFunc.toSimpleFunc", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "def_pos": [587, 5], "def_end_pos": [587, 17]}, {"full_name": "MeasureTheory.L1.SimpleFunc.negPart", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [489, 5], "def_end_pos": [489, 12]}, {"full_name": "MeasureTheory.SimpleFunc.map", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [290, 5], "def_end_pos": [290, 8]}, {"full_name": "Norm.norm", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [59, 3], "def_end_pos": [59, 7]}, {"full_name": "MeasureTheory.L1.SimpleFunc.negPart_toSimpleFunc", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [595, 9], "def_end_pos": [595, 29]}, {"full_name": "MeasureTheory.SimpleFunc.map_apply", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [294, 9], "def_end_pos": [294, 18]}, {"full_name": "MeasureTheory.SimpleFunc.negPart_map_norm", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [278, 9], "def_end_pos": [278, 25]}, {"full_name": "MeasureTheory.SimpleFunc.map_apply", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [294, 9], "def_end_pos": [294, 18]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedAddCommGroup F\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2078 : NormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\nF' : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\nE' : Type u_6\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : NormedSpace \u211d E'\ninst\u271d : NormedSpace \ud835\udd5c E'\nf : { x // x \u2208 simpleFunc \u211d 1 \u03bc }\nae_eq\u2081 : \u2191(MeasureTheory.SimpleFunc.posPart (toSimpleFunc f)) =\u1d50[\u03bc] \u2191(SimpleFunc.map norm (toSimpleFunc (posPart f)))\n\u22a2 integral f = \u2016posPart f\u2016 - \u2016negPart f\u2016", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedAddCommGroup F\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2078 : NormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\nF' : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\nE' : Type u_6\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : NormedSpace \u211d E'\ninst\u271d : NormedSpace \ud835\udd5c E'\nf : { x // x \u2208 simpleFunc \u211d 1 \u03bc }\nae_eq\u2081 : \u2191(MeasureTheory.SimpleFunc.posPart (toSimpleFunc f)) =\u1d50[\u03bc] \u2191(SimpleFunc.map norm (toSimpleFunc (posPart f)))\nae_eq\u2082 : \u2191(MeasureTheory.SimpleFunc.negPart (toSimpleFunc f)) =\u1d50[\u03bc] \u2191(SimpleFunc.map norm (toSimpleFunc (negPart f)))\n\u22a2 integral f = \u2016posPart f\u2016 - \u2016negPart f\u2016"}, {"tactic": "rw [integral, norm_eq_integral, norm_eq_integral, \u2190 SimpleFunc.integral_sub]", "annotated_tactic": ["rw [<a>integral</a>, <a>norm_eq_integral</a>, <a>norm_eq_integral</a>, \u2190 <a>SimpleFunc.integral_sub</a>]", [{"full_name": "MeasureTheory.L1.SimpleFunc.integral", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [518, 5], "def_end_pos": [518, 13]}, {"full_name": "MeasureTheory.L1.SimpleFunc.norm_eq_integral", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [470, 9], "def_end_pos": [470, 25]}, {"full_name": "MeasureTheory.L1.SimpleFunc.norm_eq_integral", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [470, 9], "def_end_pos": [470, 25]}, {"full_name": "MeasureTheory.SimpleFunc.integral_sub", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [422, 9], "def_end_pos": [422, 21]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedAddCommGroup F\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2078 : NormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\nF' : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\nE' : Type u_6\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : NormedSpace \u211d E'\ninst\u271d : NormedSpace \ud835\udd5c E'\nf : { x // x \u2208 simpleFunc \u211d 1 \u03bc }\nae_eq\u2081 : \u2191(MeasureTheory.SimpleFunc.posPart (toSimpleFunc f)) =\u1d50[\u03bc] \u2191(SimpleFunc.map norm (toSimpleFunc (posPart f)))\nae_eq\u2082 : \u2191(MeasureTheory.SimpleFunc.negPart (toSimpleFunc f)) =\u1d50[\u03bc] \u2191(SimpleFunc.map norm (toSimpleFunc (negPart f)))\n\u22a2 integral f = \u2016posPart f\u2016 - \u2016negPart f\u2016", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedAddCommGroup F\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2078 : NormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\nF' : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\nE' : Type u_6\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : NormedSpace \u211d E'\ninst\u271d : NormedSpace \ud835\udd5c E'\nf : { x // x \u2208 simpleFunc \u211d 1 \u03bc }\nae_eq\u2081 : \u2191(MeasureTheory.SimpleFunc.posPart (toSimpleFunc f)) =\u1d50[\u03bc] \u2191(SimpleFunc.map norm (toSimpleFunc (posPart f)))\nae_eq\u2082 : \u2191(MeasureTheory.SimpleFunc.negPart (toSimpleFunc f)) =\u1d50[\u03bc] \u2191(SimpleFunc.map norm (toSimpleFunc (negPart f)))\n\u22a2 MeasureTheory.SimpleFunc.integral \u03bc (toSimpleFunc f) =\n    MeasureTheory.SimpleFunc.integral \u03bc\n      (SimpleFunc.map norm (toSimpleFunc (posPart f)) - SimpleFunc.map norm (toSimpleFunc (negPart f)))\n\ncase hf\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedAddCommGroup F\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2078 : NormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\nF' : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\nE' : Type u_6\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : NormedSpace \u211d E'\ninst\u271d : NormedSpace \ud835\udd5c E'\nf : { x // x \u2208 simpleFunc \u211d 1 \u03bc }\nae_eq\u2081 : \u2191(MeasureTheory.SimpleFunc.posPart (toSimpleFunc f)) =\u1d50[\u03bc] \u2191(SimpleFunc.map norm (toSimpleFunc (posPart f)))\nae_eq\u2082 : \u2191(MeasureTheory.SimpleFunc.negPart (toSimpleFunc f)) =\u1d50[\u03bc] \u2191(SimpleFunc.map norm (toSimpleFunc (negPart f)))\n\u22a2 Integrable \u2191(SimpleFunc.map norm (toSimpleFunc (posPart f)))\n\ncase hg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedAddCommGroup F\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2078 : NormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\nF' : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\nE' : Type u_6\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : NormedSpace \u211d E'\ninst\u271d : NormedSpace \ud835\udd5c E'\nf : { x // x \u2208 simpleFunc \u211d 1 \u03bc }\nae_eq\u2081 : \u2191(MeasureTheory.SimpleFunc.posPart (toSimpleFunc f)) =\u1d50[\u03bc] \u2191(SimpleFunc.map norm (toSimpleFunc (posPart f)))\nae_eq\u2082 : \u2191(MeasureTheory.SimpleFunc.negPart (toSimpleFunc f)) =\u1d50[\u03bc] \u2191(SimpleFunc.map norm (toSimpleFunc (negPart f)))\n\u22a2 Integrable \u2191(SimpleFunc.map norm (toSimpleFunc (negPart f)))"}, {"tactic": "filter_upwards [posPart_toSimpleFunc f] with _ h", "annotated_tactic": ["filter_upwards [<a>posPart_toSimpleFunc</a> f] with _ h", [{"full_name": "MeasureTheory.L1.SimpleFunc.posPart_toSimpleFunc", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [582, 9], "def_end_pos": [582, 29]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedAddCommGroup F\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2078 : NormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\nF' : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\nE' : Type u_6\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : NormedSpace \u211d E'\ninst\u271d : NormedSpace \ud835\udd5c E'\nf : { x // x \u2208 simpleFunc \u211d 1 \u03bc }\n\u22a2 \u2191(MeasureTheory.SimpleFunc.posPart (toSimpleFunc f)) =\u1d50[\u03bc] \u2191(SimpleFunc.map norm (toSimpleFunc (posPart f)))", "state_after": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedAddCommGroup F\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2078 : NormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\nF' : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\nE' : Type u_6\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : NormedSpace \u211d E'\ninst\u271d : NormedSpace \ud835\udd5c E'\nf : { x // x \u2208 simpleFunc \u211d 1 \u03bc }\na\u271d : \u03b1\nh : \u2191(toSimpleFunc (posPart f)) a\u271d = \u2191(MeasureTheory.SimpleFunc.posPart (toSimpleFunc f)) a\u271d\n\u22a2 \u2191(MeasureTheory.SimpleFunc.posPart (toSimpleFunc f)) a\u271d = \u2191(SimpleFunc.map norm (toSimpleFunc (posPart f))) a\u271d"}, {"tactic": "rw [SimpleFunc.map_apply, h]", "annotated_tactic": ["rw [<a>SimpleFunc.map_apply</a>, h]", [{"full_name": "MeasureTheory.SimpleFunc.map_apply", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [294, 9], "def_end_pos": [294, 18]}]], "state_before": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedAddCommGroup F\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2078 : NormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\nF' : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\nE' : Type u_6\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : NormedSpace \u211d E'\ninst\u271d : NormedSpace \ud835\udd5c E'\nf : { x // x \u2208 simpleFunc \u211d 1 \u03bc }\na\u271d : \u03b1\nh : \u2191(toSimpleFunc (posPart f)) a\u271d = \u2191(MeasureTheory.SimpleFunc.posPart (toSimpleFunc f)) a\u271d\n\u22a2 \u2191(MeasureTheory.SimpleFunc.posPart (toSimpleFunc f)) a\u271d = \u2191(SimpleFunc.map norm (toSimpleFunc (posPart f))) a\u271d", "state_after": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedAddCommGroup F\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2078 : NormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\nF' : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\nE' : Type u_6\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : NormedSpace \u211d E'\ninst\u271d : NormedSpace \ud835\udd5c E'\nf : { x // x \u2208 simpleFunc \u211d 1 \u03bc }\na\u271d : \u03b1\nh : \u2191(toSimpleFunc (posPart f)) a\u271d = \u2191(MeasureTheory.SimpleFunc.posPart (toSimpleFunc f)) a\u271d\n\u22a2 \u2191(MeasureTheory.SimpleFunc.posPart (toSimpleFunc f)) a\u271d = \u2016\u2191(MeasureTheory.SimpleFunc.posPart (toSimpleFunc f)) a\u271d\u2016"}, {"tactic": "conv_lhs => rw [\u2190 SimpleFunc.posPart_map_norm, SimpleFunc.map_apply]", "annotated_tactic": ["conv_lhs => rw [\u2190 <a>SimpleFunc.posPart_map_norm</a>, <a>SimpleFunc.map_apply</a>]", [{"full_name": "MeasureTheory.SimpleFunc.posPart_map_norm", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [274, 9], "def_end_pos": [274, 25]}, {"full_name": "MeasureTheory.SimpleFunc.map_apply", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [294, 9], "def_end_pos": [294, 18]}]], "state_before": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedAddCommGroup F\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2078 : NormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\nF' : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\nE' : Type u_6\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : NormedSpace \u211d E'\ninst\u271d : NormedSpace \ud835\udd5c E'\nf : { x // x \u2208 simpleFunc \u211d 1 \u03bc }\na\u271d : \u03b1\nh : \u2191(toSimpleFunc (posPart f)) a\u271d = \u2191(MeasureTheory.SimpleFunc.posPart (toSimpleFunc f)) a\u271d\n\u22a2 \u2191(MeasureTheory.SimpleFunc.posPart (toSimpleFunc f)) a\u271d = \u2016\u2191(MeasureTheory.SimpleFunc.posPart (toSimpleFunc f)) a\u271d\u2016", "state_after": "no goals"}, {"tactic": "filter_upwards [negPart_toSimpleFunc f] with _ h", "annotated_tactic": ["filter_upwards [<a>negPart_toSimpleFunc</a> f] with _ h", [{"full_name": "MeasureTheory.L1.SimpleFunc.negPart_toSimpleFunc", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [595, 9], "def_end_pos": [595, 29]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedAddCommGroup F\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2078 : NormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\nF' : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\nE' : Type u_6\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : NormedSpace \u211d E'\ninst\u271d : NormedSpace \ud835\udd5c E'\nf : { x // x \u2208 simpleFunc \u211d 1 \u03bc }\nae_eq\u2081 : \u2191(MeasureTheory.SimpleFunc.posPart (toSimpleFunc f)) =\u1d50[\u03bc] \u2191(SimpleFunc.map norm (toSimpleFunc (posPart f)))\n\u22a2 \u2191(MeasureTheory.SimpleFunc.negPart (toSimpleFunc f)) =\u1d50[\u03bc] \u2191(SimpleFunc.map norm (toSimpleFunc (negPart f)))", "state_after": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedAddCommGroup F\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2078 : NormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\nF' : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\nE' : Type u_6\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : NormedSpace \u211d E'\ninst\u271d : NormedSpace \ud835\udd5c E'\nf : { x // x \u2208 simpleFunc \u211d 1 \u03bc }\nae_eq\u2081 : \u2191(MeasureTheory.SimpleFunc.posPart (toSimpleFunc f)) =\u1d50[\u03bc] \u2191(SimpleFunc.map norm (toSimpleFunc (posPart f)))\na\u271d : \u03b1\nh : \u2191(toSimpleFunc (negPart f)) a\u271d = \u2191(MeasureTheory.SimpleFunc.negPart (toSimpleFunc f)) a\u271d\n\u22a2 \u2191(MeasureTheory.SimpleFunc.negPart (toSimpleFunc f)) a\u271d = \u2191(SimpleFunc.map norm (toSimpleFunc (negPart f))) a\u271d"}, {"tactic": "rw [SimpleFunc.map_apply, h]", "annotated_tactic": ["rw [<a>SimpleFunc.map_apply</a>, h]", [{"full_name": "MeasureTheory.SimpleFunc.map_apply", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [294, 9], "def_end_pos": [294, 18]}]], "state_before": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedAddCommGroup F\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2078 : NormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\nF' : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\nE' : Type u_6\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : NormedSpace \u211d E'\ninst\u271d : NormedSpace \ud835\udd5c E'\nf : { x // x \u2208 simpleFunc \u211d 1 \u03bc }\nae_eq\u2081 : \u2191(MeasureTheory.SimpleFunc.posPart (toSimpleFunc f)) =\u1d50[\u03bc] \u2191(SimpleFunc.map norm (toSimpleFunc (posPart f)))\na\u271d : \u03b1\nh : \u2191(toSimpleFunc (negPart f)) a\u271d = \u2191(MeasureTheory.SimpleFunc.negPart (toSimpleFunc f)) a\u271d\n\u22a2 \u2191(MeasureTheory.SimpleFunc.negPart (toSimpleFunc f)) a\u271d = \u2191(SimpleFunc.map norm (toSimpleFunc (negPart f))) a\u271d", "state_after": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedAddCommGroup F\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2078 : NormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\nF' : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\nE' : Type u_6\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : NormedSpace \u211d E'\ninst\u271d : NormedSpace \ud835\udd5c E'\nf : { x // x \u2208 simpleFunc \u211d 1 \u03bc }\nae_eq\u2081 : \u2191(MeasureTheory.SimpleFunc.posPart (toSimpleFunc f)) =\u1d50[\u03bc] \u2191(SimpleFunc.map norm (toSimpleFunc (posPart f)))\na\u271d : \u03b1\nh : \u2191(toSimpleFunc (negPart f)) a\u271d = \u2191(MeasureTheory.SimpleFunc.negPart (toSimpleFunc f)) a\u271d\n\u22a2 \u2191(MeasureTheory.SimpleFunc.negPart (toSimpleFunc f)) a\u271d = \u2016\u2191(MeasureTheory.SimpleFunc.negPart (toSimpleFunc f)) a\u271d\u2016"}, {"tactic": "conv_lhs => rw [\u2190 SimpleFunc.negPart_map_norm, SimpleFunc.map_apply]", "annotated_tactic": ["conv_lhs => rw [\u2190 <a>SimpleFunc.negPart_map_norm</a>, <a>SimpleFunc.map_apply</a>]", [{"full_name": "MeasureTheory.SimpleFunc.negPart_map_norm", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [278, 9], "def_end_pos": [278, 25]}, {"full_name": "MeasureTheory.SimpleFunc.map_apply", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [294, 9], "def_end_pos": [294, 18]}]], "state_before": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedAddCommGroup F\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2078 : NormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\nF' : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\nE' : Type u_6\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : NormedSpace \u211d E'\ninst\u271d : NormedSpace \ud835\udd5c E'\nf : { x // x \u2208 simpleFunc \u211d 1 \u03bc }\nae_eq\u2081 : \u2191(MeasureTheory.SimpleFunc.posPart (toSimpleFunc f)) =\u1d50[\u03bc] \u2191(SimpleFunc.map norm (toSimpleFunc (posPart f)))\na\u271d : \u03b1\nh : \u2191(toSimpleFunc (negPart f)) a\u271d = \u2191(MeasureTheory.SimpleFunc.negPart (toSimpleFunc f)) a\u271d\n\u22a2 \u2191(MeasureTheory.SimpleFunc.negPart (toSimpleFunc f)) a\u271d = \u2016\u2191(MeasureTheory.SimpleFunc.negPart (toSimpleFunc f)) a\u271d\u2016", "state_after": "no goals"}, {"tactic": "apply MeasureTheory.SimpleFunc.integral_congr (SimpleFunc.integrable f)", "annotated_tactic": ["apply <a>MeasureTheory.SimpleFunc.integral_congr</a> (<a>SimpleFunc.integrable</a> f)", [{"full_name": "MeasureTheory.SimpleFunc.integral_congr", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [398, 9], "def_end_pos": [398, 23]}, {"full_name": "MeasureTheory.L1.SimpleFunc.integrable", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "def_pos": [1040, 19], "def_end_pos": [1040, 43]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedAddCommGroup F\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2078 : NormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\nF' : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\nE' : Type u_6\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : NormedSpace \u211d E'\ninst\u271d : NormedSpace \ud835\udd5c E'\nf : { x // x \u2208 simpleFunc \u211d 1 \u03bc }\nae_eq\u2081 : \u2191(MeasureTheory.SimpleFunc.posPart (toSimpleFunc f)) =\u1d50[\u03bc] \u2191(SimpleFunc.map norm (toSimpleFunc (posPart f)))\nae_eq\u2082 : \u2191(MeasureTheory.SimpleFunc.negPart (toSimpleFunc f)) =\u1d50[\u03bc] \u2191(SimpleFunc.map norm (toSimpleFunc (negPart f)))\n\u22a2 MeasureTheory.SimpleFunc.integral \u03bc (toSimpleFunc f) =\n    MeasureTheory.SimpleFunc.integral \u03bc\n      (SimpleFunc.map norm (toSimpleFunc (posPart f)) - SimpleFunc.map norm (toSimpleFunc (negPart f)))", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedAddCommGroup F\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2078 : NormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\nF' : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\nE' : Type u_6\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : NormedSpace \u211d E'\ninst\u271d : NormedSpace \ud835\udd5c E'\nf : { x // x \u2208 simpleFunc \u211d 1 \u03bc }\nae_eq\u2081 : \u2191(MeasureTheory.SimpleFunc.posPart (toSimpleFunc f)) =\u1d50[\u03bc] \u2191(SimpleFunc.map norm (toSimpleFunc (posPart f)))\nae_eq\u2082 : \u2191(MeasureTheory.SimpleFunc.negPart (toSimpleFunc f)) =\u1d50[\u03bc] \u2191(SimpleFunc.map norm (toSimpleFunc (negPart f)))\n\u22a2 \u2191(toSimpleFunc f) =\u1d50[\u03bc]\n    \u2191(SimpleFunc.map norm (toSimpleFunc (posPart f)) - SimpleFunc.map norm (toSimpleFunc (negPart f)))"}, {"tactic": "filter_upwards [ae_eq\u2081, ae_eq\u2082] with _ h\u2081 h\u2082", "annotated_tactic": ["filter_upwards [ae_eq\u2081, ae_eq\u2082] with _ h\u2081 h\u2082", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedAddCommGroup F\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2078 : NormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\nF' : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\nE' : Type u_6\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : NormedSpace \u211d E'\ninst\u271d : NormedSpace \ud835\udd5c E'\nf : { x // x \u2208 simpleFunc \u211d 1 \u03bc }\nae_eq\u2081 : \u2191(MeasureTheory.SimpleFunc.posPart (toSimpleFunc f)) =\u1d50[\u03bc] \u2191(SimpleFunc.map norm (toSimpleFunc (posPart f)))\nae_eq\u2082 : \u2191(MeasureTheory.SimpleFunc.negPart (toSimpleFunc f)) =\u1d50[\u03bc] \u2191(SimpleFunc.map norm (toSimpleFunc (negPart f)))\n\u22a2 \u2191(toSimpleFunc f) =\u1d50[\u03bc]\n    \u2191(SimpleFunc.map norm (toSimpleFunc (posPart f)) - SimpleFunc.map norm (toSimpleFunc (negPart f)))", "state_after": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedAddCommGroup F\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2078 : NormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\nF' : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\nE' : Type u_6\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : NormedSpace \u211d E'\ninst\u271d : NormedSpace \ud835\udd5c E'\nf : { x // x \u2208 simpleFunc \u211d 1 \u03bc }\nae_eq\u2081 : \u2191(MeasureTheory.SimpleFunc.posPart (toSimpleFunc f)) =\u1d50[\u03bc] \u2191(SimpleFunc.map norm (toSimpleFunc (posPart f)))\nae_eq\u2082 : \u2191(MeasureTheory.SimpleFunc.negPart (toSimpleFunc f)) =\u1d50[\u03bc] \u2191(SimpleFunc.map norm (toSimpleFunc (negPart f)))\na\u271d : \u03b1\nh\u2081 : \u2191(MeasureTheory.SimpleFunc.posPart (toSimpleFunc f)) a\u271d = \u2191(SimpleFunc.map norm (toSimpleFunc (posPart f))) a\u271d\nh\u2082 : \u2191(MeasureTheory.SimpleFunc.negPart (toSimpleFunc f)) a\u271d = \u2191(SimpleFunc.map norm (toSimpleFunc (negPart f))) a\u271d\n\u22a2 \u2191(toSimpleFunc f) a\u271d =\n    \u2191(SimpleFunc.map norm (toSimpleFunc (posPart f)) - SimpleFunc.map norm (toSimpleFunc (negPart f))) a\u271d"}, {"tactic": "show _ = _ - _", "annotated_tactic": ["show _ = _ - _", []], "state_before": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedAddCommGroup F\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2078 : NormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\nF' : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\nE' : Type u_6\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : NormedSpace \u211d E'\ninst\u271d : NormedSpace \ud835\udd5c E'\nf : { x // x \u2208 simpleFunc \u211d 1 \u03bc }\nae_eq\u2081 : \u2191(MeasureTheory.SimpleFunc.posPart (toSimpleFunc f)) =\u1d50[\u03bc] \u2191(SimpleFunc.map norm (toSimpleFunc (posPart f)))\nae_eq\u2082 : \u2191(MeasureTheory.SimpleFunc.negPart (toSimpleFunc f)) =\u1d50[\u03bc] \u2191(SimpleFunc.map norm (toSimpleFunc (negPart f)))\na\u271d : \u03b1\nh\u2081 : \u2191(MeasureTheory.SimpleFunc.posPart (toSimpleFunc f)) a\u271d = \u2191(SimpleFunc.map norm (toSimpleFunc (posPart f))) a\u271d\nh\u2082 : \u2191(MeasureTheory.SimpleFunc.negPart (toSimpleFunc f)) a\u271d = \u2191(SimpleFunc.map norm (toSimpleFunc (negPart f))) a\u271d\n\u22a2 \u2191(toSimpleFunc f) a\u271d =\n    \u2191(SimpleFunc.map norm (toSimpleFunc (posPart f)) - SimpleFunc.map norm (toSimpleFunc (negPart f))) a\u271d", "state_after": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedAddCommGroup F\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2078 : NormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\nF' : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\nE' : Type u_6\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : NormedSpace \u211d E'\ninst\u271d : NormedSpace \ud835\udd5c E'\nf : { x // x \u2208 simpleFunc \u211d 1 \u03bc }\nae_eq\u2081 : \u2191(MeasureTheory.SimpleFunc.posPart (toSimpleFunc f)) =\u1d50[\u03bc] \u2191(SimpleFunc.map norm (toSimpleFunc (posPart f)))\nae_eq\u2082 : \u2191(MeasureTheory.SimpleFunc.negPart (toSimpleFunc f)) =\u1d50[\u03bc] \u2191(SimpleFunc.map norm (toSimpleFunc (negPart f)))\na\u271d : \u03b1\nh\u2081 : \u2191(MeasureTheory.SimpleFunc.posPart (toSimpleFunc f)) a\u271d = \u2191(SimpleFunc.map norm (toSimpleFunc (posPart f))) a\u271d\nh\u2082 : \u2191(MeasureTheory.SimpleFunc.negPart (toSimpleFunc f)) a\u271d = \u2191(SimpleFunc.map norm (toSimpleFunc (negPart f))) a\u271d\n\u22a2 \u2191(toSimpleFunc f) a\u271d =\n    \u2191(SimpleFunc.map norm (toSimpleFunc (posPart f))) a\u271d - \u2191(SimpleFunc.map norm (toSimpleFunc (negPart f))) a\u271d"}, {"tactic": "rw [\u2190 h\u2081, \u2190 h\u2082]", "annotated_tactic": ["rw [\u2190 h\u2081, \u2190 h\u2082]", []], "state_before": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedAddCommGroup F\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2078 : NormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\nF' : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\nE' : Type u_6\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : NormedSpace \u211d E'\ninst\u271d : NormedSpace \ud835\udd5c E'\nf : { x // x \u2208 simpleFunc \u211d 1 \u03bc }\nae_eq\u2081 : \u2191(MeasureTheory.SimpleFunc.posPart (toSimpleFunc f)) =\u1d50[\u03bc] \u2191(SimpleFunc.map norm (toSimpleFunc (posPart f)))\nae_eq\u2082 : \u2191(MeasureTheory.SimpleFunc.negPart (toSimpleFunc f)) =\u1d50[\u03bc] \u2191(SimpleFunc.map norm (toSimpleFunc (negPart f)))\na\u271d : \u03b1\nh\u2081 : \u2191(MeasureTheory.SimpleFunc.posPart (toSimpleFunc f)) a\u271d = \u2191(SimpleFunc.map norm (toSimpleFunc (posPart f))) a\u271d\nh\u2082 : \u2191(MeasureTheory.SimpleFunc.negPart (toSimpleFunc f)) a\u271d = \u2191(SimpleFunc.map norm (toSimpleFunc (negPart f))) a\u271d\n\u22a2 \u2191(toSimpleFunc f) a\u271d =\n    \u2191(SimpleFunc.map norm (toSimpleFunc (posPart f))) a\u271d - \u2191(SimpleFunc.map norm (toSimpleFunc (negPart f))) a\u271d", "state_after": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedAddCommGroup F\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2078 : NormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\nF' : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\nE' : Type u_6\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : NormedSpace \u211d E'\ninst\u271d : NormedSpace \ud835\udd5c E'\nf : { x // x \u2208 simpleFunc \u211d 1 \u03bc }\nae_eq\u2081 : \u2191(MeasureTheory.SimpleFunc.posPart (toSimpleFunc f)) =\u1d50[\u03bc] \u2191(SimpleFunc.map norm (toSimpleFunc (posPart f)))\nae_eq\u2082 : \u2191(MeasureTheory.SimpleFunc.negPart (toSimpleFunc f)) =\u1d50[\u03bc] \u2191(SimpleFunc.map norm (toSimpleFunc (negPart f)))\na\u271d : \u03b1\nh\u2081 : \u2191(MeasureTheory.SimpleFunc.posPart (toSimpleFunc f)) a\u271d = \u2191(SimpleFunc.map norm (toSimpleFunc (posPart f))) a\u271d\nh\u2082 : \u2191(MeasureTheory.SimpleFunc.negPart (toSimpleFunc f)) a\u271d = \u2191(SimpleFunc.map norm (toSimpleFunc (negPart f))) a\u271d\n\u22a2 \u2191(toSimpleFunc f) a\u271d =\n    \u2191(MeasureTheory.SimpleFunc.posPart (toSimpleFunc f)) a\u271d - \u2191(MeasureTheory.SimpleFunc.negPart (toSimpleFunc f)) a\u271d"}, {"tactic": "have := (toSimpleFunc f).posPart_sub_negPart", "annotated_tactic": ["have := (<a>toSimpleFunc</a> f).<a>posPart_sub_negPart</a>", [{"full_name": "MeasureTheory.Lp.simpleFunc.toSimpleFunc", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "def_pos": [587, 5], "def_end_pos": [587, 17]}, {"full_name": "MeasureTheory.SimpleFunc.posPart_sub_negPart", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [282, 9], "def_end_pos": [282, 28]}]], "state_before": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedAddCommGroup F\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2078 : NormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\nF' : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\nE' : Type u_6\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : NormedSpace \u211d E'\ninst\u271d : NormedSpace \ud835\udd5c E'\nf : { x // x \u2208 simpleFunc \u211d 1 \u03bc }\nae_eq\u2081 : \u2191(MeasureTheory.SimpleFunc.posPart (toSimpleFunc f)) =\u1d50[\u03bc] \u2191(SimpleFunc.map norm (toSimpleFunc (posPart f)))\nae_eq\u2082 : \u2191(MeasureTheory.SimpleFunc.negPart (toSimpleFunc f)) =\u1d50[\u03bc] \u2191(SimpleFunc.map norm (toSimpleFunc (negPart f)))\na\u271d : \u03b1\nh\u2081 : \u2191(MeasureTheory.SimpleFunc.posPart (toSimpleFunc f)) a\u271d = \u2191(SimpleFunc.map norm (toSimpleFunc (posPart f))) a\u271d\nh\u2082 : \u2191(MeasureTheory.SimpleFunc.negPart (toSimpleFunc f)) a\u271d = \u2191(SimpleFunc.map norm (toSimpleFunc (negPart f))) a\u271d\n\u22a2 \u2191(toSimpleFunc f) a\u271d =\n    \u2191(MeasureTheory.SimpleFunc.posPart (toSimpleFunc f)) a\u271d - \u2191(MeasureTheory.SimpleFunc.negPart (toSimpleFunc f)) a\u271d", "state_after": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedAddCommGroup F\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2078 : NormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\nF' : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\nE' : Type u_6\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : NormedSpace \u211d E'\ninst\u271d : NormedSpace \ud835\udd5c E'\nf : { x // x \u2208 simpleFunc \u211d 1 \u03bc }\nae_eq\u2081 : \u2191(MeasureTheory.SimpleFunc.posPart (toSimpleFunc f)) =\u1d50[\u03bc] \u2191(SimpleFunc.map norm (toSimpleFunc (posPart f)))\nae_eq\u2082 : \u2191(MeasureTheory.SimpleFunc.negPart (toSimpleFunc f)) =\u1d50[\u03bc] \u2191(SimpleFunc.map norm (toSimpleFunc (negPart f)))\na\u271d : \u03b1\nh\u2081 : \u2191(MeasureTheory.SimpleFunc.posPart (toSimpleFunc f)) a\u271d = \u2191(SimpleFunc.map norm (toSimpleFunc (posPart f))) a\u271d\nh\u2082 : \u2191(MeasureTheory.SimpleFunc.negPart (toSimpleFunc f)) a\u271d = \u2191(SimpleFunc.map norm (toSimpleFunc (negPart f))) a\u271d\nthis :\n  MeasureTheory.SimpleFunc.posPart (toSimpleFunc f) - MeasureTheory.SimpleFunc.negPart (toSimpleFunc f) = toSimpleFunc f\n\u22a2 \u2191(toSimpleFunc f) a\u271d =\n    \u2191(MeasureTheory.SimpleFunc.posPart (toSimpleFunc f)) a\u271d - \u2191(MeasureTheory.SimpleFunc.negPart (toSimpleFunc f)) a\u271d"}, {"tactic": "conv_lhs => rw [\u2190 this]", "annotated_tactic": ["conv_lhs => rw [\u2190 this]", []], "state_before": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedAddCommGroup F\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2078 : NormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\nF' : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\nE' : Type u_6\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : NormedSpace \u211d E'\ninst\u271d : NormedSpace \ud835\udd5c E'\nf : { x // x \u2208 simpleFunc \u211d 1 \u03bc }\nae_eq\u2081 : \u2191(MeasureTheory.SimpleFunc.posPart (toSimpleFunc f)) =\u1d50[\u03bc] \u2191(SimpleFunc.map norm (toSimpleFunc (posPart f)))\nae_eq\u2082 : \u2191(MeasureTheory.SimpleFunc.negPart (toSimpleFunc f)) =\u1d50[\u03bc] \u2191(SimpleFunc.map norm (toSimpleFunc (negPart f)))\na\u271d : \u03b1\nh\u2081 : \u2191(MeasureTheory.SimpleFunc.posPart (toSimpleFunc f)) a\u271d = \u2191(SimpleFunc.map norm (toSimpleFunc (posPart f))) a\u271d\nh\u2082 : \u2191(MeasureTheory.SimpleFunc.negPart (toSimpleFunc f)) a\u271d = \u2191(SimpleFunc.map norm (toSimpleFunc (negPart f))) a\u271d\nthis :\n  MeasureTheory.SimpleFunc.posPart (toSimpleFunc f) - MeasureTheory.SimpleFunc.negPart (toSimpleFunc f) = toSimpleFunc f\n\u22a2 \u2191(toSimpleFunc f) a\u271d =\n    \u2191(MeasureTheory.SimpleFunc.posPart (toSimpleFunc f)) a\u271d - \u2191(MeasureTheory.SimpleFunc.negPart (toSimpleFunc f)) a\u271d", "state_after": "no goals"}, {"tactic": "exact (SimpleFunc.integrable f).pos_part.congr ae_eq\u2081", "annotated_tactic": ["exact (<a>SimpleFunc.integrable</a> f).pos_part.congr ae_eq\u2081", [{"full_name": "MeasureTheory.L1.SimpleFunc.integrable", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "def_pos": [1040, 19], "def_end_pos": [1040, 43]}]], "state_before": "case hf\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedAddCommGroup F\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2078 : NormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\nF' : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\nE' : Type u_6\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : NormedSpace \u211d E'\ninst\u271d : NormedSpace \ud835\udd5c E'\nf : { x // x \u2208 simpleFunc \u211d 1 \u03bc }\nae_eq\u2081 : \u2191(MeasureTheory.SimpleFunc.posPart (toSimpleFunc f)) =\u1d50[\u03bc] \u2191(SimpleFunc.map norm (toSimpleFunc (posPart f)))\nae_eq\u2082 : \u2191(MeasureTheory.SimpleFunc.negPart (toSimpleFunc f)) =\u1d50[\u03bc] \u2191(SimpleFunc.map norm (toSimpleFunc (negPart f)))\n\u22a2 Integrable \u2191(SimpleFunc.map norm (toSimpleFunc (posPart f)))", "state_after": "no goals"}, {"tactic": "exact (SimpleFunc.integrable f).neg_part.congr ae_eq\u2082", "annotated_tactic": ["exact (<a>SimpleFunc.integrable</a> f).neg_part.congr ae_eq\u2082", [{"full_name": "MeasureTheory.L1.SimpleFunc.integrable", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "def_pos": [1040, 19], "def_end_pos": [1040, 43]}]], "state_before": "case hg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedAddCommGroup F\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2078 : NormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\nF' : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\nE' : Type u_6\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : NormedSpace \u211d E'\ninst\u271d : NormedSpace \ud835\udd5c E'\nf : { x // x \u2208 simpleFunc \u211d 1 \u03bc }\nae_eq\u2081 : \u2191(MeasureTheory.SimpleFunc.posPart (toSimpleFunc f)) =\u1d50[\u03bc] \u2191(SimpleFunc.map norm (toSimpleFunc (posPart f)))\nae_eq\u2082 : \u2191(MeasureTheory.SimpleFunc.negPart (toSimpleFunc f)) =\u1d50[\u03bc] \u2191(SimpleFunc.map norm (toSimpleFunc (negPart f)))\n\u22a2 Integrable \u2191(SimpleFunc.map norm (toSimpleFunc (negPart f)))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "full_name": "MeasureTheory.iSup\u2082_lintegral_le", "start": [247, 1], "end": [250, 25], "traced_tactics": [{"tactic": "convert (monotone_lintegral \u03bc).le_map_iSup\u2082 f with a", "annotated_tactic": ["convert (<a>monotone_lintegral</a> \u03bc).<a>le_map_iSup\u2082</a> f with a", [{"full_name": "MeasureTheory.monotone_lintegral", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [131, 9], "def_end_pos": [131, 27]}, {"full_name": "Monotone.le_map_iSup\u2082", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [1007, 9], "def_end_pos": [1007, 30]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\n\u03b9 : Sort u_5\n\u03b9' : \u03b9 \u2192 Sort u_6\nf : (i : \u03b9) \u2192 \u03b9' i \u2192 \u03b1 \u2192 \u211d\u22650\u221e\n\u22a2 \u2a06 i, \u2a06 j, \u222b\u207b (a : \u03b1), f i j a \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), \u2a06 i, \u2a06 j, f i j a \u2202\u03bc", "state_after": "case h.e'_4.h.e'_4.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\n\u03b9 : Sort u_5\n\u03b9' : \u03b9 \u2192 Sort u_6\nf : (i : \u03b9) \u2192 \u03b9' i \u2192 \u03b1 \u2192 \u211d\u22650\u221e\na : \u03b1\n\u22a2 \u2a06 i, \u2a06 j, f i j a = iSup (fun i => \u2a06 j, f i j) a"}, {"tactic": "simp only [iSup_apply]", "annotated_tactic": ["simp only [<a>iSup_apply</a>]", [{"full_name": "iSup_apply", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [1844, 9], "def_end_pos": [1844, 19]}]], "state_before": "case h.e'_4.h.e'_4.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\n\u03b9 : Sort u_5\n\u03b9' : \u03b9 \u2192 Sort u_6\nf : (i : \u03b9) \u2192 \u03b9' i \u2192 \u03b1 \u2192 \u211d\u22650\u221e\na : \u03b1\n\u22a2 \u2a06 i, \u2a06 j, f i j a = iSup (fun i => \u2a06 j, f i j) a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/FundThmCalculus.lean", "full_name": "intervalIntegral.deriv_integral_of_tendsto_ae_left", "start": [797, 1], "end": [800, 61], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL2.lean", "full_name": "MeasureTheory.integral_condexpL2_eq", "start": [256, 1], "end": [273, 56], "traced_tactics": [{"tactic": "rw [\u2190 sub_eq_zero, lpMeas_coe, \u2190\n  integral_sub' (integrableOn_Lp_of_measure_ne_top _ fact_one_le_two_ennreal.elim h\u03bcs)\n    (integrableOn_Lp_of_measure_ne_top _ fact_one_le_two_ennreal.elim h\u03bcs)]", "annotated_tactic": ["rw [\u2190 <a>sub_eq_zero</a>, <a>lpMeas_coe</a>, \u2190\n    <a>integral_sub'</a> (<a>integrableOn_Lp_of_measure_ne_top</a> _ fact_one_le_two_ennreal.elim h\u03bcs)\n      (<a>integrableOn_Lp_of_measure_ne_top</a> _ fact_one_le_two_ennreal.elim h\u03bcs)]", [{"full_name": "sub_eq_zero", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [801, 3], "def_end_pos": [801, 14]}, {"full_name": "MeasureTheory.lpMeas_coe", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean", "def_pos": [257, 9], "def_end_pos": [257, 19]}, {"full_name": "MeasureTheory.integral_sub'", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [909, 9], "def_end_pos": [909, 22]}, {"full_name": "MeasureTheory.integrableOn_Lp_of_measure_ne_top", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [363, 9], "def_end_pos": [363, 42]}, {"full_name": "MeasureTheory.integrableOn_Lp_of_measure_ne_top", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [363, 9], "def_end_pos": [363, 42]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 Lp E' 2 }\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\n\u22a2 \u222b (x : \u03b1) in s, \u2191\u2191\u2191(\u2191(condexpL2 E' \ud835\udd5c hm) f) x \u2202\u03bc = \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 Lp E' 2 }\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\n\u22a2 \u222b (a : \u03b1) in s, (\u2191\u2191\u2191(\u2191(condexpL2 E' \ud835\udd5c hm) f) - \u2191\u2191f) a \u2202\u03bc = 0"}, {"tactic": "refine' integral_eq_zero_of_forall_integral_inner_eq_zero \ud835\udd5c _ _ _", "annotated_tactic": ["refine' <a>integral_eq_zero_of_forall_integral_inner_eq_zero</a> \ud835\udd5c _ _ _", [{"full_name": "integral_eq_zero_of_forall_integral_inner_eq_zero", "def_path": "Mathlib/MeasureTheory/Function/L2Space.lean", "def_pos": [107, 9], "def_end_pos": [107, 65]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 Lp E' 2 }\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\n\u22a2 \u222b (a : \u03b1) in s, (\u2191\u2191\u2191(\u2191(condexpL2 E' \ud835\udd5c hm) f) - \u2191\u2191f) a \u2202\u03bc = 0", "state_after": "case refine'_1\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 Lp E' 2 }\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\n\u22a2 Integrable fun a => (\u2191\u2191\u2191(\u2191(condexpL2 E' \ud835\udd5c hm) f) - \u2191\u2191f) a\n\ncase refine'_2\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 Lp E' 2 }\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\n\u22a2 \u2200 (c : E'), \u222b (x : \u03b1) in s, inner c ((\u2191\u2191\u2191(\u2191(condexpL2 E' \ud835\udd5c hm) f) - \u2191\u2191f) x) \u2202\u03bc = 0"}, {"tactic": "intro c", "annotated_tactic": ["intro c", []], "state_before": "case refine'_2\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 Lp E' 2 }\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\n\u22a2 \u2200 (c : E'), \u222b (x : \u03b1) in s, inner c ((\u2191\u2191\u2191(\u2191(condexpL2 E' \ud835\udd5c hm) f) - \u2191\u2191f) x) \u2202\u03bc = 0", "state_after": "case refine'_2\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 Lp E' 2 }\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nc : E'\n\u22a2 \u222b (x : \u03b1) in s, inner c ((\u2191\u2191\u2191(\u2191(condexpL2 E' \ud835\udd5c hm) f) - \u2191\u2191f) x) \u2202\u03bc = 0"}, {"tactic": "simp_rw [Pi.sub_apply, inner_sub_right]", "annotated_tactic": ["simp_rw [<a>Pi.sub_apply</a>, <a>inner_sub_right</a>]", [{"full_name": "Pi.sub_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [200, 3], "def_end_pos": [200, 14]}, {"full_name": "inner_sub_right", "def_path": "Mathlib/Analysis/InnerProductSpace/Basic.lean", "def_pos": [650, 9], "def_end_pos": [650, 24]}]], "state_before": "case refine'_2\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 Lp E' 2 }\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nc : E'\n\u22a2 \u222b (x : \u03b1) in s, inner c ((\u2191\u2191\u2191(\u2191(condexpL2 E' \ud835\udd5c hm) f) - \u2191\u2191f) x) \u2202\u03bc = 0", "state_after": "case refine'_2\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 Lp E' 2 }\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nc : E'\n\u22a2 \u222b (x : \u03b1) in s, inner c (\u2191\u2191\u2191(\u2191(condexpL2 E' \ud835\udd5c hm) f) x) - inner c (\u2191\u2191f x) \u2202\u03bc = 0"}, {"tactic": "rw [integral_sub\n    ((integrableOn_Lp_of_measure_ne_top _ fact_one_le_two_ennreal.elim h\u03bcs).const_inner c)\n    ((integrableOn_Lp_of_measure_ne_top _ fact_one_le_two_ennreal.elim h\u03bcs).const_inner c)]", "annotated_tactic": ["rw [<a>integral_sub</a>\n      ((<a>integrableOn_Lp_of_measure_ne_top</a> _ fact_one_le_two_ennreal.elim h\u03bcs).<a>const_inner</a> c)\n      ((<a>integrableOn_Lp_of_measure_ne_top</a> _ fact_one_le_two_ennreal.elim h\u03bcs).<a>const_inner</a> c)]", [{"full_name": "MeasureTheory.integral_sub", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [901, 9], "def_end_pos": [901, 21]}, {"full_name": "MeasureTheory.integrableOn_Lp_of_measure_ne_top", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [363, 9], "def_end_pos": [363, 42]}, {"full_name": "MeasureTheory.Integrable.const_inner", "def_path": "Mathlib/MeasureTheory/Function/L2Space.lean", "def_pos": [85, 9], "def_end_pos": [85, 31]}, {"full_name": "MeasureTheory.integrableOn_Lp_of_measure_ne_top", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [363, 9], "def_end_pos": [363, 42]}, {"full_name": "MeasureTheory.Integrable.const_inner", "def_path": "Mathlib/MeasureTheory/Function/L2Space.lean", "def_pos": [85, 9], "def_end_pos": [85, 31]}]], "state_before": "case refine'_2\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 Lp E' 2 }\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nc : E'\n\u22a2 \u222b (x : \u03b1) in s, inner c (\u2191\u2191\u2191(\u2191(condexpL2 E' \ud835\udd5c hm) f) x) - inner c (\u2191\u2191f x) \u2202\u03bc = 0", "state_after": "case refine'_2\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 Lp E' 2 }\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nc : E'\n\u22a2 \u222b (a : \u03b1) in s, inner c (\u2191\u2191\u2191(\u2191(condexpL2 E' \ud835\udd5c hm) f) a) \u2202\u03bc - \u222b (a : \u03b1) in s, inner c (\u2191\u2191f a) \u2202\u03bc = 0"}, {"tactic": "have h_ae_eq_f := Mem\u2112p.coeFn_toLp (E := \ud835\udd5c) ((Lp.mem\u2112p f).const_inner c)", "annotated_tactic": ["have h_ae_eq_f := <a>Mem\u2112p.coeFn_toLp</a> (E := \ud835\udd5c) ((<a>Lp.mem\u2112p</a> f).<a>const_inner</a> c)", [{"full_name": "MeasureTheory.Mem\u2112p.coeFn_toLp", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [119, 9], "def_end_pos": [119, 19]}, {"full_name": "MeasureTheory.Lp.mem\u2112p", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [216, 19], "def_end_pos": [216, 24]}, {"full_name": "MeasureTheory.Mem\u2112p.const_inner", "def_path": "Mathlib/MeasureTheory/Function/L2Space.lean", "def_pos": [73, 9], "def_end_pos": [73, 26]}]], "state_before": "case refine'_2\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 Lp E' 2 }\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nc : E'\n\u22a2 \u222b (a : \u03b1) in s, inner c (\u2191\u2191\u2191(\u2191(condexpL2 E' \ud835\udd5c hm) f) a) \u2202\u03bc - \u222b (a : \u03b1) in s, inner c (\u2191\u2191f a) \u2202\u03bc = 0", "state_after": "case refine'_2\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 Lp E' 2 }\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nc : E'\nh_ae_eq_f :\n  \u2191\u2191(Mem\u2112p.toLp (fun a => inner c (\u2191\u2191f a)) (_ : Mem\u2112p (fun a => inner c (\u2191\u2191f a)) 2)) =\u1d50[\u03bc] fun a => inner c (\u2191\u2191f a)\n\u22a2 \u222b (a : \u03b1) in s, inner c (\u2191\u2191\u2191(\u2191(condexpL2 E' \ud835\udd5c hm) f) a) \u2202\u03bc - \u222b (a : \u03b1) in s, inner c (\u2191\u2191f a) \u2202\u03bc = 0"}, {"tactic": "rw [\u2190 lpMeas_coe, sub_eq_zero, \u2190\n  set_integral_congr_ae (hm s hs) ((condexpL2_const_inner hm f c).mono fun x hx _ => hx), \u2190\n  set_integral_congr_ae (hm s hs) (h_ae_eq_f.mono fun x hx _ => hx)]", "annotated_tactic": ["rw [\u2190 <a>lpMeas_coe</a>, <a>sub_eq_zero</a>, \u2190\n    <a>set_integral_congr_ae</a> (hm s hs) ((<a>condexpL2_const_inner</a> hm f c).<a>mono</a> fun x hx _ => hx), \u2190\n    <a>set_integral_congr_ae</a> (hm s hs) (h_ae_eq_f.mono fun x hx _ => hx)]", [{"full_name": "MeasureTheory.lpMeas_coe", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean", "def_pos": [257, 9], "def_end_pos": [257, 19]}, {"full_name": "sub_eq_zero", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [801, 3], "def_end_pos": [801, 14]}, {"full_name": "MeasureTheory.set_integral_congr_ae", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [77, 9], "def_end_pos": [77, 30]}, {"full_name": "MeasureTheory.condexpL2_const_inner", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL2.lean", "def_pos": [228, 9], "def_end_pos": [228, 30]}, {"full_name": "Filter.Eventually.mono", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1140, 9], "def_end_pos": [1140, 24]}, {"full_name": "MeasureTheory.set_integral_congr_ae", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [77, 9], "def_end_pos": [77, 30]}]], "state_before": "case refine'_2\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 Lp E' 2 }\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nc : E'\nh_ae_eq_f :\n  \u2191\u2191(Mem\u2112p.toLp (fun a => inner c (\u2191\u2191f a)) (_ : Mem\u2112p (fun a => inner c (\u2191\u2191f a)) 2)) =\u1d50[\u03bc] fun a => inner c (\u2191\u2191f a)\n\u22a2 \u222b (a : \u03b1) in s, inner c (\u2191\u2191\u2191(\u2191(condexpL2 E' \ud835\udd5c hm) f) a) \u2202\u03bc - \u222b (a : \u03b1) in s, inner c (\u2191\u2191f a) \u2202\u03bc = 0", "state_after": "case refine'_2\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 Lp E' 2 }\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nc : E'\nh_ae_eq_f :\n  \u2191\u2191(Mem\u2112p.toLp (fun a => inner c (\u2191\u2191f a)) (_ : Mem\u2112p (fun a => inner c (\u2191\u2191f a)) 2)) =\u1d50[\u03bc] fun a => inner c (\u2191\u2191f a)\n\u22a2 \u222b (x : \u03b1) in s,\n      \u2191\u2191\u2191(\u2191(condexpL2 \ud835\udd5c \ud835\udd5c hm) (Mem\u2112p.toLp (fun a => inner c (\u2191\u2191f a)) (_ : Mem\u2112p (fun a => inner c (\u2191\u2191f a)) 2))) x \u2202\u03bc =\n    \u222b (x : \u03b1) in s, \u2191\u2191(Mem\u2112p.toLp (fun a => inner c (\u2191\u2191f a)) (_ : Mem\u2112p (fun a => inner c (\u2191\u2191f a)) 2)) x \u2202\u03bc"}, {"tactic": "exact integral_condexpL2_eq_of_fin_meas_real _ hs h\u03bcs", "annotated_tactic": ["exact <a>integral_condexpL2_eq_of_fin_meas_real</a> _ hs h\u03bcs", [{"full_name": "MeasureTheory.integral_condexpL2_eq_of_fin_meas_real", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL2.lean", "def_pos": [151, 9], "def_end_pos": [151, 47]}]], "state_before": "case refine'_2\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 Lp E' 2 }\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nc : E'\nh_ae_eq_f :\n  \u2191\u2191(Mem\u2112p.toLp (fun a => inner c (\u2191\u2191f a)) (_ : Mem\u2112p (fun a => inner c (\u2191\u2191f a)) 2)) =\u1d50[\u03bc] fun a => inner c (\u2191\u2191f a)\n\u22a2 \u222b (x : \u03b1) in s,\n      \u2191\u2191\u2191(\u2191(condexpL2 \ud835\udd5c \ud835\udd5c hm) (Mem\u2112p.toLp (fun a => inner c (\u2191\u2191f a)) (_ : Mem\u2112p (fun a => inner c (\u2191\u2191f a)) 2))) x \u2202\u03bc =\n    \u222b (x : \u03b1) in s, \u2191\u2191(Mem\u2112p.toLp (fun a => inner c (\u2191\u2191f a)) (_ : Mem\u2112p (fun a => inner c (\u2191\u2191f a)) 2)) x \u2202\u03bc", "state_after": "no goals"}, {"tactic": "rw [integrable_congr (ae_restrict_of_ae (Lp.coeFn_sub (\u2191(condexpL2 E' \ud835\udd5c hm f)) f).symm)]", "annotated_tactic": ["rw [<a>integrable_congr</a> (<a>ae_restrict_of_ae</a> (<a>Lp.coeFn_sub</a> (\u2191(<a>condexpL2</a> E' \ud835\udd5c hm f)) f).<a>symm</a>)]", [{"full_name": "MeasureTheory.integrable_congr", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [496, 9], "def_end_pos": [496, 25]}, {"full_name": "MeasureTheory.ae_restrict_of_ae", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2596, 9], "def_end_pos": [2596, 26]}, {"full_name": "MeasureTheory.Lp.coeFn_sub", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [236, 9], "def_end_pos": [236, 18]}, {"full_name": "MeasureTheory.condexpL2", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL2.lean", "def_pos": [71, 19], "def_end_pos": [71, 28]}, {"full_name": "Filter.EventuallyEq.symm", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1498, 9], "def_end_pos": [1498, 26]}]], "state_before": "case refine'_1\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 Lp E' 2 }\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\n\u22a2 Integrable fun a => (\u2191\u2191\u2191(\u2191(condexpL2 E' \ud835\udd5c hm) f) - \u2191\u2191f) a", "state_after": "case refine'_1\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 Lp E' 2 }\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\n\u22a2 Integrable fun x => \u2191\u2191(\u2191(\u2191(condexpL2 E' \ud835\udd5c hm) f) - f) x"}, {"tactic": "exact integrableOn_Lp_of_measure_ne_top _ fact_one_le_two_ennreal.elim h\u03bcs", "annotated_tactic": ["exact <a>integrableOn_Lp_of_measure_ne_top</a> _ fact_one_le_two_ennreal.elim h\u03bcs", [{"full_name": "MeasureTheory.integrableOn_Lp_of_measure_ne_top", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [363, 9], "def_end_pos": [363, 42]}]], "state_before": "case refine'_1\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 Lp E' 2 }\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\n\u22a2 Integrable fun x => \u2191\u2191(\u2191(\u2191(condexpL2 E' \ud835\udd5c hm) f) - f) x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "full_name": "MeasureTheory.measure_iUnion_eq_iSup", "start": [463, 1], "end": [495, 46], "traced_tactics": [{"tactic": "cases nonempty_encodable \u03b9", "annotated_tactic": ["cases <a>nonempty_encodable</a> \u03b9", [{"full_name": "nonempty_encodable", "def_path": "Mathlib/Logic/Encodable/Basic.lean", "def_pos": [472, 9], "def_end_pos": [472, 27]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\ninst\u271d : Countable \u03b9\ns : \u03b9 \u2192 Set \u03b1\nhd : Directed (fun x x_1 => x \u2286 x_1) s\n\u22a2 \u2191\u2191\u03bc (\u22c3 i, s i) = \u2a06 i, \u2191\u2191\u03bc (s i)", "state_after": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\ninst\u271d : Countable \u03b9\ns : \u03b9 \u2192 Set \u03b1\nhd : Directed (fun x x_1 => x \u2286 x_1) s\nval\u271d : Encodable \u03b9\n\u22a2 \u2191\u2191\u03bc (\u22c3 i, s i) = \u2a06 i, \u2191\u2191\u03bc (s i)"}, {"tactic": "generalize ht : Function.extend Encodable.encode s \u22a5 = t", "annotated_tactic": ["generalize ht : <a>Function.extend</a> <a>Encodable.encode</a> s \u22a5 = t", [{"full_name": "Function.extend", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [711, 5], "def_end_pos": [711, 11]}, {"full_name": "Encodable.encode", "def_path": "Mathlib/Logic/Encodable/Basic.lean", "def_pos": [47, 3], "def_end_pos": [47, 9]}]], "state_before": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\ninst\u271d : Countable \u03b9\ns : \u03b9 \u2192 Set \u03b1\nhd : Directed (fun x x_1 => x \u2286 x_1) s\nval\u271d : Encodable \u03b9\n\u22a2 \u2191\u2191\u03bc (\u22c3 i, s i) = \u2a06 i, \u2191\u2191\u03bc (s i)", "state_after": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t\u271d : Set \u03b1\ninst\u271d : Countable \u03b9\ns : \u03b9 \u2192 Set \u03b1\nhd : Directed (fun x x_1 => x \u2286 x_1) s\nval\u271d : Encodable \u03b9\nt : \u2115 \u2192 Set \u03b1\nht : Function.extend Encodable.encode s \u22a5 = t\n\u22a2 \u2191\u2191\u03bc (\u22c3 i, s i) = \u2a06 i, \u2191\u2191\u03bc (s i)"}, {"tactic": "clear! \u03b9", "annotated_tactic": ["clear! \u03b9", []], "state_before": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t\u271d : Set \u03b1\ninst\u271d : Countable \u03b9\ns : \u03b9 \u2192 Set \u03b1\nval\u271d : Encodable \u03b9\nt : \u2115 \u2192 Set \u03b1\nht : Function.extend Encodable.encode s \u22a5 = t\nhd : Directed (fun x x_1 => x \u2286 x_1) t\n\u22a2 \u2191\u2191\u03bc (\u22c3 n, t n) = \u2a06 n, \u2191\u2191\u03bc (t n)", "state_after": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns s\u2081 s\u2082 t\u271d : Set \u03b1\nt : \u2115 \u2192 Set \u03b1\nhd : Directed (fun x x_1 => x \u2286 x_1) t\n\u22a2 \u2191\u2191\u03bc (\u22c3 n, t n) = \u2a06 n, \u2191\u2191\u03bc (t n)"}, {"tactic": "refine' le_antisymm _ (iSup_le fun i => measure_mono <| subset_iUnion _ _)", "annotated_tactic": ["refine' <a>le_antisymm</a> _ (<a>iSup_le</a> fun i => <a>measure_mono</a> <| <a>subset_iUnion</a> _ _)", [{"full_name": "le_antisymm", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [188, 9], "def_end_pos": [188, 20]}, {"full_name": "iSup_le", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [875, 9], "def_end_pos": [875, 16]}, {"full_name": "MeasureTheory.measure_mono", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [193, 9], "def_end_pos": [193, 21]}, {"full_name": "Set.subset_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [431, 9], "def_end_pos": [431, 22]}]], "state_before": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns s\u2081 s\u2082 t\u271d : Set \u03b1\nt : \u2115 \u2192 Set \u03b1\nhd : Directed (fun x x_1 => x \u2286 x_1) t\n\u22a2 \u2191\u2191\u03bc (\u22c3 n, t n) = \u2a06 n, \u2191\u2191\u03bc (t n)", "state_after": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns s\u2081 s\u2082 t\u271d : Set \u03b1\nt : \u2115 \u2192 Set \u03b1\nhd : Directed (fun x x_1 => x \u2286 x_1) t\n\u22a2 \u2191\u2191\u03bc (\u22c3 n, t n) \u2264 \u2a06 n, \u2191\u2191\u03bc (t n)"}, {"tactic": "set T : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable \u03bc (t n)", "annotated_tactic": ["set T : \u2115 \u2192 <a>Set</a> \u03b1 := fun n => <a>toMeasurable</a> \u03bc (t n)", [{"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}, {"full_name": "MeasureTheory.toMeasurable", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [626, 17], "def_end_pos": [626, 29]}]], "state_before": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns s\u2081 s\u2082 t\u271d : Set \u03b1\nt : \u2115 \u2192 Set \u03b1\nhd : Directed (fun x x_1 => x \u2286 x_1) t\n\u22a2 \u2191\u2191\u03bc (\u22c3 n, t n) \u2264 \u2a06 n, \u2191\u2191\u03bc (t n)", "state_after": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns s\u2081 s\u2082 t\u271d : Set \u03b1\nt : \u2115 \u2192 Set \u03b1\nhd : Directed (fun x x_1 => x \u2286 x_1) t\nT : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable \u03bc (t n)\n\u22a2 \u2191\u2191\u03bc (\u22c3 n, t n) \u2264 \u2a06 n, \u2191\u2191\u03bc (t n)"}, {"tactic": "set Td : \u2115 \u2192 Set \u03b1 := disjointed T", "annotated_tactic": ["set Td : \u2115 \u2192 <a>Set</a> \u03b1 := <a>disjointed</a> T", [{"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}, {"full_name": "disjointed", "def_path": "Mathlib/Order/Disjointed.lean", "def_pos": [49, 5], "def_end_pos": [49, 15]}]], "state_before": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns s\u2081 s\u2082 t\u271d : Set \u03b1\nt : \u2115 \u2192 Set \u03b1\nhd : Directed (fun x x_1 => x \u2286 x_1) t\nT : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable \u03bc (t n)\n\u22a2 \u2191\u2191\u03bc (\u22c3 n, t n) \u2264 \u2a06 n, \u2191\u2191\u03bc (t n)", "state_after": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns s\u2081 s\u2082 t\u271d : Set \u03b1\nt : \u2115 \u2192 Set \u03b1\nhd : Directed (fun x x_1 => x \u2286 x_1) t\nT : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable \u03bc (t n)\nTd : \u2115 \u2192 Set \u03b1 := disjointed T\n\u22a2 \u2191\u2191\u03bc (\u22c3 n, t n) \u2264 \u2a06 n, \u2191\u2191\u03bc (t n)"}, {"tactic": "have hm : \u2200 n, MeasurableSet (Td n) :=\n  MeasurableSet.disjointed fun n => measurableSet_toMeasurable _ _", "annotated_tactic": ["have hm : \u2200 n, <a>MeasurableSet</a> (Td n) :=\n    <a>MeasurableSet.disjointed</a> fun n => <a>measurableSet_toMeasurable</a> _ _", [{"full_name": "MeasurableSet", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [64, 5], "def_end_pos": [64, 18]}, {"full_name": "MeasurableSet.disjointed", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [236, 19], "def_end_pos": [236, 43]}, {"full_name": "MeasureTheory.measurableSet_toMeasurable", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [645, 9], "def_end_pos": [645, 35]}]], "state_before": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns s\u2081 s\u2082 t\u271d : Set \u03b1\nt : \u2115 \u2192 Set \u03b1\nhd : Directed (fun x x_1 => x \u2286 x_1) t\nT : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable \u03bc (t n)\nTd : \u2115 \u2192 Set \u03b1 := disjointed T\n\u22a2 \u2191\u2191\u03bc (\u22c3 n, t n) \u2264 \u2a06 n, \u2191\u2191\u03bc (t n)", "state_after": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns s\u2081 s\u2082 t\u271d : Set \u03b1\nt : \u2115 \u2192 Set \u03b1\nhd : Directed (fun x x_1 => x \u2286 x_1) t\nT : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable \u03bc (t n)\nTd : \u2115 \u2192 Set \u03b1 := disjointed T\nhm : \u2200 (n : \u2115), MeasurableSet (Td n)\n\u22a2 \u2191\u2191\u03bc (\u22c3 n, t n) \u2264 \u2a06 n, \u2191\u2191\u03bc (t n)"}, {"tactic": "calc\n  \u03bc (\u22c3 n, t n) \u2264 \u03bc (\u22c3 n, T n) := measure_mono (iUnion_mono fun i => subset_toMeasurable _ _)\n  _ = \u03bc (\u22c3 n, Td n) := by rw [iUnion_disjointed]\n  _ \u2264 \u2211' n, \u03bc (Td n) := (measure_iUnion_le _)\n  _ = \u2a06 I : Finset \u2115, \u2211 n in I, \u03bc (Td n) := ENNReal.tsum_eq_iSup_sum\n  _ \u2264 \u2a06 n, \u03bc (t n) := iSup_le fun I => by\n    rcases hd.finset_le I with \u27e8N, hN\u27e9\n    calc\n      (\u2211 n in I, \u03bc (Td n)) = \u03bc (\u22c3 n \u2208 I, Td n) :=\n        (measure_biUnion_finset ((disjoint_disjointed T).set_pairwise I) fun n _ => hm n).symm\n      _ \u2264 \u03bc (\u22c3 n \u2208 I, T n) := (measure_mono (iUnion\u2082_mono fun n _hn => disjointed_subset _ _))\n      _ = \u03bc (\u22c3 n \u2208 I, t n) := (measure_biUnion_toMeasurable I.countable_toSet _)\n      _ \u2264 \u03bc (t N) := (measure_mono (iUnion\u2082_subset hN))\n      _ \u2264 \u2a06 n, \u03bc (t n) := le_iSup (\u03bc \u2218 t) N", "annotated_tactic": ["calc\n    \u03bc (\u22c3 n, t n) \u2264 \u03bc (\u22c3 n, T n) := <a>measure_mono</a> (<a>iUnion_mono</a> fun i => <a>subset_toMeasurable</a> _ _)\n    _ = \u03bc (\u22c3 n, Td n) := by rw [<a>iUnion_disjointed</a>]\n    _ \u2264 \u2211' n, \u03bc (Td n) := (<a>measure_iUnion_le</a> _)\n    _ = \u2a06 I : <a>Finset</a> \u2115, \u2211 n in I, \u03bc (Td n) := <a>ENNReal.tsum_eq_iSup_sum</a>\n    _ \u2264 \u2a06 n, \u03bc (t n) := <a>iSup_le</a> fun I => by\n      rcases hd.finset_le I with \u27e8N, hN\u27e9\n      calc\n        (\u2211 n in I, \u03bc (Td n)) = \u03bc (\u22c3 n \u2208 I, Td n) :=\n          (<a>measure_biUnion_finset</a> ((<a>disjoint_disjointed</a> T).<a>set_pairwise</a> I) fun n _ => hm n).<a>symm</a>\n        _ \u2264 \u03bc (\u22c3 n \u2208 I, T n) := (<a>measure_mono</a> (<a>iUnion\u2082_mono</a> fun n _hn => <a>disjointed_subset</a> _ _))\n        _ = \u03bc (\u22c3 n \u2208 I, t n) := (<a>measure_biUnion_toMeasurable</a> I.countable_toSet _)\n        _ \u2264 \u03bc (t N) := (<a>measure_mono</a> (<a>iUnion\u2082_subset</a> hN))\n        _ \u2264 \u2a06 n, \u03bc (t n) := <a>le_iSup</a> (\u03bc \u2218 t) N", [{"full_name": "MeasureTheory.measure_mono", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [193, 9], "def_end_pos": [193, 21]}, {"full_name": "Set.iUnion_mono", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [478, 9], "def_end_pos": [478, 20]}, {"full_name": "MeasureTheory.subset_toMeasurable", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [633, 9], "def_end_pos": [633, 28]}, {"full_name": "iUnion_disjointed", "def_path": "Mathlib/Order/Disjointed.lean", "def_pos": [165, 9], "def_end_pos": [165, 26]}, {"full_name": "MeasureTheory.measure_iUnion_le", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [240, 9], "def_end_pos": [240, 26]}, {"full_name": "Finset", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [138, 11], "def_end_pos": [138, 17]}, {"full_name": "ENNReal.tsum_eq_iSup_sum", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [789, 19], "def_end_pos": [789, 35]}, {"full_name": "iSup_le", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [875, 9], "def_end_pos": [875, 16]}, {"full_name": "MeasureTheory.measure_biUnion_finset", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [194, 9], "def_end_pos": [194, 31]}, {"full_name": "disjoint_disjointed", "def_path": "Mathlib/Order/Disjointed.lean", "def_pos": [74, 9], "def_end_pos": [74, 28]}, {"full_name": "Pairwise.set_pairwise", "def_path": "Mathlib/Logic/Pairwise.lean", "def_pos": [89, 9], "def_end_pos": [89, 30]}, {"full_name": "Eq.symm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [310, 9], "def_end_pos": [310, 16]}, {"full_name": "MeasureTheory.measure_mono", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [193, 9], "def_end_pos": [193, 21]}, {"full_name": "Set.iUnion\u2082_mono", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [488, 9], "def_end_pos": [488, 21]}, {"full_name": "disjointed_subset", "def_path": "Mathlib/Order/Disjointed.lean", "def_pos": [161, 9], "def_end_pos": [161, 26]}, {"full_name": "MeasureTheory.measure_biUnion_toMeasurable", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [379, 9], "def_end_pos": [379, 37]}, {"full_name": "MeasureTheory.measure_mono", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [193, 9], "def_end_pos": [193, 21]}, {"full_name": "Set.iUnion\u2082_subset", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [395, 9], "def_end_pos": [395, 23]}, {"full_name": "le_iSup", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [810, 9], "def_end_pos": [810, 16]}]], "state_before": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns s\u2081 s\u2082 t\u271d : Set \u03b1\nt : \u2115 \u2192 Set \u03b1\nhd : Directed (fun x x_1 => x \u2286 x_1) t\nT : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable \u03bc (t n)\nTd : \u2115 \u2192 Set \u03b1 := disjointed T\nhm : \u2200 (n : \u2115), MeasurableSet (Td n)\n\u22a2 \u2191\u2191\u03bc (\u22c3 n, t n) \u2264 \u2a06 n, \u2191\u2191\u03bc (t n)", "state_after": "no goals"}, {"tactic": "exact this.trans (iSup_extend_bot Encodable.encode_injective _)", "annotated_tactic": ["exact this.trans (<a>iSup_extend_bot</a> <a>Encodable.encode_injective</a> _)", [{"full_name": "iSup_extend_bot", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [1543, 9], "def_end_pos": [1543, 24]}, {"full_name": "Encodable.encode_injective", "def_path": "Mathlib/Logic/Encodable/Basic.lean", "def_pos": [64, 9], "def_end_pos": [64, 25]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t\u271d : Set \u03b1\ninst\u271d : Countable \u03b9\ns : \u03b9 \u2192 Set \u03b1\nval\u271d : Encodable \u03b9\nt : \u2115 \u2192 Set \u03b1\nht : Function.extend Encodable.encode s \u22a5 = t\nhd : Directed (fun x x_1 => x \u2286 x_1) t\nthis : \u2191\u2191\u03bc (\u2a06 i, s i) = \u2a06 n, Function.extend Encodable.encode (fun x => \u2191\u2191\u03bc (s x)) (fun x => 0) n\n\u22a2 \u2191\u2191\u03bc (\u22c3 i, s i) = \u2a06 i, \u2191\u2191\u03bc (s i)", "state_after": "no goals"}, {"tactic": "rw [iUnion_disjointed]", "annotated_tactic": ["rw [<a>iUnion_disjointed</a>]", [{"full_name": "iUnion_disjointed", "def_path": "Mathlib/Order/Disjointed.lean", "def_pos": [165, 9], "def_end_pos": [165, 26]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns s\u2081 s\u2082 t\u271d : Set \u03b1\nt : \u2115 \u2192 Set \u03b1\nhd : Directed (fun x x_1 => x \u2286 x_1) t\nT : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable \u03bc (t n)\nTd : \u2115 \u2192 Set \u03b1 := disjointed T\nhm : \u2200 (n : \u2115), MeasurableSet (Td n)\n\u22a2 \u2191\u2191\u03bc (\u22c3 n, T n) = \u2191\u2191\u03bc (\u22c3 n, Td n)", "state_after": "no goals"}, {"tactic": "rcases hd.finset_le I with \u27e8N, hN\u27e9", "annotated_tactic": ["rcases hd.finset_le I with \u27e8N, hN\u27e9", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns s\u2081 s\u2082 t\u271d : Set \u03b1\nt : \u2115 \u2192 Set \u03b1\nhd : Directed (fun x x_1 => x \u2286 x_1) t\nT : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable \u03bc (t n)\nTd : \u2115 \u2192 Set \u03b1 := disjointed T\nhm : \u2200 (n : \u2115), MeasurableSet (Td n)\nI : Finset \u2115\n\u22a2 \u2211 n in I, \u2191\u2191\u03bc (Td n) \u2264 \u2a06 n, \u2191\u2191\u03bc (t n)", "state_after": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns s\u2081 s\u2082 t\u271d : Set \u03b1\nt : \u2115 \u2192 Set \u03b1\nhd : Directed (fun x x_1 => x \u2286 x_1) t\nT : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable \u03bc (t n)\nTd : \u2115 \u2192 Set \u03b1 := disjointed T\nhm : \u2200 (n : \u2115), MeasurableSet (Td n)\nI : Finset \u2115\nN : \u2115\nhN : \u2200 (i : \u2115), i \u2208 I \u2192 t i \u2286 t N\n\u22a2 \u2211 n in I, \u2191\u2191\u03bc (Td n) \u2264 \u2a06 n, \u2191\u2191\u03bc (t n)"}, {"tactic": "calc\n  (\u2211 n in I, \u03bc (Td n)) = \u03bc (\u22c3 n \u2208 I, Td n) :=\n    (measure_biUnion_finset ((disjoint_disjointed T).set_pairwise I) fun n _ => hm n).symm\n  _ \u2264 \u03bc (\u22c3 n \u2208 I, T n) := (measure_mono (iUnion\u2082_mono fun n _hn => disjointed_subset _ _))\n  _ = \u03bc (\u22c3 n \u2208 I, t n) := (measure_biUnion_toMeasurable I.countable_toSet _)\n  _ \u2264 \u03bc (t N) := (measure_mono (iUnion\u2082_subset hN))\n  _ \u2264 \u2a06 n, \u03bc (t n) := le_iSup (\u03bc \u2218 t) N", "annotated_tactic": ["calc\n        (\u2211 n in I, \u03bc (Td n)) = \u03bc (\u22c3 n \u2208 I, Td n) :=\n          (<a>measure_biUnion_finset</a> ((<a>disjoint_disjointed</a> T).<a>set_pairwise</a> I) fun n _ => hm n).<a>symm</a>\n        _ \u2264 \u03bc (\u22c3 n \u2208 I, T n) := (<a>measure_mono</a> (<a>iUnion\u2082_mono</a> fun n _hn => <a>disjointed_subset</a> _ _))\n        _ = \u03bc (\u22c3 n \u2208 I, t n) := (<a>measure_biUnion_toMeasurable</a> I.countable_toSet _)\n        _ \u2264 \u03bc (t N) := (<a>measure_mono</a> (<a>iUnion\u2082_subset</a> hN))\n        _ \u2264 \u2a06 n, \u03bc (t n) := <a>le_iSup</a> (\u03bc \u2218 t) N", [{"full_name": "MeasureTheory.measure_biUnion_finset", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [194, 9], "def_end_pos": [194, 31]}, {"full_name": "disjoint_disjointed", "def_path": "Mathlib/Order/Disjointed.lean", "def_pos": [74, 9], "def_end_pos": [74, 28]}, {"full_name": "Pairwise.set_pairwise", "def_path": "Mathlib/Logic/Pairwise.lean", "def_pos": [89, 9], "def_end_pos": [89, 30]}, {"full_name": "Eq.symm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [310, 9], "def_end_pos": [310, 16]}, {"full_name": "MeasureTheory.measure_mono", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [193, 9], "def_end_pos": [193, 21]}, {"full_name": "Set.iUnion\u2082_mono", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [488, 9], "def_end_pos": [488, 21]}, {"full_name": "disjointed_subset", "def_path": "Mathlib/Order/Disjointed.lean", "def_pos": [161, 9], "def_end_pos": [161, 26]}, {"full_name": "MeasureTheory.measure_biUnion_toMeasurable", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [379, 9], "def_end_pos": [379, 37]}, {"full_name": "MeasureTheory.measure_mono", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [193, 9], "def_end_pos": [193, 21]}, {"full_name": "Set.iUnion\u2082_subset", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [395, 9], "def_end_pos": [395, 23]}, {"full_name": "le_iSup", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [810, 9], "def_end_pos": [810, 16]}]], "state_before": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns s\u2081 s\u2082 t\u271d : Set \u03b1\nt : \u2115 \u2192 Set \u03b1\nhd : Directed (fun x x_1 => x \u2286 x_1) t\nT : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable \u03bc (t n)\nTd : \u2115 \u2192 Set \u03b1 := disjointed T\nhm : \u2200 (n : \u2115), MeasurableSet (Td n)\nI : Finset \u2115\nN : \u2115\nhN : \u2200 (i : \u2115), i \u2208 I \u2192 t i \u2286 t N\n\u22a2 \u2211 n in I, \u2191\u2191\u03bc (Td n) \u2264 \u2a06 n, \u2191\u2191\u03bc (t n)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Logic.lean", "full_name": "Decidable.imp_iff_not_or", "start": [570, 1], "end": [571, 39], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "full_name": "intervalIntegral.norm_integral_le_of_norm_le", "start": [557, 8], "end": [561, 46], "traced_tactics": [{"tactic": "simp_rw [norm_intervalIntegral_eq, abs_intervalIntegral_eq,\n  abs_eq_self.mpr (integral_nonneg_of_ae <| h.mono fun _t ht => (norm_nonneg _).trans ht),\n  norm_integral_le_of_norm_le hbound.def h]", "annotated_tactic": ["simp_rw [<a>norm_intervalIntegral_eq</a>, <a>abs_intervalIntegral_eq</a>,\n    abs_eq_self.mpr (<a>integral_nonneg_of_ae</a> <| h.mono fun _t ht => (<a>norm_nonneg</a> _).<a>trans</a> ht),\n    <a>norm_integral_le_of_norm_le</a> hbound.def h]", [{"full_name": "intervalIntegral.norm_intervalIntegral_eq", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [489, 9], "def_end_pos": [489, 33]}, {"full_name": "intervalIntegral.abs_intervalIntegral_eq", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [495, 9], "def_end_pos": [495, 32]}, {"full_name": "MeasureTheory.integral_nonneg_of_ae", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1201, 9], "def_end_pos": [1201, 30]}, {"full_name": "norm_nonneg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [500, 30], "def_end_pos": [500, 41]}, {"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [120, 7], "def_end_pos": [120, 18]}, {"full_name": "MeasureTheory.norm_integral_le_of_norm_le", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1389, 9], "def_end_pos": [1389, 36]}]], "state_before": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\na b : \u211d\nf g\u271d : \u211d \u2192 E\n\u03bc : Measure \u211d\ng : \u211d \u2192 \u211d\nh : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict \u03bc (\u0399 a b), \u2016f t\u2016 \u2264 g t\nhbound : IntervalIntegrable g \u03bc a b\n\u22a2 \u2016\u222b (t : \u211d) in a..b, f t \u2202\u03bc\u2016 \u2264 |\u222b (t : \u211d) in a..b, g t \u2202\u03bc|", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/ProbabilityMassFunction/Constructions.lean", "full_name": "PMF.support_filter", "start": [287, 1], "end": [288, 44], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Sigma.lean", "full_name": "Set.fst_image_sigma", "start": [242, 1], "end": [245, 28], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Primrec.lean", "full_name": "Primrec.nat_mul", "start": [674, 1], "end": [675, 39], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/Basic.lean", "full_name": "MvPolynomial.eval\u2082_mul", "start": [1025, 1], "end": [1029, 97], "traced_tactics": [{"tactic": "apply MvPolynomial.induction_on q", "annotated_tactic": ["apply <a>MvPolynomial.induction_on</a> q", [{"full_name": "MvPolynomial.induction_on", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [451, 9], "def_end_pos": [451, 21]}]], "state_before": "R : Type u\nS\u2081 : Type v\nS\u2082 : Type w\nS\u2083 : Type x\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : CommSemiring S\u2081\np q : MvPolynomial \u03c3 R\nf : R \u2192+* S\u2081\ng : \u03c3 \u2192 S\u2081\n\u22a2 \u2200 {p : MvPolynomial \u03c3 R}, eval\u2082 f g (p * q) = eval\u2082 f g p * eval\u2082 f g q", "state_after": "case h_C\nR : Type u\nS\u2081 : Type v\nS\u2082 : Type w\nS\u2083 : Type x\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : CommSemiring S\u2081\np q : MvPolynomial \u03c3 R\nf : R \u2192+* S\u2081\ng : \u03c3 \u2192 S\u2081\n\u22a2 \u2200 (a : R) {p : MvPolynomial \u03c3 R}, eval\u2082 f g (p * \u2191C a) = eval\u2082 f g p * eval\u2082 f g (\u2191C a)\n\ncase h_add\nR : Type u\nS\u2081 : Type v\nS\u2082 : Type w\nS\u2083 : Type x\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : CommSemiring S\u2081\np q : MvPolynomial \u03c3 R\nf : R \u2192+* S\u2081\ng : \u03c3 \u2192 S\u2081\n\u22a2 \u2200 (p q : MvPolynomial \u03c3 R),\n    (\u2200 {p_1 : MvPolynomial \u03c3 R}, eval\u2082 f g (p_1 * p) = eval\u2082 f g p_1 * eval\u2082 f g p) \u2192\n      (\u2200 {p : MvPolynomial \u03c3 R}, eval\u2082 f g (p * q) = eval\u2082 f g p * eval\u2082 f g q) \u2192\n        \u2200 {p_1 : MvPolynomial \u03c3 R}, eval\u2082 f g (p_1 * (p + q)) = eval\u2082 f g p_1 * eval\u2082 f g (p + q)\n\ncase h_X\nR : Type u\nS\u2081 : Type v\nS\u2082 : Type w\nS\u2083 : Type x\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : CommSemiring S\u2081\np q : MvPolynomial \u03c3 R\nf : R \u2192+* S\u2081\ng : \u03c3 \u2192 S\u2081\n\u22a2 \u2200 (p : MvPolynomial \u03c3 R) (n : \u03c3),\n    (\u2200 {p_1 : MvPolynomial \u03c3 R}, eval\u2082 f g (p_1 * p) = eval\u2082 f g p_1 * eval\u2082 f g p) \u2192\n      \u2200 {p_1 : MvPolynomial \u03c3 R}, eval\u2082 f g (p_1 * (p * X n)) = eval\u2082 f g p_1 * eval\u2082 f g (p * X n)"}, {"tactic": "simp [eval\u2082_C, eval\u2082_mul_C]", "annotated_tactic": ["simp [<a>eval\u2082_C</a>, <a>eval\u2082_mul_C</a>]", [{"full_name": "MvPolynomial.eval\u2082_C", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [988, 9], "def_end_pos": [988, 16]}, {"full_name": "MvPolynomial.eval\u2082_mul_C", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [1020, 9], "def_end_pos": [1020, 20]}]], "state_before": "case h_C\nR : Type u\nS\u2081 : Type v\nS\u2082 : Type w\nS\u2083 : Type x\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : CommSemiring S\u2081\np q : MvPolynomial \u03c3 R\nf : R \u2192+* S\u2081\ng : \u03c3 \u2192 S\u2081\n\u22a2 \u2200 (a : R) {p : MvPolynomial \u03c3 R}, eval\u2082 f g (p * \u2191C a) = eval\u2082 f g p * eval\u2082 f g (\u2191C a)", "state_after": "no goals"}, {"tactic": "simp (config := { contextual := true }) [mul_add, eval\u2082_add]", "annotated_tactic": ["simp (config := { contextual := <a>true</a> }) [<a>mul_add</a>, <a>eval\u2082_add</a>]", [{"full_name": "Bool.true", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [549, 5], "def_end_pos": [549, 9]}, {"full_name": "mul_add", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [83, 7], "def_end_pos": [83, 14]}, {"full_name": "MvPolynomial.eval\u2082_add", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [978, 9], "def_end_pos": [978, 18]}]], "state_before": "case h_add\nR : Type u\nS\u2081 : Type v\nS\u2082 : Type w\nS\u2083 : Type x\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : CommSemiring S\u2081\np q : MvPolynomial \u03c3 R\nf : R \u2192+* S\u2081\ng : \u03c3 \u2192 S\u2081\n\u22a2 \u2200 (p q : MvPolynomial \u03c3 R),\n    (\u2200 {p_1 : MvPolynomial \u03c3 R}, eval\u2082 f g (p_1 * p) = eval\u2082 f g p_1 * eval\u2082 f g p) \u2192\n      (\u2200 {p : MvPolynomial \u03c3 R}, eval\u2082 f g (p * q) = eval\u2082 f g p * eval\u2082 f g q) \u2192\n        \u2200 {p_1 : MvPolynomial \u03c3 R}, eval\u2082 f g (p_1 * (p + q)) = eval\u2082 f g p_1 * eval\u2082 f g (p + q)", "state_after": "no goals"}, {"tactic": "simp (config := { contextual := true }) [X, eval\u2082_monomial, eval\u2082_mul_monomial, \u2190 mul_assoc]", "annotated_tactic": ["simp (config := { contextual := <a>true</a> }) [<a>X</a>, <a>eval\u2082_monomial</a>, <a>eval\u2082_mul_monomial</a>, \u2190 <a>mul_assoc</a>]", [{"full_name": "Bool.true", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [549, 5], "def_end_pos": [549, 9]}, {"full_name": "MvPolynomial.X", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [193, 5], "def_end_pos": [193, 6]}, {"full_name": "MvPolynomial.eval\u2082_monomial", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [983, 9], "def_end_pos": [983, 23]}, {"full_name": "MvPolynomial.eval\u2082_mul_monomial", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [1002, 9], "def_end_pos": [1002, 27]}, {"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [264, 9], "def_end_pos": [264, 18]}]], "state_before": "case h_X\nR : Type u\nS\u2081 : Type v\nS\u2082 : Type w\nS\u2083 : Type x\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : CommSemiring S\u2081\np q : MvPolynomial \u03c3 R\nf : R \u2192+* S\u2081\ng : \u03c3 \u2192 S\u2081\n\u22a2 \u2200 (p : MvPolynomial \u03c3 R) (n : \u03c3),\n    (\u2200 {p_1 : MvPolynomial \u03c3 R}, eval\u2082 f g (p_1 * p) = eval\u2082 f g p_1 * eval\u2082 f g p) \u2192\n      \u2200 {p_1 : MvPolynomial \u03c3 R}, eval\u2082 f g (p_1 * (p * X n)) = eval\u2082 f g p_1 * eval\u2082 f g (p * X n)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "full_name": "List.cons_eq_append", "start": [127, 1], "end": [129, 31], "traced_tactics": [{"tactic": "rw [eq_comm, append_eq_cons]", "annotated_tactic": ["rw [<a>eq_comm</a>, <a>append_eq_cons</a>]", [{"full_name": "eq_comm", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [104, 9], "def_end_pos": [104, 16]}, {"full_name": "List.append_eq_cons", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [122, 9], "def_end_pos": [122, 23]}]], "state_before": "\u03b1\u271d : Type u_1\nx : \u03b1\u271d\nc a b : List \u03b1\u271d\n\u22a2 x :: c = a ++ b \u2194 a = [] \u2227 b = x :: c \u2228 \u2203 a', a = x :: a' \u2227 c = a' ++ b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "full_name": "MeasureTheory.snorm'_eq_zero_of_ae_zero'", "start": [725, 1], "end": [726, 80], "traced_tactics": [{"tactic": "rw [snorm'_congr_ae hf_zero, snorm'_zero' hq0_ne h\u03bc]", "annotated_tactic": ["rw [<a>snorm'_congr_ae</a> hf_zero, <a>snorm'_zero'</a> hq0_ne h\u03bc]", [{"full_name": "MeasureTheory.snorm'_congr_ae", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [397, 9], "def_end_pos": [397, 24]}, {"full_name": "MeasureTheory.snorm'_zero'", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [188, 9], "def_end_pos": [188, 21]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nhq0_ne : q \u2260 0\nh\u03bc : \u03bc \u2260 0\nf : \u03b1 \u2192 F\nhf_zero : f =\u1d50[\u03bc] 0\n\u22a2 snorm' f q \u03bc = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Prod/Basic.lean", "full_name": "Prod.map_bijective", "start": [385, 1], "end": [389, 68], "traced_tactics": [{"tactic": "haveI := Nonempty.map f \u2039_\u203a", "annotated_tactic": ["haveI := <a>Nonempty.map</a> f \u2039_\u203a", [{"full_name": "Nonempty.map", "def_path": "Mathlib/Logic/Nonempty.lean", "def_pos": [140, 9], "def_end_pos": [140, 21]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : Nonempty \u03b1\ninst\u271d : Nonempty \u03b2\nf : \u03b1 \u2192 \u03b3\ng : \u03b2 \u2192 \u03b4\n\u22a2 Bijective (map f g) \u2194 Bijective f \u2227 Bijective g", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : Nonempty \u03b1\ninst\u271d : Nonempty \u03b2\nf : \u03b1 \u2192 \u03b3\ng : \u03b2 \u2192 \u03b4\nthis : Nonempty \u03b3\n\u22a2 Bijective (map f g) \u2194 Bijective f \u2227 Bijective g"}, {"tactic": "haveI := Nonempty.map g \u2039_\u203a", "annotated_tactic": ["haveI := <a>Nonempty.map</a> g \u2039_\u203a", [{"full_name": "Nonempty.map", "def_path": "Mathlib/Logic/Nonempty.lean", "def_pos": [140, 9], "def_end_pos": [140, 21]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : Nonempty \u03b1\ninst\u271d : Nonempty \u03b2\nf : \u03b1 \u2192 \u03b3\ng : \u03b2 \u2192 \u03b4\nthis : Nonempty \u03b3\n\u22a2 Bijective (map f g) \u2194 Bijective f \u2227 Bijective g", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : Nonempty \u03b1\ninst\u271d : Nonempty \u03b2\nf : \u03b1 \u2192 \u03b3\ng : \u03b2 \u2192 \u03b4\nthis\u271d : Nonempty \u03b3\nthis : Nonempty \u03b4\n\u22a2 Bijective (map f g) \u2194 Bijective f \u2227 Bijective g"}, {"tactic": "exact (map_injective.and map_surjective).trans (and_and_and_comm)", "annotated_tactic": ["exact (map_injective.and <a>map_surjective</a>).<a>trans</a> (<a>and_and_and_comm</a>)", [{"full_name": "Prod.map_surjective", "def_path": "Mathlib/Data/Prod/Basic.lean", "def_pos": [370, 9], "def_end_pos": [370, 23]}, {"full_name": "Iff.trans", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [666, 9], "def_end_pos": [666, 18]}, {"full_name": "and_and_and_comm", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [189, 9], "def_end_pos": [189, 25]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : Nonempty \u03b1\ninst\u271d : Nonempty \u03b2\nf : \u03b1 \u2192 \u03b3\ng : \u03b2 \u2192 \u03b4\nthis\u271d : Nonempty \u03b3\nthis : Nonempty \u03b4\n\u22a2 Bijective (map f g) \u2194 Bijective f \u2227 Bijective g", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "full_name": "MeasureTheory.Ico_ae_eq_Ioc", "start": [3188, 1], "end": [3189, 61], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/Equiv.lean", "full_name": "MvPolynomial.finSuccEquiv_eq", "start": [320, 1], "end": [328, 52], "traced_tactics": [{"tactic": "ext i : 2", "annotated_tactic": ["ext i : 2", []], "state_before": "R : Type u\nS\u2081 : Type v\nS\u2082 : Type w\nS\u2083 : Type x\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\nn : \u2115\n\u22a2 \u2191(finSuccEquiv R n) =\n    eval\u2082Hom (RingHom.comp Polynomial.C C) fun i => Fin.cases Polynomial.X (fun k => \u2191Polynomial.C (X k)) i", "state_after": "case hC.a\nR : Type u\nS\u2081 : Type v\nS\u2082 : Type w\nS\u2083 : Type x\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\nn : \u2115\ni : R\n\u22a2 \u2191(RingHom.comp (\u2191(finSuccEquiv R n)) C) i =\n    \u2191(RingHom.comp\n          (eval\u2082Hom (RingHom.comp Polynomial.C C) fun i => Fin.cases Polynomial.X (fun k => \u2191Polynomial.C (X k)) i) C)\n      i\n\ncase hX.a\nR : Type u\nS\u2081 : Type v\nS\u2082 : Type w\nS\u2083 : Type x\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\nn : \u2115\ni : Fin (n + 1)\nn\u271d : \u2115\n\u22a2 Polynomial.coeff (\u2191\u2191(finSuccEquiv R n) (X i)) n\u271d =\n    Polynomial.coeff\n      (\u2191(eval\u2082Hom (RingHom.comp Polynomial.C C) fun i => Fin.cases Polynomial.X (fun k => \u2191Polynomial.C (X k)) i) (X i))\n      n\u271d"}, {"tactic": "simp only [finSuccEquiv, optionEquivLeft_apply, aeval_C, AlgEquiv.coe_trans, RingHom.coe_coe,\n  coe_eval\u2082Hom, comp_apply, renameEquiv_apply, eval\u2082_C, RingHom.coe_comp, rename_C]", "annotated_tactic": ["simp only [<a>finSuccEquiv</a>, <a>optionEquivLeft_apply</a>, <a>aeval_C</a>, <a>AlgEquiv.coe_trans</a>, <a>RingHom.coe_coe</a>,\n      <a>coe_eval\u2082Hom</a>, <a>comp_apply</a>, <a>renameEquiv_apply</a>, <a>eval\u2082_C</a>, <a>RingHom.coe_comp</a>, <a>rename_C</a>]", [{"full_name": "MvPolynomial.finSuccEquiv", "def_path": "Mathlib/Data/MvPolynomial/Equiv.lean", "def_pos": [316, 5], "def_end_pos": [316, 17]}, {"full_name": "MvPolynomial.optionEquivLeft_apply", "def_path": "Mathlib/Data/MvPolynomial/Equiv.lean", "def_pos": [285, 3], "def_end_pos": [285, 9]}, {"full_name": "MvPolynomial.aeval_C", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [1479, 9], "def_end_pos": [1479, 16]}, {"full_name": "AlgEquiv.coe_trans", "def_path": "Mathlib/Algebra/Algebra/Equiv.lean", "def_pos": [418, 9], "def_end_pos": [418, 18]}, {"full_name": "RingHom.coe_coe", "def_path": "Mathlib/Algebra/Hom/Ring/Defs.lean", "def_pos": [454, 9], "def_end_pos": [454, 16]}, {"full_name": "MvPolynomial.coe_eval\u2082Hom", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [1051, 9], "def_end_pos": [1051, 21]}, {"full_name": "Function.comp_apply", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [33, 17], "def_end_pos": [33, 36]}, {"full_name": "MvPolynomial.renameEquiv_apply", "def_path": "Mathlib/Data/MvPolynomial/Rename.lean", "def_pos": [154, 9], "def_end_pos": [154, 14]}, {"full_name": "MvPolynomial.eval\u2082_C", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [988, 9], "def_end_pos": [988, 16]}, {"full_name": "RingHom.coe_comp", "def_path": "Mathlib/Algebra/Hom/Ring/Defs.lean", "def_pos": [668, 9], "def_end_pos": [668, 17]}, {"full_name": "MvPolynomial.rename_C", "def_path": "Mathlib/Data/MvPolynomial/Rename.lean", "def_pos": [61, 9], "def_end_pos": [61, 17]}]], "state_before": "case hC.a\nR : Type u\nS\u2081 : Type v\nS\u2082 : Type w\nS\u2083 : Type x\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\nn : \u2115\ni : R\n\u22a2 \u2191(RingHom.comp (\u2191(finSuccEquiv R n)) C) i =\n    \u2191(RingHom.comp\n          (eval\u2082Hom (RingHom.comp Polynomial.C C) fun i => Fin.cases Polynomial.X (fun k => \u2191Polynomial.C (X k)) i) C)\n      i", "state_after": "case hC.a\nR : Type u\nS\u2081 : Type v\nS\u2082 : Type w\nS\u2083 : Type x\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\nn : \u2115\ni : R\n\u22a2 \u2191(algebraMap R (MvPolynomial (Fin n) R)[X]) i = \u2191Polynomial.C (\u2191C i)"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case hC.a\nR : Type u\nS\u2081 : Type v\nS\u2082 : Type w\nS\u2083 : Type x\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\nn : \u2115\ni : R\n\u22a2 \u2191(algebraMap R (MvPolynomial (Fin n) R)[X]) i = \u2191Polynomial.C (\u2191C i)", "state_after": "no goals"}, {"tactic": "refine' Fin.cases _ _ i <;> simp [finSuccEquiv]", "annotated_tactic": ["refine' <a>Fin.cases</a> _ _ i <;> simp [<a>finSuccEquiv</a>]", [{"full_name": "Fin.cases", "def_path": "lake-packages/std/Std/Data/Fin/Lemmas.lean", "def_pos": [613, 21], "def_end_pos": [613, 26]}, {"full_name": "MvPolynomial.finSuccEquiv", "def_path": "Mathlib/Data/MvPolynomial/Equiv.lean", "def_pos": [316, 5], "def_end_pos": [316, 17]}]], "state_before": "case hX.a\nR : Type u\nS\u2081 : Type v\nS\u2082 : Type w\nS\u2083 : Type x\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\nn : \u2115\ni : Fin (n + 1)\nn\u271d : \u2115\n\u22a2 Polynomial.coeff (\u2191\u2191(finSuccEquiv R n) (X i)) n\u271d =\n    Polynomial.coeff\n      (\u2191(eval\u2082Hom (RingHom.comp Polynomial.C C) fun i => Fin.cases Polynomial.X (fun k => \u2191Polynomial.C (X k)) i) (X i))\n      n\u271d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/Division.lean", "full_name": "MvPolynomial.monomial_modMonomial", "start": [133, 1], "end": [134, 31], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Nat/Gcd.lean", "full_name": "Nat.gcd_div", "start": [117, 1], "end": [124, 54], "traced_tactics": [{"tactic": "simp [H0]", "annotated_tactic": ["simp [H0]", []], "state_before": "m n k : Nat\nH1 : k \u2223 m\nH2 : k \u2223 n\nH0 : k = 0\n\u22a2 gcd (m / k) (n / k) = gcd m n / k", "state_after": "no goals"}, {"tactic": "apply Nat.eq_of_mul_eq_mul_right H3", "annotated_tactic": ["apply <a>Nat.eq_of_mul_eq_mul_right</a> H3", [{"full_name": "Nat.eq_of_mul_eq_mul_right", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [468, 9], "def_end_pos": [468, 31]}]], "state_before": "m n k : Nat\nH1 : k \u2223 m\nH2 : k \u2223 n\nH3 : k > 0\n\u22a2 gcd (m / k) (n / k) = gcd m n / k", "state_after": "m n k : Nat\nH1 : k \u2223 m\nH2 : k \u2223 n\nH3 : k > 0\n\u22a2 gcd (m / k) (n / k) * k = gcd m n / k * k"}, {"tactic": "rw [Nat.div_mul_cancel (dvd_gcd H1 H2), \u2190 gcd_mul_right,\n    Nat.div_mul_cancel H1, Nat.div_mul_cancel H2]", "annotated_tactic": ["rw [<a>Nat.div_mul_cancel</a> (<a>dvd_gcd</a> H1 H2), \u2190 <a>gcd_mul_right</a>,\n        <a>Nat.div_mul_cancel</a> H1, <a>Nat.div_mul_cancel</a> H2]", [{"full_name": "Nat.div_mul_cancel", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [947, 19], "def_end_pos": [947, 33]}, {"full_name": "Nat.dvd_gcd", "def_path": "lake-packages/std/Std/Data/Nat/Gcd.lean", "def_pos": [50, 9], "def_end_pos": [50, 16]}, {"full_name": "Nat.gcd_mul_right", "def_path": "lake-packages/std/Std/Data/Nat/Gcd.lean", "def_pos": [89, 9], "def_end_pos": [89, 22]}, {"full_name": "Nat.div_mul_cancel", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [947, 19], "def_end_pos": [947, 33]}, {"full_name": "Nat.div_mul_cancel", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [947, 19], "def_end_pos": [947, 33]}]], "state_before": "m n k : Nat\nH1 : k \u2223 m\nH2 : k \u2223 n\nH3 : k > 0\n\u22a2 gcd (m / k) (n / k) * k = gcd m n / k * k", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "full_name": "measurable_iSup", "start": [1360, 1], "end": [1375, 13], "traced_tactics": [{"tactic": "rcases isEmpty_or_nonempty \u03b9 with h\u03b9|h\u03b9", "annotated_tactic": ["rcases <a>isEmpty_or_nonempty</a> \u03b9 with h\u03b9|h\u03b9", [{"full_name": "isEmpty_or_nonempty", "def_path": "Mathlib/Logic/IsEmpty.lean", "def_pos": [207, 9], "def_end_pos": [207, 28]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9\u271d : Sort y\ns t u : Set \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u00b9 : BorelSpace \u03b1\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b2\ninst\u271d\u2079 : MeasurableSpace \u03b2\ninst\u271d\u2078 : BorelSpace \u03b2\ninst\u271d\u2077 : TopologicalSpace \u03b3\ninst\u271d\u2076 : MeasurableSpace \u03b3\ninst\u271d\u2075 : BorelSpace \u03b3\ninst\u271d\u2074 : MeasurableSpace \u03b4\ninst\u271d\u00b3 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03b9 : Sort u_6\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nhf : \u2200 (i : \u03b9), Measurable (f i)\n\u22a2 Measurable fun b => \u2a06 i, f i b", "state_after": "case inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9\u271d : Sort y\ns t u : Set \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u00b9 : BorelSpace \u03b1\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b2\ninst\u271d\u2079 : MeasurableSpace \u03b2\ninst\u271d\u2078 : BorelSpace \u03b2\ninst\u271d\u2077 : TopologicalSpace \u03b3\ninst\u271d\u2076 : MeasurableSpace \u03b3\ninst\u271d\u2075 : BorelSpace \u03b3\ninst\u271d\u2074 : MeasurableSpace \u03b4\ninst\u271d\u00b3 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03b9 : Sort u_6\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nhf : \u2200 (i : \u03b9), Measurable (f i)\nh\u03b9 : IsEmpty \u03b9\n\u22a2 Measurable fun b => \u2a06 i, f i b\n\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9\u271d : Sort y\ns t u : Set \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u00b9 : BorelSpace \u03b1\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b2\ninst\u271d\u2079 : MeasurableSpace \u03b2\ninst\u271d\u2078 : BorelSpace \u03b2\ninst\u271d\u2077 : TopologicalSpace \u03b3\ninst\u271d\u2076 : MeasurableSpace \u03b3\ninst\u271d\u2075 : BorelSpace \u03b3\ninst\u271d\u2074 : MeasurableSpace \u03b4\ninst\u271d\u00b3 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03b9 : Sort u_6\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nhf : \u2200 (i : \u03b9), Measurable (f i)\nh\u03b9 : Nonempty \u03b9\n\u22a2 Measurable fun b => \u2a06 i, f i b"}, {"tactic": "have A : MeasurableSet {b | BddAbove (range (fun i \u21a6 f i b))} :=\n  measurableSet_bddAbove_range hf", "annotated_tactic": ["have A : <a>MeasurableSet</a> {b | <a>BddAbove</a> (<a>range</a> (fun i \u21a6 f i b))} :=\n    <a>measurableSet_bddAbove_range</a> hf", [{"full_name": "MeasurableSet", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [64, 5], "def_end_pos": [64, 18]}, {"full_name": "BddAbove", "def_path": "Mathlib/Order/Bounds/Basic.lean", "def_pos": [56, 5], "def_end_pos": [56, 13]}, {"full_name": "Set.range", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [668, 5], "def_end_pos": [668, 10]}, {"full_name": "measurableSet_bddAbove_range", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [1310, 7], "def_end_pos": [1310, 35]}]], "state_before": "case inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9\u271d : Sort y\ns t u : Set \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u00b9 : BorelSpace \u03b1\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b2\ninst\u271d\u2079 : MeasurableSpace \u03b2\ninst\u271d\u2078 : BorelSpace \u03b2\ninst\u271d\u2077 : TopologicalSpace \u03b3\ninst\u271d\u2076 : MeasurableSpace \u03b3\ninst\u271d\u2075 : BorelSpace \u03b3\ninst\u271d\u2074 : MeasurableSpace \u03b4\ninst\u271d\u00b3 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03b9 : Sort u_6\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nhf : \u2200 (i : \u03b9), Measurable (f i)\nh\u03b9 : Nonempty \u03b9\n\u22a2 Measurable fun b => \u2a06 i, f i b", "state_after": "case inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9\u271d : Sort y\ns t u : Set \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u00b9 : BorelSpace \u03b1\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b2\ninst\u271d\u2079 : MeasurableSpace \u03b2\ninst\u271d\u2078 : BorelSpace \u03b2\ninst\u271d\u2077 : TopologicalSpace \u03b3\ninst\u271d\u2076 : MeasurableSpace \u03b3\ninst\u271d\u2075 : BorelSpace \u03b3\ninst\u271d\u2074 : MeasurableSpace \u03b4\ninst\u271d\u00b3 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03b9 : Sort u_6\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nhf : \u2200 (i : \u03b9), Measurable (f i)\nh\u03b9 : Nonempty \u03b9\nA : MeasurableSet {b | BddAbove (range fun i => f i b)}\n\u22a2 Measurable fun b => \u2a06 i, f i b"}, {"tactic": "have : Measurable (fun (_b : \u03b4) \u21a6 sSup (\u2205 : Set \u03b1)) := measurable_const", "annotated_tactic": ["have : <a>Measurable</a> (fun (_b : \u03b4) \u21a6 <a>sSup</a> (\u2205 : <a>Set</a> \u03b1)) := <a>measurable_const</a>", [{"full_name": "Measurable", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [535, 5], "def_end_pos": [535, 15]}, {"full_name": "SupSet.sSup", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [55, 3], "def_end_pos": [55, 7]}, {"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}, {"full_name": "measurable_const", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [570, 9], "def_end_pos": [570, 25]}]], "state_before": "case inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9\u271d : Sort y\ns t u : Set \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u00b9 : BorelSpace \u03b1\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b2\ninst\u271d\u2079 : MeasurableSpace \u03b2\ninst\u271d\u2078 : BorelSpace \u03b2\ninst\u271d\u2077 : TopologicalSpace \u03b3\ninst\u271d\u2076 : MeasurableSpace \u03b3\ninst\u271d\u2075 : BorelSpace \u03b3\ninst\u271d\u2074 : MeasurableSpace \u03b4\ninst\u271d\u00b3 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03b9 : Sort u_6\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nhf : \u2200 (i : \u03b9), Measurable (f i)\nh\u03b9 : Nonempty \u03b9\nA : MeasurableSet {b | BddAbove (range fun i => f i b)}\n\u22a2 Measurable fun b => \u2a06 i, f i b", "state_after": "case inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9\u271d : Sort y\ns t u : Set \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u00b9 : BorelSpace \u03b1\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b2\ninst\u271d\u2079 : MeasurableSpace \u03b2\ninst\u271d\u2078 : BorelSpace \u03b2\ninst\u271d\u2077 : TopologicalSpace \u03b3\ninst\u271d\u2076 : MeasurableSpace \u03b3\ninst\u271d\u2075 : BorelSpace \u03b3\ninst\u271d\u2074 : MeasurableSpace \u03b4\ninst\u271d\u00b3 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03b9 : Sort u_6\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nhf : \u2200 (i : \u03b9), Measurable (f i)\nh\u03b9 : Nonempty \u03b9\nA : MeasurableSet {b | BddAbove (range fun i => f i b)}\nthis : Measurable fun _b => sSup \u2205\n\u22a2 Measurable fun b => \u2a06 i, f i b"}, {"tactic": "apply Measurable.isLUB_of_mem hf A _ _ this", "annotated_tactic": ["apply <a>Measurable.isLUB_of_mem</a> hf A _ _ this", [{"full_name": "Measurable.isLUB_of_mem", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [1157, 9], "def_end_pos": [1157, 32]}]], "state_before": "case inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9\u271d : Sort y\ns t u : Set \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u00b9 : BorelSpace \u03b1\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b2\ninst\u271d\u2079 : MeasurableSpace \u03b2\ninst\u271d\u2078 : BorelSpace \u03b2\ninst\u271d\u2077 : TopologicalSpace \u03b3\ninst\u271d\u2076 : MeasurableSpace \u03b3\ninst\u271d\u2075 : BorelSpace \u03b3\ninst\u271d\u2074 : MeasurableSpace \u03b4\ninst\u271d\u00b3 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03b9 : Sort u_6\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nhf : \u2200 (i : \u03b9), Measurable (f i)\nh\u03b9 : Nonempty \u03b9\nA : MeasurableSet {b | BddAbove (range fun i => f i b)}\nthis : Measurable fun _b => sSup \u2205\n\u22a2 Measurable fun b => \u2a06 i, f i b", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9\u271d : Sort y\ns t u : Set \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u00b9 : BorelSpace \u03b1\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b2\ninst\u271d\u2079 : MeasurableSpace \u03b2\ninst\u271d\u2078 : BorelSpace \u03b2\ninst\u271d\u2077 : TopologicalSpace \u03b3\ninst\u271d\u2076 : MeasurableSpace \u03b3\ninst\u271d\u2075 : BorelSpace \u03b3\ninst\u271d\u2074 : MeasurableSpace \u03b4\ninst\u271d\u00b3 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03b9 : Sort u_6\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nhf : \u2200 (i : \u03b9), Measurable (f i)\nh\u03b9 : Nonempty \u03b9\nA : MeasurableSet {b | BddAbove (range fun i => f i b)}\nthis : Measurable fun _b => sSup \u2205\n\u22a2 \u2200 (b : \u03b4), b \u2208 {b | BddAbove (range fun i => f i b)} \u2192 IsLUB {a | \u2203 i, f i b = a} (\u2a06 i, f i b)\n\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9\u271d : Sort y\ns t u : Set \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u00b9 : BorelSpace \u03b1\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b2\ninst\u271d\u2079 : MeasurableSpace \u03b2\ninst\u271d\u2078 : BorelSpace \u03b2\ninst\u271d\u2077 : TopologicalSpace \u03b3\ninst\u271d\u2076 : MeasurableSpace \u03b3\ninst\u271d\u2075 : BorelSpace \u03b3\ninst\u271d\u2074 : MeasurableSpace \u03b4\ninst\u271d\u00b3 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03b9 : Sort u_6\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nhf : \u2200 (i : \u03b9), Measurable (f i)\nh\u03b9 : Nonempty \u03b9\nA : MeasurableSet {b | BddAbove (range fun i => f i b)}\nthis : Measurable fun _b => sSup \u2205\n\u22a2 EqOn (fun b => \u2a06 i, f i b) (fun _b => sSup \u2205) {b | BddAbove (range fun i => f i b)}\u1d9c"}, {"tactic": "simp [iSup_of_empty']", "annotated_tactic": ["simp [<a>iSup_of_empty'</a>]", [{"full_name": "iSup_of_empty'", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [1559, 9], "def_end_pos": [1559, 23]}]], "state_before": "case inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9\u271d : Sort y\ns t u : Set \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u00b9 : BorelSpace \u03b1\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b2\ninst\u271d\u2079 : MeasurableSpace \u03b2\ninst\u271d\u2078 : BorelSpace \u03b2\ninst\u271d\u2077 : TopologicalSpace \u03b3\ninst\u271d\u2076 : MeasurableSpace \u03b3\ninst\u271d\u2075 : BorelSpace \u03b3\ninst\u271d\u2074 : MeasurableSpace \u03b4\ninst\u271d\u00b3 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03b9 : Sort u_6\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nhf : \u2200 (i : \u03b9), Measurable (f i)\nh\u03b9 : IsEmpty \u03b9\n\u22a2 Measurable fun b => \u2a06 i, f i b", "state_after": "no goals"}, {"tactic": "rintro b \u27e8c, hc\u27e9", "annotated_tactic": ["rintro b \u27e8c, hc\u27e9", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9\u271d : Sort y\ns t u : Set \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u00b9 : BorelSpace \u03b1\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b2\ninst\u271d\u2079 : MeasurableSpace \u03b2\ninst\u271d\u2078 : BorelSpace \u03b2\ninst\u271d\u2077 : TopologicalSpace \u03b3\ninst\u271d\u2076 : MeasurableSpace \u03b3\ninst\u271d\u2075 : BorelSpace \u03b3\ninst\u271d\u2074 : MeasurableSpace \u03b4\ninst\u271d\u00b3 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03b9 : Sort u_6\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nhf : \u2200 (i : \u03b9), Measurable (f i)\nh\u03b9 : Nonempty \u03b9\nA : MeasurableSet {b | BddAbove (range fun i => f i b)}\nthis : Measurable fun _b => sSup \u2205\n\u22a2 \u2200 (b : \u03b4), b \u2208 {b | BddAbove (range fun i => f i b)} \u2192 IsLUB {a | \u2203 i, f i b = a} (\u2a06 i, f i b)", "state_after": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9\u271d : Sort y\ns t u : Set \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u00b9 : BorelSpace \u03b1\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b2\ninst\u271d\u2079 : MeasurableSpace \u03b2\ninst\u271d\u2078 : BorelSpace \u03b2\ninst\u271d\u2077 : TopologicalSpace \u03b3\ninst\u271d\u2076 : MeasurableSpace \u03b3\ninst\u271d\u2075 : BorelSpace \u03b3\ninst\u271d\u2074 : MeasurableSpace \u03b4\ninst\u271d\u00b3 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03b9 : Sort u_6\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nhf : \u2200 (i : \u03b9), Measurable (f i)\nh\u03b9 : Nonempty \u03b9\nA : MeasurableSet {b | BddAbove (range fun i => f i b)}\nthis : Measurable fun _b => sSup \u2205\nb : \u03b4\nc : \u03b1\nhc : c \u2208 upperBounds (range fun i => f i b)\n\u22a2 IsLUB {a | \u2203 i, f i b = a} (\u2a06 i, f i b)"}, {"tactic": "apply isLUB_ciSup", "annotated_tactic": ["apply <a>isLUB_ciSup</a>", [{"full_name": "isLUB_ciSup", "def_path": "Mathlib/Order/ConditionallyCompleteLattice/Basic.lean", "def_pos": [502, 9], "def_end_pos": [502, 20]}]], "state_before": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9\u271d : Sort y\ns t u : Set \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u00b9 : BorelSpace \u03b1\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b2\ninst\u271d\u2079 : MeasurableSpace \u03b2\ninst\u271d\u2078 : BorelSpace \u03b2\ninst\u271d\u2077 : TopologicalSpace \u03b3\ninst\u271d\u2076 : MeasurableSpace \u03b3\ninst\u271d\u2075 : BorelSpace \u03b3\ninst\u271d\u2074 : MeasurableSpace \u03b4\ninst\u271d\u00b3 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03b9 : Sort u_6\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nhf : \u2200 (i : \u03b9), Measurable (f i)\nh\u03b9 : Nonempty \u03b9\nA : MeasurableSet {b | BddAbove (range fun i => f i b)}\nthis : Measurable fun _b => sSup \u2205\nb : \u03b4\nc : \u03b1\nhc : c \u2208 upperBounds (range fun i => f i b)\n\u22a2 IsLUB {a | \u2203 i, f i b = a} (\u2a06 i, f i b)", "state_after": "case intro.H\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9\u271d : Sort y\ns t u : Set \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u00b9 : BorelSpace \u03b1\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b2\ninst\u271d\u2079 : MeasurableSpace \u03b2\ninst\u271d\u2078 : BorelSpace \u03b2\ninst\u271d\u2077 : TopologicalSpace \u03b3\ninst\u271d\u2076 : MeasurableSpace \u03b3\ninst\u271d\u2075 : BorelSpace \u03b3\ninst\u271d\u2074 : MeasurableSpace \u03b4\ninst\u271d\u00b3 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03b9 : Sort u_6\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nhf : \u2200 (i : \u03b9), Measurable (f i)\nh\u03b9 : Nonempty \u03b9\nA : MeasurableSet {b | BddAbove (range fun i => f i b)}\nthis : Measurable fun _b => sSup \u2205\nb : \u03b4\nc : \u03b1\nhc : c \u2208 upperBounds (range fun i => f i b)\n\u22a2 BddAbove (range fun y => f y b)"}, {"tactic": "refine \u27e8c, ?_\u27e9", "annotated_tactic": ["refine \u27e8c, ?_\u27e9", []], "state_before": "case intro.H\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9\u271d : Sort y\ns t u : Set \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u00b9 : BorelSpace \u03b1\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b2\ninst\u271d\u2079 : MeasurableSpace \u03b2\ninst\u271d\u2078 : BorelSpace \u03b2\ninst\u271d\u2077 : TopologicalSpace \u03b3\ninst\u271d\u2076 : MeasurableSpace \u03b3\ninst\u271d\u2075 : BorelSpace \u03b3\ninst\u271d\u2074 : MeasurableSpace \u03b4\ninst\u271d\u00b3 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03b9 : Sort u_6\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nhf : \u2200 (i : \u03b9), Measurable (f i)\nh\u03b9 : Nonempty \u03b9\nA : MeasurableSet {b | BddAbove (range fun i => f i b)}\nthis : Measurable fun _b => sSup \u2205\nb : \u03b4\nc : \u03b1\nhc : c \u2208 upperBounds (range fun i => f i b)\n\u22a2 BddAbove (range fun y => f y b)", "state_after": "case intro.H\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9\u271d : Sort y\ns t u : Set \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u00b9 : BorelSpace \u03b1\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b2\ninst\u271d\u2079 : MeasurableSpace \u03b2\ninst\u271d\u2078 : BorelSpace \u03b2\ninst\u271d\u2077 : TopologicalSpace \u03b3\ninst\u271d\u2076 : MeasurableSpace \u03b3\ninst\u271d\u2075 : BorelSpace \u03b3\ninst\u271d\u2074 : MeasurableSpace \u03b4\ninst\u271d\u00b3 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03b9 : Sort u_6\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nhf : \u2200 (i : \u03b9), Measurable (f i)\nh\u03b9 : Nonempty \u03b9\nA : MeasurableSet {b | BddAbove (range fun i => f i b)}\nthis : Measurable fun _b => sSup \u2205\nb : \u03b4\nc : \u03b1\nhc : c \u2208 upperBounds (range fun i => f i b)\n\u22a2 c \u2208 upperBounds (range fun y => f y b)"}, {"tactic": "rintro d \u27e8i, rfl\u27e9", "annotated_tactic": ["rintro d \u27e8i, rfl\u27e9", []], "state_before": "case intro.H\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9\u271d : Sort y\ns t u : Set \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u00b9 : BorelSpace \u03b1\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b2\ninst\u271d\u2079 : MeasurableSpace \u03b2\ninst\u271d\u2078 : BorelSpace \u03b2\ninst\u271d\u2077 : TopologicalSpace \u03b3\ninst\u271d\u2076 : MeasurableSpace \u03b3\ninst\u271d\u2075 : BorelSpace \u03b3\ninst\u271d\u2074 : MeasurableSpace \u03b4\ninst\u271d\u00b3 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03b9 : Sort u_6\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nhf : \u2200 (i : \u03b9), Measurable (f i)\nh\u03b9 : Nonempty \u03b9\nA : MeasurableSet {b | BddAbove (range fun i => f i b)}\nthis : Measurable fun _b => sSup \u2205\nb : \u03b4\nc : \u03b1\nhc : c \u2208 upperBounds (range fun i => f i b)\n\u22a2 c \u2208 upperBounds (range fun y => f y b)", "state_after": "case intro.H.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9\u271d : Sort y\ns t u : Set \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u00b9 : BorelSpace \u03b1\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b2\ninst\u271d\u2079 : MeasurableSpace \u03b2\ninst\u271d\u2078 : BorelSpace \u03b2\ninst\u271d\u2077 : TopologicalSpace \u03b3\ninst\u271d\u2076 : MeasurableSpace \u03b3\ninst\u271d\u2075 : BorelSpace \u03b3\ninst\u271d\u2074 : MeasurableSpace \u03b4\ninst\u271d\u00b3 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03b9 : Sort u_6\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nhf : \u2200 (i : \u03b9), Measurable (f i)\nh\u03b9 : Nonempty \u03b9\nA : MeasurableSet {b | BddAbove (range fun i => f i b)}\nthis : Measurable fun _b => sSup \u2205\nb : \u03b4\nc : \u03b1\nhc : c \u2208 upperBounds (range fun i => f i b)\ni : \u03b9\n\u22a2 (fun y => f y b) i \u2264 c"}, {"tactic": "exact hc (mem_range_self i)", "annotated_tactic": ["exact hc (<a>mem_range_self</a> i)", [{"full_name": "Set.mem_range_self", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [680, 9], "def_end_pos": [680, 23]}]], "state_before": "case intro.H.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9\u271d : Sort y\ns t u : Set \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u00b9 : BorelSpace \u03b1\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b2\ninst\u271d\u2079 : MeasurableSpace \u03b2\ninst\u271d\u2078 : BorelSpace \u03b2\ninst\u271d\u2077 : TopologicalSpace \u03b3\ninst\u271d\u2076 : MeasurableSpace \u03b3\ninst\u271d\u2075 : BorelSpace \u03b3\ninst\u271d\u2074 : MeasurableSpace \u03b4\ninst\u271d\u00b3 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03b9 : Sort u_6\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nhf : \u2200 (i : \u03b9), Measurable (f i)\nh\u03b9 : Nonempty \u03b9\nA : MeasurableSet {b | BddAbove (range fun i => f i b)}\nthis : Measurable fun _b => sSup \u2205\nb : \u03b4\nc : \u03b1\nhc : c \u2208 upperBounds (range fun i => f i b)\ni : \u03b9\n\u22a2 (fun y => f y b) i \u2264 c", "state_after": "no goals"}, {"tactic": "intro b hb", "annotated_tactic": ["intro b hb", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9\u271d : Sort y\ns t u : Set \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u00b9 : BorelSpace \u03b1\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b2\ninst\u271d\u2079 : MeasurableSpace \u03b2\ninst\u271d\u2078 : BorelSpace \u03b2\ninst\u271d\u2077 : TopologicalSpace \u03b3\ninst\u271d\u2076 : MeasurableSpace \u03b3\ninst\u271d\u2075 : BorelSpace \u03b3\ninst\u271d\u2074 : MeasurableSpace \u03b4\ninst\u271d\u00b3 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03b9 : Sort u_6\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nhf : \u2200 (i : \u03b9), Measurable (f i)\nh\u03b9 : Nonempty \u03b9\nA : MeasurableSet {b | BddAbove (range fun i => f i b)}\nthis : Measurable fun _b => sSup \u2205\n\u22a2 EqOn (fun b => \u2a06 i, f i b) (fun _b => sSup \u2205) {b | BddAbove (range fun i => f i b)}\u1d9c", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9\u271d : Sort y\ns t u : Set \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u00b9 : BorelSpace \u03b1\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b2\ninst\u271d\u2079 : MeasurableSpace \u03b2\ninst\u271d\u2078 : BorelSpace \u03b2\ninst\u271d\u2077 : TopologicalSpace \u03b3\ninst\u271d\u2076 : MeasurableSpace \u03b3\ninst\u271d\u2075 : BorelSpace \u03b3\ninst\u271d\u2074 : MeasurableSpace \u03b4\ninst\u271d\u00b3 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03b9 : Sort u_6\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nhf : \u2200 (i : \u03b9), Measurable (f i)\nh\u03b9 : Nonempty \u03b9\nA : MeasurableSet {b | BddAbove (range fun i => f i b)}\nthis : Measurable fun _b => sSup \u2205\nb : \u03b4\nhb : b \u2208 {b | BddAbove (range fun i => f i b)}\u1d9c\n\u22a2 (fun b => \u2a06 i, f i b) b = (fun _b => sSup \u2205) b"}, {"tactic": "apply csSup_of_not_bddAbove", "annotated_tactic": ["apply <a>csSup_of_not_bddAbove</a>", [{"full_name": "csSup_of_not_bddAbove", "def_path": "Mathlib/Order/ConditionallyCompleteLattice/Basic.lean", "def_pos": [1025, 9], "def_end_pos": [1025, 30]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9\u271d : Sort y\ns t u : Set \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u00b9 : BorelSpace \u03b1\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b2\ninst\u271d\u2079 : MeasurableSpace \u03b2\ninst\u271d\u2078 : BorelSpace \u03b2\ninst\u271d\u2077 : TopologicalSpace \u03b3\ninst\u271d\u2076 : MeasurableSpace \u03b3\ninst\u271d\u2075 : BorelSpace \u03b3\ninst\u271d\u2074 : MeasurableSpace \u03b4\ninst\u271d\u00b3 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03b9 : Sort u_6\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nhf : \u2200 (i : \u03b9), Measurable (f i)\nh\u03b9 : Nonempty \u03b9\nA : MeasurableSet {b | BddAbove (range fun i => f i b)}\nthis : Measurable fun _b => sSup \u2205\nb : \u03b4\nhb : b \u2208 {b | BddAbove (range fun i => f i b)}\u1d9c\n\u22a2 (fun b => \u2a06 i, f i b) b = (fun _b => sSup \u2205) b", "state_after": "case hs\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9\u271d : Sort y\ns t u : Set \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u00b9 : BorelSpace \u03b1\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b2\ninst\u271d\u2079 : MeasurableSpace \u03b2\ninst\u271d\u2078 : BorelSpace \u03b2\ninst\u271d\u2077 : TopologicalSpace \u03b3\ninst\u271d\u2076 : MeasurableSpace \u03b3\ninst\u271d\u2075 : BorelSpace \u03b3\ninst\u271d\u2074 : MeasurableSpace \u03b4\ninst\u271d\u00b3 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03b9 : Sort u_6\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nhf : \u2200 (i : \u03b9), Measurable (f i)\nh\u03b9 : Nonempty \u03b9\nA : MeasurableSet {b | BddAbove (range fun i => f i b)}\nthis : Measurable fun _b => sSup \u2205\nb : \u03b4\nhb : b \u2208 {b | BddAbove (range fun i => f i b)}\u1d9c\n\u22a2 \u00acBddAbove (range fun i => f i b)"}, {"tactic": "exact hb", "annotated_tactic": ["exact hb", []], "state_before": "case hs\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b3\u2082 : Type u_4\n\u03b4 : Type u_5\n\u03b9\u271d : Sort y\ns t u : Set \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u00b9 : BorelSpace \u03b1\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b2\ninst\u271d\u2079 : MeasurableSpace \u03b2\ninst\u271d\u2078 : BorelSpace \u03b2\ninst\u271d\u2077 : TopologicalSpace \u03b3\ninst\u271d\u2076 : MeasurableSpace \u03b3\ninst\u271d\u2075 : BorelSpace \u03b3\ninst\u271d\u2074 : MeasurableSpace \u03b4\ninst\u271d\u00b3 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03b9 : Sort u_6\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nhf : \u2200 (i : \u03b9), Measurable (f i)\nh\u03b9 : Nonempty \u03b9\nA : MeasurableSet {b | BddAbove (range fun i => f i b)}\nthis : Measurable fun _b => sSup \u2205\nb : \u03b4\nhb : b \u2208 {b | BddAbove (range fun i => f i b)}\u1d9c\n\u22a2 \u00acBddAbove (range fun i => f i b)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "full_name": "intervalIntegral.integral_Iic_sub_Iic", "start": [959, 1], "end": [965, 65], "traced_tactics": [{"tactic": "wlog hab : a \u2264 b generalizing a b", "annotated_tactic": ["wlog hab : a \u2264 b generalizing a b", []], "state_before": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\na b c d : \u211d\nf g : \u211d \u2192 E\n\u03bc : Measure \u211d\nha : IntegrableOn f (Iic a)\nhb : IntegrableOn f (Iic b)\n\u22a2 \u222b (x : \u211d) in Iic b, f x \u2202\u03bc - \u222b (x : \u211d) in Iic a, f x \u2202\u03bc = \u222b (x : \u211d) in a..b, f x \u2202\u03bc", "state_after": "case inr\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\na b c d : \u211d\nf g : \u211d \u2192 E\n\u03bc : Measure \u211d\nha : IntegrableOn f (Iic a)\nhb : IntegrableOn f (Iic b)\nthis :\n  \u2200 {a b : \u211d},\n    IntegrableOn f (Iic a) \u2192\n      IntegrableOn f (Iic b) \u2192\n        a \u2264 b \u2192 \u222b (x : \u211d) in Iic b, f x \u2202\u03bc - \u222b (x : \u211d) in Iic a, f x \u2202\u03bc = \u222b (x : \u211d) in a..b, f x \u2202\u03bc\nhab : \u00aca \u2264 b\n\u22a2 \u222b (x : \u211d) in Iic b, f x \u2202\u03bc - \u222b (x : \u211d) in Iic a, f x \u2202\u03bc = \u222b 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:= Option.isSome o = true, get := Option.get o }.Dom\n\u22a2 get (\u2191o) h\u2081 = get { Dom := Option.isSome o = true, get := Option.get o } h\u2082", "state_after": "case none\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b1 : Type u_4\nh\u2081 : (\u2191Option.none).Dom\nh\u2082 : { Dom := Option.isSome Option.none = true, get := Option.get Option.none }.Dom\n\u22a2 get (\u2191Option.none) h\u2081 = get { Dom := Option.isSome Option.none = true, get := Option.get Option.none } h\u2082\n\ncase some\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b1 : Type u_4\nval\u271d : \u03b1\nh\u2081 : (\u2191(Option.some val\u271d)).Dom\nh\u2082 : { Dom := Option.isSome (Option.some val\u271d) = true, get := Option.get (Option.some val\u271d) }.Dom\n\u22a2 get (\u2191(Option.some val\u271d)) h\u2081 =\n    get { Dom := Option.isSome (Option.some val\u271d) = true, get := Option.get (Option.some val\u271d) } h\u2082"}, {"tactic": "simp at h\u2082", "annotated_tactic": ["simp at h\u2082", []], "state_before": "case none\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b1 : Type u_4\nh\u2081 : (\u2191Option.none).Dom\nh\u2082 : { Dom := Option.isSome Option.none = true, get := Option.get Option.none }.Dom\n\u22a2 get (\u2191Option.none) h\u2081 = get { Dom := Option.isSome Option.none = true, get := Option.get Option.none } h\u2082", "state_after": "no goals"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case some\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b1 : Type u_4\nval\u271d : \u03b1\nh\u2081 : (\u2191(Option.some val\u271d)).Dom\nh\u2082 : { Dom := Option.isSome (Option.some val\u271d) = true, get := Option.get (Option.some val\u271d) }.Dom\n\u22a2 get (\u2191(Option.some val\u271d)) h\u2081 =\n    get { Dom := Option.isSome (Option.some val\u271d) = true, get := Option.get (Option.some val\u271d) } h\u2082", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "full_name": "measurableSet_le", "start": [559, 1], "end": [561, 34], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Pointwise/Interval.lean", "full_name": "Set.image_affine_Icc'", "start": [784, 1], "end": [788, 50], "traced_tactics": [{"tactic": "suffices (fun x => x + b) '' ((fun x => a * x) '' Icc c d) = Icc (a * c + b) (a * d + b) by\n  rwa [Set.image_image] at this", "annotated_tactic": ["suffices (fun x => x + b) '' ((fun x => a * x) '' <a>Icc</a> c d) = <a>Icc</a> (a * c + b) (a * d + b) by\n    rwa [<a>Set.image_image</a>] at this", [{"full_name": "Set.Icc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [59, 5], "def_end_pos": [59, 8]}, {"full_name": "Set.Icc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [59, 5], "def_end_pos": [59, 8]}, {"full_name": "Set.image_image", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [299, 9], "def_end_pos": [299, 20]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na\u271d a : \u03b1\nh : 0 < a\nb c d : \u03b1\n\u22a2 (fun x => a * x + b) '' Icc c d = Icc (a * c + b) (a * d + b)", "state_after": "\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na\u271d a : \u03b1\nh : 0 < a\nb c d : \u03b1\n\u22a2 (fun x => x + b) '' ((fun x => a * x) '' Icc c d) = Icc (a * c + b) (a * d + b)"}, {"tactic": "rw [image_mul_left_Icc' h, image_add_const_Icc]", "annotated_tactic": ["rw [<a>image_mul_left_Icc'</a> h, <a>image_add_const_Icc</a>]", [{"full_name": "Set.image_mul_left_Icc'", "def_path": "Mathlib/Data/Set/Pointwise/Interval.lean", "def_pos": [739, 9], "def_end_pos": [739, 28]}, {"full_name": "Set.image_add_const_Icc", "def_path": "Mathlib/Data/Set/Intervals/Monoid.lean", "def_pos": [89, 9], "def_end_pos": [89, 28]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na\u271d a : \u03b1\nh : 0 < a\nb c d : \u03b1\n\u22a2 (fun x => x + b) '' ((fun x => a * x) '' Icc c d) = Icc (a * c + b) (a * d + b)", "state_after": "no goals"}, {"tactic": "rwa [Set.image_image] at this", "annotated_tactic": ["rwa [<a>Set.image_image</a>] at this", [{"full_name": "Set.image_image", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [299, 9], "def_end_pos": [299, 20]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na\u271d a : \u03b1\nh : 0 < a\nb c d : \u03b1\nthis : (fun x => x + b) '' ((fun x => a * x) '' Icc c d) = Icc (a * c + b) (a * d + b)\n\u22a2 (fun x => a * x + b) '' Icc c d = Icc (a * c + b) (a * d + b)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/HashMap/WF.lean", "full_name": "Std.HashMap.Imp.expand_size", "start": [79, 1], "end": [111, 50], "traced_tactics": [{"tactic": "rw [expand, go]", "annotated_tactic": ["rw [<a>expand</a>, go]", [{"full_name": "Std.HashMap.Imp.expand", "def_path": "lake-packages/std/Std/Data/HashMap/Basic.lean", "def_pos": [138, 5], "def_end_pos": [138, 11]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nsz : Nat\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\n\u22a2 Buckets.size (expand sz buckets).buckets = Buckets.size buckets", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nsz : Nat\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\n\u22a2 Nat.sum (List.map (fun x => List.length (AssocList.toList x)) buckets.val.data) +\n      Buckets.size (Buckets.mk (Array.size buckets.val * 2)) =\n    Buckets.size buckets\n\ncase hs\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nsz : Nat\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\n\u22a2 \u2200 (j : Nat), j < 0 \u2192 List.getD buckets.val.data j AssocList.nil = AssocList.nil"}, {"tactic": "rw [Buckets.mk_size]", "annotated_tactic": ["rw [<a>Buckets.mk_size</a>]", [{"full_name": "Std.HashMap.Imp.Buckets.mk_size", "def_path": "lake-packages/std/Std/Data/HashMap/WF.lean", "def_pos": [36, 9], "def_end_pos": [36, 16]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nsz : Nat\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\n\u22a2 Nat.sum (List.map (fun x => List.length (AssocList.toList x)) buckets.val.data) +\n      Buckets.size (Buckets.mk (Array.size buckets.val * 2)) =\n    Buckets.size buckets", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nsz : Nat\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\n\u22a2 Nat.sum (List.map (fun x => List.length (AssocList.toList x)) buckets.val.data) + 0 = Buckets.size buckets"}, {"tactic": "simp [Buckets.size]", "annotated_tactic": ["simp [<a>Buckets.size</a>]", [{"full_name": "Std.HashMap.Imp.Buckets.size", "def_path": "lake-packages/std/Std/Data/HashMap/Basic.lean", "def_pos": [40, 19], "def_end_pos": [40, 23]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nsz : Nat\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\n\u22a2 Nat.sum (List.map (fun x => List.length (AssocList.toList x)) buckets.val.data) + 0 = Buckets.size buckets", "state_after": "no goals"}, {"tactic": "intro.", "annotated_tactic": ["intro.", []], "state_before": "case hs\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nsz : Nat\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\n\u22a2 \u2200 (j : Nat), j < 0 \u2192 List.getD buckets.val.data j AssocList.nil = AssocList.nil", "state_after": "no goals"}, {"tactic": "unfold expand.go", "annotated_tactic": ["unfold <a>expand.go</a>", [{"full_name": "Std.HashMap.Imp.expand.go", "def_path": "lake-packages/std/Std/Data/HashMap/Basic.lean", "def_pos": [144, 3], "def_end_pos": [144, 5]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nsz : Nat\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\ni : Nat\nsource : Array (AssocList \u03b1 \u03b2)\ntarget : Buckets \u03b1 \u03b2\nhs : \u2200 (j : Nat), j < i \u2192 List.getD source.data j AssocList.nil = AssocList.nil\n\u22a2 Buckets.size (expand.go i source target) =\n    Nat.sum (List.map (fun x => List.length (AssocList.toList x)) source.data) + Buckets.size target", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nsz : Nat\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\ni : Nat\nsource : Array (AssocList \u03b1 \u03b2)\ntarget : Buckets \u03b1 \u03b2\nhs : \u2200 (j : Nat), j < i \u2192 List.getD source.data j AssocList.nil = AssocList.nil\n\u22a2 Buckets.size\n      (if h : i < Array.size source then\n        let idx := { val := i, isLt := h };\n        let es := Array.get source idx;\n        let source := Array.set source idx AssocList.nil;\n        let target := AssocList.foldl reinsertAux target es;\n        expand.go (i + 1) source target\n      else target) =\n    Nat.sum (List.map (fun x => List.length (AssocList.toList x)) source.data) + Buckets.size target"}, {"tactic": "split", "annotated_tactic": ["split", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nsz : Nat\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\ni : Nat\nsource : Array (AssocList \u03b1 \u03b2)\ntarget : Buckets \u03b1 \u03b2\nhs : \u2200 (j : Nat), j < i \u2192 List.getD source.data j AssocList.nil = AssocList.nil\n\u22a2 Buckets.size\n      (if h : i < Array.size source then\n        let idx := { val := i, isLt := h };\n        let es := Array.get source idx;\n        let source := Array.set source idx AssocList.nil;\n        let target := AssocList.foldl reinsertAux target es;\n        expand.go (i + 1) source target\n      else target) =\n    Nat.sum (List.map (fun x => List.length (AssocList.toList x)) source.data) + Buckets.size target", "state_after": "case inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nsz : Nat\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\ni : Nat\nsource : Array (AssocList \u03b1 \u03b2)\ntarget : Buckets \u03b1 \u03b2\nhs : \u2200 (j : Nat), j < i \u2192 List.getD source.data j AssocList.nil = AssocList.nil\nh\u271d : i < Array.size source\n\u22a2 Buckets.size\n      (let idx := { val := i, isLt := h\u271d };\n      let es := Array.get source idx;\n      let source := Array.set source idx AssocList.nil;\n      let target := AssocList.foldl reinsertAux target es;\n      expand.go (i + 1) source target) =\n    Nat.sum (List.map (fun x => List.length (AssocList.toList x)) source.data) + Buckets.size target\n\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nsz : Nat\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\ni : Nat\nsource : Array (AssocList \u03b1 \u03b2)\ntarget : Buckets \u03b1 \u03b2\nhs : \u2200 (j : Nat), j < i \u2192 List.getD source.data j AssocList.nil = AssocList.nil\nh\u271d : \u00aci < Array.size source\n\u22a2 Buckets.size target = Nat.sum (List.map (fun x => List.length (AssocList.toList x)) source.data) + Buckets.size target"}, {"tactic": "refine (go (i+1) _ _ fun j hj => ?a).trans ?b <;> simp", "annotated_tactic": ["refine (go (i+1) _ _ fun j hj => ?a).<a>trans</a> ?b <;> simp", [{"full_name": "Eq.trans", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [322, 9], "def_end_pos": [322, 17]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nsz : Nat\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\ni : Nat\nsource : Array (AssocList \u03b1 \u03b2)\ntarget : Buckets \u03b1 \u03b2\nhs : \u2200 (j : Nat), j < i \u2192 List.getD source.data j AssocList.nil = AssocList.nil\nH : i < Array.size source\n\u22a2 Buckets.size\n      (let idx := { val := i, isLt := H };\n      let es := Array.get source idx;\n      let source := Array.set source idx AssocList.nil;\n      let target := AssocList.foldl reinsertAux target es;\n      expand.go (i + 1) source target) =\n    Nat.sum (List.map (fun x => List.length (AssocList.toList x)) source.data) + Buckets.size target", "state_after": "case a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nsz : Nat\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\ni : Nat\nsource : Array (AssocList \u03b1 \u03b2)\ntarget : Buckets \u03b1 \u03b2\nhs : \u2200 (j : Nat), j < i \u2192 List.getD source.data j AssocList.nil = AssocList.nil\nH : i < Array.size source\nj : Nat\nhj : j < i + 1\n\u22a2 List.getD (List.set source.data i AssocList.nil) j AssocList.nil = AssocList.nil\n\ncase b\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nsz : Nat\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\ni : Nat\nsource : Array (AssocList \u03b1 \u03b2)\ntarget : Buckets \u03b1 \u03b2\nhs : \u2200 (j : Nat), j < i \u2192 List.getD source.data j AssocList.nil = AssocList.nil\nH : i < Array.size source\n\u22a2 Nat.sum (List.map (fun x => List.length (AssocList.toList x)) (List.set source.data i AssocList.nil)) +\n      Buckets.size (List.foldl (fun d x => reinsertAux d x.fst x.snd) target (AssocList.toList source[i])) =\n    Nat.sum (List.map (fun x => List.length (AssocList.toList x)) source.data) + Buckets.size target"}, {"tactic": "simp [List.getD_eq_get?, List.get?_set]", "annotated_tactic": ["simp [<a>List.getD_eq_get?</a>, <a>List.get?_set</a>]", [{"full_name": "List.getD_eq_get?", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [772, 9], "def_end_pos": [772, 21]}, {"full_name": "List.get?_set", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [922, 9], "def_end_pos": [922, 17]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nsz : Nat\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\ni : Nat\nsource : Array (AssocList \u03b1 \u03b2)\ntarget : Buckets \u03b1 \u03b2\nhs : \u2200 (j : Nat), j < i \u2192 List.getD source.data j AssocList.nil = AssocList.nil\nH : i < Array.size source\nj : Nat\nhj : j < i + 1\n\u22a2 List.getD (List.set source.data i AssocList.nil) j AssocList.nil = AssocList.nil", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nsz : Nat\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\ni : Nat\nsource : Array (AssocList \u03b1 \u03b2)\ntarget : Buckets \u03b1 \u03b2\nhs : \u2200 (j : Nat), j < i \u2192 List.getD source.data j AssocList.nil = AssocList.nil\nH : i < Array.size source\nj : Nat\nhj : j < i + 1\n\u22a2 Option.getD (if i = j then Option.map (fun x => AssocList.nil) (List.get? source.data j) else List.get? source.data j)\n      AssocList.nil =\n    AssocList.nil"}, {"tactic": "split", "annotated_tactic": ["split", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nsz : Nat\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\ni : Nat\nsource : Array (AssocList \u03b1 \u03b2)\ntarget : Buckets \u03b1 \u03b2\nhs : \u2200 (j : Nat), j < i \u2192 List.getD source.data j AssocList.nil = AssocList.nil\nH : i < Array.size source\nj : Nat\nhj : j < i + 1\n\u22a2 Option.getD (if i = j then Option.map (fun x => AssocList.nil) (List.get? source.data j) else List.get? source.data j)\n      AssocList.nil =\n    AssocList.nil", "state_after": "case inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nsz : Nat\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\ni : Nat\nsource : Array (AssocList \u03b1 \u03b2)\ntarget : Buckets \u03b1 \u03b2\nhs : \u2200 (j : Nat), j < i \u2192 List.getD source.data j AssocList.nil = AssocList.nil\nH : i < Array.size source\nj : Nat\nhj : j < i + 1\nh\u271d : i = j\n\u22a2 Option.getD (Option.map (fun x => AssocList.nil) (List.get? source.data j)) AssocList.nil = AssocList.nil\n\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nsz : Nat\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\ni : Nat\nsource : Array (AssocList \u03b1 \u03b2)\ntarget : Buckets \u03b1 \u03b2\nhs : \u2200 (j : Nat), j < i \u2192 List.getD source.data j AssocList.nil = AssocList.nil\nH : i < Array.size source\nj : Nat\nhj : j < i + 1\nh\u271d : \u00aci = j\n\u22a2 Option.getD (List.get? source.data j) AssocList.nil = AssocList.nil"}, {"tactic": "cases List.get? .. <;> rfl", "annotated_tactic": ["cases <a>List.get?</a> .. <;> rfl", [{"full_name": "List.get?", "def_path": "lake-packages/lean4/src/lean/Init/Data/List/BasicAux.lean", "def_pos": [22, 5], "def_end_pos": [22, 9]}]], "state_before": "case inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nsz : Nat\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\ni : Nat\nsource : Array (AssocList \u03b1 \u03b2)\ntarget : Buckets \u03b1 \u03b2\nhs : \u2200 (j : Nat), j < i \u2192 List.getD source.data j AssocList.nil = AssocList.nil\nH : i < Array.size source\nj : Nat\nhj : j < i + 1\nh\u271d : i = j\n\u22a2 Option.getD (Option.map (fun x => AssocList.nil) (List.get? source.data j)) AssocList.nil = AssocList.nil", "state_after": "no goals"}, {"tactic": "next H => exact hs _ (Nat.lt_of_le_of_ne (Nat.le_of_lt_succ hj) (Ne.symm H))", "annotated_tactic": ["next H => exact hs _ (<a>Nat.lt_of_le_of_ne</a> (<a>Nat.le_of_lt_succ</a> hj) (<a>Ne.symm</a> H))", [{"full_name": "Nat.lt_of_le_of_ne", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1670, 19], "def_end_pos": [1670, 37]}, {"full_name": "Nat.le_of_lt_succ", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1631, 9], "def_end_pos": [1631, 26]}, {"full_name": "Ne.symm", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [575, 9], "def_end_pos": [575, 16]}]], "state_before": "case inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nsz : Nat\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\ni : Nat\nsource : Array (AssocList \u03b1 \u03b2)\ntarget : Buckets \u03b1 \u03b2\nhs : \u2200 (j : Nat), j < i \u2192 List.getD source.data j AssocList.nil = AssocList.nil\nH : i < Array.size source\nj : Nat\nhj : j < i + 1\nh\u271d : \u00aci = j\n\u22a2 Option.getD (List.get? source.data j) AssocList.nil = AssocList.nil", "state_after": "no goals"}, {"tactic": "exact hs _ (Nat.lt_of_le_of_ne (Nat.le_of_lt_succ hj) (Ne.symm H))", "annotated_tactic": ["exact hs _ (<a>Nat.lt_of_le_of_ne</a> (<a>Nat.le_of_lt_succ</a> hj) (<a>Ne.symm</a> H))", [{"full_name": "Nat.lt_of_le_of_ne", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1670, 19], "def_end_pos": [1670, 37]}, {"full_name": "Nat.le_of_lt_succ", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1631, 9], "def_end_pos": [1631, 26]}, {"full_name": "Ne.symm", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [575, 9], "def_end_pos": [575, 16]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nsz : Nat\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\ni : Nat\nsource : Array (AssocList \u03b1 \u03b2)\ntarget : Buckets \u03b1 \u03b2\nhs : \u2200 (j : Nat), j < i \u2192 List.getD source.data j AssocList.nil = AssocList.nil\nH\u271d : i < Array.size source\nj : Nat\nhj : j < i + 1\nH : \u00aci = j\n\u22a2 Option.getD (List.get? source.data j) AssocList.nil = AssocList.nil", "state_after": "no goals"}, {"tactic": "case b =>\nrefine have \u27e8l\u2081, l\u2082, h\u2081, _, eq\u27e9 := List.exists_of_set' H; eq \u25b8 ?_\nsimp [h\u2081, Buckets.size_eq]\nrw [Nat.add_assoc, Nat.add_assoc, Nat.add_assoc]; congr 1\n(conv => rhs; rw [Nat.add_left_comm]); congr 1\nrw [\u2190 Array.getElem_eq_data_get]\nhave := @reinsertAux_size \u03b1 \u03b2 _; simp [Buckets.size] at this\ninduction source[i].toList generalizing target <;> simp [*, Nat.succ_add]; rfl", "annotated_tactic": ["case b =>\n        refine have \u27e8l\u2081, l\u2082, h\u2081, _, eq\u27e9 := <a>List.exists_of_set'</a> H; eq \u25b8 ?_\n        simp [h\u2081, <a>Buckets.size_eq</a>]\n        rw [<a>Nat.add_assoc</a>, <a>Nat.add_assoc</a>, <a>Nat.add_assoc</a>]; congr 1\n        (conv => rhs; rw [<a>Nat.add_left_comm</a>]); congr 1\n        rw [\u2190 <a>Array.getElem_eq_data_get</a>]\n        have := @<a>reinsertAux_size</a> \u03b1 \u03b2 _; simp [<a>Buckets.size</a>] at this\n        induction source[i].<a>toList</a> generalizing target <;> simp [*, <a>Nat.succ_add</a>]; rfl", [{"full_name": "List.exists_of_set'", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [909, 9], "def_end_pos": [909, 23]}, {"full_name": "Std.HashMap.Imp.Buckets.size_eq", "def_path": "lake-packages/std/Std/Data/HashMap/WF.lean", "def_pos": [33, 9], "def_end_pos": [33, 16]}, {"full_name": "Nat.add_assoc", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [138, 19], "def_end_pos": [138, 28]}, {"full_name": "Nat.add_assoc", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [138, 19], "def_end_pos": [138, 28]}, {"full_name": "Nat.add_assoc", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [138, 19], "def_end_pos": [138, 28]}, {"full_name": "Nat.add_left_comm", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [142, 19], "def_end_pos": [142, 32]}, {"full_name": "Array.getElem_eq_data_get", "def_path": "lake-packages/std/Std/Data/Array/Init/Lemmas.lean", "def_pos": [28, 9], "def_end_pos": [28, 28]}, {"full_name": "Std.HashMap.Imp.reinsertAux_size", "def_path": "lake-packages/std/Std/Data/HashMap/WF.lean", "def_pos": [64, 9], "def_end_pos": [64, 25]}, {"full_name": "Std.HashMap.Imp.Buckets.size", "def_path": "lake-packages/std/Std/Data/HashMap/Basic.lean", "def_pos": [40, 19], "def_end_pos": [40, 23]}, {"full_name": "Std.AssocList.toList", "def_path": "lake-packages/std/Std/Data/AssocList.lean", "def_pos": [30, 13], "def_end_pos": [30, 19]}, {"full_name": "Nat.succ_add", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [118, 9], "def_end_pos": [118, 17]}]], "state_before": "case b\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nsz : Nat\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\ni : Nat\nsource : Array (AssocList \u03b1 \u03b2)\ntarget : Buckets \u03b1 \u03b2\nhs : \u2200 (j : Nat), j < i \u2192 List.getD source.data j AssocList.nil = AssocList.nil\nH : i < Array.size source\n\u22a2 Nat.sum (List.map (fun x => List.length (AssocList.toList x)) (List.set source.data i AssocList.nil)) +\n      Buckets.size (List.foldl (fun d x => reinsertAux d x.fst x.snd) target (AssocList.toList source[i])) =\n    Nat.sum (List.map (fun x => List.length (AssocList.toList x)) source.data) + Buckets.size target", "state_after": "no goals"}, {"tactic": "refine have \u27e8l\u2081, l\u2082, h\u2081, _, eq\u27e9 := List.exists_of_set' H; eq \u25b8 ?_", "annotated_tactic": ["refine have \u27e8l\u2081, l\u2082, h\u2081, _, eq\u27e9 := <a>List.exists_of_set'</a> H; eq \u25b8 ?_", [{"full_name": "List.exists_of_set'", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [909, 9], "def_end_pos": [909, 23]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nsz : Nat\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\ni : Nat\nsource : Array (AssocList \u03b1 \u03b2)\ntarget : Buckets \u03b1 \u03b2\nhs : \u2200 (j : Nat), j < i \u2192 List.getD source.data j AssocList.nil = AssocList.nil\nH : i < Array.size source\n\u22a2 Nat.sum (List.map (fun x => List.length (AssocList.toList x)) (List.set source.data i AssocList.nil)) +\n      Buckets.size (List.foldl (fun d x => reinsertAux d x.fst x.snd) target (AssocList.toList source[i])) =\n    Nat.sum (List.map (fun x => List.length (AssocList.toList x)) source.data) + Buckets.size target", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nsz : Nat\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\ni : Nat\nsource : Array (AssocList \u03b1 \u03b2)\ntarget : Buckets \u03b1 \u03b2\nhs : \u2200 (j : Nat), j < i \u2192 List.getD source.data j AssocList.nil = AssocList.nil\nH : i < Array.size source\nl\u2081 l\u2082 : List (AssocList \u03b1 \u03b2)\nh\u2081 : source.data = l\u2081 ++ List.get source.data { val := i, isLt := H } :: l\u2082\nleft\u271d : List.length l\u2081 = i\neq : List.set source.data i AssocList.nil = l\u2081 ++ AssocList.nil :: l\u2082\n\u22a2 Nat.sum (List.map (fun x => List.length (AssocList.toList x)) (l\u2081 ++ AssocList.nil :: l\u2082)) +\n      Buckets.size (List.foldl (fun d x => reinsertAux d x.fst x.snd) target (AssocList.toList source[i])) =\n    Nat.sum (List.map (fun x => List.length (AssocList.toList x)) source.data) + Buckets.size target"}, {"tactic": "simp [h\u2081, Buckets.size_eq]", "annotated_tactic": ["simp [h\u2081, <a>Buckets.size_eq</a>]", [{"full_name": "Std.HashMap.Imp.Buckets.size_eq", "def_path": "lake-packages/std/Std/Data/HashMap/WF.lean", "def_pos": [33, 9], "def_end_pos": [33, 16]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nsz : Nat\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\ni : Nat\nsource : Array (AssocList \u03b1 \u03b2)\ntarget : Buckets \u03b1 \u03b2\nhs : \u2200 (j : Nat), j < i \u2192 List.getD source.data j AssocList.nil = AssocList.nil\nH : i < Array.size source\nl\u2081 l\u2082 : List (AssocList \u03b1 \u03b2)\nh\u2081 : source.data = l\u2081 ++ List.get source.data { val := i, isLt := H } :: l\u2082\nleft\u271d : List.length l\u2081 = i\neq : List.set source.data i AssocList.nil = l\u2081 ++ AssocList.nil :: l\u2082\n\u22a2 Nat.sum (List.map (fun x => List.length (AssocList.toList x)) (l\u2081 ++ AssocList.nil :: l\u2082)) +\n      Buckets.size (List.foldl (fun d x => reinsertAux d x.fst x.snd) target (AssocList.toList source[i])) =\n    Nat.sum (List.map (fun x => List.length (AssocList.toList x)) source.data) + Buckets.size target", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nsz : Nat\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\ni : Nat\nsource : Array (AssocList \u03b1 \u03b2)\ntarget : Buckets \u03b1 \u03b2\nhs : \u2200 (j : Nat), j < i \u2192 List.getD source.data j AssocList.nil = AssocList.nil\nH : i < Array.size source\nl\u2081 l\u2082 : List (AssocList \u03b1 \u03b2)\nh\u2081 : source.data = l\u2081 ++ List.get source.data { val := i, isLt := H } :: l\u2082\nleft\u271d : List.length l\u2081 = i\neq : List.set source.data i AssocList.nil = l\u2081 ++ AssocList.nil :: l\u2082\n\u22a2 Nat.sum (List.map (fun x => List.length (AssocList.toList x)) l\u2081) +\n        Nat.sum (List.map (fun x => List.length (AssocList.toList x)) l\u2082) +\n      Nat.sum\n        (List.map (fun x => List.length (AssocList.toList x))\n          (List.foldl (fun d x => reinsertAux d x.fst x.snd) target (AssocList.toList source[i])).val.data) =\n    Nat.sum (List.map (fun x => List.length (AssocList.toList x)) l\u2081) +\n        (List.length (AssocList.toList (List.get source.data { val := i, isLt := H })) +\n          Nat.sum (List.map (fun x => List.length (AssocList.toList x)) l\u2082)) +\n      Nat.sum (List.map (fun x => List.length (AssocList.toList x)) target.val.data)"}, {"tactic": "rw [Nat.add_assoc, Nat.add_assoc, Nat.add_assoc]", "annotated_tactic": ["rw [<a>Nat.add_assoc</a>, <a>Nat.add_assoc</a>, <a>Nat.add_assoc</a>]", [{"full_name": "Nat.add_assoc", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [138, 19], "def_end_pos": [138, 28]}, {"full_name": "Nat.add_assoc", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [138, 19], "def_end_pos": [138, 28]}, {"full_name": "Nat.add_assoc", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [138, 19], "def_end_pos": [138, 28]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nsz : Nat\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\ni : Nat\nsource : Array (AssocList \u03b1 \u03b2)\ntarget : Buckets \u03b1 \u03b2\nhs : \u2200 (j : Nat), j < i \u2192 List.getD source.data j AssocList.nil = AssocList.nil\nH : i < Array.size source\nl\u2081 l\u2082 : List (AssocList \u03b1 \u03b2)\nh\u2081 : source.data = l\u2081 ++ List.get source.data { val := i, isLt := H } :: l\u2082\nleft\u271d : List.length l\u2081 = i\neq : List.set source.data i AssocList.nil = l\u2081 ++ AssocList.nil :: l\u2082\n\u22a2 Nat.sum (List.map (fun x => List.length (AssocList.toList x)) l\u2081) +\n        Nat.sum (List.map (fun x => List.length (AssocList.toList x)) l\u2082) +\n      Nat.sum\n        (List.map (fun x => List.length (AssocList.toList x))\n          (List.foldl (fun d x => reinsertAux d x.fst x.snd) target (AssocList.toList source[i])).val.data) =\n    Nat.sum (List.map (fun x => List.length (AssocList.toList x)) l\u2081) +\n        (List.length (AssocList.toList (List.get source.data { val := i, isLt := H })) +\n          Nat.sum (List.map (fun x => List.length (AssocList.toList x)) l\u2082)) +\n      Nat.sum (List.map (fun x => List.length (AssocList.toList x)) target.val.data)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nsz : Nat\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\ni : Nat\nsource : Array (AssocList \u03b1 \u03b2)\ntarget : Buckets \u03b1 \u03b2\nhs : \u2200 (j : Nat), j < i \u2192 List.getD source.data j AssocList.nil = AssocList.nil\nH : i < Array.size source\nl\u2081 l\u2082 : List (AssocList \u03b1 \u03b2)\nh\u2081 : source.data = l\u2081 ++ List.get source.data { val := i, isLt := H } :: l\u2082\nleft\u271d : List.length l\u2081 = i\neq : List.set source.data i AssocList.nil = l\u2081 ++ AssocList.nil :: l\u2082\n\u22a2 Nat.sum (List.map (fun x => List.length (AssocList.toList x)) l\u2081) +\n      (Nat.sum (List.map (fun x => List.length (AssocList.toList x)) l\u2082) +\n        Nat.sum\n          (List.map (fun x => List.length (AssocList.toList x))\n            (List.foldl (fun d x => reinsertAux d x.fst x.snd) target (AssocList.toList source[i])).val.data)) =\n    Nat.sum (List.map (fun x => List.length (AssocList.toList x)) l\u2081) +\n      (List.length (AssocList.toList (List.get source.data { val := i, isLt := H })) +\n        (Nat.sum (List.map (fun x => List.length (AssocList.toList x)) l\u2082) +\n          Nat.sum (List.map (fun x => List.length (AssocList.toList x)) target.val.data)))"}, {"tactic": "congr 1", "annotated_tactic": ["congr 1", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nsz : Nat\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\ni : Nat\nsource : Array (AssocList \u03b1 \u03b2)\ntarget : Buckets \u03b1 \u03b2\nhs : \u2200 (j : Nat), j < i \u2192 List.getD source.data j AssocList.nil = AssocList.nil\nH : i < Array.size source\nl\u2081 l\u2082 : List (AssocList \u03b1 \u03b2)\nh\u2081 : source.data = l\u2081 ++ List.get source.data { val := i, isLt := H } :: l\u2082\nleft\u271d : List.length l\u2081 = i\neq : List.set source.data i AssocList.nil = l\u2081 ++ AssocList.nil :: l\u2082\n\u22a2 Nat.sum (List.map (fun x => List.length (AssocList.toList x)) l\u2081) +\n      (Nat.sum (List.map (fun x => List.length (AssocList.toList x)) l\u2082) +\n        Nat.sum\n          (List.map (fun x => List.length (AssocList.toList x))\n            (List.foldl (fun d x => reinsertAux d x.fst x.snd) target (AssocList.toList source[i])).val.data)) =\n    Nat.sum (List.map (fun x => List.length (AssocList.toList x)) l\u2081) +\n      (List.length (AssocList.toList (List.get source.data { val := i, isLt := H })) +\n        (Nat.sum (List.map (fun x => List.length (AssocList.toList x)) l\u2082) +\n          Nat.sum (List.map (fun x => List.length (AssocList.toList x)) target.val.data)))", "state_after": "case e_a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nsz : Nat\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\ni : Nat\nsource : Array (AssocList \u03b1 \u03b2)\ntarget : Buckets \u03b1 \u03b2\nhs : \u2200 (j : Nat), j < i \u2192 List.getD source.data j AssocList.nil = AssocList.nil\nH : i < Array.size source\nl\u2081 l\u2082 : List (AssocList \u03b1 \u03b2)\nh\u2081 : source.data = l\u2081 ++ List.get source.data { val := i, isLt := H } :: l\u2082\nleft\u271d : List.length l\u2081 = i\neq : List.set source.data i AssocList.nil = l\u2081 ++ AssocList.nil :: l\u2082\n\u22a2 Nat.sum (List.map (fun x => List.length (AssocList.toList x)) l\u2082) +\n      Nat.sum\n        (List.map (fun x => List.length (AssocList.toList x))\n          (List.foldl (fun d x => reinsertAux d x.fst x.snd) target (AssocList.toList source[i])).val.data) =\n    List.length (AssocList.toList (List.get source.data { val := i, isLt := H })) +\n      (Nat.sum (List.map (fun x => List.length (AssocList.toList x)) l\u2082) +\n        Nat.sum (List.map (fun x => List.length (AssocList.toList x)) target.val.data))"}, {"tactic": "(conv => rhs; rw [Nat.add_left_comm])", "annotated_tactic": ["(conv => rhs; rw [<a>Nat.add_left_comm</a>])", [{"full_name": "Nat.add_left_comm", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [142, 19], "def_end_pos": [142, 32]}]], "state_before": "case e_a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nsz : Nat\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\ni : Nat\nsource : Array (AssocList \u03b1 \u03b2)\ntarget : Buckets \u03b1 \u03b2\nhs : \u2200 (j : Nat), j < i \u2192 List.getD source.data j AssocList.nil = AssocList.nil\nH : i < Array.size source\nl\u2081 l\u2082 : List (AssocList \u03b1 \u03b2)\nh\u2081 : source.data = l\u2081 ++ List.get source.data { val := i, isLt := H } :: l\u2082\nleft\u271d : List.length l\u2081 = i\neq : List.set source.data i AssocList.nil = l\u2081 ++ AssocList.nil :: l\u2082\n\u22a2 Nat.sum (List.map (fun x => List.length (AssocList.toList x)) l\u2082) +\n      Nat.sum\n        (List.map (fun x => List.length (AssocList.toList x))\n          (List.foldl (fun d x => reinsertAux d x.fst x.snd) target (AssocList.toList source[i])).val.data) =\n    List.length (AssocList.toList (List.get source.data { val := i, isLt := H })) +\n      (Nat.sum (List.map (fun x => List.length (AssocList.toList x)) l\u2082) +\n        Nat.sum (List.map (fun x => List.length (AssocList.toList x)) target.val.data))", "state_after": "case e_a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nsz : Nat\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\ni : Nat\nsource : Array (AssocList \u03b1 \u03b2)\ntarget : Buckets \u03b1 \u03b2\nhs : \u2200 (j : Nat), j < i \u2192 List.getD source.data j AssocList.nil = AssocList.nil\nH : i < Array.size source\nl\u2081 l\u2082 : List (AssocList \u03b1 \u03b2)\nh\u2081 : source.data = l\u2081 ++ List.get source.data { val := i, isLt := H } :: l\u2082\nleft\u271d : List.length l\u2081 = i\neq : List.set source.data i AssocList.nil = l\u2081 ++ AssocList.nil :: l\u2082\n\u22a2 Nat.sum (List.map (fun x => List.length (AssocList.toList x)) l\u2082) +\n      Nat.sum\n        (List.map (fun x => List.length (AssocList.toList x))\n          (List.foldl (fun d x => reinsertAux d x.fst x.snd) target (AssocList.toList source[i])).val.data) =\n    Nat.sum (List.map (fun x => List.length (AssocList.toList x)) l\u2082) +\n      (List.length (AssocList.toList (List.get source.data { val := i, isLt := H })) +\n        Nat.sum (List.map (fun x => List.length (AssocList.toList x)) target.val.data))"}, {"tactic": "congr 1", "annotated_tactic": ["congr 1", []], "state_before": "case e_a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nsz : Nat\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\ni : Nat\nsource : Array (AssocList \u03b1 \u03b2)\ntarget : Buckets \u03b1 \u03b2\nhs : \u2200 (j : Nat), j < i \u2192 List.getD source.data j AssocList.nil = AssocList.nil\nH : i < Array.size source\nl\u2081 l\u2082 : List (AssocList \u03b1 \u03b2)\nh\u2081 : source.data = l\u2081 ++ List.get source.data { val := i, isLt := H } :: l\u2082\nleft\u271d : List.length l\u2081 = i\neq : List.set source.data i AssocList.nil = l\u2081 ++ AssocList.nil :: l\u2082\n\u22a2 Nat.sum (List.map (fun x => List.length (AssocList.toList x)) l\u2082) +\n      Nat.sum\n        (List.map (fun x => List.length (AssocList.toList x))\n          (List.foldl (fun d x => reinsertAux d x.fst x.snd) target (AssocList.toList source[i])).val.data) =\n    Nat.sum (List.map (fun x => List.length (AssocList.toList x)) l\u2082) +\n      (List.length (AssocList.toList (List.get source.data { val := i, isLt := H })) +\n        Nat.sum (List.map (fun x => List.length (AssocList.toList x)) target.val.data))", "state_after": "case e_a.e_a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nsz : Nat\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\ni : Nat\nsource : Array (AssocList \u03b1 \u03b2)\ntarget : Buckets \u03b1 \u03b2\nhs : \u2200 (j : Nat), j < i \u2192 List.getD source.data j AssocList.nil = AssocList.nil\nH : i < Array.size source\nl\u2081 l\u2082 : List (AssocList \u03b1 \u03b2)\nh\u2081 : source.data = l\u2081 ++ List.get source.data { val := i, isLt := H } :: l\u2082\nleft\u271d : List.length l\u2081 = i\neq : List.set source.data i AssocList.nil = l\u2081 ++ AssocList.nil :: l\u2082\n\u22a2 Nat.sum\n      (List.map (fun x => List.length (AssocList.toList x))\n        (List.foldl (fun d x => reinsertAux d x.fst x.snd) target (AssocList.toList source[i])).val.data) =\n    List.length (AssocList.toList (List.get source.data { val := i, isLt := H })) +\n      Nat.sum (List.map (fun x => List.length (AssocList.toList x)) target.val.data)"}, {"tactic": "rw [\u2190 Array.getElem_eq_data_get]", "annotated_tactic": ["rw [\u2190 <a>Array.getElem_eq_data_get</a>]", [{"full_name": "Array.getElem_eq_data_get", "def_path": "lake-packages/std/Std/Data/Array/Init/Lemmas.lean", "def_pos": [28, 9], "def_end_pos": [28, 28]}]], "state_before": "case e_a.e_a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nsz : Nat\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\ni : Nat\nsource : Array (AssocList \u03b1 \u03b2)\ntarget : Buckets \u03b1 \u03b2\nhs : \u2200 (j : Nat), j < i \u2192 List.getD source.data j AssocList.nil = AssocList.nil\nH : i < Array.size source\nl\u2081 l\u2082 : List (AssocList \u03b1 \u03b2)\nh\u2081 : source.data = l\u2081 ++ List.get source.data { val := i, isLt := H } :: l\u2082\nleft\u271d : List.length l\u2081 = i\neq : List.set source.data i AssocList.nil = l\u2081 ++ AssocList.nil :: l\u2082\n\u22a2 Nat.sum\n      (List.map (fun x => List.length (AssocList.toList x))\n        (List.foldl (fun d x => reinsertAux d x.fst x.snd) target (AssocList.toList source[i])).val.data) =\n    List.length (AssocList.toList (List.get source.data { val := i, isLt := H })) +\n      Nat.sum (List.map (fun x => List.length (AssocList.toList x)) target.val.data)", "state_after": "case e_a.e_a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nsz : Nat\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\ni : Nat\nsource : Array (AssocList \u03b1 \u03b2)\ntarget : Buckets \u03b1 \u03b2\nhs : \u2200 (j : Nat), j < i \u2192 List.getD source.data j AssocList.nil = AssocList.nil\nH : i < Array.size source\nl\u2081 l\u2082 : List (AssocList \u03b1 \u03b2)\nh\u2081 : source.data = l\u2081 ++ List.get source.data { val := i, isLt := H } :: l\u2082\nleft\u271d : List.length l\u2081 = i\neq : List.set source.data i AssocList.nil = l\u2081 ++ AssocList.nil :: l\u2082\n\u22a2 Nat.sum\n      (List.map (fun x => List.length (AssocList.toList x))\n        (List.foldl (fun d x => reinsertAux d x.fst x.snd) target (AssocList.toList source[i])).val.data) =\n    List.length (AssocList.toList source[i]) +\n      Nat.sum (List.map (fun x => List.length (AssocList.toList x)) target.val.data)"}, {"tactic": "have := @reinsertAux_size \u03b1 \u03b2 _", "annotated_tactic": ["have := @<a>reinsertAux_size</a> \u03b1 \u03b2 _", [{"full_name": "Std.HashMap.Imp.reinsertAux_size", "def_path": "lake-packages/std/Std/Data/HashMap/WF.lean", "def_pos": [64, 9], "def_end_pos": [64, 25]}]], "state_before": "case e_a.e_a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nsz : Nat\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\ni : Nat\nsource : Array (AssocList \u03b1 \u03b2)\ntarget : Buckets \u03b1 \u03b2\nhs : \u2200 (j : Nat), j < i \u2192 List.getD source.data j AssocList.nil = AssocList.nil\nH : i < Array.size source\nl\u2081 l\u2082 : List (AssocList \u03b1 \u03b2)\nh\u2081 : source.data = l\u2081 ++ List.get source.data { val := i, isLt := H } :: l\u2082\nleft\u271d : List.length l\u2081 = i\neq : List.set source.data i AssocList.nil = l\u2081 ++ AssocList.nil :: l\u2082\n\u22a2 Nat.sum\n      (List.map (fun x => List.length (AssocList.toList x))\n        (List.foldl (fun d x => reinsertAux d x.fst x.snd) target (AssocList.toList source[i])).val.data) =\n    List.length (AssocList.toList source[i]) +\n      Nat.sum (List.map (fun x => List.length (AssocList.toList x)) target.val.data)", "state_after": "case e_a.e_a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nsz : Nat\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\ni : Nat\nsource : Array (AssocList \u03b1 \u03b2)\ntarget : Buckets \u03b1 \u03b2\nhs : \u2200 (j : Nat), j < i \u2192 List.getD source.data j AssocList.nil = AssocList.nil\nH : i < Array.size source\nl\u2081 l\u2082 : List (AssocList \u03b1 \u03b2)\nh\u2081 : source.data = l\u2081 ++ List.get source.data { val := i, isLt := H } :: l\u2082\nleft\u271d : List.length l\u2081 = i\neq : List.set source.data i AssocList.nil = l\u2081 ++ AssocList.nil :: l\u2082\nthis : \u2200 (data : Buckets \u03b1 \u03b2) (a : \u03b1) (b : \u03b2), Buckets.size (reinsertAux data a b) = Nat.succ (Buckets.size data)\n\u22a2 Nat.sum\n      (List.map (fun x => List.length (AssocList.toList x))\n        (List.foldl (fun d x => reinsertAux d x.fst x.snd) target (AssocList.toList source[i])).val.data) =\n    List.length (AssocList.toList source[i]) +\n      Nat.sum (List.map (fun x => List.length (AssocList.toList x)) target.val.data)"}, {"tactic": "simp [Buckets.size] at this", "annotated_tactic": ["simp [<a>Buckets.size</a>] at this", [{"full_name": "Std.HashMap.Imp.Buckets.size", "def_path": "lake-packages/std/Std/Data/HashMap/Basic.lean", "def_pos": [40, 19], "def_end_pos": [40, 23]}]], "state_before": "case e_a.e_a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nsz : Nat\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\ni : Nat\nsource : Array (AssocList \u03b1 \u03b2)\ntarget : Buckets \u03b1 \u03b2\nhs : \u2200 (j : Nat), j < i \u2192 List.getD source.data j AssocList.nil = AssocList.nil\nH : i < Array.size source\nl\u2081 l\u2082 : List (AssocList \u03b1 \u03b2)\nh\u2081 : source.data = l\u2081 ++ List.get source.data { val := i, isLt := H } :: l\u2082\nleft\u271d : List.length l\u2081 = i\neq : List.set source.data i AssocList.nil = l\u2081 ++ AssocList.nil :: l\u2082\nthis : \u2200 (data : Buckets \u03b1 \u03b2) (a : \u03b1) (b : \u03b2), Buckets.size (reinsertAux data a b) = Nat.succ (Buckets.size data)\n\u22a2 Nat.sum\n      (List.map (fun x => List.length (AssocList.toList x))\n        (List.foldl (fun d x => reinsertAux d x.fst x.snd) target (AssocList.toList source[i])).val.data) =\n    List.length (AssocList.toList source[i]) +\n      Nat.sum (List.map (fun x => List.length (AssocList.toList x)) target.val.data)", "state_after": "case e_a.e_a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nsz : Nat\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\ni : Nat\nsource : Array (AssocList \u03b1 \u03b2)\ntarget : Buckets \u03b1 \u03b2\nhs : \u2200 (j : Nat), j < i \u2192 List.getD source.data j AssocList.nil = AssocList.nil\nH : i < Array.size source\nl\u2081 l\u2082 : List (AssocList \u03b1 \u03b2)\nh\u2081 : source.data = l\u2081 ++ List.get source.data { val := i, isLt := H } :: l\u2082\nleft\u271d : List.length l\u2081 = i\neq : List.set source.data i AssocList.nil = l\u2081 ++ AssocList.nil :: l\u2082\nthis :\n  \u2200 (data : Buckets \u03b1 \u03b2) (a : \u03b1) (b : \u03b2),\n    Nat.sum (List.map (fun x => List.length (AssocList.toList x)) (reinsertAux data a b).val.data) =\n      Nat.succ (Nat.sum (List.map (fun x => List.length (AssocList.toList x)) data.val.data))\n\u22a2 Nat.sum\n      (List.map (fun x => List.length (AssocList.toList x))\n        (List.foldl (fun d x => reinsertAux d x.fst x.snd) target (AssocList.toList source[i])).val.data) =\n    List.length (AssocList.toList source[i]) +\n      Nat.sum (List.map (fun x => List.length (AssocList.toList x)) target.val.data)"}, {"tactic": "induction source[i].toList generalizing target <;> simp [*, Nat.succ_add]", "annotated_tactic": ["induction source[i].<a>toList</a> generalizing target <;> simp [*, <a>Nat.succ_add</a>]", [{"full_name": "Std.AssocList.toList", "def_path": "lake-packages/std/Std/Data/AssocList.lean", "def_pos": [30, 13], "def_end_pos": [30, 19]}, {"full_name": "Nat.succ_add", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [118, 9], "def_end_pos": [118, 17]}]], "state_before": "case e_a.e_a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nsz : Nat\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\ni : Nat\nsource : Array (AssocList \u03b1 \u03b2)\ntarget : Buckets \u03b1 \u03b2\nhs : \u2200 (j : Nat), j < i \u2192 List.getD source.data j AssocList.nil = AssocList.nil\nH : i < Array.size source\nl\u2081 l\u2082 : List (AssocList \u03b1 \u03b2)\nh\u2081 : source.data = l\u2081 ++ List.get source.data { val := i, isLt := H } :: l\u2082\nleft\u271d : List.length l\u2081 = i\neq : List.set source.data i AssocList.nil = l\u2081 ++ AssocList.nil :: l\u2082\nthis :\n  \u2200 (data : Buckets \u03b1 \u03b2) (a : \u03b1) (b : \u03b2),\n    Nat.sum (List.map (fun x => List.length (AssocList.toList x)) (reinsertAux data a b).val.data) =\n      Nat.succ (Nat.sum (List.map (fun x => List.length (AssocList.toList x)) data.val.data))\n\u22a2 Nat.sum\n      (List.map (fun x => List.length (AssocList.toList x))\n        (List.foldl (fun d x => reinsertAux d x.fst x.snd) target (AssocList.toList source[i])).val.data) =\n    List.length (AssocList.toList source[i]) +\n      Nat.sum (List.map (fun x => List.length (AssocList.toList x)) target.val.data)", "state_after": "case e_a.e_a.cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nsz : Nat\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\ni : Nat\nsource : Array (AssocList \u03b1 \u03b2)\nhs : \u2200 (j : Nat), j < i \u2192 List.getD source.data j AssocList.nil = AssocList.nil\nH : i < Array.size source\nl\u2081 l\u2082 : List (AssocList \u03b1 \u03b2)\nh\u2081 : source.data = l\u2081 ++ List.get source.data { val := i, isLt := H } :: l\u2082\nleft\u271d : List.length l\u2081 = i\neq : List.set source.data i AssocList.nil = l\u2081 ++ AssocList.nil :: l\u2082\nthis :\n  \u2200 (data : Buckets \u03b1 \u03b2) (a : \u03b1) (b : \u03b2),\n    Nat.sum (List.map (fun x => List.length (AssocList.toList x)) (reinsertAux data a b).val.data) =\n      Nat.succ (Nat.sum (List.map (fun x => List.length (AssocList.toList x)) data.val.data))\nhead\u271d : \u03b1 \u00d7 \u03b2\ntail\u271d : List (\u03b1 \u00d7 \u03b2)\ntail_ih\u271d :\n  \u2200 (target : Buckets \u03b1 \u03b2),\n    Nat.sum\n        (List.map (fun x => List.length (AssocList.toList x))\n          (List.foldl (fun d x => reinsertAux d x.fst x.snd) target tail\u271d).val.data) =\n      List.length tail\u271d + Nat.sum (List.map (fun x => List.length (AssocList.toList x)) target.val.data)\ntarget : Buckets \u03b1 \u03b2\n\u22a2 List.length tail\u271d + Nat.succ (Nat.sum (List.map (fun x => List.length (AssocList.toList x)) target.val.data)) =\n    Nat.succ (List.length tail\u271d + Nat.sum (List.map (fun x => List.length (AssocList.toList x)) target.val.data))"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case e_a.e_a.cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nsz : Nat\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\ni : Nat\nsource : Array (AssocList \u03b1 \u03b2)\nhs : \u2200 (j : Nat), j < i \u2192 List.getD source.data j AssocList.nil = AssocList.nil\nH : i < Array.size source\nl\u2081 l\u2082 : List (AssocList \u03b1 \u03b2)\nh\u2081 : source.data = l\u2081 ++ List.get source.data { val := i, isLt := H } :: l\u2082\nleft\u271d : List.length l\u2081 = i\neq : List.set source.data i AssocList.nil = l\u2081 ++ AssocList.nil :: l\u2082\nthis :\n  \u2200 (data : Buckets \u03b1 \u03b2) (a : \u03b1) (b : \u03b2),\n    Nat.sum (List.map (fun x => List.length (AssocList.toList x)) (reinsertAux data a b).val.data) =\n      Nat.succ (Nat.sum (List.map (fun x => List.length (AssocList.toList x)) data.val.data))\nhead\u271d : \u03b1 \u00d7 \u03b2\ntail\u271d : List (\u03b1 \u00d7 \u03b2)\ntail_ih\u271d :\n  \u2200 (target : Buckets \u03b1 \u03b2),\n    Nat.sum\n        (List.map (fun x => List.length (AssocList.toList x))\n          (List.foldl (fun d x => reinsertAux d x.fst x.snd) target tail\u271d).val.data) =\n      List.length tail\u271d + Nat.sum (List.map (fun x => List.length (AssocList.toList x)) target.val.data)\ntarget : Buckets \u03b1 \u03b2\n\u22a2 List.length tail\u271d + Nat.succ (Nat.sum (List.map (fun x => List.length (AssocList.toList x)) target.val.data)) =\n    Nat.succ (List.length tail\u271d + Nat.sum (List.map (fun x => List.length (AssocList.toList x)) target.val.data))", "state_after": "no goals"}, {"tactic": "conv => rhs; rw [Nat.add_left_comm]", "annotated_tactic": ["conv => rhs; rw [<a>Nat.add_left_comm</a>]", [{"full_name": "Nat.add_left_comm", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [142, 19], "def_end_pos": [142, 32]}]], "state_before": "case e_a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nsz : Nat\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\ni : Nat\nsource : Array (AssocList \u03b1 \u03b2)\ntarget : Buckets \u03b1 \u03b2\nhs : \u2200 (j : Nat), j < i \u2192 List.getD source.data j AssocList.nil = AssocList.nil\nH : i < Array.size source\nl\u2081 l\u2082 : List (AssocList \u03b1 \u03b2)\nh\u2081 : source.data = l\u2081 ++ List.get source.data { val := i, isLt := H } :: l\u2082\nleft\u271d : List.length l\u2081 = i\neq : List.set source.data i AssocList.nil = l\u2081 ++ AssocList.nil :: l\u2082\n\u22a2 Nat.sum (List.map (fun x => List.length (AssocList.toList x)) l\u2082) +\n      Nat.sum\n        (List.map (fun x => List.length (AssocList.toList x))\n          (List.foldl (fun d x => reinsertAux d x.fst x.snd) target (AssocList.toList source[i])).val.data) =\n    List.length (AssocList.toList (List.get source.data { val := i, isLt := H })) +\n      (Nat.sum (List.map (fun x => List.length (AssocList.toList x)) l\u2082) +\n        Nat.sum (List.map (fun x => List.length (AssocList.toList x)) target.val.data))", "state_after": "case e_a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nsz : Nat\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\ni : Nat\nsource : Array (AssocList \u03b1 \u03b2)\ntarget : Buckets \u03b1 \u03b2\nhs : \u2200 (j : Nat), j < i \u2192 List.getD source.data j AssocList.nil = AssocList.nil\nH : i < Array.size source\nl\u2081 l\u2082 : List (AssocList \u03b1 \u03b2)\nh\u2081 : source.data = l\u2081 ++ List.get source.data { val := i, isLt := H } :: l\u2082\nleft\u271d : List.length l\u2081 = i\neq : List.set source.data i AssocList.nil = l\u2081 ++ AssocList.nil :: l\u2082\n\u22a2 Nat.sum (List.map (fun x => List.length (AssocList.toList x)) l\u2082) +\n      Nat.sum\n        (List.map (fun x => List.length (AssocList.toList x))\n          (List.foldl (fun d x => reinsertAux d x.fst x.snd) target (AssocList.toList source[i])).val.data) =\n    Nat.sum (List.map (fun x => List.length (AssocList.toList x)) l\u2082) +\n      (List.length (AssocList.toList (List.get source.data { val := i, isLt := H })) +\n        Nat.sum (List.map (fun x => List.length (AssocList.toList x)) target.val.data))"}, {"tactic": "rw [(_ : Nat.sum _ = 0), Nat.zero_add]", "annotated_tactic": ["rw [(_ : <a>Nat.sum</a> _ = 0), <a>Nat.zero_add</a>]", [{"full_name": "Nat.sum", "def_path": "lake-packages/std/Std/Data/Nat/Basic.lean", "def_pos": [111, 15], "def_end_pos": [111, 18]}, {"full_name": "Nat.zero_add", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [114, 27], "def_end_pos": [114, 35]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nsz : Nat\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\ni : Nat\nsource : Array (AssocList \u03b1 \u03b2)\ntarget : Buckets \u03b1 \u03b2\nhs : \u2200 (j : Nat), j < i \u2192 List.getD source.data j AssocList.nil = AssocList.nil\nH : \u00aci < Array.size source\n\u22a2 Buckets.size target = Nat.sum (List.map (fun x => List.length (AssocList.toList x)) source.data) + Buckets.size target", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nsz : Nat\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\ni : Nat\nsource : Array (AssocList \u03b1 \u03b2)\ntarget : Buckets \u03b1 \u03b2\nhs : \u2200 (j : Nat), j < i \u2192 List.getD source.data j AssocList.nil = AssocList.nil\nH : \u00aci < Array.size source\n\u22a2 Nat.sum (List.map (fun x => List.length (AssocList.toList x)) source.data) = 0"}, {"tactic": "rw [\u2190 (_ : source.data.map (fun _ => .nil) = source.data)]", "annotated_tactic": ["rw [\u2190 (_ : source.data.map (fun _ => .nil) = source.data)]", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nsz : Nat\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\ni : Nat\nsource : Array (AssocList \u03b1 \u03b2)\ntarget : Buckets \u03b1 \u03b2\nhs : \u2200 (j : Nat), j < i \u2192 List.getD source.data j AssocList.nil = AssocList.nil\nH : \u00aci < Array.size source\n\u22a2 Nat.sum (List.map (fun x => List.length (AssocList.toList x)) source.data) = 0", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nsz : Nat\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\ni : Nat\nsource : Array (AssocList \u03b1 \u03b2)\ntarget : Buckets \u03b1 \u03b2\nhs : \u2200 (j : Nat), j < i \u2192 List.getD source.data j AssocList.nil = AssocList.nil\nH : \u00aci < Array.size source\n\u22a2 Nat.sum (List.map (fun x => List.length (AssocList.toList x)) (List.map (fun x => AssocList.nil) source.data)) = 0\n\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nsz : Nat\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\ni : Nat\nsource : Array (AssocList \u03b1 \u03b2)\ntarget : Buckets \u03b1 \u03b2\nhs : \u2200 (j : Nat), j < i \u2192 List.getD source.data j AssocList.nil = AssocList.nil\nH : \u00aci < Array.size source\n\u22a2 List.map (fun x => AssocList.nil) source.data = source.data"}, {"tactic": "refine List.ext_get (by simp) fun j h\u2081 h\u2082 => ?_", "annotated_tactic": ["refine <a>List.ext_get</a> (by simp) fun j h\u2081 h\u2082 => ?_", [{"full_name": "List.ext_get", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [744, 9], "def_end_pos": [744, 16]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nsz : Nat\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\ni : Nat\nsource : Array (AssocList \u03b1 \u03b2)\ntarget : Buckets \u03b1 \u03b2\nhs : \u2200 (j : Nat), j < i \u2192 List.getD source.data j AssocList.nil = AssocList.nil\nH : \u00aci < Array.size source\n\u22a2 List.map (fun x => AssocList.nil) source.data = source.data", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nsz : Nat\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\ni : Nat\nsource : Array (AssocList \u03b1 \u03b2)\ntarget : Buckets \u03b1 \u03b2\nhs : \u2200 (j : Nat), j < i \u2192 List.getD source.data j AssocList.nil = AssocList.nil\nH : \u00aci < Array.size source\nj : Nat\nh\u2081 : j < List.length (List.map (fun x => AssocList.nil) source.data)\nh\u2082 : j < List.length source.data\n\u22a2 List.get (List.map (fun x => AssocList.nil) source.data) { val := j, isLt := h\u2081 } =\n    List.get source.data { val := j, isLt := h\u2082 }"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nsz : Nat\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\ni : Nat\nsource : Array (AssocList \u03b1 \u03b2)\ntarget : Buckets \u03b1 \u03b2\nhs : \u2200 (j : Nat), j < i \u2192 List.getD source.data j AssocList.nil = AssocList.nil\nH : \u00aci < Array.size source\nj : Nat\nh\u2081 : j < List.length (List.map (fun x => AssocList.nil) source.data)\nh\u2082 : j < List.length source.data\n\u22a2 List.get (List.map (fun x => AssocList.nil) source.data) { val := j, isLt := h\u2081 } =\n    List.get source.data { val := j, isLt := h\u2082 }", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nsz : Nat\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\ni : Nat\nsource : Array (AssocList \u03b1 \u03b2)\ntarget : Buckets \u03b1 \u03b2\nhs : \u2200 (j : Nat), j < i \u2192 List.getD source.data j AssocList.nil = AssocList.nil\nH : \u00aci < Array.size source\nj : Nat\nh\u2081 : j < List.length (List.map (fun x => AssocList.nil) source.data)\nh\u2082 : j < List.length source.data\n\u22a2 AssocList.nil = List.get source.data { val := j, isLt := h\u2082 }"}, {"tactic": "have := (hs j (Nat.lt_of_lt_of_le h\u2082 (Nat.not_lt.1 H))).symm", "annotated_tactic": ["have := (hs j (<a>Nat.lt_of_lt_of_le</a> h\u2082 (<a>Nat.not_lt</a>.1 H))).<a>symm</a>", [{"full_name": "Nat.lt_of_lt_of_le", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [259, 19], "def_end_pos": [259, 33]}, {"full_name": "Nat.not_lt", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [150, 27], "def_end_pos": [150, 33]}, {"full_name": "Eq.symm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [310, 9], "def_end_pos": [310, 16]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nsz : Nat\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\ni : Nat\nsource : Array (AssocList \u03b1 \u03b2)\ntarget : Buckets \u03b1 \u03b2\nhs : \u2200 (j : Nat), j < i \u2192 List.getD source.data j AssocList.nil = AssocList.nil\nH : \u00aci < Array.size source\nj : Nat\nh\u2081 : j < List.length (List.map (fun x => AssocList.nil) source.data)\nh\u2082 : j < List.length source.data\n\u22a2 AssocList.nil = List.get source.data { val := j, isLt := h\u2082 }", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nsz : Nat\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\ni : Nat\nsource : Array (AssocList \u03b1 \u03b2)\ntarget : Buckets \u03b1 \u03b2\nhs : \u2200 (j : Nat), j < i \u2192 List.getD source.data j AssocList.nil = AssocList.nil\nH : \u00aci < Array.size source\nj : Nat\nh\u2081 : j < List.length (List.map (fun x => AssocList.nil) source.data)\nh\u2082 : j < List.length source.data\nthis : AssocList.nil = List.getD source.data j AssocList.nil\n\u22a2 AssocList.nil = List.get source.data { val := j, isLt := h\u2082 }"}, {"tactic": "rwa [List.getD_eq_get?, List.get?_eq_get, Option.getD_some] at this", "annotated_tactic": ["rwa [<a>List.getD_eq_get?</a>, <a>List.get?_eq_get</a>, <a>Option.getD_some</a>] at this", [{"full_name": "List.getD_eq_get?", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [772, 9], "def_end_pos": [772, 21]}, {"full_name": "List.get?_eq_get", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [581, 9], "def_end_pos": [581, 20]}, {"full_name": "Option.getD_some", "def_path": "lake-packages/std/Std/Data/Option/Init/Lemmas.lean", "def_pos": [17, 17], "def_end_pos": [17, 26]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nsz : Nat\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\ni : Nat\nsource : Array (AssocList \u03b1 \u03b2)\ntarget : Buckets \u03b1 \u03b2\nhs : \u2200 (j : Nat), j < i \u2192 List.getD source.data j AssocList.nil = AssocList.nil\nH : \u00aci < Array.size source\nj : Nat\nh\u2081 : j < List.length (List.map (fun x => AssocList.nil) source.data)\nh\u2082 : j < List.length source.data\nthis : AssocList.nil = List.getD source.data j AssocList.nil\n\u22a2 AssocList.nil = List.get source.data { val := j, isLt := h\u2082 }", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nsz : Nat\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\ni : Nat\nsource : Array (AssocList \u03b1 \u03b2)\ntarget : Buckets \u03b1 \u03b2\nhs : \u2200 (j : Nat), j < i \u2192 List.getD source.data j AssocList.nil = AssocList.nil\nH : \u00aci < Array.size source\n\u22a2 Nat.sum (List.map (fun x => List.length (AssocList.toList x)) (List.map (fun x => AssocList.nil) source.data)) = 0", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nsz : Nat\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\ni : Nat\nsource : Array (AssocList \u03b1 \u03b2)\ntarget : Buckets \u03b1 \u03b2\nhs : \u2200 (j : Nat), j < i \u2192 List.getD source.data j AssocList.nil = AssocList.nil\nH : \u00aci < Array.size source\n\u22a2 Nat.sum (List.map ((fun x => List.length (AssocList.toList x)) \u2218 fun x => AssocList.nil) source.data) = 0"}, {"tactic": "induction source.data <;> simp [*]", "annotated_tactic": ["induction source.data <;> simp [*]", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nsz : Nat\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\ni : Nat\nsource : Array (AssocList \u03b1 \u03b2)\ntarget : Buckets \u03b1 \u03b2\nhs : \u2200 (j : Nat), j < i \u2192 List.getD source.data j AssocList.nil = AssocList.nil\nH : \u00aci < Array.size source\n\u22a2 Nat.sum (List.map ((fun x => List.length (AssocList.toList x)) \u2218 fun x => AssocList.nil) source.data) = 0", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nsz : Nat\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\ni : Nat\nsource : Array (AssocList \u03b1 \u03b2)\ntarget : Buckets \u03b1 \u03b2\nhs : \u2200 (j : Nat), j < i \u2192 List.getD source.data j AssocList.nil = AssocList.nil\nH : \u00aci < Array.size source\n\u22a2 List.length (List.map (fun x => AssocList.nil) source.data) = List.length source.data", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/TypeVec.lean", "full_name": "TypeVec.typevecCasesCons\u2082_appendFun", "start": [371, 1], "end": [376, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/MeanInequalities.lean", "full_name": "ENNReal.lintegral_Lp_add_le_aux", "start": [312, 9], "end": [342, 82], "traced_tactics": [{"tactic": "have hp_not_nonpos : \u00acp \u2264 0 := by simp [hpq.pos]", "annotated_tactic": ["have hp_not_nonpos : \u00acp \u2264 0 := by simp [hpq.pos]", []], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhg : AEMeasurable g\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc \u2260 \u22a4\nh_add_zero : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 0\nh_add_top : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 \u22a4\n\u22a2 (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)", "state_after": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhg : AEMeasurable g\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc \u2260 \u22a4\nh_add_zero : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 0\nh_add_top : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 \u22a4\nhp_not_nonpos : \u00acp \u2264 0\n\u22a2 (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)"}, {"tactic": "have htop_rpow : (\u222b\u207b a, (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 \u22a4 := by\n  by_contra h\n  exact h_add_top (@ENNReal.rpow_eq_top_of_nonneg _ (1 / p) (by simp [hpq.nonneg]) h)", "annotated_tactic": ["have htop_rpow : (\u222b\u207b a, (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 \u22a4 := by\n    by_contra h\n    exact h_add_top (@<a>ENNReal.rpow_eq_top_of_nonneg</a> _ (1 / p) (by simp [hpq.nonneg]) h)", [{"full_name": "ENNReal.rpow_eq_top_of_nonneg", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [490, 9], "def_end_pos": [490, 30]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhg : AEMeasurable g\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc \u2260 \u22a4\nh_add_zero : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 0\nh_add_top : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 \u22a4\nhp_not_nonpos : \u00acp \u2264 0\n\u22a2 (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)", "state_after": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhg : AEMeasurable g\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc \u2260 \u22a4\nh_add_zero : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 0\nh_add_top : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 \u22a4\nhp_not_nonpos : \u00acp \u2264 0\nhtop_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 \u22a4\n\u22a2 (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)"}, {"tactic": "have h0_rpow : (\u222b\u207b a, (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 0 := by\n  simp [h_add_zero, h_add_top, hpq.nonneg, hp_not_nonpos, -Pi.add_apply]", "annotated_tactic": ["have h0_rpow : (\u222b\u207b a, (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 0 := by\n    simp [h_add_zero, h_add_top, hpq.nonneg, hp_not_nonpos, -<a>Pi.add_apply</a>]", [{"full_name": "Pi.add_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [82, 3], "def_end_pos": [82, 14]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhg : AEMeasurable g\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc \u2260 \u22a4\nh_add_zero : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 0\nh_add_top : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 \u22a4\nhp_not_nonpos : \u00acp \u2264 0\nhtop_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 \u22a4\n\u22a2 (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)", "state_after": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhg : AEMeasurable g\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc \u2260 \u22a4\nh_add_zero : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 0\nh_add_top : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 \u22a4\nhp_not_nonpos : \u00acp \u2264 0\nhtop_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 \u22a4\nh0_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 0\n\u22a2 (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)"}, {"tactic": "suffices h :\n  1 \u2264\n    (\u222b\u207b a : \u03b1, (f + g) a ^ p \u2202\u03bc) ^ (-(1 / p)) *\n      ((\u222b\u207b a : \u03b1, f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b a : \u03b1, g a ^ p \u2202\u03bc) ^ (1 / p))", "annotated_tactic": ["suffices h :\n    1 \u2264\n      (\u222b\u207b a : \u03b1, (f + g) a ^ p \u2202\u03bc) ^ (-(1 / p)) *\n        ((\u222b\u207b a : \u03b1, f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b a : \u03b1, g a ^ p \u2202\u03bc) ^ (1 / p))", []], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhg : AEMeasurable g\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc \u2260 \u22a4\nh_add_zero : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 0\nh_add_top : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 \u22a4\nhp_not_nonpos : \u00acp \u2264 0\nhtop_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 \u22a4\nh0_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 0\n\u22a2 (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)", "state_after": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhg : AEMeasurable g\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc \u2260 \u22a4\nh_add_zero : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 0\nh_add_top : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 \u22a4\nhp_not_nonpos : \u00acp \u2264 0\nhtop_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 \u22a4\nh0_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 0\nh :\n  1 \u2264\n    (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (-(1 / p)) *\n      ((\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p))\n\u22a2 (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)\n\ncase h\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhg : AEMeasurable g\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc \u2260 \u22a4\nh_add_zero : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 0\nh_add_top : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 \u22a4\nhp_not_nonpos : \u00acp \u2264 0\nhtop_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 \u22a4\nh0_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 0\n\u22a2 1 \u2264\n    (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (-(1 / p)) *\n      ((\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p))"}, {"tactic": "have h :\n  (\u222b\u207b a : \u03b1, (f + g) a ^ p \u2202\u03bc) \u2264\n    ((\u222b\u207b a : \u03b1, f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b a : \u03b1, g a ^ p \u2202\u03bc) ^ (1 / p)) *\n      (\u222b\u207b a : \u03b1, (f + g) a ^ p \u2202\u03bc) ^ (1 / q) :=\n  lintegral_rpow_add_le_add_snorm_mul_lintegral_rpow_add hpq hf hf_top hg hg_top", "annotated_tactic": ["have h :\n    (\u222b\u207b a : \u03b1, (f + g) a ^ p \u2202\u03bc) \u2264\n      ((\u222b\u207b a : \u03b1, f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b a : \u03b1, g a ^ p \u2202\u03bc) ^ (1 / p)) *\n        (\u222b\u207b a : \u03b1, (f + g) a ^ p \u2202\u03bc) ^ (1 / q) :=\n    <a>lintegral_rpow_add_le_add_snorm_mul_lintegral_rpow_add</a> hpq hf hf_top hg hg_top", [{"full_name": "ENNReal.lintegral_rpow_add_le_add_snorm_mul_lintegral_rpow_add", "def_path": "Mathlib/MeasureTheory/Integral/MeanInequalities.lean", "def_pos": [274, 9], "def_end_pos": [274, 63]}]], "state_before": "case h\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhg : AEMeasurable g\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc \u2260 \u22a4\nh_add_zero : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 0\nh_add_top : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 \u22a4\nhp_not_nonpos : \u00acp \u2264 0\nhtop_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 \u22a4\nh0_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 0\n\u22a2 1 \u2264\n    (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (-(1 / p)) *\n      ((\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p))", "state_after": "case h\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhg : AEMeasurable g\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc \u2260 \u22a4\nh_add_zero : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 0\nh_add_top : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 \u22a4\nhp_not_nonpos : \u00acp \u2264 0\nhtop_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 \u22a4\nh0_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 0\nh :\n  \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2264\n    ((\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)) * (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / q)\n\u22a2 1 \u2264\n    (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (-(1 / p)) *\n      ((\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p))"}, {"tactic": "have h_one_div_q : 1 / q = 1 - 1 / p := by\n  nth_rw 2 [\u2190 hpq.inv_add_inv_conj]\n  ring", "annotated_tactic": ["have h_one_div_q : 1 / q = 1 - 1 / p := by\n    nth_rw 2 [\u2190 hpq.inv_add_inv_conj]\n    ring", []], "state_before": "case h\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhg : AEMeasurable g\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc \u2260 \u22a4\nh_add_zero : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 0\nh_add_top : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 \u22a4\nhp_not_nonpos : \u00acp \u2264 0\nhtop_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 \u22a4\nh0_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 0\nh :\n  \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2264\n    ((\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)) * (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / q)\n\u22a2 1 \u2264\n    (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (-(1 / p)) *\n      ((\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p))", "state_after": "case h\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhg : AEMeasurable g\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc \u2260 \u22a4\nh_add_zero : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 0\nh_add_top : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 \u22a4\nhp_not_nonpos : \u00acp \u2264 0\nhtop_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 \u22a4\nh0_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 0\nh :\n  \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2264\n    ((\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)) * (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / q)\nh_one_div_q : 1 / q = 1 - 1 / p\n\u22a2 1 \u2264\n    (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (-(1 / p)) *\n      ((\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p))"}, {"tactic": "simp_rw [h_one_div_q, sub_eq_add_neg 1 (1 / p), ENNReal.rpow_add _ _ h_add_zero h_add_top,\n  rpow_one] at h", "annotated_tactic": ["simp_rw [h_one_div_q, <a>sub_eq_add_neg</a> 1 (1 / p), <a>ENNReal.rpow_add</a> _ _ h_add_zero h_add_top,\n    <a>rpow_one</a>] at h", [{"full_name": "sub_eq_add_neg", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [975, 3], "def_end_pos": [975, 14]}, {"full_name": "ENNReal.rpow_add", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [507, 9], "def_end_pos": [507, 17]}, {"full_name": "ENNReal.rpow_one", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [450, 9], "def_end_pos": [450, 17]}]], "state_before": "case h\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhg : AEMeasurable g\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc \u2260 \u22a4\nh_add_zero : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 0\nh_add_top : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 \u22a4\nhp_not_nonpos : \u00acp \u2264 0\nhtop_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 \u22a4\nh0_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 0\nh :\n  \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2264\n    ((\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)) * (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / q)\nh_one_div_q : 1 / q = 1 - 1 / p\n\u22a2 1 \u2264\n    (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (-(1 / p)) *\n      ((\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p))", "state_after": "case h\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhg : AEMeasurable g\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc \u2260 \u22a4\nh_add_zero : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 0\nh_add_top : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 \u22a4\nhp_not_nonpos : \u00acp \u2264 0\nhtop_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 \u22a4\nh0_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 0\nh_one_div_q : 1 / q = 1 - 1 / p\nh :\n  \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2264\n    ((\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)) *\n      ((\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) * (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (-(1 / p)))\n\u22a2 1 \u2264\n    (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (-(1 / p)) *\n      ((\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p))"}, {"tactic": "conv_rhs at h => enter [2]; rw [mul_comm]", "annotated_tactic": ["conv_rhs at h => enter [2]; rw [<a>mul_comm</a>]", [{"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [302, 9], "def_end_pos": [302, 17]}]], "state_before": "case h\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhg : AEMeasurable g\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc \u2260 \u22a4\nh_add_zero : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 0\nh_add_top : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 \u22a4\nhp_not_nonpos : \u00acp \u2264 0\nhtop_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 \u22a4\nh0_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 0\nh_one_div_q : 1 / q = 1 - 1 / p\nh :\n  \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2264\n    ((\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)) *\n      ((\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) * (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (-(1 / p)))\n\u22a2 1 \u2264\n    (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (-(1 / p)) *\n      ((\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p))", "state_after": "case h\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhg : AEMeasurable g\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc \u2260 \u22a4\nh_add_zero : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 0\nh_add_top : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 \u22a4\nhp_not_nonpos : \u00acp \u2264 0\nhtop_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 \u22a4\nh0_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 0\nh_one_div_q : 1 / q = 1 - 1 / p\nh :\n  \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2264\n    ((\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)) *\n      ((\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (-(1 / p)) * \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc)\n\u22a2 1 \u2264\n    (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (-(1 / p)) *\n      ((\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p))"}, {"tactic": "conv_lhs at h => rw [\u2190 one_mul (\u222b\u207b a : \u03b1, (f + g) a ^ p \u2202\u03bc)]", "annotated_tactic": ["conv_lhs at h => rw [\u2190 <a>one_mul</a> (\u222b\u207b a : \u03b1, (f + g) a ^ p \u2202\u03bc)]", [{"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [464, 9], "def_end_pos": [464, 16]}]], "state_before": "case h\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhg : AEMeasurable g\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc \u2260 \u22a4\nh_add_zero : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 0\nh_add_top : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 \u22a4\nhp_not_nonpos : \u00acp \u2264 0\nhtop_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 \u22a4\nh0_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 0\nh_one_div_q : 1 / q = 1 - 1 / p\nh :\n  \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2264\n    ((\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)) *\n      ((\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (-(1 / p)) * \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc)\n\u22a2 1 \u2264\n    (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (-(1 / p)) *\n      ((\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p))", "state_after": "case h\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhg : AEMeasurable g\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc \u2260 \u22a4\nh_add_zero : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 0\nh_add_top : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 \u22a4\nhp_not_nonpos : \u00acp \u2264 0\nhtop_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 \u22a4\nh0_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 0\nh_one_div_q : 1 / q = 1 - 1 / p\nh :\n  1 * \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2264\n    ((\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)) *\n      ((\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (-(1 / p)) * \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc)\n\u22a2 1 \u2264\n    (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (-(1 / p)) *\n      ((\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p))"}, {"tactic": "rwa [\u2190 mul_assoc, ENNReal.mul_le_mul_right h_add_zero h_add_top, mul_comm] at h", "annotated_tactic": ["rwa [\u2190 <a>mul_assoc</a>, <a>ENNReal.mul_le_mul_right</a> h_add_zero h_add_top, <a>mul_comm</a>] at h", [{"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [264, 9], "def_end_pos": [264, 18]}, {"full_name": "ENNReal.mul_le_mul_right", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1070, 9], "def_end_pos": [1070, 25]}, {"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [302, 9], "def_end_pos": [302, 17]}]], "state_before": "case h\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhg : AEMeasurable g\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc \u2260 \u22a4\nh_add_zero : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 0\nh_add_top : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 \u22a4\nhp_not_nonpos : \u00acp \u2264 0\nhtop_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 \u22a4\nh0_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 0\nh_one_div_q : 1 / q = 1 - 1 / p\nh :\n  1 * \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2264\n    ((\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)) *\n      ((\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (-(1 / p)) * \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc)\n\u22a2 1 \u2264\n    (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (-(1 / p)) *\n      ((\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p))", "state_after": "no goals"}, {"tactic": "simp [hpq.pos]", "annotated_tactic": ["simp [hpq.pos]", []], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhg : AEMeasurable g\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc \u2260 \u22a4\nh_add_zero : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 0\nh_add_top : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 \u22a4\n\u22a2 \u00acp \u2264 0", "state_after": "no goals"}, {"tactic": "by_contra h", "annotated_tactic": ["by_contra h", []], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhg : AEMeasurable g\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc \u2260 \u22a4\nh_add_zero : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 0\nh_add_top : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 \u22a4\nhp_not_nonpos : \u00acp \u2264 0\n\u22a2 (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 \u22a4", "state_after": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhg : AEMeasurable g\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc \u2260 \u22a4\nh_add_zero : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 0\nh_add_top : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 \u22a4\nhp_not_nonpos : \u00acp \u2264 0\nh : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) = \u22a4\n\u22a2 False"}, {"tactic": "exact h_add_top (@ENNReal.rpow_eq_top_of_nonneg _ (1 / p) (by simp [hpq.nonneg]) h)", "annotated_tactic": ["exact h_add_top (@<a>ENNReal.rpow_eq_top_of_nonneg</a> _ (1 / p) (by simp [hpq.nonneg]) h)", [{"full_name": "ENNReal.rpow_eq_top_of_nonneg", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [490, 9], "def_end_pos": [490, 30]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhg : AEMeasurable g\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc \u2260 \u22a4\nh_add_zero : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 0\nh_add_top : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 \u22a4\nhp_not_nonpos : \u00acp \u2264 0\nh : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) = \u22a4\n\u22a2 False", "state_after": "no goals"}, {"tactic": "simp [hpq.nonneg]", "annotated_tactic": ["simp [hpq.nonneg]", []], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhg : AEMeasurable g\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc \u2260 \u22a4\nh_add_zero : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 0\nh_add_top : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 \u22a4\nhp_not_nonpos : \u00acp \u2264 0\nh : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) = \u22a4\n\u22a2 0 \u2264 1 / p", "state_after": "no goals"}, {"tactic": "simp [h_add_zero, h_add_top, hpq.nonneg, hp_not_nonpos, -Pi.add_apply]", "annotated_tactic": ["simp [h_add_zero, h_add_top, hpq.nonneg, hp_not_nonpos, -<a>Pi.add_apply</a>]", [{"full_name": "Pi.add_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [82, 3], "def_end_pos": [82, 14]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhg : AEMeasurable g\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc \u2260 \u22a4\nh_add_zero : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 0\nh_add_top : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 \u22a4\nhp_not_nonpos : \u00acp \u2264 0\nhtop_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 \u22a4\n\u22a2 (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 0", "state_after": "no goals"}, {"tactic": "rwa [\u2190 mul_le_mul_left h0_rpow htop_rpow, \u2190 mul_assoc, \u2190 rpow_add _ _ h_add_zero h_add_top, \u2190\n  sub_eq_add_neg, _root_.sub_self, rpow_zero, one_mul, mul_one] at h", "annotated_tactic": ["rwa [\u2190 <a>mul_le_mul_left</a> h0_rpow htop_rpow, \u2190 <a>mul_assoc</a>, \u2190 <a>rpow_add</a> _ _ h_add_zero h_add_top, \u2190\n      <a>sub_eq_add_neg</a>, <a>_root_.sub_self</a>, <a>rpow_zero</a>, <a>one_mul</a>, <a>mul_one</a>] at h", [{"full_name": "ENNReal.mul_le_mul_left", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1065, 9], "def_end_pos": [1065, 24]}, {"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [264, 9], "def_end_pos": [264, 18]}, {"full_name": "ENNReal.rpow_add", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [507, 9], "def_end_pos": [507, 17]}, {"full_name": "sub_eq_add_neg", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [975, 3], "def_end_pos": [975, 14]}, {"full_name": "sub_self", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [734, 30], "def_end_pos": [734, 38]}, {"full_name": "ENNReal.rpow_zero", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [381, 9], "def_end_pos": [381, 18]}, {"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [464, 9], "def_end_pos": [464, 16]}, {"full_name": "mul_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [470, 9], "def_end_pos": [470, 16]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhg : AEMeasurable g\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc \u2260 \u22a4\nh_add_zero : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 0\nh_add_top : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 \u22a4\nhp_not_nonpos : \u00acp \u2264 0\nhtop_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 \u22a4\nh0_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 0\nh :\n  1 \u2264\n    (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (-(1 / p)) *\n      ((\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p))\n\u22a2 (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)", "state_after": "no goals"}, {"tactic": "nth_rw 2 [\u2190 hpq.inv_add_inv_conj]", "annotated_tactic": ["nth_rw 2 [\u2190 hpq.inv_add_inv_conj]", []], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhg : AEMeasurable g\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc \u2260 \u22a4\nh_add_zero : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 0\nh_add_top : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 \u22a4\nhp_not_nonpos : \u00acp \u2264 0\nhtop_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 \u22a4\nh0_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 0\nh :\n  \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2264\n    ((\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)) * (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / q)\n\u22a2 1 / q = 1 - 1 / p", "state_after": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhg : AEMeasurable g\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc \u2260 \u22a4\nh_add_zero : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 0\nh_add_top : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 \u22a4\nhp_not_nonpos : \u00acp \u2264 0\nhtop_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 \u22a4\nh0_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 0\nh :\n  \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2264\n    ((\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)) * (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / q)\n\u22a2 1 / q = 1 / p + 1 / q - 1 / p"}, {"tactic": "ring", "annotated_tactic": ["ring", []], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhg : AEMeasurable g\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc \u2260 \u22a4\nh_add_zero : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 0\nh_add_top : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 \u22a4\nhp_not_nonpos : \u00acp \u2264 0\nhtop_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 \u22a4\nh0_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 0\nh :\n  \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2264\n    ((\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)) * (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / q)\n\u22a2 1 / q = 1 / p + 1 / q - 1 / p", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/ZMod/Basic.lean", "full_name": "ZMod.natAbs_min_of_le_div_two", "start": [1143, 1], "end": [1156, 44], "traced_tactics": [{"tactic": "rw [int_cast_eq_int_cast_iff_dvd_sub] at he", "annotated_tactic": ["rw [<a>int_cast_eq_int_cast_iff_dvd_sub</a>] at he", [{"full_name": "ZMod.int_cast_eq_int_cast_iff_dvd_sub", "def_path": "Mathlib/Data/ZMod/Basic.lean", "def_pos": [481, 9], "def_end_pos": [481, 41]}]], "state_before": "n\u271d a n : \u2115\nx y : \u2124\nhe : \u2191x = \u2191y\nhl : Int.natAbs x \u2264 n / 2\n\u22a2 Int.natAbs x \u2264 Int.natAbs y", "state_after": "n\u271d a n : \u2115\nx y : \u2124\nhe : \u2191n \u2223 y - x\nhl : Int.natAbs x \u2264 n / 2\n\u22a2 Int.natAbs x \u2264 Int.natAbs y"}, {"tactic": "obtain \u27e8m, he\u27e9 := he", "annotated_tactic": ["obtain \u27e8m, he\u27e9 := he", []], "state_before": "n\u271d a n : \u2115\nx y : \u2124\nhe : \u2191n \u2223 y - x\nhl : Int.natAbs x \u2264 n / 2\n\u22a2 Int.natAbs x \u2264 Int.natAbs y", "state_after": "case intro\nn\u271d a n : \u2115\nx y : \u2124\nhl : Int.natAbs x \u2264 n / 2\nm : \u2124\nhe : y - x = \u2191n * m\n\u22a2 Int.natAbs x \u2264 Int.natAbs y"}, {"tactic": "rw [sub_eq_iff_eq_add] at he", "annotated_tactic": ["rw [<a>sub_eq_iff_eq_add</a>] at he", [{"full_name": "sub_eq_iff_eq_add", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [829, 3], "def_end_pos": [829, 14]}]], "state_before": "case intro\nn\u271d a n : \u2115\nx y : \u2124\nhl : Int.natAbs x \u2264 n / 2\nm : \u2124\nhe : y - x = \u2191n * m\n\u22a2 Int.natAbs x \u2264 Int.natAbs y", "state_after": "case intro\nn\u271d a n : \u2115\nx y : \u2124\nhl : Int.natAbs x \u2264 n / 2\nm : \u2124\nhe : y = \u2191n * m + x\n\u22a2 Int.natAbs x \u2264 Int.natAbs y"}, {"tactic": "subst he", "annotated_tactic": ["subst he", []], "state_before": "case intro\nn\u271d a n : \u2115\nx y : \u2124\nhl : Int.natAbs x \u2264 n / 2\nm : \u2124\nhe : y = \u2191n * m + x\n\u22a2 Int.natAbs x \u2264 Int.natAbs y", "state_after": "case intro\nn\u271d a n : \u2115\nx : \u2124\nhl : Int.natAbs x \u2264 n / 2\nm : \u2124\n\u22a2 Int.natAbs x \u2264 Int.natAbs (\u2191n * m + x)"}, {"tactic": "obtain rfl | hm := eq_or_ne m 0", "annotated_tactic": ["obtain rfl | hm := <a>eq_or_ne</a> m 0", [{"full_name": "eq_or_ne", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [209, 9], "def_end_pos": [209, 17]}]], "state_before": "case intro\nn\u271d a n : \u2115\nx : \u2124\nhl : Int.natAbs x \u2264 n / 2\nm : \u2124\n\u22a2 Int.natAbs x \u2264 Int.natAbs (\u2191n * m + x)", "state_after": "case intro.inl\nn\u271d a n : \u2115\nx : \u2124\nhl : Int.natAbs x \u2264 n / 2\n\u22a2 Int.natAbs x \u2264 Int.natAbs (\u2191n * 0 + x)\n\ncase intro.inr\nn\u271d a n : \u2115\nx : \u2124\nhl : Int.natAbs x \u2264 n / 2\nm : \u2124\nhm : m \u2260 0\n\u22a2 Int.natAbs x \u2264 Int.natAbs (\u2191n * m + x)"}, {"tactic": "apply hl.trans", "annotated_tactic": ["apply hl.trans", []], "state_before": "case intro.inr\nn\u271d a n : \u2115\nx : \u2124\nhl : Int.natAbs x \u2264 n / 2\nm : \u2124\nhm : m \u2260 0\n\u22a2 Int.natAbs x \u2264 Int.natAbs (\u2191n * m + x)", "state_after": "case intro.inr\nn\u271d a n : \u2115\nx : \u2124\nhl : Int.natAbs x \u2264 n / 2\nm : \u2124\nhm : m \u2260 0\n\u22a2 n / 2 \u2264 Int.natAbs (\u2191n * m + x)"}, {"tactic": "rw [\u2190 add_le_add_iff_right x.natAbs]", "annotated_tactic": ["rw [\u2190 <a>add_le_add_iff_right</a> x.natAbs]", [{"full_name": "add_le_add_iff_right", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [90, 3], "def_end_pos": [90, 14]}]], "state_before": "case intro.inr\nn\u271d a n : \u2115\nx : \u2124\nhl : Int.natAbs x \u2264 n / 2\nm : \u2124\nhm : m \u2260 0\n\u22a2 n / 2 \u2264 Int.natAbs (\u2191n * m + x)", "state_after": "case intro.inr\nn\u271d a n : \u2115\nx : \u2124\nhl : Int.natAbs x \u2264 n / 2\nm : \u2124\nhm : m \u2260 0\n\u22a2 n / 2 + Int.natAbs x \u2264 Int.natAbs (\u2191n * m + x) + Int.natAbs x"}, {"tactic": "refine' le_trans (le_trans ((add_le_add_iff_left _).2 hl) _) (Int.natAbs_sub_le _ _)", "annotated_tactic": ["refine' <a>le_trans</a> (<a>le_trans</a> ((<a>add_le_add_iff_left</a> _).2 hl) _) (<a>Int.natAbs_sub_le</a> _ _)", [{"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}, {"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}, {"full_name": "add_le_add_iff_left", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [82, 3], "def_end_pos": [82, 14]}, {"full_name": "Int.natAbs_sub_le", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [1354, 9], "def_end_pos": [1354, 22]}]], "state_before": "case intro.inr\nn\u271d a n : \u2115\nx : \u2124\nhl : Int.natAbs x \u2264 n / 2\nm : \u2124\nhm : m \u2260 0\n\u22a2 n / 2 + Int.natAbs x \u2264 Int.natAbs (\u2191n * m + x) + Int.natAbs x", "state_after": "case intro.inr\nn\u271d a n : \u2115\nx : \u2124\nhl : Int.natAbs x \u2264 n / 2\nm : \u2124\nhm : m \u2260 0\n\u22a2 n / 2 + n / 2 \u2264 Int.natAbs (\u2191n * m + x - x)"}, {"tactic": "rw [add_sub_cancel, Int.natAbs_mul, Int.natAbs_ofNat]", "annotated_tactic": ["rw [<a>add_sub_cancel</a>, <a>Int.natAbs_mul</a>, <a>Int.natAbs_ofNat</a>]", [{"full_name": "add_sub_cancel", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [739, 30], "def_end_pos": [739, 44]}, {"full_name": "Int.natAbs_mul", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [171, 9], "def_end_pos": [171, 19]}, {"full_name": "Int.natAbs_ofNat", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [138, 17], "def_end_pos": [138, 29]}]], "state_before": "case intro.inr\nn\u271d a n : \u2115\nx : \u2124\nhl : Int.natAbs x \u2264 n / 2\nm : \u2124\nhm : m \u2260 0\n\u22a2 n / 2 + n / 2 \u2264 Int.natAbs (\u2191n * m + x - x)", "state_after": "case intro.inr\nn\u271d a n : \u2115\nx : \u2124\nhl : Int.natAbs x \u2264 n / 2\nm : \u2124\nhm : m \u2260 0\n\u22a2 n / 2 + n / 2 \u2264 n * Int.natAbs m"}, {"tactic": "refine' le_trans _ (Nat.le_mul_of_pos_right <| Int.natAbs_pos.2 hm)", "annotated_tactic": ["refine' <a>le_trans</a> _ (<a>Nat.le_mul_of_pos_right</a> <| <a>Int.natAbs_pos</a>.2 hm)", [{"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}, {"full_name": "Nat.le_mul_of_pos_right", "def_path": "Mathlib/Data/Nat/Order/Basic.lean", "def_pos": [300, 9], "def_end_pos": [300, 28]}, {"full_name": "Int.natAbs_pos", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [151, 9], "def_end_pos": [151, 19]}]], "state_before": "case intro.inr\nn\u271d a n : \u2115\nx : \u2124\nhl : Int.natAbs x \u2264 n / 2\nm : \u2124\nhm : m \u2260 0\n\u22a2 n / 2 + n / 2 \u2264 n * Int.natAbs m", "state_after": "case intro.inr\nn\u271d a n : \u2115\nx : \u2124\nhl : Int.natAbs x \u2264 n / 2\nm : \u2124\nhm : m \u2260 0\n\u22a2 n / 2 + n / 2 \u2264 n"}, {"tactic": "rw [\u2190 mul_two]", "annotated_tactic": ["rw [\u2190 <a>mul_two</a>]", [{"full_name": "mul_two", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [188, 9], "def_end_pos": [188, 16]}]], "state_before": "case intro.inr\nn\u271d a n : \u2115\nx : \u2124\nhl : Int.natAbs x \u2264 n / 2\nm : \u2124\nhm : m \u2260 0\n\u22a2 n / 2 + n / 2 \u2264 n", "state_after": "case intro.inr\nn\u271d a n : \u2115\nx : \u2124\nhl : Int.natAbs x \u2264 n / 2\nm : \u2124\nhm : m \u2260 0\n\u22a2 n / 2 * 2 \u2264 n"}, {"tactic": "apply Nat.div_mul_le_self", "annotated_tactic": ["apply <a>Nat.div_mul_le_self</a>", [{"full_name": "Nat.div_mul_le_self", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [613, 9], "def_end_pos": [613, 24]}]], "state_before": "case intro.inr\nn\u271d a n : \u2115\nx : \u2124\nhl : Int.natAbs x \u2264 n / 2\nm : \u2124\nhm : m \u2260 0\n\u22a2 n / 2 * 2 \u2264 n", "state_after": "no goals"}, {"tactic": "rw [mul_zero, zero_add]", "annotated_tactic": ["rw [<a>mul_zero</a>, <a>zero_add</a>]", [{"full_name": "MulZeroClass.mul_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [38, 3], "def_end_pos": [38, 11]}, {"full_name": "zero_add", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [463, 3], "def_end_pos": [463, 14]}]], "state_before": "case intro.inl\nn\u271d a n : \u2115\nx : \u2124\nhl : Int.natAbs x \u2264 n / 2\n\u22a2 Int.natAbs x \u2264 Int.natAbs (\u2191n * 0 + x)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/FundThmCalculus.lean", "full_name": "intervalIntegral.integral_sub_integral_sub_linear_isLittleO_of_tendsto_ae_left", "start": [559, 1], "end": [564, 93], "traced_tactics": [{"tactic": "simpa only [integral_const, smul_eq_mul, mul_one] using\n  measure_integral_sub_integral_sub_linear_isLittleO_of_tendsto_ae_left hab hmeas hf hu hv", "annotated_tactic": ["simpa only [<a>integral_const</a>, <a>smul_eq_mul</a>, <a>mul_one</a>] using\n    <a>measure_integral_sub_integral_sub_linear_isLittleO_of_tendsto_ae_left</a> hab hmeas hf hu hv", [{"full_name": "intervalIntegral.integral_const", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [638, 9], "def_end_pos": [638, 23]}, {"full_name": "smul_eq_mul", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [93, 9], "def_end_pos": [93, 20]}, {"full_name": "mul_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [470, 9], "def_end_pos": [470, 16]}, {"full_name": "intervalIntegral.measure_integral_sub_integral_sub_linear_isLittleO_of_tendsto_ae_left", "def_path": "Mathlib/MeasureTheory/Integral/FundThmCalculus.lean", "def_pos": [467, 9], "def_end_pos": [467, 78]}]], "state_before": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedSpace \u211d E\nf : \u211d \u2192 E\nc ca cb : E\nl l' la la' lb lb' : Filter \u211d\nlt : Filter \u03b9\na b z : \u211d\nu v ua ub va vb : \u03b9 \u2192 \u211d\ninst\u271d\u00b9 : FTCFilter a la la'\ninst\u271d : FTCFilter b lb lb'\nhab : IntervalIntegrable f volume a b\nhmeas : StronglyMeasurableAtFilter f la'\nhf : Tendsto f (la' \u2293 Measure.ae volume) (\ud835\udcdd c)\nhu : Tendsto u lt la\nhv : Tendsto v lt la\n\u22a2 (fun t => ((\u222b (x : \u211d) in v t..b, f x) - \u222b (x : \u211d) in u t..b, f x) + (v t - u t) \u2022 c) =o[lt] (v - u)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Countable.lean", "full_name": "Set.countable_iff_exists_injective", "start": [50, 11], "end": [52, 66], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/StrongLaw.lean", "full_name": "ProbabilityTheory.strong_law_Lp", "start": [852, 1], "end": [871, 41], "traced_tactics": [{"tactic": "have hmeas : \u2200 i, AEStronglyMeasurable (X i) \u2119 := fun i =>\n  (hident i).aestronglyMeasurable_iff.2 h\u2112p.1", "annotated_tactic": ["have hmeas : \u2200 i, <a>AEStronglyMeasurable</a> (X i) \u2119 := fun i =>\n    (hident i).<a>aestronglyMeasurable_iff</a>.2 h\u2112p.1", [{"full_name": "MeasureTheory.AEStronglyMeasurable", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [93, 5], "def_end_pos": [93, 25]}, {"full_name": "ProbabilityTheory.IdentDistrib.aestronglyMeasurable_iff", "def_path": "Mathlib/Probability/IdentDistrib.lean", "def_pos": [171, 9], "def_end_pos": [171, 33]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u2076 : MeasureSpace \u03a9\ninst\u271d\u2075 : IsProbabilityMeasure \u2119\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\np : \u211d\u22650\u221e\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nX : \u2115 \u2192 \u03a9 \u2192 E\nh\u2112p : Mem\u2112p (X 0) p\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\n\u22a2 Tendsto (fun n => snorm (fun \u03c9 => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in range n, X i \u03c9 - \u222b (a : \u03a9), X 0 a) p \u2119) atTop (\ud835\udcdd 0)", "state_after": "\u03a9 : Type u_1\ninst\u271d\u2076 : MeasureSpace \u03a9\ninst\u271d\u2075 : IsProbabilityMeasure \u2119\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\np : \u211d\u22650\u221e\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nX : \u2115 \u2192 \u03a9 \u2192 E\nh\u2112p : Mem\u2112p (X 0) p\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhmeas : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\n\u22a2 Tendsto (fun n => snorm (fun \u03c9 => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in range n, X i \u03c9 - \u222b (a : \u03a9), X 0 a) p \u2119) atTop (\ud835\udcdd 0)"}, {"tactic": "have hint : Integrable (X 0) \u2119 := h\u2112p.integrable hp", "annotated_tactic": ["have hint : <a>Integrable</a> (X 0) \u2119 := h\u2112p.integrable hp", [{"full_name": "MeasureTheory.Integrable", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [442, 5], "def_end_pos": [442, 15]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u2076 : MeasureSpace \u03a9\ninst\u271d\u2075 : IsProbabilityMeasure \u2119\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\np : \u211d\u22650\u221e\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nX : \u2115 \u2192 \u03a9 \u2192 E\nh\u2112p : Mem\u2112p (X 0) p\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhmeas : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\n\u22a2 Tendsto (fun n => snorm (fun \u03c9 => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in range n, X i \u03c9 - \u222b (a : \u03a9), X 0 a) p \u2119) atTop (\ud835\udcdd 0)", "state_after": "\u03a9 : Type u_1\ninst\u271d\u2076 : MeasureSpace \u03a9\ninst\u271d\u2075 : IsProbabilityMeasure \u2119\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\np : \u211d\u22650\u221e\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nX : \u2115 \u2192 \u03a9 \u2192 E\nh\u2112p : Mem\u2112p (X 0) p\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhmeas : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nhint : Integrable (X 0)\n\u22a2 Tendsto (fun n => snorm (fun \u03c9 => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in range n, X i \u03c9 - \u222b (a : \u03a9), X 0 a) p \u2119) atTop (\ud835\udcdd 0)"}, {"tactic": "have havg : \u2200 (n : \u2115),\n    AEStronglyMeasurable (fun \u03c9 => (n : \u211d) \u207b\u00b9 \u2022 (\u2211 i in range n, X i \u03c9)) \u2119 := by\n  intro n\n  exact AEStronglyMeasurable.const_smul (aestronglyMeasurable_sum _ fun i _ => hmeas i) _", "annotated_tactic": ["have havg : \u2200 (n : \u2115),\n      <a>AEStronglyMeasurable</a> (fun \u03c9 => (n : \u211d) \u207b\u00b9 \u2022 (\u2211 i in <a>range</a> n, X i \u03c9)) \u2119 := by\n    intro n\n    exact <a>AEStronglyMeasurable.const_smul</a> (<a>aestronglyMeasurable_sum</a> _ fun i _ => hmeas i) _", [{"full_name": "MeasureTheory.AEStronglyMeasurable", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [93, 5], "def_end_pos": [93, 25]}, {"full_name": "Finset.range", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3027, 5], "def_end_pos": [3027, 10]}, {"full_name": "MeasureTheory.AEStronglyMeasurable.const_smul", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1339, 19], "def_end_pos": [1339, 29]}, {"full_name": "Finset.aestronglyMeasurable_sum", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1435, 3], "def_end_pos": [1435, 14]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u2076 : MeasureSpace \u03a9\ninst\u271d\u2075 : IsProbabilityMeasure \u2119\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\np : \u211d\u22650\u221e\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nX : \u2115 \u2192 \u03a9 \u2192 E\nh\u2112p : Mem\u2112p (X 0) p\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhmeas : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nhint : Integrable (X 0)\n\u22a2 Tendsto (fun n => snorm (fun \u03c9 => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in range n, X i \u03c9 - \u222b (a : \u03a9), X 0 a) p \u2119) atTop (\ud835\udcdd 0)", "state_after": "\u03a9 : Type u_1\ninst\u271d\u2076 : MeasureSpace \u03a9\ninst\u271d\u2075 : IsProbabilityMeasure \u2119\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\np : \u211d\u22650\u221e\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nX : \u2115 \u2192 \u03a9 \u2192 E\nh\u2112p : Mem\u2112p (X 0) p\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhmeas : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nhint : Integrable (X 0)\nhavg : \u2200 (n : \u2115), AEStronglyMeasurable (fun \u03c9 => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in range n, X i \u03c9) \u2119\n\u22a2 Tendsto (fun n => snorm (fun \u03c9 => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in range n, X i \u03c9 - \u222b (a : \u03a9), X 0 a) p \u2119) atTop (\ud835\udcdd 0)"}, {"tactic": "refine' tendsto_Lp_of_tendstoInMeasure _ hp hp' havg (mem\u2112p_const _) _\n  (tendstoInMeasure_of_tendsto_ae havg (strong_law_ae _ hint hindep hident))", "annotated_tactic": ["refine' <a>tendsto_Lp_of_tendstoInMeasure</a> _ hp hp' havg (<a>mem\u2112p_const</a> _) _\n    (<a>tendstoInMeasure_of_tendsto_ae</a> havg (<a>strong_law_ae</a> _ hint hindep hident))", [{"full_name": "MeasureTheory.tendsto_Lp_of_tendstoInMeasure", "def_path": "Mathlib/MeasureTheory/Function/UniformIntegrable.lean", "def_pos": [608, 9], "def_end_pos": [608, 39]}, {"full_name": "MeasureTheory.mem\u2112p_const", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [344, 9], "def_end_pos": [344, 20]}, {"full_name": "MeasureTheory.tendstoInMeasure_of_tendsto_ae", "def_path": "Mathlib/MeasureTheory/Function/ConvergenceInMeasure.lean", "def_pos": [129, 9], "def_end_pos": [129, 39]}, {"full_name": "ProbabilityTheory.strong_law_ae", "def_path": "Mathlib/Probability/StrongLaw.lean", "def_pos": [819, 9], "def_end_pos": [819, 22]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u2076 : MeasureSpace \u03a9\ninst\u271d\u2075 : IsProbabilityMeasure \u2119\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\np : \u211d\u22650\u221e\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nX : \u2115 \u2192 \u03a9 \u2192 E\nh\u2112p : Mem\u2112p (X 0) p\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhmeas : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nhint : Integrable (X 0)\nhavg : \u2200 (n : \u2115), AEStronglyMeasurable (fun \u03c9 => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in range n, X i \u03c9) \u2119\n\u22a2 Tendsto (fun n => snorm (fun \u03c9 => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in range n, X i \u03c9 - \u222b (a : \u03a9), X 0 a) p \u2119) atTop (\ud835\udcdd 0)", "state_after": "\u03a9 : Type u_1\ninst\u271d\u2076 : MeasureSpace \u03a9\ninst\u271d\u2075 : IsProbabilityMeasure \u2119\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\np : \u211d\u22650\u221e\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nX : \u2115 \u2192 \u03a9 \u2192 E\nh\u2112p : Mem\u2112p (X 0) p\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhmeas : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nhint : Integrable (X 0)\nhavg : \u2200 (n : \u2115), AEStronglyMeasurable (fun \u03c9 => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in range n, X i \u03c9) \u2119\n\u22a2 UnifIntegrable (fun n \u03c9 => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in range n, X i \u03c9) p \u2119"}, {"tactic": "rw [(_ : (fun (n : \u2115) \u03c9 => (n : \u211d)\u207b\u00b9 \u2022 (\u2211 i in range n, X i \u03c9))\n          = fun (n : \u2115) => (n : \u211d)\u207b\u00b9 \u2022 (\u2211 i in range n, X i))]", "annotated_tactic": ["rw [(_ : (fun (n : \u2115) \u03c9 => (n : \u211d)\u207b\u00b9 \u2022 (\u2211 i in <a>range</a> n, X i \u03c9))\n            = fun (n : \u2115) => (n : \u211d)\u207b\u00b9 \u2022 (\u2211 i in <a>range</a> n, X i))]", [{"full_name": "Finset.range", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3027, 5], "def_end_pos": [3027, 10]}, {"full_name": "Finset.range", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3027, 5], "def_end_pos": [3027, 10]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u2076 : MeasureSpace \u03a9\ninst\u271d\u2075 : IsProbabilityMeasure \u2119\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\np : \u211d\u22650\u221e\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nX : \u2115 \u2192 \u03a9 \u2192 E\nh\u2112p : Mem\u2112p (X 0) p\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhmeas : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nhint : Integrable (X 0)\nhavg : \u2200 (n : \u2115), AEStronglyMeasurable (fun \u03c9 => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in range n, X i \u03c9) \u2119\n\u22a2 UnifIntegrable (fun n \u03c9 => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in range n, X i \u03c9) p \u2119", "state_after": "\u03a9 : Type u_1\ninst\u271d\u2076 : MeasureSpace \u03a9\ninst\u271d\u2075 : IsProbabilityMeasure \u2119\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\np : \u211d\u22650\u221e\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nX : \u2115 \u2192 \u03a9 \u2192 E\nh\u2112p : Mem\u2112p (X 0) p\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhmeas : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nhint : Integrable (X 0)\nhavg : \u2200 (n : \u2115), AEStronglyMeasurable (fun \u03c9 => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in range n, X i \u03c9) \u2119\n\u22a2 UnifIntegrable (fun n => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in range n, X i) p \u2119\n\n\u03a9 : Type u_1\ninst\u271d\u2076 : MeasureSpace \u03a9\ninst\u271d\u2075 : IsProbabilityMeasure \u2119\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\np : \u211d\u22650\u221e\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nX : \u2115 \u2192 \u03a9 \u2192 E\nh\u2112p : Mem\u2112p (X 0) p\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhmeas : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nhint : Integrable (X 0)\nhavg : \u2200 (n : \u2115), AEStronglyMeasurable (fun \u03c9 => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in range n, X i \u03c9) \u2119\n\u22a2 (fun n \u03c9 => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in range n, X i \u03c9) = fun n => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in range n, X i"}, {"tactic": "intro n", "annotated_tactic": ["intro n", []], "state_before": "\u03a9 : Type u_1\ninst\u271d\u2076 : MeasureSpace \u03a9\ninst\u271d\u2075 : IsProbabilityMeasure \u2119\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\np : \u211d\u22650\u221e\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nX : \u2115 \u2192 \u03a9 \u2192 E\nh\u2112p : Mem\u2112p (X 0) p\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhmeas : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nhint : Integrable (X 0)\n\u22a2 \u2200 (n : \u2115), AEStronglyMeasurable (fun \u03c9 => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in range n, X i \u03c9) \u2119", "state_after": "\u03a9 : Type u_1\ninst\u271d\u2076 : MeasureSpace \u03a9\ninst\u271d\u2075 : IsProbabilityMeasure \u2119\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\np : \u211d\u22650\u221e\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nX : \u2115 \u2192 \u03a9 \u2192 E\nh\u2112p : Mem\u2112p (X 0) p\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhmeas : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nhint : Integrable (X 0)\nn : \u2115\n\u22a2 AEStronglyMeasurable (fun \u03c9 => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in range n, X i \u03c9) \u2119"}, {"tactic": "exact AEStronglyMeasurable.const_smul (aestronglyMeasurable_sum _ fun i _ => hmeas i) _", "annotated_tactic": ["exact <a>AEStronglyMeasurable.const_smul</a> (<a>aestronglyMeasurable_sum</a> _ fun i _ => hmeas i) _", [{"full_name": "MeasureTheory.AEStronglyMeasurable.const_smul", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1339, 19], "def_end_pos": [1339, 29]}, {"full_name": "Finset.aestronglyMeasurable_sum", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1435, 3], "def_end_pos": [1435, 14]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u2076 : MeasureSpace \u03a9\ninst\u271d\u2075 : IsProbabilityMeasure \u2119\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\np : \u211d\u22650\u221e\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nX : \u2115 \u2192 \u03a9 \u2192 E\nh\u2112p : Mem\u2112p (X 0) p\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhmeas : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nhint : Integrable (X 0)\nn : \u2115\n\u22a2 AEStronglyMeasurable (fun \u03c9 => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in range n, X i \u03c9) \u2119", "state_after": "no goals"}, {"tactic": "apply UniformIntegrable.unifIntegrable", "annotated_tactic": ["apply <a>UniformIntegrable.unifIntegrable</a>", [{"full_name": "MeasureTheory.UniformIntegrable.unifIntegrable", "def_path": "Mathlib/MeasureTheory/Function/UniformIntegrable.lean", "def_pos": [83, 19], "def_end_pos": [83, 33]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u2076 : MeasureSpace \u03a9\ninst\u271d\u2075 : IsProbabilityMeasure \u2119\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\np : \u211d\u22650\u221e\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nX : \u2115 \u2192 \u03a9 \u2192 E\nh\u2112p : Mem\u2112p (X 0) p\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhmeas : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nhint : Integrable (X 0)\nhavg : \u2200 (n : \u2115), AEStronglyMeasurable (fun \u03c9 => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in range n, X i \u03c9) \u2119\n\u22a2 UnifIntegrable (fun n => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in range n, X i) p \u2119", "state_after": "case hf\n\u03a9 : Type u_1\ninst\u271d\u2076 : MeasureSpace \u03a9\ninst\u271d\u2075 : IsProbabilityMeasure \u2119\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\np : \u211d\u22650\u221e\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nX : \u2115 \u2192 \u03a9 \u2192 E\nh\u2112p : Mem\u2112p (X 0) p\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhmeas : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nhint : Integrable (X 0)\nhavg : \u2200 (n : \u2115), AEStronglyMeasurable (fun \u03c9 => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in range n, X i \u03c9) \u2119\n\u22a2 UniformIntegrable (fun n => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in range n, X i) p \u2119"}, {"tactic": "apply uniformIntegrable_average hp", "annotated_tactic": ["apply <a>uniformIntegrable_average</a> hp", [{"full_name": "MeasureTheory.uniformIntegrable_average", "def_path": "Mathlib/MeasureTheory/Function/UniformIntegrable.lean", "def_pos": [920, 9], "def_end_pos": [920, 34]}]], "state_before": "case hf\n\u03a9 : Type u_1\ninst\u271d\u2076 : MeasureSpace \u03a9\ninst\u271d\u2075 : IsProbabilityMeasure \u2119\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\np : \u211d\u22650\u221e\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nX : \u2115 \u2192 \u03a9 \u2192 E\nh\u2112p : Mem\u2112p (X 0) p\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhmeas : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nhint : Integrable (X 0)\nhavg : \u2200 (n : \u2115), AEStronglyMeasurable (fun \u03c9 => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in range n, X i \u03c9) \u2119\n\u22a2 UniformIntegrable (fun n => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in range n, X i) p \u2119", "state_after": "case hf\n\u03a9 : Type u_1\ninst\u271d\u2076 : MeasureSpace \u03a9\ninst\u271d\u2075 : IsProbabilityMeasure \u2119\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\np : \u211d\u22650\u221e\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nX : \u2115 \u2192 \u03a9 \u2192 E\nh\u2112p : Mem\u2112p (X 0) p\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhmeas : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nhint : Integrable (X 0)\nhavg : \u2200 (n : \u2115), AEStronglyMeasurable (fun \u03c9 => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in range n, X i \u03c9) \u2119\n\u22a2 UniformIntegrable (fun i => X i) p \u2119"}, {"tactic": "exact Mem\u2112p.uniformIntegrable_of_identDistrib hp hp' h\u2112p hident", "annotated_tactic": ["exact <a>Mem\u2112p.uniformIntegrable_of_identDistrib</a> hp hp' h\u2112p hident", [{"full_name": "ProbabilityTheory.Mem\u2112p.uniformIntegrable_of_identDistrib", "def_path": "Mathlib/Probability/IdentDistrib.lean", "def_pos": [348, 9], "def_end_pos": [348, 48]}]], "state_before": "case hf\n\u03a9 : Type u_1\ninst\u271d\u2076 : MeasureSpace \u03a9\ninst\u271d\u2075 : IsProbabilityMeasure \u2119\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\np : \u211d\u22650\u221e\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nX : \u2115 \u2192 \u03a9 \u2192 E\nh\u2112p : Mem\u2112p (X 0) p\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhmeas : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nhint : Integrable (X 0)\nhavg : \u2200 (n : \u2115), AEStronglyMeasurable (fun \u03c9 => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in range n, X i \u03c9) \u2119\n\u22a2 UniformIntegrable (fun i => X i) p \u2119", "state_after": "no goals"}, {"tactic": "ext n \u03c9", "annotated_tactic": ["ext n \u03c9", []], "state_before": "\u03a9 : Type u_1\ninst\u271d\u2076 : MeasureSpace \u03a9\ninst\u271d\u2075 : IsProbabilityMeasure \u2119\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\np : \u211d\u22650\u221e\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nX : \u2115 \u2192 \u03a9 \u2192 E\nh\u2112p : Mem\u2112p (X 0) p\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhmeas : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nhint : Integrable (X 0)\nhavg : \u2200 (n : \u2115), AEStronglyMeasurable (fun \u03c9 => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in range n, X i \u03c9) \u2119\n\u22a2 (fun n \u03c9 => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in range n, X i \u03c9) = fun n => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in range n, X i", "state_after": "case h.h\n\u03a9 : Type u_1\ninst\u271d\u2076 : MeasureSpace \u03a9\ninst\u271d\u2075 : IsProbabilityMeasure \u2119\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\np : \u211d\u22650\u221e\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nX : \u2115 \u2192 \u03a9 \u2192 E\nh\u2112p : Mem\u2112p (X 0) p\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhmeas : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nhint : Integrable (X 0)\nhavg : \u2200 (n : \u2115), AEStronglyMeasurable (fun \u03c9 => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in range n, X i \u03c9) \u2119\nn : \u2115\n\u03c9 : \u03a9\n\u22a2 (\u2191n)\u207b\u00b9 \u2022 \u2211 i in range n, X i \u03c9 = ((\u2191n)\u207b\u00b9 \u2022 \u2211 i in range n, X i) \u03c9"}, {"tactic": "simp only [Pi.smul_apply, sum_apply]", "annotated_tactic": ["simp only [<a>Pi.smul_apply</a>, <a>sum_apply</a>]", [{"full_name": "Pi.smul_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [116, 60], "def_end_pos": [116, 70]}, {"full_name": "Finset.sum_apply", "def_path": "Mathlib/Algebra/BigOperators/Pi.lean", "def_pos": [41, 3], "def_end_pos": [41, 14]}]], "state_before": "case h.h\n\u03a9 : Type u_1\ninst\u271d\u2076 : MeasureSpace \u03a9\ninst\u271d\u2075 : IsProbabilityMeasure \u2119\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\np : \u211d\u22650\u221e\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nX : \u2115 \u2192 \u03a9 \u2192 E\nh\u2112p : Mem\u2112p (X 0) p\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhmeas : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nhint : Integrable (X 0)\nhavg : \u2200 (n : \u2115), AEStronglyMeasurable (fun \u03c9 => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in range n, X i \u03c9) \u2119\nn : \u2115\n\u03c9 : \u03a9\n\u22a2 (\u2191n)\u207b\u00b9 \u2022 \u2211 i in range n, X i \u03c9 = ((\u2191n)\u207b\u00b9 \u2022 \u2211 i in range n, X i) \u03c9", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Constructions/Prod/Integral.lean", "full_name": "MeasureTheory.integrable_prod_iff", "start": [272, 1], "end": [276, 22], "traced_tactics": [{"tactic": "simp [Integrable, h1f, hasFiniteIntegral_prod_iff', h1f.norm.integral_prod_right',\n  h1f.prod_mk_left]", "annotated_tactic": ["simp [<a>Integrable</a>, h1f, <a>hasFiniteIntegral_prod_iff'</a>, h1f.norm.integral_prod_right',\n    h1f.prod_mk_left]", [{"full_name": "MeasureTheory.Integrable", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [442, 5], "def_end_pos": [442, 15]}, {"full_name": "MeasureTheory.hasFiniteIntegral_prod_iff'", "def_path": "Mathlib/MeasureTheory/Constructions/Prod/Integral.lean", "def_pos": [254, 9], "def_end_pos": [254, 36]}]], "state_before": "\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u00d7 \u03b2 \u2192 E\nh1f : AEStronglyMeasurable f (Measure.prod \u03bc \u03bd)\n\u22a2 Integrable f \u2194 (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, Integrable fun y => f (x, y)) \u2227 Integrable fun x => \u222b (y : \u03b2), \u2016f (x, y)\u2016 \u2202\u03bd", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/HashMap/WF.lean", "full_name": "Std.HashMap.Imp.erase_size", "start": [241, 1], "end": [255, 12], "traced_tactics": [{"tactic": "dsimp [erase, cond]", "annotated_tactic": ["dsimp [<a>erase</a>, <a>cond</a>]", [{"full_name": "Std.HashMap.Imp.erase", "def_path": "lake-packages/std/Std/Data/HashMap/Basic.lean", "def_pos": [177, 5], "def_end_pos": [177, 10]}, {"full_name": "cond", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [971, 21], "def_end_pos": [971, 25]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nk : \u03b1\nh : m.size = Buckets.size m.buckets\n\u22a2 (erase m k).size = Buckets.size (erase m k).buckets", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nk : \u03b1\nh : m.size = Buckets.size m.buckets\n\u22a2 (match\n        AssocList.contains k\n          m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] with\n      | true =>\n        { size := m.size - 1,\n          buckets :=\n            Buckets.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n              (AssocList.erase k\n                m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n              (_ :\n                USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n                  Array.size m.buckets.val) }\n      | false => m).size =\n    Buckets.size\n      (match\n          AssocList.contains k\n            m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] with\n        | true =>\n          { size := m.size - 1,\n            buckets :=\n              Buckets.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n                (AssocList.erase k\n                  m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n                (_ :\n                  USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n                    Array.size m.buckets.val) }\n        | false => m).buckets"}, {"tactic": "split", "annotated_tactic": ["split", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nk : \u03b1\nh : m.size = Buckets.size m.buckets\n\u22a2 (match\n        AssocList.contains k\n          m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] with\n      | true =>\n        { size := m.size - 1,\n          buckets :=\n            Buckets.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n              (AssocList.erase k\n                m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n              (_ :\n                USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n                  Array.size m.buckets.val) }\n      | false => m).size =\n    Buckets.size\n      (match\n          AssocList.contains k\n            m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] with\n        | true =>\n          { size := m.size - 1,\n            buckets :=\n              Buckets.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n                (AssocList.erase k\n                  m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n                (_ :\n                  USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n                    Array.size m.buckets.val) }\n        | false => m).buckets", "state_after": "case h_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nk : \u03b1\nh : m.size = Buckets.size m.buckets\nc\u271d : Bool\nheq\u271d :\n  AssocList.contains k\n      m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] =\n    true\n\u22a2 { size := m.size - 1,\n        buckets :=\n          Buckets.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n            (AssocList.erase k\n              m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n            (_ :\n              USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n                Array.size m.buckets.val) }.size =\n    Buckets.size\n      { size := m.size - 1,\n          buckets :=\n            Buckets.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n              (AssocList.erase k\n                m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n              (_ :\n                USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n                  Array.size m.buckets.val) }.buckets\n\ncase h_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nk : \u03b1\nh : m.size = Buckets.size m.buckets\nc\u271d : Bool\nheq\u271d :\n  AssocList.contains k\n      m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] =\n    false\n\u22a2 m.size = Buckets.size m.buckets"}, {"tactic": "next H =>\nsimp [h, Buckets.size]\nrefine have \u27e8_, _, h\u2081, _, eq\u27e9 := Buckets.exists_of_update ..; eq \u25b8 ?_\nsimp [h, h\u2081, Buckets.size_eq]\nrw [(_ : List.length _ = _ + 1), Nat.add_right_comm]; {rfl}\nclear h\u2081 eq\nsimp [AssocList.contains_eq] at H\nhave \u27e8a, h\u2081, h\u2082\u27e9 := H\nrefine have \u27e8_, _, _, _, _, h, eq\u27e9 := List.exists_of_eraseP h\u2081 h\u2082; eq \u25b8 ?_\nsimp [h]; rfl", "annotated_tactic": ["next H =>\n    simp [h, <a>Buckets.size</a>]\n    refine have \u27e8_, _, h\u2081, _, eq\u27e9 := <a>Buckets.exists_of_update</a> ..; eq \u25b8 ?_\n    simp [h, h\u2081, <a>Buckets.size_eq</a>]\n    rw [(_ : <a>List.length</a> _ = _ + 1), <a>Nat.add_right_comm</a>]; {rfl}\n    clear h\u2081 eq\n    simp [<a>AssocList.contains_eq</a>] at H\n    have \u27e8a, h\u2081, h\u2082\u27e9 := H\n    refine have \u27e8_, _, _, _, _, h, eq\u27e9 := <a>List.exists_of_eraseP</a> h\u2081 h\u2082; eq \u25b8 ?_\n    simp [h]; rfl", [{"full_name": "Std.HashMap.Imp.Buckets.size", "def_path": "lake-packages/std/Std/Data/HashMap/Basic.lean", "def_pos": [40, 19], "def_end_pos": [40, 23]}, {"full_name": "Std.HashMap.Imp.Buckets.exists_of_update", "def_path": "lake-packages/std/Std/Data/HashMap/WF.lean", "def_pos": [24, 9], "def_end_pos": [24, 25]}, {"full_name": "Std.HashMap.Imp.Buckets.size_eq", "def_path": "lake-packages/std/Std/Data/HashMap/WF.lean", "def_pos": [33, 9], "def_end_pos": [33, 16]}, {"full_name": "List.length", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2232, 5], "def_end_pos": [2232, 16]}, {"full_name": "Nat.add_right_comm", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [145, 19], "def_end_pos": [145, 33]}, {"full_name": "Std.AssocList.contains_eq", "def_path": "lake-packages/std/Std/Data/AssocList.lean", "def_pos": [150, 17], "def_end_pos": [150, 28]}, {"full_name": "List.exists_of_eraseP", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [1066, 9], "def_end_pos": [1066, 25]}]], "state_before": "case h_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nk : \u03b1\nh : m.size = Buckets.size m.buckets\nc\u271d : Bool\nheq\u271d :\n  AssocList.contains k\n      m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] =\n    true\n\u22a2 { size := m.size - 1,\n        buckets :=\n          Buckets.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n            (AssocList.erase k\n              m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n            (_ :\n              USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n                Array.size m.buckets.val) }.size =\n    Buckets.size\n      { size := m.size - 1,\n          buckets :=\n            Buckets.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n              (AssocList.erase k\n                m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n              (_ :\n                USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n                  Array.size m.buckets.val) }.buckets", "state_after": "no goals"}, {"tactic": "simp [h, Buckets.size]", "annotated_tactic": ["simp [h, <a>Buckets.size</a>]", [{"full_name": "Std.HashMap.Imp.Buckets.size", "def_path": "lake-packages/std/Std/Data/HashMap/Basic.lean", "def_pos": [40, 19], "def_end_pos": [40, 23]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nk : \u03b1\nh : m.size = Buckets.size m.buckets\nc\u271d : Bool\nH :\n  AssocList.contains k\n      m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] =\n    true\n\u22a2 { size := m.size - 1,\n        buckets :=\n          Buckets.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n            (AssocList.erase k\n              m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n            (_ :\n              USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n                Array.size m.buckets.val) }.size =\n    Buckets.size\n      { size := m.size - 1,\n          buckets :=\n            Buckets.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n              (AssocList.erase k\n                m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n              (_ :\n                USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n                  Array.size m.buckets.val) }.buckets", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nk : \u03b1\nh : m.size = Buckets.size m.buckets\nc\u271d : Bool\nH :\n  AssocList.contains k\n      m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] =\n    true\n\u22a2 Nat.sum (List.map (fun x => List.length (AssocList.toList x)) m.buckets.val.data) - 1 =\n    Nat.sum\n      (List.map (fun x => List.length (AssocList.toList x))\n        (Buckets.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n              (AssocList.erase k\n                m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n              (_ :\n                USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n                  Array.size m.buckets.val)).val.data)"}, {"tactic": "refine have \u27e8_, _, h\u2081, _, eq\u27e9 := Buckets.exists_of_update ..; eq \u25b8 ?_", "annotated_tactic": ["refine have \u27e8_, _, h\u2081, _, eq\u27e9 := <a>Buckets.exists_of_update</a> ..; eq \u25b8 ?_", [{"full_name": "Std.HashMap.Imp.Buckets.exists_of_update", "def_path": "lake-packages/std/Std/Data/HashMap/WF.lean", "def_pos": [24, 9], "def_end_pos": [24, 25]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nk : \u03b1\nh : m.size = Buckets.size m.buckets\nc\u271d : Bool\nH :\n  AssocList.contains k\n      m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] =\n    true\n\u22a2 Nat.sum (List.map (fun x => List.length (AssocList.toList x)) m.buckets.val.data) - 1 =\n    Nat.sum\n      (List.map (fun x => List.length (AssocList.toList x))\n        (Buckets.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n              (AssocList.erase k\n                m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n              (_ :\n                USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n                  Array.size m.buckets.val)).val.data)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nk : \u03b1\nh : m.size = Buckets.size m.buckets\nc\u271d : Bool\nH :\n  AssocList.contains k\n      m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] =\n    true\nw\u271d\u00b9 w\u271d : List (AssocList \u03b1 \u03b2)\nh\u2081 :\n  m.buckets.val.data =\n    w\u271d\u00b9 ++ m.buckets.val[(mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] :: w\u271d\nleft\u271d : List.length w\u271d\u00b9 = USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\neq :\n  (Buckets.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n          (AssocList.erase k\n            m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n          (_ :\n            USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n              Array.size m.buckets.val)).val.data =\n    w\u271d\u00b9 ++\n      AssocList.erase k\n          m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] ::\n        w\u271d\n\u22a2 Nat.sum (List.map (fun x => List.length (AssocList.toList x)) m.buckets.val.data) - 1 =\n    Nat.sum\n      (List.map (fun x => List.length (AssocList.toList x))\n        (w\u271d\u00b9 ++\n          AssocList.erase k\n              m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] ::\n            w\u271d))"}, {"tactic": "simp [h, h\u2081, Buckets.size_eq]", "annotated_tactic": ["simp [h, h\u2081, <a>Buckets.size_eq</a>]", [{"full_name": "Std.HashMap.Imp.Buckets.size_eq", "def_path": "lake-packages/std/Std/Data/HashMap/WF.lean", "def_pos": [33, 9], "def_end_pos": [33, 16]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nk : \u03b1\nh : m.size = Buckets.size m.buckets\nc\u271d : Bool\nH :\n  AssocList.contains k\n      m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] =\n    true\nw\u271d\u00b9 w\u271d : List (AssocList \u03b1 \u03b2)\nh\u2081 :\n  m.buckets.val.data =\n    w\u271d\u00b9 ++ m.buckets.val[(mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] :: w\u271d\nleft\u271d : List.length w\u271d\u00b9 = USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\neq :\n  (Buckets.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n          (AssocList.erase k\n            m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n          (_ :\n            USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n              Array.size m.buckets.val)).val.data =\n    w\u271d\u00b9 ++\n      AssocList.erase k\n          m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] ::\n        w\u271d\n\u22a2 Nat.sum (List.map (fun x => List.length (AssocList.toList x)) m.buckets.val.data) - 1 =\n    Nat.sum\n      (List.map (fun x => List.length (AssocList.toList x))\n        (w\u271d\u00b9 ++\n          AssocList.erase k\n              m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] ::\n            w\u271d))", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nk : \u03b1\nh : m.size = Buckets.size m.buckets\nc\u271d : Bool\nH :\n  AssocList.contains k\n      m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] =\n    true\nw\u271d\u00b9 w\u271d : List (AssocList \u03b1 \u03b2)\nh\u2081 :\n  m.buckets.val.data =\n    w\u271d\u00b9 ++ m.buckets.val[(mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] :: w\u271d\nleft\u271d : List.length w\u271d\u00b9 = USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\neq :\n  (Buckets.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n          (AssocList.erase k\n            m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n          (_ :\n            USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n              Array.size m.buckets.val)).val.data =\n    w\u271d\u00b9 ++\n      AssocList.erase k\n          m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] ::\n        w\u271d\n\u22a2 Nat.sum (List.map (fun x => List.length (AssocList.toList x)) w\u271d\u00b9) +\n        (List.length\n            (AssocList.toList\n              m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val]) +\n          Nat.sum (List.map (fun x => List.length (AssocList.toList x)) w\u271d)) -\n      1 =\n    Nat.sum (List.map (fun x => List.length (AssocList.toList x)) w\u271d\u00b9) +\n      (List.length\n          (List.eraseP (fun x => x.fst == k)\n            (AssocList.toList\n              m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])) +\n        Nat.sum (List.map (fun x => List.length (AssocList.toList x)) w\u271d))"}, {"tactic": "rw [(_ : List.length _ = _ + 1), Nat.add_right_comm]", "annotated_tactic": ["rw [(_ : <a>List.length</a> _ = _ + 1), <a>Nat.add_right_comm</a>]", [{"full_name": "List.length", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2232, 5], "def_end_pos": [2232, 16]}, {"full_name": "Nat.add_right_comm", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [145, 19], "def_end_pos": [145, 33]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nk : \u03b1\nh : m.size = Buckets.size m.buckets\nc\u271d : Bool\nH :\n  AssocList.contains k\n      m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] =\n    true\nw\u271d\u00b9 w\u271d : List (AssocList \u03b1 \u03b2)\nh\u2081 :\n  m.buckets.val.data =\n    w\u271d\u00b9 ++ m.buckets.val[(mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] :: w\u271d\nleft\u271d : List.length w\u271d\u00b9 = USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\neq :\n  (Buckets.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n          (AssocList.erase k\n            m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n          (_ :\n            USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n              Array.size m.buckets.val)).val.data =\n    w\u271d\u00b9 ++\n      AssocList.erase k\n          m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] ::\n        w\u271d\n\u22a2 Nat.sum (List.map (fun x => List.length (AssocList.toList x)) w\u271d\u00b9) +\n        (List.length\n            (AssocList.toList\n              m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val]) +\n          Nat.sum (List.map (fun x => List.length (AssocList.toList x)) w\u271d)) -\n      1 =\n    Nat.sum (List.map (fun x => List.length (AssocList.toList x)) w\u271d\u00b9) +\n      (List.length\n          (List.eraseP (fun x => x.fst == k)\n            (AssocList.toList\n              m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])) +\n        Nat.sum (List.map (fun x => List.length (AssocList.toList x)) w\u271d))", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nk : \u03b1\nh : m.size = Buckets.size m.buckets\nc\u271d : Bool\nH :\n  AssocList.contains k\n      m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] =\n    true\nw\u271d\u00b9 w\u271d : List (AssocList \u03b1 \u03b2)\nh\u2081 :\n  m.buckets.val.data =\n    w\u271d\u00b9 ++ m.buckets.val[(mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] :: w\u271d\nleft\u271d : List.length w\u271d\u00b9 = USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\neq :\n  (Buckets.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n          (AssocList.erase k\n            m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n          (_ :\n            USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n              Array.size m.buckets.val)).val.data =\n    w\u271d\u00b9 ++\n      AssocList.erase k\n          m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] ::\n        w\u271d\n\u22a2 Nat.sum (List.map (fun x => List.length (AssocList.toList x)) w\u271d\u00b9) +\n        (?n + Nat.sum (List.map (fun x => List.length (AssocList.toList x)) w\u271d) + 1) -\n      1 =\n    Nat.sum (List.map (fun x => List.length (AssocList.toList x)) w\u271d\u00b9) +\n      (List.length\n          (List.eraseP (fun x => x.fst == k)\n            (AssocList.toList\n              m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])) +\n        Nat.sum (List.map (fun x => List.length (AssocList.toList x)) w\u271d))\n\ncase n\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nk : \u03b1\nh : m.size = Buckets.size m.buckets\nc\u271d : Bool\nH :\n  AssocList.contains k\n      m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] =\n    true\nw\u271d\u00b9 w\u271d : List (AssocList \u03b1 \u03b2)\nh\u2081 :\n  m.buckets.val.data =\n    w\u271d\u00b9 ++ m.buckets.val[(mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] :: w\u271d\nleft\u271d : List.length w\u271d\u00b9 = USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\neq :\n  (Buckets.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n          (AssocList.erase k\n            m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n          (_ :\n            USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n              Array.size m.buckets.val)).val.data =\n    w\u271d\u00b9 ++\n      AssocList.erase k\n          m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] ::\n        w\u271d\n\u22a2 Nat\n\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nk : \u03b1\nh : m.size = Buckets.size m.buckets\nc\u271d : Bool\nH :\n  AssocList.contains k\n      m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] =\n    true\nw\u271d\u00b9 w\u271d : List (AssocList \u03b1 \u03b2)\nh\u2081 :\n  m.buckets.val.data =\n    w\u271d\u00b9 ++ m.buckets.val[(mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] :: w\u271d\nleft\u271d : List.length w\u271d\u00b9 = USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\neq :\n  (Buckets.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n          (AssocList.erase k\n            m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n          (_ :\n            USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n              Array.size m.buckets.val)).val.data =\n    w\u271d\u00b9 ++\n      AssocList.erase k\n          m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] ::\n        w\u271d\n\u22a2 List.length\n      (AssocList.toList\n        m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val]) =\n    ?n + 1"}, {"tactic": "{rfl}", "annotated_tactic": ["{rfl}", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nk : \u03b1\nh : m.size = Buckets.size m.buckets\nc\u271d : Bool\nH :\n  AssocList.contains k\n      m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] =\n    true\nw\u271d\u00b9 w\u271d : List (AssocList \u03b1 \u03b2)\nh\u2081 :\n  m.buckets.val.data =\n    w\u271d\u00b9 ++ m.buckets.val[(mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] :: w\u271d\nleft\u271d : List.length w\u271d\u00b9 = USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\neq :\n  (Buckets.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n          (AssocList.erase k\n            m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n          (_ :\n            USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n              Array.size m.buckets.val)).val.data =\n    w\u271d\u00b9 ++\n      AssocList.erase k\n          m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] ::\n        w\u271d\n\u22a2 Nat.sum (List.map (fun x => List.length (AssocList.toList x)) w\u271d\u00b9) +\n        (?n + Nat.sum (List.map (fun x => List.length (AssocList.toList x)) w\u271d) + 1) -\n      1 =\n    Nat.sum (List.map (fun x => List.length (AssocList.toList x)) w\u271d\u00b9) +\n      (List.length\n          (List.eraseP (fun x => x.fst == k)\n            (AssocList.toList\n              m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])) +\n        Nat.sum (List.map (fun x => List.length (AssocList.toList x)) w\u271d))\n\ncase n\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nk : \u03b1\nh : m.size = Buckets.size m.buckets\nc\u271d : Bool\nH :\n  AssocList.contains k\n      m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] =\n    true\nw\u271d\u00b9 w\u271d : List (AssocList \u03b1 \u03b2)\nh\u2081 :\n  m.buckets.val.data =\n    w\u271d\u00b9 ++ m.buckets.val[(mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] :: w\u271d\nleft\u271d : List.length w\u271d\u00b9 = USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\neq :\n  (Buckets.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n          (AssocList.erase k\n            m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n          (_ :\n            USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n              Array.size m.buckets.val)).val.data =\n    w\u271d\u00b9 ++\n      AssocList.erase k\n          m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] ::\n        w\u271d\n\u22a2 Nat\n\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nk : \u03b1\nh : m.size = Buckets.size m.buckets\nc\u271d : Bool\nH :\n  AssocList.contains k\n      m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] =\n    true\nw\u271d\u00b9 w\u271d : List (AssocList \u03b1 \u03b2)\nh\u2081 :\n  m.buckets.val.data =\n    w\u271d\u00b9 ++ m.buckets.val[(mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] :: w\u271d\nleft\u271d : List.length w\u271d\u00b9 = USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\neq :\n  (Buckets.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n          (AssocList.erase k\n            m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n          (_ :\n            USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n              Array.size m.buckets.val)).val.data =\n    w\u271d\u00b9 ++\n      AssocList.erase k\n          m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] ::\n        w\u271d\n\u22a2 List.length\n      (AssocList.toList\n        m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val]) =\n    ?n + 1", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nk : \u03b1\nh : m.size = Buckets.size m.buckets\nc\u271d : Bool\nH :\n  AssocList.contains k\n      m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] =\n    true\nw\u271d\u00b9 w\u271d : List (AssocList \u03b1 \u03b2)\nh\u2081 :\n  m.buckets.val.data =\n    w\u271d\u00b9 ++ m.buckets.val[(mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] :: w\u271d\nleft\u271d : List.length w\u271d\u00b9 = USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\neq :\n  (Buckets.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n          (AssocList.erase k\n            m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n          (_ :\n            USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n              Array.size m.buckets.val)).val.data =\n    w\u271d\u00b9 ++\n      AssocList.erase k\n          m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] ::\n        w\u271d\n\u22a2 List.length\n      (AssocList.toList\n        m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val]) =\n    List.length\n        (List.eraseP (fun x => x.fst == k)\n          (AssocList.toList\n            m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])) +\n      1"}, {"tactic": "clear h\u2081 eq", "annotated_tactic": ["clear h\u2081 eq", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nk : \u03b1\nh : m.size = Buckets.size m.buckets\nc\u271d : Bool\nH :\n  AssocList.contains k\n      m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] =\n    true\nw\u271d\u00b9 w\u271d : List (AssocList \u03b1 \u03b2)\nh\u2081 :\n  m.buckets.val.data =\n    w\u271d\u00b9 ++ m.buckets.val[(mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] :: w\u271d\nleft\u271d : List.length w\u271d\u00b9 = USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\neq :\n  (Buckets.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n          (AssocList.erase k\n            m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n          (_ :\n            USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n              Array.size m.buckets.val)).val.data =\n    w\u271d\u00b9 ++\n      AssocList.erase k\n          m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] ::\n        w\u271d\n\u22a2 List.length\n      (AssocList.toList\n        m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val]) =\n    List.length\n        (List.eraseP (fun x => x.fst == k)\n          (AssocList.toList\n            m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])) +\n      1", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nk : \u03b1\nh : m.size = Buckets.size m.buckets\nc\u271d : Bool\nH :\n  AssocList.contains k\n      m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] =\n    true\nw\u271d\u00b9 w\u271d : List (AssocList \u03b1 \u03b2)\nleft\u271d : List.length w\u271d\u00b9 = USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n\u22a2 List.length\n      (AssocList.toList\n        m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val]) =\n    List.length\n        (List.eraseP (fun x => x.fst == k)\n          (AssocList.toList\n            m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])) +\n      1"}, {"tactic": "simp [AssocList.contains_eq] at H", "annotated_tactic": ["simp [<a>AssocList.contains_eq</a>] at H", [{"full_name": "Std.AssocList.contains_eq", "def_path": "lake-packages/std/Std/Data/AssocList.lean", "def_pos": [150, 17], "def_end_pos": [150, 28]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nk : \u03b1\nh : m.size = Buckets.size m.buckets\nc\u271d : Bool\nH :\n  AssocList.contains k\n      m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] =\n    true\nw\u271d\u00b9 w\u271d : List (AssocList \u03b1 \u03b2)\nleft\u271d : List.length w\u271d\u00b9 = USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n\u22a2 List.length\n      (AssocList.toList\n        m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val]) =\n    List.length\n        (List.eraseP (fun x => x.fst == k)\n          (AssocList.toList\n            m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])) +\n      1", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nk : \u03b1\nh : m.size = Buckets.size m.buckets\nc\u271d : Bool\nw\u271d\u00b9 w\u271d : List (AssocList \u03b1 \u03b2)\nleft\u271d : List.length w\u271d\u00b9 = USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\nH :\n  \u2203 x,\n    x \u2208\n        AssocList.toList\n          m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] \u2227\n      (x.fst == k) = true\n\u22a2 List.length\n      (AssocList.toList\n        m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val]) =\n    List.length\n        (List.eraseP (fun x => x.fst == k)\n          (AssocList.toList\n            m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])) +\n      1"}, {"tactic": "have \u27e8a, h\u2081, h\u2082\u27e9 := H", "annotated_tactic": ["have \u27e8a, h\u2081, h\u2082\u27e9 := H", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nk : \u03b1\nh : m.size = Buckets.size m.buckets\nc\u271d : Bool\nw\u271d\u00b9 w\u271d : List (AssocList \u03b1 \u03b2)\nleft\u271d : List.length w\u271d\u00b9 = USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\nH :\n  \u2203 x,\n    x \u2208\n        AssocList.toList\n          m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] \u2227\n      (x.fst == k) = true\n\u22a2 List.length\n      (AssocList.toList\n        m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val]) =\n    List.length\n        (List.eraseP (fun x => x.fst == k)\n          (AssocList.toList\n            m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])) +\n      1", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nk : \u03b1\nh : m.size = Buckets.size m.buckets\nc\u271d : Bool\nw\u271d\u00b9 w\u271d : List (AssocList \u03b1 \u03b2)\nleft\u271d : List.length w\u271d\u00b9 = USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\nH :\n  \u2203 x,\n    x \u2208\n        AssocList.toList\n          m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] \u2227\n      (x.fst == k) = true\na : \u03b1 \u00d7 \u03b2\nh\u2081 :\n  a \u2208\n    AssocList.toList m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val]\nh\u2082 : (a.fst == k) = true\n\u22a2 List.length\n      (AssocList.toList\n        m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val]) =\n    List.length\n        (List.eraseP (fun x => x.fst == k)\n          (AssocList.toList\n            m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])) +\n      1"}, {"tactic": "refine have \u27e8_, _, _, _, _, h, eq\u27e9 := List.exists_of_eraseP h\u2081 h\u2082; eq \u25b8 ?_", "annotated_tactic": ["refine have \u27e8_, _, _, _, _, h, eq\u27e9 := <a>List.exists_of_eraseP</a> h\u2081 h\u2082; eq \u25b8 ?_", [{"full_name": "List.exists_of_eraseP", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [1066, 9], "def_end_pos": [1066, 25]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nk : \u03b1\nh : m.size = Buckets.size m.buckets\nc\u271d : Bool\nw\u271d\u00b9 w\u271d : List (AssocList \u03b1 \u03b2)\nleft\u271d : List.length w\u271d\u00b9 = USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\nH :\n  \u2203 x,\n    x \u2208\n        AssocList.toList\n          m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] \u2227\n      (x.fst == k) = true\na : \u03b1 \u00d7 \u03b2\nh\u2081 :\n  a \u2208\n    AssocList.toList m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val]\nh\u2082 : (a.fst == k) = true\n\u22a2 List.length\n      (AssocList.toList\n        m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val]) =\n    List.length\n        (List.eraseP (fun x => x.fst == k)\n          (AssocList.toList\n            m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])) +\n      1", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nk : \u03b1\nh\u271d : m.size = Buckets.size m.buckets\nc\u271d : Bool\nw\u271d\u2074 w\u271d\u00b3 : List (AssocList \u03b1 \u03b2)\nleft\u271d\u00b2 : List.length w\u271d\u2074 = USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\nH :\n  \u2203 x,\n    x \u2208\n        AssocList.toList\n          m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] \u2227\n      (x.fst == k) = true\na : \u03b1 \u00d7 \u03b2\nh\u2081 :\n  a \u2208\n    AssocList.toList m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val]\nh\u2082 : (a.fst == k) = true\nw\u271d\u00b2 : \u03b1 \u00d7 \u03b2\nw\u271d\u00b9 w\u271d : List (\u03b1 \u00d7 \u03b2)\nleft\u271d\u00b9 : \u2200 (b : \u03b1 \u00d7 \u03b2), b \u2208 w\u271d\u00b9 \u2192 \u00ac(b.fst == k) = true\nleft\u271d : (w\u271d\u00b2.fst == k) = true\nh :\n  AssocList.toList m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] =\n    w\u271d\u00b9 ++ w\u271d\u00b2 :: w\u271d\neq :\n  List.eraseP (fun x => x.fst == k)\n      (AssocList.toList\n        m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val]) =\n    w\u271d\u00b9 ++ w\u271d\n\u22a2 List.length\n      (AssocList.toList\n        m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val]) =\n    List.length (w\u271d\u00b9 ++ w\u271d) + 1"}, {"tactic": "simp [h]", "annotated_tactic": ["simp [h]", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nk : \u03b1\nh\u271d : m.size = Buckets.size m.buckets\nc\u271d : Bool\nw\u271d\u2074 w\u271d\u00b3 : List (AssocList \u03b1 \u03b2)\nleft\u271d\u00b2 : List.length w\u271d\u2074 = USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\nH :\n  \u2203 x,\n    x \u2208\n        AssocList.toList\n          m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] \u2227\n      (x.fst == k) = true\na : \u03b1 \u00d7 \u03b2\nh\u2081 :\n  a \u2208\n    AssocList.toList m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val]\nh\u2082 : (a.fst == k) = true\nw\u271d\u00b2 : \u03b1 \u00d7 \u03b2\nw\u271d\u00b9 w\u271d : List (\u03b1 \u00d7 \u03b2)\nleft\u271d\u00b9 : \u2200 (b : \u03b1 \u00d7 \u03b2), b \u2208 w\u271d\u00b9 \u2192 \u00ac(b.fst == k) = true\nleft\u271d : (w\u271d\u00b2.fst == k) = true\nh :\n  AssocList.toList m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] =\n    w\u271d\u00b9 ++ w\u271d\u00b2 :: w\u271d\neq :\n  List.eraseP (fun x => x.fst == k)\n      (AssocList.toList\n        m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val]) =\n    w\u271d\u00b9 ++ w\u271d\n\u22a2 List.length\n      (AssocList.toList\n        m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val]) =\n    List.length (w\u271d\u00b9 ++ w\u271d) + 1", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nk : \u03b1\nh\u271d : m.size = Buckets.size m.buckets\nc\u271d : Bool\nw\u271d\u2074 w\u271d\u00b3 : List (AssocList \u03b1 \u03b2)\nleft\u271d\u00b2 : List.length w\u271d\u2074 = USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\nH :\n  \u2203 x,\n    x \u2208\n        AssocList.toList\n          m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] \u2227\n      (x.fst == k) = true\na : \u03b1 \u00d7 \u03b2\nh\u2081 :\n  a \u2208\n    AssocList.toList m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val]\nh\u2082 : (a.fst == k) = true\nw\u271d\u00b2 : \u03b1 \u00d7 \u03b2\nw\u271d\u00b9 w\u271d : List (\u03b1 \u00d7 \u03b2)\nleft\u271d\u00b9 : \u2200 (b : \u03b1 \u00d7 \u03b2), b \u2208 w\u271d\u00b9 \u2192 \u00ac(b.fst == k) = true\nleft\u271d : (w\u271d\u00b2.fst == k) = true\nh :\n  AssocList.toList m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] =\n    w\u271d\u00b9 ++ w\u271d\u00b2 :: w\u271d\neq :\n  List.eraseP (fun x => x.fst == k)\n      (AssocList.toList\n        m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val]) =\n    w\u271d\u00b9 ++ w\u271d\n\u22a2 List.length w\u271d\u00b9 + Nat.succ (List.length w\u271d) = List.length w\u271d\u00b9 + List.length w\u271d + 1"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nk : \u03b1\nh\u271d : m.size = Buckets.size m.buckets\nc\u271d : Bool\nw\u271d\u2074 w\u271d\u00b3 : List (AssocList \u03b1 \u03b2)\nleft\u271d\u00b2 : List.length w\u271d\u2074 = USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\nH :\n  \u2203 x,\n    x \u2208\n        AssocList.toList\n          m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] \u2227\n      (x.fst == k) = true\na : \u03b1 \u00d7 \u03b2\nh\u2081 :\n  a \u2208\n    AssocList.toList m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val]\nh\u2082 : (a.fst == k) = true\nw\u271d\u00b2 : \u03b1 \u00d7 \u03b2\nw\u271d\u00b9 w\u271d : List (\u03b1 \u00d7 \u03b2)\nleft\u271d\u00b9 : \u2200 (b : \u03b1 \u00d7 \u03b2), b \u2208 w\u271d\u00b9 \u2192 \u00ac(b.fst == k) = true\nleft\u271d : (w\u271d\u00b2.fst == k) = true\nh :\n  AssocList.toList m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] =\n    w\u271d\u00b9 ++ w\u271d\u00b2 :: w\u271d\neq :\n  List.eraseP (fun x => x.fst == k)\n      (AssocList.toList\n        m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val]) =\n    w\u271d\u00b9 ++ w\u271d\n\u22a2 List.length w\u271d\u00b9 + Nat.succ (List.length w\u271d) = List.length w\u271d\u00b9 + List.length w\u271d + 1", "state_after": "no goals"}, {"tactic": "exact h", "annotated_tactic": ["exact h", []], "state_before": "case h_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nk : \u03b1\nh : m.size = Buckets.size m.buckets\nc\u271d : Bool\nheq\u271d :\n  AssocList.contains k\n      m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] =\n    false\n\u22a2 m.size = Buckets.size m.buckets", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/DivergenceTheorem.lean", "full_name": "MeasureTheory.integral_divergence_of_hasFDerivWithinAt_off_countable", "start": [268, 1], "end": [295, 12], "traced_tactics": [{"tactic": "rcases em (\u2203 i, a i = b i) with (\u27e8i, hi\u27e9 | hne)", "annotated_tactic": ["rcases <a>em</a> (\u2203 i, a i = b i) with (\u27e8i, hi\u27e9 | hne)", [{"full_name": "em", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [194, 7], "def_end_pos": [194, 9]}]], "state_before": "E : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\na b : Fin (n + 1) \u2192 \u211d\nhle : a \u2264 b\nf : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E\nf' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E\ns : Set (Fin (n + 1) \u2192 \u211d)\nhs : Set.Countable s\nHc : ContinuousOn f (Set.Icc a b)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 (Set.pi Set.univ fun i => Set.Ioo (a i) (b i)) \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (Set.Icc a b)\n\u22a2 \u222b (x : Fin (n + 1) \u2192 \u211d) in Set.Icc a b, \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i =\n    \u2211 i : Fin (n + 1),\n      ((\u222b (x : Fin n \u2192 \u211d) in Set.Icc (a \u2218 Fin.succAbove i) (b \u2218 Fin.succAbove i), f (Fin.insertNth i (b i) x) i) -\n        \u222b (x : Fin n \u2192 \u211d) in Set.Icc (a \u2218 Fin.succAbove i) (b \u2218 Fin.succAbove i), f (Fin.insertNth i (a i) x) i)", "state_after": "case inl.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\na b : Fin (n + 1) \u2192 \u211d\nhle : a \u2264 b\nf : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E\nf' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E\ns : Set (Fin (n + 1) \u2192 \u211d)\nhs : Set.Countable s\nHc : ContinuousOn f (Set.Icc a b)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 (Set.pi Set.univ fun i => Set.Ioo (a i) (b i)) \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (Set.Icc a b)\ni : Fin (n + 1)\nhi : a i = b i\n\u22a2 \u222b (x : Fin (n + 1) \u2192 \u211d) in Set.Icc a b, \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i =\n    \u2211 i : Fin (n + 1),\n      ((\u222b (x : Fin n \u2192 \u211d) in Set.Icc (a \u2218 Fin.succAbove i) (b \u2218 Fin.succAbove i), f (Fin.insertNth i (b i) x) i) -\n        \u222b (x : Fin n \u2192 \u211d) in Set.Icc (a \u2218 Fin.succAbove i) (b \u2218 Fin.succAbove i), f (Fin.insertNth i (a i) x) i)\n\ncase inr\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\na b : Fin (n + 1) \u2192 \u211d\nhle : a \u2264 b\nf : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E\nf' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E\ns : Set (Fin (n + 1) \u2192 \u211d)\nhs : Set.Countable s\nHc : ContinuousOn f (Set.Icc a b)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 (Set.pi Set.univ fun i => Set.Ioo (a i) (b i)) \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (Set.Icc a b)\nhne : \u00ac\u2203 i, a i = b i\n\u22a2 \u222b (x : Fin (n + 1) \u2192 \u211d) in Set.Icc a b, \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i =\n    \u2211 i : Fin (n + 1),\n      ((\u222b (x : Fin n \u2192 \u211d) in Set.Icc (a \u2218 Fin.succAbove i) (b \u2218 Fin.succAbove i), f (Fin.insertNth i (b i) x) i) -\n        \u222b (x : Fin n \u2192 \u211d) in Set.Icc (a \u2218 Fin.succAbove i) (b \u2218 Fin.succAbove i), f (Fin.insertNth i (a i) x) i)"}, {"tactic": "rw [volume_pi, \u2190 set_integral_congr_set_ae Measure.univ_pi_Ioc_ae_eq_Icc]", "annotated_tactic": ["rw [<a>volume_pi</a>, \u2190 <a>set_integral_congr_set_ae</a> <a>Measure.univ_pi_Ioc_ae_eq_Icc</a>]", [{"full_name": "MeasureTheory.volume_pi", "def_path": "Mathlib/MeasureTheory/Constructions/Pi.lean", "def_pos": [671, 9], "def_end_pos": [671, 18]}, {"full_name": "MeasureTheory.set_integral_congr_set_ae", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [92, 9], "def_end_pos": [92, 34]}, {"full_name": "MeasureTheory.Measure.univ_pi_Ioc_ae_eq_Icc", "def_path": "Mathlib/MeasureTheory/Constructions/Pi.lean", "def_pos": [525, 9], "def_end_pos": [525, 30]}]], "state_before": "case inl.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\na b : Fin (n + 1) \u2192 \u211d\nhle : a \u2264 b\nf : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E\nf' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E\ns : Set (Fin (n + 1) \u2192 \u211d)\nhs : Set.Countable s\nHc : ContinuousOn f (Set.Icc a b)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 (Set.pi Set.univ fun i => Set.Ioo (a i) (b i)) \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (Set.Icc a b)\ni : Fin (n + 1)\nhi : a i = b i\n\u22a2 \u222b (x : Fin (n + 1) \u2192 \u211d) in Set.Icc a b, \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i =\n    \u2211 i : Fin (n + 1),\n      ((\u222b (x : Fin n \u2192 \u211d) in Set.Icc (a \u2218 Fin.succAbove i) (b \u2218 Fin.succAbove i), f (Fin.insertNth i (b i) x) i) -\n        \u222b (x : Fin n \u2192 \u211d) in Set.Icc (a \u2218 Fin.succAbove i) (b \u2218 Fin.succAbove i), f (Fin.insertNth i (a i) x) i)", "state_after": "case inl.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\na b : Fin (n + 1) \u2192 \u211d\nhle : a \u2264 b\nf : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E\nf' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E\ns : Set (Fin (n + 1) \u2192 \u211d)\nhs : Set.Countable s\nHc : ContinuousOn f (Set.Icc a b)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 (Set.pi Set.univ fun i => Set.Ioo (a i) (b i)) \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (Set.Icc a b)\ni : Fin (n + 1)\nhi : a i = b i\n\u22a2 (\u222b (x : Fin (n + 1) \u2192 \u211d) in Set.pi Set.univ fun i => Set.Ioc (a i) (b i),\n      \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i \u2202Measure.pi fun x => volume) =\n    \u2211 i : Fin (n + 1),\n      ((\u222b (x : Fin n \u2192 \u211d) in Set.Icc (a \u2218 Fin.succAbove i) (b \u2218 Fin.succAbove i), f (Fin.insertNth i (b i) x) i) -\n        \u222b (x : Fin n \u2192 \u211d) in Set.Icc (a \u2218 Fin.succAbove i) (b \u2218 Fin.succAbove i), f (Fin.insertNth i (a i) x) i)"}, {"tactic": "have hi' : Ioc (a i) (b i) = \u2205 := Ioc_eq_empty hi.not_lt", "annotated_tactic": ["have hi' : <a>Ioc</a> (a i) (b i) = \u2205 := <a>Ioc_eq_empty</a> hi.not_lt", [{"full_name": "Set.Ioc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [69, 5], "def_end_pos": [69, 8]}, {"full_name": "Set.Ioc_eq_empty", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [368, 9], "def_end_pos": [368, 21]}]], "state_before": "case inl.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\na b : Fin (n + 1) \u2192 \u211d\nhle : a \u2264 b\nf : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E\nf' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E\ns : Set (Fin (n + 1) \u2192 \u211d)\nhs : Set.Countable s\nHc : ContinuousOn f (Set.Icc a b)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 (Set.pi Set.univ fun i => Set.Ioo (a i) (b i)) \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (Set.Icc a b)\ni : Fin (n + 1)\nhi : a i = b i\n\u22a2 (\u222b (x : Fin (n + 1) \u2192 \u211d) in Set.pi Set.univ fun i => Set.Ioc (a i) (b i),\n      \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i \u2202Measure.pi fun x => volume) =\n    \u2211 i : Fin (n + 1),\n      ((\u222b (x : Fin n \u2192 \u211d) in Set.Icc (a \u2218 Fin.succAbove i) (b \u2218 Fin.succAbove i), f (Fin.insertNth i (b i) x) i) -\n        \u222b (x : Fin n \u2192 \u211d) in Set.Icc (a \u2218 Fin.succAbove i) (b \u2218 Fin.succAbove i), f (Fin.insertNth i (a i) x) i)", "state_after": "case inl.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\na b : Fin (n + 1) \u2192 \u211d\nhle : a \u2264 b\nf : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E\nf' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E\ns : Set (Fin (n + 1) \u2192 \u211d)\nhs : Set.Countable s\nHc : ContinuousOn f (Set.Icc a b)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 (Set.pi Set.univ fun i => Set.Ioo (a i) (b i)) \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (Set.Icc a b)\ni : Fin (n + 1)\nhi : a i = b i\nhi' : Set.Ioc (a i) (b i) = \u2205\n\u22a2 (\u222b (x : Fin (n + 1) \u2192 \u211d) in Set.pi Set.univ fun i => Set.Ioc (a i) (b i),\n      \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i \u2202Measure.pi fun x => volume) =\n    \u2211 i : Fin (n + 1),\n      ((\u222b (x : Fin n \u2192 \u211d) in Set.Icc (a \u2218 Fin.succAbove i) (b \u2218 Fin.succAbove i), f (Fin.insertNth i (b i) x) i) -\n        \u222b (x : Fin n \u2192 \u211d) in Set.Icc (a \u2218 Fin.succAbove i) (b \u2218 Fin.succAbove i), f (Fin.insertNth i (a i) x) i)"}, {"tactic": "have : (pi Set.univ fun j => Ioc (a j) (b j)) = \u2205 := univ_pi_eq_empty hi'", "annotated_tactic": ["have : (<a>pi</a> <a>Set.univ</a> fun j => <a>Ioc</a> (a j) (b j)) = \u2205 := <a>univ_pi_eq_empty</a> hi'", [{"full_name": "Set.pi", "def_path": "Mathlib/Data/Set/Prod.lean", "def_pos": [665, 5], "def_end_pos": [665, 7]}, {"full_name": "Set.univ", "def_path": "Mathlib/Init/Set.lean", "def_pos": [90, 5], "def_end_pos": [90, 9]}, {"full_name": "Set.Ioc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [69, 5], "def_end_pos": [69, 8]}, {"full_name": "Set.univ_pi_eq_empty", "def_path": "Mathlib/Data/Set/Prod.lean", "def_pos": [715, 9], "def_end_pos": [715, 25]}]], "state_before": "case inl.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\na b : Fin (n + 1) \u2192 \u211d\nhle : a \u2264 b\nf : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E\nf' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E\ns : Set (Fin (n + 1) \u2192 \u211d)\nhs : Set.Countable s\nHc : ContinuousOn f (Set.Icc a b)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 (Set.pi Set.univ fun i => Set.Ioo (a i) (b i)) \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (Set.Icc a b)\ni : Fin (n + 1)\nhi : a i = b i\nhi' : Set.Ioc (a i) (b i) = \u2205\n\u22a2 (\u222b (x : Fin (n + 1) \u2192 \u211d) in Set.pi Set.univ fun i => Set.Ioc (a i) (b i),\n      \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i \u2202Measure.pi fun x => volume) =\n    \u2211 i : Fin (n + 1),\n      ((\u222b (x : Fin n \u2192 \u211d) in Set.Icc (a \u2218 Fin.succAbove i) (b \u2218 Fin.succAbove i), f (Fin.insertNth i (b i) x) i) -\n        \u222b (x : Fin n \u2192 \u211d) in Set.Icc (a \u2218 Fin.succAbove i) (b \u2218 Fin.succAbove i), f (Fin.insertNth i (a i) x) i)", "state_after": "case inl.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\na b : Fin (n + 1) \u2192 \u211d\nhle : a \u2264 b\nf : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E\nf' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E\ns : Set (Fin (n + 1) \u2192 \u211d)\nhs : Set.Countable s\nHc : ContinuousOn f (Set.Icc a b)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 (Set.pi Set.univ fun i => Set.Ioo (a i) (b i)) \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (Set.Icc a b)\ni : Fin (n + 1)\nhi : a i = b i\nhi' : Set.Ioc (a i) (b i) = \u2205\nthis : (Set.pi Set.univ fun j => Set.Ioc (a j) (b j)) = \u2205\n\u22a2 (\u222b (x : Fin (n + 1) \u2192 \u211d) in Set.pi Set.univ fun i => Set.Ioc (a i) (b i),\n      \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i \u2202Measure.pi fun x => volume) =\n    \u2211 i : Fin (n + 1),\n      ((\u222b (x : Fin n \u2192 \u211d) in Set.Icc (a \u2218 Fin.succAbove i) (b \u2218 Fin.succAbove i), f (Fin.insertNth i (b i) x) i) -\n        \u222b (x : Fin n \u2192 \u211d) in Set.Icc (a \u2218 Fin.succAbove i) (b \u2218 Fin.succAbove i), f (Fin.insertNth i (a i) x) i)"}, {"tactic": "rw [this, integral_empty, sum_eq_zero]", "annotated_tactic": ["rw [this, <a>integral_empty</a>, <a>sum_eq_zero</a>]", [{"full_name": "MeasureTheory.integral_empty", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [148, 9], "def_end_pos": [148, 23]}, {"full_name": "Finset.sum_eq_zero", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [728, 3], "def_end_pos": [728, 14]}]], "state_before": "case inl.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\na b : Fin (n + 1) \u2192 \u211d\nhle : a \u2264 b\nf : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E\nf' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E\ns : Set (Fin (n + 1) \u2192 \u211d)\nhs : Set.Countable s\nHc : ContinuousOn f (Set.Icc a b)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 (Set.pi Set.univ fun i => Set.Ioo (a i) (b i)) \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (Set.Icc a b)\ni : Fin (n + 1)\nhi : a i = b i\nhi' : Set.Ioc (a i) (b i) = \u2205\nthis : (Set.pi Set.univ fun j => Set.Ioc (a j) (b j)) = \u2205\n\u22a2 (\u222b (x : Fin (n + 1) \u2192 \u211d) in Set.pi Set.univ fun i => Set.Ioc (a i) (b i),\n      \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i \u2202Measure.pi fun x => volume) =\n    \u2211 i : Fin (n + 1),\n      ((\u222b (x : Fin n \u2192 \u211d) in Set.Icc (a \u2218 Fin.succAbove i) (b \u2218 Fin.succAbove i), f (Fin.insertNth i (b i) x) i) -\n        \u222b (x : Fin n \u2192 \u211d) in Set.Icc (a \u2218 Fin.succAbove i) (b \u2218 Fin.succAbove i), f (Fin.insertNth i (a i) x) i)", "state_after": "case inl.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\na b : Fin (n + 1) \u2192 \u211d\nhle : a \u2264 b\nf : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E\nf' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E\ns : Set (Fin (n + 1) \u2192 \u211d)\nhs : Set.Countable s\nHc : ContinuousOn f (Set.Icc a b)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 (Set.pi Set.univ fun i => Set.Ioo (a i) (b i)) \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (Set.Icc a b)\ni : Fin (n + 1)\nhi : a i = b i\nhi' : Set.Ioc (a i) (b i) = \u2205\nthis : (Set.pi Set.univ fun j => Set.Ioc (a j) (b j)) = \u2205\n\u22a2 \u2200 (x : Fin (n + 1)),\n    x \u2208 Finset.univ \u2192\n      (\u222b (x_1 : Fin n \u2192 \u211d) in Set.Icc (a \u2218 Fin.succAbove x) (b \u2218 Fin.succAbove x), f (Fin.insertNth x (b x) x_1) x) -\n          \u222b (x_1 : Fin n \u2192 \u211d) in Set.Icc (a \u2218 Fin.succAbove x) (b \u2218 Fin.succAbove x), f (Fin.insertNth x (a x) x_1) x =\n        0"}, {"tactic": "rintro j -", "annotated_tactic": ["rintro j -", []], "state_before": "case inl.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\na b : Fin (n + 1) \u2192 \u211d\nhle : a \u2264 b\nf : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E\nf' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E\ns : Set (Fin (n + 1) \u2192 \u211d)\nhs : Set.Countable s\nHc : ContinuousOn f (Set.Icc a b)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 (Set.pi Set.univ fun i => Set.Ioo (a i) (b i)) \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (Set.Icc a b)\ni : Fin (n + 1)\nhi : a i = b i\nhi' : Set.Ioc (a i) (b i) = \u2205\nthis : (Set.pi Set.univ fun j => Set.Ioc (a j) (b j)) = \u2205\n\u22a2 \u2200 (x : Fin (n + 1)),\n    x \u2208 Finset.univ \u2192\n      (\u222b (x_1 : Fin n \u2192 \u211d) in Set.Icc (a \u2218 Fin.succAbove x) (b \u2218 Fin.succAbove x), f (Fin.insertNth x (b x) x_1) x) -\n          \u222b (x_1 : Fin n \u2192 \u211d) in Set.Icc (a \u2218 Fin.succAbove x) (b \u2218 Fin.succAbove x), f (Fin.insertNth x (a x) x_1) x =\n        0", "state_after": "case inl.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\na b : Fin (n + 1) \u2192 \u211d\nhle : a \u2264 b\nf : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E\nf' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E\ns : Set (Fin (n + 1) \u2192 \u211d)\nhs : Set.Countable s\nHc : ContinuousOn f (Set.Icc a b)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 (Set.pi Set.univ fun i => Set.Ioo (a i) (b i)) \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (Set.Icc a b)\ni : Fin (n + 1)\nhi : a i = b i\nhi' : Set.Ioc (a i) (b i) = \u2205\nthis : (Set.pi Set.univ fun j => Set.Ioc (a j) (b j)) = \u2205\nj : Fin (n + 1)\n\u22a2 (\u222b (x : Fin n \u2192 \u211d) in Set.Icc (a \u2218 Fin.succAbove j) (b \u2218 Fin.succAbove j), f (Fin.insertNth j (b j) x) j) -\n      \u222b (x : Fin n \u2192 \u211d) in Set.Icc (a \u2218 Fin.succAbove j) (b \u2218 Fin.succAbove j), f (Fin.insertNth j (a j) x) j =\n    0"}, {"tactic": "rcases eq_or_ne i j with (rfl | hne)", "annotated_tactic": ["rcases <a>eq_or_ne</a> i j with (rfl | hne)", [{"full_name": "eq_or_ne", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [209, 9], "def_end_pos": [209, 17]}]], "state_before": "case inl.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\na b : Fin (n + 1) \u2192 \u211d\nhle : a \u2264 b\nf : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E\nf' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E\ns : Set (Fin (n + 1) \u2192 \u211d)\nhs : Set.Countable s\nHc : ContinuousOn f (Set.Icc a b)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 (Set.pi Set.univ fun i => Set.Ioo (a i) (b i)) \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (Set.Icc a b)\ni : Fin (n + 1)\nhi : a i = b i\nhi' : Set.Ioc (a i) (b i) = \u2205\nthis : (Set.pi Set.univ fun j => Set.Ioc (a j) (b j)) = \u2205\nj : Fin (n + 1)\n\u22a2 (\u222b (x : Fin n \u2192 \u211d) in Set.Icc (a \u2218 Fin.succAbove j) (b \u2218 Fin.succAbove j), f (Fin.insertNth j (b j) x) j) -\n      \u222b (x : Fin n \u2192 \u211d) in Set.Icc (a \u2218 Fin.succAbove j) (b \u2218 Fin.succAbove j), f (Fin.insertNth j (a j) x) j =\n    0", "state_after": "case inl.intro.inl\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\na b : Fin (n + 1) \u2192 \u211d\nhle : a \u2264 b\nf : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E\nf' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E\ns : Set (Fin (n + 1) \u2192 \u211d)\nhs : Set.Countable s\nHc : ContinuousOn f (Set.Icc a b)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 (Set.pi Set.univ fun i => Set.Ioo (a i) (b i)) \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (Set.Icc a b)\ni : Fin (n + 1)\nhi : a i = b i\nhi' : Set.Ioc (a i) (b i) = \u2205\nthis : (Set.pi Set.univ fun j => Set.Ioc (a j) (b j)) = \u2205\n\u22a2 (\u222b (x : Fin n \u2192 \u211d) in Set.Icc (a \u2218 Fin.succAbove i) (b \u2218 Fin.succAbove i), f (Fin.insertNth i (b i) x) i) -\n      \u222b (x : Fin n \u2192 \u211d) in Set.Icc (a \u2218 Fin.succAbove i) (b \u2218 Fin.succAbove i), f (Fin.insertNth i (a i) x) i =\n    0\n\ncase inl.intro.inr\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\na b : Fin (n + 1) \u2192 \u211d\nhle : a \u2264 b\nf : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E\nf' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E\ns : Set (Fin (n + 1) \u2192 \u211d)\nhs : Set.Countable s\nHc : ContinuousOn f (Set.Icc a b)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 (Set.pi Set.univ fun i => Set.Ioo (a i) (b i)) \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (Set.Icc a b)\ni : Fin (n + 1)\nhi : a i = b i\nhi' : Set.Ioc (a i) (b i) = \u2205\nthis : (Set.pi Set.univ fun j => Set.Ioc (a j) (b j)) = \u2205\nj : Fin (n + 1)\nhne : i \u2260 j\n\u22a2 (\u222b (x : Fin n \u2192 \u211d) in Set.Icc (a \u2218 Fin.succAbove j) (b \u2218 Fin.succAbove j), f (Fin.insertNth j (b j) x) j) -\n      \u222b (x : Fin n \u2192 \u211d) in Set.Icc (a \u2218 Fin.succAbove j) (b \u2218 Fin.succAbove j), f (Fin.insertNth j (a j) x) j =\n    0"}, {"tactic": "simp [hi]", "annotated_tactic": ["simp [hi]", []], "state_before": "case inl.intro.inl\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\na b : Fin (n + 1) \u2192 \u211d\nhle : a \u2264 b\nf : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E\nf' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E\ns : Set (Fin (n + 1) \u2192 \u211d)\nhs : Set.Countable s\nHc : ContinuousOn f (Set.Icc a b)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 (Set.pi Set.univ fun i => Set.Ioo (a i) (b i)) \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (Set.Icc a b)\ni : Fin (n + 1)\nhi : a i = b i\nhi' : Set.Ioc (a i) (b i) = \u2205\nthis : (Set.pi Set.univ fun j => Set.Ioc (a j) (b j)) = \u2205\n\u22a2 (\u222b (x : Fin n \u2192 \u211d) in Set.Icc (a \u2218 Fin.succAbove i) (b \u2218 Fin.succAbove i), f (Fin.insertNth i (b i) x) i) -\n      \u222b (x : Fin n \u2192 \u211d) in Set.Icc (a \u2218 Fin.succAbove i) (b \u2218 Fin.succAbove i), f (Fin.insertNth i (a i) x) i =\n    0", "state_after": "no goals"}, {"tactic": "rcases Fin.exists_succAbove_eq hne with \u27e8i, rfl\u27e9", "annotated_tactic": ["rcases <a>Fin.exists_succAbove_eq</a> hne with \u27e8i, rfl\u27e9", [{"full_name": "Fin.exists_succAbove_eq", "def_path": "Mathlib/Data/Fin/Basic.lean", "def_pos": [1440, 9], "def_end_pos": [1440, 28]}]], "state_before": "case inl.intro.inr\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\na b : Fin (n + 1) \u2192 \u211d\nhle : a \u2264 b\nf : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E\nf' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E\ns : Set (Fin (n + 1) \u2192 \u211d)\nhs : Set.Countable s\nHc : ContinuousOn f (Set.Icc a b)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 (Set.pi Set.univ fun i => Set.Ioo (a i) (b i)) \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (Set.Icc a b)\ni : Fin (n + 1)\nhi : a i = b i\nhi' : Set.Ioc (a i) (b i) = \u2205\nthis : (Set.pi Set.univ fun j => Set.Ioc (a j) (b j)) = \u2205\nj : Fin (n + 1)\nhne : i \u2260 j\n\u22a2 (\u222b (x : Fin n \u2192 \u211d) in Set.Icc (a \u2218 Fin.succAbove j) (b \u2218 Fin.succAbove j), f (Fin.insertNth j (b j) x) j) -\n      \u222b (x : Fin n \u2192 \u211d) in Set.Icc (a \u2218 Fin.succAbove j) (b \u2218 Fin.succAbove j), f (Fin.insertNth j (a j) x) j =\n    0", "state_after": "case inl.intro.inr.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\na b : Fin (n + 1) \u2192 \u211d\nhle : a \u2264 b\nf : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E\nf' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E\ns : Set (Fin (n + 1) \u2192 \u211d)\nhs : Set.Countable s\nHc : ContinuousOn f (Set.Icc a b)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 (Set.pi Set.univ fun i => Set.Ioo (a i) (b i)) \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (Set.Icc a b)\nthis : (Set.pi Set.univ fun j => Set.Ioc (a j) (b j)) = \u2205\nj : Fin (n + 1)\ni : Fin n\nhi : a (Fin.succAbove j i) = b (Fin.succAbove j i)\nhi' : Set.Ioc (a (Fin.succAbove j i)) (b (Fin.succAbove j i)) = \u2205\nhne : Fin.succAbove j i \u2260 j\n\u22a2 (\u222b (x : Fin n \u2192 \u211d) in Set.Icc (a \u2218 Fin.succAbove j) (b \u2218 Fin.succAbove j), f (Fin.insertNth j (b j) x) j) -\n      \u222b (x : Fin n \u2192 \u211d) in Set.Icc (a \u2218 Fin.succAbove j) (b \u2218 Fin.succAbove j), f (Fin.insertNth j (a j) x) j =\n    0"}, {"tactic": "have : Icc (a \u2218 j.succAbove) (b \u2218 j.succAbove) =\u1d50[volume] (\u2205 : Set \u211d\u207f)", "annotated_tactic": ["have : <a>Icc</a> (a \u2218 j.succAbove) (b \u2218 j.succAbove) =\u1d50[<a>volume</a>] (\u2205 : <a>Set</a> \u211d\u207f)", [{"full_name": "Set.Icc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [59, 5], "def_end_pos": [59, 8]}, {"full_name": "MeasureTheory.MeasureSpace.volume", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [663, 3], "def_end_pos": [663, 9]}, {"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}]], "state_before": "case inl.intro.inr.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\na b : Fin (n + 1) \u2192 \u211d\nhle : a \u2264 b\nf : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E\nf' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E\ns : Set (Fin (n + 1) \u2192 \u211d)\nhs : Set.Countable s\nHc : ContinuousOn f (Set.Icc a b)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 (Set.pi Set.univ fun i => Set.Ioo (a i) (b i)) \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (Set.Icc a b)\nthis : (Set.pi Set.univ fun j => Set.Ioc (a j) (b j)) = \u2205\nj : Fin (n + 1)\ni : Fin n\nhi : a (Fin.succAbove j i) = b (Fin.succAbove j i)\nhi' : Set.Ioc (a (Fin.succAbove j i)) (b (Fin.succAbove j i)) = \u2205\nhne : Fin.succAbove j i \u2260 j\n\u22a2 (\u222b (x : Fin n \u2192 \u211d) in Set.Icc (a \u2218 Fin.succAbove j) (b \u2218 Fin.succAbove j), f (Fin.insertNth j (b j) x) j) -\n      \u222b (x : Fin n \u2192 \u211d) in Set.Icc (a \u2218 Fin.succAbove j) (b \u2218 Fin.succAbove j), f (Fin.insertNth j (a j) x) j =\n    0", "state_after": "case this\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\na b : Fin (n + 1) \u2192 \u211d\nhle : a \u2264 b\nf : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E\nf' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E\ns : Set (Fin (n + 1) \u2192 \u211d)\nhs : Set.Countable s\nHc : ContinuousOn f (Set.Icc a b)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 (Set.pi Set.univ fun i => Set.Ioo (a i) (b i)) \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (Set.Icc a b)\nthis : (Set.pi Set.univ fun j => Set.Ioc (a j) (b j)) = \u2205\nj : Fin (n + 1)\ni : Fin n\nhi : a (Fin.succAbove j i) = b (Fin.succAbove j i)\nhi' : Set.Ioc (a (Fin.succAbove j i)) (b (Fin.succAbove j i)) = \u2205\nhne : Fin.succAbove j i \u2260 j\n\u22a2 Set.Icc (a \u2218 Fin.succAbove j) (b \u2218 Fin.succAbove j) =\u1d50[volume] \u2205\n\ncase inl.intro.inr.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\na b : Fin (n + 1) \u2192 \u211d\nhle : a \u2264 b\nf : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E\nf' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E\ns : Set (Fin (n + 1) \u2192 \u211d)\nhs : Set.Countable s\nHc : ContinuousOn f (Set.Icc a b)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 (Set.pi Set.univ fun i => Set.Ioo (a i) (b i)) \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (Set.Icc a b)\nthis\u271d : (Set.pi Set.univ fun j => Set.Ioc (a j) (b j)) = \u2205\nj : Fin (n + 1)\ni : Fin n\nhi : a (Fin.succAbove j i) = b (Fin.succAbove j i)\nhi' : Set.Ioc (a (Fin.succAbove j i)) (b (Fin.succAbove j i)) = \u2205\nhne : Fin.succAbove j i \u2260 j\nthis : Set.Icc (a \u2218 Fin.succAbove j) (b \u2218 Fin.succAbove j) =\u1d50[volume] \u2205\n\u22a2 (\u222b (x : Fin n \u2192 \u211d) in Set.Icc (a \u2218 Fin.succAbove j) (b \u2218 Fin.succAbove j), f (Fin.insertNth j (b j) x) j) -\n      \u222b (x : Fin n \u2192 \u211d) in Set.Icc (a \u2218 Fin.succAbove j) (b \u2218 Fin.succAbove j), f (Fin.insertNth j (a j) x) j =\n    0"}, {"tactic": "rw [set_integral_congr_set_ae this, set_integral_congr_set_ae this, integral_empty,\n  integral_empty, sub_self]", "annotated_tactic": ["rw [<a>set_integral_congr_set_ae</a> this, <a>set_integral_congr_set_ae</a> this, <a>integral_empty</a>,\n        <a>integral_empty</a>, <a>sub_self</a>]", [{"full_name": "MeasureTheory.set_integral_congr_set_ae", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [92, 9], "def_end_pos": [92, 34]}, {"full_name": "MeasureTheory.set_integral_congr_set_ae", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [92, 9], "def_end_pos": [92, 34]}, {"full_name": "MeasureTheory.integral_empty", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [148, 9], "def_end_pos": [148, 23]}, {"full_name": "MeasureTheory.integral_empty", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [148, 9], "def_end_pos": [148, 23]}, {"full_name": "sub_self", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [734, 30], "def_end_pos": [734, 38]}]], "state_before": "case inl.intro.inr.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\na b : Fin (n + 1) \u2192 \u211d\nhle : a \u2264 b\nf : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E\nf' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E\ns : Set (Fin (n + 1) \u2192 \u211d)\nhs : Set.Countable s\nHc : ContinuousOn f (Set.Icc a b)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 (Set.pi Set.univ fun i => Set.Ioo (a i) (b i)) \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (Set.Icc a b)\nthis\u271d : (Set.pi Set.univ fun j => Set.Ioc (a j) (b j)) = \u2205\nj : Fin (n + 1)\ni : Fin n\nhi : a (Fin.succAbove j i) = b (Fin.succAbove j i)\nhi' : Set.Ioc (a (Fin.succAbove j i)) (b (Fin.succAbove j i)) = \u2205\nhne : Fin.succAbove j i \u2260 j\nthis : Set.Icc (a \u2218 Fin.succAbove j) (b \u2218 Fin.succAbove j) =\u1d50[volume] \u2205\n\u22a2 (\u222b (x : Fin n \u2192 \u211d) in Set.Icc (a \u2218 Fin.succAbove j) (b \u2218 Fin.succAbove j), f (Fin.insertNth j (b j) x) j) -\n      \u222b (x : Fin n \u2192 \u211d) in Set.Icc (a \u2218 Fin.succAbove j) (b \u2218 Fin.succAbove j), f (Fin.insertNth j (a j) x) j =\n    0", "state_after": "no goals"}, {"tactic": "rw [ae_eq_empty, Real.volume_Icc_pi, prod_eq_zero (Finset.mem_univ i)]", "annotated_tactic": ["rw [<a>ae_eq_empty</a>, <a>Real.volume_Icc_pi</a>, <a>prod_eq_zero</a> (<a>Finset.mem_univ</a> i)]", [{"full_name": "MeasureTheory.ae_eq_empty", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [456, 9], "def_end_pos": [456, 20]}, {"full_name": "Real.volume_Icc_pi", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean", "def_pos": [212, 9], "def_end_pos": [212, 22]}, {"full_name": "Finset.prod_eq_zero", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [1914, 9], "def_end_pos": [1914, 21]}, {"full_name": "Finset.mem_univ", "def_path": "Mathlib/Data/Fintype/Basic.lean", "def_pos": [72, 9], "def_end_pos": [72, 17]}]], "state_before": "case this\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\na b : Fin (n + 1) \u2192 \u211d\nhle : a \u2264 b\nf : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E\nf' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E\ns : Set (Fin (n + 1) \u2192 \u211d)\nhs : Set.Countable s\nHc : ContinuousOn f (Set.Icc a b)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 (Set.pi Set.univ fun i => Set.Ioo (a i) (b i)) \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (Set.Icc a b)\nthis : (Set.pi Set.univ fun j => Set.Ioc (a j) (b j)) = \u2205\nj : Fin (n + 1)\ni : Fin n\nhi : a (Fin.succAbove j i) = b (Fin.succAbove j i)\nhi' : Set.Ioc (a (Fin.succAbove j i)) (b (Fin.succAbove j i)) = \u2205\nhne : Fin.succAbove j i \u2260 j\n\u22a2 Set.Icc (a \u2218 Fin.succAbove j) (b \u2218 Fin.succAbove j) =\u1d50[volume] \u2205", "state_after": "case this\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\na b : Fin (n + 1) \u2192 \u211d\nhle : a \u2264 b\nf : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E\nf' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E\ns : Set (Fin (n + 1) \u2192 \u211d)\nhs : Set.Countable s\nHc : ContinuousOn f (Set.Icc a b)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 (Set.pi Set.univ fun i => Set.Ioo (a i) (b i)) \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (Set.Icc a b)\nthis : (Set.pi Set.univ fun j => Set.Ioc (a j) (b j)) = \u2205\nj : Fin (n + 1)\ni : Fin n\nhi : a (Fin.succAbove j i) = b (Fin.succAbove j i)\nhi' : Set.Ioc (a (Fin.succAbove j i)) (b (Fin.succAbove j i)) = \u2205\nhne : Fin.succAbove j i \u2260 j\n\u22a2 ENNReal.ofReal ((b \u2218 Fin.succAbove j) i - (a \u2218 Fin.succAbove j) i) = 0"}, {"tactic": "simp [hi]", "annotated_tactic": ["simp [hi]", []], "state_before": "case this\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\na b : Fin (n + 1) \u2192 \u211d\nhle : a \u2264 b\nf : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E\nf' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E\ns : Set (Fin (n + 1) \u2192 \u211d)\nhs : Set.Countable s\nHc : ContinuousOn f (Set.Icc a b)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 (Set.pi Set.univ fun i => Set.Ioo (a i) (b i)) \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (Set.Icc a b)\nthis : (Set.pi Set.univ fun j => Set.Ioc (a j) (b j)) = \u2205\nj : Fin (n + 1)\ni : Fin n\nhi : a (Fin.succAbove j i) = b (Fin.succAbove j i)\nhi' : Set.Ioc (a (Fin.succAbove j i)) (b (Fin.succAbove j i)) = \u2205\nhne : Fin.succAbove j i \u2260 j\n\u22a2 ENNReal.ofReal ((b \u2218 Fin.succAbove j) i - (a \u2218 Fin.succAbove j) i) = 0", "state_after": "no goals"}, {"tactic": "have hlt : \u2200 i, a i < b i := fun i => (hle i).lt_of_ne fun hi => hne \u27e8i, hi\u27e9", "annotated_tactic": ["have hlt : \u2200 i, a i < b i := fun i => (hle i).<a>lt_of_ne</a> fun hi => hne \u27e8i, hi\u27e9", [{"full_name": "LE.le.lt_of_ne", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [132, 7], "def_end_pos": [132, 21]}]], "state_before": "case inr\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\na b : Fin (n + 1) \u2192 \u211d\nhle : a \u2264 b\nf : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E\nf' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E\ns : Set (Fin (n + 1) \u2192 \u211d)\nhs : Set.Countable s\nHc : ContinuousOn f (Set.Icc a b)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 (Set.pi Set.univ fun i => Set.Ioo (a i) (b i)) \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (Set.Icc a b)\nhne : \u00ac\u2203 i, a i = b i\n\u22a2 \u222b (x : Fin (n + 1) \u2192 \u211d) in Set.Icc a b, \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i =\n    \u2211 i : Fin (n + 1),\n      ((\u222b (x : Fin n \u2192 \u211d) in Set.Icc (a \u2218 Fin.succAbove i) (b \u2218 Fin.succAbove i), f (Fin.insertNth i (b i) x) i) -\n        \u222b (x : Fin n \u2192 \u211d) in Set.Icc (a \u2218 Fin.succAbove i) (b \u2218 Fin.succAbove i), f (Fin.insertNth i (a i) x) i)", "state_after": "case inr\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\na b : Fin (n + 1) \u2192 \u211d\nhle : a \u2264 b\nf : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E\nf' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E\ns : Set (Fin (n + 1) \u2192 \u211d)\nhs : Set.Countable s\nHc : ContinuousOn f (Set.Icc a b)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 (Set.pi Set.univ fun i => Set.Ioo (a i) (b i)) \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (Set.Icc a b)\nhne : \u00ac\u2203 i, a i = b i\nhlt : \u2200 (i : Fin (n + 1)), a i < b i\n\u22a2 \u222b (x : Fin (n + 1) \u2192 \u211d) in Set.Icc a b, \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i =\n    \u2211 i : Fin (n + 1),\n      ((\u222b (x : Fin n \u2192 \u211d) in Set.Icc (a \u2218 Fin.succAbove i) (b \u2218 Fin.succAbove i), f (Fin.insertNth i (b i) x) i) -\n        \u222b (x : Fin n \u2192 \u211d) in Set.Icc (a \u2218 Fin.succAbove i) (b \u2218 Fin.succAbove i), f (Fin.insertNth i (a i) x) i)"}, {"tactic": "exact integral_divergence_of_hasFDerivWithinAt_off_countable_aux\u2082 \u27e8a, b, hlt\u27e9 f f' s hs Hc\n  Hd Hi", "annotated_tactic": ["exact <a>integral_divergence_of_hasFDerivWithinAt_off_countable_aux\u2082</a> \u27e8a, b, hlt\u27e9 f f' s hs Hc\n      Hd Hi", [{"full_name": "MeasureTheory.integral_divergence_of_hasFDerivWithinAt_off_countable_aux\u2082", "def_path": "Mathlib/MeasureTheory/Integral/DivergenceTheorem.lean", "def_pos": [141, 9], "def_end_pos": [141, 68]}]], "state_before": "case inr\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\na b : Fin (n + 1) \u2192 \u211d\nhle : a \u2264 b\nf : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E\nf' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E\ns : Set (Fin (n + 1) \u2192 \u211d)\nhs : Set.Countable s\nHc : ContinuousOn f (Set.Icc a b)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 (Set.pi Set.univ fun i => Set.Ioo (a i) (b i)) \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (Set.Icc a b)\nhne : \u00ac\u2203 i, a i = b i\nhlt : \u2200 (i : Fin (n + 1)), a i < b i\n\u22a2 \u222b (x : Fin (n + 1) \u2192 \u211d) in Set.Icc a b, \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i =\n    \u2211 i : Fin (n + 1),\n      ((\u222b (x : Fin n \u2192 \u211d) in Set.Icc (a \u2218 Fin.succAbove i) (b \u2218 Fin.succAbove i), f (Fin.insertNth i (b i) x) i) -\n        \u222b (x : Fin n \u2192 \u211d) in Set.Icc (a \u2218 Fin.succAbove i) (b \u2218 Fin.succAbove i), f (Fin.insertNth i (a i) x) i)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "full_name": "Nat.eq_or_lt_of_not_lt", "start": [180, 11], "end": [181, 44], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/AEMeasurable.lean", "full_name": "AEMeasurable.iUnion", "start": [134, 11], "end": [136, 53], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/Basic.lean", "full_name": "MvPolynomial.eval\u2082_eq'", "start": [964, 1], "end": [967, 6], "traced_tactics": [{"tactic": "simp only [eval\u2082_eq, \u2190 Finsupp.prod_pow]", "annotated_tactic": ["simp only [<a>eval\u2082_eq</a>, \u2190 <a>Finsupp.prod_pow</a>]", [{"full_name": "MvPolynomial.eval\u2082_eq", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [959, 9], "def_end_pos": [959, 17]}, {"full_name": "Finsupp.prod_pow", "def_path": "Mathlib/Algebra/BigOperators/Finsupp.lean", "def_pos": [154, 9], "def_end_pos": [154, 17]}]], "state_before": "R : Type u\nS\u2081 : Type v\nS\u2082 : Type w\nS\u2083 : Type x\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\u2081\np q : MvPolynomial \u03c3 R\nf\u271d : R \u2192+* S\u2081\ng\u271d : \u03c3 \u2192 S\u2081\ninst\u271d : Fintype \u03c3\ng : R \u2192+* S\u2081\nX : \u03c3 \u2192 S\u2081\nf : MvPolynomial \u03c3 R\n\u22a2 eval\u2082 g X f = \u2211 d in support f, \u2191g (coeff d f) * \u220f i : \u03c3, X i ^ \u2191d i", "state_after": "R : Type u\nS\u2081 : Type v\nS\u2082 : Type w\nS\u2083 : Type x\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\u2081\np q : MvPolynomial \u03c3 R\nf\u271d : R \u2192+* S\u2081\ng\u271d : \u03c3 \u2192 S\u2081\ninst\u271d : Fintype \u03c3\ng : R \u2192+* S\u2081\nX : \u03c3 \u2192 S\u2081\nf : MvPolynomial \u03c3 R\n\u22a2 \u2211 x in support f, \u2191g (coeff x f) * \u220f x_1 in x.support, X x_1 ^ \u2191x x_1 =\n    \u2211 x in support f, \u2191g (coeff x f) * Finsupp.prod x fun a b => X a ^ b"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "R : Type u\nS\u2081 : Type v\nS\u2082 : Type w\nS\u2083 : Type x\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\u2081\np q : MvPolynomial \u03c3 R\nf\u271d : R \u2192+* S\u2081\ng\u271d : \u03c3 \u2192 S\u2081\ninst\u271d : Fintype \u03c3\ng : R \u2192+* S\u2081\nX : \u03c3 \u2192 S\u2081\nf : MvPolynomial \u03c3 R\n\u22a2 \u2211 x in support f, \u2191g (coeff x f) * \u220f x_1 in x.support, X x_1 ^ \u2191x x_1 =\n    \u2211 x in support f, \u2191g (coeff x f) * Finsupp.prod x fun a b => X a ^ b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "full_name": "List.get_of_append", "start": [699, 1], "end": [701, 86], "traced_tactics": [{"tactic": "rw [\u2190 get?_eq_get, eq, get?_append_right (h \u25b8 Nat.le_refl _), h, Nat.sub_self]", "annotated_tactic": ["rw [\u2190 <a>get?_eq_get</a>, eq, <a>get?_append_right</a> (h \u25b8 <a>Nat.le_refl</a> _), h, <a>Nat.sub_self</a>]", [{"full_name": "List.get?_eq_get", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [581, 9], "def_end_pos": [581, 20]}, {"full_name": "List.get?_append_right", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [682, 9], "def_end_pos": [682, 26]}, {"full_name": "Nat.le_refl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1605, 19], "def_end_pos": [1605, 30]}, {"full_name": "Nat.sub_self", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [250, 27], "def_end_pos": [250, 35]}]], "state_before": "\u03b1 : Type u_1\nl\u2081 : List \u03b1\na : \u03b1\nl\u2082 : List \u03b1\nn : Nat\nl : List \u03b1\neq : l = l\u2081 ++ a :: l\u2082\nh : length l\u2081 = n\n\u22a2 some (get l { val := n, isLt := (_ : n < length l) }) = some a", "state_after": "\u03b1 : Type u_1\nl\u2081 : List \u03b1\na : \u03b1\nl\u2082 : List \u03b1\nn : Nat\nl : List \u03b1\neq : l = l\u2081 ++ a :: l\u2082\nh : length l\u2081 = n\n\u22a2 get? (a :: l\u2082) 0 = some a"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\u03b1 : Type u_1\nl\u2081 : List \u03b1\na : \u03b1\nl\u2082 : List \u03b1\nn : Nat\nl : List \u03b1\neq : l = l\u2081 ++ a :: l\u2082\nh : length l\u2081 = n\n\u22a2 get? (a :: l\u2082) 0 = some a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "full_name": "MeasureTheory.SimpleFunc.simpleFunc_bot", "start": [171, 1], "end": [174, 84], "traced_tactics": [{"tactic": "have hf_meas := @SimpleFunc.measurableSet_fiber \u03b1 _ \u22a5 f", "annotated_tactic": ["have hf_meas := @<a>SimpleFunc.measurableSet_fiber</a> \u03b1 _ \u22a5 f", [{"full_name": "MeasureTheory.SimpleFunc.measurableSet_fiber", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [80, 9], "def_end_pos": [80, 28]}]], "state_before": "\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : MeasurableSpace \u03b1\u271d\n\u03b1 : Type u_5\nf : \u03b1 \u2192\u209b \u03b2\ninst\u271d : Nonempty \u03b2\n\u22a2 \u2203 c, \u2200 (x : \u03b1), \u2191f x = c", "state_after": "\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : MeasurableSpace \u03b1\u271d\n\u03b1 : Type u_5\nf : \u03b1 \u2192\u209b \u03b2\ninst\u271d : Nonempty \u03b2\nhf_meas : \u2200 (x : \u03b2), MeasurableSet (\u2191f \u207b\u00b9' {x})\n\u22a2 \u2203 c, \u2200 (x : \u03b1), \u2191f x = c"}, {"tactic": "simp_rw [MeasurableSpace.measurableSet_bot_iff] at hf_meas", "annotated_tactic": ["simp_rw [<a>MeasurableSpace.measurableSet_bot_iff</a>] at hf_meas", [{"full_name": "MeasurableSpace.measurableSet_bot_iff", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [467, 9], "def_end_pos": [467, 30]}]], "state_before": "\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : MeasurableSpace \u03b1\u271d\n\u03b1 : Type u_5\nf : \u03b1 \u2192\u209b \u03b2\ninst\u271d : Nonempty \u03b2\nhf_meas : \u2200 (x : \u03b2), MeasurableSet (\u2191f \u207b\u00b9' {x})\n\u22a2 \u2203 c, \u2200 (x : \u03b1), \u2191f x = c", "state_after": "\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : MeasurableSpace \u03b1\u271d\n\u03b1 : Type u_5\nf : \u03b1 \u2192\u209b \u03b2\ninst\u271d : Nonempty \u03b2\nhf_meas : \u2200 (x : \u03b2), \u2191f \u207b\u00b9' {x} = \u2205 \u2228 \u2191f \u207b\u00b9' {x} = univ\n\u22a2 \u2203 c, \u2200 (x : \u03b1), \u2191f x = c"}, {"tactic": "exact (exists_eq_const_of_preimage_singleton hf_meas).imp fun c hc \u21a6 congr_fun hc", "annotated_tactic": ["exact (<a>exists_eq_const_of_preimage_singleton</a> hf_meas).<a>imp</a> fun c hc \u21a6 <a>congr_fun</a> hc", [{"full_name": "Set.exists_eq_const_of_preimage_singleton", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [155, 7], "def_end_pos": [155, 44]}, {"full_name": "Exists.imp", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [360, 9], "def_end_pos": [360, 19]}, {"full_name": "congr_fun", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [42, 7], "def_end_pos": [42, 16]}]], "state_before": "\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : MeasurableSpace \u03b1\u271d\n\u03b1 : Type u_5\nf : \u03b1 \u2192\u209b \u03b2\ninst\u271d : Nonempty \u03b2\nhf_meas : \u2200 (x : \u03b2), \u2191f \u207b\u00b9' {x} = \u2205 \u2228 \u2191f \u207b\u00b9' {x} = univ\n\u22a2 \u2203 c, \u2200 (x : \u03b1), \u2191f x = c", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Lattice.lean", "full_name": "Finset.inf_sdiff_right", "start": [621, 1], "end": [625, 42], "traced_tactics": [{"tactic": "induction' hs using Finset.Nonempty.cons_induction with b b t _ _ h", "annotated_tactic": ["induction' hs using <a>Finset.Nonempty.cons_induction</a> with b b t _ _ h", [{"full_name": "Finset.Nonempty.cons_induction", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1275, 9], "def_end_pos": [1275, 32]}]], "state_before": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Type u_5\n\u03ba : Type u_6\ninst\u271d : BooleanAlgebra \u03b1\ns : Finset \u03b9\nhs : Finset.Nonempty s\nf : \u03b9 \u2192 \u03b1\na : \u03b1\n\u22a2 (inf s fun b => f b \\ a) = inf s f \\ a", "state_after": "case h\u2080\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Type u_5\n\u03ba : Type u_6\ninst\u271d : BooleanAlgebra \u03b1\ns : Finset \u03b9\nf : \u03b9 \u2192 \u03b1\na : \u03b1\nb : \u03b9\n\u22a2 (inf {b} fun b => f b \\ a) = inf {b} f \\ a\n\ncase h\u2081\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Type u_5\n\u03ba : Type u_6\ninst\u271d : BooleanAlgebra \u03b1\ns : Finset \u03b9\nf : \u03b9 \u2192 \u03b1\na : \u03b1\nb : \u03b9\nt : Finset \u03b9\nh\u271d : \u00acb \u2208 t\nhs\u271d : Finset.Nonempty t\nh : (inf t fun b => f b \\ a) = inf t f \\ a\n\u22a2 (inf (cons b t h\u271d) fun b => f b \\ a) = inf (cons b t h\u271d) f \\ a"}, {"tactic": "rw [inf_singleton, inf_singleton]", "annotated_tactic": ["rw [<a>inf_singleton</a>, <a>inf_singleton</a>]", [{"full_name": "Finset.inf_singleton", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [343, 9], "def_end_pos": [343, 22]}, {"full_name": "Finset.inf_singleton", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [343, 9], "def_end_pos": [343, 22]}]], "state_before": "case h\u2080\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Type u_5\n\u03ba : Type u_6\ninst\u271d : BooleanAlgebra \u03b1\ns : Finset \u03b9\nf : \u03b9 \u2192 \u03b1\na : \u03b1\nb : \u03b9\n\u22a2 (inf {b} fun b => f b \\ a) = inf {b} f \\ a", "state_after": "no goals"}, {"tactic": "rw [inf_cons, inf_cons, h, inf_sdiff]", "annotated_tactic": ["rw [<a>inf_cons</a>, <a>inf_cons</a>, h, <a>inf_sdiff</a>]", [{"full_name": "Finset.inf_cons", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [323, 9], "def_end_pos": [323, 17]}, {"full_name": "Finset.inf_cons", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [323, 9], "def_end_pos": [323, 17]}, {"full_name": "inf_sdiff", "def_path": "Mathlib/Order/BooleanAlgebra.lean", "def_pos": [420, 9], "def_end_pos": [420, 18]}]], "state_before": "case h\u2081\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Type u_5\n\u03ba : Type u_6\ninst\u271d : BooleanAlgebra \u03b1\ns : Finset \u03b9\nf : \u03b9 \u2192 \u03b1\na : \u03b1\nb : \u03b9\nt : Finset \u03b9\nh\u271d : \u00acb \u2208 t\nhs\u271d : Finset.Nonempty t\nh : (inf t fun b => f b \\ a) = inf t f \\ a\n\u22a2 (inf (cons b t h\u271d) fun b => f b \\ a) = inf (cons b t h\u271d) f \\ a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "full_name": "String.next'_eq", "start": [324, 9], "end": [324, 84], "traced_tactics": []}]